LMFDB - Embedded newform 1225.2.b.l.99.1

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:	$6$
Coefficient field:	6.0.4227136.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} + 9x^{4} + 22x^{2} + 9 $ x^6 + 9x^4 + 22x^2 + 9
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 175)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			99.1
Root			$-2.19869i$ of defining polynomial
Character	$\chi$	$=$	1225.99
Dual form			1225.2.b.l.99.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.19869i q^{2} -2.83424i q^{3} -2.83424 q^{4} -6.23163 q^{6} +1.83424i q^{8} -5.03293 q^{9} +O(q^{10})$	\(q-2.19869i q^{2} -2.83424i q^{3} -2.83424 q^{4} -6.23163 q^{6} +1.83424i q^{8} -5.03293 q^{9} -2.56314 q^{11} +8.03293i q^{12} +0.563139i q^{13} -1.63555 q^{16} +5.19869i q^{17} +11.0659i q^{18} -0.469795 q^{19} +5.63555i q^{22} -4.03293i q^{23} +5.19869 q^{24} +1.23817 q^{26} +5.76183i q^{27} -2.86718 q^{29} -3.86718 q^{31} +7.26456i q^{32} +7.26456i q^{33} +11.4303 q^{34} +14.2646 q^{36} +2.23163i q^{37} +1.03293i q^{38} +1.59607 q^{39} +3.30404 q^{41} -10.4303i q^{43} +7.26456 q^{44} -8.86718 q^{46} +9.36445i q^{47} +4.63555i q^{48} +14.7344 q^{51} -1.59607i q^{52} -10.7014i q^{53} +12.6685 q^{54} +1.33151i q^{57} +6.30404i q^{58} -3.96052 q^{59} -3.86718 q^{61} +8.50273i q^{62} +12.7014 q^{64} +15.9725 q^{66} -7.66849i q^{67} -14.7344i q^{68} -11.4303 q^{69} -8.73436 q^{71} -9.23163i q^{72} +11.3370i q^{73} +4.90666 q^{74} +1.33151 q^{76} -3.50927i q^{78} -5.15921 q^{79} +1.23163 q^{81} -7.26456i q^{82} -3.13282i q^{83} -22.9330 q^{86} +8.12628i q^{87} -4.70142i q^{88} +5.57514 q^{89} +11.4303i q^{92} +10.9605i q^{93} +20.5895 q^{94} +20.5895 q^{96} -15.2975i q^{97} +12.9001 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 6 q^{4} - 4 q^{6} - 8 q^{9}+O(q^{10}) $ 6 * q - 6 * q^4 - 4 * q^6 - 8 * q^9	$ 6 q - 6 q^{4} - 4 q^{6} - 8 q^{9} - 4 q^{11} - 10 q^{16} + 8 q^{19} + 20 q^{24} + 30 q^{26} + 16 q^{29} + 10 q^{31} + 24 q^{34} + 30 q^{36} - 24 q^{39} - 2 q^{41} - 12 q^{44} - 20 q^{46} + 22 q^{51} + 54 q^{54} + 10 q^{59} + 10 q^{61} + 32 q^{64} + 52 q^{66} - 24 q^{69} + 14 q^{71} + 30 q^{74} + 30 q^{76} + 14 q^{79} - 26 q^{81} - 60 q^{86} + 12 q^{89} + 34 q^{94} + 34 q^{96} + 22 q^{99}+O(q^{100}) $ 6 * q - 6 * q^4 - 4 * q^6 - 8 * q^9 - 4 * q^11 - 10 * q^16 + 8 * q^19 + 20 * q^24 + 30 * q^26 + 16 * q^29 + 10 * q^31 + 24 * q^34 + 30 * q^36 - 24 * q^39 - 2 * q^41 - 12 * q^44 - 20 * q^46 + 22 * q^51 + 54 * q^54 + 10 * q^59 + 10 * q^61 + 32 * q^64 + 52 * q^66 - 24 * q^69 + 14 * q^71 + 30 * q^74 + 30 * q^76 + 14 * q^79 - 26 * q^81 - 60 * q^86 + 12 * q^89 + 34 * q^94 + 34 * q^96 + 22 * 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 $

$2$	− 2.19869i	− 1.55471i	−0.629062	−	0.777355i	$-0.716561\pi$
$2$	− 2.19869i	− 1.55471i	0.629062	−	0.777355i	$-0.283439\pi$
$3$	− 2.83424i	− 1.63635i	−0.574969	−	0.818176i	$-0.694986\pi$
$3$	− 2.83424i	− 1.63635i	0.574969	−	0.818176i	$-0.305014\pi$
$4$	−2.83424	−1.41712
$5$	0	0
$5$	0	0
$6$	−6.23163	−2.54405
$7$	0	0
$7$	0	0
$8$	1.83424i	0.648503i
$9$	−5.03293	−1.67764
$10$	0	0
$11$	−2.56314	−0.772816	−0.386408	−	0.922328i	$-0.626284\pi$
$11$	−2.56314	−0.772816	−0.386408	+	0.922328i	$0.626284\pi$
$12$	8.03293i	2.31891i
$13$	0.563139i	0.156187i	0.996946	+	0.0780934i	$0.0248832\pi$
$13$	0.563139i	0.156187i	−0.996946	+	0.0780934i	$0.975117\pi$
$14$	0	0
$15$	0	0
$16$	−1.63555	−0.408888
$17$	5.19869i	1.26087i	0.776243	+	0.630434i	$0.217123\pi$
$17$	5.19869i	1.26087i	−0.776243	+	0.630434i	$0.782877\pi$
$18$	11.0659i	2.60825i
$19$	−0.469795	−0.107778	−0.0538892	−	0.998547i	$-0.517162\pi$
$19$	−0.469795	−0.107778	−0.0538892	+	0.998547i	$0.517162\pi$
$20$	0	0
$21$	0	0
$22$	5.63555i	1.20150i
$23$	− 4.03293i	− 0.840925i	−0.907310	−	0.420462i	$-0.861868\pi$
$23$	− 4.03293i	− 0.840925i	0.907310	−	0.420462i	$-0.138132\pi$
$24$	5.19869	1.06118
$25$	0	0
$26$	1.23817	0.242825
$27$	5.76183i	1.10886i
$28$	0	0
$29$	−2.86718	−0.532422	−0.266211	−	0.963915i	$-0.585772\pi$
$29$	−2.86718	−0.532422	−0.266211	+	0.963915i	$0.585772\pi$
$30$	0	0
$31$	−3.86718	−0.694566	−0.347283	−	0.937760i	$-0.612896\pi$
$31$	−3.86718	−0.694566	−0.347283	+	0.937760i	$0.612896\pi$
$32$	7.26456i	1.28420i
$33$	7.26456i	1.26460i
$34$	11.4303	1.96028
$35$	0	0
$36$	14.2646	2.37743
$37$	2.23163i	0.366877i	0.983031	+	0.183439i	$0.0587228\pi$
$37$	2.23163i	0.366877i	−0.983031	+	0.183439i	$0.941277\pi$
$38$	1.03293i	0.167564i
$39$	1.59607	0.255576
$40$	0	0
$41$	3.30404	0.516004	0.258002	−	0.966144i	$-0.416936\pi$
$41$	3.30404	0.516004	0.258002	+	0.966144i	$0.416936\pi$
$42$	0	0
$43$	− 10.4303i	− 1.59061i	−0.606211	−	0.795304i	$-0.707311\pi$
$43$	− 10.4303i	− 1.59061i	0.606211	−	0.795304i	$-0.292689\pi$
$44$	7.26456	1.09517
$45$	0	0
$46$	−8.86718	−1.30739
$47$	9.36445i	1.36595i	0.730444	+	0.682973i	$0.239313\pi$
$47$	9.36445i	1.36595i	−0.730444	+	0.682973i	$0.760687\pi$
$48$	4.63555i	0.669084i
$49$	0	0
$50$	0	0
$51$	14.7344	2.06322
$52$	− 1.59607i	− 0.221336i
$53$	− 10.7014i	− 1.46995i	−0.678092	−	0.734977i	$-0.737193\pi$
$53$	− 10.7014i	− 1.46995i	0.678092	−	0.734977i	$-0.262807\pi$
$54$	12.6685	1.72396
$55$	0	0
$56$	0	0
$57$	1.33151i	0.176363i
$58$	6.30404i	0.827761i
$59$	−3.96052	−0.515616	−0.257808	−	0.966196i	$-0.583000\pi$
$59$	−3.96052	−0.515616	−0.257808	+	0.966196i	$0.583000\pi$
$60$	0	0
$61$	−3.86718	−0.495141	−0.247571	−	0.968870i	$-0.579632\pi$
$61$	−3.86718	−0.495141	−0.247571	+	0.968870i	$0.579632\pi$
$62$	8.50273i	1.07985i
$63$	0	0
$64$	12.7014	1.58768
$65$	0	0
$66$	15.9725	1.96608
$67$	− 7.66849i	− 0.936855i	−0.883502	−	0.468427i	$-0.844821\pi$
$67$	− 7.66849i	− 0.936855i	0.883502	−	0.468427i	$-0.155179\pi$
$68$	− 14.7344i	− 1.78680i
$69$	−11.4303	−1.37605
$70$	0	0
$71$	−8.73436	−1.03658	−0.518289	−	0.855206i	$-0.673431\pi$
$71$	−8.73436	−1.03658	−0.518289	+	0.855206i	$0.673431\pi$
$72$	− 9.23163i	− 1.08796i
$73$	11.3370i	1.32689i	0.748224	+	0.663446i	$0.230907\pi$
$73$	11.3370i	1.32689i	−0.748224	+	0.663446i	$0.769093\pi$
$74$	4.90666	0.570387
$75$	0	0
$76$	1.33151	0.152735
$77$	0	0
$78$	− 3.50927i	− 0.397347i
$79$	−5.15921	−0.580457	−0.290228	−	0.956957i	$-0.593731\pi$
$79$	−5.15921	−0.580457	−0.290228	+	0.956957i	$0.593731\pi$
$80$	0	0
$81$	1.23163	0.136847
$82$	− 7.26456i	− 0.802236i
$83$	− 3.13282i	− 0.343872i	−0.985108	−	0.171936i	$-0.944998\pi$
$83$	− 3.13282i	− 0.343872i	0.985108	−	0.171936i	$-0.0550022\pi$
$84$	0	0
$85$	0	0
$86$	−22.9330	−2.47293
$87$	8.12628i	0.871229i
$88$	− 4.70142i	− 0.501173i
$89$	5.57514	0.590964	0.295482	−	0.955348i	$-0.404520\pi$
$89$	5.57514	0.590964	0.295482	+	0.955348i	$0.404520\pi$
$90$	0	0
$91$	0	0
$92$	11.4303i	1.19169i
$93$	10.9605i	1.13655i
$94$	20.5895	2.12365
$95$	0	0
$96$	20.5895	2.10141
$97$	− 15.2975i	− 1.55323i	−0.629978	−	0.776613i	$-0.716936\pi$
$97$	− 15.2975i	− 1.55323i	0.629978	−	0.776613i	$-0.283064\pi$
$98$	0	0
$99$	12.9001	1.29651
$100$	0	0
$101$	4.99346	0.496867	0.248434	−	0.968649i	$-0.420084\pi$
$101$	4.99346	0.496867	0.248434	+	0.968649i	$0.420084\pi$
$102$	− 32.3963i	− 3.20771i
$103$	− 20.0264i	− 1.97326i	−0.162980	−	0.986629i	$-0.552110\pi$
$103$	− 20.0264i	− 1.97326i	0.162980	−	0.986629i	$-0.447890\pi$
$104$	−1.03293	−0.101288
$105$	0	0
$106$	−23.5291	−2.28535
$107$	− 11.3699i	− 1.09917i	−0.835438	−	0.549585i	$-0.814786\pi$
$107$	− 11.3699i	− 1.09917i	0.835438	−	0.549585i	$-0.185214\pi$
$108$	− 16.3304i	− 1.57140i
$109$	−8.96052	−0.858262	−0.429131	−	0.903242i	$-0.641180\pi$
$109$	−8.96052	−0.858262	−0.429131	+	0.903242i	$0.641180\pi$
$110$	0	0
$111$	6.32497	0.600340
$112$	0	0
$113$	11.8672i	1.11637i	0.829716	+	0.558185i	$0.188502\pi$
$113$	11.8672i	1.11637i	−0.829716	+	0.558185i	$0.811498\pi$
$114$	2.92759	0.274194
$115$	0	0
$116$	8.12628	0.754506
$117$	− 2.83424i	− 0.262026i
$118$	8.70796i	0.801633i
$119$	0	0
$120$	0	0
$121$	−4.43032	−0.402756
$122$	8.50273i	0.769801i
$123$	− 9.36445i	− 0.844364i
$124$	10.9605	0.984284
$125$	0	0
$126$	0	0
$127$	− 12.4303i	− 1.10301i	−0.834171	−	0.551506i	$-0.814053\pi$
$127$	− 12.4303i	− 1.10301i	0.834171	−	0.551506i	$-0.185947\pi$
$128$	− 13.3974i	− 1.18417i
$129$	−29.5621	−2.60279
$130$	0	0
$131$	4.99346	0.436280	0.218140	−	0.975917i	$-0.430001\pi$
$131$	4.99346	0.436280	0.218140	+	0.975917i	$0.430001\pi$
$132$	− 20.5895i	− 1.79209i
$133$	0	0
$134$	−16.8606	−1.45654
$135$	0	0
$136$	−9.53566	−0.817676
$137$	0.509273i	0.0435102i	0.999763	+	0.0217551i	$0.00692540\pi$
$137$	0.509273i	0.0435102i	−0.999763	+	0.0217551i	$0.993075\pi$
$138$	25.1317i	2.13936i
$139$	−22.2975	−1.89125	−0.945624	−	0.325261i	$-0.894548\pi$
$139$	−22.2975	−1.89125	−0.945624	+	0.325261i	$0.894548\pi$
$140$	0	0
$141$	26.5411	2.23517
$142$	19.2042i	1.61158i
$143$	− 1.44340i	− 0.120704i
$144$	8.23163	0.685969
$145$	0	0
$146$	24.9265	2.06293
$147$	0	0
$148$	− 6.32497i	− 0.519909i
$149$	−10.4698	−0.857719	−0.428860	−	0.903371i	$-0.641084\pi$
$149$	−10.4698	−0.857719	−0.428860	+	0.903371i	$0.641084\pi$
$150$	0	0
$151$	9.99346	0.813256	0.406628	−	0.913594i	$-0.366705\pi$
$151$	9.99346	0.813256	0.406628	+	0.913594i	$0.366705\pi$
$152$	− 0.861719i	− 0.0698946i
$153$	− 26.1647i	− 2.11529i
$154$	0	0
$155$	0	0
$156$	−4.52366	−0.362183
$157$	− 6.62355i	− 0.528617i	−0.964438	−	0.264308i	$-0.914856\pi$
$157$	− 6.62355i	− 0.528617i	0.964438	−	0.264308i	$-0.0851437\pi$
$158$	11.3435i	0.902442i
$159$	−30.3304	−2.40536
$160$	0	0
$161$	0	0
$162$	− 2.70796i	− 0.212758i
$163$	7.21962i	0.565484i	0.959196	+	0.282742i	$0.0912441\pi$
$163$	7.21962i	0.565484i	−0.959196	+	0.282742i	$0.908756\pi$
$164$	−9.36445	−0.731241
$165$	0	0
$166$	−6.88811	−0.534621
$167$	21.1712i	1.63828i	0.573595	+	0.819139i	$0.305548\pi$
$167$	21.1712i	1.63828i	−0.573595	+	0.819139i	$0.694452\pi$
$168$	0	0
$169$	12.6829	0.975606
$170$	0	0
$171$	2.36445	0.180814
$172$	29.5621i	2.25409i
$173$	− 7.46980i	− 0.567918i	−0.958836	−	0.283959i	$-0.908352\pi$
$173$	− 7.46980i	− 0.567918i	0.958836	−	0.283959i	$-0.0916480\pi$
$174$	17.8672	1.35451
$175$	0	0
$176$	4.19215	0.315995
$177$	11.2251i	0.843729i
$178$	− 12.2580i	− 0.918777i
$179$	0.728896	0.0544803	0.0272401	−	0.999629i	$-0.491328\pi$
$179$	0.728896	0.0544803	0.0272401	+	0.999629i	$0.491328\pi$
$180$	0	0
$181$	−2.00654	−0.149145	−0.0745726	−	0.997216i	$-0.523759\pi$
$181$	−2.00654	−0.149145	−0.0745726	+	0.997216i	$0.523759\pi$
$182$	0	0
$183$	10.9605i	0.810225i
$184$	7.39738	0.545342
$185$	0	0
$186$	24.0988	1.76701
$187$	− 13.3250i	− 0.974418i
$188$	− 26.5411i	− 1.93571i
$189$	0	0
$190$	0	0
$191$	2.12628	0.153852	0.0769261	−	0.997037i	$-0.475489\pi$
$191$	2.12628	0.153852	0.0769261	+	0.997037i	$0.475489\pi$
$192$	− 35.9989i	− 2.59800i
$193$	2.64101i	0.190104i	0.995472	+	0.0950521i	$0.0303018\pi$
$193$	2.64101i	0.190104i	−0.995472	+	0.0950521i	$0.969698\pi$
$194$	−33.6345	−2.41481
$195$	0	0
$196$	0	0
$197$	6.17122i	0.439681i	0.975536	+	0.219840i	$0.0705537\pi$
$197$	6.17122i	0.439681i	−0.975536	+	0.219840i	$0.929446\pi$
$198$	− 28.3634i	− 2.01570i
$199$	22.7673	1.61393	0.806965	−	0.590599i	$-0.201108\pi$
$199$	22.7673	1.61393	0.806965	+	0.590599i	$0.201108\pi$
$200$	0	0
$201$	−21.7344	−1.53302
$202$	− 10.9791i	− 0.772485i
$203$	0	0
$204$	−41.7607	−2.92384
$205$	0	0
$206$	−44.0318	−3.06784
$207$	20.2975i	1.41077i
$208$	− 0.921044i	− 0.0638629i
$209$	1.20415	0.0832928
$210$	0	0
$211$	−15.8672	−1.09234	−0.546171	−	0.837674i	$-0.683915\pi$
$211$	−15.8672	−1.09234	−0.546171	+	0.837674i	$0.683915\pi$
$212$	30.3304i	2.08310i
$213$	24.7553i	1.69620i
$214$	−24.9989	−1.70889
$215$	0	0
$216$	−10.5686	−0.719102
$217$	0	0
$218$	19.7014i	1.33435i
$219$	32.1317	2.17126
$220$	0	0
$221$	−2.92759	−0.196931
$222$	− 13.9067i	− 0.933354i
$223$	− 7.56314i	− 0.506465i	−0.967405	−	0.253233i	$-0.918506\pi$
$223$	− 7.56314i	− 0.506465i	0.967405	−	0.253233i	$-0.0814938\pi$
$224$	0	0
$225$	0	0
$226$	26.0923	1.73563
$227$	13.3250i	0.884409i	0.896914	+	0.442205i	$0.145804\pi$
$227$	13.3250i	0.884409i	−0.896914	+	0.442205i	$0.854196\pi$
$228$	− 3.77383i	− 0.249928i
$229$	12.8212	0.847246	0.423623	−	0.905839i	$-0.360758\pi$
$229$	12.8212	0.847246	0.423623	+	0.905839i	$0.360758\pi$
$230$	0	0
$231$	0	0
$232$	− 5.25910i	− 0.345277i
$233$	20.6135i	1.35044i	0.737617	+	0.675219i	$0.235951\pi$
$233$	20.6135i	1.35044i	−0.737617	+	0.675219i	$0.764049\pi$
$234$	−6.23163	−0.407374
$235$	0	0
$236$	11.2251	0.730691
$237$	14.6225i	0.949831i
$238$	0	0
$239$	15.4753	1.00101	0.500505	−	0.865733i	$-0.333148\pi$
$239$	15.4753	1.00101	0.500505	+	0.865733i	$0.333148\pi$
$240$	0	0
$241$	11.0384	0.711045	0.355523	−	0.934668i	$-0.384303\pi$
$241$	11.0384	0.711045	0.355523	+	0.934668i	$0.384303\pi$
$242$	9.74090i	0.626169i
$243$	13.7948i	0.884935i
$244$	10.9605	0.701676
$245$	0	0
$246$	−20.5895	−1.31274
$247$	− 0.264560i	− 0.0168336i
$248$	− 7.09334i	− 0.450428i
$249$	−8.87918	−0.562695
$250$	0	0
$251$	−27.8541	−1.75813	−0.879067	−	0.476698i	$-0.841834\pi$
$251$	−27.8541	−1.75813	−0.879067	+	0.476698i	$0.841834\pi$
$252$	0	0
$253$	10.3370i	0.649880i
$254$	−27.3304	−1.71486
$255$	0	0
$256$	−4.05387	−0.253367
$257$	14.5291i	0.906302i	0.891434	+	0.453151i	$0.149700\pi$
$257$	14.5291i	0.906302i	−0.891434	+	0.453151i	$0.850300\pi$
$258$	64.9978i	4.04659i
$259$	0	0
$260$	0	0
$261$	14.4303	0.893214
$262$	− 10.9791i	− 0.678289i
$263$	0.292035i	0.0180077i	0.999959	+	0.00900384i	$0.00286605\pi$
$263$	0.292035i	0.0180077i	−0.999959	+	0.00900384i	$0.997134\pi$
$264$	−13.3250	−0.820095
$265$	0	0
$266$	0	0
$267$	− 15.8013i	− 0.967024i
$268$	21.7344i	1.32764i
$269$	−8.53675	−0.520495	−0.260247	−	0.965542i	$-0.583804\pi$
$269$	−8.53675	−0.520495	−0.260247	+	0.965542i	$0.583804\pi$
$270$	0	0
$271$	2.91211	0.176898	0.0884492	−	0.996081i	$-0.471809\pi$
$271$	2.91211	0.176898	0.0884492	+	0.996081i	$0.471809\pi$
$272$	− 8.50273i	− 0.515554i
$273$	0	0
$274$	1.11973	0.0676457
$275$	0	0
$276$	32.3963	1.95003
$277$	− 1.83970i	− 0.110537i	−0.998472	−	0.0552685i	$-0.982399\pi$
$277$	− 1.83970i	− 0.110537i	0.998472	−	0.0552685i	$-0.0176015\pi$
$278$	49.0253i	2.94034i
$279$	19.4633	1.16523
$280$	0	0
$281$	−0.608077	−0.0362748	−0.0181374	−	0.999836i	$-0.505774\pi$
$281$	−0.608077	−0.0362748	−0.0181374	+	0.999836i	$0.505774\pi$
$282$	− 58.3557i	− 3.47503i
$283$	− 11.6949i	− 0.695188i	−0.937645	−	0.347594i	$-0.886999\pi$
$283$	− 11.6949i	− 0.695188i	0.937645	−	0.347594i	$-0.113001\pi$
$284$	24.7553	1.46896
$285$	0	0
$286$	−3.17360	−0.187659
$287$	0	0
$288$	− 36.5621i	− 2.15444i
$289$	−10.0264	−0.589788
$290$	0	0
$291$	−43.3568	−2.54162
$292$	− 32.1317i	− 1.88037i
$293$	28.0253i	1.63726i	0.574324	+	0.818628i	$0.305265\pi$
$293$	28.0253i	1.63726i	−0.574324	+	0.818628i	$0.694735\pi$
$294$	0	0
$295$	0	0
$296$	−4.09334	−0.237921
$297$	− 14.7684i	− 0.856948i
$298$	23.0198i	1.33350i
$299$	2.27110	0.131341
$300$	0	0
$301$	0	0
$302$	− 21.9725i	− 1.26438i
$303$	− 14.1527i	− 0.813050i
$304$	0.768374	0.0440693
$305$	0	0
$306$	−57.5280	−3.28866
$307$	− 5.55660i	− 0.317132i	−0.987348	−	0.158566i	$-0.949313\pi$
$307$	− 5.55660i	− 0.317132i	0.987348	−	0.158566i	$-0.0506870\pi$
$308$	0	0
$309$	−56.7597	−3.22894
$310$	0	0
$311$	6.60808	0.374710	0.187355	−	0.982292i	$-0.440009\pi$
$311$	6.60808	0.374710	0.187355	+	0.982292i	$0.440009\pi$
$312$	2.92759i	0.165742i
$313$	− 11.9001i	− 0.672634i	−0.941749	−	0.336317i	$-0.890819\pi$
$313$	− 11.9001i	− 0.672634i	0.941749	−	0.336317i	$-0.109181\pi$
$314$	−14.5631	−0.821845
$315$	0	0
$316$	14.6225	0.822578
$317$	− 8.71035i	− 0.489222i	−0.969621	−	0.244611i	$-0.921340\pi$
$317$	− 8.71035i	− 0.489222i	0.969621	−	0.244611i	$-0.0786603\pi$
$318$	66.6872i	3.73964i
$319$	7.34898	0.411464
$320$	0	0
$321$	−32.2251	−1.79863
$322$	0	0
$323$	− 2.44232i	− 0.135894i
$324$	−3.49073	−0.193929
$325$	0	0
$326$	15.8737	0.879164
$327$	25.3963i	1.40442i
$328$	6.06041i	0.334630i
$329$	0	0
$330$	0	0
$331$	−12.4687	−0.685342	−0.342671	−	0.939455i	$-0.611332\pi$
$331$	−12.4687	−0.685342	−0.342671	+	0.939455i	$0.611332\pi$
$332$	8.87918i	0.487308i
$333$	− 11.2316i	− 0.615489i
$334$	46.5490	2.54705
$335$	0	0
$336$	0	0
$337$	3.61462i	0.196901i	0.995142	+	0.0984505i	$0.0313886\pi$
$337$	3.61462i	0.196901i	−0.995142	+	0.0984505i	$0.968611\pi$
$338$	− 27.8857i	− 1.51678i
$339$	33.6345	1.82677
$340$	0	0
$341$	9.91211	0.536771
$342$	− 5.19869i	− 0.281113i
$343$	0	0
$344$	19.1317	1.03151
$345$	0	0
$346$	−16.4238	−0.882948
$347$	− 21.8397i	− 1.17242i	−0.810160	−	0.586208i	$-0.800620\pi$
$347$	− 21.8397i	− 1.17242i	0.810160	−	0.586208i	$-0.199380\pi$
$348$	− 23.0318i	− 1.23464i
$349$	3.69596	0.197840	0.0989201	−	0.995095i	$-0.468461\pi$
$349$	3.69596	0.197840	0.0989201	+	0.995095i	$0.468461\pi$
$350$	0	0
$351$	−3.24471	−0.173190
$352$	− 18.6201i	− 0.992454i
$353$	− 31.6434i	− 1.68421i	−0.539315	−	0.842104i	$-0.681317\pi$
$353$	− 31.6434i	− 1.68421i	0.539315	−	0.842104i	$-0.318683\pi$
$354$	24.6805	1.31175
$355$	0	0
$356$	−15.8013	−0.837468
$357$	0	0
$358$	− 1.60262i	− 0.0847010i
$359$	17.1383	0.904524	0.452262	−	0.891885i	$-0.350617\pi$
$359$	17.1383	0.904524	0.452262	+	0.891885i	$0.350617\pi$
$360$	0	0
$361$	−18.7793	−0.988384
$362$	4.41177i	0.231877i
$363$	12.5566i	0.659050i
$364$	0	0
$365$	0	0
$366$	24.0988	1.25966
$367$	− 11.7882i	− 0.615340i	−0.951493	−	0.307670i	$-0.900451\pi$
$367$	− 11.7882i	− 0.615340i	0.951493	−	0.307670i	$-0.0995493\pi$
$368$	6.59607i	0.343844i
$369$	−16.6290	−0.865672
$370$	0	0
$371$	0	0
$372$	− 31.0648i	− 1.61063i
$373$	9.40393i	0.486917i	0.969911	+	0.243458i	$0.0782819\pi$
$373$	9.40393i	0.486917i	−0.969911	+	0.243458i	$0.921718\pi$
$374$	−29.2975	−1.51494
$375$	0	0
$376$	−17.1767	−0.885819
$377$	− 1.61462i	− 0.0831572i
$378$	0	0
$379$	17.3490	0.891157	0.445579	−	0.895243i	$-0.352998\pi$
$379$	17.3490	0.891157	0.445579	+	0.895243i	$0.352998\pi$
$380$	0	0
$381$	−35.2305	−1.80492
$382$	− 4.67503i	− 0.239195i
$383$	5.82116i	0.297447i	0.988879	+	0.148724i	$0.0475165\pi$
$383$	5.82116i	0.297447i	−0.988879	+	0.148724i	$0.952484\pi$
$384$	−37.9714	−1.93772
$385$	0	0
$386$	5.80677	0.295557
$387$	52.4951i	2.66848i
$388$	43.3568i	2.20111i
$389$	26.6279	1.35009	0.675045	−	0.737777i	$-0.264124\pi$
$389$	26.6279	1.35009	0.675045	+	0.737777i	$0.264124\pi$
$390$	0	0
$391$	20.9660	1.06030
$392$	0	0
$393$	− 14.1527i	− 0.713908i
$394$	13.5686	0.683576
$395$	0	0
$396$	−36.5621	−1.83731
$397$	− 35.2305i	− 1.76817i	−0.467326	−	0.884085i	$-0.654783\pi$
$397$	− 35.2305i	− 1.76817i	0.467326	−	0.884085i	$-0.345217\pi$
$398$	− 50.0582i	− 2.50919i
$399$	0	0
$400$	0	0
$401$	−24.4369	−1.22032	−0.610159	−	0.792279i	$-0.708895\pi$
$401$	−24.4369	−1.22032	−0.610159	+	0.792279i	$0.708895\pi$
$402$	47.7871i	2.38341i
$403$	− 2.17776i	− 0.108482i
$404$	−14.1527	−0.704122
$405$	0	0
$406$	0	0
$407$	− 5.71997i	− 0.283528i
$408$	27.0264i	1.33801i
$409$	−29.4873	−1.45805	−0.729026	−	0.684487i	$-0.760026\pi$
$409$	−29.4873	−1.45805	−0.729026	+	0.684487i	$0.760026\pi$
$410$	0	0
$411$	1.44340	0.0711979
$412$	56.7597i	2.79635i
$413$	0	0
$414$	44.6279	2.19334
$415$	0	0
$416$	−4.09096	−0.200576
$417$	63.1965i	3.09475i
$418$	− 2.64755i	− 0.129496i
$419$	−9.74090	−0.475874	−0.237937	−	0.971281i	$-0.576471\pi$
$419$	−9.74090	−0.475874	−0.237937	+	0.971281i	$0.576471\pi$
$420$	0	0
$421$	−2.68942	−0.131074	−0.0655371	−	0.997850i	$-0.520876\pi$
$421$	−2.68942	−0.131074	−0.0655371	+	0.997850i	$0.520876\pi$
$422$	34.8870i	1.69827i
$423$	− 47.1307i	− 2.29157i
$424$	19.6290	0.953269
$425$	0	0
$426$	54.4292	2.63710
$427$	0	0
$428$	32.2251i	1.55766i
$429$	−4.09096	−0.197513
$430$	0	0
$431$	−9.43686	−0.454558	−0.227279	−	0.973830i	$-0.572983\pi$
$431$	−9.43686	−0.454558	−0.227279	+	0.973830i	$0.572983\pi$
$432$	− 9.42377i	− 0.453401i
$433$	− 20.8541i	− 1.00218i	−0.865394	−	0.501092i	$-0.832932\pi$
$433$	− 20.8541i	− 1.00218i	0.865394	−	0.501092i	$-0.167068\pi$
$434$	0	0
$435$	0	0
$436$	25.3963	1.21626
$437$	1.89465i	0.0906335i
$438$	− 70.6478i	− 3.37568i
$439$	22.5621	1.07683	0.538414	−	0.842680i	$-0.319024\pi$
$439$	22.5621	1.07683	0.538414	+	0.842680i	$0.319024\pi$
$440$	0	0
$441$	0	0
$442$	6.43686i	0.306170i
$443$	− 30.5961i	− 1.45366i	−0.686816	−	0.726832i	$-0.740992\pi$
$443$	− 30.5961i	− 1.45366i	0.686816	−	0.726832i	$-0.259008\pi$
$444$	−17.9265	−0.850754
$445$	0	0
$446$	−16.6290	−0.787406
$447$	29.6739i	1.40353i
$448$	0	0
$449$	−1.25256	−0.0591118	−0.0295559	−	0.999563i	$-0.509409\pi$
$449$	−1.25256	−0.0591118	−0.0295559	+	0.999563i	$0.509409\pi$
$450$	0	0
$451$	−8.46871	−0.398776
$452$	− 33.6345i	− 1.58203i
$453$	− 28.3239i	− 1.33077i
$454$	29.2975	1.37500
$455$	0	0
$456$	−2.44232	−0.114372
$457$	− 23.7618i	− 1.11153i	−0.831339	−	0.555766i	$-0.812425\pi$
$457$	− 23.7618i	− 1.11153i	0.831339	−	0.555766i	$-0.187575\pi$
$458$	− 28.1898i	− 1.31722i
$459$	−29.9540	−1.39813
$460$	0	0
$461$	−31.0833	−1.44770	−0.723848	−	0.689960i	$-0.757628\pi$
$461$	−31.0833	−1.44770	−0.723848	+	0.689960i	$0.757628\pi$
$462$	0	0
$463$	− 24.8606i	− 1.15537i	−0.816259	−	0.577686i	$-0.803956\pi$
$463$	− 24.8606i	− 1.15537i	0.816259	−	0.577686i	$-0.196044\pi$
$464$	4.68942	0.217701
$465$	0	0
$466$	45.3228	2.09954
$467$	− 13.6674i	− 0.632452i	−0.948684	−	0.316226i	$-0.897584\pi$
$467$	− 13.6674i	− 0.632452i	0.948684	−	0.316226i	$-0.102416\pi$
$468$	8.03293i	0.371323i
$469$	0	0
$470$	0	0
$471$	−18.7727	−0.865003
$472$	− 7.26456i	− 0.334378i
$473$	26.7344i	1.22925i
$474$	32.1503	1.47671
$475$	0	0
$476$	0	0
$477$	53.8595i	2.46606i
$478$	− 34.0253i	− 1.55628i
$479$	4.16576	0.190338	0.0951691	−	0.995461i	$-0.469661\pi$
$479$	4.16576	0.190338	0.0951691	+	0.995461i	$0.469661\pi$
$480$	0	0
$481$	−1.25672	−0.0573013
$482$	− 24.2700i	− 1.10547i
$483$	0	0
$484$	12.5566	0.570754
$485$	0	0
$486$	30.3304	1.37582
$487$	− 14.1801i	− 0.642564i	−0.946984	−	0.321282i	$-0.895886\pi$
$487$	− 14.1801i	− 0.642564i	0.946984	−	0.321282i	$-0.104114\pi$
$488$	− 7.09334i	− 0.321101i
$489$	20.4622	0.925331
$490$	0	0
$491$	18.8737	0.851759	0.425880	−	0.904780i	$-0.359965\pi$
$491$	18.8737	0.851759	0.425880	+	0.904780i	$0.359965\pi$
$492$	26.5411i	1.19657i
$493$	− 14.9056i	− 0.671313i
$494$	−0.581686	−0.0261713
$495$	0	0
$496$	6.32497	0.284000
$497$	0	0
$498$	19.5226i	0.874828i
$499$	10.5237	0.471104	0.235552	−	0.971862i	$-0.424310\pi$
$499$	10.5237	0.471104	0.235552	+	0.971862i	$0.424310\pi$
$500$	0	0
$501$	60.0044	2.68080
$502$	61.2425i	2.73339i
$503$	− 19.3854i	− 0.864351i	−0.901789	−	0.432176i	$-0.857746\pi$
$503$	− 19.3854i	− 0.864351i	0.901789	−	0.432176i	$-0.142254\pi$
$504$	0	0
$505$	0	0
$506$	22.7278	1.01037
$507$	− 35.9463i	− 1.59643i
$508$	35.2305i	1.56310i
$509$	−19.0713	−0.845322	−0.422661	−	0.906288i	$-0.638904\pi$
$509$	−19.0713	−0.845322	−0.422661	+	0.906288i	$0.638904\pi$
$510$	0	0
$511$	0	0
$512$	− 17.8816i	− 0.790261i
$513$	− 2.70688i	− 0.119512i
$514$	31.9450	1.40904
$515$	0	0
$516$	83.7861	3.68848
$517$	− 24.0024i	− 1.05562i
$518$	0	0
$519$	−21.1712	−0.929313
$520$	0	0
$521$	−9.64448	−0.422532	−0.211266	−	0.977429i	$-0.567759\pi$
$521$	−9.64448	−0.422532	−0.211266	+	0.977429i	$0.567759\pi$
$522$	− 31.7278i	− 1.38869i
$523$	30.6135i	1.33864i	0.742976	+	0.669318i	$0.233414\pi$
$523$	30.6135i	1.33864i	−0.742976	+	0.669318i	$0.766586\pi$
$524$	−14.1527	−0.618262
$525$	0	0
$526$	0.642096	0.0279967
$527$	− 20.1043i	− 0.875755i
$528$	− 11.8816i	− 0.517079i
$529$	6.73544	0.292845
$530$	0	0
$531$	19.9330	0.865021
$532$	0	0
$533$	1.86063i	0.0805930i
$534$	−34.7422	−1.50344
$535$	0	0
$536$	14.0659	0.607553
$537$	− 2.06587i	− 0.0891488i
$538$	18.7697i	0.809218i
$539$	0	0
$540$	0	0
$541$	−31.6829	−1.36215	−0.681077	−	0.732212i	$-0.738488\pi$
$541$	−31.6829	−1.36215	−0.681077	+	0.732212i	$0.738488\pi$
$542$	− 6.40284i	− 0.275026i
$543$	5.68703i	0.244054i
$544$	−37.7662	−1.61921
$545$	0	0
$546$	0	0
$547$	− 5.91866i	− 0.253064i	−0.991963	−	0.126532i	$-0.959615\pi$
$547$	− 5.91866i	− 0.253064i	0.991963	−	0.126532i	$-0.0403846\pi$
$548$	− 1.44340i	− 0.0616592i
$549$	19.4633	0.830671
$550$	0	0
$551$	1.34699	0.0573835
$552$	− 20.9660i	− 0.892371i
$553$	0	0
$554$	−4.04494	−0.171853
$555$	0	0
$556$	63.1965	2.68013
$557$	− 31.5806i	− 1.33811i	−0.743212	−	0.669057i	$-0.766698\pi$
$557$	− 31.5806i	− 1.33811i	0.743212	−	0.669057i	$-0.233302\pi$
$558$	− 42.7937i	− 1.81160i
$559$	5.87372	0.248432
$560$	0	0
$561$	−37.7662	−1.59449
$562$	1.33697i	0.0563968i
$563$	20.9320i	0.882177i	0.897464	+	0.441089i	$0.145408\pi$
$563$	20.9320i	0.882177i	−0.897464	+	0.441089i	$0.854592\pi$
$564$	−75.2240	−3.16750
$565$	0	0
$566$	−25.7134	−1.08082
$567$	0	0
$568$	− 16.0209i	− 0.672223i
$569$	−26.7224	−1.12026	−0.560130	−	0.828405i	$-0.689249\pi$
$569$	−26.7224	−1.12026	−0.560130	+	0.828405i	$0.689249\pi$
$570$	0	0
$571$	23.2844	0.974422	0.487211	−	0.873284i	$-0.338014\pi$
$571$	23.2844	0.974422	0.487211	+	0.873284i	$0.338014\pi$
$572$	4.09096i	0.171052i
$573$	− 6.02639i	− 0.251756i
$574$	0	0
$575$	0	0
$576$	−63.9254	−2.66356
$577$	14.7937i	0.615869i	0.951408	+	0.307934i	$0.0996378\pi$
$577$	14.7937i	0.615869i	−0.951408	+	0.307934i	$0.900362\pi$
$578$	22.0449i	0.916949i
$579$	7.48527	0.311077
$580$	0	0
$581$	0	0
$582$	95.3283i	3.95148i
$583$	27.4292i	1.13600i
$584$	−20.7948	−0.860493
$585$	0	0
$586$	61.6190	2.54546
$587$	− 34.6334i	− 1.42947i	−0.699394	−	0.714736i	$-0.746547\pi$
$587$	− 34.6334i	− 1.42947i	0.699394	−	0.714736i	$-0.253453\pi$
$588$	0	0
$589$	1.81678	0.0748592
$590$	0	0
$591$	17.4907	0.719472
$592$	− 3.64994i	− 0.150012i
$593$	− 5.85517i	− 0.240443i	−0.992747	−	0.120222i	$-0.961639\pi$
$593$	− 5.85517i	− 0.240443i	0.992747	−	0.120222i	$-0.0383605\pi$
$594$	−32.4711	−1.33231
$595$	0	0
$596$	29.6739	1.21549
$597$	− 64.5280i	− 2.64096i
$598$	− 4.99346i	− 0.204198i
$599$	−32.7267	−1.33718	−0.668589	−	0.743632i	$-0.733101\pi$
$599$	−32.7267	−1.33718	−0.668589	+	0.743632i	$0.733101\pi$
$600$	0	0
$601$	26.9485	1.09925	0.549627	−	0.835410i	$-0.314770\pi$
$601$	26.9485	1.09925	0.549627	+	0.835410i	$0.314770\pi$
$602$	0	0
$603$	38.5950i	1.57171i
$604$	−28.3239	−1.15248
$605$	0	0
$606$	−31.1173	−1.26406
$607$	15.4094i	0.625448i	0.949844	+	0.312724i	$0.101241\pi$
$607$	15.4094i	0.625448i	−0.949844	+	0.312724i	$0.898759\pi$
$608$	− 3.41285i	− 0.138410i
$609$	0	0
$610$	0	0
$611$	−5.27349	−0.213343
$612$	74.1570i	2.99762i
$613$	− 39.8726i	− 1.61044i	−0.592976	−	0.805220i	$-0.702047\pi$
$613$	− 39.8726i	− 1.61044i	0.592976	−	0.805220i	$-0.297953\pi$
$614$	−12.2172	−0.493048
$615$	0	0
$616$	0	0
$617$	36.7409i	1.47913i	0.673084	+	0.739566i	$0.264969\pi$
$617$	36.7409i	1.47913i	−0.673084	+	0.739566i	$0.735031\pi$
$618$	124.797i	5.02007i
$619$	25.7289	1.03413	0.517066	−	0.855946i	$-0.327024\pi$
$619$	25.7289	1.03413	0.517066	+	0.855946i	$0.327024\pi$
$620$	0	0
$621$	23.2371	0.932472
$622$	− 14.5291i	− 0.582565i
$623$	0	0
$624$	−2.61046	−0.104502
$625$	0	0
$626$	−26.1647	−1.04575
$627$	− 3.41285i	− 0.136296i
$628$	18.7727i	0.749114i
$629$	−11.6015	−0.462583
$630$	0	0
$631$	2.47525	0.0985383	0.0492692	−	0.998786i	$-0.484311\pi$
$631$	2.47525	0.0985383	0.0492692	+	0.998786i	$0.484311\pi$
$632$	− 9.46325i	− 0.376428i
$633$	44.9714i	1.78745i
$634$	−19.1514	−0.760598
$635$	0	0
$636$	85.9638	3.40869
$637$	0	0
$638$	− 16.1581i	− 0.639706i
$639$	43.9594	1.73901
$640$	0	0
$641$	−35.3359	−1.39568	−0.697842	−	0.716252i	$-0.745856\pi$
$641$	−35.3359	−1.39568	−0.697842	+	0.716252i	$0.745856\pi$
$642$	70.8530i	2.79635i
$643$	− 2.98691i	− 0.117792i	−0.998264	−	0.0588962i	$-0.981242\pi$
$643$	− 2.98691i	− 0.117792i	0.998264	−	0.0588962i	$-0.0187581\pi$
$644$	0	0
$645$	0	0
$646$	−5.36991	−0.211276
$647$	26.0275i	1.02325i	0.859210	+	0.511623i	$0.170955\pi$
$647$	26.0275i	1.02325i	−0.859210	+	0.511623i	$0.829045\pi$
$648$	2.25910i	0.0887459i
$649$	10.1514	0.398476
$650$	0	0
$651$	0	0
$652$	− 20.4622i	− 0.801360i
$653$	− 17.8936i	− 0.700229i	−0.936707	−	0.350115i	$-0.886143\pi$
$653$	− 17.8936i	− 0.700229i	0.936707	−	0.350115i	$-0.113857\pi$
$654$	55.8386	2.18346
$655$	0	0
$656$	−5.40393	−0.210988
$657$	− 57.0582i	− 2.22605i
$658$	0	0
$659$	15.2076	0.592405	0.296202	−	0.955125i	$-0.404280\pi$
$659$	15.2076	0.592405	0.296202	+	0.955125i	$0.404280\pi$
$660$	0	0
$661$	−5.48180	−0.213217	−0.106609	−	0.994301i	$-0.533999\pi$
$661$	−5.48180	−0.213217	−0.106609	+	0.994301i	$0.533999\pi$
$662$	27.4148i	1.06551i
$663$	8.29749i	0.322248i
$664$	5.74636	0.223002
$665$	0	0
$666$	−24.6949	−0.956907
$667$	11.5631i	0.447727i
$668$	− 60.0044i	− 2.32164i
$669$	−21.4358	−0.828755
$670$	0	0
$671$	9.91211	0.382653
$672$	0	0
$673$	30.6763i	1.18249i	0.806494	+	0.591243i	$0.201363\pi$
$673$	30.6763i	1.18249i	−0.806494	+	0.591243i	$0.798637\pi$
$674$	7.94743	0.306124
$675$	0	0
$676$	−35.9463	−1.38255
$677$	− 5.54112i	− 0.212963i	−0.994315	−	0.106481i	$-0.966042\pi$
$677$	− 5.54112i	− 0.212963i	0.994315	−	0.106481i	$-0.0339584\pi$
$678$	− 73.9518i	− 2.84010i
$679$	0	0
$680$	0	0
$681$	37.7662	1.44720
$682$	− 21.7937i	− 0.834523i
$683$	1.90666i	0.0729562i	0.999334	+	0.0364781i	$0.0116139\pi$
$683$	1.90666i	0.0729562i	−0.999334	+	0.0364781i	$0.988386\pi$
$684$	−6.70142	−0.256235
$685$	0	0
$686$	0	0
$687$	− 36.3383i	− 1.38639i
$688$	17.0593i	0.650381i
$689$	6.02639	0.229587
$690$	0	0
$691$	−0.905571	−0.0344495	−0.0172248	−	0.999852i	$-0.505483\pi$
$691$	−0.905571	−0.0344495	−0.0172248	+	0.999852i	$0.505483\pi$
$692$	21.1712i	0.804809i
$693$	0	0
$694$	−48.0188	−1.82277
$695$	0	0
$696$	−14.9056	−0.564994
$697$	17.1767i	0.650613i
$698$	− 8.12628i	− 0.307584i
$699$	58.4238	2.20979
$700$	0	0
$701$	36.8541	1.39196	0.695980	−	0.718061i	$-0.254970\pi$
$701$	36.8541	1.39196	0.695980	+	0.718061i	$0.254970\pi$
$702$	7.13412i	0.269260i
$703$	− 1.04841i	− 0.0395414i
$704$	−32.5555	−1.22698
$705$	0	0
$706$	−69.5741	−2.61845
$707$	0	0
$708$	− 31.8146i	− 1.19567i
$709$	41.6818	1.56539	0.782696	−	0.622404i	$-0.213844\pi$
$709$	41.6818	1.56539	0.782696	+	0.622404i	$0.213844\pi$
$710$	0	0
$711$	25.9660	0.973800
$712$	10.2262i	0.383242i
$713$	15.5961i	0.584078i
$714$	0	0
$715$	0	0
$716$	−2.06587	−0.0772051
$717$	− 43.8606i	− 1.63801i
$718$	− 37.6818i	− 1.40627i
$719$	44.8595	1.67298	0.836489	−	0.547983i	$-0.184604\pi$
$719$	44.8595	1.67298	0.836489	+	0.547983i	$0.184604\pi$
$720$	0	0
$721$	0	0
$722$	41.2899i	1.53665i
$723$	− 31.2855i	− 1.16352i
$724$	5.68703	0.211357
$725$	0	0
$726$	27.6081	1.02463
$727$	9.08134i	0.336808i	0.985718	+	0.168404i	$0.0538614\pi$
$727$	9.08134i	0.336808i	−0.985718	+	0.168404i	$0.946139\pi$
$728$	0	0
$729$	42.7926	1.58491
$730$	0	0
$731$	54.2240	2.00555
$732$	− 31.0648i	− 1.14819i
$733$	− 28.4203i	− 1.04973i	−0.851186	−	0.524864i	$-0.824116\pi$
$733$	− 28.4203i	− 1.04973i	0.851186	−	0.524864i	$-0.175884\pi$
$734$	−25.9187	−0.956675
$735$	0	0
$736$	29.2975	1.07992
$737$	19.6554i	0.724016i
$738$	36.5621i	1.34587i
$739$	−5.41047	−0.199027	−0.0995137	−	0.995036i	$-0.531729\pi$
$739$	−5.41047	−0.199027	−0.0995137	+	0.995036i	$0.531729\pi$
$740$	0	0
$741$	−0.749828	−0.0275456
$742$	0	0
$743$	− 30.1712i	− 1.10687i	−0.832891	−	0.553437i	$-0.813316\pi$
$743$	− 30.1712i	− 1.10687i	0.832891	−	0.553437i	$-0.186684\pi$
$744$	−20.1043	−0.737058
$745$	0	0
$746$	20.6763	0.757014
$747$	15.7673i	0.576895i
$748$	37.7662i	1.38087i
$749$	0	0
$750$	0	0
$751$	43.9056	1.60214	0.801069	−	0.598573i	$-0.204265\pi$
$751$	43.9056	1.60214	0.801069	+	0.598573i	$0.204265\pi$
$752$	− 15.3160i	− 0.558519i
$753$	78.9453i	2.87693i
$754$	−3.55005	−0.129285
$755$	0	0
$756$	0	0
$757$	− 38.6914i	− 1.40626i	−0.711060	−	0.703132i	$-0.751784\pi$
$757$	− 38.6914i	− 1.40626i	0.711060	−	0.703132i	$-0.248216\pi$
$758$	− 38.1450i	− 1.38549i
$759$	29.2975	1.06343
$760$	0	0
$761$	−31.3293	−1.13569	−0.567844	−	0.823136i	$-0.692222\pi$
$761$	−31.3293	−1.13569	−0.567844	+	0.823136i	$0.692222\pi$
$762$	77.4611i	2.80612i
$763$	0	0
$764$	−6.02639	−0.218027
$765$	0	0
$766$	12.7989	0.462444
$767$	− 2.23033i	− 0.0805324i
$768$	11.4896i	0.414597i
$769$	−9.00654	−0.324784	−0.162392	−	0.986726i	$-0.551921\pi$
$769$	−9.00654	−0.324784	−0.162392	+	0.986726i	$0.551921\pi$
$770$	0	0
$771$	41.1791	1.48303
$772$	− 7.48527i	− 0.269401i
$773$	15.7673i	0.567110i	0.958956	+	0.283555i	$0.0915139\pi$
$773$	15.7673i	0.567110i	−0.958956	+	0.283555i	$0.908486\pi$
$774$	115.421	4.14870
$775$	0	0
$776$	28.0593	1.00727
$777$	0	0
$778$	− 58.5466i	− 2.09900i
$779$	−1.55222	−0.0556141
$780$	0	0
$781$	22.3874	0.801083
$782$	− 46.0977i	− 1.64845i
$783$	− 16.5202i	− 0.590383i
$784$	0	0
$785$	0	0
$786$	−31.1173	−1.10992
$787$	35.6225i	1.26980i	0.772593	+	0.634902i	$0.218959\pi$
$787$	35.6225i	1.26980i	−0.772593	+	0.634902i	$0.781041\pi$
$788$	− 17.4907i	− 0.623081i
$789$	0.827699	0.0294669
$790$	0	0
$791$	0	0
$792$	23.6619i	0.840791i
$793$	− 2.17776i	− 0.0773345i
$794$	−77.4611	−2.74899
$795$	0	0
$796$	−64.5280	−2.28714
$797$	39.6268i	1.40365i	0.712347	+	0.701827i	$0.247632\pi$
$797$	39.6268i	1.40365i	−0.712347	+	0.701827i	$0.752368\pi$
$798$	0	0
$799$	−48.6829	−1.72228
$800$	0	0
$801$	−28.0593	−0.991428
$802$	53.7291i	1.89724i
$803$	− 29.0582i	− 1.02544i
$804$	61.6004	2.17248
$805$	0	0
$806$	−4.78822	−0.168658
$807$	24.1952i	0.851712i
$808$	9.15921i	0.322220i
$809$	6.82531	0.239965	0.119983	−	0.992776i	$-0.461716\pi$
$809$	6.82531	0.239965	0.119983	+	0.992776i	$0.461716\pi$
$810$	0	0
$811$	46.4303	1.63039	0.815194	−	0.579187i	$-0.196630\pi$
$811$	46.4303	1.63039	0.815194	+	0.579187i	$0.196630\pi$
$812$	0	0
$813$	− 8.25364i	− 0.289468i
$814$	−12.5764	−0.440804
$815$	0	0
$816$	−24.0988	−0.843627
$817$	4.90011i	0.171433i
$818$	64.8334i	2.26685i
$819$	0	0
$820$	0	0
$821$	38.4491	1.34188	0.670941	−	0.741511i	$-0.265890\pi$
$821$	38.4491	1.34188	0.670941	+	0.741511i	$0.265890\pi$
$822$	− 3.17360i	− 0.110692i
$823$	− 22.5939i	− 0.787574i	−0.919202	−	0.393787i	$-0.871165\pi$
$823$	− 22.5939i	− 0.787574i	0.919202	−	0.393787i	$-0.128835\pi$
$824$	36.7333	1.27966
$825$	0	0
$826$	0	0
$827$	− 15.2460i	− 0.530156i	−0.964227	−	0.265078i	$-0.914602\pi$
$827$	− 15.2460i	− 0.530156i	0.964227	−	0.265078i	$-0.0853977\pi$
$828$	− 57.5280i	− 1.99924i
$829$	−12.9671	−0.450365	−0.225182	−	0.974317i	$-0.572298\pi$
$829$	−12.9671	−0.450365	−0.225182	+	0.974317i	$0.572298\pi$
$830$	0	0
$831$	−5.21416	−0.180877
$832$	7.15267i	0.247974i
$833$	0	0
$834$	138.950	4.81143
$835$	0	0
$836$	−3.41285	−0.118036
$837$	− 22.2820i	− 0.770179i
$838$	21.4172i	0.739846i
$839$	−19.4818	−0.672586	−0.336293	−	0.941757i	$-0.609173\pi$
$839$	−19.4818	−0.672586	−0.336293	+	0.941757i	$0.609173\pi$
$840$	0	0
$841$	−20.7793	−0.716527
$842$	5.91320i	0.203782i
$843$	1.72344i	0.0593583i
$844$	44.9714	1.54798
$845$	0	0
$846$	−103.626	−3.56273
$847$	0	0
$848$	17.5027i	0.601046i
$849$	−33.1461	−1.13757
$850$	0	0
$851$	9.00000	0.308516
$852$	− 70.1625i	− 2.40373i
$853$	20.0449i	0.686326i	0.939276	+	0.343163i	$0.111498\pi$
$853$	20.0449i	0.686326i	−0.939276	+	0.343163i	$0.888502\pi$
$854$	0	0
$855$	0	0
$856$	20.8552	0.712815
$857$	− 50.3383i	− 1.71952i	−0.510696	−	0.859761i	$-0.670612\pi$
$857$	− 50.3383i	− 1.71952i	0.510696	−	0.859761i	$-0.329388\pi$
$858$	8.99476i	0.307076i
$859$	−38.1396	−1.30131	−0.650653	−	0.759375i	$-0.725505\pi$
$859$	−38.1396	−1.30131	−0.650653	+	0.759375i	$0.725505\pi$
$860$	0	0
$861$	0	0
$862$	20.7487i	0.706705i
$863$	0.864103i	0.0294144i	0.999892	+	0.0147072i	$0.00468162\pi$
$863$	0.864103i	0.0294144i	−0.999892	+	0.0147072i	$0.995318\pi$
$864$	−41.8572	−1.42401
$865$	0	0
$866$	−45.8517	−1.55810
$867$	28.4172i	0.965100i
$868$	0	0
$869$	13.2238	0.448586
$870$	0	0
$871$	4.31843	0.146324
$872$	− 16.4358i	− 0.556586i
$873$	76.9913i	2.60576i
$874$	4.16576	0.140909
$875$	0	0
$876$	−91.0692	−3.07694
$877$	9.31297i	0.314477i	0.987561	+	0.157238i	$0.0502591\pi$
$877$	9.31297i	0.314477i	−0.987561	+	0.157238i	$0.949741\pi$
$878$	− 49.6070i	− 1.67415i
$879$	79.4305	2.67913
$880$	0	0
$881$	−19.4602	−0.655630	−0.327815	−	0.944742i	$-0.606312\pi$
$881$	−19.4602	−0.655630	−0.327815	+	0.944742i	$0.606312\pi$
$882$	0	0
$883$	43.1712i	1.45283i	0.687258	+	0.726414i	$0.258814\pi$
$883$	43.1712i	1.45283i	−0.687258	+	0.726414i	$0.741186\pi$
$884$	8.29749	0.279075
$885$	0	0
$886$	−67.2713	−2.26002
$887$	17.4907i	0.587281i	0.955916	+	0.293641i	$0.0948669\pi$
$887$	17.4907i	0.587281i	−0.955916	+	0.293641i	$0.905133\pi$
$888$	11.6015i	0.389322i
$889$	0	0
$890$	0	0
$891$	−3.15683	−0.105758
$892$	21.4358i	0.717723i
$893$	− 4.39937i	− 0.147219i
$894$	65.2438	2.18208
$895$	0	0
$896$	0	0
$897$	− 6.43686i	− 0.214921i
$898$	2.75399i	0.0919017i
$899$	11.0879	0.369802
$900$	0	0
$901$	55.6334	1.85342
$902$	18.6201i	0.619981i
$903$	0	0
$904$	−21.7673	−0.723969
$905$	0	0
$906$	−62.2755	−2.06896
$907$	− 16.7453i	− 0.556018i	−0.960579	−	0.278009i	$-0.910326\pi$
$907$	− 16.7453i	− 0.556018i	0.960579	−	0.278009i	$-0.0896745\pi$
$908$	− 37.7662i	− 1.25332i
$909$	−25.1317	−0.833567
$910$	0	0
$911$	−27.9485	−0.925976	−0.462988	−	0.886365i	$-0.653223\pi$
$911$	−27.9485	−0.925976	−0.462988	+	0.886365i	$0.653223\pi$
$912$	− 2.17776i	− 0.0721128i
$913$	8.02986i	0.265750i
$914$	−52.2449	−1.72811
$915$	0	0
$916$	−36.3383	−1.20065
$917$	0	0
$918$	65.8595i	2.17369i
$919$	−23.2940	−0.768399	−0.384199	−	0.923250i	$-0.625523\pi$
$919$	−23.2940	−0.768399	−0.384199	+	0.923250i	$0.625523\pi$
$920$	0	0
$921$	−15.7487	−0.518939
$922$	68.3426i	2.25075i
$923$	− 4.91866i	− 0.161900i
$924$	0	0
$925$	0	0
$926$	−54.6609	−1.79627
$927$	100.792i	3.31043i
$928$	− 20.8288i	− 0.683738i
$929$	39.6949	1.30235	0.651173	−	0.758929i	$-0.274277\pi$
$929$	39.6949	1.30235	0.651173	+	0.758929i	$0.274277\pi$
$930$	0	0
$931$	0	0
$932$	− 58.4238i	− 1.91373i
$933$	− 18.7289i	− 0.613157i
$934$	−30.0504	−0.983279
$935$	0	0
$936$	5.19869	0.169925
$937$	− 2.00654i	− 0.0655509i	−0.999463	−	0.0327755i	$-0.989565\pi$
$937$	− 2.00654i	− 0.0655509i	0.999463	−	0.0327755i	$-0.0104346\pi$
$938$	0	0
$939$	−33.7278	−1.10067
$940$	0	0
$941$	−44.0318	−1.43540	−0.717699	−	0.696354i	$-0.754804\pi$
$941$	−44.0318	−1.43540	−0.717699	+	0.696354i	$0.754804\pi$
$942$	41.2755i	1.34483i
$943$	− 13.3250i	− 0.433921i
$944$	6.47764	0.210829
$945$	0	0
$946$	58.7806	1.91112
$947$	− 24.6609i	− 0.801370i	−0.916216	−	0.400685i	$-0.868772\pi$
$947$	− 24.6609i	− 0.801370i	0.916216	−	0.400685i	$-0.131228\pi$
$948$	− 41.4436i	− 1.34603i
$949$	−6.38429	−0.207243
$950$	0	0
$951$	−24.6872	−0.800539
$952$	0	0
$953$	− 35.5435i	− 1.15137i	−0.817673	−	0.575684i	$-0.804736\pi$
$953$	− 35.5435i	− 1.15137i	0.817673	−	0.575684i	$-0.195264\pi$
$954$	118.421	3.83401
$955$	0	0
$956$	−43.8606	−1.41855
$957$	− 20.8288i	− 0.673299i
$958$	− 9.15921i	− 0.295921i
$959$	0	0
$960$	0	0
$961$	−16.0449	−0.517579
$962$	2.76313i	0.0890869i
$963$	57.2240i	1.84402i
$964$	−31.2855	−1.00764
$965$	0	0
$966$	0	0
$967$	38.3228i	1.23238i	0.787598	+	0.616189i	$0.211324\pi$
$967$	38.3228i	1.23238i	−0.787598	+	0.616189i	$0.788676\pi$
$968$	− 8.12628i	− 0.261188i
$969$	−6.92213	−0.222371
$970$	0	0
$971$	−47.5819	−1.52698	−0.763488	−	0.645822i	$-0.776515\pi$
$971$	−47.5819	−1.52698	−0.763488	+	0.645822i	$0.776515\pi$
$972$	− 39.0977i	− 1.25406i
$973$	0	0
$974$	−31.1778	−0.999000
$975$	0	0
$976$	6.32497	0.202457
$977$	− 10.4842i	− 0.335419i	−0.985836	−	0.167709i	$-0.946363\pi$
$977$	− 10.4842i	− 0.335419i	0.985836	−	0.167709i	$-0.0536370\pi$
$978$	− 44.9900i	− 1.43862i
$979$	−14.2899	−0.456706
$980$	0	0
$981$	45.0977	1.43986
$982$	− 41.4975i	− 1.32424i
$983$	− 10.2262i	− 0.326164i	−0.986613	−	0.163082i	$-0.947856\pi$
$983$	− 10.2262i	− 0.326164i	0.986613	−	0.163082i	$-0.0521435\pi$
$984$	17.1767	0.547572
$985$	0	0
$986$	−32.7727	−1.04370
$987$	0	0
$988$	0.749828i	0.0238552i
$989$	−42.0648	−1.33758
$990$	0	0
$991$	1.27873	0.0406203	0.0203101	−	0.999794i	$-0.493535\pi$
$991$	1.27873	0.0406203	0.0203101	+	0.999794i	$0.493535\pi$
$992$	− 28.0933i	− 0.891965i
$993$	35.3394i	1.12146i
$994$	0	0
$995$	0	0
$996$	25.1658	0.797408
$997$	− 27.7200i	− 0.877900i	−0.898511	−	0.438950i	$-0.855350\pi$
$997$	− 27.7200i	− 0.877900i	0.898511	−	0.438950i	$-0.144650\pi$
$998$	− 23.1383i	− 0.732430i
$999$	−12.8582	−0.406817

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1225.2.b.l.99.1		6
5.2	odd	4		1225.2.a.y.1.3		3
5.3	odd	4		1225.2.a.x.1.1		3
5.4	even	2	inner	1225.2.b.l.99.6		6
7.2	even	3		175.2.k.b.74.1		12
7.4	even	3		175.2.k.b.149.6		12
7.6	odd	2		1225.2.b.m.99.1		6
35.2	odd	12		175.2.e.d.151.1	yes	6
35.4	even	6		175.2.k.b.149.1		12
35.9	even	6		175.2.k.b.74.6		12
35.13	even	4		1225.2.a.w.1.1		3
35.18	odd	12		175.2.e.e.51.3	yes	6
35.23	odd	12		175.2.e.e.151.3	yes	6
35.27	even	4		1225.2.a.z.1.3		3
35.32	odd	12		175.2.e.d.51.1	✓	6
35.34	odd	2		1225.2.b.m.99.6		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
175.2.e.d.51.1	✓	6	35.32	odd	12
175.2.e.d.151.1	yes	6	35.2	odd	12
175.2.e.e.51.3	yes	6	35.18	odd	12
175.2.e.e.151.3	yes	6	35.23	odd	12
175.2.k.b.74.1		12	7.2	even	3
175.2.k.b.74.6		12	35.9	even	6
175.2.k.b.149.1		12	35.4	even	6
175.2.k.b.149.6		12	7.4	even	3
1225.2.a.w.1.1		3	35.13	even	4
1225.2.a.x.1.1		3	5.3	odd	4
1225.2.a.y.1.3		3	5.2	odd	4
1225.2.a.z.1.3		3	35.27	even	4
1225.2.b.l.99.1		6	1.1	even	1	trivial
1225.2.b.l.99.6		6	5.4	even	2	inner
1225.2.b.m.99.1		6	7.6	odd	2
1225.2.b.m.99.6		6	35.34	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.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-2.19869i q^{2} -2.83424i q^{3} -2.83424 q^{4} -6.23163 q^{6} +1.83424i q^{8} -5.03293 q^{9} +O(q^{10})\)	\(q-2.19869i q^{2} -2.83424i q^{3} -2.83424 q^{4} -6.23163 q^{6} +1.83424i q^{8} -5.03293 q^{9} -2.56314 q^{11} +8.03293i q^{12} +0.563139i q^{13} -1.63555 q^{16} +5.19869i q^{17} +11.0659i q^{18} -0.469795 q^{19} +5.63555i q^{22} -4.03293i q^{23} +5.19869 q^{24} +1.23817 q^{26} +5.76183i q^{27} -2.86718 q^{29} -3.86718 q^{31} +7.26456i q^{32} +7.26456i q^{33} +11.4303 q^{34} +14.2646 q^{36} +2.23163i q^{37} +1.03293i q^{38} +1.59607 q^{39} +3.30404 q^{41} -10.4303i q^{43} +7.26456 q^{44} -8.86718 q^{46} +9.36445i q^{47} +4.63555i q^{48} +14.7344 q^{51} -1.59607i q^{52} -10.7014i q^{53} +12.6685 q^{54} +1.33151i q^{57} +6.30404i q^{58} -3.96052 q^{59} -3.86718 q^{61} +8.50273i q^{62} +12.7014 q^{64} +15.9725 q^{66} -7.66849i q^{67} -14.7344i q^{68} -11.4303 q^{69} -8.73436 q^{71} -9.23163i q^{72} +11.3370i q^{73} +4.90666 q^{74} +1.33151 q^{76} -3.50927i q^{78} -5.15921 q^{79} +1.23163 q^{81} -7.26456i q^{82} -3.13282i q^{83} -22.9330 q^{86} +8.12628i q^{87} -4.70142i q^{88} +5.57514 q^{89} +11.4303i q^{92} +10.9605i q^{93} +20.5895 q^{94} +20.5895 q^{96} -15.2975i q^{97} +12.9001 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 6 q^{4} - 4 q^{6} - 8 q^{9}+O(q^{10}) \) 6 * q - 6 * q^4 - 4 * q^6 - 8 * q^9	\( 6 q - 6 q^{4} - 4 q^{6} - 8 q^{9} - 4 q^{11} - 10 q^{16} + 8 q^{19} + 20 q^{24} + 30 q^{26} + 16 q^{29} + 10 q^{31} + 24 q^{34} + 30 q^{36} - 24 q^{39} - 2 q^{41} - 12 q^{44} - 20 q^{46} + 22 q^{51} + 54 q^{54} + 10 q^{59} + 10 q^{61} + 32 q^{64} + 52 q^{66} - 24 q^{69} + 14 q^{71} + 30 q^{74} + 30 q^{76} + 14 q^{79} - 26 q^{81} - 60 q^{86} + 12 q^{89} + 34 q^{94} + 34 q^{96} + 22 q^{99}+O(q^{100}) \) 6 * q - 6 * q^4 - 4 * q^6 - 8 * q^9 - 4 * q^11 - 10 * q^16 + 8 * q^19 + 20 * q^24 + 30 * q^26 + 16 * q^29 + 10 * q^31 + 24 * q^34 + 30 * q^36 - 24 * q^39 - 2 * q^41 - 12 * q^44 - 20 * q^46 + 22 * q^51 + 54 * q^54 + 10 * q^59 + 10 * q^61 + 32 * q^64 + 52 * q^66 - 24 * q^69 + 14 * q^71 + 30 * q^74 + 30 * q^76 + 14 * q^79 - 26 * q^81 - 60 * q^86 + 12 * q^89 + 34 * q^94 + 34 * q^96 + 22 * q^99

Introduction

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