LMFDB - Embedded newform 225.5.c.b.26.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [225,5,Mod(26,225)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(225, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0]))

N = Newforms(chi, 5, names="a")

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

chi := DirichletCharacter("225.26");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 225 = 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	225.c (of order $2$, degree $1$, 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:	$23.2582416939$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-2}, \sqrt{-5})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 4x^{2} + 9 $ x^4 - 4*x^2 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 45)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			26.4
Root			$1.58114 + 0.707107i$ of defining polynomial
Character	$\chi$	$=$	225.26
Dual form			225.5.c.b.26.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+6.47871i q^{2} -25.9737 q^{4} +78.4078 q^{7} -64.6165i q^{8} +O(q^{10})$	$q+6.47871i q^{2} -25.9737 q^{4} +78.4078 q^{7} -64.6165i q^{8} +11.4687i q^{11} -249.434 q^{13} +507.981i q^{14} +3.05267 q^{16} +437.116i q^{17} -421.026 q^{19} -74.3025 q^{22} +648.919i q^{23} -1616.01i q^{26} -2036.54 q^{28} -477.154i q^{29} +667.499 q^{31} -1014.09i q^{32} -2831.95 q^{34} -1385.70 q^{37} -2727.71i q^{38} +244.633i q^{41} -590.185 q^{43} -297.885i q^{44} -4204.16 q^{46} +2067.67i q^{47} +3746.79 q^{49} +6478.72 q^{52} -2811.33i q^{53} -5066.44i q^{56} +3091.34 q^{58} +1605.43i q^{59} -4871.24 q^{61} +4324.53i q^{62} +6618.81 q^{64} -1273.68 q^{67} -11353.5i q^{68} +4652.03i q^{71} -3759.81 q^{73} -8977.52i q^{74} +10935.6 q^{76} +899.237i q^{77} +5705.24 q^{79} -1584.90 q^{82} +467.210i q^{83} -3823.64i q^{86} +741.068 q^{88} -9901.61i q^{89} -19557.6 q^{91} -16854.8i q^{92} -13395.8 q^{94} -9453.39 q^{97} +24274.3i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 28 q^{4} + 48 q^{7}+O(q^{10}) $ 4 * q - 28 * q^4 + 48 * q^7	$ 4 q - 28 q^{4} + 48 q^{7} - 808 q^{13} + 164 q^{16} - 1760 q^{19} + 272 q^{22} - 5376 q^{28} - 1656 q^{31} - 5408 q^{34} - 344 q^{37} - 5776 q^{43} - 5736 q^{46} + 8612 q^{49} + 9256 q^{52} + 1816 q^{58} - 20168 q^{61} + 11524 q^{64} + 1584 q^{67} - 88 q^{73} + 10880 q^{76} + 17736 q^{79} + 6904 q^{82} + 13248 q^{88} - 22296 q^{91} - 32864 q^{94} - 27416 q^{97}+O(q^{100}) $ 4 * q - 28 * q^4 + 48 * q^7 - 808 * q^13 + 164 * q^16 - 1760 * q^19 + 272 * q^22 - 5376 * q^28 - 1656 * q^31 - 5408 * q^34 - 344 * q^37 - 5776 * q^43 - 5736 * q^46 + 8612 * q^49 + 9256 * q^52 + 1816 * q^58 - 20168 * q^61 + 11524 * q^64 + 1584 * q^67 - 88 * q^73 + 10880 * q^76 + 17736 * q^79 + 6904 * q^82 + 13248 * q^88 - 22296 * q^91 - 32864 * q^94 - 27416 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/225\mathbb{Z}\right)^\times$.

$n$	$101$	$127$
$\chi(n)$	$-1$	$1$

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 $

$2$	6.47871i	1.61968i	0.586653	+	0.809839i	$0.300445\pi$
$2$	6.47871i	1.61968i	−0.586653	+	0.809839i	$0.699555\pi$
$3$	0	0
$3$	0	0
$4$	−25.9737	−1.62335
$5$	0	0
$5$	0	0
$6$	0	0
$7$	78.4078	1.60016	0.800080	−	0.599893i	$-0.204790\pi$
$7$	78.4078	1.60016	0.800080	+	0.599893i	$0.204790\pi$
$8$	− 64.6165i	− 1.00963i
$9$	0	0
$10$	0	0
$11$	11.4687i	0.0947828i	0.998876	+	0.0473914i	$0.0150908\pi$
$11$	11.4687i	0.0947828i	−0.998876	+	0.0473914i	$0.984909\pi$
$12$	0	0
$13$	−249.434	−1.47594	−0.737971	−	0.674833i	$-0.764216\pi$
$13$	−249.434	−1.47594	−0.737971	+	0.674833i	$0.764216\pi$
$14$	507.981i	2.59174i
$15$	0	0
$16$	3.05267	0.0119245
$17$	437.116i	1.51251i	0.654276	+	0.756256i	$0.272973\pi$
$17$	437.116i	1.51251i	−0.654276	+	0.756256i	$0.727027\pi$
$18$	0	0
$19$	−421.026	−1.16628	−0.583139	−	0.812372i	$-0.698176\pi$
$19$	−421.026	−1.16628	−0.583139	+	0.812372i	$0.698176\pi$
$20$	0	0
$21$	0	0
$22$	−74.3025	−0.153518
$23$	648.919i	1.22669i	0.789815	+	0.613345i	$0.210176\pi$
$23$	648.919i	1.22669i	−0.789815	+	0.613345i	$0.789824\pi$
$24$	0	0
$25$	0	0
$26$	− 1616.01i	− 2.39055i
$27$	0	0
$28$	−2036.54	−2.59763
$29$	− 477.154i	− 0.567365i	−0.958918	−	0.283682i	$-0.908444\pi$
$29$	− 477.154i	− 0.567365i	0.958918	−	0.283682i	$-0.0915561\pi$
$30$	0	0
$31$	667.499	0.694588	0.347294	−	0.937756i	$-0.387101\pi$
$31$	667.499	0.694588	0.347294	+	0.937756i	$0.387101\pi$
$32$	− 1014.09i	− 0.990319i
$33$	0	0
$34$	−2831.95	−2.44978
$35$	0	0
$36$	0	0
$37$	−1385.70	−1.01220	−0.506098	−	0.862476i	$-0.668913\pi$
$37$	−1385.70	−1.01220	−0.506098	+	0.862476i	$0.668913\pi$
$38$	− 2727.71i	− 1.88899i
$39$	0	0
$40$	0	0
$41$	244.633i	0.145528i	0.997349	+	0.0727641i	$0.0231820\pi$
$41$	244.633i	0.145528i	−0.997349	+	0.0727641i	$0.976818\pi$
$42$	0	0
$43$	−590.185	−0.319191	−0.159596	−	0.987182i	$-0.551019\pi$
$43$	−590.185	−0.319191	−0.159596	+	0.987182i	$0.551019\pi$
$44$	− 297.885i	− 0.153866i
$45$	0	0
$46$	−4204.16	−1.98684
$47$	2067.67i	0.936019i	0.883723	+	0.468010i	$0.155029\pi$
$47$	2067.67i	0.936019i	−0.883723	+	0.468010i	$0.844971\pi$
$48$	0	0
$49$	3746.79	1.56051
$50$	0	0
$51$	0	0
$52$	6478.72	2.39598
$53$	− 2811.33i	− 1.00083i	−0.865785	−	0.500415i	$-0.833181\pi$
$53$	− 2811.33i	− 1.00083i	0.865785	−	0.500415i	$-0.166819\pi$
$54$	0	0
$55$	0	0
$56$	− 5066.44i	− 1.61557i
$57$	0	0
$58$	3091.34	0.918948
$59$	1605.43i	0.461199i	0.973049	+	0.230599i	$0.0740687\pi$
$59$	1605.43i	0.461199i	−0.973049	+	0.230599i	$0.925931\pi$
$60$	0	0
$61$	−4871.24	−1.30912	−0.654560	−	0.756010i	$-0.727146\pi$
$61$	−4871.24	−1.30912	−0.654560	+	0.756010i	$0.727146\pi$
$62$	4324.53i	1.12501i
$63$	0	0
$64$	6618.81	1.61592
$65$	0	0
$66$	0	0
$67$	−1273.68	−0.283734	−0.141867	−	0.989886i	$-0.545311\pi$
$67$	−1273.68	−0.283734	−0.141867	+	0.989886i	$0.545311\pi$
$68$	− 11353.5i	− 2.45534i
$69$	0	0
$70$	0	0
$71$	4652.03i	0.922839i	0.887182	+	0.461419i	$0.152660\pi$
$71$	4652.03i	0.922839i	−0.887182	+	0.461419i	$0.847340\pi$
$72$	0	0
$73$	−3759.81	−0.705538	−0.352769	−	0.935710i	$-0.614760\pi$
$73$	−3759.81	−0.705538	−0.352769	+	0.935710i	$0.614760\pi$
$74$	− 8977.52i	− 1.63943i
$75$	0	0
$76$	10935.6	1.89328
$77$	899.237i	0.151668i
$78$	0	0
$79$	5705.24	0.914154	0.457077	−	0.889427i	$-0.348896\pi$
$79$	5705.24	0.914154	0.457077	+	0.889427i	$0.348896\pi$
$80$	0	0
$81$	0	0
$82$	−1584.90	−0.235709
$83$	467.210i	0.0678197i	0.999425	+	0.0339098i	$0.0107959\pi$
$83$	467.210i	0.0678197i	−0.999425	+	0.0339098i	$0.989204\pi$
$84$	0	0
$85$	0	0
$86$	− 3823.64i	− 0.516987i
$87$	0	0
$88$	741.068	0.0956958
$89$	− 9901.61i	− 1.25005i	−0.780607	−	0.625023i	$-0.785090\pi$
$89$	− 9901.61i	− 1.25005i	0.780607	−	0.625023i	$-0.214910\pi$
$90$	0	0
$91$	−19557.6	−2.36174
$92$	− 16854.8i	− 1.99135i
$93$	0	0
$94$	−13395.8	−1.51605
$95$	0	0
$96$	0	0
$97$	−9453.39	−1.00472	−0.502359	−	0.864659i	$-0.667534\pi$
$97$	−9453.39	−1.00472	−0.502359	+	0.864659i	$0.667534\pi$
$98$	24274.3i	2.52752i
$99$	0	0
$100$	0	0
$101$	1291.16i	0.126572i	0.997995	+	0.0632860i	$0.0201580\pi$
$101$	1291.16i	0.126572i	−0.997995	+	0.0632860i	$0.979842\pi$
$102$	0	0
$103$	16730.4	1.57700	0.788501	−	0.615033i	$-0.210857\pi$
$103$	16730.4	1.57700	0.788501	+	0.615033i	$0.210857\pi$
$104$	16117.6i	1.49016i
$105$	0	0
$106$	18213.8	1.62102
$107$	13833.1i	1.20824i	0.796895	+	0.604118i	$0.206474\pi$
$107$	13833.1i	1.20824i	−0.796895	+	0.604118i	$0.793526\pi$
$108$	0	0
$109$	−11325.5	−0.953248	−0.476624	−	0.879107i	$-0.658140\pi$
$109$	−11325.5	−0.953248	−0.476624	+	0.879107i	$0.658140\pi$
$110$	0	0
$111$	0	0
$112$	239.353	0.0190811
$113$	2158.75i	0.169061i	0.996421	+	0.0845307i	$0.0269391\pi$
$113$	2158.75i	0.169061i	−0.996421	+	0.0845307i	$0.973061\pi$
$114$	0	0
$115$	0	0
$116$	12393.4i	0.921034i
$117$	0	0
$118$	−10401.1	−0.746993
$119$	34273.3i	2.42026i
$120$	0	0
$121$	14509.5	0.991016
$122$	− 31559.3i	− 2.12035i
$123$	0	0
$124$	−17337.4	−1.12756
$125$	0	0
$126$	0	0
$127$	20814.0	1.29047	0.645234	−	0.763985i	$-0.276760\pi$
$127$	20814.0	1.29047	0.645234	+	0.763985i	$0.276760\pi$
$128$	26656.0i	1.62695i
$129$	0	0
$130$	0	0
$131$	17049.5i	0.993503i	0.867893	+	0.496752i	$0.165474\pi$
$131$	17049.5i	0.993503i	−0.867893	+	0.496752i	$0.834526\pi$
$132$	0	0
$133$	−33011.8	−1.86623
$134$	− 8251.82i	− 0.459558i
$135$	0	0
$136$	28244.9	1.52708
$137$	33398.1i	1.77943i	0.456519	+	0.889714i	$0.349096\pi$
$137$	33398.1i	1.77943i	−0.456519	+	0.889714i	$0.650904\pi$
$138$	0	0
$139$	16681.2	0.863372	0.431686	−	0.902024i	$-0.357919\pi$
$139$	16681.2	0.863372	0.431686	+	0.902024i	$0.357919\pi$
$140$	0	0
$141$	0	0
$142$	−30139.1	−1.49470
$143$	− 2860.69i	− 0.139894i
$144$	0	0
$145$	0	0
$146$	− 24358.7i	− 1.14274i
$147$	0	0
$148$	35991.6	1.64315
$149$	− 3647.99i	− 0.164317i	−0.996619	−	0.0821583i	$-0.973819\pi$
$149$	− 3647.99i	− 0.164317i	0.996619	−	0.0821583i	$-0.0261813\pi$
$150$	0	0
$151$	−14487.7	−0.635399	−0.317699	−	0.948192i	$-0.602910\pi$
$151$	−14487.7	−0.635399	−0.317699	+	0.948192i	$0.602910\pi$
$152$	27205.2i	1.17751i
$153$	0	0
$154$	−5825.90	−0.245653
$155$	0	0
$156$	0	0
$157$	−6810.47	−0.276298	−0.138149	−	0.990411i	$-0.544115\pi$
$157$	−6810.47	−0.276298	−0.138149	+	0.990411i	$0.544115\pi$
$158$	36962.6i	1.48063i
$159$	0	0
$160$	0	0
$161$	50880.3i	1.96290i
$162$	0	0
$163$	−1581.87	−0.0595380	−0.0297690	−	0.999557i	$-0.509477\pi$
$163$	−1581.87	−0.0595380	−0.0297690	+	0.999557i	$0.509477\pi$
$164$	− 6354.01i	− 0.236244i
$165$	0	0
$166$	−3026.92	−0.109846
$167$	− 15536.8i	− 0.557096i	−0.960422	−	0.278548i	$-0.910147\pi$
$167$	− 15536.8i	− 0.557096i	0.960422	−	0.278548i	$-0.0898531\pi$
$168$	0	0
$169$	33656.4	1.17840
$170$	0	0
$171$	0	0
$172$	15329.3	0.518161
$173$	− 7313.91i	− 0.244375i	−0.992507	−	0.122188i	$-0.961009\pi$
$173$	− 7313.91i	− 0.244375i	0.992507	−	0.122188i	$-0.0389910\pi$
$174$	0	0
$175$	0	0
$176$	35.0102i	0.00113024i
$177$	0	0
$178$	64149.6	2.02467
$179$	− 10021.0i	− 0.312756i	−0.987697	−	0.156378i	$-0.950018\pi$
$179$	− 10021.0i	− 0.312756i	0.987697	−	0.156378i	$-0.0499817\pi$
$180$	0	0
$181$	21167.6	0.646123	0.323062	−	0.946378i	$-0.395288\pi$
$181$	21167.6	0.646123	0.323062	+	0.946378i	$0.395288\pi$
$182$	− 126708.i	− 3.82526i
$183$	0	0
$184$	41930.8	1.23851
$185$	0	0
$186$	0	0
$187$	−5013.16	−0.143360
$188$	− 53704.9i	− 1.51949i
$189$	0	0
$190$	0	0
$191$	− 66359.5i	− 1.81902i	−0.415686	−	0.909508i	$-0.636459\pi$
$191$	− 66359.5i	− 1.81902i	0.415686	−	0.909508i	$-0.363541\pi$
$192$	0	0
$193$	21656.5	0.581399	0.290699	−	0.956814i	$-0.406112\pi$
$193$	21656.5	0.581399	0.290699	+	0.956814i	$0.406112\pi$
$194$	− 61245.8i	− 1.62732i
$195$	0	0
$196$	−97317.8	−2.53326
$197$	37698.4i	0.971383i	0.874130	+	0.485691i	$0.161432\pi$
$197$	37698.4i	0.971383i	−0.874130	+	0.485691i	$0.838568\pi$
$198$	0	0
$199$	−7444.18	−0.187980	−0.0939898	−	0.995573i	$-0.529962\pi$
$199$	−7444.18	−0.187980	−0.0939898	+	0.995573i	$0.529962\pi$
$200$	0	0
$201$	0	0
$202$	−8365.05	−0.205006
$203$	− 37412.6i	− 0.907874i
$204$	0	0
$205$	0	0
$206$	108392.i	2.55423i
$207$	0	0
$208$	−761.440	−0.0175998
$209$	− 4828.63i	− 0.110543i
$210$	0	0
$211$	−2928.31	−0.0657738	−0.0328869	−	0.999459i	$-0.510470\pi$
$211$	−2928.31	−0.0657738	−0.0328869	+	0.999459i	$0.510470\pi$
$212$	73020.6i	1.62470i
$213$	0	0
$214$	−89620.6	−1.95695
$215$	0	0
$216$	0	0
$217$	52337.1	1.11145
$218$	− 73374.9i	− 1.54395i
$219$	0	0
$220$	0	0
$221$	− 109032.i	− 2.23238i
$222$	0	0
$223$	−3379.41	−0.0679566	−0.0339783	−	0.999423i	$-0.510818\pi$
$223$	−3379.41	−0.0679566	−0.0339783	+	0.999423i	$0.510818\pi$
$224$	− 79512.3i	− 1.58467i
$225$	0	0
$226$	−13985.9	−0.273825
$227$	19081.5i	0.370306i	0.982710	+	0.185153i	$0.0592780\pi$
$227$	19081.5i	0.370306i	−0.982710	+	0.185153i	$0.940722\pi$
$228$	0	0
$229$	89379.7	1.70439	0.852193	−	0.523227i	$-0.175272\pi$
$229$	89379.7	1.70439	0.852193	+	0.523227i	$0.175272\pi$
$230$	0	0
$231$	0	0
$232$	−30832.0	−0.572830
$233$	27743.2i	0.511028i	0.966805	+	0.255514i	$0.0822447\pi$
$233$	27743.2i	0.511028i	−0.966805	+	0.255514i	$0.917755\pi$
$234$	0	0
$235$	0	0
$236$	− 41699.0i	− 0.748689i
$237$	0	0
$238$	−222047.	−3.92004
$239$	102355.i	1.79189i	0.444161	+	0.895947i	$0.353502\pi$
$239$	102355.i	1.79189i	−0.444161	+	0.895947i	$0.646498\pi$
$240$	0	0
$241$	65463.3	1.12710	0.563551	−	0.826081i	$-0.309435\pi$
$241$	65463.3	1.12710	0.563551	+	0.826081i	$0.309435\pi$
$242$	94002.6i	1.60513i
$243$	0	0
$244$	126524.	2.12517
$245$	0	0
$246$	0	0
$247$	105018.	1.72136
$248$	− 43131.4i	− 0.701278i
$249$	0	0
$250$	0	0
$251$	37799.5i	0.599983i	0.953942	+	0.299991i	$0.0969838\pi$
$251$	37799.5i	0.599983i	−0.953942	+	0.299991i	$0.903016\pi$
$252$	0	0
$253$	−7442.27	−0.116269
$254$	134848.i	2.09014i
$255$	0	0
$256$	−66795.3	−1.01922
$257$	− 15693.2i	− 0.237599i	−0.992918	−	0.118800i	$-0.962095\pi$
$257$	− 15693.2i	− 0.237599i	0.992918	−	0.118800i	$-0.0379046\pi$
$258$	0	0
$259$	−108649.	−1.61968
$260$	0	0
$261$	0	0
$262$	−110459.	−1.60915
$263$	− 30574.3i	− 0.442024i	−0.975271	−	0.221012i	$-0.929064\pi$
$263$	− 30574.3i	− 0.442024i	0.975271	−	0.221012i	$-0.0709359\pi$
$264$	0	0
$265$	0	0
$266$	− 213874.i	− 3.02269i
$267$	0	0
$268$	33082.2	0.460601
$269$	122138.i	1.68790i	0.536420	+	0.843951i	$0.319776\pi$
$269$	122138.i	1.68790i	−0.536420	+	0.843951i	$0.680224\pi$
$270$	0	0
$271$	54195.6	0.737948	0.368974	−	0.929440i	$-0.379709\pi$
$271$	54195.6	0.737948	0.368974	+	0.929440i	$0.379709\pi$
$272$	1334.37i	0.0180359i
$273$	0	0
$274$	−216376.	−2.88210
$275$	0	0
$276$	0	0
$277$	8143.33	0.106131	0.0530655	−	0.998591i	$-0.483101\pi$
$277$	8143.33	0.106131	0.0530655	+	0.998591i	$0.483101\pi$
$278$	108073.i	1.39838i
$279$	0	0
$280$	0	0
$281$	8128.63i	0.102945i	0.998674	+	0.0514724i	$0.0163914\pi$
$281$	8128.63i	0.102945i	−0.998674	+	0.0514724i	$0.983609\pi$
$282$	0	0
$283$	−145176.	−1.81269	−0.906343	−	0.422542i	$-0.861138\pi$
$283$	−145176.	−1.81269	−0.906343	+	0.422542i	$0.861138\pi$
$284$	− 120830.i	− 1.49809i
$285$	0	0
$286$	18533.6	0.226583
$287$	19181.1i	0.232868i
$288$	0	0
$289$	−107549.	−1.28769
$290$	0	0
$291$	0	0
$292$	97656.1	1.14534
$293$	34558.8i	0.402554i	0.979534	+	0.201277i	$0.0645091\pi$
$293$	34558.8i	0.402554i	−0.979534	+	0.201277i	$0.935491\pi$
$294$	0	0
$295$	0	0
$296$	89538.8i	1.02195i
$297$	0	0
$298$	23634.3	0.266140
$299$	− 161862.i	− 1.81052i
$300$	0	0
$301$	−46275.1	−0.510757
$302$	− 93861.8i	− 1.02914i
$303$	0	0
$304$	−1285.25	−0.0139073
$305$	0	0
$306$	0	0
$307$	30490.3	0.323508	0.161754	−	0.986831i	$-0.448285\pi$
$307$	30490.3	0.323508	0.161754	+	0.986831i	$0.448285\pi$
$308$	− 23356.5i	− 0.246210i
$309$	0	0
$310$	0	0
$311$	76145.3i	0.787268i	0.919267	+	0.393634i	$0.128782\pi$
$311$	76145.3i	0.787268i	−0.919267	+	0.393634i	$0.871218\pi$
$312$	0	0
$313$	94538.2	0.964981	0.482490	−	0.875901i	$-0.339732\pi$
$313$	94538.2	0.964981	0.482490	+	0.875901i	$0.339732\pi$
$314$	− 44123.0i	− 0.447513i
$315$	0	0
$316$	−148186.	−1.48400
$317$	− 178303.i	− 1.77435i	−0.461429	−	0.887177i	$-0.652663\pi$
$317$	− 178303.i	− 1.77435i	0.461429	−	0.887177i	$-0.347337\pi$
$318$	0	0
$319$	5472.34	0.0537764
$320$	0	0
$321$	0	0
$322$	−329639.	−3.17926
$323$	− 184037.i	− 1.76401i
$324$	0	0
$325$	0	0
$326$	− 10248.5i	− 0.0964324i
$327$	0	0
$328$	15807.3	0.146930
$329$	162121.i	1.49778i
$330$	0	0
$331$	178693.	1.63099	0.815495	−	0.578764i	$-0.196465\pi$
$331$	178693.	1.63099	0.815495	+	0.578764i	$0.196465\pi$
$332$	− 12135.2i	− 0.110095i
$333$	0	0
$334$	100659.	0.902316
$335$	0	0
$336$	0	0
$337$	86734.0	0.763712	0.381856	−	0.924222i	$-0.375285\pi$
$337$	86734.0	0.763712	0.381856	+	0.924222i	$0.375285\pi$
$338$	218050.i	1.90863i
$339$	0	0
$340$	0	0
$341$	7655.36i	0.0658350i
$342$	0	0
$343$	105520.	0.896908
$344$	38135.7i	0.322266i
$345$	0	0
$346$	47384.7	0.395809
$347$	125769.i	1.04451i	0.852789	+	0.522256i	$0.174909\pi$
$347$	125769.i	1.04451i	−0.852789	+	0.522256i	$0.825091\pi$
$348$	0	0
$349$	49961.3	0.410188	0.205094	−	0.978742i	$-0.434250\pi$
$349$	49961.3	0.410188	0.205094	+	0.978742i	$0.434250\pi$
$350$	0	0
$351$	0	0
$352$	11630.3	0.0938652
$353$	124585.i	0.999806i	0.866081	+	0.499903i	$0.166631\pi$
$353$	124585.i	0.999806i	−0.866081	+	0.499903i	$0.833369\pi$
$354$	0	0
$355$	0	0
$356$	257181.i	2.02927i
$357$	0	0
$358$	64923.2	0.506563
$359$	− 116328.i	− 0.902597i	−0.892373	−	0.451298i	$-0.850961\pi$
$359$	− 116328.i	− 0.902597i	0.892373	−	0.451298i	$-0.149039\pi$
$360$	0	0
$361$	46942.2	0.360204
$362$	137139.i	1.04651i
$363$	0	0
$364$	507982.	3.83394
$365$	0	0
$366$	0	0
$367$	62914.7	0.467111	0.233555	−	0.972344i	$-0.424964\pi$
$367$	62914.7	0.467111	0.233555	+	0.972344i	$0.424964\pi$
$368$	1980.93i	0.0146276i
$369$	0	0
$370$	0	0
$371$	− 220431.i	− 1.60149i
$372$	0	0
$373$	241787.	1.73786	0.868931	−	0.494933i	$-0.164807\pi$
$373$	241787.	1.73786	0.868931	+	0.494933i	$0.164807\pi$
$374$	− 32478.8i	− 0.232197i
$375$	0	0
$376$	133605.	0.945036
$377$	119018.i	0.837397i
$378$	0	0
$379$	226464.	1.57660	0.788298	−	0.615294i	$-0.210963\pi$
$379$	226464.	1.57660	0.788298	+	0.615294i	$0.210963\pi$
$380$	0	0
$381$	0	0
$382$	429924.	2.94622
$383$	− 3347.55i	− 0.0228207i	−0.999935	−	0.0114104i	$-0.996368\pi$
$383$	− 3347.55i	− 0.0228207i	0.999935	−	0.0114104i	$-0.00363211\pi$
$384$	0	0
$385$	0	0
$386$	140306.i	0.941678i
$387$	0	0
$388$	245539.	1.63101
$389$	− 60248.5i	− 0.398150i	−0.979984	−	0.199075i	$-0.936206\pi$
$389$	− 60248.5i	− 0.398150i	0.979984	−	0.199075i	$-0.0637938\pi$
$390$	0	0
$391$	−283653.	−1.85538
$392$	− 242104.i	− 1.57554i
$393$	0	0
$394$	−244237.	−1.57333
$395$	0	0
$396$	0	0
$397$	−173966.	−1.10378	−0.551892	−	0.833916i	$-0.686094\pi$
$397$	−173966.	−1.10378	−0.551892	+	0.833916i	$0.686094\pi$
$398$	− 48228.7i	− 0.304466i
$399$	0	0
$400$	0	0
$401$	233704.i	1.45337i	0.686970	+	0.726686i	$0.258940\pi$
$401$	233704.i	1.45337i	−0.686970	+	0.726686i	$0.741060\pi$
$402$	0	0
$403$	−166497.	−1.02517
$404$	− 33536.2i	− 0.205471i
$405$	0	0
$406$	242385.	1.47046
$407$	− 15892.2i	− 0.0959388i
$408$	0	0
$409$	−35297.1	−0.211005	−0.105502	−	0.994419i	$-0.533645\pi$
$409$	−35297.1	−0.211005	−0.105502	+	0.994419i	$0.533645\pi$
$410$	0	0
$411$	0	0
$412$	−434550.	−2.56003
$413$	125879.i	0.737992i
$414$	0	0
$415$	0	0
$416$	252948.i	1.46165i
$417$	0	0
$418$	31283.3	0.179044
$419$	− 49457.6i	− 0.281712i	−0.990030	−	0.140856i	$-0.955015\pi$
$419$	− 49457.6i	− 0.281712i	0.990030	−	0.140856i	$-0.0449854\pi$
$420$	0	0
$421$	−274153.	−1.54678	−0.773391	−	0.633929i	$-0.781441\pi$
$421$	−274153.	−1.54678	−0.773391	+	0.633929i	$0.781441\pi$
$422$	− 18971.7i	− 0.106532i
$423$	0	0
$424$	−181658.	−1.01047
$425$	0	0
$426$	0	0
$427$	−381943.	−2.09480
$428$	− 359296.i	− 1.96139i
$429$	0	0
$430$	0	0
$431$	− 99897.7i	− 0.537775i	−0.963172	−	0.268888i	$-0.913344\pi$
$431$	− 99897.7i	− 0.537775i	0.963172	−	0.268888i	$-0.0866560\pi$
$432$	0	0
$433$	−201191.	−1.07308	−0.536542	−	0.843874i	$-0.680270\pi$
$433$	−201191.	−1.07308	−0.536542	+	0.843874i	$0.680270\pi$
$434$	339077.i	1.80019i
$435$	0	0
$436$	294166.	1.54746
$437$	− 273212.i	− 1.43066i
$438$	0	0
$439$	−159785.	−0.829101	−0.414551	−	0.910026i	$-0.636061\pi$
$439$	−159785.	−0.829101	−0.414551	+	0.910026i	$0.636061\pi$
$440$	0	0
$441$	0	0
$442$	706384.	3.61573
$443$	200114.i	1.01969i	0.860265	+	0.509846i	$0.170298\pi$
$443$	200114.i	1.01969i	−0.860265	+	0.509846i	$0.829702\pi$
$444$	0	0
$445$	0	0
$446$	− 21894.2i	− 0.110068i
$447$	0	0
$448$	518967.	2.58573
$449$	− 288109.i	− 1.42910i	−0.699582	−	0.714552i	$-0.746631\pi$
$449$	− 288109.i	− 1.42910i	0.699582	−	0.714552i	$-0.253369\pi$
$450$	0	0
$451$	−2805.63	−0.0137936
$452$	− 56070.5i	− 0.274447i
$453$	0	0
$454$	−123623.	−0.599776
$455$	0	0
$456$	0	0
$457$	−64753.4	−0.310049	−0.155024	−	0.987911i	$-0.549546\pi$
$457$	−64753.4	−0.310049	−0.155024	+	0.987911i	$0.549546\pi$
$458$	579065.i	2.76056i
$459$	0	0
$460$	0	0
$461$	175178.i	0.824284i	0.911120	+	0.412142i	$0.135219\pi$
$461$	175178.i	0.824284i	−0.911120	+	0.412142i	$0.864781\pi$
$462$	0	0
$463$	−249339.	−1.16313	−0.581565	−	0.813500i	$-0.697559\pi$
$463$	−249339.	−1.16313	−0.581565	+	0.813500i	$0.697559\pi$
$464$	− 1456.59i	− 0.00676553i
$465$	0	0
$466$	−179740.	−0.827700
$467$	− 272334.i	− 1.24873i	−0.781134	−	0.624363i	$-0.785358\pi$
$467$	− 272334.i	− 1.24873i	0.781134	−	0.624363i	$-0.214642\pi$
$468$	0	0
$469$	−99866.7	−0.454020
$470$	0	0
$471$	0	0
$472$	103737.	0.465641
$473$	− 6768.67i	− 0.0302539i
$474$	0	0
$475$	0	0
$476$	− 890203.i	− 3.92894i
$477$	0	0
$478$	−663127.	−2.90229
$479$	554.420i	0.00241639i	0.999999	+	0.00120820i	$0.000384581\pi$
$479$	554.420i	0.00241639i	−0.999999	+	0.00120820i	$0.999615\pi$
$480$	0	0
$481$	345640.	1.49394
$482$	424117.i	1.82554i
$483$	0	0
$484$	−376864.	−1.60877
$485$	0	0
$486$	0	0
$487$	−433285.	−1.82690	−0.913451	−	0.406948i	$-0.866593\pi$
$487$	−433285.	−1.82690	−0.913451	+	0.406948i	$0.866593\pi$
$488$	314762.i	1.32173i
$489$	0	0
$490$	0	0
$491$	− 314344.i	− 1.30389i	−0.758265	−	0.651947i	$-0.773953\pi$
$491$	− 314344.i	− 1.30389i	0.758265	−	0.651947i	$-0.226047\pi$
$492$	0	0
$493$	208571.	0.858146
$494$	680383.i	2.78804i
$495$	0	0
$496$	2037.65	0.00828260
$497$	364756.i	1.47669i
$498$	0	0
$499$	109748.	0.440754	0.220377	−	0.975415i	$-0.429271\pi$
$499$	109748.	0.440754	0.220377	+	0.975415i	$0.429271\pi$
$500$	0	0
$501$	0	0
$502$	−244892.	−0.971779
$503$	− 266878.i	− 1.05482i	−0.849612	−	0.527408i	$-0.823164\pi$
$503$	− 266878.i	− 1.05482i	0.849612	−	0.527408i	$-0.176836\pi$
$504$	0	0
$505$	0	0
$506$	− 48216.3i	− 0.188318i
$507$	0	0
$508$	−540615.	−2.09489
$509$	176389.i	0.680827i	0.940276	+	0.340413i	$0.110567\pi$
$509$	176389.i	0.680827i	−0.940276	+	0.340413i	$0.889433\pi$
$510$	0	0
$511$	−294799.	−1.12897
$512$	− 6251.71i	− 0.0238484i
$513$	0	0
$514$	101672.	0.384834
$515$	0	0
$516$	0	0
$517$	−23713.5	−0.0887185
$518$	− 703908.i	− 2.62335i
$519$	0	0
$520$	0	0
$521$	508848.i	1.87462i	0.348502	+	0.937308i	$0.386691\pi$
$521$	508848.i	1.87462i	−0.348502	+	0.937308i	$0.613309\pi$
$522$	0	0
$523$	−93132.6	−0.340485	−0.170243	−	0.985402i	$-0.554455\pi$
$523$	−93132.6	−0.340485	−0.170243	+	0.985402i	$0.554455\pi$
$524$	− 442838.i	− 1.61281i
$525$	0	0
$526$	198082.	0.715935
$527$	291774.i	1.05057i
$528$	0	0
$529$	−141254.	−0.504767
$530$	0	0
$531$	0	0
$532$	857436.	3.02955
$533$	− 61019.8i	− 0.214791i
$534$	0	0
$535$	0	0
$536$	82300.9i	0.286467i
$537$	0	0
$538$	−791298.	−2.73386
$539$	42970.9i	0.147910i
$540$	0	0
$541$	1427.95	0.00487886	0.00243943	−	0.999997i	$-0.499224\pi$
$541$	1427.95	0.00487886	0.00243943	+	0.999997i	$0.499224\pi$
$542$	351118.i	1.19524i
$543$	0	0
$544$	443273.	1.49787
$545$	0	0
$546$	0	0
$547$	106401.	0.355608	0.177804	−	0.984066i	$-0.443101\pi$
$547$	106401.	0.355608	0.177804	+	0.984066i	$0.443101\pi$
$548$	− 867471.i	− 2.88864i
$549$	0	0
$550$	0	0
$551$	200894.i	0.661705i
$552$	0	0
$553$	447335.	1.46279
$554$	52758.3i	0.171898i
$555$	0	0
$556$	−433272.	−1.40156
$557$	− 488896.i	− 1.57582i	−0.615792	−	0.787909i	$-0.711164\pi$
$557$	− 488896.i	− 1.57582i	0.615792	−	0.787909i	$-0.288836\pi$
$558$	0	0
$559$	147212.	0.471108
$560$	0	0
$561$	0	0
$562$	−52663.0	−0.166737
$563$	561395.i	1.77114i	0.464510	+	0.885568i	$0.346231\pi$
$563$	561395.i	1.77114i	−0.464510	+	0.885568i	$0.653769\pi$
$564$	0	0
$565$	0	0
$566$	− 940555.i	− 2.93597i
$567$	0	0
$568$	300598.	0.931728
$569$	− 584844.i	− 1.80641i	−0.429213	−	0.903203i	$-0.641209\pi$
$569$	− 584844.i	− 1.80641i	0.429213	−	0.903203i	$-0.358791\pi$
$570$	0	0
$571$	−196411.	−0.602412	−0.301206	−	0.953559i	$-0.597389\pi$
$571$	−196411.	−0.602412	−0.301206	+	0.953559i	$0.597389\pi$
$572$	74302.6i	0.227097i
$573$	0	0
$574$	−124269.	−0.377171
$575$	0	0
$576$	0	0
$577$	96911.5	0.291088	0.145544	−	0.989352i	$-0.453507\pi$
$577$	96911.5	0.291088	0.145544	+	0.989352i	$0.453507\pi$
$578$	− 696780.i	− 2.08564i
$579$	0	0
$580$	0	0
$581$	36632.9i	0.108522i
$582$	0	0
$583$	32242.4	0.0948616
$584$	242946.i	0.712334i
$585$	0	0
$586$	−223897.	−0.652007
$587$	435488.i	1.26386i	0.775024	+	0.631932i	$0.217738\pi$
$587$	435488.i	1.26386i	−0.775024	+	0.631932i	$0.782262\pi$
$588$	0	0
$589$	−281035.	−0.810083
$590$	0	0
$591$	0	0
$592$	−4230.07	−0.0120699
$593$	32748.3i	0.0931278i	0.998915	+	0.0465639i	$0.0148271\pi$
$593$	32748.3i	0.0931278i	−0.998915	+	0.0465639i	$0.985173\pi$
$594$	0	0
$595$	0	0
$596$	94751.7i	0.266744i
$597$	0	0
$598$	1.04866e6	2.93246
$599$	− 5265.63i	− 0.0146756i	−0.999973	−	0.00733781i	$-0.997664\pi$
$599$	− 5265.63i	− 0.0146756i	0.999973	−	0.00733781i	$-0.00233572\pi$
$600$	0	0
$601$	−449825.	−1.24536	−0.622680	−	0.782477i	$-0.713956\pi$
$601$	−449825.	−1.24536	−0.622680	+	0.782477i	$0.713956\pi$
$602$	− 299803.i	− 0.827262i
$603$	0	0
$604$	376299.	1.03148
$605$	0	0
$606$	0	0
$607$	504065.	1.36807	0.684037	−	0.729448i	$-0.260223\pi$
$607$	504065.	1.36807	0.684037	+	0.729448i	$0.260223\pi$
$608$	426957.i	1.15499i
$609$	0	0
$610$	0	0
$611$	− 515747.i	− 1.38151i
$612$	0	0
$613$	37067.1	0.0986433	0.0493216	−	0.998783i	$-0.484294\pi$
$613$	37067.1	0.0986433	0.0493216	+	0.998783i	$0.484294\pi$
$614$	197538.i	0.523978i
$615$	0	0
$616$	58105.6	0.153129
$617$	− 610714.i	− 1.60423i	−0.597168	−	0.802116i	$-0.703707\pi$
$617$	− 610714.i	− 1.60423i	0.597168	−	0.802116i	$-0.296293\pi$
$618$	0	0
$619$	20926.8	0.0546161	0.0273080	−	0.999627i	$-0.491306\pi$
$619$	20926.8	0.0546161	0.0273080	+	0.999627i	$0.491306\pi$
$620$	0	0
$621$	0	0
$622$	−493323.	−1.27512
$623$	− 776364.i	− 2.00027i
$624$	0	0
$625$	0	0
$626$	612485.i	1.56296i
$627$	0	0
$628$	176893.	0.448529
$629$	− 605710.i	− 1.53096i
$630$	0	0
$631$	−526407.	−1.32210	−0.661048	−	0.750343i	$-0.729888\pi$
$631$	−526407.	−1.32210	−0.661048	+	0.750343i	$0.729888\pi$
$632$	− 368652.i	− 0.922960i
$633$	0	0
$634$	1.15517e6	2.87388
$635$	0	0
$636$	0	0
$637$	−934577.	−2.30322
$638$	35453.7i	0.0871004i
$639$	0	0
$640$	0	0
$641$	467100.i	1.13683i	0.822743	+	0.568413i	$0.192442\pi$
$641$	467100.i	1.13683i	−0.822743	+	0.568413i	$0.807558\pi$
$642$	0	0
$643$	77720.4	0.187981	0.0939903	−	0.995573i	$-0.470038\pi$
$643$	77720.4	0.187981	0.0939903	+	0.995573i	$0.470038\pi$
$644$	− 1.32155e6i	− 3.18648i
$645$	0	0
$646$	1.19232e6	2.85712
$647$	− 534746.i	− 1.27744i	−0.769441	−	0.638718i	$-0.779465\pi$
$647$	− 534746.i	− 1.27744i	0.769441	−	0.638718i	$-0.220535\pi$
$648$	0	0
$649$	−18412.3	−0.0437137
$650$	0	0
$651$	0	0
$652$	41086.9	0.0966513
$653$	714703.i	1.67610i	0.545596	+	0.838048i	$0.316303\pi$
$653$	714703.i	1.67610i	−0.545596	+	0.838048i	$0.683697\pi$
$654$	0	0
$655$	0	0
$656$	746.783i	0.00173535i
$657$	0	0
$658$	−1.05034e6	−2.42592
$659$	− 46610.3i	− 0.107327i	−0.998559	−	0.0536637i	$-0.982910\pi$
$659$	− 46610.3i	− 0.107327i	0.998559	−	0.0536637i	$-0.0170899\pi$
$660$	0	0
$661$	585494.	1.34004	0.670022	−	0.742341i	$-0.266285\pi$
$661$	585494.	1.34004	0.670022	+	0.742341i	$0.266285\pi$
$662$	1.15770e6i	2.64168i
$663$	0	0
$664$	30189.5	0.0684730
$665$	0	0
$666$	0	0
$667$	309634.	0.695980
$668$	403549.i	0.904364i
$669$	0	0
$670$	0	0
$671$	− 55866.9i	− 0.124082i
$672$	0	0
$673$	2713.76	0.00599157	0.00299579	−	0.999996i	$-0.499046\pi$
$673$	2713.76	0.00599157	0.00299579	+	0.999996i	$0.499046\pi$
$674$	561924.i	1.23697i
$675$	0	0
$676$	−874180.	−1.91297
$677$	− 187601.i	− 0.409316i	−0.978834	−	0.204658i	$-0.934392\pi$
$677$	− 187601.i	− 0.409316i	0.978834	−	0.204658i	$-0.0656082\pi$
$678$	0	0
$679$	−741220.	−1.60771
$680$	0	0
$681$	0	0
$682$	−49596.8	−0.106631
$683$	− 595822.i	− 1.27725i	−0.769519	−	0.638624i	$-0.779504\pi$
$683$	− 595822.i	− 1.27725i	0.769519	−	0.638624i	$-0.220496\pi$
$684$	0	0
$685$	0	0
$686$	683635.i	1.45270i
$687$	0	0
$688$	−1801.64	−0.00380619
$689$	701243.i	1.47717i
$690$	0	0
$691$	598492.	1.25344	0.626719	−	0.779246i	$-0.284398\pi$
$691$	598492.	1.25344	0.626719	+	0.779246i	$0.284398\pi$
$692$	189969.i	0.396708i
$693$	0	0
$694$	−814818.	−1.69177
$695$	0	0
$696$	0	0
$697$	−106933.	−0.220113
$698$	323685.i	0.664372i
$699$	0	0
$700$	0	0
$701$	− 979747.i	− 1.99378i	−0.0787858	−	0.996892i	$-0.525104\pi$
$701$	− 979747.i	− 1.99378i	0.0787858	−	0.996892i	$-0.474896\pi$
$702$	0	0
$703$	583415.	1.18050
$704$	75909.3i	0.153162i
$705$	0	0
$706$	−807149.	−1.61936
$707$	101237.i	0.202535i
$708$	0	0
$709$	528474.	1.05131	0.525656	−	0.850697i	$-0.323820\pi$
$709$	528474.	1.05131	0.525656	+	0.850697i	$0.323820\pi$
$710$	0	0
$711$	0	0
$712$	−639807.	−1.26209
$713$	433153.i	0.852044i
$714$	0	0
$715$	0	0
$716$	260282.i	0.507713i
$717$	0	0
$718$	753652.	1.46192
$719$	348908.i	0.674921i	0.941340	+	0.337460i	$0.109568\pi$
$719$	348908.i	0.674921i	−0.941340	+	0.337460i	$0.890432\pi$
$720$	0	0
$721$	1.31180e6	2.52346
$722$	304125.i	0.583415i
$723$	0	0
$724$	−549801.	−1.04889
$725$	0	0
$726$	0	0
$727$	710913.	1.34508	0.672539	−	0.740062i	$-0.265204\pi$
$727$	710913.	1.34508	0.672539	+	0.740062i	$0.265204\pi$
$728$	1.26374e6i	2.38449i
$729$	0	0
$730$	0	0
$731$	− 257979.i	− 0.482781i
$732$	0	0
$733$	−990839.	−1.84415	−0.922073	−	0.387016i	$-0.873506\pi$
$733$	−990839.	−1.84415	−0.922073	+	0.387016i	$0.873506\pi$
$734$	407606.i	0.756568i
$735$	0	0
$736$	658060.	1.21481
$737$	− 14607.5i	− 0.0268931i
$738$	0	0
$739$	467455.	0.855955	0.427978	−	0.903789i	$-0.359226\pi$
$739$	467455.	0.855955	0.427978	+	0.903789i	$0.359226\pi$
$740$	0	0
$741$	0	0
$742$	1.42811e6	2.59390
$743$	− 529656.i	− 0.959437i	−0.877422	−	0.479719i	$-0.840739\pi$
$743$	− 529656.i	− 0.959437i	0.877422	−	0.479719i	$-0.159261\pi$
$744$	0	0
$745$	0	0
$746$	1.56647e6i	2.81478i
$747$	0	0
$748$	130210.	0.232724
$749$	1.08462e6i	1.93337i
$750$	0	0
$751$	57653.7	0.102223	0.0511113	−	0.998693i	$-0.483724\pi$
$751$	57653.7	0.102223	0.0511113	+	0.998693i	$0.483724\pi$
$752$	6311.90i	0.0111615i
$753$	0	0
$754$	−771086.	−1.35631
$755$	0	0
$756$	0	0
$757$	148763.	0.259599	0.129799	−	0.991540i	$-0.458567\pi$
$757$	148763.	0.259599	0.129799	+	0.991540i	$0.458567\pi$
$758$	1.46719e6i	2.55358i
$759$	0	0
$760$	0	0
$761$	− 359378.i	− 0.620558i	−0.950646	−	0.310279i	$-0.899578\pi$
$761$	− 359378.i	− 0.620558i	0.950646	−	0.310279i	$-0.100422\pi$
$762$	0	0
$763$	−888011.	−1.52535
$764$	1.72360e6i	2.95291i
$765$	0	0
$766$	21687.8	0.0369622
$767$	− 400450.i	− 0.680703i
$768$	0	0
$769$	−15933.2	−0.0269432	−0.0134716	−	0.999909i	$-0.504288\pi$
$769$	−15933.2	−0.0269432	−0.0134716	+	0.999909i	$0.504288\pi$
$770$	0	0
$771$	0	0
$772$	−562499.	−0.943816
$773$	606895.i	1.01567i	0.861453	+	0.507837i	$0.169555\pi$
$773$	606895.i	1.01567i	−0.861453	+	0.507837i	$0.830445\pi$
$774$	0	0
$775$	0	0
$776$	610845.i	1.01440i
$777$	0	0
$778$	390333.	0.644875
$779$	− 102997.i	− 0.169726i
$780$	0	0
$781$	−53352.8	−0.0874692
$782$	− 1.83770e6i	− 3.00512i
$783$	0	0
$784$	11437.7	0.0186083
$785$	0	0
$786$	0	0
$787$	−699027.	−1.12861	−0.564306	−	0.825566i	$-0.690856\pi$
$787$	−699027.	−1.12861	−0.564306	+	0.825566i	$0.690856\pi$
$788$	− 979165.i	− 1.57690i
$789$	0	0
$790$	0	0
$791$	169263.i	0.270525i
$792$	0	0
$793$	1.21505e6	1.93219
$794$	− 1.12708e6i	− 1.78777i
$795$	0	0
$796$	193353.	0.305157
$797$	702353.i	1.10570i	0.833280	+	0.552852i	$0.186460\pi$
$797$	702353.i	1.10570i	−0.833280	+	0.552852i	$0.813540\pi$
$798$	0	0
$799$	−903810.	−1.41574
$800$	0	0
$801$	0	0
$802$	−1.51410e6	−2.35399
$803$	− 43120.2i	− 0.0668729i
$804$	0	0
$805$	0	0
$806$	− 1.07869e6i	− 1.66045i
$807$	0	0
$808$	83430.3	0.127791
$809$	− 403133.i	− 0.615958i	−0.951393	−	0.307979i	$-0.900347\pi$
$809$	− 403133.i	− 0.615958i	0.951393	−	0.307979i	$-0.0996527\pi$
$810$	0	0
$811$	−406803.	−0.618504	−0.309252	−	0.950980i	$-0.600079\pi$
$811$	−406803.	−0.618504	−0.309252	+	0.950980i	$0.600079\pi$
$812$	971742.i	1.47380i
$813$	0	0
$814$	102961.	0.155390
$815$	0	0
$816$	0	0
$817$	248483.	0.372266
$818$	− 228679.i	− 0.341759i
$819$	0	0
$820$	0	0
$821$	788579.i	1.16993i	0.811060	+	0.584963i	$0.198891\pi$
$821$	788579.i	1.16993i	−0.811060	+	0.584963i	$0.801109\pi$
$822$	0	0
$823$	−1.06144e6	−1.56709	−0.783547	−	0.621333i	$-0.786592\pi$
$823$	−1.06144e6	−1.56709	−0.783547	+	0.621333i	$0.786592\pi$
$824$	− 1.08106e6i	− 1.59219i
$825$	0	0
$826$	−815530.	−1.19531
$827$	399147.i	0.583608i	0.956478	+	0.291804i	$0.0942555\pi$
$827$	399147.i	0.583608i	−0.956478	+	0.291804i	$0.905744\pi$
$828$	0	0
$829$	−50667.4	−0.0737259	−0.0368629	−	0.999320i	$-0.511736\pi$
$829$	−50667.4	−0.0737259	−0.0368629	+	0.999320i	$0.511736\pi$
$830$	0	0
$831$	0	0
$832$	−1.65096e6	−2.38501
$833$	1.63778e6i	2.36029i
$834$	0	0
$835$	0	0
$836$	125417.i	0.179451i
$837$	0	0
$838$	320421.	0.456282
$839$	− 996476.i	− 1.41561i	−0.706409	−	0.707804i	$-0.749686\pi$
$839$	− 996476.i	− 1.41561i	0.706409	−	0.707804i	$-0.250314\pi$
$840$	0	0
$841$	479605.	0.678097
$842$	− 1.77616e6i	− 2.50529i
$843$	0	0
$844$	76059.1	0.106774
$845$	0	0
$846$	0	0
$847$	1.13766e6	1.58578
$848$	− 8582.07i	− 0.0119344i
$849$	0	0
$850$	0	0
$851$	− 899204.i	− 1.24165i
$852$	0	0
$853$	893233.	1.22763	0.613814	−	0.789451i	$-0.289635\pi$
$853$	893233.	1.22763	0.613814	+	0.789451i	$0.289635\pi$
$854$	− 2.47450e6i	− 3.39290i
$855$	0	0
$856$	893846.	1.21987
$857$	− 489824.i	− 0.666927i	−0.942763	−	0.333464i	$-0.891783\pi$
$857$	− 489824.i	− 0.666927i	0.942763	−	0.333464i	$-0.108217\pi$
$858$	0	0
$859$	−237919.	−0.322435	−0.161218	−	0.986919i	$-0.551542\pi$
$859$	−237919.	−0.322435	−0.161218	+	0.986919i	$0.551542\pi$
$860$	0	0
$861$	0	0
$862$	647208.	0.871022
$863$	− 1.15717e6i	− 1.55373i	−0.629665	−	0.776867i	$-0.716808\pi$
$863$	− 1.15717e6i	− 1.55373i	0.629665	−	0.776867i	$-0.283192\pi$
$864$	0	0
$865$	0	0
$866$	− 1.30346e6i	− 1.73805i
$867$	0	0
$868$	−1.35939e6	−1.80428
$869$	65431.7i	0.0866461i
$870$	0	0
$871$	317700.	0.418775
$872$	731817.i	0.962430i
$873$	0	0
$874$	1.77006e6	2.31721
$875$	0	0
$876$	0	0
$877$	1.34800e6	1.75264	0.876318	−	0.481734i	$-0.159993\pi$
$877$	1.34800e6	1.75264	0.876318	+	0.481734i	$0.159993\pi$
$878$	− 1.03520e6i	− 1.34288i
$879$	0	0
$880$	0	0
$881$	668434.i	0.861206i	0.902541	+	0.430603i	$0.141699\pi$
$881$	668434.i	0.861206i	−0.902541	+	0.430603i	$0.858301\pi$
$882$	0	0
$883$	192331.	0.246677	0.123338	−	0.992365i	$-0.460640\pi$
$883$	192331.	0.246677	0.123338	+	0.992365i	$0.460640\pi$
$884$	2.83195e6i	3.62394i
$885$	0	0
$886$	−1.29648e6	−1.65157
$887$	425786.i	0.541183i	0.962694	+	0.270591i	$0.0872192\pi$
$887$	425786.i	0.541183i	−0.962694	+	0.270591i	$0.912781\pi$
$888$	0	0
$889$	1.63198e6	2.06496
$890$	0	0
$891$	0	0
$892$	87775.7	0.110318
$893$	− 870542.i	− 1.09166i
$894$	0	0
$895$	0	0
$896$	2.09004e6i	2.60338i
$897$	0	0
$898$	1.86657e6	2.31469
$899$	− 318500.i	− 0.394085i
$900$	0	0
$901$	1.22888e6	1.51377
$902$	− 18176.8i	− 0.0223411i
$903$	0	0
$904$	139491.	0.170690
$905$	0	0
$906$	0	0
$907$	246863.	0.300083	0.150042	−	0.988680i	$-0.452059\pi$
$907$	246863.	0.300083	0.150042	+	0.988680i	$0.452059\pi$
$908$	− 495616.i	− 0.601137i
$909$	0	0
$910$	0	0
$911$	− 691646.i	− 0.833387i	−0.909047	−	0.416694i	$-0.863189\pi$
$911$	− 691646.i	− 0.833387i	0.909047	−	0.416694i	$-0.136811\pi$
$912$	0	0
$913$	−5358.30	−0.00642814
$914$	− 419518.i	− 0.502179i
$915$	0	0
$916$	−2.32152e6	−2.76682
$917$	1.33682e6i	1.58976i
$918$	0	0
$919$	−491310.	−0.581735	−0.290867	−	0.956763i	$-0.593944\pi$
$919$	−491310.	−0.581735	−0.290867	+	0.956763i	$0.593944\pi$
$920$	0	0
$921$	0	0
$922$	−1.13492e6	−1.33507
$923$	− 1.16038e6i	− 1.36206i
$924$	0	0
$925$	0	0
$926$	− 1.61539e6i	− 1.88389i
$927$	0	0
$928$	−483875.	−0.561872
$929$	676020.i	0.783300i	0.920114	+	0.391650i	$0.128096\pi$
$929$	676020.i	0.783300i	−0.920114	+	0.391650i	$0.871904\pi$
$930$	0	0
$931$	−1.57750e6	−1.81999
$932$	− 720592.i	− 0.829579i
$933$	0	0
$934$	1.76437e6	2.02253
$935$	0	0
$936$	0	0
$937$	−678920.	−0.773285	−0.386643	−	0.922230i	$-0.626365\pi$
$937$	−678920.	−0.773285	−0.386643	+	0.922230i	$0.626365\pi$
$938$	− 647007.i	− 0.735366i
$939$	0	0
$940$	0	0
$941$	886709.i	1.00139i	0.865625	+	0.500693i	$0.166922\pi$
$941$	886709.i	1.00139i	−0.865625	+	0.500693i	$0.833078\pi$
$942$	0	0
$943$	−158747.	−0.178518
$944$	4900.86i	0.00549956i
$945$	0	0
$946$	43852.2	0.0490015
$947$	− 134677.i	− 0.150174i	−0.997177	−	0.0750870i	$-0.976077\pi$
$947$	− 134677.i	− 0.150174i	0.997177	−	0.0750870i	$-0.0239235\pi$
$948$	0	0
$949$	937826.	1.04133
$950$	0	0
$951$	0	0
$952$	2.21462e6	2.44357
$953$	− 50865.8i	− 0.0560067i	−0.999608	−	0.0280034i	$-0.991085\pi$
$953$	− 50865.8i	− 0.0560067i	0.999608	−	0.0280034i	$-0.00891491\pi$
$954$	0	0
$955$	0	0
$956$	− 2.65853e6i	− 2.90888i
$957$	0	0
$958$	−3591.92	−0.00391378
$959$	2.61867e6i	2.84737i
$960$	0	0
$961$	−477966.	−0.517548
$962$	2.23930e6i	2.41970i
$963$	0	0
$964$	−1.70032e6	−1.82969
$965$	0	0
$966$	0	0
$967$	516167.	0.551997	0.275999	−	0.961158i	$-0.410991\pi$
$967$	516167.	0.551997	0.275999	+	0.961158i	$0.410991\pi$
$968$	− 937551.i	− 1.00056i
$969$	0	0
$970$	0	0
$971$	1.16276e6i	1.23325i	0.787256	+	0.616627i	$0.211501\pi$
$971$	1.16276e6i	1.23325i	−0.787256	+	0.616627i	$0.788499\pi$
$972$	0	0
$973$	1.30794e6	1.38153
$974$	− 2.80712e6i	− 2.95899i
$975$	0	0
$976$	−14870.3	−0.0156106
$977$	732143.i	0.767020i	0.923537	+	0.383510i	$0.125285\pi$
$977$	732143.i	0.767020i	−0.923537	+	0.383510i	$0.874715\pi$
$978$	0	0
$979$	113559.	0.118483
$980$	0	0
$981$	0	0
$982$	2.03654e6	2.11189
$983$	− 1.38729e6i	− 1.43568i	−0.696206	−	0.717842i	$-0.745130\pi$
$983$	− 1.38729e6i	− 1.43568i	0.696206	−	0.717842i	$-0.254870\pi$
$984$	0	0
$985$	0	0
$986$	1.35127e6i	1.38992i
$987$	0	0
$988$	−2.72771e6	−2.79437
$989$	− 382982.i	− 0.391549i
$990$	0	0
$991$	−960576.	−0.978103	−0.489052	−	0.872255i	$-0.662657\pi$
$991$	−960576.	−0.978103	−0.489052	+	0.872255i	$0.662657\pi$
$992$	− 676902.i	− 0.687863i
$993$	0	0
$994$	−2.36315e6	−2.39176
$995$	0	0
$996$	0	0
$997$	−457517.	−0.460274	−0.230137	−	0.973158i	$-0.573917\pi$
$997$	−457517.	−0.460274	−0.230137	+	0.973158i	$0.573917\pi$
$998$	711026.i	0.713879i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	225.5.c.b.26.4		4
3.2	odd	2	inner	225.5.c.b.26.1		4
5.2	odd	4		225.5.d.b.224.2		8
5.3	odd	4		225.5.d.b.224.7		8
5.4	even	2		45.5.c.a.26.1	✓	4
15.2	even	4		225.5.d.b.224.8		8
15.8	even	4		225.5.d.b.224.1		8
15.14	odd	2		45.5.c.a.26.4	yes	4
20.19	odd	2		720.5.l.c.161.2		4
60.59	even	2		720.5.l.c.161.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
45.5.c.a.26.1	✓	4	5.4	even	2
45.5.c.a.26.4	yes	4	15.14	odd	2
225.5.c.b.26.1		4	3.2	odd	2	inner
225.5.c.b.26.4		4	1.1	even	1	trivial
225.5.d.b.224.1		8	15.8	even	4
225.5.d.b.224.2		8	5.2	odd	4
225.5.d.b.224.7		8	5.3	odd	4
225.5.d.b.224.8		8	15.2	even	4
720.5.l.c.161.2		4	20.19	odd	2
720.5.l.c.161.4		4	60.59	even	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:	\( N \)	\(=\)	\( 225 = 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	225.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+6.47871i q^{2} -25.9737 q^{4} +78.4078 q^{7} -64.6165i q^{8} +O(q^{10})\)	\(q+6.47871i q^{2} -25.9737 q^{4} +78.4078 q^{7} -64.6165i q^{8} +11.4687i q^{11} -249.434 q^{13} +507.981i q^{14} +3.05267 q^{16} +437.116i q^{17} -421.026 q^{19} -74.3025 q^{22} +648.919i q^{23} -1616.01i q^{26} -2036.54 q^{28} -477.154i q^{29} +667.499 q^{31} -1014.09i q^{32} -2831.95 q^{34} -1385.70 q^{37} -2727.71i q^{38} +244.633i q^{41} -590.185 q^{43} -297.885i q^{44} -4204.16 q^{46} +2067.67i q^{47} +3746.79 q^{49} +6478.72 q^{52} -2811.33i q^{53} -5066.44i q^{56} +3091.34 q^{58} +1605.43i q^{59} -4871.24 q^{61} +4324.53i q^{62} +6618.81 q^{64} -1273.68 q^{67} -11353.5i q^{68} +4652.03i q^{71} -3759.81 q^{73} -8977.52i q^{74} +10935.6 q^{76} +899.237i q^{77} +5705.24 q^{79} -1584.90 q^{82} +467.210i q^{83} -3823.64i q^{86} +741.068 q^{88} -9901.61i q^{89} -19557.6 q^{91} -16854.8i q^{92} -13395.8 q^{94} -9453.39 q^{97} +24274.3i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 28 q^{4} + 48 q^{7}+O(q^{10}) \) 4 * q - 28 * q^4 + 48 * q^7	\( 4 q - 28 q^{4} + 48 q^{7} - 808 q^{13} + 164 q^{16} - 1760 q^{19} + 272 q^{22} - 5376 q^{28} - 1656 q^{31} - 5408 q^{34} - 344 q^{37} - 5776 q^{43} - 5736 q^{46} + 8612 q^{49} + 9256 q^{52} + 1816 q^{58} - 20168 q^{61} + 11524 q^{64} + 1584 q^{67} - 88 q^{73} + 10880 q^{76} + 17736 q^{79} + 6904 q^{82} + 13248 q^{88} - 22296 q^{91} - 32864 q^{94} - 27416 q^{97}+O(q^{100}) \) 4 * q - 28 * q^4 + 48 * q^7 - 808 * q^13 + 164 * q^16 - 1760 * q^19 + 272 * q^22 - 5376 * q^28 - 1656 * q^31 - 5408 * q^34 - 344 * q^37 - 5776 * q^43 - 5736 * q^46 + 8612 * q^49 + 9256 * q^52 + 1816 * q^58 - 20168 * q^61 + 11524 * q^64 + 1584 * q^67 - 88 * q^73 + 10880 * q^76 + 17736 * q^79 + 6904 * q^82 + 13248 * q^88 - 22296 * q^91 - 32864 * q^94 - 27416 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

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Database

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