LMFDB - Embedded newform 425.2.b.b.324.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [425,2,Mod(324,425)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(425, 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("425.324");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 425 = 5^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	425.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:	$3.39364208590$
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 17)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			324.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	425.324
Dual form			425.2.b.b.324.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{2} +1.00000 q^{4} +4.00000i q^{7} -3.00000i q^{8} +3.00000 q^{9} +2.00000i q^{13} +4.00000 q^{14} -1.00000 q^{16} +1.00000i q^{17} -3.00000i q^{18} +4.00000 q^{19} -4.00000i q^{23} +2.00000 q^{26} +4.00000i q^{28} -6.00000 q^{29} +4.00000 q^{31} -5.00000i q^{32} +1.00000 q^{34} +3.00000 q^{36} -2.00000i q^{37} -4.00000i q^{38} -6.00000 q^{41} -4.00000i q^{43} -4.00000 q^{46} -9.00000 q^{49} +2.00000i q^{52} -6.00000i q^{53} +12.0000 q^{56} +6.00000i q^{58} +12.0000 q^{59} -10.0000 q^{61} -4.00000i q^{62} +12.0000i q^{63} -7.00000 q^{64} +4.00000i q^{67} +1.00000i q^{68} -4.00000 q^{71} -9.00000i q^{72} +6.00000i q^{73} -2.00000 q^{74} +4.00000 q^{76} -12.0000 q^{79} +9.00000 q^{81} +6.00000i q^{82} +4.00000i q^{83} -4.00000 q^{86} -10.0000 q^{89} -8.00000 q^{91} -4.00000i q^{92} +2.00000i q^{97} +9.00000i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{4} + 6 q^{9} + 8 q^{14} - 2 q^{16} + 8 q^{19} + 4 q^{26} - 12 q^{29} + 8 q^{31} + 2 q^{34} + 6 q^{36} - 12 q^{41} - 8 q^{46} - 18 q^{49} + 24 q^{56} + 24 q^{59} - 20 q^{61} - 14 q^{64} - 8 q^{71}+ \cdots - 16 q^{91}+O(q^{100}) $ 2 * q + 2 * q^4 + 6 * q^9 + 8 * q^14 - 2 * q^16 + 8 * q^19 + 4 * q^26 - 12 * q^29 + 8 * q^31 + 2 * q^34 + 6 * q^36 - 12 * q^41 - 8 * q^46 - 18 * q^49 + 24 * q^56 + 24 * q^59 - 20 * q^61 - 14 * q^64 - 8 * q^71 - 4 * q^74 + 8 * q^76 - 24 * q^79 + 18 * q^81 - 8 * q^86 - 20 * q^89 - 16 * q^91

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/425\mathbb{Z}\right)^\times$.

$n$	$52$	$326$
$\chi(n)$	$-1$	$1$

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	−0.935414	−	0.353553i	$-0.884973\pi$
$2$	− 1.00000i	− 0.707107i	0.935414	−	0.353553i	$-0.115027\pi$
$3$	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
$7$	4.00000i	1.51186i	0.654654	+	0.755929i	$0.272814\pi$
$7$	4.00000i	1.51186i	−0.654654	+	0.755929i	$0.727186\pi$
$8$	− 3.00000i	− 1.06066i
$9$	3.00000	1.00000
$10$	0	0
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	0	0
$13$	2.00000i	0.554700i	0.960769	+	0.277350i	$0.0894562\pi$
$13$	2.00000i	0.554700i	−0.960769	+	0.277350i	$0.910544\pi$
$14$	4.00000	1.06904
$15$	0	0
$16$	−1.00000	−0.250000
$17$	1.00000i	0.242536i
$17$	1.00000i	0.242536i
$18$	− 3.00000i	− 0.707107i
$19$	4.00000	0.917663	0.458831	−	0.888523i	$-0.348268\pi$
$19$	4.00000	0.917663	0.458831	+	0.888523i	$0.348268\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 4.00000i	− 0.834058i	−0.908893	−	0.417029i	$-0.863071\pi$
$23$	− 4.00000i	− 0.834058i	0.908893	−	0.417029i	$-0.136929\pi$
$24$	0	0
$25$	0	0
$26$	2.00000	0.392232
$27$	0	0
$28$	4.00000i	0.755929i
$29$	−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	$-0.383046\pi$
$31$	4.00000	0.718421	0.359211	+	0.933257i	$0.383046\pi$
$32$	− 5.00000i	− 0.883883i
$33$	0	0
$34$	1.00000	0.171499
$35$	0	0
$36$	3.00000	0.500000
$37$	− 2.00000i	− 0.328798i	−0.986394	−	0.164399i	$-0.947432\pi$
$37$	− 2.00000i	− 0.328798i	0.986394	−	0.164399i	$-0.0525685\pi$
$38$	− 4.00000i	− 0.648886i
$39$	0	0
$40$	0	0
$41$	−6.00000	−0.937043	−0.468521	−	0.883452i	$-0.655213\pi$
$41$	−6.00000	−0.937043	−0.468521	+	0.883452i	$0.655213\pi$
$42$	0	0
$43$	− 4.00000i	− 0.609994i	−0.952353	−	0.304997i	$-0.901344\pi$
$43$	− 4.00000i	− 0.609994i	0.952353	−	0.304997i	$-0.0986555\pi$
$44$	0	0
$45$	0	0
$46$	−4.00000	−0.589768
$47$	0	0	1.00000			$0$
$47$	0	0	−1.00000			$\pi$
$48$	0	0
$49$	−9.00000	−1.28571
$50$	0	0
$51$	0	0
$52$	2.00000i	0.277350i
$53$	− 6.00000i	− 0.824163i	−0.911147	−	0.412082i	$-0.864802\pi$
$53$	− 6.00000i	− 0.824163i	0.911147	−	0.412082i	$-0.135198\pi$
$54$	0	0
$55$	0	0
$56$	12.0000	1.60357
$57$	0	0
$58$	6.00000i	0.787839i
$59$	12.0000	1.56227	0.781133	−	0.624364i	$-0.214642\pi$
$59$	12.0000	1.56227	0.781133	+	0.624364i	$0.214642\pi$
$60$	0	0
$61$	−10.0000	−1.28037	−0.640184	−	0.768221i	$-0.721142\pi$
$61$	−10.0000	−1.28037	−0.640184	+	0.768221i	$0.721142\pi$
$62$	− 4.00000i	− 0.508001i
$63$	12.0000i	1.51186i
$64$	−7.00000	−0.875000
$65$	0	0
$66$	0	0
$67$	4.00000i	0.488678i	0.969690	+	0.244339i	$0.0785709\pi$
$67$	4.00000i	0.488678i	−0.969690	+	0.244339i	$0.921429\pi$
$68$	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	$0.576281\pi$
$72$	− 9.00000i	− 1.06066i
$73$	6.00000i	0.702247i	0.936329	+	0.351123i	$0.114200\pi$
$73$	6.00000i	0.702247i	−0.936329	+	0.351123i	$0.885800\pi$
$74$	−2.00000	−0.232495
$75$	0	0
$76$	4.00000	0.458831
$77$	0	0
$78$	0	0
$79$	−12.0000	−1.35011	−0.675053	−	0.737769i	$-0.735879\pi$
$79$	−12.0000	−1.35011	−0.675053	+	0.737769i	$0.735879\pi$
$80$	0	0
$81$	9.00000	1.00000
$82$	6.00000i	0.662589i
$83$	4.00000i	0.439057i	0.975606	+	0.219529i	$0.0704519\pi$
$83$	4.00000i	0.439057i	−0.975606	+	0.219529i	$0.929548\pi$
$84$	0	0
$85$	0	0
$86$	−4.00000	−0.431331
$87$	0	0
$88$	0	0
$89$	−10.0000	−1.06000	−0.529999	−	0.847998i	$-0.677808\pi$
$89$	−10.0000	−1.06000	−0.529999	+	0.847998i	$0.677808\pi$
$90$	0	0
$91$	−8.00000	−0.838628
$92$	− 4.00000i	− 0.417029i
$93$	0	0
$94$	0	0
$95$	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	$0.967625\pi$
$98$	9.00000i	0.909137i
$99$	0	0
$100$	0	0
$101$	−10.0000	−0.995037	−0.497519	−	0.867453i	$-0.665755\pi$
$101$	−10.0000	−0.995037	−0.497519	+	0.867453i	$0.665755\pi$
$102$	0	0
$103$	− 8.00000i	− 0.788263i	−0.919054	−	0.394132i	$-0.871045\pi$
$103$	− 8.00000i	− 0.788263i	0.919054	−	0.394132i	$-0.128955\pi$
$104$	6.00000	0.588348
$105$	0	0
$106$	−6.00000	−0.582772
$107$	8.00000i	0.773389i	0.922208	+	0.386695i	$0.126383\pi$
$107$	8.00000i	0.773389i	−0.922208	+	0.386695i	$0.873617\pi$
$108$	0	0
$109$	−6.00000	−0.574696	−0.287348	−	0.957826i	$-0.592774\pi$
$109$	−6.00000	−0.574696	−0.287348	+	0.957826i	$0.592774\pi$
$110$	0	0
$111$	0	0
$112$	− 4.00000i	− 0.377964i
$113$	14.0000i	1.31701i	0.752577	+	0.658505i	$0.228811\pi$
$113$	14.0000i	1.31701i	−0.752577	+	0.658505i	$0.771189\pi$
$114$	0	0
$115$	0	0
$116$	−6.00000	−0.557086
$117$	6.00000i	0.554700i
$118$	− 12.0000i	− 1.10469i
$119$	−4.00000	−0.366679
$120$	0	0
$121$	−11.0000	−1.00000
$122$	10.0000i	0.905357i
$123$	0	0
$124$	4.00000	0.359211
$125$	0	0
$126$	12.0000	1.06904
$127$	8.00000i	0.709885i	0.934888	+	0.354943i	$0.115500\pi$
$127$	8.00000i	0.709885i	−0.934888	+	0.354943i	$0.884500\pi$
$128$	− 3.00000i	− 0.265165i
$129$	0	0
$130$	0	0
$131$	16.0000	1.39793	0.698963	−	0.715158i	$-0.253645\pi$
$131$	16.0000	1.39793	0.698963	+	0.715158i	$0.253645\pi$
$132$	0	0
$133$	16.0000i	1.38738i
$134$	4.00000	0.345547
$135$	0	0
$136$	3.00000	0.257248
$137$	− 6.00000i	− 0.512615i	−0.966595	−	0.256307i	$-0.917494\pi$
$137$	− 6.00000i	− 0.512615i	0.966595	−	0.256307i	$-0.0825059\pi$
$138$	0	0
$139$	8.00000	0.678551	0.339276	−	0.940687i	$-0.389818\pi$
$139$	8.00000	0.678551	0.339276	+	0.940687i	$0.389818\pi$
$140$	0	0
$141$	0	0
$142$	4.00000i	0.335673i
$143$	0	0
$144$	−3.00000	−0.250000
$145$	0	0
$146$	6.00000	0.496564
$147$	0	0
$148$	− 2.00000i	− 0.164399i
$149$	10.0000	0.819232	0.409616	−	0.912258i	$-0.365663\pi$
$149$	10.0000	0.819232	0.409616	+	0.912258i	$0.365663\pi$
$150$	0	0
$151$	−16.0000	−1.30206	−0.651031	−	0.759051i	$-0.725663\pi$
$151$	−16.0000	−1.30206	−0.651031	+	0.759051i	$0.725663\pi$
$152$	− 12.0000i	− 0.973329i
$153$	3.00000i	0.242536i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 2.00000i	− 0.159617i	−0.996810	−	0.0798087i	$-0.974569\pi$
$157$	− 2.00000i	− 0.159617i	0.996810	−	0.0798087i	$-0.0254309\pi$
$158$	12.0000i	0.954669i
$159$	0	0
$160$	0	0
$161$	16.0000	1.26098
$162$	− 9.00000i	− 0.707107i
$163$	− 24.0000i	− 1.87983i	−0.341415	−	0.939913i	$-0.610906\pi$
$163$	− 24.0000i	− 1.87983i	0.341415	−	0.939913i	$-0.389094\pi$
$164$	−6.00000	−0.468521
$165$	0	0
$166$	4.00000	0.310460
$167$	− 4.00000i	− 0.309529i	−0.987951	−	0.154765i	$-0.950538\pi$
$167$	− 4.00000i	− 0.309529i	0.987951	−	0.154765i	$-0.0494619\pi$
$168$	0	0
$169$	9.00000	0.692308
$170$	0	0
$171$	12.0000	0.917663
$172$	− 4.00000i	− 0.304997i
$173$	− 22.0000i	− 1.67263i	−0.548250	−	0.836315i	$-0.684706\pi$
$173$	− 22.0000i	− 1.67263i	0.548250	−	0.836315i	$-0.315294\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	10.0000i	0.749532i
$179$	−12.0000	−0.896922	−0.448461	−	0.893802i	$-0.648028\pi$
$179$	−12.0000	−0.896922	−0.448461	+	0.893802i	$0.648028\pi$
$180$	0	0
$181$	−2.00000	−0.148659	−0.0743294	−	0.997234i	$-0.523682\pi$
$181$	−2.00000	−0.148659	−0.0743294	+	0.997234i	$0.523682\pi$
$182$	8.00000i	0.592999i
$183$	0	0
$184$	−12.0000	−0.884652
$185$	0	0
$186$	0	0
$187$	0	0
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−16.0000	−1.15772	−0.578860	−	0.815427i	$-0.696502\pi$
$191$	−16.0000	−1.15772	−0.578860	+	0.815427i	$0.696502\pi$
$192$	0	0
$193$	− 2.00000i	− 0.143963i	−0.997406	−	0.0719816i	$-0.977068\pi$
$193$	− 2.00000i	− 0.143963i	0.997406	−	0.0719816i	$-0.0229323\pi$
$194$	2.00000	0.143592
$195$	0	0
$196$	−9.00000	−0.642857
$197$	− 18.0000i	− 1.28245i	−0.767354	−	0.641223i	$-0.778427\pi$
$197$	− 18.0000i	− 1.28245i	0.767354	−	0.641223i	$-0.221573\pi$
$198$	0	0
$199$	20.0000	1.41776	0.708881	−	0.705328i	$-0.249200\pi$
$199$	20.0000	1.41776	0.708881	+	0.705328i	$0.249200\pi$
$200$	0	0
$201$	0	0
$202$	10.0000i	0.703598i
$203$	− 24.0000i	− 1.68447i
$204$	0	0
$205$	0	0
$206$	−8.00000	−0.557386
$207$	− 12.0000i	− 0.834058i
$208$	− 2.00000i	− 0.138675i
$209$	0	0
$210$	0	0
$211$	8.00000	0.550743	0.275371	−	0.961338i	$-0.411199\pi$
$211$	8.00000	0.550743	0.275371	+	0.961338i	$0.411199\pi$
$212$	− 6.00000i	− 0.412082i
$213$	0	0
$214$	8.00000	0.546869
$215$	0	0
$216$	0	0
$217$	16.0000i	1.08615i
$218$	6.00000i	0.406371i
$219$	0	0
$220$	0	0
$221$	−2.00000	−0.134535
$222$	0	0
$223$	− 24.0000i	− 1.60716i	−0.595198	−	0.803579i	$-0.702926\pi$
$223$	− 24.0000i	− 1.60716i	0.595198	−	0.803579i	$-0.297074\pi$
$224$	20.0000	1.33631
$225$	0	0
$226$	14.0000	0.931266
$227$	− 24.0000i	− 1.59294i	−0.604681	−	0.796468i	$-0.706699\pi$
$227$	− 24.0000i	− 1.59294i	0.604681	−	0.796468i	$-0.293301\pi$
$228$	0	0
$229$	−6.00000	−0.396491	−0.198246	−	0.980152i	$-0.563524\pi$
$229$	−6.00000	−0.396491	−0.198246	+	0.980152i	$0.563524\pi$
$230$	0	0
$231$	0	0
$232$	18.0000i	1.18176i
$233$	6.00000i	0.393073i	0.980497	+	0.196537i	$0.0629694\pi$
$233$	6.00000i	0.393073i	−0.980497	+	0.196537i	$0.937031\pi$
$234$	6.00000	0.392232
$235$	0	0
$236$	12.0000	0.781133
$237$	0	0
$238$	4.00000i	0.259281i
$239$	16.0000	1.03495	0.517477	−	0.855697i	$-0.326871\pi$
$239$	16.0000	1.03495	0.517477	+	0.855697i	$0.326871\pi$
$240$	0	0
$241$	18.0000	1.15948	0.579741	−	0.814801i	$-0.303154\pi$
$241$	18.0000	1.15948	0.579741	+	0.814801i	$0.303154\pi$
$242$	11.0000i	0.707107i
$243$	0	0
$244$	−10.0000	−0.640184
$245$	0	0
$246$	0	0
$247$	8.00000i	0.509028i
$248$	− 12.0000i	− 0.762001i
$249$	0	0
$250$	0	0
$251$	12.0000	0.757433	0.378717	−	0.925513i	$-0.376365\pi$
$251$	12.0000	0.757433	0.378717	+	0.925513i	$0.376365\pi$
$252$	12.0000i	0.755929i
$253$	0	0
$254$	8.00000	0.501965
$255$	0	0
$256$	−17.0000	−1.06250
$257$	18.0000i	1.12281i	0.827541	+	0.561405i	$0.189739\pi$
$257$	18.0000i	1.12281i	−0.827541	+	0.561405i	$0.810261\pi$
$258$	0	0
$259$	8.00000	0.497096
$260$	0	0
$261$	−18.0000	−1.11417
$262$	− 16.0000i	− 0.988483i
$263$	16.0000i	0.986602i	0.869859	+	0.493301i	$0.164210\pi$
$263$	16.0000i	0.986602i	−0.869859	+	0.493301i	$0.835790\pi$
$264$	0	0
$265$	0	0
$266$	16.0000	0.981023
$267$	0	0
$268$	4.00000i	0.244339i
$269$	−22.0000	−1.34136	−0.670682	−	0.741745i	$-0.733998\pi$
$269$	−22.0000	−1.34136	−0.670682	+	0.741745i	$0.733998\pi$
$270$	0	0
$271$	−16.0000	−0.971931	−0.485965	−	0.873978i	$-0.661532\pi$
$271$	−16.0000	−0.971931	−0.485965	+	0.873978i	$0.661532\pi$
$272$	− 1.00000i	− 0.0606339i
$273$	0	0
$274$	−6.00000	−0.362473
$275$	0	0
$276$	0	0
$277$	14.0000i	0.841178i	0.907251	+	0.420589i	$0.138177\pi$
$277$	14.0000i	0.841178i	−0.907251	+	0.420589i	$0.861823\pi$
$278$	− 8.00000i	− 0.479808i
$279$	12.0000	0.718421
$280$	0	0
$281$	−6.00000	−0.357930	−0.178965	−	0.983855i	$-0.557275\pi$
$281$	−6.00000	−0.357930	−0.178965	+	0.983855i	$0.557275\pi$
$282$	0	0
$283$	16.0000i	0.951101i	0.879688	+	0.475551i	$0.157751\pi$
$283$	16.0000i	0.951101i	−0.879688	+	0.475551i	$0.842249\pi$
$284$	−4.00000	−0.237356
$285$	0	0
$286$	0	0
$287$	− 24.0000i	− 1.41668i
$288$	− 15.0000i	− 0.883883i
$289$	−1.00000	−0.0588235
$290$	0	0
$291$	0	0
$292$	6.00000i	0.351123i
$293$	− 6.00000i	− 0.350524i	−0.984522	−	0.175262i	$-0.943923\pi$
$293$	− 6.00000i	− 0.350524i	0.984522	−	0.175262i	$-0.0560772\pi$
$294$	0	0
$295$	0	0
$296$	−6.00000	−0.348743
$297$	0	0
$298$	− 10.0000i	− 0.579284i
$299$	8.00000	0.462652
$300$	0	0
$301$	16.0000	0.922225
$302$	16.0000i	0.920697i
$303$	0	0
$304$	−4.00000	−0.229416
$305$	0	0
$306$	3.00000	0.171499
$307$	− 12.0000i	− 0.684876i	−0.939540	−	0.342438i	$-0.888747\pi$
$307$	− 12.0000i	− 0.684876i	0.939540	−	0.342438i	$-0.111253\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	28.0000	1.58773	0.793867	−	0.608091i	$-0.208065\pi$
$311$	28.0000	1.58773	0.793867	+	0.608091i	$0.208065\pi$
$312$	0	0
$313$	22.0000i	1.24351i	0.783210	+	0.621757i	$0.213581\pi$
$313$	22.0000i	1.24351i	−0.783210	+	0.621757i	$0.786419\pi$
$314$	−2.00000	−0.112867
$315$	0	0
$316$	−12.0000	−0.675053
$317$	− 10.0000i	− 0.561656i	−0.959758	−	0.280828i	$-0.909391\pi$
$317$	− 10.0000i	− 0.561656i	0.959758	−	0.280828i	$-0.0906090\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	0	0
$322$	− 16.0000i	− 0.891645i
$323$	4.00000i	0.222566i
$324$	9.00000	0.500000
$325$	0	0
$326$	−24.0000	−1.32924
$327$	0	0
$328$	18.0000i	0.993884i
$329$	0	0
$330$	0	0
$331$	4.00000	0.219860	0.109930	−	0.993939i	$-0.464937\pi$
$331$	4.00000	0.219860	0.109930	+	0.993939i	$0.464937\pi$
$332$	4.00000i	0.219529i
$333$	− 6.00000i	− 0.328798i
$334$	−4.00000	−0.218870
$335$	0	0
$336$	0	0
$337$	− 14.0000i	− 0.762629i	−0.924445	−	0.381314i	$-0.875472\pi$
$337$	− 14.0000i	− 0.762629i	0.924445	−	0.381314i	$-0.124528\pi$
$338$	− 9.00000i	− 0.489535i
$339$	0	0
$340$	0	0
$341$	0	0
$342$	− 12.0000i	− 0.648886i
$343$	− 8.00000i	− 0.431959i
$344$	−12.0000	−0.646997
$345$	0	0
$346$	−22.0000	−1.18273
$347$	32.0000i	1.71785i	0.512101	+	0.858925i	$0.328867\pi$
$347$	32.0000i	1.71785i	−0.512101	+	0.858925i	$0.671133\pi$
$348$	0	0
$349$	18.0000	0.963518	0.481759	−	0.876304i	$-0.339998\pi$
$349$	18.0000	0.963518	0.481759	+	0.876304i	$0.339998\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	30.0000i	1.59674i	0.602168	+	0.798369i	$0.294304\pi$
$353$	30.0000i	1.59674i	−0.602168	+	0.798369i	$0.705696\pi$
$354$	0	0
$355$	0	0
$356$	−10.0000	−0.529999
$357$	0	0
$358$	12.0000i	0.634220i
$359$	0	0		−	1.00000i	$-0.5\pi$
$359$	0	0			1.00000i	$0.5\pi$
$360$	0	0
$361$	−3.00000	−0.157895
$362$	2.00000i	0.105118i
$363$	0	0
$364$	−8.00000	−0.419314
$365$	0	0
$366$	0	0
$367$	28.0000i	1.46159i	0.682598	+	0.730794i	$0.260850\pi$
$367$	28.0000i	1.46159i	−0.682598	+	0.730794i	$0.739150\pi$
$368$	4.00000i	0.208514i
$369$	−18.0000	−0.937043
$370$	0	0
$371$	24.0000	1.24602
$372$	0	0
$373$	− 6.00000i	− 0.310668i	−0.987862	−	0.155334i	$-0.950355\pi$
$373$	− 6.00000i	− 0.310668i	0.987862	−	0.155334i	$-0.0496454\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 12.0000i	− 0.618031i
$378$	0	0
$379$	8.00000	0.410932	0.205466	−	0.978664i	$-0.434129\pi$
$379$	8.00000	0.410932	0.205466	+	0.978664i	$0.434129\pi$
$380$	0	0
$381$	0	0
$382$	16.0000i	0.818631i
$383$	24.0000i	1.22634i	0.789950	+	0.613171i	$0.210106\pi$
$383$	24.0000i	1.22634i	−0.789950	+	0.613171i	$0.789894\pi$
$384$	0	0
$385$	0	0
$386$	−2.00000	−0.101797
$387$	− 12.0000i	− 0.609994i
$388$	2.00000i	0.101535i
$389$	−6.00000	−0.304212	−0.152106	−	0.988364i	$-0.548606\pi$
$389$	−6.00000	−0.304212	−0.152106	+	0.988364i	$0.548606\pi$
$390$	0	0
$391$	4.00000	0.202289
$392$	27.0000i	1.36371i
$393$	0	0
$394$	−18.0000	−0.906827
$395$	0	0
$396$	0	0
$397$	6.00000i	0.301131i	0.988600	+	0.150566i	$0.0481095\pi$
$397$	6.00000i	0.301131i	−0.988600	+	0.150566i	$0.951890\pi$
$398$	− 20.0000i	− 1.00251i
$399$	0	0
$400$	0	0
$401$	−14.0000	−0.699127	−0.349563	−	0.936913i	$-0.613670\pi$
$401$	−14.0000	−0.699127	−0.349563	+	0.936913i	$0.613670\pi$
$402$	0	0
$403$	8.00000i	0.398508i
$404$	−10.0000	−0.497519
$405$	0	0
$406$	−24.0000	−1.19110
$407$	0	0
$408$	0	0
$409$	−26.0000	−1.28562	−0.642809	−	0.766027i	$-0.722231\pi$
$409$	−26.0000	−1.28562	−0.642809	+	0.766027i	$0.722231\pi$
$410$	0	0
$411$	0	0
$412$	− 8.00000i	− 0.394132i
$413$	48.0000i	2.36193i
$414$	−12.0000	−0.589768
$415$	0	0
$416$	10.0000	0.490290
$417$	0	0
$418$	0	0
$419$	−8.00000	−0.390826	−0.195413	−	0.980721i	$-0.562605\pi$
$419$	−8.00000	−0.390826	−0.195413	+	0.980721i	$0.562605\pi$
$420$	0	0
$421$	22.0000	1.07221	0.536107	−	0.844150i	$-0.319894\pi$
$421$	22.0000	1.07221	0.536107	+	0.844150i	$0.319894\pi$
$422$	− 8.00000i	− 0.389434i
$423$	0	0
$424$	−18.0000	−0.874157
$425$	0	0
$426$	0	0
$427$	− 40.0000i	− 1.93574i
$428$	8.00000i	0.386695i
$429$	0	0
$430$	0	0
$431$	12.0000	0.578020	0.289010	−	0.957326i	$-0.406674\pi$
$431$	12.0000	0.578020	0.289010	+	0.957326i	$0.406674\pi$
$432$	0	0
$433$	− 2.00000i	− 0.0961139i	−0.998845	−	0.0480569i	$-0.984697\pi$
$433$	− 2.00000i	− 0.0961139i	0.998845	−	0.0480569i	$-0.0153029\pi$
$434$	16.0000	0.768025
$435$	0	0
$436$	−6.00000	−0.287348
$437$	− 16.0000i	− 0.765384i
$438$	0	0
$439$	20.0000	0.954548	0.477274	−	0.878755i	$-0.341625\pi$
$439$	20.0000	0.954548	0.477274	+	0.878755i	$0.341625\pi$
$440$	0	0
$441$	−27.0000	−1.28571
$442$	2.00000i	0.0951303i
$443$	− 28.0000i	− 1.33032i	−0.746701	−	0.665160i	$-0.768363\pi$
$443$	− 28.0000i	− 1.33032i	0.746701	−	0.665160i	$-0.231637\pi$
$444$	0	0
$445$	0	0
$446$	−24.0000	−1.13643
$447$	0	0
$448$	− 28.0000i	− 1.32288i
$449$	−34.0000	−1.60456	−0.802280	−	0.596948i	$-0.796380\pi$
$449$	−34.0000	−1.60456	−0.802280	+	0.596948i	$0.796380\pi$
$450$	0	0
$451$	0	0
$452$	14.0000i	0.658505i
$453$	0	0
$454$	−24.0000	−1.12638
$455$	0	0
$456$	0	0
$457$	− 6.00000i	− 0.280668i	−0.990104	−	0.140334i	$-0.955182\pi$
$457$	− 6.00000i	− 0.280668i	0.990104	−	0.140334i	$-0.0448177\pi$
$458$	6.00000i	0.280362i
$459$	0	0
$460$	0	0
$461$	−2.00000	−0.0931493	−0.0465746	−	0.998915i	$-0.514831\pi$
$461$	−2.00000	−0.0931493	−0.0465746	+	0.998915i	$0.514831\pi$
$462$	0	0
$463$	− 32.0000i	− 1.48717i	−0.668644	−	0.743583i	$-0.733125\pi$
$463$	− 32.0000i	− 1.48717i	0.668644	−	0.743583i	$-0.266875\pi$
$464$	6.00000	0.278543
$465$	0	0
$466$	6.00000	0.277945
$467$	12.0000i	0.555294i	0.960683	+	0.277647i	$0.0895545\pi$
$467$	12.0000i	0.555294i	−0.960683	+	0.277647i	$0.910445\pi$
$468$	6.00000i	0.277350i
$469$	−16.0000	−0.738811
$470$	0	0
$471$	0	0
$472$	− 36.0000i	− 1.65703i
$473$	0	0
$474$	0	0
$475$	0	0
$476$	−4.00000	−0.183340
$477$	− 18.0000i	− 0.824163i
$478$	− 16.0000i	− 0.731823i
$479$	−36.0000	−1.64488	−0.822441	−	0.568850i	$-0.807388\pi$
$479$	−36.0000	−1.64488	−0.822441	+	0.568850i	$0.807388\pi$
$480$	0	0
$481$	4.00000	0.182384
$482$	− 18.0000i	− 0.819878i
$483$	0	0
$484$	−11.0000	−0.500000
$485$	0	0
$486$	0	0
$487$	20.0000i	0.906287i	0.891438	+	0.453143i	$0.149697\pi$
$487$	20.0000i	0.906287i	−0.891438	+	0.453143i	$0.850303\pi$
$488$	30.0000i	1.35804i
$489$	0	0
$490$	0	0
$491$	20.0000	0.902587	0.451294	−	0.892375i	$-0.350963\pi$
$491$	20.0000	0.902587	0.451294	+	0.892375i	$0.350963\pi$
$492$	0	0
$493$	− 6.00000i	− 0.270226i
$494$	8.00000	0.359937
$495$	0	0
$496$	−4.00000	−0.179605
$497$	− 16.0000i	− 0.717698i
$498$	0	0
$499$	40.0000	1.79065	0.895323	−	0.445418i	$-0.146945\pi$
$499$	40.0000	1.79065	0.895323	+	0.445418i	$0.146945\pi$
$500$	0	0
$501$	0	0
$502$	− 12.0000i	− 0.535586i
$503$	12.0000i	0.535054i	0.963550	+	0.267527i	$0.0862064\pi$
$503$	12.0000i	0.535054i	−0.963550	+	0.267527i	$0.913794\pi$
$504$	36.0000	1.60357
$505$	0	0
$506$	0	0
$507$	0	0
$508$	8.00000i	0.354943i
$509$	2.00000	0.0886484	0.0443242	−	0.999017i	$-0.485887\pi$
$509$	2.00000	0.0886484	0.0443242	+	0.999017i	$0.485887\pi$
$510$	0	0
$511$	−24.0000	−1.06170
$512$	11.0000i	0.486136i
$513$	0	0
$514$	18.0000	0.793946
$515$	0	0
$516$	0	0
$517$	0	0
$518$	− 8.00000i	− 0.351500i
$519$	0	0
$520$	0	0
$521$	26.0000	1.13908	0.569540	−	0.821963i	$-0.307121\pi$
$521$	26.0000	1.13908	0.569540	+	0.821963i	$0.307121\pi$
$522$	18.0000i	0.787839i
$523$	36.0000i	1.57417i	0.616844	+	0.787085i	$0.288411\pi$
$523$	36.0000i	1.57417i	−0.616844	+	0.787085i	$0.711589\pi$
$524$	16.0000	0.698963
$525$	0	0
$526$	16.0000	0.697633
$527$	4.00000i	0.174243i
$528$	0	0
$529$	7.00000	0.304348
$530$	0	0
$531$	36.0000	1.56227
$532$	16.0000i	0.693688i
$533$	− 12.0000i	− 0.519778i
$534$	0	0
$535$	0	0
$536$	12.0000	0.518321
$537$	0	0
$538$	22.0000i	0.948487i
$539$	0	0
$540$	0	0
$541$	6.00000	0.257960	0.128980	−	0.991647i	$-0.458830\pi$
$541$	6.00000	0.257960	0.128980	+	0.991647i	$0.458830\pi$
$542$	16.0000i	0.687259i
$543$	0	0
$544$	5.00000	0.214373
$545$	0	0
$546$	0	0
$547$	− 32.0000i	− 1.36822i	−0.729378	−	0.684111i	$-0.760191\pi$
$547$	− 32.0000i	− 1.36822i	0.729378	−	0.684111i	$-0.239809\pi$
$548$	− 6.00000i	− 0.256307i
$549$	−30.0000	−1.28037
$550$	0	0
$551$	−24.0000	−1.02243
$552$	0	0
$553$	− 48.0000i	− 2.04117i
$554$	14.0000	0.594803
$555$	0	0
$556$	8.00000	0.339276
$557$	30.0000i	1.27114i	0.772043	+	0.635570i	$0.219235\pi$
$557$	30.0000i	1.27114i	−0.772043	+	0.635570i	$0.780765\pi$
$558$	− 12.0000i	− 0.508001i
$559$	8.00000	0.338364
$560$	0	0
$561$	0	0
$562$	6.00000i	0.253095i
$563$	4.00000i	0.168580i	0.996441	+	0.0842900i	$0.0268622\pi$
$563$	4.00000i	0.168580i	−0.996441	+	0.0842900i	$0.973138\pi$
$564$	0	0
$565$	0	0
$566$	16.0000	0.672530
$567$	36.0000i	1.51186i
$568$	12.0000i	0.503509i
$569$	38.0000	1.59304	0.796521	−	0.604610i	$-0.206671\pi$
$569$	38.0000	1.59304	0.796521	+	0.604610i	$0.206671\pi$
$570$	0	0
$571$	−32.0000	−1.33916	−0.669579	−	0.742741i	$-0.733526\pi$
$571$	−32.0000	−1.33916	−0.669579	+	0.742741i	$0.733526\pi$
$572$	0	0
$573$	0	0
$574$	−24.0000	−1.00174
$575$	0	0
$576$	−21.0000	−0.875000
$577$	− 14.0000i	− 0.582828i	−0.956597	−	0.291414i	$-0.905874\pi$
$577$	− 14.0000i	− 0.582828i	0.956597	−	0.291414i	$-0.0941257\pi$
$578$	1.00000i	0.0415945i
$579$	0	0
$580$	0	0
$581$	−16.0000	−0.663792
$582$	0	0
$583$	0	0
$584$	18.0000	0.744845
$585$	0	0
$586$	−6.00000	−0.247858
$587$	4.00000i	0.165098i	0.996587	+	0.0825488i	$0.0263060\pi$
$587$	4.00000i	0.165098i	−0.996587	+	0.0825488i	$0.973694\pi$
$588$	0	0
$589$	16.0000	0.659269
$590$	0	0
$591$	0	0
$592$	2.00000i	0.0821995i
$593$	− 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
$596$	10.0000	0.409616
$597$	0	0
$598$	− 8.00000i	− 0.327144i
$599$	24.0000	0.980613	0.490307	−	0.871550i	$-0.336885\pi$
$599$	24.0000	0.980613	0.490307	+	0.871550i	$0.336885\pi$
$600$	0	0
$601$	10.0000	0.407909	0.203954	−	0.978980i	$-0.434621\pi$
$601$	10.0000	0.407909	0.203954	+	0.978980i	$0.434621\pi$
$602$	− 16.0000i	− 0.652111i
$603$	12.0000i	0.488678i
$604$	−16.0000	−0.651031
$605$	0	0
$606$	0	0
$607$	20.0000i	0.811775i	0.913923	+	0.405887i	$0.133038\pi$
$607$	20.0000i	0.811775i	−0.913923	+	0.405887i	$0.866962\pi$
$608$	− 20.0000i	− 0.811107i
$609$	0	0
$610$	0	0
$611$	0	0
$612$	3.00000i	0.121268i
$613$	26.0000i	1.05013i	0.851062	+	0.525065i	$0.175959\pi$
$613$	26.0000i	1.05013i	−0.851062	+	0.525065i	$0.824041\pi$
$614$	−12.0000	−0.484281
$615$	0	0
$616$	0	0
$617$	− 6.00000i	− 0.241551i	−0.992680	−	0.120775i	$-0.961462\pi$
$617$	− 6.00000i	− 0.241551i	0.992680	−	0.120775i	$-0.0385381\pi$
$618$	0	0
$619$	48.0000	1.92928	0.964641	−	0.263566i	$-0.0848986\pi$
$619$	48.0000	1.92928	0.964641	+	0.263566i	$0.0848986\pi$
$620$	0	0
$621$	0	0
$622$	− 28.0000i	− 1.12270i
$623$	− 40.0000i	− 1.60257i
$624$	0	0
$625$	0	0
$626$	22.0000	0.879297
$627$	0	0
$628$	− 2.00000i	− 0.0798087i
$629$	2.00000	0.0797452
$630$	0	0
$631$	16.0000	0.636950	0.318475	−	0.947931i	$-0.396829\pi$
$631$	16.0000	0.636950	0.318475	+	0.947931i	$0.396829\pi$
$632$	36.0000i	1.43200i
$633$	0	0
$634$	−10.0000	−0.397151
$635$	0	0
$636$	0	0
$637$	− 18.0000i	− 0.713186i
$638$	0	0
$639$	−12.0000	−0.474713
$640$	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
$643$	− 32.0000i	− 1.26196i	−0.775800	−	0.630978i	$-0.782654\pi$
$643$	− 32.0000i	− 1.26196i	0.775800	−	0.630978i	$-0.217346\pi$
$644$	16.0000	0.630488
$645$	0	0
$646$	4.00000	0.157378
$647$	8.00000i	0.314512i	0.987558	+	0.157256i	$0.0502649\pi$
$647$	8.00000i	0.314512i	−0.987558	+	0.157256i	$0.949735\pi$
$648$	− 27.0000i	− 1.06066i
$649$	0	0
$650$	0	0
$651$	0	0
$652$	− 24.0000i	− 0.939913i
$653$	− 6.00000i	− 0.234798i	−0.993085	−	0.117399i	$-0.962544\pi$
$653$	− 6.00000i	− 0.234798i	0.993085	−	0.117399i	$-0.0374557\pi$
$654$	0	0
$655$	0	0
$656$	6.00000	0.234261
$657$	18.0000i	0.702247i
$658$	0	0
$659$	−4.00000	−0.155818	−0.0779089	−	0.996960i	$-0.524824\pi$
$659$	−4.00000	−0.155818	−0.0779089	+	0.996960i	$0.524824\pi$
$660$	0	0
$661$	38.0000	1.47803	0.739014	−	0.673690i	$-0.235292\pi$
$661$	38.0000	1.47803	0.739014	+	0.673690i	$0.235292\pi$
$662$	− 4.00000i	− 0.155464i
$663$	0	0
$664$	12.0000	0.465690
$665$	0	0
$666$	−6.00000	−0.232495
$667$	24.0000i	0.929284i
$668$	− 4.00000i	− 0.154765i
$669$	0	0
$670$	0	0
$671$	0	0
$672$	0	0
$673$	− 2.00000i	− 0.0770943i	−0.999257	−	0.0385472i	$-0.987727\pi$
$673$	− 2.00000i	− 0.0770943i	0.999257	−	0.0385472i	$-0.0122730\pi$
$674$	−14.0000	−0.539260
$675$	0	0
$676$	9.00000	0.346154
$677$	30.0000i	1.15299i	0.817099	+	0.576497i	$0.195581\pi$
$677$	30.0000i	1.15299i	−0.817099	+	0.576497i	$0.804419\pi$
$678$	0	0
$679$	−8.00000	−0.307012
$680$	0	0
$681$	0	0
$682$	0	0
$683$	40.0000i	1.53056i	0.643699	+	0.765279i	$0.277399\pi$
$683$	40.0000i	1.53056i	−0.643699	+	0.765279i	$0.722601\pi$
$684$	12.0000	0.458831
$685$	0	0
$686$	−8.00000	−0.305441
$687$	0	0
$688$	4.00000i	0.152499i
$689$	12.0000	0.457164
$690$	0	0
$691$	−8.00000	−0.304334	−0.152167	−	0.988355i	$-0.548625\pi$
$691$	−8.00000	−0.304334	−0.152167	+	0.988355i	$0.548625\pi$
$692$	− 22.0000i	− 0.836315i
$693$	0	0
$694$	32.0000	1.21470
$695$	0	0
$696$	0	0
$697$	− 6.00000i	− 0.227266i
$698$	− 18.0000i	− 0.681310i
$699$	0	0
$700$	0	0
$701$	−18.0000	−0.679851	−0.339925	−	0.940452i	$-0.610402\pi$
$701$	−18.0000	−0.679851	−0.339925	+	0.940452i	$0.610402\pi$
$702$	0	0
$703$	− 8.00000i	− 0.301726i
$704$	0	0
$705$	0	0
$706$	30.0000	1.12906
$707$	− 40.0000i	− 1.50435i
$708$	0	0
$709$	34.0000	1.27690	0.638448	−	0.769665i	$-0.279577\pi$
$709$	34.0000	1.27690	0.638448	+	0.769665i	$0.279577\pi$
$710$	0	0
$711$	−36.0000	−1.35011
$712$	30.0000i	1.12430i
$713$	− 16.0000i	− 0.599205i
$714$	0	0
$715$	0	0
$716$	−12.0000	−0.448461
$717$	0	0
$718$	0	0
$719$	−4.00000	−0.149175	−0.0745874	−	0.997214i	$-0.523764\pi$
$719$	−4.00000	−0.149175	−0.0745874	+	0.997214i	$0.523764\pi$
$720$	0	0
$721$	32.0000	1.19174
$722$	3.00000i	0.111648i
$723$	0	0
$724$	−2.00000	−0.0743294
$725$	0	0
$726$	0	0
$727$	40.0000i	1.48352i	0.670667	+	0.741759i	$0.266008\pi$
$727$	40.0000i	1.48352i	−0.670667	+	0.741759i	$0.733992\pi$
$728$	24.0000i	0.889499i
$729$	27.0000	1.00000
$730$	0	0
$731$	4.00000	0.147945
$732$	0	0
$733$	50.0000i	1.84679i	0.383849	+	0.923396i	$0.374598\pi$
$733$	50.0000i	1.84679i	−0.383849	+	0.923396i	$0.625402\pi$
$734$	28.0000	1.03350
$735$	0	0
$736$	−20.0000	−0.737210
$737$	0	0
$738$	18.0000i	0.662589i
$739$	−28.0000	−1.03000	−0.514998	−	0.857191i	$-0.672207\pi$
$739$	−28.0000	−1.03000	−0.514998	+	0.857191i	$0.672207\pi$
$740$	0	0
$741$	0	0
$742$	− 24.0000i	− 0.881068i
$743$	− 12.0000i	− 0.440237i	−0.975473	−	0.220119i	$-0.929356\pi$
$743$	− 12.0000i	− 0.440237i	0.975473	−	0.220119i	$-0.0706445\pi$
$744$	0	0
$745$	0	0
$746$	−6.00000	−0.219676
$747$	12.0000i	0.439057i
$748$	0	0
$749$	−32.0000	−1.16925
$750$	0	0
$751$	20.0000	0.729810	0.364905	−	0.931045i	$-0.381101\pi$
$751$	20.0000	0.729810	0.364905	+	0.931045i	$0.381101\pi$
$752$	0	0
$753$	0	0
$754$	−12.0000	−0.437014
$755$	0	0
$756$	0	0
$757$	22.0000i	0.799604i	0.916602	+	0.399802i	$0.130921\pi$
$757$	22.0000i	0.799604i	−0.916602	+	0.399802i	$0.869079\pi$
$758$	− 8.00000i	− 0.290573i
$759$	0	0
$760$	0	0
$761$	−22.0000	−0.797499	−0.398750	−	0.917060i	$-0.630556\pi$
$761$	−22.0000	−0.797499	−0.398750	+	0.917060i	$0.630556\pi$
$762$	0	0
$763$	− 24.0000i	− 0.868858i
$764$	−16.0000	−0.578860
$765$	0	0
$766$	24.0000	0.867155
$767$	24.0000i	0.866590i
$768$	0	0
$769$	14.0000	0.504853	0.252426	−	0.967616i	$-0.418771\pi$
$769$	14.0000	0.504853	0.252426	+	0.967616i	$0.418771\pi$
$770$	0	0
$771$	0	0
$772$	− 2.00000i	− 0.0719816i
$773$	26.0000i	0.935155i	0.883952	+	0.467578i	$0.154873\pi$
$773$	26.0000i	0.935155i	−0.883952	+	0.467578i	$0.845127\pi$
$774$	−12.0000	−0.431331
$775$	0	0
$776$	6.00000	0.215387
$777$	0	0
$778$	6.00000i	0.215110i
$779$	−24.0000	−0.859889
$780$	0	0
$781$	0	0
$782$	− 4.00000i	− 0.143040i
$783$	0	0
$784$	9.00000	0.321429
$785$	0	0
$786$	0	0
$787$	− 32.0000i	− 1.14068i	−0.821410	−	0.570338i	$-0.806812\pi$
$787$	− 32.0000i	− 1.14068i	0.821410	−	0.570338i	$-0.193188\pi$
$788$	− 18.0000i	− 0.641223i
$789$	0	0
$790$	0	0
$791$	−56.0000	−1.99113
$792$	0	0
$793$	− 20.0000i	− 0.710221i
$794$	6.00000	0.212932
$795$	0	0
$796$	20.0000	0.708881
$797$	− 50.0000i	− 1.77109i	−0.464553	−	0.885545i	$-0.653785\pi$
$797$	− 50.0000i	− 1.77109i	0.464553	−	0.885545i	$-0.346215\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	−30.0000	−1.06000
$802$	14.0000i	0.494357i
$803$	0	0
$804$	0	0
$805$	0	0
$806$	8.00000	0.281788
$807$	0	0
$808$	30.0000i	1.05540i
$809$	−26.0000	−0.914111	−0.457056	−	0.889438i	$-0.651096\pi$
$809$	−26.0000	−0.914111	−0.457056	+	0.889438i	$0.651096\pi$
$810$	0	0
$811$	40.0000	1.40459	0.702295	−	0.711886i	$-0.252159\pi$
$811$	40.0000	1.40459	0.702295	+	0.711886i	$0.252159\pi$
$812$	− 24.0000i	− 0.842235i
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 16.0000i	− 0.559769i
$818$	26.0000i	0.909069i
$819$	−24.0000	−0.838628
$820$	0	0
$821$	−18.0000	−0.628204	−0.314102	−	0.949389i	$-0.601703\pi$
$821$	−18.0000	−0.628204	−0.314102	+	0.949389i	$0.601703\pi$
$822$	0	0
$823$	− 20.0000i	− 0.697156i	−0.937280	−	0.348578i	$-0.886665\pi$
$823$	− 20.0000i	− 0.697156i	0.937280	−	0.348578i	$-0.113335\pi$
$824$	−24.0000	−0.836080
$825$	0	0
$826$	48.0000	1.67013
$827$	− 48.0000i	− 1.66912i	−0.550914	−	0.834562i	$-0.685721\pi$
$827$	− 48.0000i	− 1.66912i	0.550914	−	0.834562i	$-0.314279\pi$
$828$	− 12.0000i	− 0.417029i
$829$	34.0000	1.18087	0.590434	−	0.807086i	$-0.298956\pi$
$829$	34.0000	1.18087	0.590434	+	0.807086i	$0.298956\pi$
$830$	0	0
$831$	0	0
$832$	− 14.0000i	− 0.485363i
$833$	− 9.00000i	− 0.311832i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	8.00000i	0.276355i
$839$	−20.0000	−0.690477	−0.345238	−	0.938515i	$-0.612202\pi$
$839$	−20.0000	−0.690477	−0.345238	+	0.938515i	$0.612202\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	− 22.0000i	− 0.758170i
$843$	0	0
$844$	8.00000	0.275371
$845$	0	0
$846$	0	0
$847$	− 44.0000i	− 1.51186i
$848$	6.00000i	0.206041i
$849$	0	0
$850$	0	0
$851$	−8.00000	−0.274236
$852$	0	0
$853$	− 14.0000i	− 0.479351i	−0.970853	−	0.239675i	$-0.922959\pi$
$853$	− 14.0000i	− 0.479351i	0.970853	−	0.239675i	$-0.0770410\pi$
$854$	−40.0000	−1.36877
$855$	0	0
$856$	24.0000	0.820303
$857$	10.0000i	0.341593i	0.985306	+	0.170797i	$0.0546341\pi$
$857$	10.0000i	0.341593i	−0.985306	+	0.170797i	$0.945366\pi$
$858$	0	0
$859$	−52.0000	−1.77422	−0.887109	−	0.461561i	$-0.847290\pi$
$859$	−52.0000	−1.77422	−0.887109	+	0.461561i	$0.847290\pi$
$860$	0	0
$861$	0	0
$862$	− 12.0000i	− 0.408722i
$863$	− 16.0000i	− 0.544646i	−0.962206	−	0.272323i	$-0.912208\pi$
$863$	− 16.0000i	− 0.544646i	0.962206	−	0.272323i	$-0.0877920\pi$
$864$	0	0
$865$	0	0
$866$	−2.00000	−0.0679628
$867$	0	0
$868$	16.0000i	0.543075i
$869$	0	0
$870$	0	0
$871$	−8.00000	−0.271070
$872$	18.0000i	0.609557i
$873$	6.00000i	0.203069i
$874$	−16.0000	−0.541208
$875$	0	0
$876$	0	0
$877$	6.00000i	0.202606i	0.994856	+	0.101303i	$0.0323011\pi$
$877$	6.00000i	0.202606i	−0.994856	+	0.101303i	$0.967699\pi$
$878$	− 20.0000i	− 0.674967i
$879$	0	0
$880$	0	0
$881$	−46.0000	−1.54978	−0.774890	−	0.632096i	$-0.782195\pi$
$881$	−46.0000	−1.54978	−0.774890	+	0.632096i	$0.782195\pi$
$882$	27.0000i	0.909137i
$883$	12.0000i	0.403832i	0.979403	+	0.201916i	$0.0647168\pi$
$883$	12.0000i	0.403832i	−0.979403	+	0.201916i	$0.935283\pi$
$884$	−2.00000	−0.0672673
$885$	0	0
$886$	−28.0000	−0.940678
$887$	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
$889$	−32.0000	−1.07325
$890$	0	0
$891$	0	0
$892$	− 24.0000i	− 0.803579i
$893$	0	0
$894$	0	0
$895$	0	0
$896$	12.0000	0.400892
$897$	0	0
$898$	34.0000i	1.13459i
$899$	−24.0000	−0.800445
$900$	0	0
$901$	6.00000	0.199889
$902$	0	0
$903$	0	0
$904$	42.0000	1.39690
$905$	0	0
$906$	0	0
$907$	32.0000i	1.06254i	0.847202	+	0.531271i	$0.178286\pi$
$907$	32.0000i	1.06254i	−0.847202	+	0.531271i	$0.821714\pi$
$908$	− 24.0000i	− 0.796468i
$909$	−30.0000	−0.995037
$910$	0	0
$911$	−4.00000	−0.132526	−0.0662630	−	0.997802i	$-0.521108\pi$
$911$	−4.00000	−0.132526	−0.0662630	+	0.997802i	$0.521108\pi$
$912$	0	0
$913$	0	0
$914$	−6.00000	−0.198462
$915$	0	0
$916$	−6.00000	−0.198246
$917$	64.0000i	2.11347i
$918$	0	0
$919$	−24.0000	−0.791687	−0.395843	−	0.918318i	$-0.629548\pi$
$919$	−24.0000	−0.791687	−0.395843	+	0.918318i	$0.629548\pi$
$920$	0	0
$921$	0	0
$922$	2.00000i	0.0658665i
$923$	− 8.00000i	− 0.263323i
$924$	0	0
$925$	0	0
$926$	−32.0000	−1.05159
$927$	− 24.0000i	− 0.788263i
$928$	30.0000i	0.984798i
$929$	30.0000	0.984268	0.492134	−	0.870519i	$-0.336217\pi$
$929$	30.0000	0.984268	0.492134	+	0.870519i	$0.336217\pi$
$930$	0	0
$931$	−36.0000	−1.17985
$932$	6.00000i	0.196537i
$933$	0	0
$934$	12.0000	0.392652
$935$	0	0
$936$	18.0000	0.588348
$937$	10.0000i	0.326686i	0.986569	+	0.163343i	$0.0522277\pi$
$937$	10.0000i	0.326686i	−0.986569	+	0.163343i	$0.947772\pi$
$938$	16.0000i	0.522419i
$939$	0	0
$940$	0	0
$941$	6.00000	0.195594	0.0977972	−	0.995206i	$-0.468820\pi$
$941$	6.00000	0.195594	0.0977972	+	0.995206i	$0.468820\pi$
$942$	0	0
$943$	24.0000i	0.781548i
$944$	−12.0000	−0.390567
$945$	0	0
$946$	0	0
$947$	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
$949$	−12.0000	−0.389536
$950$	0	0
$951$	0	0
$952$	12.0000i	0.388922i
$953$	22.0000i	0.712650i	0.934362	+	0.356325i	$0.115970\pi$
$953$	22.0000i	0.712650i	−0.934362	+	0.356325i	$0.884030\pi$
$954$	−18.0000	−0.582772
$955$	0	0
$956$	16.0000	0.517477
$957$	0	0
$958$	36.0000i	1.16311i
$959$	24.0000	0.775000
$960$	0	0
$961$	−15.0000	−0.483871
$962$	− 4.00000i	− 0.128965i
$963$	24.0000i	0.773389i
$964$	18.0000	0.579741
$965$	0	0
$966$	0	0
$967$	0	0	1.00000			$0$
$967$	0	0	−1.00000			$\pi$
$968$	33.0000i	1.06066i
$969$	0	0
$970$	0	0
$971$	−12.0000	−0.385098	−0.192549	−	0.981287i	$-0.561675\pi$
$971$	−12.0000	−0.385098	−0.192549	+	0.981287i	$0.561675\pi$
$972$	0	0
$973$	32.0000i	1.02587i
$974$	20.0000	0.640841
$975$	0	0
$976$	10.0000	0.320092
$977$	18.0000i	0.575871i	0.957650	+	0.287936i	$0.0929689\pi$
$977$	18.0000i	0.575871i	−0.957650	+	0.287936i	$0.907031\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	−18.0000	−0.574696
$982$	− 20.0000i	− 0.638226i
$983$	− 12.0000i	− 0.382741i	−0.981518	−	0.191370i	$-0.938707\pi$
$983$	− 12.0000i	− 0.382741i	0.981518	−	0.191370i	$-0.0612931\pi$
$984$	0	0
$985$	0	0
$986$	−6.00000	−0.191079
$987$	0	0
$988$	8.00000i	0.254514i
$989$	−16.0000	−0.508770
$990$	0	0
$991$	−12.0000	−0.381193	−0.190596	−	0.981669i	$-0.561042\pi$
$991$	−12.0000	−0.381193	−0.190596	+	0.981669i	$0.561042\pi$
$992$	− 20.0000i	− 0.635001i
$993$	0	0
$994$	−16.0000	−0.507489
$995$	0	0
$996$	0	0
$997$	46.0000i	1.45683i	0.685134	+	0.728417i	$0.259744\pi$
$997$	46.0000i	1.45683i	−0.685134	+	0.728417i	$0.740256\pi$
$998$	− 40.0000i	− 1.26618i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	425.2.b.b.324.1		2
5.2	odd	4		425.2.a.d.1.1		1
5.3	odd	4		17.2.a.a.1.1	✓	1
5.4	even	2	inner	425.2.b.b.324.2		2
15.2	even	4		3825.2.a.d.1.1		1
15.8	even	4		153.2.a.c.1.1		1
20.3	even	4		272.2.a.b.1.1		1
20.7	even	4		6800.2.a.n.1.1		1
35.3	even	12		833.2.e.a.324.1		2
35.13	even	4		833.2.a.a.1.1		1
35.18	odd	12		833.2.e.b.324.1		2
35.23	odd	12		833.2.e.b.18.1		2
35.33	even	12		833.2.e.a.18.1		2
40.3	even	4		1088.2.a.h.1.1		1
40.13	odd	4		1088.2.a.i.1.1		1
55.43	even	4		2057.2.a.e.1.1		1
60.23	odd	4		2448.2.a.o.1.1		1
65.38	odd	4		2873.2.a.c.1.1		1
85.3	even	16		289.2.d.d.179.2		8
85.8	odd	8		289.2.c.a.251.1		4
85.13	odd	4		289.2.b.a.288.1		2
85.23	even	16		289.2.d.d.155.2		8
85.28	even	16		289.2.d.d.155.1		8
85.33	odd	4		289.2.a.a.1.1		1
85.38	odd	4		289.2.b.a.288.2		2
85.43	odd	8		289.2.c.a.251.2		4
85.48	even	16		289.2.d.d.179.1		8
85.53	odd	8		289.2.c.a.38.2		4
85.58	even	16		289.2.d.d.134.2		8
85.63	even	16		289.2.d.d.110.1		8
85.67	odd	4		7225.2.a.g.1.1		1
85.73	even	16		289.2.d.d.110.2		8
85.78	even	16		289.2.d.d.134.1		8
85.83	odd	8		289.2.c.a.38.1		4
95.18	even	4		6137.2.a.b.1.1		1
105.83	odd	4		7497.2.a.l.1.1		1
115.68	even	4		8993.2.a.a.1.1		1
120.53	even	4		9792.2.a.n.1.1		1
120.83	odd	4		9792.2.a.i.1.1		1
255.203	even	4		2601.2.a.g.1.1		1
340.203	even	4		4624.2.a.d.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
17.2.a.a.1.1	✓	1	5.3	odd	4
153.2.a.c.1.1		1	15.8	even	4
272.2.a.b.1.1		1	20.3	even	4
289.2.a.a.1.1		1	85.33	odd	4
289.2.b.a.288.1		2	85.13	odd	4
289.2.b.a.288.2		2	85.38	odd	4
289.2.c.a.38.1		4	85.83	odd	8
289.2.c.a.38.2		4	85.53	odd	8
289.2.c.a.251.1		4	85.8	odd	8
289.2.c.a.251.2		4	85.43	odd	8
289.2.d.d.110.1		8	85.63	even	16
289.2.d.d.110.2		8	85.73	even	16
289.2.d.d.134.1		8	85.78	even	16
289.2.d.d.134.2		8	85.58	even	16
289.2.d.d.155.1		8	85.28	even	16
289.2.d.d.155.2		8	85.23	even	16
289.2.d.d.179.1		8	85.48	even	16
289.2.d.d.179.2		8	85.3	even	16
425.2.a.d.1.1		1	5.2	odd	4
425.2.b.b.324.1		2	1.1	even	1	trivial
425.2.b.b.324.2		2	5.4	even	2	inner
833.2.a.a.1.1		1	35.13	even	4
833.2.e.a.18.1		2	35.33	even	12
833.2.e.a.324.1		2	35.3	even	12
833.2.e.b.18.1		2	35.23	odd	12
833.2.e.b.324.1		2	35.18	odd	12
1088.2.a.h.1.1		1	40.3	even	4
1088.2.a.i.1.1		1	40.13	odd	4
2057.2.a.e.1.1		1	55.43	even	4
2448.2.a.o.1.1		1	60.23	odd	4
2601.2.a.g.1.1		1	255.203	even	4
2873.2.a.c.1.1		1	65.38	odd	4
3825.2.a.d.1.1		1	15.2	even	4
4624.2.a.d.1.1		1	340.203	even	4
6137.2.a.b.1.1		1	95.18	even	4
6800.2.a.n.1.1		1	20.7	even	4
7225.2.a.g.1.1		1	85.67	odd	4
7497.2.a.l.1.1		1	105.83	odd	4
8993.2.a.a.1.1		1	115.68	even	4
9792.2.a.i.1.1		1	120.83	odd	4
9792.2.a.n.1.1		1	120.53	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 425 = 5^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	425.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} +1.00000 q^{4} +4.00000i q^{7} -3.00000i q^{8} +3.00000 q^{9} +2.00000i q^{13} +4.00000 q^{14} -1.00000 q^{16} +1.00000i q^{17} -3.00000i q^{18} +4.00000 q^{19} -4.00000i q^{23} +2.00000 q^{26} +4.00000i q^{28} -6.00000 q^{29} +4.00000 q^{31} -5.00000i q^{32} +1.00000 q^{34} +3.00000 q^{36} -2.00000i q^{37} -4.00000i q^{38} -6.00000 q^{41} -4.00000i q^{43} -4.00000 q^{46} -9.00000 q^{49} +2.00000i q^{52} -6.00000i q^{53} +12.0000 q^{56} +6.00000i q^{58} +12.0000 q^{59} -10.0000 q^{61} -4.00000i q^{62} +12.0000i q^{63} -7.00000 q^{64} +4.00000i q^{67} +1.00000i q^{68} -4.00000 q^{71} -9.00000i q^{72} +6.00000i q^{73} -2.00000 q^{74} +4.00000 q^{76} -12.0000 q^{79} +9.00000 q^{81} +6.00000i q^{82} +4.00000i q^{83} -4.00000 q^{86} -10.0000 q^{89} -8.00000 q^{91} -4.00000i q^{92} +2.00000i q^{97} +9.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{4} + 6 q^{9} + 8 q^{14} - 2 q^{16} + 8 q^{19} + 4 q^{26} - 12 q^{29} + 8 q^{31} + 2 q^{34} + 6 q^{36} - 12 q^{41} - 8 q^{46} - 18 q^{49} + 24 q^{56} + 24 q^{59} - 20 q^{61} - 14 q^{64} - 8 q^{71}+ \cdots - 16 q^{91}+O(q^{100}) \) 2 * q + 2 * q^4 + 6 * q^9 + 8 * q^14 - 2 * q^16 + 8 * q^19 + 4 * q^26 - 12 * q^29 + 8 * q^31 + 2 * q^34 + 6 * q^36 - 12 * q^41 - 8 * q^46 - 18 * q^49 + 24 * q^56 + 24 * q^59 - 20 * q^61 - 14 * q^64 - 8 * q^71 - 4 * q^74 + 8 * q^76 - 24 * q^79 + 18 * q^81 - 8 * q^86 - 20 * q^89 - 16 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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