LMFDB - Embedded newform 378.2.a.a.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:	$0$
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)

$f(q)$

$=$

$q-1.00000 q^{2} +1.00000 q^{4} -4.00000 q^{5} -1.00000 q^{7} -1.00000 q^{8} +O(q^{10})$

\(q-1.00000 q^{2} +1.00000 q^{4} -4.00000 q^{5} -1.00000 q^{7} -1.00000 q^{8} +4.00000 q^{10} +4.00000 q^{11} +3.00000 q^{13} +1.00000 q^{14} +1.00000 q^{16} +7.00000 q^{17} +2.00000 q^{19} -4.00000 q^{20} -4.00000 q^{22} +1.00000 q^{23} +11.0000 q^{25} -3.00000 q^{26} -1.00000 q^{28} -1.00000 q^{29} -9.00000 q^{31} -1.00000 q^{32} -7.00000 q^{34} +4.00000 q^{35} +2.00000 q^{37} -2.00000 q^{38} +4.00000 q^{40} -6.00000 q^{41} +11.0000 q^{43} +4.00000 q^{44} -1.00000 q^{46} +6.00000 q^{47} +1.00000 q^{49} -11.0000 q^{50} +3.00000 q^{52} +9.00000 q^{53} -16.0000 q^{55} +1.00000 q^{56} +1.00000 q^{58} +5.00000 q^{59} -6.00000 q^{61} +9.00000 q^{62} +1.00000 q^{64} -12.0000 q^{65} +7.00000 q^{67} +7.00000 q^{68} -4.00000 q^{70} +7.00000 q^{71} -14.0000 q^{73} -2.00000 q^{74} +2.00000 q^{76} -4.00000 q^{77} -6.00000 q^{79} -4.00000 q^{80} +6.00000 q^{82} +4.00000 q^{83} -28.0000 q^{85} -11.0000 q^{86} -4.00000 q^{88} +3.00000 q^{89} -3.00000 q^{91} +1.00000 q^{92} -6.00000 q^{94} -8.00000 q^{95} -8.00000 q^{97} -1.00000 q^{98} +O(q^{100})\)

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
$5$	−4.00000	−1.78885	−0.894427	−	0.447214i	$-0.852416\pi$
$5$	−4.00000	−1.78885	−0.894427	+	0.447214i	$0.852416\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	−1.00000	−0.353553
$9$	0	0
$10$	4.00000	1.26491
$11$	4.00000	1.20605	0.603023	−	0.797724i	$-0.293963\pi$
$11$	4.00000	1.20605	0.603023	+	0.797724i	$0.293963\pi$
$12$	0	0
$13$	3.00000	0.832050	0.416025	−	0.909353i	$-0.363423\pi$
$13$	3.00000	0.832050	0.416025	+	0.909353i	$0.363423\pi$
$14$	1.00000	0.267261
$15$	0	0
$16$	1.00000	0.250000
$17$	7.00000	1.69775	0.848875	−	0.528594i	$-0.177281\pi$
$17$	7.00000	1.69775	0.848875	+	0.528594i	$0.177281\pi$
$18$	0	0
$19$	2.00000	0.458831	0.229416	−	0.973329i	$-0.426318\pi$
$19$	2.00000	0.458831	0.229416	+	0.973329i	$0.426318\pi$
$20$	−4.00000	−0.894427
$21$	0	0
$22$	−4.00000	−0.852803
$23$	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
$25$	11.0000	2.20000
$26$	−3.00000	−0.588348
$27$	0	0
$28$	−1.00000	−0.188982
$29$	−1.00000	−0.185695	−0.0928477	−	0.995680i	$-0.529597\pi$
$29$	−1.00000	−0.185695	−0.0928477	+	0.995680i	$0.529597\pi$
$30$	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
$34$	−7.00000	−1.20049
$35$	4.00000	0.676123
$36$	0	0
$37$	2.00000	0.328798	0.164399	−	0.986394i	$-0.447432\pi$
$37$	2.00000	0.328798	0.164399	+	0.986394i	$0.447432\pi$
$38$	−2.00000	−0.324443
$39$	0	0
$40$	4.00000	0.632456
$41$	−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
$43$	11.0000	1.67748	0.838742	−	0.544529i	$-0.183292\pi$
$43$	11.0000	1.67748	0.838742	+	0.544529i	$0.183292\pi$
$44$	4.00000	0.603023
$45$	0	0
$46$	−1.00000	−0.147442
$47$	6.00000	0.875190	0.437595	−	0.899172i	$-0.355830\pi$
$47$	6.00000	0.875190	0.437595	+	0.899172i	$0.355830\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−11.0000	−1.55563
$51$	0	0
$52$	3.00000	0.416025
$53$	9.00000	1.23625	0.618123	−	0.786082i	$-0.287894\pi$
$53$	9.00000	1.23625	0.618123	+	0.786082i	$0.287894\pi$
$54$	0	0
$55$	−16.0000	−2.15744
$56$	1.00000	0.133631
$57$	0	0
$58$	1.00000	0.131306
$59$	5.00000	0.650945	0.325472	−	0.945552i	$-0.394477\pi$
$59$	5.00000	0.650945	0.325472	+	0.945552i	$0.394477\pi$
$60$	0	0
$61$	−6.00000	−0.768221	−0.384111	−	0.923287i	$-0.625492\pi$
$61$	−6.00000	−0.768221	−0.384111	+	0.923287i	$0.625492\pi$
$62$	9.00000	1.14300
$63$	0	0
$64$	1.00000	0.125000
$65$	−12.0000	−1.48842
$66$	0	0
$67$	7.00000	0.855186	0.427593	−	0.903971i	$-0.359362\pi$
$67$	7.00000	0.855186	0.427593	+	0.903971i	$0.359362\pi$
$68$	7.00000	0.848875
$69$	0	0
$70$	−4.00000	−0.478091
$71$	7.00000	0.830747	0.415374	−	0.909651i	$-0.363651\pi$
$71$	7.00000	0.830747	0.415374	+	0.909651i	$0.363651\pi$
$72$	0	0
$73$	−14.0000	−1.63858	−0.819288	−	0.573382i	$-0.805631\pi$
$73$	−14.0000	−1.63858	−0.819288	+	0.573382i	$0.805631\pi$
$74$	−2.00000	−0.232495
$75$	0	0
$76$	2.00000	0.229416
$77$	−4.00000	−0.455842
$78$	0	0
$79$	−6.00000	−0.675053	−0.337526	−	0.941316i	$-0.609590\pi$
$79$	−6.00000	−0.675053	−0.337526	+	0.941316i	$0.609590\pi$
$80$	−4.00000	−0.447214
$81$	0	0
$82$	6.00000	0.662589
$83$	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
$85$	−28.0000	−3.03703
$86$	−11.0000	−1.18616
$87$	0	0
$88$	−4.00000	−0.426401
$89$	3.00000	0.317999	0.159000	−	0.987279i	$-0.449173\pi$
$89$	3.00000	0.317999	0.159000	+	0.987279i	$0.449173\pi$
$90$	0	0
$91$	−3.00000	−0.314485
$92$	1.00000	0.104257
$93$	0	0
$94$	−6.00000	−0.618853
$95$	−8.00000	−0.820783
$96$	0	0
$97$	−8.00000	−0.812277	−0.406138	−	0.913812i	$-0.633125\pi$
$97$	−8.00000	−0.812277	−0.406138	+	0.913812i	$0.633125\pi$
$98$	−1.00000	−0.101015
$99$	0	0
$100$	11.0000	1.10000
$101$	−10.0000	−0.995037	−0.497519	−	0.867453i	$-0.665755\pi$
$101$	−10.0000	−0.995037	−0.497519	+	0.867453i	$0.665755\pi$
$102$	0	0
$103$	17.0000	1.67506	0.837530	−	0.546392i	$-0.183999\pi$
$103$	17.0000	1.67506	0.837530	+	0.546392i	$0.183999\pi$
$104$	−3.00000	−0.294174
$105$	0	0
$106$	−9.00000	−0.874157
$107$	−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
$109$	16.0000	1.53252	0.766261	−	0.642529i	$-0.222115\pi$
$109$	16.0000	1.53252	0.766261	+	0.642529i	$0.222115\pi$
$110$	16.0000	1.52554
$111$	0	0
$112$	−1.00000	−0.0944911
$113$	8.00000	0.752577	0.376288	−	0.926503i	$-0.377200\pi$
$113$	8.00000	0.752577	0.376288	+	0.926503i	$0.377200\pi$
$114$	0	0
$115$	−4.00000	−0.373002
$116$	−1.00000	−0.0928477
$117$	0	0
$118$	−5.00000	−0.460287
$119$	−7.00000	−0.641689
$120$	0	0
$121$	5.00000	0.454545
$122$	6.00000	0.543214
$123$	0	0
$124$	−9.00000	−0.808224
$125$	−24.0000	−2.14663
$126$	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
$130$	12.0000	1.05247
$131$	−1.00000	−0.0873704	−0.0436852	−	0.999045i	$-0.513910\pi$
$131$	−1.00000	−0.0873704	−0.0436852	+	0.999045i	$0.513910\pi$
$132$	0	0
$133$	−2.00000	−0.173422
$134$	−7.00000	−0.604708
$135$	0	0
$136$	−7.00000	−0.600245
$137$	−22.0000	−1.87959	−0.939793	−	0.341743i	$-0.888983\pi$
$137$	−22.0000	−1.87959	−0.939793	+	0.341743i	$0.888983\pi$
$138$	0	0
$139$	10.0000	0.848189	0.424094	−	0.905618i	$-0.360592\pi$
$139$	10.0000	0.848189	0.424094	+	0.905618i	$0.360592\pi$
$140$	4.00000	0.338062
$141$	0	0
$142$	−7.00000	−0.587427
$143$	12.0000	1.00349
$144$	0	0
$145$	4.00000	0.332182
$146$	14.0000	1.15865
$147$	0	0
$148$	2.00000	0.164399
$149$	3.00000	0.245770	0.122885	−	0.992421i	$-0.460785\pi$
$149$	3.00000	0.245770	0.122885	+	0.992421i	$0.460785\pi$
$150$	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
$154$	4.00000	0.322329
$155$	36.0000	2.89159
$156$	0	0
$157$	17.0000	1.35675	0.678374	−	0.734717i	$-0.262685\pi$
$157$	17.0000	1.35675	0.678374	+	0.734717i	$0.262685\pi$
$158$	6.00000	0.477334
$159$	0	0
$160$	4.00000	0.316228
$161$	−1.00000	−0.0788110
$162$	0	0
$163$	−1.00000	−0.0783260	−0.0391630	−	0.999233i	$-0.512469\pi$
$163$	−1.00000	−0.0783260	−0.0391630	+	0.999233i	$0.512469\pi$
$164$	−6.00000	−0.468521
$165$	0	0
$166$	−4.00000	−0.310460
$167$	8.00000	0.619059	0.309529	−	0.950890i	$-0.399829\pi$
$167$	8.00000	0.619059	0.309529	+	0.950890i	$0.399829\pi$
$168$	0	0
$169$	−4.00000	−0.307692
$170$	28.0000	2.14750
$171$	0	0
$172$	11.0000	0.838742
$173$	8.00000	0.608229	0.304114	−	0.952636i	$-0.401639\pi$
$173$	8.00000	0.608229	0.304114	+	0.952636i	$0.401639\pi$
$174$	0	0
$175$	−11.0000	−0.831522
$176$	4.00000	0.301511
$177$	0	0
$178$	−3.00000	−0.224860
$179$	12.0000	0.896922	0.448461	−	0.893802i	$-0.351972\pi$
$179$	12.0000	0.896922	0.448461	+	0.893802i	$0.351972\pi$
$180$	0	0
$181$	−9.00000	−0.668965	−0.334482	−	0.942402i	$-0.608561\pi$
$181$	−9.00000	−0.668965	−0.334482	+	0.942402i	$0.608561\pi$
$182$	3.00000	0.222375
$183$	0	0
$184$	−1.00000	−0.0737210
$185$	−8.00000	−0.588172
$186$	0	0
$187$	28.0000	2.04756
$188$	6.00000	0.437595
$189$	0	0
$190$	8.00000	0.580381
$191$	−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
$193$	5.00000	0.359908	0.179954	−	0.983675i	$-0.442405\pi$
$193$	5.00000	0.359908	0.179954	+	0.983675i	$0.442405\pi$
$194$	8.00000	0.574367
$195$	0	0
$196$	1.00000	0.0714286
$197$	−26.0000	−1.85242	−0.926212	−	0.377004i	$-0.876954\pi$
$197$	−26.0000	−1.85242	−0.926212	+	0.377004i	$0.876954\pi$
$198$	0	0
$199$	−7.00000	−0.496217	−0.248108	−	0.968732i	$-0.579809\pi$
$199$	−7.00000	−0.496217	−0.248108	+	0.968732i	$0.579809\pi$
$200$	−11.0000	−0.777817
$201$	0	0
$202$	10.0000	0.703598
$203$	1.00000	0.0701862
$204$	0	0
$205$	24.0000	1.67623
$206$	−17.0000	−1.18445
$207$	0	0
$208$	3.00000	0.208013
$209$	8.00000	0.553372
$210$	0	0
$211$	−1.00000	−0.0688428	−0.0344214	−	0.999407i	$-0.510959\pi$
$211$	−1.00000	−0.0688428	−0.0344214	+	0.999407i	$0.510959\pi$
$212$	9.00000	0.618123
$213$	0	0
$214$	12.0000	0.820303
$215$	−44.0000	−3.00078
$216$	0	0
$217$	9.00000	0.610960
$218$	−16.0000	−1.08366
$219$	0	0
$220$	−16.0000	−1.07872
$221$	21.0000	1.41261
$222$	0	0
$223$	8.00000	0.535720	0.267860	−	0.963458i	$-0.413684\pi$
$223$	8.00000	0.535720	0.267860	+	0.963458i	$0.413684\pi$
$224$	1.00000	0.0668153
$225$	0	0
$226$	−8.00000	−0.532152
$227$	9.00000	0.597351	0.298675	−	0.954355i	$-0.403455\pi$
$227$	9.00000	0.597351	0.298675	+	0.954355i	$0.403455\pi$
$228$	0	0
$229$	−14.0000	−0.925146	−0.462573	−	0.886581i	$-0.653074\pi$
$229$	−14.0000	−0.925146	−0.462573	+	0.886581i	$0.653074\pi$
$230$	4.00000	0.263752
$231$	0	0
$232$	1.00000	0.0656532
$233$	4.00000	0.262049	0.131024	−	0.991379i	$-0.458173\pi$
$233$	4.00000	0.262049	0.131024	+	0.991379i	$0.458173\pi$
$234$	0	0
$235$	−24.0000	−1.56559
$236$	5.00000	0.325472
$237$	0	0
$238$	7.00000	0.453743
$239$	−12.0000	−0.776215	−0.388108	−	0.921614i	$-0.626871\pi$
$239$	−12.0000	−0.776215	−0.388108	+	0.921614i	$0.626871\pi$
$240$	0	0
$241$	−10.0000	−0.644157	−0.322078	−	0.946713i	$-0.604381\pi$
$241$	−10.0000	−0.644157	−0.322078	+	0.946713i	$0.604381\pi$
$242$	−5.00000	−0.321412
$243$	0	0
$244$	−6.00000	−0.384111
$245$	−4.00000	−0.255551
$246$	0	0
$247$	6.00000	0.381771
$248$	9.00000	0.571501
$249$	0	0
$250$	24.0000	1.51789
$251$	0	0		−	1.00000i	$-0.5\pi$
$251$	0	0			1.00000i	$0.5\pi$
$252$	0	0
$253$	4.00000	0.251478
$254$	0	0
$255$	0	0
$256$	1.00000	0.0625000
$257$	−30.0000	−1.87135	−0.935674	−	0.352865i	$-0.885208\pi$
$257$	−30.0000	−1.87135	−0.935674	+	0.352865i	$0.885208\pi$
$258$	0	0
$259$	−2.00000	−0.124274
$260$	−12.0000	−0.744208
$261$	0	0
$262$	1.00000	0.0617802
$263$	−9.00000	−0.554964	−0.277482	−	0.960731i	$-0.589500\pi$
$263$	−9.00000	−0.554964	−0.277482	+	0.960731i	$0.589500\pi$
$264$	0	0
$265$	−36.0000	−2.21146
$266$	2.00000	0.122628
$267$	0	0
$268$	7.00000	0.427593
$269$	−4.00000	−0.243884	−0.121942	−	0.992537i	$-0.538912\pi$
$269$	−4.00000	−0.243884	−0.121942	+	0.992537i	$0.538912\pi$
$270$	0	0
$271$	3.00000	0.182237	0.0911185	−	0.995840i	$-0.470956\pi$
$271$	3.00000	0.182237	0.0911185	+	0.995840i	$0.470956\pi$
$272$	7.00000	0.424437
$273$	0	0
$274$	22.0000	1.32907
$275$	44.0000	2.65330
$276$	0	0
$277$	8.00000	0.480673	0.240337	−	0.970690i	$-0.422742\pi$
$277$	8.00000	0.480673	0.240337	+	0.970690i	$0.422742\pi$
$278$	−10.0000	−0.599760
$279$	0	0
$280$	−4.00000	−0.239046
$281$	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
$283$	28.0000	1.66443	0.832214	−	0.554455i	$-0.187073\pi$
$283$	28.0000	1.66443	0.832214	+	0.554455i	$0.187073\pi$
$284$	7.00000	0.415374
$285$	0	0
$286$	−12.0000	−0.709575
$287$	6.00000	0.354169
$288$	0	0
$289$	32.0000	1.88235
$290$	−4.00000	−0.234888
$291$	0	0
$292$	−14.0000	−0.819288
$293$	−6.00000	−0.350524	−0.175262	−	0.984522i	$-0.556077\pi$
$293$	−6.00000	−0.350524	−0.175262	+	0.984522i	$0.556077\pi$
$294$	0	0
$295$	−20.0000	−1.16445
$296$	−2.00000	−0.116248
$297$	0	0
$298$	−3.00000	−0.173785
$299$	3.00000	0.173494
$300$	0	0
$301$	−11.0000	−0.634029
$302$	−10.0000	−0.575435
$303$	0	0
$304$	2.00000	0.114708
$305$	24.0000	1.37424
$306$	0	0
$307$	−2.00000	−0.114146	−0.0570730	−	0.998370i	$-0.518177\pi$
$307$	−2.00000	−0.114146	−0.0570730	+	0.998370i	$0.518177\pi$
$308$	−4.00000	−0.227921
$309$	0	0
$310$	−36.0000	−2.04466
$311$	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
$313$	−26.0000	−1.46961	−0.734803	−	0.678280i	$-0.762726\pi$
$313$	−26.0000	−1.46961	−0.734803	+	0.678280i	$0.762726\pi$
$314$	−17.0000	−0.959366
$315$	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
$319$	−4.00000	−0.223957
$320$	−4.00000	−0.223607
$321$	0	0
$322$	1.00000	0.0557278
$323$	14.0000	0.778981
$324$	0	0
$325$	33.0000	1.83051
$326$	1.00000	0.0553849
$327$	0	0
$328$	6.00000	0.331295
$329$	−6.00000	−0.330791
$330$	0	0
$331$	−31.0000	−1.70391	−0.851957	−	0.523612i	$-0.824584\pi$
$331$	−31.0000	−1.70391	−0.851957	+	0.523612i	$0.824584\pi$
$332$	4.00000	0.219529
$333$	0	0
$334$	−8.00000	−0.437741
$335$	−28.0000	−1.52980
$336$	0	0
$337$	27.0000	1.47078	0.735392	−	0.677642i	$-0.236998\pi$
$337$	27.0000	1.47078	0.735392	+	0.677642i	$0.236998\pi$
$338$	4.00000	0.217571
$339$	0	0
$340$	−28.0000	−1.51851
$341$	−36.0000	−1.94951
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	−11.0000	−0.593080
$345$	0	0
$346$	−8.00000	−0.430083
$347$	−18.0000	−0.966291	−0.483145	−	0.875540i	$-0.660506\pi$
$347$	−18.0000	−0.966291	−0.483145	+	0.875540i	$0.660506\pi$
$348$	0	0
$349$	25.0000	1.33822	0.669110	−	0.743164i	$-0.266676\pi$
$349$	25.0000	1.33822	0.669110	+	0.743164i	$0.266676\pi$
$350$	11.0000	0.587975
$351$	0	0
$352$	−4.00000	−0.213201
$353$	−15.0000	−0.798369	−0.399185	−	0.916871i	$-0.630707\pi$
$353$	−15.0000	−0.798369	−0.399185	+	0.916871i	$0.630707\pi$
$354$	0	0
$355$	−28.0000	−1.48609
$356$	3.00000	0.159000
$357$	0	0
$358$	−12.0000	−0.634220
$359$	11.0000	0.580558	0.290279	−	0.956942i	$-0.406252\pi$
$359$	11.0000	0.580558	0.290279	+	0.956942i	$0.406252\pi$
$360$	0	0
$361$	−15.0000	−0.789474
$362$	9.00000	0.473029
$363$	0	0
$364$	−3.00000	−0.157243
$365$	56.0000	2.93117
$366$	0	0
$367$	−25.0000	−1.30499	−0.652495	−	0.757793i	$-0.726278\pi$
$367$	−25.0000	−1.30499	−0.652495	+	0.757793i	$0.726278\pi$
$368$	1.00000	0.0521286
$369$	0	0
$370$	8.00000	0.415900
$371$	−9.00000	−0.467257
$372$	0	0
$373$	−38.0000	−1.96757	−0.983783	−	0.179364i	$-0.942596\pi$
$373$	−38.0000	−1.96757	−0.983783	+	0.179364i	$0.942596\pi$
$374$	−28.0000	−1.44785
$375$	0	0
$376$	−6.00000	−0.309426
$377$	−3.00000	−0.154508
$378$	0	0
$379$	−8.00000	−0.410932	−0.205466	−	0.978664i	$-0.565871\pi$
$379$	−8.00000	−0.410932	−0.205466	+	0.978664i	$0.565871\pi$
$380$	−8.00000	−0.410391
$381$	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
$385$	16.0000	0.815436
$386$	−5.00000	−0.254493
$387$	0	0
$388$	−8.00000	−0.406138
$389$	14.0000	0.709828	0.354914	−	0.934899i	$-0.384510\pi$
$389$	14.0000	0.709828	0.354914	+	0.934899i	$0.384510\pi$
$390$	0	0
$391$	7.00000	0.354005
$392$	−1.00000	−0.0505076
$393$	0	0
$394$	26.0000	1.30986
$395$	24.0000	1.20757
$396$	0	0
$397$	6.00000	0.301131	0.150566	−	0.988600i	$-0.451890\pi$
$397$	6.00000	0.301131	0.150566	+	0.988600i	$0.451890\pi$
$398$	7.00000	0.350878
$399$	0	0
$400$	11.0000	0.550000
$401$	−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
$403$	−27.0000	−1.34497
$404$	−10.0000	−0.497519
$405$	0	0
$406$	−1.00000	−0.0496292
$407$	8.00000	0.396545
$408$	0	0
$409$	2.00000	0.0988936	0.0494468	−	0.998777i	$-0.484254\pi$
$409$	2.00000	0.0988936	0.0494468	+	0.998777i	$0.484254\pi$
$410$	−24.0000	−1.18528
$411$	0	0
$412$	17.0000	0.837530
$413$	−5.00000	−0.246034
$414$	0	0
$415$	−16.0000	−0.785409
$416$	−3.00000	−0.147087
$417$	0	0
$418$	−8.00000	−0.391293
$419$	33.0000	1.61216	0.806078	−	0.591810i	$-0.201586\pi$
$419$	33.0000	1.61216	0.806078	+	0.591810i	$0.201586\pi$
$420$	0	0
$421$	−6.00000	−0.292422	−0.146211	−	0.989253i	$-0.546708\pi$
$421$	−6.00000	−0.292422	−0.146211	+	0.989253i	$0.546708\pi$
$422$	1.00000	0.0486792
$423$	0	0
$424$	−9.00000	−0.437079
$425$	77.0000	3.73505
$426$	0	0
$427$	6.00000	0.290360
$428$	−12.0000	−0.580042
$429$	0	0
$430$	44.0000	2.12187
$431$	−24.0000	−1.15604	−0.578020	−	0.816023i	$-0.696174\pi$
$431$	−24.0000	−1.15604	−0.578020	+	0.816023i	$0.696174\pi$
$432$	0	0
$433$	8.00000	0.384455	0.192228	−	0.981350i	$-0.438429\pi$
$433$	8.00000	0.384455	0.192228	+	0.981350i	$0.438429\pi$
$434$	−9.00000	−0.432014
$435$	0	0
$436$	16.0000	0.766261
$437$	2.00000	0.0956730
$438$	0	0
$439$	9.00000	0.429547	0.214773	−	0.976664i	$-0.431099\pi$
$439$	9.00000	0.429547	0.214773	+	0.976664i	$0.431099\pi$
$440$	16.0000	0.762770
$441$	0	0
$442$	−21.0000	−0.998868
$443$	−26.0000	−1.23530	−0.617649	−	0.786454i	$-0.711915\pi$
$443$	−26.0000	−1.23530	−0.617649	+	0.786454i	$0.711915\pi$
$444$	0	0
$445$	−12.0000	−0.568855
$446$	−8.00000	−0.378811
$447$	0	0
$448$	−1.00000	−0.0472456
$449$	−4.00000	−0.188772	−0.0943858	−	0.995536i	$-0.530089\pi$
$449$	−4.00000	−0.188772	−0.0943858	+	0.995536i	$0.530089\pi$
$450$	0	0
$451$	−24.0000	−1.13012
$452$	8.00000	0.376288
$453$	0	0
$454$	−9.00000	−0.422391
$455$	12.0000	0.562569
$456$	0	0
$457$	−29.0000	−1.35656	−0.678281	−	0.734802i	$-0.737275\pi$
$457$	−29.0000	−1.35656	−0.678281	+	0.734802i	$0.737275\pi$
$458$	14.0000	0.654177
$459$	0	0
$460$	−4.00000	−0.186501
$461$	14.0000	0.652045	0.326023	−	0.945362i	$-0.394291\pi$
$461$	14.0000	0.652045	0.326023	+	0.945362i	$0.394291\pi$
$462$	0	0
$463$	22.0000	1.02243	0.511213	−	0.859454i	$-0.329196\pi$
$463$	22.0000	1.02243	0.511213	+	0.859454i	$0.329196\pi$
$464$	−1.00000	−0.0464238
$465$	0	0
$466$	−4.00000	−0.185296
$467$	8.00000	0.370196	0.185098	−	0.982720i	$-0.440740\pi$
$467$	8.00000	0.370196	0.185098	+	0.982720i	$0.440740\pi$
$468$	0	0
$469$	−7.00000	−0.323230
$470$	24.0000	1.10704
$471$	0	0
$472$	−5.00000	−0.230144
$473$	44.0000	2.02312
$474$	0	0
$475$	22.0000	1.00943
$476$	−7.00000	−0.320844
$477$	0	0
$478$	12.0000	0.548867
$479$	−8.00000	−0.365529	−0.182765	−	0.983157i	$-0.558505\pi$
$479$	−8.00000	−0.365529	−0.182765	+	0.983157i	$0.558505\pi$
$480$	0	0
$481$	6.00000	0.273576
$482$	10.0000	0.455488
$483$	0	0
$484$	5.00000	0.227273
$485$	32.0000	1.45305
$486$	0	0
$487$	26.0000	1.17817	0.589086	−	0.808070i	$-0.299488\pi$
$487$	26.0000	1.17817	0.589086	+	0.808070i	$0.299488\pi$
$488$	6.00000	0.271607
$489$	0	0
$490$	4.00000	0.180702
$491$	−24.0000	−1.08310	−0.541552	−	0.840667i	$-0.682163\pi$
$491$	−24.0000	−1.08310	−0.541552	+	0.840667i	$0.682163\pi$
$492$	0	0
$493$	−7.00000	−0.315264
$494$	−6.00000	−0.269953
$495$	0	0
$496$	−9.00000	−0.404112
$497$	−7.00000	−0.313993
$498$	0	0
$499$	28.0000	1.25345	0.626726	−	0.779240i	$-0.284395\pi$
$499$	28.0000	1.25345	0.626726	+	0.779240i	$0.284395\pi$
$500$	−24.0000	−1.07331
$501$	0	0
$502$	0	0
$503$	24.0000	1.07011	0.535054	−	0.844818i	$-0.320291\pi$
$503$	24.0000	1.07011	0.535054	+	0.844818i	$0.320291\pi$
$504$	0	0
$505$	40.0000	1.77998
$506$	−4.00000	−0.177822
$507$	0	0
$508$	0	0
$509$	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
$511$	14.0000	0.619324
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	30.0000	1.32324
$515$	−68.0000	−2.99644
$516$	0	0
$517$	24.0000	1.05552
$518$	2.00000	0.0878750
$519$	0	0
$520$	12.0000	0.526235
$521$	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
$523$	−16.0000	−0.699631	−0.349816	−	0.936819i	$-0.613756\pi$
$523$	−16.0000	−0.699631	−0.349816	+	0.936819i	$0.613756\pi$
$524$	−1.00000	−0.0436852
$525$	0	0
$526$	9.00000	0.392419
$527$	−63.0000	−2.74432
$528$	0	0
$529$	−22.0000	−0.956522
$530$	36.0000	1.56374
$531$	0	0
$532$	−2.00000	−0.0867110
$533$	−18.0000	−0.779667
$534$	0	0
$535$	48.0000	2.07522
$536$	−7.00000	−0.302354
$537$	0	0
$538$	4.00000	0.172452
$539$	4.00000	0.172292
$540$	0	0
$541$	−36.0000	−1.54776	−0.773880	−	0.633332i	$-0.781687\pi$
$541$	−36.0000	−1.54776	−0.773880	+	0.633332i	$0.781687\pi$
$542$	−3.00000	−0.128861
$543$	0	0
$544$	−7.00000	−0.300123
$545$	−64.0000	−2.74146
$546$	0	0
$547$	36.0000	1.53925	0.769624	−	0.638497i	$-0.220443\pi$
$547$	36.0000	1.53925	0.769624	+	0.638497i	$0.220443\pi$
$548$	−22.0000	−0.939793
$549$	0	0
$550$	−44.0000	−1.87617
$551$	−2.00000	−0.0852029
$552$	0	0
$553$	6.00000	0.255146
$554$	−8.00000	−0.339887
$555$	0	0
$556$	10.0000	0.424094
$557$	−1.00000	−0.0423714	−0.0211857	−	0.999776i	$-0.506744\pi$
$557$	−1.00000	−0.0423714	−0.0211857	+	0.999776i	$0.506744\pi$
$558$	0	0
$559$	33.0000	1.39575
$560$	4.00000	0.169031
$561$	0	0
$562$	−10.0000	−0.421825
$563$	−11.0000	−0.463595	−0.231797	−	0.972764i	$-0.574461\pi$
$563$	−11.0000	−0.463595	−0.231797	+	0.972764i	$0.574461\pi$
$564$	0	0
$565$	−32.0000	−1.34625
$566$	−28.0000	−1.17693
$567$	0	0
$568$	−7.00000	−0.293713
$569$	−12.0000	−0.503066	−0.251533	−	0.967849i	$-0.580935\pi$
$569$	−12.0000	−0.503066	−0.251533	+	0.967849i	$0.580935\pi$
$570$	0	0
$571$	−39.0000	−1.63210	−0.816050	−	0.577982i	$-0.803840\pi$
$571$	−39.0000	−1.63210	−0.816050	+	0.577982i	$0.803840\pi$
$572$	12.0000	0.501745
$573$	0	0
$574$	−6.00000	−0.250435
$575$	11.0000	0.458732
$576$	0	0
$577$	10.0000	0.416305	0.208153	−	0.978096i	$-0.433255\pi$
$577$	10.0000	0.416305	0.208153	+	0.978096i	$0.433255\pi$
$578$	−32.0000	−1.33102
$579$	0	0
$580$	4.00000	0.166091
$581$	−4.00000	−0.165948
$582$	0	0
$583$	36.0000	1.49097
$584$	14.0000	0.579324
$585$	0	0
$586$	6.00000	0.247858
$587$	25.0000	1.03186	0.515930	−	0.856631i	$-0.327446\pi$
$587$	25.0000	1.03186	0.515930	+	0.856631i	$0.327446\pi$
$588$	0	0
$589$	−18.0000	−0.741677
$590$	20.0000	0.823387
$591$	0	0
$592$	2.00000	0.0821995
$593$	18.0000	0.739171	0.369586	−	0.929197i	$-0.379500\pi$
$593$	18.0000	0.739171	0.369586	+	0.929197i	$0.379500\pi$
$594$	0	0
$595$	28.0000	1.14789
$596$	3.00000	0.122885
$597$	0	0
$598$	−3.00000	−0.122679
$599$	−21.0000	−0.858037	−0.429018	−	0.903296i	$-0.641140\pi$
$599$	−21.0000	−0.858037	−0.429018	+	0.903296i	$0.641140\pi$
$600$	0	0
$601$	−28.0000	−1.14214	−0.571072	−	0.820900i	$-0.693472\pi$
$601$	−28.0000	−1.14214	−0.571072	+	0.820900i	$0.693472\pi$
$602$	11.0000	0.448327
$603$	0	0
$604$	10.0000	0.406894
$605$	−20.0000	−0.813116
$606$	0	0
$607$	39.0000	1.58296	0.791481	−	0.611194i	$-0.209311\pi$
$607$	39.0000	1.58296	0.791481	+	0.611194i	$0.209311\pi$
$608$	−2.00000	−0.0811107
$609$	0	0
$610$	−24.0000	−0.971732
$611$	18.0000	0.728202
$612$	0	0
$613$	−6.00000	−0.242338	−0.121169	−	0.992632i	$-0.538664\pi$
$613$	−6.00000	−0.242338	−0.121169	+	0.992632i	$0.538664\pi$
$614$	2.00000	0.0807134
$615$	0	0
$616$	4.00000	0.161165
$617$	−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
$619$	−14.0000	−0.562708	−0.281354	−	0.959604i	$-0.590783\pi$
$619$	−14.0000	−0.562708	−0.281354	+	0.959604i	$0.590783\pi$
$620$	36.0000	1.44579
$621$	0	0
$622$	−8.00000	−0.320771
$623$	−3.00000	−0.120192
$624$	0	0
$625$	41.0000	1.64000
$626$	26.0000	1.03917
$627$	0	0
$628$	17.0000	0.678374
$629$	14.0000	0.558217
$630$	0	0
$631$	−4.00000	−0.159237	−0.0796187	−	0.996825i	$-0.525370\pi$
$631$	−4.00000	−0.159237	−0.0796187	+	0.996825i	$0.525370\pi$
$632$	6.00000	0.238667
$633$	0	0
$634$	−18.0000	−0.714871
$635$	0	0
$636$	0	0
$637$	3.00000	0.118864
$638$	4.00000	0.158362
$639$	0	0
$640$	4.00000	0.158114
$641$	28.0000	1.10593	0.552967	−	0.833203i	$-0.313496\pi$
$641$	28.0000	1.10593	0.552967	+	0.833203i	$0.313496\pi$
$642$	0	0
$643$	−4.00000	−0.157745	−0.0788723	−	0.996885i	$-0.525132\pi$
$643$	−4.00000	−0.157745	−0.0788723	+	0.996885i	$0.525132\pi$
$644$	−1.00000	−0.0394055
$645$	0	0
$646$	−14.0000	−0.550823
$647$	−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
$649$	20.0000	0.785069
$650$	−33.0000	−1.29437
$651$	0	0
$652$	−1.00000	−0.0391630
$653$	−3.00000	−0.117399	−0.0586995	−	0.998276i	$-0.518695\pi$
$653$	−3.00000	−0.117399	−0.0586995	+	0.998276i	$0.518695\pi$
$654$	0	0
$655$	4.00000	0.156293
$656$	−6.00000	−0.234261
$657$	0	0
$658$	6.00000	0.233904
$659$	28.0000	1.09073	0.545363	−	0.838200i	$-0.316392\pi$
$659$	28.0000	1.09073	0.545363	+	0.838200i	$0.316392\pi$
$660$	0	0
$661$	−14.0000	−0.544537	−0.272268	−	0.962221i	$-0.587774\pi$
$661$	−14.0000	−0.544537	−0.272268	+	0.962221i	$0.587774\pi$
$662$	31.0000	1.20485
$663$	0	0
$664$	−4.00000	−0.155230
$665$	8.00000	0.310227
$666$	0	0
$667$	−1.00000	−0.0387202
$668$	8.00000	0.309529
$669$	0	0
$670$	28.0000	1.08173
$671$	−24.0000	−0.926510
$672$	0	0
$673$	23.0000	0.886585	0.443292	−	0.896377i	$-0.353810\pi$
$673$	23.0000	0.886585	0.443292	+	0.896377i	$0.353810\pi$
$674$	−27.0000	−1.04000
$675$	0	0
$676$	−4.00000	−0.153846
$677$	0	0		−	1.00000i	$-0.5\pi$
$677$	0	0			1.00000i	$0.5\pi$
$678$	0	0
$679$	8.00000	0.307012
$680$	28.0000	1.07375
$681$	0	0
$682$	36.0000	1.37851
$683$	48.0000	1.83667	0.918334	−	0.395805i	$-0.129534\pi$
$683$	48.0000	1.83667	0.918334	+	0.395805i	$0.129534\pi$
$684$	0	0
$685$	88.0000	3.36231
$686$	1.00000	0.0381802
$687$	0	0
$688$	11.0000	0.419371
$689$	27.0000	1.02862
$690$	0	0
$691$	−40.0000	−1.52167	−0.760836	−	0.648944i	$-0.775211\pi$
$691$	−40.0000	−1.52167	−0.760836	+	0.648944i	$0.775211\pi$
$692$	8.00000	0.304114
$693$	0	0
$694$	18.0000	0.683271
$695$	−40.0000	−1.51729
$696$	0	0
$697$	−42.0000	−1.59086
$698$	−25.0000	−0.946264
$699$	0	0
$700$	−11.0000	−0.415761
$701$	2.00000	0.0755390	0.0377695	−	0.999286i	$-0.487975\pi$
$701$	2.00000	0.0755390	0.0377695	+	0.999286i	$0.487975\pi$
$702$	0	0
$703$	4.00000	0.150863
$704$	4.00000	0.150756
$705$	0	0
$706$	15.0000	0.564532
$707$	10.0000	0.376089
$708$	0	0
$709$	−40.0000	−1.50223	−0.751116	−	0.660171i	$-0.770484\pi$
$709$	−40.0000	−1.50223	−0.751116	+	0.660171i	$0.770484\pi$
$710$	28.0000	1.05082
$711$	0	0
$712$	−3.00000	−0.112430
$713$	−9.00000	−0.337053
$714$	0	0
$715$	−48.0000	−1.79510
$716$	12.0000	0.448461
$717$	0	0
$718$	−11.0000	−0.410516
$719$	−48.0000	−1.79010	−0.895049	−	0.445968i	$-0.852860\pi$
$719$	−48.0000	−1.79010	−0.895049	+	0.445968i	$0.852860\pi$
$720$	0	0
$721$	−17.0000	−0.633113
$722$	15.0000	0.558242
$723$	0	0
$724$	−9.00000	−0.334482
$725$	−11.0000	−0.408530
$726$	0	0
$727$	−29.0000	−1.07555	−0.537775	−	0.843088i	$-0.680735\pi$
$727$	−29.0000	−1.07555	−0.537775	+	0.843088i	$0.680735\pi$
$728$	3.00000	0.111187
$729$	0	0
$730$	−56.0000	−2.07265
$731$	77.0000	2.84795
$732$	0	0
$733$	−15.0000	−0.554038	−0.277019	−	0.960864i	$-0.589346\pi$
$733$	−15.0000	−0.554038	−0.277019	+	0.960864i	$0.589346\pi$
$734$	25.0000	0.922767
$735$	0	0
$736$	−1.00000	−0.0368605
$737$	28.0000	1.03139
$738$	0	0
$739$	12.0000	0.441427	0.220714	−	0.975339i	$-0.429161\pi$
$739$	12.0000	0.441427	0.220714	+	0.975339i	$0.429161\pi$
$740$	−8.00000	−0.294086
$741$	0	0
$742$	9.00000	0.330400
$743$	−51.0000	−1.87101	−0.935504	−	0.353315i	$-0.885054\pi$
$743$	−51.0000	−1.87101	−0.935504	+	0.353315i	$0.885054\pi$
$744$	0	0
$745$	−12.0000	−0.439646
$746$	38.0000	1.39128
$747$	0	0
$748$	28.0000	1.02378
$749$	12.0000	0.438470
$750$	0	0
$751$	6.00000	0.218943	0.109472	−	0.993990i	$-0.465084\pi$
$751$	6.00000	0.218943	0.109472	+	0.993990i	$0.465084\pi$
$752$	6.00000	0.218797
$753$	0	0
$754$	3.00000	0.109254
$755$	−40.0000	−1.45575
$756$	0	0
$757$	−12.0000	−0.436147	−0.218074	−	0.975932i	$-0.569977\pi$
$757$	−12.0000	−0.436147	−0.218074	+	0.975932i	$0.569977\pi$
$758$	8.00000	0.290573
$759$	0	0
$760$	8.00000	0.290191
$761$	13.0000	0.471250	0.235625	−	0.971844i	$-0.424286\pi$
$761$	13.0000	0.471250	0.235625	+	0.971844i	$0.424286\pi$
$762$	0	0
$763$	−16.0000	−0.579239
$764$	−12.0000	−0.434145
$765$	0	0
$766$	−4.00000	−0.144526
$767$	15.0000	0.541619
$768$	0	0
$769$	4.00000	0.144244	0.0721218	−	0.997396i	$-0.477023\pi$
$769$	4.00000	0.144244	0.0721218	+	0.997396i	$0.477023\pi$
$770$	−16.0000	−0.576600
$771$	0	0
$772$	5.00000	0.179954
$773$	−4.00000	−0.143870	−0.0719350	−	0.997409i	$-0.522917\pi$
$773$	−4.00000	−0.143870	−0.0719350	+	0.997409i	$0.522917\pi$
$774$	0	0
$775$	−99.0000	−3.55618
$776$	8.00000	0.287183
$777$	0	0
$778$	−14.0000	−0.501924
$779$	−12.0000	−0.429945
$780$	0	0
$781$	28.0000	1.00192
$782$	−7.00000	−0.250319
$783$	0	0
$784$	1.00000	0.0357143
$785$	−68.0000	−2.42702
$786$	0	0
$787$	18.0000	0.641631	0.320815	−	0.947142i	$-0.396043\pi$
$787$	18.0000	0.641631	0.320815	+	0.947142i	$0.396043\pi$
$788$	−26.0000	−0.926212
$789$	0	0
$790$	−24.0000	−0.853882
$791$	−8.00000	−0.284447
$792$	0	0
$793$	−18.0000	−0.639199
$794$	−6.00000	−0.212932
$795$	0	0
$796$	−7.00000	−0.248108
$797$	18.0000	0.637593	0.318796	−	0.947823i	$-0.396721\pi$
$797$	18.0000	0.637593	0.318796	+	0.947823i	$0.396721\pi$
$798$	0	0
$799$	42.0000	1.48585
$800$	−11.0000	−0.388909
$801$	0	0
$802$	30.0000	1.05934
$803$	−56.0000	−1.97620
$804$	0	0
$805$	4.00000	0.140981
$806$	27.0000	0.951034
$807$	0	0
$808$	10.0000	0.351799
$809$	32.0000	1.12506	0.562530	−	0.826777i	$-0.309828\pi$
$809$	32.0000	1.12506	0.562530	+	0.826777i	$0.309828\pi$
$810$	0	0
$811$	−26.0000	−0.912983	−0.456492	−	0.889728i	$-0.650894\pi$
$811$	−26.0000	−0.912983	−0.456492	+	0.889728i	$0.650894\pi$
$812$	1.00000	0.0350931
$813$	0	0
$814$	−8.00000	−0.280400
$815$	4.00000	0.140114
$816$	0	0
$817$	22.0000	0.769683
$818$	−2.00000	−0.0699284
$819$	0	0
$820$	24.0000	0.838116
$821$	13.0000	0.453703	0.226852	−	0.973929i	$-0.427157\pi$
$821$	13.0000	0.453703	0.226852	+	0.973929i	$0.427157\pi$
$822$	0	0
$823$	−38.0000	−1.32460	−0.662298	−	0.749240i	$-0.730419\pi$
$823$	−38.0000	−1.32460	−0.662298	+	0.749240i	$0.730419\pi$
$824$	−17.0000	−0.592223
$825$	0	0
$826$	5.00000	0.173972
$827$	−42.0000	−1.46048	−0.730242	−	0.683189i	$-0.760592\pi$
$827$	−42.0000	−1.46048	−0.730242	+	0.683189i	$0.760592\pi$
$828$	0	0
$829$	−26.0000	−0.903017	−0.451509	−	0.892267i	$-0.649114\pi$
$829$	−26.0000	−0.903017	−0.451509	+	0.892267i	$0.649114\pi$
$830$	16.0000	0.555368
$831$	0	0
$832$	3.00000	0.104006
$833$	7.00000	0.242536
$834$	0	0
$835$	−32.0000	−1.10741
$836$	8.00000	0.276686
$837$	0	0
$838$	−33.0000	−1.13997
$839$	10.0000	0.345238	0.172619	−	0.984989i	$-0.444777\pi$
$839$	10.0000	0.345238	0.172619	+	0.984989i	$0.444777\pi$
$840$	0	0
$841$	−28.0000	−0.965517
$842$	6.00000	0.206774
$843$	0	0
$844$	−1.00000	−0.0344214
$845$	16.0000	0.550417
$846$	0	0
$847$	−5.00000	−0.171802
$848$	9.00000	0.309061
$849$	0	0
$850$	−77.0000	−2.64108
$851$	2.00000	0.0685591
$852$	0	0
$853$	53.0000	1.81469	0.907343	−	0.420392i	$-0.138107\pi$
$853$	53.0000	1.81469	0.907343	+	0.420392i	$0.138107\pi$
$854$	−6.00000	−0.205316
$855$	0	0
$856$	12.0000	0.410152
$857$	−27.0000	−0.922302	−0.461151	−	0.887322i	$-0.652563\pi$
$857$	−27.0000	−0.922302	−0.461151	+	0.887322i	$0.652563\pi$
$858$	0	0
$859$	2.00000	0.0682391	0.0341196	−	0.999418i	$-0.489137\pi$
$859$	2.00000	0.0682391	0.0341196	+	0.999418i	$0.489137\pi$
$860$	−44.0000	−1.50039
$861$	0	0
$862$	24.0000	0.817443
$863$	−1.00000	−0.0340404	−0.0170202	−	0.999855i	$-0.505418\pi$
$863$	−1.00000	−0.0340404	−0.0170202	+	0.999855i	$0.505418\pi$
$864$	0	0
$865$	−32.0000	−1.08803
$866$	−8.00000	−0.271851
$867$	0	0
$868$	9.00000	0.305480
$869$	−24.0000	−0.814144
$870$	0	0
$871$	21.0000	0.711558
$872$	−16.0000	−0.541828
$873$	0	0
$874$	−2.00000	−0.0676510
$875$	24.0000	0.811348
$876$	0	0
$877$	−8.00000	−0.270141	−0.135070	−	0.990836i	$-0.543126\pi$
$877$	−8.00000	−0.270141	−0.135070	+	0.990836i	$0.543126\pi$
$878$	−9.00000	−0.303735
$879$	0	0
$880$	−16.0000	−0.539360
$881$	−15.0000	−0.505363	−0.252681	−	0.967550i	$-0.581312\pi$
$881$	−15.0000	−0.505363	−0.252681	+	0.967550i	$0.581312\pi$
$882$	0	0
$883$	−19.0000	−0.639401	−0.319700	−	0.947519i	$-0.603582\pi$
$883$	−19.0000	−0.639401	−0.319700	+	0.947519i	$0.603582\pi$
$884$	21.0000	0.706306
$885$	0	0
$886$	26.0000	0.873487
$887$	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
$889$	0	0
$890$	12.0000	0.402241
$891$	0	0
$892$	8.00000	0.267860
$893$	12.0000	0.401565
$894$	0	0
$895$	−48.0000	−1.60446
$896$	1.00000	0.0334077
$897$	0	0
$898$	4.00000	0.133482
$899$	9.00000	0.300167
$900$	0	0
$901$	63.0000	2.09883
$902$	24.0000	0.799113
$903$	0	0
$904$	−8.00000	−0.266076
$905$	36.0000	1.19668
$906$	0	0
$907$	12.0000	0.398453	0.199227	−	0.979953i	$-0.436157\pi$
$907$	12.0000	0.398453	0.199227	+	0.979953i	$0.436157\pi$
$908$	9.00000	0.298675
$909$	0	0
$910$	−12.0000	−0.397796
$911$	48.0000	1.59031	0.795155	−	0.606406i	$-0.207389\pi$
$911$	48.0000	1.59031	0.795155	+	0.606406i	$0.207389\pi$
$912$	0	0
$913$	16.0000	0.529523
$914$	29.0000	0.959235
$915$	0	0
$916$	−14.0000	−0.462573
$917$	1.00000	0.0330229
$918$	0	0
$919$	22.0000	0.725713	0.362857	−	0.931845i	$-0.381802\pi$
$919$	22.0000	0.725713	0.362857	+	0.931845i	$0.381802\pi$
$920$	4.00000	0.131876
$921$	0	0
$922$	−14.0000	−0.461065
$923$	21.0000	0.691223
$924$	0	0
$925$	22.0000	0.723356
$926$	−22.0000	−0.722965
$927$	0	0
$928$	1.00000	0.0328266
$929$	30.0000	0.984268	0.492134	−	0.870519i	$-0.336217\pi$
$929$	30.0000	0.984268	0.492134	+	0.870519i	$0.336217\pi$
$930$	0	0
$931$	2.00000	0.0655474
$932$	4.00000	0.131024
$933$	0	0
$934$	−8.00000	−0.261768
$935$	−112.000	−3.66279
$936$	0	0
$937$	−52.0000	−1.69877	−0.849383	−	0.527777i	$-0.823026\pi$
$937$	−52.0000	−1.69877	−0.849383	+	0.527777i	$0.823026\pi$
$938$	7.00000	0.228558
$939$	0	0
$940$	−24.0000	−0.782794
$941$	2.00000	0.0651981	0.0325991	−	0.999469i	$-0.489622\pi$
$941$	2.00000	0.0651981	0.0325991	+	0.999469i	$0.489622\pi$
$942$	0	0
$943$	−6.00000	−0.195387
$944$	5.00000	0.162736
$945$	0	0
$946$	−44.0000	−1.43056
$947$	2.00000	0.0649913	0.0324956	−	0.999472i	$-0.489654\pi$
$947$	2.00000	0.0649913	0.0324956	+	0.999472i	$0.489654\pi$
$948$	0	0
$949$	−42.0000	−1.36338
$950$	−22.0000	−0.713774
$951$	0	0
$952$	7.00000	0.226871
$953$	−8.00000	−0.259145	−0.129573	−	0.991570i	$-0.541361\pi$
$953$	−8.00000	−0.259145	−0.129573	+	0.991570i	$0.541361\pi$
$954$	0	0
$955$	48.0000	1.55324
$956$	−12.0000	−0.388108
$957$	0	0
$958$	8.00000	0.258468
$959$	22.0000	0.710417
$960$	0	0
$961$	50.0000	1.61290
$962$	−6.00000	−0.193448
$963$	0	0
$964$	−10.0000	−0.322078
$965$	−20.0000	−0.643823
$966$	0	0
$967$	−40.0000	−1.28631	−0.643157	−	0.765735i	$-0.722376\pi$
$967$	−40.0000	−1.28631	−0.643157	+	0.765735i	$0.722376\pi$
$968$	−5.00000	−0.160706
$969$	0	0
$970$	−32.0000	−1.02746
$971$	15.0000	0.481373	0.240686	−	0.970603i	$-0.422627\pi$
$971$	15.0000	0.481373	0.240686	+	0.970603i	$0.422627\pi$
$972$	0	0
$973$	−10.0000	−0.320585
$974$	−26.0000	−0.833094
$975$	0	0
$976$	−6.00000	−0.192055
$977$	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
$979$	12.0000	0.383522
$980$	−4.00000	−0.127775
$981$	0	0
$982$	24.0000	0.765871
$983$	−36.0000	−1.14822	−0.574111	−	0.818778i	$-0.694652\pi$
$983$	−36.0000	−1.14822	−0.574111	+	0.818778i	$0.694652\pi$
$984$	0	0
$985$	104.000	3.31372
$986$	7.00000	0.222925
$987$	0	0
$988$	6.00000	0.190885
$989$	11.0000	0.349780
$990$	0	0
$991$	26.0000	0.825917	0.412959	−	0.910750i	$-0.364495\pi$
$991$	26.0000	0.825917	0.412959	+	0.910750i	$0.364495\pi$
$992$	9.00000	0.285750
$993$	0	0
$994$	7.00000	0.222027
$995$	28.0000	0.887660
$996$	0	0
$997$	35.0000	1.10846	0.554231	−	0.832363i	$-0.313013\pi$
$997$	35.0000	1.10846	0.554231	+	0.832363i	$0.313013\pi$
$998$	−28.0000	−0.886325
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	378.2.a.a.1.1	✓	1
3.2	odd	2		378.2.a.h.1.1	yes	1
4.3	odd	2		3024.2.a.a.1.1		1
5.4	even	2		9450.2.a.dv.1.1		1
7.6	odd	2		2646.2.a.o.1.1		1
9.2	odd	6		1134.2.f.a.757.1		2
9.4	even	3		1134.2.f.p.379.1		2
9.5	odd	6		1134.2.f.a.379.1		2
9.7	even	3		1134.2.f.p.757.1		2
12.11	even	2		3024.2.a.bd.1.1		1
15.14	odd	2		9450.2.a.bc.1.1		1
21.20	even	2		2646.2.a.p.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
378.2.a.a.1.1	✓	1	1.1	even	1	trivial
378.2.a.h.1.1	yes	1	3.2	odd	2
1134.2.f.a.379.1		2	9.5	odd	6
1134.2.f.a.757.1		2	9.2	odd	6
1134.2.f.p.379.1		2	9.4	even	3
1134.2.f.p.757.1		2	9.7	even	3
2646.2.a.o.1.1		1	7.6	odd	2
2646.2.a.p.1.1		1	21.20	even	2
3024.2.a.a.1.1		1	4.3	odd	2
3024.2.a.bd.1.1		1	12.11	even	2
9450.2.a.bc.1.1		1	15.14	odd	2
9450.2.a.dv.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

Varieties

Fields

Representations

Groups

Database

Properties

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