LMFDB - Embedded newform 507.4.b.j.337.15

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} + 97 x^{16} + 3906 x^{14} + 84743 x^{12} + 1077128 x^{10} + 8187552 x^{8} + 36483705 x^{6} + \cdots + 26460736 $ x^18 + 97x^16 + 3906x^14 + 84743x^12 + 1077128x^10 + 8187552x^8 + 36483705x^6 + 88861676x^4 + 98825392x^2 + 26460736
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 13^{10} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			337.15
Root			$2.37739i$ of defining polynomial
Character	$\chi$	$=$	507.337
Dual form			507.4.b.j.337.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.37739i q^{2} -3.00000 q^{3} -3.40677 q^{4} -15.7127i q^{5} -10.1322i q^{6} +17.1681i q^{7} +15.5131i q^{8} +9.00000 q^{9} +O(q^{10})$	\(q+3.37739i q^{2} -3.00000 q^{3} -3.40677 q^{4} -15.7127i q^{5} -10.1322i q^{6} +17.1681i q^{7} +15.5131i q^{8} +9.00000 q^{9} +53.0679 q^{10} -52.8187i q^{11} +10.2203 q^{12} -57.9835 q^{14} +47.1380i q^{15} -79.6481 q^{16} +71.0654 q^{17} +30.3965i q^{18} +92.6916i q^{19} +53.5295i q^{20} -51.5044i q^{21} +178.390 q^{22} -190.712 q^{23} -46.5394i q^{24} -121.888 q^{25} -27.0000 q^{27} -58.4878i q^{28} -128.204 q^{29} -159.204 q^{30} +3.29674i q^{31} -144.898i q^{32} +158.456i q^{33} +240.016i q^{34} +269.757 q^{35} -30.6609 q^{36} -241.546i q^{37} -313.056 q^{38} +243.753 q^{40} -97.1824i q^{41} +173.950 q^{42} -376.151 q^{43} +179.941i q^{44} -141.414i q^{45} -644.109i q^{46} -577.354i q^{47} +238.944 q^{48} +48.2555 q^{49} -411.664i q^{50} -213.196 q^{51} -307.686 q^{53} -91.1896i q^{54} -829.924 q^{55} -266.331 q^{56} -278.075i q^{57} -432.995i q^{58} -349.914i q^{59} -160.588i q^{60} +127.467 q^{61} -11.1344 q^{62} +154.513i q^{63} -147.809 q^{64} -535.169 q^{66} -903.564i q^{67} -242.104 q^{68} +572.136 q^{69} +911.076i q^{70} +826.106i q^{71} +139.618i q^{72} +131.760i q^{73} +815.796 q^{74} +365.665 q^{75} -315.779i q^{76} +906.799 q^{77} -556.244 q^{79} +1251.48i q^{80} +81.0000 q^{81} +328.223 q^{82} -254.664i q^{83} +175.464i q^{84} -1116.63i q^{85} -1270.41i q^{86} +384.612 q^{87} +819.384 q^{88} -183.410i q^{89} +477.611 q^{90} +649.712 q^{92} -9.89023i q^{93} +1949.95 q^{94} +1456.43 q^{95} +434.693i q^{96} -780.498i q^{97} +162.978i q^{98} -475.369i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 18 q - 54 q^{3} - 64 q^{4} + 162 q^{9}+O(q^{10}) $ 18 * q - 54 * q^3 - 64 * q^4 + 162 * q^9	$ 18 q - 54 q^{3} - 64 q^{4} + 162 q^{9} - 396 q^{10} + 192 q^{12} + 196 q^{14} + 64 q^{16} + 268 q^{17} + 548 q^{22} - 452 q^{23} - 1224 q^{25} - 486 q^{27} - 1094 q^{29} + 1188 q^{30} + 276 q^{35} - 576 q^{36} + 832 q^{38} + 2684 q^{40} - 588 q^{42} - 316 q^{43} - 192 q^{48} - 1284 q^{49} - 804 q^{51} + 2798 q^{53} - 2816 q^{55} + 1232 q^{56} + 4184 q^{61} + 586 q^{62} - 4962 q^{64} - 1644 q^{66} - 3158 q^{68} + 1356 q^{69} + 2074 q^{74} + 3672 q^{75} + 3372 q^{77} - 230 q^{79} + 1458 q^{81} + 10294 q^{82} + 3282 q^{87} + 968 q^{88} - 3564 q^{90} + 4174 q^{92} - 936 q^{94} + 444 q^{95}+O(q^{100}) $ 18 * q - 54 * q^3 - 64 * q^4 + 162 * q^9 - 396 * q^10 + 192 * q^12 + 196 * q^14 + 64 * q^16 + 268 * q^17 + 548 * q^22 - 452 * q^23 - 1224 * q^25 - 486 * q^27 - 1094 * q^29 + 1188 * q^30 + 276 * q^35 - 576 * q^36 + 832 * q^38 + 2684 * q^40 - 588 * q^42 - 316 * q^43 - 192 * q^48 - 1284 * q^49 - 804 * q^51 + 2798 * q^53 - 2816 * q^55 + 1232 * q^56 + 4184 * q^61 + 586 * q^62 - 4962 * q^64 - 1644 * q^66 - 3158 * q^68 + 1356 * q^69 + 2074 * q^74 + 3672 * q^75 + 3372 * q^77 - 230 * q^79 + 1458 * q^81 + 10294 * q^82 + 3282 * q^87 + 968 * q^88 - 3564 * q^90 + 4174 * q^92 - 936 * q^94 + 444 * 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 $

$2$	3.37739i	1.19409i	0.802208	+	0.597044i	$0.203658\pi$
$2$	3.37739i	1.19409i	−0.802208	+	0.597044i	$0.796342\pi$
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	−3.40677	−0.425846
$5$	− 15.7127i	− 1.40538i	−0.711494	−	0.702692i	$-0.751981\pi$
$5$	− 15.7127i	− 1.40538i	0.711494	−	0.702692i	$-0.248019\pi$
$6$	− 10.1322i	− 0.689407i
$7$	17.1681i	0.926992i	0.886099	+	0.463496i	$0.153405\pi$
$7$	17.1681i	0.926992i	−0.886099	+	0.463496i	$0.846595\pi$
$8$	15.5131i	0.685590i
$9$	9.00000	0.333333
$10$	53.0679	1.67815
$11$	− 52.8187i	− 1.44777i	−0.689922	−	0.723884i	$-0.742355\pi$
$11$	− 52.8187i	− 1.44777i	0.689922	−	0.723884i	$-0.257645\pi$
$12$	10.2203	0.245862
$13$	0	0
$13$	0	0
$14$	−57.9835	−1.10691
$15$	47.1380i	0.811399i
$16$	−79.6481	−1.24450
$17$	71.0654	1.01388	0.506938	−	0.861982i	$-0.330777\pi$
$17$	71.0654	1.01388	0.506938	+	0.861982i	$0.330777\pi$
$18$	30.3965i	0.398029i
$19$	92.6916i	1.11921i	0.828761	+	0.559603i	$0.189046\pi$
$19$	92.6916i	1.11921i	−0.828761	+	0.559603i	$0.810954\pi$
$20$	53.5295i	0.598478i
$21$	− 51.5044i	− 0.535199i
$22$	178.390	1.72876
$23$	−190.712	−1.72896	−0.864482	−	0.502663i	$-0.832354\pi$
$23$	−190.712	−1.72896	−0.864482	+	0.502663i	$0.832354\pi$
$24$	− 46.5394i	− 0.395826i
$25$	−121.888	−0.975106
$26$	0	0
$27$	−27.0000	−0.192450
$28$	− 58.4878i	− 0.394756i
$29$	−128.204	−0.820927	−0.410463	−	0.911877i	$-0.634633\pi$
$29$	−128.204	−0.820927	−0.410463	+	0.911877i	$0.634633\pi$
$30$	−159.204	−0.968882
$31$	3.29674i	0.0191004i	0.999954	+	0.00955020i	$0.00303997\pi$
$31$	3.29674i	0.0191004i	−0.999954	+	0.00955020i	$0.996960\pi$
$32$	− 144.898i	− 0.800454i
$33$	158.456i	0.835869i
$34$	240.016i	1.21066i
$35$	269.757	1.30278
$36$	−30.6609	−0.141949
$37$	− 241.546i	− 1.07324i	−0.843823	−	0.536621i	$-0.819700\pi$
$37$	− 241.546i	− 1.07324i	0.843823	−	0.536621i	$-0.180300\pi$
$38$	−313.056	−1.33643
$39$	0	0
$40$	243.753	0.963518
$41$	− 97.1824i	− 0.370179i	−0.982722	−	0.185090i	$-0.940742\pi$
$41$	− 97.1824i	− 0.370179i	0.982722	−	0.185090i	$-0.0592575\pi$
$42$	173.950	0.639075
$43$	−376.151	−1.33401	−0.667006	−	0.745052i	$-0.732425\pi$
$43$	−376.151	−1.33401	−0.667006	+	0.745052i	$0.732425\pi$
$44$	179.941i	0.616527i
$45$	− 141.414i	− 0.468462i
$46$	− 644.109i	− 2.06454i
$47$	− 577.354i	− 1.79182i	−0.444231	−	0.895912i	$-0.646523\pi$
$47$	− 577.354i	− 1.79182i	0.444231	−	0.895912i	$-0.353477\pi$
$48$	238.944	0.718513
$49$	48.2555	0.140686
$50$	− 411.664i	− 1.16436i
$51$	−213.196	−0.585362
$52$	0	0
$53$	−307.686	−0.797433	−0.398716	−	0.917074i	$-0.630544\pi$
$53$	−307.686	−0.797433	−0.398716	+	0.917074i	$0.630544\pi$
$54$	− 91.1896i	− 0.229802i
$55$	−829.924	−2.03467
$56$	−266.331	−0.635536
$57$	− 278.075i	− 0.646174i
$58$	− 432.995i	− 0.980259i
$59$	− 349.914i	− 0.772117i	−0.922474	−	0.386059i	$-0.873836\pi$
$59$	− 349.914i	− 0.772117i	0.922474	−	0.386059i	$-0.126164\pi$
$60$	− 160.588i	− 0.345531i
$61$	127.467	0.267548	0.133774	−	0.991012i	$-0.457290\pi$
$61$	127.467	0.267548	0.133774	+	0.991012i	$0.457290\pi$
$62$	−11.1344	−0.0228076
$63$	154.513i	0.308997i
$64$	−147.809	−0.288689
$65$	0	0
$66$	−535.169	−0.998102
$67$	− 903.564i	− 1.64758i	−0.566894	−	0.823791i	$-0.691855\pi$
$67$	− 903.564i	− 1.64758i	0.566894	−	0.823791i	$-0.308145\pi$
$68$	−242.104	−0.431755
$69$	572.136	0.998218
$70$	911.076i	1.55563i
$71$	826.106i	1.38086i	0.723401	+	0.690428i	$0.242578\pi$
$71$	826.106i	1.38086i	−0.723401	+	0.690428i	$0.757422\pi$
$72$	139.618i	0.228530i
$73$	131.760i	0.211252i	0.994406	+	0.105626i	$0.0336846\pi$
$73$	131.760i	0.211252i	−0.994406	+	0.105626i	$0.966315\pi$
$74$	815.796	1.28155
$75$	365.665	0.562978
$76$	− 315.779i	− 0.476610i
$77$	906.799	1.34207
$78$	0	0
$79$	−556.244	−0.792182	−0.396091	−	0.918211i	$-0.629633\pi$
$79$	−556.244	−0.792182	−0.396091	+	0.918211i	$0.629633\pi$
$80$	1251.48i	1.74900i
$81$	81.0000	0.111111
$82$	328.223	0.442026
$83$	− 254.664i	− 0.336783i	−0.985720	−	0.168391i	$-0.946143\pi$
$83$	− 254.664i	− 0.336783i	0.985720	−	0.168391i	$-0.0538573\pi$
$84$	175.464i	0.227912i
$85$	− 1116.63i	− 1.42489i
$86$	− 1270.41i	− 1.59293i
$87$	384.612	0.473962
$88$	819.384	0.992576
$89$	− 183.410i	− 0.218443i	−0.994017	−	0.109222i	$-0.965164\pi$
$89$	− 183.410i	− 0.218443i	0.994017	−	0.109222i	$-0.0348358\pi$
$90$	477.611	0.559384
$91$	0	0
$92$	649.712	0.736273
$93$	− 9.89023i	− 0.0110276i
$94$	1949.95	2.13960
$95$	1456.43	1.57292
$96$	434.693i	0.462142i
$97$	− 780.498i	− 0.816986i	−0.912762	−	0.408493i	$-0.866054\pi$
$97$	− 780.498i	− 0.816986i	0.912762	−	0.408493i	$-0.133946\pi$
$98$	162.978i	0.167992i
$99$	− 475.369i	− 0.482589i
$100$	415.245	0.415245
$101$	1807.67	1.78089	0.890444	−	0.455093i	$-0.150394\pi$
$101$	1807.67	1.78089	0.890444	+	0.455093i	$0.150394\pi$
$102$	− 720.047i	− 0.698974i
$103$	−1560.78	−1.49308	−0.746542	−	0.665338i	$-0.768288\pi$
$103$	−1560.78	−1.49308	−0.746542	+	0.665338i	$0.768288\pi$
$104$	0	0
$105$	−809.272	−0.752160
$106$	− 1039.18i	− 0.952205i
$107$	322.266	0.291165	0.145582	−	0.989346i	$-0.453494\pi$
$107$	322.266	0.291165	0.145582	+	0.989346i	$0.453494\pi$
$108$	91.9828	0.0819541
$109$	1607.72i	1.41277i	0.707827	+	0.706386i	$0.249676\pi$
$109$	1607.72i	1.41277i	−0.707827	+	0.706386i	$0.750324\pi$
$110$	− 2802.98i	− 2.42958i
$111$	724.639i	0.619637i
$112$	− 1367.41i	− 1.15364i
$113$	−429.905	−0.357895	−0.178947	−	0.983859i	$-0.557269\pi$
$113$	−429.905	−0.357895	−0.178947	+	0.983859i	$0.557269\pi$
$114$	939.167	0.771589
$115$	2996.60i	2.42986i
$116$	436.761	0.349589
$117$	0	0
$118$	1181.80	0.921976
$119$	1220.06i	0.939855i
$120$	−731.259	−0.556287
$121$	−1458.82	−1.09603
$122$	430.505i	0.319476i
$123$	291.547i	0.213723i
$124$	− 11.2312i	− 0.00813383i
$125$	− 48.8932i	− 0.0349851i
$126$	−521.851	−0.368970
$127$	−2371.36	−1.65688	−0.828442	−	0.560076i	$-0.810772\pi$
$127$	−2371.36	−1.65688	−0.828442	+	0.560076i	$0.810772\pi$
$128$	− 1658.39i	− 1.14517i
$129$	1128.45	0.770193
$130$	0	0
$131$	169.200	0.112848	0.0564239	−	0.998407i	$-0.482030\pi$
$131$	169.200	0.112848	0.0564239	+	0.998407i	$0.482030\pi$
$132$	− 539.824i	− 0.355952i
$133$	−1591.34	−1.03749
$134$	3051.69	1.96736
$135$	424.242i	0.270466i
$136$	1102.45i	0.695104i
$137$	− 2976.21i	− 1.85602i	−0.372558	−	0.928009i	$-0.621519\pi$
$137$	− 2976.21i	− 1.85602i	0.372558	−	0.928009i	$-0.378481\pi$
$138$	1932.33i	1.19196i
$139$	−2555.15	−1.55917	−0.779585	−	0.626296i	$-0.784570\pi$
$139$	−2555.15	−1.55917	−0.779585	+	0.626296i	$0.784570\pi$
$140$	−919.001	−0.554784
$141$	1732.06i	1.03451i
$142$	−2790.08	−1.64886
$143$	0	0
$144$	−716.833	−0.414834
$145$	2014.43i	1.15372i
$146$	−445.006	−0.252253
$147$	−144.766	−0.0812254
$148$	822.892i	0.457036i
$149$	424.529i	0.233415i	0.993166	+	0.116707i	$0.0372339\pi$
$149$	424.529i	0.233415i	−0.993166	+	0.116707i	$0.962766\pi$
$150$	1234.99i	0.672245i
$151$	− 42.3232i	− 0.0228093i	−0.999935	−	0.0114047i	$-0.996370\pi$
$151$	− 42.3232i	− 0.0228093i	0.999935	−	0.0114047i	$-0.00363030\pi$
$152$	−1437.94	−0.767317
$153$	639.589	0.337959
$154$	3062.61i	1.60255i
$155$	51.8007	0.0268434
$156$	0	0
$157$	−990.187	−0.503347	−0.251674	−	0.967812i	$-0.580981\pi$
$157$	−990.187	−0.503347	−0.251674	+	0.967812i	$0.580981\pi$
$158$	− 1878.65i	− 0.945935i
$159$	923.058	0.460398
$160$	−2276.73	−1.12495
$161$	− 3274.17i	− 1.60274i
$162$	273.569i	0.132676i
$163$	− 2645.82i	− 1.27139i	−0.771940	−	0.635695i	$-0.780714\pi$
$163$	− 2645.82i	− 1.27139i	0.771940	−	0.635695i	$-0.219286\pi$
$164$	331.078i	0.157639i
$165$	2489.77	1.17472
$166$	860.099	0.402148
$167$	− 295.428i	− 0.136892i	−0.997655	−	0.0684458i	$-0.978196\pi$
$167$	− 295.428i	− 0.136892i	0.997655	−	0.0684458i	$-0.0218040\pi$
$168$	798.994	0.366927
$169$	0	0
$170$	3771.29	1.70144
$171$	834.225i	0.373069i
$172$	1281.46	0.568084
$173$	−1495.46	−0.657213	−0.328606	−	0.944467i	$-0.606579\pi$
$173$	−1495.46	−0.657213	−0.328606	+	0.944467i	$0.606579\pi$
$174$	1298.98i	0.565953i
$175$	− 2092.59i	− 0.903915i
$176$	4206.91i	1.80175i
$177$	1049.74i	0.445782i
$178$	619.448	0.260840
$179$	−785.097	−0.327826	−0.163913	−	0.986475i	$-0.552412\pi$
$179$	−785.097	−0.327826	−0.163913	+	0.986475i	$0.552412\pi$
$180$	481.765i	0.199493i
$181$	1287.01	0.528522	0.264261	−	0.964451i	$-0.414872\pi$
$181$	1287.01	0.528522	0.264261	+	0.964451i	$0.414872\pi$
$182$	0	0
$183$	−382.400	−0.154469
$184$	− 2958.54i	− 1.18536i
$185$	−3795.34	−1.50832
$186$	33.4032	0.0131680
$187$	− 3753.59i	− 1.46786i
$188$	1966.91i	0.763042i
$189$	− 463.539i	− 0.178400i
$190$	4918.95i	1.87820i
$191$	466.452	0.176708	0.0883542	−	0.996089i	$-0.471839\pi$
$191$	466.452	0.176708	0.0883542	+	0.996089i	$0.471839\pi$
$192$	443.426	0.166675
$193$	− 2636.99i	− 0.983495i	−0.870738	−	0.491747i	$-0.836358\pi$
$193$	− 2636.99i	− 0.983495i	0.870738	−	0.491747i	$-0.163642\pi$
$194$	2636.05	0.975553
$195$	0	0
$196$	−164.395	−0.0599108
$197$	3676.90i	1.32979i	0.746937	+	0.664894i	$0.231523\pi$
$197$	3676.90i	1.32979i	−0.746937	+	0.664894i	$0.768477\pi$
$198$	1605.51	0.576254
$199$	−220.557	−0.0785670	−0.0392835	−	0.999228i	$-0.512508\pi$
$199$	−220.557	−0.0785670	−0.0392835	+	0.999228i	$0.512508\pi$
$200$	− 1890.87i	− 0.668523i
$201$	2710.69i	0.951231i
$202$	6105.20i	2.12654i
$203$	− 2201.02i	− 0.760992i
$204$	726.311	0.249274
$205$	−1527.00	−0.520244
$206$	− 5271.35i	− 1.78287i
$207$	−1716.41	−0.576322
$208$	0	0
$209$	4895.85	1.62035
$210$	− 2733.23i	− 0.898146i
$211$	1385.93	0.452186	0.226093	−	0.974106i	$-0.427405\pi$
$211$	1385.93	0.452186	0.226093	+	0.974106i	$0.427405\pi$
$212$	1048.22	0.339584
$213$	− 2478.32i	− 0.797237i
$214$	1088.42i	0.347676i
$215$	5910.35i	1.87480i
$216$	− 418.855i	− 0.131942i
$217$	−56.5989	−0.0177059
$218$	−5429.91	−1.68697
$219$	− 395.281i	− 0.121966i
$220$	2827.36	0.866457
$221$	0	0
$222$	−2447.39	−0.739901
$223$	− 111.847i	− 0.0335867i	−0.999859	−	0.0167933i	$-0.994654\pi$
$223$	− 111.847i	− 0.0335867i	0.999859	−	0.0167933i	$-0.00534574\pi$
$224$	2487.62	0.742014
$225$	−1096.99	−0.325035
$226$	− 1451.96i	− 0.427358i
$227$	4990.96i	1.45930i	0.683819	+	0.729651i	$0.260318\pi$
$227$	4990.96i	1.45930i	−0.683819	+	0.729651i	$0.739682\pi$
$228$	947.337i	0.275171i
$229$	5787.56i	1.67010i	0.550174	+	0.835050i	$0.314561\pi$
$229$	5787.56i	1.67010i	−0.550174	+	0.835050i	$0.685439\pi$
$230$	−10120.7	−2.90147
$231$	−2720.40	−0.774844
$232$	− 1988.85i	− 0.562819i
$233$	723.707	0.203483	0.101742	−	0.994811i	$-0.467559\pi$
$233$	723.707	0.203483	0.101742	+	0.994811i	$0.467559\pi$
$234$	0	0
$235$	−9071.78	−2.51820
$236$	1192.08i	0.328803i
$237$	1668.73	0.457366
$238$	−4120.62	−1.12227
$239$	− 1239.74i	− 0.335531i	−0.985827	−	0.167766i	$-0.946345\pi$
$239$	− 1239.74i	− 0.335531i	0.985827	−	0.167766i	$-0.0536552\pi$
$240$	− 3754.45i	− 1.00979i
$241$	755.393i	0.201905i	0.994891	+	0.100953i	$0.0321890\pi$
$241$	755.393i	0.201905i	−0.994891	+	0.100953i	$0.967811\pi$
$242$	− 4927.00i	− 1.30876i
$243$	−243.000	−0.0641500
$244$	−434.249	−0.113934
$245$	− 758.222i	− 0.197719i
$246$	−984.669	−0.255204
$247$	0	0
$248$	−51.1428	−0.0130950
$249$	763.991i	0.194442i
$250$	165.131	0.0417753
$251$	−2252.06	−0.566330	−0.283165	−	0.959071i	$-0.591384\pi$
$251$	−2252.06	−0.566330	−0.283165	+	0.959071i	$0.591384\pi$
$252$	− 526.391i	− 0.131585i
$253$	10073.2i	2.50314i
$254$	− 8009.01i	− 1.97846i
$255$	3349.89i	0.822659i
$256$	4418.56	1.07875
$257$	−3716.75	−0.902118	−0.451059	−	0.892494i	$-0.648954\pi$
$257$	−3716.75	−0.902118	−0.451059	+	0.892494i	$0.648954\pi$
$258$	3811.23i	0.919678i
$259$	4146.90	0.994887
$260$	0	0
$261$	−1153.84	−0.273642
$262$	571.454i	0.134750i
$263$	−49.9521	−0.0117117	−0.00585585	−	0.999983i	$-0.501864\pi$
$263$	−49.9521	−0.0117117	−0.00585585	+	0.999983i	$0.501864\pi$
$264$	−2458.15	−0.573064
$265$	4834.57i	1.12070i
$266$	− 5374.58i	− 1.23886i
$267$	550.231i	0.126118i
$268$	3078.23i	0.701616i
$269$	3421.44	0.775497	0.387748	−	0.921765i	$-0.373253\pi$
$269$	3421.44	0.775497	0.387748	+	0.921765i	$0.373253\pi$
$270$	−1432.83	−0.322961
$271$	2958.89i	0.663247i	0.943412	+	0.331624i	$0.107596\pi$
$271$	2958.89i	0.663247i	−0.943412	+	0.331624i	$0.892404\pi$
$272$	−5660.23	−1.26177
$273$	0	0
$274$	10051.8	2.21625
$275$	6437.99i	1.41173i
$276$	−1949.13	−0.425087
$277$	7461.08	1.61839	0.809193	−	0.587543i	$-0.199905\pi$
$277$	7461.08	1.61839	0.809193	+	0.587543i	$0.199905\pi$
$278$	− 8629.73i	− 1.86179i
$279$	29.6707i	0.00636680i
$280$	4184.78i	0.893173i
$281$	4728.04i	1.00374i	0.864943	+	0.501871i	$0.167355\pi$
$281$	4728.04i	1.00374i	−0.864943	+	0.501871i	$0.832645\pi$
$282$	−5849.85	−1.23530
$283$	2879.69	0.604876	0.302438	−	0.953169i	$-0.402199\pi$
$283$	2879.69	0.604876	0.302438	+	0.953169i	$0.402199\pi$
$284$	− 2814.35i	− 0.588032i
$285$	−4369.30	−0.908123
$286$	0	0
$287$	1668.44	0.343153
$288$	− 1304.08i	− 0.266818i
$289$	137.298	0.0279458
$290$	−6803.51	−1.37764
$291$	2341.50i	0.471687i
$292$	− 448.877i	− 0.0899607i
$293$	− 8200.94i	− 1.63517i	−0.575810	−	0.817583i	$-0.695313\pi$
$293$	− 8200.94i	− 1.63517i	0.575810	−	0.817583i	$-0.304687\pi$
$294$	− 488.933i	− 0.0969902i
$295$	−5498.09	−1.08512
$296$	3747.14	0.735804
$297$	1426.11i	0.278623i
$298$	−1433.80	−0.278718
$299$	0	0
$300$	−1245.74	−0.239742
$301$	− 6457.81i	− 1.23662i
$302$	142.942	0.0272364
$303$	−5423.00	−1.02820
$304$	− 7382.71i	− 1.39285i
$305$	− 2002.84i	− 0.376008i
$306$	2160.14i	0.403553i
$307$	− 2058.82i	− 0.382747i	−0.981517	−	0.191374i	$-0.938706\pi$
$307$	− 2058.82i	− 0.382747i	0.981517	−	0.191374i	$-0.0612942\pi$
$308$	−3089.25	−0.571515
$309$	4682.33	0.862033
$310$	174.951i	0.0320534i
$311$	−1862.08	−0.339514	−0.169757	−	0.985486i	$-0.554298\pi$
$311$	−1862.08	−0.339514	−0.169757	+	0.985486i	$0.554298\pi$
$312$	0	0
$313$	−255.896	−0.0462112	−0.0231056	−	0.999733i	$-0.507355\pi$
$313$	−255.896	−0.0462112	−0.0231056	+	0.999733i	$0.507355\pi$
$314$	− 3344.25i	− 0.601041i
$315$	2427.82	0.434260
$316$	1895.00	0.337348
$317$	6265.68i	1.11014i	0.831802	+	0.555072i	$0.187309\pi$
$317$	6265.68i	1.11014i	−0.831802	+	0.555072i	$0.812691\pi$
$318$	3117.53i	0.549756i
$319$	6771.57i	1.18851i
$320$	2322.47i	0.405719i
$321$	−966.798	−0.168104
$322$	11058.1	1.91381
$323$	6587.17i	1.13474i
$324$	−275.948	−0.0473162
$325$	0	0
$326$	8935.96	1.51815
$327$	− 4823.17i	− 0.815664i
$328$	1507.60	0.253791
$329$	9912.09	1.66101
$330$	8408.93i	1.40272i
$331$	1071.93i	0.178002i	0.996032	+	0.0890011i	$0.0283674\pi$
$331$	1071.93i	0.178002i	−0.996032	+	0.0890011i	$0.971633\pi$
$332$	867.581i	0.143418i
$333$	− 2173.92i	− 0.357747i
$334$	997.775	0.163461
$335$	−14197.4	−2.31549
$336$	4102.22i	0.666056i
$337$	5572.02	0.900675	0.450337	−	0.892858i	$-0.351304\pi$
$337$	5572.02	0.900675	0.450337	+	0.892858i	$0.351304\pi$
$338$	0	0
$339$	1289.72	0.206631
$340$	3804.10i	0.606783i
$341$	174.130	0.0276530
$342$	−2817.50	−0.445477
$343$	6717.12i	1.05741i
$344$	− 5835.29i	− 0.914586i
$345$	− 8989.79i	− 1.40288i
$346$	− 5050.76i	− 0.784770i
$347$	−1903.44	−0.294472	−0.147236	−	0.989101i	$-0.547038\pi$
$347$	−1903.44	−0.294472	−0.147236	+	0.989101i	$0.547038\pi$
$348$	−1310.28	−0.201835
$349$	− 1097.17i	− 0.168282i	−0.996454	−	0.0841409i	$-0.973185\pi$
$349$	− 1097.17i	− 0.168282i	0.996454	−	0.0841409i	$-0.0268146\pi$
$350$	7067.51	1.07935
$351$	0	0
$352$	−7653.31	−1.15887
$353$	5420.90i	0.817352i	0.912679	+	0.408676i	$0.134009\pi$
$353$	5420.90i	0.817352i	−0.912679	+	0.408676i	$0.865991\pi$
$354$	−3545.39	−0.532303
$355$	12980.3	1.94063
$356$	624.836i	0.0930232i
$357$	− 3660.18i	− 0.542626i
$358$	− 2651.58i	− 0.391453i
$359$	− 3885.40i	− 0.571208i	−0.958348	−	0.285604i	$-0.907806\pi$
$359$	− 3885.40i	− 0.571208i	0.958348	−	0.285604i	$-0.0921942\pi$
$360$	2193.78	0.321173
$361$	−1732.74	−0.252622
$362$	4346.72i	0.631102i
$363$	4376.46	0.632795
$364$	0	0
$365$	2070.31	0.296890
$366$	− 1291.51i	− 0.184450i
$367$	−3888.26	−0.553039	−0.276520	−	0.961008i	$-0.589181\pi$
$367$	−3888.26	−0.553039	−0.276520	+	0.961008i	$0.589181\pi$
$368$	15189.8	2.15170
$369$	− 874.642i	− 0.123393i
$370$	− 12818.3i	− 1.80106i
$371$	− 5282.39i	− 0.739213i
$372$	33.6937i	0.00469607i
$373$	9960.88	1.38272	0.691361	−	0.722510i	$-0.257012\pi$
$373$	9960.88	1.38272	0.691361	+	0.722510i	$0.257012\pi$
$374$	12677.3	1.75275
$375$	146.680i	0.0201987i
$376$	8956.57	1.22846
$377$	0	0
$378$	1565.55	0.213025
$379$	5288.25i	0.716726i	0.933582	+	0.358363i	$0.116665\pi$
$379$	5288.25i	0.716726i	−0.933582	+	0.358363i	$0.883335\pi$
$380$	−4961.73	−0.669820
$381$	7114.08	0.956602
$382$	1575.39i	0.211005i
$383$	688.944i	0.0919149i	0.998943	+	0.0459575i	$0.0146339\pi$
$383$	688.944i	0.0919149i	−0.998943	+	0.0459575i	$0.985366\pi$
$384$	4975.17i	0.661166i
$385$	− 14248.2i	− 1.88612i
$386$	8906.14	1.17438
$387$	−3385.36	−0.444671
$388$	2658.98i	0.347910i
$389$	−2102.57	−0.274048	−0.137024	−	0.990568i	$-0.543754\pi$
$389$	−2102.57	−0.274048	−0.137024	+	0.990568i	$0.543754\pi$
$390$	0	0
$391$	−13553.0	−1.75296
$392$	748.593i	0.0964533i
$393$	−507.599	−0.0651527
$394$	−12418.3	−1.58788
$395$	8740.09i	1.11332i
$396$	1619.47i	0.205509i
$397$	11254.8i	1.42283i	0.702774	+	0.711413i	$0.251944\pi$
$397$	11254.8i	1.42283i	−0.702774	+	0.711413i	$0.748056\pi$
$398$	− 744.906i	− 0.0938159i
$399$	4774.02	0.598998
$400$	9708.17	1.21352
$401$	3523.22i	0.438756i	0.975640	+	0.219378i	$0.0704028\pi$
$401$	3523.22i	0.438756i	−0.975640	+	0.219378i	$0.929597\pi$
$402$	−9155.07	−1.13585
$403$	0	0
$404$	−6158.31	−0.758384
$405$	− 1272.73i	− 0.156154i
$406$	7433.71	0.908692
$407$	−12758.2	−1.55381
$408$	− 3307.34i	− 0.401318i
$409$	− 6366.65i	− 0.769708i	−0.922978	−	0.384854i	$-0.874252\pi$
$409$	− 6366.65i	− 0.769708i	0.922978	−	0.384854i	$-0.125748\pi$
$410$	− 5157.26i	− 0.621217i
$411$	8928.62i	1.07157i
$412$	5317.20	0.635824
$413$	6007.37	0.715746
$414$	− 5796.98i	− 0.688179i
$415$	−4001.45	−0.473310
$416$	0	0
$417$	7665.44	0.900187
$418$	16535.2i	1.93484i
$419$	8919.77	1.04000	0.519999	−	0.854167i	$-0.325932\pi$
$419$	8919.77	1.04000	0.519999	+	0.854167i	$0.325932\pi$
$420$	2757.00	0.320305
$421$	− 1090.71i	− 0.126266i	−0.998005	−	0.0631328i	$-0.979891\pi$
$421$	− 1090.71i	− 0.126266i	0.998005	−	0.0631328i	$-0.0201092\pi$
$422$	4680.82i	0.539949i
$423$	− 5196.19i	− 0.597275i
$424$	− 4773.17i	− 0.546712i
$425$	−8662.05	−0.988638
$426$	8370.25	0.951971
$427$	2188.36i	0.248015i
$428$	−1097.89	−0.123991
$429$	0	0
$430$	−19961.5	−2.23868
$431$	− 3760.43i	− 0.420263i	−0.977673	−	0.210132i	$-0.932611\pi$
$431$	− 3760.43i	− 0.420263i	0.977673	−	0.210132i	$-0.0673892\pi$
$432$	2150.50	0.239504
$433$	4085.06	0.453384	0.226692	−	0.973967i	$-0.427209\pi$
$433$	4085.06	0.453384	0.226692	+	0.973967i	$0.427209\pi$
$434$	− 191.157i	− 0.0211424i
$435$	− 6043.28i	− 0.666099i
$436$	− 5477.15i	− 0.601623i
$437$	− 17677.4i	− 1.93507i
$438$	1335.02	0.145638
$439$	−9385.35	−1.02036	−0.510180	−	0.860068i	$-0.670421\pi$
$439$	−9385.35	−1.02036	−0.510180	+	0.860068i	$0.670421\pi$
$440$	− 12874.7i	− 1.39495i
$441$	434.299	0.0468955
$442$	0	0
$443$	5755.41	0.617264	0.308632	−	0.951182i	$-0.400129\pi$
$443$	5755.41	0.617264	0.308632	+	0.951182i	$0.400129\pi$
$444$	− 2468.68i	− 0.263870i
$445$	−2881.87	−0.306997
$446$	377.751	0.0401055
$447$	− 1273.59i	− 0.134762i
$448$	− 2537.60i	− 0.267612i
$449$	− 1104.67i	− 0.116108i	−0.998313	−	0.0580541i	$-0.981510\pi$
$449$	− 1104.67i	− 0.116108i	0.998313	−	0.0580541i	$-0.0184896\pi$
$450$	− 3704.98i	− 0.388121i
$451$	−5133.05	−0.535934
$452$	1464.59	0.152408
$453$	126.969i	0.0131690i
$454$	−16856.4	−1.74254
$455$	0	0
$456$	4313.81	0.443011
$457$	− 13548.5i	− 1.38681i	−0.720548	−	0.693405i	$-0.756110\pi$
$457$	− 13548.5i	− 1.38681i	0.720548	−	0.693405i	$-0.243890\pi$
$458$	−19546.9	−1.99425
$459$	−1918.77	−0.195121
$460$	− 10208.7i	− 1.03475i
$461$	2996.03i	0.302688i	0.988481	+	0.151344i	$0.0483601\pi$
$461$	2996.03i	0.302688i	−0.988481	+	0.151344i	$0.951640\pi$
$462$	− 9187.84i	− 0.925232i
$463$	2026.10i	0.203371i	0.994817	+	0.101686i	$0.0324236\pi$
$463$	2026.10i	0.203371i	−0.994817	+	0.101686i	$0.967576\pi$
$464$	10211.2	1.02164
$465$	−155.402	−0.0154981
$466$	2444.24i	0.242977i
$467$	3284.20	0.325428	0.162714	−	0.986673i	$-0.447975\pi$
$467$	3284.20	0.325428	0.162714	+	0.986673i	$0.447975\pi$
$468$	0	0
$469$	15512.5	1.52729
$470$	− 30638.9i	− 3.00696i
$471$	2970.56	0.290608
$472$	5428.26	0.529356
$473$	19867.8i	1.93134i
$474$	5635.96i	0.546136i
$475$	− 11298.0i	− 1.09134i
$476$	− 4156.46i	− 0.400234i
$477$	−2769.17	−0.265811
$478$	4187.08	0.400654
$479$	− 13208.5i	− 1.25994i	−0.776621	−	0.629969i	$-0.783068\pi$
$479$	− 13208.5i	− 1.25994i	0.776621	−	0.629969i	$-0.216932\pi$
$480$	6830.19	0.649488
$481$	0	0
$482$	−2551.26	−0.241092
$483$	9822.50i	0.925340i
$484$	4969.86	0.466741
$485$	−12263.7	−1.14818
$486$	− 820.706i	− 0.0766008i
$487$	8974.09i	0.835020i	0.908672	+	0.417510i	$0.137097\pi$
$487$	8974.09i	0.835020i	−0.908672	+	0.417510i	$0.862903\pi$
$488$	1977.41i	0.183428i
$489$	7937.46i	0.734037i
$490$	2560.81	0.236093
$491$	12889.8	1.18474	0.592370	−	0.805666i	$-0.298192\pi$
$491$	12889.8	1.18474	0.592370	+	0.805666i	$0.298192\pi$
$492$	− 993.234i	− 0.0910131i
$493$	−9110.87	−0.832319
$494$	0	0
$495$	−7469.32	−0.678224
$496$	− 262.579i	− 0.0237705i
$497$	−14182.7	−1.28004
$498$	−2580.30	−0.232181
$499$	7371.92i	0.661348i	0.943745	+	0.330674i	$0.107276\pi$
$499$	7371.92i	0.661348i	−0.943745	+	0.330674i	$0.892724\pi$
$500$	166.568i	0.0148983i
$501$	886.284i	0.0790344i
$502$	− 7606.09i	− 0.676248i
$503$	19788.2	1.75410	0.877049	−	0.480400i	$-0.159509\pi$
$503$	19788.2	1.75410	0.877049	+	0.480400i	$0.159509\pi$
$504$	−2396.98	−0.211845
$505$	− 28403.3i	− 2.50283i
$506$	−34021.0	−2.98897
$507$	0	0
$508$	8078.67	0.705577
$509$	11682.4i	1.01732i	0.860968	+	0.508658i	$0.169858\pi$
$509$	11682.4i	1.01732i	−0.860968	+	0.508658i	$0.830142\pi$
$510$	−11313.9	−0.982327
$511$	−2262.08	−0.195829
$512$	1656.08i	0.142948i
$513$	− 2502.67i	− 0.215391i
$514$	− 12552.9i	− 1.07721i
$515$	24524.0i	2.09836i
$516$	−3844.38	−0.327984
$517$	−30495.1	−2.59415
$518$	14005.7i	1.18798i
$519$	4486.38	0.379442
$520$	0	0
$521$	−19025.1	−1.59982	−0.799908	−	0.600122i	$-0.795119\pi$
$521$	−19025.1	−1.59982	−0.799908	+	0.600122i	$0.795119\pi$
$522$	− 3896.95i	− 0.326753i
$523$	5345.69	0.446942	0.223471	−	0.974711i	$-0.428261\pi$
$523$	5345.69	0.446942	0.223471	+	0.974711i	$0.428261\pi$
$524$	−576.425	−0.0480558
$525$	6277.78i	0.521876i
$526$	− 168.708i	− 0.0139848i
$527$	234.284i	0.0193655i
$528$	− 12620.7i	− 1.04024i
$529$	24204.0	1.98932
$530$	−16328.2	−1.33821
$531$	− 3149.23i	− 0.257372i
$532$	5421.33	0.441813
$533$	0	0
$534$	−1858.34	−0.150596
$535$	− 5063.66i	− 0.409199i
$536$	14017.1	1.12957
$537$	2355.29	0.189271
$538$	11555.5i	0.926012i
$539$	− 2548.79i	− 0.203681i
$540$	− 1445.30i	− 0.115177i
$541$	3058.01i	0.243021i	0.992590	+	0.121510i	$0.0387737\pi$
$541$	3058.01i	0.243021i	−0.992590	+	0.121510i	$0.961226\pi$
$542$	−9993.34	−0.791975
$543$	−3861.02	−0.305142
$544$	− 10297.2i	− 0.811561i
$545$	25261.7	1.98549
$546$	0	0
$547$	17921.1	1.40082	0.700410	−	0.713740i	$-0.253000\pi$
$547$	17921.1	1.40082	0.700410	+	0.713740i	$0.253000\pi$
$548$	10139.2i	0.790378i
$549$	1147.20	0.0891827
$550$	−21743.6	−1.68573
$551$	− 11883.4i	− 0.918786i
$552$	8875.62i	0.684369i
$553$	− 9549.67i	− 0.734346i
$554$	25199.0i	1.93250i
$555$	11386.0	0.870828
$556$	8704.79	0.663967
$557$	− 9106.04i	− 0.692702i	−0.938105	−	0.346351i	$-0.887421\pi$
$557$	− 9106.04i	− 0.692702i	0.938105	−	0.346351i	$-0.112579\pi$
$558$	−100.209	−0.00760252
$559$	0	0
$560$	−21485.6	−1.62131
$561$	11260.8i	0.847468i
$562$	−15968.4	−1.19856
$563$	−11327.9	−0.847985	−0.423992	−	0.905666i	$-0.639372\pi$
$563$	−11327.9	−0.847985	−0.423992	+	0.905666i	$0.639372\pi$
$564$	− 5900.74i	− 0.440542i
$565$	6754.97i	0.502980i
$566$	9725.84i	0.722275i
$567$	1390.62i	0.102999i
$568$	−12815.5	−0.946701
$569$	−1963.27	−0.144647	−0.0723237	−	0.997381i	$-0.523041\pi$
$569$	−1963.27	−0.144647	−0.0723237	+	0.997381i	$0.523041\pi$
$570$	− 14756.8i	− 1.08438i
$571$	2270.35	0.166394	0.0831971	−	0.996533i	$-0.473487\pi$
$571$	2270.35	0.166394	0.0831971	+	0.996533i	$0.473487\pi$
$572$	0	0
$573$	−1399.36	−0.102023
$574$	5634.97i	0.409755i
$575$	23245.6	1.68592
$576$	−1330.28	−0.0962297
$577$	− 13949.6i	− 1.00646i	−0.864152	−	0.503231i	$-0.832145\pi$
$577$	− 13949.6i	− 1.00646i	0.864152	−	0.503231i	$-0.167855\pi$
$578$	463.708i	0.0333697i
$579$	7910.96i	0.567821i
$580$	− 6862.69i	− 0.491306i
$581$	4372.10	0.312195
$582$	−7908.14	−0.563236
$583$	16251.6i	1.15450i
$584$	−2044.02	−0.144832
$585$	0	0
$586$	27697.8	1.95253
$587$	− 26731.4i	− 1.87960i	−0.341727	−	0.939799i	$-0.611012\pi$
$587$	− 26731.4i	− 1.87960i	0.341727	−	0.939799i	$-0.388988\pi$
$588$	493.186	0.0345895
$589$	−305.580	−0.0213773
$590$	− 18569.2i	− 1.29573i
$591$	− 11030.7i	− 0.767754i
$592$	19238.7i	1.33565i
$593$	− 10508.6i	− 0.727715i	−0.931455	−	0.363858i	$-0.881459\pi$
$593$	− 10508.6i	− 0.727715i	0.931455	−	0.363858i	$-0.118541\pi$
$594$	−4816.52	−0.332701
$595$	19170.4	1.32086
$596$	− 1446.27i	− 0.0993987i
$597$	661.670	0.0453607
$598$	0	0
$599$	−1935.87	−0.132049	−0.0660246	−	0.997818i	$-0.521032\pi$
$599$	−1935.87	−0.132049	−0.0660246	+	0.997818i	$0.521032\pi$
$600$	5672.61i	0.385972i
$601$	−16155.5	−1.09650	−0.548249	−	0.836315i	$-0.684705\pi$
$601$	−16155.5	−1.09650	−0.548249	+	0.836315i	$0.684705\pi$
$602$	21810.6	1.47663
$603$	− 8132.08i	− 0.549194i
$604$	144.185i	0.00971327i
$605$	22922.0i	1.54035i
$606$	− 18315.6i	− 1.22776i
$607$	−1698.75	−0.113592	−0.0567959	−	0.998386i	$-0.518088\pi$
$607$	−1698.75	−0.113592	−0.0567959	+	0.998386i	$0.518088\pi$
$608$	13430.8	0.895873
$609$	6603.07i	0.439359i
$610$	6764.38	0.448987
$611$	0	0
$612$	−2178.93	−0.143918
$613$	− 8627.42i	− 0.568448i	−0.958758	−	0.284224i	$-0.908264\pi$
$613$	− 8627.42i	− 0.568448i	0.958758	−	0.284224i	$-0.0917359\pi$
$614$	6953.46	0.457034
$615$	4580.99	0.300363
$616$	14067.3i	0.920109i
$617$	− 21415.4i	− 1.39733i	−0.715450	−	0.698664i	$-0.753778\pi$
$617$	− 21415.4i	− 1.39733i	0.715450	−	0.698664i	$-0.246222\pi$
$618$	15814.0i	1.02934i
$619$	17396.0i	1.12957i	0.825238	+	0.564784i	$0.191041\pi$
$619$	17396.0i	1.12957i	−0.825238	+	0.564784i	$0.808959\pi$
$620$	−176.473	−0.0114312
$621$	5149.22	0.332739
$622$	− 6288.96i	− 0.405409i
$623$	3148.81	0.202495
$624$	0	0
$625$	−16004.3	−1.02427
$626$	− 864.261i	− 0.0551802i
$627$	−14687.6	−0.935510
$628$	3373.34	0.214349
$629$	− 17165.6i	− 1.08814i
$630$	8199.68i	0.518545i
$631$	13059.2i	0.823898i	0.911207	+	0.411949i	$0.135152\pi$
$631$	13059.2i	0.823898i	−0.911207	+	0.411949i	$0.864848\pi$
$632$	− 8629.09i	− 0.543112i
$633$	−4157.78	−0.261070
$634$	−21161.6	−1.32561
$635$	37260.4i	2.32856i
$636$	−3144.65	−0.196059
$637$	0	0
$638$	−22870.2	−1.41919
$639$	7434.95i	0.460285i
$640$	−26057.7	−1.60941
$641$	−22397.7	−1.38012	−0.690060	−	0.723752i	$-0.742416\pi$
$641$	−22397.7	−1.38012	−0.690060	+	0.723752i	$0.742416\pi$
$642$	− 3265.26i	− 0.200731i
$643$	5878.08i	0.360511i	0.983620	+	0.180256i	$0.0576925\pi$
$643$	5878.08i	0.360511i	−0.983620	+	0.180256i	$0.942307\pi$
$644$	11154.3i	0.682519i
$645$	− 17731.0i	− 1.08242i
$646$	−22247.5	−1.35498
$647$	20768.3	1.26196	0.630979	−	0.775800i	$-0.282653\pi$
$647$	20768.3	1.26196	0.630979	+	0.775800i	$0.282653\pi$
$648$	1256.56i	0.0761767i
$649$	−18482.0	−1.11785
$650$	0	0
$651$	169.797	0.0102225
$652$	9013.69i	0.541416i
$653$	15516.4	0.929865	0.464933	−	0.885346i	$-0.346079\pi$
$653$	15516.4	0.929865	0.464933	+	0.885346i	$0.346079\pi$
$654$	16289.7	0.973975
$655$	− 2658.58i	− 0.158594i
$656$	7740.39i	0.460688i
$657$	1185.84i	0.0704172i
$658$	33477.0i	1.98339i
$659$	24025.0	1.42015	0.710077	−	0.704124i	$-0.248660\pi$
$659$	24025.0	1.42015	0.710077	+	0.704124i	$0.248660\pi$
$660$	−8482.08	−0.500249
$661$	10778.8i	0.634262i	0.948382	+	0.317131i	$0.102719\pi$
$661$	10778.8i	0.634262i	−0.948382	+	0.317131i	$0.897281\pi$
$662$	−3620.33	−0.212550
$663$	0	0
$664$	3950.63	0.230895
$665$	25004.2i	1.45808i
$666$	7342.17	0.427182
$667$	24450.0	1.41935
$668$	1006.45i	0.0582948i
$669$	335.541i	0.0193913i
$670$	− 47950.2i	− 2.76489i
$671$	− 6732.63i	− 0.387348i
$672$	−7462.86	−0.428402
$673$	−22345.9	−1.27990	−0.639950	−	0.768417i	$-0.721045\pi$
$673$	−22345.9	−1.27990	−0.639950	+	0.768417i	$0.721045\pi$
$674$	18818.9i	1.07549i
$675$	3290.98	0.187659
$676$	0	0
$677$	12326.7	0.699782	0.349891	−	0.936790i	$-0.386219\pi$
$677$	12326.7	0.699782	0.349891	+	0.936790i	$0.386219\pi$
$678$	4355.88i	0.246735i
$679$	13399.7	0.757339
$680$	17322.4	0.976888
$681$	− 14972.9i	− 0.842529i
$682$	588.104i	0.0330201i
$683$	27452.5i	1.53798i	0.639262	+	0.768989i	$0.279240\pi$
$683$	27452.5i	1.53798i	−0.639262	+	0.768989i	$0.720760\pi$
$684$	− 2842.01i	− 0.158870i
$685$	−46764.2	−2.60842
$686$	−22686.3	−1.26264
$687$	− 17362.7i	− 0.964232i
$688$	29959.7	1.66018
$689$	0	0
$690$	30362.0	1.67516
$691$	− 18663.6i	− 1.02749i	−0.857942	−	0.513746i	$-0.828257\pi$
$691$	− 18663.6i	− 1.02749i	0.857942	−	0.513746i	$-0.171743\pi$
$692$	5094.69	0.279871
$693$	8161.19	0.447356
$694$	− 6428.65i	− 0.351626i
$695$	40148.2i	2.19123i
$696$	5966.54i	0.324944i
$697$	− 6906.31i	− 0.375316i
$698$	3705.59	0.200943
$699$	−2171.12	−0.117481
$700$	7128.98i	0.384929i
$701$	4131.00	0.222576	0.111288	−	0.993788i	$-0.464502\pi$
$701$	4131.00	0.222576	0.111288	+	0.993788i	$0.464502\pi$
$702$	0	0
$703$	22389.3	1.20118
$704$	7807.07i	0.417955i
$705$	27215.3	1.45389
$706$	−18308.5	−0.975990
$707$	31034.3i	1.65087i
$708$	− 3576.23i	− 0.189835i
$709$	− 33578.1i	− 1.77863i	−0.457291	−	0.889317i	$-0.651180\pi$
$709$	− 33578.1i	− 1.77863i	0.457291	−	0.889317i	$-0.348820\pi$
$710$	43839.7i	2.31729i
$711$	−5006.20	−0.264061
$712$	2845.27	0.149762
$713$	− 628.728i	− 0.0330239i
$714$	12361.9	0.647943
$715$	0	0
$716$	2674.64	0.139603
$717$	3719.21i	0.193719i
$718$	13122.5	0.682073
$719$	−22527.2	−1.16846	−0.584229	−	0.811589i	$-0.698603\pi$
$719$	−22527.2	−1.16846	−0.584229	+	0.811589i	$0.698603\pi$
$720$	11263.4i	0.583001i
$721$	− 26795.6i	− 1.38408i
$722$	− 5852.12i	− 0.301653i
$723$	− 2266.18i	− 0.116570i
$724$	−4384.53	−0.225069
$725$	15626.6	0.800491
$726$	14781.0i	0.755613i
$727$	−241.718	−0.0123312	−0.00616562	−	0.999981i	$-0.501963\pi$
$727$	−241.718	−0.0123312	−0.00616562	+	0.999981i	$0.501963\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	6992.24i	0.354513i
$731$	−26731.4	−1.35252
$732$	1302.75	0.0657800
$733$	− 17235.4i	− 0.868490i	−0.900795	−	0.434245i	$-0.857015\pi$
$733$	− 17235.4i	− 0.868490i	0.900795	−	0.434245i	$-0.142985\pi$
$734$	− 13132.2i	− 0.660377i
$735$	2274.67i	0.114153i
$736$	27633.7i	1.38396i
$737$	−47725.1	−2.38532
$738$	2954.01	0.147342
$739$	34304.3i	1.70758i	0.520615	+	0.853792i	$0.325703\pi$
$739$	34304.3i	1.70758i	−0.520615	+	0.853792i	$0.674297\pi$
$740$	12929.8	0.642312
$741$	0	0
$742$	17840.7	0.882686
$743$	2114.59i	0.104410i	0.998636	+	0.0522052i	$0.0166250\pi$
$743$	2114.59i	0.104410i	−0.998636	+	0.0522052i	$0.983375\pi$
$744$	153.428	0.00756043
$745$	6670.49	0.328037
$746$	33641.8i	1.65109i
$747$	− 2291.97i	− 0.112261i
$748$	12787.6i	0.625082i
$749$	5532.70i	0.269907i
$750$	−495.394	−0.0241190
$751$	40958.7	1.99015	0.995077	−	0.0991024i	$-0.0315971\pi$
$751$	40958.7	1.99015	0.995077	+	0.0991024i	$0.0315971\pi$
$752$	45985.1i	2.22993i
$753$	6756.18	0.326971
$754$	0	0
$755$	−665.010	−0.0320559
$756$	1579.17i	0.0759708i
$757$	10530.8	0.505613	0.252807	−	0.967517i	$-0.418646\pi$
$757$	10530.8	0.505613	0.252807	+	0.967517i	$0.418646\pi$
$758$	−17860.5	−0.855834
$759$	− 30219.5i	− 1.44519i
$760$	22593.9i	1.07838i
$761$	787.955i	0.0375340i	0.999824	+	0.0187670i	$0.00597407\pi$
$761$	787.955i	0.0375340i	−0.999824	+	0.0187670i	$0.994026\pi$
$762$	24027.0i	1.14227i
$763$	−27601.6	−1.30963
$764$	−1589.10	−0.0752506
$765$	− 10049.7i	− 0.474962i
$766$	−2326.83	−0.109755
$767$	0	0
$768$	−13255.7	−0.622816
$769$	5227.62i	0.245140i	0.992460	+	0.122570i	$0.0391136\pi$
$769$	5227.62i	0.245140i	−0.992460	+	0.122570i	$0.960886\pi$
$770$	48121.9	2.25220
$771$	11150.2	0.520838
$772$	8983.61i	0.418818i
$773$	4516.79i	0.210165i	0.994464	+	0.105082i	$0.0335107\pi$
$773$	4516.79i	0.210165i	−0.994464	+	0.105082i	$0.966489\pi$
$774$	− 11433.7i	− 0.530976i
$775$	− 401.834i	− 0.0186249i
$776$	12108.0	0.560117
$777$	−12440.7	−0.574398
$778$	− 7101.21i	− 0.327237i
$779$	9008.00	0.414307
$780$	0	0
$781$	43633.9	1.99916
$782$	− 45773.9i	− 2.09318i
$783$	3461.51	0.157987
$784$	−3843.45	−0.175084
$785$	15558.5i	0.707397i
$786$	− 1714.36i	− 0.0777980i
$787$	− 6800.09i	− 0.308001i	−0.988071	−	0.154001i	$-0.950784\pi$
$787$	− 6800.09i	− 0.308001i	0.988071	−	0.154001i	$-0.0492158\pi$
$788$	− 12526.4i	− 0.566285i
$789$	149.856	0.00676176
$790$	−29518.7	−1.32940
$791$	− 7380.67i	− 0.331765i
$792$	7374.46	0.330859
$793$	0	0
$794$	−38011.8	−1.69898
$795$	− 14503.7i	− 0.647036i
$796$	751.385	0.0334575
$797$	40553.3	1.80235	0.901174	−	0.433457i	$-0.142707\pi$
$797$	40553.3	1.80235	0.901174	+	0.433457i	$0.142707\pi$
$798$	16123.7i	0.715256i
$799$	− 41029.9i	− 1.81669i
$800$	17661.3i	0.780528i
$801$	− 1650.69i	− 0.0728144i
$802$	−11899.3	−0.523913
$803$	6959.41	0.305844
$804$	− 9234.70i	− 0.405078i
$805$	−51445.9	−2.25246
$806$	0	0
$807$	−10264.3	−0.447733
$808$	28042.6i	1.22096i
$809$	−6517.83	−0.283257	−0.141628	−	0.989920i	$-0.545234\pi$
$809$	−6517.83	−0.283257	−0.141628	+	0.989920i	$0.545234\pi$
$810$	4298.50	0.186461
$811$	2898.99i	0.125521i	0.998029	+	0.0627603i	$0.0199903\pi$
$811$	2898.99i	0.125521i	−0.998029	+	0.0627603i	$0.980010\pi$
$812$	7498.37i	0.324066i
$813$	− 8876.68i	− 0.382926i
$814$	− 43089.3i	− 1.85538i
$815$	−41572.9	−1.78679
$816$	16980.7	0.728484
$817$	− 34866.1i	− 1.49303i
$818$	21502.7	0.919099
$819$	0	0
$820$	5202.12	0.221544
$821$	716.621i	0.0304632i	0.999884	+	0.0152316i	$0.00484855\pi$
$821$	716.621i	0.0304632i	−0.999884	+	0.0152316i	$0.995151\pi$
$822$	−30155.4	−1.27955
$823$	−15510.0	−0.656920	−0.328460	−	0.944518i	$-0.606530\pi$
$823$	−15510.0	−0.656920	−0.328460	+	0.944518i	$0.606530\pi$
$824$	− 24212.5i	− 1.02364i
$825$	− 19314.0i	− 0.815062i
$826$	20289.2i	0.854664i
$827$	24063.2i	1.01180i	0.862591	+	0.505901i	$0.168840\pi$
$827$	24063.2i	1.01180i	−0.862591	+	0.505901i	$0.831160\pi$
$828$	5847.40	0.245424
$829$	−10037.8	−0.420540	−0.210270	−	0.977643i	$-0.567434\pi$
$829$	−10037.8	−0.420540	−0.210270	+	0.977643i	$0.567434\pi$
$830$	− 13514.5i	− 0.565173i
$831$	−22383.2	−0.934376
$832$	0	0
$833$	3429.30	0.142639
$834$	25889.2i	1.07490i
$835$	−4641.96	−0.192385
$836$	−16679.0	−0.690020
$837$	− 89.0121i	− 0.00367587i
$838$	30125.6i	1.24185i
$839$	4005.19i	0.164809i	0.996599	+	0.0824044i	$0.0262599\pi$
$839$	4005.19i	0.164809i	−0.996599	+	0.0824044i	$0.973740\pi$
$840$	− 12554.3i	− 0.515674i
$841$	−7952.74	−0.326079
$842$	3683.74	0.150772
$843$	− 14184.1i	− 0.579510i
$844$	−4721.53	−0.192562
$845$	0	0
$846$	17549.6	0.713199
$847$	− 25045.2i	− 1.01601i
$848$	24506.6	0.992406
$849$	−8639.08	−0.349225
$850$	− 29255.1i	− 1.18052i
$851$	46065.8i	1.85560i
$852$	8443.06i	0.339500i
$853$	− 44062.6i	− 1.76867i	−0.466856	−	0.884334i	$-0.654613\pi$
$853$	− 44062.6i	− 1.76867i	0.466856	−	0.884334i	$-0.345387\pi$
$854$	−7390.96	−0.296152
$855$	13107.9	0.524305
$856$	4999.36i	0.199620i
$857$	−3365.44	−0.134144	−0.0670718	−	0.997748i	$-0.521366\pi$
$857$	−3365.44	−0.134144	−0.0670718	+	0.997748i	$0.521366\pi$
$858$	0	0
$859$	−40969.3	−1.62731	−0.813653	−	0.581351i	$-0.802524\pi$
$859$	−40969.3	−1.62731	−0.813653	+	0.581351i	$0.802524\pi$
$860$	− 20135.2i	− 0.798377i
$861$	−5005.32	−0.198119
$862$	12700.4	0.501831
$863$	− 18132.8i	− 0.715236i	−0.933868	−	0.357618i	$-0.883589\pi$
$863$	− 18132.8i	− 0.715236i	0.933868	−	0.357618i	$-0.116411\pi$
$864$	3912.24i	0.154047i
$865$	23497.7i	0.923637i
$866$	13796.8i	0.541380i
$867$	−411.893	−0.0161345
$868$	192.819	0.00753999
$869$	29380.1i	1.14690i
$870$	20410.5	0.795381
$871$	0	0
$872$	−24940.9	−0.968582
$873$	− 7024.49i	− 0.272329i
$874$	59703.5	2.31064
$875$	839.404	0.0324309
$876$	1346.63i	0.0519388i
$877$	33047.6i	1.27245i	0.771503	+	0.636226i	$0.219505\pi$
$877$	33047.6i	1.27245i	−0.771503	+	0.636226i	$0.780495\pi$
$878$	− 31698.0i	− 1.21840i
$879$	24602.8i	0.944064i
$880$	66101.9	2.53215
$881$	−1349.22	−0.0515965	−0.0257982	−	0.999667i	$-0.508213\pi$
$881$	−1349.22	−0.0515965	−0.0257982	+	0.999667i	$0.508213\pi$
$882$	1466.80i	0.0559973i
$883$	33934.5	1.29330	0.646651	−	0.762786i	$-0.276169\pi$
$883$	33934.5	1.29330	0.646651	+	0.762786i	$0.276169\pi$
$884$	0	0
$885$	16494.3	0.626495
$886$	19438.3i	0.737067i
$887$	−45609.5	−1.72651	−0.863256	−	0.504766i	$-0.831579\pi$
$887$	−45609.5	−1.72651	−0.863256	+	0.504766i	$0.831579\pi$
$888$	−11241.4	−0.424817
$889$	− 40711.8i	− 1.53592i
$890$	− 9733.19i	− 0.366581i
$891$	− 4278.32i	− 0.160863i
$892$	381.037i	0.0143028i
$893$	53515.9	2.00542
$894$	4301.40	0.160918
$895$	12336.0i	0.460722i
$896$	28471.4	1.06157
$897$	0	0
$898$	3730.90	0.138643
$899$	− 422.655i	− 0.0156800i
$900$	3737.21	0.138415
$901$	−21865.8	−0.808498
$902$	− 17336.3i	− 0.639952i
$903$	19373.4i	0.713962i
$904$	− 6669.18i	− 0.245369i
$905$	− 20222.3i	− 0.742777i
$906$	−428.826	−0.0157249
$907$	−8620.99	−0.315607	−0.157803	−	0.987471i	$-0.550441\pi$
$907$	−8620.99	−0.315607	−0.157803	+	0.987471i	$0.550441\pi$
$908$	− 17003.1i	− 0.621438i
$909$	16269.0	0.593629
$910$	0	0
$911$	−43993.6	−1.59997	−0.799986	−	0.600019i	$-0.795160\pi$
$911$	−43993.6	−1.59997	−0.799986	+	0.600019i	$0.795160\pi$
$912$	22148.1i	0.804164i
$913$	−13451.0	−0.487584
$914$	45758.6	1.65597
$915$	6008.53i	0.217088i
$916$	− 19716.9i	− 0.711205i
$917$	2904.84i	0.104609i
$918$	− 6480.43i	− 0.232991i
$919$	15176.0	0.544733	0.272366	−	0.962194i	$-0.412194\pi$
$919$	15176.0	0.544733	0.272366	+	0.962194i	$0.412194\pi$
$920$	−46486.6	−1.66589
$921$	6176.47i	0.220979i
$922$	−10118.8	−0.361436
$923$	0	0
$924$	9267.76	0.329964
$925$	29441.7i	1.04653i
$926$	−6842.94	−0.242843
$927$	−14047.0	−0.497695
$928$	18576.4i	0.657114i
$929$	− 29749.7i	− 1.05065i	−0.850901	−	0.525326i	$-0.823943\pi$
$929$	− 29749.7i	− 1.05065i	0.850901	−	0.525326i	$-0.176057\pi$
$930$	− 524.853i	− 0.0185060i
$931$	4472.88i	0.157457i
$932$	−2465.50	−0.0866526
$933$	5586.23	0.196018
$934$	11092.0i	0.388589i
$935$	−58978.9	−2.06291
$936$	0	0
$937$	−46456.0	−1.61969	−0.809846	−	0.586643i	$-0.800449\pi$
$937$	−46456.0	−1.61969	−0.809846	+	0.586643i	$0.800449\pi$
$938$	52391.8i	1.82372i
$939$	767.688	0.0266800
$940$	30905.5	1.07237
$941$	3243.36i	0.112360i	0.998421	+	0.0561799i	$0.0178920\pi$
$941$	3243.36i	0.112360i	−0.998421	+	0.0561799i	$0.982108\pi$
$942$	10032.7i	0.347011i
$943$	18533.9i	0.640027i
$944$	27870.0i	0.960901i
$945$	−7283.45	−0.250720
$946$	−67101.5	−2.30619
$947$	− 53344.1i	− 1.83047i	−0.402926	−	0.915233i	$-0.632007\pi$
$947$	− 53344.1i	− 1.83047i	0.402926	−	0.915233i	$-0.367993\pi$
$948$	−5684.99	−0.194768
$949$	0	0
$950$	38157.8	1.30316
$951$	− 18797.0i	− 0.640942i
$952$	−18927.0	−0.644356
$953$	−53496.9	−1.81840	−0.909200	−	0.416360i	$-0.863306\pi$
$953$	−53496.9	−1.81840	−0.909200	+	0.416360i	$0.863306\pi$
$954$	− 9352.58i	− 0.317402i
$955$	− 7329.22i	− 0.248343i
$956$	4223.50i	0.142885i
$957$	− 20314.7i	− 0.686188i
$958$	44610.1	1.50448
$959$	51095.9	1.72051
$960$	− 6967.42i	− 0.234242i
$961$	29780.1	0.999635
$962$	0	0
$963$	2900.39	0.0970549
$964$	− 2573.45i	− 0.0859805i
$965$	−41434.1	−1.38219
$966$	−33174.4	−1.10494
$967$	48822.0i	1.62359i	0.583944	+	0.811794i	$0.301509\pi$
$967$	48822.0i	1.62359i	−0.583944	+	0.811794i	$0.698491\pi$
$968$	− 22630.9i	− 0.751429i
$969$	− 19761.5i	− 0.655141i
$970$	− 41419.4i	− 1.37103i
$971$	36228.1	1.19734	0.598670	−	0.800996i	$-0.295696\pi$
$971$	36228.1	1.19734	0.598670	+	0.800996i	$0.295696\pi$
$972$	827.845	0.0273180
$973$	− 43867.1i	− 1.44534i
$974$	−30309.0	−0.997087
$975$	0	0
$976$	−10152.5	−0.332964
$977$	− 46689.8i	− 1.52890i	−0.644681	−	0.764451i	$-0.723010\pi$
$977$	− 46689.8i	− 1.52890i	0.644681	−	0.764451i	$-0.276990\pi$
$978$	−26807.9	−0.876505
$979$	−9687.50	−0.316255
$980$	2583.09i	0.0841977i
$981$	14469.5i	0.470924i
$982$	43533.8i	1.41468i
$983$	11501.9i	0.373197i	0.982436	+	0.186598i	$0.0597463\pi$
$983$	11501.9i	0.373197i	−0.982436	+	0.186598i	$0.940254\pi$
$984$	−4522.81	−0.146526
$985$	57774.0	1.86886
$986$	− 30771.0i	− 0.993862i
$987$	−29736.3	−0.958983
$988$	0	0
$989$	71736.6	2.30646
$990$	− 25226.8i	− 0.809859i
$991$	7137.46	0.228788	0.114394	−	0.993435i	$-0.463507\pi$
$991$	7137.46	0.228788	0.114394	+	0.993435i	$0.463507\pi$
$992$	477.690	0.0152890
$993$	− 3215.80i	− 0.102770i
$994$	− 47900.5i	− 1.52848i
$995$	3465.53i	0.110417i
$996$	− 2602.74i	− 0.0828022i
$997$	−18389.5	−0.584152	−0.292076	−	0.956395i	$-0.594346\pi$
$997$	−18389.5	−0.584152	−0.292076	+	0.956395i	$0.594346\pi$
$998$	−24897.9	−0.789708
$999$	6521.75i	0.206546i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	507.4.b.j.337.15		18
13.5	odd	4		507.4.a.q.1.7	yes	9
13.8	odd	4		507.4.a.n.1.3	✓	9
13.12	even	2	inner	507.4.b.j.337.4		18
39.5	even	4		1521.4.a.be.1.3		9
39.8	even	4		1521.4.a.bj.1.7		9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
507.4.a.n.1.3	✓	9	13.8	odd	4
507.4.a.q.1.7	yes	9	13.5	odd	4
507.4.b.j.337.4		18	13.12	even	2	inner
507.4.b.j.337.15		18	1.1	even	1	trivial
1521.4.a.be.1.3		9	39.5	even	4
1521.4.a.bj.1.7		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)

\(f(q)\)	\(=\)	\(q+3.37739i q^{2} -3.00000 q^{3} -3.40677 q^{4} -15.7127i q^{5} -10.1322i q^{6} +17.1681i q^{7} +15.5131i q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q+3.37739i q^{2} -3.00000 q^{3} -3.40677 q^{4} -15.7127i q^{5} -10.1322i q^{6} +17.1681i q^{7} +15.5131i q^{8} +9.00000 q^{9} +53.0679 q^{10} -52.8187i q^{11} +10.2203 q^{12} -57.9835 q^{14} +47.1380i q^{15} -79.6481 q^{16} +71.0654 q^{17} +30.3965i q^{18} +92.6916i q^{19} +53.5295i q^{20} -51.5044i q^{21} +178.390 q^{22} -190.712 q^{23} -46.5394i q^{24} -121.888 q^{25} -27.0000 q^{27} -58.4878i q^{28} -128.204 q^{29} -159.204 q^{30} +3.29674i q^{31} -144.898i q^{32} +158.456i q^{33} +240.016i q^{34} +269.757 q^{35} -30.6609 q^{36} -241.546i q^{37} -313.056 q^{38} +243.753 q^{40} -97.1824i q^{41} +173.950 q^{42} -376.151 q^{43} +179.941i q^{44} -141.414i q^{45} -644.109i q^{46} -577.354i q^{47} +238.944 q^{48} +48.2555 q^{49} -411.664i q^{50} -213.196 q^{51} -307.686 q^{53} -91.1896i q^{54} -829.924 q^{55} -266.331 q^{56} -278.075i q^{57} -432.995i q^{58} -349.914i q^{59} -160.588i q^{60} +127.467 q^{61} -11.1344 q^{62} +154.513i q^{63} -147.809 q^{64} -535.169 q^{66} -903.564i q^{67} -242.104 q^{68} +572.136 q^{69} +911.076i q^{70} +826.106i q^{71} +139.618i q^{72} +131.760i q^{73} +815.796 q^{74} +365.665 q^{75} -315.779i q^{76} +906.799 q^{77} -556.244 q^{79} +1251.48i q^{80} +81.0000 q^{81} +328.223 q^{82} -254.664i q^{83} +175.464i q^{84} -1116.63i q^{85} -1270.41i q^{86} +384.612 q^{87} +819.384 q^{88} -183.410i q^{89} +477.611 q^{90} +649.712 q^{92} -9.89023i q^{93} +1949.95 q^{94} +1456.43 q^{95} +434.693i q^{96} -780.498i q^{97} +162.978i q^{98} -475.369i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q - 54 q^{3} - 64 q^{4} + 162 q^{9}+O(q^{10}) \) 18 * q - 54 * q^3 - 64 * q^4 + 162 * q^9	\( 18 q - 54 q^{3} - 64 q^{4} + 162 q^{9} - 396 q^{10} + 192 q^{12} + 196 q^{14} + 64 q^{16} + 268 q^{17} + 548 q^{22} - 452 q^{23} - 1224 q^{25} - 486 q^{27} - 1094 q^{29} + 1188 q^{30} + 276 q^{35} - 576 q^{36} + 832 q^{38} + 2684 q^{40} - 588 q^{42} - 316 q^{43} - 192 q^{48} - 1284 q^{49} - 804 q^{51} + 2798 q^{53} - 2816 q^{55} + 1232 q^{56} + 4184 q^{61} + 586 q^{62} - 4962 q^{64} - 1644 q^{66} - 3158 q^{68} + 1356 q^{69} + 2074 q^{74} + 3672 q^{75} + 3372 q^{77} - 230 q^{79} + 1458 q^{81} + 10294 q^{82} + 3282 q^{87} + 968 q^{88} - 3564 q^{90} + 4174 q^{92} - 936 q^{94} + 444 q^{95}+O(q^{100}) \) 18 * q - 54 * q^3 - 64 * q^4 + 162 * q^9 - 396 * q^10 + 192 * q^12 + 196 * q^14 + 64 * q^16 + 268 * q^17 + 548 * q^22 - 452 * q^23 - 1224 * q^25 - 486 * q^27 - 1094 * q^29 + 1188 * q^30 + 276 * q^35 - 576 * q^36 + 832 * q^38 + 2684 * q^40 - 588 * q^42 - 316 * q^43 - 192 * q^48 - 1284 * q^49 - 804 * q^51 + 2798 * q^53 - 2816 * q^55 + 1232 * q^56 + 4184 * q^61 + 586 * q^62 - 4962 * q^64 - 1644 * q^66 - 3158 * q^68 + 1356 * q^69 + 2074 * q^74 + 3672 * q^75 + 3372 * q^77 - 230 * q^79 + 1458 * q^81 + 10294 * q^82 + 3282 * q^87 + 968 * q^88 - 3564 * q^90 + 4174 * q^92 - 936 * q^94 + 444 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

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