LMFDB - Embedded newform 3872.2.a.bo.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3872,2,Mod(1,3872)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3872, 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("3872.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3872 = 2^{5} \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3872.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:	$30.9180756626$
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	$\chi$	$=$	3872.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.05906 q^{3} +3.00593 q^{5} -1.56490 q^{7} +6.35786 q^{9} +O(q^{10})$	$q-3.05906 q^{3} +3.00593 q^{5} -1.56490 q^{7} +6.35786 q^{9} -3.65733 q^{13} -9.19533 q^{15} +4.47580 q^{17} +2.74643 q^{19} +4.78714 q^{21} -4.77580 q^{23} +4.03563 q^{25} -10.2719 q^{27} -4.96996 q^{29} -0.487696 q^{31} -4.70400 q^{35} -10.6449 q^{37} +11.1880 q^{39} +12.5417 q^{41} +7.45745 q^{43} +19.1113 q^{45} -10.0370 q^{47} -4.55107 q^{49} -13.6918 q^{51} -5.26661 q^{53} -8.40151 q^{57} +8.93411 q^{59} -2.16446 q^{61} -9.94944 q^{63} -10.9937 q^{65} -0.709392 q^{67} +14.6095 q^{69} -0.808454 q^{71} +0.528076 q^{73} -12.3452 q^{75} +5.81422 q^{79} +12.3488 q^{81} -7.09133 q^{83} +13.4540 q^{85} +15.2034 q^{87} +7.76282 q^{89} +5.72337 q^{91} +1.49189 q^{93} +8.25559 q^{95} +3.62990 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 5 q^{3} + 2 q^{7} + 7 q^{9}+O(q^{10}) $ 6 * q - 5 * q^3 + 2 * q^7 + 7 * q^9	$ 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}+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

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 $

$2$	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	$-0.265372\pi$
$5$	3.00593	1.34429	0.672147	+	0.740418i	$0.265372\pi$
$6$	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	$-0.317375\pi$
$17$	4.47580	1.08554	0.542771	+	0.839881i	$0.317375\pi$
$18$	0	0
$19$	2.74643	0.630075	0.315037	−	0.949079i	$-0.397983\pi$
$19$	2.74643	0.630075	0.315037	+	0.949079i	$0.397983\pi$
$20$	0	0
$21$	4.78714	1.04464
$22$	0	0
$23$	−4.77580	−0.995823	−0.497911	−	0.867228i	$-0.665899\pi$
$23$	−4.77580	−0.995823	−0.497911	+	0.867228i	$0.665899\pi$
$24$	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	$-0.513945\pi$
$31$	−0.487696	−0.0875927	−0.0437964	+	0.999040i	$0.513945\pi$
$32$	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	$-0.839143\pi$
$37$	−10.6449	−1.75001	−0.875006	+	0.484112i	$0.839143\pi$
$38$	0	0
$39$	11.1880	1.79151
$40$	0	0
$41$	12.5417	1.95868	0.979338	−	0.202228i	$-0.0648182\pi$
$41$	12.5417	1.95868	0.979338	+	0.202228i	$0.0648182\pi$
$42$	0	0
$43$	7.45745	1.13725	0.568625	−	0.822597i	$-0.307476\pi$
$43$	7.45745	1.13725	0.568625	+	0.822597i	$0.307476\pi$
$44$	0	0
$45$	19.1113	2.84894
$46$	0	0
$47$	−10.0370	−1.46405	−0.732026	−	0.681276i	$-0.761425\pi$
$47$	−10.0370	−1.46405	−0.732026	+	0.681276i	$0.761425\pi$
$48$	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	$-0.617808\pi$
$53$	−5.26661	−0.723424	−0.361712	+	0.932290i	$0.617808\pi$
$54$	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	$-0.515276\pi$
$71$	−0.808454	−0.0959459	−0.0479729	+	0.998849i	$0.515276\pi$
$72$	0	0
$73$	0.528076	0.0618066	0.0309033	−	0.999522i	$-0.490162\pi$
$73$	0.528076	0.0618066	0.0309033	+	0.999522i	$0.490162\pi$
$74$	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	$-0.627244\pi$
$83$	−7.09133	−0.778375	−0.389187	+	0.921159i	$0.627244\pi$
$84$	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	$-0.459667\pi$
$103$	2.56508	0.252745	0.126373	+	0.991983i	$0.459667\pi$
$104$	0	0
$105$	14.3898	1.40430
$106$	0	0
$107$	−13.6612	−1.32068	−0.660338	−	0.750968i	$-0.729587\pi$
$107$	−13.6612	−1.32068	−0.660338	+	0.750968i	$0.729587\pi$
$108$	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	$-0.542026\pi$
$131$	−3.01348	−0.263289	−0.131644	+	0.991297i	$0.542026\pi$
$132$	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	$-0.539722\pi$
$139$	−2.93490	−0.248935	−0.124467	+	0.992224i	$0.539722\pi$
$140$	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	$-0.676652\pi$
$157$	−13.2045	−1.05383	−0.526917	+	0.849917i	$0.676652\pi$
$158$	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	$-0.570833\pi$
$181$	−5.93834	−0.441393	−0.220697	+	0.975342i	$0.570833\pi$
$182$	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	$-0.580802\pi$
$191$	−6.94136	−0.502259	−0.251130	+	0.967953i	$0.580802\pi$
$192$	0	0
$193$	−3.65943	−0.263412	−0.131706	−	0.991289i	$-0.542045\pi$
$193$	−3.65943	−0.263412	−0.131706	+	0.991289i	$0.542045\pi$
$194$	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	$-0.581289\pi$
$199$	−7.12699	−0.505219	−0.252609	+	0.967568i	$0.581289\pi$
$200$	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	$-0.749557\pi$
$211$	−20.5140	−1.41224	−0.706121	+	0.708091i	$0.749557\pi$
$212$	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	$-0.567053\pi$
$223$	−6.24503	−0.418198	−0.209099	+	0.977894i	$0.567053\pi$
$224$	0	0
$225$	25.6580	1.71053
$226$	0	0
$227$	15.7568	1.04581	0.522907	−	0.852390i	$-0.324848\pi$
$227$	15.7568	1.04581	0.522907	+	0.852390i	$0.324848\pi$
$228$	0	0
$229$	−16.7277	−1.10540	−0.552699	−	0.833381i	$-0.686402\pi$
$229$	−16.7277	−1.10540	−0.552699	+	0.833381i	$0.686402\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	11.7221	0.767940	0.383970	−	0.923345i	$-0.374557\pi$
$233$	11.7221	0.767940	0.383970	+	0.923345i	$0.374557\pi$
$234$	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	$-0.512887\pi$
$241$	−1.25667	−0.0809490	−0.0404745	+	0.999181i	$0.512887\pi$
$242$	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	$-0.818960\pi$
$269$	−27.6384	−1.68514	−0.842572	+	0.538583i	$0.818960\pi$
$270$	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	$-0.611281\pi$
$281$	−11.4834	−0.685044	−0.342522	+	0.939510i	$0.611281\pi$
$282$	0	0
$283$	−16.4096	−0.975447	−0.487724	−	0.872998i	$-0.662173\pi$
$283$	−16.4096	−0.975447	−0.487724	+	0.872998i	$0.662173\pi$
$284$	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	$-0.385773\pi$
$307$	12.3071	0.702403	0.351201	+	0.936300i	$0.385773\pi$
$308$	0	0
$309$	−7.84675	−0.446386
$310$	0	0
$311$	16.5968	0.941119	0.470559	−	0.882368i	$-0.344052\pi$
$311$	16.5968	0.941119	0.470559	+	0.882368i	$0.344052\pi$
$312$	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	$-0.542743\pi$
$317$	−4.76722	−0.267754	−0.133877	+	0.990998i	$0.542743\pi$
$318$	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	$-0.416708\pi$
$337$	9.49799	0.517388	0.258694	+	0.965959i	$0.416708\pi$
$338$	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	$-0.911955\pi$
$347$	−35.8397	−1.92398	−0.961989	+	0.273088i	$0.911955\pi$
$348$	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	$-0.447960\pi$
$367$	6.23615	0.325525	0.162762	+	0.986665i	$0.447960\pi$
$368$	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	$-0.594957\pi$
$383$	−11.5039	−0.587823	−0.293911	+	0.955833i	$0.594957\pi$
$384$	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	$-0.424248\pi$
$389$	9.29913	0.471485	0.235742	+	0.971816i	$0.424248\pi$
$390$	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	$-0.0879589\pi$
$397$	38.3379	1.92413	0.962063	+	0.272828i	$0.0879589\pi$
$398$	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	$-0.521205\pi$
$409$	−2.69257	−0.133139	−0.0665696	+	0.997782i	$0.521205\pi$
$410$	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	$-0.327340\pi$
$421$	21.1838	1.03243	0.516217	+	0.856458i	$0.327340\pi$
$422$	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	$-0.770877\pi$
$457$	−32.1488	−1.50386	−0.751930	+	0.659243i	$0.770877\pi$
$458$	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	$-0.527597\pi$
$463$	−3.72640	−0.173181	−0.0865903	+	0.996244i	$0.527597\pi$
$464$	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	$-0.414919\pi$
$487$	11.6571	0.528236	0.264118	+	0.964490i	$0.414919\pi$
$488$	0	0
$489$	−10.7400	−0.485678
$490$	0	0
$491$	18.5772	0.838377	0.419189	−	0.907899i	$-0.362315\pi$
$491$	18.5772	0.838377	0.419189	+	0.907899i	$0.362315\pi$
$492$	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	$-0.168303\pi$
$509$	38.9603	1.72689	0.863443	+	0.504446i	$0.168303\pi$
$510$	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	$-0.626881\pi$
$523$	−17.7528	−0.776275	−0.388137	+	0.921602i	$0.626881\pi$
$524$	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	$-0.556720\pi$
$547$	−8.29110	−0.354502	−0.177251	+	0.984166i	$0.556720\pi$
$548$	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	$-0.246749\pi$
$563$	33.8969	1.42859	0.714293	+	0.699847i	$0.246749\pi$
$564$	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	$-0.381841\pi$
$569$	17.3055	0.725485	0.362742	+	0.931889i	$0.381841\pi$
$570$	0	0
$571$	15.8559	0.663551	0.331775	−	0.943358i	$-0.392352\pi$
$571$	15.8559	0.663551	0.331775	+	0.943358i	$0.392352\pi$
$572$	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	$-0.258860\pi$
$593$	33.4666	1.37431	0.687154	+	0.726512i	$0.258860\pi$
$594$	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	$-0.387215\pi$
$599$	16.9832	0.693915	0.346957	+	0.937881i	$0.387215\pi$
$600$	0	0
$601$	−0.0293746	−0.00119821	−0.000599107	−	1.00000i	$-0.500191\pi$
$601$	−0.0293746	−0.00119821	−0.000599107		1.00000i	$0.500191\pi$
$602$	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	$-0.704074\pi$
$631$	−30.0478	−1.19618	−0.598092	+	0.801427i	$0.704074\pi$
$632$	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	$-0.570182\pi$
$647$	−11.1258	−0.437401	−0.218701	+	0.975792i	$0.570182\pi$
$648$	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	$-0.604907\pi$
$653$	−16.5405	−0.647282	−0.323641	+	0.946180i	$0.604907\pi$
$654$	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	$-0.669716\pi$
$659$	−26.0957	−1.01655	−0.508273	+	0.861196i	$0.669716\pi$
$660$	0	0
$661$	15.3262	0.596121	0.298061	−	0.954547i	$-0.403660\pi$
$661$	15.3262	0.596121	0.298061	+	0.954547i	$0.403660\pi$
$662$	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	$-0.711283\pi$
$673$	−31.9654	−1.23218	−0.616088	+	0.787678i	$0.711283\pi$
$674$	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	$-0.534767\pi$
$709$	−5.80505	−0.218013	−0.109007	+	0.994041i	$0.534767\pi$
$710$	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	$-0.298522\pi$
$719$	31.7231	1.18307	0.591537	+	0.806278i	$0.298522\pi$
$720$	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	$-0.899181\pi$
$727$	−51.2435	−1.90052	−0.950258	+	0.311464i	$0.899181\pi$
$728$	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	$-0.306663\pi$
$739$	31.0297	1.14145	0.570723	+	0.821142i	$0.306663\pi$
$740$	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	$-0.215800\pi$
$751$	42.6882	1.55771	0.778857	+	0.627202i	$0.215800\pi$
$752$	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	$-0.355371\pi$
$757$	24.1511	0.877785	0.438893	+	0.898540i	$0.355371\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−13.6328	−0.494190	−0.247095	−	0.968991i	$-0.579476\pi$
$761$	−13.6328	−0.494190	−0.247095	+	0.968991i	$0.579476\pi$
$762$	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	$-0.603370\pi$
$769$	−17.6960	−0.638135	−0.319068	+	0.947732i	$0.603370\pi$
$770$	0	0
$771$	16.5063	0.594458
$772$	0	0
$773$	−27.6029	−0.992809	−0.496404	−	0.868091i	$-0.665347\pi$
$773$	−27.6029	−0.992809	−0.496404	+	0.868091i	$0.665347\pi$
$774$	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	$-0.616631\pi$
$787$	−20.1012	−0.716529	−0.358265	+	0.933620i	$0.616631\pi$
$788$	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	$-0.423342\pi$
$797$	13.4666	0.477011	0.238506	+	0.971141i	$0.423342\pi$
$798$	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	$-0.548857\pi$
$809$	−8.69714	−0.305775	−0.152888	+	0.988244i	$0.548857\pi$
$810$	0	0
$811$	12.5832	0.441855	0.220928	−	0.975290i	$-0.429092\pi$
$811$	12.5832	0.441855	0.220928	+	0.975290i	$0.429092\pi$
$812$	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	$-0.798077\pi$
$823$	−46.2135	−1.61090	−0.805451	+	0.592662i	$0.798077\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−12.5933	−0.437913	−0.218957	−	0.975735i	$-0.570265\pi$
$827$	−12.5933	−0.437913	−0.218957	+	0.975735i	$0.570265\pi$
$828$	0	0
$829$	−32.3556	−1.12376	−0.561879	−	0.827220i	$-0.689921\pi$
$829$	−32.3556	−1.12376	−0.561879	+	0.827220i	$0.689921\pi$
$830$	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	$-0.369662\pi$
$839$	23.0636	0.796245	0.398123	+	0.917332i	$0.369662\pi$
$840$	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	$-0.644055\pi$
$857$	−25.6018	−0.874541	−0.437271	+	0.899330i	$0.644055\pi$
$858$	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	$-0.353148\pi$
$863$	26.1546	0.890313	0.445156	+	0.895453i	$0.353148\pi$
$864$	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	$-0.494796\pi$
$911$	0.986775	0.0326933	0.0163467	+	0.999866i	$0.494796\pi$
$912$	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	$-0.723477\pi$
$937$	−39.5367	−1.29161	−0.645803	+	0.763504i	$0.723477\pi$
$938$	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	$-0.504726\pi$
$953$	−0.916697	−0.0296947	−0.0148474	+	0.999890i	$0.504726\pi$
$954$	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	$-0.529606\pi$
$983$	−5.82382	−0.185751	−0.0928755	+	0.995678i	$0.529606\pi$
$984$	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	$-0.427062\pi$
$991$	14.3008	0.454280	0.227140	+	0.973862i	$0.427062\pi$
$992$	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

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3872.2.a.bo.1.2		6
4.3	odd	2		3872.2.a.bp.1.5		6
8.3	odd	2		7744.2.a.dt.1.2		6
8.5	even	2		7744.2.a.dw.1.5		6
11.2	odd	10		352.2.m.f.257.1	yes	12
11.6	odd	10		352.2.m.f.289.1	yes	12
11.10	odd	2		3872.2.a.bn.1.2		6
44.35	even	10		352.2.m.e.257.3	✓	12
44.39	even	10		352.2.m.e.289.3	yes	12
44.43	even	2		3872.2.a.bq.1.5		6
88.13	odd	10		704.2.m.n.257.3		12
88.21	odd	2		7744.2.a.dv.1.5		6
88.35	even	10		704.2.m.m.257.1		12
88.43	even	2		7744.2.a.du.1.2		6
88.61	odd	10		704.2.m.n.641.3		12
88.83	even	10		704.2.m.m.641.1		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
352.2.m.e.257.3	✓	12	44.35	even	10
352.2.m.e.289.3	yes	12	44.39	even	10
352.2.m.f.257.1	yes	12	11.2	odd	10
352.2.m.f.289.1	yes	12	11.6	odd	10
704.2.m.m.257.1		12	88.35	even	10
704.2.m.m.641.1		12	88.83	even	10
704.2.m.n.257.3		12	88.13	odd	10
704.2.m.n.641.3		12	88.61	odd	10
3872.2.a.bn.1.2		6	11.10	odd	2
3872.2.a.bo.1.2		6	1.1	even	1	trivial
3872.2.a.bp.1.5		6	4.3	odd	2
3872.2.a.bq.1.5		6	44.43	even	2
7744.2.a.dt.1.2		6	8.3	odd	2
7744.2.a.du.1.2		6	88.43	even	2
7744.2.a.dv.1.5		6	88.21	odd	2
7744.2.a.dw.1.5		6	8.5	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:	\( N \)	\(=\)	\( 3872 = 2^{5} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3872.a (trivial)

\(f(q)\)	\(=\)	\(q-3.05906 q^{3} +3.00593 q^{5} -1.56490 q^{7} +6.35786 q^{9} +O(q^{10})\)	\(q-3.05906 q^{3} +3.00593 q^{5} -1.56490 q^{7} +6.35786 q^{9} -3.65733 q^{13} -9.19533 q^{15} +4.47580 q^{17} +2.74643 q^{19} +4.78714 q^{21} -4.77580 q^{23} +4.03563 q^{25} -10.2719 q^{27} -4.96996 q^{29} -0.487696 q^{31} -4.70400 q^{35} -10.6449 q^{37} +11.1880 q^{39} +12.5417 q^{41} +7.45745 q^{43} +19.1113 q^{45} -10.0370 q^{47} -4.55107 q^{49} -13.6918 q^{51} -5.26661 q^{53} -8.40151 q^{57} +8.93411 q^{59} -2.16446 q^{61} -9.94944 q^{63} -10.9937 q^{65} -0.709392 q^{67} +14.6095 q^{69} -0.808454 q^{71} +0.528076 q^{73} -12.3452 q^{75} +5.81422 q^{79} +12.3488 q^{81} -7.09133 q^{83} +13.4540 q^{85} +15.2034 q^{87} +7.76282 q^{89} +5.72337 q^{91} +1.49189 q^{93} +8.25559 q^{95} +3.62990 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 5 q^{3} + 2 q^{7} + 7 q^{9}+O(q^{10}) \) 6 * q - 5 * q^3 + 2 * q^7 + 7 * q^9	\( 6 q - 5 q^{3} + 2 q^{7} + 7 q^{9} - 4 q^{13} - 8 q^{15} + 9 q^{17} + 5 q^{19} - 12 q^{21} - 6 q^{23} + 4 q^{25} - 26 q^{27} - 10 q^{29} - 12 q^{31} - 26 q^{35} - 12 q^{37} + 2 q^{39} + 17 q^{41} + 11 q^{43} + 14 q^{45} - 22 q^{47} - 6 q^{51} + 18 q^{53} + 6 q^{57} - 7 q^{59} - 18 q^{61} + 26 q^{63} + 26 q^{65} - 31 q^{67} + 16 q^{69} - 22 q^{71} - q^{73} - 11 q^{75} - 28 q^{79} + 34 q^{81} - 13 q^{83} - 2 q^{85} + 38 q^{87} - q^{89} - 26 q^{91} + 34 q^{93} + 2 q^{95} - 21 q^{97}+O(q^{100}) \) 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

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