LMFDB - Embedded newform 3150.2.g.v.2899.2

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

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{2} -1.00000 q^{4} +1.00000i q^{7} -1.00000i q^{8} +O(q^{10})$	$q+1.00000i q^{2} -1.00000 q^{4} +1.00000i q^{7} -1.00000i q^{8} +5.00000 q^{11} -6.00000i q^{13} -1.00000 q^{14} +1.00000 q^{16} -1.00000i q^{17} +3.00000 q^{19} +5.00000i q^{22} +6.00000 q^{26} -1.00000i q^{28} -6.00000 q^{29} -4.00000 q^{31} +1.00000i q^{32} +1.00000 q^{34} -8.00000i q^{37} +3.00000i q^{38} -11.0000 q^{41} -8.00000i q^{43} -5.00000 q^{44} +2.00000i q^{47} -1.00000 q^{49} +6.00000i q^{52} -4.00000i q^{53} +1.00000 q^{56} -6.00000i q^{58} +4.00000 q^{59} -2.00000 q^{61} -4.00000i q^{62} -1.00000 q^{64} -9.00000i q^{67} +1.00000i q^{68} +10.0000 q^{71} -7.00000i q^{73} +8.00000 q^{74} -3.00000 q^{76} +5.00000i q^{77} +2.00000 q^{79} -11.0000i q^{82} -11.0000i q^{83} +8.00000 q^{86} -5.00000i q^{88} -11.0000 q^{89} +6.00000 q^{91} -2.00000 q^{94} +10.0000i q^{97} -1.00000i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{4}+O(q^{10}) $ 2 * q - 2 * q^4	$ 2 q - 2 q^{4} + 10 q^{11} - 2 q^{14} + 2 q^{16} + 6 q^{19} + 12 q^{26} - 12 q^{29} - 8 q^{31} + 2 q^{34} - 22 q^{41} - 10 q^{44} - 2 q^{49} + 2 q^{56} + 8 q^{59} - 4 q^{61} - 2 q^{64} + 20 q^{71} + 16 q^{74} - 6 q^{76} + 4 q^{79} + 16 q^{86} - 22 q^{89} + 12 q^{91} - 4 q^{94}+O(q^{100}) $ 2 * q - 2 * q^4 + 10 * q^11 - 2 * q^14 + 2 * q^16 + 6 * q^19 + 12 * q^26 - 12 * q^29 - 8 * q^31 + 2 * q^34 - 22 * q^41 - 10 * q^44 - 2 * q^49 + 2 * q^56 + 8 * q^59 - 4 * q^61 - 2 * q^64 + 20 * q^71 + 16 * q^74 - 6 * q^76 + 4 * q^79 + 16 * q^86 - 22 * q^89 + 12 * q^91 - 4 * 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
$11$	5.00000	1.50756	0.753778	−	0.657129i	$-0.228229\pi$
$11$	5.00000	1.50756	0.753778	+	0.657129i	$0.228229\pi$
$12$	0	0
$13$	− 6.00000i	− 1.66410i	−0.554700	−	0.832050i	$-0.687167\pi$
$13$	− 6.00000i	− 1.66410i	0.554700	−	0.832050i	$-0.312833\pi$
$14$	−1.00000	−0.267261
$15$	0	0
$16$	1.00000	0.250000
$17$	− 1.00000i	− 0.242536i	−0.992620	−	0.121268i	$-0.961304\pi$
$17$	− 1.00000i	− 0.242536i	0.992620	−	0.121268i	$-0.0386960\pi$
$18$	0	0
$19$	3.00000	0.688247	0.344124	−	0.938924i	$-0.388176\pi$
$19$	3.00000	0.688247	0.344124	+	0.938924i	$0.388176\pi$
$20$	0	0
$21$	0	0
$22$	5.00000i	1.06600i
$23$	0	0	1.00000			$0$
$23$	0	0	−1.00000			$\pi$
$24$	0	0
$25$	0	0
$26$	6.00000	1.17670
$27$	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
$34$	1.00000	0.171499
$35$	0	0
$36$	0	0
$37$	− 8.00000i	− 1.31519i	−0.753371	−	0.657596i	$-0.771573\pi$
$37$	− 8.00000i	− 1.31519i	0.753371	−	0.657596i	$-0.228427\pi$
$38$	3.00000i	0.486664i
$39$	0	0
$40$	0	0
$41$	−11.0000	−1.71791	−0.858956	−	0.512050i	$-0.828886\pi$
$41$	−11.0000	−1.71791	−0.858956	+	0.512050i	$0.828886\pi$
$42$	0	0
$43$	− 8.00000i	− 1.21999i	−0.792406	−	0.609994i	$-0.791172\pi$
$43$	− 8.00000i	− 1.21999i	0.792406	−	0.609994i	$-0.208828\pi$
$44$	−5.00000	−0.753778
$45$	0	0
$46$	0	0
$47$	2.00000i	0.291730i	0.989305	+	0.145865i	$0.0465965\pi$
$47$	2.00000i	0.291730i	−0.989305	+	0.145865i	$0.953403\pi$
$48$	0	0
$49$	−1.00000	−0.142857
$50$	0	0
$51$	0	0
$52$	6.00000i	0.832050i
$53$	− 4.00000i	− 0.549442i	−0.961524	−	0.274721i	$-0.911414\pi$
$53$	− 4.00000i	− 0.549442i	0.961524	−	0.274721i	$-0.0885855\pi$
$54$	0	0
$55$	0	0
$56$	1.00000	0.133631
$57$	0	0
$58$	− 6.00000i	− 0.787839i
$59$	4.00000	0.520756	0.260378	−	0.965507i	$-0.416153\pi$
$59$	4.00000	0.520756	0.260378	+	0.965507i	$0.416153\pi$
$60$	0	0
$61$	−2.00000	−0.256074	−0.128037	−	0.991769i	$-0.540868\pi$
$61$	−2.00000	−0.256074	−0.128037	+	0.991769i	$0.540868\pi$
$62$	− 4.00000i	− 0.508001i
$63$	0	0
$64$	−1.00000	−0.125000
$65$	0	0
$66$	0	0
$67$	− 9.00000i	− 1.09952i	−0.835321	−	0.549762i	$-0.814718\pi$
$67$	− 9.00000i	− 1.09952i	0.835321	−	0.549762i	$-0.185282\pi$
$68$	1.00000i	0.121268i
$69$	0	0
$70$	0	0
$71$	10.0000	1.18678	0.593391	−	0.804914i	$-0.297789\pi$
$71$	10.0000	1.18678	0.593391	+	0.804914i	$0.297789\pi$
$72$	0	0
$73$	− 7.00000i	− 0.819288i	−0.912245	−	0.409644i	$-0.865653\pi$
$73$	− 7.00000i	− 0.819288i	0.912245	−	0.409644i	$-0.134347\pi$
$74$	8.00000	0.929981
$75$	0	0
$76$	−3.00000	−0.344124
$77$	5.00000i	0.569803i
$78$	0	0
$79$	2.00000	0.225018	0.112509	−	0.993651i	$-0.464111\pi$
$79$	2.00000	0.225018	0.112509	+	0.993651i	$0.464111\pi$
$80$	0	0
$81$	0	0
$82$	− 11.0000i	− 1.21475i
$83$	− 11.0000i	− 1.20741i	−0.797209	−	0.603703i	$-0.793691\pi$
$83$	− 11.0000i	− 1.20741i	0.797209	−	0.603703i	$-0.206309\pi$
$84$	0	0
$85$	0	0
$86$	8.00000	0.862662
$87$	0	0
$88$	− 5.00000i	− 0.533002i
$89$	−11.0000	−1.16600	−0.582999	−	0.812473i	$-0.698121\pi$
$89$	−11.0000	−1.16600	−0.582999	+	0.812473i	$0.698121\pi$
$90$	0	0
$91$	6.00000	0.628971
$92$	0	0
$93$	0	0
$94$	−2.00000	−0.206284
$95$	0	0
$96$	0	0
$97$	10.0000i	1.01535i	0.861550	+	0.507673i	$0.169494\pi$
$97$	10.0000i	1.01535i	−0.861550	+	0.507673i	$0.830506\pi$
$98$	− 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
$103$	4.00000i	0.394132i	0.980390	+	0.197066i	$0.0631413\pi$
$103$	4.00000i	0.394132i	−0.980390	+	0.197066i	$0.936859\pi$
$104$	−6.00000	−0.588348
$105$	0	0
$106$	4.00000	0.388514
$107$	3.00000i	0.290021i	0.989430	+	0.145010i	$0.0463216\pi$
$107$	3.00000i	0.290021i	−0.989430	+	0.145010i	$0.953678\pi$
$108$	0	0
$109$	18.0000	1.72409	0.862044	−	0.506834i	$-0.169184\pi$
$109$	18.0000	1.72409	0.862044	+	0.506834i	$0.169184\pi$
$110$	0	0
$111$	0	0
$112$	1.00000i	0.0944911i
$113$	1.00000i	0.0940721i	0.998893	+	0.0470360i	$0.0149776\pi$
$113$	1.00000i	0.0940721i	−0.998893	+	0.0470360i	$0.985022\pi$
$114$	0	0
$115$	0	0
$116$	6.00000	0.557086
$117$	0	0
$118$	4.00000i	0.368230i
$119$	1.00000	0.0916698
$120$	0	0
$121$	14.0000	1.27273
$122$	− 2.00000i	− 0.181071i
$123$	0	0
$124$	4.00000	0.359211
$125$	0	0
$126$	0	0
$127$	14.0000i	1.24230i	0.783692	+	0.621150i	$0.213334\pi$
$127$	14.0000i	1.24230i	−0.783692	+	0.621150i	$0.786666\pi$
$128$	− 1.00000i	− 0.0883883i
$129$	0	0
$130$	0	0
$131$	−8.00000	−0.698963	−0.349482	−	0.936943i	$-0.613642\pi$
$131$	−8.00000	−0.698963	−0.349482	+	0.936943i	$0.613642\pi$
$132$	0	0
$133$	3.00000i	0.260133i
$134$	9.00000	0.777482
$135$	0	0
$136$	−1.00000	−0.0857493
$137$	− 3.00000i	− 0.256307i	−0.991754	−	0.128154i	$-0.959095\pi$
$137$	− 3.00000i	− 0.256307i	0.991754	−	0.128154i	$-0.0409051\pi$
$138$	0	0
$139$	11.0000	0.933008	0.466504	−	0.884519i	$-0.345513\pi$
$139$	11.0000	0.933008	0.466504	+	0.884519i	$0.345513\pi$
$140$	0	0
$141$	0	0
$142$	10.0000i	0.839181i
$143$	− 30.0000i	− 2.50873i
$144$	0	0
$145$	0	0
$146$	7.00000	0.579324
$147$	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	$0.394462\pi$
$152$	− 3.00000i	− 0.243332i
$153$	0	0
$154$	−5.00000	−0.402911
$155$	0	0
$156$	0	0
$157$	− 4.00000i	− 0.319235i	−0.987179	−	0.159617i	$-0.948974\pi$
$157$	− 4.00000i	− 0.319235i	0.987179	−	0.159617i	$-0.0510260\pi$
$158$	2.00000i	0.159111i
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	− 19.0000i	− 1.48819i	−0.668071	−	0.744097i	$-0.732880\pi$
$163$	− 19.0000i	− 1.48819i	0.668071	−	0.744097i	$-0.267120\pi$
$164$	11.0000	0.858956
$165$	0	0
$166$	11.0000	0.853766
$167$	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
$169$	−23.0000	−1.76923
$170$	0	0
$171$	0	0
$172$	8.00000i	0.609994i
$173$	− 2.00000i	− 0.152057i	−0.997106	−	0.0760286i	$-0.975776\pi$
$173$	− 2.00000i	− 0.152057i	0.997106	−	0.0760286i	$-0.0242240\pi$
$174$	0	0
$175$	0	0
$176$	5.00000	0.376889
$177$	0	0
$178$	− 11.0000i	− 0.824485i
$179$	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
$181$	10.0000	0.743294	0.371647	−	0.928374i	$-0.378793\pi$
$181$	10.0000	0.743294	0.371647	+	0.928374i	$0.378793\pi$
$182$	6.00000i	0.444750i
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 5.00000i	− 0.365636i
$188$	− 2.00000i	− 0.145865i
$189$	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
$193$	− 19.0000i	− 1.36765i	−0.729646	−	0.683825i	$-0.760315\pi$
$193$	− 19.0000i	− 1.36765i	0.729646	−	0.683825i	$-0.239685\pi$
$194$	−10.0000	−0.717958
$195$	0	0
$196$	1.00000	0.0714286
$197$	− 22.0000i	− 1.56744i	−0.621117	−	0.783718i	$-0.713321\pi$
$197$	− 22.0000i	− 1.56744i	0.621117	−	0.783718i	$-0.286679\pi$
$198$	0	0
$199$	10.0000	0.708881	0.354441	−	0.935079i	$-0.384671\pi$
$199$	10.0000	0.708881	0.354441	+	0.935079i	$0.384671\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 6.00000i	− 0.421117i
$204$	0	0
$205$	0	0
$206$	−4.00000	−0.278693
$207$	0	0
$208$	− 6.00000i	− 0.416025i
$209$	15.0000	1.03757
$210$	0	0
$211$	1.00000	0.0688428	0.0344214	−	0.999407i	$-0.489041\pi$
$211$	1.00000	0.0688428	0.0344214	+	0.999407i	$0.489041\pi$
$212$	4.00000i	0.274721i
$213$	0	0
$214$	−3.00000	−0.205076
$215$	0	0
$216$	0	0
$217$	− 4.00000i	− 0.271538i
$218$	18.0000i	1.21911i
$219$	0	0
$220$	0	0
$221$	−6.00000	−0.403604
$222$	0	0
$223$	22.0000i	1.47323i	0.676313	+	0.736614i	$0.263577\pi$
$223$	22.0000i	1.47323i	−0.676313	+	0.736614i	$0.736423\pi$
$224$	−1.00000	−0.0668153
$225$	0	0
$226$	−1.00000	−0.0665190
$227$	− 28.0000i	− 1.85843i	−0.369546	−	0.929213i	$-0.620487\pi$
$227$	− 28.0000i	− 1.85843i	0.369546	−	0.929213i	$-0.379513\pi$
$228$	0	0
$229$	14.0000	0.925146	0.462573	−	0.886581i	$-0.346926\pi$
$229$	14.0000	0.925146	0.462573	+	0.886581i	$0.346926\pi$
$230$	0	0
$231$	0	0
$232$	6.00000i	0.393919i
$233$	− 6.00000i	− 0.393073i	−0.980497	−	0.196537i	$-0.937031\pi$
$233$	− 6.00000i	− 0.393073i	0.980497	−	0.196537i	$-0.0629694\pi$
$234$	0	0
$235$	0	0
$236$	−4.00000	−0.260378
$237$	0	0
$238$	1.00000i	0.0648204i
$239$	4.00000	0.258738	0.129369	−	0.991596i	$-0.458705\pi$
$239$	4.00000	0.258738	0.129369	+	0.991596i	$0.458705\pi$
$240$	0	0
$241$	−5.00000	−0.322078	−0.161039	−	0.986948i	$-0.551485\pi$
$241$	−5.00000	−0.322078	−0.161039	+	0.986948i	$0.551485\pi$
$242$	14.0000i	0.899954i
$243$	0	0
$244$	2.00000	0.128037
$245$	0	0
$246$	0	0
$247$	− 18.0000i	− 1.14531i
$248$	4.00000i	0.254000i
$249$	0	0
$250$	0	0
$251$	−27.0000	−1.70422	−0.852112	−	0.523359i	$-0.824679\pi$
$251$	−27.0000	−1.70422	−0.852112	+	0.523359i	$0.824679\pi$
$252$	0	0
$253$	0	0
$254$	−14.0000	−0.878438
$255$	0	0
$256$	1.00000	0.0625000
$257$	2.00000i	0.124757i	0.998053	+	0.0623783i	$0.0198685\pi$
$257$	2.00000i	0.124757i	−0.998053	+	0.0623783i	$0.980131\pi$
$258$	0	0
$259$	8.00000	0.497096
$260$	0	0
$261$	0	0
$262$	− 8.00000i	− 0.494242i
$263$	10.0000i	0.616626i	0.951285	+	0.308313i	$0.0997645\pi$
$263$	10.0000i	0.616626i	−0.951285	+	0.308313i	$0.900236\pi$
$264$	0	0
$265$	0	0
$266$	−3.00000	−0.183942
$267$	0	0
$268$	9.00000i	0.549762i
$269$	18.0000	1.09748	0.548740	−	0.835993i	$-0.315108\pi$
$269$	18.0000	1.09748	0.548740	+	0.835993i	$0.315108\pi$
$270$	0	0
$271$	−6.00000	−0.364474	−0.182237	−	0.983255i	$-0.558334\pi$
$271$	−6.00000	−0.364474	−0.182237	+	0.983255i	$0.558334\pi$
$272$	− 1.00000i	− 0.0606339i
$273$	0	0
$274$	3.00000	0.181237
$275$	0	0
$276$	0	0
$277$	30.0000i	1.80253i	0.433273	+	0.901263i	$0.357359\pi$
$277$	30.0000i	1.80253i	−0.433273	+	0.901263i	$0.642641\pi$
$278$	11.0000i	0.659736i
$279$	0	0
$280$	0	0
$281$	−14.0000	−0.835170	−0.417585	−	0.908638i	$-0.637123\pi$
$281$	−14.0000	−0.835170	−0.417585	+	0.908638i	$0.637123\pi$
$282$	0	0
$283$	− 13.0000i	− 0.772770i	−0.922338	−	0.386385i	$-0.873724\pi$
$283$	− 13.0000i	− 0.772770i	0.922338	−	0.386385i	$-0.126276\pi$
$284$	−10.0000	−0.593391
$285$	0	0
$286$	30.0000	1.77394
$287$	− 11.0000i	− 0.649309i
$288$	0	0
$289$	16.0000	0.941176
$290$	0	0
$291$	0	0
$292$	7.00000i	0.409644i
$293$	14.0000i	0.817889i	0.912559	+	0.408944i	$0.134103\pi$
$293$	14.0000i	0.817889i	−0.912559	+	0.408944i	$0.865897\pi$
$294$	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
$304$	3.00000	0.172062
$305$	0	0
$306$	0	0
$307$	− 13.0000i	− 0.741949i	−0.928643	−	0.370975i	$-0.879024\pi$
$307$	− 13.0000i	− 0.741949i	0.928643	−	0.370975i	$-0.120976\pi$
$308$	− 5.00000i	− 0.284901i
$309$	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
$313$	− 10.0000i	− 0.565233i	−0.959233	−	0.282617i	$-0.908798\pi$
$313$	− 10.0000i	− 0.565233i	0.959233	−	0.282617i	$-0.0912024\pi$
$314$	4.00000	0.225733
$315$	0	0
$316$	−2.00000	−0.112509
$317$	4.00000i	0.224662i	0.993671	+	0.112331i	$0.0358318\pi$
$317$	4.00000i	0.224662i	−0.993671	+	0.112331i	$0.964168\pi$
$318$	0	0
$319$	−30.0000	−1.67968
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 3.00000i	− 0.166924i
$324$	0	0
$325$	0	0
$326$	19.0000	1.05231
$327$	0	0
$328$	11.0000i	0.607373i
$329$	−2.00000	−0.110264
$330$	0	0
$331$	−17.0000	−0.934405	−0.467202	−	0.884150i	$-0.654738\pi$
$331$	−17.0000	−0.934405	−0.467202	+	0.884150i	$0.654738\pi$
$332$	11.0000i	0.603703i
$333$	0	0
$334$	−12.0000	−0.656611
$335$	0	0
$336$	0	0
$337$	29.0000i	1.57973i	0.613280	+	0.789865i	$0.289850\pi$
$337$	29.0000i	1.57973i	−0.613280	+	0.789865i	$0.710150\pi$
$338$	− 23.0000i	− 1.25104i
$339$	0	0
$340$	0	0
$341$	−20.0000	−1.08306
$342$	0	0
$343$	− 1.00000i	− 0.0539949i
$344$	−8.00000	−0.431331
$345$	0	0
$346$	2.00000	0.107521
$347$	19.0000i	1.01997i	0.860182	+	0.509987i	$0.170350\pi$
$347$	19.0000i	1.01997i	−0.860182	+	0.509987i	$0.829650\pi$
$348$	0	0
$349$	8.00000	0.428230	0.214115	−	0.976808i	$-0.431313\pi$
$349$	8.00000	0.428230	0.214115	+	0.976808i	$0.431313\pi$
$350$	0	0
$351$	0	0
$352$	5.00000i	0.266501i
$353$	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
$356$	11.0000	0.582999
$357$	0	0
$358$	3.00000i	0.158555i
$359$	26.0000	1.37223	0.686114	−	0.727494i	$-0.259315\pi$
$359$	26.0000	1.37223	0.686114	+	0.727494i	$0.259315\pi$
$360$	0	0
$361$	−10.0000	−0.526316
$362$	10.0000i	0.525588i
$363$	0	0
$364$	−6.00000	−0.314485
$365$	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
$371$	4.00000	0.207670
$372$	0	0
$373$	− 4.00000i	− 0.207112i	−0.994624	−	0.103556i	$-0.966978\pi$
$373$	− 4.00000i	− 0.207112i	0.994624	−	0.103556i	$-0.0330221\pi$
$374$	5.00000	0.258544
$375$	0	0
$376$	2.00000	0.103142
$377$	36.0000i	1.85409i
$378$	0	0
$379$	−9.00000	−0.462299	−0.231149	−	0.972918i	$-0.574249\pi$
$379$	−9.00000	−0.462299	−0.231149	+	0.972918i	$0.574249\pi$
$380$	0	0
$381$	0	0
$382$	6.00000i	0.306987i
$383$	6.00000i	0.306586i	0.988181	+	0.153293i	$0.0489878\pi$
$383$	6.00000i	0.306586i	−0.988181	+	0.153293i	$0.951012\pi$
$384$	0	0
$385$	0	0
$386$	19.0000	0.967075
$387$	0	0
$388$	− 10.0000i	− 0.507673i
$389$	8.00000	0.405616	0.202808	−	0.979219i	$-0.434993\pi$
$389$	8.00000	0.405616	0.202808	+	0.979219i	$0.434993\pi$
$390$	0	0
$391$	0	0
$392$	1.00000i	0.0505076i
$393$	0	0
$394$	22.0000	1.10834
$395$	0	0
$396$	0	0
$397$	− 10.0000i	− 0.501886i	−0.968002	−	0.250943i	$-0.919259\pi$
$397$	− 10.0000i	− 0.501886i	0.968002	−	0.250943i	$-0.0807406\pi$
$398$	10.0000i	0.501255i
$399$	0	0
$400$	0	0
$401$	−37.0000	−1.84769	−0.923846	−	0.382765i	$-0.874972\pi$
$401$	−37.0000	−1.84769	−0.923846	+	0.382765i	$0.874972\pi$
$402$	0	0
$403$	24.0000i	1.19553i
$404$	0	0
$405$	0	0
$406$	6.00000	0.297775
$407$	− 40.0000i	− 1.98273i
$408$	0	0
$409$	21.0000	1.03838	0.519192	−	0.854658i	$-0.326233\pi$
$409$	21.0000	1.03838	0.519192	+	0.854658i	$0.326233\pi$
$410$	0	0
$411$	0	0
$412$	− 4.00000i	− 0.197066i
$413$	4.00000i	0.196827i
$414$	0	0
$415$	0	0
$416$	6.00000	0.294174
$417$	0	0
$418$	15.0000i	0.733674i
$419$	−39.0000	−1.90527	−0.952637	−	0.304109i	$-0.901641\pi$
$419$	−39.0000	−1.90527	−0.952637	+	0.304109i	$0.901641\pi$
$420$	0	0
$421$	20.0000	0.974740	0.487370	−	0.873195i	$-0.337956\pi$
$421$	20.0000	0.974740	0.487370	+	0.873195i	$0.337956\pi$
$422$	1.00000i	0.0486792i
$423$	0	0
$424$	−4.00000	−0.194257
$425$	0	0
$426$	0	0
$427$	− 2.00000i	− 0.0967868i
$428$	− 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
$433$	− 1.00000i	− 0.0480569i	−0.999711	−	0.0240285i	$-0.992351\pi$
$433$	− 1.00000i	− 0.0480569i	0.999711	−	0.0240285i	$-0.00764923\pi$
$434$	4.00000	0.192006
$435$	0	0
$436$	−18.0000	−0.862044
$437$	0	0
$438$	0	0
$439$	−28.0000	−1.33637	−0.668184	−	0.743996i	$-0.732928\pi$
$439$	−28.0000	−1.33637	−0.668184	+	0.743996i	$0.732928\pi$
$440$	0	0
$441$	0	0
$442$	− 6.00000i	− 0.285391i
$443$	− 37.0000i	− 1.75792i	−0.476893	−	0.878962i	$-0.658237\pi$
$443$	− 37.0000i	− 1.75792i	0.476893	−	0.878962i	$-0.341763\pi$
$444$	0	0
$445$	0	0
$446$	−22.0000	−1.04173
$447$	0	0
$448$	− 1.00000i	− 0.0472456i
$449$	33.0000	1.55737	0.778683	−	0.627417i	$-0.215888\pi$
$449$	33.0000	1.55737	0.778683	+	0.627417i	$0.215888\pi$
$450$	0	0
$451$	−55.0000	−2.58985
$452$	− 1.00000i	− 0.0470360i
$453$	0	0
$454$	28.0000	1.31411
$455$	0	0
$456$	0	0
$457$	− 25.0000i	− 1.16945i	−0.811231	−	0.584725i	$-0.801202\pi$
$457$	− 25.0000i	− 1.16945i	0.811231	−	0.584725i	$-0.198798\pi$
$458$	14.0000i	0.654177i
$459$	0	0
$460$	0	0
$461$	38.0000	1.76984	0.884918	−	0.465746i	$-0.154214\pi$
$461$	38.0000	1.76984	0.884918	+	0.465746i	$0.154214\pi$
$462$	0	0
$463$	8.00000i	0.371792i	0.982569	+	0.185896i	$0.0595187\pi$
$463$	8.00000i	0.371792i	−0.982569	+	0.185896i	$0.940481\pi$
$464$	−6.00000	−0.278543
$465$	0	0
$466$	6.00000	0.277945
$467$	− 4.00000i	− 0.185098i	−0.995708	−	0.0925490i	$-0.970499\pi$
$467$	− 4.00000i	− 0.185098i	0.995708	−	0.0925490i	$-0.0295015\pi$
$468$	0	0
$469$	9.00000	0.415581
$470$	0	0
$471$	0	0
$472$	− 4.00000i	− 0.184115i
$473$	− 40.0000i	− 1.83920i
$474$	0	0
$475$	0	0
$476$	−1.00000	−0.0458349
$477$	0	0
$478$	4.00000i	0.182956i
$479$	−6.00000	−0.274147	−0.137073	−	0.990561i	$-0.543770\pi$
$479$	−6.00000	−0.274147	−0.137073	+	0.990561i	$0.543770\pi$
$480$	0	0
$481$	−48.0000	−2.18861
$482$	− 5.00000i	− 0.227744i
$483$	0	0
$484$	−14.0000	−0.636364
$485$	0	0
$486$	0	0
$487$	34.0000i	1.54069i	0.637629	+	0.770344i	$0.279915\pi$
$487$	34.0000i	1.54069i	−0.637629	+	0.770344i	$0.720085\pi$
$488$	2.00000i	0.0905357i
$489$	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
$493$	6.00000i	0.270226i
$494$	18.0000	0.809858
$495$	0	0
$496$	−4.00000	−0.179605
$497$	10.0000i	0.448561i
$498$	0	0
$499$	−36.0000	−1.61158	−0.805791	−	0.592200i	$-0.798259\pi$
$499$	−36.0000	−1.61158	−0.805791	+	0.592200i	$0.798259\pi$
$500$	0	0
$501$	0	0
$502$	− 27.0000i	− 1.20507i
$503$	30.0000i	1.33763i	0.743427	+	0.668817i	$0.233199\pi$
$503$	30.0000i	1.33763i	−0.743427	+	0.668817i	$0.766801\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0	0
$508$	− 14.0000i	− 0.621150i
$509$	−14.0000	−0.620539	−0.310270	−	0.950649i	$-0.600419\pi$
$509$	−14.0000	−0.620539	−0.310270	+	0.950649i	$0.600419\pi$
$510$	0	0
$511$	7.00000	0.309662
$512$	1.00000i	0.0441942i
$513$	0	0
$514$	−2.00000	−0.0882162
$515$	0	0
$516$	0	0
$517$	10.0000i	0.439799i
$518$	8.00000i	0.351500i
$519$	0	0
$520$	0	0
$521$	−11.0000	−0.481919	−0.240959	−	0.970535i	$-0.577462\pi$
$521$	−11.0000	−0.481919	−0.240959	+	0.970535i	$0.577462\pi$
$522$	0	0
$523$	13.0000i	0.568450i	0.958758	+	0.284225i	$0.0917363\pi$
$523$	13.0000i	0.568450i	−0.958758	+	0.284225i	$0.908264\pi$
$524$	8.00000	0.349482
$525$	0	0
$526$	−10.0000	−0.436021
$527$	4.00000i	0.174243i
$528$	0	0
$529$	23.0000	1.00000
$530$	0	0
$531$	0	0
$532$	− 3.00000i	− 0.130066i
$533$	66.0000i	2.85878i
$534$	0	0
$535$	0	0
$536$	−9.00000	−0.388741
$537$	0	0
$538$	18.0000i	0.776035i
$539$	−5.00000	−0.215365
$540$	0	0
$541$	−42.0000	−1.80572	−0.902861	−	0.429934i	$-0.858537\pi$
$541$	−42.0000	−1.80572	−0.902861	+	0.429934i	$0.858537\pi$
$542$	− 6.00000i	− 0.257722i
$543$	0	0
$544$	1.00000	0.0428746
$545$	0	0
$546$	0	0
$547$	− 27.0000i	− 1.15444i	−0.816590	−	0.577218i	$-0.804138\pi$
$547$	− 27.0000i	− 1.15444i	0.816590	−	0.577218i	$-0.195862\pi$
$548$	3.00000i	0.128154i
$549$	0	0
$550$	0	0
$551$	−18.0000	−0.766826
$552$	0	0
$553$	2.00000i	0.0850487i
$554$	−30.0000	−1.27458
$555$	0	0
$556$	−11.0000	−0.466504
$557$	4.00000i	0.169485i	0.996403	+	0.0847427i	$0.0270068\pi$
$557$	4.00000i	0.169485i	−0.996403	+	0.0847427i	$0.972993\pi$
$558$	0	0
$559$	−48.0000	−2.03018
$560$	0	0
$561$	0	0
$562$	− 14.0000i	− 0.590554i
$563$	− 20.0000i	− 0.842900i	−0.906852	−	0.421450i	$-0.861521\pi$
$563$	− 20.0000i	− 0.842900i	0.906852	−	0.421450i	$-0.138479\pi$
$564$	0	0
$565$	0	0
$566$	13.0000	0.546431
$567$	0	0
$568$	− 10.0000i	− 0.419591i
$569$	21.0000	0.880366	0.440183	−	0.897908i	$-0.354914\pi$
$569$	21.0000	0.880366	0.440183	+	0.897908i	$0.354914\pi$
$570$	0	0
$571$	12.0000	0.502184	0.251092	−	0.967963i	$-0.419210\pi$
$571$	12.0000	0.502184	0.251092	+	0.967963i	$0.419210\pi$
$572$	30.0000i	1.25436i
$573$	0	0
$574$	11.0000	0.459131
$575$	0	0
$576$	0	0
$577$	− 13.0000i	− 0.541197i	−0.962692	−	0.270599i	$-0.912778\pi$
$577$	− 13.0000i	− 0.541197i	0.962692	−	0.270599i	$-0.0872216\pi$
$578$	16.0000i	0.665512i
$579$	0	0
$580$	0	0
$581$	11.0000	0.456357
$582$	0	0
$583$	− 20.0000i	− 0.828315i
$584$	−7.00000	−0.289662
$585$	0	0
$586$	−14.0000	−0.578335
$587$	− 13.0000i	− 0.536567i	−0.963340	−	0.268284i	$-0.913544\pi$
$587$	− 13.0000i	− 0.536567i	0.963340	−	0.268284i	$-0.0864565\pi$
$588$	0	0
$589$	−12.0000	−0.494451
$590$	0	0
$591$	0	0
$592$	− 8.00000i	− 0.328798i
$593$	39.0000i	1.60154i	0.598973	+	0.800769i	$0.295576\pi$
$593$	39.0000i	1.60154i	−0.598973	+	0.800769i	$0.704424\pi$
$594$	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	$-0.663115\pi$
$599$	−24.0000	−0.980613	−0.490307	+	0.871550i	$0.663115\pi$
$600$	0	0
$601$	21.0000	0.856608	0.428304	−	0.903635i	$-0.359111\pi$
$601$	21.0000	0.856608	0.428304	+	0.903635i	$0.359111\pi$
$602$	8.00000i	0.326056i
$603$	0	0
$604$	−8.00000	−0.325515
$605$	0	0
$606$	0	0
$607$	− 28.0000i	− 1.13648i	−0.822861	−	0.568242i	$-0.807624\pi$
$607$	− 28.0000i	− 1.13648i	0.822861	−	0.568242i	$-0.192376\pi$
$608$	3.00000i	0.121666i
$609$	0	0
$610$	0	0
$611$	12.0000	0.485468
$612$	0	0
$613$	18.0000i	0.727013i	0.931592	+	0.363507i	$0.118421\pi$
$613$	18.0000i	0.727013i	−0.931592	+	0.363507i	$0.881579\pi$
$614$	13.0000	0.524637
$615$	0	0
$616$	5.00000	0.201456
$617$	14.0000i	0.563619i	0.959470	+	0.281809i	$0.0909346\pi$
$617$	14.0000i	0.563619i	−0.959470	+	0.281809i	$0.909065\pi$
$618$	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
$623$	− 11.0000i	− 0.440706i
$624$	0	0
$625$	0	0
$626$	10.0000	0.399680
$627$	0	0
$628$	4.00000i	0.159617i
$629$	−8.00000	−0.318981
$630$	0	0
$631$	−16.0000	−0.636950	−0.318475	−	0.947931i	$-0.603171\pi$
$631$	−16.0000	−0.636950	−0.318475	+	0.947931i	$0.603171\pi$
$632$	− 2.00000i	− 0.0795557i
$633$	0	0
$634$	−4.00000	−0.158860
$635$	0	0
$636$	0	0
$637$	6.00000i	0.237729i
$638$	− 30.0000i	− 1.18771i
$639$	0	0
$640$	0	0
$641$	−2.00000	−0.0789953	−0.0394976	−	0.999220i	$-0.512576\pi$
$641$	−2.00000	−0.0789953	−0.0394976	+	0.999220i	$0.512576\pi$
$642$	0	0
$643$	16.0000i	0.630978i	0.948929	+	0.315489i	$0.102169\pi$
$643$	16.0000i	0.630978i	−0.948929	+	0.315489i	$0.897831\pi$
$644$	0	0
$645$	0	0
$646$	3.00000	0.118033
$647$	− 8.00000i	− 0.314512i	−0.987558	−	0.157256i	$-0.949735\pi$
$647$	− 8.00000i	− 0.314512i	0.987558	−	0.157256i	$-0.0502649\pi$
$648$	0	0
$649$	20.0000	0.785069
$650$	0	0
$651$	0	0
$652$	19.0000i	0.744097i
$653$	28.0000i	1.09572i	0.836569	+	0.547862i	$0.184558\pi$
$653$	28.0000i	1.09572i	−0.836569	+	0.547862i	$0.815442\pi$
$654$	0	0
$655$	0	0
$656$	−11.0000	−0.429478
$657$	0	0
$658$	− 2.00000i	− 0.0779681i
$659$	1.00000	0.0389545	0.0194772	−	0.999810i	$-0.493800\pi$
$659$	1.00000	0.0389545	0.0194772	+	0.999810i	$0.493800\pi$
$660$	0	0
$661$	−50.0000	−1.94477	−0.972387	−	0.233373i	$-0.925024\pi$
$661$	−50.0000	−1.94477	−0.972387	+	0.233373i	$0.925024\pi$
$662$	− 17.0000i	− 0.660724i
$663$	0	0
$664$	−11.0000	−0.426883
$665$	0	0
$666$	0	0
$667$	0	0
$668$	− 12.0000i	− 0.464294i
$669$	0	0
$670$	0	0
$671$	−10.0000	−0.386046
$672$	0	0
$673$	34.0000i	1.31060i	0.755367	+	0.655302i	$0.227459\pi$
$673$	34.0000i	1.31060i	−0.755367	+	0.655302i	$0.772541\pi$
$674$	−29.0000	−1.11704
$675$	0	0
$676$	23.0000	0.884615
$677$	− 48.0000i	− 1.84479i	−0.386248	−	0.922395i	$-0.626229\pi$
$677$	− 48.0000i	− 1.84479i	0.386248	−	0.922395i	$-0.373771\pi$
$678$	0	0
$679$	−10.0000	−0.383765
$680$	0	0
$681$	0	0
$682$	− 20.0000i	− 0.765840i
$683$	13.0000i	0.497431i	0.968577	+	0.248716i	$0.0800084\pi$
$683$	13.0000i	0.497431i	−0.968577	+	0.248716i	$0.919992\pi$
$684$	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
$691$	49.0000	1.86405	0.932024	−	0.362397i	$-0.118041\pi$
$691$	49.0000	1.86405	0.932024	+	0.362397i	$0.118041\pi$
$692$	2.00000i	0.0760286i
$693$	0	0
$694$	−19.0000	−0.721230
$695$	0	0
$696$	0	0
$697$	11.0000i	0.416655i
$698$	8.00000i	0.302804i
$699$	0	0
$700$	0	0
$701$	32.0000	1.20862	0.604312	−	0.796748i	$-0.293448\pi$
$701$	32.0000	1.20862	0.604312	+	0.796748i	$0.293448\pi$
$702$	0	0
$703$	− 24.0000i	− 0.905177i
$704$	−5.00000	−0.188445
$705$	0	0
$706$	−18.0000	−0.677439
$707$	0	0
$708$	0	0
$709$	−4.00000	−0.150223	−0.0751116	−	0.997175i	$-0.523931\pi$
$709$	−4.00000	−0.150223	−0.0751116	+	0.997175i	$0.523931\pi$
$710$	0	0
$711$	0	0
$712$	11.0000i	0.412242i
$713$	0	0
$714$	0	0
$715$	0	0
$716$	−3.00000	−0.112115
$717$	0	0
$718$	26.0000i	0.970311i
$719$	−48.0000	−1.79010	−0.895049	−	0.445968i	$-0.852860\pi$
$719$	−48.0000	−1.79010	−0.895049	+	0.445968i	$0.852860\pi$
$720$	0	0
$721$	−4.00000	−0.148968
$722$	− 10.0000i	− 0.372161i
$723$	0	0
$724$	−10.0000	−0.371647
$725$	0	0
$726$	0	0
$727$	− 6.00000i	− 0.222528i	−0.993791	−	0.111264i	$-0.964510\pi$
$727$	− 6.00000i	− 0.222528i	0.993791	−	0.111264i	$-0.0354899\pi$
$728$	− 6.00000i	− 0.222375i
$729$	0	0
$730$	0	0
$731$	−8.00000	−0.295891
$732$	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
$737$	− 45.0000i	− 1.65760i
$738$	0	0
$739$	−4.00000	−0.147142	−0.0735712	−	0.997290i	$-0.523440\pi$
$739$	−4.00000	−0.147142	−0.0735712	+	0.997290i	$0.523440\pi$
$740$	0	0
$741$	0	0
$742$	4.00000i	0.146845i
$743$	24.0000i	0.880475i	0.897881	+	0.440237i	$0.145106\pi$
$743$	24.0000i	0.880475i	−0.897881	+	0.440237i	$0.854894\pi$
$744$	0	0
$745$	0	0
$746$	4.00000	0.146450
$747$	0	0
$748$	5.00000i	0.182818i
$749$	−3.00000	−0.109618
$750$	0	0
$751$	50.0000	1.82453	0.912263	−	0.409605i	$-0.134333\pi$
$751$	50.0000	1.82453	0.912263	+	0.409605i	$0.134333\pi$
$752$	2.00000i	0.0729325i
$753$	0	0
$754$	−36.0000	−1.31104
$755$	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	$0.988428\pi$
$758$	− 9.00000i	− 0.326895i
$759$	0	0
$760$	0	0
$761$	−27.0000	−0.978749	−0.489375	−	0.872074i	$-0.662775\pi$
$761$	−27.0000	−0.978749	−0.489375	+	0.872074i	$0.662775\pi$
$762$	0	0
$763$	18.0000i	0.651644i
$764$	−6.00000	−0.217072
$765$	0	0
$766$	−6.00000	−0.216789
$767$	− 24.0000i	− 0.866590i
$768$	0	0
$769$	−19.0000	−0.685158	−0.342579	−	0.939489i	$-0.611300\pi$
$769$	−19.0000	−0.685158	−0.342579	+	0.939489i	$0.611300\pi$
$770$	0	0
$771$	0	0
$772$	19.0000i	0.683825i
$773$	− 36.0000i	− 1.29483i	−0.762138	−	0.647415i	$-0.775850\pi$
$773$	− 36.0000i	− 1.29483i	0.762138	−	0.647415i	$-0.224150\pi$
$774$	0	0
$775$	0	0
$776$	10.0000	0.358979
$777$	0	0
$778$	8.00000i	0.286814i
$779$	−33.0000	−1.18235
$780$	0	0
$781$	50.0000	1.78914
$782$	0	0
$783$	0	0
$784$	−1.00000	−0.0357143
$785$	0	0
$786$	0	0
$787$	− 52.0000i	− 1.85360i	−0.375555	−	0.926800i	$-0.622548\pi$
$787$	− 52.0000i	− 1.85360i	0.375555	−	0.926800i	$-0.377452\pi$
$788$	22.0000i	0.783718i
$789$	0	0
$790$	0	0
$791$	−1.00000	−0.0355559
$792$	0	0
$793$	12.0000i	0.426132i
$794$	10.0000	0.354887
$795$	0	0
$796$	−10.0000	−0.354441
$797$	42.0000i	1.48772i	0.668338	+	0.743858i	$0.267006\pi$
$797$	42.0000i	1.48772i	−0.668338	+	0.743858i	$0.732994\pi$
$798$	0	0
$799$	2.00000	0.0707549
$800$	0	0
$801$	0	0
$802$	− 37.0000i	− 1.30652i
$803$	− 35.0000i	− 1.23512i
$804$	0	0
$805$	0	0
$806$	−24.0000	−0.845364
$807$	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
$811$	−28.0000	−0.983213	−0.491606	−	0.870817i	$-0.663590\pi$
$811$	−28.0000	−0.983213	−0.491606	+	0.870817i	$0.663590\pi$
$812$	6.00000i	0.210559i
$813$	0	0
$814$	40.0000	1.40200
$815$	0	0
$816$	0	0
$817$	− 24.0000i	− 0.839654i
$818$	21.0000i	0.734248i
$819$	0	0
$820$	0	0
$821$	24.0000	0.837606	0.418803	−	0.908077i	$-0.362450\pi$
$821$	24.0000	0.837606	0.418803	+	0.908077i	$0.362450\pi$
$822$	0	0
$823$	10.0000i	0.348578i	0.984695	+	0.174289i	$0.0557627\pi$
$823$	10.0000i	0.348578i	−0.984695	+	0.174289i	$0.944237\pi$
$824$	4.00000	0.139347
$825$	0	0
$826$	−4.00000	−0.139178
$827$	41.0000i	1.42571i	0.701312	+	0.712855i	$0.252598\pi$
$827$	41.0000i	1.42571i	−0.701312	+	0.712855i	$0.747402\pi$
$828$	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
$832$	6.00000i	0.208013i
$833$	1.00000i	0.0346479i
$834$	0	0
$835$	0	0
$836$	−15.0000	−0.518786
$837$	0	0
$838$	− 39.0000i	− 1.34723i
$839$	2.00000	0.0690477	0.0345238	−	0.999404i	$-0.489009\pi$
$839$	2.00000	0.0690477	0.0345238	+	0.999404i	$0.489009\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	20.0000i	0.689246i
$843$	0	0
$844$	−1.00000	−0.0344214
$845$	0	0
$846$	0	0
$847$	14.0000i	0.481046i
$848$	− 4.00000i	− 0.137361i
$849$	0	0
$850$	0	0
$851$	0	0
$852$	0	0
$853$	− 34.0000i	− 1.16414i	−0.813139	−	0.582069i	$-0.802243\pi$
$853$	− 34.0000i	− 1.16414i	0.813139	−	0.582069i	$-0.197757\pi$
$854$	2.00000	0.0684386
$855$	0	0
$856$	3.00000	0.102538
$857$	3.00000i	0.102478i	0.998686	+	0.0512390i	$0.0163170\pi$
$857$	3.00000i	0.102478i	−0.998686	+	0.0512390i	$0.983683\pi$
$858$	0	0
$859$	51.0000	1.74010	0.870049	−	0.492966i	$-0.164087\pi$
$859$	51.0000	1.74010	0.870049	+	0.492966i	$0.164087\pi$
$860$	0	0
$861$	0	0
$862$	36.0000i	1.22616i
$863$	− 4.00000i	− 0.136162i	−0.997680	−	0.0680808i	$-0.978312\pi$
$863$	− 4.00000i	− 0.136162i	0.997680	−	0.0680808i	$-0.0216876\pi$
$864$	0	0
$865$	0	0
$866$	1.00000	0.0339814
$867$	0	0
$868$	4.00000i	0.135769i
$869$	10.0000	0.339227
$870$	0	0
$871$	−54.0000	−1.82972
$872$	− 18.0000i	− 0.609557i
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 32.0000i	− 1.08056i	−0.841484	−	0.540282i	$-0.818318\pi$
$877$	− 32.0000i	− 1.08056i	0.841484	−	0.540282i	$-0.181682\pi$
$878$	− 28.0000i	− 0.944954i
$879$	0	0
$880$	0	0
$881$	−26.0000	−0.875962	−0.437981	−	0.898984i	$-0.644306\pi$
$881$	−26.0000	−0.875962	−0.437981	+	0.898984i	$0.644306\pi$
$882$	0	0
$883$	15.0000i	0.504790i	0.967624	+	0.252395i	$0.0812183\pi$
$883$	15.0000i	0.504790i	−0.967624	+	0.252395i	$0.918782\pi$
$884$	6.00000	0.201802
$885$	0	0
$886$	37.0000	1.24304
$887$	− 34.0000i	− 1.14161i	−0.821086	−	0.570804i	$-0.806632\pi$
$887$	− 34.0000i	− 1.14161i	0.821086	−	0.570804i	$-0.193368\pi$
$888$	0	0
$889$	−14.0000	−0.469545
$890$	0	0
$891$	0	0
$892$	− 22.0000i	− 0.736614i
$893$	6.00000i	0.200782i
$894$	0	0
$895$	0	0
$896$	1.00000	0.0334077
$897$	0	0
$898$	33.0000i	1.10122i
$899$	24.0000	0.800445
$900$	0	0
$901$	−4.00000	−0.133259
$902$	− 55.0000i	− 1.83130i
$903$	0	0
$904$	1.00000	0.0332595
$905$	0	0
$906$	0	0
$907$	4.00000i	0.132818i	0.997792	+	0.0664089i	$0.0211542\pi$
$907$	4.00000i	0.132818i	−0.997792	+	0.0664089i	$0.978846\pi$
$908$	28.0000i	0.929213i
$909$	0	0
$910$	0	0
$911$	12.0000	0.397578	0.198789	−	0.980042i	$-0.436299\pi$
$911$	12.0000	0.397578	0.198789	+	0.980042i	$0.436299\pi$
$912$	0	0
$913$	− 55.0000i	− 1.82023i
$914$	25.0000	0.826927
$915$	0	0
$916$	−14.0000	−0.462573
$917$	− 8.00000i	− 0.264183i
$918$	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
$922$	38.0000i	1.25146i
$923$	− 60.0000i	− 1.97492i
$924$	0	0
$925$	0	0
$926$	−8.00000	−0.262896
$927$	0	0
$928$	− 6.00000i	− 0.196960i
$929$	46.0000	1.50921	0.754606	−	0.656179i	$-0.227828\pi$
$929$	46.0000	1.50921	0.754606	+	0.656179i	$0.227828\pi$
$930$	0	0
$931$	−3.00000	−0.0983210
$932$	6.00000i	0.196537i
$933$	0	0
$934$	4.00000	0.130884
$935$	0	0
$936$	0	0
$937$	7.00000i	0.228680i	0.993442	+	0.114340i	$0.0364753\pi$
$937$	7.00000i	0.228680i	−0.993442	+	0.114340i	$0.963525\pi$
$938$	9.00000i	0.293860i
$939$	0	0
$940$	0	0
$941$	−56.0000	−1.82555	−0.912774	−	0.408465i	$-0.866064\pi$
$941$	−56.0000	−1.82555	−0.912774	+	0.408465i	$0.866064\pi$
$942$	0	0
$943$	0	0
$944$	4.00000	0.130189
$945$	0	0
$946$	40.0000	1.30051
$947$	− 4.00000i	− 0.129983i	−0.997886	−	0.0649913i	$-0.979298\pi$
$947$	− 4.00000i	− 0.129983i	0.997886	−	0.0649913i	$-0.0207020\pi$
$948$	0	0
$949$	−42.0000	−1.36338
$950$	0	0
$951$	0	0
$952$	− 1.00000i	− 0.0324102i
$953$	− 9.00000i	− 0.291539i	−0.989319	−	0.145769i	$-0.953434\pi$
$953$	− 9.00000i	− 0.291539i	0.989319	−	0.145769i	$-0.0465657\pi$
$954$	0	0
$955$	0	0
$956$	−4.00000	−0.129369
$957$	0	0
$958$	− 6.00000i	− 0.193851i
$959$	3.00000	0.0968751
$960$	0	0
$961$	−15.0000	−0.483871
$962$	− 48.0000i	− 1.54758i
$963$	0	0
$964$	5.00000	0.161039
$965$	0	0
$966$	0	0
$967$	2.00000i	0.0643157i	0.999483	+	0.0321578i	$0.0102379\pi$
$967$	2.00000i	0.0643157i	−0.999483	+	0.0321578i	$0.989762\pi$
$968$	− 14.0000i	− 0.449977i
$969$	0	0
$970$	0	0
$971$	51.0000	1.63667	0.818334	−	0.574743i	$-0.194898\pi$
$971$	51.0000	1.63667	0.818334	+	0.574743i	$0.194898\pi$
$972$	0	0
$973$	11.0000i	0.352644i
$974$	−34.0000	−1.08943
$975$	0	0
$976$	−2.00000	−0.0640184
$977$	21.0000i	0.671850i	0.941889	+	0.335925i	$0.109049\pi$
$977$	21.0000i	0.671850i	−0.941889	+	0.335925i	$0.890951\pi$
$978$	0	0
$979$	−55.0000	−1.75781
$980$	0	0
$981$	0	0
$982$	12.0000i	0.382935i
$983$	4.00000i	0.127580i	0.997963	+	0.0637901i	$0.0203188\pi$
$983$	4.00000i	0.127580i	−0.997963	+	0.0637901i	$0.979681\pi$
$984$	0	0
$985$	0	0
$986$	−6.00000	−0.191079
$987$	0	0
$988$	18.0000i	0.572656i
$989$	0	0
$990$	0	0
$991$	4.00000	0.127064	0.0635321	−	0.997980i	$-0.479763\pi$
$991$	4.00000	0.127064	0.0635321	+	0.997980i	$0.479763\pi$
$992$	− 4.00000i	− 0.127000i
$993$	0	0
$994$	−10.0000	−0.317181
$995$	0	0
$996$	0	0
$997$	58.0000i	1.83688i	0.395562	+	0.918439i	$0.370550\pi$
$997$	58.0000i	1.83688i	−0.395562	+	0.918439i	$0.629450\pi$
$998$	− 36.0000i	− 1.13956i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3150.2.g.v.2899.2		2
3.2	odd	2		350.2.c.a.99.1		2
5.2	odd	4		3150.2.a.j.1.1		1
5.3	odd	4		3150.2.a.bq.1.1		1
5.4	even	2	inner	3150.2.g.v.2899.1		2
12.11	even	2		2800.2.g.a.449.2		2
15.2	even	4		350.2.a.d.1.1	yes	1
15.8	even	4		350.2.a.c.1.1	✓	1
15.14	odd	2		350.2.c.a.99.2		2
21.20	even	2		2450.2.c.r.99.1		2
60.23	odd	4		2800.2.a.b.1.1		1
60.47	odd	4		2800.2.a.bg.1.1		1
60.59	even	2		2800.2.g.a.449.1		2
105.62	odd	4		2450.2.a.bg.1.1		1
105.83	odd	4		2450.2.a.a.1.1		1
105.104	even	2		2450.2.c.r.99.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
350.2.a.c.1.1	✓	1	15.8	even	4
350.2.a.d.1.1	yes	1	15.2	even	4
350.2.c.a.99.1		2	3.2	odd	2
350.2.c.a.99.2		2	15.14	odd	2
2450.2.a.a.1.1		1	105.83	odd	4
2450.2.a.bg.1.1		1	105.62	odd	4
2450.2.c.r.99.1		2	21.20	even	2
2450.2.c.r.99.2		2	105.104	even	2
2800.2.a.b.1.1		1	60.23	odd	4
2800.2.a.bg.1.1		1	60.47	odd	4
2800.2.g.a.449.1		2	60.59	even	2
2800.2.g.a.449.2		2	12.11	even	2
3150.2.a.j.1.1		1	5.2	odd	4
3150.2.a.bq.1.1		1	5.3	odd	4
3150.2.g.v.2899.1		2	5.4	even	2	inner
3150.2.g.v.2899.2		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:	\( N \)	\(=\)	\( 3150 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3150.g (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} -1.00000 q^{4} +1.00000i q^{7} -1.00000i q^{8} +O(q^{10})\)	\(q+1.00000i q^{2} -1.00000 q^{4} +1.00000i q^{7} -1.00000i q^{8} +5.00000 q^{11} -6.00000i q^{13} -1.00000 q^{14} +1.00000 q^{16} -1.00000i q^{17} +3.00000 q^{19} +5.00000i q^{22} +6.00000 q^{26} -1.00000i q^{28} -6.00000 q^{29} -4.00000 q^{31} +1.00000i q^{32} +1.00000 q^{34} -8.00000i q^{37} +3.00000i q^{38} -11.0000 q^{41} -8.00000i q^{43} -5.00000 q^{44} +2.00000i q^{47} -1.00000 q^{49} +6.00000i q^{52} -4.00000i q^{53} +1.00000 q^{56} -6.00000i q^{58} +4.00000 q^{59} -2.00000 q^{61} -4.00000i q^{62} -1.00000 q^{64} -9.00000i q^{67} +1.00000i q^{68} +10.0000 q^{71} -7.00000i q^{73} +8.00000 q^{74} -3.00000 q^{76} +5.00000i q^{77} +2.00000 q^{79} -11.0000i q^{82} -11.0000i q^{83} +8.00000 q^{86} -5.00000i q^{88} -11.0000 q^{89} +6.00000 q^{91} -2.00000 q^{94} +10.0000i q^{97} -1.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4}+O(q^{10}) \) 2 * q - 2 * q^4	\( 2 q - 2 q^{4} + 10 q^{11} - 2 q^{14} + 2 q^{16} + 6 q^{19} + 12 q^{26} - 12 q^{29} - 8 q^{31} + 2 q^{34} - 22 q^{41} - 10 q^{44} - 2 q^{49} + 2 q^{56} + 8 q^{59} - 4 q^{61} - 2 q^{64} + 20 q^{71} + 16 q^{74} - 6 q^{76} + 4 q^{79} + 16 q^{86} - 22 q^{89} + 12 q^{91} - 4 q^{94}+O(q^{100}) \) 2 * q - 2 * q^4 + 10 * q^11 - 2 * q^14 + 2 * q^16 + 6 * q^19 + 12 * q^26 - 12 * q^29 - 8 * q^31 + 2 * q^34 - 22 * q^41 - 10 * q^44 - 2 * q^49 + 2 * q^56 + 8 * q^59 - 4 * q^61 - 2 * q^64 + 20 * q^71 + 16 * q^74 - 6 * q^76 + 4 * q^79 + 16 * q^86 - 22 * q^89 + 12 * q^91 - 4 * q^94

Introduction

L-functions

Modular forms

Varieties

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Database

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