LMFDB - Embedded newform 2925.2.c.h.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 65)
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.h.2224.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{2} +1.00000 q^{4} -4.00000i q^{7} +3.00000i q^{8} +O(q^{10})$	$q+1.00000i q^{2} +1.00000 q^{4} -4.00000i q^{7} +3.00000i q^{8} -2.00000 q^{11} +1.00000i q^{13} +4.00000 q^{14} -1.00000 q^{16} -2.00000i q^{17} +6.00000 q^{19} -2.00000i q^{22} -6.00000i q^{23} -1.00000 q^{26} -4.00000i q^{28} +2.00000 q^{29} -10.0000 q^{31} +5.00000i q^{32} +2.00000 q^{34} -2.00000i q^{37} +6.00000i q^{38} +6.00000 q^{41} -10.0000i q^{43} -2.00000 q^{44} +6.00000 q^{46} -4.00000i q^{47} -9.00000 q^{49} +1.00000i q^{52} +2.00000i q^{53} +12.0000 q^{56} +2.00000i q^{58} +6.00000 q^{59} +2.00000 q^{61} -10.0000i q^{62} -7.00000 q^{64} -4.00000i q^{67} -2.00000i q^{68} -6.00000 q^{71} +6.00000i q^{73} +2.00000 q^{74} +6.00000 q^{76} +8.00000i q^{77} +12.0000 q^{79} +6.00000i q^{82} -16.0000i q^{83} +10.0000 q^{86} -6.00000i q^{88} +2.00000 q^{89} +4.00000 q^{91} -6.00000i q^{92} +4.00000 q^{94} -2.00000i q^{97} -9.00000i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{4}+O(q^{10}) $ 2 * q + 2 * q^4	$ 2 q + 2 q^{4} - 4 q^{11} + 8 q^{14} - 2 q^{16} + 12 q^{19} - 2 q^{26} + 4 q^{29} - 20 q^{31} + 4 q^{34} + 12 q^{41} - 4 q^{44} + 12 q^{46} - 18 q^{49} + 24 q^{56} + 12 q^{59} + 4 q^{61} - 14 q^{64} - 12 q^{71} + 4 q^{74} + 12 q^{76} + 24 q^{79} + 20 q^{86} + 4 q^{89} + 8 q^{91} + 8 q^{94}+O(q^{100}) $ 2 * q + 2 * q^4 - 4 * q^11 + 8 * q^14 - 2 * q^16 + 12 * q^19 - 2 * q^26 + 4 * q^29 - 20 * q^31 + 4 * q^34 + 12 * q^41 - 4 * q^44 + 12 * q^46 - 18 * q^49 + 24 * q^56 + 12 * q^59 + 4 * q^61 - 14 * q^64 - 12 * q^71 + 4 * q^74 + 12 * q^76 + 24 * q^79 + 20 * q^86 + 4 * q^89 + 8 * q^91 + 8 * 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
$7$	− 4.00000i	− 1.51186i	−0.654654	−	0.755929i	$-0.727186\pi$
$7$	− 4.00000i	− 1.51186i	0.654654	−	0.755929i	$-0.272814\pi$
$8$	3.00000i	1.06066i
$9$	0	0
$10$	0	0
$11$	−2.00000	−0.603023	−0.301511	−	0.953463i	$-0.597491\pi$
$11$	−2.00000	−0.603023	−0.301511	+	0.953463i	$0.597491\pi$
$12$	0	0
$13$	1.00000i	0.277350i
$13$	1.00000i	0.277350i
$14$	4.00000	1.06904
$15$	0	0
$16$	−1.00000	−0.250000
$17$	− 2.00000i	− 0.485071i	−0.970143	−	0.242536i	$-0.922021\pi$
$17$	− 2.00000i	− 0.485071i	0.970143	−	0.242536i	$-0.0779791\pi$
$18$	0	0
$19$	6.00000	1.37649	0.688247	−	0.725476i	$-0.258380\pi$
$19$	6.00000	1.37649	0.688247	+	0.725476i	$0.258380\pi$
$20$	0	0
$21$	0	0
$22$	− 2.00000i	− 0.426401i
$23$	− 6.00000i	− 1.25109i	−0.780189	−	0.625543i	$-0.784877\pi$
$23$	− 6.00000i	− 1.25109i	0.780189	−	0.625543i	$-0.215123\pi$
$24$	0	0
$25$	0	0
$26$	−1.00000	−0.196116
$27$	0	0
$28$	− 4.00000i	− 0.755929i
$29$	2.00000	0.371391	0.185695	−	0.982607i	$-0.440546\pi$
$29$	2.00000	0.371391	0.185695	+	0.982607i	$0.440546\pi$
$30$	0	0
$31$	−10.0000	−1.79605	−0.898027	−	0.439941i	$-0.854999\pi$
$31$	−10.0000	−1.79605	−0.898027	+	0.439941i	$0.854999\pi$
$32$	5.00000i	0.883883i
$33$	0	0
$34$	2.00000	0.342997
$35$	0	0
$36$	0	0
$37$	− 2.00000i	− 0.328798i	−0.986394	−	0.164399i	$-0.947432\pi$
$37$	− 2.00000i	− 0.328798i	0.986394	−	0.164399i	$-0.0525685\pi$
$38$	6.00000i	0.973329i
$39$	0	0
$40$	0	0
$41$	6.00000	0.937043	0.468521	−	0.883452i	$-0.344787\pi$
$41$	6.00000	0.937043	0.468521	+	0.883452i	$0.344787\pi$
$42$	0	0
$43$	− 10.0000i	− 1.52499i	−0.646997	−	0.762493i	$-0.723975\pi$
$43$	− 10.0000i	− 1.52499i	0.646997	−	0.762493i	$-0.276025\pi$
$44$	−2.00000	−0.301511
$45$	0	0
$46$	6.00000	0.884652
$47$	− 4.00000i	− 0.583460i	−0.956501	−	0.291730i	$-0.905769\pi$
$47$	− 4.00000i	− 0.583460i	0.956501	−	0.291730i	$-0.0942309\pi$
$48$	0	0
$49$	−9.00000	−1.28571
$50$	0	0
$51$	0	0
$52$	1.00000i	0.138675i
$53$	2.00000i	0.274721i	0.990521	+	0.137361i	$0.0438619\pi$
$53$	2.00000i	0.274721i	−0.990521	+	0.137361i	$0.956138\pi$
$54$	0	0
$55$	0	0
$56$	12.0000	1.60357
$57$	0	0
$58$	2.00000i	0.262613i
$59$	6.00000	0.781133	0.390567	−	0.920575i	$-0.372279\pi$
$59$	6.00000	0.781133	0.390567	+	0.920575i	$0.372279\pi$
$60$	0	0
$61$	2.00000	0.256074	0.128037	−	0.991769i	$-0.459132\pi$
$61$	2.00000	0.256074	0.128037	+	0.991769i	$0.459132\pi$
$62$	− 10.0000i	− 1.27000i
$63$	0	0
$64$	−7.00000	−0.875000
$65$	0	0
$66$	0	0
$67$	− 4.00000i	− 0.488678i	−0.969690	−	0.244339i	$-0.921429\pi$
$67$	− 4.00000i	− 0.488678i	0.969690	−	0.244339i	$-0.0785709\pi$
$68$	− 2.00000i	− 0.242536i
$69$	0	0
$70$	0	0
$71$	−6.00000	−0.712069	−0.356034	−	0.934473i	$-0.615871\pi$
$71$	−6.00000	−0.712069	−0.356034	+	0.934473i	$0.615871\pi$
$72$	0	0
$73$	6.00000i	0.702247i	0.936329	+	0.351123i	$0.114200\pi$
$73$	6.00000i	0.702247i	−0.936329	+	0.351123i	$0.885800\pi$
$74$	2.00000	0.232495
$75$	0	0
$76$	6.00000	0.688247
$77$	8.00000i	0.911685i
$78$	0	0
$79$	12.0000	1.35011	0.675053	−	0.737769i	$-0.264121\pi$
$79$	12.0000	1.35011	0.675053	+	0.737769i	$0.264121\pi$
$80$	0	0
$81$	0	0
$82$	6.00000i	0.662589i
$83$	− 16.0000i	− 1.75623i	−0.478451	−	0.878114i	$-0.658802\pi$
$83$	− 16.0000i	− 1.75623i	0.478451	−	0.878114i	$-0.341198\pi$
$84$	0	0
$85$	0	0
$86$	10.0000	1.07833
$87$	0	0
$88$	− 6.00000i	− 0.639602i
$89$	2.00000	0.212000	0.106000	−	0.994366i	$-0.466196\pi$
$89$	2.00000	0.212000	0.106000	+	0.994366i	$0.466196\pi$
$90$	0	0
$91$	4.00000	0.419314
$92$	− 6.00000i	− 0.625543i
$93$	0	0
$94$	4.00000	0.412568
$95$	0	0
$96$	0	0
$97$	− 2.00000i	− 0.203069i	−0.994832	−	0.101535i	$-0.967625\pi$
$97$	− 2.00000i	− 0.203069i	0.994832	−	0.101535i	$-0.0323753\pi$
$98$	− 9.00000i	− 0.909137i
$99$	0	0
$100$	0	0
$101$	18.0000	1.79107	0.895533	−	0.444994i	$-0.146794\pi$
$101$	18.0000	1.79107	0.895533	+	0.444994i	$0.146794\pi$
$102$	0	0
$103$	− 2.00000i	− 0.197066i	−0.995134	−	0.0985329i	$-0.968585\pi$
$103$	− 2.00000i	− 0.197066i	0.995134	−	0.0985329i	$-0.0314150\pi$
$104$	−3.00000	−0.294174
$105$	0	0
$106$	−2.00000	−0.194257
$107$	− 10.0000i	− 0.966736i	−0.875417	−	0.483368i	$-0.839413\pi$
$107$	− 10.0000i	− 0.966736i	0.875417	−	0.483368i	$-0.160587\pi$
$108$	0	0
$109$	−10.0000	−0.957826	−0.478913	−	0.877862i	$-0.658969\pi$
$109$	−10.0000	−0.957826	−0.478913	+	0.877862i	$0.658969\pi$
$110$	0	0
$111$	0	0
$112$	4.00000i	0.377964i
$113$	− 14.0000i	− 1.31701i	−0.752577	−	0.658505i	$-0.771189\pi$
$113$	− 14.0000i	− 1.31701i	0.752577	−	0.658505i	$-0.228811\pi$
$114$	0	0
$115$	0	0
$116$	2.00000	0.185695
$117$	0	0
$118$	6.00000i	0.552345i
$119$	−8.00000	−0.733359
$120$	0	0
$121$	−7.00000	−0.636364
$122$	2.00000i	0.181071i
$123$	0	0
$124$	−10.0000	−0.898027
$125$	0	0
$126$	0	0
$127$	− 2.00000i	− 0.177471i	−0.996055	−	0.0887357i	$-0.971717\pi$
$127$	− 2.00000i	− 0.177471i	0.996055	−	0.0887357i	$-0.0282826\pi$
$128$	3.00000i	0.265165i
$129$	0	0
$130$	0	0
$131$	−20.0000	−1.74741	−0.873704	−	0.486458i	$-0.838289\pi$
$131$	−20.0000	−1.74741	−0.873704	+	0.486458i	$0.838289\pi$
$132$	0	0
$133$	− 24.0000i	− 2.08106i
$134$	4.00000	0.345547
$135$	0	0
$136$	6.00000	0.514496
$137$	2.00000i	0.170872i	0.996344	+	0.0854358i	$0.0272282\pi$
$137$	2.00000i	0.170872i	−0.996344	+	0.0854358i	$0.972772\pi$
$138$	0	0
$139$	0	0		−	1.00000i	$-0.5\pi$
$139$	0	0			1.00000i	$0.5\pi$
$140$	0	0
$141$	0	0
$142$	− 6.00000i	− 0.503509i
$143$	− 2.00000i	− 0.167248i
$144$	0	0
$145$	0	0
$146$	−6.00000	−0.496564
$147$	0	0
$148$	− 2.00000i	− 0.164399i
$149$	18.0000	1.47462	0.737309	−	0.675556i	$-0.236096\pi$
$149$	18.0000	1.47462	0.737309	+	0.675556i	$0.236096\pi$
$150$	0	0
$151$	10.0000	0.813788	0.406894	−	0.913475i	$-0.366612\pi$
$151$	10.0000	0.813788	0.406894	+	0.913475i	$0.366612\pi$
$152$	18.0000i	1.45999i
$153$	0	0
$154$	−8.00000	−0.644658
$155$	0	0
$156$	0	0
$157$	− 6.00000i	− 0.478852i	−0.970915	−	0.239426i	$-0.923041\pi$
$157$	− 6.00000i	− 0.478852i	0.970915	−	0.239426i	$-0.0769593\pi$
$158$	12.0000i	0.954669i
$159$	0	0
$160$	0	0
$161$	−24.0000	−1.89146
$162$	0	0
$163$	12.0000i	0.939913i	0.882690	+	0.469956i	$0.155730\pi$
$163$	12.0000i	0.939913i	−0.882690	+	0.469956i	$0.844270\pi$
$164$	6.00000	0.468521
$165$	0	0
$166$	16.0000	1.24184
$167$	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
$172$	− 10.0000i	− 0.762493i
$173$	− 6.00000i	− 0.456172i	−0.973641	−	0.228086i	$-0.926753\pi$
$173$	− 6.00000i	− 0.456172i	0.973641	−	0.228086i	$-0.0732467\pi$
$174$	0	0
$175$	0	0
$176$	2.00000	0.150756
$177$	0	0
$178$	2.00000i	0.149906i
$179$	12.0000	0.896922	0.448461	−	0.893802i	$-0.351972\pi$
$179$	12.0000	0.896922	0.448461	+	0.893802i	$0.351972\pi$
$180$	0	0
$181$	−22.0000	−1.63525	−0.817624	−	0.575753i	$-0.804709\pi$
$181$	−22.0000	−1.63525	−0.817624	+	0.575753i	$0.804709\pi$
$182$	4.00000i	0.296500i
$183$	0	0
$184$	18.0000	1.32698
$185$	0	0
$186$	0	0
$187$	4.00000i	0.292509i
$188$	− 4.00000i	− 0.291730i
$189$	0	0
$190$	0	0
$191$	0	0		−	1.00000i	$-0.5\pi$
$191$	0	0			1.00000i	$0.5\pi$
$192$	0	0
$193$	2.00000i	0.143963i	0.997406	+	0.0719816i	$0.0229323\pi$
$193$	2.00000i	0.143963i	−0.997406	+	0.0719816i	$0.977068\pi$
$194$	2.00000	0.143592
$195$	0	0
$196$	−9.00000	−0.642857
$197$	6.00000i	0.427482i	0.976890	+	0.213741i	$0.0685649\pi$
$197$	6.00000i	0.427482i	−0.976890	+	0.213741i	$0.931435\pi$
$198$	0	0
$199$	16.0000	1.13421	0.567105	−	0.823646i	$-0.308063\pi$
$199$	16.0000	1.13421	0.567105	+	0.823646i	$0.308063\pi$
$200$	0	0
$201$	0	0
$202$	18.0000i	1.26648i
$203$	− 8.00000i	− 0.561490i
$204$	0	0
$205$	0	0
$206$	2.00000	0.139347
$207$	0	0
$208$	− 1.00000i	− 0.0693375i
$209$	−12.0000	−0.830057
$210$	0	0
$211$	12.0000	0.826114	0.413057	−	0.910705i	$-0.364461\pi$
$211$	12.0000	0.826114	0.413057	+	0.910705i	$0.364461\pi$
$212$	2.00000i	0.137361i
$213$	0	0
$214$	10.0000	0.683586
$215$	0	0
$216$	0	0
$217$	40.0000i	2.71538i
$218$	− 10.0000i	− 0.677285i
$219$	0	0
$220$	0	0
$221$	2.00000	0.134535
$222$	0	0
$223$	− 4.00000i	− 0.267860i	−0.990991	−	0.133930i	$-0.957240\pi$
$223$	− 4.00000i	− 0.267860i	0.990991	−	0.133930i	$-0.0427597\pi$
$224$	20.0000	1.33631
$225$	0	0
$226$	14.0000	0.931266
$227$	− 4.00000i	− 0.265489i	−0.991150	−	0.132745i	$-0.957621\pi$
$227$	− 4.00000i	− 0.265489i	0.991150	−	0.132745i	$-0.0423790\pi$
$228$	0	0
$229$	22.0000	1.45380	0.726900	−	0.686743i	$-0.240960\pi$
$229$	22.0000	1.45380	0.726900	+	0.686743i	$0.240960\pi$
$230$	0	0
$231$	0	0
$232$	6.00000i	0.393919i
$233$	10.0000i	0.655122i	0.944830	+	0.327561i	$0.106227\pi$
$233$	10.0000i	0.655122i	−0.944830	+	0.327561i	$0.893773\pi$
$234$	0	0
$235$	0	0
$236$	6.00000	0.390567
$237$	0	0
$238$	− 8.00000i	− 0.518563i
$239$	−6.00000	−0.388108	−0.194054	−	0.980991i	$-0.562164\pi$
$239$	−6.00000	−0.388108	−0.194054	+	0.980991i	$0.562164\pi$
$240$	0	0
$241$	10.0000	0.644157	0.322078	−	0.946713i	$-0.395619\pi$
$241$	10.0000	0.644157	0.322078	+	0.946713i	$0.395619\pi$
$242$	− 7.00000i	− 0.449977i
$243$	0	0
$244$	2.00000	0.128037
$245$	0	0
$246$	0	0
$247$	6.00000i	0.381771i
$248$	− 30.0000i	− 1.90500i
$249$	0	0
$250$	0	0
$251$	−24.0000	−1.51487	−0.757433	−	0.652913i	$-0.773547\pi$
$251$	−24.0000	−1.51487	−0.757433	+	0.652913i	$0.773547\pi$
$252$	0	0
$253$	12.0000i	0.754434i
$254$	2.00000	0.125491
$255$	0	0
$256$	−17.0000	−1.06250
$257$	− 2.00000i	− 0.124757i	−0.998053	−	0.0623783i	$-0.980131\pi$
$257$	− 2.00000i	− 0.124757i	0.998053	−	0.0623783i	$-0.0198685\pi$
$258$	0	0
$259$	−8.00000	−0.497096
$260$	0	0
$261$	0	0
$262$	− 20.0000i	− 1.23560i
$263$	14.0000i	0.863277i	0.902047	+	0.431638i	$0.142064\pi$
$263$	14.0000i	0.863277i	−0.902047	+	0.431638i	$0.857936\pi$
$264$	0	0
$265$	0	0
$266$	24.0000	1.47153
$267$	0	0
$268$	− 4.00000i	− 0.244339i
$269$	6.00000	0.365826	0.182913	−	0.983129i	$-0.441447\pi$
$269$	6.00000	0.365826	0.182913	+	0.983129i	$0.441447\pi$
$270$	0	0
$271$	2.00000	0.121491	0.0607457	−	0.998153i	$-0.480652\pi$
$271$	2.00000	0.121491	0.0607457	+	0.998153i	$0.480652\pi$
$272$	2.00000i	0.121268i
$273$	0	0
$274$	−2.00000	−0.120824
$275$	0	0
$276$	0	0
$277$	− 14.0000i	− 0.841178i	−0.907251	−	0.420589i	$-0.861823\pi$
$277$	− 14.0000i	− 0.841178i	0.907251	−	0.420589i	$-0.138177\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	6.00000	0.357930	0.178965	−	0.983855i	$-0.442725\pi$
$281$	6.00000	0.357930	0.178965	+	0.983855i	$0.442725\pi$
$282$	0	0
$283$	− 2.00000i	− 0.118888i	−0.998232	−	0.0594438i	$-0.981067\pi$
$283$	− 2.00000i	− 0.118888i	0.998232	−	0.0594438i	$-0.0189327\pi$
$284$	−6.00000	−0.356034
$285$	0	0
$286$	2.00000	0.118262
$287$	− 24.0000i	− 1.41668i
$288$	0	0
$289$	13.0000	0.764706
$290$	0	0
$291$	0	0
$292$	6.00000i	0.351123i
$293$	22.0000i	1.28525i	0.766179	+	0.642627i	$0.222155\pi$
$293$	22.0000i	1.28525i	−0.766179	+	0.642627i	$0.777845\pi$
$294$	0	0
$295$	0	0
$296$	6.00000	0.348743
$297$	0	0
$298$	18.0000i	1.04271i
$299$	6.00000	0.346989
$300$	0	0
$301$	−40.0000	−2.30556
$302$	10.0000i	0.575435i
$303$	0	0
$304$	−6.00000	−0.344124
$305$	0	0
$306$	0	0
$307$	8.00000i	0.456584i	0.973593	+	0.228292i	$0.0733141\pi$
$307$	8.00000i	0.456584i	−0.973593	+	0.228292i	$0.926686\pi$
$308$	8.00000i	0.455842i
$309$	0	0
$310$	0	0
$311$	4.00000	0.226819	0.113410	−	0.993548i	$-0.463823\pi$
$311$	4.00000	0.226819	0.113410	+	0.993548i	$0.463823\pi$
$312$	0	0
$313$	22.0000i	1.24351i	0.783210	+	0.621757i	$0.213581\pi$
$313$	22.0000i	1.24351i	−0.783210	+	0.621757i	$0.786419\pi$
$314$	6.00000	0.338600
$315$	0	0
$316$	12.0000	0.675053
$317$	18.0000i	1.01098i	0.862832	+	0.505490i	$0.168688\pi$
$317$	18.0000i	1.01098i	−0.862832	+	0.505490i	$0.831312\pi$
$318$	0	0
$319$	−4.00000	−0.223957
$320$	0	0
$321$	0	0
$322$	− 24.0000i	− 1.33747i
$323$	− 12.0000i	− 0.667698i
$324$	0	0
$325$	0	0
$326$	−12.0000	−0.664619
$327$	0	0
$328$	18.0000i	0.993884i
$329$	−16.0000	−0.882109
$330$	0	0
$331$	−18.0000	−0.989369	−0.494685	−	0.869072i	$-0.664716\pi$
$331$	−18.0000	−0.989369	−0.494685	+	0.869072i	$0.664716\pi$
$332$	− 16.0000i	− 0.878114i
$333$	0	0
$334$	−12.0000	−0.656611
$335$	0	0
$336$	0	0
$337$	26.0000i	1.41631i	0.706057	+	0.708155i	$0.250472\pi$
$337$	26.0000i	1.41631i	−0.706057	+	0.708155i	$0.749528\pi$
$338$	− 1.00000i	− 0.0543928i
$339$	0	0
$340$	0	0
$341$	20.0000	1.08306
$342$	0	0
$343$	8.00000i	0.431959i
$344$	30.0000	1.61749
$345$	0	0
$346$	6.00000	0.322562
$347$	22.0000i	1.18102i	0.807030	+	0.590511i	$0.201074\pi$
$347$	22.0000i	1.18102i	−0.807030	+	0.590511i	$0.798926\pi$
$348$	0	0
$349$	30.0000	1.60586	0.802932	−	0.596071i	$-0.203272\pi$
$349$	30.0000	1.60586	0.802932	+	0.596071i	$0.203272\pi$
$350$	0	0
$351$	0	0
$352$	− 10.0000i	− 0.533002i
$353$	18.0000i	0.958043i	0.877803	+	0.479022i	$0.159008\pi$
$353$	18.0000i	0.958043i	−0.877803	+	0.479022i	$0.840992\pi$
$354$	0	0
$355$	0	0
$356$	2.00000	0.106000
$357$	0	0
$358$	12.0000i	0.634220i
$359$	−10.0000	−0.527780	−0.263890	−	0.964553i	$-0.585006\pi$
$359$	−10.0000	−0.527780	−0.263890	+	0.964553i	$0.585006\pi$
$360$	0	0
$361$	17.0000	0.894737
$362$	− 22.0000i	− 1.15629i
$363$	0	0
$364$	4.00000	0.209657
$365$	0	0
$366$	0	0
$367$	− 14.0000i	− 0.730794i	−0.930852	−	0.365397i	$-0.880933\pi$
$367$	− 14.0000i	− 0.730794i	0.930852	−	0.365397i	$-0.119067\pi$
$368$	6.00000i	0.312772i
$369$	0	0
$370$	0	0
$371$	8.00000	0.415339
$372$	0	0
$373$	− 34.0000i	− 1.76045i	−0.474554	−	0.880227i	$-0.657390\pi$
$373$	− 34.0000i	− 1.76045i	0.474554	−	0.880227i	$-0.342610\pi$
$374$	−4.00000	−0.206835
$375$	0	0
$376$	12.0000	0.618853
$377$	2.00000i	0.103005i
$378$	0	0
$379$	−10.0000	−0.513665	−0.256833	−	0.966456i	$-0.582679\pi$
$379$	−10.0000	−0.513665	−0.256833	+	0.966456i	$0.582679\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	12.0000i	0.613171i	0.951843	+	0.306586i	$0.0991866\pi$
$383$	12.0000i	0.613171i	−0.951843	+	0.306586i	$0.900813\pi$
$384$	0	0
$385$	0	0
$386$	−2.00000	−0.101797
$387$	0	0
$388$	− 2.00000i	− 0.101535i
$389$	−10.0000	−0.507020	−0.253510	−	0.967333i	$-0.581585\pi$
$389$	−10.0000	−0.507020	−0.253510	+	0.967333i	$0.581585\pi$
$390$	0	0
$391$	−12.0000	−0.606866
$392$	− 27.0000i	− 1.36371i
$393$	0	0
$394$	−6.00000	−0.302276
$395$	0	0
$396$	0	0
$397$	6.00000i	0.301131i	0.988600	+	0.150566i	$0.0481095\pi$
$397$	6.00000i	0.301131i	−0.988600	+	0.150566i	$0.951890\pi$
$398$	16.0000i	0.802008i
$399$	0	0
$400$	0	0
$401$	−10.0000	−0.499376	−0.249688	−	0.968326i	$-0.580328\pi$
$401$	−10.0000	−0.499376	−0.249688	+	0.968326i	$0.580328\pi$
$402$	0	0
$403$	− 10.0000i	− 0.498135i
$404$	18.0000	0.895533
$405$	0	0
$406$	8.00000	0.397033
$407$	4.00000i	0.198273i
$408$	0	0
$409$	−18.0000	−0.890043	−0.445021	−	0.895520i	$-0.646804\pi$
$409$	−18.0000	−0.890043	−0.445021	+	0.895520i	$0.646804\pi$
$410$	0	0
$411$	0	0
$412$	− 2.00000i	− 0.0985329i
$413$	− 24.0000i	− 1.18096i
$414$	0	0
$415$	0	0
$416$	−5.00000	−0.245145
$417$	0	0
$418$	− 12.0000i	− 0.586939i
$419$	−40.0000	−1.95413	−0.977064	−	0.212946i	$-0.931694\pi$
$419$	−40.0000	−1.95413	−0.977064	+	0.212946i	$0.931694\pi$
$420$	0	0
$421$	10.0000	0.487370	0.243685	−	0.969854i	$-0.421644\pi$
$421$	10.0000	0.487370	0.243685	+	0.969854i	$0.421644\pi$
$422$	12.0000i	0.584151i
$423$	0	0
$424$	−6.00000	−0.291386
$425$	0	0
$426$	0	0
$427$	− 8.00000i	− 0.387147i
$428$	− 10.0000i	− 0.483368i
$429$	0	0
$430$	0	0
$431$	−14.0000	−0.674356	−0.337178	−	0.941441i	$-0.609472\pi$
$431$	−14.0000	−0.674356	−0.337178	+	0.941441i	$0.609472\pi$
$432$	0	0
$433$	− 10.0000i	− 0.480569i	−0.970702	−	0.240285i	$-0.922759\pi$
$433$	− 10.0000i	− 0.480569i	0.970702	−	0.240285i	$-0.0772408\pi$
$434$	−40.0000	−1.92006
$435$	0	0
$436$	−10.0000	−0.478913
$437$	− 36.0000i	− 1.72211i
$438$	0	0
$439$	0	0		−	1.00000i	$-0.5\pi$
$439$	0	0			1.00000i	$0.5\pi$
$440$	0	0
$441$	0	0
$442$	2.00000i	0.0951303i
$443$	14.0000i	0.665160i	0.943075	+	0.332580i	$0.107919\pi$
$443$	14.0000i	0.665160i	−0.943075	+	0.332580i	$0.892081\pi$
$444$	0	0
$445$	0	0
$446$	4.00000	0.189405
$447$	0	0
$448$	28.0000i	1.32288i
$449$	−6.00000	−0.283158	−0.141579	−	0.989927i	$-0.545218\pi$
$449$	−6.00000	−0.283158	−0.141579	+	0.989927i	$0.545218\pi$
$450$	0	0
$451$	−12.0000	−0.565058
$452$	− 14.0000i	− 0.658505i
$453$	0	0
$454$	4.00000	0.187729
$455$	0	0
$456$	0	0
$457$	38.0000i	1.77757i	0.458329	+	0.888783i	$0.348448\pi$
$457$	38.0000i	1.77757i	−0.458329	+	0.888783i	$0.651552\pi$
$458$	22.0000i	1.02799i
$459$	0	0
$460$	0	0
$461$	−10.0000	−0.465746	−0.232873	−	0.972507i	$-0.574813\pi$
$461$	−10.0000	−0.465746	−0.232873	+	0.972507i	$0.574813\pi$
$462$	0	0
$463$	24.0000i	1.11537i	0.830051	+	0.557687i	$0.188311\pi$
$463$	24.0000i	1.11537i	−0.830051	+	0.557687i	$0.811689\pi$
$464$	−2.00000	−0.0928477
$465$	0	0
$466$	−10.0000	−0.463241
$467$	10.0000i	0.462745i	0.972865	+	0.231372i	$0.0743216\pi$
$467$	10.0000i	0.462745i	−0.972865	+	0.231372i	$0.925678\pi$
$468$	0	0
$469$	−16.0000	−0.738811
$470$	0	0
$471$	0	0
$472$	18.0000i	0.828517i
$473$	20.0000i	0.919601i
$474$	0	0
$475$	0	0
$476$	−8.00000	−0.366679
$477$	0	0
$478$	− 6.00000i	− 0.274434i
$479$	30.0000	1.37073	0.685367	−	0.728197i	$-0.259642\pi$
$479$	30.0000	1.37073	0.685367	+	0.728197i	$0.259642\pi$
$480$	0	0
$481$	2.00000	0.0911922
$482$	10.0000i	0.455488i
$483$	0	0
$484$	−7.00000	−0.318182
$485$	0	0
$486$	0	0
$487$	− 40.0000i	− 1.81257i	−0.422664	−	0.906287i	$-0.638905\pi$
$487$	− 40.0000i	− 1.81257i	0.422664	−	0.906287i	$-0.361095\pi$
$488$	6.00000i	0.271607i
$489$	0	0
$490$	0	0
$491$	24.0000	1.08310	0.541552	−	0.840667i	$-0.317837\pi$
$491$	24.0000	1.08310	0.541552	+	0.840667i	$0.317837\pi$
$492$	0	0
$493$	− 4.00000i	− 0.180151i
$494$	−6.00000	−0.269953
$495$	0	0
$496$	10.0000	0.449013
$497$	24.0000i	1.07655i
$498$	0	0
$499$	−10.0000	−0.447661	−0.223831	−	0.974628i	$-0.571856\pi$
$499$	−10.0000	−0.447661	−0.223831	+	0.974628i	$0.571856\pi$
$500$	0	0
$501$	0	0
$502$	− 24.0000i	− 1.07117i
$503$	− 18.0000i	− 0.802580i	−0.915951	−	0.401290i	$-0.868562\pi$
$503$	− 18.0000i	− 0.802580i	0.915951	−	0.401290i	$-0.131438\pi$
$504$	0	0
$505$	0	0
$506$	−12.0000	−0.533465
$507$	0	0
$508$	− 2.00000i	− 0.0887357i
$509$	−6.00000	−0.265945	−0.132973	−	0.991120i	$-0.542452\pi$
$509$	−6.00000	−0.265945	−0.132973	+	0.991120i	$0.542452\pi$
$510$	0	0
$511$	24.0000	1.06170
$512$	− 11.0000i	− 0.486136i
$513$	0	0
$514$	2.00000	0.0882162
$515$	0	0
$516$	0	0
$517$	8.00000i	0.351840i
$518$	− 8.00000i	− 0.351500i
$519$	0	0
$520$	0	0
$521$	−6.00000	−0.262865	−0.131432	−	0.991325i	$-0.541958\pi$
$521$	−6.00000	−0.262865	−0.131432	+	0.991325i	$0.541958\pi$
$522$	0	0
$523$	− 6.00000i	− 0.262362i	−0.991358	−	0.131181i	$-0.958123\pi$
$523$	− 6.00000i	− 0.262362i	0.991358	−	0.131181i	$-0.0418769\pi$
$524$	−20.0000	−0.873704
$525$	0	0
$526$	−14.0000	−0.610429
$527$	20.0000i	0.871214i
$528$	0	0
$529$	−13.0000	−0.565217
$530$	0	0
$531$	0	0
$532$	− 24.0000i	− 1.04053i
$533$	6.00000i	0.259889i
$534$	0	0
$535$	0	0
$536$	12.0000	0.518321
$537$	0	0
$538$	6.00000i	0.258678i
$539$	18.0000	0.775315
$540$	0	0
$541$	−22.0000	−0.945854	−0.472927	−	0.881102i	$-0.656803\pi$
$541$	−22.0000	−0.945854	−0.472927	+	0.881102i	$0.656803\pi$
$542$	2.00000i	0.0859074i
$543$	0	0
$544$	10.0000	0.428746
$545$	0	0
$546$	0	0
$547$	− 6.00000i	− 0.256541i	−0.991739	−	0.128271i	$-0.959057\pi$
$547$	− 6.00000i	− 0.256541i	0.991739	−	0.128271i	$-0.0409426\pi$
$548$	2.00000i	0.0854358i
$549$	0	0
$550$	0	0
$551$	12.0000	0.511217
$552$	0	0
$553$	− 48.0000i	− 2.04117i
$554$	14.0000	0.594803
$555$	0	0
$556$	0	0
$557$	26.0000i	1.10166i	0.834619	+	0.550828i	$0.185688\pi$
$557$	26.0000i	1.10166i	−0.834619	+	0.550828i	$0.814312\pi$
$558$	0	0
$559$	10.0000	0.422955
$560$	0	0
$561$	0	0
$562$	6.00000i	0.253095i
$563$	− 22.0000i	− 0.927189i	−0.886047	−	0.463595i	$-0.846559\pi$
$563$	− 22.0000i	− 0.927189i	0.886047	−	0.463595i	$-0.153441\pi$
$564$	0	0
$565$	0	0
$566$	2.00000	0.0840663
$567$	0	0
$568$	− 18.0000i	− 0.755263i
$569$	−26.0000	−1.08998	−0.544988	−	0.838444i	$-0.683466\pi$
$569$	−26.0000	−1.08998	−0.544988	+	0.838444i	$0.683466\pi$
$570$	0	0
$571$	24.0000	1.00437	0.502184	−	0.864761i	$-0.332530\pi$
$571$	24.0000	1.00437	0.502184	+	0.864761i	$0.332530\pi$
$572$	− 2.00000i	− 0.0836242i
$573$	0	0
$574$	24.0000	1.00174
$575$	0	0
$576$	0	0
$577$	2.00000i	0.0832611i	0.999133	+	0.0416305i	$0.0132552\pi$
$577$	2.00000i	0.0832611i	−0.999133	+	0.0416305i	$0.986745\pi$
$578$	13.0000i	0.540729i
$579$	0	0
$580$	0	0
$581$	−64.0000	−2.65517
$582$	0	0
$583$	− 4.00000i	− 0.165663i
$584$	−18.0000	−0.744845
$585$	0	0
$586$	−22.0000	−0.908812
$587$	44.0000i	1.81607i	0.418890	+	0.908037i	$0.362419\pi$
$587$	44.0000i	1.81607i	−0.418890	+	0.908037i	$0.637581\pi$
$588$	0	0
$589$	−60.0000	−2.47226
$590$	0	0
$591$	0	0
$592$	2.00000i	0.0821995i
$593$	14.0000i	0.574911i	0.957794	+	0.287456i	$0.0928094\pi$
$593$	14.0000i	0.574911i	−0.957794	+	0.287456i	$0.907191\pi$
$594$	0	0
$595$	0	0
$596$	18.0000	0.737309
$597$	0	0
$598$	6.00000i	0.245358i
$599$	−4.00000	−0.163436	−0.0817178	−	0.996656i	$-0.526041\pi$
$599$	−4.00000	−0.163436	−0.0817178	+	0.996656i	$0.526041\pi$
$600$	0	0
$601$	10.0000	0.407909	0.203954	−	0.978980i	$-0.434621\pi$
$601$	10.0000	0.407909	0.203954	+	0.978980i	$0.434621\pi$
$602$	− 40.0000i	− 1.63028i
$603$	0	0
$604$	10.0000	0.406894
$605$	0	0
$606$	0	0
$607$	34.0000i	1.38002i	0.723801	+	0.690009i	$0.242393\pi$
$607$	34.0000i	1.38002i	−0.723801	+	0.690009i	$0.757607\pi$
$608$	30.0000i	1.21666i
$609$	0	0
$610$	0	0
$611$	4.00000	0.161823
$612$	0	0
$613$	6.00000i	0.242338i	0.992632	+	0.121169i	$0.0386643\pi$
$613$	6.00000i	0.242338i	−0.992632	+	0.121169i	$0.961336\pi$
$614$	−8.00000	−0.322854
$615$	0	0
$616$	−24.0000	−0.966988
$617$	42.0000i	1.69086i	0.534089	+	0.845428i	$0.320655\pi$
$617$	42.0000i	1.69086i	−0.534089	+	0.845428i	$0.679345\pi$
$618$	0	0
$619$	2.00000	0.0803868	0.0401934	−	0.999192i	$-0.487203\pi$
$619$	2.00000	0.0803868	0.0401934	+	0.999192i	$0.487203\pi$
$620$	0	0
$621$	0	0
$622$	4.00000i	0.160385i
$623$	− 8.00000i	− 0.320513i
$624$	0	0
$625$	0	0
$626$	−22.0000	−0.879297
$627$	0	0
$628$	− 6.00000i	− 0.239426i
$629$	−4.00000	−0.159490
$630$	0	0
$631$	−14.0000	−0.557331	−0.278666	−	0.960388i	$-0.589892\pi$
$631$	−14.0000	−0.557331	−0.278666	+	0.960388i	$0.589892\pi$
$632$	36.0000i	1.43200i
$633$	0	0
$634$	−18.0000	−0.714871
$635$	0	0
$636$	0	0
$637$	− 9.00000i	− 0.356593i
$638$	− 4.00000i	− 0.158362i
$639$	0	0
$640$	0	0
$641$	46.0000	1.81689	0.908445	−	0.418004i	$-0.137270\pi$
$641$	46.0000	1.81689	0.908445	+	0.418004i	$0.137270\pi$
$642$	0	0
$643$	16.0000i	0.630978i	0.948929	+	0.315489i	$0.102169\pi$
$643$	16.0000i	0.630978i	−0.948929	+	0.315489i	$0.897831\pi$
$644$	−24.0000	−0.945732
$645$	0	0
$646$	12.0000	0.472134
$647$	− 38.0000i	− 1.49393i	−0.664861	−	0.746967i	$-0.731509\pi$
$647$	− 38.0000i	− 1.49393i	0.664861	−	0.746967i	$-0.268491\pi$
$648$	0	0
$649$	−12.0000	−0.471041
$650$	0	0
$651$	0	0
$652$	12.0000i	0.469956i
$653$	− 6.00000i	− 0.234798i	−0.993085	−	0.117399i	$-0.962544\pi$
$653$	− 6.00000i	− 0.234798i	0.993085	−	0.117399i	$-0.0374557\pi$
$654$	0	0
$655$	0	0
$656$	−6.00000	−0.234261
$657$	0	0
$658$	− 16.0000i	− 0.623745i
$659$	0	0		−	1.00000i	$-0.5\pi$
$659$	0	0			1.00000i	$0.5\pi$
$660$	0	0
$661$	2.00000	0.0777910	0.0388955	−	0.999243i	$-0.487616\pi$
$661$	2.00000	0.0777910	0.0388955	+	0.999243i	$0.487616\pi$
$662$	− 18.0000i	− 0.699590i
$663$	0	0
$664$	48.0000	1.86276
$665$	0	0
$666$	0	0
$667$	− 12.0000i	− 0.464642i
$668$	12.0000i	0.464294i
$669$	0	0
$670$	0	0
$671$	−4.00000	−0.154418
$672$	0	0
$673$	30.0000i	1.15642i	0.815890	+	0.578208i	$0.196248\pi$
$673$	30.0000i	1.15642i	−0.815890	+	0.578208i	$0.803752\pi$
$674$	−26.0000	−1.00148
$675$	0	0
$676$	−1.00000	−0.0384615
$677$	6.00000i	0.230599i	0.993331	+	0.115299i	$0.0367827\pi$
$677$	6.00000i	0.230599i	−0.993331	+	0.115299i	$0.963217\pi$
$678$	0	0
$679$	−8.00000	−0.307012
$680$	0	0
$681$	0	0
$682$	20.0000i	0.765840i
$683$	− 20.0000i	− 0.765279i	−0.923898	−	0.382639i	$-0.875015\pi$
$683$	− 20.0000i	− 0.765279i	0.923898	−	0.382639i	$-0.124985\pi$
$684$	0	0
$685$	0	0
$686$	−8.00000	−0.305441
$687$	0	0
$688$	10.0000i	0.381246i
$689$	−2.00000	−0.0761939
$690$	0	0
$691$	22.0000	0.836919	0.418460	−	0.908235i	$-0.362570\pi$
$691$	22.0000	0.836919	0.418460	+	0.908235i	$0.362570\pi$
$692$	− 6.00000i	− 0.228086i
$693$	0	0
$694$	−22.0000	−0.835109
$695$	0	0
$696$	0	0
$697$	− 12.0000i	− 0.454532i
$698$	30.0000i	1.13552i
$699$	0	0
$700$	0	0
$701$	26.0000	0.982006	0.491003	−	0.871158i	$-0.336630\pi$
$701$	26.0000	0.982006	0.491003	+	0.871158i	$0.336630\pi$
$702$	0	0
$703$	− 12.0000i	− 0.452589i
$704$	14.0000	0.527645
$705$	0	0
$706$	−18.0000	−0.677439
$707$	− 72.0000i	− 2.70784i
$708$	0	0
$709$	−10.0000	−0.375558	−0.187779	−	0.982211i	$-0.560129\pi$
$709$	−10.0000	−0.375558	−0.187779	+	0.982211i	$0.560129\pi$
$710$	0	0
$711$	0	0
$712$	6.00000i	0.224860i
$713$	60.0000i	2.24702i
$714$	0	0
$715$	0	0
$716$	12.0000	0.448461
$717$	0	0
$718$	− 10.0000i	− 0.373197i
$719$	−32.0000	−1.19340	−0.596699	−	0.802465i	$-0.703521\pi$
$719$	−32.0000	−1.19340	−0.596699	+	0.802465i	$0.703521\pi$
$720$	0	0
$721$	−8.00000	−0.297936
$722$	17.0000i	0.632674i
$723$	0	0
$724$	−22.0000	−0.817624
$725$	0	0
$726$	0	0
$727$	− 18.0000i	− 0.667583i	−0.942647	−	0.333792i	$-0.891672\pi$
$727$	− 18.0000i	− 0.667583i	0.942647	−	0.333792i	$-0.108328\pi$
$728$	12.0000i	0.444750i
$729$	0	0
$730$	0	0
$731$	−20.0000	−0.739727
$732$	0	0
$733$	− 34.0000i	− 1.25582i	−0.778287	−	0.627909i	$-0.783911\pi$
$733$	− 34.0000i	− 1.25582i	0.778287	−	0.627909i	$-0.216089\pi$
$734$	14.0000	0.516749
$735$	0	0
$736$	30.0000	1.10581
$737$	8.00000i	0.294684i
$738$	0	0
$739$	−6.00000	−0.220714	−0.110357	−	0.993892i	$-0.535199\pi$
$739$	−6.00000	−0.220714	−0.110357	+	0.993892i	$0.535199\pi$
$740$	0	0
$741$	0	0
$742$	8.00000i	0.293689i
$743$	− 12.0000i	− 0.440237i	−0.975473	−	0.220119i	$-0.929356\pi$
$743$	− 12.0000i	− 0.440237i	0.975473	−	0.220119i	$-0.0706445\pi$
$744$	0	0
$745$	0	0
$746$	34.0000	1.24483
$747$	0	0
$748$	4.00000i	0.146254i
$749$	−40.0000	−1.46157
$750$	0	0
$751$	−28.0000	−1.02173	−0.510867	−	0.859660i	$-0.670676\pi$
$751$	−28.0000	−1.02173	−0.510867	+	0.859660i	$0.670676\pi$
$752$	4.00000i	0.145865i
$753$	0	0
$754$	−2.00000	−0.0728357
$755$	0	0
$756$	0	0
$757$	2.00000i	0.0726912i	0.999339	+	0.0363456i	$0.0115717\pi$
$757$	2.00000i	0.0726912i	−0.999339	+	0.0363456i	$0.988428\pi$
$758$	− 10.0000i	− 0.363216i
$759$	0	0
$760$	0	0
$761$	−18.0000	−0.652499	−0.326250	−	0.945284i	$-0.605785\pi$
$761$	−18.0000	−0.652499	−0.326250	+	0.945284i	$0.605785\pi$
$762$	0	0
$763$	40.0000i	1.44810i
$764$	0	0
$765$	0	0
$766$	−12.0000	−0.433578
$767$	6.00000i	0.216647i
$768$	0	0
$769$	−26.0000	−0.937584	−0.468792	−	0.883309i	$-0.655311\pi$
$769$	−26.0000	−0.937584	−0.468792	+	0.883309i	$0.655311\pi$
$770$	0	0
$771$	0	0
$772$	2.00000i	0.0719816i
$773$	− 2.00000i	− 0.0719350i	−0.999353	−	0.0359675i	$-0.988549\pi$
$773$	− 2.00000i	− 0.0719350i	0.999353	−	0.0359675i	$-0.0114513\pi$
$774$	0	0
$775$	0	0
$776$	6.00000	0.215387
$777$	0	0
$778$	− 10.0000i	− 0.358517i
$779$	36.0000	1.28983
$780$	0	0
$781$	12.0000	0.429394
$782$	− 12.0000i	− 0.429119i
$783$	0	0
$784$	9.00000	0.321429
$785$	0	0
$786$	0	0
$787$	8.00000i	0.285169i	0.989783	+	0.142585i	$0.0455413\pi$
$787$	8.00000i	0.285169i	−0.989783	+	0.142585i	$0.954459\pi$
$788$	6.00000i	0.213741i
$789$	0	0
$790$	0	0
$791$	−56.0000	−1.99113
$792$	0	0
$793$	2.00000i	0.0710221i
$794$	−6.00000	−0.212932
$795$	0	0
$796$	16.0000	0.567105
$797$	− 42.0000i	− 1.48772i	−0.668338	−	0.743858i	$-0.732994\pi$
$797$	− 42.0000i	− 1.48772i	0.668338	−	0.743858i	$-0.267006\pi$
$798$	0	0
$799$	−8.00000	−0.283020
$800$	0	0
$801$	0	0
$802$	− 10.0000i	− 0.353112i
$803$	− 12.0000i	− 0.423471i
$804$	0	0
$805$	0	0
$806$	10.0000	0.352235
$807$	0	0
$808$	54.0000i	1.89971i
$809$	−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
$811$	−42.0000	−1.47482	−0.737410	−	0.675446i	$-0.763951\pi$
$811$	−42.0000	−1.47482	−0.737410	+	0.675446i	$0.763951\pi$
$812$	− 8.00000i	− 0.280745i
$813$	0	0
$814$	−4.00000	−0.140200
$815$	0	0
$816$	0	0
$817$	− 60.0000i	− 2.09913i
$818$	− 18.0000i	− 0.629355i
$819$	0	0
$820$	0	0
$821$	−50.0000	−1.74501	−0.872506	−	0.488603i	$-0.837507\pi$
$821$	−50.0000	−1.74501	−0.872506	+	0.488603i	$0.837507\pi$
$822$	0	0
$823$	46.0000i	1.60346i	0.597687	+	0.801730i	$0.296087\pi$
$823$	46.0000i	1.60346i	−0.597687	+	0.801730i	$0.703913\pi$
$824$	6.00000	0.209020
$825$	0	0
$826$	24.0000	0.835067
$827$	32.0000i	1.11275i	0.830932	+	0.556375i	$0.187808\pi$
$827$	32.0000i	1.11275i	−0.830932	+	0.556375i	$0.812192\pi$
$828$	0	0
$829$	−34.0000	−1.18087	−0.590434	−	0.807086i	$-0.701044\pi$
$829$	−34.0000	−1.18087	−0.590434	+	0.807086i	$0.701044\pi$
$830$	0	0
$831$	0	0
$832$	− 7.00000i	− 0.242681i
$833$	18.0000i	0.623663i
$834$	0	0
$835$	0	0
$836$	−12.0000	−0.415029
$837$	0	0
$838$	− 40.0000i	− 1.38178i
$839$	38.0000	1.31191	0.655953	−	0.754802i	$-0.272267\pi$
$839$	38.0000	1.31191	0.655953	+	0.754802i	$0.272267\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	10.0000i	0.344623i
$843$	0	0
$844$	12.0000	0.413057
$845$	0	0
$846$	0	0
$847$	28.0000i	0.962091i
$848$	− 2.00000i	− 0.0686803i
$849$	0	0
$850$	0	0
$851$	−12.0000	−0.411355
$852$	0	0
$853$	− 6.00000i	− 0.205436i	−0.994711	−	0.102718i	$-0.967246\pi$
$853$	− 6.00000i	− 0.205436i	0.994711	−	0.102718i	$-0.0327539\pi$
$854$	8.00000	0.273754
$855$	0	0
$856$	30.0000	1.02538
$857$	− 18.0000i	− 0.614868i	−0.951569	−	0.307434i	$-0.900530\pi$
$857$	− 18.0000i	− 0.614868i	0.951569	−	0.307434i	$-0.0994704\pi$
$858$	0	0
$859$	−40.0000	−1.36478	−0.682391	−	0.730987i	$-0.739060\pi$
$859$	−40.0000	−1.36478	−0.682391	+	0.730987i	$0.739060\pi$
$860$	0	0
$861$	0	0
$862$	− 14.0000i	− 0.476842i
$863$	12.0000i	0.408485i	0.978920	+	0.204242i	$0.0654731\pi$
$863$	12.0000i	0.408485i	−0.978920	+	0.204242i	$0.934527\pi$
$864$	0	0
$865$	0	0
$866$	10.0000	0.339814
$867$	0	0
$868$	40.0000i	1.35769i
$869$	−24.0000	−0.814144
$870$	0	0
$871$	4.00000	0.135535
$872$	− 30.0000i	− 1.01593i
$873$	0	0
$874$	36.0000	1.21772
$875$	0	0
$876$	0	0
$877$	18.0000i	0.607817i	0.952701	+	0.303908i	$0.0982917\pi$
$877$	18.0000i	0.607817i	−0.952701	+	0.303908i	$0.901708\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−38.0000	−1.28025	−0.640126	−	0.768270i	$-0.721118\pi$
$881$	−38.0000	−1.28025	−0.640126	+	0.768270i	$0.721118\pi$
$882$	0	0
$883$	22.0000i	0.740359i	0.928960	+	0.370179i	$0.120704\pi$
$883$	22.0000i	0.740359i	−0.928960	+	0.370179i	$0.879296\pi$
$884$	2.00000	0.0672673
$885$	0	0
$886$	−14.0000	−0.470339
$887$	− 58.0000i	− 1.94745i	−0.227728	−	0.973725i	$-0.573130\pi$
$887$	− 58.0000i	− 1.94745i	0.227728	−	0.973725i	$-0.426870\pi$
$888$	0	0
$889$	−8.00000	−0.268311
$890$	0	0
$891$	0	0
$892$	− 4.00000i	− 0.133930i
$893$	− 24.0000i	− 0.803129i
$894$	0	0
$895$	0	0
$896$	12.0000	0.400892
$897$	0	0
$898$	− 6.00000i	− 0.200223i
$899$	−20.0000	−0.667037
$900$	0	0
$901$	4.00000	0.133259
$902$	− 12.0000i	− 0.399556i
$903$	0	0
$904$	42.0000	1.39690
$905$	0	0
$906$	0	0
$907$	− 34.0000i	− 1.12895i	−0.825450	−	0.564476i	$-0.809078\pi$
$907$	− 34.0000i	− 1.12895i	0.825450	−	0.564476i	$-0.190922\pi$
$908$	− 4.00000i	− 0.132745i
$909$	0	0
$910$	0	0
$911$	48.0000	1.59031	0.795155	−	0.606406i	$-0.207389\pi$
$911$	48.0000	1.59031	0.795155	+	0.606406i	$0.207389\pi$
$912$	0	0
$913$	32.0000i	1.05905i
$914$	−38.0000	−1.25693
$915$	0	0
$916$	22.0000	0.726900
$917$	80.0000i	2.64183i
$918$	0	0
$919$	52.0000	1.71532	0.857661	−	0.514216i	$-0.171917\pi$
$919$	52.0000	1.71532	0.857661	+	0.514216i	$0.171917\pi$
$920$	0	0
$921$	0	0
$922$	− 10.0000i	− 0.329332i
$923$	− 6.00000i	− 0.197492i
$924$	0	0
$925$	0	0
$926$	−24.0000	−0.788689
$927$	0	0
$928$	10.0000i	0.328266i
$929$	−38.0000	−1.24674	−0.623370	−	0.781927i	$-0.714237\pi$
$929$	−38.0000	−1.24674	−0.623370	+	0.781927i	$0.714237\pi$
$930$	0	0
$931$	−54.0000	−1.76978
$932$	10.0000i	0.327561i
$933$	0	0
$934$	−10.0000	−0.327210
$935$	0	0
$936$	0	0
$937$	− 30.0000i	− 0.980057i	−0.871706	−	0.490029i	$-0.836986\pi$
$937$	− 30.0000i	− 0.980057i	0.871706	−	0.490029i	$-0.163014\pi$
$938$	− 16.0000i	− 0.522419i
$939$	0	0
$940$	0	0
$941$	−18.0000	−0.586783	−0.293392	−	0.955992i	$-0.594784\pi$
$941$	−18.0000	−0.586783	−0.293392	+	0.955992i	$0.594784\pi$
$942$	0	0
$943$	− 36.0000i	− 1.17232i
$944$	−6.00000	−0.195283
$945$	0	0
$946$	−20.0000	−0.650256
$947$	− 24.0000i	− 0.779895i	−0.920837	−	0.389948i	$-0.872493\pi$
$947$	− 24.0000i	− 0.779895i	0.920837	−	0.389948i	$-0.127507\pi$
$948$	0	0
$949$	−6.00000	−0.194768
$950$	0	0
$951$	0	0
$952$	− 24.0000i	− 0.777844i
$953$	18.0000i	0.583077i	0.956559	+	0.291539i	$0.0941672\pi$
$953$	18.0000i	0.583077i	−0.956559	+	0.291539i	$0.905833\pi$
$954$	0	0
$955$	0	0
$956$	−6.00000	−0.194054
$957$	0	0
$958$	30.0000i	0.969256i
$959$	8.00000	0.258333
$960$	0	0
$961$	69.0000	2.22581
$962$	2.00000i	0.0644826i
$963$	0	0
$964$	10.0000	0.322078
$965$	0	0
$966$	0	0
$967$	− 8.00000i	− 0.257263i	−0.991692	−	0.128631i	$-0.958942\pi$
$967$	− 8.00000i	− 0.257263i	0.991692	−	0.128631i	$-0.0410584\pi$
$968$	− 21.0000i	− 0.674966i
$969$	0	0
$970$	0	0
$971$	28.0000	0.898563	0.449281	−	0.893390i	$-0.351680\pi$
$971$	28.0000	0.898563	0.449281	+	0.893390i	$0.351680\pi$
$972$	0	0
$973$	0	0
$974$	40.0000	1.28168
$975$	0	0
$976$	−2.00000	−0.0640184
$977$	− 18.0000i	− 0.575871i	−0.957650	−	0.287936i	$-0.907031\pi$
$977$	− 18.0000i	− 0.575871i	0.957650	−	0.287936i	$-0.0929689\pi$
$978$	0	0
$979$	−4.00000	−0.127841
$980$	0	0
$981$	0	0
$982$	24.0000i	0.765871i
$983$	56.0000i	1.78612i	0.449935	+	0.893061i	$0.351447\pi$
$983$	56.0000i	1.78612i	−0.449935	+	0.893061i	$0.648553\pi$
$984$	0	0
$985$	0	0
$986$	4.00000	0.127386
$987$	0	0
$988$	6.00000i	0.190885i
$989$	−60.0000	−1.90789
$990$	0	0
$991$	48.0000	1.52477	0.762385	−	0.647124i	$-0.224028\pi$
$991$	48.0000	1.52477	0.762385	+	0.647124i	$0.224028\pi$
$992$	− 50.0000i	− 1.58750i
$993$	0	0
$994$	−24.0000	−0.761234
$995$	0	0
$996$	0	0
$997$	− 22.0000i	− 0.696747i	−0.937356	−	0.348373i	$-0.886734\pi$
$997$	− 22.0000i	− 0.696747i	0.937356	−	0.348373i	$-0.113266\pi$
$998$	− 10.0000i	− 0.316544i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2925.2.c.h.2224.2		2
3.2	odd	2		325.2.b.b.274.1		2
5.2	odd	4		2925.2.a.f.1.1		1
5.3	odd	4		585.2.a.h.1.1		1
5.4	even	2	inner	2925.2.c.h.2224.1		2
15.2	even	4		325.2.a.d.1.1		1
15.8	even	4		65.2.a.a.1.1	✓	1
15.14	odd	2		325.2.b.b.274.2		2
20.3	even	4		9360.2.a.ca.1.1		1
60.23	odd	4		1040.2.a.f.1.1		1
60.47	odd	4		5200.2.a.d.1.1		1
65.38	odd	4		7605.2.a.f.1.1		1
105.83	odd	4		3185.2.a.e.1.1		1
120.53	even	4		4160.2.a.q.1.1		1
120.83	odd	4		4160.2.a.f.1.1		1
165.98	odd	4		7865.2.a.c.1.1		1
195.8	odd	4		845.2.c.a.506.1		2
195.23	even	12		845.2.e.a.191.1		2
195.38	even	4		845.2.a.a.1.1		1
195.68	even	12		845.2.e.b.191.1		2
195.77	even	4		4225.2.a.g.1.1		1
195.83	odd	4		845.2.c.a.506.2		2
195.98	odd	12		845.2.m.b.361.1		4
195.113	even	12		845.2.e.b.146.1		2
195.128	odd	12		845.2.m.b.316.2		4
195.158	odd	12		845.2.m.b.316.1		4
195.173	even	12		845.2.e.a.146.1		2
195.188	odd	12		845.2.m.b.361.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
65.2.a.a.1.1	✓	1	15.8	even	4
325.2.a.d.1.1		1	15.2	even	4
325.2.b.b.274.1		2	3.2	odd	2
325.2.b.b.274.2		2	15.14	odd	2
585.2.a.h.1.1		1	5.3	odd	4
845.2.a.a.1.1		1	195.38	even	4
845.2.c.a.506.1		2	195.8	odd	4
845.2.c.a.506.2		2	195.83	odd	4
845.2.e.a.146.1		2	195.173	even	12
845.2.e.a.191.1		2	195.23	even	12
845.2.e.b.146.1		2	195.113	even	12
845.2.e.b.191.1		2	195.68	even	12
845.2.m.b.316.1		4	195.158	odd	12
845.2.m.b.316.2		4	195.128	odd	12
845.2.m.b.361.1		4	195.98	odd	12
845.2.m.b.361.2		4	195.188	odd	12
1040.2.a.f.1.1		1	60.23	odd	4
2925.2.a.f.1.1		1	5.2	odd	4
2925.2.c.h.2224.1		2	5.4	even	2	inner
2925.2.c.h.2224.2		2	1.1	even	1	trivial
3185.2.a.e.1.1		1	105.83	odd	4
4160.2.a.f.1.1		1	120.83	odd	4
4160.2.a.q.1.1		1	120.53	even	4
4225.2.a.g.1.1		1	195.77	even	4
5200.2.a.d.1.1		1	60.47	odd	4
7605.2.a.f.1.1		1	65.38	odd	4
7865.2.a.c.1.1		1	165.98	odd	4
9360.2.a.ca.1.1		1	20.3	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:	\( N \)	\(=\)	\( 2925 = 3^{2} \cdot 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2925.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} +1.00000 q^{4} -4.00000i q^{7} +3.00000i q^{8} +O(q^{10})\)	\(q+1.00000i q^{2} +1.00000 q^{4} -4.00000i q^{7} +3.00000i q^{8} -2.00000 q^{11} +1.00000i q^{13} +4.00000 q^{14} -1.00000 q^{16} -2.00000i q^{17} +6.00000 q^{19} -2.00000i q^{22} -6.00000i q^{23} -1.00000 q^{26} -4.00000i q^{28} +2.00000 q^{29} -10.0000 q^{31} +5.00000i q^{32} +2.00000 q^{34} -2.00000i q^{37} +6.00000i q^{38} +6.00000 q^{41} -10.0000i q^{43} -2.00000 q^{44} +6.00000 q^{46} -4.00000i q^{47} -9.00000 q^{49} +1.00000i q^{52} +2.00000i q^{53} +12.0000 q^{56} +2.00000i q^{58} +6.00000 q^{59} +2.00000 q^{61} -10.0000i q^{62} -7.00000 q^{64} -4.00000i q^{67} -2.00000i q^{68} -6.00000 q^{71} +6.00000i q^{73} +2.00000 q^{74} +6.00000 q^{76} +8.00000i q^{77} +12.0000 q^{79} +6.00000i q^{82} -16.0000i q^{83} +10.0000 q^{86} -6.00000i q^{88} +2.00000 q^{89} +4.00000 q^{91} -6.00000i q^{92} +4.00000 q^{94} -2.00000i q^{97} -9.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{4}+O(q^{10}) \) 2 * q + 2 * q^4	\( 2 q + 2 q^{4} - 4 q^{11} + 8 q^{14} - 2 q^{16} + 12 q^{19} - 2 q^{26} + 4 q^{29} - 20 q^{31} + 4 q^{34} + 12 q^{41} - 4 q^{44} + 12 q^{46} - 18 q^{49} + 24 q^{56} + 12 q^{59} + 4 q^{61} - 14 q^{64} - 12 q^{71} + 4 q^{74} + 12 q^{76} + 24 q^{79} + 20 q^{86} + 4 q^{89} + 8 q^{91} + 8 q^{94}+O(q^{100}) \) 2 * q + 2 * q^4 - 4 * q^11 + 8 * q^14 - 2 * q^16 + 12 * q^19 - 2 * q^26 + 4 * q^29 - 20 * q^31 + 4 * q^34 + 12 * q^41 - 4 * q^44 + 12 * q^46 - 18 * q^49 + 24 * q^56 + 12 * q^59 + 4 * q^61 - 14 * q^64 - 12 * q^71 + 4 * q^74 + 12 * q^76 + 24 * q^79 + 20 * q^86 + 4 * q^89 + 8 * q^91 + 8 * q^94

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more