LMFDB - Embedded newform 1225.2.a.m.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:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 35)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.41421$ 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)

$f(q)$	$=$	$q+0.414214 q^{2} +2.41421 q^{3} -1.82843 q^{4} +1.00000 q^{6} -1.58579 q^{8} +2.82843 q^{9} +O(q^{10})$	\(q+0.414214 q^{2} +2.41421 q^{3} -1.82843 q^{4} +1.00000 q^{6} -1.58579 q^{8} +2.82843 q^{9} +4.82843 q^{11} -4.41421 q^{12} +0.828427 q^{13} +3.00000 q^{16} -0.828427 q^{17} +1.17157 q^{18} +2.82843 q^{19} +2.00000 q^{22} +2.41421 q^{23} -3.82843 q^{24} +0.343146 q^{26} -0.414214 q^{27} -1.00000 q^{29} +6.00000 q^{31} +4.41421 q^{32} +11.6569 q^{33} -0.343146 q^{34} -5.17157 q^{36} +1.17157 q^{38} +2.00000 q^{39} +2.17157 q^{41} -6.41421 q^{43} -8.82843 q^{44} +1.00000 q^{46} +2.00000 q^{47} +7.24264 q^{48} -2.00000 q^{51} -1.51472 q^{52} +6.82843 q^{53} -0.171573 q^{54} +6.82843 q^{57} -0.414214 q^{58} +12.4853 q^{59} +11.4853 q^{61} +2.48528 q^{62} -4.17157 q^{64} +4.82843 q^{66} -12.4142 q^{67} +1.51472 q^{68} +5.82843 q^{69} -12.4853 q^{71} -4.48528 q^{72} +4.82843 q^{73} -5.17157 q^{76} +0.828427 q^{78} +9.17157 q^{79} -9.48528 q^{81} +0.899495 q^{82} -11.7279 q^{83} -2.65685 q^{86} -2.41421 q^{87} -7.65685 q^{88} -2.65685 q^{89} -4.41421 q^{92} +14.4853 q^{93} +0.828427 q^{94} +10.6569 q^{96} +0.343146 q^{97} +13.6569 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} + 2 q^{6} - 6 q^{8}+O(q^{10}) $ 2 * q - 2 * q^2 + 2 * q^3 + 2 * q^4 + 2 * q^6 - 6 * q^8	$ 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} + 2 q^{6} - 6 q^{8} + 4 q^{11} - 6 q^{12} - 4 q^{13} + 6 q^{16} + 4 q^{17} + 8 q^{18} + 4 q^{22} + 2 q^{23} - 2 q^{24} + 12 q^{26} + 2 q^{27} - 2 q^{29} + 12 q^{31} + 6 q^{32} + 12 q^{33} - 12 q^{34} - 16 q^{36} + 8 q^{38} + 4 q^{39} + 10 q^{41} - 10 q^{43} - 12 q^{44} + 2 q^{46} + 4 q^{47} + 6 q^{48} - 4 q^{51} - 20 q^{52} + 8 q^{53} - 6 q^{54} + 8 q^{57} + 2 q^{58} + 8 q^{59} + 6 q^{61} - 12 q^{62} - 14 q^{64} + 4 q^{66} - 22 q^{67} + 20 q^{68} + 6 q^{69} - 8 q^{71} + 8 q^{72} + 4 q^{73} - 16 q^{76} - 4 q^{78} + 24 q^{79} - 2 q^{81} - 18 q^{82} + 2 q^{83} + 6 q^{86} - 2 q^{87} - 4 q^{88} + 6 q^{89} - 6 q^{92} + 12 q^{93} - 4 q^{94} + 10 q^{96} + 12 q^{97} + 16 q^{99}+O(q^{100}) $ 2 * q - 2 * q^2 + 2 * q^3 + 2 * q^4 + 2 * q^6 - 6 * q^8 + 4 * q^11 - 6 * q^12 - 4 * q^13 + 6 * q^16 + 4 * q^17 + 8 * q^18 + 4 * q^22 + 2 * q^23 - 2 * q^24 + 12 * q^26 + 2 * q^27 - 2 * q^29 + 12 * q^31 + 6 * q^32 + 12 * q^33 - 12 * q^34 - 16 * q^36 + 8 * q^38 + 4 * q^39 + 10 * q^41 - 10 * q^43 - 12 * q^44 + 2 * q^46 + 4 * q^47 + 6 * q^48 - 4 * q^51 - 20 * q^52 + 8 * q^53 - 6 * q^54 + 8 * q^57 + 2 * q^58 + 8 * q^59 + 6 * q^61 - 12 * q^62 - 14 * q^64 + 4 * q^66 - 22 * q^67 + 20 * q^68 + 6 * q^69 - 8 * q^71 + 8 * q^72 + 4 * q^73 - 16 * q^76 - 4 * q^78 + 24 * q^79 - 2 * q^81 - 18 * q^82 + 2 * q^83 + 6 * q^86 - 2 * q^87 - 4 * q^88 + 6 * q^89 - 6 * q^92 + 12 * q^93 - 4 * q^94 + 10 * q^96 + 12 * q^97 + 16 * 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 $

$2$	0.414214	0.292893	0.146447	−	0.989219i	$-0.453216\pi$
$2$	0.414214	0.292893	0.146447	+	0.989219i	$0.453216\pi$
$3$	2.41421	1.39385	0.696923	−	0.717146i	$-0.254552\pi$
$3$	2.41421	1.39385	0.696923	+	0.717146i	$0.254552\pi$
$4$	−1.82843	−0.914214
$5$	0	0
$5$	0	0
$6$	1.00000	0.408248
$7$	0	0
$7$	0	0
$8$	−1.58579	−0.560660
$9$	2.82843	0.942809
$10$	0	0
$11$	4.82843	1.45583	0.727913	−	0.685670i	$-0.240491\pi$
$11$	4.82843	1.45583	0.727913	+	0.685670i	$0.240491\pi$
$12$	−4.41421	−1.27427
$13$	0.828427	0.229764	0.114882	−	0.993379i	$-0.463351\pi$
$13$	0.828427	0.229764	0.114882	+	0.993379i	$0.463351\pi$
$14$	0	0
$15$	0	0
$16$	3.00000	0.750000
$17$	−0.828427	−0.200923	−0.100462	−	0.994941i	$-0.532032\pi$
$17$	−0.828427	−0.200923	−0.100462	+	0.994941i	$0.532032\pi$
$18$	1.17157	0.276142
$19$	2.82843	0.648886	0.324443	−	0.945905i	$-0.394823\pi$
$19$	2.82843	0.648886	0.324443	+	0.945905i	$0.394823\pi$
$20$	0	0
$21$	0	0
$22$	2.00000	0.426401
$23$	2.41421	0.503398	0.251699	−	0.967806i	$-0.419011\pi$
$23$	2.41421	0.503398	0.251699	+	0.967806i	$0.419011\pi$
$24$	−3.82843	−0.781474
$25$	0	0
$26$	0.343146	0.0672964
$27$	−0.414214	−0.0797154
$28$	0	0
$29$	−1.00000	−0.185695	−0.0928477	−	0.995680i	$-0.529597\pi$
$29$	−1.00000	−0.185695	−0.0928477	+	0.995680i	$0.529597\pi$
$30$	0	0
$31$	6.00000	1.07763	0.538816	−	0.842424i	$-0.318872\pi$
$31$	6.00000	1.07763	0.538816	+	0.842424i	$0.318872\pi$
$32$	4.41421	0.780330
$33$	11.6569	2.02920
$34$	−0.343146	−0.0588490
$35$	0	0
$36$	−5.17157	−0.861929
$37$	0	0		−	1.00000i	$-0.5\pi$
$37$	0	0			1.00000i	$0.5\pi$
$38$	1.17157	0.190054
$39$	2.00000	0.320256
$40$	0	0
$41$	2.17157	0.339143	0.169571	−	0.985518i	$-0.445762\pi$
$41$	2.17157	0.339143	0.169571	+	0.985518i	$0.445762\pi$
$42$	0	0
$43$	−6.41421	−0.978158	−0.489079	−	0.872239i	$-0.662667\pi$
$43$	−6.41421	−0.978158	−0.489079	+	0.872239i	$0.662667\pi$
$44$	−8.82843	−1.33094
$45$	0	0
$46$	1.00000	0.147442
$47$	2.00000	0.291730	0.145865	−	0.989305i	$-0.453403\pi$
$47$	2.00000	0.291730	0.145865	+	0.989305i	$0.453403\pi$
$48$	7.24264	1.04539
$49$	0	0
$50$	0	0
$51$	−2.00000	−0.280056
$52$	−1.51472	−0.210054
$53$	6.82843	0.937957	0.468978	−	0.883210i	$-0.344622\pi$
$53$	6.82843	0.937957	0.468978	+	0.883210i	$0.344622\pi$
$54$	−0.171573	−0.0233481
$55$	0	0
$56$	0	0
$57$	6.82843	0.904447
$58$	−0.414214	−0.0543889
$59$	12.4853	1.62545	0.812723	−	0.582651i	$-0.197984\pi$
$59$	12.4853	1.62545	0.812723	+	0.582651i	$0.197984\pi$
$60$	0	0
$61$	11.4853	1.47054	0.735270	−	0.677775i	$-0.237055\pi$
$61$	11.4853	1.47054	0.735270	+	0.677775i	$0.237055\pi$
$62$	2.48528	0.315631
$63$	0	0
$64$	−4.17157	−0.521447
$65$	0	0
$66$	4.82843	0.594338
$67$	−12.4142	−1.51664	−0.758319	−	0.651884i	$-0.773979\pi$
$67$	−12.4142	−1.51664	−0.758319	+	0.651884i	$0.773979\pi$
$68$	1.51472	0.183687
$69$	5.82843	0.701660
$70$	0	0
$71$	−12.4853	−1.48173	−0.740865	−	0.671654i	$-0.765584\pi$
$71$	−12.4853	−1.48173	−0.740865	+	0.671654i	$0.765584\pi$
$72$	−4.48528	−0.528595
$73$	4.82843	0.565125	0.282562	−	0.959249i	$-0.408816\pi$
$73$	4.82843	0.565125	0.282562	+	0.959249i	$0.408816\pi$
$74$	0	0
$75$	0	0
$76$	−5.17157	−0.593220
$77$	0	0
$78$	0.828427	0.0938009
$79$	9.17157	1.03188	0.515941	−	0.856624i	$-0.327442\pi$
$79$	9.17157	1.03188	0.515941	+	0.856624i	$0.327442\pi$
$80$	0	0
$81$	−9.48528	−1.05392
$82$	0.899495	0.0993326
$83$	−11.7279	−1.28731	−0.643653	−	0.765317i	$-0.722582\pi$
$83$	−11.7279	−1.28731	−0.643653	+	0.765317i	$0.722582\pi$
$84$	0	0
$85$	0	0
$86$	−2.65685	−0.286496
$87$	−2.41421	−0.258831
$88$	−7.65685	−0.816223
$89$	−2.65685	−0.281626	−0.140813	−	0.990036i	$-0.544972\pi$
$89$	−2.65685	−0.281626	−0.140813	+	0.990036i	$0.544972\pi$
$90$	0	0
$91$	0	0
$92$	−4.41421	−0.460214
$93$	14.4853	1.50205
$94$	0.828427	0.0854457
$95$	0	0
$96$	10.6569	1.08766
$97$	0.343146	0.0348412	0.0174206	−	0.999848i	$-0.494455\pi$
$97$	0.343146	0.0348412	0.0174206	+	0.999848i	$0.494455\pi$
$98$	0	0
$99$	13.6569	1.37257
$100$	0	0
$101$	−12.3137	−1.22526	−0.612630	−	0.790370i	$-0.709888\pi$
$101$	−12.3137	−1.22526	−0.612630	+	0.790370i	$0.709888\pi$
$102$	−0.828427	−0.0820265
$103$	0.414214	0.0408137	0.0204068	−	0.999792i	$-0.493504\pi$
$103$	0.414214	0.0408137	0.0204068	+	0.999792i	$0.493504\pi$
$104$	−1.31371	−0.128820
$105$	0	0
$106$	2.82843	0.274721
$107$	−2.75736	−0.266564	−0.133282	−	0.991078i	$-0.542552\pi$
$107$	−2.75736	−0.266564	−0.133282	+	0.991078i	$0.542552\pi$
$108$	0.757359	0.0728769
$109$	3.48528	0.333829	0.166915	−	0.985971i	$-0.446620\pi$
$109$	3.48528	0.333829	0.166915	+	0.985971i	$0.446620\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−12.4853	−1.17452	−0.587258	−	0.809400i	$-0.699793\pi$
$113$	−12.4853	−1.17452	−0.587258	+	0.809400i	$0.699793\pi$
$114$	2.82843	0.264906
$115$	0	0
$116$	1.82843	0.169765
$117$	2.34315	0.216624
$118$	5.17157	0.476082
$119$	0	0
$120$	0	0
$121$	12.3137	1.11943
$122$	4.75736	0.430711
$123$	5.24264	0.472713
$124$	−10.9706	−0.985186
$125$	0	0
$126$	0	0
$127$	−13.3137	−1.18140	−0.590700	−	0.806891i	$-0.701148\pi$
$127$	−13.3137	−1.18140	−0.590700	+	0.806891i	$0.701148\pi$
$128$	−10.5563	−0.933058
$129$	−15.4853	−1.36340
$130$	0	0
$131$	−3.31371	−0.289520	−0.144760	−	0.989467i	$-0.546241\pi$
$131$	−3.31371	−0.289520	−0.144760	+	0.989467i	$0.546241\pi$
$132$	−21.3137	−1.85512
$133$	0	0
$134$	−5.14214	−0.444213
$135$	0	0
$136$	1.31371	0.112650
$137$	−1.65685	−0.141555	−0.0707773	−	0.997492i	$-0.522548\pi$
$137$	−1.65685	−0.141555	−0.0707773	+	0.997492i	$0.522548\pi$
$138$	2.41421	0.205512
$139$	−12.1421	−1.02988	−0.514941	−	0.857225i	$-0.672186\pi$
$139$	−12.1421	−1.02988	−0.514941	+	0.857225i	$0.672186\pi$
$140$	0	0
$141$	4.82843	0.406627
$142$	−5.17157	−0.433989
$143$	4.00000	0.334497
$144$	8.48528	0.707107
$145$	0	0
$146$	2.00000	0.165521
$147$	0	0
$148$	0	0
$149$	−7.82843	−0.641330	−0.320665	−	0.947193i	$-0.603906\pi$
$149$	−7.82843	−0.641330	−0.320665	+	0.947193i	$0.603906\pi$
$150$	0	0
$151$	0.343146	0.0279248	0.0139624	−	0.999903i	$-0.495555\pi$
$151$	0.343146	0.0279248	0.0139624	+	0.999903i	$0.495555\pi$
$152$	−4.48528	−0.363804
$153$	−2.34315	−0.189432
$154$	0	0
$155$	0	0
$156$	−3.65685	−0.292783
$157$	−5.31371	−0.424080	−0.212040	−	0.977261i	$-0.568011\pi$
$157$	−5.31371	−0.424080	−0.212040	+	0.977261i	$0.568011\pi$
$158$	3.79899	0.302231
$159$	16.4853	1.30737
$160$	0	0
$161$	0	0
$162$	−3.92893	−0.308686
$163$	−23.6569	−1.85295	−0.926474	−	0.376359i	$-0.877176\pi$
$163$	−23.6569	−1.85295	−0.926474	+	0.376359i	$0.877176\pi$
$164$	−3.97056	−0.310049
$165$	0	0
$166$	−4.85786	−0.377043
$167$	19.5858	1.51559	0.757797	−	0.652491i	$-0.226276\pi$
$167$	19.5858	1.51559	0.757797	+	0.652491i	$0.226276\pi$
$168$	0	0
$169$	−12.3137	−0.947208
$170$	0	0
$171$	8.00000	0.611775
$172$	11.7279	0.894246
$173$	−19.3137	−1.46839	−0.734197	−	0.678936i	$-0.762441\pi$
$173$	−19.3137	−1.46839	−0.734197	+	0.678936i	$0.762441\pi$
$174$	−1.00000	−0.0758098
$175$	0	0
$176$	14.4853	1.09187
$177$	30.1421	2.26562
$178$	−1.10051	−0.0824863
$179$	−10.0000	−0.747435	−0.373718	−	0.927543i	$-0.621917\pi$
$179$	−10.0000	−0.747435	−0.373718	+	0.927543i	$0.621917\pi$
$180$	0	0
$181$	8.65685	0.643459	0.321729	−	0.946832i	$-0.395736\pi$
$181$	8.65685	0.643459	0.321729	+	0.946832i	$0.395736\pi$
$182$	0	0
$183$	27.7279	2.04971
$184$	−3.82843	−0.282235
$185$	0	0
$186$	6.00000	0.439941
$187$	−4.00000	−0.292509
$188$	−3.65685	−0.266704
$189$	0	0
$190$	0	0
$191$	−7.17157	−0.518917	−0.259458	−	0.965754i	$-0.583544\pi$
$191$	−7.17157	−0.518917	−0.259458	+	0.965754i	$0.583544\pi$
$192$	−10.0711	−0.726817
$193$	2.00000	0.143963	0.0719816	−	0.997406i	$-0.477068\pi$
$193$	2.00000	0.143963	0.0719816	+	0.997406i	$0.477068\pi$
$194$	0.142136	0.0102047
$195$	0	0
$196$	0	0
$197$	23.6569	1.68548	0.842741	−	0.538320i	$-0.180941\pi$
$197$	23.6569	1.68548	0.842741	+	0.538320i	$0.180941\pi$
$198$	5.65685	0.402015
$199$	−1.65685	−0.117451	−0.0587256	−	0.998274i	$-0.518704\pi$
$199$	−1.65685	−0.117451	−0.0587256	+	0.998274i	$0.518704\pi$
$200$	0	0
$201$	−29.9706	−2.11396
$202$	−5.10051	−0.358870
$203$	0	0
$204$	3.65685	0.256031
$205$	0	0
$206$	0.171573	0.0119540
$207$	6.82843	0.474608
$208$	2.48528	0.172323
$209$	13.6569	0.944664
$210$	0	0
$211$	3.51472	0.241963	0.120982	−	0.992655i	$-0.461396\pi$
$211$	3.51472	0.241963	0.120982	+	0.992655i	$0.461396\pi$
$212$	−12.4853	−0.857493
$213$	−30.1421	−2.06531
$214$	−1.14214	−0.0780748
$215$	0	0
$216$	0.656854	0.0446933
$217$	0	0
$218$	1.44365	0.0977764
$219$	11.6569	0.787697
$220$	0	0
$221$	−0.686292	−0.0461650
$222$	0	0
$223$	11.6569	0.780601	0.390300	−	0.920688i	$-0.372371\pi$
$223$	11.6569	0.780601	0.390300	+	0.920688i	$0.372371\pi$
$224$	0	0
$225$	0	0
$226$	−5.17157	−0.344008
$227$	26.9706	1.79010	0.895050	−	0.445967i	$-0.147140\pi$
$227$	26.9706	1.79010	0.895050	+	0.445967i	$0.147140\pi$
$228$	−12.4853	−0.826858
$229$	0.343146	0.0226757	0.0113379	−	0.999936i	$-0.496391\pi$
$229$	0.343146	0.0226757	0.0113379	+	0.999936i	$0.496391\pi$
$230$	0	0
$231$	0	0
$232$	1.58579	0.104112
$233$	11.1716	0.731874	0.365937	−	0.930640i	$-0.380749\pi$
$233$	11.1716	0.731874	0.365937	+	0.930640i	$0.380749\pi$
$234$	0.970563	0.0634477
$235$	0	0
$236$	−22.8284	−1.48600
$237$	22.1421	1.43829
$238$	0	0
$239$	1.31371	0.0849767	0.0424884	−	0.999097i	$-0.486471\pi$
$239$	1.31371	0.0849767	0.0424884	+	0.999097i	$0.486471\pi$
$240$	0	0
$241$	−16.3431	−1.05275	−0.526377	−	0.850251i	$-0.676450\pi$
$241$	−16.3431	−1.05275	−0.526377	+	0.850251i	$0.676450\pi$
$242$	5.10051	0.327873
$243$	−21.6569	−1.38929
$244$	−21.0000	−1.34439
$245$	0	0
$246$	2.17157	0.138454
$247$	2.34315	0.149091
$248$	−9.51472	−0.604185
$249$	−28.3137	−1.79431
$250$	0	0
$251$	−13.3137	−0.840354	−0.420177	−	0.907442i	$-0.638032\pi$
$251$	−13.3137	−0.840354	−0.420177	+	0.907442i	$0.638032\pi$
$252$	0	0
$253$	11.6569	0.732860
$254$	−5.51472	−0.346024
$255$	0	0
$256$	3.97056	0.248160
$257$	17.6569	1.10140	0.550702	−	0.834702i	$-0.314360\pi$
$257$	17.6569	1.10140	0.550702	+	0.834702i	$0.314360\pi$
$258$	−6.41421	−0.399331
$259$	0	0
$260$	0	0
$261$	−2.82843	−0.175075
$262$	−1.37258	−0.0847985
$263$	−19.0416	−1.17416	−0.587079	−	0.809530i	$-0.699722\pi$
$263$	−19.0416	−1.17416	−0.587079	+	0.809530i	$0.699722\pi$
$264$	−18.4853	−1.13769
$265$	0	0
$266$	0	0
$267$	−6.41421	−0.392543
$268$	22.6985	1.38653
$269$	30.4558	1.85693	0.928463	−	0.371425i	$-0.121131\pi$
$269$	30.4558	1.85693	0.928463	+	0.371425i	$0.121131\pi$
$270$	0	0
$271$	0.485281	0.0294787	0.0147394	−	0.999891i	$-0.495308\pi$
$271$	0.485281	0.0294787	0.0147394	+	0.999891i	$0.495308\pi$
$272$	−2.48528	−0.150692
$273$	0	0
$274$	−0.686292	−0.0414604
$275$	0	0
$276$	−10.6569	−0.641467
$277$	12.1421	0.729550	0.364775	−	0.931096i	$-0.381146\pi$
$277$	12.1421	0.729550	0.364775	+	0.931096i	$0.381146\pi$
$278$	−5.02944	−0.301646
$279$	16.9706	1.01600
$280$	0	0
$281$	26.2843	1.56799	0.783994	−	0.620768i	$-0.213179\pi$
$281$	26.2843	1.56799	0.783994	+	0.620768i	$0.213179\pi$
$282$	2.00000	0.119098
$283$	14.0000	0.832214	0.416107	−	0.909316i	$-0.363394\pi$
$283$	14.0000	0.832214	0.416107	+	0.909316i	$0.363394\pi$
$284$	22.8284	1.35462
$285$	0	0
$286$	1.65685	0.0979718
$287$	0	0
$288$	12.4853	0.735702
$289$	−16.3137	−0.959630
$290$	0	0
$291$	0.828427	0.0485633
$292$	−8.82843	−0.516645
$293$	−16.0000	−0.934730	−0.467365	−	0.884064i	$-0.654797\pi$
$293$	−16.0000	−0.934730	−0.467365	+	0.884064i	$0.654797\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−2.00000	−0.116052
$298$	−3.24264	−0.187841
$299$	2.00000	0.115663
$300$	0	0
$301$	0	0
$302$	0.142136	0.00817899
$303$	−29.7279	−1.70782
$304$	8.48528	0.486664
$305$	0	0
$306$	−0.970563	−0.0554834
$307$	−13.2426	−0.755797	−0.377899	−	0.925847i	$-0.623353\pi$
$307$	−13.2426	−0.755797	−0.377899	+	0.925847i	$0.623353\pi$
$308$	0	0
$309$	1.00000	0.0568880
$310$	0	0
$311$	18.8284	1.06766	0.533831	−	0.845591i	$-0.320752\pi$
$311$	18.8284	1.06766	0.533831	+	0.845591i	$0.320752\pi$
$312$	−3.17157	−0.179555
$313$	−17.6569	−0.998024	−0.499012	−	0.866595i	$-0.666304\pi$
$313$	−17.6569	−0.998024	−0.499012	+	0.866595i	$0.666304\pi$
$314$	−2.20101	−0.124210
$315$	0	0
$316$	−16.7696	−0.943361
$317$	−25.7990	−1.44902	−0.724508	−	0.689267i	$-0.757933\pi$
$317$	−25.7990	−1.44902	−0.724508	+	0.689267i	$0.757933\pi$
$318$	6.82843	0.382919
$319$	−4.82843	−0.270340
$320$	0	0
$321$	−6.65685	−0.371549
$322$	0	0
$323$	−2.34315	−0.130376
$324$	17.3431	0.963508
$325$	0	0
$326$	−9.79899	−0.542716
$327$	8.41421	0.465307
$328$	−3.44365	−0.190144
$329$	0	0
$330$	0	0
$331$	−10.9706	−0.602997	−0.301498	−	0.953467i	$-0.597487\pi$
$331$	−10.9706	−0.602997	−0.301498	+	0.953467i	$0.597487\pi$
$332$	21.4437	1.17687
$333$	0	0
$334$	8.11270	0.443907
$335$	0	0
$336$	0	0
$337$	−14.8284	−0.807756	−0.403878	−	0.914813i	$-0.632338\pi$
$337$	−14.8284	−0.807756	−0.403878	+	0.914813i	$0.632338\pi$
$338$	−5.10051	−0.277431
$339$	−30.1421	−1.63710
$340$	0	0
$341$	28.9706	1.56884
$342$	3.31371	0.179185
$343$	0	0
$344$	10.1716	0.548414
$345$	0	0
$346$	−8.00000	−0.430083
$347$	−22.0711	−1.18484	−0.592418	−	0.805630i	$-0.701827\pi$
$347$	−22.0711	−1.18484	−0.592418	+	0.805630i	$0.701827\pi$
$348$	4.41421	0.236627
$349$	26.6569	1.42691	0.713454	−	0.700702i	$-0.247130\pi$
$349$	26.6569	1.42691	0.713454	+	0.700702i	$0.247130\pi$
$350$	0	0
$351$	−0.343146	−0.0183158
$352$	21.3137	1.13602
$353$	−21.1716	−1.12685	−0.563425	−	0.826168i	$-0.690516\pi$
$353$	−21.1716	−1.12685	−0.563425	+	0.826168i	$0.690516\pi$
$354$	12.4853	0.663585
$355$	0	0
$356$	4.85786	0.257466
$357$	0	0
$358$	−4.14214	−0.218919
$359$	−10.0000	−0.527780	−0.263890	−	0.964553i	$-0.585006\pi$
$359$	−10.0000	−0.527780	−0.263890	+	0.964553i	$0.585006\pi$
$360$	0	0
$361$	−11.0000	−0.578947
$362$	3.58579	0.188465
$363$	29.7279	1.56031
$364$	0	0
$365$	0	0
$366$	11.4853	0.600345
$367$	−11.2426	−0.586861	−0.293431	−	0.955980i	$-0.594797\pi$
$367$	−11.2426	−0.586861	−0.293431	+	0.955980i	$0.594797\pi$
$368$	7.24264	0.377549
$369$	6.14214	0.319747
$370$	0	0
$371$	0	0
$372$	−26.4853	−1.37320
$373$	−12.9706	−0.671590	−0.335795	−	0.941935i	$-0.609005\pi$
$373$	−12.9706	−0.671590	−0.335795	+	0.941935i	$0.609005\pi$
$374$	−1.65685	−0.0856739
$375$	0	0
$376$	−3.17157	−0.163561
$377$	−0.828427	−0.0426662
$378$	0	0
$379$	21.1716	1.08751	0.543755	−	0.839244i	$-0.317002\pi$
$379$	21.1716	1.08751	0.543755	+	0.839244i	$0.317002\pi$
$380$	0	0
$381$	−32.1421	−1.64669
$382$	−2.97056	−0.151987
$383$	16.8995	0.863524	0.431762	−	0.901988i	$-0.357892\pi$
$383$	16.8995	0.863524	0.431762	+	0.901988i	$0.357892\pi$
$384$	−25.4853	−1.30054
$385$	0	0
$386$	0.828427	0.0421658
$387$	−18.1421	−0.922217
$388$	−0.627417	−0.0318523
$389$	12.3431	0.625822	0.312911	−	0.949782i	$-0.398696\pi$
$389$	12.3431	0.625822	0.312911	+	0.949782i	$0.398696\pi$
$390$	0	0
$391$	−2.00000	−0.101144
$392$	0	0
$393$	−8.00000	−0.403547
$394$	9.79899	0.493666
$395$	0	0
$396$	−24.9706	−1.25482
$397$	28.6274	1.43677	0.718384	−	0.695646i	$-0.244882\pi$
$397$	28.6274	1.43677	0.718384	+	0.695646i	$0.244882\pi$
$398$	−0.686292	−0.0344007
$399$	0	0
$400$	0	0
$401$	7.68629	0.383835	0.191918	−	0.981411i	$-0.438529\pi$
$401$	7.68629	0.383835	0.191918	+	0.981411i	$0.438529\pi$
$402$	−12.4142	−0.619165
$403$	4.97056	0.247601
$404$	22.5147	1.12015
$405$	0	0
$406$	0	0
$407$	0	0
$408$	3.17157	0.157016
$409$	−24.7990	−1.22623	−0.613116	−	0.789993i	$-0.710084\pi$
$409$	−24.7990	−1.22623	−0.613116	+	0.789993i	$0.710084\pi$
$410$	0	0
$411$	−4.00000	−0.197305
$412$	−0.757359	−0.0373124
$413$	0	0
$414$	2.82843	0.139010
$415$	0	0
$416$	3.65685	0.179292
$417$	−29.3137	−1.43550
$418$	5.65685	0.276686
$419$	23.3137	1.13895	0.569475	−	0.822009i	$-0.307147\pi$
$419$	23.3137	1.13895	0.569475	+	0.822009i	$0.307147\pi$
$420$	0	0
$421$	−3.48528	−0.169862	−0.0849311	−	0.996387i	$-0.527067\pi$
$421$	−3.48528	−0.169862	−0.0849311	+	0.996387i	$0.527067\pi$
$422$	1.45584	0.0708694
$423$	5.65685	0.275046
$424$	−10.8284	−0.525875
$425$	0	0
$426$	−12.4853	−0.604914
$427$	0	0
$428$	5.04163	0.243696
$429$	9.65685	0.466237
$430$	0	0
$431$	−21.7990	−1.05002	−0.525010	−	0.851096i	$-0.675938\pi$
$431$	−21.7990	−1.05002	−0.525010	+	0.851096i	$0.675938\pi$
$432$	−1.24264	−0.0597866
$433$	−31.7990	−1.52816	−0.764081	−	0.645120i	$-0.776807\pi$
$433$	−31.7990	−1.52816	−0.764081	+	0.645120i	$0.776807\pi$
$434$	0	0
$435$	0	0
$436$	−6.37258	−0.305191
$437$	6.82843	0.326648
$438$	4.82843	0.230711
$439$	−33.9411	−1.61992	−0.809961	−	0.586484i	$-0.800512\pi$
$439$	−33.9411	−1.61992	−0.809961	+	0.586484i	$0.800512\pi$
$440$	0	0
$441$	0	0
$442$	−0.284271	−0.0135214
$443$	12.2132	0.580267	0.290133	−	0.956986i	$-0.406300\pi$
$443$	12.2132	0.580267	0.290133	+	0.956986i	$0.406300\pi$
$444$	0	0
$445$	0	0
$446$	4.82843	0.228633
$447$	−18.8995	−0.893915
$448$	0	0
$449$	−1.82843	−0.0862888	−0.0431444	−	0.999069i	$-0.513738\pi$
$449$	−1.82843	−0.0862888	−0.0431444	+	0.999069i	$0.513738\pi$
$450$	0	0
$451$	10.4853	0.493733
$452$	22.8284	1.07376
$453$	0.828427	0.0389229
$454$	11.1716	0.524308
$455$	0	0
$456$	−10.8284	−0.507088
$457$	32.2843	1.51019	0.755097	−	0.655613i	$-0.227590\pi$
$457$	32.2843	1.51019	0.755097	+	0.655613i	$0.227590\pi$
$458$	0.142136	0.00664156
$459$	0.343146	0.0160167
$460$	0	0
$461$	−18.6863	−0.870307	−0.435154	−	0.900356i	$-0.643306\pi$
$461$	−18.6863	−0.870307	−0.435154	+	0.900356i	$0.643306\pi$
$462$	0	0
$463$	−11.0416	−0.513148	−0.256574	−	0.966525i	$-0.582594\pi$
$463$	−11.0416	−0.513148	−0.256574	+	0.966525i	$0.582594\pi$
$464$	−3.00000	−0.139272
$465$	0	0
$466$	4.62742	0.214361
$467$	−22.8995	−1.05966	−0.529831	−	0.848103i	$-0.677745\pi$
$467$	−22.8995	−1.05966	−0.529831	+	0.848103i	$0.677745\pi$
$468$	−4.28427	−0.198041
$469$	0	0
$470$	0	0
$471$	−12.8284	−0.591103
$472$	−19.7990	−0.911322
$473$	−30.9706	−1.42403
$474$	9.17157	0.421264
$475$	0	0
$476$	0	0
$477$	19.3137	0.884314
$478$	0.544156	0.0248891
$479$	24.3431	1.11227	0.556133	−	0.831093i	$-0.312284\pi$
$479$	24.3431	1.11227	0.556133	+	0.831093i	$0.312284\pi$
$480$	0	0
$481$	0	0
$482$	−6.76955	−0.308345
$483$	0	0
$484$	−22.5147	−1.02340
$485$	0	0
$486$	−8.97056	−0.406913
$487$	−15.6569	−0.709480	−0.354740	−	0.934965i	$-0.615431\pi$
$487$	−15.6569	−0.709480	−0.354740	+	0.934965i	$0.615431\pi$
$488$	−18.2132	−0.824473
$489$	−57.1127	−2.58273
$490$	0	0
$491$	−13.3137	−0.600839	−0.300420	−	0.953807i	$-0.597127\pi$
$491$	−13.3137	−0.600839	−0.300420	+	0.953807i	$0.597127\pi$
$492$	−9.58579	−0.432161
$493$	0.828427	0.0373105
$494$	0.970563	0.0436677
$495$	0	0
$496$	18.0000	0.808224
$497$	0	0
$498$	−11.7279	−0.525541
$499$	4.82843	0.216150	0.108075	−	0.994143i	$-0.465531\pi$
$499$	4.82843	0.216150	0.108075	+	0.994143i	$0.465531\pi$
$500$	0	0
$501$	47.2843	2.11251
$502$	−5.51472	−0.246134
$503$	37.8701	1.68854	0.844271	−	0.535916i	$-0.180034\pi$
$503$	37.8701	1.68854	0.844271	+	0.535916i	$0.180034\pi$
$504$	0	0
$505$	0	0
$506$	4.82843	0.214650
$507$	−29.7279	−1.32026
$508$	24.3431	1.08005
$509$	−24.6569	−1.09290	−0.546448	−	0.837493i	$-0.684020\pi$
$509$	−24.6569	−1.09290	−0.546448	+	0.837493i	$0.684020\pi$
$510$	0	0
$511$	0	0
$512$	22.7574	1.00574
$513$	−1.17157	−0.0517262
$514$	7.31371	0.322594
$515$	0	0
$516$	28.3137	1.24644
$517$	9.65685	0.424708
$518$	0	0
$519$	−46.6274	−2.04672
$520$	0	0
$521$	18.9706	0.831115	0.415558	−	0.909567i	$-0.363586\pi$
$521$	18.9706	0.831115	0.415558	+	0.909567i	$0.363586\pi$
$522$	−1.17157	−0.0512784
$523$	24.3431	1.06445	0.532226	−	0.846602i	$-0.321356\pi$
$523$	24.3431	1.06445	0.532226	+	0.846602i	$0.321356\pi$
$524$	6.05887	0.264683
$525$	0	0
$526$	−7.88730	−0.343903
$527$	−4.97056	−0.216521
$528$	34.9706	1.52190
$529$	−17.1716	−0.746590
$530$	0	0
$531$	35.3137	1.53248
$532$	0	0
$533$	1.79899	0.0779229
$534$	−2.65685	−0.114973
$535$	0	0
$536$	19.6863	0.850318
$537$	−24.1421	−1.04181
$538$	12.6152	0.543881
$539$	0	0
$540$	0	0
$541$	18.6569	0.802121	0.401060	−	0.916052i	$-0.368642\pi$
$541$	18.6569	0.802121	0.401060	+	0.916052i	$0.368642\pi$
$542$	0.201010	0.00863412
$543$	20.8995	0.896883
$544$	−3.65685	−0.156786
$545$	0	0
$546$	0	0
$547$	−5.10051	−0.218082	−0.109041	−	0.994037i	$-0.534778\pi$
$547$	−5.10051	−0.218082	−0.109041	+	0.994037i	$0.534778\pi$
$548$	3.02944	0.129411
$549$	32.4853	1.38644
$550$	0	0
$551$	−2.82843	−0.120495
$552$	−9.24264	−0.393393
$553$	0	0
$554$	5.02944	0.213680
$555$	0	0
$556$	22.2010	0.941533
$557$	34.2843	1.45267	0.726336	−	0.687340i	$-0.241222\pi$
$557$	34.2843	1.45267	0.726336	+	0.687340i	$0.241222\pi$
$558$	7.02944	0.297580
$559$	−5.31371	−0.224746
$560$	0	0
$561$	−9.65685	−0.407713
$562$	10.8873	0.459253
$563$	16.2721	0.685786	0.342893	−	0.939374i	$-0.388593\pi$
$563$	16.2721	0.685786	0.342893	+	0.939374i	$0.388593\pi$
$564$	−8.82843	−0.371744
$565$	0	0
$566$	5.79899	0.243750
$567$	0	0
$568$	19.7990	0.830747
$569$	−3.65685	−0.153303	−0.0766517	−	0.997058i	$-0.524423\pi$
$569$	−3.65685	−0.153303	−0.0766517	+	0.997058i	$0.524423\pi$
$570$	0	0
$571$	14.8284	0.620550	0.310275	−	0.950647i	$-0.399579\pi$
$571$	14.8284	0.620550	0.310275	+	0.950647i	$0.399579\pi$
$572$	−7.31371	−0.305802
$573$	−17.3137	−0.723291
$574$	0	0
$575$	0	0
$576$	−11.7990	−0.491625
$577$	23.9411	0.996682	0.498341	−	0.866981i	$-0.333943\pi$
$577$	23.9411	0.996682	0.498341	+	0.866981i	$0.333943\pi$
$578$	−6.75736	−0.281069
$579$	4.82843	0.200663
$580$	0	0
$581$	0	0
$582$	0.343146	0.0142238
$583$	32.9706	1.36550
$584$	−7.65685	−0.316843
$585$	0	0
$586$	−6.62742	−0.273776
$587$	22.2843	0.919770	0.459885	−	0.887978i	$-0.347891\pi$
$587$	22.2843	0.919770	0.459885	+	0.887978i	$0.347891\pi$
$588$	0	0
$589$	16.9706	0.699260
$590$	0	0
$591$	57.1127	2.34930
$592$	0	0
$593$	43.7990	1.79861	0.899304	−	0.437323i	$-0.144073\pi$
$593$	43.7990	1.79861	0.899304	+	0.437323i	$0.144073\pi$
$594$	−0.828427	−0.0339908
$595$	0	0
$596$	14.3137	0.586312
$597$	−4.00000	−0.163709
$598$	0.828427	0.0338769
$599$	17.6569	0.721440	0.360720	−	0.932674i	$-0.382531\pi$
$599$	17.6569	0.721440	0.360720	+	0.932674i	$0.382531\pi$
$600$	0	0
$601$	−8.34315	−0.340324	−0.170162	−	0.985416i	$-0.554429\pi$
$601$	−8.34315	−0.340324	−0.170162	+	0.985416i	$0.554429\pi$
$602$	0	0
$603$	−35.1127	−1.42990
$604$	−0.627417	−0.0255292
$605$	0	0
$606$	−12.3137	−0.500210
$607$	−4.21320	−0.171009	−0.0855043	−	0.996338i	$-0.527250\pi$
$607$	−4.21320	−0.171009	−0.0855043	+	0.996338i	$0.527250\pi$
$608$	12.4853	0.506345
$609$	0	0
$610$	0	0
$611$	1.65685	0.0670291
$612$	4.28427	0.173181
$613$	15.4558	0.624256	0.312128	−	0.950040i	$-0.398958\pi$
$613$	15.4558	0.624256	0.312128	+	0.950040i	$0.398958\pi$
$614$	−5.48528	−0.221368
$615$	0	0
$616$	0	0
$617$	11.3137	0.455473	0.227736	−	0.973723i	$-0.426868\pi$
$617$	11.3137	0.455473	0.227736	+	0.973723i	$0.426868\pi$
$618$	0.414214	0.0166621
$619$	−42.4853	−1.70763	−0.853814	−	0.520578i	$-0.825716\pi$
$619$	−42.4853	−1.70763	−0.853814	+	0.520578i	$0.825716\pi$
$620$	0	0
$621$	−1.00000	−0.0401286
$622$	7.79899	0.312711
$623$	0	0
$624$	6.00000	0.240192
$625$	0	0
$626$	−7.31371	−0.292315
$627$	32.9706	1.31672
$628$	9.71573	0.387700
$629$	0	0
$630$	0	0
$631$	8.14214	0.324133	0.162067	−	0.986780i	$-0.448184\pi$
$631$	8.14214	0.324133	0.162067	+	0.986780i	$0.448184\pi$
$632$	−14.5442	−0.578535
$633$	8.48528	0.337260
$634$	−10.6863	−0.424407
$635$	0	0
$636$	−30.1421	−1.19521
$637$	0	0
$638$	−2.00000	−0.0791808
$639$	−35.3137	−1.39699
$640$	0	0
$641$	14.5147	0.573297	0.286648	−	0.958036i	$-0.407459\pi$
$641$	14.5147	0.573297	0.286648	+	0.958036i	$0.407459\pi$
$642$	−2.75736	−0.108824
$643$	−30.2843	−1.19430	−0.597148	−	0.802131i	$-0.703699\pi$
$643$	−30.2843	−1.19430	−0.597148	+	0.802131i	$0.703699\pi$
$644$	0	0
$645$	0	0
$646$	−0.970563	−0.0381863
$647$	17.0416	0.669976	0.334988	−	0.942222i	$-0.391268\pi$
$647$	17.0416	0.669976	0.334988	+	0.942222i	$0.391268\pi$
$648$	15.0416	0.590891
$649$	60.2843	2.36636
$650$	0	0
$651$	0	0
$652$	43.2548	1.69399
$653$	24.8284	0.971611	0.485806	−	0.874067i	$-0.338526\pi$
$653$	24.8284	0.971611	0.485806	+	0.874067i	$0.338526\pi$
$654$	3.48528	0.136285
$655$	0	0
$656$	6.51472	0.254357
$657$	13.6569	0.532805
$658$	0	0
$659$	26.8284	1.04509	0.522544	−	0.852613i	$-0.324983\pi$
$659$	26.8284	1.04509	0.522544	+	0.852613i	$0.324983\pi$
$660$	0	0
$661$	−26.1716	−1.01796	−0.508978	−	0.860779i	$-0.669977\pi$
$661$	−26.1716	−1.01796	−0.508978	+	0.860779i	$0.669977\pi$
$662$	−4.54416	−0.176614
$663$	−1.65685	−0.0643469
$664$	18.5980	0.721742
$665$	0	0
$666$	0	0
$667$	−2.41421	−0.0934787
$668$	−35.8112	−1.38558
$669$	28.1421	1.08804
$670$	0	0
$671$	55.4558	2.14085
$672$	0	0
$673$	−18.3431	−0.707076	−0.353538	−	0.935420i	$-0.615022\pi$
$673$	−18.3431	−0.707076	−0.353538	+	0.935420i	$0.615022\pi$
$674$	−6.14214	−0.236586
$675$	0	0
$676$	22.5147	0.865951
$677$	0.142136	0.00546272	0.00273136	−	0.999996i	$-0.499131\pi$
$677$	0.142136	0.00546272	0.00273136	+	0.999996i	$0.499131\pi$
$678$	−12.4853	−0.479494
$679$	0	0
$680$	0	0
$681$	65.1127	2.49512
$682$	12.0000	0.459504
$683$	43.2426	1.65463	0.827317	−	0.561736i	$-0.189866\pi$
$683$	43.2426	1.65463	0.827317	+	0.561736i	$0.189866\pi$
$684$	−14.6274	−0.559293
$685$	0	0
$686$	0	0
$687$	0.828427	0.0316065
$688$	−19.2426	−0.733619
$689$	5.65685	0.215509
$690$	0	0
$691$	4.82843	0.183682	0.0918410	−	0.995774i	$-0.470725\pi$
$691$	4.82843	0.183682	0.0918410	+	0.995774i	$0.470725\pi$
$692$	35.3137	1.34243
$693$	0	0
$694$	−9.14214	−0.347031
$695$	0	0
$696$	3.82843	0.145116
$697$	−1.79899	−0.0681416
$698$	11.0416	0.417932
$699$	26.9706	1.02012
$700$	0	0
$701$	−42.7990	−1.61650	−0.808248	−	0.588843i	$-0.799584\pi$
$701$	−42.7990	−1.61650	−0.808248	+	0.588843i	$0.799584\pi$
$702$	−0.142136	−0.00536456
$703$	0	0
$704$	−20.1421	−0.759135
$705$	0	0
$706$	−8.76955	−0.330046
$707$	0	0
$708$	−55.1127	−2.07126
$709$	38.3137	1.43890	0.719451	−	0.694543i	$-0.244394\pi$
$709$	38.3137	1.43890	0.719451	+	0.694543i	$0.244394\pi$
$710$	0	0
$711$	25.9411	0.972868
$712$	4.21320	0.157896
$713$	14.4853	0.542478
$714$	0	0
$715$	0	0
$716$	18.2843	0.683315
$717$	3.17157	0.118445
$718$	−4.14214	−0.154583
$719$	41.1127	1.53324	0.766622	−	0.642098i	$-0.221936\pi$
$719$	41.1127	1.53324	0.766622	+	0.642098i	$0.221936\pi$
$720$	0	0
$721$	0	0
$722$	−4.55635	−0.169570
$723$	−39.4558	−1.46738
$724$	−15.8284	−0.588259
$725$	0	0
$726$	12.3137	0.457005
$727$	−40.4142	−1.49888	−0.749440	−	0.662072i	$-0.769677\pi$
$727$	−40.4142	−1.49888	−0.749440	+	0.662072i	$0.769677\pi$
$728$	0	0
$729$	−23.8284	−0.882534
$730$	0	0
$731$	5.31371	0.196535
$732$	−50.6985	−1.87387
$733$	−22.0000	−0.812589	−0.406294	−	0.913742i	$-0.633179\pi$
$733$	−22.0000	−0.812589	−0.406294	+	0.913742i	$0.633179\pi$
$734$	−4.65685	−0.171888
$735$	0	0
$736$	10.6569	0.392817
$737$	−59.9411	−2.20796
$738$	2.54416	0.0936517
$739$	41.1127	1.51236	0.756178	−	0.654367i	$-0.227065\pi$
$739$	41.1127	1.51236	0.756178	+	0.654367i	$0.227065\pi$
$740$	0	0
$741$	5.65685	0.207810
$742$	0	0
$743$	1.92893	0.0707657	0.0353828	−	0.999374i	$-0.488735\pi$
$743$	1.92893	0.0707657	0.0353828	+	0.999374i	$0.488735\pi$
$744$	−22.9706	−0.842142
$745$	0	0
$746$	−5.37258	−0.196704
$747$	−33.1716	−1.21368
$748$	7.31371	0.267416
$749$	0	0
$750$	0	0
$751$	−41.6569	−1.52008	−0.760040	−	0.649876i	$-0.774821\pi$
$751$	−41.6569	−1.52008	−0.760040	+	0.649876i	$0.774821\pi$
$752$	6.00000	0.218797
$753$	−32.1421	−1.17132
$754$	−0.343146	−0.0124966
$755$	0	0
$756$	0	0
$757$	−19.4558	−0.707135	−0.353567	−	0.935409i	$-0.615031\pi$
$757$	−19.4558	−0.707135	−0.353567	+	0.935409i	$0.615031\pi$
$758$	8.76955	0.318524
$759$	28.1421	1.02149
$760$	0	0
$761$	13.3137	0.482622	0.241311	−	0.970448i	$-0.422423\pi$
$761$	13.3137	0.482622	0.241311	+	0.970448i	$0.422423\pi$
$762$	−13.3137	−0.482305
$763$	0	0
$764$	13.1127	0.474401
$765$	0	0
$766$	7.00000	0.252920
$767$	10.3431	0.373469
$768$	9.58579	0.345897
$769$	−44.6274	−1.60931	−0.804653	−	0.593745i	$-0.797649\pi$
$769$	−44.6274	−1.60931	−0.804653	+	0.593745i	$0.797649\pi$
$770$	0	0
$771$	42.6274	1.53519
$772$	−3.65685	−0.131613
$773$	25.1127	0.903241	0.451620	−	0.892210i	$-0.350846\pi$
$773$	25.1127	0.903241	0.451620	+	0.892210i	$0.350846\pi$
$774$	−7.51472	−0.270111
$775$	0	0
$776$	−0.544156	−0.0195341
$777$	0	0
$778$	5.11270	0.183299
$779$	6.14214	0.220065
$780$	0	0
$781$	−60.2843	−2.15714
$782$	−0.828427	−0.0296245
$783$	0.414214	0.0148028
$784$	0	0
$785$	0	0
$786$	−3.31371	−0.118196
$787$	−28.5563	−1.01792	−0.508962	−	0.860789i	$-0.669971\pi$
$787$	−28.5563	−1.01792	−0.508962	+	0.860789i	$0.669971\pi$
$788$	−43.2548	−1.54089
$789$	−45.9706	−1.63660
$790$	0	0
$791$	0	0
$792$	−21.6569	−0.769543
$793$	9.51472	0.337878
$794$	11.8579	0.420820
$795$	0	0
$796$	3.02944	0.107376
$797$	−8.00000	−0.283375	−0.141687	−	0.989911i	$-0.545253\pi$
$797$	−8.00000	−0.283375	−0.141687	+	0.989911i	$0.545253\pi$
$798$	0	0
$799$	−1.65685	−0.0586153
$800$	0	0
$801$	−7.51472	−0.265520
$802$	3.18377	0.112423
$803$	23.3137	0.822723
$804$	54.7990	1.93261
$805$	0	0
$806$	2.05887	0.0725208
$807$	73.5269	2.58827
$808$	19.5269	0.686954
$809$	−9.62742	−0.338482	−0.169241	−	0.985575i	$-0.554132\pi$
$809$	−9.62742	−0.338482	−0.169241	+	0.985575i	$0.554132\pi$
$810$	0	0
$811$	−24.6274	−0.864786	−0.432393	−	0.901685i	$-0.642331\pi$
$811$	−24.6274	−0.864786	−0.432393	+	0.901685i	$0.642331\pi$
$812$	0	0
$813$	1.17157	0.0410889
$814$	0	0
$815$	0	0
$816$	−6.00000	−0.210042
$817$	−18.1421	−0.634713
$818$	−10.2721	−0.359155
$819$	0	0
$820$	0	0
$821$	−19.9411	−0.695950	−0.347975	−	0.937504i	$-0.613131\pi$
$821$	−19.9411	−0.695950	−0.347975	+	0.937504i	$0.613131\pi$
$822$	−1.65685	−0.0577894
$823$	12.0711	0.420771	0.210385	−	0.977619i	$-0.432528\pi$
$823$	12.0711	0.420771	0.210385	+	0.977619i	$0.432528\pi$
$824$	−0.656854	−0.0228826
$825$	0	0
$826$	0	0
$827$	−16.2132	−0.563788	−0.281894	−	0.959446i	$-0.590963\pi$
$827$	−16.2132	−0.563788	−0.281894	+	0.959446i	$0.590963\pi$
$828$	−12.4853	−0.433894
$829$	−6.68629	−0.232225	−0.116112	−	0.993236i	$-0.537043\pi$
$829$	−6.68629	−0.232225	−0.116112	+	0.993236i	$0.537043\pi$
$830$	0	0
$831$	29.3137	1.01688
$832$	−3.45584	−0.119810
$833$	0	0
$834$	−12.1421	−0.420448
$835$	0	0
$836$	−24.9706	−0.863625
$837$	−2.48528	−0.0859039
$838$	9.65685	0.333590
$839$	−20.8284	−0.719077	−0.359539	−	0.933130i	$-0.617066\pi$
$839$	−20.8284	−0.719077	−0.359539	+	0.933130i	$0.617066\pi$
$840$	0	0
$841$	−28.0000	−0.965517
$842$	−1.44365	−0.0497515
$843$	63.4558	2.18554
$844$	−6.42641	−0.221206
$845$	0	0
$846$	2.34315	0.0805590
$847$	0	0
$848$	20.4853	0.703467
$849$	33.7990	1.15998
$850$	0	0
$851$	0	0
$852$	55.1127	1.88813
$853$	53.4558	1.83029	0.915147	−	0.403121i	$-0.132075\pi$
$853$	53.4558	1.83029	0.915147	+	0.403121i	$0.132075\pi$
$854$	0	0
$855$	0	0
$856$	4.37258	0.149452
$857$	−22.2843	−0.761216	−0.380608	−	0.924736i	$-0.624285\pi$
$857$	−22.2843	−0.761216	−0.380608	+	0.924736i	$0.624285\pi$
$858$	4.00000	0.136558
$859$	−46.6274	−1.59091	−0.795453	−	0.606015i	$-0.792767\pi$
$859$	−46.6274	−1.59091	−0.795453	+	0.606015i	$0.792767\pi$
$860$	0	0
$861$	0	0
$862$	−9.02944	−0.307544
$863$	16.5563	0.563585	0.281792	−	0.959475i	$-0.409071\pi$
$863$	16.5563	0.563585	0.281792	+	0.959475i	$0.409071\pi$
$864$	−1.82843	−0.0622044
$865$	0	0
$866$	−13.1716	−0.447588
$867$	−39.3848	−1.33758
$868$	0	0
$869$	44.2843	1.50224
$870$	0	0
$871$	−10.2843	−0.348469
$872$	−5.52691	−0.187165
$873$	0.970563	0.0328486
$874$	2.82843	0.0956730
$875$	0	0
$876$	−21.3137	−0.720123
$877$	−30.8284	−1.04100	−0.520501	−	0.853861i	$-0.674255\pi$
$877$	−30.8284	−1.04100	−0.520501	+	0.853861i	$0.674255\pi$
$878$	−14.0589	−0.474464
$879$	−38.6274	−1.30287
$880$	0	0
$881$	3.82843	0.128983	0.0644915	−	0.997918i	$-0.479457\pi$
$881$	3.82843	0.128983	0.0644915	+	0.997918i	$0.479457\pi$
$882$	0	0
$883$	38.2843	1.28837	0.644184	−	0.764870i	$-0.277197\pi$
$883$	38.2843	1.28837	0.644184	+	0.764870i	$0.277197\pi$
$884$	1.25483	0.0422046
$885$	0	0
$886$	5.05887	0.169956
$887$	44.0711	1.47976	0.739881	−	0.672738i	$-0.234882\pi$
$887$	44.0711	1.47976	0.739881	+	0.672738i	$0.234882\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−45.7990	−1.53432
$892$	−21.3137	−0.713636
$893$	5.65685	0.189299
$894$	−7.82843	−0.261822
$895$	0	0
$896$	0	0
$897$	4.82843	0.161216
$898$	−0.757359	−0.0252734
$899$	−6.00000	−0.200111
$900$	0	0
$901$	−5.65685	−0.188457
$902$	4.34315	0.144611
$903$	0	0
$904$	19.7990	0.658505
$905$	0	0
$906$	0.343146	0.0114003
$907$	−28.2132	−0.936804	−0.468402	−	0.883515i	$-0.655170\pi$
$907$	−28.2132	−0.936804	−0.468402	+	0.883515i	$0.655170\pi$
$908$	−49.3137	−1.63653
$909$	−34.8284	−1.15519
$910$	0	0
$911$	−49.7990	−1.64991	−0.824957	−	0.565195i	$-0.808801\pi$
$911$	−49.7990	−1.64991	−0.824957	+	0.565195i	$0.808801\pi$
$912$	20.4853	0.678335
$913$	−56.6274	−1.87409
$914$	13.3726	0.442326
$915$	0	0
$916$	−0.627417	−0.0207304
$917$	0	0
$918$	0.142136	0.00469117
$919$	19.1127	0.630470	0.315235	−	0.949014i	$-0.397917\pi$
$919$	19.1127	0.630470	0.315235	+	0.949014i	$0.397917\pi$
$920$	0	0
$921$	−31.9706	−1.05347
$922$	−7.74012	−0.254907
$923$	−10.3431	−0.340449
$924$	0	0
$925$	0	0
$926$	−4.57359	−0.150298
$927$	1.17157	0.0384795
$928$	−4.41421	−0.144904
$929$	11.4853	0.376820	0.188410	−	0.982090i	$-0.439667\pi$
$929$	11.4853	0.376820	0.188410	+	0.982090i	$0.439667\pi$
$930$	0	0
$931$	0	0
$932$	−20.4264	−0.669089
$933$	45.4558	1.48816
$934$	−9.48528	−0.310368
$935$	0	0
$936$	−3.71573	−0.121452
$937$	−10.6274	−0.347183	−0.173591	−	0.984818i	$-0.555537\pi$
$937$	−10.6274	−0.347183	−0.173591	+	0.984818i	$0.555537\pi$
$938$	0	0
$939$	−42.6274	−1.39109
$940$	0	0
$941$	−10.2843	−0.335258	−0.167629	−	0.985850i	$-0.553611\pi$
$941$	−10.2843	−0.335258	−0.167629	+	0.985850i	$0.553611\pi$
$942$	−5.31371	−0.173130
$943$	5.24264	0.170724
$944$	37.4558	1.21908
$945$	0	0
$946$	−12.8284	−0.417088
$947$	43.1838	1.40328	0.701642	−	0.712530i	$-0.252451\pi$
$947$	43.1838	1.40328	0.701642	+	0.712530i	$0.252451\pi$
$948$	−40.4853	−1.31490
$949$	4.00000	0.129845
$950$	0	0
$951$	−62.2843	−2.01971
$952$	0	0
$953$	2.34315	0.0759019	0.0379510	−	0.999280i	$-0.487917\pi$
$953$	2.34315	0.0759019	0.0379510	+	0.999280i	$0.487917\pi$
$954$	8.00000	0.259010
$955$	0	0
$956$	−2.40202	−0.0776869
$957$	−11.6569	−0.376813
$958$	10.0833	0.325775
$959$	0	0
$960$	0	0
$961$	5.00000	0.161290
$962$	0	0
$963$	−7.79899	−0.251319
$964$	29.8823	0.962442
$965$	0	0
$966$	0	0
$967$	27.5269	0.885206	0.442603	−	0.896718i	$-0.354055\pi$
$967$	27.5269	0.885206	0.442603	+	0.896718i	$0.354055\pi$
$968$	−19.5269	−0.627619
$969$	−5.65685	−0.181724
$970$	0	0
$971$	24.0000	0.770197	0.385098	−	0.922876i	$-0.374168\pi$
$971$	24.0000	0.770197	0.385098	+	0.922876i	$0.374168\pi$
$972$	39.5980	1.27011
$973$	0	0
$974$	−6.48528	−0.207802
$975$	0	0
$976$	34.4558	1.10290
$977$	−21.3137	−0.681886	−0.340943	−	0.940084i	$-0.610746\pi$
$977$	−21.3137	−0.681886	−0.340943	+	0.940084i	$0.610746\pi$
$978$	−23.6569	−0.756463
$979$	−12.8284	−0.409998
$980$	0	0
$981$	9.85786	0.314737
$982$	−5.51472	−0.175982
$983$	−14.2132	−0.453331	−0.226665	−	0.973973i	$-0.572782\pi$
$983$	−14.2132	−0.453331	−0.226665	+	0.973973i	$0.572782\pi$
$984$	−8.31371	−0.265031
$985$	0	0
$986$	0.343146	0.0109280
$987$	0	0
$988$	−4.28427	−0.136301
$989$	−15.4853	−0.492403
$990$	0	0
$991$	−15.6569	−0.497356	−0.248678	−	0.968586i	$-0.579996\pi$
$991$	−15.6569	−0.497356	−0.248678	+	0.968586i	$0.579996\pi$
$992$	26.4853	0.840909
$993$	−26.4853	−0.840485
$994$	0	0
$995$	0	0
$996$	51.7696	1.64038
$997$	−17.4558	−0.552832	−0.276416	−	0.961038i	$-0.589147\pi$
$997$	−17.4558	−0.552832	−0.276416	+	0.961038i	$0.589147\pi$
$998$	2.00000	0.0633089
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1225.2.a.m.1.2		2
5.2	odd	4		1225.2.b.h.99.3		4
5.3	odd	4		1225.2.b.h.99.2		4
5.4	even	2		245.2.a.g.1.1		2
7.3	odd	6		175.2.e.c.51.1		4
7.5	odd	6		175.2.e.c.151.1		4
7.6	odd	2		1225.2.a.k.1.2		2
15.14	odd	2		2205.2.a.q.1.2		2
20.19	odd	2		3920.2.a.bv.1.2		2
35.3	even	12		175.2.k.a.149.3		8
35.4	even	6		245.2.e.e.226.2		4
35.9	even	6		245.2.e.e.116.2		4
35.12	even	12		175.2.k.a.74.3		8
35.13	even	4		1225.2.b.g.99.2		4
35.17	even	12		175.2.k.a.149.2		8
35.19	odd	6		35.2.e.a.11.2	✓	4
35.24	odd	6		35.2.e.a.16.2	yes	4
35.27	even	4		1225.2.b.g.99.3		4
35.33	even	12		175.2.k.a.74.2		8
35.34	odd	2		245.2.a.h.1.1		2
105.59	even	6		315.2.j.e.226.1		4
105.89	even	6		315.2.j.e.46.1		4
105.104	even	2		2205.2.a.n.1.2		2
140.19	even	6		560.2.q.k.81.2		4
140.59	even	6		560.2.q.k.401.2		4
140.139	even	2		3920.2.a.bq.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
35.2.e.a.11.2	✓	4	35.19	odd	6
35.2.e.a.16.2	yes	4	35.24	odd	6
175.2.e.c.51.1		4	7.3	odd	6
175.2.e.c.151.1		4	7.5	odd	6
175.2.k.a.74.2		8	35.33	even	12
175.2.k.a.74.3		8	35.12	even	12
175.2.k.a.149.2		8	35.17	even	12
175.2.k.a.149.3		8	35.3	even	12
245.2.a.g.1.1		2	5.4	even	2
245.2.a.h.1.1		2	35.34	odd	2
245.2.e.e.116.2		4	35.9	even	6
245.2.e.e.226.2		4	35.4	even	6
315.2.j.e.46.1		4	105.89	even	6
315.2.j.e.226.1		4	105.59	even	6
560.2.q.k.81.2		4	140.19	even	6
560.2.q.k.401.2		4	140.59	even	6
1225.2.a.k.1.2		2	7.6	odd	2
1225.2.a.m.1.2		2	1.1	even	1	trivial
1225.2.b.g.99.2		4	35.13	even	4
1225.2.b.g.99.3		4	35.27	even	4
1225.2.b.h.99.2		4	5.3	odd	4
1225.2.b.h.99.3		4	5.2	odd	4
2205.2.a.n.1.2		2	105.104	even	2
2205.2.a.q.1.2		2	15.14	odd	2
3920.2.a.bq.1.1		2	140.139	even	2
3920.2.a.bv.1.2		2	20.19	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

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

\(f(q)\)	\(=\)	\(q+0.414214 q^{2} +2.41421 q^{3} -1.82843 q^{4} +1.00000 q^{6} -1.58579 q^{8} +2.82843 q^{9} +O(q^{10})\)	\(q+0.414214 q^{2} +2.41421 q^{3} -1.82843 q^{4} +1.00000 q^{6} -1.58579 q^{8} +2.82843 q^{9} +4.82843 q^{11} -4.41421 q^{12} +0.828427 q^{13} +3.00000 q^{16} -0.828427 q^{17} +1.17157 q^{18} +2.82843 q^{19} +2.00000 q^{22} +2.41421 q^{23} -3.82843 q^{24} +0.343146 q^{26} -0.414214 q^{27} -1.00000 q^{29} +6.00000 q^{31} +4.41421 q^{32} +11.6569 q^{33} -0.343146 q^{34} -5.17157 q^{36} +1.17157 q^{38} +2.00000 q^{39} +2.17157 q^{41} -6.41421 q^{43} -8.82843 q^{44} +1.00000 q^{46} +2.00000 q^{47} +7.24264 q^{48} -2.00000 q^{51} -1.51472 q^{52} +6.82843 q^{53} -0.171573 q^{54} +6.82843 q^{57} -0.414214 q^{58} +12.4853 q^{59} +11.4853 q^{61} +2.48528 q^{62} -4.17157 q^{64} +4.82843 q^{66} -12.4142 q^{67} +1.51472 q^{68} +5.82843 q^{69} -12.4853 q^{71} -4.48528 q^{72} +4.82843 q^{73} -5.17157 q^{76} +0.828427 q^{78} +9.17157 q^{79} -9.48528 q^{81} +0.899495 q^{82} -11.7279 q^{83} -2.65685 q^{86} -2.41421 q^{87} -7.65685 q^{88} -2.65685 q^{89} -4.41421 q^{92} +14.4853 q^{93} +0.828427 q^{94} +10.6569 q^{96} +0.343146 q^{97} +13.6569 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} + 2 q^{6} - 6 q^{8}+O(q^{10}) \) 2 * q - 2 * q^2 + 2 * q^3 + 2 * q^4 + 2 * q^6 - 6 * q^8	\( 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} + 2 q^{6} - 6 q^{8} + 4 q^{11} - 6 q^{12} - 4 q^{13} + 6 q^{16} + 4 q^{17} + 8 q^{18} + 4 q^{22} + 2 q^{23} - 2 q^{24} + 12 q^{26} + 2 q^{27} - 2 q^{29} + 12 q^{31} + 6 q^{32} + 12 q^{33} - 12 q^{34} - 16 q^{36} + 8 q^{38} + 4 q^{39} + 10 q^{41} - 10 q^{43} - 12 q^{44} + 2 q^{46} + 4 q^{47} + 6 q^{48} - 4 q^{51} - 20 q^{52} + 8 q^{53} - 6 q^{54} + 8 q^{57} + 2 q^{58} + 8 q^{59} + 6 q^{61} - 12 q^{62} - 14 q^{64} + 4 q^{66} - 22 q^{67} + 20 q^{68} + 6 q^{69} - 8 q^{71} + 8 q^{72} + 4 q^{73} - 16 q^{76} - 4 q^{78} + 24 q^{79} - 2 q^{81} - 18 q^{82} + 2 q^{83} + 6 q^{86} - 2 q^{87} - 4 q^{88} + 6 q^{89} - 6 q^{92} + 12 q^{93} - 4 q^{94} + 10 q^{96} + 12 q^{97} + 16 q^{99}+O(q^{100}) \) 2 * q - 2 * q^2 + 2 * q^3 + 2 * q^4 + 2 * q^6 - 6 * q^8 + 4 * q^11 - 6 * q^12 - 4 * q^13 + 6 * q^16 + 4 * q^17 + 8 * q^18 + 4 * q^22 + 2 * q^23 - 2 * q^24 + 12 * q^26 + 2 * q^27 - 2 * q^29 + 12 * q^31 + 6 * q^32 + 12 * q^33 - 12 * q^34 - 16 * q^36 + 8 * q^38 + 4 * q^39 + 10 * q^41 - 10 * q^43 - 12 * q^44 + 2 * q^46 + 4 * q^47 + 6 * q^48 - 4 * q^51 - 20 * q^52 + 8 * q^53 - 6 * q^54 + 8 * q^57 + 2 * q^58 + 8 * q^59 + 6 * q^61 - 12 * q^62 - 14 * q^64 + 4 * q^66 - 22 * q^67 + 20 * q^68 + 6 * q^69 - 8 * q^71 + 8 * q^72 + 4 * q^73 - 16 * q^76 - 4 * q^78 + 24 * q^79 - 2 * q^81 - 18 * q^82 + 2 * q^83 + 6 * q^86 - 2 * q^87 - 4 * q^88 + 6 * q^89 - 6 * q^92 + 12 * q^93 - 4 * q^94 + 10 * q^96 + 12 * q^97 + 16 * q^99

Introduction

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