LMFDB - Embedded newform 1225.2.b.g.99.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1225,2,Mod(99,1225)]

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([1, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1225.99");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1225 = 5^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1225.b (of order $2$, degree $1$, not minimal)

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:	no
Analytic conductor:	$9.78167424761$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{8})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 1 $ x^4 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 35)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			99.3
Root			$-0.707107 + 0.707107i$ of defining polynomial
Character	$\chi$	$=$	1225.99
Dual form			1225.2.b.g.99.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.414214i q^{2} +2.41421i q^{3} +1.82843 q^{4} -1.00000 q^{6} +1.58579i q^{8} -2.82843 q^{9} +O(q^{10})$	\(q+0.414214i q^{2} +2.41421i q^{3} +1.82843 q^{4} -1.00000 q^{6} +1.58579i q^{8} -2.82843 q^{9} +4.82843 q^{11} +4.41421i q^{12} +0.828427i q^{13} +3.00000 q^{16} +0.828427i q^{17} -1.17157i q^{18} +2.82843 q^{19} +2.00000i q^{22} -2.41421i q^{23} -3.82843 q^{24} -0.343146 q^{26} +0.414214i q^{27} +1.00000 q^{29} -6.00000 q^{31} +4.41421i q^{32} +11.6569i q^{33} -0.343146 q^{34} -5.17157 q^{36} +1.17157i q^{38} -2.00000 q^{39} -2.17157 q^{41} +6.41421i q^{43} +8.82843 q^{44} +1.00000 q^{46} -2.00000i q^{47} +7.24264i q^{48} -2.00000 q^{51} +1.51472i q^{52} -6.82843i q^{53} -0.171573 q^{54} +6.82843i q^{57} +0.414214i q^{58} +12.4853 q^{59} -11.4853 q^{61} -2.48528i q^{62} +4.17157 q^{64} -4.82843 q^{66} -12.4142i q^{67} +1.51472i q^{68} +5.82843 q^{69} -12.4853 q^{71} -4.48528i q^{72} +4.82843i q^{73} +5.17157 q^{76} -0.828427i q^{78} -9.17157 q^{79} -9.48528 q^{81} -0.899495i q^{82} -11.7279i q^{83} -2.65685 q^{86} +2.41421i q^{87} +7.65685i q^{88} -2.65685 q^{89} -4.41421i q^{92} -14.4853i q^{93} +0.828427 q^{94} -10.6569 q^{96} -0.343146i q^{97} -13.6569 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{4} - 4 q^{6}+O(q^{10}) $ 4 * q - 4 * q^4 - 4 * q^6	$ 4 q - 4 q^{4} - 4 q^{6} + 8 q^{11} + 12 q^{16} - 4 q^{24} - 24 q^{26} + 4 q^{29} - 24 q^{31} - 24 q^{34} - 32 q^{36} - 8 q^{39} - 20 q^{41} + 24 q^{44} + 4 q^{46} - 8 q^{51} - 12 q^{54} + 16 q^{59} - 12 q^{61} + 28 q^{64} - 8 q^{66} + 12 q^{69} - 16 q^{71} + 32 q^{76} - 48 q^{79} - 4 q^{81} + 12 q^{86} + 12 q^{89} - 8 q^{94} - 20 q^{96} - 32 q^{99}+O(q^{100}) $ 4 * q - 4 * q^4 - 4 * q^6 + 8 * q^11 + 12 * q^16 - 4 * q^24 - 24 * q^26 + 4 * q^29 - 24 * q^31 - 24 * q^34 - 32 * q^36 - 8 * q^39 - 20 * q^41 + 24 * q^44 + 4 * q^46 - 8 * q^51 - 12 * q^54 + 16 * q^59 - 12 * q^61 + 28 * q^64 - 8 * q^66 + 12 * q^69 - 16 * q^71 + 32 * q^76 - 48 * q^79 - 4 * q^81 + 12 * q^86 + 12 * q^89 - 8 * q^94 - 20 * q^96 - 32 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1225\mathbb{Z}\right)^\times$.

$n$	$101$	$1177$
$\chi(n)$	$1$	$-1$

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 $

$2$	0.414214i	0.292893i	0.989219	+	0.146447i	$0.0467837\pi$
$2$	0.414214i	0.292893i	−0.989219	+	0.146447i	$0.953216\pi$
$3$	2.41421i	1.39385i	0.717146	+	0.696923i	$0.245448\pi$
$3$	2.41421i	1.39385i	−0.717146	+	0.696923i	$0.754552\pi$
$4$	1.82843	0.914214
$5$	0	0
$5$	0	0
$6$	−1.00000	−0.408248
$7$	0	0
$7$	0	0
$8$	1.58579i	0.560660i
$9$	−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	$0.240491\pi$
$12$	4.41421i	1.27427i
$13$	0.828427i	0.229764i	0.993379	+	0.114882i	$0.0366490\pi$
$13$	0.828427i	0.229764i	−0.993379	+	0.114882i	$0.963351\pi$
$14$	0	0
$15$	0	0
$16$	3.00000	0.750000
$17$	0.828427i	0.200923i	0.994941	+	0.100462i	$0.0320319\pi$
$17$	0.828427i	0.200923i	−0.994941	+	0.100462i	$0.967968\pi$
$18$	− 1.17157i	− 0.276142i
$19$	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
$22$	2.00000i	0.426401i
$23$	− 2.41421i	− 0.503398i	−0.967806	−	0.251699i	$-0.919011\pi$
$23$	− 2.41421i	− 0.503398i	0.967806	−	0.251699i	$-0.0809893\pi$
$24$	−3.82843	−0.781474
$25$	0	0
$26$	−0.343146	−0.0672964
$27$	0.414214i	0.0797154i
$28$	0	0
$29$	1.00000	0.185695	0.0928477	−	0.995680i	$-0.470403\pi$
$29$	1.00000	0.185695	0.0928477	+	0.995680i	$0.470403\pi$
$30$	0	0
$31$	−6.00000	−1.07763	−0.538816	−	0.842424i	$-0.681128\pi$
$31$	−6.00000	−1.07763	−0.538816	+	0.842424i	$0.681128\pi$
$32$	4.41421i	0.780330i
$33$	11.6569i	2.02920i
$34$	−0.343146	−0.0588490
$35$	0	0
$36$	−5.17157	−0.861929
$37$	0	0	1.00000			$0$
$37$	0	0	−1.00000			$\pi$
$38$	1.17157i	0.190054i
$39$	−2.00000	−0.320256
$40$	0	0
$41$	−2.17157	−0.339143	−0.169571	−	0.985518i	$-0.554238\pi$
$41$	−2.17157	−0.339143	−0.169571	+	0.985518i	$0.554238\pi$
$42$	0	0
$43$	6.41421i	0.978158i	0.872239	+	0.489079i	$0.162667\pi$
$43$	6.41421i	0.978158i	−0.872239	+	0.489079i	$0.837333\pi$
$44$	8.82843	1.33094
$45$	0	0
$46$	1.00000	0.147442
$47$	− 2.00000i	− 0.291730i	−0.989305	−	0.145865i	$-0.953403\pi$
$47$	− 2.00000i	− 0.291730i	0.989305	−	0.145865i	$-0.0465965\pi$
$48$	7.24264i	1.04539i
$49$	0	0
$50$	0	0
$51$	−2.00000	−0.280056
$52$	1.51472i	0.210054i
$53$	− 6.82843i	− 0.937957i	−0.883210	−	0.468978i	$-0.844622\pi$
$53$	− 6.82843i	− 0.937957i	0.883210	−	0.468978i	$-0.155378\pi$
$54$	−0.171573	−0.0233481
$55$	0	0
$56$	0	0
$57$	6.82843i	0.904447i
$58$	0.414214i	0.0543889i
$59$	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	$-0.762945\pi$
$61$	−11.4853	−1.47054	−0.735270	+	0.677775i	$0.762945\pi$
$62$	− 2.48528i	− 0.315631i
$63$	0	0
$64$	4.17157	0.521447
$65$	0	0
$66$	−4.82843	−0.594338
$67$	− 12.4142i	− 1.51664i	−0.651884	−	0.758319i	$-0.726021\pi$
$67$	− 12.4142i	− 1.51664i	0.651884	−	0.758319i	$-0.273979\pi$
$68$	1.51472i	0.183687i
$69$	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	$0.765584\pi$
$72$	− 4.48528i	− 0.528595i
$73$	4.82843i	0.565125i	0.959249	+	0.282562i	$0.0911844\pi$
$73$	4.82843i	0.565125i	−0.959249	+	0.282562i	$0.908816\pi$
$74$	0	0
$75$	0	0
$76$	5.17157	0.593220
$77$	0	0
$78$	− 0.828427i	− 0.0938009i
$79$	−9.17157	−1.03188	−0.515941	−	0.856624i	$-0.672558\pi$
$79$	−9.17157	−1.03188	−0.515941	+	0.856624i	$0.672558\pi$
$80$	0	0
$81$	−9.48528	−1.05392
$82$	− 0.899495i	− 0.0993326i
$83$	− 11.7279i	− 1.28731i	−0.765317	−	0.643653i	$-0.777418\pi$
$83$	− 11.7279i	− 1.28731i	0.765317	−	0.643653i	$-0.222582\pi$
$84$	0	0
$85$	0	0
$86$	−2.65685	−0.286496
$87$	2.41421i	0.258831i
$88$	7.65685i	0.816223i
$89$	−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
$92$	− 4.41421i	− 0.460214i
$93$	− 14.4853i	− 1.50205i
$94$	0.828427	0.0854457
$95$	0	0
$96$	−10.6569	−1.08766
$97$	− 0.343146i	− 0.0348412i	−0.999848	−	0.0174206i	$-0.994455\pi$
$97$	− 0.343146i	− 0.0348412i	0.999848	−	0.0174206i	$-0.00554543\pi$
$98$	0	0
$99$	−13.6569	−1.37257
$100$	0	0
$101$	12.3137	1.22526	0.612630	−	0.790370i	$-0.290112\pi$
$101$	12.3137	1.22526	0.612630	+	0.790370i	$0.290112\pi$
$102$	− 0.828427i	− 0.0820265i
$103$	0.414214i	0.0408137i	0.999792	+	0.0204068i	$0.00649615\pi$
$103$	0.414214i	0.0408137i	−0.999792	+	0.0204068i	$0.993504\pi$
$104$	−1.31371	−0.128820
$105$	0	0
$106$	2.82843	0.274721
$107$	− 2.75736i	− 0.266564i	−0.991078	−	0.133282i	$-0.957448\pi$
$107$	− 2.75736i	− 0.266564i	0.991078	−	0.133282i	$-0.0425516\pi$
$108$	0.757359i	0.0728769i
$109$	−3.48528	−0.333829	−0.166915	−	0.985971i	$-0.553380\pi$
$109$	−3.48528	−0.333829	−0.166915	+	0.985971i	$0.553380\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	12.4853i	1.17452i	0.809400	+	0.587258i	$0.199793\pi$
$113$	12.4853i	1.17452i	−0.809400	+	0.587258i	$0.800207\pi$
$114$	−2.82843	−0.264906
$115$	0	0
$116$	1.82843	0.169765
$117$	− 2.34315i	− 0.216624i
$118$	5.17157i	0.476082i
$119$	0	0
$120$	0	0
$121$	12.3137	1.11943
$122$	− 4.75736i	− 0.430711i
$123$	− 5.24264i	− 0.472713i
$124$	−10.9706	−0.985186
$125$	0	0
$126$	0	0
$127$	− 13.3137i	− 1.18140i	−0.806891	−	0.590700i	$-0.798852\pi$
$127$	− 13.3137i	− 1.18140i	0.806891	−	0.590700i	$-0.201148\pi$
$128$	10.5563i	0.933058i
$129$	−15.4853	−1.36340
$130$	0	0
$131$	3.31371	0.289520	0.144760	−	0.989467i	$-0.453759\pi$
$131$	3.31371	0.289520	0.144760	+	0.989467i	$0.453759\pi$
$132$	21.3137i	1.85512i
$133$	0	0
$134$	5.14214	0.444213
$135$	0	0
$136$	−1.31371	−0.112650
$137$	− 1.65685i	− 0.141555i	−0.997492	−	0.0707773i	$-0.977452\pi$
$137$	− 1.65685i	− 0.141555i	0.997492	−	0.0707773i	$-0.0225480\pi$
$138$	2.41421i	0.205512i
$139$	−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
$142$	− 5.17157i	− 0.433989i
$143$	4.00000i	0.334497i
$144$	−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	$-0.396094\pi$
$149$	7.82843	0.641330	0.320665	+	0.947193i	$0.396094\pi$
$150$	0	0
$151$	0.343146	0.0279248	0.0139624	−	0.999903i	$-0.495555\pi$
$151$	0.343146	0.0279248	0.0139624	+	0.999903i	$0.495555\pi$
$152$	4.48528i	0.363804i
$153$	− 2.34315i	− 0.189432i
$154$	0	0
$155$	0	0
$156$	−3.65685	−0.292783
$157$	5.31371i	0.424080i	0.977261	+	0.212040i	$0.0680107\pi$
$157$	5.31371i	0.424080i	−0.977261	+	0.212040i	$0.931989\pi$
$158$	− 3.79899i	− 0.302231i
$159$	16.4853	1.30737
$160$	0	0
$161$	0	0
$162$	− 3.92893i	− 0.308686i
$163$	23.6569i	1.85295i	0.376359	+	0.926474i	$0.377176\pi$
$163$	23.6569i	1.85295i	−0.376359	+	0.926474i	$0.622824\pi$
$164$	−3.97056	−0.310049
$165$	0	0
$166$	4.85786	0.377043
$167$	− 19.5858i	− 1.51559i	−0.652491	−	0.757797i	$-0.726276\pi$
$167$	− 19.5858i	− 1.51559i	0.652491	−	0.757797i	$-0.273724\pi$
$168$	0	0
$169$	12.3137	0.947208
$170$	0	0
$171$	−8.00000	−0.611775
$172$	11.7279i	0.894246i
$173$	− 19.3137i	− 1.46839i	−0.678936	−	0.734197i	$-0.737559\pi$
$173$	− 19.3137i	− 1.46839i	0.678936	−	0.734197i	$-0.262441\pi$
$174$	−1.00000	−0.0758098
$175$	0	0
$176$	14.4853	1.09187
$177$	30.1421i	2.26562i
$178$	− 1.10051i	− 0.0824863i
$179$	10.0000	0.747435	0.373718	−	0.927543i	$-0.378083\pi$
$179$	10.0000	0.747435	0.373718	+	0.927543i	$0.378083\pi$
$180$	0	0
$181$	−8.65685	−0.643459	−0.321729	−	0.946832i	$-0.604264\pi$
$181$	−8.65685	−0.643459	−0.321729	+	0.946832i	$0.604264\pi$
$182$	0	0
$183$	− 27.7279i	− 2.04971i
$184$	3.82843	0.282235
$185$	0	0
$186$	6.00000	0.439941
$187$	4.00000i	0.292509i
$188$	− 3.65685i	− 0.266704i
$189$	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	$0.583544\pi$
$192$	10.0711i	0.726817i
$193$	− 2.00000i	− 0.143963i	−0.997406	−	0.0719816i	$-0.977068\pi$
$193$	− 2.00000i	− 0.143963i	0.997406	−	0.0719816i	$-0.0229323\pi$
$194$	0.142136	0.0102047
$195$	0	0
$196$	0	0
$197$	23.6569i	1.68548i	0.538320	+	0.842741i	$0.319059\pi$
$197$	23.6569i	1.68548i	−0.538320	+	0.842741i	$0.680941\pi$
$198$	− 5.65685i	− 0.402015i
$199$	−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
$202$	5.10051i	0.358870i
$203$	0	0
$204$	−3.65685	−0.256031
$205$	0	0
$206$	−0.171573	−0.0119540
$207$	6.82843i	0.474608i
$208$	2.48528i	0.172323i
$209$	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	$0.461396\pi$
$212$	− 12.4853i	− 0.857493i
$213$	− 30.1421i	− 2.06531i
$214$	1.14214	0.0780748
$215$	0	0
$216$	−0.656854	−0.0446933
$217$	0	0
$218$	− 1.44365i	− 0.0977764i
$219$	−11.6569	−0.787697
$220$	0	0
$221$	−0.686292	−0.0461650
$222$	0	0
$223$	11.6569i	0.780601i	0.920688	+	0.390300i	$0.127629\pi$
$223$	11.6569i	0.780601i	−0.920688	+	0.390300i	$0.872371\pi$
$224$	0	0
$225$	0	0
$226$	−5.17157	−0.344008
$227$	− 26.9706i	− 1.79010i	−0.445967	−	0.895050i	$-0.647140\pi$
$227$	− 26.9706i	− 1.79010i	0.445967	−	0.895050i	$-0.352860\pi$
$228$	12.4853i	0.826858i
$229$	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
$232$	1.58579i	0.104112i
$233$	− 11.1716i	− 0.731874i	−0.930640	−	0.365937i	$-0.880749\pi$
$233$	− 11.1716i	− 0.731874i	0.930640	−	0.365937i	$-0.119251\pi$
$234$	0.970563	0.0634477
$235$	0	0
$236$	22.8284	1.48600
$237$	− 22.1421i	− 1.43829i
$238$	0	0
$239$	−1.31371	−0.0849767	−0.0424884	−	0.999097i	$-0.513529\pi$
$239$	−1.31371	−0.0849767	−0.0424884	+	0.999097i	$0.513529\pi$
$240$	0	0
$241$	16.3431	1.05275	0.526377	−	0.850251i	$-0.323550\pi$
$241$	16.3431	1.05275	0.526377	+	0.850251i	$0.323550\pi$
$242$	5.10051i	0.327873i
$243$	− 21.6569i	− 1.38929i
$244$	−21.0000	−1.34439
$245$	0	0
$246$	2.17157	0.138454
$247$	2.34315i	0.149091i
$248$	− 9.51472i	− 0.604185i
$249$	28.3137	1.79431
$250$	0	0
$251$	13.3137	0.840354	0.420177	−	0.907442i	$-0.361968\pi$
$251$	13.3137	0.840354	0.420177	+	0.907442i	$0.361968\pi$
$252$	0	0
$253$	− 11.6569i	− 0.732860i
$254$	5.51472	0.346024
$255$	0	0
$256$	3.97056	0.248160
$257$	− 17.6569i	− 1.10140i	−0.834702	−	0.550702i	$-0.814360\pi$
$257$	− 17.6569i	− 1.10140i	0.834702	−	0.550702i	$-0.185640\pi$
$258$	− 6.41421i	− 0.399331i
$259$	0	0
$260$	0	0
$261$	−2.82843	−0.175075
$262$	1.37258i	0.0847985i
$263$	19.0416i	1.17416i	0.809530	+	0.587079i	$0.199722\pi$
$263$	19.0416i	1.17416i	−0.809530	+	0.587079i	$0.800278\pi$
$264$	−18.4853	−1.13769
$265$	0	0
$266$	0	0
$267$	− 6.41421i	− 0.392543i
$268$	− 22.6985i	− 1.38653i
$269$	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	$-0.504692\pi$
$271$	−0.485281	−0.0294787	−0.0147394	+	0.999891i	$0.504692\pi$
$272$	2.48528i	0.150692i
$273$	0	0
$274$	0.686292	0.0414604
$275$	0	0
$276$	10.6569	0.641467
$277$	12.1421i	0.729550i	0.931096	+	0.364775i	$0.118854\pi$
$277$	12.1421i	0.729550i	−0.931096	+	0.364775i	$0.881146\pi$
$278$	− 5.02944i	− 0.301646i
$279$	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	$0.213179\pi$
$282$	2.00000i	0.119098i
$283$	14.0000i	0.832214i	0.909316	+	0.416107i	$0.136606\pi$
$283$	14.0000i	0.832214i	−0.909316	+	0.416107i	$0.863394\pi$
$284$	−22.8284	−1.35462
$285$	0	0
$286$	−1.65685	−0.0979718
$287$	0	0
$288$	− 12.4853i	− 0.735702i
$289$	16.3137	0.959630
$290$	0	0
$291$	0.828427	0.0485633
$292$	8.82843i	0.516645i
$293$	− 16.0000i	− 0.934730i	−0.884064	−	0.467365i	$-0.845203\pi$
$293$	− 16.0000i	− 0.934730i	0.884064	−	0.467365i	$-0.154797\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	2.00000i	0.116052i
$298$	3.24264i	0.187841i
$299$	2.00000	0.115663
$300$	0	0
$301$	0	0
$302$	0.142136i	0.00817899i
$303$	29.7279i	1.70782i
$304$	8.48528	0.486664
$305$	0	0
$306$	0.970563	0.0554834
$307$	13.2426i	0.755797i	0.925847	+	0.377899i	$0.123353\pi$
$307$	13.2426i	0.755797i	−0.925847	+	0.377899i	$0.876647\pi$
$308$	0	0
$309$	−1.00000	−0.0568880
$310$	0	0
$311$	−18.8284	−1.06766	−0.533831	−	0.845591i	$-0.679248\pi$
$311$	−18.8284	−1.06766	−0.533831	+	0.845591i	$0.679248\pi$
$312$	− 3.17157i	− 0.179555i
$313$	− 17.6569i	− 0.998024i	−0.866595	−	0.499012i	$-0.833696\pi$
$313$	− 17.6569i	− 0.998024i	0.866595	−	0.499012i	$-0.166304\pi$
$314$	−2.20101	−0.124210
$315$	0	0
$316$	−16.7696	−0.943361
$317$	− 25.7990i	− 1.44902i	−0.689267	−	0.724508i	$-0.742067\pi$
$317$	− 25.7990i	− 1.44902i	0.689267	−	0.724508i	$-0.257933\pi$
$318$	6.82843i	0.382919i
$319$	4.82843	0.270340
$320$	0	0
$321$	6.65685	0.371549
$322$	0	0
$323$	2.34315i	0.130376i
$324$	−17.3431	−0.963508
$325$	0	0
$326$	−9.79899	−0.542716
$327$	− 8.41421i	− 0.465307i
$328$	− 3.44365i	− 0.190144i
$329$	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	$0.597487\pi$
$332$	− 21.4437i	− 1.17687i
$333$	0	0
$334$	8.11270	0.443907
$335$	0	0
$336$	0	0
$337$	− 14.8284i	− 0.807756i	−0.914813	−	0.403878i	$-0.867662\pi$
$337$	− 14.8284i	− 0.807756i	0.914813	−	0.403878i	$-0.132338\pi$
$338$	5.10051i	0.277431i
$339$	−30.1421	−1.63710
$340$	0	0
$341$	−28.9706	−1.56884
$342$	− 3.31371i	− 0.179185i
$343$	0	0
$344$	−10.1716	−0.548414
$345$	0	0
$346$	8.00000	0.430083
$347$	− 22.0711i	− 1.18484i	−0.805630	−	0.592418i	$-0.798173\pi$
$347$	− 22.0711i	− 1.18484i	0.805630	−	0.592418i	$-0.201827\pi$
$348$	4.41421i	0.236627i
$349$	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
$352$	21.3137i	1.13602i
$353$	− 21.1716i	− 1.12685i	−0.826168	−	0.563425i	$-0.809484\pi$
$353$	− 21.1716i	− 1.12685i	0.826168	−	0.563425i	$-0.190516\pi$
$354$	−12.4853	−0.663585
$355$	0	0
$356$	−4.85786	−0.257466
$357$	0	0
$358$	4.14214i	0.218919i
$359$	10.0000	0.527780	0.263890	−	0.964553i	$-0.414994\pi$
$359$	10.0000	0.527780	0.263890	+	0.964553i	$0.414994\pi$
$360$	0	0
$361$	−11.0000	−0.578947
$362$	− 3.58579i	− 0.188465i
$363$	29.7279i	1.56031i
$364$	0	0
$365$	0	0
$366$	11.4853	0.600345
$367$	11.2426i	0.586861i	0.955980	+	0.293431i	$0.0947969\pi$
$367$	11.2426i	0.586861i	−0.955980	+	0.293431i	$0.905203\pi$
$368$	− 7.24264i	− 0.377549i
$369$	6.14214	0.319747
$370$	0	0
$371$	0	0
$372$	− 26.4853i	− 1.37320i
$373$	12.9706i	0.671590i	0.941935	+	0.335795i	$0.109005\pi$
$373$	12.9706i	0.671590i	−0.941935	+	0.335795i	$0.890995\pi$
$374$	−1.65685	−0.0856739
$375$	0	0
$376$	3.17157	0.163561
$377$	0.828427i	0.0426662i
$378$	0	0
$379$	−21.1716	−1.08751	−0.543755	−	0.839244i	$-0.682998\pi$
$379$	−21.1716	−1.08751	−0.543755	+	0.839244i	$0.682998\pi$
$380$	0	0
$381$	32.1421	1.64669
$382$	− 2.97056i	− 0.151987i
$383$	16.8995i	0.863524i	0.901988	+	0.431762i	$0.142108\pi$
$383$	16.8995i	0.863524i	−0.901988	+	0.431762i	$0.857892\pi$
$384$	−25.4853	−1.30054
$385$	0	0
$386$	0.828427	0.0421658
$387$	− 18.1421i	− 0.922217i
$388$	− 0.627417i	− 0.0318523i
$389$	−12.3431	−0.625822	−0.312911	−	0.949782i	$-0.601304\pi$
$389$	−12.3431	−0.625822	−0.312911	+	0.949782i	$0.601304\pi$
$390$	0	0
$391$	2.00000	0.101144
$392$	0	0
$393$	8.00000i	0.403547i
$394$	−9.79899	−0.493666
$395$	0	0
$396$	−24.9706	−1.25482
$397$	− 28.6274i	− 1.43677i	−0.695646	−	0.718384i	$-0.744882\pi$
$397$	− 28.6274i	− 1.43677i	0.695646	−	0.718384i	$-0.255118\pi$
$398$	− 0.686292i	− 0.0344007i
$399$	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	$0.438529\pi$
$402$	12.4142i	0.619165i
$403$	− 4.97056i	− 0.247601i
$404$	22.5147	1.12015
$405$	0	0
$406$	0	0
$407$	0	0
$408$	− 3.17157i	− 0.157016i
$409$	−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
$412$	0.757359i	0.0373124i
$413$	0	0
$414$	−2.82843	−0.139010
$415$	0	0
$416$	−3.65685	−0.179292
$417$	− 29.3137i	− 1.43550i
$418$	5.65685i	0.276686i
$419$	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	$0.527067\pi$
$422$	1.45584i	0.0708694i
$423$	5.65685i	0.275046i
$424$	10.8284	0.525875
$425$	0	0
$426$	12.4853	0.604914
$427$	0	0
$428$	− 5.04163i	− 0.243696i
$429$	−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	$0.675938\pi$
$432$	1.24264i	0.0597866i
$433$	− 31.7990i	− 1.52816i	−0.645120	−	0.764081i	$-0.723193\pi$
$433$	− 31.7990i	− 1.52816i	0.645120	−	0.764081i	$-0.276807\pi$
$434$	0	0
$435$	0	0
$436$	−6.37258	−0.305191
$437$	− 6.82843i	− 0.326648i
$438$	− 4.82843i	− 0.230711i
$439$	−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
$442$	− 0.284271i	− 0.0135214i
$443$	− 12.2132i	− 0.580267i	−0.956986	−	0.290133i	$-0.906300\pi$
$443$	− 12.2132i	− 0.580267i	0.956986	−	0.290133i	$-0.0936997\pi$
$444$	0	0
$445$	0	0
$446$	−4.82843	−0.228633
$447$	18.8995i	0.893915i
$448$	0	0
$449$	1.82843	0.0862888	0.0431444	−	0.999069i	$-0.486262\pi$
$449$	1.82843	0.0862888	0.0431444	+	0.999069i	$0.486262\pi$
$450$	0	0
$451$	−10.4853	−0.493733
$452$	22.8284i	1.07376i
$453$	0.828427i	0.0389229i
$454$	11.1716	0.524308
$455$	0	0
$456$	−10.8284	−0.507088
$457$	32.2843i	1.51019i	0.655613	+	0.755097i	$0.272410\pi$
$457$	32.2843i	1.51019i	−0.655613	+	0.755097i	$0.727590\pi$
$458$	0.142136i	0.00664156i
$459$	−0.343146	−0.0160167
$460$	0	0
$461$	18.6863	0.870307	0.435154	−	0.900356i	$-0.356694\pi$
$461$	18.6863	0.870307	0.435154	+	0.900356i	$0.356694\pi$
$462$	0	0
$463$	11.0416i	0.513148i	0.966525	+	0.256574i	$0.0825937\pi$
$463$	11.0416i	0.513148i	−0.966525	+	0.256574i	$0.917406\pi$
$464$	3.00000	0.139272
$465$	0	0
$466$	4.62742	0.214361
$467$	22.8995i	1.05966i	0.848103	+	0.529831i	$0.177745\pi$
$467$	22.8995i	1.05966i	−0.848103	+	0.529831i	$0.822255\pi$
$468$	− 4.28427i	− 0.198041i
$469$	0	0
$470$	0	0
$471$	−12.8284	−0.591103
$472$	19.7990i	0.911322i
$473$	30.9706i	1.42403i
$474$	9.17157	0.421264
$475$	0	0
$476$	0	0
$477$	19.3137i	0.884314i
$478$	− 0.544156i	− 0.0248891i
$479$	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
$482$	6.76955i	0.308345i
$483$	0	0
$484$	22.5147	1.02340
$485$	0	0
$486$	8.97056	0.406913
$487$	− 15.6569i	− 0.709480i	−0.934965	−	0.354740i	$-0.884569\pi$
$487$	− 15.6569i	− 0.709480i	0.934965	−	0.354740i	$-0.115431\pi$
$488$	− 18.2132i	− 0.824473i
$489$	−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	$0.597127\pi$
$492$	− 9.58579i	− 0.432161i
$493$	0.828427i	0.0373105i
$494$	−0.970563	−0.0436677
$495$	0	0
$496$	−18.0000	−0.808224
$497$	0	0
$498$	11.7279i	0.525541i
$499$	−4.82843	−0.216150	−0.108075	−	0.994143i	$-0.534469\pi$
$499$	−4.82843	−0.216150	−0.108075	+	0.994143i	$0.534469\pi$
$500$	0	0
$501$	47.2843	2.11251
$502$	5.51472i	0.246134i
$503$	37.8701i	1.68854i	0.535916	+	0.844271i	$0.319966\pi$
$503$	37.8701i	1.68854i	−0.535916	+	0.844271i	$0.680034\pi$
$504$	0	0
$505$	0	0
$506$	4.82843	0.214650
$507$	29.7279i	1.32026i
$508$	− 24.3431i	− 1.08005i
$509$	−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
$512$	22.7574i	1.00574i
$513$	1.17157i	0.0517262i
$514$	7.31371	0.322594
$515$	0	0
$516$	−28.3137	−1.24644
$517$	− 9.65685i	− 0.424708i
$518$	0	0
$519$	46.6274	2.04672
$520$	0	0
$521$	−18.9706	−0.831115	−0.415558	−	0.909567i	$-0.636414\pi$
$521$	−18.9706	−0.831115	−0.415558	+	0.909567i	$0.636414\pi$
$522$	− 1.17157i	− 0.0512784i
$523$	24.3431i	1.06445i	0.846602	+	0.532226i	$0.178644\pi$
$523$	24.3431i	1.06445i	−0.846602	+	0.532226i	$0.821356\pi$
$524$	6.05887	0.264683
$525$	0	0
$526$	−7.88730	−0.343903
$527$	− 4.97056i	− 0.216521i
$528$	34.9706i	1.52190i
$529$	17.1716	0.746590
$530$	0	0
$531$	−35.3137	−1.53248
$532$	0	0
$533$	− 1.79899i	− 0.0779229i
$534$	2.65685	0.114973
$535$	0	0
$536$	19.6863	0.850318
$537$	24.1421i	1.04181i
$538$	12.6152i	0.543881i
$539$	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	$0.368642\pi$
$542$	− 0.201010i	− 0.00863412i
$543$	− 20.8995i	− 0.896883i
$544$	−3.65685	−0.156786
$545$	0	0
$546$	0	0
$547$	− 5.10051i	− 0.218082i	−0.994037	−	0.109041i	$-0.965222\pi$
$547$	− 5.10051i	− 0.218082i	0.994037	−	0.109041i	$-0.0347780\pi$
$548$	− 3.02944i	− 0.129411i
$549$	32.4853	1.38644
$550$	0	0
$551$	2.82843	0.120495
$552$	9.24264i	0.393393i
$553$	0	0
$554$	−5.02944	−0.213680
$555$	0	0
$556$	−22.2010	−0.941533
$557$	34.2843i	1.45267i	0.687340	+	0.726336i	$0.258778\pi$
$557$	34.2843i	1.45267i	−0.687340	+	0.726336i	$0.741222\pi$
$558$	7.02944i	0.297580i
$559$	−5.31371	−0.224746
$560$	0	0
$561$	−9.65685	−0.407713
$562$	10.8873i	0.459253i
$563$	16.2721i	0.685786i	0.939374	+	0.342893i	$0.111407\pi$
$563$	16.2721i	0.685786i	−0.939374	+	0.342893i	$0.888593\pi$
$564$	8.82843	0.371744
$565$	0	0
$566$	−5.79899	−0.243750
$567$	0	0
$568$	− 19.7990i	− 0.830747i
$569$	3.65685	0.153303	0.0766517	−	0.997058i	$-0.475577\pi$
$569$	3.65685	0.153303	0.0766517	+	0.997058i	$0.475577\pi$
$570$	0	0
$571$	14.8284	0.620550	0.310275	−	0.950647i	$-0.399579\pi$
$571$	14.8284	0.620550	0.310275	+	0.950647i	$0.399579\pi$
$572$	7.31371i	0.305802i
$573$	− 17.3137i	− 0.723291i
$574$	0	0
$575$	0	0
$576$	−11.7990	−0.491625
$577$	− 23.9411i	− 0.996682i	−0.866981	−	0.498341i	$-0.833943\pi$
$577$	− 23.9411i	− 0.996682i	0.866981	−	0.498341i	$-0.166057\pi$
$578$	6.75736i	0.281069i
$579$	4.82843	0.200663
$580$	0	0
$581$	0	0
$582$	0.343146i	0.0142238i
$583$	− 32.9706i	− 1.36550i
$584$	−7.65685	−0.316843
$585$	0	0
$586$	6.62742	0.273776
$587$	− 22.2843i	− 0.919770i	−0.887978	−	0.459885i	$-0.847891\pi$
$587$	− 22.2843i	− 0.919770i	0.887978	−	0.459885i	$-0.152109\pi$
$588$	0	0
$589$	−16.9706	−0.699260
$590$	0	0
$591$	−57.1127	−2.34930
$592$	0	0
$593$	43.7990i	1.79861i	0.437323	+	0.899304i	$0.355927\pi$
$593$	43.7990i	1.79861i	−0.437323	+	0.899304i	$0.644073\pi$
$594$	−0.828427	−0.0339908
$595$	0	0
$596$	14.3137	0.586312
$597$	− 4.00000i	− 0.163709i
$598$	0.828427i	0.0338769i
$599$	−17.6569	−0.721440	−0.360720	−	0.932674i	$-0.617469\pi$
$599$	−17.6569	−0.721440	−0.360720	+	0.932674i	$0.617469\pi$
$600$	0	0
$601$	8.34315	0.340324	0.170162	−	0.985416i	$-0.445571\pi$
$601$	8.34315	0.340324	0.170162	+	0.985416i	$0.445571\pi$
$602$	0	0
$603$	35.1127i	1.42990i
$604$	0.627417	0.0255292
$605$	0	0
$606$	−12.3137	−0.500210
$607$	4.21320i	0.171009i	0.996338	+	0.0855043i	$0.0272501\pi$
$607$	4.21320i	0.171009i	−0.996338	+	0.0855043i	$0.972750\pi$
$608$	12.4853i	0.506345i
$609$	0	0
$610$	0	0
$611$	1.65685	0.0670291
$612$	− 4.28427i	− 0.173181i
$613$	− 15.4558i	− 0.624256i	−0.950040	−	0.312128i	$-0.898958\pi$
$613$	− 15.4558i	− 0.624256i	0.950040	−	0.312128i	$-0.101042\pi$
$614$	−5.48528	−0.221368
$615$	0	0
$616$	0	0
$617$	11.3137i	0.455473i	0.973723	+	0.227736i	$0.0731324\pi$
$617$	11.3137i	0.455473i	−0.973723	+	0.227736i	$0.926868\pi$
$618$	− 0.414214i	− 0.0166621i
$619$	−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
$622$	− 7.79899i	− 0.312711i
$623$	0	0
$624$	−6.00000	−0.240192
$625$	0	0
$626$	7.31371	0.292315
$627$	32.9706i	1.31672i
$628$	9.71573i	0.387700i
$629$	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	$0.448184\pi$
$632$	− 14.5442i	− 0.578535i
$633$	8.48528i	0.337260i
$634$	10.6863	0.424407
$635$	0	0
$636$	30.1421	1.19521
$637$	0	0
$638$	2.00000i	0.0791808i
$639$	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	$0.407459\pi$
$642$	2.75736i	0.108824i
$643$	− 30.2843i	− 1.19430i	−0.802131	−	0.597148i	$-0.796301\pi$
$643$	− 30.2843i	− 1.19430i	0.802131	−	0.597148i	$-0.203699\pi$
$644$	0	0
$645$	0	0
$646$	−0.970563	−0.0381863
$647$	− 17.0416i	− 0.669976i	−0.942222	−	0.334988i	$-0.891268\pi$
$647$	− 17.0416i	− 0.669976i	0.942222	−	0.334988i	$-0.108732\pi$
$648$	− 15.0416i	− 0.590891i
$649$	60.2843	2.36636
$650$	0	0
$651$	0	0
$652$	43.2548i	1.69399i
$653$	− 24.8284i	− 0.971611i	−0.874067	−	0.485806i	$-0.838526\pi$
$653$	− 24.8284i	− 0.971611i	0.874067	−	0.485806i	$-0.161474\pi$
$654$	3.48528	0.136285
$655$	0	0
$656$	−6.51472	−0.254357
$657$	− 13.6569i	− 0.532805i
$658$	0	0
$659$	−26.8284	−1.04509	−0.522544	−	0.852613i	$-0.675017\pi$
$659$	−26.8284	−1.04509	−0.522544	+	0.852613i	$0.675017\pi$
$660$	0	0
$661$	26.1716	1.01796	0.508978	−	0.860779i	$-0.330023\pi$
$661$	26.1716	1.01796	0.508978	+	0.860779i	$0.330023\pi$
$662$	− 4.54416i	− 0.176614i
$663$	− 1.65685i	− 0.0643469i
$664$	18.5980	0.721742
$665$	0	0
$666$	0	0
$667$	− 2.41421i	− 0.0934787i
$668$	− 35.8112i	− 1.38558i
$669$	−28.1421	−1.08804
$670$	0	0
$671$	−55.4558	−2.14085
$672$	0	0
$673$	18.3431i	0.707076i	0.935420	+	0.353538i	$0.115022\pi$
$673$	18.3431i	0.707076i	−0.935420	+	0.353538i	$0.884978\pi$
$674$	6.14214	0.236586
$675$	0	0
$676$	22.5147	0.865951
$677$	− 0.142136i	− 0.00546272i	−0.999996	−	0.00273136i	$-0.999131\pi$
$677$	− 0.142136i	− 0.00546272i	0.999996	−	0.00273136i	$-0.000869419\pi$
$678$	− 12.4853i	− 0.479494i
$679$	0	0
$680$	0	0
$681$	65.1127	2.49512
$682$	− 12.0000i	− 0.459504i
$683$	− 43.2426i	− 1.65463i	−0.561736	−	0.827317i	$-0.689866\pi$
$683$	− 43.2426i	− 1.65463i	0.561736	−	0.827317i	$-0.310134\pi$
$684$	−14.6274	−0.559293
$685$	0	0
$686$	0	0
$687$	0.828427i	0.0316065i
$688$	19.2426i	0.733619i
$689$	5.65685	0.215509
$690$	0	0
$691$	−4.82843	−0.183682	−0.0918410	−	0.995774i	$-0.529275\pi$
$691$	−4.82843	−0.183682	−0.0918410	+	0.995774i	$0.529275\pi$
$692$	− 35.3137i	− 1.34243i
$693$	0	0
$694$	9.14214	0.347031
$695$	0	0
$696$	−3.82843	−0.145116
$697$	− 1.79899i	− 0.0681416i
$698$	11.0416i	0.417932i
$699$	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	$0.799584\pi$
$702$	− 0.142136i	− 0.00536456i
$703$	0	0
$704$	20.1421	0.759135
$705$	0	0
$706$	8.76955	0.330046
$707$	0	0
$708$	55.1127i	2.07126i
$709$	−38.3137	−1.43890	−0.719451	−	0.694543i	$-0.755606\pi$
$709$	−38.3137	−1.43890	−0.719451	+	0.694543i	$0.755606\pi$
$710$	0	0
$711$	25.9411	0.972868
$712$	− 4.21320i	− 0.157896i
$713$	14.4853i	0.542478i
$714$	0	0
$715$	0	0
$716$	18.2843	0.683315
$717$	− 3.17157i	− 0.118445i
$718$	4.14214i	0.154583i
$719$	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
$722$	− 4.55635i	− 0.169570i
$723$	39.4558i	1.46738i
$724$	−15.8284	−0.588259
$725$	0	0
$726$	−12.3137	−0.457005
$727$	40.4142i	1.49888i	0.662072	+	0.749440i	$0.269677\pi$
$727$	40.4142i	1.49888i	−0.662072	+	0.749440i	$0.730323\pi$
$728$	0	0
$729$	23.8284	0.882534
$730$	0	0
$731$	−5.31371	−0.196535
$732$	− 50.6985i	− 1.87387i
$733$	− 22.0000i	− 0.812589i	−0.913742	−	0.406294i	$-0.866821\pi$
$733$	− 22.0000i	− 0.812589i	0.913742	−	0.406294i	$-0.133179\pi$
$734$	−4.65685	−0.171888
$735$	0	0
$736$	10.6569	0.392817
$737$	− 59.9411i	− 2.20796i
$738$	2.54416i	0.0936517i
$739$	−41.1127	−1.51236	−0.756178	−	0.654367i	$-0.772935\pi$
$739$	−41.1127	−1.51236	−0.756178	+	0.654367i	$0.772935\pi$
$740$	0	0
$741$	−5.65685	−0.207810
$742$	0	0
$743$	− 1.92893i	− 0.0707657i	−0.999374	−	0.0353828i	$-0.988735\pi$
$743$	− 1.92893i	− 0.0707657i	0.999374	−	0.0353828i	$-0.0112651\pi$
$744$	22.9706	0.842142
$745$	0	0
$746$	−5.37258	−0.196704
$747$	33.1716i	1.21368i
$748$	7.31371i	0.267416i
$749$	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	$0.774821\pi$
$752$	− 6.00000i	− 0.218797i
$753$	32.1421i	1.17132i
$754$	−0.343146	−0.0124966
$755$	0	0
$756$	0	0
$757$	− 19.4558i	− 0.707135i	−0.935409	−	0.353567i	$-0.884969\pi$
$757$	− 19.4558i	− 0.707135i	0.935409	−	0.353567i	$-0.115031\pi$
$758$	− 8.76955i	− 0.318524i
$759$	28.1421	1.02149
$760$	0	0
$761$	−13.3137	−0.482622	−0.241311	−	0.970448i	$-0.577577\pi$
$761$	−13.3137	−0.482622	−0.241311	+	0.970448i	$0.577577\pi$
$762$	13.3137i	0.482305i
$763$	0	0
$764$	−13.1127	−0.474401
$765$	0	0
$766$	−7.00000	−0.252920
$767$	10.3431i	0.373469i
$768$	9.58579i	0.345897i
$769$	−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
$772$	− 3.65685i	− 0.131613i
$773$	25.1127i	0.903241i	0.892210	+	0.451620i	$0.149154\pi$
$773$	25.1127i	0.903241i	−0.892210	+	0.451620i	$0.850846\pi$
$774$	7.51472	0.270111
$775$	0	0
$776$	0.544156	0.0195341
$777$	0	0
$778$	− 5.11270i	− 0.183299i
$779$	−6.14214	−0.220065
$780$	0	0
$781$	−60.2843	−2.15714
$782$	0.828427i	0.0296245i
$783$	0.414214i	0.0148028i
$784$	0	0
$785$	0	0
$786$	−3.31371	−0.118196
$787$	28.5563i	1.01792i	0.860789	+	0.508962i	$0.169971\pi$
$787$	28.5563i	1.01792i	−0.860789	+	0.508962i	$0.830029\pi$
$788$	43.2548i	1.54089i
$789$	−45.9706	−1.63660
$790$	0	0
$791$	0	0
$792$	− 21.6569i	− 0.769543i
$793$	− 9.51472i	− 0.337878i
$794$	11.8579	0.420820
$795$	0	0
$796$	−3.02944	−0.107376
$797$	8.00000i	0.283375i	0.989911	+	0.141687i	$0.0452527\pi$
$797$	8.00000i	0.283375i	−0.989911	+	0.141687i	$0.954747\pi$
$798$	0	0
$799$	1.65685	0.0586153
$800$	0	0
$801$	7.51472	0.265520
$802$	3.18377i	0.112423i
$803$	23.3137i	0.822723i
$804$	54.7990	1.93261
$805$	0	0
$806$	2.05887	0.0725208
$807$	73.5269i	2.58827i
$808$	19.5269i	0.686954i
$809$	9.62742	0.338482	0.169241	−	0.985575i	$-0.445868\pi$
$809$	9.62742	0.338482	0.169241	+	0.985575i	$0.445868\pi$
$810$	0	0
$811$	24.6274	0.864786	0.432393	−	0.901685i	$-0.357669\pi$
$811$	24.6274	0.864786	0.432393	+	0.901685i	$0.357669\pi$
$812$	0	0
$813$	− 1.17157i	− 0.0410889i
$814$	0	0
$815$	0	0
$816$	−6.00000	−0.210042
$817$	18.1421i	0.634713i
$818$	− 10.2721i	− 0.359155i
$819$	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	$0.613131\pi$
$822$	1.65685i	0.0577894i
$823$	− 12.0711i	− 0.420771i	−0.977619	−	0.210385i	$-0.932528\pi$
$823$	− 12.0711i	− 0.420771i	0.977619	−	0.210385i	$-0.0674719\pi$
$824$	−0.656854	−0.0228826
$825$	0	0
$826$	0	0
$827$	− 16.2132i	− 0.563788i	−0.959446	−	0.281894i	$-0.909037\pi$
$827$	− 16.2132i	− 0.563788i	0.959446	−	0.281894i	$-0.0909627\pi$
$828$	12.4853i	0.433894i
$829$	−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
$832$	3.45584i	0.119810i
$833$	0	0
$834$	12.1421	0.420448
$835$	0	0
$836$	24.9706	0.863625
$837$	− 2.48528i	− 0.0859039i
$838$	9.65685i	0.333590i
$839$	−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
$842$	− 1.44365i	− 0.0497515i
$843$	63.4558i	2.18554i
$844$	6.42641	0.221206
$845$	0	0
$846$	−2.34315	−0.0805590
$847$	0	0
$848$	− 20.4853i	− 0.703467i
$849$	−33.7990	−1.15998
$850$	0	0
$851$	0	0
$852$	− 55.1127i	− 1.88813i
$853$	53.4558i	1.83029i	0.403121	+	0.915147i	$0.367925\pi$
$853$	53.4558i	1.83029i	−0.403121	+	0.915147i	$0.632075\pi$
$854$	0	0
$855$	0	0
$856$	4.37258	0.149452
$857$	22.2843i	0.761216i	0.924736	+	0.380608i	$0.124285\pi$
$857$	22.2843i	0.761216i	−0.924736	+	0.380608i	$0.875715\pi$
$858$	− 4.00000i	− 0.136558i
$859$	−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
$862$	− 9.02944i	− 0.307544i
$863$	− 16.5563i	− 0.563585i	−0.959475	−	0.281792i	$-0.909071\pi$
$863$	− 16.5563i	− 0.563585i	0.959475	−	0.281792i	$-0.0909289\pi$
$864$	−1.82843	−0.0622044
$865$	0	0
$866$	13.1716	0.447588
$867$	39.3848i	1.33758i
$868$	0	0
$869$	−44.2843	−1.50224
$870$	0	0
$871$	10.2843	0.348469
$872$	− 5.52691i	− 0.187165i
$873$	0.970563i	0.0328486i
$874$	2.82843	0.0956730
$875$	0	0
$876$	−21.3137	−0.720123
$877$	− 30.8284i	− 1.04100i	−0.853861	−	0.520501i	$-0.825745\pi$
$877$	− 30.8284i	− 1.04100i	0.853861	−	0.520501i	$-0.174255\pi$
$878$	− 14.0589i	− 0.474464i
$879$	38.6274	1.30287
$880$	0	0
$881$	−3.82843	−0.128983	−0.0644915	−	0.997918i	$-0.520543\pi$
$881$	−3.82843	−0.128983	−0.0644915	+	0.997918i	$0.520543\pi$
$882$	0	0
$883$	− 38.2843i	− 1.28837i	−0.764870	−	0.644184i	$-0.777197\pi$
$883$	− 38.2843i	− 1.28837i	0.764870	−	0.644184i	$-0.222803\pi$
$884$	−1.25483	−0.0422046
$885$	0	0
$886$	5.05887	0.169956
$887$	− 44.0711i	− 1.47976i	−0.672738	−	0.739881i	$-0.734882\pi$
$887$	− 44.0711i	− 1.47976i	0.672738	−	0.739881i	$-0.265118\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−45.7990	−1.53432
$892$	21.3137i	0.713636i
$893$	− 5.65685i	− 0.189299i
$894$	−7.82843	−0.261822
$895$	0	0
$896$	0	0
$897$	4.82843i	0.161216i
$898$	0.757359i	0.0252734i
$899$	−6.00000	−0.200111
$900$	0	0
$901$	5.65685	0.188457
$902$	− 4.34315i	− 0.144611i
$903$	0	0
$904$	−19.7990	−0.658505
$905$	0	0
$906$	−0.343146	−0.0114003
$907$	− 28.2132i	− 0.936804i	−0.883515	−	0.468402i	$-0.844830\pi$
$907$	− 28.2132i	− 0.936804i	0.883515	−	0.468402i	$-0.155170\pi$
$908$	− 49.3137i	− 1.63653i
$909$	−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	$0.808801\pi$
$912$	20.4853i	0.678335i
$913$	− 56.6274i	− 1.87409i
$914$	−13.3726	−0.442326
$915$	0	0
$916$	0.627417	0.0207304
$917$	0	0
$918$	− 0.142136i	− 0.00469117i
$919$	−19.1127	−0.630470	−0.315235	−	0.949014i	$-0.602083\pi$
$919$	−19.1127	−0.630470	−0.315235	+	0.949014i	$0.602083\pi$
$920$	0	0
$921$	−31.9706	−1.05347
$922$	7.74012i	0.254907i
$923$	− 10.3431i	− 0.340449i
$924$	0	0
$925$	0	0
$926$	−4.57359	−0.150298
$927$	− 1.17157i	− 0.0384795i
$928$	4.41421i	0.144904i
$929$	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
$932$	− 20.4264i	− 0.669089i
$933$	− 45.4558i	− 1.48816i
$934$	−9.48528	−0.310368
$935$	0	0
$936$	3.71573	0.121452
$937$	10.6274i	0.347183i	0.984818	+	0.173591i	$0.0555372\pi$
$937$	10.6274i	0.347183i	−0.984818	+	0.173591i	$0.944463\pi$
$938$	0	0
$939$	42.6274	1.39109
$940$	0	0
$941$	10.2843	0.335258	0.167629	−	0.985850i	$-0.446389\pi$
$941$	10.2843	0.335258	0.167629	+	0.985850i	$0.446389\pi$
$942$	− 5.31371i	− 0.173130i
$943$	5.24264i	0.170724i
$944$	37.4558	1.21908
$945$	0	0
$946$	−12.8284	−0.417088
$947$	43.1838i	1.40328i	0.712530	+	0.701642i	$0.247549\pi$
$947$	43.1838i	1.40328i	−0.712530	+	0.701642i	$0.752451\pi$
$948$	− 40.4853i	− 1.31490i
$949$	−4.00000	−0.129845
$950$	0	0
$951$	62.2843	2.01971
$952$	0	0
$953$	− 2.34315i	− 0.0759019i	−0.999280	−	0.0379510i	$-0.987917\pi$
$953$	− 2.34315i	− 0.0759019i	0.999280	−	0.0379510i	$-0.0120831\pi$
$954$	−8.00000	−0.259010
$955$	0	0
$956$	−2.40202	−0.0776869
$957$	11.6569i	0.376813i
$958$	10.0833i	0.325775i
$959$	0	0
$960$	0	0
$961$	5.00000	0.161290
$962$	0	0
$963$	7.79899i	0.251319i
$964$	29.8823	0.962442
$965$	0	0
$966$	0	0
$967$	27.5269i	0.885206i	0.896718	+	0.442603i	$0.145945\pi$
$967$	27.5269i	0.885206i	−0.896718	+	0.442603i	$0.854055\pi$
$968$	19.5269i	0.627619i
$969$	−5.65685	−0.181724
$970$	0	0
$971$	−24.0000	−0.770197	−0.385098	−	0.922876i	$-0.625832\pi$
$971$	−24.0000	−0.770197	−0.385098	+	0.922876i	$0.625832\pi$
$972$	− 39.5980i	− 1.27011i
$973$	0	0
$974$	6.48528	0.207802
$975$	0	0
$976$	−34.4558	−1.10290
$977$	− 21.3137i	− 0.681886i	−0.940084	−	0.340943i	$-0.889254\pi$
$977$	− 21.3137i	− 0.681886i	0.940084	−	0.340943i	$-0.110746\pi$
$978$	− 23.6569i	− 0.756463i
$979$	−12.8284	−0.409998
$980$	0	0
$981$	9.85786	0.314737
$982$	− 5.51472i	− 0.175982i
$983$	− 14.2132i	− 0.453331i	−0.973973	−	0.226665i	$-0.927218\pi$
$983$	− 14.2132i	− 0.453331i	0.973973	−	0.226665i	$-0.0727824\pi$
$984$	8.31371	0.265031
$985$	0	0
$986$	−0.343146	−0.0109280
$987$	0	0
$988$	4.28427i	0.136301i
$989$	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	$0.579996\pi$
$992$	− 26.4853i	− 0.840909i
$993$	− 26.4853i	− 0.840485i
$994$	0	0
$995$	0	0
$996$	51.7696	1.64038
$997$	17.4558i	0.552832i	0.961038	+	0.276416i	$0.0891468\pi$
$997$	17.4558i	0.552832i	−0.961038	+	0.276416i	$0.910853\pi$
$998$	− 2.00000i	− 0.0633089i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1225.2.b.g.99.3		4
5.2	odd	4		245.2.a.h.1.1		2
5.3	odd	4		1225.2.a.k.1.2		2
5.4	even	2	inner	1225.2.b.g.99.2		4
7.2	even	3		175.2.k.a.74.3		8
7.4	even	3		175.2.k.a.149.2		8
7.6	odd	2		1225.2.b.h.99.3		4
15.2	even	4		2205.2.a.n.1.2		2
20.7	even	4		3920.2.a.bq.1.1		2
35.2	odd	12		35.2.e.a.11.2	✓	4
35.4	even	6		175.2.k.a.149.3		8
35.9	even	6		175.2.k.a.74.2		8
35.12	even	12		245.2.e.e.116.2		4
35.13	even	4		1225.2.a.m.1.2		2
35.17	even	12		245.2.e.e.226.2		4
35.18	odd	12		175.2.e.c.51.1		4
35.23	odd	12		175.2.e.c.151.1		4
35.27	even	4		245.2.a.g.1.1		2
35.32	odd	12		35.2.e.a.16.2	yes	4
35.34	odd	2		1225.2.b.h.99.2		4
105.2	even	12		315.2.j.e.46.1		4
105.32	even	12		315.2.j.e.226.1		4
105.62	odd	4		2205.2.a.q.1.2		2
140.27	odd	4		3920.2.a.bv.1.2		2
140.67	even	12		560.2.q.k.401.2		4
140.107	even	12		560.2.q.k.81.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
35.2.e.a.11.2	✓	4	35.2	odd	12
35.2.e.a.16.2	yes	4	35.32	odd	12
175.2.e.c.51.1		4	35.18	odd	12
175.2.e.c.151.1		4	35.23	odd	12
175.2.k.a.74.2		8	35.9	even	6
175.2.k.a.74.3		8	7.2	even	3
175.2.k.a.149.2		8	7.4	even	3
175.2.k.a.149.3		8	35.4	even	6
245.2.a.g.1.1		2	35.27	even	4
245.2.a.h.1.1		2	5.2	odd	4
245.2.e.e.116.2		4	35.12	even	12
245.2.e.e.226.2		4	35.17	even	12
315.2.j.e.46.1		4	105.2	even	12
315.2.j.e.226.1		4	105.32	even	12
560.2.q.k.81.2		4	140.107	even	12
560.2.q.k.401.2		4	140.67	even	12
1225.2.a.k.1.2		2	5.3	odd	4
1225.2.a.m.1.2		2	35.13	even	4
1225.2.b.g.99.2		4	5.4	even	2	inner
1225.2.b.g.99.3		4	1.1	even	1	trivial
1225.2.b.h.99.2		4	35.34	odd	2
1225.2.b.h.99.3		4	7.6	odd	2
2205.2.a.n.1.2		2	15.2	even	4
2205.2.a.q.1.2		2	105.62	odd	4
3920.2.a.bq.1.1		2	20.7	even	4
3920.2.a.bv.1.2		2	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.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+0.414214i q^{2} +2.41421i q^{3} +1.82843 q^{4} -1.00000 q^{6} +1.58579i q^{8} -2.82843 q^{9} +O(q^{10})\)	\(q+0.414214i q^{2} +2.41421i q^{3} +1.82843 q^{4} -1.00000 q^{6} +1.58579i q^{8} -2.82843 q^{9} +4.82843 q^{11} +4.41421i q^{12} +0.828427i q^{13} +3.00000 q^{16} +0.828427i q^{17} -1.17157i q^{18} +2.82843 q^{19} +2.00000i q^{22} -2.41421i q^{23} -3.82843 q^{24} -0.343146 q^{26} +0.414214i q^{27} +1.00000 q^{29} -6.00000 q^{31} +4.41421i q^{32} +11.6569i q^{33} -0.343146 q^{34} -5.17157 q^{36} +1.17157i q^{38} -2.00000 q^{39} -2.17157 q^{41} +6.41421i q^{43} +8.82843 q^{44} +1.00000 q^{46} -2.00000i q^{47} +7.24264i q^{48} -2.00000 q^{51} +1.51472i q^{52} -6.82843i q^{53} -0.171573 q^{54} +6.82843i q^{57} +0.414214i q^{58} +12.4853 q^{59} -11.4853 q^{61} -2.48528i q^{62} +4.17157 q^{64} -4.82843 q^{66} -12.4142i q^{67} +1.51472i q^{68} +5.82843 q^{69} -12.4853 q^{71} -4.48528i q^{72} +4.82843i q^{73} +5.17157 q^{76} -0.828427i q^{78} -9.17157 q^{79} -9.48528 q^{81} -0.899495i q^{82} -11.7279i q^{83} -2.65685 q^{86} +2.41421i q^{87} +7.65685i q^{88} -2.65685 q^{89} -4.41421i q^{92} -14.4853i q^{93} +0.828427 q^{94} -10.6569 q^{96} -0.343146i q^{97} -13.6569 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{4} - 4 q^{6}+O(q^{10}) \) 4 * q - 4 * q^4 - 4 * q^6	\( 4 q - 4 q^{4} - 4 q^{6} + 8 q^{11} + 12 q^{16} - 4 q^{24} - 24 q^{26} + 4 q^{29} - 24 q^{31} - 24 q^{34} - 32 q^{36} - 8 q^{39} - 20 q^{41} + 24 q^{44} + 4 q^{46} - 8 q^{51} - 12 q^{54} + 16 q^{59} - 12 q^{61} + 28 q^{64} - 8 q^{66} + 12 q^{69} - 16 q^{71} + 32 q^{76} - 48 q^{79} - 4 q^{81} + 12 q^{86} + 12 q^{89} - 8 q^{94} - 20 q^{96} - 32 q^{99}+O(q^{100}) \) 4 * q - 4 * q^4 - 4 * q^6 + 8 * q^11 + 12 * q^16 - 4 * q^24 - 24 * q^26 + 4 * q^29 - 24 * q^31 - 24 * q^34 - 32 * q^36 - 8 * q^39 - 20 * q^41 + 24 * q^44 + 4 * q^46 - 8 * q^51 - 12 * q^54 + 16 * q^59 - 12 * q^61 + 28 * q^64 - 8 * q^66 + 12 * q^69 - 16 * q^71 + 32 * q^76 - 48 * q^79 - 4 * q^81 + 12 * q^86 + 12 * q^89 - 8 * q^94 - 20 * q^96 - 32 * q^99

Introduction

L-functions

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