LMFDB - Embedded newform 2112.3.j.d.769.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2112,3,Mod(769,2112)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2112.769"); S:= CuspForms(chi, 3); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2112, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 1])) N = Newforms(chi, 3, names="a")

Level:	$ N $	$=$	$ 2112 = 2^{6} \cdot 3 \cdot 11 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	2112.j (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,-4,0,0,0,12,0,20,0,0,0,36,0,0,0,0,0,0,0,92,0,12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(25)] == traces)

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

Self dual:	no
Analytic conductor:	$57.5478318329$
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:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 33)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			769.4
Root			$1.36603 + 3.21405i$ of defining polynomial
Character	$\chi$	$=$	2112.769
Dual form			2112.3.j.d.769.3

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.73205 q^{3} +4.19615 q^{5} +6.42810i q^{7} +3.00000 q^{9} +(3.26795 - 10.5034i) q^{11} +7.68899i q^{13} +7.26795 q^{15} +8.15051i q^{17} +30.4181i q^{19} +11.1338i q^{21} +31.6603 q^{23} -7.39230 q^{25} +5.19615 q^{27} +6.88962i q^{29} -51.1769 q^{31} +(5.66025 - 18.1923i) q^{33} +26.9733i q^{35} +19.4641 q^{37} +13.3177i q^{39} -47.9800i q^{41} +81.8429i q^{43} +12.5885 q^{45} -30.1962 q^{47} +7.67949 q^{49} +14.1171i q^{51} -26.0526 q^{53} +(13.7128 - 44.0737i) q^{55} +52.6857i q^{57} +82.7461 q^{59} +75.4148i q^{61} +19.2843i q^{63} +32.2642i q^{65} -34.0000 q^{67} +54.8372 q^{69} +72.7321 q^{71} -54.8696i q^{73} -12.8038 q^{75} +(67.5167 + 21.0067i) q^{77} +24.3279i q^{79} +9.00000 q^{81} -0.923034i q^{83} +34.2008i q^{85} +11.9332i q^{87} +44.8231 q^{89} -49.4256 q^{91} -88.6410 q^{93} +127.639i q^{95} -21.2539 q^{97} +(9.80385 - 31.5101i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{5} + 12 q^{9} + 20 q^{11} + 36 q^{15} + 92 q^{23} + 12 q^{25} - 80 q^{31} - 12 q^{33} + 64 q^{37} - 12 q^{45} - 100 q^{47} + 100 q^{49} - 28 q^{53} - 56 q^{55} + 40 q^{59} - 136 q^{67} + 60 q^{69}+ \cdots + 60 q^{99}+O(q^{100}) $ 4 * q - 4 * q^5 + 12 * q^9 + 20 * q^11 + 36 * q^15 + 92 * q^23 + 12 * q^25 - 80 * q^31 - 12 * q^33 + 64 * q^37 - 12 * q^45 - 100 * q^47 + 100 * q^49 - 28 * q^53 - 56 * q^55 + 40 * q^59 - 136 * q^67 + 60 * q^69 + 284 * q^71 - 72 * q^75 + 180 * q^77 + 36 * q^81 + 304 * q^89 + 24 * q^91 - 216 * q^93 - 376 * q^97 + 60 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2112\mathbb{Z}\right)^\times$.

$n$	$133$	$1409$	$1729$	$2047$
$\chi(n)$	$1$	$1$	$-1$	$1$

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 $

$2$		0			0
$2$		0			0
$3$	1.73205			0.577350
$3$	1.73205			0.577350
$4$		0			0
$5$	4.19615			0.839230			0.419615	−	0.907702i	$-0.362165\pi$
$5$	4.19615			0.839230			0.419615	+	0.907702i	$0.362165\pi$
$6$		0			0
$7$			6.42810i			0.918300i	0.888359	+	0.459150i	$0.151846\pi$
$7$			6.42810i			0.918300i	−0.888359	+	0.459150i	$0.848154\pi$
$8$		0			0
$9$	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
$12$		0			0
$13$			7.68899i			0.591461i	0.955271	+	0.295730i	$0.0955630\pi$
$13$			7.68899i			0.591461i	−0.955271	+	0.295730i	$0.904437\pi$
$14$		0			0
$15$	7.26795			0.484530
$16$		0			0
$17$			8.15051i			0.479442i	0.970842	+	0.239721i	$0.0770559\pi$
$17$			8.15051i			0.479442i	−0.970842	+	0.239721i	$0.922944\pi$
$18$		0			0
$19$			30.4181i			1.60095i	0.599364	+	0.800477i	$0.295420\pi$
$19$			30.4181i			1.60095i	−0.599364	+	0.800477i	$0.704580\pi$
$20$		0			0
$21$			11.1338i			0.530181i
$22$		0			0
$23$	31.6603			1.37653			0.688266	−	0.725458i	$-0.258372\pi$
$23$	31.6603			1.37653			0.688266	+	0.725458i	$0.258372\pi$
$24$		0			0
$25$	−7.39230			−0.295692
$26$		0			0
$27$	5.19615			0.192450
$28$		0			0
$29$			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	$-0.809070\pi$
$31$	−51.1769			−1.65087			−0.825434	+	0.564498i	$0.809070\pi$
$32$		0			0
$33$	5.66025	−	18.1923i	0.171523	−	0.551283i
$34$		0			0
$35$			26.9733i			0.770666i
$36$		0			0
$37$	19.4641			0.526057			0.263028	−	0.964788i	$-0.415279\pi$
$37$	19.4641			0.526057			0.263028	+	0.964788i	$0.415279\pi$
$38$		0			0
$39$			13.3177i			0.341480i
$40$		0			0
$41$		−	47.9800i		−	1.17024i	−0.810945	−	0.585122i	$-0.801047\pi$
$41$		−	47.9800i		−	1.17024i	0.810945	−	0.585122i	$-0.198953\pi$
$42$		0			0
$43$			81.8429i			1.90332i	0.307147	+	0.951662i	$0.400626\pi$
$43$			81.8429i			1.90332i	−0.307147	+	0.951662i	$0.599374\pi$
$44$		0			0
$45$	12.5885			0.279743
$46$		0			0
$47$	−30.1962			−0.642471			−0.321236	−	0.946999i	$-0.604098\pi$
$47$	−30.1962			−0.642471			−0.321236	+	0.946999i	$0.604098\pi$
$48$		0			0
$49$	7.67949			0.156724
$50$		0			0
$51$			14.1171i			0.276806i
$52$		0			0
$53$	−26.0526			−0.491558			−0.245779	−	0.969326i	$-0.579044\pi$
$53$	−26.0526			−0.491558			−0.245779	+	0.969326i	$0.579044\pi$
$54$		0			0
$55$	13.7128	−	44.0737i	0.249324	−	0.801340i
$56$		0			0
$57$			52.6857i			0.924311i
$58$		0			0
$59$	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	$0.787881\pi$
$62$		0			0
$63$			19.2843i			0.306100i
$64$		0			0
$65$			32.2642i			0.496372i
$66$		0			0
$67$	−34.0000			−0.507463			−0.253731	−	0.967275i	$-0.581658\pi$
$67$	−34.0000			−0.507463			−0.253731	+	0.967275i	$0.581658\pi$
$68$		0			0
$69$	54.8372			0.794742
$70$		0			0
$71$	72.7321			1.02440			0.512198	−	0.858868i	$-0.328832\pi$
$71$	72.7321			1.02440			0.512198	+	0.858868i	$0.328832\pi$
$72$		0			0
$73$		−	54.8696i		−	0.751639i	−0.926693	−	0.375819i	$-0.877361\pi$
$73$		−	54.8696i		−	0.751639i	0.926693	−	0.375819i	$-0.122639\pi$
$74$		0			0
$75$	−12.8038			−0.170718
$76$		0			0
$77$	67.5167	+	21.0067i	0.876840	+	0.272814i
$78$		0			0
$79$			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
$82$		0			0
$83$		−	0.923034i		−	0.0111209i	−0.999985	−	0.00556045i	$-0.998230\pi$
$83$		−	0.923034i		−	0.0111209i	0.999985	−	0.00556045i	$-0.00176995\pi$
$84$		0			0
$85$			34.2008i			0.402362i
$86$		0			0
$87$			11.9332i			0.137163i
$88$		0			0
$89$	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
$92$		0			0
$93$	−88.6410			−0.953129
$94$		0			0
$95$			127.639i			1.34357i
$96$		0			0
$97$	−21.2539			−0.219112			−0.109556	−	0.993981i	$-0.534943\pi$
$97$	−21.2539			−0.219112			−0.109556	+	0.993981i	$0.534943\pi$
$98$		0			0
$99$	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	$-0.0834417\pi$
$102$		0			0
$103$	4.74613			0.0460790			0.0230395	−	0.999735i	$-0.492666\pi$
$103$	4.74613			0.0460790			0.0230395	+	0.999735i	$0.492666\pi$
$104$		0			0
$105$			46.7191i			0.444944i
$106$		0			0
$107$			147.723i			1.38059i	0.723530	+	0.690293i	$0.242518\pi$
$107$			147.723i			1.38059i	−0.723530	+	0.690293i	$0.757482\pi$
$108$		0			0
$109$			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
$112$		0			0
$113$	−62.3154			−0.551463			−0.275732	−	0.961235i	$-0.588920\pi$
$113$	−62.3154			−0.551463			−0.275732	+	0.961235i	$0.588920\pi$
$114$		0			0
$115$	132.851			1.15523
$116$		0			0
$117$			23.0670i			0.197154i
$118$		0			0
$119$	−52.3923			−0.440271
$120$		0			0
$121$	−99.6410	−	68.6489i	−0.823479	−	0.567346i
$122$		0			0
$123$		−	83.1038i		−	0.675641i
$124$		0			0
$125$	−135.923			−1.08738
$126$		0			0
$127$		−	28.0200i		−	0.220630i	−0.993897	−	0.110315i	$-0.964814\pi$
$127$		−	28.0200i		−	0.220630i	0.993897	−	0.110315i	$-0.0351860\pi$
$128$		0			0
$129$			141.756i			1.09888i
$130$		0			0
$131$		−	113.184i		−	0.864001i	−0.901873	−	0.432000i	$-0.857808\pi$
$131$		−	113.184i		−	0.864001i	0.901873	−	0.432000i	$-0.142192\pi$
$132$		0			0
$133$	−195.531			−1.47016
$134$		0			0
$135$	21.8038			0.161510
$136$		0			0
$137$	−70.7846			−0.516676			−0.258338	−	0.966055i	$-0.583175\pi$
$137$	−70.7846			−0.516676			−0.258338	+	0.966055i	$0.583175\pi$
$138$		0			0
$139$			130.499i			0.938839i	0.882975	+	0.469420i	$0.155537\pi$
$139$			130.499i			0.938839i	−0.882975	+	0.469420i	$0.844463\pi$
$140$		0			0
$141$	−52.3013			−0.370931
$142$		0			0
$143$	80.7602	+	25.1272i	0.564757	+	0.175715i
$144$		0			0
$145$			28.9099i			0.199379i
$146$		0			0
$147$	13.3013			0.0904848
$148$		0			0
$149$		−	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	$0.635015\pi$
$152$		0			0
$153$			24.4515i			0.159814i
$154$		0			0
$155$	−214.746			−1.38546
$156$		0			0
$157$	273.138			1.73974			0.869868	−	0.493285i	$-0.164204\pi$
$157$	273.138			1.73974			0.869868	+	0.493285i	$0.164204\pi$
$158$		0			0
$159$	−45.1244			−0.283801
$160$		0			0
$161$			203.515i			1.26407i
$162$		0			0
$163$	62.0666			0.380777			0.190388	−	0.981709i	$-0.439025\pi$
$163$	62.0666			0.380777			0.190388	+	0.981709i	$0.439025\pi$
$164$		0			0
$165$	23.7513	−	76.3379i	0.143947	−	0.462654i
$166$		0			0
$167$			47.9800i			0.287305i	0.989628	+	0.143653i	$0.0458848\pi$
$167$			47.9800i			0.287305i	−0.989628	+	0.143653i	$0.954115\pi$
$168$		0			0
$169$	109.879			0.650174
$170$		0			0
$171$			91.2543i			0.533651i
$172$		0			0
$173$		−	143.017i		−	0.826688i	−0.910575	−	0.413344i	$-0.864361\pi$
$173$		−	143.017i		−	0.826688i	0.910575	−	0.413344i	$-0.135639\pi$
$174$		0			0
$175$		−	47.5185i		−	0.271534i
$176$		0			0
$177$	143.321			0.809720
$178$		0			0
$179$	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	$-0.425303\pi$
$181$	84.1718			0.465037			0.232519	+	0.972592i	$0.425303\pi$
$182$		0			0
$183$			130.622i			0.713783i
$184$		0			0
$185$	81.6743			0.441483
$186$		0			0
$187$	85.6077	+	26.6354i	0.457795	+	0.142436i
$188$		0			0
$189$			33.4014i			0.176727i
$190$		0			0
$191$	−44.2346			−0.231595			−0.115797	−	0.993273i	$-0.536942\pi$
$191$	−44.2346			−0.231595			−0.115797	+	0.993273i	$0.536942\pi$
$192$		0			0
$193$		−	136.127i		−	0.705323i	−0.935751	−	0.352662i	$-0.885277\pi$
$193$		−	136.127i		−	0.705323i	0.935751	−	0.352662i	$-0.114723\pi$
$194$		0			0
$195$			55.8832i			0.286580i
$196$		0			0
$197$		−	5.96659i		−	0.0302872i	−0.999885	−	0.0151436i	$-0.995179\pi$
$197$		−	5.96659i		−	0.0302872i	0.999885	−	0.0151436i	$-0.00482055\pi$
$198$		0			0
$199$	234.056			1.17616			0.588081	−	0.808802i	$-0.299884\pi$
$199$	234.056			1.17616			0.588081	+	0.808802i	$0.299884\pi$
$200$		0			0
$201$	−58.8897			−0.292984
$202$		0			0
$203$	−44.2872			−0.218163
$204$		0			0
$205$		−	201.331i		−	0.982105i
$206$		0			0
$207$	94.9808			0.458844
$208$		0			0
$209$	319.492	+	99.4048i	1.52867	+	0.475621i
$210$		0			0
$211$			167.469i			0.793690i	0.917886	+	0.396845i	$0.129895\pi$
$211$			167.469i			0.793690i	−0.917886	+	0.396845i	$0.870105\pi$
$212$		0			0
$213$	125.976			0.591435
$214$		0			0
$215$			343.425i			1.59733i
$216$		0			0
$217$		−	328.970i		−	1.51599i
$218$		0			0
$219$		−	95.0370i		−	0.433959i
$220$		0			0
$221$	−62.6692			−0.283571
$222$		0			0
$223$	−82.2872			−0.369001			−0.184500	−	0.982832i	$-0.559067\pi$
$223$	−82.2872			−0.369001			−0.184500	+	0.982832i	$0.559067\pi$
$224$		0			0
$225$	−22.1769			−0.0985641
$226$		0			0
$227$			40.9999i			0.180616i	0.995914	+	0.0903081i	$0.0287852\pi$
$227$			40.9999i			0.180616i	−0.995914	+	0.0903081i	$0.971215\pi$
$228$		0			0
$229$	−160.718			−0.701825			−0.350913	−	0.936408i	$-0.614129\pi$
$229$	−160.718			−0.701825			−0.350913	+	0.936408i	$0.614129\pi$
$230$		0			0
$231$	116.942	+	36.3847i	0.506244	+	0.157510i
$232$		0			0
$233$		−	407.954i		−	1.75087i	−0.483332	−	0.875437i	$-0.660573\pi$
$233$		−	407.954i		−	1.75087i	0.483332	−	0.875437i	$-0.339427\pi$
$234$		0			0
$235$	−126.708			−0.539182
$236$		0			0
$237$			42.1371i			0.177794i
$238$		0			0
$239$			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	$-0.982238\pi$
$241$		−	26.8828i		−	0.111547i	0.998443	−	0.0557734i	$-0.0177624\pi$
$242$		0			0
$243$	15.5885			0.0641500
$244$		0			0
$245$	32.2243			0.131528
$246$		0			0
$247$	−233.885			−0.946901
$248$		0			0
$249$		−	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	$0.208319\pi$
$252$		0			0
$253$	103.464	−	332.539i	0.408949	−	1.31438i
$254$		0			0
$255$			59.2375i			0.232304i
$256$		0			0
$257$	−337.818			−1.31447			−0.657233	−	0.753687i	$-0.728273\pi$
$257$	−337.818			−1.31447			−0.657233	+	0.753687i	$0.728273\pi$
$258$		0			0
$259$			125.117i			0.483078i
$260$		0			0
$261$			20.6689i			0.0791910i
$262$		0			0
$263$			134.529i			0.511516i	0.966741	+	0.255758i	$0.0823250\pi$
$263$			134.529i			0.511516i	−0.966741	+	0.255758i	$0.917675\pi$
$264$		0			0
$265$	−109.321			−0.412530
$266$		0			0
$267$	77.6359			0.290771
$268$		0			0
$269$	341.870			1.27089			0.635447	−	0.772144i	$-0.280816\pi$
$269$	341.870			1.27089			0.635447	+	0.772144i	$0.280816\pi$
$270$		0			0
$271$		−	213.298i		−	0.787077i	−0.919308	−	0.393538i	$-0.871251\pi$
$271$		−	213.298i		−	0.787077i	0.919308	−	0.393538i	$-0.128749\pi$
$272$		0			0
$273$	−85.6077			−0.313581
$274$		0			0
$275$	−24.1577	+	77.6440i	−0.0878461	+	0.282342i
$276$		0			0
$277$			84.5789i			0.305339i	0.988277	+	0.152669i	$0.0487870\pi$
$277$			84.5789i			0.305339i	−0.988277	+	0.152669i	$0.951213\pi$
$278$		0			0
$279$	−153.531			−0.550289
$280$		0			0
$281$			351.148i			1.24964i	0.780771	+	0.624818i	$0.214827\pi$
$281$			351.148i			1.24964i	−0.780771	+	0.624818i	$0.785173\pi$
$282$		0			0
$283$			285.943i			1.01040i	0.863002	+	0.505201i	$0.168581\pi$
$283$			285.943i			1.01040i	−0.863002	+	0.505201i	$0.831419\pi$
$284$		0			0
$285$			221.077i			0.775710i
$286$		0			0
$287$	308.420			1.07464
$288$		0			0
$289$	222.569			0.770136
$290$		0			0
$291$	−36.8128			−0.126504
$292$		0			0
$293$			393.927i			1.34446i	0.740342	+	0.672231i	$0.234664\pi$
$293$			393.927i			1.34446i	−0.740342	+	0.672231i	$0.765336\pi$
$294$		0			0
$295$	347.215			1.17700
$296$		0			0
$297$	16.9808	−	54.5770i	0.0571743	−	0.183761i
$298$		0			0
$299$			243.435i			0.814165i
$300$		0			0
$301$	−526.095			−1.74782
$302$		0			0
$303$		−	90.6691i		−	0.299238i
$304$		0			0
$305$			316.452i			1.03755i
$306$		0			0
$307$		−	58.2239i		−	0.189654i	−0.995494	−	0.0948272i	$-0.969770\pi$
$307$		−	58.2239i		−	0.189654i	0.995494	−	0.0948272i	$-0.0302299\pi$
$308$		0			0
$309$	8.22055			0.0266037
$310$		0			0
$311$	−207.611			−0.667561			−0.333780	−	0.942651i	$-0.608324\pi$
$311$	−207.611			−0.667561			−0.333780	+	0.942651i	$0.608324\pi$
$312$		0			0
$313$	−376.697			−1.20351			−0.601753	−	0.798682i	$-0.705531\pi$
$313$	−376.697			−1.20351			−0.601753	+	0.798682i	$0.705531\pi$
$314$		0			0
$315$			80.9199i			0.256889i
$316$		0			0
$317$	−246.809			−0.778577			−0.389289	−	0.921116i	$-0.627279\pi$
$317$	−246.809			−0.778577			−0.389289	+	0.921116i	$0.627279\pi$
$318$		0			0
$319$	72.3641	+	22.5149i	0.226847	+	0.0705797i
$320$		0			0
$321$			255.863i			0.797082i
$322$		0			0
$323$	−247.923			−0.767564
$324$		0			0
$325$		−	56.8394i		−	0.174890i
$326$		0			0
$327$			6.18078i			0.0189015i
$328$		0			0
$329$		−	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	$0.429004\pi$
$332$		0			0
$333$	58.3923			0.175352
$334$		0			0
$335$	−142.669			−0.425878
$336$		0			0
$337$		−	354.007i		−	1.05047i	−0.850958	−	0.525233i	$-0.823978\pi$
$337$		−	354.007i		−	1.05047i	0.850958	−	0.525233i	$-0.176022\pi$
$338$		0			0
$339$	−107.933			−0.318387
$340$		0			0
$341$	−167.244	+	537.529i	−0.490450	+	1.57633i
$342$		0			0
$343$			364.342i			1.06222i
$344$		0			0
$345$	230.105			0.666971
$346$		0			0
$347$			529.807i			1.52682i	0.645913	+	0.763411i	$0.276477\pi$
$347$			529.807i			1.52682i	−0.645913	+	0.763411i	$0.723523\pi$
$348$		0			0
$349$		−	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
$352$		0			0
$353$	191.990			0.543880			0.271940	−	0.962314i	$-0.412335\pi$
$353$	191.990			0.543880			0.271940	+	0.962314i	$0.412335\pi$
$354$		0			0
$355$	305.195			0.859704
$356$		0			0
$357$	−90.7461			−0.254191
$358$		0			0
$359$		−	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
$362$		0			0
$363$	−172.583	−	118.903i	−0.475436	−	0.327557i
$364$		0			0
$365$		−	230.241i		−	0.630798i
$366$		0			0
$367$	139.415			0.379878			0.189939	−	0.981796i	$-0.439171\pi$
$367$	139.415			0.379878			0.189939	+	0.981796i	$0.439171\pi$
$368$		0			0
$369$		−	143.940i		−	0.390081i
$370$		0			0
$371$		−	167.469i		−	0.451398i
$372$		0			0
$373$		−	303.629i		−	0.814019i	−0.913424	−	0.407009i	$-0.866572\pi$
$373$		−	303.629i		−	0.814019i	0.913424	−	0.407009i	$-0.133428\pi$
$374$		0			0
$375$	−235.426			−0.627802
$376$		0			0
$377$	−52.9742			−0.140515
$378$		0			0
$379$	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
$382$		0			0
$383$	53.2576			0.139054			0.0695269	−	0.997580i	$-0.477851\pi$
$383$	53.2576			0.139054			0.0695269	+	0.997580i	$0.477851\pi$
$384$		0			0
$385$	283.310	+	88.1474i	0.735871	+	0.228954i
$386$		0			0
$387$			245.529i			0.634441i
$388$		0			0
$389$	587.027			1.50907			0.754533	−	0.656262i	$-0.227863\pi$
$389$	587.027			1.50907			0.754533	+	0.656262i	$0.227863\pi$
$390$		0			0
$391$			258.047i			0.659967i
$392$		0			0
$393$		−	196.041i		−	0.498831i
$394$		0			0
$395$			102.083i			0.258439i
$396$		0			0
$397$	−485.797			−1.22367			−0.611835	−	0.790985i	$-0.709569\pi$
$397$	−485.797			−1.22367			−0.611835	+	0.790985i	$0.709569\pi$
$398$		0			0
$399$	−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	$0.566555\pi$
$402$		0			0
$403$		−	393.499i		−	0.976424i
$404$		0			0
$405$	37.7654			0.0932478
$406$		0			0
$407$	63.6077	−	204.438i	0.156284	−	0.502306i
$408$		0			0
$409$		−	270.161i		−	0.660541i	−0.943886	−	0.330271i	$-0.892860\pi$
$409$		−	270.161i		−	0.660541i	0.943886	−	0.330271i	$-0.107140\pi$
$410$		0			0
$411$	−122.603			−0.298303
$412$		0			0
$413$			531.901i			1.28790i
$414$		0			0
$415$		−	3.87319i		−	0.00933299i
$416$		0			0
$417$			226.030i			0.542039i
$418$		0			0
$419$	−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	$-0.315018\pi$
$421$	462.238			1.09795			0.548977	+	0.835838i	$0.315018\pi$
$422$		0			0
$423$	−90.5885			−0.214157
$424$		0			0
$425$		−	60.2510i		−	0.141767i
$426$		0			0
$427$	−484.774			−1.13530
$428$		0			0
$429$	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	$-0.0743628\pi$
$432$		0			0
$433$	−45.3693			−0.104779			−0.0523895	−	0.998627i	$-0.516684\pi$
$433$	−45.3693			−0.104779			−0.0523895	+	0.998627i	$0.516684\pi$
$434$		0			0
$435$			50.0734i			0.115111i
$436$		0			0
$437$			963.045i			2.20376i
$438$		0			0
$439$		−	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
$442$		0			0
$443$	375.538			0.847716			0.423858	−	0.905729i	$-0.360676\pi$
$443$	375.538			0.847716			0.423858	+	0.905729i	$0.360676\pi$
$444$		0			0
$445$	188.084			0.422662
$446$		0			0
$447$		−	217.880i		−	0.487427i
$448$		0			0
$449$	−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
$452$		0			0
$453$			476.726i			1.05238i
$454$		0			0
$455$	−207.397			−0.455819
$456$		0			0
$457$		−	378.368i		−	0.827939i	−0.910291	−	0.413970i	$-0.864142\pi$
$457$		−	378.368i		−	0.827939i	0.910291	−	0.413970i	$-0.135858\pi$
$458$		0			0
$459$			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	$-0.0783010\pi$
$462$		0			0
$463$	52.3820			0.113136			0.0565680	−	0.998399i	$-0.481984\pi$
$463$	52.3820			0.113136			0.0565680	+	0.998399i	$0.481984\pi$
$464$		0			0
$465$	−371.951			−0.799895
$466$		0			0
$467$	513.387			1.09933			0.549665	−	0.835385i	$-0.314755\pi$
$467$	513.387			1.09933			0.549665	+	0.835385i	$0.314755\pi$
$468$		0			0
$469$		−	218.556i		−	0.466003i
$470$		0			0
$471$	473.090			1.00444
$472$		0			0
$473$	859.626	+	267.459i	1.81739	+	0.565451i
$474$		0			0
$475$		−	224.860i		−	0.473389i
$476$		0			0
$477$	−78.1577			−0.163853
$478$		0			0
$479$			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
$482$		0			0
$483$			352.499i			0.729812i
$484$		0			0
$485$	−89.1845			−0.183885
$486$		0			0
$487$	−593.251			−1.21817			−0.609087	−	0.793103i	$-0.708464\pi$
$487$	−593.251			−1.21817			−0.609087	+	0.793103i	$0.708464\pi$
$488$		0			0
$489$	107.503			0.219842
$490$		0			0
$491$		−	858.935i		−	1.74936i	−0.484703	−	0.874679i	$-0.661072\pi$
$491$		−	858.935i		−	1.74936i	0.484703	−	0.874679i	$-0.338928\pi$
$492$		0			0
$493$	−56.1539			−0.113902
$494$		0			0
$495$	41.1384	−	132.221i	0.0831080	−	0.267113i
$496$		0			0
$497$			467.529i			0.940702i
$498$		0			0
$499$	−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
$502$		0			0
$503$		−	752.302i		−	1.49563i	−0.663907	−	0.747815i	$-0.731103\pi$
$503$		−	752.302i		−	1.49563i	0.663907	−	0.747815i	$-0.268897\pi$
$504$		0			0
$505$		−	219.660i		−	0.434969i
$506$		0			0
$507$	190.317			0.375378
$508$		0			0
$509$	249.268			0.489721			0.244860	−	0.969558i	$-0.421258\pi$
$509$	249.268			0.489721			0.244860	+	0.969558i	$0.421258\pi$
$510$		0			0
$511$	352.708			0.690230
$512$		0			0
$513$			158.057i			0.308104i
$514$		0			0
$515$	19.9155			0.0386709
$516$		0			0
$517$	−98.6795	+	317.161i	−0.190869	+	0.613464i
$518$		0			0
$519$		−	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	$0.349037\pi$
$522$		0			0
$523$		−	362.833i		−	0.693754i	−0.937911	−	0.346877i	$-0.887242\pi$
$523$		−	362.833i		−	0.693754i	0.937911	−	0.346877i	$-0.112758\pi$
$524$		0			0
$525$		−	82.3045i		−	0.156770i
$526$		0			0
$527$		−	417.118i		−	0.791495i
$528$		0			0
$529$	473.372			0.894843
$530$		0			0
$531$	248.238			0.467492
$532$		0			0
$533$	368.918			0.692154
$534$		0			0
$535$			619.867i			1.15863i
$536$		0			0
$537$	63.3975			0.118059
$538$		0			0
$539$	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	$-0.278606\pi$
$542$		0			0
$543$	145.790			0.268489
$544$		0			0
$545$			14.9739i			0.0274750i
$546$		0			0
$547$		−	456.091i		−	0.833804i	−0.908952	−	0.416902i	$-0.863116\pi$
$547$		−	456.091i		−	0.833804i	0.908952	−	0.416902i	$-0.136884\pi$
$548$		0			0
$549$			226.244i			0.412103i
$550$		0			0
$551$	−209.569			−0.380343
$552$		0			0
$553$	−156.382			−0.282788
$554$		0			0
$555$	141.464			0.254890
$556$		0			0
$557$		−	338.810i		−	0.608277i	−0.952628	−	0.304138i	$-0.901631\pi$
$557$		−	338.810i		−	0.608277i	0.952628	−	0.304138i	$-0.0983686\pi$
$558$		0			0
$559$	−629.290			−1.12574
$560$		0			0
$561$	148.277	+	46.1339i	0.264308	+	0.0822352i
$562$		0			0
$563$		−	447.041i		−	0.794034i	−0.917811	−	0.397017i	$-0.870045\pi$
$563$		−	447.041i		−	0.794034i	0.917811	−	0.397017i	$-0.129955\pi$
$564$		0			0
$565$	−261.485			−0.462805
$566$		0			0
$567$			57.8529i			0.102033i
$568$		0			0
$569$			522.893i			0.918969i	0.888186	+	0.459485i	$0.151966\pi$
$569$			522.893i			0.918969i	−0.888186	+	0.459485i	$0.848034\pi$
$570$		0			0
$571$		−	904.574i		−	1.58419i	−0.610396	−	0.792096i	$-0.708990\pi$
$571$		−	904.574i		−	1.58419i	0.610396	−	0.792096i	$-0.291010\pi$
$572$		0			0
$573$	−76.6166			−0.133711
$574$		0			0
$575$	−234.042			−0.407030
$576$		0			0
$577$	237.674			0.411914			0.205957	−	0.978561i	$-0.433969\pi$
$577$	237.674			0.411914			0.205957	+	0.978561i	$0.433969\pi$
$578$		0			0
$579$		−	235.780i		−	0.407219i
$580$		0			0
$581$	5.93336			0.0102123
$582$		0			0
$583$	−85.1384	+	273.639i	−0.146035	+	0.469364i
$584$		0			0
$585$			96.7925i			0.165457i
$586$		0			0
$587$	−578.515			−0.985546			−0.492773	−	0.870158i	$-0.664017\pi$
$587$	−578.515			−0.985546			−0.492773	+	0.870158i	$0.664017\pi$
$588$		0			0
$589$		−	1556.71i		−	2.64296i
$590$		0			0
$591$		−	10.3344i		−	0.0174863i
$592$		0			0
$593$			333.857i			0.562997i	0.959562	+	0.281499i	$0.0908314\pi$
$593$			333.857i			0.562997i	−0.959562	+	0.281499i	$0.909169\pi$
$594$		0			0
$595$	−219.846			−0.369489
$596$		0			0
$597$	405.397			0.679058
$598$		0			0
$599$	846.483			1.41316			0.706580	−	0.707633i	$-0.250237\pi$
$599$	846.483			1.41316			0.706580	+	0.707633i	$0.250237\pi$
$600$		0			0
$601$			455.011i			0.757089i	0.925583	+	0.378545i	$0.123575\pi$
$601$			455.011i			0.757089i	−0.925583	+	0.378545i	$0.876425\pi$
$602$		0			0
$603$	−102.000			−0.169154
$604$		0			0
$605$	−418.109	−	288.061i	−0.691089	−	0.476134i
$606$		0			0
$607$		−	726.557i		−	1.19696i	−0.801137	−	0.598482i	$-0.795771\pi$
$607$		−	726.557i		−	1.19696i	0.801137	−	0.598482i	$-0.204229\pi$
$608$		0			0
$609$	−76.7077			−0.125957
$610$		0			0
$611$		−	232.178i		−	0.379997i
$612$		0			0
$613$		−	574.532i		−	0.937247i	−0.883398	−	0.468624i	$-0.844750\pi$
$613$		−	574.532i		−	0.937247i	0.883398	−	0.468624i	$-0.155250\pi$
$614$		0			0
$615$		−	348.716i		−	0.567018i
$616$		0			0
$617$	−278.946			−0.452101			−0.226050	−	0.974116i	$-0.572581\pi$
$617$	−278.946			−0.452101			−0.226050	+	0.974116i	$0.572581\pi$
$618$		0			0
$619$	−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
$622$		0			0
$623$			288.127i			0.462484i
$624$		0			0
$625$	−385.546			−0.616874
$626$		0			0
$627$	553.377	+	172.174i	0.882579	+	0.274600i
$628$		0			0
$629$			158.642i			0.252214i
$630$		0			0
$631$	361.664			0.573160			0.286580	−	0.958056i	$-0.407482\pi$
$631$	361.664			0.573160			0.286580	+	0.958056i	$0.407482\pi$
$632$		0			0
$633$			290.064i			0.458237i
$634$		0			0
$635$		−	117.576i		−	0.185159i
$636$		0			0
$637$			59.0475i			0.0926963i
$638$		0			0
$639$	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	$0.290429\pi$
$642$		0			0
$643$	362.764			0.564174			0.282087	−	0.959389i	$-0.408973\pi$
$643$	362.764			0.564174			0.282087	+	0.959389i	$0.408973\pi$
$644$		0			0
$645$			594.830i			0.922217i
$646$		0			0
$647$	−921.727			−1.42462			−0.712308	−	0.701867i	$-0.752350\pi$
$647$	−921.727			−1.42462			−0.712308	+	0.701867i	$0.752350\pi$
$648$		0			0
$649$	270.410	−	869.112i	0.416657	−	1.33916i
$650$		0			0
$651$		−	569.794i		−	0.875259i
$652$		0			0
$653$	−26.8653			−0.0411414			−0.0205707	−	0.999788i	$-0.506548\pi$
$653$	−26.8653			−0.0411414			−0.0205707	+	0.999788i	$0.506548\pi$
$654$		0			0
$655$		−	474.938i		−	0.725096i
$656$		0			0
$657$		−	164.609i		−	0.250546i
$658$		0			0
$659$		−	320.820i		−	0.486828i	−0.969922	−	0.243414i	$-0.921733\pi$
$659$		−	320.820i		−	0.486828i	0.969922	−	0.243414i	$-0.0782674\pi$
$660$		0			0
$661$	−331.969			−0.502222			−0.251111	−	0.967958i	$-0.580796\pi$
$661$	−331.969			−0.502222			−0.251111	+	0.967958i	$0.580796\pi$
$662$		0			0
$663$	−108.546			−0.163720
$664$		0			0
$665$	−820.477			−1.23380
$666$		0			0
$667$			218.127i			0.327027i
$668$		0			0
$669$	−142.526			−0.213043
$670$		0			0
$671$	792.109	+	246.452i	1.18049	+	0.367290i
$672$		0			0
$673$			797.019i			1.18428i	0.805836	+	0.592139i	$0.201716\pi$
$673$			797.019i			1.18428i	−0.805836	+	0.592139i	$0.798284\pi$
$674$		0			0
$675$	−38.4115			−0.0569060
$676$		0			0
$677$			534.175i			0.789033i	0.918889	+	0.394516i	$0.129088\pi$
$677$			534.175i			0.789033i	−0.918889	+	0.394516i	$0.870912\pi$
$678$		0			0
$679$		−	136.622i		−	0.201211i
$680$		0			0
$681$			71.0139i			0.104279i
$682$		0			0
$683$	−843.097			−1.23440			−0.617201	−	0.786805i	$-0.711734\pi$
$683$	−843.097			−1.23440			−0.617201	+	0.786805i	$0.711734\pi$
$684$		0			0
$685$	−297.023			−0.433610
$686$		0			0
$687$	−278.372			−0.405199
$688$		0			0
$689$		−	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	$0.544742\pi$
$692$		0			0
$693$	202.550	+	63.0201i	0.292280	+	0.0909382i
$694$		0			0
$695$			547.592i			0.787903i
$696$		0			0
$697$	391.061			0.561064
$698$		0			0
$699$		−	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	$0.896264\pi$
$702$		0			0
$703$			592.061i			0.842192i
$704$		0			0
$705$	−219.464			−0.311297
$706$		0			0
$707$	336.497			0.475951
$708$		0			0
$709$	311.031			0.438689			0.219345	−	0.975647i	$-0.429608\pi$
$709$	311.031			0.438689			0.219345	+	0.975647i	$0.429608\pi$
$710$		0			0
$711$			72.9836i			0.102649i
$712$		0			0
$713$	−1620.27			−2.27247
$714$		0			0
$715$	338.882	+	105.438i	0.473961	+	0.147465i
$716$		0			0
$717$			350.315i			0.488584i
$718$		0			0
$719$	−486.014			−0.675958			−0.337979	−	0.941154i	$-0.609743\pi$
$719$	−486.014			−0.675958			−0.337979	+	0.941154i	$0.609743\pi$
$720$		0			0
$721$			30.5086i			0.0423143i
$722$		0			0
$723$		−	46.5623i		−	0.0644016i
$724$		0			0
$725$		−	50.9302i		−	0.0702485i
$726$		0			0
$727$	−38.8616			−0.0534547			−0.0267273	−	0.999643i	$-0.508509\pi$
$727$	−38.8616			−0.0534547			−0.0267273	+	0.999643i	$0.508509\pi$
$728$		0			0
$729$	27.0000			0.0370370
$730$		0			0
$731$	−667.061			−0.912533
$732$		0			0
$733$		−	175.562i		−	0.239511i	−0.992803	−	0.119756i	$-0.961789\pi$
$733$		−	175.562i		−	0.239511i	0.992803	−	0.119756i	$-0.0382111\pi$
$734$		0			0
$735$	55.8142			0.0759376
$736$		0			0
$737$	−111.110	+	357.114i	−0.150760	+	0.484551i
$738$		0			0
$739$		−	494.527i		−	0.669184i	−0.942363	−	0.334592i	$-0.891402\pi$
$739$		−	494.527i		−	0.669184i	0.942363	−	0.334592i	$-0.108598\pi$
$740$		0			0
$741$	−405.100			−0.546694
$742$		0			0
$743$		−	1150.84i		−	1.54892i	−0.632625	−	0.774458i	$-0.718023\pi$
$743$		−	1150.84i		−	1.54892i	0.632625	−	0.774458i	$-0.281977\pi$
$744$		0			0
$745$		−	527.846i		−	0.708519i
$746$		0			0
$747$		−	2.76910i		−	0.00370696i
$748$		0			0
$749$	−949.577			−1.26779
$750$		0			0
$751$	−1343.73			−1.78926			−0.894628	−	0.446813i	$-0.852559\pi$
$751$	−1343.73			−1.78926			−0.894628	+	0.446813i	$0.852559\pi$
$752$		0			0
$753$	689.836			0.916117
$754$		0			0
$755$			1154.94i			1.52972i
$756$		0			0
$757$	−1034.58			−1.36669			−0.683343	−	0.730097i	$-0.739475\pi$
$757$	−1034.58			−1.36669			−0.683343	+	0.730097i	$0.739475\pi$
$758$		0			0
$759$	179.205	−	575.974i	0.236107	−	0.758859i
$760$		0			0
$761$		−	798.527i		−	1.04931i	−0.851314	−	0.524656i	$-0.824194\pi$
$761$		−	798.527i		−	1.04931i	0.851314	−	0.524656i	$-0.175806\pi$
$762$		0			0
$763$	−22.9385			−0.0300636
$764$		0			0
$765$			102.602i			0.134121i
$766$		0			0
$767$			636.234i			0.829510i
$768$		0			0
$769$		−	261.311i		−	0.339806i	−0.985461	−	0.169903i	$-0.945655\pi$
$769$		−	261.311i		−	0.339806i	0.985461	−	0.169903i	$-0.0543455\pi$
$770$		0			0
$771$	−585.118			−0.758908
$772$		0			0
$773$	327.734			0.423977			0.211989	−	0.977272i	$-0.432006\pi$
$773$	327.734			0.423977			0.211989	+	0.977272i	$0.432006\pi$
$774$		0			0
$775$	378.315			0.488149
$776$		0			0
$777$			216.709i			0.278905i
$778$		0			0
$779$	1459.46			1.87351
$780$		0			0
$781$	237.685	−	763.931i	0.304334	−	0.978144i
$782$		0			0
$783$			35.7995i			0.0457210i
$784$		0			0
$785$	1146.13			1.46004
$786$		0			0
$787$		−	289.388i		−	0.367711i	−0.982953	−	0.183855i	$-0.941142\pi$
$787$		−	289.388i		−	0.367711i	0.982953	−	0.183855i	$-0.0588578\pi$
$788$		0			0
$789$			233.010i			0.295324i
$790$		0			0
$791$		−	400.570i		−	0.506409i
$792$		0			0
$793$	−579.864			−0.731228
$794$		0			0
$795$	−189.349			−0.238174
$796$		0			0
$797$	−113.681			−0.142636			−0.0713180	−	0.997454i	$-0.522721\pi$
$797$	−113.681			−0.142636			−0.0713180	+	0.997454i	$0.522721\pi$
$798$		0			0
$799$		−	246.114i		−	0.308028i
$800$		0			0
$801$	134.469			0.167877
$802$		0			0
$803$	−576.315	−	179.311i	−0.717703	−	0.223302i
$804$		0			0
$805$			853.982i			1.06085i
$806$		0			0
$807$	592.137			0.733751
$808$		0			0
$809$		−	854.163i		−	1.05583i	−0.849299	−	0.527913i	$-0.822975\pi$
$809$		−	854.163i		−	1.05583i	0.849299	−	0.527913i	$-0.177025\pi$
$810$		0			0
$811$			486.533i			0.599917i	0.953952	+	0.299959i	$0.0969729\pi$
$811$			486.533i			0.599917i	−0.953952	+	0.299959i	$0.903027\pi$
$812$		0			0
$813$		−	369.443i		−	0.454419i
$814$		0			0
$815$	260.441			0.319560
$816$		0			0
$817$	−2489.51			−3.04713
$818$		0			0
$819$	−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	$-0.104024\pi$
$822$		0			0
$823$	314.805			0.382509			0.191255	−	0.981540i	$-0.438744\pi$
$823$	314.805			0.382509			0.191255	+	0.981540i	$0.438744\pi$
$824$		0			0
$825$	−41.8423	+	134.483i	−0.0507180	+	0.163010i
$826$		0			0
$827$		−	111.652i		−	0.135008i	−0.997719	−	0.0675040i	$-0.978496\pi$
$827$		−	111.652i		−	0.135008i	0.997719	−	0.0675040i	$-0.0215035\pi$
$828$		0			0
$829$	695.395			0.838836			0.419418	−	0.907793i	$-0.362234\pi$
$829$	695.395			0.838836			0.419418	+	0.907793i	$0.362234\pi$
$830$		0			0
$831$			146.495i			0.176288i
$832$		0			0
$833$			62.5918i			0.0751402i
$834$		0			0
$835$			201.331i			0.241116i
$836$		0			0
$837$	−265.923			−0.317710
$838$		0			0
$839$	383.968			0.457650			0.228825	−	0.973468i	$-0.426512\pi$
$839$	383.968			0.457650			0.228825	+	0.973468i	$0.426512\pi$
$840$		0			0
$841$	793.533			0.943559
$842$		0			0
$843$			608.205i			0.721477i
$844$		0			0
$845$	461.071			0.545646
$846$		0			0
$847$	441.282	−	640.503i	0.520994	−	0.756202i
$848$		0			0
$849$			495.269i			0.583355i
$850$		0			0
$851$	616.238			0.724134
$852$		0			0
$853$			570.527i			0.668847i	0.942423	+	0.334424i	$0.108542\pi$
$853$			570.527i			0.668847i	−0.942423	+	0.334424i	$0.891458\pi$
$854$		0			0
$855$			382.917i			0.447856i
$856$		0			0
$857$		−	315.553i		−	0.368207i	−0.982907	−	0.184103i	$-0.941062\pi$
$857$		−	315.553i		−	0.368207i	0.982907	−	0.184103i	$-0.0589382\pi$
$858$		0			0
$859$	−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
$862$		0			0
$863$	−1431.55			−1.65881			−0.829403	−	0.558651i	$-0.811319\pi$
$863$	−1431.55			−1.65881			−0.829403	+	0.558651i	$0.811319\pi$
$864$		0			0
$865$		−	600.121i		−	0.693782i
$866$		0			0
$867$	385.501			0.444638
$868$		0			0
$869$	255.524	+	79.5022i	0.294044	+	0.0914870i
$870$		0			0
$871$		−	261.426i		−	0.300144i
$872$		0			0
$873$	−63.7616			−0.0730373
$874$		0			0
$875$		−	873.727i		−	0.998546i
$876$		0			0
$877$		−	567.758i		−	0.647386i	−0.946162	−	0.323693i	$-0.895075\pi$
$877$		−	567.758i		−	0.647386i	0.946162	−	0.323693i	$-0.104925\pi$
$878$		0			0
$879$			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	$0.410553\pi$
$882$		0			0
$883$	840.144			0.951465			0.475732	−	0.879590i	$-0.342183\pi$
$883$	840.144			0.951465			0.475732	+	0.879590i	$0.342183\pi$
$884$		0			0
$885$	601.395			0.679542
$886$		0			0
$887$		−	981.439i		−	1.10647i	−0.833025	−	0.553235i	$-0.813393\pi$
$887$		−	981.439i		−	1.10647i	0.833025	−	0.553235i	$-0.186607\pi$
$888$		0			0
$889$	180.115			0.202605
$890$		0			0
$891$	29.4115	−	94.5302i	0.0330096	−	0.106095i
$892$		0			0
$893$		−	918.510i		−	1.02857i
$894$		0			0
$895$	153.590			0.171609
$896$		0			0
$897$			421.642i			0.470059i
$898$		0			0
$899$		−	352.589i		−	0.392202i
$900$		0			0
$901$		−	212.342i		−	0.235673i
$902$		0			0
$903$	−911.223			−1.00911
$904$		0			0
$905$	353.198			0.390274
$906$		0			0
$907$	438.610			0.483583			0.241792	−	0.970328i	$-0.422265\pi$
$907$	438.610			0.483583			0.241792	+	0.970328i	$0.422265\pi$
$908$		0			0
$909$		−	157.044i		−	0.172765i
$910$		0			0
$911$	170.042			0.186654			0.0933272	−	0.995635i	$-0.470250\pi$
$911$	170.042			0.186654			0.0933272	+	0.995635i	$0.470250\pi$
$912$		0			0
$913$	−9.69496	−	3.01643i	−0.0106188	−	0.00330386i
$914$		0			0
$915$			548.111i			0.599029i
$916$		0			0
$917$	727.559			0.793412
$918$		0			0
$919$		−	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
$922$		0			0
$923$			559.236i			0.605890i
$924$		0			0
$925$	−143.885			−0.155551
$926$		0			0
$927$	14.2384			0.0153597
$928$		0			0
$929$	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
$932$		0			0
$933$	−359.594			−0.385417
$934$		0			0
$935$	359.223	+	111.766i	0.384196	+	0.119536i
$936$		0			0
$937$		−	1606.78i		−	1.71481i	−0.514641	−	0.857406i	$-0.672075\pi$
$937$		−	1606.78i		−	1.71481i	0.514641	−	0.857406i	$-0.327925\pi$
$938$		0			0
$939$	−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	$-0.237923\pi$
$942$		0			0
$943$		−	1519.06i		−	1.61088i
$944$		0			0
$945$			140.157i			0.148315i
$946$		0			0
$947$	−985.800			−1.04097			−0.520486	−	0.853870i	$-0.674249\pi$
$947$	−985.800			−1.04097			−0.520486	+	0.853870i	$0.674249\pi$
$948$		0			0
$949$	421.892			0.444565
$950$		0			0
$951$	−427.486			−0.449512
$952$		0			0
$953$		−	761.038i		−	0.798571i	−0.916827	−	0.399285i	$-0.869258\pi$
$953$		−	761.038i		−	0.798571i	0.916827	−	0.399285i	$-0.130742\pi$
$954$		0			0
$955$	−185.615			−0.194362
$956$		0			0
$957$	125.338	+	38.9970i	0.130970	+	0.0407492i
$958$		0			0
$959$		−	455.011i		−	0.474464i
$960$		0			0
$961$	1658.08			1.72537
$962$		0			0
$963$			443.168i			0.460195i
$964$		0			0
$965$		−	571.211i		−	0.591929i
$966$		0			0
$967$			702.938i			0.726926i	0.931609	+	0.363463i	$0.118406\pi$
$967$			702.938i			0.726926i	−0.931609	+	0.363463i	$0.881594\pi$
$968$		0			0
$969$	−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	$0.731127\pi$
$972$		0			0
$973$	−838.859			−0.862136
$974$		0			0
$975$		−	98.4487i		−	0.100973i
$976$		0			0
$977$	−690.526			−0.706782			−0.353391	−	0.935476i	$-0.614971\pi$
$977$	−690.526			−0.706782			−0.353391	+	0.935476i	$0.614971\pi$
$978$		0			0
$979$	146.480	−	470.793i	0.149622	−	0.480892i
$980$		0			0
$981$			10.7054i			0.0109128i
$982$		0			0
$983$	−1080.40			−1.09908			−0.549540	−	0.835467i	$-0.685197\pi$
$983$	−1080.40			−1.09908			−0.549540	+	0.835467i	$0.685197\pi$
$984$		0			0
$985$		−	25.0367i		−	0.0254180i
$986$		0			0
$987$		−	336.198i		−	0.340626i
$988$		0			0
$989$			2591.17i			2.61999i
$990$		0			0
$991$	686.561			0.692796			0.346398	−	0.938088i	$-0.387405\pi$
$991$	686.561			0.692796			0.346398	+	0.938088i	$0.387405\pi$
$992$		0			0
$993$	253.626			0.255413
$994$		0			0
$995$	982.136			0.987071
$996$		0			0
$997$			1413.50i			1.41775i	0.705333	+	0.708876i	$0.250797\pi$
$997$			1413.50i			1.41775i	−0.705333	+	0.708876i	$0.749203\pi$
$998$		0			0
$999$	101.138			0.101240

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2112.3.j.d.769.4		4
4.3	odd	2		2112.3.j.a.769.1		4
8.3	odd	2		33.3.c.a.10.3	yes	4
8.5	even	2		528.3.j.c.241.2		4
11.10	odd	2	inner	2112.3.j.d.769.3		4
24.5	odd	2		1584.3.j.f.1297.4		4
24.11	even	2		99.3.c.b.10.2		4
40.3	even	4		825.3.h.a.274.6		8
40.19	odd	2		825.3.b.a.76.2		4
40.27	even	4		825.3.h.a.274.3		8
44.43	even	2		2112.3.j.a.769.2		4
88.3	odd	10		363.3.g.e.112.3		16
88.19	even	10		363.3.g.e.112.2		16
88.21	odd	2		528.3.j.c.241.1		4
88.27	odd	10		363.3.g.e.118.2		16
88.35	even	10		363.3.g.e.40.2		16
88.43	even	2		33.3.c.a.10.2	✓	4
88.51	even	10		363.3.g.e.94.3		16
88.59	odd	10		363.3.g.e.94.2		16
88.75	odd	10		363.3.g.e.40.3		16
88.83	even	10		363.3.g.e.118.3		16
264.131	odd	2		99.3.c.b.10.3		4
264.197	even	2		1584.3.j.f.1297.3		4
440.43	odd	4		825.3.h.a.274.4		8
440.219	even	2		825.3.b.a.76.3		4
440.307	odd	4		825.3.h.a.274.5		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
33.3.c.a.10.2	✓	4	88.43	even	2
33.3.c.a.10.3	yes	4	8.3	odd	2
99.3.c.b.10.2		4	24.11	even	2
99.3.c.b.10.3		4	264.131	odd	2
363.3.g.e.40.2		16	88.35	even	10
363.3.g.e.40.3		16	88.75	odd	10
363.3.g.e.94.2		16	88.59	odd	10
363.3.g.e.94.3		16	88.51	even	10
363.3.g.e.112.2		16	88.19	even	10
363.3.g.e.112.3		16	88.3	odd	10
363.3.g.e.118.2		16	88.27	odd	10
363.3.g.e.118.3		16	88.83	even	10
528.3.j.c.241.1		4	88.21	odd	2
528.3.j.c.241.2		4	8.5	even	2
825.3.b.a.76.2		4	40.19	odd	2
825.3.b.a.76.3		4	440.219	even	2
825.3.h.a.274.3		8	40.27	even	4
825.3.h.a.274.4		8	440.43	odd	4
825.3.h.a.274.5		8	440.307	odd	4
825.3.h.a.274.6		8	40.3	even	4
1584.3.j.f.1297.3		4	264.197	even	2
1584.3.j.f.1297.4		4	24.5	odd	2
2112.3.j.a.769.1		4	4.3	odd	2
2112.3.j.a.769.2		4	44.43	even	2
2112.3.j.d.769.3		4	11.10	odd	2	inner
2112.3.j.d.769.4		4	1.1	even	1	trivial

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2112 = 2^{6} \cdot 3 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	2112.j (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.73205 q^{3} +4.19615 q^{5} +6.42810i q^{7} +3.00000 q^{9} +(3.26795 - 10.5034i) q^{11} +7.68899i q^{13} +7.26795 q^{15} +8.15051i q^{17} +30.4181i q^{19} +11.1338i q^{21} +31.6603 q^{23} -7.39230 q^{25} +5.19615 q^{27} +6.88962i q^{29} -51.1769 q^{31} +(5.66025 - 18.1923i) q^{33} +26.9733i q^{35} +19.4641 q^{37} +13.3177i q^{39} -47.9800i q^{41} +81.8429i q^{43} +12.5885 q^{45} -30.1962 q^{47} +7.67949 q^{49} +14.1171i q^{51} -26.0526 q^{53} +(13.7128 - 44.0737i) q^{55} +52.6857i q^{57} +82.7461 q^{59} +75.4148i q^{61} +19.2843i q^{63} +32.2642i q^{65} -34.0000 q^{67} +54.8372 q^{69} +72.7321 q^{71} -54.8696i q^{73} -12.8038 q^{75} +(67.5167 + 21.0067i) q^{77} +24.3279i q^{79} +9.00000 q^{81} -0.923034i q^{83} +34.2008i q^{85} +11.9332i q^{87} +44.8231 q^{89} -49.4256 q^{91} -88.6410 q^{93} +127.639i q^{95} -21.2539 q^{97} +(9.80385 - 31.5101i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{5} + 12 q^{9} + 20 q^{11} + 36 q^{15} + 92 q^{23} + 12 q^{25} - 80 q^{31} - 12 q^{33} + 64 q^{37} - 12 q^{45} - 100 q^{47} + 100 q^{49} - 28 q^{53} - 56 q^{55} + 40 q^{59} - 136 q^{67} + 60 q^{69}+ \cdots + 60 q^{99}+O(q^{100}) \) 4 * q - 4 * q^5 + 12 * q^9 + 20 * q^11 + 36 * q^15 + 92 * q^23 + 12 * q^25 - 80 * q^31 - 12 * q^33 + 64 * q^37 - 12 * q^45 - 100 * q^47 + 100 * q^49 - 28 * q^53 - 56 * q^55 + 40 * q^59 - 136 * q^67 + 60 * q^69 + 284 * q^71 - 72 * q^75 + 180 * q^77 + 36 * q^81 + 304 * q^89 + 24 * q^91 - 216 * q^93 - 376 * q^97 + 60 * q^99

Introduction

L-functions

Modular forms

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Database

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