LMFDB - Embedded newform 1339.2.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1339,2,Mod(1,1339)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1339, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1339.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1339 = 13 \cdot 103 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1339.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:	$10.6919688306$
Analytic rank:	$0$
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	$\chi$	$=$	1339.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+1.00000 q^{2} -1.00000 q^{4} -4.00000 q^{7} -3.00000 q^{8} -3.00000 q^{9} +O(q^{10})$

$q+1.00000 q^{2} -1.00000 q^{4} -4.00000 q^{7} -3.00000 q^{8} -3.00000 q^{9} +6.00000 q^{11} +1.00000 q^{13} -4.00000 q^{14} -1.00000 q^{16} +6.00000 q^{17} -3.00000 q^{18} +4.00000 q^{19} +6.00000 q^{22} -5.00000 q^{25} +1.00000 q^{26} +4.00000 q^{28} +6.00000 q^{29} +6.00000 q^{31} +5.00000 q^{32} +6.00000 q^{34} +3.00000 q^{36} +4.00000 q^{37} +4.00000 q^{38} -10.0000 q^{41} +8.00000 q^{43} -6.00000 q^{44} +6.00000 q^{47} +9.00000 q^{49} -5.00000 q^{50} -1.00000 q^{52} -2.00000 q^{53} +12.0000 q^{56} +6.00000 q^{58} -4.00000 q^{59} +2.00000 q^{61} +6.00000 q^{62} +12.0000 q^{63} +7.00000 q^{64} -2.00000 q^{67} -6.00000 q^{68} -2.00000 q^{71} +9.00000 q^{72} +8.00000 q^{73} +4.00000 q^{74} -4.00000 q^{76} -24.0000 q^{77} -8.00000 q^{79} +9.00000 q^{81} -10.0000 q^{82} -8.00000 q^{83} +8.00000 q^{86} -18.0000 q^{88} -12.0000 q^{89} -4.00000 q^{91} +6.00000 q^{94} +2.00000 q^{97} +9.00000 q^{98} -18.0000 q^{99} +O(q^{100})$

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	0.353553	−	0.935414i	$-0.384973\pi$
$2$	1.00000	0.707107	0.353553	+	0.935414i	$0.384973\pi$
$3$	0	0		−	1.00000i	$-0.5\pi$
$3$	0	0			1.00000i	$0.5\pi$
$4$	−1.00000	−0.500000
$5$	0	0		−	1.00000i	$-0.5\pi$
$5$	0	0			1.00000i	$0.5\pi$
$6$	0	0
$7$	−4.00000	−1.51186	−0.755929	−	0.654654i	$-0.772814\pi$
$7$	−4.00000	−1.51186	−0.755929	+	0.654654i	$0.772814\pi$
$8$	−3.00000	−1.06066
$9$	−3.00000	−1.00000
$10$	0	0
$11$	6.00000	1.80907	0.904534	−	0.426401i	$-0.140219\pi$
$11$	6.00000	1.80907	0.904534	+	0.426401i	$0.140219\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	−4.00000	−1.06904
$15$	0	0
$16$	−1.00000	−0.250000
$17$	6.00000	1.45521	0.727607	−	0.685994i	$-0.240633\pi$
$17$	6.00000	1.45521	0.727607	+	0.685994i	$0.240633\pi$
$18$	−3.00000	−0.707107
$19$	4.00000	0.917663	0.458831	−	0.888523i	$-0.348268\pi$
$19$	4.00000	0.917663	0.458831	+	0.888523i	$0.348268\pi$
$20$	0	0
$21$	0	0
$22$	6.00000	1.27920
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	0	0
$25$	−5.00000	−1.00000
$26$	1.00000	0.196116
$27$	0	0
$28$	4.00000	0.755929
$29$	6.00000	1.11417	0.557086	−	0.830455i	$-0.311919\pi$
$29$	6.00000	1.11417	0.557086	+	0.830455i	$0.311919\pi$
$30$	0	0
$31$	6.00000	1.07763	0.538816	−	0.842424i	$-0.318872\pi$
$31$	6.00000	1.07763	0.538816	+	0.842424i	$0.318872\pi$
$32$	5.00000	0.883883
$33$	0	0
$34$	6.00000	1.02899
$35$	0	0
$36$	3.00000	0.500000
$37$	4.00000	0.657596	0.328798	−	0.944400i	$-0.393356\pi$
$37$	4.00000	0.657596	0.328798	+	0.944400i	$0.393356\pi$
$38$	4.00000	0.648886
$39$	0	0
$40$	0	0
$41$	−10.0000	−1.56174	−0.780869	−	0.624695i	$-0.785223\pi$
$41$	−10.0000	−1.56174	−0.780869	+	0.624695i	$0.785223\pi$
$42$	0	0
$43$	8.00000	1.21999	0.609994	−	0.792406i	$-0.291172\pi$
$43$	8.00000	1.21999	0.609994	+	0.792406i	$0.291172\pi$
$44$	−6.00000	−0.904534
$45$	0	0
$46$	0	0
$47$	6.00000	0.875190	0.437595	−	0.899172i	$-0.355830\pi$
$47$	6.00000	0.875190	0.437595	+	0.899172i	$0.355830\pi$
$48$	0	0
$49$	9.00000	1.28571
$50$	−5.00000	−0.707107
$51$	0	0
$52$	−1.00000	−0.138675
$53$	−2.00000	−0.274721	−0.137361	−	0.990521i	$-0.543862\pi$
$53$	−2.00000	−0.274721	−0.137361	+	0.990521i	$0.543862\pi$
$54$	0	0
$55$	0	0
$56$	12.0000	1.60357
$57$	0	0
$58$	6.00000	0.787839
$59$	−4.00000	−0.520756	−0.260378	−	0.965507i	$-0.583847\pi$
$59$	−4.00000	−0.520756	−0.260378	+	0.965507i	$0.583847\pi$
$60$	0	0
$61$	2.00000	0.256074	0.128037	−	0.991769i	$-0.459132\pi$
$61$	2.00000	0.256074	0.128037	+	0.991769i	$0.459132\pi$
$62$	6.00000	0.762001
$63$	12.0000	1.51186
$64$	7.00000	0.875000
$65$	0	0
$66$	0	0
$67$	−2.00000	−0.244339	−0.122169	−	0.992509i	$-0.538985\pi$
$67$	−2.00000	−0.244339	−0.122169	+	0.992509i	$0.538985\pi$
$68$	−6.00000	−0.727607
$69$	0	0
$70$	0	0
$71$	−2.00000	−0.237356	−0.118678	−	0.992933i	$-0.537866\pi$
$71$	−2.00000	−0.237356	−0.118678	+	0.992933i	$0.537866\pi$
$72$	9.00000	1.06066
$73$	8.00000	0.936329	0.468165	−	0.883641i	$-0.344915\pi$
$73$	8.00000	0.936329	0.468165	+	0.883641i	$0.344915\pi$
$74$	4.00000	0.464991
$75$	0	0
$76$	−4.00000	−0.458831
$77$	−24.0000	−2.73505
$78$	0	0
$79$	−8.00000	−0.900070	−0.450035	−	0.893011i	$-0.648589\pi$
$79$	−8.00000	−0.900070	−0.450035	+	0.893011i	$0.648589\pi$
$80$	0	0
$81$	9.00000	1.00000
$82$	−10.0000	−1.10432
$83$	−8.00000	−0.878114	−0.439057	−	0.898459i	$-0.644687\pi$
$83$	−8.00000	−0.878114	−0.439057	+	0.898459i	$0.644687\pi$
$84$	0	0
$85$	0	0
$86$	8.00000	0.862662
$87$	0	0
$88$	−18.0000	−1.91881
$89$	−12.0000	−1.27200	−0.635999	−	0.771690i	$-0.719412\pi$
$89$	−12.0000	−1.27200	−0.635999	+	0.771690i	$0.719412\pi$
$90$	0	0
$91$	−4.00000	−0.419314
$92$	0	0
$93$	0	0
$94$	6.00000	0.618853
$95$	0	0
$96$	0	0
$97$	2.00000	0.203069	0.101535	−	0.994832i	$-0.467625\pi$
$97$	2.00000	0.203069	0.101535	+	0.994832i	$0.467625\pi$
$98$	9.00000	0.909137
$99$	−18.0000	−1.80907
$100$	5.00000	0.500000
$101$	18.0000	1.79107	0.895533	−	0.444994i	$-0.146794\pi$
$101$	18.0000	1.79107	0.895533	+	0.444994i	$0.146794\pi$
$102$	0	0
$103$	−1.00000	−0.0985329
$103$	−1.00000	−0.0985329
$104$	−3.00000	−0.294174
$105$	0	0
$106$	−2.00000	−0.194257
$107$	−4.00000	−0.386695	−0.193347	−	0.981130i	$-0.561934\pi$
$107$	−4.00000	−0.386695	−0.193347	+	0.981130i	$0.561934\pi$
$108$	0	0
$109$	0	0		−	1.00000i	$-0.5\pi$
$109$	0	0			1.00000i	$0.5\pi$
$110$	0	0
$111$	0	0
$112$	4.00000	0.377964
$113$	−2.00000	−0.188144	−0.0940721	−	0.995565i	$-0.529988\pi$
$113$	−2.00000	−0.188144	−0.0940721	+	0.995565i	$0.529988\pi$
$114$	0	0
$115$	0	0
$116$	−6.00000	−0.557086
$117$	−3.00000	−0.277350
$118$	−4.00000	−0.368230
$119$	−24.0000	−2.20008
$120$	0	0
$121$	25.0000	2.27273
$122$	2.00000	0.181071
$123$	0	0
$124$	−6.00000	−0.538816
$125$	0	0
$126$	12.0000	1.06904
$127$	12.0000	1.06483	0.532414	−	0.846484i	$-0.321285\pi$
$127$	12.0000	1.06483	0.532414	+	0.846484i	$0.321285\pi$
$128$	−3.00000	−0.265165
$129$	0	0
$130$	0	0
$131$	−4.00000	−0.349482	−0.174741	−	0.984614i	$-0.555909\pi$
$131$	−4.00000	−0.349482	−0.174741	+	0.984614i	$0.555909\pi$
$132$	0	0
$133$	−16.0000	−1.38738
$134$	−2.00000	−0.172774
$135$	0	0
$136$	−18.0000	−1.54349
$137$	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
$139$	−4.00000	−0.339276	−0.169638	−	0.985506i	$-0.554260\pi$
$139$	−4.00000	−0.339276	−0.169638	+	0.985506i	$0.554260\pi$
$140$	0	0
$141$	0	0
$142$	−2.00000	−0.167836
$143$	6.00000	0.501745
$144$	3.00000	0.250000
$145$	0	0
$146$	8.00000	0.662085
$147$	0	0
$148$	−4.00000	−0.328798
$149$	22.0000	1.80231	0.901155	−	0.433497i	$-0.142720\pi$
$149$	22.0000	1.80231	0.901155	+	0.433497i	$0.142720\pi$
$150$	0	0
$151$	14.0000	1.13930	0.569652	−	0.821886i	$-0.307078\pi$
$151$	14.0000	1.13930	0.569652	+	0.821886i	$0.307078\pi$
$152$	−12.0000	−0.973329
$153$	−18.0000	−1.45521
$154$	−24.0000	−1.93398
$155$	0	0
$156$	0	0
$157$	2.00000	0.159617	0.0798087	−	0.996810i	$-0.474569\pi$
$157$	2.00000	0.159617	0.0798087	+	0.996810i	$0.474569\pi$
$158$	−8.00000	−0.636446
$159$	0	0
$160$	0	0
$161$	0	0
$162$	9.00000	0.707107
$163$	−4.00000	−0.313304	−0.156652	−	0.987654i	$-0.550070\pi$
$163$	−4.00000	−0.313304	−0.156652	+	0.987654i	$0.550070\pi$
$164$	10.0000	0.780869
$165$	0	0
$166$	−8.00000	−0.620920
$167$	12.0000	0.928588	0.464294	−	0.885681i	$-0.346308\pi$
$167$	12.0000	0.928588	0.464294	+	0.885681i	$0.346308\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−12.0000	−0.917663
$172$	−8.00000	−0.609994
$173$	−6.00000	−0.456172	−0.228086	−	0.973641i	$-0.573247\pi$
$173$	−6.00000	−0.456172	−0.228086	+	0.973641i	$0.573247\pi$
$174$	0	0
$175$	20.0000	1.51186
$176$	−6.00000	−0.452267
$177$	0	0
$178$	−12.0000	−0.899438
$179$	4.00000	0.298974	0.149487	−	0.988764i	$-0.452238\pi$
$179$	4.00000	0.298974	0.149487	+	0.988764i	$0.452238\pi$
$180$	0	0
$181$	6.00000	0.445976	0.222988	−	0.974821i	$-0.428419\pi$
$181$	6.00000	0.445976	0.222988	+	0.974821i	$0.428419\pi$
$182$	−4.00000	−0.296500
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	36.0000	2.63258
$188$	−6.00000	−0.437595
$189$	0	0
$190$	0	0
$191$	20.0000	1.44715	0.723575	−	0.690246i	$-0.242498\pi$
$191$	20.0000	1.44715	0.723575	+	0.690246i	$0.242498\pi$
$192$	0	0
$193$	4.00000	0.287926	0.143963	−	0.989583i	$-0.454015\pi$
$193$	4.00000	0.287926	0.143963	+	0.989583i	$0.454015\pi$
$194$	2.00000	0.143592
$195$	0	0
$196$	−9.00000	−0.642857
$197$	−8.00000	−0.569976	−0.284988	−	0.958531i	$-0.591990\pi$
$197$	−8.00000	−0.569976	−0.284988	+	0.958531i	$0.591990\pi$
$198$	−18.0000	−1.27920
$199$	4.00000	0.283552	0.141776	−	0.989899i	$-0.454719\pi$
$199$	4.00000	0.283552	0.141776	+	0.989899i	$0.454719\pi$
$200$	15.0000	1.06066
$201$	0	0
$202$	18.0000	1.26648
$203$	−24.0000	−1.68447
$204$	0	0
$205$	0	0
$206$	−1.00000	−0.0696733
$207$	0	0
$208$	−1.00000	−0.0693375
$209$	24.0000	1.66011
$210$	0	0
$211$	4.00000	0.275371	0.137686	−	0.990476i	$-0.456034\pi$
$211$	4.00000	0.275371	0.137686	+	0.990476i	$0.456034\pi$
$212$	2.00000	0.137361
$213$	0	0
$214$	−4.00000	−0.273434
$215$	0	0
$216$	0	0
$217$	−24.0000	−1.62923
$218$	0	0
$219$	0	0
$220$	0	0
$221$	6.00000	0.403604
$222$	0	0
$223$	0	0		−	1.00000i	$-0.5\pi$
$223$	0	0			1.00000i	$0.5\pi$
$224$	−20.0000	−1.33631
$225$	15.0000	1.00000
$226$	−2.00000	−0.133038
$227$	−10.0000	−0.663723	−0.331862	−	0.943328i	$-0.607677\pi$
$227$	−10.0000	−0.663723	−0.331862	+	0.943328i	$0.607677\pi$
$228$	0	0
$229$	−10.0000	−0.660819	−0.330409	−	0.943838i	$-0.607187\pi$
$229$	−10.0000	−0.660819	−0.330409	+	0.943838i	$0.607187\pi$
$230$	0	0
$231$	0	0
$232$	−18.0000	−1.18176
$233$	−18.0000	−1.17922	−0.589610	−	0.807688i	$-0.700718\pi$
$233$	−18.0000	−1.17922	−0.589610	+	0.807688i	$0.700718\pi$
$234$	−3.00000	−0.196116
$235$	0	0
$236$	4.00000	0.260378
$237$	0	0
$238$	−24.0000	−1.55569
$239$	−24.0000	−1.55243	−0.776215	−	0.630468i	$-0.782863\pi$
$239$	−24.0000	−1.55243	−0.776215	+	0.630468i	$0.782863\pi$
$240$	0	0
$241$	28.0000	1.80364	0.901819	−	0.432113i	$-0.142232\pi$
$241$	28.0000	1.80364	0.901819	+	0.432113i	$0.142232\pi$
$242$	25.0000	1.60706
$243$	0	0
$244$	−2.00000	−0.128037
$245$	0	0
$246$	0	0
$247$	4.00000	0.254514
$248$	−18.0000	−1.14300
$249$	0	0
$250$	0	0
$251$	−20.0000	−1.26239	−0.631194	−	0.775625i	$-0.717435\pi$
$251$	−20.0000	−1.26239	−0.631194	+	0.775625i	$0.717435\pi$
$252$	−12.0000	−0.755929
$253$	0	0
$254$	12.0000	0.752947
$255$	0	0
$256$	−17.0000	−1.06250
$257$	−18.0000	−1.12281	−0.561405	−	0.827541i	$-0.689739\pi$
$257$	−18.0000	−1.12281	−0.561405	+	0.827541i	$0.689739\pi$
$258$	0	0
$259$	−16.0000	−0.994192
$260$	0	0
$261$	−18.0000	−1.11417
$262$	−4.00000	−0.247121
$263$	−28.0000	−1.72655	−0.863277	−	0.504730i	$-0.831592\pi$
$263$	−28.0000	−1.72655	−0.863277	+	0.504730i	$0.831592\pi$
$264$	0	0
$265$	0	0
$266$	−16.0000	−0.981023
$267$	0	0
$268$	2.00000	0.122169
$269$	18.0000	1.09748	0.548740	−	0.835993i	$-0.315108\pi$
$269$	18.0000	1.09748	0.548740	+	0.835993i	$0.315108\pi$
$270$	0	0
$271$	−2.00000	−0.121491	−0.0607457	−	0.998153i	$-0.519348\pi$
$271$	−2.00000	−0.121491	−0.0607457	+	0.998153i	$0.519348\pi$
$272$	−6.00000	−0.363803
$273$	0	0
$274$	18.0000	1.08742
$275$	−30.0000	−1.80907
$276$	0	0
$277$	−22.0000	−1.32185	−0.660926	−	0.750451i	$-0.729836\pi$
$277$	−22.0000	−1.32185	−0.660926	+	0.750451i	$0.729836\pi$
$278$	−4.00000	−0.239904
$279$	−18.0000	−1.07763
$280$	0	0
$281$	24.0000	1.43172	0.715860	−	0.698244i	$-0.246035\pi$
$281$	24.0000	1.43172	0.715860	+	0.698244i	$0.246035\pi$
$282$	0	0
$283$	−20.0000	−1.18888	−0.594438	−	0.804141i	$-0.702626\pi$
$283$	−20.0000	−1.18888	−0.594438	+	0.804141i	$0.702626\pi$
$284$	2.00000	0.118678
$285$	0	0
$286$	6.00000	0.354787
$287$	40.0000	2.36113
$288$	−15.0000	−0.883883
$289$	19.0000	1.11765
$290$	0	0
$291$	0	0
$292$	−8.00000	−0.468165
$293$	24.0000	1.40209	0.701047	−	0.713115i	$-0.252716\pi$
$293$	24.0000	1.40209	0.701047	+	0.713115i	$0.252716\pi$
$294$	0	0
$295$	0	0
$296$	−12.0000	−0.697486
$297$	0	0
$298$	22.0000	1.27443
$299$	0	0
$300$	0	0
$301$	−32.0000	−1.84445
$302$	14.0000	0.805609
$303$	0	0
$304$	−4.00000	−0.229416
$305$	0	0
$306$	−18.0000	−1.02899
$307$	−26.0000	−1.48390	−0.741949	−	0.670456i	$-0.766098\pi$
$307$	−26.0000	−1.48390	−0.741949	+	0.670456i	$0.766098\pi$
$308$	24.0000	1.36753
$309$	0	0
$310$	0	0
$311$	24.0000	1.36092	0.680458	−	0.732787i	$-0.261781\pi$
$311$	24.0000	1.36092	0.680458	+	0.732787i	$0.261781\pi$
$312$	0	0
$313$	10.0000	0.565233	0.282617	−	0.959233i	$-0.408798\pi$
$313$	10.0000	0.565233	0.282617	+	0.959233i	$0.408798\pi$
$314$	2.00000	0.112867
$315$	0	0
$316$	8.00000	0.450035
$317$	2.00000	0.112331	0.0561656	−	0.998421i	$-0.482113\pi$
$317$	2.00000	0.112331	0.0561656	+	0.998421i	$0.482113\pi$
$318$	0	0
$319$	36.0000	2.01561
$320$	0	0
$321$	0	0
$322$	0	0
$323$	24.0000	1.33540
$324$	−9.00000	−0.500000
$325$	−5.00000	−0.277350
$326$	−4.00000	−0.221540
$327$	0	0
$328$	30.0000	1.65647
$329$	−24.0000	−1.32316
$330$	0	0
$331$	−34.0000	−1.86881	−0.934405	−	0.356214i	$-0.884068\pi$
$331$	−34.0000	−1.86881	−0.934405	+	0.356214i	$0.884068\pi$
$332$	8.00000	0.439057
$333$	−12.0000	−0.657596
$334$	12.0000	0.656611
$335$	0	0
$336$	0	0
$337$	2.00000	0.108947	0.0544735	−	0.998515i	$-0.482652\pi$
$337$	2.00000	0.108947	0.0544735	+	0.998515i	$0.482652\pi$
$338$	1.00000	0.0543928
$339$	0	0
$340$	0	0
$341$	36.0000	1.94951
$342$	−12.0000	−0.648886
$343$	−8.00000	−0.431959
$344$	−24.0000	−1.29399
$345$	0	0
$346$	−6.00000	−0.322562
$347$	28.0000	1.50312	0.751559	−	0.659665i	$-0.229302\pi$
$347$	28.0000	1.50312	0.751559	+	0.659665i	$0.229302\pi$
$348$	0	0
$349$	4.00000	0.214115	0.107058	−	0.994253i	$-0.465857\pi$
$349$	4.00000	0.214115	0.107058	+	0.994253i	$0.465857\pi$
$350$	20.0000	1.06904
$351$	0	0
$352$	30.0000	1.59901
$353$	−4.00000	−0.212899	−0.106449	−	0.994318i	$-0.533948\pi$
$353$	−4.00000	−0.212899	−0.106449	+	0.994318i	$0.533948\pi$
$354$	0	0
$355$	0	0
$356$	12.0000	0.635999
$357$	0	0
$358$	4.00000	0.211407
$359$	−4.00000	−0.211112	−0.105556	−	0.994413i	$-0.533662\pi$
$359$	−4.00000	−0.211112	−0.105556	+	0.994413i	$0.533662\pi$
$360$	0	0
$361$	−3.00000	−0.157895
$362$	6.00000	0.315353
$363$	0	0
$364$	4.00000	0.209657
$365$	0	0
$366$	0	0
$367$	−8.00000	−0.417597	−0.208798	−	0.977959i	$-0.566955\pi$
$367$	−8.00000	−0.417597	−0.208798	+	0.977959i	$0.566955\pi$
$368$	0	0
$369$	30.0000	1.56174
$370$	0	0
$371$	8.00000	0.415339
$372$	0	0
$373$	10.0000	0.517780	0.258890	−	0.965907i	$-0.416643\pi$
$373$	10.0000	0.517780	0.258890	+	0.965907i	$0.416643\pi$
$374$	36.0000	1.86152
$375$	0	0
$376$	−18.0000	−0.928279
$377$	6.00000	0.309016
$378$	0	0
$379$	18.0000	0.924598	0.462299	−	0.886724i	$-0.347025\pi$
$379$	18.0000	0.924598	0.462299	+	0.886724i	$0.347025\pi$
$380$	0	0
$381$	0	0
$382$	20.0000	1.02329
$383$	−34.0000	−1.73732	−0.868659	−	0.495410i	$-0.835018\pi$
$383$	−34.0000	−1.73732	−0.868659	+	0.495410i	$0.835018\pi$
$384$	0	0
$385$	0	0
$386$	4.00000	0.203595
$387$	−24.0000	−1.21999
$388$	−2.00000	−0.101535
$389$	−26.0000	−1.31825	−0.659126	−	0.752032i	$-0.729074\pi$
$389$	−26.0000	−1.31825	−0.659126	+	0.752032i	$0.729074\pi$
$390$	0	0
$391$	0	0
$392$	−27.0000	−1.36371
$393$	0	0
$394$	−8.00000	−0.403034
$395$	0	0
$396$	18.0000	0.904534
$397$	32.0000	1.60603	0.803017	−	0.595956i	$-0.203227\pi$
$397$	32.0000	1.60603	0.803017	+	0.595956i	$0.203227\pi$
$398$	4.00000	0.200502
$399$	0	0
$400$	5.00000	0.250000
$401$	22.0000	1.09863	0.549314	−	0.835616i	$-0.314889\pi$
$401$	22.0000	1.09863	0.549314	+	0.835616i	$0.314889\pi$
$402$	0	0
$403$	6.00000	0.298881
$404$	−18.0000	−0.895533
$405$	0	0
$406$	−24.0000	−1.19110
$407$	24.0000	1.18964
$408$	0	0
$409$	−18.0000	−0.890043	−0.445021	−	0.895520i	$-0.646804\pi$
$409$	−18.0000	−0.890043	−0.445021	+	0.895520i	$0.646804\pi$
$410$	0	0
$411$	0	0
$412$	1.00000	0.0492665
$413$	16.0000	0.787309
$414$	0	0
$415$	0	0
$416$	5.00000	0.245145
$417$	0	0
$418$	24.0000	1.17388
$419$	−4.00000	−0.195413	−0.0977064	−	0.995215i	$-0.531151\pi$
$419$	−4.00000	−0.195413	−0.0977064	+	0.995215i	$0.531151\pi$
$420$	0	0
$421$	26.0000	1.26716	0.633581	−	0.773676i	$-0.281584\pi$
$421$	26.0000	1.26716	0.633581	+	0.773676i	$0.281584\pi$
$422$	4.00000	0.194717
$423$	−18.0000	−0.875190
$424$	6.00000	0.291386
$425$	−30.0000	−1.45521
$426$	0	0
$427$	−8.00000	−0.387147
$428$	4.00000	0.193347
$429$	0	0
$430$	0	0
$431$	36.0000	1.73406	0.867029	−	0.498257i	$-0.166026\pi$
$431$	36.0000	1.73406	0.867029	+	0.498257i	$0.166026\pi$
$432$	0	0
$433$	−26.0000	−1.24948	−0.624740	−	0.780833i	$-0.714795\pi$
$433$	−26.0000	−1.24948	−0.624740	+	0.780833i	$0.714795\pi$
$434$	−24.0000	−1.15204
$435$	0	0
$436$	0	0
$437$	0	0
$438$	0	0
$439$	28.0000	1.33637	0.668184	−	0.743996i	$-0.267072\pi$
$439$	28.0000	1.33637	0.668184	+	0.743996i	$0.267072\pi$
$440$	0	0
$441$	−27.0000	−1.28571
$442$	6.00000	0.285391
$443$	−24.0000	−1.14027	−0.570137	−	0.821549i	$-0.693110\pi$
$443$	−24.0000	−1.14027	−0.570137	+	0.821549i	$0.693110\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	−28.0000	−1.32288
$449$	−24.0000	−1.13263	−0.566315	−	0.824189i	$-0.691631\pi$
$449$	−24.0000	−1.13263	−0.566315	+	0.824189i	$0.691631\pi$
$450$	15.0000	0.707107
$451$	−60.0000	−2.82529
$452$	2.00000	0.0940721
$453$	0	0
$454$	−10.0000	−0.469323
$455$	0	0
$456$	0	0
$457$	−8.00000	−0.374224	−0.187112	−	0.982339i	$-0.559913\pi$
$457$	−8.00000	−0.374224	−0.187112	+	0.982339i	$0.559913\pi$
$458$	−10.0000	−0.467269
$459$	0	0
$460$	0	0
$461$	22.0000	1.02464	0.512321	−	0.858794i	$-0.328786\pi$
$461$	22.0000	1.02464	0.512321	+	0.858794i	$0.328786\pi$
$462$	0	0
$463$	−18.0000	−0.836531	−0.418265	−	0.908325i	$-0.637362\pi$
$463$	−18.0000	−0.836531	−0.418265	+	0.908325i	$0.637362\pi$
$464$	−6.00000	−0.278543
$465$	0	0
$466$	−18.0000	−0.833834
$467$	−36.0000	−1.66588	−0.832941	−	0.553362i	$-0.813345\pi$
$467$	−36.0000	−1.66588	−0.832941	+	0.553362i	$0.813345\pi$
$468$	3.00000	0.138675
$469$	8.00000	0.369406
$470$	0	0
$471$	0	0
$472$	12.0000	0.552345
$473$	48.0000	2.20704
$474$	0	0
$475$	−20.0000	−0.917663
$476$	24.0000	1.10004
$477$	6.00000	0.274721
$478$	−24.0000	−1.09773
$479$	30.0000	1.37073	0.685367	−	0.728197i	$-0.259642\pi$
$479$	30.0000	1.37073	0.685367	+	0.728197i	$0.259642\pi$
$480$	0	0
$481$	4.00000	0.182384
$482$	28.0000	1.27537
$483$	0	0
$484$	−25.0000	−1.13636
$485$	0	0
$486$	0	0
$487$	−30.0000	−1.35943	−0.679715	−	0.733476i	$-0.737896\pi$
$487$	−30.0000	−1.35943	−0.679715	+	0.733476i	$0.737896\pi$
$488$	−6.00000	−0.271607
$489$	0	0
$490$	0	0
$491$	12.0000	0.541552	0.270776	−	0.962642i	$-0.412720\pi$
$491$	12.0000	0.541552	0.270776	+	0.962642i	$0.412720\pi$
$492$	0	0
$493$	36.0000	1.62136
$494$	4.00000	0.179969
$495$	0	0
$496$	−6.00000	−0.269408
$497$	8.00000	0.358849
$498$	0	0
$499$	10.0000	0.447661	0.223831	−	0.974628i	$-0.428144\pi$
$499$	10.0000	0.447661	0.223831	+	0.974628i	$0.428144\pi$
$500$	0	0
$501$	0	0
$502$	−20.0000	−0.892644
$503$	8.00000	0.356702	0.178351	−	0.983967i	$-0.442924\pi$
$503$	8.00000	0.356702	0.178351	+	0.983967i	$0.442924\pi$
$504$	−36.0000	−1.60357
$505$	0	0
$506$	0	0
$507$	0	0
$508$	−12.0000	−0.532414
$509$	−6.00000	−0.265945	−0.132973	−	0.991120i	$-0.542452\pi$
$509$	−6.00000	−0.265945	−0.132973	+	0.991120i	$0.542452\pi$
$510$	0	0
$511$	−32.0000	−1.41560
$512$	−11.0000	−0.486136
$513$	0	0
$514$	−18.0000	−0.793946
$515$	0	0
$516$	0	0
$517$	36.0000	1.58328
$518$	−16.0000	−0.703000
$519$	0	0
$520$	0	0
$521$	−26.0000	−1.13908	−0.569540	−	0.821963i	$-0.692879\pi$
$521$	−26.0000	−1.13908	−0.569540	+	0.821963i	$0.692879\pi$
$522$	−18.0000	−0.787839
$523$	36.0000	1.57417	0.787085	−	0.616844i	$-0.211589\pi$
$523$	36.0000	1.57417	0.787085	+	0.616844i	$0.211589\pi$
$524$	4.00000	0.174741
$525$	0	0
$526$	−28.0000	−1.22086
$527$	36.0000	1.56818
$528$	0	0
$529$	−23.0000	−1.00000
$530$	0	0
$531$	12.0000	0.520756
$532$	16.0000	0.693688
$533$	−10.0000	−0.433148
$534$	0	0
$535$	0	0
$536$	6.00000	0.259161
$537$	0	0
$538$	18.0000	0.776035
$539$	54.0000	2.32594
$540$	0	0
$541$	−30.0000	−1.28980	−0.644900	−	0.764267i	$-0.723101\pi$
$541$	−30.0000	−1.28980	−0.644900	+	0.764267i	$0.723101\pi$
$542$	−2.00000	−0.0859074
$543$	0	0
$544$	30.0000	1.28624
$545$	0	0
$546$	0	0
$547$	−44.0000	−1.88130	−0.940652	−	0.339372i	$-0.889785\pi$
$547$	−44.0000	−1.88130	−0.940652	+	0.339372i	$0.889785\pi$
$548$	−18.0000	−0.768922
$549$	−6.00000	−0.256074
$550$	−30.0000	−1.27920
$551$	24.0000	1.02243
$552$	0	0
$553$	32.0000	1.36078
$554$	−22.0000	−0.934690
$555$	0	0
$556$	4.00000	0.169638
$557$	−4.00000	−0.169485	−0.0847427	−	0.996403i	$-0.527007\pi$
$557$	−4.00000	−0.169485	−0.0847427	+	0.996403i	$0.527007\pi$
$558$	−18.0000	−0.762001
$559$	8.00000	0.338364
$560$	0	0
$561$	0	0
$562$	24.0000	1.01238
$563$	−44.0000	−1.85438	−0.927189	−	0.374593i	$-0.877783\pi$
$563$	−44.0000	−1.85438	−0.927189	+	0.374593i	$0.877783\pi$
$564$	0	0
$565$	0	0
$566$	−20.0000	−0.840663
$567$	−36.0000	−1.51186
$568$	6.00000	0.251754
$569$	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
$571$	36.0000	1.50655	0.753277	−	0.657704i	$-0.228472\pi$
$571$	36.0000	1.50655	0.753277	+	0.657704i	$0.228472\pi$
$572$	−6.00000	−0.250873
$573$	0	0
$574$	40.0000	1.66957
$575$	0	0
$576$	−21.0000	−0.875000
$577$	−16.0000	−0.666089	−0.333044	−	0.942911i	$-0.608076\pi$
$577$	−16.0000	−0.666089	−0.333044	+	0.942911i	$0.608076\pi$
$578$	19.0000	0.790296
$579$	0	0
$580$	0	0
$581$	32.0000	1.32758
$582$	0	0
$583$	−12.0000	−0.496989
$584$	−24.0000	−0.993127
$585$	0	0
$586$	24.0000	0.991431
$587$	−40.0000	−1.65098	−0.825488	−	0.564419i	$-0.809100\pi$
$587$	−40.0000	−1.65098	−0.825488	+	0.564419i	$0.809100\pi$
$588$	0	0
$589$	24.0000	0.988903
$590$	0	0
$591$	0	0
$592$	−4.00000	−0.164399
$593$	28.0000	1.14982	0.574911	−	0.818216i	$-0.305037\pi$
$593$	28.0000	1.14982	0.574911	+	0.818216i	$0.305037\pi$
$594$	0	0
$595$	0	0
$596$	−22.0000	−0.901155
$597$	0	0
$598$	0	0
$599$	−16.0000	−0.653742	−0.326871	−	0.945069i	$-0.605994\pi$
$599$	−16.0000	−0.653742	−0.326871	+	0.945069i	$0.605994\pi$
$600$	0	0
$601$	22.0000	0.897399	0.448699	−	0.893683i	$-0.351887\pi$
$601$	22.0000	0.897399	0.448699	+	0.893683i	$0.351887\pi$
$602$	−32.0000	−1.30422
$603$	6.00000	0.244339
$604$	−14.0000	−0.569652
$605$	0	0
$606$	0	0
$607$	8.00000	0.324710	0.162355	−	0.986732i	$-0.448091\pi$
$607$	8.00000	0.324710	0.162355	+	0.986732i	$0.448091\pi$
$608$	20.0000	0.811107
$609$	0	0
$610$	0	0
$611$	6.00000	0.242734
$612$	18.0000	0.727607
$613$	34.0000	1.37325	0.686624	−	0.727013i	$-0.259092\pi$
$613$	34.0000	1.37325	0.686624	+	0.727013i	$0.259092\pi$
$614$	−26.0000	−1.04927
$615$	0	0
$616$	72.0000	2.90096
$617$	20.0000	0.805170	0.402585	−	0.915383i	$-0.368112\pi$
$617$	20.0000	0.805170	0.402585	+	0.915383i	$0.368112\pi$
$618$	0	0
$619$	32.0000	1.28619	0.643094	−	0.765787i	$-0.277650\pi$
$619$	32.0000	1.28619	0.643094	+	0.765787i	$0.277650\pi$
$620$	0	0
$621$	0	0
$622$	24.0000	0.962312
$623$	48.0000	1.92308
$624$	0	0
$625$	25.0000	1.00000
$626$	10.0000	0.399680
$627$	0	0
$628$	−2.00000	−0.0798087
$629$	24.0000	0.956943
$630$	0	0
$631$	−20.0000	−0.796187	−0.398094	−	0.917345i	$-0.630328\pi$
$631$	−20.0000	−0.796187	−0.398094	+	0.917345i	$0.630328\pi$
$632$	24.0000	0.954669
$633$	0	0
$634$	2.00000	0.0794301
$635$	0	0
$636$	0	0
$637$	9.00000	0.356593
$638$	36.0000	1.42525
$639$	6.00000	0.237356
$640$	0	0
$641$	2.00000	0.0789953	0.0394976	−	0.999220i	$-0.487424\pi$
$641$	2.00000	0.0789953	0.0394976	+	0.999220i	$0.487424\pi$
$642$	0	0
$643$	−32.0000	−1.26196	−0.630978	−	0.775800i	$-0.717346\pi$
$643$	−32.0000	−1.26196	−0.630978	+	0.775800i	$0.717346\pi$
$644$	0	0
$645$	0	0
$646$	24.0000	0.944267
$647$	−8.00000	−0.314512	−0.157256	−	0.987558i	$-0.550265\pi$
$647$	−8.00000	−0.314512	−0.157256	+	0.987558i	$0.550265\pi$
$648$	−27.0000	−1.06066
$649$	−24.0000	−0.942082
$650$	−5.00000	−0.196116
$651$	0	0
$652$	4.00000	0.156652
$653$	−6.00000	−0.234798	−0.117399	−	0.993085i	$-0.537456\pi$
$653$	−6.00000	−0.234798	−0.117399	+	0.993085i	$0.537456\pi$
$654$	0	0
$655$	0	0
$656$	10.0000	0.390434
$657$	−24.0000	−0.936329
$658$	−24.0000	−0.935617
$659$	12.0000	0.467454	0.233727	−	0.972302i	$-0.424908\pi$
$659$	12.0000	0.467454	0.233727	+	0.972302i	$0.424908\pi$
$660$	0	0
$661$	−16.0000	−0.622328	−0.311164	−	0.950356i	$-0.600719\pi$
$661$	−16.0000	−0.622328	−0.311164	+	0.950356i	$0.600719\pi$
$662$	−34.0000	−1.32145
$663$	0	0
$664$	24.0000	0.931381
$665$	0	0
$666$	−12.0000	−0.464991
$667$	0	0
$668$	−12.0000	−0.464294
$669$	0	0
$670$	0	0
$671$	12.0000	0.463255
$672$	0	0
$673$	−14.0000	−0.539660	−0.269830	−	0.962908i	$-0.586968\pi$
$673$	−14.0000	−0.539660	−0.269830	+	0.962908i	$0.586968\pi$
$674$	2.00000	0.0770371
$675$	0	0
$676$	−1.00000	−0.0384615
$677$	6.00000	0.230599	0.115299	−	0.993331i	$-0.463217\pi$
$677$	6.00000	0.230599	0.115299	+	0.993331i	$0.463217\pi$
$678$	0	0
$679$	−8.00000	−0.307012
$680$	0	0
$681$	0	0
$682$	36.0000	1.37851
$683$	18.0000	0.688751	0.344375	−	0.938832i	$-0.388091\pi$
$683$	18.0000	0.688751	0.344375	+	0.938832i	$0.388091\pi$
$684$	12.0000	0.458831
$685$	0	0
$686$	−8.00000	−0.305441
$687$	0	0
$688$	−8.00000	−0.304997
$689$	−2.00000	−0.0761939
$690$	0	0
$691$	−18.0000	−0.684752	−0.342376	−	0.939563i	$-0.611232\pi$
$691$	−18.0000	−0.684752	−0.342376	+	0.939563i	$0.611232\pi$
$692$	6.00000	0.228086
$693$	72.0000	2.73505
$694$	28.0000	1.06287
$695$	0	0
$696$	0	0
$697$	−60.0000	−2.27266
$698$	4.00000	0.151402
$699$	0	0
$700$	−20.0000	−0.755929
$701$	−30.0000	−1.13308	−0.566542	−	0.824033i	$-0.691719\pi$
$701$	−30.0000	−1.13308	−0.566542	+	0.824033i	$0.691719\pi$
$702$	0	0
$703$	16.0000	0.603451
$704$	42.0000	1.58293
$705$	0	0
$706$	−4.00000	−0.150542
$707$	−72.0000	−2.70784
$708$	0	0
$709$	−50.0000	−1.87779	−0.938895	−	0.344204i	$-0.888149\pi$
$709$	−50.0000	−1.87779	−0.938895	+	0.344204i	$0.888149\pi$
$710$	0	0
$711$	24.0000	0.900070
$712$	36.0000	1.34916
$713$	0	0
$714$	0	0
$715$	0	0
$716$	−4.00000	−0.149487
$717$	0	0
$718$	−4.00000	−0.149279
$719$	8.00000	0.298350	0.149175	−	0.988811i	$-0.452338\pi$
$719$	8.00000	0.298350	0.149175	+	0.988811i	$0.452338\pi$
$720$	0	0
$721$	4.00000	0.148968
$722$	−3.00000	−0.111648
$723$	0	0
$724$	−6.00000	−0.222988
$725$	−30.0000	−1.11417
$726$	0	0
$727$	44.0000	1.63187	0.815935	−	0.578144i	$-0.196223\pi$
$727$	44.0000	1.63187	0.815935	+	0.578144i	$0.196223\pi$
$728$	12.0000	0.444750
$729$	−27.0000	−1.00000
$730$	0	0
$731$	48.0000	1.77534
$732$	0	0
$733$	−4.00000	−0.147743	−0.0738717	−	0.997268i	$-0.523536\pi$
$733$	−4.00000	−0.147743	−0.0738717	+	0.997268i	$0.523536\pi$
$734$	−8.00000	−0.295285
$735$	0	0
$736$	0	0
$737$	−12.0000	−0.442026
$738$	30.0000	1.10432
$739$	−52.0000	−1.91285	−0.956425	−	0.291977i	$-0.905687\pi$
$739$	−52.0000	−1.91285	−0.956425	+	0.291977i	$0.905687\pi$
$740$	0	0
$741$	0	0
$742$	8.00000	0.293689
$743$	−46.0000	−1.68758	−0.843788	−	0.536676i	$-0.819680\pi$
$743$	−46.0000	−1.68758	−0.843788	+	0.536676i	$0.819680\pi$
$744$	0	0
$745$	0	0
$746$	10.0000	0.366126
$747$	24.0000	0.878114
$748$	−36.0000	−1.31629
$749$	16.0000	0.584627
$750$	0	0
$751$	0	0		−	1.00000i	$-0.5\pi$
$751$	0	0			1.00000i	$0.5\pi$
$752$	−6.00000	−0.218797
$753$	0	0
$754$	6.00000	0.218507
$755$	0	0
$756$	0	0
$757$	−26.0000	−0.944986	−0.472493	−	0.881334i	$-0.656646\pi$
$757$	−26.0000	−0.944986	−0.472493	+	0.881334i	$0.656646\pi$
$758$	18.0000	0.653789
$759$	0	0
$760$	0	0
$761$	40.0000	1.45000	0.724999	−	0.688749i	$-0.241840\pi$
$761$	40.0000	1.45000	0.724999	+	0.688749i	$0.241840\pi$
$762$	0	0
$763$	0	0
$764$	−20.0000	−0.723575
$765$	0	0
$766$	−34.0000	−1.22847
$767$	−4.00000	−0.144432
$768$	0	0
$769$	−4.00000	−0.144244	−0.0721218	−	0.997396i	$-0.522977\pi$
$769$	−4.00000	−0.144244	−0.0721218	+	0.997396i	$0.522977\pi$
$770$	0	0
$771$	0	0
$772$	−4.00000	−0.143963
$773$	46.0000	1.65451	0.827253	−	0.561830i	$-0.189903\pi$
$773$	46.0000	1.65451	0.827253	+	0.561830i	$0.189903\pi$
$774$	−24.0000	−0.862662
$775$	−30.0000	−1.07763
$776$	−6.00000	−0.215387
$777$	0	0
$778$	−26.0000	−0.932145
$779$	−40.0000	−1.43315
$780$	0	0
$781$	−12.0000	−0.429394
$782$	0	0
$783$	0	0
$784$	−9.00000	−0.321429
$785$	0	0
$786$	0	0
$787$	12.0000	0.427754	0.213877	−	0.976861i	$-0.431391\pi$
$787$	12.0000	0.427754	0.213877	+	0.976861i	$0.431391\pi$
$788$	8.00000	0.284988
$789$	0	0
$790$	0	0
$791$	8.00000	0.284447
$792$	54.0000	1.91881
$793$	2.00000	0.0710221
$794$	32.0000	1.13564
$795$	0	0
$796$	−4.00000	−0.141776
$797$	30.0000	1.06265	0.531327	−	0.847167i	$-0.321693\pi$
$797$	30.0000	1.06265	0.531327	+	0.847167i	$0.321693\pi$
$798$	0	0
$799$	36.0000	1.27359
$800$	−25.0000	−0.883883
$801$	36.0000	1.27200
$802$	22.0000	0.776847
$803$	48.0000	1.69388
$804$	0	0
$805$	0	0
$806$	6.00000	0.211341
$807$	0	0
$808$	−54.0000	−1.89971
$809$	−34.0000	−1.19538	−0.597688	−	0.801729i	$-0.703914\pi$
$809$	−34.0000	−1.19538	−0.597688	+	0.801729i	$0.703914\pi$
$810$	0	0
$811$	6.00000	0.210688	0.105344	−	0.994436i	$-0.466406\pi$
$811$	6.00000	0.210688	0.105344	+	0.994436i	$0.466406\pi$
$812$	24.0000	0.842235
$813$	0	0
$814$	24.0000	0.841200
$815$	0	0
$816$	0	0
$817$	32.0000	1.11954
$818$	−18.0000	−0.629355
$819$	12.0000	0.419314
$820$	0	0
$821$	30.0000	1.04701	0.523504	−	0.852023i	$-0.324625\pi$
$821$	30.0000	1.04701	0.523504	+	0.852023i	$0.324625\pi$
$822$	0	0
$823$	−4.00000	−0.139431	−0.0697156	−	0.997567i	$-0.522209\pi$
$823$	−4.00000	−0.139431	−0.0697156	+	0.997567i	$0.522209\pi$
$824$	3.00000	0.104510
$825$	0	0
$826$	16.0000	0.556711
$827$	46.0000	1.59958	0.799788	−	0.600282i	$-0.204945\pi$
$827$	46.0000	1.59958	0.799788	+	0.600282i	$0.204945\pi$
$828$	0	0
$829$	26.0000	0.903017	0.451509	−	0.892267i	$-0.350886\pi$
$829$	26.0000	0.903017	0.451509	+	0.892267i	$0.350886\pi$
$830$	0	0
$831$	0	0
$832$	7.00000	0.242681
$833$	54.0000	1.87099
$834$	0	0
$835$	0	0
$836$	−24.0000	−0.830057
$837$	0	0
$838$	−4.00000	−0.138178
$839$	40.0000	1.38095	0.690477	−	0.723355i	$-0.257401\pi$
$839$	40.0000	1.38095	0.690477	+	0.723355i	$0.257401\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	26.0000	0.896019
$843$	0	0
$844$	−4.00000	−0.137686
$845$	0	0
$846$	−18.0000	−0.618853
$847$	−100.000	−3.43604
$848$	2.00000	0.0686803
$849$	0	0
$850$	−30.0000	−1.02899
$851$	0	0
$852$	0	0
$853$	46.0000	1.57501	0.787505	−	0.616308i	$-0.211372\pi$
$853$	46.0000	1.57501	0.787505	+	0.616308i	$0.211372\pi$
$854$	−8.00000	−0.273754
$855$	0	0
$856$	12.0000	0.410152
$857$	−42.0000	−1.43469	−0.717346	−	0.696717i	$-0.754643\pi$
$857$	−42.0000	−1.43469	−0.717346	+	0.696717i	$0.754643\pi$
$858$	0	0
$859$	4.00000	0.136478	0.0682391	−	0.997669i	$-0.478262\pi$
$859$	4.00000	0.136478	0.0682391	+	0.997669i	$0.478262\pi$
$860$	0	0
$861$	0	0
$862$	36.0000	1.22616
$863$	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
$865$	0	0
$866$	−26.0000	−0.883516
$867$	0	0
$868$	24.0000	0.814613
$869$	−48.0000	−1.62829
$870$	0	0
$871$	−2.00000	−0.0677674
$872$	0	0
$873$	−6.00000	−0.203069
$874$	0	0
$875$	0	0
$876$	0	0
$877$	4.00000	0.135070	0.0675352	−	0.997717i	$-0.478487\pi$
$877$	4.00000	0.135070	0.0675352	+	0.997717i	$0.478487\pi$
$878$	28.0000	0.944954
$879$	0	0
$880$	0	0
$881$	−2.00000	−0.0673817	−0.0336909	−	0.999432i	$-0.510726\pi$
$881$	−2.00000	−0.0673817	−0.0336909	+	0.999432i	$0.510726\pi$
$882$	−27.0000	−0.909137
$883$	20.0000	0.673054	0.336527	−	0.941674i	$-0.390748\pi$
$883$	20.0000	0.673054	0.336527	+	0.941674i	$0.390748\pi$
$884$	−6.00000	−0.201802
$885$	0	0
$886$	−24.0000	−0.806296
$887$	−8.00000	−0.268614	−0.134307	−	0.990940i	$-0.542881\pi$
$887$	−8.00000	−0.268614	−0.134307	+	0.990940i	$0.542881\pi$
$888$	0	0
$889$	−48.0000	−1.60987
$890$	0	0
$891$	54.0000	1.80907
$892$	0	0
$893$	24.0000	0.803129
$894$	0	0
$895$	0	0
$896$	12.0000	0.400892
$897$	0	0
$898$	−24.0000	−0.800890
$899$	36.0000	1.20067
$900$	−15.0000	−0.500000
$901$	−12.0000	−0.399778
$902$	−60.0000	−1.99778
$903$	0	0
$904$	6.00000	0.199557
$905$	0	0
$906$	0	0
$907$	−12.0000	−0.398453	−0.199227	−	0.979953i	$-0.563843\pi$
$907$	−12.0000	−0.398453	−0.199227	+	0.979953i	$0.563843\pi$
$908$	10.0000	0.331862
$909$	−54.0000	−1.79107
$910$	0	0
$911$	−4.00000	−0.132526	−0.0662630	−	0.997802i	$-0.521108\pi$
$911$	−4.00000	−0.132526	−0.0662630	+	0.997802i	$0.521108\pi$
$912$	0	0
$913$	−48.0000	−1.58857
$914$	−8.00000	−0.264616
$915$	0	0
$916$	10.0000	0.330409
$917$	16.0000	0.528367
$918$	0	0
$919$	24.0000	0.791687	0.395843	−	0.918318i	$-0.370452\pi$
$919$	24.0000	0.791687	0.395843	+	0.918318i	$0.370452\pi$
$920$	0	0
$921$	0	0
$922$	22.0000	0.724531
$923$	−2.00000	−0.0658308
$924$	0	0
$925$	−20.0000	−0.657596
$926$	−18.0000	−0.591517
$927$	3.00000	0.0985329
$928$	30.0000	0.984798
$929$	−6.00000	−0.196854	−0.0984268	−	0.995144i	$-0.531381\pi$
$929$	−6.00000	−0.196854	−0.0984268	+	0.995144i	$0.531381\pi$
$930$	0	0
$931$	36.0000	1.17985
$932$	18.0000	0.589610
$933$	0	0
$934$	−36.0000	−1.17796
$935$	0	0
$936$	9.00000	0.294174
$937$	−22.0000	−0.718709	−0.359354	−	0.933201i	$-0.617003\pi$
$937$	−22.0000	−0.718709	−0.359354	+	0.933201i	$0.617003\pi$
$938$	8.00000	0.261209
$939$	0	0
$940$	0	0
$941$	−2.00000	−0.0651981	−0.0325991	−	0.999469i	$-0.510378\pi$
$941$	−2.00000	−0.0651981	−0.0325991	+	0.999469i	$0.510378\pi$
$942$	0	0
$943$	0	0
$944$	4.00000	0.130189
$945$	0	0
$946$	48.0000	1.56061
$947$	−10.0000	−0.324956	−0.162478	−	0.986712i	$-0.551949\pi$
$947$	−10.0000	−0.324956	−0.162478	+	0.986712i	$0.551949\pi$
$948$	0	0
$949$	8.00000	0.259691
$950$	−20.0000	−0.648886
$951$	0	0
$952$	72.0000	2.33353
$953$	54.0000	1.74923	0.874616	−	0.484817i	$-0.161114\pi$
$953$	54.0000	1.74923	0.874616	+	0.484817i	$0.161114\pi$
$954$	6.00000	0.194257
$955$	0	0
$956$	24.0000	0.776215
$957$	0	0
$958$	30.0000	0.969256
$959$	−72.0000	−2.32500
$960$	0	0
$961$	5.00000	0.161290
$962$	4.00000	0.128965
$963$	12.0000	0.386695
$964$	−28.0000	−0.901819
$965$	0	0
$966$	0	0
$967$	−18.0000	−0.578841	−0.289420	−	0.957202i	$-0.593463\pi$
$967$	−18.0000	−0.578841	−0.289420	+	0.957202i	$0.593463\pi$
$968$	−75.0000	−2.41059
$969$	0	0
$970$	0	0
$971$	−4.00000	−0.128366	−0.0641831	−	0.997938i	$-0.520444\pi$
$971$	−4.00000	−0.128366	−0.0641831	+	0.997938i	$0.520444\pi$
$972$	0	0
$973$	16.0000	0.512936
$974$	−30.0000	−0.961262
$975$	0	0
$976$	−2.00000	−0.0640184
$977$	54.0000	1.72761	0.863807	−	0.503824i	$-0.168074\pi$
$977$	54.0000	1.72761	0.863807	+	0.503824i	$0.168074\pi$
$978$	0	0
$979$	−72.0000	−2.30113
$980$	0	0
$981$	0	0
$982$	12.0000	0.382935
$983$	−4.00000	−0.127580	−0.0637901	−	0.997963i	$-0.520319\pi$
$983$	−4.00000	−0.127580	−0.0637901	+	0.997963i	$0.520319\pi$
$984$	0	0
$985$	0	0
$986$	36.0000	1.14647
$987$	0	0
$988$	−4.00000	−0.127257
$989$	0	0
$990$	0	0
$991$	−40.0000	−1.27064	−0.635321	−	0.772248i	$-0.719132\pi$
$991$	−40.0000	−1.27064	−0.635321	+	0.772248i	$0.719132\pi$
$992$	30.0000	0.952501
$993$	0	0
$994$	8.00000	0.253745
$995$	0	0
$996$	0	0
$997$	−42.0000	−1.33015	−0.665077	−	0.746775i	$-0.731601\pi$
$997$	−42.0000	−1.33015	−0.665077	+	0.746775i	$0.731601\pi$
$998$	10.0000	0.316544
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1339.2.a.b.1.1	✓	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1339.2.a.b.1.1	✓	1	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:	\( N \)	\(=\)	\( 1339 = 13 \cdot 103 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1339.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more