LMFDB - Embedded newform 4368.2.a.bk.1.1

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:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{17}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 4 $ x^2 - x - 4
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 2184)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.56155$ 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)

$f(q)$	$=$	$q+1.00000 q^{3} -1.56155 q^{5} -1.00000 q^{7} +1.00000 q^{9} +2.00000 q^{11} -1.00000 q^{13} -1.56155 q^{15} +5.12311 q^{17} +2.43845 q^{19} -1.00000 q^{21} -4.68466 q^{23} -2.56155 q^{25} +1.00000 q^{27} -3.56155 q^{29} -1.56155 q^{31} +2.00000 q^{33} +1.56155 q^{35} +1.12311 q^{37} -1.00000 q^{39} +7.12311 q^{41} +9.56155 q^{43} -1.56155 q^{45} +6.68466 q^{47} +1.00000 q^{49} +5.12311 q^{51} +0.438447 q^{53} -3.12311 q^{55} +2.43845 q^{57} -5.12311 q^{59} -6.00000 q^{61} -1.00000 q^{63} +1.56155 q^{65} +13.3693 q^{67} -4.68466 q^{69} +2.87689 q^{71} -5.80776 q^{73} -2.56155 q^{75} -2.00000 q^{77} +11.8078 q^{79} +1.00000 q^{81} -9.80776 q^{83} -8.00000 q^{85} -3.56155 q^{87} +5.56155 q^{89} +1.00000 q^{91} -1.56155 q^{93} -3.80776 q^{95} -7.56155 q^{97} +2.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} + q^{5} - 2 q^{7} + 2 q^{9} + 4 q^{11} - 2 q^{13} + q^{15} + 2 q^{17} + 9 q^{19} - 2 q^{21} + 3 q^{23} - q^{25} + 2 q^{27} - 3 q^{29} + q^{31} + 4 q^{33} - q^{35} - 6 q^{37} - 2 q^{39}+ \cdots + 4 q^{99}+O(q^{100}) $ 2 * q + 2 * q^3 + q^5 - 2 * q^7 + 2 * q^9 + 4 * q^11 - 2 * q^13 + q^15 + 2 * q^17 + 9 * q^19 - 2 * q^21 + 3 * q^23 - q^25 + 2 * q^27 - 3 * q^29 + q^31 + 4 * q^33 - q^35 - 6 * q^37 - 2 * q^39 + 6 * q^41 + 15 * q^43 + q^45 + q^47 + 2 * q^49 + 2 * q^51 + 5 * q^53 + 2 * q^55 + 9 * q^57 - 2 * q^59 - 12 * q^61 - 2 * q^63 - q^65 + 2 * q^67 + 3 * q^69 + 14 * q^71 + 9 * q^73 - q^75 - 4 * q^77 + 3 * q^79 + 2 * q^81 + q^83 - 16 * q^85 - 3 * q^87 + 7 * q^89 + 2 * q^91 + q^93 + 13 * q^95 - 11 * q^97 + 4 * q^99

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
$5$	−1.56155	−0.698348	−0.349174	−	0.937058i	$-0.613538\pi$
$5$	−1.56155	−0.698348	−0.349174	+	0.937058i	$0.613538\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	2.00000	0.603023	0.301511	−	0.953463i	$-0.402509\pi$
$11$	2.00000	0.603023	0.301511	+	0.953463i	$0.402509\pi$
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	−1.56155	−0.403191
$16$	0	0
$17$	5.12311	1.24254	0.621268	−	0.783598i	$-0.286618\pi$
$17$	5.12311	1.24254	0.621268	+	0.783598i	$0.286618\pi$
$18$	0	0
$19$	2.43845	0.559418	0.279709	−	0.960085i	$-0.409762\pi$
$19$	2.43845	0.559418	0.279709	+	0.960085i	$0.409762\pi$
$20$	0	0
$21$	−1.00000	−0.218218
$22$	0	0
$23$	−4.68466	−0.976819	−0.488409	−	0.872615i	$-0.662423\pi$
$23$	−4.68466	−0.976819	−0.488409	+	0.872615i	$0.662423\pi$
$24$	0	0
$25$	−2.56155	−0.512311
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−3.56155	−0.661364	−0.330682	−	0.943742i	$-0.607279\pi$
$29$	−3.56155	−0.661364	−0.330682	+	0.943742i	$0.607279\pi$
$30$	0	0
$31$	−1.56155	−0.280463	−0.140232	−	0.990119i	$-0.544785\pi$
$31$	−1.56155	−0.280463	−0.140232	+	0.990119i	$0.544785\pi$
$32$	0	0
$33$	2.00000	0.348155
$34$	0	0
$35$	1.56155	0.263951
$36$	0	0
$37$	1.12311	0.184637	0.0923187	−	0.995730i	$-0.470572\pi$
$37$	1.12311	0.184637	0.0923187	+	0.995730i	$0.470572\pi$
$38$	0	0
$39$	−1.00000	−0.160128
$40$	0	0
$41$	7.12311	1.11244	0.556221	−	0.831034i	$-0.312251\pi$
$41$	7.12311	1.11244	0.556221	+	0.831034i	$0.312251\pi$
$42$	0	0
$43$	9.56155	1.45812	0.729062	−	0.684448i	$-0.239957\pi$
$43$	9.56155	1.45812	0.729062	+	0.684448i	$0.239957\pi$
$44$	0	0
$45$	−1.56155	−0.232783
$46$	0	0
$47$	6.68466	0.975058	0.487529	−	0.873107i	$-0.337898\pi$
$47$	6.68466	0.975058	0.487529	+	0.873107i	$0.337898\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	5.12311	0.717378
$52$	0	0
$53$	0.438447	0.0602254	0.0301127	−	0.999547i	$-0.490413\pi$
$53$	0.438447	0.0602254	0.0301127	+	0.999547i	$0.490413\pi$
$54$	0	0
$55$	−3.12311	−0.421119
$56$	0	0
$57$	2.43845	0.322980
$58$	0	0
$59$	−5.12311	−0.666972	−0.333486	−	0.942755i	$-0.608225\pi$
$59$	−5.12311	−0.666972	−0.333486	+	0.942755i	$0.608225\pi$
$60$	0	0
$61$	−6.00000	−0.768221	−0.384111	−	0.923287i	$-0.625492\pi$
$61$	−6.00000	−0.768221	−0.384111	+	0.923287i	$0.625492\pi$
$62$	0	0
$63$	−1.00000	−0.125988
$64$	0	0
$65$	1.56155	0.193687
$66$	0	0
$67$	13.3693	1.63332	0.816661	−	0.577118i	$-0.195823\pi$
$67$	13.3693	1.63332	0.816661	+	0.577118i	$0.195823\pi$
$68$	0	0
$69$	−4.68466	−0.563967
$70$	0	0
$71$	2.87689	0.341425	0.170712	−	0.985321i	$-0.445393\pi$
$71$	2.87689	0.341425	0.170712	+	0.985321i	$0.445393\pi$
$72$	0	0
$73$	−5.80776	−0.679747	−0.339874	−	0.940471i	$-0.610384\pi$
$73$	−5.80776	−0.679747	−0.339874	+	0.940471i	$0.610384\pi$
$74$	0	0
$75$	−2.56155	−0.295783
$76$	0	0
$77$	−2.00000	−0.227921
$78$	0	0
$79$	11.8078	1.32848	0.664239	−	0.747521i	$-0.268756\pi$
$79$	11.8078	1.32848	0.664239	+	0.747521i	$0.268756\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−9.80776	−1.07654	−0.538271	−	0.842772i	$-0.680922\pi$
$83$	−9.80776	−1.07654	−0.538271	+	0.842772i	$0.680922\pi$
$84$	0	0
$85$	−8.00000	−0.867722
$86$	0	0
$87$	−3.56155	−0.381839
$88$	0	0
$89$	5.56155	0.589523	0.294762	−	0.955571i	$-0.404760\pi$
$89$	5.56155	0.589523	0.294762	+	0.955571i	$0.404760\pi$
$90$	0	0
$91$	1.00000	0.104828
$92$	0	0
$93$	−1.56155	−0.161925
$94$	0	0
$95$	−3.80776	−0.390668
$96$	0	0
$97$	−7.56155	−0.767759	−0.383880	−	0.923383i	$-0.625412\pi$
$97$	−7.56155	−0.767759	−0.383880	+	0.923383i	$0.625412\pi$
$98$	0	0
$99$	2.00000	0.201008
$100$	0	0
$101$	−4.24621	−0.422514	−0.211257	−	0.977431i	$-0.567756\pi$
$101$	−4.24621	−0.422514	−0.211257	+	0.977431i	$0.567756\pi$
$102$	0	0
$103$	8.00000	0.788263	0.394132	−	0.919054i	$-0.371045\pi$
$103$	8.00000	0.788263	0.394132	+	0.919054i	$0.371045\pi$
$104$	0	0
$105$	1.56155	0.152392
$106$	0	0
$107$	14.2462	1.37723	0.688617	−	0.725126i	$-0.258218\pi$
$107$	14.2462	1.37723	0.688617	+	0.725126i	$0.258218\pi$
$108$	0	0
$109$	16.2462	1.55610	0.778052	−	0.628199i	$-0.216208\pi$
$109$	16.2462	1.55610	0.778052	+	0.628199i	$0.216208\pi$
$110$	0	0
$111$	1.12311	0.106600
$112$	0	0
$113$	−10.6847	−1.00513	−0.502564	−	0.864540i	$-0.667610\pi$
$113$	−10.6847	−1.00513	−0.502564	+	0.864540i	$0.667610\pi$
$114$	0	0
$115$	7.31534	0.682159
$116$	0	0
$117$	−1.00000	−0.0924500
$118$	0	0
$119$	−5.12311	−0.469634
$120$	0	0
$121$	−7.00000	−0.636364
$122$	0	0
$123$	7.12311	0.642269
$124$	0	0
$125$	11.8078	1.05612
$126$	0	0
$127$	1.75379	0.155624	0.0778118	−	0.996968i	$-0.475207\pi$
$127$	1.75379	0.155624	0.0778118	+	0.996968i	$0.475207\pi$
$128$	0	0
$129$	9.56155	0.841848
$130$	0	0
$131$	20.4924	1.79043	0.895216	−	0.445633i	$-0.147021\pi$
$131$	20.4924	1.79043	0.895216	+	0.445633i	$0.147021\pi$
$132$	0	0
$133$	−2.43845	−0.211440
$134$	0	0
$135$	−1.56155	−0.134397
$136$	0	0
$137$	−2.24621	−0.191907	−0.0959534	−	0.995386i	$-0.530590\pi$
$137$	−2.24621	−0.191907	−0.0959534	+	0.995386i	$0.530590\pi$
$138$	0	0
$139$	0.876894	0.0743772	0.0371886	−	0.999308i	$-0.488160\pi$
$139$	0.876894	0.0743772	0.0371886	+	0.999308i	$0.488160\pi$
$140$	0	0
$141$	6.68466	0.562950
$142$	0	0
$143$	−2.00000	−0.167248
$144$	0	0
$145$	5.56155	0.461862
$146$	0	0
$147$	1.00000	0.0824786
$148$	0	0
$149$	−4.87689	−0.399531	−0.199765	−	0.979844i	$-0.564018\pi$
$149$	−4.87689	−0.399531	−0.199765	+	0.979844i	$0.564018\pi$
$150$	0	0
$151$	10.2462	0.833825	0.416912	−	0.908947i	$-0.363112\pi$
$151$	10.2462	0.833825	0.416912	+	0.908947i	$0.363112\pi$
$152$	0	0
$153$	5.12311	0.414179
$154$	0	0
$155$	2.43845	0.195861
$156$	0	0
$157$	15.3693	1.22661	0.613303	−	0.789848i	$-0.289841\pi$
$157$	15.3693	1.22661	0.613303	+	0.789848i	$0.289841\pi$
$158$	0	0
$159$	0.438447	0.0347711
$160$	0	0
$161$	4.68466	0.369203
$162$	0	0
$163$	22.2462	1.74246	0.871229	−	0.490877i	$-0.163324\pi$
$163$	22.2462	1.74246	0.871229	+	0.490877i	$0.163324\pi$
$164$	0	0
$165$	−3.12311	−0.243133
$166$	0	0
$167$	9.80776	0.758948	0.379474	−	0.925202i	$-0.376105\pi$
$167$	9.80776	0.758948	0.379474	+	0.925202i	$0.376105\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	2.43845	0.186473
$172$	0	0
$173$	−7.36932	−0.560279	−0.280139	−	0.959959i	$-0.590381\pi$
$173$	−7.36932	−0.560279	−0.280139	+	0.959959i	$0.590381\pi$
$174$	0	0
$175$	2.56155	0.193635
$176$	0	0
$177$	−5.12311	−0.385076
$178$	0	0
$179$	7.80776	0.583580	0.291790	−	0.956482i	$-0.405749\pi$
$179$	7.80776	0.583580	0.291790	+	0.956482i	$0.405749\pi$
$180$	0	0
$181$	−10.0000	−0.743294	−0.371647	−	0.928374i	$-0.621207\pi$
$181$	−10.0000	−0.743294	−0.371647	+	0.928374i	$0.621207\pi$
$182$	0	0
$183$	−6.00000	−0.443533
$184$	0	0
$185$	−1.75379	−0.128941
$186$	0	0
$187$	10.2462	0.749277
$188$	0	0
$189$	−1.00000	−0.0727393
$190$	0	0
$191$	8.00000	0.578860	0.289430	−	0.957199i	$-0.406534\pi$
$191$	8.00000	0.578860	0.289430	+	0.957199i	$0.406534\pi$
$192$	0	0
$193$	10.0000	0.719816	0.359908	−	0.932988i	$-0.382808\pi$
$193$	10.0000	0.719816	0.359908	+	0.932988i	$0.382808\pi$
$194$	0	0
$195$	1.56155	0.111825
$196$	0	0
$197$	−8.87689	−0.632453	−0.316226	−	0.948684i	$-0.602416\pi$
$197$	−8.87689	−0.632453	−0.316226	+	0.948684i	$0.602416\pi$
$198$	0	0
$199$	20.4924	1.45267	0.726335	−	0.687341i	$-0.241222\pi$
$199$	20.4924	1.45267	0.726335	+	0.687341i	$0.241222\pi$
$200$	0	0
$201$	13.3693	0.942999
$202$	0	0
$203$	3.56155	0.249972
$204$	0	0
$205$	−11.1231	−0.776871
$206$	0	0
$207$	−4.68466	−0.325606
$208$	0	0
$209$	4.87689	0.337342
$210$	0	0
$211$	−17.1771	−1.18252	−0.591260	−	0.806481i	$-0.701369\pi$
$211$	−17.1771	−1.18252	−0.591260	+	0.806481i	$0.701369\pi$
$212$	0	0
$213$	2.87689	0.197122
$214$	0	0
$215$	−14.9309	−1.01828
$216$	0	0
$217$	1.56155	0.106005
$218$	0	0
$219$	−5.80776	−0.392452
$220$	0	0
$221$	−5.12311	−0.344617
$222$	0	0
$223$	3.31534	0.222012	0.111006	−	0.993820i	$-0.464593\pi$
$223$	3.31534	0.222012	0.111006	+	0.993820i	$0.464593\pi$
$224$	0	0
$225$	−2.56155	−0.170770
$226$	0	0
$227$	−19.3693	−1.28559	−0.642793	−	0.766040i	$-0.722225\pi$
$227$	−19.3693	−1.28559	−0.642793	+	0.766040i	$0.722225\pi$
$228$	0	0
$229$	28.7386	1.89910	0.949551	−	0.313612i	$-0.101539\pi$
$229$	28.7386	1.89910	0.949551	+	0.313612i	$0.101539\pi$
$230$	0	0
$231$	−2.00000	−0.131590
$232$	0	0
$233$	−21.8078	−1.42867	−0.714337	−	0.699802i	$-0.753272\pi$
$233$	−21.8078	−1.42867	−0.714337	+	0.699802i	$0.753272\pi$
$234$	0	0
$235$	−10.4384	−0.680929
$236$	0	0
$237$	11.8078	0.766997
$238$	0	0
$239$	−13.1231	−0.848863	−0.424432	−	0.905460i	$-0.639526\pi$
$239$	−13.1231	−0.848863	−0.424432	+	0.905460i	$0.639526\pi$
$240$	0	0
$241$	12.9309	0.832951	0.416475	−	0.909147i	$-0.363265\pi$
$241$	12.9309	0.832951	0.416475	+	0.909147i	$0.363265\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−1.56155	−0.0997639
$246$	0	0
$247$	−2.43845	−0.155155
$248$	0	0
$249$	−9.80776	−0.621542
$250$	0	0
$251$	9.36932	0.591386	0.295693	−	0.955283i	$-0.404449\pi$
$251$	9.36932	0.591386	0.295693	+	0.955283i	$0.404449\pi$
$252$	0	0
$253$	−9.36932	−0.589044
$254$	0	0
$255$	−8.00000	−0.500979
$256$	0	0
$257$	22.0000	1.37232	0.686161	−	0.727450i	$-0.259294\pi$
$257$	22.0000	1.37232	0.686161	+	0.727450i	$0.259294\pi$
$258$	0	0
$259$	−1.12311	−0.0697864
$260$	0	0
$261$	−3.56155	−0.220455
$262$	0	0
$263$	−17.5616	−1.08289	−0.541446	−	0.840736i	$-0.682123\pi$
$263$	−17.5616	−1.08289	−0.541446	+	0.840736i	$0.682123\pi$
$264$	0	0
$265$	−0.684658	−0.0420582
$266$	0	0
$267$	5.56155	0.340362
$268$	0	0
$269$	−9.12311	−0.556246	−0.278123	−	0.960546i	$-0.589712\pi$
$269$	−9.12311	−0.556246	−0.278123	+	0.960546i	$0.589712\pi$
$270$	0	0
$271$	16.0000	0.971931	0.485965	−	0.873978i	$-0.338468\pi$
$271$	16.0000	0.971931	0.485965	+	0.873978i	$0.338468\pi$
$272$	0	0
$273$	1.00000	0.0605228
$274$	0	0
$275$	−5.12311	−0.308935
$276$	0	0
$277$	11.5616	0.694666	0.347333	−	0.937742i	$-0.387087\pi$
$277$	11.5616	0.694666	0.347333	+	0.937742i	$0.387087\pi$
$278$	0	0
$279$	−1.56155	−0.0934877
$280$	0	0
$281$	−7.12311	−0.424929	−0.212464	−	0.977169i	$-0.568149\pi$
$281$	−7.12311	−0.424929	−0.212464	+	0.977169i	$0.568149\pi$
$282$	0	0
$283$	12.0000	0.713326	0.356663	−	0.934233i	$-0.383914\pi$
$283$	12.0000	0.713326	0.356663	+	0.934233i	$0.383914\pi$
$284$	0	0
$285$	−3.80776	−0.225552
$286$	0	0
$287$	−7.12311	−0.420464
$288$	0	0
$289$	9.24621	0.543895
$290$	0	0
$291$	−7.56155	−0.443266
$292$	0	0
$293$	32.6847	1.90946	0.954729	−	0.297477i	$-0.0961451\pi$
$293$	32.6847	1.90946	0.954729	+	0.297477i	$0.0961451\pi$
$294$	0	0
$295$	8.00000	0.465778
$296$	0	0
$297$	2.00000	0.116052
$298$	0	0
$299$	4.68466	0.270921
$300$	0	0
$301$	−9.56155	−0.551119
$302$	0	0
$303$	−4.24621	−0.243938
$304$	0	0
$305$	9.36932	0.536486
$306$	0	0
$307$	14.0540	0.802103	0.401051	−	0.916056i	$-0.368645\pi$
$307$	14.0540	0.802103	0.401051	+	0.916056i	$0.368645\pi$
$308$	0	0
$309$	8.00000	0.455104
$310$	0	0
$311$	−24.8769	−1.41064	−0.705320	−	0.708889i	$-0.749197\pi$
$311$	−24.8769	−1.41064	−0.705320	+	0.708889i	$0.749197\pi$
$312$	0	0
$313$	−0.246211	−0.0139167	−0.00695834	−	0.999976i	$-0.502215\pi$
$313$	−0.246211	−0.0139167	−0.00695834	+	0.999976i	$0.502215\pi$
$314$	0	0
$315$	1.56155	0.0879835
$316$	0	0
$317$	16.4924	0.926307	0.463153	−	0.886278i	$-0.346718\pi$
$317$	16.4924	0.926307	0.463153	+	0.886278i	$0.346718\pi$
$318$	0	0
$319$	−7.12311	−0.398817
$320$	0	0
$321$	14.2462	0.795146
$322$	0	0
$323$	12.4924	0.695097
$324$	0	0
$325$	2.56155	0.142089
$326$	0	0
$327$	16.2462	0.898418
$328$	0	0
$329$	−6.68466	−0.368537
$330$	0	0
$331$	2.63068	0.144595	0.0722977	−	0.997383i	$-0.476967\pi$
$331$	2.63068	0.144595	0.0722977	+	0.997383i	$0.476967\pi$
$332$	0	0
$333$	1.12311	0.0615458
$334$	0	0
$335$	−20.8769	−1.14063
$336$	0	0
$337$	−10.6847	−0.582030	−0.291015	−	0.956718i	$-0.593993\pi$
$337$	−10.6847	−0.582030	−0.291015	+	0.956718i	$0.593993\pi$
$338$	0	0
$339$	−10.6847	−0.580311
$340$	0	0
$341$	−3.12311	−0.169126
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	7.31534	0.393845
$346$	0	0
$347$	−30.7386	−1.65014	−0.825068	−	0.565033i	$-0.808863\pi$
$347$	−30.7386	−1.65014	−0.825068	+	0.565033i	$0.808863\pi$
$348$	0	0
$349$	8.43845	0.451700	0.225850	−	0.974162i	$-0.427484\pi$
$349$	8.43845	0.451700	0.225850	+	0.974162i	$0.427484\pi$
$350$	0	0
$351$	−1.00000	−0.0533761
$352$	0	0
$353$	−2.63068	−0.140017	−0.0700086	−	0.997546i	$-0.522303\pi$
$353$	−2.63068	−0.140017	−0.0700086	+	0.997546i	$0.522303\pi$
$354$	0	0
$355$	−4.49242	−0.238433
$356$	0	0
$357$	−5.12311	−0.271144
$358$	0	0
$359$	−0.630683	−0.0332862	−0.0166431	−	0.999861i	$-0.505298\pi$
$359$	−0.630683	−0.0332862	−0.0166431	+	0.999861i	$0.505298\pi$
$360$	0	0
$361$	−13.0540	−0.687051
$362$	0	0
$363$	−7.00000	−0.367405
$364$	0	0
$365$	9.06913	0.474700
$366$	0	0
$367$	−19.1231	−0.998218	−0.499109	−	0.866539i	$-0.666339\pi$
$367$	−19.1231	−0.998218	−0.499109	+	0.866539i	$0.666339\pi$
$368$	0	0
$369$	7.12311	0.370814
$370$	0	0
$371$	−0.438447	−0.0227630
$372$	0	0
$373$	−15.7538	−0.815700	−0.407850	−	0.913049i	$-0.633721\pi$
$373$	−15.7538	−0.815700	−0.407850	+	0.913049i	$0.633721\pi$
$374$	0	0
$375$	11.8078	0.609750
$376$	0	0
$377$	3.56155	0.183429
$378$	0	0
$379$	31.6155	1.62398	0.811990	−	0.583671i	$-0.198384\pi$
$379$	31.6155	1.62398	0.811990	+	0.583671i	$0.198384\pi$
$380$	0	0
$381$	1.75379	0.0898493
$382$	0	0
$383$	11.3693	0.580945	0.290472	−	0.956883i	$-0.406188\pi$
$383$	11.3693	0.580945	0.290472	+	0.956883i	$0.406188\pi$
$384$	0	0
$385$	3.12311	0.159168
$386$	0	0
$387$	9.56155	0.486041
$388$	0	0
$389$	−10.0000	−0.507020	−0.253510	−	0.967333i	$-0.581585\pi$
$389$	−10.0000	−0.507020	−0.253510	+	0.967333i	$0.581585\pi$
$390$	0	0
$391$	−24.0000	−1.21373
$392$	0	0
$393$	20.4924	1.03371
$394$	0	0
$395$	−18.4384	−0.927739
$396$	0	0
$397$	−4.05398	−0.203463	−0.101732	−	0.994812i	$-0.532438\pi$
$397$	−4.05398	−0.203463	−0.101732	+	0.994812i	$0.532438\pi$
$398$	0	0
$399$	−2.43845	−0.122075
$400$	0	0
$401$	−22.2462	−1.11092	−0.555461	−	0.831542i	$-0.687458\pi$
$401$	−22.2462	−1.11092	−0.555461	+	0.831542i	$0.687458\pi$
$402$	0	0
$403$	1.56155	0.0777865
$404$	0	0
$405$	−1.56155	−0.0775942
$406$	0	0
$407$	2.24621	0.111341
$408$	0	0
$409$	−13.8078	−0.682750	−0.341375	−	0.939927i	$-0.610893\pi$
$409$	−13.8078	−0.682750	−0.341375	+	0.939927i	$0.610893\pi$
$410$	0	0
$411$	−2.24621	−0.110797
$412$	0	0
$413$	5.12311	0.252092
$414$	0	0
$415$	15.3153	0.751801
$416$	0	0
$417$	0.876894	0.0429417
$418$	0	0
$419$	−12.8769	−0.629077	−0.314539	−	0.949245i	$-0.601850\pi$
$419$	−12.8769	−0.629077	−0.314539	+	0.949245i	$0.601850\pi$
$420$	0	0
$421$	−10.8769	−0.530107	−0.265054	−	0.964234i	$-0.585390\pi$
$421$	−10.8769	−0.530107	−0.265054	+	0.964234i	$0.585390\pi$
$422$	0	0
$423$	6.68466	0.325019
$424$	0	0
$425$	−13.1231	−0.636564
$426$	0	0
$427$	6.00000	0.290360
$428$	0	0
$429$	−2.00000	−0.0965609
$430$	0	0
$431$	8.24621	0.397206	0.198603	−	0.980080i	$-0.436360\pi$
$431$	8.24621	0.397206	0.198603	+	0.980080i	$0.436360\pi$
$432$	0	0
$433$	31.3693	1.50751	0.753757	−	0.657154i	$-0.228240\pi$
$433$	31.3693	1.50751	0.753757	+	0.657154i	$0.228240\pi$
$434$	0	0
$435$	5.56155	0.266656
$436$	0	0
$437$	−11.4233	−0.546450
$438$	0	0
$439$	−23.6155	−1.12711	−0.563554	−	0.826079i	$-0.690566\pi$
$439$	−23.6155	−1.12711	−0.563554	+	0.826079i	$0.690566\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	−21.1771	−1.00615	−0.503077	−	0.864242i	$-0.667799\pi$
$443$	−21.1771	−1.00615	−0.503077	+	0.864242i	$0.667799\pi$
$444$	0	0
$445$	−8.68466	−0.411692
$446$	0	0
$447$	−4.87689	−0.230669
$448$	0	0
$449$	−1.75379	−0.0827664	−0.0413832	−	0.999143i	$-0.513176\pi$
$449$	−1.75379	−0.0827664	−0.0413832	+	0.999143i	$0.513176\pi$
$450$	0	0
$451$	14.2462	0.670828
$452$	0	0
$453$	10.2462	0.481409
$454$	0	0
$455$	−1.56155	−0.0732067
$456$	0	0
$457$	−5.12311	−0.239649	−0.119824	−	0.992795i	$-0.538233\pi$
$457$	−5.12311	−0.239649	−0.119824	+	0.992795i	$0.538233\pi$
$458$	0	0
$459$	5.12311	0.239126
$460$	0	0
$461$	2.63068	0.122523	0.0612616	−	0.998122i	$-0.480488\pi$
$461$	2.63068	0.122523	0.0612616	+	0.998122i	$0.480488\pi$
$462$	0	0
$463$	−21.8617	−1.01600	−0.508001	−	0.861357i	$-0.669615\pi$
$463$	−21.8617	−1.01600	−0.508001	+	0.861357i	$0.669615\pi$
$464$	0	0
$465$	2.43845	0.113080
$466$	0	0
$467$	17.3693	0.803756	0.401878	−	0.915693i	$-0.368358\pi$
$467$	17.3693	0.803756	0.401878	+	0.915693i	$0.368358\pi$
$468$	0	0
$469$	−13.3693	−0.617338
$470$	0	0
$471$	15.3693	0.708181
$472$	0	0
$473$	19.1231	0.879281
$474$	0	0
$475$	−6.24621	−0.286596
$476$	0	0
$477$	0.438447	0.0200751
$478$	0	0
$479$	−8.43845	−0.385562	−0.192781	−	0.981242i	$-0.561751\pi$
$479$	−8.43845	−0.385562	−0.192781	+	0.981242i	$0.561751\pi$
$480$	0	0
$481$	−1.12311	−0.0512092
$482$	0	0
$483$	4.68466	0.213159
$484$	0	0
$485$	11.8078	0.536163
$486$	0	0
$487$	−33.8617	−1.53442	−0.767211	−	0.641395i	$-0.778356\pi$
$487$	−33.8617	−1.53442	−0.767211	+	0.641395i	$0.778356\pi$
$488$	0	0
$489$	22.2462	1.00601
$490$	0	0
$491$	−1.75379	−0.0791474	−0.0395737	−	0.999217i	$-0.512600\pi$
$491$	−1.75379	−0.0791474	−0.0395737	+	0.999217i	$0.512600\pi$
$492$	0	0
$493$	−18.2462	−0.821768
$494$	0	0
$495$	−3.12311	−0.140373
$496$	0	0
$497$	−2.87689	−0.129046
$498$	0	0
$499$	−21.7538	−0.973833	−0.486917	−	0.873448i	$-0.661878\pi$
$499$	−21.7538	−0.973833	−0.486917	+	0.873448i	$0.661878\pi$
$500$	0	0
$501$	9.80776	0.438179
$502$	0	0
$503$	−2.63068	−0.117296	−0.0586482	−	0.998279i	$-0.518679\pi$
$503$	−2.63068	−0.117296	−0.0586482	+	0.998279i	$0.518679\pi$
$504$	0	0
$505$	6.63068	0.295062
$506$	0	0
$507$	1.00000	0.0444116
$508$	0	0
$509$	−13.0691	−0.579279	−0.289640	−	0.957136i	$-0.593535\pi$
$509$	−13.0691	−0.579279	−0.289640	+	0.957136i	$0.593535\pi$
$510$	0	0
$511$	5.80776	0.256920
$512$	0	0
$513$	2.43845	0.107660
$514$	0	0
$515$	−12.4924	−0.550482
$516$	0	0
$517$	13.3693	0.587982
$518$	0	0
$519$	−7.36932	−0.323477
$520$	0	0
$521$	−6.87689	−0.301282	−0.150641	−	0.988589i	$-0.548134\pi$
$521$	−6.87689	−0.301282	−0.150641	+	0.988589i	$0.548134\pi$
$522$	0	0
$523$	11.6155	0.507912	0.253956	−	0.967216i	$-0.418268\pi$
$523$	11.6155	0.507912	0.253956	+	0.967216i	$0.418268\pi$
$524$	0	0
$525$	2.56155	0.111795
$526$	0	0
$527$	−8.00000	−0.348485
$528$	0	0
$529$	−1.05398	−0.0458250
$530$	0	0
$531$	−5.12311	−0.222324
$532$	0	0
$533$	−7.12311	−0.308536
$534$	0	0
$535$	−22.2462	−0.961788
$536$	0	0
$537$	7.80776	0.336930
$538$	0	0
$539$	2.00000	0.0861461
$540$	0	0
$541$	−13.6155	−0.585377	−0.292689	−	0.956208i	$-0.594550\pi$
$541$	−13.6155	−0.585377	−0.292689	+	0.956208i	$0.594550\pi$
$542$	0	0
$543$	−10.0000	−0.429141
$544$	0	0
$545$	−25.3693	−1.08670
$546$	0	0
$547$	−22.0540	−0.942960	−0.471480	−	0.881877i	$-0.656280\pi$
$547$	−22.0540	−0.942960	−0.471480	+	0.881877i	$0.656280\pi$
$548$	0	0
$549$	−6.00000	−0.256074
$550$	0	0
$551$	−8.68466	−0.369979
$552$	0	0
$553$	−11.8078	−0.502117
$554$	0	0
$555$	−1.75379	−0.0744442
$556$	0	0
$557$	20.4924	0.868292	0.434146	−	0.900843i	$-0.357050\pi$
$557$	20.4924	0.868292	0.434146	+	0.900843i	$0.357050\pi$
$558$	0	0
$559$	−9.56155	−0.404411
$560$	0	0
$561$	10.2462	0.432595
$562$	0	0
$563$	−22.7386	−0.958319	−0.479160	−	0.877728i	$-0.659058\pi$
$563$	−22.7386	−0.958319	−0.479160	+	0.877728i	$0.659058\pi$
$564$	0	0
$565$	16.6847	0.701929
$566$	0	0
$567$	−1.00000	−0.0419961
$568$	0	0
$569$	−34.3002	−1.43794	−0.718969	−	0.695042i	$-0.755386\pi$
$569$	−34.3002	−1.43794	−0.718969	+	0.695042i	$0.755386\pi$
$570$	0	0
$571$	5.06913	0.212137	0.106068	−	0.994359i	$-0.466174\pi$
$571$	5.06913	0.212137	0.106068	+	0.994359i	$0.466174\pi$
$572$	0	0
$573$	8.00000	0.334205
$574$	0	0
$575$	12.0000	0.500435
$576$	0	0
$577$	18.0000	0.749350	0.374675	−	0.927156i	$-0.377754\pi$
$577$	18.0000	0.749350	0.374675	+	0.927156i	$0.377754\pi$
$578$	0	0
$579$	10.0000	0.415586
$580$	0	0
$581$	9.80776	0.406895
$582$	0	0
$583$	0.876894	0.0363173
$584$	0	0
$585$	1.56155	0.0645623
$586$	0	0
$587$	−1.80776	−0.0746144	−0.0373072	−	0.999304i	$-0.511878\pi$
$587$	−1.80776	−0.0746144	−0.0373072	+	0.999304i	$0.511878\pi$
$588$	0	0
$589$	−3.80776	−0.156896
$590$	0	0
$591$	−8.87689	−0.365147
$592$	0	0
$593$	18.4384	0.757176	0.378588	−	0.925565i	$-0.376410\pi$
$593$	18.4384	0.757176	0.378588	+	0.925565i	$0.376410\pi$
$594$	0	0
$595$	8.00000	0.327968
$596$	0	0
$597$	20.4924	0.838699
$598$	0	0
$599$	−36.6847	−1.49889	−0.749447	−	0.662064i	$-0.769681\pi$
$599$	−36.6847	−1.49889	−0.749447	+	0.662064i	$0.769681\pi$
$600$	0	0
$601$	−3.75379	−0.153120	−0.0765601	−	0.997065i	$-0.524394\pi$
$601$	−3.75379	−0.153120	−0.0765601	+	0.997065i	$0.524394\pi$
$602$	0	0
$603$	13.3693	0.544441
$604$	0	0
$605$	10.9309	0.444403
$606$	0	0
$607$	15.6155	0.633815	0.316907	−	0.948456i	$-0.397356\pi$
$607$	15.6155	0.633815	0.316907	+	0.948456i	$0.397356\pi$
$608$	0	0
$609$	3.56155	0.144321
$610$	0	0
$611$	−6.68466	−0.270432
$612$	0	0
$613$	15.3693	0.620761	0.310380	−	0.950612i	$-0.399544\pi$
$613$	15.3693	0.620761	0.310380	+	0.950612i	$0.399544\pi$
$614$	0	0
$615$	−11.1231	−0.448527
$616$	0	0
$617$	29.3693	1.18236	0.591182	−	0.806538i	$-0.298661\pi$
$617$	29.3693	1.18236	0.591182	+	0.806538i	$0.298661\pi$
$618$	0	0
$619$	36.9848	1.48655	0.743273	−	0.668988i	$-0.233272\pi$
$619$	36.9848	1.48655	0.743273	+	0.668988i	$0.233272\pi$
$620$	0	0
$621$	−4.68466	−0.187989
$622$	0	0
$623$	−5.56155	−0.222819
$624$	0	0
$625$	−5.63068	−0.225227
$626$	0	0
$627$	4.87689	0.194764
$628$	0	0
$629$	5.75379	0.229419
$630$	0	0
$631$	23.1231	0.920516	0.460258	−	0.887785i	$-0.347757\pi$
$631$	23.1231	0.920516	0.460258	+	0.887785i	$0.347757\pi$
$632$	0	0
$633$	−17.1771	−0.682728
$634$	0	0
$635$	−2.73863	−0.108679
$636$	0	0
$637$	−1.00000	−0.0396214
$638$	0	0
$639$	2.87689	0.113808
$640$	0	0
$641$	−10.1922	−0.402569	−0.201285	−	0.979533i	$-0.564512\pi$
$641$	−10.1922	−0.402569	−0.201285	+	0.979533i	$0.564512\pi$
$642$	0	0
$643$	5.75379	0.226907	0.113454	−	0.993543i	$-0.463809\pi$
$643$	5.75379	0.226907	0.113454	+	0.993543i	$0.463809\pi$
$644$	0	0
$645$	−14.9309	−0.587902
$646$	0	0
$647$	−33.3693	−1.31188	−0.655942	−	0.754812i	$-0.727728\pi$
$647$	−33.3693	−1.31188	−0.655942	+	0.754812i	$0.727728\pi$
$648$	0	0
$649$	−10.2462	−0.402199
$650$	0	0
$651$	1.56155	0.0612021
$652$	0	0
$653$	22.4924	0.880197	0.440098	−	0.897950i	$-0.354944\pi$
$653$	22.4924	0.880197	0.440098	+	0.897950i	$0.354944\pi$
$654$	0	0
$655$	−32.0000	−1.25034
$656$	0	0
$657$	−5.80776	−0.226582
$658$	0	0
$659$	16.1922	0.630760	0.315380	−	0.948965i	$-0.397868\pi$
$659$	16.1922	0.630760	0.315380	+	0.948965i	$0.397868\pi$
$660$	0	0
$661$	−33.3153	−1.29582	−0.647908	−	0.761718i	$-0.724356\pi$
$661$	−33.3153	−1.29582	−0.647908	+	0.761718i	$0.724356\pi$
$662$	0	0
$663$	−5.12311	−0.198965
$664$	0	0
$665$	3.80776	0.147659
$666$	0	0
$667$	16.6847	0.646033
$668$	0	0
$669$	3.31534	0.128179
$670$	0	0
$671$	−12.0000	−0.463255
$672$	0	0
$673$	−0.438447	−0.0169009	−0.00845045	−	0.999964i	$-0.502690\pi$
$673$	−0.438447	−0.0169009	−0.00845045	+	0.999964i	$0.502690\pi$
$674$	0	0
$675$	−2.56155	−0.0985942
$676$	0	0
$677$	10.4924	0.403257	0.201628	−	0.979462i	$-0.435377\pi$
$677$	10.4924	0.403257	0.201628	+	0.979462i	$0.435377\pi$
$678$	0	0
$679$	7.56155	0.290186
$680$	0	0
$681$	−19.3693	−0.742234
$682$	0	0
$683$	25.6155	0.980151	0.490075	−	0.871680i	$-0.336969\pi$
$683$	25.6155	0.980151	0.490075	+	0.871680i	$0.336969\pi$
$684$	0	0
$685$	3.50758	0.134018
$686$	0	0
$687$	28.7386	1.09645
$688$	0	0
$689$	−0.438447	−0.0167035
$690$	0	0
$691$	27.3153	1.03912	0.519562	−	0.854433i	$-0.326095\pi$
$691$	27.3153	1.03912	0.519562	+	0.854433i	$0.326095\pi$
$692$	0	0
$693$	−2.00000	−0.0759737
$694$	0	0
$695$	−1.36932	−0.0519411
$696$	0	0
$697$	36.4924	1.38225
$698$	0	0
$699$	−21.8078	−0.824845
$700$	0	0
$701$	−28.4384	−1.07411	−0.537053	−	0.843549i	$-0.680462\pi$
$701$	−28.4384	−1.07411	−0.537053	+	0.843549i	$0.680462\pi$
$702$	0	0
$703$	2.73863	0.103290
$704$	0	0
$705$	−10.4384	−0.393135
$706$	0	0
$707$	4.24621	0.159695
$708$	0	0
$709$	20.7386	0.778856	0.389428	−	0.921057i	$-0.372673\pi$
$709$	20.7386	0.778856	0.389428	+	0.921057i	$0.372673\pi$
$710$	0	0
$711$	11.8078	0.442826
$712$	0	0
$713$	7.31534	0.273962
$714$	0	0
$715$	3.12311	0.116798
$716$	0	0
$717$	−13.1231	−0.490091
$718$	0	0
$719$	43.6155	1.62658	0.813292	−	0.581855i	$-0.197673\pi$
$719$	43.6155	1.62658	0.813292	+	0.581855i	$0.197673\pi$
$720$	0	0
$721$	−8.00000	−0.297936
$722$	0	0
$723$	12.9309	0.480904
$724$	0	0
$725$	9.12311	0.338824
$726$	0	0
$727$	3.12311	0.115830	0.0579148	−	0.998322i	$-0.481555\pi$
$727$	3.12311	0.115830	0.0579148	+	0.998322i	$0.481555\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	48.9848	1.81177
$732$	0	0
$733$	39.1771	1.44704	0.723519	−	0.690304i	$-0.242523\pi$
$733$	39.1771	1.44704	0.723519	+	0.690304i	$0.242523\pi$
$734$	0	0
$735$	−1.56155	−0.0575987
$736$	0	0
$737$	26.7386	0.984930
$738$	0	0
$739$	−38.3542	−1.41088	−0.705440	−	0.708769i	$-0.749251\pi$
$739$	−38.3542	−1.41088	−0.705440	+	0.708769i	$0.749251\pi$
$740$	0	0
$741$	−2.43845	−0.0895786
$742$	0	0
$743$	5.61553	0.206014	0.103007	−	0.994681i	$-0.467154\pi$
$743$	5.61553	0.206014	0.103007	+	0.994681i	$0.467154\pi$
$744$	0	0
$745$	7.61553	0.279011
$746$	0	0
$747$	−9.80776	−0.358847
$748$	0	0
$749$	−14.2462	−0.520545
$750$	0	0
$751$	10.0540	0.366875	0.183437	−	0.983031i	$-0.441278\pi$
$751$	10.0540	0.366875	0.183437	+	0.983031i	$0.441278\pi$
$752$	0	0
$753$	9.36932	0.341437
$754$	0	0
$755$	−16.0000	−0.582300
$756$	0	0
$757$	−30.6847	−1.11525	−0.557626	−	0.830092i	$-0.688288\pi$
$757$	−30.6847	−1.11525	−0.557626	+	0.830092i	$0.688288\pi$
$758$	0	0
$759$	−9.36932	−0.340085
$760$	0	0
$761$	10.9309	0.396244	0.198122	−	0.980177i	$-0.436516\pi$
$761$	10.9309	0.396244	0.198122	+	0.980177i	$0.436516\pi$
$762$	0	0
$763$	−16.2462	−0.588152
$764$	0	0
$765$	−8.00000	−0.289241
$766$	0	0
$767$	5.12311	0.184985
$768$	0	0
$769$	−14.6847	−0.529542	−0.264771	−	0.964311i	$-0.585296\pi$
$769$	−14.6847	−0.529542	−0.264771	+	0.964311i	$0.585296\pi$
$770$	0	0
$771$	22.0000	0.792311
$772$	0	0
$773$	22.6307	0.813969	0.406985	−	0.913435i	$-0.366580\pi$
$773$	22.6307	0.813969	0.406985	+	0.913435i	$0.366580\pi$
$774$	0	0
$775$	4.00000	0.143684
$776$	0	0
$777$	−1.12311	−0.0402912
$778$	0	0
$779$	17.3693	0.622320
$780$	0	0
$781$	5.75379	0.205887
$782$	0	0
$783$	−3.56155	−0.127280
$784$	0	0
$785$	−24.0000	−0.856597
$786$	0	0
$787$	−3.31534	−0.118179	−0.0590896	−	0.998253i	$-0.518820\pi$
$787$	−3.31534	−0.118179	−0.0590896	+	0.998253i	$0.518820\pi$
$788$	0	0
$789$	−17.5616	−0.625208
$790$	0	0
$791$	10.6847	0.379903
$792$	0	0
$793$	6.00000	0.213066
$794$	0	0
$795$	−0.684658	−0.0242823
$796$	0	0
$797$	−18.9848	−0.672478	−0.336239	−	0.941777i	$-0.609155\pi$
$797$	−18.9848	−0.672478	−0.336239	+	0.941777i	$0.609155\pi$
$798$	0	0
$799$	34.2462	1.21154
$800$	0	0
$801$	5.56155	0.196508
$802$	0	0
$803$	−11.6155	−0.409903
$804$	0	0
$805$	−7.31534	−0.257832
$806$	0	0
$807$	−9.12311	−0.321149
$808$	0	0
$809$	−44.4384	−1.56237	−0.781186	−	0.624298i	$-0.785385\pi$
$809$	−44.4384	−1.56237	−0.781186	+	0.624298i	$0.785385\pi$
$810$	0	0
$811$	−22.2462	−0.781170	−0.390585	−	0.920567i	$-0.627727\pi$
$811$	−22.2462	−0.781170	−0.390585	+	0.920567i	$0.627727\pi$
$812$	0	0
$813$	16.0000	0.561144
$814$	0	0
$815$	−34.7386	−1.21684
$816$	0	0
$817$	23.3153	0.815701
$818$	0	0
$819$	1.00000	0.0349428
$820$	0	0
$821$	−51.1231	−1.78421	−0.892104	−	0.451829i	$-0.850772\pi$
$821$	−51.1231	−1.78421	−0.892104	+	0.451829i	$0.850772\pi$
$822$	0	0
$823$	−19.5076	−0.679991	−0.339996	−	0.940427i	$-0.610426\pi$
$823$	−19.5076	−0.679991	−0.339996	+	0.940427i	$0.610426\pi$
$824$	0	0
$825$	−5.12311	−0.178364
$826$	0	0
$827$	−42.1080	−1.46424	−0.732118	−	0.681177i	$-0.761468\pi$
$827$	−42.1080	−1.46424	−0.732118	+	0.681177i	$0.761468\pi$
$828$	0	0
$829$	14.4924	0.503343	0.251671	−	0.967813i	$-0.419020\pi$
$829$	14.4924	0.503343	0.251671	+	0.967813i	$0.419020\pi$
$830$	0	0
$831$	11.5616	0.401066
$832$	0	0
$833$	5.12311	0.177505
$834$	0	0
$835$	−15.3153	−0.530009
$836$	0	0
$837$	−1.56155	−0.0539752
$838$	0	0
$839$	−10.8769	−0.375512	−0.187756	−	0.982216i	$-0.560121\pi$
$839$	−10.8769	−0.375512	−0.187756	+	0.982216i	$0.560121\pi$
$840$	0	0
$841$	−16.3153	−0.562598
$842$	0	0
$843$	−7.12311	−0.245333
$844$	0	0
$845$	−1.56155	−0.0537190
$846$	0	0
$847$	7.00000	0.240523
$848$	0	0
$849$	12.0000	0.411839
$850$	0	0
$851$	−5.26137	−0.180357
$852$	0	0
$853$	41.8078	1.43147	0.715735	−	0.698372i	$-0.246092\pi$
$853$	41.8078	1.43147	0.715735	+	0.698372i	$0.246092\pi$
$854$	0	0
$855$	−3.80776	−0.130223
$856$	0	0
$857$	−48.2462	−1.64806	−0.824030	−	0.566547i	$-0.808279\pi$
$857$	−48.2462	−1.64806	−0.824030	+	0.566547i	$0.808279\pi$
$858$	0	0
$859$	15.1231	0.515994	0.257997	−	0.966146i	$-0.416938\pi$
$859$	15.1231	0.515994	0.257997	+	0.966146i	$0.416938\pi$
$860$	0	0
$861$	−7.12311	−0.242755
$862$	0	0
$863$	−53.6155	−1.82509	−0.912547	−	0.408972i	$-0.865887\pi$
$863$	−53.6155	−1.82509	−0.912547	+	0.408972i	$0.865887\pi$
$864$	0	0
$865$	11.5076	0.391269
$866$	0	0
$867$	9.24621	0.314018
$868$	0	0
$869$	23.6155	0.801102
$870$	0	0
$871$	−13.3693	−0.453002
$872$	0	0
$873$	−7.56155	−0.255920
$874$	0	0
$875$	−11.8078	−0.399175
$876$	0	0
$877$	−33.6155	−1.13512	−0.567558	−	0.823334i	$-0.692112\pi$
$877$	−33.6155	−1.13512	−0.567558	+	0.823334i	$0.692112\pi$
$878$	0	0
$879$	32.6847	1.10243
$880$	0	0
$881$	0.738634	0.0248852	0.0124426	−	0.999923i	$-0.496039\pi$
$881$	0.738634	0.0248852	0.0124426	+	0.999923i	$0.496039\pi$
$882$	0	0
$883$	52.0000	1.74994	0.874970	−	0.484178i	$-0.160881\pi$
$883$	52.0000	1.74994	0.874970	+	0.484178i	$0.160881\pi$
$884$	0	0
$885$	8.00000	0.268917
$886$	0	0
$887$	40.9848	1.37614	0.688068	−	0.725646i	$-0.258459\pi$
$887$	40.9848	1.37614	0.688068	+	0.725646i	$0.258459\pi$
$888$	0	0
$889$	−1.75379	−0.0588202
$890$	0	0
$891$	2.00000	0.0670025
$892$	0	0
$893$	16.3002	0.545465
$894$	0	0
$895$	−12.1922	−0.407542
$896$	0	0
$897$	4.68466	0.156416
$898$	0	0
$899$	5.56155	0.185488
$900$	0	0
$901$	2.24621	0.0748321
$902$	0	0
$903$	−9.56155	−0.318189
$904$	0	0
$905$	15.6155	0.519078
$906$	0	0
$907$	14.0540	0.466655	0.233327	−	0.972398i	$-0.425039\pi$
$907$	14.0540	0.466655	0.233327	+	0.972398i	$0.425039\pi$
$908$	0	0
$909$	−4.24621	−0.140838
$910$	0	0
$911$	−31.3153	−1.03752	−0.518762	−	0.854919i	$-0.673607\pi$
$911$	−31.3153	−1.03752	−0.518762	+	0.854919i	$0.673607\pi$
$912$	0	0
$913$	−19.6155	−0.649179
$914$	0	0
$915$	9.36932	0.309740
$916$	0	0
$917$	−20.4924	−0.676719
$918$	0	0
$919$	−19.5076	−0.643496	−0.321748	−	0.946825i	$-0.604270\pi$
$919$	−19.5076	−0.643496	−0.321748	+	0.946825i	$0.604270\pi$
$920$	0	0
$921$	14.0540	0.463094
$922$	0	0
$923$	−2.87689	−0.0946941
$924$	0	0
$925$	−2.87689	−0.0945917
$926$	0	0
$927$	8.00000	0.262754
$928$	0	0
$929$	−48.7926	−1.60083	−0.800417	−	0.599444i	$-0.795388\pi$
$929$	−48.7926	−1.60083	−0.800417	+	0.599444i	$0.795388\pi$
$930$	0	0
$931$	2.43845	0.0799169
$932$	0	0
$933$	−24.8769	−0.814433
$934$	0	0
$935$	−16.0000	−0.523256
$936$	0	0
$937$	−51.3693	−1.67816	−0.839081	−	0.544006i	$-0.816907\pi$
$937$	−51.3693	−1.67816	−0.839081	+	0.544006i	$0.816907\pi$
$938$	0	0
$939$	−0.246211	−0.00803480
$940$	0	0
$941$	−6.93087	−0.225940	−0.112970	−	0.993598i	$-0.536036\pi$
$941$	−6.93087	−0.225940	−0.112970	+	0.993598i	$0.536036\pi$
$942$	0	0
$943$	−33.3693	−1.08665
$944$	0	0
$945$	1.56155	0.0507973
$946$	0	0
$947$	34.4924	1.12085	0.560427	−	0.828204i	$-0.310637\pi$
$947$	34.4924	1.12085	0.560427	+	0.828204i	$0.310637\pi$
$948$	0	0
$949$	5.80776	0.188528
$950$	0	0
$951$	16.4924	0.534803
$952$	0	0
$953$	−12.4384	−0.402921	−0.201460	−	0.979497i	$-0.564569\pi$
$953$	−12.4384	−0.402921	−0.201460	+	0.979497i	$0.564569\pi$
$954$	0	0
$955$	−12.4924	−0.404245
$956$	0	0
$957$	−7.12311	−0.230257
$958$	0	0
$959$	2.24621	0.0725339
$960$	0	0
$961$	−28.5616	−0.921340
$962$	0	0
$963$	14.2462	0.459078
$964$	0	0
$965$	−15.6155	−0.502682
$966$	0	0
$967$	52.6004	1.69151	0.845757	−	0.533568i	$-0.179149\pi$
$967$	52.6004	1.69151	0.845757	+	0.533568i	$0.179149\pi$
$968$	0	0
$969$	12.4924	0.401314
$970$	0	0
$971$	10.6307	0.341155	0.170577	−	0.985344i	$-0.445437\pi$
$971$	10.6307	0.341155	0.170577	+	0.985344i	$0.445437\pi$
$972$	0	0
$973$	−0.876894	−0.0281119
$974$	0	0
$975$	2.56155	0.0820353
$976$	0	0
$977$	−49.3693	−1.57946	−0.789732	−	0.613452i	$-0.789781\pi$
$977$	−49.3693	−1.57946	−0.789732	+	0.613452i	$0.789781\pi$
$978$	0	0
$979$	11.1231	0.355496
$980$	0	0
$981$	16.2462	0.518702
$982$	0	0
$983$	−18.1922	−0.580242	−0.290121	−	0.956990i	$-0.593696\pi$
$983$	−18.1922	−0.580242	−0.290121	+	0.956990i	$0.593696\pi$
$984$	0	0
$985$	13.8617	0.441672
$986$	0	0
$987$	−6.68466	−0.212775
$988$	0	0
$989$	−44.7926	−1.42432
$990$	0	0
$991$	−38.2462	−1.21493	−0.607465	−	0.794346i	$-0.707814\pi$
$991$	−38.2462	−1.21493	−0.607465	+	0.794346i	$0.707814\pi$
$992$	0	0
$993$	2.63068	0.0834822
$994$	0	0
$995$	−32.0000	−1.01447
$996$	0	0
$997$	−33.6155	−1.06461	−0.532307	−	0.846551i	$-0.678675\pi$
$997$	−33.6155	−1.06461	−0.532307	+	0.846551i	$0.678675\pi$
$998$	0	0
$999$	1.12311	0.0355335

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4368.2.a.bk.1.1		2
4.3	odd	2		2184.2.a.p.1.1	✓	2
12.11	even	2		6552.2.a.be.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2184.2.a.p.1.1	✓	2	4.3	odd	2
4368.2.a.bk.1.1		2	1.1	even	1	trivial
6552.2.a.be.1.2		2	12.11	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:	\( N \)	\(=\)	\( 4368 = 2^{4} \cdot 3 \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4368.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -1.56155 q^{5} -1.00000 q^{7} +1.00000 q^{9} +2.00000 q^{11} -1.00000 q^{13} -1.56155 q^{15} +5.12311 q^{17} +2.43845 q^{19} -1.00000 q^{21} -4.68466 q^{23} -2.56155 q^{25} +1.00000 q^{27} -3.56155 q^{29} -1.56155 q^{31} +2.00000 q^{33} +1.56155 q^{35} +1.12311 q^{37} -1.00000 q^{39} +7.12311 q^{41} +9.56155 q^{43} -1.56155 q^{45} +6.68466 q^{47} +1.00000 q^{49} +5.12311 q^{51} +0.438447 q^{53} -3.12311 q^{55} +2.43845 q^{57} -5.12311 q^{59} -6.00000 q^{61} -1.00000 q^{63} +1.56155 q^{65} +13.3693 q^{67} -4.68466 q^{69} +2.87689 q^{71} -5.80776 q^{73} -2.56155 q^{75} -2.00000 q^{77} +11.8078 q^{79} +1.00000 q^{81} -9.80776 q^{83} -8.00000 q^{85} -3.56155 q^{87} +5.56155 q^{89} +1.00000 q^{91} -1.56155 q^{93} -3.80776 q^{95} -7.56155 q^{97} +2.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} + q^{5} - 2 q^{7} + 2 q^{9} + 4 q^{11} - 2 q^{13} + q^{15} + 2 q^{17} + 9 q^{19} - 2 q^{21} + 3 q^{23} - q^{25} + 2 q^{27} - 3 q^{29} + q^{31} + 4 q^{33} - q^{35} - 6 q^{37} - 2 q^{39}+ \cdots + 4 q^{99}+O(q^{100}) \) 2 * q + 2 * q^3 + q^5 - 2 * q^7 + 2 * q^9 + 4 * q^11 - 2 * q^13 + q^15 + 2 * q^17 + 9 * q^19 - 2 * q^21 + 3 * q^23 - q^25 + 2 * q^27 - 3 * q^29 + q^31 + 4 * q^33 - q^35 - 6 * q^37 - 2 * q^39 + 6 * q^41 + 15 * q^43 + q^45 + q^47 + 2 * q^49 + 2 * q^51 + 5 * q^53 + 2 * q^55 + 9 * q^57 - 2 * q^59 - 12 * q^61 - 2 * q^63 - q^65 + 2 * q^67 + 3 * q^69 + 14 * q^71 + 9 * q^73 - q^75 - 4 * q^77 + 3 * q^79 + 2 * q^81 + q^83 - 16 * q^85 - 3 * q^87 + 7 * q^89 + 2 * q^91 + q^93 + 13 * q^95 - 11 * q^97 + 4 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more