LMFDB - Embedded newform 825.3.b.a.76.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [825,3,Mod(76,825)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(825, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1]))

N = Newforms(chi, 3, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("825.76");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 825 = 3 \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	825.b (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$22.4796218097$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.0.39744.5
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} + 12x^{2} + 4x + 22 $ x^4 - 2x^3 + 12x^2 + 4*x + 22
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 33)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			76.3
Root			$1.36603 - 3.21405i$ of defining polynomial
Character	$\chi$	$=$	825.76
Dual form			825.3.b.a.76.2

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.35285i q^{2} -1.73205 q^{3} -1.53590 q^{4} -4.07525i q^{6} -6.42810i q^{7} +5.79766i q^{8} +3.00000 q^{9} +O(q^{10})$	\(q+2.35285i q^{2} -1.73205 q^{3} -1.53590 q^{4} -4.07525i q^{6} -6.42810i q^{7} +5.79766i q^{8} +3.00000 q^{9} +(3.26795 + 10.5034i) q^{11} +2.66025 q^{12} -7.68899i q^{13} +15.1244 q^{14} -19.7846 q^{16} +8.15051i q^{17} +7.05855i q^{18} -30.4181i q^{19} +11.1338i q^{21} +(-24.7128 + 7.68899i) q^{22} +31.6603 q^{23} -10.0418i q^{24} +18.0910 q^{26} -5.19615 q^{27} +9.87291i q^{28} +6.88962i q^{29} +51.1769 q^{31} -23.3596i q^{32} +(-5.66025 - 18.1923i) q^{33} -19.1769 q^{34} -4.60770 q^{36} +19.4641 q^{37} +71.5692 q^{38} +13.3177i q^{39} +47.9800i q^{41} -26.1962 q^{42} +81.8429i q^{43} +(-5.01924 - 16.1321i) q^{44} +74.4918i q^{46} -30.1962 q^{47} +34.2679 q^{48} +7.67949 q^{49} -14.1171i q^{51} +11.8095i q^{52} -26.0526 q^{53} -12.2258i q^{54} +37.2679 q^{56} +52.6857i q^{57} -16.2102 q^{58} +82.7461 q^{59} +75.4148i q^{61} +120.412i q^{62} -19.2843i q^{63} -24.1769 q^{64} +(42.8038 - 13.3177i) q^{66} +34.0000 q^{67} -12.5184i q^{68} -54.8372 q^{69} -72.7321 q^{71} +17.3930i q^{72} -54.8696i q^{73} +45.7961i q^{74} +46.7191i q^{76} +(67.5167 - 21.0067i) q^{77} -31.3346 q^{78} +24.3279i q^{79} +9.00000 q^{81} -112.890 q^{82} -0.923034i q^{83} -17.1004i q^{84} -192.564 q^{86} -11.9332i q^{87} +(-60.8949 + 18.9465i) q^{88} +44.8231 q^{89} -49.4256 q^{91} -48.6269 q^{92} -88.6410 q^{93} -71.0470i q^{94} +40.4599i q^{96} +21.2539 q^{97} +18.0687i q^{98} +(9.80385 + 31.5101i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 20 q^{4} + 12 q^{9}+O(q^{10}) $ 4 * q - 20 * q^4 + 12 * q^9	$ 4 q - 20 q^{4} + 12 q^{9} + 20 q^{11} - 24 q^{12} + 12 q^{14} + 4 q^{16} + 12 q^{22} + 92 q^{23} + 204 q^{26} + 80 q^{31} + 12 q^{33} + 48 q^{34} - 60 q^{36} + 64 q^{37} + 120 q^{38} - 84 q^{42} - 124 q^{44} - 100 q^{47} + 144 q^{48} + 100 q^{49} - 28 q^{53} + 156 q^{56} + 240 q^{58} + 40 q^{59} + 28 q^{64} + 192 q^{66} + 136 q^{67} - 60 q^{69} - 284 q^{71} + 180 q^{77} + 228 q^{78} + 36 q^{81} - 216 q^{82} - 216 q^{86} - 396 q^{88} + 304 q^{89} + 24 q^{91} - 340 q^{92} - 216 q^{93} + 376 q^{97} + 60 q^{99}+O(q^{100}) $ 4 * q - 20 * q^4 + 12 * q^9 + 20 * q^11 - 24 * q^12 + 12 * q^14 + 4 * q^16 + 12 * q^22 + 92 * q^23 + 204 * q^26 + 80 * q^31 + 12 * q^33 + 48 * q^34 - 60 * q^36 + 64 * q^37 + 120 * q^38 - 84 * q^42 - 124 * q^44 - 100 * q^47 + 144 * q^48 + 100 * q^49 - 28 * q^53 + 156 * q^56 + 240 * q^58 + 40 * q^59 + 28 * q^64 + 192 * q^66 + 136 * q^67 - 60 * q^69 - 284 * q^71 + 180 * q^77 + 228 * q^78 + 36 * q^81 - 216 * q^82 - 216 * q^86 - 396 * q^88 + 304 * q^89 + 24 * q^91 - 340 * q^92 - 216 * q^93 + 376 * q^97 + 60 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/825\mathbb{Z}\right)^\times$.

$n$	$376$	$551$	$727$
$\chi(n)$	$-1$	$1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$			2.35285i			1.17642i	0.808707	+	0.588212i	$0.200168\pi$
$2$			2.35285i			1.17642i	−0.808707	+	0.588212i	$0.799832\pi$
$3$	−1.73205			−0.577350
$3$	−1.73205			−0.577350
$4$	−1.53590			−0.383975
$5$		0			0
$5$		0			0
$6$		−	4.07525i		−	0.679209i
$7$		−	6.42810i		−	0.918300i	−0.888359	−	0.459150i	$-0.848154\pi$
$7$		−	6.42810i		−	0.918300i	0.888359	−	0.459150i	$-0.151846\pi$
$8$			5.79766i			0.724707i
$9$	3.00000			0.333333
$10$		0			0
$11$	3.26795	+	10.5034i	0.297086	+	0.954851i
$11$	3.26795	+	10.5034i	0.297086	+	0.954851i
$12$	2.66025			0.221688
$13$		−	7.68899i		−	0.591461i	−0.955271	−	0.295730i	$-0.904437\pi$
$13$		−	7.68899i		−	0.591461i	0.955271	−	0.295730i	$-0.0955630\pi$
$14$	15.1244			1.08031
$15$		0			0
$16$	−19.7846			−1.23654
$17$			8.15051i			0.479442i	0.970842	+	0.239721i	$0.0770559\pi$
$17$			8.15051i			0.479442i	−0.970842	+	0.239721i	$0.922944\pi$
$18$			7.05855i			0.392141i
$19$		−	30.4181i		−	1.60095i	−0.599364	−	0.800477i	$-0.704580\pi$
$19$		−	30.4181i		−	1.60095i	0.599364	−	0.800477i	$-0.295420\pi$
$20$		0			0
$21$			11.1338i			0.530181i
$22$	−24.7128	+	7.68899i	−1.12331	+	0.349500i
$23$	31.6603			1.37653			0.688266	−	0.725458i	$-0.258372\pi$
$23$	31.6603			1.37653			0.688266	+	0.725458i	$0.258372\pi$
$24$		−	10.0418i		−	0.418410i
$25$		0			0
$26$	18.0910			0.695809
$27$	−5.19615			−0.192450
$28$			9.87291i			0.352604i
$29$			6.88962i			0.237573i	0.992920	+	0.118787i	$0.0379004\pi$
$29$			6.88962i			0.237573i	−0.992920	+	0.118787i	$0.962100\pi$
$30$		0			0
$31$	51.1769			1.65087			0.825434	−	0.564498i	$-0.190930\pi$
$31$	51.1769			1.65087			0.825434	+	0.564498i	$0.190930\pi$
$32$		−	23.3596i		−	0.729986i
$33$	−5.66025	−	18.1923i	−0.171523	−	0.551283i
$34$	−19.1769			−0.564027
$35$		0			0
$36$	−4.60770			−0.127992
$37$	19.4641			0.526057			0.263028	−	0.964788i	$-0.415279\pi$
$37$	19.4641			0.526057			0.263028	+	0.964788i	$0.415279\pi$
$38$	71.5692			1.88340
$39$			13.3177i			0.341480i
$40$		0			0
$41$			47.9800i			1.17024i	0.810945	+	0.585122i	$0.198953\pi$
$41$			47.9800i			1.17024i	−0.810945	+	0.585122i	$0.801047\pi$
$42$	−26.1962			−0.623718
$43$			81.8429i			1.90332i	0.307147	+	0.951662i	$0.400626\pi$
$43$			81.8429i			1.90332i	−0.307147	+	0.951662i	$0.599374\pi$
$44$	−5.01924	−	16.1321i	−0.114074	−	0.366638i
$45$		0			0
$46$			74.4918i			1.61939i
$47$	−30.1962			−0.642471			−0.321236	−	0.946999i	$-0.604098\pi$
$47$	−30.1962			−0.642471			−0.321236	+	0.946999i	$0.604098\pi$
$48$	34.2679			0.713916
$49$	7.67949			0.156724
$50$		0			0
$51$		−	14.1171i		−	0.276806i
$52$			11.8095i			0.227106i
$53$	−26.0526			−0.491558			−0.245779	−	0.969326i	$-0.579044\pi$
$53$	−26.0526			−0.491558			−0.245779	+	0.969326i	$0.579044\pi$
$54$		−	12.2258i		−	0.226403i
$55$		0			0
$56$	37.2679			0.665499
$57$			52.6857i			0.924311i
$58$	−16.2102			−0.279487
$59$	82.7461			1.40248			0.701238	−	0.712927i	$-0.252631\pi$
$59$	82.7461			1.40248			0.701238	+	0.712927i	$0.252631\pi$
$60$		0			0
$61$			75.4148i			1.23631i	0.786057	+	0.618154i	$0.212119\pi$
$61$			75.4148i			1.23631i	−0.786057	+	0.618154i	$0.787881\pi$
$62$			120.412i			1.94212i
$63$		−	19.2843i		−	0.306100i
$64$	−24.1769			−0.377764
$65$		0			0
$66$	42.8038	−	13.3177i	0.648543	−	0.201784i
$67$	34.0000			0.507463			0.253731	−	0.967275i	$-0.418342\pi$
$67$	34.0000			0.507463			0.253731	+	0.967275i	$0.418342\pi$
$68$		−	12.5184i		−	0.184093i
$69$	−54.8372			−0.794742
$70$		0			0
$71$	−72.7321			−1.02440			−0.512198	−	0.858868i	$-0.671168\pi$
$71$	−72.7321			−1.02440			−0.512198	+	0.858868i	$0.671168\pi$
$72$			17.3930i			0.241569i
$73$		−	54.8696i		−	0.751639i	−0.926693	−	0.375819i	$-0.877361\pi$
$73$		−	54.8696i		−	0.751639i	0.926693	−	0.375819i	$-0.122639\pi$
$74$			45.7961i			0.618866i
$75$		0			0
$76$			46.7191i			0.614725i
$77$	67.5167	−	21.0067i	0.876840	−	0.272814i
$78$	−31.3346			−0.401726
$79$			24.3279i			0.307948i	0.988075	+	0.153974i	$0.0492071\pi$
$79$			24.3279i			0.307948i	−0.988075	+	0.153974i	$0.950793\pi$
$80$		0			0
$81$	9.00000			0.111111
$82$	−112.890			−1.37670
$83$		−	0.923034i		−	0.0111209i	−0.999985	−	0.00556045i	$-0.998230\pi$
$83$		−	0.923034i		−	0.0111209i	0.999985	−	0.00556045i	$-0.00176995\pi$
$84$		−	17.1004i		−	0.203576i
$85$		0			0
$86$	−192.564			−2.23912
$87$		−	11.9332i		−	0.137163i
$88$	−60.8949	+	18.9465i	−0.691987	+	0.215301i
$89$	44.8231			0.503630			0.251815	−	0.967775i	$-0.418973\pi$
$89$	44.8231			0.503630			0.251815	+	0.967775i	$0.418973\pi$
$90$		0			0
$91$	−49.4256			−0.543139
$92$	−48.6269			−0.528554
$93$	−88.6410			−0.953129
$94$		−	71.0470i		−	0.755819i
$95$		0			0
$96$			40.4599i			0.421458i
$97$	21.2539			0.219112			0.109556	−	0.993981i	$-0.465057\pi$
$97$	21.2539			0.219112			0.109556	+	0.993981i	$0.465057\pi$
$98$			18.0687i			0.184374i
$99$	9.80385	+	31.5101i	0.0990288	+	0.318284i
$100$		0			0
$101$		−	52.3479i		−	0.518296i	−0.965838	−	0.259148i	$-0.916558\pi$
$101$		−	52.3479i		−	0.518296i	0.965838	−	0.259148i	$-0.0834417\pi$
$102$	33.2154			0.325641
$103$	4.74613			0.0460790			0.0230395	−	0.999735i	$-0.492666\pi$
$103$	4.74613			0.0460790			0.0230395	+	0.999735i	$0.492666\pi$
$104$	44.5781			0.428636
$105$		0			0
$106$		−	61.2977i		−	0.578281i
$107$			147.723i			1.38059i	0.723530	+	0.690293i	$0.242518\pi$
$107$			147.723i			1.38059i	−0.723530	+	0.690293i	$0.757482\pi$
$108$	7.98076			0.0738959
$109$			3.56847i			0.0327383i	0.999866	+	0.0163691i	$0.00521069\pi$
$109$			3.56847i			0.0327383i	−0.999866	+	0.0163691i	$0.994789\pi$
$110$		0			0
$111$	−33.7128			−0.303719
$112$			127.178i			1.13551i
$113$	62.3154			0.551463			0.275732	−	0.961235i	$-0.411080\pi$
$113$	62.3154			0.551463			0.275732	+	0.961235i	$0.411080\pi$
$114$	−123.962			−1.08738
$115$		0			0
$116$		−	10.5818i		−	0.0912220i
$117$		−	23.0670i		−	0.197154i
$118$			194.689i			1.64991i
$119$	52.3923			0.440271
$120$		0			0
$121$	−99.6410	+	68.6489i	−0.823479	+	0.567346i
$122$	−177.440			−1.45442
$123$		−	83.1038i		−	0.675641i
$124$	−78.6025			−0.633891
$125$		0			0
$126$	45.3731			0.360104
$127$			28.0200i			0.220630i	0.993897	+	0.110315i	$0.0351860\pi$
$127$			28.0200i			0.220630i	−0.993897	+	0.110315i	$0.964814\pi$
$128$		−	150.323i		−	1.17440i
$129$		−	141.756i		−	1.09888i
$130$		0			0
$131$			113.184i			0.864001i	0.901873	+	0.432000i	$0.142192\pi$
$131$			113.184i			0.864001i	−0.901873	+	0.432000i	$0.857808\pi$
$132$	8.69358	+	27.9416i	0.0658604	+	0.211679i
$133$	−195.531			−1.47016
$134$			79.9969i			0.596992i
$135$		0			0
$136$	−47.2539			−0.347455
$137$	70.7846			0.516676			0.258338	−	0.966055i	$-0.416825\pi$
$137$	70.7846			0.516676			0.258338	+	0.966055i	$0.416825\pi$
$138$		−	129.024i		−	0.934953i
$139$		−	130.499i		−	0.938839i	−0.882975	−	0.469420i	$-0.844463\pi$
$139$		−	130.499i		−	0.938839i	0.882975	−	0.469420i	$-0.155537\pi$
$140$		0			0
$141$	52.3013			0.370931
$142$		−	171.128i		−	1.20512i
$143$	80.7602	−	25.1272i	0.564757	−	0.175715i
$144$	−59.3538			−0.412179
$145$		0			0
$146$	129.100			0.884246
$147$	−13.3013			−0.0904848
$148$	−29.8949			−0.201992
$149$		−	125.793i		−	0.844248i	−0.906538	−	0.422124i	$-0.861285\pi$
$149$		−	125.793i		−	0.844248i	0.906538	−	0.422124i	$-0.138715\pi$
$150$		0			0
$151$			275.238i			1.82277i	0.411556	+	0.911384i	$0.364985\pi$
$151$			275.238i			1.82277i	−0.411556	+	0.911384i	$0.635015\pi$
$152$	176.354			1.16022
$153$			24.4515i			0.159814i
$154$	49.4256	+	158.857i	0.320946	+	1.03154i
$155$		0			0
$156$		−	20.4547i		−	0.131120i
$157$	273.138			1.73974			0.869868	−	0.493285i	$-0.164204\pi$
$157$	273.138			1.73974			0.869868	+	0.493285i	$0.164204\pi$
$158$	−57.2398			−0.362277
$159$	45.1244			0.283801
$160$		0			0
$161$		−	203.515i		−	1.26407i
$162$			21.1756i			0.130714i
$163$	−62.0666			−0.380777			−0.190388	−	0.981709i	$-0.560975\pi$
$163$	−62.0666			−0.380777			−0.190388	+	0.981709i	$0.560975\pi$
$164$		−	73.6924i		−	0.449344i
$165$		0			0
$166$	2.17176			0.0130829
$167$		−	47.9800i		−	0.287305i	−0.989628	−	0.143653i	$-0.954115\pi$
$167$		−	47.9800i		−	0.287305i	0.989628	−	0.143653i	$-0.0458848\pi$
$168$	−64.5500			−0.384226
$169$	109.879			0.650174
$170$		0			0
$171$		−	91.2543i		−	0.533651i
$172$		−	125.702i		−	0.730828i
$173$			143.017i			0.826688i	0.910575	+	0.413344i	$0.135639\pi$
$173$			143.017i			0.826688i	−0.910575	+	0.413344i	$0.864361\pi$
$174$	28.0770			0.161362
$175$		0			0
$176$	−64.6551	−	207.805i	−0.367359	−	1.18071i
$177$	−143.321			−0.809720
$178$			105.462i			0.592483i
$179$	36.6025			0.204483			0.102242	−	0.994760i	$-0.467398\pi$
$179$	36.6025			0.204483			0.102242	+	0.994760i	$0.467398\pi$
$180$		0			0
$181$	−84.1718			−0.465037			−0.232519	−	0.972592i	$-0.574697\pi$
$181$	−84.1718			−0.465037			−0.232519	+	0.972592i	$0.574697\pi$
$182$		−	116.291i		−	0.638962i
$183$		−	130.622i		−	0.713783i
$184$			183.555i			0.997583i
$185$		0			0
$186$		−	208.559i		−	1.12128i
$187$	−85.6077	+	26.6354i	−0.457795	+	0.142436i
$188$	46.3782			0.246693
$189$			33.4014i			0.176727i
$190$		0			0
$191$	44.2346			0.231595			0.115797	−	0.993273i	$-0.463058\pi$
$191$	44.2346			0.231595			0.115797	+	0.993273i	$0.463058\pi$
$192$	41.8756			0.218102
$193$		−	136.127i		−	0.705323i	−0.935751	−	0.352662i	$-0.885277\pi$
$193$		−	136.127i		−	0.705323i	0.935751	−	0.352662i	$-0.114723\pi$
$194$			50.0071i			0.257769i
$195$		0			0
$196$	−11.7949			−0.0601782
$197$			5.96659i			0.0302872i	0.999885	+	0.0151436i	$0.00482055\pi$
$197$			5.96659i			0.0302872i	−0.999885	+	0.0151436i	$0.995179\pi$
$198$	−74.1384	+	23.0670i	−0.374437	+	0.116500i
$199$	−234.056			−1.17616			−0.588081	−	0.808802i	$-0.700116\pi$
$199$	−234.056			−1.17616			−0.588081	+	0.808802i	$0.700116\pi$
$200$		0			0
$201$	−58.8897			−0.292984
$202$	123.167			0.609736
$203$	44.2872			0.218163
$204$			21.6824i			0.106286i
$205$		0			0
$206$			11.1669i			0.0542084i
$207$	94.9808			0.458844
$208$			152.124i			0.731364i
$209$	319.492	−	99.4048i	1.52867	−	0.475621i
$210$		0			0
$211$		−	167.469i		−	0.793690i	−0.917886	−	0.396845i	$-0.870105\pi$
$211$		−	167.469i		−	0.793690i	0.917886	−	0.396845i	$-0.129895\pi$
$212$	40.0141			0.188746
$213$	125.976			0.591435
$214$	−347.569			−1.62416
$215$		0			0
$216$		−	30.1255i		−	0.139470i
$217$		−	328.970i		−	1.51599i
$218$	−8.39608			−0.0385141
$219$			95.0370i			0.433959i
$220$		0			0
$221$	62.6692			0.283571
$222$		−	79.3212i		−	0.357303i
$223$	−82.2872			−0.369001			−0.184500	−	0.982832i	$-0.559067\pi$
$223$	−82.2872			−0.369001			−0.184500	+	0.982832i	$0.559067\pi$
$224$	−150.158			−0.670347
$225$		0			0
$226$			146.619i			0.648755i
$227$			40.9999i			0.180616i	0.995914	+	0.0903081i	$0.0287852\pi$
$227$			40.9999i			0.180616i	−0.995914	+	0.0903081i	$0.971215\pi$
$228$		−	80.9199i		−	0.354912i
$229$	160.718			0.701825			0.350913	−	0.936408i	$-0.385871\pi$
$229$	160.718			0.701825			0.350913	+	0.936408i	$0.385871\pi$
$230$		0			0
$231$	−116.942	+	36.3847i	−0.506244	+	0.157510i
$232$	−39.9437			−0.172171
$233$		−	407.954i		−	1.75087i	−0.483332	−	0.875437i	$-0.660573\pi$
$233$		−	407.954i		−	1.75087i	0.483332	−	0.875437i	$-0.339427\pi$
$234$	54.2731			0.231936
$235$		0			0
$236$	−127.090			−0.538515
$237$		−	42.1371i		−	0.177794i
$238$			123.271i			0.517946i
$239$			202.254i			0.846253i	0.906071	+	0.423127i	$0.139067\pi$
$239$			202.254i			0.846253i	−0.906071	+	0.423127i	$0.860933\pi$
$240$		0			0
$241$			26.8828i			0.111547i	0.998443	+	0.0557734i	$0.0177624\pi$
$241$			26.8828i			0.111547i	−0.998443	+	0.0557734i	$0.982238\pi$
$242$	−161.520	−	234.440i	−0.667440	−	0.968761i
$243$	−15.5885			−0.0641500
$244$		−	115.830i		−	0.474711i
$245$		0			0
$246$	195.531			0.794840
$247$	−233.885			−0.946901
$248$			296.706i			1.19640i
$249$			1.59874i			0.00642065i
$250$		0			0
$251$	398.277			1.58676			0.793380	−	0.608726i	$-0.208319\pi$
$251$	398.277			1.58676			0.793380	+	0.608726i	$0.208319\pi$
$252$			29.6187i			0.117535i
$253$	103.464	+	332.539i	0.408949	+	1.31438i
$254$	−65.9268			−0.259554
$255$		0			0
$256$	256.979			1.00383
$257$	337.818			1.31447			0.657233	−	0.753687i	$-0.271727\pi$
$257$	337.818			1.31447			0.657233	+	0.753687i	$0.271727\pi$
$258$	333.531			1.29275
$259$		−	125.117i		−	0.483078i
$260$		0			0
$261$			20.6689i			0.0791910i
$262$	−266.305			−1.01643
$263$		−	134.529i		−	0.511516i	−0.966741	−	0.255758i	$-0.917675\pi$
$263$		−	134.529i		−	0.511516i	0.966741	−	0.255758i	$-0.0823250\pi$
$264$	105.473	−	32.8162i	0.399519	−	0.124304i
$265$		0			0
$266$		−	460.054i		−	1.72953i
$267$	−77.6359			−0.290771
$268$	−52.2205			−0.194853
$269$	−341.870			−1.27089			−0.635447	−	0.772144i	$-0.719184\pi$
$269$	−341.870			−1.27089			−0.635447	+	0.772144i	$0.719184\pi$
$270$		0			0
$271$		−	213.298i		−	0.787077i	−0.919308	−	0.393538i	$-0.871251\pi$
$271$		−	213.298i		−	0.787077i	0.919308	−	0.393538i	$-0.128749\pi$
$272$		−	161.255i		−	0.592848i
$273$	85.6077			0.313581
$274$			166.545i			0.607830i
$275$		0			0
$276$	84.2243			0.305161
$277$		−	84.5789i		−	0.305339i	−0.988277	−	0.152669i	$-0.951213\pi$
$277$		−	84.5789i		−	0.305339i	0.988277	−	0.152669i	$-0.0487870\pi$
$278$	307.044			1.10447
$279$	153.531			0.550289
$280$		0			0
$281$		−	351.148i		−	1.24964i	−0.780771	−	0.624818i	$-0.785173\pi$
$281$		−	351.148i		−	1.24964i	0.780771	−	0.624818i	$-0.214827\pi$
$282$			123.057i			0.436372i
$283$			285.943i			1.01040i	0.863002	+	0.505201i	$0.168581\pi$
$283$			285.943i			1.01040i	−0.863002	+	0.505201i	$0.831419\pi$
$284$	111.709			0.393342
$285$		0			0
$286$	59.1206	+	190.017i	0.206715	+	0.664394i
$287$	308.420			1.07464
$288$		−	70.0787i		−	0.243329i
$289$	222.569			0.770136
$290$		0			0
$291$	−36.8128			−0.126504
$292$			84.2742i			0.288610i
$293$		−	393.927i		−	1.34446i	−0.740342	−	0.672231i	$-0.765336\pi$
$293$		−	393.927i		−	1.34446i	0.740342	−	0.672231i	$-0.234664\pi$
$294$		−	31.2959i		−	0.106449i
$295$		0			0
$296$			112.846i			0.381237i
$297$	−16.9808	−	54.5770i	−0.0571743	−	0.183761i
$298$	295.972			0.993194
$299$		−	243.435i		−	0.814165i
$300$		0			0
$301$	526.095			1.74782
$302$	−647.594			−2.14435
$303$			90.6691i			0.299238i
$304$			601.810i			1.97964i
$305$		0			0
$306$	−57.5307			−0.188009
$307$		−	58.2239i		−	0.189654i	−0.995494	−	0.0948272i	$-0.969770\pi$
$307$		−	58.2239i		−	0.189654i	0.995494	−	0.0948272i	$-0.0302299\pi$
$308$	−103.699	+	32.2642i	−0.336684	+	0.104754i
$309$	−8.22055			−0.0266037
$310$		0			0
$311$	207.611			0.667561			0.333780	−	0.942651i	$-0.391676\pi$
$311$	207.611			0.667561			0.333780	+	0.942651i	$0.391676\pi$
$312$	−77.2116			−0.247473
$313$	376.697			1.20351			0.601753	−	0.798682i	$-0.294469\pi$
$313$	376.697			1.20351			0.601753	+	0.798682i	$0.294469\pi$
$314$			642.654i			2.04667i
$315$		0			0
$316$		−	37.3651i		−	0.118244i
$317$	−246.809			−0.778577			−0.389289	−	0.921116i	$-0.627279\pi$
$317$	−246.809			−0.778577			−0.389289	+	0.921116i	$0.627279\pi$
$318$			106.171i			0.333870i
$319$	−72.3641	+	22.5149i	−0.226847	+	0.0705797i
$320$		0			0
$321$		−	255.863i		−	0.797082i
$322$	478.841			1.48708
$323$	247.923			0.767564
$324$	−13.8231			−0.0426638
$325$		0			0
$326$		−	146.033i		−	0.447955i
$327$		−	6.18078i		−	0.0189015i
$328$	−278.172			−0.848085
$329$			194.104i			0.589982i
$330$		0			0
$331$	146.431			0.442389			0.221195	−	0.975230i	$-0.429004\pi$
$331$	146.431			0.442389			0.221195	+	0.975230i	$0.429004\pi$
$332$			1.41769i			0.00427014i
$333$	58.3923			0.175352
$334$	112.890			0.337993
$335$		0			0
$336$		−	220.278i		−	0.655589i
$337$		−	354.007i		−	1.05047i	−0.850958	−	0.525233i	$-0.823978\pi$
$337$		−	354.007i		−	1.05047i	0.850958	−	0.525233i	$-0.176022\pi$
$338$			258.530i			0.764881i
$339$	−107.933			−0.318387
$340$		0			0
$341$	167.244	+	537.529i	0.490450	+	1.57633i
$342$	214.708			0.627800
$343$		−	364.342i		−	1.06222i
$344$	−474.497			−1.37935
$345$		0			0
$346$	−336.497			−0.972536
$347$			529.807i			1.52682i	0.645913	+	0.763411i	$0.276477\pi$
$347$			529.807i			1.52682i	−0.645913	+	0.763411i	$0.723523\pi$
$348$			18.3281i			0.0526671i
$349$		−	342.873i		−	0.982445i	−0.871034	−	0.491223i	$-0.836550\pi$
$349$		−	342.873i		−	0.982445i	0.871034	−	0.491223i	$-0.163450\pi$
$350$		0			0
$351$			39.9532i			0.113827i
$352$	245.354	−	76.3379i	0.697028	−	0.216869i
$353$	−191.990			−0.543880			−0.271940	−	0.962314i	$-0.587665\pi$
$353$	−191.990			−0.543880			−0.271940	+	0.962314i	$0.587665\pi$
$354$		−	337.212i		−	0.952575i
$355$		0			0
$356$	−68.8437			−0.193381
$357$	−90.7461			−0.254191
$358$			86.1203i			0.240559i
$359$		−	601.654i		−	1.67592i	−0.545735	−	0.837958i	$-0.683750\pi$
$359$		−	601.654i		−	1.67592i	0.545735	−	0.837958i	$-0.316250\pi$
$360$		0			0
$361$	−564.261			−1.56305
$362$		−	198.043i		−	0.547081i
$363$	172.583	−	118.903i	0.475436	−	0.327557i
$364$	75.9127			0.208551
$365$		0			0
$366$	307.335			0.839712
$367$	139.415			0.379878			0.189939	−	0.981796i	$-0.439171\pi$
$367$	139.415			0.379878			0.189939	+	0.981796i	$0.439171\pi$
$368$	−626.386			−1.70214
$369$			143.940i			0.390081i
$370$		0			0
$371$			167.469i			0.451398i
$372$	136.144			0.365977
$373$			303.629i			0.814019i	0.913424	+	0.407009i	$0.133428\pi$
$373$			303.629i			0.814019i	−0.913424	+	0.407009i	$0.866572\pi$
$374$	−62.6692	−	201.422i	−0.167565	−	0.538561i
$375$		0			0
$376$		−	175.067i		−	0.465604i
$377$	52.9742			0.140515
$378$	−78.5885			−0.207906
$379$	690.046			1.82070			0.910351	−	0.413837i	$-0.135812\pi$
$379$	690.046			1.82070			0.910351	+	0.413837i	$0.135812\pi$
$380$		0			0
$381$		−	48.5321i		−	0.127381i
$382$			104.077i			0.272454i
$383$	53.2576			0.139054			0.0695269	−	0.997580i	$-0.477851\pi$
$383$	53.2576			0.139054			0.0695269	+	0.997580i	$0.477851\pi$
$384$			260.367i			0.678039i
$385$		0			0
$386$	320.287			0.829760
$387$			245.529i			0.634441i
$388$	−32.6438			−0.0841334
$389$	−587.027			−1.50907			−0.754533	−	0.656262i	$-0.772137\pi$
$389$	−587.027			−1.50907			−0.754533	+	0.656262i	$0.772137\pi$
$390$		0			0
$391$			258.047i			0.659967i
$392$			44.5231i			0.113579i
$393$		−	196.041i		−	0.498831i
$394$	−14.0385			−0.0356306
$395$		0			0
$396$	−15.0577	−	48.3963i	−0.0380245	−	0.122213i
$397$	−485.797			−1.22367			−0.611835	−	0.790985i	$-0.709569\pi$
$397$	−485.797			−1.22367			−0.611835	+	0.790985i	$0.709569\pi$
$398$		−	550.699i		−	1.38367i
$399$	338.669			0.848795
$400$		0			0
$401$	−166.469			−0.415135			−0.207568	−	0.978221i	$-0.566555\pi$
$401$	−166.469			−0.415135			−0.207568	+	0.978221i	$0.566555\pi$
$402$		−	138.559i		−	0.344673i
$403$		−	393.499i		−	0.976424i
$404$			80.4010i			0.199012i
$405$		0			0
$406$			104.201i			0.256653i
$407$	63.6077	+	204.438i	0.156284	+	0.502306i
$408$	81.8461			0.200603
$409$			270.161i			0.660541i	0.943886	+	0.330271i	$0.107140\pi$
$409$			270.161i			0.660541i	−0.943886	+	0.330271i	$0.892860\pi$
$410$		0			0
$411$	−122.603			−0.298303
$412$	−7.28958			−0.0176932
$413$		−	531.901i		−	1.28790i
$414$			223.475i			0.539796i
$415$		0			0
$416$	−179.611			−0.431758
$417$			226.030i			0.542039i
$418$	233.885	+	751.717i	0.559532	+	1.79837i
$419$	−356.631			−0.851147			−0.425574	−	0.904924i	$-0.639928\pi$
$419$	−356.631			−0.851147			−0.425574	+	0.904924i	$0.639928\pi$
$420$		0			0
$421$	−462.238			−1.09795			−0.548977	−	0.835838i	$-0.684982\pi$
$421$	−462.238			−1.09795			−0.548977	+	0.835838i	$0.684982\pi$
$422$	394.028			0.933716
$423$	−90.5885			−0.214157
$424$		−	151.044i		−	0.356236i
$425$		0			0
$426$			296.402i			0.695778i
$427$	484.774			1.13530
$428$		−	226.887i		−	0.530110i
$429$	−139.881	+	43.5216i	−0.326062	+	0.101449i
$430$		0			0
$431$		−	199.552i		−	0.462997i	−0.972835	−	0.231498i	$-0.925637\pi$
$431$		−	199.552i		−	0.462997i	0.972835	−	0.231498i	$-0.0743628\pi$
$432$	102.804			0.237972
$433$	45.3693			0.104779			0.0523895	−	0.998627i	$-0.483316\pi$
$433$	45.3693			0.104779			0.0523895	+	0.998627i	$0.483316\pi$
$434$	774.018			1.78345
$435$		0			0
$436$		−	5.48081i		−	0.0125707i
$437$		−	963.045i		−	2.20376i
$438$	−223.608			−0.510520
$439$		−	484.630i		−	1.10394i	−0.833864	−	0.551970i	$-0.813876\pi$
$439$		−	484.630i		−	1.10394i	0.833864	−	0.551970i	$-0.186124\pi$
$440$		0			0
$441$	23.0385			0.0522414
$442$			147.451i			0.333600i
$443$	−375.538			−0.847716			−0.423858	−	0.905729i	$-0.639324\pi$
$443$	−375.538			−0.847716			−0.423858	+	0.905729i	$0.639324\pi$
$444$	51.7795			0.116620
$445$		0			0
$446$		−	193.609i		−	0.434102i
$447$			217.880i			0.487427i
$448$			155.412i			0.346901i
$449$	−385.636			−0.858877			−0.429439	−	0.903096i	$-0.641289\pi$
$449$	−385.636			−0.858877			−0.429439	+	0.903096i	$0.641289\pi$
$450$		0			0
$451$	−503.951	+	156.796i	−1.11741	+	0.347664i
$452$	−95.7101			−0.211748
$453$		−	476.726i		−	1.05238i
$454$	−96.4665			−0.212481
$455$		0			0
$456$	−305.454			−0.669855
$457$		−	378.368i		−	0.827939i	−0.910291	−	0.413970i	$-0.864142\pi$
$457$		−	378.368i		−	0.827939i	0.910291	−	0.413970i	$-0.135858\pi$
$458$			378.145i			0.825644i
$459$		−	42.3513i		−	0.0922686i
$460$		0			0
$461$		−	224.522i		−	0.487033i	−0.969897	−	0.243516i	$-0.921699\pi$
$461$		−	224.522i		−	0.487033i	0.969897	−	0.243516i	$-0.0783010\pi$
$462$	−85.6077	−	275.148i	−0.185298	−	0.595557i
$463$	52.3820			0.113136			0.0565680	−	0.998399i	$-0.481984\pi$
$463$	52.3820			0.113136			0.0565680	+	0.998399i	$0.481984\pi$
$464$		−	136.308i		−	0.293768i
$465$		0			0
$466$	959.854			2.05977
$467$	−513.387			−1.09933			−0.549665	−	0.835385i	$-0.685245\pi$
$467$	−513.387			−1.09933			−0.549665	+	0.835385i	$0.685245\pi$
$468$			35.4285i			0.0757020i
$469$		−	218.556i		−	0.466003i
$470$		0			0
$471$	−473.090			−1.00444
$472$			479.734i			1.01639i
$473$	−859.626	+	267.459i	−1.81739	+	0.565451i
$474$	99.1422			0.209161
$475$		0			0
$476$	−80.4693			−0.169053
$477$	−78.1577			−0.163853
$478$	−475.874			−0.995553
$479$			238.730i			0.498392i	0.968453	+	0.249196i	$0.0801663\pi$
$479$			238.730i			0.498392i	−0.968453	+	0.249196i	$0.919834\pi$
$480$		0			0
$481$		−	149.659i		−	0.311142i
$482$	−63.2511			−0.131226
$483$			352.499i			0.729812i
$484$	153.038	−	105.438i	0.316195	−	0.217846i
$485$		0			0
$486$		−	36.6773i		−	0.0754677i
$487$	−593.251			−1.21817			−0.609087	−	0.793103i	$-0.708464\pi$
$487$	−593.251			−1.21817			−0.609087	+	0.793103i	$0.708464\pi$
$488$	−437.229			−0.895962
$489$	107.503			0.219842
$490$		0			0
$491$			858.935i			1.74936i	0.484703	+	0.874679i	$0.338928\pi$
$491$			858.935i			1.74936i	−0.484703	+	0.874679i	$0.661072\pi$
$492$			127.639i			0.259429i
$493$	−56.1539			−0.113902
$494$		−	550.295i		−	1.11396i
$495$		0			0
$496$	−1012.52			−2.04136
$497$			467.529i			0.940702i
$498$	−3.76160			−0.00755341
$499$	−803.692			−1.61061			−0.805303	−	0.592864i	$-0.797997\pi$
$499$	−803.692			−1.61061			−0.805303	+	0.592864i	$0.797997\pi$
$500$		0			0
$501$			83.1038i			0.165876i
$502$			937.085i			1.86670i
$503$			752.302i			1.49563i	0.663907	+	0.747815i	$0.268897\pi$
$503$			752.302i			1.49563i	−0.663907	+	0.747815i	$0.731103\pi$
$504$	111.804			0.221833
$505$		0			0
$506$	−782.414	+	243.435i	−1.54627	+	0.481098i
$507$	−190.317			−0.375378
$508$		−	43.0359i		−	0.0847163i
$509$	−249.268			−0.489721			−0.244860	−	0.969558i	$-0.578742\pi$
$509$	−249.268			−0.489721			−0.244860	+	0.969558i	$0.578742\pi$
$510$		0			0
$511$	−352.708			−0.690230
$512$			3.34215i			0.00652764i
$513$			158.057i			0.308104i
$514$			794.835i			1.54637i
$515$		0			0
$516$			217.723i			0.421944i
$517$	−98.6795	−	317.161i	−0.190869	−	0.613464i
$518$	294.382			0.568305
$519$		−	247.713i		−	0.477288i
$520$		0			0
$521$	475.864			0.913367			0.456683	−	0.889629i	$-0.349037\pi$
$521$	475.864			0.913367			0.456683	+	0.889629i	$0.349037\pi$
$522$	−48.6307			−0.0931623
$523$		−	362.833i		−	0.693754i	−0.937911	−	0.346877i	$-0.887242\pi$
$523$		−	362.833i		−	0.693754i	0.937911	−	0.346877i	$-0.112758\pi$
$524$		−	173.839i		−	0.331754i
$525$		0			0
$526$	316.526			0.601760
$527$			417.118i			0.791495i
$528$	111.986	+	359.929i	0.212095	+	0.681683i
$529$	473.372			0.894843
$530$		0			0
$531$	248.238			0.467492
$532$	300.315			0.564503
$533$	368.918			0.692154
$534$		−	182.665i		−	0.342070i
$535$		0			0
$536$			197.120i			0.367762i
$537$	−63.3975			−0.118059
$538$		−	804.370i		−	1.49511i
$539$	25.0962	+	80.6604i	0.0465606	+	0.149648i
$540$		0			0
$541$		−	830.667i		−	1.53543i	−0.640792	−	0.767715i	$-0.721394\pi$
$541$		−	830.667i		−	1.53543i	0.640792	−	0.767715i	$-0.278606\pi$
$542$	501.857			0.925936
$543$	145.790			0.268489
$544$	190.392			0.349986
$545$		0			0
$546$			201.422i			0.368905i
$547$		−	456.091i		−	0.833804i	−0.908952	−	0.416902i	$-0.863116\pi$
$547$		−	456.091i		−	0.833804i	0.908952	−	0.416902i	$-0.136884\pi$
$548$	−108.718			−0.198390
$549$			226.244i			0.412103i
$550$		0			0
$551$	209.569			0.380343
$552$		−	317.927i		−	0.575955i
$553$	156.382			0.282788
$554$	199.001			0.359208
$555$		0			0
$556$			200.433i			0.360490i
$557$			338.810i			0.608277i	0.952628	+	0.304138i	$0.0983686\pi$
$557$			338.810i			0.608277i	−0.952628	+	0.304138i	$0.901631\pi$
$558$			361.235i			0.647374i
$559$	629.290			1.12574
$560$		0			0
$561$	148.277	−	46.1339i	0.264308	−	0.0822352i
$562$	826.197			1.47010
$563$		−	447.041i		−	0.794034i	−0.917811	−	0.397017i	$-0.870045\pi$
$563$		−	447.041i		−	0.794034i	0.917811	−	0.397017i	$-0.129955\pi$
$564$	−80.3294			−0.142428
$565$		0			0
$566$	−672.782			−1.18866
$567$		−	57.8529i		−	0.102033i
$568$		−	421.676i		−	0.742387i
$569$		−	522.893i		−	0.918969i	−0.888186	−	0.459485i	$-0.848034\pi$
$569$		−	522.893i		−	0.918969i	0.888186	−	0.459485i	$-0.151966\pi$
$570$		0			0
$571$			904.574i			1.58419i	0.610396	+	0.792096i	$0.291010\pi$
$571$			904.574i			1.58419i	−0.610396	+	0.792096i	$0.708990\pi$
$572$	−124.039	+	38.5929i	−0.216852	+	0.0674701i
$573$	−76.6166			−0.133711
$574$			725.667i			1.26423i
$575$		0			0
$576$	−72.5307			−0.125921
$577$	−237.674			−0.411914			−0.205957	−	0.978561i	$-0.566031\pi$
$577$	−237.674			−0.411914			−0.205957	+	0.978561i	$0.566031\pi$
$578$			523.672i			0.906007i
$579$			235.780i			0.407219i
$580$		0			0
$581$	−5.93336			−0.0102123
$582$		−	86.6149i		−	0.148823i
$583$	−85.1384	−	273.639i	−0.146035	−	0.469364i
$584$	318.115			0.544718
$585$		0			0
$586$	926.851			1.58166
$587$	578.515			0.985546			0.492773	−	0.870158i	$-0.335983\pi$
$587$	578.515			0.985546			0.492773	+	0.870158i	$0.335983\pi$
$588$	20.4294			0.0347439
$589$		−	1556.71i		−	2.64296i
$590$		0			0
$591$		−	10.3344i		−	0.0174863i
$592$	−385.090			−0.650489
$593$			333.857i			0.562997i	0.959562	+	0.281499i	$0.0908314\pi$
$593$			333.857i			0.562997i	−0.959562	+	0.281499i	$0.909169\pi$
$594$	128.412	−	39.9532i	0.216181	−	0.0672612i
$595$		0			0
$596$			193.205i			0.324170i
$597$	405.397			0.679058
$598$	572.767			0.957804
$599$	−846.483			−1.41316			−0.706580	−	0.707633i	$-0.749763\pi$
$599$	−846.483			−1.41316			−0.706580	+	0.707633i	$0.749763\pi$
$600$		0			0
$601$		−	455.011i		−	0.757089i	−0.925583	−	0.378545i	$-0.876425\pi$
$601$		−	455.011i		−	0.757089i	0.925583	−	0.378545i	$-0.123575\pi$
$602$			1237.82i			2.05618i
$603$	102.000			0.169154
$604$		−	422.738i		−	0.699897i
$605$		0			0
$606$	−213.331			−0.352031
$607$			726.557i			1.19696i	0.801137	+	0.598482i	$0.204229\pi$
$607$			726.557i			1.19696i	−0.801137	+	0.598482i	$0.795771\pi$
$608$	−710.554			−1.16867
$609$	−76.7077			−0.125957
$610$		0			0
$611$			232.178i			0.379997i
$612$		−	37.5551i		−	0.0613645i
$613$			574.532i			0.937247i	0.883398	+	0.468624i	$0.155250\pi$
$613$			574.532i			0.937247i	−0.883398	+	0.468624i	$0.844750\pi$
$614$	136.992			0.223114
$615$		0			0
$616$	121.790	+	391.439i	0.197711	+	0.635452i
$617$	278.946			0.452101			0.226050	−	0.974116i	$-0.427419\pi$
$617$	278.946			0.452101			0.226050	+	0.974116i	$0.427419\pi$
$618$		−	19.3417i		−	0.0312973i
$619$	−593.061			−0.958096			−0.479048	−	0.877789i	$-0.659018\pi$
$619$	−593.061			−0.958096			−0.479048	+	0.877789i	$0.659018\pi$
$620$		0			0
$621$	−164.512			−0.264914
$622$			488.478i			0.785335i
$623$		−	288.127i		−	0.462484i
$624$		−	263.486i		−	0.422253i
$625$		0			0
$626$			886.312i			1.41583i
$627$	−553.377	+	172.174i	−0.882579	+	0.274600i
$628$	−419.513			−0.668014
$629$			158.642i			0.252214i
$630$		0			0
$631$	−361.664			−0.573160			−0.286580	−	0.958056i	$-0.592518\pi$
$631$	−361.664			−0.573160			−0.286580	+	0.958056i	$0.592518\pi$
$632$	−141.045			−0.223172
$633$			290.064i			0.458237i
$634$		−	580.704i		−	0.915937i
$635$		0			0
$636$	−69.3064			−0.108972
$637$		−	59.0475i		−	0.0926963i
$638$	−52.9742	−	170.262i	−0.0830317	−	0.266868i
$639$	−218.196			−0.341465
$640$		0			0
$641$	784.382			1.22368			0.611842	−	0.790980i	$-0.290429\pi$
$641$	784.382			1.22368			0.611842	+	0.790980i	$0.290429\pi$
$642$	602.008			0.937706
$643$	−362.764			−0.564174			−0.282087	−	0.959389i	$-0.591027\pi$
$643$	−362.764			−0.564174			−0.282087	+	0.959389i	$0.591027\pi$
$644$			312.579i			0.485371i
$645$		0			0
$646$			583.325i			0.902981i
$647$	−921.727			−1.42462			−0.712308	−	0.701867i	$-0.752350\pi$
$647$	−921.727			−1.42462			−0.712308	+	0.701867i	$0.752350\pi$
$648$			52.1789i			0.0805230i
$649$	270.410	+	869.112i	0.416657	+	1.33916i
$650$		0			0
$651$			569.794i			0.875259i
$652$	95.3281			0.146209
$653$	−26.8653			−0.0411414			−0.0205707	−	0.999788i	$-0.506548\pi$
$653$	−26.8653			−0.0411414			−0.0205707	+	0.999788i	$0.506548\pi$
$654$	14.5424			0.0222361
$655$		0			0
$656$		−	949.266i		−	1.44705i
$657$		−	164.609i		−	0.250546i
$658$	−456.697			−0.694069
$659$			320.820i			0.486828i	0.969922	+	0.243414i	$0.0782674\pi$
$659$			320.820i			0.486828i	−0.969922	+	0.243414i	$0.921733\pi$
$660$		0			0
$661$	331.969			0.502222			0.251111	−	0.967958i	$-0.419204\pi$
$661$	331.969			0.502222			0.251111	+	0.967958i	$0.419204\pi$
$662$			344.530i			0.520437i
$663$	−108.546			−0.163720
$664$	5.35144			0.00805939
$665$		0			0
$666$			137.388i			0.206289i
$667$			218.127i			0.327027i
$668$			73.6924i			0.110318i
$669$	142.526			0.213043
$670$		0			0
$671$	−792.109	+	246.452i	−1.18049	+	0.367290i
$672$	260.081			0.387025
$673$			797.019i			1.18428i	0.805836	+	0.592139i	$0.201716\pi$
$673$			797.019i			1.18428i	−0.805836	+	0.592139i	$0.798284\pi$
$674$	832.925			1.23579
$675$		0			0
$676$	−168.764			−0.249650
$677$		−	534.175i		−	0.789033i	−0.918889	−	0.394516i	$-0.870912\pi$
$677$		−	534.175i		−	0.789033i	0.918889	−	0.394516i	$-0.129088\pi$
$678$		−	253.951i		−	0.374559i
$679$		−	136.622i		−	0.201211i
$680$		0			0
$681$		−	71.0139i		−	0.104279i
$682$	−1264.73	+	393.499i	−1.85444	+	0.576978i
$683$	843.097			1.23440			0.617201	−	0.786805i	$-0.288266\pi$
$683$	843.097			1.23440			0.617201	+	0.786805i	$0.288266\pi$
$684$			140.157i			0.204908i
$685$		0			0
$686$	857.241			1.24962
$687$	−278.372			−0.405199
$688$		−	1619.23i		−	2.35353i
$689$			200.318i			0.290737i
$690$		0			0
$691$	−193.615			−0.280196			−0.140098	−	0.990138i	$-0.544742\pi$
$691$	−193.615			−0.280196			−0.140098	+	0.990138i	$0.544742\pi$
$692$		−	219.660i		−	0.317427i
$693$	202.550	−	63.0201i	0.292280	−	0.0909382i
$694$	−1246.56			−1.79619
$695$		0			0
$696$	69.1845			0.0994030
$697$	−391.061			−0.561064
$698$	806.729			1.15577
$699$			706.597i			1.01087i
$700$		0			0
$701$			448.863i			0.640318i	0.947364	+	0.320159i	$0.103736\pi$
$701$			448.863i			0.640318i	−0.947364	+	0.320159i	$0.896264\pi$
$702$	−94.0038			−0.133909
$703$		−	592.061i		−	0.842192i
$704$	−79.0089	−	253.939i	−0.112229	−	0.360708i
$705$		0			0
$706$		−	451.723i		−	0.639834i
$707$	−336.497			−0.475951
$708$	220.126			0.310912
$709$	−311.031			−0.438689			−0.219345	−	0.975647i	$-0.570392\pi$
$709$	−311.031			−0.438689			−0.219345	+	0.975647i	$0.570392\pi$
$710$		0			0
$711$			72.9836i			0.102649i
$712$			259.869i			0.364985i
$713$	1620.27			2.27247
$714$		−	213.512i		−	0.299036i
$715$		0			0
$716$	−56.2178			−0.0785165
$717$		−	350.315i		−	0.488584i
$718$	1415.60			1.97159
$719$	486.014			0.675958			0.337979	−	0.941154i	$-0.390257\pi$
$719$	486.014			0.675958			0.337979	+	0.941154i	$0.390257\pi$
$720$		0			0
$721$		−	30.5086i		−	0.0423143i
$722$		−	1327.62i		−	1.83881i
$723$		−	46.5623i		−	0.0644016i
$724$	129.279			0.178563
$725$		0			0
$726$	279.762	+	406.062i	0.385347	+	0.559315i
$727$	−38.8616			−0.0534547			−0.0267273	−	0.999643i	$-0.508509\pi$
$727$	−38.8616			−0.0534547			−0.0267273	+	0.999643i	$0.508509\pi$
$728$		−	286.553i		−	0.393617i
$729$	27.0000			0.0370370
$730$		0			0
$731$	−667.061			−0.912533
$732$			200.623i			0.274075i
$733$			175.562i			0.239511i	0.992803	+	0.119756i	$0.0382111\pi$
$733$			175.562i			0.239511i	−0.992803	+	0.119756i	$0.961789\pi$
$734$			328.023i			0.446898i
$735$		0			0
$736$		−	739.570i		−	1.00485i
$737$	111.110	+	357.114i	0.150760	+	0.484551i
$738$	−338.669			−0.458901
$739$			494.527i			0.669184i	0.942363	+	0.334592i	$0.108598\pi$
$739$			494.527i			0.669184i	−0.942363	+	0.334592i	$0.891402\pi$
$740$		0			0
$741$	405.100			0.546694
$742$	−394.028			−0.531035
$743$			1150.84i			1.54892i	0.632625	+	0.774458i	$0.281977\pi$
$743$			1150.84i			1.54892i	−0.632625	+	0.774458i	$0.718023\pi$
$744$		−	513.910i		−	0.690740i
$745$		0			0
$746$	−714.393			−0.957632
$747$		−	2.76910i		−	0.00370696i
$748$	131.485	−	40.9093i	0.175782	−	0.0546916i
$749$	949.577			1.26779
$750$		0			0
$751$	1343.73			1.78926			0.894628	−	0.446813i	$-0.147441\pi$
$751$	1343.73			1.78926			0.894628	+	0.446813i	$0.147441\pi$
$752$	597.419			0.794440
$753$	−689.836			−0.916117
$754$			124.640i			0.165306i
$755$		0			0
$756$		−	51.3012i		−	0.0678587i
$757$	−1034.58			−1.36669			−0.683343	−	0.730097i	$-0.739475\pi$
$757$	−1034.58			−1.36669			−0.683343	+	0.730097i	$0.739475\pi$
$758$			1623.57i			2.14192i
$759$	−179.205	−	575.974i	−0.236107	−	0.758859i
$760$		0			0
$761$			798.527i			1.04931i	0.851314	+	0.524656i	$0.175806\pi$
$761$			798.527i			1.04931i	−0.851314	+	0.524656i	$0.824194\pi$
$762$	114.189			0.149854
$763$	22.9385			0.0300636
$764$	−67.9399			−0.0889266
$765$		0			0
$766$			125.307i			0.163586i
$767$		−	636.234i		−	0.829510i
$768$	−445.101			−0.579559
$769$			261.311i			0.339806i	0.985461	+	0.169903i	$0.0543455\pi$
$769$			261.311i			0.339806i	−0.985461	+	0.169903i	$0.945655\pi$
$770$		0			0
$771$	−585.118			−0.758908
$772$			209.078i			0.270826i
$773$	327.734			0.423977			0.211989	−	0.977272i	$-0.432006\pi$
$773$	327.734			0.423977			0.211989	+	0.977272i	$0.432006\pi$
$774$	−577.692			−0.746372
$775$		0			0
$776$			123.223i			0.158792i
$777$			216.709i			0.278905i
$778$		−	1381.19i		−	1.77530i
$779$	1459.46			1.87351
$780$		0			0
$781$	−237.685	−	763.931i	−0.304334	−	0.978144i
$782$	−607.146			−0.776402
$783$		−	35.7995i		−	0.0457210i
$784$	−151.936			−0.193796
$785$		0			0
$786$	461.254			0.586837
$787$		−	289.388i		−	0.367711i	−0.982953	−	0.183855i	$-0.941142\pi$
$787$		−	289.388i		−	0.367711i	0.982953	−	0.183855i	$-0.0588578\pi$
$788$		−	9.16407i		−	0.0116295i
$789$			233.010i			0.295324i
$790$		0			0
$791$		−	400.570i		−	0.506409i
$792$	−182.685	+	56.8394i	−0.230662	+	0.0717669i
$793$	579.864			0.731228
$794$		−	1143.01i		−	1.43956i
$795$		0			0
$796$	359.487			0.451617
$797$	−113.681			−0.142636			−0.0713180	−	0.997454i	$-0.522721\pi$
$797$	−113.681			−0.142636			−0.0713180	+	0.997454i	$0.522721\pi$
$798$			796.837i			0.998543i
$799$		−	246.114i		−	0.308028i
$800$		0			0
$801$	134.469			0.167877
$802$		−	391.677i		−	0.488375i
$803$	576.315	−	179.311i	0.717703	−	0.223302i
$804$	90.4486			0.112498
$805$		0			0
$806$	925.843			1.14869
$807$	592.137			0.733751
$808$	303.495			0.375613
$809$			854.163i			1.05583i	0.849299	+	0.527913i	$0.177025\pi$
$809$			854.163i			1.05583i	−0.849299	+	0.527913i	$0.822975\pi$
$810$		0			0
$811$		−	486.533i		−	0.599917i	−0.953952	−	0.299959i	$-0.903027\pi$
$811$		−	486.533i		−	0.599917i	0.953952	−	0.299959i	$-0.0969729\pi$
$812$	−68.0206			−0.0837692
$813$			369.443i			0.454419i
$814$	−481.013	+	149.659i	−0.590925	+	0.183857i
$815$		0			0
$816$			279.301i			0.342281i
$817$	2489.51			3.04713
$818$	−635.649			−0.777077
$819$	−148.277			−0.181046
$820$		0			0
$821$		−	527.104i		−	0.642027i	−0.947075	−	0.321014i	$-0.895976\pi$
$821$		−	527.104i		−	0.642027i	0.947075	−	0.321014i	$-0.104024\pi$
$822$		−	288.465i		−	0.350931i
$823$	314.805			0.382509			0.191255	−	0.981540i	$-0.438744\pi$
$823$	314.805			0.382509			0.191255	+	0.981540i	$0.438744\pi$
$824$			27.5165i			0.0333938i
$825$		0			0
$826$	1251.48			1.51511
$827$		−	111.652i		−	0.135008i	−0.997719	−	0.0675040i	$-0.978496\pi$
$827$		−	111.652i		−	0.135008i	0.997719	−	0.0675040i	$-0.0215035\pi$
$828$	−145.881			−0.176185
$829$	−695.395			−0.838836			−0.419418	−	0.907793i	$-0.637766\pi$
$829$	−695.395			−0.838836			−0.419418	+	0.907793i	$0.637766\pi$
$830$		0			0
$831$			146.495i			0.176288i
$832$			185.896i			0.223433i
$833$			62.5918i			0.0751402i
$834$	−531.815			−0.637668
$835$		0			0
$836$	−490.708	+	152.676i	−0.586971	+	0.182626i
$837$	−265.923			−0.317710
$838$		−	839.098i		−	1.00131i
$839$	−383.968			−0.457650			−0.228825	−	0.973468i	$-0.573488\pi$
$839$	−383.968			−0.457650			−0.228825	+	0.973468i	$0.573488\pi$
$840$		0			0
$841$	793.533			0.943559
$842$		−	1087.58i		−	1.29166i
$843$			608.205i			0.721477i
$844$			257.215i			0.304757i
$845$		0			0
$846$		−	213.141i		−	0.251940i
$847$	441.282	+	640.503i	0.520994	+	0.756202i
$848$	515.440			0.607830
$849$		−	495.269i		−	0.583355i
$850$		0			0
$851$	616.238			0.724134
$852$	−193.486			−0.227096
$853$		−	570.527i		−	0.668847i	−0.942423	−	0.334424i	$-0.891458\pi$
$853$		−	570.527i		−	0.668847i	0.942423	−	0.334424i	$-0.108542\pi$
$854$			1140.60i			1.33560i
$855$		0			0
$856$	−856.446			−1.00052
$857$		−	315.553i		−	0.368207i	−0.982907	−	0.184103i	$-0.941062\pi$
$857$		−	315.553i		−	0.368207i	0.982907	−	0.184103i	$-0.0589382\pi$
$858$	−102.400	−	329.118i	−0.119347	−	0.383588i
$859$	−107.923			−0.125638			−0.0628190	−	0.998025i	$-0.520009\pi$
$859$	−107.923			−0.125638			−0.0628190	+	0.998025i	$0.520009\pi$
$860$		0			0
$861$	−534.200			−0.620441
$862$	469.515			0.544681
$863$	−1431.55			−1.65881			−0.829403	−	0.558651i	$-0.811319\pi$
$863$	−1431.55			−1.65881			−0.829403	+	0.558651i	$0.811319\pi$
$864$			121.380i			0.140486i
$865$		0			0
$866$			106.747i			0.123265i
$867$	−385.501			−0.444638
$868$			505.265i			0.582103i
$869$	−255.524	+	79.5022i	−0.294044	+	0.0914870i
$870$		0			0
$871$		−	261.426i		−	0.300144i
$872$	−20.6888			−0.0237257
$873$	63.7616			0.0730373
$874$	2265.90			2.59256
$875$		0			0
$876$		−	145.967i		−	0.166629i
$877$			567.758i			0.647386i	0.946162	+	0.323693i	$0.104925\pi$
$877$			567.758i			0.647386i	−0.946162	+	0.323693i	$0.895075\pi$
$878$	1140.26			1.29870
$879$			682.302i			0.776225i
$880$		0			0
$881$	488.641			0.554644			0.277322	−	0.960777i	$-0.410553\pi$
$881$	488.641			0.554644			0.277322	+	0.960777i	$0.410553\pi$
$882$			54.2061i			0.0614581i
$883$	−840.144			−0.951465			−0.475732	−	0.879590i	$-0.657817\pi$
$883$	−840.144			−0.951465			−0.475732	+	0.879590i	$0.657817\pi$
$884$	−96.2535			−0.108884
$885$		0			0
$886$		−	883.585i		−	0.997274i
$887$			981.439i			1.10647i	0.833025	+	0.553235i	$0.186607\pi$
$887$			981.439i			1.10647i	−0.833025	+	0.553235i	$0.813393\pi$
$888$		−	195.455i		−	0.220107i
$889$	180.115			0.202605
$890$		0			0
$891$	29.4115	+	94.5302i	0.0330096	+	0.106095i
$892$	126.385			0.141687
$893$			918.510i			1.02857i
$894$	−512.638			−0.573421
$895$		0			0
$896$	−966.291			−1.07845
$897$			421.642i			0.470059i
$898$		−	907.343i		−	1.01040i
$899$			352.589i			0.392202i
$900$		0			0
$901$		−	212.342i		−	0.235673i
$902$	−368.918	−	1185.72i	−0.409000	−	1.31455i
$903$	−911.223			−1.00911
$904$			361.283i			0.399650i
$905$		0			0
$906$	1121.67			1.23804
$907$	−438.610			−0.483583			−0.241792	−	0.970328i	$-0.577735\pi$
$907$	−438.610			−0.483583			−0.241792	+	0.970328i	$0.577735\pi$
$908$		−	62.9716i		−	0.0693520i
$909$		−	157.044i		−	0.172765i
$910$		0			0
$911$	−170.042			−0.186654			−0.0933272	−	0.995635i	$-0.529750\pi$
$911$	−170.042			−0.186654			−0.0933272	+	0.995635i	$0.529750\pi$
$912$		−	1042.37i		−	1.14295i
$913$	9.69496	−	3.01643i	0.0106188	−	0.00330386i
$914$	890.243			0.974008
$915$		0			0
$916$	−246.846			−0.269483
$917$	727.559			0.793412
$918$	99.6462			0.108547
$919$		−	1089.73i		−	1.18578i	−0.805285	−	0.592888i	$-0.797988\pi$
$919$		−	1089.73i		−	1.18578i	0.805285	−	0.592888i	$-0.202012\pi$
$920$		0			0
$921$			100.847i			0.109497i
$922$	528.267			0.572957
$923$			559.236i			0.605890i
$924$	179.611	−	55.8832i	0.194385	−	0.0604796i
$925$		0			0
$926$			123.247i			0.133096i
$927$	14.2384			0.0153597
$928$	160.939			0.173425
$929$	769.446			0.828252			0.414126	−	0.910220i	$-0.364087\pi$
$929$	769.446			0.828252			0.414126	+	0.910220i	$0.364087\pi$
$930$		0			0
$931$		−	233.596i		−	0.250908i
$932$			626.576i			0.672291i
$933$	−359.594			−0.385417
$934$		−	1207.92i		−	1.29328i
$935$		0			0
$936$	133.734			0.142879
$937$		−	1606.78i		−	1.71481i	−0.514641	−	0.857406i	$-0.672075\pi$
$937$		−	1606.78i		−	1.71481i	0.514641	−	0.857406i	$-0.327925\pi$
$938$	514.228			0.548218
$939$	−652.459			−0.694844
$940$		0			0
$941$		−	1279.34i		−	1.35955i	−0.733419	−	0.679777i	$-0.762077\pi$
$941$		−	1279.34i		−	1.35955i	0.733419	−	0.679777i	$-0.237923\pi$
$942$		−	1113.11i		−	1.18164i
$943$			1519.06i			1.61088i
$944$	−1637.10			−1.73422
$945$		0			0
$946$	−629.290	−	2022.57i	−0.665211	−	2.13802i
$947$	985.800			1.04097			0.520486	−	0.853870i	$-0.325751\pi$
$947$	985.800			1.04097			0.520486	+	0.853870i	$0.325751\pi$
$948$			64.7183i			0.0682682i
$949$	−421.892			−0.444565
$950$		0			0
$951$	427.486			0.449512
$952$			303.753i			0.319068i
$953$		−	761.038i		−	0.798571i	−0.916827	−	0.399285i	$-0.869258\pi$
$953$		−	761.038i		−	0.798571i	0.916827	−	0.399285i	$-0.130742\pi$
$954$		−	183.893i		−	0.192760i
$955$		0			0
$956$		−	310.642i		−	0.324940i
$957$	125.338	−	38.9970i	0.130970	−	0.0407492i
$958$	−561.695			−0.586320
$959$		−	455.011i		−	0.474464i
$960$		0			0
$961$	1658.08			1.72537
$962$	352.126			0.366035
$963$			443.168i			0.460195i
$964$		−	41.2892i		−	0.0428311i
$965$		0			0
$966$	−829.377			−0.858568
$967$		−	702.938i		−	0.726926i	−0.931609	−	0.363463i	$-0.881594\pi$
$967$		−	702.938i		−	0.726926i	0.931609	−	0.363463i	$-0.118406\pi$
$968$	−398.003	−	577.685i	−0.411160	−	0.596782i
$969$	−429.415			−0.443153
$970$		0			0
$971$	−1289.42			−1.32793			−0.663963	−	0.747766i	$-0.731127\pi$
$971$	−1289.42			−1.32793			−0.663963	+	0.747766i	$0.731127\pi$
$972$	23.9423			0.0246320
$973$	−838.859			−0.862136
$974$		−	1395.83i		−	1.43309i
$975$		0			0
$976$		−	1492.05i		−	1.52874i
$977$	690.526			0.706782			0.353391	−	0.935476i	$-0.385029\pi$
$977$	690.526			0.706782			0.353391	+	0.935476i	$0.385029\pi$
$978$			252.937i			0.258627i
$979$	146.480	+	470.793i	0.149622	+	0.480892i
$980$		0			0
$981$			10.7054i			0.0109128i
$982$	−2020.94			−2.05799
$983$	−1080.40			−1.09908			−0.549540	−	0.835467i	$-0.685197\pi$
$983$	−1080.40			−1.09908			−0.549540	+	0.835467i	$0.685197\pi$
$984$	481.808			0.489642
$985$		0			0
$986$		−	132.122i		−	0.133998i
$987$		−	336.198i		−	0.340626i
$988$	359.223			0.363586
$989$			2591.17i			2.61999i
$990$		0			0
$991$	−686.561			−0.692796			−0.346398	−	0.938088i	$-0.612595\pi$
$991$	−686.561			−0.692796			−0.346398	+	0.938088i	$0.612595\pi$
$992$		−	1195.47i		−	1.20511i
$993$	−253.626			−0.255413
$994$	−1100.03			−1.10667
$995$		0			0
$996$		−	2.45551i		−	0.00246537i
$997$		−	1413.50i		−	1.41775i	−0.705333	−	0.708876i	$-0.749203\pi$
$997$		−	1413.50i		−	1.41775i	0.705333	−	0.708876i	$-0.250797\pi$
$998$		−	1890.97i		−	1.89476i
$999$	−101.138			−0.101240

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	825.3.b.a.76.3		4
5.2	odd	4		825.3.h.a.274.4		8
5.3	odd	4		825.3.h.a.274.5		8
5.4	even	2		33.3.c.a.10.2	✓	4
11.10	odd	2	inner	825.3.b.a.76.2		4
15.14	odd	2		99.3.c.b.10.3		4
20.19	odd	2		528.3.j.c.241.1		4
40.19	odd	2		2112.3.j.d.769.3		4
40.29	even	2		2112.3.j.a.769.2		4
55.4	even	10		363.3.g.e.94.3		16
55.9	even	10		363.3.g.e.40.2		16
55.14	even	10		363.3.g.e.112.2		16
55.19	odd	10		363.3.g.e.112.3		16
55.24	odd	10		363.3.g.e.40.3		16
55.29	odd	10		363.3.g.e.94.2		16
55.32	even	4		825.3.h.a.274.6		8
55.39	odd	10		363.3.g.e.118.2		16
55.43	even	4		825.3.h.a.274.3		8
55.49	even	10		363.3.g.e.118.3		16
55.54	odd	2		33.3.c.a.10.3	yes	4
60.59	even	2		1584.3.j.f.1297.3		4
165.164	even	2		99.3.c.b.10.2		4
220.219	even	2		528.3.j.c.241.2		4
440.109	odd	2		2112.3.j.a.769.1		4
440.219	even	2		2112.3.j.d.769.4		4
660.659	odd	2		1584.3.j.f.1297.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
33.3.c.a.10.2	✓	4	5.4	even	2
33.3.c.a.10.3	yes	4	55.54	odd	2
99.3.c.b.10.2		4	165.164	even	2
99.3.c.b.10.3		4	15.14	odd	2
363.3.g.e.40.2		16	55.9	even	10
363.3.g.e.40.3		16	55.24	odd	10
363.3.g.e.94.2		16	55.29	odd	10
363.3.g.e.94.3		16	55.4	even	10
363.3.g.e.112.2		16	55.14	even	10
363.3.g.e.112.3		16	55.19	odd	10
363.3.g.e.118.2		16	55.39	odd	10
363.3.g.e.118.3		16	55.49	even	10
528.3.j.c.241.1		4	20.19	odd	2
528.3.j.c.241.2		4	220.219	even	2
825.3.b.a.76.2		4	11.10	odd	2	inner
825.3.b.a.76.3		4	1.1	even	1	trivial
825.3.h.a.274.3		8	55.43	even	4
825.3.h.a.274.4		8	5.2	odd	4
825.3.h.a.274.5		8	5.3	odd	4
825.3.h.a.274.6		8	55.32	even	4
1584.3.j.f.1297.3		4	60.59	even	2
1584.3.j.f.1297.4		4	660.659	odd	2
2112.3.j.a.769.1		4	440.109	odd	2
2112.3.j.a.769.2		4	40.29	even	2
2112.3.j.d.769.3		4	40.19	odd	2
2112.3.j.d.769.4		4	440.219	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 825 = 3 \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	825.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+2.35285i q^{2} -1.73205 q^{3} -1.53590 q^{4} -4.07525i q^{6} -6.42810i q^{7} +5.79766i q^{8} +3.00000 q^{9} +O(q^{10})\)	\(q+2.35285i q^{2} -1.73205 q^{3} -1.53590 q^{4} -4.07525i q^{6} -6.42810i q^{7} +5.79766i q^{8} +3.00000 q^{9} +(3.26795 + 10.5034i) q^{11} +2.66025 q^{12} -7.68899i q^{13} +15.1244 q^{14} -19.7846 q^{16} +8.15051i q^{17} +7.05855i q^{18} -30.4181i q^{19} +11.1338i q^{21} +(-24.7128 + 7.68899i) q^{22} +31.6603 q^{23} -10.0418i q^{24} +18.0910 q^{26} -5.19615 q^{27} +9.87291i q^{28} +6.88962i q^{29} +51.1769 q^{31} -23.3596i q^{32} +(-5.66025 - 18.1923i) q^{33} -19.1769 q^{34} -4.60770 q^{36} +19.4641 q^{37} +71.5692 q^{38} +13.3177i q^{39} +47.9800i q^{41} -26.1962 q^{42} +81.8429i q^{43} +(-5.01924 - 16.1321i) q^{44} +74.4918i q^{46} -30.1962 q^{47} +34.2679 q^{48} +7.67949 q^{49} -14.1171i q^{51} +11.8095i q^{52} -26.0526 q^{53} -12.2258i q^{54} +37.2679 q^{56} +52.6857i q^{57} -16.2102 q^{58} +82.7461 q^{59} +75.4148i q^{61} +120.412i q^{62} -19.2843i q^{63} -24.1769 q^{64} +(42.8038 - 13.3177i) q^{66} +34.0000 q^{67} -12.5184i q^{68} -54.8372 q^{69} -72.7321 q^{71} +17.3930i q^{72} -54.8696i q^{73} +45.7961i q^{74} +46.7191i q^{76} +(67.5167 - 21.0067i) q^{77} -31.3346 q^{78} +24.3279i q^{79} +9.00000 q^{81} -112.890 q^{82} -0.923034i q^{83} -17.1004i q^{84} -192.564 q^{86} -11.9332i q^{87} +(-60.8949 + 18.9465i) q^{88} +44.8231 q^{89} -49.4256 q^{91} -48.6269 q^{92} -88.6410 q^{93} -71.0470i q^{94} +40.4599i q^{96} +21.2539 q^{97} +18.0687i q^{98} +(9.80385 + 31.5101i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 20 q^{4} + 12 q^{9}+O(q^{10}) \) 4 * q - 20 * q^4 + 12 * q^9	\( 4 q - 20 q^{4} + 12 q^{9} + 20 q^{11} - 24 q^{12} + 12 q^{14} + 4 q^{16} + 12 q^{22} + 92 q^{23} + 204 q^{26} + 80 q^{31} + 12 q^{33} + 48 q^{34} - 60 q^{36} + 64 q^{37} + 120 q^{38} - 84 q^{42} - 124 q^{44} - 100 q^{47} + 144 q^{48} + 100 q^{49} - 28 q^{53} + 156 q^{56} + 240 q^{58} + 40 q^{59} + 28 q^{64} + 192 q^{66} + 136 q^{67} - 60 q^{69} - 284 q^{71} + 180 q^{77} + 228 q^{78} + 36 q^{81} - 216 q^{82} - 216 q^{86} - 396 q^{88} + 304 q^{89} + 24 q^{91} - 340 q^{92} - 216 q^{93} + 376 q^{97} + 60 q^{99}+O(q^{100}) \) 4 * q - 20 * q^4 + 12 * q^9 + 20 * q^11 - 24 * q^12 + 12 * q^14 + 4 * q^16 + 12 * q^22 + 92 * q^23 + 204 * q^26 + 80 * q^31 + 12 * q^33 + 48 * q^34 - 60 * q^36 + 64 * q^37 + 120 * q^38 - 84 * q^42 - 124 * q^44 - 100 * q^47 + 144 * q^48 + 100 * q^49 - 28 * q^53 + 156 * q^56 + 240 * q^58 + 40 * q^59 + 28 * q^64 + 192 * q^66 + 136 * q^67 - 60 * q^69 - 284 * q^71 + 180 * q^77 + 228 * q^78 + 36 * q^81 - 216 * q^82 - 216 * q^86 - 396 * q^88 + 304 * q^89 + 24 * q^91 - 340 * q^92 - 216 * q^93 + 376 * q^97 + 60 * q^99

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