LMFDB - Embedded newform 475.2.b.a.324.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [475,2,Mod(324,475)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(475, 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("475.324");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 475 = 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	475.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:	$3.79289409601$
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:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 19)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			324.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	475.324
Dual form			475.2.b.a.324.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.00000i q^{3} +2.00000 q^{4} -1.00000i q^{7} -1.00000 q^{9} +O(q^{10})$	$q+2.00000i q^{3} +2.00000 q^{4} -1.00000i q^{7} -1.00000 q^{9} +3.00000 q^{11} +4.00000i q^{12} +4.00000i q^{13} +4.00000 q^{16} -3.00000i q^{17} -1.00000 q^{19} +2.00000 q^{21} +4.00000i q^{27} -2.00000i q^{28} -6.00000 q^{29} -4.00000 q^{31} +6.00000i q^{33} -2.00000 q^{36} +2.00000i q^{37} -8.00000 q^{39} -6.00000 q^{41} +1.00000i q^{43} +6.00000 q^{44} -3.00000i q^{47} +8.00000i q^{48} +6.00000 q^{49} +6.00000 q^{51} +8.00000i q^{52} -12.0000i q^{53} -2.00000i q^{57} +6.00000 q^{59} -1.00000 q^{61} +1.00000i q^{63} +8.00000 q^{64} -4.00000i q^{67} -6.00000i q^{68} +6.00000 q^{71} +7.00000i q^{73} -2.00000 q^{76} -3.00000i q^{77} -8.00000 q^{79} -11.0000 q^{81} -12.0000i q^{83} +4.00000 q^{84} -12.0000i q^{87} -12.0000 q^{89} +4.00000 q^{91} -8.00000i q^{93} +8.00000i q^{97} -3.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 4 q^{4} - 2 q^{9}+O(q^{10}) $ 2 * q + 4 * q^4 - 2 * q^9	$ 2 q + 4 q^{4} - 2 q^{9} + 6 q^{11} + 8 q^{16} - 2 q^{19} + 4 q^{21} - 12 q^{29} - 8 q^{31} - 4 q^{36} - 16 q^{39} - 12 q^{41} + 12 q^{44} + 12 q^{49} + 12 q^{51} + 12 q^{59} - 2 q^{61} + 16 q^{64} + 12 q^{71} - 4 q^{76} - 16 q^{79} - 22 q^{81} + 8 q^{84} - 24 q^{89} + 8 q^{91} - 6 q^{99}+O(q^{100}) $ 2 * q + 4 * q^4 - 2 * q^9 + 6 * q^11 + 8 * q^16 - 2 * q^19 + 4 * q^21 - 12 * q^29 - 8 * q^31 - 4 * q^36 - 16 * q^39 - 12 * q^41 + 12 * q^44 + 12 * q^49 + 12 * q^51 + 12 * q^59 - 2 * q^61 + 16 * q^64 + 12 * q^71 - 4 * q^76 - 16 * q^79 - 22 * q^81 + 8 * q^84 - 24 * q^89 + 8 * q^91 - 6 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/475\mathbb{Z}\right)^\times$.

$n$	$77$	$401$
$\chi(n)$	$-1$	$1$

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 $

$2$	0	0	1.00000			$0$
$2$	0	0	−1.00000			$\pi$
$3$	2.00000i	1.15470i	0.816497	+	0.577350i	$0.195913\pi$
$3$	2.00000i	1.15470i	−0.816497	+	0.577350i	$0.804087\pi$
$4$	2.00000	1.00000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 1.00000i	− 0.377964i	−0.981981	−	0.188982i	$-0.939481\pi$
$7$	− 1.00000i	− 0.377964i	0.981981	−	0.188982i	$-0.0605189\pi$
$8$	0	0
$9$	−1.00000	−0.333333
$10$	0	0
$11$	3.00000	0.904534	0.452267	−	0.891883i	$-0.350615\pi$
$11$	3.00000	0.904534	0.452267	+	0.891883i	$0.350615\pi$
$12$	4.00000i	1.15470i
$13$	4.00000i	1.10940i	0.832050	+	0.554700i	$0.187167\pi$
$13$	4.00000i	1.10940i	−0.832050	+	0.554700i	$0.812833\pi$
$14$	0	0
$15$	0	0
$16$	4.00000	1.00000
$17$	− 3.00000i	− 0.727607i	−0.931476	−	0.363803i	$-0.881478\pi$
$17$	− 3.00000i	− 0.727607i	0.931476	−	0.363803i	$-0.118522\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	2.00000	0.436436
$22$	0	0
$23$	0	0	1.00000			$0$
$23$	0	0	−1.00000			$\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	4.00000i	0.769800i
$28$	− 2.00000i	− 0.377964i
$29$	−6.00000	−1.11417	−0.557086	−	0.830455i	$-0.688081\pi$
$29$	−6.00000	−1.11417	−0.557086	+	0.830455i	$0.688081\pi$
$30$	0	0
$31$	−4.00000	−0.718421	−0.359211	−	0.933257i	$-0.616954\pi$
$31$	−4.00000	−0.718421	−0.359211	+	0.933257i	$0.616954\pi$
$32$	0	0
$33$	6.00000i	1.04447i
$34$	0	0
$35$	0	0
$36$	−2.00000	−0.333333
$37$	2.00000i	0.328798i	0.986394	+	0.164399i	$0.0525685\pi$
$37$	2.00000i	0.328798i	−0.986394	+	0.164399i	$0.947432\pi$
$38$	0	0
$39$	−8.00000	−1.28103
$40$	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
$43$	1.00000i	0.152499i	0.997089	+	0.0762493i	$0.0242945\pi$
$43$	1.00000i	0.152499i	−0.997089	+	0.0762493i	$0.975706\pi$
$44$	6.00000	0.904534
$45$	0	0
$46$	0	0
$47$	− 3.00000i	− 0.437595i	−0.975770	−	0.218797i	$-0.929787\pi$
$47$	− 3.00000i	− 0.437595i	0.975770	−	0.218797i	$-0.0702134\pi$
$48$	8.00000i	1.15470i
$49$	6.00000	0.857143
$50$	0	0
$51$	6.00000	0.840168
$52$	8.00000i	1.10940i
$53$	− 12.0000i	− 1.64833i	−0.566352	−	0.824163i	$-0.691646\pi$
$53$	− 12.0000i	− 1.64833i	0.566352	−	0.824163i	$-0.308354\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 2.00000i	− 0.264906i
$58$	0	0
$59$	6.00000	0.781133	0.390567	−	0.920575i	$-0.372279\pi$
$59$	6.00000	0.781133	0.390567	+	0.920575i	$0.372279\pi$
$60$	0	0
$61$	−1.00000	−0.128037	−0.0640184	−	0.997949i	$-0.520392\pi$
$61$	−1.00000	−0.128037	−0.0640184	+	0.997949i	$0.520392\pi$
$62$	0	0
$63$	1.00000i	0.125988i
$64$	8.00000	1.00000
$65$	0	0
$66$	0	0
$67$	− 4.00000i	− 0.488678i	−0.969690	−	0.244339i	$-0.921429\pi$
$67$	− 4.00000i	− 0.488678i	0.969690	−	0.244339i	$-0.0785709\pi$
$68$	− 6.00000i	− 0.727607i
$69$	0	0
$70$	0	0
$71$	6.00000	0.712069	0.356034	−	0.934473i	$-0.384129\pi$
$71$	6.00000	0.712069	0.356034	+	0.934473i	$0.384129\pi$
$72$	0	0
$73$	7.00000i	0.819288i	0.912245	+	0.409644i	$0.134347\pi$
$73$	7.00000i	0.819288i	−0.912245	+	0.409644i	$0.865653\pi$
$74$	0	0
$75$	0	0
$76$	−2.00000	−0.229416
$77$	− 3.00000i	− 0.341882i
$78$	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
$81$	−11.0000	−1.22222
$82$	0	0
$83$	− 12.0000i	− 1.31717i	−0.752506	−	0.658586i	$-0.771155\pi$
$83$	− 12.0000i	− 1.31717i	0.752506	−	0.658586i	$-0.228845\pi$
$84$	4.00000	0.436436
$85$	0	0
$86$	0	0
$87$	− 12.0000i	− 1.28654i
$88$	0	0
$89$	−12.0000	−1.27200	−0.635999	−	0.771690i	$-0.719412\pi$
$89$	−12.0000	−1.27200	−0.635999	+	0.771690i	$0.719412\pi$
$90$	0	0
$91$	4.00000	0.419314
$92$	0	0
$93$	− 8.00000i	− 0.829561i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	8.00000i	0.812277i	0.913812	+	0.406138i	$0.133125\pi$
$97$	8.00000i	0.812277i	−0.913812	+	0.406138i	$0.866875\pi$
$98$	0	0
$99$	−3.00000	−0.301511
$100$	0	0
$101$	6.00000	0.597022	0.298511	−	0.954406i	$-0.403510\pi$
$101$	6.00000	0.597022	0.298511	+	0.954406i	$0.403510\pi$
$102$	0	0
$103$	− 14.0000i	− 1.37946i	−0.724066	−	0.689730i	$-0.757729\pi$
$103$	− 14.0000i	− 1.37946i	0.724066	−	0.689730i	$-0.242271\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 18.0000i	− 1.74013i	−0.492941	−	0.870063i	$-0.664078\pi$
$107$	− 18.0000i	− 1.74013i	0.492941	−	0.870063i	$-0.335922\pi$
$108$	8.00000i	0.769800i
$109$	16.0000	1.53252	0.766261	−	0.642529i	$-0.222115\pi$
$109$	16.0000	1.53252	0.766261	+	0.642529i	$0.222115\pi$
$110$	0	0
$111$	−4.00000	−0.379663
$112$	− 4.00000i	− 0.377964i
$113$	− 6.00000i	− 0.564433i	−0.959351	−	0.282216i	$-0.908930\pi$
$113$	− 6.00000i	− 0.564433i	0.959351	−	0.282216i	$-0.0910696\pi$
$114$	0	0
$115$	0	0
$116$	−12.0000	−1.11417
$117$	− 4.00000i	− 0.369800i
$118$	0	0
$119$	−3.00000	−0.275010
$120$	0	0
$121$	−2.00000	−0.181818
$122$	0	0
$123$	− 12.0000i	− 1.08200i
$124$	−8.00000	−0.718421
$125$	0	0
$126$	0	0
$127$	2.00000i	0.177471i	0.996055	+	0.0887357i	$0.0282826\pi$
$127$	2.00000i	0.177471i	−0.996055	+	0.0887357i	$0.971717\pi$
$128$	0	0
$129$	−2.00000	−0.176090
$130$	0	0
$131$	−15.0000	−1.31056	−0.655278	−	0.755388i	$-0.727449\pi$
$131$	−15.0000	−1.31056	−0.655278	+	0.755388i	$0.727449\pi$
$132$	12.0000i	1.04447i
$133$	1.00000i	0.0867110i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 3.00000i	− 0.256307i	−0.991754	−	0.128154i	$-0.959095\pi$
$137$	− 3.00000i	− 0.256307i	0.991754	−	0.128154i	$-0.0409051\pi$
$138$	0	0
$139$	13.0000	1.10265	0.551323	−	0.834292i	$-0.314123\pi$
$139$	13.0000	1.10265	0.551323	+	0.834292i	$0.314123\pi$
$140$	0	0
$141$	6.00000	0.505291
$142$	0	0
$143$	12.0000i	1.00349i
$144$	−4.00000	−0.333333
$145$	0	0
$146$	0	0
$147$	12.0000i	0.989743i
$148$	4.00000i	0.328798i
$149$	−21.0000	−1.72039	−0.860194	−	0.509968i	$-0.829657\pi$
$149$	−21.0000	−1.72039	−0.860194	+	0.509968i	$0.829657\pi$
$150$	0	0
$151$	−10.0000	−0.813788	−0.406894	−	0.913475i	$-0.633388\pi$
$151$	−10.0000	−0.813788	−0.406894	+	0.913475i	$0.633388\pi$
$152$	0	0
$153$	3.00000i	0.242536i
$154$	0	0
$155$	0	0
$156$	−16.0000	−1.28103
$157$	14.0000i	1.11732i	0.829396	+	0.558661i	$0.188685\pi$
$157$	14.0000i	1.11732i	−0.829396	+	0.558661i	$0.811315\pi$
$158$	0	0
$159$	24.0000	1.90332
$160$	0	0
$161$	0	0
$162$	0	0
$163$	− 20.0000i	− 1.56652i	−0.621694	−	0.783260i	$-0.713555\pi$
$163$	− 20.0000i	− 1.56652i	0.621694	−	0.783260i	$-0.286445\pi$
$164$	−12.0000	−0.937043
$165$	0	0
$166$	0	0
$167$	− 18.0000i	− 1.39288i	−0.717614	−	0.696441i	$-0.754766\pi$
$167$	− 18.0000i	− 1.39288i	0.717614	−	0.696441i	$-0.245234\pi$
$168$	0	0
$169$	−3.00000	−0.230769
$170$	0	0
$171$	1.00000	0.0764719
$172$	2.00000i	0.152499i
$173$	18.0000i	1.36851i	0.729241	+	0.684257i	$0.239873\pi$
$173$	18.0000i	1.36851i	−0.729241	+	0.684257i	$0.760127\pi$
$174$	0	0
$175$	0	0
$176$	12.0000	0.904534
$177$	12.0000i	0.901975i
$178$	0	0
$179$	18.0000	1.34538	0.672692	−	0.739923i	$-0.265138\pi$
$179$	18.0000	1.34538	0.672692	+	0.739923i	$0.265138\pi$
$180$	0	0
$181$	2.00000	0.148659	0.0743294	−	0.997234i	$-0.476318\pi$
$181$	2.00000	0.148659	0.0743294	+	0.997234i	$0.476318\pi$
$182$	0	0
$183$	− 2.00000i	− 0.147844i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 9.00000i	− 0.658145i
$188$	− 6.00000i	− 0.437595i
$189$	4.00000	0.290957
$190$	0	0
$191$	3.00000	0.217072	0.108536	−	0.994092i	$-0.465384\pi$
$191$	3.00000	0.217072	0.108536	+	0.994092i	$0.465384\pi$
$192$	16.0000i	1.15470i
$193$	4.00000i	0.287926i	0.989583	+	0.143963i	$0.0459847\pi$
$193$	4.00000i	0.287926i	−0.989583	+	0.143963i	$0.954015\pi$
$194$	0	0
$195$	0	0
$196$	12.0000	0.857143
$197$	18.0000i	1.28245i	0.767354	+	0.641223i	$0.221573\pi$
$197$	18.0000i	1.28245i	−0.767354	+	0.641223i	$0.778427\pi$
$198$	0	0
$199$	−11.0000	−0.779769	−0.389885	−	0.920864i	$-0.627485\pi$
$199$	−11.0000	−0.779769	−0.389885	+	0.920864i	$0.627485\pi$
$200$	0	0
$201$	8.00000	0.564276
$202$	0	0
$203$	6.00000i	0.421117i
$204$	12.0000	0.840168
$205$	0	0
$206$	0	0
$207$	0	0
$208$	16.0000i	1.10940i
$209$	−3.00000	−0.207514
$210$	0	0
$211$	14.0000	0.963800	0.481900	−	0.876226i	$-0.339947\pi$
$211$	14.0000	0.963800	0.481900	+	0.876226i	$0.339947\pi$
$212$	− 24.0000i	− 1.64833i
$213$	12.0000i	0.822226i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	4.00000i	0.271538i
$218$	0	0
$219$	−14.0000	−0.946032
$220$	0	0
$221$	12.0000	0.807207
$222$	0	0
$223$	10.0000i	0.669650i	0.942280	+	0.334825i	$0.108677\pi$
$223$	10.0000i	0.669650i	−0.942280	+	0.334825i	$0.891323\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	12.0000i	0.796468i	0.917284	+	0.398234i	$0.130377\pi$
$227$	12.0000i	0.796468i	−0.917284	+	0.398234i	$0.869623\pi$
$228$	− 4.00000i	− 0.264906i
$229$	−5.00000	−0.330409	−0.165205	−	0.986259i	$-0.552828\pi$
$229$	−5.00000	−0.330409	−0.165205	+	0.986259i	$0.552828\pi$
$230$	0	0
$231$	6.00000	0.394771
$232$	0	0
$233$	21.0000i	1.37576i	0.725826	+	0.687878i	$0.241458\pi$
$233$	21.0000i	1.37576i	−0.725826	+	0.687878i	$0.758542\pi$
$234$	0	0
$235$	0	0
$236$	12.0000	0.781133
$237$	− 16.0000i	− 1.03931i
$238$	0	0
$239$	−15.0000	−0.970269	−0.485135	−	0.874439i	$-0.661229\pi$
$239$	−15.0000	−0.970269	−0.485135	+	0.874439i	$0.661229\pi$
$240$	0	0
$241$	−10.0000	−0.644157	−0.322078	−	0.946713i	$-0.604381\pi$
$241$	−10.0000	−0.644157	−0.322078	+	0.946713i	$0.604381\pi$
$242$	0	0
$243$	− 10.0000i	− 0.641500i
$244$	−2.00000	−0.128037
$245$	0	0
$246$	0	0
$247$	− 4.00000i	− 0.254514i
$248$	0	0
$249$	24.0000	1.52094
$250$	0	0
$251$	21.0000	1.32551	0.662754	−	0.748837i	$-0.269387\pi$
$251$	21.0000	1.32551	0.662754	+	0.748837i	$0.269387\pi$
$252$	2.00000i	0.125988i
$253$	0	0
$254$	0	0
$255$	0	0
$256$	16.0000	1.00000
$257$	0	0	1.00000			$0$
$257$	0	0	−1.00000			$\pi$
$258$	0	0
$259$	2.00000	0.124274
$260$	0	0
$261$	6.00000	0.371391
$262$	0	0
$263$	− 9.00000i	− 0.554964i	−0.960731	−	0.277482i	$-0.910500\pi$
$263$	− 9.00000i	− 0.554964i	0.960731	−	0.277482i	$-0.0894999\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 24.0000i	− 1.46878i
$268$	− 8.00000i	− 0.488678i
$269$	−24.0000	−1.46331	−0.731653	−	0.681677i	$-0.761251\pi$
$269$	−24.0000	−1.46331	−0.731653	+	0.681677i	$0.761251\pi$
$270$	0	0
$271$	−16.0000	−0.971931	−0.485965	−	0.873978i	$-0.661532\pi$
$271$	−16.0000	−0.971931	−0.485965	+	0.873978i	$0.661532\pi$
$272$	− 12.0000i	− 0.727607i
$273$	8.00000i	0.484182i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 19.0000i	− 1.14160i	−0.821089	−	0.570800i	$-0.806633\pi$
$277$	− 19.0000i	− 1.14160i	0.821089	−	0.570800i	$-0.193367\pi$
$278$	0	0
$279$	4.00000	0.239474
$280$	0	0
$281$	6.00000	0.357930	0.178965	−	0.983855i	$-0.442725\pi$
$281$	6.00000	0.357930	0.178965	+	0.983855i	$0.442725\pi$
$282$	0	0
$283$	13.0000i	0.772770i	0.922338	+	0.386385i	$0.126276\pi$
$283$	13.0000i	0.772770i	−0.922338	+	0.386385i	$0.873724\pi$
$284$	12.0000	0.712069
$285$	0	0
$286$	0	0
$287$	6.00000i	0.354169i
$288$	0	0
$289$	8.00000	0.470588
$290$	0	0
$291$	−16.0000	−0.937937
$292$	14.0000i	0.819288i
$293$	12.0000i	0.701047i	0.936554	+	0.350524i	$0.113996\pi$
$293$	12.0000i	0.701047i	−0.936554	+	0.350524i	$0.886004\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	12.0000i	0.696311i
$298$	0	0
$299$	0	0
$300$	0	0
$301$	1.00000	0.0576390
$302$	0	0
$303$	12.0000i	0.689382i
$304$	−4.00000	−0.229416
$305$	0	0
$306$	0	0
$307$	20.0000i	1.14146i	0.821138	+	0.570730i	$0.193340\pi$
$307$	20.0000i	1.14146i	−0.821138	+	0.570730i	$0.806660\pi$
$308$	− 6.00000i	− 0.341882i
$309$	28.0000	1.59286
$310$	0	0
$311$	−3.00000	−0.170114	−0.0850572	−	0.996376i	$-0.527107\pi$
$311$	−3.00000	−0.170114	−0.0850572	+	0.996376i	$0.527107\pi$
$312$	0	0
$313$	10.0000i	0.565233i	0.959233	+	0.282617i	$0.0912024\pi$
$313$	10.0000i	0.565233i	−0.959233	+	0.282617i	$0.908798\pi$
$314$	0	0
$315$	0	0
$316$	−16.0000	−0.900070
$317$	6.00000i	0.336994i	0.985702	+	0.168497i	$0.0538913\pi$
$317$	6.00000i	0.336994i	−0.985702	+	0.168497i	$0.946109\pi$
$318$	0	0
$319$	−18.0000	−1.00781
$320$	0	0
$321$	36.0000	2.00932
$322$	0	0
$323$	3.00000i	0.166924i
$324$	−22.0000	−1.22222
$325$	0	0
$326$	0	0
$327$	32.0000i	1.76960i
$328$	0	0
$329$	−3.00000	−0.165395
$330$	0	0
$331$	−28.0000	−1.53902	−0.769510	−	0.638635i	$-0.779499\pi$
$331$	−28.0000	−1.53902	−0.769510	+	0.638635i	$0.779499\pi$
$332$	− 24.0000i	− 1.31717i
$333$	− 2.00000i	− 0.109599i
$334$	0	0
$335$	0	0
$336$	8.00000	0.436436
$337$	32.0000i	1.74315i	0.490261	+	0.871576i	$0.336901\pi$
$337$	32.0000i	1.74315i	−0.490261	+	0.871576i	$0.663099\pi$
$338$	0	0
$339$	12.0000	0.651751
$340$	0	0
$341$	−12.0000	−0.649836
$342$	0	0
$343$	− 13.0000i	− 0.701934i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	21.0000i	1.12734i	0.826000	+	0.563670i	$0.190611\pi$
$347$	21.0000i	1.12734i	−0.826000	+	0.563670i	$0.809389\pi$
$348$	− 24.0000i	− 1.28654i
$349$	−17.0000	−0.909989	−0.454995	−	0.890494i	$-0.650359\pi$
$349$	−17.0000	−0.909989	−0.454995	+	0.890494i	$0.650359\pi$
$350$	0	0
$351$	−16.0000	−0.854017
$352$	0	0
$353$	6.00000i	0.319348i	0.987170	+	0.159674i	$0.0510443\pi$
$353$	6.00000i	0.319348i	−0.987170	+	0.159674i	$0.948956\pi$
$354$	0	0
$355$	0	0
$356$	−24.0000	−1.27200
$357$	− 6.00000i	− 0.317554i
$358$	0	0
$359$	−15.0000	−0.791670	−0.395835	−	0.918322i	$-0.629545\pi$
$359$	−15.0000	−0.791670	−0.395835	+	0.918322i	$0.629545\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	− 4.00000i	− 0.209946i
$364$	8.00000	0.419314
$365$	0	0
$366$	0	0
$367$	8.00000i	0.417597i	0.977959	+	0.208798i	$0.0669552\pi$
$367$	8.00000i	0.417597i	−0.977959	+	0.208798i	$0.933045\pi$
$368$	0	0
$369$	6.00000	0.312348
$370$	0	0
$371$	−12.0000	−0.623009
$372$	− 16.0000i	− 0.829561i
$373$	4.00000i	0.207112i	0.994624	+	0.103556i	$0.0330221\pi$
$373$	4.00000i	0.207112i	−0.994624	+	0.103556i	$0.966978\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 24.0000i	− 1.23606i
$378$	0	0
$379$	34.0000	1.74646	0.873231	−	0.487306i	$-0.162020\pi$
$379$	34.0000	1.74646	0.873231	+	0.487306i	$0.162020\pi$
$380$	0	0
$381$	−4.00000	−0.204926
$382$	0	0
$383$	− 12.0000i	− 0.613171i	−0.951843	−	0.306586i	$-0.900813\pi$
$383$	− 12.0000i	− 0.613171i	0.951843	−	0.306586i	$-0.0991866\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 1.00000i	− 0.0508329i
$388$	16.0000i	0.812277i
$389$	−15.0000	−0.760530	−0.380265	−	0.924878i	$-0.624167\pi$
$389$	−15.0000	−0.760530	−0.380265	+	0.924878i	$0.624167\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	− 30.0000i	− 1.51330i
$394$	0	0
$395$	0	0
$396$	−6.00000	−0.301511
$397$	− 7.00000i	− 0.351320i	−0.984451	−	0.175660i	$-0.943794\pi$
$397$	− 7.00000i	− 0.351320i	0.984451	−	0.175660i	$-0.0562059\pi$
$398$	0	0
$399$	−2.00000	−0.100125
$400$	0	0
$401$	12.0000	0.599251	0.299626	−	0.954057i	$-0.403138\pi$
$401$	12.0000	0.599251	0.299626	+	0.954057i	$0.403138\pi$
$402$	0	0
$403$	− 16.0000i	− 0.797017i
$404$	12.0000	0.597022
$405$	0	0
$406$	0	0
$407$	6.00000i	0.297409i
$408$	0	0
$409$	4.00000	0.197787	0.0988936	−	0.995098i	$-0.468470\pi$
$409$	4.00000	0.197787	0.0988936	+	0.995098i	$0.468470\pi$
$410$	0	0
$411$	6.00000	0.295958
$412$	− 28.0000i	− 1.37946i
$413$	− 6.00000i	− 0.295241i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	26.0000i	1.27323i
$418$	0	0
$419$	12.0000	0.586238	0.293119	−	0.956076i	$-0.405307\pi$
$419$	12.0000	0.586238	0.293119	+	0.956076i	$0.405307\pi$
$420$	0	0
$421$	8.00000	0.389896	0.194948	−	0.980814i	$-0.437546\pi$
$421$	8.00000	0.389896	0.194948	+	0.980814i	$0.437546\pi$
$422$	0	0
$423$	3.00000i	0.145865i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	1.00000i	0.0483934i
$428$	− 36.0000i	− 1.74013i
$429$	−24.0000	−1.15873
$430$	0	0
$431$	−24.0000	−1.15604	−0.578020	−	0.816023i	$-0.696174\pi$
$431$	−24.0000	−1.15604	−0.578020	+	0.816023i	$0.696174\pi$
$432$	16.0000i	0.769800i
$433$	− 2.00000i	− 0.0961139i	−0.998845	−	0.0480569i	$-0.984697\pi$
$433$	− 2.00000i	− 0.0961139i	0.998845	−	0.0480569i	$-0.0153029\pi$
$434$	0	0
$435$	0	0
$436$	32.0000	1.53252
$437$	0	0
$438$	0	0
$439$	10.0000	0.477274	0.238637	−	0.971109i	$-0.423299\pi$
$439$	10.0000	0.477274	0.238637	+	0.971109i	$0.423299\pi$
$440$	0	0
$441$	−6.00000	−0.285714
$442$	0	0
$443$	3.00000i	0.142534i	0.997457	+	0.0712672i	$0.0227043\pi$
$443$	3.00000i	0.142534i	−0.997457	+	0.0712672i	$0.977296\pi$
$444$	−8.00000	−0.379663
$445$	0	0
$446$	0	0
$447$	− 42.0000i	− 1.98653i
$448$	− 8.00000i	− 0.377964i
$449$	0	0		−	1.00000i	$-0.5\pi$
$449$	0	0			1.00000i	$0.5\pi$
$450$	0	0
$451$	−18.0000	−0.847587
$452$	− 12.0000i	− 0.564433i
$453$	− 20.0000i	− 0.939682i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 37.0000i	− 1.73079i	−0.501093	−	0.865393i	$-0.667069\pi$
$457$	− 37.0000i	− 1.73079i	0.501093	−	0.865393i	$-0.332931\pi$
$458$	0	0
$459$	12.0000	0.560112
$460$	0	0
$461$	9.00000	0.419172	0.209586	−	0.977790i	$-0.432788\pi$
$461$	9.00000	0.419172	0.209586	+	0.977790i	$0.432788\pi$
$462$	0	0
$463$	31.0000i	1.44069i	0.693615	+	0.720346i	$0.256017\pi$
$463$	31.0000i	1.44069i	−0.693615	+	0.720346i	$0.743983\pi$
$464$	−24.0000	−1.11417
$465$	0	0
$466$	0	0
$467$	− 27.0000i	− 1.24941i	−0.780860	−	0.624705i	$-0.785219\pi$
$467$	− 27.0000i	− 1.24941i	0.780860	−	0.624705i	$-0.214781\pi$
$468$	− 8.00000i	− 0.369800i
$469$	−4.00000	−0.184703
$470$	0	0
$471$	−28.0000	−1.29017
$472$	0	0
$473$	3.00000i	0.137940i
$474$	0	0
$475$	0	0
$476$	−6.00000	−0.275010
$477$	12.0000i	0.549442i
$478$	0	0
$479$	12.0000	0.548294	0.274147	−	0.961688i	$-0.411605\pi$
$479$	12.0000	0.548294	0.274147	+	0.961688i	$0.411605\pi$
$480$	0	0
$481$	−8.00000	−0.364769
$482$	0	0
$483$	0	0
$484$	−4.00000	−0.181818
$485$	0	0
$486$	0	0
$487$	2.00000i	0.0906287i	0.998973	+	0.0453143i	$0.0144289\pi$
$487$	2.00000i	0.0906287i	−0.998973	+	0.0453143i	$0.985571\pi$
$488$	0	0
$489$	40.0000	1.80886
$490$	0	0
$491$	12.0000	0.541552	0.270776	−	0.962642i	$-0.412720\pi$
$491$	12.0000	0.541552	0.270776	+	0.962642i	$0.412720\pi$
$492$	− 24.0000i	− 1.08200i
$493$	18.0000i	0.810679i
$494$	0	0
$495$	0	0
$496$	−16.0000	−0.718421
$497$	− 6.00000i	− 0.269137i
$498$	0	0
$499$	−5.00000	−0.223831	−0.111915	−	0.993718i	$-0.535699\pi$
$499$	−5.00000	−0.223831	−0.111915	+	0.993718i	$0.535699\pi$
$500$	0	0
$501$	36.0000	1.60836
$502$	0	0
$503$	− 12.0000i	− 0.535054i	−0.963550	−	0.267527i	$-0.913794\pi$
$503$	− 12.0000i	− 0.535054i	0.963550	−	0.267527i	$-0.0862064\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 6.00000i	− 0.266469i
$508$	4.00000i	0.177471i
$509$	0	0		−	1.00000i	$-0.5\pi$
$509$	0	0			1.00000i	$0.5\pi$
$510$	0	0
$511$	7.00000	0.309662
$512$	0	0
$513$	− 4.00000i	− 0.176604i
$514$	0	0
$515$	0	0
$516$	−4.00000	−0.176090
$517$	− 9.00000i	− 0.395820i
$518$	0	0
$519$	−36.0000	−1.58022
$520$	0	0
$521$	0	0		−	1.00000i	$-0.5\pi$
$521$	0	0			1.00000i	$0.5\pi$
$522$	0	0
$523$	− 38.0000i	− 1.66162i	−0.556553	−	0.830812i	$-0.687876\pi$
$523$	− 38.0000i	− 1.66162i	0.556553	−	0.830812i	$-0.312124\pi$
$524$	−30.0000	−1.31056
$525$	0	0
$526$	0	0
$527$	12.0000i	0.522728i
$528$	24.0000i	1.04447i
$529$	23.0000	1.00000
$530$	0	0
$531$	−6.00000	−0.260378
$532$	2.00000i	0.0867110i
$533$	− 24.0000i	− 1.03956i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	36.0000i	1.55351i
$538$	0	0
$539$	18.0000	0.775315
$540$	0	0
$541$	−25.0000	−1.07483	−0.537417	−	0.843317i	$-0.680600\pi$
$541$	−25.0000	−1.07483	−0.537417	+	0.843317i	$0.680600\pi$
$542$	0	0
$543$	4.00000i	0.171656i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 28.0000i	− 1.19719i	−0.801050	−	0.598597i	$-0.795725\pi$
$547$	− 28.0000i	− 1.19719i	0.801050	−	0.598597i	$-0.204275\pi$
$548$	− 6.00000i	− 0.256307i
$549$	1.00000	0.0426790
$550$	0	0
$551$	6.00000	0.255609
$552$	0	0
$553$	8.00000i	0.340195i
$554$	0	0
$555$	0	0
$556$	26.0000	1.10265
$557$	21.0000i	0.889799i	0.895581	+	0.444899i	$0.146761\pi$
$557$	21.0000i	0.889799i	−0.895581	+	0.444899i	$0.853239\pi$
$558$	0	0
$559$	−4.00000	−0.169182
$560$	0	0
$561$	18.0000	0.759961
$562$	0	0
$563$	− 6.00000i	− 0.252870i	−0.991975	−	0.126435i	$-0.959647\pi$
$563$	− 6.00000i	− 0.252870i	0.991975	−	0.126435i	$-0.0403535\pi$
$564$	12.0000	0.505291
$565$	0	0
$566$	0	0
$567$	11.0000i	0.461957i
$568$	0	0
$569$	24.0000	1.00613	0.503066	−	0.864248i	$-0.332205\pi$
$569$	24.0000	1.00613	0.503066	+	0.864248i	$0.332205\pi$
$570$	0	0
$571$	−4.00000	−0.167395	−0.0836974	−	0.996491i	$-0.526673\pi$
$571$	−4.00000	−0.167395	−0.0836974	+	0.996491i	$0.526673\pi$
$572$	24.0000i	1.00349i
$573$	6.00000i	0.250654i
$574$	0	0
$575$	0	0
$576$	−8.00000	−0.333333
$577$	11.0000i	0.457936i	0.973434	+	0.228968i	$0.0735351\pi$
$577$	11.0000i	0.457936i	−0.973434	+	0.228968i	$0.926465\pi$
$578$	0	0
$579$	−8.00000	−0.332469
$580$	0	0
$581$	−12.0000	−0.497844
$582$	0	0
$583$	− 36.0000i	− 1.49097i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	45.0000i	1.85735i	0.370896	+	0.928674i	$0.379051\pi$
$587$	45.0000i	1.85735i	−0.370896	+	0.928674i	$0.620949\pi$
$588$	24.0000i	0.989743i
$589$	4.00000	0.164817
$590$	0	0
$591$	−36.0000	−1.48084
$592$	8.00000i	0.328798i
$593$	42.0000i	1.72473i	0.506284	+	0.862367i	$0.331019\pi$
$593$	42.0000i	1.72473i	−0.506284	+	0.862367i	$0.668981\pi$
$594$	0	0
$595$	0	0
$596$	−42.0000	−1.72039
$597$	− 22.0000i	− 0.900400i
$598$	0	0
$599$	36.0000	1.47092	0.735460	−	0.677568i	$-0.236966\pi$
$599$	36.0000	1.47092	0.735460	+	0.677568i	$0.236966\pi$
$600$	0	0
$601$	26.0000	1.06056	0.530281	−	0.847822i	$-0.322086\pi$
$601$	26.0000	1.06056	0.530281	+	0.847822i	$0.322086\pi$
$602$	0	0
$603$	4.00000i	0.162893i
$604$	−20.0000	−0.813788
$605$	0	0
$606$	0	0
$607$	32.0000i	1.29884i	0.760430	+	0.649420i	$0.224988\pi$
$607$	32.0000i	1.29884i	−0.760430	+	0.649420i	$0.775012\pi$
$608$	0	0
$609$	−12.0000	−0.486265
$610$	0	0
$611$	12.0000	0.485468
$612$	6.00000i	0.242536i
$613$	− 29.0000i	− 1.17130i	−0.810564	−	0.585649i	$-0.800840\pi$
$613$	− 29.0000i	− 1.17130i	0.810564	−	0.585649i	$-0.199160\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	9.00000i	0.362326i	0.983453	+	0.181163i	$0.0579862\pi$
$617$	9.00000i	0.362326i	−0.983453	+	0.181163i	$0.942014\pi$
$618$	0	0
$619$	−44.0000	−1.76851	−0.884255	−	0.467005i	$-0.845333\pi$
$619$	−44.0000	−1.76851	−0.884255	+	0.467005i	$0.845333\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	12.0000i	0.480770i
$624$	−32.0000	−1.28103
$625$	0	0
$626$	0	0
$627$	− 6.00000i	− 0.239617i
$628$	28.0000i	1.11732i
$629$	6.00000	0.239236
$630$	0	0
$631$	11.0000	0.437903	0.218952	−	0.975736i	$-0.429736\pi$
$631$	11.0000	0.437903	0.218952	+	0.975736i	$0.429736\pi$
$632$	0	0
$633$	28.0000i	1.11290i
$634$	0	0
$635$	0	0
$636$	48.0000	1.90332
$637$	24.0000i	0.950915i
$638$	0	0
$639$	−6.00000	−0.237356
$640$	0	0
$641$	0	0		−	1.00000i	$-0.5\pi$
$641$	0	0			1.00000i	$0.5\pi$
$642$	0	0
$643$	13.0000i	0.512670i	0.966588	+	0.256335i	$0.0825150\pi$
$643$	13.0000i	0.512670i	−0.966588	+	0.256335i	$0.917485\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	27.0000i	1.06148i	0.847535	+	0.530740i	$0.178086\pi$
$647$	27.0000i	1.06148i	−0.847535	+	0.530740i	$0.821914\pi$
$648$	0	0
$649$	18.0000	0.706562
$650$	0	0
$651$	−8.00000	−0.313545
$652$	− 40.0000i	− 1.56652i
$653$	39.0000i	1.52619i	0.646288	+	0.763094i	$0.276321\pi$
$653$	39.0000i	1.52619i	−0.646288	+	0.763094i	$0.723679\pi$
$654$	0	0
$655$	0	0
$656$	−24.0000	−0.937043
$657$	− 7.00000i	− 0.273096i
$658$	0	0
$659$	30.0000	1.16863	0.584317	−	0.811525i	$-0.301362\pi$
$659$	30.0000	1.16863	0.584317	+	0.811525i	$0.301362\pi$
$660$	0	0
$661$	32.0000	1.24466	0.622328	−	0.782757i	$-0.286187\pi$
$661$	32.0000	1.24466	0.622328	+	0.782757i	$0.286187\pi$
$662$	0	0
$663$	24.0000i	0.932083i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	0	0
$668$	− 36.0000i	− 1.39288i
$669$	−20.0000	−0.773245
$670$	0	0
$671$	−3.00000	−0.115814
$672$	0	0
$673$	10.0000i	0.385472i	0.981251	+	0.192736i	$0.0617360\pi$
$673$	10.0000i	0.385472i	−0.981251	+	0.192736i	$0.938264\pi$
$674$	0	0
$675$	0	0
$676$	−6.00000	−0.230769
$677$	− 42.0000i	− 1.61419i	−0.590421	−	0.807096i	$-0.701038\pi$
$677$	− 42.0000i	− 1.61419i	0.590421	−	0.807096i	$-0.298962\pi$
$678$	0	0
$679$	8.00000	0.307012
$680$	0	0
$681$	−24.0000	−0.919682
$682$	0	0
$683$	− 36.0000i	− 1.37750i	−0.724998	−	0.688751i	$-0.758159\pi$
$683$	− 36.0000i	− 1.37750i	0.724998	−	0.688751i	$-0.241841\pi$
$684$	2.00000	0.0764719
$685$	0	0
$686$	0	0
$687$	− 10.0000i	− 0.381524i
$688$	4.00000i	0.152499i
$689$	48.0000	1.82865
$690$	0	0
$691$	17.0000	0.646710	0.323355	−	0.946278i	$-0.395189\pi$
$691$	17.0000	0.646710	0.323355	+	0.946278i	$0.395189\pi$
$692$	36.0000i	1.36851i
$693$	3.00000i	0.113961i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	18.0000i	0.681799i
$698$	0	0
$699$	−42.0000	−1.58859
$700$	0	0
$701$	6.00000	0.226617	0.113308	−	0.993560i	$-0.463855\pi$
$701$	6.00000	0.226617	0.113308	+	0.993560i	$0.463855\pi$
$702$	0	0
$703$	− 2.00000i	− 0.0754314i
$704$	24.0000	0.904534
$705$	0	0
$706$	0	0
$707$	− 6.00000i	− 0.225653i
$708$	24.0000i	0.901975i
$709$	−26.0000	−0.976450	−0.488225	−	0.872718i	$-0.662356\pi$
$709$	−26.0000	−0.976450	−0.488225	+	0.872718i	$0.662356\pi$
$710$	0	0
$711$	8.00000	0.300023
$712$	0	0
$713$	0	0
$714$	0	0
$715$	0	0
$716$	36.0000	1.34538
$717$	− 30.0000i	− 1.12037i
$718$	0	0
$719$	−15.0000	−0.559406	−0.279703	−	0.960087i	$-0.590236\pi$
$719$	−15.0000	−0.559406	−0.279703	+	0.960087i	$0.590236\pi$
$720$	0	0
$721$	−14.0000	−0.521387
$722$	0	0
$723$	− 20.0000i	− 0.743808i
$724$	4.00000	0.148659
$725$	0	0
$726$	0	0
$727$	− 19.0000i	− 0.704671i	−0.935874	−	0.352335i	$-0.885388\pi$
$727$	− 19.0000i	− 0.704671i	0.935874	−	0.352335i	$-0.114612\pi$
$728$	0	0
$729$	−13.0000	−0.481481
$730$	0	0
$731$	3.00000	0.110959
$732$	− 4.00000i	− 0.147844i
$733$	22.0000i	0.812589i	0.913742	+	0.406294i	$0.133179\pi$
$733$	22.0000i	0.812589i	−0.913742	+	0.406294i	$0.866821\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 12.0000i	− 0.442026i
$738$	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
$741$	8.00000	0.293887
$742$	0	0
$743$	24.0000i	0.880475i	0.897881	+	0.440237i	$0.145106\pi$
$743$	24.0000i	0.880475i	−0.897881	+	0.440237i	$0.854894\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	12.0000i	0.439057i
$748$	− 18.0000i	− 0.658145i
$749$	−18.0000	−0.657706
$750$	0	0
$751$	32.0000	1.16770	0.583848	−	0.811863i	$-0.301546\pi$
$751$	32.0000	1.16770	0.583848	+	0.811863i	$0.301546\pi$
$752$	− 12.0000i	− 0.437595i
$753$	42.0000i	1.53057i
$754$	0	0
$755$	0	0
$756$	8.00000	0.290957
$757$	− 25.0000i	− 0.908640i	−0.890838	−	0.454320i	$-0.849882\pi$
$757$	− 25.0000i	− 0.908640i	0.890838	−	0.454320i	$-0.150118\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	33.0000	1.19625	0.598125	−	0.801403i	$-0.295913\pi$
$761$	33.0000	1.19625	0.598125	+	0.801403i	$0.295913\pi$
$762$	0	0
$763$	− 16.0000i	− 0.579239i
$764$	6.00000	0.217072
$765$	0	0
$766$	0	0
$767$	24.0000i	0.866590i
$768$	32.0000i	1.15470i
$769$	−23.0000	−0.829401	−0.414701	−	0.909958i	$-0.636114\pi$
$769$	−23.0000	−0.829401	−0.414701	+	0.909958i	$0.636114\pi$
$770$	0	0
$771$	0	0
$772$	8.00000i	0.287926i
$773$	6.00000i	0.215805i	0.994161	+	0.107903i	$0.0344134\pi$
$773$	6.00000i	0.215805i	−0.994161	+	0.107903i	$0.965587\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	4.00000i	0.143499i
$778$	0	0
$779$	6.00000	0.214972
$780$	0	0
$781$	18.0000	0.644091
$782$	0	0
$783$	− 24.0000i	− 0.857690i
$784$	24.0000	0.857143
$785$	0	0
$786$	0	0
$787$	− 4.00000i	− 0.142585i	−0.997455	−	0.0712923i	$-0.977288\pi$
$787$	− 4.00000i	− 0.142585i	0.997455	−	0.0712923i	$-0.0227123\pi$
$788$	36.0000i	1.28245i
$789$	18.0000	0.640817
$790$	0	0
$791$	−6.00000	−0.213335
$792$	0	0
$793$	− 4.00000i	− 0.142044i
$794$	0	0
$795$	0	0
$796$	−22.0000	−0.779769
$797$	− 12.0000i	− 0.425062i	−0.977154	−	0.212531i	$-0.931829\pi$
$797$	− 12.0000i	− 0.425062i	0.977154	−	0.212531i	$-0.0681706\pi$
$798$	0	0
$799$	−9.00000	−0.318397
$800$	0	0
$801$	12.0000	0.423999
$802$	0	0
$803$	21.0000i	0.741074i
$804$	16.0000	0.564276
$805$	0	0
$806$	0	0
$807$	− 48.0000i	− 1.68968i
$808$	0	0
$809$	9.00000	0.316423	0.158212	−	0.987405i	$-0.449427\pi$
$809$	9.00000	0.316423	0.158212	+	0.987405i	$0.449427\pi$
$810$	0	0
$811$	−16.0000	−0.561836	−0.280918	−	0.959732i	$-0.590639\pi$
$811$	−16.0000	−0.561836	−0.280918	+	0.959732i	$0.590639\pi$
$812$	12.0000i	0.421117i
$813$	− 32.0000i	− 1.12229i
$814$	0	0
$815$	0	0
$816$	24.0000	0.840168
$817$	− 1.00000i	− 0.0349856i
$818$	0	0
$819$	−4.00000	−0.139771
$820$	0	0
$821$	33.0000	1.15171	0.575854	−	0.817553i	$-0.304670\pi$
$821$	33.0000	1.15171	0.575854	+	0.817553i	$0.304670\pi$
$822$	0	0
$823$	49.0000i	1.70803i	0.520246	+	0.854016i	$0.325840\pi$
$823$	49.0000i	1.70803i	−0.520246	+	0.854016i	$0.674160\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	12.0000i	0.417281i	0.977992	+	0.208640i	$0.0669038\pi$
$827$	12.0000i	0.417281i	−0.977992	+	0.208640i	$0.933096\pi$
$828$	0	0
$829$	16.0000	0.555703	0.277851	−	0.960624i	$-0.410378\pi$
$829$	16.0000	0.555703	0.277851	+	0.960624i	$0.410378\pi$
$830$	0	0
$831$	38.0000	1.31821
$832$	32.0000i	1.10940i
$833$	− 18.0000i	− 0.623663i
$834$	0	0
$835$	0	0
$836$	−6.00000	−0.207514
$837$	− 16.0000i	− 0.553041i
$838$	0	0
$839$	−18.0000	−0.621429	−0.310715	−	0.950503i	$-0.600568\pi$
$839$	−18.0000	−0.621429	−0.310715	+	0.950503i	$0.600568\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	0	0
$843$	12.0000i	0.413302i
$844$	28.0000	0.963800
$845$	0	0
$846$	0	0
$847$	2.00000i	0.0687208i
$848$	− 48.0000i	− 1.64833i
$849$	−26.0000	−0.892318
$850$	0	0
$851$	0	0
$852$	24.0000i	0.822226i
$853$	− 26.0000i	− 0.890223i	−0.895475	−	0.445112i	$-0.853164\pi$
$853$	− 26.0000i	− 0.890223i	0.895475	−	0.445112i	$-0.146836\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	18.0000i	0.614868i	0.951569	+	0.307434i	$0.0994704\pi$
$857$	18.0000i	0.614868i	−0.951569	+	0.307434i	$0.900530\pi$
$858$	0	0
$859$	49.0000	1.67186	0.835929	−	0.548837i	$-0.184929\pi$
$859$	49.0000	1.67186	0.835929	+	0.548837i	$0.184929\pi$
$860$	0	0
$861$	−12.0000	−0.408959
$862$	0	0
$863$	− 18.0000i	− 0.612727i	−0.951915	−	0.306364i	$-0.900888\pi$
$863$	− 18.0000i	− 0.612727i	0.951915	−	0.306364i	$-0.0991123\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	16.0000i	0.543388i
$868$	8.00000i	0.271538i
$869$	−24.0000	−0.814144
$870$	0	0
$871$	16.0000	0.542139
$872$	0	0
$873$	− 8.00000i	− 0.270759i
$874$	0	0
$875$	0	0
$876$	−28.0000	−0.946032
$877$	− 22.0000i	− 0.742887i	−0.928456	−	0.371444i	$-0.878863\pi$
$877$	− 22.0000i	− 0.742887i	0.928456	−	0.371444i	$-0.121137\pi$
$878$	0	0
$879$	−24.0000	−0.809500
$880$	0	0
$881$	−27.0000	−0.909653	−0.454827	−	0.890580i	$-0.650299\pi$
$881$	−27.0000	−0.909653	−0.454827	+	0.890580i	$0.650299\pi$
$882$	0	0
$883$	− 47.0000i	− 1.58168i	−0.612026	−	0.790838i	$-0.709645\pi$
$883$	− 47.0000i	− 1.58168i	0.612026	−	0.790838i	$-0.290355\pi$
$884$	24.0000	0.807207
$885$	0	0
$886$	0	0
$887$	18.0000i	0.604381i	0.953248	+	0.302190i	$0.0977178\pi$
$887$	18.0000i	0.604381i	−0.953248	+	0.302190i	$0.902282\pi$
$888$	0	0
$889$	2.00000	0.0670778
$890$	0	0
$891$	−33.0000	−1.10554
$892$	20.0000i	0.669650i
$893$	3.00000i	0.100391i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	24.0000	0.800445
$900$	0	0
$901$	−36.0000	−1.19933
$902$	0	0
$903$	2.00000i	0.0665558i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	8.00000i	0.265636i	0.991140	+	0.132818i	$0.0424025\pi$
$907$	8.00000i	0.265636i	−0.991140	+	0.132818i	$0.957597\pi$
$908$	24.0000i	0.796468i
$909$	−6.00000	−0.199007
$910$	0	0
$911$	−6.00000	−0.198789	−0.0993944	−	0.995048i	$-0.531691\pi$
$911$	−6.00000	−0.198789	−0.0993944	+	0.995048i	$0.531691\pi$
$912$	− 8.00000i	− 0.264906i
$913$	− 36.0000i	− 1.19143i
$914$	0	0
$915$	0	0
$916$	−10.0000	−0.330409
$917$	15.0000i	0.495344i
$918$	0	0
$919$	−20.0000	−0.659739	−0.329870	−	0.944027i	$-0.607005\pi$
$919$	−20.0000	−0.659739	−0.329870	+	0.944027i	$0.607005\pi$
$920$	0	0
$921$	−40.0000	−1.31804
$922$	0	0
$923$	24.0000i	0.789970i
$924$	12.0000	0.394771
$925$	0	0
$926$	0	0
$927$	14.0000i	0.459820i
$928$	0	0
$929$	18.0000	0.590561	0.295280	−	0.955411i	$-0.404587\pi$
$929$	18.0000	0.590561	0.295280	+	0.955411i	$0.404587\pi$
$930$	0	0
$931$	−6.00000	−0.196642
$932$	42.0000i	1.37576i
$933$	− 6.00000i	− 0.196431i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 7.00000i	− 0.228680i	−0.993442	−	0.114340i	$-0.963525\pi$
$937$	− 7.00000i	− 0.228680i	0.993442	−	0.114340i	$-0.0364753\pi$
$938$	0	0
$939$	−20.0000	−0.652675
$940$	0	0
$941$	−18.0000	−0.586783	−0.293392	−	0.955992i	$-0.594784\pi$
$941$	−18.0000	−0.586783	−0.293392	+	0.955992i	$0.594784\pi$
$942$	0	0
$943$	0	0
$944$	24.0000	0.781133
$945$	0	0
$946$	0	0
$947$	− 36.0000i	− 1.16984i	−0.811090	−	0.584921i	$-0.801125\pi$
$947$	− 36.0000i	− 1.16984i	0.811090	−	0.584921i	$-0.198875\pi$
$948$	− 32.0000i	− 1.03931i
$949$	−28.0000	−0.908918
$950$	0	0
$951$	−12.0000	−0.389127
$952$	0	0
$953$	48.0000i	1.55487i	0.628962	+	0.777436i	$0.283480\pi$
$953$	48.0000i	1.55487i	−0.628962	+	0.777436i	$0.716520\pi$
$954$	0	0
$955$	0	0
$956$	−30.0000	−0.970269
$957$	− 36.0000i	− 1.16371i
$958$	0	0
$959$	−3.00000	−0.0968751
$960$	0	0
$961$	−15.0000	−0.483871
$962$	0	0
$963$	18.0000i	0.580042i
$964$	−20.0000	−0.644157
$965$	0	0
$966$	0	0
$967$	− 40.0000i	− 1.28631i	−0.765735	−	0.643157i	$-0.777624\pi$
$967$	− 40.0000i	− 1.28631i	0.765735	−	0.643157i	$-0.222376\pi$
$968$	0	0
$969$	−6.00000	−0.192748
$970$	0	0
$971$	60.0000	1.92549	0.962746	−	0.270408i	$-0.0871586\pi$
$971$	60.0000	1.92549	0.962746	+	0.270408i	$0.0871586\pi$
$972$	− 20.0000i	− 0.641500i
$973$	− 13.0000i	− 0.416761i
$974$	0	0
$975$	0	0
$976$	−4.00000	−0.128037
$977$	24.0000i	0.767828i	0.923369	+	0.383914i	$0.125424\pi$
$977$	24.0000i	0.767828i	−0.923369	+	0.383914i	$0.874576\pi$
$978$	0	0
$979$	−36.0000	−1.15056
$980$	0	0
$981$	−16.0000	−0.510841
$982$	0	0
$983$	36.0000i	1.14822i	0.818778	+	0.574111i	$0.194652\pi$
$983$	36.0000i	1.14822i	−0.818778	+	0.574111i	$0.805348\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	− 6.00000i	− 0.190982i
$988$	− 8.00000i	− 0.254514i
$989$	0	0
$990$	0	0
$991$	−34.0000	−1.08005	−0.540023	−	0.841650i	$-0.681584\pi$
$991$	−34.0000	−1.08005	−0.540023	+	0.841650i	$0.681584\pi$
$992$	0	0
$993$	− 56.0000i	− 1.77711i
$994$	0	0
$995$	0	0
$996$	48.0000	1.52094
$997$	17.0000i	0.538395i	0.963085	+	0.269198i	$0.0867585\pi$
$997$	17.0000i	0.538395i	−0.963085	+	0.269198i	$0.913241\pi$
$998$	0	0
$999$	−8.00000	−0.253109

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	475.2.b.a.324.2		2
5.2	odd	4		475.2.a.b.1.1		1
5.3	odd	4		19.2.a.a.1.1	✓	1
5.4	even	2	inner	475.2.b.a.324.1		2
15.2	even	4		4275.2.a.i.1.1		1
15.8	even	4		171.2.a.b.1.1		1
20.3	even	4		304.2.a.f.1.1		1
20.7	even	4		7600.2.a.c.1.1		1
35.3	even	12		931.2.f.b.324.1		2
35.13	even	4		931.2.a.a.1.1		1
35.18	odd	12		931.2.f.c.324.1		2
35.23	odd	12		931.2.f.c.704.1		2
35.33	even	12		931.2.f.b.704.1		2
40.3	even	4		1216.2.a.b.1.1		1
40.13	odd	4		1216.2.a.o.1.1		1
55.43	even	4		2299.2.a.b.1.1		1
60.23	odd	4		2736.2.a.c.1.1		1
65.38	odd	4		3211.2.a.a.1.1		1
85.33	odd	4		5491.2.a.b.1.1		1
95.3	even	36		361.2.e.e.28.1		6
95.8	even	12		361.2.c.a.292.1		2
95.13	even	36		361.2.e.e.245.1		6
95.18	even	4		361.2.a.b.1.1		1
95.23	odd	36		361.2.e.d.54.1		6
95.28	odd	36		361.2.e.d.62.1		6
95.33	even	36		361.2.e.e.234.1		6
95.37	even	4		9025.2.a.d.1.1		1
95.43	odd	36		361.2.e.d.234.1		6
95.48	even	36		361.2.e.e.62.1		6
95.53	even	36		361.2.e.e.54.1		6
95.63	odd	36		361.2.e.d.245.1		6
95.68	odd	12		361.2.c.c.292.1		2
95.73	odd	36		361.2.e.d.28.1		6
95.78	even	36		361.2.e.e.99.1		6
95.83	odd	12		361.2.c.c.68.1		2
95.88	even	12		361.2.c.a.68.1		2
95.93	odd	36		361.2.e.d.99.1		6
105.83	odd	4		8379.2.a.j.1.1		1
285.113	odd	4		3249.2.a.d.1.1		1
380.303	odd	4		5776.2.a.c.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
19.2.a.a.1.1	✓	1	5.3	odd	4
171.2.a.b.1.1		1	15.8	even	4
304.2.a.f.1.1		1	20.3	even	4
361.2.a.b.1.1		1	95.18	even	4
361.2.c.a.68.1		2	95.88	even	12
361.2.c.a.292.1		2	95.8	even	12
361.2.c.c.68.1		2	95.83	odd	12
361.2.c.c.292.1		2	95.68	odd	12
361.2.e.d.28.1		6	95.73	odd	36
361.2.e.d.54.1		6	95.23	odd	36
361.2.e.d.62.1		6	95.28	odd	36
361.2.e.d.99.1		6	95.93	odd	36
361.2.e.d.234.1		6	95.43	odd	36
361.2.e.d.245.1		6	95.63	odd	36
361.2.e.e.28.1		6	95.3	even	36
361.2.e.e.54.1		6	95.53	even	36
361.2.e.e.62.1		6	95.48	even	36
361.2.e.e.99.1		6	95.78	even	36
361.2.e.e.234.1		6	95.33	even	36
361.2.e.e.245.1		6	95.13	even	36
475.2.a.b.1.1		1	5.2	odd	4
475.2.b.a.324.1		2	5.4	even	2	inner
475.2.b.a.324.2		2	1.1	even	1	trivial
931.2.a.a.1.1		1	35.13	even	4
931.2.f.b.324.1		2	35.3	even	12
931.2.f.b.704.1		2	35.33	even	12
931.2.f.c.324.1		2	35.18	odd	12
931.2.f.c.704.1		2	35.23	odd	12
1216.2.a.b.1.1		1	40.3	even	4
1216.2.a.o.1.1		1	40.13	odd	4
2299.2.a.b.1.1		1	55.43	even	4
2736.2.a.c.1.1		1	60.23	odd	4
3211.2.a.a.1.1		1	65.38	odd	4
3249.2.a.d.1.1		1	285.113	odd	4
4275.2.a.i.1.1		1	15.2	even	4
5491.2.a.b.1.1		1	85.33	odd	4
5776.2.a.c.1.1		1	380.303	odd	4
7600.2.a.c.1.1		1	20.7	even	4
8379.2.a.j.1.1		1	105.83	odd	4
9025.2.a.d.1.1		1	95.37	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:	\( N \)	\(=\)	\( 475 = 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	475.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+2.00000i q^{3} +2.00000 q^{4} -1.00000i q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q+2.00000i q^{3} +2.00000 q^{4} -1.00000i q^{7} -1.00000 q^{9} +3.00000 q^{11} +4.00000i q^{12} +4.00000i q^{13} +4.00000 q^{16} -3.00000i q^{17} -1.00000 q^{19} +2.00000 q^{21} +4.00000i q^{27} -2.00000i q^{28} -6.00000 q^{29} -4.00000 q^{31} +6.00000i q^{33} -2.00000 q^{36} +2.00000i q^{37} -8.00000 q^{39} -6.00000 q^{41} +1.00000i q^{43} +6.00000 q^{44} -3.00000i q^{47} +8.00000i q^{48} +6.00000 q^{49} +6.00000 q^{51} +8.00000i q^{52} -12.0000i q^{53} -2.00000i q^{57} +6.00000 q^{59} -1.00000 q^{61} +1.00000i q^{63} +8.00000 q^{64} -4.00000i q^{67} -6.00000i q^{68} +6.00000 q^{71} +7.00000i q^{73} -2.00000 q^{76} -3.00000i q^{77} -8.00000 q^{79} -11.0000 q^{81} -12.0000i q^{83} +4.00000 q^{84} -12.0000i q^{87} -12.0000 q^{89} +4.00000 q^{91} -8.00000i q^{93} +8.00000i q^{97} -3.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 4 q^{4} - 2 q^{9}+O(q^{10}) \) 2 * q + 4 * q^4 - 2 * q^9	\( 2 q + 4 q^{4} - 2 q^{9} + 6 q^{11} + 8 q^{16} - 2 q^{19} + 4 q^{21} - 12 q^{29} - 8 q^{31} - 4 q^{36} - 16 q^{39} - 12 q^{41} + 12 q^{44} + 12 q^{49} + 12 q^{51} + 12 q^{59} - 2 q^{61} + 16 q^{64} + 12 q^{71} - 4 q^{76} - 16 q^{79} - 22 q^{81} + 8 q^{84} - 24 q^{89} + 8 q^{91} - 6 q^{99}+O(q^{100}) \) 2 * q + 4 * q^4 - 2 * q^9 + 6 * q^11 + 8 * q^16 - 2 * q^19 + 4 * q^21 - 12 * q^29 - 8 * q^31 - 4 * q^36 - 16 * q^39 - 12 * q^41 + 12 * q^44 + 12 * q^49 + 12 * q^51 + 12 * q^59 - 2 * q^61 + 16 * q^64 + 12 * q^71 - 4 * q^76 - 16 * q^79 - 22 * q^81 + 8 * q^84 - 24 * q^89 + 8 * q^91 - 6 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

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Database

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