LMFDB - Embedded newform 6900.2.f.r.6349.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6900,2,Mod(6349,6900)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6900, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("6900.6349");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6900 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6900.f (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:	$55.0967773947$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.158155776.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} + 2x^{4} + 24x^{3} + 81x^{2} + 54x + 18 $ x^6 - 2x^5 + 2x^4 + 24x^3 + 81x^2 + 54*x + 18
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 1380)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			6349.4
Root			$-0.376763 - 0.376763i$ of defining polynomial
Character	$\chi$	$=$	6900.6349
Dual form			6900.2.f.r.6349.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{3} -4.20905i q^{7} -1.00000 q^{9} +O(q^{10})$	$q+1.00000i q^{3} -4.20905i q^{7} -1.00000 q^{9} +2.75353 q^{11} -0.753525i q^{13} +4.96257i q^{17} -4.75353 q^{19} +4.20905 q^{21} +1.00000i q^{23} -1.00000i q^{27} -4.96257 q^{29} -0.209050 q^{31} +2.75353i q^{33} +5.71610i q^{37} +0.753525 q^{39} -9.38067 q^{41} +12.4181i q^{43} -7.17162i q^{47} -10.7161 q^{49} -4.96257 q^{51} -9.38067i q^{53} -4.75353i q^{57} +4.96257 q^{59} -5.17162 q^{61} +4.20905i q^{63} +0.209050i q^{67} -1.00000 q^{69} +9.38067 q^{71} +10.2606i q^{73} -11.5897i q^{77} +2.41810 q^{79} +1.00000 q^{81} +5.45552i q^{83} -4.96257i q^{87} -4.91105 q^{89} -3.17162 q^{91} -0.209050i q^{93} -10.3432i q^{97} -2.75353 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 6 q^{9}+O(q^{10}) $ 6 * q - 6 * q^9	$ 6 q - 6 q^{9} + 8 q^{11} - 20 q^{19} + 4 q^{21} + 20 q^{31} - 4 q^{39} + 16 q^{41} - 26 q^{49} + 20 q^{61} - 6 q^{69} - 16 q^{71} - 28 q^{79} + 6 q^{81} - 4 q^{89} + 32 q^{91} - 8 q^{99}+O(q^{100}) $ 6 * q - 6 * q^9 + 8 * q^11 - 20 * q^19 + 4 * q^21 + 20 * q^31 - 4 * q^39 + 16 * q^41 - 26 * q^49 + 20 * q^61 - 6 * q^69 - 16 * q^71 - 28 * q^79 + 6 * q^81 - 4 * q^89 + 32 * q^91 - 8 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/6900\mathbb{Z}\right)^\times$.

$n$	$277$	$1201$	$3451$	$4601$
$\chi(n)$	$-1$	$1$	$1$	$1$

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 $

$2$	0	0
$2$	0	0
$3$	1.00000i	0.577350i
$3$	1.00000i	0.577350i
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 4.20905i	− 1.59087i	−0.606038	−	0.795436i	$-0.707242\pi$
$7$	− 4.20905i	− 1.59087i	0.606038	−	0.795436i	$-0.292758\pi$
$8$	0	0
$9$	−1.00000	−0.333333
$10$	0	0
$11$	2.75353	0.830219	0.415110	−	0.909771i	$-0.363743\pi$
$11$	2.75353	0.830219	0.415110	+	0.909771i	$0.363743\pi$
$12$	0	0
$13$	− 0.753525i	− 0.208990i	−0.994525	−	0.104495i	$-0.966677\pi$
$13$	− 0.753525i	− 0.208990i	0.994525	−	0.104495i	$-0.0333227\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	4.96257i	1.20360i	0.798646	+	0.601801i	$0.205550\pi$
$17$	4.96257i	1.20360i	−0.798646	+	0.601801i	$0.794450\pi$
$18$	0	0
$19$	−4.75353	−1.09053	−0.545267	−	0.838263i	$-0.683572\pi$
$19$	−4.75353	−1.09053	−0.545267	+	0.838263i	$0.683572\pi$
$20$	0	0
$21$	4.20905	0.918490
$22$	0	0
$23$	1.00000i	0.208514i
$23$	1.00000i	0.208514i
$24$	0	0
$25$	0	0
$26$	0	0
$27$	− 1.00000i	− 0.192450i
$28$	0	0
$29$	−4.96257	−0.921527	−0.460764	−	0.887523i	$-0.652424\pi$
$29$	−4.96257	−0.921527	−0.460764	+	0.887523i	$0.652424\pi$
$30$	0	0
$31$	−0.209050	−0.0375464	−0.0187732	−	0.999824i	$-0.505976\pi$
$31$	−0.209050	−0.0375464	−0.0187732	+	0.999824i	$0.505976\pi$
$32$	0	0
$33$	2.75353i	0.479327i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	5.71610i	0.939721i	0.882741	+	0.469861i	$0.155696\pi$
$37$	5.71610i	0.939721i	−0.882741	+	0.469861i	$0.844304\pi$
$38$	0	0
$39$	0.753525	0.120661
$40$	0	0
$41$	−9.38067	−1.46502	−0.732508	−	0.680759i	$-0.761650\pi$
$41$	−9.38067	−1.46502	−0.732508	+	0.680759i	$0.761650\pi$
$42$	0	0
$43$	12.4181i	1.89374i	0.321613	+	0.946871i	$0.395775\pi$
$43$	12.4181i	1.89374i	−0.321613	+	0.946871i	$0.604225\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 7.17162i	− 1.04609i	−0.852305	−	0.523044i	$-0.824796\pi$
$47$	− 7.17162i	− 1.04609i	0.852305	−	0.523044i	$-0.175204\pi$
$48$	0	0
$49$	−10.7161	−1.53087
$50$	0	0
$51$	−4.96257	−0.694899
$52$	0	0
$53$	− 9.38067i	− 1.28853i	−0.764800	−	0.644267i	$-0.777162\pi$
$53$	− 9.38067i	− 1.28853i	0.764800	−	0.644267i	$-0.222838\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 4.75353i	− 0.629620i
$58$	0	0
$59$	4.96257	0.646072	0.323036	−	0.946387i	$-0.395296\pi$
$59$	4.96257	0.646072	0.323036	+	0.946387i	$0.395296\pi$
$60$	0	0
$61$	−5.17162	−0.662159	−0.331079	−	0.943603i	$-0.607413\pi$
$61$	−5.17162	−0.662159	−0.331079	+	0.943603i	$0.607413\pi$
$62$	0	0
$63$	4.20905i	0.530290i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	0.209050i	0.0255395i	0.999918	+	0.0127697i	$0.00406485\pi$
$67$	0.209050i	0.0255395i	−0.999918	+	0.0127697i	$0.995935\pi$
$68$	0	0
$69$	−1.00000	−0.120386
$70$	0	0
$71$	9.38067	1.11328	0.556641	−	0.830753i	$-0.312090\pi$
$71$	9.38067	1.11328	0.556641	+	0.830753i	$0.312090\pi$
$72$	0	0
$73$	10.2606i	1.20091i	0.799659	+	0.600455i	$0.205014\pi$
$73$	10.2606i	1.20091i	−0.799659	+	0.600455i	$0.794986\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 11.5897i	− 1.32077i
$78$	0	0
$79$	2.41810	0.272057	0.136029	−	0.990705i	$-0.456566\pi$
$79$	2.41810	0.272057	0.136029	+	0.990705i	$0.456566\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	5.45552i	0.598822i	0.954124	+	0.299411i	$0.0967901\pi$
$83$	5.45552i	0.598822i	−0.954124	+	0.299411i	$0.903210\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	− 4.96257i	− 0.532044i
$88$	0	0
$89$	−4.91105	−0.520570	−0.260285	−	0.965532i	$-0.583817\pi$
$89$	−4.91105	−0.520570	−0.260285	+	0.965532i	$0.583817\pi$
$90$	0	0
$91$	−3.17162	−0.332477
$92$	0	0
$93$	− 0.209050i	− 0.0216775i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 10.3432i	− 1.05020i	−0.851041	−	0.525099i	$-0.824028\pi$
$97$	− 10.3432i	− 1.05020i	0.851041	−	0.525099i	$-0.175972\pi$
$98$	0	0
$99$	−2.75353	−0.276740
$100$	0	0
$101$	14.8877	1.48138	0.740692	−	0.671845i	$-0.234498\pi$
$101$	14.8877	1.48138	0.740692	+	0.671845i	$0.234498\pi$
$102$	0	0
$103$	17.9251i	1.76622i	0.469168	+	0.883109i	$0.344554\pi$
$103$	17.9251i	1.76622i	−0.469168	+	0.883109i	$0.655446\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	16.4696i	1.59218i	0.605179	+	0.796089i	$0.293102\pi$
$107$	16.4696i	1.59218i	−0.605179	+	0.796089i	$0.706898\pi$
$108$	0	0
$109$	6.26058	0.599654	0.299827	−	0.953994i	$-0.403071\pi$
$109$	6.26058	0.599654	0.299827	+	0.953994i	$0.403071\pi$
$110$	0	0
$111$	−5.71610	−0.542548
$112$	0	0
$113$	− 9.38067i	− 0.882460i	−0.897394	−	0.441230i	$-0.854542\pi$
$113$	− 9.38067i	− 0.882460i	0.897394	−	0.441230i	$-0.145458\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0.753525i	0.0696634i
$118$	0	0
$119$	20.8877	1.91477
$120$	0	0
$121$	−3.41810	−0.310736
$122$	0	0
$123$	− 9.38067i	− 0.845827i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	22.0827i	1.95952i	0.200175	+	0.979760i	$0.435849\pi$
$127$	22.0827i	1.95952i	−0.200175	+	0.979760i	$0.564151\pi$
$128$	0	0
$129$	−12.4181	−1.09335
$130$	0	0
$131$	−1.58190	−0.138211	−0.0691056	−	0.997609i	$-0.522015\pi$
$131$	−1.58190	−0.138211	−0.0691056	+	0.997609i	$0.522015\pi$
$132$	0	0
$133$	20.0078i	1.73490i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	4.91105i	0.419579i	0.977747	+	0.209790i	$0.0672779\pi$
$137$	4.91105i	0.419579i	−0.977747	+	0.209790i	$0.932722\pi$
$138$	0	0
$139$	−14.1342	−1.19885	−0.599424	−	0.800432i	$-0.704603\pi$
$139$	−14.1342	−1.19885	−0.599424	+	0.800432i	$0.704603\pi$
$140$	0	0
$141$	7.17162	0.603960
$142$	0	0
$143$	− 2.07485i	− 0.173508i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 10.7161i	− 0.883849i
$148$	0	0
$149$	−3.24647	−0.265962	−0.132981	−	0.991119i	$-0.542455\pi$
$149$	−3.24647	−0.265962	−0.132981	+	0.991119i	$0.542455\pi$
$150$	0	0
$151$	0.418100	0.0340245	0.0170122	−	0.999855i	$-0.494585\pi$
$151$	0.418100	0.0340245	0.0170122	+	0.999855i	$0.494585\pi$
$152$	0	0
$153$	− 4.96257i	− 0.401200i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	21.0452i	1.67959i	0.542901	+	0.839797i	$0.317326\pi$
$157$	21.0452i	1.67959i	−0.542901	+	0.839797i	$0.682674\pi$
$158$	0	0
$159$	9.38067	0.743936
$160$	0	0
$161$	4.20905	0.331720
$162$	0	0
$163$	− 7.43220i	− 0.582135i	−0.956703	−	0.291067i	$-0.905990\pi$
$163$	− 7.43220i	− 0.582135i	0.956703	−	0.291067i	$-0.0940104\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	19.1716i	1.48354i	0.670652	+	0.741772i	$0.266015\pi$
$167$	19.1716i	1.48354i	−0.670652	+	0.741772i	$0.733985\pi$
$168$	0	0
$169$	12.4322	0.956323
$170$	0	0
$171$	4.75353	0.363511
$172$	0	0
$173$	0	0	1.00000			$0$
$173$	0	0	−1.00000			$\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	4.96257i	0.373010i
$178$	0	0
$179$	−13.8503	−1.03522	−0.517610	−	0.855617i	$-0.673178\pi$
$179$	−13.8503	−1.03522	−0.517610	+	0.855617i	$0.673178\pi$
$180$	0	0
$181$	12.9110	0.959671	0.479835	−	0.877359i	$-0.340696\pi$
$181$	12.9110	0.959671	0.479835	+	0.877359i	$0.340696\pi$
$182$	0	0
$183$	− 5.17162i	− 0.382297i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	13.6646i	0.999253i
$188$	0	0
$189$	−4.20905	−0.306163
$190$	0	0
$191$	−8.15752	−0.590258	−0.295129	−	0.955457i	$-0.595363\pi$
$191$	−8.15752	−0.590258	−0.295129	+	0.955457i	$0.595363\pi$
$192$	0	0
$193$	− 16.7613i	− 1.20651i	−0.797549	−	0.603254i	$-0.793870\pi$
$193$	− 16.7613i	− 1.20651i	0.797549	−	0.603254i	$-0.206130\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	10.9110i	0.777380i	0.921369	+	0.388690i	$0.127072\pi$
$197$	10.9110i	0.777380i	−0.921369	+	0.388690i	$0.872928\pi$
$198$	0	0
$199$	4.49295	0.318497	0.159248	−	0.987239i	$-0.449093\pi$
$199$	4.49295	0.318497	0.159248	+	0.987239i	$0.449093\pi$
$200$	0	0
$201$	−0.209050	−0.0147452
$202$	0	0
$203$	20.8877i	1.46603i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 1.00000i	− 0.0695048i
$208$	0	0
$209$	−13.0890	−0.905382
$210$	0	0
$211$	9.71610	0.668884	0.334442	−	0.942416i	$-0.391452\pi$
$211$	9.71610	0.668884	0.334442	+	0.942416i	$0.391452\pi$
$212$	0	0
$213$	9.38067i	0.642753i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	0.879901i	0.0597316i
$218$	0	0
$219$	−10.2606	−0.693345
$220$	0	0
$221$	3.73942	0.251541
$222$	0	0
$223$	− 19.4322i	− 1.30128i	−0.759388	−	0.650638i	$-0.774501\pi$
$223$	− 19.4322i	− 1.30128i	0.759388	−	0.650638i	$-0.225499\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	0.985900i	0.0654365i	0.999465	+	0.0327182i	$0.0104164\pi$
$227$	0.985900i	0.0654365i	−0.999465	+	0.0327182i	$0.989584\pi$
$228$	0	0
$229$	−3.08895	−0.204124	−0.102062	−	0.994778i	$-0.532544\pi$
$229$	−3.08895	−0.204124	−0.102062	+	0.994778i	$0.532544\pi$
$230$	0	0
$231$	11.5897	0.762548
$232$	0	0
$233$	8.83620i	0.578879i	0.957196	+	0.289439i	$0.0934688\pi$
$233$	8.83620i	0.578879i	−0.957196	+	0.289439i	$0.906531\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	2.41810i	0.157072i
$238$	0	0
$239$	21.3807	1.38300	0.691500	−	0.722376i	$-0.256950\pi$
$239$	21.3807	1.38300	0.691500	+	0.722376i	$0.256950\pi$
$240$	0	0
$241$	22.2606	1.43393	0.716965	−	0.697109i	$-0.245531\pi$
$241$	22.2606	1.43393	0.716965	+	0.697109i	$0.245531\pi$
$242$	0	0
$243$	1.00000i	0.0641500i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	3.58190i	0.227911i
$248$	0	0
$249$	−5.45552	−0.345730
$250$	0	0
$251$	−6.59600	−0.416336	−0.208168	−	0.978093i	$-0.566750\pi$
$251$	−6.59600	−0.416336	−0.208168	+	0.978093i	$0.566750\pi$
$252$	0	0
$253$	2.75353i	0.173113i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	18.1857i	1.13439i	0.823582	+	0.567197i	$0.191972\pi$
$257$	18.1857i	1.13439i	−0.823582	+	0.567197i	$0.808028\pi$
$258$	0	0
$259$	24.0593	1.49498
$260$	0	0
$261$	4.96257	0.307176
$262$	0	0
$263$	− 15.3807i	− 0.948413i	−0.880414	−	0.474207i	$-0.842735\pi$
$263$	− 15.3807i	− 0.948413i	0.880414	−	0.474207i	$-0.157265\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 4.91105i	− 0.300551i
$268$	0	0
$269$	−0.544475	−0.0331972	−0.0165986	−	0.999862i	$-0.505284\pi$
$269$	−0.544475	−0.0331972	−0.0165986	+	0.999862i	$0.505284\pi$
$270$	0	0
$271$	−12.2090	−0.741647	−0.370823	−	0.928703i	$-0.620925\pi$
$271$	−12.2090	−0.741647	−0.370823	+	0.928703i	$0.620925\pi$
$272$	0	0
$273$	− 3.17162i	− 0.191955i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	21.0141i	1.26261i	0.775533	+	0.631307i	$0.217481\pi$
$277$	21.0141i	1.26261i	−0.775533	+	0.631307i	$0.782519\pi$
$278$	0	0
$279$	0.209050	0.0125155
$280$	0	0
$281$	−24.1857	−1.44280	−0.721400	−	0.692519i	$-0.756501\pi$
$281$	−24.1857	−1.44280	−0.721400	+	0.692519i	$0.756501\pi$
$282$	0	0
$283$	15.1201i	0.898797i	0.893331	+	0.449398i	$0.148362\pi$
$283$	15.1201i	0.898797i	−0.893331	+	0.449398i	$0.851638\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	39.4837i	2.33065i
$288$	0	0
$289$	−7.62715	−0.448656
$290$	0	0
$291$	10.3432	0.606332
$292$	0	0
$293$	10.4696i	0.611642i	0.952089	+	0.305821i	$0.0989309\pi$
$293$	10.4696i	0.611642i	−0.952089	+	0.305821i	$0.901069\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 2.75353i	− 0.159776i
$298$	0	0
$299$	0.753525	0.0435775
$300$	0	0
$301$	52.2684	3.01270
$302$	0	0
$303$	14.8877i	0.855277i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	12.1575i	0.693867i	0.937890	+	0.346933i	$0.112777\pi$
$307$	12.1575i	0.693867i	−0.937890	+	0.346933i	$0.887223\pi$
$308$	0	0
$309$	−17.9251	−1.01973
$310$	0	0
$311$	−34.4181	−1.95167	−0.975836	−	0.218506i	$-0.929882\pi$
$311$	−34.4181	−1.95167	−0.975836	+	0.218506i	$0.929882\pi$
$312$	0	0
$313$	29.5664i	1.67119i	0.549345	+	0.835596i	$0.314877\pi$
$313$	29.5664i	1.67119i	−0.549345	+	0.835596i	$0.685123\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 1.66457i	− 0.0934918i	−0.998907	−	0.0467459i	$-0.985115\pi$
$317$	− 1.66457i	− 0.0934918i	0.998907	−	0.0467459i	$-0.0148851\pi$
$318$	0	0
$319$	−13.6646	−0.765069
$320$	0	0
$321$	−16.4696	−0.919245
$322$	0	0
$323$	− 23.5897i	− 1.31257i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	6.26058i	0.346211i
$328$	0	0
$329$	−30.1857	−1.66419
$330$	0	0
$331$	3.12010	0.171496	0.0857481	−	0.996317i	$-0.472672\pi$
$331$	3.12010	0.171496	0.0857481	+	0.996317i	$0.472672\pi$
$332$	0	0
$333$	− 5.71610i	− 0.313240i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	11.5819i	0.630906i	0.948941	+	0.315453i	$0.102157\pi$
$337$	11.5819i	0.630906i	−0.948941	+	0.315453i	$0.897843\pi$
$338$	0	0
$339$	9.38067	0.509488
$340$	0	0
$341$	−0.575624	−0.0311718
$342$	0	0
$343$	15.6412i	0.844548i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 26.3432i	− 1.41418i	−0.707124	−	0.707090i	$-0.750008\pi$
$347$	− 26.3432i	− 1.41418i	0.707124	−	0.707090i	$-0.249992\pi$
$348$	0	0
$349$	−9.22315	−0.493704	−0.246852	−	0.969053i	$-0.579396\pi$
$349$	−9.22315	−0.493704	−0.246852	+	0.969053i	$0.579396\pi$
$350$	0	0
$351$	−0.753525	−0.0402202
$352$	0	0
$353$	− 18.1857i	− 0.967928i	−0.875088	−	0.483964i	$-0.839196\pi$
$353$	− 18.1857i	− 0.967928i	0.875088	−	0.483964i	$-0.160804\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	20.8877i	1.10550i
$358$	0	0
$359$	−7.17162	−0.378504	−0.189252	−	0.981929i	$-0.560606\pi$
$359$	−7.17162	−0.378504	−0.189252	+	0.981929i	$0.560606\pi$
$360$	0	0
$361$	3.59600	0.189263
$362$	0	0
$363$	− 3.41810i	− 0.179404i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 17.5664i	− 0.916959i	−0.888705	−	0.458479i	$-0.848394\pi$
$367$	− 17.5664i	− 0.916959i	0.888705	−	0.458479i	$-0.151606\pi$
$368$	0	0
$369$	9.38067	0.488338
$370$	0	0
$371$	−39.4837	−2.04989
$372$	0	0
$373$	− 33.6724i	− 1.74349i	−0.489959	−	0.871745i	$-0.662988\pi$
$373$	− 33.6724i	− 1.74349i	0.489959	−	0.871745i	$-0.337012\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	3.73942i	0.192590i
$378$	0	0
$379$	−18.4181	−0.946074	−0.473037	−	0.881043i	$-0.656842\pi$
$379$	−18.4181	−0.946074	−0.473037	+	0.881043i	$0.656842\pi$
$380$	0	0
$381$	−22.0827	−1.13133
$382$	0	0
$383$	− 20.7847i	− 1.06205i	−0.847357	−	0.531024i	$-0.821808\pi$
$383$	− 20.7847i	− 1.06205i	0.847357	−	0.531024i	$-0.178192\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 12.4181i	− 0.631247i
$388$	0	0
$389$	−22.5212	−1.14187	−0.570934	−	0.820996i	$-0.693419\pi$
$389$	−22.5212	−1.14187	−0.570934	+	0.820996i	$0.693419\pi$
$390$	0	0
$391$	−4.96257	−0.250968
$392$	0	0
$393$	− 1.58190i	− 0.0797963i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	15.4040i	0.773105i	0.922267	+	0.386552i	$0.126334\pi$
$397$	15.4040i	0.773105i	−0.922267	+	0.386552i	$0.873666\pi$
$398$	0	0
$399$	−20.0078	−1.00164
$400$	0	0
$401$	−10.5212	−0.525401	−0.262701	−	0.964877i	$-0.584613\pi$
$401$	−10.5212	−0.525401	−0.262701	+	0.964877i	$0.584613\pi$
$402$	0	0
$403$	0.157524i	0.00784684i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	15.7394i	0.780174i
$408$	0	0
$409$	26.0593	1.28855	0.644276	−	0.764793i	$-0.277159\pi$
$409$	26.0593	1.28855	0.644276	+	0.764793i	$0.277159\pi$
$410$	0	0
$411$	−4.91105	−0.242244
$412$	0	0
$413$	− 20.8877i	− 1.02782i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 14.1342i	− 0.692155i
$418$	0	0
$419$	3.73942	0.182683	0.0913414	−	0.995820i	$-0.470885\pi$
$419$	3.73942	0.182683	0.0913414	+	0.995820i	$0.470885\pi$
$420$	0	0
$421$	−3.09677	−0.150928	−0.0754638	−	0.997149i	$-0.524044\pi$
$421$	−3.09677	−0.150928	−0.0754638	+	0.997149i	$0.524044\pi$
$422$	0	0
$423$	7.17162i	0.348696i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	21.7676i	1.05341i
$428$	0	0
$429$	2.07485	0.100175
$430$	0	0
$431$	−36.1653	−1.74202	−0.871012	−	0.491262i	$-0.836536\pi$
$431$	−36.1653	−1.74202	−0.871012	+	0.491262i	$0.836536\pi$
$432$	0	0
$433$	− 16.6271i	− 0.799050i	−0.916722	−	0.399525i	$-0.869175\pi$
$433$	− 16.6271i	− 0.799050i	0.916722	−	0.399525i	$-0.130825\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 4.75353i	− 0.227392i
$438$	0	0
$439$	−38.7613	−1.84998	−0.924989	−	0.379994i	$-0.875926\pi$
$439$	−38.7613	−1.84998	−0.924989	+	0.379994i	$0.875926\pi$
$440$	0	0
$441$	10.7161	0.510290
$442$	0	0
$443$	33.5149i	1.59234i	0.605073	+	0.796170i	$0.293144\pi$
$443$	33.5149i	1.59234i	−0.605073	+	0.796170i	$0.706856\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 3.24647i	− 0.153553i
$448$	0	0
$449$	13.9018	0.656068	0.328034	−	0.944666i	$-0.393614\pi$
$449$	13.9018	0.656068	0.328034	+	0.944666i	$0.393614\pi$
$450$	0	0
$451$	−25.8299	−1.21628
$452$	0	0
$453$	0.418100i	0.0196440i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	2.28390i	0.106836i	0.998572	+	0.0534182i	$0.0170116\pi$
$457$	2.28390i	0.106836i	−0.998572	+	0.0534182i	$0.982988\pi$
$458$	0	0
$459$	4.96257	0.231633
$460$	0	0
$461$	8.34325	0.388584	0.194292	−	0.980944i	$-0.437759\pi$
$461$	8.34325	0.388584	0.194292	+	0.980944i	$0.437759\pi$
$462$	0	0
$463$	− 27.6928i	− 1.28699i	−0.765449	−	0.643496i	$-0.777483\pi$
$463$	− 27.6928i	− 1.28699i	0.765449	−	0.643496i	$-0.222517\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 7.53037i	− 0.348464i	−0.984705	−	0.174232i	$-0.944256\pi$
$467$	− 7.53037i	− 0.348464i	0.984705	−	0.174232i	$-0.0557443\pi$
$468$	0	0
$469$	0.879901	0.0406300
$470$	0	0
$471$	−21.0452	−0.969714
$472$	0	0
$473$	34.1935i	1.57222i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	9.38067i	0.429512i
$478$	0	0
$479$	18.1857	0.830927	0.415463	−	0.909610i	$-0.363619\pi$
$479$	18.1857	0.830927	0.415463	+	0.909610i	$0.363619\pi$
$480$	0	0
$481$	4.30722	0.196393
$482$	0	0
$483$	4.20905i	0.191518i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	24.2606i	1.09935i	0.835378	+	0.549676i	$0.185249\pi$
$487$	24.2606i	1.09935i	−0.835378	+	0.549676i	$0.814751\pi$
$488$	0	0
$489$	7.43220	0.336096
$490$	0	0
$491$	14.8877	0.671874	0.335937	−	0.941885i	$-0.390947\pi$
$491$	14.8877	0.671874	0.335937	+	0.941885i	$0.390947\pi$
$492$	0	0
$493$	− 24.6271i	− 1.10915i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 39.4837i	− 1.77109i
$498$	0	0
$499$	0.312101	0.0139716	0.00698578	−	0.999976i	$-0.497776\pi$
$499$	0.312101	0.0139716	0.00698578	+	0.999976i	$0.497776\pi$
$500$	0	0
$501$	−19.1716	−0.856525
$502$	0	0
$503$	20.8877i	0.931338i	0.884959	+	0.465669i	$0.154186\pi$
$503$	20.8877i	0.931338i	−0.884959	+	0.465669i	$0.845814\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	12.4322i	0.552133i
$508$	0	0
$509$	−27.9251	−1.23776	−0.618880	−	0.785485i	$-0.712413\pi$
$509$	−27.9251	−1.23776	−0.618880	+	0.785485i	$0.712413\pi$
$510$	0	0
$511$	43.1873	1.91049
$512$	0	0
$513$	4.75353i	0.209873i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 19.7472i	− 0.868483i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	32.0360	1.40352	0.701762	−	0.712412i	$-0.252397\pi$
$521$	32.0360	1.40352	0.701762	+	0.712412i	$0.252397\pi$
$522$	0	0
$523$	38.8644i	1.69942i	0.527249	+	0.849711i	$0.323223\pi$
$523$	38.8644i	1.69942i	−0.527249	+	0.849711i	$0.676777\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 1.03743i	− 0.0451909i
$528$	0	0
$529$	−1.00000	−0.0434783
$530$	0	0
$531$	−4.96257	−0.215357
$532$	0	0
$533$	7.06857i	0.306174i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 13.8503i	− 0.597685i
$538$	0	0
$539$	−29.5071	−1.27096
$540$	0	0
$541$	4.07485	0.175191	0.0875957	−	0.996156i	$-0.472082\pi$
$541$	4.07485	0.175191	0.0875957	+	0.996156i	$0.472082\pi$
$542$	0	0
$543$	12.9110i	0.554066i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	28.0000i	1.19719i	0.801050	+	0.598597i	$0.204275\pi$
$547$	28.0000i	1.19719i	−0.801050	+	0.598597i	$0.795725\pi$
$548$	0	0
$549$	5.17162	0.220720
$550$	0	0
$551$	23.5897	1.00496
$552$	0	0
$553$	− 10.1779i	− 0.432808i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 1.63343i	− 0.0692105i	−0.999401	−	0.0346052i	$-0.988983\pi$
$557$	− 1.63343i	− 0.0692105i	0.999401	−	0.0346052i	$-0.0110174\pi$
$558$	0	0
$559$	9.35735	0.395774
$560$	0	0
$561$	−13.6646	−0.576919
$562$	0	0
$563$	− 23.2310i	− 0.979069i	−0.871984	−	0.489534i	$-0.837167\pi$
$563$	− 23.2310i	− 0.979069i	0.871984	−	0.489534i	$-0.162833\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 4.20905i	− 0.176763i
$568$	0	0
$569$	−21.3291	−0.894164	−0.447082	−	0.894493i	$-0.647537\pi$
$569$	−21.3291	−0.894164	−0.447082	+	0.894493i	$0.647537\pi$
$570$	0	0
$571$	−4.18573	−0.175167	−0.0875836	−	0.996157i	$-0.527914\pi$
$571$	−4.18573	−0.175167	−0.0875836	+	0.996157i	$0.527914\pi$
$572$	0	0
$573$	− 8.15752i	− 0.340785i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 10.5678i	− 0.439943i	−0.975506	−	0.219972i	$-0.929404\pi$
$577$	− 10.5678i	− 0.439943i	0.975506	−	0.219972i	$-0.0705965\pi$
$578$	0	0
$579$	16.7613	0.696578
$580$	0	0
$581$	22.9626	0.952648
$582$	0	0
$583$	− 25.8299i	− 1.06977i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 28.4181i	− 1.17294i	−0.809971	−	0.586470i	$-0.800517\pi$
$587$	− 28.4181i	− 1.17294i	0.809971	−	0.586470i	$-0.199483\pi$
$588$	0	0
$589$	0.993723	0.0409457
$590$	0	0
$591$	−10.9110	−0.448821
$592$	0	0
$593$	28.9937i	1.19063i	0.803493	+	0.595315i	$0.202973\pi$
$593$	28.9937i	1.19063i	−0.803493	+	0.595315i	$0.797027\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	4.49295i	0.183884i
$598$	0	0
$599$	6.10305	0.249364	0.124682	−	0.992197i	$-0.460209\pi$
$599$	6.10305	0.249364	0.124682	+	0.992197i	$0.460209\pi$
$600$	0	0
$601$	−29.3885	−1.19878	−0.599391	−	0.800456i	$-0.704590\pi$
$601$	−29.3885	−1.19878	−0.599391	+	0.800456i	$0.704590\pi$
$602$	0	0
$603$	− 0.209050i	− 0.00851316i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	3.58972i	0.145702i	0.997343	+	0.0728512i	$0.0232098\pi$
$607$	3.58972i	0.145702i	−0.997343	+	0.0728512i	$0.976790\pi$
$608$	0	0
$609$	−20.8877	−0.846413
$610$	0	0
$611$	−5.40400	−0.218622
$612$	0	0
$613$	− 21.4040i	− 0.864499i	−0.901754	−	0.432250i	$-0.857720\pi$
$613$	− 21.4040i	− 0.864499i	0.901754	−	0.432250i	$-0.142280\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 32.2917i	− 1.30002i	−0.759927	−	0.650008i	$-0.774766\pi$
$617$	− 32.2917i	− 1.30002i	0.759927	−	0.650008i	$-0.225234\pi$
$618$	0	0
$619$	−47.1046	−1.89329	−0.946647	−	0.322273i	$-0.895553\pi$
$619$	−47.1046	−1.89329	−0.946647	+	0.322273i	$0.895553\pi$
$620$	0	0
$621$	1.00000	0.0401286
$622$	0	0
$623$	20.6709i	0.828160i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	− 13.0890i	− 0.522722i
$628$	0	0
$629$	−28.3666	−1.13105
$630$	0	0
$631$	23.5149	0.936112	0.468056	−	0.883699i	$-0.344954\pi$
$631$	23.5149	0.936112	0.468056	+	0.883699i	$0.344954\pi$
$632$	0	0
$633$	9.71610i	0.386180i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	8.07485i	0.319937i
$638$	0	0
$639$	−9.38067	−0.371094
$640$	0	0
$641$	−6.67867	−0.263792	−0.131896	−	0.991264i	$-0.542106\pi$
$641$	−6.67867	−0.263792	−0.131896	+	0.991264i	$0.542106\pi$
$642$	0	0
$643$	27.1201i	1.06951i	0.845006	+	0.534756i	$0.179597\pi$
$643$	27.1201i	1.06951i	−0.845006	+	0.534756i	$0.820403\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	27.7394i	1.09055i	0.838257	+	0.545275i	$0.183575\pi$
$647$	27.7394i	1.09055i	−0.838257	+	0.545275i	$0.816425\pi$
$648$	0	0
$649$	13.6646	0.536381
$650$	0	0
$651$	−0.879901	−0.0344860
$652$	0	0
$653$	− 2.75353i	− 0.107754i	−0.998548	−	0.0538769i	$-0.982842\pi$
$653$	− 2.75353i	− 0.107754i	0.998548	−	0.0538769i	$-0.0171579\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 10.2606i	− 0.400303i
$658$	0	0
$659$	−11.5897	−0.451472	−0.225736	−	0.974189i	$-0.572479\pi$
$659$	−11.5897	−0.451472	−0.225736	+	0.974189i	$0.572479\pi$
$660$	0	0
$661$	28.3432	1.10242	0.551212	−	0.834365i	$-0.314165\pi$
$661$	28.3432	1.10242	0.551212	+	0.834365i	$0.314165\pi$
$662$	0	0
$663$	3.73942i	0.145227i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 4.96257i	− 0.192152i
$668$	0	0
$669$	19.4322	0.751292
$670$	0	0
$671$	−14.2402	−0.549737
$672$	0	0
$673$	− 48.0360i	− 1.85165i	−0.377949	−	0.925826i	$-0.623371\pi$
$673$	− 48.0360i	− 1.85165i	0.377949	−	0.925826i	$-0.376629\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 2.51627i	− 0.0967083i	−0.998830	−	0.0483541i	$-0.984602\pi$
$677$	− 2.51627i	− 0.0967083i	0.998830	−	0.0483541i	$-0.0153976\pi$
$678$	0	0
$679$	−43.5353	−1.67073
$680$	0	0
$681$	−0.985900	−0.0377798
$682$	0	0
$683$	− 3.84248i	− 0.147028i	−0.997294	−	0.0735141i	$-0.976579\pi$
$683$	− 3.84248i	− 0.147028i	0.997294	−	0.0735141i	$-0.0234214\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 3.08895i	− 0.117851i
$688$	0	0
$689$	−7.06857	−0.269291
$690$	0	0
$691$	2.59600	0.0987565	0.0493783	−	0.998780i	$-0.484276\pi$
$691$	2.59600	0.0987565	0.0493783	+	0.998780i	$0.484276\pi$
$692$	0	0
$693$	11.5897i	0.440257i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 46.5523i	− 1.76329i
$698$	0	0
$699$	−8.83620	−0.334216
$700$	0	0
$701$	41.5897	1.57082	0.785411	−	0.618974i	$-0.212452\pi$
$701$	41.5897	1.57082	0.785411	+	0.618974i	$0.212452\pi$
$702$	0	0
$703$	− 27.1716i	− 1.02480i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 62.6632i	− 2.35669i
$708$	0	0
$709$	53.1716	1.99690	0.998451	−	0.0556356i	$-0.0177185\pi$
$709$	53.1716	1.99690	0.998451	+	0.0556356i	$0.0177185\pi$
$710$	0	0
$711$	−2.41810	−0.0906858
$712$	0	0
$713$	− 0.209050i	− 0.00782898i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	21.3807i	0.798476i
$718$	0	0
$719$	−45.3807	−1.69241	−0.846207	−	0.532855i	$-0.821119\pi$
$719$	−45.3807	−1.69241	−0.846207	+	0.532855i	$0.821119\pi$
$720$	0	0
$721$	75.4478	2.80982
$722$	0	0
$723$	22.2606i	0.827880i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	46.4026i	1.72098i	0.509470	+	0.860489i	$0.329842\pi$
$727$	46.4026i	1.72098i	−0.509470	+	0.860489i	$0.670158\pi$
$728$	0	0
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	−61.6257	−2.27931
$732$	0	0
$733$	22.8051i	0.842324i	0.906985	+	0.421162i	$0.138378\pi$
$733$	22.8051i	0.842324i	−0.906985	+	0.421162i	$0.861622\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0.575624i	0.0212034i
$738$	0	0
$739$	16.5241	0.607849	0.303924	−	0.952696i	$-0.401703\pi$
$739$	16.5241	0.607849	0.303924	+	0.952696i	$0.401703\pi$
$740$	0	0
$741$	−3.58190	−0.131584
$742$	0	0
$743$	43.8503i	1.60871i	0.594148	+	0.804356i	$0.297489\pi$
$743$	43.8503i	1.60871i	−0.594148	+	0.804356i	$0.702511\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 5.45552i	− 0.199607i
$748$	0	0
$749$	69.3215	2.53295
$750$	0	0
$751$	5.73942	0.209435	0.104717	−	0.994502i	$-0.466606\pi$
$751$	5.73942	0.209435	0.104717	+	0.994502i	$0.466606\pi$
$752$	0	0
$753$	− 6.59600i	− 0.240372i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	9.86580i	0.358579i	0.983796	+	0.179289i	$0.0573798\pi$
$757$	9.86580i	0.358579i	−0.983796	+	0.179289i	$0.942620\pi$
$758$	0	0
$759$	−2.75353	−0.0999466
$760$	0	0
$761$	4.69418	0.170164	0.0850819	−	0.996374i	$-0.472885\pi$
$761$	4.69418	0.170164	0.0850819	+	0.996374i	$0.472885\pi$
$762$	0	0
$763$	− 26.3511i	− 0.953973i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 3.73942i	− 0.135023i
$768$	0	0
$769$	−45.5431	−1.64233	−0.821163	−	0.570694i	$-0.806674\pi$
$769$	−45.5431	−1.64233	−0.821163	+	0.570694i	$0.806674\pi$
$770$	0	0
$771$	−18.1857	−0.654943
$772$	0	0
$773$	12.4929i	0.449340i	0.974435	+	0.224670i	$0.0721304\pi$
$773$	12.4929i	0.449340i	−0.974435	+	0.224670i	$0.927870\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	24.0593i	0.863124i
$778$	0	0
$779$	44.5913	1.59765
$780$	0	0
$781$	25.8299	0.924267
$782$	0	0
$783$	4.96257i	0.177348i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	5.71610i	0.203757i	0.994797	+	0.101878i	$0.0324853\pi$
$787$	5.71610i	0.203757i	−0.994797	+	0.101878i	$0.967515\pi$
$788$	0	0
$789$	15.3807	0.547567
$790$	0	0
$791$	−39.4837	−1.40388
$792$	0	0
$793$	3.89695i	0.138385i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	9.48373i	0.335931i	0.985793	+	0.167965i	$0.0537197\pi$
$797$	9.48373i	0.335931i	−0.985793	+	0.167965i	$0.946280\pi$
$798$	0	0
$799$	35.5897	1.25907
$800$	0	0
$801$	4.91105	0.173523
$802$	0	0
$803$	28.2528i	0.997018i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 0.544475i	− 0.0191664i
$808$	0	0
$809$	−50.0672	−1.76027	−0.880134	−	0.474725i	$-0.842547\pi$
$809$	−50.0672	−1.76027	−0.880134	+	0.474725i	$0.842547\pi$
$810$	0	0
$811$	−9.14830	−0.321240	−0.160620	−	0.987016i	$-0.551349\pi$
$811$	−9.14830	−0.321240	−0.160620	+	0.987016i	$0.551349\pi$
$812$	0	0
$813$	− 12.2090i	− 0.428190i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 59.0297i	− 2.06519i
$818$	0	0
$819$	3.17162	0.110826
$820$	0	0
$821$	−4.91105	−0.171397	−0.0856984	−	0.996321i	$-0.527312\pi$
$821$	−4.91105	−0.171397	−0.0856984	+	0.996321i	$0.527312\pi$
$822$	0	0
$823$	23.7006i	0.826151i	0.910697	+	0.413075i	$0.135545\pi$
$823$	23.7006i	0.826151i	−0.910697	+	0.413075i	$0.864455\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 13.1405i	− 0.456939i	−0.973551	−	0.228470i	$-0.926628\pi$
$827$	− 13.1405i	− 0.456939i	0.973551	−	0.228470i	$-0.0733721\pi$
$828$	0	0
$829$	45.6412	1.58519	0.792593	−	0.609751i	$-0.208731\pi$
$829$	45.6412	1.58519	0.792593	+	0.609751i	$0.208731\pi$
$830$	0	0
$831$	−21.0141	−0.728971
$832$	0	0
$833$	− 53.1794i	− 1.84256i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0.209050i	0.00722582i
$838$	0	0
$839$	5.40400	0.186567	0.0932834	−	0.995640i	$-0.470264\pi$
$839$	5.40400	0.186567	0.0932834	+	0.995640i	$0.470264\pi$
$840$	0	0
$841$	−4.37285	−0.150788
$842$	0	0
$843$	− 24.1857i	− 0.833001i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	14.3870i	0.494341i
$848$	0	0
$849$	−15.1201	−0.518920
$850$	0	0
$851$	−5.71610	−0.195945
$852$	0	0
$853$	− 53.0297i	− 1.81570i	−0.419291	−	0.907852i	$-0.637721\pi$
$853$	− 53.0297i	− 1.81570i	0.419291	−	0.907852i	$-0.362279\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	14.4463i	0.493476i	0.969082	+	0.246738i	$0.0793587\pi$
$857$	14.4463i	0.493476i	−0.969082	+	0.246738i	$0.920641\pi$
$858$	0	0
$859$	−13.8658	−0.473095	−0.236548	−	0.971620i	$-0.576016\pi$
$859$	−13.8658	−0.473095	−0.236548	+	0.971620i	$0.576016\pi$
$860$	0	0
$861$	−39.4837	−1.34560
$862$	0	0
$863$	52.5834i	1.78996i	0.446105	+	0.894981i	$0.352811\pi$
$863$	52.5834i	1.78996i	−0.446105	+	0.894981i	$0.647189\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 7.62715i	− 0.259032i
$868$	0	0
$869$	6.65830	0.225867
$870$	0	0
$871$	0.157524	0.00533751
$872$	0	0
$873$	10.3432i	0.350066i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	25.3291i	0.855305i	0.903943	+	0.427652i	$0.140659\pi$
$877$	25.3291i	0.855305i	−0.903943	+	0.427652i	$0.859341\pi$
$878$	0	0
$879$	−10.4696	−0.353132
$880$	0	0
$881$	14.2606	0.480451	0.240225	−	0.970717i	$-0.422779\pi$
$881$	14.2606	0.480451	0.240225	+	0.970717i	$0.422779\pi$
$882$	0	0
$883$	40.6320i	1.36738i	0.729774	+	0.683688i	$0.239625\pi$
$883$	40.6320i	1.36738i	−0.729774	+	0.683688i	$0.760375\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	19.9534i	0.669968i	0.942224	+	0.334984i	$0.108731\pi$
$887$	19.9534i	0.669968i	−0.942224	+	0.334984i	$0.891269\pi$
$888$	0	0
$889$	92.9471	3.11734
$890$	0	0
$891$	2.75353	0.0922466
$892$	0	0
$893$	34.0905i	1.14080i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0.753525i	0.0251595i
$898$	0	0
$899$	1.03743	0.0346001
$900$	0	0
$901$	46.5523	1.55088
$902$	0	0
$903$	52.2684i	1.73938i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	29.8814i	0.992197i	0.868266	+	0.496099i	$0.165235\pi$
$907$	29.8814i	0.992197i	−0.868266	+	0.496099i	$0.834765\pi$
$908$	0	0
$909$	−14.8877	−0.493795
$910$	0	0
$911$	18.8644	0.625005	0.312503	−	0.949917i	$-0.398833\pi$
$911$	18.8644	0.625005	0.312503	+	0.949917i	$0.398833\pi$
$912$	0	0
$913$	15.0219i	0.497153i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	6.65830i	0.219876i
$918$	0	0
$919$	−16.2402	−0.535715	−0.267857	−	0.963459i	$-0.586316\pi$
$919$	−16.2402	−0.535715	−0.267857	+	0.963459i	$0.586316\pi$
$920$	0	0
$921$	−12.1575	−0.400604
$922$	0	0
$923$	− 7.06857i	− 0.232665i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 17.9251i	− 0.588739i
$928$	0	0
$929$	−29.3340	−0.962418	−0.481209	−	0.876606i	$-0.659802\pi$
$929$	−29.3340	−0.962418	−0.481209	+	0.876606i	$0.659802\pi$
$930$	0	0
$931$	50.9393	1.66947
$932$	0	0
$933$	− 34.4181i	− 1.12680i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	41.3573i	1.35109i	0.737321	+	0.675543i	$0.236091\pi$
$937$	41.3573i	1.35109i	−0.737321	+	0.675543i	$0.763909\pi$
$938$	0	0
$939$	−29.5664	−0.964863
$940$	0	0
$941$	50.4259	1.64384	0.821919	−	0.569604i	$-0.192904\pi$
$941$	50.4259	1.64384	0.821919	+	0.569604i	$0.192904\pi$
$942$	0	0
$943$	− 9.38067i	− 0.305477i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 5.50705i	− 0.178955i	−0.995989	−	0.0894775i	$-0.971480\pi$
$947$	− 5.50705i	− 0.178955i	0.995989	−	0.0894775i	$-0.0285197\pi$
$948$	0	0
$949$	7.73160	0.250978
$950$	0	0
$951$	1.66457	0.0539775
$952$	0	0
$953$	0.864400i	0.0280007i	0.999902	+	0.0140003i	$0.00445659\pi$
$953$	0.864400i	0.0280007i	−0.999902	+	0.0140003i	$0.995543\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	− 13.6646i	− 0.441713i
$958$	0	0
$959$	20.6709	0.667497
$960$	0	0
$961$	−30.9563	−0.998590
$962$	0	0
$963$	− 16.4696i	− 0.530726i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 10.7535i	− 0.345810i	−0.984939	−	0.172905i	$-0.944685\pi$
$967$	− 10.7535i	− 0.345810i	0.984939	−	0.172905i	$-0.0553154\pi$
$968$	0	0
$969$	23.5897	0.757811
$970$	0	0
$971$	35.1794	1.12896	0.564481	−	0.825446i	$-0.309076\pi$
$971$	35.1794	1.12896	0.564481	+	0.825446i	$0.309076\pi$
$972$	0	0
$973$	59.4915i	1.90721i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 15.9767i	− 0.511139i	−0.966791	−	0.255570i	$-0.917737\pi$
$977$	− 15.9767i	− 0.511139i	0.966791	−	0.255570i	$-0.0822630\pi$
$978$	0	0
$979$	−13.5227	−0.432187
$980$	0	0
$981$	−6.26058	−0.199885
$982$	0	0
$983$	− 57.2592i	− 1.82628i	−0.407642	−	0.913142i	$-0.633649\pi$
$983$	− 57.2592i	− 1.82628i	0.407642	−	0.913142i	$-0.366351\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	− 30.1857i	− 0.960822i
$988$	0	0
$989$	−12.4181	−0.394873
$990$	0	0
$991$	12.8799	0.409144	0.204572	−	0.978852i	$-0.434420\pi$
$991$	12.8799	0.409144	0.204572	+	0.978852i	$0.434420\pi$
$992$	0	0
$993$	3.12010i	0.0990134i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	39.7754i	1.25970i	0.776716	+	0.629851i	$0.216884\pi$
$997$	39.7754i	1.25970i	−0.776716	+	0.629851i	$0.783116\pi$
$998$	0	0
$999$	5.71610	0.180849

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6900.2.f.r.6349.4		6
5.2	odd	4		1380.2.a.j.1.3	✓	3
5.3	odd	4		6900.2.a.x.1.1		3
5.4	even	2	inner	6900.2.f.r.6349.3		6
15.2	even	4		4140.2.a.s.1.3		3
20.7	even	4		5520.2.a.bv.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1380.2.a.j.1.3	✓	3	5.2	odd	4
4140.2.a.s.1.3		3	15.2	even	4
5520.2.a.bv.1.1		3	20.7	even	4
6900.2.a.x.1.1		3	5.3	odd	4
6900.2.f.r.6349.3		6	5.4	even	2	inner
6900.2.f.r.6349.4		6	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:	\( N \)	\(=\)	\( 6900 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6900.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{3} -4.20905i q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q+1.00000i q^{3} -4.20905i q^{7} -1.00000 q^{9} +2.75353 q^{11} -0.753525i q^{13} +4.96257i q^{17} -4.75353 q^{19} +4.20905 q^{21} +1.00000i q^{23} -1.00000i q^{27} -4.96257 q^{29} -0.209050 q^{31} +2.75353i q^{33} +5.71610i q^{37} +0.753525 q^{39} -9.38067 q^{41} +12.4181i q^{43} -7.17162i q^{47} -10.7161 q^{49} -4.96257 q^{51} -9.38067i q^{53} -4.75353i q^{57} +4.96257 q^{59} -5.17162 q^{61} +4.20905i q^{63} +0.209050i q^{67} -1.00000 q^{69} +9.38067 q^{71} +10.2606i q^{73} -11.5897i q^{77} +2.41810 q^{79} +1.00000 q^{81} +5.45552i q^{83} -4.96257i q^{87} -4.91105 q^{89} -3.17162 q^{91} -0.209050i q^{93} -10.3432i q^{97} -2.75353 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 6 q^{9}+O(q^{10}) \) 6 * q - 6 * q^9	\( 6 q - 6 q^{9} + 8 q^{11} - 20 q^{19} + 4 q^{21} + 20 q^{31} - 4 q^{39} + 16 q^{41} - 26 q^{49} + 20 q^{61} - 6 q^{69} - 16 q^{71} - 28 q^{79} + 6 q^{81} - 4 q^{89} + 32 q^{91} - 8 q^{99}+O(q^{100}) \) 6 * q - 6 * q^9 + 8 * q^11 - 20 * q^19 + 4 * q^21 + 20 * q^31 - 4 * q^39 + 16 * q^41 - 26 * q^49 + 20 * q^61 - 6 * q^69 - 16 * q^71 - 28 * q^79 + 6 * q^81 - 4 * q^89 + 32 * q^91 - 8 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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