LMFDB - Embedded newform 2475.2.c.j.199.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2475,2,Mod(199,2475)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2475, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("2475.199");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2475 = 3^{2} \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2475.c (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:	$19.7629745003$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{17})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 9x^{2} + 16 $ x^4 + 9*x^2 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			199.4
Root			$2.56155i$ of defining polynomial
Character	$\chi$	$=$	2475.199
Dual form			2475.2.c.j.199.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.56155i q^{2} -4.56155 q^{4} -1.00000i q^{7} -6.56155i q^{8} +O(q^{10})$	$q+2.56155i q^{2} -4.56155 q^{4} -1.00000i q^{7} -6.56155i q^{8} +1.00000 q^{11} -0.438447i q^{13} +2.56155 q^{14} +7.68466 q^{16} +1.43845i q^{17} -3.00000 q^{19} +2.56155i q^{22} -6.56155i q^{23} +1.12311 q^{26} +4.56155i q^{28} +5.12311 q^{29} +4.68466 q^{31} +6.56155i q^{32} -3.68466 q^{34} +10.8078i q^{37} -7.68466i q^{38} +7.68466 q^{41} +4.68466i q^{43} -4.56155 q^{44} +16.8078 q^{46} +5.43845i q^{47} +6.00000 q^{49} +2.00000i q^{52} +1.12311i q^{53} -6.56155 q^{56} +13.1231i q^{58} +10.5616 q^{59} -7.56155 q^{61} +12.0000i q^{62} -1.43845 q^{64} +6.68466i q^{67} -6.56155i q^{68} -8.80776 q^{71} +16.2462i q^{73} -27.6847 q^{74} +13.6847 q^{76} -1.00000i q^{77} +15.6847 q^{79} +19.6847i q^{82} -12.0000 q^{86} -6.56155i q^{88} -17.1231 q^{89} -0.438447 q^{91} +29.9309i q^{92} -13.9309 q^{94} -2.12311i q^{97} +15.3693i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 10 q^{4}+O(q^{10}) $ 4 * q - 10 * q^4	$ 4 q - 10 q^{4} + 4 q^{11} + 2 q^{14} + 6 q^{16} - 12 q^{19} - 12 q^{26} + 4 q^{29} - 6 q^{31} + 10 q^{34} + 6 q^{41} - 10 q^{44} + 26 q^{46} + 24 q^{49} - 18 q^{56} + 34 q^{59} - 22 q^{61} - 14 q^{64} + 6 q^{71} - 86 q^{74} + 30 q^{76} + 38 q^{79} - 48 q^{86} - 52 q^{89} - 10 q^{91} + 2 q^{94}+O(q^{100}) $ 4 * q - 10 * q^4 + 4 * q^11 + 2 * q^14 + 6 * q^16 - 12 * q^19 - 12 * q^26 + 4 * q^29 - 6 * q^31 + 10 * q^34 + 6 * q^41 - 10 * q^44 + 26 * q^46 + 24 * q^49 - 18 * q^56 + 34 * q^59 - 22 * q^61 - 14 * q^64 + 6 * q^71 - 86 * q^74 + 30 * q^76 + 38 * q^79 - 48 * q^86 - 52 * q^89 - 10 * q^91 + 2 * q^94

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2475\mathbb{Z}\right)^\times$.

$n$	$551$	$2026$	$2377$
$\chi(n)$	$1$	$1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	2.56155i	1.81129i	0.424035	+	0.905646i	$0.360613\pi$
$2$	2.56155i	1.81129i	−0.424035	+	0.905646i	$0.639387\pi$
$3$	0	0
$3$	0	0
$4$	−4.56155	−2.28078
$5$	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	$-0.0605189\pi$
$8$	− 6.56155i	− 2.31986i
$9$	0	0
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	− 0.438447i	− 0.121603i	−0.998150	−	0.0608017i	$-0.980634\pi$
$13$	− 0.438447i	− 0.121603i	0.998150	−	0.0608017i	$-0.0193657\pi$
$14$	2.56155	0.684604
$15$	0	0
$16$	7.68466	1.92116
$17$	1.43845i	0.348875i	0.984668	+	0.174437i	$0.0558106\pi$
$17$	1.43845i	0.348875i	−0.984668	+	0.174437i	$0.944189\pi$
$18$	0	0
$19$	−3.00000	−0.688247	−0.344124	−	0.938924i	$-0.611824\pi$
$19$	−3.00000	−0.688247	−0.344124	+	0.938924i	$0.611824\pi$
$20$	0	0
$21$	0	0
$22$	2.56155i	0.546125i
$23$	− 6.56155i	− 1.36818i	−0.729398	−	0.684089i	$-0.760200\pi$
$23$	− 6.56155i	− 1.36818i	0.729398	−	0.684089i	$-0.239800\pi$
$24$	0	0
$25$	0	0
$26$	1.12311	0.220259
$27$	0	0
$28$	4.56155i	0.862052i
$29$	5.12311	0.951337	0.475668	−	0.879625i	$-0.342206\pi$
$29$	5.12311	0.951337	0.475668	+	0.879625i	$0.342206\pi$
$30$	0	0
$31$	4.68466	0.841389	0.420695	−	0.907202i	$-0.361786\pi$
$31$	4.68466	0.841389	0.420695	+	0.907202i	$0.361786\pi$
$32$	6.56155i	1.15993i
$33$	0	0
$34$	−3.68466	−0.631914
$35$	0	0
$36$	0	0
$37$	10.8078i	1.77679i	0.459084	+	0.888393i	$0.348178\pi$
$37$	10.8078i	1.77679i	−0.459084	+	0.888393i	$0.651822\pi$
$38$	− 7.68466i	− 1.24662i
$39$	0	0
$40$	0	0
$41$	7.68466	1.20014	0.600071	−	0.799947i	$-0.295139\pi$
$41$	7.68466	1.20014	0.600071	+	0.799947i	$0.295139\pi$
$42$	0	0
$43$	4.68466i	0.714404i	0.934027	+	0.357202i	$0.116269\pi$
$43$	4.68466i	0.714404i	−0.934027	+	0.357202i	$0.883731\pi$
$44$	−4.56155	−0.687680
$45$	0	0
$46$	16.8078	2.47817
$47$	5.43845i	0.793279i	0.917974	+	0.396640i	$0.129824\pi$
$47$	5.43845i	0.793279i	−0.917974	+	0.396640i	$0.870176\pi$
$48$	0	0
$49$	6.00000	0.857143
$50$	0	0
$51$	0	0
$52$	2.00000i	0.277350i
$53$	1.12311i	0.154270i	0.997021	+	0.0771352i	$0.0245773\pi$
$53$	1.12311i	0.154270i	−0.997021	+	0.0771352i	$0.975423\pi$
$54$	0	0
$55$	0	0
$56$	−6.56155	−0.876824
$57$	0	0
$58$	13.1231i	1.72315i
$59$	10.5616	1.37500	0.687499	−	0.726186i	$-0.258709\pi$
$59$	10.5616	1.37500	0.687499	+	0.726186i	$0.258709\pi$
$60$	0	0
$61$	−7.56155	−0.968158	−0.484079	−	0.875024i	$-0.660845\pi$
$61$	−7.56155	−0.968158	−0.484079	+	0.875024i	$0.660845\pi$
$62$	12.0000i	1.52400i
$63$	0	0
$64$	−1.43845	−0.179806
$65$	0	0
$66$	0	0
$67$	6.68466i	0.816661i	0.912834	+	0.408331i	$0.133889\pi$
$67$	6.68466i	0.816661i	−0.912834	+	0.408331i	$0.866111\pi$
$68$	− 6.56155i	− 0.795705i
$69$	0	0
$70$	0	0
$71$	−8.80776	−1.04529	−0.522645	−	0.852551i	$-0.675055\pi$
$71$	−8.80776	−1.04529	−0.522645	+	0.852551i	$0.675055\pi$
$72$	0	0
$73$	16.2462i	1.90148i	0.309997	+	0.950738i	$0.399672\pi$
$73$	16.2462i	1.90148i	−0.309997	+	0.950738i	$0.600328\pi$
$74$	−27.6847	−3.21828
$75$	0	0
$76$	13.6847	1.56974
$77$	− 1.00000i	− 0.113961i
$78$	0	0
$79$	15.6847	1.76466	0.882331	−	0.470629i	$-0.155973\pi$
$79$	15.6847	1.76466	0.882331	+	0.470629i	$0.155973\pi$
$80$	0	0
$81$	0	0
$82$	19.6847i	2.17381i
$83$	0	0	1.00000			$0$
$83$	0	0	−1.00000			$\pi$
$84$	0	0
$85$	0	0
$86$	−12.0000	−1.29399
$87$	0	0
$88$	− 6.56155i	− 0.699464i
$89$	−17.1231	−1.81505	−0.907523	−	0.420003i	$-0.862029\pi$
$89$	−17.1231	−1.81505	−0.907523	+	0.420003i	$0.862029\pi$
$90$	0	0
$91$	−0.438447	−0.0459618
$92$	29.9309i	3.12051i
$93$	0	0
$94$	−13.9309	−1.43686
$95$	0	0
$96$	0	0
$97$	− 2.12311i	− 0.215569i	−0.994174	−	0.107784i	$-0.965624\pi$
$97$	− 2.12311i	− 0.215569i	0.994174	−	0.107784i	$-0.0343756\pi$
$98$	15.3693i	1.55254i
$99$	0	0
$100$	0	0
$101$	−1.43845	−0.143131	−0.0715654	−	0.997436i	$-0.522799\pi$
$101$	−1.43845	−0.143131	−0.0715654	+	0.997436i	$0.522799\pi$
$102$	0	0
$103$	− 9.12311i	− 0.898926i	−0.893299	−	0.449463i	$-0.851615\pi$
$103$	− 9.12311i	− 0.898926i	0.893299	−	0.449463i	$-0.148385\pi$
$104$	−2.87689	−0.282103
$105$	0	0
$106$	−2.87689	−0.279429
$107$	− 9.12311i	− 0.881964i	−0.897516	−	0.440982i	$-0.854630\pi$
$107$	− 9.12311i	− 0.881964i	0.897516	−	0.440982i	$-0.145370\pi$
$108$	0	0
$109$	7.80776	0.747848	0.373924	−	0.927459i	$-0.378012\pi$
$109$	7.80776	0.747848	0.373924	+	0.927459i	$0.378012\pi$
$110$	0	0
$111$	0	0
$112$	− 7.68466i	− 0.726132i
$113$	9.12311i	0.858230i	0.903250	+	0.429115i	$0.141174\pi$
$113$	9.12311i	0.858230i	−0.903250	+	0.429115i	$0.858826\pi$
$114$	0	0
$115$	0	0
$116$	−23.3693	−2.16979
$117$	0	0
$118$	27.0540i	2.49052i
$119$	1.43845	0.131862
$120$	0	0
$121$	1.00000	0.0909091
$122$	− 19.3693i	− 1.75362i
$123$	0	0
$124$	−21.3693	−1.91902
$125$	0	0
$126$	0	0
$127$	4.80776i	0.426620i	0.976985	+	0.213310i	$0.0684244\pi$
$127$	4.80776i	0.426620i	−0.976985	+	0.213310i	$0.931576\pi$
$128$	9.43845i	0.834249i
$129$	0	0
$130$	0	0
$131$	16.4924	1.44095	0.720475	−	0.693481i	$-0.243924\pi$
$131$	16.4924	1.44095	0.720475	+	0.693481i	$0.243924\pi$
$132$	0	0
$133$	3.00000i	0.260133i
$134$	−17.1231	−1.47921
$135$	0	0
$136$	9.43845	0.809340
$137$	− 3.36932i	− 0.287860i	−0.989588	−	0.143930i	$-0.954026\pi$
$137$	− 3.36932i	− 0.287860i	0.989588	−	0.143930i	$-0.0459740\pi$
$138$	0	0
$139$	−8.00000	−0.678551	−0.339276	−	0.940687i	$-0.610182\pi$
$139$	−8.00000	−0.678551	−0.339276	+	0.940687i	$0.610182\pi$
$140$	0	0
$141$	0	0
$142$	− 22.5616i	− 1.89332i
$143$	− 0.438447i	− 0.0366648i
$144$	0	0
$145$	0	0
$146$	−41.6155	−3.44413
$147$	0	0
$148$	− 49.3002i	− 4.05245i
$149$	8.80776	0.721560	0.360780	−	0.932651i	$-0.382510\pi$
$149$	8.80776	0.721560	0.360780	+	0.932651i	$0.382510\pi$
$150$	0	0
$151$	−16.6847	−1.35778	−0.678889	−	0.734241i	$-0.737538\pi$
$151$	−16.6847	−1.35778	−0.678889	+	0.734241i	$0.737538\pi$
$152$	19.6847i	1.59664i
$153$	0	0
$154$	2.56155	0.206416
$155$	0	0
$156$	0	0
$157$	− 2.68466i	− 0.214259i	−0.994245	−	0.107130i	$-0.965834\pi$
$157$	− 2.68466i	− 0.214259i	0.994245	−	0.107130i	$-0.0341660\pi$
$158$	40.1771i	3.19632i
$159$	0	0
$160$	0	0
$161$	−6.56155	−0.517123
$162$	0	0
$163$	− 5.31534i	− 0.416330i	−0.978094	−	0.208165i	$-0.933251\pi$
$163$	− 5.31534i	− 0.416330i	0.978094	−	0.208165i	$-0.0667490\pi$
$164$	−35.0540	−2.73726
$165$	0	0
$166$	0	0
$167$	− 15.3693i	− 1.18931i	−0.803980	−	0.594657i	$-0.797288\pi$
$167$	− 15.3693i	− 1.18931i	0.803980	−	0.594657i	$-0.202712\pi$
$168$	0	0
$169$	12.8078	0.985213
$170$	0	0
$171$	0	0
$172$	− 21.3693i	− 1.62940i
$173$	− 13.4384i	− 1.02171i	−0.859668	−	0.510853i	$-0.829330\pi$
$173$	− 13.4384i	− 1.02171i	0.859668	−	0.510853i	$-0.170670\pi$
$174$	0	0
$175$	0	0
$176$	7.68466	0.579253
$177$	0	0
$178$	− 43.8617i	− 3.28758i
$179$	14.5616	1.08838	0.544191	−	0.838961i	$-0.316837\pi$
$179$	14.5616	1.08838	0.544191	+	0.838961i	$0.316837\pi$
$180$	0	0
$181$	11.8769	0.882803	0.441401	−	0.897310i	$-0.354482\pi$
$181$	11.8769	0.882803	0.441401	+	0.897310i	$0.354482\pi$
$182$	− 1.12311i	− 0.0832501i
$183$	0	0
$184$	−43.0540	−3.17398
$185$	0	0
$186$	0	0
$187$	1.43845i	0.105190i
$188$	− 24.8078i	− 1.80929i
$189$	0	0
$190$	0	0
$191$	1.93087	0.139713	0.0698564	−	0.997557i	$-0.477746\pi$
$191$	1.93087	0.139713	0.0698564	+	0.997557i	$0.477746\pi$
$192$	0	0
$193$	17.5616i	1.26411i	0.774924	+	0.632054i	$0.217788\pi$
$193$	17.5616i	1.26411i	−0.774924	+	0.632054i	$0.782212\pi$
$194$	5.43845	0.390458
$195$	0	0
$196$	−27.3693	−1.95495
$197$	17.9309i	1.27752i	0.769405	+	0.638761i	$0.220553\pi$
$197$	17.9309i	1.27752i	−0.769405	+	0.638761i	$0.779447\pi$
$198$	0	0
$199$	13.8078	0.978806	0.489403	−	0.872058i	$-0.337215\pi$
$199$	13.8078	0.978806	0.489403	+	0.872058i	$0.337215\pi$
$200$	0	0
$201$	0	0
$202$	− 3.68466i	− 0.259252i
$203$	− 5.12311i	− 0.359572i
$204$	0	0
$205$	0	0
$206$	23.3693	1.62822
$207$	0	0
$208$	− 3.36932i	− 0.233620i
$209$	−3.00000	−0.207514
$210$	0	0
$211$	3.31534	0.228238	0.114119	−	0.993467i	$-0.463596\pi$
$211$	3.31534	0.228238	0.114119	+	0.993467i	$0.463596\pi$
$212$	− 5.12311i	− 0.351856i
$213$	0	0
$214$	23.3693	1.59749
$215$	0	0
$216$	0	0
$217$	− 4.68466i	− 0.318015i
$218$	20.0000i	1.35457i
$219$	0	0
$220$	0	0
$221$	0.630683	0.0424243
$222$	0	0
$223$	21.8078i	1.46036i	0.683257	+	0.730178i	$0.260563\pi$
$223$	21.8078i	1.46036i	−0.683257	+	0.730178i	$0.739437\pi$
$224$	6.56155	0.438412
$225$	0	0
$226$	−23.3693	−1.55450
$227$	20.4924i	1.36013i	0.733152	+	0.680065i	$0.238048\pi$
$227$	20.4924i	1.36013i	−0.733152	+	0.680065i	$0.761952\pi$
$228$	0	0
$229$	8.12311	0.536790	0.268395	−	0.963309i	$-0.413507\pi$
$229$	8.12311	0.536790	0.268395	+	0.963309i	$0.413507\pi$
$230$	0	0
$231$	0	0
$232$	− 33.6155i	− 2.20697i
$233$	− 24.8078i	− 1.62521i	−0.582814	−	0.812605i	$-0.698049\pi$
$233$	− 24.8078i	− 1.62521i	0.582814	−	0.812605i	$-0.301951\pi$
$234$	0	0
$235$	0	0
$236$	−48.1771	−3.13606
$237$	0	0
$238$	3.68466i	0.238841i
$239$	−16.4924	−1.06681	−0.533403	−	0.845861i	$-0.679087\pi$
$239$	−16.4924	−1.06681	−0.533403	+	0.845861i	$0.679087\pi$
$240$	0	0
$241$	−14.0540	−0.905296	−0.452648	−	0.891689i	$-0.649521\pi$
$241$	−14.0540	−0.905296	−0.452648	+	0.891689i	$0.649521\pi$
$242$	2.56155i	0.164663i
$243$	0	0
$244$	34.4924	2.20815
$245$	0	0
$246$	0	0
$247$	1.31534i	0.0836932i
$248$	− 30.7386i	− 1.95191i
$249$	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	$0.376365\pi$
$252$	0	0
$253$	− 6.56155i	− 0.412521i
$254$	−12.3153	−0.772733
$255$	0	0
$256$	−27.0540	−1.69087
$257$	18.2462i	1.13817i	0.822280	+	0.569084i	$0.192702\pi$
$257$	18.2462i	1.13817i	−0.822280	+	0.569084i	$0.807298\pi$
$258$	0	0
$259$	10.8078	0.671562
$260$	0	0
$261$	0	0
$262$	42.2462i	2.60998i
$263$	3.36932i	0.207761i	0.994590	+	0.103880i	$0.0331259\pi$
$263$	3.36932i	0.207761i	−0.994590	+	0.103880i	$0.966874\pi$
$264$	0	0
$265$	0	0
$266$	−7.68466	−0.471177
$267$	0	0
$268$	− 30.4924i	− 1.86262i
$269$	28.4924	1.73721	0.868607	−	0.495502i	$-0.165016\pi$
$269$	28.4924	1.73721	0.868607	+	0.495502i	$0.165016\pi$
$270$	0	0
$271$	−28.1771	−1.71164	−0.855818	−	0.517277i	$-0.826946\pi$
$271$	−28.1771	−1.71164	−0.855818	+	0.517277i	$0.826946\pi$
$272$	11.0540i	0.670246i
$273$	0	0
$274$	8.63068	0.521399
$275$	0	0
$276$	0	0
$277$	23.1771i	1.39258i	0.717763	+	0.696288i	$0.245166\pi$
$277$	23.1771i	1.39258i	−0.717763	+	0.696288i	$0.754834\pi$
$278$	− 20.4924i	− 1.22905i
$279$	0	0
$280$	0	0
$281$	3.19224	0.190433	0.0952164	−	0.995457i	$-0.469646\pi$
$281$	3.19224	0.190433	0.0952164	+	0.995457i	$0.469646\pi$
$282$	0	0
$283$	32.1231i	1.90952i	0.297377	+	0.954760i	$0.403888\pi$
$283$	32.1231i	1.90952i	−0.297377	+	0.954760i	$0.596112\pi$
$284$	40.1771	2.38407
$285$	0	0
$286$	1.12311	0.0664106
$287$	− 7.68466i	− 0.453611i
$288$	0	0
$289$	14.9309	0.878286
$290$	0	0
$291$	0	0
$292$	− 74.1080i	− 4.33684i
$293$	− 2.06913i	− 0.120880i	−0.998172	−	0.0604399i	$-0.980750\pi$
$293$	− 2.06913i	− 0.120880i	0.998172	−	0.0604399i	$-0.0192504\pi$
$294$	0	0
$295$	0	0
$296$	70.9157	4.12189
$297$	0	0
$298$	22.5616i	1.30696i
$299$	−2.87689	−0.166375
$300$	0	0
$301$	4.68466	0.270019
$302$	− 42.7386i	− 2.45933i
$303$	0	0
$304$	−23.0540	−1.32224
$305$	0	0
$306$	0	0
$307$	22.9309i	1.30873i	0.756177	+	0.654367i	$0.227065\pi$
$307$	22.9309i	1.30873i	−0.756177	+	0.654367i	$0.772935\pi$
$308$	4.56155i	0.259919i
$309$	0	0
$310$	0	0
$311$	13.7538	0.779906	0.389953	−	0.920835i	$-0.372491\pi$
$311$	13.7538	0.779906	0.389953	+	0.920835i	$0.372491\pi$
$312$	0	0
$313$	− 31.2462i	− 1.76614i	−0.469241	−	0.883070i	$-0.655472\pi$
$313$	− 31.2462i	− 1.76614i	0.469241	−	0.883070i	$-0.344528\pi$
$314$	6.87689	0.388086
$315$	0	0
$316$	−71.5464	−4.02480
$317$	6.87689i	0.386245i	0.981175	+	0.193122i	$0.0618615\pi$
$317$	6.87689i	0.386245i	−0.981175	+	0.193122i	$0.938139\pi$
$318$	0	0
$319$	5.12311	0.286839
$320$	0	0
$321$	0	0
$322$	− 16.8078i	− 0.936660i
$323$	− 4.31534i	− 0.240112i
$324$	0	0
$325$	0	0
$326$	13.6155	0.754094
$327$	0	0
$328$	− 50.4233i	− 2.78416i
$329$	5.43845	0.299831
$330$	0	0
$331$	−10.2462	−0.563183	−0.281591	−	0.959534i	$-0.590862\pi$
$331$	−10.2462	−0.563183	−0.281591	+	0.959534i	$0.590862\pi$
$332$	0	0
$333$	0	0
$334$	39.3693	2.15419
$335$	0	0
$336$	0	0
$337$	10.6847i	0.582030i	0.956718	+	0.291015i	$0.0939930\pi$
$337$	10.6847i	0.582030i	−0.956718	+	0.291015i	$0.906007\pi$
$338$	32.8078i	1.78451i
$339$	0	0
$340$	0	0
$341$	4.68466	0.253688
$342$	0	0
$343$	− 13.0000i	− 0.701934i
$344$	30.7386	1.65732
$345$	0	0
$346$	34.4233	1.85061
$347$	− 27.3693i	− 1.46926i	−0.678467	−	0.734631i	$-0.737355\pi$
$347$	− 27.3693i	− 1.46926i	0.678467	−	0.734631i	$-0.262645\pi$
$348$	0	0
$349$	−6.63068	−0.354932	−0.177466	−	0.984127i	$-0.556790\pi$
$349$	−6.63068	−0.354932	−0.177466	+	0.984127i	$0.556790\pi$
$350$	0	0
$351$	0	0
$352$	6.56155i	0.349732i
$353$	− 27.8617i	− 1.48293i	−0.670991	−	0.741465i	$-0.734131\pi$
$353$	− 27.8617i	− 1.48293i	0.670991	−	0.741465i	$-0.265869\pi$
$354$	0	0
$355$	0	0
$356$	78.1080	4.13971
$357$	0	0
$358$	37.3002i	1.97138i
$359$	26.2462	1.38522	0.692611	−	0.721311i	$-0.256460\pi$
$359$	26.2462	1.38522	0.692611	+	0.721311i	$0.256460\pi$
$360$	0	0
$361$	−10.0000	−0.526316
$362$	30.4233i	1.59901i
$363$	0	0
$364$	2.00000	0.104828
$365$	0	0
$366$	0	0
$367$	− 14.0540i	− 0.733612i	−0.930298	−	0.366806i	$-0.880451\pi$
$367$	− 14.0540i	− 0.733612i	0.930298	−	0.366806i	$-0.119549\pi$
$368$	− 50.4233i	− 2.62850i
$369$	0	0
$370$	0	0
$371$	1.12311	0.0583087
$372$	0	0
$373$	− 13.8078i	− 0.714939i	−0.933925	−	0.357469i	$-0.883640\pi$
$373$	− 13.8078i	− 0.714939i	0.933925	−	0.357469i	$-0.116360\pi$
$374$	−3.68466	−0.190529
$375$	0	0
$376$	35.6847	1.84030
$377$	− 2.24621i	− 0.115686i
$378$	0	0
$379$	24.3002	1.24822	0.624108	−	0.781338i	$-0.285462\pi$
$379$	24.3002	1.24822	0.624108	+	0.781338i	$0.285462\pi$
$380$	0	0
$381$	0	0
$382$	4.94602i	0.253061i
$383$	− 22.2462i	− 1.13673i	−0.822777	−	0.568364i	$-0.807576\pi$
$383$	− 22.2462i	− 1.13673i	0.822777	−	0.568364i	$-0.192424\pi$
$384$	0	0
$385$	0	0
$386$	−44.9848	−2.28967
$387$	0	0
$388$	9.68466i	0.491664i
$389$	−23.8617	−1.20984	−0.604919	−	0.796287i	$-0.706795\pi$
$389$	−23.8617	−1.20984	−0.604919	+	0.796287i	$0.706795\pi$
$390$	0	0
$391$	9.43845	0.477323
$392$	− 39.3693i	− 1.98845i
$393$	0	0
$394$	−45.9309	−2.31396
$395$	0	0
$396$	0	0
$397$	− 28.4384i	− 1.42728i	−0.700510	−	0.713642i	$-0.747044\pi$
$397$	− 28.4384i	− 1.42728i	0.700510	−	0.713642i	$-0.252956\pi$
$398$	35.3693i	1.77290i
$399$	0	0
$400$	0	0
$401$	−5.61553	−0.280426	−0.140213	−	0.990121i	$-0.544779\pi$
$401$	−5.61553	−0.280426	−0.140213	+	0.990121i	$0.544779\pi$
$402$	0	0
$403$	− 2.05398i	− 0.102316i
$404$	6.56155	0.326449
$405$	0	0
$406$	13.1231	0.651289
$407$	10.8078i	0.535721i
$408$	0	0
$409$	−8.05398	−0.398243	−0.199122	−	0.979975i	$-0.563809\pi$
$409$	−8.05398	−0.398243	−0.199122	+	0.979975i	$0.563809\pi$
$410$	0	0
$411$	0	0
$412$	41.6155i	2.05025i
$413$	− 10.5616i	− 0.519700i
$414$	0	0
$415$	0	0
$416$	2.87689	0.141051
$417$	0	0
$418$	− 7.68466i	− 0.375869i
$419$	25.4384	1.24275	0.621375	−	0.783514i	$-0.286574\pi$
$419$	25.4384	1.24275	0.621375	+	0.783514i	$0.286574\pi$
$420$	0	0
$421$	−17.0540	−0.831160	−0.415580	−	0.909557i	$-0.636421\pi$
$421$	−17.0540	−0.831160	−0.415580	+	0.909557i	$0.636421\pi$
$422$	8.49242i	0.413405i
$423$	0	0
$424$	7.36932	0.357886
$425$	0	0
$426$	0	0
$427$	7.56155i	0.365929i
$428$	41.6155i	2.01156i
$429$	0	0
$430$	0	0
$431$	21.7538	1.04784	0.523922	−	0.851767i	$-0.324468\pi$
$431$	21.7538	1.04784	0.523922	+	0.851767i	$0.324468\pi$
$432$	0	0
$433$	16.9309i	0.813646i	0.913507	+	0.406823i	$0.133363\pi$
$433$	16.9309i	0.813646i	−0.913507	+	0.406823i	$0.866637\pi$
$434$	12.0000	0.576018
$435$	0	0
$436$	−35.6155	−1.70567
$437$	19.6847i	0.941645i
$438$	0	0
$439$	−11.7386	−0.560254	−0.280127	−	0.959963i	$-0.590377\pi$
$439$	−11.7386	−0.560254	−0.280127	+	0.959963i	$0.590377\pi$
$440$	0	0
$441$	0	0
$442$	1.61553i	0.0768428i
$443$	− 25.3002i	− 1.20205i	−0.799231	−	0.601024i	$-0.794760\pi$
$443$	− 25.3002i	− 1.20205i	0.799231	−	0.601024i	$-0.205240\pi$
$444$	0	0
$445$	0	0
$446$	−55.8617	−2.64513
$447$	0	0
$448$	1.43845i	0.0679602i
$449$	−28.4924	−1.34464	−0.672320	−	0.740260i	$-0.734702\pi$
$449$	−28.4924	−1.34464	−0.672320	+	0.740260i	$0.734702\pi$
$450$	0	0
$451$	7.68466	0.361856
$452$	− 41.6155i	− 1.95743i
$453$	0	0
$454$	−52.4924	−2.46359
$455$	0	0
$456$	0	0
$457$	− 13.3693i	− 0.625390i	−0.949854	−	0.312695i	$-0.898768\pi$
$457$	− 13.3693i	− 0.625390i	0.949854	−	0.312695i	$-0.101232\pi$
$458$	20.8078i	0.972283i
$459$	0	0
$460$	0	0
$461$	15.8617	0.738755	0.369377	−	0.929279i	$-0.379571\pi$
$461$	15.8617	0.738755	0.369377	+	0.929279i	$0.379571\pi$
$462$	0	0
$463$	− 1.75379i	− 0.0815055i	−0.999169	−	0.0407527i	$-0.987024\pi$
$463$	− 1.75379i	− 0.0815055i	0.999169	−	0.0407527i	$-0.0129756\pi$
$464$	39.3693	1.82767
$465$	0	0
$466$	63.5464	2.94373
$467$	2.24621i	0.103942i	0.998649	+	0.0519711i	$0.0165504\pi$
$467$	2.24621i	0.103942i	−0.998649	+	0.0519711i	$0.983450\pi$
$468$	0	0
$469$	6.68466	0.308669
$470$	0	0
$471$	0	0
$472$	− 69.3002i	− 3.18980i
$473$	4.68466i	0.215401i
$474$	0	0
$475$	0	0
$476$	−6.56155	−0.300748
$477$	0	0
$478$	− 42.2462i	− 1.93230i
$479$	3.36932	0.153948	0.0769740	−	0.997033i	$-0.475474\pi$
$479$	3.36932	0.153948	0.0769740	+	0.997033i	$0.475474\pi$
$480$	0	0
$481$	4.73863	0.216063
$482$	− 36.0000i	− 1.63976i
$483$	0	0
$484$	−4.56155	−0.207343
$485$	0	0
$486$	0	0
$487$	12.0540i	0.546218i	0.961983	+	0.273109i	$0.0880519\pi$
$487$	12.0540i	0.546218i	−0.961983	+	0.273109i	$0.911948\pi$
$488$	49.6155i	2.24599i
$489$	0	0
$490$	0	0
$491$	−16.4924	−0.744293	−0.372146	−	0.928174i	$-0.621378\pi$
$491$	−16.4924	−0.744293	−0.372146	+	0.928174i	$0.621378\pi$
$492$	0	0
$493$	7.36932i	0.331897i
$494$	−3.36932	−0.151593
$495$	0	0
$496$	36.0000	1.61645
$497$	8.80776i	0.395082i
$498$	0	0
$499$	−3.80776	−0.170459	−0.0852295	−	0.996361i	$-0.527162\pi$
$499$	−3.80776	−0.170459	−0.0852295	+	0.996361i	$0.527162\pi$
$500$	0	0
$501$	0	0
$502$	30.7386i	1.37193i
$503$	16.4924i	0.735361i	0.929952	+	0.367680i	$0.119848\pi$
$503$	16.4924i	0.735361i	−0.929952	+	0.367680i	$0.880152\pi$
$504$	0	0
$505$	0	0
$506$	16.8078	0.747196
$507$	0	0
$508$	− 21.9309i	− 0.973025i
$509$	−17.1231	−0.758968	−0.379484	−	0.925198i	$-0.623899\pi$
$509$	−17.1231	−0.758968	−0.379484	+	0.925198i	$0.623899\pi$
$510$	0	0
$511$	16.2462	0.718690
$512$	− 50.4233i	− 2.22842i
$513$	0	0
$514$	−46.7386	−2.06155
$515$	0	0
$516$	0	0
$517$	5.43845i	0.239183i
$518$	27.6847i	1.21639i
$519$	0	0
$520$	0	0
$521$	25.6155	1.12224	0.561118	−	0.827736i	$-0.310371\pi$
$521$	25.6155	1.12224	0.561118	+	0.827736i	$0.310371\pi$
$522$	0	0
$523$	9.24621i	0.404309i	0.979354	+	0.202154i	$0.0647942\pi$
$523$	9.24621i	0.404309i	−0.979354	+	0.202154i	$0.935206\pi$
$524$	−75.2311	−3.28648
$525$	0	0
$526$	−8.63068	−0.376316
$527$	6.73863i	0.293539i
$528$	0	0
$529$	−20.0540	−0.871912
$530$	0	0
$531$	0	0
$532$	− 13.6847i	− 0.593305i
$533$	− 3.36932i	− 0.145941i
$534$	0	0
$535$	0	0
$536$	43.8617	1.89454
$537$	0	0
$538$	72.9848i	3.14660i
$539$	6.00000	0.258438
$540$	0	0
$541$	−7.31534	−0.314511	−0.157256	−	0.987558i	$-0.550265\pi$
$541$	−7.31534	−0.314511	−0.157256	+	0.987558i	$0.550265\pi$
$542$	− 72.1771i	− 3.10027i
$543$	0	0
$544$	−9.43845	−0.404670
$545$	0	0
$546$	0	0
$547$	− 3.68466i	− 0.157545i	−0.996893	−	0.0787723i	$-0.974900\pi$
$547$	− 3.68466i	− 0.157545i	0.996893	−	0.0787723i	$-0.0251000\pi$
$548$	15.3693i	0.656545i
$549$	0	0
$550$	0	0
$551$	−15.3693	−0.654755
$552$	0	0
$553$	− 15.6847i	− 0.666980i
$554$	−59.3693	−2.52236
$555$	0	0
$556$	36.4924	1.54762
$557$	1.12311i	0.0475875i	0.999717	+	0.0237938i	$0.00757450\pi$
$557$	1.12311i	0.0475875i	−0.999717	+	0.0237938i	$0.992425\pi$
$558$	0	0
$559$	2.05398	0.0868739
$560$	0	0
$561$	0	0
$562$	8.17708i	0.344929i
$563$	10.8769i	0.458406i	0.973379	+	0.229203i	$0.0736120\pi$
$563$	10.8769i	0.458406i	−0.973379	+	0.229203i	$0.926388\pi$
$564$	0	0
$565$	0	0
$566$	−82.2850	−3.45870
$567$	0	0
$568$	57.7926i	2.42492i
$569$	37.9309	1.59014	0.795072	−	0.606515i	$-0.207433\pi$
$569$	37.9309	1.59014	0.795072	+	0.606515i	$0.207433\pi$
$570$	0	0
$571$	38.0540	1.59251	0.796255	−	0.604962i	$-0.206812\pi$
$571$	38.0540	1.59251	0.796255	+	0.604962i	$0.206812\pi$
$572$	2.00000i	0.0836242i
$573$	0	0
$574$	19.6847	0.821622
$575$	0	0
$576$	0	0
$577$	28.3693i	1.18103i	0.807027	+	0.590515i	$0.201075\pi$
$577$	28.3693i	1.18103i	−0.807027	+	0.590515i	$0.798925\pi$
$578$	38.2462i	1.59083i
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	1.12311i	0.0465143i
$584$	106.600	4.41115
$585$	0	0
$586$	5.30019	0.218949
$587$	− 27.1922i	− 1.12234i	−0.827699	−	0.561172i	$-0.810351\pi$
$587$	− 27.1922i	− 1.12234i	0.827699	−	0.561172i	$-0.189649\pi$
$588$	0	0
$589$	−14.0540	−0.579084
$590$	0	0
$591$	0	0
$592$	83.0540i	3.41350i
$593$	34.1080i	1.40065i	0.713826	+	0.700323i	$0.246961\pi$
$593$	34.1080i	1.40065i	−0.713826	+	0.700323i	$0.753039\pi$
$594$	0	0
$595$	0	0
$596$	−40.1771	−1.64572
$597$	0	0
$598$	− 7.36932i	− 0.301354i
$599$	24.1771	0.987849	0.493924	−	0.869505i	$-0.335562\pi$
$599$	24.1771	0.987849	0.493924	+	0.869505i	$0.335562\pi$
$600$	0	0
$601$	37.4233	1.52653	0.763264	−	0.646087i	$-0.223596\pi$
$601$	37.4233	1.52653	0.763264	+	0.646087i	$0.223596\pi$
$602$	12.0000i	0.489083i
$603$	0	0
$604$	76.1080	3.09679
$605$	0	0
$606$	0	0
$607$	− 6.24621i	− 0.253526i	−0.991933	−	0.126763i	$-0.959541\pi$
$607$	− 6.24621i	− 0.253526i	0.991933	−	0.126763i	$-0.0404588\pi$
$608$	− 19.6847i	− 0.798318i
$609$	0	0
$610$	0	0
$611$	2.38447	0.0964654
$612$	0	0
$613$	28.1080i	1.13527i	0.823281	+	0.567635i	$0.192141\pi$
$613$	28.1080i	1.13527i	−0.823281	+	0.567635i	$0.807859\pi$
$614$	−58.7386	−2.37050
$615$	0	0
$616$	−6.56155	−0.264372
$617$	21.7538i	0.875775i	0.899030	+	0.437887i	$0.144273\pi$
$617$	21.7538i	0.875775i	−0.899030	+	0.437887i	$0.855727\pi$
$618$	0	0
$619$	−7.17708	−0.288471	−0.144236	−	0.989543i	$-0.546072\pi$
$619$	−7.17708	−0.288471	−0.144236	+	0.989543i	$0.546072\pi$
$620$	0	0
$621$	0	0
$622$	35.2311i	1.41264i
$623$	17.1231i	0.686023i
$624$	0	0
$625$	0	0
$626$	80.0388	3.19899
$627$	0	0
$628$	12.2462i	0.488677i
$629$	−15.5464	−0.619875
$630$	0	0
$631$	−22.6847	−0.903062	−0.451531	−	0.892255i	$-0.649122\pi$
$631$	−22.6847	−0.903062	−0.451531	+	0.892255i	$0.649122\pi$
$632$	− 102.916i	− 4.09377i
$633$	0	0
$634$	−17.6155	−0.699602
$635$	0	0
$636$	0	0
$637$	− 2.63068i	− 0.104231i
$638$	13.1231i	0.519549i
$639$	0	0
$640$	0	0
$641$	−42.7386	−1.68807	−0.844037	−	0.536285i	$-0.819827\pi$
$641$	−42.7386	−1.68807	−0.844037	+	0.536285i	$0.819827\pi$
$642$	0	0
$643$	15.3693i	0.606107i	0.952974	+	0.303053i	$0.0980060\pi$
$643$	15.3693i	0.606107i	−0.952974	+	0.303053i	$0.901994\pi$
$644$	29.9309	1.17944
$645$	0	0
$646$	11.0540	0.434913
$647$	− 40.8078i	− 1.60432i	−0.597110	−	0.802159i	$-0.703684\pi$
$647$	− 40.8078i	− 1.60432i	0.597110	−	0.802159i	$-0.296316\pi$
$648$	0	0
$649$	10.5616	0.414577
$650$	0	0
$651$	0	0
$652$	24.2462i	0.949555i
$653$	− 7.50758i	− 0.293794i	−0.989152	−	0.146897i	$-0.953071\pi$
$653$	− 7.50758i	− 0.293794i	0.989152	−	0.146897i	$-0.0469286\pi$
$654$	0	0
$655$	0	0
$656$	59.0540	2.30567
$657$	0	0
$658$	13.9309i	0.543082i
$659$	8.63068	0.336204	0.168102	−	0.985770i	$-0.446236\pi$
$659$	8.63068	0.336204	0.168102	+	0.985770i	$0.446236\pi$
$660$	0	0
$661$	6.80776	0.264791	0.132396	−	0.991197i	$-0.457733\pi$
$661$	6.80776	0.264791	0.132396	+	0.991197i	$0.457733\pi$
$662$	− 26.2462i	− 1.02009i
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 33.6155i	− 1.30160i
$668$	70.1080i	2.71256i
$669$	0	0
$670$	0	0
$671$	−7.56155	−0.291911
$672$	0	0
$673$	10.6307i	0.409783i	0.978785	+	0.204891i	$0.0656841\pi$
$673$	10.6307i	0.409783i	−0.978785	+	0.204891i	$0.934316\pi$
$674$	−27.3693	−1.05423
$675$	0	0
$676$	−58.4233	−2.24705
$677$	29.1231i	1.11929i	0.828732	+	0.559646i	$0.189063\pi$
$677$	29.1231i	1.11929i	−0.828732	+	0.559646i	$0.810937\pi$
$678$	0	0
$679$	−2.12311	−0.0814773
$680$	0	0
$681$	0	0
$682$	12.0000i	0.459504i
$683$	20.8078i	0.796187i	0.917345	+	0.398093i	$0.130328\pi$
$683$	20.8078i	0.796187i	−0.917345	+	0.398093i	$0.869672\pi$
$684$	0	0
$685$	0	0
$686$	33.3002	1.27141
$687$	0	0
$688$	36.0000i	1.37249i
$689$	0.492423	0.0187598
$690$	0	0
$691$	12.4924	0.475234	0.237617	−	0.971359i	$-0.423634\pi$
$691$	12.4924	0.475234	0.237617	+	0.971359i	$0.423634\pi$
$692$	61.3002i	2.33028i
$693$	0	0
$694$	70.1080	2.66126
$695$	0	0
$696$	0	0
$697$	11.0540i	0.418699i
$698$	− 16.9848i	− 0.642886i
$699$	0	0
$700$	0	0
$701$	1.93087	0.0729279	0.0364640	−	0.999335i	$-0.488391\pi$
$701$	1.93087	0.0729279	0.0364640	+	0.999335i	$0.488391\pi$
$702$	0	0
$703$	− 32.4233i	− 1.22287i
$704$	−1.43845	−0.0542135
$705$	0	0
$706$	71.3693	2.68602
$707$	1.43845i	0.0540984i
$708$	0	0
$709$	−6.12311	−0.229958	−0.114979	−	0.993368i	$-0.536680\pi$
$709$	−6.12311	−0.229958	−0.114979	+	0.993368i	$0.536680\pi$
$710$	0	0
$711$	0	0
$712$	112.354i	4.21065i
$713$	− 30.7386i	− 1.15117i
$714$	0	0
$715$	0	0
$716$	−66.4233	−2.48235
$717$	0	0
$718$	67.2311i	2.50904i
$719$	−32.9848	−1.23013	−0.615064	−	0.788478i	$-0.710870\pi$
$719$	−32.9848	−1.23013	−0.615064	+	0.788478i	$0.710870\pi$
$720$	0	0
$721$	−9.12311	−0.339762
$722$	− 25.6155i	− 0.953311i
$723$	0	0
$724$	−54.1771	−2.01348
$725$	0	0
$726$	0	0
$727$	30.0540i	1.11464i	0.830298	+	0.557320i	$0.188170\pi$
$727$	30.0540i	1.11464i	−0.830298	+	0.557320i	$0.811830\pi$
$728$	2.87689i	0.106625i
$729$	0	0
$730$	0	0
$731$	−6.73863	−0.249237
$732$	0	0
$733$	2.00000i	0.0738717i	0.999318	+	0.0369358i	$0.0117597\pi$
$733$	2.00000i	0.0738717i	−0.999318	+	0.0369358i	$0.988240\pi$
$734$	36.0000	1.32878
$735$	0	0
$736$	43.0540	1.58699
$737$	6.68466i	0.246233i
$738$	0	0
$739$	11.6847	0.429827	0.214914	−	0.976633i	$-0.431053\pi$
$739$	11.6847	0.429827	0.214914	+	0.976633i	$0.431053\pi$
$740$	0	0
$741$	0	0
$742$	2.87689i	0.105614i
$743$	− 47.8617i	− 1.75588i	−0.478773	−	0.877938i	$-0.658918\pi$
$743$	− 47.8617i	− 1.75588i	0.478773	−	0.877938i	$-0.341082\pi$
$744$	0	0
$745$	0	0
$746$	35.3693	1.29496
$747$	0	0
$748$	− 6.56155i	− 0.239914i
$749$	−9.12311	−0.333351
$750$	0	0
$751$	26.7386	0.975707	0.487853	−	0.872926i	$-0.337780\pi$
$751$	26.7386	0.975707	0.487853	+	0.872926i	$0.337780\pi$
$752$	41.7926i	1.52402i
$753$	0	0
$754$	5.75379	0.209541
$755$	0	0
$756$	0	0
$757$	− 6.30019i	− 0.228984i	−0.993424	−	0.114492i	$-0.963476\pi$
$757$	− 6.30019i	− 0.228984i	0.993424	−	0.114492i	$-0.0365241\pi$
$758$	62.2462i	2.26088i
$759$	0	0
$760$	0	0
$761$	−20.6307	−0.747862	−0.373931	−	0.927457i	$-0.621990\pi$
$761$	−20.6307	−0.747862	−0.373931	+	0.927457i	$0.621990\pi$
$762$	0	0
$763$	− 7.80776i	− 0.282660i
$764$	−8.80776	−0.318654
$765$	0	0
$766$	56.9848	2.05895
$767$	− 4.63068i	− 0.167204i
$768$	0	0
$769$	−6.05398	−0.218312	−0.109156	−	0.994025i	$-0.534815\pi$
$769$	−6.05398	−0.218312	−0.109156	+	0.994025i	$0.534815\pi$
$770$	0	0
$771$	0	0
$772$	− 80.1080i	− 2.88315i
$773$	− 48.4924i	− 1.74415i	−0.489371	−	0.872076i	$-0.662774\pi$
$773$	− 48.4924i	− 1.74415i	0.489371	−	0.872076i	$-0.337226\pi$
$774$	0	0
$775$	0	0
$776$	−13.9309	−0.500089
$777$	0	0
$778$	− 61.1231i	− 2.19137i
$779$	−23.0540	−0.825994
$780$	0	0
$781$	−8.80776	−0.315167
$782$	24.1771i	0.864571i
$783$	0	0
$784$	46.1080	1.64671
$785$	0	0
$786$	0	0
$787$	− 4.26137i	− 0.151901i	−0.997112	−	0.0759507i	$-0.975801\pi$
$787$	− 4.26137i	− 0.151901i	0.997112	−	0.0759507i	$-0.0241991\pi$
$788$	− 81.7926i	− 2.91374i
$789$	0	0
$790$	0	0
$791$	9.12311	0.324380
$792$	0	0
$793$	3.31534i	0.117731i
$794$	72.8466	2.58523
$795$	0	0
$796$	−62.9848	−2.23244
$797$	42.8769i	1.51878i	0.650637	+	0.759389i	$0.274502\pi$
$797$	42.8769i	1.51878i	−0.650637	+	0.759389i	$0.725498\pi$
$798$	0	0
$799$	−7.82292	−0.276755
$800$	0	0
$801$	0	0
$802$	− 14.3845i	− 0.507933i
$803$	16.2462i	0.573316i
$804$	0	0
$805$	0	0
$806$	5.26137	0.185324
$807$	0	0
$808$	9.43845i	0.332043i
$809$	19.6847	0.692076	0.346038	−	0.938221i	$-0.387527\pi$
$809$	19.6847	0.692076	0.346038	+	0.938221i	$0.387527\pi$
$810$	0	0
$811$	−1.00000	−0.0351147	−0.0175574	−	0.999846i	$-0.505589\pi$
$811$	−1.00000	−0.0351147	−0.0175574	+	0.999846i	$0.505589\pi$
$812$	23.3693i	0.820102i
$813$	0	0
$814$	−27.6847	−0.970347
$815$	0	0
$816$	0	0
$817$	− 14.0540i	− 0.491686i
$818$	− 20.6307i	− 0.721335i
$819$	0	0
$820$	0	0
$821$	25.1231	0.876802	0.438401	−	0.898779i	$-0.355545\pi$
$821$	25.1231	0.876802	0.438401	+	0.898779i	$0.355545\pi$
$822$	0	0
$823$	0.0539753i	0.00188146i	1.00000		0.000940731i	$0.000299444\pi$
$823$	0.0539753i	0.00188146i	−1.00000		0.000940731i	$0.999701\pi$
$824$	−59.8617	−2.08538
$825$	0	0
$826$	27.0540	0.941328
$827$	9.75379i	0.339172i	0.985515	+	0.169586i	$0.0542431\pi$
$827$	9.75379i	0.339172i	−0.985515	+	0.169586i	$0.945757\pi$
$828$	0	0
$829$	−48.7386	−1.69276	−0.846381	−	0.532577i	$-0.821224\pi$
$829$	−48.7386	−1.69276	−0.846381	+	0.532577i	$0.821224\pi$
$830$	0	0
$831$	0	0
$832$	0.630683i	0.0218650i
$833$	8.63068i	0.299035i
$834$	0	0
$835$	0	0
$836$	13.6847	0.473294
$837$	0	0
$838$	65.1619i	2.25098i
$839$	20.4924	0.707477	0.353738	−	0.935344i	$-0.384910\pi$
$839$	20.4924	0.707477	0.353738	+	0.935344i	$0.384910\pi$
$840$	0	0
$841$	−2.75379	−0.0949582
$842$	− 43.6847i	− 1.50547i
$843$	0	0
$844$	−15.1231	−0.520559
$845$	0	0
$846$	0	0
$847$	− 1.00000i	− 0.0343604i
$848$	8.63068i	0.296379i
$849$	0	0
$850$	0	0
$851$	70.9157	2.43096
$852$	0	0
$853$	− 35.4233i	− 1.21287i	−0.795133	−	0.606435i	$-0.792599\pi$
$853$	− 35.4233i	− 1.21287i	0.795133	−	0.606435i	$-0.207401\pi$
$854$	−19.3693	−0.662804
$855$	0	0
$856$	−59.8617	−2.04603
$857$	5.43845i	0.185774i	0.995677	+	0.0928869i	$0.0296095\pi$
$857$	5.43845i	0.185774i	−0.995677	+	0.0928869i	$0.970390\pi$
$858$	0	0
$859$	−56.3542	−1.92278	−0.961390	−	0.275191i	$-0.911259\pi$
$859$	−56.3542	−1.92278	−0.961390	+	0.275191i	$0.911259\pi$
$860$	0	0
$861$	0	0
$862$	55.7235i	1.89795i
$863$	2.24621i	0.0764619i	0.999269	+	0.0382310i	$0.0121723\pi$
$863$	2.24621i	0.0764619i	−0.999269	+	0.0382310i	$0.987828\pi$
$864$	0	0
$865$	0	0
$866$	−43.3693	−1.47375
$867$	0	0
$868$	21.3693i	0.725322i
$869$	15.6847	0.532066
$870$	0	0
$871$	2.93087	0.0993087
$872$	− 51.2311i	− 1.73490i
$873$	0	0
$874$	−50.4233	−1.70559
$875$	0	0
$876$	0	0
$877$	− 14.0540i	− 0.474569i	−0.971440	−	0.237285i	$-0.923743\pi$
$877$	− 14.0540i	− 0.474569i	0.971440	−	0.237285i	$-0.0762574\pi$
$878$	− 30.0691i	− 1.01478i
$879$	0	0
$880$	0	0
$881$	9.75379	0.328613	0.164307	−	0.986409i	$-0.447461\pi$
$881$	9.75379	0.328613	0.164307	+	0.986409i	$0.447461\pi$
$882$	0	0
$883$	− 44.9309i	− 1.51204i	−0.654546	−	0.756022i	$-0.727140\pi$
$883$	− 44.9309i	− 1.51204i	0.654546	−	0.756022i	$-0.272860\pi$
$884$	−2.87689	−0.0967604
$885$	0	0
$886$	64.8078	2.17726
$887$	27.8617i	0.935506i	0.883859	+	0.467753i	$0.154936\pi$
$887$	27.8617i	0.935506i	−0.883859	+	0.467753i	$0.845064\pi$
$888$	0	0
$889$	4.80776	0.161247
$890$	0	0
$891$	0	0
$892$	− 99.4773i	− 3.33075i
$893$	− 16.3153i	− 0.545972i
$894$	0	0
$895$	0	0
$896$	9.43845	0.315316
$897$	0	0
$898$	− 72.9848i	− 2.43554i
$899$	24.0000	0.800445
$900$	0	0
$901$	−1.61553	−0.0538210
$902$	19.6847i	0.655427i
$903$	0	0
$904$	59.8617	1.99097
$905$	0	0
$906$	0	0
$907$	− 46.1080i	− 1.53099i	−0.643442	−	0.765495i	$-0.722494\pi$
$907$	− 46.1080i	− 1.53099i	0.643442	−	0.765495i	$-0.277506\pi$
$908$	− 93.4773i	− 3.10215i
$909$	0	0
$910$	0	0
$911$	57.1619	1.89386	0.946930	−	0.321441i	$-0.104167\pi$
$911$	57.1619	1.89386	0.946930	+	0.321441i	$0.104167\pi$
$912$	0	0
$913$	0	0
$914$	34.2462	1.13276
$915$	0	0
$916$	−37.0540	−1.22430
$917$	− 16.4924i	− 0.544628i
$918$	0	0
$919$	−41.1080	−1.35603	−0.678013	−	0.735050i	$-0.737159\pi$
$919$	−41.1080	−1.35603	−0.678013	+	0.735050i	$0.737159\pi$
$920$	0	0
$921$	0	0
$922$	40.6307i	1.33810i
$923$	3.86174i	0.127111i
$924$	0	0
$925$	0	0
$926$	4.49242	0.147630
$927$	0	0
$928$	33.6155i	1.10348i
$929$	23.2311	0.762186	0.381093	−	0.924537i	$-0.375548\pi$
$929$	23.2311	0.762186	0.381093	+	0.924537i	$0.375548\pi$
$930$	0	0
$931$	−18.0000	−0.589926
$932$	113.162i	3.70674i
$933$	0	0
$934$	−5.75379	−0.188270
$935$	0	0
$936$	0	0
$937$	− 34.6847i	− 1.13310i	−0.824028	−	0.566549i	$-0.808278\pi$
$937$	− 34.6847i	− 1.13310i	0.824028	−	0.566549i	$-0.191722\pi$
$938$	17.1231i	0.559089i
$939$	0	0
$940$	0	0
$941$	−60.1771	−1.96172	−0.980858	−	0.194722i	$-0.937619\pi$
$941$	−60.1771	−1.96172	−0.980858	+	0.194722i	$0.937619\pi$
$942$	0	0
$943$	− 50.4233i	− 1.64201i
$944$	81.1619	2.64160
$945$	0	0
$946$	−12.0000	−0.390154
$947$	− 32.1771i	− 1.04561i	−0.852451	−	0.522807i	$-0.824885\pi$
$947$	− 32.1771i	− 1.04561i	0.852451	−	0.522807i	$-0.175115\pi$
$948$	0	0
$949$	7.12311	0.231226
$950$	0	0
$951$	0	0
$952$	− 9.43845i	− 0.305902i
$953$	4.80776i	0.155739i	0.996964	+	0.0778694i	$0.0248117\pi$
$953$	4.80776i	0.155739i	−0.996964	+	0.0778694i	$0.975188\pi$
$954$	0	0
$955$	0	0
$956$	75.2311	2.43315
$957$	0	0
$958$	8.63068i	0.278845i
$959$	−3.36932	−0.108801
$960$	0	0
$961$	−9.05398	−0.292064
$962$	12.1383i	0.391353i
$963$	0	0
$964$	64.1080	2.06478
$965$	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	$-0.250165\pi$
$968$	− 6.56155i	− 0.210896i
$969$	0	0
$970$	0	0
$971$	−8.17708	−0.262415	−0.131208	−	0.991355i	$-0.541885\pi$
$971$	−8.17708	−0.262415	−0.131208	+	0.991355i	$0.541885\pi$
$972$	0	0
$973$	8.00000i	0.256468i
$974$	−30.8769	−0.989360
$975$	0	0
$976$	−58.1080	−1.85999
$977$	− 25.6155i	− 0.819513i	−0.912195	−	0.409757i	$-0.865614\pi$
$977$	− 25.6155i	− 0.819513i	0.912195	−	0.409757i	$-0.134386\pi$
$978$	0	0
$979$	−17.1231	−0.547257
$980$	0	0
$981$	0	0
$982$	− 42.2462i	− 1.34813i
$983$	− 21.9309i	− 0.699486i	−0.936846	−	0.349743i	$-0.886269\pi$
$983$	− 21.9309i	− 0.699486i	0.936846	−	0.349743i	$-0.113731\pi$
$984$	0	0
$985$	0	0
$986$	−18.8769	−0.601163
$987$	0	0
$988$	− 6.00000i	− 0.190885i
$989$	30.7386	0.977432
$990$	0	0
$991$	37.3153	1.18536	0.592680	−	0.805438i	$-0.298070\pi$
$991$	37.3153	1.18536	0.592680	+	0.805438i	$0.298070\pi$
$992$	30.7386i	0.975953i
$993$	0	0
$994$	−22.5616	−0.715609
$995$	0	0
$996$	0	0
$997$	− 20.7386i	− 0.656799i	−0.944539	−	0.328400i	$-0.893491\pi$
$997$	− 20.7386i	− 0.656799i	0.944539	−	0.328400i	$-0.106509\pi$
$998$	− 9.75379i	− 0.308751i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2475.2.c.j.199.4		4
3.2	odd	2		2475.2.c.i.199.1		4
5.2	odd	4		2475.2.a.q.1.1	yes	2
5.3	odd	4		2475.2.a.u.1.2	yes	2
5.4	even	2	inner	2475.2.c.j.199.1		4
15.2	even	4		2475.2.a.v.1.2	yes	2
15.8	even	4		2475.2.a.p.1.1	✓	2
15.14	odd	2		2475.2.c.i.199.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2475.2.a.p.1.1	✓	2	15.8	even	4
2475.2.a.q.1.1	yes	2	5.2	odd	4
2475.2.a.u.1.2	yes	2	5.3	odd	4
2475.2.a.v.1.2	yes	2	15.2	even	4
2475.2.c.i.199.1		4	3.2	odd	2
2475.2.c.i.199.4		4	15.14	odd	2
2475.2.c.j.199.1		4	5.4	even	2	inner
2475.2.c.j.199.4		4	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2475 = 3^{2} \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2475.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+2.56155i q^{2} -4.56155 q^{4} -1.00000i q^{7} -6.56155i q^{8} +O(q^{10})\)	\(q+2.56155i q^{2} -4.56155 q^{4} -1.00000i q^{7} -6.56155i q^{8} +1.00000 q^{11} -0.438447i q^{13} +2.56155 q^{14} +7.68466 q^{16} +1.43845i q^{17} -3.00000 q^{19} +2.56155i q^{22} -6.56155i q^{23} +1.12311 q^{26} +4.56155i q^{28} +5.12311 q^{29} +4.68466 q^{31} +6.56155i q^{32} -3.68466 q^{34} +10.8078i q^{37} -7.68466i q^{38} +7.68466 q^{41} +4.68466i q^{43} -4.56155 q^{44} +16.8078 q^{46} +5.43845i q^{47} +6.00000 q^{49} +2.00000i q^{52} +1.12311i q^{53} -6.56155 q^{56} +13.1231i q^{58} +10.5616 q^{59} -7.56155 q^{61} +12.0000i q^{62} -1.43845 q^{64} +6.68466i q^{67} -6.56155i q^{68} -8.80776 q^{71} +16.2462i q^{73} -27.6847 q^{74} +13.6847 q^{76} -1.00000i q^{77} +15.6847 q^{79} +19.6847i q^{82} -12.0000 q^{86} -6.56155i q^{88} -17.1231 q^{89} -0.438447 q^{91} +29.9309i q^{92} -13.9309 q^{94} -2.12311i q^{97} +15.3693i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 10 q^{4}+O(q^{10}) \) 4 * q - 10 * q^4	\( 4 q - 10 q^{4} + 4 q^{11} + 2 q^{14} + 6 q^{16} - 12 q^{19} - 12 q^{26} + 4 q^{29} - 6 q^{31} + 10 q^{34} + 6 q^{41} - 10 q^{44} + 26 q^{46} + 24 q^{49} - 18 q^{56} + 34 q^{59} - 22 q^{61} - 14 q^{64} + 6 q^{71} - 86 q^{74} + 30 q^{76} + 38 q^{79} - 48 q^{86} - 52 q^{89} - 10 q^{91} + 2 q^{94}+O(q^{100}) \) 4 * q - 10 * q^4 + 4 * q^11 + 2 * q^14 + 6 * q^16 - 12 * q^19 - 12 * q^26 + 4 * q^29 - 6 * q^31 + 10 * q^34 + 6 * q^41 - 10 * q^44 + 26 * q^46 + 24 * q^49 - 18 * q^56 + 34 * q^59 - 22 * q^61 - 14 * q^64 + 6 * q^71 - 86 * q^74 + 30 * q^76 + 38 * q^79 - 48 * q^86 - 52 * q^89 - 10 * q^91 + 2 * q^94

Introduction

L-functions

Modular forms

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Database

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