LMFDB - Embedded newform 2601.2.a.q.1.2

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:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 153)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.41421$ 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)

$f(q)$	$=$	$q+1.41421 q^{2} +1.00000 q^{5} -1.41421 q^{7} -2.82843 q^{8} +O(q^{10})$	$q+1.41421 q^{2} +1.00000 q^{5} -1.41421 q^{7} -2.82843 q^{8} +1.41421 q^{10} +1.00000 q^{11} +1.00000 q^{13} -2.00000 q^{14} -4.00000 q^{16} -3.00000 q^{19} +1.41421 q^{22} -5.00000 q^{23} -4.00000 q^{25} +1.41421 q^{26} -8.00000 q^{29} +5.65685 q^{31} -1.41421 q^{35} +1.41421 q^{37} -4.24264 q^{38} -2.82843 q^{40} +11.0000 q^{41} -9.00000 q^{43} -7.07107 q^{46} -8.48528 q^{47} -5.00000 q^{49} -5.65685 q^{50} -2.82843 q^{53} +1.00000 q^{55} +4.00000 q^{56} -11.3137 q^{58} -9.89949 q^{59} +7.07107 q^{61} +8.00000 q^{62} +8.00000 q^{64} +1.00000 q^{65} -10.0000 q^{67} -2.00000 q^{70} -10.0000 q^{71} +11.3137 q^{73} +2.00000 q^{74} -1.41421 q^{77} -7.07107 q^{79} -4.00000 q^{80} +15.5563 q^{82} +14.1421 q^{83} -12.7279 q^{86} -2.82843 q^{88} -4.24264 q^{89} -1.41421 q^{91} -12.0000 q^{94} -3.00000 q^{95} -14.1421 q^{97} -7.07107 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{5}+O(q^{10}) $ 2 * q + 2 * q^5	$ 2 q + 2 q^{5} + 2 q^{11} + 2 q^{13} - 4 q^{14} - 8 q^{16} - 6 q^{19} - 10 q^{23} - 8 q^{25} - 16 q^{29} + 22 q^{41} - 18 q^{43} - 10 q^{49} + 2 q^{55} + 8 q^{56} + 16 q^{62} + 16 q^{64} + 2 q^{65} - 20 q^{67} - 4 q^{70} - 20 q^{71} + 4 q^{74} - 8 q^{80} - 24 q^{94} - 6 q^{95}+O(q^{100}) $ 2 * q + 2 * q^5 + 2 * q^11 + 2 * q^13 - 4 * q^14 - 8 * q^16 - 6 * q^19 - 10 * q^23 - 8 * q^25 - 16 * q^29 + 22 * q^41 - 18 * q^43 - 10 * q^49 + 2 * q^55 + 8 * q^56 + 16 * q^62 + 16 * q^64 + 2 * q^65 - 20 * q^67 - 4 * q^70 - 20 * q^71 + 4 * q^74 - 8 * q^80 - 24 * q^94 - 6 * q^95

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 $

$2$	1.41421	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$2$	1.41421	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$3$	0	0
$3$	0	0
$4$	0	0
$5$	1.00000	0.447214	0.223607	−	0.974679i	$-0.428217\pi$
$5$	1.00000	0.447214	0.223607	+	0.974679i	$0.428217\pi$
$6$	0	0
$7$	−1.41421	−0.534522	−0.267261	−	0.963624i	$-0.586119\pi$
$7$	−1.41421	−0.534522	−0.267261	+	0.963624i	$0.586119\pi$
$8$	−2.82843	−1.00000
$9$	0	0
$10$	1.41421	0.447214
$11$	1.00000	0.301511	0.150756	−	0.988571i	$-0.451829\pi$
$11$	1.00000	0.301511	0.150756	+	0.988571i	$0.451829\pi$
$12$	0	0
$13$	1.00000	0.277350	0.138675	−	0.990338i	$-0.455716\pi$
$13$	1.00000	0.277350	0.138675	+	0.990338i	$0.455716\pi$
$14$	−2.00000	−0.534522
$15$	0	0
$16$	−4.00000	−1.00000
$17$	0	0
$17$	0	0
$18$	0	0
$19$	−3.00000	−0.688247	−0.344124	−	0.938924i	$-0.611824\pi$
$19$	−3.00000	−0.688247	−0.344124	+	0.938924i	$0.611824\pi$
$20$	0	0
$21$	0	0
$22$	1.41421	0.301511
$23$	−5.00000	−1.04257	−0.521286	−	0.853382i	$-0.674548\pi$
$23$	−5.00000	−1.04257	−0.521286	+	0.853382i	$0.674548\pi$
$24$	0	0
$25$	−4.00000	−0.800000
$26$	1.41421	0.277350
$27$	0	0
$28$	0	0
$29$	−8.00000	−1.48556	−0.742781	−	0.669534i	$-0.766494\pi$
$29$	−8.00000	−1.48556	−0.742781	+	0.669534i	$0.766494\pi$
$30$	0	0
$31$	5.65685	1.01600	0.508001	−	0.861357i	$-0.330385\pi$
$31$	5.65685	1.01600	0.508001	+	0.861357i	$0.330385\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−1.41421	−0.239046
$36$	0	0
$37$	1.41421	0.232495	0.116248	−	0.993220i	$-0.462913\pi$
$37$	1.41421	0.232495	0.116248	+	0.993220i	$0.462913\pi$
$38$	−4.24264	−0.688247
$39$	0	0
$40$	−2.82843	−0.447214
$41$	11.0000	1.71791	0.858956	−	0.512050i	$-0.171114\pi$
$41$	11.0000	1.71791	0.858956	+	0.512050i	$0.171114\pi$
$42$	0	0
$43$	−9.00000	−1.37249	−0.686244	−	0.727372i	$-0.740742\pi$
$43$	−9.00000	−1.37249	−0.686244	+	0.727372i	$0.740742\pi$
$44$	0	0
$45$	0	0
$46$	−7.07107	−1.04257
$47$	−8.48528	−1.23771	−0.618853	−	0.785507i	$-0.712402\pi$
$47$	−8.48528	−1.23771	−0.618853	+	0.785507i	$0.712402\pi$
$48$	0	0
$49$	−5.00000	−0.714286
$50$	−5.65685	−0.800000
$51$	0	0
$52$	0	0
$53$	−2.82843	−0.388514	−0.194257	−	0.980951i	$-0.562230\pi$
$53$	−2.82843	−0.388514	−0.194257	+	0.980951i	$0.562230\pi$
$54$	0	0
$55$	1.00000	0.134840
$56$	4.00000	0.534522
$57$	0	0
$58$	−11.3137	−1.48556
$59$	−9.89949	−1.28880	−0.644402	−	0.764687i	$-0.722894\pi$
$59$	−9.89949	−1.28880	−0.644402	+	0.764687i	$0.722894\pi$
$60$	0	0
$61$	7.07107	0.905357	0.452679	−	0.891674i	$-0.350468\pi$
$61$	7.07107	0.905357	0.452679	+	0.891674i	$0.350468\pi$
$62$	8.00000	1.01600
$63$	0	0
$64$	8.00000	1.00000
$65$	1.00000	0.124035
$66$	0	0
$67$	−10.0000	−1.22169	−0.610847	−	0.791748i	$-0.709171\pi$
$67$	−10.0000	−1.22169	−0.610847	+	0.791748i	$0.709171\pi$
$68$	0	0
$69$	0	0
$70$	−2.00000	−0.239046
$71$	−10.0000	−1.18678	−0.593391	−	0.804914i	$-0.702211\pi$
$71$	−10.0000	−1.18678	−0.593391	+	0.804914i	$0.702211\pi$
$72$	0	0
$73$	11.3137	1.32417	0.662085	−	0.749429i	$-0.269672\pi$
$73$	11.3137	1.32417	0.662085	+	0.749429i	$0.269672\pi$
$74$	2.00000	0.232495
$75$	0	0
$76$	0	0
$77$	−1.41421	−0.161165
$78$	0	0
$79$	−7.07107	−0.795557	−0.397779	−	0.917481i	$-0.630219\pi$
$79$	−7.07107	−0.795557	−0.397779	+	0.917481i	$0.630219\pi$
$80$	−4.00000	−0.447214
$81$	0	0
$82$	15.5563	1.71791
$83$	14.1421	1.55230	0.776151	−	0.630548i	$-0.217170\pi$
$83$	14.1421	1.55230	0.776151	+	0.630548i	$0.217170\pi$
$84$	0	0
$85$	0	0
$86$	−12.7279	−1.37249
$87$	0	0
$88$	−2.82843	−0.301511
$89$	−4.24264	−0.449719	−0.224860	−	0.974391i	$-0.572192\pi$
$89$	−4.24264	−0.449719	−0.224860	+	0.974391i	$0.572192\pi$
$90$	0	0
$91$	−1.41421	−0.148250
$92$	0	0
$93$	0	0
$94$	−12.0000	−1.23771
$95$	−3.00000	−0.307794
$96$	0	0
$97$	−14.1421	−1.43592	−0.717958	−	0.696086i	$-0.754923\pi$
$97$	−14.1421	−1.43592	−0.717958	+	0.696086i	$0.754923\pi$
$98$	−7.07107	−0.714286
$99$	0	0
$100$	0	0
$101$	0	0		−	1.00000i	$-0.5\pi$
$101$	0	0			1.00000i	$0.5\pi$
$102$	0	0
$103$	5.00000	0.492665	0.246332	−	0.969185i	$-0.420775\pi$
$103$	5.00000	0.492665	0.246332	+	0.969185i	$0.420775\pi$
$104$	−2.82843	−0.277350
$105$	0	0
$106$	−4.00000	−0.388514
$107$	7.00000	0.676716	0.338358	−	0.941018i	$-0.390129\pi$
$107$	7.00000	0.676716	0.338358	+	0.941018i	$0.390129\pi$
$108$	0	0
$109$	−14.1421	−1.35457	−0.677285	−	0.735720i	$-0.736844\pi$
$109$	−14.1421	−1.35457	−0.677285	+	0.735720i	$0.736844\pi$
$110$	1.41421	0.134840
$111$	0	0
$112$	5.65685	0.534522
$113$	−5.00000	−0.470360	−0.235180	−	0.971952i	$-0.575568\pi$
$113$	−5.00000	−0.470360	−0.235180	+	0.971952i	$0.575568\pi$
$114$	0	0
$115$	−5.00000	−0.466252
$116$	0	0
$117$	0	0
$118$	−14.0000	−1.28880
$119$	0	0
$120$	0	0
$121$	−10.0000	−0.909091
$122$	10.0000	0.905357
$123$	0	0
$124$	0	0
$125$	−9.00000	−0.804984
$126$	0	0
$127$	−15.0000	−1.33103	−0.665517	−	0.746382i	$-0.731789\pi$
$127$	−15.0000	−1.33103	−0.665517	+	0.746382i	$0.731789\pi$
$128$	11.3137	1.00000
$129$	0	0
$130$	1.41421	0.124035
$131$	7.00000	0.611593	0.305796	−	0.952097i	$-0.401077\pi$
$131$	7.00000	0.611593	0.305796	+	0.952097i	$0.401077\pi$
$132$	0	0
$133$	4.24264	0.367884
$134$	−14.1421	−1.22169
$135$	0	0
$136$	0	0
$137$	12.7279	1.08742	0.543710	−	0.839273i	$-0.317019\pi$
$137$	12.7279	1.08742	0.543710	+	0.839273i	$0.317019\pi$
$138$	0	0
$139$	9.89949	0.839664	0.419832	−	0.907602i	$-0.362089\pi$
$139$	9.89949	0.839664	0.419832	+	0.907602i	$0.362089\pi$
$140$	0	0
$141$	0	0
$142$	−14.1421	−1.18678
$143$	1.00000	0.0836242
$144$	0	0
$145$	−8.00000	−0.664364
$146$	16.0000	1.32417
$147$	0	0
$148$	0	0
$149$	4.24264	0.347571	0.173785	−	0.984784i	$-0.444400\pi$
$149$	4.24264	0.347571	0.173785	+	0.984784i	$0.444400\pi$
$150$	0	0
$151$	−6.00000	−0.488273	−0.244137	−	0.969741i	$-0.578505\pi$
$151$	−6.00000	−0.488273	−0.244137	+	0.969741i	$0.578505\pi$
$152$	8.48528	0.688247
$153$	0	0
$154$	−2.00000	−0.161165
$155$	5.65685	0.454369
$156$	0	0
$157$	7.00000	0.558661	0.279330	−	0.960195i	$-0.409888\pi$
$157$	7.00000	0.558661	0.279330	+	0.960195i	$0.409888\pi$
$158$	−10.0000	−0.795557
$159$	0	0
$160$	0	0
$161$	7.07107	0.557278
$162$	0	0
$163$	2.82843	0.221540	0.110770	−	0.993846i	$-0.464668\pi$
$163$	2.82843	0.221540	0.110770	+	0.993846i	$0.464668\pi$
$164$	0	0
$165$	0	0
$166$	20.0000	1.55230
$167$	5.00000	0.386912	0.193456	−	0.981109i	$-0.438030\pi$
$167$	5.00000	0.386912	0.193456	+	0.981109i	$0.438030\pi$
$168$	0	0
$169$	−12.0000	−0.923077
$170$	0	0
$171$	0	0
$172$	0	0
$173$	5.00000	0.380143	0.190071	−	0.981770i	$-0.439128\pi$
$173$	5.00000	0.380143	0.190071	+	0.981770i	$0.439128\pi$
$174$	0	0
$175$	5.65685	0.427618
$176$	−4.00000	−0.301511
$177$	0	0
$178$	−6.00000	−0.449719
$179$	−14.1421	−1.05703	−0.528516	−	0.848923i	$-0.677252\pi$
$179$	−14.1421	−1.05703	−0.528516	+	0.848923i	$0.677252\pi$
$180$	0	0
$181$	−7.07107	−0.525588	−0.262794	−	0.964852i	$-0.584644\pi$
$181$	−7.07107	−0.525588	−0.262794	+	0.964852i	$0.584644\pi$
$182$	−2.00000	−0.148250
$183$	0	0
$184$	14.1421	1.04257
$185$	1.41421	0.103975
$186$	0	0
$187$	0	0
$188$	0	0
$189$	0	0
$190$	−4.24264	−0.307794
$191$	21.2132	1.53493	0.767467	−	0.641089i	$-0.221517\pi$
$191$	21.2132	1.53493	0.767467	+	0.641089i	$0.221517\pi$
$192$	0	0
$193$	14.1421	1.01797	0.508987	−	0.860774i	$-0.330020\pi$
$193$	14.1421	1.01797	0.508987	+	0.860774i	$0.330020\pi$
$194$	−20.0000	−1.43592
$195$	0	0
$196$	0	0
$197$	−7.00000	−0.498729	−0.249365	−	0.968410i	$-0.580222\pi$
$197$	−7.00000	−0.498729	−0.249365	+	0.968410i	$0.580222\pi$
$198$	0	0
$199$	2.82843	0.200502	0.100251	−	0.994962i	$-0.468035\pi$
$199$	2.82843	0.200502	0.100251	+	0.994962i	$0.468035\pi$
$200$	11.3137	0.800000
$201$	0	0
$202$	0	0
$203$	11.3137	0.794067
$204$	0	0
$205$	11.0000	0.768273
$206$	7.07107	0.492665
$207$	0	0
$208$	−4.00000	−0.277350
$209$	−3.00000	−0.207514
$210$	0	0
$211$	−14.1421	−0.973585	−0.486792	−	0.873518i	$-0.661833\pi$
$211$	−14.1421	−0.973585	−0.486792	+	0.873518i	$0.661833\pi$
$212$	0	0
$213$	0	0
$214$	9.89949	0.676716
$215$	−9.00000	−0.613795
$216$	0	0
$217$	−8.00000	−0.543075
$218$	−20.0000	−1.35457
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0	0
$223$	−15.0000	−1.00447	−0.502237	−	0.864730i	$-0.667490\pi$
$223$	−15.0000	−1.00447	−0.502237	+	0.864730i	$0.667490\pi$
$224$	0	0
$225$	0	0
$226$	−7.07107	−0.470360
$227$	25.0000	1.65931	0.829654	−	0.558278i	$-0.188538\pi$
$227$	25.0000	1.65931	0.829654	+	0.558278i	$0.188538\pi$
$228$	0	0
$229$	12.0000	0.792982	0.396491	−	0.918039i	$-0.370228\pi$
$229$	12.0000	0.792982	0.396491	+	0.918039i	$0.370228\pi$
$230$	−7.07107	−0.466252
$231$	0	0
$232$	22.6274	1.48556
$233$	25.0000	1.63780	0.818902	−	0.573933i	$-0.194583\pi$
$233$	25.0000	1.63780	0.818902	+	0.573933i	$0.194583\pi$
$234$	0	0
$235$	−8.48528	−0.553519
$236$	0	0
$237$	0	0
$238$	0	0
$239$	16.9706	1.09773	0.548867	−	0.835910i	$-0.315059\pi$
$239$	16.9706	1.09773	0.548867	+	0.835910i	$0.315059\pi$
$240$	0	0
$241$	−7.07107	−0.455488	−0.227744	−	0.973721i	$-0.573135\pi$
$241$	−7.07107	−0.455488	−0.227744	+	0.973721i	$0.573135\pi$
$242$	−14.1421	−0.909091
$243$	0	0
$244$	0	0
$245$	−5.00000	−0.319438
$246$	0	0
$247$	−3.00000	−0.190885
$248$	−16.0000	−1.01600
$249$	0	0
$250$	−12.7279	−0.804984
$251$	29.6985	1.87455	0.937276	−	0.348589i	$-0.113339\pi$
$251$	29.6985	1.87455	0.937276	+	0.348589i	$0.113339\pi$
$252$	0	0
$253$	−5.00000	−0.314347
$254$	−21.2132	−1.33103
$255$	0	0
$256$	0	0
$257$	−7.07107	−0.441081	−0.220541	−	0.975378i	$-0.570782\pi$
$257$	−7.07107	−0.441081	−0.220541	+	0.975378i	$0.570782\pi$
$258$	0	0
$259$	−2.00000	−0.124274
$260$	0	0
$261$	0	0
$262$	9.89949	0.611593
$263$	24.0416	1.48247	0.741235	−	0.671245i	$-0.234240\pi$
$263$	24.0416	1.48247	0.741235	+	0.671245i	$0.234240\pi$
$264$	0	0
$265$	−2.82843	−0.173749
$266$	6.00000	0.367884
$267$	0	0
$268$	0	0
$269$	23.0000	1.40233	0.701167	−	0.712997i	$-0.252663\pi$
$269$	23.0000	1.40233	0.701167	+	0.712997i	$0.252663\pi$
$270$	0	0
$271$	5.00000	0.303728	0.151864	−	0.988401i	$-0.451472\pi$
$271$	5.00000	0.303728	0.151864	+	0.988401i	$0.451472\pi$
$272$	0	0
$273$	0	0
$274$	18.0000	1.08742
$275$	−4.00000	−0.241209
$276$	0	0
$277$	28.2843	1.69944	0.849719	−	0.527237i	$-0.176772\pi$
$277$	28.2843	1.69944	0.849719	+	0.527237i	$0.176772\pi$
$278$	14.0000	0.839664
$279$	0	0
$280$	4.00000	0.239046
$281$	7.07107	0.421825	0.210912	−	0.977505i	$-0.432357\pi$
$281$	7.07107	0.421825	0.210912	+	0.977505i	$0.432357\pi$
$282$	0	0
$283$	−24.0416	−1.42913	−0.714563	−	0.699571i	$-0.753375\pi$
$283$	−24.0416	−1.42913	−0.714563	+	0.699571i	$0.753375\pi$
$284$	0	0
$285$	0	0
$286$	1.41421	0.0836242
$287$	−15.5563	−0.918262
$288$	0	0
$289$	0	0
$290$	−11.3137	−0.664364
$291$	0	0
$292$	0	0
$293$	4.24264	0.247858	0.123929	−	0.992291i	$-0.460451\pi$
$293$	4.24264	0.247858	0.123929	+	0.992291i	$0.460451\pi$
$294$	0	0
$295$	−9.89949	−0.576371
$296$	−4.00000	−0.232495
$297$	0	0
$298$	6.00000	0.347571
$299$	−5.00000	−0.289157
$300$	0	0
$301$	12.7279	0.733625
$302$	−8.48528	−0.488273
$303$	0	0
$304$	12.0000	0.688247
$305$	7.07107	0.404888
$306$	0	0
$307$	−10.0000	−0.570730	−0.285365	−	0.958419i	$-0.592115\pi$
$307$	−10.0000	−0.570730	−0.285365	+	0.958419i	$0.592115\pi$
$308$	0	0
$309$	0	0
$310$	8.00000	0.454369
$311$	−26.0000	−1.47432	−0.737162	−	0.675716i	$-0.763835\pi$
$311$	−26.0000	−1.47432	−0.737162	+	0.675716i	$0.763835\pi$
$312$	0	0
$313$	11.3137	0.639489	0.319744	−	0.947504i	$-0.396403\pi$
$313$	11.3137	0.639489	0.319744	+	0.947504i	$0.396403\pi$
$314$	9.89949	0.558661
$315$	0	0
$316$	0	0
$317$	−32.0000	−1.79730	−0.898650	−	0.438667i	$-0.855451\pi$
$317$	−32.0000	−1.79730	−0.898650	+	0.438667i	$0.855451\pi$
$318$	0	0
$319$	−8.00000	−0.447914
$320$	8.00000	0.447214
$321$	0	0
$322$	10.0000	0.557278
$323$	0	0
$324$	0	0
$325$	−4.00000	−0.221880
$326$	4.00000	0.221540
$327$	0	0
$328$	−31.1127	−1.71791
$329$	12.0000	0.661581
$330$	0	0
$331$	−21.0000	−1.15426	−0.577132	−	0.816651i	$-0.695828\pi$
$331$	−21.0000	−1.15426	−0.577132	+	0.816651i	$0.695828\pi$
$332$	0	0
$333$	0	0
$334$	7.07107	0.386912
$335$	−10.0000	−0.546358
$336$	0	0
$337$	−15.5563	−0.847408	−0.423704	−	0.905801i	$-0.639270\pi$
$337$	−15.5563	−0.847408	−0.423704	+	0.905801i	$0.639270\pi$
$338$	−16.9706	−0.923077
$339$	0	0
$340$	0	0
$341$	5.65685	0.306336
$342$	0	0
$343$	16.9706	0.916324
$344$	25.4558	1.37249
$345$	0	0
$346$	7.07107	0.380143
$347$	−10.0000	−0.536828	−0.268414	−	0.963304i	$-0.586500\pi$
$347$	−10.0000	−0.536828	−0.268414	+	0.963304i	$0.586500\pi$
$348$	0	0
$349$	3.00000	0.160586	0.0802932	−	0.996771i	$-0.474414\pi$
$349$	3.00000	0.160586	0.0802932	+	0.996771i	$0.474414\pi$
$350$	8.00000	0.427618
$351$	0	0
$352$	0	0
$353$	4.24264	0.225813	0.112906	−	0.993606i	$-0.463984\pi$
$353$	4.24264	0.225813	0.112906	+	0.993606i	$0.463984\pi$
$354$	0	0
$355$	−10.0000	−0.530745
$356$	0	0
$357$	0	0
$358$	−20.0000	−1.05703
$359$	−24.0416	−1.26887	−0.634434	−	0.772977i	$-0.718767\pi$
$359$	−24.0416	−1.26887	−0.634434	+	0.772977i	$0.718767\pi$
$360$	0	0
$361$	−10.0000	−0.526316
$362$	−10.0000	−0.525588
$363$	0	0
$364$	0	0
$365$	11.3137	0.592187
$366$	0	0
$367$	−35.3553	−1.84553	−0.922767	−	0.385359i	$-0.874078\pi$
$367$	−35.3553	−1.84553	−0.922767	+	0.385359i	$0.874078\pi$
$368$	20.0000	1.04257
$369$	0	0
$370$	2.00000	0.103975
$371$	4.00000	0.207670
$372$	0	0
$373$	20.0000	1.03556	0.517780	−	0.855514i	$-0.326758\pi$
$373$	20.0000	1.03556	0.517780	+	0.855514i	$0.326758\pi$
$374$	0	0
$375$	0	0
$376$	24.0000	1.23771
$377$	−8.00000	−0.412021
$378$	0	0
$379$	−9.89949	−0.508503	−0.254251	−	0.967138i	$-0.581829\pi$
$379$	−9.89949	−0.508503	−0.254251	+	0.967138i	$0.581829\pi$
$380$	0	0
$381$	0	0
$382$	30.0000	1.53493
$383$	2.82843	0.144526	0.0722629	−	0.997386i	$-0.476978\pi$
$383$	2.82843	0.144526	0.0722629	+	0.997386i	$0.476978\pi$
$384$	0	0
$385$	−1.41421	−0.0720750
$386$	20.0000	1.01797
$387$	0	0
$388$	0	0
$389$	18.3848	0.932145	0.466073	−	0.884746i	$-0.345669\pi$
$389$	18.3848	0.932145	0.466073	+	0.884746i	$0.345669\pi$
$390$	0	0
$391$	0	0
$392$	14.1421	0.714286
$393$	0	0
$394$	−9.89949	−0.498729
$395$	−7.07107	−0.355784
$396$	0	0
$397$	5.65685	0.283909	0.141955	−	0.989873i	$-0.454661\pi$
$397$	5.65685	0.283909	0.141955	+	0.989873i	$0.454661\pi$
$398$	4.00000	0.200502
$399$	0	0
$400$	16.0000	0.800000
$401$	−1.00000	−0.0499376	−0.0249688	−	0.999688i	$-0.507949\pi$
$401$	−1.00000	−0.0499376	−0.0249688	+	0.999688i	$0.507949\pi$
$402$	0	0
$403$	5.65685	0.281788
$404$	0	0
$405$	0	0
$406$	16.0000	0.794067
$407$	1.41421	0.0701000
$408$	0	0
$409$	17.0000	0.840596	0.420298	−	0.907386i	$-0.361926\pi$
$409$	17.0000	0.840596	0.420298	+	0.907386i	$0.361926\pi$
$410$	15.5563	0.768273
$411$	0	0
$412$	0	0
$413$	14.0000	0.688895
$414$	0	0
$415$	14.1421	0.694210
$416$	0	0
$417$	0	0
$418$	−4.24264	−0.207514
$419$	−2.00000	−0.0977064	−0.0488532	−	0.998806i	$-0.515557\pi$
$419$	−2.00000	−0.0977064	−0.0488532	+	0.998806i	$0.515557\pi$
$420$	0	0
$421$	−29.0000	−1.41337	−0.706687	−	0.707527i	$-0.749811\pi$
$421$	−29.0000	−1.41337	−0.706687	+	0.707527i	$0.749811\pi$
$422$	−20.0000	−0.973585
$423$	0	0
$424$	8.00000	0.388514
$425$	0	0
$426$	0	0
$427$	−10.0000	−0.483934
$428$	0	0
$429$	0	0
$430$	−12.7279	−0.613795
$431$	−26.0000	−1.25238	−0.626188	−	0.779672i	$-0.715386\pi$
$431$	−26.0000	−1.25238	−0.626188	+	0.779672i	$0.715386\pi$
$432$	0	0
$433$	15.0000	0.720854	0.360427	−	0.932787i	$-0.382631\pi$
$433$	15.0000	0.720854	0.360427	+	0.932787i	$0.382631\pi$
$434$	−11.3137	−0.543075
$435$	0	0
$436$	0	0
$437$	15.0000	0.717547
$438$	0	0
$439$	−28.2843	−1.34993	−0.674967	−	0.737848i	$-0.735842\pi$
$439$	−28.2843	−1.34993	−0.674967	+	0.737848i	$0.735842\pi$
$440$	−2.82843	−0.134840
$441$	0	0
$442$	0	0
$443$	−8.48528	−0.403148	−0.201574	−	0.979473i	$-0.564606\pi$
$443$	−8.48528	−0.403148	−0.201574	+	0.979473i	$0.564606\pi$
$444$	0	0
$445$	−4.24264	−0.201120
$446$	−21.2132	−1.00447
$447$	0	0
$448$	−11.3137	−0.534522
$449$	−10.0000	−0.471929	−0.235965	−	0.971762i	$-0.575825\pi$
$449$	−10.0000	−0.471929	−0.235965	+	0.971762i	$0.575825\pi$
$450$	0	0
$451$	11.0000	0.517970
$452$	0	0
$453$	0	0
$454$	35.3553	1.65931
$455$	−1.41421	−0.0662994
$456$	0	0
$457$	15.0000	0.701670	0.350835	−	0.936437i	$-0.385898\pi$
$457$	15.0000	0.701670	0.350835	+	0.936437i	$0.385898\pi$
$458$	16.9706	0.792982
$459$	0	0
$460$	0	0
$461$	35.3553	1.64666	0.823331	−	0.567561i	$-0.192113\pi$
$461$	35.3553	1.64666	0.823331	+	0.567561i	$0.192113\pi$
$462$	0	0
$463$	10.0000	0.464739	0.232370	−	0.972628i	$-0.425352\pi$
$463$	10.0000	0.464739	0.232370	+	0.972628i	$0.425352\pi$
$464$	32.0000	1.48556
$465$	0	0
$466$	35.3553	1.63780
$467$	−26.8701	−1.24340	−0.621699	−	0.783256i	$-0.713557\pi$
$467$	−26.8701	−1.24340	−0.621699	+	0.783256i	$0.713557\pi$
$468$	0	0
$469$	14.1421	0.653023
$470$	−12.0000	−0.553519
$471$	0	0
$472$	28.0000	1.28880
$473$	−9.00000	−0.413820
$474$	0	0
$475$	12.0000	0.550598
$476$	0	0
$477$	0	0
$478$	24.0000	1.09773
$479$	29.0000	1.32504	0.662522	−	0.749043i	$-0.269486\pi$
$479$	29.0000	1.32504	0.662522	+	0.749043i	$0.269486\pi$
$480$	0	0
$481$	1.41421	0.0644826
$482$	−10.0000	−0.455488
$483$	0	0
$484$	0	0
$485$	−14.1421	−0.642161
$486$	0	0
$487$	9.89949	0.448589	0.224294	−	0.974521i	$-0.427992\pi$
$487$	9.89949	0.448589	0.224294	+	0.974521i	$0.427992\pi$
$488$	−20.0000	−0.905357
$489$	0	0
$490$	−7.07107	−0.319438
$491$	22.6274	1.02116	0.510581	−	0.859830i	$-0.329431\pi$
$491$	22.6274	1.02116	0.510581	+	0.859830i	$0.329431\pi$
$492$	0	0
$493$	0	0
$494$	−4.24264	−0.190885
$495$	0	0
$496$	−22.6274	−1.01600
$497$	14.1421	0.634361
$498$	0	0
$499$	31.1127	1.39280	0.696398	−	0.717656i	$-0.254785\pi$
$499$	31.1127	1.39280	0.696398	+	0.717656i	$0.254785\pi$
$500$	0	0
$501$	0	0
$502$	42.0000	1.87455
$503$	11.0000	0.490466	0.245233	−	0.969464i	$-0.421136\pi$
$503$	11.0000	0.490466	0.245233	+	0.969464i	$0.421136\pi$
$504$	0	0
$505$	0	0
$506$	−7.07107	−0.314347
$507$	0	0
$508$	0	0
$509$	0	0		−	1.00000i	$-0.5\pi$
$509$	0	0			1.00000i	$0.5\pi$
$510$	0	0
$511$	−16.0000	−0.707798
$512$	−22.6274	−1.00000
$513$	0	0
$514$	−10.0000	−0.441081
$515$	5.00000	0.220326
$516$	0	0
$517$	−8.48528	−0.373182
$518$	−2.82843	−0.124274
$519$	0	0
$520$	−2.82843	−0.124035
$521$	31.0000	1.35813	0.679067	−	0.734076i	$-0.262384\pi$
$521$	31.0000	1.35813	0.679067	+	0.734076i	$0.262384\pi$
$522$	0	0
$523$	−26.0000	−1.13690	−0.568450	−	0.822718i	$-0.692457\pi$
$523$	−26.0000	−1.13690	−0.568450	+	0.822718i	$0.692457\pi$
$524$	0	0
$525$	0	0
$526$	34.0000	1.48247
$527$	0	0
$528$	0	0
$529$	2.00000	0.0869565
$530$	−4.00000	−0.173749
$531$	0	0
$532$	0	0
$533$	11.0000	0.476463
$534$	0	0
$535$	7.00000	0.302636
$536$	28.2843	1.22169
$537$	0	0
$538$	32.5269	1.40233
$539$	−5.00000	−0.215365
$540$	0	0
$541$	−41.0122	−1.76325	−0.881626	−	0.471949i	$-0.843551\pi$
$541$	−41.0122	−1.76325	−0.881626	+	0.471949i	$0.843551\pi$
$542$	7.07107	0.303728
$543$	0	0
$544$	0	0
$545$	−14.1421	−0.605783
$546$	0	0
$547$	26.8701	1.14888	0.574440	−	0.818546i	$-0.305220\pi$
$547$	26.8701	1.14888	0.574440	+	0.818546i	$0.305220\pi$
$548$	0	0
$549$	0	0
$550$	−5.65685	−0.241209
$551$	24.0000	1.02243
$552$	0	0
$553$	10.0000	0.425243
$554$	40.0000	1.69944
$555$	0	0
$556$	0	0
$557$	−21.2132	−0.898832	−0.449416	−	0.893323i	$-0.648368\pi$
$557$	−21.2132	−0.898832	−0.449416	+	0.893323i	$0.648368\pi$
$558$	0	0
$559$	−9.00000	−0.380659
$560$	5.65685	0.239046
$561$	0	0
$562$	10.0000	0.421825
$563$	−2.82843	−0.119204	−0.0596020	−	0.998222i	$-0.518983\pi$
$563$	−2.82843	−0.119204	−0.0596020	+	0.998222i	$0.518983\pi$
$564$	0	0
$565$	−5.00000	−0.210352
$566$	−34.0000	−1.42913
$567$	0	0
$568$	28.2843	1.18678
$569$	−31.1127	−1.30431	−0.652156	−	0.758085i	$-0.726135\pi$
$569$	−31.1127	−1.30431	−0.652156	+	0.758085i	$0.726135\pi$
$570$	0	0
$571$	−22.6274	−0.946928	−0.473464	−	0.880813i	$-0.656997\pi$
$571$	−22.6274	−0.946928	−0.473464	+	0.880813i	$0.656997\pi$
$572$	0	0
$573$	0	0
$574$	−22.0000	−0.918262
$575$	20.0000	0.834058
$576$	0	0
$577$	17.0000	0.707719	0.353860	−	0.935299i	$-0.384869\pi$
$577$	17.0000	0.707719	0.353860	+	0.935299i	$0.384869\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−20.0000	−0.829740
$582$	0	0
$583$	−2.82843	−0.117141
$584$	−32.0000	−1.32417
$585$	0	0
$586$	6.00000	0.247858
$587$	19.7990	0.817192	0.408596	−	0.912715i	$-0.366019\pi$
$587$	19.7990	0.817192	0.408596	+	0.912715i	$0.366019\pi$
$588$	0	0
$589$	−16.9706	−0.699260
$590$	−14.0000	−0.576371
$591$	0	0
$592$	−5.65685	−0.232495
$593$	−11.3137	−0.464598	−0.232299	−	0.972644i	$-0.574625\pi$
$593$	−11.3137	−0.464598	−0.232299	+	0.972644i	$0.574625\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	−7.07107	−0.289157
$599$	−21.2132	−0.866748	−0.433374	−	0.901214i	$-0.642677\pi$
$599$	−21.2132	−0.866748	−0.433374	+	0.901214i	$0.642677\pi$
$600$	0	0
$601$	22.6274	0.922992	0.461496	−	0.887142i	$-0.347313\pi$
$601$	22.6274	0.922992	0.461496	+	0.887142i	$0.347313\pi$
$602$	18.0000	0.733625
$603$	0	0
$604$	0	0
$605$	−10.0000	−0.406558
$606$	0	0
$607$	41.0122	1.66463	0.832317	−	0.554300i	$-0.187014\pi$
$607$	41.0122	1.66463	0.832317	+	0.554300i	$0.187014\pi$
$608$	0	0
$609$	0	0
$610$	10.0000	0.404888
$611$	−8.48528	−0.343278
$612$	0	0
$613$	11.0000	0.444286	0.222143	−	0.975014i	$-0.428695\pi$
$613$	11.0000	0.444286	0.222143	+	0.975014i	$0.428695\pi$
$614$	−14.1421	−0.570730
$615$	0	0
$616$	4.00000	0.161165
$617$	40.0000	1.61034	0.805170	−	0.593045i	$-0.202074\pi$
$617$	40.0000	1.61034	0.805170	+	0.593045i	$0.202074\pi$
$618$	0	0
$619$	49.4975	1.98947	0.994736	−	0.102473i	$-0.0326757\pi$
$619$	49.4975	1.98947	0.994736	+	0.102473i	$0.0326757\pi$
$620$	0	0
$621$	0	0
$622$	−36.7696	−1.47432
$623$	6.00000	0.240385
$624$	0	0
$625$	11.0000	0.440000
$626$	16.0000	0.639489
$627$	0	0
$628$	0	0
$629$	0	0
$630$	0	0
$631$	−9.00000	−0.358284	−0.179142	−	0.983823i	$-0.557332\pi$
$631$	−9.00000	−0.358284	−0.179142	+	0.983823i	$0.557332\pi$
$632$	20.0000	0.795557
$633$	0	0
$634$	−45.2548	−1.79730
$635$	−15.0000	−0.595257
$636$	0	0
$637$	−5.00000	−0.198107
$638$	−11.3137	−0.447914
$639$	0	0
$640$	11.3137	0.447214
$641$	−41.0000	−1.61940	−0.809701	−	0.586842i	$-0.800371\pi$
$641$	−41.0000	−1.61940	−0.809701	+	0.586842i	$0.800371\pi$
$642$	0	0
$643$	−11.3137	−0.446169	−0.223085	−	0.974799i	$-0.571613\pi$
$643$	−11.3137	−0.446169	−0.223085	+	0.974799i	$0.571613\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−21.2132	−0.833977	−0.416989	−	0.908912i	$-0.636914\pi$
$647$	−21.2132	−0.833977	−0.416989	+	0.908912i	$0.636914\pi$
$648$	0	0
$649$	−9.89949	−0.388589
$650$	−5.65685	−0.221880
$651$	0	0
$652$	0	0
$653$	35.0000	1.36966	0.684828	−	0.728705i	$-0.259877\pi$
$653$	35.0000	1.36966	0.684828	+	0.728705i	$0.259877\pi$
$654$	0	0
$655$	7.00000	0.273513
$656$	−44.0000	−1.71791
$657$	0	0
$658$	16.9706	0.661581
$659$	−38.1838	−1.48743	−0.743714	−	0.668498i	$-0.766938\pi$
$659$	−38.1838	−1.48743	−0.743714	+	0.668498i	$0.766938\pi$
$660$	0	0
$661$	−9.00000	−0.350059	−0.175030	−	0.984563i	$-0.556002\pi$
$661$	−9.00000	−0.350059	−0.175030	+	0.984563i	$0.556002\pi$
$662$	−29.6985	−1.15426
$663$	0	0
$664$	−40.0000	−1.55230
$665$	4.24264	0.164523
$666$	0	0
$667$	40.0000	1.54881
$668$	0	0
$669$	0	0
$670$	−14.1421	−0.546358
$671$	7.07107	0.272976
$672$	0	0
$673$	19.7990	0.763195	0.381597	−	0.924329i	$-0.375374\pi$
$673$	19.7990	0.763195	0.381597	+	0.924329i	$0.375374\pi$
$674$	−22.0000	−0.847408
$675$	0	0
$676$	0	0
$677$	−37.0000	−1.42203	−0.711013	−	0.703179i	$-0.751763\pi$
$677$	−37.0000	−1.42203	−0.711013	+	0.703179i	$0.751763\pi$
$678$	0	0
$679$	20.0000	0.767530
$680$	0	0
$681$	0	0
$682$	8.00000	0.306336
$683$	−31.0000	−1.18618	−0.593091	−	0.805135i	$-0.702093\pi$
$683$	−31.0000	−1.18618	−0.593091	+	0.805135i	$0.702093\pi$
$684$	0	0
$685$	12.7279	0.486309
$686$	24.0000	0.916324
$687$	0	0
$688$	36.0000	1.37249
$689$	−2.82843	−0.107754
$690$	0	0
$691$	−19.7990	−0.753189	−0.376595	−	0.926378i	$-0.622905\pi$
$691$	−19.7990	−0.753189	−0.376595	+	0.926378i	$0.622905\pi$
$692$	0	0
$693$	0	0
$694$	−14.1421	−0.536828
$695$	9.89949	0.375509
$696$	0	0
$697$	0	0
$698$	4.24264	0.160586
$699$	0	0
$700$	0	0
$701$	−42.4264	−1.60242	−0.801212	−	0.598381i	$-0.795811\pi$
$701$	−42.4264	−1.60242	−0.801212	+	0.598381i	$0.795811\pi$
$702$	0	0
$703$	−4.24264	−0.160014
$704$	8.00000	0.301511
$705$	0	0
$706$	6.00000	0.225813
$707$	0	0
$708$	0	0
$709$	−14.1421	−0.531119	−0.265560	−	0.964094i	$-0.585557\pi$
$709$	−14.1421	−0.531119	−0.265560	+	0.964094i	$0.585557\pi$
$710$	−14.1421	−0.530745
$711$	0	0
$712$	12.0000	0.449719
$713$	−28.2843	−1.05925
$714$	0	0
$715$	1.00000	0.0373979
$716$	0	0
$717$	0	0
$718$	−34.0000	−1.26887
$719$	13.0000	0.484818	0.242409	−	0.970174i	$-0.422062\pi$
$719$	13.0000	0.484818	0.242409	+	0.970174i	$0.422062\pi$
$720$	0	0
$721$	−7.07107	−0.263340
$722$	−14.1421	−0.526316
$723$	0	0
$724$	0	0
$725$	32.0000	1.18845
$726$	0	0
$727$	−20.0000	−0.741759	−0.370879	−	0.928681i	$-0.620944\pi$
$727$	−20.0000	−0.741759	−0.370879	+	0.928681i	$0.620944\pi$
$728$	4.00000	0.148250
$729$	0	0
$730$	16.0000	0.592187
$731$	0	0
$732$	0	0
$733$	24.0000	0.886460	0.443230	−	0.896408i	$-0.353832\pi$
$733$	24.0000	0.886460	0.443230	+	0.896408i	$0.353832\pi$
$734$	−50.0000	−1.84553
$735$	0	0
$736$	0	0
$737$	−10.0000	−0.368355
$738$	0	0
$739$	−15.0000	−0.551784	−0.275892	−	0.961189i	$-0.588973\pi$
$739$	−15.0000	−0.551784	−0.275892	+	0.961189i	$0.588973\pi$
$740$	0	0
$741$	0	0
$742$	5.65685	0.207670
$743$	−20.0000	−0.733729	−0.366864	−	0.930274i	$-0.619569\pi$
$743$	−20.0000	−0.733729	−0.366864	+	0.930274i	$0.619569\pi$
$744$	0	0
$745$	4.24264	0.155438
$746$	28.2843	1.03556
$747$	0	0
$748$	0	0
$749$	−9.89949	−0.361720
$750$	0	0
$751$	22.6274	0.825686	0.412843	−	0.910802i	$-0.364536\pi$
$751$	22.6274	0.825686	0.412843	+	0.910802i	$0.364536\pi$
$752$	33.9411	1.23771
$753$	0	0
$754$	−11.3137	−0.412021
$755$	−6.00000	−0.218362
$756$	0	0
$757$	33.0000	1.19941	0.599703	−	0.800223i	$-0.295286\pi$
$757$	33.0000	1.19941	0.599703	+	0.800223i	$0.295286\pi$
$758$	−14.0000	−0.508503
$759$	0	0
$760$	8.48528	0.307794
$761$	8.48528	0.307591	0.153796	−	0.988103i	$-0.450850\pi$
$761$	8.48528	0.307591	0.153796	+	0.988103i	$0.450850\pi$
$762$	0	0
$763$	20.0000	0.724049
$764$	0	0
$765$	0	0
$766$	4.00000	0.144526
$767$	−9.89949	−0.357450
$768$	0	0
$769$	−17.0000	−0.613036	−0.306518	−	0.951865i	$-0.599164\pi$
$769$	−17.0000	−0.613036	−0.306518	+	0.951865i	$0.599164\pi$
$770$	−2.00000	−0.0720750
$771$	0	0
$772$	0	0
$773$	−39.5980	−1.42424	−0.712120	−	0.702058i	$-0.752265\pi$
$773$	−39.5980	−1.42424	−0.712120	+	0.702058i	$0.752265\pi$
$774$	0	0
$775$	−22.6274	−0.812801
$776$	40.0000	1.43592
$777$	0	0
$778$	26.0000	0.932145
$779$	−33.0000	−1.18235
$780$	0	0
$781$	−10.0000	−0.357828
$782$	0	0
$783$	0	0
$784$	20.0000	0.714286
$785$	7.00000	0.249841
$786$	0	0
$787$	−5.65685	−0.201645	−0.100823	−	0.994904i	$-0.532147\pi$
$787$	−5.65685	−0.201645	−0.100823	+	0.994904i	$0.532147\pi$
$788$	0	0
$789$	0	0
$790$	−10.0000	−0.355784
$791$	7.07107	0.251418
$792$	0	0
$793$	7.07107	0.251101
$794$	8.00000	0.283909
$795$	0	0
$796$	0	0
$797$	−19.7990	−0.701316	−0.350658	−	0.936504i	$-0.614042\pi$
$797$	−19.7990	−0.701316	−0.350658	+	0.936504i	$0.614042\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	0	0
$802$	−1.41421	−0.0499376
$803$	11.3137	0.399252
$804$	0	0
$805$	7.07107	0.249222
$806$	8.00000	0.281788
$807$	0	0
$808$	0	0
$809$	5.00000	0.175791	0.0878953	−	0.996130i	$-0.471986\pi$
$809$	5.00000	0.175791	0.0878953	+	0.996130i	$0.471986\pi$
$810$	0	0
$811$	−14.1421	−0.496598	−0.248299	−	0.968683i	$-0.579871\pi$
$811$	−14.1421	−0.496598	−0.248299	+	0.968683i	$0.579871\pi$
$812$	0	0
$813$	0	0
$814$	2.00000	0.0701000
$815$	2.82843	0.0990755
$816$	0	0
$817$	27.0000	0.944610
$818$	24.0416	0.840596
$819$	0	0
$820$	0	0
$821$	−35.0000	−1.22151	−0.610754	−	0.791820i	$-0.709134\pi$
$821$	−35.0000	−1.22151	−0.610754	+	0.791820i	$0.709134\pi$
$822$	0	0
$823$	−9.89949	−0.345075	−0.172537	−	0.985003i	$-0.555197\pi$
$823$	−9.89949	−0.345075	−0.172537	+	0.985003i	$0.555197\pi$
$824$	−14.1421	−0.492665
$825$	0	0
$826$	19.7990	0.688895
$827$	−5.00000	−0.173867	−0.0869335	−	0.996214i	$-0.527707\pi$
$827$	−5.00000	−0.173867	−0.0869335	+	0.996214i	$0.527707\pi$
$828$	0	0
$829$	−32.0000	−1.11141	−0.555703	−	0.831381i	$-0.687551\pi$
$829$	−32.0000	−1.11141	−0.555703	+	0.831381i	$0.687551\pi$
$830$	20.0000	0.694210
$831$	0	0
$832$	8.00000	0.277350
$833$	0	0
$834$	0	0
$835$	5.00000	0.173032
$836$	0	0
$837$	0	0
$838$	−2.82843	−0.0977064
$839$	7.00000	0.241667	0.120833	−	0.992673i	$-0.461443\pi$
$839$	7.00000	0.241667	0.120833	+	0.992673i	$0.461443\pi$
$840$	0	0
$841$	35.0000	1.20690
$842$	−41.0122	−1.41337
$843$	0	0
$844$	0	0
$845$	−12.0000	−0.412813
$846$	0	0
$847$	14.1421	0.485930
$848$	11.3137	0.388514
$849$	0	0
$850$	0	0
$851$	−7.07107	−0.242393
$852$	0	0
$853$	−49.4975	−1.69476	−0.847381	−	0.530986i	$-0.821822\pi$
$853$	−49.4975	−1.69476	−0.847381	+	0.530986i	$0.821822\pi$
$854$	−14.1421	−0.483934
$855$	0	0
$856$	−19.7990	−0.676716
$857$	8.00000	0.273275	0.136637	−	0.990621i	$-0.456370\pi$
$857$	8.00000	0.273275	0.136637	+	0.990621i	$0.456370\pi$
$858$	0	0
$859$	−42.0000	−1.43302	−0.716511	−	0.697576i	$-0.754262\pi$
$859$	−42.0000	−1.43302	−0.716511	+	0.697576i	$0.754262\pi$
$860$	0	0
$861$	0	0
$862$	−36.7696	−1.25238
$863$	−4.24264	−0.144421	−0.0722106	−	0.997389i	$-0.523005\pi$
$863$	−4.24264	−0.144421	−0.0722106	+	0.997389i	$0.523005\pi$
$864$	0	0
$865$	5.00000	0.170005
$866$	21.2132	0.720854
$867$	0	0
$868$	0	0
$869$	−7.07107	−0.239870
$870$	0	0
$871$	−10.0000	−0.338837
$872$	40.0000	1.35457
$873$	0	0
$874$	21.2132	0.717547
$875$	12.7279	0.430282
$876$	0	0
$877$	−1.41421	−0.0477546	−0.0238773	−	0.999715i	$-0.507601\pi$
$877$	−1.41421	−0.0477546	−0.0238773	+	0.999715i	$0.507601\pi$
$878$	−40.0000	−1.34993
$879$	0	0
$880$	−4.00000	−0.134840
$881$	26.0000	0.875962	0.437981	−	0.898984i	$-0.355694\pi$
$881$	26.0000	0.875962	0.437981	+	0.898984i	$0.355694\pi$
$882$	0	0
$883$	−1.00000	−0.0336527	−0.0168263	−	0.999858i	$-0.505356\pi$
$883$	−1.00000	−0.0336527	−0.0168263	+	0.999858i	$0.505356\pi$
$884$	0	0
$885$	0	0
$886$	−12.0000	−0.403148
$887$	35.0000	1.17518	0.587592	−	0.809157i	$-0.300076\pi$
$887$	35.0000	1.17518	0.587592	+	0.809157i	$0.300076\pi$
$888$	0	0
$889$	21.2132	0.711468
$890$	−6.00000	−0.201120
$891$	0	0
$892$	0	0
$893$	25.4558	0.851847
$894$	0	0
$895$	−14.1421	−0.472719
$896$	−16.0000	−0.534522
$897$	0	0
$898$	−14.1421	−0.471929
$899$	−45.2548	−1.50933
$900$	0	0
$901$	0	0
$902$	15.5563	0.517970
$903$	0	0
$904$	14.1421	0.470360
$905$	−7.07107	−0.235050
$906$	0	0
$907$	−15.5563	−0.516540	−0.258270	−	0.966073i	$-0.583152\pi$
$907$	−15.5563	−0.516540	−0.258270	+	0.966073i	$0.583152\pi$
$908$	0	0
$909$	0	0
$910$	−2.00000	−0.0662994
$911$	29.0000	0.960813	0.480406	−	0.877046i	$-0.340489\pi$
$911$	29.0000	0.960813	0.480406	+	0.877046i	$0.340489\pi$
$912$	0	0
$913$	14.1421	0.468036
$914$	21.2132	0.701670
$915$	0	0
$916$	0	0
$917$	−9.89949	−0.326910
$918$	0	0
$919$	−37.0000	−1.22052	−0.610259	−	0.792202i	$-0.708935\pi$
$919$	−37.0000	−1.22052	−0.610259	+	0.792202i	$0.708935\pi$
$920$	14.1421	0.466252
$921$	0	0
$922$	50.0000	1.64666
$923$	−10.0000	−0.329154
$924$	0	0
$925$	−5.65685	−0.185996
$926$	14.1421	0.464739
$927$	0	0
$928$	0	0
$929$	−17.0000	−0.557752	−0.278876	−	0.960327i	$-0.589962\pi$
$929$	−17.0000	−0.557752	−0.278876	+	0.960327i	$0.589962\pi$
$930$	0	0
$931$	15.0000	0.491605
$932$	0	0
$933$	0	0
$934$	−38.0000	−1.24340
$935$	0	0
$936$	0	0
$937$	−30.0000	−0.980057	−0.490029	−	0.871706i	$-0.663014\pi$
$937$	−30.0000	−0.980057	−0.490029	+	0.871706i	$0.663014\pi$
$938$	20.0000	0.653023
$939$	0	0
$940$	0	0
$941$	−20.0000	−0.651981	−0.325991	−	0.945373i	$-0.605698\pi$
$941$	−20.0000	−0.651981	−0.325991	+	0.945373i	$0.605698\pi$
$942$	0	0
$943$	−55.0000	−1.79105
$944$	39.5980	1.28880
$945$	0	0
$946$	−12.7279	−0.413820
$947$	10.0000	0.324956	0.162478	−	0.986712i	$-0.448051\pi$
$947$	10.0000	0.324956	0.162478	+	0.986712i	$0.448051\pi$
$948$	0	0
$949$	11.3137	0.367259
$950$	16.9706	0.550598
$951$	0	0
$952$	0	0
$953$	25.4558	0.824596	0.412298	−	0.911049i	$-0.364726\pi$
$953$	25.4558	0.824596	0.412298	+	0.911049i	$0.364726\pi$
$954$	0	0
$955$	21.2132	0.686443
$956$	0	0
$957$	0	0
$958$	41.0122	1.32504
$959$	−18.0000	−0.581250
$960$	0	0
$961$	1.00000	0.0322581
$962$	2.00000	0.0644826
$963$	0	0
$964$	0	0
$965$	14.1421	0.455251
$966$	0	0
$967$	−57.0000	−1.83300	−0.916498	−	0.400039i	$-0.868997\pi$
$967$	−57.0000	−1.83300	−0.916498	+	0.400039i	$0.868997\pi$
$968$	28.2843	0.909091
$969$	0	0
$970$	−20.0000	−0.642161
$971$	14.1421	0.453843	0.226921	−	0.973913i	$-0.427134\pi$
$971$	14.1421	0.453843	0.226921	+	0.973913i	$0.427134\pi$
$972$	0	0
$973$	−14.0000	−0.448819
$974$	14.0000	0.448589
$975$	0	0
$976$	−28.2843	−0.905357
$977$	32.5269	1.04063	0.520314	−	0.853975i	$-0.325815\pi$
$977$	32.5269	1.04063	0.520314	+	0.853975i	$0.325815\pi$
$978$	0	0
$979$	−4.24264	−0.135595
$980$	0	0
$981$	0	0
$982$	32.0000	1.02116
$983$	−19.0000	−0.606006	−0.303003	−	0.952990i	$-0.597989\pi$
$983$	−19.0000	−0.606006	−0.303003	+	0.952990i	$0.597989\pi$
$984$	0	0
$985$	−7.00000	−0.223039
$986$	0	0
$987$	0	0
$988$	0	0
$989$	45.0000	1.43092
$990$	0	0
$991$	−14.1421	−0.449240	−0.224620	−	0.974446i	$-0.572114\pi$
$991$	−14.1421	−0.449240	−0.224620	+	0.974446i	$0.572114\pi$
$992$	0	0
$993$	0	0
$994$	20.0000	0.634361
$995$	2.82843	0.0896672
$996$	0	0
$997$	26.8701	0.850983	0.425492	−	0.904962i	$-0.360101\pi$
$997$	26.8701	0.850983	0.425492	+	0.904962i	$0.360101\pi$
$998$	44.0000	1.39280
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2601.2.a.q.1.2		2
3.2	odd	2		2601.2.a.p.1.1		2
17.2	even	8		153.2.f.a.55.2	yes	4
17.9	even	8		153.2.f.a.64.1	yes	4
17.16	even	2		2601.2.a.p.1.2		2
51.2	odd	8		153.2.f.a.55.1	✓	4
51.26	odd	8		153.2.f.a.64.2	yes	4
51.50	odd	2	inner	2601.2.a.q.1.1		2
68.19	odd	8		2448.2.be.r.1585.1		4
68.43	odd	8		2448.2.be.r.1441.1		4
204.155	even	8		2448.2.be.r.1585.2		4
204.179	even	8		2448.2.be.r.1441.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
153.2.f.a.55.1	✓	4	51.2	odd	8
153.2.f.a.55.2	yes	4	17.2	even	8
153.2.f.a.64.1	yes	4	17.9	even	8
153.2.f.a.64.2	yes	4	51.26	odd	8
2448.2.be.r.1441.1		4	68.43	odd	8
2448.2.be.r.1441.2		4	204.179	even	8
2448.2.be.r.1585.1		4	68.19	odd	8
2448.2.be.r.1585.2		4	204.155	even	8
2601.2.a.p.1.1		2	3.2	odd	2
2601.2.a.p.1.2		2	17.16	even	2
2601.2.a.q.1.1		2	51.50	odd	2	inner
2601.2.a.q.1.2		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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 2601 = 3^{2} \cdot 17^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2601.a (trivial)

\(f(q)\)	\(=\)	\(q+1.41421 q^{2} +1.00000 q^{5} -1.41421 q^{7} -2.82843 q^{8} +O(q^{10})\)	\(q+1.41421 q^{2} +1.00000 q^{5} -1.41421 q^{7} -2.82843 q^{8} +1.41421 q^{10} +1.00000 q^{11} +1.00000 q^{13} -2.00000 q^{14} -4.00000 q^{16} -3.00000 q^{19} +1.41421 q^{22} -5.00000 q^{23} -4.00000 q^{25} +1.41421 q^{26} -8.00000 q^{29} +5.65685 q^{31} -1.41421 q^{35} +1.41421 q^{37} -4.24264 q^{38} -2.82843 q^{40} +11.0000 q^{41} -9.00000 q^{43} -7.07107 q^{46} -8.48528 q^{47} -5.00000 q^{49} -5.65685 q^{50} -2.82843 q^{53} +1.00000 q^{55} +4.00000 q^{56} -11.3137 q^{58} -9.89949 q^{59} +7.07107 q^{61} +8.00000 q^{62} +8.00000 q^{64} +1.00000 q^{65} -10.0000 q^{67} -2.00000 q^{70} -10.0000 q^{71} +11.3137 q^{73} +2.00000 q^{74} -1.41421 q^{77} -7.07107 q^{79} -4.00000 q^{80} +15.5563 q^{82} +14.1421 q^{83} -12.7279 q^{86} -2.82843 q^{88} -4.24264 q^{89} -1.41421 q^{91} -12.0000 q^{94} -3.00000 q^{95} -14.1421 q^{97} -7.07107 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{5}+O(q^{10}) \) 2 * q + 2 * q^5	\( 2 q + 2 q^{5} + 2 q^{11} + 2 q^{13} - 4 q^{14} - 8 q^{16} - 6 q^{19} - 10 q^{23} - 8 q^{25} - 16 q^{29} + 22 q^{41} - 18 q^{43} - 10 q^{49} + 2 q^{55} + 8 q^{56} + 16 q^{62} + 16 q^{64} + 2 q^{65} - 20 q^{67} - 4 q^{70} - 20 q^{71} + 4 q^{74} - 8 q^{80} - 24 q^{94} - 6 q^{95}+O(q^{100}) \) 2 * q + 2 * q^5 + 2 * q^11 + 2 * q^13 - 4 * q^14 - 8 * q^16 - 6 * q^19 - 10 * q^23 - 8 * q^25 - 16 * q^29 + 22 * q^41 - 18 * q^43 - 10 * q^49 + 2 * q^55 + 8 * q^56 + 16 * q^62 + 16 * q^64 + 2 * q^65 - 20 * q^67 - 4 * q^70 - 20 * q^71 + 4 * q^74 - 8 * q^80 - 24 * q^94 - 6 * q^95

Introduction

L-functions

Modular forms

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