LMFDB - Embedded newform 2475.2.c.m.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:	$\Q(\zeta_{8})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 1 $ x^4 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 165)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			199.4
Root			$0.707107 + 0.707107i$ of defining polynomial
Character	$\chi$	$=$	2475.199
Dual form			2475.2.c.m.199.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.41421i q^{2} -3.82843 q^{4} +0.828427i q^{7} -4.41421i q^{8} +O(q^{10})$	$q+2.41421i q^{2} -3.82843 q^{4} +0.828427i q^{7} -4.41421i q^{8} +1.00000 q^{11} +5.65685i q^{13} -2.00000 q^{14} +3.00000 q^{16} +1.17157i q^{17} +6.82843 q^{19} +2.41421i q^{22} -4.00000i q^{23} -13.6569 q^{26} -3.17157i q^{28} -4.82843 q^{29} -1.58579i q^{32} -2.82843 q^{34} +11.6569i q^{37} +16.4853i q^{38} -4.82843 q^{41} +8.82843i q^{43} -3.82843 q^{44} +9.65685 q^{46} +4.00000i q^{47} +6.31371 q^{49} -21.6569i q^{52} +9.31371i q^{53} +3.65685 q^{56} -11.6569i q^{58} -4.00000 q^{59} -11.6569 q^{61} +9.82843 q^{64} -5.65685i q^{67} -4.48528i q^{68} -2.34315 q^{71} -11.3137i q^{73} -28.1421 q^{74} -26.1421 q^{76} +0.828427i q^{77} -8.48528 q^{79} -11.6569i q^{82} -10.0000i q^{83} -21.3137 q^{86} -4.41421i q^{88} +3.65685 q^{89} -4.68629 q^{91} +15.3137i q^{92} -9.65685 q^{94} +11.6569i q^{97} +15.2426i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{4}+O(q^{10}) $ 4 * q - 4 * q^4	$ 4 q - 4 q^{4} + 4 q^{11} - 8 q^{14} + 12 q^{16} + 16 q^{19} - 32 q^{26} - 8 q^{29} - 8 q^{41} - 4 q^{44} + 16 q^{46} - 20 q^{49} - 8 q^{56} - 16 q^{59} - 24 q^{61} + 28 q^{64} - 32 q^{71} - 56 q^{74} - 48 q^{76} - 40 q^{86} - 8 q^{89} - 64 q^{91} - 16 q^{94}+O(q^{100}) $ 4 * q - 4 * q^4 + 4 * q^11 - 8 * q^14 + 12 * q^16 + 16 * q^19 - 32 * q^26 - 8 * q^29 - 8 * q^41 - 4 * q^44 + 16 * q^46 - 20 * q^49 - 8 * q^56 - 16 * q^59 - 24 * q^61 + 28 * q^64 - 32 * q^71 - 56 * q^74 - 48 * q^76 - 40 * q^86 - 8 * q^89 - 64 * q^91 - 16 * 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 $

$2$	2.41421i	1.70711i	0.521005	+	0.853553i	$0.325557\pi$
$2$	2.41421i	1.70711i	−0.521005	+	0.853553i	$0.674443\pi$
$3$	0	0
$3$	0	0
$4$	−3.82843	−1.91421
$5$	0	0
$5$	0	0
$6$	0	0
$7$	0.828427i	0.313116i	0.987669	+	0.156558i	$0.0500398\pi$
$7$	0.828427i	0.313116i	−0.987669	+	0.156558i	$0.949960\pi$
$8$	− 4.41421i	− 1.56066i
$9$	0	0
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	5.65685i	1.56893i	0.620174	+	0.784465i	$0.287062\pi$
$13$	5.65685i	1.56893i	−0.620174	+	0.784465i	$0.712938\pi$
$14$	−2.00000	−0.534522
$15$	0	0
$16$	3.00000	0.750000
$17$	1.17157i	0.284148i	0.989856	+	0.142074i	$0.0453771\pi$
$17$	1.17157i	0.284148i	−0.989856	+	0.142074i	$0.954623\pi$
$18$	0	0
$19$	6.82843	1.56655	0.783274	−	0.621676i	$-0.213548\pi$
$19$	6.82843	1.56655	0.783274	+	0.621676i	$0.213548\pi$
$20$	0	0
$21$	0	0
$22$	2.41421i	0.514712i
$23$	− 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
$26$	−13.6569	−2.67833
$27$	0	0
$28$	− 3.17157i	− 0.599371i
$29$	−4.82843	−0.896616	−0.448308	−	0.893879i	$-0.647973\pi$
$29$	−4.82843	−0.896616	−0.448308	+	0.893879i	$0.647973\pi$
$30$	0	0
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	− 1.58579i	− 0.280330i
$33$	0	0
$34$	−2.82843	−0.485071
$35$	0	0
$36$	0	0
$37$	11.6569i	1.91638i	0.286141	+	0.958188i	$0.407627\pi$
$37$	11.6569i	1.91638i	−0.286141	+	0.958188i	$0.592373\pi$
$38$	16.4853i	2.67427i
$39$	0	0
$40$	0	0
$41$	−4.82843	−0.754074	−0.377037	−	0.926198i	$-0.623057\pi$
$41$	−4.82843	−0.754074	−0.377037	+	0.926198i	$0.623057\pi$
$42$	0	0
$43$	8.82843i	1.34632i	0.739496	+	0.673161i	$0.235064\pi$
$43$	8.82843i	1.34632i	−0.739496	+	0.673161i	$0.764936\pi$
$44$	−3.82843	−0.577157
$45$	0	0
$46$	9.65685	1.42383
$47$	4.00000i	0.583460i	0.956501	+	0.291730i	$0.0942309\pi$
$47$	4.00000i	0.583460i	−0.956501	+	0.291730i	$0.905769\pi$
$48$	0	0
$49$	6.31371	0.901958
$50$	0	0
$51$	0	0
$52$	− 21.6569i	− 3.00327i
$53$	9.31371i	1.27934i	0.768651	+	0.639668i	$0.220928\pi$
$53$	9.31371i	1.27934i	−0.768651	+	0.639668i	$0.779072\pi$
$54$	0	0
$55$	0	0
$56$	3.65685	0.488668
$57$	0	0
$58$	− 11.6569i	− 1.53062i
$59$	−4.00000	−0.520756	−0.260378	−	0.965507i	$-0.583847\pi$
$59$	−4.00000	−0.520756	−0.260378	+	0.965507i	$0.583847\pi$
$60$	0	0
$61$	−11.6569	−1.49251	−0.746254	−	0.665662i	$-0.768149\pi$
$61$	−11.6569	−1.49251	−0.746254	+	0.665662i	$0.768149\pi$
$62$	0	0
$63$	0	0
$64$	9.82843	1.22855
$65$	0	0
$66$	0	0
$67$	− 5.65685i	− 0.691095i	−0.938401	−	0.345547i	$-0.887693\pi$
$67$	− 5.65685i	− 0.691095i	0.938401	−	0.345547i	$-0.112307\pi$
$68$	− 4.48528i	− 0.543920i
$69$	0	0
$70$	0	0
$71$	−2.34315	−0.278080	−0.139040	−	0.990287i	$-0.544402\pi$
$71$	−2.34315	−0.278080	−0.139040	+	0.990287i	$0.544402\pi$
$72$	0	0
$73$	− 11.3137i	− 1.32417i	−0.749429	−	0.662085i	$-0.769672\pi$
$73$	− 11.3137i	− 1.32417i	0.749429	−	0.662085i	$-0.230328\pi$
$74$	−28.1421	−3.27146
$75$	0	0
$76$	−26.1421	−2.99871
$77$	0.828427i	0.0944080i
$78$	0	0
$79$	−8.48528	−0.954669	−0.477334	−	0.878722i	$-0.658397\pi$
$79$	−8.48528	−0.954669	−0.477334	+	0.878722i	$0.658397\pi$
$80$	0	0
$81$	0	0
$82$	− 11.6569i	− 1.28728i
$83$	− 10.0000i	− 1.09764i	−0.835940	−	0.548821i	$-0.815077\pi$
$83$	− 10.0000i	− 1.09764i	0.835940	−	0.548821i	$-0.184923\pi$
$84$	0	0
$85$	0	0
$86$	−21.3137	−2.29832
$87$	0	0
$88$	− 4.41421i	− 0.470557i
$89$	3.65685	0.387626	0.193813	−	0.981039i	$-0.437915\pi$
$89$	3.65685	0.387626	0.193813	+	0.981039i	$0.437915\pi$
$90$	0	0
$91$	−4.68629	−0.491257
$92$	15.3137i	1.59656i
$93$	0	0
$94$	−9.65685	−0.996028
$95$	0	0
$96$	0	0
$97$	11.6569i	1.18357i	0.806094	+	0.591787i	$0.201577\pi$
$97$	11.6569i	1.18357i	−0.806094	+	0.591787i	$0.798423\pi$
$98$	15.2426i	1.53974i
$99$	0	0
$100$	0	0
$101$	0.828427	0.0824316	0.0412158	−	0.999150i	$-0.486877\pi$
$101$	0.828427	0.0824316	0.0412158	+	0.999150i	$0.486877\pi$
$102$	0	0
$103$	3.31371i	0.326509i	0.986584	+	0.163255i	$0.0521992\pi$
$103$	3.31371i	0.326509i	−0.986584	+	0.163255i	$0.947801\pi$
$104$	24.9706	2.44857
$105$	0	0
$106$	−22.4853	−2.18396
$107$	− 17.3137i	− 1.67378i	−0.547372	−	0.836890i	$-0.684372\pi$
$107$	− 17.3137i	− 1.67378i	0.547372	−	0.836890i	$-0.315628\pi$
$108$	0	0
$109$	−17.3137	−1.65835	−0.829176	−	0.558987i	$-0.811190\pi$
$109$	−17.3137	−1.65835	−0.829176	+	0.558987i	$0.811190\pi$
$110$	0	0
$111$	0	0
$112$	2.48528i	0.234837i
$113$	− 18.9706i	− 1.78460i	−0.451442	−	0.892300i	$-0.649090\pi$
$113$	− 18.9706i	− 1.78460i	0.451442	−	0.892300i	$-0.350910\pi$
$114$	0	0
$115$	0	0
$116$	18.4853	1.71632
$117$	0	0
$118$	− 9.65685i	− 0.888985i
$119$	−0.970563	−0.0889713
$120$	0	0
$121$	1.00000	0.0909091
$122$	− 28.1421i	− 2.54787i
$123$	0	0
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 14.4853i	− 1.28536i	−0.766134	−	0.642680i	$-0.777822\pi$
$127$	− 14.4853i	− 1.28536i	0.766134	−	0.642680i	$-0.222178\pi$
$128$	20.5563i	1.81694i
$129$	0	0
$130$	0	0
$131$	−3.31371	−0.289520	−0.144760	−	0.989467i	$-0.546241\pi$
$131$	−3.31371	−0.289520	−0.144760	+	0.989467i	$0.546241\pi$
$132$	0	0
$133$	5.65685i	0.490511i
$134$	13.6569	1.17977
$135$	0	0
$136$	5.17157	0.443459
$137$	13.3137i	1.13747i	0.822522	+	0.568733i	$0.192566\pi$
$137$	13.3137i	1.13747i	−0.822522	+	0.568733i	$0.807434\pi$
$138$	0	0
$139$	−0.485281	−0.0411610	−0.0205805	−	0.999788i	$-0.506551\pi$
$139$	−0.485281	−0.0411610	−0.0205805	+	0.999788i	$0.506551\pi$
$140$	0	0
$141$	0	0
$142$	− 5.65685i	− 0.474713i
$143$	5.65685i	0.473050i
$144$	0	0
$145$	0	0
$146$	27.3137	2.26050
$147$	0	0
$148$	− 44.6274i	− 3.66835i
$149$	1.51472	0.124091	0.0620453	−	0.998073i	$-0.480238\pi$
$149$	1.51472	0.124091	0.0620453	+	0.998073i	$0.480238\pi$
$150$	0	0
$151$	16.4853	1.34155	0.670777	−	0.741659i	$-0.265961\pi$
$151$	16.4853	1.34155	0.670777	+	0.741659i	$0.265961\pi$
$152$	− 30.1421i	− 2.44485i
$153$	0	0
$154$	−2.00000	−0.161165
$155$	0	0
$156$	0	0
$157$	18.0000i	1.43656i	0.695756	+	0.718278i	$0.255069\pi$
$157$	18.0000i	1.43656i	−0.695756	+	0.718278i	$0.744931\pi$
$158$	− 20.4853i	− 1.62972i
$159$	0	0
$160$	0	0
$161$	3.31371	0.261157
$162$	0	0
$163$	7.31371i	0.572854i	0.958102	+	0.286427i	$0.0924676\pi$
$163$	7.31371i	0.572854i	−0.958102	+	0.286427i	$0.907532\pi$
$164$	18.4853	1.44346
$165$	0	0
$166$	24.1421	1.87379
$167$	13.3137i	1.03025i	0.857116	+	0.515123i	$0.172254\pi$
$167$	13.3137i	1.03025i	−0.857116	+	0.515123i	$0.827746\pi$
$168$	0	0
$169$	−19.0000	−1.46154
$170$	0	0
$171$	0	0
$172$	− 33.7990i	− 2.57715i
$173$	2.82843i	0.215041i	0.994203	+	0.107521i	$0.0342912\pi$
$173$	2.82843i	0.215041i	−0.994203	+	0.107521i	$0.965709\pi$
$174$	0	0
$175$	0	0
$176$	3.00000	0.226134
$177$	0	0
$178$	8.82843i	0.661719i
$179$	−17.6569	−1.31974	−0.659868	−	0.751382i	$-0.729388\pi$
$179$	−17.6569	−1.31974	−0.659868	+	0.751382i	$0.729388\pi$
$180$	0	0
$181$	−14.0000	−1.04061	−0.520306	−	0.853980i	$-0.674182\pi$
$181$	−14.0000	−1.04061	−0.520306	+	0.853980i	$0.674182\pi$
$182$	− 11.3137i	− 0.838628i
$183$	0	0
$184$	−17.6569	−1.30168
$185$	0	0
$186$	0	0
$187$	1.17157i	0.0856739i
$188$	− 15.3137i	− 1.11687i
$189$	0	0
$190$	0	0
$191$	−5.65685	−0.409316	−0.204658	−	0.978834i	$-0.565608\pi$
$191$	−5.65685	−0.409316	−0.204658	+	0.978834i	$0.565608\pi$
$192$	0	0
$193$	− 13.6569i	− 0.983042i	−0.870866	−	0.491521i	$-0.836441\pi$
$193$	− 13.6569i	− 0.983042i	0.870866	−	0.491521i	$-0.163559\pi$
$194$	−28.1421	−2.02049
$195$	0	0
$196$	−24.1716	−1.72654
$197$	8.48528i	0.604551i	0.953221	+	0.302276i	$0.0977463\pi$
$197$	8.48528i	0.604551i	−0.953221	+	0.302276i	$0.902254\pi$
$198$	0	0
$199$	21.6569	1.53521	0.767607	−	0.640921i	$-0.221447\pi$
$199$	21.6569	1.53521	0.767607	+	0.640921i	$0.221447\pi$
$200$	0	0
$201$	0	0
$202$	2.00000i	0.140720i
$203$	− 4.00000i	− 0.280745i
$204$	0	0
$205$	0	0
$206$	−8.00000	−0.557386
$207$	0	0
$208$	16.9706i	1.17670i
$209$	6.82843	0.472332
$210$	0	0
$211$	1.17157	0.0806544	0.0403272	−	0.999187i	$-0.487160\pi$
$211$	1.17157	0.0806544	0.0403272	+	0.999187i	$0.487160\pi$
$212$	− 35.6569i	− 2.44892i
$213$	0	0
$214$	41.7990	2.85732
$215$	0	0
$216$	0	0
$217$	0	0
$218$	− 41.7990i	− 2.83098i
$219$	0	0
$220$	0	0
$221$	−6.62742	−0.445808
$222$	0	0
$223$	6.34315i	0.424768i	0.977186	+	0.212384i	$0.0681228\pi$
$223$	6.34315i	0.424768i	−0.977186	+	0.212384i	$0.931877\pi$
$224$	1.31371	0.0877758
$225$	0	0
$226$	45.7990	3.04650
$227$	14.0000i	0.929213i	0.885517	+	0.464606i	$0.153804\pi$
$227$	14.0000i	0.929213i	−0.885517	+	0.464606i	$0.846196\pi$
$228$	0	0
$229$	2.00000	0.132164	0.0660819	−	0.997814i	$-0.478950\pi$
$229$	2.00000	0.132164	0.0660819	+	0.997814i	$0.478950\pi$
$230$	0	0
$231$	0	0
$232$	21.3137i	1.39931i
$233$	− 18.8284i	− 1.23349i	−0.787163	−	0.616746i	$-0.788451\pi$
$233$	− 18.8284i	− 1.23349i	0.787163	−	0.616746i	$-0.211549\pi$
$234$	0	0
$235$	0	0
$236$	15.3137	0.996838
$237$	0	0
$238$	− 2.34315i	− 0.151884i
$239$	17.6569	1.14213	0.571063	−	0.820906i	$-0.306531\pi$
$239$	17.6569	1.14213	0.571063	+	0.820906i	$0.306531\pi$
$240$	0	0
$241$	−12.3431	−0.795092	−0.397546	−	0.917582i	$-0.630138\pi$
$241$	−12.3431	−0.795092	−0.397546	+	0.917582i	$0.630138\pi$
$242$	2.41421i	0.155192i
$243$	0	0
$244$	44.6274	2.85698
$245$	0	0
$246$	0	0
$247$	38.6274i	2.45780i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−20.9706	−1.32365	−0.661825	−	0.749658i	$-0.730218\pi$
$251$	−20.9706	−1.32365	−0.661825	+	0.749658i	$0.730218\pi$
$252$	0	0
$253$	− 4.00000i	− 0.251478i
$254$	34.9706	2.19425
$255$	0	0
$256$	−29.9706	−1.87316
$257$	16.3431i	1.01946i	0.860335	+	0.509729i	$0.170254\pi$
$257$	16.3431i	1.01946i	−0.860335	+	0.509729i	$0.829746\pi$
$258$	0	0
$259$	−9.65685	−0.600048
$260$	0	0
$261$	0	0
$262$	− 8.00000i	− 0.494242i
$263$	18.0000i	1.10993i	0.831875	+	0.554964i	$0.187268\pi$
$263$	18.0000i	1.10993i	−0.831875	+	0.554964i	$0.812732\pi$
$264$	0	0
$265$	0	0
$266$	−13.6569	−0.837355
$267$	0	0
$268$	21.6569i	1.32290i
$269$	20.6274	1.25768	0.628838	−	0.777536i	$-0.283531\pi$
$269$	20.6274	1.25768	0.628838	+	0.777536i	$0.283531\pi$
$270$	0	0
$271$	−11.7990	−0.716738	−0.358369	−	0.933580i	$-0.616667\pi$
$271$	−11.7990	−0.716738	−0.358369	+	0.933580i	$0.616667\pi$
$272$	3.51472i	0.213111i
$273$	0	0
$274$	−32.1421	−1.94178
$275$	0	0
$276$	0	0
$277$	− 2.34315i	− 0.140786i	−0.997519	−	0.0703930i	$-0.977575\pi$
$277$	− 2.34315i	− 0.140786i	0.997519	−	0.0703930i	$-0.0224253\pi$
$278$	− 1.17157i	− 0.0702663i
$279$	0	0
$280$	0	0
$281$	11.1716	0.666440	0.333220	−	0.942849i	$-0.391865\pi$
$281$	11.1716	0.666440	0.333220	+	0.942849i	$0.391865\pi$
$282$	0	0
$283$	8.82843i	0.524796i	0.964960	+	0.262398i	$0.0845132\pi$
$283$	8.82843i	0.524796i	−0.964960	+	0.262398i	$0.915487\pi$
$284$	8.97056	0.532305
$285$	0	0
$286$	−13.6569	−0.807547
$287$	− 4.00000i	− 0.236113i
$288$	0	0
$289$	15.6274	0.919260
$290$	0	0
$291$	0	0
$292$	43.3137i	2.53474i
$293$	− 6.82843i	− 0.398921i	−0.979906	−	0.199460i	$-0.936081\pi$
$293$	− 6.82843i	− 0.398921i	0.979906	−	0.199460i	$-0.0639190\pi$
$294$	0	0
$295$	0	0
$296$	51.4558	2.99081
$297$	0	0
$298$	3.65685i	0.211836i
$299$	22.6274	1.30858
$300$	0	0
$301$	−7.31371	−0.421555
$302$	39.7990i	2.29017i
$303$	0	0
$304$	20.4853	1.17491
$305$	0	0
$306$	0	0
$307$	− 3.17157i	− 0.181011i	−0.995896	−	0.0905056i	$-0.971152\pi$
$307$	− 3.17157i	− 0.181011i	0.995896	−	0.0905056i	$-0.0288483\pi$
$308$	− 3.17157i	− 0.180717i
$309$	0	0
$310$	0	0
$311$	3.31371	0.187903	0.0939516	−	0.995577i	$-0.470050\pi$
$311$	3.31371	0.187903	0.0939516	+	0.995577i	$0.470050\pi$
$312$	0	0
$313$	− 15.6569i	− 0.884978i	−0.896774	−	0.442489i	$-0.854096\pi$
$313$	− 15.6569i	− 0.884978i	0.896774	−	0.442489i	$-0.145904\pi$
$314$	−43.4558	−2.45236
$315$	0	0
$316$	32.4853	1.82744
$317$	26.2843i	1.47627i	0.674652	+	0.738136i	$0.264294\pi$
$317$	26.2843i	1.47627i	−0.674652	+	0.738136i	$0.735706\pi$
$318$	0	0
$319$	−4.82843	−0.270340
$320$	0	0
$321$	0	0
$322$	8.00000i	0.445823i
$323$	8.00000i	0.445132i
$324$	0	0
$325$	0	0
$326$	−17.6569	−0.977923
$327$	0	0
$328$	21.3137i	1.17685i
$329$	−3.31371	−0.182691
$330$	0	0
$331$	6.34315	0.348651	0.174325	−	0.984688i	$-0.444226\pi$
$331$	6.34315	0.348651	0.174325	+	0.984688i	$0.444226\pi$
$332$	38.2843i	2.10112i
$333$	0	0
$334$	−32.1421	−1.75874
$335$	0	0
$336$	0	0
$337$	3.31371i	0.180509i	0.995919	+	0.0902546i	$0.0287681\pi$
$337$	3.31371i	0.180509i	−0.995919	+	0.0902546i	$0.971232\pi$
$338$	− 45.8701i	− 2.49500i
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	11.0294i	0.595534i
$344$	38.9706	2.10115
$345$	0	0
$346$	−6.82843	−0.367099
$347$	29.3137i	1.57364i	0.617181	+	0.786821i	$0.288275\pi$
$347$	29.3137i	1.57364i	−0.617181	+	0.786821i	$0.711725\pi$
$348$	0	0
$349$	−10.9706	−0.587241	−0.293620	−	0.955922i	$-0.594860\pi$
$349$	−10.9706	−0.587241	−0.293620	+	0.955922i	$0.594860\pi$
$350$	0	0
$351$	0	0
$352$	− 1.58579i	− 0.0845227i
$353$	26.0000i	1.38384i	0.721974	+	0.691920i	$0.243235\pi$
$353$	26.0000i	1.38384i	−0.721974	+	0.691920i	$0.756765\pi$
$354$	0	0
$355$	0	0
$356$	−14.0000	−0.741999
$357$	0	0
$358$	− 42.6274i	− 2.25293i
$359$	12.0000	0.633336	0.316668	−	0.948536i	$-0.397436\pi$
$359$	12.0000	0.633336	0.316668	+	0.948536i	$0.397436\pi$
$360$	0	0
$361$	27.6274	1.45407
$362$	− 33.7990i	− 1.77644i
$363$	0	0
$364$	17.9411	0.940370
$365$	0	0
$366$	0	0
$367$	9.65685i	0.504084i	0.967716	+	0.252042i	$0.0811021\pi$
$367$	9.65685i	0.504084i	−0.967716	+	0.252042i	$0.918898\pi$
$368$	− 12.0000i	− 0.625543i
$369$	0	0
$370$	0	0
$371$	−7.71573	−0.400581
$372$	0	0
$373$	− 10.6274i	− 0.550267i	−0.961406	−	0.275133i	$-0.911278\pi$
$373$	− 10.6274i	− 0.550267i	0.961406	−	0.275133i	$-0.0887220\pi$
$374$	−2.82843	−0.146254
$375$	0	0
$376$	17.6569	0.910583
$377$	− 27.3137i	− 1.40673i
$378$	0	0
$379$	23.3137	1.19754	0.598772	−	0.800919i	$-0.295655\pi$
$379$	23.3137	1.19754	0.598772	+	0.800919i	$0.295655\pi$
$380$	0	0
$381$	0	0
$382$	− 13.6569i	− 0.698745i
$383$	− 8.00000i	− 0.408781i	−0.978889	−	0.204390i	$-0.934479\pi$
$383$	− 8.00000i	− 0.408781i	0.978889	−	0.204390i	$-0.0655212\pi$
$384$	0	0
$385$	0	0
$386$	32.9706	1.67816
$387$	0	0
$388$	− 44.6274i	− 2.26561i
$389$	−23.6569	−1.19945	−0.599725	−	0.800206i	$-0.704723\pi$
$389$	−23.6569	−1.19945	−0.599725	+	0.800206i	$0.704723\pi$
$390$	0	0
$391$	4.68629	0.236996
$392$	− 27.8701i	− 1.40765i
$393$	0	0
$394$	−20.4853	−1.03203
$395$	0	0
$396$	0	0
$397$	− 14.9706i	− 0.751351i	−0.926751	−	0.375676i	$-0.877411\pi$
$397$	− 14.9706i	− 0.751351i	0.926751	−	0.375676i	$-0.122589\pi$
$398$	52.2843i	2.62077i
$399$	0	0
$400$	0	0
$401$	6.68629	0.333897	0.166949	−	0.985966i	$-0.446609\pi$
$401$	6.68629	0.333897	0.166949	+	0.985966i	$0.446609\pi$
$402$	0	0
$403$	0	0
$404$	−3.17157	−0.157792
$405$	0	0
$406$	9.65685	0.479262
$407$	11.6569i	0.577809i
$408$	0	0
$409$	19.6569	0.971969	0.485984	−	0.873967i	$-0.338461\pi$
$409$	19.6569	0.971969	0.485984	+	0.873967i	$0.338461\pi$
$410$	0	0
$411$	0	0
$412$	− 12.6863i	− 0.625009i
$413$	− 3.31371i	− 0.163057i
$414$	0	0
$415$	0	0
$416$	8.97056	0.439818
$417$	0	0
$418$	16.4853i	0.806321i
$419$	−36.9706	−1.80613	−0.903065	−	0.429504i	$-0.858689\pi$
$419$	−36.9706	−1.80613	−0.903065	+	0.429504i	$0.858689\pi$
$420$	0	0
$421$	−6.00000	−0.292422	−0.146211	−	0.989253i	$-0.546708\pi$
$421$	−6.00000	−0.292422	−0.146211	+	0.989253i	$0.546708\pi$
$422$	2.82843i	0.137686i
$423$	0	0
$424$	41.1127	1.99661
$425$	0	0
$426$	0	0
$427$	− 9.65685i	− 0.467328i
$428$	66.2843i	3.20397i
$429$	0	0
$430$	0	0
$431$	−21.6569	−1.04317	−0.521587	−	0.853198i	$-0.674660\pi$
$431$	−21.6569	−1.04317	−0.521587	+	0.853198i	$0.674660\pi$
$432$	0	0
$433$	15.6569i	0.752420i	0.926534	+	0.376210i	$0.122773\pi$
$433$	15.6569i	0.752420i	−0.926534	+	0.376210i	$0.877227\pi$
$434$	0	0
$435$	0	0
$436$	66.2843	3.17444
$437$	− 27.3137i	− 1.30659i
$438$	0	0
$439$	20.4853	0.977709	0.488855	−	0.872365i	$-0.337415\pi$
$439$	20.4853	0.977709	0.488855	+	0.872365i	$0.337415\pi$
$440$	0	0
$441$	0	0
$442$	− 16.0000i	− 0.761042i
$443$	12.0000i	0.570137i	0.958507	+	0.285069i	$0.0920164\pi$
$443$	12.0000i	0.570137i	−0.958507	+	0.285069i	$0.907984\pi$
$444$	0	0
$445$	0	0
$446$	−15.3137	−0.725125
$447$	0	0
$448$	8.14214i	0.384680i
$449$	30.9706	1.46159	0.730796	−	0.682596i	$-0.239149\pi$
$449$	30.9706	1.46159	0.730796	+	0.682596i	$0.239149\pi$
$450$	0	0
$451$	−4.82843	−0.227362
$452$	72.6274i	3.41611i
$453$	0	0
$454$	−33.7990	−1.58627
$455$	0	0
$456$	0	0
$457$	− 23.3137i	− 1.09057i	−0.838251	−	0.545285i	$-0.816422\pi$
$457$	− 23.3137i	− 1.09057i	0.838251	−	0.545285i	$-0.183578\pi$
$458$	4.82843i	0.225618i
$459$	0	0
$460$	0	0
$461$	−0.142136	−0.00661992	−0.00330996	−	0.999995i	$-0.501054\pi$
$461$	−0.142136	−0.00661992	−0.00330996	+	0.999995i	$0.501054\pi$
$462$	0	0
$463$	− 4.97056i	− 0.231002i	−0.993307	−	0.115501i	$-0.963153\pi$
$463$	− 4.97056i	− 0.231002i	0.993307	−	0.115501i	$-0.0368473\pi$
$464$	−14.4853	−0.672462
$465$	0	0
$466$	45.4558	2.10570
$467$	22.6274i	1.04707i	0.852004	+	0.523536i	$0.175387\pi$
$467$	22.6274i	1.04707i	−0.852004	+	0.523536i	$0.824613\pi$
$468$	0	0
$469$	4.68629	0.216393
$470$	0	0
$471$	0	0
$472$	17.6569i	0.812723i
$473$	8.82843i	0.405932i
$474$	0	0
$475$	0	0
$476$	3.71573	0.170310
$477$	0	0
$478$	42.6274i	1.94973i
$479$	36.9706	1.68923	0.844614	−	0.535376i	$-0.179830\pi$
$479$	36.9706	1.68923	0.844614	+	0.535376i	$0.179830\pi$
$480$	0	0
$481$	−65.9411	−3.00666
$482$	− 29.7990i	− 1.35731i
$483$	0	0
$484$	−3.82843	−0.174019
$485$	0	0
$486$	0	0
$487$	− 12.9706i	− 0.587752i	−0.955844	−	0.293876i	$-0.905055\pi$
$487$	− 12.9706i	− 0.587752i	0.955844	−	0.293876i	$-0.0949453\pi$
$488$	51.4558i	2.32930i
$489$	0	0
$490$	0	0
$491$	−14.3431	−0.647297	−0.323649	−	0.946177i	$-0.604910\pi$
$491$	−14.3431	−0.647297	−0.323649	+	0.946177i	$0.604910\pi$
$492$	0	0
$493$	− 5.65685i	− 0.254772i
$494$	−93.2548	−4.19573
$495$	0	0
$496$	0	0
$497$	− 1.94113i	− 0.0870714i
$498$	0	0
$499$	22.3431	1.00022	0.500108	−	0.865963i	$-0.333294\pi$
$499$	22.3431	1.00022	0.500108	+	0.865963i	$0.333294\pi$
$500$	0	0
$501$	0	0
$502$	− 50.6274i	− 2.25961i
$503$	17.3137i	0.771980i	0.922503	+	0.385990i	$0.126140\pi$
$503$	17.3137i	0.771980i	−0.922503	+	0.385990i	$0.873860\pi$
$504$	0	0
$505$	0	0
$506$	9.65685	0.429300
$507$	0	0
$508$	55.4558i	2.46046i
$509$	18.6863	0.828255	0.414128	−	0.910219i	$-0.364087\pi$
$509$	18.6863	0.828255	0.414128	+	0.910219i	$0.364087\pi$
$510$	0	0
$511$	9.37258	0.414619
$512$	− 31.2426i	− 1.38074i
$513$	0	0
$514$	−39.4558	−1.74032
$515$	0	0
$516$	0	0
$517$	4.00000i	0.175920i
$518$	− 23.3137i	− 1.02435i
$519$	0	0
$520$	0	0
$521$	32.6274	1.42943	0.714717	−	0.699414i	$-0.246556\pi$
$521$	32.6274	1.42943	0.714717	+	0.699414i	$0.246556\pi$
$522$	0	0
$523$	9.51472i	0.416050i	0.978124	+	0.208025i	$0.0667035\pi$
$523$	9.51472i	0.416050i	−0.978124	+	0.208025i	$0.933297\pi$
$524$	12.6863	0.554203
$525$	0	0
$526$	−43.4558	−1.89476
$527$	0	0
$528$	0	0
$529$	7.00000	0.304348
$530$	0	0
$531$	0	0
$532$	− 21.6569i	− 0.938944i
$533$	− 27.3137i	− 1.18309i
$534$	0	0
$535$	0	0
$536$	−24.9706	−1.07856
$537$	0	0
$538$	49.7990i	2.14699i
$539$	6.31371	0.271951
$540$	0	0
$541$	17.3137	0.744374	0.372187	−	0.928158i	$-0.378608\pi$
$541$	17.3137	0.744374	0.372187	+	0.928158i	$0.378608\pi$
$542$	− 28.4853i	− 1.22355i
$543$	0	0
$544$	1.85786	0.0796553
$545$	0	0
$546$	0	0
$547$	− 8.14214i	− 0.348133i	−0.984734	−	0.174066i	$-0.944309\pi$
$547$	− 8.14214i	− 0.348133i	0.984734	−	0.174066i	$-0.0556907\pi$
$548$	− 50.9706i	− 2.17735i
$549$	0	0
$550$	0	0
$551$	−32.9706	−1.40459
$552$	0	0
$553$	− 7.02944i	− 0.298922i
$554$	5.65685	0.240337
$555$	0	0
$556$	1.85786	0.0787910
$557$	− 5.17157i	− 0.219127i	−0.993980	−	0.109563i	$-0.965055\pi$
$557$	− 5.17157i	− 0.219127i	0.993980	−	0.109563i	$-0.0349452\pi$
$558$	0	0
$559$	−49.9411	−2.11228
$560$	0	0
$561$	0	0
$562$	26.9706i	1.13768i
$563$	31.6569i	1.33418i	0.744978	+	0.667089i	$0.232460\pi$
$563$	31.6569i	1.33418i	−0.744978	+	0.667089i	$0.767540\pi$
$564$	0	0
$565$	0	0
$566$	−21.3137	−0.895882
$567$	0	0
$568$	10.3431i	0.433989i
$569$	−35.4558	−1.48639	−0.743193	−	0.669077i	$-0.766690\pi$
$569$	−35.4558	−1.48639	−0.743193	+	0.669077i	$0.766690\pi$
$570$	0	0
$571$	−16.4853	−0.689888	−0.344944	−	0.938623i	$-0.612102\pi$
$571$	−16.4853	−0.689888	−0.344944	+	0.938623i	$0.612102\pi$
$572$	− 21.6569i	− 0.905519i
$573$	0	0
$574$	9.65685	0.403069
$575$	0	0
$576$	0	0
$577$	14.0000i	0.582828i	0.956597	+	0.291414i	$0.0941257\pi$
$577$	14.0000i	0.582828i	−0.956597	+	0.291414i	$0.905874\pi$
$578$	37.7279i	1.56927i
$579$	0	0
$580$	0	0
$581$	8.28427	0.343689
$582$	0	0
$583$	9.31371i	0.385734i
$584$	−49.9411	−2.06658
$585$	0	0
$586$	16.4853	0.681001
$587$	− 14.6274i	− 0.603738i	−0.953349	−	0.301869i	$-0.902389\pi$
$587$	− 14.6274i	− 0.603738i	0.953349	−	0.301869i	$-0.0976105\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	0	0
$592$	34.9706i	1.43728i
$593$	− 22.8284i	− 0.937451i	−0.883344	−	0.468726i	$-0.844713\pi$
$593$	− 22.8284i	− 0.937451i	0.883344	−	0.468726i	$-0.155287\pi$
$594$	0	0
$595$	0	0
$596$	−5.79899	−0.237536
$597$	0	0
$598$	54.6274i	2.23388i
$599$	27.3137	1.11601	0.558004	−	0.829838i	$-0.311567\pi$
$599$	27.3137	1.11601	0.558004	+	0.829838i	$0.311567\pi$
$600$	0	0
$601$	−5.31371	−0.216751	−0.108375	−	0.994110i	$-0.534565\pi$
$601$	−5.31371	−0.216751	−0.108375	+	0.994110i	$0.534565\pi$
$602$	− 17.6569i	− 0.719640i
$603$	0	0
$604$	−63.1127	−2.56802
$605$	0	0
$606$	0	0
$607$	1.51472i	0.0614805i	0.999527	+	0.0307403i	$0.00978647\pi$
$607$	1.51472i	0.0614805i	−0.999527	+	0.0307403i	$0.990214\pi$
$608$	− 10.8284i	− 0.439151i
$609$	0	0
$610$	0	0
$611$	−22.6274	−0.915407
$612$	0	0
$613$	45.9411i	1.85554i	0.373147	+	0.927772i	$0.378279\pi$
$613$	45.9411i	1.85554i	−0.373147	+	0.927772i	$0.621721\pi$
$614$	7.65685	0.309005
$615$	0	0
$616$	3.65685	0.147339
$617$	− 0.343146i	− 0.0138145i	−0.999976	−	0.00690726i	$-0.997801\pi$
$617$	− 0.343146i	− 0.0138145i	0.999976	−	0.00690726i	$-0.00219867\pi$
$618$	0	0
$619$	14.3431	0.576500	0.288250	−	0.957555i	$-0.406927\pi$
$619$	14.3431	0.576500	0.288250	+	0.957555i	$0.406927\pi$
$620$	0	0
$621$	0	0
$622$	8.00000i	0.320771i
$623$	3.02944i	0.121372i
$624$	0	0
$625$	0	0
$626$	37.7990	1.51075
$627$	0	0
$628$	− 68.9117i	− 2.74988i
$629$	−13.6569	−0.544534
$630$	0	0
$631$	45.6569	1.81757	0.908785	−	0.417264i	$-0.137011\pi$
$631$	45.6569	1.81757	0.908785	+	0.417264i	$0.137011\pi$
$632$	37.4558i	1.48991i
$633$	0	0
$634$	−63.4558	−2.52015
$635$	0	0
$636$	0	0
$637$	35.7157i	1.41511i
$638$	− 11.6569i	− 0.461499i
$639$	0	0
$640$	0	0
$641$	−6.97056	−0.275321	−0.137660	−	0.990479i	$-0.543958\pi$
$641$	−6.97056	−0.275321	−0.137660	+	0.990479i	$0.543958\pi$
$642$	0	0
$643$	− 37.9411i	− 1.49625i	−0.663557	−	0.748126i	$-0.730954\pi$
$643$	− 37.9411i	− 1.49625i	0.663557	−	0.748126i	$-0.269046\pi$
$644$	−12.6863	−0.499910
$645$	0	0
$646$	−19.3137	−0.759888
$647$	− 4.68629i	− 0.184237i	−0.995748	−	0.0921186i	$-0.970636\pi$
$647$	− 4.68629i	− 0.184237i	0.995748	−	0.0921186i	$-0.0293639\pi$
$648$	0	0
$649$	−4.00000	−0.157014
$650$	0	0
$651$	0	0
$652$	− 28.0000i	− 1.09656i
$653$	6.97056i	0.272779i	0.990655	+	0.136390i	$0.0435499\pi$
$653$	6.97056i	0.272779i	−0.990655	+	0.136390i	$0.956450\pi$
$654$	0	0
$655$	0	0
$656$	−14.4853	−0.565555
$657$	0	0
$658$	− 8.00000i	− 0.311872i
$659$	−15.3137	−0.596537	−0.298269	−	0.954482i	$-0.596409\pi$
$659$	−15.3137	−0.596537	−0.298269	+	0.954482i	$0.596409\pi$
$660$	0	0
$661$	9.31371	0.362261	0.181131	−	0.983459i	$-0.442024\pi$
$661$	9.31371	0.362261	0.181131	+	0.983459i	$0.442024\pi$
$662$	15.3137i	0.595184i
$663$	0	0
$664$	−44.1421	−1.71305
$665$	0	0
$666$	0	0
$667$	19.3137i	0.747830i
$668$	− 50.9706i	− 1.97211i
$669$	0	0
$670$	0	0
$671$	−11.6569	−0.450008
$672$	0	0
$673$	− 18.3431i	− 0.707076i	−0.935420	−	0.353538i	$-0.884978\pi$
$673$	− 18.3431i	− 0.707076i	0.935420	−	0.353538i	$-0.115022\pi$
$674$	−8.00000	−0.308148
$675$	0	0
$676$	72.7401	2.79770
$677$	− 29.4558i	− 1.13208i	−0.824378	−	0.566040i	$-0.808475\pi$
$677$	− 29.4558i	− 1.13208i	0.824378	−	0.566040i	$-0.191525\pi$
$678$	0	0
$679$	−9.65685	−0.370596
$680$	0	0
$681$	0	0
$682$	0	0
$683$	24.0000i	0.918334i	0.888350	+	0.459167i	$0.151852\pi$
$683$	24.0000i	0.918334i	−0.888350	+	0.459167i	$0.848148\pi$
$684$	0	0
$685$	0	0
$686$	−26.6274	−1.01664
$687$	0	0
$688$	26.4853i	1.00974i
$689$	−52.6863	−2.00719
$690$	0	0
$691$	20.0000	0.760836	0.380418	−	0.924815i	$-0.375780\pi$
$691$	20.0000	0.760836	0.380418	+	0.924815i	$0.375780\pi$
$692$	− 10.8284i	− 0.411635i
$693$	0	0
$694$	−70.7696	−2.68638
$695$	0	0
$696$	0	0
$697$	− 5.65685i	− 0.214269i
$698$	− 26.4853i	− 1.00248i
$699$	0	0
$700$	0	0
$701$	−36.1421	−1.36507	−0.682535	−	0.730853i	$-0.739122\pi$
$701$	−36.1421	−1.36507	−0.682535	+	0.730853i	$0.739122\pi$
$702$	0	0
$703$	79.5980i	3.00209i
$704$	9.82843	0.370423
$705$	0	0
$706$	−62.7696	−2.36236
$707$	0.686292i	0.0258106i
$708$	0	0
$709$	−6.68629	−0.251109	−0.125554	−	0.992087i	$-0.540071\pi$
$709$	−6.68629	−0.251109	−0.125554	+	0.992087i	$0.540071\pi$
$710$	0	0
$711$	0	0
$712$	− 16.1421i	− 0.604952i
$713$	0	0
$714$	0	0
$715$	0	0
$716$	67.5980	2.52626
$717$	0	0
$718$	28.9706i	1.08117i
$719$	−47.5980	−1.77511	−0.887553	−	0.460706i	$-0.847596\pi$
$719$	−47.5980	−1.77511	−0.887553	+	0.460706i	$0.847596\pi$
$720$	0	0
$721$	−2.74517	−0.102235
$722$	66.6985i	2.48226i
$723$	0	0
$724$	53.5980	1.99195
$725$	0	0
$726$	0	0
$727$	33.9411i	1.25881i	0.777079	+	0.629403i	$0.216701\pi$
$727$	33.9411i	1.25881i	−0.777079	+	0.629403i	$0.783299\pi$
$728$	20.6863i	0.766685i
$729$	0	0
$730$	0	0
$731$	−10.3431	−0.382555
$732$	0	0
$733$	6.34315i	0.234289i	0.993115	+	0.117145i	$0.0373741\pi$
$733$	6.34315i	0.234289i	−0.993115	+	0.117145i	$0.962626\pi$
$734$	−23.3137	−0.860525
$735$	0	0
$736$	−6.34315	−0.233811
$737$	− 5.65685i	− 0.208373i
$738$	0	0
$739$	15.1127	0.555930	0.277965	−	0.960591i	$-0.410340\pi$
$739$	15.1127	0.555930	0.277965	+	0.960591i	$0.410340\pi$
$740$	0	0
$741$	0	0
$742$	− 18.6274i	− 0.683834i
$743$	36.3431i	1.33330i	0.745371	+	0.666650i	$0.232273\pi$
$743$	36.3431i	1.33330i	−0.745371	+	0.666650i	$0.767727\pi$
$744$	0	0
$745$	0	0
$746$	25.6569	0.939364
$747$	0	0
$748$	− 4.48528i	− 0.163998i
$749$	14.3431	0.524087
$750$	0	0
$751$	20.2843	0.740184	0.370092	−	0.928995i	$-0.379326\pi$
$751$	20.2843	0.740184	0.370092	+	0.928995i	$0.379326\pi$
$752$	12.0000i	0.437595i
$753$	0	0
$754$	65.9411	2.40143
$755$	0	0
$756$	0	0
$757$	− 36.6274i	− 1.33125i	−0.746288	−	0.665623i	$-0.768166\pi$
$757$	− 36.6274i	− 1.33125i	0.746288	−	0.665623i	$-0.231834\pi$
$758$	56.2843i	2.04434i
$759$	0	0
$760$	0	0
$761$	28.8284	1.04503	0.522515	−	0.852630i	$-0.324994\pi$
$761$	28.8284	1.04503	0.522515	+	0.852630i	$0.324994\pi$
$762$	0	0
$763$	− 14.3431i	− 0.519257i
$764$	21.6569	0.783517
$765$	0	0
$766$	19.3137	0.697833
$767$	− 22.6274i	− 0.817029i
$768$	0	0
$769$	−10.6863	−0.385358	−0.192679	−	0.981262i	$-0.561718\pi$
$769$	−10.6863	−0.385358	−0.192679	+	0.981262i	$0.561718\pi$
$770$	0	0
$771$	0	0
$772$	52.2843i	1.88175i
$773$	3.65685i	0.131528i	0.997835	+	0.0657640i	$0.0209484\pi$
$773$	3.65685i	0.131528i	−0.997835	+	0.0657640i	$0.979052\pi$
$774$	0	0
$775$	0	0
$776$	51.4558	1.84716
$777$	0	0
$778$	− 57.1127i	− 2.04759i
$779$	−32.9706	−1.18129
$780$	0	0
$781$	−2.34315	−0.0838443
$782$	11.3137i	0.404577i
$783$	0	0
$784$	18.9411	0.676469
$785$	0	0
$786$	0	0
$787$	− 20.1421i	− 0.717990i	−0.933340	−	0.358995i	$-0.883120\pi$
$787$	− 20.1421i	− 0.717990i	0.933340	−	0.358995i	$-0.116880\pi$
$788$	− 32.4853i	− 1.15724i
$789$	0	0
$790$	0	0
$791$	15.7157	0.558787
$792$	0	0
$793$	− 65.9411i	− 2.34164i
$794$	36.1421	1.28264
$795$	0	0
$796$	−82.9117	−2.93873
$797$	− 34.9706i	− 1.23872i	−0.785107	−	0.619360i	$-0.787392\pi$
$797$	− 34.9706i	− 1.23872i	0.785107	−	0.619360i	$-0.212608\pi$
$798$	0	0
$799$	−4.68629	−0.165789
$800$	0	0
$801$	0	0
$802$	16.1421i	0.569999i
$803$	− 11.3137i	− 0.399252i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	− 3.65685i	− 0.128648i
$809$	28.4264	0.999419	0.499710	−	0.866193i	$-0.333440\pi$
$809$	28.4264	0.999419	0.499710	+	0.866193i	$0.333440\pi$
$810$	0	0
$811$	0.485281	0.0170405	0.00852027	−	0.999964i	$-0.497288\pi$
$811$	0.485281	0.0170405	0.00852027	+	0.999964i	$0.497288\pi$
$812$	15.3137i	0.537406i
$813$	0	0
$814$	−28.1421	−0.986381
$815$	0	0
$816$	0	0
$817$	60.2843i	2.10908i
$818$	47.4558i	1.65925i
$819$	0	0
$820$	0	0
$821$	12.8284	0.447715	0.223858	−	0.974622i	$-0.428135\pi$
$821$	12.8284	0.447715	0.223858	+	0.974622i	$0.428135\pi$
$822$	0	0
$823$	− 16.0000i	− 0.557725i	−0.960331	−	0.278862i	$-0.910043\pi$
$823$	− 16.0000i	− 0.557725i	0.960331	−	0.278862i	$-0.0899574\pi$
$824$	14.6274	0.509570
$825$	0	0
$826$	8.00000	0.278356
$827$	− 41.3137i	− 1.43662i	−0.695724	−	0.718309i	$-0.744916\pi$
$827$	− 41.3137i	− 1.43662i	0.695724	−	0.718309i	$-0.255084\pi$
$828$	0	0
$829$	38.0000	1.31979	0.659897	−	0.751356i	$-0.270600\pi$
$829$	38.0000	1.31979	0.659897	+	0.751356i	$0.270600\pi$
$830$	0	0
$831$	0	0
$832$	55.5980i	1.92751i
$833$	7.39697i	0.256290i
$834$	0	0
$835$	0	0
$836$	−26.1421	−0.904145
$837$	0	0
$838$	− 89.2548i	− 3.08326i
$839$	−22.6274	−0.781185	−0.390593	−	0.920564i	$-0.627730\pi$
$839$	−22.6274	−0.781185	−0.390593	+	0.920564i	$0.627730\pi$
$840$	0	0
$841$	−5.68629	−0.196079
$842$	− 14.4853i	− 0.499196i
$843$	0	0
$844$	−4.48528	−0.154390
$845$	0	0
$846$	0	0
$847$	0.828427i	0.0284651i
$848$	27.9411i	0.959502i
$849$	0	0
$850$	0	0
$851$	46.6274	1.59837
$852$	0	0
$853$	8.68629i	0.297413i	0.988881	+	0.148706i	$0.0475110\pi$
$853$	8.68629i	0.297413i	−0.988881	+	0.148706i	$0.952489\pi$
$854$	23.3137	0.797779
$855$	0	0
$856$	−76.4264	−2.61220
$857$	28.4853i	0.973039i	0.873670	+	0.486519i	$0.161734\pi$
$857$	28.4853i	0.973039i	−0.873670	+	0.486519i	$0.838266\pi$
$858$	0	0
$859$	52.9706	1.80733	0.903666	−	0.428238i	$-0.140865\pi$
$859$	52.9706	1.80733	0.903666	+	0.428238i	$0.140865\pi$
$860$	0	0
$861$	0	0
$862$	− 52.2843i	− 1.78081i
$863$	− 20.6863i	− 0.704170i	−0.935968	−	0.352085i	$-0.885473\pi$
$863$	− 20.6863i	− 0.704170i	0.935968	−	0.352085i	$-0.114527\pi$
$864$	0	0
$865$	0	0
$866$	−37.7990	−1.28446
$867$	0	0
$868$	0	0
$869$	−8.48528	−0.287843
$870$	0	0
$871$	32.0000	1.08428
$872$	76.4264i	2.58812i
$873$	0	0
$874$	65.9411	2.23049
$875$	0	0
$876$	0	0
$877$	2.62742i	0.0887216i	0.999016	+	0.0443608i	$0.0141251\pi$
$877$	2.62742i	0.0887216i	−0.999016	+	0.0443608i	$0.985875\pi$
$878$	49.4558i	1.66905i
$879$	0	0
$880$	0	0
$881$	46.9706	1.58248	0.791239	−	0.611507i	$-0.209436\pi$
$881$	46.9706	1.58248	0.791239	+	0.611507i	$0.209436\pi$
$882$	0	0
$883$	5.37258i	0.180802i	0.995905	+	0.0904009i	$0.0288148\pi$
$883$	5.37258i	0.180802i	−0.995905	+	0.0904009i	$0.971185\pi$
$884$	25.3726	0.853372
$885$	0	0
$886$	−28.9706	−0.973285
$887$	15.6569i	0.525706i	0.964836	+	0.262853i	$0.0846634\pi$
$887$	15.6569i	0.525706i	−0.964836	+	0.262853i	$0.915337\pi$
$888$	0	0
$889$	12.0000	0.402467
$890$	0	0
$891$	0	0
$892$	− 24.2843i	− 0.813098i
$893$	27.3137i	0.914018i
$894$	0	0
$895$	0	0
$896$	−17.0294	−0.568914
$897$	0	0
$898$	74.7696i	2.49509i
$899$	0	0
$900$	0	0
$901$	−10.9117	−0.363521
$902$	− 11.6569i	− 0.388131i
$903$	0	0
$904$	−83.7401	−2.78515
$905$	0	0
$906$	0	0
$907$	40.9706i	1.36041i	0.733024	+	0.680203i	$0.238108\pi$
$907$	40.9706i	1.36041i	−0.733024	+	0.680203i	$0.761892\pi$
$908$	− 53.5980i	− 1.77871i
$909$	0	0
$910$	0	0
$911$	48.9706	1.62247	0.811234	−	0.584722i	$-0.198797\pi$
$911$	48.9706	1.62247	0.811234	+	0.584722i	$0.198797\pi$
$912$	0	0
$913$	− 10.0000i	− 0.330952i
$914$	56.2843	1.86172
$915$	0	0
$916$	−7.65685	−0.252990
$917$	− 2.74517i	− 0.0906534i
$918$	0	0
$919$	−11.5147	−0.379836	−0.189918	−	0.981800i	$-0.560822\pi$
$919$	−11.5147	−0.379836	−0.189918	+	0.981800i	$0.560822\pi$
$920$	0	0
$921$	0	0
$922$	− 0.343146i	− 0.0113009i
$923$	− 13.2548i	− 0.436288i
$924$	0	0
$925$	0	0
$926$	12.0000	0.394344
$927$	0	0
$928$	7.65685i	0.251349i
$929$	−45.5980	−1.49602	−0.748011	−	0.663687i	$-0.768991\pi$
$929$	−45.5980	−1.49602	−0.748011	+	0.663687i	$0.768991\pi$
$930$	0	0
$931$	43.1127	1.41296
$932$	72.0833i	2.36117i
$933$	0	0
$934$	−54.6274	−1.78746
$935$	0	0
$936$	0	0
$937$	11.0294i	0.360316i	0.983638	+	0.180158i	$0.0576609\pi$
$937$	11.0294i	0.360316i	−0.983638	+	0.180158i	$0.942339\pi$
$938$	11.3137i	0.369406i
$939$	0	0
$940$	0	0
$941$	34.7696	1.13346	0.566728	−	0.823905i	$-0.308209\pi$
$941$	34.7696	1.13346	0.566728	+	0.823905i	$0.308209\pi$
$942$	0	0
$943$	19.3137i	0.628941i
$944$	−12.0000	−0.390567
$945$	0	0
$946$	−21.3137	−0.692968
$947$	− 6.62742i	− 0.215362i	−0.994185	−	0.107681i	$-0.965657\pi$
$947$	− 6.62742i	− 0.215362i	0.994185	−	0.107681i	$-0.0343425\pi$
$948$	0	0
$949$	64.0000	2.07753
$950$	0	0
$951$	0	0
$952$	4.28427i	0.138854i
$953$	− 11.7990i	− 0.382207i	−0.981570	−	0.191103i	$-0.938793\pi$
$953$	− 11.7990i	− 0.382207i	0.981570	−	0.191103i	$-0.0612066\pi$
$954$	0	0
$955$	0	0
$956$	−67.5980	−2.18627
$957$	0	0
$958$	89.2548i	2.88369i
$959$	−11.0294	−0.356159
$960$	0	0
$961$	−31.0000	−1.00000
$962$	− 159.196i	− 5.13268i
$963$	0	0
$964$	47.2548	1.52198
$965$	0	0
$966$	0	0
$967$	11.4558i	0.368395i	0.982889	+	0.184198i	$0.0589686\pi$
$967$	11.4558i	0.368395i	−0.982889	+	0.184198i	$0.941031\pi$
$968$	− 4.41421i	− 0.141878i
$969$	0	0
$970$	0	0
$971$	34.6274	1.11125	0.555623	−	0.831434i	$-0.312480\pi$
$971$	34.6274	1.11125	0.555623	+	0.831434i	$0.312480\pi$
$972$	0	0
$973$	− 0.402020i	− 0.0128882i
$974$	31.3137	1.00336
$975$	0	0
$976$	−34.9706	−1.11938
$977$	− 2.68629i	− 0.0859421i	−0.999076	−	0.0429710i	$-0.986318\pi$
$977$	− 2.68629i	− 0.0859421i	0.999076	−	0.0429710i	$-0.0136823\pi$
$978$	0	0
$979$	3.65685	0.116874
$980$	0	0
$981$	0	0
$982$	− 34.6274i	− 1.10501i
$983$	− 30.6274i	− 0.976863i	−0.872602	−	0.488431i	$-0.837569\pi$
$983$	− 30.6274i	− 0.976863i	0.872602	−	0.488431i	$-0.162431\pi$
$984$	0	0
$985$	0	0
$986$	13.6569	0.434923
$987$	0	0
$988$	− 147.882i	− 4.70476i
$989$	35.3137	1.12291
$990$	0	0
$991$	−30.6274	−0.972912	−0.486456	−	0.873705i	$-0.661711\pi$
$991$	−30.6274	−0.972912	−0.486456	+	0.873705i	$0.661711\pi$
$992$	0	0
$993$	0	0
$994$	4.68629	0.148640
$995$	0	0
$996$	0	0
$997$	− 39.3137i	− 1.24508i	−0.782589	−	0.622539i	$-0.786101\pi$
$997$	− 39.3137i	− 1.24508i	0.782589	−	0.622539i	$-0.213899\pi$
$998$	53.9411i	1.70748i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2475.2.c.m.199.4		4
3.2	odd	2		825.2.c.e.199.1		4
5.2	odd	4		2475.2.a.m.1.1		2
5.3	odd	4		495.2.a.d.1.2		2
5.4	even	2	inner	2475.2.c.m.199.1		4
15.2	even	4		825.2.a.g.1.2		2
15.8	even	4		165.2.a.a.1.1	✓	2
15.14	odd	2		825.2.c.e.199.4		4
20.3	even	4		7920.2.a.cg.1.1		2
55.43	even	4		5445.2.a.m.1.1		2
60.23	odd	4		2640.2.a.bb.1.1		2
105.83	odd	4		8085.2.a.ba.1.1		2
165.32	odd	4		9075.2.a.v.1.1		2
165.98	odd	4		1815.2.a.k.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
165.2.a.a.1.1	✓	2	15.8	even	4
495.2.a.d.1.2		2	5.3	odd	4
825.2.a.g.1.2		2	15.2	even	4
825.2.c.e.199.1		4	3.2	odd	2
825.2.c.e.199.4		4	15.14	odd	2
1815.2.a.k.1.2		2	165.98	odd	4
2475.2.a.m.1.1		2	5.2	odd	4
2475.2.c.m.199.1		4	5.4	even	2	inner
2475.2.c.m.199.4		4	1.1	even	1	trivial
2640.2.a.bb.1.1		2	60.23	odd	4
5445.2.a.m.1.1		2	55.43	even	4
7920.2.a.cg.1.1		2	20.3	even	4
8085.2.a.ba.1.1		2	105.83	odd	4
9075.2.a.v.1.1		2	165.32	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+2.41421i q^{2} -3.82843 q^{4} +0.828427i q^{7} -4.41421i q^{8} +O(q^{10})\)	\(q+2.41421i q^{2} -3.82843 q^{4} +0.828427i q^{7} -4.41421i q^{8} +1.00000 q^{11} +5.65685i q^{13} -2.00000 q^{14} +3.00000 q^{16} +1.17157i q^{17} +6.82843 q^{19} +2.41421i q^{22} -4.00000i q^{23} -13.6569 q^{26} -3.17157i q^{28} -4.82843 q^{29} -1.58579i q^{32} -2.82843 q^{34} +11.6569i q^{37} +16.4853i q^{38} -4.82843 q^{41} +8.82843i q^{43} -3.82843 q^{44} +9.65685 q^{46} +4.00000i q^{47} +6.31371 q^{49} -21.6569i q^{52} +9.31371i q^{53} +3.65685 q^{56} -11.6569i q^{58} -4.00000 q^{59} -11.6569 q^{61} +9.82843 q^{64} -5.65685i q^{67} -4.48528i q^{68} -2.34315 q^{71} -11.3137i q^{73} -28.1421 q^{74} -26.1421 q^{76} +0.828427i q^{77} -8.48528 q^{79} -11.6569i q^{82} -10.0000i q^{83} -21.3137 q^{86} -4.41421i q^{88} +3.65685 q^{89} -4.68629 q^{91} +15.3137i q^{92} -9.65685 q^{94} +11.6569i q^{97} +15.2426i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{4}+O(q^{10}) \) 4 * q - 4 * q^4	\( 4 q - 4 q^{4} + 4 q^{11} - 8 q^{14} + 12 q^{16} + 16 q^{19} - 32 q^{26} - 8 q^{29} - 8 q^{41} - 4 q^{44} + 16 q^{46} - 20 q^{49} - 8 q^{56} - 16 q^{59} - 24 q^{61} + 28 q^{64} - 32 q^{71} - 56 q^{74} - 48 q^{76} - 40 q^{86} - 8 q^{89} - 64 q^{91} - 16 q^{94}+O(q^{100}) \) 4 * q - 4 * q^4 + 4 * q^11 - 8 * q^14 + 12 * q^16 + 16 * q^19 - 32 * q^26 - 8 * q^29 - 8 * q^41 - 4 * q^44 + 16 * q^46 - 20 * q^49 - 8 * q^56 - 16 * q^59 - 24 * q^61 + 28 * q^64 - 32 * q^71 - 56 * q^74 - 48 * q^76 - 40 * q^86 - 8 * q^89 - 64 * q^91 - 16 * q^94

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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