LMFDB - Embedded newform 4368.2.a.br.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4368,2,Mod(1,4368)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4368, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("4368.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4368 = 2^{4} \cdot 3 \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4368.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:	$34.8786556029$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.17428.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 6x^{2} + 4x + 6 $ x^4 - x^3 - 6x^2 + 4x + 6
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 273)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$1.52616$ of defining polynomial
Character	$\chi$	$=$	4368.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} +2.54997 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +2.54997 q^{5} -1.00000 q^{7} +1.00000 q^{9} +3.05232 q^{11} +1.00000 q^{13} -2.54997 q^{15} +1.34169 q^{17} -7.60228 q^{19} +1.00000 q^{21} -1.84403 q^{23} +1.50235 q^{25} -1.00000 q^{27} -5.60228 q^{29} -10.2857 q^{31} -3.05232 q^{33} -2.54997 q^{35} -9.20457 q^{37} -1.00000 q^{39} -8.04762 q^{41} -1.49765 q^{43} +2.54997 q^{45} +3.89166 q^{47} +1.00000 q^{49} -1.34169 q^{51} +0.502345 q^{53} +7.78331 q^{55} +7.60228 q^{57} +4.28937 q^{59} -0.683372 q^{61} -1.00000 q^{63} +2.54997 q^{65} +7.68806 q^{67} +1.84403 q^{69} +14.2569 q^{71} -12.0189 q^{73} -1.50235 q^{75} -3.05232 q^{77} -4.91891 q^{79} +1.00000 q^{81} +1.20828 q^{83} +3.42126 q^{85} +5.60228 q^{87} +13.7545 q^{89} -1.00000 q^{91} +10.2857 q^{93} -19.3856 q^{95} -7.18572 q^{97} +3.05232 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{3} - 3 q^{5} - 4 q^{7} + 4 q^{9}+O(q^{10}) $ 4 * q - 4 * q^3 - 3 * q^5 - 4 * q^7 + 4 * q^9	$ 4 q - 4 q^{3} - 3 q^{5} - 4 q^{7} + 4 q^{9} + 2 q^{11} + 4 q^{13} + 3 q^{15} - 2 q^{17} - 7 q^{19} + 4 q^{21} - 3 q^{23} + 9 q^{25} - 4 q^{27} + q^{29} - 3 q^{31} - 2 q^{33} + 3 q^{35} + 10 q^{37} - 4 q^{39} - 16 q^{41} - 3 q^{43} - 3 q^{45} - 5 q^{47} + 4 q^{49} + 2 q^{51} + 5 q^{53} - 10 q^{55} + 7 q^{57} + 20 q^{59} + 12 q^{61} - 4 q^{63} - 3 q^{65} + 22 q^{67} + 3 q^{69} - 13 q^{73} - 9 q^{75} - 2 q^{77} - 11 q^{79} + 4 q^{81} - q^{83} + 8 q^{85} - q^{87} - 5 q^{89} - 4 q^{91} + 3 q^{93} - 13 q^{95} - 17 q^{97} + 2 q^{99}+O(q^{100}) $ 4 * q - 4 * q^3 - 3 * q^5 - 4 * q^7 + 4 * q^9 + 2 * q^11 + 4 * q^13 + 3 * q^15 - 2 * q^17 - 7 * q^19 + 4 * q^21 - 3 * q^23 + 9 * q^25 - 4 * q^27 + q^29 - 3 * q^31 - 2 * q^33 + 3 * q^35 + 10 * q^37 - 4 * q^39 - 16 * q^41 - 3 * q^43 - 3 * q^45 - 5 * q^47 + 4 * q^49 + 2 * q^51 + 5 * q^53 - 10 * q^55 + 7 * q^57 + 20 * q^59 + 12 * q^61 - 4 * q^63 - 3 * q^65 + 22 * q^67 + 3 * q^69 - 13 * q^73 - 9 * q^75 - 2 * q^77 - 11 * q^79 + 4 * q^81 - q^83 + 8 * q^85 - q^87 - 5 * q^89 - 4 * q^91 + 3 * q^93 - 13 * q^95 - 17 * q^97 + 2 * 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
$5$	2.54997	1.14038	0.570191	−	0.821512i	$-0.306869\pi$
$5$	2.54997	1.14038	0.570191	+	0.821512i	$0.306869\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	3.05232	0.920308	0.460154	−	0.887839i	$-0.347794\pi$
$11$	3.05232	0.920308	0.460154	+	0.887839i	$0.347794\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	−2.54997	−0.658399
$16$	0	0
$17$	1.34169	0.325407	0.162703	−	0.986675i	$-0.447979\pi$
$17$	1.34169	0.325407	0.162703	+	0.986675i	$0.447979\pi$
$18$	0	0
$19$	−7.60228	−1.74408	−0.872042	−	0.489431i	$-0.837204\pi$
$19$	−7.60228	−1.74408	−0.872042	+	0.489431i	$0.837204\pi$
$20$	0	0
$21$	1.00000	0.218218
$22$	0	0
$23$	−1.84403	−0.384507	−0.192254	−	0.981345i	$-0.561580\pi$
$23$	−1.84403	−0.384507	−0.192254	+	0.981345i	$0.561580\pi$
$24$	0	0
$25$	1.50235	0.300469
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−5.60228	−1.04032	−0.520159	−	0.854069i	$-0.674127\pi$
$29$	−5.60228	−1.04032	−0.520159	+	0.854069i	$0.674127\pi$
$30$	0	0
$31$	−10.2857	−1.84736	−0.923679	−	0.383167i	$-0.874834\pi$
$31$	−10.2857	−1.84736	−0.923679	+	0.383167i	$0.874834\pi$
$32$	0	0
$33$	−3.05232	−0.531340
$34$	0	0
$35$	−2.54997	−0.431024
$36$	0	0
$37$	−9.20457	−1.51322	−0.756611	−	0.653865i	$-0.773146\pi$
$37$	−9.20457	−1.51322	−0.756611	+	0.653865i	$0.773146\pi$
$38$	0	0
$39$	−1.00000	−0.160128
$40$	0	0
$41$	−8.04762	−1.25683	−0.628414	−	0.777879i	$-0.716296\pi$
$41$	−8.04762	−1.25683	−0.628414	+	0.777879i	$0.716296\pi$
$42$	0	0
$43$	−1.49765	−0.228390	−0.114195	−	0.993458i	$-0.536429\pi$
$43$	−1.49765	−0.228390	−0.114195	+	0.993458i	$0.536429\pi$
$44$	0	0
$45$	2.54997	0.380127
$46$	0	0
$47$	3.89166	0.567656	0.283828	−	0.958875i	$-0.408395\pi$
$47$	3.89166	0.567656	0.283828	+	0.958875i	$0.408395\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−1.34169	−0.187874
$52$	0	0
$53$	0.502345	0.0690025	0.0345012	−	0.999405i	$-0.489016\pi$
$53$	0.502345	0.0690025	0.0345012	+	0.999405i	$0.489016\pi$
$54$	0	0
$55$	7.78331	1.04950
$56$	0	0
$57$	7.60228	1.00695
$58$	0	0
$59$	4.28937	0.558429	0.279214	−	0.960229i	$-0.409926\pi$
$59$	4.28937	0.558429	0.279214	+	0.960229i	$0.409926\pi$
$60$	0	0
$61$	−0.683372	−0.0874968	−0.0437484	−	0.999043i	$-0.513930\pi$
$61$	−0.683372	−0.0874968	−0.0437484	+	0.999043i	$0.513930\pi$
$62$	0	0
$63$	−1.00000	−0.125988
$64$	0	0
$65$	2.54997	0.316285
$66$	0	0
$67$	7.68806	0.939246	0.469623	−	0.882867i	$-0.344390\pi$
$67$	7.68806	0.939246	0.469623	+	0.882867i	$0.344390\pi$
$68$	0	0
$69$	1.84403	0.221995
$70$	0	0
$71$	14.2569	1.69198	0.845990	−	0.533198i	$-0.179010\pi$
$71$	14.2569	1.69198	0.845990	+	0.533198i	$0.179010\pi$
$72$	0	0
$73$	−12.0189	−1.40670	−0.703350	−	0.710844i	$-0.748313\pi$
$73$	−12.0189	−1.40670	−0.703350	+	0.710844i	$0.748313\pi$
$74$	0	0
$75$	−1.50235	−0.173476
$76$	0	0
$77$	−3.05232	−0.347844
$78$	0	0
$79$	−4.91891	−0.553421	−0.276710	−	0.960953i	$-0.589244\pi$
$79$	−4.91891	−0.553421	−0.276710	+	0.960953i	$0.589244\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	1.20828	0.132626	0.0663132	−	0.997799i	$-0.478876\pi$
$83$	1.20828	0.132626	0.0663132	+	0.997799i	$0.478876\pi$
$84$	0	0
$85$	3.42126	0.371088
$86$	0	0
$87$	5.60228	0.600628
$88$	0	0
$89$	13.7545	1.45798	0.728989	−	0.684525i	$-0.239990\pi$
$89$	13.7545	1.45798	0.728989	+	0.684525i	$0.239990\pi$
$90$	0	0
$91$	−1.00000	−0.104828
$92$	0	0
$93$	10.2857	1.06657
$94$	0	0
$95$	−19.3856	−1.98892
$96$	0	0
$97$	−7.18572	−0.729599	−0.364800	−	0.931086i	$-0.618862\pi$
$97$	−7.18572	−0.729599	−0.364800	+	0.931086i	$0.618862\pi$
$98$	0	0
$99$	3.05232	0.306769
$100$	0	0
$101$	−18.4416	−1.83501	−0.917505	−	0.397724i	$-0.869800\pi$
$101$	−18.4416	−1.83501	−0.917505	+	0.397724i	$0.869800\pi$
$102$	0	0
$103$	−1.89537	−0.186756	−0.0933782	−	0.995631i	$-0.529767\pi$
$103$	−1.89537	−0.186756	−0.0933782	+	0.995631i	$0.529767\pi$
$104$	0	0
$105$	2.54997	0.248852
$106$	0	0
$107$	−18.5463	−1.79293	−0.896467	−	0.443110i	$-0.853875\pi$
$107$	−18.5463	−1.79293	−0.896467	+	0.443110i	$0.853875\pi$
$108$	0	0
$109$	−1.42126	−0.136132	−0.0680659	−	0.997681i	$-0.521683\pi$
$109$	−1.42126	−0.136132	−0.0680659	+	0.997681i	$0.521683\pi$
$110$	0	0
$111$	9.20457	0.873659
$112$	0	0
$113$	5.60228	0.527019	0.263509	−	0.964657i	$-0.415120\pi$
$113$	5.60228	0.527019	0.263509	+	0.964657i	$0.415120\pi$
$114$	0	0
$115$	−4.70222	−0.438485
$116$	0	0
$117$	1.00000	0.0924500
$118$	0	0
$119$	−1.34169	−0.122992
$120$	0	0
$121$	−1.68337	−0.153034
$122$	0	0
$123$	8.04762	0.725630
$124$	0	0
$125$	−8.91891	−0.797732
$126$	0	0
$127$	−14.1046	−1.25158	−0.625792	−	0.779990i	$-0.715224\pi$
$127$	−14.1046	−1.25158	−0.625792	+	0.779990i	$0.715224\pi$
$128$	0	0
$129$	1.49765	0.131861
$130$	0	0
$131$	8.78800	0.767811	0.383906	−	0.923372i	$-0.374579\pi$
$131$	8.78800	0.767811	0.383906	+	0.923372i	$0.374579\pi$
$132$	0	0
$133$	7.60228	0.659202
$134$	0	0
$135$	−2.54997	−0.219466
$136$	0	0
$137$	−1.97743	−0.168944	−0.0844718	−	0.996426i	$-0.526920\pi$
$137$	−1.97743	−0.168944	−0.0844718	+	0.996426i	$0.526920\pi$
$138$	0	0
$139$	−13.0999	−1.11112	−0.555561	−	0.831476i	$-0.687497\pi$
$139$	−13.0999	−1.11112	−0.555561	+	0.831476i	$0.687497\pi$
$140$	0	0
$141$	−3.89166	−0.327737
$142$	0	0
$143$	3.05232	0.255247
$144$	0	0
$145$	−14.2857	−1.18636
$146$	0	0
$147$	−1.00000	−0.0824786
$148$	0	0
$149$	14.5939	1.19558	0.597789	−	0.801654i	$-0.296046\pi$
$149$	14.5939	1.19558	0.597789	+	0.801654i	$0.296046\pi$
$150$	0	0
$151$	5.41188	0.440412	0.220206	−	0.975453i	$-0.429327\pi$
$151$	5.41188	0.440412	0.220206	+	0.975453i	$0.429327\pi$
$152$	0	0
$153$	1.34169	0.108469
$154$	0	0
$155$	−26.2281	−2.10669
$156$	0	0
$157$	23.4044	1.86788	0.933939	−	0.357432i	$-0.116348\pi$
$157$	23.4044	1.86788	0.933939	+	0.357432i	$0.116348\pi$
$158$	0	0
$159$	−0.502345	−0.0398386
$160$	0	0
$161$	1.84403	0.145330
$162$	0	0
$163$	10.1046	0.791456	0.395728	−	0.918368i	$-0.370492\pi$
$163$	10.1046	0.791456	0.395728	+	0.918368i	$0.370492\pi$
$164$	0	0
$165$	−7.78331	−0.605930
$166$	0	0
$167$	3.09623	0.239593	0.119797	−	0.992798i	$-0.461776\pi$
$167$	3.09623	0.239593	0.119797	+	0.992798i	$0.461776\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−7.60228	−0.581361
$172$	0	0
$173$	2.75356	0.209349	0.104675	−	0.994507i	$-0.466620\pi$
$173$	2.75356	0.209349	0.104675	+	0.994507i	$0.466620\pi$
$174$	0	0
$175$	−1.50235	−0.113567
$176$	0	0
$177$	−4.28937	−0.322409
$178$	0	0
$179$	1.57723	0.117887	0.0589437	−	0.998261i	$-0.481227\pi$
$179$	1.57723	0.117887	0.0589437	+	0.998261i	$0.481227\pi$
$180$	0	0
$181$	9.51651	0.707356	0.353678	−	0.935367i	$-0.384931\pi$
$181$	9.51651	0.707356	0.353678	+	0.935367i	$0.384931\pi$
$182$	0	0
$183$	0.683372	0.0505163
$184$	0	0
$185$	−23.4714	−1.72565
$186$	0	0
$187$	4.09525	0.299474
$188$	0	0
$189$	1.00000	0.0727393
$190$	0	0
$191$	22.8508	1.65342	0.826712	−	0.562626i	$-0.190209\pi$
$191$	22.8508	1.65342	0.826712	+	0.562626i	$0.190209\pi$
$192$	0	0
$193$	19.5760	1.40911	0.704556	−	0.709649i	$-0.251146\pi$
$193$	19.5760	1.40911	0.704556	+	0.709649i	$0.251146\pi$
$194$	0	0
$195$	−2.54997	−0.182607
$196$	0	0
$197$	−21.2271	−1.51237	−0.756185	−	0.654357i	$-0.772939\pi$
$197$	−21.2271	−1.51237	−0.756185	+	0.654357i	$0.772939\pi$
$198$	0	0
$199$	6.68337	0.473772	0.236886	−	0.971537i	$-0.423873\pi$
$199$	6.68337	0.473772	0.236886	+	0.971537i	$0.423873\pi$
$200$	0	0
$201$	−7.68806	−0.542274
$202$	0	0
$203$	5.60228	0.393203
$204$	0	0
$205$	−20.5212	−1.43326
$206$	0	0
$207$	−1.84403	−0.128169
$208$	0	0
$209$	−23.2046	−1.60509
$210$	0	0
$211$	4.60698	0.317157	0.158579	−	0.987346i	$-0.449309\pi$
$211$	4.60698	0.317157	0.158579	+	0.987346i	$0.449309\pi$
$212$	0	0
$213$	−14.2569	−0.976866
$214$	0	0
$215$	−3.81897	−0.260452
$216$	0	0
$217$	10.2857	0.698236
$218$	0	0
$219$	12.0189	0.812159
$220$	0	0
$221$	1.34169	0.0902516
$222$	0	0
$223$	−8.60698	−0.576366	−0.288183	−	0.957575i	$-0.593051\pi$
$223$	−8.60698	−0.576366	−0.288183	+	0.957575i	$0.593051\pi$
$224$	0	0
$225$	1.50235	0.100156
$226$	0	0
$227$	14.4892	0.961685	0.480843	−	0.876807i	$-0.340331\pi$
$227$	14.4892	0.961685	0.480843	+	0.876807i	$0.340331\pi$
$228$	0	0
$229$	−20.1999	−1.33485	−0.667423	−	0.744679i	$-0.732603\pi$
$229$	−20.1999	−1.33485	−0.667423	+	0.744679i	$0.732603\pi$
$230$	0	0
$231$	3.05232	0.200828
$232$	0	0
$233$	7.28097	0.476992	0.238496	−	0.971143i	$-0.423346\pi$
$233$	7.28097	0.476992	0.238496	+	0.971143i	$0.423346\pi$
$234$	0	0
$235$	9.92360	0.647345
$236$	0	0
$237$	4.91891	0.319518
$238$	0	0
$239$	−27.6688	−1.78974	−0.894872	−	0.446323i	$-0.852733\pi$
$239$	−27.6688	−1.78974	−0.894872	+	0.446323i	$0.852733\pi$
$240$	0	0
$241$	−22.2187	−1.43123	−0.715617	−	0.698493i	$-0.753854\pi$
$241$	−22.2187	−1.43123	−0.715617	+	0.698493i	$0.753854\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	2.54997	0.162912
$246$	0	0
$247$	−7.60228	−0.483722
$248$	0	0
$249$	−1.20828	−0.0765719
$250$	0	0
$251$	−17.3092	−1.09255	−0.546274	−	0.837607i	$-0.683954\pi$
$251$	−17.3092	−1.09255	−0.546274	+	0.837607i	$0.683954\pi$
$252$	0	0
$253$	−5.62857	−0.353865
$254$	0	0
$255$	−3.42126	−0.214248
$256$	0	0
$257$	−5.55837	−0.346722	−0.173361	−	0.984858i	$-0.555463\pi$
$257$	−5.55837	−0.346722	−0.173361	+	0.984858i	$0.555463\pi$
$258$	0	0
$259$	9.20457	0.571944
$260$	0	0
$261$	−5.60228	−0.346773
$262$	0	0
$263$	−25.0486	−1.54456	−0.772281	−	0.635281i	$-0.780884\pi$
$263$	−25.0486	−1.54456	−0.772281	+	0.635281i	$0.780884\pi$
$264$	0	0
$265$	1.28097	0.0786891
$266$	0	0
$267$	−13.7545	−0.841764
$268$	0	0
$269$	−4.55368	−0.277643	−0.138821	−	0.990317i	$-0.544331\pi$
$269$	−4.55368	−0.277643	−0.138821	+	0.990317i	$0.544331\pi$
$270$	0	0
$271$	−8.88325	−0.539619	−0.269810	−	0.962914i	$-0.586961\pi$
$271$	−8.88325	−0.539619	−0.269810	+	0.962914i	$0.586961\pi$
$272$	0	0
$273$	1.00000	0.0605228
$274$	0	0
$275$	4.58563	0.276524
$276$	0	0
$277$	−5.38560	−0.323589	−0.161795	−	0.986824i	$-0.551728\pi$
$277$	−5.38560	−0.323589	−0.161795	+	0.986824i	$0.551728\pi$
$278$	0	0
$279$	−10.2857	−0.615786
$280$	0	0
$281$	−12.9727	−0.773889	−0.386944	−	0.922103i	$-0.626469\pi$
$281$	−12.9727	−0.773889	−0.386944	+	0.922103i	$0.626469\pi$
$282$	0	0
$283$	1.52589	0.0907047	0.0453523	−	0.998971i	$-0.485559\pi$
$283$	1.52589	0.0907047	0.0453523	+	0.998971i	$0.485559\pi$
$284$	0	0
$285$	19.3856	1.14830
$286$	0	0
$287$	8.04762	0.475036
$288$	0	0
$289$	−15.1999	−0.894111
$290$	0	0
$291$	7.18572	0.421234
$292$	0	0
$293$	18.8545	1.10149	0.550745	−	0.834673i	$-0.314344\pi$
$293$	18.8545	1.10149	0.550745	+	0.834673i	$0.314344\pi$
$294$	0	0
$295$	10.9378	0.636821
$296$	0	0
$297$	−3.05232	−0.177113
$298$	0	0
$299$	−1.84403	−0.106643
$300$	0	0
$301$	1.49765	0.0863234
$302$	0	0
$303$	18.4416	1.05944
$304$	0	0
$305$	−1.74258	−0.0997797
$306$	0	0
$307$	−15.3856	−0.878102	−0.439051	−	0.898462i	$-0.644685\pi$
$307$	−15.3856	−0.878102	−0.439051	+	0.898462i	$0.644685\pi$
$308$	0	0
$309$	1.89537	0.107824
$310$	0	0
$311$	17.0999	0.969649	0.484824	−	0.874612i	$-0.338884\pi$
$311$	17.0999	0.969649	0.484824	+	0.874612i	$0.338884\pi$
$312$	0	0
$313$	4.89263	0.276548	0.138274	−	0.990394i	$-0.455845\pi$
$313$	4.89263	0.276548	0.138274	+	0.990394i	$0.455845\pi$
$314$	0	0
$315$	−2.54997	−0.143675
$316$	0	0
$317$	1.44382	0.0810933	0.0405466	−	0.999178i	$-0.487090\pi$
$317$	1.44382	0.0810933	0.0405466	+	0.999178i	$0.487090\pi$
$318$	0	0
$319$	−17.0999	−0.957413
$320$	0	0
$321$	18.5463	1.03515
$322$	0	0
$323$	−10.1999	−0.567536
$324$	0	0
$325$	1.50235	0.0833351
$326$	0	0
$327$	1.42126	0.0785958
$328$	0	0
$329$	−3.89166	−0.214554
$330$	0	0
$331$	14.9953	0.824217	0.412108	−	0.911135i	$-0.364793\pi$
$331$	14.9953	0.824217	0.412108	+	0.911135i	$0.364793\pi$
$332$	0	0
$333$	−9.20457	−0.504407
$334$	0	0
$335$	19.6043	1.07110
$336$	0	0
$337$	−14.9115	−0.812280	−0.406140	−	0.913811i	$-0.633126\pi$
$337$	−14.9115	−0.812280	−0.406140	+	0.913811i	$0.633126\pi$
$338$	0	0
$339$	−5.60228	−0.304274
$340$	0	0
$341$	−31.3951	−1.70014
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	4.70222	0.253159
$346$	0	0
$347$	−7.07488	−0.379800	−0.189900	−	0.981803i	$-0.560816\pi$
$347$	−7.07488	−0.379800	−0.189900	+	0.981803i	$0.560816\pi$
$348$	0	0
$349$	21.1188	1.13046	0.565232	−	0.824932i	$-0.308787\pi$
$349$	21.1188	1.13046	0.565232	+	0.824932i	$0.308787\pi$
$350$	0	0
$351$	−1.00000	−0.0533761
$352$	0	0
$353$	−17.3643	−0.924206	−0.462103	−	0.886826i	$-0.652905\pi$
$353$	−17.3643	−0.924206	−0.462103	+	0.886826i	$0.652905\pi$
$354$	0	0
$355$	36.3546	1.92950
$356$	0	0
$357$	1.34169	0.0710096
$358$	0	0
$359$	−18.3521	−0.968589	−0.484294	−	0.874905i	$-0.660924\pi$
$359$	−18.3521	−0.968589	−0.484294	+	0.874905i	$0.660924\pi$
$360$	0	0
$361$	38.7947	2.04183
$362$	0	0
$363$	1.68337	0.0883541
$364$	0	0
$365$	−30.6477	−1.60417
$366$	0	0
$367$	−8.40719	−0.438852	−0.219426	−	0.975629i	$-0.570418\pi$
$367$	−8.40719	−0.438852	−0.219426	+	0.975629i	$0.570418\pi$
$368$	0	0
$369$	−8.04762	−0.418943
$370$	0	0
$371$	−0.502345	−0.0260805
$372$	0	0
$373$	27.0925	1.40280	0.701399	−	0.712769i	$-0.252559\pi$
$373$	27.0925	1.40280	0.701399	+	0.712769i	$0.252559\pi$
$374$	0	0
$375$	8.91891	0.460571
$376$	0	0
$377$	−5.60228	−0.288532
$378$	0	0
$379$	6.41657	0.329597	0.164798	−	0.986327i	$-0.447303\pi$
$379$	6.41657	0.329597	0.164798	+	0.986327i	$0.447303\pi$
$380$	0	0
$381$	14.1046	0.722602
$382$	0	0
$383$	−32.5939	−1.66547	−0.832735	−	0.553672i	$-0.813226\pi$
$383$	−32.5939	−1.66547	−0.832735	+	0.553672i	$0.813226\pi$
$384$	0	0
$385$	−7.78331	−0.396674
$386$	0	0
$387$	−1.49765	−0.0761301
$388$	0	0
$389$	6.00000	0.304212	0.152106	−	0.988364i	$-0.451394\pi$
$389$	6.00000	0.304212	0.152106	+	0.988364i	$0.451394\pi$
$390$	0	0
$391$	−2.47411	−0.125121
$392$	0	0
$393$	−8.78800	−0.443296
$394$	0	0
$395$	−12.5431	−0.631111
$396$	0	0
$397$	15.7520	0.790573	0.395286	−	0.918558i	$-0.370645\pi$
$397$	15.7520	0.790573	0.395286	+	0.918558i	$0.370645\pi$
$398$	0	0
$399$	−7.60228	−0.380590
$400$	0	0
$401$	26.1867	1.30770	0.653851	−	0.756624i	$-0.273152\pi$
$401$	26.1867	1.30770	0.653851	+	0.756624i	$0.273152\pi$
$402$	0	0
$403$	−10.2857	−0.512365
$404$	0	0
$405$	2.54997	0.126709
$406$	0	0
$407$	−28.0952	−1.39263
$408$	0	0
$409$	−17.3856	−0.859662	−0.429831	−	0.902909i	$-0.641427\pi$
$409$	−17.3856	−0.859662	−0.429831	+	0.902909i	$0.641427\pi$
$410$	0	0
$411$	1.97743	0.0975396
$412$	0	0
$413$	−4.28937	−0.211066
$414$	0	0
$415$	3.08109	0.151245
$416$	0	0
$417$	13.0999	0.641507
$418$	0	0
$419$	12.4761	0.609496	0.304748	−	0.952433i	$-0.401428\pi$
$419$	12.4761	0.609496	0.304748	+	0.952433i	$0.401428\pi$
$420$	0	0
$421$	9.57405	0.466611	0.233305	−	0.972404i	$-0.425046\pi$
$421$	9.57405	0.466611	0.233305	+	0.972404i	$0.425046\pi$
$422$	0	0
$423$	3.89166	0.189219
$424$	0	0
$425$	2.01568	0.0977746
$426$	0	0
$427$	0.683372	0.0330707
$428$	0	0
$429$	−3.05232	−0.147367
$430$	0	0
$431$	−5.20208	−0.250575	−0.125288	−	0.992120i	$-0.539985\pi$
$431$	−5.20208	−0.250575	−0.125288	+	0.992120i	$0.539985\pi$
$432$	0	0
$433$	−17.2547	−0.829207	−0.414604	−	0.910002i	$-0.636080\pi$
$433$	−17.2547	−0.829207	−0.414604	+	0.910002i	$0.636080\pi$
$434$	0	0
$435$	14.2857	0.684945
$436$	0	0
$437$	14.0189	0.670613
$438$	0	0
$439$	22.1925	1.05919	0.529594	−	0.848251i	$-0.322344\pi$
$439$	22.1925	1.05919	0.529594	+	0.848251i	$0.322344\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	37.9319	1.80220	0.901098	−	0.433615i	$-0.142762\pi$
$443$	37.9319	1.80220	0.901098	+	0.433615i	$0.142762\pi$
$444$	0	0
$445$	35.0737	1.66265
$446$	0	0
$447$	−14.5939	−0.690267
$448$	0	0
$449$	−33.3150	−1.57223	−0.786115	−	0.618080i	$-0.787911\pi$
$449$	−33.3150	−1.57223	−0.786115	+	0.618080i	$0.787911\pi$
$450$	0	0
$451$	−24.5639	−1.15667
$452$	0	0
$453$	−5.41188	−0.254272
$454$	0	0
$455$	−2.54997	−0.119544
$456$	0	0
$457$	−7.79269	−0.364527	−0.182263	−	0.983250i	$-0.558342\pi$
$457$	−7.79269	−0.364527	−0.182263	+	0.983250i	$0.558342\pi$
$458$	0	0
$459$	−1.34169	−0.0626245
$460$	0	0
$461$	30.4568	1.41851	0.709256	−	0.704951i	$-0.249031\pi$
$461$	30.4568	1.41851	0.709256	+	0.704951i	$0.249031\pi$
$462$	0	0
$463$	−39.4639	−1.83405	−0.917023	−	0.398835i	$-0.869415\pi$
$463$	−39.4639	−1.83405	−0.917023	+	0.398835i	$0.869415\pi$
$464$	0	0
$465$	26.2281	1.21630
$466$	0	0
$467$	9.93307	0.459648	0.229824	−	0.973232i	$-0.426185\pi$
$467$	9.93307	0.459648	0.229824	+	0.973232i	$0.426185\pi$
$468$	0	0
$469$	−7.68806	−0.355002
$470$	0	0
$471$	−23.4044	−1.07842
$472$	0	0
$473$	−4.57131	−0.210189
$474$	0	0
$475$	−11.4213	−0.524043
$476$	0	0
$477$	0.502345	0.0230008
$478$	0	0
$479$	0.941479	0.0430173	0.0215086	−	0.999769i	$-0.493153\pi$
$479$	0.941479	0.0430173	0.0215086	+	0.999769i	$0.493153\pi$
$480$	0	0
$481$	−9.20457	−0.419692
$482$	0	0
$483$	−1.84403	−0.0839063
$484$	0	0
$485$	−18.3234	−0.832021
$486$	0	0
$487$	39.3499	1.78312	0.891558	−	0.452907i	$-0.149613\pi$
$487$	39.3499	1.78312	0.891558	+	0.452907i	$0.149613\pi$
$488$	0	0
$489$	−10.1046	−0.456947
$490$	0	0
$491$	−5.45374	−0.246124	−0.123062	−	0.992399i	$-0.539271\pi$
$491$	−5.45374	−0.246124	−0.123062	+	0.992399i	$0.539271\pi$
$492$	0	0
$493$	−7.51651	−0.338526
$494$	0	0
$495$	7.78331	0.349834
$496$	0	0
$497$	−14.2569	−0.639509
$498$	0	0
$499$	31.3574	1.40375	0.701874	−	0.712301i	$-0.252347\pi$
$499$	31.3574	1.40375	0.701874	+	0.712301i	$0.252347\pi$
$500$	0	0
$501$	−3.09623	−0.138329
$502$	0	0
$503$	−43.9926	−1.96153	−0.980766	−	0.195188i	$-0.937468\pi$
$503$	−43.9926	−1.96153	−0.980766	+	0.195188i	$0.937468\pi$
$504$	0	0
$505$	−47.0256	−2.09261
$506$	0	0
$507$	−1.00000	−0.0444116
$508$	0	0
$509$	−41.8498	−1.85496	−0.927480	−	0.373874i	$-0.878029\pi$
$509$	−41.8498	−1.85496	−0.927480	+	0.373874i	$0.878029\pi$
$510$	0	0
$511$	12.0189	0.531683
$512$	0	0
$513$	7.60228	0.335649
$514$	0	0
$515$	−4.83314	−0.212973
$516$	0	0
$517$	11.8786	0.522418
$518$	0	0
$519$	−2.75356	−0.120868
$520$	0	0
$521$	8.32957	0.364925	0.182462	−	0.983213i	$-0.441593\pi$
$521$	8.32957	0.364925	0.182462	+	0.983213i	$0.441593\pi$
$522$	0	0
$523$	−25.3092	−1.10669	−0.553347	−	0.832951i	$-0.686650\pi$
$523$	−25.3092	−1.10669	−0.553347	+	0.832951i	$0.686650\pi$
$524$	0	0
$525$	1.50235	0.0655677
$526$	0	0
$527$	−13.8001	−0.601143
$528$	0	0
$529$	−19.5995	−0.852154
$530$	0	0
$531$	4.28937	0.186143
$532$	0	0
$533$	−8.04762	−0.348581
$534$	0	0
$535$	−47.2924	−2.04463
$536$	0	0
$537$	−1.57723	−0.0680624
$538$	0	0
$539$	3.05232	0.131473
$540$	0	0
$541$	27.1501	1.16727	0.583636	−	0.812015i	$-0.301630\pi$
$541$	27.1501	1.16727	0.583636	+	0.812015i	$0.301630\pi$
$542$	0	0
$543$	−9.51651	−0.408392
$544$	0	0
$545$	−3.62417	−0.155242
$546$	0	0
$547$	−27.5949	−1.17987	−0.589935	−	0.807450i	$-0.700847\pi$
$547$	−27.5949	−1.17987	−0.589935	+	0.807450i	$0.700847\pi$
$548$	0	0
$549$	−0.683372	−0.0291656
$550$	0	0
$551$	42.5902	1.81440
$552$	0	0
$553$	4.91891	0.209173
$554$	0	0
$555$	23.4714	0.996304
$556$	0	0
$557$	7.19881	0.305024	0.152512	−	0.988302i	$-0.451264\pi$
$557$	7.19881	0.305024	0.152512	+	0.988302i	$0.451264\pi$
$558$	0	0
$559$	−1.49765	−0.0633440
$560$	0	0
$561$	−4.09525	−0.172902
$562$	0	0
$563$	20.2594	0.853831	0.426915	−	0.904292i	$-0.359600\pi$
$563$	20.2594	0.853831	0.426915	+	0.904292i	$0.359600\pi$
$564$	0	0
$565$	14.2857	0.601002
$566$	0	0
$567$	−1.00000	−0.0419961
$568$	0	0
$569$	7.28097	0.305234	0.152617	−	0.988285i	$-0.451230\pi$
$569$	7.28097	0.305234	0.152617	+	0.988285i	$0.451230\pi$
$570$	0	0
$571$	22.1141	0.925446	0.462723	−	0.886503i	$-0.346872\pi$
$571$	22.1141	0.925446	0.462723	+	0.886503i	$0.346872\pi$
$572$	0	0
$573$	−22.8508	−0.954604
$574$	0	0
$575$	−2.77037	−0.115533
$576$	0	0
$577$	20.3139	0.845678	0.422839	−	0.906205i	$-0.361034\pi$
$577$	20.3139	0.845678	0.422839	+	0.906205i	$0.361034\pi$
$578$	0	0
$579$	−19.5760	−0.813551
$580$	0	0
$581$	−1.20828	−0.0501281
$582$	0	0
$583$	1.53332	0.0635035
$584$	0	0
$585$	2.54997	0.105428
$586$	0	0
$587$	−35.8842	−1.48110	−0.740550	−	0.672001i	$-0.765435\pi$
$587$	−35.8842	−1.48110	−0.740550	+	0.672001i	$0.765435\pi$
$588$	0	0
$589$	78.1945	3.22195
$590$	0	0
$591$	21.2271	0.873168
$592$	0	0
$593$	20.2664	0.832239	0.416120	−	0.909310i	$-0.363390\pi$
$593$	20.2664	0.832239	0.416120	+	0.909310i	$0.363390\pi$
$594$	0	0
$595$	−3.42126	−0.140258
$596$	0	0
$597$	−6.68337	−0.273532
$598$	0	0
$599$	−14.9365	−0.610291	−0.305145	−	0.952306i	$-0.598705\pi$
$599$	−14.9365	−0.610291	−0.305145	+	0.952306i	$0.598705\pi$
$600$	0	0
$601$	4.89263	0.199575	0.0997873	−	0.995009i	$-0.468184\pi$
$601$	4.89263	0.199575	0.0997873	+	0.995009i	$0.468184\pi$
$602$	0	0
$603$	7.68806	0.313082
$604$	0	0
$605$	−4.29255	−0.174517
$606$	0	0
$607$	−20.6164	−0.836796	−0.418398	−	0.908264i	$-0.637408\pi$
$607$	−20.6164	−0.836796	−0.418398	+	0.908264i	$0.637408\pi$
$608$	0	0
$609$	−5.60228	−0.227016
$610$	0	0
$611$	3.89166	0.157440
$612$	0	0
$613$	−9.73818	−0.393321	−0.196661	−	0.980472i	$-0.563010\pi$
$613$	−9.73818	−0.393321	−0.196661	+	0.980472i	$0.563010\pi$
$614$	0	0
$615$	20.5212	0.827495
$616$	0	0
$617$	−0.763482	−0.0307366	−0.0153683	−	0.999882i	$-0.504892\pi$
$617$	−0.763482	−0.0307366	−0.0153683	+	0.999882i	$0.504892\pi$
$618$	0	0
$619$	41.0925	1.65165	0.825824	−	0.563928i	$-0.190711\pi$
$619$	41.0925	1.65165	0.825824	+	0.563928i	$0.190711\pi$
$620$	0	0
$621$	1.84403	0.0739984
$622$	0	0
$623$	−13.7545	−0.551064
$624$	0	0
$625$	−30.2547	−1.21019
$626$	0	0
$627$	23.2046	0.926701
$628$	0	0
$629$	−12.3496	−0.492412
$630$	0	0
$631$	14.4667	0.575910	0.287955	−	0.957644i	$-0.407025\pi$
$631$	14.4667	0.575910	0.287955	+	0.957644i	$0.407025\pi$
$632$	0	0
$633$	−4.60698	−0.183111
$634$	0	0
$635$	−35.9664	−1.42728
$636$	0	0
$637$	1.00000	0.0396214
$638$	0	0
$639$	14.2569	0.563994
$640$	0	0
$641$	35.7069	1.41034	0.705169	−	0.709039i	$-0.250871\pi$
$641$	35.7069	1.41034	0.705169	+	0.709039i	$0.250871\pi$
$642$	0	0
$643$	26.4091	1.04147	0.520737	−	0.853717i	$-0.325657\pi$
$643$	26.4091	1.04147	0.520737	+	0.853717i	$0.325657\pi$
$644$	0	0
$645$	3.81897	0.150372
$646$	0	0
$647$	−37.3543	−1.46855	−0.734275	−	0.678852i	$-0.762478\pi$
$647$	−37.3543	−1.46855	−0.734275	+	0.678852i	$0.762478\pi$
$648$	0	0
$649$	13.0925	0.513926
$650$	0	0
$651$	−10.2857	−0.403127
$652$	0	0
$653$	−0.987881	−0.0386588	−0.0193294	−	0.999813i	$-0.506153\pi$
$653$	−0.987881	−0.0386588	−0.0193294	+	0.999813i	$0.506153\pi$
$654$	0	0
$655$	22.4091	0.875598
$656$	0	0
$657$	−12.0189	−0.468900
$658$	0	0
$659$	1.24653	0.0485578	0.0242789	−	0.999705i	$-0.492271\pi$
$659$	1.24653	0.0485578	0.0242789	+	0.999705i	$0.492271\pi$
$660$	0	0
$661$	−33.8994	−1.31853	−0.659266	−	0.751910i	$-0.729133\pi$
$661$	−33.8994	−1.31853	−0.659266	+	0.751910i	$0.729133\pi$
$662$	0	0
$663$	−1.34169	−0.0521068
$664$	0	0
$665$	19.3856	0.751741
$666$	0	0
$667$	10.3308	0.400010
$668$	0	0
$669$	8.60698	0.332765
$670$	0	0
$671$	−2.08587	−0.0805240
$672$	0	0
$673$	42.5808	1.64137	0.820684	−	0.571382i	$-0.193592\pi$
$673$	42.5808	1.64137	0.820684	+	0.571382i	$0.193592\pi$
$674$	0	0
$675$	−1.50235	−0.0578253
$676$	0	0
$677$	−33.8629	−1.30146	−0.650728	−	0.759311i	$-0.725536\pi$
$677$	−33.8629	−1.30146	−0.650728	+	0.759311i	$0.725536\pi$
$678$	0	0
$679$	7.18572	0.275763
$680$	0	0
$681$	−14.4892	−0.555229
$682$	0	0
$683$	34.5163	1.32073	0.660364	−	0.750946i	$-0.270402\pi$
$683$	34.5163	1.32073	0.660364	+	0.750946i	$0.270402\pi$
$684$	0	0
$685$	−5.04240	−0.192660
$686$	0	0
$687$	20.1999	0.770673
$688$	0	0
$689$	0.502345	0.0191378
$690$	0	0
$691$	7.90679	0.300789	0.150394	−	0.988626i	$-0.451946\pi$
$691$	7.90679	0.300789	0.150394	+	0.988626i	$0.451946\pi$
$692$	0	0
$693$	−3.05232	−0.115948
$694$	0	0
$695$	−33.4044	−1.26710
$696$	0	0
$697$	−10.7974	−0.408980
$698$	0	0
$699$	−7.28097	−0.275391
$700$	0	0
$701$	−5.19510	−0.196216	−0.0981081	−	0.995176i	$-0.531279\pi$
$701$	−5.19510	−0.196216	−0.0981081	+	0.995176i	$0.531279\pi$
$702$	0	0
$703$	69.9758	2.63919
$704$	0	0
$705$	−9.92360	−0.373745
$706$	0	0
$707$	18.4416	0.693569
$708$	0	0
$709$	33.1971	1.24674	0.623372	−	0.781925i	$-0.285762\pi$
$709$	33.1971	1.24674	0.623372	+	0.781925i	$0.285762\pi$
$710$	0	0
$711$	−4.91891	−0.184474
$712$	0	0
$713$	18.9671	0.710323
$714$	0	0
$715$	7.78331	0.291079
$716$	0	0
$717$	27.6688	1.03331
$718$	0	0
$719$	−30.7261	−1.14589	−0.572944	−	0.819594i	$-0.694199\pi$
$719$	−30.7261	−1.14589	−0.572944	+	0.819594i	$0.694199\pi$
$720$	0	0
$721$	1.89537	0.0705873
$722$	0	0
$723$	22.2187	0.826324
$724$	0	0
$725$	−8.41657	−0.312583
$726$	0	0
$727$	−1.83783	−0.0681612	−0.0340806	−	0.999419i	$-0.510850\pi$
$727$	−1.83783	−0.0681612	−0.0340806	+	0.999419i	$0.510850\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−2.00938	−0.0743197
$732$	0	0
$733$	−19.1857	−0.708641	−0.354320	−	0.935124i	$-0.615288\pi$
$733$	−19.1857	−0.708641	−0.354320	+	0.935124i	$0.615288\pi$
$734$	0	0
$735$	−2.54997	−0.0940570
$736$	0	0
$737$	23.4664	0.864396
$738$	0	0
$739$	21.4882	0.790456	0.395228	−	0.918583i	$-0.370666\pi$
$739$	21.4882	0.790456	0.395228	+	0.918583i	$0.370666\pi$
$740$	0	0
$741$	7.60228	0.279277
$742$	0	0
$743$	−19.9430	−0.731637	−0.365819	−	0.930686i	$-0.619211\pi$
$743$	−19.9430	−0.731637	−0.365819	+	0.930686i	$0.619211\pi$
$744$	0	0
$745$	37.2140	1.36341
$746$	0	0
$747$	1.20828	0.0442088
$748$	0	0
$749$	18.5463	0.677665
$750$	0	0
$751$	−38.4498	−1.40305	−0.701526	−	0.712644i	$-0.747498\pi$
$751$	−38.4498	−1.40305	−0.701526	+	0.712644i	$0.747498\pi$
$752$	0	0
$753$	17.3092	0.630782
$754$	0	0
$755$	13.8001	0.502238
$756$	0	0
$757$	−16.2931	−0.592182	−0.296091	−	0.955160i	$-0.595683\pi$
$757$	−16.2931	−0.592182	−0.296091	+	0.955160i	$0.595683\pi$
$758$	0	0
$759$	5.62857	0.204304
$760$	0	0
$761$	35.3380	1.28100	0.640500	−	0.767958i	$-0.278727\pi$
$761$	35.3380	1.28100	0.640500	+	0.767958i	$0.278727\pi$
$762$	0	0
$763$	1.42126	0.0514530
$764$	0	0
$765$	3.42126	0.123696
$766$	0	0
$767$	4.28937	0.154880
$768$	0	0
$769$	−11.5428	−0.416244	−0.208122	−	0.978103i	$-0.566735\pi$
$769$	−11.5428	−0.416244	−0.208122	+	0.978103i	$0.566735\pi$
$770$	0	0
$771$	5.55837	0.200180
$772$	0	0
$773$	−5.89786	−0.212131	−0.106066	−	0.994359i	$-0.533825\pi$
$773$	−5.89786	−0.212131	−0.106066	+	0.994359i	$0.533825\pi$
$774$	0	0
$775$	−15.4526	−0.555074
$776$	0	0
$777$	−9.20457	−0.330212
$778$	0	0
$779$	61.1803	2.19201
$780$	0	0
$781$	43.5165	1.55714
$782$	0	0
$783$	5.60228	0.200209
$784$	0	0
$785$	59.6806	2.13009
$786$	0	0
$787$	−6.33079	−0.225668	−0.112834	−	0.993614i	$-0.535993\pi$
$787$	−6.33079	−0.225668	−0.112834	+	0.993614i	$0.535993\pi$
$788$	0	0
$789$	25.0486	0.891754
$790$	0	0
$791$	−5.60228	−0.199194
$792$	0	0
$793$	−0.683372	−0.0242672
$794$	0	0
$795$	−1.28097	−0.0454312
$796$	0	0
$797$	−22.0721	−0.781835	−0.390918	−	0.920426i	$-0.627842\pi$
$797$	−22.0721	−0.781835	−0.390918	+	0.920426i	$0.627842\pi$
$798$	0	0
$799$	5.22138	0.184719
$800$	0	0
$801$	13.7545	0.485993
$802$	0	0
$803$	−36.6853	−1.29460
$804$	0	0
$805$	4.70222	0.165732
$806$	0	0
$807$	4.55368	0.160297
$808$	0	0
$809$	36.2019	1.27279	0.636396	−	0.771363i	$-0.280424\pi$
$809$	36.2019	1.27279	0.636396	+	0.771363i	$0.280424\pi$
$810$	0	0
$811$	52.4950	1.84335	0.921674	−	0.387964i	$-0.126822\pi$
$811$	52.4950	1.84335	0.921674	+	0.387964i	$0.126822\pi$
$812$	0	0
$813$	8.88325	0.311549
$814$	0	0
$815$	25.7665	0.902561
$816$	0	0
$817$	11.3856	0.398332
$818$	0	0
$819$	−1.00000	−0.0349428
$820$	0	0
$821$	−0.972743	−0.0339490	−0.0169745	−	0.999856i	$-0.505403\pi$
$821$	−0.972743	−0.0339490	−0.0169745	+	0.999856i	$0.505403\pi$
$822$	0	0
$823$	29.5572	1.03030	0.515150	−	0.857100i	$-0.327736\pi$
$823$	29.5572	1.03030	0.515150	+	0.857100i	$0.327736\pi$
$824$	0	0
$825$	−4.58563	−0.159651
$826$	0	0
$827$	−15.3191	−0.532698	−0.266349	−	0.963877i	$-0.585817\pi$
$827$	−15.3191	−0.532698	−0.266349	+	0.963877i	$0.585817\pi$
$828$	0	0
$829$	27.0236	0.938570	0.469285	−	0.883047i	$-0.344512\pi$
$829$	27.0236	0.938570	0.469285	+	0.883047i	$0.344512\pi$
$830$	0	0
$831$	5.38560	0.186824
$832$	0	0
$833$	1.34169	0.0464867
$834$	0	0
$835$	7.89528	0.273227
$836$	0	0
$837$	10.2857	0.355524
$838$	0	0
$839$	−46.2200	−1.59569	−0.797846	−	0.602862i	$-0.794027\pi$
$839$	−46.2200	−1.59569	−0.797846	+	0.602862i	$0.794027\pi$
$840$	0	0
$841$	2.38560	0.0822619
$842$	0	0
$843$	12.9727	0.446805
$844$	0	0
$845$	2.54997	0.0877216
$846$	0	0
$847$	1.68337	0.0578413
$848$	0	0
$849$	−1.52589	−0.0523684
$850$	0	0
$851$	16.9735	0.581845
$852$	0	0
$853$	−15.1094	−0.517336	−0.258668	−	0.965966i	$-0.583284\pi$
$853$	−15.1094	−0.517336	−0.258668	+	0.965966i	$0.583284\pi$
$854$	0	0
$855$	−19.3856	−0.662973
$856$	0	0
$857$	−0.191631	−0.00654599	−0.00327299	−	0.999995i	$-0.501042\pi$
$857$	−0.191631	−0.00654599	−0.00327299	+	0.999995i	$0.501042\pi$
$858$	0	0
$859$	−7.06419	−0.241027	−0.120514	−	0.992712i	$-0.538454\pi$
$859$	−7.06419	−0.241027	−0.120514	+	0.992712i	$0.538454\pi$
$860$	0	0
$861$	−8.04762	−0.274262
$862$	0	0
$863$	−41.6142	−1.41657	−0.708283	−	0.705929i	$-0.750530\pi$
$863$	−41.6142	−1.41657	−0.708283	+	0.705929i	$0.750530\pi$
$864$	0	0
$865$	7.02150	0.238738
$866$	0	0
$867$	15.1999	0.516215
$868$	0	0
$869$	−15.0141	−0.509318
$870$	0	0
$871$	7.68806	0.260500
$872$	0	0
$873$	−7.18572	−0.243200
$874$	0	0
$875$	8.91891	0.301514
$876$	0	0
$877$	−52.5471	−1.77439	−0.887194	−	0.461396i	$-0.847349\pi$
$877$	−52.5471	−1.77439	−0.887194	+	0.461396i	$0.847349\pi$
$878$	0	0
$879$	−18.8545	−0.635946
$880$	0	0
$881$	0.934500	0.0314841	0.0157421	−	0.999876i	$-0.494989\pi$
$881$	0.934500	0.0314841	0.0157421	+	0.999876i	$0.494989\pi$
$882$	0	0
$883$	42.6448	1.43511	0.717555	−	0.696501i	$-0.245261\pi$
$883$	42.6448	1.43511	0.717555	+	0.696501i	$0.245261\pi$
$884$	0	0
$885$	−10.9378	−0.367669
$886$	0	0
$887$	−24.8833	−0.835498	−0.417749	−	0.908563i	$-0.637181\pi$
$887$	−24.8833	−0.835498	−0.417749	+	0.908563i	$0.637181\pi$
$888$	0	0
$889$	14.1046	0.473054
$890$	0	0
$891$	3.05232	0.102256
$892$	0	0
$893$	−29.5855	−0.990040
$894$	0	0
$895$	4.02188	0.134437
$896$	0	0
$897$	1.84403	0.0615704
$898$	0	0
$899$	57.6232	1.92184
$900$	0	0
$901$	0.673990	0.0224539
$902$	0	0
$903$	−1.49765	−0.0498388
$904$	0	0
$905$	24.2668	0.806656
$906$	0	0
$907$	−42.2207	−1.40191	−0.700957	−	0.713203i	$-0.747244\pi$
$907$	−42.2207	−1.40191	−0.700957	+	0.713203i	$0.747244\pi$
$908$	0	0
$909$	−18.4416	−0.611670
$910$	0	0
$911$	−14.1108	−0.467513	−0.233756	−	0.972295i	$-0.575102\pi$
$911$	−14.1108	−0.467513	−0.233756	+	0.972295i	$0.575102\pi$
$912$	0	0
$913$	3.68806	0.122057
$914$	0	0
$915$	1.74258	0.0576078
$916$	0	0
$917$	−8.78800	−0.290205
$918$	0	0
$919$	18.1547	0.598870	0.299435	−	0.954117i	$-0.403202\pi$
$919$	18.1547	0.598870	0.299435	+	0.954117i	$0.403202\pi$
$920$	0	0
$921$	15.3856	0.506973
$922$	0	0
$923$	14.2569	0.469271
$924$	0	0
$925$	−13.8284	−0.454676
$926$	0	0
$927$	−1.89537	−0.0622521
$928$	0	0
$929$	−8.24741	−0.270589	−0.135294	−	0.990805i	$-0.543198\pi$
$929$	−8.24741	−0.270589	−0.135294	+	0.990805i	$0.543198\pi$
$930$	0	0
$931$	−7.60228	−0.249155
$932$	0	0
$933$	−17.0999	−0.559827
$934$	0	0
$935$	10.4428	0.341515
$936$	0	0
$937$	−21.6693	−0.707905	−0.353953	−	0.935263i	$-0.615163\pi$
$937$	−21.6693	−0.707905	−0.353953	+	0.935263i	$0.615163\pi$
$938$	0	0
$939$	−4.89263	−0.159665
$940$	0	0
$941$	42.5238	1.38624	0.693118	−	0.720824i	$-0.256237\pi$
$941$	42.5238	1.38624	0.693118	+	0.720824i	$0.256237\pi$
$942$	0	0
$943$	14.8401	0.483259
$944$	0	0
$945$	2.54997	0.0829505
$946$	0	0
$947$	6.79792	0.220903	0.110451	−	0.993882i	$-0.464770\pi$
$947$	6.79792	0.220903	0.110451	+	0.993882i	$0.464770\pi$
$948$	0	0
$949$	−12.0189	−0.390148
$950$	0	0
$951$	−1.44382	−0.0468192
$952$	0	0
$953$	57.3782	1.85866	0.929331	−	0.369249i	$-0.120385\pi$
$953$	57.3782	1.85866	0.929331	+	0.369249i	$0.120385\pi$
$954$	0	0
$955$	58.2688	1.88553
$956$	0	0
$957$	17.0999	0.552763
$958$	0	0
$959$	1.97743	0.0638547
$960$	0	0
$961$	74.7947	2.41273
$962$	0	0
$963$	−18.5463	−0.597645
$964$	0	0
$965$	49.9182	1.60692
$966$	0	0
$967$	−7.52393	−0.241953	−0.120977	−	0.992655i	$-0.538603\pi$
$967$	−7.52393	−0.241953	−0.120977	+	0.992655i	$0.538603\pi$
$968$	0	0
$969$	10.1999	0.327667
$970$	0	0
$971$	18.5807	0.596283	0.298141	−	0.954522i	$-0.403633\pi$
$971$	18.5807	0.596283	0.298141	+	0.954522i	$0.403633\pi$
$972$	0	0
$973$	13.0999	0.419965
$974$	0	0
$975$	−1.50235	−0.0481136
$976$	0	0
$977$	−38.5939	−1.23473	−0.617364	−	0.786678i	$-0.711799\pi$
$977$	−38.5939	−1.23473	−0.617364	+	0.786678i	$0.711799\pi$
$978$	0	0
$979$	41.9832	1.34179
$980$	0	0
$981$	−1.42126	−0.0453773
$982$	0	0
$983$	0.591837	0.0188767	0.00943834	−	0.999955i	$-0.496996\pi$
$983$	0.591837	0.0188767	0.00943834	+	0.999955i	$0.496996\pi$
$984$	0	0
$985$	−54.1286	−1.72468
$986$	0	0
$987$	3.89166	0.123873
$988$	0	0
$989$	2.76172	0.0878177
$990$	0	0
$991$	−9.41686	−0.299136	−0.149568	−	0.988751i	$-0.547788\pi$
$991$	−9.41686	−0.299136	−0.149568	+	0.988751i	$0.547788\pi$
$992$	0	0
$993$	−14.9953	−0.475862
$994$	0	0
$995$	17.0424	0.540280
$996$	0	0
$997$	22.1716	0.702180	0.351090	−	0.936342i	$-0.385811\pi$
$997$	22.1716	0.702180	0.351090	+	0.936342i	$0.385811\pi$
$998$	0	0
$999$	9.20457	0.291220

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4368.2.a.br.1.4		4
4.3	odd	2		273.2.a.e.1.2	✓	4
12.11	even	2		819.2.a.k.1.3		4
20.19	odd	2		6825.2.a.bg.1.3		4
28.27	even	2		1911.2.a.s.1.2		4
52.51	odd	2		3549.2.a.w.1.3		4
84.83	odd	2		5733.2.a.bf.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
273.2.a.e.1.2	✓	4	4.3	odd	2
819.2.a.k.1.3		4	12.11	even	2
1911.2.a.s.1.2		4	28.27	even	2
3549.2.a.w.1.3		4	52.51	odd	2
4368.2.a.br.1.4		4	1.1	even	1	trivial
5733.2.a.bf.1.3		4	84.83	odd	2
6825.2.a.bg.1.3		4	20.19	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:	\( N \)	\(=\)	\( 4368 = 2^{4} \cdot 3 \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4368.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +2.54997 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +2.54997 q^{5} -1.00000 q^{7} +1.00000 q^{9} +3.05232 q^{11} +1.00000 q^{13} -2.54997 q^{15} +1.34169 q^{17} -7.60228 q^{19} +1.00000 q^{21} -1.84403 q^{23} +1.50235 q^{25} -1.00000 q^{27} -5.60228 q^{29} -10.2857 q^{31} -3.05232 q^{33} -2.54997 q^{35} -9.20457 q^{37} -1.00000 q^{39} -8.04762 q^{41} -1.49765 q^{43} +2.54997 q^{45} +3.89166 q^{47} +1.00000 q^{49} -1.34169 q^{51} +0.502345 q^{53} +7.78331 q^{55} +7.60228 q^{57} +4.28937 q^{59} -0.683372 q^{61} -1.00000 q^{63} +2.54997 q^{65} +7.68806 q^{67} +1.84403 q^{69} +14.2569 q^{71} -12.0189 q^{73} -1.50235 q^{75} -3.05232 q^{77} -4.91891 q^{79} +1.00000 q^{81} +1.20828 q^{83} +3.42126 q^{85} +5.60228 q^{87} +13.7545 q^{89} -1.00000 q^{91} +10.2857 q^{93} -19.3856 q^{95} -7.18572 q^{97} +3.05232 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{3} - 3 q^{5} - 4 q^{7} + 4 q^{9}+O(q^{10}) \) 4 * q - 4 * q^3 - 3 * q^5 - 4 * q^7 + 4 * q^9	\( 4 q - 4 q^{3} - 3 q^{5} - 4 q^{7} + 4 q^{9} + 2 q^{11} + 4 q^{13} + 3 q^{15} - 2 q^{17} - 7 q^{19} + 4 q^{21} - 3 q^{23} + 9 q^{25} - 4 q^{27} + q^{29} - 3 q^{31} - 2 q^{33} + 3 q^{35} + 10 q^{37} - 4 q^{39} - 16 q^{41} - 3 q^{43} - 3 q^{45} - 5 q^{47} + 4 q^{49} + 2 q^{51} + 5 q^{53} - 10 q^{55} + 7 q^{57} + 20 q^{59} + 12 q^{61} - 4 q^{63} - 3 q^{65} + 22 q^{67} + 3 q^{69} - 13 q^{73} - 9 q^{75} - 2 q^{77} - 11 q^{79} + 4 q^{81} - q^{83} + 8 q^{85} - q^{87} - 5 q^{89} - 4 q^{91} + 3 q^{93} - 13 q^{95} - 17 q^{97} + 2 q^{99}+O(q^{100}) \) 4 * q - 4 * q^3 - 3 * q^5 - 4 * q^7 + 4 * q^9 + 2 * q^11 + 4 * q^13 + 3 * q^15 - 2 * q^17 - 7 * q^19 + 4 * q^21 - 3 * q^23 + 9 * q^25 - 4 * q^27 + q^29 - 3 * q^31 - 2 * q^33 + 3 * q^35 + 10 * q^37 - 4 * q^39 - 16 * q^41 - 3 * q^43 - 3 * q^45 - 5 * q^47 + 4 * q^49 + 2 * q^51 + 5 * q^53 - 10 * q^55 + 7 * q^57 + 20 * q^59 + 12 * q^61 - 4 * q^63 - 3 * q^65 + 22 * q^67 + 3 * q^69 - 13 * q^73 - 9 * q^75 - 2 * q^77 - 11 * q^79 + 4 * q^81 - q^83 + 8 * q^85 - q^87 - 5 * q^89 - 4 * q^91 + 3 * q^93 - 13 * q^95 - 17 * q^97 + 2 * q^99

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