LMFDB - Embedded newform 7744.2.a.bs.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7744,2,Mod(1,7744)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7744, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("7744.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$7744 = 2^{6} \cdot 11^{2}$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	7744.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:	$61.8361513253$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{2} - x - 1$ x^2 - x - 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$2$
Twist minimal:	no (minimal twist has level 968)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$1.61803$ of defining polynomial
Character	$\chi$	$=$	7744.1

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.23607 q^{3} -2.23607 q^{5} +3.23607 q^{7} +7.47214 q^{9} +1.76393 q^{13} +7.23607 q^{15} -1.00000 q^{17} +5.70820 q^{19} -10.4721 q^{21} -0.763932 q^{23} -14.4721 q^{27} +1.76393 q^{29} -4.76393 q^{31} -7.23607 q^{35} +0.236068 q^{37} -5.70820 q^{39} -7.47214 q^{41} -10.4721 q^{43} -16.7082 q^{45} -5.70820 q^{47} +3.47214 q^{49} +3.23607 q^{51} +13.1803 q^{53} -18.4721 q^{57} -5.52786 q^{59} +14.9443 q^{61} +24.1803 q^{63} -3.94427 q^{65} -0.763932 q^{67} +2.47214 q^{69} +4.00000 q^{71} -3.52786 q^{73} -7.23607 q^{79} +24.4164 q^{81} -12.1803 q^{83} +2.23607 q^{85} -5.70820 q^{87} -12.4164 q^{89} +5.70820 q^{91} +15.4164 q^{93} -12.7639 q^{95} +11.9443 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$2 q - 2 q^{3} + 2 q^{7} + 6 q^{9} + 8 q^{13} + 10 q^{15} - 2 q^{17} - 2 q^{19} - 12 q^{21} - 6 q^{23} - 20 q^{27} + 8 q^{29} - 14 q^{31} - 10 q^{35} - 4 q^{37} + 2 q^{39} - 6 q^{41} - 12 q^{43} - 20 q^{45}+ \cdots + 6 q^{97}+O(q^{100})$ 2 * q - 2 * q^3 + 2 * q^7 + 6 * q^9 + 8 * q^13 + 10 * q^15 - 2 * q^17 - 2 * q^19 - 12 * q^21 - 6 * q^23 - 20 * q^27 + 8 * q^29 - 14 * q^31 - 10 * q^35 - 4 * q^37 + 2 * q^39 - 6 * q^41 - 12 * q^43 - 20 * q^45 + 2 * q^47 - 2 * q^49 + 2 * q^51 + 4 * q^53 - 28 * q^57 - 20 * q^59 + 12 * q^61 + 26 * q^63 + 10 * q^65 - 6 * q^67 - 4 * q^69 + 8 * q^71 - 16 * q^73 - 10 * q^79 + 22 * q^81 - 2 * q^83 + 2 * q^87 + 2 * q^89 - 2 * q^91 + 4 * q^93 - 30 * q^95 + 6 * q^97

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

)

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

$2$	0	0
$2$	0	0
$3$	−3.23607	−1.86834	−0.934172	−	0.356822i	$-0.883860\pi$
$3$	−3.23607	−1.86834	−0.934172	+	0.356822i	$0.883860\pi$
$4$	0	0
$5$	−2.23607	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$5$	−2.23607	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$6$	0	0
$7$	3.23607	1.22312	0.611559	−	0.791199i	$-0.290543\pi$
$7$	3.23607	1.22312	0.611559	+	0.791199i	$0.290543\pi$
$8$	0	0
$9$	7.47214	2.49071
$10$	0	0
$11$	0	0
$11$	0	0
$12$	0	0
$13$	1.76393	0.489227	0.244613	−	0.969621i	$-0.421339\pi$
$13$	1.76393	0.489227	0.244613	+	0.969621i	$0.421339\pi$
$14$	0	0
$15$	7.23607	1.86834
$16$	0	0
$17$	−1.00000	−0.242536	−0.121268	−	0.992620i	$-0.538696\pi$
$17$	−1.00000	−0.242536	−0.121268	+	0.992620i	$0.538696\pi$
$18$	0	0
$19$	5.70820	1.30955	0.654776	−	0.755823i	$-0.272763\pi$
$19$	5.70820	1.30955	0.654776	+	0.755823i	$0.272763\pi$
$20$	0	0
$21$	−10.4721	−2.28521
$22$	0	0
$23$	−0.763932	−0.159291	−0.0796454	−	0.996823i	$-0.525379\pi$
$23$	−0.763932	−0.159291	−0.0796454	+	0.996823i	$0.525379\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−14.4721	−2.78516
$28$	0	0
$29$	1.76393	0.327554	0.163777	−	0.986497i	$-0.447632\pi$
$29$	1.76393	0.327554	0.163777	+	0.986497i	$0.447632\pi$
$30$	0	0
$31$	−4.76393	−0.855627	−0.427814	−	0.903867i	$-0.640716\pi$
$31$	−4.76393	−0.855627	−0.427814	+	0.903867i	$0.640716\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−7.23607	−1.22312
$36$	0	0
$37$	0.236068	0.0388093	0.0194047	−	0.999812i	$-0.493823\pi$
$37$	0.236068	0.0388093	0.0194047	+	0.999812i	$0.493823\pi$
$38$	0	0
$39$	−5.70820	−0.914044
$40$	0	0
$41$	−7.47214	−1.16695	−0.583476	−	0.812131i	$-0.698308\pi$
$41$	−7.47214	−1.16695	−0.583476	+	0.812131i	$0.698308\pi$
$42$	0	0
$43$	−10.4721	−1.59699	−0.798493	−	0.602004i	$-0.794369\pi$
$43$	−10.4721	−1.59699	−0.798493	+	0.602004i	$0.794369\pi$
$44$	0	0
$45$	−16.7082	−2.49071
$46$	0	0
$47$	−5.70820	−0.832627	−0.416314	−	0.909221i	$-0.636678\pi$
$47$	−5.70820	−0.832627	−0.416314	+	0.909221i	$0.636678\pi$
$48$	0	0
$49$	3.47214	0.496019
$50$	0	0
$51$	3.23607	0.453140
$52$	0	0
$53$	13.1803	1.81046	0.905229	−	0.424923i	$-0.139699\pi$
$53$	13.1803	1.81046	0.905229	+	0.424923i	$0.139699\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−18.4721	−2.44669
$58$	0	0
$59$	−5.52786	−0.719667	−0.359833	−	0.933017i	$-0.617166\pi$
$59$	−5.52786	−0.719667	−0.359833	+	0.933017i	$0.617166\pi$
$60$	0	0
$61$	14.9443	1.91342	0.956709	−	0.291046i	$-0.0940034\pi$
$61$	14.9443	1.91342	0.956709	+	0.291046i	$0.0940034\pi$
$62$	0	0
$63$	24.1803	3.04644
$64$	0	0
$65$	−3.94427	−0.489227
$66$	0	0
$67$	−0.763932	−0.0933292	−0.0466646	−	0.998911i	$-0.514859\pi$
$67$	−0.763932	−0.0933292	−0.0466646	+	0.998911i	$0.514859\pi$
$68$	0	0
$69$	2.47214	0.297610
$70$	0	0
$71$	4.00000	0.474713	0.237356	−	0.971423i	$-0.423719\pi$
$71$	4.00000	0.474713	0.237356	+	0.971423i	$0.423719\pi$
$72$	0	0
$73$	−3.52786	−0.412905	−0.206453	−	0.978457i	$-0.566192\pi$
$73$	−3.52786	−0.412905	−0.206453	+	0.978457i	$0.566192\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−7.23607	−0.814121	−0.407061	−	0.913401i	$-0.633446\pi$
$79$	−7.23607	−0.814121	−0.407061	+	0.913401i	$0.633446\pi$
$80$	0	0
$81$	24.4164	2.71293
$82$	0	0
$83$	−12.1803	−1.33697	−0.668483	−	0.743727i	$-0.733056\pi$
$83$	−12.1803	−1.33697	−0.668483	+	0.743727i	$0.733056\pi$
$84$	0	0
$85$	2.23607	0.242536
$86$	0	0
$87$	−5.70820	−0.611984
$88$	0	0
$89$	−12.4164	−1.31614	−0.658068	−	0.752958i	$-0.728626\pi$
$89$	−12.4164	−1.31614	−0.658068	+	0.752958i	$0.728626\pi$
$90$	0	0
$91$	5.70820	0.598382
$92$	0	0
$93$	15.4164	1.59861
$94$	0	0
$95$	−12.7639	−1.30955
$96$	0	0
$97$	11.9443	1.21276	0.606379	−	0.795176i	$-0.292622\pi$
$97$	11.9443	1.21276	0.606379	+	0.795176i	$0.292622\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	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
$103$	0	0		−	1.00000i	$-0.5\pi$
$103$	0	0			1.00000i	$0.5\pi$
$104$	0	0
$105$	23.4164	2.28521
$106$	0	0
$107$	−1.70820	−0.165138	−0.0825692	−	0.996585i	$-0.526313\pi$
$107$	−1.70820	−0.165138	−0.0825692	+	0.996585i	$0.526313\pi$
$108$	0	0
$109$	−6.23607	−0.597307	−0.298653	−	0.954362i	$-0.596537\pi$
$109$	−6.23607	−0.597307	−0.298653	+	0.954362i	$0.596537\pi$
$110$	0	0
$111$	−0.763932	−0.0725092
$112$	0	0
$113$	19.9443	1.87620	0.938100	−	0.346366i	$-0.112584\pi$
$113$	19.9443	1.87620	0.938100	+	0.346366i	$0.112584\pi$
$114$	0	0
$115$	1.70820	0.159291
$116$	0	0
$117$	13.1803	1.21852
$118$	0	0
$119$	−3.23607	−0.296650
$120$	0	0
$121$	0	0
$122$	0	0
$123$	24.1803	2.18027
$124$	0	0
$125$	11.1803	1.00000
$126$	0	0
$127$	18.4721	1.63914	0.819569	−	0.572981i	$-0.194213\pi$
$127$	18.4721	1.63914	0.819569	+	0.572981i	$0.194213\pi$
$128$	0	0
$129$	33.8885	2.98372
$130$	0	0
$131$	−8.00000	−0.698963	−0.349482	−	0.936943i	$-0.613642\pi$
$131$	−8.00000	−0.698963	−0.349482	+	0.936943i	$0.613642\pi$
$132$	0	0
$133$	18.4721	1.60174
$134$	0	0
$135$	32.3607	2.78516
$136$	0	0
$137$	7.52786	0.643149	0.321574	−	0.946884i	$-0.395788\pi$
$137$	7.52786	0.643149	0.321574	+	0.946884i	$0.395788\pi$
$138$	0	0
$139$	−15.2361	−1.29231	−0.646153	−	0.763208i	$-0.723623\pi$
$139$	−15.2361	−1.29231	−0.646153	+	0.763208i	$0.723623\pi$
$140$	0	0
$141$	18.4721	1.55563
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−3.94427	−0.327554
$146$	0	0
$147$	−11.2361	−0.926735
$148$	0	0
$149$	−10.2361	−0.838571	−0.419286	−	0.907854i	$-0.637719\pi$
$149$	−10.2361	−0.838571	−0.419286	+	0.907854i	$0.637719\pi$
$150$	0	0
$151$	−7.41641	−0.603539	−0.301769	−	0.953381i	$-0.597577\pi$
$151$	−7.41641	−0.603539	−0.301769	+	0.953381i	$0.597577\pi$
$152$	0	0
$153$	−7.47214	−0.604086
$154$	0	0
$155$	10.6525	0.855627
$156$	0	0
$157$	12.4721	0.995385	0.497692	−	0.867354i	$-0.334181\pi$
$157$	12.4721	0.995385	0.497692	+	0.867354i	$0.334181\pi$
$158$	0	0
$159$	−42.6525	−3.38256
$160$	0	0
$161$	−2.47214	−0.194832
$162$	0	0
$163$	5.70820	0.447101	0.223551	−	0.974692i	$-0.428235\pi$
$163$	5.70820	0.447101	0.223551	+	0.974692i	$0.428235\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	11.4164	0.883428	0.441714	−	0.897156i	$-0.354371\pi$
$167$	11.4164	0.883428	0.441714	+	0.897156i	$0.354371\pi$
$168$	0	0
$169$	−9.88854	−0.760657
$170$	0	0
$171$	42.6525	3.26172
$172$	0	0
$173$	−18.9443	−1.44031	−0.720153	−	0.693815i	$-0.755928\pi$
$173$	−18.9443	−1.44031	−0.720153	+	0.693815i	$0.755928\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	17.8885	1.34459
$178$	0	0
$179$	−2.47214	−0.184776	−0.0923881	−	0.995723i	$-0.529450\pi$
$179$	−2.47214	−0.184776	−0.0923881	+	0.995723i	$0.529450\pi$
$180$	0	0
$181$	−12.7082	−0.944593	−0.472297	−	0.881440i	$-0.656575\pi$
$181$	−12.7082	−0.944593	−0.472297	+	0.881440i	$0.656575\pi$
$182$	0	0
$183$	−48.3607	−3.57492
$184$	0	0
$185$	−0.527864	−0.0388093
$186$	0	0
$187$	0	0
$188$	0	0
$189$	−46.8328	−3.40659
$190$	0	0
$191$	−5.52786	−0.399982	−0.199991	−	0.979798i	$-0.564091\pi$
$191$	−5.52786	−0.399982	−0.199991	+	0.979798i	$0.564091\pi$
$192$	0	0
$193$	3.94427	0.283915	0.141957	−	0.989873i	$-0.454660\pi$
$193$	3.94427	0.283915	0.141957	+	0.989873i	$0.454660\pi$
$194$	0	0
$195$	12.7639	0.914044
$196$	0	0
$197$	−17.6525	−1.25769	−0.628843	−	0.777532i	$-0.716471\pi$
$197$	−17.6525	−1.25769	−0.628843	+	0.777532i	$0.716471\pi$
$198$	0	0
$199$	18.4721	1.30945	0.654727	−	0.755865i	$-0.272783\pi$
$199$	18.4721	1.30945	0.654727	+	0.755865i	$0.272783\pi$
$200$	0	0
$201$	2.47214	0.174371
$202$	0	0
$203$	5.70820	0.400637
$204$	0	0
$205$	16.7082	1.16695
$206$	0	0
$207$	−5.70820	−0.396748
$208$	0	0
$209$	0	0
$210$	0	0
$211$	7.05573	0.485736	0.242868	−	0.970059i	$-0.421912\pi$
$211$	7.05573	0.485736	0.242868	+	0.970059i	$0.421912\pi$
$212$	0	0
$213$	−12.9443	−0.886927
$214$	0	0
$215$	23.4164	1.59699
$216$	0	0
$217$	−15.4164	−1.04653
$218$	0	0
$219$	11.4164	0.771449
$220$	0	0
$221$	−1.76393	−0.118655
$222$	0	0
$223$	−8.94427	−0.598953	−0.299476	−	0.954104i	$-0.596812\pi$
$223$	−8.94427	−0.598953	−0.299476	+	0.954104i	$0.596812\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	2.47214	0.164081	0.0820407	−	0.996629i	$-0.473856\pi$
$227$	2.47214	0.164081	0.0820407	+	0.996629i	$0.473856\pi$
$228$	0	0
$229$	2.70820	0.178963	0.0894816	−	0.995988i	$-0.471479\pi$
$229$	2.70820	0.178963	0.0894816	+	0.995988i	$0.471479\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	11.0000	0.720634	0.360317	−	0.932830i	$-0.382669\pi$
$233$	11.0000	0.720634	0.360317	+	0.932830i	$0.382669\pi$
$234$	0	0
$235$	12.7639	0.832627
$236$	0	0
$237$	23.4164	1.52106
$238$	0	0
$239$	−20.1803	−1.30536	−0.652679	−	0.757635i	$-0.726355\pi$
$239$	−20.1803	−1.30536	−0.652679	+	0.757635i	$0.726355\pi$
$240$	0	0
$241$	−3.52786	−0.227250	−0.113625	−	0.993524i	$-0.536246\pi$
$241$	−3.52786	−0.227250	−0.113625	+	0.993524i	$0.536246\pi$
$242$	0	0
$243$	−35.5967	−2.28353
$244$	0	0
$245$	−7.76393	−0.496019
$246$	0	0
$247$	10.0689	0.640668
$248$	0	0
$249$	39.4164	2.49791
$250$	0	0
$251$	−20.7639	−1.31061	−0.655304	−	0.755365i	$-0.727459\pi$
$251$	−20.7639	−1.31061	−0.655304	+	0.755365i	$0.727459\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	−7.23607	−0.453140
$256$	0	0
$257$	−14.5279	−0.906223	−0.453112	−	0.891454i	$-0.649686\pi$
$257$	−14.5279	−0.906223	−0.453112	+	0.891454i	$0.649686\pi$
$258$	0	0
$259$	0.763932	0.0474684
$260$	0	0
$261$	13.1803	0.815843
$262$	0	0
$263$	6.65248	0.410209	0.205105	−	0.978740i	$-0.434247\pi$
$263$	6.65248	0.410209	0.205105	+	0.978740i	$0.434247\pi$
$264$	0	0
$265$	−29.4721	−1.81046
$266$	0	0
$267$	40.1803	2.45900
$268$	0	0
$269$	−16.7082	−1.01872	−0.509359	−	0.860554i	$-0.670117\pi$
$269$	−16.7082	−1.01872	−0.509359	+	0.860554i	$0.670117\pi$
$270$	0	0
$271$	29.8885	1.81560	0.907800	−	0.419404i	$-0.137761\pi$
$271$	29.8885	1.81560	0.907800	+	0.419404i	$0.137761\pi$
$272$	0	0
$273$	−18.4721	−1.11798
$274$	0	0
$275$	0	0
$276$	0	0
$277$	26.1246	1.56968	0.784838	−	0.619701i	$-0.212746\pi$
$277$	26.1246	1.56968	0.784838	+	0.619701i	$0.212746\pi$
$278$	0	0
$279$	−35.5967	−2.13112
$280$	0	0
$281$	15.5279	0.926315	0.463157	−	0.886276i	$-0.346716\pi$
$281$	15.5279	0.926315	0.463157	+	0.886276i	$0.346716\pi$
$282$	0	0
$283$	−11.4164	−0.678635	−0.339318	−	0.940672i	$-0.610196\pi$
$283$	−11.4164	−0.678635	−0.339318	+	0.940672i	$0.610196\pi$
$284$	0	0
$285$	41.3050	2.44669
$286$	0	0
$287$	−24.1803	−1.42732
$288$	0	0
$289$	−16.0000	−0.941176
$290$	0	0
$291$	−38.6525	−2.26585
$292$	0	0
$293$	−5.29180	−0.309150	−0.154575	−	0.987981i	$-0.549401\pi$
$293$	−5.29180	−0.309150	−0.154575	+	0.987981i	$0.549401\pi$
$294$	0	0
$295$	12.3607	0.719667
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−1.34752	−0.0779293
$300$	0	0
$301$	−33.8885	−1.95330
$302$	0	0
$303$	−6.47214	−0.371814
$304$	0	0
$305$	−33.4164	−1.91342
$306$	0	0
$307$	−24.1803	−1.38004	−0.690022	−	0.723788i	$-0.742399\pi$
$307$	−24.1803	−1.38004	−0.690022	+	0.723788i	$0.742399\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	13.5279	0.767095	0.383547	−	0.923521i	$-0.374702\pi$
$311$	13.5279	0.767095	0.383547	+	0.923521i	$0.374702\pi$
$312$	0	0
$313$	7.94427	0.449037	0.224518	−	0.974470i	$-0.427919\pi$
$313$	7.94427	0.449037	0.224518	+	0.974470i	$0.427919\pi$
$314$	0	0
$315$	−54.0689	−3.04644
$316$	0	0
$317$	−15.8885	−0.892390	−0.446195	−	0.894936i	$-0.647221\pi$
$317$	−15.8885	−0.892390	−0.446195	+	0.894936i	$0.647221\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	5.52786	0.308535
$322$	0	0
$323$	−5.70820	−0.317613
$324$	0	0
$325$	0	0
$326$	0	0
$327$	20.1803	1.11598
$328$	0	0
$329$	−18.4721	−1.01840
$330$	0	0
$331$	−6.47214	−0.355741	−0.177870	−	0.984054i	$-0.556921\pi$
$331$	−6.47214	−0.355741	−0.177870	+	0.984054i	$0.556921\pi$
$332$	0	0
$333$	1.76393	0.0966629
$334$	0	0
$335$	1.70820	0.0933292
$336$	0	0
$337$	6.41641	0.349524	0.174762	−	0.984611i	$-0.444084\pi$
$337$	6.41641	0.349524	0.174762	+	0.984611i	$0.444084\pi$
$338$	0	0
$339$	−64.5410	−3.50539
$340$	0	0
$341$	0	0
$342$	0	0
$343$	−11.4164	−0.616428
$344$	0	0
$345$	−5.52786	−0.297610
$346$	0	0
$347$	11.4164	0.612865	0.306432	−	0.951892i	$-0.400865\pi$
$347$	11.4164	0.612865	0.306432	+	0.951892i	$0.400865\pi$
$348$	0	0
$349$	27.6525	1.48020	0.740102	−	0.672495i	$-0.234777\pi$
$349$	27.6525	1.48020	0.740102	+	0.672495i	$0.234777\pi$
$350$	0	0
$351$	−25.5279	−1.36258
$352$	0	0
$353$	23.0000	1.22417	0.612083	−	0.790793i	$-0.290332\pi$
$353$	23.0000	1.22417	0.612083	+	0.790793i	$0.290332\pi$
$354$	0	0
$355$	−8.94427	−0.474713
$356$	0	0
$357$	10.4721	0.554244
$358$	0	0
$359$	−12.7639	−0.673655	−0.336827	−	0.941566i	$-0.609354\pi$
$359$	−12.7639	−0.673655	−0.336827	+	0.941566i	$0.609354\pi$
$360$	0	0
$361$	13.5836	0.714926
$362$	0	0
$363$	0	0
$364$	0	0
$365$	7.88854	0.412905
$366$	0	0
$367$	4.18034	0.218212	0.109106	−	0.994030i	$-0.465201\pi$
$367$	4.18034	0.218212	0.109106	+	0.994030i	$0.465201\pi$
$368$	0	0
$369$	−55.8328	−2.90654
$370$	0	0
$371$	42.6525	2.21441
$372$	0	0
$373$	0.111456	0.00577098	0.00288549	−	0.999996i	$-0.499082\pi$
$373$	0.111456	0.00577098	0.00288549	+	0.999996i	$0.499082\pi$
$374$	0	0
$375$	−36.1803	−1.86834
$376$	0	0
$377$	3.11146	0.160248
$378$	0	0
$379$	−3.05573	−0.156962	−0.0784811	−	0.996916i	$-0.525007\pi$
$379$	−3.05573	−0.156962	−0.0784811	+	0.996916i	$0.525007\pi$
$380$	0	0
$381$	−59.7771	−3.06247
$382$	0	0
$383$	−19.4164	−0.992132	−0.496066	−	0.868285i	$-0.665223\pi$
$383$	−19.4164	−0.992132	−0.496066	+	0.868285i	$0.665223\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−78.2492	−3.97763
$388$	0	0
$389$	21.7639	1.10348	0.551738	−	0.834018i	$-0.313965\pi$
$389$	21.7639	1.10348	0.551738	+	0.834018i	$0.313965\pi$
$390$	0	0
$391$	0.763932	0.0386337
$392$	0	0
$393$	25.8885	1.30590
$394$	0	0
$395$	16.1803	0.814121
$396$	0	0
$397$	20.2361	1.01562	0.507810	−	0.861469i	$-0.330455\pi$
$397$	20.2361	1.01562	0.507810	+	0.861469i	$0.330455\pi$
$398$	0	0
$399$	−59.7771	−2.99260
$400$	0	0
$401$	−14.5279	−0.725487	−0.362743	−	0.931889i	$-0.618160\pi$
$401$	−14.5279	−0.725487	−0.362743	+	0.931889i	$0.618160\pi$
$402$	0	0
$403$	−8.40325	−0.418596
$404$	0	0
$405$	−54.5967	−2.71293
$406$	0	0
$407$	0	0
$408$	0	0
$409$	28.3050	1.39959	0.699795	−	0.714344i	$-0.253275\pi$
$409$	28.3050	1.39959	0.699795	+	0.714344i	$0.253275\pi$
$410$	0	0
$411$	−24.3607	−1.20162
$412$	0	0
$413$	−17.8885	−0.880238
$414$	0	0
$415$	27.2361	1.33697
$416$	0	0
$417$	49.3050	2.41447
$418$	0	0
$419$	−24.1803	−1.18129	−0.590643	−	0.806933i	$-0.701126\pi$
$419$	−24.1803	−1.18129	−0.590643	+	0.806933i	$0.701126\pi$
$420$	0	0
$421$	8.81966	0.429844	0.214922	−	0.976631i	$-0.431050\pi$
$421$	8.81966	0.429844	0.214922	+	0.976631i	$0.431050\pi$
$422$	0	0
$423$	−42.6525	−2.07383
$424$	0	0
$425$	0	0
$426$	0	0
$427$	48.3607	2.34034
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−21.3050	−1.02622	−0.513112	−	0.858322i	$-0.671507\pi$
$431$	−21.3050	−1.02622	−0.513112	+	0.858322i	$0.671507\pi$
$432$	0	0
$433$	−9.00000	−0.432512	−0.216256	−	0.976337i	$-0.569385\pi$
$433$	−9.00000	−0.432512	−0.216256	+	0.976337i	$0.569385\pi$
$434$	0	0
$435$	12.7639	0.611984
$436$	0	0
$437$	−4.36068	−0.208600
$438$	0	0
$439$	−12.7639	−0.609189	−0.304595	−	0.952482i	$-0.598521\pi$
$439$	−12.7639	−0.609189	−0.304595	+	0.952482i	$0.598521\pi$
$440$	0	0
$441$	25.9443	1.23544
$442$	0	0
$443$	−27.4164	−1.30259	−0.651296	−	0.758823i	$-0.725775\pi$
$443$	−27.4164	−1.30259	−0.651296	+	0.758823i	$0.725775\pi$
$444$	0	0
$445$	27.7639	1.31614
$446$	0	0
$447$	33.1246	1.56674
$448$	0	0
$449$	−26.8885	−1.26895	−0.634474	−	0.772944i	$-0.718783\pi$
$449$	−26.8885	−1.26895	−0.634474	+	0.772944i	$0.718783\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	24.0000	1.12762
$454$	0	0
$455$	−12.7639	−0.598382
$456$	0	0
$457$	−25.9443	−1.21362	−0.606811	−	0.794846i	$-0.707551\pi$
$457$	−25.9443	−1.21362	−0.606811	+	0.794846i	$0.707551\pi$
$458$	0	0
$459$	14.4721	0.675501
$460$	0	0
$461$	−35.1803	−1.63851	−0.819256	−	0.573428i	$-0.805613\pi$
$461$	−35.1803	−1.63851	−0.819256	+	0.573428i	$0.805613\pi$
$462$	0	0
$463$	−34.8328	−1.61882	−0.809409	−	0.587245i	$-0.800212\pi$
$463$	−34.8328	−1.61882	−0.809409	+	0.587245i	$0.800212\pi$
$464$	0	0
$465$	−34.4721	−1.59861
$466$	0	0
$467$	−22.4721	−1.03989	−0.519943	−	0.854201i	$-0.674047\pi$
$467$	−22.4721	−1.03989	−0.519943	+	0.854201i	$0.674047\pi$
$468$	0	0
$469$	−2.47214	−0.114153
$470$	0	0
$471$	−40.3607	−1.85972
$472$	0	0
$473$	0	0
$474$	0	0
$475$	0	0
$476$	0	0
$477$	98.4853	4.50933
$478$	0	0
$479$	−22.8328	−1.04326	−0.521629	−	0.853172i	$-0.674675\pi$
$479$	−22.8328	−1.04326	−0.521629	+	0.853172i	$0.674675\pi$
$480$	0	0
$481$	0.416408	0.0189866
$482$	0	0
$483$	8.00000	0.364013
$484$	0	0
$485$	−26.7082	−1.21276
$486$	0	0
$487$	19.5967	0.888013	0.444007	−	0.896023i	$-0.353557\pi$
$487$	19.5967	0.888013	0.444007	+	0.896023i	$0.353557\pi$
$488$	0	0
$489$	−18.4721	−0.835339
$490$	0	0
$491$	24.1803	1.09124	0.545622	−	0.838032i	$-0.316294\pi$
$491$	24.1803	1.09124	0.545622	+	0.838032i	$0.316294\pi$
$492$	0	0
$493$	−1.76393	−0.0794435
$494$	0	0
$495$	0	0
$496$	0	0
$497$	12.9443	0.580630
$498$	0	0
$499$	41.3050	1.84906	0.924532	−	0.381105i	$-0.124456\pi$
$499$	41.3050	1.84906	0.924532	+	0.381105i	$0.124456\pi$
$500$	0	0
$501$	−36.9443	−1.65055
$502$	0	0
$503$	−31.5967	−1.40883	−0.704415	−	0.709789i	$-0.748790\pi$
$503$	−31.5967	−1.40883	−0.704415	+	0.709789i	$0.748790\pi$
$504$	0	0
$505$	−4.47214	−0.199007
$506$	0	0
$507$	32.0000	1.42117
$508$	0	0
$509$	−1.05573	−0.0467943	−0.0233972	−	0.999726i	$-0.507448\pi$
$509$	−1.05573	−0.0467943	−0.0233972	+	0.999726i	$0.507448\pi$
$510$	0	0
$511$	−11.4164	−0.505032
$512$	0	0
$513$	−82.6099	−3.64732
$514$	0	0
$515$	0	0
$516$	0	0
$517$	0	0
$518$	0	0
$519$	61.3050	2.69099
$520$	0	0
$521$	−13.4164	−0.587784	−0.293892	−	0.955839i	$-0.594951\pi$
$521$	−13.4164	−0.587784	−0.293892	+	0.955839i	$0.594951\pi$
$522$	0	0
$523$	0	0		−	1.00000i	$-0.5\pi$
$523$	0	0			1.00000i	$0.5\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	4.76393	0.207520
$528$	0	0
$529$	−22.4164	−0.974626
$530$	0	0
$531$	−41.3050	−1.79248
$532$	0	0
$533$	−13.1803	−0.570904
$534$	0	0
$535$	3.81966	0.165138
$536$	0	0
$537$	8.00000	0.345225
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−3.52786	−0.151675	−0.0758374	−	0.997120i	$-0.524163\pi$
$541$	−3.52786	−0.151675	−0.0758374	+	0.997120i	$0.524163\pi$
$542$	0	0
$543$	41.1246	1.76483
$544$	0	0
$545$	13.9443	0.597307
$546$	0	0
$547$	18.4721	0.789812	0.394906	−	0.918722i	$-0.370777\pi$
$547$	18.4721	0.789812	0.394906	+	0.918722i	$0.370777\pi$
$548$	0	0
$549$	111.666	4.76577
$550$	0	0
$551$	10.0689	0.428949
$552$	0	0
$553$	−23.4164	−0.995767
$554$	0	0
$555$	1.70820	0.0725092
$556$	0	0
$557$	35.8885	1.52065	0.760323	−	0.649545i	$-0.225041\pi$
$557$	35.8885	1.52065	0.760323	+	0.649545i	$0.225041\pi$
$558$	0	0
$559$	−18.4721	−0.781288
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−19.8197	−0.835299	−0.417650	−	0.908608i	$-0.637146\pi$
$563$	−19.8197	−0.835299	−0.417650	+	0.908608i	$0.637146\pi$
$564$	0	0
$565$	−44.5967	−1.87620
$566$	0	0
$567$	79.0132	3.31824
$568$	0	0
$569$	−26.3607	−1.10510	−0.552549	−	0.833481i	$-0.686345\pi$
$569$	−26.3607	−1.10510	−0.552549	+	0.833481i	$0.686345\pi$
$570$	0	0
$571$	12.7639	0.534154	0.267077	−	0.963675i	$-0.413942\pi$
$571$	12.7639	0.534154	0.267077	+	0.963675i	$0.413942\pi$
$572$	0	0
$573$	17.8885	0.747305
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−34.3050	−1.42813	−0.714067	−	0.700077i	$-0.753149\pi$
$577$	−34.3050	−1.42813	−0.714067	+	0.700077i	$0.753149\pi$
$578$	0	0
$579$	−12.7639	−0.530451
$580$	0	0
$581$	−39.4164	−1.63527
$582$	0	0
$583$	0	0
$584$	0	0
$585$	−29.4721	−1.21852
$586$	0	0
$587$	−1.34752	−0.0556183	−0.0278091	−	0.999613i	$-0.508853\pi$
$587$	−1.34752	−0.0556183	−0.0278091	+	0.999613i	$0.508853\pi$
$588$	0	0
$589$	−27.1935	−1.12049
$590$	0	0
$591$	57.1246	2.34979
$592$	0	0
$593$	22.4164	0.920532	0.460266	−	0.887781i	$-0.347754\pi$
$593$	22.4164	0.920532	0.460266	+	0.887781i	$0.347754\pi$
$594$	0	0
$595$	7.23607	0.296650
$596$	0	0
$597$	−59.7771	−2.44651
$598$	0	0
$599$	−5.70820	−0.233231	−0.116615	−	0.993177i	$-0.537205\pi$
$599$	−5.70820	−0.233231	−0.116615	+	0.993177i	$0.537205\pi$
$600$	0	0
$601$	−32.7771	−1.33701	−0.668503	−	0.743710i	$-0.733064\pi$
$601$	−32.7771	−1.33701	−0.668503	+	0.743710i	$0.733064\pi$
$602$	0	0
$603$	−5.70820	−0.232456
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−22.6525	−0.919436	−0.459718	−	0.888065i	$-0.652049\pi$
$607$	−22.6525	−0.919436	−0.459718	+	0.888065i	$0.652049\pi$
$608$	0	0
$609$	−18.4721	−0.748529
$610$	0	0
$611$	−10.0689	−0.407343
$612$	0	0
$613$	−46.5967	−1.88202	−0.941012	−	0.338372i	$-0.890124\pi$
$613$	−46.5967	−1.88202	−0.941012	+	0.338372i	$0.890124\pi$
$614$	0	0
$615$	−54.0689	−2.18027
$616$	0	0
$617$	15.3607	0.618398	0.309199	−	0.950997i	$-0.399939\pi$
$617$	15.3607	0.618398	0.309199	+	0.950997i	$0.399939\pi$
$618$	0	0
$619$	38.0689	1.53012	0.765059	−	0.643960i	$-0.222710\pi$
$619$	38.0689	1.53012	0.765059	+	0.643960i	$0.222710\pi$
$620$	0	0
$621$	11.0557	0.443651
$622$	0	0
$623$	−40.1803	−1.60979
$624$	0	0
$625$	−25.0000	−1.00000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−0.236068	−0.00941265
$630$	0	0
$631$	−29.7082	−1.18266	−0.591332	−	0.806428i	$-0.701398\pi$
$631$	−29.7082	−1.18266	−0.591332	+	0.806428i	$0.701398\pi$
$632$	0	0
$633$	−22.8328	−0.907523
$634$	0	0
$635$	−41.3050	−1.63914
$636$	0	0
$637$	6.12461	0.242666
$638$	0	0
$639$	29.8885	1.18237
$640$	0	0
$641$	1.47214	0.0581459	0.0290729	−	0.999577i	$-0.490744\pi$
$641$	1.47214	0.0581459	0.0290729	+	0.999577i	$0.490744\pi$
$642$	0	0
$643$	−20.1803	−0.795835	−0.397917	−	0.917421i	$-0.630267\pi$
$643$	−20.1803	−0.795835	−0.397917	+	0.917421i	$0.630267\pi$
$644$	0	0
$645$	−75.7771	−2.98372
$646$	0	0
$647$	37.8885	1.48955	0.744776	−	0.667314i	$-0.232556\pi$
$647$	37.8885	1.48955	0.744776	+	0.667314i	$0.232556\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	49.8885	1.95529
$652$	0	0
$653$	−39.3050	−1.53812	−0.769061	−	0.639176i	$-0.779276\pi$
$653$	−39.3050	−1.53812	−0.769061	+	0.639176i	$0.779276\pi$
$654$	0	0
$655$	17.8885	0.698963
$656$	0	0
$657$	−26.3607	−1.02843
$658$	0	0
$659$	32.1803	1.25357	0.626784	−	0.779193i	$-0.284371\pi$
$659$	32.1803	1.25357	0.626784	+	0.779193i	$0.284371\pi$
$660$	0	0
$661$	−23.1803	−0.901611	−0.450805	−	0.892622i	$-0.648863\pi$
$661$	−23.1803	−0.901611	−0.450805	+	0.892622i	$0.648863\pi$
$662$	0	0
$663$	5.70820	0.221688
$664$	0	0
$665$	−41.3050	−1.60174
$666$	0	0
$667$	−1.34752	−0.0521763
$668$	0	0
$669$	28.9443	1.11905
$670$	0	0
$671$	0	0
$672$	0	0
$673$	20.4721	0.789143	0.394571	−	0.918865i	$-0.370893\pi$
$673$	20.4721	0.789143	0.394571	+	0.918865i	$0.370893\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−28.7082	−1.10335	−0.551673	−	0.834061i	$-0.686010\pi$
$677$	−28.7082	−1.10335	−0.551673	+	0.834061i	$0.686010\pi$
$678$	0	0
$679$	38.6525	1.48335
$680$	0	0
$681$	−8.00000	−0.306561
$682$	0	0
$683$	−6.87539	−0.263079	−0.131540	−	0.991311i	$-0.541992\pi$
$683$	−6.87539	−0.263079	−0.131540	+	0.991311i	$0.541992\pi$
$684$	0	0
$685$	−16.8328	−0.643149
$686$	0	0
$687$	−8.76393	−0.334365
$688$	0	0
$689$	23.2492	0.885725
$690$	0	0
$691$	19.4164	0.738635	0.369317	−	0.929303i	$-0.379591\pi$
$691$	19.4164	0.738635	0.369317	+	0.929303i	$0.379591\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	34.0689	1.29231
$696$	0	0
$697$	7.47214	0.283027
$698$	0	0
$699$	−35.5967	−1.34639
$700$	0	0
$701$	35.6525	1.34658	0.673288	−	0.739381i	$-0.264882\pi$
$701$	35.6525	1.34658	0.673288	+	0.739381i	$0.264882\pi$
$702$	0	0
$703$	1.34752	0.0508228
$704$	0	0
$705$	−41.3050	−1.55563
$706$	0	0
$707$	6.47214	0.243410
$708$	0	0
$709$	3.88854	0.146037	0.0730187	−	0.997331i	$-0.476737\pi$
$709$	3.88854	0.146037	0.0730187	+	0.997331i	$0.476737\pi$
$710$	0	0
$711$	−54.0689	−2.02774
$712$	0	0
$713$	3.63932	0.136294
$714$	0	0
$715$	0	0
$716$	0	0
$717$	65.3050	2.43886
$718$	0	0
$719$	−48.5410	−1.81027	−0.905137	−	0.425119i	$-0.860232\pi$
$719$	−48.5410	−1.81027	−0.905137	+	0.425119i	$0.860232\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	11.4164	0.424581
$724$	0	0
$725$	0	0
$726$	0	0
$727$	6.29180	0.233350	0.116675	−	0.993170i	$-0.462776\pi$
$727$	6.29180	0.233350	0.116675	+	0.993170i	$0.462776\pi$
$728$	0	0
$729$	41.9443	1.55349
$730$	0	0
$731$	10.4721	0.387326
$732$	0	0
$733$	20.2361	0.747436	0.373718	−	0.927542i	$-0.378083\pi$
$733$	20.2361	0.747436	0.373718	+	0.927542i	$0.378083\pi$
$734$	0	0
$735$	25.1246	0.926735
$736$	0	0
$737$	0	0
$738$	0	0
$739$	14.2918	0.525732	0.262866	−	0.964832i	$-0.415332\pi$
$739$	14.2918	0.525732	0.262866	+	0.964832i	$0.415332\pi$
$740$	0	0
$741$	−32.5836	−1.19699
$742$	0	0
$743$	−28.5410	−1.04707	−0.523534	−	0.852005i	$-0.675387\pi$
$743$	−28.5410	−1.04707	−0.523534	+	0.852005i	$0.675387\pi$
$744$	0	0
$745$	22.8885	0.838571
$746$	0	0
$747$	−91.0132	−3.33000
$748$	0	0
$749$	−5.52786	−0.201984
$750$	0	0
$751$	45.8885	1.67450	0.837248	−	0.546823i	$-0.184163\pi$
$751$	45.8885	1.67450	0.837248	+	0.546823i	$0.184163\pi$
$752$	0	0
$753$	67.1935	2.44867
$754$	0	0
$755$	16.5836	0.603539
$756$	0	0
$757$	15.6525	0.568899	0.284449	−	0.958691i	$-0.408189\pi$
$757$	15.6525	0.568899	0.284449	+	0.958691i	$0.408189\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	29.4721	1.06836	0.534182	−	0.845369i	$-0.320620\pi$
$761$	29.4721	1.06836	0.534182	+	0.845369i	$0.320620\pi$
$762$	0	0
$763$	−20.1803	−0.730577
$764$	0	0
$765$	16.7082	0.604086
$766$	0	0
$767$	−9.75078	−0.352080
$768$	0	0
$769$	−18.8885	−0.681138	−0.340569	−	0.940219i	$-0.610620\pi$
$769$	−18.8885	−0.681138	−0.340569	+	0.940219i	$0.610620\pi$
$770$	0	0
$771$	47.0132	1.69314
$772$	0	0
$773$	−22.0000	−0.791285	−0.395643	−	0.918405i	$-0.629478\pi$
$773$	−22.0000	−0.791285	−0.395643	+	0.918405i	$0.629478\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−2.47214	−0.0886874
$778$	0	0
$779$	−42.6525	−1.52818
$780$	0	0
$781$	0	0
$782$	0	0
$783$	−25.5279	−0.912291
$784$	0	0
$785$	−27.8885	−0.995385
$786$	0	0
$787$	15.7771	0.562392	0.281196	−	0.959650i	$-0.409269\pi$
$787$	15.7771	0.562392	0.281196	+	0.959650i	$0.409269\pi$
$788$	0	0
$789$	−21.5279	−0.766412
$790$	0	0
$791$	64.5410	2.29481
$792$	0	0
$793$	26.3607	0.936095
$794$	0	0
$795$	95.3738	3.38256
$796$	0	0
$797$	−25.0557	−0.887519	−0.443760	−	0.896146i	$-0.646356\pi$
$797$	−25.0557	−0.887519	−0.443760	+	0.896146i	$0.646356\pi$
$798$	0	0
$799$	5.70820	0.201942
$800$	0	0
$801$	−92.7771	−3.27812
$802$	0	0
$803$	0	0
$804$	0	0
$805$	5.52786	0.194832
$806$	0	0
$807$	54.0689	1.90331
$808$	0	0
$809$	−26.3607	−0.926792	−0.463396	−	0.886151i	$-0.653369\pi$
$809$	−26.3607	−0.926792	−0.463396	+	0.886151i	$0.653369\pi$
$810$	0	0
$811$	−22.4721	−0.789103	−0.394552	−	0.918874i	$-0.629100\pi$
$811$	−22.4721	−0.789103	−0.394552	+	0.918874i	$0.629100\pi$
$812$	0	0
$813$	−96.7214	−3.39217
$814$	0	0
$815$	−12.7639	−0.447101
$816$	0	0
$817$	−59.7771	−2.09134
$818$	0	0
$819$	42.6525	1.49040
$820$	0	0
$821$	30.9443	1.07996	0.539981	−	0.841677i	$-0.318431\pi$
$821$	30.9443	1.07996	0.539981	+	0.841677i	$0.318431\pi$
$822$	0	0
$823$	10.4721	0.365036	0.182518	−	0.983203i	$-0.441575\pi$
$823$	10.4721	0.365036	0.182518	+	0.983203i	$0.441575\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	5.70820	0.198494	0.0992468	−	0.995063i	$-0.468357\pi$
$827$	5.70820	0.198494	0.0992468	+	0.995063i	$0.468357\pi$
$828$	0	0
$829$	11.6525	0.404707	0.202354	−	0.979313i	$-0.435141\pi$
$829$	11.6525	0.404707	0.202354	+	0.979313i	$0.435141\pi$
$830$	0	0
$831$	−84.5410	−2.93270
$832$	0	0
$833$	−3.47214	−0.120302
$834$	0	0
$835$	−25.5279	−0.883428
$836$	0	0
$837$	68.9443	2.38306
$838$	0	0
$839$	40.1803	1.38718	0.693590	−	0.720370i	$-0.256028\pi$
$839$	40.1803	1.38718	0.693590	+	0.720370i	$0.256028\pi$
$840$	0	0
$841$	−25.8885	−0.892708
$842$	0	0
$843$	−50.2492	−1.73068
$844$	0	0
$845$	22.1115	0.760657
$846$	0	0
$847$	0	0
$848$	0	0
$849$	36.9443	1.26792
$850$	0	0
$851$	−0.180340	−0.00618197
$852$	0	0
$853$	12.5967	0.431304	0.215652	−	0.976470i	$-0.430812\pi$
$853$	12.5967	0.431304	0.215652	+	0.976470i	$0.430812\pi$
$854$	0	0
$855$	−95.3738	−3.26172
$856$	0	0
$857$	−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
$859$	−0.583592	−0.0199119	−0.00995595	−	0.999950i	$-0.503169\pi$
$859$	−0.583592	−0.0199119	−0.00995595	+	0.999950i	$0.503169\pi$
$860$	0	0
$861$	78.2492	2.66673
$862$	0	0
$863$	35.5967	1.21173	0.605864	−	0.795568i	$-0.292828\pi$
$863$	35.5967	1.21173	0.605864	+	0.795568i	$0.292828\pi$
$864$	0	0
$865$	42.3607	1.44031
$866$	0	0
$867$	51.7771	1.75844
$868$	0	0
$869$	0	0
$870$	0	0
$871$	−1.34752	−0.0456591
$872$	0	0
$873$	89.2492	3.02063
$874$	0	0
$875$	36.1803	1.22312
$876$	0	0
$877$	−16.7082	−0.564196	−0.282098	−	0.959386i	$-0.591030\pi$
$877$	−16.7082	−0.564196	−0.282098	+	0.959386i	$0.591030\pi$
$878$	0	0
$879$	17.1246	0.577599
$880$	0	0
$881$	4.52786	0.152548	0.0762738	−	0.997087i	$-0.475698\pi$
$881$	4.52786	0.152548	0.0762738	+	0.997087i	$0.475698\pi$
$882$	0	0
$883$	7.05573	0.237444	0.118722	−	0.992928i	$-0.462120\pi$
$883$	7.05573	0.237444	0.118722	+	0.992928i	$0.462120\pi$
$884$	0	0
$885$	−40.0000	−1.34459
$886$	0	0
$887$	19.5967	0.657994	0.328997	−	0.944331i	$-0.393289\pi$
$887$	19.5967	0.657994	0.328997	+	0.944331i	$0.393289\pi$
$888$	0	0
$889$	59.7771	2.00486
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−32.5836	−1.09037
$894$	0	0
$895$	5.52786	0.184776
$896$	0	0
$897$	4.36068	0.145599
$898$	0	0
$899$	−8.40325	−0.280264
$900$	0	0
$901$	−13.1803	−0.439101
$902$	0	0
$903$	109.666	3.64944
$904$	0	0
$905$	28.4164	0.944593
$906$	0	0
$907$	29.8885	0.992433	0.496216	−	0.868199i	$-0.334722\pi$
$907$	29.8885	0.992433	0.496216	+	0.868199i	$0.334722\pi$
$908$	0	0
$909$	14.9443	0.495670
$910$	0	0
$911$	−16.0000	−0.530104	−0.265052	−	0.964234i	$-0.585389\pi$
$911$	−16.0000	−0.530104	−0.265052	+	0.964234i	$0.585389\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	108.138	3.57492
$916$	0	0
$917$	−25.8885	−0.854915
$918$	0	0
$919$	−20.5836	−0.678990	−0.339495	−	0.940608i	$-0.610256\pi$
$919$	−20.5836	−0.678990	−0.339495	+	0.940608i	$0.610256\pi$
$920$	0	0
$921$	78.2492	2.57840
$922$	0	0
$923$	7.05573	0.232242
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	18.0557	0.592389	0.296195	−	0.955128i	$-0.404282\pi$
$929$	18.0557	0.592389	0.296195	+	0.955128i	$0.404282\pi$
$930$	0	0
$931$	19.8197	0.649563
$932$	0	0
$933$	−43.7771	−1.43320
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−21.5836	−0.705105	−0.352553	−	0.935792i	$-0.614686\pi$
$937$	−21.5836	−0.705105	−0.352553	+	0.935792i	$0.614686\pi$
$938$	0	0
$939$	−25.7082	−0.838956
$940$	0	0
$941$	11.0689	0.360835	0.180418	−	0.983590i	$-0.442255\pi$
$941$	11.0689	0.360835	0.180418	+	0.983590i	$0.442255\pi$
$942$	0	0
$943$	5.70820	0.185885
$944$	0	0
$945$	104.721	3.40659
$946$	0	0
$947$	−49.1246	−1.59634	−0.798168	−	0.602435i	$-0.794197\pi$
$947$	−49.1246	−1.59634	−0.798168	+	0.602435i	$0.794197\pi$
$948$	0	0
$949$	−6.22291	−0.202004
$950$	0	0
$951$	51.4164	1.66729
$952$	0	0
$953$	54.7771	1.77440	0.887202	−	0.461381i	$-0.152646\pi$
$953$	54.7771	1.77440	0.887202	+	0.461381i	$0.152646\pi$
$954$	0	0
$955$	12.3607	0.399982
$956$	0	0
$957$	0	0
$958$	0	0
$959$	24.3607	0.786647
$960$	0	0
$961$	−8.30495	−0.267902
$962$	0	0
$963$	−12.7639	−0.411312
$964$	0	0
$965$	−8.81966	−0.283915
$966$	0	0
$967$	−39.9574	−1.28494	−0.642472	−	0.766309i	$-0.722091\pi$
$967$	−39.9574	−1.28494	−0.642472	+	0.766309i	$0.722091\pi$
$968$	0	0
$969$	18.4721	0.593411
$970$	0	0
$971$	−40.7639	−1.30818	−0.654088	−	0.756418i	$-0.726948\pi$
$971$	−40.7639	−1.30818	−0.654088	+	0.756418i	$0.726948\pi$
$972$	0	0
$973$	−49.3050	−1.58064
$974$	0	0
$975$	0	0
$976$	0	0
$977$	16.3050	0.521642	0.260821	−	0.965387i	$-0.416007\pi$
$977$	16.3050	0.521642	0.260821	+	0.965387i	$0.416007\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	−46.5967	−1.48772
$982$	0	0
$983$	53.8885	1.71878	0.859389	−	0.511323i	$-0.170844\pi$
$983$	53.8885	1.71878	0.859389	+	0.511323i	$0.170844\pi$
$984$	0	0
$985$	39.4721	1.25769
$986$	0	0
$987$	59.7771	1.90273
$988$	0	0
$989$	8.00000	0.254385
$990$	0	0
$991$	29.8885	0.949441	0.474720	−	0.880137i	$-0.342549\pi$
$991$	29.8885	0.949441	0.474720	+	0.880137i	$0.342549\pi$
$992$	0	0
$993$	20.9443	0.664646
$994$	0	0
$995$	−41.3050	−1.30945
$996$	0	0
$997$	−2.81966	−0.0892995	−0.0446498	−	0.999003i	$-0.514217\pi$
$997$	−2.81966	−0.0892995	−0.0446498	+	0.999003i	$0.514217\pi$
$998$	0	0
$999$	−3.41641	−0.108090

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7744.2.a.bs.1.1		2
4.3	odd	2		7744.2.a.ct.1.2		2
8.3	odd	2		968.2.a.f.1.1	✓	2
8.5	even	2		1936.2.a.x.1.2		2
11.10	odd	2		7744.2.a.br.1.1		2
24.11	even	2		8712.2.a.bj.1.1		2
44.43	even	2		7744.2.a.cu.1.2		2
88.3	odd	10		968.2.i.m.9.1		4
88.19	even	10		968.2.i.l.9.1		4
88.21	odd	2		1936.2.a.w.1.2		2
88.27	odd	10		968.2.i.a.729.1		4
88.35	even	10		968.2.i.b.81.1		4
88.43	even	2		968.2.a.g.1.1	yes	2
88.51	even	10		968.2.i.l.753.1		4
88.59	odd	10		968.2.i.m.753.1		4
88.75	odd	10		968.2.i.a.81.1		4
88.83	even	10		968.2.i.b.729.1		4
264.131	odd	2		8712.2.a.bk.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
968.2.a.f.1.1	✓	2	8.3	odd	2
968.2.a.g.1.1	yes	2	88.43	even	2
968.2.i.a.81.1		4	88.75	odd	10
968.2.i.a.729.1		4	88.27	odd	10
968.2.i.b.81.1		4	88.35	even	10
968.2.i.b.729.1		4	88.83	even	10
968.2.i.l.9.1		4	88.19	even	10
968.2.i.l.753.1		4	88.51	even	10
968.2.i.m.9.1		4	88.3	odd	10
968.2.i.m.753.1		4	88.59	odd	10
1936.2.a.w.1.2		2	88.21	odd	2
1936.2.a.x.1.2		2	8.5	even	2
7744.2.a.br.1.1		2	11.10	odd	2
7744.2.a.bs.1.1		2	1.1	even	1	trivial
7744.2.a.ct.1.2		2	4.3	odd	2
7744.2.a.cu.1.2		2	44.43	even	2
8712.2.a.bj.1.1		2	24.11	even	2
8712.2.a.bk.1.1		2	264.131	odd	2

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Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

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Newspace parameters

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Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	7744.2.a.bs.1.1

Level	$7744$
Weight	$2$
Character	7744.1
Self dual	yes
Analytic conductor	$61.836$
Analytic rank	$1$
Dimension	$2$
CM	no
Inner twists	$1$