LMFDB - Embedded newform 507.4.b.k.337.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [507,4,Mod(337,507)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(507, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("507.337");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 507 = 3 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	507.b (of order $2$, degree $1$, not 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:	$29.9139683729$
Analytic rank:	$0$
Dimension:	$18$
Coefficient field:	$\mathbb{Q}[x]/(x^{18} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{18} + 112 x^{16} + 5026 x^{14} + 114847 x^{12} + 1397921 x^{10} + 8545747 x^{8} + 21033277 x^{6} + \cdots + 729 $ x^18 + 112x^16 + 5026x^14 + 114847x^12 + 1397921x^10 + 8545747x^8 + 21033277x^6 + 6703200x^4 + 137781x^2 + 729
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 13^{8} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			337.9
Root			$0.588238i$ of defining polynomial
Character	$\chi$	$=$	507.337
Dual form			507.4.b.k.337.10

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.213700i q^{2} +3.00000 q^{3} +7.95433 q^{4} -15.3391i q^{5} -0.641100i q^{6} -32.3928i q^{7} -3.40944i q^{8} +9.00000 q^{9} +O(q^{10})$	\(q-0.213700i q^{2} +3.00000 q^{3} +7.95433 q^{4} -15.3391i q^{5} -0.641100i q^{6} -32.3928i q^{7} -3.40944i q^{8} +9.00000 q^{9} -3.27797 q^{10} +29.5925i q^{11} +23.8630 q^{12} -6.92233 q^{14} -46.0174i q^{15} +62.9061 q^{16} +78.1958 q^{17} -1.92330i q^{18} -10.6600i q^{19} -122.013i q^{20} -97.1783i q^{21} +6.32392 q^{22} +26.8789 q^{23} -10.2283i q^{24} -110.289 q^{25} +27.0000 q^{27} -257.663i q^{28} -190.785 q^{29} -9.83392 q^{30} +128.108i q^{31} -40.7185i q^{32} +88.7776i q^{33} -16.7104i q^{34} -496.877 q^{35} +71.5890 q^{36} -379.934i q^{37} -2.27805 q^{38} -52.2979 q^{40} +464.631i q^{41} -20.7670 q^{42} -322.758 q^{43} +235.389i q^{44} -138.052i q^{45} -5.74401i q^{46} -248.529i q^{47} +188.718 q^{48} -706.292 q^{49} +23.5688i q^{50} +234.587 q^{51} +740.167 q^{53} -5.76990i q^{54} +453.924 q^{55} -110.441 q^{56} -31.9801i q^{57} +40.7706i q^{58} -340.673i q^{59} -366.038i q^{60} -590.834 q^{61} +27.3767 q^{62} -291.535i q^{63} +494.547 q^{64} +18.9718 q^{66} +340.777i q^{67} +621.995 q^{68} +80.6366 q^{69} +106.183i q^{70} +36.2243i q^{71} -30.6850i q^{72} -164.572i q^{73} -81.1919 q^{74} -330.868 q^{75} -84.7935i q^{76} +958.584 q^{77} -327.242 q^{79} -964.925i q^{80} +81.0000 q^{81} +99.2916 q^{82} +1404.48i q^{83} -772.989i q^{84} -1199.46i q^{85} +68.9734i q^{86} -572.354 q^{87} +100.894 q^{88} +736.986i q^{89} -29.5018 q^{90} +213.803 q^{92} +384.324i q^{93} -53.1107 q^{94} -163.516 q^{95} -122.156i q^{96} -1494.87i q^{97} +150.935i q^{98} +266.333i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 18 q + 54 q^{3} - 88 q^{4} + 162 q^{9}+O(q^{10}) $ 18 * q + 54 * q^3 - 88 * q^4 + 162 * q^9	$ 18 q + 54 q^{3} - 88 q^{4} + 162 q^{9} + 108 q^{10} - 264 q^{12} + 316 q^{14} + 432 q^{16} - 356 q^{17} - 1260 q^{22} - 300 q^{23} + 40 q^{25} + 486 q^{27} - 194 q^{29} + 324 q^{30} - 836 q^{35} - 792 q^{36} + 1320 q^{38} - 3012 q^{40} + 948 q^{42} - 484 q^{43} + 1296 q^{48} + 76 q^{49} - 1068 q^{51} - 302 q^{53} + 4128 q^{55} - 4552 q^{56} - 2680 q^{61} - 694 q^{62} - 1786 q^{64} - 3780 q^{66} + 5570 q^{68} - 900 q^{69} - 2382 q^{74} + 120 q^{75} + 4284 q^{77} - 3182 q^{79} + 1458 q^{81} - 3034 q^{82} - 582 q^{87} + 7432 q^{88} + 972 q^{90} + 1030 q^{92} - 1384 q^{94} - 8316 q^{95}+O(q^{100}) $ 18 * q + 54 * q^3 - 88 * q^4 + 162 * q^9 + 108 * q^10 - 264 * q^12 + 316 * q^14 + 432 * q^16 - 356 * q^17 - 1260 * q^22 - 300 * q^23 + 40 * q^25 + 486 * q^27 - 194 * q^29 + 324 * q^30 - 836 * q^35 - 792 * q^36 + 1320 * q^38 - 3012 * q^40 + 948 * q^42 - 484 * q^43 + 1296 * q^48 + 76 * q^49 - 1068 * q^51 - 302 * q^53 + 4128 * q^55 - 4552 * q^56 - 2680 * q^61 - 694 * q^62 - 1786 * q^64 - 3780 * q^66 + 5570 * q^68 - 900 * q^69 - 2382 * q^74 + 120 * q^75 + 4284 * q^77 - 3182 * q^79 + 1458 * q^81 - 3034 * q^82 - 582 * q^87 + 7432 * q^88 + 972 * q^90 + 1030 * q^92 - 1384 * q^94 - 8316 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/507\mathbb{Z}\right)^\times$.

$n$	$170$	$340$
$\chi(n)$	$1$	$-1$

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 $

$2$	− 0.213700i	− 0.0755543i	−0.999286	−	0.0377772i	$-0.987972\pi$
$2$	− 0.213700i	− 0.0755543i	0.999286	−	0.0377772i	$-0.0120277\pi$
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	7.95433	0.994292
$5$	− 15.3391i	− 1.37197i	−0.727614	−	0.685987i	$-0.759371\pi$
$5$	− 15.3391i	− 1.37197i	0.727614	−	0.685987i	$-0.240629\pi$
$6$	− 0.641100i	− 0.0436213i
$7$	− 32.3928i	− 1.74905i	−0.484984	−	0.874523i	$-0.661175\pi$
$7$	− 32.3928i	− 1.74905i	0.484984	−	0.874523i	$-0.338825\pi$
$8$	− 3.40944i	− 0.150677i
$9$	9.00000	0.333333
$10$	−3.27797	−0.103659
$11$	29.5925i	0.811135i	0.914065	+	0.405567i	$0.132926\pi$
$11$	29.5925i	0.811135i	−0.914065	+	0.405567i	$0.867074\pi$
$12$	23.8630	0.574054
$13$	0	0
$13$	0	0
$14$	−6.92233	−0.132148
$15$	− 46.0174i	− 0.792110i
$16$	62.9061	0.982907
$17$	78.1958	1.11560	0.557802	−	0.829974i	$-0.311645\pi$
$17$	78.1958	1.11560	0.557802	+	0.829974i	$0.311645\pi$
$18$	− 1.92330i	− 0.0251848i
$19$	− 10.6600i	− 0.128715i	−0.997927	−	0.0643574i	$-0.979500\pi$
$19$	− 10.6600i	− 0.128715i	0.997927	−	0.0643574i	$-0.0204998\pi$
$20$	− 122.013i	− 1.36414i
$21$	− 97.1783i	− 1.00981i
$22$	6.32392	0.0612847
$23$	26.8789	0.243680	0.121840	−	0.992550i	$-0.461121\pi$
$23$	26.8789	0.243680	0.121840	+	0.992550i	$0.461121\pi$
$24$	− 10.2283i	− 0.0869936i
$25$	−110.289	−0.882314
$26$	0	0
$27$	27.0000	0.192450
$28$	− 257.663i	− 1.73906i
$29$	−190.785	−1.22165	−0.610824	−	0.791766i	$-0.709162\pi$
$29$	−190.785	−1.22165	−0.610824	+	0.791766i	$0.709162\pi$
$30$	−9.83392	−0.0598473
$31$	128.108i	0.742222i	0.928589	+	0.371111i	$0.121023\pi$
$31$	128.108i	0.742222i	−0.928589	+	0.371111i	$0.878977\pi$
$32$	− 40.7185i	− 0.224940i
$33$	88.7776i	0.468309i
$34$	− 16.7104i	− 0.0842887i
$35$	−496.877	−2.39965
$36$	71.5890	0.331431
$37$	− 379.934i	− 1.68813i	−0.536241	−	0.844065i	$-0.680156\pi$
$37$	− 379.934i	− 1.68813i	0.536241	−	0.844065i	$-0.319844\pi$
$38$	−2.27805	−0.00972496
$39$	0	0
$40$	−52.2979	−0.206725
$41$	464.631i	1.76983i	0.465749	+	0.884917i	$0.345785\pi$
$41$	464.631i	1.76983i	−0.465749	+	0.884917i	$0.654215\pi$
$42$	−20.7670	−0.0762957
$43$	−322.758	−1.14466	−0.572328	−	0.820025i	$-0.693959\pi$
$43$	−322.758	−1.14466	−0.572328	+	0.820025i	$0.693959\pi$
$44$	235.389i	0.806504i
$45$	− 138.052i	− 0.457325i
$46$	− 5.74401i	− 0.0184110i
$47$	− 248.529i	− 0.771314i	−0.922642	−	0.385657i	$-0.873975\pi$
$47$	− 248.529i	− 0.771314i	0.922642	−	0.385657i	$-0.126025\pi$
$48$	188.718	0.567482
$49$	−706.292	−2.05916
$50$	23.5688i	0.0666626i
$51$	234.587	0.644094
$52$	0	0
$53$	740.167	1.91830	0.959148	−	0.282904i	$-0.0912976\pi$
$53$	740.167	1.91830	0.959148	+	0.282904i	$0.0912976\pi$
$54$	− 5.76990i	− 0.0145404i
$55$	453.924	1.11286
$56$	−110.441	−0.263542
$57$	− 31.9801i	− 0.0743135i
$58$	40.7706i	0.0923008i
$59$	− 340.673i	− 0.751727i	−0.926675	−	0.375864i	$-0.877346\pi$
$59$	− 340.673i	− 0.751727i	0.926675	−	0.375864i	$-0.122654\pi$
$60$	− 366.038i	− 0.787588i
$61$	−590.834	−1.24014	−0.620070	−	0.784547i	$-0.712896\pi$
$61$	−590.834	−1.24014	−0.620070	+	0.784547i	$0.712896\pi$
$62$	27.3767	0.0560781
$63$	− 291.535i	− 0.583015i
$64$	494.547	0.965912
$65$	0	0
$66$	18.9718	0.0353828
$67$	340.777i	0.621382i	0.950511	+	0.310691i	$0.100560\pi$
$67$	340.777i	0.621382i	−0.950511	+	0.310691i	$0.899440\pi$
$68$	621.995	1.10924
$69$	80.6366	0.140688
$70$	106.183i	0.181304i
$71$	36.2243i	0.0605498i	0.999542	+	0.0302749i	$0.00963827\pi$
$71$	36.2243i	0.0605498i	−0.999542	+	0.0302749i	$0.990362\pi$
$72$	− 30.6850i	− 0.0502258i
$73$	− 164.572i	− 0.263859i	−0.991259	−	0.131929i	$-0.957883\pi$
$73$	− 164.572i	− 0.263859i	0.991259	−	0.131929i	$-0.0421172\pi$
$74$	−81.1919	−0.127546
$75$	−330.868	−0.509404
$76$	− 84.7935i	− 0.127980i
$77$	958.584	1.41871
$78$	0	0
$79$	−327.242	−0.466046	−0.233023	−	0.972471i	$-0.574862\pi$
$79$	−327.242	−0.466046	−0.233023	+	0.972471i	$0.574862\pi$
$80$	− 964.925i	− 1.34852i
$81$	81.0000	0.111111
$82$	99.2916	0.133719
$83$	1404.48i	1.85737i	0.370871	+	0.928684i	$0.379059\pi$
$83$	1404.48i	1.85737i	−0.370871	+	0.928684i	$0.620941\pi$
$84$	− 772.989i	− 1.00405i
$85$	− 1199.46i	− 1.53058i
$86$	68.9734i	0.0864837i
$87$	−572.354	−0.705319
$88$	100.894	0.122220
$89$	736.986i	0.877756i	0.898547	+	0.438878i	$0.144624\pi$
$89$	736.986i	0.877756i	−0.898547	+	0.438878i	$0.855376\pi$
$90$	−29.5018	−0.0345529
$91$	0	0
$92$	213.803	0.242289
$93$	384.324i	0.428522i
$94$	−53.1107	−0.0582761
$95$	−163.516	−0.176593
$96$	− 122.156i	− 0.129869i
$97$	− 1494.87i	− 1.56476i	−0.622804	−	0.782378i	$-0.714007\pi$
$97$	− 1494.87i	− 1.56476i	0.622804	−	0.782378i	$-0.285993\pi$
$98$	150.935i	0.155578i
$99$	266.333i	0.270378i
$100$	−877.277	−0.877277
$101$	484.010	0.476839	0.238420	−	0.971162i	$-0.423371\pi$
$101$	484.010	0.476839	0.238420	+	0.971162i	$0.423371\pi$
$102$	− 50.1313i	− 0.0486641i
$103$	−214.049	−0.204766	−0.102383	−	0.994745i	$-0.532647\pi$
$103$	−214.049	−0.204766	−0.102383	+	0.994745i	$0.532647\pi$
$104$	0	0
$105$	−1490.63	−1.38544
$106$	− 158.174i	− 0.144936i
$107$	148.288	0.133977	0.0669884	−	0.997754i	$-0.478661\pi$
$107$	148.288	0.133977	0.0669884	+	0.997754i	$0.478661\pi$
$108$	214.767	0.191351
$109$	504.881i	0.443659i	0.975086	+	0.221829i	$0.0712028\pi$
$109$	504.881i	0.443659i	−0.975086	+	0.221829i	$0.928797\pi$
$110$	− 97.0035i	− 0.0840811i
$111$	− 1139.80i	− 0.974642i
$112$	− 2037.70i	− 1.71915i
$113$	−1136.49	−0.946120	−0.473060	−	0.881030i	$-0.656851\pi$
$113$	−1136.49	−0.946120	−0.473060	+	0.881030i	$0.656851\pi$
$114$	−6.83415	−0.00561471
$115$	− 412.299i	− 0.334322i
$116$	−1517.56	−1.21467
$117$	0	0
$118$	−72.8019	−0.0567962
$119$	− 2532.98i	− 1.95124i
$120$	−156.894	−0.119353
$121$	455.282	0.342060
$122$	126.261i	0.0936979i
$123$	1393.89i	1.02181i
$124$	1019.01i	0.737985i
$125$	− 225.651i	− 0.161463i
$126$	−62.3010	−0.0440493
$127$	1286.84	0.899122	0.449561	−	0.893250i	$-0.351580\pi$
$127$	1286.84	0.899122	0.449561	+	0.893250i	$0.351580\pi$
$128$	− 431.433i	− 0.297919i
$129$	−968.275	−0.660867
$130$	0	0
$131$	2229.88	1.48722	0.743610	−	0.668614i	$-0.233112\pi$
$131$	2229.88	1.48722	0.743610	+	0.668614i	$0.233112\pi$
$132$	706.166i	0.465636i
$133$	−345.308	−0.225128
$134$	72.8241	0.0469481
$135$	− 414.157i	− 0.264037i
$136$	− 266.604i	− 0.168096i
$137$	474.745i	0.296060i	0.988983	+	0.148030i	$0.0472932\pi$
$137$	474.745i	0.296060i	−0.988983	+	0.148030i	$0.952707\pi$
$138$	− 17.2320i	− 0.0106296i
$139$	1927.02	1.17588	0.587941	−	0.808904i	$-0.299939\pi$
$139$	1927.02	1.17588	0.587941	+	0.808904i	$0.299939\pi$
$140$	−3952.33	−2.38595
$141$	− 745.588i	− 0.445318i
$142$	7.74113	0.00457480
$143$	0	0
$144$	566.155	0.327636
$145$	2926.47i	1.67607i
$146$	−35.1690	−0.0199357
$147$	−2118.88	−1.18886
$148$	− 3022.12i	− 1.67849i
$149$	1065.73i	0.585959i	0.956119	+	0.292979i	$0.0946467\pi$
$149$	1065.73i	0.585959i	−0.956119	+	0.292979i	$0.905353\pi$
$150$	70.7064i	0.0384877i
$151$	− 877.888i	− 0.473123i	−0.971617	−	0.236561i	$-0.923980\pi$
$151$	− 877.888i	− 0.473123i	0.971617	−	0.236561i	$-0.0760204\pi$
$152$	−36.3448	−0.0193944
$153$	703.762	0.371868
$154$	− 204.849i	− 0.107190i
$155$	1965.07	1.01831
$156$	0	0
$157$	2314.94	1.17676	0.588382	−	0.808583i	$-0.299765\pi$
$157$	2314.94	1.17676	0.588382	+	0.808583i	$0.299765\pi$
$158$	69.9317i	0.0352118i
$159$	2220.50	1.10753
$160$	−624.587	−0.308612
$161$	− 870.681i	− 0.426207i
$162$	− 17.3097i	− 0.00839492i
$163$	− 331.399i	− 0.159246i	−0.996825	−	0.0796232i	$-0.974628\pi$
$163$	− 331.399i	− 0.159246i	0.996825	−	0.0796232i	$-0.0253717\pi$
$164$	3695.83i	1.75973i
$165$	1361.77	0.642508
$166$	300.137	0.140332
$167$	2171.62i	1.00626i	0.864212	+	0.503129i	$0.167818\pi$
$167$	2171.62i	1.00626i	−0.864212	+	0.503129i	$0.832182\pi$
$168$	−331.324	−0.152156
$169$	0	0
$170$	−256.324	−0.115642
$171$	− 95.9404i	− 0.0429049i
$172$	−2567.33	−1.13812
$173$	2936.69	1.29059	0.645295	−	0.763933i	$-0.276734\pi$
$173$	2936.69	1.29059	0.645295	+	0.763933i	$0.276734\pi$
$174$	122.312i	0.0532899i
$175$	3572.57i	1.54321i
$176$	1861.55i	0.797270i
$177$	− 1022.02i	− 0.434010i
$178$	157.494	0.0663183
$179$	−1615.57	−0.674599	−0.337300	−	0.941397i	$-0.609514\pi$
$179$	−1615.57	−0.674599	−0.337300	+	0.941397i	$0.609514\pi$
$180$	− 1098.11i	− 0.454714i
$181$	−725.019	−0.297736	−0.148868	−	0.988857i	$-0.547563\pi$
$181$	−725.019	−0.297736	−0.148868	+	0.988857i	$0.547563\pi$
$182$	0	0
$183$	−1772.50	−0.715995
$184$	− 91.6419i	− 0.0367170i
$185$	−5827.87	−2.31607
$186$	82.1300	0.0323767
$187$	2314.01i	0.904905i
$188$	− 1976.89i	− 0.766911i
$189$	− 874.605i	− 0.336604i
$190$	34.9433i	0.0133424i
$191$	1717.08	0.650490	0.325245	−	0.945630i	$-0.394553\pi$
$191$	1717.08	0.650490	0.325245	+	0.945630i	$0.394553\pi$
$192$	1483.64	0.557670
$193$	− 435.830i	− 0.162548i	−0.996692	−	0.0812740i	$-0.974101\pi$
$193$	− 435.830i	− 0.162548i	0.996692	−	0.0812740i	$-0.0258989\pi$
$194$	−319.454	−0.118224
$195$	0	0
$196$	−5618.08	−2.04741
$197$	694.556i	0.251193i	0.992081	+	0.125597i	$0.0400845\pi$
$197$	694.556i	0.251193i	−0.992081	+	0.125597i	$0.959915\pi$
$198$	56.9153	0.0204282
$199$	−2899.24	−1.03277	−0.516386	−	0.856356i	$-0.672723\pi$
$199$	−2899.24	−1.03277	−0.516386	+	0.856356i	$0.672723\pi$
$200$	376.024i	0.132945i
$201$	1022.33i	0.358755i
$202$	− 103.433i	− 0.0360273i
$203$	6180.04i	2.13672i
$204$	1865.99	0.640417
$205$	7127.04	2.42817
$206$	45.7423i	0.0154709i
$207$	241.910	0.0812265
$208$	0	0
$209$	315.457	0.104405
$210$	318.548i	0.104676i
$211$	5250.30	1.71301	0.856507	−	0.516136i	$-0.172630\pi$
$211$	5250.30	1.71301	0.856507	+	0.516136i	$0.172630\pi$
$212$	5887.53	1.90735
$213$	108.673i	0.0349584i
$214$	− 31.6891i	− 0.0101225i
$215$	4950.84i	1.57044i
$216$	− 92.0549i	− 0.0289979i
$217$	4149.77	1.29818
$218$	107.893	0.0335203
$219$	− 493.716i	− 0.152339i
$220$	3610.66	1.10650
$221$	0	0
$222$	−243.576	−0.0736384
$223$	− 1382.46i	− 0.415141i	−0.978220	−	0.207570i	$-0.933444\pi$
$223$	− 1382.46i	− 0.415141i	0.978220	−	0.207570i	$-0.0665556\pi$
$224$	−1318.99	−0.393431
$225$	−992.603	−0.294105
$226$	242.867i	0.0714835i
$227$	1223.57i	0.357760i	0.983871	+	0.178880i	$0.0572474\pi$
$227$	1223.57i	0.357760i	−0.983871	+	0.178880i	$0.942753\pi$
$228$	− 254.380i	− 0.0738893i
$229$	4019.61i	1.15993i	0.814642	+	0.579964i	$0.196933\pi$
$229$	4019.61i	1.15993i	−0.814642	+	0.579964i	$0.803067\pi$
$230$	−88.1082	−0.0252595
$231$	2875.75	0.819094
$232$	650.468i	0.184075i
$233$	−1658.40	−0.466290	−0.233145	−	0.972442i	$-0.574902\pi$
$233$	−1658.40	−0.466290	−0.233145	+	0.972442i	$0.574902\pi$
$234$	0	0
$235$	−3812.23	−1.05822
$236$	− 2709.83i	− 0.747436i
$237$	−981.727	−0.269072
$238$	−541.297	−0.147425
$239$	− 618.554i	− 0.167410i	−0.996491	−	0.0837049i	$-0.973325\pi$
$239$	− 618.554i	− 0.167410i	0.996491	−	0.0837049i	$-0.0266753\pi$
$240$	− 2894.77i	− 0.778570i
$241$	1135.25i	0.303434i	0.988424	+	0.151717i	$0.0484803\pi$
$241$	1135.25i	0.303434i	−0.988424	+	0.151717i	$0.951520\pi$
$242$	− 97.2938i	− 0.0258441i
$243$	243.000	0.0641500
$244$	−4699.69	−1.23306
$245$	10833.9i	2.82512i
$246$	297.875	0.0772025
$247$	0	0
$248$	436.777	0.111836
$249$	4213.44i	1.07235i
$250$	−48.2215	−0.0121992
$251$	−2287.83	−0.575326	−0.287663	−	0.957732i	$-0.592878\pi$
$251$	−2287.83	−0.575326	−0.287663	+	0.957732i	$0.592878\pi$
$252$	− 2318.97i	− 0.579687i
$253$	795.414i	0.197657i
$254$	− 274.997i	− 0.0679326i
$255$	− 3598.37i	− 0.883681i
$256$	3864.18	0.943403
$257$	5113.49	1.24113	0.620566	−	0.784154i	$-0.286903\pi$
$257$	5113.49	1.24113	0.620566	+	0.784154i	$0.286903\pi$
$258$	206.920i	0.0499314i
$259$	−12307.1	−2.95262
$260$	0	0
$261$	−1717.06	−0.407216
$262$	− 476.526i	− 0.112366i
$263$	2760.17	0.647145	0.323573	−	0.946203i	$-0.395116\pi$
$263$	2760.17	0.647145	0.323573	+	0.946203i	$0.395116\pi$
$264$	302.682	0.0705635
$265$	− 11353.5i	− 2.63185i
$266$	73.7923i	0.0170094i
$267$	2210.96i	0.506773i
$268$	2710.66i	0.617834i
$269$	−3310.32	−0.750312	−0.375156	−	0.926962i	$-0.622411\pi$
$269$	−3310.32	−0.750312	−0.375156	+	0.926962i	$0.622411\pi$
$270$	−88.5053	−0.0199491
$271$	− 2522.26i	− 0.565374i	−0.959212	−	0.282687i	$-0.908774\pi$
$271$	− 2522.26i	− 0.565374i	0.959212	−	0.282687i	$-0.0912258\pi$
$272$	4918.99	1.09654
$273$	0	0
$274$	101.453	0.0223686
$275$	− 3263.74i	− 0.715675i
$276$	641.410	0.139885
$277$	3455.48	0.749530	0.374765	−	0.927120i	$-0.377723\pi$
$277$	3455.48	0.749530	0.374765	+	0.927120i	$0.377723\pi$
$278$	− 411.804i	− 0.0888430i
$279$	1152.97i	0.247407i
$280$	1694.07i	0.361572i
$281$	981.649i	0.208400i	0.994556	+	0.104200i	$0.0332281\pi$
$281$	981.649i	0.208400i	−0.994556	+	0.104200i	$0.966772\pi$
$282$	−159.332	−0.0336457
$283$	−5331.10	−1.11979	−0.559896	−	0.828563i	$-0.689159\pi$
$283$	−5331.10	−1.11979	−0.559896	+	0.828563i	$0.689159\pi$
$284$	288.140i	0.0602041i
$285$	−490.547	−0.101956
$286$	0	0
$287$	15050.7	3.09552
$288$	− 366.467i	− 0.0749801i
$289$	1201.58	0.244572
$290$	625.387	0.126634
$291$	− 4484.62i	− 0.903412i
$292$	− 1309.06i	− 0.262353i
$293$	2420.39i	0.482597i	0.970451	+	0.241299i	$0.0775733\pi$
$293$	2420.39i	0.482597i	−0.970451	+	0.241299i	$0.922427\pi$
$294$	452.804i	0.0898233i
$295$	−5225.64	−1.03135
$296$	−1295.36	−0.254363
$297$	798.998i	0.156103i
$298$	227.746	0.0442717
$299$	0	0
$300$	−2631.83	−0.506496
$301$	10455.0i	2.00205i
$302$	−187.605	−0.0357465
$303$	1452.03	0.275303
$304$	− 670.581i	− 0.126515i
$305$	9062.88i	1.70144i
$306$	− 150.394i	− 0.0280962i
$307$	875.509i	0.162762i	0.996683	+	0.0813810i	$0.0259330\pi$
$307$	875.509i	0.162762i	−0.996683	+	0.0813810i	$0.974067\pi$
$308$	7624.90	1.41061
$309$	−642.147	−0.118222
$310$	− 419.935i	− 0.0769377i
$311$	−6419.10	−1.17040	−0.585199	−	0.810890i	$-0.698984\pi$
$311$	−6419.10	−1.17040	−0.585199	+	0.810890i	$0.698984\pi$
$312$	0	0
$313$	−5015.58	−0.905742	−0.452871	−	0.891576i	$-0.649600\pi$
$313$	−5015.58	−0.905742	−0.452871	+	0.891576i	$0.649600\pi$
$314$	− 494.701i	− 0.0889096i
$315$	−4471.90	−0.799882
$316$	−2602.99	−0.463386
$317$	5228.81i	0.926433i	0.886245	+	0.463217i	$0.153305\pi$
$317$	5228.81i	0.926433i	−0.886245	+	0.463217i	$0.846695\pi$
$318$	− 474.521i	− 0.0836786i
$319$	− 5645.80i	− 0.990922i
$320$	− 7585.92i	− 1.32521i
$321$	444.863	0.0773515
$322$	−186.064	−0.0322018
$323$	− 833.570i	− 0.143595i
$324$	644.301	0.110477
$325$	0	0
$326$	−70.8199	−0.0120318
$327$	1514.64i	0.256146i
$328$	1584.13	0.266674
$329$	−8050.56	−1.34906
$330$	− 291.010i	− 0.0485442i
$331$	− 3186.81i	− 0.529193i	−0.964359	−	0.264596i	$-0.914761\pi$
$331$	− 3186.81i	− 0.529193i	0.964359	−	0.264596i	$-0.0852387\pi$
$332$	11171.7i	1.84677i
$333$	− 3419.41i	− 0.562710i
$334$	464.075	0.0760271
$335$	5227.23	0.852520
$336$	− 6113.11i	− 0.992551i
$337$	5600.69	0.905309	0.452655	−	0.891686i	$-0.350477\pi$
$337$	5600.69	0.905309	0.452655	+	0.891686i	$0.350477\pi$
$338$	0	0
$339$	−3409.46	−0.546243
$340$	− 9540.87i	− 1.52184i
$341$	−3791.04	−0.602042
$342$	−20.5024	−0.00324165
$343$	11768.0i	1.85252i
$344$	1100.42i	0.172474i
$345$	− 1236.90i	− 0.193021i
$346$	− 627.570i	− 0.0975097i
$347$	−6982.33	−1.08020	−0.540102	−	0.841599i	$-0.681614\pi$
$347$	−6982.33	−1.08020	−0.540102	+	0.841599i	$0.681614\pi$
$348$	−4552.69	−0.701293
$349$	− 2872.20i	− 0.440531i	−0.975440	−	0.220265i	$-0.929308\pi$
$349$	− 2872.20i	− 0.440531i	0.975440	−	0.220265i	$-0.0706924\pi$
$350$	763.459	0.116596
$351$	0	0
$352$	1204.96	0.182457
$353$	− 5104.55i	− 0.769655i	−0.922989	−	0.384827i	$-0.874261\pi$
$353$	− 5104.55i	− 0.769655i	0.922989	−	0.384827i	$-0.125739\pi$
$354$	−218.406	−0.0327913
$355$	555.650	0.0830728
$356$	5862.23i	0.872746i
$357$	− 7598.94i	− 1.12655i
$358$	345.247i	0.0509689i
$359$	10771.0i	1.58348i	0.610857	+	0.791741i	$0.290825\pi$
$359$	10771.0i	1.58348i	−0.610857	+	0.791741i	$0.709175\pi$
$360$	−470.681	−0.0689085
$361$	6745.36	0.983433
$362$	154.936i	0.0224952i
$363$	1365.85	0.197489
$364$	0	0
$365$	−2524.39	−0.362008
$366$	378.783i	0.0540965i
$367$	−11380.5	−1.61868	−0.809340	−	0.587340i	$-0.800175\pi$
$367$	−11380.5	−1.61868	−0.809340	+	0.587340i	$0.800175\pi$
$368$	1690.84	0.239514
$369$	4181.68i	0.589945i
$370$	1245.41i	0.174989i
$371$	− 23976.1i	− 3.35519i
$372$	3057.04i	0.426076i
$373$	3196.75	0.443758	0.221879	−	0.975074i	$-0.428781\pi$
$373$	3196.75	0.443758	0.221879	+	0.975074i	$0.428781\pi$
$374$	494.504	0.0683695
$375$	− 676.952i	− 0.0932204i
$376$	−847.346	−0.116220
$377$	0	0
$378$	−186.903	−0.0254319
$379$	2050.75i	0.277942i	0.990296	+	0.138971i	$0.0443795\pi$
$379$	2050.75i	0.277942i	−0.990296	+	0.138971i	$0.955621\pi$
$380$	−1300.66	−0.175585
$381$	3860.52	0.519108
$382$	− 366.940i	− 0.0491473i
$383$	− 3507.54i	− 0.467955i	−0.972242	−	0.233978i	$-0.924826\pi$
$383$	− 3507.54i	− 0.467955i	0.972242	−	0.233978i	$-0.0751742\pi$
$384$	− 1294.30i	− 0.172004i
$385$	− 14703.9i	− 1.94644i
$386$	−93.1369	−0.0122812
$387$	−2904.83	−0.381552
$388$	− 11890.7i	− 1.55582i
$389$	6572.66	0.856676	0.428338	−	0.903618i	$-0.359099\pi$
$389$	6572.66	0.856676	0.428338	+	0.903618i	$0.359099\pi$
$390$	0	0
$391$	2101.81	0.271850
$392$	2408.06i	0.310269i
$393$	6689.65	0.858646
$394$	148.427	0.0189787
$395$	5019.62i	0.639403i
$396$	2118.50i	0.268835i
$397$	− 5285.98i	− 0.668251i	−0.942529	−	0.334126i	$-0.891559\pi$
$397$	− 5285.98i	− 0.668251i	0.942529	−	0.334126i	$-0.108441\pi$
$398$	619.567i	0.0780304i
$399$	−1035.92	−0.129978
$400$	−6937.86	−0.867233
$401$	− 919.541i	− 0.114513i	−0.998360	−	0.0572565i	$-0.981765\pi$
$401$	− 919.541i	− 0.114513i	0.998360	−	0.0572565i	$-0.0182353\pi$
$402$	218.472	0.0271055
$403$	0	0
$404$	3849.97	0.474117
$405$	− 1242.47i	− 0.152442i
$406$	1320.67	0.161438
$407$	11243.2	1.36930
$408$	− 799.811i	− 0.0970504i
$409$	1539.17i	0.186081i	0.995662	+	0.0930403i	$0.0296585\pi$
$409$	1539.17i	0.186081i	−0.995662	+	0.0930403i	$0.970341\pi$
$410$	− 1523.05i	− 0.183458i
$411$	1424.23i	0.170930i
$412$	−1702.62	−0.203597
$413$	−11035.4	−1.31481
$414$	− 51.6961i	− 0.00613702i
$415$	21543.5	2.54826
$416$	0	0
$417$	5781.06	0.678896
$418$	− 67.4132i	− 0.00788825i
$419$	10579.9	1.23357	0.616783	−	0.787133i	$-0.288436\pi$
$419$	10579.9	1.23357	0.616783	+	0.787133i	$0.288436\pi$
$420$	−11857.0	−1.37753
$421$	− 74.9351i	− 0.00867485i	−0.999991	−	0.00433743i	$-0.998619\pi$
$421$	− 74.9351i	− 0.00867485i	0.999991	−	0.00433743i	$-0.00138065\pi$
$422$	− 1121.99i	− 0.129426i
$423$	− 2236.77i	− 0.257105i
$424$	− 2523.55i	− 0.289044i
$425$	−8624.15	−0.984313
$426$	23.2234	0.00264126
$427$	19138.8i	2.16906i
$428$	1179.53	0.133212
$429$	0	0
$430$	1057.99	0.118653
$431$	12165.6i	1.35962i	0.733388	+	0.679810i	$0.237938\pi$
$431$	12165.6i	1.35962i	−0.733388	+	0.679810i	$0.762062\pi$
$432$	1698.46	0.189161
$433$	2869.23	0.318445	0.159222	−	0.987243i	$-0.449101\pi$
$433$	2869.23	0.318445	0.159222	+	0.987243i	$0.449101\pi$
$434$	− 886.806i	− 0.0980831i
$435$	8779.41i	0.967680i
$436$	4015.99i	0.441126i
$437$	− 286.530i	− 0.0313652i
$438$	−105.507	−0.0115099
$439$	3845.12	0.418035	0.209018	−	0.977912i	$-0.432973\pi$
$439$	3845.12	0.418035	0.209018	+	0.977912i	$0.432973\pi$
$440$	− 1547.63i	− 0.167682i
$441$	−6356.63	−0.686387
$442$	0	0
$443$	−3858.30	−0.413799	−0.206900	−	0.978362i	$-0.566337\pi$
$443$	−3858.30	−0.413799	−0.206900	+	0.978362i	$0.566337\pi$
$444$	− 9066.37i	− 0.969079i
$445$	11304.7	1.20426
$446$	−295.432	−0.0313657
$447$	3197.18i	0.338303i
$448$	− 16019.7i	− 1.68942i
$449$	− 5550.51i	− 0.583395i	−0.956511	−	0.291698i	$-0.905780\pi$
$449$	− 5550.51i	− 0.583395i	0.956511	−	0.291698i	$-0.0942201\pi$
$450$	212.119i	0.0222209i
$451$	−13749.6	−1.43557
$452$	−9039.99	−0.940719
$453$	− 2633.66i	− 0.273157i
$454$	261.478	0.0270303
$455$	0	0
$456$	−109.034	−0.0111974
$457$	− 8102.42i	− 0.829355i	−0.909968	−	0.414677i	$-0.863894\pi$
$457$	− 8102.42i	− 0.829355i	0.909968	−	0.414677i	$-0.136106\pi$
$458$	858.991	0.0876376
$459$	2111.29	0.214698
$460$	− 3279.56i	− 0.332414i
$461$	− 9230.86i	− 0.932590i	−0.884629	−	0.466295i	$-0.845588\pi$
$461$	− 9230.86i	− 0.932590i	0.884629	−	0.466295i	$-0.154412\pi$
$462$	− 614.548i	− 0.0618861i
$463$	− 13934.7i	− 1.39870i	−0.714777	−	0.699352i	$-0.753472\pi$
$463$	− 13934.7i	− 1.39870i	0.714777	−	0.699352i	$-0.246528\pi$
$464$	−12001.5	−1.20077
$465$	5895.20	0.587921
$466$	354.400i	0.0352302i
$467$	−10918.0	−1.08185	−0.540926	−	0.841070i	$-0.681926\pi$
$467$	−10918.0	−1.08185	−0.540926	+	0.841070i	$0.681926\pi$
$468$	0	0
$469$	11038.7	1.08682
$470$	814.673i	0.0799533i
$471$	6944.81	0.679405
$472$	−1161.51	−0.113268
$473$	− 9551.23i	− 0.928470i
$474$	209.795i	0.0203295i
$475$	1175.69i	0.113567i
$476$	− 20148.2i	− 1.94010i
$477$	6661.50	0.639432
$478$	−132.185	−0.0126485
$479$	− 11028.4i	− 1.05198i	−0.850490	−	0.525992i	$-0.823694\pi$
$479$	− 11028.4i	− 1.05198i	0.850490	−	0.525992i	$-0.176306\pi$
$480$	−1873.76	−0.178177
$481$	0	0
$482$	242.602	0.0229258
$483$	− 2612.04i	− 0.246071i
$484$	3621.47	0.340108
$485$	−22930.0	−2.14680
$486$	− 51.9291i	− 0.00484681i
$487$	1078.10i	0.100315i	0.998741	+	0.0501576i	$0.0159724\pi$
$487$	1078.10i	0.100315i	−0.998741	+	0.0501576i	$0.984028\pi$
$488$	2014.41i	0.186861i
$489$	− 994.197i	− 0.0919410i
$490$	2315.21	0.213450
$491$	−6572.88	−0.604134	−0.302067	−	0.953287i	$-0.597677\pi$
$491$	−6572.88	−0.604134	−0.302067	+	0.953287i	$0.597677\pi$
$492$	11087.5i	1.01598i
$493$	−14918.6	−1.36288
$494$	0	0
$495$	4085.32	0.370952
$496$	8058.77i	0.729535i
$497$	1173.41	0.105904
$498$	900.411	0.0810208
$499$	9956.79i	0.893241i	0.894724	+	0.446620i	$0.147373\pi$
$499$	9956.79i	0.893241i	−0.894724	+	0.446620i	$0.852627\pi$
$500$	− 1794.90i	− 0.160541i
$501$	6514.86i	0.580963i
$502$	488.910i	0.0434683i
$503$	−11965.0	−1.06062	−0.530310	−	0.847804i	$-0.677924\pi$
$503$	−11965.0	−1.06062	−0.530310	+	0.847804i	$0.677924\pi$
$504$	−993.971	−0.0878472
$505$	− 7424.29i	− 0.654211i
$506$	169.980	0.0149338
$507$	0	0
$508$	10235.9	0.893990
$509$	− 7160.44i	− 0.623538i	−0.950158	−	0.311769i	$-0.899078\pi$
$509$	− 7160.44i	− 0.623538i	0.950158	−	0.311769i	$-0.100922\pi$
$510$	−768.971	−0.0667659
$511$	−5330.94	−0.461501
$512$	− 4277.24i	− 0.369197i
$513$	− 287.821i	− 0.0247712i
$514$	− 1092.75i	− 0.0937729i
$515$	3283.33i	0.280933i
$516$	−7701.98	−0.657095
$517$	7354.61	0.625639
$518$	2630.03i	0.223083i
$519$	8810.06	0.745123
$520$	0	0
$521$	−17213.1	−1.44745	−0.723725	−	0.690089i	$-0.757572\pi$
$521$	−17213.1	−1.44745	−0.723725	+	0.690089i	$0.757572\pi$
$522$	366.936i	0.0307669i
$523$	−10110.1	−0.845281	−0.422640	−	0.906297i	$-0.638897\pi$
$523$	−10110.1	−0.845281	−0.422640	+	0.906297i	$0.638897\pi$
$524$	17737.2	1.47873
$525$	10717.7i	0.890971i
$526$	− 589.847i	− 0.0488946i
$527$	10017.5i	0.828026i
$528$	5584.65i	0.460304i
$529$	−11444.5	−0.940620
$530$	−2426.25	−0.198848
$531$	− 3066.06i	− 0.250576i
$532$	−2746.70	−0.223843
$533$	0	0
$534$	472.481	0.0382889
$535$	− 2274.61i	− 0.183813i
$536$	1161.86	0.0936281
$537$	−4846.71	−0.389480
$538$	707.416i	0.0566893i
$539$	− 20901.0i	− 1.67026i
$540$	− 3294.34i	− 0.262529i
$541$	5951.54i	0.472970i	0.971635	+	0.236485i	$0.0759954\pi$
$541$	5951.54i	0.472970i	−0.971635	+	0.236485i	$0.924005\pi$
$542$	−539.007	−0.0427165
$543$	−2175.06	−0.171898
$544$	− 3184.02i	− 0.250944i
$545$	7744.43	0.608688
$546$	0	0
$547$	5157.62	0.403152	0.201576	−	0.979473i	$-0.435394\pi$
$547$	5157.62	0.403152	0.201576	+	0.979473i	$0.435394\pi$
$548$	3776.28i	0.294370i
$549$	−5317.51	−0.413380
$550$	−697.460	−0.0540724
$551$	2033.77i	0.157244i
$552$	− 274.926i	− 0.0211986i
$553$	10600.3i	0.815136i
$554$	− 738.436i	− 0.0566302i
$555$	−17483.6	−1.33718
$556$	15328.2	1.16917
$557$	− 23314.4i	− 1.77354i	−0.462206	−	0.886772i	$-0.652942\pi$
$557$	− 23314.4i	− 1.77354i	0.462206	−	0.886772i	$-0.347058\pi$
$558$	246.390	0.0186927
$559$	0	0
$560$	−31256.6	−2.35863
$561$	6942.03i	0.522447i
$562$	209.778	0.0157455
$563$	−2329.12	−0.174353	−0.0871764	−	0.996193i	$-0.527784\pi$
$563$	−2329.12	−0.174353	−0.0871764	+	0.996193i	$0.527784\pi$
$564$	− 5930.66i	− 0.442776i
$565$	17432.7i	1.29805i
$566$	1139.26i	0.0846051i
$567$	− 2623.81i	− 0.194338i
$568$	123.505	0.00912348
$569$	17446.6	1.28541	0.642706	−	0.766113i	$-0.277812\pi$
$569$	17446.6	1.28541	0.642706	+	0.766113i	$0.277812\pi$
$570$	104.830i	0.00770323i
$571$	413.526	0.0303074	0.0151537	−	0.999885i	$-0.495176\pi$
$571$	413.526	0.0303074	0.0151537	+	0.999885i	$0.495176\pi$
$572$	0	0
$573$	5151.24	0.375560
$574$	− 3216.33i	− 0.233880i
$575$	−2964.45	−0.215002
$576$	4450.92	0.321971
$577$	21258.8i	1.53382i	0.641754	+	0.766911i	$0.278207\pi$
$577$	21258.8i	1.53382i	−0.641754	+	0.766911i	$0.721793\pi$
$578$	− 256.778i	− 0.0184785i
$579$	− 1307.49i	− 0.0938471i
$580$	23278.1i	1.66650i
$581$	45495.0	3.24862
$582$	−958.362	−0.0682567
$583$	21903.4i	1.55600i
$584$	−561.098	−0.0397575
$585$	0	0
$586$	517.238	0.0364623
$587$	− 2965.69i	− 0.208530i	−0.994550	−	0.104265i	$-0.966751\pi$
$587$	− 2965.69i	− 0.208530i	0.994550	−	0.104265i	$-0.0332490\pi$
$588$	−16854.2	−1.18207
$589$	1365.64	0.0955349
$590$	1116.72i	0.0779230i
$591$	2083.67i	0.145027i
$592$	− 23900.2i	− 1.65928i
$593$	− 17786.8i	− 1.23173i	−0.787852	−	0.615865i	$-0.788807\pi$
$593$	− 17786.8i	− 1.23173i	0.787852	−	0.615865i	$-0.211193\pi$
$594$	170.746	0.0117943
$595$	−38853.7	−2.67705
$596$	8477.16i	0.582614i
$597$	−8697.72	−0.596271
$598$	0	0
$599$	21796.4	1.48677	0.743387	−	0.668861i	$-0.233218\pi$
$599$	21796.4	1.48677	0.743387	+	0.668861i	$0.233218\pi$
$600$	1128.07i	0.0767557i
$601$	20468.7	1.38924	0.694622	−	0.719375i	$-0.255572\pi$
$601$	20468.7	1.38924	0.694622	+	0.719375i	$0.255572\pi$
$602$	2234.24	0.151264
$603$	3066.99i	0.207127i
$604$	− 6983.01i	− 0.470422i
$605$	− 6983.64i	− 0.469298i
$606$	− 310.298i	− 0.0208004i
$607$	7852.53	0.525081	0.262541	−	0.964921i	$-0.415440\pi$
$607$	7852.53	0.525081	0.262541	+	0.964921i	$0.415440\pi$
$608$	−434.061	−0.0289531
$609$	18540.1i	1.23364i
$610$	1936.74	0.128551
$611$	0	0
$612$	5597.96	0.369745
$613$	2460.21i	0.162099i	0.996710	+	0.0810497i	$0.0258273\pi$
$613$	2460.21i	0.162099i	−0.996710	+	0.0810497i	$0.974173\pi$
$614$	187.096	0.0122974
$615$	21381.1	1.40190
$616$	− 3268.23i	− 0.213768i
$617$	17829.3i	1.16334i	0.813425	+	0.581670i	$0.197600\pi$
$617$	17829.3i	1.16334i	−0.813425	+	0.581670i	$0.802400\pi$
$618$	137.227i	0.00893215i
$619$	16901.7i	1.09748i	0.835994	+	0.548739i	$0.184892\pi$
$619$	16901.7i	1.09748i	−0.835994	+	0.548739i	$0.815108\pi$
$620$	15630.8	1.01250
$621$	725.729	0.0468962
$622$	1371.76i	0.0884286i
$623$	23873.0	1.53524
$624$	0	0
$625$	−17247.4	−1.10384
$626$	1071.83i	0.0684327i
$627$	946.372	0.0602783
$628$	18413.8	1.17005
$629$	− 29709.3i	− 1.88328i
$630$	955.644i	0.0604345i
$631$	− 1833.82i	− 0.115695i	−0.998325	−	0.0578473i	$-0.981576\pi$
$631$	− 1833.82i	− 0.115695i	0.998325	−	0.0578473i	$-0.0184237\pi$
$632$	1115.71i	0.0702226i
$633$	15750.9	0.989009
$634$	1117.40	0.0699960
$635$	− 19739.0i	− 1.23357i
$636$	17662.6	1.10121
$637$	0	0
$638$	−1206.51	−0.0748684
$639$	326.019i	0.0201833i
$640$	−6617.81	−0.408737
$641$	−29172.1	−1.79755	−0.898773	−	0.438414i	$-0.855540\pi$
$641$	−29172.1	−1.79755	−0.898773	+	0.438414i	$0.855540\pi$
$642$	− 95.0672i	− 0.00584424i
$643$	25103.8i	1.53965i	0.638254	+	0.769826i	$0.279657\pi$
$643$	25103.8i	1.53965i	−0.638254	+	0.769826i	$0.720343\pi$
$644$	− 6925.69i	− 0.423774i
$645$	14852.5i	0.906693i
$646$	−178.134	−0.0108492
$647$	24471.6	1.48698	0.743492	−	0.668745i	$-0.233168\pi$
$647$	24471.6	1.48698	0.743492	+	0.668745i	$0.233168\pi$
$648$	− 276.165i	− 0.0167419i
$649$	10081.4	0.609752
$650$	0	0
$651$	12449.3	0.749505
$652$	− 2636.06i	− 0.158337i
$653$	6632.68	0.397484	0.198742	−	0.980052i	$-0.436314\pi$
$653$	6632.68	0.397484	0.198742	+	0.980052i	$0.436314\pi$
$654$	323.679	0.0193530
$655$	− 34204.5i	− 2.04043i
$656$	29228.1i	1.73958i
$657$	− 1481.15i	− 0.0879529i
$658$	1720.40i	0.101928i
$659$	26436.9	1.56272	0.781361	−	0.624080i	$-0.214526\pi$
$659$	26436.9	1.56272	0.781361	+	0.624080i	$0.214526\pi$
$660$	10832.0	0.638840
$661$	− 27596.7i	− 1.62389i	−0.583737	−	0.811943i	$-0.698410\pi$
$661$	− 27596.7i	− 1.62389i	0.583737	−	0.811943i	$-0.301590\pi$
$662$	−681.020	−0.0399828
$663$	0	0
$664$	4788.49	0.279863
$665$	5296.73i	0.308870i
$666$	−730.727	−0.0425152
$667$	−5128.07	−0.297691
$668$	17273.8i	1.00051i
$669$	− 4147.38i	− 0.239682i
$670$	− 1117.06i	− 0.0644115i
$671$	− 17484.3i	− 1.00592i
$672$	−3956.96	−0.227147
$673$	−18860.1	−1.08024	−0.540122	−	0.841587i	$-0.681622\pi$
$673$	−18860.1	−1.08024	−0.540122	+	0.841587i	$0.681622\pi$
$674$	− 1196.87i	− 0.0684000i
$675$	−2977.81	−0.169801
$676$	0	0
$677$	−14705.5	−0.834825	−0.417412	−	0.908717i	$-0.637063\pi$
$677$	−14705.5	−0.834825	−0.417412	+	0.908717i	$0.637063\pi$
$678$	728.601i	0.0412710i
$679$	−48423.1	−2.73683
$680$	−4089.47	−0.230624
$681$	3670.72i	0.206553i
$682$	810.145i	0.0454869i
$683$	17950.2i	1.00563i	0.864395	+	0.502814i	$0.167702\pi$
$683$	17950.2i	1.00563i	−0.864395	+	0.502814i	$0.832298\pi$
$684$	− 763.141i	− 0.0426600i
$685$	7282.18	0.406186
$686$	2514.83	0.139966
$687$	12058.8i	0.669685i
$688$	−20303.5	−1.12509
$689$	0	0
$690$	−264.325	−0.0145836
$691$	− 8692.38i	− 0.478544i	−0.970953	−	0.239272i	$-0.923091\pi$
$691$	− 8692.38i	− 0.478544i	0.970953	−	0.239272i	$-0.0769087\pi$
$692$	23359.4	1.28322
$693$	8627.26	0.472904
$694$	1492.12i	0.0816141i
$695$	− 29558.8i	− 1.61328i
$696$	1951.41i	0.106276i
$697$	36332.2i	1.97443i
$698$	−613.789	−0.0332840
$699$	−4975.20	−0.269212
$700$	28417.4i	1.53440i
$701$	−23275.9	−1.25409	−0.627047	−	0.778982i	$-0.715736\pi$
$701$	−23275.9	−1.25409	−0.627047	+	0.778982i	$0.715736\pi$
$702$	0	0
$703$	−4050.11	−0.217287
$704$	14634.9i	0.783485i
$705$	−11436.7	−0.610965
$706$	−1090.84	−0.0581507
$707$	− 15678.4i	− 0.834014i
$708$	− 8129.49i	− 0.431532i
$709$	34164.7i	1.80971i	0.425722	+	0.904854i	$0.360020\pi$
$709$	34164.7i	1.80971i	−0.425722	+	0.904854i	$0.639980\pi$
$710$	− 118.742i	− 0.00627651i
$711$	−2945.18	−0.155349
$712$	2512.71	0.132258
$713$	3443.40i	0.180864i
$714$	−1623.89	−0.0851157
$715$	0	0
$716$	−12850.8	−0.670748
$717$	− 1855.66i	− 0.0966541i
$718$	2301.75	0.119639
$719$	8391.74	0.435270	0.217635	−	0.976030i	$-0.430166\pi$
$719$	8391.74	0.435270	0.217635	+	0.976030i	$0.430166\pi$
$720$	− 8684.32i	− 0.449508i
$721$	6933.64i	0.358145i
$722$	− 1441.48i	− 0.0743026i
$723$	3405.74i	0.175188i
$724$	−5767.04	−0.296036
$725$	21041.5	1.07788
$726$	− 291.881i	− 0.0149211i
$727$	−23500.4	−1.19887	−0.599436	−	0.800423i	$-0.704608\pi$
$727$	−23500.4	−1.19887	−0.599436	+	0.800423i	$0.704608\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	539.462i	0.0273512i
$731$	−25238.3	−1.27698
$732$	−14099.1	−0.711908
$733$	1289.07i	0.0649563i	0.999472	+	0.0324781i	$0.0103399\pi$
$733$	1289.07i	0.0649563i	−0.999472	+	0.0324781i	$0.989660\pi$
$734$	2432.01i	0.122298i
$735$	32501.7i	1.63108i
$736$	− 1094.47i	− 0.0548133i
$737$	−10084.5	−0.504024
$738$	893.624	0.0445729
$739$	18197.8i	0.905844i	0.891550	+	0.452922i	$0.149618\pi$
$739$	18197.8i	0.905844i	−0.891550	+	0.452922i	$0.850382\pi$
$740$	−46356.8	−2.30285
$741$	0	0
$742$	−5123.68	−0.253499
$743$	21277.2i	1.05059i	0.850921	+	0.525294i	$0.176044\pi$
$743$	21277.2i	1.05059i	−0.850921	+	0.525294i	$0.823956\pi$
$744$	1310.33	0.0645686
$745$	16347.4	0.803920
$746$	− 683.146i	− 0.0335278i
$747$	12640.3i	0.619123i
$748$	18406.4i	0.899739i
$749$	− 4803.45i	− 0.234331i
$750$	−144.665	−0.00704321
$751$	25653.4	1.24648	0.623240	−	0.782030i	$-0.285816\pi$
$751$	25653.4	1.24648	0.623240	+	0.782030i	$0.285816\pi$
$752$	− 15634.0i	− 0.758130i
$753$	−6863.50	−0.332164
$754$	0	0
$755$	−13466.0	−0.649112
$756$	− 6956.90i	− 0.334682i
$757$	26521.2	1.27336	0.636678	−	0.771130i	$-0.280308\pi$
$757$	26521.2	1.27336	0.636678	+	0.771130i	$0.280308\pi$
$758$	438.245	0.0209997
$759$	2386.24i	0.114117i
$760$	557.497i	0.0266086i
$761$	− 21607.2i	− 1.02925i	−0.857415	−	0.514625i	$-0.827931\pi$
$761$	− 21607.2i	− 1.02925i	0.857415	−	0.514625i	$-0.172069\pi$
$762$	− 824.992i	− 0.0392209i
$763$	16354.5	0.775979
$764$	13658.2	0.646776
$765$	− 10795.1i	− 0.510193i
$766$	−749.560	−0.0353560
$767$	0	0
$768$	11592.5	0.544674
$769$	− 17179.2i	− 0.805591i	−0.915290	−	0.402795i	$-0.868039\pi$
$769$	− 17179.2i	− 0.805591i	0.915290	−	0.402795i	$-0.131961\pi$
$770$	−3142.21	−0.147062
$771$	15340.5	0.716568
$772$	− 3466.74i	− 0.161620i
$773$	− 12822.1i	− 0.596609i	−0.954471	−	0.298304i	$-0.903579\pi$
$773$	− 12822.1i	− 0.596609i	0.954471	−	0.298304i	$-0.0964211\pi$
$774$	620.761i	0.0288279i
$775$	− 14128.9i	− 0.654873i
$776$	−5096.67	−0.235773
$777$	−36921.4	−1.70469
$778$	− 1404.58i	− 0.0647256i
$779$	4952.98	0.227804
$780$	0	0
$781$	−1071.97	−0.0491140
$782$	− 449.157i	− 0.0205394i
$783$	−5151.18	−0.235106
$784$	−44430.0	−2.02396
$785$	− 35509.1i	− 1.61449i
$786$	− 1429.58i	− 0.0648744i
$787$	35213.4i	1.59495i	0.603354	+	0.797474i	$0.293831\pi$
$787$	35213.4i	1.59495i	−0.603354	+	0.797474i	$0.706169\pi$
$788$	5524.73i	0.249759i
$789$	8280.50	0.373629
$790$	1072.69	0.0483097
$791$	36813.9i	1.65481i
$792$	908.045	0.0407399
$793$	0	0
$794$	−1129.61	−0.0504893
$795$	− 34060.6i	− 1.51950i
$796$	−23061.5	−1.02688
$797$	−7500.15	−0.333336	−0.166668	−	0.986013i	$-0.553301\pi$
$797$	−7500.15	−0.333336	−0.166668	+	0.986013i	$0.553301\pi$
$798$	221.377i	0.00982038i
$799$	− 19434.0i	− 0.860481i
$800$	4490.82i	0.198468i
$801$	6632.87i	0.292585i
$802$	−196.506	−0.00865195
$803$	4870.10	0.214025
$804$	8131.97i	0.356707i
$805$	−13355.5	−0.584745
$806$	0	0
$807$	−9930.97	−0.433193
$808$	− 1650.20i	− 0.0718489i
$809$	−22020.5	−0.956982	−0.478491	−	0.878092i	$-0.658816\pi$
$809$	−22020.5	−0.956982	−0.478491	+	0.878092i	$0.658816\pi$
$810$	−265.516	−0.0115176
$811$	20444.6i	0.885210i	0.896717	+	0.442605i	$0.145946\pi$
$811$	20444.6i	0.885210i	−0.896717	+	0.442605i	$0.854054\pi$
$812$	49158.1i	2.12452i
$813$	− 7566.78i	− 0.326419i
$814$	− 2402.67i	− 0.103457i
$815$	−5083.37	−0.218482
$816$	14757.0	0.633085
$817$	3440.62i	0.147334i
$818$	328.920	0.0140592
$819$	0	0
$820$	56690.8	2.41431
$821$	40353.0i	1.71538i	0.514164	+	0.857692i	$0.328102\pi$
$821$	40353.0i	1.71538i	−0.514164	+	0.857692i	$0.671898\pi$
$822$	304.359	0.0129145
$823$	33110.7	1.40239	0.701194	−	0.712971i	$-0.252651\pi$
$823$	33110.7	1.40239	0.701194	+	0.712971i	$0.252651\pi$
$824$	729.787i	0.0308536i
$825$	− 9791.21i	− 0.413195i
$826$	2358.26i	0.0993392i
$827$	43410.4i	1.82531i	0.408735	+	0.912653i	$0.365970\pi$
$827$	43410.4i	1.82531i	−0.408735	+	0.912653i	$0.634030\pi$
$828$	1924.23	0.0807629
$829$	5502.87	0.230546	0.115273	−	0.993334i	$-0.463226\pi$
$829$	5502.87	0.230546	0.115273	+	0.993334i	$0.463226\pi$
$830$	− 4603.84i	− 0.192532i
$831$	10366.4	0.432741
$832$	0	0
$833$	−55229.1	−2.29721
$834$	− 1235.41i	− 0.0512935i
$835$	33310.8	1.38056
$836$	2509.25	0.103809
$837$	3458.92i	0.142841i
$838$	− 2260.93i	− 0.0932013i
$839$	1698.27i	0.0698819i	0.999389	+	0.0349409i	$0.0111243\pi$
$839$	1698.27i	0.0698819i	−0.999389	+	0.0349409i	$0.988876\pi$
$840$	5082.22i	0.208754i
$841$	12009.8	0.492425
$842$	−16.0136	−0.000655423 0
$843$	2944.95i	0.120320i
$844$	41762.7	1.70323
$845$	0	0
$846$	−477.996	−0.0194254
$847$	− 14747.9i	− 0.598279i
$848$	46561.0	1.88551
$849$	−15993.3	−0.646512
$850$	1842.98i	0.0743691i
$851$	− 10212.2i	− 0.411363i
$852$	864.421i	0.0347589i
$853$	− 18686.8i	− 0.750087i	−0.927007	−	0.375044i	$-0.877628\pi$
$853$	− 18686.8i	− 0.750087i	0.927007	−	0.375044i	$-0.122372\pi$
$854$	4089.95	0.163882
$855$	−1471.64	−0.0588645
$856$	− 505.578i	− 0.0201873i
$857$	9659.16	0.385007	0.192503	−	0.981296i	$-0.438339\pi$
$857$	9659.16	0.385007	0.192503	+	0.981296i	$0.438339\pi$
$858$	0	0
$859$	−1559.90	−0.0619595	−0.0309798	−	0.999520i	$-0.509863\pi$
$859$	−1559.90	−0.0619595	−0.0309798	+	0.999520i	$0.509863\pi$
$860$	39380.6i	1.56147i
$861$	45152.1	1.78720
$862$	2599.79	0.102725
$863$	− 11358.7i	− 0.448034i	−0.974585	−	0.224017i	$-0.928083\pi$
$863$	− 11358.7i	− 0.448034i	0.974585	−	0.224017i	$-0.0719171\pi$
$864$	− 1099.40i	− 0.0432898i
$865$	− 45046.3i	− 1.77066i
$866$	− 613.155i	− 0.0240599i
$867$	3604.74	0.141204
$868$	33008.7	1.29077
$869$	− 9683.93i	− 0.378026i
$870$	1876.16	0.0731124
$871$	0	0
$872$	1721.36	0.0668493
$873$	− 13453.8i	− 0.521585i
$874$	−61.2314	−0.00236977
$875$	−7309.45	−0.282405
$876$	− 3927.18i	− 0.151469i
$877$	− 33301.6i	− 1.28223i	−0.767444	−	0.641115i	$-0.778472\pi$
$877$	− 33301.6i	− 1.28223i	0.767444	−	0.641115i	$-0.221528\pi$
$878$	− 821.701i	− 0.0315844i
$879$	7261.18i	0.278628i
$880$	28554.6	1.09383
$881$	−28912.3	−1.10565	−0.552826	−	0.833296i	$-0.686451\pi$
$881$	−28912.3	−1.10565	−0.552826	+	0.833296i	$0.686451\pi$
$882$	1358.41i	0.0518595i
$883$	49100.9	1.87132	0.935661	−	0.352901i	$-0.114805\pi$
$883$	49100.9	1.87132	0.935661	+	0.352901i	$0.114805\pi$
$884$	0	0
$885$	−15676.9	−0.595451
$886$	824.517i	0.0312643i
$887$	4566.49	0.172861	0.0864304	−	0.996258i	$-0.472454\pi$
$887$	4566.49	0.172861	0.0864304	+	0.996258i	$0.472454\pi$
$888$	−3886.09	−0.146857
$889$	− 41684.3i	− 1.57261i
$890$	− 2415.82i	− 0.0909870i
$891$	2396.99i	0.0901261i
$892$	− 10996.5i	− 0.412771i
$893$	−2649.33	−0.0992795
$894$	683.238	0.0255603
$895$	24781.4i	0.925533i
$896$	−13975.3	−0.521074
$897$	0	0
$898$	−1186.14	−0.0440780
$899$	− 24441.0i	− 0.906734i
$900$	−7895.49	−0.292426
$901$	57877.9	2.14006
$902$	2938.29i	0.108464i
$903$	31365.1i	1.15589i
$904$	3874.78i	0.142559i
$905$	11121.2i	0.408486i
$906$	−562.814	−0.0206382
$907$	−48873.3	−1.78921	−0.894604	−	0.446860i	$-0.852542\pi$
$907$	−48873.3	−1.78921	−0.894604	+	0.446860i	$0.852542\pi$
$908$	9732.72i	0.355718i
$909$	4356.09	0.158946
$910$	0	0
$911$	31550.1	1.14742	0.573710	−	0.819058i	$-0.305503\pi$
$911$	31550.1	1.14742	0.573710	+	0.819058i	$0.305503\pi$
$912$	− 2011.74i	− 0.0730433i
$913$	−41562.1	−1.50658
$914$	−1731.49	−0.0626613
$915$	27188.7i	0.982327i
$916$	31973.3i	1.15331i
$917$	− 72232.1i	− 2.60121i
$918$	− 451.182i	− 0.0162214i
$919$	−11720.3	−0.420694	−0.210347	−	0.977627i	$-0.567459\pi$
$919$	−11720.3	−0.420694	−0.210347	+	0.977627i	$0.567459\pi$
$920$	−1405.71	−0.0503748
$921$	2626.53i	0.0939707i
$922$	−1972.63	−0.0704612
$923$	0	0
$924$	22874.7	0.814418
$925$	41902.7i	1.48946i
$926$	−2977.84	−0.105678
$927$	−1926.44	−0.0682553
$928$	7768.47i	0.274798i
$929$	− 6911.23i	− 0.244080i	−0.992525	−	0.122040i	$-0.961056\pi$
$929$	− 6911.23i	− 0.244080i	0.992525	−	0.122040i	$-0.0389436\pi$
$930$	− 1259.80i	− 0.0444200i
$931$	7529.10i	0.265044i
$932$	−13191.5	−0.463628
$933$	−19257.3	−0.675730
$934$	2333.18i	0.0817386i
$935$	35494.9	1.24151
$936$	0	0
$937$	−28673.9	−0.999717	−0.499858	−	0.866107i	$-0.666615\pi$
$937$	−28673.9	−0.999717	−0.499858	+	0.866107i	$0.666615\pi$
$938$	− 2358.97i	− 0.0821143i
$939$	−15046.7	−0.522930
$940$	−30323.7	−1.05218
$941$	22463.6i	0.778207i	0.921194	+	0.389103i	$0.127215\pi$
$941$	22463.6i	0.778207i	−0.921194	+	0.389103i	$0.872785\pi$
$942$	− 1484.10i	− 0.0513320i
$943$	12488.8i	0.431272i
$944$	− 21430.4i	− 0.738878i
$945$	−13415.7	−0.461812
$946$	−2041.10	−0.0701499
$947$	− 36806.6i	− 1.26299i	−0.775379	−	0.631497i	$-0.782441\pi$
$947$	− 36806.6i	− 1.26299i	0.775379	−	0.631497i	$-0.217559\pi$
$948$	−7808.98	−0.267536
$949$	0	0
$950$	251.244	0.00858046
$951$	15686.4i	0.534876i
$952$	−8636.04	−0.294008
$953$	1781.07	0.0605400	0.0302700	−	0.999542i	$-0.490363\pi$
$953$	1781.07	0.0605400	0.0302700	+	0.999542i	$0.490363\pi$
$954$	− 1423.56i	− 0.0483119i
$955$	− 26338.5i	− 0.892455i
$956$	− 4920.19i	− 0.166454i
$957$	− 16937.4i	− 0.572109i
$958$	−2356.77	−0.0794819
$959$	15378.3	0.517822
$960$	− 22757.8i	− 0.765108i
$961$	13379.3	0.449107
$962$	0	0
$963$	1334.59	0.0446589
$964$	9030.12i	0.301702i
$965$	−6685.26	−0.223012
$966$	−558.193	−0.0185917
$967$	54777.2i	1.82163i	0.412815	+	0.910815i	$0.364545\pi$
$967$	54777.2i	1.82163i	−0.412815	+	0.910815i	$0.635455\pi$
$968$	− 1552.26i	− 0.0515408i
$969$	− 2500.71i	− 0.0829044i
$970$	4900.15i	0.162200i
$971$	−2391.54	−0.0790403	−0.0395201	−	0.999219i	$-0.512583\pi$
$971$	−2391.54	−0.0790403	−0.0395201	+	0.999219i	$0.512583\pi$
$972$	1932.90	0.0637838
$973$	− 62421.5i	− 2.05667i
$974$	230.391	0.00757925
$975$	0	0
$976$	−37167.0	−1.21894
$977$	− 43738.5i	− 1.43226i	−0.697967	−	0.716130i	$-0.745912\pi$
$977$	− 43738.5i	− 1.43226i	0.697967	−	0.716130i	$-0.254088\pi$
$978$	−212.460	−0.00694654
$979$	−21809.3	−0.711979
$980$	86176.5i	2.80899i
$981$	4543.92i	0.147886i
$982$	1404.62i	0.0456449i
$983$	− 27712.4i	− 0.899174i	−0.893236	−	0.449587i	$-0.851571\pi$
$983$	− 27712.4i	− 0.899174i	0.893236	−	0.449587i	$-0.148429\pi$
$984$	4752.39	0.153964
$985$	10653.9	0.344631
$986$	3188.09i	0.102971i
$987$	−24151.7	−0.778882
$988$	0	0
$989$	−8675.38	−0.278929
$990$	− 873.031i	− 0.0280270i
$991$	3550.23	0.113801	0.0569005	−	0.998380i	$-0.481878\pi$
$991$	3550.23	0.113801	0.0569005	+	0.998380i	$0.481878\pi$
$992$	5216.37	0.166956
$993$	− 9560.42i	− 0.305529i
$994$	− 250.757i	− 0.00800153i
$995$	44471.8i	1.41694i
$996$	33515.1i	1.06623i
$997$	−28338.4	−0.900188	−0.450094	−	0.892981i	$-0.648610\pi$
$997$	−28338.4	−0.900188	−0.450094	+	0.892981i	$0.648610\pi$
$998$	2127.76	0.0674882
$999$	− 10258.2i	− 0.324881i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	507.4.b.k.337.9		18
13.5	odd	4		507.4.a.p.1.4	yes	9
13.8	odd	4		507.4.a.o.1.6	✓	9
13.12	even	2	inner	507.4.b.k.337.10		18
39.5	even	4		1521.4.a.bf.1.6		9
39.8	even	4		1521.4.a.bi.1.4		9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
507.4.a.o.1.6	✓	9	13.8	odd	4
507.4.a.p.1.4	yes	9	13.5	odd	4
507.4.b.k.337.9		18	1.1	even	1	trivial
507.4.b.k.337.10		18	13.12	even	2	inner
1521.4.a.bf.1.6		9	39.5	even	4
1521.4.a.bi.1.4		9	39.8	even	4

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:	\( N \)	\(=\)	\( 507 = 3 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	507.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-0.213700i q^{2} +3.00000 q^{3} +7.95433 q^{4} -15.3391i q^{5} -0.641100i q^{6} -32.3928i q^{7} -3.40944i q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q-0.213700i q^{2} +3.00000 q^{3} +7.95433 q^{4} -15.3391i q^{5} -0.641100i q^{6} -32.3928i q^{7} -3.40944i q^{8} +9.00000 q^{9} -3.27797 q^{10} +29.5925i q^{11} +23.8630 q^{12} -6.92233 q^{14} -46.0174i q^{15} +62.9061 q^{16} +78.1958 q^{17} -1.92330i q^{18} -10.6600i q^{19} -122.013i q^{20} -97.1783i q^{21} +6.32392 q^{22} +26.8789 q^{23} -10.2283i q^{24} -110.289 q^{25} +27.0000 q^{27} -257.663i q^{28} -190.785 q^{29} -9.83392 q^{30} +128.108i q^{31} -40.7185i q^{32} +88.7776i q^{33} -16.7104i q^{34} -496.877 q^{35} +71.5890 q^{36} -379.934i q^{37} -2.27805 q^{38} -52.2979 q^{40} +464.631i q^{41} -20.7670 q^{42} -322.758 q^{43} +235.389i q^{44} -138.052i q^{45} -5.74401i q^{46} -248.529i q^{47} +188.718 q^{48} -706.292 q^{49} +23.5688i q^{50} +234.587 q^{51} +740.167 q^{53} -5.76990i q^{54} +453.924 q^{55} -110.441 q^{56} -31.9801i q^{57} +40.7706i q^{58} -340.673i q^{59} -366.038i q^{60} -590.834 q^{61} +27.3767 q^{62} -291.535i q^{63} +494.547 q^{64} +18.9718 q^{66} +340.777i q^{67} +621.995 q^{68} +80.6366 q^{69} +106.183i q^{70} +36.2243i q^{71} -30.6850i q^{72} -164.572i q^{73} -81.1919 q^{74} -330.868 q^{75} -84.7935i q^{76} +958.584 q^{77} -327.242 q^{79} -964.925i q^{80} +81.0000 q^{81} +99.2916 q^{82} +1404.48i q^{83} -772.989i q^{84} -1199.46i q^{85} +68.9734i q^{86} -572.354 q^{87} +100.894 q^{88} +736.986i q^{89} -29.5018 q^{90} +213.803 q^{92} +384.324i q^{93} -53.1107 q^{94} -163.516 q^{95} -122.156i q^{96} -1494.87i q^{97} +150.935i q^{98} +266.333i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q + 54 q^{3} - 88 q^{4} + 162 q^{9}+O(q^{10}) \) 18 * q + 54 * q^3 - 88 * q^4 + 162 * q^9	\( 18 q + 54 q^{3} - 88 q^{4} + 162 q^{9} + 108 q^{10} - 264 q^{12} + 316 q^{14} + 432 q^{16} - 356 q^{17} - 1260 q^{22} - 300 q^{23} + 40 q^{25} + 486 q^{27} - 194 q^{29} + 324 q^{30} - 836 q^{35} - 792 q^{36} + 1320 q^{38} - 3012 q^{40} + 948 q^{42} - 484 q^{43} + 1296 q^{48} + 76 q^{49} - 1068 q^{51} - 302 q^{53} + 4128 q^{55} - 4552 q^{56} - 2680 q^{61} - 694 q^{62} - 1786 q^{64} - 3780 q^{66} + 5570 q^{68} - 900 q^{69} - 2382 q^{74} + 120 q^{75} + 4284 q^{77} - 3182 q^{79} + 1458 q^{81} - 3034 q^{82} - 582 q^{87} + 7432 q^{88} + 972 q^{90} + 1030 q^{92} - 1384 q^{94} - 8316 q^{95}+O(q^{100}) \) 18 * q + 54 * q^3 - 88 * q^4 + 162 * q^9 + 108 * q^10 - 264 * q^12 + 316 * q^14 + 432 * q^16 - 356 * q^17 - 1260 * q^22 - 300 * q^23 + 40 * q^25 + 486 * q^27 - 194 * q^29 + 324 * q^30 - 836 * q^35 - 792 * q^36 + 1320 * q^38 - 3012 * q^40 + 948 * q^42 - 484 * q^43 + 1296 * q^48 + 76 * q^49 - 1068 * q^51 - 302 * q^53 + 4128 * q^55 - 4552 * q^56 - 2680 * q^61 - 694 * q^62 - 1786 * q^64 - 3780 * q^66 + 5570 * q^68 - 900 * q^69 - 2382 * q^74 + 120 * q^75 + 4284 * q^77 - 3182 * q^79 + 1458 * q^81 - 3034 * q^82 - 582 * q^87 + 7432 * q^88 + 972 * q^90 + 1030 * q^92 - 1384 * q^94 - 8316 * q^95

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