LMFDB - Embedded newform 8025.2.a.s.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8025,2,Mod(1,8025)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8025, 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("8025.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8025 = 3 \cdot 5^{2} \cdot 107 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8025.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:	$64.0799476221$
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:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 321)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-0.618034$ of defining polynomial
Character	$\chi$	$=$	8025.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.618034 q^{2} +1.00000 q^{3} -1.61803 q^{4} -0.618034 q^{6} +2.00000 q^{7} +2.23607 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-0.618034 q^{2} +1.00000 q^{3} -1.61803 q^{4} -0.618034 q^{6} +2.00000 q^{7} +2.23607 q^{8} +1.00000 q^{9} -5.23607 q^{11} -1.61803 q^{12} +1.00000 q^{13} -1.23607 q^{14} +1.85410 q^{16} +3.47214 q^{17} -0.618034 q^{18} -1.76393 q^{19} +2.00000 q^{21} +3.23607 q^{22} -2.47214 q^{23} +2.23607 q^{24} -0.618034 q^{26} +1.00000 q^{27} -3.23607 q^{28} +1.23607 q^{29} -6.00000 q^{31} -5.61803 q^{32} -5.23607 q^{33} -2.14590 q^{34} -1.61803 q^{36} -1.00000 q^{37} +1.09017 q^{38} +1.00000 q^{39} +8.94427 q^{41} -1.23607 q^{42} +5.23607 q^{43} +8.47214 q^{44} +1.52786 q^{46} -5.23607 q^{47} +1.85410 q^{48} -3.00000 q^{49} +3.47214 q^{51} -1.61803 q^{52} -9.23607 q^{53} -0.618034 q^{54} +4.47214 q^{56} -1.76393 q^{57} -0.763932 q^{58} -4.94427 q^{59} +14.4164 q^{61} +3.70820 q^{62} +2.00000 q^{63} -0.236068 q^{64} +3.23607 q^{66} -7.23607 q^{67} -5.61803 q^{68} -2.47214 q^{69} -1.76393 q^{71} +2.23607 q^{72} +10.1803 q^{73} +0.618034 q^{74} +2.85410 q^{76} -10.4721 q^{77} -0.618034 q^{78} -15.4164 q^{79} +1.00000 q^{81} -5.52786 q^{82} +1.52786 q^{83} -3.23607 q^{84} -3.23607 q^{86} +1.23607 q^{87} -11.7082 q^{88} -7.23607 q^{89} +2.00000 q^{91} +4.00000 q^{92} -6.00000 q^{93} +3.23607 q^{94} -5.61803 q^{96} +7.70820 q^{97} +1.85410 q^{98} -5.23607 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + q^{2} + 2 q^{3} - q^{4} + q^{6} + 4 q^{7} + 2 q^{9}+O(q^{10}) $ 2 * q + q^2 + 2 * q^3 - q^4 + q^6 + 4 * q^7 + 2 * q^9	$ 2 q + q^{2} + 2 q^{3} - q^{4} + q^{6} + 4 q^{7} + 2 q^{9} - 6 q^{11} - q^{12} + 2 q^{13} + 2 q^{14} - 3 q^{16} - 2 q^{17} + q^{18} - 8 q^{19} + 4 q^{21} + 2 q^{22} + 4 q^{23} + q^{26} + 2 q^{27} - 2 q^{28} - 2 q^{29} - 12 q^{31} - 9 q^{32} - 6 q^{33} - 11 q^{34} - q^{36} - 2 q^{37} - 9 q^{38} + 2 q^{39} + 2 q^{42} + 6 q^{43} + 8 q^{44} + 12 q^{46} - 6 q^{47} - 3 q^{48} - 6 q^{49} - 2 q^{51} - q^{52} - 14 q^{53} + q^{54} - 8 q^{57} - 6 q^{58} + 8 q^{59} + 2 q^{61} - 6 q^{62} + 4 q^{63} + 4 q^{64} + 2 q^{66} - 10 q^{67} - 9 q^{68} + 4 q^{69} - 8 q^{71} - 2 q^{73} - q^{74} - q^{76} - 12 q^{77} + q^{78} - 4 q^{79} + 2 q^{81} - 20 q^{82} + 12 q^{83} - 2 q^{84} - 2 q^{86} - 2 q^{87} - 10 q^{88} - 10 q^{89} + 4 q^{91} + 8 q^{92} - 12 q^{93} + 2 q^{94} - 9 q^{96} + 2 q^{97} - 3 q^{98} - 6 q^{99}+O(q^{100}) $ 2 * q + q^2 + 2 * q^3 - q^4 + q^6 + 4 * q^7 + 2 * q^9 - 6 * q^11 - q^12 + 2 * q^13 + 2 * q^14 - 3 * q^16 - 2 * q^17 + q^18 - 8 * q^19 + 4 * q^21 + 2 * q^22 + 4 * q^23 + q^26 + 2 * q^27 - 2 * q^28 - 2 * q^29 - 12 * q^31 - 9 * q^32 - 6 * q^33 - 11 * q^34 - q^36 - 2 * q^37 - 9 * q^38 + 2 * q^39 + 2 * q^42 + 6 * q^43 + 8 * q^44 + 12 * q^46 - 6 * q^47 - 3 * q^48 - 6 * q^49 - 2 * q^51 - q^52 - 14 * q^53 + q^54 - 8 * q^57 - 6 * q^58 + 8 * q^59 + 2 * q^61 - 6 * q^62 + 4 * q^63 + 4 * q^64 + 2 * q^66 - 10 * q^67 - 9 * q^68 + 4 * q^69 - 8 * q^71 - 2 * q^73 - q^74 - q^76 - 12 * q^77 + q^78 - 4 * q^79 + 2 * q^81 - 20 * q^82 + 12 * q^83 - 2 * q^84 - 2 * q^86 - 2 * q^87 - 10 * q^88 - 10 * q^89 + 4 * q^91 + 8 * q^92 - 12 * q^93 + 2 * q^94 - 9 * q^96 + 2 * q^97 - 3 * q^98 - 6 * 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 $

$2$	−0.618034	−0.437016	−0.218508	−	0.975835i	$-0.570119\pi$
$2$	−0.618034	−0.437016	−0.218508	+	0.975835i	$0.570119\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	−1.61803	−0.809017
$5$	0	0
$5$	0	0
$6$	−0.618034	−0.252311
$7$	2.00000	0.755929	0.377964	−	0.925820i	$-0.376624\pi$
$7$	2.00000	0.755929	0.377964	+	0.925820i	$0.376624\pi$
$8$	2.23607	0.790569
$9$	1.00000	0.333333
$10$	0	0
$11$	−5.23607	−1.57873	−0.789367	−	0.613922i	$-0.789591\pi$
$11$	−5.23607	−1.57873	−0.789367	+	0.613922i	$0.789591\pi$
$12$	−1.61803	−0.467086
$13$	1.00000	0.277350	0.138675	−	0.990338i	$-0.455716\pi$
$13$	1.00000	0.277350	0.138675	+	0.990338i	$0.455716\pi$
$14$	−1.23607	−0.330353
$15$	0	0
$16$	1.85410	0.463525
$17$	3.47214	0.842117	0.421058	−	0.907034i	$-0.361659\pi$
$17$	3.47214	0.842117	0.421058	+	0.907034i	$0.361659\pi$
$18$	−0.618034	−0.145672
$19$	−1.76393	−0.404674	−0.202337	−	0.979316i	$-0.564854\pi$
$19$	−1.76393	−0.404674	−0.202337	+	0.979316i	$0.564854\pi$
$20$	0	0
$21$	2.00000	0.436436
$22$	3.23607	0.689932
$23$	−2.47214	−0.515476	−0.257738	−	0.966215i	$-0.582977\pi$
$23$	−2.47214	−0.515476	−0.257738	+	0.966215i	$0.582977\pi$
$24$	2.23607	0.456435
$25$	0	0
$26$	−0.618034	−0.121206
$27$	1.00000	0.192450
$28$	−3.23607	−0.611559
$29$	1.23607	0.229532	0.114766	−	0.993393i	$-0.463388\pi$
$29$	1.23607	0.229532	0.114766	+	0.993393i	$0.463388\pi$
$30$	0	0
$31$	−6.00000	−1.07763	−0.538816	−	0.842424i	$-0.681128\pi$
$31$	−6.00000	−1.07763	−0.538816	+	0.842424i	$0.681128\pi$
$32$	−5.61803	−0.993137
$33$	−5.23607	−0.911482
$34$	−2.14590	−0.368018
$35$	0	0
$36$	−1.61803	−0.269672
$37$	−1.00000	−0.164399	−0.0821995	−	0.996616i	$-0.526194\pi$
$37$	−1.00000	−0.164399	−0.0821995	+	0.996616i	$0.526194\pi$
$38$	1.09017	0.176849
$39$	1.00000	0.160128
$40$	0	0
$41$	8.94427	1.39686	0.698430	−	0.715678i	$-0.253882\pi$
$41$	8.94427	1.39686	0.698430	+	0.715678i	$0.253882\pi$
$42$	−1.23607	−0.190729
$43$	5.23607	0.798493	0.399246	−	0.916844i	$-0.369272\pi$
$43$	5.23607	0.798493	0.399246	+	0.916844i	$0.369272\pi$
$44$	8.47214	1.27722
$45$	0	0
$46$	1.52786	0.225271
$47$	−5.23607	−0.763759	−0.381880	−	0.924212i	$-0.624723\pi$
$47$	−5.23607	−0.763759	−0.381880	+	0.924212i	$0.624723\pi$
$48$	1.85410	0.267617
$49$	−3.00000	−0.428571
$50$	0	0
$51$	3.47214	0.486196
$52$	−1.61803	−0.224381
$53$	−9.23607	−1.26867	−0.634336	−	0.773058i	$-0.718726\pi$
$53$	−9.23607	−1.26867	−0.634336	+	0.773058i	$0.718726\pi$
$54$	−0.618034	−0.0841038
$55$	0	0
$56$	4.47214	0.597614
$57$	−1.76393	−0.233639
$58$	−0.763932	−0.100309
$59$	−4.94427	−0.643689	−0.321845	−	0.946792i	$-0.604303\pi$
$59$	−4.94427	−0.643689	−0.321845	+	0.946792i	$0.604303\pi$
$60$	0	0
$61$	14.4164	1.84583	0.922916	−	0.385002i	$-0.125799\pi$
$61$	14.4164	1.84583	0.922916	+	0.385002i	$0.125799\pi$
$62$	3.70820	0.470942
$63$	2.00000	0.251976
$64$	−0.236068	−0.0295085
$65$	0	0
$66$	3.23607	0.398332
$67$	−7.23607	−0.884026	−0.442013	−	0.897009i	$-0.645736\pi$
$67$	−7.23607	−0.884026	−0.442013	+	0.897009i	$0.645736\pi$
$68$	−5.61803	−0.681287
$69$	−2.47214	−0.297610
$70$	0	0
$71$	−1.76393	−0.209340	−0.104670	−	0.994507i	$-0.533379\pi$
$71$	−1.76393	−0.209340	−0.104670	+	0.994507i	$0.533379\pi$
$72$	2.23607	0.263523
$73$	10.1803	1.19152	0.595759	−	0.803163i	$-0.296851\pi$
$73$	10.1803	1.19152	0.595759	+	0.803163i	$0.296851\pi$
$74$	0.618034	0.0718450
$75$	0	0
$76$	2.85410	0.327388
$77$	−10.4721	−1.19341
$78$	−0.618034	−0.0699786
$79$	−15.4164	−1.73448	−0.867241	−	0.497889i	$-0.834109\pi$
$79$	−15.4164	−1.73448	−0.867241	+	0.497889i	$0.834109\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	−5.52786	−0.610450
$83$	1.52786	0.167705	0.0838524	−	0.996478i	$-0.473278\pi$
$83$	1.52786	0.167705	0.0838524	+	0.996478i	$0.473278\pi$
$84$	−3.23607	−0.353084
$85$	0	0
$86$	−3.23607	−0.348954
$87$	1.23607	0.132520
$88$	−11.7082	−1.24810
$89$	−7.23607	−0.767022	−0.383511	−	0.923536i	$-0.625285\pi$
$89$	−7.23607	−0.767022	−0.383511	+	0.923536i	$0.625285\pi$
$90$	0	0
$91$	2.00000	0.209657
$92$	4.00000	0.417029
$93$	−6.00000	−0.622171
$94$	3.23607	0.333775
$95$	0	0
$96$	−5.61803	−0.573388
$97$	7.70820	0.782650	0.391325	−	0.920253i	$-0.372017\pi$
$97$	7.70820	0.782650	0.391325	+	0.920253i	$0.372017\pi$
$98$	1.85410	0.187293
$99$	−5.23607	−0.526245
$100$	0	0
$101$	−11.7082	−1.16501	−0.582505	−	0.812827i	$-0.697927\pi$
$101$	−11.7082	−1.16501	−0.582505	+	0.812827i	$0.697927\pi$
$102$	−2.14590	−0.212476
$103$	6.76393	0.666470	0.333235	−	0.942844i	$-0.391860\pi$
$103$	6.76393	0.666470	0.333235	+	0.942844i	$0.391860\pi$
$104$	2.23607	0.219265
$105$	0	0
$106$	5.70820	0.554430
$107$	1.00000	0.0966736
$107$	1.00000	0.0966736
$108$	−1.61803	−0.155695
$109$	−14.9443	−1.43140	−0.715701	−	0.698407i	$-0.753893\pi$
$109$	−14.9443	−1.43140	−0.715701	+	0.698407i	$0.753893\pi$
$110$	0	0
$111$	−1.00000	−0.0949158
$112$	3.70820	0.350392
$113$	−5.47214	−0.514775	−0.257388	−	0.966308i	$-0.582862\pi$
$113$	−5.47214	−0.514775	−0.257388	+	0.966308i	$0.582862\pi$
$114$	1.09017	0.102104
$115$	0	0
$116$	−2.00000	−0.185695
$117$	1.00000	0.0924500
$118$	3.05573	0.281303
$119$	6.94427	0.636580
$120$	0	0
$121$	16.4164	1.49240
$122$	−8.90983	−0.806658
$123$	8.94427	0.806478
$124$	9.70820	0.871822
$125$	0	0
$126$	−1.23607	−0.110118
$127$	−7.70820	−0.683992	−0.341996	−	0.939701i	$-0.611103\pi$
$127$	−7.70820	−0.683992	−0.341996	+	0.939701i	$0.611103\pi$
$128$	11.3820	1.00603
$129$	5.23607	0.461010
$130$	0	0
$131$	12.2361	1.06907	0.534535	−	0.845146i	$-0.320487\pi$
$131$	12.2361	1.06907	0.534535	+	0.845146i	$0.320487\pi$
$132$	8.47214	0.737405
$133$	−3.52786	−0.305905
$134$	4.47214	0.386334
$135$	0	0
$136$	7.76393	0.665752
$137$	4.18034	0.357151	0.178575	−	0.983926i	$-0.442851\pi$
$137$	4.18034	0.357151	0.178575	+	0.983926i	$0.442851\pi$
$138$	1.52786	0.130060
$139$	−8.47214	−0.718597	−0.359299	−	0.933223i	$-0.616984\pi$
$139$	−8.47214	−0.718597	−0.359299	+	0.933223i	$0.616984\pi$
$140$	0	0
$141$	−5.23607	−0.440956
$142$	1.09017	0.0914850
$143$	−5.23607	−0.437862
$144$	1.85410	0.154508
$145$	0	0
$146$	−6.29180	−0.520713
$147$	−3.00000	−0.247436
$148$	1.61803	0.133002
$149$	12.4721	1.02176	0.510879	−	0.859653i	$-0.329320\pi$
$149$	12.4721	1.02176	0.510879	+	0.859653i	$0.329320\pi$
$150$	0	0
$151$	7.18034	0.584328	0.292164	−	0.956368i	$-0.405625\pi$
$151$	7.18034	0.584328	0.292164	+	0.956368i	$0.405625\pi$
$152$	−3.94427	−0.319923
$153$	3.47214	0.280706
$154$	6.47214	0.521540
$155$	0	0
$156$	−1.61803	−0.129546
$157$	15.7082	1.25365	0.626826	−	0.779160i	$-0.284354\pi$
$157$	15.7082	1.25365	0.626826	+	0.779160i	$0.284354\pi$
$158$	9.52786	0.757996
$159$	−9.23607	−0.732468
$160$	0	0
$161$	−4.94427	−0.389663
$162$	−0.618034	−0.0485573
$163$	12.9443	1.01387	0.506937	−	0.861983i	$-0.330778\pi$
$163$	12.9443	1.01387	0.506937	+	0.861983i	$0.330778\pi$
$164$	−14.4721	−1.13008
$165$	0	0
$166$	−0.944272	−0.0732897
$167$	−14.7082	−1.13815	−0.569077	−	0.822284i	$-0.692700\pi$
$167$	−14.7082	−1.13815	−0.569077	+	0.822284i	$0.692700\pi$
$168$	4.47214	0.345033
$169$	−12.0000	−0.923077
$170$	0	0
$171$	−1.76393	−0.134891
$172$	−8.47214	−0.645994
$173$	−5.00000	−0.380143	−0.190071	−	0.981770i	$-0.560872\pi$
$173$	−5.00000	−0.380143	−0.190071	+	0.981770i	$0.560872\pi$
$174$	−0.763932	−0.0579135
$175$	0	0
$176$	−9.70820	−0.731783
$177$	−4.94427	−0.371634
$178$	4.47214	0.335201
$179$	−21.7639	−1.62671	−0.813356	−	0.581766i	$-0.802362\pi$
$179$	−21.7639	−1.62671	−0.813356	+	0.581766i	$0.802362\pi$
$180$	0	0
$181$	−11.7082	−0.870264	−0.435132	−	0.900367i	$-0.643298\pi$
$181$	−11.7082	−0.870264	−0.435132	+	0.900367i	$0.643298\pi$
$182$	−1.23607	−0.0916235
$183$	14.4164	1.06569
$184$	−5.52786	−0.407520
$185$	0	0
$186$	3.70820	0.271899
$187$	−18.1803	−1.32948
$188$	8.47214	0.617894
$189$	2.00000	0.145479
$190$	0	0
$191$	−14.2361	−1.03009	−0.515043	−	0.857164i	$-0.672224\pi$
$191$	−14.2361	−1.03009	−0.515043	+	0.857164i	$0.672224\pi$
$192$	−0.236068	−0.0170367
$193$	−2.94427	−0.211933	−0.105967	−	0.994370i	$-0.533794\pi$
$193$	−2.94427	−0.211933	−0.105967	+	0.994370i	$0.533794\pi$
$194$	−4.76393	−0.342030
$195$	0	0
$196$	4.85410	0.346722
$197$	−14.0000	−0.997459	−0.498729	−	0.866758i	$-0.666200\pi$
$197$	−14.0000	−0.997459	−0.498729	+	0.866758i	$0.666200\pi$
$198$	3.23607	0.229977
$199$	−13.6525	−0.967798	−0.483899	−	0.875124i	$-0.660780\pi$
$199$	−13.6525	−0.967798	−0.483899	+	0.875124i	$0.660780\pi$
$200$	0	0
$201$	−7.23607	−0.510393
$202$	7.23607	0.509128
$203$	2.47214	0.173510
$204$	−5.61803	−0.393341
$205$	0	0
$206$	−4.18034	−0.291258
$207$	−2.47214	−0.171825
$208$	1.85410	0.128559
$209$	9.23607	0.638872
$210$	0	0
$211$	9.05573	0.623422	0.311711	−	0.950177i	$-0.399098\pi$
$211$	9.05573	0.623422	0.311711	+	0.950177i	$0.399098\pi$
$212$	14.9443	1.02638
$213$	−1.76393	−0.120863
$214$	−0.618034	−0.0422479
$215$	0	0
$216$	2.23607	0.152145
$217$	−12.0000	−0.814613
$218$	9.23607	0.625545
$219$	10.1803	0.687924
$220$	0	0
$221$	3.47214	0.233561
$222$	0.618034	0.0414797
$223$	−7.18034	−0.480831	−0.240416	−	0.970670i	$-0.577284\pi$
$223$	−7.18034	−0.480831	−0.240416	+	0.970670i	$0.577284\pi$
$224$	−11.2361	−0.750741
$225$	0	0
$226$	3.38197	0.224965
$227$	−12.2918	−0.815835	−0.407918	−	0.913019i	$-0.633745\pi$
$227$	−12.2918	−0.815835	−0.407918	+	0.913019i	$0.633745\pi$
$228$	2.85410	0.189018
$229$	22.1803	1.46572	0.732859	−	0.680380i	$-0.238185\pi$
$229$	22.1803	1.46572	0.732859	+	0.680380i	$0.238185\pi$
$230$	0	0
$231$	−10.4721	−0.689016
$232$	2.76393	0.181461
$233$	15.4164	1.00996	0.504981	−	0.863130i	$-0.331499\pi$
$233$	15.4164	1.00996	0.504981	+	0.863130i	$0.331499\pi$
$234$	−0.618034	−0.0404021
$235$	0	0
$236$	8.00000	0.520756
$237$	−15.4164	−1.00140
$238$	−4.29180	−0.278196
$239$	−19.1246	−1.23707	−0.618534	−	0.785758i	$-0.712273\pi$
$239$	−19.1246	−1.23707	−0.618534	+	0.785758i	$0.712273\pi$
$240$	0	0
$241$	24.4164	1.57280	0.786400	−	0.617718i	$-0.211943\pi$
$241$	24.4164	1.57280	0.786400	+	0.617718i	$0.211943\pi$
$242$	−10.1459	−0.652203
$243$	1.00000	0.0641500
$244$	−23.3262	−1.49331
$245$	0	0
$246$	−5.52786	−0.352444
$247$	−1.76393	−0.112236
$248$	−13.4164	−0.851943
$249$	1.52786	0.0968244
$250$	0	0
$251$	−9.23607	−0.582975	−0.291488	−	0.956575i	$-0.594150\pi$
$251$	−9.23607	−0.582975	−0.291488	+	0.956575i	$0.594150\pi$
$252$	−3.23607	−0.203853
$253$	12.9443	0.813799
$254$	4.76393	0.298916
$255$	0	0
$256$	−6.56231	−0.410144
$257$	1.41641	0.0883531	0.0441765	−	0.999024i	$-0.485934\pi$
$257$	1.41641	0.0883531	0.0441765	+	0.999024i	$0.485934\pi$
$258$	−3.23607	−0.201469
$259$	−2.00000	−0.124274
$260$	0	0
$261$	1.23607	0.0765107
$262$	−7.56231	−0.467201
$263$	4.00000	0.246651	0.123325	−	0.992366i	$-0.460644\pi$
$263$	4.00000	0.246651	0.123325	+	0.992366i	$0.460644\pi$
$264$	−11.7082	−0.720590
$265$	0	0
$266$	2.18034	0.133685
$267$	−7.23607	−0.442840
$268$	11.7082	0.715192
$269$	16.4164	1.00093	0.500463	−	0.865758i	$-0.333163\pi$
$269$	16.4164	1.00093	0.500463	+	0.865758i	$0.333163\pi$
$270$	0	0
$271$	−19.6525	−1.19380	−0.596901	−	0.802315i	$-0.703602\pi$
$271$	−19.6525	−1.19380	−0.596901	+	0.802315i	$0.703602\pi$
$272$	6.43769	0.390343
$273$	2.00000	0.121046
$274$	−2.58359	−0.156081
$275$	0	0
$276$	4.00000	0.240772
$277$	11.7082	0.703478	0.351739	−	0.936098i	$-0.385590\pi$
$277$	11.7082	0.703478	0.351739	+	0.936098i	$0.385590\pi$
$278$	5.23607	0.314038
$279$	−6.00000	−0.359211
$280$	0	0
$281$	8.47214	0.505405	0.252703	−	0.967544i	$-0.418681\pi$
$281$	8.47214	0.505405	0.252703	+	0.967544i	$0.418681\pi$
$282$	3.23607	0.192705
$283$	−20.2361	−1.20291	−0.601455	−	0.798907i	$-0.705412\pi$
$283$	−20.2361	−1.20291	−0.601455	+	0.798907i	$0.705412\pi$
$284$	2.85410	0.169360
$285$	0	0
$286$	3.23607	0.191353
$287$	17.8885	1.05593
$288$	−5.61803	−0.331046
$289$	−4.94427	−0.290840
$290$	0	0
$291$	7.70820	0.451863
$292$	−16.4721	−0.963959
$293$	−29.7082	−1.73557	−0.867786	−	0.496938i	$-0.834458\pi$
$293$	−29.7082	−1.73557	−0.867786	+	0.496938i	$0.834458\pi$
$294$	1.85410	0.108133
$295$	0	0
$296$	−2.23607	−0.129969
$297$	−5.23607	−0.303827
$298$	−7.70820	−0.446524
$299$	−2.47214	−0.142967
$300$	0	0
$301$	10.4721	0.603604
$302$	−4.43769	−0.255361
$303$	−11.7082	−0.672619
$304$	−3.27051	−0.187577
$305$	0	0
$306$	−2.14590	−0.122673
$307$	−15.8885	−0.906807	−0.453404	−	0.891305i	$-0.649790\pi$
$307$	−15.8885	−0.906807	−0.453404	+	0.891305i	$0.649790\pi$
$308$	16.9443	0.965489
$309$	6.76393	0.384787
$310$	0	0
$311$	−1.18034	−0.0669309	−0.0334655	−	0.999440i	$-0.510654\pi$
$311$	−1.18034	−0.0669309	−0.0334655	+	0.999440i	$0.510654\pi$
$312$	2.23607	0.126592
$313$	−10.8885	−0.615457	−0.307728	−	0.951474i	$-0.599569\pi$
$313$	−10.8885	−0.615457	−0.307728	+	0.951474i	$0.599569\pi$
$314$	−9.70820	−0.547866
$315$	0	0
$316$	24.9443	1.40322
$317$	30.3607	1.70523	0.852613	−	0.522543i	$-0.175017\pi$
$317$	30.3607	1.70523	0.852613	+	0.522543i	$0.175017\pi$
$318$	5.70820	0.320100
$319$	−6.47214	−0.362370
$320$	0	0
$321$	1.00000	0.0558146
$322$	3.05573	0.170289
$323$	−6.12461	−0.340783
$324$	−1.61803	−0.0898908
$325$	0	0
$326$	−8.00000	−0.443079
$327$	−14.9443	−0.826420
$328$	20.0000	1.10432
$329$	−10.4721	−0.577348
$330$	0	0
$331$	32.5967	1.79168	0.895840	−	0.444377i	$-0.146575\pi$
$331$	32.5967	1.79168	0.895840	+	0.444377i	$0.146575\pi$
$332$	−2.47214	−0.135676
$333$	−1.00000	−0.0547997
$334$	9.09017	0.497392
$335$	0	0
$336$	3.70820	0.202299
$337$	−10.4164	−0.567418	−0.283709	−	0.958910i	$-0.591565\pi$
$337$	−10.4164	−0.567418	−0.283709	+	0.958910i	$0.591565\pi$
$338$	7.41641	0.403399
$339$	−5.47214	−0.297206
$340$	0	0
$341$	31.4164	1.70129
$342$	1.09017	0.0589496
$343$	−20.0000	−1.07990
$344$	11.7082	0.631264
$345$	0	0
$346$	3.09017	0.166129
$347$	6.70820	0.360115	0.180058	−	0.983656i	$-0.442372\pi$
$347$	6.70820	0.360115	0.180058	+	0.983656i	$0.442372\pi$
$348$	−2.00000	−0.107211
$349$	−32.0689	−1.71661	−0.858304	−	0.513142i	$-0.828482\pi$
$349$	−32.0689	−1.71661	−0.858304	+	0.513142i	$0.828482\pi$
$350$	0	0
$351$	1.00000	0.0533761
$352$	29.4164	1.56790
$353$	−26.3607	−1.40304	−0.701519	−	0.712651i	$-0.747494\pi$
$353$	−26.3607	−1.40304	−0.701519	+	0.712651i	$0.747494\pi$
$354$	3.05573	0.162410
$355$	0	0
$356$	11.7082	0.620534
$357$	6.94427	0.367530
$358$	13.4508	0.710899
$359$	−10.8197	−0.571040	−0.285520	−	0.958373i	$-0.592166\pi$
$359$	−10.8197	−0.571040	−0.285520	+	0.958373i	$0.592166\pi$
$360$	0	0
$361$	−15.8885	−0.836239
$362$	7.23607	0.380319
$363$	16.4164	0.861638
$364$	−3.23607	−0.169616
$365$	0	0
$366$	−8.90983	−0.465724
$367$	−11.1246	−0.580700	−0.290350	−	0.956921i	$-0.593772\pi$
$367$	−11.1246	−0.580700	−0.290350	+	0.956921i	$0.593772\pi$
$368$	−4.58359	−0.238936
$369$	8.94427	0.465620
$370$	0	0
$371$	−18.4721	−0.959026
$372$	9.70820	0.503347
$373$	7.58359	0.392664	0.196332	−	0.980538i	$-0.437097\pi$
$373$	7.58359	0.392664	0.196332	+	0.980538i	$0.437097\pi$
$374$	11.2361	0.581003
$375$	0	0
$376$	−11.7082	−0.603805
$377$	1.23607	0.0636607
$378$	−1.23607	−0.0635765
$379$	16.1803	0.831128	0.415564	−	0.909564i	$-0.363584\pi$
$379$	16.1803	0.831128	0.415564	+	0.909564i	$0.363584\pi$
$380$	0	0
$381$	−7.70820	−0.394903
$382$	8.79837	0.450164
$383$	−18.8328	−0.962312	−0.481156	−	0.876635i	$-0.659783\pi$
$383$	−18.8328	−0.962312	−0.481156	+	0.876635i	$0.659783\pi$
$384$	11.3820	0.580834
$385$	0	0
$386$	1.81966	0.0926183
$387$	5.23607	0.266164
$388$	−12.4721	−0.633177
$389$	10.9443	0.554897	0.277448	−	0.960741i	$-0.410511\pi$
$389$	10.9443	0.554897	0.277448	+	0.960741i	$0.410511\pi$
$390$	0	0
$391$	−8.58359	−0.434091
$392$	−6.70820	−0.338815
$393$	12.2361	0.617228
$394$	8.65248	0.435905
$395$	0	0
$396$	8.47214	0.425741
$397$	−24.8328	−1.24632	−0.623162	−	0.782093i	$-0.714152\pi$
$397$	−24.8328	−1.24632	−0.623162	+	0.782093i	$0.714152\pi$
$398$	8.43769	0.422943
$399$	−3.52786	−0.176614
$400$	0	0
$401$	−18.0000	−0.898877	−0.449439	−	0.893311i	$-0.648376\pi$
$401$	−18.0000	−0.898877	−0.449439	+	0.893311i	$0.648376\pi$
$402$	4.47214	0.223050
$403$	−6.00000	−0.298881
$404$	18.9443	0.942513
$405$	0	0
$406$	−1.52786	−0.0758266
$407$	5.23607	0.259542
$408$	7.76393	0.384372
$409$	20.7639	1.02671	0.513355	−	0.858176i	$-0.328402\pi$
$409$	20.7639	1.02671	0.513355	+	0.858176i	$0.328402\pi$
$410$	0	0
$411$	4.18034	0.206201
$412$	−10.9443	−0.539186
$413$	−9.88854	−0.486583
$414$	1.52786	0.0750904
$415$	0	0
$416$	−5.61803	−0.275447
$417$	−8.47214	−0.414882
$418$	−5.70820	−0.279197
$419$	25.6525	1.25321	0.626603	−	0.779339i	$-0.284445\pi$
$419$	25.6525	1.25321	0.626603	+	0.779339i	$0.284445\pi$
$420$	0	0
$421$	−29.0000	−1.41337	−0.706687	−	0.707527i	$-0.749811\pi$
$421$	−29.0000	−1.41337	−0.706687	+	0.707527i	$0.749811\pi$
$422$	−5.59675	−0.272445
$423$	−5.23607	−0.254586
$424$	−20.6525	−1.00297
$425$	0	0
$426$	1.09017	0.0528189
$427$	28.8328	1.39532
$428$	−1.61803	−0.0782106
$429$	−5.23607	−0.252800
$430$	0	0
$431$	−9.70820	−0.467628	−0.233814	−	0.972281i	$-0.575121\pi$
$431$	−9.70820	−0.467628	−0.233814	+	0.972281i	$0.575121\pi$
$432$	1.85410	0.0892055
$433$	21.1246	1.01518	0.507592	−	0.861598i	$-0.330536\pi$
$433$	21.1246	1.01518	0.507592	+	0.861598i	$0.330536\pi$
$434$	7.41641	0.355999
$435$	0	0
$436$	24.1803	1.15803
$437$	4.36068	0.208600
$438$	−6.29180	−0.300634
$439$	−25.5279	−1.21838	−0.609189	−	0.793025i	$-0.708505\pi$
$439$	−25.5279	−1.21838	−0.609189	+	0.793025i	$0.708505\pi$
$440$	0	0
$441$	−3.00000	−0.142857
$442$	−2.14590	−0.102070
$443$	26.8328	1.27487	0.637433	−	0.770506i	$-0.279996\pi$
$443$	26.8328	1.27487	0.637433	+	0.770506i	$0.279996\pi$
$444$	1.61803	0.0767885
$445$	0	0
$446$	4.43769	0.210131
$447$	12.4721	0.589912
$448$	−0.472136	−0.0223063
$449$	−35.8328	−1.69106	−0.845528	−	0.533932i	$-0.820714\pi$
$449$	−35.8328	−1.69106	−0.845528	+	0.533932i	$0.820714\pi$
$450$	0	0
$451$	−46.8328	−2.20527
$452$	8.85410	0.416462
$453$	7.18034	0.337362
$454$	7.59675	0.356533
$455$	0	0
$456$	−3.94427	−0.184707
$457$	40.3050	1.88539	0.942693	−	0.333661i	$-0.108284\pi$
$457$	40.3050	1.88539	0.942693	+	0.333661i	$0.108284\pi$
$458$	−13.7082	−0.640542
$459$	3.47214	0.162065
$460$	0	0
$461$	10.2918	0.479337	0.239668	−	0.970855i	$-0.422961\pi$
$461$	10.2918	0.479337	0.239668	+	0.970855i	$0.422961\pi$
$462$	6.47214	0.301111
$463$	33.8885	1.57493	0.787467	−	0.616357i	$-0.211392\pi$
$463$	33.8885	1.57493	0.787467	+	0.616357i	$0.211392\pi$
$464$	2.29180	0.106394
$465$	0	0
$466$	−9.52786	−0.441370
$467$	22.6525	1.04823	0.524116	−	0.851647i	$-0.324396\pi$
$467$	22.6525	1.04823	0.524116	+	0.851647i	$0.324396\pi$
$468$	−1.61803	−0.0747936
$469$	−14.4721	−0.668261
$470$	0	0
$471$	15.7082	0.723796
$472$	−11.0557	−0.508881
$473$	−27.4164	−1.26061
$474$	9.52786	0.437629
$475$	0	0
$476$	−11.2361	−0.515004
$477$	−9.23607	−0.422891
$478$	11.8197	0.540619
$479$	−20.1246	−0.919517	−0.459758	−	0.888044i	$-0.652064\pi$
$479$	−20.1246	−0.919517	−0.459758	+	0.888044i	$0.652064\pi$
$480$	0	0
$481$	−1.00000	−0.0455961
$482$	−15.0902	−0.687338
$483$	−4.94427	−0.224972
$484$	−26.5623	−1.20738
$485$	0	0
$486$	−0.618034	−0.0280346
$487$	−0.111456	−0.00505056	−0.00252528	−	0.999997i	$-0.500804\pi$
$487$	−0.111456	−0.00505056	−0.00252528	+	0.999997i	$0.500804\pi$
$488$	32.2361	1.45926
$489$	12.9443	0.585360
$490$	0	0
$491$	−18.2361	−0.822982	−0.411491	−	0.911414i	$-0.634992\pi$
$491$	−18.2361	−0.822982	−0.411491	+	0.911414i	$0.634992\pi$
$492$	−14.4721	−0.652454
$493$	4.29180	0.193293
$494$	1.09017	0.0490491
$495$	0	0
$496$	−11.1246	−0.499510
$497$	−3.52786	−0.158246
$498$	−0.944272	−0.0423138
$499$	−1.81966	−0.0814592	−0.0407296	−	0.999170i	$-0.512968\pi$
$499$	−1.81966	−0.0814592	−0.0407296	+	0.999170i	$0.512968\pi$
$500$	0	0
$501$	−14.7082	−0.657114
$502$	5.70820	0.254770
$503$	−30.0689	−1.34071	−0.670353	−	0.742043i	$-0.733857\pi$
$503$	−30.0689	−1.34071	−0.670353	+	0.742043i	$0.733857\pi$
$504$	4.47214	0.199205
$505$	0	0
$506$	−8.00000	−0.355643
$507$	−12.0000	−0.532939
$508$	12.4721	0.553362
$509$	−10.9443	−0.485096	−0.242548	−	0.970139i	$-0.577983\pi$
$509$	−10.9443	−0.485096	−0.242548	+	0.970139i	$0.577983\pi$
$510$	0	0
$511$	20.3607	0.900703
$512$	−18.7082	−0.826794
$513$	−1.76393	−0.0778795
$514$	−0.875388	−0.0386117
$515$	0	0
$516$	−8.47214	−0.372965
$517$	27.4164	1.20577
$518$	1.23607	0.0543097
$519$	−5.00000	−0.219476
$520$	0	0
$521$	6.41641	0.281108	0.140554	−	0.990073i	$-0.455112\pi$
$521$	6.41641	0.281108	0.140554	+	0.990073i	$0.455112\pi$
$522$	−0.763932	−0.0334364
$523$	−34.6525	−1.51525	−0.757623	−	0.652692i	$-0.773640\pi$
$523$	−34.6525	−1.51525	−0.757623	+	0.652692i	$0.773640\pi$
$524$	−19.7984	−0.864896
$525$	0	0
$526$	−2.47214	−0.107790
$527$	−20.8328	−0.907492
$528$	−9.70820	−0.422495
$529$	−16.8885	−0.734285
$530$	0	0
$531$	−4.94427	−0.214563
$532$	5.70820	0.247482
$533$	8.94427	0.387419
$534$	4.47214	0.193528
$535$	0	0
$536$	−16.1803	−0.698884
$537$	−21.7639	−0.939183
$538$	−10.1459	−0.437421
$539$	15.7082	0.676600
$540$	0	0
$541$	−13.5279	−0.581608	−0.290804	−	0.956783i	$-0.593923\pi$
$541$	−13.5279	−0.581608	−0.290804	+	0.956783i	$0.593923\pi$
$542$	12.1459	0.521711
$543$	−11.7082	−0.502447
$544$	−19.5066	−0.836338
$545$	0	0
$546$	−1.23607	−0.0528988
$547$	−30.2361	−1.29280	−0.646400	−	0.762998i	$-0.723726\pi$
$547$	−30.2361	−1.29280	−0.646400	+	0.762998i	$0.723726\pi$
$548$	−6.76393	−0.288941
$549$	14.4164	0.615277
$550$	0	0
$551$	−2.18034	−0.0928856
$552$	−5.52786	−0.235282
$553$	−30.8328	−1.31114
$554$	−7.23607	−0.307431
$555$	0	0
$556$	13.7082	0.581357
$557$	−44.8885	−1.90199	−0.950994	−	0.309208i	$-0.899936\pi$
$557$	−44.8885	−1.90199	−0.950994	+	0.309208i	$0.899936\pi$
$558$	3.70820	0.156981
$559$	5.23607	0.221462
$560$	0	0
$561$	−18.1803	−0.767575
$562$	−5.23607	−0.220870
$563$	−27.7639	−1.17011	−0.585055	−	0.810994i	$-0.698927\pi$
$563$	−27.7639	−1.17011	−0.585055	+	0.810994i	$0.698927\pi$
$564$	8.47214	0.356741
$565$	0	0
$566$	12.5066	0.525691
$567$	2.00000	0.0839921
$568$	−3.94427	−0.165498
$569$	−28.2918	−1.18605	−0.593027	−	0.805183i	$-0.702067\pi$
$569$	−28.2918	−1.18605	−0.593027	+	0.805183i	$0.702067\pi$
$570$	0	0
$571$	−7.18034	−0.300488	−0.150244	−	0.988649i	$-0.548006\pi$
$571$	−7.18034	−0.300488	−0.150244	+	0.988649i	$0.548006\pi$
$572$	8.47214	0.354238
$573$	−14.2361	−0.594720
$574$	−11.0557	−0.461457
$575$	0	0
$576$	−0.236068	−0.00983617
$577$	28.0557	1.16798	0.583988	−	0.811763i	$-0.301492\pi$
$577$	28.0557	1.16798	0.583988	+	0.811763i	$0.301492\pi$
$578$	3.05573	0.127102
$579$	−2.94427	−0.122360
$580$	0	0
$581$	3.05573	0.126773
$582$	−4.76393	−0.197471
$583$	48.3607	2.00289
$584$	22.7639	0.941978
$585$	0	0
$586$	18.3607	0.758473
$587$	−28.6525	−1.18261	−0.591307	−	0.806446i	$-0.701388\pi$
$587$	−28.6525	−1.18261	−0.591307	+	0.806446i	$0.701388\pi$
$588$	4.85410	0.200180
$589$	10.5836	0.436089
$590$	0	0
$591$	−14.0000	−0.575883
$592$	−1.85410	−0.0762031
$593$	33.9443	1.39392	0.696962	−	0.717108i	$-0.254535\pi$
$593$	33.9443	1.39392	0.696962	+	0.717108i	$0.254535\pi$
$594$	3.23607	0.132777
$595$	0	0
$596$	−20.1803	−0.826619
$597$	−13.6525	−0.558759
$598$	1.52786	0.0624790
$599$	10.0000	0.408589	0.204294	−	0.978909i	$-0.434510\pi$
$599$	10.0000	0.408589	0.204294	+	0.978909i	$0.434510\pi$
$600$	0	0
$601$	−38.1803	−1.55741	−0.778704	−	0.627391i	$-0.784123\pi$
$601$	−38.1803	−1.55741	−0.778704	+	0.627391i	$0.784123\pi$
$602$	−6.47214	−0.263785
$603$	−7.23607	−0.294675
$604$	−11.6180	−0.472731
$605$	0	0
$606$	7.23607	0.293945
$607$	−10.7639	−0.436895	−0.218447	−	0.975849i	$-0.570099\pi$
$607$	−10.7639	−0.436895	−0.218447	+	0.975849i	$0.570099\pi$
$608$	9.90983	0.401897
$609$	2.47214	0.100176
$610$	0	0
$611$	−5.23607	−0.211829
$612$	−5.61803	−0.227096
$613$	−34.2492	−1.38331	−0.691657	−	0.722227i	$-0.743119\pi$
$613$	−34.2492	−1.38331	−0.691657	+	0.722227i	$0.743119\pi$
$614$	9.81966	0.396289
$615$	0	0
$616$	−23.4164	−0.943474
$617$	−37.8328	−1.52309	−0.761546	−	0.648111i	$-0.775559\pi$
$617$	−37.8328	−1.52309	−0.761546	+	0.648111i	$0.775559\pi$
$618$	−4.18034	−0.168158
$619$	1.34752	0.0541616	0.0270808	−	0.999633i	$-0.491379\pi$
$619$	1.34752	0.0541616	0.0270808	+	0.999633i	$0.491379\pi$
$620$	0	0
$621$	−2.47214	−0.0992034
$622$	0.729490	0.0292499
$623$	−14.4721	−0.579814
$624$	1.85410	0.0742235
$625$	0	0
$626$	6.72949	0.268965
$627$	9.23607	0.368853
$628$	−25.4164	−1.01423
$629$	−3.47214	−0.138443
$630$	0	0
$631$	28.1803	1.12184	0.560921	−	0.827869i	$-0.310447\pi$
$631$	28.1803	1.12184	0.560921	+	0.827869i	$0.310447\pi$
$632$	−34.4721	−1.37123
$633$	9.05573	0.359933
$634$	−18.7639	−0.745211
$635$	0	0
$636$	14.9443	0.592579
$637$	−3.00000	−0.118864
$638$	4.00000	0.158362
$639$	−1.76393	−0.0697801
$640$	0	0
$641$	−8.05573	−0.318182	−0.159091	−	0.987264i	$-0.550856\pi$
$641$	−8.05573	−0.318182	−0.159091	+	0.987264i	$0.550856\pi$
$642$	−0.618034	−0.0243919
$643$	35.0689	1.38298	0.691491	−	0.722385i	$-0.256954\pi$
$643$	35.0689	1.38298	0.691491	+	0.722385i	$0.256954\pi$
$644$	8.00000	0.315244
$645$	0	0
$646$	3.78522	0.148927
$647$	−22.8328	−0.897651	−0.448825	−	0.893620i	$-0.648157\pi$
$647$	−22.8328	−0.897651	−0.448825	+	0.893620i	$0.648157\pi$
$648$	2.23607	0.0878410
$649$	25.8885	1.01621
$650$	0	0
$651$	−12.0000	−0.470317
$652$	−20.9443	−0.820241
$653$	16.7639	0.656023	0.328012	−	0.944674i	$-0.393621\pi$
$653$	16.7639	0.656023	0.328012	+	0.944674i	$0.393621\pi$
$654$	9.23607	0.361159
$655$	0	0
$656$	16.5836	0.647480
$657$	10.1803	0.397173
$658$	6.47214	0.252310
$659$	−45.8885	−1.78756	−0.893782	−	0.448502i	$-0.851958\pi$
$659$	−45.8885	−1.78756	−0.893782	+	0.448502i	$0.851958\pi$
$660$	0	0
$661$	−6.94427	−0.270101	−0.135050	−	0.990839i	$-0.543120\pi$
$661$	−6.94427	−0.270101	−0.135050	+	0.990839i	$0.543120\pi$
$662$	−20.1459	−0.782993
$663$	3.47214	0.134847
$664$	3.41641	0.132582
$665$	0	0
$666$	0.618034	0.0239483
$667$	−3.05573	−0.118318
$668$	23.7984	0.920787
$669$	−7.18034	−0.277608
$670$	0	0
$671$	−75.4853	−2.91408
$672$	−11.2361	−0.433441
$673$	−27.2361	−1.04987	−0.524937	−	0.851141i	$-0.675911\pi$
$673$	−27.2361	−1.04987	−0.524937	+	0.851141i	$0.675911\pi$
$674$	6.43769	0.247971
$675$	0	0
$676$	19.4164	0.746785
$677$	−19.5279	−0.750517	−0.375258	−	0.926920i	$-0.622446\pi$
$677$	−19.5279	−0.750517	−0.375258	+	0.926920i	$0.622446\pi$
$678$	3.38197	0.129884
$679$	15.4164	0.591627
$680$	0	0
$681$	−12.2918	−0.471023
$682$	−19.4164	−0.743493
$683$	−14.9443	−0.571827	−0.285913	−	0.958255i	$-0.592297\pi$
$683$	−14.9443	−0.571827	−0.285913	+	0.958255i	$0.592297\pi$
$684$	2.85410	0.109129
$685$	0	0
$686$	12.3607	0.471933
$687$	22.1803	0.846233
$688$	9.70820	0.370122
$689$	−9.23607	−0.351866
$690$	0	0
$691$	14.8197	0.563766	0.281883	−	0.959449i	$-0.409041\pi$
$691$	14.8197	0.563766	0.281883	+	0.959449i	$0.409041\pi$
$692$	8.09017	0.307542
$693$	−10.4721	−0.397804
$694$	−4.14590	−0.157376
$695$	0	0
$696$	2.76393	0.104767
$697$	31.0557	1.17632
$698$	19.8197	0.750185
$699$	15.4164	0.583102
$700$	0	0
$701$	−26.3050	−0.993524	−0.496762	−	0.867887i	$-0.665478\pi$
$701$	−26.3050	−0.993524	−0.496762	+	0.867887i	$0.665478\pi$
$702$	−0.618034	−0.0233262
$703$	1.76393	0.0665280
$704$	1.23607	0.0465861
$705$	0	0
$706$	16.2918	0.613150
$707$	−23.4164	−0.880665
$708$	8.00000	0.300658
$709$	41.3050	1.55124	0.775620	−	0.631200i	$-0.217437\pi$
$709$	41.3050	1.55124	0.775620	+	0.631200i	$0.217437\pi$
$710$	0	0
$711$	−15.4164	−0.578160
$712$	−16.1803	−0.606384
$713$	14.8328	0.555493
$714$	−4.29180	−0.160616
$715$	0	0
$716$	35.2148	1.31604
$717$	−19.1246	−0.714222
$718$	6.68692	0.249554
$719$	−26.1246	−0.974284	−0.487142	−	0.873323i	$-0.661961\pi$
$719$	−26.1246	−0.974284	−0.487142	+	0.873323i	$0.661961\pi$
$720$	0	0
$721$	13.5279	0.503804
$722$	9.81966	0.365450
$723$	24.4164	0.908056
$724$	18.9443	0.704058
$725$	0	0
$726$	−10.1459	−0.376550
$727$	1.87539	0.0695543	0.0347771	−	0.999395i	$-0.488928\pi$
$727$	1.87539	0.0695543	0.0347771	+	0.999395i	$0.488928\pi$
$728$	4.47214	0.165748
$729$	1.00000	0.0370370
$730$	0	0
$731$	18.1803	0.672424
$732$	−23.3262	−0.862163
$733$	34.0689	1.25836	0.629181	−	0.777258i	$-0.283390\pi$
$733$	34.0689	1.25836	0.629181	+	0.777258i	$0.283390\pi$
$734$	6.87539	0.253775
$735$	0	0
$736$	13.8885	0.511939
$737$	37.8885	1.39564
$738$	−5.52786	−0.203483
$739$	12.2918	0.452161	0.226081	−	0.974109i	$-0.427409\pi$
$739$	12.2918	0.452161	0.226081	+	0.974109i	$0.427409\pi$
$740$	0	0
$741$	−1.76393	−0.0647997
$742$	11.4164	0.419110
$743$	29.0557	1.06595	0.532976	−	0.846131i	$-0.321074\pi$
$743$	29.0557	1.06595	0.532976	+	0.846131i	$0.321074\pi$
$744$	−13.4164	−0.491869
$745$	0	0
$746$	−4.68692	−0.171600
$747$	1.52786	0.0559016
$748$	29.4164	1.07557
$749$	2.00000	0.0730784
$750$	0	0
$751$	25.5967	0.934039	0.467019	−	0.884247i	$-0.345328\pi$
$751$	25.5967	0.934039	0.467019	+	0.884247i	$0.345328\pi$
$752$	−9.70820	−0.354022
$753$	−9.23607	−0.336581
$754$	−0.763932	−0.0278208
$755$	0	0
$756$	−3.23607	−0.117695
$757$	0.291796	0.0106055	0.00530275	−	0.999986i	$-0.498312\pi$
$757$	0.291796	0.0106055	0.00530275	+	0.999986i	$0.498312\pi$
$758$	−10.0000	−0.363216
$759$	12.9443	0.469847
$760$	0	0
$761$	−3.41641	−0.123845	−0.0619223	−	0.998081i	$-0.519723\pi$
$761$	−3.41641	−0.123845	−0.0619223	+	0.998081i	$0.519723\pi$
$762$	4.76393	0.172579
$763$	−29.8885	−1.08204
$764$	23.0344	0.833357
$765$	0	0
$766$	11.6393	0.420546
$767$	−4.94427	−0.178527
$768$	−6.56231	−0.236797
$769$	48.8328	1.76096	0.880478	−	0.474087i	$-0.157222\pi$
$769$	48.8328	1.76096	0.880478	+	0.474087i	$0.157222\pi$
$770$	0	0
$771$	1.41641	0.0510107
$772$	4.76393	0.171458
$773$	38.7214	1.39271	0.696355	−	0.717697i	$-0.254804\pi$
$773$	38.7214	1.39271	0.696355	+	0.717697i	$0.254804\pi$
$774$	−3.23607	−0.116318
$775$	0	0
$776$	17.2361	0.618739
$777$	−2.00000	−0.0717496
$778$	−6.76393	−0.242499
$779$	−15.7771	−0.565273
$780$	0	0
$781$	9.23607	0.330492
$782$	5.30495	0.189705
$783$	1.23607	0.0441735
$784$	−5.56231	−0.198654
$785$	0	0
$786$	−7.56231	−0.269739
$787$	4.58359	0.163387	0.0816937	−	0.996657i	$-0.473967\pi$
$787$	4.58359	0.163387	0.0816937	+	0.996657i	$0.473967\pi$
$788$	22.6525	0.806961
$789$	4.00000	0.142404
$790$	0	0
$791$	−10.9443	−0.389134
$792$	−11.7082	−0.416033
$793$	14.4164	0.511942
$794$	15.3475	0.544663
$795$	0	0
$796$	22.0902	0.782965
$797$	4.65248	0.164799	0.0823996	−	0.996599i	$-0.473742\pi$
$797$	4.65248	0.164799	0.0823996	+	0.996599i	$0.473742\pi$
$798$	2.18034	0.0771832
$799$	−18.1803	−0.643174
$800$	0	0
$801$	−7.23607	−0.255674
$802$	11.1246	0.392824
$803$	−53.3050	−1.88109
$804$	11.7082	0.412917
$805$	0	0
$806$	3.70820	0.130616
$807$	16.4164	0.577885
$808$	−26.1803	−0.921021
$809$	39.2492	1.37993	0.689965	−	0.723843i	$-0.257626\pi$
$809$	39.2492	1.37993	0.689965	+	0.723843i	$0.257626\pi$
$810$	0	0
$811$	36.0000	1.26413	0.632065	−	0.774915i	$-0.282207\pi$
$811$	36.0000	1.26413	0.632065	+	0.774915i	$0.282207\pi$
$812$	−4.00000	−0.140372
$813$	−19.6525	−0.689242
$814$	−3.23607	−0.113424
$815$	0	0
$816$	6.43769	0.225364
$817$	−9.23607	−0.323129
$818$	−12.8328	−0.448689
$819$	2.00000	0.0698857
$820$	0	0
$821$	−20.0557	−0.699950	−0.349975	−	0.936759i	$-0.613810\pi$
$821$	−20.0557	−0.699950	−0.349975	+	0.936759i	$0.613810\pi$
$822$	−2.58359	−0.0901131
$823$	15.0557	0.524810	0.262405	−	0.964958i	$-0.415484\pi$
$823$	15.0557	0.524810	0.262405	+	0.964958i	$0.415484\pi$
$824$	15.1246	0.526891
$825$	0	0
$826$	6.11146	0.212645
$827$	−6.12461	−0.212974	−0.106487	−	0.994314i	$-0.533960\pi$
$827$	−6.12461	−0.212974	−0.106487	+	0.994314i	$0.533960\pi$
$828$	4.00000	0.139010
$829$	−33.2361	−1.15434	−0.577168	−	0.816625i	$-0.695842\pi$
$829$	−33.2361	−1.15434	−0.577168	+	0.816625i	$0.695842\pi$
$830$	0	0
$831$	11.7082	0.406153
$832$	−0.236068	−0.00818418
$833$	−10.4164	−0.360907
$834$	5.23607	0.181310
$835$	0	0
$836$	−14.9443	−0.516858
$837$	−6.00000	−0.207390
$838$	−15.8541	−0.547671
$839$	−30.6525	−1.05824	−0.529120	−	0.848547i	$-0.677478\pi$
$839$	−30.6525	−1.05824	−0.529120	+	0.848547i	$0.677478\pi$
$840$	0	0
$841$	−27.4721	−0.947315
$842$	17.9230	0.617667
$843$	8.47214	0.291796
$844$	−14.6525	−0.504359
$845$	0	0
$846$	3.23607	0.111258
$847$	32.8328	1.12815
$848$	−17.1246	−0.588062
$849$	−20.2361	−0.694500
$850$	0	0
$851$	2.47214	0.0847437
$852$	2.85410	0.0977799
$853$	−8.87539	−0.303888	−0.151944	−	0.988389i	$-0.548553\pi$
$853$	−8.87539	−0.303888	−0.151944	+	0.988389i	$0.548553\pi$
$854$	−17.8197	−0.609776
$855$	0	0
$856$	2.23607	0.0764272
$857$	29.7082	1.01481	0.507406	−	0.861707i	$-0.330604\pi$
$857$	29.7082	1.01481	0.507406	+	0.861707i	$0.330604\pi$
$858$	3.23607	0.110478
$859$	42.7082	1.45719	0.728593	−	0.684947i	$-0.240175\pi$
$859$	42.7082	1.45719	0.728593	+	0.684947i	$0.240175\pi$
$860$	0	0
$861$	17.8885	0.609640
$862$	6.00000	0.204361
$863$	26.2361	0.893086	0.446543	−	0.894762i	$-0.352655\pi$
$863$	26.2361	0.893086	0.446543	+	0.894762i	$0.352655\pi$
$864$	−5.61803	−0.191129
$865$	0	0
$866$	−13.0557	−0.443652
$867$	−4.94427	−0.167916
$868$	19.4164	0.659036
$869$	80.7214	2.73828
$870$	0	0
$871$	−7.23607	−0.245185
$872$	−33.4164	−1.13162
$873$	7.70820	0.260883
$874$	−2.69505	−0.0911614
$875$	0	0
$876$	−16.4721	−0.556542
$877$	−27.0132	−0.912169	−0.456085	−	0.889936i	$-0.650749\pi$
$877$	−27.0132	−0.912169	−0.456085	+	0.889936i	$0.650749\pi$
$878$	15.7771	0.532451
$879$	−29.7082	−1.00203
$880$	0	0
$881$	42.2492	1.42341	0.711706	−	0.702477i	$-0.247923\pi$
$881$	42.2492	1.42341	0.711706	+	0.702477i	$0.247923\pi$
$882$	1.85410	0.0624309
$883$	−50.7082	−1.70647	−0.853233	−	0.521529i	$-0.825362\pi$
$883$	−50.7082	−1.70647	−0.853233	+	0.521529i	$0.825362\pi$
$884$	−5.61803	−0.188955
$885$	0	0
$886$	−16.5836	−0.557137
$887$	14.2361	0.478000	0.239000	−	0.971020i	$-0.423180\pi$
$887$	14.2361	0.478000	0.239000	+	0.971020i	$0.423180\pi$
$888$	−2.23607	−0.0750375
$889$	−15.4164	−0.517050
$890$	0	0
$891$	−5.23607	−0.175415
$892$	11.6180	0.389001
$893$	9.23607	0.309073
$894$	−7.70820	−0.257801
$895$	0	0
$896$	22.7639	0.760490
$897$	−2.47214	−0.0825422
$898$	22.1459	0.739018
$899$	−7.41641	−0.247351
$900$	0	0
$901$	−32.0689	−1.06837
$902$	28.9443	0.963739
$903$	10.4721	0.348491
$904$	−12.2361	−0.406966
$905$	0	0
$906$	−4.43769	−0.147433
$907$	24.3607	0.808883	0.404442	−	0.914564i	$-0.367466\pi$
$907$	24.3607	0.808883	0.404442	+	0.914564i	$0.367466\pi$
$908$	19.8885	0.660025
$909$	−11.7082	−0.388337
$910$	0	0
$911$	−27.0557	−0.896396	−0.448198	−	0.893934i	$-0.647934\pi$
$911$	−27.0557	−0.896396	−0.448198	+	0.893934i	$0.647934\pi$
$912$	−3.27051	−0.108297
$913$	−8.00000	−0.264761
$914$	−24.9098	−0.823944
$915$	0	0
$916$	−35.8885	−1.18579
$917$	24.4721	0.808141
$918$	−2.14590	−0.0708252
$919$	8.47214	0.279470	0.139735	−	0.990189i	$-0.455375\pi$
$919$	8.47214	0.279470	0.139735	+	0.990189i	$0.455375\pi$
$920$	0	0
$921$	−15.8885	−0.523545
$922$	−6.36068	−0.209478
$923$	−1.76393	−0.0580605
$924$	16.9443	0.557426
$925$	0	0
$926$	−20.9443	−0.688271
$927$	6.76393	0.222157
$928$	−6.94427	−0.227957
$929$	−27.8328	−0.913165	−0.456583	−	0.889681i	$-0.650927\pi$
$929$	−27.8328	−0.913165	−0.456583	+	0.889681i	$0.650927\pi$
$930$	0	0
$931$	5.29180	0.173432
$932$	−24.9443	−0.817077
$933$	−1.18034	−0.0386426
$934$	−14.0000	−0.458094
$935$	0	0
$936$	2.23607	0.0730882
$937$	−1.47214	−0.0480926	−0.0240463	−	0.999711i	$-0.507655\pi$
$937$	−1.47214	−0.0480926	−0.0240463	+	0.999711i	$0.507655\pi$
$938$	8.94427	0.292041
$939$	−10.8885	−0.355334
$940$	0	0
$941$	−30.2918	−0.987484	−0.493742	−	0.869608i	$-0.664371\pi$
$941$	−30.2918	−0.987484	−0.493742	+	0.869608i	$0.664371\pi$
$942$	−9.70820	−0.316310
$943$	−22.1115	−0.720048
$944$	−9.16718	−0.298366
$945$	0	0
$946$	16.9443	0.550906
$947$	−37.8885	−1.23121	−0.615606	−	0.788054i	$-0.711089\pi$
$947$	−37.8885	−1.23121	−0.615606	+	0.788054i	$0.711089\pi$
$948$	24.9443	0.810152
$949$	10.1803	0.330468
$950$	0	0
$951$	30.3607	0.984512
$952$	15.5279	0.503261
$953$	−41.7771	−1.35329	−0.676646	−	0.736308i	$-0.736567\pi$
$953$	−41.7771	−1.35329	−0.676646	+	0.736308i	$0.736567\pi$
$954$	5.70820	0.184810
$955$	0	0
$956$	30.9443	1.00081
$957$	−6.47214	−0.209214
$958$	12.4377	0.401844
$959$	8.36068	0.269980
$960$	0	0
$961$	5.00000	0.161290
$962$	0.618034	0.0199262
$963$	1.00000	0.0322245
$964$	−39.5066	−1.27242
$965$	0	0
$966$	3.05573	0.0983164
$967$	22.1246	0.711480	0.355740	−	0.934585i	$-0.384229\pi$
$967$	22.1246	0.711480	0.355740	+	0.934585i	$0.384229\pi$
$968$	36.7082	1.17985
$969$	−6.12461	−0.196751
$970$	0	0
$971$	−33.2918	−1.06838	−0.534192	−	0.845363i	$-0.679384\pi$
$971$	−33.2918	−1.06838	−0.534192	+	0.845363i	$0.679384\pi$
$972$	−1.61803	−0.0518985
$973$	−16.9443	−0.543208
$974$	0.0688837	0.00220718
$975$	0	0
$976$	26.7295	0.855590
$977$	−17.7771	−0.568739	−0.284370	−	0.958715i	$-0.591784\pi$
$977$	−17.7771	−0.568739	−0.284370	+	0.958715i	$0.591784\pi$
$978$	−8.00000	−0.255812
$979$	37.8885	1.21092
$980$	0	0
$981$	−14.9443	−0.477134
$982$	11.2705	0.359656
$983$	−40.7082	−1.29839	−0.649195	−	0.760622i	$-0.724894\pi$
$983$	−40.7082	−1.29839	−0.649195	+	0.760622i	$0.724894\pi$
$984$	20.0000	0.637577
$985$	0	0
$986$	−2.65248	−0.0844720
$987$	−10.4721	−0.333332
$988$	2.85410	0.0908011
$989$	−12.9443	−0.411604
$990$	0	0
$991$	24.0689	0.764573	0.382287	−	0.924044i	$-0.375137\pi$
$991$	24.0689	0.764573	0.382287	+	0.924044i	$0.375137\pi$
$992$	33.7082	1.07024
$993$	32.5967	1.03443
$994$	2.18034	0.0691562
$995$	0	0
$996$	−2.47214	−0.0783326
$997$	50.5279	1.60023	0.800117	−	0.599844i	$-0.204771\pi$
$997$	50.5279	1.60023	0.800117	+	0.599844i	$0.204771\pi$
$998$	1.12461	0.0355990
$999$	−1.00000	−0.0316386

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8025.2.a.s.1.1		2
5.4	even	2		321.2.a.a.1.2	✓	2
15.14	odd	2		963.2.a.a.1.1		2
20.19	odd	2		5136.2.a.y.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
321.2.a.a.1.2	✓	2	5.4	even	2
963.2.a.a.1.1		2	15.14	odd	2
5136.2.a.y.1.2		2	20.19	odd	2
8025.2.a.s.1.1		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:	\( N \)	\(=\)	\( 8025 = 3 \cdot 5^{2} \cdot 107 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8025.a (trivial)

\(f(q)\)	\(=\)	\(q-0.618034 q^{2} +1.00000 q^{3} -1.61803 q^{4} -0.618034 q^{6} +2.00000 q^{7} +2.23607 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-0.618034 q^{2} +1.00000 q^{3} -1.61803 q^{4} -0.618034 q^{6} +2.00000 q^{7} +2.23607 q^{8} +1.00000 q^{9} -5.23607 q^{11} -1.61803 q^{12} +1.00000 q^{13} -1.23607 q^{14} +1.85410 q^{16} +3.47214 q^{17} -0.618034 q^{18} -1.76393 q^{19} +2.00000 q^{21} +3.23607 q^{22} -2.47214 q^{23} +2.23607 q^{24} -0.618034 q^{26} +1.00000 q^{27} -3.23607 q^{28} +1.23607 q^{29} -6.00000 q^{31} -5.61803 q^{32} -5.23607 q^{33} -2.14590 q^{34} -1.61803 q^{36} -1.00000 q^{37} +1.09017 q^{38} +1.00000 q^{39} +8.94427 q^{41} -1.23607 q^{42} +5.23607 q^{43} +8.47214 q^{44} +1.52786 q^{46} -5.23607 q^{47} +1.85410 q^{48} -3.00000 q^{49} +3.47214 q^{51} -1.61803 q^{52} -9.23607 q^{53} -0.618034 q^{54} +4.47214 q^{56} -1.76393 q^{57} -0.763932 q^{58} -4.94427 q^{59} +14.4164 q^{61} +3.70820 q^{62} +2.00000 q^{63} -0.236068 q^{64} +3.23607 q^{66} -7.23607 q^{67} -5.61803 q^{68} -2.47214 q^{69} -1.76393 q^{71} +2.23607 q^{72} +10.1803 q^{73} +0.618034 q^{74} +2.85410 q^{76} -10.4721 q^{77} -0.618034 q^{78} -15.4164 q^{79} +1.00000 q^{81} -5.52786 q^{82} +1.52786 q^{83} -3.23607 q^{84} -3.23607 q^{86} +1.23607 q^{87} -11.7082 q^{88} -7.23607 q^{89} +2.00000 q^{91} +4.00000 q^{92} -6.00000 q^{93} +3.23607 q^{94} -5.61803 q^{96} +7.70820 q^{97} +1.85410 q^{98} -5.23607 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{2} + 2 q^{3} - q^{4} + q^{6} + 4 q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q + q^2 + 2 * q^3 - q^4 + q^6 + 4 * q^7 + 2 * q^9	\( 2 q + q^{2} + 2 q^{3} - q^{4} + q^{6} + 4 q^{7} + 2 q^{9} - 6 q^{11} - q^{12} + 2 q^{13} + 2 q^{14} - 3 q^{16} - 2 q^{17} + q^{18} - 8 q^{19} + 4 q^{21} + 2 q^{22} + 4 q^{23} + q^{26} + 2 q^{27} - 2 q^{28} - 2 q^{29} - 12 q^{31} - 9 q^{32} - 6 q^{33} - 11 q^{34} - q^{36} - 2 q^{37} - 9 q^{38} + 2 q^{39} + 2 q^{42} + 6 q^{43} + 8 q^{44} + 12 q^{46} - 6 q^{47} - 3 q^{48} - 6 q^{49} - 2 q^{51} - q^{52} - 14 q^{53} + q^{54} - 8 q^{57} - 6 q^{58} + 8 q^{59} + 2 q^{61} - 6 q^{62} + 4 q^{63} + 4 q^{64} + 2 q^{66} - 10 q^{67} - 9 q^{68} + 4 q^{69} - 8 q^{71} - 2 q^{73} - q^{74} - q^{76} - 12 q^{77} + q^{78} - 4 q^{79} + 2 q^{81} - 20 q^{82} + 12 q^{83} - 2 q^{84} - 2 q^{86} - 2 q^{87} - 10 q^{88} - 10 q^{89} + 4 q^{91} + 8 q^{92} - 12 q^{93} + 2 q^{94} - 9 q^{96} + 2 q^{97} - 3 q^{98} - 6 q^{99}+O(q^{100}) \) 2 * q + q^2 + 2 * q^3 - q^4 + q^6 + 4 * q^7 + 2 * q^9 - 6 * q^11 - q^12 + 2 * q^13 + 2 * q^14 - 3 * q^16 - 2 * q^17 + q^18 - 8 * q^19 + 4 * q^21 + 2 * q^22 + 4 * q^23 + q^26 + 2 * q^27 - 2 * q^28 - 2 * q^29 - 12 * q^31 - 9 * q^32 - 6 * q^33 - 11 * q^34 - q^36 - 2 * q^37 - 9 * q^38 + 2 * q^39 + 2 * q^42 + 6 * q^43 + 8 * q^44 + 12 * q^46 - 6 * q^47 - 3 * q^48 - 6 * q^49 - 2 * q^51 - q^52 - 14 * q^53 + q^54 - 8 * q^57 - 6 * q^58 + 8 * q^59 + 2 * q^61 - 6 * q^62 + 4 * q^63 + 4 * q^64 + 2 * q^66 - 10 * q^67 - 9 * q^68 + 4 * q^69 - 8 * q^71 - 2 * q^73 - q^74 - q^76 - 12 * q^77 + q^78 - 4 * q^79 + 2 * q^81 - 20 * q^82 + 12 * q^83 - 2 * q^84 - 2 * q^86 - 2 * q^87 - 10 * q^88 - 10 * q^89 + 4 * q^91 + 8 * q^92 - 12 * q^93 + 2 * q^94 - 9 * q^96 + 2 * q^97 - 3 * q^98 - 6 * q^99

Introduction

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