LMFDB - Embedded newform 3520.2.g.f.1761.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3520,2,Mod(1761,3520)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3520, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("3520.1761");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3520 = 2^{6} \cdot 5 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3520.g (of order $2$, degree $1$, 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:	$28.1073415115$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1761.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	3520.1761
Dual form			3520.2.g.f.1761.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{3} +1.00000i q^{5} +3.00000 q^{7} +2.00000 q^{9} +O(q^{10})$	$q+1.00000i q^{3} +1.00000i q^{5} +3.00000 q^{7} +2.00000 q^{9} +1.00000i q^{11} +6.00000i q^{13} -1.00000 q^{15} -3.00000 q^{17} +7.00000i q^{19} +3.00000i q^{21} -1.00000 q^{25} +5.00000i q^{27} -3.00000i q^{29} -9.00000 q^{31} -1.00000 q^{33} +3.00000i q^{35} -3.00000i q^{37} -6.00000 q^{39} -10.0000i q^{43} +2.00000i q^{45} +2.00000 q^{49} -3.00000i q^{51} +9.00000i q^{53} -1.00000 q^{55} -7.00000 q^{57} -12.0000i q^{59} +15.0000i q^{61} +6.00000 q^{63} -6.00000 q^{65} -4.00000i q^{67} +3.00000 q^{71} +2.00000 q^{73} -1.00000i q^{75} +3.00000i q^{77} +1.00000 q^{81} +6.00000i q^{83} -3.00000i q^{85} +3.00000 q^{87} +15.0000 q^{89} +18.0000i q^{91} -9.00000i q^{93} -7.00000 q^{95} +10.0000 q^{97} +2.00000i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 6 q^{7} + 4 q^{9}+O(q^{10}) $ 2 * q + 6 * q^7 + 4 * q^9	$ 2 q + 6 q^{7} + 4 q^{9} - 2 q^{15} - 6 q^{17} - 2 q^{25} - 18 q^{31} - 2 q^{33} - 12 q^{39} + 4 q^{49} - 2 q^{55} - 14 q^{57} + 12 q^{63} - 12 q^{65} + 6 q^{71} + 4 q^{73} + 2 q^{81} + 6 q^{87} + 30 q^{89} - 14 q^{95} + 20 q^{97}+O(q^{100}) $ 2 * q + 6 * q^7 + 4 * q^9 - 2 * q^15 - 6 * q^17 - 2 * q^25 - 18 * q^31 - 2 * q^33 - 12 * q^39 + 4 * q^49 - 2 * q^55 - 14 * q^57 + 12 * q^63 - 12 * q^65 + 6 * q^71 + 4 * q^73 + 2 * q^81 + 6 * q^87 + 30 * q^89 - 14 * q^95 + 20 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3520\mathbb{Z}\right)^\times$.

$n$	$321$	$1541$	$2751$	$2817$
$\chi(n)$	$1$	$-1$	$1$	$1$

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	0.957427	+	0.288675i	$0.0932147\pi$
$3$	1.00000i	0.577350i	−0.957427	+	0.288675i	$0.906785\pi$
$4$	0	0
$5$	1.00000i	0.447214i
$5$	1.00000i	0.447214i
$6$	0	0
$7$	3.00000	1.13389	0.566947	−	0.823754i	$-0.308125\pi$
$7$	3.00000	1.13389	0.566947	+	0.823754i	$0.308125\pi$
$8$	0	0
$9$	2.00000	0.666667
$10$	0	0
$11$	1.00000i	0.301511i
$11$	1.00000i	0.301511i
$12$	0	0
$13$	6.00000i	1.66410i	0.554700	+	0.832050i	$0.312833\pi$
$13$	6.00000i	1.66410i	−0.554700	+	0.832050i	$0.687167\pi$
$14$	0	0
$15$	−1.00000	−0.258199
$16$	0	0
$17$	−3.00000	−0.727607	−0.363803	−	0.931476i	$-0.618522\pi$
$17$	−3.00000	−0.727607	−0.363803	+	0.931476i	$0.618522\pi$
$18$	0	0
$19$	7.00000i	1.60591i	0.596040	+	0.802955i	$0.296740\pi$
$19$	7.00000i	1.60591i	−0.596040	+	0.802955i	$0.703260\pi$
$20$	0	0
$21$	3.00000i	0.654654i
$22$	0	0
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	0	0
$25$	−1.00000	−0.200000
$26$	0	0
$27$	5.00000i	0.962250i
$28$	0	0
$29$	− 3.00000i	− 0.557086i	−0.960424	−	0.278543i	$-0.910149\pi$
$29$	− 3.00000i	− 0.557086i	0.960424	−	0.278543i	$-0.0898515\pi$
$30$	0	0
$31$	−9.00000	−1.61645	−0.808224	−	0.588875i	$-0.799571\pi$
$31$	−9.00000	−1.61645	−0.808224	+	0.588875i	$0.799571\pi$
$32$	0	0
$33$	−1.00000	−0.174078
$34$	0	0
$35$	3.00000i	0.507093i
$36$	0	0
$37$	− 3.00000i	− 0.493197i	−0.969118	−	0.246598i	$-0.920687\pi$
$37$	− 3.00000i	− 0.493197i	0.969118	−	0.246598i	$-0.0793129\pi$
$38$	0	0
$39$	−6.00000	−0.960769
$40$	0	0
$41$	0	0		−	1.00000i	$-0.5\pi$
$41$	0	0			1.00000i	$0.5\pi$
$42$	0	0
$43$	− 10.0000i	− 1.52499i	−0.646997	−	0.762493i	$-0.723975\pi$
$43$	− 10.0000i	− 1.52499i	0.646997	−	0.762493i	$-0.276025\pi$
$44$	0	0
$45$	2.00000i	0.298142i
$46$	0	0
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	0	0
$49$	2.00000	0.285714
$50$	0	0
$51$	− 3.00000i	− 0.420084i
$52$	0	0
$53$	9.00000i	1.23625i	0.786082	+	0.618123i	$0.212106\pi$
$53$	9.00000i	1.23625i	−0.786082	+	0.618123i	$0.787894\pi$
$54$	0	0
$55$	−1.00000	−0.134840
$56$	0	0
$57$	−7.00000	−0.927173
$58$	0	0
$59$	− 12.0000i	− 1.56227i	−0.624364	−	0.781133i	$-0.714642\pi$
$59$	− 12.0000i	− 1.56227i	0.624364	−	0.781133i	$-0.285358\pi$
$60$	0	0
$61$	15.0000i	1.92055i	0.279050	+	0.960277i	$0.409981\pi$
$61$	15.0000i	1.92055i	−0.279050	+	0.960277i	$0.590019\pi$
$62$	0	0
$63$	6.00000	0.755929
$64$	0	0
$65$	−6.00000	−0.744208
$66$	0	0
$67$	− 4.00000i	− 0.488678i	−0.969690	−	0.244339i	$-0.921429\pi$
$67$	− 4.00000i	− 0.488678i	0.969690	−	0.244339i	$-0.0785709\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	3.00000	0.356034	0.178017	−	0.984027i	$-0.443032\pi$
$71$	3.00000	0.356034	0.178017	+	0.984027i	$0.443032\pi$
$72$	0	0
$73$	2.00000	0.234082	0.117041	−	0.993127i	$-0.462659\pi$
$73$	2.00000	0.234082	0.117041	+	0.993127i	$0.462659\pi$
$74$	0	0
$75$	− 1.00000i	− 0.115470i
$76$	0	0
$77$	3.00000i	0.341882i
$78$	0	0
$79$	0	0		−	1.00000i	$-0.5\pi$
$79$	0	0			1.00000i	$0.5\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	6.00000i	0.658586i	0.944228	+	0.329293i	$0.106810\pi$
$83$	6.00000i	0.658586i	−0.944228	+	0.329293i	$0.893190\pi$
$84$	0	0
$85$	− 3.00000i	− 0.325396i
$86$	0	0
$87$	3.00000	0.321634
$88$	0	0
$89$	15.0000	1.59000	0.794998	−	0.606612i	$-0.207472\pi$
$89$	15.0000	1.59000	0.794998	+	0.606612i	$0.207472\pi$
$90$	0	0
$91$	18.0000i	1.88691i
$92$	0	0
$93$	− 9.00000i	− 0.933257i
$94$	0	0
$95$	−7.00000	−0.718185
$96$	0	0
$97$	10.0000	1.01535	0.507673	−	0.861550i	$-0.330506\pi$
$97$	10.0000	1.01535	0.507673	+	0.861550i	$0.330506\pi$
$98$	0	0
$99$	2.00000i	0.201008i
$100$	0	0
$101$	6.00000i	0.597022i	0.954406	+	0.298511i	$0.0964900\pi$
$101$	6.00000i	0.597022i	−0.954406	+	0.298511i	$0.903510\pi$
$102$	0	0
$103$	0	0		−	1.00000i	$-0.5\pi$
$103$	0	0			1.00000i	$0.5\pi$
$104$	0	0
$105$	−3.00000	−0.292770
$106$	0	0
$107$	− 18.0000i	− 1.74013i	−0.492941	−	0.870063i	$-0.664078\pi$
$107$	− 18.0000i	− 1.74013i	0.492941	−	0.870063i	$-0.335922\pi$
$108$	0	0
$109$	6.00000i	0.574696i	0.957826	+	0.287348i	$0.0927736\pi$
$109$	6.00000i	0.574696i	−0.957826	+	0.287348i	$0.907226\pi$
$110$	0	0
$111$	3.00000	0.284747
$112$	0	0
$113$	−12.0000	−1.12887	−0.564433	−	0.825479i	$-0.690905\pi$
$113$	−12.0000	−1.12887	−0.564433	+	0.825479i	$0.690905\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	12.0000i	1.10940i
$118$	0	0
$119$	−9.00000	−0.825029
$120$	0	0
$121$	−1.00000	−0.0909091
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 1.00000i	− 0.0894427i
$126$	0	0
$127$	−12.0000	−1.06483	−0.532414	−	0.846484i	$-0.678715\pi$
$127$	−12.0000	−1.06483	−0.532414	+	0.846484i	$0.678715\pi$
$128$	0	0
$129$	10.0000	0.880451
$130$	0	0
$131$	− 9.00000i	− 0.786334i	−0.919467	−	0.393167i	$-0.871379\pi$
$131$	− 9.00000i	− 0.786334i	0.919467	−	0.393167i	$-0.128621\pi$
$132$	0	0
$133$	21.0000i	1.82093i
$134$	0	0
$135$	−5.00000	−0.430331
$136$	0	0
$137$	18.0000	1.53784	0.768922	−	0.639343i	$-0.220793\pi$
$137$	18.0000	1.53784	0.768922	+	0.639343i	$0.220793\pi$
$138$	0	0
$139$	4.00000i	0.339276i	0.985506	+	0.169638i	$0.0542598\pi$
$139$	4.00000i	0.339276i	−0.985506	+	0.169638i	$0.945740\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−6.00000	−0.501745
$144$	0	0
$145$	3.00000	0.249136
$146$	0	0
$147$	2.00000i	0.164957i
$148$	0	0
$149$	− 15.0000i	− 1.22885i	−0.788976	−	0.614424i	$-0.789388\pi$
$149$	− 15.0000i	− 1.22885i	0.788976	−	0.614424i	$-0.210612\pi$
$150$	0	0
$151$	0	0		−	1.00000i	$-0.5\pi$
$151$	0	0			1.00000i	$0.5\pi$
$152$	0	0
$153$	−6.00000	−0.485071
$154$	0	0
$155$	− 9.00000i	− 0.722897i
$156$	0	0
$157$	− 9.00000i	− 0.718278i	−0.933284	−	0.359139i	$-0.883070\pi$
$157$	− 9.00000i	− 0.718278i	0.933284	−	0.359139i	$-0.116930\pi$
$158$	0	0
$159$	−9.00000	−0.713746
$160$	0	0
$161$	0	0
$162$	0	0
$163$	− 11.0000i	− 0.861586i	−0.902451	−	0.430793i	$-0.858234\pi$
$163$	− 11.0000i	− 0.861586i	0.902451	−	0.430793i	$-0.141766\pi$
$164$	0	0
$165$	− 1.00000i	− 0.0778499i
$166$	0	0
$167$	3.00000	0.232147	0.116073	−	0.993241i	$-0.462969\pi$
$167$	3.00000	0.232147	0.116073	+	0.993241i	$0.462969\pi$
$168$	0	0
$169$	−23.0000	−1.76923
$170$	0	0
$171$	14.0000i	1.07061i
$172$	0	0
$173$	− 6.00000i	− 0.456172i	−0.973641	−	0.228086i	$-0.926753\pi$
$173$	− 6.00000i	− 0.456172i	0.973641	−	0.228086i	$-0.0732467\pi$
$174$	0	0
$175$	−3.00000	−0.226779
$176$	0	0
$177$	12.0000	0.901975
$178$	0	0
$179$	6.00000i	0.448461i	0.974536	+	0.224231i	$0.0719869\pi$
$179$	6.00000i	0.448461i	−0.974536	+	0.224231i	$0.928013\pi$
$180$	0	0
$181$	24.0000i	1.78391i	0.452128	+	0.891953i	$0.350665\pi$
$181$	24.0000i	1.78391i	−0.452128	+	0.891953i	$0.649335\pi$
$182$	0	0
$183$	−15.0000	−1.10883
$184$	0	0
$185$	3.00000	0.220564
$186$	0	0
$187$	− 3.00000i	− 0.219382i
$188$	0	0
$189$	15.0000i	1.09109i
$190$	0	0
$191$	−24.0000	−1.73658	−0.868290	−	0.496058i	$-0.834780\pi$
$191$	−24.0000	−1.73658	−0.868290	+	0.496058i	$0.834780\pi$
$192$	0	0
$193$	5.00000	0.359908	0.179954	−	0.983675i	$-0.442405\pi$
$193$	5.00000	0.359908	0.179954	+	0.983675i	$0.442405\pi$
$194$	0	0
$195$	− 6.00000i	− 0.429669i
$196$	0	0
$197$	24.0000i	1.70993i	0.518686	+	0.854965i	$0.326421\pi$
$197$	24.0000i	1.70993i	−0.518686	+	0.854965i	$0.673579\pi$
$198$	0	0
$199$	−9.00000	−0.637993	−0.318997	−	0.947756i	$-0.603346\pi$
$199$	−9.00000	−0.637993	−0.318997	+	0.947756i	$0.603346\pi$
$200$	0	0
$201$	4.00000	0.282138
$202$	0	0
$203$	− 9.00000i	− 0.631676i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−7.00000	−0.484200
$210$	0	0
$211$	23.0000i	1.58339i	0.610920	+	0.791693i	$0.290800\pi$
$211$	23.0000i	1.58339i	−0.610920	+	0.791693i	$0.709200\pi$
$212$	0	0
$213$	3.00000i	0.205557i
$214$	0	0
$215$	10.0000	0.681994
$216$	0	0
$217$	−27.0000	−1.83288
$218$	0	0
$219$	2.00000i	0.135147i
$220$	0	0
$221$	− 18.0000i	− 1.21081i
$222$	0	0
$223$	0	0		−	1.00000i	$-0.5\pi$
$223$	0	0			1.00000i	$0.5\pi$
$224$	0	0
$225$	−2.00000	−0.133333
$226$	0	0
$227$	− 6.00000i	− 0.398234i	−0.979976	−	0.199117i	$-0.936193\pi$
$227$	− 6.00000i	− 0.398234i	0.979976	−	0.199117i	$-0.0638074\pi$
$228$	0	0
$229$	− 6.00000i	− 0.396491i	−0.980152	−	0.198246i	$-0.936476\pi$
$229$	− 6.00000i	− 0.396491i	0.980152	−	0.198246i	$-0.0635244\pi$
$230$	0	0
$231$	−3.00000	−0.197386
$232$	0	0
$233$	−21.0000	−1.37576	−0.687878	−	0.725826i	$-0.741458\pi$
$233$	−21.0000	−1.37576	−0.687878	+	0.725826i	$0.741458\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−12.0000	−0.776215	−0.388108	−	0.921614i	$-0.626871\pi$
$239$	−12.0000	−0.776215	−0.388108	+	0.921614i	$0.626871\pi$
$240$	0	0
$241$	−10.0000	−0.644157	−0.322078	−	0.946713i	$-0.604381\pi$
$241$	−10.0000	−0.644157	−0.322078	+	0.946713i	$0.604381\pi$
$242$	0	0
$243$	16.0000i	1.02640i
$244$	0	0
$245$	2.00000i	0.127775i
$246$	0	0
$247$	−42.0000	−2.67240
$248$	0	0
$249$	−6.00000	−0.380235
$250$	0	0
$251$	0	0	1.00000			$0$
$251$	0	0	−1.00000			$\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	3.00000	0.187867
$256$	0	0
$257$	24.0000	1.49708	0.748539	−	0.663090i	$-0.230755\pi$
$257$	24.0000	1.49708	0.748539	+	0.663090i	$0.230755\pi$
$258$	0	0
$259$	− 9.00000i	− 0.559233i
$260$	0	0
$261$	− 6.00000i	− 0.371391i
$262$	0	0
$263$	21.0000	1.29492	0.647458	−	0.762101i	$-0.275832\pi$
$263$	21.0000	1.29492	0.647458	+	0.762101i	$0.275832\pi$
$264$	0	0
$265$	−9.00000	−0.552866
$266$	0	0
$267$	15.0000i	0.917985i
$268$	0	0
$269$	6.00000i	0.365826i	0.983129	+	0.182913i	$0.0585527\pi$
$269$	6.00000i	0.365826i	−0.983129	+	0.182913i	$0.941447\pi$
$270$	0	0
$271$	12.0000	0.728948	0.364474	−	0.931214i	$-0.381249\pi$
$271$	12.0000	0.728948	0.364474	+	0.931214i	$0.381249\pi$
$272$	0	0
$273$	−18.0000	−1.08941
$274$	0	0
$275$	− 1.00000i	− 0.0603023i
$276$	0	0
$277$	18.0000i	1.08152i	0.841178	+	0.540758i	$0.181862\pi$
$277$	18.0000i	1.08152i	−0.841178	+	0.540758i	$0.818138\pi$
$278$	0	0
$279$	−18.0000	−1.07763
$280$	0	0
$281$	18.0000	1.07379	0.536895	−	0.843649i	$-0.319597\pi$
$281$	18.0000	1.07379	0.536895	+	0.843649i	$0.319597\pi$
$282$	0	0
$283$	− 14.0000i	− 0.832214i	−0.909316	−	0.416107i	$-0.863394\pi$
$283$	− 14.0000i	− 0.832214i	0.909316	−	0.416107i	$-0.136606\pi$
$284$	0	0
$285$	− 7.00000i	− 0.414644i
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−8.00000	−0.470588
$290$	0	0
$291$	10.0000i	0.586210i
$292$	0	0
$293$	18.0000i	1.05157i	0.850617	+	0.525786i	$0.176229\pi$
$293$	18.0000i	1.05157i	−0.850617	+	0.525786i	$0.823771\pi$
$294$	0	0
$295$	12.0000	0.698667
$296$	0	0
$297$	−5.00000	−0.290129
$298$	0	0
$299$	0	0
$300$	0	0
$301$	− 30.0000i	− 1.72917i
$302$	0	0
$303$	−6.00000	−0.344691
$304$	0	0
$305$	−15.0000	−0.858898
$306$	0	0
$307$	2.00000i	0.114146i	0.998370	+	0.0570730i	$0.0181768\pi$
$307$	2.00000i	0.114146i	−0.998370	+	0.0570730i	$0.981823\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	33.0000	1.87126	0.935629	−	0.352985i	$-0.114833\pi$
$311$	33.0000	1.87126	0.935629	+	0.352985i	$0.114833\pi$
$312$	0	0
$313$	10.0000	0.565233	0.282617	−	0.959233i	$-0.408798\pi$
$313$	10.0000	0.565233	0.282617	+	0.959233i	$0.408798\pi$
$314$	0	0
$315$	6.00000i	0.338062i
$316$	0	0
$317$	− 3.00000i	− 0.168497i	−0.996445	−	0.0842484i	$-0.973151\pi$
$317$	− 3.00000i	− 0.168497i	0.996445	−	0.0842484i	$-0.0268489\pi$
$318$	0	0
$319$	3.00000	0.167968
$320$	0	0
$321$	18.0000	1.00466
$322$	0	0
$323$	− 21.0000i	− 1.16847i
$324$	0	0
$325$	− 6.00000i	− 0.332820i
$326$	0	0
$327$	−6.00000	−0.331801
$328$	0	0
$329$	0	0
$330$	0	0
$331$	10.0000i	0.549650i	0.961494	+	0.274825i	$0.0886199\pi$
$331$	10.0000i	0.549650i	−0.961494	+	0.274825i	$0.911380\pi$
$332$	0	0
$333$	− 6.00000i	− 0.328798i
$334$	0	0
$335$	4.00000	0.218543
$336$	0	0
$337$	31.0000	1.68868	0.844339	−	0.535810i	$-0.179994\pi$
$337$	31.0000	1.68868	0.844339	+	0.535810i	$0.179994\pi$
$338$	0	0
$339$	− 12.0000i	− 0.651751i
$340$	0	0
$341$	− 9.00000i	− 0.487377i
$342$	0	0
$343$	−15.0000	−0.809924
$344$	0	0
$345$	0	0
$346$	0	0
$347$	12.0000i	0.644194i	0.946707	+	0.322097i	$0.104388\pi$
$347$	12.0000i	0.644194i	−0.946707	+	0.322097i	$0.895612\pi$
$348$	0	0
$349$	6.00000i	0.321173i	0.987022	+	0.160586i	$0.0513385\pi$
$349$	6.00000i	0.321173i	−0.987022	+	0.160586i	$0.948662\pi$
$350$	0	0
$351$	−30.0000	−1.60128
$352$	0	0
$353$	6.00000	0.319348	0.159674	−	0.987170i	$-0.448956\pi$
$353$	6.00000	0.319348	0.159674	+	0.987170i	$0.448956\pi$
$354$	0	0
$355$	3.00000i	0.159223i
$356$	0	0
$357$	− 9.00000i	− 0.476331i
$358$	0	0
$359$	30.0000	1.58334	0.791670	−	0.610949i	$-0.209212\pi$
$359$	30.0000	1.58334	0.791670	+	0.610949i	$0.209212\pi$
$360$	0	0
$361$	−30.0000	−1.57895
$362$	0	0
$363$	− 1.00000i	− 0.0524864i
$364$	0	0
$365$	2.00000i	0.104685i
$366$	0	0
$367$	6.00000	0.313197	0.156599	−	0.987662i	$-0.449947\pi$
$367$	6.00000	0.313197	0.156599	+	0.987662i	$0.449947\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	27.0000i	1.40177i
$372$	0	0
$373$	6.00000i	0.310668i	0.987862	+	0.155334i	$0.0496454\pi$
$373$	6.00000i	0.310668i	−0.987862	+	0.155334i	$0.950355\pi$
$374$	0	0
$375$	1.00000	0.0516398
$376$	0	0
$377$	18.0000	0.927047
$378$	0	0
$379$	2.00000i	0.102733i	0.998680	+	0.0513665i	$0.0163577\pi$
$379$	2.00000i	0.102733i	−0.998680	+	0.0513665i	$0.983642\pi$
$380$	0	0
$381$	− 12.0000i	− 0.614779i
$382$	0	0
$383$	−6.00000	−0.306586	−0.153293	−	0.988181i	$-0.548988\pi$
$383$	−6.00000	−0.306586	−0.153293	+	0.988181i	$0.548988\pi$
$384$	0	0
$385$	−3.00000	−0.152894
$386$	0	0
$387$	− 20.0000i	− 1.01666i
$388$	0	0
$389$	30.0000i	1.52106i	0.649303	+	0.760530i	$0.275061\pi$
$389$	30.0000i	1.52106i	−0.649303	+	0.760530i	$0.724939\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	9.00000	0.453990
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 30.0000i	− 1.50566i	−0.658217	−	0.752828i	$-0.728689\pi$
$397$	− 30.0000i	− 1.50566i	0.658217	−	0.752828i	$-0.271311\pi$
$398$	0	0
$399$	−21.0000	−1.05131
$400$	0	0
$401$	15.0000	0.749064	0.374532	−	0.927214i	$-0.377803\pi$
$401$	15.0000	0.749064	0.374532	+	0.927214i	$0.377803\pi$
$402$	0	0
$403$	− 54.0000i	− 2.68993i
$404$	0	0
$405$	1.00000i	0.0496904i
$406$	0	0
$407$	3.00000	0.148704
$408$	0	0
$409$	32.0000	1.58230	0.791149	−	0.611623i	$-0.209483\pi$
$409$	32.0000	1.58230	0.791149	+	0.611623i	$0.209483\pi$
$410$	0	0
$411$	18.0000i	0.887875i
$412$	0	0
$413$	− 36.0000i	− 1.77144i
$414$	0	0
$415$	−6.00000	−0.294528
$416$	0	0
$417$	−4.00000	−0.195881
$418$	0	0
$419$	30.0000i	1.46560i	0.680446	+	0.732798i	$0.261786\pi$
$419$	30.0000i	1.46560i	−0.680446	+	0.732798i	$0.738214\pi$
$420$	0	0
$421$	12.0000i	0.584844i	0.956289	+	0.292422i	$0.0944612\pi$
$421$	12.0000i	0.584844i	−0.956289	+	0.292422i	$0.905539\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	3.00000	0.145521
$426$	0	0
$427$	45.0000i	2.17770i
$428$	0	0
$429$	− 6.00000i	− 0.289683i
$430$	0	0
$431$	18.0000	0.867029	0.433515	−	0.901146i	$-0.357273\pi$
$431$	18.0000	0.867029	0.433515	+	0.901146i	$0.357273\pi$
$432$	0	0
$433$	−38.0000	−1.82616	−0.913082	−	0.407777i	$-0.866304\pi$
$433$	−38.0000	−1.82616	−0.913082	+	0.407777i	$0.866304\pi$
$434$	0	0
$435$	3.00000i	0.143839i
$436$	0	0
$437$	0	0
$438$	0	0
$439$	30.0000	1.43182	0.715911	−	0.698192i	$-0.246012\pi$
$439$	30.0000	1.43182	0.715911	+	0.698192i	$0.246012\pi$
$440$	0	0
$441$	4.00000	0.190476
$442$	0	0
$443$	− 12.0000i	− 0.570137i	−0.958507	−	0.285069i	$-0.907984\pi$
$443$	− 12.0000i	− 0.570137i	0.958507	−	0.285069i	$-0.0920164\pi$
$444$	0	0
$445$	15.0000i	0.711068i
$446$	0	0
$447$	15.0000	0.709476
$448$	0	0
$449$	18.0000	0.849473	0.424736	−	0.905317i	$-0.360367\pi$
$449$	18.0000	0.849473	0.424736	+	0.905317i	$0.360367\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−18.0000	−0.843853
$456$	0	0
$457$	17.0000	0.795226	0.397613	−	0.917553i	$-0.369839\pi$
$457$	17.0000	0.795226	0.397613	+	0.917553i	$0.369839\pi$
$458$	0	0
$459$	− 15.0000i	− 0.700140i
$460$	0	0
$461$	− 33.0000i	− 1.53696i	−0.639872	−	0.768482i	$-0.721013\pi$
$461$	− 33.0000i	− 1.53696i	0.639872	−	0.768482i	$-0.278987\pi$
$462$	0	0
$463$	−30.0000	−1.39422	−0.697109	−	0.716965i	$-0.745531\pi$
$463$	−30.0000	−1.39422	−0.697109	+	0.716965i	$0.745531\pi$
$464$	0	0
$465$	9.00000	0.417365
$466$	0	0
$467$	21.0000i	0.971764i	0.874024	+	0.485882i	$0.161502\pi$
$467$	21.0000i	0.971764i	−0.874024	+	0.485882i	$0.838498\pi$
$468$	0	0
$469$	− 12.0000i	− 0.554109i
$470$	0	0
$471$	9.00000	0.414698
$472$	0	0
$473$	10.0000	0.459800
$474$	0	0
$475$	− 7.00000i	− 0.321182i
$476$	0	0
$477$	18.0000i	0.824163i
$478$	0	0
$479$	24.0000	1.09659	0.548294	−	0.836286i	$-0.315277\pi$
$479$	24.0000	1.09659	0.548294	+	0.836286i	$0.315277\pi$
$480$	0	0
$481$	18.0000	0.820729
$482$	0	0
$483$	0	0
$484$	0	0
$485$	10.0000i	0.454077i
$486$	0	0
$487$	−6.00000	−0.271886	−0.135943	−	0.990717i	$-0.543406\pi$
$487$	−6.00000	−0.271886	−0.135943	+	0.990717i	$0.543406\pi$
$488$	0	0
$489$	11.0000	0.497437
$490$	0	0
$491$	21.0000i	0.947717i	0.880601	+	0.473858i	$0.157139\pi$
$491$	21.0000i	0.947717i	−0.880601	+	0.473858i	$0.842861\pi$
$492$	0	0
$493$	9.00000i	0.405340i
$494$	0	0
$495$	−2.00000	−0.0898933
$496$	0	0
$497$	9.00000	0.403705
$498$	0	0
$499$	− 40.0000i	− 1.79065i	−0.445418	−	0.895323i	$-0.646945\pi$
$499$	− 40.0000i	− 1.79065i	0.445418	−	0.895323i	$-0.353055\pi$
$500$	0	0
$501$	3.00000i	0.134030i
$502$	0	0
$503$	24.0000	1.07011	0.535054	−	0.844818i	$-0.320291\pi$
$503$	24.0000	1.07011	0.535054	+	0.844818i	$0.320291\pi$
$504$	0	0
$505$	−6.00000	−0.266996
$506$	0	0
$507$	− 23.0000i	− 1.02147i
$508$	0	0
$509$	24.0000i	1.06378i	0.846813	+	0.531891i	$0.178518\pi$
$509$	24.0000i	1.06378i	−0.846813	+	0.531891i	$0.821482\pi$
$510$	0	0
$511$	6.00000	0.265424
$512$	0	0
$513$	−35.0000	−1.54529
$514$	0	0
$515$	0	0
$516$	0	0
$517$	0	0
$518$	0	0
$519$	6.00000	0.263371
$520$	0	0
$521$	6.00000	0.262865	0.131432	−	0.991325i	$-0.458042\pi$
$521$	6.00000	0.262865	0.131432	+	0.991325i	$0.458042\pi$
$522$	0	0
$523$	− 16.0000i	− 0.699631i	−0.936819	−	0.349816i	$-0.886244\pi$
$523$	− 16.0000i	− 0.699631i	0.936819	−	0.349816i	$-0.113756\pi$
$524$	0	0
$525$	− 3.00000i	− 0.130931i
$526$	0	0
$527$	27.0000	1.17614
$528$	0	0
$529$	−23.0000	−1.00000
$530$	0	0
$531$	− 24.0000i	− 1.04151i
$532$	0	0
$533$	0	0
$534$	0	0
$535$	18.0000	0.778208
$536$	0	0
$537$	−6.00000	−0.258919
$538$	0	0
$539$	2.00000i	0.0861461i
$540$	0	0
$541$	45.0000i	1.93470i	0.253442	+	0.967351i	$0.418437\pi$
$541$	45.0000i	1.93470i	−0.253442	+	0.967351i	$0.581563\pi$
$542$	0	0
$543$	−24.0000	−1.02994
$544$	0	0
$545$	−6.00000	−0.257012
$546$	0	0
$547$	10.0000i	0.427569i	0.976881	+	0.213785i	$0.0685791\pi$
$547$	10.0000i	0.427569i	−0.976881	+	0.213785i	$0.931421\pi$
$548$	0	0
$549$	30.0000i	1.28037i
$550$	0	0
$551$	21.0000	0.894630
$552$	0	0
$553$	0	0
$554$	0	0
$555$	3.00000i	0.127343i
$556$	0	0
$557$	− 12.0000i	− 0.508456i	−0.967144	−	0.254228i	$-0.918179\pi$
$557$	− 12.0000i	− 0.508456i	0.967144	−	0.254228i	$-0.0818214\pi$
$558$	0	0
$559$	60.0000	2.53773
$560$	0	0
$561$	3.00000	0.126660
$562$	0	0
$563$	− 24.0000i	− 1.01148i	−0.862686	−	0.505740i	$-0.831220\pi$
$563$	− 24.0000i	− 1.01148i	0.862686	−	0.505740i	$-0.168780\pi$
$564$	0	0
$565$	− 12.0000i	− 0.504844i
$566$	0	0
$567$	3.00000	0.125988
$568$	0	0
$569$	30.0000	1.25767	0.628833	−	0.777541i	$-0.283533\pi$
$569$	30.0000	1.25767	0.628833	+	0.777541i	$0.283533\pi$
$570$	0	0
$571$	− 5.00000i	− 0.209243i	−0.994512	−	0.104622i	$-0.966637\pi$
$571$	− 5.00000i	− 0.209243i	0.994512	−	0.104622i	$-0.0333632\pi$
$572$	0	0
$573$	− 24.0000i	− 1.00261i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−38.0000	−1.58196	−0.790980	−	0.611842i	$-0.790429\pi$
$577$	−38.0000	−1.58196	−0.790980	+	0.611842i	$0.790429\pi$
$578$	0	0
$579$	5.00000i	0.207793i
$580$	0	0
$581$	18.0000i	0.746766i
$582$	0	0
$583$	−9.00000	−0.372742
$584$	0	0
$585$	−12.0000	−0.496139
$586$	0	0
$587$	21.0000i	0.866763i	0.901211	+	0.433381i	$0.142680\pi$
$587$	21.0000i	0.866763i	−0.901211	+	0.433381i	$0.857320\pi$
$588$	0	0
$589$	− 63.0000i	− 2.59587i
$590$	0	0
$591$	−24.0000	−0.987228
$592$	0	0
$593$	−42.0000	−1.72473	−0.862367	−	0.506284i	$-0.831019\pi$
$593$	−42.0000	−1.72473	−0.862367	+	0.506284i	$0.831019\pi$
$594$	0	0
$595$	− 9.00000i	− 0.368964i
$596$	0	0
$597$	− 9.00000i	− 0.368345i
$598$	0	0
$599$	−27.0000	−1.10319	−0.551595	−	0.834112i	$-0.685981\pi$
$599$	−27.0000	−1.10319	−0.551595	+	0.834112i	$0.685981\pi$
$600$	0	0
$601$	44.0000	1.79480	0.897399	−	0.441221i	$-0.145454\pi$
$601$	44.0000	1.79480	0.897399	+	0.441221i	$0.145454\pi$
$602$	0	0
$603$	− 8.00000i	− 0.325785i
$604$	0	0
$605$	− 1.00000i	− 0.0406558i
$606$	0	0
$607$	15.0000	0.608831	0.304416	−	0.952539i	$-0.401539\pi$
$607$	15.0000	0.608831	0.304416	+	0.952539i	$0.401539\pi$
$608$	0	0
$609$	9.00000	0.364698
$610$	0	0
$611$	0	0
$612$	0	0
$613$	− 36.0000i	− 1.45403i	−0.686624	−	0.727013i	$-0.740908\pi$
$613$	− 36.0000i	− 1.45403i	0.686624	−	0.727013i	$-0.259092\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−12.0000	−0.483102	−0.241551	−	0.970388i	$-0.577656\pi$
$617$	−12.0000	−0.483102	−0.241551	+	0.970388i	$0.577656\pi$
$618$	0	0
$619$	− 8.00000i	− 0.321547i	−0.986991	−	0.160774i	$-0.948601\pi$
$619$	− 8.00000i	− 0.321547i	0.986991	−	0.160774i	$-0.0513989\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	45.0000	1.80289
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	− 7.00000i	− 0.279553i
$628$	0	0
$629$	9.00000i	0.358854i
$630$	0	0
$631$	−21.0000	−0.835997	−0.417998	−	0.908448i	$-0.637268\pi$
$631$	−21.0000	−0.835997	−0.417998	+	0.908448i	$0.637268\pi$
$632$	0	0
$633$	−23.0000	−0.914168
$634$	0	0
$635$	− 12.0000i	− 0.476205i
$636$	0	0
$637$	12.0000i	0.475457i
$638$	0	0
$639$	6.00000	0.237356
$640$	0	0
$641$	−21.0000	−0.829450	−0.414725	−	0.909947i	$-0.636122\pi$
$641$	−21.0000	−0.829450	−0.414725	+	0.909947i	$0.636122\pi$
$642$	0	0
$643$	− 23.0000i	− 0.907031i	−0.891248	−	0.453516i	$-0.850170\pi$
$643$	− 23.0000i	− 0.907031i	0.891248	−	0.453516i	$-0.149830\pi$
$644$	0	0
$645$	10.0000i	0.393750i
$646$	0	0
$647$	30.0000	1.17942	0.589711	−	0.807614i	$-0.299242\pi$
$647$	30.0000	1.17942	0.589711	+	0.807614i	$0.299242\pi$
$648$	0	0
$649$	12.0000	0.471041
$650$	0	0
$651$	− 27.0000i	− 1.05821i
$652$	0	0
$653$	− 39.0000i	− 1.52619i	−0.646288	−	0.763094i	$-0.723679\pi$
$653$	− 39.0000i	− 1.52619i	0.646288	−	0.763094i	$-0.276321\pi$
$654$	0	0
$655$	9.00000	0.351659
$656$	0	0
$657$	4.00000	0.156055
$658$	0	0
$659$	21.0000i	0.818044i	0.912525	+	0.409022i	$0.134130\pi$
$659$	21.0000i	0.818044i	−0.912525	+	0.409022i	$0.865870\pi$
$660$	0	0
$661$	36.0000i	1.40024i	0.714026	+	0.700119i	$0.246870\pi$
$661$	36.0000i	1.40024i	−0.714026	+	0.700119i	$0.753130\pi$
$662$	0	0
$663$	18.0000	0.699062
$664$	0	0
$665$	−21.0000	−0.814345
$666$	0	0
$667$	0	0
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−15.0000	−0.579069
$672$	0	0
$673$	1.00000	0.0385472	0.0192736	−	0.999814i	$-0.493865\pi$
$673$	1.00000	0.0385472	0.0192736	+	0.999814i	$0.493865\pi$
$674$	0	0
$675$	− 5.00000i	− 0.192450i
$676$	0	0
$677$	− 42.0000i	− 1.61419i	−0.590421	−	0.807096i	$-0.701038\pi$
$677$	− 42.0000i	− 1.61419i	0.590421	−	0.807096i	$-0.298962\pi$
$678$	0	0
$679$	30.0000	1.15129
$680$	0	0
$681$	6.00000	0.229920
$682$	0	0
$683$	9.00000i	0.344375i	0.985064	+	0.172188i	$0.0550836\pi$
$683$	9.00000i	0.344375i	−0.985064	+	0.172188i	$0.944916\pi$
$684$	0	0
$685$	18.0000i	0.687745i
$686$	0	0
$687$	6.00000	0.228914
$688$	0	0
$689$	−54.0000	−2.05724
$690$	0	0
$691$	46.0000i	1.74992i	0.484193	+	0.874961i	$0.339113\pi$
$691$	46.0000i	1.74992i	−0.484193	+	0.874961i	$0.660887\pi$
$692$	0	0
$693$	6.00000i	0.227921i
$694$	0	0
$695$	−4.00000	−0.151729
$696$	0	0
$697$	0	0
$698$	0	0
$699$	− 21.0000i	− 0.794293i
$700$	0	0
$701$	− 33.0000i	− 1.24639i	−0.782065	−	0.623196i	$-0.785834\pi$
$701$	− 33.0000i	− 1.24639i	0.782065	−	0.623196i	$-0.214166\pi$
$702$	0	0
$703$	21.0000	0.792030
$704$	0	0
$705$	0	0
$706$	0	0
$707$	18.0000i	0.676960i
$708$	0	0
$709$	18.0000i	0.676004i	0.941145	+	0.338002i	$0.109751\pi$
$709$	18.0000i	0.676004i	−0.941145	+	0.338002i	$0.890249\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	0	0
$714$	0	0
$715$	− 6.00000i	− 0.224387i
$716$	0	0
$717$	− 12.0000i	− 0.448148i
$718$	0	0
$719$	15.0000	0.559406	0.279703	−	0.960087i	$-0.409764\pi$
$719$	15.0000	0.559406	0.279703	+	0.960087i	$0.409764\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	− 10.0000i	− 0.371904i
$724$	0	0
$725$	3.00000i	0.111417i
$726$	0	0
$727$	−42.0000	−1.55769	−0.778847	−	0.627214i	$-0.784195\pi$
$727$	−42.0000	−1.55769	−0.778847	+	0.627214i	$0.784195\pi$
$728$	0	0
$729$	−13.0000	−0.481481
$730$	0	0
$731$	30.0000i	1.10959i
$732$	0	0
$733$	18.0000i	0.664845i	0.943131	+	0.332423i	$0.107866\pi$
$733$	18.0000i	0.664845i	−0.943131	+	0.332423i	$0.892134\pi$
$734$	0	0
$735$	−2.00000	−0.0737711
$736$	0	0
$737$	4.00000	0.147342
$738$	0	0
$739$	20.0000i	0.735712i	0.929883	+	0.367856i	$0.119908\pi$
$739$	20.0000i	0.735712i	−0.929883	+	0.367856i	$0.880092\pi$
$740$	0	0
$741$	− 42.0000i	− 1.54291i
$742$	0	0
$743$	−45.0000	−1.65089	−0.825445	−	0.564483i	$-0.809076\pi$
$743$	−45.0000	−1.65089	−0.825445	+	0.564483i	$0.809076\pi$
$744$	0	0
$745$	15.0000	0.549557
$746$	0	0
$747$	12.0000i	0.439057i
$748$	0	0
$749$	− 54.0000i	− 1.97312i
$750$	0	0
$751$	27.0000	0.985244	0.492622	−	0.870243i	$-0.336039\pi$
$751$	27.0000	0.985244	0.492622	+	0.870243i	$0.336039\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	42.0000i	1.52652i	0.646094	+	0.763258i	$0.276401\pi$
$757$	42.0000i	1.52652i	−0.646094	+	0.763258i	$0.723599\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	12.0000	0.435000	0.217500	−	0.976060i	$-0.430210\pi$
$761$	12.0000	0.435000	0.217500	+	0.976060i	$0.430210\pi$
$762$	0	0
$763$	18.0000i	0.651644i
$764$	0	0
$765$	− 6.00000i	− 0.216930i
$766$	0	0
$767$	72.0000	2.59977
$768$	0	0
$769$	14.0000	0.504853	0.252426	−	0.967616i	$-0.418771\pi$
$769$	14.0000	0.504853	0.252426	+	0.967616i	$0.418771\pi$
$770$	0	0
$771$	24.0000i	0.864339i
$772$	0	0
$773$	− 51.0000i	− 1.83434i	−0.398493	−	0.917171i	$-0.630467\pi$
$773$	− 51.0000i	− 1.83434i	0.398493	−	0.917171i	$-0.369533\pi$
$774$	0	0
$775$	9.00000	0.323290
$776$	0	0
$777$	9.00000	0.322873
$778$	0	0
$779$	0	0
$780$	0	0
$781$	3.00000i	0.107348i
$782$	0	0
$783$	15.0000	0.536056
$784$	0	0
$785$	9.00000	0.321224
$786$	0	0
$787$	22.0000i	0.784215i	0.919919	+	0.392108i	$0.128254\pi$
$787$	22.0000i	0.784215i	−0.919919	+	0.392108i	$0.871746\pi$
$788$	0	0
$789$	21.0000i	0.747620i
$790$	0	0
$791$	−36.0000	−1.28001
$792$	0	0
$793$	−90.0000	−3.19599
$794$	0	0
$795$	− 9.00000i	− 0.319197i
$796$	0	0
$797$	− 30.0000i	− 1.06265i	−0.847167	−	0.531327i	$-0.821693\pi$
$797$	− 30.0000i	− 1.06265i	0.847167	−	0.531327i	$-0.178307\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	30.0000	1.06000
$802$	0	0
$803$	2.00000i	0.0705785i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−6.00000	−0.211210
$808$	0	0
$809$	12.0000	0.421898	0.210949	−	0.977497i	$-0.432345\pi$
$809$	12.0000	0.421898	0.210949	+	0.977497i	$0.432345\pi$
$810$	0	0
$811$	11.0000i	0.386262i	0.981173	+	0.193131i	$0.0618643\pi$
$811$	11.0000i	0.386262i	−0.981173	+	0.193131i	$0.938136\pi$
$812$	0	0
$813$	12.0000i	0.420858i
$814$	0	0
$815$	11.0000	0.385313
$816$	0	0
$817$	70.0000	2.44899
$818$	0	0
$819$	36.0000i	1.25794i
$820$	0	0
$821$	6.00000i	0.209401i	0.994504	+	0.104701i	$0.0333885\pi$
$821$	6.00000i	0.209401i	−0.994504	+	0.104701i	$0.966612\pi$
$822$	0	0
$823$	0	0		−	1.00000i	$-0.5\pi$
$823$	0	0			1.00000i	$0.5\pi$
$824$	0	0
$825$	1.00000	0.0348155
$826$	0	0
$827$	− 30.0000i	− 1.04320i	−0.853189	−	0.521601i	$-0.825335\pi$
$827$	− 30.0000i	− 1.04320i	0.853189	−	0.521601i	$-0.174665\pi$
$828$	0	0
$829$	− 42.0000i	− 1.45872i	−0.684130	−	0.729360i	$-0.739818\pi$
$829$	− 42.0000i	− 1.45872i	0.684130	−	0.729360i	$-0.260182\pi$
$830$	0	0
$831$	−18.0000	−0.624413
$832$	0	0
$833$	−6.00000	−0.207888
$834$	0	0
$835$	3.00000i	0.103819i
$836$	0	0
$837$	− 45.0000i	− 1.55543i
$838$	0	0
$839$	12.0000	0.414286	0.207143	−	0.978311i	$-0.433583\pi$
$839$	12.0000	0.414286	0.207143	+	0.978311i	$0.433583\pi$
$840$	0	0
$841$	20.0000	0.689655
$842$	0	0
$843$	18.0000i	0.619953i
$844$	0	0
$845$	− 23.0000i	− 0.791224i
$846$	0	0
$847$	−3.00000	−0.103081
$848$	0	0
$849$	14.0000	0.480479
$850$	0	0
$851$	0	0
$852$	0	0
$853$	48.0000i	1.64349i	0.569856	+	0.821744i	$0.306999\pi$
$853$	48.0000i	1.64349i	−0.569856	+	0.821744i	$0.693001\pi$
$854$	0	0
$855$	−14.0000	−0.478790
$856$	0	0
$857$	3.00000	0.102478	0.0512390	−	0.998686i	$-0.483683\pi$
$857$	3.00000	0.102478	0.0512390	+	0.998686i	$0.483683\pi$
$858$	0	0
$859$	14.0000i	0.477674i	0.971060	+	0.238837i	$0.0767661\pi$
$859$	14.0000i	0.477674i	−0.971060	+	0.238837i	$0.923234\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	42.0000	1.42970	0.714848	−	0.699280i	$-0.246496\pi$
$863$	42.0000	1.42970	0.714848	+	0.699280i	$0.246496\pi$
$864$	0	0
$865$	6.00000	0.204006
$866$	0	0
$867$	− 8.00000i	− 0.271694i
$868$	0	0
$869$	0	0
$870$	0	0
$871$	24.0000	0.813209
$872$	0	0
$873$	20.0000	0.676897
$874$	0	0
$875$	− 3.00000i	− 0.101419i
$876$	0	0
$877$	− 6.00000i	− 0.202606i	−0.994856	−	0.101303i	$-0.967699\pi$
$877$	− 6.00000i	− 0.202606i	0.994856	−	0.101303i	$-0.0323011\pi$
$878$	0	0
$879$	−18.0000	−0.607125
$880$	0	0
$881$	18.0000	0.606435	0.303218	−	0.952921i	$-0.401939\pi$
$881$	18.0000	0.606435	0.303218	+	0.952921i	$0.401939\pi$
$882$	0	0
$883$	− 29.0000i	− 0.975928i	−0.872864	−	0.487964i	$-0.837740\pi$
$883$	− 29.0000i	− 0.975928i	0.872864	−	0.487964i	$-0.162260\pi$
$884$	0	0
$885$	12.0000i	0.403376i
$886$	0	0
$887$	12.0000	0.402921	0.201460	−	0.979497i	$-0.435431\pi$
$887$	12.0000	0.402921	0.201460	+	0.979497i	$0.435431\pi$
$888$	0	0
$889$	−36.0000	−1.20740
$890$	0	0
$891$	1.00000i	0.0335013i
$892$	0	0
$893$	0	0
$894$	0	0
$895$	−6.00000	−0.200558
$896$	0	0
$897$	0	0
$898$	0	0
$899$	27.0000i	0.900500i
$900$	0	0
$901$	− 27.0000i	− 0.899500i
$902$	0	0
$903$	30.0000	0.998337
$904$	0	0
$905$	−24.0000	−0.797787
$906$	0	0
$907$	− 37.0000i	− 1.22856i	−0.789086	−	0.614282i	$-0.789446\pi$
$907$	− 37.0000i	− 1.22856i	0.789086	−	0.614282i	$-0.210554\pi$
$908$	0	0
$909$	12.0000i	0.398015i
$910$	0	0
$911$	9.00000	0.298183	0.149092	−	0.988823i	$-0.452365\pi$
$911$	9.00000	0.298183	0.149092	+	0.988823i	$0.452365\pi$
$912$	0	0
$913$	−6.00000	−0.198571
$914$	0	0
$915$	− 15.0000i	− 0.495885i
$916$	0	0
$917$	− 27.0000i	− 0.891619i
$918$	0	0
$919$	−6.00000	−0.197922	−0.0989609	−	0.995091i	$-0.531552\pi$
$919$	−6.00000	−0.197922	−0.0989609	+	0.995091i	$0.531552\pi$
$920$	0	0
$921$	−2.00000	−0.0659022
$922$	0	0
$923$	18.0000i	0.592477i
$924$	0	0
$925$	3.00000i	0.0986394i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−45.0000	−1.47640	−0.738201	−	0.674581i	$-0.764324\pi$
$929$	−45.0000	−1.47640	−0.738201	+	0.674581i	$0.764324\pi$
$930$	0	0
$931$	14.0000i	0.458831i
$932$	0	0
$933$	33.0000i	1.08037i
$934$	0	0
$935$	3.00000	0.0981105
$936$	0	0
$937$	−2.00000	−0.0653372	−0.0326686	−	0.999466i	$-0.510401\pi$
$937$	−2.00000	−0.0653372	−0.0326686	+	0.999466i	$0.510401\pi$
$938$	0	0
$939$	10.0000i	0.326338i
$940$	0	0
$941$	− 27.0000i	− 0.880175i	−0.897955	−	0.440087i	$-0.854947\pi$
$941$	− 27.0000i	− 0.880175i	0.897955	−	0.440087i	$-0.145053\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	−15.0000	−0.487950
$946$	0	0
$947$	− 33.0000i	− 1.07236i	−0.844105	−	0.536178i	$-0.819868\pi$
$947$	− 33.0000i	− 1.07236i	0.844105	−	0.536178i	$-0.180132\pi$
$948$	0	0
$949$	12.0000i	0.389536i
$950$	0	0
$951$	3.00000	0.0972817
$952$	0	0
$953$	21.0000	0.680257	0.340128	−	0.940379i	$-0.389529\pi$
$953$	21.0000	0.680257	0.340128	+	0.940379i	$0.389529\pi$
$954$	0	0
$955$	− 24.0000i	− 0.776622i
$956$	0	0
$957$	3.00000i	0.0969762i
$958$	0	0
$959$	54.0000	1.74375
$960$	0	0
$961$	50.0000	1.61290
$962$	0	0
$963$	− 36.0000i	− 1.16008i
$964$	0	0
$965$	5.00000i	0.160956i
$966$	0	0
$967$	57.0000	1.83300	0.916498	−	0.400039i	$-0.131003\pi$
$967$	57.0000	1.83300	0.916498	+	0.400039i	$0.131003\pi$
$968$	0	0
$969$	21.0000	0.674617
$970$	0	0
$971$	− 12.0000i	− 0.385098i	−0.981287	−	0.192549i	$-0.938325\pi$
$971$	− 12.0000i	− 0.385098i	0.981287	−	0.192549i	$-0.0616755\pi$
$972$	0	0
$973$	12.0000i	0.384702i
$974$	0	0
$975$	6.00000	0.192154
$976$	0	0
$977$	18.0000	0.575871	0.287936	−	0.957650i	$-0.407031\pi$
$977$	18.0000	0.575871	0.287936	+	0.957650i	$0.407031\pi$
$978$	0	0
$979$	15.0000i	0.479402i
$980$	0	0
$981$	12.0000i	0.383131i
$982$	0	0
$983$	12.0000	0.382741	0.191370	−	0.981518i	$-0.438707\pi$
$983$	12.0000	0.382741	0.191370	+	0.981518i	$0.438707\pi$
$984$	0	0
$985$	−24.0000	−0.764704
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0	0
$990$	0	0
$991$	0	0		−	1.00000i	$-0.5\pi$
$991$	0	0			1.00000i	$0.5\pi$
$992$	0	0
$993$	−10.0000	−0.317340
$994$	0	0
$995$	− 9.00000i	− 0.285319i
$996$	0	0
$997$	6.00000i	0.190022i	0.995476	+	0.0950110i	$0.0302886\pi$
$997$	6.00000i	0.190022i	−0.995476	+	0.0950110i	$0.969711\pi$
$998$	0	0
$999$	15.0000	0.474579

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3520.2.g.f.1761.2	yes	2
4.3	odd	2		3520.2.g.b.1761.1	✓	2
8.3	odd	2		3520.2.g.b.1761.2	yes	2
8.5	even	2	inner	3520.2.g.f.1761.1	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3520.2.g.b.1761.1	✓	2	4.3	odd	2
3520.2.g.b.1761.2	yes	2	8.3	odd	2
3520.2.g.f.1761.1	yes	2	8.5	even	2	inner
3520.2.g.f.1761.2	yes	2	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:	\( N \)	\(=\)	\( 3520 = 2^{6} \cdot 5 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3520.g (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{3} +1.00000i q^{5} +3.00000 q^{7} +2.00000 q^{9} +O(q^{10})\)	\(q+1.00000i q^{3} +1.00000i q^{5} +3.00000 q^{7} +2.00000 q^{9} +1.00000i q^{11} +6.00000i q^{13} -1.00000 q^{15} -3.00000 q^{17} +7.00000i q^{19} +3.00000i q^{21} -1.00000 q^{25} +5.00000i q^{27} -3.00000i q^{29} -9.00000 q^{31} -1.00000 q^{33} +3.00000i q^{35} -3.00000i q^{37} -6.00000 q^{39} -10.0000i q^{43} +2.00000i q^{45} +2.00000 q^{49} -3.00000i q^{51} +9.00000i q^{53} -1.00000 q^{55} -7.00000 q^{57} -12.0000i q^{59} +15.0000i q^{61} +6.00000 q^{63} -6.00000 q^{65} -4.00000i q^{67} +3.00000 q^{71} +2.00000 q^{73} -1.00000i q^{75} +3.00000i q^{77} +1.00000 q^{81} +6.00000i q^{83} -3.00000i q^{85} +3.00000 q^{87} +15.0000 q^{89} +18.0000i q^{91} -9.00000i q^{93} -7.00000 q^{95} +10.0000 q^{97} +2.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 6 q^{7} + 4 q^{9}+O(q^{10}) \) 2 * q + 6 * q^7 + 4 * q^9	\( 2 q + 6 q^{7} + 4 q^{9} - 2 q^{15} - 6 q^{17} - 2 q^{25} - 18 q^{31} - 2 q^{33} - 12 q^{39} + 4 q^{49} - 2 q^{55} - 14 q^{57} + 12 q^{63} - 12 q^{65} + 6 q^{71} + 4 q^{73} + 2 q^{81} + 6 q^{87} + 30 q^{89} - 14 q^{95} + 20 q^{97}+O(q^{100}) \) 2 * q + 6 * q^7 + 4 * q^9 - 2 * q^15 - 6 * q^17 - 2 * q^25 - 18 * q^31 - 2 * q^33 - 12 * q^39 + 4 * q^49 - 2 * q^55 - 14 * q^57 + 12 * q^63 - 12 * q^65 + 6 * q^71 + 4 * q^73 + 2 * q^81 + 6 * q^87 + 30 * q^89 - 14 * q^95 + 20 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more