LMFDB - Embedded newform 3150.2.g.f.2899.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3150,2,Mod(2899,3150)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3150, 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("3150.2899");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3150 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3150.g (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:	$25.1528766367$
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 350)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2899.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	3150.2899
Dual form			3150.2.g.f.2899.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{2} -1.00000 q^{4} +1.00000i q^{7} +1.00000i q^{8} +O(q^{10})$	$q-1.00000i q^{2} -1.00000 q^{4} +1.00000i q^{7} +1.00000i q^{8} -3.00000 q^{11} -2.00000i q^{13} +1.00000 q^{14} +1.00000 q^{16} -3.00000i q^{17} +7.00000 q^{19} +3.00000i q^{22} -2.00000 q^{26} -1.00000i q^{28} -6.00000 q^{29} -4.00000 q^{31} -1.00000i q^{32} -3.00000 q^{34} +8.00000i q^{37} -7.00000i q^{38} +9.00000 q^{41} -8.00000i q^{43} +3.00000 q^{44} +6.00000i q^{47} -1.00000 q^{49} +2.00000i q^{52} -12.0000i q^{53} -1.00000 q^{56} +6.00000i q^{58} +12.0000 q^{59} -10.0000 q^{61} +4.00000i q^{62} -1.00000 q^{64} -7.00000i q^{67} +3.00000i q^{68} -6.00000 q^{71} -5.00000i q^{73} +8.00000 q^{74} -7.00000 q^{76} -3.00000i q^{77} -14.0000 q^{79} -9.00000i q^{82} -9.00000i q^{83} -8.00000 q^{86} -3.00000i q^{88} -15.0000 q^{89} +2.00000 q^{91} +6.00000 q^{94} -10.0000i q^{97} +1.00000i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{4}+O(q^{10}) $ 2 * q - 2 * q^4	$ 2 q - 2 q^{4} - 6 q^{11} + 2 q^{14} + 2 q^{16} + 14 q^{19} - 4 q^{26} - 12 q^{29} - 8 q^{31} - 6 q^{34} + 18 q^{41} + 6 q^{44} - 2 q^{49} - 2 q^{56} + 24 q^{59} - 20 q^{61} - 2 q^{64} - 12 q^{71} + 16 q^{74} - 14 q^{76} - 28 q^{79} - 16 q^{86} - 30 q^{89} + 4 q^{91} + 12 q^{94}+O(q^{100}) $ 2 * q - 2 * q^4 - 6 * q^11 + 2 * q^14 + 2 * q^16 + 14 * q^19 - 4 * q^26 - 12 * q^29 - 8 * q^31 - 6 * q^34 + 18 * q^41 + 6 * q^44 - 2 * q^49 - 2 * q^56 + 24 * q^59 - 20 * q^61 - 2 * q^64 - 12 * q^71 + 16 * q^74 - 14 * q^76 - 28 * q^79 - 16 * q^86 - 30 * q^89 + 4 * q^91 + 12 * q^94

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3150\mathbb{Z}\right)^\times$.

$n$	$127$	$451$	$2801$
$\chi(n)$	$-1$	$1$	$1$

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
$2$	− 1.00000i	− 0.707107i
$3$	0	0
$3$	0	0
$4$	−1.00000	−0.500000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	1.00000i	0.377964i
$7$	1.00000i	0.377964i
$8$	1.00000i	0.353553i
$9$	0	0
$10$	0	0
$11$	−3.00000	−0.904534	−0.452267	−	0.891883i	$-0.649385\pi$
$11$	−3.00000	−0.904534	−0.452267	+	0.891883i	$0.649385\pi$
$12$	0	0
$13$	− 2.00000i	− 0.554700i	−0.960769	−	0.277350i	$-0.910544\pi$
$13$	− 2.00000i	− 0.554700i	0.960769	−	0.277350i	$-0.0894562\pi$
$14$	1.00000	0.267261
$15$	0	0
$16$	1.00000	0.250000
$17$	− 3.00000i	− 0.727607i	−0.931476	−	0.363803i	$-0.881478\pi$
$17$	− 3.00000i	− 0.727607i	0.931476	−	0.363803i	$-0.118522\pi$
$18$	0	0
$19$	7.00000	1.60591	0.802955	−	0.596040i	$-0.203260\pi$
$19$	7.00000	1.60591	0.802955	+	0.596040i	$0.203260\pi$
$20$	0	0
$21$	0	0
$22$	3.00000i	0.639602i
$23$	0	0	1.00000			$0$
$23$	0	0	−1.00000			$\pi$
$24$	0	0
$25$	0	0
$26$	−2.00000	−0.392232
$27$	0	0
$28$	− 1.00000i	− 0.188982i
$29$	−6.00000	−1.11417	−0.557086	−	0.830455i	$-0.688081\pi$
$29$	−6.00000	−1.11417	−0.557086	+	0.830455i	$0.688081\pi$
$30$	0	0
$31$	−4.00000	−0.718421	−0.359211	−	0.933257i	$-0.616954\pi$
$31$	−4.00000	−0.718421	−0.359211	+	0.933257i	$0.616954\pi$
$32$	− 1.00000i	− 0.176777i
$33$	0	0
$34$	−3.00000	−0.514496
$35$	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	$0.771573\pi$
$38$	− 7.00000i	− 1.13555i
$39$	0	0
$40$	0	0
$41$	9.00000	1.40556	0.702782	−	0.711405i	$-0.251941\pi$
$41$	9.00000	1.40556	0.702782	+	0.711405i	$0.251941\pi$
$42$	0	0
$43$	− 8.00000i	− 1.21999i	−0.792406	−	0.609994i	$-0.791172\pi$
$43$	− 8.00000i	− 1.21999i	0.792406	−	0.609994i	$-0.208828\pi$
$44$	3.00000	0.452267
$45$	0	0
$46$	0	0
$47$	6.00000i	0.875190i	0.899172	+	0.437595i	$0.144170\pi$
$47$	6.00000i	0.875190i	−0.899172	+	0.437595i	$0.855830\pi$
$48$	0	0
$49$	−1.00000	−0.142857
$50$	0	0
$51$	0	0
$52$	2.00000i	0.277350i
$53$	− 12.0000i	− 1.64833i	−0.566352	−	0.824163i	$-0.691646\pi$
$53$	− 12.0000i	− 1.64833i	0.566352	−	0.824163i	$-0.308354\pi$
$54$	0	0
$55$	0	0
$56$	−1.00000	−0.133631
$57$	0	0
$58$	6.00000i	0.787839i
$59$	12.0000	1.56227	0.781133	−	0.624364i	$-0.214642\pi$
$59$	12.0000	1.56227	0.781133	+	0.624364i	$0.214642\pi$
$60$	0	0
$61$	−10.0000	−1.28037	−0.640184	−	0.768221i	$-0.721142\pi$
$61$	−10.0000	−1.28037	−0.640184	+	0.768221i	$0.721142\pi$
$62$	4.00000i	0.508001i
$63$	0	0
$64$	−1.00000	−0.125000
$65$	0	0
$66$	0	0
$67$	− 7.00000i	− 0.855186i	−0.903971	−	0.427593i	$-0.859362\pi$
$67$	− 7.00000i	− 0.855186i	0.903971	−	0.427593i	$-0.140638\pi$
$68$	3.00000i	0.363803i
$69$	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
$73$	− 5.00000i	− 0.585206i	−0.956234	−	0.292603i	$-0.905479\pi$
$73$	− 5.00000i	− 0.585206i	0.956234	−	0.292603i	$-0.0945214\pi$
$74$	8.00000	0.929981
$75$	0	0
$76$	−7.00000	−0.802955
$77$	− 3.00000i	− 0.341882i
$78$	0	0
$79$	−14.0000	−1.57512	−0.787562	−	0.616236i	$-0.788657\pi$
$79$	−14.0000	−1.57512	−0.787562	+	0.616236i	$0.788657\pi$
$80$	0	0
$81$	0	0
$82$	− 9.00000i	− 0.993884i
$83$	− 9.00000i	− 0.987878i	−0.869496	−	0.493939i	$-0.835557\pi$
$83$	− 9.00000i	− 0.987878i	0.869496	−	0.493939i	$-0.164443\pi$
$84$	0	0
$85$	0	0
$86$	−8.00000	−0.862662
$87$	0	0
$88$	− 3.00000i	− 0.319801i
$89$	−15.0000	−1.59000	−0.794998	−	0.606612i	$-0.792528\pi$
$89$	−15.0000	−1.59000	−0.794998	+	0.606612i	$0.792528\pi$
$90$	0	0
$91$	2.00000	0.209657
$92$	0	0
$93$	0	0
$94$	6.00000	0.618853
$95$	0	0
$96$	0	0
$97$	− 10.0000i	− 1.01535i	−0.861550	−	0.507673i	$-0.830506\pi$
$97$	− 10.0000i	− 1.01535i	0.861550	−	0.507673i	$-0.169494\pi$
$98$	1.00000i	0.101015i
$99$	0	0
$100$	0	0
$101$	0	0		−	1.00000i	$-0.5\pi$
$101$	0	0			1.00000i	$0.5\pi$
$102$	0	0
$103$	− 20.0000i	− 1.97066i	−0.170664	−	0.985329i	$-0.554591\pi$
$103$	− 20.0000i	− 1.97066i	0.170664	−	0.985329i	$-0.445409\pi$
$104$	2.00000	0.196116
$105$	0	0
$106$	−12.0000	−1.16554
$107$	− 3.00000i	− 0.290021i	−0.989430	−	0.145010i	$-0.953678\pi$
$107$	− 3.00000i	− 0.290021i	0.989430	−	0.145010i	$-0.0463216\pi$
$108$	0	0
$109$	−14.0000	−1.34096	−0.670478	−	0.741929i	$-0.733911\pi$
$109$	−14.0000	−1.34096	−0.670478	+	0.741929i	$0.733911\pi$
$110$	0	0
$111$	0	0
$112$	1.00000i	0.0944911i
$113$	− 9.00000i	− 0.846649i	−0.905978	−	0.423324i	$-0.860863\pi$
$113$	− 9.00000i	− 0.846649i	0.905978	−	0.423324i	$-0.139137\pi$
$114$	0	0
$115$	0	0
$116$	6.00000	0.557086
$117$	0	0
$118$	− 12.0000i	− 1.10469i
$119$	3.00000	0.275010
$120$	0	0
$121$	−2.00000	−0.181818
$122$	10.0000i	0.905357i
$123$	0	0
$124$	4.00000	0.359211
$125$	0	0
$126$	0	0
$127$	2.00000i	0.177471i	0.996055	+	0.0887357i	$0.0282826\pi$
$127$	2.00000i	0.177471i	−0.996055	+	0.0887357i	$0.971717\pi$
$128$	1.00000i	0.0883883i
$129$	0	0
$130$	0	0
$131$	0	0		−	1.00000i	$-0.5\pi$
$131$	0	0			1.00000i	$0.5\pi$
$132$	0	0
$133$	7.00000i	0.606977i
$134$	−7.00000	−0.604708
$135$	0	0
$136$	3.00000	0.257248
$137$	− 21.0000i	− 1.79415i	−0.441877	−	0.897076i	$-0.645687\pi$
$137$	− 21.0000i	− 1.79415i	0.441877	−	0.897076i	$-0.354313\pi$
$138$	0	0
$139$	7.00000	0.593732	0.296866	−	0.954919i	$-0.404058\pi$
$139$	7.00000	0.593732	0.296866	+	0.954919i	$0.404058\pi$
$140$	0	0
$141$	0	0
$142$	6.00000i	0.503509i
$143$	6.00000i	0.501745i
$144$	0	0
$145$	0	0
$146$	−5.00000	−0.413803
$147$	0	0
$148$	− 8.00000i	− 0.657596i
$149$	12.0000	0.983078	0.491539	−	0.870855i	$-0.336434\pi$
$149$	12.0000	0.983078	0.491539	+	0.870855i	$0.336434\pi$
$150$	0	0
$151$	8.00000	0.651031	0.325515	−	0.945537i	$-0.394462\pi$
$151$	8.00000	0.651031	0.325515	+	0.945537i	$0.394462\pi$
$152$	7.00000i	0.567775i
$153$	0	0
$154$	−3.00000	−0.241747
$155$	0	0
$156$	0	0
$157$	20.0000i	1.59617i	0.602542	+	0.798087i	$0.294154\pi$
$157$	20.0000i	1.59617i	−0.602542	+	0.798087i	$0.705846\pi$
$158$	14.0000i	1.11378i
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	− 5.00000i	− 0.391630i	−0.980641	−	0.195815i	$-0.937265\pi$
$163$	− 5.00000i	− 0.391630i	0.980641	−	0.195815i	$-0.0627352\pi$
$164$	−9.00000	−0.702782
$165$	0	0
$166$	−9.00000	−0.698535
$167$	− 12.0000i	− 0.928588i	−0.885681	−	0.464294i	$-0.846308\pi$
$167$	− 12.0000i	− 0.928588i	0.885681	−	0.464294i	$-0.153692\pi$
$168$	0	0
$169$	9.00000	0.692308
$170$	0	0
$171$	0	0
$172$	8.00000i	0.609994i
$173$	− 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
$176$	−3.00000	−0.226134
$177$	0	0
$178$	15.0000i	1.12430i
$179$	3.00000	0.224231	0.112115	−	0.993695i	$-0.464237\pi$
$179$	3.00000	0.224231	0.112115	+	0.993695i	$0.464237\pi$
$180$	0	0
$181$	2.00000	0.148659	0.0743294	−	0.997234i	$-0.476318\pi$
$181$	2.00000	0.148659	0.0743294	+	0.997234i	$0.476318\pi$
$182$	− 2.00000i	− 0.148250i
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	9.00000i	0.658145i
$188$	− 6.00000i	− 0.437595i
$189$	0	0
$190$	0	0
$191$	6.00000	0.434145	0.217072	−	0.976156i	$-0.430349\pi$
$191$	6.00000	0.434145	0.217072	+	0.976156i	$0.430349\pi$
$192$	0	0
$193$	− 5.00000i	− 0.359908i	−0.983675	−	0.179954i	$-0.942405\pi$
$193$	− 5.00000i	− 0.359908i	0.983675	−	0.179954i	$-0.0575949\pi$
$194$	−10.0000	−0.717958
$195$	0	0
$196$	1.00000	0.0714286
$197$	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
$199$	−14.0000	−0.992434	−0.496217	−	0.868199i	$-0.665278\pi$
$199$	−14.0000	−0.992434	−0.496217	+	0.868199i	$0.665278\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 6.00000i	− 0.421117i
$204$	0	0
$205$	0	0
$206$	−20.0000	−1.39347
$207$	0	0
$208$	− 2.00000i	− 0.138675i
$209$	−21.0000	−1.45260
$210$	0	0
$211$	17.0000	1.17033	0.585164	−	0.810915i	$-0.301030\pi$
$211$	17.0000	1.17033	0.585164	+	0.810915i	$0.301030\pi$
$212$	12.0000i	0.824163i
$213$	0	0
$214$	−3.00000	−0.205076
$215$	0	0
$216$	0	0
$217$	− 4.00000i	− 0.271538i
$218$	14.0000i	0.948200i
$219$	0	0
$220$	0	0
$221$	−6.00000	−0.403604
$222$	0	0
$223$	− 14.0000i	− 0.937509i	−0.883328	−	0.468755i	$-0.844703\pi$
$223$	− 14.0000i	− 0.937509i	0.883328	−	0.468755i	$-0.155297\pi$
$224$	1.00000	0.0668153
$225$	0	0
$226$	−9.00000	−0.598671
$227$	− 12.0000i	− 0.796468i	−0.917284	−	0.398234i	$-0.869623\pi$
$227$	− 12.0000i	− 0.796468i	0.917284	−	0.398234i	$-0.130377\pi$
$228$	0	0
$229$	−26.0000	−1.71813	−0.859064	−	0.511868i	$-0.828954\pi$
$229$	−26.0000	−1.71813	−0.859064	+	0.511868i	$0.828954\pi$
$230$	0	0
$231$	0	0
$232$	− 6.00000i	− 0.393919i
$233$	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
$236$	−12.0000	−0.781133
$237$	0	0
$238$	− 3.00000i	− 0.194461i
$239$	−12.0000	−0.776215	−0.388108	−	0.921614i	$-0.626871\pi$
$239$	−12.0000	−0.776215	−0.388108	+	0.921614i	$0.626871\pi$
$240$	0	0
$241$	−25.0000	−1.61039	−0.805196	−	0.593009i	$-0.797940\pi$
$241$	−25.0000	−1.61039	−0.805196	+	0.593009i	$0.797940\pi$
$242$	2.00000i	0.128565i
$243$	0	0
$244$	10.0000	0.640184
$245$	0	0
$246$	0	0
$247$	− 14.0000i	− 0.890799i
$248$	− 4.00000i	− 0.254000i
$249$	0	0
$250$	0	0
$251$	−15.0000	−0.946792	−0.473396	−	0.880850i	$-0.656972\pi$
$251$	−15.0000	−0.946792	−0.473396	+	0.880850i	$0.656972\pi$
$252$	0	0
$253$	0	0
$254$	2.00000	0.125491
$255$	0	0
$256$	1.00000	0.0625000
$257$	30.0000i	1.87135i	0.352865	+	0.935674i	$0.385208\pi$
$257$	30.0000i	1.87135i	−0.352865	+	0.935674i	$0.614792\pi$
$258$	0	0
$259$	−8.00000	−0.497096
$260$	0	0
$261$	0	0
$262$	0	0
$263$	6.00000i	0.369976i	0.982741	+	0.184988i	$0.0592246\pi$
$263$	6.00000i	0.369976i	−0.982741	+	0.184988i	$0.940775\pi$
$264$	0	0
$265$	0	0
$266$	7.00000	0.429198
$267$	0	0
$268$	7.00000i	0.427593i
$269$	−6.00000	−0.365826	−0.182913	−	0.983129i	$-0.558553\pi$
$269$	−6.00000	−0.365826	−0.182913	+	0.983129i	$0.558553\pi$
$270$	0	0
$271$	2.00000	0.121491	0.0607457	−	0.998153i	$-0.480652\pi$
$271$	2.00000	0.121491	0.0607457	+	0.998153i	$0.480652\pi$
$272$	− 3.00000i	− 0.181902i
$273$	0	0
$274$	−21.0000	−1.26866
$275$	0	0
$276$	0	0
$277$	2.00000i	0.120168i	0.998193	+	0.0600842i	$0.0191369\pi$
$277$	2.00000i	0.120168i	−0.998193	+	0.0600842i	$0.980863\pi$
$278$	− 7.00000i	− 0.419832i
$279$	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
$283$	1.00000i	0.0594438i	0.999558	+	0.0297219i	$0.00946217\pi$
$283$	1.00000i	0.0594438i	−0.999558	+	0.0297219i	$0.990538\pi$
$284$	6.00000	0.356034
$285$	0	0
$286$	6.00000	0.354787
$287$	9.00000i	0.531253i
$288$	0	0
$289$	8.00000	0.470588
$290$	0	0
$291$	0	0
$292$	5.00000i	0.292603i
$293$	− 6.00000i	− 0.350524i	−0.984522	−	0.175262i	$-0.943923\pi$
$293$	− 6.00000i	− 0.350524i	0.984522	−	0.175262i	$-0.0560772\pi$
$294$	0	0
$295$	0	0
$296$	−8.00000	−0.464991
$297$	0	0
$298$	− 12.0000i	− 0.695141i
$299$	0	0
$300$	0	0
$301$	8.00000	0.461112
$302$	− 8.00000i	− 0.460348i
$303$	0	0
$304$	7.00000	0.401478
$305$	0	0
$306$	0	0
$307$	− 7.00000i	− 0.399511i	−0.979846	−	0.199756i	$-0.935985\pi$
$307$	− 7.00000i	− 0.399511i	0.979846	−	0.199756i	$-0.0640148\pi$
$308$	3.00000i	0.170941i
$309$	0	0
$310$	0	0
$311$	18.0000	1.02069	0.510343	−	0.859971i	$-0.329518\pi$
$311$	18.0000	1.02069	0.510343	+	0.859971i	$0.329518\pi$
$312$	0	0
$313$	10.0000i	0.565233i	0.959233	+	0.282617i	$0.0912024\pi$
$313$	10.0000i	0.565233i	−0.959233	+	0.282617i	$0.908798\pi$
$314$	20.0000	1.12867
$315$	0	0
$316$	14.0000	0.787562
$317$	12.0000i	0.673987i	0.941507	+	0.336994i	$0.109410\pi$
$317$	12.0000i	0.673987i	−0.941507	+	0.336994i	$0.890590\pi$
$318$	0	0
$319$	18.0000	1.00781
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 21.0000i	− 1.16847i
$324$	0	0
$325$	0	0
$326$	−5.00000	−0.276924
$327$	0	0
$328$	9.00000i	0.496942i
$329$	−6.00000	−0.330791
$330$	0	0
$331$	−25.0000	−1.37412	−0.687062	−	0.726599i	$-0.741100\pi$
$331$	−25.0000	−1.37412	−0.687062	+	0.726599i	$0.741100\pi$
$332$	9.00000i	0.493939i
$333$	0	0
$334$	−12.0000	−0.656611
$335$	0	0
$336$	0	0
$337$	− 13.0000i	− 0.708155i	−0.935216	−	0.354078i	$-0.884795\pi$
$337$	− 13.0000i	− 0.708155i	0.935216	−	0.354078i	$-0.115205\pi$
$338$	− 9.00000i	− 0.489535i
$339$	0	0
$340$	0	0
$341$	12.0000	0.649836
$342$	0	0
$343$	− 1.00000i	− 0.0539949i
$344$	8.00000	0.431331
$345$	0	0
$346$	−6.00000	−0.322562
$347$	21.0000i	1.12734i	0.826000	+	0.563670i	$0.190611\pi$
$347$	21.0000i	1.12734i	−0.826000	+	0.563670i	$0.809389\pi$
$348$	0	0
$349$	−8.00000	−0.428230	−0.214115	−	0.976808i	$-0.568687\pi$
$349$	−8.00000	−0.428230	−0.214115	+	0.976808i	$0.568687\pi$
$350$	0	0
$351$	0	0
$352$	3.00000i	0.159901i
$353$	30.0000i	1.59674i	0.602168	+	0.798369i	$0.294304\pi$
$353$	30.0000i	1.59674i	−0.602168	+	0.798369i	$0.705696\pi$
$354$	0	0
$355$	0	0
$356$	15.0000	0.794998
$357$	0	0
$358$	− 3.00000i	− 0.158555i
$359$	−6.00000	−0.316668	−0.158334	−	0.987386i	$-0.550612\pi$
$359$	−6.00000	−0.316668	−0.158334	+	0.987386i	$0.550612\pi$
$360$	0	0
$361$	30.0000	1.57895
$362$	− 2.00000i	− 0.105118i
$363$	0	0
$364$	−2.00000	−0.104828
$365$	0	0
$366$	0	0
$367$	8.00000i	0.417597i	0.977959	+	0.208798i	$0.0669552\pi$
$367$	8.00000i	0.417597i	−0.977959	+	0.208798i	$0.933045\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	12.0000	0.623009
$372$	0	0
$373$	4.00000i	0.207112i	0.994624	+	0.103556i	$0.0330221\pi$
$373$	4.00000i	0.207112i	−0.994624	+	0.103556i	$0.966978\pi$
$374$	9.00000	0.465379
$375$	0	0
$376$	−6.00000	−0.309426
$377$	12.0000i	0.618031i
$378$	0	0
$379$	−17.0000	−0.873231	−0.436616	−	0.899648i	$-0.643823\pi$
$379$	−17.0000	−0.873231	−0.436616	+	0.899648i	$0.643823\pi$
$380$	0	0
$381$	0	0
$382$	− 6.00000i	− 0.306987i
$383$	− 30.0000i	− 1.53293i	−0.642287	−	0.766464i	$-0.722014\pi$
$383$	− 30.0000i	− 1.53293i	0.642287	−	0.766464i	$-0.277986\pi$
$384$	0	0
$385$	0	0
$386$	−5.00000	−0.254493
$387$	0	0
$388$	10.0000i	0.507673i
$389$	−24.0000	−1.21685	−0.608424	−	0.793612i	$-0.708198\pi$
$389$	−24.0000	−1.21685	−0.608424	+	0.793612i	$0.708198\pi$
$390$	0	0
$391$	0	0
$392$	− 1.00000i	− 0.0505076i
$393$	0	0
$394$	6.00000	0.302276
$395$	0	0
$396$	0	0
$397$	2.00000i	0.100377i	0.998740	+	0.0501886i	$0.0159822\pi$
$397$	2.00000i	0.100377i	−0.998740	+	0.0501886i	$0.984018\pi$
$398$	14.0000i	0.701757i
$399$	0	0
$400$	0	0
$401$	27.0000	1.34832	0.674158	−	0.738587i	$-0.264507\pi$
$401$	27.0000	1.34832	0.674158	+	0.738587i	$0.264507\pi$
$402$	0	0
$403$	8.00000i	0.398508i
$404$	0	0
$405$	0	0
$406$	−6.00000	−0.297775
$407$	− 24.0000i	− 1.18964i
$408$	0	0
$409$	25.0000	1.23617	0.618085	−	0.786111i	$-0.287909\pi$
$409$	25.0000	1.23617	0.618085	+	0.786111i	$0.287909\pi$
$410$	0	0
$411$	0	0
$412$	20.0000i	0.985329i
$413$	12.0000i	0.590481i
$414$	0	0
$415$	0	0
$416$	−2.00000	−0.0980581
$417$	0	0
$418$	21.0000i	1.02714i
$419$	−3.00000	−0.146560	−0.0732798	−	0.997311i	$-0.523347\pi$
$419$	−3.00000	−0.146560	−0.0732798	+	0.997311i	$0.523347\pi$
$420$	0	0
$421$	20.0000	0.974740	0.487370	−	0.873195i	$-0.337956\pi$
$421$	20.0000	0.974740	0.487370	+	0.873195i	$0.337956\pi$
$422$	− 17.0000i	− 0.827547i
$423$	0	0
$424$	12.0000	0.582772
$425$	0	0
$426$	0	0
$427$	− 10.0000i	− 0.483934i
$428$	3.00000i	0.145010i
$429$	0	0
$430$	0	0
$431$	36.0000	1.73406	0.867029	−	0.498257i	$-0.166026\pi$
$431$	36.0000	1.73406	0.867029	+	0.498257i	$0.166026\pi$
$432$	0	0
$433$	− 11.0000i	− 0.528626i	−0.964437	−	0.264313i	$-0.914855\pi$
$433$	− 11.0000i	− 0.528626i	0.964437	−	0.264313i	$-0.0851452\pi$
$434$	−4.00000	−0.192006
$435$	0	0
$436$	14.0000	0.670478
$437$	0	0
$438$	0	0
$439$	4.00000	0.190910	0.0954548	−	0.995434i	$-0.469569\pi$
$439$	4.00000	0.190910	0.0954548	+	0.995434i	$0.469569\pi$
$440$	0	0
$441$	0	0
$442$	6.00000i	0.285391i
$443$	21.0000i	0.997740i	0.866677	+	0.498870i	$0.166252\pi$
$443$	21.0000i	0.997740i	−0.866677	+	0.498870i	$0.833748\pi$
$444$	0	0
$445$	0	0
$446$	−14.0000	−0.662919
$447$	0	0
$448$	− 1.00000i	− 0.0472456i
$449$	−15.0000	−0.707894	−0.353947	−	0.935266i	$-0.615161\pi$
$449$	−15.0000	−0.707894	−0.353947	+	0.935266i	$0.615161\pi$
$450$	0	0
$451$	−27.0000	−1.27138
$452$	9.00000i	0.423324i
$453$	0	0
$454$	−12.0000	−0.563188
$455$	0	0
$456$	0	0
$457$	17.0000i	0.795226i	0.917553	+	0.397613i	$0.130161\pi$
$457$	17.0000i	0.795226i	−0.917553	+	0.397613i	$0.869839\pi$
$458$	26.0000i	1.21490i
$459$	0	0
$460$	0	0
$461$	−18.0000	−0.838344	−0.419172	−	0.907907i	$-0.637680\pi$
$461$	−18.0000	−0.838344	−0.419172	+	0.907907i	$0.637680\pi$
$462$	0	0
$463$	− 8.00000i	− 0.371792i	−0.982569	−	0.185896i	$-0.940481\pi$
$463$	− 8.00000i	− 0.371792i	0.982569	−	0.185896i	$-0.0595187\pi$
$464$	−6.00000	−0.278543
$465$	0	0
$466$	6.00000	0.277945
$467$	− 36.0000i	− 1.66588i	−0.553362	−	0.832941i	$-0.686655\pi$
$467$	− 36.0000i	− 1.66588i	0.553362	−	0.832941i	$-0.313345\pi$
$468$	0	0
$469$	7.00000	0.323230
$470$	0	0
$471$	0	0
$472$	12.0000i	0.552345i
$473$	24.0000i	1.10352i
$474$	0	0
$475$	0	0
$476$	−3.00000	−0.137505
$477$	0	0
$478$	12.0000i	0.548867i
$479$	18.0000	0.822441	0.411220	−	0.911536i	$-0.365103\pi$
$479$	18.0000	0.822441	0.411220	+	0.911536i	$0.365103\pi$
$480$	0	0
$481$	16.0000	0.729537
$482$	25.0000i	1.13872i
$483$	0	0
$484$	2.00000	0.0909091
$485$	0	0
$486$	0	0
$487$	− 34.0000i	− 1.54069i	−0.637629	−	0.770344i	$-0.720085\pi$
$487$	− 34.0000i	− 1.54069i	0.637629	−	0.770344i	$-0.279915\pi$
$488$	− 10.0000i	− 0.452679i
$489$	0	0
$490$	0	0
$491$	12.0000	0.541552	0.270776	−	0.962642i	$-0.412720\pi$
$491$	12.0000	0.541552	0.270776	+	0.962642i	$0.412720\pi$
$492$	0	0
$493$	18.0000i	0.810679i
$494$	−14.0000	−0.629890
$495$	0	0
$496$	−4.00000	−0.179605
$497$	− 6.00000i	− 0.269137i
$498$	0	0
$499$	28.0000	1.25345	0.626726	−	0.779240i	$-0.284395\pi$
$499$	28.0000	1.25345	0.626726	+	0.779240i	$0.284395\pi$
$500$	0	0
$501$	0	0
$502$	15.0000i	0.669483i
$503$	− 6.00000i	− 0.267527i	−0.991013	−	0.133763i	$-0.957294\pi$
$503$	− 6.00000i	− 0.267527i	0.991013	−	0.133763i	$-0.0427062\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0	0
$508$	− 2.00000i	− 0.0887357i
$509$	42.0000	1.86162	0.930809	−	0.365507i	$-0.119104\pi$
$509$	42.0000	1.86162	0.930809	+	0.365507i	$0.119104\pi$
$510$	0	0
$511$	5.00000	0.221187
$512$	− 1.00000i	− 0.0441942i
$513$	0	0
$514$	30.0000	1.32324
$515$	0	0
$516$	0	0
$517$	− 18.0000i	− 0.791639i
$518$	8.00000i	0.351500i
$519$	0	0
$520$	0	0
$521$	−39.0000	−1.70862	−0.854311	−	0.519763i	$-0.826020\pi$
$521$	−39.0000	−1.70862	−0.854311	+	0.519763i	$0.826020\pi$
$522$	0	0
$523$	7.00000i	0.306089i	0.988219	+	0.153044i	$0.0489077\pi$
$523$	7.00000i	0.306089i	−0.988219	+	0.153044i	$0.951092\pi$
$524$	0	0
$525$	0	0
$526$	6.00000	0.261612
$527$	12.0000i	0.522728i
$528$	0	0
$529$	23.0000	1.00000
$530$	0	0
$531$	0	0
$532$	− 7.00000i	− 0.303488i
$533$	− 18.0000i	− 0.779667i
$534$	0	0
$535$	0	0
$536$	7.00000	0.302354
$537$	0	0
$538$	6.00000i	0.258678i
$539$	3.00000	0.129219
$540$	0	0
$541$	38.0000	1.63375	0.816874	−	0.576816i	$-0.195705\pi$
$541$	38.0000	1.63375	0.816874	+	0.576816i	$0.195705\pi$
$542$	− 2.00000i	− 0.0859074i
$543$	0	0
$544$	−3.00000	−0.128624
$545$	0	0
$546$	0	0
$547$	35.0000i	1.49649i	0.663421	+	0.748246i	$0.269104\pi$
$547$	35.0000i	1.49649i	−0.663421	+	0.748246i	$0.730896\pi$
$548$	21.0000i	0.897076i
$549$	0	0
$550$	0	0
$551$	−42.0000	−1.78926
$552$	0	0
$553$	− 14.0000i	− 0.595341i
$554$	2.00000	0.0849719
$555$	0	0
$556$	−7.00000	−0.296866
$557$	− 36.0000i	− 1.52537i	−0.646771	−	0.762684i	$-0.723881\pi$
$557$	− 36.0000i	− 1.52537i	0.646771	−	0.762684i	$-0.276119\pi$
$558$	0	0
$559$	−16.0000	−0.676728
$560$	0	0
$561$	0	0
$562$	− 18.0000i	− 0.759284i
$563$	12.0000i	0.505740i	0.967500	+	0.252870i	$0.0813744\pi$
$563$	12.0000i	0.505740i	−0.967500	+	0.252870i	$0.918626\pi$
$564$	0	0
$565$	0	0
$566$	1.00000	0.0420331
$567$	0	0
$568$	− 6.00000i	− 0.251754i
$569$	−27.0000	−1.13190	−0.565949	−	0.824440i	$-0.691490\pi$
$569$	−27.0000	−1.13190	−0.565949	+	0.824440i	$0.691490\pi$
$570$	0	0
$571$	−4.00000	−0.167395	−0.0836974	−	0.996491i	$-0.526673\pi$
$571$	−4.00000	−0.167395	−0.0836974	+	0.996491i	$0.526673\pi$
$572$	− 6.00000i	− 0.250873i
$573$	0	0
$574$	9.00000	0.375653
$575$	0	0
$576$	0	0
$577$	− 7.00000i	− 0.291414i	−0.989328	−	0.145707i	$-0.953454\pi$
$577$	− 7.00000i	− 0.291414i	0.989328	−	0.145707i	$-0.0465456\pi$
$578$	− 8.00000i	− 0.332756i
$579$	0	0
$580$	0	0
$581$	9.00000	0.373383
$582$	0	0
$583$	36.0000i	1.49097i
$584$	5.00000	0.206901
$585$	0	0
$586$	−6.00000	−0.247858
$587$	− 39.0000i	− 1.60970i	−0.593477	−	0.804851i	$-0.702245\pi$
$587$	− 39.0000i	− 1.60970i	0.593477	−	0.804851i	$-0.297755\pi$
$588$	0	0
$589$	−28.0000	−1.15372
$590$	0	0
$591$	0	0
$592$	8.00000i	0.328798i
$593$	− 27.0000i	− 1.10876i	−0.832265	−	0.554379i	$-0.812956\pi$
$593$	− 27.0000i	− 1.10876i	0.832265	−	0.554379i	$-0.187044\pi$
$594$	0	0
$595$	0	0
$596$	−12.0000	−0.491539
$597$	0	0
$598$	0	0
$599$	24.0000	0.980613	0.490307	−	0.871550i	$-0.336885\pi$
$599$	24.0000	0.980613	0.490307	+	0.871550i	$0.336885\pi$
$600$	0	0
$601$	−7.00000	−0.285536	−0.142768	−	0.989756i	$-0.545600\pi$
$601$	−7.00000	−0.285536	−0.142768	+	0.989756i	$0.545600\pi$
$602$	− 8.00000i	− 0.326056i
$603$	0	0
$604$	−8.00000	−0.325515
$605$	0	0
$606$	0	0
$607$	44.0000i	1.78590i	0.450151	+	0.892952i	$0.351370\pi$
$607$	44.0000i	1.78590i	−0.450151	+	0.892952i	$0.648630\pi$
$608$	− 7.00000i	− 0.283887i
$609$	0	0
$610$	0	0
$611$	12.0000	0.485468
$612$	0	0
$613$	− 2.00000i	− 0.0807792i	−0.999184	−	0.0403896i	$-0.987140\pi$
$613$	− 2.00000i	− 0.0807792i	0.999184	−	0.0403896i	$-0.0128599\pi$
$614$	−7.00000	−0.282497
$615$	0	0
$616$	3.00000	0.120873
$617$	18.0000i	0.724653i	0.932051	+	0.362326i	$0.118017\pi$
$617$	18.0000i	0.724653i	−0.932051	+	0.362326i	$0.881983\pi$
$618$	0	0
$619$	28.0000	1.12542	0.562708	−	0.826656i	$-0.309760\pi$
$619$	28.0000	1.12542	0.562708	+	0.826656i	$0.309760\pi$
$620$	0	0
$621$	0	0
$622$	− 18.0000i	− 0.721734i
$623$	− 15.0000i	− 0.600962i
$624$	0	0
$625$	0	0
$626$	10.0000	0.399680
$627$	0	0
$628$	− 20.0000i	− 0.798087i
$629$	24.0000	0.956943
$630$	0	0
$631$	32.0000	1.27390	0.636950	−	0.770905i	$-0.280196\pi$
$631$	32.0000	1.27390	0.636950	+	0.770905i	$0.280196\pi$
$632$	− 14.0000i	− 0.556890i
$633$	0	0
$634$	12.0000	0.476581
$635$	0	0
$636$	0	0
$637$	2.00000i	0.0792429i
$638$	− 18.0000i	− 0.712627i
$639$	0	0
$640$	0	0
$641$	30.0000	1.18493	0.592464	−	0.805597i	$-0.298155\pi$
$641$	30.0000	1.18493	0.592464	+	0.805597i	$0.298155\pi$
$642$	0	0
$643$	40.0000i	1.57745i	0.614749	+	0.788723i	$0.289257\pi$
$643$	40.0000i	1.57745i	−0.614749	+	0.788723i	$0.710743\pi$
$644$	0	0
$645$	0	0
$646$	−21.0000	−0.826234
$647$	24.0000i	0.943537i	0.881722	+	0.471769i	$0.156384\pi$
$647$	24.0000i	0.943537i	−0.881722	+	0.471769i	$0.843616\pi$
$648$	0	0
$649$	−36.0000	−1.41312
$650$	0	0
$651$	0	0
$652$	5.00000i	0.195815i
$653$	36.0000i	1.40879i	0.709809	+	0.704394i	$0.248781\pi$
$653$	36.0000i	1.40879i	−0.709809	+	0.704394i	$0.751219\pi$
$654$	0	0
$655$	0	0
$656$	9.00000	0.351391
$657$	0	0
$658$	6.00000i	0.233904i
$659$	9.00000	0.350590	0.175295	−	0.984516i	$-0.443912\pi$
$659$	9.00000	0.350590	0.175295	+	0.984516i	$0.443912\pi$
$660$	0	0
$661$	−10.0000	−0.388955	−0.194477	−	0.980907i	$-0.562301\pi$
$661$	−10.0000	−0.388955	−0.194477	+	0.980907i	$0.562301\pi$
$662$	25.0000i	0.971653i
$663$	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
$671$	30.0000	1.15814
$672$	0	0
$673$	− 2.00000i	− 0.0770943i	−0.999257	−	0.0385472i	$-0.987727\pi$
$673$	− 2.00000i	− 0.0770943i	0.999257	−	0.0385472i	$-0.0122730\pi$
$674$	−13.0000	−0.500741
$675$	0	0
$676$	−9.00000	−0.346154
$677$	0	0	1.00000			$0$
$677$	0	0	−1.00000			$\pi$
$678$	0	0
$679$	10.0000	0.383765
$680$	0	0
$681$	0	0
$682$	− 12.0000i	− 0.459504i
$683$	3.00000i	0.114792i	0.998351	+	0.0573959i	$0.0182797\pi$
$683$	3.00000i	0.114792i	−0.998351	+	0.0573959i	$0.981720\pi$
$684$	0	0
$685$	0	0
$686$	−1.00000	−0.0381802
$687$	0	0
$688$	− 8.00000i	− 0.304997i
$689$	−24.0000	−0.914327
$690$	0	0
$691$	−19.0000	−0.722794	−0.361397	−	0.932412i	$-0.617700\pi$
$691$	−19.0000	−0.722794	−0.361397	+	0.932412i	$0.617700\pi$
$692$	6.00000i	0.228086i
$693$	0	0
$694$	21.0000	0.797149
$695$	0	0
$696$	0	0
$697$	− 27.0000i	− 1.02270i
$698$	8.00000i	0.302804i
$699$	0	0
$700$	0	0
$701$	0	0		−	1.00000i	$-0.5\pi$
$701$	0	0			1.00000i	$0.5\pi$
$702$	0	0
$703$	56.0000i	2.11208i
$704$	3.00000	0.113067
$705$	0	0
$706$	30.0000	1.12906
$707$	0	0
$708$	0	0
$709$	28.0000	1.05156	0.525781	−	0.850620i	$-0.323773\pi$
$709$	28.0000	1.05156	0.525781	+	0.850620i	$0.323773\pi$
$710$	0	0
$711$	0	0
$712$	− 15.0000i	− 0.562149i
$713$	0	0
$714$	0	0
$715$	0	0
$716$	−3.00000	−0.112115
$717$	0	0
$718$	6.00000i	0.223918i
$719$	0	0		−	1.00000i	$-0.5\pi$
$719$	0	0			1.00000i	$0.5\pi$
$720$	0	0
$721$	20.0000	0.744839
$722$	− 30.0000i	− 1.11648i
$723$	0	0
$724$	−2.00000	−0.0743294
$725$	0	0
$726$	0	0
$727$	− 34.0000i	− 1.26099i	−0.776193	−	0.630495i	$-0.782852\pi$
$727$	− 34.0000i	− 1.26099i	0.776193	−	0.630495i	$-0.217148\pi$
$728$	2.00000i	0.0741249i
$729$	0	0
$730$	0	0
$731$	−24.0000	−0.887672
$732$	0	0
$733$	40.0000i	1.47743i	0.674016	+	0.738717i	$0.264568\pi$
$733$	40.0000i	1.47743i	−0.674016	+	0.738717i	$0.735432\pi$
$734$	8.00000	0.295285
$735$	0	0
$736$	0	0
$737$	21.0000i	0.773545i
$738$	0	0
$739$	−20.0000	−0.735712	−0.367856	−	0.929883i	$-0.619908\pi$
$739$	−20.0000	−0.735712	−0.367856	+	0.929883i	$0.619908\pi$
$740$	0	0
$741$	0	0
$742$	− 12.0000i	− 0.440534i
$743$	− 24.0000i	− 0.880475i	−0.897881	−	0.440237i	$-0.854894\pi$
$743$	− 24.0000i	− 0.880475i	0.897881	−	0.440237i	$-0.145106\pi$
$744$	0	0
$745$	0	0
$746$	4.00000	0.146450
$747$	0	0
$748$	− 9.00000i	− 0.329073i
$749$	3.00000	0.109618
$750$	0	0
$751$	−46.0000	−1.67856	−0.839282	−	0.543696i	$-0.817024\pi$
$751$	−46.0000	−1.67856	−0.839282	+	0.543696i	$0.817024\pi$
$752$	6.00000i	0.218797i
$753$	0	0
$754$	12.0000	0.437014
$755$	0	0
$756$	0	0
$757$	− 34.0000i	− 1.23575i	−0.786276	−	0.617876i	$-0.787994\pi$
$757$	− 34.0000i	− 1.23575i	0.786276	−	0.617876i	$-0.212006\pi$
$758$	17.0000i	0.617468i
$759$	0	0
$760$	0	0
$761$	9.00000	0.326250	0.163125	−	0.986605i	$-0.447843\pi$
$761$	9.00000	0.326250	0.163125	+	0.986605i	$0.447843\pi$
$762$	0	0
$763$	− 14.0000i	− 0.506834i
$764$	−6.00000	−0.217072
$765$	0	0
$766$	−30.0000	−1.08394
$767$	− 24.0000i	− 0.866590i
$768$	0	0
$769$	−23.0000	−0.829401	−0.414701	−	0.909958i	$-0.636114\pi$
$769$	−23.0000	−0.829401	−0.414701	+	0.909958i	$0.636114\pi$
$770$	0	0
$771$	0	0
$772$	5.00000i	0.179954i
$773$	− 12.0000i	− 0.431610i	−0.976436	−	0.215805i	$-0.930762\pi$
$773$	− 12.0000i	− 0.431610i	0.976436	−	0.215805i	$-0.0692376\pi$
$774$	0	0
$775$	0	0
$776$	10.0000	0.358979
$777$	0	0
$778$	24.0000i	0.860442i
$779$	63.0000	2.25721
$780$	0	0
$781$	18.0000	0.644091
$782$	0	0
$783$	0	0
$784$	−1.00000	−0.0357143
$785$	0	0
$786$	0	0
$787$	− 4.00000i	− 0.142585i	−0.997455	−	0.0712923i	$-0.977288\pi$
$787$	− 4.00000i	− 0.142585i	0.997455	−	0.0712923i	$-0.0227123\pi$
$788$	− 6.00000i	− 0.213741i
$789$	0	0
$790$	0	0
$791$	9.00000	0.320003
$792$	0	0
$793$	20.0000i	0.710221i
$794$	2.00000	0.0709773
$795$	0	0
$796$	14.0000	0.496217
$797$	30.0000i	1.06265i	0.847167	+	0.531327i	$0.178307\pi$
$797$	30.0000i	1.06265i	−0.847167	+	0.531327i	$0.821693\pi$
$798$	0	0
$799$	18.0000	0.636794
$800$	0	0
$801$	0	0
$802$	− 27.0000i	− 0.953403i
$803$	15.0000i	0.529339i
$804$	0	0
$805$	0	0
$806$	8.00000	0.281788
$807$	0	0
$808$	0	0
$809$	6.00000	0.210949	0.105474	−	0.994422i	$-0.466364\pi$
$809$	6.00000	0.210949	0.105474	+	0.994422i	$0.466364\pi$
$810$	0	0
$811$	44.0000	1.54505	0.772524	−	0.634985i	$-0.218994\pi$
$811$	44.0000	1.54505	0.772524	+	0.634985i	$0.218994\pi$
$812$	6.00000i	0.210559i
$813$	0	0
$814$	−24.0000	−0.841200
$815$	0	0
$816$	0	0
$817$	− 56.0000i	− 1.95919i
$818$	− 25.0000i	− 0.874105i
$819$	0	0
$820$	0	0
$821$	−24.0000	−0.837606	−0.418803	−	0.908077i	$-0.637550\pi$
$821$	−24.0000	−0.837606	−0.418803	+	0.908077i	$0.637550\pi$
$822$	0	0
$823$	− 26.0000i	− 0.906303i	−0.891434	−	0.453152i	$-0.850300\pi$
$823$	− 26.0000i	− 0.906303i	0.891434	−	0.453152i	$-0.149700\pi$
$824$	20.0000	0.696733
$825$	0	0
$826$	12.0000	0.417533
$827$	− 9.00000i	− 0.312961i	−0.987681	−	0.156480i	$-0.949985\pi$
$827$	− 9.00000i	− 0.312961i	0.987681	−	0.156480i	$-0.0500148\pi$
$828$	0	0
$829$	4.00000	0.138926	0.0694629	−	0.997585i	$-0.477871\pi$
$829$	4.00000	0.138926	0.0694629	+	0.997585i	$0.477871\pi$
$830$	0	0
$831$	0	0
$832$	2.00000i	0.0693375i
$833$	3.00000i	0.103944i
$834$	0	0
$835$	0	0
$836$	21.0000	0.726300
$837$	0	0
$838$	3.00000i	0.103633i
$839$	−6.00000	−0.207143	−0.103572	−	0.994622i	$-0.533027\pi$
$839$	−6.00000	−0.207143	−0.103572	+	0.994622i	$0.533027\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	− 20.0000i	− 0.689246i
$843$	0	0
$844$	−17.0000	−0.585164
$845$	0	0
$846$	0	0
$847$	− 2.00000i	− 0.0687208i
$848$	− 12.0000i	− 0.412082i
$849$	0	0
$850$	0	0
$851$	0	0
$852$	0	0
$853$	10.0000i	0.342393i	0.985237	+	0.171197i	$0.0547634\pi$
$853$	10.0000i	0.342393i	−0.985237	+	0.171197i	$0.945237\pi$
$854$	−10.0000	−0.342193
$855$	0	0
$856$	3.00000	0.102538
$857$	− 15.0000i	− 0.512390i	−0.966625	−	0.256195i	$-0.917531\pi$
$857$	− 15.0000i	− 0.512390i	0.966625	−	0.256195i	$-0.0824690\pi$
$858$	0	0
$859$	31.0000	1.05771	0.528853	−	0.848713i	$-0.322622\pi$
$859$	31.0000	1.05771	0.528853	+	0.848713i	$0.322622\pi$
$860$	0	0
$861$	0	0
$862$	− 36.0000i	− 1.22616i
$863$	− 12.0000i	− 0.408485i	−0.978920	−	0.204242i	$-0.934527\pi$
$863$	− 12.0000i	− 0.408485i	0.978920	−	0.204242i	$-0.0654731\pi$
$864$	0	0
$865$	0	0
$866$	−11.0000	−0.373795
$867$	0	0
$868$	4.00000i	0.135769i
$869$	42.0000	1.42475
$870$	0	0
$871$	−14.0000	−0.474372
$872$	− 14.0000i	− 0.474100i
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	32.0000i	1.08056i	0.841484	+	0.540282i	$0.181682\pi$
$877$	32.0000i	1.08056i	−0.841484	+	0.540282i	$0.818318\pi$
$878$	− 4.00000i	− 0.134993i
$879$	0	0
$880$	0	0
$881$	6.00000	0.202145	0.101073	−	0.994879i	$-0.467773\pi$
$881$	6.00000	0.202145	0.101073	+	0.994879i	$0.467773\pi$
$882$	0	0
$883$	− 47.0000i	− 1.58168i	−0.612026	−	0.790838i	$-0.709645\pi$
$883$	− 47.0000i	− 1.58168i	0.612026	−	0.790838i	$-0.290355\pi$
$884$	6.00000	0.201802
$885$	0	0
$886$	21.0000	0.705509
$887$	− 6.00000i	− 0.201460i	−0.994914	−	0.100730i	$-0.967882\pi$
$887$	− 6.00000i	− 0.201460i	0.994914	−	0.100730i	$-0.0321179\pi$
$888$	0	0
$889$	−2.00000	−0.0670778
$890$	0	0
$891$	0	0
$892$	14.0000i	0.468755i
$893$	42.0000i	1.40548i
$894$	0	0
$895$	0	0
$896$	−1.00000	−0.0334077
$897$	0	0
$898$	15.0000i	0.500556i
$899$	24.0000	0.800445
$900$	0	0
$901$	−36.0000	−1.19933
$902$	27.0000i	0.899002i
$903$	0	0
$904$	9.00000	0.299336
$905$	0	0
$906$	0	0
$907$	− 4.00000i	− 0.132818i	−0.997792	−	0.0664089i	$-0.978846\pi$
$907$	− 4.00000i	− 0.132818i	0.997792	−	0.0664089i	$-0.0211542\pi$
$908$	12.0000i	0.398234i
$909$	0	0
$910$	0	0
$911$	−36.0000	−1.19273	−0.596367	−	0.802712i	$-0.703390\pi$
$911$	−36.0000	−1.19273	−0.596367	+	0.802712i	$0.703390\pi$
$912$	0	0
$913$	27.0000i	0.893570i
$914$	17.0000	0.562310
$915$	0	0
$916$	26.0000	0.859064
$917$	0	0
$918$	0	0
$919$	34.0000	1.12156	0.560778	−	0.827966i	$-0.310502\pi$
$919$	34.0000	1.12156	0.560778	+	0.827966i	$0.310502\pi$
$920$	0	0
$921$	0	0
$922$	18.0000i	0.592798i
$923$	12.0000i	0.394985i
$924$	0	0
$925$	0	0
$926$	−8.00000	−0.262896
$927$	0	0
$928$	6.00000i	0.196960i
$929$	−18.0000	−0.590561	−0.295280	−	0.955411i	$-0.595413\pi$
$929$	−18.0000	−0.590561	−0.295280	+	0.955411i	$0.595413\pi$
$930$	0	0
$931$	−7.00000	−0.229416
$932$	− 6.00000i	− 0.196537i
$933$	0	0
$934$	−36.0000	−1.17796
$935$	0	0
$936$	0	0
$937$	29.0000i	0.947389i	0.880689	+	0.473694i	$0.157080\pi$
$937$	29.0000i	0.947389i	−0.880689	+	0.473694i	$0.842920\pi$
$938$	− 7.00000i	− 0.228558i
$939$	0	0
$940$	0	0
$941$	−24.0000	−0.782378	−0.391189	−	0.920310i	$-0.627936\pi$
$941$	−24.0000	−0.782378	−0.391189	+	0.920310i	$0.627936\pi$
$942$	0	0
$943$	0	0
$944$	12.0000	0.390567
$945$	0	0
$946$	24.0000	0.780307
$947$	− 12.0000i	− 0.389948i	−0.980808	−	0.194974i	$-0.937538\pi$
$947$	− 12.0000i	− 0.389948i	0.980808	−	0.194974i	$-0.0624622\pi$
$948$	0	0
$949$	−10.0000	−0.324614
$950$	0	0
$951$	0	0
$952$	3.00000i	0.0972306i
$953$	57.0000i	1.84641i	0.384307	+	0.923206i	$0.374441\pi$
$953$	57.0000i	1.84641i	−0.384307	+	0.923206i	$0.625559\pi$
$954$	0	0
$955$	0	0
$956$	12.0000	0.388108
$957$	0	0
$958$	− 18.0000i	− 0.581554i
$959$	21.0000	0.678125
$960$	0	0
$961$	−15.0000	−0.483871
$962$	− 16.0000i	− 0.515861i
$963$	0	0
$964$	25.0000	0.805196
$965$	0	0
$966$	0	0
$967$	− 34.0000i	− 1.09337i	−0.837340	−	0.546683i	$-0.815890\pi$
$967$	− 34.0000i	− 1.09337i	0.837340	−	0.546683i	$-0.184110\pi$
$968$	− 2.00000i	− 0.0642824i
$969$	0	0
$970$	0	0
$971$	−9.00000	−0.288824	−0.144412	−	0.989518i	$-0.546129\pi$
$971$	−9.00000	−0.288824	−0.144412	+	0.989518i	$0.546129\pi$
$972$	0	0
$973$	7.00000i	0.224410i
$974$	−34.0000	−1.08943
$975$	0	0
$976$	−10.0000	−0.320092
$977$	3.00000i	0.0959785i	0.998848	+	0.0479893i	$0.0152813\pi$
$977$	3.00000i	0.0959785i	−0.998848	+	0.0479893i	$0.984719\pi$
$978$	0	0
$979$	45.0000	1.43821
$980$	0	0
$981$	0	0
$982$	− 12.0000i	− 0.382935i
$983$	12.0000i	0.382741i	0.981518	+	0.191370i	$0.0612931\pi$
$983$	12.0000i	0.382741i	−0.981518	+	0.191370i	$0.938707\pi$
$984$	0	0
$985$	0	0
$986$	18.0000	0.573237
$987$	0	0
$988$	14.0000i	0.445399i
$989$	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	$0.397103\pi$
$992$	4.00000i	0.127000i
$993$	0	0
$994$	−6.00000	−0.190308
$995$	0	0
$996$	0	0
$997$	62.0000i	1.96356i	0.190022	+	0.981780i	$0.439144\pi$
$997$	62.0000i	1.96356i	−0.190022	+	0.981780i	$0.560856\pi$
$998$	− 28.0000i	− 0.886325i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3150.2.g.f.2899.1		2
3.2	odd	2		350.2.c.c.99.2		2
5.2	odd	4		3150.2.a.x.1.1		1
5.3	odd	4		3150.2.a.m.1.1		1
5.4	even	2	inner	3150.2.g.f.2899.2		2
12.11	even	2		2800.2.g.i.449.2		2
15.2	even	4		350.2.a.a.1.1	✓	1
15.8	even	4		350.2.a.e.1.1	yes	1
15.14	odd	2		350.2.c.c.99.1		2
21.20	even	2		2450.2.c.h.99.2		2
60.23	odd	4		2800.2.a.h.1.1		1
60.47	odd	4		2800.2.a.x.1.1		1
60.59	even	2		2800.2.g.i.449.1		2
105.62	odd	4		2450.2.a.m.1.1		1
105.83	odd	4		2450.2.a.x.1.1		1
105.104	even	2		2450.2.c.h.99.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
350.2.a.a.1.1	✓	1	15.2	even	4
350.2.a.e.1.1	yes	1	15.8	even	4
350.2.c.c.99.1		2	15.14	odd	2
350.2.c.c.99.2		2	3.2	odd	2
2450.2.a.m.1.1		1	105.62	odd	4
2450.2.a.x.1.1		1	105.83	odd	4
2450.2.c.h.99.1		2	105.104	even	2
2450.2.c.h.99.2		2	21.20	even	2
2800.2.a.h.1.1		1	60.23	odd	4
2800.2.a.x.1.1		1	60.47	odd	4
2800.2.g.i.449.1		2	60.59	even	2
2800.2.g.i.449.2		2	12.11	even	2
3150.2.a.m.1.1		1	5.3	odd	4
3150.2.a.x.1.1		1	5.2	odd	4
3150.2.g.f.2899.1		2	1.1	even	1	trivial
3150.2.g.f.2899.2		2	5.4	even	2	inner

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:	\( N \)	\(=\)	\( 3150 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3150.g (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} -1.00000 q^{4} +1.00000i q^{7} +1.00000i q^{8} +O(q^{10})\)	\(q-1.00000i q^{2} -1.00000 q^{4} +1.00000i q^{7} +1.00000i q^{8} -3.00000 q^{11} -2.00000i q^{13} +1.00000 q^{14} +1.00000 q^{16} -3.00000i q^{17} +7.00000 q^{19} +3.00000i q^{22} -2.00000 q^{26} -1.00000i q^{28} -6.00000 q^{29} -4.00000 q^{31} -1.00000i q^{32} -3.00000 q^{34} +8.00000i q^{37} -7.00000i q^{38} +9.00000 q^{41} -8.00000i q^{43} +3.00000 q^{44} +6.00000i q^{47} -1.00000 q^{49} +2.00000i q^{52} -12.0000i q^{53} -1.00000 q^{56} +6.00000i q^{58} +12.0000 q^{59} -10.0000 q^{61} +4.00000i q^{62} -1.00000 q^{64} -7.00000i q^{67} +3.00000i q^{68} -6.00000 q^{71} -5.00000i q^{73} +8.00000 q^{74} -7.00000 q^{76} -3.00000i q^{77} -14.0000 q^{79} -9.00000i q^{82} -9.00000i q^{83} -8.00000 q^{86} -3.00000i q^{88} -15.0000 q^{89} +2.00000 q^{91} +6.00000 q^{94} -10.0000i q^{97} +1.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4}+O(q^{10}) \) 2 * q - 2 * q^4	\( 2 q - 2 q^{4} - 6 q^{11} + 2 q^{14} + 2 q^{16} + 14 q^{19} - 4 q^{26} - 12 q^{29} - 8 q^{31} - 6 q^{34} + 18 q^{41} + 6 q^{44} - 2 q^{49} - 2 q^{56} + 24 q^{59} - 20 q^{61} - 2 q^{64} - 12 q^{71} + 16 q^{74} - 14 q^{76} - 28 q^{79} - 16 q^{86} - 30 q^{89} + 4 q^{91} + 12 q^{94}+O(q^{100}) \) 2 * q - 2 * q^4 - 6 * q^11 + 2 * q^14 + 2 * q^16 + 14 * q^19 - 4 * q^26 - 12 * q^29 - 8 * q^31 - 6 * q^34 + 18 * q^41 + 6 * q^44 - 2 * q^49 - 2 * q^56 + 24 * q^59 - 20 * q^61 - 2 * q^64 - 12 * q^71 + 16 * q^74 - 14 * q^76 - 28 * q^79 - 16 * q^86 - 30 * q^89 + 4 * q^91 + 12 * q^94

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