LMFDB - Embedded newform 378.2.a.c.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [378,2,Mod(1,378)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(378, 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("378.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 378 = 2 \cdot 3^{3} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	378.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:	$3.01834519640$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	378.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{7} -1.00000 q^{8} +O(q^{10})$

$q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{10} -5.00000 q^{11} +1.00000 q^{14} +1.00000 q^{16} -2.00000 q^{17} -1.00000 q^{19} -1.00000 q^{20} +5.00000 q^{22} +1.00000 q^{23} -4.00000 q^{25} -1.00000 q^{28} -4.00000 q^{29} -9.00000 q^{31} -1.00000 q^{32} +2.00000 q^{34} +1.00000 q^{35} +5.00000 q^{37} +1.00000 q^{38} +1.00000 q^{40} +9.00000 q^{41} -10.0000 q^{43} -5.00000 q^{44} -1.00000 q^{46} -6.00000 q^{47} +1.00000 q^{49} +4.00000 q^{50} -12.0000 q^{53} +5.00000 q^{55} +1.00000 q^{56} +4.00000 q^{58} +14.0000 q^{59} +9.00000 q^{62} +1.00000 q^{64} -8.00000 q^{67} -2.00000 q^{68} -1.00000 q^{70} +13.0000 q^{71} -2.00000 q^{73} -5.00000 q^{74} -1.00000 q^{76} +5.00000 q^{77} +6.00000 q^{79} -1.00000 q^{80} -9.00000 q^{82} +4.00000 q^{83} +2.00000 q^{85} +10.0000 q^{86} +5.00000 q^{88} +9.00000 q^{89} +1.00000 q^{92} +6.00000 q^{94} +1.00000 q^{95} +16.0000 q^{97} -1.00000 q^{98} +O(q^{100})$

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	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
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	−1.00000	−0.353553
$9$	0	0
$10$	1.00000	0.316228
$11$	−5.00000	−1.50756	−0.753778	−	0.657129i	$-0.771771\pi$
$11$	−5.00000	−1.50756	−0.753778	+	0.657129i	$0.771771\pi$
$12$	0	0
$13$	0	0		−	1.00000i	$-0.5\pi$
$13$	0	0			1.00000i	$0.5\pi$
$14$	1.00000	0.267261
$15$	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
$19$	−1.00000	−0.229416	−0.114708	−	0.993399i	$-0.536593\pi$
$19$	−1.00000	−0.229416	−0.114708	+	0.993399i	$0.536593\pi$
$20$	−1.00000	−0.223607
$21$	0	0
$22$	5.00000	1.06600
$23$	1.00000	0.208514	0.104257	−	0.994550i	$-0.466753\pi$
$23$	1.00000	0.208514	0.104257	+	0.994550i	$0.466753\pi$
$24$	0	0
$25$	−4.00000	−0.800000
$26$	0	0
$27$	0	0
$28$	−1.00000	−0.188982
$29$	−4.00000	−0.742781	−0.371391	−	0.928477i	$-0.621119\pi$
$29$	−4.00000	−0.742781	−0.371391	+	0.928477i	$0.621119\pi$
$30$	0	0
$31$	−9.00000	−1.61645	−0.808224	−	0.588875i	$-0.799571\pi$
$31$	−9.00000	−1.61645	−0.808224	+	0.588875i	$0.799571\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	2.00000	0.342997
$35$	1.00000	0.169031
$36$	0	0
$37$	5.00000	0.821995	0.410997	−	0.911636i	$-0.365181\pi$
$37$	5.00000	0.821995	0.410997	+	0.911636i	$0.365181\pi$
$38$	1.00000	0.162221
$39$	0	0
$40$	1.00000	0.158114
$41$	9.00000	1.40556	0.702782	−	0.711405i	$-0.251941\pi$
$41$	9.00000	1.40556	0.702782	+	0.711405i	$0.251941\pi$
$42$	0	0
$43$	−10.0000	−1.52499	−0.762493	−	0.646997i	$-0.776025\pi$
$43$	−10.0000	−1.52499	−0.762493	+	0.646997i	$0.776025\pi$
$44$	−5.00000	−0.753778
$45$	0	0
$46$	−1.00000	−0.147442
$47$	−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
$49$	1.00000	0.142857
$50$	4.00000	0.565685
$51$	0	0
$52$	0	0
$53$	−12.0000	−1.64833	−0.824163	−	0.566352i	$-0.808354\pi$
$53$	−12.0000	−1.64833	−0.824163	+	0.566352i	$0.808354\pi$
$54$	0	0
$55$	5.00000	0.674200
$56$	1.00000	0.133631
$57$	0	0
$58$	4.00000	0.525226
$59$	14.0000	1.82264	0.911322	−	0.411693i	$-0.135063\pi$
$59$	14.0000	1.82264	0.911322	+	0.411693i	$0.135063\pi$
$60$	0	0
$61$	0	0		−	1.00000i	$-0.5\pi$
$61$	0	0			1.00000i	$0.5\pi$
$62$	9.00000	1.14300
$63$	0	0
$64$	1.00000	0.125000
$65$	0	0
$66$	0	0
$67$	−8.00000	−0.977356	−0.488678	−	0.872464i	$-0.662521\pi$
$67$	−8.00000	−0.977356	−0.488678	+	0.872464i	$0.662521\pi$
$68$	−2.00000	−0.242536
$69$	0	0
$70$	−1.00000	−0.119523
$71$	13.0000	1.54282	0.771408	−	0.636341i	$-0.219553\pi$
$71$	13.0000	1.54282	0.771408	+	0.636341i	$0.219553\pi$
$72$	0	0
$73$	−2.00000	−0.234082	−0.117041	−	0.993127i	$-0.537341\pi$
$73$	−2.00000	−0.234082	−0.117041	+	0.993127i	$0.537341\pi$
$74$	−5.00000	−0.581238
$75$	0	0
$76$	−1.00000	−0.114708
$77$	5.00000	0.569803
$78$	0	0
$79$	6.00000	0.675053	0.337526	−	0.941316i	$-0.390410\pi$
$79$	6.00000	0.675053	0.337526	+	0.941316i	$0.390410\pi$
$80$	−1.00000	−0.111803
$81$	0	0
$82$	−9.00000	−0.993884
$83$	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
$85$	2.00000	0.216930
$86$	10.0000	1.07833
$87$	0	0
$88$	5.00000	0.533002
$89$	9.00000	0.953998	0.476999	−	0.878904i	$-0.341725\pi$
$89$	9.00000	0.953998	0.476999	+	0.878904i	$0.341725\pi$
$90$	0	0
$91$	0	0
$92$	1.00000	0.104257
$93$	0	0
$94$	6.00000	0.618853
$95$	1.00000	0.102598
$96$	0	0
$97$	16.0000	1.62455	0.812277	−	0.583272i	$-0.198228\pi$
$97$	16.0000	1.62455	0.812277	+	0.583272i	$0.198228\pi$
$98$	−1.00000	−0.101015
$99$	0	0
$100$	−4.00000	−0.400000
$101$	14.0000	1.39305	0.696526	−	0.717532i	$-0.254728\pi$
$101$	14.0000	1.39305	0.696526	+	0.717532i	$0.254728\pi$
$102$	0	0
$103$	−1.00000	−0.0985329	−0.0492665	−	0.998786i	$-0.515688\pi$
$103$	−1.00000	−0.0985329	−0.0492665	+	0.998786i	$0.515688\pi$
$104$	0	0
$105$	0	0
$106$	12.0000	1.16554
$107$	−12.0000	−1.16008	−0.580042	−	0.814587i	$-0.696964\pi$
$107$	−12.0000	−1.16008	−0.580042	+	0.814587i	$0.696964\pi$
$108$	0	0
$109$	7.00000	0.670478	0.335239	−	0.942133i	$-0.391183\pi$
$109$	7.00000	0.670478	0.335239	+	0.942133i	$0.391183\pi$
$110$	−5.00000	−0.476731
$111$	0	0
$112$	−1.00000	−0.0944911
$113$	2.00000	0.188144	0.0940721	−	0.995565i	$-0.470012\pi$
$113$	2.00000	0.188144	0.0940721	+	0.995565i	$0.470012\pi$
$114$	0	0
$115$	−1.00000	−0.0932505
$116$	−4.00000	−0.371391
$117$	0	0
$118$	−14.0000	−1.28880
$119$	2.00000	0.183340
$120$	0	0
$121$	14.0000	1.27273
$122$	0	0
$123$	0	0
$124$	−9.00000	−0.808224
$125$	9.00000	0.804984
$126$	0	0
$127$	0	0		−	1.00000i	$-0.5\pi$
$127$	0	0			1.00000i	$0.5\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	0	0
$131$	−22.0000	−1.92215	−0.961074	−	0.276289i	$-0.910895\pi$
$131$	−22.0000	−1.92215	−0.961074	+	0.276289i	$0.910895\pi$
$132$	0	0
$133$	1.00000	0.0867110
$134$	8.00000	0.691095
$135$	0	0
$136$	2.00000	0.171499
$137$	−16.0000	−1.36697	−0.683486	−	0.729964i	$-0.739537\pi$
$137$	−16.0000	−1.36697	−0.683486	+	0.729964i	$0.739537\pi$
$138$	0	0
$139$	−20.0000	−1.69638	−0.848189	−	0.529694i	$-0.822307\pi$
$139$	−20.0000	−1.69638	−0.848189	+	0.529694i	$0.822307\pi$
$140$	1.00000	0.0845154
$141$	0	0
$142$	−13.0000	−1.09094
$143$	0	0
$144$	0	0
$145$	4.00000	0.332182
$146$	2.00000	0.165521
$147$	0	0
$148$	5.00000	0.410997
$149$	−6.00000	−0.491539	−0.245770	−	0.969328i	$-0.579041\pi$
$149$	−6.00000	−0.491539	−0.245770	+	0.969328i	$0.579041\pi$
$150$	0	0
$151$	10.0000	0.813788	0.406894	−	0.913475i	$-0.366612\pi$
$151$	10.0000	0.813788	0.406894	+	0.913475i	$0.366612\pi$
$152$	1.00000	0.0811107
$153$	0	0
$154$	−5.00000	−0.402911
$155$	9.00000	0.722897
$156$	0	0
$157$	8.00000	0.638470	0.319235	−	0.947676i	$-0.396574\pi$
$157$	8.00000	0.638470	0.319235	+	0.947676i	$0.396574\pi$
$158$	−6.00000	−0.477334
$159$	0	0
$160$	1.00000	0.0790569
$161$	−1.00000	−0.0788110
$162$	0	0
$163$	−4.00000	−0.313304	−0.156652	−	0.987654i	$-0.550070\pi$
$163$	−4.00000	−0.313304	−0.156652	+	0.987654i	$0.550070\pi$
$164$	9.00000	0.702782
$165$	0	0
$166$	−4.00000	−0.310460
$167$	−10.0000	−0.773823	−0.386912	−	0.922117i	$-0.626458\pi$
$167$	−10.0000	−0.773823	−0.386912	+	0.922117i	$0.626458\pi$
$168$	0	0
$169$	−13.0000	−1.00000
$170$	−2.00000	−0.153393
$171$	0	0
$172$	−10.0000	−0.762493
$173$	−7.00000	−0.532200	−0.266100	−	0.963945i	$-0.585735\pi$
$173$	−7.00000	−0.532200	−0.266100	+	0.963945i	$0.585735\pi$
$174$	0	0
$175$	4.00000	0.302372
$176$	−5.00000	−0.376889
$177$	0	0
$178$	−9.00000	−0.674579
$179$	−24.0000	−1.79384	−0.896922	−	0.442189i	$-0.854202\pi$
$179$	−24.0000	−1.79384	−0.896922	+	0.442189i	$0.854202\pi$
$180$	0	0
$181$	18.0000	1.33793	0.668965	−	0.743294i	$-0.266738\pi$
$181$	18.0000	1.33793	0.668965	+	0.743294i	$0.266738\pi$
$182$	0	0
$183$	0	0
$184$	−1.00000	−0.0737210
$185$	−5.00000	−0.367607
$186$	0	0
$187$	10.0000	0.731272
$188$	−6.00000	−0.437595
$189$	0	0
$190$	−1.00000	−0.0725476
$191$	3.00000	0.217072	0.108536	−	0.994092i	$-0.465384\pi$
$191$	3.00000	0.217072	0.108536	+	0.994092i	$0.465384\pi$
$192$	0	0
$193$	−10.0000	−0.719816	−0.359908	−	0.932988i	$-0.617192\pi$
$193$	−10.0000	−0.719816	−0.359908	+	0.932988i	$0.617192\pi$
$194$	−16.0000	−1.14873
$195$	0	0
$196$	1.00000	0.0714286
$197$	10.0000	0.712470	0.356235	−	0.934396i	$-0.384060\pi$
$197$	10.0000	0.712470	0.356235	+	0.934396i	$0.384060\pi$
$198$	0	0
$199$	−13.0000	−0.921546	−0.460773	−	0.887518i	$-0.652428\pi$
$199$	−13.0000	−0.921546	−0.460773	+	0.887518i	$0.652428\pi$
$200$	4.00000	0.282843
$201$	0	0
$202$	−14.0000	−0.985037
$203$	4.00000	0.280745
$204$	0	0
$205$	−9.00000	−0.628587
$206$	1.00000	0.0696733
$207$	0	0
$208$	0	0
$209$	5.00000	0.345857
$210$	0	0
$211$	−22.0000	−1.51454	−0.757271	−	0.653101i	$-0.773468\pi$
$211$	−22.0000	−1.51454	−0.757271	+	0.653101i	$0.773468\pi$
$212$	−12.0000	−0.824163
$213$	0	0
$214$	12.0000	0.820303
$215$	10.0000	0.681994
$216$	0	0
$217$	9.00000	0.610960
$218$	−7.00000	−0.474100
$219$	0	0
$220$	5.00000	0.337100
$221$	0	0
$222$	0	0
$223$	5.00000	0.334825	0.167412	−	0.985887i	$-0.446459\pi$
$223$	5.00000	0.334825	0.167412	+	0.985887i	$0.446459\pi$
$224$	1.00000	0.0668153
$225$	0	0
$226$	−2.00000	−0.133038
$227$	−6.00000	−0.398234	−0.199117	−	0.979976i	$-0.563807\pi$
$227$	−6.00000	−0.398234	−0.199117	+	0.979976i	$0.563807\pi$
$228$	0	0
$229$	28.0000	1.85029	0.925146	−	0.379611i	$-0.123942\pi$
$229$	28.0000	1.85029	0.925146	+	0.379611i	$0.123942\pi$
$230$	1.00000	0.0659380
$231$	0	0
$232$	4.00000	0.262613
$233$	−14.0000	−0.917170	−0.458585	−	0.888650i	$-0.651644\pi$
$233$	−14.0000	−0.917170	−0.458585	+	0.888650i	$0.651644\pi$
$234$	0	0
$235$	6.00000	0.391397
$236$	14.0000	0.911322
$237$	0	0
$238$	−2.00000	−0.129641
$239$	−24.0000	−1.55243	−0.776215	−	0.630468i	$-0.782863\pi$
$239$	−24.0000	−1.55243	−0.776215	+	0.630468i	$0.782863\pi$
$240$	0	0
$241$	14.0000	0.901819	0.450910	−	0.892570i	$-0.351100\pi$
$241$	14.0000	0.901819	0.450910	+	0.892570i	$0.351100\pi$
$242$	−14.0000	−0.899954
$243$	0	0
$244$	0	0
$245$	−1.00000	−0.0638877
$246$	0	0
$247$	0	0
$248$	9.00000	0.571501
$249$	0	0
$250$	−9.00000	−0.569210
$251$	24.0000	1.51487	0.757433	−	0.652913i	$-0.226453\pi$
$251$	24.0000	1.51487	0.757433	+	0.652913i	$0.226453\pi$
$252$	0	0
$253$	−5.00000	−0.314347
$254$	0	0
$255$	0	0
$256$	1.00000	0.0625000
$257$	−27.0000	−1.68421	−0.842107	−	0.539311i	$-0.818685\pi$
$257$	−27.0000	−1.68421	−0.842107	+	0.539311i	$0.818685\pi$
$258$	0	0
$259$	−5.00000	−0.310685
$260$	0	0
$261$	0	0
$262$	22.0000	1.35916
$263$	21.0000	1.29492	0.647458	−	0.762101i	$-0.275832\pi$
$263$	21.0000	1.29492	0.647458	+	0.762101i	$0.275832\pi$
$264$	0	0
$265$	12.0000	0.737154
$266$	−1.00000	−0.0613139
$267$	0	0
$268$	−8.00000	−0.488678
$269$	−13.0000	−0.792624	−0.396312	−	0.918116i	$-0.629710\pi$
$269$	−13.0000	−0.792624	−0.396312	+	0.918116i	$0.629710\pi$
$270$	0	0
$271$	24.0000	1.45790	0.728948	−	0.684569i	$-0.240010\pi$
$271$	24.0000	1.45790	0.728948	+	0.684569i	$0.240010\pi$
$272$	−2.00000	−0.121268
$273$	0	0
$274$	16.0000	0.966595
$275$	20.0000	1.20605
$276$	0	0
$277$	−19.0000	−1.14160	−0.570800	−	0.821089i	$-0.693367\pi$
$277$	−19.0000	−1.14160	−0.570800	+	0.821089i	$0.693367\pi$
$278$	20.0000	1.19952
$279$	0	0
$280$	−1.00000	−0.0597614
$281$	10.0000	0.596550	0.298275	−	0.954480i	$-0.403589\pi$
$281$	10.0000	0.596550	0.298275	+	0.954480i	$0.403589\pi$
$282$	0	0
$283$	−20.0000	−1.18888	−0.594438	−	0.804141i	$-0.702626\pi$
$283$	−20.0000	−1.18888	−0.594438	+	0.804141i	$0.702626\pi$
$284$	13.0000	0.771408
$285$	0	0
$286$	0	0
$287$	−9.00000	−0.531253
$288$	0	0
$289$	−13.0000	−0.764706
$290$	−4.00000	−0.234888
$291$	0	0
$292$	−2.00000	−0.117041
$293$	−18.0000	−1.05157	−0.525786	−	0.850617i	$-0.676229\pi$
$293$	−18.0000	−1.05157	−0.525786	+	0.850617i	$0.676229\pi$
$294$	0	0
$295$	−14.0000	−0.815112
$296$	−5.00000	−0.290619
$297$	0	0
$298$	6.00000	0.347571
$299$	0	0
$300$	0	0
$301$	10.0000	0.576390
$302$	−10.0000	−0.575435
$303$	0	0
$304$	−1.00000	−0.0573539
$305$	0	0
$306$	0	0
$307$	−5.00000	−0.285365	−0.142683	−	0.989769i	$-0.545573\pi$
$307$	−5.00000	−0.285365	−0.142683	+	0.989769i	$0.545573\pi$
$308$	5.00000	0.284901
$309$	0	0
$310$	−9.00000	−0.511166
$311$	8.00000	0.453638	0.226819	−	0.973937i	$-0.427167\pi$
$311$	8.00000	0.453638	0.226819	+	0.973937i	$0.427167\pi$
$312$	0	0
$313$	−8.00000	−0.452187	−0.226093	−	0.974106i	$-0.572595\pi$
$313$	−8.00000	−0.452187	−0.226093	+	0.974106i	$0.572595\pi$
$314$	−8.00000	−0.451466
$315$	0	0
$316$	6.00000	0.337526
$317$	18.0000	1.01098	0.505490	−	0.862832i	$-0.331312\pi$
$317$	18.0000	1.01098	0.505490	+	0.862832i	$0.331312\pi$
$318$	0	0
$319$	20.0000	1.11979
$320$	−1.00000	−0.0559017
$321$	0	0
$322$	1.00000	0.0557278
$323$	2.00000	0.111283
$324$	0	0
$325$	0	0
$326$	4.00000	0.221540
$327$	0	0
$328$	−9.00000	−0.496942
$329$	6.00000	0.330791
$330$	0	0
$331$	−4.00000	−0.219860	−0.109930	−	0.993939i	$-0.535063\pi$
$331$	−4.00000	−0.219860	−0.109930	+	0.993939i	$0.535063\pi$
$332$	4.00000	0.219529
$333$	0	0
$334$	10.0000	0.547176
$335$	8.00000	0.437087
$336$	0	0
$337$	27.0000	1.47078	0.735392	−	0.677642i	$-0.236998\pi$
$337$	27.0000	1.47078	0.735392	+	0.677642i	$0.236998\pi$
$338$	13.0000	0.707107
$339$	0	0
$340$	2.00000	0.108465
$341$	45.0000	2.43689
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	10.0000	0.539164
$345$	0	0
$346$	7.00000	0.376322
$347$	3.00000	0.161048	0.0805242	−	0.996753i	$-0.474341\pi$
$347$	3.00000	0.161048	0.0805242	+	0.996753i	$0.474341\pi$
$348$	0	0
$349$	−26.0000	−1.39175	−0.695874	−	0.718164i	$-0.744983\pi$
$349$	−26.0000	−1.39175	−0.695874	+	0.718164i	$0.744983\pi$
$350$	−4.00000	−0.213809
$351$	0	0
$352$	5.00000	0.266501
$353$	3.00000	0.159674	0.0798369	−	0.996808i	$-0.474560\pi$
$353$	3.00000	0.159674	0.0798369	+	0.996808i	$0.474560\pi$
$354$	0	0
$355$	−13.0000	−0.689968
$356$	9.00000	0.476999
$357$	0	0
$358$	24.0000	1.26844
$359$	−4.00000	−0.211112	−0.105556	−	0.994413i	$-0.533662\pi$
$359$	−4.00000	−0.211112	−0.105556	+	0.994413i	$0.533662\pi$
$360$	0	0
$361$	−18.0000	−0.947368
$362$	−18.0000	−0.946059
$363$	0	0
$364$	0	0
$365$	2.00000	0.104685
$366$	0	0
$367$	35.0000	1.82699	0.913493	−	0.406855i	$-0.133375\pi$
$367$	35.0000	1.82699	0.913493	+	0.406855i	$0.133375\pi$
$368$	1.00000	0.0521286
$369$	0	0
$370$	5.00000	0.259938
$371$	12.0000	0.623009
$372$	0	0
$373$	−17.0000	−0.880227	−0.440113	−	0.897942i	$-0.645062\pi$
$373$	−17.0000	−0.880227	−0.440113	+	0.897942i	$0.645062\pi$
$374$	−10.0000	−0.517088
$375$	0	0
$376$	6.00000	0.309426
$377$	0	0
$378$	0	0
$379$	−14.0000	−0.719132	−0.359566	−	0.933120i	$-0.617075\pi$
$379$	−14.0000	−0.719132	−0.359566	+	0.933120i	$0.617075\pi$
$380$	1.00000	0.0512989
$381$	0	0
$382$	−3.00000	−0.153493
$383$	10.0000	0.510976	0.255488	−	0.966812i	$-0.417764\pi$
$383$	10.0000	0.510976	0.255488	+	0.966812i	$0.417764\pi$
$384$	0	0
$385$	−5.00000	−0.254824
$386$	10.0000	0.508987
$387$	0	0
$388$	16.0000	0.812277
$389$	20.0000	1.01404	0.507020	−	0.861934i	$-0.330747\pi$
$389$	20.0000	1.01404	0.507020	+	0.861934i	$0.330747\pi$
$390$	0	0
$391$	−2.00000	−0.101144
$392$	−1.00000	−0.0505076
$393$	0	0
$394$	−10.0000	−0.503793
$395$	−6.00000	−0.301893
$396$	0	0
$397$	−30.0000	−1.50566	−0.752828	−	0.658217i	$-0.771311\pi$
$397$	−30.0000	−1.50566	−0.752828	+	0.658217i	$0.771311\pi$
$398$	13.0000	0.651631
$399$	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
$403$	0	0
$404$	14.0000	0.696526
$405$	0	0
$406$	−4.00000	−0.198517
$407$	−25.0000	−1.23920
$408$	0	0
$409$	−10.0000	−0.494468	−0.247234	−	0.968956i	$-0.579522\pi$
$409$	−10.0000	−0.494468	−0.247234	+	0.968956i	$0.579522\pi$
$410$	9.00000	0.444478
$411$	0	0
$412$	−1.00000	−0.0492665
$413$	−14.0000	−0.688895
$414$	0	0
$415$	−4.00000	−0.196352
$416$	0	0
$417$	0	0
$418$	−5.00000	−0.244558
$419$	−6.00000	−0.293119	−0.146560	−	0.989202i	$-0.546820\pi$
$419$	−6.00000	−0.293119	−0.146560	+	0.989202i	$0.546820\pi$
$420$	0	0
$421$	−27.0000	−1.31590	−0.657950	−	0.753062i	$-0.728576\pi$
$421$	−27.0000	−1.31590	−0.657950	+	0.753062i	$0.728576\pi$
$422$	22.0000	1.07094
$423$	0	0
$424$	12.0000	0.582772
$425$	8.00000	0.388057
$426$	0	0
$427$	0	0
$428$	−12.0000	−0.580042
$429$	0	0
$430$	−10.0000	−0.482243
$431$	15.0000	0.722525	0.361262	−	0.932464i	$-0.382346\pi$
$431$	15.0000	0.722525	0.361262	+	0.932464i	$0.382346\pi$
$432$	0	0
$433$	−4.00000	−0.192228	−0.0961139	−	0.995370i	$-0.530641\pi$
$433$	−4.00000	−0.192228	−0.0961139	+	0.995370i	$0.530641\pi$
$434$	−9.00000	−0.432014
$435$	0	0
$436$	7.00000	0.335239
$437$	−1.00000	−0.0478365
$438$	0	0
$439$	24.0000	1.14546	0.572729	−	0.819745i	$-0.305885\pi$
$439$	24.0000	1.14546	0.572729	+	0.819745i	$0.305885\pi$
$440$	−5.00000	−0.238366
$441$	0	0
$442$	0	0
$443$	−11.0000	−0.522626	−0.261313	−	0.965254i	$-0.584155\pi$
$443$	−11.0000	−0.522626	−0.261313	+	0.965254i	$0.584155\pi$
$444$	0	0
$445$	−9.00000	−0.426641
$446$	−5.00000	−0.236757
$447$	0	0
$448$	−1.00000	−0.0472456
$449$	−10.0000	−0.471929	−0.235965	−	0.971762i	$-0.575825\pi$
$449$	−10.0000	−0.471929	−0.235965	+	0.971762i	$0.575825\pi$
$450$	0	0
$451$	−45.0000	−2.11897
$452$	2.00000	0.0940721
$453$	0	0
$454$	6.00000	0.281594
$455$	0	0
$456$	0	0
$457$	13.0000	0.608114	0.304057	−	0.952654i	$-0.401659\pi$
$457$	13.0000	0.608114	0.304057	+	0.952654i	$0.401659\pi$
$458$	−28.0000	−1.30835
$459$	0	0
$460$	−1.00000	−0.0466252
$461$	−31.0000	−1.44381	−0.721907	−	0.691990i	$-0.756734\pi$
$461$	−31.0000	−1.44381	−0.721907	+	0.691990i	$0.756734\pi$
$462$	0	0
$463$	−14.0000	−0.650635	−0.325318	−	0.945605i	$-0.605471\pi$
$463$	−14.0000	−0.650635	−0.325318	+	0.945605i	$0.605471\pi$
$464$	−4.00000	−0.185695
$465$	0	0
$466$	14.0000	0.648537
$467$	26.0000	1.20314	0.601568	−	0.798821i	$-0.294543\pi$
$467$	26.0000	1.20314	0.601568	+	0.798821i	$0.294543\pi$
$468$	0	0
$469$	8.00000	0.369406
$470$	−6.00000	−0.276759
$471$	0	0
$472$	−14.0000	−0.644402
$473$	50.0000	2.29900
$474$	0	0
$475$	4.00000	0.183533
$476$	2.00000	0.0916698
$477$	0	0
$478$	24.0000	1.09773
$479$	−32.0000	−1.46212	−0.731059	−	0.682315i	$-0.760973\pi$
$479$	−32.0000	−1.46212	−0.731059	+	0.682315i	$0.760973\pi$
$480$	0	0
$481$	0	0
$482$	−14.0000	−0.637683
$483$	0	0
$484$	14.0000	0.636364
$485$	−16.0000	−0.726523
$486$	0	0
$487$	26.0000	1.17817	0.589086	−	0.808070i	$-0.299488\pi$
$487$	26.0000	1.17817	0.589086	+	0.808070i	$0.299488\pi$
$488$	0	0
$489$	0	0
$490$	1.00000	0.0451754
$491$	−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
$493$	8.00000	0.360302
$494$	0	0
$495$	0	0
$496$	−9.00000	−0.404112
$497$	−13.0000	−0.583130
$498$	0	0
$499$	10.0000	0.447661	0.223831	−	0.974628i	$-0.428144\pi$
$499$	10.0000	0.447661	0.223831	+	0.974628i	$0.428144\pi$
$500$	9.00000	0.402492
$501$	0	0
$502$	−24.0000	−1.07117
$503$	−36.0000	−1.60516	−0.802580	−	0.596544i	$-0.796540\pi$
$503$	−36.0000	−1.60516	−0.802580	+	0.596544i	$0.796540\pi$
$504$	0	0
$505$	−14.0000	−0.622992
$506$	5.00000	0.222277
$507$	0	0
$508$	0	0
$509$	−18.0000	−0.797836	−0.398918	−	0.916987i	$-0.630614\pi$
$509$	−18.0000	−0.797836	−0.398918	+	0.916987i	$0.630614\pi$
$510$	0	0
$511$	2.00000	0.0884748
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	27.0000	1.19092
$515$	1.00000	0.0440653
$516$	0	0
$517$	30.0000	1.31940
$518$	5.00000	0.219687
$519$	0	0
$520$	0	0
$521$	15.0000	0.657162	0.328581	−	0.944476i	$-0.393430\pi$
$521$	15.0000	0.657162	0.328581	+	0.944476i	$0.393430\pi$
$522$	0	0
$523$	11.0000	0.480996	0.240498	−	0.970650i	$-0.422689\pi$
$523$	11.0000	0.480996	0.240498	+	0.970650i	$0.422689\pi$
$524$	−22.0000	−0.961074
$525$	0	0
$526$	−21.0000	−0.915644
$527$	18.0000	0.784092
$528$	0	0
$529$	−22.0000	−0.956522
$530$	−12.0000	−0.521247
$531$	0	0
$532$	1.00000	0.0433555
$533$	0	0
$534$	0	0
$535$	12.0000	0.518805
$536$	8.00000	0.345547
$537$	0	0
$538$	13.0000	0.560470
$539$	−5.00000	−0.215365
$540$	0	0
$541$	−3.00000	−0.128980	−0.0644900	−	0.997918i	$-0.520542\pi$
$541$	−3.00000	−0.128980	−0.0644900	+	0.997918i	$0.520542\pi$
$542$	−24.0000	−1.03089
$543$	0	0
$544$	2.00000	0.0857493
$545$	−7.00000	−0.299847
$546$	0	0
$547$	12.0000	0.513083	0.256541	−	0.966533i	$-0.417417\pi$
$547$	12.0000	0.513083	0.256541	+	0.966533i	$0.417417\pi$
$548$	−16.0000	−0.683486
$549$	0	0
$550$	−20.0000	−0.852803
$551$	4.00000	0.170406
$552$	0	0
$553$	−6.00000	−0.255146
$554$	19.0000	0.807233
$555$	0	0
$556$	−20.0000	−0.848189
$557$	−22.0000	−0.932170	−0.466085	−	0.884740i	$-0.654336\pi$
$557$	−22.0000	−0.932170	−0.466085	+	0.884740i	$0.654336\pi$
$558$	0	0
$559$	0	0
$560$	1.00000	0.0422577
$561$	0	0
$562$	−10.0000	−0.421825
$563$	4.00000	0.168580	0.0842900	−	0.996441i	$-0.473138\pi$
$563$	4.00000	0.168580	0.0842900	+	0.996441i	$0.473138\pi$
$564$	0	0
$565$	−2.00000	−0.0841406
$566$	20.0000	0.840663
$567$	0	0
$568$	−13.0000	−0.545468
$569$	−30.0000	−1.25767	−0.628833	−	0.777541i	$-0.716467\pi$
$569$	−30.0000	−1.25767	−0.628833	+	0.777541i	$0.716467\pi$
$570$	0	0
$571$	18.0000	0.753277	0.376638	−	0.926360i	$-0.377080\pi$
$571$	18.0000	0.753277	0.376638	+	0.926360i	$0.377080\pi$
$572$	0	0
$573$	0	0
$574$	9.00000	0.375653
$575$	−4.00000	−0.166812
$576$	0	0
$577$	−14.0000	−0.582828	−0.291414	−	0.956597i	$-0.594126\pi$
$577$	−14.0000	−0.582828	−0.291414	+	0.956597i	$0.594126\pi$
$578$	13.0000	0.540729
$579$	0	0
$580$	4.00000	0.166091
$581$	−4.00000	−0.165948
$582$	0	0
$583$	60.0000	2.48495
$584$	2.00000	0.0827606
$585$	0	0
$586$	18.0000	0.743573
$587$	−14.0000	−0.577842	−0.288921	−	0.957353i	$-0.593296\pi$
$587$	−14.0000	−0.577842	−0.288921	+	0.957353i	$0.593296\pi$
$588$	0	0
$589$	9.00000	0.370839
$590$	14.0000	0.576371
$591$	0	0
$592$	5.00000	0.205499
$593$	−9.00000	−0.369586	−0.184793	−	0.982777i	$-0.559161\pi$
$593$	−9.00000	−0.369586	−0.184793	+	0.982777i	$0.559161\pi$
$594$	0	0
$595$	−2.00000	−0.0819920
$596$	−6.00000	−0.245770
$597$	0	0
$598$	0	0
$599$	27.0000	1.10319	0.551595	−	0.834112i	$-0.314019\pi$
$599$	27.0000	1.10319	0.551595	+	0.834112i	$0.314019\pi$
$600$	0	0
$601$	8.00000	0.326327	0.163163	−	0.986599i	$-0.447830\pi$
$601$	8.00000	0.326327	0.163163	+	0.986599i	$0.447830\pi$
$602$	−10.0000	−0.407570
$603$	0	0
$604$	10.0000	0.406894
$605$	−14.0000	−0.569181
$606$	0	0
$607$	12.0000	0.487065	0.243532	−	0.969893i	$-0.421694\pi$
$607$	12.0000	0.487065	0.243532	+	0.969893i	$0.421694\pi$
$608$	1.00000	0.0405554
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0	0
$613$	−15.0000	−0.605844	−0.302922	−	0.953015i	$-0.597962\pi$
$613$	−15.0000	−0.605844	−0.302922	+	0.953015i	$0.597962\pi$
$614$	5.00000	0.201784
$615$	0	0
$616$	−5.00000	−0.201456
$617$	−14.0000	−0.563619	−0.281809	−	0.959470i	$-0.590935\pi$
$617$	−14.0000	−0.563619	−0.281809	+	0.959470i	$0.590935\pi$
$618$	0	0
$619$	−11.0000	−0.442127	−0.221064	−	0.975259i	$-0.570953\pi$
$619$	−11.0000	−0.442127	−0.221064	+	0.975259i	$0.570953\pi$
$620$	9.00000	0.361449
$621$	0	0
$622$	−8.00000	−0.320771
$623$	−9.00000	−0.360577
$624$	0	0
$625$	11.0000	0.440000
$626$	8.00000	0.319744
$627$	0	0
$628$	8.00000	0.319235
$629$	−10.0000	−0.398726
$630$	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
$634$	−18.0000	−0.714871
$635$	0	0
$636$	0	0
$637$	0	0
$638$	−20.0000	−0.791808
$639$	0	0
$640$	1.00000	0.0395285
$641$	22.0000	0.868948	0.434474	−	0.900684i	$-0.356934\pi$
$641$	22.0000	0.868948	0.434474	+	0.900684i	$0.356934\pi$
$642$	0	0
$643$	−13.0000	−0.512670	−0.256335	−	0.966588i	$-0.582515\pi$
$643$	−13.0000	−0.512670	−0.256335	+	0.966588i	$0.582515\pi$
$644$	−1.00000	−0.0394055
$645$	0	0
$646$	−2.00000	−0.0786889
$647$	−42.0000	−1.65119	−0.825595	−	0.564263i	$-0.809160\pi$
$647$	−42.0000	−1.65119	−0.825595	+	0.564263i	$0.809160\pi$
$648$	0	0
$649$	−70.0000	−2.74774
$650$	0	0
$651$	0	0
$652$	−4.00000	−0.156652
$653$	30.0000	1.17399	0.586995	−	0.809590i	$-0.300311\pi$
$653$	30.0000	1.17399	0.586995	+	0.809590i	$0.300311\pi$
$654$	0	0
$655$	22.0000	0.859611
$656$	9.00000	0.351391
$657$	0	0
$658$	−6.00000	−0.233904
$659$	−17.0000	−0.662226	−0.331113	−	0.943591i	$-0.607424\pi$
$659$	−17.0000	−0.662226	−0.331113	+	0.943591i	$0.607424\pi$
$660$	0	0
$661$	28.0000	1.08907	0.544537	−	0.838737i	$-0.316705\pi$
$661$	28.0000	1.08907	0.544537	+	0.838737i	$0.316705\pi$
$662$	4.00000	0.155464
$663$	0	0
$664$	−4.00000	−0.155230
$665$	−1.00000	−0.0387783
$666$	0	0
$667$	−4.00000	−0.154881
$668$	−10.0000	−0.386912
$669$	0	0
$670$	−8.00000	−0.309067
$671$	0	0
$672$	0	0
$673$	26.0000	1.00223	0.501113	−	0.865382i	$-0.332924\pi$
$673$	26.0000	1.00223	0.501113	+	0.865382i	$0.332924\pi$
$674$	−27.0000	−1.04000
$675$	0	0
$676$	−13.0000	−0.500000
$677$	−27.0000	−1.03769	−0.518847	−	0.854867i	$-0.673639\pi$
$677$	−27.0000	−1.03769	−0.518847	+	0.854867i	$0.673639\pi$
$678$	0	0
$679$	−16.0000	−0.614024
$680$	−2.00000	−0.0766965
$681$	0	0
$682$	−45.0000	−1.72314
$683$	9.00000	0.344375	0.172188	−	0.985064i	$-0.444916\pi$
$683$	9.00000	0.344375	0.172188	+	0.985064i	$0.444916\pi$
$684$	0	0
$685$	16.0000	0.611329
$686$	1.00000	0.0381802
$687$	0	0
$688$	−10.0000	−0.381246
$689$	0	0
$690$	0	0
$691$	−4.00000	−0.152167	−0.0760836	−	0.997101i	$-0.524242\pi$
$691$	−4.00000	−0.152167	−0.0760836	+	0.997101i	$0.524242\pi$
$692$	−7.00000	−0.266100
$693$	0	0
$694$	−3.00000	−0.113878
$695$	20.0000	0.758643
$696$	0	0
$697$	−18.0000	−0.681799
$698$	26.0000	0.984115
$699$	0	0
$700$	4.00000	0.151186
$701$	20.0000	0.755390	0.377695	−	0.925930i	$-0.376717\pi$
$701$	20.0000	0.755390	0.377695	+	0.925930i	$0.376717\pi$
$702$	0	0
$703$	−5.00000	−0.188579
$704$	−5.00000	−0.188445
$705$	0	0
$706$	−3.00000	−0.112906
$707$	−14.0000	−0.526524
$708$	0	0
$709$	−25.0000	−0.938895	−0.469447	−	0.882960i	$-0.655547\pi$
$709$	−25.0000	−0.938895	−0.469447	+	0.882960i	$0.655547\pi$
$710$	13.0000	0.487881
$711$	0	0
$712$	−9.00000	−0.337289
$713$	−9.00000	−0.337053
$714$	0	0
$715$	0	0
$716$	−24.0000	−0.896922
$717$	0	0
$718$	4.00000	0.149279
$719$	30.0000	1.11881	0.559406	−	0.828894i	$-0.311029\pi$
$719$	30.0000	1.11881	0.559406	+	0.828894i	$0.311029\pi$
$720$	0	0
$721$	1.00000	0.0372419
$722$	18.0000	0.669891
$723$	0	0
$724$	18.0000	0.668965
$725$	16.0000	0.594225
$726$	0	0
$727$	−32.0000	−1.18681	−0.593407	−	0.804902i	$-0.702218\pi$
$727$	−32.0000	−1.18681	−0.593407	+	0.804902i	$0.702218\pi$
$728$	0	0
$729$	0	0
$730$	−2.00000	−0.0740233
$731$	20.0000	0.739727
$732$	0	0
$733$	−18.0000	−0.664845	−0.332423	−	0.943131i	$-0.607866\pi$
$733$	−18.0000	−0.664845	−0.332423	+	0.943131i	$0.607866\pi$
$734$	−35.0000	−1.29187
$735$	0	0
$736$	−1.00000	−0.0368605
$737$	40.0000	1.47342
$738$	0	0
$739$	18.0000	0.662141	0.331070	−	0.943606i	$-0.392590\pi$
$739$	18.0000	0.662141	0.331070	+	0.943606i	$0.392590\pi$
$740$	−5.00000	−0.183804
$741$	0	0
$742$	−12.0000	−0.440534
$743$	21.0000	0.770415	0.385208	−	0.922830i	$-0.374130\pi$
$743$	21.0000	0.770415	0.385208	+	0.922830i	$0.374130\pi$
$744$	0	0
$745$	6.00000	0.219823
$746$	17.0000	0.622414
$747$	0	0
$748$	10.0000	0.365636
$749$	12.0000	0.438470
$750$	0	0
$751$	18.0000	0.656829	0.328415	−	0.944534i	$-0.393486\pi$
$751$	18.0000	0.656829	0.328415	+	0.944534i	$0.393486\pi$
$752$	−6.00000	−0.218797
$753$	0	0
$754$	0	0
$755$	−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	$0.223599\pi$
$758$	14.0000	0.508503
$759$	0	0
$760$	−1.00000	−0.0362738
$761$	22.0000	0.797499	0.398750	−	0.917060i	$-0.369444\pi$
$761$	22.0000	0.797499	0.398750	+	0.917060i	$0.369444\pi$
$762$	0	0
$763$	−7.00000	−0.253417
$764$	3.00000	0.108536
$765$	0	0
$766$	−10.0000	−0.361315
$767$	0	0
$768$	0	0
$769$	40.0000	1.44244	0.721218	−	0.692708i	$-0.243582\pi$
$769$	40.0000	1.44244	0.721218	+	0.692708i	$0.243582\pi$
$770$	5.00000	0.180187
$771$	0	0
$772$	−10.0000	−0.359908
$773$	−19.0000	−0.683383	−0.341691	−	0.939812i	$-0.611000\pi$
$773$	−19.0000	−0.683383	−0.341691	+	0.939812i	$0.611000\pi$
$774$	0	0
$775$	36.0000	1.29316
$776$	−16.0000	−0.574367
$777$	0	0
$778$	−20.0000	−0.717035
$779$	−9.00000	−0.322458
$780$	0	0
$781$	−65.0000	−2.32588
$782$	2.00000	0.0715199
$783$	0	0
$784$	1.00000	0.0357143
$785$	−8.00000	−0.285532
$786$	0	0
$787$	0	0		−	1.00000i	$-0.5\pi$
$787$	0	0			1.00000i	$0.5\pi$
$788$	10.0000	0.356235
$789$	0	0
$790$	6.00000	0.213470
$791$	−2.00000	−0.0711118
$792$	0	0
$793$	0	0
$794$	30.0000	1.06466
$795$	0	0
$796$	−13.0000	−0.460773
$797$	33.0000	1.16892	0.584460	−	0.811423i	$-0.301306\pi$
$797$	33.0000	1.16892	0.584460	+	0.811423i	$0.301306\pi$
$798$	0	0
$799$	12.0000	0.424529
$800$	4.00000	0.141421
$801$	0	0
$802$	−12.0000	−0.423735
$803$	10.0000	0.352892
$804$	0	0
$805$	1.00000	0.0352454
$806$	0	0
$807$	0	0
$808$	−14.0000	−0.492518
$809$	20.0000	0.703163	0.351581	−	0.936157i	$-0.385644\pi$
$809$	20.0000	0.703163	0.351581	+	0.936157i	$0.385644\pi$
$810$	0	0
$811$	1.00000	0.0351147	0.0175574	−	0.999846i	$-0.494411\pi$
$811$	1.00000	0.0351147	0.0175574	+	0.999846i	$0.494411\pi$
$812$	4.00000	0.140372
$813$	0	0
$814$	25.0000	0.876250
$815$	4.00000	0.140114
$816$	0	0
$817$	10.0000	0.349856
$818$	10.0000	0.349642
$819$	0	0
$820$	−9.00000	−0.314294
$821$	10.0000	0.349002	0.174501	−	0.984657i	$-0.444169\pi$
$821$	10.0000	0.349002	0.174501	+	0.984657i	$0.444169\pi$
$822$	0	0
$823$	34.0000	1.18517	0.592583	−	0.805510i	$-0.298108\pi$
$823$	34.0000	1.18517	0.592583	+	0.805510i	$0.298108\pi$
$824$	1.00000	0.0348367
$825$	0	0
$826$	14.0000	0.487122
$827$	33.0000	1.14752	0.573761	−	0.819023i	$-0.305484\pi$
$827$	33.0000	1.14752	0.573761	+	0.819023i	$0.305484\pi$
$828$	0	0
$829$	10.0000	0.347314	0.173657	−	0.984806i	$-0.444442\pi$
$829$	10.0000	0.347314	0.173657	+	0.984806i	$0.444442\pi$
$830$	4.00000	0.138842
$831$	0	0
$832$	0	0
$833$	−2.00000	−0.0692959
$834$	0	0
$835$	10.0000	0.346064
$836$	5.00000	0.172929
$837$	0	0
$838$	6.00000	0.207267
$839$	4.00000	0.138095	0.0690477	−	0.997613i	$-0.478004\pi$
$839$	4.00000	0.138095	0.0690477	+	0.997613i	$0.478004\pi$
$840$	0	0
$841$	−13.0000	−0.448276
$842$	27.0000	0.930481
$843$	0	0
$844$	−22.0000	−0.757271
$845$	13.0000	0.447214
$846$	0	0
$847$	−14.0000	−0.481046
$848$	−12.0000	−0.412082
$849$	0	0
$850$	−8.00000	−0.274398
$851$	5.00000	0.171398
$852$	0	0
$853$	−16.0000	−0.547830	−0.273915	−	0.961754i	$-0.588319\pi$
$853$	−16.0000	−0.547830	−0.273915	+	0.961754i	$0.588319\pi$
$854$	0	0
$855$	0	0
$856$	12.0000	0.410152
$857$	−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
$859$	−25.0000	−0.852989	−0.426494	−	0.904490i	$-0.640252\pi$
$859$	−25.0000	−0.852989	−0.426494	+	0.904490i	$0.640252\pi$
$860$	10.0000	0.340997
$861$	0	0
$862$	−15.0000	−0.510902
$863$	−16.0000	−0.544646	−0.272323	−	0.962206i	$-0.587792\pi$
$863$	−16.0000	−0.544646	−0.272323	+	0.962206i	$0.587792\pi$
$864$	0	0
$865$	7.00000	0.238007
$866$	4.00000	0.135926
$867$	0	0
$868$	9.00000	0.305480
$869$	−30.0000	−1.01768
$870$	0	0
$871$	0	0
$872$	−7.00000	−0.237050
$873$	0	0
$874$	1.00000	0.0338255
$875$	−9.00000	−0.304256
$876$	0	0
$877$	−50.0000	−1.68838	−0.844190	−	0.536044i	$-0.819918\pi$
$877$	−50.0000	−1.68838	−0.844190	+	0.536044i	$0.819918\pi$
$878$	−24.0000	−0.809961
$879$	0	0
$880$	5.00000	0.168550
$881$	15.0000	0.505363	0.252681	−	0.967550i	$-0.418688\pi$
$881$	15.0000	0.505363	0.252681	+	0.967550i	$0.418688\pi$
$882$	0	0
$883$	32.0000	1.07689	0.538443	−	0.842662i	$-0.319013\pi$
$883$	32.0000	1.07689	0.538443	+	0.842662i	$0.319013\pi$
$884$	0	0
$885$	0	0
$886$	11.0000	0.369552
$887$	−36.0000	−1.20876	−0.604381	−	0.796696i	$-0.706579\pi$
$887$	−36.0000	−1.20876	−0.604381	+	0.796696i	$0.706579\pi$
$888$	0	0
$889$	0	0
$890$	9.00000	0.301681
$891$	0	0
$892$	5.00000	0.167412
$893$	6.00000	0.200782
$894$	0	0
$895$	24.0000	0.802232
$896$	1.00000	0.0334077
$897$	0	0
$898$	10.0000	0.333704
$899$	36.0000	1.20067
$900$	0	0
$901$	24.0000	0.799556
$902$	45.0000	1.49834
$903$	0	0
$904$	−2.00000	−0.0665190
$905$	−18.0000	−0.598340
$906$	0	0
$907$	30.0000	0.996134	0.498067	−	0.867139i	$-0.334043\pi$
$907$	30.0000	0.996134	0.498067	+	0.867139i	$0.334043\pi$
$908$	−6.00000	−0.199117
$909$	0	0
$910$	0	0
$911$	24.0000	0.795155	0.397578	−	0.917568i	$-0.369851\pi$
$911$	24.0000	0.795155	0.397578	+	0.917568i	$0.369851\pi$
$912$	0	0
$913$	−20.0000	−0.661903
$914$	−13.0000	−0.430002
$915$	0	0
$916$	28.0000	0.925146
$917$	22.0000	0.726504
$918$	0	0
$919$	4.00000	0.131948	0.0659739	−	0.997821i	$-0.478985\pi$
$919$	4.00000	0.131948	0.0659739	+	0.997821i	$0.478985\pi$
$920$	1.00000	0.0329690
$921$	0	0
$922$	31.0000	1.02093
$923$	0	0
$924$	0	0
$925$	−20.0000	−0.657596
$926$	14.0000	0.460069
$927$	0	0
$928$	4.00000	0.131306
$929$	42.0000	1.37798	0.688988	−	0.724773i	$-0.258055\pi$
$929$	42.0000	1.37798	0.688988	+	0.724773i	$0.258055\pi$
$930$	0	0
$931$	−1.00000	−0.0327737
$932$	−14.0000	−0.458585
$933$	0	0
$934$	−26.0000	−0.850746
$935$	−10.0000	−0.327035
$936$	0	0
$937$	−28.0000	−0.914720	−0.457360	−	0.889282i	$-0.651205\pi$
$937$	−28.0000	−0.914720	−0.457360	+	0.889282i	$0.651205\pi$
$938$	−8.00000	−0.261209
$939$	0	0
$940$	6.00000	0.195698
$941$	−13.0000	−0.423788	−0.211894	−	0.977293i	$-0.567963\pi$
$941$	−13.0000	−0.423788	−0.211894	+	0.977293i	$0.567963\pi$
$942$	0	0
$943$	9.00000	0.293080
$944$	14.0000	0.455661
$945$	0	0
$946$	−50.0000	−1.62564
$947$	17.0000	0.552426	0.276213	−	0.961096i	$-0.410921\pi$
$947$	17.0000	0.552426	0.276213	+	0.961096i	$0.410921\pi$
$948$	0	0
$949$	0	0
$950$	−4.00000	−0.129777
$951$	0	0
$952$	−2.00000	−0.0648204
$953$	−44.0000	−1.42530	−0.712650	−	0.701520i	$-0.752505\pi$
$953$	−44.0000	−1.42530	−0.712650	+	0.701520i	$0.752505\pi$
$954$	0	0
$955$	−3.00000	−0.0970777
$956$	−24.0000	−0.776215
$957$	0	0
$958$	32.0000	1.03387
$959$	16.0000	0.516667
$960$	0	0
$961$	50.0000	1.61290
$962$	0	0
$963$	0	0
$964$	14.0000	0.450910
$965$	10.0000	0.321911
$966$	0	0
$967$	2.00000	0.0643157	0.0321578	−	0.999483i	$-0.489762\pi$
$967$	2.00000	0.0643157	0.0321578	+	0.999483i	$0.489762\pi$
$968$	−14.0000	−0.449977
$969$	0	0
$970$	16.0000	0.513729
$971$	−36.0000	−1.15529	−0.577647	−	0.816286i	$-0.696029\pi$
$971$	−36.0000	−1.15529	−0.577647	+	0.816286i	$0.696029\pi$
$972$	0	0
$973$	20.0000	0.641171
$974$	−26.0000	−0.833094
$975$	0	0
$976$	0	0
$977$	−60.0000	−1.91957	−0.959785	−	0.280736i	$-0.909421\pi$
$977$	−60.0000	−1.91957	−0.959785	+	0.280736i	$0.909421\pi$
$978$	0	0
$979$	−45.0000	−1.43821
$980$	−1.00000	−0.0319438
$981$	0	0
$982$	9.00000	0.287202
$983$	30.0000	0.956851	0.478426	−	0.878128i	$-0.341208\pi$
$983$	30.0000	0.956851	0.478426	+	0.878128i	$0.341208\pi$
$984$	0	0
$985$	−10.0000	−0.318626
$986$	−8.00000	−0.254772
$987$	0	0
$988$	0	0
$989$	−10.0000	−0.317982
$990$	0	0
$991$	−4.00000	−0.127064	−0.0635321	−	0.997980i	$-0.520237\pi$
$991$	−4.00000	−0.127064	−0.0635321	+	0.997980i	$0.520237\pi$
$992$	9.00000	0.285750
$993$	0	0
$994$	13.0000	0.412335
$995$	13.0000	0.412128
$996$	0	0
$997$	38.0000	1.20347	0.601736	−	0.798695i	$-0.294476\pi$
$997$	38.0000	1.20347	0.601736	+	0.798695i	$0.294476\pi$
$998$	−10.0000	−0.316544
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	378.2.a.c.1.1	✓	1
3.2	odd	2		378.2.a.f.1.1	yes	1
4.3	odd	2		3024.2.a.m.1.1		1
5.4	even	2		9450.2.a.dc.1.1		1
7.6	odd	2		2646.2.a.i.1.1		1
9.2	odd	6		1134.2.f.c.757.1		2
9.4	even	3		1134.2.f.n.379.1		2
9.5	odd	6		1134.2.f.c.379.1		2
9.7	even	3		1134.2.f.n.757.1		2
12.11	even	2		3024.2.a.t.1.1		1
15.14	odd	2		9450.2.a.bx.1.1		1
21.20	even	2		2646.2.a.v.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
378.2.a.c.1.1	✓	1	1.1	even	1	trivial
378.2.a.f.1.1	yes	1	3.2	odd	2
1134.2.f.c.379.1		2	9.5	odd	6
1134.2.f.c.757.1		2	9.2	odd	6
1134.2.f.n.379.1		2	9.4	even	3
1134.2.f.n.757.1		2	9.7	even	3
2646.2.a.i.1.1		1	7.6	odd	2
2646.2.a.v.1.1		1	21.20	even	2
3024.2.a.m.1.1		1	4.3	odd	2
3024.2.a.t.1.1		1	12.11	even	2
9450.2.a.bx.1.1		1	15.14	odd	2
9450.2.a.dc.1.1		1	5.4	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:	\( N \)	\(=\)	\( 378 = 2 \cdot 3^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	378.a (trivial)

Introduction

L-functions

Modular forms

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Fields

Representations

Groups

Database

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