LMFDB - Embedded newform 3360.2.g.a.1681.6

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3360,2,Mod(1681,3360)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3360, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0, 0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("3360.1681"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 3360 = 2^{5} \cdot 3 \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3360.g (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [8,0,0,0,0,0,-8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$26.8297350792$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\Q(\zeta_{16})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 1 $ x^8 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{7} $
Twist minimal:	no (minimal twist has level 840)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1681.6
Root			$0.382683 + 0.923880i$ of defining polynomial
Character	$\chi$	$=$	3360.1681
Dual form			3360.2.g.a.1681.3

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{3} -1.00000i q^{5} -1.00000 q^{7} -1.00000 q^{9} -1.08239i q^{11} -1.74603i q^{13} +1.00000 q^{15} -3.69552 q^{17} +8.26998i q^{19} -1.00000i q^{21} +7.69552 q^{23} -1.00000 q^{25} -1.00000i q^{27} -6.35916i q^{29} -3.55166 q^{31} +1.08239 q^{33} +1.00000i q^{35} +3.69552i q^{37} +1.74603 q^{39} -4.99321 q^{41} +0.469266i q^{43} +1.00000i q^{45} +1.69552 q^{47} +1.00000 q^{49} -3.69552i q^{51} +10.3086i q^{53} -1.08239 q^{55} -8.26998 q^{57} +9.39104i q^{59} -0.236226i q^{61} +1.00000 q^{63} -1.74603 q^{65} +5.42063i q^{67} +7.69552i q^{69} +2.40968 q^{71} -7.17574 q^{73} -1.00000i q^{75} +1.08239i q^{77} +9.11615 q^{79} +1.00000 q^{81} +5.28093i q^{83} +3.69552i q^{85} +6.35916 q^{87} +8.28772 q^{89} +1.74603i q^{91} -3.55166i q^{93} +8.26998 q^{95} +0.681382 q^{97} +1.08239i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{7} - 8 q^{9} + 8 q^{15} + 32 q^{23} - 8 q^{25} - 32 q^{31} - 16 q^{47} + 8 q^{49} + 8 q^{63} - 16 q^{79} + 8 q^{81} + 16 q^{87}+O(q^{100}) $ 8 * q - 8 * q^7 - 8 * q^9 + 8 * q^15 + 32 * q^23 - 8 * q^25 - 32 * q^31 - 16 * q^47 + 8 * q^49 + 8 * q^63 - 16 * q^79 + 8 * q^81 + 16 * q^87

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3360\mathbb{Z}\right)^\times$.

$n$	$421$	$1121$	$1471$	$1921$	$2017$
$\chi(n)$	$-1$	$1$	$1$	$1$	$1$

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
$5$	− 1.00000i	− 0.447214i
$5$	− 1.00000i	− 0.447214i
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	−1.00000	−0.333333
$10$	0	0
$11$	− 1.08239i	− 0.326354i	−0.986597	−	0.163177i	$-0.947826\pi$
$11$	− 1.08239i	− 0.326354i	0.986597	−	0.163177i	$-0.0521741\pi$
$12$	0	0
$13$	− 1.74603i	− 0.484263i	−0.970243	−	0.242131i	$-0.922153\pi$
$13$	− 1.74603i	− 0.484263i	0.970243	−	0.242131i	$-0.0778465\pi$
$14$	0	0
$15$	1.00000	0.258199
$16$	0	0
$17$	−3.69552	−0.896295	−0.448147	−	0.893960i	$-0.647916\pi$
$17$	−3.69552	−0.896295	−0.448147	+	0.893960i	$0.647916\pi$
$18$	0	0
$19$	8.26998i	1.89726i	0.316382	+	0.948632i	$0.397532\pi$
$19$	8.26998i	1.89726i	−0.316382	+	0.948632i	$0.602468\pi$
$20$	0	0
$21$	− 1.00000i	− 0.218218i
$22$	0	0
$23$	7.69552	1.60463	0.802313	−	0.596903i	$-0.203602\pi$
$23$	7.69552	1.60463	0.802313	+	0.596903i	$0.203602\pi$
$24$	0	0
$25$	−1.00000	−0.200000
$26$	0	0
$27$	− 1.00000i	− 0.192450i
$28$	0	0
$29$	− 6.35916i	− 1.18087i	−0.807086	−	0.590433i	$-0.798957\pi$
$29$	− 6.35916i	− 1.18087i	0.807086	−	0.590433i	$-0.201043\pi$
$30$	0	0
$31$	−3.55166	−0.637897	−0.318948	−	0.947772i	$-0.603330\pi$
$31$	−3.55166	−0.637897	−0.318948	+	0.947772i	$0.603330\pi$
$32$	0	0
$33$	1.08239	0.188420
$34$	0	0
$35$	1.00000i	0.169031i
$36$	0	0
$37$	3.69552i	0.607539i	0.952745	+	0.303770i	$0.0982453\pi$
$37$	3.69552i	0.607539i	−0.952745	+	0.303770i	$0.901755\pi$
$38$	0	0
$39$	1.74603	0.279589
$40$	0	0
$41$	−4.99321	−0.779809	−0.389904	−	0.920855i	$-0.627492\pi$
$41$	−4.99321	−0.779809	−0.389904	+	0.920855i	$0.627492\pi$
$42$	0	0
$43$	0.469266i	0.0715624i	0.999360	+	0.0357812i	$0.0113919\pi$
$43$	0.469266i	0.0715624i	−0.999360	+	0.0357812i	$0.988608\pi$
$44$	0	0
$45$	1.00000i	0.149071i
$46$	0	0
$47$	1.69552	0.247317	0.123658	−	0.992325i	$-0.460537\pi$
$47$	1.69552	0.247317	0.123658	+	0.992325i	$0.460537\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	− 3.69552i	− 0.517476i
$52$	0	0
$53$	10.3086i	1.41600i	0.706212	+	0.708001i	$0.250403\pi$
$53$	10.3086i	1.41600i	−0.706212	+	0.708001i	$0.749597\pi$
$54$	0	0
$55$	−1.08239	−0.145950
$56$	0	0
$57$	−8.26998	−1.09539
$58$	0	0
$59$	9.39104i	1.22261i	0.791396	+	0.611304i	$0.209355\pi$
$59$	9.39104i	1.22261i	−0.791396	+	0.611304i	$0.790645\pi$
$60$	0	0
$61$	− 0.236226i	− 0.0302456i	−0.999886	−	0.0151228i	$-0.995186\pi$
$61$	− 0.236226i	− 0.0302456i	0.999886	−	0.0151228i	$-0.00481393\pi$
$62$	0	0
$63$	1.00000	0.125988
$64$	0	0
$65$	−1.74603	−0.216569
$66$	0	0
$67$	5.42063i	0.662235i	0.943590	+	0.331118i	$0.107426\pi$
$67$	5.42063i	0.662235i	−0.943590	+	0.331118i	$0.892574\pi$
$68$	0	0
$69$	7.69552i	0.926432i
$70$	0	0
$71$	2.40968	0.285976	0.142988	−	0.989724i	$-0.454329\pi$
$71$	2.40968	0.285976	0.142988	+	0.989724i	$0.454329\pi$
$72$	0	0
$73$	−7.17574	−0.839856	−0.419928	−	0.907557i	$-0.637945\pi$
$73$	−7.17574	−0.839856	−0.419928	+	0.907557i	$0.637945\pi$
$74$	0	0
$75$	− 1.00000i	− 0.115470i
$76$	0	0
$77$	1.08239i	0.123350i
$78$	0	0
$79$	9.11615	1.02565	0.512823	−	0.858494i	$-0.328600\pi$
$79$	9.11615	1.02565	0.512823	+	0.858494i	$0.328600\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	5.28093i	0.579657i	0.957079	+	0.289829i	$0.0935983\pi$
$83$	5.28093i	0.579657i	−0.957079	+	0.289829i	$0.906402\pi$
$84$	0	0
$85$	3.69552i	0.400835i
$86$	0	0
$87$	6.35916	0.681774
$88$	0	0
$89$	8.28772	0.878496	0.439248	−	0.898366i	$-0.355245\pi$
$89$	8.28772	0.878496	0.439248	+	0.898366i	$0.355245\pi$
$90$	0	0
$91$	1.74603i	0.183034i
$92$	0	0
$93$	− 3.55166i	− 0.368290i
$94$	0	0
$95$	8.26998	0.848482
$96$	0	0
$97$	0.681382	0.0691838	0.0345919	−	0.999402i	$-0.488987\pi$
$97$	0.681382	0.0691838	0.0345919	+	0.999402i	$0.488987\pi$
$98$	0	0
$99$	1.08239i	0.108785i
$100$	0	0
$101$	16.2195i	1.61390i	0.590622	+	0.806948i	$0.298883\pi$
$101$	16.2195i	1.61390i	−0.590622	+	0.806948i	$0.701117\pi$
$102$	0	0
$103$	−14.2741	−1.40647	−0.703237	−	0.710956i	$-0.748263\pi$
$103$	−14.2741	−1.40647	−0.703237	+	0.710956i	$0.748263\pi$
$104$	0	0
$105$	−1.00000	−0.0975900
$106$	0	0
$107$	14.6560i	1.41684i	0.705789	+	0.708422i	$0.250593\pi$
$107$	14.6560i	1.41684i	−0.705789	+	0.708422i	$0.749407\pi$
$108$	0	0
$109$	5.96722i	0.571556i	0.958296	+	0.285778i	$0.0922520\pi$
$109$	5.96722i	0.571556i	−0.958296	+	0.285778i	$0.907748\pi$
$110$	0	0
$111$	−3.69552	−0.350763
$112$	0	0
$113$	−3.98226	−0.374620	−0.187310	−	0.982301i	$-0.559977\pi$
$113$	−3.98226	−0.374620	−0.187310	+	0.982301i	$0.559977\pi$
$114$	0	0
$115$	− 7.69552i	− 0.717611i
$116$	0	0
$117$	1.74603i	0.161421i
$118$	0	0
$119$	3.69552	0.338768
$120$	0	0
$121$	9.82843	0.893493
$122$	0	0
$123$	− 4.99321i	− 0.450223i
$124$	0	0
$125$	1.00000i	0.0894427i
$126$	0	0
$127$	−5.96722	−0.529505	−0.264753	−	0.964316i	$-0.585290\pi$
$127$	−5.96722	−0.529505	−0.264753	+	0.964316i	$0.585290\pi$
$128$	0	0
$129$	−0.469266	−0.0413166
$130$	0	0
$131$	0.164784i	0.0143973i	0.999974	+	0.00719864i	$0.00229142\pi$
$131$	0.164784i	0.0143973i	−0.999974	+	0.00719864i	$0.997709\pi$
$132$	0	0
$133$	− 8.26998i	− 0.717098i
$134$	0	0
$135$	−1.00000	−0.0860663
$136$	0	0
$137$	−8.59955	−0.734709	−0.367355	−	0.930081i	$-0.619736\pi$
$137$	−8.59955	−0.734709	−0.367355	+	0.930081i	$0.619736\pi$
$138$	0	0
$139$	19.0521i	1.61597i	0.589200	+	0.807987i	$0.299443\pi$
$139$	19.0521i	1.61597i	−0.589200	+	0.807987i	$0.700557\pi$
$140$	0	0
$141$	1.69552i	0.142788i
$142$	0	0
$143$	−1.88989	−0.158041
$144$	0	0
$145$	−6.35916	−0.528100
$146$	0	0
$147$	1.00000i	0.0824786i
$148$	0	0
$149$	− 11.2558i	− 0.922115i	−0.887370	−	0.461057i	$-0.847470\pi$
$149$	− 11.2558i	− 0.922115i	0.887370	−	0.461057i	$-0.152530\pi$
$150$	0	0
$151$	−8.54903	−0.695710	−0.347855	−	0.937548i	$-0.613090\pi$
$151$	−8.54903	−0.695710	−0.347855	+	0.937548i	$0.613090\pi$
$152$	0	0
$153$	3.69552	0.298765
$154$	0	0
$155$	3.55166i	0.285276i
$156$	0	0
$157$	− 3.48022i	− 0.277752i	−0.990310	−	0.138876i	$-0.955651\pi$
$157$	− 3.48022i	− 0.277752i	0.990310	−	0.138876i	$-0.0443489\pi$
$158$	0	0
$159$	−10.3086	−0.817529
$160$	0	0
$161$	−7.69552	−0.606492
$162$	0	0
$163$	− 2.92177i	− 0.228851i	−0.993432	−	0.114425i	$-0.963497\pi$
$163$	− 2.92177i	− 0.228851i	0.993432	−	0.114425i	$-0.0365027\pi$
$164$	0	0
$165$	− 1.08239i	− 0.0842641i
$166$	0	0
$167$	12.9127	0.999215	0.499607	−	0.866252i	$-0.333478\pi$
$167$	12.9127	0.999215	0.499607	+	0.866252i	$0.333478\pi$
$168$	0	0
$169$	9.95136	0.765489
$170$	0	0
$171$	− 8.26998i	− 0.632421i
$172$	0	0
$173$	− 17.2900i	− 1.31453i	−0.753658	−	0.657267i	$-0.771712\pi$
$173$	− 17.2900i	− 1.31453i	0.753658	−	0.657267i	$-0.228288\pi$
$174$	0	0
$175$	1.00000	0.0755929
$176$	0	0
$177$	−9.39104	−0.705874
$178$	0	0
$179$	14.5609i	1.08833i	0.838978	+	0.544166i	$0.183154\pi$
$179$	14.5609i	1.08833i	−0.838978	+	0.544166i	$0.816846\pi$
$180$	0	0
$181$	19.4425i	1.44515i	0.691292	+	0.722576i	$0.257042\pi$
$181$	19.4425i	1.44515i	−0.691292	+	0.722576i	$0.742958\pi$
$182$	0	0
$183$	0.236226	0.0174623
$184$	0	0
$185$	3.69552	0.271700
$186$	0	0
$187$	4.00000i	0.292509i
$188$	0	0
$189$	1.00000i	0.0727393i
$190$	0	0
$191$	−22.7392	−1.64535	−0.822677	−	0.568509i	$-0.807520\pi$
$191$	−22.7392	−1.64535	−0.822677	+	0.568509i	$0.807520\pi$
$192$	0	0
$193$	−26.9050	−1.93666	−0.968332	−	0.249664i	$-0.919680\pi$
$193$	−26.9050	−1.93666	−0.968332	+	0.249664i	$0.919680\pi$
$194$	0	0
$195$	− 1.74603i	− 0.125036i
$196$	0	0
$197$	2.40968i	0.171682i	0.996309	+	0.0858412i	$0.0273578\pi$
$197$	2.40968i	0.171682i	−0.996309	+	0.0858412i	$0.972642\pi$
$198$	0	0
$199$	−12.1927	−0.864314	−0.432157	−	0.901798i	$-0.642247\pi$
$199$	−12.1927	−0.864314	−0.432157	+	0.901798i	$0.642247\pi$
$200$	0	0
$201$	−5.42063	−0.382342
$202$	0	0
$203$	6.35916i	0.446326i
$204$	0	0
$205$	4.99321i	0.348741i
$206$	0	0
$207$	−7.69552	−0.534875
$208$	0	0
$209$	8.95136	0.619179
$210$	0	0
$211$	26.8368i	1.84752i	0.382974	+	0.923759i	$0.374900\pi$
$211$	26.8368i	1.84752i	−0.382974	+	0.923759i	$0.625100\pi$
$212$	0	0
$213$	2.40968i	0.165108i
$214$	0	0
$215$	0.469266	0.0320037
$216$	0	0
$217$	3.55166	0.241102
$218$	0	0
$219$	− 7.17574i	− 0.484891i
$220$	0	0
$221$	6.45250i	0.434042i
$222$	0	0
$223$	−8.76017	−0.586624	−0.293312	−	0.956017i	$-0.594758\pi$
$223$	−8.76017	−0.586624	−0.293312	+	0.956017i	$0.594758\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	− 20.7375i	− 1.37640i	−0.725522	−	0.688199i	$-0.758402\pi$
$227$	− 20.7375i	− 1.37640i	0.725522	−	0.688199i	$-0.241598\pi$
$228$	0	0
$229$	− 11.1193i	− 0.734786i	−0.930066	−	0.367393i	$-0.880250\pi$
$229$	− 11.1193i	− 0.734786i	0.930066	−	0.367393i	$-0.119750\pi$
$230$	0	0
$231$	−1.08239	−0.0712162
$232$	0	0
$233$	17.3179	1.13453	0.567266	−	0.823535i	$-0.308001\pi$
$233$	17.3179	1.13453	0.567266	+	0.823535i	$0.308001\pi$
$234$	0	0
$235$	− 1.69552i	− 0.110603i
$236$	0	0
$237$	9.11615i	0.592157i
$238$	0	0
$239$	14.5964	0.944160	0.472080	−	0.881556i	$-0.343503\pi$
$239$	14.5964	0.944160	0.472080	+	0.881556i	$0.343503\pi$
$240$	0	0
$241$	22.3616	1.44044	0.720219	−	0.693747i	$-0.244041\pi$
$241$	22.3616	1.44044	0.720219	+	0.693747i	$0.244041\pi$
$242$	0	0
$243$	1.00000i	0.0641500i
$244$	0	0
$245$	− 1.00000i	− 0.0638877i
$246$	0	0
$247$	14.4397	0.918774
$248$	0	0
$249$	−5.28093	−0.334665
$250$	0	0
$251$	23.7662i	1.50011i	0.661375	+	0.750055i	$0.269973\pi$
$251$	23.7662i	1.50011i	−0.661375	+	0.750055i	$0.730027\pi$
$252$	0	0
$253$	− 8.32957i	− 0.523676i
$254$	0	0
$255$	−3.69552	−0.231422
$256$	0	0
$257$	17.3105	1.07980	0.539900	−	0.841729i	$-0.318462\pi$
$257$	17.3105	1.07980	0.539900	+	0.841729i	$0.318462\pi$
$258$	0	0
$259$	− 3.69552i	− 0.229628i
$260$	0	0
$261$	6.35916i	0.393622i
$262$	0	0
$263$	−6.97720	−0.430232	−0.215116	−	0.976588i	$-0.569013\pi$
$263$	−6.97720	−0.430232	−0.215116	+	0.976588i	$0.569013\pi$
$264$	0	0
$265$	10.3086	0.633255
$266$	0	0
$267$	8.28772i	0.507200i
$268$	0	0
$269$	7.61050i	0.464020i	0.972713	+	0.232010i	$0.0745303\pi$
$269$	7.61050i	0.464020i	−0.972713	+	0.232010i	$0.925470\pi$
$270$	0	0
$271$	−32.0581	−1.94739	−0.973695	−	0.227854i	$-0.926829\pi$
$271$	−32.0581	−1.94739	−0.973695	+	0.227854i	$0.926829\pi$
$272$	0	0
$273$	−1.74603	−0.105675
$274$	0	0
$275$	1.08239i	0.0652707i
$276$	0	0
$277$	− 22.4776i	− 1.35055i	−0.737567	−	0.675274i	$-0.764025\pi$
$277$	− 22.4776i	− 1.35055i	0.737567	−	0.675274i	$-0.235975\pi$
$278$	0	0
$279$	3.55166	0.212632
$280$	0	0
$281$	6.40690	0.382203	0.191102	−	0.981570i	$-0.438794\pi$
$281$	6.40690	0.382203	0.191102	+	0.981570i	$0.438794\pi$
$282$	0	0
$283$	− 0.371418i	− 0.0220785i	−0.999939	−	0.0110393i	$-0.996486\pi$
$283$	− 0.371418i	− 0.0220785i	0.999939	−	0.0110393i	$-0.00351398\pi$
$284$	0	0
$285$	8.26998i	0.489871i
$286$	0	0
$287$	4.99321	0.294740
$288$	0	0
$289$	−3.34315	−0.196656
$290$	0	0
$291$	0.681382i	0.0399433i
$292$	0	0
$293$	10.5079i	0.613880i	0.951729	+	0.306940i	$0.0993051\pi$
$293$	10.5079i	0.613880i	−0.951729	+	0.306940i	$0.900695\pi$
$294$	0	0
$295$	9.39104	0.546767
$296$	0	0
$297$	−1.08239	−0.0628068
$298$	0	0
$299$	− 13.4366i	− 0.777061i
$300$	0	0
$301$	− 0.469266i	− 0.0270481i
$302$	0	0
$303$	−16.2195	−0.931784
$304$	0	0
$305$	−0.236226	−0.0135263
$306$	0	0
$307$	− 14.8831i	− 0.849424i	−0.905329	−	0.424712i	$-0.860375\pi$
$307$	− 14.8831i	− 0.849424i	0.905329	−	0.424712i	$-0.139625\pi$
$308$	0	0
$309$	− 14.2741i	− 0.812028i
$310$	0	0
$311$	19.0615	1.08088	0.540438	−	0.841384i	$-0.318258\pi$
$311$	19.0615	1.08088	0.540438	+	0.841384i	$0.318258\pi$
$312$	0	0
$313$	13.6165	0.769648	0.384824	−	0.922990i	$-0.374262\pi$
$313$	13.6165	0.769648	0.384824	+	0.922990i	$0.374262\pi$
$314$	0	0
$315$	− 1.00000i	− 0.0563436i
$316$	0	0
$317$	32.5473i	1.82804i	0.405670	+	0.914019i	$0.367038\pi$
$317$	32.5473i	1.82804i	−0.405670	+	0.914019i	$0.632962\pi$
$318$	0	0
$319$	−6.88311	−0.385380
$320$	0	0
$321$	−14.6560	−0.818015
$322$	0	0
$323$	− 30.5619i	− 1.70051i
$324$	0	0
$325$	1.74603i	0.0968526i
$326$	0	0
$327$	−5.96722	−0.329988
$328$	0	0
$329$	−1.69552	−0.0934769
$330$	0	0
$331$	− 32.4936i	− 1.78601i	−0.450047	−	0.893005i	$-0.648593\pi$
$331$	− 32.4936i	− 1.78601i	0.450047	−	0.893005i	$-0.351407\pi$
$332$	0	0
$333$	− 3.69552i	− 0.202513i
$334$	0	0
$335$	5.42063	0.296161
$336$	0	0
$337$	−23.4366	−1.27668	−0.638338	−	0.769756i	$-0.720378\pi$
$337$	−23.4366	−1.27668	−0.638338	+	0.769756i	$0.720378\pi$
$338$	0	0
$339$	− 3.98226i	− 0.216287i
$340$	0	0
$341$	3.84429i	0.208180i
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	7.69552	0.414313
$346$	0	0
$347$	− 8.04879i	− 0.432082i	−0.976384	−	0.216041i	$-0.930686\pi$
$347$	− 8.04879i	− 0.432082i	0.976384	−	0.216041i	$-0.0693145\pi$
$348$	0	0
$349$	15.9150i	0.851909i	0.904744	+	0.425955i	$0.140062\pi$
$349$	15.9150i	0.851909i	−0.904744	+	0.425955i	$0.859938\pi$
$350$	0	0
$351$	−1.74603	−0.0931964
$352$	0	0
$353$	6.18155	0.329011	0.164505	−	0.986376i	$-0.447397\pi$
$353$	6.18155	0.329011	0.164505	+	0.986376i	$0.447397\pi$
$354$	0	0
$355$	− 2.40968i	− 0.127892i
$356$	0	0
$357$	3.69552i	0.195588i
$358$	0	0
$359$	19.1816	1.01237	0.506184	−	0.862426i	$-0.331056\pi$
$359$	19.1816	1.01237	0.506184	+	0.862426i	$0.331056\pi$
$360$	0	0
$361$	−49.3926	−2.59961
$362$	0	0
$363$	9.82843i	0.515859i
$364$	0	0
$365$	7.17574i	0.375595i
$366$	0	0
$367$	−36.4389	−1.90210	−0.951048	−	0.309042i	$-0.899991\pi$
$367$	−36.4389	−1.90210	−0.951048	+	0.309042i	$0.899991\pi$
$368$	0	0
$369$	4.99321	0.259936
$370$	0	0
$371$	− 10.3086i	− 0.535198i
$372$	0	0
$373$	− 11.0447i	− 0.571873i	−0.958249	−	0.285937i	$-0.907695\pi$
$373$	− 11.0447i	− 0.571873i	0.958249	−	0.285937i	$-0.0923047\pi$
$374$	0	0
$375$	−1.00000	−0.0516398
$376$	0	0
$377$	−11.1033	−0.571850
$378$	0	0
$379$	14.8605i	0.763330i	0.924301	+	0.381665i	$0.124649\pi$
$379$	14.8605i	0.763330i	−0.924301	+	0.381665i	$0.875351\pi$
$380$	0	0
$381$	− 5.96722i	− 0.305710i
$382$	0	0
$383$	−32.3037	−1.65064	−0.825322	−	0.564663i	$-0.809006\pi$
$383$	−32.3037	−1.65064	−0.825322	+	0.564663i	$0.809006\pi$
$384$	0	0
$385$	1.08239	0.0551638
$386$	0	0
$387$	− 0.469266i	− 0.0238541i
$388$	0	0
$389$	16.7525i	0.849384i	0.905338	+	0.424692i	$0.139618\pi$
$389$	16.7525i	0.849384i	−0.905338	+	0.424692i	$0.860382\pi$
$390$	0	0
$391$	−28.4389	−1.43822
$392$	0	0
$393$	−0.164784	−0.00831227
$394$	0	0
$395$	− 9.11615i	− 0.458683i
$396$	0	0
$397$	− 9.03604i	− 0.453506i	−0.973952	−	0.226753i	$-0.927189\pi$
$397$	− 9.03604i	− 0.453506i	0.973952	−	0.226753i	$-0.0728110\pi$
$398$	0	0
$399$	8.26998	0.414017
$400$	0	0
$401$	10.4306	0.520879	0.260440	−	0.965490i	$-0.416132\pi$
$401$	10.4306	0.520879	0.260440	+	0.965490i	$0.416132\pi$
$402$	0	0
$403$	6.20132i	0.308910i
$404$	0	0
$405$	− 1.00000i	− 0.0496904i
$406$	0	0
$407$	4.00000	0.198273
$408$	0	0
$409$	16.4762	0.814696	0.407348	−	0.913273i	$-0.366454\pi$
$409$	16.4762	0.814696	0.407348	+	0.913273i	$0.366454\pi$
$410$	0	0
$411$	− 8.59955i	− 0.424184i
$412$	0	0
$413$	− 9.39104i	− 0.462103i
$414$	0	0
$415$	5.28093	0.259231
$416$	0	0
$417$	−19.0521	−0.932983
$418$	0	0
$419$	32.6810i	1.59657i	0.602278	+	0.798287i	$0.294260\pi$
$419$	32.6810i	1.59657i	−0.602278	+	0.798287i	$0.705740\pi$
$420$	0	0
$421$	− 10.7821i	− 0.525486i	−0.964866	−	0.262743i	$-0.915373\pi$
$421$	− 10.7821i	− 0.525486i	0.964866	−	0.262743i	$-0.0846271\pi$
$422$	0	0
$423$	−1.69552	−0.0824389
$424$	0	0
$425$	3.69552	0.179259
$426$	0	0
$427$	0.236226i	0.0114318i
$428$	0	0
$429$	− 1.88989i	− 0.0912450i
$430$	0	0
$431$	38.1978	1.83992	0.919961	−	0.392009i	$-0.128220\pi$
$431$	38.1978	1.83992	0.919961	+	0.392009i	$0.128220\pi$
$432$	0	0
$433$	−12.5113	−0.601257	−0.300628	−	0.953741i	$-0.597196\pi$
$433$	−12.5113	−0.601257	−0.300628	+	0.953741i	$0.597196\pi$
$434$	0	0
$435$	− 6.35916i	− 0.304898i
$436$	0	0
$437$	63.6418i	3.04440i
$438$	0	0
$439$	7.07921	0.337872	0.168936	−	0.985627i	$-0.445967\pi$
$439$	7.07921	0.337872	0.168936	+	0.985627i	$0.445967\pi$
$440$	0	0
$441$	−1.00000	−0.0476190
$442$	0	0
$443$	− 2.60578i	− 0.123804i	−0.998082	−	0.0619021i	$-0.980283\pi$
$443$	− 2.60578i	− 0.123804i	0.998082	−	0.0619021i	$-0.0197167\pi$
$444$	0	0
$445$	− 8.28772i	− 0.392876i
$446$	0	0
$447$	11.2558	0.532383
$448$	0	0
$449$	36.2470	1.71060	0.855301	−	0.518132i	$-0.173372\pi$
$449$	36.2470	1.71060	0.855301	+	0.518132i	$0.173372\pi$
$450$	0	0
$451$	5.40461i	0.254493i
$452$	0	0
$453$	− 8.54903i	− 0.401669i
$454$	0	0
$455$	1.74603	0.0818554
$456$	0	0
$457$	−17.8808	−0.836430	−0.418215	−	0.908348i	$-0.637344\pi$
$457$	−17.8808	−0.836430	−0.418215	+	0.908348i	$0.637344\pi$
$458$	0	0
$459$	3.69552i	0.172492i
$460$	0	0
$461$	− 21.9227i	− 1.02104i	−0.859866	−	0.510520i	$-0.829453\pi$
$461$	− 21.9227i	− 1.02104i	0.859866	−	0.510520i	$-0.170547\pi$
$462$	0	0
$463$	18.2696	0.849062	0.424531	−	0.905413i	$-0.360439\pi$
$463$	18.2696	0.849062	0.424531	+	0.905413i	$0.360439\pi$
$464$	0	0
$465$	−3.55166	−0.164704
$466$	0	0
$467$	0.892178i	0.0412851i	0.999787	+	0.0206425i	$0.00657119\pi$
$467$	0.892178i	0.0412851i	−0.999787	+	0.0206425i	$0.993429\pi$
$468$	0	0
$469$	− 5.42063i	− 0.250301i
$470$	0	0
$471$	3.48022	0.160360
$472$	0	0
$473$	0.507930	0.0233547
$474$	0	0
$475$	− 8.26998i	− 0.379453i
$476$	0	0
$477$	− 10.3086i	− 0.472000i
$478$	0	0
$479$	−29.9091	−1.36658	−0.683291	−	0.730146i	$-0.739452\pi$
$479$	−29.9091	−1.36658	−0.683291	+	0.730146i	$0.739452\pi$
$480$	0	0
$481$	6.45250	0.294209
$482$	0	0
$483$	− 7.69552i	− 0.350158i
$484$	0	0
$485$	− 0.681382i	− 0.0309400i
$486$	0	0
$487$	30.4125	1.37812	0.689061	−	0.724703i	$-0.258023\pi$
$487$	30.4125	1.37812	0.689061	+	0.724703i	$0.258023\pi$
$488$	0	0
$489$	2.92177	0.132127
$490$	0	0
$491$	− 23.1043i	− 1.04268i	−0.853348	−	0.521341i	$-0.825432\pi$
$491$	− 23.1043i	− 1.04268i	0.853348	−	0.521341i	$-0.174568\pi$
$492$	0	0
$493$	23.5004i	1.05840i
$494$	0	0
$495$	1.08239	0.0486499
$496$	0	0
$497$	−2.40968	−0.108089
$498$	0	0
$499$	− 6.75110i	− 0.302221i	−0.988517	−	0.151110i	$-0.951715\pi$
$499$	− 6.75110i	− 0.302221i	0.988517	−	0.151110i	$-0.0482849\pi$
$500$	0	0
$501$	12.9127i	0.576897i
$502$	0	0
$503$	−5.24227	−0.233741	−0.116871	−	0.993147i	$-0.537286\pi$
$503$	−5.24227	−0.233741	−0.116871	+	0.993147i	$0.537286\pi$
$504$	0	0
$505$	16.2195	0.721757
$506$	0	0
$507$	9.95136i	0.441956i
$508$	0	0
$509$	19.6761i	0.872126i	0.899916	+	0.436063i	$0.143627\pi$
$509$	19.6761i	0.872126i	−0.899916	+	0.436063i	$0.856373\pi$
$510$	0	0
$511$	7.17574	0.317436
$512$	0	0
$513$	8.26998	0.365129
$514$	0	0
$515$	14.2741i	0.628994i
$516$	0	0
$517$	− 1.83522i	− 0.0807127i
$518$	0	0
$519$	17.2900	0.758947
$520$	0	0
$521$	−30.3978	−1.33175	−0.665876	−	0.746062i	$-0.731942\pi$
$521$	−30.3978	−1.33175	−0.665876	+	0.746062i	$0.731942\pi$
$522$	0	0
$523$	− 18.2150i	− 0.796485i	−0.917280	−	0.398242i	$-0.869620\pi$
$523$	− 18.2150i	− 0.796485i	0.917280	−	0.398242i	$-0.130380\pi$
$524$	0	0
$525$	1.00000i	0.0436436i
$526$	0	0
$527$	13.1252	0.571743
$528$	0	0
$529$	36.2210	1.57483
$530$	0	0
$531$	− 9.39104i	− 0.407536i
$532$	0	0
$533$	8.71832i	0.377632i
$534$	0	0
$535$	14.6560	0.633632
$536$	0	0
$537$	−14.5609	−0.628349
$538$	0	0
$539$	− 1.08239i	− 0.0466219i
$540$	0	0
$541$	− 9.20255i	− 0.395648i	−0.980238	−	0.197824i	$-0.936612\pi$
$541$	− 9.20255i	− 0.395648i	0.980238	−	0.197824i	$-0.0633875\pi$
$542$	0	0
$543$	−19.4425	−0.834359
$544$	0	0
$545$	5.96722	0.255608
$546$	0	0
$547$	− 29.7167i	− 1.27059i	−0.772268	−	0.635297i	$-0.780878\pi$
$547$	− 29.7167i	− 1.27059i	0.772268	−	0.635297i	$-0.219122\pi$
$548$	0	0
$549$	0.236226i	0.0100819i
$550$	0	0
$551$	52.5901	2.24042
$552$	0	0
$553$	−9.11615	−0.387658
$554$	0	0
$555$	3.69552i	0.156866i
$556$	0	0
$557$	− 14.3678i	− 0.608784i	−0.952547	−	0.304392i	$-0.901547\pi$
$557$	− 14.3678i	− 0.608784i	0.952547	−	0.304392i	$-0.0984533\pi$
$558$	0	0
$559$	0.819355	0.0346550
$560$	0	0
$561$	−4.00000	−0.168880
$562$	0	0
$563$	− 40.4710i	− 1.70565i	−0.522199	−	0.852824i	$-0.674888\pi$
$563$	− 40.4710i	− 1.70565i	0.522199	−	0.852824i	$-0.325112\pi$
$564$	0	0
$565$	3.98226i	0.167535i
$566$	0	0
$567$	−1.00000	−0.0419961
$568$	0	0
$569$	39.0562	1.63732	0.818661	−	0.574277i	$-0.194717\pi$
$569$	39.0562	1.63732	0.818661	+	0.574277i	$0.194717\pi$
$570$	0	0
$571$	− 2.82843i	− 0.118366i	−0.998247	−	0.0591830i	$-0.981150\pi$
$571$	− 2.82843i	− 0.118366i	0.998247	−	0.0591830i	$-0.0188495\pi$
$572$	0	0
$573$	− 22.7392i	− 0.949946i
$574$	0	0
$575$	−7.69552	−0.320925
$576$	0	0
$577$	14.3801	0.598651	0.299325	−	0.954151i	$-0.403238\pi$
$577$	14.3801	0.598651	0.299325	+	0.954151i	$0.403238\pi$
$578$	0	0
$579$	− 26.9050i	− 1.11813i
$580$	0	0
$581$	− 5.28093i	− 0.219090i
$582$	0	0
$583$	11.1580	0.462117
$584$	0	0
$585$	1.74603	0.0721897
$586$	0	0
$587$	14.2658i	0.588813i	0.955680	+	0.294407i	$0.0951220\pi$
$587$	14.2658i	0.588813i	−0.955680	+	0.294407i	$0.904878\pi$
$588$	0	0
$589$	− 29.3721i	− 1.21026i
$590$	0	0
$591$	−2.40968	−0.0991209
$592$	0	0
$593$	18.9500	0.778185	0.389092	−	0.921199i	$-0.372789\pi$
$593$	18.9500	0.778185	0.389092	+	0.921199i	$0.372789\pi$
$594$	0	0
$595$	− 3.69552i	− 0.151501i
$596$	0	0
$597$	− 12.1927i	− 0.499012i
$598$	0	0
$599$	2.01455	0.0823124	0.0411562	−	0.999153i	$-0.486896\pi$
$599$	2.01455	0.0823124	0.0411562	+	0.999153i	$0.486896\pi$
$600$	0	0
$601$	13.9582	0.569365	0.284682	−	0.958622i	$-0.408112\pi$
$601$	13.9582	0.569365	0.284682	+	0.958622i	$0.408112\pi$
$602$	0	0
$603$	− 5.42063i	− 0.220745i
$604$	0	0
$605$	− 9.82843i	− 0.399582i
$606$	0	0
$607$	21.4868	0.872123	0.436062	−	0.899917i	$-0.356373\pi$
$607$	21.4868	0.872123	0.436062	+	0.899917i	$0.356373\pi$
$608$	0	0
$609$	−6.35916	−0.257686
$610$	0	0
$611$	− 2.96043i	− 0.119766i
$612$	0	0
$613$	− 18.7634i	− 0.757845i	−0.925428	−	0.378922i	$-0.876295\pi$
$613$	− 18.7634i	− 0.757845i	0.925428	−	0.378922i	$-0.123705\pi$
$614$	0	0
$615$	−4.99321	−0.201346
$616$	0	0
$617$	11.5008	0.463005	0.231502	−	0.972834i	$-0.425636\pi$
$617$	11.5008	0.463005	0.231502	+	0.972834i	$0.425636\pi$
$618$	0	0
$619$	25.1577i	1.01117i	0.862776	+	0.505586i	$0.168724\pi$
$619$	25.1577i	1.01117i	−0.862776	+	0.505586i	$0.831276\pi$
$620$	0	0
$621$	− 7.69552i	− 0.308811i
$622$	0	0
$623$	−8.28772	−0.332040
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	8.95136i	0.357483i
$628$	0	0
$629$	− 13.6569i	− 0.544534i
$630$	0	0
$631$	−1.87977	−0.0748323	−0.0374161	−	0.999300i	$-0.511913\pi$
$631$	−1.87977	−0.0748323	−0.0374161	+	0.999300i	$0.511913\pi$
$632$	0	0
$633$	−26.8368	−1.06667
$634$	0	0
$635$	5.96722i	0.236802i
$636$	0	0
$637$	− 1.74603i	− 0.0691804i
$638$	0	0
$639$	−2.40968	−0.0953254
$640$	0	0
$641$	32.7165	1.29222	0.646112	−	0.763242i	$-0.276394\pi$
$641$	32.7165	1.29222	0.646112	+	0.763242i	$0.276394\pi$
$642$	0	0
$643$	42.7402i	1.68551i	0.538297	+	0.842755i	$0.319068\pi$
$643$	42.7402i	1.68551i	−0.538297	+	0.842755i	$0.680932\pi$
$644$	0	0
$645$	0.469266i	0.0184773i
$646$	0	0
$647$	−24.1080	−0.947785	−0.473892	−	0.880583i	$-0.657151\pi$
$647$	−24.1080	−0.947785	−0.473892	+	0.880583i	$0.657151\pi$
$648$	0	0
$649$	10.1648	0.399003
$650$	0	0
$651$	3.55166i	0.139200i
$652$	0	0
$653$	− 28.1759i	− 1.10261i	−0.834305	−	0.551304i	$-0.814130\pi$
$653$	− 28.1759i	− 1.10261i	0.834305	−	0.551304i	$-0.185870\pi$
$654$	0	0
$655$	0.164784	0.00643866
$656$	0	0
$657$	7.17574	0.279952
$658$	0	0
$659$	9.51937i	0.370822i	0.982661	+	0.185411i	$0.0593616\pi$
$659$	9.51937i	0.370822i	−0.982661	+	0.185411i	$0.940638\pi$
$660$	0	0
$661$	− 18.6232i	− 0.724358i	−0.932109	−	0.362179i	$-0.882033\pi$
$661$	− 18.6232i	− 0.724358i	0.932109	−	0.362179i	$-0.117967\pi$
$662$	0	0
$663$	−6.45250	−0.250594
$664$	0	0
$665$	−8.26998	−0.320696
$666$	0	0
$667$	− 48.9370i	− 1.89485i
$668$	0	0
$669$	− 8.76017i	− 0.338688i
$670$	0	0
$671$	−0.255689	−0.00987077
$672$	0	0
$673$	−17.5241	−0.675504	−0.337752	−	0.941235i	$-0.609667\pi$
$673$	−17.5241	−0.675504	−0.337752	+	0.941235i	$0.609667\pi$
$674$	0	0
$675$	1.00000i	0.0384900i
$676$	0	0
$677$	− 6.75380i	− 0.259570i	−0.991542	−	0.129785i	$-0.958571\pi$
$677$	− 6.75380i	− 0.259570i	0.991542	−	0.129785i	$-0.0414287\pi$
$678$	0	0
$679$	−0.681382	−0.0261490
$680$	0	0
$681$	20.7375	0.794664
$682$	0	0
$683$	19.1521i	0.732835i	0.930451	+	0.366418i	$0.119416\pi$
$683$	19.1521i	0.732835i	−0.930451	+	0.366418i	$0.880584\pi$
$684$	0	0
$685$	8.59955i	0.328572i
$686$	0	0
$687$	11.1193	0.424229
$688$	0	0
$689$	17.9993	0.685717
$690$	0	0
$691$	− 0.584853i	− 0.0222489i	−0.999938	−	0.0111244i	$-0.996459\pi$
$691$	− 0.584853i	− 0.0222489i	0.999938	−	0.0111244i	$-0.00354109\pi$
$692$	0	0
$693$	− 1.08239i	− 0.0411167i
$694$	0	0
$695$	19.0521	0.722686
$696$	0	0
$697$	18.4525	0.698938
$698$	0	0
$699$	17.3179i	0.655022i
$700$	0	0
$701$	− 18.8218i	− 0.710889i	−0.934697	−	0.355445i	$-0.884329\pi$
$701$	− 18.8218i	− 0.710889i	0.934697	−	0.355445i	$-0.115671\pi$
$702$	0	0
$703$	−30.5619	−1.15266
$704$	0	0
$705$	1.69552	0.0638569
$706$	0	0
$707$	− 16.2195i	− 0.609996i
$708$	0	0
$709$	46.1278i	1.73237i	0.499727	+	0.866183i	$0.333434\pi$
$709$	46.1278i	1.73237i	−0.499727	+	0.866183i	$0.666566\pi$
$710$	0	0
$711$	−9.11615	−0.341882
$712$	0	0
$713$	−27.3319	−1.02359
$714$	0	0
$715$	1.88989i	0.0706781i
$716$	0	0
$717$	14.5964i	0.545111i
$718$	0	0
$719$	32.3699	1.20719	0.603597	−	0.797289i	$-0.293734\pi$
$719$	32.3699	1.20719	0.603597	+	0.797289i	$0.293734\pi$
$720$	0	0
$721$	14.2741	0.531597
$722$	0	0
$723$	22.3616i	0.831637i
$724$	0	0
$725$	6.35916i	0.236173i
$726$	0	0
$727$	23.1945	0.860238	0.430119	−	0.902772i	$-0.358472\pi$
$727$	23.1945	0.860238	0.430119	+	0.902772i	$0.358472\pi$
$728$	0	0
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	− 1.73418i	− 0.0641410i
$732$	0	0
$733$	11.3290i	0.418447i	0.977868	+	0.209223i	$0.0670935\pi$
$733$	11.3290i	0.418447i	−0.977868	+	0.209223i	$0.932906\pi$
$734$	0	0
$735$	1.00000	0.0368856
$736$	0	0
$737$	5.86725	0.216123
$738$	0	0
$739$	6.97131i	0.256444i	0.991746	+	0.128222i	$0.0409270\pi$
$739$	6.97131i	0.256444i	−0.991746	+	0.128222i	$0.959073\pi$
$740$	0	0
$741$	14.4397i	0.530455i
$742$	0	0
$743$	−5.10846	−0.187411	−0.0937055	−	0.995600i	$-0.529871\pi$
$743$	−5.10846	−0.187411	−0.0937055	+	0.995600i	$0.529871\pi$
$744$	0	0
$745$	−11.2558	−0.412382
$746$	0	0
$747$	− 5.28093i	− 0.193219i
$748$	0	0
$749$	− 14.6560i	− 0.535517i
$750$	0	0
$751$	−35.6278	−1.30008	−0.650039	−	0.759901i	$-0.725247\pi$
$751$	−35.6278	−1.30008	−0.650039	+	0.759901i	$0.725247\pi$
$752$	0	0
$753$	−23.7662	−0.866089
$754$	0	0
$755$	8.54903i	0.311131i
$756$	0	0
$757$	− 33.2178i	− 1.20732i	−0.797241	−	0.603661i	$-0.793708\pi$
$757$	− 33.2178i	− 1.20732i	0.797241	−	0.603661i	$-0.206292\pi$
$758$	0	0
$759$	8.32957	0.302344
$760$	0	0
$761$	−47.6335	−1.72671	−0.863356	−	0.504596i	$-0.831642\pi$
$761$	−47.6335	−1.72671	−0.863356	+	0.504596i	$0.831642\pi$
$762$	0	0
$763$	− 5.96722i	− 0.216028i
$764$	0	0
$765$	− 3.69552i	− 0.133612i
$766$	0	0
$767$	16.3971	0.592064
$768$	0	0
$769$	−25.9374	−0.935325	−0.467663	−	0.883907i	$-0.654904\pi$
$769$	−25.9374	−0.935325	−0.467663	+	0.883907i	$0.654904\pi$
$770$	0	0
$771$	17.3105i	0.623423i
$772$	0	0
$773$	− 31.7154i	− 1.14072i	−0.821394	−	0.570361i	$-0.806803\pi$
$773$	− 31.7154i	− 1.14072i	0.821394	−	0.570361i	$-0.193197\pi$
$774$	0	0
$775$	3.55166	0.127579
$776$	0	0
$777$	3.69552	0.132576
$778$	0	0
$779$	− 41.2938i	− 1.47950i
$780$	0	0
$781$	− 2.60822i	− 0.0933293i
$782$	0	0
$783$	−6.35916	−0.227258
$784$	0	0
$785$	−3.48022	−0.124214
$786$	0	0
$787$	− 18.4107i	− 0.656269i	−0.944631	−	0.328134i	$-0.893580\pi$
$787$	− 18.4107i	− 0.656269i	0.944631	−	0.328134i	$-0.106420\pi$
$788$	0	0
$789$	− 6.97720i	− 0.248395i
$790$	0	0
$791$	3.98226	0.141593
$792$	0	0
$793$	−0.412459	−0.0146468
$794$	0	0
$795$	10.3086i	0.365610i
$796$	0	0
$797$	18.8632i	0.668167i	0.942543	+	0.334084i	$0.108427\pi$
$797$	18.8632i	0.668167i	−0.942543	+	0.334084i	$0.891573\pi$
$798$	0	0
$799$	−6.26582	−0.221669
$800$	0	0
$801$	−8.28772	−0.292832
$802$	0	0
$803$	7.76696i	0.274090i
$804$	0	0
$805$	7.69552i	0.271231i
$806$	0	0
$807$	−7.61050	−0.267902
$808$	0	0
$809$	50.0487	1.75962	0.879809	−	0.475328i	$-0.157670\pi$
$809$	50.0487	1.75962	0.879809	+	0.475328i	$0.157670\pi$
$810$	0	0
$811$	32.0936i	1.12696i	0.826130	+	0.563479i	$0.190537\pi$
$811$	32.0936i	1.12696i	−0.826130	+	0.563479i	$0.809463\pi$
$812$	0	0
$813$	− 32.0581i	− 1.12433i
$814$	0	0
$815$	−2.92177	−0.102345
$816$	0	0
$817$	−3.88082	−0.135773
$818$	0	0
$819$	− 1.74603i	− 0.0610114i
$820$	0	0
$821$	− 48.2849i	− 1.68515i	−0.538575	−	0.842577i	$-0.681037\pi$
$821$	− 48.2849i	− 1.68515i	0.538575	−	0.842577i	$-0.318963\pi$
$822$	0	0
$823$	17.0169	0.593172	0.296586	−	0.955006i	$-0.404152\pi$
$823$	17.0169	0.593172	0.296586	+	0.955006i	$0.404152\pi$
$824$	0	0
$825$	−1.08239	−0.0376841
$826$	0	0
$827$	− 30.4120i	− 1.05753i	−0.848768	−	0.528765i	$-0.822655\pi$
$827$	− 30.4120i	− 1.05753i	0.848768	−	0.528765i	$-0.177345\pi$
$828$	0	0
$829$	− 13.0639i	− 0.453728i	−0.973926	−	0.226864i	$-0.927153\pi$
$829$	− 13.0639i	− 0.453728i	0.973926	−	0.226864i	$-0.0728473\pi$
$830$	0	0
$831$	22.4776	0.779739
$832$	0	0
$833$	−3.69552	−0.128042
$834$	0	0
$835$	− 12.9127i	− 0.446862i
$836$	0	0
$837$	3.55166i	0.122763i
$838$	0	0
$839$	27.2127	0.939486	0.469743	−	0.882803i	$-0.344347\pi$
$839$	27.2127	0.939486	0.469743	+	0.882803i	$0.344347\pi$
$840$	0	0
$841$	−11.4389	−0.394446
$842$	0	0
$843$	6.40690i	0.220665i
$844$	0	0
$845$	− 9.95136i	− 0.342337i
$846$	0	0
$847$	−9.82843	−0.337709
$848$	0	0
$849$	0.371418	0.0127470
$850$	0	0
$851$	28.4389i	0.974874i
$852$	0	0
$853$	2.22044i	0.0760264i	0.999277	+	0.0380132i	$0.0121029\pi$
$853$	2.22044i	0.0760264i	−0.999277	+	0.0380132i	$0.987897\pi$
$854$	0	0
$855$	−8.26998	−0.282827
$856$	0	0
$857$	38.0352	1.29926	0.649629	−	0.760251i	$-0.274924\pi$
$857$	38.0352	1.29926	0.649629	+	0.760251i	$0.274924\pi$
$858$	0	0
$859$	− 31.0521i	− 1.05948i	−0.848159	−	0.529741i	$-0.822289\pi$
$859$	− 31.0521i	− 1.05948i	0.848159	−	0.529741i	$-0.177711\pi$
$860$	0	0
$861$	4.99321i	0.170168i
$862$	0	0
$863$	19.7812	0.673359	0.336679	−	0.941619i	$-0.390696\pi$
$863$	19.7812	0.673359	0.336679	+	0.941619i	$0.390696\pi$
$864$	0	0
$865$	−17.2900	−0.587878
$866$	0	0
$867$	− 3.34315i	− 0.113539i
$868$	0	0
$869$	− 9.86725i	− 0.334723i
$870$	0	0
$871$	9.46461	0.320696
$872$	0	0
$873$	−0.681382	−0.0230613
$874$	0	0
$875$	− 1.00000i	− 0.0338062i
$876$	0	0
$877$	− 49.0576i	− 1.65656i	−0.560316	−	0.828279i	$-0.689320\pi$
$877$	− 49.0576i	− 1.65656i	0.560316	−	0.828279i	$-0.310680\pi$
$878$	0	0
$879$	−10.5079	−0.354424
$880$	0	0
$881$	−27.8383	−0.937896	−0.468948	−	0.883226i	$-0.655367\pi$
$881$	−27.8383	−0.937896	−0.468948	+	0.883226i	$0.655367\pi$
$882$	0	0
$883$	− 44.2091i	− 1.48775i	−0.668317	−	0.743877i	$-0.732985\pi$
$883$	− 44.2091i	− 1.48775i	0.668317	−	0.743877i	$-0.267015\pi$
$884$	0	0
$885$	9.39104i	0.315676i
$886$	0	0
$887$	−10.9455	−0.367513	−0.183757	−	0.982972i	$-0.558826\pi$
$887$	−10.9455	−0.367513	−0.183757	+	0.982972i	$0.558826\pi$
$888$	0	0
$889$	5.96722	0.200134
$890$	0	0
$891$	− 1.08239i	− 0.0362615i
$892$	0	0
$893$	14.0219i	0.469225i
$894$	0	0
$895$	14.5609	0.486717
$896$	0	0
$897$	13.4366	0.448636
$898$	0	0
$899$	22.5856i	0.753271i
$900$	0	0
$901$	− 38.0958i	− 1.26915i
$902$	0	0
$903$	0.469266	0.0156162
$904$	0	0
$905$	19.4425	0.646292
$906$	0	0
$907$	50.1891i	1.66650i	0.552895	+	0.833251i	$0.313523\pi$
$907$	50.1891i	1.66650i	−0.552895	+	0.833251i	$0.686477\pi$
$908$	0	0
$909$	− 16.2195i	− 0.537966i
$910$	0	0
$911$	−19.6714	−0.651743	−0.325871	−	0.945414i	$-0.605658\pi$
$911$	−19.6714	−0.651743	−0.325871	+	0.945414i	$0.605658\pi$
$912$	0	0
$913$	5.71604	0.189173
$914$	0	0
$915$	− 0.236226i	− 0.00780939i
$916$	0	0
$917$	− 0.164784i	− 0.00544166i
$918$	0	0
$919$	5.79067	0.191016	0.0955082	−	0.995429i	$-0.469552\pi$
$919$	5.79067	0.191016	0.0955082	+	0.995429i	$0.469552\pi$
$920$	0	0
$921$	14.8831	0.490415
$922$	0	0
$923$	− 4.20738i	− 0.138488i
$924$	0	0
$925$	− 3.69552i	− 0.121508i
$926$	0	0
$927$	14.2741	0.468824
$928$	0	0
$929$	15.0106	0.492482	0.246241	−	0.969209i	$-0.420805\pi$
$929$	15.0106	0.492482	0.246241	+	0.969209i	$0.420805\pi$
$930$	0	0
$931$	8.26998i	0.271038i
$932$	0	0
$933$	19.0615i	0.624044i
$934$	0	0
$935$	4.00000	0.130814
$936$	0	0
$937$	−34.9374	−1.14135	−0.570677	−	0.821174i	$-0.693319\pi$
$937$	−34.9374	−1.14135	−0.570677	+	0.821174i	$0.693319\pi$
$938$	0	0
$939$	13.6165i	0.444357i
$940$	0	0
$941$	− 32.2504i	− 1.05133i	−0.850690	−	0.525667i	$-0.823816\pi$
$941$	− 32.2504i	− 1.05133i	0.850690	−	0.525667i	$-0.176184\pi$
$942$	0	0
$943$	−38.4253	−1.25130
$944$	0	0
$945$	1.00000	0.0325300
$946$	0	0
$947$	− 29.8686i	− 0.970600i	−0.874348	−	0.485300i	$-0.838710\pi$
$947$	− 29.8686i	− 0.970600i	0.874348	−	0.485300i	$-0.161290\pi$
$948$	0	0
$949$	12.5291i	0.406711i
$950$	0	0
$951$	−32.5473	−1.05542
$952$	0	0
$953$	−6.85900	−0.222185	−0.111092	−	0.993810i	$-0.535435\pi$
$953$	−6.85900	−0.222185	−0.111092	+	0.993810i	$0.535435\pi$
$954$	0	0
$955$	22.7392i	0.735825i
$956$	0	0
$957$	− 6.88311i	− 0.222499i
$958$	0	0
$959$	8.59955	0.277694
$960$	0	0
$961$	−18.3857	−0.593088
$962$	0	0
$963$	− 14.6560i	− 0.472281i
$964$	0	0
$965$	26.9050i	0.866103i
$966$	0	0
$967$	35.0343	1.12663	0.563314	−	0.826243i	$-0.309526\pi$
$967$	35.0343	1.12663	0.563314	+	0.826243i	$0.309526\pi$
$968$	0	0
$969$	30.5619	0.981788
$970$	0	0
$971$	26.2843i	0.843502i	0.906712	+	0.421751i	$0.138584\pi$
$971$	26.2843i	0.843502i	−0.906712	+	0.421751i	$0.861416\pi$
$972$	0	0
$973$	− 19.0521i	− 0.610781i
$974$	0	0
$975$	−1.74603	−0.0559179
$976$	0	0
$977$	−13.3597	−0.427415	−0.213708	−	0.976898i	$-0.568554\pi$
$977$	−13.3597	−0.427415	−0.213708	+	0.976898i	$0.568554\pi$
$978$	0	0
$979$	− 8.97056i	− 0.286700i
$980$	0	0
$981$	− 5.96722i	− 0.190519i
$982$	0	0
$983$	−5.62996	−0.179568	−0.0897840	−	0.995961i	$-0.528618\pi$
$983$	−5.62996	−0.179568	−0.0897840	+	0.995961i	$0.528618\pi$
$984$	0	0
$985$	2.40968	0.0767787
$986$	0	0
$987$	− 1.69552i	− 0.0539689i
$988$	0	0
$989$	3.61125i	0.114831i
$990$	0	0
$991$	−42.8232	−1.36032	−0.680161	−	0.733062i	$-0.738090\pi$
$991$	−42.8232	−1.36032	−0.680161	+	0.733062i	$0.738090\pi$
$992$	0	0
$993$	32.4936	1.03115
$994$	0	0
$995$	12.1927i	0.386533i
$996$	0	0
$997$	4.68276i	0.148305i	0.997247	+	0.0741523i	$0.0236251\pi$
$997$	4.68276i	0.148305i	−0.997247	+	0.0741523i	$0.976375\pi$
$998$	0	0
$999$	3.69552	0.116921

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3360.2.g.a.1681.6		8
4.3	odd	2		840.2.g.a.421.3	✓	8
8.3	odd	2		840.2.g.a.421.4	yes	8
8.5	even	2	inner	3360.2.g.a.1681.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
840.2.g.a.421.3	✓	8	4.3	odd	2
840.2.g.a.421.4	yes	8	8.3	odd	2
3360.2.g.a.1681.3		8	8.5	even	2	inner
3360.2.g.a.1681.6		8	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 3360 = 2^{5} \cdot 3 \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3360.g (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{3} -1.00000i q^{5} -1.00000 q^{7} -1.00000 q^{9} -1.08239i q^{11} -1.74603i q^{13} +1.00000 q^{15} -3.69552 q^{17} +8.26998i q^{19} -1.00000i q^{21} +7.69552 q^{23} -1.00000 q^{25} -1.00000i q^{27} -6.35916i q^{29} -3.55166 q^{31} +1.08239 q^{33} +1.00000i q^{35} +3.69552i q^{37} +1.74603 q^{39} -4.99321 q^{41} +0.469266i q^{43} +1.00000i q^{45} +1.69552 q^{47} +1.00000 q^{49} -3.69552i q^{51} +10.3086i q^{53} -1.08239 q^{55} -8.26998 q^{57} +9.39104i q^{59} -0.236226i q^{61} +1.00000 q^{63} -1.74603 q^{65} +5.42063i q^{67} +7.69552i q^{69} +2.40968 q^{71} -7.17574 q^{73} -1.00000i q^{75} +1.08239i q^{77} +9.11615 q^{79} +1.00000 q^{81} +5.28093i q^{83} +3.69552i q^{85} +6.35916 q^{87} +8.28772 q^{89} +1.74603i q^{91} -3.55166i q^{93} +8.26998 q^{95} +0.681382 q^{97} +1.08239i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{7} - 8 q^{9} + 8 q^{15} + 32 q^{23} - 8 q^{25} - 32 q^{31} - 16 q^{47} + 8 q^{49} + 8 q^{63} - 16 q^{79} + 8 q^{81} + 16 q^{87}+O(q^{100}) \) 8 * q - 8 * q^7 - 8 * q^9 + 8 * q^15 + 32 * q^23 - 8 * q^25 - 32 * q^31 - 16 * q^47 + 8 * q^49 + 8 * q^63 - 16 * q^79 + 8 * q^81 + 16 * q^87

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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