LMFDB - Embedded newform 2940.2.a.r.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2940,2,Mod(1,2940)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2940, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("2940.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2940 = 2^{2} \cdot 3 \cdot 5 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2940.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:	$23.4760181943$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 3 $
Twist minimal:	no (minimal twist has level 420)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.41421$ of defining polynomial
Character	$\chi$	$=$	2940.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} -1.00000 q^{5} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -1.00000 q^{5} +1.00000 q^{9} -4.24264 q^{11} +3.24264 q^{13} -1.00000 q^{15} +4.24264 q^{17} -7.00000 q^{19} -4.24264 q^{23} +1.00000 q^{25} +1.00000 q^{27} -1.75736 q^{29} -9.48528 q^{31} -4.24264 q^{33} +3.24264 q^{37} +3.24264 q^{39} -4.24264 q^{41} +3.24264 q^{43} -1.00000 q^{45} -6.00000 q^{47} +4.24264 q^{51} +8.48528 q^{53} +4.24264 q^{55} -7.00000 q^{57} -10.2426 q^{59} +4.48528 q^{61} -3.24264 q^{65} -5.24264 q^{67} -4.24264 q^{69} -12.7279 q^{71} +9.24264 q^{73} +1.00000 q^{75} +11.0000 q^{79} +1.00000 q^{81} -10.2426 q^{83} -4.24264 q^{85} -1.75736 q^{87} -10.2426 q^{89} -9.48528 q^{93} +7.00000 q^{95} -0.485281 q^{97} -4.24264 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} - 2 q^{5} + 2 q^{9}+O(q^{10}) $ 2 * q + 2 * q^3 - 2 * q^5 + 2 * q^9	$ 2 q + 2 q^{3} - 2 q^{5} + 2 q^{9} - 2 q^{13} - 2 q^{15} - 14 q^{19} + 2 q^{25} + 2 q^{27} - 12 q^{29} - 2 q^{31} - 2 q^{37} - 2 q^{39} - 2 q^{43} - 2 q^{45} - 12 q^{47} - 14 q^{57} - 12 q^{59} - 8 q^{61} + 2 q^{65} - 2 q^{67} + 10 q^{73} + 2 q^{75} + 22 q^{79} + 2 q^{81} - 12 q^{83} - 12 q^{87} - 12 q^{89} - 2 q^{93} + 14 q^{95} + 16 q^{97}+O(q^{100}) $ 2 * q + 2 * q^3 - 2 * q^5 + 2 * q^9 - 2 * q^13 - 2 * q^15 - 14 * q^19 + 2 * q^25 + 2 * q^27 - 12 * q^29 - 2 * q^31 - 2 * q^37 - 2 * q^39 - 2 * q^43 - 2 * q^45 - 12 * q^47 - 14 * q^57 - 12 * q^59 - 8 * q^61 + 2 * q^65 - 2 * q^67 + 10 * q^73 + 2 * q^75 + 22 * q^79 + 2 * q^81 - 12 * q^83 - 12 * q^87 - 12 * q^89 - 2 * q^93 + 14 * q^95 + 16 * q^97

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	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
$7$	0	0
$7$	0	0
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−4.24264	−1.27920	−0.639602	−	0.768706i	$-0.720901\pi$
$11$	−4.24264	−1.27920	−0.639602	+	0.768706i	$0.720901\pi$
$12$	0	0
$13$	3.24264	0.899347	0.449673	−	0.893193i	$-0.351540\pi$
$13$	3.24264	0.899347	0.449673	+	0.893193i	$0.351540\pi$
$14$	0	0
$15$	−1.00000	−0.258199
$16$	0	0
$17$	4.24264	1.02899	0.514496	−	0.857493i	$-0.327979\pi$
$17$	4.24264	1.02899	0.514496	+	0.857493i	$0.327979\pi$
$18$	0	0
$19$	−7.00000	−1.60591	−0.802955	−	0.596040i	$-0.796740\pi$
$19$	−7.00000	−1.60591	−0.802955	+	0.596040i	$0.796740\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−4.24264	−0.884652	−0.442326	−	0.896854i	$-0.645847\pi$
$23$	−4.24264	−0.884652	−0.442326	+	0.896854i	$0.645847\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−1.75736	−0.326333	−0.163167	−	0.986599i	$-0.552171\pi$
$29$	−1.75736	−0.326333	−0.163167	+	0.986599i	$0.552171\pi$
$30$	0	0
$31$	−9.48528	−1.70361	−0.851803	−	0.523862i	$-0.824491\pi$
$31$	−9.48528	−1.70361	−0.851803	+	0.523862i	$0.824491\pi$
$32$	0	0
$33$	−4.24264	−0.738549
$34$	0	0
$35$	0	0
$36$	0	0
$37$	3.24264	0.533087	0.266543	−	0.963823i	$-0.414118\pi$
$37$	3.24264	0.533087	0.266543	+	0.963823i	$0.414118\pi$
$38$	0	0
$39$	3.24264	0.519238
$40$	0	0
$41$	−4.24264	−0.662589	−0.331295	−	0.943527i	$-0.607485\pi$
$41$	−4.24264	−0.662589	−0.331295	+	0.943527i	$0.607485\pi$
$42$	0	0
$43$	3.24264	0.494498	0.247249	−	0.968952i	$-0.420473\pi$
$43$	3.24264	0.494498	0.247249	+	0.968952i	$0.420473\pi$
$44$	0	0
$45$	−1.00000	−0.149071
$46$	0	0
$47$	−6.00000	−0.875190	−0.437595	−	0.899172i	$-0.644170\pi$
$47$	−6.00000	−0.875190	−0.437595	+	0.899172i	$0.644170\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	4.24264	0.594089
$52$	0	0
$53$	8.48528	1.16554	0.582772	−	0.812636i	$-0.301968\pi$
$53$	8.48528	1.16554	0.582772	+	0.812636i	$0.301968\pi$
$54$	0	0
$55$	4.24264	0.572078
$56$	0	0
$57$	−7.00000	−0.927173
$58$	0	0
$59$	−10.2426	−1.33348	−0.666739	−	0.745291i	$-0.732310\pi$
$59$	−10.2426	−1.33348	−0.666739	+	0.745291i	$0.732310\pi$
$60$	0	0
$61$	4.48528	0.574281	0.287141	−	0.957888i	$-0.407295\pi$
$61$	4.48528	0.574281	0.287141	+	0.957888i	$0.407295\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−3.24264	−0.402200
$66$	0	0
$67$	−5.24264	−0.640490	−0.320245	−	0.947335i	$-0.603765\pi$
$67$	−5.24264	−0.640490	−0.320245	+	0.947335i	$0.603765\pi$
$68$	0	0
$69$	−4.24264	−0.510754
$70$	0	0
$71$	−12.7279	−1.51053	−0.755263	−	0.655422i	$-0.772491\pi$
$71$	−12.7279	−1.51053	−0.755263	+	0.655422i	$0.772491\pi$
$72$	0	0
$73$	9.24264	1.08177	0.540885	−	0.841097i	$-0.318090\pi$
$73$	9.24264	1.08177	0.540885	+	0.841097i	$0.318090\pi$
$74$	0	0
$75$	1.00000	0.115470
$76$	0	0
$77$	0	0
$78$	0	0
$79$	11.0000	1.23760	0.618798	−	0.785550i	$-0.287620\pi$
$79$	11.0000	1.23760	0.618798	+	0.785550i	$0.287620\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−10.2426	−1.12428	−0.562138	−	0.827043i	$-0.690021\pi$
$83$	−10.2426	−1.12428	−0.562138	+	0.827043i	$0.690021\pi$
$84$	0	0
$85$	−4.24264	−0.460179
$86$	0	0
$87$	−1.75736	−0.188409
$88$	0	0
$89$	−10.2426	−1.08572	−0.542859	−	0.839824i	$-0.682658\pi$
$89$	−10.2426	−1.08572	−0.542859	+	0.839824i	$0.682658\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−9.48528	−0.983578
$94$	0	0
$95$	7.00000	0.718185
$96$	0	0
$97$	−0.485281	−0.0492729	−0.0246364	−	0.999696i	$-0.507843\pi$
$97$	−0.485281	−0.0492729	−0.0246364	+	0.999696i	$0.507843\pi$
$98$	0	0
$99$	−4.24264	−0.426401
$100$	0	0
$101$	−7.75736	−0.771886	−0.385943	−	0.922523i	$-0.626124\pi$
$101$	−7.75736	−0.771886	−0.385943	+	0.922523i	$0.626124\pi$
$102$	0	0
$103$	−17.2426	−1.69897	−0.849484	−	0.527614i	$-0.823087\pi$
$103$	−17.2426	−1.69897	−0.849484	+	0.527614i	$0.823087\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	12.7279	1.23045	0.615227	−	0.788350i	$-0.289064\pi$
$107$	12.7279	1.23045	0.615227	+	0.788350i	$0.289064\pi$
$108$	0	0
$109$	−9.48528	−0.908525	−0.454263	−	0.890868i	$-0.650097\pi$
$109$	−9.48528	−0.908525	−0.454263	+	0.890868i	$0.650097\pi$
$110$	0	0
$111$	3.24264	0.307778
$112$	0	0
$113$	−18.0000	−1.69330	−0.846649	−	0.532152i	$-0.821383\pi$
$113$	−18.0000	−1.69330	−0.846649	+	0.532152i	$0.821383\pi$
$114$	0	0
$115$	4.24264	0.395628
$116$	0	0
$117$	3.24264	0.299782
$118$	0	0
$119$	0	0
$120$	0	0
$121$	7.00000	0.636364
$122$	0	0
$123$	−4.24264	−0.382546
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−2.75736	−0.244676	−0.122338	−	0.992488i	$-0.539039\pi$
$127$	−2.75736	−0.244676	−0.122338	+	0.992488i	$0.539039\pi$
$128$	0	0
$129$	3.24264	0.285499
$130$	0	0
$131$	14.4853	1.26558	0.632792	−	0.774321i	$-0.281909\pi$
$131$	14.4853	1.26558	0.632792	+	0.774321i	$0.281909\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−1.00000	−0.0860663
$136$	0	0
$137$	4.24264	0.362473	0.181237	−	0.983440i	$-0.441990\pi$
$137$	4.24264	0.362473	0.181237	+	0.983440i	$0.441990\pi$
$138$	0	0
$139$	−15.4853	−1.31344	−0.656722	−	0.754133i	$-0.728058\pi$
$139$	−15.4853	−1.31344	−0.656722	+	0.754133i	$0.728058\pi$
$140$	0	0
$141$	−6.00000	−0.505291
$142$	0	0
$143$	−13.7574	−1.15045
$144$	0	0
$145$	1.75736	0.145941
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−12.0000	−0.983078	−0.491539	−	0.870855i	$-0.663566\pi$
$149$	−12.0000	−0.983078	−0.491539	+	0.870855i	$0.663566\pi$
$150$	0	0
$151$	22.4853	1.82983	0.914913	−	0.403651i	$-0.132259\pi$
$151$	22.4853	1.82983	0.914913	+	0.403651i	$0.132259\pi$
$152$	0	0
$153$	4.24264	0.342997
$154$	0	0
$155$	9.48528	0.761876
$156$	0	0
$157$	−6.48528	−0.517582	−0.258791	−	0.965933i	$-0.583324\pi$
$157$	−6.48528	−0.517582	−0.258791	+	0.965933i	$0.583324\pi$
$158$	0	0
$159$	8.48528	0.672927
$160$	0	0
$161$	0	0
$162$	0	0
$163$	8.00000	0.626608	0.313304	−	0.949653i	$-0.398564\pi$
$163$	8.00000	0.626608	0.313304	+	0.949653i	$0.398564\pi$
$164$	0	0
$165$	4.24264	0.330289
$166$	0	0
$167$	18.7279	1.44921	0.724605	−	0.689164i	$-0.242022\pi$
$167$	18.7279	1.44921	0.724605	+	0.689164i	$0.242022\pi$
$168$	0	0
$169$	−2.48528	−0.191175
$170$	0	0
$171$	−7.00000	−0.535303
$172$	0	0
$173$	−20.4853	−1.55747	−0.778734	−	0.627355i	$-0.784138\pi$
$173$	−20.4853	−1.55747	−0.778734	+	0.627355i	$0.784138\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−10.2426	−0.769884
$178$	0	0
$179$	−6.00000	−0.448461	−0.224231	−	0.974536i	$-0.571987\pi$
$179$	−6.00000	−0.448461	−0.224231	+	0.974536i	$0.571987\pi$
$180$	0	0
$181$	−13.0000	−0.966282	−0.483141	−	0.875542i	$-0.660504\pi$
$181$	−13.0000	−0.966282	−0.483141	+	0.875542i	$0.660504\pi$
$182$	0	0
$183$	4.48528	0.331562
$184$	0	0
$185$	−3.24264	−0.238404
$186$	0	0
$187$	−18.0000	−1.31629
$188$	0	0
$189$	0	0
$190$	0	0
$191$	6.00000	0.434145	0.217072	−	0.976156i	$-0.430349\pi$
$191$	6.00000	0.434145	0.217072	+	0.976156i	$0.430349\pi$
$192$	0	0
$193$	6.75736	0.486405	0.243203	−	0.969975i	$-0.421802\pi$
$193$	6.75736	0.486405	0.243203	+	0.969975i	$0.421802\pi$
$194$	0	0
$195$	−3.24264	−0.232210
$196$	0	0
$197$	−16.2426	−1.15724	−0.578620	−	0.815597i	$-0.696409\pi$
$197$	−16.2426	−1.15724	−0.578620	+	0.815597i	$0.696409\pi$
$198$	0	0
$199$	10.4853	0.743282	0.371641	−	0.928377i	$-0.378795\pi$
$199$	10.4853	0.743282	0.371641	+	0.928377i	$0.378795\pi$
$200$	0	0
$201$	−5.24264	−0.369787
$202$	0	0
$203$	0	0
$204$	0	0
$205$	4.24264	0.296319
$206$	0	0
$207$	−4.24264	−0.294884
$208$	0	0
$209$	29.6985	2.05429
$210$	0	0
$211$	22.4853	1.54795	0.773975	−	0.633216i	$-0.218265\pi$
$211$	22.4853	1.54795	0.773975	+	0.633216i	$0.218265\pi$
$212$	0	0
$213$	−12.7279	−0.872103
$214$	0	0
$215$	−3.24264	−0.221146
$216$	0	0
$217$	0	0
$218$	0	0
$219$	9.24264	0.624560
$220$	0	0
$221$	13.7574	0.925420
$222$	0	0
$223$	−7.51472	−0.503223	−0.251611	−	0.967828i	$-0.580960\pi$
$223$	−7.51472	−0.503223	−0.251611	+	0.967828i	$0.580960\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	−15.2132	−1.00974	−0.504868	−	0.863197i	$-0.668459\pi$
$227$	−15.2132	−1.00974	−0.504868	+	0.863197i	$0.668459\pi$
$228$	0	0
$229$	−7.00000	−0.462573	−0.231287	−	0.972886i	$-0.574293\pi$
$229$	−7.00000	−0.462573	−0.231287	+	0.972886i	$0.574293\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	14.4853	0.948962	0.474481	−	0.880266i	$-0.342636\pi$
$233$	14.4853	0.948962	0.474481	+	0.880266i	$0.342636\pi$
$234$	0	0
$235$	6.00000	0.391397
$236$	0	0
$237$	11.0000	0.714527
$238$	0	0
$239$	10.9706	0.709627	0.354813	−	0.934937i	$-0.384544\pi$
$239$	10.9706	0.709627	0.354813	+	0.934937i	$0.384544\pi$
$240$	0	0
$241$	−4.00000	−0.257663	−0.128831	−	0.991667i	$-0.541123\pi$
$241$	−4.00000	−0.257663	−0.128831	+	0.991667i	$0.541123\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−22.6985	−1.44427
$248$	0	0
$249$	−10.2426	−0.649101
$250$	0	0
$251$	−18.7279	−1.18210	−0.591048	−	0.806636i	$-0.701286\pi$
$251$	−18.7279	−1.18210	−0.591048	+	0.806636i	$0.701286\pi$
$252$	0	0
$253$	18.0000	1.13165
$254$	0	0
$255$	−4.24264	−0.265684
$256$	0	0
$257$	10.2426	0.638918	0.319459	−	0.947600i	$-0.396499\pi$
$257$	10.2426	0.638918	0.319459	+	0.947600i	$0.396499\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−1.75736	−0.108778
$262$	0	0
$263$	−8.48528	−0.523225	−0.261612	−	0.965173i	$-0.584254\pi$
$263$	−8.48528	−0.523225	−0.261612	+	0.965173i	$0.584254\pi$
$264$	0	0
$265$	−8.48528	−0.521247
$266$	0	0
$267$	−10.2426	−0.626839
$268$	0	0
$269$	−2.48528	−0.151530	−0.0757651	−	0.997126i	$-0.524140\pi$
$269$	−2.48528	−0.151530	−0.0757651	+	0.997126i	$0.524140\pi$
$270$	0	0
$271$	−23.4558	−1.42484	−0.712421	−	0.701753i	$-0.752401\pi$
$271$	−23.4558	−1.42484	−0.712421	+	0.701753i	$0.752401\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−4.24264	−0.255841
$276$	0	0
$277$	17.7279	1.06517	0.532584	−	0.846377i	$-0.321221\pi$
$277$	17.7279	1.06517	0.532584	+	0.846377i	$0.321221\pi$
$278$	0	0
$279$	−9.48528	−0.567869
$280$	0	0
$281$	28.9706	1.72824	0.864119	−	0.503287i	$-0.167876\pi$
$281$	28.9706	1.72824	0.864119	+	0.503287i	$0.167876\pi$
$282$	0	0
$283$	23.7279	1.41048	0.705239	−	0.708969i	$-0.250840\pi$
$283$	23.7279	1.41048	0.705239	+	0.708969i	$0.250840\pi$
$284$	0	0
$285$	7.00000	0.414644
$286$	0	0
$287$	0	0
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	−0.485281	−0.0284477
$292$	0	0
$293$	−4.97056	−0.290383	−0.145192	−	0.989404i	$-0.546380\pi$
$293$	−4.97056	−0.290383	−0.145192	+	0.989404i	$0.546380\pi$
$294$	0	0
$295$	10.2426	0.596350
$296$	0	0
$297$	−4.24264	−0.246183
$298$	0	0
$299$	−13.7574	−0.795609
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−7.75736	−0.445649
$304$	0	0
$305$	−4.48528	−0.256826
$306$	0	0
$307$	3.24264	0.185067	0.0925336	−	0.995710i	$-0.470503\pi$
$307$	3.24264	0.185067	0.0925336	+	0.995710i	$0.470503\pi$
$308$	0	0
$309$	−17.2426	−0.980900
$310$	0	0
$311$	21.2132	1.20289	0.601445	−	0.798914i	$-0.294592\pi$
$311$	21.2132	1.20289	0.601445	+	0.798914i	$0.294592\pi$
$312$	0	0
$313$	23.7279	1.34118	0.670591	−	0.741828i	$-0.266041\pi$
$313$	23.7279	1.34118	0.670591	+	0.741828i	$0.266041\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−24.7279	−1.38886	−0.694429	−	0.719561i	$-0.744343\pi$
$317$	−24.7279	−1.38886	−0.694429	+	0.719561i	$0.744343\pi$
$318$	0	0
$319$	7.45584	0.417447
$320$	0	0
$321$	12.7279	0.710403
$322$	0	0
$323$	−29.6985	−1.65247
$324$	0	0
$325$	3.24264	0.179869
$326$	0	0
$327$	−9.48528	−0.524537
$328$	0	0
$329$	0	0
$330$	0	0
$331$	17.0000	0.934405	0.467202	−	0.884150i	$-0.345262\pi$
$331$	17.0000	0.934405	0.467202	+	0.884150i	$0.345262\pi$
$332$	0	0
$333$	3.24264	0.177696
$334$	0	0
$335$	5.24264	0.286436
$336$	0	0
$337$	−13.7279	−0.747808	−0.373904	−	0.927467i	$-0.621981\pi$
$337$	−13.7279	−0.747808	−0.373904	+	0.927467i	$0.621981\pi$
$338$	0	0
$339$	−18.0000	−0.977626
$340$	0	0
$341$	40.2426	2.17926
$342$	0	0
$343$	0	0
$344$	0	0
$345$	4.24264	0.228416
$346$	0	0
$347$	−24.0000	−1.28839	−0.644194	−	0.764862i	$-0.722807\pi$
$347$	−24.0000	−1.28839	−0.644194	+	0.764862i	$0.722807\pi$
$348$	0	0
$349$	−10.0000	−0.535288	−0.267644	−	0.963518i	$-0.586245\pi$
$349$	−10.0000	−0.535288	−0.267644	+	0.963518i	$0.586245\pi$
$350$	0	0
$351$	3.24264	0.173079
$352$	0	0
$353$	−10.2426	−0.545161	−0.272580	−	0.962133i	$-0.587877\pi$
$353$	−10.2426	−0.545161	−0.272580	+	0.962133i	$0.587877\pi$
$354$	0	0
$355$	12.7279	0.675528
$356$	0	0
$357$	0	0
$358$	0	0
$359$	1.75736	0.0927499	0.0463749	−	0.998924i	$-0.485233\pi$
$359$	1.75736	0.0927499	0.0463749	+	0.998924i	$0.485233\pi$
$360$	0	0
$361$	30.0000	1.57895
$362$	0	0
$363$	7.00000	0.367405
$364$	0	0
$365$	−9.24264	−0.483782
$366$	0	0
$367$	35.7279	1.86498	0.932491	−	0.361193i	$-0.117630\pi$
$367$	35.7279	1.86498	0.932491	+	0.361193i	$0.117630\pi$
$368$	0	0
$369$	−4.24264	−0.220863
$370$	0	0
$371$	0	0
$372$	0	0
$373$	17.7279	0.917917	0.458959	−	0.888458i	$-0.348223\pi$
$373$	17.7279	0.917917	0.458959	+	0.888458i	$0.348223\pi$
$374$	0	0
$375$	−1.00000	−0.0516398
$376$	0	0
$377$	−5.69848	−0.293487
$378$	0	0
$379$	−20.4558	−1.05075	−0.525373	−	0.850872i	$-0.676074\pi$
$379$	−20.4558	−1.05075	−0.525373	+	0.850872i	$0.676074\pi$
$380$	0	0
$381$	−2.75736	−0.141264
$382$	0	0
$383$	25.4558	1.30073	0.650366	−	0.759621i	$-0.274615\pi$
$383$	25.4558	1.30073	0.650366	+	0.759621i	$0.274615\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	3.24264	0.164833
$388$	0	0
$389$	−21.2132	−1.07555	−0.537776	−	0.843088i	$-0.680735\pi$
$389$	−21.2132	−1.07555	−0.537776	+	0.843088i	$0.680735\pi$
$390$	0	0
$391$	−18.0000	−0.910299
$392$	0	0
$393$	14.4853	0.730686
$394$	0	0
$395$	−11.0000	−0.553470
$396$	0	0
$397$	−8.75736	−0.439519	−0.219760	−	0.975554i	$-0.570527\pi$
$397$	−8.75736	−0.439519	−0.219760	+	0.975554i	$0.570527\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−3.51472	−0.175517	−0.0877583	−	0.996142i	$-0.527970\pi$
$401$	−3.51472	−0.175517	−0.0877583	+	0.996142i	$0.527970\pi$
$402$	0	0
$403$	−30.7574	−1.53213
$404$	0	0
$405$	−1.00000	−0.0496904
$406$	0	0
$407$	−13.7574	−0.681927
$408$	0	0
$409$	17.0000	0.840596	0.420298	−	0.907386i	$-0.361926\pi$
$409$	17.0000	0.840596	0.420298	+	0.907386i	$0.361926\pi$
$410$	0	0
$411$	4.24264	0.209274
$412$	0	0
$413$	0	0
$414$	0	0
$415$	10.2426	0.502791
$416$	0	0
$417$	−15.4853	−0.758317
$418$	0	0
$419$	−14.4853	−0.707652	−0.353826	−	0.935311i	$-0.615120\pi$
$419$	−14.4853	−0.707652	−0.353826	+	0.935311i	$0.615120\pi$
$420$	0	0
$421$	31.4853	1.53450	0.767249	−	0.641349i	$-0.221625\pi$
$421$	31.4853	1.53450	0.767249	+	0.641349i	$0.221625\pi$
$422$	0	0
$423$	−6.00000	−0.291730
$424$	0	0
$425$	4.24264	0.205798
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−13.7574	−0.664212
$430$	0	0
$431$	19.4558	0.937155	0.468578	−	0.883422i	$-0.344767\pi$
$431$	19.4558	0.937155	0.468578	+	0.883422i	$0.344767\pi$
$432$	0	0
$433$	33.2426	1.59754	0.798770	−	0.601637i	$-0.205485\pi$
$433$	33.2426	1.59754	0.798770	+	0.601637i	$0.205485\pi$
$434$	0	0
$435$	1.75736	0.0842589
$436$	0	0
$437$	29.6985	1.42067
$438$	0	0
$439$	−10.0000	−0.477274	−0.238637	−	0.971109i	$-0.576701\pi$
$439$	−10.0000	−0.477274	−0.238637	+	0.971109i	$0.576701\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	20.4853	0.973285	0.486643	−	0.873601i	$-0.338221\pi$
$443$	20.4853	0.973285	0.486643	+	0.873601i	$0.338221\pi$
$444$	0	0
$445$	10.2426	0.485548
$446$	0	0
$447$	−12.0000	−0.567581
$448$	0	0
$449$	−6.00000	−0.283158	−0.141579	−	0.989927i	$-0.545218\pi$
$449$	−6.00000	−0.283158	−0.141579	+	0.989927i	$0.545218\pi$
$450$	0	0
$451$	18.0000	0.847587
$452$	0	0
$453$	22.4853	1.05645
$454$	0	0
$455$	0	0
$456$	0	0
$457$	32.2132	1.50687	0.753435	−	0.657522i	$-0.228395\pi$
$457$	32.2132	1.50687	0.753435	+	0.657522i	$0.228395\pi$
$458$	0	0
$459$	4.24264	0.198030
$460$	0	0
$461$	18.7279	0.872246	0.436123	−	0.899887i	$-0.356351\pi$
$461$	18.7279	0.872246	0.436123	+	0.899887i	$0.356351\pi$
$462$	0	0
$463$	10.2721	0.477384	0.238692	−	0.971095i	$-0.423281\pi$
$463$	10.2721	0.477384	0.238692	+	0.971095i	$0.423281\pi$
$464$	0	0
$465$	9.48528	0.439869
$466$	0	0
$467$	−10.9706	−0.507657	−0.253829	−	0.967249i	$-0.581690\pi$
$467$	−10.9706	−0.507657	−0.253829	+	0.967249i	$0.581690\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−6.48528	−0.298826
$472$	0	0
$473$	−13.7574	−0.632564
$474$	0	0
$475$	−7.00000	−0.321182
$476$	0	0
$477$	8.48528	0.388514
$478$	0	0
$479$	12.0000	0.548294	0.274147	−	0.961688i	$-0.411605\pi$
$479$	12.0000	0.548294	0.274147	+	0.961688i	$0.411605\pi$
$480$	0	0
$481$	10.5147	0.479430
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0.485281	0.0220355
$486$	0	0
$487$	−37.7279	−1.70962	−0.854808	−	0.518945i	$-0.826325\pi$
$487$	−37.7279	−1.70962	−0.854808	+	0.518945i	$0.826325\pi$
$488$	0	0
$489$	8.00000	0.361773
$490$	0	0
$491$	15.5147	0.700169	0.350085	−	0.936718i	$-0.386153\pi$
$491$	15.5147	0.700169	0.350085	+	0.936718i	$0.386153\pi$
$492$	0	0
$493$	−7.45584	−0.335794
$494$	0	0
$495$	4.24264	0.190693
$496$	0	0
$497$	0	0
$498$	0	0
$499$	19.4853	0.872281	0.436140	−	0.899879i	$-0.356345\pi$
$499$	19.4853	0.872281	0.436140	+	0.899879i	$0.356345\pi$
$500$	0	0
$501$	18.7279	0.836702
$502$	0	0
$503$	−26.4853	−1.18092	−0.590460	−	0.807067i	$-0.701054\pi$
$503$	−26.4853	−1.18092	−0.590460	+	0.807067i	$0.701054\pi$
$504$	0	0
$505$	7.75736	0.345198
$506$	0	0
$507$	−2.48528	−0.110375
$508$	0	0
$509$	−19.4558	−0.862365	−0.431183	−	0.902265i	$-0.641904\pi$
$509$	−19.4558	−0.862365	−0.431183	+	0.902265i	$0.641904\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−7.00000	−0.309058
$514$	0	0
$515$	17.2426	0.759802
$516$	0	0
$517$	25.4558	1.11955
$518$	0	0
$519$	−20.4853	−0.899204
$520$	0	0
$521$	−13.4558	−0.589511	−0.294756	−	0.955573i	$-0.595238\pi$
$521$	−13.4558	−0.589511	−0.294756	+	0.955573i	$0.595238\pi$
$522$	0	0
$523$	−5.24264	−0.229245	−0.114622	−	0.993409i	$-0.536566\pi$
$523$	−5.24264	−0.229245	−0.114622	+	0.993409i	$0.536566\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−40.2426	−1.75300
$528$	0	0
$529$	−5.00000	−0.217391
$530$	0	0
$531$	−10.2426	−0.444493
$532$	0	0
$533$	−13.7574	−0.595897
$534$	0	0
$535$	−12.7279	−0.550276
$536$	0	0
$537$	−6.00000	−0.258919
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−40.9411	−1.76020	−0.880098	−	0.474792i	$-0.842523\pi$
$541$	−40.9411	−1.76020	−0.880098	+	0.474792i	$0.842523\pi$
$542$	0	0
$543$	−13.0000	−0.557883
$544$	0	0
$545$	9.48528	0.406305
$546$	0	0
$547$	33.4558	1.43047	0.715234	−	0.698885i	$-0.246320\pi$
$547$	33.4558	1.43047	0.715234	+	0.698885i	$0.246320\pi$
$548$	0	0
$549$	4.48528	0.191427
$550$	0	0
$551$	12.3015	0.524062
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−3.24264	−0.137642
$556$	0	0
$557$	−30.0000	−1.27114	−0.635570	−	0.772043i	$-0.719235\pi$
$557$	−30.0000	−1.27114	−0.635570	+	0.772043i	$0.719235\pi$
$558$	0	0
$559$	10.5147	0.444725
$560$	0	0
$561$	−18.0000	−0.759961
$562$	0	0
$563$	−6.00000	−0.252870	−0.126435	−	0.991975i	$-0.540353\pi$
$563$	−6.00000	−0.252870	−0.126435	+	0.991975i	$0.540353\pi$
$564$	0	0
$565$	18.0000	0.757266
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−41.6985	−1.74809	−0.874046	−	0.485844i	$-0.838512\pi$
$569$	−41.6985	−1.74809	−0.874046	+	0.485844i	$0.838512\pi$
$570$	0	0
$571$	−28.9411	−1.21115	−0.605574	−	0.795789i	$-0.707057\pi$
$571$	−28.9411	−1.21115	−0.605574	+	0.795789i	$0.707057\pi$
$572$	0	0
$573$	6.00000	0.250654
$574$	0	0
$575$	−4.24264	−0.176930
$576$	0	0
$577$	12.7574	0.531096	0.265548	−	0.964098i	$-0.414447\pi$
$577$	12.7574	0.531096	0.265548	+	0.964098i	$0.414447\pi$
$578$	0	0
$579$	6.75736	0.280826
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−36.0000	−1.49097
$584$	0	0
$585$	−3.24264	−0.134067
$586$	0	0
$587$	45.2132	1.86615	0.933074	−	0.359684i	$-0.117115\pi$
$587$	45.2132	1.86615	0.933074	+	0.359684i	$0.117115\pi$
$588$	0	0
$589$	66.3970	2.73584
$590$	0	0
$591$	−16.2426	−0.668133
$592$	0	0
$593$	3.21320	0.131950	0.0659752	−	0.997821i	$-0.478984\pi$
$593$	3.21320	0.131950	0.0659752	+	0.997821i	$0.478984\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	10.4853	0.429134
$598$	0	0
$599$	−32.4853	−1.32731	−0.663656	−	0.748038i	$-0.730996\pi$
$599$	−32.4853	−1.32731	−0.663656	+	0.748038i	$0.730996\pi$
$600$	0	0
$601$	−3.48528	−0.142168	−0.0710838	−	0.997470i	$-0.522646\pi$
$601$	−3.48528	−0.142168	−0.0710838	+	0.997470i	$0.522646\pi$
$602$	0	0
$603$	−5.24264	−0.213497
$604$	0	0
$605$	−7.00000	−0.284590
$606$	0	0
$607$	−29.2426	−1.18692	−0.593461	−	0.804863i	$-0.702239\pi$
$607$	−29.2426	−1.18692	−0.593461	+	0.804863i	$0.702239\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−19.4558	−0.787099
$612$	0	0
$613$	−5.45584	−0.220359	−0.110180	−	0.993912i	$-0.535143\pi$
$613$	−5.45584	−0.220359	−0.110180	+	0.993912i	$0.535143\pi$
$614$	0	0
$615$	4.24264	0.171080
$616$	0	0
$617$	−26.4853	−1.06626	−0.533129	−	0.846034i	$-0.678984\pi$
$617$	−26.4853	−1.06626	−0.533129	+	0.846034i	$0.678984\pi$
$618$	0	0
$619$	−11.9706	−0.481138	−0.240569	−	0.970632i	$-0.577334\pi$
$619$	−11.9706	−0.481138	−0.240569	+	0.970632i	$0.577334\pi$
$620$	0	0
$621$	−4.24264	−0.170251
$622$	0	0
$623$	0	0
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	29.6985	1.18604
$628$	0	0
$629$	13.7574	0.548542
$630$	0	0
$631$	8.00000	0.318475	0.159237	−	0.987240i	$-0.449096\pi$
$631$	8.00000	0.318475	0.159237	+	0.987240i	$0.449096\pi$
$632$	0	0
$633$	22.4853	0.893710
$634$	0	0
$635$	2.75736	0.109422
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−12.7279	−0.503509
$640$	0	0
$641$	34.2426	1.35250	0.676251	−	0.736671i	$-0.263603\pi$
$641$	34.2426	1.35250	0.676251	+	0.736671i	$0.263603\pi$
$642$	0	0
$643$	−19.7279	−0.777993	−0.388997	−	0.921239i	$-0.627178\pi$
$643$	−19.7279	−0.777993	−0.388997	+	0.921239i	$0.627178\pi$
$644$	0	0
$645$	−3.24264	−0.127679
$646$	0	0
$647$	10.2426	0.402680	0.201340	−	0.979521i	$-0.435470\pi$
$647$	10.2426	0.402680	0.201340	+	0.979521i	$0.435470\pi$
$648$	0	0
$649$	43.4558	1.70579
$650$	0	0
$651$	0	0
$652$	0	0
$653$	10.2426	0.400826	0.200413	−	0.979712i	$-0.435772\pi$
$653$	10.2426	0.400826	0.200413	+	0.979712i	$0.435772\pi$
$654$	0	0
$655$	−14.4853	−0.565987
$656$	0	0
$657$	9.24264	0.360590
$658$	0	0
$659$	40.9706	1.59599	0.797993	−	0.602666i	$-0.205895\pi$
$659$	40.9706	1.59599	0.797993	+	0.602666i	$0.205895\pi$
$660$	0	0
$661$	−2.02944	−0.0789360	−0.0394680	−	0.999221i	$-0.512566\pi$
$661$	−2.02944	−0.0789360	−0.0394680	+	0.999221i	$0.512566\pi$
$662$	0	0
$663$	13.7574	0.534292
$664$	0	0
$665$	0	0
$666$	0	0
$667$	7.45584	0.288691
$668$	0	0
$669$	−7.51472	−0.290536
$670$	0	0
$671$	−19.0294	−0.734623
$672$	0	0
$673$	29.7279	1.14593	0.572964	−	0.819581i	$-0.305794\pi$
$673$	29.7279	1.14593	0.572964	+	0.819581i	$0.305794\pi$
$674$	0	0
$675$	1.00000	0.0384900
$676$	0	0
$677$	12.7279	0.489174	0.244587	−	0.969627i	$-0.421348\pi$
$677$	12.7279	0.489174	0.244587	+	0.969627i	$0.421348\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−15.2132	−0.582971
$682$	0	0
$683$	33.2132	1.27087	0.635434	−	0.772155i	$-0.280821\pi$
$683$	33.2132	1.27087	0.635434	+	0.772155i	$0.280821\pi$
$684$	0	0
$685$	−4.24264	−0.162103
$686$	0	0
$687$	−7.00000	−0.267067
$688$	0	0
$689$	27.5147	1.04823
$690$	0	0
$691$	26.9411	1.02489	0.512444	−	0.858720i	$-0.328740\pi$
$691$	26.9411	1.02489	0.512444	+	0.858720i	$0.328740\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	15.4853	0.587390
$696$	0	0
$697$	−18.0000	−0.681799
$698$	0	0
$699$	14.4853	0.547884
$700$	0	0
$701$	8.78680	0.331873	0.165936	−	0.986136i	$-0.446935\pi$
$701$	8.78680	0.331873	0.165936	+	0.986136i	$0.446935\pi$
$702$	0	0
$703$	−22.6985	−0.856090
$704$	0	0
$705$	6.00000	0.225973
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−36.4853	−1.37023	−0.685117	−	0.728433i	$-0.740249\pi$
$709$	−36.4853	−1.37023	−0.685117	+	0.728433i	$0.740249\pi$
$710$	0	0
$711$	11.0000	0.412532
$712$	0	0
$713$	40.2426	1.50710
$714$	0	0
$715$	13.7574	0.514496
$716$	0	0
$717$	10.9706	0.409703
$718$	0	0
$719$	26.4853	0.987734	0.493867	−	0.869537i	$-0.335583\pi$
$719$	26.4853	0.987734	0.493867	+	0.869537i	$0.335583\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−4.00000	−0.148762
$724$	0	0
$725$	−1.75736	−0.0652667
$726$	0	0
$727$	0.757359	0.0280889	0.0140445	−	0.999901i	$-0.495529\pi$
$727$	0.757359	0.0280889	0.0140445	+	0.999901i	$0.495529\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	13.7574	0.508834
$732$	0	0
$733$	1.78680	0.0659968	0.0329984	−	0.999455i	$-0.489494\pi$
$733$	1.78680	0.0659968	0.0329984	+	0.999455i	$0.489494\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	22.2426	0.819318
$738$	0	0
$739$	−15.4853	−0.569635	−0.284818	−	0.958582i	$-0.591933\pi$
$739$	−15.4853	−0.569635	−0.284818	+	0.958582i	$0.591933\pi$
$740$	0	0
$741$	−22.6985	−0.833850
$742$	0	0
$743$	15.2132	0.558118	0.279059	−	0.960274i	$-0.409977\pi$
$743$	15.2132	0.558118	0.279059	+	0.960274i	$0.409977\pi$
$744$	0	0
$745$	12.0000	0.439646
$746$	0	0
$747$	−10.2426	−0.374759
$748$	0	0
$749$	0	0
$750$	0	0
$751$	44.9411	1.63992	0.819962	−	0.572417i	$-0.193994\pi$
$751$	44.9411	1.63992	0.819962	+	0.572417i	$0.193994\pi$
$752$	0	0
$753$	−18.7279	−0.682483
$754$	0	0
$755$	−22.4853	−0.818323
$756$	0	0
$757$	9.02944	0.328180	0.164090	−	0.986445i	$-0.447531\pi$
$757$	9.02944	0.328180	0.164090	+	0.986445i	$0.447531\pi$
$758$	0	0
$759$	18.0000	0.653359
$760$	0	0
$761$	46.2426	1.67629	0.838147	−	0.545444i	$-0.183639\pi$
$761$	46.2426	1.67629	0.838147	+	0.545444i	$0.183639\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−4.24264	−0.153393
$766$	0	0
$767$	−33.2132	−1.19926
$768$	0	0
$769$	5.00000	0.180305	0.0901523	−	0.995928i	$-0.471265\pi$
$769$	5.00000	0.180305	0.0901523	+	0.995928i	$0.471265\pi$
$770$	0	0
$771$	10.2426	0.368880
$772$	0	0
$773$	11.6985	0.420765	0.210383	−	0.977619i	$-0.432529\pi$
$773$	11.6985	0.420765	0.210383	+	0.977619i	$0.432529\pi$
$774$	0	0
$775$	−9.48528	−0.340721
$776$	0	0
$777$	0	0
$778$	0	0
$779$	29.6985	1.06406
$780$	0	0
$781$	54.0000	1.93227
$782$	0	0
$783$	−1.75736	−0.0628029
$784$	0	0
$785$	6.48528	0.231470
$786$	0	0
$787$	28.4853	1.01539	0.507695	−	0.861537i	$-0.330498\pi$
$787$	28.4853	1.01539	0.507695	+	0.861537i	$0.330498\pi$
$788$	0	0
$789$	−8.48528	−0.302084
$790$	0	0
$791$	0	0
$792$	0	0
$793$	14.5442	0.516478
$794$	0	0
$795$	−8.48528	−0.300942
$796$	0	0
$797$	−31.7574	−1.12490	−0.562452	−	0.826830i	$-0.690142\pi$
$797$	−31.7574	−1.12490	−0.562452	+	0.826830i	$0.690142\pi$
$798$	0	0
$799$	−25.4558	−0.900563
$800$	0	0
$801$	−10.2426	−0.361906
$802$	0	0
$803$	−39.2132	−1.38380
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−2.48528	−0.0874860
$808$	0	0
$809$	27.9411	0.982358	0.491179	−	0.871059i	$-0.336566\pi$
$809$	27.9411	0.982358	0.491179	+	0.871059i	$0.336566\pi$
$810$	0	0
$811$	11.9411	0.419310	0.209655	−	0.977775i	$-0.432766\pi$
$811$	11.9411	0.419310	0.209655	+	0.977775i	$0.432766\pi$
$812$	0	0
$813$	−23.4558	−0.822632
$814$	0	0
$815$	−8.00000	−0.280228
$816$	0	0
$817$	−22.6985	−0.794119
$818$	0	0
$819$	0	0
$820$	0	0
$821$	5.69848	0.198878	0.0994392	−	0.995044i	$-0.468295\pi$
$821$	5.69848	0.198878	0.0994392	+	0.995044i	$0.468295\pi$
$822$	0	0
$823$	−38.9706	−1.35843	−0.679214	−	0.733940i	$-0.737679\pi$
$823$	−38.9706	−1.35843	−0.679214	+	0.733940i	$0.737679\pi$
$824$	0	0
$825$	−4.24264	−0.147710
$826$	0	0
$827$	43.4558	1.51111	0.755554	−	0.655087i	$-0.227368\pi$
$827$	43.4558	1.51111	0.755554	+	0.655087i	$0.227368\pi$
$828$	0	0
$829$	11.0000	0.382046	0.191023	−	0.981586i	$-0.438820\pi$
$829$	11.0000	0.382046	0.191023	+	0.981586i	$0.438820\pi$
$830$	0	0
$831$	17.7279	0.614975
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−18.7279	−0.648106
$836$	0	0
$837$	−9.48528	−0.327859
$838$	0	0
$839$	25.7574	0.889243	0.444621	−	0.895719i	$-0.353338\pi$
$839$	25.7574	0.889243	0.444621	+	0.895719i	$0.353338\pi$
$840$	0	0
$841$	−25.9117	−0.893506
$842$	0	0
$843$	28.9706	0.997799
$844$	0	0
$845$	2.48528	0.0854963
$846$	0	0
$847$	0	0
$848$	0	0
$849$	23.7279	0.814340
$850$	0	0
$851$	−13.7574	−0.471596
$852$	0	0
$853$	−25.7279	−0.880907	−0.440454	−	0.897775i	$-0.645182\pi$
$853$	−25.7279	−0.880907	−0.440454	+	0.897775i	$0.645182\pi$
$854$	0	0
$855$	7.00000	0.239395
$856$	0	0
$857$	4.24264	0.144926	0.0724629	−	0.997371i	$-0.476914\pi$
$857$	4.24264	0.144926	0.0724629	+	0.997371i	$0.476914\pi$
$858$	0	0
$859$	−22.0000	−0.750630	−0.375315	−	0.926897i	$-0.622466\pi$
$859$	−22.0000	−0.750630	−0.375315	+	0.926897i	$0.622466\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−21.5147	−0.732370	−0.366185	−	0.930542i	$-0.619336\pi$
$863$	−21.5147	−0.732370	−0.366185	+	0.930542i	$0.619336\pi$
$864$	0	0
$865$	20.4853	0.696520
$866$	0	0
$867$	1.00000	0.0339618
$868$	0	0
$869$	−46.6690	−1.58314
$870$	0	0
$871$	−17.0000	−0.576023
$872$	0	0
$873$	−0.485281	−0.0164243
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−10.0000	−0.337676	−0.168838	−	0.985644i	$-0.554001\pi$
$877$	−10.0000	−0.337676	−0.168838	+	0.985644i	$0.554001\pi$
$878$	0	0
$879$	−4.97056	−0.167653
$880$	0	0
$881$	40.9706	1.38033	0.690167	−	0.723650i	$-0.257537\pi$
$881$	40.9706	1.38033	0.690167	+	0.723650i	$0.257537\pi$
$882$	0	0
$883$	5.72792	0.192760	0.0963800	−	0.995345i	$-0.469274\pi$
$883$	5.72792	0.192760	0.0963800	+	0.995345i	$0.469274\pi$
$884$	0	0
$885$	10.2426	0.344303
$886$	0	0
$887$	15.2132	0.510809	0.255405	−	0.966834i	$-0.417791\pi$
$887$	15.2132	0.510809	0.255405	+	0.966834i	$0.417791\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−4.24264	−0.142134
$892$	0	0
$893$	42.0000	1.40548
$894$	0	0
$895$	6.00000	0.200558
$896$	0	0
$897$	−13.7574	−0.459345
$898$	0	0
$899$	16.6690	0.555944
$900$	0	0
$901$	36.0000	1.19933
$902$	0	0
$903$	0	0
$904$	0	0
$905$	13.0000	0.432135
$906$	0	0
$907$	−30.2721	−1.00517	−0.502584	−	0.864528i	$-0.667617\pi$
$907$	−30.2721	−1.00517	−0.502584	+	0.864528i	$0.667617\pi$
$908$	0	0
$909$	−7.75736	−0.257295
$910$	0	0
$911$	51.2132	1.69677	0.848385	−	0.529380i	$-0.177576\pi$
$911$	51.2132	1.69677	0.848385	+	0.529380i	$0.177576\pi$
$912$	0	0
$913$	43.4558	1.43818
$914$	0	0
$915$	−4.48528	−0.148279
$916$	0	0
$917$	0	0
$918$	0	0
$919$	33.9706	1.12059	0.560293	−	0.828295i	$-0.310689\pi$
$919$	33.9706	1.12059	0.560293	+	0.828295i	$0.310689\pi$
$920$	0	0
$921$	3.24264	0.106849
$922$	0	0
$923$	−41.2721	−1.35849
$924$	0	0
$925$	3.24264	0.106617
$926$	0	0
$927$	−17.2426	−0.566323
$928$	0	0
$929$	36.7279	1.20500	0.602502	−	0.798117i	$-0.294171\pi$
$929$	36.7279	1.20500	0.602502	+	0.798117i	$0.294171\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	21.2132	0.694489
$934$	0	0
$935$	18.0000	0.588663
$936$	0	0
$937$	−18.6985	−0.610853	−0.305426	−	0.952216i	$-0.598799\pi$
$937$	−18.6985	−0.610853	−0.305426	+	0.952216i	$0.598799\pi$
$938$	0	0
$939$	23.7279	0.774331
$940$	0	0
$941$	−47.6985	−1.55493	−0.777463	−	0.628929i	$-0.783494\pi$
$941$	−47.6985	−1.55493	−0.777463	+	0.628929i	$0.783494\pi$
$942$	0	0
$943$	18.0000	0.586161
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−3.21320	−0.104415	−0.0522075	−	0.998636i	$-0.516626\pi$
$947$	−3.21320	−0.104415	−0.0522075	+	0.998636i	$0.516626\pi$
$948$	0	0
$949$	29.9706	0.972886
$950$	0	0
$951$	−24.7279	−0.801858
$952$	0	0
$953$	−27.5147	−0.891289	−0.445645	−	0.895210i	$-0.647025\pi$
$953$	−27.5147	−0.891289	−0.445645	+	0.895210i	$0.647025\pi$
$954$	0	0
$955$	−6.00000	−0.194155
$956$	0	0
$957$	7.45584	0.241013
$958$	0	0
$959$	0	0
$960$	0	0
$961$	58.9706	1.90228
$962$	0	0
$963$	12.7279	0.410152
$964$	0	0
$965$	−6.75736	−0.217527
$966$	0	0
$967$	−35.2426	−1.13333	−0.566663	−	0.823949i	$-0.691766\pi$
$967$	−35.2426	−1.13333	−0.566663	+	0.823949i	$0.691766\pi$
$968$	0	0
$969$	−29.6985	−0.954053
$970$	0	0
$971$	52.9706	1.69991	0.849953	−	0.526858i	$-0.176630\pi$
$971$	52.9706	1.69991	0.849953	+	0.526858i	$0.176630\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	3.24264	0.103848
$976$	0	0
$977$	−8.78680	−0.281115	−0.140557	−	0.990073i	$-0.544889\pi$
$977$	−8.78680	−0.281115	−0.140557	+	0.990073i	$0.544889\pi$
$978$	0	0
$979$	43.4558	1.38885
$980$	0	0
$981$	−9.48528	−0.302842
$982$	0	0
$983$	−41.6985	−1.32998	−0.664988	−	0.746854i	$-0.731563\pi$
$983$	−41.6985	−1.32998	−0.664988	+	0.746854i	$0.731563\pi$
$984$	0	0
$985$	16.2426	0.517534
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−13.7574	−0.437459
$990$	0	0
$991$	14.9411	0.474620	0.237310	−	0.971434i	$-0.423734\pi$
$991$	14.9411	0.474620	0.237310	+	0.971434i	$0.423734\pi$
$992$	0	0
$993$	17.0000	0.539479
$994$	0	0
$995$	−10.4853	−0.332406
$996$	0	0
$997$	−25.7279	−0.814811	−0.407406	−	0.913247i	$-0.633567\pi$
$997$	−25.7279	−0.814811	−0.407406	+	0.913247i	$0.633567\pi$
$998$	0	0
$999$	3.24264	0.102593

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2940.2.a.r.1.1		2
3.2	odd	2		8820.2.a.bk.1.2		2
7.2	even	3		420.2.q.d.361.2	yes	4
7.3	odd	6		2940.2.q.q.961.2		4
7.4	even	3		420.2.q.d.121.2	✓	4
7.5	odd	6		2940.2.q.q.361.2		4
7.6	odd	2		2940.2.a.p.1.1		2
21.2	odd	6		1260.2.s.e.361.2		4
21.11	odd	6		1260.2.s.e.541.2		4
21.20	even	2		8820.2.a.bf.1.2		2
28.11	odd	6		1680.2.bg.t.961.1		4
28.23	odd	6		1680.2.bg.t.1201.1		4
35.2	odd	12		2100.2.bc.f.949.4		8
35.4	even	6		2100.2.q.k.1801.1		4
35.9	even	6		2100.2.q.k.1201.1		4
35.18	odd	12		2100.2.bc.f.1549.4		8
35.23	odd	12		2100.2.bc.f.949.1		8
35.32	odd	12		2100.2.bc.f.1549.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
420.2.q.d.121.2	✓	4	7.4	even	3
420.2.q.d.361.2	yes	4	7.2	even	3
1260.2.s.e.361.2		4	21.2	odd	6
1260.2.s.e.541.2		4	21.11	odd	6
1680.2.bg.t.961.1		4	28.11	odd	6
1680.2.bg.t.1201.1		4	28.23	odd	6
2100.2.q.k.1201.1		4	35.9	even	6
2100.2.q.k.1801.1		4	35.4	even	6
2100.2.bc.f.949.1		8	35.23	odd	12
2100.2.bc.f.949.4		8	35.2	odd	12
2100.2.bc.f.1549.1		8	35.32	odd	12
2100.2.bc.f.1549.4		8	35.18	odd	12
2940.2.a.p.1.1		2	7.6	odd	2
2940.2.a.r.1.1		2	1.1	even	1	trivial
2940.2.q.q.361.2		4	7.5	odd	6
2940.2.q.q.961.2		4	7.3	odd	6
8820.2.a.bf.1.2		2	21.20	even	2
8820.2.a.bk.1.2		2	3.2	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:	\( N \)	\(=\)	\( 2940 = 2^{2} \cdot 3 \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2940.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -1.00000 q^{5} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -1.00000 q^{5} +1.00000 q^{9} -4.24264 q^{11} +3.24264 q^{13} -1.00000 q^{15} +4.24264 q^{17} -7.00000 q^{19} -4.24264 q^{23} +1.00000 q^{25} +1.00000 q^{27} -1.75736 q^{29} -9.48528 q^{31} -4.24264 q^{33} +3.24264 q^{37} +3.24264 q^{39} -4.24264 q^{41} +3.24264 q^{43} -1.00000 q^{45} -6.00000 q^{47} +4.24264 q^{51} +8.48528 q^{53} +4.24264 q^{55} -7.00000 q^{57} -10.2426 q^{59} +4.48528 q^{61} -3.24264 q^{65} -5.24264 q^{67} -4.24264 q^{69} -12.7279 q^{71} +9.24264 q^{73} +1.00000 q^{75} +11.0000 q^{79} +1.00000 q^{81} -10.2426 q^{83} -4.24264 q^{85} -1.75736 q^{87} -10.2426 q^{89} -9.48528 q^{93} +7.00000 q^{95} -0.485281 q^{97} -4.24264 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} - 2 q^{5} + 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^3 - 2 * q^5 + 2 * q^9	\( 2 q + 2 q^{3} - 2 q^{5} + 2 q^{9} - 2 q^{13} - 2 q^{15} - 14 q^{19} + 2 q^{25} + 2 q^{27} - 12 q^{29} - 2 q^{31} - 2 q^{37} - 2 q^{39} - 2 q^{43} - 2 q^{45} - 12 q^{47} - 14 q^{57} - 12 q^{59} - 8 q^{61} + 2 q^{65} - 2 q^{67} + 10 q^{73} + 2 q^{75} + 22 q^{79} + 2 q^{81} - 12 q^{83} - 12 q^{87} - 12 q^{89} - 2 q^{93} + 14 q^{95} + 16 q^{97}+O(q^{100}) \) 2 * q + 2 * q^3 - 2 * q^5 + 2 * q^9 - 2 * q^13 - 2 * q^15 - 14 * q^19 + 2 * q^25 + 2 * q^27 - 12 * q^29 - 2 * q^31 - 2 * q^37 - 2 * q^39 - 2 * q^43 - 2 * q^45 - 12 * q^47 - 14 * q^57 - 12 * q^59 - 8 * q^61 + 2 * q^65 - 2 * q^67 + 10 * q^73 + 2 * q^75 + 22 * q^79 + 2 * q^81 - 12 * q^83 - 12 * q^87 - 12 * q^89 - 2 * q^93 + 14 * q^95 + 16 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more