LMFDB - Embedded newform 3150.2.g.g.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 1050)
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.g.2899.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{2} -1.00000 q^{4} -1.00000i q^{7} +1.00000i q^{8} +O(q^{10})$	$q-1.00000i q^{2} -1.00000 q^{4} -1.00000i q^{7} +1.00000i q^{8} -2.00000 q^{11} -1.00000i q^{13} -1.00000 q^{14} +1.00000 q^{16} +1.00000i q^{17} -4.00000 q^{19} +2.00000i q^{22} +7.00000i q^{23} -1.00000 q^{26} +1.00000i q^{28} +1.00000 q^{29} +3.00000 q^{31} -1.00000i q^{32} +1.00000 q^{34} -6.00000i q^{37} +4.00000i q^{38} +3.00000 q^{41} +1.00000i q^{43} +2.00000 q^{44} +7.00000 q^{46} +12.0000i q^{47} -1.00000 q^{49} +1.00000i q^{52} +11.0000i q^{53} +1.00000 q^{56} -1.00000i q^{58} -3.00000 q^{59} +5.00000 q^{61} -3.00000i q^{62} -1.00000 q^{64} -12.0000i q^{67} -1.00000i q^{68} -4.00000 q^{71} +14.0000i q^{73} -6.00000 q^{74} +4.00000 q^{76} +2.00000i q^{77} +2.00000 q^{79} -3.00000i q^{82} -3.00000i q^{83} +1.00000 q^{86} -2.00000i q^{88} +10.0000 q^{89} -1.00000 q^{91} -7.00000i q^{92} +12.0000 q^{94} -10.0000i q^{97} +1.00000i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{4}+O(q^{10}) $ 2 * q - 2 * q^4	$ 2 q - 2 q^{4} - 4 q^{11} - 2 q^{14} + 2 q^{16} - 8 q^{19} - 2 q^{26} + 2 q^{29} + 6 q^{31} + 2 q^{34} + 6 q^{41} + 4 q^{44} + 14 q^{46} - 2 q^{49} + 2 q^{56} - 6 q^{59} + 10 q^{61} - 2 q^{64} - 8 q^{71} - 12 q^{74} + 8 q^{76} + 4 q^{79} + 2 q^{86} + 20 q^{89} - 2 q^{91} + 24 q^{94}+O(q^{100}) $ 2 * q - 2 * q^4 - 4 * q^11 - 2 * q^14 + 2 * q^16 - 8 * q^19 - 2 * q^26 + 2 * q^29 + 6 * q^31 + 2 * q^34 + 6 * q^41 + 4 * q^44 + 14 * q^46 - 2 * q^49 + 2 * q^56 - 6 * q^59 + 10 * q^61 - 2 * q^64 - 8 * q^71 - 12 * q^74 + 8 * q^76 + 4 * q^79 + 2 * q^86 + 20 * q^89 - 2 * q^91 + 24 * 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
$11$	−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	−0.990338	−	0.138675i	$-0.955716\pi$
$13$	− 1.00000i	− 0.277350i	0.990338	−	0.138675i	$-0.0442844\pi$
$14$	−1.00000	−0.267261
$15$	0	0
$16$	1.00000	0.250000
$17$	1.00000i	0.242536i	0.992620	+	0.121268i	$0.0386960\pi$
$17$	1.00000i	0.242536i	−0.992620	+	0.121268i	$0.961304\pi$
$18$	0	0
$19$	−4.00000	−0.917663	−0.458831	−	0.888523i	$-0.651732\pi$
$19$	−4.00000	−0.917663	−0.458831	+	0.888523i	$0.651732\pi$
$20$	0	0
$21$	0	0
$22$	2.00000i	0.426401i
$23$	7.00000i	1.45960i	0.683660	+	0.729800i	$0.260387\pi$
$23$	7.00000i	1.45960i	−0.683660	+	0.729800i	$0.739613\pi$
$24$	0	0
$25$	0	0
$26$	−1.00000	−0.196116
$27$	0	0
$28$	1.00000i	0.188982i
$29$	1.00000	0.185695	0.0928477	−	0.995680i	$-0.470403\pi$
$29$	1.00000	0.185695	0.0928477	+	0.995680i	$0.470403\pi$
$30$	0	0
$31$	3.00000	0.538816	0.269408	−	0.963026i	$-0.413172\pi$
$31$	3.00000	0.538816	0.269408	+	0.963026i	$0.413172\pi$
$32$	− 1.00000i	− 0.176777i
$33$	0	0
$34$	1.00000	0.171499
$35$	0	0
$36$	0	0
$37$	− 6.00000i	− 0.986394i	−0.869918	−	0.493197i	$-0.835828\pi$
$37$	− 6.00000i	− 0.986394i	0.869918	−	0.493197i	$-0.164172\pi$
$38$	4.00000i	0.648886i
$39$	0	0
$40$	0	0
$41$	3.00000	0.468521	0.234261	−	0.972174i	$-0.424733\pi$
$41$	3.00000	0.468521	0.234261	+	0.972174i	$0.424733\pi$
$42$	0	0
$43$	1.00000i	0.152499i	0.997089	+	0.0762493i	$0.0242945\pi$
$43$	1.00000i	0.152499i	−0.997089	+	0.0762493i	$0.975706\pi$
$44$	2.00000	0.301511
$45$	0	0
$46$	7.00000	1.03209
$47$	12.0000i	1.75038i	0.483779	+	0.875190i	$0.339264\pi$
$47$	12.0000i	1.75038i	−0.483779	+	0.875190i	$0.660736\pi$
$48$	0	0
$49$	−1.00000	−0.142857
$50$	0	0
$51$	0	0
$52$	1.00000i	0.138675i
$53$	11.0000i	1.51097i	0.655168	+	0.755483i	$0.272598\pi$
$53$	11.0000i	1.51097i	−0.655168	+	0.755483i	$0.727402\pi$
$54$	0	0
$55$	0	0
$56$	1.00000	0.133631
$57$	0	0
$58$	− 1.00000i	− 0.131306i
$59$	−3.00000	−0.390567	−0.195283	−	0.980747i	$-0.562563\pi$
$59$	−3.00000	−0.390567	−0.195283	+	0.980747i	$0.562563\pi$
$60$	0	0
$61$	5.00000	0.640184	0.320092	−	0.947386i	$-0.396286\pi$
$61$	5.00000	0.640184	0.320092	+	0.947386i	$0.396286\pi$
$62$	− 3.00000i	− 0.381000i
$63$	0	0
$64$	−1.00000	−0.125000
$65$	0	0
$66$	0	0
$67$	− 12.0000i	− 1.46603i	−0.680211	−	0.733017i	$-0.738112\pi$
$67$	− 12.0000i	− 1.46603i	0.680211	−	0.733017i	$-0.261888\pi$
$68$	− 1.00000i	− 0.121268i
$69$	0	0
$70$	0	0
$71$	−4.00000	−0.474713	−0.237356	−	0.971423i	$-0.576281\pi$
$71$	−4.00000	−0.474713	−0.237356	+	0.971423i	$0.576281\pi$
$72$	0	0
$73$	14.0000i	1.63858i	0.573382	+	0.819288i	$0.305631\pi$
$73$	14.0000i	1.63858i	−0.573382	+	0.819288i	$0.694369\pi$
$74$	−6.00000	−0.697486
$75$	0	0
$76$	4.00000	0.458831
$77$	2.00000i	0.227921i
$78$	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
$82$	− 3.00000i	− 0.331295i
$83$	− 3.00000i	− 0.329293i	−0.986353	−	0.164646i	$-0.947352\pi$
$83$	− 3.00000i	− 0.329293i	0.986353	−	0.164646i	$-0.0526483\pi$
$84$	0	0
$85$	0	0
$86$	1.00000	0.107833
$87$	0	0
$88$	− 2.00000i	− 0.213201i
$89$	10.0000	1.06000	0.529999	−	0.847998i	$-0.322192\pi$
$89$	10.0000	1.06000	0.529999	+	0.847998i	$0.322192\pi$
$90$	0	0
$91$	−1.00000	−0.104828
$92$	− 7.00000i	− 0.729800i
$93$	0	0
$94$	12.0000	1.23771
$95$	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
$103$	17.0000i	1.67506i	0.546392	+	0.837530i	$0.316001\pi$
$103$	17.0000i	1.67506i	−0.546392	+	0.837530i	$0.683999\pi$
$104$	1.00000	0.0980581
$105$	0	0
$106$	11.0000	1.06841
$107$	18.0000i	1.74013i	0.492941	+	0.870063i	$0.335922\pi$
$107$	18.0000i	1.74013i	−0.492941	+	0.870063i	$0.664078\pi$
$108$	0	0
$109$	4.00000	0.383131	0.191565	−	0.981480i	$-0.438644\pi$
$109$	4.00000	0.383131	0.191565	+	0.981480i	$0.438644\pi$
$110$	0	0
$111$	0	0
$112$	− 1.00000i	− 0.0944911i
$113$	6.00000i	0.564433i	0.959351	+	0.282216i	$0.0910696\pi$
$113$	6.00000i	0.564433i	−0.959351	+	0.282216i	$0.908930\pi$
$114$	0	0
$115$	0	0
$116$	−1.00000	−0.0928477
$117$	0	0
$118$	3.00000i	0.276172i
$119$	1.00000	0.0916698
$120$	0	0
$121$	−7.00000	−0.636364
$122$	− 5.00000i	− 0.452679i
$123$	0	0
$124$	−3.00000	−0.269408
$125$	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
$133$	4.00000i	0.346844i
$134$	−12.0000	−1.03664
$135$	0	0
$136$	−1.00000	−0.0857493
$137$	− 4.00000i	− 0.341743i	−0.985293	−	0.170872i	$-0.945342\pi$
$137$	− 4.00000i	− 0.341743i	0.985293	−	0.170872i	$-0.0546583\pi$
$138$	0	0
$139$	4.00000	0.339276	0.169638	−	0.985506i	$-0.445740\pi$
$139$	4.00000	0.339276	0.169638	+	0.985506i	$0.445740\pi$
$140$	0	0
$141$	0	0
$142$	4.00000i	0.335673i
$143$	2.00000i	0.167248i
$144$	0	0
$145$	0	0
$146$	14.0000	1.15865
$147$	0	0
$148$	6.00000i	0.493197i
$149$	5.00000	0.409616	0.204808	−	0.978802i	$-0.434343\pi$
$149$	5.00000	0.409616	0.204808	+	0.978802i	$0.434343\pi$
$150$	0	0
$151$	22.0000	1.79033	0.895167	−	0.445730i	$-0.147056\pi$
$151$	22.0000	1.79033	0.895167	+	0.445730i	$0.147056\pi$
$152$	− 4.00000i	− 0.324443i
$153$	0	0
$154$	2.00000	0.161165
$155$	0	0
$156$	0	0
$157$	18.0000i	1.43656i	0.695756	+	0.718278i	$0.255069\pi$
$157$	18.0000i	1.43656i	−0.695756	+	0.718278i	$0.744931\pi$
$158$	− 2.00000i	− 0.159111i
$159$	0	0
$160$	0	0
$161$	7.00000	0.551677
$162$	0	0
$163$	19.0000i	1.48819i	0.668071	+	0.744097i	$0.267120\pi$
$163$	19.0000i	1.48819i	−0.668071	+	0.744097i	$0.732880\pi$
$164$	−3.00000	−0.234261
$165$	0	0
$166$	−3.00000	−0.232845
$167$	2.00000i	0.154765i	0.997001	+	0.0773823i	$0.0246562\pi$
$167$	2.00000i	0.154765i	−0.997001	+	0.0773823i	$0.975344\pi$
$168$	0	0
$169$	12.0000	0.923077
$170$	0	0
$171$	0	0
$172$	− 1.00000i	− 0.0762493i
$173$	− 12.0000i	− 0.912343i	−0.889892	−	0.456172i	$-0.849220\pi$
$173$	− 12.0000i	− 0.912343i	0.889892	−	0.456172i	$-0.150780\pi$
$174$	0	0
$175$	0	0
$176$	−2.00000	−0.150756
$177$	0	0
$178$	− 10.0000i	− 0.749532i
$179$	10.0000	0.747435	0.373718	−	0.927543i	$-0.378083\pi$
$179$	10.0000	0.747435	0.373718	+	0.927543i	$0.378083\pi$
$180$	0	0
$181$	−18.0000	−1.33793	−0.668965	−	0.743294i	$-0.733262\pi$
$181$	−18.0000	−1.33793	−0.668965	+	0.743294i	$0.733262\pi$
$182$	1.00000i	0.0741249i
$183$	0	0
$184$	−7.00000	−0.516047
$185$	0	0
$186$	0	0
$187$	− 2.00000i	− 0.146254i
$188$	− 12.0000i	− 0.875190i
$189$	0	0
$190$	0	0
$191$	13.0000	0.940647	0.470323	−	0.882494i	$-0.344137\pi$
$191$	13.0000	0.940647	0.470323	+	0.882494i	$0.344137\pi$
$192$	0	0
$193$	− 2.00000i	− 0.143963i	−0.997406	−	0.0719816i	$-0.977068\pi$
$193$	− 2.00000i	− 0.143963i	0.997406	−	0.0719816i	$-0.0229323\pi$
$194$	−10.0000	−0.717958
$195$	0	0
$196$	1.00000	0.0714286
$197$	− 27.0000i	− 1.92367i	−0.273629	−	0.961835i	$-0.588224\pi$
$197$	− 27.0000i	− 1.92367i	0.273629	−	0.961835i	$-0.411776\pi$
$198$	0	0
$199$	24.0000	1.70131	0.850657	−	0.525720i	$-0.176204\pi$
$199$	24.0000	1.70131	0.850657	+	0.525720i	$0.176204\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 1.00000i	− 0.0701862i
$204$	0	0
$205$	0	0
$206$	17.0000	1.18445
$207$	0	0
$208$	− 1.00000i	− 0.0693375i
$209$	8.00000	0.553372
$210$	0	0
$211$	15.0000	1.03264	0.516321	−	0.856395i	$-0.327301\pi$
$211$	15.0000	1.03264	0.516321	+	0.856395i	$0.327301\pi$
$212$	− 11.0000i	− 0.755483i
$213$	0	0
$214$	18.0000	1.23045
$215$	0	0
$216$	0	0
$217$	− 3.00000i	− 0.203653i
$218$	− 4.00000i	− 0.270914i
$219$	0	0
$220$	0	0
$221$	1.00000	0.0672673
$222$	0	0
$223$	13.0000i	0.870544i	0.900299	+	0.435272i	$0.143348\pi$
$223$	13.0000i	0.870544i	−0.900299	+	0.435272i	$0.856652\pi$
$224$	−1.00000	−0.0668153
$225$	0	0
$226$	6.00000	0.399114
$227$	− 7.00000i	− 0.464606i	−0.972643	−	0.232303i	$-0.925374\pi$
$227$	− 7.00000i	− 0.464606i	0.972643	−	0.232303i	$-0.0746261\pi$
$228$	0	0
$229$	−14.0000	−0.925146	−0.462573	−	0.886581i	$-0.653074\pi$
$229$	−14.0000	−0.925146	−0.462573	+	0.886581i	$0.653074\pi$
$230$	0	0
$231$	0	0
$232$	1.00000i	0.0656532i
$233$	20.0000i	1.31024i	0.755523	+	0.655122i	$0.227383\pi$
$233$	20.0000i	1.31024i	−0.755523	+	0.655122i	$0.772617\pi$
$234$	0	0
$235$	0	0
$236$	3.00000	0.195283
$237$	0	0
$238$	− 1.00000i	− 0.0648204i
$239$	−24.0000	−1.55243	−0.776215	−	0.630468i	$-0.782863\pi$
$239$	−24.0000	−1.55243	−0.776215	+	0.630468i	$0.782863\pi$
$240$	0	0
$241$	−26.0000	−1.67481	−0.837404	−	0.546585i	$-0.815928\pi$
$241$	−26.0000	−1.67481	−0.837404	+	0.546585i	$0.815928\pi$
$242$	7.00000i	0.449977i
$243$	0	0
$244$	−5.00000	−0.320092
$245$	0	0
$246$	0	0
$247$	4.00000i	0.254514i
$248$	3.00000i	0.190500i
$249$	0	0
$250$	0	0
$251$	15.0000	0.946792	0.473396	−	0.880850i	$-0.343028\pi$
$251$	15.0000	0.946792	0.473396	+	0.880850i	$0.343028\pi$
$252$	0	0
$253$	− 14.0000i	− 0.880172i
$254$	14.0000	0.878438
$255$	0	0
$256$	1.00000	0.0625000
$257$	5.00000i	0.311891i	0.987766	+	0.155946i	$0.0498425\pi$
$257$	5.00000i	0.311891i	−0.987766	+	0.155946i	$0.950158\pi$
$258$	0	0
$259$	−6.00000	−0.372822
$260$	0	0
$261$	0	0
$262$	8.00000i	0.494242i
$263$	11.0000i	0.678289i	0.940734	+	0.339145i	$0.110138\pi$
$263$	11.0000i	0.678289i	−0.940734	+	0.339145i	$0.889862\pi$
$264$	0	0
$265$	0	0
$266$	4.00000	0.245256
$267$	0	0
$268$	12.0000i	0.733017i
$269$	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
$271$	−20.0000	−1.21491	−0.607457	−	0.794353i	$-0.707810\pi$
$271$	−20.0000	−1.21491	−0.607457	+	0.794353i	$0.707810\pi$
$272$	1.00000i	0.0606339i
$273$	0	0
$274$	−4.00000	−0.241649
$275$	0	0
$276$	0	0
$277$	− 2.00000i	− 0.120168i	−0.998193	−	0.0600842i	$-0.980863\pi$
$277$	− 2.00000i	− 0.120168i	0.998193	−	0.0600842i	$-0.0191369\pi$
$278$	− 4.00000i	− 0.239904i
$279$	0	0
$280$	0	0
$281$	0	0		−	1.00000i	$-0.5\pi$
$281$	0	0			1.00000i	$0.5\pi$
$282$	0	0
$283$	20.0000i	1.18888i	0.804141	+	0.594438i	$0.202626\pi$
$283$	20.0000i	1.18888i	−0.804141	+	0.594438i	$0.797374\pi$
$284$	4.00000	0.237356
$285$	0	0
$286$	2.00000	0.118262
$287$	− 3.00000i	− 0.177084i
$288$	0	0
$289$	16.0000	0.941176
$290$	0	0
$291$	0	0
$292$	− 14.0000i	− 0.819288i
$293$	0	0	1.00000			$0$
$293$	0	0	−1.00000			$\pi$
$294$	0	0
$295$	0	0
$296$	6.00000	0.348743
$297$	0	0
$298$	− 5.00000i	− 0.289642i
$299$	7.00000	0.404820
$300$	0	0
$301$	1.00000	0.0576390
$302$	− 22.0000i	− 1.26596i
$303$	0	0
$304$	−4.00000	−0.229416
$305$	0	0
$306$	0	0
$307$	− 22.0000i	− 1.25561i	−0.778372	−	0.627803i	$-0.783954\pi$
$307$	− 22.0000i	− 1.25561i	0.778372	−	0.627803i	$-0.216046\pi$
$308$	− 2.00000i	− 0.113961i
$309$	0	0
$310$	0	0
$311$	−34.0000	−1.92796	−0.963982	−	0.265969i	$-0.914308\pi$
$311$	−34.0000	−1.92796	−0.963982	+	0.265969i	$0.914308\pi$
$312$	0	0
$313$	− 18.0000i	− 1.01742i	−0.860938	−	0.508710i	$-0.830123\pi$
$313$	− 18.0000i	− 1.01742i	0.860938	−	0.508710i	$-0.169877\pi$
$314$	18.0000	1.01580
$315$	0	0
$316$	−2.00000	−0.112509
$317$	17.0000i	0.954815i	0.878682	+	0.477408i	$0.158423\pi$
$317$	17.0000i	0.954815i	−0.878682	+	0.477408i	$0.841577\pi$
$318$	0	0
$319$	−2.00000	−0.111979
$320$	0	0
$321$	0	0
$322$	− 7.00000i	− 0.390095i
$323$	− 4.00000i	− 0.222566i
$324$	0	0
$325$	0	0
$326$	19.0000	1.05231
$327$	0	0
$328$	3.00000i	0.165647i
$329$	12.0000	0.661581
$330$	0	0
$331$	−3.00000	−0.164895	−0.0824475	−	0.996595i	$-0.526274\pi$
$331$	−3.00000	−0.164895	−0.0824475	+	0.996595i	$0.526274\pi$
$332$	3.00000i	0.164646i
$333$	0	0
$334$	2.00000	0.109435
$335$	0	0
$336$	0	0
$337$	27.0000i	1.47078i	0.677642	+	0.735392i	$0.263002\pi$
$337$	27.0000i	1.47078i	−0.677642	+	0.735392i	$0.736998\pi$
$338$	− 12.0000i	− 0.652714i
$339$	0	0
$340$	0	0
$341$	−6.00000	−0.324918
$342$	0	0
$343$	1.00000i	0.0539949i
$344$	−1.00000	−0.0539164
$345$	0	0
$346$	−12.0000	−0.645124
$347$	− 26.0000i	− 1.39575i	−0.716218	−	0.697877i	$-0.754128\pi$
$347$	− 26.0000i	− 1.39575i	0.716218	−	0.697877i	$-0.245872\pi$
$348$	0	0
$349$	1.00000	0.0535288	0.0267644	−	0.999642i	$-0.491480\pi$
$349$	1.00000	0.0535288	0.0267644	+	0.999642i	$0.491480\pi$
$350$	0	0
$351$	0	0
$352$	2.00000i	0.106600i
$353$	10.0000i	0.532246i	0.963939	+	0.266123i	$0.0857428\pi$
$353$	10.0000i	0.532246i	−0.963939	+	0.266123i	$0.914257\pi$
$354$	0	0
$355$	0	0
$356$	−10.0000	−0.529999
$357$	0	0
$358$	− 10.0000i	− 0.528516i
$359$	−9.00000	−0.475002	−0.237501	−	0.971387i	$-0.576328\pi$
$359$	−9.00000	−0.475002	−0.237501	+	0.971387i	$0.576328\pi$
$360$	0	0
$361$	−3.00000	−0.157895
$362$	18.0000i	0.946059i
$363$	0	0
$364$	1.00000	0.0524142
$365$	0	0
$366$	0	0
$367$	13.0000i	0.678594i	0.940679	+	0.339297i	$0.110189\pi$
$367$	13.0000i	0.678594i	−0.940679	+	0.339297i	$0.889811\pi$
$368$	7.00000i	0.364900i
$369$	0	0
$370$	0	0
$371$	11.0000	0.571092
$372$	0	0
$373$	4.00000i	0.207112i	0.994624	+	0.103556i	$0.0330221\pi$
$373$	4.00000i	0.207112i	−0.994624	+	0.103556i	$0.966978\pi$
$374$	−2.00000	−0.103418
$375$	0	0
$376$	−12.0000	−0.618853
$377$	− 1.00000i	− 0.0515026i
$378$	0	0
$379$	5.00000	0.256833	0.128416	−	0.991720i	$-0.459011\pi$
$379$	5.00000	0.256833	0.128416	+	0.991720i	$0.459011\pi$
$380$	0	0
$381$	0	0
$382$	− 13.0000i	− 0.665138i
$383$	− 20.0000i	− 1.02195i	−0.859595	−	0.510976i	$-0.829284\pi$
$383$	− 20.0000i	− 1.02195i	0.859595	−	0.510976i	$-0.170716\pi$
$384$	0	0
$385$	0	0
$386$	−2.00000	−0.101797
$387$	0	0
$388$	10.0000i	0.507673i
$389$	22.0000	1.11544	0.557722	−	0.830028i	$-0.311675\pi$
$389$	22.0000	1.11544	0.557722	+	0.830028i	$0.311675\pi$
$390$	0	0
$391$	−7.00000	−0.354005
$392$	− 1.00000i	− 0.0505076i
$393$	0	0
$394$	−27.0000	−1.36024
$395$	0	0
$396$	0	0
$397$	3.00000i	0.150566i	0.997162	+	0.0752828i	$0.0239860\pi$
$397$	3.00000i	0.150566i	−0.997162	+	0.0752828i	$0.976014\pi$
$398$	− 24.0000i	− 1.20301i
$399$	0	0
$400$	0	0
$401$	−16.0000	−0.799002	−0.399501	−	0.916733i	$-0.630817\pi$
$401$	−16.0000	−0.799002	−0.399501	+	0.916733i	$0.630817\pi$
$402$	0	0
$403$	− 3.00000i	− 0.149441i
$404$	0	0
$405$	0	0
$406$	−1.00000	−0.0496292
$407$	12.0000i	0.594818i
$408$	0	0
$409$	14.0000	0.692255	0.346128	−	0.938187i	$-0.387496\pi$
$409$	14.0000	0.692255	0.346128	+	0.938187i	$0.387496\pi$
$410$	0	0
$411$	0	0
$412$	− 17.0000i	− 0.837530i
$413$	3.00000i	0.147620i
$414$	0	0
$415$	0	0
$416$	−1.00000	−0.0490290
$417$	0	0
$418$	− 8.00000i	− 0.391293i
$419$	−25.0000	−1.22133	−0.610665	−	0.791889i	$-0.709098\pi$
$419$	−25.0000	−1.22133	−0.610665	+	0.791889i	$0.709098\pi$
$420$	0	0
$421$	34.0000	1.65706	0.828529	−	0.559946i	$-0.189178\pi$
$421$	34.0000	1.65706	0.828529	+	0.559946i	$0.189178\pi$
$422$	− 15.0000i	− 0.730189i
$423$	0	0
$424$	−11.0000	−0.534207
$425$	0	0
$426$	0	0
$427$	− 5.00000i	− 0.241967i
$428$	− 18.0000i	− 0.870063i
$429$	0	0
$430$	0	0
$431$	−27.0000	−1.30054	−0.650272	−	0.759701i	$-0.725345\pi$
$431$	−27.0000	−1.30054	−0.650272	+	0.759701i	$0.725345\pi$
$432$	0	0
$433$	− 34.0000i	− 1.63394i	−0.576683	−	0.816968i	$-0.695653\pi$
$433$	− 34.0000i	− 1.63394i	0.576683	−	0.816968i	$-0.304347\pi$
$434$	−3.00000	−0.144005
$435$	0	0
$436$	−4.00000	−0.191565
$437$	− 28.0000i	− 1.33942i
$438$	0	0
$439$	−7.00000	−0.334092	−0.167046	−	0.985949i	$-0.553423\pi$
$439$	−7.00000	−0.334092	−0.167046	+	0.985949i	$0.553423\pi$
$440$	0	0
$441$	0	0
$442$	− 1.00000i	− 0.0475651i
$443$	− 12.0000i	− 0.570137i	−0.958507	−	0.285069i	$-0.907984\pi$
$443$	− 12.0000i	− 0.570137i	0.958507	−	0.285069i	$-0.0920164\pi$
$444$	0	0
$445$	0	0
$446$	13.0000	0.615568
$447$	0	0
$448$	1.00000i	0.0472456i
$449$	−16.0000	−0.755087	−0.377543	−	0.925992i	$-0.623231\pi$
$449$	−16.0000	−0.755087	−0.377543	+	0.925992i	$0.623231\pi$
$450$	0	0
$451$	−6.00000	−0.282529
$452$	− 6.00000i	− 0.282216i
$453$	0	0
$454$	−7.00000	−0.328526
$455$	0	0
$456$	0	0
$457$	− 17.0000i	− 0.795226i	−0.917553	−	0.397613i	$-0.869839\pi$
$457$	− 17.0000i	− 0.795226i	0.917553	−	0.397613i	$-0.130161\pi$
$458$	14.0000i	0.654177i
$459$	0	0
$460$	0	0
$461$	−32.0000	−1.49039	−0.745194	−	0.666847i	$-0.767643\pi$
$461$	−32.0000	−1.49039	−0.745194	+	0.666847i	$0.767643\pi$
$462$	0	0
$463$	− 36.0000i	− 1.67306i	−0.547920	−	0.836531i	$-0.684580\pi$
$463$	− 36.0000i	− 1.67306i	0.547920	−	0.836531i	$-0.315420\pi$
$464$	1.00000	0.0464238
$465$	0	0
$466$	20.0000	0.926482
$467$	39.0000i	1.80470i	0.430999	+	0.902352i	$0.358161\pi$
$467$	39.0000i	1.80470i	−0.430999	+	0.902352i	$0.641839\pi$
$468$	0	0
$469$	−12.0000	−0.554109
$470$	0	0
$471$	0	0
$472$	− 3.00000i	− 0.138086i
$473$	− 2.00000i	− 0.0919601i
$474$	0	0
$475$	0	0
$476$	−1.00000	−0.0458349
$477$	0	0
$478$	24.0000i	1.09773i
$479$	8.00000	0.365529	0.182765	−	0.983157i	$-0.441495\pi$
$479$	8.00000	0.365529	0.182765	+	0.983157i	$0.441495\pi$
$480$	0	0
$481$	−6.00000	−0.273576
$482$	26.0000i	1.18427i
$483$	0	0
$484$	7.00000	0.318182
$485$	0	0
$486$	0	0
$487$	22.0000i	0.996915i	0.866914	+	0.498458i	$0.166100\pi$
$487$	22.0000i	0.996915i	−0.866914	+	0.498458i	$0.833900\pi$
$488$	5.00000i	0.226339i
$489$	0	0
$490$	0	0
$491$	−30.0000	−1.35388	−0.676941	−	0.736038i	$-0.736695\pi$
$491$	−30.0000	−1.35388	−0.676941	+	0.736038i	$0.736695\pi$
$492$	0	0
$493$	1.00000i	0.0450377i
$494$	4.00000	0.179969
$495$	0	0
$496$	3.00000	0.134704
$497$	4.00000i	0.179425i
$498$	0	0
$499$	27.0000	1.20869	0.604343	−	0.796724i	$-0.293436\pi$
$499$	27.0000	1.20869	0.604343	+	0.796724i	$0.293436\pi$
$500$	0	0
$501$	0	0
$502$	− 15.0000i	− 0.669483i
$503$	− 2.00000i	− 0.0891756i	−0.999005	−	0.0445878i	$-0.985803\pi$
$503$	− 2.00000i	− 0.0891756i	0.999005	−	0.0445878i	$-0.0141974\pi$
$504$	0	0
$505$	0	0
$506$	−14.0000	−0.622376
$507$	0	0
$508$	− 14.0000i	− 0.621150i
$509$	0	0		−	1.00000i	$-0.5\pi$
$509$	0	0			1.00000i	$0.5\pi$
$510$	0	0
$511$	14.0000	0.619324
$512$	− 1.00000i	− 0.0441942i
$513$	0	0
$514$	5.00000	0.220541
$515$	0	0
$516$	0	0
$517$	− 24.0000i	− 1.05552i
$518$	6.00000i	0.263625i
$519$	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
$523$	8.00000i	0.349816i	0.984585	+	0.174908i	$0.0559627\pi$
$523$	8.00000i	0.349816i	−0.984585	+	0.174908i	$0.944037\pi$
$524$	8.00000	0.349482
$525$	0	0
$526$	11.0000	0.479623
$527$	3.00000i	0.130682i
$528$	0	0
$529$	−26.0000	−1.13043
$530$	0	0
$531$	0	0
$532$	− 4.00000i	− 0.173422i
$533$	− 3.00000i	− 0.129944i
$534$	0	0
$535$	0	0
$536$	12.0000	0.518321
$537$	0	0
$538$	− 18.0000i	− 0.776035i
$539$	2.00000	0.0861461
$540$	0	0
$541$	−28.0000	−1.20381	−0.601907	−	0.798566i	$-0.705592\pi$
$541$	−28.0000	−1.20381	−0.601907	+	0.798566i	$0.705592\pi$
$542$	20.0000i	0.859074i
$543$	0	0
$544$	1.00000	0.0428746
$545$	0	0
$546$	0	0
$547$	− 15.0000i	− 0.641354i	−0.947189	−	0.320677i	$-0.896090\pi$
$547$	− 15.0000i	− 0.641354i	0.947189	−	0.320677i	$-0.103910\pi$
$548$	4.00000i	0.170872i
$549$	0	0
$550$	0	0
$551$	−4.00000	−0.170406
$552$	0	0
$553$	− 2.00000i	− 0.0850487i
$554$	−2.00000	−0.0849719
$555$	0	0
$556$	−4.00000	−0.169638
$557$	− 18.0000i	− 0.762684i	−0.924434	−	0.381342i	$-0.875462\pi$
$557$	− 18.0000i	− 0.762684i	0.924434	−	0.381342i	$-0.124538\pi$
$558$	0	0
$559$	1.00000	0.0422955
$560$	0	0
$561$	0	0
$562$	0	0
$563$	41.0000i	1.72794i	0.503540	+	0.863972i	$0.332031\pi$
$563$	41.0000i	1.72794i	−0.503540	+	0.863972i	$0.667969\pi$
$564$	0	0
$565$	0	0
$566$	20.0000	0.840663
$567$	0	0
$568$	− 4.00000i	− 0.167836i
$569$	14.0000	0.586911	0.293455	−	0.955973i	$-0.405195\pi$
$569$	14.0000	0.586911	0.293455	+	0.955973i	$0.405195\pi$
$570$	0	0
$571$	19.0000	0.795125	0.397563	−	0.917575i	$-0.369856\pi$
$571$	19.0000	0.795125	0.397563	+	0.917575i	$0.369856\pi$
$572$	− 2.00000i	− 0.0836242i
$573$	0	0
$574$	−3.00000	−0.125218
$575$	0	0
$576$	0	0
$577$	34.0000i	1.41544i	0.706494	+	0.707719i	$0.250276\pi$
$577$	34.0000i	1.41544i	−0.706494	+	0.707719i	$0.749724\pi$
$578$	− 16.0000i	− 0.665512i
$579$	0	0
$580$	0	0
$581$	−3.00000	−0.124461
$582$	0	0
$583$	− 22.0000i	− 0.911147i
$584$	−14.0000	−0.579324
$585$	0	0
$586$	0	0
$587$	27.0000i	1.11441i	0.830375	+	0.557205i	$0.188126\pi$
$587$	27.0000i	1.11441i	−0.830375	+	0.557205i	$0.811874\pi$
$588$	0	0
$589$	−12.0000	−0.494451
$590$	0	0
$591$	0	0
$592$	− 6.00000i	− 0.246598i
$593$	− 18.0000i	− 0.739171i	−0.929197	−	0.369586i	$-0.879500\pi$
$593$	− 18.0000i	− 0.739171i	0.929197	−	0.369586i	$-0.120500\pi$
$594$	0	0
$595$	0	0
$596$	−5.00000	−0.204808
$597$	0	0
$598$	− 7.00000i	− 0.286251i
$599$	−45.0000	−1.83865	−0.919325	−	0.393499i	$-0.871265\pi$
$599$	−45.0000	−1.83865	−0.919325	+	0.393499i	$0.871265\pi$
$600$	0	0
$601$	28.0000	1.14214	0.571072	−	0.820900i	$-0.306528\pi$
$601$	28.0000	1.14214	0.571072	+	0.820900i	$0.306528\pi$
$602$	− 1.00000i	− 0.0407570i
$603$	0	0
$604$	−22.0000	−0.895167
$605$	0	0
$606$	0	0
$607$	28.0000i	1.13648i	0.822861	+	0.568242i	$0.192376\pi$
$607$	28.0000i	1.13648i	−0.822861	+	0.568242i	$0.807624\pi$
$608$	4.00000i	0.162221i
$609$	0	0
$610$	0	0
$611$	12.0000	0.485468
$612$	0	0
$613$	− 18.0000i	− 0.727013i	−0.931592	−	0.363507i	$-0.881579\pi$
$613$	− 18.0000i	− 0.727013i	0.931592	−	0.363507i	$-0.118421\pi$
$614$	−22.0000	−0.887848
$615$	0	0
$616$	−2.00000	−0.0805823
$617$	− 14.0000i	− 0.563619i	−0.959470	−	0.281809i	$-0.909065\pi$
$617$	− 14.0000i	− 0.563619i	0.959470	−	0.281809i	$-0.0909346\pi$
$618$	0	0
$619$	−38.0000	−1.52735	−0.763674	−	0.645601i	$-0.776607\pi$
$619$	−38.0000	−1.52735	−0.763674	+	0.645601i	$0.776607\pi$
$620$	0	0
$621$	0	0
$622$	34.0000i	1.36328i
$623$	− 10.0000i	− 0.400642i
$624$	0	0
$625$	0	0
$626$	−18.0000	−0.719425
$627$	0	0
$628$	− 18.0000i	− 0.718278i
$629$	6.00000	0.239236
$630$	0	0
$631$	26.0000	1.03504	0.517522	−	0.855670i	$-0.326855\pi$
$631$	26.0000	1.03504	0.517522	+	0.855670i	$0.326855\pi$
$632$	2.00000i	0.0795557i
$633$	0	0
$634$	17.0000	0.675156
$635$	0	0
$636$	0	0
$637$	1.00000i	0.0396214i
$638$	2.00000i	0.0791808i
$639$	0	0
$640$	0	0
$641$	26.0000	1.02694	0.513469	−	0.858108i	$-0.328360\pi$
$641$	26.0000	1.02694	0.513469	+	0.858108i	$0.328360\pi$
$642$	0	0
$643$	− 2.00000i	− 0.0788723i	−0.999222	−	0.0394362i	$-0.987444\pi$
$643$	− 2.00000i	− 0.0788723i	0.999222	−	0.0394362i	$-0.0125562\pi$
$644$	−7.00000	−0.275839
$645$	0	0
$646$	−4.00000	−0.157378
$647$	− 6.00000i	− 0.235884i	−0.993020	−	0.117942i	$-0.962370\pi$
$647$	− 6.00000i	− 0.235884i	0.993020	−	0.117942i	$-0.0376297\pi$
$648$	0	0
$649$	6.00000	0.235521
$650$	0	0
$651$	0	0
$652$	− 19.0000i	− 0.744097i
$653$	14.0000i	0.547862i	0.961749	+	0.273931i	$0.0883240\pi$
$653$	14.0000i	0.547862i	−0.961749	+	0.273931i	$0.911676\pi$
$654$	0	0
$655$	0	0
$656$	3.00000	0.117130
$657$	0	0
$658$	− 12.0000i	− 0.467809i
$659$	−6.00000	−0.233727	−0.116863	−	0.993148i	$-0.537284\pi$
$659$	−6.00000	−0.233727	−0.116863	+	0.993148i	$0.537284\pi$
$660$	0	0
$661$	−22.0000	−0.855701	−0.427850	−	0.903850i	$-0.640729\pi$
$661$	−22.0000	−0.855701	−0.427850	+	0.903850i	$0.640729\pi$
$662$	3.00000i	0.116598i
$663$	0	0
$664$	3.00000	0.116423
$665$	0	0
$666$	0	0
$667$	7.00000i	0.271041i
$668$	− 2.00000i	− 0.0773823i
$669$	0	0
$670$	0	0
$671$	−10.0000	−0.386046
$672$	0	0
$673$	− 27.0000i	− 1.04077i	−0.853931	−	0.520387i	$-0.825788\pi$
$673$	− 27.0000i	− 1.04077i	0.853931	−	0.520387i	$-0.174212\pi$
$674$	27.0000	1.04000
$675$	0	0
$676$	−12.0000	−0.461538
$677$	20.0000i	0.768662i	0.923195	+	0.384331i	$0.125568\pi$
$677$	20.0000i	0.768662i	−0.923195	+	0.384331i	$0.874432\pi$
$678$	0	0
$679$	−10.0000	−0.383765
$680$	0	0
$681$	0	0
$682$	6.00000i	0.229752i
$683$	− 6.00000i	− 0.229584i	−0.993390	−	0.114792i	$-0.963380\pi$
$683$	− 6.00000i	− 0.229584i	0.993390	−	0.114792i	$-0.0366201\pi$
$684$	0	0
$685$	0	0
$686$	1.00000	0.0381802
$687$	0	0
$688$	1.00000i	0.0381246i
$689$	11.0000	0.419067
$690$	0	0
$691$	−42.0000	−1.59776	−0.798878	−	0.601494i	$-0.794573\pi$
$691$	−42.0000	−1.59776	−0.798878	+	0.601494i	$0.794573\pi$
$692$	12.0000i	0.456172i
$693$	0	0
$694$	−26.0000	−0.986947
$695$	0	0
$696$	0	0
$697$	3.00000i	0.113633i
$698$	− 1.00000i	− 0.0378506i
$699$	0	0
$700$	0	0
$701$	39.0000	1.47301	0.736505	−	0.676432i	$-0.236475\pi$
$701$	39.0000	1.47301	0.736505	+	0.676432i	$0.236475\pi$
$702$	0	0
$703$	24.0000i	0.905177i
$704$	2.00000	0.0753778
$705$	0	0
$706$	10.0000	0.376355
$707$	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
$712$	10.0000i	0.374766i
$713$	21.0000i	0.786456i
$714$	0	0
$715$	0	0
$716$	−10.0000	−0.373718
$717$	0	0
$718$	9.00000i	0.335877i
$719$	−34.0000	−1.26799	−0.633993	−	0.773339i	$-0.718585\pi$
$719$	−34.0000	−1.26799	−0.633993	+	0.773339i	$0.718585\pi$
$720$	0	0
$721$	17.0000	0.633113
$722$	3.00000i	0.111648i
$723$	0	0
$724$	18.0000	0.668965
$725$	0	0
$726$	0	0
$727$	− 1.00000i	− 0.0370879i	−0.999828	−	0.0185440i	$-0.994097\pi$
$727$	− 1.00000i	− 0.0370879i	0.999828	−	0.0185440i	$-0.00590307\pi$
$728$	− 1.00000i	− 0.0370625i
$729$	0	0
$730$	0	0
$731$	−1.00000	−0.0369863
$732$	0	0
$733$	− 19.0000i	− 0.701781i	−0.936416	−	0.350891i	$-0.885879\pi$
$733$	− 19.0000i	− 0.701781i	0.936416	−	0.350891i	$-0.114121\pi$
$734$	13.0000	0.479839
$735$	0	0
$736$	7.00000	0.258023
$737$	24.0000i	0.884051i
$738$	0	0
$739$	−11.0000	−0.404642	−0.202321	−	0.979319i	$-0.564848\pi$
$739$	−11.0000	−0.404642	−0.202321	+	0.979319i	$0.564848\pi$
$740$	0	0
$741$	0	0
$742$	− 11.0000i	− 0.403823i
$743$	− 3.00000i	− 0.110059i	−0.998485	−	0.0550297i	$-0.982475\pi$
$743$	− 3.00000i	− 0.110059i	0.998485	−	0.0550297i	$-0.0175253\pi$
$744$	0	0
$745$	0	0
$746$	4.00000	0.146450
$747$	0	0
$748$	2.00000i	0.0731272i
$749$	18.0000	0.657706
$750$	0	0
$751$	−20.0000	−0.729810	−0.364905	−	0.931045i	$-0.618899\pi$
$751$	−20.0000	−0.729810	−0.364905	+	0.931045i	$0.618899\pi$
$752$	12.0000i	0.437595i
$753$	0	0
$754$	−1.00000	−0.0364179
$755$	0	0
$756$	0	0
$757$	− 2.00000i	− 0.0726912i	−0.999339	−	0.0363456i	$-0.988428\pi$
$757$	− 2.00000i	− 0.0726912i	0.999339	−	0.0363456i	$-0.0115717\pi$
$758$	− 5.00000i	− 0.181608i
$759$	0	0
$760$	0	0
$761$	−34.0000	−1.23250	−0.616250	−	0.787551i	$-0.711349\pi$
$761$	−34.0000	−1.23250	−0.616250	+	0.787551i	$0.711349\pi$
$762$	0	0
$763$	− 4.00000i	− 0.144810i
$764$	−13.0000	−0.470323
$765$	0	0
$766$	−20.0000	−0.722629
$767$	3.00000i	0.108324i
$768$	0	0
$769$	−40.0000	−1.44244	−0.721218	−	0.692708i	$-0.756418\pi$
$769$	−40.0000	−1.44244	−0.721218	+	0.692708i	$0.756418\pi$
$770$	0	0
$771$	0	0
$772$	2.00000i	0.0719816i
$773$	− 20.0000i	− 0.719350i	−0.933078	−	0.359675i	$-0.882888\pi$
$773$	− 20.0000i	− 0.719350i	0.933078	−	0.359675i	$-0.117112\pi$
$774$	0	0
$775$	0	0
$776$	10.0000	0.358979
$777$	0	0
$778$	− 22.0000i	− 0.788738i
$779$	−12.0000	−0.429945
$780$	0	0
$781$	8.00000	0.286263
$782$	7.00000i	0.250319i
$783$	0	0
$784$	−1.00000	−0.0357143
$785$	0	0
$786$	0	0
$787$	38.0000i	1.35455i	0.735728	+	0.677277i	$0.236840\pi$
$787$	38.0000i	1.35455i	−0.735728	+	0.677277i	$0.763160\pi$
$788$	27.0000i	0.961835i
$789$	0	0
$790$	0	0
$791$	6.00000	0.213335
$792$	0	0
$793$	− 5.00000i	− 0.177555i
$794$	3.00000	0.106466
$795$	0	0
$796$	−24.0000	−0.850657
$797$	− 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
$799$	−12.0000	−0.424529
$800$	0	0
$801$	0	0
$802$	16.0000i	0.564980i
$803$	− 28.0000i	− 0.988099i
$804$	0	0
$805$	0	0
$806$	−3.00000	−0.105670
$807$	0	0
$808$	0	0
$809$	20.0000	0.703163	0.351581	−	0.936157i	$-0.385644\pi$
$809$	20.0000	0.703163	0.351581	+	0.936157i	$0.385644\pi$
$810$	0	0
$811$	−14.0000	−0.491606	−0.245803	−	0.969320i	$-0.579052\pi$
$811$	−14.0000	−0.491606	−0.245803	+	0.969320i	$0.579052\pi$
$812$	1.00000i	0.0350931i
$813$	0	0
$814$	12.0000	0.420600
$815$	0	0
$816$	0	0
$817$	− 4.00000i	− 0.139942i
$818$	− 14.0000i	− 0.489499i
$819$	0	0
$820$	0	0
$821$	38.0000	1.32621	0.663105	−	0.748527i	$-0.269238\pi$
$821$	38.0000	1.32621	0.663105	+	0.748527i	$0.269238\pi$
$822$	0	0
$823$	18.0000i	0.627441i	0.949515	+	0.313720i	$0.101575\pi$
$823$	18.0000i	0.627441i	−0.949515	+	0.313720i	$0.898425\pi$
$824$	−17.0000	−0.592223
$825$	0	0
$826$	3.00000	0.104383
$827$	50.0000i	1.73867i	0.494223	+	0.869335i	$0.335453\pi$
$827$	50.0000i	1.73867i	−0.494223	+	0.869335i	$0.664547\pi$
$828$	0	0
$829$	−3.00000	−0.104194	−0.0520972	−	0.998642i	$-0.516591\pi$
$829$	−3.00000	−0.104194	−0.0520972	+	0.998642i	$0.516591\pi$
$830$	0	0
$831$	0	0
$832$	1.00000i	0.0346688i
$833$	− 1.00000i	− 0.0346479i
$834$	0	0
$835$	0	0
$836$	−8.00000	−0.276686
$837$	0	0
$838$	25.0000i	0.863611i
$839$	−40.0000	−1.38095	−0.690477	−	0.723355i	$-0.742599\pi$
$839$	−40.0000	−1.38095	−0.690477	+	0.723355i	$0.742599\pi$
$840$	0	0
$841$	−28.0000	−0.965517
$842$	− 34.0000i	− 1.17172i
$843$	0	0
$844$	−15.0000	−0.516321
$845$	0	0
$846$	0	0
$847$	7.00000i	0.240523i
$848$	11.0000i	0.377742i
$849$	0	0
$850$	0	0
$851$	42.0000	1.43974
$852$	0	0
$853$	− 43.0000i	− 1.47229i	−0.676823	−	0.736146i	$-0.736644\pi$
$853$	− 43.0000i	− 1.47229i	0.676823	−	0.736146i	$-0.263356\pi$
$854$	−5.00000	−0.171096
$855$	0	0
$856$	−18.0000	−0.615227
$857$	46.0000i	1.57133i	0.618652	+	0.785665i	$0.287679\pi$
$857$	46.0000i	1.57133i	−0.618652	+	0.785665i	$0.712321\pi$
$858$	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
$862$	27.0000i	0.919624i
$863$	32.0000i	1.08929i	0.838666	+	0.544646i	$0.183336\pi$
$863$	32.0000i	1.08929i	−0.838666	+	0.544646i	$0.816664\pi$
$864$	0	0
$865$	0	0
$866$	−34.0000	−1.15537
$867$	0	0
$868$	3.00000i	0.101827i
$869$	−4.00000	−0.135691
$870$	0	0
$871$	−12.0000	−0.406604
$872$	4.00000i	0.135457i
$873$	0	0
$874$	−28.0000	−0.947114
$875$	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	$0.818318\pi$
$878$	7.00000i	0.236239i
$879$	0	0
$880$	0	0
$881$	−5.00000	−0.168454	−0.0842271	−	0.996447i	$-0.526842\pi$
$881$	−5.00000	−0.168454	−0.0842271	+	0.996447i	$0.526842\pi$
$882$	0	0
$883$	− 29.0000i	− 0.975928i	−0.872864	−	0.487964i	$-0.837740\pi$
$883$	− 29.0000i	− 0.975928i	0.872864	−	0.487964i	$-0.162260\pi$
$884$	−1.00000	−0.0336336
$885$	0	0
$886$	−12.0000	−0.403148
$887$	− 50.0000i	− 1.67884i	−0.543487	−	0.839418i	$-0.682896\pi$
$887$	− 50.0000i	− 1.67884i	0.543487	−	0.839418i	$-0.317104\pi$
$888$	0	0
$889$	14.0000	0.469545
$890$	0	0
$891$	0	0
$892$	− 13.0000i	− 0.435272i
$893$	− 48.0000i	− 1.60626i
$894$	0	0
$895$	0	0
$896$	1.00000	0.0334077
$897$	0	0
$898$	16.0000i	0.533927i
$899$	3.00000	0.100056
$900$	0	0
$901$	−11.0000	−0.366463
$902$	6.00000i	0.199778i
$903$	0	0
$904$	−6.00000	−0.199557
$905$	0	0
$906$	0	0
$907$	− 11.0000i	− 0.365249i	−0.983183	−	0.182625i	$-0.941541\pi$
$907$	− 11.0000i	− 0.365249i	0.983183	−	0.182625i	$-0.0584593\pi$
$908$	7.00000i	0.232303i
$909$	0	0
$910$	0	0
$911$	19.0000	0.629498	0.314749	−	0.949175i	$-0.398080\pi$
$911$	19.0000	0.629498	0.314749	+	0.949175i	$0.398080\pi$
$912$	0	0
$913$	6.00000i	0.198571i
$914$	−17.0000	−0.562310
$915$	0	0
$916$	14.0000	0.462573
$917$	8.00000i	0.264183i
$918$	0	0
$919$	20.0000	0.659739	0.329870	−	0.944027i	$-0.392995\pi$
$919$	20.0000	0.659739	0.329870	+	0.944027i	$0.392995\pi$
$920$	0	0
$921$	0	0
$922$	32.0000i	1.05386i
$923$	4.00000i	0.131662i
$924$	0	0
$925$	0	0
$926$	−36.0000	−1.18303
$927$	0	0
$928$	− 1.00000i	− 0.0328266i
$929$	25.0000	0.820223	0.410112	−	0.912035i	$-0.365490\pi$
$929$	25.0000	0.820223	0.410112	+	0.912035i	$0.365490\pi$
$930$	0	0
$931$	4.00000	0.131095
$932$	− 20.0000i	− 0.655122i
$933$	0	0
$934$	39.0000	1.27612
$935$	0	0
$936$	0	0
$937$	− 14.0000i	− 0.457360i	−0.973502	−	0.228680i	$-0.926559\pi$
$937$	− 14.0000i	− 0.457360i	0.973502	−	0.228680i	$-0.0734410\pi$
$938$	12.0000i	0.391814i
$939$	0	0
$940$	0	0
$941$	0	0		−	1.00000i	$-0.5\pi$
$941$	0	0			1.00000i	$0.5\pi$
$942$	0	0
$943$	21.0000i	0.683854i
$944$	−3.00000	−0.0976417
$945$	0	0
$946$	−2.00000	−0.0650256
$947$	32.0000i	1.03986i	0.854209	+	0.519930i	$0.174042\pi$
$947$	32.0000i	1.03986i	−0.854209	+	0.519930i	$0.825958\pi$
$948$	0	0
$949$	14.0000	0.454459
$950$	0	0
$951$	0	0
$952$	1.00000i	0.0324102i
$953$	− 40.0000i	− 1.29573i	−0.761756	−	0.647864i	$-0.775663\pi$
$953$	− 40.0000i	− 1.29573i	0.761756	−	0.647864i	$-0.224337\pi$
$954$	0	0
$955$	0	0
$956$	24.0000	0.776215
$957$	0	0
$958$	− 8.00000i	− 0.258468i
$959$	−4.00000	−0.129167
$960$	0	0
$961$	−22.0000	−0.709677
$962$	6.00000i	0.193448i
$963$	0	0
$964$	26.0000	0.837404
$965$	0	0
$966$	0	0
$967$	− 44.0000i	− 1.41494i	−0.706741	−	0.707472i	$-0.749835\pi$
$967$	− 44.0000i	− 1.41494i	0.706741	−	0.707472i	$-0.250165\pi$
$968$	− 7.00000i	− 0.224989i
$969$	0	0
$970$	0	0
$971$	44.0000	1.41203	0.706014	−	0.708198i	$-0.250492\pi$
$971$	44.0000	1.41203	0.706014	+	0.708198i	$0.250492\pi$
$972$	0	0
$973$	− 4.00000i	− 0.128234i
$974$	22.0000	0.704925
$975$	0	0
$976$	5.00000	0.160046
$977$	− 42.0000i	− 1.34370i	−0.740688	−	0.671850i	$-0.765500\pi$
$977$	− 42.0000i	− 1.34370i	0.740688	−	0.671850i	$-0.234500\pi$
$978$	0	0
$979$	−20.0000	−0.639203
$980$	0	0
$981$	0	0
$982$	30.0000i	0.957338i
$983$	24.0000i	0.765481i	0.923856	+	0.382741i	$0.125020\pi$
$983$	24.0000i	0.765481i	−0.923856	+	0.382741i	$0.874980\pi$
$984$	0	0
$985$	0	0
$986$	1.00000	0.0318465
$987$	0	0
$988$	− 4.00000i	− 0.127257i
$989$	−7.00000	−0.222587
$990$	0	0
$991$	4.00000	0.127064	0.0635321	−	0.997980i	$-0.479763\pi$
$991$	4.00000	0.127064	0.0635321	+	0.997980i	$0.479763\pi$
$992$	− 3.00000i	− 0.0952501i
$993$	0	0
$994$	4.00000	0.126872
$995$	0	0
$996$	0	0
$997$	− 2.00000i	− 0.0633406i	−0.999498	−	0.0316703i	$-0.989917\pi$
$997$	− 2.00000i	− 0.0633406i	0.999498	−	0.0316703i	$-0.0100827\pi$
$998$	− 27.0000i	− 0.854670i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3150.2.g.g.2899.1		2
3.2	odd	2		1050.2.g.j.799.2		2
5.2	odd	4		3150.2.a.bl.1.1		1
5.3	odd	4		3150.2.a.c.1.1		1
5.4	even	2	inner	3150.2.g.g.2899.2		2
15.2	even	4		1050.2.a.e.1.1	✓	1
15.8	even	4		1050.2.a.o.1.1	yes	1
15.14	odd	2		1050.2.g.j.799.1		2
60.23	odd	4		8400.2.a.t.1.1		1
60.47	odd	4		8400.2.a.bt.1.1		1
105.62	odd	4		7350.2.a.bj.1.1		1
105.83	odd	4		7350.2.a.ca.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1050.2.a.e.1.1	✓	1	15.2	even	4
1050.2.a.o.1.1	yes	1	15.8	even	4
1050.2.g.j.799.1		2	15.14	odd	2
1050.2.g.j.799.2		2	3.2	odd	2
3150.2.a.c.1.1		1	5.3	odd	4
3150.2.a.bl.1.1		1	5.2	odd	4
3150.2.g.g.2899.1		2	1.1	even	1	trivial
3150.2.g.g.2899.2		2	5.4	even	2	inner
7350.2.a.bj.1.1		1	105.62	odd	4
7350.2.a.ca.1.1		1	105.83	odd	4
8400.2.a.t.1.1		1	60.23	odd	4
8400.2.a.bt.1.1		1	60.47	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} -1.00000 q^{4} -1.00000i q^{7} +1.00000i q^{8} +O(q^{10})\)	\(q-1.00000i q^{2} -1.00000 q^{4} -1.00000i q^{7} +1.00000i q^{8} -2.00000 q^{11} -1.00000i q^{13} -1.00000 q^{14} +1.00000 q^{16} +1.00000i q^{17} -4.00000 q^{19} +2.00000i q^{22} +7.00000i q^{23} -1.00000 q^{26} +1.00000i q^{28} +1.00000 q^{29} +3.00000 q^{31} -1.00000i q^{32} +1.00000 q^{34} -6.00000i q^{37} +4.00000i q^{38} +3.00000 q^{41} +1.00000i q^{43} +2.00000 q^{44} +7.00000 q^{46} +12.0000i q^{47} -1.00000 q^{49} +1.00000i q^{52} +11.0000i q^{53} +1.00000 q^{56} -1.00000i q^{58} -3.00000 q^{59} +5.00000 q^{61} -3.00000i q^{62} -1.00000 q^{64} -12.0000i q^{67} -1.00000i q^{68} -4.00000 q^{71} +14.0000i q^{73} -6.00000 q^{74} +4.00000 q^{76} +2.00000i q^{77} +2.00000 q^{79} -3.00000i q^{82} -3.00000i q^{83} +1.00000 q^{86} -2.00000i q^{88} +10.0000 q^{89} -1.00000 q^{91} -7.00000i q^{92} +12.0000 q^{94} -10.0000i q^{97} +1.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4}+O(q^{10}) \) 2 * q - 2 * q^4	\( 2 q - 2 q^{4} - 4 q^{11} - 2 q^{14} + 2 q^{16} - 8 q^{19} - 2 q^{26} + 2 q^{29} + 6 q^{31} + 2 q^{34} + 6 q^{41} + 4 q^{44} + 14 q^{46} - 2 q^{49} + 2 q^{56} - 6 q^{59} + 10 q^{61} - 2 q^{64} - 8 q^{71} - 12 q^{74} + 8 q^{76} + 4 q^{79} + 2 q^{86} + 20 q^{89} - 2 q^{91} + 24 q^{94}+O(q^{100}) \) 2 * q - 2 * q^4 - 4 * q^11 - 2 * q^14 + 2 * q^16 - 8 * q^19 - 2 * q^26 + 2 * q^29 + 6 * q^31 + 2 * q^34 + 6 * q^41 + 4 * q^44 + 14 * q^46 - 2 * q^49 + 2 * q^56 - 6 * q^59 + 10 * q^61 - 2 * q^64 - 8 * q^71 - 12 * q^74 + 8 * q^76 + 4 * q^79 + 2 * q^86 + 20 * q^89 - 2 * q^91 + 24 * q^94

Introduction

L-functions

Modular forms

Varieties

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Representations

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Database

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