LMFDB - Embedded newform 7440.2.a.bd.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [7440,2,Mod(1,7440)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("7440.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(7440, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 7440 = 2^{4} \cdot 3 \cdot 5 \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7440.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,-2,0,-2,0,1,0,2,0,-5,0,12,0,2,0,-8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$59.4086991038$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{65}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 16 $ x^2 - x - 16
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 930)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-3.53113$ of defining polynomial
Character	$\chi$	$=$	7440.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} -1.00000 q^{5} -3.53113 q^{7} +1.00000 q^{9} +1.53113 q^{11} +6.00000 q^{13} +1.00000 q^{15} -4.00000 q^{17} +3.53113 q^{19} +3.53113 q^{21} +1.53113 q^{23} +1.00000 q^{25} -1.00000 q^{27} -1.00000 q^{31} -1.53113 q^{33} +3.53113 q^{35} +9.06226 q^{37} -6.00000 q^{39} -9.06226 q^{41} +0.468871 q^{43} -1.00000 q^{45} -11.0623 q^{47} +5.46887 q^{49} +4.00000 q^{51} +5.53113 q^{53} -1.53113 q^{55} -3.53113 q^{57} -7.06226 q^{59} -11.0623 q^{61} -3.53113 q^{63} -6.00000 q^{65} +11.0623 q^{67} -1.53113 q^{69} +4.46887 q^{71} -0.468871 q^{73} -1.00000 q^{75} -5.40661 q^{77} +0.468871 q^{79} +1.00000 q^{81} +8.00000 q^{83} +4.00000 q^{85} +1.53113 q^{89} -21.1868 q^{91} +1.00000 q^{93} -3.53113 q^{95} -16.1245 q^{97} +1.53113 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} - 2 q^{5} + q^{7} + 2 q^{9} - 5 q^{11} + 12 q^{13} + 2 q^{15} - 8 q^{17} - q^{19} - q^{21} - 5 q^{23} + 2 q^{25} - 2 q^{27} - 2 q^{31} + 5 q^{33} - q^{35} + 2 q^{37} - 12 q^{39} - 2 q^{41}+ \cdots - 5 q^{99}+O(q^{100}) $ 2 * q - 2 * q^3 - 2 * q^5 + q^7 + 2 * q^9 - 5 * q^11 + 12 * q^13 + 2 * q^15 - 8 * q^17 - q^19 - q^21 - 5 * q^23 + 2 * q^25 - 2 * q^27 - 2 * q^31 + 5 * q^33 - q^35 + 2 * q^37 - 12 * q^39 - 2 * q^41 + 9 * q^43 - 2 * q^45 - 6 * q^47 + 19 * q^49 + 8 * q^51 + 3 * q^53 + 5 * q^55 + q^57 + 2 * q^59 - 6 * q^61 + q^63 - 12 * q^65 + 6 * q^67 + 5 * q^69 + 17 * q^71 - 9 * q^73 - 2 * q^75 - 35 * q^77 + 9 * q^79 + 2 * q^81 + 16 * q^83 + 8 * q^85 - 5 * q^89 + 6 * q^91 + 2 * q^93 + q^95 - 5 * 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
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−3.53113	−1.33464	−0.667321	−	0.744771i	$-0.732559\pi$
$7$	−3.53113	−1.33464	−0.667321	+	0.744771i	$0.732559\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	1.53113	0.461653	0.230826	−	0.972995i	$-0.425857\pi$
$11$	1.53113	0.461653	0.230826	+	0.972995i	$0.425857\pi$
$12$	0	0
$13$	6.00000	1.66410	0.832050	−	0.554700i	$-0.187167\pi$
$13$	6.00000	1.66410	0.832050	+	0.554700i	$0.187167\pi$
$14$	0	0
$15$	1.00000	0.258199
$16$	0	0
$17$	−4.00000	−0.970143	−0.485071	−	0.874475i	$-0.661206\pi$
$17$	−4.00000	−0.970143	−0.485071	+	0.874475i	$0.661206\pi$
$18$	0	0
$19$	3.53113	0.810097	0.405048	−	0.914295i	$-0.367255\pi$
$19$	3.53113	0.810097	0.405048	+	0.914295i	$0.367255\pi$
$20$	0	0
$21$	3.53113	0.770555
$22$	0	0
$23$	1.53113	0.319262	0.159631	−	0.987177i	$-0.448969\pi$
$23$	1.53113	0.319262	0.159631	+	0.987177i	$0.448969\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	0	0		−	1.00000i	$-0.5\pi$
$29$	0	0			1.00000i	$0.5\pi$
$30$	0	0
$31$	−1.00000	−0.179605
$31$	−1.00000	−0.179605
$32$	0	0
$33$	−1.53113	−0.266535
$34$	0	0
$35$	3.53113	0.596870
$36$	0	0
$37$	9.06226	1.48983	0.744913	−	0.667162i	$-0.232491\pi$
$37$	9.06226	1.48983	0.744913	+	0.667162i	$0.232491\pi$
$38$	0	0
$39$	−6.00000	−0.960769
$40$	0	0
$41$	−9.06226	−1.41529	−0.707643	−	0.706570i	$-0.750242\pi$
$41$	−9.06226	−1.41529	−0.707643	+	0.706570i	$0.750242\pi$
$42$	0	0
$43$	0.468871	0.0715022	0.0357511	−	0.999361i	$-0.488618\pi$
$43$	0.468871	0.0715022	0.0357511	+	0.999361i	$0.488618\pi$
$44$	0	0
$45$	−1.00000	−0.149071
$46$	0	0
$47$	−11.0623	−1.61360	−0.806798	−	0.590827i	$-0.798801\pi$
$47$	−11.0623	−1.61360	−0.806798	+	0.590827i	$0.798801\pi$
$48$	0	0
$49$	5.46887	0.781267
$50$	0	0
$51$	4.00000	0.560112
$52$	0	0
$53$	5.53113	0.759759	0.379879	−	0.925036i	$-0.375965\pi$
$53$	5.53113	0.759759	0.379879	+	0.925036i	$0.375965\pi$
$54$	0	0
$55$	−1.53113	−0.206457
$56$	0	0
$57$	−3.53113	−0.467709
$58$	0	0
$59$	−7.06226	−0.919428	−0.459714	−	0.888067i	$-0.652048\pi$
$59$	−7.06226	−0.919428	−0.459714	+	0.888067i	$0.652048\pi$
$60$	0	0
$61$	−11.0623	−1.41638	−0.708188	−	0.706023i	$-0.750487\pi$
$61$	−11.0623	−1.41638	−0.708188	+	0.706023i	$0.750487\pi$
$62$	0	0
$63$	−3.53113	−0.444880
$64$	0	0
$65$	−6.00000	−0.744208
$66$	0	0
$67$	11.0623	1.35147	0.675735	−	0.737145i	$-0.263826\pi$
$67$	11.0623	1.35147	0.675735	+	0.737145i	$0.263826\pi$
$68$	0	0
$69$	−1.53113	−0.184326
$70$	0	0
$71$	4.46887	0.530357	0.265179	−	0.964199i	$-0.414569\pi$
$71$	4.46887	0.530357	0.265179	+	0.964199i	$0.414569\pi$
$72$	0	0
$73$	−0.468871	−0.0548772	−0.0274386	−	0.999623i	$-0.508735\pi$
$73$	−0.468871	−0.0548772	−0.0274386	+	0.999623i	$0.508735\pi$
$74$	0	0
$75$	−1.00000	−0.115470
$76$	0	0
$77$	−5.40661	−0.616141
$78$	0	0
$79$	0.468871	0.0527521	0.0263761	−	0.999652i	$-0.491603\pi$
$79$	0.468871	0.0527521	0.0263761	+	0.999652i	$0.491603\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	8.00000	0.878114	0.439057	−	0.898459i	$-0.355313\pi$
$83$	8.00000	0.878114	0.439057	+	0.898459i	$0.355313\pi$
$84$	0	0
$85$	4.00000	0.433861
$86$	0	0
$87$	0	0
$88$	0	0
$89$	1.53113	0.162299	0.0811497	−	0.996702i	$-0.474141\pi$
$89$	1.53113	0.162299	0.0811497	+	0.996702i	$0.474141\pi$
$90$	0	0
$91$	−21.1868	−2.22098
$92$	0	0
$93$	1.00000	0.103695
$94$	0	0
$95$	−3.53113	−0.362286
$96$	0	0
$97$	−16.1245	−1.63720	−0.818598	−	0.574367i	$-0.805248\pi$
$97$	−16.1245	−1.63720	−0.818598	+	0.574367i	$0.805248\pi$
$98$	0	0
$99$	1.53113	0.153884
$100$	0	0
$101$	−17.5311	−1.74441	−0.872206	−	0.489138i	$-0.837311\pi$
$101$	−17.5311	−1.74441	−0.872206	+	0.489138i	$0.837311\pi$
$102$	0	0
$103$	0	0		−	1.00000i	$-0.5\pi$
$103$	0	0			1.00000i	$0.5\pi$
$104$	0	0
$105$	−3.53113	−0.344603
$106$	0	0
$107$	14.5934	1.41080	0.705398	−	0.708811i	$-0.250768\pi$
$107$	14.5934	1.41080	0.705398	+	0.708811i	$0.250768\pi$
$108$	0	0
$109$	−1.06226	−0.101746	−0.0508729	−	0.998705i	$-0.516200\pi$
$109$	−1.06226	−0.101746	−0.0508729	+	0.998705i	$0.516200\pi$
$110$	0	0
$111$	−9.06226	−0.860151
$112$	0	0
$113$	16.5934	1.56097	0.780487	−	0.625172i	$-0.214971\pi$
$113$	16.5934	1.56097	0.780487	+	0.625172i	$0.214971\pi$
$114$	0	0
$115$	−1.53113	−0.142779
$116$	0	0
$117$	6.00000	0.554700
$118$	0	0
$119$	14.1245	1.29479
$120$	0	0
$121$	−8.65564	−0.786877
$122$	0	0
$123$	9.06226	0.817116
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	9.06226	0.804145	0.402073	−	0.915608i	$-0.368290\pi$
$127$	9.06226	0.804145	0.402073	+	0.915608i	$0.368290\pi$
$128$	0	0
$129$	−0.468871	−0.0412818
$130$	0	0
$131$	−7.06226	−0.617032	−0.308516	−	0.951219i	$-0.599832\pi$
$131$	−7.06226	−0.617032	−0.308516	+	0.951219i	$0.599832\pi$
$132$	0	0
$133$	−12.4689	−1.08119
$134$	0	0
$135$	1.00000	0.0860663
$136$	0	0
$137$	0.937742	0.0801167	0.0400584	−	0.999197i	$-0.487246\pi$
$137$	0.937742	0.0801167	0.0400584	+	0.999197i	$0.487246\pi$
$138$	0	0
$139$	−6.00000	−0.508913	−0.254457	−	0.967084i	$-0.581897\pi$
$139$	−6.00000	−0.508913	−0.254457	+	0.967084i	$0.581897\pi$
$140$	0	0
$141$	11.0623	0.931610
$142$	0	0
$143$	9.18677	0.768237
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−5.46887	−0.451065
$148$	0	0
$149$	−10.4689	−0.857643	−0.428822	−	0.903389i	$-0.641071\pi$
$149$	−10.4689	−0.857643	−0.428822	+	0.903389i	$0.641071\pi$
$150$	0	0
$151$	18.1245	1.47495	0.737476	−	0.675373i	$-0.236017\pi$
$151$	18.1245	1.47495	0.737476	+	0.675373i	$0.236017\pi$
$152$	0	0
$153$	−4.00000	−0.323381
$154$	0	0
$155$	1.00000	0.0803219
$156$	0	0
$157$	−14.4689	−1.15474	−0.577371	−	0.816482i	$-0.695921\pi$
$157$	−14.4689	−1.15474	−0.577371	+	0.816482i	$0.695921\pi$
$158$	0	0
$159$	−5.53113	−0.438647
$160$	0	0
$161$	−5.40661	−0.426101
$162$	0	0
$163$	11.0623	0.866463	0.433231	−	0.901283i	$-0.357373\pi$
$163$	11.0623	0.866463	0.433231	+	0.901283i	$0.357373\pi$
$164$	0	0
$165$	1.53113	0.119198
$166$	0	0
$167$	−0.593387	−0.0459176	−0.0229588	−	0.999736i	$-0.507309\pi$
$167$	−0.593387	−0.0459176	−0.0229588	+	0.999736i	$0.507309\pi$
$168$	0	0
$169$	23.0000	1.76923
$170$	0	0
$171$	3.53113	0.270032
$172$	0	0
$173$	14.0000	1.06440	0.532200	−	0.846619i	$-0.321365\pi$
$173$	14.0000	1.06440	0.532200	+	0.846619i	$0.321365\pi$
$174$	0	0
$175$	−3.53113	−0.266928
$176$	0	0
$177$	7.06226	0.530832
$178$	0	0
$179$	−13.0623	−0.976319	−0.488159	−	0.872754i	$-0.662332\pi$
$179$	−13.0623	−0.976319	−0.488159	+	0.872754i	$0.662332\pi$
$180$	0	0
$181$	26.5934	1.97667	0.988335	−	0.152293i	$-0.0486657\pi$
$181$	26.5934	1.97667	0.988335	+	0.152293i	$0.0486657\pi$
$182$	0	0
$183$	11.0623	0.817746
$184$	0	0
$185$	−9.06226	−0.666270
$186$	0	0
$187$	−6.12452	−0.447869
$188$	0	0
$189$	3.53113	0.256852
$190$	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
$193$	14.0000	1.00774	0.503871	−	0.863779i	$-0.331909\pi$
$193$	14.0000	1.00774	0.503871	+	0.863779i	$0.331909\pi$
$194$	0	0
$195$	6.00000	0.429669
$196$	0	0
$197$	6.00000	0.427482	0.213741	−	0.976890i	$-0.431435\pi$
$197$	6.00000	0.427482	0.213741	+	0.976890i	$0.431435\pi$
$198$	0	0
$199$	0.468871	0.0332374	0.0166187	−	0.999862i	$-0.494710\pi$
$199$	0.468871	0.0332374	0.0166187	+	0.999862i	$0.494710\pi$
$200$	0	0
$201$	−11.0623	−0.780272
$202$	0	0
$203$	0	0
$204$	0	0
$205$	9.06226	0.632936
$206$	0	0
$207$	1.53113	0.106421
$208$	0	0
$209$	5.40661	0.373983
$210$	0	0
$211$	−22.5934	−1.55539	−0.777696	−	0.628640i	$-0.783612\pi$
$211$	−22.5934	−1.55539	−0.777696	+	0.628640i	$0.783612\pi$
$212$	0	0
$213$	−4.46887	−0.306202
$214$	0	0
$215$	−0.468871	−0.0319767
$216$	0	0
$217$	3.53113	0.239709
$218$	0	0
$219$	0.468871	0.0316834
$220$	0	0
$221$	−24.0000	−1.61441
$222$	0	0
$223$	−2.00000	−0.133930	−0.0669650	−	0.997755i	$-0.521332\pi$
$223$	−2.00000	−0.133930	−0.0669650	+	0.997755i	$0.521332\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	16.4689	1.09308	0.546539	−	0.837434i	$-0.315945\pi$
$227$	16.4689	1.09308	0.546539	+	0.837434i	$0.315945\pi$
$228$	0	0
$229$	18.5934	1.22869	0.614343	−	0.789039i	$-0.289421\pi$
$229$	18.5934	1.22869	0.614343	+	0.789039i	$0.289421\pi$
$230$	0	0
$231$	5.40661	0.355729
$232$	0	0
$233$	9.53113	0.624405	0.312203	−	0.950016i	$-0.398933\pi$
$233$	9.53113	0.624405	0.312203	+	0.950016i	$0.398933\pi$
$234$	0	0
$235$	11.0623	0.721622
$236$	0	0
$237$	−0.468871	−0.0304565
$238$	0	0
$239$	8.00000	0.517477	0.258738	−	0.965947i	$-0.416693\pi$
$239$	8.00000	0.517477	0.258738	+	0.965947i	$0.416693\pi$
$240$	0	0
$241$	2.00000	0.128831	0.0644157	−	0.997923i	$-0.479482\pi$
$241$	2.00000	0.128831	0.0644157	+	0.997923i	$0.479482\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	−5.46887	−0.349393
$246$	0	0
$247$	21.1868	1.34808
$248$	0	0
$249$	−8.00000	−0.506979
$250$	0	0
$251$	13.0623	0.824482	0.412241	−	0.911075i	$-0.364746\pi$
$251$	13.0623	0.824482	0.412241	+	0.911075i	$0.364746\pi$
$252$	0	0
$253$	2.34436	0.147388
$254$	0	0
$255$	−4.00000	−0.250490
$256$	0	0
$257$	−27.6556	−1.72511	−0.862556	−	0.505962i	$-0.831138\pi$
$257$	−27.6556	−1.72511	−0.862556	+	0.505962i	$0.831138\pi$
$258$	0	0
$259$	−32.0000	−1.98838
$260$	0	0
$261$	0	0
$262$	0	0
$263$	6.93774	0.427800	0.213900	−	0.976856i	$-0.431383\pi$
$263$	6.93774	0.427800	0.213900	+	0.976856i	$0.431383\pi$
$264$	0	0
$265$	−5.53113	−0.339775
$266$	0	0
$267$	−1.53113	−0.0937036
$268$	0	0
$269$	29.1868	1.77955	0.889774	−	0.456400i	$-0.150862\pi$
$269$	29.1868	1.77955	0.889774	+	0.456400i	$0.150862\pi$
$270$	0	0
$271$	19.5311	1.18643	0.593216	−	0.805043i	$-0.297858\pi$
$271$	19.5311	1.18643	0.593216	+	0.805043i	$0.297858\pi$
$272$	0	0
$273$	21.1868	1.28228
$274$	0	0
$275$	1.53113	0.0923305
$276$	0	0
$277$	1.06226	0.0638249	0.0319124	−	0.999491i	$-0.489840\pi$
$277$	1.06226	0.0638249	0.0319124	+	0.999491i	$0.489840\pi$
$278$	0	0
$279$	−1.00000	−0.0598684
$280$	0	0
$281$	−1.06226	−0.0633690	−0.0316845	−	0.999498i	$-0.510087\pi$
$281$	−1.06226	−0.0633690	−0.0316845	+	0.999498i	$0.510087\pi$
$282$	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
$285$	3.53113	0.209166
$286$	0	0
$287$	32.0000	1.88890
$288$	0	0
$289$	−1.00000	−0.0588235
$290$	0	0
$291$	16.1245	0.945236
$292$	0	0
$293$	−14.0000	−0.817889	−0.408944	−	0.912559i	$-0.634103\pi$
$293$	−14.0000	−0.817889	−0.408944	+	0.912559i	$0.634103\pi$
$294$	0	0
$295$	7.06226	0.411181
$296$	0	0
$297$	−1.53113	−0.0888451
$298$	0	0
$299$	9.18677	0.531285
$300$	0	0
$301$	−1.65564	−0.0954298
$302$	0	0
$303$	17.5311	1.00714
$304$	0	0
$305$	11.0623	0.633423
$306$	0	0
$307$	−2.12452	−0.121253	−0.0606263	−	0.998161i	$-0.519310\pi$
$307$	−2.12452	−0.121253	−0.0606263	+	0.998161i	$0.519310\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	8.00000	0.453638	0.226819	−	0.973937i	$-0.427167\pi$
$311$	8.00000	0.453638	0.226819	+	0.973937i	$0.427167\pi$
$312$	0	0
$313$	27.0623	1.52965	0.764825	−	0.644239i	$-0.222826\pi$
$313$	27.0623	1.52965	0.764825	+	0.644239i	$0.222826\pi$
$314$	0	0
$315$	3.53113	0.198957
$316$	0	0
$317$	29.0623	1.63230	0.816150	−	0.577841i	$-0.196105\pi$
$317$	29.0623	1.63230	0.816150	+	0.577841i	$0.196105\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	−14.5934	−0.814523
$322$	0	0
$323$	−14.1245	−0.785909
$324$	0	0
$325$	6.00000	0.332820
$326$	0	0
$327$	1.06226	0.0587430
$328$	0	0
$329$	39.0623	2.15357
$330$	0	0
$331$	−29.0623	−1.59741	−0.798703	−	0.601725i	$-0.794480\pi$
$331$	−29.0623	−1.59741	−0.798703	+	0.601725i	$0.794480\pi$
$332$	0	0
$333$	9.06226	0.496609
$334$	0	0
$335$	−11.0623	−0.604396
$336$	0	0
$337$	29.1868	1.58990	0.794952	−	0.606672i	$-0.207496\pi$
$337$	29.1868	1.58990	0.794952	+	0.606672i	$0.207496\pi$
$338$	0	0
$339$	−16.5934	−0.901229
$340$	0	0
$341$	−1.53113	−0.0829153
$342$	0	0
$343$	5.40661	0.291930
$344$	0	0
$345$	1.53113	0.0824332
$346$	0	0
$347$	22.1245	1.18771	0.593853	−	0.804573i	$-0.297606\pi$
$347$	22.1245	1.18771	0.593853	+	0.804573i	$0.297606\pi$
$348$	0	0
$349$	25.0623	1.34155	0.670776	−	0.741660i	$-0.265961\pi$
$349$	25.0623	1.34155	0.670776	+	0.741660i	$0.265961\pi$
$350$	0	0
$351$	−6.00000	−0.320256
$352$	0	0
$353$	−23.0623	−1.22748	−0.613740	−	0.789508i	$-0.710336\pi$
$353$	−23.0623	−1.22748	−0.613740	+	0.789508i	$0.710336\pi$
$354$	0	0
$355$	−4.46887	−0.237183
$356$	0	0
$357$	−14.1245	−0.747549
$358$	0	0
$359$	−11.5311	−0.608590	−0.304295	−	0.952578i	$-0.598421\pi$
$359$	−11.5311	−0.608590	−0.304295	+	0.952578i	$0.598421\pi$
$360$	0	0
$361$	−6.53113	−0.343744
$362$	0	0
$363$	8.65564	0.454304
$364$	0	0
$365$	0.468871	0.0245418
$366$	0	0
$367$	10.0000	0.521996	0.260998	−	0.965339i	$-0.415948\pi$
$367$	10.0000	0.521996	0.260998	+	0.965339i	$0.415948\pi$
$368$	0	0
$369$	−9.06226	−0.471762
$370$	0	0
$371$	−19.5311	−1.01401
$372$	0	0
$373$	−10.4689	−0.542058	−0.271029	−	0.962571i	$-0.587364\pi$
$373$	−10.4689	−0.542058	−0.271029	+	0.962571i	$0.587364\pi$
$374$	0	0
$375$	1.00000	0.0516398
$376$	0	0
$377$	0	0
$378$	0	0
$379$	2.59339	0.133213	0.0666067	−	0.997779i	$-0.478783\pi$
$379$	2.59339	0.133213	0.0666067	+	0.997779i	$0.478783\pi$
$380$	0	0
$381$	−9.06226	−0.464274
$382$	0	0
$383$	−13.0623	−0.667450	−0.333725	−	0.942670i	$-0.608306\pi$
$383$	−13.0623	−0.667450	−0.333725	+	0.942670i	$0.608306\pi$
$384$	0	0
$385$	5.40661	0.275547
$386$	0	0
$387$	0.468871	0.0238341
$388$	0	0
$389$	27.0623	1.37211	0.686055	−	0.727549i	$-0.259341\pi$
$389$	27.0623	1.37211	0.686055	+	0.727549i	$0.259341\pi$
$390$	0	0
$391$	−6.12452	−0.309730
$392$	0	0
$393$	7.06226	0.356244
$394$	0	0
$395$	−0.468871	−0.0235915
$396$	0	0
$397$	−18.7179	−0.939425	−0.469712	−	0.882820i	$-0.655642\pi$
$397$	−18.7179	−0.939425	−0.469712	+	0.882820i	$0.655642\pi$
$398$	0	0
$399$	12.4689	0.624224
$400$	0	0
$401$	−34.7179	−1.73373	−0.866865	−	0.498544i	$-0.833868\pi$
$401$	−34.7179	−1.73373	−0.866865	+	0.498544i	$0.833868\pi$
$402$	0	0
$403$	−6.00000	−0.298881
$404$	0	0
$405$	−1.00000	−0.0496904
$406$	0	0
$407$	13.8755	0.687782
$408$	0	0
$409$	−9.06226	−0.448100	−0.224050	−	0.974578i	$-0.571928\pi$
$409$	−9.06226	−0.448100	−0.224050	+	0.974578i	$0.571928\pi$
$410$	0	0
$411$	−0.937742	−0.0462554
$412$	0	0
$413$	24.9377	1.22711
$414$	0	0
$415$	−8.00000	−0.392705
$416$	0	0
$417$	6.00000	0.293821
$418$	0	0
$419$	0	0		−	1.00000i	$-0.5\pi$
$419$	0	0			1.00000i	$0.5\pi$
$420$	0	0
$421$	38.2490	1.86414	0.932072	−	0.362273i	$-0.117999\pi$
$421$	38.2490	1.86414	0.932072	+	0.362273i	$0.117999\pi$
$422$	0	0
$423$	−11.0623	−0.537865
$424$	0	0
$425$	−4.00000	−0.194029
$426$	0	0
$427$	39.0623	1.89036
$428$	0	0
$429$	−9.18677	−0.443542
$430$	0	0
$431$	24.0000	1.15604	0.578020	−	0.816023i	$-0.303826\pi$
$431$	24.0000	1.15604	0.578020	+	0.816023i	$0.303826\pi$
$432$	0	0
$433$	−1.40661	−0.0675975	−0.0337988	−	0.999429i	$-0.510761\pi$
$433$	−1.40661	−0.0675975	−0.0337988	+	0.999429i	$0.510761\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	5.40661	0.258633
$438$	0	0
$439$	8.93774	0.426575	0.213288	−	0.976989i	$-0.431583\pi$
$439$	8.93774	0.426575	0.213288	+	0.976989i	$0.431583\pi$
$440$	0	0
$441$	5.46887	0.260422
$442$	0	0
$443$	−38.5934	−1.83363	−0.916814	−	0.399316i	$-0.869248\pi$
$443$	−38.5934	−1.83363	−0.916814	+	0.399316i	$0.869248\pi$
$444$	0	0
$445$	−1.53113	−0.0725825
$446$	0	0
$447$	10.4689	0.495161
$448$	0	0
$449$	28.1245	1.32728	0.663639	−	0.748053i	$-0.269011\pi$
$449$	28.1245	1.32728	0.663639	+	0.748053i	$0.269011\pi$
$450$	0	0
$451$	−13.8755	−0.653371
$452$	0	0
$453$	−18.1245	−0.851564
$454$	0	0
$455$	21.1868	0.993251
$456$	0	0
$457$	31.0623	1.45303	0.726516	−	0.687150i	$-0.241138\pi$
$457$	31.0623	1.45303	0.726516	+	0.687150i	$0.241138\pi$
$458$	0	0
$459$	4.00000	0.186704
$460$	0	0
$461$	24.0000	1.11779	0.558896	−	0.829238i	$-0.311225\pi$
$461$	24.0000	1.11779	0.558896	+	0.829238i	$0.311225\pi$
$462$	0	0
$463$	35.1868	1.63527	0.817634	−	0.575738i	$-0.195285\pi$
$463$	35.1868	1.63527	0.817634	+	0.575738i	$0.195285\pi$
$464$	0	0
$465$	−1.00000	−0.0463739
$466$	0	0
$467$	−10.1245	−0.468507	−0.234253	−	0.972176i	$-0.575265\pi$
$467$	−10.1245	−0.468507	−0.234253	+	0.972176i	$0.575265\pi$
$468$	0	0
$469$	−39.0623	−1.80373
$470$	0	0
$471$	14.4689	0.666690
$472$	0	0
$473$	0.717902	0.0330092
$474$	0	0
$475$	3.53113	0.162019
$476$	0	0
$477$	5.53113	0.253253
$478$	0	0
$479$	−41.6556	−1.90329	−0.951647	−	0.307192i	$-0.900611\pi$
$479$	−41.6556	−1.90329	−0.951647	+	0.307192i	$0.900611\pi$
$480$	0	0
$481$	54.3735	2.47922
$482$	0	0
$483$	5.40661	0.246009
$484$	0	0
$485$	16.1245	0.732177
$486$	0	0
$487$	−6.93774	−0.314379	−0.157190	−	0.987568i	$-0.550243\pi$
$487$	−6.93774	−0.314379	−0.157190	+	0.987568i	$0.550243\pi$
$488$	0	0
$489$	−11.0623	−0.500253
$490$	0	0
$491$	−16.5934	−0.748849	−0.374425	−	0.927257i	$-0.622160\pi$
$491$	−16.5934	−0.748849	−0.374425	+	0.927257i	$0.622160\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	−1.53113	−0.0688191
$496$	0	0
$497$	−15.7802	−0.707837
$498$	0	0
$499$	−9.06226	−0.405682	−0.202841	−	0.979212i	$-0.565018\pi$
$499$	−9.06226	−0.405682	−0.202841	+	0.979212i	$0.565018\pi$
$500$	0	0
$501$	0.593387	0.0265106
$502$	0	0
$503$	−16.0000	−0.713405	−0.356702	−	0.934218i	$-0.616099\pi$
$503$	−16.0000	−0.713405	−0.356702	+	0.934218i	$0.616099\pi$
$504$	0	0
$505$	17.5311	0.780125
$506$	0	0
$507$	−23.0000	−1.02147
$508$	0	0
$509$	5.87548	0.260426	0.130213	−	0.991486i	$-0.458434\pi$
$509$	5.87548	0.260426	0.130213	+	0.991486i	$0.458434\pi$
$510$	0	0
$511$	1.65564	0.0732414
$512$	0	0
$513$	−3.53113	−0.155903
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−16.9377	−0.744921
$518$	0	0
$519$	−14.0000	−0.614532
$520$	0	0
$521$	20.1245	0.881671	0.440836	−	0.897588i	$-0.354682\pi$
$521$	20.1245	0.881671	0.440836	+	0.897588i	$0.354682\pi$
$522$	0	0
$523$	14.5934	0.638124	0.319062	−	0.947734i	$-0.396632\pi$
$523$	14.5934	0.638124	0.319062	+	0.947734i	$0.396632\pi$
$524$	0	0
$525$	3.53113	0.154111
$526$	0	0
$527$	4.00000	0.174243
$528$	0	0
$529$	−20.6556	−0.898071
$530$	0	0
$531$	−7.06226	−0.306476
$532$	0	0
$533$	−54.3735	−2.35518
$534$	0	0
$535$	−14.5934	−0.630927
$536$	0	0
$537$	13.0623	0.563678
$538$	0	0
$539$	8.37355	0.360674
$540$	0	0
$541$	28.1245	1.20917	0.604584	−	0.796542i	$-0.293340\pi$
$541$	28.1245	1.20917	0.604584	+	0.796542i	$0.293340\pi$
$542$	0	0
$543$	−26.5934	−1.14123
$544$	0	0
$545$	1.06226	0.0455021
$546$	0	0
$547$	−41.1868	−1.76102	−0.880510	−	0.474028i	$-0.842799\pi$
$547$	−41.1868	−1.76102	−0.880510	+	0.474028i	$0.842799\pi$
$548$	0	0
$549$	−11.0623	−0.472126
$550$	0	0
$551$	0	0
$552$	0	0
$553$	−1.65564	−0.0704052
$554$	0	0
$555$	9.06226	0.384671
$556$	0	0
$557$	45.7802	1.93977	0.969884	−	0.243568i	$-0.0783179\pi$
$557$	45.7802	1.93977	0.969884	+	0.243568i	$0.0783179\pi$
$558$	0	0
$559$	2.81323	0.118987
$560$	0	0
$561$	6.12452	0.258577
$562$	0	0
$563$	4.00000	0.168580	0.0842900	−	0.996441i	$-0.473138\pi$
$563$	4.00000	0.168580	0.0842900	+	0.996441i	$0.473138\pi$
$564$	0	0
$565$	−16.5934	−0.698089
$566$	0	0
$567$	−3.53113	−0.148293
$568$	0	0
$569$	17.5311	0.734943	0.367472	−	0.930035i	$-0.380224\pi$
$569$	17.5311	0.734943	0.367472	+	0.930035i	$0.380224\pi$
$570$	0	0
$571$	−3.87548	−0.162184	−0.0810920	−	0.996707i	$-0.525841\pi$
$571$	−3.87548	−0.162184	−0.0810920	+	0.996707i	$0.525841\pi$
$572$	0	0
$573$	−8.00000	−0.334205
$574$	0	0
$575$	1.53113	0.0638525
$576$	0	0
$577$	−11.1868	−0.465711	−0.232856	−	0.972511i	$-0.574807\pi$
$577$	−11.1868	−0.465711	−0.232856	+	0.972511i	$0.574807\pi$
$578$	0	0
$579$	−14.0000	−0.581820
$580$	0	0
$581$	−28.2490	−1.17197
$582$	0	0
$583$	8.46887	0.350745
$584$	0	0
$585$	−6.00000	−0.248069
$586$	0	0
$587$	−20.2490	−0.835767	−0.417883	−	0.908501i	$-0.637228\pi$
$587$	−20.2490	−0.835767	−0.417883	+	0.908501i	$0.637228\pi$
$588$	0	0
$589$	−3.53113	−0.145498
$590$	0	0
$591$	−6.00000	−0.246807
$592$	0	0
$593$	−14.0000	−0.574911	−0.287456	−	0.957794i	$-0.592809\pi$
$593$	−14.0000	−0.574911	−0.287456	+	0.957794i	$0.592809\pi$
$594$	0	0
$595$	−14.1245	−0.579049
$596$	0	0
$597$	−0.468871	−0.0191896
$598$	0	0
$599$	−10.5934	−0.432834	−0.216417	−	0.976301i	$-0.569437\pi$
$599$	−10.5934	−0.432834	−0.216417	+	0.976301i	$0.569437\pi$
$600$	0	0
$601$	30.0000	1.22373	0.611863	−	0.790964i	$-0.290420\pi$
$601$	30.0000	1.22373	0.611863	+	0.790964i	$0.290420\pi$
$602$	0	0
$603$	11.0623	0.450490
$604$	0	0
$605$	8.65564	0.351902
$606$	0	0
$607$	38.5934	1.56646	0.783229	−	0.621734i	$-0.213571\pi$
$607$	38.5934	1.56646	0.783229	+	0.621734i	$0.213571\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−66.3735	−2.68519
$612$	0	0
$613$	−30.0000	−1.21169	−0.605844	−	0.795583i	$-0.707165\pi$
$613$	−30.0000	−1.21169	−0.605844	+	0.795583i	$0.707165\pi$
$614$	0	0
$615$	−9.06226	−0.365426
$616$	0	0
$617$	20.5934	0.829059	0.414529	−	0.910036i	$-0.363946\pi$
$617$	20.5934	0.829059	0.414529	+	0.910036i	$0.363946\pi$
$618$	0	0
$619$	34.0000	1.36658	0.683288	−	0.730149i	$-0.260549\pi$
$619$	34.0000	1.36658	0.683288	+	0.730149i	$0.260549\pi$
$620$	0	0
$621$	−1.53113	−0.0614421
$622$	0	0
$623$	−5.40661	−0.216611
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−5.40661	−0.215919
$628$	0	0
$629$	−36.2490	−1.44534
$630$	0	0
$631$	9.65564	0.384385	0.192193	−	0.981357i	$-0.438440\pi$
$631$	9.65564	0.384385	0.192193	+	0.981357i	$0.438440\pi$
$632$	0	0
$633$	22.5934	0.898006
$634$	0	0
$635$	−9.06226	−0.359625
$636$	0	0
$637$	32.8132	1.30011
$638$	0	0
$639$	4.46887	0.176786
$640$	0	0
$641$	−22.0000	−0.868948	−0.434474	−	0.900684i	$-0.643066\pi$
$641$	−22.0000	−0.868948	−0.434474	+	0.900684i	$0.643066\pi$
$642$	0	0
$643$	−6.59339	−0.260018	−0.130009	−	0.991513i	$-0.541501\pi$
$643$	−6.59339	−0.260018	−0.130009	+	0.991513i	$0.541501\pi$
$644$	0	0
$645$	0.468871	0.0184618
$646$	0	0
$647$	−1.53113	−0.0601949	−0.0300974	−	0.999547i	$-0.509582\pi$
$647$	−1.53113	−0.0601949	−0.0300974	+	0.999547i	$0.509582\pi$
$648$	0	0
$649$	−10.8132	−0.424456
$650$	0	0
$651$	−3.53113	−0.138396
$652$	0	0
$653$	−26.2490	−1.02720	−0.513602	−	0.858029i	$-0.671689\pi$
$653$	−26.2490	−1.02720	−0.513602	+	0.858029i	$0.671689\pi$
$654$	0	0
$655$	7.06226	0.275945
$656$	0	0
$657$	−0.468871	−0.0182924
$658$	0	0
$659$	−16.0000	−0.623272	−0.311636	−	0.950202i	$-0.600877\pi$
$659$	−16.0000	−0.623272	−0.311636	+	0.950202i	$0.600877\pi$
$660$	0	0
$661$	−10.0000	−0.388955	−0.194477	−	0.980907i	$-0.562301\pi$
$661$	−10.0000	−0.388955	−0.194477	+	0.980907i	$0.562301\pi$
$662$	0	0
$663$	24.0000	0.932083
$664$	0	0
$665$	12.4689	0.483522
$666$	0	0
$667$	0	0
$668$	0	0
$669$	2.00000	0.0773245
$670$	0	0
$671$	−16.9377	−0.653874
$672$	0	0
$673$	7.06226	0.272230	0.136115	−	0.990693i	$-0.456538\pi$
$673$	7.06226	0.272230	0.136115	+	0.990693i	$0.456538\pi$
$674$	0	0
$675$	−1.00000	−0.0384900
$676$	0	0
$677$	3.40661	0.130927	0.0654634	−	0.997855i	$-0.479147\pi$
$677$	3.40661	0.130927	0.0654634	+	0.997855i	$0.479147\pi$
$678$	0	0
$679$	56.9377	2.18507
$680$	0	0
$681$	−16.4689	−0.631089
$682$	0	0
$683$	−15.5311	−0.594282	−0.297141	−	0.954834i	$-0.596033\pi$
$683$	−15.5311	−0.594282	−0.297141	+	0.954834i	$0.596033\pi$
$684$	0	0
$685$	−0.937742	−0.0358293
$686$	0	0
$687$	−18.5934	−0.709382
$688$	0	0
$689$	33.1868	1.26432
$690$	0	0
$691$	7.53113	0.286498	0.143249	−	0.989687i	$-0.454245\pi$
$691$	7.53113	0.286498	0.143249	+	0.989687i	$0.454245\pi$
$692$	0	0
$693$	−5.40661	−0.205380
$694$	0	0
$695$	6.00000	0.227593
$696$	0	0
$697$	36.2490	1.37303
$698$	0	0
$699$	−9.53113	−0.360500
$700$	0	0
$701$	−21.5311	−0.813220	−0.406610	−	0.913602i	$-0.633289\pi$
$701$	−21.5311	−0.813220	−0.406610	+	0.913602i	$0.633289\pi$
$702$	0	0
$703$	32.0000	1.20690
$704$	0	0
$705$	−11.0623	−0.416629
$706$	0	0
$707$	61.9047	2.32816
$708$	0	0
$709$	25.4066	0.954165	0.477083	−	0.878858i	$-0.341694\pi$
$709$	25.4066	0.954165	0.477083	+	0.878858i	$0.341694\pi$
$710$	0	0
$711$	0.468871	0.0175840
$712$	0	0
$713$	−1.53113	−0.0573412
$714$	0	0
$715$	−9.18677	−0.343566
$716$	0	0
$717$	−8.00000	−0.298765
$718$	0	0
$719$	33.1868	1.23766	0.618829	−	0.785526i	$-0.287607\pi$
$719$	33.1868	1.23766	0.618829	+	0.785526i	$0.287607\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−2.00000	−0.0743808
$724$	0	0
$725$	0	0
$726$	0	0
$727$	50.8424	1.88564	0.942820	−	0.333301i	$-0.108163\pi$
$727$	50.8424	1.88564	0.942820	+	0.333301i	$0.108163\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−1.87548	−0.0693673
$732$	0	0
$733$	4.12452	0.152342	0.0761712	−	0.997095i	$-0.475730\pi$
$733$	4.12452	0.152342	0.0761712	+	0.997095i	$0.475730\pi$
$734$	0	0
$735$	5.46887	0.201722
$736$	0	0
$737$	16.9377	0.623910
$738$	0	0
$739$	−27.1868	−1.00008	−0.500041	−	0.866002i	$-0.666682\pi$
$739$	−27.1868	−1.00008	−0.500041	+	0.866002i	$0.666682\pi$
$740$	0	0
$741$	−21.1868	−0.778316
$742$	0	0
$743$	−11.6556	−0.427604	−0.213802	−	0.976877i	$-0.568585\pi$
$743$	−11.6556	−0.427604	−0.213802	+	0.976877i	$0.568585\pi$
$744$	0	0
$745$	10.4689	0.383550
$746$	0	0
$747$	8.00000	0.292705
$748$	0	0
$749$	−51.5311	−1.88291
$750$	0	0
$751$	−47.0623	−1.71733	−0.858663	−	0.512540i	$-0.828704\pi$
$751$	−47.0623	−1.71733	−0.858663	+	0.512540i	$0.828704\pi$
$752$	0	0
$753$	−13.0623	−0.476015
$754$	0	0
$755$	−18.1245	−0.659619
$756$	0	0
$757$	26.0000	0.944986	0.472493	−	0.881334i	$-0.343354\pi$
$757$	26.0000	0.944986	0.472493	+	0.881334i	$0.343354\pi$
$758$	0	0
$759$	−2.34436	−0.0850947
$760$	0	0
$761$	−12.5934	−0.456510	−0.228255	−	0.973601i	$-0.573302\pi$
$761$	−12.5934	−0.456510	−0.228255	+	0.973601i	$0.573302\pi$
$762$	0	0
$763$	3.75097	0.135794
$764$	0	0
$765$	4.00000	0.144620
$766$	0	0
$767$	−42.3735	−1.53002
$768$	0	0
$769$	−28.5934	−1.03110	−0.515552	−	0.856858i	$-0.672413\pi$
$769$	−28.5934	−1.03110	−0.515552	+	0.856858i	$0.672413\pi$
$770$	0	0
$771$	27.6556	0.995994
$772$	0	0
$773$	14.4689	0.520409	0.260205	−	0.965554i	$-0.416210\pi$
$773$	14.4689	0.520409	0.260205	+	0.965554i	$0.416210\pi$
$774$	0	0
$775$	−1.00000	−0.0359211
$776$	0	0
$777$	32.0000	1.14799
$778$	0	0
$779$	−32.0000	−1.14652
$780$	0	0
$781$	6.84242	0.244841
$782$	0	0
$783$	0	0
$784$	0	0
$785$	14.4689	0.516416
$786$	0	0
$787$	28.4689	1.01481	0.507403	−	0.861709i	$-0.330606\pi$
$787$	28.4689	1.01481	0.507403	+	0.861709i	$0.330606\pi$
$788$	0	0
$789$	−6.93774	−0.246990
$790$	0	0
$791$	−58.5934	−2.08334
$792$	0	0
$793$	−66.3735	−2.35699
$794$	0	0
$795$	5.53113	0.196169
$796$	0	0
$797$	−20.1245	−0.712847	−0.356423	−	0.934325i	$-0.616004\pi$
$797$	−20.1245	−0.712847	−0.356423	+	0.934325i	$0.616004\pi$
$798$	0	0
$799$	44.2490	1.56542
$800$	0	0
$801$	1.53113	0.0540998
$802$	0	0
$803$	−0.717902	−0.0253342
$804$	0	0
$805$	5.40661	0.190558
$806$	0	0
$807$	−29.1868	−1.02742
$808$	0	0
$809$	−8.59339	−0.302127	−0.151064	−	0.988524i	$-0.548270\pi$
$809$	−8.59339	−0.302127	−0.151064	+	0.988524i	$0.548270\pi$
$810$	0	0
$811$	39.7802	1.39687	0.698435	−	0.715673i	$-0.253880\pi$
$811$	39.7802	1.39687	0.698435	+	0.715673i	$0.253880\pi$
$812$	0	0
$813$	−19.5311	−0.684987
$814$	0	0
$815$	−11.0623	−0.387494
$816$	0	0
$817$	1.65564	0.0579237
$818$	0	0
$819$	−21.1868	−0.740326
$820$	0	0
$821$	9.87548	0.344657	0.172328	−	0.985040i	$-0.444871\pi$
$821$	9.87548	0.344657	0.172328	+	0.985040i	$0.444871\pi$
$822$	0	0
$823$	−7.87548	−0.274522	−0.137261	−	0.990535i	$-0.543830\pi$
$823$	−7.87548	−0.274522	−0.137261	+	0.990535i	$0.543830\pi$
$824$	0	0
$825$	−1.53113	−0.0533071
$826$	0	0
$827$	−8.00000	−0.278187	−0.139094	−	0.990279i	$-0.544419\pi$
$827$	−8.00000	−0.278187	−0.139094	+	0.990279i	$0.544419\pi$
$828$	0	0
$829$	−9.65564	−0.335354	−0.167677	−	0.985842i	$-0.553627\pi$
$829$	−9.65564	−0.335354	−0.167677	+	0.985842i	$0.553627\pi$
$830$	0	0
$831$	−1.06226	−0.0368493
$832$	0	0
$833$	−21.8755	−0.757941
$834$	0	0
$835$	0.593387	0.0205350
$836$	0	0
$837$	1.00000	0.0345651
$838$	0	0
$839$	54.8424	1.89337	0.946685	−	0.322160i	$-0.104409\pi$
$839$	54.8424	1.89337	0.946685	+	0.322160i	$0.104409\pi$
$840$	0	0
$841$	−29.0000	−1.00000
$842$	0	0
$843$	1.06226	0.0365861
$844$	0	0
$845$	−23.0000	−0.791224
$846$	0	0
$847$	30.5642	1.05020
$848$	0	0
$849$	−12.0000	−0.411839
$850$	0	0
$851$	13.8755	0.475645
$852$	0	0
$853$	1.28210	0.0438982	0.0219491	−	0.999759i	$-0.493013\pi$
$853$	1.28210	0.0438982	0.0219491	+	0.999759i	$0.493013\pi$
$854$	0	0
$855$	−3.53113	−0.120762
$856$	0	0
$857$	14.0000	0.478231	0.239115	−	0.970991i	$-0.423143\pi$
$857$	14.0000	0.478231	0.239115	+	0.970991i	$0.423143\pi$
$858$	0	0
$859$	6.93774	0.236713	0.118356	−	0.992971i	$-0.462237\pi$
$859$	6.93774	0.236713	0.118356	+	0.992971i	$0.462237\pi$
$860$	0	0
$861$	−32.0000	−1.09056
$862$	0	0
$863$	−41.5311	−1.41374	−0.706868	−	0.707345i	$-0.749893\pi$
$863$	−41.5311	−1.41374	−0.706868	+	0.707345i	$0.749893\pi$
$864$	0	0
$865$	−14.0000	−0.476014
$866$	0	0
$867$	1.00000	0.0339618
$868$	0	0
$869$	0.717902	0.0243532
$870$	0	0
$871$	66.3735	2.24898
$872$	0	0
$873$	−16.1245	−0.545732
$874$	0	0
$875$	3.53113	0.119374
$876$	0	0
$877$	44.1245	1.48998	0.744990	−	0.667076i	$-0.232454\pi$
$877$	44.1245	1.48998	0.744990	+	0.667076i	$0.232454\pi$
$878$	0	0
$879$	14.0000	0.472208
$880$	0	0
$881$	36.1245	1.21707	0.608533	−	0.793529i	$-0.291758\pi$
$881$	36.1245	1.21707	0.608533	+	0.793529i	$0.291758\pi$
$882$	0	0
$883$	53.6556	1.80566	0.902828	−	0.430002i	$-0.141487\pi$
$883$	53.6556	1.80566	0.902828	+	0.430002i	$0.141487\pi$
$884$	0	0
$885$	−7.06226	−0.237395
$886$	0	0
$887$	25.1868	0.845689	0.422845	−	0.906202i	$-0.361032\pi$
$887$	25.1868	0.845689	0.422845	+	0.906202i	$0.361032\pi$
$888$	0	0
$889$	−32.0000	−1.07325
$890$	0	0
$891$	1.53113	0.0512947
$892$	0	0
$893$	−39.0623	−1.30717
$894$	0	0
$895$	13.0623	0.436623
$896$	0	0
$897$	−9.18677	−0.306737
$898$	0	0
$899$	0	0
$900$	0	0
$901$	−22.1245	−0.737074
$902$	0	0
$903$	1.65564	0.0550964
$904$	0	0
$905$	−26.5934	−0.883994
$906$	0	0
$907$	32.2490	1.07081	0.535406	−	0.844595i	$-0.320159\pi$
$907$	32.2490	1.07081	0.535406	+	0.844595i	$0.320159\pi$
$908$	0	0
$909$	−17.5311	−0.581471
$910$	0	0
$911$	16.0000	0.530104	0.265052	−	0.964234i	$-0.414611\pi$
$911$	16.0000	0.530104	0.265052	+	0.964234i	$0.414611\pi$
$912$	0	0
$913$	12.2490	0.405384
$914$	0	0
$915$	−11.0623	−0.365707
$916$	0	0
$917$	24.9377	0.823517
$918$	0	0
$919$	8.93774	0.294829	0.147414	−	0.989075i	$-0.452905\pi$
$919$	8.93774	0.294829	0.147414	+	0.989075i	$0.452905\pi$
$920$	0	0
$921$	2.12452	0.0700052
$922$	0	0
$923$	26.8132	0.882568
$924$	0	0
$925$	9.06226	0.297965
$926$	0	0
$927$	0	0
$928$	0	0
$929$	36.8424	1.20876	0.604380	−	0.796696i	$-0.293421\pi$
$929$	36.8424	1.20876	0.604380	+	0.796696i	$0.293421\pi$
$930$	0	0
$931$	19.3113	0.632902
$932$	0	0
$933$	−8.00000	−0.261908
$934$	0	0
$935$	6.12452	0.200293
$936$	0	0
$937$	−25.0623	−0.818748	−0.409374	−	0.912367i	$-0.634253\pi$
$937$	−25.0623	−0.818748	−0.409374	+	0.912367i	$0.634253\pi$
$938$	0	0
$939$	−27.0623	−0.883143
$940$	0	0
$941$	−21.8755	−0.713120	−0.356560	−	0.934272i	$-0.616051\pi$
$941$	−21.8755	−0.713120	−0.356560	+	0.934272i	$0.616051\pi$
$942$	0	0
$943$	−13.8755	−0.451848
$944$	0	0
$945$	−3.53113	−0.114868
$946$	0	0
$947$	−35.0623	−1.13937	−0.569685	−	0.821863i	$-0.692935\pi$
$947$	−35.0623	−1.13937	−0.569685	+	0.821863i	$0.692935\pi$
$948$	0	0
$949$	−2.81323	−0.0913212
$950$	0	0
$951$	−29.0623	−0.942408
$952$	0	0
$953$	54.1245	1.75327	0.876633	−	0.481161i	$-0.159785\pi$
$953$	54.1245	1.75327	0.876633	+	0.481161i	$0.159785\pi$
$954$	0	0
$955$	−8.00000	−0.258874
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−3.31129	−0.106927
$960$	0	0
$961$	1.00000	0.0322581
$962$	0	0
$963$	14.5934	0.470265
$964$	0	0
$965$	−14.0000	−0.450676
$966$	0	0
$967$	17.0623	0.548685	0.274343	−	0.961632i	$-0.411540\pi$
$967$	17.0623	0.548685	0.274343	+	0.961632i	$0.411540\pi$
$968$	0	0
$969$	14.1245	0.453745
$970$	0	0
$971$	33.1868	1.06501	0.532507	−	0.846426i	$-0.321250\pi$
$971$	33.1868	1.06501	0.532507	+	0.846426i	$0.321250\pi$
$972$	0	0
$973$	21.1868	0.679217
$974$	0	0
$975$	−6.00000	−0.192154
$976$	0	0
$977$	−38.0000	−1.21573	−0.607864	−	0.794041i	$-0.707973\pi$
$977$	−38.0000	−1.21573	−0.607864	+	0.794041i	$0.707973\pi$
$978$	0	0
$979$	2.34436	0.0749259
$980$	0	0
$981$	−1.06226	−0.0339153
$982$	0	0
$983$	−17.0623	−0.544202	−0.272101	−	0.962269i	$-0.587718\pi$
$983$	−17.0623	−0.544202	−0.272101	+	0.962269i	$0.587718\pi$
$984$	0	0
$985$	−6.00000	−0.191176
$986$	0	0
$987$	−39.0623	−1.24337
$988$	0	0
$989$	0.717902	0.0228280
$990$	0	0
$991$	24.4689	0.777279	0.388640	−	0.921390i	$-0.372945\pi$
$991$	24.4689	0.777279	0.388640	+	0.921390i	$0.372945\pi$
$992$	0	0
$993$	29.0623	0.922263
$994$	0	0
$995$	−0.468871	−0.0148642
$996$	0	0
$997$	62.2490	1.97145	0.985723	−	0.168373i	$-0.0538514\pi$
$997$	62.2490	1.97145	0.985723	+	0.168373i	$0.0538514\pi$
$998$	0	0
$999$	−9.06226	−0.286717

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7440.2.a.bd.1.1		2
4.3	odd	2		930.2.a.q.1.2	✓	2
12.11	even	2		2790.2.a.bf.1.2		2
20.3	even	4		4650.2.d.bg.3349.1		4
20.7	even	4		4650.2.d.bg.3349.4		4
20.19	odd	2		4650.2.a.bz.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
930.2.a.q.1.2	✓	2	4.3	odd	2
2790.2.a.bf.1.2		2	12.11	even	2
4650.2.a.bz.1.1		2	20.19	odd	2
4650.2.d.bg.3349.1		4	20.3	even	4
4650.2.d.bg.3349.4		4	20.7	even	4
7440.2.a.bd.1.1		2	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 7440 = 2^{4} \cdot 3 \cdot 5 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7440.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -1.00000 q^{5} -3.53113 q^{7} +1.00000 q^{9} +1.53113 q^{11} +6.00000 q^{13} +1.00000 q^{15} -4.00000 q^{17} +3.53113 q^{19} +3.53113 q^{21} +1.53113 q^{23} +1.00000 q^{25} -1.00000 q^{27} -1.00000 q^{31} -1.53113 q^{33} +3.53113 q^{35} +9.06226 q^{37} -6.00000 q^{39} -9.06226 q^{41} +0.468871 q^{43} -1.00000 q^{45} -11.0623 q^{47} +5.46887 q^{49} +4.00000 q^{51} +5.53113 q^{53} -1.53113 q^{55} -3.53113 q^{57} -7.06226 q^{59} -11.0623 q^{61} -3.53113 q^{63} -6.00000 q^{65} +11.0623 q^{67} -1.53113 q^{69} +4.46887 q^{71} -0.468871 q^{73} -1.00000 q^{75} -5.40661 q^{77} +0.468871 q^{79} +1.00000 q^{81} +8.00000 q^{83} +4.00000 q^{85} +1.53113 q^{89} -21.1868 q^{91} +1.00000 q^{93} -3.53113 q^{95} -16.1245 q^{97} +1.53113 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} - 2 q^{5} + q^{7} + 2 q^{9} - 5 q^{11} + 12 q^{13} + 2 q^{15} - 8 q^{17} - q^{19} - q^{21} - 5 q^{23} + 2 q^{25} - 2 q^{27} - 2 q^{31} + 5 q^{33} - q^{35} + 2 q^{37} - 12 q^{39} - 2 q^{41}+ \cdots - 5 q^{99}+O(q^{100}) \) 2 * q - 2 * q^3 - 2 * q^5 + q^7 + 2 * q^9 - 5 * q^11 + 12 * q^13 + 2 * q^15 - 8 * q^17 - q^19 - q^21 - 5 * q^23 + 2 * q^25 - 2 * q^27 - 2 * q^31 + 5 * q^33 - q^35 + 2 * q^37 - 12 * q^39 - 2 * q^41 + 9 * q^43 - 2 * q^45 - 6 * q^47 + 19 * q^49 + 8 * q^51 + 3 * q^53 + 5 * q^55 + q^57 + 2 * q^59 - 6 * q^61 + q^63 - 12 * q^65 + 6 * q^67 + 5 * q^69 + 17 * q^71 - 9 * q^73 - 2 * q^75 - 35 * q^77 + 9 * q^79 + 2 * q^81 + 16 * q^83 + 8 * q^85 - 5 * q^89 + 6 * q^91 + 2 * q^93 + q^95 - 5 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more