LMFDB - Embedded newform 142.2.a.d.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [142,2,Mod(1,142)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(142, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("142.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 142 = 2 \cdot 71 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	142.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:	$1.13387570870$
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	$\chi$	$=$	142.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+1.00000 q^{2} -3.00000 q^{3} +1.00000 q^{4} -4.00000 q^{5} -3.00000 q^{6} -3.00000 q^{7} +1.00000 q^{8} +6.00000 q^{9} +O(q^{10})$

\(q+1.00000 q^{2} -3.00000 q^{3} +1.00000 q^{4} -4.00000 q^{5} -3.00000 q^{6} -3.00000 q^{7} +1.00000 q^{8} +6.00000 q^{9} -4.00000 q^{10} -3.00000 q^{12} +1.00000 q^{13} -3.00000 q^{14} +12.0000 q^{15} +1.00000 q^{16} +6.00000 q^{18} -5.00000 q^{19} -4.00000 q^{20} +9.00000 q^{21} -7.00000 q^{23} -3.00000 q^{24} +11.0000 q^{25} +1.00000 q^{26} -9.00000 q^{27} -3.00000 q^{28} -8.00000 q^{29} +12.0000 q^{30} +7.00000 q^{31} +1.00000 q^{32} +12.0000 q^{35} +6.00000 q^{36} +4.00000 q^{37} -5.00000 q^{38} -3.00000 q^{39} -4.00000 q^{40} +4.00000 q^{41} +9.00000 q^{42} -5.00000 q^{43} -24.0000 q^{45} -7.00000 q^{46} -13.0000 q^{47} -3.00000 q^{48} +2.00000 q^{49} +11.0000 q^{50} +1.00000 q^{52} -6.00000 q^{53} -9.00000 q^{54} -3.00000 q^{56} +15.0000 q^{57} -8.00000 q^{58} +10.0000 q^{59} +12.0000 q^{60} -2.00000 q^{61} +7.00000 q^{62} -18.0000 q^{63} +1.00000 q^{64} -4.00000 q^{65} -4.00000 q^{67} +21.0000 q^{69} +12.0000 q^{70} +1.00000 q^{71} +6.00000 q^{72} +7.00000 q^{73} +4.00000 q^{74} -33.0000 q^{75} -5.00000 q^{76} -3.00000 q^{78} -4.00000 q^{80} +9.00000 q^{81} +4.00000 q^{82} -4.00000 q^{83} +9.00000 q^{84} -5.00000 q^{86} +24.0000 q^{87} -3.00000 q^{89} -24.0000 q^{90} -3.00000 q^{91} -7.00000 q^{92} -21.0000 q^{93} -13.0000 q^{94} +20.0000 q^{95} -3.00000 q^{96} -4.00000 q^{97} +2.00000 q^{98} +O(q^{100})\)

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
$3$	−3.00000	−1.73205	−0.866025	−	0.500000i	$-0.833333\pi$
$3$	−3.00000	−1.73205	−0.866025	+	0.500000i	$0.833333\pi$
$4$	1.00000	0.500000
$5$	−4.00000	−1.78885	−0.894427	−	0.447214i	$-0.852416\pi$
$5$	−4.00000	−1.78885	−0.894427	+	0.447214i	$0.852416\pi$
$6$	−3.00000	−1.22474
$7$	−3.00000	−1.13389	−0.566947	−	0.823754i	$-0.691875\pi$
$7$	−3.00000	−1.13389	−0.566947	+	0.823754i	$0.691875\pi$
$8$	1.00000	0.353553
$9$	6.00000	2.00000
$10$	−4.00000	−1.26491
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	−3.00000	−0.866025
$13$	1.00000	0.277350	0.138675	−	0.990338i	$-0.455716\pi$
$13$	1.00000	0.277350	0.138675	+	0.990338i	$0.455716\pi$
$14$	−3.00000	−0.801784
$15$	12.0000	3.09839
$16$	1.00000	0.250000
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	6.00000	1.41421
$19$	−5.00000	−1.14708	−0.573539	−	0.819178i	$-0.694430\pi$
$19$	−5.00000	−1.14708	−0.573539	+	0.819178i	$0.694430\pi$
$20$	−4.00000	−0.894427
$21$	9.00000	1.96396
$22$	0	0
$23$	−7.00000	−1.45960	−0.729800	−	0.683660i	$-0.760387\pi$
$23$	−7.00000	−1.45960	−0.729800	+	0.683660i	$0.760387\pi$
$24$	−3.00000	−0.612372
$25$	11.0000	2.20000
$26$	1.00000	0.196116
$27$	−9.00000	−1.73205
$28$	−3.00000	−0.566947
$29$	−8.00000	−1.48556	−0.742781	−	0.669534i	$-0.766494\pi$
$29$	−8.00000	−1.48556	−0.742781	+	0.669534i	$0.766494\pi$
$30$	12.0000	2.19089
$31$	7.00000	1.25724	0.628619	−	0.777714i	$-0.283621\pi$
$31$	7.00000	1.25724	0.628619	+	0.777714i	$0.283621\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	0	0
$35$	12.0000	2.02837
$36$	6.00000	1.00000
$37$	4.00000	0.657596	0.328798	−	0.944400i	$-0.393356\pi$
$37$	4.00000	0.657596	0.328798	+	0.944400i	$0.393356\pi$
$38$	−5.00000	−0.811107
$39$	−3.00000	−0.480384
$40$	−4.00000	−0.632456
$41$	4.00000	0.624695	0.312348	−	0.949968i	$-0.398885\pi$
$41$	4.00000	0.624695	0.312348	+	0.949968i	$0.398885\pi$
$42$	9.00000	1.38873
$43$	−5.00000	−0.762493	−0.381246	−	0.924473i	$-0.624505\pi$
$43$	−5.00000	−0.762493	−0.381246	+	0.924473i	$0.624505\pi$
$44$	0	0
$45$	−24.0000	−3.57771
$46$	−7.00000	−1.03209
$47$	−13.0000	−1.89624	−0.948122	−	0.317905i	$-0.897021\pi$
$47$	−13.0000	−1.89624	−0.948122	+	0.317905i	$0.897021\pi$
$48$	−3.00000	−0.433013
$49$	2.00000	0.285714
$50$	11.0000	1.55563
$51$	0	0
$52$	1.00000	0.138675
$53$	−6.00000	−0.824163	−0.412082	−	0.911147i	$-0.635198\pi$
$53$	−6.00000	−0.824163	−0.412082	+	0.911147i	$0.635198\pi$
$54$	−9.00000	−1.22474
$55$	0	0
$56$	−3.00000	−0.400892
$57$	15.0000	1.98680
$58$	−8.00000	−1.05045
$59$	10.0000	1.30189	0.650945	−	0.759125i	$-0.274373\pi$
$59$	10.0000	1.30189	0.650945	+	0.759125i	$0.274373\pi$
$60$	12.0000	1.54919
$61$	−2.00000	−0.256074	−0.128037	−	0.991769i	$-0.540868\pi$
$61$	−2.00000	−0.256074	−0.128037	+	0.991769i	$0.540868\pi$
$62$	7.00000	0.889001
$63$	−18.0000	−2.26779
$64$	1.00000	0.125000
$65$	−4.00000	−0.496139
$66$	0	0
$67$	−4.00000	−0.488678	−0.244339	−	0.969690i	$-0.578571\pi$
$67$	−4.00000	−0.488678	−0.244339	+	0.969690i	$0.578571\pi$
$68$	0	0
$69$	21.0000	2.52810
$70$	12.0000	1.43427
$71$	1.00000	0.118678
$71$	1.00000	0.118678
$72$	6.00000	0.707107
$73$	7.00000	0.819288	0.409644	−	0.912245i	$-0.365653\pi$
$73$	7.00000	0.819288	0.409644	+	0.912245i	$0.365653\pi$
$74$	4.00000	0.464991
$75$	−33.0000	−3.81051
$76$	−5.00000	−0.573539
$77$	0	0
$78$	−3.00000	−0.339683
$79$	0	0		−	1.00000i	$-0.5\pi$
$79$	0	0			1.00000i	$0.5\pi$
$80$	−4.00000	−0.447214
$81$	9.00000	1.00000
$82$	4.00000	0.441726
$83$	−4.00000	−0.439057	−0.219529	−	0.975606i	$-0.570452\pi$
$83$	−4.00000	−0.439057	−0.219529	+	0.975606i	$0.570452\pi$
$84$	9.00000	0.981981
$85$	0	0
$86$	−5.00000	−0.539164
$87$	24.0000	2.57307
$88$	0	0
$89$	−3.00000	−0.317999	−0.159000	−	0.987279i	$-0.550827\pi$
$89$	−3.00000	−0.317999	−0.159000	+	0.987279i	$0.550827\pi$
$90$	−24.0000	−2.52982
$91$	−3.00000	−0.314485
$92$	−7.00000	−0.729800
$93$	−21.0000	−2.17760
$94$	−13.0000	−1.34085
$95$	20.0000	2.05196
$96$	−3.00000	−0.306186
$97$	−4.00000	−0.406138	−0.203069	−	0.979164i	$-0.565092\pi$
$97$	−4.00000	−0.406138	−0.203069	+	0.979164i	$0.565092\pi$
$98$	2.00000	0.202031
$99$	0	0
$100$	11.0000	1.10000
$101$	0	0		−	1.00000i	$-0.5\pi$
$101$	0	0			1.00000i	$0.5\pi$
$102$	0	0
$103$	8.00000	0.788263	0.394132	−	0.919054i	$-0.371045\pi$
$103$	8.00000	0.788263	0.394132	+	0.919054i	$0.371045\pi$
$104$	1.00000	0.0980581
$105$	−36.0000	−3.51324
$106$	−6.00000	−0.582772
$107$	17.0000	1.64345	0.821726	−	0.569883i	$-0.193011\pi$
$107$	17.0000	1.64345	0.821726	+	0.569883i	$0.193011\pi$
$108$	−9.00000	−0.866025
$109$	0	0		−	1.00000i	$-0.5\pi$
$109$	0	0			1.00000i	$0.5\pi$
$110$	0	0
$111$	−12.0000	−1.13899
$112$	−3.00000	−0.283473
$113$	−14.0000	−1.31701	−0.658505	−	0.752577i	$-0.728811\pi$
$113$	−14.0000	−1.31701	−0.658505	+	0.752577i	$0.728811\pi$
$114$	15.0000	1.40488
$115$	28.0000	2.61101
$116$	−8.00000	−0.742781
$117$	6.00000	0.554700
$118$	10.0000	0.920575
$119$	0	0
$120$	12.0000	1.09545
$121$	−11.0000	−1.00000
$122$	−2.00000	−0.181071
$123$	−12.0000	−1.08200
$124$	7.00000	0.628619
$125$	−24.0000	−2.14663
$126$	−18.0000	−1.60357
$127$	8.00000	0.709885	0.354943	−	0.934888i	$-0.384500\pi$
$127$	8.00000	0.709885	0.354943	+	0.934888i	$0.384500\pi$
$128$	1.00000	0.0883883
$129$	15.0000	1.32068
$130$	−4.00000	−0.350823
$131$	1.00000	0.0873704	0.0436852	−	0.999045i	$-0.486090\pi$
$131$	1.00000	0.0873704	0.0436852	+	0.999045i	$0.486090\pi$
$132$	0	0
$133$	15.0000	1.30066
$134$	−4.00000	−0.345547
$135$	36.0000	3.09839
$136$	0	0
$137$	4.00000	0.341743	0.170872	−	0.985293i	$-0.445342\pi$
$137$	4.00000	0.341743	0.170872	+	0.985293i	$0.445342\pi$
$138$	21.0000	1.78764
$139$	8.00000	0.678551	0.339276	−	0.940687i	$-0.389818\pi$
$139$	8.00000	0.678551	0.339276	+	0.940687i	$0.389818\pi$
$140$	12.0000	1.01419
$141$	39.0000	3.28439
$142$	1.00000	0.0839181
$143$	0	0
$144$	6.00000	0.500000
$145$	32.0000	2.65746
$146$	7.00000	0.579324
$147$	−6.00000	−0.494872
$148$	4.00000	0.328798
$149$	−17.0000	−1.39269	−0.696347	−	0.717705i	$-0.745193\pi$
$149$	−17.0000	−1.39269	−0.696347	+	0.717705i	$0.745193\pi$
$150$	−33.0000	−2.69444
$151$	4.00000	0.325515	0.162758	−	0.986666i	$-0.447961\pi$
$151$	4.00000	0.325515	0.162758	+	0.986666i	$0.447961\pi$
$152$	−5.00000	−0.405554
$153$	0	0
$154$	0	0
$155$	−28.0000	−2.24901
$156$	−3.00000	−0.240192
$157$	10.0000	0.798087	0.399043	−	0.916932i	$-0.369342\pi$
$157$	10.0000	0.798087	0.399043	+	0.916932i	$0.369342\pi$
$158$	0	0
$159$	18.0000	1.42749
$160$	−4.00000	−0.316228
$161$	21.0000	1.65503
$162$	9.00000	0.707107
$163$	−8.00000	−0.626608	−0.313304	−	0.949653i	$-0.601436\pi$
$163$	−8.00000	−0.626608	−0.313304	+	0.949653i	$0.601436\pi$
$164$	4.00000	0.312348
$165$	0	0
$166$	−4.00000	−0.310460
$167$	−2.00000	−0.154765	−0.0773823	−	0.997001i	$-0.524656\pi$
$167$	−2.00000	−0.154765	−0.0773823	+	0.997001i	$0.524656\pi$
$168$	9.00000	0.694365
$169$	−12.0000	−0.923077
$170$	0	0
$171$	−30.0000	−2.29416
$172$	−5.00000	−0.381246
$173$	−21.0000	−1.59660	−0.798300	−	0.602260i	$-0.794267\pi$
$173$	−21.0000	−1.59660	−0.798300	+	0.602260i	$0.794267\pi$
$174$	24.0000	1.81944
$175$	−33.0000	−2.49457
$176$	0	0
$177$	−30.0000	−2.25494
$178$	−3.00000	−0.224860
$179$	3.00000	0.224231	0.112115	−	0.993695i	$-0.464237\pi$
$179$	3.00000	0.224231	0.112115	+	0.993695i	$0.464237\pi$
$180$	−24.0000	−1.78885
$181$	−3.00000	−0.222988	−0.111494	−	0.993765i	$-0.535564\pi$
$181$	−3.00000	−0.222988	−0.111494	+	0.993765i	$0.535564\pi$
$182$	−3.00000	−0.222375
$183$	6.00000	0.443533
$184$	−7.00000	−0.516047
$185$	−16.0000	−1.17634
$186$	−21.0000	−1.53979
$187$	0	0
$188$	−13.0000	−0.948122
$189$	27.0000	1.96396
$190$	20.0000	1.45095
$191$	10.0000	0.723575	0.361787	−	0.932261i	$-0.382167\pi$
$191$	10.0000	0.723575	0.361787	+	0.932261i	$0.382167\pi$
$192$	−3.00000	−0.216506
$193$	4.00000	0.287926	0.143963	−	0.989583i	$-0.454015\pi$
$193$	4.00000	0.287926	0.143963	+	0.989583i	$0.454015\pi$
$194$	−4.00000	−0.287183
$195$	12.0000	0.859338
$196$	2.00000	0.142857
$197$	27.0000	1.92367	0.961835	−	0.273629i	$-0.0882242\pi$
$197$	27.0000	1.92367	0.961835	+	0.273629i	$0.0882242\pi$
$198$	0	0
$199$	−20.0000	−1.41776	−0.708881	−	0.705328i	$-0.750800\pi$
$199$	−20.0000	−1.41776	−0.708881	+	0.705328i	$0.750800\pi$
$200$	11.0000	0.777817
$201$	12.0000	0.846415
$202$	0	0
$203$	24.0000	1.68447
$204$	0	0
$205$	−16.0000	−1.11749
$206$	8.00000	0.557386
$207$	−42.0000	−2.91920
$208$	1.00000	0.0693375
$209$	0	0
$210$	−36.0000	−2.48424
$211$	−6.00000	−0.413057	−0.206529	−	0.978441i	$-0.566217\pi$
$211$	−6.00000	−0.413057	−0.206529	+	0.978441i	$0.566217\pi$
$212$	−6.00000	−0.412082
$213$	−3.00000	−0.205557
$214$	17.0000	1.16210
$215$	20.0000	1.36399
$216$	−9.00000	−0.612372
$217$	−21.0000	−1.42557
$218$	0	0
$219$	−21.0000	−1.41905
$220$	0	0
$221$	0	0
$222$	−12.0000	−0.805387
$223$	−20.0000	−1.33930	−0.669650	−	0.742677i	$-0.733556\pi$
$223$	−20.0000	−1.33930	−0.669650	+	0.742677i	$0.733556\pi$
$224$	−3.00000	−0.200446
$225$	66.0000	4.40000
$226$	−14.0000	−0.931266
$227$	−8.00000	−0.530979	−0.265489	−	0.964114i	$-0.585534\pi$
$227$	−8.00000	−0.530979	−0.265489	+	0.964114i	$0.585534\pi$
$228$	15.0000	0.993399
$229$	−22.0000	−1.45380	−0.726900	−	0.686743i	$-0.759040\pi$
$229$	−22.0000	−1.45380	−0.726900	+	0.686743i	$0.759040\pi$
$230$	28.0000	1.84627
$231$	0	0
$232$	−8.00000	−0.525226
$233$	−1.00000	−0.0655122	−0.0327561	−	0.999463i	$-0.510428\pi$
$233$	−1.00000	−0.0655122	−0.0327561	+	0.999463i	$0.510428\pi$
$234$	6.00000	0.392232
$235$	52.0000	3.39211
$236$	10.0000	0.650945
$237$	0	0
$238$	0	0
$239$	11.0000	0.711531	0.355765	−	0.934575i	$-0.384220\pi$
$239$	11.0000	0.711531	0.355765	+	0.934575i	$0.384220\pi$
$240$	12.0000	0.774597
$241$	−10.0000	−0.644157	−0.322078	−	0.946713i	$-0.604381\pi$
$241$	−10.0000	−0.644157	−0.322078	+	0.946713i	$0.604381\pi$
$242$	−11.0000	−0.707107
$243$	0	0
$244$	−2.00000	−0.128037
$245$	−8.00000	−0.511101
$246$	−12.0000	−0.765092
$247$	−5.00000	−0.318142
$248$	7.00000	0.444500
$249$	12.0000	0.760469
$250$	−24.0000	−1.51789
$251$	13.0000	0.820553	0.410276	−	0.911961i	$-0.365432\pi$
$251$	13.0000	0.820553	0.410276	+	0.911961i	$0.365432\pi$
$252$	−18.0000	−1.13389
$253$	0	0
$254$	8.00000	0.501965
$255$	0	0
$256$	1.00000	0.0625000
$257$	−2.00000	−0.124757	−0.0623783	−	0.998053i	$-0.519869\pi$
$257$	−2.00000	−0.124757	−0.0623783	+	0.998053i	$0.519869\pi$
$258$	15.0000	0.933859
$259$	−12.0000	−0.745644
$260$	−4.00000	−0.248069
$261$	−48.0000	−2.97113
$262$	1.00000	0.0617802
$263$	−24.0000	−1.47990	−0.739952	−	0.672660i	$-0.765152\pi$
$263$	−24.0000	−1.47990	−0.739952	+	0.672660i	$0.765152\pi$
$264$	0	0
$265$	24.0000	1.47431
$266$	15.0000	0.919709
$267$	9.00000	0.550791
$268$	−4.00000	−0.244339
$269$	−21.0000	−1.28039	−0.640196	−	0.768211i	$-0.721147\pi$
$269$	−21.0000	−1.28039	−0.640196	+	0.768211i	$0.721147\pi$
$270$	36.0000	2.19089
$271$	−22.0000	−1.33640	−0.668202	−	0.743980i	$-0.732936\pi$
$271$	−22.0000	−1.33640	−0.668202	+	0.743980i	$0.732936\pi$
$272$	0	0
$273$	9.00000	0.544705
$274$	4.00000	0.241649
$275$	0	0
$276$	21.0000	1.26405
$277$	−16.0000	−0.961347	−0.480673	−	0.876900i	$-0.659608\pi$
$277$	−16.0000	−0.961347	−0.480673	+	0.876900i	$0.659608\pi$
$278$	8.00000	0.479808
$279$	42.0000	2.51447
$280$	12.0000	0.717137
$281$	−10.0000	−0.596550	−0.298275	−	0.954480i	$-0.596411\pi$
$281$	−10.0000	−0.596550	−0.298275	+	0.954480i	$0.596411\pi$
$282$	39.0000	2.32242
$283$	6.00000	0.356663	0.178331	−	0.983970i	$-0.442930\pi$
$283$	6.00000	0.356663	0.178331	+	0.983970i	$0.442930\pi$
$284$	1.00000	0.0593391
$285$	−60.0000	−3.55409
$286$	0	0
$287$	−12.0000	−0.708338
$288$	6.00000	0.353553
$289$	−17.0000	−1.00000
$290$	32.0000	1.87910
$291$	12.0000	0.703452
$292$	7.00000	0.409644
$293$	30.0000	1.75262	0.876309	−	0.481749i	$-0.159998\pi$
$293$	30.0000	1.75262	0.876309	+	0.481749i	$0.159998\pi$
$294$	−6.00000	−0.349927
$295$	−40.0000	−2.32889
$296$	4.00000	0.232495
$297$	0	0
$298$	−17.0000	−0.984784
$299$	−7.00000	−0.404820
$300$	−33.0000	−1.90526
$301$	15.0000	0.864586
$302$	4.00000	0.230174
$303$	0	0
$304$	−5.00000	−0.286770
$305$	8.00000	0.458079
$306$	0	0
$307$	26.0000	1.48390	0.741949	−	0.670456i	$-0.233902\pi$
$307$	26.0000	1.48390	0.741949	+	0.670456i	$0.233902\pi$
$308$	0	0
$309$	−24.0000	−1.36531
$310$	−28.0000	−1.59029
$311$	−16.0000	−0.907277	−0.453638	−	0.891186i	$-0.649874\pi$
$311$	−16.0000	−0.907277	−0.453638	+	0.891186i	$0.649874\pi$
$312$	−3.00000	−0.169842
$313$	15.0000	0.847850	0.423925	−	0.905697i	$-0.360652\pi$
$313$	15.0000	0.847850	0.423925	+	0.905697i	$0.360652\pi$
$314$	10.0000	0.564333
$315$	72.0000	4.05674
$316$	0	0
$317$	30.0000	1.68497	0.842484	−	0.538721i	$-0.181092\pi$
$317$	30.0000	1.68497	0.842484	+	0.538721i	$0.181092\pi$
$318$	18.0000	1.00939
$319$	0	0
$320$	−4.00000	−0.223607
$321$	−51.0000	−2.84654
$322$	21.0000	1.17028
$323$	0	0
$324$	9.00000	0.500000
$325$	11.0000	0.610170
$326$	−8.00000	−0.443079
$327$	0	0
$328$	4.00000	0.220863
$329$	39.0000	2.15014
$330$	0	0
$331$	16.0000	0.879440	0.439720	−	0.898135i	$-0.355078\pi$
$331$	16.0000	0.879440	0.439720	+	0.898135i	$0.355078\pi$
$332$	−4.00000	−0.219529
$333$	24.0000	1.31519
$334$	−2.00000	−0.109435
$335$	16.0000	0.874173
$336$	9.00000	0.490990
$337$	−16.0000	−0.871576	−0.435788	−	0.900049i	$-0.643530\pi$
$337$	−16.0000	−0.871576	−0.435788	+	0.900049i	$0.643530\pi$
$338$	−12.0000	−0.652714
$339$	42.0000	2.28113
$340$	0	0
$341$	0	0
$342$	−30.0000	−1.62221
$343$	15.0000	0.809924
$344$	−5.00000	−0.269582
$345$	−84.0000	−4.52241
$346$	−21.0000	−1.12897
$347$	−4.00000	−0.214731	−0.107366	−	0.994220i	$-0.534242\pi$
$347$	−4.00000	−0.214731	−0.107366	+	0.994220i	$0.534242\pi$
$348$	24.0000	1.28654
$349$	−11.0000	−0.588817	−0.294408	−	0.955680i	$-0.595123\pi$
$349$	−11.0000	−0.588817	−0.294408	+	0.955680i	$0.595123\pi$
$350$	−33.0000	−1.76392
$351$	−9.00000	−0.480384
$352$	0	0
$353$	18.0000	0.958043	0.479022	−	0.877803i	$-0.340992\pi$
$353$	18.0000	0.958043	0.479022	+	0.877803i	$0.340992\pi$
$354$	−30.0000	−1.59448
$355$	−4.00000	−0.212298
$356$	−3.00000	−0.159000
$357$	0	0
$358$	3.00000	0.158555
$359$	0	0		−	1.00000i	$-0.5\pi$
$359$	0	0			1.00000i	$0.5\pi$
$360$	−24.0000	−1.26491
$361$	6.00000	0.315789
$362$	−3.00000	−0.157676
$363$	33.0000	1.73205
$364$	−3.00000	−0.157243
$365$	−28.0000	−1.46559
$366$	6.00000	0.313625
$367$	−22.0000	−1.14839	−0.574195	−	0.818718i	$-0.694685\pi$
$367$	−22.0000	−1.14839	−0.574195	+	0.818718i	$0.694685\pi$
$368$	−7.00000	−0.364900
$369$	24.0000	1.24939
$370$	−16.0000	−0.831800
$371$	18.0000	0.934513
$372$	−21.0000	−1.08880
$373$	−10.0000	−0.517780	−0.258890	−	0.965907i	$-0.583357\pi$
$373$	−10.0000	−0.517780	−0.258890	+	0.965907i	$0.583357\pi$
$374$	0	0
$375$	72.0000	3.71806
$376$	−13.0000	−0.670424
$377$	−8.00000	−0.412021
$378$	27.0000	1.38873
$379$	4.00000	0.205466	0.102733	−	0.994709i	$-0.467241\pi$
$379$	4.00000	0.205466	0.102733	+	0.994709i	$0.467241\pi$
$380$	20.0000	1.02598
$381$	−24.0000	−1.22956
$382$	10.0000	0.511645
$383$	−1.00000	−0.0510976	−0.0255488	−	0.999674i	$-0.508133\pi$
$383$	−1.00000	−0.0510976	−0.0255488	+	0.999674i	$0.508133\pi$
$384$	−3.00000	−0.153093
$385$	0	0
$386$	4.00000	0.203595
$387$	−30.0000	−1.52499
$388$	−4.00000	−0.203069
$389$	19.0000	0.963338	0.481669	−	0.876353i	$-0.340031\pi$
$389$	19.0000	0.963338	0.481669	+	0.876353i	$0.340031\pi$
$390$	12.0000	0.607644
$391$	0	0
$392$	2.00000	0.101015
$393$	−3.00000	−0.151330
$394$	27.0000	1.36024
$395$	0	0
$396$	0	0
$397$	30.0000	1.50566	0.752828	−	0.658217i	$-0.228689\pi$
$397$	30.0000	1.50566	0.752828	+	0.658217i	$0.228689\pi$
$398$	−20.0000	−1.00251
$399$	−45.0000	−2.25282
$400$	11.0000	0.550000
$401$	24.0000	1.19850	0.599251	−	0.800561i	$-0.295465\pi$
$401$	24.0000	1.19850	0.599251	+	0.800561i	$0.295465\pi$
$402$	12.0000	0.598506
$403$	7.00000	0.348695
$404$	0	0
$405$	−36.0000	−1.78885
$406$	24.0000	1.19110
$407$	0	0
$408$	0	0
$409$	−7.00000	−0.346128	−0.173064	−	0.984911i	$-0.555367\pi$
$409$	−7.00000	−0.346128	−0.173064	+	0.984911i	$0.555367\pi$
$410$	−16.0000	−0.790184
$411$	−12.0000	−0.591916
$412$	8.00000	0.394132
$413$	−30.0000	−1.47620
$414$	−42.0000	−2.06419
$415$	16.0000	0.785409
$416$	1.00000	0.0490290
$417$	−24.0000	−1.17529
$418$	0	0
$419$	12.0000	0.586238	0.293119	−	0.956076i	$-0.405307\pi$
$419$	12.0000	0.586238	0.293119	+	0.956076i	$0.405307\pi$
$420$	−36.0000	−1.75662
$421$	29.0000	1.41337	0.706687	−	0.707527i	$-0.250189\pi$
$421$	29.0000	1.41337	0.706687	+	0.707527i	$0.250189\pi$
$422$	−6.00000	−0.292075
$423$	−78.0000	−3.79249
$424$	−6.00000	−0.291386
$425$	0	0
$426$	−3.00000	−0.145350
$427$	6.00000	0.290360
$428$	17.0000	0.821726
$429$	0	0
$430$	20.0000	0.964486
$431$	8.00000	0.385346	0.192673	−	0.981263i	$-0.438284\pi$
$431$	8.00000	0.385346	0.192673	+	0.981263i	$0.438284\pi$
$432$	−9.00000	−0.433013
$433$	2.00000	0.0961139	0.0480569	−	0.998845i	$-0.484697\pi$
$433$	2.00000	0.0961139	0.0480569	+	0.998845i	$0.484697\pi$
$434$	−21.0000	−1.00803
$435$	−96.0000	−4.60285
$436$	0	0
$437$	35.0000	1.67428
$438$	−21.0000	−1.00342
$439$	16.0000	0.763638	0.381819	−	0.924237i	$-0.375298\pi$
$439$	16.0000	0.763638	0.381819	+	0.924237i	$0.375298\pi$
$440$	0	0
$441$	12.0000	0.571429
$442$	0	0
$443$	−18.0000	−0.855206	−0.427603	−	0.903967i	$-0.640642\pi$
$443$	−18.0000	−0.855206	−0.427603	+	0.903967i	$0.640642\pi$
$444$	−12.0000	−0.569495
$445$	12.0000	0.568855
$446$	−20.0000	−0.947027
$447$	51.0000	2.41222
$448$	−3.00000	−0.141737
$449$	12.0000	0.566315	0.283158	−	0.959073i	$-0.408618\pi$
$449$	12.0000	0.566315	0.283158	+	0.959073i	$0.408618\pi$
$450$	66.0000	3.11127
$451$	0	0
$452$	−14.0000	−0.658505
$453$	−12.0000	−0.563809
$454$	−8.00000	−0.375459
$455$	12.0000	0.562569
$456$	15.0000	0.702439
$457$	−22.0000	−1.02912	−0.514558	−	0.857455i	$-0.672044\pi$
$457$	−22.0000	−1.02912	−0.514558	+	0.857455i	$0.672044\pi$
$458$	−22.0000	−1.02799
$459$	0	0
$460$	28.0000	1.30551
$461$	5.00000	0.232873	0.116437	−	0.993198i	$-0.462853\pi$
$461$	5.00000	0.232873	0.116437	+	0.993198i	$0.462853\pi$
$462$	0	0
$463$	6.00000	0.278844	0.139422	−	0.990233i	$-0.455476\pi$
$463$	6.00000	0.278844	0.139422	+	0.990233i	$0.455476\pi$
$464$	−8.00000	−0.371391
$465$	84.0000	3.89541
$466$	−1.00000	−0.0463241
$467$	−2.00000	−0.0925490	−0.0462745	−	0.998929i	$-0.514735\pi$
$467$	−2.00000	−0.0925490	−0.0462745	+	0.998929i	$0.514735\pi$
$468$	6.00000	0.277350
$469$	12.0000	0.554109
$470$	52.0000	2.39858
$471$	−30.0000	−1.38233
$472$	10.0000	0.460287
$473$	0	0
$474$	0	0
$475$	−55.0000	−2.52357
$476$	0	0
$477$	−36.0000	−1.64833
$478$	11.0000	0.503128
$479$	−3.00000	−0.137073	−0.0685367	−	0.997649i	$-0.521833\pi$
$479$	−3.00000	−0.137073	−0.0685367	+	0.997649i	$0.521833\pi$
$480$	12.0000	0.547723
$481$	4.00000	0.182384
$482$	−10.0000	−0.455488
$483$	−63.0000	−2.86660
$484$	−11.0000	−0.500000
$485$	16.0000	0.726523
$486$	0	0
$487$	32.0000	1.45006	0.725029	−	0.688718i	$-0.241826\pi$
$487$	32.0000	1.45006	0.725029	+	0.688718i	$0.241826\pi$
$488$	−2.00000	−0.0905357
$489$	24.0000	1.08532
$490$	−8.00000	−0.361403
$491$	24.0000	1.08310	0.541552	−	0.840667i	$-0.317837\pi$
$491$	24.0000	1.08310	0.541552	+	0.840667i	$0.317837\pi$
$492$	−12.0000	−0.541002
$493$	0	0
$494$	−5.00000	−0.224961
$495$	0	0
$496$	7.00000	0.314309
$497$	−3.00000	−0.134568
$498$	12.0000	0.537733
$499$	−11.0000	−0.492428	−0.246214	−	0.969216i	$-0.579187\pi$
$499$	−11.0000	−0.492428	−0.246214	+	0.969216i	$0.579187\pi$
$500$	−24.0000	−1.07331
$501$	6.00000	0.268060
$502$	13.0000	0.580218
$503$	−6.00000	−0.267527	−0.133763	−	0.991013i	$-0.542706\pi$
$503$	−6.00000	−0.267527	−0.133763	+	0.991013i	$0.542706\pi$
$504$	−18.0000	−0.801784
$505$	0	0
$506$	0	0
$507$	36.0000	1.59882
$508$	8.00000	0.354943
$509$	−36.0000	−1.59567	−0.797836	−	0.602875i	$-0.794022\pi$
$509$	−36.0000	−1.59567	−0.797836	+	0.602875i	$0.794022\pi$
$510$	0	0
$511$	−21.0000	−0.928985
$512$	1.00000	0.0441942
$513$	45.0000	1.98680
$514$	−2.00000	−0.0882162
$515$	−32.0000	−1.41009
$516$	15.0000	0.660338
$517$	0	0
$518$	−12.0000	−0.527250
$519$	63.0000	2.76539
$520$	−4.00000	−0.175412
$521$	26.0000	1.13908	0.569540	−	0.821963i	$-0.307121\pi$
$521$	26.0000	1.13908	0.569540	+	0.821963i	$0.307121\pi$
$522$	−48.0000	−2.10090
$523$	−26.0000	−1.13690	−0.568450	−	0.822718i	$-0.692457\pi$
$523$	−26.0000	−1.13690	−0.568450	+	0.822718i	$0.692457\pi$
$524$	1.00000	0.0436852
$525$	99.0000	4.32071
$526$	−24.0000	−1.04645
$527$	0	0
$528$	0	0
$529$	26.0000	1.13043
$530$	24.0000	1.04249
$531$	60.0000	2.60378
$532$	15.0000	0.650332
$533$	4.00000	0.173259
$534$	9.00000	0.389468
$535$	−68.0000	−2.93990
$536$	−4.00000	−0.172774
$537$	−9.00000	−0.388379
$538$	−21.0000	−0.905374
$539$	0	0
$540$	36.0000	1.54919
$541$	−42.0000	−1.80572	−0.902861	−	0.429934i	$-0.858537\pi$
$541$	−42.0000	−1.80572	−0.902861	+	0.429934i	$0.858537\pi$
$542$	−22.0000	−0.944981
$543$	9.00000	0.386227
$544$	0	0
$545$	0	0
$546$	9.00000	0.385164
$547$	−19.0000	−0.812381	−0.406191	−	0.913788i	$-0.633143\pi$
$547$	−19.0000	−0.812381	−0.406191	+	0.913788i	$0.633143\pi$
$548$	4.00000	0.170872
$549$	−12.0000	−0.512148
$550$	0	0
$551$	40.0000	1.70406
$552$	21.0000	0.893819
$553$	0	0
$554$	−16.0000	−0.679775
$555$	48.0000	2.03749
$556$	8.00000	0.339276
$557$	−44.0000	−1.86434	−0.932170	−	0.362021i	$-0.882087\pi$
$557$	−44.0000	−1.86434	−0.932170	+	0.362021i	$0.882087\pi$
$558$	42.0000	1.77800
$559$	−5.00000	−0.211477
$560$	12.0000	0.507093
$561$	0	0
$562$	−10.0000	−0.421825
$563$	−34.0000	−1.43293	−0.716465	−	0.697623i	$-0.754241\pi$
$563$	−34.0000	−1.43293	−0.716465	+	0.697623i	$0.754241\pi$
$564$	39.0000	1.64220
$565$	56.0000	2.35594
$566$	6.00000	0.252199
$567$	−27.0000	−1.13389
$568$	1.00000	0.0419591
$569$	−33.0000	−1.38343	−0.691716	−	0.722170i	$-0.743145\pi$
$569$	−33.0000	−1.38343	−0.691716	+	0.722170i	$0.743145\pi$
$570$	−60.0000	−2.51312
$571$	20.0000	0.836974	0.418487	−	0.908223i	$-0.362561\pi$
$571$	20.0000	0.836974	0.418487	+	0.908223i	$0.362561\pi$
$572$	0	0
$573$	−30.0000	−1.25327
$574$	−12.0000	−0.500870
$575$	−77.0000	−3.21112
$576$	6.00000	0.250000
$577$	−42.0000	−1.74848	−0.874241	−	0.485491i	$-0.838641\pi$
$577$	−42.0000	−1.74848	−0.874241	+	0.485491i	$0.838641\pi$
$578$	−17.0000	−0.707107
$579$	−12.0000	−0.498703
$580$	32.0000	1.32873
$581$	12.0000	0.497844
$582$	12.0000	0.497416
$583$	0	0
$584$	7.00000	0.289662
$585$	−24.0000	−0.992278
$586$	30.0000	1.23929
$587$	36.0000	1.48588	0.742940	−	0.669359i	$-0.233431\pi$
$587$	36.0000	1.48588	0.742940	+	0.669359i	$0.233431\pi$
$588$	−6.00000	−0.247436
$589$	−35.0000	−1.44215
$590$	−40.0000	−1.64677
$591$	−81.0000	−3.33189
$592$	4.00000	0.164399
$593$	33.0000	1.35515	0.677574	−	0.735455i	$-0.263031\pi$
$593$	33.0000	1.35515	0.677574	+	0.735455i	$0.263031\pi$
$594$	0	0
$595$	0	0
$596$	−17.0000	−0.696347
$597$	60.0000	2.45564
$598$	−7.00000	−0.286251
$599$	11.0000	0.449448	0.224724	−	0.974422i	$-0.427852\pi$
$599$	11.0000	0.449448	0.224724	+	0.974422i	$0.427852\pi$
$600$	−33.0000	−1.34722
$601$	16.0000	0.652654	0.326327	−	0.945257i	$-0.394189\pi$
$601$	16.0000	0.652654	0.326327	+	0.945257i	$0.394189\pi$
$602$	15.0000	0.611354
$603$	−24.0000	−0.977356
$604$	4.00000	0.162758
$605$	44.0000	1.78885
$606$	0	0
$607$	20.0000	0.811775	0.405887	−	0.913923i	$-0.366962\pi$
$607$	20.0000	0.811775	0.405887	+	0.913923i	$0.366962\pi$
$608$	−5.00000	−0.202777
$609$	−72.0000	−2.91759
$610$	8.00000	0.323911
$611$	−13.0000	−0.525924
$612$	0	0
$613$	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
$615$	48.0000	1.93555
$616$	0	0
$617$	15.0000	0.603877	0.301939	−	0.953327i	$-0.402366\pi$
$617$	15.0000	0.603877	0.301939	+	0.953327i	$0.402366\pi$
$618$	−24.0000	−0.965422
$619$	−36.0000	−1.44696	−0.723481	−	0.690344i	$-0.757459\pi$
$619$	−36.0000	−1.44696	−0.723481	+	0.690344i	$0.757459\pi$
$620$	−28.0000	−1.12451
$621$	63.0000	2.52810
$622$	−16.0000	−0.641542
$623$	9.00000	0.360577
$624$	−3.00000	−0.120096
$625$	41.0000	1.64000
$626$	15.0000	0.599521
$627$	0	0
$628$	10.0000	0.399043
$629$	0	0
$630$	72.0000	2.86855
$631$	8.00000	0.318475	0.159237	−	0.987240i	$-0.449096\pi$
$631$	8.00000	0.318475	0.159237	+	0.987240i	$0.449096\pi$
$632$	0	0
$633$	18.0000	0.715436
$634$	30.0000	1.19145
$635$	−32.0000	−1.26988
$636$	18.0000	0.713746
$637$	2.00000	0.0792429
$638$	0	0
$639$	6.00000	0.237356
$640$	−4.00000	−0.158114
$641$	−11.0000	−0.434474	−0.217237	−	0.976119i	$-0.569704\pi$
$641$	−11.0000	−0.434474	−0.217237	+	0.976119i	$0.569704\pi$
$642$	−51.0000	−2.01281
$643$	−27.0000	−1.06478	−0.532388	−	0.846500i	$-0.678705\pi$
$643$	−27.0000	−1.06478	−0.532388	+	0.846500i	$0.678705\pi$
$644$	21.0000	0.827516
$645$	−60.0000	−2.36250
$646$	0	0
$647$	−22.0000	−0.864909	−0.432455	−	0.901656i	$-0.642352\pi$
$647$	−22.0000	−0.864909	−0.432455	+	0.901656i	$0.642352\pi$
$648$	9.00000	0.353553
$649$	0	0
$650$	11.0000	0.431455
$651$	63.0000	2.46916
$652$	−8.00000	−0.313304
$653$	38.0000	1.48705	0.743527	−	0.668705i	$-0.233151\pi$
$653$	38.0000	1.48705	0.743527	+	0.668705i	$0.233151\pi$
$654$	0	0
$655$	−4.00000	−0.156293
$656$	4.00000	0.156174
$657$	42.0000	1.63858
$658$	39.0000	1.52038
$659$	−9.00000	−0.350590	−0.175295	−	0.984516i	$-0.556088\pi$
$659$	−9.00000	−0.350590	−0.175295	+	0.984516i	$0.556088\pi$
$660$	0	0
$661$	−45.0000	−1.75030	−0.875149	−	0.483854i	$-0.839236\pi$
$661$	−45.0000	−1.75030	−0.875149	+	0.483854i	$0.839236\pi$
$662$	16.0000	0.621858
$663$	0	0
$664$	−4.00000	−0.155230
$665$	−60.0000	−2.32670
$666$	24.0000	0.929981
$667$	56.0000	2.16833
$668$	−2.00000	−0.0773823
$669$	60.0000	2.31973
$670$	16.0000	0.618134
$671$	0	0
$672$	9.00000	0.347183
$673$	0	0		−	1.00000i	$-0.5\pi$
$673$	0	0			1.00000i	$0.5\pi$
$674$	−16.0000	−0.616297
$675$	−99.0000	−3.81051
$676$	−12.0000	−0.461538
$677$	20.0000	0.768662	0.384331	−	0.923195i	$-0.374432\pi$
$677$	20.0000	0.768662	0.384331	+	0.923195i	$0.374432\pi$
$678$	42.0000	1.61300
$679$	12.0000	0.460518
$680$	0	0
$681$	24.0000	0.919682
$682$	0	0
$683$	16.0000	0.612223	0.306111	−	0.951996i	$-0.400972\pi$
$683$	16.0000	0.612223	0.306111	+	0.951996i	$0.400972\pi$
$684$	−30.0000	−1.14708
$685$	−16.0000	−0.611329
$686$	15.0000	0.572703
$687$	66.0000	2.51806
$688$	−5.00000	−0.190623
$689$	−6.00000	−0.228582
$690$	−84.0000	−3.19783
$691$	−24.0000	−0.913003	−0.456502	−	0.889723i	$-0.650898\pi$
$691$	−24.0000	−0.913003	−0.456502	+	0.889723i	$0.650898\pi$
$692$	−21.0000	−0.798300
$693$	0	0
$694$	−4.00000	−0.151838
$695$	−32.0000	−1.21383
$696$	24.0000	0.909718
$697$	0	0
$698$	−11.0000	−0.416356
$699$	3.00000	0.113470
$700$	−33.0000	−1.24728
$701$	−9.00000	−0.339925	−0.169963	−	0.985451i	$-0.554365\pi$
$701$	−9.00000	−0.339925	−0.169963	+	0.985451i	$0.554365\pi$
$702$	−9.00000	−0.339683
$703$	−20.0000	−0.754314
$704$	0	0
$705$	−156.000	−5.87530
$706$	18.0000	0.677439
$707$	0	0
$708$	−30.0000	−1.12747
$709$	5.00000	0.187779	0.0938895	−	0.995583i	$-0.470070\pi$
$709$	5.00000	0.187779	0.0938895	+	0.995583i	$0.470070\pi$
$710$	−4.00000	−0.150117
$711$	0	0
$712$	−3.00000	−0.112430
$713$	−49.0000	−1.83506
$714$	0	0
$715$	0	0
$716$	3.00000	0.112115
$717$	−33.0000	−1.23241
$718$	0	0
$719$	−6.00000	−0.223762	−0.111881	−	0.993722i	$-0.535688\pi$
$719$	−6.00000	−0.223762	−0.111881	+	0.993722i	$0.535688\pi$
$720$	−24.0000	−0.894427
$721$	−24.0000	−0.893807
$722$	6.00000	0.223297
$723$	30.0000	1.11571
$724$	−3.00000	−0.111494
$725$	−88.0000	−3.26824
$726$	33.0000	1.22474
$727$	−3.00000	−0.111264	−0.0556319	−	0.998451i	$-0.517717\pi$
$727$	−3.00000	−0.111264	−0.0556319	+	0.998451i	$0.517717\pi$
$728$	−3.00000	−0.111187
$729$	−27.0000	−1.00000
$730$	−28.0000	−1.03633
$731$	0	0
$732$	6.00000	0.221766
$733$	49.0000	1.80986	0.904928	−	0.425564i	$-0.139924\pi$
$733$	49.0000	1.80986	0.904928	+	0.425564i	$0.139924\pi$
$734$	−22.0000	−0.812035
$735$	24.0000	0.885253
$736$	−7.00000	−0.258023
$737$	0	0
$738$	24.0000	0.883452
$739$	23.0000	0.846069	0.423034	−	0.906114i	$-0.360965\pi$
$739$	23.0000	0.846069	0.423034	+	0.906114i	$0.360965\pi$
$740$	−16.0000	−0.588172
$741$	15.0000	0.551039
$742$	18.0000	0.660801
$743$	−41.0000	−1.50414	−0.752072	−	0.659081i	$-0.770945\pi$
$743$	−41.0000	−1.50414	−0.752072	+	0.659081i	$0.770945\pi$
$744$	−21.0000	−0.769897
$745$	68.0000	2.49133
$746$	−10.0000	−0.366126
$747$	−24.0000	−0.878114
$748$	0	0
$749$	−51.0000	−1.86350
$750$	72.0000	2.62907
$751$	−24.0000	−0.875772	−0.437886	−	0.899030i	$-0.644273\pi$
$751$	−24.0000	−0.875772	−0.437886	+	0.899030i	$0.644273\pi$
$752$	−13.0000	−0.474061
$753$	−39.0000	−1.42124
$754$	−8.00000	−0.291343
$755$	−16.0000	−0.582300
$756$	27.0000	0.981981
$757$	7.00000	0.254419	0.127210	−	0.991876i	$-0.459398\pi$
$757$	7.00000	0.254419	0.127210	+	0.991876i	$0.459398\pi$
$758$	4.00000	0.145287
$759$	0	0
$760$	20.0000	0.725476
$761$	42.0000	1.52250	0.761249	−	0.648459i	$-0.224586\pi$
$761$	42.0000	1.52250	0.761249	+	0.648459i	$0.224586\pi$
$762$	−24.0000	−0.869428
$763$	0	0
$764$	10.0000	0.361787
$765$	0	0
$766$	−1.00000	−0.0361315
$767$	10.0000	0.361079
$768$	−3.00000	−0.108253
$769$	44.0000	1.58668	0.793340	−	0.608778i	$-0.208340\pi$
$769$	44.0000	1.58668	0.793340	+	0.608778i	$0.208340\pi$
$770$	0	0
$771$	6.00000	0.216085
$772$	4.00000	0.143963
$773$	−7.00000	−0.251773	−0.125886	−	0.992045i	$-0.540177\pi$
$773$	−7.00000	−0.251773	−0.125886	+	0.992045i	$0.540177\pi$
$774$	−30.0000	−1.07833
$775$	77.0000	2.76592
$776$	−4.00000	−0.143592
$777$	36.0000	1.29149
$778$	19.0000	0.681183
$779$	−20.0000	−0.716574
$780$	12.0000	0.429669
$781$	0	0
$782$	0	0
$783$	72.0000	2.57307
$784$	2.00000	0.0714286
$785$	−40.0000	−1.42766
$786$	−3.00000	−0.107006
$787$	−23.0000	−0.819861	−0.409931	−	0.912117i	$-0.634447\pi$
$787$	−23.0000	−0.819861	−0.409931	+	0.912117i	$0.634447\pi$
$788$	27.0000	0.961835
$789$	72.0000	2.56327
$790$	0	0
$791$	42.0000	1.49335
$792$	0	0
$793$	−2.00000	−0.0710221
$794$	30.0000	1.06466
$795$	−72.0000	−2.55358
$796$	−20.0000	−0.708881
$797$	−46.0000	−1.62940	−0.814702	−	0.579880i	$-0.803099\pi$
$797$	−46.0000	−1.62940	−0.814702	+	0.579880i	$0.803099\pi$
$798$	−45.0000	−1.59298
$799$	0	0
$800$	11.0000	0.388909
$801$	−18.0000	−0.635999
$802$	24.0000	0.847469
$803$	0	0
$804$	12.0000	0.423207
$805$	−84.0000	−2.96061
$806$	7.00000	0.246564
$807$	63.0000	2.21771
$808$	0	0
$809$	24.0000	0.843795	0.421898	−	0.906644i	$-0.361364\pi$
$809$	24.0000	0.843795	0.421898	+	0.906644i	$0.361364\pi$
$810$	−36.0000	−1.26491
$811$	−40.0000	−1.40459	−0.702295	−	0.711886i	$-0.747841\pi$
$811$	−40.0000	−1.40459	−0.702295	+	0.711886i	$0.747841\pi$
$812$	24.0000	0.842235
$813$	66.0000	2.31472
$814$	0	0
$815$	32.0000	1.12091
$816$	0	0
$817$	25.0000	0.874639
$818$	−7.00000	−0.244749
$819$	−18.0000	−0.628971
$820$	−16.0000	−0.558744
$821$	−28.0000	−0.977207	−0.488603	−	0.872506i	$-0.662493\pi$
$821$	−28.0000	−0.977207	−0.488603	+	0.872506i	$0.662493\pi$
$822$	−12.0000	−0.418548
$823$	−41.0000	−1.42917	−0.714585	−	0.699549i	$-0.753384\pi$
$823$	−41.0000	−1.42917	−0.714585	+	0.699549i	$0.753384\pi$
$824$	8.00000	0.278693
$825$	0	0
$826$	−30.0000	−1.04383
$827$	54.0000	1.87776	0.938882	−	0.344239i	$-0.111863\pi$
$827$	54.0000	1.87776	0.938882	+	0.344239i	$0.111863\pi$
$828$	−42.0000	−1.45960
$829$	−16.0000	−0.555703	−0.277851	−	0.960624i	$-0.589622\pi$
$829$	−16.0000	−0.555703	−0.277851	+	0.960624i	$0.589622\pi$
$830$	16.0000	0.555368
$831$	48.0000	1.66510
$832$	1.00000	0.0346688
$833$	0	0
$834$	−24.0000	−0.831052
$835$	8.00000	0.276851
$836$	0	0
$837$	−63.0000	−2.17760
$838$	12.0000	0.414533
$839$	−12.0000	−0.414286	−0.207143	−	0.978311i	$-0.566417\pi$
$839$	−12.0000	−0.414286	−0.207143	+	0.978311i	$0.566417\pi$
$840$	−36.0000	−1.24212
$841$	35.0000	1.20690
$842$	29.0000	0.999406
$843$	30.0000	1.03325
$844$	−6.00000	−0.206529
$845$	48.0000	1.65125
$846$	−78.0000	−2.68170
$847$	33.0000	1.13389
$848$	−6.00000	−0.206041
$849$	−18.0000	−0.617758
$850$	0	0
$851$	−28.0000	−0.959828
$852$	−3.00000	−0.102778
$853$	44.0000	1.50653	0.753266	−	0.657716i	$-0.228477\pi$
$853$	44.0000	1.50653	0.753266	+	0.657716i	$0.228477\pi$
$854$	6.00000	0.205316
$855$	120.000	4.10391
$856$	17.0000	0.581048
$857$	5.00000	0.170797	0.0853984	−	0.996347i	$-0.472784\pi$
$857$	5.00000	0.170797	0.0853984	+	0.996347i	$0.472784\pi$
$858$	0	0
$859$	4.00000	0.136478	0.0682391	−	0.997669i	$-0.478262\pi$
$859$	4.00000	0.136478	0.0682391	+	0.997669i	$0.478262\pi$
$860$	20.0000	0.681994
$861$	36.0000	1.22688
$862$	8.00000	0.272481
$863$	47.0000	1.59990	0.799949	−	0.600068i	$-0.204860\pi$
$863$	47.0000	1.59990	0.799949	+	0.600068i	$0.204860\pi$
$864$	−9.00000	−0.306186
$865$	84.0000	2.85609
$866$	2.00000	0.0679628
$867$	51.0000	1.73205
$868$	−21.0000	−0.712786
$869$	0	0
$870$	−96.0000	−3.25470
$871$	−4.00000	−0.135535
$872$	0	0
$873$	−24.0000	−0.812277
$874$	35.0000	1.18389
$875$	72.0000	2.43404
$876$	−21.0000	−0.709524
$877$	−44.0000	−1.48577	−0.742887	−	0.669417i	$-0.766544\pi$
$877$	−44.0000	−1.48577	−0.742887	+	0.669417i	$0.766544\pi$
$878$	16.0000	0.539974
$879$	−90.0000	−3.03562
$880$	0	0
$881$	−23.0000	−0.774890	−0.387445	−	0.921893i	$-0.626642\pi$
$881$	−23.0000	−0.774890	−0.387445	+	0.921893i	$0.626642\pi$
$882$	12.0000	0.404061
$883$	6.00000	0.201916	0.100958	−	0.994891i	$-0.467809\pi$
$883$	6.00000	0.201916	0.100958	+	0.994891i	$0.467809\pi$
$884$	0	0
$885$	120.000	4.03376
$886$	−18.0000	−0.604722
$887$	12.0000	0.402921	0.201460	−	0.979497i	$-0.435431\pi$
$887$	12.0000	0.402921	0.201460	+	0.979497i	$0.435431\pi$
$888$	−12.0000	−0.402694
$889$	−24.0000	−0.804934
$890$	12.0000	0.402241
$891$	0	0
$892$	−20.0000	−0.669650
$893$	65.0000	2.17514
$894$	51.0000	1.70570
$895$	−12.0000	−0.401116
$896$	−3.00000	−0.100223
$897$	21.0000	0.701170
$898$	12.0000	0.400445
$899$	−56.0000	−1.86770
$900$	66.0000	2.20000
$901$	0	0
$902$	0	0
$903$	−45.0000	−1.49751
$904$	−14.0000	−0.465633
$905$	12.0000	0.398893
$906$	−12.0000	−0.398673
$907$	−32.0000	−1.06254	−0.531271	−	0.847202i	$-0.678286\pi$
$907$	−32.0000	−1.06254	−0.531271	+	0.847202i	$0.678286\pi$
$908$	−8.00000	−0.265489
$909$	0	0
$910$	12.0000	0.397796
$911$	−45.0000	−1.49092	−0.745458	−	0.666552i	$-0.767769\pi$
$911$	−45.0000	−1.49092	−0.745458	+	0.666552i	$0.767769\pi$
$912$	15.0000	0.496700
$913$	0	0
$914$	−22.0000	−0.727695
$915$	−24.0000	−0.793416
$916$	−22.0000	−0.726900
$917$	−3.00000	−0.0990687
$918$	0	0
$919$	33.0000	1.08857	0.544285	−	0.838901i	$-0.316801\pi$
$919$	33.0000	1.08857	0.544285	+	0.838901i	$0.316801\pi$
$920$	28.0000	0.923133
$921$	−78.0000	−2.57019
$922$	5.00000	0.164666
$923$	1.00000	0.0329154
$924$	0	0
$925$	44.0000	1.44671
$926$	6.00000	0.197172
$927$	48.0000	1.57653
$928$	−8.00000	−0.262613
$929$	−30.0000	−0.984268	−0.492134	−	0.870519i	$-0.663783\pi$
$929$	−30.0000	−0.984268	−0.492134	+	0.870519i	$0.663783\pi$
$930$	84.0000	2.75447
$931$	−10.0000	−0.327737
$932$	−1.00000	−0.0327561
$933$	48.0000	1.57145
$934$	−2.00000	−0.0654420
$935$	0	0
$936$	6.00000	0.196116
$937$	8.00000	0.261349	0.130674	−	0.991425i	$-0.458286\pi$
$937$	8.00000	0.261349	0.130674	+	0.991425i	$0.458286\pi$
$938$	12.0000	0.391814
$939$	−45.0000	−1.46852
$940$	52.0000	1.69605
$941$	48.0000	1.56476	0.782378	−	0.622804i	$-0.214007\pi$
$941$	48.0000	1.56476	0.782378	+	0.622804i	$0.214007\pi$
$942$	−30.0000	−0.977453
$943$	−28.0000	−0.911805
$944$	10.0000	0.325472
$945$	−108.000	−3.51324
$946$	0	0
$947$	12.0000	0.389948	0.194974	−	0.980808i	$-0.437538\pi$
$947$	12.0000	0.389948	0.194974	+	0.980808i	$0.437538\pi$
$948$	0	0
$949$	7.00000	0.227230
$950$	−55.0000	−1.78444
$951$	−90.0000	−2.91845
$952$	0	0
$953$	26.0000	0.842223	0.421111	−	0.907009i	$-0.361640\pi$
$953$	26.0000	0.842223	0.421111	+	0.907009i	$0.361640\pi$
$954$	−36.0000	−1.16554
$955$	−40.0000	−1.29437
$956$	11.0000	0.355765
$957$	0	0
$958$	−3.00000	−0.0969256
$959$	−12.0000	−0.387500
$960$	12.0000	0.387298
$961$	18.0000	0.580645
$962$	4.00000	0.128965
$963$	102.000	3.28690
$964$	−10.0000	−0.322078
$965$	−16.0000	−0.515058
$966$	−63.0000	−2.02699
$967$	−15.0000	−0.482367	−0.241184	−	0.970479i	$-0.577536\pi$
$967$	−15.0000	−0.482367	−0.241184	+	0.970479i	$0.577536\pi$
$968$	−11.0000	−0.353553
$969$	0	0
$970$	16.0000	0.513729
$971$	43.0000	1.37994	0.689968	−	0.723840i	$-0.257625\pi$
$971$	43.0000	1.37994	0.689968	+	0.723840i	$0.257625\pi$
$972$	0	0
$973$	−24.0000	−0.769405
$974$	32.0000	1.02535
$975$	−33.0000	−1.05685
$976$	−2.00000	−0.0640184
$977$	−42.0000	−1.34370	−0.671850	−	0.740688i	$-0.734500\pi$
$977$	−42.0000	−1.34370	−0.671850	+	0.740688i	$0.734500\pi$
$978$	24.0000	0.767435
$979$	0	0
$980$	−8.00000	−0.255551
$981$	0	0
$982$	24.0000	0.765871
$983$	0	0		−	1.00000i	$-0.5\pi$
$983$	0	0			1.00000i	$0.5\pi$
$984$	−12.0000	−0.382546
$985$	−108.000	−3.44117
$986$	0	0
$987$	−117.000	−3.72415
$988$	−5.00000	−0.159071
$989$	35.0000	1.11294
$990$	0	0
$991$	−25.0000	−0.794151	−0.397076	−	0.917786i	$-0.629975\pi$
$991$	−25.0000	−0.794151	−0.397076	+	0.917786i	$0.629975\pi$
$992$	7.00000	0.222250
$993$	−48.0000	−1.52323
$994$	−3.00000	−0.0951542
$995$	80.0000	2.53617
$996$	12.0000	0.380235
$997$	−28.0000	−0.886769	−0.443384	−	0.896332i	$-0.646222\pi$
$997$	−28.0000	−0.886769	−0.443384	+	0.896332i	$0.646222\pi$
$998$	−11.0000	−0.348199
$999$	−36.0000	−1.13899

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	142.2.a.d.1.1	✓	1
3.2	odd	2		1278.2.a.e.1.1		1
4.3	odd	2		1136.2.a.f.1.1		1
5.4	even	2		3550.2.a.i.1.1		1
7.6	odd	2		6958.2.a.p.1.1		1
8.3	odd	2		4544.2.a.c.1.1		1
8.5	even	2		4544.2.a.p.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
142.2.a.d.1.1	✓	1	1.1	even	1	trivial
1136.2.a.f.1.1		1	4.3	odd	2
1278.2.a.e.1.1		1	3.2	odd	2
3550.2.a.i.1.1		1	5.4	even	2
4544.2.a.c.1.1		1	8.3	odd	2
4544.2.a.p.1.1		1	8.5	even	2
6958.2.a.p.1.1		1	7.6	odd	2

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:	\( N \)	\(=\)	\( 142 = 2 \cdot 71 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	142.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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