LMFDB - Embedded newform 3042.2.a.c.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3042,2,Mod(1,3042)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3042, 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("3042.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3042 = 2 \cdot 3^{2} \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3042.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:	$24.2904922949$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 78)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	3042.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-1.00000 q^{2} +1.00000 q^{4} -2.00000 q^{5} +2.00000 q^{7} -1.00000 q^{8} +2.00000 q^{10} -2.00000 q^{14} +1.00000 q^{16} -2.00000 q^{17} -6.00000 q^{19} -2.00000 q^{20} +4.00000 q^{23} -1.00000 q^{25} +2.00000 q^{28} +10.0000 q^{29} +10.0000 q^{31} -1.00000 q^{32} +2.00000 q^{34} -4.00000 q^{35} -8.00000 q^{37} +6.00000 q^{38} +2.00000 q^{40} -10.0000 q^{41} -4.00000 q^{43} -4.00000 q^{46} -12.0000 q^{47} -3.00000 q^{49} +1.00000 q^{50} +6.00000 q^{53} -2.00000 q^{56} -10.0000 q^{58} +4.00000 q^{59} +2.00000 q^{61} -10.0000 q^{62} +1.00000 q^{64} -2.00000 q^{67} -2.00000 q^{68} +4.00000 q^{70} +4.00000 q^{73} +8.00000 q^{74} -6.00000 q^{76} -2.00000 q^{80} +10.0000 q^{82} +4.00000 q^{83} +4.00000 q^{85} +4.00000 q^{86} -6.00000 q^{89} +4.00000 q^{92} +12.0000 q^{94} +12.0000 q^{95} -12.0000 q^{97} +3.00000 q^{98} +O(q^{100})$

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
$5$	−2.00000	−0.894427	−0.447214	−	0.894427i	$-0.647584\pi$
$5$	−2.00000	−0.894427	−0.447214	+	0.894427i	$0.647584\pi$
$6$	0	0
$7$	2.00000	0.755929	0.377964	−	0.925820i	$-0.376624\pi$
$7$	2.00000	0.755929	0.377964	+	0.925820i	$0.376624\pi$
$8$	−1.00000	−0.353553
$9$	0	0
$10$	2.00000	0.632456
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	−2.00000	−0.534522
$15$	0	0
$16$	1.00000	0.250000
$17$	−2.00000	−0.485071	−0.242536	−	0.970143i	$-0.577979\pi$
$17$	−2.00000	−0.485071	−0.242536	+	0.970143i	$0.577979\pi$
$18$	0	0
$19$	−6.00000	−1.37649	−0.688247	−	0.725476i	$-0.741620\pi$
$19$	−6.00000	−1.37649	−0.688247	+	0.725476i	$0.741620\pi$
$20$	−2.00000	−0.447214
$21$	0	0
$22$	0	0
$23$	4.00000	0.834058	0.417029	−	0.908893i	$-0.363071\pi$
$23$	4.00000	0.834058	0.417029	+	0.908893i	$0.363071\pi$
$24$	0	0
$25$	−1.00000	−0.200000
$26$	0	0
$27$	0	0
$28$	2.00000	0.377964
$29$	10.0000	1.85695	0.928477	−	0.371391i	$-0.121119\pi$
$29$	10.0000	1.85695	0.928477	+	0.371391i	$0.121119\pi$
$30$	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
$34$	2.00000	0.342997
$35$	−4.00000	−0.676123
$36$	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
$40$	2.00000	0.316228
$41$	−10.0000	−1.56174	−0.780869	−	0.624695i	$-0.785223\pi$
$41$	−10.0000	−1.56174	−0.780869	+	0.624695i	$0.785223\pi$
$42$	0	0
$43$	−4.00000	−0.609994	−0.304997	−	0.952353i	$-0.598656\pi$
$43$	−4.00000	−0.609994	−0.304997	+	0.952353i	$0.598656\pi$
$44$	0	0
$45$	0	0
$46$	−4.00000	−0.589768
$47$	−12.0000	−1.75038	−0.875190	−	0.483779i	$-0.839264\pi$
$47$	−12.0000	−1.75038	−0.875190	+	0.483779i	$0.839264\pi$
$48$	0	0
$49$	−3.00000	−0.428571
$50$	1.00000	0.141421
$51$	0	0
$52$	0	0
$53$	6.00000	0.824163	0.412082	−	0.911147i	$-0.364802\pi$
$53$	6.00000	0.824163	0.412082	+	0.911147i	$0.364802\pi$
$54$	0	0
$55$	0	0
$56$	−2.00000	−0.267261
$57$	0	0
$58$	−10.0000	−1.31306
$59$	4.00000	0.520756	0.260378	−	0.965507i	$-0.416153\pi$
$59$	4.00000	0.520756	0.260378	+	0.965507i	$0.416153\pi$
$60$	0	0
$61$	2.00000	0.256074	0.128037	−	0.991769i	$-0.459132\pi$
$61$	2.00000	0.256074	0.128037	+	0.991769i	$0.459132\pi$
$62$	−10.0000	−1.27000
$63$	0	0
$64$	1.00000	0.125000
$65$	0	0
$66$	0	0
$67$	−2.00000	−0.244339	−0.122169	−	0.992509i	$-0.538985\pi$
$67$	−2.00000	−0.244339	−0.122169	+	0.992509i	$0.538985\pi$
$68$	−2.00000	−0.242536
$69$	0	0
$70$	4.00000	0.478091
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	0	0
$73$	4.00000	0.468165	0.234082	−	0.972217i	$-0.424791\pi$
$73$	4.00000	0.468165	0.234082	+	0.972217i	$0.424791\pi$
$74$	8.00000	0.929981
$75$	0	0
$76$	−6.00000	−0.688247
$77$	0	0
$78$	0	0
$79$	0	0		−	1.00000i	$-0.5\pi$
$79$	0	0			1.00000i	$0.5\pi$
$80$	−2.00000	−0.223607
$81$	0	0
$82$	10.0000	1.10432
$83$	4.00000	0.439057	0.219529	−	0.975606i	$-0.429548\pi$
$83$	4.00000	0.439057	0.219529	+	0.975606i	$0.429548\pi$
$84$	0	0
$85$	4.00000	0.433861
$86$	4.00000	0.431331
$87$	0	0
$88$	0	0
$89$	−6.00000	−0.635999	−0.317999	−	0.948091i	$-0.603011\pi$
$89$	−6.00000	−0.635999	−0.317999	+	0.948091i	$0.603011\pi$
$90$	0	0
$91$	0	0
$92$	4.00000	0.417029
$93$	0	0
$94$	12.0000	1.23771
$95$	12.0000	1.23117
$96$	0	0
$97$	−12.0000	−1.21842	−0.609208	−	0.793011i	$-0.708512\pi$
$97$	−12.0000	−1.21842	−0.609208	+	0.793011i	$0.708512\pi$
$98$	3.00000	0.303046
$99$	0	0
$100$	−1.00000	−0.100000
$101$	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	$0.210980\pi$
$104$	0	0
$105$	0	0
$106$	−6.00000	−0.582772
$107$	−8.00000	−0.773389	−0.386695	−	0.922208i	$-0.626383\pi$
$107$	−8.00000	−0.773389	−0.386695	+	0.922208i	$0.626383\pi$
$108$	0	0
$109$	4.00000	0.383131	0.191565	−	0.981480i	$-0.438644\pi$
$109$	4.00000	0.383131	0.191565	+	0.981480i	$0.438644\pi$
$110$	0	0
$111$	0	0
$112$	2.00000	0.188982
$113$	−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
$115$	−8.00000	−0.746004
$116$	10.0000	0.928477
$117$	0	0
$118$	−4.00000	−0.368230
$119$	−4.00000	−0.366679
$120$	0	0
$121$	−11.0000	−1.00000
$122$	−2.00000	−0.181071
$123$	0	0
$124$	10.0000	0.898027
$125$	12.0000	1.07331
$126$	0	0
$127$	−8.00000	−0.709885	−0.354943	−	0.934888i	$-0.615500\pi$
$127$	−8.00000	−0.709885	−0.354943	+	0.934888i	$0.615500\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	0	0
$131$	8.00000	0.698963	0.349482	−	0.936943i	$-0.386358\pi$
$131$	8.00000	0.698963	0.349482	+	0.936943i	$0.386358\pi$
$132$	0	0
$133$	−12.0000	−1.04053
$134$	2.00000	0.172774
$135$	0	0
$136$	2.00000	0.171499
$137$	−2.00000	−0.170872	−0.0854358	−	0.996344i	$-0.527228\pi$
$137$	−2.00000	−0.170872	−0.0854358	+	0.996344i	$0.527228\pi$
$138$	0	0
$139$	−20.0000	−1.69638	−0.848189	−	0.529694i	$-0.822307\pi$
$139$	−20.0000	−1.69638	−0.848189	+	0.529694i	$0.822307\pi$
$140$	−4.00000	−0.338062
$141$	0	0
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−20.0000	−1.66091
$146$	−4.00000	−0.331042
$147$	0	0
$148$	−8.00000	−0.657596
$149$	−14.0000	−1.14692	−0.573462	−	0.819232i	$-0.694400\pi$
$149$	−14.0000	−1.14692	−0.573462	+	0.819232i	$0.694400\pi$
$150$	0	0
$151$	−10.0000	−0.813788	−0.406894	−	0.913475i	$-0.633388\pi$
$151$	−10.0000	−0.813788	−0.406894	+	0.913475i	$0.633388\pi$
$152$	6.00000	0.486664
$153$	0	0
$154$	0	0
$155$	−20.0000	−1.60644
$156$	0	0
$157$	−2.00000	−0.159617	−0.0798087	−	0.996810i	$-0.525431\pi$
$157$	−2.00000	−0.159617	−0.0798087	+	0.996810i	$0.525431\pi$
$158$	0	0
$159$	0	0
$160$	2.00000	0.158114
$161$	8.00000	0.630488
$162$	0	0
$163$	14.0000	1.09656	0.548282	−	0.836293i	$-0.315282\pi$
$163$	14.0000	1.09656	0.548282	+	0.836293i	$0.315282\pi$
$164$	−10.0000	−0.780869
$165$	0	0
$166$	−4.00000	−0.310460
$167$	−12.0000	−0.928588	−0.464294	−	0.885681i	$-0.653692\pi$
$167$	−12.0000	−0.928588	−0.464294	+	0.885681i	$0.653692\pi$
$168$	0	0
$169$	0	0
$170$	−4.00000	−0.306786
$171$	0	0
$172$	−4.00000	−0.304997
$173$	−6.00000	−0.456172	−0.228086	−	0.973641i	$-0.573247\pi$
$173$	−6.00000	−0.456172	−0.228086	+	0.973641i	$0.573247\pi$
$174$	0	0
$175$	−2.00000	−0.151186
$176$	0	0
$177$	0	0
$178$	6.00000	0.449719
$179$	0	0		−	1.00000i	$-0.5\pi$
$179$	0	0			1.00000i	$0.5\pi$
$180$	0	0
$181$	−22.0000	−1.63525	−0.817624	−	0.575753i	$-0.804709\pi$
$181$	−22.0000	−1.63525	−0.817624	+	0.575753i	$0.804709\pi$
$182$	0	0
$183$	0	0
$184$	−4.00000	−0.294884
$185$	16.0000	1.17634
$186$	0	0
$187$	0	0
$188$	−12.0000	−0.875190
$189$	0	0
$190$	−12.0000	−0.870572
$191$	−12.0000	−0.868290	−0.434145	−	0.900843i	$-0.642949\pi$
$191$	−12.0000	−0.868290	−0.434145	+	0.900843i	$0.642949\pi$
$192$	0	0
$193$	−16.0000	−1.15171	−0.575853	−	0.817554i	$-0.695330\pi$
$193$	−16.0000	−1.15171	−0.575853	+	0.817554i	$0.695330\pi$
$194$	12.0000	0.861550
$195$	0	0
$196$	−3.00000	−0.214286
$197$	22.0000	1.56744	0.783718	−	0.621117i	$-0.213321\pi$
$197$	22.0000	1.56744	0.783718	+	0.621117i	$0.213321\pi$
$198$	0	0
$199$	0	0		−	1.00000i	$-0.5\pi$
$199$	0	0			1.00000i	$0.5\pi$
$200$	1.00000	0.0707107
$201$	0	0
$202$	−2.00000	−0.140720
$203$	20.0000	1.40372
$204$	0	0
$205$	20.0000	1.39686
$206$	−16.0000	−1.11477
$207$	0	0
$208$	0	0
$209$	0	0
$210$	0	0
$211$	12.0000	0.826114	0.413057	−	0.910705i	$-0.364461\pi$
$211$	12.0000	0.826114	0.413057	+	0.910705i	$0.364461\pi$
$212$	6.00000	0.412082
$213$	0	0
$214$	8.00000	0.546869
$215$	8.00000	0.545595
$216$	0	0
$217$	20.0000	1.35769
$218$	−4.00000	−0.270914
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0	0
$223$	−14.0000	−0.937509	−0.468755	−	0.883328i	$-0.655297\pi$
$223$	−14.0000	−0.937509	−0.468755	+	0.883328i	$0.655297\pi$
$224$	−2.00000	−0.133631
$225$	0	0
$226$	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	$0.585534\pi$
$228$	0	0
$229$	−4.00000	−0.264327	−0.132164	−	0.991228i	$-0.542192\pi$
$229$	−4.00000	−0.264327	−0.132164	+	0.991228i	$0.542192\pi$
$230$	8.00000	0.527504
$231$	0	0
$232$	−10.0000	−0.656532
$233$	−6.00000	−0.393073	−0.196537	−	0.980497i	$-0.562969\pi$
$233$	−6.00000	−0.393073	−0.196537	+	0.980497i	$0.562969\pi$
$234$	0	0
$235$	24.0000	1.56559
$236$	4.00000	0.260378
$237$	0	0
$238$	4.00000	0.259281
$239$	16.0000	1.03495	0.517477	−	0.855697i	$-0.326871\pi$
$239$	16.0000	1.03495	0.517477	+	0.855697i	$0.326871\pi$
$240$	0	0
$241$	−20.0000	−1.28831	−0.644157	−	0.764894i	$-0.722792\pi$
$241$	−20.0000	−1.28831	−0.644157	+	0.764894i	$0.722792\pi$
$242$	11.0000	0.707107
$243$	0	0
$244$	2.00000	0.128037
$245$	6.00000	0.383326
$246$	0	0
$247$	0	0
$248$	−10.0000	−0.635001
$249$	0	0
$250$	−12.0000	−0.758947
$251$	−28.0000	−1.76734	−0.883672	−	0.468106i	$-0.844936\pi$
$251$	−28.0000	−1.76734	−0.883672	+	0.468106i	$0.844936\pi$
$252$	0	0
$253$	0	0
$254$	8.00000	0.501965
$255$	0	0
$256$	1.00000	0.0625000
$257$	18.0000	1.12281	0.561405	−	0.827541i	$-0.310261\pi$
$257$	18.0000	1.12281	0.561405	+	0.827541i	$0.310261\pi$
$258$	0	0
$259$	−16.0000	−0.994192
$260$	0	0
$261$	0	0
$262$	−8.00000	−0.494242
$263$	−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
$265$	−12.0000	−0.737154
$266$	12.0000	0.735767
$267$	0	0
$268$	−2.00000	−0.122169
$269$	10.0000	0.609711	0.304855	−	0.952399i	$-0.401392\pi$
$269$	10.0000	0.609711	0.304855	+	0.952399i	$0.401392\pi$
$270$	0	0
$271$	10.0000	0.607457	0.303728	−	0.952759i	$-0.401768\pi$
$271$	10.0000	0.607457	0.303728	+	0.952759i	$0.401768\pi$
$272$	−2.00000	−0.121268
$273$	0	0
$274$	2.00000	0.120824
$275$	0	0
$276$	0	0
$277$	2.00000	0.120168	0.0600842	−	0.998193i	$-0.480863\pi$
$277$	2.00000	0.120168	0.0600842	+	0.998193i	$0.480863\pi$
$278$	20.0000	1.19952
$279$	0	0
$280$	4.00000	0.239046
$281$	−10.0000	−0.596550	−0.298275	−	0.954480i	$-0.596411\pi$
$281$	−10.0000	−0.596550	−0.298275	+	0.954480i	$0.596411\pi$
$282$	0	0
$283$	−4.00000	−0.237775	−0.118888	−	0.992908i	$-0.537933\pi$
$283$	−4.00000	−0.237775	−0.118888	+	0.992908i	$0.537933\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−20.0000	−1.18056
$288$	0	0
$289$	−13.0000	−0.764706
$290$	20.0000	1.17444
$291$	0	0
$292$	4.00000	0.234082
$293$	−14.0000	−0.817889	−0.408944	−	0.912559i	$-0.634103\pi$
$293$	−14.0000	−0.817889	−0.408944	+	0.912559i	$0.634103\pi$
$294$	0	0
$295$	−8.00000	−0.465778
$296$	8.00000	0.464991
$297$	0	0
$298$	14.0000	0.810998
$299$	0	0
$300$	0	0
$301$	−8.00000	−0.461112
$302$	10.0000	0.575435
$303$	0	0
$304$	−6.00000	−0.344124
$305$	−4.00000	−0.229039
$306$	0	0
$307$	2.00000	0.114146	0.0570730	−	0.998370i	$-0.481823\pi$
$307$	2.00000	0.114146	0.0570730	+	0.998370i	$0.481823\pi$
$308$	0	0
$309$	0	0
$310$	20.0000	1.13592
$311$	−28.0000	−1.58773	−0.793867	−	0.608091i	$-0.791935\pi$
$311$	−28.0000	−1.58773	−0.793867	+	0.608091i	$0.791935\pi$
$312$	0	0
$313$	−26.0000	−1.46961	−0.734803	−	0.678280i	$-0.762726\pi$
$313$	−26.0000	−1.46961	−0.734803	+	0.678280i	$0.762726\pi$
$314$	2.00000	0.112867
$315$	0	0
$316$	0	0
$317$	−18.0000	−1.01098	−0.505490	−	0.862832i	$-0.668688\pi$
$317$	−18.0000	−1.01098	−0.505490	+	0.862832i	$0.668688\pi$
$318$	0	0
$319$	0	0
$320$	−2.00000	−0.111803
$321$	0	0
$322$	−8.00000	−0.445823
$323$	12.0000	0.667698
$324$	0	0
$325$	0	0
$326$	−14.0000	−0.775388
$327$	0	0
$328$	10.0000	0.552158
$329$	−24.0000	−1.32316
$330$	0	0
$331$	−10.0000	−0.549650	−0.274825	−	0.961494i	$-0.588620\pi$
$331$	−10.0000	−0.549650	−0.274825	+	0.961494i	$0.588620\pi$
$332$	4.00000	0.219529
$333$	0	0
$334$	12.0000	0.656611
$335$	4.00000	0.218543
$336$	0	0
$337$	2.00000	0.108947	0.0544735	−	0.998515i	$-0.482652\pi$
$337$	2.00000	0.108947	0.0544735	+	0.998515i	$0.482652\pi$
$338$	0	0
$339$	0	0
$340$	4.00000	0.216930
$341$	0	0
$342$	0	0
$343$	−20.0000	−1.07990
$344$	4.00000	0.215666
$345$	0	0
$346$	6.00000	0.322562
$347$	12.0000	0.644194	0.322097	−	0.946707i	$-0.395612\pi$
$347$	12.0000	0.644194	0.322097	+	0.946707i	$0.395612\pi$
$348$	0	0
$349$	16.0000	0.856460	0.428230	−	0.903670i	$-0.359137\pi$
$349$	16.0000	0.856460	0.428230	+	0.903670i	$0.359137\pi$
$350$	2.00000	0.106904
$351$	0	0
$352$	0	0
$353$	−26.0000	−1.38384	−0.691920	−	0.721974i	$-0.743235\pi$
$353$	−26.0000	−1.38384	−0.691920	+	0.721974i	$0.743235\pi$
$354$	0	0
$355$	0	0
$356$	−6.00000	−0.317999
$357$	0	0
$358$	0	0
$359$	4.00000	0.211112	0.105556	−	0.994413i	$-0.466338\pi$
$359$	4.00000	0.211112	0.105556	+	0.994413i	$0.466338\pi$
$360$	0	0
$361$	17.0000	0.894737
$362$	22.0000	1.15629
$363$	0	0
$364$	0	0
$365$	−8.00000	−0.418739
$366$	0	0
$367$	8.00000	0.417597	0.208798	−	0.977959i	$-0.433045\pi$
$367$	8.00000	0.417597	0.208798	+	0.977959i	$0.433045\pi$
$368$	4.00000	0.208514
$369$	0	0
$370$	−16.0000	−0.831800
$371$	12.0000	0.623009
$372$	0	0
$373$	−6.00000	−0.310668	−0.155334	−	0.987862i	$-0.549645\pi$
$373$	−6.00000	−0.310668	−0.155334	+	0.987862i	$0.549645\pi$
$374$	0	0
$375$	0	0
$376$	12.0000	0.618853
$377$	0	0
$378$	0	0
$379$	34.0000	1.74646	0.873231	−	0.487306i	$-0.162020\pi$
$379$	34.0000	1.74646	0.873231	+	0.487306i	$0.162020\pi$
$380$	12.0000	0.615587
$381$	0	0
$382$	12.0000	0.613973
$383$	4.00000	0.204390	0.102195	−	0.994764i	$-0.467413\pi$
$383$	4.00000	0.204390	0.102195	+	0.994764i	$0.467413\pi$
$384$	0	0
$385$	0	0
$386$	16.0000	0.814379
$387$	0	0
$388$	−12.0000	−0.609208
$389$	30.0000	1.52106	0.760530	−	0.649303i	$-0.224939\pi$
$389$	30.0000	1.52106	0.760530	+	0.649303i	$0.224939\pi$
$390$	0	0
$391$	−8.00000	−0.404577
$392$	3.00000	0.151523
$393$	0	0
$394$	−22.0000	−1.10834
$395$	0	0
$396$	0	0
$397$	−8.00000	−0.401508	−0.200754	−	0.979642i	$-0.564339\pi$
$397$	−8.00000	−0.401508	−0.200754	+	0.979642i	$0.564339\pi$
$398$	0	0
$399$	0	0
$400$	−1.00000	−0.0500000
$401$	−30.0000	−1.49813	−0.749064	−	0.662497i	$-0.769497\pi$
$401$	−30.0000	−1.49813	−0.749064	+	0.662497i	$0.769497\pi$
$402$	0	0
$403$	0	0
$404$	2.00000	0.0995037
$405$	0	0
$406$	−20.0000	−0.992583
$407$	0	0
$408$	0	0
$409$	4.00000	0.197787	0.0988936	−	0.995098i	$-0.468470\pi$
$409$	4.00000	0.197787	0.0988936	+	0.995098i	$0.468470\pi$
$410$	−20.0000	−0.987730
$411$	0	0
$412$	16.0000	0.788263
$413$	8.00000	0.393654
$414$	0	0
$415$	−8.00000	−0.392705
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−40.0000	−1.95413	−0.977064	−	0.212946i	$-0.931694\pi$
$419$	−40.0000	−1.95413	−0.977064	+	0.212946i	$0.931694\pi$
$420$	0	0
$421$	20.0000	0.974740	0.487370	−	0.873195i	$-0.337956\pi$
$421$	20.0000	0.974740	0.487370	+	0.873195i	$0.337956\pi$
$422$	−12.0000	−0.584151
$423$	0	0
$424$	−6.00000	−0.291386
$425$	2.00000	0.0970143
$426$	0	0
$427$	4.00000	0.193574
$428$	−8.00000	−0.386695
$429$	0	0
$430$	−8.00000	−0.385794
$431$	20.0000	0.963366	0.481683	−	0.876346i	$-0.340026\pi$
$431$	20.0000	0.963366	0.481683	+	0.876346i	$0.340026\pi$
$432$	0	0
$433$	26.0000	1.24948	0.624740	−	0.780833i	$-0.285205\pi$
$433$	26.0000	1.24948	0.624740	+	0.780833i	$0.285205\pi$
$434$	−20.0000	−0.960031
$435$	0	0
$436$	4.00000	0.191565
$437$	−24.0000	−1.14808
$438$	0	0
$439$	0	0		−	1.00000i	$-0.5\pi$
$439$	0	0			1.00000i	$0.5\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	16.0000	0.760183	0.380091	−	0.924949i	$-0.375893\pi$
$443$	16.0000	0.760183	0.380091	+	0.924949i	$0.375893\pi$
$444$	0	0
$445$	12.0000	0.568855
$446$	14.0000	0.662919
$447$	0	0
$448$	2.00000	0.0944911
$449$	−6.00000	−0.283158	−0.141579	−	0.989927i	$-0.545218\pi$
$449$	−6.00000	−0.283158	−0.141579	+	0.989927i	$0.545218\pi$
$450$	0	0
$451$	0	0
$452$	−14.0000	−0.658505
$453$	0	0
$454$	8.00000	0.375459
$455$	0	0
$456$	0	0
$457$	28.0000	1.30978	0.654892	−	0.755722i	$-0.272714\pi$
$457$	28.0000	1.30978	0.654892	+	0.755722i	$0.272714\pi$
$458$	4.00000	0.186908
$459$	0	0
$460$	−8.00000	−0.373002
$461$	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
$463$	−6.00000	−0.278844	−0.139422	−	0.990233i	$-0.544524\pi$
$463$	−6.00000	−0.278844	−0.139422	+	0.990233i	$0.544524\pi$
$464$	10.0000	0.464238
$465$	0	0
$466$	6.00000	0.277945
$467$	−12.0000	−0.555294	−0.277647	−	0.960683i	$-0.589555\pi$
$467$	−12.0000	−0.555294	−0.277647	+	0.960683i	$0.589555\pi$
$468$	0	0
$469$	−4.00000	−0.184703
$470$	−24.0000	−1.10704
$471$	0	0
$472$	−4.00000	−0.184115
$473$	0	0
$474$	0	0
$475$	6.00000	0.275299
$476$	−4.00000	−0.183340
$477$	0	0
$478$	−16.0000	−0.731823
$479$	24.0000	1.09659	0.548294	−	0.836286i	$-0.315277\pi$
$479$	24.0000	1.09659	0.548294	+	0.836286i	$0.315277\pi$
$480$	0	0
$481$	0	0
$482$	20.0000	0.910975
$483$	0	0
$484$	−11.0000	−0.500000
$485$	24.0000	1.08978
$486$	0	0
$487$	18.0000	0.815658	0.407829	−	0.913058i	$-0.366286\pi$
$487$	18.0000	0.815658	0.407829	+	0.913058i	$0.366286\pi$
$488$	−2.00000	−0.0905357
$489$	0	0
$490$	−6.00000	−0.271052
$491$	−28.0000	−1.26362	−0.631811	−	0.775122i	$-0.717688\pi$
$491$	−28.0000	−1.26362	−0.631811	+	0.775122i	$0.717688\pi$
$492$	0	0
$493$	−20.0000	−0.900755
$494$	0	0
$495$	0	0
$496$	10.0000	0.449013
$497$	0	0
$498$	0	0
$499$	14.0000	0.626726	0.313363	−	0.949633i	$-0.398544\pi$
$499$	14.0000	0.626726	0.313363	+	0.949633i	$0.398544\pi$
$500$	12.0000	0.536656
$501$	0	0
$502$	28.0000	1.24970
$503$	−24.0000	−1.07011	−0.535054	−	0.844818i	$-0.679709\pi$
$503$	−24.0000	−1.07011	−0.535054	+	0.844818i	$0.679709\pi$
$504$	0	0
$505$	−4.00000	−0.177998
$506$	0	0
$507$	0	0
$508$	−8.00000	−0.354943
$509$	6.00000	0.265945	0.132973	−	0.991120i	$-0.457548\pi$
$509$	6.00000	0.265945	0.132973	+	0.991120i	$0.457548\pi$
$510$	0	0
$511$	8.00000	0.353899
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	−18.0000	−0.793946
$515$	−32.0000	−1.41009
$516$	0	0
$517$	0	0
$518$	16.0000	0.703000
$519$	0	0
$520$	0	0
$521$	18.0000	0.788594	0.394297	−	0.918983i	$-0.370988\pi$
$521$	18.0000	0.788594	0.394297	+	0.918983i	$0.370988\pi$
$522$	0	0
$523$	4.00000	0.174908	0.0874539	−	0.996169i	$-0.472127\pi$
$523$	4.00000	0.174908	0.0874539	+	0.996169i	$0.472127\pi$
$524$	8.00000	0.349482
$525$	0	0
$526$	24.0000	1.04645
$527$	−20.0000	−0.871214
$528$	0	0
$529$	−7.00000	−0.304348
$530$	12.0000	0.521247
$531$	0	0
$532$	−12.0000	−0.520266
$533$	0	0
$534$	0	0
$535$	16.0000	0.691740
$536$	2.00000	0.0863868
$537$	0	0
$538$	−10.0000	−0.431131
$539$	0	0
$540$	0	0
$541$	20.0000	0.859867	0.429934	−	0.902861i	$-0.358537\pi$
$541$	20.0000	0.859867	0.429934	+	0.902861i	$0.358537\pi$
$542$	−10.0000	−0.429537
$543$	0	0
$544$	2.00000	0.0857493
$545$	−8.00000	−0.342682
$546$	0	0
$547$	28.0000	1.19719	0.598597	−	0.801050i	$-0.295725\pi$
$547$	28.0000	1.19719	0.598597	+	0.801050i	$0.295725\pi$
$548$	−2.00000	−0.0854358
$549$	0	0
$550$	0	0
$551$	−60.0000	−2.55609
$552$	0	0
$553$	0	0
$554$	−2.00000	−0.0849719
$555$	0	0
$556$	−20.0000	−0.848189
$557$	18.0000	0.762684	0.381342	−	0.924434i	$-0.375462\pi$
$557$	18.0000	0.762684	0.381342	+	0.924434i	$0.375462\pi$
$558$	0	0
$559$	0	0
$560$	−4.00000	−0.169031
$561$	0	0
$562$	10.0000	0.421825
$563$	−16.0000	−0.674320	−0.337160	−	0.941447i	$-0.609466\pi$
$563$	−16.0000	−0.674320	−0.337160	+	0.941447i	$0.609466\pi$
$564$	0	0
$565$	28.0000	1.17797
$566$	4.00000	0.168133
$567$	0	0
$568$	0	0
$569$	10.0000	0.419222	0.209611	−	0.977785i	$-0.432780\pi$
$569$	10.0000	0.419222	0.209611	+	0.977785i	$0.432780\pi$
$570$	0	0
$571$	28.0000	1.17176	0.585882	−	0.810397i	$-0.300748\pi$
$571$	28.0000	1.17176	0.585882	+	0.810397i	$0.300748\pi$
$572$	0	0
$573$	0	0
$574$	20.0000	0.834784
$575$	−4.00000	−0.166812
$576$	0	0
$577$	8.00000	0.333044	0.166522	−	0.986038i	$-0.446746\pi$
$577$	8.00000	0.333044	0.166522	+	0.986038i	$0.446746\pi$
$578$	13.0000	0.540729
$579$	0	0
$580$	−20.0000	−0.830455
$581$	8.00000	0.331896
$582$	0	0
$583$	0	0
$584$	−4.00000	−0.165521
$585$	0	0
$586$	14.0000	0.578335
$587$	−28.0000	−1.15568	−0.577842	−	0.816149i	$-0.696105\pi$
$587$	−28.0000	−1.15568	−0.577842	+	0.816149i	$0.696105\pi$
$588$	0	0
$589$	−60.0000	−2.47226
$590$	8.00000	0.329355
$591$	0	0
$592$	−8.00000	−0.328798
$593$	26.0000	1.06769	0.533846	−	0.845582i	$-0.320746\pi$
$593$	26.0000	1.06769	0.533846	+	0.845582i	$0.320746\pi$
$594$	0	0
$595$	8.00000	0.327968
$596$	−14.0000	−0.573462
$597$	0	0
$598$	0	0
$599$	0	0		−	1.00000i	$-0.5\pi$
$599$	0	0			1.00000i	$0.5\pi$
$600$	0	0
$601$	22.0000	0.897399	0.448699	−	0.893683i	$-0.351887\pi$
$601$	22.0000	0.897399	0.448699	+	0.893683i	$0.351887\pi$
$602$	8.00000	0.326056
$603$	0	0
$604$	−10.0000	−0.406894
$605$	22.0000	0.894427
$606$	0	0
$607$	−32.0000	−1.29884	−0.649420	−	0.760430i	$-0.724988\pi$
$607$	−32.0000	−1.29884	−0.649420	+	0.760430i	$0.724988\pi$
$608$	6.00000	0.243332
$609$	0	0
$610$	4.00000	0.161955
$611$	0	0
$612$	0	0
$613$	16.0000	0.646234	0.323117	−	0.946359i	$-0.395269\pi$
$613$	16.0000	0.646234	0.323117	+	0.946359i	$0.395269\pi$
$614$	−2.00000	−0.0807134
$615$	0	0
$616$	0	0
$617$	22.0000	0.885687	0.442843	−	0.896599i	$-0.353970\pi$
$617$	22.0000	0.885687	0.442843	+	0.896599i	$0.353970\pi$
$618$	0	0
$619$	26.0000	1.04503	0.522514	−	0.852631i	$-0.324994\pi$
$619$	26.0000	1.04503	0.522514	+	0.852631i	$0.324994\pi$
$620$	−20.0000	−0.803219
$621$	0	0
$622$	28.0000	1.12270
$623$	−12.0000	−0.480770
$624$	0	0
$625$	−19.0000	−0.760000
$626$	26.0000	1.03917
$627$	0	0
$628$	−2.00000	−0.0798087
$629$	16.0000	0.637962
$630$	0	0
$631$	−10.0000	−0.398094	−0.199047	−	0.979990i	$-0.563785\pi$
$631$	−10.0000	−0.398094	−0.199047	+	0.979990i	$0.563785\pi$
$632$	0	0
$633$	0	0
$634$	18.0000	0.714871
$635$	16.0000	0.634941
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	2.00000	0.0790569
$641$	−18.0000	−0.710957	−0.355479	−	0.934684i	$-0.615682\pi$
$641$	−18.0000	−0.710957	−0.355479	+	0.934684i	$0.615682\pi$
$642$	0	0
$643$	6.00000	0.236617	0.118308	−	0.992977i	$-0.462253\pi$
$643$	6.00000	0.236617	0.118308	+	0.992977i	$0.462253\pi$
$644$	8.00000	0.315244
$645$	0	0
$646$	−12.0000	−0.472134
$647$	−32.0000	−1.25805	−0.629025	−	0.777385i	$-0.716546\pi$
$647$	−32.0000	−1.25805	−0.629025	+	0.777385i	$0.716546\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	0	0
$652$	14.0000	0.548282
$653$	26.0000	1.01746	0.508729	−	0.860927i	$-0.330115\pi$
$653$	26.0000	1.01746	0.508729	+	0.860927i	$0.330115\pi$
$654$	0	0
$655$	−16.0000	−0.625172
$656$	−10.0000	−0.390434
$657$	0	0
$658$	24.0000	0.935617
$659$	−20.0000	−0.779089	−0.389545	−	0.921008i	$-0.627368\pi$
$659$	−20.0000	−0.779089	−0.389545	+	0.921008i	$0.627368\pi$
$660$	0	0
$661$	40.0000	1.55582	0.777910	−	0.628376i	$-0.216280\pi$
$661$	40.0000	1.55582	0.777910	+	0.628376i	$0.216280\pi$
$662$	10.0000	0.388661
$663$	0	0
$664$	−4.00000	−0.155230
$665$	24.0000	0.930680
$666$	0	0
$667$	40.0000	1.54881
$668$	−12.0000	−0.464294
$669$	0	0
$670$	−4.00000	−0.154533
$671$	0	0
$672$	0	0
$673$	6.00000	0.231283	0.115642	−	0.993291i	$-0.463108\pi$
$673$	6.00000	0.231283	0.115642	+	0.993291i	$0.463108\pi$
$674$	−2.00000	−0.0770371
$675$	0	0
$676$	0	0
$677$	−18.0000	−0.691796	−0.345898	−	0.938272i	$-0.612426\pi$
$677$	−18.0000	−0.691796	−0.345898	+	0.938272i	$0.612426\pi$
$678$	0	0
$679$	−24.0000	−0.921035
$680$	−4.00000	−0.153393
$681$	0	0
$682$	0	0
$683$	−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
$685$	4.00000	0.152832
$686$	20.0000	0.763604
$687$	0	0
$688$	−4.00000	−0.152499
$689$	0	0
$690$	0	0
$691$	−10.0000	−0.380418	−0.190209	−	0.981744i	$-0.560917\pi$
$691$	−10.0000	−0.380418	−0.190209	+	0.981744i	$0.560917\pi$
$692$	−6.00000	−0.228086
$693$	0	0
$694$	−12.0000	−0.455514
$695$	40.0000	1.51729
$696$	0	0
$697$	20.0000	0.757554
$698$	−16.0000	−0.605609
$699$	0	0
$700$	−2.00000	−0.0755929
$701$	22.0000	0.830929	0.415464	−	0.909610i	$-0.363619\pi$
$701$	22.0000	0.830929	0.415464	+	0.909610i	$0.363619\pi$
$702$	0	0
$703$	48.0000	1.81035
$704$	0	0
$705$	0	0
$706$	26.0000	0.978523
$707$	4.00000	0.150435
$708$	0	0
$709$	36.0000	1.35201	0.676004	−	0.736898i	$-0.263710\pi$
$709$	36.0000	1.35201	0.676004	+	0.736898i	$0.263710\pi$
$710$	0	0
$711$	0	0
$712$	6.00000	0.224860
$713$	40.0000	1.49801
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	−4.00000	−0.149279
$719$	20.0000	0.745874	0.372937	−	0.927857i	$-0.378351\pi$
$719$	20.0000	0.745874	0.372937	+	0.927857i	$0.378351\pi$
$720$	0	0
$721$	32.0000	1.19174
$722$	−17.0000	−0.632674
$723$	0	0
$724$	−22.0000	−0.817624
$725$	−10.0000	−0.371391
$726$	0	0
$727$	−8.00000	−0.296704	−0.148352	−	0.988935i	$-0.547397\pi$
$727$	−8.00000	−0.296704	−0.148352	+	0.988935i	$0.547397\pi$
$728$	0	0
$729$	0	0
$730$	8.00000	0.296093
$731$	8.00000	0.295891
$732$	0	0
$733$	−44.0000	−1.62518	−0.812589	−	0.582838i	$-0.801942\pi$
$733$	−44.0000	−1.62518	−0.812589	+	0.582838i	$0.801942\pi$
$734$	−8.00000	−0.295285
$735$	0	0
$736$	−4.00000	−0.147442
$737$	0	0
$738$	0	0
$739$	26.0000	0.956425	0.478213	−	0.878244i	$-0.341285\pi$
$739$	26.0000	0.956425	0.478213	+	0.878244i	$0.341285\pi$
$740$	16.0000	0.588172
$741$	0	0
$742$	−12.0000	−0.440534
$743$	−16.0000	−0.586983	−0.293492	−	0.955962i	$-0.594817\pi$
$743$	−16.0000	−0.586983	−0.293492	+	0.955962i	$0.594817\pi$
$744$	0	0
$745$	28.0000	1.02584
$746$	6.00000	0.219676
$747$	0	0
$748$	0	0
$749$	−16.0000	−0.584627
$750$	0	0
$751$	−32.0000	−1.16770	−0.583848	−	0.811863i	$-0.698454\pi$
$751$	−32.0000	−1.16770	−0.583848	+	0.811863i	$0.698454\pi$
$752$	−12.0000	−0.437595
$753$	0	0
$754$	0	0
$755$	20.0000	0.727875
$756$	0	0
$757$	−22.0000	−0.799604	−0.399802	−	0.916602i	$-0.630921\pi$
$757$	−22.0000	−0.799604	−0.399802	+	0.916602i	$0.630921\pi$
$758$	−34.0000	−1.23494
$759$	0	0
$760$	−12.0000	−0.435286
$761$	30.0000	1.08750	0.543750	−	0.839248i	$-0.317004\pi$
$761$	30.0000	1.08750	0.543750	+	0.839248i	$0.317004\pi$
$762$	0	0
$763$	8.00000	0.289619
$764$	−12.0000	−0.434145
$765$	0	0
$766$	−4.00000	−0.144526
$767$	0	0
$768$	0	0
$769$	24.0000	0.865462	0.432731	−	0.901523i	$-0.357550\pi$
$769$	24.0000	0.865462	0.432731	+	0.901523i	$0.357550\pi$
$770$	0	0
$771$	0	0
$772$	−16.0000	−0.575853
$773$	−6.00000	−0.215805	−0.107903	−	0.994161i	$-0.534413\pi$
$773$	−6.00000	−0.215805	−0.107903	+	0.994161i	$0.534413\pi$
$774$	0	0
$775$	−10.0000	−0.359211
$776$	12.0000	0.430775
$777$	0	0
$778$	−30.0000	−1.07555
$779$	60.0000	2.14972
$780$	0	0
$781$	0	0
$782$	8.00000	0.286079
$783$	0	0
$784$	−3.00000	−0.107143
$785$	4.00000	0.142766
$786$	0	0
$787$	−38.0000	−1.35455	−0.677277	−	0.735728i	$-0.736840\pi$
$787$	−38.0000	−1.35455	−0.677277	+	0.735728i	$0.736840\pi$
$788$	22.0000	0.783718
$789$	0	0
$790$	0	0
$791$	−28.0000	−0.995565
$792$	0	0
$793$	0	0
$794$	8.00000	0.283909
$795$	0	0
$796$	0	0
$797$	−2.00000	−0.0708436	−0.0354218	−	0.999372i	$-0.511277\pi$
$797$	−2.00000	−0.0708436	−0.0354218	+	0.999372i	$0.511277\pi$
$798$	0	0
$799$	24.0000	0.849059
$800$	1.00000	0.0353553
$801$	0	0
$802$	30.0000	1.05934
$803$	0	0
$804$	0	0
$805$	−16.0000	−0.563926
$806$	0	0
$807$	0	0
$808$	−2.00000	−0.0703598
$809$	50.0000	1.75791	0.878953	−	0.476908i	$-0.158243\pi$
$809$	50.0000	1.75791	0.878953	+	0.476908i	$0.158243\pi$
$810$	0	0
$811$	10.0000	0.351147	0.175574	−	0.984466i	$-0.443822\pi$
$811$	10.0000	0.351147	0.175574	+	0.984466i	$0.443822\pi$
$812$	20.0000	0.701862
$813$	0	0
$814$	0	0
$815$	−28.0000	−0.980797
$816$	0	0
$817$	24.0000	0.839654
$818$	−4.00000	−0.139857
$819$	0	0
$820$	20.0000	0.698430
$821$	−30.0000	−1.04701	−0.523504	−	0.852023i	$-0.675375\pi$
$821$	−30.0000	−1.04701	−0.523504	+	0.852023i	$0.675375\pi$
$822$	0	0
$823$	−24.0000	−0.836587	−0.418294	−	0.908312i	$-0.637372\pi$
$823$	−24.0000	−0.836587	−0.418294	+	0.908312i	$0.637372\pi$
$824$	−16.0000	−0.557386
$825$	0	0
$826$	−8.00000	−0.278356
$827$	48.0000	1.66912	0.834562	−	0.550914i	$-0.185721\pi$
$827$	48.0000	1.66912	0.834562	+	0.550914i	$0.185721\pi$
$828$	0	0
$829$	30.0000	1.04194	0.520972	−	0.853574i	$-0.325570\pi$
$829$	30.0000	1.04194	0.520972	+	0.853574i	$0.325570\pi$
$830$	8.00000	0.277684
$831$	0	0
$832$	0	0
$833$	6.00000	0.207888
$834$	0	0
$835$	24.0000	0.830554
$836$	0	0
$837$	0	0
$838$	40.0000	1.38178
$839$	−16.0000	−0.552381	−0.276191	−	0.961103i	$-0.589072\pi$
$839$	−16.0000	−0.552381	−0.276191	+	0.961103i	$0.589072\pi$
$840$	0	0
$841$	71.0000	2.44828
$842$	−20.0000	−0.689246
$843$	0	0
$844$	12.0000	0.413057
$845$	0	0
$846$	0	0
$847$	−22.0000	−0.755929
$848$	6.00000	0.206041
$849$	0	0
$850$	−2.00000	−0.0685994
$851$	−32.0000	−1.09695
$852$	0	0
$853$	−56.0000	−1.91740	−0.958702	−	0.284413i	$-0.908201\pi$
$853$	−56.0000	−1.91740	−0.958702	+	0.284413i	$0.908201\pi$
$854$	−4.00000	−0.136877
$855$	0	0
$856$	8.00000	0.273434
$857$	−22.0000	−0.751506	−0.375753	−	0.926720i	$-0.622616\pi$
$857$	−22.0000	−0.751506	−0.375753	+	0.926720i	$0.622616\pi$
$858$	0	0
$859$	−20.0000	−0.682391	−0.341196	−	0.939992i	$-0.610832\pi$
$859$	−20.0000	−0.682391	−0.341196	+	0.939992i	$0.610832\pi$
$860$	8.00000	0.272798
$861$	0	0
$862$	−20.0000	−0.681203
$863$	44.0000	1.49778	0.748889	−	0.662696i	$-0.230588\pi$
$863$	44.0000	1.49778	0.748889	+	0.662696i	$0.230588\pi$
$864$	0	0
$865$	12.0000	0.408012
$866$	−26.0000	−0.883516
$867$	0	0
$868$	20.0000	0.678844
$869$	0	0
$870$	0	0
$871$	0	0
$872$	−4.00000	−0.135457
$873$	0	0
$874$	24.0000	0.811812
$875$	24.0000	0.811348
$876$	0	0
$877$	8.00000	0.270141	0.135070	−	0.990836i	$-0.456874\pi$
$877$	8.00000	0.270141	0.135070	+	0.990836i	$0.456874\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	42.0000	1.41502	0.707508	−	0.706705i	$-0.249819\pi$
$881$	42.0000	1.41502	0.707508	+	0.706705i	$0.249819\pi$
$882$	0	0
$883$	36.0000	1.21150	0.605748	−	0.795656i	$-0.292874\pi$
$883$	36.0000	1.21150	0.605748	+	0.795656i	$0.292874\pi$
$884$	0	0
$885$	0	0
$886$	−16.0000	−0.537531
$887$	12.0000	0.402921	0.201460	−	0.979497i	$-0.435431\pi$
$887$	12.0000	0.402921	0.201460	+	0.979497i	$0.435431\pi$
$888$	0	0
$889$	−16.0000	−0.536623
$890$	−12.0000	−0.402241
$891$	0	0
$892$	−14.0000	−0.468755
$893$	72.0000	2.40939
$894$	0	0
$895$	0	0
$896$	−2.00000	−0.0668153
$897$	0	0
$898$	6.00000	0.200223
$899$	100.000	3.33519
$900$	0	0
$901$	−12.0000	−0.399778
$902$	0	0
$903$	0	0
$904$	14.0000	0.465633
$905$	44.0000	1.46261
$906$	0	0
$907$	−28.0000	−0.929725	−0.464862	−	0.885383i	$-0.653896\pi$
$907$	−28.0000	−0.929725	−0.464862	+	0.885383i	$0.653896\pi$
$908$	−8.00000	−0.265489
$909$	0	0
$910$	0	0
$911$	−12.0000	−0.397578	−0.198789	−	0.980042i	$-0.563701\pi$
$911$	−12.0000	−0.397578	−0.198789	+	0.980042i	$0.563701\pi$
$912$	0	0
$913$	0	0
$914$	−28.0000	−0.926158
$915$	0	0
$916$	−4.00000	−0.132164
$917$	16.0000	0.528367
$918$	0	0
$919$	0	0		−	1.00000i	$-0.5\pi$
$919$	0	0			1.00000i	$0.5\pi$
$920$	8.00000	0.263752
$921$	0	0
$922$	−30.0000	−0.987997
$923$	0	0
$924$	0	0
$925$	8.00000	0.263038
$926$	6.00000	0.197172
$927$	0	0
$928$	−10.0000	−0.328266
$929$	6.00000	0.196854	0.0984268	−	0.995144i	$-0.468619\pi$
$929$	6.00000	0.196854	0.0984268	+	0.995144i	$0.468619\pi$
$930$	0	0
$931$	18.0000	0.589926
$932$	−6.00000	−0.196537
$933$	0	0
$934$	12.0000	0.392652
$935$	0	0
$936$	0	0
$937$	−2.00000	−0.0653372	−0.0326686	−	0.999466i	$-0.510401\pi$
$937$	−2.00000	−0.0653372	−0.0326686	+	0.999466i	$0.510401\pi$
$938$	4.00000	0.130605
$939$	0	0
$940$	24.0000	0.782794
$941$	10.0000	0.325991	0.162995	−	0.986627i	$-0.447884\pi$
$941$	10.0000	0.325991	0.162995	+	0.986627i	$0.447884\pi$
$942$	0	0
$943$	−40.0000	−1.30258
$944$	4.00000	0.130189
$945$	0	0
$946$	0	0
$947$	−52.0000	−1.68977	−0.844886	−	0.534946i	$-0.820332\pi$
$947$	−52.0000	−1.68977	−0.844886	+	0.534946i	$0.820332\pi$
$948$	0	0
$949$	0	0
$950$	−6.00000	−0.194666
$951$	0	0
$952$	4.00000	0.129641
$953$	−6.00000	−0.194359	−0.0971795	−	0.995267i	$-0.530982\pi$
$953$	−6.00000	−0.194359	−0.0971795	+	0.995267i	$0.530982\pi$
$954$	0	0
$955$	24.0000	0.776622
$956$	16.0000	0.517477
$957$	0	0
$958$	−24.0000	−0.775405
$959$	−4.00000	−0.129167
$960$	0	0
$961$	69.0000	2.22581
$962$	0	0
$963$	0	0
$964$	−20.0000	−0.644157
$965$	32.0000	1.03012
$966$	0	0
$967$	−22.0000	−0.707472	−0.353736	−	0.935345i	$-0.615089\pi$
$967$	−22.0000	−0.707472	−0.353736	+	0.935345i	$0.615089\pi$
$968$	11.0000	0.353553
$969$	0	0
$970$	−24.0000	−0.770594
$971$	−12.0000	−0.385098	−0.192549	−	0.981287i	$-0.561675\pi$
$971$	−12.0000	−0.385098	−0.192549	+	0.981287i	$0.561675\pi$
$972$	0	0
$973$	−40.0000	−1.28234
$974$	−18.0000	−0.576757
$975$	0	0
$976$	2.00000	0.0640184
$977$	42.0000	1.34370	0.671850	−	0.740688i	$-0.265500\pi$
$977$	42.0000	1.34370	0.671850	+	0.740688i	$0.265500\pi$
$978$	0	0
$979$	0	0
$980$	6.00000	0.191663
$981$	0	0
$982$	28.0000	0.893516
$983$	−24.0000	−0.765481	−0.382741	−	0.923856i	$-0.625020\pi$
$983$	−24.0000	−0.765481	−0.382741	+	0.923856i	$0.625020\pi$
$984$	0	0
$985$	−44.0000	−1.40196
$986$	20.0000	0.636930
$987$	0	0
$988$	0	0
$989$	−16.0000	−0.508770
$990$	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
$995$	0	0
$996$	0	0
$997$	−42.0000	−1.33015	−0.665077	−	0.746775i	$-0.731601\pi$
$997$	−42.0000	−1.33015	−0.665077	+	0.746775i	$0.731601\pi$
$998$	−14.0000	−0.443162
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3042.2.a.c.1.1		1
3.2	odd	2		1014.2.a.g.1.1		1
12.11	even	2		8112.2.a.j.1.1		1
13.5	odd	4		234.2.b.a.181.2		2
13.8	odd	4		234.2.b.a.181.1		2
13.12	even	2		3042.2.a.n.1.1		1
39.2	even	12		1014.2.i.c.823.2		4
39.5	even	4		78.2.b.a.25.1	✓	2
39.8	even	4		78.2.b.a.25.2	yes	2
39.11	even	12		1014.2.i.c.823.1		4
39.17	odd	6		1014.2.e.e.991.1		2
39.20	even	12		1014.2.i.c.361.2		4
39.23	odd	6		1014.2.e.e.529.1		2
39.29	odd	6		1014.2.e.b.529.1		2
39.32	even	12		1014.2.i.c.361.1		4
39.35	odd	6		1014.2.e.b.991.1		2
39.38	odd	2		1014.2.a.b.1.1		1
52.31	even	4		1872.2.c.b.1585.2		2
52.47	even	4		1872.2.c.b.1585.1		2
156.47	odd	4		624.2.c.a.337.2		2
156.83	odd	4		624.2.c.a.337.1		2
156.155	even	2		8112.2.a.g.1.1		1
195.8	odd	4		1950.2.f.g.649.2		2
195.44	even	4		1950.2.b.c.1351.2		2
195.47	odd	4		1950.2.f.d.649.1		2
195.83	odd	4		1950.2.f.d.649.2		2
195.122	odd	4		1950.2.f.g.649.1		2
195.164	even	4		1950.2.b.c.1351.1		2
273.83	odd	4		3822.2.c.d.883.1		2
273.125	odd	4		3822.2.c.d.883.2		2
312.5	even	4		2496.2.c.f.961.2		2
312.83	odd	4		2496.2.c.m.961.2		2
312.125	even	4		2496.2.c.f.961.1		2
312.203	odd	4		2496.2.c.m.961.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
78.2.b.a.25.1	✓	2	39.5	even	4
78.2.b.a.25.2	yes	2	39.8	even	4
234.2.b.a.181.1		2	13.8	odd	4
234.2.b.a.181.2		2	13.5	odd	4
624.2.c.a.337.1		2	156.83	odd	4
624.2.c.a.337.2		2	156.47	odd	4
1014.2.a.b.1.1		1	39.38	odd	2
1014.2.a.g.1.1		1	3.2	odd	2
1014.2.e.b.529.1		2	39.29	odd	6
1014.2.e.b.991.1		2	39.35	odd	6
1014.2.e.e.529.1		2	39.23	odd	6
1014.2.e.e.991.1		2	39.17	odd	6
1014.2.i.c.361.1		4	39.32	even	12
1014.2.i.c.361.2		4	39.20	even	12
1014.2.i.c.823.1		4	39.11	even	12
1014.2.i.c.823.2		4	39.2	even	12
1872.2.c.b.1585.1		2	52.47	even	4
1872.2.c.b.1585.2		2	52.31	even	4
1950.2.b.c.1351.1		2	195.164	even	4
1950.2.b.c.1351.2		2	195.44	even	4
1950.2.f.d.649.1		2	195.47	odd	4
1950.2.f.d.649.2		2	195.83	odd	4
1950.2.f.g.649.1		2	195.122	odd	4
1950.2.f.g.649.2		2	195.8	odd	4
2496.2.c.f.961.1		2	312.125	even	4
2496.2.c.f.961.2		2	312.5	even	4
2496.2.c.m.961.1		2	312.203	odd	4
2496.2.c.m.961.2		2	312.83	odd	4
3042.2.a.c.1.1		1	1.1	even	1	trivial
3042.2.a.n.1.1		1	13.12	even	2
3822.2.c.d.883.1		2	273.83	odd	4
3822.2.c.d.883.2		2	273.125	odd	4
8112.2.a.g.1.1		1	156.155	even	2
8112.2.a.j.1.1		1	12.11	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:	\( N \)	\(=\)	\( 3042 = 2 \cdot 3^{2} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3042.a (trivial)

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