LMFDB - Embedded newform 1450.2.b.c.349.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1450,2,Mod(349,1450)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1450, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1450.349");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1450 = 2 \cdot 5^{2} \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1450.b (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:	$11.5783082931$
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:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			349.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	1450.349
Dual form			1450.2.b.c.349.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{2} -1.00000 q^{4} +1.00000i q^{8} +3.00000 q^{9} +O(q^{10})$	$q-1.00000i q^{2} -1.00000 q^{4} +1.00000i q^{8} +3.00000 q^{9} -2.00000 q^{11} -4.00000i q^{13} +1.00000 q^{16} +6.00000i q^{17} -3.00000i q^{18} +8.00000 q^{19} +2.00000i q^{22} +2.00000i q^{23} -4.00000 q^{26} -1.00000 q^{29} -1.00000 q^{31} -1.00000i q^{32} +6.00000 q^{34} -3.00000 q^{36} -7.00000i q^{37} -8.00000i q^{38} +2.00000 q^{41} -4.00000i q^{43} +2.00000 q^{44} +2.00000 q^{46} -9.00000i q^{47} +7.00000 q^{49} +4.00000i q^{52} +12.0000i q^{53} +1.00000i q^{58} +15.0000 q^{59} +11.0000 q^{61} +1.00000i q^{62} -1.00000 q^{64} -3.00000i q^{67} -6.00000i q^{68} -6.00000 q^{71} +3.00000i q^{72} -12.0000i q^{73} -7.00000 q^{74} -8.00000 q^{76} -8.00000 q^{79} +9.00000 q^{81} -2.00000i q^{82} -8.00000i q^{83} -4.00000 q^{86} -2.00000i q^{88} +16.0000 q^{89} -2.00000i q^{92} -9.00000 q^{94} +2.00000i q^{97} -7.00000i q^{98} -6.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{4} + 6 q^{9}+O(q^{10}) $ 2 * q - 2 * q^4 + 6 * q^9	$ 2 q - 2 q^{4} + 6 q^{9} - 4 q^{11} + 2 q^{16} + 16 q^{19} - 8 q^{26} - 2 q^{29} - 2 q^{31} + 12 q^{34} - 6 q^{36} + 4 q^{41} + 4 q^{44} + 4 q^{46} + 14 q^{49} + 30 q^{59} + 22 q^{61} - 2 q^{64} - 12 q^{71} - 14 q^{74} - 16 q^{76} - 16 q^{79} + 18 q^{81} - 8 q^{86} + 32 q^{89} - 18 q^{94} - 12 q^{99}+O(q^{100}) $ 2 * q - 2 * q^4 + 6 * q^9 - 4 * q^11 + 2 * q^16 + 16 * q^19 - 8 * q^26 - 2 * q^29 - 2 * q^31 + 12 * q^34 - 6 * q^36 + 4 * q^41 + 4 * q^44 + 4 * q^46 + 14 * q^49 + 30 * q^59 + 22 * q^61 - 2 * q^64 - 12 * q^71 - 14 * q^74 - 16 * q^76 - 16 * q^79 + 18 * q^81 - 8 * q^86 + 32 * q^89 - 18 * q^94 - 12 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1450\mathbb{Z}\right)^\times$.

$n$	$901$	$1277$
$\chi(n)$	$1$	$-1$

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	1.00000			$0$
$3$	0	0	−1.00000			$\pi$
$4$	−1.00000	−0.500000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	0	0	1.00000			$0$
$7$	0	0	−1.00000			$\pi$
$8$	1.00000i	0.353553i
$9$	3.00000	1.00000
$10$	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
$13$	− 4.00000i	− 1.10940i	−0.832050	−	0.554700i	$-0.812833\pi$
$13$	− 4.00000i	− 1.10940i	0.832050	−	0.554700i	$-0.187167\pi$
$14$	0	0
$15$	0	0
$16$	1.00000	0.250000
$17$	6.00000i	1.45521i	0.685994	+	0.727607i	$0.259367\pi$
$17$	6.00000i	1.45521i	−0.685994	+	0.727607i	$0.740633\pi$
$18$	− 3.00000i	− 0.707107i
$19$	8.00000	1.83533	0.917663	−	0.397360i	$-0.130073\pi$
$19$	8.00000	1.83533	0.917663	+	0.397360i	$0.130073\pi$
$20$	0	0
$21$	0	0
$22$	2.00000i	0.426401i
$23$	2.00000i	0.417029i	0.978019	+	0.208514i	$0.0668628\pi$
$23$	2.00000i	0.417029i	−0.978019	+	0.208514i	$0.933137\pi$
$24$	0	0
$25$	0	0
$26$	−4.00000	−0.784465
$27$	0	0
$28$	0	0
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0	0
$31$	−1.00000	−0.179605	−0.0898027	−	0.995960i	$-0.528624\pi$
$31$	−1.00000	−0.179605	−0.0898027	+	0.995960i	$0.528624\pi$
$32$	− 1.00000i	− 0.176777i
$33$	0	0
$34$	6.00000	1.02899
$35$	0	0
$36$	−3.00000	−0.500000
$37$	− 7.00000i	− 1.15079i	−0.817875	−	0.575396i	$-0.804848\pi$
$37$	− 7.00000i	− 1.15079i	0.817875	−	0.575396i	$-0.195152\pi$
$38$	− 8.00000i	− 1.29777i
$39$	0	0
$40$	0	0
$41$	2.00000	0.312348	0.156174	−	0.987730i	$-0.450084\pi$
$41$	2.00000	0.312348	0.156174	+	0.987730i	$0.450084\pi$
$42$	0	0
$43$	− 4.00000i	− 0.609994i	−0.952353	−	0.304997i	$-0.901344\pi$
$43$	− 4.00000i	− 0.609994i	0.952353	−	0.304997i	$-0.0986555\pi$
$44$	2.00000	0.301511
$45$	0	0
$46$	2.00000	0.294884
$47$	− 9.00000i	− 1.31278i	−0.754420	−	0.656392i	$-0.772082\pi$
$47$	− 9.00000i	− 1.31278i	0.754420	−	0.656392i	$-0.227918\pi$
$48$	0	0
$49$	7.00000	1.00000
$50$	0	0
$51$	0	0
$52$	4.00000i	0.554700i
$53$	12.0000i	1.64833i	0.566352	+	0.824163i	$0.308354\pi$
$53$	12.0000i	1.64833i	−0.566352	+	0.824163i	$0.691646\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	1.00000i	0.131306i
$59$	15.0000	1.95283	0.976417	−	0.215894i	$-0.0692665\pi$
$59$	15.0000	1.95283	0.976417	+	0.215894i	$0.0692665\pi$
$60$	0	0
$61$	11.0000	1.40841	0.704203	−	0.709999i	$-0.251305\pi$
$61$	11.0000	1.40841	0.704203	+	0.709999i	$0.251305\pi$
$62$	1.00000i	0.127000i
$63$	0	0
$64$	−1.00000	−0.125000
$65$	0	0
$66$	0	0
$67$	− 3.00000i	− 0.366508i	−0.983066	−	0.183254i	$-0.941337\pi$
$67$	− 3.00000i	− 0.366508i	0.983066	−	0.183254i	$-0.0586631\pi$
$68$	− 6.00000i	− 0.727607i
$69$	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	$0.615871\pi$
$72$	3.00000i	0.353553i
$73$	− 12.0000i	− 1.40449i	−0.711934	−	0.702247i	$-0.752180\pi$
$73$	− 12.0000i	− 1.40449i	0.711934	−	0.702247i	$-0.247820\pi$
$74$	−7.00000	−0.813733
$75$	0	0
$76$	−8.00000	−0.917663
$77$	0	0
$78$	0	0
$79$	−8.00000	−0.900070	−0.450035	−	0.893011i	$-0.648589\pi$
$79$	−8.00000	−0.900070	−0.450035	+	0.893011i	$0.648589\pi$
$80$	0	0
$81$	9.00000	1.00000
$82$	− 2.00000i	− 0.220863i
$83$	− 8.00000i	− 0.878114i	−0.898459	−	0.439057i	$-0.855313\pi$
$83$	− 8.00000i	− 0.878114i	0.898459	−	0.439057i	$-0.144687\pi$
$84$	0	0
$85$	0	0
$86$	−4.00000	−0.431331
$87$	0	0
$88$	− 2.00000i	− 0.213201i
$89$	16.0000	1.69600	0.847998	−	0.529999i	$-0.177808\pi$
$89$	16.0000	1.69600	0.847998	+	0.529999i	$0.177808\pi$
$90$	0	0
$91$	0	0
$92$	− 2.00000i	− 0.208514i
$93$	0	0
$94$	−9.00000	−0.928279
$95$	0	0
$96$	0	0
$97$	2.00000i	0.203069i	0.994832	+	0.101535i	$0.0323753\pi$
$97$	2.00000i	0.203069i	−0.994832	+	0.101535i	$0.967625\pi$
$98$	− 7.00000i	− 0.707107i
$99$	−6.00000	−0.603023
$100$	0	0
$101$	−9.00000	−0.895533	−0.447767	−	0.894150i	$-0.647781\pi$
$101$	−9.00000	−0.895533	−0.447767	+	0.894150i	$0.647781\pi$
$102$	0	0
$103$	6.00000i	0.591198i	0.955312	+	0.295599i	$0.0955191\pi$
$103$	6.00000i	0.591198i	−0.955312	+	0.295599i	$0.904481\pi$
$104$	4.00000	0.392232
$105$	0	0
$106$	12.0000	1.16554
$107$	− 17.0000i	− 1.64345i	−0.569883	−	0.821726i	$-0.693011\pi$
$107$	− 17.0000i	− 1.64345i	0.569883	−	0.821726i	$-0.306989\pi$
$108$	0	0
$109$	6.00000	0.574696	0.287348	−	0.957826i	$-0.407226\pi$
$109$	6.00000	0.574696	0.287348	+	0.957826i	$0.407226\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	4.00000i	0.376288i	0.982141	+	0.188144i	$0.0602472\pi$
$113$	4.00000i	0.376288i	−0.982141	+	0.188144i	$0.939753\pi$
$114$	0	0
$115$	0	0
$116$	1.00000	0.0928477
$117$	− 12.0000i	− 1.10940i
$118$	− 15.0000i	− 1.38086i
$119$	0	0
$120$	0	0
$121$	−7.00000	−0.636364
$122$	− 11.0000i	− 0.995893i
$123$	0	0
$124$	1.00000	0.0898027
$125$	0	0
$126$	0	0
$127$	− 1.00000i	− 0.0887357i	−0.999015	−	0.0443678i	$-0.985873\pi$
$127$	− 1.00000i	− 0.0887357i	0.999015	−	0.0443678i	$-0.0141274\pi$
$128$	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
$133$	0	0
$134$	−3.00000	−0.259161
$135$	0	0
$136$	−6.00000	−0.514496
$137$	0	0	1.00000			$0$
$137$	0	0	−1.00000			$\pi$
$138$	0	0
$139$	−15.0000	−1.27228	−0.636142	−	0.771572i	$-0.719471\pi$
$139$	−15.0000	−1.27228	−0.636142	+	0.771572i	$0.719471\pi$
$140$	0	0
$141$	0	0
$142$	6.00000i	0.503509i
$143$	8.00000i	0.668994i
$144$	3.00000	0.250000
$145$	0	0
$146$	−12.0000	−0.993127
$147$	0	0
$148$	7.00000i	0.575396i
$149$	18.0000	1.47462	0.737309	−	0.675556i	$-0.236096\pi$
$149$	18.0000	1.47462	0.737309	+	0.675556i	$0.236096\pi$
$150$	0	0
$151$	−18.0000	−1.46482	−0.732410	−	0.680864i	$-0.761604\pi$
$151$	−18.0000	−1.46482	−0.732410	+	0.680864i	$0.761604\pi$
$152$	8.00000i	0.648886i
$153$	18.0000i	1.45521i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	1.00000i	0.0798087i	0.999204	+	0.0399043i	$0.0127053\pi$
$157$	1.00000i	0.0798087i	−0.999204	+	0.0399043i	$0.987295\pi$
$158$	8.00000i	0.636446i
$159$	0	0
$160$	0	0
$161$	0	0
$162$	− 9.00000i	− 0.707107i
$163$	6.00000i	0.469956i	0.972001	+	0.234978i	$0.0755019\pi$
$163$	6.00000i	0.469956i	−0.972001	+	0.234978i	$0.924498\pi$
$164$	−2.00000	−0.156174
$165$	0	0
$166$	−8.00000	−0.620920
$167$	16.0000i	1.23812i	0.785345	+	0.619059i	$0.212486\pi$
$167$	16.0000i	1.23812i	−0.785345	+	0.619059i	$0.787514\pi$
$168$	0	0
$169$	−3.00000	−0.230769
$170$	0	0
$171$	24.0000	1.83533
$172$	4.00000i	0.304997i
$173$	12.0000i	0.912343i	0.889892	+	0.456172i	$0.150780\pi$
$173$	12.0000i	0.912343i	−0.889892	+	0.456172i	$0.849220\pi$
$174$	0	0
$175$	0	0
$176$	−2.00000	−0.150756
$177$	0	0
$178$	− 16.0000i	− 1.19925i
$179$	9.00000	0.672692	0.336346	−	0.941739i	$-0.390809\pi$
$179$	9.00000	0.672692	0.336346	+	0.941739i	$0.390809\pi$
$180$	0	0
$181$	−8.00000	−0.594635	−0.297318	−	0.954779i	$-0.596092\pi$
$181$	−8.00000	−0.594635	−0.297318	+	0.954779i	$0.596092\pi$
$182$	0	0
$183$	0	0
$184$	−2.00000	−0.147442
$185$	0	0
$186$	0	0
$187$	− 12.0000i	− 0.877527i
$188$	9.00000i	0.656392i
$189$	0	0
$190$	0	0
$191$	−3.00000	−0.217072	−0.108536	−	0.994092i	$-0.534616\pi$
$191$	−3.00000	−0.217072	−0.108536	+	0.994092i	$0.534616\pi$
$192$	0	0
$193$	4.00000i	0.287926i	0.989583	+	0.143963i	$0.0459847\pi$
$193$	4.00000i	0.287926i	−0.989583	+	0.143963i	$0.954015\pi$
$194$	2.00000	0.143592
$195$	0	0
$196$	−7.00000	−0.500000
$197$	18.0000i	1.28245i	0.767354	+	0.641223i	$0.221573\pi$
$197$	18.0000i	1.28245i	−0.767354	+	0.641223i	$0.778427\pi$
$198$	6.00000i	0.426401i
$199$	−26.0000	−1.84309	−0.921546	−	0.388270i	$-0.873073\pi$
$199$	−26.0000	−1.84309	−0.921546	+	0.388270i	$0.873073\pi$
$200$	0	0
$201$	0	0
$202$	9.00000i	0.633238i
$203$	0	0
$204$	0	0
$205$	0	0
$206$	6.00000	0.418040
$207$	6.00000i	0.417029i
$208$	− 4.00000i	− 0.277350i
$209$	−16.0000	−1.10674
$210$	0	0
$211$	14.0000	0.963800	0.481900	−	0.876226i	$-0.339947\pi$
$211$	14.0000	0.963800	0.481900	+	0.876226i	$0.339947\pi$
$212$	− 12.0000i	− 0.824163i
$213$	0	0
$214$	−17.0000	−1.16210
$215$	0	0
$216$	0	0
$217$	0	0
$218$	− 6.00000i	− 0.406371i
$219$	0	0
$220$	0	0
$221$	24.0000	1.61441
$222$	0	0
$223$	8.00000i	0.535720i	0.963458	+	0.267860i	$0.0863164\pi$
$223$	8.00000i	0.535720i	−0.963458	+	0.267860i	$0.913684\pi$
$224$	0	0
$225$	0	0
$226$	4.00000	0.266076
$227$	− 11.0000i	− 0.730096i	−0.930989	−	0.365048i	$-0.881053\pi$
$227$	− 11.0000i	− 0.730096i	0.930989	−	0.365048i	$-0.118947\pi$
$228$	0	0
$229$	10.0000	0.660819	0.330409	−	0.943838i	$-0.392813\pi$
$229$	10.0000	0.660819	0.330409	+	0.943838i	$0.392813\pi$
$230$	0	0
$231$	0	0
$232$	− 1.00000i	− 0.0656532i
$233$	− 23.0000i	− 1.50678i	−0.657574	−	0.753390i	$-0.728417\pi$
$233$	− 23.0000i	− 1.50678i	0.657574	−	0.753390i	$-0.271583\pi$
$234$	−12.0000	−0.784465
$235$	0	0
$236$	−15.0000	−0.976417
$237$	0	0
$238$	0	0
$239$	6.00000	0.388108	0.194054	−	0.980991i	$-0.437836\pi$
$239$	6.00000	0.388108	0.194054	+	0.980991i	$0.437836\pi$
$240$	0	0
$241$	7.00000	0.450910	0.225455	−	0.974254i	$-0.427613\pi$
$241$	7.00000	0.450910	0.225455	+	0.974254i	$0.427613\pi$
$242$	7.00000i	0.449977i
$243$	0	0
$244$	−11.0000	−0.704203
$245$	0	0
$246$	0	0
$247$	− 32.0000i	− 2.03611i
$248$	− 1.00000i	− 0.0635001i
$249$	0	0
$250$	0	0
$251$	2.00000	0.126239	0.0631194	−	0.998006i	$-0.479895\pi$
$251$	2.00000	0.126239	0.0631194	+	0.998006i	$0.479895\pi$
$252$	0	0
$253$	− 4.00000i	− 0.251478i
$254$	−1.00000	−0.0627456
$255$	0	0
$256$	1.00000	0.0625000
$257$	26.0000i	1.62184i	0.585160	+	0.810918i	$0.301032\pi$
$257$	26.0000i	1.62184i	−0.585160	+	0.810918i	$0.698968\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−3.00000	−0.185695
$262$	0	0
$263$	19.0000i	1.17159i	0.810459	+	0.585795i	$0.199218\pi$
$263$	19.0000i	1.17159i	−0.810459	+	0.585795i	$0.800782\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	3.00000i	0.183254i
$269$	−23.0000	−1.40233	−0.701167	−	0.712997i	$-0.747337\pi$
$269$	−23.0000	−1.40233	−0.701167	+	0.712997i	$0.747337\pi$
$270$	0	0
$271$	−5.00000	−0.303728	−0.151864	−	0.988401i	$-0.548528\pi$
$271$	−5.00000	−0.303728	−0.151864	+	0.988401i	$0.548528\pi$
$272$	6.00000i	0.363803i
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	8.00000i	0.480673i	0.970690	+	0.240337i	$0.0772579\pi$
$277$	8.00000i	0.480673i	−0.970690	+	0.240337i	$0.922742\pi$
$278$	15.0000i	0.899640i
$279$	−3.00000	−0.179605
$280$	0	0
$281$	−13.0000	−0.775515	−0.387757	−	0.921761i	$-0.626750\pi$
$281$	−13.0000	−0.775515	−0.387757	+	0.921761i	$0.626750\pi$
$282$	0	0
$283$	20.0000i	1.18888i	0.804141	+	0.594438i	$0.202626\pi$
$283$	20.0000i	1.18888i	−0.804141	+	0.594438i	$0.797374\pi$
$284$	6.00000	0.356034
$285$	0	0
$286$	8.00000	0.473050
$287$	0	0
$288$	− 3.00000i	− 0.176777i
$289$	−19.0000	−1.11765
$290$	0	0
$291$	0	0
$292$	12.0000i	0.702247i
$293$	− 9.00000i	− 0.525786i	−0.964825	−	0.262893i	$-0.915323\pi$
$293$	− 9.00000i	− 0.525786i	0.964825	−	0.262893i	$-0.0846766\pi$
$294$	0	0
$295$	0	0
$296$	7.00000	0.406867
$297$	0	0
$298$	− 18.0000i	− 1.04271i
$299$	8.00000	0.462652
$300$	0	0
$301$	0	0
$302$	18.0000i	1.03578i
$303$	0	0
$304$	8.00000	0.458831
$305$	0	0
$306$	18.0000	1.02899
$307$	28.0000i	1.59804i	0.601302	+	0.799022i	$0.294649\pi$
$307$	28.0000i	1.59804i	−0.601302	+	0.799022i	$0.705351\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	0	0		−	1.00000i	$-0.5\pi$
$311$	0	0			1.00000i	$0.5\pi$
$312$	0	0
$313$	− 27.0000i	− 1.52613i	−0.646322	−	0.763065i	$-0.723694\pi$
$313$	− 27.0000i	− 1.52613i	0.646322	−	0.763065i	$-0.276306\pi$
$314$	1.00000	0.0564333
$315$	0	0
$316$	8.00000	0.450035
$317$	− 11.0000i	− 0.617822i	−0.951091	−	0.308911i	$-0.900036\pi$
$317$	− 11.0000i	− 0.617822i	0.951091	−	0.308911i	$-0.0999645\pi$
$318$	0	0
$319$	2.00000	0.111979
$320$	0	0
$321$	0	0
$322$	0	0
$323$	48.0000i	2.67079i
$324$	−9.00000	−0.500000
$325$	0	0
$326$	6.00000	0.332309
$327$	0	0
$328$	2.00000i	0.110432i
$329$	0	0
$330$	0	0
$331$	−6.00000	−0.329790	−0.164895	−	0.986311i	$-0.552728\pi$
$331$	−6.00000	−0.329790	−0.164895	+	0.986311i	$0.552728\pi$
$332$	8.00000i	0.439057i
$333$	− 21.0000i	− 1.15079i
$334$	16.0000	0.875481
$335$	0	0
$336$	0	0
$337$	18.0000i	0.980522i	0.871576	+	0.490261i	$0.163099\pi$
$337$	18.0000i	0.980522i	−0.871576	+	0.490261i	$0.836901\pi$
$338$	3.00000i	0.163178i
$339$	0	0
$340$	0	0
$341$	2.00000	0.108306
$342$	− 24.0000i	− 1.29777i
$343$	0	0
$344$	4.00000	0.215666
$345$	0	0
$346$	12.0000	0.645124
$347$	27.0000i	1.44944i	0.689046	+	0.724718i	$0.258030\pi$
$347$	27.0000i	1.44944i	−0.689046	+	0.724718i	$0.741970\pi$
$348$	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
$352$	2.00000i	0.106600i
$353$	25.0000i	1.33062i	0.746569	+	0.665308i	$0.231700\pi$
$353$	25.0000i	1.33062i	−0.746569	+	0.665308i	$0.768300\pi$
$354$	0	0
$355$	0	0
$356$	−16.0000	−0.847998
$357$	0	0
$358$	− 9.00000i	− 0.475665i
$359$	−11.0000	−0.580558	−0.290279	−	0.956942i	$-0.593748\pi$
$359$	−11.0000	−0.580558	−0.290279	+	0.956942i	$0.593748\pi$
$360$	0	0
$361$	45.0000	2.36842
$362$	8.00000i	0.420471i
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 17.0000i	− 0.887393i	−0.896177	−	0.443696i	$-0.853667\pi$
$367$	− 17.0000i	− 0.887393i	0.896177	−	0.443696i	$-0.146333\pi$
$368$	2.00000i	0.104257i
$369$	6.00000	0.312348
$370$	0	0
$371$	0	0
$372$	0	0
$373$	16.0000i	0.828449i	0.910175	+	0.414224i	$0.135947\pi$
$373$	16.0000i	0.828449i	−0.910175	+	0.414224i	$0.864053\pi$
$374$	−12.0000	−0.620505
$375$	0	0
$376$	9.00000	0.464140
$377$	4.00000i	0.206010i
$378$	0	0
$379$	−28.0000	−1.43826	−0.719132	−	0.694874i	$-0.755460\pi$
$379$	−28.0000	−1.43826	−0.719132	+	0.694874i	$0.755460\pi$
$380$	0	0
$381$	0	0
$382$	3.00000i	0.153493i
$383$	− 10.0000i	− 0.510976i	−0.966812	−	0.255488i	$-0.917764\pi$
$383$	− 10.0000i	− 0.510976i	0.966812	−	0.255488i	$-0.0822362\pi$
$384$	0	0
$385$	0	0
$386$	4.00000	0.203595
$387$	− 12.0000i	− 0.609994i
$388$	− 2.00000i	− 0.101535i
$389$	−3.00000	−0.152106	−0.0760530	−	0.997104i	$-0.524232\pi$
$389$	−3.00000	−0.152106	−0.0760530	+	0.997104i	$0.524232\pi$
$390$	0	0
$391$	−12.0000	−0.606866
$392$	7.00000i	0.353553i
$393$	0	0
$394$	18.0000	0.906827
$395$	0	0
$396$	6.00000	0.301511
$397$	32.0000i	1.60603i	0.595956	+	0.803017i	$0.296773\pi$
$397$	32.0000i	1.60603i	−0.595956	+	0.803017i	$0.703227\pi$
$398$	26.0000i	1.30326i
$399$	0	0
$400$	0	0
$401$	5.00000	0.249688	0.124844	−	0.992176i	$-0.460157\pi$
$401$	5.00000	0.249688	0.124844	+	0.992176i	$0.460157\pi$
$402$	0	0
$403$	4.00000i	0.199254i
$404$	9.00000	0.447767
$405$	0	0
$406$	0	0
$407$	14.0000i	0.693954i
$408$	0	0
$409$	0	0		−	1.00000i	$-0.5\pi$
$409$	0	0			1.00000i	$0.5\pi$
$410$	0	0
$411$	0	0
$412$	− 6.00000i	− 0.295599i
$413$	0	0
$414$	6.00000	0.294884
$415$	0	0
$416$	−4.00000	−0.196116
$417$	0	0
$418$	16.0000i	0.782586i
$419$	−13.0000	−0.635092	−0.317546	−	0.948243i	$-0.602859\pi$
$419$	−13.0000	−0.635092	−0.317546	+	0.948243i	$0.602859\pi$
$420$	0	0
$421$	17.0000	0.828529	0.414265	−	0.910156i	$-0.364039\pi$
$421$	17.0000	0.828529	0.414265	+	0.910156i	$0.364039\pi$
$422$	− 14.0000i	− 0.681509i
$423$	− 27.0000i	− 1.31278i
$424$	−12.0000	−0.582772
$425$	0	0
$426$	0	0
$427$	0	0
$428$	17.0000i	0.821726i
$429$	0	0
$430$	0	0
$431$	−2.00000	−0.0963366	−0.0481683	−	0.998839i	$-0.515338\pi$
$431$	−2.00000	−0.0963366	−0.0481683	+	0.998839i	$0.515338\pi$
$432$	0	0
$433$	− 30.0000i	− 1.44171i	−0.693087	−	0.720854i	$-0.743750\pi$
$433$	− 30.0000i	− 1.44171i	0.693087	−	0.720854i	$-0.256250\pi$
$434$	0	0
$435$	0	0
$436$	−6.00000	−0.287348
$437$	16.0000i	0.765384i
$438$	0	0
$439$	−20.0000	−0.954548	−0.477274	−	0.878755i	$-0.658375\pi$
$439$	−20.0000	−0.954548	−0.477274	+	0.878755i	$0.658375\pi$
$440$	0	0
$441$	21.0000	1.00000
$442$	− 24.0000i	− 1.14156i
$443$	− 26.0000i	− 1.23530i	−0.786454	−	0.617649i	$-0.788085\pi$
$443$	− 26.0000i	− 1.23530i	0.786454	−	0.617649i	$-0.211915\pi$
$444$	0	0
$445$	0	0
$446$	8.00000	0.378811
$447$	0	0
$448$	0	0
$449$	8.00000	0.377543	0.188772	−	0.982021i	$-0.439549\pi$
$449$	8.00000	0.377543	0.188772	+	0.982021i	$0.439549\pi$
$450$	0	0
$451$	−4.00000	−0.188353
$452$	− 4.00000i	− 0.188144i
$453$	0	0
$454$	−11.0000	−0.516256
$455$	0	0
$456$	0	0
$457$	2.00000i	0.0935561i	0.998905	+	0.0467780i	$0.0148953\pi$
$457$	2.00000i	0.0935561i	−0.998905	+	0.0467780i	$0.985105\pi$
$458$	− 10.0000i	− 0.467269i
$459$	0	0
$460$	0	0
$461$	−6.00000	−0.279448	−0.139724	−	0.990190i	$-0.544622\pi$
$461$	−6.00000	−0.279448	−0.139724	+	0.990190i	$0.544622\pi$
$462$	0	0
$463$	8.00000i	0.371792i	0.982569	+	0.185896i	$0.0595187\pi$
$463$	8.00000i	0.371792i	−0.982569	+	0.185896i	$0.940481\pi$
$464$	−1.00000	−0.0464238
$465$	0	0
$466$	−23.0000	−1.06545
$467$	0	0	1.00000			$0$
$467$	0	0	−1.00000			$\pi$
$468$	12.0000i	0.554700i
$469$	0	0
$470$	0	0
$471$	0	0
$472$	15.0000i	0.690431i
$473$	8.00000i	0.367840i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	36.0000i	1.64833i
$478$	− 6.00000i	− 0.274434i
$479$	24.0000	1.09659	0.548294	−	0.836286i	$-0.315277\pi$
$479$	24.0000	1.09659	0.548294	+	0.836286i	$0.315277\pi$
$480$	0	0
$481$	−28.0000	−1.27669
$482$	− 7.00000i	− 0.318841i
$483$	0	0
$484$	7.00000	0.318182
$485$	0	0
$486$	0	0
$487$	− 38.0000i	− 1.72194i	−0.508652	−	0.860972i	$-0.669856\pi$
$487$	− 38.0000i	− 1.72194i	0.508652	−	0.860972i	$-0.330144\pi$
$488$	11.0000i	0.497947i
$489$	0	0
$490$	0	0
$491$	−42.0000	−1.89543	−0.947717	−	0.319113i	$-0.896615\pi$
$491$	−42.0000	−1.89543	−0.947717	+	0.319113i	$0.896615\pi$
$492$	0	0
$493$	− 6.00000i	− 0.270226i
$494$	−32.0000	−1.43975
$495$	0	0
$496$	−1.00000	−0.0449013
$497$	0	0
$498$	0	0
$499$	31.0000	1.38775	0.693875	−	0.720095i	$-0.255902\pi$
$499$	31.0000	1.38775	0.693875	+	0.720095i	$0.255902\pi$
$500$	0	0
$501$	0	0
$502$	− 2.00000i	− 0.0892644i
$503$	− 33.0000i	− 1.47140i	−0.677309	−	0.735699i	$-0.736854\pi$
$503$	− 33.0000i	− 1.47140i	0.677309	−	0.735699i	$-0.263146\pi$
$504$	0	0
$505$	0	0
$506$	−4.00000	−0.177822
$507$	0	0
$508$	1.00000i	0.0443678i
$509$	0	0		−	1.00000i	$-0.5\pi$
$509$	0	0			1.00000i	$0.5\pi$
$510$	0	0
$511$	0	0
$512$	− 1.00000i	− 0.0441942i
$513$	0	0
$514$	26.0000	1.14681
$515$	0	0
$516$	0	0
$517$	18.0000i	0.791639i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−23.0000	−1.00765	−0.503824	−	0.863806i	$-0.668074\pi$
$521$	−23.0000	−1.00765	−0.503824	+	0.863806i	$0.668074\pi$
$522$	3.00000i	0.131306i
$523$	− 11.0000i	− 0.480996i	−0.970650	−	0.240498i	$-0.922689\pi$
$523$	− 11.0000i	− 0.480996i	0.970650	−	0.240498i	$-0.0773108\pi$
$524$	0	0
$525$	0	0
$526$	19.0000	0.828439
$527$	− 6.00000i	− 0.261364i
$528$	0	0
$529$	19.0000	0.826087
$530$	0	0
$531$	45.0000	1.95283
$532$	0	0
$533$	− 8.00000i	− 0.346518i
$534$	0	0
$535$	0	0
$536$	3.00000	0.129580
$537$	0	0
$538$	23.0000i	0.991600i
$539$	−14.0000	−0.603023
$540$	0	0
$541$	43.0000	1.84871	0.924357	−	0.381528i	$-0.124602\pi$
$541$	43.0000	1.84871	0.924357	+	0.381528i	$0.124602\pi$
$542$	5.00000i	0.214768i
$543$	0	0
$544$	6.00000	0.257248
$545$	0	0
$546$	0	0
$547$	− 44.0000i	− 1.88130i	−0.339372	−	0.940652i	$-0.610215\pi$
$547$	− 44.0000i	− 1.88130i	0.339372	−	0.940652i	$-0.389785\pi$
$548$	0	0
$549$	33.0000	1.40841
$550$	0	0
$551$	−8.00000	−0.340811
$552$	0	0
$553$	0	0
$554$	8.00000	0.339887
$555$	0	0
$556$	15.0000	0.636142
$557$	18.0000i	0.762684i	0.924434	+	0.381342i	$0.124538\pi$
$557$	18.0000i	0.762684i	−0.924434	+	0.381342i	$0.875462\pi$
$558$	3.00000i	0.127000i
$559$	−16.0000	−0.676728
$560$	0	0
$561$	0	0
$562$	13.0000i	0.548372i
$563$	24.0000i	1.01148i	0.862686	+	0.505740i	$0.168780\pi$
$563$	24.0000i	1.01148i	−0.862686	+	0.505740i	$0.831220\pi$
$564$	0	0
$565$	0	0
$566$	20.0000	0.840663
$567$	0	0
$568$	− 6.00000i	− 0.251754i
$569$	−6.00000	−0.251533	−0.125767	−	0.992060i	$-0.540139\pi$
$569$	−6.00000	−0.251533	−0.125767	+	0.992060i	$0.540139\pi$
$570$	0	0
$571$	−29.0000	−1.21361	−0.606806	−	0.794850i	$-0.707550\pi$
$571$	−29.0000	−1.21361	−0.606806	+	0.794850i	$0.707550\pi$
$572$	− 8.00000i	− 0.334497i
$573$	0	0
$574$	0	0
$575$	0	0
$576$	−3.00000	−0.125000
$577$	− 24.0000i	− 0.999133i	−0.866276	−	0.499567i	$-0.833493\pi$
$577$	− 24.0000i	− 0.999133i	0.866276	−	0.499567i	$-0.166507\pi$
$578$	19.0000i	0.790296i
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	− 24.0000i	− 0.993978i
$584$	12.0000	0.496564
$585$	0	0
$586$	−9.00000	−0.371787
$587$	− 7.00000i	− 0.288921i	−0.989511	−	0.144460i	$-0.953855\pi$
$587$	− 7.00000i	− 0.288921i	0.989511	−	0.144460i	$-0.0461446\pi$
$588$	0	0
$589$	−8.00000	−0.329634
$590$	0	0
$591$	0	0
$592$	− 7.00000i	− 0.287698i
$593$	21.0000i	0.862367i	0.902264	+	0.431183i	$0.141904\pi$
$593$	21.0000i	0.862367i	−0.902264	+	0.431183i	$0.858096\pi$
$594$	0	0
$595$	0	0
$596$	−18.0000	−0.737309
$597$	0	0
$598$	− 8.00000i	− 0.327144i
$599$	−27.0000	−1.10319	−0.551595	−	0.834112i	$-0.685981\pi$
$599$	−27.0000	−1.10319	−0.551595	+	0.834112i	$0.685981\pi$
$600$	0	0
$601$	−12.0000	−0.489490	−0.244745	−	0.969587i	$-0.578704\pi$
$601$	−12.0000	−0.489490	−0.244745	+	0.969587i	$0.578704\pi$
$602$	0	0
$603$	− 9.00000i	− 0.366508i
$604$	18.0000	0.732410
$605$	0	0
$606$	0	0
$607$	− 1.00000i	− 0.0405887i	−0.999794	−	0.0202944i	$-0.993540\pi$
$607$	− 1.00000i	− 0.0405887i	0.999794	−	0.0202944i	$-0.00646034\pi$
$608$	− 8.00000i	− 0.324443i
$609$	0	0
$610$	0	0
$611$	−36.0000	−1.45640
$612$	− 18.0000i	− 0.727607i
$613$	2.00000i	0.0807792i	0.999184	+	0.0403896i	$0.0128599\pi$
$613$	2.00000i	0.0807792i	−0.999184	+	0.0403896i	$0.987140\pi$
$614$	28.0000	1.12999
$615$	0	0
$616$	0	0
$617$	6.00000i	0.241551i	0.992680	+	0.120775i	$0.0385381\pi$
$617$	6.00000i	0.241551i	−0.992680	+	0.120775i	$0.961462\pi$
$618$	0	0
$619$	−42.0000	−1.68812	−0.844061	−	0.536247i	$-0.819842\pi$
$619$	−42.0000	−1.68812	−0.844061	+	0.536247i	$0.819842\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	0	0
$626$	−27.0000	−1.07914
$627$	0	0
$628$	− 1.00000i	− 0.0399043i
$629$	42.0000	1.67465
$630$	0	0
$631$	−48.0000	−1.91085	−0.955425	−	0.295234i	$-0.904602\pi$
$631$	−48.0000	−1.91085	−0.955425	+	0.295234i	$0.904602\pi$
$632$	− 8.00000i	− 0.318223i
$633$	0	0
$634$	−11.0000	−0.436866
$635$	0	0
$636$	0	0
$637$	− 28.0000i	− 1.10940i
$638$	− 2.00000i	− 0.0791808i
$639$	−18.0000	−0.712069
$640$	0	0
$641$	−30.0000	−1.18493	−0.592464	−	0.805597i	$-0.701845\pi$
$641$	−30.0000	−1.18493	−0.592464	+	0.805597i	$0.701845\pi$
$642$	0	0
$643$	11.0000i	0.433798i	0.976194	+	0.216899i	$0.0695942\pi$
$643$	11.0000i	0.433798i	−0.976194	+	0.216899i	$0.930406\pi$
$644$	0	0
$645$	0	0
$646$	48.0000	1.88853
$647$	− 12.0000i	− 0.471769i	−0.971781	−	0.235884i	$-0.924201\pi$
$647$	− 12.0000i	− 0.471769i	0.971781	−	0.235884i	$-0.0757987\pi$
$648$	9.00000i	0.353553i
$649$	−30.0000	−1.17760
$650$	0	0
$651$	0	0
$652$	− 6.00000i	− 0.234978i
$653$	9.00000i	0.352197i	0.984373	+	0.176099i	$0.0563478\pi$
$653$	9.00000i	0.352197i	−0.984373	+	0.176099i	$0.943652\pi$
$654$	0	0
$655$	0	0
$656$	2.00000	0.0780869
$657$	− 36.0000i	− 1.40449i
$658$	0	0
$659$	36.0000	1.40236	0.701180	−	0.712984i	$-0.252657\pi$
$659$	36.0000	1.40236	0.701180	+	0.712984i	$0.252657\pi$
$660$	0	0
$661$	−10.0000	−0.388955	−0.194477	−	0.980907i	$-0.562301\pi$
$661$	−10.0000	−0.388955	−0.194477	+	0.980907i	$0.562301\pi$
$662$	6.00000i	0.233197i
$663$	0	0
$664$	8.00000	0.310460
$665$	0	0
$666$	−21.0000	−0.813733
$667$	− 2.00000i	− 0.0774403i
$668$	− 16.0000i	− 0.619059i
$669$	0	0
$670$	0	0
$671$	−22.0000	−0.849301
$672$	0	0
$673$	1.00000i	0.0385472i	0.999814	+	0.0192736i	$0.00613535\pi$
$673$	1.00000i	0.0385472i	−0.999814	+	0.0192736i	$0.993865\pi$
$674$	18.0000	0.693334
$675$	0	0
$676$	3.00000	0.115385
$677$	25.0000i	0.960828i	0.877042	+	0.480414i	$0.159514\pi$
$677$	25.0000i	0.960828i	−0.877042	+	0.480414i	$0.840486\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	− 2.00000i	− 0.0765840i
$683$	12.0000i	0.459167i	0.973289	+	0.229584i	$0.0737364\pi$
$683$	12.0000i	0.459167i	−0.973289	+	0.229584i	$0.926264\pi$
$684$	−24.0000	−0.917663
$685$	0	0
$686$	0	0
$687$	0	0
$688$	− 4.00000i	− 0.152499i
$689$	48.0000	1.82865
$690$	0	0
$691$	5.00000	0.190209	0.0951045	−	0.995467i	$-0.469681\pi$
$691$	5.00000	0.190209	0.0951045	+	0.995467i	$0.469681\pi$
$692$	− 12.0000i	− 0.456172i
$693$	0	0
$694$	27.0000	1.02491
$695$	0	0
$696$	0	0
$697$	12.0000i	0.454532i
$698$	8.00000i	0.302804i
$699$	0	0
$700$	0	0
$701$	−32.0000	−1.20862	−0.604312	−	0.796748i	$-0.706552\pi$
$701$	−32.0000	−1.20862	−0.604312	+	0.796748i	$0.706552\pi$
$702$	0	0
$703$	− 56.0000i	− 2.11208i
$704$	2.00000	0.0753778
$705$	0	0
$706$	25.0000	0.940887
$707$	0	0
$708$	0	0
$709$	14.0000	0.525781	0.262891	−	0.964826i	$-0.415324\pi$
$709$	14.0000	0.525781	0.262891	+	0.964826i	$0.415324\pi$
$710$	0	0
$711$	−24.0000	−0.900070
$712$	16.0000i	0.599625i
$713$	− 2.00000i	− 0.0749006i
$714$	0	0
$715$	0	0
$716$	−9.00000	−0.336346
$717$	0	0
$718$	11.0000i	0.410516i
$719$	−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
$721$	0	0
$722$	− 45.0000i	− 1.67473i
$723$	0	0
$724$	8.00000	0.297318
$725$	0	0
$726$	0	0
$727$	48.0000i	1.78022i	0.455744	+	0.890111i	$0.349373\pi$
$727$	48.0000i	1.78022i	−0.455744	+	0.890111i	$0.650627\pi$
$728$	0	0
$729$	27.0000	1.00000
$730$	0	0
$731$	24.0000	0.887672
$732$	0	0
$733$	− 42.0000i	− 1.55131i	−0.631160	−	0.775653i	$-0.717421\pi$
$733$	− 42.0000i	− 1.55131i	0.631160	−	0.775653i	$-0.282579\pi$
$734$	−17.0000	−0.627481
$735$	0	0
$736$	2.00000	0.0737210
$737$	6.00000i	0.221013i
$738$	− 6.00000i	− 0.220863i
$739$	−22.0000	−0.809283	−0.404642	−	0.914475i	$-0.632604\pi$
$739$	−22.0000	−0.809283	−0.404642	+	0.914475i	$0.632604\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 32.0000i	− 1.17397i	−0.809599	−	0.586983i	$-0.800316\pi$
$743$	− 32.0000i	− 1.17397i	0.809599	−	0.586983i	$-0.199684\pi$
$744$	0	0
$745$	0	0
$746$	16.0000	0.585802
$747$	− 24.0000i	− 0.878114i
$748$	12.0000i	0.438763i
$749$	0	0
$750$	0	0
$751$	13.0000	0.474377	0.237188	−	0.971464i	$-0.423774\pi$
$751$	13.0000	0.474377	0.237188	+	0.971464i	$0.423774\pi$
$752$	− 9.00000i	− 0.328196i
$753$	0	0
$754$	4.00000	0.145671
$755$	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	$-0.212006\pi$
$758$	28.0000i	1.01701i
$759$	0	0
$760$	0	0
$761$	15.0000	0.543750	0.271875	−	0.962333i	$-0.412356\pi$
$761$	15.0000	0.543750	0.271875	+	0.962333i	$0.412356\pi$
$762$	0	0
$763$	0	0
$764$	3.00000	0.108536
$765$	0	0
$766$	−10.0000	−0.361315
$767$	− 60.0000i	− 2.16647i
$768$	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
$772$	− 4.00000i	− 0.143963i
$773$	− 17.0000i	− 0.611448i	−0.952120	−	0.305724i	$-0.901102\pi$
$773$	− 17.0000i	− 0.611448i	0.952120	−	0.305724i	$-0.0988984\pi$
$774$	−12.0000	−0.431331
$775$	0	0
$776$	−2.00000	−0.0717958
$777$	0	0
$778$	3.00000i	0.107555i
$779$	16.0000	0.573259
$780$	0	0
$781$	12.0000	0.429394
$782$	12.0000i	0.429119i
$783$	0	0
$784$	7.00000	0.250000
$785$	0	0
$786$	0	0
$787$	− 29.0000i	− 1.03374i	−0.856064	−	0.516869i	$-0.827097\pi$
$787$	− 29.0000i	− 1.03374i	0.856064	−	0.516869i	$-0.172903\pi$
$788$	− 18.0000i	− 0.641223i
$789$	0	0
$790$	0	0
$791$	0	0
$792$	− 6.00000i	− 0.213201i
$793$	− 44.0000i	− 1.56249i
$794$	32.0000	1.13564
$795$	0	0
$796$	26.0000	0.921546
$797$	− 18.0000i	− 0.637593i	−0.947823	−	0.318796i	$-0.896721\pi$
$797$	− 18.0000i	− 0.637593i	0.947823	−	0.318796i	$-0.103279\pi$
$798$	0	0
$799$	54.0000	1.91038
$800$	0	0
$801$	48.0000	1.69600
$802$	− 5.00000i	− 0.176556i
$803$	24.0000i	0.846942i
$804$	0	0
$805$	0	0
$806$	4.00000	0.140894
$807$	0	0
$808$	− 9.00000i	− 0.316619i
$809$	16.0000	0.562530	0.281265	−	0.959630i	$-0.409246\pi$
$809$	16.0000	0.562530	0.281265	+	0.959630i	$0.409246\pi$
$810$	0	0
$811$	−17.0000	−0.596951	−0.298475	−	0.954417i	$-0.596478\pi$
$811$	−17.0000	−0.596951	−0.298475	+	0.954417i	$0.596478\pi$
$812$	0	0
$813$	0	0
$814$	14.0000	0.490700
$815$	0	0
$816$	0	0
$817$	− 32.0000i	− 1.11954i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	44.0000	1.53561	0.767805	−	0.640683i	$-0.221349\pi$
$821$	44.0000	1.53561	0.767805	+	0.640683i	$0.221349\pi$
$822$	0	0
$823$	− 56.0000i	− 1.95204i	−0.217687	−	0.976019i	$-0.569851\pi$
$823$	− 56.0000i	− 1.95204i	0.217687	−	0.976019i	$-0.430149\pi$
$824$	−6.00000	−0.209020
$825$	0	0
$826$	0	0
$827$	14.0000i	0.486828i	0.969923	+	0.243414i	$0.0782673\pi$
$827$	14.0000i	0.486828i	−0.969923	+	0.243414i	$0.921733\pi$
$828$	− 6.00000i	− 0.208514i
$829$	14.0000	0.486240	0.243120	−	0.969996i	$-0.421829\pi$
$829$	14.0000	0.486240	0.243120	+	0.969996i	$0.421829\pi$
$830$	0	0
$831$	0	0
$832$	4.00000i	0.138675i
$833$	42.0000i	1.45521i
$834$	0	0
$835$	0	0
$836$	16.0000	0.553372
$837$	0	0
$838$	13.0000i	0.449078i
$839$	−43.0000	−1.48452	−0.742262	−	0.670109i	$-0.766247\pi$
$839$	−43.0000	−1.48452	−0.742262	+	0.670109i	$0.766247\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	− 17.0000i	− 0.585859i
$843$	0	0
$844$	−14.0000	−0.481900
$845$	0	0
$846$	−27.0000	−0.928279
$847$	0	0
$848$	12.0000i	0.412082i
$849$	0	0
$850$	0	0
$851$	14.0000	0.479914
$852$	0	0
$853$	− 22.0000i	− 0.753266i	−0.926363	−	0.376633i	$-0.877082\pi$
$853$	− 22.0000i	− 0.753266i	0.926363	−	0.376633i	$-0.122918\pi$
$854$	0	0
$855$	0	0
$856$	17.0000	0.581048
$857$	14.0000i	0.478231i	0.970991	+	0.239115i	$0.0768574\pi$
$857$	14.0000i	0.478231i	−0.970991	+	0.239115i	$0.923143\pi$
$858$	0	0
$859$	54.0000	1.84246	0.921228	−	0.389023i	$-0.127187\pi$
$859$	54.0000	1.84246	0.921228	+	0.389023i	$0.127187\pi$
$860$	0	0
$861$	0	0
$862$	2.00000i	0.0681203i
$863$	18.0000i	0.612727i	0.951915	+	0.306364i	$0.0991123\pi$
$863$	18.0000i	0.612727i	−0.951915	+	0.306364i	$0.900888\pi$
$864$	0	0
$865$	0	0
$866$	−30.0000	−1.01944
$867$	0	0
$868$	0	0
$869$	16.0000	0.542763
$870$	0	0
$871$	−12.0000	−0.406604
$872$	6.00000i	0.203186i
$873$	6.00000i	0.203069i
$874$	16.0000	0.541208
$875$	0	0
$876$	0	0
$877$	36.0000i	1.21563i	0.794077	+	0.607817i	$0.207955\pi$
$877$	36.0000i	1.21563i	−0.794077	+	0.607817i	$0.792045\pi$
$878$	20.0000i	0.674967i
$879$	0	0
$880$	0	0
$881$	−32.0000	−1.07811	−0.539054	−	0.842271i	$-0.681218\pi$
$881$	−32.0000	−1.07811	−0.539054	+	0.842271i	$0.681218\pi$
$882$	− 21.0000i	− 0.707107i
$883$	41.0000i	1.37976i	0.723924	+	0.689880i	$0.242337\pi$
$883$	41.0000i	1.37976i	−0.723924	+	0.689880i	$0.757663\pi$
$884$	−24.0000	−0.807207
$885$	0	0
$886$	−26.0000	−0.873487
$887$	12.0000i	0.402921i	0.979497	+	0.201460i	$0.0645687\pi$
$887$	12.0000i	0.402921i	−0.979497	+	0.201460i	$0.935431\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−18.0000	−0.603023
$892$	− 8.00000i	− 0.267860i
$893$	− 72.0000i	− 2.40939i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	− 8.00000i	− 0.266963i
$899$	1.00000	0.0333519
$900$	0	0
$901$	−72.0000	−2.39867
$902$	4.00000i	0.133185i
$903$	0	0
$904$	−4.00000	−0.133038
$905$	0	0
$906$	0	0
$907$	− 42.0000i	− 1.39459i	−0.716786	−	0.697294i	$-0.754387\pi$
$907$	− 42.0000i	− 1.39459i	0.716786	−	0.697294i	$-0.245613\pi$
$908$	11.0000i	0.365048i
$909$	−27.0000	−0.895533
$910$	0	0
$911$	−7.00000	−0.231920	−0.115960	−	0.993254i	$-0.536994\pi$
$911$	−7.00000	−0.231920	−0.115960	+	0.993254i	$0.536994\pi$
$912$	0	0
$913$	16.0000i	0.529523i
$914$	2.00000	0.0661541
$915$	0	0
$916$	−10.0000	−0.330409
$917$	0	0
$918$	0	0
$919$	28.0000	0.923635	0.461817	−	0.886975i	$-0.347198\pi$
$919$	28.0000	0.923635	0.461817	+	0.886975i	$0.347198\pi$
$920$	0	0
$921$	0	0
$922$	6.00000i	0.197599i
$923$	24.0000i	0.789970i
$924$	0	0
$925$	0	0
$926$	8.00000	0.262896
$927$	18.0000i	0.591198i
$928$	1.00000i	0.0328266i
$929$	13.0000	0.426516	0.213258	−	0.976996i	$-0.431592\pi$
$929$	13.0000	0.426516	0.213258	+	0.976996i	$0.431592\pi$
$930$	0	0
$931$	56.0000	1.83533
$932$	23.0000i	0.753390i
$933$	0	0
$934$	0	0
$935$	0	0
$936$	12.0000	0.392232
$937$	− 13.0000i	− 0.424691i	−0.977195	−	0.212346i	$-0.931890\pi$
$937$	− 13.0000i	− 0.424691i	0.977195	−	0.212346i	$-0.0681103\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−42.0000	−1.36916	−0.684580	−	0.728937i	$-0.740015\pi$
$941$	−42.0000	−1.36916	−0.684580	+	0.728937i	$0.740015\pi$
$942$	0	0
$943$	4.00000i	0.130258i
$944$	15.0000	0.488208
$945$	0	0
$946$	8.00000	0.260102
$947$	− 2.00000i	− 0.0649913i	−0.999472	−	0.0324956i	$-0.989654\pi$
$947$	− 2.00000i	− 0.0649913i	0.999472	−	0.0324956i	$-0.0103455\pi$
$948$	0	0
$949$	−48.0000	−1.55815
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 50.0000i	− 1.61966i	−0.586665	−	0.809829i	$-0.699560\pi$
$953$	− 50.0000i	− 1.61966i	0.586665	−	0.809829i	$-0.300440\pi$
$954$	36.0000	1.16554
$955$	0	0
$956$	−6.00000	−0.194054
$957$	0	0
$958$	− 24.0000i	− 0.775405i
$959$	0	0
$960$	0	0
$961$	−30.0000	−0.967742
$962$	28.0000i	0.902756i
$963$	− 51.0000i	− 1.64345i
$964$	−7.00000	−0.225455
$965$	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
$971$	−24.0000	−0.770197	−0.385098	−	0.922876i	$-0.625832\pi$
$971$	−24.0000	−0.770197	−0.385098	+	0.922876i	$0.625832\pi$
$972$	0	0
$973$	0	0
$974$	−38.0000	−1.21760
$975$	0	0
$976$	11.0000	0.352101
$977$	51.0000i	1.63163i	0.578310	+	0.815817i	$0.303713\pi$
$977$	51.0000i	1.63163i	−0.578310	+	0.815817i	$0.696287\pi$
$978$	0	0
$979$	−32.0000	−1.02272
$980$	0	0
$981$	18.0000	0.574696
$982$	42.0000i	1.34027i
$983$	1.00000i	0.0318950i	0.999873	+	0.0159475i	$0.00507647\pi$
$983$	1.00000i	0.0318950i	−0.999873	+	0.0159475i	$0.994924\pi$
$984$	0	0
$985$	0	0
$986$	−6.00000	−0.191079
$987$	0	0
$988$	32.0000i	1.01806i
$989$	8.00000	0.254385
$990$	0	0
$991$	48.0000	1.52477	0.762385	−	0.647124i	$-0.224028\pi$
$991$	48.0000	1.52477	0.762385	+	0.647124i	$0.224028\pi$
$992$	1.00000i	0.0317500i
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 29.0000i	− 0.918439i	−0.888323	−	0.459220i	$-0.848129\pi$
$997$	− 29.0000i	− 0.918439i	0.888323	−	0.459220i	$-0.151871\pi$
$998$	− 31.0000i	− 0.981288i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1450.2.b.c.349.1		2
5.2	odd	4		1450.2.a.f.1.1	yes	1
5.3	odd	4		1450.2.a.b.1.1	✓	1
5.4	even	2	inner	1450.2.b.c.349.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1450.2.a.b.1.1	✓	1	5.3	odd	4
1450.2.a.f.1.1	yes	1	5.2	odd	4
1450.2.b.c.349.1		2	1.1	even	1	trivial
1450.2.b.c.349.2		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:	\( N \)	\(=\)	\( 1450 = 2 \cdot 5^{2} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1450.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} -1.00000 q^{4} +1.00000i q^{8} +3.00000 q^{9} +O(q^{10})\)	\(q-1.00000i q^{2} -1.00000 q^{4} +1.00000i q^{8} +3.00000 q^{9} -2.00000 q^{11} -4.00000i q^{13} +1.00000 q^{16} +6.00000i q^{17} -3.00000i q^{18} +8.00000 q^{19} +2.00000i q^{22} +2.00000i q^{23} -4.00000 q^{26} -1.00000 q^{29} -1.00000 q^{31} -1.00000i q^{32} +6.00000 q^{34} -3.00000 q^{36} -7.00000i q^{37} -8.00000i q^{38} +2.00000 q^{41} -4.00000i q^{43} +2.00000 q^{44} +2.00000 q^{46} -9.00000i q^{47} +7.00000 q^{49} +4.00000i q^{52} +12.0000i q^{53} +1.00000i q^{58} +15.0000 q^{59} +11.0000 q^{61} +1.00000i q^{62} -1.00000 q^{64} -3.00000i q^{67} -6.00000i q^{68} -6.00000 q^{71} +3.00000i q^{72} -12.0000i q^{73} -7.00000 q^{74} -8.00000 q^{76} -8.00000 q^{79} +9.00000 q^{81} -2.00000i q^{82} -8.00000i q^{83} -4.00000 q^{86} -2.00000i q^{88} +16.0000 q^{89} -2.00000i q^{92} -9.00000 q^{94} +2.00000i q^{97} -7.00000i q^{98} -6.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4} + 6 q^{9}+O(q^{10}) \) 2 * q - 2 * q^4 + 6 * q^9	\( 2 q - 2 q^{4} + 6 q^{9} - 4 q^{11} + 2 q^{16} + 16 q^{19} - 8 q^{26} - 2 q^{29} - 2 q^{31} + 12 q^{34} - 6 q^{36} + 4 q^{41} + 4 q^{44} + 4 q^{46} + 14 q^{49} + 30 q^{59} + 22 q^{61} - 2 q^{64} - 12 q^{71} - 14 q^{74} - 16 q^{76} - 16 q^{79} + 18 q^{81} - 8 q^{86} + 32 q^{89} - 18 q^{94} - 12 q^{99}+O(q^{100}) \) 2 * q - 2 * q^4 + 6 * q^9 - 4 * q^11 + 2 * q^16 + 16 * q^19 - 8 * q^26 - 2 * q^29 - 2 * q^31 + 12 * q^34 - 6 * q^36 + 4 * q^41 + 4 * q^44 + 4 * q^46 + 14 * q^49 + 30 * q^59 + 22 * q^61 - 2 * q^64 - 12 * q^71 - 14 * q^74 - 16 * q^76 - 16 * q^79 + 18 * q^81 - 8 * q^86 + 32 * q^89 - 18 * q^94 - 12 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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