LMFDB - Embedded newform 2601.2.a.r.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2601,2,Mod(1,2601)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2601, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("2601.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2601 = 3^{2} \cdot 17^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2601.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:	$20.7690895657$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{13}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 3 $ x^2 - x - 3
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 289)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.30278$ of defining polynomial
Character	$\chi$	$=$	2601.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.30278 q^{2} -0.302776 q^{4} +1.30278 q^{5} +3.30278 q^{7} +3.00000 q^{8} +O(q^{10})$	$q-1.30278 q^{2} -0.302776 q^{4} +1.30278 q^{5} +3.30278 q^{7} +3.00000 q^{8} -1.69722 q^{10} -3.00000 q^{11} +0.302776 q^{13} -4.30278 q^{14} -3.30278 q^{16} -4.90833 q^{19} -0.394449 q^{20} +3.90833 q^{22} +1.30278 q^{23} -3.30278 q^{25} -0.394449 q^{26} -1.00000 q^{28} +0.908327 q^{29} -3.60555 q^{31} -1.69722 q^{32} +4.30278 q^{35} -6.60555 q^{37} +6.39445 q^{38} +3.90833 q^{40} -6.00000 q^{41} -9.60555 q^{43} +0.908327 q^{44} -1.69722 q^{46} -3.00000 q^{47} +3.90833 q^{49} +4.30278 q^{50} -0.0916731 q^{52} -12.9083 q^{53} -3.90833 q^{55} +9.90833 q^{56} -1.18335 q^{58} -6.00000 q^{59} +12.8167 q^{61} +4.69722 q^{62} +8.81665 q^{64} +0.394449 q^{65} +5.39445 q^{67} -5.60555 q^{70} -11.2111 q^{71} +7.60555 q^{73} +8.60555 q^{74} +1.48612 q^{76} -9.90833 q^{77} -3.21110 q^{79} -4.30278 q^{80} +7.81665 q^{82} +15.5139 q^{83} +12.5139 q^{86} -9.00000 q^{88} +11.2111 q^{89} +1.00000 q^{91} -0.394449 q^{92} +3.90833 q^{94} -6.39445 q^{95} -0.0916731 q^{97} -5.09167 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + q^{2} + 3 q^{4} - q^{5} + 3 q^{7} + 6 q^{8}+O(q^{10}) $ 2 * q + q^2 + 3 * q^4 - q^5 + 3 * q^7 + 6 * q^8	$ 2 q + q^{2} + 3 q^{4} - q^{5} + 3 q^{7} + 6 q^{8} - 7 q^{10} - 6 q^{11} - 3 q^{13} - 5 q^{14} - 3 q^{16} + q^{19} - 8 q^{20} - 3 q^{22} - q^{23} - 3 q^{25} - 8 q^{26} - 2 q^{28} - 9 q^{29} - 7 q^{32} + 5 q^{35} - 6 q^{37} + 20 q^{38} - 3 q^{40} - 12 q^{41} - 12 q^{43} - 9 q^{44} - 7 q^{46} - 6 q^{47} - 3 q^{49} + 5 q^{50} - 11 q^{52} - 15 q^{53} + 3 q^{55} + 9 q^{56} - 24 q^{58} - 12 q^{59} + 4 q^{61} + 13 q^{62} - 4 q^{64} + 8 q^{65} + 18 q^{67} - 4 q^{70} - 8 q^{71} + 8 q^{73} + 10 q^{74} + 21 q^{76} - 9 q^{77} + 8 q^{79} - 5 q^{80} - 6 q^{82} + 13 q^{83} + 7 q^{86} - 18 q^{88} + 8 q^{89} + 2 q^{91} - 8 q^{92} - 3 q^{94} - 20 q^{95} - 11 q^{97} - 21 q^{98}+O(q^{100}) $ 2 * q + q^2 + 3 * q^4 - q^5 + 3 * q^7 + 6 * q^8 - 7 * q^10 - 6 * q^11 - 3 * q^13 - 5 * q^14 - 3 * q^16 + q^19 - 8 * q^20 - 3 * q^22 - q^23 - 3 * q^25 - 8 * q^26 - 2 * q^28 - 9 * q^29 - 7 * q^32 + 5 * q^35 - 6 * q^37 + 20 * q^38 - 3 * q^40 - 12 * q^41 - 12 * q^43 - 9 * q^44 - 7 * q^46 - 6 * q^47 - 3 * q^49 + 5 * q^50 - 11 * q^52 - 15 * q^53 + 3 * q^55 + 9 * q^56 - 24 * q^58 - 12 * q^59 + 4 * q^61 + 13 * q^62 - 4 * q^64 + 8 * q^65 + 18 * q^67 - 4 * q^70 - 8 * q^71 + 8 * q^73 + 10 * q^74 + 21 * q^76 - 9 * q^77 + 8 * q^79 - 5 * q^80 - 6 * q^82 + 13 * q^83 + 7 * q^86 - 18 * q^88 + 8 * q^89 + 2 * q^91 - 8 * q^92 - 3 * q^94 - 20 * q^95 - 11 * q^97 - 21 * q^98

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 $

$2$	−1.30278	−0.921201	−0.460601	−	0.887607i	$-0.652366\pi$
$2$	−1.30278	−0.921201	−0.460601	+	0.887607i	$0.652366\pi$
$3$	0	0
$3$	0	0
$4$	−0.302776	−0.151388
$5$	1.30278	0.582619	0.291309	−	0.956629i	$-0.405909\pi$
$5$	1.30278	0.582619	0.291309	+	0.956629i	$0.405909\pi$
$6$	0	0
$7$	3.30278	1.24833	0.624166	−	0.781292i	$-0.285439\pi$
$7$	3.30278	1.24833	0.624166	+	0.781292i	$0.285439\pi$
$8$	3.00000	1.06066
$9$	0	0
$10$	−1.69722	−0.536709
$11$	−3.00000	−0.904534	−0.452267	−	0.891883i	$-0.649385\pi$
$11$	−3.00000	−0.904534	−0.452267	+	0.891883i	$0.649385\pi$
$12$	0	0
$13$	0.302776	0.0839749	0.0419874	−	0.999118i	$-0.486631\pi$
$13$	0.302776	0.0839749	0.0419874	+	0.999118i	$0.486631\pi$
$14$	−4.30278	−1.14997
$15$	0	0
$16$	−3.30278	−0.825694
$17$	0	0
$17$	0	0
$18$	0	0
$19$	−4.90833	−1.12605	−0.563024	−	0.826441i	$-0.690362\pi$
$19$	−4.90833	−1.12605	−0.563024	+	0.826441i	$0.690362\pi$
$20$	−0.394449	−0.0882014
$21$	0	0
$22$	3.90833	0.833258
$23$	1.30278	0.271647	0.135824	−	0.990733i	$-0.456632\pi$
$23$	1.30278	0.271647	0.135824	+	0.990733i	$0.456632\pi$
$24$	0	0
$25$	−3.30278	−0.660555
$26$	−0.394449	−0.0773578
$27$	0	0
$28$	−1.00000	−0.188982
$29$	0.908327	0.168672	0.0843360	−	0.996437i	$-0.473123\pi$
$29$	0.908327	0.168672	0.0843360	+	0.996437i	$0.473123\pi$
$30$	0	0
$31$	−3.60555	−0.647576	−0.323788	−	0.946130i	$-0.604956\pi$
$31$	−3.60555	−0.647576	−0.323788	+	0.946130i	$0.604956\pi$
$32$	−1.69722	−0.300030
$33$	0	0
$34$	0	0
$35$	4.30278	0.727302
$36$	0	0
$37$	−6.60555	−1.08595	−0.542973	−	0.839750i	$-0.682701\pi$
$37$	−6.60555	−1.08595	−0.542973	+	0.839750i	$0.682701\pi$
$38$	6.39445	1.03732
$39$	0	0
$40$	3.90833	0.617961
$41$	−6.00000	−0.937043	−0.468521	−	0.883452i	$-0.655213\pi$
$41$	−6.00000	−0.937043	−0.468521	+	0.883452i	$0.655213\pi$
$42$	0	0
$43$	−9.60555	−1.46483	−0.732416	−	0.680857i	$-0.761608\pi$
$43$	−9.60555	−1.46483	−0.732416	+	0.680857i	$0.761608\pi$
$44$	0.908327	0.136935
$45$	0	0
$46$	−1.69722	−0.250242
$47$	−3.00000	−0.437595	−0.218797	−	0.975770i	$-0.570213\pi$
$47$	−3.00000	−0.437595	−0.218797	+	0.975770i	$0.570213\pi$
$48$	0	0
$49$	3.90833	0.558332
$50$	4.30278	0.608504
$51$	0	0
$52$	−0.0916731	−0.0127128
$53$	−12.9083	−1.77310	−0.886548	−	0.462638i	$-0.846903\pi$
$53$	−12.9083	−1.77310	−0.886548	+	0.462638i	$0.846903\pi$
$54$	0	0
$55$	−3.90833	−0.526999
$56$	9.90833	1.32406
$57$	0	0
$58$	−1.18335	−0.155381
$59$	−6.00000	−0.781133	−0.390567	−	0.920575i	$-0.627721\pi$
$59$	−6.00000	−0.781133	−0.390567	+	0.920575i	$0.627721\pi$
$60$	0	0
$61$	12.8167	1.64100	0.820502	−	0.571643i	$-0.193694\pi$
$61$	12.8167	1.64100	0.820502	+	0.571643i	$0.193694\pi$
$62$	4.69722	0.596548
$63$	0	0
$64$	8.81665	1.10208
$65$	0.394449	0.0489253
$66$	0	0
$67$	5.39445	0.659037	0.329518	−	0.944149i	$-0.393114\pi$
$67$	5.39445	0.659037	0.329518	+	0.944149i	$0.393114\pi$
$68$	0	0
$69$	0	0
$70$	−5.60555	−0.669992
$71$	−11.2111	−1.33051	−0.665257	−	0.746615i	$-0.731678\pi$
$71$	−11.2111	−1.33051	−0.665257	+	0.746615i	$0.731678\pi$
$72$	0	0
$73$	7.60555	0.890162	0.445081	−	0.895490i	$-0.353175\pi$
$73$	7.60555	0.890162	0.445081	+	0.895490i	$0.353175\pi$
$74$	8.60555	1.00038
$75$	0	0
$76$	1.48612	0.170470
$77$	−9.90833	−1.12916
$78$	0	0
$79$	−3.21110	−0.361277	−0.180639	−	0.983550i	$-0.557816\pi$
$79$	−3.21110	−0.361277	−0.180639	+	0.983550i	$0.557816\pi$
$80$	−4.30278	−0.481065
$81$	0	0
$82$	7.81665	0.863205
$83$	15.5139	1.70287	0.851435	−	0.524461i	$-0.175733\pi$
$83$	15.5139	1.70287	0.851435	+	0.524461i	$0.175733\pi$
$84$	0	0
$85$	0	0
$86$	12.5139	1.34941
$87$	0	0
$88$	−9.00000	−0.959403
$89$	11.2111	1.18837	0.594187	−	0.804327i	$-0.297474\pi$
$89$	11.2111	1.18837	0.594187	+	0.804327i	$0.297474\pi$
$90$	0	0
$91$	1.00000	0.104828
$92$	−0.394449	−0.0411241
$93$	0	0
$94$	3.90833	0.403113
$95$	−6.39445	−0.656057
$96$	0	0
$97$	−0.0916731	−0.00930799	−0.00465400	−	0.999989i	$-0.501481\pi$
$97$	−0.0916731	−0.00930799	−0.00465400	+	0.999989i	$0.501481\pi$
$98$	−5.09167	−0.514337
$99$	0	0
$100$	1.00000	0.100000
$101$	11.6056	1.15480	0.577398	−	0.816463i	$-0.304068\pi$
$101$	11.6056	1.15480	0.577398	+	0.816463i	$0.304068\pi$
$102$	0	0
$103$	2.00000	0.197066	0.0985329	−	0.995134i	$-0.468585\pi$
$103$	2.00000	0.197066	0.0985329	+	0.995134i	$0.468585\pi$
$104$	0.908327	0.0890688
$105$	0	0
$106$	16.8167	1.63338
$107$	−2.21110	−0.213755	−0.106878	−	0.994272i	$-0.534085\pi$
$107$	−2.21110	−0.213755	−0.106878	+	0.994272i	$0.534085\pi$
$108$	0	0
$109$	−0.211103	−0.0202200	−0.0101100	−	0.999949i	$-0.503218\pi$
$109$	−0.211103	−0.0202200	−0.0101100	+	0.999949i	$0.503218\pi$
$110$	5.09167	0.485472
$111$	0	0
$112$	−10.9083	−1.03074
$113$	−6.51388	−0.612774	−0.306387	−	0.951907i	$-0.599120\pi$
$113$	−6.51388	−0.612774	−0.306387	+	0.951907i	$0.599120\pi$
$114$	0	0
$115$	1.69722	0.158267
$116$	−0.275019	−0.0255349
$117$	0	0
$118$	7.81665	0.719581
$119$	0	0
$120$	0	0
$121$	−2.00000	−0.181818
$122$	−16.6972	−1.51170
$123$	0	0
$124$	1.09167	0.0980351
$125$	−10.8167	−0.967471
$126$	0	0
$127$	6.81665	0.604880	0.302440	−	0.953168i	$-0.402199\pi$
$127$	6.81665	0.604880	0.302440	+	0.953168i	$0.402199\pi$
$128$	−8.09167	−0.715210
$129$	0	0
$130$	−0.513878	−0.0450701
$131$	−2.60555	−0.227648	−0.113824	−	0.993501i	$-0.536310\pi$
$131$	−2.60555	−0.227648	−0.113824	+	0.993501i	$0.536310\pi$
$132$	0	0
$133$	−16.2111	−1.40568
$134$	−7.02776	−0.607106
$135$	0	0
$136$	0	0
$137$	−19.8167	−1.69305	−0.846525	−	0.532348i	$-0.821310\pi$
$137$	−19.8167	−1.69305	−0.846525	+	0.532348i	$0.821310\pi$
$138$	0	0
$139$	1.48612	0.126051	0.0630256	−	0.998012i	$-0.479925\pi$
$139$	1.48612	0.126051	0.0630256	+	0.998012i	$0.479925\pi$
$140$	−1.30278	−0.110105
$141$	0	0
$142$	14.6056	1.22567
$143$	−0.908327	−0.0759581
$144$	0	0
$145$	1.18335	0.0982716
$146$	−9.90833	−0.820019
$147$	0	0
$148$	2.00000	0.164399
$149$	8.60555	0.704994	0.352497	−	0.935813i	$-0.385333\pi$
$149$	8.60555	0.704994	0.352497	+	0.935813i	$0.385333\pi$
$150$	0	0
$151$	5.00000	0.406894	0.203447	−	0.979086i	$-0.434786\pi$
$151$	5.00000	0.406894	0.203447	+	0.979086i	$0.434786\pi$
$152$	−14.7250	−1.19435
$153$	0	0
$154$	12.9083	1.04018
$155$	−4.69722	−0.377290
$156$	0	0
$157$	9.30278	0.742442	0.371221	−	0.928544i	$-0.378939\pi$
$157$	9.30278	0.742442	0.371221	+	0.928544i	$0.378939\pi$
$158$	4.18335	0.332809
$159$	0	0
$160$	−2.21110	−0.174803
$161$	4.30278	0.339106
$162$	0	0
$163$	−10.3944	−0.814156	−0.407078	−	0.913393i	$-0.633452\pi$
$163$	−10.3944	−0.814156	−0.407078	+	0.913393i	$0.633452\pi$
$164$	1.81665	0.141857
$165$	0	0
$166$	−20.2111	−1.56869
$167$	−22.3028	−1.72584	−0.862920	−	0.505340i	$-0.831367\pi$
$167$	−22.3028	−1.72584	−0.862920	+	0.505340i	$0.831367\pi$
$168$	0	0
$169$	−12.9083	−0.992948
$170$	0	0
$171$	0	0
$172$	2.90833	0.221758
$173$	0.394449	0.0299894	0.0149947	−	0.999888i	$-0.495227\pi$
$173$	0.394449	0.0299894	0.0149947	+	0.999888i	$0.495227\pi$
$174$	0	0
$175$	−10.9083	−0.824592
$176$	9.90833	0.746868
$177$	0	0
$178$	−14.6056	−1.09473
$179$	−11.7250	−0.876366	−0.438183	−	0.898886i	$-0.644378\pi$
$179$	−11.7250	−0.876366	−0.438183	+	0.898886i	$0.644378\pi$
$180$	0	0
$181$	−7.78890	−0.578944	−0.289472	−	0.957186i	$-0.593480\pi$
$181$	−7.78890	−0.578944	−0.289472	+	0.957186i	$0.593480\pi$
$182$	−1.30278	−0.0965682
$183$	0	0
$184$	3.90833	0.288126
$185$	−8.60555	−0.632693
$186$	0	0
$187$	0	0
$188$	0.908327	0.0662465
$189$	0	0
$190$	8.33053	0.604360
$191$	7.69722	0.556952	0.278476	−	0.960443i	$-0.410171\pi$
$191$	7.69722	0.556952	0.278476	+	0.960443i	$0.410171\pi$
$192$	0	0
$193$	−22.6333	−1.62918	−0.814591	−	0.580036i	$-0.803038\pi$
$193$	−22.6333	−1.62918	−0.814591	+	0.580036i	$0.803038\pi$
$194$	0.119429	0.00857454
$195$	0	0
$196$	−1.18335	−0.0845247
$197$	−3.90833	−0.278457	−0.139228	−	0.990260i	$-0.544462\pi$
$197$	−3.90833	−0.278457	−0.139228	+	0.990260i	$0.544462\pi$
$198$	0	0
$199$	−3.60555	−0.255591	−0.127795	−	0.991801i	$-0.540790\pi$
$199$	−3.60555	−0.255591	−0.127795	+	0.991801i	$0.540790\pi$
$200$	−9.90833	−0.700625
$201$	0	0
$202$	−15.1194	−1.06380
$203$	3.00000	0.210559
$204$	0	0
$205$	−7.81665	−0.545939
$206$	−2.60555	−0.181537
$207$	0	0
$208$	−1.00000	−0.0693375
$209$	14.7250	1.01855
$210$	0	0
$211$	−19.9083	−1.37055	−0.685273	−	0.728286i	$-0.740317\pi$
$211$	−19.9083	−1.37055	−0.685273	+	0.728286i	$0.740317\pi$
$212$	3.90833	0.268425
$213$	0	0
$214$	2.88057	0.196912
$215$	−12.5139	−0.853439
$216$	0	0
$217$	−11.9083	−0.808390
$218$	0.275019	0.0186267
$219$	0	0
$220$	1.18335	0.0797812
$221$	0	0
$222$	0	0
$223$	−9.48612	−0.635238	−0.317619	−	0.948218i	$-0.602883\pi$
$223$	−9.48612	−0.635238	−0.317619	+	0.948218i	$0.602883\pi$
$224$	−5.60555	−0.374537
$225$	0	0
$226$	8.48612	0.564488
$227$	4.81665	0.319693	0.159846	−	0.987142i	$-0.448900\pi$
$227$	4.81665	0.319693	0.159846	+	0.987142i	$0.448900\pi$
$228$	0	0
$229$	18.4222	1.21737	0.608687	−	0.793411i	$-0.291697\pi$
$229$	18.4222	1.21737	0.608687	+	0.793411i	$0.291697\pi$
$230$	−2.21110	−0.145796
$231$	0	0
$232$	2.72498	0.178904
$233$	−25.8167	−1.69131	−0.845653	−	0.533734i	$-0.820788\pi$
$233$	−25.8167	−1.69131	−0.845653	+	0.533734i	$0.820788\pi$
$234$	0	0
$235$	−3.90833	−0.254951
$236$	1.81665	0.118254
$237$	0	0
$238$	0	0
$239$	−8.21110	−0.531132	−0.265566	−	0.964093i	$-0.585559\pi$
$239$	−8.21110	−0.531132	−0.265566	+	0.964093i	$0.585559\pi$
$240$	0	0
$241$	8.51388	0.548427	0.274214	−	0.961669i	$-0.411582\pi$
$241$	8.51388	0.548427	0.274214	+	0.961669i	$0.411582\pi$
$242$	2.60555	0.167491
$243$	0	0
$244$	−3.88057	−0.248428
$245$	5.09167	0.325295
$246$	0	0
$247$	−1.48612	−0.0945597
$248$	−10.8167	−0.686858
$249$	0	0
$250$	14.0917	0.891236
$251$	5.09167	0.321384	0.160692	−	0.987005i	$-0.448627\pi$
$251$	5.09167	0.321384	0.160692	+	0.987005i	$0.448627\pi$
$252$	0	0
$253$	−3.90833	−0.245714
$254$	−8.88057	−0.557217
$255$	0	0
$256$	−7.09167	−0.443230
$257$	5.72498	0.357114	0.178557	−	0.983930i	$-0.442857\pi$
$257$	5.72498	0.357114	0.178557	+	0.983930i	$0.442857\pi$
$258$	0	0
$259$	−21.8167	−1.35562
$260$	−0.119429	−0.00740670
$261$	0	0
$262$	3.39445	0.209710
$263$	−2.48612	−0.153301	−0.0766504	−	0.997058i	$-0.524423\pi$
$263$	−2.48612	−0.153301	−0.0766504	+	0.997058i	$0.524423\pi$
$264$	0	0
$265$	−16.8167	−1.03304
$266$	21.1194	1.29492
$267$	0	0
$268$	−1.63331	−0.0997701
$269$	24.1194	1.47059	0.735294	−	0.677749i	$-0.237044\pi$
$269$	24.1194	1.47059	0.735294	+	0.677749i	$0.237044\pi$
$270$	0	0
$271$	27.5416	1.67304	0.836518	−	0.547940i	$-0.184588\pi$
$271$	27.5416	1.67304	0.836518	+	0.547940i	$0.184588\pi$
$272$	0	0
$273$	0	0
$274$	25.8167	1.55964
$275$	9.90833	0.597495
$276$	0	0
$277$	−4.00000	−0.240337	−0.120168	−	0.992754i	$-0.538343\pi$
$277$	−4.00000	−0.240337	−0.120168	+	0.992754i	$0.538343\pi$
$278$	−1.93608	−0.116119
$279$	0	0
$280$	12.9083	0.771420
$281$	25.5416	1.52369	0.761843	−	0.647762i	$-0.224295\pi$
$281$	25.5416	1.52369	0.761843	+	0.647762i	$0.224295\pi$
$282$	0	0
$283$	20.0000	1.18888	0.594438	−	0.804141i	$-0.297374\pi$
$283$	20.0000	1.18888	0.594438	+	0.804141i	$0.297374\pi$
$284$	3.39445	0.201423
$285$	0	0
$286$	1.18335	0.0699727
$287$	−19.8167	−1.16974
$288$	0	0
$289$	0	0
$290$	−1.54163	−0.0905279
$291$	0	0
$292$	−2.30278	−0.134760
$293$	4.18335	0.244394	0.122197	−	0.992506i	$-0.461006\pi$
$293$	4.18335	0.244394	0.122197	+	0.992506i	$0.461006\pi$
$294$	0	0
$295$	−7.81665	−0.455103
$296$	−19.8167	−1.15182
$297$	0	0
$298$	−11.2111	−0.649442
$299$	0.394449	0.0228116
$300$	0	0
$301$	−31.7250	−1.82860
$302$	−6.51388	−0.374832
$303$	0	0
$304$	16.2111	0.929770
$305$	16.6972	0.956080
$306$	0	0
$307$	−13.1194	−0.748765	−0.374383	−	0.927274i	$-0.622145\pi$
$307$	−13.1194	−0.748765	−0.374383	+	0.927274i	$0.622145\pi$
$308$	3.00000	0.170941
$309$	0	0
$310$	6.11943	0.347560
$311$	−33.5139	−1.90040	−0.950199	−	0.311644i	$-0.899120\pi$
$311$	−33.5139	−1.90040	−0.950199	+	0.311644i	$0.899120\pi$
$312$	0	0
$313$	−11.8167	−0.667917	−0.333958	−	0.942588i	$-0.608385\pi$
$313$	−11.8167	−0.667917	−0.333958	+	0.942588i	$0.608385\pi$
$314$	−12.1194	−0.683939
$315$	0	0
$316$	0.972244	0.0546930
$317$	27.2389	1.52989	0.764943	−	0.644098i	$-0.222767\pi$
$317$	27.2389	1.52989	0.764943	+	0.644098i	$0.222767\pi$
$318$	0	0
$319$	−2.72498	−0.152570
$320$	11.4861	0.642094
$321$	0	0
$322$	−5.60555	−0.312385
$323$	0	0
$324$	0	0
$325$	−1.00000	−0.0554700
$326$	13.5416	0.750002
$327$	0	0
$328$	−18.0000	−0.993884
$329$	−9.90833	−0.546264
$330$	0	0
$331$	3.30278	0.181537	0.0907685	−	0.995872i	$-0.471068\pi$
$331$	3.30278	0.181537	0.0907685	+	0.995872i	$0.471068\pi$
$332$	−4.69722	−0.257794
$333$	0	0
$334$	29.0555	1.58985
$335$	7.02776	0.383967
$336$	0	0
$337$	4.48612	0.244375	0.122187	−	0.992507i	$-0.461009\pi$
$337$	4.48612	0.244375	0.122187	+	0.992507i	$0.461009\pi$
$338$	16.8167	0.914705
$339$	0	0
$340$	0	0
$341$	10.8167	0.585755
$342$	0	0
$343$	−10.2111	−0.551348
$344$	−28.8167	−1.55369
$345$	0	0
$346$	−0.513878	−0.0276263
$347$	13.6972	0.735306	0.367653	−	0.929963i	$-0.380162\pi$
$347$	13.6972	0.735306	0.367653	+	0.929963i	$0.380162\pi$
$348$	0	0
$349$	13.6056	0.728288	0.364144	−	0.931343i	$-0.381362\pi$
$349$	13.6056	0.728288	0.364144	+	0.931343i	$0.381362\pi$
$350$	14.2111	0.759615
$351$	0	0
$352$	5.09167	0.271387
$353$	−3.00000	−0.159674	−0.0798369	−	0.996808i	$-0.525440\pi$
$353$	−3.00000	−0.159674	−0.0798369	+	0.996808i	$0.525440\pi$
$354$	0	0
$355$	−14.6056	−0.775182
$356$	−3.39445	−0.179905
$357$	0	0
$358$	15.2750	0.807310
$359$	21.9083	1.15628	0.578139	−	0.815939i	$-0.303779\pi$
$359$	21.9083	1.15628	0.578139	+	0.815939i	$0.303779\pi$
$360$	0	0
$361$	5.09167	0.267983
$362$	10.1472	0.533324
$363$	0	0
$364$	−0.302776	−0.0158698
$365$	9.90833	0.518626
$366$	0	0
$367$	17.3944	0.907983	0.453991	−	0.891006i	$-0.350000\pi$
$367$	17.3944	0.907983	0.453991	+	0.891006i	$0.350000\pi$
$368$	−4.30278	−0.224298
$369$	0	0
$370$	11.2111	0.582837
$371$	−42.6333	−2.21341
$372$	0	0
$373$	−6.21110	−0.321599	−0.160799	−	0.986987i	$-0.551407\pi$
$373$	−6.21110	−0.321599	−0.160799	+	0.986987i	$0.551407\pi$
$374$	0	0
$375$	0	0
$376$	−9.00000	−0.464140
$377$	0.275019	0.0141642
$378$	0	0
$379$	−15.6056	−0.801603	−0.400802	−	0.916165i	$-0.631268\pi$
$379$	−15.6056	−0.801603	−0.400802	+	0.916165i	$0.631268\pi$
$380$	1.93608	0.0993190
$381$	0	0
$382$	−10.0278	−0.513065
$383$	6.90833	0.352999	0.176500	−	0.984301i	$-0.443523\pi$
$383$	6.90833	0.352999	0.176500	+	0.984301i	$0.443523\pi$
$384$	0	0
$385$	−12.9083	−0.657869
$386$	29.4861	1.50080
$387$	0	0
$388$	0.0277564	0.00140912
$389$	30.6333	1.55317	0.776585	−	0.630012i	$-0.216950\pi$
$389$	30.6333	1.55317	0.776585	+	0.630012i	$0.216950\pi$
$390$	0	0
$391$	0	0
$392$	11.7250	0.592201
$393$	0	0
$394$	5.09167	0.256515
$395$	−4.18335	−0.210487
$396$	0	0
$397$	7.60555	0.381712	0.190856	−	0.981618i	$-0.438874\pi$
$397$	7.60555	0.381712	0.190856	+	0.981618i	$0.438874\pi$
$398$	4.69722	0.235451
$399$	0	0
$400$	10.9083	0.545416
$401$	19.3028	0.963935	0.481967	−	0.876189i	$-0.339922\pi$
$401$	19.3028	0.963935	0.481967	+	0.876189i	$0.339922\pi$
$402$	0	0
$403$	−1.09167	−0.0543801
$404$	−3.51388	−0.174822
$405$	0	0
$406$	−3.90833	−0.193967
$407$	19.8167	0.982275
$408$	0	0
$409$	−32.8167	−1.62268	−0.811340	−	0.584575i	$-0.801261\pi$
$409$	−32.8167	−1.62268	−0.811340	+	0.584575i	$0.801261\pi$
$410$	10.1833	0.502920
$411$	0	0
$412$	−0.605551	−0.0298334
$413$	−19.8167	−0.975114
$414$	0	0
$415$	20.2111	0.992124
$416$	−0.513878	−0.0251950
$417$	0	0
$418$	−19.1833	−0.938288
$419$	−26.2111	−1.28050	−0.640248	−	0.768168i	$-0.721168\pi$
$419$	−26.2111	−1.28050	−0.640248	+	0.768168i	$0.721168\pi$
$420$	0	0
$421$	18.9361	0.922888	0.461444	−	0.887169i	$-0.347331\pi$
$421$	18.9361	0.922888	0.461444	+	0.887169i	$0.347331\pi$
$422$	25.9361	1.26255
$423$	0	0
$424$	−38.7250	−1.88065
$425$	0	0
$426$	0	0
$427$	42.3305	2.04852
$428$	0.669468	0.0323600
$429$	0	0
$430$	16.3028	0.786190
$431$	−21.9083	−1.05529	−0.527643	−	0.849466i	$-0.676924\pi$
$431$	−21.9083	−1.05529	−0.527643	+	0.849466i	$0.676924\pi$
$432$	0	0
$433$	−28.0000	−1.34559	−0.672797	−	0.739827i	$-0.734907\pi$
$433$	−28.0000	−1.34559	−0.672797	+	0.739827i	$0.734907\pi$
$434$	15.5139	0.744690
$435$	0	0
$436$	0.0639167	0.00306106
$437$	−6.39445	−0.305888
$438$	0	0
$439$	15.0278	0.717236	0.358618	−	0.933484i	$-0.383248\pi$
$439$	15.0278	0.717236	0.358618	+	0.933484i	$0.383248\pi$
$440$	−11.7250	−0.558967
$441$	0	0
$442$	0	0
$443$	−15.2389	−0.724020	−0.362010	−	0.932174i	$-0.617909\pi$
$443$	−15.2389	−0.724020	−0.362010	+	0.932174i	$0.617909\pi$
$444$	0	0
$445$	14.6056	0.692370
$446$	12.3583	0.585182
$447$	0	0
$448$	29.1194	1.37576
$449$	−9.51388	−0.448988	−0.224494	−	0.974476i	$-0.572073\pi$
$449$	−9.51388	−0.448988	−0.224494	+	0.974476i	$0.572073\pi$
$450$	0	0
$451$	18.0000	0.847587
$452$	1.97224	0.0927665
$453$	0	0
$454$	−6.27502	−0.294501
$455$	1.30278	0.0610751
$456$	0	0
$457$	8.51388	0.398262	0.199131	−	0.979973i	$-0.436188\pi$
$457$	8.51388	0.398262	0.199131	+	0.979973i	$0.436188\pi$
$458$	−24.0000	−1.12145
$459$	0	0
$460$	−0.513878	−0.0239597
$461$	−16.4222	−0.764858	−0.382429	−	0.923985i	$-0.624912\pi$
$461$	−16.4222	−0.764858	−0.382429	+	0.923985i	$0.624912\pi$
$462$	0	0
$463$	−5.81665	−0.270323	−0.135161	−	0.990824i	$-0.543155\pi$
$463$	−5.81665	−0.270323	−0.135161	+	0.990824i	$0.543155\pi$
$464$	−3.00000	−0.139272
$465$	0	0
$466$	33.6333	1.55803
$467$	−14.0917	−0.652085	−0.326042	−	0.945355i	$-0.605715\pi$
$467$	−14.0917	−0.652085	−0.326042	+	0.945355i	$0.605715\pi$
$468$	0	0
$469$	17.8167	0.822697
$470$	5.09167	0.234861
$471$	0	0
$472$	−18.0000	−0.828517
$473$	28.8167	1.32499
$474$	0	0
$475$	16.2111	0.743816
$476$	0	0
$477$	0	0
$478$	10.6972	0.489280
$479$	30.6333	1.39967	0.699836	−	0.714304i	$-0.253257\pi$
$479$	30.6333	1.39967	0.699836	+	0.714304i	$0.253257\pi$
$480$	0	0
$481$	−2.00000	−0.0911922
$482$	−11.0917	−0.505212
$483$	0	0
$484$	0.605551	0.0275251
$485$	−0.119429	−0.00542301
$486$	0	0
$487$	−22.9083	−1.03808	−0.519038	−	0.854751i	$-0.673710\pi$
$487$	−22.9083	−1.03808	−0.519038	+	0.854751i	$0.673710\pi$
$488$	38.4500	1.74055
$489$	0	0
$490$	−6.63331	−0.299662
$491$	−3.11943	−0.140778	−0.0703889	−	0.997520i	$-0.522424\pi$
$491$	−3.11943	−0.140778	−0.0703889	+	0.997520i	$0.522424\pi$
$492$	0	0
$493$	0	0
$494$	1.93608	0.0871085
$495$	0	0
$496$	11.9083	0.534700
$497$	−37.0278	−1.66092
$498$	0	0
$499$	−12.0917	−0.541298	−0.270649	−	0.962678i	$-0.587238\pi$
$499$	−12.0917	−0.541298	−0.270649	+	0.962678i	$0.587238\pi$
$500$	3.27502	0.146463
$501$	0	0
$502$	−6.63331	−0.296059
$503$	6.63331	0.295765	0.147882	−	0.989005i	$-0.452754\pi$
$503$	6.63331	0.295765	0.147882	+	0.989005i	$0.452754\pi$
$504$	0	0
$505$	15.1194	0.672806
$506$	5.09167	0.226352
$507$	0	0
$508$	−2.06392	−0.0915715
$509$	−11.6056	−0.514407	−0.257204	−	0.966357i	$-0.582801\pi$
$509$	−11.6056	−0.514407	−0.257204	+	0.966357i	$0.582801\pi$
$510$	0	0
$511$	25.1194	1.11122
$512$	25.4222	1.12351
$513$	0	0
$514$	−7.45837	−0.328974
$515$	2.60555	0.114814
$516$	0	0
$517$	9.00000	0.395820
$518$	28.4222	1.24880
$519$	0	0
$520$	1.18335	0.0518932
$521$	−44.8444	−1.96467	−0.982335	−	0.187133i	$-0.940081\pi$
$521$	−44.8444	−1.96467	−0.982335	+	0.187133i	$0.940081\pi$
$522$	0	0
$523$	15.4222	0.674366	0.337183	−	0.941439i	$-0.390526\pi$
$523$	15.4222	0.674366	0.337183	+	0.941439i	$0.390526\pi$
$524$	0.788897	0.0344631
$525$	0	0
$526$	3.23886	0.141221
$527$	0	0
$528$	0	0
$529$	−21.3028	−0.926208
$530$	21.9083	0.951637
$531$	0	0
$532$	4.90833	0.212803
$533$	−1.81665	−0.0786880
$534$	0	0
$535$	−2.88057	−0.124538
$536$	16.1833	0.699014
$537$	0	0
$538$	−31.4222	−1.35471
$539$	−11.7250	−0.505031
$540$	0	0
$541$	11.5139	0.495020	0.247510	−	0.968885i	$-0.420388\pi$
$541$	11.5139	0.495020	0.247510	+	0.968885i	$0.420388\pi$
$542$	−35.8806	−1.54120
$543$	0	0
$544$	0	0
$545$	−0.275019	−0.0117805
$546$	0	0
$547$	−25.2389	−1.07914	−0.539568	−	0.841942i	$-0.681412\pi$
$547$	−25.2389	−1.07914	−0.539568	+	0.841942i	$0.681412\pi$
$548$	6.00000	0.256307
$549$	0	0
$550$	−12.9083	−0.550413
$551$	−4.45837	−0.189933
$552$	0	0
$553$	−10.6056	−0.450994
$554$	5.21110	0.221399
$555$	0	0
$556$	−0.449961	−0.0190826
$557$	15.6333	0.662405	0.331202	−	0.943560i	$-0.392546\pi$
$557$	15.6333	0.662405	0.331202	+	0.943560i	$0.392546\pi$
$558$	0	0
$559$	−2.90833	−0.123009
$560$	−14.2111	−0.600529
$561$	0	0
$562$	−33.2750	−1.40362
$563$	−9.39445	−0.395929	−0.197964	−	0.980209i	$-0.563433\pi$
$563$	−9.39445	−0.395929	−0.197964	+	0.980209i	$0.563433\pi$
$564$	0	0
$565$	−8.48612	−0.357014
$566$	−26.0555	−1.09519
$567$	0	0
$568$	−33.6333	−1.41122
$569$	−17.6056	−0.738063	−0.369032	−	0.929417i	$-0.620311\pi$
$569$	−17.6056	−0.738063	−0.369032	+	0.929417i	$0.620311\pi$
$570$	0	0
$571$	−25.5139	−1.06772	−0.533861	−	0.845572i	$-0.679260\pi$
$571$	−25.5139	−1.06772	−0.533861	+	0.845572i	$0.679260\pi$
$572$	0.275019	0.0114991
$573$	0	0
$574$	25.8167	1.07757
$575$	−4.30278	−0.179438
$576$	0	0
$577$	−13.2389	−0.551141	−0.275570	−	0.961281i	$-0.588867\pi$
$577$	−13.2389	−0.551141	−0.275570	+	0.961281i	$0.588867\pi$
$578$	0	0
$579$	0	0
$580$	−0.358288	−0.0148771
$581$	51.2389	2.12575
$582$	0	0
$583$	38.7250	1.60382
$584$	22.8167	0.944160
$585$	0	0
$586$	−5.44996	−0.225136
$587$	30.1194	1.24316	0.621581	−	0.783350i	$-0.286491\pi$
$587$	30.1194	1.24316	0.621581	+	0.783350i	$0.286491\pi$
$588$	0	0
$589$	17.6972	0.729201
$590$	10.1833	0.419242
$591$	0	0
$592$	21.8167	0.896659
$593$	−6.63331	−0.272397	−0.136199	−	0.990682i	$-0.543489\pi$
$593$	−6.63331	−0.272397	−0.136199	+	0.990682i	$0.543489\pi$
$594$	0	0
$595$	0	0
$596$	−2.60555	−0.106728
$597$	0	0
$598$	−0.513878	−0.0210140
$599$	−30.6333	−1.25164	−0.625822	−	0.779966i	$-0.715236\pi$
$599$	−30.6333	−1.25164	−0.625822	+	0.779966i	$0.715236\pi$
$600$	0	0
$601$	−23.6972	−0.966630	−0.483315	−	0.875447i	$-0.660567\pi$
$601$	−23.6972	−0.966630	−0.483315	+	0.875447i	$0.660567\pi$
$602$	41.3305	1.68451
$603$	0	0
$604$	−1.51388	−0.0615988
$605$	−2.60555	−0.105931
$606$	0	0
$607$	26.6333	1.08101	0.540506	−	0.841340i	$-0.318233\pi$
$607$	26.6333	1.08101	0.540506	+	0.841340i	$0.318233\pi$
$608$	8.33053	0.337848
$609$	0	0
$610$	−21.7527	−0.880743
$611$	−0.908327	−0.0367470
$612$	0	0
$613$	9.02776	0.364628	0.182314	−	0.983240i	$-0.441641\pi$
$613$	9.02776	0.364628	0.182314	+	0.983240i	$0.441641\pi$
$614$	17.0917	0.689764
$615$	0	0
$616$	−29.7250	−1.19765
$617$	21.0000	0.845428	0.422714	−	0.906263i	$-0.361077\pi$
$617$	21.0000	0.845428	0.422714	+	0.906263i	$0.361077\pi$
$618$	0	0
$619$	37.0555	1.48939	0.744693	−	0.667407i	$-0.232596\pi$
$619$	37.0555	1.48939	0.744693	+	0.667407i	$0.232596\pi$
$620$	1.42221	0.0571171
$621$	0	0
$622$	43.6611	1.75065
$623$	37.0278	1.48349
$624$	0	0
$625$	2.42221	0.0968882
$626$	15.3944	0.615286
$627$	0	0
$628$	−2.81665	−0.112397
$629$	0	0
$630$	0	0
$631$	−35.1472	−1.39919	−0.699594	−	0.714541i	$-0.746636\pi$
$631$	−35.1472	−1.39919	−0.699594	+	0.714541i	$0.746636\pi$
$632$	−9.63331	−0.383192
$633$	0	0
$634$	−35.4861	−1.40933
$635$	8.88057	0.352415
$636$	0	0
$637$	1.18335	0.0468859
$638$	3.55004	0.140547
$639$	0	0
$640$	−10.5416	−0.416695
$641$	0.908327	0.0358768	0.0179384	−	0.999839i	$-0.494290\pi$
$641$	0.908327	0.0358768	0.0179384	+	0.999839i	$0.494290\pi$
$642$	0	0
$643$	−8.18335	−0.322720	−0.161360	−	0.986896i	$-0.551588\pi$
$643$	−8.18335	−0.322720	−0.161360	+	0.986896i	$0.551588\pi$
$644$	−1.30278	−0.0513366
$645$	0	0
$646$	0	0
$647$	29.0555	1.14229	0.571145	−	0.820849i	$-0.306499\pi$
$647$	29.0555	1.14229	0.571145	+	0.820849i	$0.306499\pi$
$648$	0	0
$649$	18.0000	0.706562
$650$	1.30278	0.0510991
$651$	0	0
$652$	3.14719	0.123253
$653$	28.8167	1.12768	0.563841	−	0.825883i	$-0.309323\pi$
$653$	28.8167	1.12768	0.563841	+	0.825883i	$0.309323\pi$
$654$	0	0
$655$	−3.39445	−0.132632
$656$	19.8167	0.773710
$657$	0	0
$658$	12.9083	0.503219
$659$	36.3944	1.41773	0.708863	−	0.705346i	$-0.249208\pi$
$659$	36.3944	1.41773	0.708863	+	0.705346i	$0.249208\pi$
$660$	0	0
$661$	31.7250	1.23396	0.616979	−	0.786979i	$-0.288356\pi$
$661$	31.7250	1.23396	0.616979	+	0.786979i	$0.288356\pi$
$662$	−4.30278	−0.167232
$663$	0	0
$664$	46.5416	1.80617
$665$	−21.1194	−0.818976
$666$	0	0
$667$	1.18335	0.0458193
$668$	6.75274	0.261271
$669$	0	0
$670$	−9.15559	−0.353711
$671$	−38.4500	−1.48434
$672$	0	0
$673$	47.6333	1.83613	0.918065	−	0.396431i	$-0.129751\pi$
$673$	47.6333	1.83613	0.918065	+	0.396431i	$0.129751\pi$
$674$	−5.84441	−0.225118
$675$	0	0
$676$	3.90833	0.150320
$677$	0.908327	0.0349098	0.0174549	−	0.999848i	$-0.494444\pi$
$677$	0.908327	0.0349098	0.0174549	+	0.999848i	$0.494444\pi$
$678$	0	0
$679$	−0.302776	−0.0116195
$680$	0	0
$681$	0	0
$682$	−14.0917	−0.539598
$683$	−21.7527	−0.832345	−0.416173	−	0.909286i	$-0.636629\pi$
$683$	−21.7527	−0.832345	−0.416173	+	0.909286i	$0.636629\pi$
$684$	0	0
$685$	−25.8167	−0.986404
$686$	13.3028	0.507902
$687$	0	0
$688$	31.7250	1.20950
$689$	−3.90833	−0.148895
$690$	0	0
$691$	−26.4222	−1.00515	−0.502574	−	0.864534i	$-0.667614\pi$
$691$	−26.4222	−1.00515	−0.502574	+	0.864534i	$0.667614\pi$
$692$	−0.119429	−0.00454003
$693$	0	0
$694$	−17.8444	−0.677365
$695$	1.93608	0.0734398
$696$	0	0
$697$	0	0
$698$	−17.7250	−0.670900
$699$	0	0
$700$	3.30278	0.124833
$701$	6.63331	0.250537	0.125268	−	0.992123i	$-0.460021\pi$
$701$	6.63331	0.250537	0.125268	+	0.992123i	$0.460021\pi$
$702$	0	0
$703$	32.4222	1.22283
$704$	−26.4500	−0.996870
$705$	0	0
$706$	3.90833	0.147092
$707$	38.3305	1.44157
$708$	0	0
$709$	34.7250	1.30412	0.652062	−	0.758166i	$-0.273904\pi$
$709$	34.7250	1.30412	0.652062	+	0.758166i	$0.273904\pi$
$710$	19.0278	0.714099
$711$	0	0
$712$	33.6333	1.26046
$713$	−4.69722	−0.175912
$714$	0	0
$715$	−1.18335	−0.0442546
$716$	3.55004	0.132671
$717$	0	0
$718$	−28.5416	−1.06516
$719$	−41.6056	−1.55163	−0.775813	−	0.630963i	$-0.782660\pi$
$719$	−41.6056	−1.55163	−0.775813	+	0.630963i	$0.782660\pi$
$720$	0	0
$721$	6.60555	0.246004
$722$	−6.63331	−0.246866
$723$	0	0
$724$	2.35829	0.0876451
$725$	−3.00000	−0.111417
$726$	0	0
$727$	43.2111	1.60261	0.801306	−	0.598255i	$-0.204139\pi$
$727$	43.2111	1.60261	0.801306	+	0.598255i	$0.204139\pi$
$728$	3.00000	0.111187
$729$	0	0
$730$	−12.9083	−0.477759
$731$	0	0
$732$	0	0
$733$	18.3028	0.676028	0.338014	−	0.941141i	$-0.390245\pi$
$733$	18.3028	0.676028	0.338014	+	0.941141i	$0.390245\pi$
$734$	−22.6611	−0.836435
$735$	0	0
$736$	−2.21110	−0.0815023
$737$	−16.1833	−0.596121
$738$	0	0
$739$	−23.4222	−0.861600	−0.430800	−	0.902447i	$-0.641769\pi$
$739$	−23.4222	−0.861600	−0.430800	+	0.902447i	$0.641769\pi$
$740$	2.60555	0.0957820
$741$	0	0
$742$	55.5416	2.03900
$743$	−7.18335	−0.263531	−0.131766	−	0.991281i	$-0.542065\pi$
$743$	−7.18335	−0.263531	−0.131766	+	0.991281i	$0.542065\pi$
$744$	0	0
$745$	11.2111	0.410743
$746$	8.09167	0.296257
$747$	0	0
$748$	0	0
$749$	−7.30278	−0.266838
$750$	0	0
$751$	42.2666	1.54233	0.771165	−	0.636635i	$-0.219674\pi$
$751$	42.2666	1.54233	0.771165	+	0.636635i	$0.219674\pi$
$752$	9.90833	0.361320
$753$	0	0
$754$	−0.358288	−0.0130481
$755$	6.51388	0.237064
$756$	0	0
$757$	−25.7889	−0.937313	−0.468657	−	0.883380i	$-0.655262\pi$
$757$	−25.7889	−0.937313	−0.468657	+	0.883380i	$0.655262\pi$
$758$	20.3305	0.738438
$759$	0	0
$760$	−19.1833	−0.695853
$761$	−26.2111	−0.950152	−0.475076	−	0.879945i	$-0.657579\pi$
$761$	−26.2111	−0.950152	−0.475076	+	0.879945i	$0.657579\pi$
$762$	0	0
$763$	−0.697224	−0.0252412
$764$	−2.33053	−0.0843157
$765$	0	0
$766$	−9.00000	−0.325183
$767$	−1.81665	−0.0655956
$768$	0	0
$769$	27.9361	1.00740	0.503700	−	0.863878i	$-0.331972\pi$
$769$	27.9361	1.00740	0.503700	+	0.863878i	$0.331972\pi$
$770$	16.8167	0.606030
$771$	0	0
$772$	6.85281	0.246638
$773$	35.8444	1.28923	0.644617	−	0.764506i	$-0.277017\pi$
$773$	35.8444	1.28923	0.644617	+	0.764506i	$0.277017\pi$
$774$	0	0
$775$	11.9083	0.427760
$776$	−0.275019	−0.00987262
$777$	0	0
$778$	−39.9083	−1.43078
$779$	29.4500	1.05515
$780$	0	0
$781$	33.6333	1.20349
$782$	0	0
$783$	0	0
$784$	−12.9083	−0.461012
$785$	12.1194	0.432561
$786$	0	0
$787$	3.02776	0.107928	0.0539639	−	0.998543i	$-0.482814\pi$
$787$	3.02776	0.107928	0.0539639	+	0.998543i	$0.482814\pi$
$788$	1.18335	0.0421550
$789$	0	0
$790$	5.44996	0.193901
$791$	−21.5139	−0.764945
$792$	0	0
$793$	3.88057	0.137803
$794$	−9.90833	−0.351633
$795$	0	0
$796$	1.09167	0.0386933
$797$	16.5778	0.587216	0.293608	−	0.955926i	$-0.405144\pi$
$797$	16.5778	0.587216	0.293608	+	0.955926i	$0.405144\pi$
$798$	0	0
$799$	0	0
$800$	5.60555	0.198186
$801$	0	0
$802$	−25.1472	−0.887978
$803$	−22.8167	−0.805182
$804$	0	0
$805$	5.60555	0.197570
$806$	1.42221	0.0500950
$807$	0	0
$808$	34.8167	1.22485
$809$	−19.0278	−0.668980	−0.334490	−	0.942399i	$-0.608564\pi$
$809$	−19.0278	−0.668980	−0.334490	+	0.942399i	$0.608564\pi$
$810$	0	0
$811$	−22.2389	−0.780912	−0.390456	−	0.920622i	$-0.627683\pi$
$811$	−22.2389	−0.780912	−0.390456	+	0.920622i	$0.627683\pi$
$812$	−0.908327	−0.0318760
$813$	0	0
$814$	−25.8167	−0.904873
$815$	−13.5416	−0.474343
$816$	0	0
$817$	47.1472	1.64947
$818$	42.7527	1.49481
$819$	0	0
$820$	2.36669	0.0826485
$821$	4.81665	0.168102	0.0840512	−	0.996461i	$-0.473214\pi$
$821$	4.81665	0.168102	0.0840512	+	0.996461i	$0.473214\pi$
$822$	0	0
$823$	−30.4500	−1.06142	−0.530709	−	0.847554i	$-0.678074\pi$
$823$	−30.4500	−1.06142	−0.530709	+	0.847554i	$0.678074\pi$
$824$	6.00000	0.209020
$825$	0	0
$826$	25.8167	0.898276
$827$	50.4500	1.75432	0.877159	−	0.480201i	$-0.159436\pi$
$827$	50.4500	1.75432	0.877159	+	0.480201i	$0.159436\pi$
$828$	0	0
$829$	−33.4500	−1.16177	−0.580883	−	0.813987i	$-0.697292\pi$
$829$	−33.4500	−1.16177	−0.580883	+	0.813987i	$0.697292\pi$
$830$	−26.3305	−0.913946
$831$	0	0
$832$	2.66947	0.0925472
$833$	0	0
$834$	0	0
$835$	−29.0555	−1.00551
$836$	−4.45837	−0.154196
$837$	0	0
$838$	34.1472	1.17959
$839$	0.238859	0.00824633	0.00412316	−	0.999991i	$-0.498688\pi$
$839$	0.238859	0.00824633	0.00412316	+	0.999991i	$0.498688\pi$
$840$	0	0
$841$	−28.1749	−0.971550
$842$	−24.6695	−0.850166
$843$	0	0
$844$	6.02776	0.207484
$845$	−16.8167	−0.578510
$846$	0	0
$847$	−6.60555	−0.226969
$848$	42.6333	1.46403
$849$	0	0
$850$	0	0
$851$	−8.60555	−0.294994
$852$	0	0
$853$	5.00000	0.171197	0.0855984	−	0.996330i	$-0.472720\pi$
$853$	5.00000	0.171197	0.0855984	+	0.996330i	$0.472720\pi$
$854$	−55.1472	−1.88710
$855$	0	0
$856$	−6.63331	−0.226722
$857$	52.0278	1.77723	0.888617	−	0.458650i	$-0.151667\pi$
$857$	52.0278	1.77723	0.888617	+	0.458650i	$0.151667\pi$
$858$	0	0
$859$	41.3944	1.41236	0.706180	−	0.708032i	$-0.250417\pi$
$859$	41.3944	1.41236	0.706180	+	0.708032i	$0.250417\pi$
$860$	3.78890	0.129200
$861$	0	0
$862$	28.5416	0.972132
$863$	9.66947	0.329153	0.164576	−	0.986364i	$-0.447374\pi$
$863$	9.66947	0.329153	0.164576	+	0.986364i	$0.447374\pi$
$864$	0	0
$865$	0.513878	0.0174724
$866$	36.4777	1.23956
$867$	0	0
$868$	3.60555	0.122380
$869$	9.63331	0.326788
$870$	0	0
$871$	1.63331	0.0553425
$872$	−0.633308	−0.0214465
$873$	0	0
$874$	8.33053	0.281784
$875$	−35.7250	−1.20772
$876$	0	0
$877$	−37.1194	−1.25343	−0.626717	−	0.779247i	$-0.715602\pi$
$877$	−37.1194	−1.25343	−0.626717	+	0.779247i	$0.715602\pi$
$878$	−19.5778	−0.660719
$879$	0	0
$880$	12.9083	0.435140
$881$	−6.11943	−0.206169	−0.103084	−	0.994673i	$-0.532871\pi$
$881$	−6.11943	−0.206169	−0.103084	+	0.994673i	$0.532871\pi$
$882$	0	0
$883$	−14.3028	−0.481327	−0.240663	−	0.970609i	$-0.577365\pi$
$883$	−14.3028	−0.481327	−0.240663	+	0.970609i	$0.577365\pi$
$884$	0	0
$885$	0	0
$886$	19.8528	0.666968
$887$	−12.2750	−0.412155	−0.206077	−	0.978536i	$-0.566070\pi$
$887$	−12.2750	−0.412155	−0.206077	+	0.978536i	$0.566070\pi$
$888$	0	0
$889$	22.5139	0.755091
$890$	−19.0278	−0.637812
$891$	0	0
$892$	2.87217	0.0961673
$893$	14.7250	0.492753
$894$	0	0
$895$	−15.2750	−0.510588
$896$	−26.7250	−0.892819
$897$	0	0
$898$	12.3944	0.413608
$899$	−3.27502	−0.109228
$900$	0	0
$901$	0	0
$902$	−23.4500	−0.780798
$903$	0	0
$904$	−19.5416	−0.649945
$905$	−10.1472	−0.337304
$906$	0	0
$907$	6.54163	0.217211	0.108606	−	0.994085i	$-0.465361\pi$
$907$	6.54163	0.217211	0.108606	+	0.994085i	$0.465361\pi$
$908$	−1.45837	−0.0483976
$909$	0	0
$910$	−1.69722	−0.0562624
$911$	−27.1194	−0.898507	−0.449253	−	0.893404i	$-0.648310\pi$
$911$	−27.1194	−0.898507	−0.449253	+	0.893404i	$0.648310\pi$
$912$	0	0
$913$	−46.5416	−1.54030
$914$	−11.0917	−0.366880
$915$	0	0
$916$	−5.57779	−0.184296
$917$	−8.60555	−0.284180
$918$	0	0
$919$	21.6972	0.715725	0.357863	−	0.933774i	$-0.383506\pi$
$919$	21.6972	0.715725	0.357863	+	0.933774i	$0.383506\pi$
$920$	5.09167	0.167867
$921$	0	0
$922$	21.3944	0.704589
$923$	−3.39445	−0.111730
$924$	0	0
$925$	21.8167	0.717327
$926$	7.57779	0.249022
$927$	0	0
$928$	−1.54163	−0.0506066
$929$	−27.2750	−0.894864	−0.447432	−	0.894318i	$-0.647661\pi$
$929$	−27.2750	−0.894864	−0.447432	+	0.894318i	$0.647661\pi$
$930$	0	0
$931$	−19.1833	−0.628709
$932$	7.81665	0.256043
$933$	0	0
$934$	18.3583	0.600702
$935$	0	0
$936$	0	0
$937$	20.7527	0.677962	0.338981	−	0.940793i	$-0.389918\pi$
$937$	20.7527	0.677962	0.338981	+	0.940793i	$0.389918\pi$
$938$	−23.2111	−0.757869
$939$	0	0
$940$	1.18335	0.0385965
$941$	−38.0917	−1.24175	−0.620877	−	0.783908i	$-0.713223\pi$
$941$	−38.0917	−1.24175	−0.620877	+	0.783908i	$0.713223\pi$
$942$	0	0
$943$	−7.81665	−0.254545
$944$	19.8167	0.644977
$945$	0	0
$946$	−37.5416	−1.22058
$947$	55.4222	1.80098	0.900490	−	0.434877i	$-0.143208\pi$
$947$	55.4222	1.80098	0.900490	+	0.434877i	$0.143208\pi$
$948$	0	0
$949$	2.30278	0.0747513
$950$	−21.1194	−0.685205
$951$	0	0
$952$	0	0
$953$	18.5139	0.599723	0.299862	−	0.953983i	$-0.403059\pi$
$953$	18.5139	0.599723	0.299862	+	0.953983i	$0.403059\pi$
$954$	0	0
$955$	10.0278	0.324491
$956$	2.48612	0.0804069
$957$	0	0
$958$	−39.9083	−1.28938
$959$	−65.4500	−2.11349
$960$	0	0
$961$	−18.0000	−0.580645
$962$	2.60555	0.0840063
$963$	0	0
$964$	−2.57779	−0.0830252
$965$	−29.4861	−0.949192
$966$	0	0
$967$	−44.9361	−1.44505	−0.722524	−	0.691346i	$-0.757018\pi$
$967$	−44.9361	−1.44505	−0.722524	+	0.691346i	$0.757018\pi$
$968$	−6.00000	−0.192847
$969$	0	0
$970$	0.155590	0.00499569
$971$	−47.7250	−1.53157	−0.765784	−	0.643098i	$-0.777649\pi$
$971$	−47.7250	−1.53157	−0.765784	+	0.643098i	$0.777649\pi$
$972$	0	0
$973$	4.90833	0.157354
$974$	29.8444	0.956277
$975$	0	0
$976$	−42.3305	−1.35497
$977$	−16.5416	−0.529214	−0.264607	−	0.964356i	$-0.585242\pi$
$977$	−16.5416	−0.529214	−0.264607	+	0.964356i	$0.585242\pi$
$978$	0	0
$979$	−33.6333	−1.07493
$980$	−1.54163	−0.0492457
$981$	0	0
$982$	4.06392	0.129685
$983$	−9.63331	−0.307255	−0.153627	−	0.988129i	$-0.549096\pi$
$983$	−9.63331	−0.307255	−0.153627	+	0.988129i	$0.549096\pi$
$984$	0	0
$985$	−5.09167	−0.162234
$986$	0	0
$987$	0	0
$988$	0.449961	0.0143152
$989$	−12.5139	−0.397918
$990$	0	0
$991$	−37.7527	−1.19926	−0.599628	−	0.800279i	$-0.704685\pi$
$991$	−37.7527	−1.19926	−0.599628	+	0.800279i	$0.704685\pi$
$992$	6.11943	0.194292
$993$	0	0
$994$	48.2389	1.53004
$995$	−4.69722	−0.148912
$996$	0	0
$997$	−15.8806	−0.502943	−0.251471	−	0.967865i	$-0.580914\pi$
$997$	−15.8806	−0.502943	−0.251471	+	0.967865i	$0.580914\pi$
$998$	15.7527	0.498644
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2601.2.a.r.1.1		2
3.2	odd	2		289.2.a.c.1.2	yes	2
12.11	even	2		4624.2.a.j.1.1		2
15.14	odd	2		7225.2.a.m.1.1		2
17.16	even	2		2601.2.a.s.1.1		2
51.2	odd	8		289.2.c.b.38.4		8
51.5	even	16		289.2.d.e.110.1		16
51.8	odd	8		289.2.c.b.251.1		8
51.11	even	16		289.2.d.e.155.3		16
51.14	even	16		289.2.d.e.179.3		16
51.20	even	16		289.2.d.e.179.4		16
51.23	even	16		289.2.d.e.155.4		16
51.26	odd	8		289.2.c.b.251.2		8
51.29	even	16		289.2.d.e.110.2		16
51.32	odd	8		289.2.c.b.38.3		8
51.38	odd	4		289.2.b.c.288.1		4
51.41	even	16		289.2.d.e.134.1		16
51.44	even	16		289.2.d.e.134.2		16
51.47	odd	4		289.2.b.c.288.2		4
51.50	odd	2		289.2.a.b.1.2	✓	2
204.203	even	2		4624.2.a.v.1.2		2
255.254	odd	2		7225.2.a.n.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
289.2.a.b.1.2	✓	2	51.50	odd	2
289.2.a.c.1.2	yes	2	3.2	odd	2
289.2.b.c.288.1		4	51.38	odd	4
289.2.b.c.288.2		4	51.47	odd	4
289.2.c.b.38.3		8	51.32	odd	8
289.2.c.b.38.4		8	51.2	odd	8
289.2.c.b.251.1		8	51.8	odd	8
289.2.c.b.251.2		8	51.26	odd	8
289.2.d.e.110.1		16	51.5	even	16
289.2.d.e.110.2		16	51.29	even	16
289.2.d.e.134.1		16	51.41	even	16
289.2.d.e.134.2		16	51.44	even	16
289.2.d.e.155.3		16	51.11	even	16
289.2.d.e.155.4		16	51.23	even	16
289.2.d.e.179.3		16	51.14	even	16
289.2.d.e.179.4		16	51.20	even	16
2601.2.a.r.1.1		2	1.1	even	1	trivial
2601.2.a.s.1.1		2	17.16	even	2
4624.2.a.j.1.1		2	12.11	even	2
4624.2.a.v.1.2		2	204.203	even	2
7225.2.a.m.1.1		2	15.14	odd	2
7225.2.a.n.1.1		2	255.254	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:	\( N \)	\(=\)	\( 2601 = 3^{2} \cdot 17^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2601.a (trivial)

\(f(q)\)	\(=\)	\(q-1.30278 q^{2} -0.302776 q^{4} +1.30278 q^{5} +3.30278 q^{7} +3.00000 q^{8} +O(q^{10})\)	\(q-1.30278 q^{2} -0.302776 q^{4} +1.30278 q^{5} +3.30278 q^{7} +3.00000 q^{8} -1.69722 q^{10} -3.00000 q^{11} +0.302776 q^{13} -4.30278 q^{14} -3.30278 q^{16} -4.90833 q^{19} -0.394449 q^{20} +3.90833 q^{22} +1.30278 q^{23} -3.30278 q^{25} -0.394449 q^{26} -1.00000 q^{28} +0.908327 q^{29} -3.60555 q^{31} -1.69722 q^{32} +4.30278 q^{35} -6.60555 q^{37} +6.39445 q^{38} +3.90833 q^{40} -6.00000 q^{41} -9.60555 q^{43} +0.908327 q^{44} -1.69722 q^{46} -3.00000 q^{47} +3.90833 q^{49} +4.30278 q^{50} -0.0916731 q^{52} -12.9083 q^{53} -3.90833 q^{55} +9.90833 q^{56} -1.18335 q^{58} -6.00000 q^{59} +12.8167 q^{61} +4.69722 q^{62} +8.81665 q^{64} +0.394449 q^{65} +5.39445 q^{67} -5.60555 q^{70} -11.2111 q^{71} +7.60555 q^{73} +8.60555 q^{74} +1.48612 q^{76} -9.90833 q^{77} -3.21110 q^{79} -4.30278 q^{80} +7.81665 q^{82} +15.5139 q^{83} +12.5139 q^{86} -9.00000 q^{88} +11.2111 q^{89} +1.00000 q^{91} -0.394449 q^{92} +3.90833 q^{94} -6.39445 q^{95} -0.0916731 q^{97} -5.09167 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{2} + 3 q^{4} - q^{5} + 3 q^{7} + 6 q^{8}+O(q^{10}) \) 2 * q + q^2 + 3 * q^4 - q^5 + 3 * q^7 + 6 * q^8	\( 2 q + q^{2} + 3 q^{4} - q^{5} + 3 q^{7} + 6 q^{8} - 7 q^{10} - 6 q^{11} - 3 q^{13} - 5 q^{14} - 3 q^{16} + q^{19} - 8 q^{20} - 3 q^{22} - q^{23} - 3 q^{25} - 8 q^{26} - 2 q^{28} - 9 q^{29} - 7 q^{32} + 5 q^{35} - 6 q^{37} + 20 q^{38} - 3 q^{40} - 12 q^{41} - 12 q^{43} - 9 q^{44} - 7 q^{46} - 6 q^{47} - 3 q^{49} + 5 q^{50} - 11 q^{52} - 15 q^{53} + 3 q^{55} + 9 q^{56} - 24 q^{58} - 12 q^{59} + 4 q^{61} + 13 q^{62} - 4 q^{64} + 8 q^{65} + 18 q^{67} - 4 q^{70} - 8 q^{71} + 8 q^{73} + 10 q^{74} + 21 q^{76} - 9 q^{77} + 8 q^{79} - 5 q^{80} - 6 q^{82} + 13 q^{83} + 7 q^{86} - 18 q^{88} + 8 q^{89} + 2 q^{91} - 8 q^{92} - 3 q^{94} - 20 q^{95} - 11 q^{97} - 21 q^{98}+O(q^{100}) \) 2 * q + q^2 + 3 * q^4 - q^5 + 3 * q^7 + 6 * q^8 - 7 * q^10 - 6 * q^11 - 3 * q^13 - 5 * q^14 - 3 * q^16 + q^19 - 8 * q^20 - 3 * q^22 - q^23 - 3 * q^25 - 8 * q^26 - 2 * q^28 - 9 * q^29 - 7 * q^32 + 5 * q^35 - 6 * q^37 + 20 * q^38 - 3 * q^40 - 12 * q^41 - 12 * q^43 - 9 * q^44 - 7 * q^46 - 6 * q^47 - 3 * q^49 + 5 * q^50 - 11 * q^52 - 15 * q^53 + 3 * q^55 + 9 * q^56 - 24 * q^58 - 12 * q^59 + 4 * q^61 + 13 * q^62 - 4 * q^64 + 8 * q^65 + 18 * q^67 - 4 * q^70 - 8 * q^71 + 8 * q^73 + 10 * q^74 + 21 * q^76 - 9 * q^77 + 8 * q^79 - 5 * q^80 - 6 * q^82 + 13 * q^83 + 7 * q^86 - 18 * q^88 + 8 * q^89 + 2 * q^91 - 8 * q^92 - 3 * q^94 - 20 * q^95 - 11 * q^97 - 21 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more