LMFDB - Embedded newform 7938.2.a.w.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7938,2,Mod(1,7938)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7938, 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("7938.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7938 = 2 \cdot 3^{4} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7938.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:	$63.3852491245$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1134)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	7938.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{8} -1.00000 q^{10} +2.00000 q^{11} -3.00000 q^{13} +1.00000 q^{16} +1.00000 q^{17} -2.00000 q^{19} -1.00000 q^{20} +2.00000 q^{22} +2.00000 q^{23} -4.00000 q^{25} -3.00000 q^{26} +7.00000 q^{29} -6.00000 q^{31} +1.00000 q^{32} +1.00000 q^{34} -7.00000 q^{37} -2.00000 q^{38} -1.00000 q^{40} -6.00000 q^{41} -4.00000 q^{43} +2.00000 q^{44} +2.00000 q^{46} -6.00000 q^{47} -4.00000 q^{50} -3.00000 q^{52} -6.00000 q^{53} -2.00000 q^{55} +7.00000 q^{58} -10.0000 q^{59} +9.00000 q^{61} -6.00000 q^{62} +1.00000 q^{64} +3.00000 q^{65} +10.0000 q^{67} +1.00000 q^{68} -4.00000 q^{71} +11.0000 q^{73} -7.00000 q^{74} -2.00000 q^{76} -6.00000 q^{79} -1.00000 q^{80} -6.00000 q^{82} +10.0000 q^{83} -1.00000 q^{85} -4.00000 q^{86} +2.00000 q^{88} -15.0000 q^{89} +2.00000 q^{92} -6.00000 q^{94} +2.00000 q^{95} +2.00000 q^{97} +O(q^{100})$

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	$-0.571783\pi$
$5$	−1.00000	−0.447214	−0.223607	+	0.974679i	$0.571783\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	1.00000	0.353553
$9$	0	0
$10$	−1.00000	−0.316228
$11$	2.00000	0.603023	0.301511	−	0.953463i	$-0.402509\pi$
$11$	2.00000	0.603023	0.301511	+	0.953463i	$0.402509\pi$
$12$	0	0
$13$	−3.00000	−0.832050	−0.416025	−	0.909353i	$-0.636577\pi$
$13$	−3.00000	−0.832050	−0.416025	+	0.909353i	$0.636577\pi$
$14$	0	0
$15$	0	0
$16$	1.00000	0.250000
$17$	1.00000	0.242536	0.121268	−	0.992620i	$-0.461304\pi$
$17$	1.00000	0.242536	0.121268	+	0.992620i	$0.461304\pi$
$18$	0	0
$19$	−2.00000	−0.458831	−0.229416	−	0.973329i	$-0.573682\pi$
$19$	−2.00000	−0.458831	−0.229416	+	0.973329i	$0.573682\pi$
$20$	−1.00000	−0.223607
$21$	0	0
$22$	2.00000	0.426401
$23$	2.00000	0.417029	0.208514	−	0.978019i	$-0.433137\pi$
$23$	2.00000	0.417029	0.208514	+	0.978019i	$0.433137\pi$
$24$	0	0
$25$	−4.00000	−0.800000
$26$	−3.00000	−0.588348
$27$	0	0
$28$	0	0
$29$	7.00000	1.29987	0.649934	−	0.759991i	$-0.274797\pi$
$29$	7.00000	1.29987	0.649934	+	0.759991i	$0.274797\pi$
$30$	0	0
$31$	−6.00000	−1.07763	−0.538816	−	0.842424i	$-0.681128\pi$
$31$	−6.00000	−1.07763	−0.538816	+	0.842424i	$0.681128\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	1.00000	0.171499
$35$	0	0
$36$	0	0
$37$	−7.00000	−1.15079	−0.575396	−	0.817875i	$-0.695152\pi$
$37$	−7.00000	−1.15079	−0.575396	+	0.817875i	$0.695152\pi$
$38$	−2.00000	−0.324443
$39$	0	0
$40$	−1.00000	−0.158114
$41$	−6.00000	−0.937043	−0.468521	−	0.883452i	$-0.655213\pi$
$41$	−6.00000	−0.937043	−0.468521	+	0.883452i	$0.655213\pi$
$42$	0	0
$43$	−4.00000	−0.609994	−0.304997	−	0.952353i	$-0.598656\pi$
$43$	−4.00000	−0.609994	−0.304997	+	0.952353i	$0.598656\pi$
$44$	2.00000	0.301511
$45$	0	0
$46$	2.00000	0.294884
$47$	−6.00000	−0.875190	−0.437595	−	0.899172i	$-0.644170\pi$
$47$	−6.00000	−0.875190	−0.437595	+	0.899172i	$0.644170\pi$
$48$	0	0
$49$	0	0
$50$	−4.00000	−0.565685
$51$	0	0
$52$	−3.00000	−0.416025
$53$	−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
$55$	−2.00000	−0.269680
$56$	0	0
$57$	0	0
$58$	7.00000	0.919145
$59$	−10.0000	−1.30189	−0.650945	−	0.759125i	$-0.725627\pi$
$59$	−10.0000	−1.30189	−0.650945	+	0.759125i	$0.725627\pi$
$60$	0	0
$61$	9.00000	1.15233	0.576166	−	0.817333i	$-0.304548\pi$
$61$	9.00000	1.15233	0.576166	+	0.817333i	$0.304548\pi$
$62$	−6.00000	−0.762001
$63$	0	0
$64$	1.00000	0.125000
$65$	3.00000	0.372104
$66$	0	0
$67$	10.0000	1.22169	0.610847	−	0.791748i	$-0.290829\pi$
$67$	10.0000	1.22169	0.610847	+	0.791748i	$0.290829\pi$
$68$	1.00000	0.121268
$69$	0	0
$70$	0	0
$71$	−4.00000	−0.474713	−0.237356	−	0.971423i	$-0.576281\pi$
$71$	−4.00000	−0.474713	−0.237356	+	0.971423i	$0.576281\pi$
$72$	0	0
$73$	11.0000	1.28745	0.643726	−	0.765256i	$-0.277388\pi$
$73$	11.0000	1.28745	0.643726	+	0.765256i	$0.277388\pi$
$74$	−7.00000	−0.813733
$75$	0	0
$76$	−2.00000	−0.229416
$77$	0	0
$78$	0	0
$79$	−6.00000	−0.675053	−0.337526	−	0.941316i	$-0.609590\pi$
$79$	−6.00000	−0.675053	−0.337526	+	0.941316i	$0.609590\pi$
$80$	−1.00000	−0.111803
$81$	0	0
$82$	−6.00000	−0.662589
$83$	10.0000	1.09764	0.548821	−	0.835940i	$-0.315077\pi$
$83$	10.0000	1.09764	0.548821	+	0.835940i	$0.315077\pi$
$84$	0	0
$85$	−1.00000	−0.108465
$86$	−4.00000	−0.431331
$87$	0	0
$88$	2.00000	0.213201
$89$	−15.0000	−1.59000	−0.794998	−	0.606612i	$-0.792528\pi$
$89$	−15.0000	−1.59000	−0.794998	+	0.606612i	$0.792528\pi$
$90$	0	0
$91$	0	0
$92$	2.00000	0.208514
$93$	0	0
$94$	−6.00000	−0.618853
$95$	2.00000	0.205196
$96$	0	0
$97$	2.00000	0.203069	0.101535	−	0.994832i	$-0.467625\pi$
$97$	2.00000	0.203069	0.101535	+	0.994832i	$0.467625\pi$
$98$	0	0
$99$	0	0
$100$	−4.00000	−0.400000
$101$	2.00000	0.199007	0.0995037	−	0.995037i	$-0.468274\pi$
$101$	2.00000	0.199007	0.0995037	+	0.995037i	$0.468274\pi$
$102$	0	0
$103$	16.0000	1.57653	0.788263	−	0.615338i	$-0.210980\pi$
$103$	16.0000	1.57653	0.788263	+	0.615338i	$0.210980\pi$
$104$	−3.00000	−0.294174
$105$	0	0
$106$	−6.00000	−0.582772
$107$	0	0		−	1.00000i	$-0.5\pi$
$107$	0	0			1.00000i	$0.5\pi$
$108$	0	0
$109$	−11.0000	−1.05361	−0.526804	−	0.849987i	$-0.676610\pi$
$109$	−11.0000	−1.05361	−0.526804	+	0.849987i	$0.676610\pi$
$110$	−2.00000	−0.190693
$111$	0	0
$112$	0	0
$113$	1.00000	0.0940721	0.0470360	−	0.998893i	$-0.485022\pi$
$113$	1.00000	0.0940721	0.0470360	+	0.998893i	$0.485022\pi$
$114$	0	0
$115$	−2.00000	−0.186501
$116$	7.00000	0.649934
$117$	0	0
$118$	−10.0000	−0.920575
$119$	0	0
$120$	0	0
$121$	−7.00000	−0.636364
$122$	9.00000	0.814822
$123$	0	0
$124$	−6.00000	−0.538816
$125$	9.00000	0.804984
$126$	0	0
$127$	12.0000	1.06483	0.532414	−	0.846484i	$-0.321285\pi$
$127$	12.0000	1.06483	0.532414	+	0.846484i	$0.321285\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	3.00000	0.263117
$131$	−16.0000	−1.39793	−0.698963	−	0.715158i	$-0.746355\pi$
$131$	−16.0000	−1.39793	−0.698963	+	0.715158i	$0.746355\pi$
$132$	0	0
$133$	0	0
$134$	10.0000	0.863868
$135$	0	0
$136$	1.00000	0.0857493
$137$	−11.0000	−0.939793	−0.469897	−	0.882721i	$-0.655709\pi$
$137$	−11.0000	−0.939793	−0.469897	+	0.882721i	$0.655709\pi$
$138$	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
$142$	−4.00000	−0.335673
$143$	−6.00000	−0.501745
$144$	0	0
$145$	−7.00000	−0.581318
$146$	11.0000	0.910366
$147$	0	0
$148$	−7.00000	−0.575396
$149$	−21.0000	−1.72039	−0.860194	−	0.509968i	$-0.829657\pi$
$149$	−21.0000	−1.72039	−0.860194	+	0.509968i	$0.829657\pi$
$150$	0	0
$151$	10.0000	0.813788	0.406894	−	0.913475i	$-0.366612\pi$
$151$	10.0000	0.813788	0.406894	+	0.913475i	$0.366612\pi$
$152$	−2.00000	−0.162221
$153$	0	0
$154$	0	0
$155$	6.00000	0.481932
$156$	0	0
$157$	13.0000	1.03751	0.518756	−	0.854922i	$-0.326395\pi$
$157$	13.0000	1.03751	0.518756	+	0.854922i	$0.326395\pi$
$158$	−6.00000	−0.477334
$159$	0	0
$160$	−1.00000	−0.0790569
$161$	0	0
$162$	0	0
$163$	−16.0000	−1.25322	−0.626608	−	0.779334i	$-0.715557\pi$
$163$	−16.0000	−1.25322	−0.626608	+	0.779334i	$0.715557\pi$
$164$	−6.00000	−0.468521
$165$	0	0
$166$	10.0000	0.776151
$167$	−22.0000	−1.70241	−0.851206	−	0.524832i	$-0.824128\pi$
$167$	−22.0000	−1.70241	−0.851206	+	0.524832i	$0.824128\pi$
$168$	0	0
$169$	−4.00000	−0.307692
$170$	−1.00000	−0.0766965
$171$	0	0
$172$	−4.00000	−0.304997
$173$	−13.0000	−0.988372	−0.494186	−	0.869356i	$-0.664534\pi$
$173$	−13.0000	−0.988372	−0.494186	+	0.869356i	$0.664534\pi$
$174$	0	0
$175$	0	0
$176$	2.00000	0.150756
$177$	0	0
$178$	−15.0000	−1.12430
$179$	−18.0000	−1.34538	−0.672692	−	0.739923i	$-0.734862\pi$
$179$	−18.0000	−1.34538	−0.672692	+	0.739923i	$0.734862\pi$
$180$	0	0
$181$	−18.0000	−1.33793	−0.668965	−	0.743294i	$-0.733262\pi$
$181$	−18.0000	−1.33793	−0.668965	+	0.743294i	$0.733262\pi$
$182$	0	0
$183$	0	0
$184$	2.00000	0.147442
$185$	7.00000	0.514650
$186$	0	0
$187$	2.00000	0.146254
$188$	−6.00000	−0.437595
$189$	0	0
$190$	2.00000	0.145095
$191$	−18.0000	−1.30243	−0.651217	−	0.758891i	$-0.725741\pi$
$191$	−18.0000	−1.30243	−0.651217	+	0.758891i	$0.725741\pi$
$192$	0	0
$193$	11.0000	0.791797	0.395899	−	0.918294i	$-0.370433\pi$
$193$	11.0000	0.791797	0.395899	+	0.918294i	$0.370433\pi$
$194$	2.00000	0.143592
$195$	0	0
$196$	0	0
$197$	−1.00000	−0.0712470	−0.0356235	−	0.999365i	$-0.511342\pi$
$197$	−1.00000	−0.0712470	−0.0356235	+	0.999365i	$0.511342\pi$
$198$	0	0
$199$	10.0000	0.708881	0.354441	−	0.935079i	$-0.384671\pi$
$199$	10.0000	0.708881	0.354441	+	0.935079i	$0.384671\pi$
$200$	−4.00000	−0.282843
$201$	0	0
$202$	2.00000	0.140720
$203$	0	0
$204$	0	0
$205$	6.00000	0.419058
$206$	16.0000	1.11477
$207$	0	0
$208$	−3.00000	−0.208013
$209$	−4.00000	−0.276686
$210$	0	0
$211$	−22.0000	−1.51454	−0.757271	−	0.653101i	$-0.773468\pi$
$211$	−22.0000	−1.51454	−0.757271	+	0.653101i	$0.773468\pi$
$212$	−6.00000	−0.412082
$213$	0	0
$214$	0	0
$215$	4.00000	0.272798
$216$	0	0
$217$	0	0
$218$	−11.0000	−0.745014
$219$	0	0
$220$	−2.00000	−0.134840
$221$	−3.00000	−0.201802
$222$	0	0
$223$	−20.0000	−1.33930	−0.669650	−	0.742677i	$-0.733556\pi$
$223$	−20.0000	−1.33930	−0.669650	+	0.742677i	$0.733556\pi$
$224$	0	0
$225$	0	0
$226$	1.00000	0.0665190
$227$	24.0000	1.59294	0.796468	−	0.604681i	$-0.206699\pi$
$227$	24.0000	1.59294	0.796468	+	0.604681i	$0.206699\pi$
$228$	0	0
$229$	29.0000	1.91637	0.958187	−	0.286143i	$-0.0923732\pi$
$229$	29.0000	1.91637	0.958187	+	0.286143i	$0.0923732\pi$
$230$	−2.00000	−0.131876
$231$	0	0
$232$	7.00000	0.459573
$233$	−19.0000	−1.24473	−0.622366	−	0.782727i	$-0.713828\pi$
$233$	−19.0000	−1.24473	−0.622366	+	0.782727i	$0.713828\pi$
$234$	0	0
$235$	6.00000	0.391397
$236$	−10.0000	−0.650945
$237$	0	0
$238$	0	0
$239$	−6.00000	−0.388108	−0.194054	−	0.980991i	$-0.562164\pi$
$239$	−6.00000	−0.388108	−0.194054	+	0.980991i	$0.562164\pi$
$240$	0	0
$241$	7.00000	0.450910	0.225455	−	0.974254i	$-0.427613\pi$
$241$	7.00000	0.450910	0.225455	+	0.974254i	$0.427613\pi$
$242$	−7.00000	−0.449977
$243$	0	0
$244$	9.00000	0.576166
$245$	0	0
$246$	0	0
$247$	6.00000	0.381771
$248$	−6.00000	−0.381000
$249$	0	0
$250$	9.00000	0.569210
$251$	6.00000	0.378717	0.189358	−	0.981908i	$-0.439359\pi$
$251$	6.00000	0.378717	0.189358	+	0.981908i	$0.439359\pi$
$252$	0	0
$253$	4.00000	0.251478
$254$	12.0000	0.752947
$255$	0	0
$256$	1.00000	0.0625000
$257$	9.00000	0.561405	0.280702	−	0.959795i	$-0.409433\pi$
$257$	9.00000	0.561405	0.280702	+	0.959795i	$0.409433\pi$
$258$	0	0
$259$	0	0
$260$	3.00000	0.186052
$261$	0	0
$262$	−16.0000	−0.988483
$263$	−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
$268$	10.0000	0.610847
$269$	−1.00000	−0.0609711	−0.0304855	−	0.999535i	$-0.509705\pi$
$269$	−1.00000	−0.0609711	−0.0304855	+	0.999535i	$0.509705\pi$
$270$	0	0
$271$	−30.0000	−1.82237	−0.911185	−	0.411997i	$-0.864831\pi$
$271$	−30.0000	−1.82237	−0.911185	+	0.411997i	$0.864831\pi$
$272$	1.00000	0.0606339
$273$	0	0
$274$	−11.0000	−0.664534
$275$	−8.00000	−0.482418
$276$	0	0
$277$	14.0000	0.841178	0.420589	−	0.907251i	$-0.361823\pi$
$277$	14.0000	0.841178	0.420589	+	0.907251i	$0.361823\pi$
$278$	2.00000	0.119952
$279$	0	0
$280$	0	0
$281$	29.0000	1.72999	0.864997	−	0.501776i	$-0.167320\pi$
$281$	29.0000	1.72999	0.864997	+	0.501776i	$0.167320\pi$
$282$	0	0
$283$	−16.0000	−0.951101	−0.475551	−	0.879688i	$-0.657751\pi$
$283$	−16.0000	−0.951101	−0.475551	+	0.879688i	$0.657751\pi$
$284$	−4.00000	−0.237356
$285$	0	0
$286$	−6.00000	−0.354787
$287$	0	0
$288$	0	0
$289$	−16.0000	−0.941176
$290$	−7.00000	−0.411054
$291$	0	0
$292$	11.0000	0.643726
$293$	15.0000	0.876309	0.438155	−	0.898900i	$-0.355632\pi$
$293$	15.0000	0.876309	0.438155	+	0.898900i	$0.355632\pi$
$294$	0	0
$295$	10.0000	0.582223
$296$	−7.00000	−0.406867
$297$	0	0
$298$	−21.0000	−1.21650
$299$	−6.00000	−0.346989
$300$	0	0
$301$	0	0
$302$	10.0000	0.575435
$303$	0	0
$304$	−2.00000	−0.114708
$305$	−9.00000	−0.515339
$306$	0	0
$307$	−16.0000	−0.913168	−0.456584	−	0.889680i	$-0.650927\pi$
$307$	−16.0000	−0.913168	−0.456584	+	0.889680i	$0.650927\pi$
$308$	0	0
$309$	0	0
$310$	6.00000	0.340777
$311$	14.0000	0.793867	0.396934	−	0.917847i	$-0.370074\pi$
$311$	14.0000	0.793867	0.396934	+	0.917847i	$0.370074\pi$
$312$	0	0
$313$	−1.00000	−0.0565233	−0.0282617	−	0.999601i	$-0.508997\pi$
$313$	−1.00000	−0.0565233	−0.0282617	+	0.999601i	$0.508997\pi$
$314$	13.0000	0.733632
$315$	0	0
$316$	−6.00000	−0.337526
$317$	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
$319$	14.0000	0.783850
$320$	−1.00000	−0.0559017
$321$	0	0
$322$	0	0
$323$	−2.00000	−0.111283
$324$	0	0
$325$	12.0000	0.665640
$326$	−16.0000	−0.886158
$327$	0	0
$328$	−6.00000	−0.331295
$329$	0	0
$330$	0	0
$331$	20.0000	1.09930	0.549650	−	0.835395i	$-0.314761\pi$
$331$	20.0000	1.09930	0.549650	+	0.835395i	$0.314761\pi$
$332$	10.0000	0.548821
$333$	0	0
$334$	−22.0000	−1.20379
$335$	−10.0000	−0.546358
$336$	0	0
$337$	−18.0000	−0.980522	−0.490261	−	0.871576i	$-0.663099\pi$
$337$	−18.0000	−0.980522	−0.490261	+	0.871576i	$0.663099\pi$
$338$	−4.00000	−0.217571
$339$	0	0
$340$	−1.00000	−0.0542326
$341$	−12.0000	−0.649836
$342$	0	0
$343$	0	0
$344$	−4.00000	−0.215666
$345$	0	0
$346$	−13.0000	−0.698884
$347$	−12.0000	−0.644194	−0.322097	−	0.946707i	$-0.604388\pi$
$347$	−12.0000	−0.644194	−0.322097	+	0.946707i	$0.604388\pi$
$348$	0	0
$349$	−10.0000	−0.535288	−0.267644	−	0.963518i	$-0.586245\pi$
$349$	−10.0000	−0.535288	−0.267644	+	0.963518i	$0.586245\pi$
$350$	0	0
$351$	0	0
$352$	2.00000	0.106600
$353$	30.0000	1.59674	0.798369	−	0.602168i	$-0.205696\pi$
$353$	30.0000	1.59674	0.798369	+	0.602168i	$0.205696\pi$
$354$	0	0
$355$	4.00000	0.212298
$356$	−15.0000	−0.794998
$357$	0	0
$358$	−18.0000	−0.951330
$359$	16.0000	0.844448	0.422224	−	0.906492i	$-0.361250\pi$
$359$	16.0000	0.844448	0.422224	+	0.906492i	$0.361250\pi$
$360$	0	0
$361$	−15.0000	−0.789474
$362$	−18.0000	−0.946059
$363$	0	0
$364$	0	0
$365$	−11.0000	−0.575766
$366$	0	0
$367$	10.0000	0.521996	0.260998	−	0.965339i	$-0.415948\pi$
$367$	10.0000	0.521996	0.260998	+	0.965339i	$0.415948\pi$
$368$	2.00000	0.104257
$369$	0	0
$370$	7.00000	0.363913
$371$	0	0
$372$	0	0
$373$	10.0000	0.517780	0.258890	−	0.965907i	$-0.416643\pi$
$373$	10.0000	0.517780	0.258890	+	0.965907i	$0.416643\pi$
$374$	2.00000	0.103418
$375$	0	0
$376$	−6.00000	−0.309426
$377$	−21.0000	−1.08156
$378$	0	0
$379$	10.0000	0.513665	0.256833	−	0.966456i	$-0.417321\pi$
$379$	10.0000	0.513665	0.256833	+	0.966456i	$0.417321\pi$
$380$	2.00000	0.102598
$381$	0	0
$382$	−18.0000	−0.920960
$383$	−32.0000	−1.63512	−0.817562	−	0.575841i	$-0.804675\pi$
$383$	−32.0000	−1.63512	−0.817562	+	0.575841i	$0.804675\pi$
$384$	0	0
$385$	0	0
$386$	11.0000	0.559885
$387$	0	0
$388$	2.00000	0.101535
$389$	34.0000	1.72387	0.861934	−	0.507020i	$-0.169253\pi$
$389$	34.0000	1.72387	0.861934	+	0.507020i	$0.169253\pi$
$390$	0	0
$391$	2.00000	0.101144
$392$	0	0
$393$	0	0
$394$	−1.00000	−0.0503793
$395$	6.00000	0.301893
$396$	0	0
$397$	−39.0000	−1.95735	−0.978677	−	0.205406i	$-0.934149\pi$
$397$	−39.0000	−1.95735	−0.978677	+	0.205406i	$0.934149\pi$
$398$	10.0000	0.501255
$399$	0	0
$400$	−4.00000	−0.200000
$401$	−3.00000	−0.149813	−0.0749064	−	0.997191i	$-0.523866\pi$
$401$	−3.00000	−0.149813	−0.0749064	+	0.997191i	$0.523866\pi$
$402$	0	0
$403$	18.0000	0.896644
$404$	2.00000	0.0995037
$405$	0	0
$406$	0	0
$407$	−14.0000	−0.693954
$408$	0	0
$409$	7.00000	0.346128	0.173064	−	0.984911i	$-0.444633\pi$
$409$	7.00000	0.346128	0.173064	+	0.984911i	$0.444633\pi$
$410$	6.00000	0.296319
$411$	0	0
$412$	16.0000	0.788263
$413$	0	0
$414$	0	0
$415$	−10.0000	−0.490881
$416$	−3.00000	−0.147087
$417$	0	0
$418$	−4.00000	−0.195646
$419$	−12.0000	−0.586238	−0.293119	−	0.956076i	$-0.594693\pi$
$419$	−12.0000	−0.586238	−0.293119	+	0.956076i	$0.594693\pi$
$420$	0	0
$421$	9.00000	0.438633	0.219317	−	0.975654i	$-0.429617\pi$
$421$	9.00000	0.438633	0.219317	+	0.975654i	$0.429617\pi$
$422$	−22.0000	−1.07094
$423$	0	0
$424$	−6.00000	−0.291386
$425$	−4.00000	−0.194029
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0	0
$430$	4.00000	0.192897
$431$	12.0000	0.578020	0.289010	−	0.957326i	$-0.406674\pi$
$431$	12.0000	0.578020	0.289010	+	0.957326i	$0.406674\pi$
$432$	0	0
$433$	−5.00000	−0.240285	−0.120142	−	0.992757i	$-0.538335\pi$
$433$	−5.00000	−0.240285	−0.120142	+	0.992757i	$0.538335\pi$
$434$	0	0
$435$	0	0
$436$	−11.0000	−0.526804
$437$	−4.00000	−0.191346
$438$	0	0
$439$	12.0000	0.572729	0.286364	−	0.958121i	$-0.407553\pi$
$439$	12.0000	0.572729	0.286364	+	0.958121i	$0.407553\pi$
$440$	−2.00000	−0.0953463
$441$	0	0
$442$	−3.00000	−0.142695
$443$	−22.0000	−1.04525	−0.522626	−	0.852562i	$-0.675047\pi$
$443$	−22.0000	−1.04525	−0.522626	+	0.852562i	$0.675047\pi$
$444$	0	0
$445$	15.0000	0.711068
$446$	−20.0000	−0.947027
$447$	0	0
$448$	0	0
$449$	34.0000	1.60456	0.802280	−	0.596948i	$-0.203620\pi$
$449$	34.0000	1.60456	0.802280	+	0.596948i	$0.203620\pi$
$450$	0	0
$451$	−12.0000	−0.565058
$452$	1.00000	0.0470360
$453$	0	0
$454$	24.0000	1.12638
$455$	0	0
$456$	0	0
$457$	−41.0000	−1.91790	−0.958950	−	0.283577i	$-0.908479\pi$
$457$	−41.0000	−1.91790	−0.958950	+	0.283577i	$0.908479\pi$
$458$	29.0000	1.35508
$459$	0	0
$460$	−2.00000	−0.0932505
$461$	−22.0000	−1.02464	−0.512321	−	0.858794i	$-0.671214\pi$
$461$	−22.0000	−1.02464	−0.512321	+	0.858794i	$0.671214\pi$
$462$	0	0
$463$	34.0000	1.58011	0.790057	−	0.613033i	$-0.210051\pi$
$463$	34.0000	1.58011	0.790057	+	0.613033i	$0.210051\pi$
$464$	7.00000	0.324967
$465$	0	0
$466$	−19.0000	−0.880158
$467$	−34.0000	−1.57333	−0.786666	−	0.617379i	$-0.788195\pi$
$467$	−34.0000	−1.57333	−0.786666	+	0.617379i	$0.788195\pi$
$468$	0	0
$469$	0	0
$470$	6.00000	0.276759
$471$	0	0
$472$	−10.0000	−0.460287
$473$	−8.00000	−0.367840
$474$	0	0
$475$	8.00000	0.367065
$476$	0	0
$477$	0	0
$478$	−6.00000	−0.274434
$479$	22.0000	1.00521	0.502603	−	0.864517i	$-0.332376\pi$
$479$	22.0000	1.00521	0.502603	+	0.864517i	$0.332376\pi$
$480$	0	0
$481$	21.0000	0.957518
$482$	7.00000	0.318841
$483$	0	0
$484$	−7.00000	−0.318182
$485$	−2.00000	−0.0908153
$486$	0	0
$487$	−10.0000	−0.453143	−0.226572	−	0.973995i	$-0.572752\pi$
$487$	−10.0000	−0.453143	−0.226572	+	0.973995i	$0.572752\pi$
$488$	9.00000	0.407411
$489$	0	0
$490$	0	0
$491$	36.0000	1.62466	0.812329	−	0.583200i	$-0.198200\pi$
$491$	36.0000	1.62466	0.812329	+	0.583200i	$0.198200\pi$
$492$	0	0
$493$	7.00000	0.315264
$494$	6.00000	0.269953
$495$	0	0
$496$	−6.00000	−0.269408
$497$	0	0
$498$	0	0
$499$	22.0000	0.984855	0.492428	−	0.870353i	$-0.336110\pi$
$499$	22.0000	0.984855	0.492428	+	0.870353i	$0.336110\pi$
$500$	9.00000	0.402492
$501$	0	0
$502$	6.00000	0.267793
$503$	6.00000	0.267527	0.133763	−	0.991013i	$-0.457294\pi$
$503$	6.00000	0.267527	0.133763	+	0.991013i	$0.457294\pi$
$504$	0	0
$505$	−2.00000	−0.0889988
$506$	4.00000	0.177822
$507$	0	0
$508$	12.0000	0.532414
$509$	18.0000	0.797836	0.398918	−	0.916987i	$-0.369386\pi$
$509$	18.0000	0.797836	0.398918	+	0.916987i	$0.369386\pi$
$510$	0	0
$511$	0	0
$512$	1.00000	0.0441942
$513$	0	0
$514$	9.00000	0.396973
$515$	−16.0000	−0.705044
$516$	0	0
$517$	−12.0000	−0.527759
$518$	0	0
$519$	0	0
$520$	3.00000	0.131559
$521$	−42.0000	−1.84005	−0.920027	−	0.391856i	$-0.871833\pi$
$521$	−42.0000	−1.84005	−0.920027	+	0.391856i	$0.871833\pi$
$522$	0	0
$523$	4.00000	0.174908	0.0874539	−	0.996169i	$-0.472127\pi$
$523$	4.00000	0.174908	0.0874539	+	0.996169i	$0.472127\pi$
$524$	−16.0000	−0.698963
$525$	0	0
$526$	−18.0000	−0.784837
$527$	−6.00000	−0.261364
$528$	0	0
$529$	−19.0000	−0.826087
$530$	6.00000	0.260623
$531$	0	0
$532$	0	0
$533$	18.0000	0.779667
$534$	0	0
$535$	0	0
$536$	10.0000	0.431934
$537$	0	0
$538$	−1.00000	−0.0431131
$539$	0	0
$540$	0	0
$541$	−3.00000	−0.128980	−0.0644900	−	0.997918i	$-0.520542\pi$
$541$	−3.00000	−0.128980	−0.0644900	+	0.997918i	$0.520542\pi$
$542$	−30.0000	−1.28861
$543$	0	0
$544$	1.00000	0.0428746
$545$	11.0000	0.471188
$546$	0	0
$547$	−6.00000	−0.256541	−0.128271	−	0.991739i	$-0.540943\pi$
$547$	−6.00000	−0.256541	−0.128271	+	0.991739i	$0.540943\pi$
$548$	−11.0000	−0.469897
$549$	0	0
$550$	−8.00000	−0.341121
$551$	−14.0000	−0.596420
$552$	0	0
$553$	0	0
$554$	14.0000	0.594803
$555$	0	0
$556$	2.00000	0.0848189
$557$	−17.0000	−0.720313	−0.360157	−	0.932892i	$-0.617277\pi$
$557$	−17.0000	−0.720313	−0.360157	+	0.932892i	$0.617277\pi$
$558$	0	0
$559$	12.0000	0.507546
$560$	0	0
$561$	0	0
$562$	29.0000	1.22329
$563$	−8.00000	−0.337160	−0.168580	−	0.985688i	$-0.553918\pi$
$563$	−8.00000	−0.337160	−0.168580	+	0.985688i	$0.553918\pi$
$564$	0	0
$565$	−1.00000	−0.0420703
$566$	−16.0000	−0.672530
$567$	0	0
$568$	−4.00000	−0.167836
$569$	−27.0000	−1.13190	−0.565949	−	0.824440i	$-0.691490\pi$
$569$	−27.0000	−1.13190	−0.565949	+	0.824440i	$0.691490\pi$
$570$	0	0
$571$	18.0000	0.753277	0.376638	−	0.926360i	$-0.377080\pi$
$571$	18.0000	0.753277	0.376638	+	0.926360i	$0.377080\pi$
$572$	−6.00000	−0.250873
$573$	0	0
$574$	0	0
$575$	−8.00000	−0.333623
$576$	0	0
$577$	35.0000	1.45707	0.728535	−	0.685009i	$-0.240202\pi$
$577$	35.0000	1.45707	0.728535	+	0.685009i	$0.240202\pi$
$578$	−16.0000	−0.665512
$579$	0	0
$580$	−7.00000	−0.290659
$581$	0	0
$582$	0	0
$583$	−12.0000	−0.496989
$584$	11.0000	0.455183
$585$	0	0
$586$	15.0000	0.619644
$587$	−2.00000	−0.0825488	−0.0412744	−	0.999148i	$-0.513142\pi$
$587$	−2.00000	−0.0825488	−0.0412744	+	0.999148i	$0.513142\pi$
$588$	0	0
$589$	12.0000	0.494451
$590$	10.0000	0.411693
$591$	0	0
$592$	−7.00000	−0.287698
$593$	−39.0000	−1.60154	−0.800769	−	0.598973i	$-0.795576\pi$
$593$	−39.0000	−1.60154	−0.800769	+	0.598973i	$0.795576\pi$
$594$	0	0
$595$	0	0
$596$	−21.0000	−0.860194
$597$	0	0
$598$	−6.00000	−0.245358
$599$	−6.00000	−0.245153	−0.122577	−	0.992459i	$-0.539116\pi$
$599$	−6.00000	−0.245153	−0.122577	+	0.992459i	$0.539116\pi$
$600$	0	0
$601$	19.0000	0.775026	0.387513	−	0.921864i	$-0.373334\pi$
$601$	19.0000	0.775026	0.387513	+	0.921864i	$0.373334\pi$
$602$	0	0
$603$	0	0
$604$	10.0000	0.406894
$605$	7.00000	0.284590
$606$	0	0
$607$	6.00000	0.243532	0.121766	−	0.992559i	$-0.461144\pi$
$607$	6.00000	0.243532	0.121766	+	0.992559i	$0.461144\pi$
$608$	−2.00000	−0.0811107
$609$	0	0
$610$	−9.00000	−0.364399
$611$	18.0000	0.728202
$612$	0	0
$613$	−18.0000	−0.727013	−0.363507	−	0.931592i	$-0.618421\pi$
$613$	−18.0000	−0.727013	−0.363507	+	0.931592i	$0.618421\pi$
$614$	−16.0000	−0.645707
$615$	0	0
$616$	0	0
$617$	−7.00000	−0.281809	−0.140905	−	0.990023i	$-0.545001\pi$
$617$	−7.00000	−0.281809	−0.140905	+	0.990023i	$0.545001\pi$
$618$	0	0
$619$	20.0000	0.803868	0.401934	−	0.915669i	$-0.368338\pi$
$619$	20.0000	0.803868	0.401934	+	0.915669i	$0.368338\pi$
$620$	6.00000	0.240966
$621$	0	0
$622$	14.0000	0.561349
$623$	0	0
$624$	0	0
$625$	11.0000	0.440000
$626$	−1.00000	−0.0399680
$627$	0	0
$628$	13.0000	0.518756
$629$	−7.00000	−0.279108
$630$	0	0
$631$	38.0000	1.51276	0.756378	−	0.654135i	$-0.226967\pi$
$631$	38.0000	1.51276	0.756378	+	0.654135i	$0.226967\pi$
$632$	−6.00000	−0.238667
$633$	0	0
$634$	3.00000	0.119145
$635$	−12.0000	−0.476205
$636$	0	0
$637$	0	0
$638$	14.0000	0.554265
$639$	0	0
$640$	−1.00000	−0.0395285
$641$	17.0000	0.671460	0.335730	−	0.941958i	$-0.391017\pi$
$641$	17.0000	0.671460	0.335730	+	0.941958i	$0.391017\pi$
$642$	0	0
$643$	−26.0000	−1.02534	−0.512670	−	0.858586i	$-0.671344\pi$
$643$	−26.0000	−1.02534	−0.512670	+	0.858586i	$0.671344\pi$
$644$	0	0
$645$	0	0
$646$	−2.00000	−0.0786889
$647$	−24.0000	−0.943537	−0.471769	−	0.881722i	$-0.656384\pi$
$647$	−24.0000	−0.943537	−0.471769	+	0.881722i	$0.656384\pi$
$648$	0	0
$649$	−20.0000	−0.785069
$650$	12.0000	0.470679
$651$	0	0
$652$	−16.0000	−0.626608
$653$	18.0000	0.704394	0.352197	−	0.935926i	$-0.385435\pi$
$653$	18.0000	0.704394	0.352197	+	0.935926i	$0.385435\pi$
$654$	0	0
$655$	16.0000	0.625172
$656$	−6.00000	−0.234261
$657$	0	0
$658$	0	0
$659$	32.0000	1.24654	0.623272	−	0.782006i	$-0.285803\pi$
$659$	32.0000	1.24654	0.623272	+	0.782006i	$0.285803\pi$
$660$	0	0
$661$	−7.00000	−0.272268	−0.136134	−	0.990690i	$-0.543468\pi$
$661$	−7.00000	−0.272268	−0.136134	+	0.990690i	$0.543468\pi$
$662$	20.0000	0.777322
$663$	0	0
$664$	10.0000	0.388075
$665$	0	0
$666$	0	0
$667$	14.0000	0.542082
$668$	−22.0000	−0.851206
$669$	0	0
$670$	−10.0000	−0.386334
$671$	18.0000	0.694882
$672$	0	0
$673$	23.0000	0.886585	0.443292	−	0.896377i	$-0.353810\pi$
$673$	23.0000	0.886585	0.443292	+	0.896377i	$0.353810\pi$
$674$	−18.0000	−0.693334
$675$	0	0
$676$	−4.00000	−0.153846
$677$	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
$679$	0	0
$680$	−1.00000	−0.0383482
$681$	0	0
$682$	−12.0000	−0.459504
$683$	−12.0000	−0.459167	−0.229584	−	0.973289i	$-0.573736\pi$
$683$	−12.0000	−0.459167	−0.229584	+	0.973289i	$0.573736\pi$
$684$	0	0
$685$	11.0000	0.420288
$686$	0	0
$687$	0	0
$688$	−4.00000	−0.152499
$689$	18.0000	0.685745
$690$	0	0
$691$	4.00000	0.152167	0.0760836	−	0.997101i	$-0.475758\pi$
$691$	4.00000	0.152167	0.0760836	+	0.997101i	$0.475758\pi$
$692$	−13.0000	−0.494186
$693$	0	0
$694$	−12.0000	−0.455514
$695$	−2.00000	−0.0758643
$696$	0	0
$697$	−6.00000	−0.227266
$698$	−10.0000	−0.378506
$699$	0	0
$700$	0	0
$701$	−5.00000	−0.188847	−0.0944237	−	0.995532i	$-0.530101\pi$
$701$	−5.00000	−0.188847	−0.0944237	+	0.995532i	$0.530101\pi$
$702$	0	0
$703$	14.0000	0.528020
$704$	2.00000	0.0753778
$705$	0	0
$706$	30.0000	1.12906
$707$	0	0
$708$	0	0
$709$	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
$712$	−15.0000	−0.562149
$713$	−12.0000	−0.449404
$714$	0	0
$715$	6.00000	0.224387
$716$	−18.0000	−0.672692
$717$	0	0
$718$	16.0000	0.597115
$719$	24.0000	0.895049	0.447524	−	0.894272i	$-0.352306\pi$
$719$	24.0000	0.895049	0.447524	+	0.894272i	$0.352306\pi$
$720$	0	0
$721$	0	0
$722$	−15.0000	−0.558242
$723$	0	0
$724$	−18.0000	−0.668965
$725$	−28.0000	−1.03989
$726$	0	0
$727$	8.00000	0.296704	0.148352	−	0.988935i	$-0.452603\pi$
$727$	8.00000	0.296704	0.148352	+	0.988935i	$0.452603\pi$
$728$	0	0
$729$	0	0
$730$	−11.0000	−0.407128
$731$	−4.00000	−0.147945
$732$	0	0
$733$	−42.0000	−1.55131	−0.775653	−	0.631160i	$-0.782579\pi$
$733$	−42.0000	−1.55131	−0.775653	+	0.631160i	$0.782579\pi$
$734$	10.0000	0.369107
$735$	0	0
$736$	2.00000	0.0737210
$737$	20.0000	0.736709
$738$	0	0
$739$	−6.00000	−0.220714	−0.110357	−	0.993892i	$-0.535199\pi$
$739$	−6.00000	−0.220714	−0.110357	+	0.993892i	$0.535199\pi$
$740$	7.00000	0.257325
$741$	0	0
$742$	0	0
$743$	24.0000	0.880475	0.440237	−	0.897881i	$-0.354894\pi$
$743$	24.0000	0.880475	0.440237	+	0.897881i	$0.354894\pi$
$744$	0	0
$745$	21.0000	0.769380
$746$	10.0000	0.366126
$747$	0	0
$748$	2.00000	0.0731272
$749$	0	0
$750$	0	0
$751$	12.0000	0.437886	0.218943	−	0.975738i	$-0.429739\pi$
$751$	12.0000	0.437886	0.218943	+	0.975738i	$0.429739\pi$
$752$	−6.00000	−0.218797
$753$	0	0
$754$	−21.0000	−0.764775
$755$	−10.0000	−0.363937
$756$	0	0
$757$	42.0000	1.52652	0.763258	−	0.646094i	$-0.223599\pi$
$757$	42.0000	1.52652	0.763258	+	0.646094i	$0.223599\pi$
$758$	10.0000	0.363216
$759$	0	0
$760$	2.00000	0.0725476
$761$	−11.0000	−0.398750	−0.199375	−	0.979923i	$-0.563891\pi$
$761$	−11.0000	−0.398750	−0.199375	+	0.979923i	$0.563891\pi$
$762$	0	0
$763$	0	0
$764$	−18.0000	−0.651217
$765$	0	0
$766$	−32.0000	−1.15621
$767$	30.0000	1.08324
$768$	0	0
$769$	23.0000	0.829401	0.414701	−	0.909958i	$-0.363886\pi$
$769$	23.0000	0.829401	0.414701	+	0.909958i	$0.363886\pi$
$770$	0	0
$771$	0	0
$772$	11.0000	0.395899
$773$	35.0000	1.25886	0.629431	−	0.777056i	$-0.283288\pi$
$773$	35.0000	1.25886	0.629431	+	0.777056i	$0.283288\pi$
$774$	0	0
$775$	24.0000	0.862105
$776$	2.00000	0.0717958
$777$	0	0
$778$	34.0000	1.21896
$779$	12.0000	0.429945
$780$	0	0
$781$	−8.00000	−0.286263
$782$	2.00000	0.0715199
$783$	0	0
$784$	0	0
$785$	−13.0000	−0.463990
$786$	0	0
$787$	24.0000	0.855508	0.427754	−	0.903895i	$-0.359305\pi$
$787$	24.0000	0.855508	0.427754	+	0.903895i	$0.359305\pi$
$788$	−1.00000	−0.0356235
$789$	0	0
$790$	6.00000	0.213470
$791$	0	0
$792$	0	0
$793$	−27.0000	−0.958798
$794$	−39.0000	−1.38406
$795$	0	0
$796$	10.0000	0.354441
$797$	3.00000	0.106265	0.0531327	−	0.998587i	$-0.483079\pi$
$797$	3.00000	0.106265	0.0531327	+	0.998587i	$0.483079\pi$
$798$	0	0
$799$	−6.00000	−0.212265
$800$	−4.00000	−0.141421
$801$	0	0
$802$	−3.00000	−0.105934
$803$	22.0000	0.776363
$804$	0	0
$805$	0	0
$806$	18.0000	0.634023
$807$	0	0
$808$	2.00000	0.0703598
$809$	1.00000	0.0351581	0.0175791	−	0.999845i	$-0.494404\pi$
$809$	1.00000	0.0351581	0.0175791	+	0.999845i	$0.494404\pi$
$810$	0	0
$811$	−34.0000	−1.19390	−0.596951	−	0.802278i	$-0.703621\pi$
$811$	−34.0000	−1.19390	−0.596951	+	0.802278i	$0.703621\pi$
$812$	0	0
$813$	0	0
$814$	−14.0000	−0.490700
$815$	16.0000	0.560456
$816$	0	0
$817$	8.00000	0.279885
$818$	7.00000	0.244749
$819$	0	0
$820$	6.00000	0.209529
$821$	11.0000	0.383903	0.191951	−	0.981404i	$-0.438518\pi$
$821$	11.0000	0.383903	0.191951	+	0.981404i	$0.438518\pi$
$822$	0	0
$823$	16.0000	0.557725	0.278862	−	0.960331i	$-0.410043\pi$
$823$	16.0000	0.557725	0.278862	+	0.960331i	$0.410043\pi$
$824$	16.0000	0.557386
$825$	0	0
$826$	0	0
$827$	−48.0000	−1.66912	−0.834562	−	0.550914i	$-0.814279\pi$
$827$	−48.0000	−1.66912	−0.834562	+	0.550914i	$0.814279\pi$
$828$	0	0
$829$	2.00000	0.0694629	0.0347314	−	0.999397i	$-0.488942\pi$
$829$	2.00000	0.0694629	0.0347314	+	0.999397i	$0.488942\pi$
$830$	−10.0000	−0.347105
$831$	0	0
$832$	−3.00000	−0.104006
$833$	0	0
$834$	0	0
$835$	22.0000	0.761341
$836$	−4.00000	−0.138343
$837$	0	0
$838$	−12.0000	−0.414533
$839$	−32.0000	−1.10476	−0.552381	−	0.833592i	$-0.686281\pi$
$839$	−32.0000	−1.10476	−0.552381	+	0.833592i	$0.686281\pi$
$840$	0	0
$841$	20.0000	0.689655
$842$	9.00000	0.310160
$843$	0	0
$844$	−22.0000	−0.757271
$845$	4.00000	0.137604
$846$	0	0
$847$	0	0
$848$	−6.00000	−0.206041
$849$	0	0
$850$	−4.00000	−0.137199
$851$	−14.0000	−0.479914
$852$	0	0
$853$	10.0000	0.342393	0.171197	−	0.985237i	$-0.445237\pi$
$853$	10.0000	0.342393	0.171197	+	0.985237i	$0.445237\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−3.00000	−0.102478	−0.0512390	−	0.998686i	$-0.516317\pi$
$857$	−3.00000	−0.102478	−0.0512390	+	0.998686i	$0.516317\pi$
$858$	0	0
$859$	52.0000	1.77422	0.887109	−	0.461561i	$-0.152710\pi$
$859$	52.0000	1.77422	0.887109	+	0.461561i	$0.152710\pi$
$860$	4.00000	0.136399
$861$	0	0
$862$	12.0000	0.408722
$863$	−14.0000	−0.476566	−0.238283	−	0.971196i	$-0.576585\pi$
$863$	−14.0000	−0.476566	−0.238283	+	0.971196i	$0.576585\pi$
$864$	0	0
$865$	13.0000	0.442013
$866$	−5.00000	−0.169907
$867$	0	0
$868$	0	0
$869$	−12.0000	−0.407072
$870$	0	0
$871$	−30.0000	−1.01651
$872$	−11.0000	−0.372507
$873$	0	0
$874$	−4.00000	−0.135302
$875$	0	0
$876$	0	0
$877$	−23.0000	−0.776655	−0.388327	−	0.921521i	$-0.626947\pi$
$877$	−23.0000	−0.776655	−0.388327	+	0.921521i	$0.626947\pi$
$878$	12.0000	0.404980
$879$	0	0
$880$	−2.00000	−0.0674200
$881$	18.0000	0.606435	0.303218	−	0.952921i	$-0.401939\pi$
$881$	18.0000	0.606435	0.303218	+	0.952921i	$0.401939\pi$
$882$	0	0
$883$	−10.0000	−0.336527	−0.168263	−	0.985742i	$-0.553816\pi$
$883$	−10.0000	−0.336527	−0.168263	+	0.985742i	$0.553816\pi$
$884$	−3.00000	−0.100901
$885$	0	0
$886$	−22.0000	−0.739104
$887$	−42.0000	−1.41022	−0.705111	−	0.709097i	$-0.749103\pi$
$887$	−42.0000	−1.41022	−0.705111	+	0.709097i	$0.749103\pi$
$888$	0	0
$889$	0	0
$890$	15.0000	0.502801
$891$	0	0
$892$	−20.0000	−0.669650
$893$	12.0000	0.401565
$894$	0	0
$895$	18.0000	0.601674
$896$	0	0
$897$	0	0
$898$	34.0000	1.13459
$899$	−42.0000	−1.40078
$900$	0	0
$901$	−6.00000	−0.199889
$902$	−12.0000	−0.399556
$903$	0	0
$904$	1.00000	0.0332595
$905$	18.0000	0.598340
$906$	0	0
$907$	−12.0000	−0.398453	−0.199227	−	0.979953i	$-0.563843\pi$
$907$	−12.0000	−0.398453	−0.199227	+	0.979953i	$0.563843\pi$
$908$	24.0000	0.796468
$909$	0	0
$910$	0	0
$911$	12.0000	0.397578	0.198789	−	0.980042i	$-0.436299\pi$
$911$	12.0000	0.397578	0.198789	+	0.980042i	$0.436299\pi$
$912$	0	0
$913$	20.0000	0.661903
$914$	−41.0000	−1.35616
$915$	0	0
$916$	29.0000	0.958187
$917$	0	0
$918$	0	0
$919$	10.0000	0.329870	0.164935	−	0.986304i	$-0.447259\pi$
$919$	10.0000	0.329870	0.164935	+	0.986304i	$0.447259\pi$
$920$	−2.00000	−0.0659380
$921$	0	0
$922$	−22.0000	−0.724531
$923$	12.0000	0.394985
$924$	0	0
$925$	28.0000	0.920634
$926$	34.0000	1.11731
$927$	0	0
$928$	7.00000	0.229786
$929$	33.0000	1.08269	0.541347	−	0.840799i	$-0.317914\pi$
$929$	33.0000	1.08269	0.541347	+	0.840799i	$0.317914\pi$
$930$	0	0
$931$	0	0
$932$	−19.0000	−0.622366
$933$	0	0
$934$	−34.0000	−1.11251
$935$	−2.00000	−0.0654070
$936$	0	0
$937$	43.0000	1.40475	0.702374	−	0.711808i	$-0.252123\pi$
$937$	43.0000	1.40475	0.702374	+	0.711808i	$0.252123\pi$
$938$	0	0
$939$	0	0
$940$	6.00000	0.195698
$941$	−25.0000	−0.814977	−0.407488	−	0.913210i	$-0.633595\pi$
$941$	−25.0000	−0.814977	−0.407488	+	0.913210i	$0.633595\pi$
$942$	0	0
$943$	−12.0000	−0.390774
$944$	−10.0000	−0.325472
$945$	0	0
$946$	−8.00000	−0.260102
$947$	4.00000	0.129983	0.0649913	−	0.997886i	$-0.479298\pi$
$947$	4.00000	0.129983	0.0649913	+	0.997886i	$0.479298\pi$
$948$	0	0
$949$	−33.0000	−1.07123
$950$	8.00000	0.259554
$951$	0	0
$952$	0	0
$953$	−19.0000	−0.615470	−0.307735	−	0.951472i	$-0.599571\pi$
$953$	−19.0000	−0.615470	−0.307735	+	0.951472i	$0.599571\pi$
$954$	0	0
$955$	18.0000	0.582466
$956$	−6.00000	−0.194054
$957$	0	0
$958$	22.0000	0.710788
$959$	0	0
$960$	0	0
$961$	5.00000	0.161290
$962$	21.0000	0.677067
$963$	0	0
$964$	7.00000	0.225455
$965$	−11.0000	−0.354103
$966$	0	0
$967$	−22.0000	−0.707472	−0.353736	−	0.935345i	$-0.615089\pi$
$967$	−22.0000	−0.707472	−0.353736	+	0.935345i	$0.615089\pi$
$968$	−7.00000	−0.224989
$969$	0	0
$970$	−2.00000	−0.0642161
$971$	42.0000	1.34784	0.673922	−	0.738802i	$-0.264608\pi$
$971$	42.0000	1.34784	0.673922	+	0.738802i	$0.264608\pi$
$972$	0	0
$973$	0	0
$974$	−10.0000	−0.320421
$975$	0	0
$976$	9.00000	0.288083
$977$	18.0000	0.575871	0.287936	−	0.957650i	$-0.407031\pi$
$977$	18.0000	0.575871	0.287936	+	0.957650i	$0.407031\pi$
$978$	0	0
$979$	−30.0000	−0.958804
$980$	0	0
$981$	0	0
$982$	36.0000	1.14881
$983$	60.0000	1.91370	0.956851	−	0.290578i	$-0.0938475\pi$
$983$	60.0000	1.91370	0.956851	+	0.290578i	$0.0938475\pi$
$984$	0	0
$985$	1.00000	0.0318626
$986$	7.00000	0.222925
$987$	0	0
$988$	6.00000	0.190885
$989$	−8.00000	−0.254385
$990$	0	0
$991$	62.0000	1.96949	0.984747	−	0.173990i	$-0.0556660\pi$
$991$	62.0000	1.96949	0.984747	+	0.173990i	$0.0556660\pi$
$992$	−6.00000	−0.190500
$993$	0	0
$994$	0	0
$995$	−10.0000	−0.317021
$996$	0	0
$997$	−23.0000	−0.728417	−0.364209	−	0.931317i	$-0.618661\pi$
$997$	−23.0000	−0.728417	−0.364209	+	0.931317i	$0.618661\pi$
$998$	22.0000	0.696398
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7938.2.a.w.1.1		1
3.2	odd	2		7938.2.a.j.1.1		1
7.6	odd	2		1134.2.a.g.1.1	yes	1
21.20	even	2		1134.2.a.b.1.1	✓	1
28.27	even	2		9072.2.a.p.1.1		1
63.13	odd	6		1134.2.f.d.379.1		2
63.20	even	6		1134.2.f.m.757.1		2
63.34	odd	6		1134.2.f.d.757.1		2
63.41	even	6		1134.2.f.m.379.1		2
84.83	odd	2		9072.2.a.k.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1134.2.a.b.1.1	✓	1	21.20	even	2
1134.2.a.g.1.1	yes	1	7.6	odd	2
1134.2.f.d.379.1		2	63.13	odd	6
1134.2.f.d.757.1		2	63.34	odd	6
1134.2.f.m.379.1		2	63.41	even	6
1134.2.f.m.757.1		2	63.20	even	6
7938.2.a.j.1.1		1	3.2	odd	2
7938.2.a.w.1.1		1	1.1	even	1	trivial
9072.2.a.k.1.1		1	84.83	odd	2
9072.2.a.p.1.1		1	28.27	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:	\( N \)	\(=\)	\( 7938 = 2 \cdot 3^{4} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7938.a (trivial)

Introduction

L-functions

Modular forms

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Representations

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Database

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