LMFDB - Embedded newform 4416.2.a.bp.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4416,2,Mod(1,4416)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4416, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("4416.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4416 = 2^{6} \cdot 3 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4416.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:	$35.2619375326$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.148.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 3x + 1 $ x^3 - x^2 - 3*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 552)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.48119$ of defining polynomial
Character	$\chi$	$=$	4416.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} -3.35026 q^{5} -2.96239 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} -3.35026 q^{5} -2.96239 q^{7} +1.00000 q^{9} -1.61213 q^{11} -2.00000 q^{13} +3.35026 q^{15} +4.96239 q^{17} +1.35026 q^{19} +2.96239 q^{21} -1.00000 q^{23} +6.22425 q^{25} -1.00000 q^{27} +7.92478 q^{29} +5.92478 q^{31} +1.61213 q^{33} +9.92478 q^{35} +2.31265 q^{37} +2.00000 q^{39} -1.22425 q^{41} +4.57452 q^{43} -3.35026 q^{45} +1.92478 q^{47} +1.77575 q^{49} -4.96239 q^{51} +4.12601 q^{53} +5.40105 q^{55} -1.35026 q^{57} -2.70052 q^{59} -14.3127 q^{61} -2.96239 q^{63} +6.70052 q^{65} -4.57452 q^{67} +1.00000 q^{69} -16.6253 q^{71} +2.00000 q^{73} -6.22425 q^{75} +4.77575 q^{77} -5.03761 q^{79} +1.00000 q^{81} +15.0132 q^{83} -16.6253 q^{85} -7.92478 q^{87} -5.73813 q^{89} +5.92478 q^{91} -5.92478 q^{93} -4.52373 q^{95} +12.7005 q^{97} -1.61213 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{3} + 2 q^{7} + 3 q^{9}+O(q^{10}) $ 3 * q - 3 * q^3 + 2 * q^7 + 3 * q^9	$ 3 q - 3 q^{3} + 2 q^{7} + 3 q^{9} - 4 q^{11} - 6 q^{13} + 4 q^{17} - 6 q^{19} - 2 q^{21} - 3 q^{23} + 17 q^{25} - 3 q^{27} + 2 q^{29} - 4 q^{31} + 4 q^{33} + 8 q^{35} - 14 q^{37} + 6 q^{39} - 2 q^{41} + 2 q^{43} - 16 q^{47} + 7 q^{49} - 4 q^{51} + 4 q^{53} - 24 q^{55} + 6 q^{57} + 12 q^{59} - 22 q^{61} + 2 q^{63} - 2 q^{67} + 3 q^{69} - 8 q^{71} + 6 q^{73} - 17 q^{75} + 16 q^{77} - 26 q^{79} + 3 q^{81} + 4 q^{83} - 8 q^{85} - 2 q^{87} - 8 q^{89} - 4 q^{91} + 4 q^{93} - 32 q^{95} + 18 q^{97} - 4 q^{99}+O(q^{100}) $ 3 * q - 3 * q^3 + 2 * q^7 + 3 * q^9 - 4 * q^11 - 6 * q^13 + 4 * q^17 - 6 * q^19 - 2 * q^21 - 3 * q^23 + 17 * q^25 - 3 * q^27 + 2 * q^29 - 4 * q^31 + 4 * q^33 + 8 * q^35 - 14 * q^37 + 6 * q^39 - 2 * q^41 + 2 * q^43 - 16 * q^47 + 7 * q^49 - 4 * q^51 + 4 * q^53 - 24 * q^55 + 6 * q^57 + 12 * q^59 - 22 * q^61 + 2 * q^63 - 2 * q^67 + 3 * q^69 - 8 * q^71 + 6 * q^73 - 17 * q^75 + 16 * q^77 - 26 * q^79 + 3 * q^81 + 4 * q^83 - 8 * q^85 - 2 * q^87 - 8 * q^89 - 4 * q^91 + 4 * q^93 - 32 * q^95 + 18 * q^97 - 4 * q^99

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
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	−3.35026	−1.49828	−0.749141	−	0.662410i	$-0.769534\pi$
$5$	−3.35026	−1.49828	−0.749141	+	0.662410i	$0.769534\pi$
$6$	0	0
$7$	−2.96239	−1.11968	−0.559839	−	0.828602i	$-0.689137\pi$
$7$	−2.96239	−1.11968	−0.559839	+	0.828602i	$0.689137\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−1.61213	−0.486075	−0.243037	−	0.970017i	$-0.578144\pi$
$11$	−1.61213	−0.486075	−0.243037	+	0.970017i	$0.578144\pi$
$12$	0	0
$13$	−2.00000	−0.554700	−0.277350	−	0.960769i	$-0.589456\pi$
$13$	−2.00000	−0.554700	−0.277350	+	0.960769i	$0.589456\pi$
$14$	0	0
$15$	3.35026	0.865034
$16$	0	0
$17$	4.96239	1.20356	0.601778	−	0.798663i	$-0.294459\pi$
$17$	4.96239	1.20356	0.601778	+	0.798663i	$0.294459\pi$
$18$	0	0
$19$	1.35026	0.309771	0.154886	−	0.987932i	$-0.450499\pi$
$19$	1.35026	0.309771	0.154886	+	0.987932i	$0.450499\pi$
$20$	0	0
$21$	2.96239	0.646446
$22$	0	0
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	6.22425	1.24485
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	7.92478	1.47159	0.735797	−	0.677202i	$-0.236808\pi$
$29$	7.92478	1.47159	0.735797	+	0.677202i	$0.236808\pi$
$30$	0	0
$31$	5.92478	1.06412	0.532061	−	0.846706i	$-0.321418\pi$
$31$	5.92478	1.06412	0.532061	+	0.846706i	$0.321418\pi$
$32$	0	0
$33$	1.61213	0.280635
$34$	0	0
$35$	9.92478	1.67759
$36$	0	0
$37$	2.31265	0.380197	0.190099	−	0.981765i	$-0.439119\pi$
$37$	2.31265	0.380197	0.190099	+	0.981765i	$0.439119\pi$
$38$	0	0
$39$	2.00000	0.320256
$40$	0	0
$41$	−1.22425	−0.191196	−0.0955982	−	0.995420i	$-0.530476\pi$
$41$	−1.22425	−0.191196	−0.0955982	+	0.995420i	$0.530476\pi$
$42$	0	0
$43$	4.57452	0.697607	0.348804	−	0.937196i	$-0.386588\pi$
$43$	4.57452	0.697607	0.348804	+	0.937196i	$0.386588\pi$
$44$	0	0
$45$	−3.35026	−0.499428
$46$	0	0
$47$	1.92478	0.280758	0.140379	−	0.990098i	$-0.455168\pi$
$47$	1.92478	0.280758	0.140379	+	0.990098i	$0.455168\pi$
$48$	0	0
$49$	1.77575	0.253678
$50$	0	0
$51$	−4.96239	−0.694873
$52$	0	0
$53$	4.12601	0.566751	0.283375	−	0.959009i	$-0.408546\pi$
$53$	4.12601	0.566751	0.283375	+	0.959009i	$0.408546\pi$
$54$	0	0
$55$	5.40105	0.728277
$56$	0	0
$57$	−1.35026	−0.178847
$58$	0	0
$59$	−2.70052	−0.351578	−0.175789	−	0.984428i	$-0.556248\pi$
$59$	−2.70052	−0.351578	−0.175789	+	0.984428i	$0.556248\pi$
$60$	0	0
$61$	−14.3127	−1.83255	−0.916274	−	0.400553i	$-0.868818\pi$
$61$	−14.3127	−1.83255	−0.916274	+	0.400553i	$0.868818\pi$
$62$	0	0
$63$	−2.96239	−0.373226
$64$	0	0
$65$	6.70052	0.831098
$66$	0	0
$67$	−4.57452	−0.558866	−0.279433	−	0.960165i	$-0.590146\pi$
$67$	−4.57452	−0.558866	−0.279433	+	0.960165i	$0.590146\pi$
$68$	0	0
$69$	1.00000	0.120386
$70$	0	0
$71$	−16.6253	−1.97306	−0.986530	−	0.163580i	$-0.947696\pi$
$71$	−16.6253	−1.97306	−0.986530	+	0.163580i	$0.947696\pi$
$72$	0	0
$73$	2.00000	0.234082	0.117041	−	0.993127i	$-0.462659\pi$
$73$	2.00000	0.234082	0.117041	+	0.993127i	$0.462659\pi$
$74$	0	0
$75$	−6.22425	−0.718715
$76$	0	0
$77$	4.77575	0.544247
$78$	0	0
$79$	−5.03761	−0.566776	−0.283388	−	0.959005i	$-0.591458\pi$
$79$	−5.03761	−0.566776	−0.283388	+	0.959005i	$0.591458\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	15.0132	1.64791	0.823955	−	0.566655i	$-0.191763\pi$
$83$	15.0132	1.64791	0.823955	+	0.566655i	$0.191763\pi$
$84$	0	0
$85$	−16.6253	−1.80327
$86$	0	0
$87$	−7.92478	−0.849625
$88$	0	0
$89$	−5.73813	−0.608241	−0.304121	−	0.952634i	$-0.598363\pi$
$89$	−5.73813	−0.608241	−0.304121	+	0.952634i	$0.598363\pi$
$90$	0	0
$91$	5.92478	0.621085
$92$	0	0
$93$	−5.92478	−0.614371
$94$	0	0
$95$	−4.52373	−0.464125
$96$	0	0
$97$	12.7005	1.28954	0.644771	−	0.764375i	$-0.276953\pi$
$97$	12.7005	1.28954	0.644771	+	0.764375i	$0.276953\pi$
$98$	0	0
$99$	−1.61213	−0.162025
$100$	0	0
$101$	15.4010	1.53246	0.766231	−	0.642566i	$-0.222130\pi$
$101$	15.4010	1.53246	0.766231	+	0.642566i	$0.222130\pi$
$102$	0	0
$103$	−13.6629	−1.34625	−0.673123	−	0.739530i	$-0.735048\pi$
$103$	−13.6629	−1.34625	−0.673123	+	0.739530i	$0.735048\pi$
$104$	0	0
$105$	−9.92478	−0.968559
$106$	0	0
$107$	−18.2374	−1.76308	−0.881539	−	0.472111i	$-0.843492\pi$
$107$	−18.2374	−1.76308	−0.881539	+	0.472111i	$0.843492\pi$
$108$	0	0
$109$	19.4617	1.86409	0.932045	−	0.362341i	$-0.118023\pi$
$109$	19.4617	1.86409	0.932045	+	0.362341i	$0.118023\pi$
$110$	0	0
$111$	−2.31265	−0.219507
$112$	0	0
$113$	7.66291	0.720866	0.360433	−	0.932785i	$-0.382629\pi$
$113$	7.66291	0.720866	0.360433	+	0.932785i	$0.382629\pi$
$114$	0	0
$115$	3.35026	0.312414
$116$	0	0
$117$	−2.00000	−0.184900
$118$	0	0
$119$	−14.7005	−1.34759
$120$	0	0
$121$	−8.40105	−0.763732
$122$	0	0
$123$	1.22425	0.110387
$124$	0	0
$125$	−4.10157	−0.366856
$126$	0	0
$127$	−15.4763	−1.37330	−0.686648	−	0.726990i	$-0.740919\pi$
$127$	−15.4763	−1.37330	−0.686648	+	0.726990i	$0.740919\pi$
$128$	0	0
$129$	−4.57452	−0.402764
$130$	0	0
$131$	−0.775746	−0.0677773	−0.0338886	−	0.999426i	$-0.510789\pi$
$131$	−0.775746	−0.0677773	−0.0338886	+	0.999426i	$0.510789\pi$
$132$	0	0
$133$	−4.00000	−0.346844
$134$	0	0
$135$	3.35026	0.288345
$136$	0	0
$137$	−8.96239	−0.765709	−0.382854	−	0.923809i	$-0.625059\pi$
$137$	−8.96239	−0.765709	−0.382854	+	0.923809i	$0.625059\pi$
$138$	0	0
$139$	4.00000	0.339276	0.169638	−	0.985506i	$-0.445740\pi$
$139$	4.00000	0.339276	0.169638	+	0.985506i	$0.445740\pi$
$140$	0	0
$141$	−1.92478	−0.162095
$142$	0	0
$143$	3.22425	0.269626
$144$	0	0
$145$	−26.5501	−2.20486
$146$	0	0
$147$	−1.77575	−0.146461
$148$	0	0
$149$	10.0508	0.823392	0.411696	−	0.911321i	$-0.364936\pi$
$149$	10.0508	0.823392	0.411696	+	0.911321i	$0.364936\pi$
$150$	0	0
$151$	11.8496	0.964303	0.482152	−	0.876088i	$-0.339855\pi$
$151$	11.8496	0.964303	0.482152	+	0.876088i	$0.339855\pi$
$152$	0	0
$153$	4.96239	0.401185
$154$	0	0
$155$	−19.8496	−1.59435
$156$	0	0
$157$	−22.9380	−1.83065	−0.915324	−	0.402718i	$-0.868065\pi$
$157$	−22.9380	−1.83065	−0.915324	+	0.402718i	$0.868065\pi$
$158$	0	0
$159$	−4.12601	−0.327214
$160$	0	0
$161$	2.96239	0.233469
$162$	0	0
$163$	−6.70052	−0.524826	−0.262413	−	0.964956i	$-0.584518\pi$
$163$	−6.70052	−0.524826	−0.262413	+	0.964956i	$0.584518\pi$
$164$	0	0
$165$	−5.40105	−0.420471
$166$	0	0
$167$	24.6253	1.90556	0.952781	−	0.303657i	$-0.0982076\pi$
$167$	24.6253	1.90556	0.952781	+	0.303657i	$0.0982076\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	0	0
$171$	1.35026	0.103257
$172$	0	0
$173$	23.9248	1.81897	0.909484	−	0.415740i	$-0.136477\pi$
$173$	23.9248	1.81897	0.909484	+	0.415740i	$0.136477\pi$
$174$	0	0
$175$	−18.4387	−1.39383
$176$	0	0
$177$	2.70052	0.202984
$178$	0	0
$179$	−19.3258	−1.44448	−0.722240	−	0.691643i	$-0.756887\pi$
$179$	−19.3258	−1.44448	−0.722240	+	0.691643i	$0.756887\pi$
$180$	0	0
$181$	−20.2374	−1.50424	−0.752118	−	0.659028i	$-0.770968\pi$
$181$	−20.2374	−1.50424	−0.752118	+	0.659028i	$0.770968\pi$
$182$	0	0
$183$	14.3127	1.05802
$184$	0	0
$185$	−7.74798	−0.569643
$186$	0	0
$187$	−8.00000	−0.585018
$188$	0	0
$189$	2.96239	0.215482
$190$	0	0
$191$	−9.14903	−0.662001	−0.331000	−	0.943631i	$-0.607386\pi$
$191$	−9.14903	−0.662001	−0.331000	+	0.943631i	$0.607386\pi$
$192$	0	0
$193$	−6.77575	−0.487729	−0.243864	−	0.969809i	$-0.578415\pi$
$193$	−6.77575	−0.487729	−0.243864	+	0.969809i	$0.578415\pi$
$194$	0	0
$195$	−6.70052	−0.479834
$196$	0	0
$197$	24.5501	1.74912	0.874560	−	0.484917i	$-0.161150\pi$
$197$	24.5501	1.74912	0.874560	+	0.484917i	$0.161150\pi$
$198$	0	0
$199$	−15.3357	−1.08712	−0.543559	−	0.839371i	$-0.682923\pi$
$199$	−15.3357	−1.08712	−0.543559	+	0.839371i	$0.682923\pi$
$200$	0	0
$201$	4.57452	0.322661
$202$	0	0
$203$	−23.4763	−1.64771
$204$	0	0
$205$	4.10157	0.286466
$206$	0	0
$207$	−1.00000	−0.0695048
$208$	0	0
$209$	−2.17679	−0.150572
$210$	0	0
$211$	5.14903	0.354474	0.177237	−	0.984168i	$-0.443284\pi$
$211$	5.14903	0.354474	0.177237	+	0.984168i	$0.443284\pi$
$212$	0	0
$213$	16.6253	1.13915
$214$	0	0
$215$	−15.3258	−1.04521
$216$	0	0
$217$	−17.5515	−1.19147
$218$	0	0
$219$	−2.00000	−0.135147
$220$	0	0
$221$	−9.92478	−0.667613
$222$	0	0
$223$	9.55149	0.639615	0.319808	−	0.947482i	$-0.396382\pi$
$223$	9.55149	0.639615	0.319808	+	0.947482i	$0.396382\pi$
$224$	0	0
$225$	6.22425	0.414950
$226$	0	0
$227$	−3.68735	−0.244738	−0.122369	−	0.992485i	$-0.539049\pi$
$227$	−3.68735	−0.244738	−0.122369	+	0.992485i	$0.539049\pi$
$228$	0	0
$229$	−12.2374	−0.808672	−0.404336	−	0.914611i	$-0.632497\pi$
$229$	−12.2374	−0.808672	−0.404336	+	0.914611i	$0.632497\pi$
$230$	0	0
$231$	−4.77575	−0.314221
$232$	0	0
$233$	−3.40105	−0.222810	−0.111405	−	0.993775i	$-0.535535\pi$
$233$	−3.40105	−0.222810	−0.111405	+	0.993775i	$0.535535\pi$
$234$	0	0
$235$	−6.44851	−0.420654
$236$	0	0
$237$	5.03761	0.327228
$238$	0	0
$239$	−1.55149	−0.100358	−0.0501789	−	0.998740i	$-0.515979\pi$
$239$	−1.55149	−0.100358	−0.0501789	+	0.998740i	$0.515979\pi$
$240$	0	0
$241$	20.7005	1.33344	0.666719	−	0.745309i	$-0.267698\pi$
$241$	20.7005	1.33344	0.666719	+	0.745309i	$0.267698\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	−5.94921	−0.380081
$246$	0	0
$247$	−2.70052	−0.171830
$248$	0	0
$249$	−15.0132	−0.951421
$250$	0	0
$251$	−8.46310	−0.534186	−0.267093	−	0.963671i	$-0.586063\pi$
$251$	−8.46310	−0.534186	−0.267093	+	0.963671i	$0.586063\pi$
$252$	0	0
$253$	1.61213	0.101354
$254$	0	0
$255$	16.6253	1.04112
$256$	0	0
$257$	18.6253	1.16181	0.580907	−	0.813970i	$-0.302698\pi$
$257$	18.6253	1.16181	0.580907	+	0.813970i	$0.302698\pi$
$258$	0	0
$259$	−6.85097	−0.425699
$260$	0	0
$261$	7.92478	0.490531
$262$	0	0
$263$	−0.523730	−0.0322946	−0.0161473	−	0.999870i	$-0.505140\pi$
$263$	−0.523730	−0.0322946	−0.0161473	+	0.999870i	$0.505140\pi$
$264$	0	0
$265$	−13.8232	−0.849153
$266$	0	0
$267$	5.73813	0.351168
$268$	0	0
$269$	−0.700523	−0.0427117	−0.0213558	−	0.999772i	$-0.506798\pi$
$269$	−0.700523	−0.0427117	−0.0213558	+	0.999772i	$0.506798\pi$
$270$	0	0
$271$	−10.0752	−0.612026	−0.306013	−	0.952027i	$-0.598995\pi$
$271$	−10.0752	−0.612026	−0.306013	+	0.952027i	$0.598995\pi$
$272$	0	0
$273$	−5.92478	−0.358584
$274$	0	0
$275$	−10.0343	−0.605090
$276$	0	0
$277$	22.4749	1.35038	0.675192	−	0.737642i	$-0.264061\pi$
$277$	22.4749	1.35038	0.675192	+	0.737642i	$0.264061\pi$
$278$	0	0
$279$	5.92478	0.354707
$280$	0	0
$281$	11.8134	0.704726	0.352363	−	0.935863i	$-0.385378\pi$
$281$	11.8134	0.704726	0.352363	+	0.935863i	$0.385378\pi$
$282$	0	0
$283$	−29.1998	−1.73575	−0.867874	−	0.496784i	$-0.834514\pi$
$283$	−29.1998	−1.73575	−0.867874	+	0.496784i	$0.834514\pi$
$284$	0	0
$285$	4.52373	0.267963
$286$	0	0
$287$	3.62672	0.214078
$288$	0	0
$289$	7.62530	0.448547
$290$	0	0
$291$	−12.7005	−0.744518
$292$	0	0
$293$	−9.27504	−0.541854	−0.270927	−	0.962600i	$-0.587330\pi$
$293$	−9.27504	−0.541854	−0.270927	+	0.962600i	$0.587330\pi$
$294$	0	0
$295$	9.04746	0.526764
$296$	0	0
$297$	1.61213	0.0935451
$298$	0	0
$299$	2.00000	0.115663
$300$	0	0
$301$	−13.5515	−0.781095
$302$	0	0
$303$	−15.4010	−0.884767
$304$	0	0
$305$	47.9511	2.74567
$306$	0	0
$307$	−31.9511	−1.82355	−0.911774	−	0.410693i	$-0.865287\pi$
$307$	−31.9511	−1.82355	−0.911774	+	0.410693i	$0.865287\pi$
$308$	0	0
$309$	13.6629	0.777256
$310$	0	0
$311$	13.1490	0.745613	0.372807	−	0.927909i	$-0.378395\pi$
$311$	13.1490	0.745613	0.372807	+	0.927909i	$0.378395\pi$
$312$	0	0
$313$	24.0263	1.35805	0.679025	−	0.734115i	$-0.262403\pi$
$313$	24.0263	1.35805	0.679025	+	0.734115i	$0.262403\pi$
$314$	0	0
$315$	9.92478	0.559198
$316$	0	0
$317$	−27.9248	−1.56841	−0.784206	−	0.620501i	$-0.786929\pi$
$317$	−27.9248	−1.56841	−0.784206	+	0.620501i	$0.786929\pi$
$318$	0	0
$319$	−12.7757	−0.715304
$320$	0	0
$321$	18.2374	1.01791
$322$	0	0
$323$	6.70052	0.372827
$324$	0	0
$325$	−12.4485	−0.690519
$326$	0	0
$327$	−19.4617	−1.07623
$328$	0	0
$329$	−5.70194	−0.314358
$330$	0	0
$331$	−6.70052	−0.368294	−0.184147	−	0.982899i	$-0.558952\pi$
$331$	−6.70052	−0.368294	−0.184147	+	0.982899i	$0.558952\pi$
$332$	0	0
$333$	2.31265	0.126732
$334$	0	0
$335$	15.3258	0.837339
$336$	0	0
$337$	−10.3733	−0.565069	−0.282534	−	0.959257i	$-0.591175\pi$
$337$	−10.3733	−0.565069	−0.282534	+	0.959257i	$0.591175\pi$
$338$	0	0
$339$	−7.66291	−0.416192
$340$	0	0
$341$	−9.55149	−0.517242
$342$	0	0
$343$	15.4763	0.835640
$344$	0	0
$345$	−3.35026	−0.180372
$346$	0	0
$347$	17.7743	0.954176	0.477088	−	0.878855i	$-0.341692\pi$
$347$	17.7743	0.954176	0.477088	+	0.878855i	$0.341692\pi$
$348$	0	0
$349$	−18.6253	−0.996989	−0.498495	−	0.866893i	$-0.666114\pi$
$349$	−18.6253	−0.996989	−0.498495	+	0.866893i	$0.666114\pi$
$350$	0	0
$351$	2.00000	0.106752
$352$	0	0
$353$	−15.7743	−0.839583	−0.419791	−	0.907621i	$-0.637897\pi$
$353$	−15.7743	−0.839583	−0.419791	+	0.907621i	$0.637897\pi$
$354$	0	0
$355$	55.6991	2.95620
$356$	0	0
$357$	14.7005	0.778034
$358$	0	0
$359$	−5.29948	−0.279696	−0.139848	−	0.990173i	$-0.544661\pi$
$359$	−5.29948	−0.279696	−0.139848	+	0.990173i	$0.544661\pi$
$360$	0	0
$361$	−17.1768	−0.904042
$362$	0	0
$363$	8.40105	0.440941
$364$	0	0
$365$	−6.70052	−0.350721
$366$	0	0
$367$	4.51388	0.235623	0.117811	−	0.993036i	$-0.462412\pi$
$367$	4.51388	0.235623	0.117811	+	0.993036i	$0.462412\pi$
$368$	0	0
$369$	−1.22425	−0.0637321
$370$	0	0
$371$	−12.2228	−0.634578
$372$	0	0
$373$	−13.6873	−0.708704	−0.354352	−	0.935112i	$-0.615299\pi$
$373$	−13.6873	−0.708704	−0.354352	+	0.935112i	$0.615299\pi$
$374$	0	0
$375$	4.10157	0.211804
$376$	0	0
$377$	−15.8496	−0.816294
$378$	0	0
$379$	−30.3488	−1.55892	−0.779458	−	0.626455i	$-0.784505\pi$
$379$	−30.3488	−1.55892	−0.779458	+	0.626455i	$0.784505\pi$
$380$	0	0
$381$	15.4763	0.792873
$382$	0	0
$383$	28.3733	1.44981	0.724904	−	0.688850i	$-0.241884\pi$
$383$	28.3733	1.44981	0.724904	+	0.688850i	$0.241884\pi$
$384$	0	0
$385$	−16.0000	−0.815436
$386$	0	0
$387$	4.57452	0.232536
$388$	0	0
$389$	−19.4518	−0.986247	−0.493124	−	0.869959i	$-0.664145\pi$
$389$	−19.4518	−0.986247	−0.493124	+	0.869959i	$0.664145\pi$
$390$	0	0
$391$	−4.96239	−0.250959
$392$	0	0
$393$	0.775746	0.0391312
$394$	0	0
$395$	16.8773	0.849190
$396$	0	0
$397$	28.8021	1.44554	0.722768	−	0.691091i	$-0.242870\pi$
$397$	28.8021	1.44554	0.722768	+	0.691091i	$0.242870\pi$
$398$	0	0
$399$	4.00000	0.200250
$400$	0	0
$401$	−24.1378	−1.20538	−0.602691	−	0.797974i	$-0.705905\pi$
$401$	−24.1378	−1.20538	−0.602691	+	0.797974i	$0.705905\pi$
$402$	0	0
$403$	−11.8496	−0.590268
$404$	0	0
$405$	−3.35026	−0.166476
$406$	0	0
$407$	−3.72829	−0.184804
$408$	0	0
$409$	−28.1768	−1.39325	−0.696626	−	0.717434i	$-0.745316\pi$
$409$	−28.1768	−1.39325	−0.696626	+	0.717434i	$0.745316\pi$
$410$	0	0
$411$	8.96239	0.442082
$412$	0	0
$413$	8.00000	0.393654
$414$	0	0
$415$	−50.2981	−2.46903
$416$	0	0
$417$	−4.00000	−0.195881
$418$	0	0
$419$	21.4617	1.04847	0.524236	−	0.851573i	$-0.324351\pi$
$419$	21.4617	1.04847	0.524236	+	0.851573i	$0.324351\pi$
$420$	0	0
$421$	−13.1636	−0.641556	−0.320778	−	0.947154i	$-0.603944\pi$
$421$	−13.1636	−0.641556	−0.320778	+	0.947154i	$0.603944\pi$
$422$	0	0
$423$	1.92478	0.0935859
$424$	0	0
$425$	30.8872	1.49825
$426$	0	0
$427$	42.3996	2.05186
$428$	0	0
$429$	−3.22425	−0.155668
$430$	0	0
$431$	−21.5223	−1.03669	−0.518347	−	0.855171i	$-0.673452\pi$
$431$	−21.5223	−1.03669	−0.518347	+	0.855171i	$0.673452\pi$
$432$	0	0
$433$	−2.77575	−0.133394	−0.0666969	−	0.997773i	$-0.521246\pi$
$433$	−2.77575	−0.133394	−0.0666969	+	0.997773i	$0.521246\pi$
$434$	0	0
$435$	26.5501	1.27298
$436$	0	0
$437$	−1.35026	−0.0645918
$438$	0	0
$439$	3.84955	0.183729	0.0918646	−	0.995772i	$-0.470717\pi$
$439$	3.84955	0.183729	0.0918646	+	0.995772i	$0.470717\pi$
$440$	0	0
$441$	1.77575	0.0845593
$442$	0	0
$443$	−26.9525	−1.28055	−0.640277	−	0.768144i	$-0.721180\pi$
$443$	−26.9525	−1.28055	−0.640277	+	0.768144i	$0.721180\pi$
$444$	0	0
$445$	19.2243	0.911317
$446$	0	0
$447$	−10.0508	−0.475386
$448$	0	0
$449$	1.07381	0.0506761	0.0253381	−	0.999679i	$-0.491934\pi$
$449$	1.07381	0.0506761	0.0253381	+	0.999679i	$0.491934\pi$
$450$	0	0
$451$	1.97365	0.0929357
$452$	0	0
$453$	−11.8496	−0.556741
$454$	0	0
$455$	−19.8496	−0.930561
$456$	0	0
$457$	7.29948	0.341455	0.170728	−	0.985318i	$-0.445388\pi$
$457$	7.29948	0.341455	0.170728	+	0.985318i	$0.445388\pi$
$458$	0	0
$459$	−4.96239	−0.231624
$460$	0	0
$461$	−5.59754	−0.260703	−0.130352	−	0.991468i	$-0.541611\pi$
$461$	−5.59754	−0.260703	−0.130352	+	0.991468i	$0.541611\pi$
$462$	0	0
$463$	−3.84955	−0.178904	−0.0894520	−	0.995991i	$-0.528512\pi$
$463$	−3.84955	−0.178904	−0.0894520	+	0.995991i	$0.528512\pi$
$464$	0	0
$465$	19.8496	0.920501
$466$	0	0
$467$	3.68735	0.170630	0.0853151	−	0.996354i	$-0.472810\pi$
$467$	3.68735	0.170630	0.0853151	+	0.996354i	$0.472810\pi$
$468$	0	0
$469$	13.5515	0.625750
$470$	0	0
$471$	22.9380	1.05692
$472$	0	0
$473$	−7.37470	−0.339089
$474$	0	0
$475$	8.40437	0.385619
$476$	0	0
$477$	4.12601	0.188917
$478$	0	0
$479$	−2.59895	−0.118749	−0.0593746	−	0.998236i	$-0.518911\pi$
$479$	−2.59895	−0.118749	−0.0593746	+	0.998236i	$0.518911\pi$
$480$	0	0
$481$	−4.62530	−0.210896
$482$	0	0
$483$	−2.96239	−0.134793
$484$	0	0
$485$	−42.5501	−1.93210
$486$	0	0
$487$	33.2506	1.50673	0.753364	−	0.657603i	$-0.228430\pi$
$487$	33.2506	1.50673	0.753364	+	0.657603i	$0.228430\pi$
$488$	0	0
$489$	6.70052	0.303008
$490$	0	0
$491$	13.9248	0.628416	0.314208	−	0.949354i	$-0.398261\pi$
$491$	13.9248	0.628416	0.314208	+	0.949354i	$0.398261\pi$
$492$	0	0
$493$	39.3258	1.77115
$494$	0	0
$495$	5.40105	0.242759
$496$	0	0
$497$	49.2506	2.20919
$498$	0	0
$499$	−19.5975	−0.877306	−0.438653	−	0.898656i	$-0.644544\pi$
$499$	−19.5975	−0.877306	−0.438653	+	0.898656i	$0.644544\pi$
$500$	0	0
$501$	−24.6253	−1.10018
$502$	0	0
$503$	28.4749	1.26963	0.634816	−	0.772664i	$-0.281076\pi$
$503$	28.4749	1.26963	0.634816	+	0.772664i	$0.281076\pi$
$504$	0	0
$505$	−51.5975	−2.29606
$506$	0	0
$507$	9.00000	0.399704
$508$	0	0
$509$	39.6239	1.75630	0.878149	−	0.478387i	$-0.158778\pi$
$509$	39.6239	1.75630	0.878149	+	0.478387i	$0.158778\pi$
$510$	0	0
$511$	−5.92478	−0.262097
$512$	0	0
$513$	−1.35026	−0.0596155
$514$	0	0
$515$	45.7743	2.01706
$516$	0	0
$517$	−3.10299	−0.136469
$518$	0	0
$519$	−23.9248	−1.05018
$520$	0	0
$521$	−18.2130	−0.797926	−0.398963	−	0.916967i	$-0.630630\pi$
$521$	−18.2130	−0.797926	−0.398963	+	0.916967i	$0.630630\pi$
$522$	0	0
$523$	24.8265	1.08559	0.542794	−	0.839866i	$-0.317366\pi$
$523$	24.8265	1.08559	0.542794	+	0.839866i	$0.317366\pi$
$524$	0	0
$525$	18.4387	0.804729
$526$	0	0
$527$	29.4010	1.28073
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	−2.70052	−0.117193
$532$	0	0
$533$	2.44851	0.106057
$534$	0	0
$535$	61.1002	2.64159
$536$	0	0
$537$	19.3258	0.833971
$538$	0	0
$539$	−2.86273	−0.123306
$540$	0	0
$541$	−6.47486	−0.278376	−0.139188	−	0.990266i	$-0.544449\pi$
$541$	−6.47486	−0.278376	−0.139188	+	0.990266i	$0.544449\pi$
$542$	0	0
$543$	20.2374	0.868471
$544$	0	0
$545$	−65.2017	−2.79294
$546$	0	0
$547$	−2.55008	−0.109033	−0.0545167	−	0.998513i	$-0.517362\pi$
$547$	−2.55008	−0.109033	−0.0545167	+	0.998513i	$0.517362\pi$
$548$	0	0
$549$	−14.3127	−0.610849
$550$	0	0
$551$	10.7005	0.455858
$552$	0	0
$553$	14.9234	0.634606
$554$	0	0
$555$	7.74798	0.328884
$556$	0	0
$557$	4.75131	0.201319	0.100660	−	0.994921i	$-0.467905\pi$
$557$	4.75131	0.201319	0.100660	+	0.994921i	$0.467905\pi$
$558$	0	0
$559$	−9.14903	−0.386963
$560$	0	0
$561$	8.00000	0.337760
$562$	0	0
$563$	−19.6873	−0.829723	−0.414861	−	0.909885i	$-0.636170\pi$
$563$	−19.6873	−0.829723	−0.414861	+	0.909885i	$0.636170\pi$
$564$	0	0
$565$	−25.6728	−1.08006
$566$	0	0
$567$	−2.96239	−0.124409
$568$	0	0
$569$	−1.48612	−0.0623013	−0.0311507	−	0.999515i	$-0.509917\pi$
$569$	−1.48612	−0.0623013	−0.0311507	+	0.999515i	$0.509917\pi$
$570$	0	0
$571$	11.1246	0.465550	0.232775	−	0.972531i	$-0.425219\pi$
$571$	11.1246	0.465550	0.232775	+	0.972531i	$0.425219\pi$
$572$	0	0
$573$	9.14903	0.382206
$574$	0	0
$575$	−6.22425	−0.259569
$576$	0	0
$577$	0.176793	0.00736000	0.00368000	−	0.999993i	$-0.498829\pi$
$577$	0.176793	0.00736000	0.00368000	+	0.999993i	$0.498829\pi$
$578$	0	0
$579$	6.77575	0.281590
$580$	0	0
$581$	−44.4749	−1.84513
$582$	0	0
$583$	−6.65165	−0.275483
$584$	0	0
$585$	6.70052	0.277033
$586$	0	0
$587$	13.6728	0.564335	0.282168	−	0.959365i	$-0.408947\pi$
$587$	13.6728	0.564335	0.282168	+	0.959365i	$0.408947\pi$
$588$	0	0
$589$	8.00000	0.329634
$590$	0	0
$591$	−24.5501	−1.00986
$592$	0	0
$593$	21.2243	0.871576	0.435788	−	0.900049i	$-0.356470\pi$
$593$	21.2243	0.871576	0.435788	+	0.900049i	$0.356470\pi$
$594$	0	0
$595$	49.2506	2.01908
$596$	0	0
$597$	15.3357	0.627647
$598$	0	0
$599$	−32.0000	−1.30748	−0.653742	−	0.756717i	$-0.726802\pi$
$599$	−32.0000	−1.30748	−0.653742	+	0.756717i	$0.726802\pi$
$600$	0	0
$601$	40.1768	1.63885	0.819423	−	0.573190i	$-0.194294\pi$
$601$	40.1768	1.63885	0.819423	+	0.573190i	$0.194294\pi$
$602$	0	0
$603$	−4.57452	−0.186289
$604$	0	0
$605$	28.1457	1.14429
$606$	0	0
$607$	−24.0000	−0.974130	−0.487065	−	0.873366i	$-0.661933\pi$
$607$	−24.0000	−0.974130	−0.487065	+	0.873366i	$0.661933\pi$
$608$	0	0
$609$	23.4763	0.951306
$610$	0	0
$611$	−3.84955	−0.155736
$612$	0	0
$613$	−14.3127	−0.578083	−0.289041	−	0.957317i	$-0.593336\pi$
$613$	−14.3127	−0.578083	−0.289041	+	0.957317i	$0.593336\pi$
$614$	0	0
$615$	−4.10157	−0.165391
$616$	0	0
$617$	−20.6907	−0.832975	−0.416488	−	0.909141i	$-0.636739\pi$
$617$	−20.6907	−0.832975	−0.416488	+	0.909141i	$0.636739\pi$
$618$	0	0
$619$	−32.3028	−1.29836	−0.649180	−	0.760635i	$-0.724888\pi$
$619$	−32.3028	−1.29836	−0.649180	+	0.760635i	$0.724888\pi$
$620$	0	0
$621$	1.00000	0.0401286
$622$	0	0
$623$	16.9986	0.681034
$624$	0	0
$625$	−17.3799	−0.695197
$626$	0	0
$627$	2.17679	0.0869327
$628$	0	0
$629$	11.4763	0.457589
$630$	0	0
$631$	3.58769	0.142824	0.0714118	−	0.997447i	$-0.477250\pi$
$631$	3.58769	0.142824	0.0714118	+	0.997447i	$0.477250\pi$
$632$	0	0
$633$	−5.14903	−0.204656
$634$	0	0
$635$	51.8496	2.05759
$636$	0	0
$637$	−3.55149	−0.140715
$638$	0	0
$639$	−16.6253	−0.657687
$640$	0	0
$641$	3.71179	0.146607	0.0733034	−	0.997310i	$-0.476646\pi$
$641$	3.71179	0.146607	0.0733034	+	0.997310i	$0.476646\pi$
$642$	0	0
$643$	−43.1246	−1.70067	−0.850334	−	0.526243i	$-0.823600\pi$
$643$	−43.1246	−1.70067	−0.850334	+	0.526243i	$0.823600\pi$
$644$	0	0
$645$	15.3258	0.603454
$646$	0	0
$647$	−28.1016	−1.10479	−0.552393	−	0.833584i	$-0.686285\pi$
$647$	−28.1016	−1.10479	−0.552393	+	0.833584i	$0.686285\pi$
$648$	0	0
$649$	4.35359	0.170893
$650$	0	0
$651$	17.5515	0.687897
$652$	0	0
$653$	10.0000	0.391330	0.195665	−	0.980671i	$-0.437313\pi$
$653$	10.0000	0.391330	0.195665	+	0.980671i	$0.437313\pi$
$654$	0	0
$655$	2.59895	0.101549
$656$	0	0
$657$	2.00000	0.0780274
$658$	0	0
$659$	33.8350	1.31802	0.659012	−	0.752133i	$-0.270975\pi$
$659$	33.8350	1.31802	0.659012	+	0.752133i	$0.270975\pi$
$660$	0	0
$661$	−16.6107	−0.646082	−0.323041	−	0.946385i	$-0.604705\pi$
$661$	−16.6107	−0.646082	−0.323041	+	0.946385i	$0.604705\pi$
$662$	0	0
$663$	9.92478	0.385446
$664$	0	0
$665$	13.4010	0.519670
$666$	0	0
$667$	−7.92478	−0.306849
$668$	0	0
$669$	−9.55149	−0.369282
$670$	0	0
$671$	23.0738	0.890754
$672$	0	0
$673$	8.17679	0.315192	0.157596	−	0.987504i	$-0.449626\pi$
$673$	8.17679	0.315192	0.157596	+	0.987504i	$0.449626\pi$
$674$	0	0
$675$	−6.22425	−0.239572
$676$	0	0
$677$	−46.8989	−1.80247	−0.901236	−	0.433328i	$-0.857339\pi$
$677$	−46.8989	−1.80247	−0.901236	+	0.433328i	$0.857339\pi$
$678$	0	0
$679$	−37.6239	−1.44387
$680$	0	0
$681$	3.68735	0.141300
$682$	0	0
$683$	4.92619	0.188495	0.0942477	−	0.995549i	$-0.469955\pi$
$683$	4.92619	0.188495	0.0942477	+	0.995549i	$0.469955\pi$
$684$	0	0
$685$	30.0263	1.14725
$686$	0	0
$687$	12.2374	0.466887
$688$	0	0
$689$	−8.25202	−0.314377
$690$	0	0
$691$	36.1016	1.37337	0.686684	−	0.726956i	$-0.259066\pi$
$691$	36.1016	1.37337	0.686684	+	0.726956i	$0.259066\pi$
$692$	0	0
$693$	4.77575	0.181416
$694$	0	0
$695$	−13.4010	−0.508331
$696$	0	0
$697$	−6.07522	−0.230115
$698$	0	0
$699$	3.40105	0.128639
$700$	0	0
$701$	−46.2736	−1.74773	−0.873865	−	0.486168i	$-0.838394\pi$
$701$	−46.2736	−1.74773	−0.873865	+	0.486168i	$0.838394\pi$
$702$	0	0
$703$	3.12268	0.117774
$704$	0	0
$705$	6.44851	0.242865
$706$	0	0
$707$	−45.6239	−1.71586
$708$	0	0
$709$	−26.5647	−0.997657	−0.498828	−	0.866701i	$-0.666236\pi$
$709$	−26.5647	−0.997657	−0.498828	+	0.866701i	$0.666236\pi$
$710$	0	0
$711$	−5.03761	−0.188925
$712$	0	0
$713$	−5.92478	−0.221885
$714$	0	0
$715$	−10.8021	−0.403975
$716$	0	0
$717$	1.55149	0.0579416
$718$	0	0
$719$	−26.5501	−0.990151	−0.495075	−	0.868850i	$-0.664860\pi$
$719$	−26.5501	−0.990151	−0.495075	+	0.868850i	$0.664860\pi$
$720$	0	0
$721$	40.4749	1.50736
$722$	0	0
$723$	−20.7005	−0.769861
$724$	0	0
$725$	49.3258	1.83192
$726$	0	0
$727$	18.4387	0.683852	0.341926	−	0.939727i	$-0.388921\pi$
$727$	18.4387	0.683852	0.341926	+	0.939727i	$0.388921\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	22.7005	0.839609
$732$	0	0
$733$	−20.1162	−0.743007	−0.371504	−	0.928431i	$-0.621158\pi$
$733$	−20.1162	−0.743007	−0.371504	+	0.928431i	$0.621158\pi$
$734$	0	0
$735$	5.94921	0.219440
$736$	0	0
$737$	7.37470	0.271651
$738$	0	0
$739$	34.6516	1.27468	0.637341	−	0.770582i	$-0.280034\pi$
$739$	34.6516	1.27468	0.637341	+	0.770582i	$0.280034\pi$
$740$	0	0
$741$	2.70052	0.0992062
$742$	0	0
$743$	−3.32582	−0.122013	−0.0610063	−	0.998137i	$-0.519431\pi$
$743$	−3.32582	−0.122013	−0.0610063	+	0.998137i	$0.519431\pi$
$744$	0	0
$745$	−33.6728	−1.23367
$746$	0	0
$747$	15.0132	0.549303
$748$	0	0
$749$	54.0263	1.97408
$750$	0	0
$751$	28.1114	1.02580	0.512900	−	0.858448i	$-0.328571\pi$
$751$	28.1114	1.02580	0.512900	+	0.858448i	$0.328571\pi$
$752$	0	0
$753$	8.46310	0.308412
$754$	0	0
$755$	−39.6991	−1.44480
$756$	0	0
$757$	−42.6859	−1.55145	−0.775723	−	0.631073i	$-0.782615\pi$
$757$	−42.6859	−1.55145	−0.775723	+	0.631073i	$0.782615\pi$
$758$	0	0
$759$	−1.61213	−0.0585165
$760$	0	0
$761$	−23.7743	−0.861819	−0.430909	−	0.902395i	$-0.641807\pi$
$761$	−23.7743	−0.861819	−0.430909	+	0.902395i	$0.641807\pi$
$762$	0	0
$763$	−57.6531	−2.08718
$764$	0	0
$765$	−16.6253	−0.601089
$766$	0	0
$767$	5.40105	0.195021
$768$	0	0
$769$	−33.2243	−1.19810	−0.599049	−	0.800713i	$-0.704454\pi$
$769$	−33.2243	−1.19810	−0.599049	+	0.800713i	$0.704454\pi$
$770$	0	0
$771$	−18.6253	−0.670774
$772$	0	0
$773$	−4.27645	−0.153813	−0.0769067	−	0.997038i	$-0.524504\pi$
$773$	−4.27645	−0.153813	−0.0769067	+	0.997038i	$0.524504\pi$
$774$	0	0
$775$	36.8773	1.32467
$776$	0	0
$777$	6.85097	0.245777
$778$	0	0
$779$	−1.65306	−0.0592271
$780$	0	0
$781$	26.8021	0.959054
$782$	0	0
$783$	−7.92478	−0.283208
$784$	0	0
$785$	76.8481	2.74283
$786$	0	0
$787$	19.0230	0.678098	0.339049	−	0.940769i	$-0.389895\pi$
$787$	19.0230	0.678098	0.339049	+	0.940769i	$0.389895\pi$
$788$	0	0
$789$	0.523730	0.0186453
$790$	0	0
$791$	−22.7005	−0.807138
$792$	0	0
$793$	28.6253	1.01651
$794$	0	0
$795$	13.8232	0.490259
$796$	0	0
$797$	7.35026	0.260360	0.130180	−	0.991490i	$-0.458445\pi$
$797$	7.35026	0.260360	0.130180	+	0.991490i	$0.458445\pi$
$798$	0	0
$799$	9.55149	0.337908
$800$	0	0
$801$	−5.73813	−0.202747
$802$	0	0
$803$	−3.22425	−0.113781
$804$	0	0
$805$	−9.92478	−0.349802
$806$	0	0
$807$	0.700523	0.0246596
$808$	0	0
$809$	−26.3733	−0.927235	−0.463618	−	0.886035i	$-0.653449\pi$
$809$	−26.3733	−0.927235	−0.463618	+	0.886035i	$0.653449\pi$
$810$	0	0
$811$	8.99859	0.315983	0.157992	−	0.987440i	$-0.449498\pi$
$811$	8.99859	0.315983	0.157992	+	0.987440i	$0.449498\pi$
$812$	0	0
$813$	10.0752	0.353353
$814$	0	0
$815$	22.4485	0.786337
$816$	0	0
$817$	6.17679	0.216099
$818$	0	0
$819$	5.92478	0.207028
$820$	0	0
$821$	−9.10299	−0.317696	−0.158848	−	0.987303i	$-0.550778\pi$
$821$	−9.10299	−0.317696	−0.158848	+	0.987303i	$0.550778\pi$
$822$	0	0
$823$	−33.7743	−1.17730	−0.588650	−	0.808388i	$-0.700340\pi$
$823$	−33.7743	−1.17730	−0.588650	+	0.808388i	$0.700340\pi$
$824$	0	0
$825$	10.0343	0.349349
$826$	0	0
$827$	−23.4156	−0.814241	−0.407121	−	0.913374i	$-0.633467\pi$
$827$	−23.4156	−0.814241	−0.407121	+	0.913374i	$0.633467\pi$
$828$	0	0
$829$	10.7757	0.374257	0.187129	−	0.982335i	$-0.440082\pi$
$829$	10.7757	0.374257	0.187129	+	0.982335i	$0.440082\pi$
$830$	0	0
$831$	−22.4749	−0.779644
$832$	0	0
$833$	8.81194	0.305316
$834$	0	0
$835$	−82.5012	−2.85507
$836$	0	0
$837$	−5.92478	−0.204790
$838$	0	0
$839$	−5.92478	−0.204546	−0.102273	−	0.994756i	$-0.532612\pi$
$839$	−5.92478	−0.204546	−0.102273	+	0.994756i	$0.532612\pi$
$840$	0	0
$841$	33.8021	1.16559
$842$	0	0
$843$	−11.8134	−0.406874
$844$	0	0
$845$	30.1524	1.03727
$846$	0	0
$847$	24.8872	0.855133
$848$	0	0
$849$	29.1998	1.00214
$850$	0	0
$851$	−2.31265	−0.0792766
$852$	0	0
$853$	−38.0000	−1.30110	−0.650548	−	0.759465i	$-0.725461\pi$
$853$	−38.0000	−1.30110	−0.650548	+	0.759465i	$0.725461\pi$
$854$	0	0
$855$	−4.52373	−0.154708
$856$	0	0
$857$	18.0000	0.614868	0.307434	−	0.951569i	$-0.400530\pi$
$857$	18.0000	0.614868	0.307434	+	0.951569i	$0.400530\pi$
$858$	0	0
$859$	−20.0000	−0.682391	−0.341196	−	0.939992i	$-0.610832\pi$
$859$	−20.0000	−0.682391	−0.341196	+	0.939992i	$0.610832\pi$
$860$	0	0
$861$	−3.62672	−0.123598
$862$	0	0
$863$	25.2995	0.861204	0.430602	−	0.902542i	$-0.358301\pi$
$863$	25.2995	0.861204	0.430602	+	0.902542i	$0.358301\pi$
$864$	0	0
$865$	−80.1543	−2.72533
$866$	0	0
$867$	−7.62530	−0.258969
$868$	0	0
$869$	8.12127	0.275495
$870$	0	0
$871$	9.14903	0.310003
$872$	0	0
$873$	12.7005	0.429848
$874$	0	0
$875$	12.1504	0.410760
$876$	0	0
$877$	−30.6253	−1.03414	−0.517071	−	0.855942i	$-0.672978\pi$
$877$	−30.6253	−1.03414	−0.517071	+	0.855942i	$0.672978\pi$
$878$	0	0
$879$	9.27504	0.312839
$880$	0	0
$881$	−44.1866	−1.48869	−0.744343	−	0.667798i	$-0.767237\pi$
$881$	−44.1866	−1.48869	−0.744343	+	0.667798i	$0.767237\pi$
$882$	0	0
$883$	6.70052	0.225491	0.112745	−	0.993624i	$-0.464036\pi$
$883$	6.70052	0.225491	0.112745	+	0.993624i	$0.464036\pi$
$884$	0	0
$885$	−9.04746	−0.304127
$886$	0	0
$887$	−32.5764	−1.09381	−0.546905	−	0.837195i	$-0.684194\pi$
$887$	−32.5764	−1.09381	−0.546905	+	0.837195i	$0.684194\pi$
$888$	0	0
$889$	45.8467	1.53765
$890$	0	0
$891$	−1.61213	−0.0540083
$892$	0	0
$893$	2.59895	0.0869706
$894$	0	0
$895$	64.7466	2.16424
$896$	0	0
$897$	−2.00000	−0.0667781
$898$	0	0
$899$	46.9525	1.56595
$900$	0	0
$901$	20.4749	0.682116
$902$	0	0
$903$	13.5515	0.450965
$904$	0	0
$905$	67.8007	2.25377
$906$	0	0
$907$	1.75272	0.0581982	0.0290991	−	0.999577i	$-0.490736\pi$
$907$	1.75272	0.0581982	0.0290991	+	0.999577i	$0.490736\pi$
$908$	0	0
$909$	15.4010	0.510820
$910$	0	0
$911$	−9.67276	−0.320473	−0.160236	−	0.987079i	$-0.551226\pi$
$911$	−9.67276	−0.320473	−0.160236	+	0.987079i	$0.551226\pi$
$912$	0	0
$913$	−24.2031	−0.801007
$914$	0	0
$915$	−47.9511	−1.58522
$916$	0	0
$917$	2.29806	0.0758887
$918$	0	0
$919$	25.4109	0.838228	0.419114	−	0.907934i	$-0.362341\pi$
$919$	25.4109	0.838228	0.419114	+	0.907934i	$0.362341\pi$
$920$	0	0
$921$	31.9511	1.05283
$922$	0	0
$923$	33.2506	1.09446
$924$	0	0
$925$	14.3945	0.473289
$926$	0	0
$927$	−13.6629	−0.448749
$928$	0	0
$929$	24.0263	0.788279	0.394139	−	0.919051i	$-0.371043\pi$
$929$	24.0263	0.788279	0.394139	+	0.919051i	$0.371043\pi$
$930$	0	0
$931$	2.39772	0.0785822
$932$	0	0
$933$	−13.1490	−0.430480
$934$	0	0
$935$	26.8021	0.876522
$936$	0	0
$937$	−49.8496	−1.62851	−0.814257	−	0.580505i	$-0.802855\pi$
$937$	−49.8496	−1.62851	−0.814257	+	0.580505i	$0.802855\pi$
$938$	0	0
$939$	−24.0263	−0.784070
$940$	0	0
$941$	30.2012	0.984532	0.492266	−	0.870445i	$-0.336169\pi$
$941$	30.2012	0.984532	0.492266	+	0.870445i	$0.336169\pi$
$942$	0	0
$943$	1.22425	0.0398672
$944$	0	0
$945$	−9.92478	−0.322853
$946$	0	0
$947$	10.8218	0.351661	0.175830	−	0.984420i	$-0.443739\pi$
$947$	10.8218	0.351661	0.175830	+	0.984420i	$0.443739\pi$
$948$	0	0
$949$	−4.00000	−0.129845
$950$	0	0
$951$	27.9248	0.905523
$952$	0	0
$953$	−33.6893	−1.09130	−0.545651	−	0.838012i	$-0.683718\pi$
$953$	−33.6893	−1.09130	−0.545651	+	0.838012i	$0.683718\pi$
$954$	0	0
$955$	30.6516	0.991864
$956$	0	0
$957$	12.7757	0.412981
$958$	0	0
$959$	26.5501	0.857347
$960$	0	0
$961$	4.10299	0.132354
$962$	0	0
$963$	−18.2374	−0.587693
$964$	0	0
$965$	22.7005	0.730756
$966$	0	0
$967$	43.0249	1.38359	0.691794	−	0.722095i	$-0.256820\pi$
$967$	43.0249	1.38359	0.691794	+	0.722095i	$0.256820\pi$
$968$	0	0
$969$	−6.70052	−0.215252
$970$	0	0
$971$	−42.0381	−1.34907	−0.674534	−	0.738244i	$-0.735655\pi$
$971$	−42.0381	−1.34907	−0.674534	+	0.738244i	$0.735655\pi$
$972$	0	0
$973$	−11.8496	−0.379879
$974$	0	0
$975$	12.4485	0.398671
$976$	0	0
$977$	11.1881	0.357938	0.178969	−	0.983855i	$-0.442724\pi$
$977$	11.1881	0.357938	0.178969	+	0.983855i	$0.442724\pi$
$978$	0	0
$979$	9.25060	0.295651
$980$	0	0
$981$	19.4617	0.621364
$982$	0	0
$983$	−18.9234	−0.603562	−0.301781	−	0.953377i	$-0.597581\pi$
$983$	−18.9234	−0.603562	−0.301781	+	0.953377i	$0.597581\pi$
$984$	0	0
$985$	−82.2492	−2.62068
$986$	0	0
$987$	5.70194	0.181495
$988$	0	0
$989$	−4.57452	−0.145461
$990$	0	0
$991$	−62.1279	−1.97356	−0.986779	−	0.162071i	$-0.948183\pi$
$991$	−62.1279	−1.97356	−0.986779	+	0.162071i	$0.948183\pi$
$992$	0	0
$993$	6.70052	0.212635
$994$	0	0
$995$	51.3785	1.62881
$996$	0	0
$997$	55.8759	1.76961	0.884804	−	0.465964i	$-0.154292\pi$
$997$	55.8759	1.76961	0.884804	+	0.465964i	$0.154292\pi$
$998$	0	0
$999$	−2.31265	−0.0731690

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4416.2.a.bp.1.1		3
4.3	odd	2		4416.2.a.bs.1.1		3
8.3	odd	2		1104.2.a.o.1.3		3
8.5	even	2		552.2.a.g.1.3	✓	3
24.5	odd	2		1656.2.a.n.1.1		3
24.11	even	2		3312.2.a.bf.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
552.2.a.g.1.3	✓	3	8.5	even	2
1104.2.a.o.1.3		3	8.3	odd	2
1656.2.a.n.1.1		3	24.5	odd	2
3312.2.a.bf.1.1		3	24.11	even	2
4416.2.a.bp.1.1		3	1.1	even	1	trivial
4416.2.a.bs.1.1		3	4.3	odd	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:	\( N \)	\(=\)	\( 4416 = 2^{6} \cdot 3 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4416.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -3.35026 q^{5} -2.96239 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -3.35026 q^{5} -2.96239 q^{7} +1.00000 q^{9} -1.61213 q^{11} -2.00000 q^{13} +3.35026 q^{15} +4.96239 q^{17} +1.35026 q^{19} +2.96239 q^{21} -1.00000 q^{23} +6.22425 q^{25} -1.00000 q^{27} +7.92478 q^{29} +5.92478 q^{31} +1.61213 q^{33} +9.92478 q^{35} +2.31265 q^{37} +2.00000 q^{39} -1.22425 q^{41} +4.57452 q^{43} -3.35026 q^{45} +1.92478 q^{47} +1.77575 q^{49} -4.96239 q^{51} +4.12601 q^{53} +5.40105 q^{55} -1.35026 q^{57} -2.70052 q^{59} -14.3127 q^{61} -2.96239 q^{63} +6.70052 q^{65} -4.57452 q^{67} +1.00000 q^{69} -16.6253 q^{71} +2.00000 q^{73} -6.22425 q^{75} +4.77575 q^{77} -5.03761 q^{79} +1.00000 q^{81} +15.0132 q^{83} -16.6253 q^{85} -7.92478 q^{87} -5.73813 q^{89} +5.92478 q^{91} -5.92478 q^{93} -4.52373 q^{95} +12.7005 q^{97} -1.61213 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{3} + 2 q^{7} + 3 q^{9}+O(q^{10}) \) 3 * q - 3 * q^3 + 2 * q^7 + 3 * q^9	\( 3 q - 3 q^{3} + 2 q^{7} + 3 q^{9} - 4 q^{11} - 6 q^{13} + 4 q^{17} - 6 q^{19} - 2 q^{21} - 3 q^{23} + 17 q^{25} - 3 q^{27} + 2 q^{29} - 4 q^{31} + 4 q^{33} + 8 q^{35} - 14 q^{37} + 6 q^{39} - 2 q^{41} + 2 q^{43} - 16 q^{47} + 7 q^{49} - 4 q^{51} + 4 q^{53} - 24 q^{55} + 6 q^{57} + 12 q^{59} - 22 q^{61} + 2 q^{63} - 2 q^{67} + 3 q^{69} - 8 q^{71} + 6 q^{73} - 17 q^{75} + 16 q^{77} - 26 q^{79} + 3 q^{81} + 4 q^{83} - 8 q^{85} - 2 q^{87} - 8 q^{89} - 4 q^{91} + 4 q^{93} - 32 q^{95} + 18 q^{97} - 4 q^{99}+O(q^{100}) \) 3 * q - 3 * q^3 + 2 * q^7 + 3 * q^9 - 4 * q^11 - 6 * q^13 + 4 * q^17 - 6 * q^19 - 2 * q^21 - 3 * q^23 + 17 * q^25 - 3 * q^27 + 2 * q^29 - 4 * q^31 + 4 * q^33 + 8 * q^35 - 14 * q^37 + 6 * q^39 - 2 * q^41 + 2 * q^43 - 16 * q^47 + 7 * q^49 - 4 * q^51 + 4 * q^53 - 24 * q^55 + 6 * q^57 + 12 * q^59 - 22 * q^61 + 2 * q^63 - 2 * q^67 + 3 * q^69 - 8 * q^71 + 6 * q^73 - 17 * q^75 + 16 * q^77 - 26 * q^79 + 3 * q^81 + 4 * q^83 - 8 * q^85 - 2 * q^87 - 8 * q^89 - 4 * q^91 + 4 * q^93 - 32 * q^95 + 18 * q^97 - 4 * q^99

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