LMFDB - Embedded newform 3450.2.a.w.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3450,2,Mod(1,3450)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3450, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("3450.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3450 = 2 \cdot 3 \cdot 5^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3450.a (trivial)

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:	yes
Analytic conductor:	$27.5483886973$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	3450.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} -1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$

\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} -1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} -5.00000 q^{11} +1.00000 q^{12} -3.00000 q^{13} -1.00000 q^{14} +1.00000 q^{16} -2.00000 q^{17} +1.00000 q^{18} -3.00000 q^{19} -1.00000 q^{21} -5.00000 q^{22} -1.00000 q^{23} +1.00000 q^{24} -3.00000 q^{26} +1.00000 q^{27} -1.00000 q^{28} -1.00000 q^{29} +1.00000 q^{32} -5.00000 q^{33} -2.00000 q^{34} +1.00000 q^{36} -4.00000 q^{37} -3.00000 q^{38} -3.00000 q^{39} -3.00000 q^{41} -1.00000 q^{42} -5.00000 q^{43} -5.00000 q^{44} -1.00000 q^{46} +6.00000 q^{47} +1.00000 q^{48} -6.00000 q^{49} -2.00000 q^{51} -3.00000 q^{52} +1.00000 q^{54} -1.00000 q^{56} -3.00000 q^{57} -1.00000 q^{58} -6.00000 q^{59} +4.00000 q^{61} -1.00000 q^{63} +1.00000 q^{64} -5.00000 q^{66} +4.00000 q^{67} -2.00000 q^{68} -1.00000 q^{69} -2.00000 q^{71} +1.00000 q^{72} -9.00000 q^{73} -4.00000 q^{74} -3.00000 q^{76} +5.00000 q^{77} -3.00000 q^{78} -7.00000 q^{79} +1.00000 q^{81} -3.00000 q^{82} +1.00000 q^{83} -1.00000 q^{84} -5.00000 q^{86} -1.00000 q^{87} -5.00000 q^{88} -2.00000 q^{89} +3.00000 q^{91} -1.00000 q^{92} +6.00000 q^{94} +1.00000 q^{96} -6.00000 q^{98} -5.00000 q^{99} +O(q^{100})\)

Display 100 coefficients 10 coefficients

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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	1.00000	0.408248
$7$	−1.00000	−0.377964	−0.188982	−	0.981981i	$-0.560519\pi$
$7$	−1.00000	−0.377964	−0.188982	+	0.981981i	$0.560519\pi$
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	0	0
$11$	−5.00000	−1.50756	−0.753778	−	0.657129i	$-0.771771\pi$
$11$	−5.00000	−1.50756	−0.753778	+	0.657129i	$0.771771\pi$
$12$	1.00000	0.288675
$13$	−3.00000	−0.832050	−0.416025	−	0.909353i	$-0.636577\pi$
$13$	−3.00000	−0.832050	−0.416025	+	0.909353i	$0.636577\pi$
$14$	−1.00000	−0.267261
$15$	0	0
$16$	1.00000	0.250000
$17$	−2.00000	−0.485071	−0.242536	−	0.970143i	$-0.577979\pi$
$17$	−2.00000	−0.485071	−0.242536	+	0.970143i	$0.577979\pi$
$18$	1.00000	0.235702
$19$	−3.00000	−0.688247	−0.344124	−	0.938924i	$-0.611824\pi$
$19$	−3.00000	−0.688247	−0.344124	+	0.938924i	$0.611824\pi$
$20$	0	0
$21$	−1.00000	−0.218218
$22$	−5.00000	−1.06600
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	1.00000	0.204124
$25$	0	0
$26$	−3.00000	−0.588348
$27$	1.00000	0.192450
$28$	−1.00000	−0.188982
$29$	−1.00000	−0.185695	−0.0928477	−	0.995680i	$-0.529597\pi$
$29$	−1.00000	−0.185695	−0.0928477	+	0.995680i	$0.529597\pi$
$30$	0	0
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	1.00000	0.176777
$33$	−5.00000	−0.870388
$34$	−2.00000	−0.342997
$35$	0	0
$36$	1.00000	0.166667
$37$	−4.00000	−0.657596	−0.328798	−	0.944400i	$-0.606644\pi$
$37$	−4.00000	−0.657596	−0.328798	+	0.944400i	$0.606644\pi$
$38$	−3.00000	−0.486664
$39$	−3.00000	−0.480384
$40$	0	0
$41$	−3.00000	−0.468521	−0.234261	−	0.972174i	$-0.575267\pi$
$41$	−3.00000	−0.468521	−0.234261	+	0.972174i	$0.575267\pi$
$42$	−1.00000	−0.154303
$43$	−5.00000	−0.762493	−0.381246	−	0.924473i	$-0.624505\pi$
$43$	−5.00000	−0.762493	−0.381246	+	0.924473i	$0.624505\pi$
$44$	−5.00000	−0.753778
$45$	0	0
$46$	−1.00000	−0.147442
$47$	6.00000	0.875190	0.437595	−	0.899172i	$-0.355830\pi$
$47$	6.00000	0.875190	0.437595	+	0.899172i	$0.355830\pi$
$48$	1.00000	0.144338
$49$	−6.00000	−0.857143
$50$	0	0
$51$	−2.00000	−0.280056
$52$	−3.00000	−0.416025
$53$	0	0		−	1.00000i	$-0.5\pi$
$53$	0	0			1.00000i	$0.5\pi$
$54$	1.00000	0.136083
$55$	0	0
$56$	−1.00000	−0.133631
$57$	−3.00000	−0.397360
$58$	−1.00000	−0.131306
$59$	−6.00000	−0.781133	−0.390567	−	0.920575i	$-0.627721\pi$
$59$	−6.00000	−0.781133	−0.390567	+	0.920575i	$0.627721\pi$
$60$	0	0
$61$	4.00000	0.512148	0.256074	−	0.966657i	$-0.417571\pi$
$61$	4.00000	0.512148	0.256074	+	0.966657i	$0.417571\pi$
$62$	0	0
$63$	−1.00000	−0.125988
$64$	1.00000	0.125000
$65$	0	0
$66$	−5.00000	−0.615457
$67$	4.00000	0.488678	0.244339	−	0.969690i	$-0.421429\pi$
$67$	4.00000	0.488678	0.244339	+	0.969690i	$0.421429\pi$
$68$	−2.00000	−0.242536
$69$	−1.00000	−0.120386
$70$	0	0
$71$	−2.00000	−0.237356	−0.118678	−	0.992933i	$-0.537866\pi$
$71$	−2.00000	−0.237356	−0.118678	+	0.992933i	$0.537866\pi$
$72$	1.00000	0.117851
$73$	−9.00000	−1.05337	−0.526685	−	0.850060i	$-0.676565\pi$
$73$	−9.00000	−1.05337	−0.526685	+	0.850060i	$0.676565\pi$
$74$	−4.00000	−0.464991
$75$	0	0
$76$	−3.00000	−0.344124
$77$	5.00000	0.569803
$78$	−3.00000	−0.339683
$79$	−7.00000	−0.787562	−0.393781	−	0.919204i	$-0.628833\pi$
$79$	−7.00000	−0.787562	−0.393781	+	0.919204i	$0.628833\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	−3.00000	−0.331295
$83$	1.00000	0.109764	0.0548821	−	0.998493i	$-0.482522\pi$
$83$	1.00000	0.109764	0.0548821	+	0.998493i	$0.482522\pi$
$84$	−1.00000	−0.109109
$85$	0	0
$86$	−5.00000	−0.539164
$87$	−1.00000	−0.107211
$88$	−5.00000	−0.533002
$89$	−2.00000	−0.212000	−0.106000	−	0.994366i	$-0.533804\pi$
$89$	−2.00000	−0.212000	−0.106000	+	0.994366i	$0.533804\pi$
$90$	0	0
$91$	3.00000	0.314485
$92$	−1.00000	−0.104257
$93$	0	0
$94$	6.00000	0.618853
$95$	0	0
$96$	1.00000	0.102062
$97$	0	0		−	1.00000i	$-0.5\pi$
$97$	0	0			1.00000i	$0.5\pi$
$98$	−6.00000	−0.606092
$99$	−5.00000	−0.502519
$100$	0	0
$101$	6.00000	0.597022	0.298511	−	0.954406i	$-0.403510\pi$
$101$	6.00000	0.597022	0.298511	+	0.954406i	$0.403510\pi$
$102$	−2.00000	−0.198030
$103$	−7.00000	−0.689730	−0.344865	−	0.938652i	$-0.612075\pi$
$103$	−7.00000	−0.689730	−0.344865	+	0.938652i	$0.612075\pi$
$104$	−3.00000	−0.294174
$105$	0	0
$106$	0	0
$107$	12.0000	1.16008	0.580042	−	0.814587i	$-0.303036\pi$
$107$	12.0000	1.16008	0.580042	+	0.814587i	$0.303036\pi$
$108$	1.00000	0.0962250
$109$	−4.00000	−0.383131	−0.191565	−	0.981480i	$-0.561356\pi$
$109$	−4.00000	−0.383131	−0.191565	+	0.981480i	$0.561356\pi$
$110$	0	0
$111$	−4.00000	−0.379663
$112$	−1.00000	−0.0944911
$113$	6.00000	0.564433	0.282216	−	0.959351i	$-0.408930\pi$
$113$	6.00000	0.564433	0.282216	+	0.959351i	$0.408930\pi$
$114$	−3.00000	−0.280976
$115$	0	0
$116$	−1.00000	−0.0928477
$117$	−3.00000	−0.277350
$118$	−6.00000	−0.552345
$119$	2.00000	0.183340
$120$	0	0
$121$	14.0000	1.27273
$122$	4.00000	0.362143
$123$	−3.00000	−0.270501
$124$	0	0
$125$	0	0
$126$	−1.00000	−0.0890871
$127$	18.0000	1.59724	0.798621	−	0.601834i	$-0.205563\pi$
$127$	18.0000	1.59724	0.798621	+	0.601834i	$0.205563\pi$
$128$	1.00000	0.0883883
$129$	−5.00000	−0.440225
$130$	0	0
$131$	12.0000	1.04844	0.524222	−	0.851581i	$-0.324356\pi$
$131$	12.0000	1.04844	0.524222	+	0.851581i	$0.324356\pi$
$132$	−5.00000	−0.435194
$133$	3.00000	0.260133
$134$	4.00000	0.345547
$135$	0	0
$136$	−2.00000	−0.171499
$137$	−16.0000	−1.36697	−0.683486	−	0.729964i	$-0.739537\pi$
$137$	−16.0000	−1.36697	−0.683486	+	0.729964i	$0.739537\pi$
$138$	−1.00000	−0.0851257
$139$	−10.0000	−0.848189	−0.424094	−	0.905618i	$-0.639408\pi$
$139$	−10.0000	−0.848189	−0.424094	+	0.905618i	$0.639408\pi$
$140$	0	0
$141$	6.00000	0.505291
$142$	−2.00000	−0.167836
$143$	15.0000	1.25436
$144$	1.00000	0.0833333
$145$	0	0
$146$	−9.00000	−0.744845
$147$	−6.00000	−0.494872
$148$	−4.00000	−0.328798
$149$	−20.0000	−1.63846	−0.819232	−	0.573462i	$-0.805600\pi$
$149$	−20.0000	−1.63846	−0.819232	+	0.573462i	$0.805600\pi$
$150$	0	0
$151$	18.0000	1.46482	0.732410	−	0.680864i	$-0.238396\pi$
$151$	18.0000	1.46482	0.732410	+	0.680864i	$0.238396\pi$
$152$	−3.00000	−0.243332
$153$	−2.00000	−0.161690
$154$	5.00000	0.402911
$155$	0	0
$156$	−3.00000	−0.240192
$157$	4.00000	0.319235	0.159617	−	0.987179i	$-0.448974\pi$
$157$	4.00000	0.319235	0.159617	+	0.987179i	$0.448974\pi$
$158$	−7.00000	−0.556890
$159$	0	0
$160$	0	0
$161$	1.00000	0.0788110
$162$	1.00000	0.0785674
$163$	4.00000	0.313304	0.156652	−	0.987654i	$-0.449930\pi$
$163$	4.00000	0.313304	0.156652	+	0.987654i	$0.449930\pi$
$164$	−3.00000	−0.234261
$165$	0	0
$166$	1.00000	0.0776151
$167$	12.0000	0.928588	0.464294	−	0.885681i	$-0.346308\pi$
$167$	12.0000	0.928588	0.464294	+	0.885681i	$0.346308\pi$
$168$	−1.00000	−0.0771517
$169$	−4.00000	−0.307692
$170$	0	0
$171$	−3.00000	−0.229416
$172$	−5.00000	−0.381246
$173$	9.00000	0.684257	0.342129	−	0.939653i	$-0.388852\pi$
$173$	9.00000	0.684257	0.342129	+	0.939653i	$0.388852\pi$
$174$	−1.00000	−0.0758098
$175$	0	0
$176$	−5.00000	−0.376889
$177$	−6.00000	−0.450988
$178$	−2.00000	−0.149906
$179$	10.0000	0.747435	0.373718	−	0.927543i	$-0.378083\pi$
$179$	10.0000	0.747435	0.373718	+	0.927543i	$0.378083\pi$
$180$	0	0
$181$	−6.00000	−0.445976	−0.222988	−	0.974821i	$-0.571581\pi$
$181$	−6.00000	−0.445976	−0.222988	+	0.974821i	$0.571581\pi$
$182$	3.00000	0.222375
$183$	4.00000	0.295689
$184$	−1.00000	−0.0737210
$185$	0	0
$186$	0	0
$187$	10.0000	0.731272
$188$	6.00000	0.437595
$189$	−1.00000	−0.0727393
$190$	0	0
$191$	−3.00000	−0.217072	−0.108536	−	0.994092i	$-0.534616\pi$
$191$	−3.00000	−0.217072	−0.108536	+	0.994092i	$0.534616\pi$
$192$	1.00000	0.0721688
$193$	−10.0000	−0.719816	−0.359908	−	0.932988i	$-0.617192\pi$
$193$	−10.0000	−0.719816	−0.359908	+	0.932988i	$0.617192\pi$
$194$	0	0
$195$	0	0
$196$	−6.00000	−0.428571
$197$	−15.0000	−1.06871	−0.534353	−	0.845262i	$-0.679445\pi$
$197$	−15.0000	−1.06871	−0.534353	+	0.845262i	$0.679445\pi$
$198$	−5.00000	−0.355335
$199$	−11.0000	−0.779769	−0.389885	−	0.920864i	$-0.627485\pi$
$199$	−11.0000	−0.779769	−0.389885	+	0.920864i	$0.627485\pi$
$200$	0	0
$201$	4.00000	0.282138
$202$	6.00000	0.422159
$203$	1.00000	0.0701862
$204$	−2.00000	−0.140028
$205$	0	0
$206$	−7.00000	−0.487713
$207$	−1.00000	−0.0695048
$208$	−3.00000	−0.208013
$209$	15.0000	1.03757
$210$	0	0
$211$	−2.00000	−0.137686	−0.0688428	−	0.997628i	$-0.521931\pi$
$211$	−2.00000	−0.137686	−0.0688428	+	0.997628i	$0.521931\pi$
$212$	0	0
$213$	−2.00000	−0.137038
$214$	12.0000	0.820303
$215$	0	0
$216$	1.00000	0.0680414
$217$	0	0
$218$	−4.00000	−0.270914
$219$	−9.00000	−0.608164
$220$	0	0
$221$	6.00000	0.403604
$222$	−4.00000	−0.268462
$223$	−2.00000	−0.133930	−0.0669650	−	0.997755i	$-0.521332\pi$
$223$	−2.00000	−0.133930	−0.0669650	+	0.997755i	$0.521332\pi$
$224$	−1.00000	−0.0668153
$225$	0	0
$226$	6.00000	0.399114
$227$	24.0000	1.59294	0.796468	−	0.604681i	$-0.206699\pi$
$227$	24.0000	1.59294	0.796468	+	0.604681i	$0.206699\pi$
$228$	−3.00000	−0.198680
$229$	−8.00000	−0.528655	−0.264327	−	0.964433i	$-0.585150\pi$
$229$	−8.00000	−0.528655	−0.264327	+	0.964433i	$0.585150\pi$
$230$	0	0
$231$	5.00000	0.328976
$232$	−1.00000	−0.0656532
$233$	11.0000	0.720634	0.360317	−	0.932830i	$-0.382669\pi$
$233$	11.0000	0.720634	0.360317	+	0.932830i	$0.382669\pi$
$234$	−3.00000	−0.196116
$235$	0	0
$236$	−6.00000	−0.390567
$237$	−7.00000	−0.454699
$238$	2.00000	0.129641
$239$	−8.00000	−0.517477	−0.258738	−	0.965947i	$-0.583307\pi$
$239$	−8.00000	−0.517477	−0.258738	+	0.965947i	$0.583307\pi$
$240$	0	0
$241$	−2.00000	−0.128831	−0.0644157	−	0.997923i	$-0.520518\pi$
$241$	−2.00000	−0.128831	−0.0644157	+	0.997923i	$0.520518\pi$
$242$	14.0000	0.899954
$243$	1.00000	0.0641500
$244$	4.00000	0.256074
$245$	0	0
$246$	−3.00000	−0.191273
$247$	9.00000	0.572656
$248$	0	0
$249$	1.00000	0.0633724
$250$	0	0
$251$	−12.0000	−0.757433	−0.378717	−	0.925513i	$-0.623635\pi$
$251$	−12.0000	−0.757433	−0.378717	+	0.925513i	$0.623635\pi$
$252$	−1.00000	−0.0629941
$253$	5.00000	0.314347
$254$	18.0000	1.12942
$255$	0	0
$256$	1.00000	0.0625000
$257$	22.0000	1.37232	0.686161	−	0.727450i	$-0.259294\pi$
$257$	22.0000	1.37232	0.686161	+	0.727450i	$0.259294\pi$
$258$	−5.00000	−0.311286
$259$	4.00000	0.248548
$260$	0	0
$261$	−1.00000	−0.0618984
$262$	12.0000	0.741362
$263$	−28.0000	−1.72655	−0.863277	−	0.504730i	$-0.831592\pi$
$263$	−28.0000	−1.72655	−0.863277	+	0.504730i	$0.831592\pi$
$264$	−5.00000	−0.307729
$265$	0	0
$266$	3.00000	0.183942
$267$	−2.00000	−0.122398
$268$	4.00000	0.244339
$269$	−9.00000	−0.548740	−0.274370	−	0.961624i	$-0.588469\pi$
$269$	−9.00000	−0.548740	−0.274370	+	0.961624i	$0.588469\pi$
$270$	0	0
$271$	−28.0000	−1.70088	−0.850439	−	0.526073i	$-0.823664\pi$
$271$	−28.0000	−1.70088	−0.850439	+	0.526073i	$0.823664\pi$
$272$	−2.00000	−0.121268
$273$	3.00000	0.181568
$274$	−16.0000	−0.966595
$275$	0	0
$276$	−1.00000	−0.0601929
$277$	31.0000	1.86261	0.931305	−	0.364241i	$-0.118672\pi$
$277$	31.0000	1.86261	0.931305	+	0.364241i	$0.118672\pi$
$278$	−10.0000	−0.599760
$279$	0	0
$280$	0	0
$281$	−22.0000	−1.31241	−0.656205	−	0.754583i	$-0.727839\pi$
$281$	−22.0000	−1.31241	−0.656205	+	0.754583i	$0.727839\pi$
$282$	6.00000	0.357295
$283$	4.00000	0.237775	0.118888	−	0.992908i	$-0.462067\pi$
$283$	4.00000	0.237775	0.118888	+	0.992908i	$0.462067\pi$
$284$	−2.00000	−0.118678
$285$	0	0
$286$	15.0000	0.886969
$287$	3.00000	0.177084
$288$	1.00000	0.0589256
$289$	−13.0000	−0.764706
$290$	0	0
$291$	0	0
$292$	−9.00000	−0.526685
$293$	28.0000	1.63578	0.817889	−	0.575376i	$-0.195144\pi$
$293$	28.0000	1.63578	0.817889	+	0.575376i	$0.195144\pi$
$294$	−6.00000	−0.349927
$295$	0	0
$296$	−4.00000	−0.232495
$297$	−5.00000	−0.290129
$298$	−20.0000	−1.15857
$299$	3.00000	0.173494
$300$	0	0
$301$	5.00000	0.288195
$302$	18.0000	1.03578
$303$	6.00000	0.344691
$304$	−3.00000	−0.172062
$305$	0	0
$306$	−2.00000	−0.114332
$307$	20.0000	1.14146	0.570730	−	0.821138i	$-0.306660\pi$
$307$	20.0000	1.14146	0.570730	+	0.821138i	$0.306660\pi$
$308$	5.00000	0.284901
$309$	−7.00000	−0.398216
$310$	0	0
$311$	−20.0000	−1.13410	−0.567048	−	0.823685i	$-0.691915\pi$
$311$	−20.0000	−1.13410	−0.567048	+	0.823685i	$0.691915\pi$
$312$	−3.00000	−0.169842
$313$	−32.0000	−1.80875	−0.904373	−	0.426742i	$-0.859661\pi$
$313$	−32.0000	−1.80875	−0.904373	+	0.426742i	$0.859661\pi$
$314$	4.00000	0.225733
$315$	0	0
$316$	−7.00000	−0.393781
$317$	13.0000	0.730153	0.365076	−	0.930978i	$-0.381043\pi$
$317$	13.0000	0.730153	0.365076	+	0.930978i	$0.381043\pi$
$318$	0	0
$319$	5.00000	0.279946
$320$	0	0
$321$	12.0000	0.669775
$322$	1.00000	0.0557278
$323$	6.00000	0.333849
$324$	1.00000	0.0555556
$325$	0	0
$326$	4.00000	0.221540
$327$	−4.00000	−0.221201
$328$	−3.00000	−0.165647
$329$	−6.00000	−0.330791
$330$	0	0
$331$	20.0000	1.09930	0.549650	−	0.835395i	$-0.314761\pi$
$331$	20.0000	1.09930	0.549650	+	0.835395i	$0.314761\pi$
$332$	1.00000	0.0548821
$333$	−4.00000	−0.219199
$334$	12.0000	0.656611
$335$	0	0
$336$	−1.00000	−0.0545545
$337$	−8.00000	−0.435788	−0.217894	−	0.975972i	$-0.569919\pi$
$337$	−8.00000	−0.435788	−0.217894	+	0.975972i	$0.569919\pi$
$338$	−4.00000	−0.217571
$339$	6.00000	0.325875
$340$	0	0
$341$	0	0
$342$	−3.00000	−0.162221
$343$	13.0000	0.701934
$344$	−5.00000	−0.269582
$345$	0	0
$346$	9.00000	0.483843
$347$	20.0000	1.07366	0.536828	−	0.843692i	$-0.319622\pi$
$347$	20.0000	1.07366	0.536828	+	0.843692i	$0.319622\pi$
$348$	−1.00000	−0.0536056
$349$	−15.0000	−0.802932	−0.401466	−	0.915874i	$-0.631499\pi$
$349$	−15.0000	−0.802932	−0.401466	+	0.915874i	$0.631499\pi$
$350$	0	0
$351$	−3.00000	−0.160128
$352$	−5.00000	−0.266501
$353$	−11.0000	−0.585471	−0.292735	−	0.956193i	$-0.594566\pi$
$353$	−11.0000	−0.585471	−0.292735	+	0.956193i	$0.594566\pi$
$354$	−6.00000	−0.318896
$355$	0	0
$356$	−2.00000	−0.106000
$357$	2.00000	0.105851
$358$	10.0000	0.528516
$359$	−9.00000	−0.475002	−0.237501	−	0.971387i	$-0.576328\pi$
$359$	−9.00000	−0.475002	−0.237501	+	0.971387i	$0.576328\pi$
$360$	0	0
$361$	−10.0000	−0.526316
$362$	−6.00000	−0.315353
$363$	14.0000	0.734809
$364$	3.00000	0.157243
$365$	0	0
$366$	4.00000	0.209083
$367$	−37.0000	−1.93138	−0.965692	−	0.259690i	$-0.916380\pi$
$367$	−37.0000	−1.93138	−0.965692	+	0.259690i	$0.916380\pi$
$368$	−1.00000	−0.0521286
$369$	−3.00000	−0.156174
$370$	0	0
$371$	0	0
$372$	0	0
$373$	22.0000	1.13912	0.569558	−	0.821951i	$-0.307114\pi$
$373$	22.0000	1.13912	0.569558	+	0.821951i	$0.307114\pi$
$374$	10.0000	0.517088
$375$	0	0
$376$	6.00000	0.309426
$377$	3.00000	0.154508
$378$	−1.00000	−0.0514344
$379$	−20.0000	−1.02733	−0.513665	−	0.857991i	$-0.671713\pi$
$379$	−20.0000	−1.02733	−0.513665	+	0.857991i	$0.671713\pi$
$380$	0	0
$381$	18.0000	0.922168
$382$	−3.00000	−0.153493
$383$	9.00000	0.459879	0.229939	−	0.973205i	$-0.426147\pi$
$383$	9.00000	0.459879	0.229939	+	0.973205i	$0.426147\pi$
$384$	1.00000	0.0510310
$385$	0	0
$386$	−10.0000	−0.508987
$387$	−5.00000	−0.254164
$388$	0	0
$389$	10.0000	0.507020	0.253510	−	0.967333i	$-0.418415\pi$
$389$	10.0000	0.507020	0.253510	+	0.967333i	$0.418415\pi$
$390$	0	0
$391$	2.00000	0.101144
$392$	−6.00000	−0.303046
$393$	12.0000	0.605320
$394$	−15.0000	−0.755689
$395$	0	0
$396$	−5.00000	−0.251259
$397$	−2.00000	−0.100377	−0.0501886	−	0.998740i	$-0.515982\pi$
$397$	−2.00000	−0.100377	−0.0501886	+	0.998740i	$0.515982\pi$
$398$	−11.0000	−0.551380
$399$	3.00000	0.150188
$400$	0	0
$401$	12.0000	0.599251	0.299626	−	0.954057i	$-0.403138\pi$
$401$	12.0000	0.599251	0.299626	+	0.954057i	$0.403138\pi$
$402$	4.00000	0.199502
$403$	0	0
$404$	6.00000	0.298511
$405$	0	0
$406$	1.00000	0.0496292
$407$	20.0000	0.991363
$408$	−2.00000	−0.0990148
$409$	17.0000	0.840596	0.420298	−	0.907386i	$-0.361926\pi$
$409$	17.0000	0.840596	0.420298	+	0.907386i	$0.361926\pi$
$410$	0	0
$411$	−16.0000	−0.789222
$412$	−7.00000	−0.344865
$413$	6.00000	0.295241
$414$	−1.00000	−0.0491473
$415$	0	0
$416$	−3.00000	−0.147087
$417$	−10.0000	−0.489702
$418$	15.0000	0.733674
$419$	25.0000	1.22133	0.610665	−	0.791889i	$-0.290902\pi$
$419$	25.0000	1.22133	0.610665	+	0.791889i	$0.290902\pi$
$420$	0	0
$421$	6.00000	0.292422	0.146211	−	0.989253i	$-0.453292\pi$
$421$	6.00000	0.292422	0.146211	+	0.989253i	$0.453292\pi$
$422$	−2.00000	−0.0973585
$423$	6.00000	0.291730
$424$	0	0
$425$	0	0
$426$	−2.00000	−0.0969003
$427$	−4.00000	−0.193574
$428$	12.0000	0.580042
$429$	15.0000	0.724207
$430$	0	0
$431$	0	0		−	1.00000i	$-0.5\pi$
$431$	0	0			1.00000i	$0.5\pi$
$432$	1.00000	0.0481125
$433$	12.0000	0.576683	0.288342	−	0.957528i	$-0.406896\pi$
$433$	12.0000	0.576683	0.288342	+	0.957528i	$0.406896\pi$
$434$	0	0
$435$	0	0
$436$	−4.00000	−0.191565
$437$	3.00000	0.143509
$438$	−9.00000	−0.430037
$439$	20.0000	0.954548	0.477274	−	0.878755i	$-0.341625\pi$
$439$	20.0000	0.954548	0.477274	+	0.878755i	$0.341625\pi$
$440$	0	0
$441$	−6.00000	−0.285714
$442$	6.00000	0.285391
$443$	20.0000	0.950229	0.475114	−	0.879924i	$-0.342407\pi$
$443$	20.0000	0.950229	0.475114	+	0.879924i	$0.342407\pi$
$444$	−4.00000	−0.189832
$445$	0	0
$446$	−2.00000	−0.0947027
$447$	−20.0000	−0.945968
$448$	−1.00000	−0.0472456
$449$	14.0000	0.660701	0.330350	−	0.943858i	$-0.392833\pi$
$449$	14.0000	0.660701	0.330350	+	0.943858i	$0.392833\pi$
$450$	0	0
$451$	15.0000	0.706322
$452$	6.00000	0.282216
$453$	18.0000	0.845714
$454$	24.0000	1.12638
$455$	0	0
$456$	−3.00000	−0.140488
$457$	−18.0000	−0.842004	−0.421002	−	0.907060i	$-0.638322\pi$
$457$	−18.0000	−0.842004	−0.421002	+	0.907060i	$0.638322\pi$
$458$	−8.00000	−0.373815
$459$	−2.00000	−0.0933520
$460$	0	0
$461$	−37.0000	−1.72326	−0.861631	−	0.507535i	$-0.830557\pi$
$461$	−37.0000	−1.72326	−0.861631	+	0.507535i	$0.830557\pi$
$462$	5.00000	0.232621
$463$	−12.0000	−0.557687	−0.278844	−	0.960337i	$-0.589951\pi$
$463$	−12.0000	−0.557687	−0.278844	+	0.960337i	$0.589951\pi$
$464$	−1.00000	−0.0464238
$465$	0	0
$466$	11.0000	0.509565
$467$	27.0000	1.24941	0.624705	−	0.780860i	$-0.285219\pi$
$467$	27.0000	1.24941	0.624705	+	0.780860i	$0.285219\pi$
$468$	−3.00000	−0.138675
$469$	−4.00000	−0.184703
$470$	0	0
$471$	4.00000	0.184310
$472$	−6.00000	−0.276172
$473$	25.0000	1.14950
$474$	−7.00000	−0.321521
$475$	0	0
$476$	2.00000	0.0916698
$477$	0	0
$478$	−8.00000	−0.365911
$479$	17.0000	0.776750	0.388375	−	0.921501i	$-0.373037\pi$
$479$	17.0000	0.776750	0.388375	+	0.921501i	$0.373037\pi$
$480$	0	0
$481$	12.0000	0.547153
$482$	−2.00000	−0.0910975
$483$	1.00000	0.0455016
$484$	14.0000	0.636364
$485$	0	0
$486$	1.00000	0.0453609
$487$	−18.0000	−0.815658	−0.407829	−	0.913058i	$-0.633714\pi$
$487$	−18.0000	−0.815658	−0.407829	+	0.913058i	$0.633714\pi$
$488$	4.00000	0.181071
$489$	4.00000	0.180886
$490$	0	0
$491$	0	0		−	1.00000i	$-0.5\pi$
$491$	0	0			1.00000i	$0.5\pi$
$492$	−3.00000	−0.135250
$493$	2.00000	0.0900755
$494$	9.00000	0.404929
$495$	0	0
$496$	0	0
$497$	2.00000	0.0897123
$498$	1.00000	0.0448111
$499$	−34.0000	−1.52205	−0.761025	−	0.648723i	$-0.775303\pi$
$499$	−34.0000	−1.52205	−0.761025	+	0.648723i	$0.775303\pi$
$500$	0	0
$501$	12.0000	0.536120
$502$	−12.0000	−0.535586
$503$	11.0000	0.490466	0.245233	−	0.969464i	$-0.421136\pi$
$503$	11.0000	0.490466	0.245233	+	0.969464i	$0.421136\pi$
$504$	−1.00000	−0.0445435
$505$	0	0
$506$	5.00000	0.222277
$507$	−4.00000	−0.177646
$508$	18.0000	0.798621
$509$	−2.00000	−0.0886484	−0.0443242	−	0.999017i	$-0.514113\pi$
$509$	−2.00000	−0.0886484	−0.0443242	+	0.999017i	$0.514113\pi$
$510$	0	0
$511$	9.00000	0.398137
$512$	1.00000	0.0441942
$513$	−3.00000	−0.132453
$514$	22.0000	0.970378
$515$	0	0
$516$	−5.00000	−0.220113
$517$	−30.0000	−1.31940
$518$	4.00000	0.175750
$519$	9.00000	0.395056
$520$	0	0
$521$	−4.00000	−0.175243	−0.0876216	−	0.996154i	$-0.527927\pi$
$521$	−4.00000	−0.175243	−0.0876216	+	0.996154i	$0.527927\pi$
$522$	−1.00000	−0.0437688
$523$	−19.0000	−0.830812	−0.415406	−	0.909636i	$-0.636360\pi$
$523$	−19.0000	−0.830812	−0.415406	+	0.909636i	$0.636360\pi$
$524$	12.0000	0.524222
$525$	0	0
$526$	−28.0000	−1.22086
$527$	0	0
$528$	−5.00000	−0.217597
$529$	1.00000	0.0434783
$530$	0	0
$531$	−6.00000	−0.260378
$532$	3.00000	0.130066
$533$	9.00000	0.389833
$534$	−2.00000	−0.0865485
$535$	0	0
$536$	4.00000	0.172774
$537$	10.0000	0.431532
$538$	−9.00000	−0.388018
$539$	30.0000	1.29219
$540$	0	0
$541$	−31.0000	−1.33279	−0.666397	−	0.745597i	$-0.732164\pi$
$541$	−31.0000	−1.33279	−0.666397	+	0.745597i	$0.732164\pi$
$542$	−28.0000	−1.20270
$543$	−6.00000	−0.257485
$544$	−2.00000	−0.0857493
$545$	0	0
$546$	3.00000	0.128388
$547$	−6.00000	−0.256541	−0.128271	−	0.991739i	$-0.540943\pi$
$547$	−6.00000	−0.256541	−0.128271	+	0.991739i	$0.540943\pi$
$548$	−16.0000	−0.683486
$549$	4.00000	0.170716
$550$	0	0
$551$	3.00000	0.127804
$552$	−1.00000	−0.0425628
$553$	7.00000	0.297670
$554$	31.0000	1.31706
$555$	0	0
$556$	−10.0000	−0.424094
$557$	−2.00000	−0.0847427	−0.0423714	−	0.999102i	$-0.513491\pi$
$557$	−2.00000	−0.0847427	−0.0423714	+	0.999102i	$0.513491\pi$
$558$	0	0
$559$	15.0000	0.634432
$560$	0	0
$561$	10.0000	0.422200
$562$	−22.0000	−0.928014
$563$	−11.0000	−0.463595	−0.231797	−	0.972764i	$-0.574461\pi$
$563$	−11.0000	−0.463595	−0.231797	+	0.972764i	$0.574461\pi$
$564$	6.00000	0.252646
$565$	0	0
$566$	4.00000	0.168133
$567$	−1.00000	−0.0419961
$568$	−2.00000	−0.0839181
$569$	18.0000	0.754599	0.377300	−	0.926091i	$-0.376853\pi$
$569$	18.0000	0.754599	0.377300	+	0.926091i	$0.376853\pi$
$570$	0	0
$571$	36.0000	1.50655	0.753277	−	0.657704i	$-0.228472\pi$
$571$	36.0000	1.50655	0.753277	+	0.657704i	$0.228472\pi$
$572$	15.0000	0.627182
$573$	−3.00000	−0.125327
$574$	3.00000	0.125218
$575$	0	0
$576$	1.00000	0.0416667
$577$	−17.0000	−0.707719	−0.353860	−	0.935299i	$-0.615131\pi$
$577$	−17.0000	−0.707719	−0.353860	+	0.935299i	$0.615131\pi$
$578$	−13.0000	−0.540729
$579$	−10.0000	−0.415586
$580$	0	0
$581$	−1.00000	−0.0414870
$582$	0	0
$583$	0	0
$584$	−9.00000	−0.372423
$585$	0	0
$586$	28.0000	1.15667
$587$	−2.00000	−0.0825488	−0.0412744	−	0.999148i	$-0.513142\pi$
$587$	−2.00000	−0.0825488	−0.0412744	+	0.999148i	$0.513142\pi$
$588$	−6.00000	−0.247436
$589$	0	0
$590$	0	0
$591$	−15.0000	−0.617018
$592$	−4.00000	−0.164399
$593$	−9.00000	−0.369586	−0.184793	−	0.982777i	$-0.559161\pi$
$593$	−9.00000	−0.369586	−0.184793	+	0.982777i	$0.559161\pi$
$594$	−5.00000	−0.205152
$595$	0	0
$596$	−20.0000	−0.819232
$597$	−11.0000	−0.450200
$598$	3.00000	0.122679
$599$	−36.0000	−1.47092	−0.735460	−	0.677568i	$-0.763034\pi$
$599$	−36.0000	−1.47092	−0.735460	+	0.677568i	$0.763034\pi$
$600$	0	0
$601$	26.0000	1.06056	0.530281	−	0.847822i	$-0.322086\pi$
$601$	26.0000	1.06056	0.530281	+	0.847822i	$0.322086\pi$
$602$	5.00000	0.203785
$603$	4.00000	0.162893
$604$	18.0000	0.732410
$605$	0	0
$606$	6.00000	0.243733
$607$	14.0000	0.568242	0.284121	−	0.958788i	$-0.408298\pi$
$607$	14.0000	0.568242	0.284121	+	0.958788i	$0.408298\pi$
$608$	−3.00000	−0.121666
$609$	1.00000	0.0405220
$610$	0	0
$611$	−18.0000	−0.728202
$612$	−2.00000	−0.0808452
$613$	−14.0000	−0.565455	−0.282727	−	0.959200i	$-0.591239\pi$
$613$	−14.0000	−0.565455	−0.282727	+	0.959200i	$0.591239\pi$
$614$	20.0000	0.807134
$615$	0	0
$616$	5.00000	0.201456
$617$	−10.0000	−0.402585	−0.201292	−	0.979531i	$-0.564514\pi$
$617$	−10.0000	−0.402585	−0.201292	+	0.979531i	$0.564514\pi$
$618$	−7.00000	−0.281581
$619$	−36.0000	−1.44696	−0.723481	−	0.690344i	$-0.757459\pi$
$619$	−36.0000	−1.44696	−0.723481	+	0.690344i	$0.757459\pi$
$620$	0	0
$621$	−1.00000	−0.0401286
$622$	−20.0000	−0.801927
$623$	2.00000	0.0801283
$624$	−3.00000	−0.120096
$625$	0	0
$626$	−32.0000	−1.27898
$627$	15.0000	0.599042
$628$	4.00000	0.159617
$629$	8.00000	0.318981
$630$	0	0
$631$	−25.0000	−0.995234	−0.497617	−	0.867397i	$-0.665792\pi$
$631$	−25.0000	−0.995234	−0.497617	+	0.867397i	$0.665792\pi$
$632$	−7.00000	−0.278445
$633$	−2.00000	−0.0794929
$634$	13.0000	0.516296
$635$	0	0
$636$	0	0
$637$	18.0000	0.713186
$638$	5.00000	0.197952
$639$	−2.00000	−0.0791188
$640$	0	0
$641$	4.00000	0.157991	0.0789953	−	0.996875i	$-0.474829\pi$
$641$	4.00000	0.157991	0.0789953	+	0.996875i	$0.474829\pi$
$642$	12.0000	0.473602
$643$	19.0000	0.749287	0.374643	−	0.927169i	$-0.377765\pi$
$643$	19.0000	0.749287	0.374643	+	0.927169i	$0.377765\pi$
$644$	1.00000	0.0394055
$645$	0	0
$646$	6.00000	0.236067
$647$	−24.0000	−0.943537	−0.471769	−	0.881722i	$-0.656384\pi$
$647$	−24.0000	−0.943537	−0.471769	+	0.881722i	$0.656384\pi$
$648$	1.00000	0.0392837
$649$	30.0000	1.17760
$650$	0	0
$651$	0	0
$652$	4.00000	0.156652
$653$	−1.00000	−0.0391330	−0.0195665	−	0.999809i	$-0.506229\pi$
$653$	−1.00000	−0.0391330	−0.0195665	+	0.999809i	$0.506229\pi$
$654$	−4.00000	−0.156412
$655$	0	0
$656$	−3.00000	−0.117130
$657$	−9.00000	−0.351123
$658$	−6.00000	−0.233904
$659$	45.0000	1.75295	0.876476	−	0.481446i	$-0.159888\pi$
$659$	45.0000	1.75295	0.876476	+	0.481446i	$0.159888\pi$
$660$	0	0
$661$	24.0000	0.933492	0.466746	−	0.884391i	$-0.345426\pi$
$661$	24.0000	0.933492	0.466746	+	0.884391i	$0.345426\pi$
$662$	20.0000	0.777322
$663$	6.00000	0.233021
$664$	1.00000	0.0388075
$665$	0	0
$666$	−4.00000	−0.154997
$667$	1.00000	0.0387202
$668$	12.0000	0.464294
$669$	−2.00000	−0.0773245
$670$	0	0
$671$	−20.0000	−0.772091
$672$	−1.00000	−0.0385758
$673$	−27.0000	−1.04077	−0.520387	−	0.853931i	$-0.674212\pi$
$673$	−27.0000	−1.04077	−0.520387	+	0.853931i	$0.674212\pi$
$674$	−8.00000	−0.308148
$675$	0	0
$676$	−4.00000	−0.153846
$677$	−8.00000	−0.307465	−0.153732	−	0.988113i	$-0.549129\pi$
$677$	−8.00000	−0.307465	−0.153732	+	0.988113i	$0.549129\pi$
$678$	6.00000	0.230429
$679$	0	0
$680$	0	0
$681$	24.0000	0.919682
$682$	0	0
$683$	18.0000	0.688751	0.344375	−	0.938832i	$-0.388091\pi$
$683$	18.0000	0.688751	0.344375	+	0.938832i	$0.388091\pi$
$684$	−3.00000	−0.114708
$685$	0	0
$686$	13.0000	0.496342
$687$	−8.00000	−0.305219
$688$	−5.00000	−0.190623
$689$	0	0
$690$	0	0
$691$	−14.0000	−0.532585	−0.266293	−	0.963892i	$-0.585799\pi$
$691$	−14.0000	−0.532585	−0.266293	+	0.963892i	$0.585799\pi$
$692$	9.00000	0.342129
$693$	5.00000	0.189934
$694$	20.0000	0.759190
$695$	0	0
$696$	−1.00000	−0.0379049
$697$	6.00000	0.227266
$698$	−15.0000	−0.567758
$699$	11.0000	0.416058
$700$	0	0
$701$	−36.0000	−1.35970	−0.679851	−	0.733351i	$-0.737955\pi$
$701$	−36.0000	−1.35970	−0.679851	+	0.733351i	$0.737955\pi$
$702$	−3.00000	−0.113228
$703$	12.0000	0.452589
$704$	−5.00000	−0.188445
$705$	0	0
$706$	−11.0000	−0.413990
$707$	−6.00000	−0.225653
$708$	−6.00000	−0.225494
$709$	−44.0000	−1.65245	−0.826227	−	0.563337i	$-0.809517\pi$
$709$	−44.0000	−1.65245	−0.826227	+	0.563337i	$0.809517\pi$
$710$	0	0
$711$	−7.00000	−0.262521
$712$	−2.00000	−0.0749532
$713$	0	0
$714$	2.00000	0.0748481
$715$	0	0
$716$	10.0000	0.373718
$717$	−8.00000	−0.298765
$718$	−9.00000	−0.335877
$719$	30.0000	1.11881	0.559406	−	0.828894i	$-0.311029\pi$
$719$	30.0000	1.11881	0.559406	+	0.828894i	$0.311029\pi$
$720$	0	0
$721$	7.00000	0.260694
$722$	−10.0000	−0.372161
$723$	−2.00000	−0.0743808
$724$	−6.00000	−0.222988
$725$	0	0
$726$	14.0000	0.519589
$727$	−8.00000	−0.296704	−0.148352	−	0.988935i	$-0.547397\pi$
$727$	−8.00000	−0.296704	−0.148352	+	0.988935i	$0.547397\pi$
$728$	3.00000	0.111187
$729$	1.00000	0.0370370
$730$	0	0
$731$	10.0000	0.369863
$732$	4.00000	0.147844
$733$	24.0000	0.886460	0.443230	−	0.896408i	$-0.353832\pi$
$733$	24.0000	0.886460	0.443230	+	0.896408i	$0.353832\pi$
$734$	−37.0000	−1.36569
$735$	0	0
$736$	−1.00000	−0.0368605
$737$	−20.0000	−0.736709
$738$	−3.00000	−0.110432
$739$	24.0000	0.882854	0.441427	−	0.897297i	$-0.354472\pi$
$739$	24.0000	0.882854	0.441427	+	0.897297i	$0.354472\pi$
$740$	0	0
$741$	9.00000	0.330623
$742$	0	0
$743$	5.00000	0.183432	0.0917161	−	0.995785i	$-0.470765\pi$
$743$	5.00000	0.183432	0.0917161	+	0.995785i	$0.470765\pi$
$744$	0	0
$745$	0	0
$746$	22.0000	0.805477
$747$	1.00000	0.0365881
$748$	10.0000	0.365636
$749$	−12.0000	−0.438470
$750$	0	0
$751$	7.00000	0.255434	0.127717	−	0.991811i	$-0.459235\pi$
$751$	7.00000	0.255434	0.127717	+	0.991811i	$0.459235\pi$
$752$	6.00000	0.218797
$753$	−12.0000	−0.437304
$754$	3.00000	0.109254
$755$	0	0
$756$	−1.00000	−0.0363696
$757$	−28.0000	−1.01768	−0.508839	−	0.860862i	$-0.669925\pi$
$757$	−28.0000	−1.01768	−0.508839	+	0.860862i	$0.669925\pi$
$758$	−20.0000	−0.726433
$759$	5.00000	0.181489
$760$	0	0
$761$	23.0000	0.833749	0.416875	−	0.908964i	$-0.363125\pi$
$761$	23.0000	0.833749	0.416875	+	0.908964i	$0.363125\pi$
$762$	18.0000	0.652071
$763$	4.00000	0.144810
$764$	−3.00000	−0.108536
$765$	0	0
$766$	9.00000	0.325183
$767$	18.0000	0.649942
$768$	1.00000	0.0360844
$769$	6.00000	0.216366	0.108183	−	0.994131i	$-0.465497\pi$
$769$	6.00000	0.216366	0.108183	+	0.994131i	$0.465497\pi$
$770$	0	0
$771$	22.0000	0.792311
$772$	−10.0000	−0.359908
$773$	−12.0000	−0.431610	−0.215805	−	0.976436i	$-0.569238\pi$
$773$	−12.0000	−0.431610	−0.215805	+	0.976436i	$0.569238\pi$
$774$	−5.00000	−0.179721
$775$	0	0
$776$	0	0
$777$	4.00000	0.143499
$778$	10.0000	0.358517
$779$	9.00000	0.322458
$780$	0	0
$781$	10.0000	0.357828
$782$	2.00000	0.0715199
$783$	−1.00000	−0.0357371
$784$	−6.00000	−0.214286
$785$	0	0
$786$	12.0000	0.428026
$787$	37.0000	1.31891	0.659454	−	0.751745i	$-0.270788\pi$
$787$	37.0000	1.31891	0.659454	+	0.751745i	$0.270788\pi$
$788$	−15.0000	−0.534353
$789$	−28.0000	−0.996826
$790$	0	0
$791$	−6.00000	−0.213335
$792$	−5.00000	−0.177667
$793$	−12.0000	−0.426132
$794$	−2.00000	−0.0709773
$795$	0	0
$796$	−11.0000	−0.389885
$797$	10.0000	0.354218	0.177109	−	0.984191i	$-0.443325\pi$
$797$	10.0000	0.354218	0.177109	+	0.984191i	$0.443325\pi$
$798$	3.00000	0.106199
$799$	−12.0000	−0.424529
$800$	0	0
$801$	−2.00000	−0.0706665
$802$	12.0000	0.423735
$803$	45.0000	1.58802
$804$	4.00000	0.141069
$805$	0	0
$806$	0	0
$807$	−9.00000	−0.316815
$808$	6.00000	0.211079
$809$	39.0000	1.37117	0.685583	−	0.727994i	$-0.259547\pi$
$809$	39.0000	1.37117	0.685583	+	0.727994i	$0.259547\pi$
$810$	0	0
$811$	−2.00000	−0.0702295	−0.0351147	−	0.999383i	$-0.511180\pi$
$811$	−2.00000	−0.0702295	−0.0351147	+	0.999383i	$0.511180\pi$
$812$	1.00000	0.0350931
$813$	−28.0000	−0.982003
$814$	20.0000	0.701000
$815$	0	0
$816$	−2.00000	−0.0700140
$817$	15.0000	0.524784
$818$	17.0000	0.594391
$819$	3.00000	0.104828
$820$	0	0
$821$	53.0000	1.84971	0.924856	−	0.380317i	$-0.124185\pi$
$821$	53.0000	1.84971	0.924856	+	0.380317i	$0.124185\pi$
$822$	−16.0000	−0.558064
$823$	−2.00000	−0.0697156	−0.0348578	−	0.999392i	$-0.511098\pi$
$823$	−2.00000	−0.0697156	−0.0348578	+	0.999392i	$0.511098\pi$
$824$	−7.00000	−0.243857
$825$	0	0
$826$	6.00000	0.208767
$827$	−53.0000	−1.84299	−0.921495	−	0.388390i	$-0.873032\pi$
$827$	−53.0000	−1.84299	−0.921495	+	0.388390i	$0.873032\pi$
$828$	−1.00000	−0.0347524
$829$	−23.0000	−0.798823	−0.399412	−	0.916772i	$-0.630786\pi$
$829$	−23.0000	−0.798823	−0.399412	+	0.916772i	$0.630786\pi$
$830$	0	0
$831$	31.0000	1.07538
$832$	−3.00000	−0.104006
$833$	12.0000	0.415775
$834$	−10.0000	−0.346272
$835$	0	0
$836$	15.0000	0.518786
$837$	0	0
$838$	25.0000	0.863611
$839$	25.0000	0.863096	0.431548	−	0.902090i	$-0.357968\pi$
$839$	25.0000	0.863096	0.431548	+	0.902090i	$0.357968\pi$
$840$	0	0
$841$	−28.0000	−0.965517
$842$	6.00000	0.206774
$843$	−22.0000	−0.757720
$844$	−2.00000	−0.0688428
$845$	0	0
$846$	6.00000	0.206284
$847$	−14.0000	−0.481046
$848$	0	0
$849$	4.00000	0.137280
$850$	0	0
$851$	4.00000	0.137118
$852$	−2.00000	−0.0685189
$853$	49.0000	1.67773	0.838864	−	0.544341i	$-0.183220\pi$
$853$	49.0000	1.67773	0.838864	+	0.544341i	$0.183220\pi$
$854$	−4.00000	−0.136877
$855$	0	0
$856$	12.0000	0.410152
$857$	−6.00000	−0.204956	−0.102478	−	0.994735i	$-0.532677\pi$
$857$	−6.00000	−0.204956	−0.102478	+	0.994735i	$0.532677\pi$
$858$	15.0000	0.512092
$859$	6.00000	0.204717	0.102359	−	0.994748i	$-0.467361\pi$
$859$	6.00000	0.204717	0.102359	+	0.994748i	$0.467361\pi$
$860$	0	0
$861$	3.00000	0.102240
$862$	0	0
$863$	−18.0000	−0.612727	−0.306364	−	0.951915i	$-0.599112\pi$
$863$	−18.0000	−0.612727	−0.306364	+	0.951915i	$0.599112\pi$
$864$	1.00000	0.0340207
$865$	0	0
$866$	12.0000	0.407777
$867$	−13.0000	−0.441503
$868$	0	0
$869$	35.0000	1.18729
$870$	0	0
$871$	−12.0000	−0.406604
$872$	−4.00000	−0.135457
$873$	0	0
$874$	3.00000	0.101477
$875$	0	0
$876$	−9.00000	−0.304082
$877$	−22.0000	−0.742887	−0.371444	−	0.928456i	$-0.621137\pi$
$877$	−22.0000	−0.742887	−0.371444	+	0.928456i	$0.621137\pi$
$878$	20.0000	0.674967
$879$	28.0000	0.944417
$880$	0	0
$881$	40.0000	1.34763	0.673817	−	0.738898i	$-0.264654\pi$
$881$	40.0000	1.34763	0.673817	+	0.738898i	$0.264654\pi$
$882$	−6.00000	−0.202031
$883$	−24.0000	−0.807664	−0.403832	−	0.914833i	$-0.632322\pi$
$883$	−24.0000	−0.807664	−0.403832	+	0.914833i	$0.632322\pi$
$884$	6.00000	0.201802
$885$	0	0
$886$	20.0000	0.671913
$887$	−6.00000	−0.201460	−0.100730	−	0.994914i	$-0.532118\pi$
$887$	−6.00000	−0.201460	−0.100730	+	0.994914i	$0.532118\pi$
$888$	−4.00000	−0.134231
$889$	−18.0000	−0.603701
$890$	0	0
$891$	−5.00000	−0.167506
$892$	−2.00000	−0.0669650
$893$	−18.0000	−0.602347
$894$	−20.0000	−0.668900
$895$	0	0
$896$	−1.00000	−0.0334077
$897$	3.00000	0.100167
$898$	14.0000	0.467186
$899$	0	0
$900$	0	0
$901$	0	0
$902$	15.0000	0.499445
$903$	5.00000	0.166390
$904$	6.00000	0.199557
$905$	0	0
$906$	18.0000	0.598010
$907$	33.0000	1.09575	0.547874	−	0.836561i	$-0.315438\pi$
$907$	33.0000	1.09575	0.547874	+	0.836561i	$0.315438\pi$
$908$	24.0000	0.796468
$909$	6.00000	0.199007
$910$	0	0
$911$	−9.00000	−0.298183	−0.149092	−	0.988823i	$-0.547635\pi$
$911$	−9.00000	−0.298183	−0.149092	+	0.988823i	$0.547635\pi$
$912$	−3.00000	−0.0993399
$913$	−5.00000	−0.165476
$914$	−18.0000	−0.595387
$915$	0	0
$916$	−8.00000	−0.264327
$917$	−12.0000	−0.396275
$918$	−2.00000	−0.0660098
$919$	4.00000	0.131948	0.0659739	−	0.997821i	$-0.478985\pi$
$919$	4.00000	0.131948	0.0659739	+	0.997821i	$0.478985\pi$
$920$	0	0
$921$	20.0000	0.659022
$922$	−37.0000	−1.21853
$923$	6.00000	0.197492
$924$	5.00000	0.164488
$925$	0	0
$926$	−12.0000	−0.394344
$927$	−7.00000	−0.229910
$928$	−1.00000	−0.0328266
$929$	33.0000	1.08269	0.541347	−	0.840799i	$-0.317914\pi$
$929$	33.0000	1.08269	0.541347	+	0.840799i	$0.317914\pi$
$930$	0	0
$931$	18.0000	0.589926
$932$	11.0000	0.360317
$933$	−20.0000	−0.654771
$934$	27.0000	0.883467
$935$	0	0
$936$	−3.00000	−0.0980581
$937$	−60.0000	−1.96011	−0.980057	−	0.198715i	$-0.936323\pi$
$937$	−60.0000	−1.96011	−0.980057	+	0.198715i	$0.936323\pi$
$938$	−4.00000	−0.130605
$939$	−32.0000	−1.04428
$940$	0	0
$941$	32.0000	1.04317	0.521585	−	0.853199i	$-0.325341\pi$
$941$	32.0000	1.04317	0.521585	+	0.853199i	$0.325341\pi$
$942$	4.00000	0.130327
$943$	3.00000	0.0976934
$944$	−6.00000	−0.195283
$945$	0	0
$946$	25.0000	0.812820
$947$	−32.0000	−1.03986	−0.519930	−	0.854209i	$-0.674042\pi$
$947$	−32.0000	−1.03986	−0.519930	+	0.854209i	$0.674042\pi$
$948$	−7.00000	−0.227349
$949$	27.0000	0.876457
$950$	0	0
$951$	13.0000	0.421554
$952$	2.00000	0.0648204
$953$	−20.0000	−0.647864	−0.323932	−	0.946080i	$-0.605005\pi$
$953$	−20.0000	−0.647864	−0.323932	+	0.946080i	$0.605005\pi$
$954$	0	0
$955$	0	0
$956$	−8.00000	−0.258738
$957$	5.00000	0.161627
$958$	17.0000	0.549245
$959$	16.0000	0.516667
$960$	0	0
$961$	−31.0000	−1.00000
$962$	12.0000	0.386896
$963$	12.0000	0.386695
$964$	−2.00000	−0.0644157
$965$	0	0
$966$	1.00000	0.0321745
$967$	32.0000	1.02905	0.514525	−	0.857475i	$-0.327968\pi$
$967$	32.0000	1.02905	0.514525	+	0.857475i	$0.327968\pi$
$968$	14.0000	0.449977
$969$	6.00000	0.192748
$970$	0	0
$971$	−31.0000	−0.994837	−0.497419	−	0.867511i	$-0.665719\pi$
$971$	−31.0000	−0.994837	−0.497419	+	0.867511i	$0.665719\pi$
$972$	1.00000	0.0320750
$973$	10.0000	0.320585
$974$	−18.0000	−0.576757
$975$	0	0
$976$	4.00000	0.128037
$977$	−24.0000	−0.767828	−0.383914	−	0.923369i	$-0.625424\pi$
$977$	−24.0000	−0.767828	−0.383914	+	0.923369i	$0.625424\pi$
$978$	4.00000	0.127906
$979$	10.0000	0.319601
$980$	0	0
$981$	−4.00000	−0.127710
$982$	0	0
$983$	−9.00000	−0.287055	−0.143528	−	0.989646i	$-0.545845\pi$
$983$	−9.00000	−0.287055	−0.143528	+	0.989646i	$0.545845\pi$
$984$	−3.00000	−0.0956365
$985$	0	0
$986$	2.00000	0.0636930
$987$	−6.00000	−0.190982
$988$	9.00000	0.286328
$989$	5.00000	0.158991
$990$	0	0
$991$	−20.0000	−0.635321	−0.317660	−	0.948205i	$-0.602897\pi$
$991$	−20.0000	−0.635321	−0.317660	+	0.948205i	$0.602897\pi$
$992$	0	0
$993$	20.0000	0.634681
$994$	2.00000	0.0634361
$995$	0	0
$996$	1.00000	0.0316862
$997$	−49.0000	−1.55185	−0.775923	−	0.630828i	$-0.782715\pi$
$997$	−49.0000	−1.55185	−0.775923	+	0.630828i	$0.782715\pi$
$998$	−34.0000	−1.07625
$999$	−4.00000	−0.126554

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3450.2.a.w.1.1	yes	1
5.2	odd	4		3450.2.d.k.2899.2		2
5.3	odd	4		3450.2.d.k.2899.1		2
5.4	even	2		3450.2.a.e.1.1	✓	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3450.2.a.e.1.1	✓	1	5.4	even	2
3450.2.a.w.1.1	yes	1	1.1	even	1	trivial
3450.2.d.k.2899.1		2	5.3	odd	4
3450.2.d.k.2899.2		2	5.2	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 3450 = 2 \cdot 3 \cdot 5^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3450.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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