LMFDB - Embedded newform 2925.2.c.i.2224.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2925,2,Mod(2224,2925)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2925, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("2925.2224");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2925 = 3^{2} \cdot 5^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2925.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:	$23.3562425912$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 975)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2224.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	2925.2224
Dual form			2925.2.c.i.2224.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{2} +1.00000 q^{4} +3.00000i q^{7} +3.00000i q^{8} +O(q^{10})$	$q+1.00000i q^{2} +1.00000 q^{4} +3.00000i q^{7} +3.00000i q^{8} +1.00000 q^{11} +1.00000i q^{13} -3.00000 q^{14} -1.00000 q^{16} -5.00000i q^{17} +8.00000 q^{19} +1.00000i q^{22} -1.00000 q^{26} +3.00000i q^{28} +1.00000 q^{29} +3.00000 q^{31} +5.00000i q^{32} +5.00000 q^{34} +8.00000i q^{37} +8.00000i q^{38} +2.00000 q^{41} +8.00000i q^{43} +1.00000 q^{44} -11.0000i q^{47} -2.00000 q^{49} +1.00000i q^{52} +11.0000i q^{53} -9.00000 q^{56} +1.00000i q^{58} +5.00000 q^{59} +1.00000 q^{61} +3.00000i q^{62} -7.00000 q^{64} -3.00000i q^{67} -5.00000i q^{68} -16.0000 q^{71} -4.00000i q^{73} -8.00000 q^{74} +8.00000 q^{76} +3.00000i q^{77} -12.0000 q^{79} +2.00000i q^{82} +3.00000i q^{83} -8.00000 q^{86} +3.00000i q^{88} -3.00000 q^{91} +11.0000 q^{94} +2.00000i q^{97} -2.00000i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{4}+O(q^{10}) $ 2 * q + 2 * q^4	$ 2 q + 2 q^{4} + 2 q^{11} - 6 q^{14} - 2 q^{16} + 16 q^{19} - 2 q^{26} + 2 q^{29} + 6 q^{31} + 10 q^{34} + 4 q^{41} + 2 q^{44} - 4 q^{49} - 18 q^{56} + 10 q^{59} + 2 q^{61} - 14 q^{64} - 32 q^{71} - 16 q^{74} + 16 q^{76} - 24 q^{79} - 16 q^{86} - 6 q^{91} + 22 q^{94}+O(q^{100}) $ 2 * q + 2 * q^4 + 2 * q^11 - 6 * q^14 - 2 * q^16 + 16 * q^19 - 2 * q^26 + 2 * q^29 + 6 * q^31 + 10 * q^34 + 4 * q^41 + 2 * q^44 - 4 * q^49 - 18 * q^56 + 10 * q^59 + 2 * q^61 - 14 * q^64 - 32 * q^71 - 16 * q^74 + 16 * q^76 - 24 * q^79 - 16 * q^86 - 6 * q^91 + 22 * q^94

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2925\mathbb{Z}\right)^\times$.

$n$	$326$	$352$	$2251$
$\chi(n)$	$1$	$-1$	$1$

Coefficient data

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

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

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

$2$	1.00000i	0.707107i	0.935414	+	0.353553i	$0.115027\pi$
$2$	1.00000i	0.707107i	−0.935414	+	0.353553i	$0.884973\pi$
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	3.00000i	1.13389i	0.823754	+	0.566947i	$0.191875\pi$
$7$	3.00000i	1.13389i	−0.823754	+	0.566947i	$0.808125\pi$
$8$	3.00000i	1.06066i
$9$	0	0
$10$	0	0
$11$	1.00000	0.301511	0.150756	−	0.988571i	$-0.451829\pi$
$11$	1.00000	0.301511	0.150756	+	0.988571i	$0.451829\pi$
$12$	0	0
$13$	1.00000i	0.277350i
$13$	1.00000i	0.277350i
$14$	−3.00000	−0.801784
$15$	0	0
$16$	−1.00000	−0.250000
$17$	− 5.00000i	− 1.21268i	−0.795206	−	0.606339i	$-0.792637\pi$
$17$	− 5.00000i	− 1.21268i	0.795206	−	0.606339i	$-0.207363\pi$
$18$	0	0
$19$	8.00000	1.83533	0.917663	−	0.397360i	$-0.130073\pi$
$19$	8.00000	1.83533	0.917663	+	0.397360i	$0.130073\pi$
$20$	0	0
$21$	0	0
$22$	1.00000i	0.213201i
$23$	0	0	1.00000			$0$
$23$	0	0	−1.00000			$\pi$
$24$	0	0
$25$	0	0
$26$	−1.00000	−0.196116
$27$	0	0
$28$	3.00000i	0.566947i
$29$	1.00000	0.185695	0.0928477	−	0.995680i	$-0.470403\pi$
$29$	1.00000	0.185695	0.0928477	+	0.995680i	$0.470403\pi$
$30$	0	0
$31$	3.00000	0.538816	0.269408	−	0.963026i	$-0.413172\pi$
$31$	3.00000	0.538816	0.269408	+	0.963026i	$0.413172\pi$
$32$	5.00000i	0.883883i
$33$	0	0
$34$	5.00000	0.857493
$35$	0	0
$36$	0	0
$37$	8.00000i	1.31519i	0.753371	+	0.657596i	$0.228427\pi$
$37$	8.00000i	1.31519i	−0.753371	+	0.657596i	$0.771573\pi$
$38$	8.00000i	1.29777i
$39$	0	0
$40$	0	0
$41$	2.00000	0.312348	0.156174	−	0.987730i	$-0.450084\pi$
$41$	2.00000	0.312348	0.156174	+	0.987730i	$0.450084\pi$
$42$	0	0
$43$	8.00000i	1.21999i	0.792406	+	0.609994i	$0.208828\pi$
$43$	8.00000i	1.21999i	−0.792406	+	0.609994i	$0.791172\pi$
$44$	1.00000	0.150756
$45$	0	0
$46$	0	0
$47$	− 11.0000i	− 1.60451i	−0.596978	−	0.802257i	$-0.703632\pi$
$47$	− 11.0000i	− 1.60451i	0.596978	−	0.802257i	$-0.296368\pi$
$48$	0	0
$49$	−2.00000	−0.285714
$50$	0	0
$51$	0	0
$52$	1.00000i	0.138675i
$53$	11.0000i	1.51097i	0.655168	+	0.755483i	$0.272598\pi$
$53$	11.0000i	1.51097i	−0.655168	+	0.755483i	$0.727402\pi$
$54$	0	0
$55$	0	0
$56$	−9.00000	−1.20268
$57$	0	0
$58$	1.00000i	0.131306i
$59$	5.00000	0.650945	0.325472	−	0.945552i	$-0.394477\pi$
$59$	5.00000	0.650945	0.325472	+	0.945552i	$0.394477\pi$
$60$	0	0
$61$	1.00000	0.128037	0.0640184	−	0.997949i	$-0.479608\pi$
$61$	1.00000	0.128037	0.0640184	+	0.997949i	$0.479608\pi$
$62$	3.00000i	0.381000i
$63$	0	0
$64$	−7.00000	−0.875000
$65$	0	0
$66$	0	0
$67$	− 3.00000i	− 0.366508i	−0.983066	−	0.183254i	$-0.941337\pi$
$67$	− 3.00000i	− 0.366508i	0.983066	−	0.183254i	$-0.0586631\pi$
$68$	− 5.00000i	− 0.606339i
$69$	0	0
$70$	0	0
$71$	−16.0000	−1.89885	−0.949425	−	0.313993i	$-0.898333\pi$
$71$	−16.0000	−1.89885	−0.949425	+	0.313993i	$0.898333\pi$
$72$	0	0
$73$	− 4.00000i	− 0.468165i	−0.972217	−	0.234082i	$-0.924791\pi$
$73$	− 4.00000i	− 0.468165i	0.972217	−	0.234082i	$-0.0752085\pi$
$74$	−8.00000	−0.929981
$75$	0	0
$76$	8.00000	0.917663
$77$	3.00000i	0.341882i
$78$	0	0
$79$	−12.0000	−1.35011	−0.675053	−	0.737769i	$-0.735879\pi$
$79$	−12.0000	−1.35011	−0.675053	+	0.737769i	$0.735879\pi$
$80$	0	0
$81$	0	0
$82$	2.00000i	0.220863i
$83$	3.00000i	0.329293i	0.986353	+	0.164646i	$0.0526483\pi$
$83$	3.00000i	0.329293i	−0.986353	+	0.164646i	$0.947352\pi$
$84$	0	0
$85$	0	0
$86$	−8.00000	−0.862662
$87$	0	0
$88$	3.00000i	0.319801i
$89$	0	0		−	1.00000i	$-0.5\pi$
$89$	0	0			1.00000i	$0.5\pi$
$90$	0	0
$91$	−3.00000	−0.314485
$92$	0	0
$93$	0	0
$94$	11.0000	1.13456
$95$	0	0
$96$	0	0
$97$	2.00000i	0.203069i	0.994832	+	0.101535i	$0.0323753\pi$
$97$	2.00000i	0.203069i	−0.994832	+	0.101535i	$0.967625\pi$
$98$	− 2.00000i	− 0.202031i
$99$	0	0
$100$	0	0
$101$	−9.00000	−0.895533	−0.447767	−	0.894150i	$-0.647781\pi$
$101$	−9.00000	−0.895533	−0.447767	+	0.894150i	$0.647781\pi$
$102$	0	0
$103$	− 14.0000i	− 1.37946i	−0.724066	−	0.689730i	$-0.757729\pi$
$103$	− 14.0000i	− 1.37946i	0.724066	−	0.689730i	$-0.242271\pi$
$104$	−3.00000	−0.294174
$105$	0	0
$106$	−11.0000	−1.06841
$107$	18.0000i	1.74013i	0.492941	+	0.870063i	$0.335922\pi$
$107$	18.0000i	1.74013i	−0.492941	+	0.870063i	$0.664078\pi$
$108$	0	0
$109$	10.0000	0.957826	0.478913	−	0.877862i	$-0.341031\pi$
$109$	10.0000	0.957826	0.478913	+	0.877862i	$0.341031\pi$
$110$	0	0
$111$	0	0
$112$	− 3.00000i	− 0.283473i
$113$	2.00000i	0.188144i	0.995565	+	0.0940721i	$0.0299884\pi$
$113$	2.00000i	0.188144i	−0.995565	+	0.0940721i	$0.970012\pi$
$114$	0	0
$115$	0	0
$116$	1.00000	0.0928477
$117$	0	0
$118$	5.00000i	0.460287i
$119$	15.0000	1.37505
$120$	0	0
$121$	−10.0000	−0.909091
$122$	1.00000i	0.0905357i
$123$	0	0
$124$	3.00000	0.269408
$125$	0	0
$126$	0	0
$127$	− 8.00000i	− 0.709885i	−0.934888	−	0.354943i	$-0.884500\pi$
$127$	− 8.00000i	− 0.709885i	0.934888	−	0.354943i	$-0.115500\pi$
$128$	3.00000i	0.265165i
$129$	0	0
$130$	0	0
$131$	−6.00000	−0.524222	−0.262111	−	0.965038i	$-0.584419\pi$
$131$	−6.00000	−0.524222	−0.262111	+	0.965038i	$0.584419\pi$
$132$	0	0
$133$	24.0000i	2.08106i
$134$	3.00000	0.259161
$135$	0	0
$136$	15.0000	1.28624
$137$	− 12.0000i	− 1.02523i	−0.858619	−	0.512615i	$-0.828677\pi$
$137$	− 12.0000i	− 1.02523i	0.858619	−	0.512615i	$-0.171323\pi$
$138$	0	0
$139$	10.0000	0.848189	0.424094	−	0.905618i	$-0.360592\pi$
$139$	10.0000	0.848189	0.424094	+	0.905618i	$0.360592\pi$
$140$	0	0
$141$	0	0
$142$	− 16.0000i	− 1.34269i
$143$	1.00000i	0.0836242i
$144$	0	0
$145$	0	0
$146$	4.00000	0.331042
$147$	0	0
$148$	8.00000i	0.657596i
$149$	10.0000	0.819232	0.409616	−	0.912258i	$-0.365663\pi$
$149$	10.0000	0.819232	0.409616	+	0.912258i	$0.365663\pi$
$150$	0	0
$151$	−17.0000	−1.38344	−0.691720	−	0.722166i	$-0.743147\pi$
$151$	−17.0000	−1.38344	−0.691720	+	0.722166i	$0.743147\pi$
$152$	24.0000i	1.94666i
$153$	0	0
$154$	−3.00000	−0.241747
$155$	0	0
$156$	0	0
$157$	11.0000i	0.877896i	0.898513	+	0.438948i	$0.144649\pi$
$157$	11.0000i	0.877896i	−0.898513	+	0.438948i	$0.855351\pi$
$158$	− 12.0000i	− 0.954669i
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	− 20.0000i	− 1.56652i	−0.621694	−	0.783260i	$-0.713555\pi$
$163$	− 20.0000i	− 1.56652i	0.621694	−	0.783260i	$-0.286445\pi$
$164$	2.00000	0.156174
$165$	0	0
$166$	−3.00000	−0.232845
$167$	12.0000i	0.928588i	0.885681	+	0.464294i	$0.153692\pi$
$167$	12.0000i	0.928588i	−0.885681	+	0.464294i	$0.846308\pi$
$168$	0	0
$169$	−1.00000	−0.0769231
$170$	0	0
$171$	0	0
$172$	8.00000i	0.609994i
$173$	21.0000i	1.59660i	0.602260	+	0.798300i	$0.294267\pi$
$173$	21.0000i	1.59660i	−0.602260	+	0.798300i	$0.705733\pi$
$174$	0	0
$175$	0	0
$176$	−1.00000	−0.0753778
$177$	0	0
$178$	0	0
$179$	26.0000	1.94333	0.971666	−	0.236360i	$-0.0759544\pi$
$179$	26.0000	1.94333	0.971666	+	0.236360i	$0.0759544\pi$
$180$	0	0
$181$	5.00000	0.371647	0.185824	−	0.982583i	$-0.440505\pi$
$181$	5.00000	0.371647	0.185824	+	0.982583i	$0.440505\pi$
$182$	− 3.00000i	− 0.222375i
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 5.00000i	− 0.365636i
$188$	− 11.0000i	− 0.802257i
$189$	0	0
$190$	0	0
$191$	−10.0000	−0.723575	−0.361787	−	0.932261i	$-0.617833\pi$
$191$	−10.0000	−0.723575	−0.361787	+	0.932261i	$0.617833\pi$
$192$	0	0
$193$	10.0000i	0.719816i	0.932988	+	0.359908i	$0.117192\pi$
$193$	10.0000i	0.719816i	−0.932988	+	0.359908i	$0.882808\pi$
$194$	−2.00000	−0.143592
$195$	0	0
$196$	−2.00000	−0.142857
$197$	− 6.00000i	− 0.427482i	−0.976890	−	0.213741i	$-0.931435\pi$
$197$	− 6.00000i	− 0.427482i	0.976890	−	0.213741i	$-0.0685649\pi$
$198$	0	0
$199$	−10.0000	−0.708881	−0.354441	−	0.935079i	$-0.615329\pi$
$199$	−10.0000	−0.708881	−0.354441	+	0.935079i	$0.615329\pi$
$200$	0	0
$201$	0	0
$202$	− 9.00000i	− 0.633238i
$203$	3.00000i	0.210559i
$204$	0	0
$205$	0	0
$206$	14.0000	0.975426
$207$	0	0
$208$	− 1.00000i	− 0.0693375i
$209$	8.00000	0.553372
$210$	0	0
$211$	−16.0000	−1.10149	−0.550743	−	0.834675i	$-0.685655\pi$
$211$	−16.0000	−1.10149	−0.550743	+	0.834675i	$0.685655\pi$
$212$	11.0000i	0.755483i
$213$	0	0
$214$	−18.0000	−1.23045
$215$	0	0
$216$	0	0
$217$	9.00000i	0.610960i
$218$	10.0000i	0.677285i
$219$	0	0
$220$	0	0
$221$	5.00000	0.336336
$222$	0	0
$223$	− 8.00000i	− 0.535720i	−0.963458	−	0.267860i	$-0.913684\pi$
$223$	− 8.00000i	− 0.535720i	0.963458	−	0.267860i	$-0.0863164\pi$
$224$	−15.0000	−1.00223
$225$	0	0
$226$	−2.00000	−0.133038
$227$	23.0000i	1.52656i	0.646066	+	0.763282i	$0.276413\pi$
$227$	23.0000i	1.52656i	−0.646066	+	0.763282i	$0.723587\pi$
$228$	0	0
$229$	18.0000	1.18947	0.594737	−	0.803921i	$-0.297256\pi$
$229$	18.0000	1.18947	0.594737	+	0.803921i	$0.297256\pi$
$230$	0	0
$231$	0	0
$232$	3.00000i	0.196960i
$233$	6.00000i	0.393073i	0.980497	+	0.196537i	$0.0629694\pi$
$233$	6.00000i	0.393073i	−0.980497	+	0.196537i	$0.937031\pi$
$234$	0	0
$235$	0	0
$236$	5.00000	0.325472
$237$	0	0
$238$	15.0000i	0.972306i
$239$	−19.0000	−1.22901	−0.614504	−	0.788914i	$-0.710644\pi$
$239$	−19.0000	−1.22901	−0.614504	+	0.788914i	$0.710644\pi$
$240$	0	0
$241$	10.0000	0.644157	0.322078	−	0.946713i	$-0.395619\pi$
$241$	10.0000	0.644157	0.322078	+	0.946713i	$0.395619\pi$
$242$	− 10.0000i	− 0.642824i
$243$	0	0
$244$	1.00000	0.0640184
$245$	0	0
$246$	0	0
$247$	8.00000i	0.509028i
$248$	9.00000i	0.571501i
$249$	0	0
$250$	0	0
$251$	30.0000	1.89358	0.946792	−	0.321847i	$-0.104304\pi$
$251$	30.0000	1.89358	0.946792	+	0.321847i	$0.104304\pi$
$252$	0	0
$253$	0	0
$254$	8.00000	0.501965
$255$	0	0
$256$	−17.0000	−1.06250
$257$	− 3.00000i	− 0.187135i	−0.995613	−	0.0935674i	$-0.970173\pi$
$257$	− 3.00000i	− 0.187135i	0.995613	−	0.0935674i	$-0.0298271\pi$
$258$	0	0
$259$	−24.0000	−1.49129
$260$	0	0
$261$	0	0
$262$	− 6.00000i	− 0.370681i
$263$	− 14.0000i	− 0.863277i	−0.902047	−	0.431638i	$-0.857936\pi$
$263$	− 14.0000i	− 0.863277i	0.902047	−	0.431638i	$-0.142064\pi$
$264$	0	0
$265$	0	0
$266$	−24.0000	−1.47153
$267$	0	0
$268$	− 3.00000i	− 0.183254i
$269$	17.0000	1.03651	0.518254	−	0.855227i	$-0.326582\pi$
$269$	17.0000	1.03651	0.518254	+	0.855227i	$0.326582\pi$
$270$	0	0
$271$	−5.00000	−0.303728	−0.151864	−	0.988401i	$-0.548528\pi$
$271$	−5.00000	−0.303728	−0.151864	+	0.988401i	$0.548528\pi$
$272$	5.00000i	0.303170i
$273$	0	0
$274$	12.0000	0.724947
$275$	0	0
$276$	0	0
$277$	− 6.00000i	− 0.360505i	−0.983620	−	0.180253i	$-0.942309\pi$
$277$	− 6.00000i	− 0.360505i	0.983620	−	0.180253i	$-0.0576915\pi$
$278$	10.0000i	0.599760i
$279$	0	0
$280$	0	0
$281$	18.0000	1.07379	0.536895	−	0.843649i	$-0.319597\pi$
$281$	18.0000	1.07379	0.536895	+	0.843649i	$0.319597\pi$
$282$	0	0
$283$	− 18.0000i	− 1.06999i	−0.844856	−	0.534994i	$-0.820314\pi$
$283$	− 18.0000i	− 1.06999i	0.844856	−	0.534994i	$-0.179686\pi$
$284$	−16.0000	−0.949425
$285$	0	0
$286$	−1.00000	−0.0591312
$287$	6.00000i	0.354169i
$288$	0	0
$289$	−8.00000	−0.470588
$290$	0	0
$291$	0	0
$292$	− 4.00000i	− 0.234082i
$293$	− 30.0000i	− 1.75262i	−0.481749	−	0.876309i	$-0.659998\pi$
$293$	− 30.0000i	− 1.75262i	0.481749	−	0.876309i	$-0.340002\pi$
$294$	0	0
$295$	0	0
$296$	−24.0000	−1.39497
$297$	0	0
$298$	10.0000i	0.579284i
$299$	0	0
$300$	0	0
$301$	−24.0000	−1.38334
$302$	− 17.0000i	− 0.978240i
$303$	0	0
$304$	−8.00000	−0.458831
$305$	0	0
$306$	0	0
$307$	4.00000i	0.228292i	0.993464	+	0.114146i	$0.0364132\pi$
$307$	4.00000i	0.228292i	−0.993464	+	0.114146i	$0.963587\pi$
$308$	3.00000i	0.170941i
$309$	0	0
$310$	0	0
$311$	−6.00000	−0.340229	−0.170114	−	0.985424i	$-0.554414\pi$
$311$	−6.00000	−0.340229	−0.170114	+	0.985424i	$0.554414\pi$
$312$	0	0
$313$	− 29.0000i	− 1.63918i	−0.572953	−	0.819588i	$-0.694202\pi$
$313$	− 29.0000i	− 1.63918i	0.572953	−	0.819588i	$-0.305798\pi$
$314$	−11.0000	−0.620766
$315$	0	0
$316$	−12.0000	−0.675053
$317$	− 16.0000i	− 0.898650i	−0.893368	−	0.449325i	$-0.851665\pi$
$317$	− 16.0000i	− 0.898650i	0.893368	−	0.449325i	$-0.148335\pi$
$318$	0	0
$319$	1.00000	0.0559893
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 40.0000i	− 2.22566i
$324$	0	0
$325$	0	0
$326$	20.0000	1.10770
$327$	0	0
$328$	6.00000i	0.331295i
$329$	33.0000	1.81935
$330$	0	0
$331$	−12.0000	−0.659580	−0.329790	−	0.944054i	$-0.606978\pi$
$331$	−12.0000	−0.659580	−0.329790	+	0.944054i	$0.606978\pi$
$332$	3.00000i	0.164646i
$333$	0	0
$334$	−12.0000	−0.656611
$335$	0	0
$336$	0	0
$337$	− 5.00000i	− 0.272367i	−0.990684	−	0.136184i	$-0.956516\pi$
$337$	− 5.00000i	− 0.272367i	0.990684	−	0.136184i	$-0.0434837\pi$
$338$	− 1.00000i	− 0.0543928i
$339$	0	0
$340$	0	0
$341$	3.00000	0.162459
$342$	0	0
$343$	15.0000i	0.809924i
$344$	−24.0000	−1.29399
$345$	0	0
$346$	−21.0000	−1.12897
$347$	− 2.00000i	− 0.107366i	−0.998558	−	0.0536828i	$-0.982904\pi$
$347$	− 2.00000i	− 0.107366i	0.998558	−	0.0536828i	$-0.0170960\pi$
$348$	0	0
$349$	−16.0000	−0.856460	−0.428230	−	0.903670i	$-0.640863\pi$
$349$	−16.0000	−0.856460	−0.428230	+	0.903670i	$0.640863\pi$
$350$	0	0
$351$	0	0
$352$	5.00000i	0.266501i
$353$	− 18.0000i	− 0.958043i	−0.877803	−	0.479022i	$-0.840992\pi$
$353$	− 18.0000i	− 0.958043i	0.877803	−	0.479022i	$-0.159008\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	26.0000i	1.37414i
$359$	15.0000	0.791670	0.395835	−	0.918322i	$-0.370455\pi$
$359$	15.0000	0.791670	0.395835	+	0.918322i	$0.370455\pi$
$360$	0	0
$361$	45.0000	2.36842
$362$	5.00000i	0.262794i
$363$	0	0
$364$	−3.00000	−0.157243
$365$	0	0
$366$	0	0
$367$	12.0000i	0.626395i	0.949688	+	0.313197i	$0.101400\pi$
$367$	12.0000i	0.626395i	−0.949688	+	0.313197i	$0.898600\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−33.0000	−1.71327
$372$	0	0
$373$	− 31.0000i	− 1.60512i	−0.596572	−	0.802560i	$-0.703471\pi$
$373$	− 31.0000i	− 1.60512i	0.596572	−	0.802560i	$-0.296529\pi$
$374$	5.00000	0.258544
$375$	0	0
$376$	33.0000	1.70185
$377$	1.00000i	0.0515026i
$378$	0	0
$379$	35.0000	1.79783	0.898915	−	0.438124i	$-0.144357\pi$
$379$	35.0000	1.79783	0.898915	+	0.438124i	$0.144357\pi$
$380$	0	0
$381$	0	0
$382$	− 10.0000i	− 0.511645i
$383$	24.0000i	1.22634i	0.789950	+	0.613171i	$0.210106\pi$
$383$	24.0000i	1.22634i	−0.789950	+	0.613171i	$0.789894\pi$
$384$	0	0
$385$	0	0
$386$	−10.0000	−0.508987
$387$	0	0
$388$	2.00000i	0.101535i
$389$	−30.0000	−1.52106	−0.760530	−	0.649303i	$-0.775061\pi$
$389$	−30.0000	−1.52106	−0.760530	+	0.649303i	$0.775061\pi$
$390$	0	0
$391$	0	0
$392$	− 6.00000i	− 0.303046i
$393$	0	0
$394$	6.00000	0.302276
$395$	0	0
$396$	0	0
$397$	− 2.00000i	− 0.100377i	−0.998740	−	0.0501886i	$-0.984018\pi$
$397$	− 2.00000i	− 0.100377i	0.998740	−	0.0501886i	$-0.0159822\pi$
$398$	− 10.0000i	− 0.501255i
$399$	0	0
$400$	0	0
$401$	0	0		−	1.00000i	$-0.5\pi$
$401$	0	0			1.00000i	$0.5\pi$
$402$	0	0
$403$	3.00000i	0.149441i
$404$	−9.00000	−0.447767
$405$	0	0
$406$	−3.00000	−0.148888
$407$	8.00000i	0.396545i
$408$	0	0
$409$	−2.00000	−0.0988936	−0.0494468	−	0.998777i	$-0.515746\pi$
$409$	−2.00000	−0.0988936	−0.0494468	+	0.998777i	$0.515746\pi$
$410$	0	0
$411$	0	0
$412$	− 14.0000i	− 0.689730i
$413$	15.0000i	0.738102i
$414$	0	0
$415$	0	0
$416$	−5.00000	−0.245145
$417$	0	0
$418$	8.00000i	0.391293i
$419$	−26.0000	−1.27018	−0.635092	−	0.772437i	$-0.719038\pi$
$419$	−26.0000	−1.27018	−0.635092	+	0.772437i	$0.719038\pi$
$420$	0	0
$421$	28.0000	1.36464	0.682318	−	0.731055i	$-0.260972\pi$
$421$	28.0000	1.36464	0.682318	+	0.731055i	$0.260972\pi$
$422$	− 16.0000i	− 0.778868i
$423$	0	0
$424$	−33.0000	−1.60262
$425$	0	0
$426$	0	0
$427$	3.00000i	0.145180i
$428$	18.0000i	0.870063i
$429$	0	0
$430$	0	0
$431$	12.0000	0.578020	0.289010	−	0.957326i	$-0.406674\pi$
$431$	12.0000	0.578020	0.289010	+	0.957326i	$0.406674\pi$
$432$	0	0
$433$	− 34.0000i	− 1.63394i	−0.576683	−	0.816968i	$-0.695653\pi$
$433$	− 34.0000i	− 1.63394i	0.576683	−	0.816968i	$-0.304347\pi$
$434$	−9.00000	−0.432014
$435$	0	0
$436$	10.0000	0.478913
$437$	0	0
$438$	0	0
$439$	−4.00000	−0.190910	−0.0954548	−	0.995434i	$-0.530431\pi$
$439$	−4.00000	−0.190910	−0.0954548	+	0.995434i	$0.530431\pi$
$440$	0	0
$441$	0	0
$442$	5.00000i	0.237826i
$443$	0	0	1.00000			$0$
$443$	0	0	−1.00000			$\pi$
$444$	0	0
$445$	0	0
$446$	8.00000	0.378811
$447$	0	0
$448$	− 21.0000i	− 0.992157i
$449$	−28.0000	−1.32140	−0.660701	−	0.750649i	$-0.729741\pi$
$449$	−28.0000	−1.32140	−0.660701	+	0.750649i	$0.729741\pi$
$450$	0	0
$451$	2.00000	0.0941763
$452$	2.00000i	0.0940721i
$453$	0	0
$454$	−23.0000	−1.07944
$455$	0	0
$456$	0	0
$457$	34.0000i	1.59045i	0.606313	+	0.795226i	$0.292648\pi$
$457$	34.0000i	1.59045i	−0.606313	+	0.795226i	$0.707352\pi$
$458$	18.0000i	0.841085i
$459$	0	0
$460$	0	0
$461$	−42.0000	−1.95614	−0.978068	−	0.208288i	$-0.933211\pi$
$461$	−42.0000	−1.95614	−0.978068	+	0.208288i	$0.933211\pi$
$462$	0	0
$463$	− 9.00000i	− 0.418265i	−0.977887	−	0.209133i	$-0.932936\pi$
$463$	− 9.00000i	− 0.418265i	0.977887	−	0.209133i	$-0.0670641\pi$
$464$	−1.00000	−0.0464238
$465$	0	0
$466$	−6.00000	−0.277945
$467$	32.0000i	1.48078i	0.672176	+	0.740392i	$0.265360\pi$
$467$	32.0000i	1.48078i	−0.672176	+	0.740392i	$0.734640\pi$
$468$	0	0
$469$	9.00000	0.415581
$470$	0	0
$471$	0	0
$472$	15.0000i	0.690431i
$473$	8.00000i	0.367840i
$474$	0	0
$475$	0	0
$476$	15.0000	0.687524
$477$	0	0
$478$	− 19.0000i	− 0.869040i
$479$	−15.0000	−0.685367	−0.342684	−	0.939451i	$-0.611336\pi$
$479$	−15.0000	−0.685367	−0.342684	+	0.939451i	$0.611336\pi$
$480$	0	0
$481$	−8.00000	−0.364769
$482$	10.0000i	0.455488i
$483$	0	0
$484$	−10.0000	−0.454545
$485$	0	0
$486$	0	0
$487$	− 37.0000i	− 1.67663i	−0.545186	−	0.838315i	$-0.683541\pi$
$487$	− 37.0000i	− 1.67663i	0.545186	−	0.838315i	$-0.316459\pi$
$488$	3.00000i	0.135804i
$489$	0	0
$490$	0	0
$491$	26.0000	1.17336	0.586682	−	0.809818i	$-0.300434\pi$
$491$	26.0000	1.17336	0.586682	+	0.809818i	$0.300434\pi$
$492$	0	0
$493$	− 5.00000i	− 0.225189i
$494$	−8.00000	−0.359937
$495$	0	0
$496$	−3.00000	−0.134704
$497$	− 48.0000i	− 2.15309i
$498$	0	0
$499$	41.0000	1.83541	0.917706	−	0.397260i	$-0.130039\pi$
$499$	41.0000	1.83541	0.917706	+	0.397260i	$0.130039\pi$
$500$	0	0
$501$	0	0
$502$	30.0000i	1.33897i
$503$	− 26.0000i	− 1.15928i	−0.814872	−	0.579641i	$-0.803193\pi$
$503$	− 26.0000i	− 1.15928i	0.814872	−	0.579641i	$-0.196807\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0	0
$508$	− 8.00000i	− 0.354943i
$509$	−10.0000	−0.443242	−0.221621	−	0.975133i	$-0.571135\pi$
$509$	−10.0000	−0.443242	−0.221621	+	0.975133i	$0.571135\pi$
$510$	0	0
$511$	12.0000	0.530849
$512$	− 11.0000i	− 0.486136i
$513$	0	0
$514$	3.00000	0.132324
$515$	0	0
$516$	0	0
$517$	− 11.0000i	− 0.483779i
$518$	− 24.0000i	− 1.05450i
$519$	0	0
$520$	0	0
$521$	26.0000	1.13908	0.569540	−	0.821963i	$-0.307121\pi$
$521$	26.0000	1.13908	0.569540	+	0.821963i	$0.307121\pi$
$522$	0	0
$523$	26.0000i	1.13690i	0.822718	+	0.568450i	$0.192457\pi$
$523$	26.0000i	1.13690i	−0.822718	+	0.568450i	$0.807543\pi$
$524$	−6.00000	−0.262111
$525$	0	0
$526$	14.0000	0.610429
$527$	− 15.0000i	− 0.653410i
$528$	0	0
$529$	23.0000	1.00000
$530$	0	0
$531$	0	0
$532$	24.0000i	1.04053i
$533$	2.00000i	0.0866296i
$534$	0	0
$535$	0	0
$536$	9.00000	0.388741
$537$	0	0
$538$	17.0000i	0.732922i
$539$	−2.00000	−0.0861461
$540$	0	0
$541$	12.0000	0.515920	0.257960	−	0.966156i	$-0.416950\pi$
$541$	12.0000	0.515920	0.257960	+	0.966156i	$0.416950\pi$
$542$	− 5.00000i	− 0.214768i
$543$	0	0
$544$	25.0000	1.07187
$545$	0	0
$546$	0	0
$547$	22.0000i	0.940652i	0.882493	+	0.470326i	$0.155864\pi$
$547$	22.0000i	0.940652i	−0.882493	+	0.470326i	$0.844136\pi$
$548$	− 12.0000i	− 0.512615i
$549$	0	0
$550$	0	0
$551$	8.00000	0.340811
$552$	0	0
$553$	− 36.0000i	− 1.53088i
$554$	6.00000	0.254916
$555$	0	0
$556$	10.0000	0.424094
$557$	− 18.0000i	− 0.762684i	−0.924434	−	0.381342i	$-0.875462\pi$
$557$	− 18.0000i	− 0.762684i	0.924434	−	0.381342i	$-0.124538\pi$
$558$	0	0
$559$	−8.00000	−0.338364
$560$	0	0
$561$	0	0
$562$	18.0000i	0.759284i
$563$	24.0000i	1.01148i	0.862686	+	0.505740i	$0.168780\pi$
$563$	24.0000i	1.01148i	−0.862686	+	0.505740i	$0.831220\pi$
$564$	0	0
$565$	0	0
$566$	18.0000	0.756596
$567$	0	0
$568$	− 48.0000i	− 2.01404i
$569$	15.0000	0.628833	0.314416	−	0.949285i	$-0.398191\pi$
$569$	15.0000	0.628833	0.314416	+	0.949285i	$0.398191\pi$
$570$	0	0
$571$	32.0000	1.33916	0.669579	−	0.742741i	$-0.266474\pi$
$571$	32.0000	1.33916	0.669579	+	0.742741i	$0.266474\pi$
$572$	1.00000i	0.0418121i
$573$	0	0
$574$	−6.00000	−0.250435
$575$	0	0
$576$	0	0
$577$	− 10.0000i	− 0.416305i	−0.978096	−	0.208153i	$-0.933255\pi$
$577$	− 10.0000i	− 0.416305i	0.978096	−	0.208153i	$-0.0667451\pi$
$578$	− 8.00000i	− 0.332756i
$579$	0	0
$580$	0	0
$581$	−9.00000	−0.373383
$582$	0	0
$583$	11.0000i	0.455573i
$584$	12.0000	0.496564
$585$	0	0
$586$	30.0000	1.23929
$587$	− 45.0000i	− 1.85735i	−0.370896	−	0.928674i	$-0.620949\pi$
$587$	− 45.0000i	− 1.85735i	0.370896	−	0.928674i	$-0.379051\pi$
$588$	0	0
$589$	24.0000	0.988903
$590$	0	0
$591$	0	0
$592$	− 8.00000i	− 0.328798i
$593$	− 26.0000i	− 1.06769i	−0.845582	−	0.533846i	$-0.820746\pi$
$593$	− 26.0000i	− 1.06769i	0.845582	−	0.533846i	$-0.179254\pi$
$594$	0	0
$595$	0	0
$596$	10.0000	0.409616
$597$	0	0
$598$	0	0
$599$	46.0000	1.87951	0.939755	−	0.341850i	$-0.111053\pi$
$599$	46.0000	1.87951	0.939755	+	0.341850i	$0.111053\pi$
$600$	0	0
$601$	5.00000	0.203954	0.101977	−	0.994787i	$-0.467483\pi$
$601$	5.00000	0.203954	0.101977	+	0.994787i	$0.467483\pi$
$602$	− 24.0000i	− 0.978167i
$603$	0	0
$604$	−17.0000	−0.691720
$605$	0	0
$606$	0	0
$607$	− 38.0000i	− 1.54237i	−0.636610	−	0.771186i	$-0.719664\pi$
$607$	− 38.0000i	− 1.54237i	0.636610	−	0.771186i	$-0.280336\pi$
$608$	40.0000i	1.62221i
$609$	0	0
$610$	0	0
$611$	11.0000	0.445012
$612$	0	0
$613$	14.0000i	0.565455i	0.959200	+	0.282727i	$0.0912392\pi$
$613$	14.0000i	0.565455i	−0.959200	+	0.282727i	$0.908761\pi$
$614$	−4.00000	−0.161427
$615$	0	0
$616$	−9.00000	−0.362620
$617$	6.00000i	0.241551i	0.992680	+	0.120775i	$0.0385381\pi$
$617$	6.00000i	0.241551i	−0.992680	+	0.120775i	$0.961462\pi$
$618$	0	0
$619$	4.00000	0.160774	0.0803868	−	0.996764i	$-0.474384\pi$
$619$	4.00000	0.160774	0.0803868	+	0.996764i	$0.474384\pi$
$620$	0	0
$621$	0	0
$622$	− 6.00000i	− 0.240578i
$623$	0	0
$624$	0	0
$625$	0	0
$626$	29.0000	1.15907
$627$	0	0
$628$	11.0000i	0.438948i
$629$	40.0000	1.59490
$630$	0	0
$631$	−44.0000	−1.75161	−0.875806	−	0.482663i	$-0.839670\pi$
$631$	−44.0000	−1.75161	−0.875806	+	0.482663i	$0.839670\pi$
$632$	− 36.0000i	− 1.43200i
$633$	0	0
$634$	16.0000	0.635441
$635$	0	0
$636$	0	0
$637$	− 2.00000i	− 0.0792429i
$638$	1.00000i	0.0395904i
$639$	0	0
$640$	0	0
$641$	15.0000	0.592464	0.296232	−	0.955116i	$-0.404270\pi$
$641$	15.0000	0.592464	0.296232	+	0.955116i	$0.404270\pi$
$642$	0	0
$643$	− 12.0000i	− 0.473234i	−0.971603	−	0.236617i	$-0.923961\pi$
$643$	− 12.0000i	− 0.473234i	0.971603	−	0.236617i	$-0.0760386\pi$
$644$	0	0
$645$	0	0
$646$	40.0000	1.57378
$647$	16.0000i	0.629025i	0.949253	+	0.314512i	$0.101841\pi$
$647$	16.0000i	0.629025i	−0.949253	+	0.314512i	$0.898159\pi$
$648$	0	0
$649$	5.00000	0.196267
$650$	0	0
$651$	0	0
$652$	− 20.0000i	− 0.783260i
$653$	− 35.0000i	− 1.36966i	−0.728705	−	0.684828i	$-0.759877\pi$
$653$	− 35.0000i	− 1.36966i	0.728705	−	0.684828i	$-0.240123\pi$
$654$	0	0
$655$	0	0
$656$	−2.00000	−0.0780869
$657$	0	0
$658$	33.0000i	1.28647i
$659$	−4.00000	−0.155818	−0.0779089	−	0.996960i	$-0.524824\pi$
$659$	−4.00000	−0.155818	−0.0779089	+	0.996960i	$0.524824\pi$
$660$	0	0
$661$	−2.00000	−0.0777910	−0.0388955	−	0.999243i	$-0.512384\pi$
$661$	−2.00000	−0.0777910	−0.0388955	+	0.999243i	$0.512384\pi$
$662$	− 12.0000i	− 0.466393i
$663$	0	0
$664$	−9.00000	−0.349268
$665$	0	0
$666$	0	0
$667$	0	0
$668$	12.0000i	0.464294i
$669$	0	0
$670$	0	0
$671$	1.00000	0.0386046
$672$	0	0
$673$	45.0000i	1.73462i	0.497766	+	0.867311i	$0.334154\pi$
$673$	45.0000i	1.73462i	−0.497766	+	0.867311i	$0.665846\pi$
$674$	5.00000	0.192593
$675$	0	0
$676$	−1.00000	−0.0384615
$677$	− 18.0000i	− 0.691796i	−0.938272	−	0.345898i	$-0.887574\pi$
$677$	− 18.0000i	− 0.691796i	0.938272	−	0.345898i	$-0.112426\pi$
$678$	0	0
$679$	−6.00000	−0.230259
$680$	0	0
$681$	0	0
$682$	3.00000i	0.114876i
$683$	− 9.00000i	− 0.344375i	−0.985064	−	0.172188i	$-0.944916\pi$
$683$	− 9.00000i	− 0.344375i	0.985064	−	0.172188i	$-0.0550836\pi$
$684$	0	0
$685$	0	0
$686$	−15.0000	−0.572703
$687$	0	0
$688$	− 8.00000i	− 0.304997i
$689$	−11.0000	−0.419067
$690$	0	0
$691$	−29.0000	−1.10321	−0.551606	−	0.834105i	$-0.685985\pi$
$691$	−29.0000	−1.10321	−0.551606	+	0.834105i	$0.685985\pi$
$692$	21.0000i	0.798300i
$693$	0	0
$694$	2.00000	0.0759190
$695$	0	0
$696$	0	0
$697$	− 10.0000i	− 0.378777i
$698$	− 16.0000i	− 0.605609i
$699$	0	0
$700$	0	0
$701$	−31.0000	−1.17085	−0.585427	−	0.810725i	$-0.699073\pi$
$701$	−31.0000	−1.17085	−0.585427	+	0.810725i	$0.699073\pi$
$702$	0	0
$703$	64.0000i	2.41381i
$704$	−7.00000	−0.263822
$705$	0	0
$706$	18.0000	0.677439
$707$	− 27.0000i	− 1.01544i
$708$	0	0
$709$	14.0000	0.525781	0.262891	−	0.964826i	$-0.415324\pi$
$709$	14.0000	0.525781	0.262891	+	0.964826i	$0.415324\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	0	0
$714$	0	0
$715$	0	0
$716$	26.0000	0.971666
$717$	0	0
$718$	15.0000i	0.559795i
$719$	6.00000	0.223762	0.111881	−	0.993722i	$-0.464312\pi$
$719$	6.00000	0.223762	0.111881	+	0.993722i	$0.464312\pi$
$720$	0	0
$721$	42.0000	1.56416
$722$	45.0000i	1.67473i
$723$	0	0
$724$	5.00000	0.185824
$725$	0	0
$726$	0	0
$727$	2.00000i	0.0741759i	0.999312	+	0.0370879i	$0.0118082\pi$
$727$	2.00000i	0.0741759i	−0.999312	+	0.0370879i	$0.988192\pi$
$728$	− 9.00000i	− 0.333562i
$729$	0	0
$730$	0	0
$731$	40.0000	1.47945
$732$	0	0
$733$	− 20.0000i	− 0.738717i	−0.929287	−	0.369358i	$-0.879577\pi$
$733$	− 20.0000i	− 0.738717i	0.929287	−	0.369358i	$-0.120423\pi$
$734$	−12.0000	−0.442928
$735$	0	0
$736$	0	0
$737$	− 3.00000i	− 0.110506i
$738$	0	0
$739$	−7.00000	−0.257499	−0.128750	−	0.991677i	$-0.541096\pi$
$739$	−7.00000	−0.257499	−0.128750	+	0.991677i	$0.541096\pi$
$740$	0	0
$741$	0	0
$742$	− 33.0000i	− 1.21147i
$743$	− 13.0000i	− 0.476924i	−0.971152	−	0.238462i	$-0.923357\pi$
$743$	− 13.0000i	− 0.476924i	0.971152	−	0.238462i	$-0.0766432\pi$
$744$	0	0
$745$	0	0
$746$	31.0000	1.13499
$747$	0	0
$748$	− 5.00000i	− 0.182818i
$749$	−54.0000	−1.97312
$750$	0	0
$751$	40.0000	1.45962	0.729810	−	0.683650i	$-0.239608\pi$
$751$	40.0000	1.45962	0.729810	+	0.683650i	$0.239608\pi$
$752$	11.0000i	0.401129i
$753$	0	0
$754$	−1.00000	−0.0364179
$755$	0	0
$756$	0	0
$757$	− 23.0000i	− 0.835949i	−0.908459	−	0.417975i	$-0.862740\pi$
$757$	− 23.0000i	− 0.835949i	0.908459	−	0.417975i	$-0.137260\pi$
$758$	35.0000i	1.27126i
$759$	0	0
$760$	0	0
$761$	24.0000	0.869999	0.435000	−	0.900431i	$-0.356748\pi$
$761$	24.0000	0.869999	0.435000	+	0.900431i	$0.356748\pi$
$762$	0	0
$763$	30.0000i	1.08607i
$764$	−10.0000	−0.361787
$765$	0	0
$766$	−24.0000	−0.867155
$767$	5.00000i	0.180540i
$768$	0	0
$769$	40.0000	1.44244	0.721218	−	0.692708i	$-0.243582\pi$
$769$	40.0000	1.44244	0.721218	+	0.692708i	$0.243582\pi$
$770$	0	0
$771$	0	0
$772$	10.0000i	0.359908i
$773$	− 18.0000i	− 0.647415i	−0.946157	−	0.323708i	$-0.895071\pi$
$773$	− 18.0000i	− 0.647415i	0.946157	−	0.323708i	$-0.104929\pi$
$774$	0	0
$775$	0	0
$776$	−6.00000	−0.215387
$777$	0	0
$778$	− 30.0000i	− 1.07555i
$779$	16.0000	0.573259
$780$	0	0
$781$	−16.0000	−0.572525
$782$	0	0
$783$	0	0
$784$	2.00000	0.0714286
$785$	0	0
$786$	0	0
$787$	− 17.0000i	− 0.605985i	−0.952993	−	0.302992i	$-0.902014\pi$
$787$	− 17.0000i	− 0.605985i	0.952993	−	0.302992i	$-0.0979856\pi$
$788$	− 6.00000i	− 0.213741i
$789$	0	0
$790$	0	0
$791$	−6.00000	−0.213335
$792$	0	0
$793$	1.00000i	0.0355110i
$794$	2.00000	0.0709773
$795$	0	0
$796$	−10.0000	−0.354441
$797$	− 43.0000i	− 1.52314i	−0.648084	−	0.761569i	$-0.724429\pi$
$797$	− 43.0000i	− 1.52314i	0.648084	−	0.761569i	$-0.275571\pi$
$798$	0	0
$799$	−55.0000	−1.94576
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 4.00000i	− 0.141157i
$804$	0	0
$805$	0	0
$806$	−3.00000	−0.105670
$807$	0	0
$808$	− 27.0000i	− 0.949857i
$809$	−2.00000	−0.0703163	−0.0351581	−	0.999382i	$-0.511193\pi$
$809$	−2.00000	−0.0703163	−0.0351581	+	0.999382i	$0.511193\pi$
$810$	0	0
$811$	−33.0000	−1.15879	−0.579393	−	0.815048i	$-0.696710\pi$
$811$	−33.0000	−1.15879	−0.579393	+	0.815048i	$0.696710\pi$
$812$	3.00000i	0.105279i
$813$	0	0
$814$	−8.00000	−0.280400
$815$	0	0
$816$	0	0
$817$	64.0000i	2.23908i
$818$	− 2.00000i	− 0.0699284i
$819$	0	0
$820$	0	0
$821$	0	0		−	1.00000i	$-0.5\pi$
$821$	0	0			1.00000i	$0.5\pi$
$822$	0	0
$823$	28.0000i	0.976019i	0.872838	+	0.488009i	$0.162277\pi$
$823$	28.0000i	0.976019i	−0.872838	+	0.488009i	$0.837723\pi$
$824$	42.0000	1.46314
$825$	0	0
$826$	−15.0000	−0.521917
$827$	1.00000i	0.0347734i	0.999849	+	0.0173867i	$0.00553464\pi$
$827$	1.00000i	0.0347734i	−0.999849	+	0.0173867i	$0.994465\pi$
$828$	0	0
$829$	−51.0000	−1.77130	−0.885652	−	0.464350i	$-0.846288\pi$
$829$	−51.0000	−1.77130	−0.885652	+	0.464350i	$0.846288\pi$
$830$	0	0
$831$	0	0
$832$	− 7.00000i	− 0.242681i
$833$	10.0000i	0.346479i
$834$	0	0
$835$	0	0
$836$	8.00000	0.276686
$837$	0	0
$838$	− 26.0000i	− 0.898155i
$839$	−8.00000	−0.276191	−0.138095	−	0.990419i	$-0.544098\pi$
$839$	−8.00000	−0.276191	−0.138095	+	0.990419i	$0.544098\pi$
$840$	0	0
$841$	−28.0000	−0.965517
$842$	28.0000i	0.964944i
$843$	0	0
$844$	−16.0000	−0.550743
$845$	0	0
$846$	0	0
$847$	− 30.0000i	− 1.03081i
$848$	− 11.0000i	− 0.377742i
$849$	0	0
$850$	0	0
$851$	0	0
$852$	0	0
$853$	24.0000i	0.821744i	0.911693	+	0.410872i	$0.134776\pi$
$853$	24.0000i	0.821744i	−0.911693	+	0.410872i	$0.865224\pi$
$854$	−3.00000	−0.102658
$855$	0	0
$856$	−54.0000	−1.84568
$857$	− 38.0000i	− 1.29806i	−0.760765	−	0.649028i	$-0.775176\pi$
$857$	− 38.0000i	− 1.29806i	0.760765	−	0.649028i	$-0.224824\pi$
$858$	0	0
$859$	−4.00000	−0.136478	−0.0682391	−	0.997669i	$-0.521738\pi$
$859$	−4.00000	−0.136478	−0.0682391	+	0.997669i	$0.521738\pi$
$860$	0	0
$861$	0	0
$862$	12.0000i	0.408722i
$863$	7.00000i	0.238283i	0.992877	+	0.119141i	$0.0380142\pi$
$863$	7.00000i	0.238283i	−0.992877	+	0.119141i	$0.961986\pi$
$864$	0	0
$865$	0	0
$866$	34.0000	1.15537
$867$	0	0
$868$	9.00000i	0.305480i
$869$	−12.0000	−0.407072
$870$	0	0
$871$	3.00000	0.101651
$872$	30.0000i	1.01593i
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 24.0000i	− 0.810422i	−0.914223	−	0.405211i	$-0.867198\pi$
$877$	− 24.0000i	− 0.810422i	0.914223	−	0.405211i	$-0.132802\pi$
$878$	− 4.00000i	− 0.134993i
$879$	0	0
$880$	0	0
$881$	25.0000	0.842271	0.421136	−	0.906998i	$-0.361632\pi$
$881$	25.0000	0.842271	0.421136	+	0.906998i	$0.361632\pi$
$882$	0	0
$883$	16.0000i	0.538443i	0.963078	+	0.269221i	$0.0867663\pi$
$883$	16.0000i	0.538443i	−0.963078	+	0.269221i	$0.913234\pi$
$884$	5.00000	0.168168
$885$	0	0
$886$	0	0
$887$	12.0000i	0.402921i	0.979497	+	0.201460i	$0.0645687\pi$
$887$	12.0000i	0.402921i	−0.979497	+	0.201460i	$0.935431\pi$
$888$	0	0
$889$	24.0000	0.804934
$890$	0	0
$891$	0	0
$892$	− 8.00000i	− 0.267860i
$893$	− 88.0000i	− 2.94481i
$894$	0	0
$895$	0	0
$896$	−9.00000	−0.300669
$897$	0	0
$898$	− 28.0000i	− 0.934372i
$899$	3.00000	0.100056
$900$	0	0
$901$	55.0000	1.83232
$902$	2.00000i	0.0665927i
$903$	0	0
$904$	−6.00000	−0.199557
$905$	0	0
$906$	0	0
$907$	− 24.0000i	− 0.796907i	−0.917189	−	0.398453i	$-0.869547\pi$
$907$	− 24.0000i	− 0.796907i	0.917189	−	0.398453i	$-0.130453\pi$
$908$	23.0000i	0.763282i
$909$	0	0
$910$	0	0
$911$	−56.0000	−1.85536	−0.927681	−	0.373373i	$-0.878201\pi$
$911$	−56.0000	−1.85536	−0.927681	+	0.373373i	$0.878201\pi$
$912$	0	0
$913$	3.00000i	0.0992855i
$914$	−34.0000	−1.12462
$915$	0	0
$916$	18.0000	0.594737
$917$	− 18.0000i	− 0.594412i
$918$	0	0
$919$	−22.0000	−0.725713	−0.362857	−	0.931845i	$-0.618198\pi$
$919$	−22.0000	−0.725713	−0.362857	+	0.931845i	$0.618198\pi$
$920$	0	0
$921$	0	0
$922$	− 42.0000i	− 1.38320i
$923$	− 16.0000i	− 0.526646i
$924$	0	0
$925$	0	0
$926$	9.00000	0.295758
$927$	0	0
$928$	5.00000i	0.164133i
$929$	−26.0000	−0.853032	−0.426516	−	0.904480i	$-0.640259\pi$
$929$	−26.0000	−0.853032	−0.426516	+	0.904480i	$0.640259\pi$
$930$	0	0
$931$	−16.0000	−0.524379
$932$	6.00000i	0.196537i
$933$	0	0
$934$	−32.0000	−1.04707
$935$	0	0
$936$	0	0
$937$	− 45.0000i	− 1.47009i	−0.678021	−	0.735043i	$-0.737162\pi$
$937$	− 45.0000i	− 1.47009i	0.678021	−	0.735043i	$-0.262838\pi$
$938$	9.00000i	0.293860i
$939$	0	0
$940$	0	0
$941$	56.0000	1.82555	0.912774	−	0.408465i	$-0.133936\pi$
$941$	56.0000	1.82555	0.912774	+	0.408465i	$0.133936\pi$
$942$	0	0
$943$	0	0
$944$	−5.00000	−0.162736
$945$	0	0
$946$	−8.00000	−0.260102
$947$	21.0000i	0.682408i	0.939989	+	0.341204i	$0.110835\pi$
$947$	21.0000i	0.682408i	−0.939989	+	0.341204i	$0.889165\pi$
$948$	0	0
$949$	4.00000	0.129845
$950$	0	0
$951$	0	0
$952$	45.0000i	1.45846i
$953$	− 37.0000i	− 1.19855i	−0.800544	−	0.599274i	$-0.795456\pi$
$953$	− 37.0000i	− 1.19855i	0.800544	−	0.599274i	$-0.204544\pi$
$954$	0	0
$955$	0	0
$956$	−19.0000	−0.614504
$957$	0	0
$958$	− 15.0000i	− 0.484628i
$959$	36.0000	1.16250
$960$	0	0
$961$	−22.0000	−0.709677
$962$	− 8.00000i	− 0.257930i
$963$	0	0
$964$	10.0000	0.322078
$965$	0	0
$966$	0	0
$967$	− 37.0000i	− 1.18984i	−0.803785	−	0.594920i	$-0.797184\pi$
$967$	− 37.0000i	− 1.18984i	0.803785	−	0.594920i	$-0.202816\pi$
$968$	− 30.0000i	− 0.964237i
$969$	0	0
$970$	0	0
$971$	−54.0000	−1.73294	−0.866471	−	0.499227i	$-0.833617\pi$
$971$	−54.0000	−1.73294	−0.866471	+	0.499227i	$0.833617\pi$
$972$	0	0
$973$	30.0000i	0.961756i
$974$	37.0000	1.18556
$975$	0	0
$976$	−1.00000	−0.0320092
$977$	28.0000i	0.895799i	0.894084	+	0.447900i	$0.147828\pi$
$977$	28.0000i	0.895799i	−0.894084	+	0.447900i	$0.852172\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	0	0
$982$	26.0000i	0.829693i
$983$	21.0000i	0.669796i	0.942254	+	0.334898i	$0.108702\pi$
$983$	21.0000i	0.669796i	−0.942254	+	0.334898i	$0.891298\pi$
$984$	0	0
$985$	0	0
$986$	5.00000	0.159232
$987$	0	0
$988$	8.00000i	0.254514i
$989$	0	0
$990$	0	0
$991$	20.0000	0.635321	0.317660	−	0.948205i	$-0.397103\pi$
$991$	20.0000	0.635321	0.317660	+	0.948205i	$0.397103\pi$
$992$	15.0000i	0.476250i
$993$	0	0
$994$	48.0000	1.52247
$995$	0	0
$996$	0	0
$997$	− 3.00000i	− 0.0950110i	−0.998871	−	0.0475055i	$-0.984873\pi$
$997$	− 3.00000i	− 0.0950110i	0.998871	−	0.0475055i	$-0.0151272\pi$
$998$	41.0000i	1.29783i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2925.2.c.i.2224.2		2
3.2	odd	2		975.2.c.g.274.1		2
5.2	odd	4		2925.2.a.b.1.1		1
5.3	odd	4		2925.2.a.o.1.1		1
5.4	even	2	inner	2925.2.c.i.2224.1		2
15.2	even	4		975.2.a.k.1.1	yes	1
15.8	even	4		975.2.a.d.1.1	✓	1
15.14	odd	2		975.2.c.g.274.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
975.2.a.d.1.1	✓	1	15.8	even	4
975.2.a.k.1.1	yes	1	15.2	even	4
975.2.c.g.274.1		2	3.2	odd	2
975.2.c.g.274.2		2	15.14	odd	2
2925.2.a.b.1.1		1	5.2	odd	4
2925.2.a.o.1.1		1	5.3	odd	4
2925.2.c.i.2224.1		2	5.4	even	2	inner
2925.2.c.i.2224.2		2	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2925 = 3^{2} \cdot 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2925.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} +1.00000 q^{4} +3.00000i q^{7} +3.00000i q^{8} +O(q^{10})\)	\(q+1.00000i q^{2} +1.00000 q^{4} +3.00000i q^{7} +3.00000i q^{8} +1.00000 q^{11} +1.00000i q^{13} -3.00000 q^{14} -1.00000 q^{16} -5.00000i q^{17} +8.00000 q^{19} +1.00000i q^{22} -1.00000 q^{26} +3.00000i q^{28} +1.00000 q^{29} +3.00000 q^{31} +5.00000i q^{32} +5.00000 q^{34} +8.00000i q^{37} +8.00000i q^{38} +2.00000 q^{41} +8.00000i q^{43} +1.00000 q^{44} -11.0000i q^{47} -2.00000 q^{49} +1.00000i q^{52} +11.0000i q^{53} -9.00000 q^{56} +1.00000i q^{58} +5.00000 q^{59} +1.00000 q^{61} +3.00000i q^{62} -7.00000 q^{64} -3.00000i q^{67} -5.00000i q^{68} -16.0000 q^{71} -4.00000i q^{73} -8.00000 q^{74} +8.00000 q^{76} +3.00000i q^{77} -12.0000 q^{79} +2.00000i q^{82} +3.00000i q^{83} -8.00000 q^{86} +3.00000i q^{88} -3.00000 q^{91} +11.0000 q^{94} +2.00000i q^{97} -2.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{4}+O(q^{10}) \) 2 * q + 2 * q^4	\( 2 q + 2 q^{4} + 2 q^{11} - 6 q^{14} - 2 q^{16} + 16 q^{19} - 2 q^{26} + 2 q^{29} + 6 q^{31} + 10 q^{34} + 4 q^{41} + 2 q^{44} - 4 q^{49} - 18 q^{56} + 10 q^{59} + 2 q^{61} - 14 q^{64} - 32 q^{71} - 16 q^{74} + 16 q^{76} - 24 q^{79} - 16 q^{86} - 6 q^{91} + 22 q^{94}+O(q^{100}) \) 2 * q + 2 * q^4 + 2 * q^11 - 6 * q^14 - 2 * q^16 + 16 * q^19 - 2 * q^26 + 2 * q^29 + 6 * q^31 + 10 * q^34 + 4 * q^41 + 2 * q^44 - 4 * q^49 - 18 * q^56 + 10 * q^59 + 2 * q^61 - 14 * q^64 - 32 * q^71 - 16 * q^74 + 16 * q^76 - 24 * q^79 - 16 * q^86 - 6 * q^91 + 22 * q^94

Introduction

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