LMFDB - Embedded newform 425.4.c.c.424.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [425,4,Mod(424,425)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(425, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("425.424");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 425 = 5^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	425.c (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:	$25.0758117524$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 833x^{4} + 71824 $ x^8 + 833*x^4 + 71824
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 17)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			424.6
Root			$-2.22297 - 2.22297i$ of defining polynomial
Character	$\chi$	$=$	425.424
Dual form			425.4.c.c.424.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.37228i q^{2} +4.44593 q^{3} +2.37228 q^{4} +10.5470i q^{6} -14.9929 q^{7} +24.6060i q^{8} -7.23369 q^{9} +O(q^{10})$	\(q+2.37228i q^{2} +4.44593 q^{3} +2.37228 q^{4} +10.5470i q^{6} -14.9929 q^{7} +24.6060i q^{8} -7.23369 q^{9} +31.1215i q^{11} +10.5470 q^{12} +5.21194i q^{13} -35.5675i q^{14} -39.3940 q^{16} +(16.1286 + 68.2119i) q^{17} -17.1603i q^{18} +28.0000 q^{19} -66.6576 q^{21} -73.8290 q^{22} -167.194 q^{23} +109.396i q^{24} -12.3642 q^{26} -152.201 q^{27} -35.5675 q^{28} -136.072i q^{29} -50.5604i q^{31} +103.394i q^{32} +138.364i q^{33} +(-161.818 + 38.2616i) q^{34} -17.1603 q^{36} +260.558 q^{37} +66.4239i q^{38} +23.1719i q^{39} +183.225i q^{41} -158.130i q^{42} +348.293i q^{43} +73.8290i q^{44} -396.630i q^{46} +318.424i q^{47} -175.143 q^{48} -118.212 q^{49} +(71.7066 + 303.266i) q^{51} +12.3642i q^{52} +408.250i q^{53} -361.063i q^{54} -368.916i q^{56} +124.486 q^{57} +322.801 q^{58} -108.119 q^{59} -123.677i q^{61} +119.943 q^{62} +108.454 q^{63} -560.432 q^{64} -328.239 q^{66} -243.826i q^{67} +(38.2616 + 161.818i) q^{68} -743.331 q^{69} +42.7075i q^{71} -177.992i q^{72} +875.172 q^{73} +618.117i q^{74} +66.4239 q^{76} -466.603i q^{77} -54.9703 q^{78} -750.553i q^{79} -481.364 q^{81} -434.662 q^{82} +472.728i q^{83} -158.130 q^{84} -826.250 q^{86} -604.967i q^{87} -765.775 q^{88} -376.364 q^{89} -78.1422i q^{91} -396.630 q^{92} -224.788i q^{93} -755.391 q^{94} +459.683i q^{96} +303.169 q^{97} -280.432i q^{98} -225.123i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 4 q^{4} + 80 q^{9}+O(q^{10}) $ 8 * q - 4 * q^4 + 80 * q^9	$ 8 q - 4 q^{4} + 80 q^{9} - 476 q^{16} + 224 q^{19} + 248 q^{21} - 1064 q^{26} - 812 q^{34} - 436 q^{36} - 624 q^{49} + 2320 q^{51} + 2352 q^{59} - 2852 q^{64} + 3808 q^{66} - 2408 q^{69} - 112 q^{76} - 4816 q^{81} - 2368 q^{84} - 4496 q^{86} - 3976 q^{89} - 896 q^{94}+O(q^{100}) $ 8 * q - 4 * q^4 + 80 * q^9 - 476 * q^16 + 224 * q^19 + 248 * q^21 - 1064 * q^26 - 812 * q^34 - 436 * q^36 - 624 * q^49 + 2320 * q^51 + 2352 * q^59 - 2852 * q^64 + 3808 * q^66 - 2408 * q^69 - 112 * q^76 - 4816 * q^81 - 2368 * q^84 - 4496 * q^86 - 3976 * q^89 - 896 * q^94

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/425\mathbb{Z}\right)^\times$.

$n$	$52$	$326$
$\chi(n)$	$-1$	$-1$

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 $

$2$			2.37228i			0.838728i	0.907818	+	0.419364i	$0.137747\pi$
$2$			2.37228i			0.838728i	−0.907818	+	0.419364i	$0.862253\pi$
$3$	4.44593			0.855620			0.427810	−	0.903869i	$-0.359285\pi$
$3$	4.44593			0.855620			0.427810	+	0.903869i	$0.359285\pi$
$4$	2.37228			0.296535
$5$		0			0
$5$		0			0
$6$			10.5470i			0.717633i
$7$	−14.9929			−0.809542			−0.404771	−	0.914418i	$-0.632649\pi$
$7$	−14.9929			−0.809542			−0.404771	+	0.914418i	$0.632649\pi$
$8$			24.6060i			1.08744i
$9$	−7.23369			−0.267914
$10$		0			0
$11$			31.1215i			0.853045i	0.904477	+	0.426522i	$0.140261\pi$
$11$			31.1215i			0.853045i	−0.904477	+	0.426522i	$0.859739\pi$
$12$	10.5470			0.253721
$13$			5.21194i			0.111195i	0.998453	+	0.0555974i	$0.0177063\pi$
$13$			5.21194i			0.111195i	−0.998453	+	0.0555974i	$0.982294\pi$
$14$		−	35.5675i		−	0.678986i
$15$		0			0
$16$	−39.3940			−0.615532
$17$	16.1286	+	68.2119i	0.230103	+	0.973166i
$17$	16.1286	+	68.2119i	0.230103	+	0.973166i
$18$		−	17.1603i		−	0.224707i
$19$	28.0000			0.338086			0.169043	−	0.985609i	$-0.445932\pi$
$19$	28.0000			0.338086			0.169043	+	0.985609i	$0.445932\pi$
$20$		0			0
$21$	−66.6576			−0.692661
$22$	−73.8290			−0.715473
$23$	−167.194			−1.51575			−0.757875	−	0.652399i	$-0.773763\pi$
$23$	−167.194			−1.51575			−0.757875	+	0.652399i	$0.773763\pi$
$24$			109.396i			0.930436i
$25$		0			0
$26$	−12.3642			−0.0932622
$27$	−152.201			−1.08485
$28$	−35.5675			−0.240058
$29$		−	136.072i		−	0.871309i	−0.900114	−	0.435654i	$-0.856517\pi$
$29$		−	136.072i		−	0.871309i	0.900114	−	0.435654i	$-0.143483\pi$
$30$		0			0
$31$		−	50.5604i		−	0.292933i	−0.989216	−	0.146466i	$-0.953210\pi$
$31$		−	50.5604i		−	0.292933i	0.989216	−	0.146466i	$-0.0467900\pi$
$32$			103.394i			0.571177i
$33$			138.364i			0.729882i
$34$	−161.818	+	38.2616i	−0.816222	+	0.192994i
$35$		0			0
$36$	−17.1603			−0.0794460
$37$	260.558			1.15772			0.578858	−	0.815428i	$-0.303499\pi$
$37$	260.558			1.15772			0.578858	+	0.815428i	$0.303499\pi$
$38$			66.4239i			0.283563i
$39$			23.1719i			0.0951404i
$40$		0			0
$41$			183.225i			0.697927i	0.937136	+	0.348964i	$0.113466\pi$
$41$			183.225i			0.697927i	−0.937136	+	0.348964i	$0.886534\pi$
$42$		−	158.130i		−	0.580954i
$43$			348.293i			1.23521i	0.786486	+	0.617607i	$0.211898\pi$
$43$			348.293i			1.23521i	−0.786486	+	0.617607i	$0.788102\pi$
$44$			73.8290i			0.252958i
$45$		0			0
$46$		−	396.630i		−	1.27130i
$47$			318.424i			0.988232i	0.869396	+	0.494116i	$0.164508\pi$
$47$			318.424i			0.988232i	−0.869396	+	0.494116i	$0.835492\pi$
$48$	−175.143			−0.526661
$49$	−118.212			−0.344641
$50$		0			0
$51$	71.7066	+	303.266i	0.196881	+	0.832660i
$52$			12.3642i			0.0329732i
$53$			408.250i			1.05806i	0.848602	+	0.529032i	$0.177445\pi$
$53$			408.250i			1.05806i	−0.848602	+	0.529032i	$0.822555\pi$
$54$		−	361.063i		−	0.909897i
$55$		0			0
$56$		−	368.916i		−	0.880329i
$57$	124.486			0.289273
$58$	322.801			0.730791
$59$	−108.119			−0.238575			−0.119288	−	0.992860i	$-0.538061\pi$
$59$	−108.119			−0.238575			−0.119288	+	0.992860i	$0.538061\pi$
$60$		0			0
$61$		−	123.677i		−	0.259593i	−0.991541	−	0.129796i	$-0.958568\pi$
$61$		−	123.677i		−	0.259593i	0.991541	−	0.129796i	$-0.0414324\pi$
$62$	119.943			0.245691
$63$	108.454			0.216888
$64$	−560.432			−1.09459
$65$		0			0
$66$	−328.239			−0.612173
$67$		−	243.826i		−	0.444598i	−0.974979	−	0.222299i	$-0.928644\pi$
$67$		−	243.826i		−	0.444598i	0.974979	−	0.222299i	$-0.0713562\pi$
$68$	38.2616	+	161.818i	0.0682338	+	0.288578i
$69$	−743.331			−1.29691
$70$		0			0
$71$			42.7075i			0.0713866i	0.999363	+	0.0356933i	$0.0113639\pi$
$71$			42.7075i			0.0713866i	−0.999363	+	0.0356933i	$0.988636\pi$
$72$		−	177.992i		−	0.291341i
$73$	875.172			1.40317			0.701583	−	0.712588i	$-0.252477\pi$
$73$	875.172			1.40317			0.701583	+	0.712588i	$0.252477\pi$
$74$			618.117i			0.971009i
$75$		0			0
$76$	66.4239			0.100254
$77$		−	466.603i		−	0.690576i
$78$	−54.9703			−0.0797970
$79$		−	750.553i		−	1.06891i	−0.845197	−	0.534454i	$-0.820517\pi$
$79$		−	750.553i		−	1.06891i	0.845197	−	0.534454i	$-0.179483\pi$
$80$		0			0
$81$	−481.364			−0.660308
$82$	−434.662			−0.585371
$83$			472.728i			0.625165i	0.949891	+	0.312582i	$0.101194\pi$
$83$			472.728i			0.625165i	−0.949891	+	0.312582i	$0.898806\pi$
$84$	−158.130			−0.205398
$85$		0			0
$86$	−826.250			−1.03601
$87$		−	604.967i		−	0.745509i
$88$	−765.775			−0.927635
$89$	−376.364			−0.448253			−0.224127	−	0.974560i	$-0.571953\pi$
$89$	−376.364			−0.448253			−0.224127	+	0.974560i	$0.571953\pi$
$90$		0			0
$91$		−	78.1422i		−	0.0900169i
$92$	−396.630			−0.449473
$93$		−	224.788i		−	0.250639i
$94$	−755.391			−0.828858
$95$		0			0
$96$			459.683i			0.488710i
$97$	303.169			0.317342			0.158671	−	0.987332i	$-0.449279\pi$
$97$	303.169			0.317342			0.158671	+	0.987332i	$0.449279\pi$
$98$		−	280.432i		−	0.289060i
$99$		−	225.123i		−	0.228543i
$100$		0			0
$101$	1484.72			1.46273			0.731363	−	0.681988i	$-0.238884\pi$
$101$	1484.72			1.46273			0.731363	+	0.681988i	$0.238884\pi$
$102$	−719.431	+	170.108i	−0.698376	+	0.165130i
$103$			168.000i			0.160714i	0.996766	+	0.0803570i	$0.0256060\pi$
$103$			168.000i			0.160714i	−0.996766	+	0.0803570i	$0.974394\pi$
$104$	−128.245			−0.120918
$105$		0			0
$106$	−968.484			−0.887429
$107$	1096.53			0.990703			0.495351	−	0.868693i	$-0.335039\pi$
$107$	1096.53			0.990703			0.495351	+	0.868693i	$0.335039\pi$
$108$	−361.063			−0.321697
$109$		−	828.018i		−	0.727613i	−0.931475	−	0.363806i	$-0.881477\pi$
$109$		−	828.018i		−	0.727613i	0.931475	−	0.363806i	$-0.118523\pi$
$110$		0			0
$111$	1158.42			0.990565
$112$	590.632			0.498299
$113$	−1711.01			−1.42441			−0.712204	−	0.701973i	$-0.752303\pi$
$113$	−1711.01			−1.42441			−0.712204	+	0.701973i	$0.752303\pi$
$114$			295.316i			0.242622i
$115$		0			0
$116$		−	322.801i		−	0.258374i
$117$		−	37.7015i		−	0.0297907i
$118$		−	256.490i		−	0.200100i
$119$	−241.815	−	1022.70i	−0.186278	−	0.787819i
$120$		0			0
$121$	362.451			0.272315
$122$	293.396			0.217728
$123$			814.608i			0.597160i
$124$		−	119.943i		−	0.0868649i
$125$		0			0
$126$			257.284i			0.181910i
$127$		−	1131.46i		−	0.790555i	−0.918562	−	0.395277i	$-0.870648\pi$
$127$		−	1131.46i		−	0.790555i	0.918562	−	0.395277i	$-0.129352\pi$
$128$		−	502.350i		−	0.346890i
$129$			1548.49i			1.05687i
$130$		0			0
$131$		−	1525.41i		−	1.01737i	−0.860952	−	0.508687i	$-0.830131\pi$
$131$		−	1525.41i		−	1.01737i	0.860952	−	0.508687i	$-0.169869\pi$
$132$			328.239i			0.216436i
$133$	−419.802			−0.273695
$134$	578.424			0.372897
$135$		0			0
$136$	−1678.42	+	396.860i	−1.05826	+	0.250224i
$137$			430.495i			0.268465i	0.990950	+	0.134232i	$0.0428568\pi$
$137$			430.495i			0.268465i	−0.990950	+	0.134232i	$0.957143\pi$
$138$		−	1763.39i		−	1.08775i
$139$			1073.93i			0.655323i	0.944795	+	0.327662i	$0.106261\pi$
$139$			1073.93i			0.655323i	−0.944795	+	0.327662i	$0.893739\pi$
$140$		0			0
$141$			1415.69i			0.845551i
$142$	−101.314			−0.0598739
$143$	−162.203			−0.0948541
$144$	284.964			0.164910
$145$		0			0
$146$			2076.15i			1.17687i
$147$	−525.562			−0.294882
$148$	618.117			0.343304
$149$	1821.57			1.00153			0.500767	−	0.865582i	$-0.333052\pi$
$149$	1821.57			1.00153			0.500767	+	0.865582i	$0.333052\pi$
$150$		0			0
$151$	150.608			0.0811676			0.0405838	−	0.999176i	$-0.487078\pi$
$151$	150.608			0.0811676			0.0405838	+	0.999176i	$0.487078\pi$
$152$			688.967i			0.367649i
$153$	−116.669	−	493.424i	−0.0616480	−	0.260725i
$154$	1106.91			0.579205
$155$		0			0
$156$			54.9703i			0.0282125i
$157$			1457.75i			0.741026i	0.928827	+	0.370513i	$0.120818\pi$
$157$			1457.75i			0.741026i	−0.928827	+	0.370513i	$0.879182\pi$
$158$	1780.52			0.896524
$159$			1815.05i			0.905301i
$160$		0			0
$161$	2506.72			1.22706
$162$		−	1141.93i		−	0.553818i
$163$	2573.11			1.23645			0.618225	−	0.786001i	$-0.287852\pi$
$163$	2573.11			1.23645			0.618225	+	0.786001i	$0.287852\pi$
$164$			434.662i			0.206960i
$165$		0			0
$166$	−1121.44			−0.524343
$167$	−1563.48			−0.724466			−0.362233	−	0.932088i	$-0.617986\pi$
$167$	−1563.48			−0.724466			−0.362233	+	0.932088i	$0.617986\pi$
$168$		−	1640.17i		−	0.753227i
$169$	2169.84			0.987636
$170$		0			0
$171$	−202.543			−0.0905782
$172$			826.250i			0.366285i
$173$	−340.850			−0.149794			−0.0748970	−	0.997191i	$-0.523863\pi$
$173$	−340.850			−0.149794			−0.0748970	+	0.997191i	$0.523863\pi$
$174$	1435.15			0.625279
$175$		0			0
$176$		−	1226.00i		−	0.525076i
$177$	−480.691			−0.204130
$178$		−	892.842i		−	0.375962i
$179$	1674.72			0.699297			0.349649	−	0.936881i	$-0.386301\pi$
$179$	1674.72			0.699297			0.349649	+	0.936881i	$0.386301\pi$
$180$		0			0
$181$		−	1178.76i		−	0.484071i	−0.970267	−	0.242035i	$-0.922185\pi$
$181$		−	1178.76i		−	0.484071i	0.970267	−	0.242035i	$-0.0778150\pi$
$182$	185.375			0.0754997
$183$		−	549.857i		−	0.222113i
$184$		−	4113.96i		−	1.64829i
$185$		0			0
$186$	533.261			0.210218
$187$	−2122.86	+	501.946i	−0.830154	+	0.196289i
$188$			755.391i			0.293045i
$189$	2281.93			0.878234
$190$		0			0
$191$	3686.60			1.39661			0.698306	−	0.715799i	$-0.253938\pi$
$191$	3686.60			1.39661			0.698306	+	0.715799i	$0.253938\pi$
$192$	−2491.64			−0.936556
$193$	−1499.75			−0.559350			−0.279675	−	0.960095i	$-0.590227\pi$
$193$	−1499.75			−0.559350			−0.279675	+	0.960095i	$0.590227\pi$
$194$			719.202i			0.266163i
$195$		0			0
$196$	−280.432			−0.102198
$197$	4359.97			1.57683			0.788414	−	0.615145i	$-0.210902\pi$
$197$	4359.97			1.57683			0.788414	+	0.615145i	$0.210902\pi$
$198$	534.056			0.191685
$199$			473.286i			0.168595i	0.996441	+	0.0842974i	$0.0268646\pi$
$199$			473.286i			0.168595i	−0.996441	+	0.0842974i	$0.973135\pi$
$200$		0			0
$201$		−	1084.03i		−	0.380407i
$202$			3522.18i			1.22683i
$203$			2040.12i			0.705361i
$204$	170.108	+	719.431i	0.0583822	+	0.246913i
$205$		0			0
$206$	−398.543			−0.134795
$207$	1209.43			0.406091
$208$		−	205.319i		−	0.0684439i
$209$			871.403i			0.288403i
$210$		0			0
$211$			4192.78i			1.36798i	0.729493	+	0.683988i	$0.239756\pi$
$211$			4192.78i			1.36798i	−0.729493	+	0.683988i	$0.760244\pi$
$212$			968.484i			0.313753i
$213$			189.875i			0.0610798i
$214$			2601.27i			0.830930i
$215$		0			0
$216$		−	3745.04i		−	1.17971i
$217$			758.049i			0.237141i
$218$	1964.29			0.610269
$219$	3890.95			1.20058
$220$		0			0
$221$	−355.516	+	84.0612i	−0.108211	+	0.0255863i
$222$			2748.11i			0.830815i
$223$		−	3017.07i		−	0.906001i	−0.891510	−	0.453001i	$-0.850354\pi$
$223$		−	3017.07i		−	0.906001i	0.891510	−	0.453001i	$-0.149646\pi$
$224$		−	1550.18i		−	0.462392i
$225$		0			0
$226$		−	4058.99i		−	1.19469i
$227$	−535.758			−0.156650			−0.0783250	−	0.996928i	$-0.524957\pi$
$227$	−535.758			−0.156650			−0.0783250	+	0.996928i	$0.524957\pi$
$228$	295.316			0.0857797
$229$	4029.45			1.16277			0.581383	−	0.813630i	$-0.302512\pi$
$229$	4029.45			1.16277			0.581383	+	0.813630i	$0.302512\pi$
$230$		0			0
$231$		−	2074.49i		−	0.590871i
$232$	3348.18			0.947496
$233$	−1415.69			−0.398047			−0.199024	−	0.979995i	$-0.563777\pi$
$233$	−1415.69			−0.398047			−0.199024	+	0.979995i	$0.563777\pi$
$234$	89.4387			0.0249863
$235$		0			0
$236$	−256.490			−0.0707460
$237$		−	3336.91i		−	0.914580i
$238$	2426.13	−	573.653i	0.660766	−	0.156237i
$239$	−3584.18			−0.970049			−0.485024	−	0.874501i	$-0.661189\pi$
$239$	−3584.18			−0.970049			−0.485024	+	0.874501i	$0.661189\pi$
$240$		0			0
$241$			6042.91i			1.61518i	0.589744	+	0.807590i	$0.299228\pi$
$241$			6042.91i			1.61518i	−0.589744	+	0.807590i	$0.700772\pi$
$242$			859.835i			0.228398i
$243$	1969.31			0.519881
$244$		−	293.396i		−	0.0769784i
$245$		0			0
$246$	−1932.48			−0.500855
$247$			145.934i			0.0375934i
$248$	1244.09			0.318547
$249$			2101.72i			0.534904i
$250$		0			0
$251$	5801.33			1.45887			0.729435	−	0.684050i	$-0.239783\pi$
$251$	5801.33			1.45887			0.729435	+	0.684050i	$0.239783\pi$
$252$	257.284			0.0643149
$253$		−	5203.32i		−	1.29300i
$254$	2684.13			0.663061
$255$		0			0
$256$	−3291.74			−0.803647
$257$		−	2865.09i		−	0.695407i	−0.937605	−	0.347703i	$-0.886962\pi$
$257$		−	2865.09i		−	0.695407i	0.937605	−	0.347703i	$-0.113038\pi$
$258$	−3673.45			−0.886430
$259$	−3906.53			−0.937220
$260$		0			0
$261$			984.303i			0.233436i
$262$	3618.71			0.853300
$263$		−	542.066i		−	0.127092i	−0.997979	−	0.0635460i	$-0.979759\pi$
$263$		−	542.066i		−	0.127092i	0.997979	−	0.0635460i	$-0.0202410\pi$
$264$	−3404.58			−0.793703
$265$		0			0
$266$		−	995.889i		−	0.229556i
$267$	−1673.29			−0.383534
$268$		−	578.424i		−	0.131839i
$269$		−	1759.85i		−	0.398885i	−0.979910	−	0.199442i	$-0.936087\pi$
$269$		−	1759.85i		−	0.398885i	0.979910	−	0.199442i	$-0.0639131\pi$
$270$		0			0
$271$	−3596.70			−0.806215			−0.403107	−	0.915153i	$-0.632070\pi$
$271$	−3596.70			−0.806215			−0.403107	+	0.915153i	$0.632070\pi$
$272$	−635.370	−	2687.14i	−0.141636	−	0.599015i
$273$		−	347.415i		−	0.0770202i
$274$	−1021.25			−0.225169
$275$		0			0
$276$	−1763.39			−0.384578
$277$	−4539.43			−0.984649			−0.492325	−	0.870412i	$-0.663853\pi$
$277$	−4539.43			−0.984649			−0.492325	+	0.870412i	$0.663853\pi$
$278$	−2547.67			−0.549638
$279$			365.738i			0.0784809i
$280$		0			0
$281$	1838.52			0.390309			0.195155	−	0.980772i	$-0.437479\pi$
$281$	1838.52			0.390309			0.195155	+	0.980772i	$0.437479\pi$
$282$	−3358.42			−0.709187
$283$	−3934.41			−0.826418			−0.413209	−	0.910636i	$-0.635592\pi$
$283$	−3934.41			−0.826418			−0.413209	+	0.910636i	$0.635592\pi$
$284$			101.314i			0.0211686i
$285$		0			0
$286$		−	384.792i		−	0.0795568i
$287$		−	2747.09i		−	0.565002i
$288$		−	747.920i		−	0.153026i
$289$	−4392.74	+	2200.32i	−0.894105	+	0.447858i
$290$		0			0
$291$	1347.87			0.271524
$292$	2076.15			0.416088
$293$			4739.91i			0.945080i	0.881309	+	0.472540i	$0.156663\pi$
$293$			4739.91i			0.945080i	−0.881309	+	0.472540i	$0.843337\pi$
$294$		−	1246.78i		−	0.247326i
$295$		0			0
$296$			6411.29i			1.25895i
$297$		−	4736.72i		−	0.925428i
$298$			4321.26i			0.840014i
$299$		−	871.403i		−	0.168544i
$300$		0			0
$301$		−	5221.94i		−	0.999959i
$302$			357.285i			0.0680776i
$303$	6600.97			1.25154
$304$	−1103.03			−0.208103
$305$		0			0
$306$	1170.54	−	276.772i	0.218678	−	0.0517059i
$307$		−	4965.81i		−	0.923173i	−0.887095	−	0.461586i	$-0.847280\pi$
$307$		−	4965.81i		−	0.923173i	0.887095	−	0.461586i	$-0.152720\pi$
$308$		−	1106.91i		−	0.204780i
$309$			746.917i			0.137510i
$310$		0			0
$311$			6039.28i			1.10114i	0.834787	+	0.550572i	$0.185591\pi$
$311$			6039.28i			1.10114i	−0.834787	+	0.550572i	$0.814409\pi$
$312$	−570.168			−0.103460
$313$	−6495.68			−1.17303			−0.586513	−	0.809940i	$-0.699500\pi$
$313$	−6495.68			−1.17303			−0.586513	+	0.809940i	$0.699500\pi$
$314$	−3458.19			−0.621519
$315$		0			0
$316$		−	1780.52i		−	0.316969i
$317$	3036.97			0.538086			0.269043	−	0.963128i	$-0.413293\pi$
$317$	3036.97			0.538086			0.269043	+	0.963128i	$0.413293\pi$
$318$	−4305.81			−0.759302
$319$	4234.77			0.743265
$320$		0			0
$321$	4875.08			0.847665
$322$			5946.65i			1.02917i
$323$	451.601	+	1909.93i	0.0777948	+	0.329014i
$324$	−1141.93			−0.195804
$325$		0			0
$326$			6104.13i			1.03704i
$327$		−	3681.31i		−	0.622560i
$328$	−4508.44			−0.758954
$329$		−	4774.11i		−	0.800015i
$330$		0			0
$331$	−7769.51			−1.29018			−0.645092	−	0.764105i	$-0.723181\pi$
$331$	−7769.51			−1.29018			−0.645092	+	0.764105i	$0.723181\pi$
$332$			1121.44i			0.185383i
$333$	−1884.80			−0.310169
$334$		−	3709.02i		−	0.607630i
$335$		0			0
$336$	2625.91			0.426355
$337$	−6053.47			−0.978498			−0.489249	−	0.872144i	$-0.662729\pi$
$337$	−6053.47			−0.978498			−0.489249	+	0.872144i	$0.662729\pi$
$338$			5147.46i			0.828358i
$339$	−7607.02			−1.21875
$340$		0			0
$341$	1573.52			0.249885
$342$		−	480.490i		−	0.0759705i
$343$	6914.92			1.08854
$344$	−8570.10			−1.34322
$345$		0			0
$346$		−	808.593i		−	0.125636i
$347$	336.418			0.0520457			0.0260228	−	0.999661i	$-0.491716\pi$
$347$	336.418			0.0520457			0.0260228	+	0.999661i	$0.491716\pi$
$348$		−	1435.15i		−	0.221070i
$349$	−8360.05			−1.28224			−0.641122	−	0.767439i	$-0.721531\pi$
$349$	−8360.05			−1.28224			−0.641122	+	0.767439i	$0.721531\pi$
$350$		0			0
$351$		−	793.260i		−	0.120630i
$352$	−3217.78			−0.487239
$353$		−	2040.04i		−	0.307593i	−0.988103	−	0.153797i	$-0.950850\pi$
$353$		−	2040.04i		−	0.307593i	0.988103	−	0.153797i	$-0.0491501\pi$
$354$		−	1140.34i		−	0.171209i
$355$		0			0
$356$	−892.842			−0.132923
$357$	−1075.09	−	4546.84i	−0.159384	−	0.674074i
$358$			3972.90i			0.586520i
$359$	594.357			0.0873788			0.0436894	−	0.999045i	$-0.486089\pi$
$359$	594.357			0.0873788			0.0436894	+	0.999045i	$0.486089\pi$
$360$		0			0
$361$	−6075.00			−0.885698
$362$	2796.36			0.406004
$363$	1611.43			0.232998
$364$		−	185.375i		−	0.0266932i
$365$		0			0
$366$	1304.42			0.186292
$367$	11557.1			1.64381			0.821903	−	0.569628i	$-0.192913\pi$
$367$	11557.1			1.64381			0.821903	+	0.569628i	$0.192913\pi$
$368$	6586.43			0.932993
$369$		−	1325.40i		−	0.186985i
$370$		0			0
$371$		−	6120.86i		−	0.856548i
$372$		−	533.261i		−	0.0743233i
$373$		−	154.375i		−	0.0214296i	−0.999943	−	0.0107148i	$-0.996589\pi$
$373$		−	154.375i		−	0.0214296i	0.999943	−	0.0107148i	$-0.00341070\pi$
$374$	−1190.76	−	5036.02i	−0.164633	−	0.696274i
$375$		0			0
$376$	−7835.13			−1.07464
$377$	709.199			0.0968849
$378$			5413.39i			0.736600i
$379$		−	8975.51i		−	1.21647i	−0.793758	−	0.608234i	$-0.791878\pi$
$379$		−	8975.51i		−	1.21647i	0.793758	−	0.608234i	$-0.208122\pi$
$380$		0			0
$381$		−	5030.38i		−	0.676415i
$382$			8745.65i			1.17138i
$383$			14032.7i			1.87216i	0.351784	+	0.936081i	$0.385575\pi$
$383$			14032.7i			1.87216i	−0.351784	+	0.936081i	$0.614425\pi$
$384$		−	2233.41i		−	0.296806i
$385$		0			0
$386$		−	3557.83i		−	0.469142i
$387$		−	2519.45i		−	0.330932i
$388$	719.202			0.0941030
$389$	11432.1			1.49006			0.745029	−	0.667032i	$-0.232436\pi$
$389$	11432.1			1.49006			0.745029	+	0.667032i	$0.232436\pi$
$390$		0			0
$391$	−2696.60	−	11404.6i	−0.348779	−	1.47508i
$392$		−	2908.72i		−	0.374777i
$393$		−	6781.88i		−	0.870486i
$394$			10343.1i			1.32253i
$395$		0			0
$396$		−	534.056i		−	0.0677710i
$397$	1261.59			0.159490			0.0797450	−	0.996815i	$-0.474589\pi$
$397$	1261.59			0.159490			0.0797450	+	0.996815i	$0.474589\pi$
$398$	−1122.77			−0.141405
$399$	−1866.41			−0.234179
$400$		0			0
$401$			14316.4i			1.78286i	0.453158	+	0.891430i	$0.350297\pi$
$401$			14316.4i			1.78286i	−0.453158	+	0.891430i	$0.649703\pi$
$402$	2571.63			0.319058
$403$	263.518			0.0325726
$404$	3522.18			0.433750
$405$		0			0
$406$	−4839.74			−0.591606
$407$			8108.97i			0.987584i
$408$	−7462.15	+	1764.41i	−0.905469	+	0.214096i
$409$	−6819.29			−0.824431			−0.412215	−	0.911086i	$-0.635245\pi$
$409$	−6819.29			−0.824431			−0.412215	+	0.911086i	$0.635245\pi$
$410$		0			0
$411$			1913.95i			0.229704i
$412$			398.543i			0.0476573i
$413$	1621.03			0.193137
$414$			2869.10i			0.340600i
$415$		0			0
$416$	−538.883			−0.0635118
$417$			4774.64i			0.560708i
$418$	−2067.21			−0.241892
$419$			11987.4i			1.39767i	0.715282	+	0.698836i	$0.246298\pi$
$419$			11987.4i			1.39767i	−0.715282	+	0.698836i	$0.753702\pi$
$420$		0			0
$421$	8307.55			0.961722			0.480861	−	0.876797i	$-0.340324\pi$
$421$	8307.55			0.961722			0.480861	+	0.876797i	$0.340324\pi$
$422$	−9946.45			−1.14736
$423$		−	2303.38i		−	0.264762i
$424$	−10045.4			−1.15058
$425$		0			0
$426$	−450.436			−0.0512293
$427$			1854.27i			0.210151i
$428$	2601.27			0.293778
$429$	−721.146			−0.0811591
$430$		0			0
$431$		−	14582.2i		−	1.62970i	−0.579673	−	0.814849i	$-0.696820\pi$
$431$		−	14582.2i		−	1.62970i	0.579673	−	0.814849i	$-0.303180\pi$
$432$	5995.80			0.667761
$433$		−	14056.3i		−	1.56006i	−0.625744	−	0.780028i	$-0.715205\pi$
$433$		−	14056.3i		−	1.56006i	0.625744	−	0.780028i	$-0.284795\pi$
$434$	−1798.30			−0.198897
$435$		0			0
$436$		−	1964.29i		−	0.215763i
$437$	−4681.42			−0.512455
$438$			9230.44i			1.00696i
$439$			11386.3i			1.23790i	0.785431	+	0.618950i	$0.212441\pi$
$439$			11386.3i			1.23790i	−0.785431	+	0.618950i	$0.787559\pi$
$440$		0			0
$441$	855.108			0.0923343
$442$	−199.417	−	843.385i	−0.0214599	−	0.0907596i
$443$			11695.6i			1.25435i	0.778880	+	0.627173i	$0.215788\pi$
$443$			11695.6i			1.25435i	−0.778880	+	0.627173i	$0.784212\pi$
$444$	2748.11			0.293737
$445$		0			0
$446$	7157.35			0.759889
$447$	8098.55			0.856932
$448$	8402.52			0.886120
$449$			5124.39i			0.538608i	0.963055	+	0.269304i	$0.0867937\pi$
$449$			5124.39i			0.538608i	−0.963055	+	0.269304i	$0.913206\pi$
$450$		0			0
$451$	−5702.26			−0.595363
$452$	−4058.99			−0.422387
$453$	669.593			0.0694486
$454$		−	1270.97i		−	0.131387i
$455$		0			0
$456$			3063.10i			0.314568i
$457$			6346.71i			0.649643i	0.945775	+	0.324821i	$0.105304\pi$
$457$			6346.71i			0.649643i	−0.945775	+	0.324821i	$0.894696\pi$
$458$			9558.99i			0.975245i
$459$	−2454.78	−	10381.9i	−0.249628	−	1.05574i
$460$		0			0
$461$	−4564.08			−0.461107			−0.230553	−	0.973060i	$-0.574054\pi$
$461$	−4564.08			−0.461107			−0.230553	+	0.973060i	$0.574054\pi$
$462$	4921.26			0.495580
$463$			13669.2i			1.37206i	0.727575	+	0.686028i	$0.240647\pi$
$463$			13669.2i			1.37206i	−0.727575	+	0.686028i	$0.759353\pi$
$464$			5360.43i			0.536318i
$465$		0			0
$466$		−	3358.42i		−	0.333853i
$467$			7042.76i			0.697859i	0.937149	+	0.348929i	$0.113455\pi$
$467$			7042.76i			0.697859i	−0.937149	+	0.348929i	$0.886545\pi$
$468$		−	89.4387i		−	0.00883398i
$469$			3655.67i			0.359921i
$470$		0			0
$471$			6481.05i			0.634037i
$472$		−	2660.38i		−	0.259436i
$473$	−10839.4			−1.05369
$474$	7916.08			0.767084
$475$		0			0
$476$	−573.653	−	2426.13i	−0.0552381	−	0.233616i
$477$		−	2953.15i		−	0.283471i
$478$		−	8502.69i		−	0.813607i
$479$		−	6350.78i		−	0.605793i	−0.953024	−	0.302896i	$-0.902046\pi$
$479$		−	6350.78i		−	0.605793i	0.953024	−	0.302896i	$-0.0979536\pi$
$480$		0			0
$481$			1358.01i			0.128732i
$482$	−14335.5			−1.35470
$483$	11144.7			1.04990
$484$	859.835			0.0807508
$485$		0			0
$486$			4671.75i			0.436038i
$487$	−10100.7			−0.939849			−0.469925	−	0.882706i	$-0.655719\pi$
$487$	−10100.7			−0.939849			−0.469925	+	0.882706i	$0.655719\pi$
$488$	3043.18			0.282292
$489$	11439.9			1.05793
$490$		0			0
$491$	11896.0			1.09340			0.546699	−	0.837329i	$-0.315884\pi$
$491$	11896.0			1.09340			0.546699	+	0.837329i	$0.315884\pi$
$492$			1932.48i			0.177079i
$493$	9281.74	−	2194.65i	0.847928	−	0.200491i
$494$	−346.197			−0.0315307
$495$		0			0
$496$			1991.78i			0.180309i
$497$		−	640.311i		−	0.0577904i
$498$	−4985.87			−0.448639
$499$		−	19328.3i		−	1.73398i	−0.498328	−	0.866989i	$-0.666052\pi$
$499$		−	19328.3i		−	1.73398i	0.498328	−	0.866989i	$-0.333948\pi$
$500$		0			0
$501$	−6951.13			−0.619868
$502$			13762.4i			1.22360i
$503$	−20998.4			−1.86138			−0.930688	−	0.365815i	$-0.880790\pi$
$503$	−20998.4			−1.86138			−0.930688	+	0.365815i	$0.880790\pi$
$504$			2668.62i			0.235853i
$505$		0			0
$506$	12343.7			1.08448
$507$	9646.94			0.845041
$508$		−	2684.13i		−	0.234427i
$509$	7644.91			0.665727			0.332863	−	0.942975i	$-0.391985\pi$
$509$	7644.91			0.665727			0.332863	+	0.942975i	$0.391985\pi$
$510$		0			0
$511$	−13121.4			−1.13592
$512$		−	11827.7i		−	1.02093i
$513$	−4261.62			−0.366774
$514$	6796.81			0.583257
$515$		0			0
$516$			3673.45i			0.313400i
$517$	−9909.84			−0.843006
$518$		−	9267.39i		−	0.786073i
$519$	−1515.40			−0.128167
$520$		0			0
$521$			19875.7i			1.67134i	0.549229	+	0.835672i	$0.314922\pi$
$521$			19875.7i			1.67134i	−0.549229	+	0.835672i	$0.685078\pi$
$522$	−2335.04			−0.195789
$523$		−	12054.6i		−	1.00786i	−0.863744	−	0.503930i	$-0.831887\pi$
$523$		−	12054.6i		−	1.00786i	0.863744	−	0.503930i	$-0.168113\pi$
$524$		−	3618.71i		−	0.301687i
$525$		0			0
$526$	1285.93			0.106596
$527$	3448.82	−	815.468i	0.285072	−	0.0674048i
$528$		−	5450.72i		−	0.449266i
$529$	15786.7			1.29750
$530$		0			0
$531$	782.102			0.0639178
$532$	−995.889			−0.0811603
$533$	−954.960			−0.0776058
$534$		−	3969.51i		−	0.321681i
$535$		0			0
$536$	5999.58			0.483474
$537$	7445.68			0.598333
$538$	4174.86			0.334556
$539$		−	3678.94i		−	0.293994i
$540$		0			0
$541$		−	607.633i		−	0.0482887i	−0.999708	−	0.0241443i	$-0.992314\pi$
$541$		−	607.633i		−	0.0482887i	0.999708	−	0.0241443i	$-0.00768613\pi$
$542$		−	8532.39i		−	0.676195i
$543$		−	5240.70i		−	0.414181i
$544$	−7052.71	+	1667.60i	−0.555850	+	0.131430i
$545$		0			0
$546$	824.166			0.0645990
$547$	7671.62			0.599662			0.299831	−	0.953992i	$-0.403070\pi$
$547$	7671.62			0.599662			0.299831	+	0.953992i	$0.403070\pi$
$548$			1021.25i			0.0796092i
$549$			894.637i			0.0695486i
$550$		0			0
$551$		−	3810.02i		−	0.294578i
$552$		−	18290.4i		−	1.41031i
$553$			11253.0i			0.865327i
$554$		−	10768.8i		−	0.825853i
$555$		0			0
$556$			2547.67i			0.194326i
$557$			6613.01i			0.503056i	0.967850	+	0.251528i	$0.0809331\pi$
$557$			6613.01i			0.503056i	−0.967850	+	0.251528i	$0.919067\pi$
$558$	−867.634			−0.0658241
$559$	−1815.28			−0.137349
$560$		0			0
$561$	−9438.09	+	2231.62i	−0.710297	+	0.167948i
$562$			4361.49i			0.327363i
$563$		−	24867.1i		−	1.86150i	−0.365657	−	0.930750i	$-0.619156\pi$
$563$		−	24867.1i		−	1.86150i	0.365657	−	0.930750i	$-0.380844\pi$
$564$			3358.42i			0.250736i
$565$		0			0
$566$		−	9333.52i		−	0.693140i
$567$	7217.06			0.534547
$568$	−1050.86			−0.0776286
$569$	11933.7			0.879242			0.439621	−	0.898183i	$-0.355113\pi$
$569$	11933.7			0.879242			0.439621	+	0.898183i	$0.355113\pi$
$570$		0			0
$571$			8593.93i			0.629851i	0.949116	+	0.314925i	$0.101980\pi$
$571$			8593.93i			0.629851i	−0.949116	+	0.314925i	$0.898020\pi$
$572$	−384.792			−0.0281276
$573$	16390.4			1.19497
$574$	6516.86			0.473883
$575$		0			0
$576$	4053.99			0.293257
$577$		−	2580.53i		−	0.186185i	−0.995657	−	0.0930927i	$-0.970325\pi$
$577$		−	2580.53i		−	0.186185i	0.995657	−	0.0930927i	$-0.0296753\pi$
$578$	−5219.79	−	10420.8i	−0.375631	−	0.749911i
$579$	−6667.80			−0.478591
$580$		0			0
$581$		−	7087.59i		−	0.506097i
$582$			3197.52i			0.227735i
$583$	−12705.4			−0.902577
$584$			21534.4i			1.52586i
$585$		0			0
$586$	−11244.4			−0.792665
$587$		−	13533.0i		−	0.951560i	−0.879564	−	0.475780i	$-0.842166\pi$
$587$		−	13533.0i		−	0.951560i	0.879564	−	0.475780i	$-0.157834\pi$
$588$	−1246.78			−0.0874429
$589$		−	1415.69i		−	0.0990366i
$590$		0			0
$591$	19384.1			1.34917
$592$	−10264.4			−0.712611
$593$		−	26998.2i		−	1.86961i	−0.355156	−	0.934807i	$-0.615572\pi$
$593$		−	26998.2i		−	1.86961i	0.355156	−	0.934807i	$-0.384428\pi$
$594$	11236.8			0.776183
$595$		0			0
$596$	4321.26			0.296990
$597$			2104.20i			0.144253i
$598$	2067.21			0.141362
$599$	−26341.6			−1.79681			−0.898403	−	0.439172i	$-0.855272\pi$
$599$	−26341.6			−1.79681			−0.898403	+	0.439172i	$0.855272\pi$
$600$		0			0
$601$		−	4099.87i		−	0.278265i	−0.990274	−	0.139133i	$-0.955569\pi$
$601$		−	4099.87i		−	0.278265i	0.990274	−	0.139133i	$-0.0444314\pi$
$602$	12387.9			0.838694
$603$			1763.76i			0.119114i
$604$	357.285			0.0240690
$605$		0			0
$606$			15659.4i			1.04970i
$607$	−11101.4			−0.742328			−0.371164	−	0.928567i	$-0.621041\pi$
$607$	−11101.4			−0.742328			−0.371164	+	0.928567i	$0.621041\pi$
$608$			2895.03i			0.193107i
$609$			9070.23i			0.603521i
$610$		0			0
$611$	−1659.61			−0.109886
$612$	−276.772	−	1170.54i	−0.0182808	−	0.0773142i
$613$		−	5421.00i		−	0.357182i	−0.983923	−	0.178591i	$-0.942846\pi$
$613$		−	5421.00i		−	0.357182i	0.983923	−	0.178591i	$-0.0571538\pi$
$614$	11780.3			0.774291
$615$		0			0
$616$	11481.2			0.750960
$617$	−7246.58			−0.472830			−0.236415	−	0.971652i	$-0.575973\pi$
$617$	−7246.58			−0.472830			−0.236415	+	0.971652i	$0.575973\pi$
$618$	−1771.90			−0.115334
$619$			18816.8i			1.22183i	0.791697	+	0.610914i	$0.209198\pi$
$619$			18816.8i			1.22183i	−0.791697	+	0.610914i	$0.790802\pi$
$620$		0			0
$621$	25447.0			1.64437
$622$	−14326.9			−0.923561
$623$	5642.80			0.362880
$624$		−	912.836i		−	0.0585620i
$625$		0			0
$626$		−	15409.6i		−	0.983850i
$627$			3874.20i			0.246763i
$628$			3458.19i			0.219740i
$629$	4202.44	+	17773.2i	0.266394	+	1.12665i
$630$		0			0
$631$	−17690.3			−1.11607			−0.558036	−	0.829817i	$-0.688445\pi$
$631$	−17690.3			−1.11607			−0.558036	+	0.829817i	$0.688445\pi$
$632$	18468.1			1.16237
$633$			18640.8i			1.17047i
$634$			7204.55i			0.451308i
$635$		0			0
$636$			4305.81i			0.268454i
$637$		−	616.113i		−	0.0383223i
$638$			10046.1i			0.623397i
$639$		−	308.933i		−	0.0191255i
$640$		0			0
$641$		−	8132.02i		−	0.501085i	−0.968106	−	0.250543i	$-0.919391\pi$
$641$		−	8132.02i		−	0.501085i	0.968106	−	0.250543i	$-0.0806090\pi$
$642$			11565.1i			0.710960i
$643$	−2653.40			−0.162737			−0.0813684	−	0.996684i	$-0.525929\pi$
$643$	−2653.40			−0.162737			−0.0813684	+	0.996684i	$0.525929\pi$
$644$	5946.65			0.363868
$645$		0			0
$646$	−4530.90	+	1071.32i	−0.275953	+	0.0652487i
$647$		−	12989.6i		−	0.789297i	−0.918832	−	0.394649i	$-0.870866\pi$
$647$		−	12989.6i		−	0.789297i	0.918832	−	0.394649i	$-0.129134\pi$
$648$		−	11844.4i		−	0.718045i
$649$		−	3364.84i		−	0.203515i
$650$		0			0
$651$			3370.23i			0.202903i
$652$	6104.13			0.366651
$653$	20945.7			1.25524			0.627618	−	0.778521i	$-0.284030\pi$
$653$	20945.7			1.25524			0.627618	+	0.778521i	$0.284030\pi$
$654$	8733.11			0.522158
$655$		0			0
$656$		−	7217.99i		−	0.429596i
$657$	−6330.72			−0.375928
$658$	11325.5			0.670995
$659$	5888.03			0.348050			0.174025	−	0.984741i	$-0.444323\pi$
$659$	5888.03			0.348050			0.174025	+	0.984741i	$0.444323\pi$
$660$		0			0
$661$	−7987.91			−0.470036			−0.235018	−	0.971991i	$-0.575515\pi$
$661$	−7987.91			−0.470036			−0.235018	+	0.971991i	$0.575515\pi$
$662$		−	18431.5i		−	1.08211i
$663$	−1580.60	+	373.730i	−0.0925875	+	0.0218921i
$664$	−11631.9			−0.679830
$665$		0			0
$666$		−	4471.27i		−	0.260147i
$667$			22750.4i			1.32069i
$668$	−3709.02			−0.214830
$669$		−	13413.7i		−	0.775193i
$670$		0			0
$671$	3849.00			0.221444
$672$		−	6891.99i		−	0.395632i
$673$	16.7490			0.000959325			0.000479662	−	1.00000i	$-0.499847\pi$
$673$	16.7490			0.000959325			0.000479662		1.00000i	$0.499847\pi$
$674$		−	14360.5i		−	0.820694i
$675$		0			0
$676$	5147.46			0.292869
$677$	28456.3			1.61546			0.807728	−	0.589556i	$-0.200697\pi$
$677$	28456.3			1.61546			0.807728	+	0.589556i	$0.200697\pi$
$678$		−	18046.0i		−	1.02220i
$679$	−4545.39			−0.256902
$680$		0			0
$681$	−2381.95			−0.134033
$682$			3732.82i			0.209585i
$683$	8066.90			0.451935			0.225967	−	0.974135i	$-0.427446\pi$
$683$	8066.90			0.451935			0.225967	+	0.974135i	$0.427446\pi$
$684$	−480.490			−0.0268596
$685$		0			0
$686$			16404.1i			0.912992i
$687$	17914.7			0.994887
$688$		−	13720.7i		−	0.760314i
$689$	−2127.77			−0.117651
$690$		0			0
$691$		−	22083.7i		−	1.21578i	−0.794021	−	0.607890i	$-0.792016\pi$
$691$		−	22083.7i		−	1.21578i	0.794021	−	0.607890i	$-0.207984\pi$
$692$	−808.593			−0.0444192
$693$			3375.26i			0.185015i
$694$			798.078i			0.0436522i
$695$		0			0
$696$	14885.8			0.810697
$697$	−12498.2	+	2955.17i	−0.679199	+	0.160595i
$698$		−	19832.4i		−	1.07545i
$699$	−6294.07			−0.340577
$700$		0			0
$701$	276.484			0.0148968			0.00744839	−	0.999972i	$-0.497629\pi$
$701$	276.484			0.0148968			0.00744839	+	0.999972i	$0.497629\pi$
$702$	1881.84			0.101176
$703$	7295.63			0.391408
$704$		−	17441.5i		−	0.933737i
$705$		0			0
$706$	4839.55			0.257987
$707$	−22260.3			−1.18414
$708$	−1140.34			−0.0605317
$709$		−	24512.7i		−	1.29844i	−0.760601	−	0.649219i	$-0.775096\pi$
$709$		−	24512.7i		−	1.29844i	0.760601	−	0.649219i	$-0.224904\pi$
$710$		0			0
$711$			5429.27i			0.286376i
$712$		−	9260.81i		−	0.487449i
$713$			8453.37i			0.444013i
$714$	10786.4	−	2550.42i	0.565365	−	0.133679i
$715$		0			0
$716$	3972.90			0.207366
$717$	−15935.0			−0.829993
$718$			1409.98i			0.0732870i
$719$			10555.8i			0.547518i	0.961798	+	0.273759i	$0.0882670\pi$
$719$			10555.8i			0.547518i	−0.961798	+	0.273759i	$0.911733\pi$
$720$		0			0
$721$		−	2518.81i		−	0.130105i
$722$		−	14411.6i		−	0.742859i
$723$			26866.4i			1.38198i
$724$		−	2796.36i		−	0.143544i
$725$		0			0
$726$			3822.77i			0.195422i
$727$			26611.2i			1.35757i	0.734337	+	0.678785i	$0.237493\pi$
$727$			26611.2i			1.35757i	−0.734337	+	0.678785i	$0.762507\pi$
$728$	1922.77			0.0978880
$729$	21752.2			1.10513
$730$		0			0
$731$	−23757.8	+	5617.48i	−1.20207	+	0.284227i
$732$		−	1304.42i		−	0.0658642i
$733$		−	16288.7i		−	0.820787i	−0.911909	−	0.410393i	$-0.865391\pi$
$733$		−	16288.7i		−	0.820787i	0.911909	−	0.410393i	$-0.134609\pi$
$734$			27416.7i			1.37871i
$735$		0			0
$736$		−	17286.8i		−	0.865762i
$737$	7588.24			0.379262
$738$	3144.21			0.156829
$739$	7901.82			0.393333			0.196667	−	0.980470i	$-0.436988\pi$
$739$	7901.82			0.393333			0.196667	+	0.980470i	$0.436988\pi$
$740$		0			0
$741$			648.814i			0.0321657i
$742$	14520.4			0.718411
$743$	−28240.7			−1.39441			−0.697207	−	0.716870i	$-0.745574\pi$
$743$	−28240.7			−1.39441			−0.697207	+	0.716870i	$0.745574\pi$
$744$	5531.13			0.272555
$745$		0			0
$746$	366.222			0.0179736
$747$		−	3419.57i		−	0.167491i
$748$	−5036.02	+	1190.76i	−0.246170	+	0.0582064i
$749$	−16440.1			−0.802016
$750$		0			0
$751$			520.691i			0.0253000i	0.999920	+	0.0126500i	$0.00402673\pi$
$751$			520.691i			0.0253000i	−0.999920	+	0.0126500i	$0.995973\pi$
$752$		−	12544.0i		−	0.608288i
$753$	25792.3			1.24824
$754$			1682.42i			0.0812601i
$755$		0			0
$756$	5413.39			0.260427
$757$			26471.9i			1.27099i	0.772105	+	0.635495i	$0.219204\pi$
$757$			26471.9i			1.27099i	−0.772105	+	0.635495i	$0.780796\pi$
$758$	21292.4			1.02029
$759$		−	23133.6i		−	1.10632i
$760$		0			0
$761$	−8204.72			−0.390829			−0.195415	−	0.980721i	$-0.562605\pi$
$761$	−8204.72			−0.390829			−0.195415	+	0.980721i	$0.562605\pi$
$762$	11933.5			0.567328
$763$			12414.4i			0.589033i
$764$	8745.65			0.414145
$765$		0			0
$766$	−33289.6			−1.57024
$767$		−	563.512i		−	0.0265283i
$768$	−14634.9			−0.687617
$769$	930.048			0.0436130			0.0218065	−	0.999762i	$-0.493058\pi$
$769$	930.048			0.0436130			0.0218065	+	0.999762i	$0.493058\pi$
$770$		0			0
$771$		−	12738.0i		−	0.595004i
$772$	−3557.83			−0.165867
$773$		−	24346.7i		−	1.13285i	−0.824114	−	0.566423i	$-0.808327\pi$
$773$		−	24346.7i		−	1.13285i	0.824114	−	0.566423i	$-0.191673\pi$
$774$	5976.83			0.277562
$775$		0			0
$776$			7459.77i			0.345090i
$777$	−17368.2			−0.801904
$778$			27120.3i			1.24975i
$779$			5130.31i			0.235960i
$780$		0			0
$781$	−1329.12			−0.0608959
$782$	27054.9	−	6397.09i	1.23719	−	0.292531i
$783$			20710.3i			0.945242i
$784$	4656.84			0.212138
$785$		0			0
$786$	16088.5			0.730101
$787$	−3683.89			−0.166857			−0.0834286	−	0.996514i	$-0.526587\pi$
$787$	−3683.89			−0.166857			−0.0834286	+	0.996514i	$0.526587\pi$
$788$	10343.1			0.467585
$789$		−	2409.99i		−	0.108742i
$790$		0			0
$791$	25653.0			1.15312
$792$	5539.38			0.248527
$793$	644.594			0.0288653
$794$			2992.85i			0.133769i
$795$		0			0
$796$			1122.77i			0.0499943i
$797$			10785.2i			0.479336i	0.970855	+	0.239668i	$0.0770385\pi$
$797$			10785.2i			0.479336i	−0.970855	+	0.239668i	$0.922961\pi$
$798$		−	4427.65i		−	0.196413i
$799$	−21720.3	+	5135.73i	−0.961714	+	0.227396i
$800$		0			0
$801$	2722.50			0.120093
$802$	−33962.5			−1.49534
$803$			27236.7i			1.19696i
$804$		−	2571.63i		−	0.112804i
$805$		0			0
$806$			625.138i			0.0273195i
$807$		−	7824.18i		−	0.341294i
$808$			36533.0i			1.59063i
$809$			5287.10i			0.229771i	0.993379	+	0.114885i	$0.0366501\pi$
$809$			5287.10i			0.229771i	−0.993379	+	0.114885i	$0.963350\pi$
$810$		0			0
$811$			34185.5i			1.48017i	0.672515	+	0.740084i	$0.265214\pi$
$811$			34185.5i			1.48017i	−0.672515	+	0.740084i	$0.734786\pi$
$812$			4839.74i			0.209164i
$813$	−15990.7			−0.689814
$814$	−19236.8			−0.828314
$815$		0			0
$816$	−2824.81	−	11946.9i	−0.121187	−	0.512529i
$817$			9752.21i			0.417609i
$818$		−	16177.3i		−	0.691473i
$819$			565.257i			0.0241168i
$820$		0			0
$821$			13283.1i			0.564655i	0.959318	+	0.282327i	$0.0911065\pi$
$821$			13283.1i			0.564655i	−0.959318	+	0.282327i	$0.908894\pi$
$822$	−4540.43			−0.192659
$823$	−37925.7			−1.60633			−0.803164	−	0.595759i	$-0.796851\pi$
$823$	−37925.7			−1.60633			−0.803164	+	0.595759i	$0.796851\pi$
$824$	−4133.80			−0.174767
$825$		0			0
$826$			3845.53i			0.161989i
$827$	−29268.3			−1.23066			−0.615331	−	0.788269i	$-0.710978\pi$
$827$	−29268.3			−1.23066			−0.615331	+	0.788269i	$0.710978\pi$
$828$	2869.10			0.120420
$829$	13301.0			0.557252			0.278626	−	0.960400i	$-0.410121\pi$
$829$	13301.0			0.557252			0.278626	+	0.960400i	$0.410121\pi$
$830$		0			0
$831$	−20182.0			−0.842486
$832$		−	2920.94i		−	0.121713i
$833$	−1906.59	−	8063.47i	−0.0793031	−	0.335393i
$834$	−11326.8			−0.470281
$835$		0			0
$836$			2067.21i			0.0855216i
$837$			7695.32i			0.317789i
$838$	−28437.6			−1.17227
$839$			10311.1i			0.424288i	0.977238	+	0.212144i	$0.0680446\pi$
$839$			10311.1i			0.424288i	−0.977238	+	0.212144i	$0.931955\pi$
$840$		0			0
$841$	5873.39			0.240821
$842$			19707.8i			0.806623i
$843$	8173.94			0.333956
$844$			9946.45i			0.405653i
$845$		0			0
$846$	5464.26			0.222063
$847$	−5434.20			−0.220450
$848$		−	16082.6i		−	0.651272i
$849$	−17492.1			−0.707100
$850$		0			0
$851$	−43563.7			−1.75481
$852$			450.436i			0.0181123i
$853$	−2240.96			−0.0899518			−0.0449759	−	0.998988i	$-0.514321\pi$
$853$	−2240.96			−0.0899518			−0.0449759	+	0.998988i	$0.514321\pi$
$854$	−4398.86			−0.176260
$855$		0			0
$856$			26981.1i			1.07733i
$857$	−31870.7			−1.27034			−0.635171	−	0.772371i	$-0.719070\pi$
$857$	−31870.7			−1.27034			−0.635171	+	0.772371i	$0.719070\pi$
$858$		−	1710.76i		−	0.0680704i
$859$	−36917.9			−1.46638			−0.733191	−	0.680022i	$-0.761970\pi$
$859$	−36917.9			−1.46638			−0.733191	+	0.680022i	$0.761970\pi$
$860$		0			0
$861$		−	12213.4i		−	0.483427i
$862$	34593.1			1.36687
$863$			45364.1i			1.78936i	0.446712	+	0.894678i	$0.352595\pi$
$863$			45364.1i			1.78936i	−0.446712	+	0.894678i	$0.647405\pi$
$864$		−	15736.6i		−	0.619643i
$865$		0			0
$866$	33345.6			1.30846
$867$	−19529.8	+	9782.50i	−0.765014	+	0.383196i
$868$			1798.30i			0.0703208i
$869$	23358.4			0.911827
$870$		0			0
$871$	1270.81			0.0494370
$872$	20374.2			0.791235
$873$	−2193.03			−0.0850204
$874$		−	11105.6i		−	0.429810i
$875$		0			0
$876$	9230.44			0.356013
$877$	37081.4			1.42776			0.713882	−	0.700266i	$-0.246935\pi$
$877$	37081.4			1.42776			0.713882	+	0.700266i	$0.246935\pi$
$878$	−27011.5			−1.03826
$879$			21073.3i			0.808630i
$880$		0			0
$881$		−	7501.20i		−	0.286858i	−0.989661	−	0.143429i	$-0.954187\pi$
$881$		−	7501.20i		−	0.286858i	0.989661	−	0.143429i	$-0.0458129\pi$
$882$			2028.56i			0.0774434i
$883$			24875.8i			0.948059i	0.880509	+	0.474029i	$0.157201\pi$
$883$			24875.8i			0.948059i	−0.880509	+	0.474029i	$0.842799\pi$
$884$	−843.385	+	199.417i	−0.0320884	+	0.00758723i
$885$		0			0
$886$	−27745.3			−1.05205
$887$	−6729.37			−0.254735			−0.127368	−	0.991856i	$-0.540653\pi$
$887$	−6729.37			−0.254735			−0.127368	+	0.991856i	$0.540653\pi$
$888$			28504.1i			1.07718i
$889$			16963.8i			0.639988i
$890$		0			0
$891$		−	14980.8i		−	0.563272i
$892$		−	7157.35i		−	0.268661i
$893$			8915.87i			0.334108i
$894$			19212.0i			0.718733i
$895$		0			0
$896$			7531.70i			0.280822i
$897$		−	3874.20i		−	0.144209i
$898$	−12156.5			−0.451746
$899$	−6879.86			−0.255235
$900$		0			0
$901$	−27847.5	+	6584.50i	−1.02967	+	0.243464i
$902$		−	13527.4i		−	0.499348i
$903$		−	23216.4i		−	0.855585i
$904$		−	42101.0i		−	1.54896i
$905$		0			0
$906$			1588.46i			0.0582485i
$907$	−25375.5			−0.928976			−0.464488	−	0.885580i	$-0.653762\pi$
$907$	−25375.5			−0.928976			−0.464488	+	0.885580i	$0.653762\pi$
$908$	−1270.97			−0.0464522
$909$	−10740.0			−0.391885
$910$		0			0
$911$		−	34073.4i		−	1.23919i	−0.784922	−	0.619595i	$-0.787297\pi$
$911$		−	34073.4i		−	1.23919i	0.784922	−	0.619595i	$-0.212703\pi$
$912$	−4904.01			−0.178057
$913$	−14712.0			−0.533294
$914$	−15056.2			−0.544873
$915$		0			0
$916$	9558.99			0.344801
$917$			22870.4i			0.823607i
$918$	24628.8	−	5823.43i	0.885481	−	0.209370i
$919$	−2927.25			−0.105072			−0.0525360	−	0.998619i	$-0.516730\pi$
$919$	−2927.25			−0.105072			−0.0525360	+	0.998619i	$0.516730\pi$
$920$		0			0
$921$		−	22077.7i		−	0.789885i
$922$		−	10827.3i		−	0.386743i
$923$	−222.589			−0.00793781
$924$		−	4921.26i		−	0.175214i
$925$		0			0
$926$	−32427.2			−1.15078
$927$		−	1215.26i		−	0.0430576i
$928$	14069.0			0.497671
$929$		−	10239.9i		−	0.361637i	−0.983516	−	0.180819i	$-0.942125\pi$
$929$		−	10239.9i		−	0.361637i	0.983516	−	0.180819i	$-0.0578747\pi$
$930$		0			0
$931$	−3309.93			−0.116518
$932$	−3358.42			−0.118035
$933$			26850.2i			0.942162i
$934$	−16707.4			−0.585314
$935$		0			0
$936$	927.683			0.0323956
$937$		−	23319.4i		−	0.813031i	−0.913644	−	0.406516i	$-0.866744\pi$
$937$		−	23319.4i		−	0.813031i	0.913644	−	0.406516i	$-0.133256\pi$
$938$	−8672.27			−0.301876
$939$	−28879.3			−1.00366
$940$		0			0
$941$			11650.7i			0.403617i	0.979425	+	0.201808i	$0.0646818\pi$
$941$			11650.7i			0.403617i	−0.979425	+	0.201808i	$0.935318\pi$
$942$	−15374.9			−0.531784
$943$		−	30634.1i		−	1.05788i
$944$	4259.26			0.146851
$945$		0			0
$946$		−	25714.2i		−	0.883762i
$947$	37562.0			1.28891			0.644456	−	0.764641i	$-0.277084\pi$
$947$	37562.0			1.28891			0.644456	+	0.764641i	$0.277084\pi$
$948$		−	7916.08i		−	0.271205i
$949$			4561.34i			0.156025i
$950$		0			0
$951$	13502.2			0.460397
$952$	25164.5	−	5950.09i	0.856707	−	0.202567i
$953$		−	23737.8i		−	0.806864i	−0.915010	−	0.403432i	$-0.867817\pi$
$953$		−	23737.8i		−	0.806864i	0.915010	−	0.403432i	$-0.132183\pi$
$954$	7005.71			0.237755
$955$		0			0
$956$	−8502.69			−0.287654
$957$	18827.5			0.635953
$958$	15065.8			0.508095
$959$		−	6454.38i		−	0.217333i
$960$		0			0
$961$	27234.6			0.914190
$962$	−3221.59			−0.107971
$963$	−7931.93			−0.265423
$964$			14335.5i			0.478958i
$965$		0			0
$966$			26438.4i			0.880581i
$967$		−	5987.71i		−	0.199123i	−0.995031	−	0.0995614i	$-0.968256\pi$
$967$		−	5987.71i		−	0.199123i	0.995031	−	0.0995614i	$-0.0317440\pi$
$968$			8918.45i			0.296126i
$969$	2007.79	+	8491.44i	0.0665628	+	0.281511i
$970$		0			0
$971$	−49295.6			−1.62922			−0.814609	−	0.580011i	$-0.803048\pi$
$971$	−49295.6			−1.62922			−0.814609	+	0.580011i	$0.803048\pi$
$972$	4671.75			0.154163
$973$		−	16101.4i		−	0.530512i
$974$		−	23961.7i		−	0.788278i
$975$		0			0
$976$			4872.12i			0.159788i
$977$		−	14932.5i		−	0.488979i	−0.969652	−	0.244490i	$-0.921380\pi$
$977$		−	14932.5i		−	0.488979i	0.969652	−	0.244490i	$-0.0786204\pi$
$978$			27138.6i			0.887316i
$979$		−	11713.0i		−	0.382380i
$980$		0			0
$981$			5989.63i			0.194938i
$982$			28220.6i			0.917064i
$983$	10738.5			0.348427			0.174213	−	0.984708i	$-0.444262\pi$
$983$	10738.5			0.348427			0.174213	+	0.984708i	$0.444262\pi$
$984$	−20044.2			−0.649376
$985$		0			0
$986$	5206.33	+	22018.9i	0.168157	+	0.711181i
$987$		−	21225.4i		−	0.684509i
$988$			346.197i			0.0111478i
$989$		−	58232.4i		−	1.87228i
$990$		0			0
$991$			27948.6i			0.895878i	0.894064	+	0.447939i	$0.147842\pi$
$991$			27948.6i			0.895878i	−0.894064	+	0.447939i	$0.852158\pi$
$992$	5227.64			0.167316
$993$	−34542.7			−1.10391
$994$	1519.00			0.0484705
$995$		0			0
$996$			4985.87i			0.158618i
$997$	28711.8			0.912049			0.456025	−	0.889967i	$-0.349273\pi$
$997$	28711.8			0.912049			0.456025	+	0.889967i	$0.349273\pi$
$998$	45852.2			1.45434
$999$	−39657.1			−1.25595

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	425.4.c.c.424.6		8
5.2	odd	4		425.4.d.c.101.1		4
5.3	odd	4		17.4.b.a.16.4	yes	4
5.4	even	2	inner	425.4.c.c.424.3		8
15.8	even	4		153.4.d.b.118.2		4
17.16	even	2	inner	425.4.c.c.424.5		8
20.3	even	4		272.4.b.d.33.2		4
85.13	odd	4		289.4.a.e.1.1		4
85.33	odd	4		17.4.b.a.16.3	✓	4
85.38	odd	4		289.4.a.e.1.2		4
85.67	odd	4		425.4.d.c.101.2		4
85.84	even	2	inner	425.4.c.c.424.4		8
255.203	even	4		153.4.d.b.118.1		4
340.203	even	4		272.4.b.d.33.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
17.4.b.a.16.3	✓	4	85.33	odd	4
17.4.b.a.16.4	yes	4	5.3	odd	4
153.4.d.b.118.1		4	255.203	even	4
153.4.d.b.118.2		4	15.8	even	4
272.4.b.d.33.2		4	20.3	even	4
272.4.b.d.33.3		4	340.203	even	4
289.4.a.e.1.1		4	85.13	odd	4
289.4.a.e.1.2		4	85.38	odd	4
425.4.c.c.424.3		8	5.4	even	2	inner
425.4.c.c.424.4		8	85.84	even	2	inner
425.4.c.c.424.5		8	17.16	even	2	inner
425.4.c.c.424.6		8	1.1	even	1	trivial
425.4.d.c.101.1		4	5.2	odd	4
425.4.d.c.101.2		4	85.67	odd	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:	\( N \)	\(=\)	\( 425 = 5^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	425.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+2.37228i q^{2} +4.44593 q^{3} +2.37228 q^{4} +10.5470i q^{6} -14.9929 q^{7} +24.6060i q^{8} -7.23369 q^{9} +O(q^{10})\)	\(q+2.37228i q^{2} +4.44593 q^{3} +2.37228 q^{4} +10.5470i q^{6} -14.9929 q^{7} +24.6060i q^{8} -7.23369 q^{9} +31.1215i q^{11} +10.5470 q^{12} +5.21194i q^{13} -35.5675i q^{14} -39.3940 q^{16} +(16.1286 + 68.2119i) q^{17} -17.1603i q^{18} +28.0000 q^{19} -66.6576 q^{21} -73.8290 q^{22} -167.194 q^{23} +109.396i q^{24} -12.3642 q^{26} -152.201 q^{27} -35.5675 q^{28} -136.072i q^{29} -50.5604i q^{31} +103.394i q^{32} +138.364i q^{33} +(-161.818 + 38.2616i) q^{34} -17.1603 q^{36} +260.558 q^{37} +66.4239i q^{38} +23.1719i q^{39} +183.225i q^{41} -158.130i q^{42} +348.293i q^{43} +73.8290i q^{44} -396.630i q^{46} +318.424i q^{47} -175.143 q^{48} -118.212 q^{49} +(71.7066 + 303.266i) q^{51} +12.3642i q^{52} +408.250i q^{53} -361.063i q^{54} -368.916i q^{56} +124.486 q^{57} +322.801 q^{58} -108.119 q^{59} -123.677i q^{61} +119.943 q^{62} +108.454 q^{63} -560.432 q^{64} -328.239 q^{66} -243.826i q^{67} +(38.2616 + 161.818i) q^{68} -743.331 q^{69} +42.7075i q^{71} -177.992i q^{72} +875.172 q^{73} +618.117i q^{74} +66.4239 q^{76} -466.603i q^{77} -54.9703 q^{78} -750.553i q^{79} -481.364 q^{81} -434.662 q^{82} +472.728i q^{83} -158.130 q^{84} -826.250 q^{86} -604.967i q^{87} -765.775 q^{88} -376.364 q^{89} -78.1422i q^{91} -396.630 q^{92} -224.788i q^{93} -755.391 q^{94} +459.683i q^{96} +303.169 q^{97} -280.432i q^{98} -225.123i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 4 q^{4} + 80 q^{9}+O(q^{10}) \) 8 * q - 4 * q^4 + 80 * q^9	\( 8 q - 4 q^{4} + 80 q^{9} - 476 q^{16} + 224 q^{19} + 248 q^{21} - 1064 q^{26} - 812 q^{34} - 436 q^{36} - 624 q^{49} + 2320 q^{51} + 2352 q^{59} - 2852 q^{64} + 3808 q^{66} - 2408 q^{69} - 112 q^{76} - 4816 q^{81} - 2368 q^{84} - 4496 q^{86} - 3976 q^{89} - 896 q^{94}+O(q^{100}) \) 8 * q - 4 * q^4 + 80 * q^9 - 476 * q^16 + 224 * q^19 + 248 * q^21 - 1064 * q^26 - 812 * q^34 - 436 * q^36 - 624 * q^49 + 2320 * q^51 + 2352 * q^59 - 2852 * q^64 + 3808 * q^66 - 2408 * q^69 - 112 * q^76 - 4816 * q^81 - 2368 * q^84 - 4496 * q^86 - 3976 * q^89 - 896 * q^94

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