LMFDB - Embedded newform 1225.2.a.n.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1225,2,Mod(1,1225)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1225, 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("1225.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1225 = 5^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1225.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:	$9.78167424761$
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 175)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.618034$ of defining polynomial
Character	$\chi$	$=$	1225.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.618034 q^{2} -3.23607 q^{3} -1.61803 q^{4} -2.00000 q^{6} -2.23607 q^{8} +7.47214 q^{9} +O(q^{10})$	\(q+0.618034 q^{2} -3.23607 q^{3} -1.61803 q^{4} -2.00000 q^{6} -2.23607 q^{8} +7.47214 q^{9} -0.236068 q^{11} +5.23607 q^{12} +1.23607 q^{13} +1.85410 q^{16} +2.47214 q^{17} +4.61803 q^{18} +4.47214 q^{19} -0.145898 q^{22} -6.23607 q^{23} +7.23607 q^{24} +0.763932 q^{26} -14.4721 q^{27} +5.00000 q^{29} -3.70820 q^{31} +5.61803 q^{32} +0.763932 q^{33} +1.52786 q^{34} -12.0902 q^{36} -3.00000 q^{37} +2.76393 q^{38} -4.00000 q^{39} -4.76393 q^{41} -1.76393 q^{43} +0.381966 q^{44} -3.85410 q^{46} -2.00000 q^{47} -6.00000 q^{48} -8.00000 q^{51} -2.00000 q^{52} -8.47214 q^{53} -8.94427 q^{54} -14.4721 q^{57} +3.09017 q^{58} -11.7082 q^{59} +9.70820 q^{61} -2.29180 q^{62} -0.236068 q^{64} +0.472136 q^{66} +4.23607 q^{67} -4.00000 q^{68} +20.1803 q^{69} +8.70820 q^{71} -16.7082 q^{72} -8.76393 q^{73} -1.85410 q^{74} -7.23607 q^{76} -2.47214 q^{78} -11.1803 q^{79} +24.4164 q^{81} -2.94427 q^{82} -7.70820 q^{83} -1.09017 q^{86} -16.1803 q^{87} +0.527864 q^{88} -17.2361 q^{89} +10.0902 q^{92} +12.0000 q^{93} -1.23607 q^{94} -18.1803 q^{96} +5.23607 q^{97} -1.76393 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{2} - 2 q^{3} - q^{4} - 4 q^{6} + 6 q^{9}+O(q^{10}) $ 2 * q - q^2 - 2 * q^3 - q^4 - 4 * q^6 + 6 * q^9	$ 2 q - q^{2} - 2 q^{3} - q^{4} - 4 q^{6} + 6 q^{9} + 4 q^{11} + 6 q^{12} - 2 q^{13} - 3 q^{16} - 4 q^{17} + 7 q^{18} - 7 q^{22} - 8 q^{23} + 10 q^{24} + 6 q^{26} - 20 q^{27} + 10 q^{29} + 6 q^{31} + 9 q^{32} + 6 q^{33} + 12 q^{34} - 13 q^{36} - 6 q^{37} + 10 q^{38} - 8 q^{39} - 14 q^{41} - 8 q^{43} + 3 q^{44} - q^{46} - 4 q^{47} - 12 q^{48} - 16 q^{51} - 4 q^{52} - 8 q^{53} - 20 q^{57} - 5 q^{58} - 10 q^{59} + 6 q^{61} - 18 q^{62} + 4 q^{64} - 8 q^{66} + 4 q^{67} - 8 q^{68} + 18 q^{69} + 4 q^{71} - 20 q^{72} - 22 q^{73} + 3 q^{74} - 10 q^{76} + 4 q^{78} + 22 q^{81} + 12 q^{82} - 2 q^{83} + 9 q^{86} - 10 q^{87} + 10 q^{88} - 30 q^{89} + 9 q^{92} + 24 q^{93} + 2 q^{94} - 14 q^{96} + 6 q^{97} - 8 q^{99}+O(q^{100}) $ 2 * q - q^2 - 2 * q^3 - q^4 - 4 * q^6 + 6 * q^9 + 4 * q^11 + 6 * q^12 - 2 * q^13 - 3 * q^16 - 4 * q^17 + 7 * q^18 - 7 * q^22 - 8 * q^23 + 10 * q^24 + 6 * q^26 - 20 * q^27 + 10 * q^29 + 6 * q^31 + 9 * q^32 + 6 * q^33 + 12 * q^34 - 13 * q^36 - 6 * q^37 + 10 * q^38 - 8 * q^39 - 14 * q^41 - 8 * q^43 + 3 * q^44 - q^46 - 4 * q^47 - 12 * q^48 - 16 * q^51 - 4 * q^52 - 8 * q^53 - 20 * q^57 - 5 * q^58 - 10 * q^59 + 6 * q^61 - 18 * q^62 + 4 * q^64 - 8 * q^66 + 4 * q^67 - 8 * q^68 + 18 * q^69 + 4 * q^71 - 20 * q^72 - 22 * q^73 + 3 * q^74 - 10 * q^76 + 4 * q^78 + 22 * q^81 + 12 * q^82 - 2 * q^83 + 9 * q^86 - 10 * q^87 + 10 * q^88 - 30 * q^89 + 9 * q^92 + 24 * q^93 + 2 * q^94 - 14 * q^96 + 6 * q^97 - 8 * 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	$-0.429881\pi$
$2$	0.618034	0.437016	0.218508	+	0.975835i	$0.429881\pi$
$3$	−3.23607	−1.86834	−0.934172	−	0.356822i	$-0.883860\pi$
$3$	−3.23607	−1.86834	−0.934172	+	0.356822i	$0.883860\pi$
$4$	−1.61803	−0.809017
$5$	0	0
$5$	0	0
$6$	−2.00000	−0.816497
$7$	0	0
$7$	0	0
$8$	−2.23607	−0.790569
$9$	7.47214	2.49071
$10$	0	0
$11$	−0.236068	−0.0711772	−0.0355886	−	0.999367i	$-0.511331\pi$
$11$	−0.236068	−0.0711772	−0.0355886	+	0.999367i	$0.511331\pi$
$12$	5.23607	1.51152
$13$	1.23607	0.342824	0.171412	−	0.985199i	$-0.445167\pi$
$13$	1.23607	0.342824	0.171412	+	0.985199i	$0.445167\pi$
$14$	0	0
$15$	0	0
$16$	1.85410	0.463525
$17$	2.47214	0.599581	0.299791	−	0.954005i	$-0.403083\pi$
$17$	2.47214	0.599581	0.299791	+	0.954005i	$0.403083\pi$
$18$	4.61803	1.08848
$19$	4.47214	1.02598	0.512989	−	0.858395i	$-0.328538\pi$
$19$	4.47214	1.02598	0.512989	+	0.858395i	$0.328538\pi$
$20$	0	0
$21$	0	0
$22$	−0.145898	−0.0311056
$23$	−6.23607	−1.30031	−0.650155	−	0.759802i	$-0.725296\pi$
$23$	−6.23607	−1.30031	−0.650155	+	0.759802i	$0.725296\pi$
$24$	7.23607	1.47706
$25$	0	0
$26$	0.763932	0.149819
$27$	−14.4721	−2.78516
$28$	0	0
$29$	5.00000	0.928477	0.464238	−	0.885710i	$-0.346328\pi$
$29$	5.00000	0.928477	0.464238	+	0.885710i	$0.346328\pi$
$30$	0	0
$31$	−3.70820	−0.666013	−0.333007	−	0.942925i	$-0.608063\pi$
$31$	−3.70820	−0.666013	−0.333007	+	0.942925i	$0.608063\pi$
$32$	5.61803	0.993137
$33$	0.763932	0.132983
$34$	1.52786	0.262027
$35$	0	0
$36$	−12.0902	−2.01503
$37$	−3.00000	−0.493197	−0.246598	−	0.969118i	$-0.579313\pi$
$37$	−3.00000	−0.493197	−0.246598	+	0.969118i	$0.579313\pi$
$38$	2.76393	0.448369
$39$	−4.00000	−0.640513
$40$	0	0
$41$	−4.76393	−0.744001	−0.372001	−	0.928232i	$-0.621328\pi$
$41$	−4.76393	−0.744001	−0.372001	+	0.928232i	$0.621328\pi$
$42$	0	0
$43$	−1.76393	−0.268997	−0.134499	−	0.990914i	$-0.542942\pi$
$43$	−1.76393	−0.268997	−0.134499	+	0.990914i	$0.542942\pi$
$44$	0.381966	0.0575835
$45$	0	0
$46$	−3.85410	−0.568256
$47$	−2.00000	−0.291730	−0.145865	−	0.989305i	$-0.546597\pi$
$47$	−2.00000	−0.291730	−0.145865	+	0.989305i	$0.546597\pi$
$48$	−6.00000	−0.866025
$49$	0	0
$50$	0	0
$51$	−8.00000	−1.12022
$52$	−2.00000	−0.277350
$53$	−8.47214	−1.16374	−0.581869	−	0.813283i	$-0.697678\pi$
$53$	−8.47214	−1.16374	−0.581869	+	0.813283i	$0.697678\pi$
$54$	−8.94427	−1.21716
$55$	0	0
$56$	0	0
$57$	−14.4721	−1.91688
$58$	3.09017	0.405759
$59$	−11.7082	−1.52428	−0.762139	−	0.647413i	$-0.775851\pi$
$59$	−11.7082	−1.52428	−0.762139	+	0.647413i	$0.775851\pi$
$60$	0	0
$61$	9.70820	1.24301	0.621504	−	0.783411i	$-0.286522\pi$
$61$	9.70820	1.24301	0.621504	+	0.783411i	$0.286522\pi$
$62$	−2.29180	−0.291058
$63$	0	0
$64$	−0.236068	−0.0295085
$65$	0	0
$66$	0.472136	0.0581159
$67$	4.23607	0.517518	0.258759	−	0.965942i	$-0.416686\pi$
$67$	4.23607	0.517518	0.258759	+	0.965942i	$0.416686\pi$
$68$	−4.00000	−0.485071
$69$	20.1803	2.42943
$70$	0	0
$71$	8.70820	1.03347	0.516737	−	0.856144i	$-0.327147\pi$
$71$	8.70820	1.03347	0.516737	+	0.856144i	$0.327147\pi$
$72$	−16.7082	−1.96908
$73$	−8.76393	−1.02574	−0.512870	−	0.858466i	$-0.671418\pi$
$73$	−8.76393	−1.02574	−0.512870	+	0.858466i	$0.671418\pi$
$74$	−1.85410	−0.215535
$75$	0	0
$76$	−7.23607	−0.830034
$77$	0	0
$78$	−2.47214	−0.279914
$79$	−11.1803	−1.25789	−0.628943	−	0.777451i	$-0.716512\pi$
$79$	−11.1803	−1.25789	−0.628943	+	0.777451i	$0.716512\pi$
$80$	0	0
$81$	24.4164	2.71293
$82$	−2.94427	−0.325140
$83$	−7.70820	−0.846085	−0.423043	−	0.906110i	$-0.639038\pi$
$83$	−7.70820	−0.846085	−0.423043	+	0.906110i	$0.639038\pi$
$84$	0	0
$85$	0	0
$86$	−1.09017	−0.117556
$87$	−16.1803	−1.73471
$88$	0.527864	0.0562705
$89$	−17.2361	−1.82702	−0.913510	−	0.406817i	$-0.866639\pi$
$89$	−17.2361	−1.82702	−0.913510	+	0.406817i	$0.866639\pi$
$90$	0	0
$91$	0	0
$92$	10.0902	1.05197
$93$	12.0000	1.24434
$94$	−1.23607	−0.127491
$95$	0	0
$96$	−18.1803	−1.85552
$97$	5.23607	0.531642	0.265821	−	0.964022i	$-0.414357\pi$
$97$	5.23607	0.531642	0.265821	+	0.964022i	$0.414357\pi$
$98$	0	0
$99$	−1.76393	−0.177282
$100$	0	0
$101$	−4.76393	−0.474029	−0.237014	−	0.971506i	$-0.576169\pi$
$101$	−4.76393	−0.474029	−0.237014	+	0.971506i	$0.576169\pi$
$102$	−4.94427	−0.489556
$103$	8.47214	0.834784	0.417392	−	0.908726i	$-0.362944\pi$
$103$	8.47214	0.834784	0.417392	+	0.908726i	$0.362944\pi$
$104$	−2.76393	−0.271026
$105$	0	0
$106$	−5.23607	−0.508572
$107$	−8.00000	−0.773389	−0.386695	−	0.922208i	$-0.626383\pi$
$107$	−8.00000	−0.773389	−0.386695	+	0.922208i	$0.626383\pi$
$108$	23.4164	2.25324
$109$	8.41641	0.806146	0.403073	−	0.915168i	$-0.367942\pi$
$109$	8.41641	0.806146	0.403073	+	0.915168i	$0.367942\pi$
$110$	0	0
$111$	9.70820	0.921462
$112$	0	0
$113$	14.4164	1.35618	0.678091	−	0.734978i	$-0.262808\pi$
$113$	14.4164	1.35618	0.678091	+	0.734978i	$0.262808\pi$
$114$	−8.94427	−0.837708
$115$	0	0
$116$	−8.09017	−0.751153
$117$	9.23607	0.853875
$118$	−7.23607	−0.666134
$119$	0	0
$120$	0	0
$121$	−10.9443	−0.994934
$122$	6.00000	0.543214
$123$	15.4164	1.39005
$124$	6.00000	0.538816
$125$	0	0
$126$	0	0
$127$	−13.6525	−1.21146	−0.605731	−	0.795670i	$-0.707119\pi$
$127$	−13.6525	−1.21146	−0.605731	+	0.795670i	$0.707119\pi$
$128$	−11.3820	−1.00603
$129$	5.70820	0.502579
$130$	0	0
$131$	16.9443	1.48043	0.740214	−	0.672371i	$-0.234724\pi$
$131$	16.9443	1.48043	0.740214	+	0.672371i	$0.234724\pi$
$132$	−1.23607	−0.107586
$133$	0	0
$134$	2.61803	0.226164
$135$	0	0
$136$	−5.52786	−0.474010
$137$	10.9443	0.935032	0.467516	−	0.883985i	$-0.345149\pi$
$137$	10.9443	0.935032	0.467516	+	0.883985i	$0.345149\pi$
$138$	12.4721	1.06170
$139$	−10.6525	−0.903531	−0.451766	−	0.892137i	$-0.649206\pi$
$139$	−10.6525	−0.903531	−0.451766	+	0.892137i	$0.649206\pi$
$140$	0	0
$141$	6.47214	0.545052
$142$	5.38197	0.451645
$143$	−0.291796	−0.0244012
$144$	13.8541	1.15451
$145$	0	0
$146$	−5.41641	−0.448265
$147$	0	0
$148$	4.85410	0.399005
$149$	3.94427	0.323127	0.161564	−	0.986862i	$-0.448346\pi$
$149$	3.94427	0.323127	0.161564	+	0.986862i	$0.448346\pi$
$150$	0	0
$151$	−20.2361	−1.64679	−0.823394	−	0.567470i	$-0.807922\pi$
$151$	−20.2361	−1.64679	−0.823394	+	0.567470i	$0.807922\pi$
$152$	−10.0000	−0.811107
$153$	18.4721	1.49338
$154$	0	0
$155$	0	0
$156$	6.47214	0.518186
$157$	0.763932	0.0609684	0.0304842	−	0.999535i	$-0.490295\pi$
$157$	0.763932	0.0609684	0.0304842	+	0.999535i	$0.490295\pi$
$158$	−6.90983	−0.549717
$159$	27.4164	2.17426
$160$	0	0
$161$	0	0
$162$	15.0902	1.18560
$163$	1.52786	0.119672	0.0598358	−	0.998208i	$-0.480942\pi$
$163$	1.52786	0.119672	0.0598358	+	0.998208i	$0.480942\pi$
$164$	7.70820	0.601910
$165$	0	0
$166$	−4.76393	−0.369753
$167$	5.23607	0.405179	0.202590	−	0.979264i	$-0.435064\pi$
$167$	5.23607	0.405179	0.202590	+	0.979264i	$0.435064\pi$
$168$	0	0
$169$	−11.4721	−0.882472
$170$	0	0
$171$	33.4164	2.55542
$172$	2.85410	0.217623
$173$	−11.5279	−0.876447	−0.438224	−	0.898866i	$-0.644392\pi$
$173$	−11.5279	−0.876447	−0.438224	+	0.898866i	$0.644392\pi$
$174$	−10.0000	−0.758098
$175$	0	0
$176$	−0.437694	−0.0329924
$177$	37.8885	2.84788
$178$	−10.6525	−0.798437
$179$	−23.4164	−1.75022	−0.875112	−	0.483920i	$-0.839213\pi$
$179$	−23.4164	−1.75022	−0.875112	+	0.483920i	$0.839213\pi$
$180$	0	0
$181$	−8.18034	−0.608040	−0.304020	−	0.952666i	$-0.598329\pi$
$181$	−8.18034	−0.608040	−0.304020	+	0.952666i	$0.598329\pi$
$182$	0	0
$183$	−31.4164	−2.32237
$184$	13.9443	1.02799
$185$	0	0
$186$	7.41641	0.543797
$187$	−0.583592	−0.0426765
$188$	3.23607	0.236015
$189$	0	0
$190$	0	0
$191$	6.47214	0.468307	0.234154	−	0.972200i	$-0.424768\pi$
$191$	6.47214	0.468307	0.234154	+	0.972200i	$0.424768\pi$
$192$	0.763932	0.0551320
$193$	−12.4164	−0.893753	−0.446876	−	0.894596i	$-0.647464\pi$
$193$	−12.4164	−0.893753	−0.446876	+	0.894596i	$0.647464\pi$
$194$	3.23607	0.232336
$195$	0	0
$196$	0	0
$197$	1.47214	0.104885	0.0524427	−	0.998624i	$-0.483299\pi$
$197$	1.47214	0.104885	0.0524427	+	0.998624i	$0.483299\pi$
$198$	−1.09017	−0.0774750
$199$	−7.23607	−0.512951	−0.256476	−	0.966551i	$-0.582561\pi$
$199$	−7.23607	−0.512951	−0.256476	+	0.966551i	$0.582561\pi$
$200$	0	0
$201$	−13.7082	−0.966902
$202$	−2.94427	−0.207158
$203$	0	0
$204$	12.9443	0.906280
$205$	0	0
$206$	5.23607	0.364814
$207$	−46.5967	−3.23870
$208$	2.29180	0.158907
$209$	−1.05573	−0.0730262
$210$	0	0
$211$	12.0000	0.826114	0.413057	−	0.910705i	$-0.364461\pi$
$211$	12.0000	0.826114	0.413057	+	0.910705i	$0.364461\pi$
$212$	13.7082	0.941483
$213$	−28.1803	−1.93089
$214$	−4.94427	−0.337983
$215$	0	0
$216$	32.3607	2.20187
$217$	0	0
$218$	5.20163	0.352299
$219$	28.3607	1.91644
$220$	0	0
$221$	3.05573	0.205551
$222$	6.00000	0.402694
$223$	20.1803	1.35138	0.675688	−	0.737188i	$-0.263847\pi$
$223$	20.1803	1.35138	0.675688	+	0.737188i	$0.263847\pi$
$224$	0	0
$225$	0	0
$226$	8.90983	0.592673
$227$	21.4164	1.42146	0.710728	−	0.703466i	$-0.248365\pi$
$227$	21.4164	1.42146	0.710728	+	0.703466i	$0.248365\pi$
$228$	23.4164	1.55079
$229$	4.47214	0.295527	0.147764	−	0.989023i	$-0.452793\pi$
$229$	4.47214	0.295527	0.147764	+	0.989023i	$0.452793\pi$
$230$	0	0
$231$	0	0
$232$	−11.1803	−0.734025
$233$	−7.94427	−0.520447	−0.260223	−	0.965548i	$-0.583796\pi$
$233$	−7.94427	−0.520447	−0.260223	+	0.965548i	$0.583796\pi$
$234$	5.70820	0.373157
$235$	0	0
$236$	18.9443	1.23317
$237$	36.1803	2.35017
$238$	0	0
$239$	−5.52786	−0.357568	−0.178784	−	0.983888i	$-0.557216\pi$
$239$	−5.52786	−0.357568	−0.178784	+	0.983888i	$0.557216\pi$
$240$	0	0
$241$	3.52786	0.227250	0.113625	−	0.993524i	$-0.463754\pi$
$241$	3.52786	0.227250	0.113625	+	0.993524i	$0.463754\pi$
$242$	−6.76393	−0.434802
$243$	−35.5967	−2.28353
$244$	−15.7082	−1.00561
$245$	0	0
$246$	9.52786	0.607474
$247$	5.52786	0.351730
$248$	8.29180	0.526530
$249$	24.9443	1.58078
$250$	0	0
$251$	−6.47214	−0.408518	−0.204259	−	0.978917i	$-0.565478\pi$
$251$	−6.47214	−0.408518	−0.204259	+	0.978917i	$0.565478\pi$
$252$	0	0
$253$	1.47214	0.0925524
$254$	−8.43769	−0.529428
$255$	0	0
$256$	−6.56231	−0.410144
$257$	−12.6525	−0.789240	−0.394620	−	0.918844i	$-0.629124\pi$
$257$	−12.6525	−0.789240	−0.394620	+	0.918844i	$0.629124\pi$
$258$	3.52786	0.219635
$259$	0	0
$260$	0	0
$261$	37.3607	2.31257
$262$	10.4721	0.646971
$263$	−16.2361	−1.00116	−0.500579	−	0.865691i	$-0.666880\pi$
$263$	−16.2361	−1.00116	−0.500579	+	0.865691i	$0.666880\pi$
$264$	−1.70820	−0.105133
$265$	0	0
$266$	0	0
$267$	55.7771	3.41350
$268$	−6.85410	−0.418681
$269$	11.7082	0.713862	0.356931	−	0.934131i	$-0.383823\pi$
$269$	11.7082	0.713862	0.356931	+	0.934131i	$0.383823\pi$
$270$	0	0
$271$	−23.7082	−1.44017	−0.720085	−	0.693885i	$-0.755897\pi$
$271$	−23.7082	−1.44017	−0.720085	+	0.693885i	$0.755897\pi$
$272$	4.58359	0.277921
$273$	0	0
$274$	6.76393	0.408624
$275$	0	0
$276$	−32.6525	−1.96545
$277$	19.8885	1.19499	0.597493	−	0.801874i	$-0.296163\pi$
$277$	19.8885	1.19499	0.597493	+	0.801874i	$0.296163\pi$
$278$	−6.58359	−0.394858
$279$	−27.7082	−1.65885
$280$	0	0
$281$	−15.3607	−0.916341	−0.458171	−	0.888864i	$-0.651495\pi$
$281$	−15.3607	−0.916341	−0.458171	+	0.888864i	$0.651495\pi$
$282$	4.00000	0.238197
$283$	17.4164	1.03530	0.517649	−	0.855593i	$-0.326807\pi$
$283$	17.4164	1.03530	0.517649	+	0.855593i	$0.326807\pi$
$284$	−14.0902	−0.836098
$285$	0	0
$286$	−0.180340	−0.0106637
$287$	0	0
$288$	41.9787	2.47362
$289$	−10.8885	−0.640503
$290$	0	0
$291$	−16.9443	−0.993291
$292$	14.1803	0.829842
$293$	−31.1246	−1.81832	−0.909160	−	0.416448i	$-0.863275\pi$
$293$	−31.1246	−1.81832	−0.909160	+	0.416448i	$0.863275\pi$
$294$	0	0
$295$	0	0
$296$	6.70820	0.389906
$297$	3.41641	0.198240
$298$	2.43769	0.141212
$299$	−7.70820	−0.445777
$300$	0	0
$301$	0	0
$302$	−12.5066	−0.719673
$303$	15.4164	0.885649
$304$	8.29180	0.475567
$305$	0	0
$306$	11.4164	0.652633
$307$	4.58359	0.261599	0.130800	−	0.991409i	$-0.458246\pi$
$307$	4.58359	0.261599	0.130800	+	0.991409i	$0.458246\pi$
$308$	0	0
$309$	−27.4164	−1.55966
$310$	0	0
$311$	−24.3607	−1.38137	−0.690684	−	0.723157i	$-0.742690\pi$
$311$	−24.3607	−1.38137	−0.690684	+	0.723157i	$0.742690\pi$
$312$	8.94427	0.506370
$313$	19.5279	1.10378	0.551890	−	0.833917i	$-0.313907\pi$
$313$	19.5279	1.10378	0.551890	+	0.833917i	$0.313907\pi$
$314$	0.472136	0.0266442
$315$	0	0
$316$	18.0902	1.01765
$317$	−25.3607	−1.42440	−0.712199	−	0.701978i	$-0.752301\pi$
$317$	−25.3607	−1.42440	−0.712199	+	0.701978i	$0.752301\pi$
$318$	16.9443	0.950188
$319$	−1.18034	−0.0660863
$320$	0	0
$321$	25.8885	1.44496
$322$	0	0
$323$	11.0557	0.615157
$324$	−39.5066	−2.19481
$325$	0	0
$326$	0.944272	0.0522984
$327$	−27.2361	−1.50616
$328$	10.6525	0.588185
$329$	0	0
$330$	0	0
$331$	−24.7082	−1.35809	−0.679043	−	0.734099i	$-0.737605\pi$
$331$	−24.7082	−1.35809	−0.679043	+	0.734099i	$0.737605\pi$
$332$	12.4721	0.684497
$333$	−22.4164	−1.22841
$334$	3.23607	0.177070
$335$	0	0
$336$	0	0
$337$	16.4721	0.897294	0.448647	−	0.893709i	$-0.351906\pi$
$337$	16.4721	0.897294	0.448647	+	0.893709i	$0.351906\pi$
$338$	−7.09017	−0.385654
$339$	−46.6525	−2.53381
$340$	0	0
$341$	0.875388	0.0474049
$342$	20.6525	1.11676
$343$	0	0
$344$	3.94427	0.212661
$345$	0	0
$346$	−7.12461	−0.383022
$347$	−20.2361	−1.08633	−0.543165	−	0.839626i	$-0.682774\pi$
$347$	−20.2361	−1.08633	−0.543165	+	0.839626i	$0.682774\pi$
$348$	26.1803	1.40341
$349$	−4.47214	−0.239388	−0.119694	−	0.992811i	$-0.538191\pi$
$349$	−4.47214	−0.239388	−0.119694	+	0.992811i	$0.538191\pi$
$350$	0	0
$351$	−17.8885	−0.954820
$352$	−1.32624	−0.0706887
$353$	−2.18034	−0.116048	−0.0580239	−	0.998315i	$-0.518480\pi$
$353$	−2.18034	−0.116048	−0.0580239	+	0.998315i	$0.518480\pi$
$354$	23.4164	1.24457
$355$	0	0
$356$	27.8885	1.47809
$357$	0	0
$358$	−14.4721	−0.764876
$359$	−30.1246	−1.58992	−0.794958	−	0.606664i	$-0.792507\pi$
$359$	−30.1246	−1.58992	−0.794958	+	0.606664i	$0.792507\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	−5.05573	−0.265723
$363$	35.4164	1.85888
$364$	0	0
$365$	0	0
$366$	−19.4164	−1.01491
$367$	−37.1246	−1.93789	−0.968944	−	0.247278i	$-0.920464\pi$
$367$	−37.1246	−1.93789	−0.968944	+	0.247278i	$0.920464\pi$
$368$	−11.5623	−0.602727
$369$	−35.5967	−1.85309
$370$	0	0
$371$	0	0
$372$	−19.4164	−1.00669
$373$	37.8328	1.95891	0.979454	−	0.201665i	$-0.0646354\pi$
$373$	37.8328	1.95891	0.979454	+	0.201665i	$0.0646354\pi$
$374$	−0.360680	−0.0186503
$375$	0	0
$376$	4.47214	0.230633
$377$	6.18034	0.318304
$378$	0	0
$379$	−11.1803	−0.574295	−0.287148	−	0.957886i	$-0.592707\pi$
$379$	−11.1803	−0.574295	−0.287148	+	0.957886i	$0.592707\pi$
$380$	0	0
$381$	44.1803	2.26343
$382$	4.00000	0.204658
$383$	−33.2361	−1.69828	−0.849142	−	0.528165i	$-0.822880\pi$
$383$	−33.2361	−1.69828	−0.849142	+	0.528165i	$0.822880\pi$
$384$	36.8328	1.87962
$385$	0	0
$386$	−7.67376	−0.390584
$387$	−13.1803	−0.669994
$388$	−8.47214	−0.430108
$389$	2.88854	0.146455	0.0732275	−	0.997315i	$-0.476670\pi$
$389$	2.88854	0.146455	0.0732275	+	0.997315i	$0.476670\pi$
$390$	0	0
$391$	−15.4164	−0.779641
$392$	0	0
$393$	−54.8328	−2.76595
$394$	0.909830	0.0458366
$395$	0	0
$396$	2.85410	0.143424
$397$	9.05573	0.454494	0.227247	−	0.973837i	$-0.427028\pi$
$397$	9.05573	0.454494	0.227247	+	0.973837i	$0.427028\pi$
$398$	−4.47214	−0.224168
$399$	0	0
$400$	0	0
$401$	2.52786	0.126236	0.0631178	−	0.998006i	$-0.479896\pi$
$401$	2.52786	0.126236	0.0631178	+	0.998006i	$0.479896\pi$
$402$	−8.47214	−0.422552
$403$	−4.58359	−0.228325
$404$	7.70820	0.383497
$405$	0	0
$406$	0	0
$407$	0.708204	0.0351044
$408$	17.8885	0.885615
$409$	−24.4721	−1.21007	−0.605035	−	0.796199i	$-0.706841\pi$
$409$	−24.4721	−1.21007	−0.605035	+	0.796199i	$0.706841\pi$
$410$	0	0
$411$	−35.4164	−1.74696
$412$	−13.7082	−0.675355
$413$	0	0
$414$	−28.7984	−1.41536
$415$	0	0
$416$	6.94427	0.340471
$417$	34.4721	1.68811
$418$	−0.652476	−0.0319136
$419$	26.1803	1.27899	0.639497	−	0.768794i	$-0.279143\pi$
$419$	26.1803	1.27899	0.639497	+	0.768794i	$0.279143\pi$
$420$	0	0
$421$	−13.0000	−0.633581	−0.316791	−	0.948495i	$-0.602605\pi$
$421$	−13.0000	−0.633581	−0.316791	+	0.948495i	$0.602605\pi$
$422$	7.41641	0.361025
$423$	−14.9443	−0.726615
$424$	18.9443	0.920015
$425$	0	0
$426$	−17.4164	−0.843828
$427$	0	0
$428$	12.9443	0.625685
$429$	0.944272	0.0455899
$430$	0	0
$431$	17.5279	0.844288	0.422144	−	0.906529i	$-0.361278\pi$
$431$	17.5279	0.844288	0.422144	+	0.906529i	$0.361278\pi$
$432$	−26.8328	−1.29099
$433$	−28.3607	−1.36293	−0.681464	−	0.731852i	$-0.738656\pi$
$433$	−28.3607	−1.36293	−0.681464	+	0.731852i	$0.738656\pi$
$434$	0	0
$435$	0	0
$436$	−13.6180	−0.652186
$437$	−27.8885	−1.33409
$438$	17.5279	0.837514
$439$	8.29180	0.395746	0.197873	−	0.980228i	$-0.436597\pi$
$439$	8.29180	0.395746	0.197873	+	0.980228i	$0.436597\pi$
$440$	0	0
$441$	0	0
$442$	1.88854	0.0898289
$443$	19.4164	0.922501	0.461251	−	0.887270i	$-0.347401\pi$
$443$	19.4164	0.922501	0.461251	+	0.887270i	$0.347401\pi$
$444$	−15.7082	−0.745478
$445$	0	0
$446$	12.4721	0.590573
$447$	−12.7639	−0.603713
$448$	0	0
$449$	20.5279	0.968770	0.484385	−	0.874855i	$-0.339043\pi$
$449$	20.5279	0.968770	0.484385	+	0.874855i	$0.339043\pi$
$450$	0	0
$451$	1.12461	0.0529559
$452$	−23.3262	−1.09717
$453$	65.4853	3.07677
$454$	13.2361	0.621199
$455$	0	0
$456$	32.3607	1.51543
$457$	12.5279	0.586029	0.293014	−	0.956108i	$-0.405342\pi$
$457$	12.5279	0.586029	0.293014	+	0.956108i	$0.405342\pi$
$458$	2.76393	0.129150
$459$	−35.7771	−1.66993
$460$	0	0
$461$	14.1803	0.660444	0.330222	−	0.943903i	$-0.392876\pi$
$461$	14.1803	0.660444	0.330222	+	0.943903i	$0.392876\pi$
$462$	0	0
$463$	13.8885	0.645455	0.322728	−	0.946492i	$-0.395400\pi$
$463$	13.8885	0.645455	0.322728	+	0.946492i	$0.395400\pi$
$464$	9.27051	0.430373
$465$	0	0
$466$	−4.90983	−0.227443
$467$	6.94427	0.321343	0.160671	−	0.987008i	$-0.448634\pi$
$467$	6.94427	0.321343	0.160671	+	0.987008i	$0.448634\pi$
$468$	−14.9443	−0.690799
$469$	0	0
$470$	0	0
$471$	−2.47214	−0.113910
$472$	26.1803	1.20505
$473$	0.416408	0.0191465
$474$	22.3607	1.02706
$475$	0	0
$476$	0	0
$477$	−63.3050	−2.89853
$478$	−3.41641	−0.156263
$479$	26.1803	1.19621	0.598105	−	0.801418i	$-0.295921\pi$
$479$	26.1803	1.19621	0.598105	+	0.801418i	$0.295921\pi$
$480$	0	0
$481$	−3.70820	−0.169080
$482$	2.18034	0.0993118
$483$	0	0
$484$	17.7082	0.804918
$485$	0	0
$486$	−22.0000	−0.997940
$487$	−5.76393	−0.261189	−0.130594	−	0.991436i	$-0.541689\pi$
$487$	−5.76393	−0.261189	−0.130594	+	0.991436i	$0.541689\pi$
$488$	−21.7082	−0.982684
$489$	−4.94427	−0.223588
$490$	0	0
$491$	−5.76393	−0.260123	−0.130061	−	0.991506i	$-0.541517\pi$
$491$	−5.76393	−0.260123	−0.130061	+	0.991506i	$0.541517\pi$
$492$	−24.9443	−1.12457
$493$	12.3607	0.556697
$494$	3.41641	0.153711
$495$	0	0
$496$	−6.87539	−0.308714
$497$	0	0
$498$	15.4164	0.690826
$499$	11.0557	0.494922	0.247461	−	0.968898i	$-0.420404\pi$
$499$	11.0557	0.494922	0.247461	+	0.968898i	$0.420404\pi$
$500$	0	0
$501$	−16.9443	−0.757014
$502$	−4.00000	−0.178529
$503$	−8.11146	−0.361672	−0.180836	−	0.983513i	$-0.557880\pi$
$503$	−8.11146	−0.361672	−0.180836	+	0.983513i	$0.557880\pi$
$504$	0	0
$505$	0	0
$506$	0.909830	0.0404469
$507$	37.1246	1.64876
$508$	22.0902	0.980093
$509$	−40.6525	−1.80189	−0.900945	−	0.433934i	$-0.857125\pi$
$509$	−40.6525	−1.80189	−0.900945	+	0.433934i	$0.857125\pi$
$510$	0	0
$511$	0	0
$512$	18.7082	0.826794
$513$	−64.7214	−2.85752
$514$	−7.81966	−0.344910
$515$	0	0
$516$	−9.23607	−0.406595
$517$	0.472136	0.0207645
$518$	0	0
$519$	37.3050	1.63751
$520$	0	0
$521$	−22.0000	−0.963837	−0.481919	−	0.876216i	$-0.660060\pi$
$521$	−22.0000	−0.963837	−0.481919	+	0.876216i	$0.660060\pi$
$522$	23.0902	1.01063
$523$	16.3607	0.715403	0.357701	−	0.933836i	$-0.383561\pi$
$523$	16.3607	0.715403	0.357701	+	0.933836i	$0.383561\pi$
$524$	−27.4164	−1.19769
$525$	0	0
$526$	−10.0344	−0.437522
$527$	−9.16718	−0.399329
$528$	1.41641	0.0616412
$529$	15.8885	0.690806
$530$	0	0
$531$	−87.4853	−3.79654
$532$	0	0
$533$	−5.88854	−0.255061
$534$	34.4721	1.49176
$535$	0	0
$536$	−9.47214	−0.409134
$537$	75.7771	3.27002
$538$	7.23607	0.311969
$539$	0	0
$540$	0	0
$541$	15.9443	0.685498	0.342749	−	0.939427i	$-0.388642\pi$
$541$	15.9443	0.685498	0.342749	+	0.939427i	$0.388642\pi$
$542$	−14.6525	−0.629378
$543$	26.4721	1.13603
$544$	13.8885	0.595466
$545$	0	0
$546$	0	0
$547$	9.76393	0.417476	0.208738	−	0.977972i	$-0.433064\pi$
$547$	9.76393	0.417476	0.208738	+	0.977972i	$0.433064\pi$
$548$	−17.7082	−0.756457
$549$	72.5410	3.09598
$550$	0	0
$551$	22.3607	0.952597
$552$	−45.1246	−1.92063
$553$	0	0
$554$	12.2918	0.522228
$555$	0	0
$556$	17.2361	0.730972
$557$	9.11146	0.386065	0.193032	−	0.981192i	$-0.438168\pi$
$557$	9.11146	0.386065	0.193032	+	0.981192i	$0.438168\pi$
$558$	−17.1246	−0.724943
$559$	−2.18034	−0.0922186
$560$	0	0
$561$	1.88854	0.0797344
$562$	−9.49342	−0.400456
$563$	17.4164	0.734014	0.367007	−	0.930218i	$-0.380382\pi$
$563$	17.4164	0.734014	0.367007	+	0.930218i	$0.380382\pi$
$564$	−10.4721	−0.440956
$565$	0	0
$566$	10.7639	0.452442
$567$	0	0
$568$	−19.4721	−0.817033
$569$	−3.94427	−0.165352	−0.0826762	−	0.996576i	$-0.526347\pi$
$569$	−3.94427	−0.165352	−0.0826762	+	0.996576i	$0.526347\pi$
$570$	0	0
$571$	36.5967	1.53153	0.765763	−	0.643123i	$-0.222362\pi$
$571$	36.5967	1.53153	0.765763	+	0.643123i	$0.222362\pi$
$572$	0.472136	0.0197410
$573$	−20.9443	−0.874960
$574$	0	0
$575$	0	0
$576$	−1.76393	−0.0734972
$577$	−2.00000	−0.0832611	−0.0416305	−	0.999133i	$-0.513255\pi$
$577$	−2.00000	−0.0832611	−0.0416305	+	0.999133i	$0.513255\pi$
$578$	−6.72949	−0.279910
$579$	40.1803	1.66984
$580$	0	0
$581$	0	0
$582$	−10.4721	−0.434084
$583$	2.00000	0.0828315
$584$	19.5967	0.810919
$585$	0	0
$586$	−19.2361	−0.794635
$587$	−24.7639	−1.02212	−0.511058	−	0.859546i	$-0.670746\pi$
$587$	−24.7639	−1.02212	−0.511058	+	0.859546i	$0.670746\pi$
$588$	0	0
$589$	−16.5836	−0.683315
$590$	0	0
$591$	−4.76393	−0.195962
$592$	−5.56231	−0.228609
$593$	−37.3050	−1.53193	−0.765965	−	0.642882i	$-0.777739\pi$
$593$	−37.3050	−1.53193	−0.765965	+	0.642882i	$0.777739\pi$
$594$	2.11146	0.0866341
$595$	0	0
$596$	−6.38197	−0.261416
$597$	23.4164	0.958370
$598$	−4.76393	−0.194812
$599$	−11.1803	−0.456816	−0.228408	−	0.973565i	$-0.573352\pi$
$599$	−11.1803	−0.456816	−0.228408	+	0.973565i	$0.573352\pi$
$600$	0	0
$601$	36.9443	1.50699	0.753494	−	0.657455i	$-0.228367\pi$
$601$	36.9443	1.50699	0.753494	+	0.657455i	$0.228367\pi$
$602$	0	0
$603$	31.6525	1.28899
$604$	32.7426	1.33228
$605$	0	0
$606$	9.52786	0.387043
$607$	−7.12461	−0.289179	−0.144590	−	0.989492i	$-0.546186\pi$
$607$	−7.12461	−0.289179	−0.144590	+	0.989492i	$0.546186\pi$
$608$	25.1246	1.01894
$609$	0	0
$610$	0	0
$611$	−2.47214	−0.100012
$612$	−29.8885	−1.20817
$613$	44.4164	1.79396	0.896981	−	0.442069i	$-0.145755\pi$
$613$	44.4164	1.79396	0.896981	+	0.442069i	$0.145755\pi$
$614$	2.83282	0.114323
$615$	0	0
$616$	0	0
$617$	5.94427	0.239307	0.119654	−	0.992816i	$-0.461822\pi$
$617$	5.94427	0.239307	0.119654	+	0.992816i	$0.461822\pi$
$618$	−16.9443	−0.681599
$619$	11.7082	0.470592	0.235296	−	0.971924i	$-0.424394\pi$
$619$	11.7082	0.470592	0.235296	+	0.971924i	$0.424394\pi$
$620$	0	0
$621$	90.2492	3.62158
$622$	−15.0557	−0.603680
$623$	0	0
$624$	−7.41641	−0.296894
$625$	0	0
$626$	12.0689	0.482370
$627$	3.41641	0.136438
$628$	−1.23607	−0.0493245
$629$	−7.41641	−0.295712
$630$	0	0
$631$	27.6525	1.10083	0.550414	−	0.834892i	$-0.314470\pi$
$631$	27.6525	1.10083	0.550414	+	0.834892i	$0.314470\pi$
$632$	25.0000	0.994447
$633$	−38.8328	−1.54347
$634$	−15.6738	−0.622485
$635$	0	0
$636$	−44.3607	−1.75902
$637$	0	0
$638$	−0.729490	−0.0288808
$639$	65.0689	2.57409
$640$	0	0
$641$	43.8328	1.73129	0.865646	−	0.500656i	$-0.166908\pi$
$641$	43.8328	1.73129	0.865646	+	0.500656i	$0.166908\pi$
$642$	16.0000	0.631470
$643$	18.4721	0.728470	0.364235	−	0.931307i	$-0.381331\pi$
$643$	18.4721	0.728470	0.364235	+	0.931307i	$0.381331\pi$
$644$	0	0
$645$	0	0
$646$	6.83282	0.268834
$647$	−19.8885	−0.781899	−0.390950	−	0.920412i	$-0.627853\pi$
$647$	−19.8885	−0.781899	−0.390950	+	0.920412i	$0.627853\pi$
$648$	−54.5967	−2.14476
$649$	2.76393	0.108494
$650$	0	0
$651$	0	0
$652$	−2.47214	−0.0968163
$653$	−25.0557	−0.980506	−0.490253	−	0.871580i	$-0.663096\pi$
$653$	−25.0557	−0.980506	−0.490253	+	0.871580i	$0.663096\pi$
$654$	−16.8328	−0.658215
$655$	0	0
$656$	−8.83282	−0.344864
$657$	−65.4853	−2.55482
$658$	0	0
$659$	17.8885	0.696839	0.348419	−	0.937339i	$-0.386719\pi$
$659$	17.8885	0.696839	0.348419	+	0.937339i	$0.386719\pi$
$660$	0	0
$661$	42.7214	1.66167	0.830834	−	0.556520i	$-0.187864\pi$
$661$	42.7214	1.66167	0.830834	+	0.556520i	$0.187864\pi$
$662$	−15.2705	−0.593505
$663$	−9.88854	−0.384039
$664$	17.2361	0.668889
$665$	0	0
$666$	−13.8541	−0.536836
$667$	−31.1803	−1.20731
$668$	−8.47214	−0.327797
$669$	−65.3050	−2.52484
$670$	0	0
$671$	−2.29180	−0.0884738
$672$	0	0
$673$	−19.5279	−0.752744	−0.376372	−	0.926469i	$-0.622829\pi$
$673$	−19.5279	−0.752744	−0.376372	+	0.926469i	$0.622829\pi$
$674$	10.1803	0.392132
$675$	0	0
$676$	18.5623	0.713935
$677$	−14.3607	−0.551926	−0.275963	−	0.961168i	$-0.588997\pi$
$677$	−14.3607	−0.551926	−0.275963	+	0.961168i	$0.588997\pi$
$678$	−28.8328	−1.10732
$679$	0	0
$680$	0	0
$681$	−69.3050	−2.65577
$682$	0.541020	0.0207167
$683$	−14.1246	−0.540463	−0.270232	−	0.962795i	$-0.587100\pi$
$683$	−14.1246	−0.540463	−0.270232	+	0.962795i	$0.587100\pi$
$684$	−54.0689	−2.06738
$685$	0	0
$686$	0	0
$687$	−14.4721	−0.552146
$688$	−3.27051	−0.124687
$689$	−10.4721	−0.398957
$690$	0	0
$691$	4.18034	0.159028	0.0795138	−	0.996834i	$-0.474663\pi$
$691$	4.18034	0.159028	0.0795138	+	0.996834i	$0.474663\pi$
$692$	18.6525	0.709061
$693$	0	0
$694$	−12.5066	−0.474743
$695$	0	0
$696$	36.1803	1.37141
$697$	−11.7771	−0.446089
$698$	−2.76393	−0.104616
$699$	25.7082	0.972374
$700$	0	0
$701$	−29.0557	−1.09742	−0.548710	−	0.836013i	$-0.684881\pi$
$701$	−29.0557	−1.09742	−0.548710	+	0.836013i	$0.684881\pi$
$702$	−11.0557	−0.417272
$703$	−13.4164	−0.506009
$704$	0.0557281	0.00210033
$705$	0	0
$706$	−1.34752	−0.0507147
$707$	0	0
$708$	−61.3050	−2.30398
$709$	−12.1115	−0.454855	−0.227428	−	0.973795i	$-0.573032\pi$
$709$	−12.1115	−0.454855	−0.227428	+	0.973795i	$0.573032\pi$
$710$	0	0
$711$	−83.5410	−3.13303
$712$	38.5410	1.44439
$713$	23.1246	0.866024
$714$	0	0
$715$	0	0
$716$	37.8885	1.41596
$717$	17.8885	0.668060
$718$	−18.6180	−0.694819
$719$	−16.1803	−0.603425	−0.301712	−	0.953399i	$-0.597558\pi$
$719$	−16.1803	−0.603425	−0.301712	+	0.953399i	$0.597558\pi$
$720$	0	0
$721$	0	0
$722$	0.618034	0.0230008
$723$	−11.4164	−0.424581
$724$	13.2361	0.491915
$725$	0	0
$726$	21.8885	0.812360
$727$	−3.05573	−0.113331	−0.0566653	−	0.998393i	$-0.518047\pi$
$727$	−3.05573	−0.113331	−0.0566653	+	0.998393i	$0.518047\pi$
$728$	0	0
$729$	41.9443	1.55349
$730$	0	0
$731$	−4.36068	−0.161286
$732$	50.8328	1.87883
$733$	4.00000	0.147743	0.0738717	−	0.997268i	$-0.476464\pi$
$733$	4.00000	0.147743	0.0738717	+	0.997268i	$0.476464\pi$
$734$	−22.9443	−0.846889
$735$	0	0
$736$	−35.0344	−1.29139
$737$	−1.00000	−0.0368355
$738$	−22.0000	−0.809831
$739$	25.6525	0.943642	0.471821	−	0.881694i	$-0.343597\pi$
$739$	25.6525	0.943642	0.471821	+	0.881694i	$0.343597\pi$
$740$	0	0
$741$	−17.8885	−0.657152
$742$	0	0
$743$	10.4721	0.384185	0.192093	−	0.981377i	$-0.438473\pi$
$743$	10.4721	0.384185	0.192093	+	0.981377i	$0.438473\pi$
$744$	−26.8328	−0.983739
$745$	0	0
$746$	23.3820	0.856075
$747$	−57.5967	−2.10735
$748$	0.944272	0.0345260
$749$	0	0
$750$	0	0
$751$	3.05573	0.111505	0.0557526	−	0.998445i	$-0.482244\pi$
$751$	3.05573	0.111505	0.0557526	+	0.998445i	$0.482244\pi$
$752$	−3.70820	−0.135224
$753$	20.9443	0.763252
$754$	3.81966	0.139104
$755$	0	0
$756$	0	0
$757$	−19.5836	−0.711778	−0.355889	−	0.934528i	$-0.615822\pi$
$757$	−19.5836	−0.711778	−0.355889	+	0.934528i	$0.615822\pi$
$758$	−6.90983	−0.250976
$759$	−4.76393	−0.172920
$760$	0	0
$761$	−27.7771	−1.00692	−0.503459	−	0.864019i	$-0.667940\pi$
$761$	−27.7771	−1.00692	−0.503459	+	0.864019i	$0.667940\pi$
$762$	27.3050	0.989154
$763$	0	0
$764$	−10.4721	−0.378869
$765$	0	0
$766$	−20.5410	−0.742177
$767$	−14.4721	−0.522559
$768$	21.2361	0.766291
$769$	43.0132	1.55109	0.775547	−	0.631290i	$-0.217474\pi$
$769$	43.0132	1.55109	0.775547	+	0.631290i	$0.217474\pi$
$770$	0	0
$771$	40.9443	1.47457
$772$	20.0902	0.723061
$773$	50.1803	1.80486	0.902431	−	0.430835i	$-0.141781\pi$
$773$	50.1803	1.80486	0.902431	+	0.430835i	$0.141781\pi$
$774$	−8.14590	−0.292798
$775$	0	0
$776$	−11.7082	−0.420300
$777$	0	0
$778$	1.78522	0.0640032
$779$	−21.3050	−0.763329
$780$	0	0
$781$	−2.05573	−0.0735597
$782$	−9.52786	−0.340716
$783$	−72.3607	−2.58596
$784$	0	0
$785$	0	0
$786$	−33.8885	−1.20876
$787$	40.7639	1.45308	0.726539	−	0.687126i	$-0.241128\pi$
$787$	40.7639	1.45308	0.726539	+	0.687126i	$0.241128\pi$
$788$	−2.38197	−0.0848540
$789$	52.5410	1.87051
$790$	0	0
$791$	0	0
$792$	3.94427	0.140154
$793$	12.0000	0.426132
$794$	5.59675	0.198621
$795$	0	0
$796$	11.7082	0.414986
$797$	−35.4164	−1.25451	−0.627257	−	0.778813i	$-0.715822\pi$
$797$	−35.4164	−1.25451	−0.627257	+	0.778813i	$0.715822\pi$
$798$	0	0
$799$	−4.94427	−0.174916
$800$	0	0
$801$	−128.790	−4.55058
$802$	1.56231	0.0551669
$803$	2.06888	0.0730093
$804$	22.1803	0.782240
$805$	0	0
$806$	−2.83282	−0.0997817
$807$	−37.8885	−1.33374
$808$	10.6525	0.374753
$809$	29.4721	1.03619	0.518093	−	0.855325i	$-0.326642\pi$
$809$	29.4721	1.03619	0.518093	+	0.855325i	$0.326642\pi$
$810$	0	0
$811$	42.7214	1.50015	0.750075	−	0.661353i	$-0.230017\pi$
$811$	42.7214	1.50015	0.750075	+	0.661353i	$0.230017\pi$
$812$	0	0
$813$	76.7214	2.69074
$814$	0.437694	0.0153412
$815$	0	0
$816$	−14.8328	−0.519252
$817$	−7.88854	−0.275985
$818$	−15.1246	−0.528820
$819$	0	0
$820$	0	0
$821$	28.8328	1.00627	0.503136	−	0.864207i	$-0.332179\pi$
$821$	28.8328	1.00627	0.503136	+	0.864207i	$0.332179\pi$
$822$	−21.8885	−0.763451
$823$	31.6525	1.10334	0.551668	−	0.834064i	$-0.313992\pi$
$823$	31.6525	1.10334	0.551668	+	0.834064i	$0.313992\pi$
$824$	−18.9443	−0.659955
$825$	0	0
$826$	0	0
$827$	−41.5410	−1.44452	−0.722261	−	0.691620i	$-0.756897\pi$
$827$	−41.5410	−1.44452	−0.722261	+	0.691620i	$0.756897\pi$
$828$	75.3951	2.62016
$829$	7.63932	0.265325	0.132662	−	0.991161i	$-0.457647\pi$
$829$	7.63932	0.265325	0.132662	+	0.991161i	$0.457647\pi$
$830$	0	0
$831$	−64.3607	−2.23265
$832$	−0.291796	−0.0101162
$833$	0	0
$834$	21.3050	0.737730
$835$	0	0
$836$	1.70820	0.0590795
$837$	53.6656	1.85496
$838$	16.1803	0.558941
$839$	30.6525	1.05824	0.529120	−	0.848547i	$-0.322522\pi$
$839$	30.6525	1.05824	0.529120	+	0.848547i	$0.322522\pi$
$840$	0	0
$841$	−4.00000	−0.137931
$842$	−8.03444	−0.276885
$843$	49.7082	1.71204
$844$	−19.4164	−0.668340
$845$	0	0
$846$	−9.23607	−0.317543
$847$	0	0
$848$	−15.7082	−0.539422
$849$	−56.3607	−1.93429
$850$	0	0
$851$	18.7082	0.641309
$852$	45.5967	1.56212
$853$	27.4164	0.938720	0.469360	−	0.883007i	$-0.344485\pi$
$853$	27.4164	0.938720	0.469360	+	0.883007i	$0.344485\pi$
$854$	0	0
$855$	0	0
$856$	17.8885	0.611418
$857$	−15.8197	−0.540389	−0.270195	−	0.962806i	$-0.587088\pi$
$857$	−15.8197	−0.540389	−0.270195	+	0.962806i	$0.587088\pi$
$858$	0.583592	0.0199235
$859$	−22.3607	−0.762937	−0.381468	−	0.924382i	$-0.624581\pi$
$859$	−22.3607	−0.762937	−0.381468	+	0.924382i	$0.624581\pi$
$860$	0	0
$861$	0	0
$862$	10.8328	0.368967
$863$	−18.3475	−0.624557	−0.312278	−	0.949991i	$-0.601092\pi$
$863$	−18.3475	−0.624557	−0.312278	+	0.949991i	$0.601092\pi$
$864$	−81.3050	−2.76605
$865$	0	0
$866$	−17.5279	−0.595621
$867$	35.2361	1.19668
$868$	0	0
$869$	2.63932	0.0895328
$870$	0	0
$871$	5.23607	0.177417
$872$	−18.8197	−0.637314
$873$	39.1246	1.32417
$874$	−17.2361	−0.583019
$875$	0	0
$876$	−45.8885	−1.55043
$877$	−30.3607	−1.02521	−0.512604	−	0.858625i	$-0.671319\pi$
$877$	−30.3607	−1.02521	−0.512604	+	0.858625i	$0.671319\pi$
$878$	5.12461	0.172947
$879$	100.721	3.39725
$880$	0	0
$881$	−5.81966	−0.196069	−0.0980347	−	0.995183i	$-0.531256\pi$
$881$	−5.81966	−0.196069	−0.0980347	+	0.995183i	$0.531256\pi$
$882$	0	0
$883$	1.40325	0.0472232	0.0236116	−	0.999721i	$-0.492483\pi$
$883$	1.40325	0.0472232	0.0236116	+	0.999721i	$0.492483\pi$
$884$	−4.94427	−0.166294
$885$	0	0
$886$	12.0000	0.403148
$887$	−21.3475	−0.716780	−0.358390	−	0.933572i	$-0.616674\pi$
$887$	−21.3475	−0.716780	−0.358390	+	0.933572i	$0.616674\pi$
$888$	−21.7082	−0.728480
$889$	0	0
$890$	0	0
$891$	−5.76393	−0.193099
$892$	−32.6525	−1.09329
$893$	−8.94427	−0.299309
$894$	−7.88854	−0.263832
$895$	0	0
$896$	0	0
$897$	24.9443	0.832865
$898$	12.6869	0.423368
$899$	−18.5410	−0.618378
$900$	0	0
$901$	−20.9443	−0.697755
$902$	0.695048	0.0231426
$903$	0	0
$904$	−32.2361	−1.07216
$905$	0	0
$906$	40.4721	1.34460
$907$	−34.8328	−1.15660	−0.578302	−	0.815823i	$-0.696285\pi$
$907$	−34.8328	−1.15660	−0.578302	+	0.815823i	$0.696285\pi$
$908$	−34.6525	−1.14998
$909$	−35.5967	−1.18067
$910$	0	0
$911$	0.819660	0.0271566	0.0135783	−	0.999908i	$-0.495678\pi$
$911$	0.819660	0.0271566	0.0135783	+	0.999908i	$0.495678\pi$
$912$	−26.8328	−0.888523
$913$	1.81966	0.0602220
$914$	7.74265	0.256104
$915$	0	0
$916$	−7.23607	−0.239086
$917$	0	0
$918$	−22.1115	−0.729787
$919$	−27.7639	−0.915848	−0.457924	−	0.888991i	$-0.651407\pi$
$919$	−27.7639	−0.915848	−0.457924	+	0.888991i	$0.651407\pi$
$920$	0	0
$921$	−14.8328	−0.488758
$922$	8.76393	0.288625
$923$	10.7639	0.354299
$924$	0	0
$925$	0	0
$926$	8.58359	0.282074
$927$	63.3050	2.07921
$928$	28.0902	0.922105
$929$	−38.2918	−1.25631	−0.628157	−	0.778087i	$-0.716190\pi$
$929$	−38.2918	−1.25631	−0.628157	+	0.778087i	$0.716190\pi$
$930$	0	0
$931$	0	0
$932$	12.8541	0.421050
$933$	78.8328	2.58087
$934$	4.29180	0.140432
$935$	0	0
$936$	−20.6525	−0.675047
$937$	35.2361	1.15111	0.575556	−	0.817762i	$-0.304786\pi$
$937$	35.2361	1.15111	0.575556	+	0.817762i	$0.304786\pi$
$938$	0	0
$939$	−63.1935	−2.06224
$940$	0	0
$941$	5.23607	0.170691	0.0853455	−	0.996351i	$-0.472801\pi$
$941$	5.23607	0.170691	0.0853455	+	0.996351i	$0.472801\pi$
$942$	−1.52786	−0.0497805
$943$	29.7082	0.967432
$944$	−21.7082	−0.706542
$945$	0	0
$946$	0.257354	0.00836731
$947$	−34.8328	−1.13191	−0.565957	−	0.824435i	$-0.691493\pi$
$947$	−34.8328	−1.13191	−0.565957	+	0.824435i	$0.691493\pi$
$948$	−58.5410	−1.90132
$949$	−10.8328	−0.351648
$950$	0	0
$951$	82.0689	2.66127
$952$	0	0
$953$	−3.47214	−0.112474	−0.0562368	−	0.998417i	$-0.517910\pi$
$953$	−3.47214	−0.112474	−0.0562368	+	0.998417i	$0.517910\pi$
$954$	−39.1246	−1.26671
$955$	0	0
$956$	8.94427	0.289278
$957$	3.81966	0.123472
$958$	16.1803	0.522763
$959$	0	0
$960$	0	0
$961$	−17.2492	−0.556427
$962$	−2.29180	−0.0738905
$963$	−59.7771	−1.92629
$964$	−5.70820	−0.183849
$965$	0	0
$966$	0	0
$967$	14.1115	0.453794	0.226897	−	0.973919i	$-0.427142\pi$
$967$	14.1115	0.453794	0.226897	+	0.973919i	$0.427142\pi$
$968$	24.4721	0.786564
$969$	−35.7771	−1.14933
$970$	0	0
$971$	18.0000	0.577647	0.288824	−	0.957382i	$-0.406736\pi$
$971$	18.0000	0.577647	0.288824	+	0.957382i	$0.406736\pi$
$972$	57.5967	1.84742
$973$	0	0
$974$	−3.56231	−0.114144
$975$	0	0
$976$	18.0000	0.576166
$977$	11.4721	0.367026	0.183513	−	0.983017i	$-0.441253\pi$
$977$	11.4721	0.367026	0.183513	+	0.983017i	$0.441253\pi$
$978$	−3.05573	−0.0977114
$979$	4.06888	0.130042
$980$	0	0
$981$	62.8885	2.00788
$982$	−3.56231	−0.113678
$983$	−34.5410	−1.10169	−0.550844	−	0.834608i	$-0.685694\pi$
$983$	−34.5410	−1.10169	−0.550844	+	0.834608i	$0.685694\pi$
$984$	−34.4721	−1.09893
$985$	0	0
$986$	7.63932	0.243286
$987$	0	0
$988$	−8.94427	−0.284555
$989$	11.0000	0.349780
$990$	0	0
$991$	13.1803	0.418687	0.209344	−	0.977842i	$-0.432867\pi$
$991$	13.1803	0.418687	0.209344	+	0.977842i	$0.432867\pi$
$992$	−20.8328	−0.661443
$993$	79.9574	2.53737
$994$	0	0
$995$	0	0
$996$	−40.3607	−1.27888
$997$	−45.4164	−1.43835	−0.719176	−	0.694828i	$-0.755481\pi$
$997$	−45.4164	−1.43835	−0.719176	+	0.694828i	$0.755481\pi$
$998$	6.83282	0.216289
$999$	43.4164	1.37363

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1225.2.a.n.1.2		2
5.2	odd	4		1225.2.b.k.99.3		4
5.3	odd	4		1225.2.b.k.99.2		4
5.4	even	2		1225.2.a.u.1.1		2
7.6	odd	2		175.2.a.d.1.2	✓	2
21.20	even	2		1575.2.a.s.1.1		2
28.27	even	2		2800.2.a.bh.1.1		2
35.13	even	4		175.2.b.c.99.2		4
35.27	even	4		175.2.b.c.99.3		4
35.34	odd	2		175.2.a.e.1.1	yes	2
105.62	odd	4		1575.2.d.k.1324.2		4
105.83	odd	4		1575.2.d.k.1324.3		4
105.104	even	2		1575.2.a.n.1.2		2
140.27	odd	4		2800.2.g.s.449.4		4
140.83	odd	4		2800.2.g.s.449.1		4
140.139	even	2		2800.2.a.bp.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
175.2.a.d.1.2	✓	2	7.6	odd	2
175.2.a.e.1.1	yes	2	35.34	odd	2
175.2.b.c.99.2		4	35.13	even	4
175.2.b.c.99.3		4	35.27	even	4
1225.2.a.n.1.2		2	1.1	even	1	trivial
1225.2.a.u.1.1		2	5.4	even	2
1225.2.b.k.99.2		4	5.3	odd	4
1225.2.b.k.99.3		4	5.2	odd	4
1575.2.a.n.1.2		2	105.104	even	2
1575.2.a.s.1.1		2	21.20	even	2
1575.2.d.k.1324.2		4	105.62	odd	4
1575.2.d.k.1324.3		4	105.83	odd	4
2800.2.a.bh.1.1		2	28.27	even	2
2800.2.a.bp.1.2		2	140.139	even	2
2800.2.g.s.449.1		4	140.83	odd	4
2800.2.g.s.449.4		4	140.27	odd	4

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:	\( N \)	\(=\)	\( 1225 = 5^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1225.a (trivial)

\(f(q)\)	\(=\)	\(q+0.618034 q^{2} -3.23607 q^{3} -1.61803 q^{4} -2.00000 q^{6} -2.23607 q^{8} +7.47214 q^{9} +O(q^{10})\)	\(q+0.618034 q^{2} -3.23607 q^{3} -1.61803 q^{4} -2.00000 q^{6} -2.23607 q^{8} +7.47214 q^{9} -0.236068 q^{11} +5.23607 q^{12} +1.23607 q^{13} +1.85410 q^{16} +2.47214 q^{17} +4.61803 q^{18} +4.47214 q^{19} -0.145898 q^{22} -6.23607 q^{23} +7.23607 q^{24} +0.763932 q^{26} -14.4721 q^{27} +5.00000 q^{29} -3.70820 q^{31} +5.61803 q^{32} +0.763932 q^{33} +1.52786 q^{34} -12.0902 q^{36} -3.00000 q^{37} +2.76393 q^{38} -4.00000 q^{39} -4.76393 q^{41} -1.76393 q^{43} +0.381966 q^{44} -3.85410 q^{46} -2.00000 q^{47} -6.00000 q^{48} -8.00000 q^{51} -2.00000 q^{52} -8.47214 q^{53} -8.94427 q^{54} -14.4721 q^{57} +3.09017 q^{58} -11.7082 q^{59} +9.70820 q^{61} -2.29180 q^{62} -0.236068 q^{64} +0.472136 q^{66} +4.23607 q^{67} -4.00000 q^{68} +20.1803 q^{69} +8.70820 q^{71} -16.7082 q^{72} -8.76393 q^{73} -1.85410 q^{74} -7.23607 q^{76} -2.47214 q^{78} -11.1803 q^{79} +24.4164 q^{81} -2.94427 q^{82} -7.70820 q^{83} -1.09017 q^{86} -16.1803 q^{87} +0.527864 q^{88} -17.2361 q^{89} +10.0902 q^{92} +12.0000 q^{93} -1.23607 q^{94} -18.1803 q^{96} +5.23607 q^{97} -1.76393 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{2} - 2 q^{3} - q^{4} - 4 q^{6} + 6 q^{9}+O(q^{10}) \) 2 * q - q^2 - 2 * q^3 - q^4 - 4 * q^6 + 6 * q^9	\( 2 q - q^{2} - 2 q^{3} - q^{4} - 4 q^{6} + 6 q^{9} + 4 q^{11} + 6 q^{12} - 2 q^{13} - 3 q^{16} - 4 q^{17} + 7 q^{18} - 7 q^{22} - 8 q^{23} + 10 q^{24} + 6 q^{26} - 20 q^{27} + 10 q^{29} + 6 q^{31} + 9 q^{32} + 6 q^{33} + 12 q^{34} - 13 q^{36} - 6 q^{37} + 10 q^{38} - 8 q^{39} - 14 q^{41} - 8 q^{43} + 3 q^{44} - q^{46} - 4 q^{47} - 12 q^{48} - 16 q^{51} - 4 q^{52} - 8 q^{53} - 20 q^{57} - 5 q^{58} - 10 q^{59} + 6 q^{61} - 18 q^{62} + 4 q^{64} - 8 q^{66} + 4 q^{67} - 8 q^{68} + 18 q^{69} + 4 q^{71} - 20 q^{72} - 22 q^{73} + 3 q^{74} - 10 q^{76} + 4 q^{78} + 22 q^{81} + 12 q^{82} - 2 q^{83} + 9 q^{86} - 10 q^{87} + 10 q^{88} - 30 q^{89} + 9 q^{92} + 24 q^{93} + 2 q^{94} - 14 q^{96} + 6 q^{97} - 8 q^{99}+O(q^{100}) \) 2 * q - q^2 - 2 * q^3 - q^4 - 4 * q^6 + 6 * q^9 + 4 * q^11 + 6 * q^12 - 2 * q^13 - 3 * q^16 - 4 * q^17 + 7 * q^18 - 7 * q^22 - 8 * q^23 + 10 * q^24 + 6 * q^26 - 20 * q^27 + 10 * q^29 + 6 * q^31 + 9 * q^32 + 6 * q^33 + 12 * q^34 - 13 * q^36 - 6 * q^37 + 10 * q^38 - 8 * q^39 - 14 * q^41 - 8 * q^43 + 3 * q^44 - q^46 - 4 * q^47 - 12 * q^48 - 16 * q^51 - 4 * q^52 - 8 * q^53 - 20 * q^57 - 5 * q^58 - 10 * q^59 + 6 * q^61 - 18 * q^62 + 4 * q^64 - 8 * q^66 + 4 * q^67 - 8 * q^68 + 18 * q^69 + 4 * q^71 - 20 * q^72 - 22 * q^73 + 3 * q^74 - 10 * q^76 + 4 * q^78 + 22 * q^81 + 12 * q^82 - 2 * q^83 + 9 * q^86 - 10 * q^87 + 10 * q^88 - 30 * q^89 + 9 * q^92 + 24 * q^93 + 2 * q^94 - 14 * q^96 + 6 * q^97 - 8 * q^99

Introduction

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