LMFDB - Embedded newform 5265.2.a.q.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5265,2,Mod(1,5265)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5265, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("5265.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5265 = 3^{4} \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5265.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:	$42.0412366642$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$1.73205$ of defining polynomial
Character	$\chi$	$=$	5265.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} -1.00000 q^{4} +1.00000 q^{5} -1.73205 q^{7} +3.00000 q^{8} +O(q^{10})$	\(q-1.00000 q^{2} -1.00000 q^{4} +1.00000 q^{5} -1.73205 q^{7} +3.00000 q^{8} -1.00000 q^{10} +2.46410 q^{11} -1.00000 q^{13} +1.73205 q^{14} -1.00000 q^{16} -4.00000 q^{17} -3.73205 q^{19} -1.00000 q^{20} -2.46410 q^{22} +2.53590 q^{23} +1.00000 q^{25} +1.00000 q^{26} +1.73205 q^{28} +9.19615 q^{29} -3.46410 q^{31} -5.00000 q^{32} +4.00000 q^{34} -1.73205 q^{35} +5.46410 q^{37} +3.73205 q^{38} +3.00000 q^{40} -5.46410 q^{41} -2.00000 q^{43} -2.46410 q^{44} -2.53590 q^{46} +2.92820 q^{47} -4.00000 q^{49} -1.00000 q^{50} +1.00000 q^{52} +7.19615 q^{53} +2.46410 q^{55} -5.19615 q^{56} -9.19615 q^{58} -11.3923 q^{59} +0.0717968 q^{61} +3.46410 q^{62} +7.00000 q^{64} -1.00000 q^{65} -3.46410 q^{67} +4.00000 q^{68} +1.73205 q^{70} -7.53590 q^{71} +1.46410 q^{73} -5.46410 q^{74} +3.73205 q^{76} -4.26795 q^{77} +10.3923 q^{79} -1.00000 q^{80} +5.46410 q^{82} +3.92820 q^{83} -4.00000 q^{85} +2.00000 q^{86} +7.39230 q^{88} -10.3923 q^{89} +1.73205 q^{91} -2.53590 q^{92} -2.92820 q^{94} -3.73205 q^{95} -13.4641 q^{97} +4.00000 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} - 2 q^{4} + 2 q^{5} + 6 q^{8}+O(q^{10}) $ 2 * q - 2 * q^2 - 2 * q^4 + 2 * q^5 + 6 * q^8	$ 2 q - 2 q^{2} - 2 q^{4} + 2 q^{5} + 6 q^{8} - 2 q^{10} - 2 q^{11} - 2 q^{13} - 2 q^{16} - 8 q^{17} - 4 q^{19} - 2 q^{20} + 2 q^{22} + 12 q^{23} + 2 q^{25} + 2 q^{26} + 8 q^{29} - 10 q^{32} + 8 q^{34} + 4 q^{37} + 4 q^{38} + 6 q^{40} - 4 q^{41} - 4 q^{43} + 2 q^{44} - 12 q^{46} - 8 q^{47} - 8 q^{49} - 2 q^{50} + 2 q^{52} + 4 q^{53} - 2 q^{55} - 8 q^{58} - 2 q^{59} + 14 q^{61} + 14 q^{64} - 2 q^{65} + 8 q^{68} - 22 q^{71} - 4 q^{73} - 4 q^{74} + 4 q^{76} - 12 q^{77} - 2 q^{80} + 4 q^{82} - 6 q^{83} - 8 q^{85} + 4 q^{86} - 6 q^{88} - 12 q^{92} + 8 q^{94} - 4 q^{95} - 20 q^{97} + 8 q^{98}+O(q^{100}) $ 2 * q - 2 * q^2 - 2 * q^4 + 2 * q^5 + 6 * q^8 - 2 * q^10 - 2 * q^11 - 2 * q^13 - 2 * q^16 - 8 * q^17 - 4 * q^19 - 2 * q^20 + 2 * q^22 + 12 * q^23 + 2 * q^25 + 2 * q^26 + 8 * q^29 - 10 * q^32 + 8 * q^34 + 4 * q^37 + 4 * q^38 + 6 * q^40 - 4 * q^41 - 4 * q^43 + 2 * q^44 - 12 * q^46 - 8 * q^47 - 8 * q^49 - 2 * q^50 + 2 * q^52 + 4 * q^53 - 2 * q^55 - 8 * q^58 - 2 * q^59 + 14 * q^61 + 14 * q^64 - 2 * q^65 + 8 * q^68 - 22 * q^71 - 4 * q^73 - 4 * q^74 + 4 * q^76 - 12 * q^77 - 2 * q^80 + 4 * q^82 - 6 * q^83 - 8 * q^85 + 4 * q^86 - 6 * q^88 - 12 * q^92 + 8 * q^94 - 4 * q^95 - 20 * q^97 + 8 * 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 $

$2$	−1.00000	−0.707107	−0.353553	−	0.935414i	$-0.615027\pi$
$2$	−1.00000	−0.707107	−0.353553	+	0.935414i	$0.615027\pi$
$3$	0	0
$3$	0	0
$4$	−1.00000	−0.500000
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−1.73205	−0.654654	−0.327327	−	0.944911i	$-0.606148\pi$
$7$	−1.73205	−0.654654	−0.327327	+	0.944911i	$0.606148\pi$
$8$	3.00000	1.06066
$9$	0	0
$10$	−1.00000	−0.316228
$11$	2.46410	0.742955	0.371477	−	0.928442i	$-0.378851\pi$
$11$	2.46410	0.742955	0.371477	+	0.928442i	$0.378851\pi$
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	1.73205	0.462910
$15$	0	0
$16$	−1.00000	−0.250000
$17$	−4.00000	−0.970143	−0.485071	−	0.874475i	$-0.661206\pi$
$17$	−4.00000	−0.970143	−0.485071	+	0.874475i	$0.661206\pi$
$18$	0	0
$19$	−3.73205	−0.856191	−0.428096	−	0.903733i	$-0.640815\pi$
$19$	−3.73205	−0.856191	−0.428096	+	0.903733i	$0.640815\pi$
$20$	−1.00000	−0.223607
$21$	0	0
$22$	−2.46410	−0.525348
$23$	2.53590	0.528771	0.264386	−	0.964417i	$-0.414831\pi$
$23$	2.53590	0.528771	0.264386	+	0.964417i	$0.414831\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	1.00000	0.196116
$27$	0	0
$28$	1.73205	0.327327
$29$	9.19615	1.70768	0.853841	−	0.520533i	$-0.174267\pi$
$29$	9.19615	1.70768	0.853841	+	0.520533i	$0.174267\pi$
$30$	0	0
$31$	−3.46410	−0.622171	−0.311086	−	0.950382i	$-0.600693\pi$
$31$	−3.46410	−0.622171	−0.311086	+	0.950382i	$0.600693\pi$
$32$	−5.00000	−0.883883
$33$	0	0
$34$	4.00000	0.685994
$35$	−1.73205	−0.292770
$36$	0	0
$37$	5.46410	0.898293	0.449146	−	0.893458i	$-0.351728\pi$
$37$	5.46410	0.898293	0.449146	+	0.893458i	$0.351728\pi$
$38$	3.73205	0.605419
$39$	0	0
$40$	3.00000	0.474342
$41$	−5.46410	−0.853349	−0.426675	−	0.904405i	$-0.640315\pi$
$41$	−5.46410	−0.853349	−0.426675	+	0.904405i	$0.640315\pi$
$42$	0	0
$43$	−2.00000	−0.304997	−0.152499	−	0.988304i	$-0.548732\pi$
$43$	−2.00000	−0.304997	−0.152499	+	0.988304i	$0.548732\pi$
$44$	−2.46410	−0.371477
$45$	0	0
$46$	−2.53590	−0.373898
$47$	2.92820	0.427122	0.213561	−	0.976930i	$-0.431494\pi$
$47$	2.92820	0.427122	0.213561	+	0.976930i	$0.431494\pi$
$48$	0	0
$49$	−4.00000	−0.571429
$50$	−1.00000	−0.141421
$51$	0	0
$52$	1.00000	0.138675
$53$	7.19615	0.988468	0.494234	−	0.869329i	$-0.335449\pi$
$53$	7.19615	0.988468	0.494234	+	0.869329i	$0.335449\pi$
$54$	0	0
$55$	2.46410	0.332259
$56$	−5.19615	−0.694365
$57$	0	0
$58$	−9.19615	−1.20751
$59$	−11.3923	−1.48315	−0.741576	−	0.670869i	$-0.765921\pi$
$59$	−11.3923	−1.48315	−0.741576	+	0.670869i	$0.765921\pi$
$60$	0	0
$61$	0.0717968	0.00919263	0.00459632	−	0.999989i	$-0.498537\pi$
$61$	0.0717968	0.00919263	0.00459632	+	0.999989i	$0.498537\pi$
$62$	3.46410	0.439941
$63$	0	0
$64$	7.00000	0.875000
$65$	−1.00000	−0.124035
$66$	0	0
$67$	−3.46410	−0.423207	−0.211604	−	0.977356i	$-0.567869\pi$
$67$	−3.46410	−0.423207	−0.211604	+	0.977356i	$0.567869\pi$
$68$	4.00000	0.485071
$69$	0	0
$70$	1.73205	0.207020
$71$	−7.53590	−0.894347	−0.447173	−	0.894447i	$-0.647569\pi$
$71$	−7.53590	−0.894347	−0.447173	+	0.894447i	$0.647569\pi$
$72$	0	0
$73$	1.46410	0.171360	0.0856801	−	0.996323i	$-0.472694\pi$
$73$	1.46410	0.171360	0.0856801	+	0.996323i	$0.472694\pi$
$74$	−5.46410	−0.635189
$75$	0	0
$76$	3.73205	0.428096
$77$	−4.26795	−0.486378
$78$	0	0
$79$	10.3923	1.16923	0.584613	−	0.811312i	$-0.301246\pi$
$79$	10.3923	1.16923	0.584613	+	0.811312i	$0.301246\pi$
$80$	−1.00000	−0.111803
$81$	0	0
$82$	5.46410	0.603409
$83$	3.92820	0.431176	0.215588	−	0.976484i	$-0.430833\pi$
$83$	3.92820	0.431176	0.215588	+	0.976484i	$0.430833\pi$
$84$	0	0
$85$	−4.00000	−0.433861
$86$	2.00000	0.215666
$87$	0	0
$88$	7.39230	0.788022
$89$	−10.3923	−1.10158	−0.550791	−	0.834643i	$-0.685674\pi$
$89$	−10.3923	−1.10158	−0.550791	+	0.834643i	$0.685674\pi$
$90$	0	0
$91$	1.73205	0.181568
$92$	−2.53590	−0.264386
$93$	0	0
$94$	−2.92820	−0.302021
$95$	−3.73205	−0.382900
$96$	0	0
$97$	−13.4641	−1.36707	−0.683536	−	0.729917i	$-0.739559\pi$
$97$	−13.4641	−1.36707	−0.683536	+	0.729917i	$0.739559\pi$
$98$	4.00000	0.404061
$99$	0	0
$100$	−1.00000	−0.100000
$101$	−5.19615	−0.517036	−0.258518	−	0.966006i	$-0.583234\pi$
$101$	−5.19615	−0.517036	−0.258518	+	0.966006i	$0.583234\pi$
$102$	0	0
$103$	4.53590	0.446935	0.223468	−	0.974711i	$-0.428262\pi$
$103$	4.53590	0.446935	0.223468	+	0.974711i	$0.428262\pi$
$104$	−3.00000	−0.294174
$105$	0	0
$106$	−7.19615	−0.698952
$107$	−12.9282	−1.24982	−0.624908	−	0.780698i	$-0.714864\pi$
$107$	−12.9282	−1.24982	−0.624908	+	0.780698i	$0.714864\pi$
$108$	0	0
$109$	14.9282	1.42986	0.714931	−	0.699195i	$-0.246458\pi$
$109$	14.9282	1.42986	0.714931	+	0.699195i	$0.246458\pi$
$110$	−2.46410	−0.234943
$111$	0	0
$112$	1.73205	0.163663
$113$	−1.33975	−0.126033	−0.0630163	−	0.998012i	$-0.520072\pi$
$113$	−1.33975	−0.126033	−0.0630163	+	0.998012i	$0.520072\pi$
$114$	0	0
$115$	2.53590	0.236474
$116$	−9.19615	−0.853841
$117$	0	0
$118$	11.3923	1.04875
$119$	6.92820	0.635107
$120$	0	0
$121$	−4.92820	−0.448018
$122$	−0.0717968	−0.00650017
$123$	0	0
$124$	3.46410	0.311086
$125$	1.00000	0.0894427
$126$	0	0
$127$	10.0000	0.887357	0.443678	−	0.896186i	$-0.353673\pi$
$127$	10.0000	0.887357	0.443678	+	0.896186i	$0.353673\pi$
$128$	3.00000	0.265165
$129$	0	0
$130$	1.00000	0.0877058
$131$	−8.53590	−0.745785	−0.372892	−	0.927875i	$-0.621634\pi$
$131$	−8.53590	−0.745785	−0.372892	+	0.927875i	$0.621634\pi$
$132$	0	0
$133$	6.46410	0.560509
$134$	3.46410	0.299253
$135$	0	0
$136$	−12.0000	−1.02899
$137$	17.3205	1.47979	0.739895	−	0.672722i	$-0.234875\pi$
$137$	17.3205	1.47979	0.739895	+	0.672722i	$0.234875\pi$
$138$	0	0
$139$	−1.46410	−0.124183	−0.0620917	−	0.998070i	$-0.519777\pi$
$139$	−1.46410	−0.124183	−0.0620917	+	0.998070i	$0.519777\pi$
$140$	1.73205	0.146385
$141$	0	0
$142$	7.53590	0.632399
$143$	−2.46410	−0.206059
$144$	0	0
$145$	9.19615	0.763699
$146$	−1.46410	−0.121170
$147$	0	0
$148$	−5.46410	−0.449146
$149$	14.3923	1.17906	0.589532	−	0.807745i	$-0.299312\pi$
$149$	14.3923	1.17906	0.589532	+	0.807745i	$0.299312\pi$
$150$	0	0
$151$	17.0526	1.38772	0.693859	−	0.720111i	$-0.255909\pi$
$151$	17.0526	1.38772	0.693859	+	0.720111i	$0.255909\pi$
$152$	−11.1962	−0.908128
$153$	0	0
$154$	4.26795	0.343921
$155$	−3.46410	−0.278243
$156$	0	0
$157$	14.4641	1.15436	0.577180	−	0.816617i	$-0.304153\pi$
$157$	14.4641	1.15436	0.577180	+	0.816617i	$0.304153\pi$
$158$	−10.3923	−0.826767
$159$	0	0
$160$	−5.00000	−0.395285
$161$	−4.39230	−0.346162
$162$	0	0
$163$	11.4641	0.897938	0.448969	−	0.893547i	$-0.351791\pi$
$163$	11.4641	0.897938	0.448969	+	0.893547i	$0.351791\pi$
$164$	5.46410	0.426675
$165$	0	0
$166$	−3.92820	−0.304888
$167$	−4.85641	−0.375800	−0.187900	−	0.982188i	$-0.560168\pi$
$167$	−4.85641	−0.375800	−0.187900	+	0.982188i	$0.560168\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	4.00000	0.306786
$171$	0	0
$172$	2.00000	0.152499
$173$	−0.803848	−0.0611154	−0.0305577	−	0.999533i	$-0.509728\pi$
$173$	−0.803848	−0.0611154	−0.0305577	+	0.999533i	$0.509728\pi$
$174$	0	0
$175$	−1.73205	−0.130931
$176$	−2.46410	−0.185739
$177$	0	0
$178$	10.3923	0.778936
$179$	−6.53590	−0.488516	−0.244258	−	0.969710i	$-0.578544\pi$
$179$	−6.53590	−0.488516	−0.244258	+	0.969710i	$0.578544\pi$
$180$	0	0
$181$	−13.9282	−1.03528	−0.517638	−	0.855600i	$-0.673188\pi$
$181$	−13.9282	−1.03528	−0.517638	+	0.855600i	$0.673188\pi$
$182$	−1.73205	−0.128388
$183$	0	0
$184$	7.60770	0.560847
$185$	5.46410	0.401729
$186$	0	0
$187$	−9.85641	−0.720772
$188$	−2.92820	−0.213561
$189$	0	0
$190$	3.73205	0.270751
$191$	−25.3205	−1.83213	−0.916064	−	0.401032i	$-0.868651\pi$
$191$	−25.3205	−1.83213	−0.916064	+	0.401032i	$0.868651\pi$
$192$	0	0
$193$	3.60770	0.259688	0.129844	−	0.991534i	$-0.458552\pi$
$193$	3.60770	0.259688	0.129844	+	0.991534i	$0.458552\pi$
$194$	13.4641	0.966666
$195$	0	0
$196$	4.00000	0.285714
$197$	−18.0000	−1.28245	−0.641223	−	0.767354i	$-0.721573\pi$
$197$	−18.0000	−1.28245	−0.641223	+	0.767354i	$0.721573\pi$
$198$	0	0
$199$	6.53590	0.463318	0.231659	−	0.972797i	$-0.425585\pi$
$199$	6.53590	0.463318	0.231659	+	0.972797i	$0.425585\pi$
$200$	3.00000	0.212132
$201$	0	0
$202$	5.19615	0.365600
$203$	−15.9282	−1.11794
$204$	0	0
$205$	−5.46410	−0.381629
$206$	−4.53590	−0.316031
$207$	0	0
$208$	1.00000	0.0693375
$209$	−9.19615	−0.636111
$210$	0	0
$211$	−14.3923	−0.990807	−0.495404	−	0.868663i	$-0.664980\pi$
$211$	−14.3923	−0.990807	−0.495404	+	0.868663i	$0.664980\pi$
$212$	−7.19615	−0.494234
$213$	0	0
$214$	12.9282	0.883754
$215$	−2.00000	−0.136399
$216$	0	0
$217$	6.00000	0.407307
$218$	−14.9282	−1.01107
$219$	0	0
$220$	−2.46410	−0.166130
$221$	4.00000	0.269069
$222$	0	0
$223$	−7.58846	−0.508161	−0.254080	−	0.967183i	$-0.581773\pi$
$223$	−7.58846	−0.508161	−0.254080	+	0.967183i	$0.581773\pi$
$224$	8.66025	0.578638
$225$	0	0
$226$	1.33975	0.0891186
$227$	−24.8564	−1.64978	−0.824889	−	0.565295i	$-0.808762\pi$
$227$	−24.8564	−1.64978	−0.824889	+	0.565295i	$0.808762\pi$
$228$	0	0
$229$	−19.8564	−1.31215	−0.656074	−	0.754696i	$-0.727784\pi$
$229$	−19.8564	−1.31215	−0.656074	+	0.754696i	$0.727784\pi$
$230$	−2.53590	−0.167212
$231$	0	0
$232$	27.5885	1.81127
$233$	−4.26795	−0.279603	−0.139801	−	0.990180i	$-0.544646\pi$
$233$	−4.26795	−0.279603	−0.139801	+	0.990180i	$0.544646\pi$
$234$	0	0
$235$	2.92820	0.191015
$236$	11.3923	0.741576
$237$	0	0
$238$	−6.92820	−0.449089
$239$	−15.5359	−1.00493	−0.502467	−	0.864596i	$-0.667574\pi$
$239$	−15.5359	−1.00493	−0.502467	+	0.864596i	$0.667574\pi$
$240$	0	0
$241$	−4.53590	−0.292183	−0.146091	−	0.989271i	$-0.546669\pi$
$241$	−4.53590	−0.292183	−0.146091	+	0.989271i	$0.546669\pi$
$242$	4.92820	0.316797
$243$	0	0
$244$	−0.0717968	−0.00459632
$245$	−4.00000	−0.255551
$246$	0	0
$247$	3.73205	0.237465
$248$	−10.3923	−0.659912
$249$	0	0
$250$	−1.00000	−0.0632456
$251$	1.85641	0.117175	0.0585877	−	0.998282i	$-0.481340\pi$
$251$	1.85641	0.117175	0.0585877	+	0.998282i	$0.481340\pi$
$252$	0	0
$253$	6.24871	0.392853
$254$	−10.0000	−0.627456
$255$	0	0
$256$	−17.0000	−1.06250
$257$	−0.803848	−0.0501426	−0.0250713	−	0.999686i	$-0.507981\pi$
$257$	−0.803848	−0.0501426	−0.0250713	+	0.999686i	$0.507981\pi$
$258$	0	0
$259$	−9.46410	−0.588071
$260$	1.00000	0.0620174
$261$	0	0
$262$	8.53590	0.527350
$263$	2.39230	0.147516	0.0737579	−	0.997276i	$-0.476501\pi$
$263$	2.39230	0.147516	0.0737579	+	0.997276i	$0.476501\pi$
$264$	0	0
$265$	7.19615	0.442056
$266$	−6.46410	−0.396339
$267$	0	0
$268$	3.46410	0.211604
$269$	14.9282	0.910189	0.455094	−	0.890443i	$-0.349606\pi$
$269$	14.9282	0.910189	0.455094	+	0.890443i	$0.349606\pi$
$270$	0	0
$271$	−7.19615	−0.437135	−0.218568	−	0.975822i	$-0.570138\pi$
$271$	−7.19615	−0.437135	−0.218568	+	0.975822i	$0.570138\pi$
$272$	4.00000	0.242536
$273$	0	0
$274$	−17.3205	−1.04637
$275$	2.46410	0.148591
$276$	0	0
$277$	−12.4641	−0.748895	−0.374448	−	0.927248i	$-0.622168\pi$
$277$	−12.4641	−0.748895	−0.374448	+	0.927248i	$0.622168\pi$
$278$	1.46410	0.0878110
$279$	0	0
$280$	−5.19615	−0.310530
$281$	0.928203	0.0553720	0.0276860	−	0.999617i	$-0.491186\pi$
$281$	0.928203	0.0553720	0.0276860	+	0.999617i	$0.491186\pi$
$282$	0	0
$283$	−27.4641	−1.63257	−0.816286	−	0.577648i	$-0.803970\pi$
$283$	−27.4641	−1.63257	−0.816286	+	0.577648i	$0.803970\pi$
$284$	7.53590	0.447173
$285$	0	0
$286$	2.46410	0.145705
$287$	9.46410	0.558648
$288$	0	0
$289$	−1.00000	−0.0588235
$290$	−9.19615	−0.540017
$291$	0	0
$292$	−1.46410	−0.0856801
$293$	−23.3205	−1.36240	−0.681199	−	0.732098i	$-0.738541\pi$
$293$	−23.3205	−1.36240	−0.681199	+	0.732098i	$0.738541\pi$
$294$	0	0
$295$	−11.3923	−0.663286
$296$	16.3923	0.952783
$297$	0	0
$298$	−14.3923	−0.833724
$299$	−2.53590	−0.146655
$300$	0	0
$301$	3.46410	0.199667
$302$	−17.0526	−0.981264
$303$	0	0
$304$	3.73205	0.214048
$305$	0.0717968	0.00411107
$306$	0	0
$307$	−4.53590	−0.258877	−0.129439	−	0.991587i	$-0.541318\pi$
$307$	−4.53590	−0.258877	−0.129439	+	0.991587i	$0.541318\pi$
$308$	4.26795	0.243189
$309$	0	0
$310$	3.46410	0.196748
$311$	6.00000	0.340229	0.170114	−	0.985424i	$-0.445586\pi$
$311$	6.00000	0.340229	0.170114	+	0.985424i	$0.445586\pi$
$312$	0	0
$313$	−30.3205	−1.71382	−0.856908	−	0.515469i	$-0.827618\pi$
$313$	−30.3205	−1.71382	−0.856908	+	0.515469i	$0.827618\pi$
$314$	−14.4641	−0.816256
$315$	0	0
$316$	−10.3923	−0.584613
$317$	3.32051	0.186498	0.0932492	−	0.995643i	$-0.470275\pi$
$317$	3.32051	0.186498	0.0932492	+	0.995643i	$0.470275\pi$
$318$	0	0
$319$	22.6603	1.26873
$320$	7.00000	0.391312
$321$	0	0
$322$	4.39230	0.244774
$323$	14.9282	0.830627
$324$	0	0
$325$	−1.00000	−0.0554700
$326$	−11.4641	−0.634938
$327$	0	0
$328$	−16.3923	−0.905114
$329$	−5.07180	−0.279617
$330$	0	0
$331$	5.32051	0.292442	0.146221	−	0.989252i	$-0.453289\pi$
$331$	5.32051	0.292442	0.146221	+	0.989252i	$0.453289\pi$
$332$	−3.92820	−0.215588
$333$	0	0
$334$	4.85641	0.265731
$335$	−3.46410	−0.189264
$336$	0	0
$337$	−32.9282	−1.79371	−0.896857	−	0.442321i	$-0.854155\pi$
$337$	−32.9282	−1.79371	−0.896857	+	0.442321i	$0.854155\pi$
$338$	−1.00000	−0.0543928
$339$	0	0
$340$	4.00000	0.216930
$341$	−8.53590	−0.462245
$342$	0	0
$343$	19.0526	1.02874
$344$	−6.00000	−0.323498
$345$	0	0
$346$	0.803848	0.0432151
$347$	−24.7846	−1.33051	−0.665254	−	0.746617i	$-0.731677\pi$
$347$	−24.7846	−1.33051	−0.665254	+	0.746617i	$0.731677\pi$
$348$	0	0
$349$	14.3923	0.770402	0.385201	−	0.922833i	$-0.374132\pi$
$349$	14.3923	0.770402	0.385201	+	0.922833i	$0.374132\pi$
$350$	1.73205	0.0925820
$351$	0	0
$352$	−12.3205	−0.656685
$353$	14.5359	0.773668	0.386834	−	0.922149i	$-0.373569\pi$
$353$	14.5359	0.773668	0.386834	+	0.922149i	$0.373569\pi$
$354$	0	0
$355$	−7.53590	−0.399964
$356$	10.3923	0.550791
$357$	0	0
$358$	6.53590	0.345433
$359$	−27.7128	−1.46263	−0.731313	−	0.682042i	$-0.761092\pi$
$359$	−27.7128	−1.46263	−0.731313	+	0.682042i	$0.761092\pi$
$360$	0	0
$361$	−5.07180	−0.266937
$362$	13.9282	0.732050
$363$	0	0
$364$	−1.73205	−0.0907841
$365$	1.46410	0.0766346
$366$	0	0
$367$	25.8564	1.34969	0.674847	−	0.737958i	$-0.264210\pi$
$367$	25.8564	1.34969	0.674847	+	0.737958i	$0.264210\pi$
$368$	−2.53590	−0.132193
$369$	0	0
$370$	−5.46410	−0.284065
$371$	−12.4641	−0.647104
$372$	0	0
$373$	16.4641	0.852479	0.426239	−	0.904610i	$-0.359838\pi$
$373$	16.4641	0.852479	0.426239	+	0.904610i	$0.359838\pi$
$374$	9.85641	0.509663
$375$	0	0
$376$	8.78461	0.453032
$377$	−9.19615	−0.473626
$378$	0	0
$379$	−36.5167	−1.87573	−0.937867	−	0.346994i	$-0.887202\pi$
$379$	−36.5167	−1.87573	−0.937867	+	0.346994i	$0.887202\pi$
$380$	3.73205	0.191450
$381$	0	0
$382$	25.3205	1.29551
$383$	17.7846	0.908751	0.454376	−	0.890810i	$-0.349862\pi$
$383$	17.7846	0.908751	0.454376	+	0.890810i	$0.349862\pi$
$384$	0	0
$385$	−4.26795	−0.217515
$386$	−3.60770	−0.183627
$387$	0	0
$388$	13.4641	0.683536
$389$	13.8564	0.702548	0.351274	−	0.936273i	$-0.385749\pi$
$389$	13.8564	0.702548	0.351274	+	0.936273i	$0.385749\pi$
$390$	0	0
$391$	−10.1436	−0.512984
$392$	−12.0000	−0.606092
$393$	0	0
$394$	18.0000	0.906827
$395$	10.3923	0.522894
$396$	0	0
$397$	−20.0000	−1.00377	−0.501886	−	0.864934i	$-0.667360\pi$
$397$	−20.0000	−1.00377	−0.501886	+	0.864934i	$0.667360\pi$
$398$	−6.53590	−0.327615
$399$	0	0
$400$	−1.00000	−0.0500000
$401$	−13.6077	−0.679536	−0.339768	−	0.940509i	$-0.610349\pi$
$401$	−13.6077	−0.679536	−0.339768	+	0.940509i	$0.610349\pi$
$402$	0	0
$403$	3.46410	0.172559
$404$	5.19615	0.258518
$405$	0	0
$406$	15.9282	0.790503
$407$	13.4641	0.667391
$408$	0	0
$409$	−26.3923	−1.30502	−0.652508	−	0.757782i	$-0.726283\pi$
$409$	−26.3923	−1.30502	−0.652508	+	0.757782i	$0.726283\pi$
$410$	5.46410	0.269853
$411$	0	0
$412$	−4.53590	−0.223468
$413$	19.7321	0.970951
$414$	0	0
$415$	3.92820	0.192828
$416$	5.00000	0.245145
$417$	0	0
$418$	9.19615	0.449799
$419$	−8.92820	−0.436171	−0.218086	−	0.975930i	$-0.569981\pi$
$419$	−8.92820	−0.436171	−0.218086	+	0.975930i	$0.569981\pi$
$420$	0	0
$421$	−23.4641	−1.14357	−0.571785	−	0.820403i	$-0.693749\pi$
$421$	−23.4641	−1.14357	−0.571785	+	0.820403i	$0.693749\pi$
$422$	14.3923	0.700606
$423$	0	0
$424$	21.5885	1.04843
$425$	−4.00000	−0.194029
$426$	0	0
$427$	−0.124356	−0.00601799
$428$	12.9282	0.624908
$429$	0	0
$430$	2.00000	0.0964486
$431$	−5.53590	−0.266655	−0.133327	−	0.991072i	$-0.542566\pi$
$431$	−5.53590	−0.266655	−0.133327	+	0.991072i	$0.542566\pi$
$432$	0	0
$433$	25.2487	1.21338	0.606688	−	0.794940i	$-0.292498\pi$
$433$	25.2487	1.21338	0.606688	+	0.794940i	$0.292498\pi$
$434$	−6.00000	−0.288009
$435$	0	0
$436$	−14.9282	−0.714931
$437$	−9.46410	−0.452729
$438$	0	0
$439$	5.85641	0.279511	0.139756	−	0.990186i	$-0.455368\pi$
$439$	5.85641	0.279511	0.139756	+	0.990186i	$0.455368\pi$
$440$	7.39230	0.352414
$441$	0	0
$442$	−4.00000	−0.190261
$443$	−2.53590	−0.120484	−0.0602421	−	0.998184i	$-0.519187\pi$
$443$	−2.53590	−0.120484	−0.0602421	+	0.998184i	$0.519187\pi$
$444$	0	0
$445$	−10.3923	−0.492642
$446$	7.58846	0.359324
$447$	0	0
$448$	−12.1244	−0.572822
$449$	−20.3923	−0.962372	−0.481186	−	0.876618i	$-0.659794\pi$
$449$	−20.3923	−0.962372	−0.481186	+	0.876618i	$0.659794\pi$
$450$	0	0
$451$	−13.4641	−0.634000
$452$	1.33975	0.0630163
$453$	0	0
$454$	24.8564	1.16657
$455$	1.73205	0.0811998
$456$	0	0
$457$	34.9282	1.63387	0.816936	−	0.576728i	$-0.195671\pi$
$457$	34.9282	1.63387	0.816936	+	0.576728i	$0.195671\pi$
$458$	19.8564	0.927829
$459$	0	0
$460$	−2.53590	−0.118237
$461$	−34.6410	−1.61339	−0.806696	−	0.590966i	$-0.798747\pi$
$461$	−34.6410	−1.61339	−0.806696	+	0.590966i	$0.798747\pi$
$462$	0	0
$463$	12.1244	0.563467	0.281733	−	0.959493i	$-0.409091\pi$
$463$	12.1244	0.563467	0.281733	+	0.959493i	$0.409091\pi$
$464$	−9.19615	−0.426921
$465$	0	0
$466$	4.26795	0.197709
$467$	−27.8564	−1.28904	−0.644520	−	0.764587i	$-0.722943\pi$
$467$	−27.8564	−1.28904	−0.644520	+	0.764587i	$0.722943\pi$
$468$	0	0
$469$	6.00000	0.277054
$470$	−2.92820	−0.135068
$471$	0	0
$472$	−34.1769	−1.57312
$473$	−4.92820	−0.226599
$474$	0	0
$475$	−3.73205	−0.171238
$476$	−6.92820	−0.317554
$477$	0	0
$478$	15.5359	0.710595
$479$	−14.3205	−0.654321	−0.327160	−	0.944969i	$-0.606092\pi$
$479$	−14.3205	−0.654321	−0.327160	+	0.944969i	$0.606092\pi$
$480$	0	0
$481$	−5.46410	−0.249142
$482$	4.53590	0.206605
$483$	0	0
$484$	4.92820	0.224009
$485$	−13.4641	−0.611373
$486$	0	0
$487$	15.0526	0.682097	0.341048	−	0.940046i	$-0.389218\pi$
$487$	15.0526	0.682097	0.341048	+	0.940046i	$0.389218\pi$
$488$	0.215390	0.00975026
$489$	0	0
$490$	4.00000	0.180702
$491$	38.6410	1.74384	0.871922	−	0.489644i	$-0.162873\pi$
$491$	38.6410	1.74384	0.871922	+	0.489644i	$0.162873\pi$
$492$	0	0
$493$	−36.7846	−1.65670
$494$	−3.73205	−0.167913
$495$	0	0
$496$	3.46410	0.155543
$497$	13.0526	0.585487
$498$	0	0
$499$	15.4449	0.691407	0.345704	−	0.938344i	$-0.387640\pi$
$499$	15.4449	0.691407	0.345704	+	0.938344i	$0.387640\pi$
$500$	−1.00000	−0.0447214
$501$	0	0
$502$	−1.85641	−0.0828555
$503$	−3.60770	−0.160859	−0.0804296	−	0.996760i	$-0.525629\pi$
$503$	−3.60770	−0.160859	−0.0804296	+	0.996760i	$0.525629\pi$
$504$	0	0
$505$	−5.19615	−0.231226
$506$	−6.24871	−0.277789
$507$	0	0
$508$	−10.0000	−0.443678
$509$	−4.00000	−0.177297	−0.0886484	−	0.996063i	$-0.528255\pi$
$509$	−4.00000	−0.177297	−0.0886484	+	0.996063i	$0.528255\pi$
$510$	0	0
$511$	−2.53590	−0.112182
$512$	11.0000	0.486136
$513$	0	0
$514$	0.803848	0.0354562
$515$	4.53590	0.199876
$516$	0	0
$517$	7.21539	0.317333
$518$	9.46410	0.415829
$519$	0	0
$520$	−3.00000	−0.131559
$521$	11.0526	0.484221	0.242111	−	0.970249i	$-0.422160\pi$
$521$	11.0526	0.484221	0.242111	+	0.970249i	$0.422160\pi$
$522$	0	0
$523$	−8.92820	−0.390403	−0.195202	−	0.980763i	$-0.562536\pi$
$523$	−8.92820	−0.390403	−0.195202	+	0.980763i	$0.562536\pi$
$524$	8.53590	0.372892
$525$	0	0
$526$	−2.39230	−0.104309
$527$	13.8564	0.603595
$528$	0	0
$529$	−16.5692	−0.720401
$530$	−7.19615	−0.312581
$531$	0	0
$532$	−6.46410	−0.280254
$533$	5.46410	0.236677
$534$	0	0
$535$	−12.9282	−0.558935
$536$	−10.3923	−0.448879
$537$	0	0
$538$	−14.9282	−0.643601
$539$	−9.85641	−0.424545
$540$	0	0
$541$	−37.1769	−1.59836	−0.799180	−	0.601092i	$-0.794733\pi$
$541$	−37.1769	−1.59836	−0.799180	+	0.601092i	$0.794733\pi$
$542$	7.19615	0.309101
$543$	0	0
$544$	20.0000	0.857493
$545$	14.9282	0.639454
$546$	0	0
$547$	25.7128	1.09940	0.549700	−	0.835362i	$-0.314742\pi$
$547$	25.7128	1.09940	0.549700	+	0.835362i	$0.314742\pi$
$548$	−17.3205	−0.739895
$549$	0	0
$550$	−2.46410	−0.105070
$551$	−34.3205	−1.46210
$552$	0	0
$553$	−18.0000	−0.765438
$554$	12.4641	0.529549
$555$	0	0
$556$	1.46410	0.0620917
$557$	24.0000	1.01691	0.508456	−	0.861088i	$-0.330216\pi$
$557$	24.0000	1.01691	0.508456	+	0.861088i	$0.330216\pi$
$558$	0	0
$559$	2.00000	0.0845910
$560$	1.73205	0.0731925
$561$	0	0
$562$	−0.928203	−0.0391539
$563$	42.9282	1.80921	0.904604	−	0.426253i	$-0.140167\pi$
$563$	42.9282	1.80921	0.904604	+	0.426253i	$0.140167\pi$
$564$	0	0
$565$	−1.33975	−0.0563635
$566$	27.4641	1.15440
$567$	0	0
$568$	−22.6077	−0.948598
$569$	12.0000	0.503066	0.251533	−	0.967849i	$-0.419065\pi$
$569$	12.0000	0.503066	0.251533	+	0.967849i	$0.419065\pi$
$570$	0	0
$571$	−34.0000	−1.42286	−0.711428	−	0.702759i	$-0.751951\pi$
$571$	−34.0000	−1.42286	−0.711428	+	0.702759i	$0.751951\pi$
$572$	2.46410	0.103029
$573$	0	0
$574$	−9.46410	−0.395024
$575$	2.53590	0.105754
$576$	0	0
$577$	8.00000	0.333044	0.166522	−	0.986038i	$-0.446746\pi$
$577$	8.00000	0.333044	0.166522	+	0.986038i	$0.446746\pi$
$578$	1.00000	0.0415945
$579$	0	0
$580$	−9.19615	−0.381849
$581$	−6.80385	−0.282271
$582$	0	0
$583$	17.7321	0.734387
$584$	4.39230	0.181755
$585$	0	0
$586$	23.3205	0.963361
$587$	−36.0000	−1.48588	−0.742940	−	0.669359i	$-0.766569\pi$
$587$	−36.0000	−1.48588	−0.742940	+	0.669359i	$0.766569\pi$
$588$	0	0
$589$	12.9282	0.532697
$590$	11.3923	0.469014
$591$	0	0
$592$	−5.46410	−0.224573
$593$	−28.7846	−1.18204	−0.591021	−	0.806656i	$-0.701275\pi$
$593$	−28.7846	−1.18204	−0.591021	+	0.806656i	$0.701275\pi$
$594$	0	0
$595$	6.92820	0.284029
$596$	−14.3923	−0.589532
$597$	0	0
$598$	2.53590	0.103701
$599$	−19.3205	−0.789415	−0.394707	−	0.918807i	$-0.629154\pi$
$599$	−19.3205	−0.789415	−0.394707	+	0.918807i	$0.629154\pi$
$600$	0	0
$601$	−5.14359	−0.209812	−0.104906	−	0.994482i	$-0.533454\pi$
$601$	−5.14359	−0.209812	−0.104906	+	0.994482i	$0.533454\pi$
$602$	−3.46410	−0.141186
$603$	0	0
$604$	−17.0526	−0.693859
$605$	−4.92820	−0.200360
$606$	0	0
$607$	6.53590	0.265284	0.132642	−	0.991164i	$-0.457654\pi$
$607$	6.53590	0.265284	0.132642	+	0.991164i	$0.457654\pi$
$608$	18.6603	0.756773
$609$	0	0
$610$	−0.0717968	−0.00290697
$611$	−2.92820	−0.118462
$612$	0	0
$613$	39.3205	1.58814	0.794070	−	0.607826i	$-0.207958\pi$
$613$	39.3205	1.58814	0.794070	+	0.607826i	$0.207958\pi$
$614$	4.53590	0.183054
$615$	0	0
$616$	−12.8038	−0.515882
$617$	−3.46410	−0.139459	−0.0697297	−	0.997566i	$-0.522214\pi$
$617$	−3.46410	−0.139459	−0.0697297	+	0.997566i	$0.522214\pi$
$618$	0	0
$619$	−39.4449	−1.58542	−0.792711	−	0.609597i	$-0.791331\pi$
$619$	−39.4449	−1.58542	−0.792711	+	0.609597i	$0.791331\pi$
$620$	3.46410	0.139122
$621$	0	0
$622$	−6.00000	−0.240578
$623$	18.0000	0.721155
$624$	0	0
$625$	1.00000	0.0400000
$626$	30.3205	1.21185
$627$	0	0
$628$	−14.4641	−0.577180
$629$	−21.8564	−0.871472
$630$	0	0
$631$	−15.7321	−0.626283	−0.313142	−	0.949706i	$-0.601381\pi$
$631$	−15.7321	−0.626283	−0.313142	+	0.949706i	$0.601381\pi$
$632$	31.1769	1.24015
$633$	0	0
$634$	−3.32051	−0.131874
$635$	10.0000	0.396838
$636$	0	0
$637$	4.00000	0.158486
$638$	−22.6603	−0.897128
$639$	0	0
$640$	3.00000	0.118585
$641$	−15.3397	−0.605884	−0.302942	−	0.953009i	$-0.597969\pi$
$641$	−15.3397	−0.605884	−0.302942	+	0.953009i	$0.597969\pi$
$642$	0	0
$643$	−18.8038	−0.741551	−0.370776	−	0.928722i	$-0.620908\pi$
$643$	−18.8038	−0.741551	−0.370776	+	0.928722i	$0.620908\pi$
$644$	4.39230	0.173081
$645$	0	0
$646$	−14.9282	−0.587342
$647$	−30.7846	−1.21027	−0.605134	−	0.796124i	$-0.706880\pi$
$647$	−30.7846	−1.21027	−0.605134	+	0.796124i	$0.706880\pi$
$648$	0	0
$649$	−28.0718	−1.10191
$650$	1.00000	0.0392232
$651$	0	0
$652$	−11.4641	−0.448969
$653$	−16.7846	−0.656833	−0.328416	−	0.944533i	$-0.606515\pi$
$653$	−16.7846	−0.656833	−0.328416	+	0.944533i	$0.606515\pi$
$654$	0	0
$655$	−8.53590	−0.333525
$656$	5.46410	0.213337
$657$	0	0
$658$	5.07180	0.197719
$659$	−39.3205	−1.53171	−0.765855	−	0.643014i	$-0.777684\pi$
$659$	−39.3205	−1.53171	−0.765855	+	0.643014i	$0.777684\pi$
$660$	0	0
$661$	26.6410	1.03622	0.518108	−	0.855315i	$-0.326637\pi$
$661$	26.6410	1.03622	0.518108	+	0.855315i	$0.326637\pi$
$662$	−5.32051	−0.206787
$663$	0	0
$664$	11.7846	0.457332
$665$	6.46410	0.250667
$666$	0	0
$667$	23.3205	0.902974
$668$	4.85641	0.187900
$669$	0	0
$670$	3.46410	0.133830
$671$	0.176915	0.00682971
$672$	0	0
$673$	−2.32051	−0.0894490	−0.0447245	−	0.998999i	$-0.514241\pi$
$673$	−2.32051	−0.0894490	−0.0447245	+	0.998999i	$0.514241\pi$
$674$	32.9282	1.26835
$675$	0	0
$676$	−1.00000	−0.0384615
$677$	5.87564	0.225819	0.112910	−	0.993605i	$-0.463983\pi$
$677$	5.87564	0.225819	0.112910	+	0.993605i	$0.463983\pi$
$678$	0	0
$679$	23.3205	0.894959
$680$	−12.0000	−0.460179
$681$	0	0
$682$	8.53590	0.326856
$683$	30.9282	1.18343	0.591717	−	0.806145i	$-0.298450\pi$
$683$	30.9282	1.18343	0.591717	+	0.806145i	$0.298450\pi$
$684$	0	0
$685$	17.3205	0.661783
$686$	−19.0526	−0.727430
$687$	0	0
$688$	2.00000	0.0762493
$689$	−7.19615	−0.274152
$690$	0	0
$691$	−46.9090	−1.78450	−0.892251	−	0.451541i	$-0.850875\pi$
$691$	−46.9090	−1.78450	−0.892251	+	0.451541i	$0.850875\pi$
$692$	0.803848	0.0305577
$693$	0	0
$694$	24.7846	0.940811
$695$	−1.46410	−0.0555365
$696$	0	0
$697$	21.8564	0.827870
$698$	−14.3923	−0.544757
$699$	0	0
$700$	1.73205	0.0654654
$701$	34.8038	1.31452	0.657262	−	0.753663i	$-0.271715\pi$
$701$	34.8038	1.31452	0.657262	+	0.753663i	$0.271715\pi$
$702$	0	0
$703$	−20.3923	−0.769110
$704$	17.2487	0.650085
$705$	0	0
$706$	−14.5359	−0.547066
$707$	9.00000	0.338480
$708$	0	0
$709$	−7.32051	−0.274927	−0.137464	−	0.990507i	$-0.543895\pi$
$709$	−7.32051	−0.274927	−0.137464	+	0.990507i	$0.543895\pi$
$710$	7.53590	0.282817
$711$	0	0
$712$	−31.1769	−1.16840
$713$	−8.78461	−0.328986
$714$	0	0
$715$	−2.46410	−0.0921522
$716$	6.53590	0.244258
$717$	0	0
$718$	27.7128	1.03423
$719$	−18.9282	−0.705903	−0.352951	−	0.935642i	$-0.614822\pi$
$719$	−18.9282	−0.705903	−0.352951	+	0.935642i	$0.614822\pi$
$720$	0	0
$721$	−7.85641	−0.292588
$722$	5.07180	0.188753
$723$	0	0
$724$	13.9282	0.517638
$725$	9.19615	0.341537
$726$	0	0
$727$	−16.0000	−0.593407	−0.296704	−	0.954970i	$-0.595887\pi$
$727$	−16.0000	−0.593407	−0.296704	+	0.954970i	$0.595887\pi$
$728$	5.19615	0.192582
$729$	0	0
$730$	−1.46410	−0.0541888
$731$	8.00000	0.295891
$732$	0	0
$733$	−6.78461	−0.250595	−0.125298	−	0.992119i	$-0.539989\pi$
$733$	−6.78461	−0.250595	−0.125298	+	0.992119i	$0.539989\pi$
$734$	−25.8564	−0.954377
$735$	0	0
$736$	−12.6795	−0.467372
$737$	−8.53590	−0.314424
$738$	0	0
$739$	13.3397	0.490710	0.245355	−	0.969433i	$-0.421095\pi$
$739$	13.3397	0.490710	0.245355	+	0.969433i	$0.421095\pi$
$740$	−5.46410	−0.200864
$741$	0	0
$742$	12.4641	0.457572
$743$	6.85641	0.251537	0.125769	−	0.992060i	$-0.459860\pi$
$743$	6.85641	0.251537	0.125769	+	0.992060i	$0.459860\pi$
$744$	0	0
$745$	14.3923	0.527293
$746$	−16.4641	−0.602794
$747$	0	0
$748$	9.85641	0.360386
$749$	22.3923	0.818197
$750$	0	0
$751$	23.8564	0.870533	0.435266	−	0.900302i	$-0.356654\pi$
$751$	23.8564	0.870533	0.435266	+	0.900302i	$0.356654\pi$
$752$	−2.92820	−0.106781
$753$	0	0
$754$	9.19615	0.334904
$755$	17.0526	0.620606
$756$	0	0
$757$	−15.3923	−0.559443	−0.279721	−	0.960081i	$-0.590242\pi$
$757$	−15.3923	−0.559443	−0.279721	+	0.960081i	$0.590242\pi$
$758$	36.5167	1.32634
$759$	0	0
$760$	−11.1962	−0.406127
$761$	11.3205	0.410368	0.205184	−	0.978723i	$-0.434221\pi$
$761$	11.3205	0.410368	0.205184	+	0.978723i	$0.434221\pi$
$762$	0	0
$763$	−25.8564	−0.936065
$764$	25.3205	0.916064
$765$	0	0
$766$	−17.7846	−0.642584
$767$	11.3923	0.411352
$768$	0	0
$769$	47.0333	1.69606	0.848032	−	0.529944i	$-0.177787\pi$
$769$	47.0333	1.69606	0.848032	+	0.529944i	$0.177787\pi$
$770$	4.26795	0.153806
$771$	0	0
$772$	−3.60770	−0.129844
$773$	−44.7846	−1.61079	−0.805395	−	0.592738i	$-0.798047\pi$
$773$	−44.7846	−1.61079	−0.805395	+	0.592738i	$0.798047\pi$
$774$	0	0
$775$	−3.46410	−0.124434
$776$	−40.3923	−1.45000
$777$	0	0
$778$	−13.8564	−0.496776
$779$	20.3923	0.730630
$780$	0	0
$781$	−18.5692	−0.664459
$782$	10.1436	0.362734
$783$	0	0
$784$	4.00000	0.142857
$785$	14.4641	0.516246
$786$	0	0
$787$	38.5167	1.37297	0.686485	−	0.727144i	$-0.259153\pi$
$787$	38.5167	1.37297	0.686485	+	0.727144i	$0.259153\pi$
$788$	18.0000	0.641223
$789$	0	0
$790$	−10.3923	−0.369742
$791$	2.32051	0.0825078
$792$	0	0
$793$	−0.0717968	−0.00254958
$794$	20.0000	0.709773
$795$	0	0
$796$	−6.53590	−0.231659
$797$	−3.98076	−0.141006	−0.0705029	−	0.997512i	$-0.522460\pi$
$797$	−3.98076	−0.141006	−0.0705029	+	0.997512i	$0.522460\pi$
$798$	0	0
$799$	−11.7128	−0.414370
$800$	−5.00000	−0.176777
$801$	0	0
$802$	13.6077	0.480504
$803$	3.60770	0.127313
$804$	0	0
$805$	−4.39230	−0.154808
$806$	−3.46410	−0.122018
$807$	0	0
$808$	−15.5885	−0.548400
$809$	31.3397	1.10185	0.550923	−	0.834556i	$-0.314276\pi$
$809$	31.3397	1.10185	0.550923	+	0.834556i	$0.314276\pi$
$810$	0	0
$811$	−14.6603	−0.514791	−0.257396	−	0.966306i	$-0.582864\pi$
$811$	−14.6603	−0.514791	−0.257396	+	0.966306i	$0.582864\pi$
$812$	15.9282	0.558970
$813$	0	0
$814$	−13.4641	−0.471917
$815$	11.4641	0.401570
$816$	0	0
$817$	7.46410	0.261136
$818$	26.3923	0.922785
$819$	0	0
$820$	5.46410	0.190815
$821$	−30.2487	−1.05569	−0.527844	−	0.849342i	$-0.676999\pi$
$821$	−30.2487	−1.05569	−0.527844	+	0.849342i	$0.676999\pi$
$822$	0	0
$823$	21.6077	0.753197	0.376598	−	0.926377i	$-0.377094\pi$
$823$	21.6077	0.753197	0.376598	+	0.926377i	$0.377094\pi$
$824$	13.6077	0.474047
$825$	0	0
$826$	−19.7321	−0.686566
$827$	−12.7846	−0.444564	−0.222282	−	0.974982i	$-0.571351\pi$
$827$	−12.7846	−0.444564	−0.222282	+	0.974982i	$0.571351\pi$
$828$	0	0
$829$	6.21539	0.215869	0.107935	−	0.994158i	$-0.465576\pi$
$829$	6.21539	0.215869	0.107935	+	0.994158i	$0.465576\pi$
$830$	−3.92820	−0.136350
$831$	0	0
$832$	−7.00000	−0.242681
$833$	16.0000	0.554367
$834$	0	0
$835$	−4.85641	−0.168063
$836$	9.19615	0.318056
$837$	0	0
$838$	8.92820	0.308420
$839$	−11.2154	−0.387198	−0.193599	−	0.981081i	$-0.562016\pi$
$839$	−11.2154	−0.387198	−0.193599	+	0.981081i	$0.562016\pi$
$840$	0	0
$841$	55.5692	1.91618
$842$	23.4641	0.808626
$843$	0	0
$844$	14.3923	0.495404
$845$	1.00000	0.0344010
$846$	0	0
$847$	8.53590	0.293297
$848$	−7.19615	−0.247117
$849$	0	0
$850$	4.00000	0.137199
$851$	13.8564	0.474991
$852$	0	0
$853$	−17.3205	−0.593043	−0.296521	−	0.955026i	$-0.595827\pi$
$853$	−17.3205	−0.593043	−0.296521	+	0.955026i	$0.595827\pi$
$854$	0.124356	0.00425536
$855$	0	0
$856$	−38.7846	−1.32563
$857$	2.12436	0.0725666	0.0362833	−	0.999342i	$-0.488448\pi$
$857$	2.12436	0.0725666	0.0362833	+	0.999342i	$0.488448\pi$
$858$	0	0
$859$	−24.3923	−0.832255	−0.416127	−	0.909306i	$-0.636613\pi$
$859$	−24.3923	−0.832255	−0.416127	+	0.909306i	$0.636613\pi$
$860$	2.00000	0.0681994
$861$	0	0
$862$	5.53590	0.188553
$863$	1.92820	0.0656368	0.0328184	−	0.999461i	$-0.489552\pi$
$863$	1.92820	0.0656368	0.0328184	+	0.999461i	$0.489552\pi$
$864$	0	0
$865$	−0.803848	−0.0273316
$866$	−25.2487	−0.857986
$867$	0	0
$868$	−6.00000	−0.203653
$869$	25.6077	0.868682
$870$	0	0
$871$	3.46410	0.117377
$872$	44.7846	1.51660
$873$	0	0
$874$	9.46410	0.320128
$875$	−1.73205	−0.0585540
$876$	0	0
$877$	38.7846	1.30966	0.654832	−	0.755775i	$-0.272740\pi$
$877$	38.7846	1.30966	0.654832	+	0.755775i	$0.272740\pi$
$878$	−5.85641	−0.197644
$879$	0	0
$880$	−2.46410	−0.0830648
$881$	49.5692	1.67003	0.835015	−	0.550228i	$-0.185459\pi$
$881$	49.5692	1.67003	0.835015	+	0.550228i	$0.185459\pi$
$882$	0	0
$883$	−53.8564	−1.81241	−0.906206	−	0.422836i	$-0.861035\pi$
$883$	−53.8564	−1.81241	−0.906206	+	0.422836i	$0.861035\pi$
$884$	−4.00000	−0.134535
$885$	0	0
$886$	2.53590	0.0851952
$887$	−12.6795	−0.425736	−0.212868	−	0.977081i	$-0.568280\pi$
$887$	−12.6795	−0.425736	−0.212868	+	0.977081i	$0.568280\pi$
$888$	0	0
$889$	−17.3205	−0.580911
$890$	10.3923	0.348351
$891$	0	0
$892$	7.58846	0.254080
$893$	−10.9282	−0.365698
$894$	0	0
$895$	−6.53590	−0.218471
$896$	−5.19615	−0.173591
$897$	0	0
$898$	20.3923	0.680500
$899$	−31.8564	−1.06247
$900$	0	0
$901$	−28.7846	−0.958954
$902$	13.4641	0.448306
$903$	0	0
$904$	−4.01924	−0.133678
$905$	−13.9282	−0.462989
$906$	0	0
$907$	47.5692	1.57951	0.789755	−	0.613422i	$-0.210208\pi$
$907$	47.5692	1.57951	0.789755	+	0.613422i	$0.210208\pi$
$908$	24.8564	0.824889
$909$	0	0
$910$	−1.73205	−0.0574169
$911$	−22.2487	−0.737133	−0.368566	−	0.929601i	$-0.620151\pi$
$911$	−22.2487	−0.737133	−0.368566	+	0.929601i	$0.620151\pi$
$912$	0	0
$913$	9.67949	0.320344
$914$	−34.9282	−1.15532
$915$	0	0
$916$	19.8564	0.656074
$917$	14.7846	0.488231
$918$	0	0
$919$	46.2487	1.52560	0.762802	−	0.646632i	$-0.223823\pi$
$919$	46.2487	1.52560	0.762802	+	0.646632i	$0.223823\pi$
$920$	7.60770	0.250818
$921$	0	0
$922$	34.6410	1.14084
$923$	7.53590	0.248047
$924$	0	0
$925$	5.46410	0.179659
$926$	−12.1244	−0.398431
$927$	0	0
$928$	−45.9808	−1.50939
$929$	42.1051	1.38142	0.690712	−	0.723130i	$-0.257297\pi$
$929$	42.1051	1.38142	0.690712	+	0.723130i	$0.257297\pi$
$930$	0	0
$931$	14.9282	0.489252
$932$	4.26795	0.139801
$933$	0	0
$934$	27.8564	0.911489
$935$	−9.85641	−0.322339
$936$	0	0
$937$	23.0718	0.753723	0.376861	−	0.926270i	$-0.377003\pi$
$937$	23.0718	0.753723	0.376861	+	0.926270i	$0.377003\pi$
$938$	−6.00000	−0.195907
$939$	0	0
$940$	−2.92820	−0.0955075
$941$	−8.92820	−0.291051	−0.145526	−	0.989354i	$-0.546487\pi$
$941$	−8.92820	−0.291051	−0.145526	+	0.989354i	$0.546487\pi$
$942$	0	0
$943$	−13.8564	−0.451227
$944$	11.3923	0.370788
$945$	0	0
$946$	4.92820	0.160230
$947$	41.0718	1.33465	0.667327	−	0.744765i	$-0.267438\pi$
$947$	41.0718	1.33465	0.667327	+	0.744765i	$0.267438\pi$
$948$	0	0
$949$	−1.46410	−0.0475267
$950$	3.73205	0.121084
$951$	0	0
$952$	20.7846	0.673633
$953$	50.9282	1.64973	0.824863	−	0.565332i	$-0.191252\pi$
$953$	50.9282	1.64973	0.824863	+	0.565332i	$0.191252\pi$
$954$	0	0
$955$	−25.3205	−0.819352
$956$	15.5359	0.502467
$957$	0	0
$958$	14.3205	0.462675
$959$	−30.0000	−0.968751
$960$	0	0
$961$	−19.0000	−0.612903
$962$	5.46410	0.176170
$963$	0	0
$964$	4.53590	0.146091
$965$	3.60770	0.116136
$966$	0	0
$967$	−8.94744	−0.287730	−0.143865	−	0.989597i	$-0.545953\pi$
$967$	−8.94744	−0.287730	−0.143865	+	0.989597i	$0.545953\pi$
$968$	−14.7846	−0.475195
$969$	0	0
$970$	13.4641	0.432306
$971$	−3.21539	−0.103187	−0.0515934	−	0.998668i	$-0.516430\pi$
$971$	−3.21539	−0.103187	−0.0515934	+	0.998668i	$0.516430\pi$
$972$	0	0
$973$	2.53590	0.0812972
$974$	−15.0526	−0.482315
$975$	0	0
$976$	−0.0717968	−0.00229816
$977$	−39.3205	−1.25797	−0.628987	−	0.777416i	$-0.716530\pi$
$977$	−39.3205	−1.25797	−0.628987	+	0.777416i	$0.716530\pi$
$978$	0	0
$979$	−25.6077	−0.818425
$980$	4.00000	0.127775
$981$	0	0
$982$	−38.6410	−1.23308
$983$	−18.7128	−0.596846	−0.298423	−	0.954434i	$-0.596461\pi$
$983$	−18.7128	−0.596846	−0.298423	+	0.954434i	$0.596461\pi$
$984$	0	0
$985$	−18.0000	−0.573528
$986$	36.7846	1.17146
$987$	0	0
$988$	−3.73205	−0.118732
$989$	−5.07180	−0.161274
$990$	0	0
$991$	−32.3923	−1.02898	−0.514488	−	0.857498i	$-0.672018\pi$
$991$	−32.3923	−1.02898	−0.514488	+	0.857498i	$0.672018\pi$
$992$	17.3205	0.549927
$993$	0	0
$994$	−13.0526	−0.414002
$995$	6.53590	0.207202
$996$	0	0
$997$	9.67949	0.306553	0.153276	−	0.988183i	$-0.451018\pi$
$997$	9.67949	0.306553	0.153276	+	0.988183i	$0.451018\pi$
$998$	−15.4449	−0.488899
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5265.2.a.q.1.1	✓	2
3.2	odd	2		5265.2.a.r.1.1	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5265.2.a.q.1.1	✓	2	1.1	even	1	trivial
5265.2.a.r.1.1	yes	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:	\( N \)	\(=\)	\( 5265 = 3^{4} \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5265.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -1.00000 q^{4} +1.00000 q^{5} -1.73205 q^{7} +3.00000 q^{8} +O(q^{10})\)	\(q-1.00000 q^{2} -1.00000 q^{4} +1.00000 q^{5} -1.73205 q^{7} +3.00000 q^{8} -1.00000 q^{10} +2.46410 q^{11} -1.00000 q^{13} +1.73205 q^{14} -1.00000 q^{16} -4.00000 q^{17} -3.73205 q^{19} -1.00000 q^{20} -2.46410 q^{22} +2.53590 q^{23} +1.00000 q^{25} +1.00000 q^{26} +1.73205 q^{28} +9.19615 q^{29} -3.46410 q^{31} -5.00000 q^{32} +4.00000 q^{34} -1.73205 q^{35} +5.46410 q^{37} +3.73205 q^{38} +3.00000 q^{40} -5.46410 q^{41} -2.00000 q^{43} -2.46410 q^{44} -2.53590 q^{46} +2.92820 q^{47} -4.00000 q^{49} -1.00000 q^{50} +1.00000 q^{52} +7.19615 q^{53} +2.46410 q^{55} -5.19615 q^{56} -9.19615 q^{58} -11.3923 q^{59} +0.0717968 q^{61} +3.46410 q^{62} +7.00000 q^{64} -1.00000 q^{65} -3.46410 q^{67} +4.00000 q^{68} +1.73205 q^{70} -7.53590 q^{71} +1.46410 q^{73} -5.46410 q^{74} +3.73205 q^{76} -4.26795 q^{77} +10.3923 q^{79} -1.00000 q^{80} +5.46410 q^{82} +3.92820 q^{83} -4.00000 q^{85} +2.00000 q^{86} +7.39230 q^{88} -10.3923 q^{89} +1.73205 q^{91} -2.53590 q^{92} -2.92820 q^{94} -3.73205 q^{95} -13.4641 q^{97} +4.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} - 2 q^{4} + 2 q^{5} + 6 q^{8}+O(q^{10}) \) 2 * q - 2 * q^2 - 2 * q^4 + 2 * q^5 + 6 * q^8	\( 2 q - 2 q^{2} - 2 q^{4} + 2 q^{5} + 6 q^{8} - 2 q^{10} - 2 q^{11} - 2 q^{13} - 2 q^{16} - 8 q^{17} - 4 q^{19} - 2 q^{20} + 2 q^{22} + 12 q^{23} + 2 q^{25} + 2 q^{26} + 8 q^{29} - 10 q^{32} + 8 q^{34} + 4 q^{37} + 4 q^{38} + 6 q^{40} - 4 q^{41} - 4 q^{43} + 2 q^{44} - 12 q^{46} - 8 q^{47} - 8 q^{49} - 2 q^{50} + 2 q^{52} + 4 q^{53} - 2 q^{55} - 8 q^{58} - 2 q^{59} + 14 q^{61} + 14 q^{64} - 2 q^{65} + 8 q^{68} - 22 q^{71} - 4 q^{73} - 4 q^{74} + 4 q^{76} - 12 q^{77} - 2 q^{80} + 4 q^{82} - 6 q^{83} - 8 q^{85} + 4 q^{86} - 6 q^{88} - 12 q^{92} + 8 q^{94} - 4 q^{95} - 20 q^{97} + 8 q^{98}+O(q^{100}) \) 2 * q - 2 * q^2 - 2 * q^4 + 2 * q^5 + 6 * q^8 - 2 * q^10 - 2 * q^11 - 2 * q^13 - 2 * q^16 - 8 * q^17 - 4 * q^19 - 2 * q^20 + 2 * q^22 + 12 * q^23 + 2 * q^25 + 2 * q^26 + 8 * q^29 - 10 * q^32 + 8 * q^34 + 4 * q^37 + 4 * q^38 + 6 * q^40 - 4 * q^41 - 4 * q^43 + 2 * q^44 - 12 * q^46 - 8 * q^47 - 8 * q^49 - 2 * q^50 + 2 * q^52 + 4 * q^53 - 2 * q^55 - 8 * q^58 - 2 * q^59 + 14 * q^61 + 14 * q^64 - 2 * q^65 + 8 * q^68 - 22 * q^71 - 4 * q^73 - 4 * q^74 + 4 * q^76 - 12 * q^77 - 2 * q^80 + 4 * q^82 - 6 * q^83 - 8 * q^85 + 4 * q^86 - 6 * q^88 - 12 * q^92 + 8 * q^94 - 4 * q^95 - 20 * q^97 + 8 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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