LMFDB - Embedded newform 7744.2.a.du.1.2

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:	$6$
Coefficient field:	6.6.19898000.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{6} - x^{5} - 10x^{4} + 7x^{3} + 24x^{2} - 15x - 5$ x^6 - x^5 - 10x^4 + 7x^3 + 24x^2 - 15x - 5
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 352)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-2.05906$ 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)

$f(q)$	$=$	$q-3.05906 q^{3} -3.00593 q^{5} -1.56490 q^{7} +6.35786 q^{9} -3.65733 q^{13} +9.19533 q^{15} -4.47580 q^{17} -2.74643 q^{19} +4.78714 q^{21} +4.77580 q^{23} +4.03563 q^{25} -10.2719 q^{27} -4.96996 q^{29} +0.487696 q^{31} +4.70400 q^{35} +10.6449 q^{37} +11.1880 q^{39} -12.5417 q^{41} -7.45745 q^{43} -19.1113 q^{45} +10.0370 q^{47} -4.55107 q^{49} +13.6918 q^{51} +5.26661 q^{53} +8.40151 q^{57} +8.93411 q^{59} -2.16446 q^{61} -9.94944 q^{63} +10.9937 q^{65} -0.709392 q^{67} -14.6095 q^{69} +0.808454 q^{71} -0.528076 q^{73} -12.3452 q^{75} +5.81422 q^{79} +12.3488 q^{81} +7.09133 q^{83} +13.4540 q^{85} +15.2034 q^{87} +7.76282 q^{89} +5.72337 q^{91} -1.49189 q^{93} +8.25559 q^{95} +3.62990 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$6 q - 5 q^{3} + 2 q^{7} + 7 q^{9} - 4 q^{13} + 8 q^{15} - 9 q^{17} - 5 q^{19} - 12 q^{21} + 6 q^{23} + 4 q^{25} - 26 q^{27} - 10 q^{29} + 12 q^{31} + 26 q^{35} + 12 q^{37} + 2 q^{39} - 17 q^{41} - 11 q^{43}+ \cdots - 21 q^{97}+O(q^{100})$ 6 * q - 5 * q^3 + 2 * q^7 + 7 * q^9 - 4 * q^13 + 8 * q^15 - 9 * q^17 - 5 * q^19 - 12 * q^21 + 6 * q^23 + 4 * q^25 - 26 * q^27 - 10 * q^29 + 12 * q^31 + 26 * q^35 + 12 * q^37 + 2 * q^39 - 17 * q^41 - 11 * q^43 - 14 * q^45 + 22 * q^47 + 6 * q^51 - 18 * q^53 - 6 * q^57 - 7 * q^59 - 18 * q^61 + 26 * q^63 - 26 * q^65 - 31 * q^67 - 16 * q^69 + 22 * q^71 + q^73 - 11 * q^75 - 28 * q^79 + 34 * q^81 + 13 * q^83 - 2 * q^85 + 38 * q^87 - q^89 - 26 * q^91 - 34 * q^93 + 2 * q^95 - 21 * 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
$3$	−3.05906	−1.76615	−0.883075	−	0.469232i	$-0.844531\pi$
$3$	−3.05906	−1.76615	−0.883075	+	0.469232i	$0.844531\pi$
$4$	0	0
$5$	−3.00593	−1.34429	−0.672147	−	0.740418i	$-0.734628\pi$
$5$	−3.00593	−1.34429	−0.672147	+	0.740418i	$0.734628\pi$
$6$	0	0
$7$	−1.56490	−0.591478	−0.295739	−	0.955269i	$-0.595566\pi$
$7$	−1.56490	−0.591478	−0.295739	+	0.955269i	$0.595566\pi$
$8$	0	0
$9$	6.35786	2.11929
$10$	0	0
$11$	0	0
$11$	0	0
$12$	0	0
$13$	−3.65733	−1.01436	−0.507181	−	0.861840i	$-0.669312\pi$
$13$	−3.65733	−1.01436	−0.507181	+	0.861840i	$0.669312\pi$
$14$	0	0
$15$	9.19533	2.37422
$16$	0	0
$17$	−4.47580	−1.08554	−0.542771	−	0.839881i	$-0.682625\pi$
$17$	−4.47580	−1.08554	−0.542771	+	0.839881i	$0.682625\pi$
$18$	0	0
$19$	−2.74643	−0.630075	−0.315037	−	0.949079i	$-0.602017\pi$
$19$	−2.74643	−0.630075	−0.315037	+	0.949079i	$0.602017\pi$
$20$	0	0
$21$	4.78714	1.04464
$22$	0	0
$23$	4.77580	0.995823	0.497911	−	0.867228i	$-0.334101\pi$
$23$	4.77580	0.995823	0.497911	+	0.867228i	$0.334101\pi$
$24$	0	0
$25$	4.03563	0.807126
$26$	0	0
$27$	−10.2719	−1.97683
$28$	0	0
$29$	−4.96996	−0.922898	−0.461449	−	0.887167i	$-0.652670\pi$
$29$	−4.96996	−0.922898	−0.461449	+	0.887167i	$0.652670\pi$
$30$	0	0
$31$	0.487696	0.0875927	0.0437964	−	0.999040i	$-0.486055\pi$
$31$	0.487696	0.0875927	0.0437964	+	0.999040i	$0.486055\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	4.70400	0.795121
$36$	0	0
$37$	10.6449	1.75001	0.875006	−	0.484112i	$-0.160857\pi$
$37$	10.6449	1.75001	0.875006	+	0.484112i	$0.160857\pi$
$38$	0	0
$39$	11.1880	1.79151
$40$	0	0
$41$	−12.5417	−1.95868	−0.979338	−	0.202228i	$-0.935182\pi$
$41$	−12.5417	−1.95868	−0.979338	+	0.202228i	$0.935182\pi$
$42$	0	0
$43$	−7.45745	−1.13725	−0.568625	−	0.822597i	$-0.692524\pi$
$43$	−7.45745	−1.13725	−0.568625	+	0.822597i	$0.692524\pi$
$44$	0	0
$45$	−19.1113	−2.84894
$46$	0	0
$47$	10.0370	1.46405	0.732026	−	0.681276i	$-0.238575\pi$
$47$	10.0370	1.46405	0.732026	+	0.681276i	$0.238575\pi$
$48$	0	0
$49$	−4.55107	−0.650153
$50$	0	0
$51$	13.6918	1.91723
$52$	0	0
$53$	5.26661	0.723424	0.361712	−	0.932290i	$-0.382192\pi$
$53$	5.26661	0.723424	0.361712	+	0.932290i	$0.382192\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	8.40151	1.11281
$58$	0	0
$59$	8.93411	1.16312	0.581561	−	0.813503i	$-0.302442\pi$
$59$	8.93411	1.16312	0.581561	+	0.813503i	$0.302442\pi$
$60$	0	0
$61$	−2.16446	−0.277131	−0.138566	−	0.990353i	$-0.544249\pi$
$61$	−2.16446	−0.277131	−0.138566	+	0.990353i	$0.544249\pi$
$62$	0	0
$63$	−9.94944	−1.25351
$64$	0	0
$65$	10.9937	1.36360
$66$	0	0
$67$	−0.709392	−0.0866661	−0.0433330	−	0.999061i	$-0.513798\pi$
$67$	−0.709392	−0.0866661	−0.0433330	+	0.999061i	$0.513798\pi$
$68$	0	0
$69$	−14.6095	−1.75877
$70$	0	0
$71$	0.808454	0.0959459	0.0479729	−	0.998849i	$-0.484724\pi$
$71$	0.808454	0.0959459	0.0479729	+	0.998849i	$0.484724\pi$
$72$	0	0
$73$	−0.528076	−0.0618066	−0.0309033	−	0.999522i	$-0.509838\pi$
$73$	−0.528076	−0.0618066	−0.0309033	+	0.999522i	$0.509838\pi$
$74$	0	0
$75$	−12.3452	−1.42551
$76$	0	0
$77$	0	0
$78$	0	0
$79$	5.81422	0.654151	0.327076	−	0.944998i	$-0.393937\pi$
$79$	5.81422	0.654151	0.327076	+	0.944998i	$0.393937\pi$
$80$	0	0
$81$	12.3488	1.37209
$82$	0	0
$83$	7.09133	0.778375	0.389187	−	0.921159i	$-0.372756\pi$
$83$	7.09133	0.778375	0.389187	+	0.921159i	$0.372756\pi$
$84$	0	0
$85$	13.4540	1.45929
$86$	0	0
$87$	15.2034	1.62998
$88$	0	0
$89$	7.76282	0.822857	0.411429	−	0.911442i	$-0.365030\pi$
$89$	7.76282	0.822857	0.411429	+	0.911442i	$0.365030\pi$
$90$	0	0
$91$	5.72337	0.599973
$92$	0	0
$93$	−1.49189	−0.154702
$94$	0	0
$95$	8.25559	0.847006
$96$	0	0
$97$	3.62990	0.368560	0.184280	−	0.982874i	$-0.441005\pi$
$97$	3.62990	0.368560	0.184280	+	0.982874i	$0.441005\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	1.23031	0.122421	0.0612104	−	0.998125i	$-0.480504\pi$
$101$	1.23031	0.122421	0.0612104	+	0.998125i	$0.480504\pi$
$102$	0	0
$103$	−2.56508	−0.252745	−0.126373	−	0.991983i	$-0.540333\pi$
$103$	−2.56508	−0.252745	−0.126373	+	0.991983i	$0.540333\pi$
$104$	0	0
$105$	−14.3898	−1.40430
$106$	0	0
$107$	13.6612	1.32068	0.660338	−	0.750968i	$-0.270413\pi$
$107$	13.6612	1.32068	0.660338	+	0.750968i	$0.270413\pi$
$108$	0	0
$109$	−0.935861	−0.0896393	−0.0448196	−	0.998995i	$-0.514271\pi$
$109$	−0.935861	−0.0896393	−0.0448196	+	0.998995i	$0.514271\pi$
$110$	0	0
$111$	−32.5634	−3.09078
$112$	0	0
$113$	−3.21175	−0.302136	−0.151068	−	0.988523i	$-0.548271\pi$
$113$	−3.21175	−0.302136	−0.151068	+	0.988523i	$0.548271\pi$
$114$	0	0
$115$	−14.3557	−1.33868
$116$	0	0
$117$	−23.2528	−2.14972
$118$	0	0
$119$	7.00421	0.642074
$120$	0	0
$121$	0	0
$122$	0	0
$123$	38.3657	3.45932
$124$	0	0
$125$	2.89883	0.259279
$126$	0	0
$127$	10.1602	0.901576	0.450788	−	0.892631i	$-0.351143\pi$
$127$	10.1602	0.901576	0.450788	+	0.892631i	$0.351143\pi$
$128$	0	0
$129$	22.8128	2.00855
$130$	0	0
$131$	3.01348	0.263289	0.131644	−	0.991297i	$-0.457974\pi$
$131$	3.01348	0.263289	0.131644	+	0.991297i	$0.457974\pi$
$132$	0	0
$133$	4.29791	0.372676
$134$	0	0
$135$	30.8766	2.65744
$136$	0	0
$137$	1.03790	0.0886735	0.0443367	−	0.999017i	$-0.485883\pi$
$137$	1.03790	0.0886735	0.0443367	+	0.999017i	$0.485883\pi$
$138$	0	0
$139$	2.93490	0.248935	0.124467	−	0.992224i	$-0.460278\pi$
$139$	2.93490	0.248935	0.124467	+	0.992224i	$0.460278\pi$
$140$	0	0
$141$	−30.7039	−2.58574
$142$	0	0
$143$	0	0
$144$	0	0
$145$	14.9394	1.24065
$146$	0	0
$147$	13.9220	1.14827
$148$	0	0
$149$	−12.3003	−1.00768	−0.503842	−	0.863796i	$-0.668080\pi$
$149$	−12.3003	−1.00768	−0.503842	+	0.863796i	$0.668080\pi$
$150$	0	0
$151$	20.1834	1.64250	0.821251	−	0.570567i	$-0.193276\pi$
$151$	20.1834	1.64250	0.821251	+	0.570567i	$0.193276\pi$
$152$	0	0
$153$	−28.4565	−2.30057
$154$	0	0
$155$	−1.46598	−0.117750
$156$	0	0
$157$	13.2045	1.05383	0.526917	−	0.849917i	$-0.323348\pi$
$157$	13.2045	1.05383	0.526917	+	0.849917i	$0.323348\pi$
$158$	0	0
$159$	−16.1109	−1.27768
$160$	0	0
$161$	−7.47367	−0.589008
$162$	0	0
$163$	3.51087	0.274993	0.137496	−	0.990502i	$-0.456094\pi$
$163$	3.51087	0.274993	0.137496	+	0.990502i	$0.456094\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	10.0092	0.774537	0.387269	−	0.921967i	$-0.373419\pi$
$167$	10.0092	0.774537	0.387269	+	0.921967i	$0.373419\pi$
$168$	0	0
$169$	0.376067	0.0289283
$170$	0	0
$171$	−17.4614	−1.33531
$172$	0	0
$173$	2.50969	0.190808	0.0954040	−	0.995439i	$-0.469586\pi$
$173$	2.50969	0.190808	0.0954040	+	0.995439i	$0.469586\pi$
$174$	0	0
$175$	−6.31538	−0.477398
$176$	0	0
$177$	−27.3300	−2.05425
$178$	0	0
$179$	−16.7954	−1.25535	−0.627675	−	0.778476i	$-0.715993\pi$
$179$	−16.7954	−1.25535	−0.627675	+	0.778476i	$0.715993\pi$
$180$	0	0
$181$	5.93834	0.441393	0.220697	−	0.975342i	$-0.429167\pi$
$181$	5.93834	0.441393	0.220697	+	0.975342i	$0.429167\pi$
$182$	0	0
$183$	6.62123	0.489455
$184$	0	0
$185$	−31.9979	−2.35253
$186$	0	0
$187$	0	0
$188$	0	0
$189$	16.0745	1.16925
$190$	0	0
$191$	6.94136	0.502259	0.251130	−	0.967953i	$-0.419198\pi$
$191$	6.94136	0.502259	0.251130	+	0.967953i	$0.419198\pi$
$192$	0	0
$193$	3.65943	0.263412	0.131706	−	0.991289i	$-0.457955\pi$
$193$	3.65943	0.263412	0.131706	+	0.991289i	$0.457955\pi$
$194$	0	0
$195$	−33.6304	−2.40832
$196$	0	0
$197$	−17.5687	−1.25172	−0.625859	−	0.779936i	$-0.715251\pi$
$197$	−17.5687	−1.25172	−0.625859	+	0.779936i	$0.715251\pi$
$198$	0	0
$199$	7.12699	0.505219	0.252609	−	0.967568i	$-0.418711\pi$
$199$	7.12699	0.505219	0.252609	+	0.967568i	$0.418711\pi$
$200$	0	0
$201$	2.17007	0.153065
$202$	0	0
$203$	7.77751	0.545874
$204$	0	0
$205$	37.6994	2.63304
$206$	0	0
$207$	30.3638	2.11043
$208$	0	0
$209$	0	0
$210$	0	0
$211$	20.5140	1.41224	0.706121	−	0.708091i	$-0.250443\pi$
$211$	20.5140	1.41224	0.706121	+	0.708091i	$0.250443\pi$
$212$	0	0
$213$	−2.47311	−0.169455
$214$	0	0
$215$	22.4166	1.52880
$216$	0	0
$217$	−0.763197	−0.0518092
$218$	0	0
$219$	1.61542	0.109160
$220$	0	0
$221$	16.3695	1.10113
$222$	0	0
$223$	6.24503	0.418198	0.209099	−	0.977894i	$-0.432947\pi$
$223$	6.24503	0.418198	0.209099	+	0.977894i	$0.432947\pi$
$224$	0	0
$225$	25.6580	1.71053
$226$	0	0
$227$	−15.7568	−1.04581	−0.522907	−	0.852390i	$-0.675152\pi$
$227$	−15.7568	−1.04581	−0.522907	+	0.852390i	$0.675152\pi$
$228$	0	0
$229$	16.7277	1.10540	0.552699	−	0.833381i	$-0.313598\pi$
$229$	16.7277	1.10540	0.552699	+	0.833381i	$0.313598\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−11.7221	−0.767940	−0.383970	−	0.923345i	$-0.625443\pi$
$233$	−11.7221	−0.767940	−0.383970	+	0.923345i	$0.625443\pi$
$234$	0	0
$235$	−30.1707	−1.96812
$236$	0	0
$237$	−17.7861	−1.15533
$238$	0	0
$239$	−25.3503	−1.63978	−0.819889	−	0.572523i	$-0.805965\pi$
$239$	−25.3503	−1.63978	−0.819889	+	0.572523i	$0.805965\pi$
$240$	0	0
$241$	1.25667	0.0809490	0.0404745	−	0.999181i	$-0.487113\pi$
$241$	1.25667	0.0809490	0.0404745	+	0.999181i	$0.487113\pi$
$242$	0	0
$243$	−6.96000	−0.446484
$244$	0	0
$245$	13.6802	0.873997
$246$	0	0
$247$	10.0446	0.639123
$248$	0	0
$249$	−21.6928	−1.37473
$250$	0	0
$251$	−23.0548	−1.45521	−0.727604	−	0.685998i	$-0.759366\pi$
$251$	−23.0548	−1.45521	−0.727604	+	0.685998i	$0.759366\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	−41.1565	−2.57732
$256$	0	0
$257$	−5.39585	−0.336584	−0.168292	−	0.985737i	$-0.553825\pi$
$257$	−5.39585	−0.336584	−0.168292	+	0.985737i	$0.553825\pi$
$258$	0	0
$259$	−16.6583	−1.03509
$260$	0	0
$261$	−31.5983	−1.95589
$262$	0	0
$263$	−16.3375	−1.00742	−0.503708	−	0.863874i	$-0.668031\pi$
$263$	−16.3375	−1.00742	−0.503708	+	0.863874i	$0.668031\pi$
$264$	0	0
$265$	−15.8311	−0.972495
$266$	0	0
$267$	−23.7469	−1.45329
$268$	0	0
$269$	27.6384	1.68514	0.842572	−	0.538583i	$-0.181040\pi$
$269$	27.6384	1.68514	0.842572	+	0.538583i	$0.181040\pi$
$270$	0	0
$271$	−29.8843	−1.81534	−0.907670	−	0.419684i	$-0.862141\pi$
$271$	−29.8843	−1.81534	−0.907670	+	0.419684i	$0.862141\pi$
$272$	0	0
$273$	−17.5082	−1.05964
$274$	0	0
$275$	0	0
$276$	0	0
$277$	23.9789	1.44075	0.720375	−	0.693584i	$-0.243970\pi$
$277$	23.9789	1.44075	0.720375	+	0.693584i	$0.243970\pi$
$278$	0	0
$279$	3.10070	0.185634
$280$	0	0
$281$	11.4834	0.685044	0.342522	−	0.939510i	$-0.388719\pi$
$281$	11.4834	0.685044	0.342522	+	0.939510i	$0.388719\pi$
$282$	0	0
$283$	16.4096	0.975447	0.487724	−	0.872998i	$-0.337827\pi$
$283$	16.4096	0.975447	0.487724	+	0.872998i	$0.337827\pi$
$284$	0	0
$285$	−25.2544	−1.49594
$286$	0	0
$287$	19.6265	1.15852
$288$	0	0
$289$	3.03281	0.178400
$290$	0	0
$291$	−11.1041	−0.650933
$292$	0	0
$293$	−16.8060	−0.981818	−0.490909	−	0.871211i	$-0.663335\pi$
$293$	−16.8060	−0.981818	−0.490909	+	0.871211i	$0.663335\pi$
$294$	0	0
$295$	−26.8553	−1.56358
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−17.4667	−1.01012
$300$	0	0
$301$	11.6702	0.672659
$302$	0	0
$303$	−3.76361	−0.216213
$304$	0	0
$305$	6.50623	0.372546
$306$	0	0
$307$	−12.3071	−0.702403	−0.351201	−	0.936300i	$-0.614227\pi$
$307$	−12.3071	−0.702403	−0.351201	+	0.936300i	$0.614227\pi$
$308$	0	0
$309$	7.84675	0.446386
$310$	0	0
$311$	−16.5968	−0.941119	−0.470559	−	0.882368i	$-0.655948\pi$
$311$	−16.5968	−0.941119	−0.470559	+	0.882368i	$0.655948\pi$
$312$	0	0
$313$	−22.8326	−1.29057	−0.645287	−	0.763941i	$-0.723262\pi$
$313$	−22.8326	−1.29057	−0.645287	+	0.763941i	$0.723262\pi$
$314$	0	0
$315$	29.9074	1.68509
$316$	0	0
$317$	4.76722	0.267754	0.133877	−	0.990998i	$-0.457257\pi$
$317$	4.76722	0.267754	0.133877	+	0.990998i	$0.457257\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	−41.7904	−2.33251
$322$	0	0
$323$	12.2925	0.683973
$324$	0	0
$325$	−14.7596	−0.818717
$326$	0	0
$327$	2.86286	0.158316
$328$	0	0
$329$	−15.7070	−0.865956
$330$	0	0
$331$	32.4925	1.78595	0.892976	−	0.450105i	$-0.148613\pi$
$331$	32.4925	1.78595	0.892976	+	0.450105i	$0.148613\pi$
$332$	0	0
$333$	67.6788	3.70878
$334$	0	0
$335$	2.13239	0.116505
$336$	0	0
$337$	−9.49799	−0.517388	−0.258694	−	0.965959i	$-0.583292\pi$
$337$	−9.49799	−0.517388	−0.258694	+	0.965959i	$0.583292\pi$
$338$	0	0
$339$	9.82493	0.533617
$340$	0	0
$341$	0	0
$342$	0	0
$343$	18.0763	0.976030
$344$	0	0
$345$	43.9150	2.36431
$346$	0	0
$347$	35.8397	1.92398	0.961989	−	0.273088i	$-0.0880450\pi$
$347$	35.8397	1.92398	0.961989	+	0.273088i	$0.0880450\pi$
$348$	0	0
$349$	−25.9988	−1.39168	−0.695842	−	0.718195i	$-0.744969\pi$
$349$	−25.9988	−1.39168	−0.695842	+	0.718195i	$0.744969\pi$
$350$	0	0
$351$	37.5677	2.00522
$352$	0	0
$353$	−16.3264	−0.868969	−0.434484	−	0.900679i	$-0.643069\pi$
$353$	−16.3264	−0.868969	−0.434484	+	0.900679i	$0.643069\pi$
$354$	0	0
$355$	−2.43016	−0.128979
$356$	0	0
$357$	−21.4263	−1.13400
$358$	0	0
$359$	−7.72256	−0.407581	−0.203791	−	0.979015i	$-0.565326\pi$
$359$	−7.72256	−0.407581	−0.203791	+	0.979015i	$0.565326\pi$
$360$	0	0
$361$	−11.4571	−0.603006
$362$	0	0
$363$	0	0
$364$	0	0
$365$	1.58736	0.0830863
$366$	0	0
$367$	−6.23615	−0.325525	−0.162762	−	0.986665i	$-0.552040\pi$
$367$	−6.23615	−0.325525	−0.162762	+	0.986665i	$0.552040\pi$
$368$	0	0
$369$	−79.7380	−4.15100
$370$	0	0
$371$	−8.24174	−0.427890
$372$	0	0
$373$	−4.84081	−0.250648	−0.125324	−	0.992116i	$-0.539997\pi$
$373$	−4.84081	−0.250648	−0.125324	+	0.992116i	$0.539997\pi$
$374$	0	0
$375$	−8.86770	−0.457926
$376$	0	0
$377$	18.1768	0.936152
$378$	0	0
$379$	−13.6788	−0.702634	−0.351317	−	0.936257i	$-0.614266\pi$
$379$	−13.6788	−0.702634	−0.351317	+	0.936257i	$0.614266\pi$
$380$	0	0
$381$	−31.0808	−1.59232
$382$	0	0
$383$	11.5039	0.587823	0.293911	−	0.955833i	$-0.405043\pi$
$383$	11.5039	0.587823	0.293911	+	0.955833i	$0.405043\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−47.4134	−2.41016
$388$	0	0
$389$	−9.29913	−0.471485	−0.235742	−	0.971816i	$-0.575752\pi$
$389$	−9.29913	−0.471485	−0.235742	+	0.971816i	$0.575752\pi$
$390$	0	0
$391$	−21.3755	−1.08101
$392$	0	0
$393$	−9.21841	−0.465007
$394$	0	0
$395$	−17.4772	−0.879371
$396$	0	0
$397$	−38.3379	−1.92413	−0.962063	−	0.272828i	$-0.912041\pi$
$397$	−38.3379	−1.92413	−0.962063	+	0.272828i	$0.912041\pi$
$398$	0	0
$399$	−13.1476	−0.658201
$400$	0	0
$401$	17.3130	0.864571	0.432285	−	0.901737i	$-0.357707\pi$
$401$	17.3130	0.864571	0.432285	+	0.901737i	$0.357707\pi$
$402$	0	0
$403$	−1.78366	−0.0888506
$404$	0	0
$405$	−37.1196	−1.84449
$406$	0	0
$407$	0	0
$408$	0	0
$409$	2.69257	0.133139	0.0665696	−	0.997782i	$-0.478795\pi$
$409$	2.69257	0.133139	0.0665696	+	0.997782i	$0.478795\pi$
$410$	0	0
$411$	−3.17499	−0.156611
$412$	0	0
$413$	−13.9810	−0.687961
$414$	0	0
$415$	−21.3161	−1.04636
$416$	0	0
$417$	−8.97803	−0.439656
$418$	0	0
$419$	−13.3425	−0.651824	−0.325912	−	0.945400i	$-0.605671\pi$
$419$	−13.3425	−0.651824	−0.325912	+	0.945400i	$0.605671\pi$
$420$	0	0
$421$	−21.1838	−1.03243	−0.516217	−	0.856458i	$-0.672660\pi$
$421$	−21.1838	−1.03243	−0.516217	+	0.856458i	$0.672660\pi$
$422$	0	0
$423$	63.8141	3.10275
$424$	0	0
$425$	−18.0627	−0.876169
$426$	0	0
$427$	3.38718	0.163917
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−17.4050	−0.838367	−0.419184	−	0.907901i	$-0.637684\pi$
$431$	−17.4050	−0.838367	−0.419184	+	0.907901i	$0.637684\pi$
$432$	0	0
$433$	4.12280	0.198129	0.0990645	−	0.995081i	$-0.468415\pi$
$433$	4.12280	0.198129	0.0990645	+	0.995081i	$0.468415\pi$
$434$	0	0
$435$	−45.7004	−2.19117
$436$	0	0
$437$	−13.1164	−0.627443
$438$	0	0
$439$	30.8478	1.47229	0.736143	−	0.676826i	$-0.236645\pi$
$439$	30.8478	1.47229	0.736143	+	0.676826i	$0.236645\pi$
$440$	0	0
$441$	−28.9351	−1.37786
$442$	0	0
$443$	−9.35946	−0.444681	−0.222341	−	0.974969i	$-0.571370\pi$
$443$	−9.35946	−0.444681	−0.222341	+	0.974969i	$0.571370\pi$
$444$	0	0
$445$	−23.3345	−1.10616
$446$	0	0
$447$	37.6275	1.77972
$448$	0	0
$449$	−3.68708	−0.174004	−0.0870021	−	0.996208i	$-0.527729\pi$
$449$	−3.68708	−0.174004	−0.0870021	+	0.996208i	$0.527729\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	−61.7423	−2.90091
$454$	0	0
$455$	−17.2041	−0.806540
$456$	0	0
$457$	32.1488	1.50386	0.751930	−	0.659243i	$-0.229123\pi$
$457$	32.1488	1.50386	0.751930	+	0.659243i	$0.229123\pi$
$458$	0	0
$459$	45.9750	2.14593
$460$	0	0
$461$	3.41140	0.158885	0.0794423	−	0.996839i	$-0.474686\pi$
$461$	3.41140	0.158885	0.0794423	+	0.996839i	$0.474686\pi$
$462$	0	0
$463$	3.72640	0.173181	0.0865903	−	0.996244i	$-0.472403\pi$
$463$	3.72640	0.173181	0.0865903	+	0.996244i	$0.472403\pi$
$464$	0	0
$465$	4.48452	0.207965
$466$	0	0
$467$	1.92963	0.0892928	0.0446464	−	0.999003i	$-0.485784\pi$
$467$	1.92963	0.0892928	0.0446464	+	0.999003i	$0.485784\pi$
$468$	0	0
$469$	1.11013	0.0512611
$470$	0	0
$471$	−40.3934	−1.86123
$472$	0	0
$473$	0	0
$474$	0	0
$475$	−11.0836	−0.508550
$476$	0	0
$477$	33.4844	1.53314
$478$	0	0
$479$	−28.2867	−1.29245	−0.646226	−	0.763146i	$-0.723654\pi$
$479$	−28.2867	−1.29245	−0.646226	+	0.763146i	$0.723654\pi$
$480$	0	0
$481$	−38.9320	−1.77514
$482$	0	0
$483$	22.8624	1.04028
$484$	0	0
$485$	−10.9112	−0.495453
$486$	0	0
$487$	−11.6571	−0.528236	−0.264118	−	0.964490i	$-0.585081\pi$
$487$	−11.6571	−0.528236	−0.264118	+	0.964490i	$0.585081\pi$
$488$	0	0
$489$	−10.7400	−0.485678
$490$	0	0
$491$	−18.5772	−0.838377	−0.419189	−	0.907899i	$-0.637685\pi$
$491$	−18.5772	−0.838377	−0.419189	+	0.907899i	$0.637685\pi$
$492$	0	0
$493$	22.2446	1.00184
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−1.26515	−0.0567499
$498$	0	0
$499$	−3.70872	−0.166025	−0.0830125	−	0.996549i	$-0.526454\pi$
$499$	−3.70872	−0.166025	−0.0830125	+	0.996549i	$0.526454\pi$
$500$	0	0
$501$	−30.6188	−1.36795
$502$	0	0
$503$	31.0826	1.38590	0.692952	−	0.720984i	$-0.256310\pi$
$503$	31.0826	1.38590	0.692952	+	0.720984i	$0.256310\pi$
$504$	0	0
$505$	−3.69824	−0.164570
$506$	0	0
$507$	−1.15041	−0.0510917
$508$	0	0
$509$	−38.9603	−1.72689	−0.863443	−	0.504446i	$-0.831697\pi$
$509$	−38.9603	−1.72689	−0.863443	+	0.504446i	$0.831697\pi$
$510$	0	0
$511$	0.826389	0.0365573
$512$	0	0
$513$	28.2111	1.24555
$514$	0	0
$515$	7.71047	0.339764
$516$	0	0
$517$	0	0
$518$	0	0
$519$	−7.67729	−0.336996
$520$	0	0
$521$	−11.6240	−0.509258	−0.254629	−	0.967039i	$-0.581953\pi$
$521$	−11.6240	−0.509258	−0.254629	+	0.967039i	$0.581953\pi$
$522$	0	0
$523$	17.7528	0.776275	0.388137	−	0.921602i	$-0.373119\pi$
$523$	17.7528	0.776275	0.388137	+	0.921602i	$0.373119\pi$
$524$	0	0
$525$	19.3191	0.843156
$526$	0	0
$527$	−2.18283	−0.0950855
$528$	0	0
$529$	−0.191763	−0.00833752
$530$	0	0
$531$	56.8018	2.46499
$532$	0	0
$533$	45.8690	1.98681
$534$	0	0
$535$	−41.0646	−1.77538
$536$	0	0
$537$	51.3783	2.21714
$538$	0	0
$539$	0	0
$540$	0	0
$541$	24.2627	1.04313	0.521567	−	0.853210i	$-0.325348\pi$
$541$	24.2627	1.04313	0.521567	+	0.853210i	$0.325348\pi$
$542$	0	0
$543$	−18.1657	−0.779567
$544$	0	0
$545$	2.81314	0.120501
$546$	0	0
$547$	8.29110	0.354502	0.177251	−	0.984166i	$-0.443280\pi$
$547$	8.29110	0.354502	0.177251	+	0.984166i	$0.443280\pi$
$548$	0	0
$549$	−13.7614	−0.587320
$550$	0	0
$551$	13.6497	0.581495
$552$	0	0
$553$	−9.09870	−0.386916
$554$	0	0
$555$	97.8835	4.15492
$556$	0	0
$557$	41.9655	1.77814	0.889068	−	0.457775i	$-0.151354\pi$
$557$	41.9655	1.77814	0.889068	+	0.457775i	$0.151354\pi$
$558$	0	0
$559$	27.2744	1.15358
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−33.8969	−1.42859	−0.714293	−	0.699847i	$-0.753251\pi$
$563$	−33.8969	−1.42859	−0.714293	+	0.699847i	$0.753251\pi$
$564$	0	0
$565$	9.65430	0.406159
$566$	0	0
$567$	−19.3247	−0.811560
$568$	0	0
$569$	−17.3055	−0.725485	−0.362742	−	0.931889i	$-0.618159\pi$
$569$	−17.3055	−0.725485	−0.362742	+	0.931889i	$0.618159\pi$
$570$	0	0
$571$	−15.8559	−0.663551	−0.331775	−	0.943358i	$-0.607648\pi$
$571$	−15.8559	−0.663551	−0.331775	+	0.943358i	$0.607648\pi$
$572$	0	0
$573$	−21.2340	−0.887065
$574$	0	0
$575$	19.2734	0.803754
$576$	0	0
$577$	−28.8545	−1.20123	−0.600615	−	0.799538i	$-0.705078\pi$
$577$	−28.8545	−1.20123	−0.600615	+	0.799538i	$0.705078\pi$
$578$	0	0
$579$	−11.1944	−0.465224
$580$	0	0
$581$	−11.0973	−0.460392
$582$	0	0
$583$	0	0
$584$	0	0
$585$	69.8963	2.88986
$586$	0	0
$587$	−27.3318	−1.12811	−0.564053	−	0.825739i	$-0.690759\pi$
$587$	−27.3318	−1.12811	−0.564053	+	0.825739i	$0.690759\pi$
$588$	0	0
$589$	−1.33942	−0.0551900
$590$	0	0
$591$	53.7437	2.21072
$592$	0	0
$593$	−33.4666	−1.37431	−0.687154	−	0.726512i	$-0.741140\pi$
$593$	−33.4666	−1.37431	−0.687154	+	0.726512i	$0.741140\pi$
$594$	0	0
$595$	−21.0542	−0.863137
$596$	0	0
$597$	−21.8019	−0.892292
$598$	0	0
$599$	−16.9832	−0.693915	−0.346957	−	0.937881i	$-0.612785\pi$
$599$	−16.9832	−0.693915	−0.346957	+	0.937881i	$0.612785\pi$
$600$	0	0
$601$	0.0293746	0.00119821	0.000599107	−	1.00000i	$-0.499809\pi$
$601$	0.0293746	0.00119821	0.000599107		1.00000i	$0.499809\pi$
$602$	0	0
$603$	−4.51021	−0.183670
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−26.0033	−1.05544	−0.527721	−	0.849418i	$-0.676953\pi$
$607$	−26.0033	−1.05544	−0.527721	+	0.849418i	$0.676953\pi$
$608$	0	0
$609$	−23.7919	−0.964096
$610$	0	0
$611$	−36.7088	−1.48508
$612$	0	0
$613$	30.3947	1.22763	0.613815	−	0.789450i	$-0.289634\pi$
$613$	30.3947	1.22763	0.613815	+	0.789450i	$0.289634\pi$
$614$	0	0
$615$	−115.325	−4.65034
$616$	0	0
$617$	22.3795	0.900966	0.450483	−	0.892785i	$-0.351252\pi$
$617$	22.3795	0.900966	0.450483	+	0.892785i	$0.351252\pi$
$618$	0	0
$619$	−23.0882	−0.927994	−0.463997	−	0.885837i	$-0.653585\pi$
$619$	−23.0882	−0.927994	−0.463997	+	0.885837i	$0.653585\pi$
$620$	0	0
$621$	−49.0565	−1.96857
$622$	0	0
$623$	−12.1481	−0.486702
$624$	0	0
$625$	−28.8918	−1.15567
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−47.6445	−1.89971
$630$	0	0
$631$	30.0478	1.19618	0.598092	−	0.801427i	$-0.295926\pi$
$631$	30.0478	1.19618	0.598092	+	0.801427i	$0.295926\pi$
$632$	0	0
$633$	−62.7536	−2.49423
$634$	0	0
$635$	−30.5410	−1.21198
$636$	0	0
$637$	16.6448	0.659490
$638$	0	0
$639$	5.14004	0.203337
$640$	0	0
$641$	35.9682	1.42066	0.710328	−	0.703870i	$-0.248546\pi$
$641$	35.9682	1.42066	0.710328	+	0.703870i	$0.248546\pi$
$642$	0	0
$643$	−18.2238	−0.718675	−0.359337	−	0.933208i	$-0.616997\pi$
$643$	−18.2238	−0.718675	−0.359337	+	0.933208i	$0.616997\pi$
$644$	0	0
$645$	−68.5737	−2.70009
$646$	0	0
$647$	11.1258	0.437401	0.218701	−	0.975792i	$-0.429818\pi$
$647$	11.1258	0.437401	0.218701	+	0.975792i	$0.429818\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	2.33467	0.0915028
$652$	0	0
$653$	16.5405	0.647282	0.323641	−	0.946180i	$-0.395093\pi$
$653$	16.5405	0.647282	0.323641	+	0.946180i	$0.395093\pi$
$654$	0	0
$655$	−9.05831	−0.353937
$656$	0	0
$657$	−3.35743	−0.130986
$658$	0	0
$659$	26.0957	1.01655	0.508273	−	0.861196i	$-0.330284\pi$
$659$	26.0957	1.01655	0.508273	+	0.861196i	$0.330284\pi$
$660$	0	0
$661$	−15.3262	−0.596121	−0.298061	−	0.954547i	$-0.596340\pi$
$661$	−15.3262	−0.596121	−0.298061	+	0.954547i	$0.596340\pi$
$662$	0	0
$663$	−50.0753	−1.94476
$664$	0	0
$665$	−12.9192	−0.500986
$666$	0	0
$667$	−23.7355	−0.919043
$668$	0	0
$669$	−19.1039	−0.738600
$670$	0	0
$671$	0	0
$672$	0	0
$673$	31.9654	1.23218	0.616088	−	0.787678i	$-0.288717\pi$
$673$	31.9654	1.23218	0.616088	+	0.787678i	$0.288717\pi$
$674$	0	0
$675$	−41.4536	−1.59555
$676$	0	0
$677$	−12.4878	−0.479945	−0.239973	−	0.970780i	$-0.577138\pi$
$677$	−12.4878	−0.479945	−0.239973	+	0.970780i	$0.577138\pi$
$678$	0	0
$679$	−5.68045	−0.217996
$680$	0	0
$681$	48.2009	1.84706
$682$	0	0
$683$	26.0471	0.996665	0.498332	−	0.866986i	$-0.333946\pi$
$683$	26.0471	0.996665	0.498332	+	0.866986i	$0.333946\pi$
$684$	0	0
$685$	−3.11985	−0.119203
$686$	0	0
$687$	−51.1711	−1.95230
$688$	0	0
$689$	−19.2617	−0.733814
$690$	0	0
$691$	46.4545	1.76721	0.883605	−	0.468232i	$-0.155109\pi$
$691$	46.4545	1.76721	0.883605	+	0.468232i	$0.155109\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−8.82210	−0.334641
$696$	0	0
$697$	56.1340	2.12623
$698$	0	0
$699$	35.8586	1.35630
$700$	0	0
$701$	25.2518	0.953747	0.476873	−	0.878972i	$-0.341770\pi$
$701$	25.2518	0.953747	0.476873	+	0.878972i	$0.341770\pi$
$702$	0	0
$703$	−29.2355	−1.10264
$704$	0	0
$705$	92.2939	3.47599
$706$	0	0
$707$	−1.92532	−0.0724093
$708$	0	0
$709$	5.80505	0.218013	0.109007	−	0.994041i	$-0.465233\pi$
$709$	5.80505	0.218013	0.109007	+	0.994041i	$0.465233\pi$
$710$	0	0
$711$	36.9660	1.38633
$712$	0	0
$713$	2.32913	0.0872268
$714$	0	0
$715$	0	0
$716$	0	0
$717$	77.5483	2.89609
$718$	0	0
$719$	−31.7231	−1.18307	−0.591537	−	0.806278i	$-0.701478\pi$
$719$	−31.7231	−1.18307	−0.591537	+	0.806278i	$0.701478\pi$
$720$	0	0
$721$	4.01411	0.149493
$722$	0	0
$723$	−3.84422	−0.142968
$724$	0	0
$725$	−20.0569	−0.744895
$726$	0	0
$727$	51.2435	1.90052	0.950258	−	0.311464i	$-0.100819\pi$
$727$	51.2435	1.90052	0.950258	+	0.311464i	$0.100819\pi$
$728$	0	0
$729$	−15.7553	−0.583529
$730$	0	0
$731$	33.3781	1.23453
$732$	0	0
$733$	−43.8519	−1.61971	−0.809853	−	0.586633i	$-0.800453\pi$
$733$	−43.8519	−1.61971	−0.809853	+	0.586633i	$0.800453\pi$
$734$	0	0
$735$	−41.8486	−1.54361
$736$	0	0
$737$	0	0
$738$	0	0
$739$	−31.0297	−1.14145	−0.570723	−	0.821142i	$-0.693337\pi$
$739$	−31.0297	−1.14145	−0.570723	+	0.821142i	$0.693337\pi$
$740$	0	0
$741$	−30.7271	−1.12879
$742$	0	0
$743$	23.7687	0.871991	0.435995	−	0.899949i	$-0.356396\pi$
$743$	23.7687	0.871991	0.435995	+	0.899949i	$0.356396\pi$
$744$	0	0
$745$	36.9740	1.35462
$746$	0	0
$747$	45.0857	1.64960
$748$	0	0
$749$	−21.3785	−0.781152
$750$	0	0
$751$	−42.6882	−1.55771	−0.778857	−	0.627202i	$-0.784200\pi$
$751$	−42.6882	−1.55771	−0.778857	+	0.627202i	$0.784200\pi$
$752$	0	0
$753$	70.5261	2.57011
$754$	0	0
$755$	−60.6700	−2.20801
$756$	0	0
$757$	−24.1511	−0.877785	−0.438893	−	0.898540i	$-0.644629\pi$
$757$	−24.1511	−0.877785	−0.438893	+	0.898540i	$0.644629\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	13.6328	0.494190	0.247095	−	0.968991i	$-0.420524\pi$
$761$	13.6328	0.494190	0.247095	+	0.968991i	$0.420524\pi$
$762$	0	0
$763$	1.46453	0.0530197
$764$	0	0
$765$	85.5384	3.09265
$766$	0	0
$767$	−32.6750	−1.17983
$768$	0	0
$769$	17.6960	0.638135	0.319068	−	0.947732i	$-0.396630\pi$
$769$	17.6960	0.638135	0.319068	+	0.947732i	$0.396630\pi$
$770$	0	0
$771$	16.5063	0.594458
$772$	0	0
$773$	27.6029	0.992809	0.496404	−	0.868091i	$-0.334653\pi$
$773$	27.6029	0.992809	0.496404	+	0.868091i	$0.334653\pi$
$774$	0	0
$775$	1.96816	0.0706983
$776$	0	0
$777$	50.9587	1.82813
$778$	0	0
$779$	34.4448	1.23411
$780$	0	0
$781$	0	0
$782$	0	0
$783$	51.0509	1.82441
$784$	0	0
$785$	−39.6918	−1.41666
$786$	0	0
$787$	20.1012	0.716529	0.358265	−	0.933620i	$-0.383369\pi$
$787$	20.1012	0.716529	0.358265	+	0.933620i	$0.383369\pi$
$788$	0	0
$789$	49.9775	1.77925
$790$	0	0
$791$	5.02608	0.178707
$792$	0	0
$793$	7.91616	0.281111
$794$	0	0
$795$	48.4282	1.71757
$796$	0	0
$797$	−13.4666	−0.477011	−0.238506	−	0.971141i	$-0.576658\pi$
$797$	−13.4666	−0.477011	−0.238506	+	0.971141i	$0.576658\pi$
$798$	0	0
$799$	−44.9238	−1.58929
$800$	0	0
$801$	49.3549	1.74387
$802$	0	0
$803$	0	0
$804$	0	0
$805$	22.4653	0.791799
$806$	0	0
$807$	−84.5477	−2.97622
$808$	0	0
$809$	8.69714	0.305775	0.152888	−	0.988244i	$-0.451143\pi$
$809$	8.69714	0.305775	0.152888	+	0.988244i	$0.451143\pi$
$810$	0	0
$811$	−12.5832	−0.441855	−0.220928	−	0.975290i	$-0.570908\pi$
$811$	−12.5832	−0.441855	−0.220928	+	0.975290i	$0.570908\pi$
$812$	0	0
$813$	91.4178	3.20616
$814$	0	0
$815$	−10.5534	−0.369671
$816$	0	0
$817$	20.4814	0.716553
$818$	0	0
$819$	36.3884	1.27151
$820$	0	0
$821$	34.6064	1.20777	0.603886	−	0.797071i	$-0.293618\pi$
$821$	34.6064	1.20777	0.603886	+	0.797071i	$0.293618\pi$
$822$	0	0
$823$	46.2135	1.61090	0.805451	−	0.592662i	$-0.201923\pi$
$823$	46.2135	1.61090	0.805451	+	0.592662i	$0.201923\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	12.5933	0.437913	0.218957	−	0.975735i	$-0.429735\pi$
$827$	12.5933	0.437913	0.218957	+	0.975735i	$0.429735\pi$
$828$	0	0
$829$	32.3556	1.12376	0.561879	−	0.827220i	$-0.310079\pi$
$829$	32.3556	1.12376	0.561879	+	0.827220i	$0.310079\pi$
$830$	0	0
$831$	−73.3528	−2.54458
$832$	0	0
$833$	20.3697	0.705768
$834$	0	0
$835$	−30.0871	−1.04121
$836$	0	0
$837$	−5.00956	−0.173156
$838$	0	0
$839$	−23.0636	−0.796245	−0.398123	−	0.917332i	$-0.630338\pi$
$839$	−23.0636	−0.796245	−0.398123	+	0.917332i	$0.630338\pi$
$840$	0	0
$841$	−4.29951	−0.148259
$842$	0	0
$843$	−35.1285	−1.20989
$844$	0	0
$845$	−1.13043	−0.0388881
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−50.1979	−1.72279
$850$	0	0
$851$	50.8379	1.74270
$852$	0	0
$853$	45.6465	1.56290	0.781452	−	0.623965i	$-0.214479\pi$
$853$	45.6465	1.56290	0.781452	+	0.623965i	$0.214479\pi$
$854$	0	0
$855$	52.4879	1.79505
$856$	0	0
$857$	25.6018	0.874541	0.437271	−	0.899330i	$-0.355945\pi$
$857$	25.6018	0.874541	0.437271	+	0.899330i	$0.355945\pi$
$858$	0	0
$859$	3.93870	0.134387	0.0671934	−	0.997740i	$-0.478596\pi$
$859$	3.93870	0.134387	0.0671934	+	0.997740i	$0.478596\pi$
$860$	0	0
$861$	−60.0386	−2.04611
$862$	0	0
$863$	−26.1546	−0.890313	−0.445156	−	0.895453i	$-0.646852\pi$
$863$	−26.1546	−0.890313	−0.445156	+	0.895453i	$0.646852\pi$
$864$	0	0
$865$	−7.54395	−0.256502
$866$	0	0
$867$	−9.27754	−0.315082
$868$	0	0
$869$	0	0
$870$	0	0
$871$	2.59448	0.0879107
$872$	0	0
$873$	23.0784	0.781085
$874$	0	0
$875$	−4.53639	−0.153358
$876$	0	0
$877$	17.9876	0.607397	0.303699	−	0.952768i	$-0.401778\pi$
$877$	17.9876	0.607397	0.303699	+	0.952768i	$0.401778\pi$
$878$	0	0
$879$	51.4106	1.73404
$880$	0	0
$881$	−13.1327	−0.442453	−0.221227	−	0.975222i	$-0.571006\pi$
$881$	−13.1327	−0.442453	−0.221227	+	0.975222i	$0.571006\pi$
$882$	0	0
$883$	−24.6832	−0.830655	−0.415328	−	0.909672i	$-0.636333\pi$
$883$	−24.6832	−0.830655	−0.415328	+	0.909672i	$0.636333\pi$
$884$	0	0
$885$	82.1521	2.76151
$886$	0	0
$887$	23.5929	0.792173	0.396086	−	0.918213i	$-0.370368\pi$
$887$	23.5929	0.792173	0.396086	+	0.918213i	$0.370368\pi$
$888$	0	0
$889$	−15.8998	−0.533263
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−27.5661	−0.922463
$894$	0	0
$895$	50.4859	1.68756
$896$	0	0
$897$	53.4316	1.78403
$898$	0	0
$899$	−2.42383	−0.0808391
$900$	0	0
$901$	−23.5723	−0.785307
$902$	0	0
$903$	−35.6999	−1.18802
$904$	0	0
$905$	−17.8502	−0.593362
$906$	0	0
$907$	−47.7002	−1.58386	−0.791930	−	0.610612i	$-0.790924\pi$
$907$	−47.7002	−1.58386	−0.791930	+	0.610612i	$0.790924\pi$
$908$	0	0
$909$	7.82216	0.259445
$910$	0	0
$911$	−0.986775	−0.0326933	−0.0163467	−	0.999866i	$-0.505204\pi$
$911$	−0.986775	−0.0326933	−0.0163467	+	0.999866i	$0.505204\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	−19.9030	−0.657972
$916$	0	0
$917$	−4.71581	−0.155730
$918$	0	0
$919$	19.9561	0.658291	0.329146	−	0.944279i	$-0.393239\pi$
$919$	19.9561	0.658291	0.329146	+	0.944279i	$0.393239\pi$
$920$	0	0
$921$	37.6481	1.24055
$922$	0	0
$923$	−2.95678	−0.0973238
$924$	0	0
$925$	42.9589	1.41248
$926$	0	0
$927$	−16.3084	−0.535639
$928$	0	0
$929$	27.0216	0.886550	0.443275	−	0.896386i	$-0.353817\pi$
$929$	27.0216	0.886550	0.443275	+	0.896386i	$0.353817\pi$
$930$	0	0
$931$	12.4992	0.409645
$932$	0	0
$933$	50.7707	1.66216
$934$	0	0
$935$	0	0
$936$	0	0
$937$	39.5367	1.29161	0.645803	−	0.763504i	$-0.276523\pi$
$937$	39.5367	1.29161	0.645803	+	0.763504i	$0.276523\pi$
$938$	0	0
$939$	69.8462	2.27935
$940$	0	0
$941$	−24.0796	−0.784972	−0.392486	−	0.919758i	$-0.628385\pi$
$941$	−24.0796	−0.784972	−0.392486	+	0.919758i	$0.628385\pi$
$942$	0	0
$943$	−59.8964	−1.95049
$944$	0	0
$945$	−48.3190	−1.57182
$946$	0	0
$947$	34.6471	1.12588	0.562940	−	0.826498i	$-0.309670\pi$
$947$	34.6471	1.12588	0.562940	+	0.826498i	$0.309670\pi$
$948$	0	0
$949$	1.93135	0.0626943
$950$	0	0
$951$	−14.5832	−0.472893
$952$	0	0
$953$	0.916697	0.0296947	0.0148474	−	0.999890i	$-0.495274\pi$
$953$	0.916697	0.0296947	0.0148474	+	0.999890i	$0.495274\pi$
$954$	0	0
$955$	−20.8653	−0.675184
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−1.62421	−0.0524484
$960$	0	0
$961$	−30.7622	−0.992328
$962$	0	0
$963$	86.8559	2.79889
$964$	0	0
$965$	−11.0000	−0.354103
$966$	0	0
$967$	40.8240	1.31281	0.656406	−	0.754408i	$-0.272076\pi$
$967$	40.8240	1.31281	0.656406	+	0.754408i	$0.272076\pi$
$968$	0	0
$969$	−37.6035	−1.20800
$970$	0	0
$971$	49.4231	1.58606	0.793031	−	0.609181i	$-0.208502\pi$
$971$	49.4231	1.58606	0.793031	+	0.609181i	$0.208502\pi$
$972$	0	0
$973$	−4.59283	−0.147239
$974$	0	0
$975$	45.1506	1.44598
$976$	0	0
$977$	−23.5440	−0.753239	−0.376619	−	0.926368i	$-0.622914\pi$
$977$	−23.5440	−0.753239	−0.376619	+	0.926368i	$0.622914\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	−5.95007	−0.189971
$982$	0	0
$983$	5.82382	0.185751	0.0928755	−	0.995678i	$-0.470394\pi$
$983$	5.82382	0.185751	0.0928755	+	0.995678i	$0.470394\pi$
$984$	0	0
$985$	52.8103	1.68268
$986$	0	0
$987$	48.0487	1.52941
$988$	0	0
$989$	−35.6153	−1.13250
$990$	0	0
$991$	−14.3008	−0.454280	−0.227140	−	0.973862i	$-0.572938\pi$
$991$	−14.3008	−0.454280	−0.227140	+	0.973862i	$0.572938\pi$
$992$	0	0
$993$	−99.3967	−3.15426
$994$	0	0
$995$	−21.4232	−0.679163
$996$	0	0
$997$	28.2737	0.895437	0.447719	−	0.894174i	$-0.352237\pi$
$997$	28.2737	0.895437	0.447719	+	0.894174i	$0.352237\pi$
$998$	0	0
$999$	−109.343	−3.45947

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.du.1.2		6
4.3	odd	2		7744.2.a.dv.1.5		6
8.3	odd	2		3872.2.a.bn.1.2		6
8.5	even	2		3872.2.a.bq.1.5		6
11.5	even	5		704.2.m.m.641.1		12
11.9	even	5		704.2.m.m.257.1		12
11.10	odd	2		7744.2.a.dt.1.2		6
44.27	odd	10		704.2.m.n.641.3		12
44.31	odd	10		704.2.m.n.257.3		12
44.43	even	2		7744.2.a.dw.1.5		6
88.5	even	10		352.2.m.e.289.3	yes	12
88.21	odd	2		3872.2.a.bp.1.5		6
88.27	odd	10		352.2.m.f.289.1	yes	12
88.43	even	2		3872.2.a.bo.1.2		6
88.53	even	10		352.2.m.e.257.3	✓	12
88.75	odd	10		352.2.m.f.257.1	yes	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
352.2.m.e.257.3	✓	12	88.53	even	10
352.2.m.e.289.3	yes	12	88.5	even	10
352.2.m.f.257.1	yes	12	88.75	odd	10
352.2.m.f.289.1	yes	12	88.27	odd	10
704.2.m.m.257.1		12	11.9	even	5
704.2.m.m.641.1		12	11.5	even	5
704.2.m.n.257.3		12	44.31	odd	10
704.2.m.n.641.3		12	44.27	odd	10
3872.2.a.bn.1.2		6	8.3	odd	2
3872.2.a.bo.1.2		6	88.43	even	2
3872.2.a.bp.1.5		6	88.21	odd	2
3872.2.a.bq.1.5		6	8.5	even	2
7744.2.a.dt.1.2		6	11.10	odd	2
7744.2.a.du.1.2		6	1.1	even	1	trivial
7744.2.a.dv.1.5		6	4.3	odd	2
7744.2.a.dw.1.5		6	44.43	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 $\F_{q}$
Belyi maps

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more

Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Coefficient data

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.du.1.2

Level	$7744$
Weight	$2$
Character	7744.1
Self dual	yes
Analytic conductor	$61.836$
Analytic rank	$1$
Dimension	$6$
CM	no
Inner twists	$1$