LMFDB - Embedded newform 525.2.a.f.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [525,2,Mod(1,525)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(525, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

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

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

chi := DirichletCharacter("525.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 525 = 3 \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	525.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:	$4.19214610612$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{13}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 3 $ x^2 - x - 3
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.30278$ of defining polynomial
Character	$\chi$	$=$	525.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.30278 q^{2} -1.00000 q^{3} -0.302776 q^{4} -1.30278 q^{6} -1.00000 q^{7} -3.00000 q^{8} +1.00000 q^{9} -3.00000 q^{11} +0.302776 q^{12} -4.60555 q^{13} -1.30278 q^{14} -3.30278 q^{16} +2.60555 q^{17} +1.30278 q^{18} -0.605551 q^{19} +1.00000 q^{21} -3.90833 q^{22} -8.21110 q^{23} +3.00000 q^{24} -6.00000 q^{26} -1.00000 q^{27} +0.302776 q^{28} -0.394449 q^{29} +7.21110 q^{31} +1.69722 q^{32} +3.00000 q^{33} +3.39445 q^{34} -0.302776 q^{36} -10.2111 q^{37} -0.788897 q^{38} +4.60555 q^{39} +1.30278 q^{42} -2.39445 q^{43} +0.908327 q^{44} -10.6972 q^{46} -3.39445 q^{47} +3.30278 q^{48} +1.00000 q^{49} -2.60555 q^{51} +1.39445 q^{52} +11.2111 q^{53} -1.30278 q^{54} +3.00000 q^{56} +0.605551 q^{57} -0.513878 q^{58} -3.39445 q^{59} +13.2111 q^{61} +9.39445 q^{62} -1.00000 q^{63} +8.81665 q^{64} +3.90833 q^{66} -8.39445 q^{67} -0.788897 q^{68} +8.21110 q^{69} -3.00000 q^{71} -3.00000 q^{72} +6.60555 q^{73} -13.3028 q^{74} +0.183346 q^{76} +3.00000 q^{77} +6.00000 q^{78} +6.81665 q^{79} +1.00000 q^{81} -11.2111 q^{83} -0.302776 q^{84} -3.11943 q^{86} +0.394449 q^{87} +9.00000 q^{88} -13.8167 q^{89} +4.60555 q^{91} +2.48612 q^{92} -7.21110 q^{93} -4.42221 q^{94} -1.69722 q^{96} +15.2111 q^{97} +1.30278 q^{98} -3.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{2} - 2 q^{3} + 3 q^{4} + q^{6} - 2 q^{7} - 6 q^{8} + 2 q^{9} - 6 q^{11} - 3 q^{12} - 2 q^{13} + q^{14} - 3 q^{16} - 2 q^{17} - q^{18} + 6 q^{19} + 2 q^{21} + 3 q^{22} - 2 q^{23} + 6 q^{24}+ \cdots - 6 q^{99}+O(q^{100}) $ 2 * q - q^2 - 2 * q^3 + 3 * q^4 + q^6 - 2 * q^7 - 6 * q^8 + 2 * q^9 - 6 * q^11 - 3 * q^12 - 2 * q^13 + q^14 - 3 * q^16 - 2 * q^17 - q^18 + 6 * q^19 + 2 * q^21 + 3 * q^22 - 2 * q^23 + 6 * q^24 - 12 * q^26 - 2 * q^27 - 3 * q^28 - 8 * q^29 + 7 * q^32 + 6 * q^33 + 14 * q^34 + 3 * q^36 - 6 * q^37 - 16 * q^38 + 2 * q^39 - q^42 - 12 * q^43 - 9 * q^44 - 25 * q^46 - 14 * q^47 + 3 * q^48 + 2 * q^49 + 2 * q^51 + 10 * q^52 + 8 * q^53 + q^54 + 6 * q^56 - 6 * q^57 + 17 * q^58 - 14 * q^59 + 12 * q^61 + 26 * q^62 - 2 * q^63 - 4 * q^64 - 3 * q^66 - 24 * q^67 - 16 * q^68 + 2 * q^69 - 6 * q^71 - 6 * q^72 + 6 * q^73 - 23 * q^74 + 22 * q^76 + 6 * q^77 + 12 * q^78 - 8 * q^79 + 2 * q^81 - 8 * q^83 + 3 * q^84 + 19 * q^86 + 8 * q^87 + 18 * q^88 - 6 * q^89 + 2 * q^91 + 23 * q^92 + 20 * q^94 - 7 * q^96 + 16 * q^97 - q^98 - 6 * q^99

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 $

$2$	1.30278	0.921201	0.460601	−	0.887607i	$-0.347634\pi$
$2$	1.30278	0.921201	0.460601	+	0.887607i	$0.347634\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	−0.302776	−0.151388
$5$	0	0
$5$	0	0
$6$	−1.30278	−0.531856
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	−3.00000	−1.06066
$9$	1.00000	0.333333
$10$	0	0
$11$	−3.00000	−0.904534	−0.452267	−	0.891883i	$-0.649385\pi$
$11$	−3.00000	−0.904534	−0.452267	+	0.891883i	$0.649385\pi$
$12$	0.302776	0.0874038
$13$	−4.60555	−1.27735	−0.638675	−	0.769477i	$-0.720517\pi$
$13$	−4.60555	−1.27735	−0.638675	+	0.769477i	$0.720517\pi$
$14$	−1.30278	−0.348181
$15$	0	0
$16$	−3.30278	−0.825694
$17$	2.60555	0.631939	0.315970	−	0.948769i	$-0.397670\pi$
$17$	2.60555	0.631939	0.315970	+	0.948769i	$0.397670\pi$
$18$	1.30278	0.307067
$19$	−0.605551	−0.138923	−0.0694615	−	0.997585i	$-0.522128\pi$
$19$	−0.605551	−0.138923	−0.0694615	+	0.997585i	$0.522128\pi$
$20$	0	0
$21$	1.00000	0.218218
$22$	−3.90833	−0.833258
$23$	−8.21110	−1.71213	−0.856067	−	0.516865i	$-0.827099\pi$
$23$	−8.21110	−1.71213	−0.856067	+	0.516865i	$0.827099\pi$
$24$	3.00000	0.612372
$25$	0	0
$26$	−6.00000	−1.17670
$27$	−1.00000	−0.192450
$28$	0.302776	0.0572192
$29$	−0.394449	−0.0732473	−0.0366236	−	0.999329i	$-0.511660\pi$
$29$	−0.394449	−0.0732473	−0.0366236	+	0.999329i	$0.511660\pi$
$30$	0	0
$31$	7.21110	1.29515	0.647576	−	0.762001i	$-0.275783\pi$
$31$	7.21110	1.29515	0.647576	+	0.762001i	$0.275783\pi$
$32$	1.69722	0.300030
$33$	3.00000	0.522233
$34$	3.39445	0.582143
$35$	0	0
$36$	−0.302776	−0.0504626
$37$	−10.2111	−1.67869	−0.839347	−	0.543595i	$-0.817063\pi$
$37$	−10.2111	−1.67869	−0.839347	+	0.543595i	$0.817063\pi$
$38$	−0.788897	−0.127976
$39$	4.60555	0.737478
$40$	0	0
$41$	0	0		−	1.00000i	$-0.5\pi$
$41$	0	0			1.00000i	$0.5\pi$
$42$	1.30278	0.201023
$43$	−2.39445	−0.365150	−0.182575	−	0.983192i	$-0.558443\pi$
$43$	−2.39445	−0.365150	−0.182575	+	0.983192i	$0.558443\pi$
$44$	0.908327	0.136935
$45$	0	0
$46$	−10.6972	−1.57722
$47$	−3.39445	−0.495131	−0.247566	−	0.968871i	$-0.579631\pi$
$47$	−3.39445	−0.495131	−0.247566	+	0.968871i	$0.579631\pi$
$48$	3.30278	0.476715
$49$	1.00000	0.142857
$50$	0	0
$51$	−2.60555	−0.364850
$52$	1.39445	0.193375
$53$	11.2111	1.53996	0.769982	−	0.638066i	$-0.220265\pi$
$53$	11.2111	1.53996	0.769982	+	0.638066i	$0.220265\pi$
$54$	−1.30278	−0.177285
$55$	0	0
$56$	3.00000	0.400892
$57$	0.605551	0.0802072
$58$	−0.513878	−0.0674755
$59$	−3.39445	−0.441920	−0.220960	−	0.975283i	$-0.570919\pi$
$59$	−3.39445	−0.441920	−0.220960	+	0.975283i	$0.570919\pi$
$60$	0	0
$61$	13.2111	1.69151	0.845754	−	0.533573i	$-0.179151\pi$
$61$	13.2111	1.69151	0.845754	+	0.533573i	$0.179151\pi$
$62$	9.39445	1.19310
$63$	−1.00000	−0.125988
$64$	8.81665	1.10208
$65$	0	0
$66$	3.90833	0.481082
$67$	−8.39445	−1.02555	−0.512773	−	0.858524i	$-0.671382\pi$
$67$	−8.39445	−1.02555	−0.512773	+	0.858524i	$0.671382\pi$
$68$	−0.788897	−0.0956679
$69$	8.21110	0.988501
$70$	0	0
$71$	−3.00000	−0.356034	−0.178017	−	0.984027i	$-0.556968\pi$
$71$	−3.00000	−0.356034	−0.178017	+	0.984027i	$0.556968\pi$
$72$	−3.00000	−0.353553
$73$	6.60555	0.773121	0.386561	−	0.922264i	$-0.373663\pi$
$73$	6.60555	0.773121	0.386561	+	0.922264i	$0.373663\pi$
$74$	−13.3028	−1.54642
$75$	0	0
$76$	0.183346	0.0210312
$77$	3.00000	0.341882
$78$	6.00000	0.679366
$79$	6.81665	0.766933	0.383467	−	0.923555i	$-0.374730\pi$
$79$	6.81665	0.766933	0.383467	+	0.923555i	$0.374730\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−11.2111	−1.23058	−0.615289	−	0.788301i	$-0.710961\pi$
$83$	−11.2111	−1.23058	−0.615289	+	0.788301i	$0.710961\pi$
$84$	−0.302776	−0.0330355
$85$	0	0
$86$	−3.11943	−0.336377
$87$	0.394449	0.0422893
$88$	9.00000	0.959403
$89$	−13.8167	−1.46456	−0.732281	−	0.681002i	$-0.761544\pi$
$89$	−13.8167	−1.46456	−0.732281	+	0.681002i	$0.761544\pi$
$90$	0	0
$91$	4.60555	0.482793
$92$	2.48612	0.259196
$93$	−7.21110	−0.747757
$94$	−4.42221	−0.456116
$95$	0	0
$96$	−1.69722	−0.173222
$97$	15.2111	1.54445	0.772227	−	0.635347i	$-0.219143\pi$
$97$	15.2111	1.54445	0.772227	+	0.635347i	$0.219143\pi$
$98$	1.30278	0.131600
$99$	−3.00000	−0.301511
$100$	0	0
$101$	9.39445	0.934783	0.467391	−	0.884051i	$-0.345194\pi$
$101$	9.39445	0.934783	0.467391	+	0.884051i	$0.345194\pi$
$102$	−3.39445	−0.336101
$103$	17.8167	1.75553	0.877764	−	0.479094i	$-0.159035\pi$
$103$	17.8167	1.75553	0.877764	+	0.479094i	$0.159035\pi$
$104$	13.8167	1.35483
$105$	0	0
$106$	14.6056	1.41862
$107$	0	0		−	1.00000i	$-0.5\pi$
$107$	0	0			1.00000i	$0.5\pi$
$108$	0.302776	0.0291346
$109$	−12.2111	−1.16961	−0.584806	−	0.811173i	$-0.698829\pi$
$109$	−12.2111	−1.16961	−0.584806	+	0.811173i	$0.698829\pi$
$110$	0	0
$111$	10.2111	0.969195
$112$	3.30278	0.312083
$113$	−10.8167	−1.01755	−0.508773	−	0.860901i	$-0.669901\pi$
$113$	−10.8167	−1.01755	−0.508773	+	0.860901i	$0.669901\pi$
$114$	0.788897	0.0738870
$115$	0	0
$116$	0.119429	0.0110887
$117$	−4.60555	−0.425783
$118$	−4.42221	−0.407097
$119$	−2.60555	−0.238850
$120$	0	0
$121$	−2.00000	−0.181818
$122$	17.2111	1.55822
$123$	0	0
$124$	−2.18335	−0.196070
$125$	0	0
$126$	−1.30278	−0.116060
$127$	8.81665	0.782352	0.391176	−	0.920316i	$-0.372068\pi$
$127$	8.81665	0.782352	0.391176	+	0.920316i	$0.372068\pi$
$128$	8.09167	0.715210
$129$	2.39445	0.210819
$130$	0	0
$131$	−14.6056	−1.27609	−0.638046	−	0.769998i	$-0.720257\pi$
$131$	−14.6056	−1.27609	−0.638046	+	0.769998i	$0.720257\pi$
$132$	−0.908327	−0.0790597
$133$	0.605551	0.0525080
$134$	−10.9361	−0.944734
$135$	0	0
$136$	−7.81665	−0.670273
$137$	−11.2111	−0.957829	−0.478915	−	0.877862i	$-0.658970\pi$
$137$	−11.2111	−0.957829	−0.478915	+	0.877862i	$0.658970\pi$
$138$	10.6972	0.910608
$139$	−17.0278	−1.44428	−0.722138	−	0.691749i	$-0.756840\pi$
$139$	−17.0278	−1.44428	−0.722138	+	0.691749i	$0.756840\pi$
$140$	0	0
$141$	3.39445	0.285864
$142$	−3.90833	−0.327980
$143$	13.8167	1.15541
$144$	−3.30278	−0.275231
$145$	0	0
$146$	8.60555	0.712200
$147$	−1.00000	−0.0824786
$148$	3.09167	0.254134
$149$	−23.6056	−1.93384	−0.966921	−	0.255076i	$-0.917900\pi$
$149$	−23.6056	−1.93384	−0.966921	+	0.255076i	$0.917900\pi$
$150$	0	0
$151$	−14.8167	−1.20576	−0.602881	−	0.797831i	$-0.705981\pi$
$151$	−14.8167	−1.20576	−0.602881	+	0.797831i	$0.705981\pi$
$152$	1.81665	0.147350
$153$	2.60555	0.210646
$154$	3.90833	0.314942
$155$	0	0
$156$	−1.39445	−0.111645
$157$	14.4222	1.15102	0.575509	−	0.817796i	$-0.304804\pi$
$157$	14.4222	1.15102	0.575509	+	0.817796i	$0.304804\pi$
$158$	8.88057	0.706500
$159$	−11.2111	−0.889098
$160$	0	0
$161$	8.21110	0.647126
$162$	1.30278	0.102356
$163$	−2.78890	−0.218443	−0.109222	−	0.994017i	$-0.534836\pi$
$163$	−2.78890	−0.218443	−0.109222	+	0.994017i	$0.534836\pi$
$164$	0	0
$165$	0	0
$166$	−14.6056	−1.13361
$167$	−19.0278	−1.47241	−0.736206	−	0.676758i	$-0.763385\pi$
$167$	−19.0278	−1.47241	−0.736206	+	0.676758i	$0.763385\pi$
$168$	−3.00000	−0.231455
$169$	8.21110	0.631623
$170$	0	0
$171$	−0.605551	−0.0463077
$172$	0.724981	0.0552793
$173$	−13.8167	−1.05046	−0.525230	−	0.850960i	$-0.676021\pi$
$173$	−13.8167	−1.05046	−0.525230	+	0.850960i	$0.676021\pi$
$174$	0.513878	0.0389570
$175$	0	0
$176$	9.90833	0.746868
$177$	3.39445	0.255142
$178$	−18.0000	−1.34916
$179$	0	0		−	1.00000i	$-0.5\pi$
$179$	0	0			1.00000i	$0.5\pi$
$180$	0	0
$181$	5.39445	0.400966	0.200483	−	0.979697i	$-0.435749\pi$
$181$	5.39445	0.400966	0.200483	+	0.979697i	$0.435749\pi$
$182$	6.00000	0.444750
$183$	−13.2111	−0.976593
$184$	24.6333	1.81599
$185$	0	0
$186$	−9.39445	−0.688834
$187$	−7.81665	−0.571610
$188$	1.02776	0.0749568
$189$	1.00000	0.0727393
$190$	0	0
$191$	−17.2111	−1.24535	−0.622676	−	0.782480i	$-0.713954\pi$
$191$	−17.2111	−1.24535	−0.622676	+	0.782480i	$0.713954\pi$
$192$	−8.81665	−0.636287
$193$	6.21110	0.447085	0.223542	−	0.974694i	$-0.428238\pi$
$193$	6.21110	0.447085	0.223542	+	0.974694i	$0.428238\pi$
$194$	19.8167	1.42275
$195$	0	0
$196$	−0.302776	−0.0216268
$197$	4.81665	0.343172	0.171586	−	0.985169i	$-0.445111\pi$
$197$	4.81665	0.343172	0.171586	+	0.985169i	$0.445111\pi$
$198$	−3.90833	−0.277753
$199$	8.00000	0.567105	0.283552	−	0.958957i	$-0.408487\pi$
$199$	8.00000	0.567105	0.283552	+	0.958957i	$0.408487\pi$
$200$	0	0
$201$	8.39445	0.592099
$202$	12.2389	0.861123
$203$	0.394449	0.0276849
$204$	0.788897	0.0552339
$205$	0	0
$206$	23.2111	1.61719
$207$	−8.21110	−0.570711
$208$	15.2111	1.05470
$209$	1.81665	0.125661
$210$	0	0
$211$	−2.42221	−0.166751	−0.0833757	−	0.996518i	$-0.526570\pi$
$211$	−2.42221	−0.166751	−0.0833757	+	0.996518i	$0.526570\pi$
$212$	−3.39445	−0.233132
$213$	3.00000	0.205557
$214$	0	0
$215$	0	0
$216$	3.00000	0.204124
$217$	−7.21110	−0.489522
$218$	−15.9083	−1.07745
$219$	−6.60555	−0.446362
$220$	0	0
$221$	−12.0000	−0.807207
$222$	13.3028	0.892824
$223$	−22.6056	−1.51378	−0.756890	−	0.653542i	$-0.773282\pi$
$223$	−22.6056	−1.51378	−0.756890	+	0.653542i	$0.773282\pi$
$224$	−1.69722	−0.113401
$225$	0	0
$226$	−14.0917	−0.937364
$227$	−15.3944	−1.02177	−0.510883	−	0.859650i	$-0.670681\pi$
$227$	−15.3944	−1.02177	−0.510883	+	0.859650i	$0.670681\pi$
$228$	−0.183346	−0.0121424
$229$	7.21110	0.476523	0.238262	−	0.971201i	$-0.423422\pi$
$229$	7.21110	0.476523	0.238262	+	0.971201i	$0.423422\pi$
$230$	0	0
$231$	−3.00000	−0.197386
$232$	1.18335	0.0776905
$233$	22.8167	1.49477	0.747384	−	0.664392i	$-0.231309\pi$
$233$	22.8167	1.49477	0.747384	+	0.664392i	$0.231309\pi$
$234$	−6.00000	−0.392232
$235$	0	0
$236$	1.02776	0.0669012
$237$	−6.81665	−0.442789
$238$	−3.39445	−0.220029
$239$	29.2111	1.88951	0.944755	−	0.327779i	$-0.106300\pi$
$239$	29.2111	1.88951	0.944755	+	0.327779i	$0.106300\pi$
$240$	0	0
$241$	16.6056	1.06966	0.534829	−	0.844960i	$-0.320376\pi$
$241$	16.6056	1.06966	0.534829	+	0.844960i	$0.320376\pi$
$242$	−2.60555	−0.167491
$243$	−1.00000	−0.0641500
$244$	−4.00000	−0.256074
$245$	0	0
$246$	0	0
$247$	2.78890	0.177453
$248$	−21.6333	−1.37372
$249$	11.2111	0.710475
$250$	0	0
$251$	7.81665	0.493383	0.246691	−	0.969094i	$-0.420657\pi$
$251$	7.81665	0.493383	0.246691	+	0.969094i	$0.420657\pi$
$252$	0.302776	0.0190731
$253$	24.6333	1.54868
$254$	11.4861	0.720703
$255$	0	0
$256$	−7.09167	−0.443230
$257$	−21.6333	−1.34945	−0.674724	−	0.738070i	$-0.735737\pi$
$257$	−21.6333	−1.34945	−0.674724	+	0.738070i	$0.735737\pi$
$258$	3.11943	0.194207
$259$	10.2111	0.634487
$260$	0	0
$261$	−0.394449	−0.0244158
$262$	−19.0278	−1.17554
$263$	20.2111	1.24627	0.623135	−	0.782114i	$-0.285859\pi$
$263$	20.2111	1.24627	0.623135	+	0.782114i	$0.285859\pi$
$264$	−9.00000	−0.553912
$265$	0	0
$266$	0.788897	0.0483704
$267$	13.8167	0.845565
$268$	2.54163	0.155255
$269$	11.2111	0.683553	0.341776	−	0.939781i	$-0.388971\pi$
$269$	11.2111	0.683553	0.341776	+	0.939781i	$0.388971\pi$
$270$	0	0
$271$	−19.3944	−1.17813	−0.589064	−	0.808086i	$-0.700504\pi$
$271$	−19.3944	−1.17813	−0.589064	+	0.808086i	$0.700504\pi$
$272$	−8.60555	−0.521788
$273$	−4.60555	−0.278741
$274$	−14.6056	−0.882354
$275$	0	0
$276$	−2.48612	−0.149647
$277$	10.0000	0.600842	0.300421	−	0.953807i	$-0.402873\pi$
$277$	10.0000	0.600842	0.300421	+	0.953807i	$0.402873\pi$
$278$	−22.1833	−1.33047
$279$	7.21110	0.431717
$280$	0	0
$281$	−22.8167	−1.36113	−0.680564	−	0.732689i	$-0.738265\pi$
$281$	−22.8167	−1.36113	−0.680564	+	0.732689i	$0.738265\pi$
$282$	4.42221	0.263338
$283$	−29.6333	−1.76152	−0.880759	−	0.473565i	$-0.842967\pi$
$283$	−29.6333	−1.76152	−0.880759	+	0.473565i	$0.842967\pi$
$284$	0.908327	0.0538993
$285$	0	0
$286$	18.0000	1.06436
$287$	0	0
$288$	1.69722	0.100010
$289$	−10.2111	−0.600653
$290$	0	0
$291$	−15.2111	−0.891691
$292$	−2.00000	−0.117041
$293$	−3.39445	−0.198306	−0.0991529	−	0.995072i	$-0.531613\pi$
$293$	−3.39445	−0.198306	−0.0991529	+	0.995072i	$0.531613\pi$
$294$	−1.30278	−0.0759794
$295$	0	0
$296$	30.6333	1.78052
$297$	3.00000	0.174078
$298$	−30.7527	−1.78146
$299$	37.8167	2.18699
$300$	0	0
$301$	2.39445	0.138014
$302$	−19.3028	−1.11075
$303$	−9.39445	−0.539697
$304$	2.00000	0.114708
$305$	0	0
$306$	3.39445	0.194048
$307$	1.39445	0.0795854	0.0397927	−	0.999208i	$-0.487330\pi$
$307$	1.39445	0.0795854	0.0397927	+	0.999208i	$0.487330\pi$
$308$	−0.908327	−0.0517567
$309$	−17.8167	−1.01355
$310$	0	0
$311$	−7.81665	−0.443242	−0.221621	−	0.975133i	$-0.571135\pi$
$311$	−7.81665	−0.443242	−0.221621	+	0.975133i	$0.571135\pi$
$312$	−13.8167	−0.782214
$313$	2.18335	0.123410	0.0617050	−	0.998094i	$-0.480346\pi$
$313$	2.18335	0.123410	0.0617050	+	0.998094i	$0.480346\pi$
$314$	18.7889	1.06032
$315$	0	0
$316$	−2.06392	−0.116104
$317$	23.6056	1.32582	0.662910	−	0.748699i	$-0.269321\pi$
$317$	23.6056	1.32582	0.662910	+	0.748699i	$0.269321\pi$
$318$	−14.6056	−0.819039
$319$	1.18335	0.0662547
$320$	0	0
$321$	0	0
$322$	10.6972	0.596133
$323$	−1.57779	−0.0877909
$324$	−0.302776	−0.0168209
$325$	0	0
$326$	−3.63331	−0.201230
$327$	12.2111	0.675276
$328$	0	0
$329$	3.39445	0.187142
$330$	0	0
$331$	29.2389	1.60711	0.803557	−	0.595228i	$-0.202938\pi$
$331$	29.2389	1.60711	0.803557	+	0.595228i	$0.202938\pi$
$332$	3.39445	0.186295
$333$	−10.2111	−0.559565
$334$	−24.7889	−1.35639
$335$	0	0
$336$	−3.30278	−0.180181
$337$	−7.21110	−0.392814	−0.196407	−	0.980522i	$-0.562927\pi$
$337$	−7.21110	−0.392814	−0.196407	+	0.980522i	$0.562927\pi$
$338$	10.6972	0.581852
$339$	10.8167	0.587480
$340$	0	0
$341$	−21.6333	−1.17151
$342$	−0.788897	−0.0426587
$343$	−1.00000	−0.0539949
$344$	7.18335	0.387300
$345$	0	0
$346$	−18.0000	−0.967686
$347$	−15.7889	−0.847592	−0.423796	−	0.905758i	$-0.639303\pi$
$347$	−15.7889	−0.847592	−0.423796	+	0.905758i	$0.639303\pi$
$348$	−0.119429	−0.00640209
$349$	−33.4500	−1.79054	−0.895268	−	0.445529i	$-0.853016\pi$
$349$	−33.4500	−1.79054	−0.895268	+	0.445529i	$0.853016\pi$
$350$	0	0
$351$	4.60555	0.245826
$352$	−5.09167	−0.271387
$353$	30.0000	1.59674	0.798369	−	0.602168i	$-0.205696\pi$
$353$	30.0000	1.59674	0.798369	+	0.602168i	$0.205696\pi$
$354$	4.42221	0.235038
$355$	0	0
$356$	4.18335	0.221717
$357$	2.60555	0.137900
$358$	0	0
$359$	−18.6333	−0.983428	−0.491714	−	0.870757i	$-0.663630\pi$
$359$	−18.6333	−0.983428	−0.491714	+	0.870757i	$0.663630\pi$
$360$	0	0
$361$	−18.6333	−0.980700
$362$	7.02776	0.369371
$363$	2.00000	0.104973
$364$	−1.39445	−0.0730890
$365$	0	0
$366$	−17.2111	−0.899639
$367$	14.4222	0.752833	0.376416	−	0.926451i	$-0.377156\pi$
$367$	14.4222	0.752833	0.376416	+	0.926451i	$0.377156\pi$
$368$	27.1194	1.41370
$369$	0	0
$370$	0	0
$371$	−11.2111	−0.582051
$372$	2.18335	0.113201
$373$	1.00000	0.0517780	0.0258890	−	0.999665i	$-0.491758\pi$
$373$	1.00000	0.0517780	0.0258890	+	0.999665i	$0.491758\pi$
$374$	−10.1833	−0.526568
$375$	0	0
$376$	10.1833	0.525166
$377$	1.81665	0.0935624
$378$	1.30278	0.0670075
$379$	6.81665	0.350148	0.175074	−	0.984555i	$-0.443984\pi$
$379$	6.81665	0.350148	0.175074	+	0.984555i	$0.443984\pi$
$380$	0	0
$381$	−8.81665	−0.451691
$382$	−22.4222	−1.14722
$383$	−24.2389	−1.23855	−0.619274	−	0.785175i	$-0.712573\pi$
$383$	−24.2389	−1.23855	−0.619274	+	0.785175i	$0.712573\pi$
$384$	−8.09167	−0.412926
$385$	0	0
$386$	8.09167	0.411855
$387$	−2.39445	−0.121717
$388$	−4.60555	−0.233811
$389$	−7.18335	−0.364210	−0.182105	−	0.983279i	$-0.558291\pi$
$389$	−7.18335	−0.364210	−0.182105	+	0.983279i	$0.558291\pi$
$390$	0	0
$391$	−21.3944	−1.08196
$392$	−3.00000	−0.151523
$393$	14.6056	0.736753
$394$	6.27502	0.316131
$395$	0	0
$396$	0.908327	0.0456451
$397$	−2.00000	−0.100377	−0.0501886	−	0.998740i	$-0.515982\pi$
$397$	−2.00000	−0.100377	−0.0501886	+	0.998740i	$0.515982\pi$
$398$	10.4222	0.522418
$399$	−0.605551	−0.0303155
$400$	0	0
$401$	4.81665	0.240532	0.120266	−	0.992742i	$-0.461625\pi$
$401$	4.81665	0.240532	0.120266	+	0.992742i	$0.461625\pi$
$402$	10.9361	0.545442
$403$	−33.2111	−1.65436
$404$	−2.84441	−0.141515
$405$	0	0
$406$	0.513878	0.0255033
$407$	30.6333	1.51844
$408$	7.81665	0.386982
$409$	9.81665	0.485402	0.242701	−	0.970101i	$-0.421967\pi$
$409$	9.81665	0.485402	0.242701	+	0.970101i	$0.421967\pi$
$410$	0	0
$411$	11.2111	0.553003
$412$	−5.39445	−0.265765
$413$	3.39445	0.167030
$414$	−10.6972	−0.525740
$415$	0	0
$416$	−7.81665	−0.383243
$417$	17.0278	0.833853
$418$	2.36669	0.115759
$419$	6.00000	0.293119	0.146560	−	0.989202i	$-0.453180\pi$
$419$	6.00000	0.293119	0.146560	+	0.989202i	$0.453180\pi$
$420$	0	0
$421$	4.21110	0.205237	0.102618	−	0.994721i	$-0.467278\pi$
$421$	4.21110	0.205237	0.102618	+	0.994721i	$0.467278\pi$
$422$	−3.15559	−0.153612
$423$	−3.39445	−0.165044
$424$	−33.6333	−1.63338
$425$	0	0
$426$	3.90833	0.189359
$427$	−13.2111	−0.639330
$428$	0	0
$429$	−13.8167	−0.667074
$430$	0	0
$431$	34.4222	1.65806	0.829030	−	0.559205i	$-0.188893\pi$
$431$	34.4222	1.65806	0.829030	+	0.559205i	$0.188893\pi$
$432$	3.30278	0.158905
$433$	22.7889	1.09516	0.547582	−	0.836752i	$-0.315548\pi$
$433$	22.7889	1.09516	0.547582	+	0.836752i	$0.315548\pi$
$434$	−9.39445	−0.450948
$435$	0	0
$436$	3.69722	0.177065
$437$	4.97224	0.237855
$438$	−8.60555	−0.411189
$439$	−0.605551	−0.0289014	−0.0144507	−	0.999896i	$-0.504600\pi$
$439$	−0.605551	−0.0289014	−0.0144507	+	0.999896i	$0.504600\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	−15.6333	−0.743601
$443$	27.6333	1.31290	0.656449	−	0.754370i	$-0.272058\pi$
$443$	27.6333	1.31290	0.656449	+	0.754370i	$0.272058\pi$
$444$	−3.09167	−0.146724
$445$	0	0
$446$	−29.4500	−1.39450
$447$	23.6056	1.11650
$448$	−8.81665	−0.416548
$449$	−13.1833	−0.622161	−0.311080	−	0.950384i	$-0.600691\pi$
$449$	−13.1833	−0.622161	−0.311080	+	0.950384i	$0.600691\pi$
$450$	0	0
$451$	0	0
$452$	3.27502	0.154044
$453$	14.8167	0.696147
$454$	−20.0555	−0.941252
$455$	0	0
$456$	−1.81665	−0.0850726
$457$	18.2111	0.851879	0.425940	−	0.904752i	$-0.359944\pi$
$457$	18.2111	0.851879	0.425940	+	0.904752i	$0.359944\pi$
$458$	9.39445	0.438974
$459$	−2.60555	−0.121617
$460$	0	0
$461$	24.2389	1.12892	0.564458	−	0.825462i	$-0.309085\pi$
$461$	24.2389	1.12892	0.564458	+	0.825462i	$0.309085\pi$
$462$	−3.90833	−0.181832
$463$	−2.78890	−0.129611	−0.0648055	−	0.997898i	$-0.520643\pi$
$463$	−2.78890	−0.129611	−0.0648055	+	0.997898i	$0.520643\pi$
$464$	1.30278	0.0604798
$465$	0	0
$466$	29.7250	1.37698
$467$	−28.4222	−1.31522	−0.657611	−	0.753357i	$-0.728433\pi$
$467$	−28.4222	−1.31522	−0.657611	+	0.753357i	$0.728433\pi$
$468$	1.39445	0.0644584
$469$	8.39445	0.387620
$470$	0	0
$471$	−14.4222	−0.664540
$472$	10.1833	0.468727
$473$	7.18335	0.330291
$474$	−8.88057	−0.407898
$475$	0	0
$476$	0.788897	0.0361591
$477$	11.2111	0.513321
$478$	38.0555	1.74062
$479$	27.3944	1.25168	0.625842	−	0.779950i	$-0.284755\pi$
$479$	27.3944	1.25168	0.625842	+	0.779950i	$0.284755\pi$
$480$	0	0
$481$	47.0278	2.14428
$482$	21.6333	0.985370
$483$	−8.21110	−0.373618
$484$	0.605551	0.0275251
$485$	0	0
$486$	−1.30278	−0.0590951
$487$	−0.816654	−0.0370061	−0.0185031	−	0.999829i	$-0.505890\pi$
$487$	−0.816654	−0.0370061	−0.0185031	+	0.999829i	$0.505890\pi$
$488$	−39.6333	−1.79412
$489$	2.78890	0.126118
$490$	0	0
$491$	−3.00000	−0.135388	−0.0676941	−	0.997706i	$-0.521564\pi$
$491$	−3.00000	−0.135388	−0.0676941	+	0.997706i	$0.521564\pi$
$492$	0	0
$493$	−1.02776	−0.0462878
$494$	3.63331	0.163470
$495$	0	0
$496$	−23.8167	−1.06940
$497$	3.00000	0.134568
$498$	14.6056	0.654490
$499$	−28.0000	−1.25345	−0.626726	−	0.779240i	$-0.715605\pi$
$499$	−28.0000	−1.25345	−0.626726	+	0.779240i	$0.715605\pi$
$500$	0	0
$501$	19.0278	0.850097
$502$	10.1833	0.454505
$503$	23.2111	1.03493	0.517466	−	0.855704i	$-0.326875\pi$
$503$	23.2111	1.03493	0.517466	+	0.855704i	$0.326875\pi$
$504$	3.00000	0.133631
$505$	0	0
$506$	32.0917	1.42665
$507$	−8.21110	−0.364668
$508$	−2.66947	−0.118438
$509$	8.60555	0.381434	0.190717	−	0.981645i	$-0.438919\pi$
$509$	8.60555	0.381434	0.190717	+	0.981645i	$0.438919\pi$
$510$	0	0
$511$	−6.60555	−0.292212
$512$	−25.4222	−1.12351
$513$	0.605551	0.0267357
$514$	−28.1833	−1.24311
$515$	0	0
$516$	−0.724981	−0.0319155
$517$	10.1833	0.447863
$518$	13.3028	0.584490
$519$	13.8167	0.606484
$520$	0	0
$521$	−7.57779	−0.331989	−0.165995	−	0.986127i	$-0.553083\pi$
$521$	−7.57779	−0.331989	−0.165995	+	0.986127i	$0.553083\pi$
$522$	−0.513878	−0.0224918
$523$	16.0000	0.699631	0.349816	−	0.936819i	$-0.386244\pi$
$523$	16.0000	0.699631	0.349816	+	0.936819i	$0.386244\pi$
$524$	4.42221	0.193185
$525$	0	0
$526$	26.3305	1.14807
$527$	18.7889	0.818457
$528$	−9.90833	−0.431205
$529$	44.4222	1.93140
$530$	0	0
$531$	−3.39445	−0.147307
$532$	−0.183346	−0.00794906
$533$	0	0
$534$	18.0000	0.778936
$535$	0	0
$536$	25.1833	1.08775
$537$	0	0
$538$	14.6056	0.629690
$539$	−3.00000	−0.129219
$540$	0	0
$541$	−2.57779	−0.110828	−0.0554140	−	0.998463i	$-0.517648\pi$
$541$	−2.57779	−0.110828	−0.0554140	+	0.998463i	$0.517648\pi$
$542$	−25.2666	−1.08529
$543$	−5.39445	−0.231498
$544$	4.42221	0.189600
$545$	0	0
$546$	−6.00000	−0.256776
$547$	22.3944	0.957517	0.478759	−	0.877947i	$-0.341087\pi$
$547$	22.3944	0.957517	0.478759	+	0.877947i	$0.341087\pi$
$548$	3.39445	0.145004
$549$	13.2111	0.563836
$550$	0	0
$551$	0.238859	0.0101757
$552$	−24.6333	−1.04846
$553$	−6.81665	−0.289874
$554$	13.0278	0.553496
$555$	0	0
$556$	5.15559	0.218646
$557$	39.2389	1.66260	0.831302	−	0.555821i	$-0.187596\pi$
$557$	39.2389	1.66260	0.831302	+	0.555821i	$0.187596\pi$
$558$	9.39445	0.397699
$559$	11.0278	0.466424
$560$	0	0
$561$	7.81665	0.330019
$562$	−29.7250	−1.25387
$563$	−0.788897	−0.0332481	−0.0166240	−	0.999862i	$-0.505292\pi$
$563$	−0.788897	−0.0332481	−0.0166240	+	0.999862i	$0.505292\pi$
$564$	−1.02776	−0.0432764
$565$	0	0
$566$	−38.6056	−1.62271
$567$	−1.00000	−0.0419961
$568$	9.00000	0.377632
$569$	−22.8167	−0.956524	−0.478262	−	0.878217i	$-0.658733\pi$
$569$	−22.8167	−0.956524	−0.478262	+	0.878217i	$0.658733\pi$
$570$	0	0
$571$	1.60555	0.0671902	0.0335951	−	0.999436i	$-0.489304\pi$
$571$	1.60555	0.0671902	0.0335951	+	0.999436i	$0.489304\pi$
$572$	−4.18335	−0.174914
$573$	17.2111	0.719004
$574$	0	0
$575$	0	0
$576$	8.81665	0.367361
$577$	−23.3944	−0.973924	−0.486962	−	0.873423i	$-0.661895\pi$
$577$	−23.3944	−0.973924	−0.486962	+	0.873423i	$0.661895\pi$
$578$	−13.3028	−0.553323
$579$	−6.21110	−0.258125
$580$	0	0
$581$	11.2111	0.465115
$582$	−19.8167	−0.821427
$583$	−33.6333	−1.39295
$584$	−19.8167	−0.820019
$585$	0	0
$586$	−4.42221	−0.182680
$587$	−16.1833	−0.667958	−0.333979	−	0.942580i	$-0.608391\pi$
$587$	−16.1833	−0.667958	−0.333979	+	0.942580i	$0.608391\pi$
$588$	0.302776	0.0124863
$589$	−4.36669	−0.179926
$590$	0	0
$591$	−4.81665	−0.198131
$592$	33.7250	1.38609
$593$	15.6333	0.641983	0.320991	−	0.947082i	$-0.395984\pi$
$593$	15.6333	0.641983	0.320991	+	0.947082i	$0.395984\pi$
$594$	3.90833	0.160361
$595$	0	0
$596$	7.14719	0.292760
$597$	−8.00000	−0.327418
$598$	49.2666	2.01466
$599$	20.2111	0.825803	0.412902	−	0.910776i	$-0.364515\pi$
$599$	20.2111	0.825803	0.412902	+	0.910776i	$0.364515\pi$
$600$	0	0
$601$	2.78890	0.113761	0.0568807	−	0.998381i	$-0.481885\pi$
$601$	2.78890	0.113761	0.0568807	+	0.998381i	$0.481885\pi$
$602$	3.11943	0.127138
$603$	−8.39445	−0.341848
$604$	4.48612	0.182538
$605$	0	0
$606$	−12.2389	−0.497170
$607$	2.18335	0.0886193	0.0443096	−	0.999018i	$-0.485891\pi$
$607$	2.18335	0.0886193	0.0443096	+	0.999018i	$0.485891\pi$
$608$	−1.02776	−0.0416810
$609$	−0.394449	−0.0159839
$610$	0	0
$611$	15.6333	0.632456
$612$	−0.788897	−0.0318893
$613$	20.5778	0.831129	0.415565	−	0.909564i	$-0.363584\pi$
$613$	20.5778	0.831129	0.415565	+	0.909564i	$0.363584\pi$
$614$	1.81665	0.0733142
$615$	0	0
$616$	−9.00000	−0.362620
$617$	−1.18335	−0.0476397	−0.0238199	−	0.999716i	$-0.507583\pi$
$617$	−1.18335	−0.0476397	−0.0238199	+	0.999716i	$0.507583\pi$
$618$	−23.2111	−0.933687
$619$	4.60555	0.185113	0.0925564	−	0.995707i	$-0.470496\pi$
$619$	4.60555	0.185113	0.0925564	+	0.995707i	$0.470496\pi$
$620$	0	0
$621$	8.21110	0.329500
$622$	−10.1833	−0.408315
$623$	13.8167	0.553553
$624$	−15.2111	−0.608931
$625$	0	0
$626$	2.84441	0.113685
$627$	−1.81665	−0.0725502
$628$	−4.36669	−0.174250
$629$	−26.6056	−1.06083
$630$	0	0
$631$	−14.0278	−0.558436	−0.279218	−	0.960228i	$-0.590075\pi$
$631$	−14.0278	−0.558436	−0.279218	+	0.960228i	$0.590075\pi$
$632$	−20.4500	−0.813456
$633$	2.42221	0.0962740
$634$	30.7527	1.22135
$635$	0	0
$636$	3.39445	0.134599
$637$	−4.60555	−0.182479
$638$	1.54163	0.0610339
$639$	−3.00000	−0.118678
$640$	0	0
$641$	1.18335	0.0467394	0.0233697	−	0.999727i	$-0.492561\pi$
$641$	1.18335	0.0467394	0.0233697	+	0.999727i	$0.492561\pi$
$642$	0	0
$643$	−9.57779	−0.377711	−0.188856	−	0.982005i	$-0.560478\pi$
$643$	−9.57779	−0.377711	−0.188856	+	0.982005i	$0.560478\pi$
$644$	−2.48612	−0.0979669
$645$	0	0
$646$	−2.05551	−0.0808731
$647$	−6.00000	−0.235884	−0.117942	−	0.993020i	$-0.537630\pi$
$647$	−6.00000	−0.235884	−0.117942	+	0.993020i	$0.537630\pi$
$648$	−3.00000	−0.117851
$649$	10.1833	0.399731
$650$	0	0
$651$	7.21110	0.282625
$652$	0.844410	0.0330697
$653$	−11.2111	−0.438724	−0.219362	−	0.975643i	$-0.570398\pi$
$653$	−11.2111	−0.438724	−0.219362	+	0.975643i	$0.570398\pi$
$654$	15.9083	0.622065
$655$	0	0
$656$	0	0
$657$	6.60555	0.257707
$658$	4.42221	0.172396
$659$	−12.0000	−0.467454	−0.233727	−	0.972302i	$-0.575092\pi$
$659$	−12.0000	−0.467454	−0.233727	+	0.972302i	$0.575092\pi$
$660$	0	0
$661$	−41.8167	−1.62648	−0.813240	−	0.581929i	$-0.802298\pi$
$661$	−41.8167	−1.62648	−0.813240	+	0.581929i	$0.802298\pi$
$662$	38.0917	1.48047
$663$	12.0000	0.466041
$664$	33.6333	1.30523
$665$	0	0
$666$	−13.3028	−0.515472
$667$	3.23886	0.125409
$668$	5.76114	0.222905
$669$	22.6056	0.873981
$670$	0	0
$671$	−39.6333	−1.53003
$672$	1.69722	0.0654719
$673$	10.0000	0.385472	0.192736	−	0.981251i	$-0.438264\pi$
$673$	10.0000	0.385472	0.192736	+	0.981251i	$0.438264\pi$
$674$	−9.39445	−0.361861
$675$	0	0
$676$	−2.48612	−0.0956201
$677$	23.2111	0.892075	0.446038	−	0.895014i	$-0.352835\pi$
$677$	23.2111	0.892075	0.446038	+	0.895014i	$0.352835\pi$
$678$	14.0917	0.541187
$679$	−15.2111	−0.583749
$680$	0	0
$681$	15.3944	0.589917
$682$	−28.1833	−1.07920
$683$	−24.6333	−0.942567	−0.471284	−	0.881982i	$-0.656209\pi$
$683$	−24.6333	−0.942567	−0.471284	+	0.881982i	$0.656209\pi$
$684$	0.183346	0.00701042
$685$	0	0
$686$	−1.30278	−0.0497402
$687$	−7.21110	−0.275121
$688$	7.90833	0.301502
$689$	−51.6333	−1.96707
$690$	0	0
$691$	26.7889	1.01910	0.509549	−	0.860442i	$-0.329812\pi$
$691$	26.7889	1.01910	0.509549	+	0.860442i	$0.329812\pi$
$692$	4.18335	0.159027
$693$	3.00000	0.113961
$694$	−20.5694	−0.780803
$695$	0	0
$696$	−1.18335	−0.0448546
$697$	0	0
$698$	−43.5778	−1.64944
$699$	−22.8167	−0.863005
$700$	0	0
$701$	−35.2111	−1.32990	−0.664952	−	0.746886i	$-0.731548\pi$
$701$	−35.2111	−1.32990	−0.664952	+	0.746886i	$0.731548\pi$
$702$	6.00000	0.226455
$703$	6.18335	0.233209
$704$	−26.4500	−0.996870
$705$	0	0
$706$	39.0833	1.47092
$707$	−9.39445	−0.353315
$708$	−1.02776	−0.0386254
$709$	22.8444	0.857940	0.428970	−	0.903319i	$-0.358877\pi$
$709$	22.8444	0.857940	0.428970	+	0.903319i	$0.358877\pi$
$710$	0	0
$711$	6.81665	0.255644
$712$	41.4500	1.55340
$713$	−59.2111	−2.21747
$714$	3.39445	0.127034
$715$	0	0
$716$	0	0
$717$	−29.2111	−1.09091
$718$	−24.2750	−0.905936
$719$	41.2111	1.53691	0.768457	−	0.639901i	$-0.221025\pi$
$719$	41.2111	1.53691	0.768457	+	0.639901i	$0.221025\pi$
$720$	0	0
$721$	−17.8167	−0.663527
$722$	−24.2750	−0.903423
$723$	−16.6056	−0.617567
$724$	−1.63331	−0.0607014
$725$	0	0
$726$	2.60555	0.0967011
$727$	30.6056	1.13510	0.567549	−	0.823340i	$-0.307892\pi$
$727$	30.6056	1.13510	0.567549	+	0.823340i	$0.307892\pi$
$728$	−13.8167	−0.512079
$729$	1.00000	0.0370370
$730$	0	0
$731$	−6.23886	−0.230753
$732$	4.00000	0.147844
$733$	−20.0000	−0.738717	−0.369358	−	0.929287i	$-0.620423\pi$
$733$	−20.0000	−0.738717	−0.369358	+	0.929287i	$0.620423\pi$
$734$	18.7889	0.693511
$735$	0	0
$736$	−13.9361	−0.513691
$737$	25.1833	0.927640
$738$	0	0
$739$	24.8167	0.912895	0.456448	−	0.889750i	$-0.349122\pi$
$739$	24.8167	0.912895	0.456448	+	0.889750i	$0.349122\pi$
$740$	0	0
$741$	−2.78890	−0.102453
$742$	−14.6056	−0.536187
$743$	−15.6333	−0.573530	−0.286765	−	0.958001i	$-0.592580\pi$
$743$	−15.6333	−0.573530	−0.286765	+	0.958001i	$0.592580\pi$
$744$	21.6333	0.793116
$745$	0	0
$746$	1.30278	0.0476980
$747$	−11.2111	−0.410193
$748$	2.36669	0.0865348
$749$	0	0
$750$	0	0
$751$	−43.6333	−1.59220	−0.796101	−	0.605164i	$-0.793108\pi$
$751$	−43.6333	−1.59220	−0.796101	+	0.605164i	$0.793108\pi$
$752$	11.2111	0.408827
$753$	−7.81665	−0.284855
$754$	2.36669	0.0861899
$755$	0	0
$756$	−0.302776	−0.0110118
$757$	−41.0000	−1.49017	−0.745085	−	0.666969i	$-0.767591\pi$
$757$	−41.0000	−1.49017	−0.745085	+	0.666969i	$0.767591\pi$
$758$	8.88057	0.322557
$759$	−24.6333	−0.894132
$760$	0	0
$761$	15.6333	0.566707	0.283353	−	0.959016i	$-0.408553\pi$
$761$	15.6333	0.566707	0.283353	+	0.959016i	$0.408553\pi$
$762$	−11.4861	−0.416098
$763$	12.2111	0.442072
$764$	5.21110	0.188531
$765$	0	0
$766$	−31.5778	−1.14095
$767$	15.6333	0.564486
$768$	7.09167	0.255899
$769$	11.6333	0.419508	0.209754	−	0.977754i	$-0.432734\pi$
$769$	11.6333	0.419508	0.209754	+	0.977754i	$0.432734\pi$
$770$	0	0
$771$	21.6333	0.779105
$772$	−1.88057	−0.0676832
$773$	−6.00000	−0.215805	−0.107903	−	0.994161i	$-0.534413\pi$
$773$	−6.00000	−0.215805	−0.107903	+	0.994161i	$0.534413\pi$
$774$	−3.11943	−0.112126
$775$	0	0
$776$	−45.6333	−1.63814
$777$	−10.2111	−0.366321
$778$	−9.35829	−0.335511
$779$	0	0
$780$	0	0
$781$	9.00000	0.322045
$782$	−27.8722	−0.996707
$783$	0.394449	0.0140964
$784$	−3.30278	−0.117956
$785$	0	0
$786$	19.0278	0.678698
$787$	22.2389	0.792730	0.396365	−	0.918093i	$-0.370272\pi$
$787$	22.2389	0.792730	0.396365	+	0.918093i	$0.370272\pi$
$788$	−1.45837	−0.0519521
$789$	−20.2111	−0.719534
$790$	0	0
$791$	10.8167	0.384596
$792$	9.00000	0.319801
$793$	−60.8444	−2.16065
$794$	−2.60555	−0.0924676
$795$	0	0
$796$	−2.42221	−0.0858528
$797$	−35.4500	−1.25570	−0.627851	−	0.778334i	$-0.716065\pi$
$797$	−35.4500	−1.25570	−0.627851	+	0.778334i	$0.716065\pi$
$798$	−0.788897	−0.0279267
$799$	−8.84441	−0.312893
$800$	0	0
$801$	−13.8167	−0.488187
$802$	6.27502	0.221579
$803$	−19.8167	−0.699315
$804$	−2.54163	−0.0896365
$805$	0	0
$806$	−43.2666	−1.52400
$807$	−11.2111	−0.394650
$808$	−28.1833	−0.991487
$809$	−32.4500	−1.14088	−0.570440	−	0.821339i	$-0.693227\pi$
$809$	−32.4500	−1.14088	−0.570440	+	0.821339i	$0.693227\pi$
$810$	0	0
$811$	−42.8444	−1.50447	−0.752235	−	0.658894i	$-0.771024\pi$
$811$	−42.8444	−1.50447	−0.752235	+	0.658894i	$0.771024\pi$
$812$	−0.119429	−0.00419115
$813$	19.3944	0.680193
$814$	39.9083	1.39879
$815$	0	0
$816$	8.60555	0.301255
$817$	1.44996	0.0507277
$818$	12.7889	0.447153
$819$	4.60555	0.160931
$820$	0	0
$821$	−37.2666	−1.30061	−0.650307	−	0.759672i	$-0.725360\pi$
$821$	−37.2666	−1.30061	−0.650307	+	0.759672i	$0.725360\pi$
$822$	14.6056	0.509427
$823$	−3.18335	−0.110964	−0.0554822	−	0.998460i	$-0.517670\pi$
$823$	−3.18335	−0.110964	−0.0554822	+	0.998460i	$0.517670\pi$
$824$	−53.4500	−1.86202
$825$	0	0
$826$	4.42221	0.153868
$827$	−12.6333	−0.439303	−0.219652	−	0.975578i	$-0.570492\pi$
$827$	−12.6333	−0.439303	−0.219652	+	0.975578i	$0.570492\pi$
$828$	2.48612	0.0863987
$829$	50.2389	1.74487	0.872434	−	0.488732i	$-0.162540\pi$
$829$	50.2389	1.74487	0.872434	+	0.488732i	$0.162540\pi$
$830$	0	0
$831$	−10.0000	−0.346896
$832$	−40.6056	−1.40774
$833$	2.60555	0.0902770
$834$	22.1833	0.768146
$835$	0	0
$836$	−0.550039	−0.0190235
$837$	−7.21110	−0.249252
$838$	7.81665	0.270022
$839$	−43.8167	−1.51272	−0.756359	−	0.654156i	$-0.773024\pi$
$839$	−43.8167	−1.51272	−0.756359	+	0.654156i	$0.773024\pi$
$840$	0	0
$841$	−28.8444	−0.994635
$842$	5.48612	0.189064
$843$	22.8167	0.785847
$844$	0.733385	0.0252441
$845$	0	0
$846$	−4.42221	−0.152039
$847$	2.00000	0.0687208
$848$	−37.0278	−1.27154
$849$	29.6333	1.01701
$850$	0	0
$851$	83.8444	2.87415
$852$	−0.908327	−0.0311188
$853$	21.2111	0.726254	0.363127	−	0.931740i	$-0.381709\pi$
$853$	21.2111	0.726254	0.363127	+	0.931740i	$0.381709\pi$
$854$	−17.2111	−0.588952
$855$	0	0
$856$	0	0
$857$	15.6333	0.534024	0.267012	−	0.963693i	$-0.413964\pi$
$857$	15.6333	0.534024	0.267012	+	0.963693i	$0.413964\pi$
$858$	−18.0000	−0.614510
$859$	−6.36669	−0.217229	−0.108614	−	0.994084i	$-0.534641\pi$
$859$	−6.36669	−0.217229	−0.108614	+	0.994084i	$0.534641\pi$
$860$	0	0
$861$	0	0
$862$	44.8444	1.52741
$863$	−48.6333	−1.65550	−0.827749	−	0.561099i	$-0.810379\pi$
$863$	−48.6333	−1.65550	−0.827749	+	0.561099i	$0.810379\pi$
$864$	−1.69722	−0.0577407
$865$	0	0
$866$	29.6888	1.00887
$867$	10.2111	0.346787
$868$	2.18335	0.0741076
$869$	−20.4500	−0.693717
$870$	0	0
$871$	38.6611	1.30998
$872$	36.6333	1.24056
$873$	15.2111	0.514818
$874$	6.47772	0.219112
$875$	0	0
$876$	2.00000	0.0675737
$877$	22.0000	0.742887	0.371444	−	0.928456i	$-0.378863\pi$
$877$	22.0000	0.742887	0.371444	+	0.928456i	$0.378863\pi$
$878$	−0.788897	−0.0266240
$879$	3.39445	0.114492
$880$	0	0
$881$	−9.63331	−0.324554	−0.162277	−	0.986745i	$-0.551884\pi$
$881$	−9.63331	−0.324554	−0.162277	+	0.986745i	$0.551884\pi$
$882$	1.30278	0.0438667
$883$	−8.39445	−0.282496	−0.141248	−	0.989974i	$-0.545111\pi$
$883$	−8.39445	−0.282496	−0.141248	+	0.989974i	$0.545111\pi$
$884$	3.63331	0.122201
$885$	0	0
$886$	36.0000	1.20944
$887$	29.2111	0.980813	0.490406	−	0.871494i	$-0.336848\pi$
$887$	29.2111	0.980813	0.490406	+	0.871494i	$0.336848\pi$
$888$	−30.6333	−1.02799
$889$	−8.81665	−0.295701
$890$	0	0
$891$	−3.00000	−0.100504
$892$	6.84441	0.229168
$893$	2.05551	0.0687851
$894$	30.7527	1.02853
$895$	0	0
$896$	−8.09167	−0.270324
$897$	−37.8167	−1.26266
$898$	−17.1749	−0.573135
$899$	−2.84441	−0.0948664
$900$	0	0
$901$	29.2111	0.973163
$902$	0	0
$903$	−2.39445	−0.0796823
$904$	32.4500	1.07927
$905$	0	0
$906$	19.3028	0.641292
$907$	−1.21110	−0.0402140	−0.0201070	−	0.999798i	$-0.506401\pi$
$907$	−1.21110	−0.0402140	−0.0201070	+	0.999798i	$0.506401\pi$
$908$	4.66106	0.154683
$909$	9.39445	0.311594
$910$	0	0
$911$	15.0000	0.496972	0.248486	−	0.968635i	$-0.420067\pi$
$911$	15.0000	0.496972	0.248486	+	0.968635i	$0.420067\pi$
$912$	−2.00000	−0.0662266
$913$	33.6333	1.11310
$914$	23.7250	0.784753
$915$	0	0
$916$	−2.18335	−0.0721398
$917$	14.6056	0.482318
$918$	−3.39445	−0.112034
$919$	−10.3944	−0.342881	−0.171441	−	0.985194i	$-0.554842\pi$
$919$	−10.3944	−0.342881	−0.171441	+	0.985194i	$0.554842\pi$
$920$	0	0
$921$	−1.39445	−0.0459486
$922$	31.5778	1.03996
$923$	13.8167	0.454781
$924$	0.908327	0.0298818
$925$	0	0
$926$	−3.63331	−0.119398
$927$	17.8167	0.585176
$928$	−0.669468	−0.0219764
$929$	30.2389	0.992105	0.496052	−	0.868293i	$-0.334782\pi$
$929$	30.2389	0.992105	0.496052	+	0.868293i	$0.334782\pi$
$930$	0	0
$931$	−0.605551	−0.0198461
$932$	−6.90833	−0.226290
$933$	7.81665	0.255906
$934$	−37.0278	−1.21159
$935$	0	0
$936$	13.8167	0.451611
$937$	−44.4777	−1.45302	−0.726512	−	0.687154i	$-0.758860\pi$
$937$	−44.4777	−1.45302	−0.726512	+	0.687154i	$0.758860\pi$
$938$	10.9361	0.357076
$939$	−2.18335	−0.0712508
$940$	0	0
$941$	6.00000	0.195594	0.0977972	−	0.995206i	$-0.468820\pi$
$941$	6.00000	0.195594	0.0977972	+	0.995206i	$0.468820\pi$
$942$	−18.7889	−0.612175
$943$	0	0
$944$	11.2111	0.364890
$945$	0	0
$946$	9.35829	0.304264
$947$	−56.8444	−1.84720	−0.923598	−	0.383363i	$-0.874766\pi$
$947$	−56.8444	−1.84720	−0.923598	+	0.383363i	$0.874766\pi$
$948$	2.06392	0.0670329
$949$	−30.4222	−0.987547
$950$	0	0
$951$	−23.6056	−0.765462
$952$	7.81665	0.253339
$953$	6.39445	0.207137	0.103568	−	0.994622i	$-0.466974\pi$
$953$	6.39445	0.207137	0.103568	+	0.994622i	$0.466974\pi$
$954$	14.6056	0.472872
$955$	0	0
$956$	−8.84441	−0.286049
$957$	−1.18335	−0.0382521
$958$	35.6888	1.15305
$959$	11.2111	0.362025
$960$	0	0
$961$	21.0000	0.677419
$962$	61.2666	1.97531
$963$	0	0
$964$	−5.02776	−0.161933
$965$	0	0
$966$	−10.6972	−0.344178
$967$	−16.8444	−0.541680	−0.270840	−	0.962624i	$-0.587301\pi$
$967$	−16.8444	−0.541680	−0.270840	+	0.962624i	$0.587301\pi$
$968$	6.00000	0.192847
$969$	1.57779	0.0506861
$970$	0	0
$971$	−39.8722	−1.27956	−0.639779	−	0.768559i	$-0.720974\pi$
$971$	−39.8722	−1.27956	−0.639779	+	0.768559i	$0.720974\pi$
$972$	0.302776	0.00971153
$973$	17.0278	0.545885
$974$	−1.06392	−0.0340901
$975$	0	0
$976$	−43.6333	−1.39667
$977$	−34.0278	−1.08864	−0.544322	−	0.838876i	$-0.683213\pi$
$977$	−34.0278	−1.08864	−0.544322	+	0.838876i	$0.683213\pi$
$978$	3.63331	0.116180
$979$	41.4500	1.32475
$980$	0	0
$981$	−12.2111	−0.389870
$982$	−3.90833	−0.124720
$983$	32.0555	1.02241	0.511206	−	0.859458i	$-0.329199\pi$
$983$	32.0555	1.02241	0.511206	+	0.859458i	$0.329199\pi$
$984$	0	0
$985$	0	0
$986$	−1.33894	−0.0426404
$987$	−3.39445	−0.108046
$988$	−0.844410	−0.0268643
$989$	19.6611	0.625185
$990$	0	0
$991$	9.97224	0.316779	0.158389	−	0.987377i	$-0.449370\pi$
$991$	9.97224	0.316779	0.158389	+	0.987377i	$0.449370\pi$
$992$	12.2389	0.388584
$993$	−29.2389	−0.927867
$994$	3.90833	0.123965
$995$	0	0
$996$	−3.39445	−0.107557
$997$	−49.4500	−1.56610	−0.783048	−	0.621961i	$-0.786336\pi$
$997$	−49.4500	−1.56610	−0.783048	+	0.621961i	$0.786336\pi$
$998$	−36.4777	−1.15468
$999$	10.2111	0.323065

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	525.2.a.f.1.2	✓	2
3.2	odd	2		1575.2.a.t.1.1		2
4.3	odd	2		8400.2.a.df.1.1		2
5.2	odd	4		525.2.d.d.274.3		4
5.3	odd	4		525.2.d.d.274.2		4
5.4	even	2		525.2.a.h.1.1	yes	2
7.6	odd	2		3675.2.a.w.1.2		2
15.2	even	4		1575.2.d.g.1324.2		4
15.8	even	4		1575.2.d.g.1324.3		4
15.14	odd	2		1575.2.a.o.1.2		2
20.19	odd	2		8400.2.a.cw.1.2		2
35.34	odd	2		3675.2.a.bb.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
525.2.a.f.1.2	✓	2	1.1	even	1	trivial
525.2.a.h.1.1	yes	2	5.4	even	2
525.2.d.d.274.2		4	5.3	odd	4
525.2.d.d.274.3		4	5.2	odd	4
1575.2.a.o.1.2		2	15.14	odd	2
1575.2.a.t.1.1		2	3.2	odd	2
1575.2.d.g.1324.2		4	15.2	even	4
1575.2.d.g.1324.3		4	15.8	even	4
3675.2.a.w.1.2		2	7.6	odd	2
3675.2.a.bb.1.1		2	35.34	odd	2
8400.2.a.cw.1.2		2	20.19	odd	2
8400.2.a.df.1.1		2	4.3	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:	\( N \)	\(=\)	\( 525 = 3 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	525.a (trivial)

\(f(q)\)	\(=\)	\(q+1.30278 q^{2} -1.00000 q^{3} -0.302776 q^{4} -1.30278 q^{6} -1.00000 q^{7} -3.00000 q^{8} +1.00000 q^{9} -3.00000 q^{11} +0.302776 q^{12} -4.60555 q^{13} -1.30278 q^{14} -3.30278 q^{16} +2.60555 q^{17} +1.30278 q^{18} -0.605551 q^{19} +1.00000 q^{21} -3.90833 q^{22} -8.21110 q^{23} +3.00000 q^{24} -6.00000 q^{26} -1.00000 q^{27} +0.302776 q^{28} -0.394449 q^{29} +7.21110 q^{31} +1.69722 q^{32} +3.00000 q^{33} +3.39445 q^{34} -0.302776 q^{36} -10.2111 q^{37} -0.788897 q^{38} +4.60555 q^{39} +1.30278 q^{42} -2.39445 q^{43} +0.908327 q^{44} -10.6972 q^{46} -3.39445 q^{47} +3.30278 q^{48} +1.00000 q^{49} -2.60555 q^{51} +1.39445 q^{52} +11.2111 q^{53} -1.30278 q^{54} +3.00000 q^{56} +0.605551 q^{57} -0.513878 q^{58} -3.39445 q^{59} +13.2111 q^{61} +9.39445 q^{62} -1.00000 q^{63} +8.81665 q^{64} +3.90833 q^{66} -8.39445 q^{67} -0.788897 q^{68} +8.21110 q^{69} -3.00000 q^{71} -3.00000 q^{72} +6.60555 q^{73} -13.3028 q^{74} +0.183346 q^{76} +3.00000 q^{77} +6.00000 q^{78} +6.81665 q^{79} +1.00000 q^{81} -11.2111 q^{83} -0.302776 q^{84} -3.11943 q^{86} +0.394449 q^{87} +9.00000 q^{88} -13.8167 q^{89} +4.60555 q^{91} +2.48612 q^{92} -7.21110 q^{93} -4.42221 q^{94} -1.69722 q^{96} +15.2111 q^{97} +1.30278 q^{98} -3.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{2} - 2 q^{3} + 3 q^{4} + q^{6} - 2 q^{7} - 6 q^{8} + 2 q^{9} - 6 q^{11} - 3 q^{12} - 2 q^{13} + q^{14} - 3 q^{16} - 2 q^{17} - q^{18} + 6 q^{19} + 2 q^{21} + 3 q^{22} - 2 q^{23} + 6 q^{24}+ \cdots - 6 q^{99}+O(q^{100}) \) 2 * q - q^2 - 2 * q^3 + 3 * q^4 + q^6 - 2 * q^7 - 6 * q^8 + 2 * q^9 - 6 * q^11 - 3 * q^12 - 2 * q^13 + q^14 - 3 * q^16 - 2 * q^17 - q^18 + 6 * q^19 + 2 * q^21 + 3 * q^22 - 2 * q^23 + 6 * q^24 - 12 * q^26 - 2 * q^27 - 3 * q^28 - 8 * q^29 + 7 * q^32 + 6 * q^33 + 14 * q^34 + 3 * q^36 - 6 * q^37 - 16 * q^38 + 2 * q^39 - q^42 - 12 * q^43 - 9 * q^44 - 25 * q^46 - 14 * q^47 + 3 * q^48 + 2 * q^49 + 2 * q^51 + 10 * q^52 + 8 * q^53 + q^54 + 6 * q^56 - 6 * q^57 + 17 * q^58 - 14 * q^59 + 12 * q^61 + 26 * q^62 - 2 * q^63 - 4 * q^64 - 3 * q^66 - 24 * q^67 - 16 * q^68 + 2 * q^69 - 6 * q^71 - 6 * q^72 + 6 * q^73 - 23 * q^74 + 22 * q^76 + 6 * q^77 + 12 * q^78 - 8 * q^79 + 2 * q^81 - 8 * q^83 + 3 * q^84 + 19 * q^86 + 8 * q^87 + 18 * q^88 - 6 * q^89 + 2 * q^91 + 23 * q^92 + 20 * q^94 - 7 * q^96 + 16 * q^97 - q^98 - 6 * q^99

Introduction

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