LMFDB - Embedded newform 6025.2.a.c.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6025,2,Mod(1,6025)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6025, 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("6025.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6025 = 5^{2} \cdot 241 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6025.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:	$48.1098672178$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1205)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.618034$ of defining polynomial
Character	$\chi$	$=$	6025.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.23607 q^{2} +0.381966 q^{3} +3.00000 q^{4} +0.854102 q^{6} +1.23607 q^{7} +2.23607 q^{8} -2.85410 q^{9} +O(q^{10})$	\(q+2.23607 q^{2} +0.381966 q^{3} +3.00000 q^{4} +0.854102 q^{6} +1.23607 q^{7} +2.23607 q^{8} -2.85410 q^{9} -1.61803 q^{11} +1.14590 q^{12} +2.85410 q^{13} +2.76393 q^{14} -1.00000 q^{16} -5.09017 q^{17} -6.38197 q^{18} -5.38197 q^{19} +0.472136 q^{21} -3.61803 q^{22} -4.47214 q^{23} +0.854102 q^{24} +6.38197 q^{26} -2.23607 q^{27} +3.70820 q^{28} -7.09017 q^{29} +2.47214 q^{31} -6.70820 q^{32} -0.618034 q^{33} -11.3820 q^{34} -8.56231 q^{36} -4.47214 q^{37} -12.0344 q^{38} +1.09017 q^{39} +2.38197 q^{41} +1.05573 q^{42} +1.52786 q^{43} -4.85410 q^{44} -10.0000 q^{46} -3.38197 q^{47} -0.381966 q^{48} -5.47214 q^{49} -1.94427 q^{51} +8.56231 q^{52} -10.4721 q^{53} -5.00000 q^{54} +2.76393 q^{56} -2.05573 q^{57} -15.8541 q^{58} +7.70820 q^{59} +12.8541 q^{61} +5.52786 q^{62} -3.52786 q^{63} -13.0000 q^{64} -1.38197 q^{66} +7.09017 q^{67} -15.2705 q^{68} -1.70820 q^{69} +7.56231 q^{71} -6.38197 q^{72} -5.38197 q^{73} -10.0000 q^{74} -16.1459 q^{76} -2.00000 q^{77} +2.43769 q^{78} +16.1803 q^{79} +7.70820 q^{81} +5.32624 q^{82} +0.909830 q^{83} +1.41641 q^{84} +3.41641 q^{86} -2.70820 q^{87} -3.61803 q^{88} -14.9443 q^{89} +3.52786 q^{91} -13.4164 q^{92} +0.944272 q^{93} -7.56231 q^{94} -2.56231 q^{96} +7.23607 q^{97} -12.2361 q^{98} +4.61803 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 3 q^{3} + 6 q^{4} - 5 q^{6} - 2 q^{7} + q^{9}+O(q^{10}) $ 2 * q + 3 * q^3 + 6 * q^4 - 5 * q^6 - 2 * q^7 + q^9	$ 2 q + 3 q^{3} + 6 q^{4} - 5 q^{6} - 2 q^{7} + q^{9} - q^{11} + 9 q^{12} - q^{13} + 10 q^{14} - 2 q^{16} + q^{17} - 15 q^{18} - 13 q^{19} - 8 q^{21} - 5 q^{22} - 5 q^{24} + 15 q^{26} - 6 q^{28} - 3 q^{29} - 4 q^{31} + q^{33} - 25 q^{34} + 3 q^{36} + 5 q^{38} - 9 q^{39} + 7 q^{41} + 20 q^{42} + 12 q^{43} - 3 q^{44} - 20 q^{46} - 9 q^{47} - 3 q^{48} - 2 q^{49} + 14 q^{51} - 3 q^{52} - 12 q^{53} - 10 q^{54} + 10 q^{56} - 22 q^{57} - 25 q^{58} + 2 q^{59} + 19 q^{61} + 20 q^{62} - 16 q^{63} - 26 q^{64} - 5 q^{66} + 3 q^{67} + 3 q^{68} + 10 q^{69} - 5 q^{71} - 15 q^{72} - 13 q^{73} - 20 q^{74} - 39 q^{76} - 4 q^{77} + 25 q^{78} + 10 q^{79} + 2 q^{81} - 5 q^{82} + 13 q^{83} - 24 q^{84} - 20 q^{86} + 8 q^{87} - 5 q^{88} - 12 q^{89} + 16 q^{91} - 16 q^{93} + 5 q^{94} + 15 q^{96} + 10 q^{97} - 20 q^{98} + 7 q^{99}+O(q^{100}) $ 2 * q + 3 * q^3 + 6 * q^4 - 5 * q^6 - 2 * q^7 + q^9 - q^11 + 9 * q^12 - q^13 + 10 * q^14 - 2 * q^16 + q^17 - 15 * q^18 - 13 * q^19 - 8 * q^21 - 5 * q^22 - 5 * q^24 + 15 * q^26 - 6 * q^28 - 3 * q^29 - 4 * q^31 + q^33 - 25 * q^34 + 3 * q^36 + 5 * q^38 - 9 * q^39 + 7 * q^41 + 20 * q^42 + 12 * q^43 - 3 * q^44 - 20 * q^46 - 9 * q^47 - 3 * q^48 - 2 * q^49 + 14 * q^51 - 3 * q^52 - 12 * q^53 - 10 * q^54 + 10 * q^56 - 22 * q^57 - 25 * q^58 + 2 * q^59 + 19 * q^61 + 20 * q^62 - 16 * q^63 - 26 * q^64 - 5 * q^66 + 3 * q^67 + 3 * q^68 + 10 * q^69 - 5 * q^71 - 15 * q^72 - 13 * q^73 - 20 * q^74 - 39 * q^76 - 4 * q^77 + 25 * q^78 + 10 * q^79 + 2 * q^81 - 5 * q^82 + 13 * q^83 - 24 * q^84 - 20 * q^86 + 8 * q^87 - 5 * q^88 - 12 * q^89 + 16 * q^91 - 16 * q^93 + 5 * q^94 + 15 * q^96 + 10 * q^97 - 20 * q^98 + 7 * q^99

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 $

$2$	2.23607	1.58114	0.790569	−	0.612372i	$-0.209785\pi$
$2$	2.23607	1.58114	0.790569	+	0.612372i	$0.209785\pi$
$3$	0.381966	0.220528	0.110264	−	0.993902i	$-0.464830\pi$
$3$	0.381966	0.220528	0.110264	+	0.993902i	$0.464830\pi$
$4$	3.00000	1.50000
$5$	0	0
$5$	0	0
$6$	0.854102	0.348686
$7$	1.23607	0.467190	0.233595	−	0.972334i	$-0.424951\pi$
$7$	1.23607	0.467190	0.233595	+	0.972334i	$0.424951\pi$
$8$	2.23607	0.790569
$9$	−2.85410	−0.951367
$10$	0	0
$11$	−1.61803	−0.487856	−0.243928	−	0.969793i	$-0.578436\pi$
$11$	−1.61803	−0.487856	−0.243928	+	0.969793i	$0.578436\pi$
$12$	1.14590	0.330792
$13$	2.85410	0.791585	0.395793	−	0.918340i	$-0.370470\pi$
$13$	2.85410	0.791585	0.395793	+	0.918340i	$0.370470\pi$
$14$	2.76393	0.738692
$15$	0	0
$16$	−1.00000	−0.250000
$17$	−5.09017	−1.23455	−0.617274	−	0.786748i	$-0.711763\pi$
$17$	−5.09017	−1.23455	−0.617274	+	0.786748i	$0.711763\pi$
$18$	−6.38197	−1.50424
$19$	−5.38197	−1.23471	−0.617354	−	0.786686i	$-0.711795\pi$
$19$	−5.38197	−1.23471	−0.617354	+	0.786686i	$0.711795\pi$
$20$	0	0
$21$	0.472136	0.103029
$22$	−3.61803	−0.771367
$23$	−4.47214	−0.932505	−0.466252	−	0.884652i	$-0.654396\pi$
$23$	−4.47214	−0.932505	−0.466252	+	0.884652i	$0.654396\pi$
$24$	0.854102	0.174343
$25$	0	0
$26$	6.38197	1.25161
$27$	−2.23607	−0.430331
$28$	3.70820	0.700785
$29$	−7.09017	−1.31661	−0.658306	−	0.752751i	$-0.728727\pi$
$29$	−7.09017	−1.31661	−0.658306	+	0.752751i	$0.728727\pi$
$30$	0	0
$31$	2.47214	0.444009	0.222004	−	0.975046i	$-0.428740\pi$
$31$	2.47214	0.444009	0.222004	+	0.975046i	$0.428740\pi$
$32$	−6.70820	−1.18585
$33$	−0.618034	−0.107586
$34$	−11.3820	−1.95199
$35$	0	0
$36$	−8.56231	−1.42705
$37$	−4.47214	−0.735215	−0.367607	−	0.929981i	$-0.619823\pi$
$37$	−4.47214	−0.735215	−0.367607	+	0.929981i	$0.619823\pi$
$38$	−12.0344	−1.95224
$39$	1.09017	0.174567
$40$	0	0
$41$	2.38197	0.372001	0.186000	−	0.982550i	$-0.440447\pi$
$41$	2.38197	0.372001	0.186000	+	0.982550i	$0.440447\pi$
$42$	1.05573	0.162902
$43$	1.52786	0.232997	0.116499	−	0.993191i	$-0.462833\pi$
$43$	1.52786	0.232997	0.116499	+	0.993191i	$0.462833\pi$
$44$	−4.85410	−0.731783
$45$	0	0
$46$	−10.0000	−1.47442
$47$	−3.38197	−0.493310	−0.246655	−	0.969103i	$-0.579332\pi$
$47$	−3.38197	−0.493310	−0.246655	+	0.969103i	$0.579332\pi$
$48$	−0.381966	−0.0551320
$49$	−5.47214	−0.781734
$50$	0	0
$51$	−1.94427	−0.272253
$52$	8.56231	1.18738
$53$	−10.4721	−1.43846	−0.719229	−	0.694773i	$-0.755505\pi$
$53$	−10.4721	−1.43846	−0.719229	+	0.694773i	$0.755505\pi$
$54$	−5.00000	−0.680414
$55$	0	0
$56$	2.76393	0.369346
$57$	−2.05573	−0.272288
$58$	−15.8541	−2.08175
$59$	7.70820	1.00352	0.501761	−	0.865006i	$-0.332686\pi$
$59$	7.70820	1.00352	0.501761	+	0.865006i	$0.332686\pi$
$60$	0	0
$61$	12.8541	1.64580	0.822900	−	0.568187i	$-0.192355\pi$
$61$	12.8541	1.64580	0.822900	+	0.568187i	$0.192355\pi$
$62$	5.52786	0.702039
$63$	−3.52786	−0.444469
$64$	−13.0000	−1.62500
$65$	0	0
$66$	−1.38197	−0.170108
$67$	7.09017	0.866202	0.433101	−	0.901345i	$-0.357419\pi$
$67$	7.09017	0.866202	0.433101	+	0.901345i	$0.357419\pi$
$68$	−15.2705	−1.85182
$69$	−1.70820	−0.205644
$70$	0	0
$71$	7.56231	0.897481	0.448740	−	0.893662i	$-0.351873\pi$
$71$	7.56231	0.897481	0.448740	+	0.893662i	$0.351873\pi$
$72$	−6.38197	−0.752122
$73$	−5.38197	−0.629911	−0.314956	−	0.949106i	$-0.601990\pi$
$73$	−5.38197	−0.629911	−0.314956	+	0.949106i	$0.601990\pi$
$74$	−10.0000	−1.16248
$75$	0	0
$76$	−16.1459	−1.85206
$77$	−2.00000	−0.227921
$78$	2.43769	0.276015
$79$	16.1803	1.82043	0.910215	−	0.414136i	$-0.135916\pi$
$79$	16.1803	1.82043	0.910215	+	0.414136i	$0.135916\pi$
$80$	0	0
$81$	7.70820	0.856467
$82$	5.32624	0.588185
$83$	0.909830	0.0998668	0.0499334	−	0.998753i	$-0.484099\pi$
$83$	0.909830	0.0998668	0.0499334	+	0.998753i	$0.484099\pi$
$84$	1.41641	0.154543
$85$	0	0
$86$	3.41641	0.368401
$87$	−2.70820	−0.290350
$88$	−3.61803	−0.385684
$89$	−14.9443	−1.58409	−0.792045	−	0.610463i	$-0.790983\pi$
$89$	−14.9443	−1.58409	−0.792045	+	0.610463i	$0.790983\pi$
$90$	0	0
$91$	3.52786	0.369821
$92$	−13.4164	−1.39876
$93$	0.944272	0.0979164
$94$	−7.56231	−0.779992
$95$	0	0
$96$	−2.56231	−0.261514
$97$	7.23607	0.734711	0.367356	−	0.930081i	$-0.380263\pi$
$97$	7.23607	0.734711	0.367356	+	0.930081i	$0.380263\pi$
$98$	−12.2361	−1.23603
$99$	4.61803	0.464130
$100$	0	0
$101$	−6.18034	−0.614967	−0.307483	−	0.951553i	$-0.599487\pi$
$101$	−6.18034	−0.614967	−0.307483	+	0.951553i	$0.599487\pi$
$102$	−4.34752	−0.430469
$103$	−2.29180	−0.225817	−0.112909	−	0.993605i	$-0.536017\pi$
$103$	−2.29180	−0.225817	−0.112909	+	0.993605i	$0.536017\pi$
$104$	6.38197	0.625803
$105$	0	0
$106$	−23.4164	−2.27440
$107$	12.5623	1.21444	0.607222	−	0.794532i	$-0.292284\pi$
$107$	12.5623	1.21444	0.607222	+	0.794532i	$0.292284\pi$
$108$	−6.70820	−0.645497
$109$	12.9443	1.23984	0.619918	−	0.784666i	$-0.287166\pi$
$109$	12.9443	1.23984	0.619918	+	0.784666i	$0.287166\pi$
$110$	0	0
$111$	−1.70820	−0.162136
$112$	−1.23607	−0.116797
$113$	19.2361	1.80958	0.904789	−	0.425861i	$-0.140029\pi$
$113$	19.2361	1.80958	0.904789	+	0.425861i	$0.140029\pi$
$114$	−4.59675	−0.430525
$115$	0	0
$116$	−21.2705	−1.97492
$117$	−8.14590	−0.753089
$118$	17.2361	1.58671
$119$	−6.29180	−0.576768
$120$	0	0
$121$	−8.38197	−0.761997
$122$	28.7426	2.60224
$123$	0.909830	0.0820366
$124$	7.41641	0.666013
$125$	0	0
$126$	−7.88854	−0.702767
$127$	−2.00000	−0.177471	−0.0887357	−	0.996055i	$-0.528283\pi$
$127$	−2.00000	−0.177471	−0.0887357	+	0.996055i	$0.528283\pi$
$128$	−15.6525	−1.38350
$129$	0.583592	0.0513824
$130$	0	0
$131$	−20.7984	−1.81716	−0.908581	−	0.417708i	$-0.862833\pi$
$131$	−20.7984	−1.81716	−0.908581	+	0.417708i	$0.862833\pi$
$132$	−1.85410	−0.161379
$133$	−6.65248	−0.576843
$134$	15.8541	1.36959
$135$	0	0
$136$	−11.3820	−0.975996
$137$	−13.1459	−1.12313	−0.561565	−	0.827433i	$-0.689801\pi$
$137$	−13.1459	−1.12313	−0.561565	+	0.827433i	$0.689801\pi$
$138$	−3.81966	−0.325151
$139$	−10.3820	−0.880587	−0.440293	−	0.897854i	$-0.645126\pi$
$139$	−10.3820	−0.880587	−0.440293	+	0.897854i	$0.645126\pi$
$140$	0	0
$141$	−1.29180	−0.108789
$142$	16.9098	1.41904
$143$	−4.61803	−0.386179
$144$	2.85410	0.237842
$145$	0	0
$146$	−12.0344	−0.995977
$147$	−2.09017	−0.172394
$148$	−13.4164	−1.10282
$149$	3.41641	0.279883	0.139942	−	0.990160i	$-0.455309\pi$
$149$	3.41641	0.279883	0.139942	+	0.990160i	$0.455309\pi$
$150$	0	0
$151$	5.41641	0.440781	0.220391	−	0.975412i	$-0.429267\pi$
$151$	5.41641	0.440781	0.220391	+	0.975412i	$0.429267\pi$
$152$	−12.0344	−0.976122
$153$	14.5279	1.17451
$154$	−4.47214	−0.360375
$155$	0	0
$156$	3.27051	0.261850
$157$	−22.0344	−1.75854	−0.879270	−	0.476324i	$-0.841969\pi$
$157$	−22.0344	−1.75854	−0.879270	+	0.476324i	$0.841969\pi$
$158$	36.1803	2.87835
$159$	−4.00000	−0.317221
$160$	0	0
$161$	−5.52786	−0.435657
$162$	17.2361	1.35419
$163$	−1.52786	−0.119672	−0.0598358	−	0.998208i	$-0.519058\pi$
$163$	−1.52786	−0.119672	−0.0598358	+	0.998208i	$0.519058\pi$
$164$	7.14590	0.558001
$165$	0	0
$166$	2.03444	0.157903
$167$	−5.70820	−0.441714	−0.220857	−	0.975306i	$-0.570885\pi$
$167$	−5.70820	−0.441714	−0.220857	+	0.975306i	$0.570885\pi$
$168$	1.05573	0.0814512
$169$	−4.85410	−0.373392
$170$	0	0
$171$	15.3607	1.17466
$172$	4.58359	0.349496
$173$	−8.38197	−0.637269	−0.318635	−	0.947878i	$-0.603224\pi$
$173$	−8.38197	−0.637269	−0.318635	+	0.947878i	$0.603224\pi$
$174$	−6.05573	−0.459084
$175$	0	0
$176$	1.61803	0.121964
$177$	2.94427	0.221305
$178$	−33.4164	−2.50467
$179$	−4.00000	−0.298974	−0.149487	−	0.988764i	$-0.547762\pi$
$179$	−4.00000	−0.298974	−0.149487	+	0.988764i	$0.547762\pi$
$180$	0	0
$181$	0.854102	0.0634849	0.0317424	−	0.999496i	$-0.489894\pi$
$181$	0.854102	0.0634849	0.0317424	+	0.999496i	$0.489894\pi$
$182$	7.88854	0.584738
$183$	4.90983	0.362945
$184$	−10.0000	−0.737210
$185$	0	0
$186$	2.11146	0.154819
$187$	8.23607	0.602281
$188$	−10.1459	−0.739966
$189$	−2.76393	−0.201046
$190$	0	0
$191$	−16.1803	−1.17077	−0.585384	−	0.810756i	$-0.699056\pi$
$191$	−16.1803	−1.17077	−0.585384	+	0.810756i	$0.699056\pi$
$192$	−4.96556	−0.358358
$193$	20.4721	1.47362	0.736808	−	0.676102i	$-0.236332\pi$
$193$	20.4721	1.47362	0.736808	+	0.676102i	$0.236332\pi$
$194$	16.1803	1.16168
$195$	0	0
$196$	−16.4164	−1.17260
$197$	10.0000	0.712470	0.356235	−	0.934396i	$-0.384060\pi$
$197$	10.0000	0.712470	0.356235	+	0.934396i	$0.384060\pi$
$198$	10.3262	0.733854
$199$	−15.3262	−1.08645	−0.543224	−	0.839588i	$-0.682797\pi$
$199$	−15.3262	−1.08645	−0.543224	+	0.839588i	$0.682797\pi$
$200$	0	0
$201$	2.70820	0.191022
$202$	−13.8197	−0.972348
$203$	−8.76393	−0.615107
$204$	−5.83282	−0.408379
$205$	0	0
$206$	−5.12461	−0.357049
$207$	12.7639	0.887155
$208$	−2.85410	−0.197896
$209$	8.70820	0.602359
$210$	0	0
$211$	19.1246	1.31659	0.658296	−	0.752759i	$-0.271277\pi$
$211$	19.1246	1.31659	0.658296	+	0.752759i	$0.271277\pi$
$212$	−31.4164	−2.15769
$213$	2.88854	0.197920
$214$	28.0902	1.92020
$215$	0	0
$216$	−5.00000	−0.340207
$217$	3.05573	0.207436
$218$	28.9443	1.96035
$219$	−2.05573	−0.138913
$220$	0	0
$221$	−14.5279	−0.977250
$222$	−3.81966	−0.256359
$223$	−1.20163	−0.0804668	−0.0402334	−	0.999190i	$-0.512810\pi$
$223$	−1.20163	−0.0804668	−0.0402334	+	0.999190i	$0.512810\pi$
$224$	−8.29180	−0.554019
$225$	0	0
$226$	43.0132	2.86119
$227$	17.1246	1.13660	0.568300	−	0.822821i	$-0.307601\pi$
$227$	17.1246	1.13660	0.568300	+	0.822821i	$0.307601\pi$
$228$	−6.16718	−0.408432
$229$	4.09017	0.270286	0.135143	−	0.990826i	$-0.456851\pi$
$229$	4.09017	0.270286	0.135143	+	0.990826i	$0.456851\pi$
$230$	0	0
$231$	−0.763932	−0.0502630
$232$	−15.8541	−1.04087
$233$	−10.9443	−0.716983	−0.358492	−	0.933533i	$-0.616709\pi$
$233$	−10.9443	−0.716983	−0.358492	+	0.933533i	$0.616709\pi$
$234$	−18.2148	−1.19074
$235$	0	0
$236$	23.1246	1.50528
$237$	6.18034	0.401456
$238$	−14.0689	−0.911950
$239$	10.4721	0.677386	0.338693	−	0.940897i	$-0.390015\pi$
$239$	10.4721	0.677386	0.338693	+	0.940897i	$0.390015\pi$
$240$	0	0
$241$	1.00000	0.0644157
$241$	1.00000	0.0644157
$242$	−18.7426	−1.20482
$243$	9.65248	0.619207
$244$	38.5623	2.46870
$245$	0	0
$246$	2.03444	0.129711
$247$	−15.3607	−0.977377
$248$	5.52786	0.351020
$249$	0.347524	0.0220234
$250$	0	0
$251$	−29.2361	−1.84536	−0.922682	−	0.385562i	$-0.874008\pi$
$251$	−29.2361	−1.84536	−0.922682	+	0.385562i	$0.874008\pi$
$252$	−10.5836	−0.666704
$253$	7.23607	0.454928
$254$	−4.47214	−0.280607
$255$	0	0
$256$	−9.00000	−0.562500
$257$	23.2361	1.44943	0.724713	−	0.689051i	$-0.241972\pi$
$257$	23.2361	1.44943	0.724713	+	0.689051i	$0.241972\pi$
$258$	1.30495	0.0812427
$259$	−5.52786	−0.343485
$260$	0	0
$261$	20.2361	1.25258
$262$	−46.5066	−2.87319
$263$	−16.0000	−0.986602	−0.493301	−	0.869859i	$-0.664210\pi$
$263$	−16.0000	−0.986602	−0.493301	+	0.869859i	$0.664210\pi$
$264$	−1.38197	−0.0850541
$265$	0	0
$266$	−14.8754	−0.912069
$267$	−5.70820	−0.349336
$268$	21.2705	1.29930
$269$	18.6525	1.13726	0.568631	−	0.822593i	$-0.307473\pi$
$269$	18.6525	1.13726	0.568631	+	0.822593i	$0.307473\pi$
$270$	0	0
$271$	14.7639	0.896845	0.448423	−	0.893822i	$-0.351986\pi$
$271$	14.7639	0.896845	0.448423	+	0.893822i	$0.351986\pi$
$272$	5.09017	0.308637
$273$	1.34752	0.0815559
$274$	−29.3951	−1.77582
$275$	0	0
$276$	−5.12461	−0.308465
$277$	4.94427	0.297073	0.148536	−	0.988907i	$-0.452544\pi$
$277$	4.94427	0.297073	0.148536	+	0.988907i	$0.452544\pi$
$278$	−23.2148	−1.39233
$279$	−7.05573	−0.422415
$280$	0	0
$281$	−28.0902	−1.67572	−0.837860	−	0.545886i	$-0.816193\pi$
$281$	−28.0902	−1.67572	−0.837860	+	0.545886i	$0.816193\pi$
$282$	−2.88854	−0.172010
$283$	1.23607	0.0734766	0.0367383	−	0.999325i	$-0.488303\pi$
$283$	1.23607	0.0734766	0.0367383	+	0.999325i	$0.488303\pi$
$284$	22.6869	1.34622
$285$	0	0
$286$	−10.3262	−0.610603
$287$	2.94427	0.173795
$288$	19.1459	1.12818
$289$	8.90983	0.524108
$290$	0	0
$291$	2.76393	0.162025
$292$	−16.1459	−0.944867
$293$	−14.6180	−0.853995	−0.426997	−	0.904253i	$-0.640429\pi$
$293$	−14.6180	−0.853995	−0.426997	+	0.904253i	$0.640429\pi$
$294$	−4.67376	−0.272579
$295$	0	0
$296$	−10.0000	−0.581238
$297$	3.61803	0.209940
$298$	7.63932	0.442534
$299$	−12.7639	−0.738157
$300$	0	0
$301$	1.88854	0.108854
$302$	12.1115	0.696936
$303$	−2.36068	−0.135618
$304$	5.38197	0.308677
$305$	0	0
$306$	32.4853	1.85706
$307$	−25.1246	−1.43394	−0.716969	−	0.697105i	$-0.754471\pi$
$307$	−25.1246	−1.43394	−0.716969	+	0.697105i	$0.754471\pi$
$308$	−6.00000	−0.341882
$309$	−0.875388	−0.0497991
$310$	0	0
$311$	7.56231	0.428819	0.214410	−	0.976744i	$-0.431217\pi$
$311$	7.56231	0.428819	0.214410	+	0.976744i	$0.431217\pi$
$312$	2.43769	0.138007
$313$	−6.76393	−0.382320	−0.191160	−	0.981559i	$-0.561225\pi$
$313$	−6.76393	−0.382320	−0.191160	+	0.981559i	$0.561225\pi$
$314$	−49.2705	−2.78050
$315$	0	0
$316$	48.5410	2.73065
$317$	9.14590	0.513685	0.256842	−	0.966453i	$-0.417318\pi$
$317$	9.14590	0.513685	0.256842	+	0.966453i	$0.417318\pi$
$318$	−8.94427	−0.501570
$319$	11.4721	0.642316
$320$	0	0
$321$	4.79837	0.267819
$322$	−12.3607	−0.688834
$323$	27.3951	1.52431
$324$	23.1246	1.28470
$325$	0	0
$326$	−3.41641	−0.189217
$327$	4.94427	0.273419
$328$	5.32624	0.294092
$329$	−4.18034	−0.230470
$330$	0	0
$331$	−20.6525	−1.13516	−0.567581	−	0.823317i	$-0.692121\pi$
$331$	−20.6525	−1.13516	−0.567581	+	0.823317i	$0.692121\pi$
$332$	2.72949	0.149800
$333$	12.7639	0.699459
$334$	−12.7639	−0.698411
$335$	0	0
$336$	−0.472136	−0.0257571
$337$	−13.7082	−0.746733	−0.373367	−	0.927684i	$-0.621797\pi$
$337$	−13.7082	−0.746733	−0.373367	+	0.927684i	$0.621797\pi$
$338$	−10.8541	−0.590385
$339$	7.34752	0.399063
$340$	0	0
$341$	−4.00000	−0.216612
$342$	34.3475	1.85730
$343$	−15.4164	−0.832408
$344$	3.41641	0.184200
$345$	0	0
$346$	−18.7426	−1.00761
$347$	−28.0344	−1.50497	−0.752484	−	0.658611i	$-0.771144\pi$
$347$	−28.0344	−1.50497	−0.752484	+	0.658611i	$0.771144\pi$
$348$	−8.12461	−0.435525
$349$	11.3820	0.609263	0.304631	−	0.952470i	$-0.401467\pi$
$349$	11.3820	0.609263	0.304631	+	0.952470i	$0.401467\pi$
$350$	0	0
$351$	−6.38197	−0.340644
$352$	10.8541	0.578526
$353$	−10.2705	−0.546644	−0.273322	−	0.961923i	$-0.588122\pi$
$353$	−10.2705	−0.546644	−0.273322	+	0.961923i	$0.588122\pi$
$354$	6.58359	0.349914
$355$	0	0
$356$	−44.8328	−2.37613
$357$	−2.40325	−0.127194
$358$	−8.94427	−0.472719
$359$	−24.7639	−1.30699	−0.653495	−	0.756931i	$-0.726698\pi$
$359$	−24.7639	−1.30699	−0.653495	+	0.756931i	$0.726698\pi$
$360$	0	0
$361$	9.96556	0.524503
$362$	1.90983	0.100378
$363$	−3.20163	−0.168042
$364$	10.5836	0.554731
$365$	0	0
$366$	10.9787	0.573867
$367$	20.6525	1.07805	0.539025	−	0.842290i	$-0.318793\pi$
$367$	20.6525	1.07805	0.539025	+	0.842290i	$0.318793\pi$
$368$	4.47214	0.233126
$369$	−6.79837	−0.353909
$370$	0	0
$371$	−12.9443	−0.672033
$372$	2.83282	0.146875
$373$	−17.0902	−0.884895	−0.442448	−	0.896794i	$-0.645890\pi$
$373$	−17.0902	−0.884895	−0.442448	+	0.896794i	$0.645890\pi$
$374$	18.4164	0.952290
$375$	0	0
$376$	−7.56231	−0.389996
$377$	−20.2361	−1.04221
$378$	−6.18034	−0.317882
$379$	−3.50658	−0.180121	−0.0900604	−	0.995936i	$-0.528706\pi$
$379$	−3.50658	−0.180121	−0.0900604	+	0.995936i	$0.528706\pi$
$380$	0	0
$381$	−0.763932	−0.0391374
$382$	−36.1803	−1.85115
$383$	20.4721	1.04608	0.523039	−	0.852309i	$-0.324798\pi$
$383$	20.4721	1.04608	0.523039	+	0.852309i	$0.324798\pi$
$384$	−5.97871	−0.305100
$385$	0	0
$386$	45.7771	2.32999
$387$	−4.36068	−0.221666
$388$	21.7082	1.10207
$389$	15.1246	0.766848	0.383424	−	0.923572i	$-0.374745\pi$
$389$	15.1246	0.766848	0.383424	+	0.923572i	$0.374745\pi$
$390$	0	0
$391$	22.7639	1.15122
$392$	−12.2361	−0.618015
$393$	−7.94427	−0.400736
$394$	22.3607	1.12651
$395$	0	0
$396$	13.8541	0.696195
$397$	−28.4721	−1.42898	−0.714488	−	0.699648i	$-0.753340\pi$
$397$	−28.4721	−1.42898	−0.714488	+	0.699648i	$0.753340\pi$
$398$	−34.2705	−1.71783
$399$	−2.54102	−0.127210
$400$	0	0
$401$	−10.5066	−0.524673	−0.262337	−	0.964976i	$-0.584493\pi$
$401$	−10.5066	−0.524673	−0.262337	+	0.964976i	$0.584493\pi$
$402$	6.05573	0.302032
$403$	7.05573	0.351471
$404$	−18.5410	−0.922450
$405$	0	0
$406$	−19.5967	−0.972570
$407$	7.23607	0.358679
$408$	−4.34752	−0.215235
$409$	−3.41641	−0.168930	−0.0844652	−	0.996426i	$-0.526918\pi$
$409$	−3.41641	−0.168930	−0.0844652	+	0.996426i	$0.526918\pi$
$410$	0	0
$411$	−5.02129	−0.247682
$412$	−6.87539	−0.338726
$413$	9.52786	0.468836
$414$	28.5410	1.40271
$415$	0	0
$416$	−19.1459	−0.938705
$417$	−3.96556	−0.194194
$418$	19.4721	0.952413
$419$	26.9230	1.31527	0.657637	−	0.753335i	$-0.271556\pi$
$419$	26.9230	1.31527	0.657637	+	0.753335i	$0.271556\pi$
$420$	0	0
$421$	−28.4721	−1.38765	−0.693823	−	0.720145i	$-0.744075\pi$
$421$	−28.4721	−1.38765	−0.693823	+	0.720145i	$0.744075\pi$
$422$	42.7639	2.08172
$423$	9.65248	0.469319
$424$	−23.4164	−1.13720
$425$	0	0
$426$	6.45898	0.312939
$427$	15.8885	0.768901
$428$	37.6869	1.82167
$429$	−1.76393	−0.0851634
$430$	0	0
$431$	−29.9787	−1.44402	−0.722012	−	0.691881i	$-0.756782\pi$
$431$	−29.9787	−1.44402	−0.722012	+	0.691881i	$0.756782\pi$
$432$	2.23607	0.107583
$433$	25.7082	1.23546	0.617729	−	0.786391i	$-0.288053\pi$
$433$	25.7082	1.23546	0.617729	+	0.786391i	$0.288053\pi$
$434$	6.83282	0.327986
$435$	0	0
$436$	38.8328	1.85975
$437$	24.0689	1.15137
$438$	−4.59675	−0.219641
$439$	−15.9098	−0.759335	−0.379667	−	0.925123i	$-0.623962\pi$
$439$	−15.9098	−0.759335	−0.379667	+	0.925123i	$0.623962\pi$
$440$	0	0
$441$	15.6180	0.743716
$442$	−32.4853	−1.54517
$443$	14.0000	0.665160	0.332580	−	0.943075i	$-0.392081\pi$
$443$	14.0000	0.665160	0.332580	+	0.943075i	$0.392081\pi$
$444$	−5.12461	−0.243203
$445$	0	0
$446$	−2.68692	−0.127229
$447$	1.30495	0.0617221
$448$	−16.0689	−0.759183
$449$	11.1246	0.525003	0.262501	−	0.964932i	$-0.415453\pi$
$449$	11.1246	0.525003	0.262501	+	0.964932i	$0.415453\pi$
$450$	0	0
$451$	−3.85410	−0.181483
$452$	57.7082	2.71437
$453$	2.06888	0.0972046
$454$	38.2918	1.79712
$455$	0	0
$456$	−4.59675	−0.215262
$457$	27.7082	1.29614	0.648068	−	0.761583i	$-0.275577\pi$
$457$	27.7082	1.29614	0.648068	+	0.761583i	$0.275577\pi$
$458$	9.14590	0.427360
$459$	11.3820	0.531265
$460$	0	0
$461$	18.4721	0.860333	0.430167	−	0.902750i	$-0.358455\pi$
$461$	18.4721	0.860333	0.430167	+	0.902750i	$0.358455\pi$
$462$	−1.70820	−0.0794728
$463$	34.1803	1.58850	0.794248	−	0.607594i	$-0.207865\pi$
$463$	34.1803	1.58850	0.794248	+	0.607594i	$0.207865\pi$
$464$	7.09017	0.329153
$465$	0	0
$466$	−24.4721	−1.13365
$467$	−10.8541	−0.502268	−0.251134	−	0.967952i	$-0.580803\pi$
$467$	−10.8541	−0.502268	−0.251134	+	0.967952i	$0.580803\pi$
$468$	−24.4377	−1.12963
$469$	8.76393	0.404681
$470$	0	0
$471$	−8.41641	−0.387808
$472$	17.2361	0.793354
$473$	−2.47214	−0.113669
$474$	13.8197	0.634758
$475$	0	0
$476$	−18.8754	−0.865152
$477$	29.8885	1.36850
$478$	23.4164	1.07104
$479$	−8.47214	−0.387102	−0.193551	−	0.981090i	$-0.562000\pi$
$479$	−8.47214	−0.387102	−0.193551	+	0.981090i	$0.562000\pi$
$480$	0	0
$481$	−12.7639	−0.581985
$482$	2.23607	0.101850
$483$	−2.11146	−0.0960746
$484$	−25.1459	−1.14300
$485$	0	0
$486$	21.5836	0.979052
$487$	39.0344	1.76882	0.884410	−	0.466711i	$-0.154561\pi$
$487$	39.0344	1.76882	0.884410	+	0.466711i	$0.154561\pi$
$488$	28.7426	1.30112
$489$	−0.583592	−0.0263909
$490$	0	0
$491$	−29.3050	−1.32251	−0.661257	−	0.750159i	$-0.729977\pi$
$491$	−29.3050	−1.32251	−0.661257	+	0.750159i	$0.729977\pi$
$492$	2.72949	0.123055
$493$	36.0902	1.62542
$494$	−34.3475	−1.54537
$495$	0	0
$496$	−2.47214	−0.111002
$497$	9.34752	0.419294
$498$	0.777088	0.0348221
$499$	25.5279	1.14278	0.571392	−	0.820677i	$-0.306404\pi$
$499$	25.5279	1.14278	0.571392	+	0.820677i	$0.306404\pi$
$500$	0	0
$501$	−2.18034	−0.0974104
$502$	−65.3738	−2.91778
$503$	8.18034	0.364743	0.182372	−	0.983230i	$-0.441623\pi$
$503$	8.18034	0.364743	0.182372	+	0.983230i	$0.441623\pi$
$504$	−7.88854	−0.351384
$505$	0	0
$506$	16.1803	0.719304
$507$	−1.85410	−0.0823436
$508$	−6.00000	−0.266207
$509$	17.4377	0.772912	0.386456	−	0.922308i	$-0.373699\pi$
$509$	17.4377	0.772912	0.386456	+	0.922308i	$0.373699\pi$
$510$	0	0
$511$	−6.65248	−0.294288
$512$	11.1803	0.494106
$513$	12.0344	0.531334
$514$	51.9574	2.29174
$515$	0	0
$516$	1.75078	0.0770736
$517$	5.47214	0.240664
$518$	−12.3607	−0.543097
$519$	−3.20163	−0.140536
$520$	0	0
$521$	−14.6525	−0.641937	−0.320968	−	0.947090i	$-0.604008\pi$
$521$	−14.6525	−0.641937	−0.320968	+	0.947090i	$0.604008\pi$
$522$	45.2492	1.98050
$523$	−33.5066	−1.46514	−0.732570	−	0.680692i	$-0.761679\pi$
$523$	−33.5066	−1.46514	−0.732570	+	0.680692i	$0.761679\pi$
$524$	−62.3951	−2.72574
$525$	0	0
$526$	−35.7771	−1.55996
$527$	−12.5836	−0.548150
$528$	0.618034	0.0268965
$529$	−3.00000	−0.130435
$530$	0	0
$531$	−22.0000	−0.954719
$532$	−19.9574	−0.865264
$533$	6.79837	0.294470
$534$	−12.7639	−0.552349
$535$	0	0
$536$	15.8541	0.684793
$537$	−1.52786	−0.0659322
$538$	41.7082	1.79817
$539$	8.85410	0.381373
$540$	0	0
$541$	1.49342	0.0642072	0.0321036	−	0.999485i	$-0.489779\pi$
$541$	1.49342	0.0642072	0.0321036	+	0.999485i	$0.489779\pi$
$542$	33.0132	1.41804
$543$	0.326238	0.0140002
$544$	34.1459	1.46399
$545$	0	0
$546$	3.01316	0.128951
$547$	−37.1246	−1.58733	−0.793667	−	0.608353i	$-0.791831\pi$
$547$	−37.1246	−1.58733	−0.793667	+	0.608353i	$0.791831\pi$
$548$	−39.4377	−1.68469
$549$	−36.6869	−1.56576
$550$	0	0
$551$	38.1591	1.62563
$552$	−3.81966	−0.162576
$553$	20.0000	0.850487
$554$	11.0557	0.469713
$555$	0	0
$556$	−31.1459	−1.32088
$557$	33.8885	1.43590	0.717952	−	0.696093i	$-0.245080\pi$
$557$	33.8885	1.43590	0.717952	+	0.696093i	$0.245080\pi$
$558$	−15.7771	−0.667897
$559$	4.36068	0.184437
$560$	0	0
$561$	3.14590	0.132820
$562$	−62.8115	−2.64954
$563$	−7.97871	−0.336263	−0.168131	−	0.985765i	$-0.553773\pi$
$563$	−7.97871	−0.336263	−0.168131	+	0.985765i	$0.553773\pi$
$564$	−3.87539	−0.163183
$565$	0	0
$566$	2.76393	0.116177
$567$	9.52786	0.400133
$568$	16.9098	0.709521
$569$	−22.1591	−0.928956	−0.464478	−	0.885585i	$-0.653758\pi$
$569$	−22.1591	−0.928956	−0.464478	+	0.885585i	$0.653758\pi$
$570$	0	0
$571$	43.8541	1.83524	0.917619	−	0.397462i	$-0.130109\pi$
$571$	43.8541	1.83524	0.917619	+	0.397462i	$0.130109\pi$
$572$	−13.8541	−0.579269
$573$	−6.18034	−0.258187
$574$	6.58359	0.274794
$575$	0	0
$576$	37.1033	1.54597
$577$	27.2148	1.13297	0.566483	−	0.824073i	$-0.308304\pi$
$577$	27.2148	1.13297	0.566483	+	0.824073i	$0.308304\pi$
$578$	19.9230	0.828687
$579$	7.81966	0.324974
$580$	0	0
$581$	1.12461	0.0466568
$582$	6.18034	0.256183
$583$	16.9443	0.701760
$584$	−12.0344	−0.497989
$585$	0	0
$586$	−32.6869	−1.35028
$587$	39.8885	1.64638	0.823188	−	0.567769i	$-0.192193\pi$
$587$	39.8885	1.64638	0.823188	+	0.567769i	$0.192193\pi$
$588$	−6.27051	−0.258591
$589$	−13.3050	−0.548221
$590$	0	0
$591$	3.81966	0.157120
$592$	4.47214	0.183804
$593$	−3.88854	−0.159683	−0.0798417	−	0.996808i	$-0.525441\pi$
$593$	−3.88854	−0.159683	−0.0798417	+	0.996808i	$0.525441\pi$
$594$	8.09017	0.331944
$595$	0	0
$596$	10.2492	0.419825
$597$	−5.85410	−0.239592
$598$	−28.5410	−1.16713
$599$	−6.32624	−0.258483	−0.129242	−	0.991613i	$-0.541254\pi$
$599$	−6.32624	−0.258483	−0.129242	+	0.991613i	$0.541254\pi$
$600$	0	0
$601$	−29.0344	−1.18434	−0.592170	−	0.805813i	$-0.701729\pi$
$601$	−29.0344	−1.18434	−0.592170	+	0.805813i	$0.701729\pi$
$602$	4.22291	0.172113
$603$	−20.2361	−0.824076
$604$	16.2492	0.661172
$605$	0	0
$606$	−5.27864	−0.214430
$607$	24.3607	0.988769	0.494385	−	0.869243i	$-0.335393\pi$
$607$	24.3607	0.988769	0.494385	+	0.869243i	$0.335393\pi$
$608$	36.1033	1.46418
$609$	−3.34752	−0.135649
$610$	0	0
$611$	−9.65248	−0.390497
$612$	43.5836	1.76176
$613$	−0.729490	−0.0294638	−0.0147319	−	0.999891i	$-0.504689\pi$
$613$	−0.729490	−0.0294638	−0.0147319	+	0.999891i	$0.504689\pi$
$614$	−56.1803	−2.26725
$615$	0	0
$616$	−4.47214	−0.180187
$617$	−17.5279	−0.705645	−0.352823	−	0.935690i	$-0.614778\pi$
$617$	−17.5279	−0.705645	−0.352823	+	0.935690i	$0.614778\pi$
$618$	−1.95743	−0.0787393
$619$	−48.5066	−1.94964	−0.974822	−	0.222985i	$-0.928420\pi$
$619$	−48.5066	−1.94964	−0.974822	+	0.222985i	$0.928420\pi$
$620$	0	0
$621$	10.0000	0.401286
$622$	16.9098	0.678022
$623$	−18.4721	−0.740070
$624$	−1.09017	−0.0436417
$625$	0	0
$626$	−15.1246	−0.604501
$627$	3.32624	0.132837
$628$	−66.1033	−2.63781
$629$	22.7639	0.907657
$630$	0	0
$631$	−9.67376	−0.385106	−0.192553	−	0.981287i	$-0.561677\pi$
$631$	−9.67376	−0.385106	−0.192553	+	0.981287i	$0.561677\pi$
$632$	36.1803	1.43918
$633$	7.30495	0.290346
$634$	20.4508	0.812207
$635$	0	0
$636$	−12.0000	−0.475831
$637$	−15.6180	−0.618809
$638$	25.6525	1.01559
$639$	−21.5836	−0.853834
$640$	0	0
$641$	−8.61803	−0.340392	−0.170196	−	0.985410i	$-0.554440\pi$
$641$	−8.61803	−0.340392	−0.170196	+	0.985410i	$0.554440\pi$
$642$	10.7295	0.423459
$643$	−6.11146	−0.241012	−0.120506	−	0.992713i	$-0.538452\pi$
$643$	−6.11146	−0.241012	−0.120506	+	0.992713i	$0.538452\pi$
$644$	−16.5836	−0.653485
$645$	0	0
$646$	61.2574	2.41014
$647$	26.2918	1.03364	0.516819	−	0.856095i	$-0.327116\pi$
$647$	26.2918	1.03364	0.516819	+	0.856095i	$0.327116\pi$
$648$	17.2361	0.677097
$649$	−12.4721	−0.489574
$650$	0	0
$651$	1.16718	0.0457456
$652$	−4.58359	−0.179507
$653$	−49.0902	−1.92105	−0.960523	−	0.278199i	$-0.910262\pi$
$653$	−49.0902	−1.92105	−0.960523	+	0.278199i	$0.910262\pi$
$654$	11.0557	0.432313
$655$	0	0
$656$	−2.38197	−0.0930001
$657$	15.3607	0.599277
$658$	−9.34752	−0.364404
$659$	−0.875388	−0.0341003	−0.0170501	−	0.999855i	$-0.505427\pi$
$659$	−0.875388	−0.0341003	−0.0170501	+	0.999855i	$0.505427\pi$
$660$	0	0
$661$	20.2918	0.789259	0.394630	−	0.918840i	$-0.370873\pi$
$661$	20.2918	0.789259	0.394630	+	0.918840i	$0.370873\pi$
$662$	−46.1803	−1.79485
$663$	−5.54915	−0.215511
$664$	2.03444	0.0789517
$665$	0	0
$666$	28.5410	1.10594
$667$	31.7082	1.22775
$668$	−17.1246	−0.662571
$669$	−0.458980	−0.0177452
$670$	0	0
$671$	−20.7984	−0.802912
$672$	−3.16718	−0.122177
$673$	−19.8197	−0.763992	−0.381996	−	0.924164i	$-0.624763\pi$
$673$	−19.8197	−0.763992	−0.381996	+	0.924164i	$0.624763\pi$
$674$	−30.6525	−1.18069
$675$	0	0
$676$	−14.5623	−0.560089
$677$	15.9787	0.614112	0.307056	−	0.951691i	$-0.400656\pi$
$677$	15.9787	0.614112	0.307056	+	0.951691i	$0.400656\pi$
$678$	16.4296	0.630974
$679$	8.94427	0.343250
$680$	0	0
$681$	6.54102	0.250652
$682$	−8.94427	−0.342494
$683$	−5.52786	−0.211518	−0.105759	−	0.994392i	$-0.533727\pi$
$683$	−5.52786	−0.211518	−0.105759	+	0.994392i	$0.533727\pi$
$684$	46.0820	1.76199
$685$	0	0
$686$	−34.4721	−1.31615
$687$	1.56231	0.0596057
$688$	−1.52786	−0.0582493
$689$	−29.8885	−1.13866
$690$	0	0
$691$	−6.29180	−0.239351	−0.119676	−	0.992813i	$-0.538185\pi$
$691$	−6.29180	−0.239351	−0.119676	+	0.992813i	$0.538185\pi$
$692$	−25.1459	−0.955904
$693$	5.70820	0.216837
$694$	−62.6869	−2.37956
$695$	0	0
$696$	−6.05573	−0.229542
$697$	−12.1246	−0.459252
$698$	25.4508	0.963329
$699$	−4.18034	−0.158115
$700$	0	0
$701$	−20.2918	−0.766411	−0.383205	−	0.923663i	$-0.625180\pi$
$701$	−20.2918	−0.766411	−0.383205	+	0.923663i	$0.625180\pi$
$702$	−14.2705	−0.538606
$703$	24.0689	0.907775
$704$	21.0344	0.792765
$705$	0	0
$706$	−22.9656	−0.864320
$707$	−7.63932	−0.287306
$708$	8.83282	0.331958
$709$	10.8328	0.406835	0.203417	−	0.979092i	$-0.434795\pi$
$709$	10.8328	0.406835	0.203417	+	0.979092i	$0.434795\pi$
$710$	0	0
$711$	−46.1803	−1.73190
$712$	−33.4164	−1.25233
$713$	−11.0557	−0.414040
$714$	−5.37384	−0.201111
$715$	0	0
$716$	−12.0000	−0.448461
$717$	4.00000	0.149383
$718$	−55.3738	−2.06653
$719$	−48.7639	−1.81859	−0.909294	−	0.416155i	$-0.863378\pi$
$719$	−48.7639	−1.81859	−0.909294	+	0.416155i	$0.863378\pi$
$720$	0	0
$721$	−2.83282	−0.105500
$722$	22.2837	0.829312
$723$	0.381966	0.0142055
$724$	2.56231	0.0952273
$725$	0	0
$726$	−7.15905	−0.265697
$727$	32.9787	1.22311	0.611556	−	0.791201i	$-0.290544\pi$
$727$	32.9787	1.22311	0.611556	+	0.791201i	$0.290544\pi$
$728$	7.88854	0.292369
$729$	−19.4377	−0.719915
$730$	0	0
$731$	−7.77709	−0.287646
$732$	14.7295	0.544418
$733$	−4.29180	−0.158521	−0.0792606	−	0.996854i	$-0.525256\pi$
$733$	−4.29180	−0.158521	−0.0792606	+	0.996854i	$0.525256\pi$
$734$	46.1803	1.70455
$735$	0	0
$736$	30.0000	1.10581
$737$	−11.4721	−0.422582
$738$	−15.2016	−0.559580
$739$	12.1803	0.448061	0.224031	−	0.974582i	$-0.428078\pi$
$739$	12.1803	0.448061	0.224031	+	0.974582i	$0.428078\pi$
$740$	0	0
$741$	−5.86726	−0.215539
$742$	−28.9443	−1.06258
$743$	−0.562306	−0.0206290	−0.0103145	−	0.999947i	$-0.503283\pi$
$743$	−0.562306	−0.0206290	−0.0103145	+	0.999947i	$0.503283\pi$
$744$	2.11146	0.0774097
$745$	0	0
$746$	−38.2148	−1.39914
$747$	−2.59675	−0.0950100
$748$	24.7082	0.903421
$749$	15.5279	0.567376
$750$	0	0
$751$	26.3820	0.962692	0.481346	−	0.876531i	$-0.340148\pi$
$751$	26.3820	0.962692	0.481346	+	0.876531i	$0.340148\pi$
$752$	3.38197	0.123328
$753$	−11.1672	−0.406955
$754$	−45.2492	−1.64788
$755$	0	0
$756$	−8.29180	−0.301570
$757$	−30.3262	−1.10223	−0.551113	−	0.834431i	$-0.685797\pi$
$757$	−30.3262	−1.10223	−0.551113	+	0.834431i	$0.685797\pi$
$758$	−7.84095	−0.284796
$759$	2.76393	0.100324
$760$	0	0
$761$	−35.8885	−1.30096	−0.650479	−	0.759524i	$-0.725432\pi$
$761$	−35.8885	−1.30096	−0.650479	+	0.759524i	$0.725432\pi$
$762$	−1.70820	−0.0618817
$763$	16.0000	0.579239
$764$	−48.5410	−1.75615
$765$	0	0
$766$	45.7771	1.65399
$767$	22.0000	0.794374
$768$	−3.43769	−0.124047
$769$	−18.8328	−0.679129	−0.339564	−	0.940583i	$-0.610280\pi$
$769$	−18.8328	−0.679129	−0.339564	+	0.940583i	$0.610280\pi$
$770$	0	0
$771$	8.87539	0.319639
$772$	61.4164	2.21042
$773$	−11.1246	−0.400124	−0.200062	−	0.979783i	$-0.564114\pi$
$773$	−11.1246	−0.400124	−0.200062	+	0.979783i	$0.564114\pi$
$774$	−9.75078	−0.350484
$775$	0	0
$776$	16.1803	0.580840
$777$	−2.11146	−0.0757481
$778$	33.8197	1.21249
$779$	−12.8197	−0.459312
$780$	0	0
$781$	−12.2361	−0.437841
$782$	50.9017	1.82024
$783$	15.8541	0.566579
$784$	5.47214	0.195433
$785$	0	0
$786$	−17.7639	−0.633618
$787$	27.6180	0.984477	0.492238	−	0.870460i	$-0.336179\pi$
$787$	27.6180	0.984477	0.492238	+	0.870460i	$0.336179\pi$
$788$	30.0000	1.06871
$789$	−6.11146	−0.217574
$790$	0	0
$791$	23.7771	0.845416
$792$	10.3262	0.366927
$793$	36.6869	1.30279
$794$	−63.6656	−2.25941
$795$	0	0
$796$	−45.9787	−1.62967
$797$	11.2705	0.399222	0.199611	−	0.979875i	$-0.436032\pi$
$797$	11.2705	0.399222	0.199611	+	0.979875i	$0.436032\pi$
$798$	−5.68189	−0.201137
$799$	17.2148	0.609015
$800$	0	0
$801$	42.6525	1.50705
$802$	−23.4934	−0.829582
$803$	8.70820	0.307306
$804$	8.12461	0.286533
$805$	0	0
$806$	15.7771	0.555724
$807$	7.12461	0.250798
$808$	−13.8197	−0.486174
$809$	−16.9443	−0.595729	−0.297864	−	0.954608i	$-0.596274\pi$
$809$	−16.9443	−0.595729	−0.297864	+	0.954608i	$0.596274\pi$
$810$	0	0
$811$	7.50658	0.263592	0.131796	−	0.991277i	$-0.457926\pi$
$811$	7.50658	0.263592	0.131796	+	0.991277i	$0.457926\pi$
$812$	−26.2918	−0.922661
$813$	5.63932	0.197780
$814$	16.1803	0.567121
$815$	0	0
$816$	1.94427	0.0680631
$817$	−8.22291	−0.287683
$818$	−7.63932	−0.267103
$819$	−10.0689	−0.351835
$820$	0	0
$821$	−6.38197	−0.222732	−0.111366	−	0.993779i	$-0.535523\pi$
$821$	−6.38197	−0.222732	−0.111366	+	0.993779i	$0.535523\pi$
$822$	−11.2279	−0.391619
$823$	11.4164	0.397951	0.198975	−	0.980004i	$-0.436239\pi$
$823$	11.4164	0.397951	0.198975	+	0.980004i	$0.436239\pi$
$824$	−5.12461	−0.178524
$825$	0	0
$826$	21.3050	0.741294
$827$	23.8885	0.830686	0.415343	−	0.909665i	$-0.363662\pi$
$827$	23.8885	0.830686	0.415343	+	0.909665i	$0.363662\pi$
$828$	38.2918	1.33073
$829$	−28.6869	−0.996338	−0.498169	−	0.867080i	$-0.665994\pi$
$829$	−28.6869	−0.996338	−0.498169	+	0.867080i	$0.665994\pi$
$830$	0	0
$831$	1.88854	0.0655129
$832$	−37.1033	−1.28633
$833$	27.8541	0.965087
$834$	−8.86726	−0.307048
$835$	0	0
$836$	26.1246	0.903539
$837$	−5.52786	−0.191071
$838$	60.2016	2.07963
$839$	−8.47214	−0.292491	−0.146245	−	0.989248i	$-0.546719\pi$
$839$	−8.47214	−0.292491	−0.146245	+	0.989248i	$0.546719\pi$
$840$	0	0
$841$	21.2705	0.733466
$842$	−63.6656	−2.19406
$843$	−10.7295	−0.369543
$844$	57.3738	1.97489
$845$	0	0
$846$	21.5836	0.742059
$847$	−10.3607	−0.355997
$848$	10.4721	0.359615
$849$	0.472136	0.0162037
$850$	0	0
$851$	20.0000	0.685591
$852$	8.66563	0.296880
$853$	47.2148	1.61660	0.808302	−	0.588769i	$-0.200387\pi$
$853$	47.2148	1.61660	0.808302	+	0.588769i	$0.200387\pi$
$854$	35.5279	1.21574
$855$	0	0
$856$	28.0902	0.960102
$857$	29.8197	1.01862	0.509310	−	0.860583i	$-0.329901\pi$
$857$	29.8197	1.01862	0.509310	+	0.860583i	$0.329901\pi$
$858$	−3.94427	−0.134655
$859$	25.3820	0.866022	0.433011	−	0.901389i	$-0.357451\pi$
$859$	25.3820	0.866022	0.433011	+	0.901389i	$0.357451\pi$
$860$	0	0
$861$	1.12461	0.0383267
$862$	−67.0344	−2.28320
$863$	13.7082	0.466633	0.233316	−	0.972401i	$-0.425042\pi$
$863$	13.7082	0.466633	0.233316	+	0.972401i	$0.425042\pi$
$864$	15.0000	0.510310
$865$	0	0
$866$	57.4853	1.95343
$867$	3.40325	0.115581
$868$	9.16718	0.311155
$869$	−26.1803	−0.888107
$870$	0	0
$871$	20.2361	0.685673
$872$	28.9443	0.980177
$873$	−20.6525	−0.698980
$874$	53.8197	1.82048
$875$	0	0
$876$	−6.16718	−0.208370
$877$	−15.2361	−0.514485	−0.257243	−	0.966347i	$-0.582814\pi$
$877$	−15.2361	−0.514485	−0.257243	+	0.966347i	$0.582814\pi$
$878$	−35.5755	−1.20061
$879$	−5.58359	−0.188330
$880$	0	0
$881$	23.0902	0.777928	0.388964	−	0.921253i	$-0.372833\pi$
$881$	23.0902	0.777928	0.388964	+	0.921253i	$0.372833\pi$
$882$	34.9230	1.17592
$883$	−9.14590	−0.307784	−0.153892	−	0.988088i	$-0.549181\pi$
$883$	−9.14590	−0.307784	−0.153892	+	0.988088i	$0.549181\pi$
$884$	−43.5836	−1.46587
$885$	0	0
$886$	31.3050	1.05171
$887$	13.7295	0.460991	0.230496	−	0.973073i	$-0.425965\pi$
$887$	13.7295	0.460991	0.230496	+	0.973073i	$0.425965\pi$
$888$	−3.81966	−0.128179
$889$	−2.47214	−0.0829128
$890$	0	0
$891$	−12.4721	−0.417832
$892$	−3.60488	−0.120700
$893$	18.2016	0.609094
$894$	2.91796	0.0975912
$895$	0	0
$896$	−19.3475	−0.646355
$897$	−4.87539	−0.162784
$898$	24.8754	0.830102
$899$	−17.5279	−0.584587
$900$	0	0
$901$	53.3050	1.77585
$902$	−8.61803	−0.286949
$903$	0.721360	0.0240053
$904$	43.0132	1.43060
$905$	0	0
$906$	4.62616	0.153694
$907$	−32.7639	−1.08791	−0.543954	−	0.839115i	$-0.683073\pi$
$907$	−32.7639	−1.08791	−0.543954	+	0.839115i	$0.683073\pi$
$908$	51.3738	1.70490
$909$	17.6393	0.585059
$910$	0	0
$911$	20.0689	0.664912	0.332456	−	0.943119i	$-0.392123\pi$
$911$	20.0689	0.664912	0.332456	+	0.943119i	$0.392123\pi$
$912$	2.05573	0.0680720
$913$	−1.47214	−0.0487206
$914$	61.9574	2.04937
$915$	0	0
$916$	12.2705	0.405429
$917$	−25.7082	−0.848960
$918$	25.4508	0.840003
$919$	−35.3050	−1.16460	−0.582301	−	0.812973i	$-0.697848\pi$
$919$	−35.3050	−1.16460	−0.582301	+	0.812973i	$0.697848\pi$
$920$	0	0
$921$	−9.59675	−0.316224
$922$	41.3050	1.36031
$923$	21.5836	0.710433
$924$	−2.29180	−0.0753946
$925$	0	0
$926$	76.4296	2.51163
$927$	6.54102	0.214835
$928$	47.5623	1.56131
$929$	13.1246	0.430605	0.215302	−	0.976547i	$-0.430926\pi$
$929$	13.1246	0.430605	0.215302	+	0.976547i	$0.430926\pi$
$930$	0	0
$931$	29.4508	0.965213
$932$	−32.8328	−1.07547
$933$	2.88854	0.0945667
$934$	−24.2705	−0.794155
$935$	0	0
$936$	−18.2148	−0.595369
$937$	−20.0000	−0.653372	−0.326686	−	0.945133i	$-0.605932\pi$
$937$	−20.0000	−0.653372	−0.326686	+	0.945133i	$0.605932\pi$
$938$	19.5967	0.639856
$939$	−2.58359	−0.0843123
$940$	0	0
$941$	40.6525	1.32523	0.662616	−	0.748959i	$-0.269446\pi$
$941$	40.6525	1.32523	0.662616	+	0.748959i	$0.269446\pi$
$942$	−18.8197	−0.613178
$943$	−10.6525	−0.346892
$944$	−7.70820	−0.250881
$945$	0	0
$946$	−5.52786	−0.179726
$947$	−2.18034	−0.0708515	−0.0354258	−	0.999372i	$-0.511279\pi$
$947$	−2.18034	−0.0708515	−0.0354258	+	0.999372i	$0.511279\pi$
$948$	18.5410	0.602184
$949$	−15.3607	−0.498629
$950$	0	0
$951$	3.49342	0.113282
$952$	−14.0689	−0.455975
$953$	−31.9787	−1.03589	−0.517946	−	0.855413i	$-0.673303\pi$
$953$	−31.9787	−1.03589	−0.517946	+	0.855413i	$0.673303\pi$
$954$	66.8328	2.16379
$955$	0	0
$956$	31.4164	1.01608
$957$	4.38197	0.141649
$958$	−18.9443	−0.612062
$959$	−16.2492	−0.524715
$960$	0	0
$961$	−24.8885	−0.802856
$962$	−28.5410	−0.920199
$963$	−35.8541	−1.15538
$964$	3.00000	0.0966235
$965$	0	0
$966$	−4.72136	−0.151907
$967$	3.05573	0.0982656	0.0491328	−	0.998792i	$-0.484354\pi$
$967$	3.05573	0.0982656	0.0491328	+	0.998792i	$0.484354\pi$
$968$	−18.7426	−0.602411
$969$	10.4640	0.336152
$970$	0	0
$971$	−43.0557	−1.38172	−0.690862	−	0.722987i	$-0.742769\pi$
$971$	−43.0557	−1.38172	−0.690862	+	0.722987i	$0.742769\pi$
$972$	28.9574	0.928810
$973$	−12.8328	−0.411401
$974$	87.2837	2.79675
$975$	0	0
$976$	−12.8541	−0.411450
$977$	39.8885	1.27615	0.638074	−	0.769975i	$-0.279731\pi$
$977$	39.8885	1.27615	0.638074	+	0.769975i	$0.279731\pi$
$978$	−1.30495	−0.0417278
$979$	24.1803	0.772807
$980$	0	0
$981$	−36.9443	−1.17954
$982$	−65.5279	−2.09108
$983$	−11.1246	−0.354820	−0.177410	−	0.984137i	$-0.556772\pi$
$983$	−11.1246	−0.354820	−0.177410	+	0.984137i	$0.556772\pi$
$984$	2.03444	0.0648556
$985$	0	0
$986$	80.7001	2.57001
$987$	−1.59675	−0.0508250
$988$	−46.0820	−1.46606
$989$	−6.83282	−0.217271
$990$	0	0
$991$	−40.3607	−1.28210	−0.641050	−	0.767499i	$-0.721501\pi$
$991$	−40.3607	−1.28210	−0.641050	+	0.767499i	$0.721501\pi$
$992$	−16.5836	−0.526530
$993$	−7.88854	−0.250335
$994$	20.9017	0.662962
$995$	0	0
$996$	1.04257	0.0330352
$997$	35.7984	1.13375	0.566873	−	0.823805i	$-0.308153\pi$
$997$	35.7984	1.13375	0.566873	+	0.823805i	$0.308153\pi$
$998$	57.0820	1.80690
$999$	10.0000	0.316386

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6025.2.a.c.1.2		2
5.2	odd	4		1205.2.b.a.724.3	yes	4
5.3	odd	4		1205.2.b.a.724.2	✓	4
5.4	even	2		6025.2.a.b.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1205.2.b.a.724.2	✓	4	5.3	odd	4
1205.2.b.a.724.3	yes	4	5.2	odd	4
6025.2.a.b.1.1		2	5.4	even	2
6025.2.a.c.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:	\( N \)	\(=\)	\( 6025 = 5^{2} \cdot 241 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6025.a (trivial)

\(f(q)\)	\(=\)	\(q+2.23607 q^{2} +0.381966 q^{3} +3.00000 q^{4} +0.854102 q^{6} +1.23607 q^{7} +2.23607 q^{8} -2.85410 q^{9} +O(q^{10})\)	\(q+2.23607 q^{2} +0.381966 q^{3} +3.00000 q^{4} +0.854102 q^{6} +1.23607 q^{7} +2.23607 q^{8} -2.85410 q^{9} -1.61803 q^{11} +1.14590 q^{12} +2.85410 q^{13} +2.76393 q^{14} -1.00000 q^{16} -5.09017 q^{17} -6.38197 q^{18} -5.38197 q^{19} +0.472136 q^{21} -3.61803 q^{22} -4.47214 q^{23} +0.854102 q^{24} +6.38197 q^{26} -2.23607 q^{27} +3.70820 q^{28} -7.09017 q^{29} +2.47214 q^{31} -6.70820 q^{32} -0.618034 q^{33} -11.3820 q^{34} -8.56231 q^{36} -4.47214 q^{37} -12.0344 q^{38} +1.09017 q^{39} +2.38197 q^{41} +1.05573 q^{42} +1.52786 q^{43} -4.85410 q^{44} -10.0000 q^{46} -3.38197 q^{47} -0.381966 q^{48} -5.47214 q^{49} -1.94427 q^{51} +8.56231 q^{52} -10.4721 q^{53} -5.00000 q^{54} +2.76393 q^{56} -2.05573 q^{57} -15.8541 q^{58} +7.70820 q^{59} +12.8541 q^{61} +5.52786 q^{62} -3.52786 q^{63} -13.0000 q^{64} -1.38197 q^{66} +7.09017 q^{67} -15.2705 q^{68} -1.70820 q^{69} +7.56231 q^{71} -6.38197 q^{72} -5.38197 q^{73} -10.0000 q^{74} -16.1459 q^{76} -2.00000 q^{77} +2.43769 q^{78} +16.1803 q^{79} +7.70820 q^{81} +5.32624 q^{82} +0.909830 q^{83} +1.41641 q^{84} +3.41641 q^{86} -2.70820 q^{87} -3.61803 q^{88} -14.9443 q^{89} +3.52786 q^{91} -13.4164 q^{92} +0.944272 q^{93} -7.56231 q^{94} -2.56231 q^{96} +7.23607 q^{97} -12.2361 q^{98} +4.61803 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 3 q^{3} + 6 q^{4} - 5 q^{6} - 2 q^{7} + q^{9}+O(q^{10}) \) 2 * q + 3 * q^3 + 6 * q^4 - 5 * q^6 - 2 * q^7 + q^9	\( 2 q + 3 q^{3} + 6 q^{4} - 5 q^{6} - 2 q^{7} + q^{9} - q^{11} + 9 q^{12} - q^{13} + 10 q^{14} - 2 q^{16} + q^{17} - 15 q^{18} - 13 q^{19} - 8 q^{21} - 5 q^{22} - 5 q^{24} + 15 q^{26} - 6 q^{28} - 3 q^{29} - 4 q^{31} + q^{33} - 25 q^{34} + 3 q^{36} + 5 q^{38} - 9 q^{39} + 7 q^{41} + 20 q^{42} + 12 q^{43} - 3 q^{44} - 20 q^{46} - 9 q^{47} - 3 q^{48} - 2 q^{49} + 14 q^{51} - 3 q^{52} - 12 q^{53} - 10 q^{54} + 10 q^{56} - 22 q^{57} - 25 q^{58} + 2 q^{59} + 19 q^{61} + 20 q^{62} - 16 q^{63} - 26 q^{64} - 5 q^{66} + 3 q^{67} + 3 q^{68} + 10 q^{69} - 5 q^{71} - 15 q^{72} - 13 q^{73} - 20 q^{74} - 39 q^{76} - 4 q^{77} + 25 q^{78} + 10 q^{79} + 2 q^{81} - 5 q^{82} + 13 q^{83} - 24 q^{84} - 20 q^{86} + 8 q^{87} - 5 q^{88} - 12 q^{89} + 16 q^{91} - 16 q^{93} + 5 q^{94} + 15 q^{96} + 10 q^{97} - 20 q^{98} + 7 q^{99}+O(q^{100}) \) 2 * q + 3 * q^3 + 6 * q^4 - 5 * q^6 - 2 * q^7 + q^9 - q^11 + 9 * q^12 - q^13 + 10 * q^14 - 2 * q^16 + q^17 - 15 * q^18 - 13 * q^19 - 8 * q^21 - 5 * q^22 - 5 * q^24 + 15 * q^26 - 6 * q^28 - 3 * q^29 - 4 * q^31 + q^33 - 25 * q^34 + 3 * q^36 + 5 * q^38 - 9 * q^39 + 7 * q^41 + 20 * q^42 + 12 * q^43 - 3 * q^44 - 20 * q^46 - 9 * q^47 - 3 * q^48 - 2 * q^49 + 14 * q^51 - 3 * q^52 - 12 * q^53 - 10 * q^54 + 10 * q^56 - 22 * q^57 - 25 * q^58 + 2 * q^59 + 19 * q^61 + 20 * q^62 - 16 * q^63 - 26 * q^64 - 5 * q^66 + 3 * q^67 + 3 * q^68 + 10 * q^69 - 5 * q^71 - 15 * q^72 - 13 * q^73 - 20 * q^74 - 39 * q^76 - 4 * q^77 + 25 * q^78 + 10 * q^79 + 2 * q^81 - 5 * q^82 + 13 * q^83 - 24 * q^84 - 20 * q^86 + 8 * q^87 - 5 * q^88 - 12 * q^89 + 16 * q^91 - 16 * q^93 + 5 * q^94 + 15 * q^96 + 10 * q^97 - 20 * q^98 + 7 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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