LMFDB - Embedded newform 9702.2.a.p.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9702,2,Mod(1,9702)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9702, 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("9702.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9702 = 2 \cdot 3^{2} \cdot 7^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9702.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:	$77.4708600410$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1386)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	9702.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{8} +O(q^{10})$

$q-1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{8} -1.00000 q^{10} +1.00000 q^{11} +2.00000 q^{13} +1.00000 q^{16} -5.00000 q^{17} -6.00000 q^{19} +1.00000 q^{20} -1.00000 q^{22} -7.00000 q^{23} -4.00000 q^{25} -2.00000 q^{26} +8.00000 q^{29} +10.0000 q^{31} -1.00000 q^{32} +5.00000 q^{34} -8.00000 q^{37} +6.00000 q^{38} -1.00000 q^{40} -7.00000 q^{41} +4.00000 q^{43} +1.00000 q^{44} +7.00000 q^{46} +1.00000 q^{47} +4.00000 q^{50} +2.00000 q^{52} -6.00000 q^{53} +1.00000 q^{55} -8.00000 q^{58} -6.00000 q^{59} +1.00000 q^{61} -10.0000 q^{62} +1.00000 q^{64} +2.00000 q^{65} +3.00000 q^{67} -5.00000 q^{68} +8.00000 q^{71} +10.0000 q^{73} +8.00000 q^{74} -6.00000 q^{76} -9.00000 q^{79} +1.00000 q^{80} +7.00000 q^{82} +15.0000 q^{83} -5.00000 q^{85} -4.00000 q^{86} -1.00000 q^{88} +12.0000 q^{89} -7.00000 q^{92} -1.00000 q^{94} -6.00000 q^{95} +13.0000 q^{97} +O(q^{100})$

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
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	1.00000	0.447214	0.223607	−	0.974679i	$-0.428217\pi$
$5$	1.00000	0.447214	0.223607	+	0.974679i	$0.428217\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	−1.00000	−0.353553
$9$	0	0
$10$	−1.00000	−0.316228
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	2.00000	0.554700	0.277350	−	0.960769i	$-0.410544\pi$
$13$	2.00000	0.554700	0.277350	+	0.960769i	$0.410544\pi$
$14$	0	0
$15$	0	0
$16$	1.00000	0.250000
$17$	−5.00000	−1.21268	−0.606339	−	0.795206i	$-0.707363\pi$
$17$	−5.00000	−1.21268	−0.606339	+	0.795206i	$0.707363\pi$
$18$	0	0
$19$	−6.00000	−1.37649	−0.688247	−	0.725476i	$-0.741620\pi$
$19$	−6.00000	−1.37649	−0.688247	+	0.725476i	$0.741620\pi$
$20$	1.00000	0.223607
$21$	0	0
$22$	−1.00000	−0.213201
$23$	−7.00000	−1.45960	−0.729800	−	0.683660i	$-0.760387\pi$
$23$	−7.00000	−1.45960	−0.729800	+	0.683660i	$0.760387\pi$
$24$	0	0
$25$	−4.00000	−0.800000
$26$	−2.00000	−0.392232
$27$	0	0
$28$	0	0
$29$	8.00000	1.48556	0.742781	−	0.669534i	$-0.233506\pi$
$29$	8.00000	1.48556	0.742781	+	0.669534i	$0.233506\pi$
$30$	0	0
$31$	10.0000	1.79605	0.898027	−	0.439941i	$-0.145001\pi$
$31$	10.0000	1.79605	0.898027	+	0.439941i	$0.145001\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	5.00000	0.857493
$35$	0	0
$36$	0	0
$37$	−8.00000	−1.31519	−0.657596	−	0.753371i	$-0.728427\pi$
$37$	−8.00000	−1.31519	−0.657596	+	0.753371i	$0.728427\pi$
$38$	6.00000	0.973329
$39$	0	0
$40$	−1.00000	−0.158114
$41$	−7.00000	−1.09322	−0.546608	−	0.837389i	$-0.684081\pi$
$41$	−7.00000	−1.09322	−0.546608	+	0.837389i	$0.684081\pi$
$42$	0	0
$43$	4.00000	0.609994	0.304997	−	0.952353i	$-0.401344\pi$
$43$	4.00000	0.609994	0.304997	+	0.952353i	$0.401344\pi$
$44$	1.00000	0.150756
$45$	0	0
$46$	7.00000	1.03209
$47$	1.00000	0.145865	0.0729325	−	0.997337i	$-0.476764\pi$
$47$	1.00000	0.145865	0.0729325	+	0.997337i	$0.476764\pi$
$48$	0	0
$49$	0	0
$50$	4.00000	0.565685
$51$	0	0
$52$	2.00000	0.277350
$53$	−6.00000	−0.824163	−0.412082	−	0.911147i	$-0.635198\pi$
$53$	−6.00000	−0.824163	−0.412082	+	0.911147i	$0.635198\pi$
$54$	0	0
$55$	1.00000	0.134840
$56$	0	0
$57$	0	0
$58$	−8.00000	−1.05045
$59$	−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
$61$	1.00000	0.128037	0.0640184	−	0.997949i	$-0.479608\pi$
$61$	1.00000	0.128037	0.0640184	+	0.997949i	$0.479608\pi$
$62$	−10.0000	−1.27000
$63$	0	0
$64$	1.00000	0.125000
$65$	2.00000	0.248069
$66$	0	0
$67$	3.00000	0.366508	0.183254	−	0.983066i	$-0.441337\pi$
$67$	3.00000	0.366508	0.183254	+	0.983066i	$0.441337\pi$
$68$	−5.00000	−0.606339
$69$	0	0
$70$	0	0
$71$	8.00000	0.949425	0.474713	−	0.880141i	$-0.342552\pi$
$71$	8.00000	0.949425	0.474713	+	0.880141i	$0.342552\pi$
$72$	0	0
$73$	10.0000	1.17041	0.585206	−	0.810885i	$-0.301014\pi$
$73$	10.0000	1.17041	0.585206	+	0.810885i	$0.301014\pi$
$74$	8.00000	0.929981
$75$	0	0
$76$	−6.00000	−0.688247
$77$	0	0
$78$	0	0
$79$	−9.00000	−1.01258	−0.506290	−	0.862364i	$-0.668983\pi$
$79$	−9.00000	−1.01258	−0.506290	+	0.862364i	$0.668983\pi$
$80$	1.00000	0.111803
$81$	0	0
$82$	7.00000	0.773021
$83$	15.0000	1.64646	0.823232	−	0.567705i	$-0.192169\pi$
$83$	15.0000	1.64646	0.823232	+	0.567705i	$0.192169\pi$
$84$	0	0
$85$	−5.00000	−0.542326
$86$	−4.00000	−0.431331
$87$	0	0
$88$	−1.00000	−0.106600
$89$	12.0000	1.27200	0.635999	−	0.771690i	$-0.280588\pi$
$89$	12.0000	1.27200	0.635999	+	0.771690i	$0.280588\pi$
$90$	0	0
$91$	0	0
$92$	−7.00000	−0.729800
$93$	0	0
$94$	−1.00000	−0.103142
$95$	−6.00000	−0.615587
$96$	0	0
$97$	13.0000	1.31995	0.659975	−	0.751288i	$-0.270567\pi$
$97$	13.0000	1.31995	0.659975	+	0.751288i	$0.270567\pi$
$98$	0	0
$99$	0	0
$100$	−4.00000	−0.400000
$101$	−12.0000	−1.19404	−0.597022	−	0.802225i	$-0.703650\pi$
$101$	−12.0000	−1.19404	−0.597022	+	0.802225i	$0.703650\pi$
$102$	0	0
$103$	12.0000	1.18240	0.591198	−	0.806527i	$-0.298655\pi$
$103$	12.0000	1.18240	0.591198	+	0.806527i	$0.298655\pi$
$104$	−2.00000	−0.196116
$105$	0	0
$106$	6.00000	0.582772
$107$	13.0000	1.25676	0.628379	−	0.777908i	$-0.283719\pi$
$107$	13.0000	1.25676	0.628379	+	0.777908i	$0.283719\pi$
$108$	0	0
$109$	19.0000	1.81987	0.909935	−	0.414751i	$-0.136131\pi$
$109$	19.0000	1.81987	0.909935	+	0.414751i	$0.136131\pi$
$110$	−1.00000	−0.0953463
$111$	0	0
$112$	0	0
$113$	−14.0000	−1.31701	−0.658505	−	0.752577i	$-0.728811\pi$
$113$	−14.0000	−1.31701	−0.658505	+	0.752577i	$0.728811\pi$
$114$	0	0
$115$	−7.00000	−0.652753
$116$	8.00000	0.742781
$117$	0	0
$118$	6.00000	0.552345
$119$	0	0
$120$	0	0
$121$	1.00000	0.0909091
$122$	−1.00000	−0.0905357
$123$	0	0
$124$	10.0000	0.898027
$125$	−9.00000	−0.804984
$126$	0	0
$127$	17.0000	1.50851	0.754253	−	0.656584i	$-0.227999\pi$
$127$	17.0000	1.50851	0.754253	+	0.656584i	$0.227999\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	−2.00000	−0.175412
$131$	−12.0000	−1.04844	−0.524222	−	0.851581i	$-0.675644\pi$
$131$	−12.0000	−1.04844	−0.524222	+	0.851581i	$0.675644\pi$
$132$	0	0
$133$	0	0
$134$	−3.00000	−0.259161
$135$	0	0
$136$	5.00000	0.428746
$137$	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
$139$	2.00000	0.169638	0.0848189	−	0.996396i	$-0.472969\pi$
$139$	2.00000	0.169638	0.0848189	+	0.996396i	$0.472969\pi$
$140$	0	0
$141$	0	0
$142$	−8.00000	−0.671345
$143$	2.00000	0.167248
$144$	0	0
$145$	8.00000	0.664364
$146$	−10.0000	−0.827606
$147$	0	0
$148$	−8.00000	−0.657596
$149$	4.00000	0.327693	0.163846	−	0.986486i	$-0.447610\pi$
$149$	4.00000	0.327693	0.163846	+	0.986486i	$0.447610\pi$
$150$	0	0
$151$	9.00000	0.732410	0.366205	−	0.930534i	$-0.380657\pi$
$151$	9.00000	0.732410	0.366205	+	0.930534i	$0.380657\pi$
$152$	6.00000	0.486664
$153$	0	0
$154$	0	0
$155$	10.0000	0.803219
$156$	0	0
$157$	4.00000	0.319235	0.159617	−	0.987179i	$-0.448974\pi$
$157$	4.00000	0.319235	0.159617	+	0.987179i	$0.448974\pi$
$158$	9.00000	0.716002
$159$	0	0
$160$	−1.00000	−0.0790569
$161$	0	0
$162$	0	0
$163$	13.0000	1.01824	0.509119	−	0.860696i	$-0.329971\pi$
$163$	13.0000	1.01824	0.509119	+	0.860696i	$0.329971\pi$
$164$	−7.00000	−0.546608
$165$	0	0
$166$	−15.0000	−1.16423
$167$	8.00000	0.619059	0.309529	−	0.950890i	$-0.399829\pi$
$167$	8.00000	0.619059	0.309529	+	0.950890i	$0.399829\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	5.00000	0.383482
$171$	0	0
$172$	4.00000	0.304997
$173$	14.0000	1.06440	0.532200	−	0.846619i	$-0.321365\pi$
$173$	14.0000	1.06440	0.532200	+	0.846619i	$0.321365\pi$
$174$	0	0
$175$	0	0
$176$	1.00000	0.0753778
$177$	0	0
$178$	−12.0000	−0.899438
$179$	2.00000	0.149487	0.0747435	−	0.997203i	$-0.476186\pi$
$179$	2.00000	0.149487	0.0747435	+	0.997203i	$0.476186\pi$
$180$	0	0
$181$	20.0000	1.48659	0.743294	−	0.668965i	$-0.233262\pi$
$181$	20.0000	1.48659	0.743294	+	0.668965i	$0.233262\pi$
$182$	0	0
$183$	0	0
$184$	7.00000	0.516047
$185$	−8.00000	−0.588172
$186$	0	0
$187$	−5.00000	−0.365636
$188$	1.00000	0.0729325
$189$	0	0
$190$	6.00000	0.435286
$191$	−20.0000	−1.44715	−0.723575	−	0.690246i	$-0.757502\pi$
$191$	−20.0000	−1.44715	−0.723575	+	0.690246i	$0.757502\pi$
$192$	0	0
$193$	8.00000	0.575853	0.287926	−	0.957653i	$-0.407034\pi$
$193$	8.00000	0.575853	0.287926	+	0.957653i	$0.407034\pi$
$194$	−13.0000	−0.933346
$195$	0	0
$196$	0	0
$197$	24.0000	1.70993	0.854965	−	0.518686i	$-0.173579\pi$
$197$	24.0000	1.70993	0.854965	+	0.518686i	$0.173579\pi$
$198$	0	0
$199$	−26.0000	−1.84309	−0.921546	−	0.388270i	$-0.873073\pi$
$199$	−26.0000	−1.84309	−0.921546	+	0.388270i	$0.873073\pi$
$200$	4.00000	0.282843
$201$	0	0
$202$	12.0000	0.844317
$203$	0	0
$204$	0	0
$205$	−7.00000	−0.488901
$206$	−12.0000	−0.836080
$207$	0	0
$208$	2.00000	0.138675
$209$	−6.00000	−0.415029
$210$	0	0
$211$	−20.0000	−1.37686	−0.688428	−	0.725304i	$-0.741699\pi$
$211$	−20.0000	−1.37686	−0.688428	+	0.725304i	$0.741699\pi$
$212$	−6.00000	−0.412082
$213$	0	0
$214$	−13.0000	−0.888662
$215$	4.00000	0.272798
$216$	0	0
$217$	0	0
$218$	−19.0000	−1.28684
$219$	0	0
$220$	1.00000	0.0674200
$221$	−10.0000	−0.672673
$222$	0	0
$223$	10.0000	0.669650	0.334825	−	0.942280i	$-0.391323\pi$
$223$	10.0000	0.669650	0.334825	+	0.942280i	$0.391323\pi$
$224$	0	0
$225$	0	0
$226$	14.0000	0.931266
$227$	17.0000	1.12833	0.564165	−	0.825662i	$-0.309198\pi$
$227$	17.0000	1.12833	0.564165	+	0.825662i	$0.309198\pi$
$228$	0	0
$229$	20.0000	1.32164	0.660819	−	0.750546i	$-0.270209\pi$
$229$	20.0000	1.32164	0.660819	+	0.750546i	$0.270209\pi$
$230$	7.00000	0.461566
$231$	0	0
$232$	−8.00000	−0.525226
$233$	−21.0000	−1.37576	−0.687878	−	0.725826i	$-0.741458\pi$
$233$	−21.0000	−1.37576	−0.687878	+	0.725826i	$0.741458\pi$
$234$	0	0
$235$	1.00000	0.0652328
$236$	−6.00000	−0.390567
$237$	0	0
$238$	0	0
$239$	−16.0000	−1.03495	−0.517477	−	0.855697i	$-0.673129\pi$
$239$	−16.0000	−1.03495	−0.517477	+	0.855697i	$0.673129\pi$
$240$	0	0
$241$	12.0000	0.772988	0.386494	−	0.922292i	$-0.373686\pi$
$241$	12.0000	0.772988	0.386494	+	0.922292i	$0.373686\pi$
$242$	−1.00000	−0.0642824
$243$	0	0
$244$	1.00000	0.0640184
$245$	0	0
$246$	0	0
$247$	−12.0000	−0.763542
$248$	−10.0000	−0.635001
$249$	0	0
$250$	9.00000	0.569210
$251$	−24.0000	−1.51487	−0.757433	−	0.652913i	$-0.773547\pi$
$251$	−24.0000	−1.51487	−0.757433	+	0.652913i	$0.773547\pi$
$252$	0	0
$253$	−7.00000	−0.440086
$254$	−17.0000	−1.06667
$255$	0	0
$256$	1.00000	0.0625000
$257$	0	0		−	1.00000i	$-0.5\pi$
$257$	0	0			1.00000i	$0.5\pi$
$258$	0	0
$259$	0	0
$260$	2.00000	0.124035
$261$	0	0
$262$	12.0000	0.741362
$263$	−18.0000	−1.10993	−0.554964	−	0.831875i	$-0.687268\pi$
$263$	−18.0000	−1.10993	−0.554964	+	0.831875i	$0.687268\pi$
$264$	0	0
$265$	−6.00000	−0.368577
$266$	0	0
$267$	0	0
$268$	3.00000	0.183254
$269$	−3.00000	−0.182913	−0.0914566	−	0.995809i	$-0.529152\pi$
$269$	−3.00000	−0.182913	−0.0914566	+	0.995809i	$0.529152\pi$
$270$	0	0
$271$	16.0000	0.971931	0.485965	−	0.873978i	$-0.338468\pi$
$271$	16.0000	0.971931	0.485965	+	0.873978i	$0.338468\pi$
$272$	−5.00000	−0.303170
$273$	0	0
$274$	−18.0000	−1.08742
$275$	−4.00000	−0.241209
$276$	0	0
$277$	−18.0000	−1.08152	−0.540758	−	0.841178i	$-0.681862\pi$
$277$	−18.0000	−1.08152	−0.540758	+	0.841178i	$0.681862\pi$
$278$	−2.00000	−0.119952
$279$	0	0
$280$	0	0
$281$	19.0000	1.13344	0.566722	−	0.823909i	$-0.308211\pi$
$281$	19.0000	1.13344	0.566722	+	0.823909i	$0.308211\pi$
$282$	0	0
$283$	24.0000	1.42665	0.713326	−	0.700832i	$-0.247188\pi$
$283$	24.0000	1.42665	0.713326	+	0.700832i	$0.247188\pi$
$284$	8.00000	0.474713
$285$	0	0
$286$	−2.00000	−0.118262
$287$	0	0
$288$	0	0
$289$	8.00000	0.470588
$290$	−8.00000	−0.469776
$291$	0	0
$292$	10.0000	0.585206
$293$	−6.00000	−0.350524	−0.175262	−	0.984522i	$-0.556077\pi$
$293$	−6.00000	−0.350524	−0.175262	+	0.984522i	$0.556077\pi$
$294$	0	0
$295$	−6.00000	−0.349334
$296$	8.00000	0.464991
$297$	0	0
$298$	−4.00000	−0.231714
$299$	−14.0000	−0.809641
$300$	0	0
$301$	0	0
$302$	−9.00000	−0.517892
$303$	0	0
$304$	−6.00000	−0.344124
$305$	1.00000	0.0572598
$306$	0	0
$307$	8.00000	0.456584	0.228292	−	0.973593i	$-0.426686\pi$
$307$	8.00000	0.456584	0.228292	+	0.973593i	$0.426686\pi$
$308$	0	0
$309$	0	0
$310$	−10.0000	−0.567962
$311$	−17.0000	−0.963982	−0.481991	−	0.876176i	$-0.660086\pi$
$311$	−17.0000	−0.963982	−0.481991	+	0.876176i	$0.660086\pi$
$312$	0	0
$313$	22.0000	1.24351	0.621757	−	0.783210i	$-0.286419\pi$
$313$	22.0000	1.24351	0.621757	+	0.783210i	$0.286419\pi$
$314$	−4.00000	−0.225733
$315$	0	0
$316$	−9.00000	−0.506290
$317$	3.00000	0.168497	0.0842484	−	0.996445i	$-0.473151\pi$
$317$	3.00000	0.168497	0.0842484	+	0.996445i	$0.473151\pi$
$318$	0	0
$319$	8.00000	0.447914
$320$	1.00000	0.0559017
$321$	0	0
$322$	0	0
$323$	30.0000	1.66924
$324$	0	0
$325$	−8.00000	−0.443760
$326$	−13.0000	−0.720003
$327$	0	0
$328$	7.00000	0.386510
$329$	0	0
$330$	0	0
$331$	13.0000	0.714545	0.357272	−	0.934000i	$-0.383707\pi$
$331$	13.0000	0.714545	0.357272	+	0.934000i	$0.383707\pi$
$332$	15.0000	0.823232
$333$	0	0
$334$	−8.00000	−0.437741
$335$	3.00000	0.163908
$336$	0	0
$337$	18.0000	0.980522	0.490261	−	0.871576i	$-0.336901\pi$
$337$	18.0000	0.980522	0.490261	+	0.871576i	$0.336901\pi$
$338$	9.00000	0.489535
$339$	0	0
$340$	−5.00000	−0.271163
$341$	10.0000	0.541530
$342$	0	0
$343$	0	0
$344$	−4.00000	−0.215666
$345$	0	0
$346$	−14.0000	−0.752645
$347$	13.0000	0.697877	0.348938	−	0.937146i	$-0.386542\pi$
$347$	13.0000	0.697877	0.348938	+	0.937146i	$0.386542\pi$
$348$	0	0
$349$	−13.0000	−0.695874	−0.347937	−	0.937518i	$-0.613118\pi$
$349$	−13.0000	−0.695874	−0.347937	+	0.937518i	$0.613118\pi$
$350$	0	0
$351$	0	0
$352$	−1.00000	−0.0533002
$353$	6.00000	0.319348	0.159674	−	0.987170i	$-0.448956\pi$
$353$	6.00000	0.319348	0.159674	+	0.987170i	$0.448956\pi$
$354$	0	0
$355$	8.00000	0.424596
$356$	12.0000	0.635999
$357$	0	0
$358$	−2.00000	−0.105703
$359$	−20.0000	−1.05556	−0.527780	−	0.849381i	$-0.676975\pi$
$359$	−20.0000	−1.05556	−0.527780	+	0.849381i	$0.676975\pi$
$360$	0	0
$361$	17.0000	0.894737
$362$	−20.0000	−1.05118
$363$	0	0
$364$	0	0
$365$	10.0000	0.523424
$366$	0	0
$367$	−2.00000	−0.104399	−0.0521996	−	0.998637i	$-0.516623\pi$
$367$	−2.00000	−0.104399	−0.0521996	+	0.998637i	$0.516623\pi$
$368$	−7.00000	−0.364900
$369$	0	0
$370$	8.00000	0.415900
$371$	0	0
$372$	0	0
$373$	−13.0000	−0.673114	−0.336557	−	0.941663i	$-0.609263\pi$
$373$	−13.0000	−0.673114	−0.336557	+	0.941663i	$0.609263\pi$
$374$	5.00000	0.258544
$375$	0	0
$376$	−1.00000	−0.0515711
$377$	16.0000	0.824042
$378$	0	0
$379$	−17.0000	−0.873231	−0.436616	−	0.899648i	$-0.643823\pi$
$379$	−17.0000	−0.873231	−0.436616	+	0.899648i	$0.643823\pi$
$380$	−6.00000	−0.307794
$381$	0	0
$382$	20.0000	1.02329
$383$	16.0000	0.817562	0.408781	−	0.912633i	$-0.365954\pi$
$383$	16.0000	0.817562	0.408781	+	0.912633i	$0.365954\pi$
$384$	0	0
$385$	0	0
$386$	−8.00000	−0.407189
$387$	0	0
$388$	13.0000	0.659975
$389$	−15.0000	−0.760530	−0.380265	−	0.924878i	$-0.624167\pi$
$389$	−15.0000	−0.760530	−0.380265	+	0.924878i	$0.624167\pi$
$390$	0	0
$391$	35.0000	1.77003
$392$	0	0
$393$	0	0
$394$	−24.0000	−1.20910
$395$	−9.00000	−0.452839
$396$	0	0
$397$	12.0000	0.602263	0.301131	−	0.953583i	$-0.402636\pi$
$397$	12.0000	0.602263	0.301131	+	0.953583i	$0.402636\pi$
$398$	26.0000	1.30326
$399$	0	0
$400$	−4.00000	−0.200000
$401$	12.0000	0.599251	0.299626	−	0.954057i	$-0.403138\pi$
$401$	12.0000	0.599251	0.299626	+	0.954057i	$0.403138\pi$
$402$	0	0
$403$	20.0000	0.996271
$404$	−12.0000	−0.597022
$405$	0	0
$406$	0	0
$407$	−8.00000	−0.396545
$408$	0	0
$409$	10.0000	0.494468	0.247234	−	0.968956i	$-0.420478\pi$
$409$	10.0000	0.494468	0.247234	+	0.968956i	$0.420478\pi$
$410$	7.00000	0.345705
$411$	0	0
$412$	12.0000	0.591198
$413$	0	0
$414$	0	0
$415$	15.0000	0.736321
$416$	−2.00000	−0.0980581
$417$	0	0
$418$	6.00000	0.293470
$419$	−10.0000	−0.488532	−0.244266	−	0.969708i	$-0.578547\pi$
$419$	−10.0000	−0.488532	−0.244266	+	0.969708i	$0.578547\pi$
$420$	0	0
$421$	16.0000	0.779792	0.389896	−	0.920859i	$-0.372511\pi$
$421$	16.0000	0.779792	0.389896	+	0.920859i	$0.372511\pi$
$422$	20.0000	0.973585
$423$	0	0
$424$	6.00000	0.291386
$425$	20.0000	0.970143
$426$	0	0
$427$	0	0
$428$	13.0000	0.628379
$429$	0	0
$430$	−4.00000	−0.192897
$431$	−10.0000	−0.481683	−0.240842	−	0.970564i	$-0.577423\pi$
$431$	−10.0000	−0.481683	−0.240842	+	0.970564i	$0.577423\pi$
$432$	0	0
$433$	−13.0000	−0.624740	−0.312370	−	0.949960i	$-0.601123\pi$
$433$	−13.0000	−0.624740	−0.312370	+	0.949960i	$0.601123\pi$
$434$	0	0
$435$	0	0
$436$	19.0000	0.909935
$437$	42.0000	2.00913
$438$	0	0
$439$	7.00000	0.334092	0.167046	−	0.985949i	$-0.446577\pi$
$439$	7.00000	0.334092	0.167046	+	0.985949i	$0.446577\pi$
$440$	−1.00000	−0.0476731
$441$	0	0
$442$	10.0000	0.475651
$443$	−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
$445$	12.0000	0.568855
$446$	−10.0000	−0.473514
$447$	0	0
$448$	0	0
$449$	−36.0000	−1.69895	−0.849473	−	0.527633i	$-0.823080\pi$
$449$	−36.0000	−1.69895	−0.849473	+	0.527633i	$0.823080\pi$
$450$	0	0
$451$	−7.00000	−0.329617
$452$	−14.0000	−0.658505
$453$	0	0
$454$	−17.0000	−0.797850
$455$	0	0
$456$	0	0
$457$	8.00000	0.374224	0.187112	−	0.982339i	$-0.440087\pi$
$457$	8.00000	0.374224	0.187112	+	0.982339i	$0.440087\pi$
$458$	−20.0000	−0.934539
$459$	0	0
$460$	−7.00000	−0.326377
$461$	30.0000	1.39724	0.698620	−	0.715493i	$-0.253798\pi$
$461$	30.0000	1.39724	0.698620	+	0.715493i	$0.253798\pi$
$462$	0	0
$463$	4.00000	0.185896	0.0929479	−	0.995671i	$-0.470371\pi$
$463$	4.00000	0.185896	0.0929479	+	0.995671i	$0.470371\pi$
$464$	8.00000	0.371391
$465$	0	0
$466$	21.0000	0.972806
$467$	−6.00000	−0.277647	−0.138823	−	0.990317i	$-0.544332\pi$
$467$	−6.00000	−0.277647	−0.138823	+	0.990317i	$0.544332\pi$
$468$	0	0
$469$	0	0
$470$	−1.00000	−0.0461266
$471$	0	0
$472$	6.00000	0.276172
$473$	4.00000	0.183920
$474$	0	0
$475$	24.0000	1.10120
$476$	0	0
$477$	0	0
$478$	16.0000	0.731823
$479$	4.00000	0.182765	0.0913823	−	0.995816i	$-0.470871\pi$
$479$	4.00000	0.182765	0.0913823	+	0.995816i	$0.470871\pi$
$480$	0	0
$481$	−16.0000	−0.729537
$482$	−12.0000	−0.546585
$483$	0	0
$484$	1.00000	0.0454545
$485$	13.0000	0.590300
$486$	0	0
$487$	−10.0000	−0.453143	−0.226572	−	0.973995i	$-0.572752\pi$
$487$	−10.0000	−0.453143	−0.226572	+	0.973995i	$0.572752\pi$
$488$	−1.00000	−0.0452679
$489$	0	0
$490$	0	0
$491$	−9.00000	−0.406164	−0.203082	−	0.979162i	$-0.565096\pi$
$491$	−9.00000	−0.406164	−0.203082	+	0.979162i	$0.565096\pi$
$492$	0	0
$493$	−40.0000	−1.80151
$494$	12.0000	0.539906
$495$	0	0
$496$	10.0000	0.449013
$497$	0	0
$498$	0	0
$499$	−28.0000	−1.25345	−0.626726	−	0.779240i	$-0.715605\pi$
$499$	−28.0000	−1.25345	−0.626726	+	0.779240i	$0.715605\pi$
$500$	−9.00000	−0.402492
$501$	0	0
$502$	24.0000	1.07117
$503$	0	0		−	1.00000i	$-0.5\pi$
$503$	0	0			1.00000i	$0.5\pi$
$504$	0	0
$505$	−12.0000	−0.533993
$506$	7.00000	0.311188
$507$	0	0
$508$	17.0000	0.754253
$509$	22.0000	0.975133	0.487566	−	0.873086i	$-0.337885\pi$
$509$	22.0000	0.975133	0.487566	+	0.873086i	$0.337885\pi$
$510$	0	0
$511$	0	0
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	0	0
$515$	12.0000	0.528783
$516$	0	0
$517$	1.00000	0.0439799
$518$	0	0
$519$	0	0
$520$	−2.00000	−0.0877058
$521$	−10.0000	−0.438108	−0.219054	−	0.975713i	$-0.570297\pi$
$521$	−10.0000	−0.438108	−0.219054	+	0.975713i	$0.570297\pi$
$522$	0	0
$523$	−34.0000	−1.48672	−0.743358	−	0.668894i	$-0.766768\pi$
$523$	−34.0000	−1.48672	−0.743358	+	0.668894i	$0.766768\pi$
$524$	−12.0000	−0.524222
$525$	0	0
$526$	18.0000	0.784837
$527$	−50.0000	−2.17803
$528$	0	0
$529$	26.0000	1.13043
$530$	6.00000	0.260623
$531$	0	0
$532$	0	0
$533$	−14.0000	−0.606407
$534$	0	0
$535$	13.0000	0.562039
$536$	−3.00000	−0.129580
$537$	0	0
$538$	3.00000	0.129339
$539$	0	0
$540$	0	0
$541$	−1.00000	−0.0429934	−0.0214967	−	0.999769i	$-0.506843\pi$
$541$	−1.00000	−0.0429934	−0.0214967	+	0.999769i	$0.506843\pi$
$542$	−16.0000	−0.687259
$543$	0	0
$544$	5.00000	0.214373
$545$	19.0000	0.813871
$546$	0	0
$547$	6.00000	0.256541	0.128271	−	0.991739i	$-0.459057\pi$
$547$	6.00000	0.256541	0.128271	+	0.991739i	$0.459057\pi$
$548$	18.0000	0.768922
$549$	0	0
$550$	4.00000	0.170561
$551$	−48.0000	−2.04487
$552$	0	0
$553$	0	0
$554$	18.0000	0.764747
$555$	0	0
$556$	2.00000	0.0848189
$557$	16.0000	0.677942	0.338971	−	0.940797i	$-0.389921\pi$
$557$	16.0000	0.677942	0.338971	+	0.940797i	$0.389921\pi$
$558$	0	0
$559$	8.00000	0.338364
$560$	0	0
$561$	0	0
$562$	−19.0000	−0.801467
$563$	28.0000	1.18006	0.590030	−	0.807382i	$-0.299116\pi$
$563$	28.0000	1.18006	0.590030	+	0.807382i	$0.299116\pi$
$564$	0	0
$565$	−14.0000	−0.588984
$566$	−24.0000	−1.00880
$567$	0	0
$568$	−8.00000	−0.335673
$569$	42.0000	1.76073	0.880366	−	0.474295i	$-0.157297\pi$
$569$	42.0000	1.76073	0.880366	+	0.474295i	$0.157297\pi$
$570$	0	0
$571$	16.0000	0.669579	0.334790	−	0.942293i	$-0.391335\pi$
$571$	16.0000	0.669579	0.334790	+	0.942293i	$0.391335\pi$
$572$	2.00000	0.0836242
$573$	0	0
$574$	0	0
$575$	28.0000	1.16768
$576$	0	0
$577$	−7.00000	−0.291414	−0.145707	−	0.989328i	$-0.546546\pi$
$577$	−7.00000	−0.291414	−0.145707	+	0.989328i	$0.546546\pi$
$578$	−8.00000	−0.332756
$579$	0	0
$580$	8.00000	0.332182
$581$	0	0
$582$	0	0
$583$	−6.00000	−0.248495
$584$	−10.0000	−0.413803
$585$	0	0
$586$	6.00000	0.247858
$587$	−42.0000	−1.73353	−0.866763	−	0.498721i	$-0.833803\pi$
$587$	−42.0000	−1.73353	−0.866763	+	0.498721i	$0.833803\pi$
$588$	0	0
$589$	−60.0000	−2.47226
$590$	6.00000	0.247016
$591$	0	0
$592$	−8.00000	−0.328798
$593$	6.00000	0.246390	0.123195	−	0.992382i	$-0.460686\pi$
$593$	6.00000	0.246390	0.123195	+	0.992382i	$0.460686\pi$
$594$	0	0
$595$	0	0
$596$	4.00000	0.163846
$597$	0	0
$598$	14.0000	0.572503
$599$	−3.00000	−0.122577	−0.0612883	−	0.998120i	$-0.519521\pi$
$599$	−3.00000	−0.122577	−0.0612883	+	0.998120i	$0.519521\pi$
$600$	0	0
$601$	8.00000	0.326327	0.163163	−	0.986599i	$-0.447830\pi$
$601$	8.00000	0.326327	0.163163	+	0.986599i	$0.447830\pi$
$602$	0	0
$603$	0	0
$604$	9.00000	0.366205
$605$	1.00000	0.0406558
$606$	0	0
$607$	29.0000	1.17707	0.588537	−	0.808470i	$-0.299704\pi$
$607$	29.0000	1.17707	0.588537	+	0.808470i	$0.299704\pi$
$608$	6.00000	0.243332
$609$	0	0
$610$	−1.00000	−0.0404888
$611$	2.00000	0.0809113
$612$	0	0
$613$	−11.0000	−0.444286	−0.222143	−	0.975014i	$-0.571305\pi$
$613$	−11.0000	−0.444286	−0.222143	+	0.975014i	$0.571305\pi$
$614$	−8.00000	−0.322854
$615$	0	0
$616$	0	0
$617$	24.0000	0.966204	0.483102	−	0.875564i	$-0.339510\pi$
$617$	24.0000	0.966204	0.483102	+	0.875564i	$0.339510\pi$
$618$	0	0
$619$	−7.00000	−0.281354	−0.140677	−	0.990056i	$-0.544928\pi$
$619$	−7.00000	−0.281354	−0.140677	+	0.990056i	$0.544928\pi$
$620$	10.0000	0.401610
$621$	0	0
$622$	17.0000	0.681638
$623$	0	0
$624$	0	0
$625$	11.0000	0.440000
$626$	−22.0000	−0.879297
$627$	0	0
$628$	4.00000	0.159617
$629$	40.0000	1.59490
$630$	0	0
$631$	−42.0000	−1.67199	−0.835997	−	0.548734i	$-0.815110\pi$
$631$	−42.0000	−1.67199	−0.835997	+	0.548734i	$0.815110\pi$
$632$	9.00000	0.358001
$633$	0	0
$634$	−3.00000	−0.119145
$635$	17.0000	0.674624
$636$	0	0
$637$	0	0
$638$	−8.00000	−0.316723
$639$	0	0
$640$	−1.00000	−0.0395285
$641$	12.0000	0.473972	0.236986	−	0.971513i	$-0.423841\pi$
$641$	12.0000	0.473972	0.236986	+	0.971513i	$0.423841\pi$
$642$	0	0
$643$	4.00000	0.157745	0.0788723	−	0.996885i	$-0.474868\pi$
$643$	4.00000	0.157745	0.0788723	+	0.996885i	$0.474868\pi$
$644$	0	0
$645$	0	0
$646$	−30.0000	−1.18033
$647$	45.0000	1.76913	0.884566	−	0.466415i	$-0.154454\pi$
$647$	45.0000	1.76913	0.884566	+	0.466415i	$0.154454\pi$
$648$	0	0
$649$	−6.00000	−0.235521
$650$	8.00000	0.313786
$651$	0	0
$652$	13.0000	0.509119
$653$	−13.0000	−0.508729	−0.254365	−	0.967108i	$-0.581866\pi$
$653$	−13.0000	−0.508729	−0.254365	+	0.967108i	$0.581866\pi$
$654$	0	0
$655$	−12.0000	−0.468879
$656$	−7.00000	−0.273304
$657$	0	0
$658$	0	0
$659$	−19.0000	−0.740135	−0.370067	−	0.929005i	$-0.620665\pi$
$659$	−19.0000	−0.740135	−0.370067	+	0.929005i	$0.620665\pi$
$660$	0	0
$661$	14.0000	0.544537	0.272268	−	0.962221i	$-0.412226\pi$
$661$	14.0000	0.544537	0.272268	+	0.962221i	$0.412226\pi$
$662$	−13.0000	−0.505259
$663$	0	0
$664$	−15.0000	−0.582113
$665$	0	0
$666$	0	0
$667$	−56.0000	−2.16833
$668$	8.00000	0.309529
$669$	0	0
$670$	−3.00000	−0.115900
$671$	1.00000	0.0386046
$672$	0	0
$673$	22.0000	0.848038	0.424019	−	0.905653i	$-0.360619\pi$
$673$	22.0000	0.848038	0.424019	+	0.905653i	$0.360619\pi$
$674$	−18.0000	−0.693334
$675$	0	0
$676$	−9.00000	−0.346154
$677$	−2.00000	−0.0768662	−0.0384331	−	0.999261i	$-0.512237\pi$
$677$	−2.00000	−0.0768662	−0.0384331	+	0.999261i	$0.512237\pi$
$678$	0	0
$679$	0	0
$680$	5.00000	0.191741
$681$	0	0
$682$	−10.0000	−0.382920
$683$	−24.0000	−0.918334	−0.459167	−	0.888350i	$-0.651852\pi$
$683$	−24.0000	−0.918334	−0.459167	+	0.888350i	$0.651852\pi$
$684$	0	0
$685$	18.0000	0.687745
$686$	0	0
$687$	0	0
$688$	4.00000	0.152499
$689$	−12.0000	−0.457164
$690$	0	0
$691$	15.0000	0.570627	0.285313	−	0.958434i	$-0.407902\pi$
$691$	15.0000	0.570627	0.285313	+	0.958434i	$0.407902\pi$
$692$	14.0000	0.532200
$693$	0	0
$694$	−13.0000	−0.493473
$695$	2.00000	0.0758643
$696$	0	0
$697$	35.0000	1.32572
$698$	13.0000	0.492057
$699$	0	0
$700$	0	0
$701$	−42.0000	−1.58632	−0.793159	−	0.609015i	$-0.791565\pi$
$701$	−42.0000	−1.58632	−0.793159	+	0.609015i	$0.791565\pi$
$702$	0	0
$703$	48.0000	1.81035
$704$	1.00000	0.0376889
$705$	0	0
$706$	−6.00000	−0.225813
$707$	0	0
$708$	0	0
$709$	−42.0000	−1.57734	−0.788672	−	0.614815i	$-0.789231\pi$
$709$	−42.0000	−1.57734	−0.788672	+	0.614815i	$0.789231\pi$
$710$	−8.00000	−0.300235
$711$	0	0
$712$	−12.0000	−0.449719
$713$	−70.0000	−2.62152
$714$	0	0
$715$	2.00000	0.0747958
$716$	2.00000	0.0747435
$717$	0	0
$718$	20.0000	0.746393
$719$	5.00000	0.186469	0.0932343	−	0.995644i	$-0.470279\pi$
$719$	5.00000	0.186469	0.0932343	+	0.995644i	$0.470279\pi$
$720$	0	0
$721$	0	0
$722$	−17.0000	−0.632674
$723$	0	0
$724$	20.0000	0.743294
$725$	−32.0000	−1.18845
$726$	0	0
$727$	20.0000	0.741759	0.370879	−	0.928681i	$-0.379056\pi$
$727$	20.0000	0.741759	0.370879	+	0.928681i	$0.379056\pi$
$728$	0	0
$729$	0	0
$730$	−10.0000	−0.370117
$731$	−20.0000	−0.739727
$732$	0	0
$733$	−37.0000	−1.36663	−0.683313	−	0.730125i	$-0.739462\pi$
$733$	−37.0000	−1.36663	−0.683313	+	0.730125i	$0.739462\pi$
$734$	2.00000	0.0738213
$735$	0	0
$736$	7.00000	0.258023
$737$	3.00000	0.110506
$738$	0	0
$739$	6.00000	0.220714	0.110357	−	0.993892i	$-0.464801\pi$
$739$	6.00000	0.220714	0.110357	+	0.993892i	$0.464801\pi$
$740$	−8.00000	−0.294086
$741$	0	0
$742$	0	0
$743$	12.0000	0.440237	0.220119	−	0.975473i	$-0.429356\pi$
$743$	12.0000	0.440237	0.220119	+	0.975473i	$0.429356\pi$
$744$	0	0
$745$	4.00000	0.146549
$746$	13.0000	0.475964
$747$	0	0
$748$	−5.00000	−0.182818
$749$	0	0
$750$	0	0
$751$	−38.0000	−1.38664	−0.693320	−	0.720630i	$-0.743853\pi$
$751$	−38.0000	−1.38664	−0.693320	+	0.720630i	$0.743853\pi$
$752$	1.00000	0.0364662
$753$	0	0
$754$	−16.0000	−0.582686
$755$	9.00000	0.327544
$756$	0	0
$757$	−10.0000	−0.363456	−0.181728	−	0.983349i	$-0.558169\pi$
$757$	−10.0000	−0.363456	−0.181728	+	0.983349i	$0.558169\pi$
$758$	17.0000	0.617468
$759$	0	0
$760$	6.00000	0.217643
$761$	39.0000	1.41375	0.706874	−	0.707339i	$-0.250105\pi$
$761$	39.0000	1.41375	0.706874	+	0.707339i	$0.250105\pi$
$762$	0	0
$763$	0	0
$764$	−20.0000	−0.723575
$765$	0	0
$766$	−16.0000	−0.578103
$767$	−12.0000	−0.433295
$768$	0	0
$769$	34.0000	1.22607	0.613036	−	0.790055i	$-0.289948\pi$
$769$	34.0000	1.22607	0.613036	+	0.790055i	$0.289948\pi$
$770$	0	0
$771$	0	0
$772$	8.00000	0.287926
$773$	33.0000	1.18693	0.593464	−	0.804861i	$-0.297760\pi$
$773$	33.0000	1.18693	0.593464	+	0.804861i	$0.297760\pi$
$774$	0	0
$775$	−40.0000	−1.43684
$776$	−13.0000	−0.466673
$777$	0	0
$778$	15.0000	0.537776
$779$	42.0000	1.50481
$780$	0	0
$781$	8.00000	0.286263
$782$	−35.0000	−1.25160
$783$	0	0
$784$	0	0
$785$	4.00000	0.142766
$786$	0	0
$787$	14.0000	0.499046	0.249523	−	0.968369i	$-0.419726\pi$
$787$	14.0000	0.499046	0.249523	+	0.968369i	$0.419726\pi$
$788$	24.0000	0.854965
$789$	0	0
$790$	9.00000	0.320206
$791$	0	0
$792$	0	0
$793$	2.00000	0.0710221
$794$	−12.0000	−0.425864
$795$	0	0
$796$	−26.0000	−0.921546
$797$	−23.0000	−0.814702	−0.407351	−	0.913272i	$-0.633547\pi$
$797$	−23.0000	−0.814702	−0.407351	+	0.913272i	$0.633547\pi$
$798$	0	0
$799$	−5.00000	−0.176887
$800$	4.00000	0.141421
$801$	0	0
$802$	−12.0000	−0.423735
$803$	10.0000	0.352892
$804$	0	0
$805$	0	0
$806$	−20.0000	−0.704470
$807$	0	0
$808$	12.0000	0.422159
$809$	−9.00000	−0.316423	−0.158212	−	0.987405i	$-0.550573\pi$
$809$	−9.00000	−0.316423	−0.158212	+	0.987405i	$0.550573\pi$
$810$	0	0
$811$	2.00000	0.0702295	0.0351147	−	0.999383i	$-0.488820\pi$
$811$	2.00000	0.0702295	0.0351147	+	0.999383i	$0.488820\pi$
$812$	0	0
$813$	0	0
$814$	8.00000	0.280400
$815$	13.0000	0.455370
$816$	0	0
$817$	−24.0000	−0.839654
$818$	−10.0000	−0.349642
$819$	0	0
$820$	−7.00000	−0.244451
$821$	−6.00000	−0.209401	−0.104701	−	0.994504i	$-0.533388\pi$
$821$	−6.00000	−0.209401	−0.104701	+	0.994504i	$0.533388\pi$
$822$	0	0
$823$	−10.0000	−0.348578	−0.174289	−	0.984695i	$-0.555763\pi$
$823$	−10.0000	−0.348578	−0.174289	+	0.984695i	$0.555763\pi$
$824$	−12.0000	−0.418040
$825$	0	0
$826$	0	0
$827$	43.0000	1.49526	0.747628	−	0.664117i	$-0.231193\pi$
$827$	43.0000	1.49526	0.747628	+	0.664117i	$0.231193\pi$
$828$	0	0
$829$	−20.0000	−0.694629	−0.347314	−	0.937749i	$-0.612906\pi$
$829$	−20.0000	−0.694629	−0.347314	+	0.937749i	$0.612906\pi$
$830$	−15.0000	−0.520658
$831$	0	0
$832$	2.00000	0.0693375
$833$	0	0
$834$	0	0
$835$	8.00000	0.276851
$836$	−6.00000	−0.207514
$837$	0	0
$838$	10.0000	0.345444
$839$	−21.0000	−0.725001	−0.362500	−	0.931984i	$-0.618077\pi$
$839$	−21.0000	−0.725001	−0.362500	+	0.931984i	$0.618077\pi$
$840$	0	0
$841$	35.0000	1.20690
$842$	−16.0000	−0.551396
$843$	0	0
$844$	−20.0000	−0.688428
$845$	−9.00000	−0.309609
$846$	0	0
$847$	0	0
$848$	−6.00000	−0.206041
$849$	0	0
$850$	−20.0000	−0.685994
$851$	56.0000	1.91966
$852$	0	0
$853$	−7.00000	−0.239675	−0.119838	−	0.992793i	$-0.538237\pi$
$853$	−7.00000	−0.239675	−0.119838	+	0.992793i	$0.538237\pi$
$854$	0	0
$855$	0	0
$856$	−13.0000	−0.444331
$857$	7.00000	0.239115	0.119558	−	0.992827i	$-0.461852\pi$
$857$	7.00000	0.239115	0.119558	+	0.992827i	$0.461852\pi$
$858$	0	0
$859$	25.0000	0.852989	0.426494	−	0.904490i	$-0.359748\pi$
$859$	25.0000	0.852989	0.426494	+	0.904490i	$0.359748\pi$
$860$	4.00000	0.136399
$861$	0	0
$862$	10.0000	0.340601
$863$	5.00000	0.170202	0.0851010	−	0.996372i	$-0.472879\pi$
$863$	5.00000	0.170202	0.0851010	+	0.996372i	$0.472879\pi$
$864$	0	0
$865$	14.0000	0.476014
$866$	13.0000	0.441758
$867$	0	0
$868$	0	0
$869$	−9.00000	−0.305304
$870$	0	0
$871$	6.00000	0.203302
$872$	−19.0000	−0.643421
$873$	0	0
$874$	−42.0000	−1.42067
$875$	0	0
$876$	0	0
$877$	−41.0000	−1.38447	−0.692236	−	0.721671i	$-0.743374\pi$
$877$	−41.0000	−1.38447	−0.692236	+	0.721671i	$0.743374\pi$
$878$	−7.00000	−0.236239
$879$	0	0
$880$	1.00000	0.0337100
$881$	54.0000	1.81931	0.909653	−	0.415369i	$-0.136347\pi$
$881$	54.0000	1.81931	0.909653	+	0.415369i	$0.136347\pi$
$882$	0	0
$883$	−47.0000	−1.58168	−0.790838	−	0.612026i	$-0.790355\pi$
$883$	−47.0000	−1.58168	−0.790838	+	0.612026i	$0.790355\pi$
$884$	−10.0000	−0.336336
$885$	0	0
$886$	24.0000	0.806296
$887$	2.00000	0.0671534	0.0335767	−	0.999436i	$-0.489310\pi$
$887$	2.00000	0.0671534	0.0335767	+	0.999436i	$0.489310\pi$
$888$	0	0
$889$	0	0
$890$	−12.0000	−0.402241
$891$	0	0
$892$	10.0000	0.334825
$893$	−6.00000	−0.200782
$894$	0	0
$895$	2.00000	0.0668526
$896$	0	0
$897$	0	0
$898$	36.0000	1.20134
$899$	80.0000	2.66815
$900$	0	0
$901$	30.0000	0.999445
$902$	7.00000	0.233075
$903$	0	0
$904$	14.0000	0.465633
$905$	20.0000	0.664822
$906$	0	0
$907$	17.0000	0.564476	0.282238	−	0.959344i	$-0.408923\pi$
$907$	17.0000	0.564476	0.282238	+	0.959344i	$0.408923\pi$
$908$	17.0000	0.564165
$909$	0	0
$910$	0	0
$911$	39.0000	1.29213	0.646064	−	0.763283i	$-0.276414\pi$
$911$	39.0000	1.29213	0.646064	+	0.763283i	$0.276414\pi$
$912$	0	0
$913$	15.0000	0.496428
$914$	−8.00000	−0.264616
$915$	0	0
$916$	20.0000	0.660819
$917$	0	0
$918$	0	0
$919$	39.0000	1.28649	0.643246	−	0.765660i	$-0.277587\pi$
$919$	39.0000	1.28649	0.643246	+	0.765660i	$0.277587\pi$
$920$	7.00000	0.230783
$921$	0	0
$922$	−30.0000	−0.987997
$923$	16.0000	0.526646
$924$	0	0
$925$	32.0000	1.05215
$926$	−4.00000	−0.131448
$927$	0	0
$928$	−8.00000	−0.262613
$929$	30.0000	0.984268	0.492134	−	0.870519i	$-0.336217\pi$
$929$	30.0000	0.984268	0.492134	+	0.870519i	$0.336217\pi$
$930$	0	0
$931$	0	0
$932$	−21.0000	−0.687878
$933$	0	0
$934$	6.00000	0.196326
$935$	−5.00000	−0.163517
$936$	0	0
$937$	42.0000	1.37208	0.686040	−	0.727564i	$-0.259347\pi$
$937$	42.0000	1.37208	0.686040	+	0.727564i	$0.259347\pi$
$938$	0	0
$939$	0	0
$940$	1.00000	0.0326164
$941$	14.0000	0.456387	0.228193	−	0.973616i	$-0.426718\pi$
$941$	14.0000	0.456387	0.228193	+	0.973616i	$0.426718\pi$
$942$	0	0
$943$	49.0000	1.59566
$944$	−6.00000	−0.195283
$945$	0	0
$946$	−4.00000	−0.130051
$947$	58.0000	1.88475	0.942373	−	0.334563i	$-0.108589\pi$
$947$	58.0000	1.88475	0.942373	+	0.334563i	$0.108589\pi$
$948$	0	0
$949$	20.0000	0.649227
$950$	−24.0000	−0.778663
$951$	0	0
$952$	0	0
$953$	19.0000	0.615470	0.307735	−	0.951472i	$-0.400429\pi$
$953$	19.0000	0.615470	0.307735	+	0.951472i	$0.400429\pi$
$954$	0	0
$955$	−20.0000	−0.647185
$956$	−16.0000	−0.517477
$957$	0	0
$958$	−4.00000	−0.129234
$959$	0	0
$960$	0	0
$961$	69.0000	2.22581
$962$	16.0000	0.515861
$963$	0	0
$964$	12.0000	0.386494
$965$	8.00000	0.257529
$966$	0	0
$967$	35.0000	1.12552	0.562762	−	0.826619i	$-0.309739\pi$
$967$	35.0000	1.12552	0.562762	+	0.826619i	$0.309739\pi$
$968$	−1.00000	−0.0321412
$969$	0	0
$970$	−13.0000	−0.417405
$971$	22.0000	0.706014	0.353007	−	0.935621i	$-0.385159\pi$
$971$	22.0000	0.706014	0.353007	+	0.935621i	$0.385159\pi$
$972$	0	0
$973$	0	0
$974$	10.0000	0.320421
$975$	0	0
$976$	1.00000	0.0320092
$977$	10.0000	0.319928	0.159964	−	0.987123i	$-0.448862\pi$
$977$	10.0000	0.319928	0.159964	+	0.987123i	$0.448862\pi$
$978$	0	0
$979$	12.0000	0.383522
$980$	0	0
$981$	0	0
$982$	9.00000	0.287202
$983$	−15.0000	−0.478426	−0.239213	−	0.970967i	$-0.576889\pi$
$983$	−15.0000	−0.478426	−0.239213	+	0.970967i	$0.576889\pi$
$984$	0	0
$985$	24.0000	0.764704
$986$	40.0000	1.27386
$987$	0	0
$988$	−12.0000	−0.381771
$989$	−28.0000	−0.890348
$990$	0	0
$991$	−8.00000	−0.254128	−0.127064	−	0.991894i	$-0.540555\pi$
$991$	−8.00000	−0.254128	−0.127064	+	0.991894i	$0.540555\pi$
$992$	−10.0000	−0.317500
$993$	0	0
$994$	0	0
$995$	−26.0000	−0.824255
$996$	0	0
$997$	−30.0000	−0.950110	−0.475055	−	0.879956i	$-0.657572\pi$
$997$	−30.0000	−0.950110	−0.475055	+	0.879956i	$0.657572\pi$
$998$	28.0000	0.886325
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9702.2.a.p.1.1		1
3.2	odd	2		9702.2.a.bj.1.1		1
7.2	even	3		1386.2.k.m.991.1	yes	2
7.4	even	3		1386.2.k.m.793.1	yes	2
7.6	odd	2		9702.2.a.k.1.1		1
21.2	odd	6		1386.2.k.g.991.1	yes	2
21.11	odd	6		1386.2.k.g.793.1	✓	2
21.20	even	2		9702.2.a.bw.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1386.2.k.g.793.1	✓	2	21.11	odd	6
1386.2.k.g.991.1	yes	2	21.2	odd	6
1386.2.k.m.793.1	yes	2	7.4	even	3
1386.2.k.m.991.1	yes	2	7.2	even	3
9702.2.a.k.1.1		1	7.6	odd	2
9702.2.a.p.1.1		1	1.1	even	1	trivial
9702.2.a.bj.1.1		1	3.2	odd	2
9702.2.a.bw.1.1		1	21.20	even	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:	\( N \)	\(=\)	\( 9702 = 2 \cdot 3^{2} \cdot 7^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9702.a (trivial)

Introduction

L-functions

Modular forms

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Fields

Representations

Groups

Database

Properties

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