LMFDB - Embedded newform 450.4.a.u.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [450,4,Mod(1,450)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(450, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("450.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 450 = 2 \cdot 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	450.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:	$26.5508595026$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{31}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 31 $ x^2 - 31
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 90)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-5.56776$ of defining polynomial
Character	$\chi$	$=$	450.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.00000 q^{2} +4.00000 q^{4} -22.2711 q^{7} -8.00000 q^{8} +O(q^{10})$	$q-2.00000 q^{2} +4.00000 q^{4} -22.2711 q^{7} -8.00000 q^{8} +22.2711 q^{11} +66.8132 q^{13} +44.5421 q^{14} +16.0000 q^{16} -62.0000 q^{17} +84.0000 q^{19} -44.5421 q^{22} -140.000 q^{23} -133.626 q^{26} -89.0842 q^{28} -200.440 q^{29} +16.0000 q^{31} -32.0000 q^{32} +124.000 q^{34} +244.982 q^{37} -168.000 q^{38} +222.711 q^{41} -356.337 q^{43} +89.0842 q^{44} +280.000 q^{46} -100.000 q^{47} +153.000 q^{49} +267.253 q^{52} -738.000 q^{53} +178.168 q^{56} +400.879 q^{58} +645.861 q^{59} -358.000 q^{61} -32.0000 q^{62} +64.0000 q^{64} -846.300 q^{67} -248.000 q^{68} -935.384 q^{71} +445.421 q^{73} -489.963 q^{74} +336.000 q^{76} -496.000 q^{77} -936.000 q^{79} -445.421 q^{82} -1304.00 q^{83} +712.674 q^{86} -178.168 q^{88} -712.674 q^{89} -1488.00 q^{91} -560.000 q^{92} +200.000 q^{94} +757.216 q^{97} -306.000 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 4 q^{2} + 8 q^{4} - 16 q^{8}+O(q^{10}) $ 2 * q - 4 * q^2 + 8 * q^4 - 16 * q^8	$ 2 q - 4 q^{2} + 8 q^{4} - 16 q^{8} + 32 q^{16} - 124 q^{17} + 168 q^{19} - 280 q^{23} + 32 q^{31} - 64 q^{32} + 248 q^{34} - 336 q^{38} + 560 q^{46} - 200 q^{47} + 306 q^{49} - 1476 q^{53} - 716 q^{61} - 64 q^{62} + 128 q^{64} - 496 q^{68} + 672 q^{76} - 992 q^{77} - 1872 q^{79} - 2608 q^{83} - 2976 q^{91} - 1120 q^{92} + 400 q^{94} - 612 q^{98}+O(q^{100}) $ 2 * q - 4 * q^2 + 8 * q^4 - 16 * q^8 + 32 * q^16 - 124 * q^17 + 168 * q^19 - 280 * q^23 + 32 * q^31 - 64 * q^32 + 248 * q^34 - 336 * q^38 + 560 * q^46 - 200 * q^47 + 306 * q^49 - 1476 * q^53 - 716 * q^61 - 64 * q^62 + 128 * q^64 - 496 * q^68 + 672 * q^76 - 992 * q^77 - 1872 * q^79 - 2608 * q^83 - 2976 * q^91 - 1120 * q^92 + 400 * q^94 - 612 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−2.00000	−0.707107
$2$	−2.00000	−0.707107
$3$	0	0
$3$	0	0
$4$	4.00000	0.500000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−22.2711	−1.20252	−0.601262	−	0.799052i	$-0.705335\pi$
$7$	−22.2711	−1.20252	−0.601262	+	0.799052i	$0.705335\pi$
$8$	−8.00000	−0.353553
$9$	0	0
$10$	0	0
$11$	22.2711	0.610452	0.305226	−	0.952280i	$-0.401268\pi$
$11$	22.2711	0.610452	0.305226	+	0.952280i	$0.401268\pi$
$12$	0	0
$13$	66.8132	1.42543	0.712717	−	0.701452i	$-0.247464\pi$
$13$	66.8132	1.42543	0.712717	+	0.701452i	$0.247464\pi$
$14$	44.5421	0.850313
$15$	0	0
$16$	16.0000	0.250000
$17$	−62.0000	−0.884542	−0.442271	−	0.896882i	$-0.645827\pi$
$17$	−62.0000	−0.884542	−0.442271	+	0.896882i	$0.645827\pi$
$18$	0	0
$19$	84.0000	1.01426	0.507130	−	0.861870i	$-0.330707\pi$
$19$	84.0000	1.01426	0.507130	+	0.861870i	$0.330707\pi$
$20$	0	0
$21$	0	0
$22$	−44.5421	−0.431655
$23$	−140.000	−1.26922	−0.634609	−	0.772833i	$-0.718839\pi$
$23$	−140.000	−1.26922	−0.634609	+	0.772833i	$0.718839\pi$
$24$	0	0
$25$	0	0
$26$	−133.626	−1.00793
$27$	0	0
$28$	−89.0842	−0.601262
$29$	−200.440	−1.28347	−0.641736	−	0.766926i	$-0.721785\pi$
$29$	−200.440	−1.28347	−0.641736	+	0.766926i	$0.721785\pi$
$30$	0	0
$31$	16.0000	0.0926995	0.0463498	−	0.998925i	$-0.485241\pi$
$31$	16.0000	0.0926995	0.0463498	+	0.998925i	$0.485241\pi$
$32$	−32.0000	−0.176777
$33$	0	0
$34$	124.000	0.625465
$35$	0	0
$36$	0	0
$37$	244.982	1.08851	0.544253	−	0.838921i	$-0.316813\pi$
$37$	244.982	1.08851	0.544253	+	0.838921i	$0.316813\pi$
$38$	−168.000	−0.717189
$39$	0	0
$40$	0	0
$41$	222.711	0.848330	0.424165	−	0.905585i	$-0.360568\pi$
$41$	222.711	0.848330	0.424165	+	0.905585i	$0.360568\pi$
$42$	0	0
$43$	−356.337	−1.26374	−0.631871	−	0.775074i	$-0.717713\pi$
$43$	−356.337	−1.26374	−0.631871	+	0.775074i	$0.717713\pi$
$44$	89.0842	0.305226
$45$	0	0
$46$	280.000	0.897473
$47$	−100.000	−0.310351	−0.155176	−	0.987887i	$-0.549594\pi$
$47$	−100.000	−0.310351	−0.155176	+	0.987887i	$0.549594\pi$
$48$	0	0
$49$	153.000	0.446064
$50$	0	0
$51$	0	0
$52$	267.253	0.712717
$53$	−738.000	−1.91268	−0.956341	−	0.292255i	$-0.905595\pi$
$53$	−738.000	−1.91268	−0.956341	+	0.292255i	$0.905595\pi$
$54$	0	0
$55$	0	0
$56$	178.168	0.425156
$57$	0	0
$58$	400.879	0.907552
$59$	645.861	1.42515	0.712575	−	0.701596i	$-0.247529\pi$
$59$	645.861	1.42515	0.712575	+	0.701596i	$0.247529\pi$
$60$	0	0
$61$	−358.000	−0.751430	−0.375715	−	0.926735i	$-0.622603\pi$
$61$	−358.000	−0.751430	−0.375715	+	0.926735i	$0.622603\pi$
$62$	−32.0000	−0.0655485
$63$	0	0
$64$	64.0000	0.125000
$65$	0	0
$66$	0	0
$67$	−846.300	−1.54316	−0.771582	−	0.636130i	$-0.780534\pi$
$67$	−846.300	−1.54316	−0.771582	+	0.636130i	$0.780534\pi$
$68$	−248.000	−0.442271
$69$	0	0
$70$	0	0
$71$	−935.384	−1.56352	−0.781758	−	0.623581i	$-0.785677\pi$
$71$	−935.384	−1.56352	−0.781758	+	0.623581i	$0.785677\pi$
$72$	0	0
$73$	445.421	0.714145	0.357073	−	0.934077i	$-0.383775\pi$
$73$	445.421	0.714145	0.357073	+	0.934077i	$0.383775\pi$
$74$	−489.963	−0.769690
$75$	0	0
$76$	336.000	0.507130
$77$	−496.000	−0.734084
$78$	0	0
$79$	−936.000	−1.33302	−0.666508	−	0.745498i	$-0.732212\pi$
$79$	−936.000	−1.33302	−0.666508	+	0.745498i	$0.732212\pi$
$80$	0	0
$81$	0	0
$82$	−445.421	−0.599860
$83$	−1304.00	−1.72449	−0.862245	−	0.506492i	$-0.830942\pi$
$83$	−1304.00	−1.72449	−0.862245	+	0.506492i	$0.830942\pi$
$84$	0	0
$85$	0	0
$86$	712.674	0.893600
$87$	0	0
$88$	−178.168	−0.215828
$89$	−712.674	−0.848801	−0.424400	−	0.905475i	$-0.639515\pi$
$89$	−712.674	−0.848801	−0.424400	+	0.905475i	$0.639515\pi$
$90$	0	0
$91$	−1488.00	−1.71412
$92$	−560.000	−0.634609
$93$	0	0
$94$	200.000	0.219451
$95$	0	0
$96$	0	0
$97$	757.216	0.792615	0.396307	−	0.918118i	$-0.370291\pi$
$97$	757.216	0.792615	0.396307	+	0.918118i	$0.370291\pi$
$98$	−306.000	−0.315415
$99$	0	0
$100$	0	0
$101$	645.861	0.636292	0.318146	−	0.948042i	$-0.396940\pi$
$101$	645.861	0.636292	0.318146	+	0.948042i	$0.396940\pi$
$102$	0	0
$103$	690.403	0.660460	0.330230	−	0.943900i	$-0.392874\pi$
$103$	690.403	0.660460	0.330230	+	0.943900i	$0.392874\pi$
$104$	−534.505	−0.503967
$105$	0	0
$106$	1476.00	1.35247
$107$	−696.000	−0.628830	−0.314415	−	0.949286i	$-0.601808\pi$
$107$	−696.000	−0.628830	−0.314415	+	0.949286i	$0.601808\pi$
$108$	0	0
$109$	−1142.00	−1.00352	−0.501760	−	0.865007i	$-0.667314\pi$
$109$	−1142.00	−1.00352	−0.501760	+	0.865007i	$0.667314\pi$
$110$	0	0
$111$	0	0
$112$	−356.337	−0.300631
$113$	78.0000	0.0649347	0.0324674	−	0.999473i	$-0.489664\pi$
$113$	78.0000	0.0649347	0.0324674	+	0.999473i	$0.489664\pi$
$114$	0	0
$115$	0	0
$116$	−801.758	−0.641736
$117$	0	0
$118$	−1291.72	−1.00773
$119$	1380.81	1.06368
$120$	0	0
$121$	−835.000	−0.627348
$122$	716.000	0.531341
$123$	0	0
$124$	64.0000	0.0463498
$125$	0	0
$126$	0	0
$127$	1313.99	0.918094	0.459047	−	0.888412i	$-0.348191\pi$
$127$	1313.99	0.918094	0.459047	+	0.888412i	$0.348191\pi$
$128$	−128.000	−0.0883883
$129$	0	0
$130$	0	0
$131$	−467.692	−0.311927	−0.155964	−	0.987763i	$-0.549848\pi$
$131$	−467.692	−0.311927	−0.155964	+	0.987763i	$0.549848\pi$
$132$	0	0
$133$	−1870.77	−1.21967
$134$	1692.60	1.09118
$135$	0	0
$136$	496.000	0.312733
$137$	1062.00	0.662283	0.331142	−	0.943581i	$-0.392566\pi$
$137$	1062.00	0.662283	0.331142	+	0.943581i	$0.392566\pi$
$138$	0	0
$139$	−860.000	−0.524779	−0.262389	−	0.964962i	$-0.584510\pi$
$139$	−860.000	−0.524779	−0.262389	+	0.964962i	$0.584510\pi$
$140$	0	0
$141$	0	0
$142$	1870.77	1.10557
$143$	1488.00	0.870160
$144$	0	0
$145$	0	0
$146$	−890.842	−0.504977
$147$	0	0
$148$	979.927	0.544253
$149$	−1358.53	−0.746950	−0.373475	−	0.927640i	$-0.621834\pi$
$149$	−1358.53	−0.746950	−0.373475	+	0.927640i	$0.621834\pi$
$150$	0	0
$151$	−1376.00	−0.741571	−0.370786	−	0.928718i	$-0.620912\pi$
$151$	−1376.00	−0.741571	−0.370786	+	0.928718i	$0.620912\pi$
$152$	−672.000	−0.358595
$153$	0	0
$154$	992.000	0.519076
$155$	0	0
$156$	0	0
$157$	−1848.50	−0.939657	−0.469829	−	0.882758i	$-0.655684\pi$
$157$	−1848.50	−0.939657	−0.469829	+	0.882758i	$0.655684\pi$
$158$	1872.00	0.942584
$159$	0	0
$160$	0	0
$161$	3117.95	1.52627
$162$	0	0
$163$	222.711	0.107019	0.0535093	−	0.998567i	$-0.482959\pi$
$163$	222.711	0.107019	0.0535093	+	0.998567i	$0.482959\pi$
$164$	890.842	0.424165
$165$	0	0
$166$	2608.00	1.21940
$167$	2172.00	1.00643	0.503217	−	0.864160i	$-0.332150\pi$
$167$	2172.00	1.00643	0.503217	+	0.864160i	$0.332150\pi$
$168$	0	0
$169$	2267.00	1.03186
$170$	0	0
$171$	0	0
$172$	−1425.35	−0.631871
$173$	3434.00	1.50915	0.754573	−	0.656216i	$-0.227844\pi$
$173$	3434.00	1.50915	0.754573	+	0.656216i	$0.227844\pi$
$174$	0	0
$175$	0	0
$176$	356.337	0.152613
$177$	0	0
$178$	1425.35	0.600193
$179$	−3362.93	−1.40423	−0.702115	−	0.712064i	$-0.747761\pi$
$179$	−3362.93	−1.40423	−0.702115	+	0.712064i	$0.747761\pi$
$180$	0	0
$181$	−1678.00	−0.689087	−0.344544	−	0.938770i	$-0.611966\pi$
$181$	−1678.00	−0.689087	−0.344544	+	0.938770i	$0.611966\pi$
$182$	2976.00	1.21206
$183$	0	0
$184$	1120.00	0.448736
$185$	0	0
$186$	0	0
$187$	−1380.81	−0.539971
$188$	−400.000	−0.155176
$189$	0	0
$190$	0	0
$191$	−1737.14	−0.658090	−0.329045	−	0.944314i	$-0.606727\pi$
$191$	−1737.14	−0.658090	−0.329045	+	0.944314i	$0.606727\pi$
$192$	0	0
$193$	2449.82	0.913687	0.456844	−	0.889547i	$-0.348980\pi$
$193$	2449.82	0.913687	0.456844	+	0.889547i	$0.348980\pi$
$194$	−1514.43	−0.560463
$195$	0	0
$196$	612.000	0.223032
$197$	−526.000	−0.190233	−0.0951166	−	0.995466i	$-0.530322\pi$
$197$	−526.000	−0.190233	−0.0951166	+	0.995466i	$0.530322\pi$
$198$	0	0
$199$	−744.000	−0.265029	−0.132514	−	0.991181i	$-0.542305\pi$
$199$	−744.000	−0.265029	−0.132514	+	0.991181i	$0.542305\pi$
$200$	0	0
$201$	0	0
$202$	−1291.72	−0.449927
$203$	4464.00	1.54341
$204$	0	0
$205$	0	0
$206$	−1380.81	−0.467016
$207$	0	0
$208$	1069.01	0.356358
$209$	1870.77	0.619157
$210$	0	0
$211$	1476.00	0.481574	0.240787	−	0.970578i	$-0.422595\pi$
$211$	1476.00	0.481574	0.240787	+	0.970578i	$0.422595\pi$
$212$	−2952.00	−0.956341
$213$	0	0
$214$	1392.00	0.444650
$215$	0	0
$216$	0	0
$217$	−356.337	−0.111473
$218$	2284.00	0.709596
$219$	0	0
$220$	0	0
$221$	−4142.42	−1.26086
$222$	0	0
$223$	−824.029	−0.247449	−0.123724	−	0.992317i	$-0.539484\pi$
$223$	−824.029	−0.247449	−0.123724	+	0.992317i	$0.539484\pi$
$224$	712.674	0.212578
$225$	0	0
$226$	−156.000	−0.0459158
$227$	2200.00	0.643256	0.321628	−	0.946866i	$-0.395770\pi$
$227$	2200.00	0.643256	0.321628	+	0.946866i	$0.395770\pi$
$228$	0	0
$229$	5246.00	1.51382	0.756911	−	0.653517i	$-0.226707\pi$
$229$	5246.00	1.51382	0.756911	+	0.653517i	$0.226707\pi$
$230$	0	0
$231$	0	0
$232$	1603.52	0.453776
$233$	842.000	0.236744	0.118372	−	0.992969i	$-0.462233\pi$
$233$	842.000	0.236744	0.118372	+	0.992969i	$0.462233\pi$
$234$	0	0
$235$	0	0
$236$	2583.44	0.712575
$237$	0	0
$238$	−2761.61	−0.752137
$239$	5835.02	1.57923	0.789615	−	0.613603i	$-0.210280\pi$
$239$	5835.02	1.57923	0.789615	+	0.613603i	$0.210280\pi$
$240$	0	0
$241$	2530.00	0.676231	0.338115	−	0.941105i	$-0.390211\pi$
$241$	2530.00	0.676231	0.338115	+	0.941105i	$0.390211\pi$
$242$	1670.00	0.443602
$243$	0	0
$244$	−1432.00	−0.375715
$245$	0	0
$246$	0	0
$247$	5612.31	1.44576
$248$	−128.000	−0.0327742
$249$	0	0
$250$	0	0
$251$	6748.13	1.69696	0.848482	−	0.529223i	$-0.177517\pi$
$251$	6748.13	1.69696	0.848482	+	0.529223i	$0.177517\pi$
$252$	0	0
$253$	−3117.95	−0.774797
$254$	−2627.98	−0.649191
$255$	0	0
$256$	256.000	0.0625000
$257$	−258.000	−0.0626210	−0.0313105	−	0.999510i	$-0.509968\pi$
$257$	−258.000	−0.0626210	−0.0313105	+	0.999510i	$0.509968\pi$
$258$	0	0
$259$	−5456.00	−1.30895
$260$	0	0
$261$	0	0
$262$	935.384	0.220566
$263$	772.000	0.181002	0.0905011	−	0.995896i	$-0.471153\pi$
$263$	772.000	0.181002	0.0905011	+	0.995896i	$0.471153\pi$
$264$	0	0
$265$	0	0
$266$	3741.54	0.862438
$267$	0	0
$268$	−3385.20	−0.771582
$269$	2605.71	0.590607	0.295303	−	0.955404i	$-0.404579\pi$
$269$	2605.71	0.590607	0.295303	+	0.955404i	$0.404579\pi$
$270$	0	0
$271$	−6392.00	−1.43279	−0.716395	−	0.697694i	$-0.754209\pi$
$271$	−6392.00	−1.43279	−0.716395	+	0.697694i	$0.754209\pi$
$272$	−992.000	−0.221135
$273$	0	0
$274$	−2124.00	−0.468305
$275$	0	0
$276$	0	0
$277$	−1759.41	−0.381635	−0.190818	−	0.981626i	$-0.561114\pi$
$277$	−1759.41	−0.381635	−0.190818	+	0.981626i	$0.561114\pi$
$278$	1720.00	0.371075
$279$	0	0
$280$	0	0
$281$	−6414.06	−1.36168	−0.680838	−	0.732434i	$-0.738384\pi$
$281$	−6414.06	−1.36168	−0.680838	+	0.732434i	$0.738384\pi$
$282$	0	0
$283$	−1380.81	−0.290037	−0.145018	−	0.989429i	$-0.546324\pi$
$283$	−1380.81	−0.290037	−0.145018	+	0.989429i	$0.546324\pi$
$284$	−3741.54	−0.781758
$285$	0	0
$286$	−2976.00	−0.615296
$287$	−4960.00	−1.02014
$288$	0	0
$289$	−1069.00	−0.217586
$290$	0	0
$291$	0	0
$292$	1781.68	0.357073
$293$	−1374.00	−0.273959	−0.136979	−	0.990574i	$-0.543739\pi$
$293$	−1374.00	−0.273959	−0.136979	+	0.990574i	$0.543739\pi$
$294$	0	0
$295$	0	0
$296$	−1959.85	−0.384845
$297$	0	0
$298$	2717.07	0.528173
$299$	−9353.84	−1.80919
$300$	0	0
$301$	7936.00	1.51968
$302$	2752.00	0.524370
$303$	0	0
$304$	1344.00	0.253565
$305$	0	0
$306$	0	0
$307$	−267.253	−0.0496838	−0.0248419	−	0.999691i	$-0.507908\pi$
$307$	−267.253	−0.0496838	−0.0248419	+	0.999691i	$0.507908\pi$
$308$	−1984.00	−0.367042
$309$	0	0
$310$	0	0
$311$	−3607.91	−0.657832	−0.328916	−	0.944359i	$-0.606683\pi$
$311$	−3607.91	−0.657832	−0.328916	+	0.944359i	$0.606683\pi$
$312$	0	0
$313$	7794.87	1.40764	0.703821	−	0.710377i	$-0.251476\pi$
$313$	7794.87	1.40764	0.703821	+	0.710377i	$0.251476\pi$
$314$	3697.00	0.664438
$315$	0	0
$316$	−3744.00	−0.666508
$317$	9334.00	1.65378	0.826892	−	0.562360i	$-0.190107\pi$
$317$	9334.00	1.65378	0.826892	+	0.562360i	$0.190107\pi$
$318$	0	0
$319$	−4464.00	−0.783498
$320$	0	0
$321$	0	0
$322$	−6235.90	−1.07923
$323$	−5208.00	−0.897154
$324$	0	0
$325$	0	0
$326$	−445.421	−0.0756736
$327$	0	0
$328$	−1781.68	−0.299930
$329$	2227.11	0.373205
$330$	0	0
$331$	1636.00	0.271670	0.135835	−	0.990731i	$-0.456628\pi$
$331$	1636.00	0.271670	0.135835	+	0.990731i	$0.456628\pi$
$332$	−5216.00	−0.862245
$333$	0	0
$334$	−4344.00	−0.711656
$335$	0	0
$336$	0	0
$337$	178.168	0.0287996	0.0143998	−	0.999896i	$-0.495416\pi$
$337$	178.168	0.0287996	0.0143998	+	0.999896i	$0.495416\pi$
$338$	−4534.00	−0.729636
$339$	0	0
$340$	0	0
$341$	356.337	0.0565886
$342$	0	0
$343$	4231.50	0.666121
$344$	2850.70	0.446800
$345$	0	0
$346$	−6868.00	−1.06713
$347$	−3840.00	−0.594069	−0.297035	−	0.954867i	$-0.595998\pi$
$347$	−3840.00	−0.594069	−0.297035	+	0.954867i	$0.595998\pi$
$348$	0	0
$349$	6682.00	1.02487	0.512434	−	0.858726i	$-0.328744\pi$
$349$	6682.00	1.02487	0.512434	+	0.858726i	$0.328744\pi$
$350$	0	0
$351$	0	0
$352$	−712.674	−0.107914
$353$	−2110.00	−0.318142	−0.159071	−	0.987267i	$-0.550850\pi$
$353$	−2110.00	−0.318142	−0.159071	+	0.987267i	$0.550850\pi$
$354$	0	0
$355$	0	0
$356$	−2850.70	−0.424400
$357$	0	0
$358$	6725.86	0.992941
$359$	−1870.77	−0.275029	−0.137514	−	0.990500i	$-0.543911\pi$
$359$	−1870.77	−0.275029	−0.137514	+	0.990500i	$0.543911\pi$
$360$	0	0
$361$	197.000	0.0287214
$362$	3356.00	0.487258
$363$	0	0
$364$	−5952.00	−0.857059
$365$	0	0
$366$	0	0
$367$	−9643.37	−1.37161	−0.685803	−	0.727787i	$-0.740549\pi$
$367$	−9643.37	−1.37161	−0.685803	+	0.727787i	$0.740549\pi$
$368$	−2240.00	−0.317305
$369$	0	0
$370$	0	0
$371$	16436.0	2.30005
$372$	0	0
$373$	−11558.7	−1.60452	−0.802260	−	0.596975i	$-0.796369\pi$
$373$	−11558.7	−1.60452	−0.802260	+	0.596975i	$0.796369\pi$
$374$	2761.61	0.381817
$375$	0	0
$376$	800.000	0.109726
$377$	−13392.0	−1.82950
$378$	0	0
$379$	−6220.00	−0.843008	−0.421504	−	0.906827i	$-0.638498\pi$
$379$	−6220.00	−0.843008	−0.421504	+	0.906827i	$0.638498\pi$
$380$	0	0
$381$	0	0
$382$	3474.28	0.465340
$383$	−13900.0	−1.85446	−0.927228	−	0.374497i	$-0.877815\pi$
$383$	−13900.0	−1.85446	−0.927228	+	0.374497i	$0.877815\pi$
$384$	0	0
$385$	0	0
$386$	−4899.63	−0.646074
$387$	0	0
$388$	3028.86	0.396307
$389$	3630.18	0.473156	0.236578	−	0.971613i	$-0.423974\pi$
$389$	3630.18	0.473156	0.236578	+	0.971613i	$0.423974\pi$
$390$	0	0
$391$	8680.00	1.12268
$392$	−1224.00	−0.157707
$393$	0	0
$394$	1052.00	0.134515
$395$	0	0
$396$	0	0
$397$	512.234	0.0647564	0.0323782	−	0.999476i	$-0.489692\pi$
$397$	512.234	0.0647564	0.0323782	+	0.999476i	$0.489692\pi$
$398$	1488.00	0.187404
$399$	0	0
$400$	0	0
$401$	2984.32	0.371646	0.185823	−	0.982583i	$-0.440505\pi$
$401$	2984.32	0.371646	0.185823	+	0.982583i	$0.440505\pi$
$402$	0	0
$403$	1069.01	0.132137
$404$	2583.44	0.318146
$405$	0	0
$406$	−8928.00	−1.09135
$407$	5456.00	0.664481
$408$	0	0
$409$	14822.0	1.79193	0.895967	−	0.444121i	$-0.146484\pi$
$409$	14822.0	1.79193	0.895967	+	0.444121i	$0.146484\pi$
$410$	0	0
$411$	0	0
$412$	2761.61	0.330230
$413$	−14384.0	−1.71378
$414$	0	0
$415$	0	0
$416$	−2138.02	−0.251983
$417$	0	0
$418$	−3741.54	−0.437810
$419$	−8663.44	−1.01011	−0.505056	−	0.863087i	$-0.668528\pi$
$419$	−8663.44	−1.01011	−0.505056	+	0.863087i	$0.668528\pi$
$420$	0	0
$421$	4894.00	0.566553	0.283277	−	0.959038i	$-0.408579\pi$
$421$	4894.00	0.566553	0.283277	+	0.959038i	$0.408579\pi$
$422$	−2952.00	−0.340524
$423$	0	0
$424$	5904.00	0.676235
$425$	0	0
$426$	0	0
$427$	7973.04	0.903612
$428$	−2784.00	−0.314415
$429$	0	0
$430$	0	0
$431$	−9799.27	−1.09516	−0.547580	−	0.836753i	$-0.684451\pi$
$431$	−9799.27	−1.09516	−0.547580	+	0.836753i	$0.684451\pi$
$432$	0	0
$433$	−3697.00	−0.410315	−0.205157	−	0.978729i	$-0.565771\pi$
$433$	−3697.00	−0.410315	−0.205157	+	0.978729i	$0.565771\pi$
$434$	712.674	0.0788236
$435$	0	0
$436$	−4568.00	−0.501760
$437$	−11760.0	−1.28732
$438$	0	0
$439$	−9288.00	−1.00978	−0.504888	−	0.863185i	$-0.668466\pi$
$439$	−9288.00	−1.00978	−0.504888	+	0.863185i	$0.668466\pi$
$440$	0	0
$441$	0	0
$442$	8284.83	0.891560
$443$	7368.00	0.790213	0.395106	−	0.918635i	$-0.370708\pi$
$443$	7368.00	0.790213	0.395106	+	0.918635i	$0.370708\pi$
$444$	0	0
$445$	0	0
$446$	1648.06	0.174973
$447$	0	0
$448$	−1425.35	−0.150316
$449$	−400.879	−0.0421351	−0.0210675	−	0.999778i	$-0.506707\pi$
$449$	−400.879	−0.0421351	−0.0210675	+	0.999778i	$0.506707\pi$
$450$	0	0
$451$	4960.00	0.517865
$452$	312.000	0.0324674
$453$	0	0
$454$	−4400.00	−0.454851
$455$	0	0
$456$	0	0
$457$	14743.4	1.50912	0.754561	−	0.656230i	$-0.227850\pi$
$457$	14743.4	1.50912	0.754561	+	0.656230i	$0.227850\pi$
$458$	−10492.0	−1.07043
$459$	0	0
$460$	0	0
$461$	−4743.74	−0.479258	−0.239629	−	0.970865i	$-0.577026\pi$
$461$	−4743.74	−0.479258	−0.239629	+	0.970865i	$0.577026\pi$
$462$	0	0
$463$	−7282.64	−0.731000	−0.365500	−	0.930811i	$-0.619102\pi$
$463$	−7282.64	−0.731000	−0.365500	+	0.930811i	$0.619102\pi$
$464$	−3207.03	−0.320868
$465$	0	0
$466$	−1684.00	−0.167403
$467$	−5664.00	−0.561239	−0.280620	−	0.959819i	$-0.590540\pi$
$467$	−5664.00	−0.561239	−0.280620	+	0.959819i	$0.590540\pi$
$468$	0	0
$469$	18848.0	1.85569
$470$	0	0
$471$	0	0
$472$	−5166.89	−0.503867
$473$	−7936.00	−0.771454
$474$	0	0
$475$	0	0
$476$	5523.22	0.531841
$477$	0	0
$478$	−11670.0	−1.11668
$479$	10111.1	0.964480	0.482240	−	0.876039i	$-0.339823\pi$
$479$	10111.1	0.964480	0.482240	+	0.876039i	$0.339823\pi$
$480$	0	0
$481$	16368.0	1.55159
$482$	−5060.00	−0.478167
$483$	0	0
$484$	−3340.00	−0.313674
$485$	0	0
$486$	0	0
$487$	−11558.7	−1.07551	−0.537755	−	0.843101i	$-0.680728\pi$
$487$	−11558.7	−1.07551	−0.537755	+	0.843101i	$0.680728\pi$
$488$	2864.00	0.265670
$489$	0	0
$490$	0	0
$491$	−467.692	−0.0429871	−0.0214935	−	0.999769i	$-0.506842\pi$
$491$	−467.692	−0.0429871	−0.0214935	+	0.999769i	$0.506842\pi$
$492$	0	0
$493$	12427.3	1.13528
$494$	−11224.6	−1.02231
$495$	0	0
$496$	256.000	0.0231749
$497$	20832.0	1.88017
$498$	0	0
$499$	9204.00	0.825707	0.412853	−	0.910798i	$-0.364532\pi$
$499$	9204.00	0.825707	0.412853	+	0.910798i	$0.364532\pi$
$500$	0	0
$501$	0	0
$502$	−13496.3	−1.19994
$503$	1260.00	0.111691	0.0558455	−	0.998439i	$-0.482215\pi$
$503$	1260.00	0.111691	0.0558455	+	0.998439i	$0.482215\pi$
$504$	0	0
$505$	0	0
$506$	6235.90	0.547864
$507$	0	0
$508$	5255.97	0.459047
$509$	−2071.21	−0.180363	−0.0901814	−	0.995925i	$-0.528745\pi$
$509$	−2071.21	−0.180363	−0.0901814	+	0.995925i	$0.528745\pi$
$510$	0	0
$511$	−9920.00	−0.858777
$512$	−512.000	−0.0441942
$513$	0	0
$514$	516.000	0.0442797
$515$	0	0
$516$	0	0
$517$	−2227.11	−0.189455
$518$	10912.0	0.925571
$519$	0	0
$520$	0	0
$521$	−5300.51	−0.445719	−0.222860	−	0.974851i	$-0.571539\pi$
$521$	−5300.51	−0.445719	−0.222860	+	0.974851i	$0.571539\pi$
$522$	0	0
$523$	9309.30	0.778331	0.389166	−	0.921168i	$-0.372763\pi$
$523$	9309.30	0.778331	0.389166	+	0.921168i	$0.372763\pi$
$524$	−1870.77	−0.155964
$525$	0	0
$526$	−1544.00	−0.127988
$527$	−992.000	−0.0819966
$528$	0	0
$529$	7433.00	0.610915
$530$	0	0
$531$	0	0
$532$	−7483.08	−0.609835
$533$	14880.0	1.20924
$534$	0	0
$535$	0	0
$536$	6770.40	0.545591
$537$	0	0
$538$	−5211.43	−0.417622
$539$	3407.47	0.272301
$540$	0	0
$541$	9658.00	0.767523	0.383761	−	0.923432i	$-0.374629\pi$
$541$	9658.00	0.767523	0.383761	+	0.923432i	$0.374629\pi$
$542$	12784.0	1.01314
$543$	0	0
$544$	1984.00	0.156366
$545$	0	0
$546$	0	0
$547$	−890.842	−0.0696338	−0.0348169	−	0.999394i	$-0.511085\pi$
$547$	−890.842	−0.0696338	−0.0348169	+	0.999394i	$0.511085\pi$
$548$	4248.00	0.331142
$549$	0	0
$550$	0	0
$551$	−16836.9	−1.30177
$552$	0	0
$553$	20845.7	1.60298
$554$	3518.83	0.269857
$555$	0	0
$556$	−3440.00	−0.262389
$557$	−9066.00	−0.689657	−0.344828	−	0.938666i	$-0.612063\pi$
$557$	−9066.00	−0.689657	−0.344828	+	0.938666i	$0.612063\pi$
$558$	0	0
$559$	−23808.0	−1.80138
$560$	0	0
$561$	0	0
$562$	12828.1	0.962850
$563$	1568.00	0.117377	0.0586886	−	0.998276i	$-0.481308\pi$
$563$	1568.00	0.117377	0.0586886	+	0.998276i	$0.481308\pi$
$564$	0	0
$565$	0	0
$566$	2761.61	0.205087
$567$	0	0
$568$	7483.08	0.552787
$569$	2093.48	0.154241	0.0771206	−	0.997022i	$-0.475427\pi$
$569$	2093.48	0.154241	0.0771206	+	0.997022i	$0.475427\pi$
$570$	0	0
$571$	−13916.0	−1.01991	−0.509953	−	0.860202i	$-0.670337\pi$
$571$	−13916.0	−1.01991	−0.509953	+	0.860202i	$0.670337\pi$
$572$	5952.00	0.435080
$573$	0	0
$574$	9920.00	0.721346
$575$	0	0
$576$	0	0
$577$	−8106.66	−0.584896	−0.292448	−	0.956281i	$-0.594470\pi$
$577$	−8106.66	−0.584896	−0.292448	+	0.956281i	$0.594470\pi$
$578$	2138.00	0.153857
$579$	0	0
$580$	0	0
$581$	29041.5	2.07374
$582$	0	0
$583$	−16436.0	−1.16760
$584$	−3563.37	−0.252488
$585$	0	0
$586$	2748.00	0.193718
$587$	20808.0	1.46310	0.731549	−	0.681789i	$-0.238798\pi$
$587$	20808.0	1.46310	0.731549	+	0.681789i	$0.238798\pi$
$588$	0	0
$589$	1344.00	0.0940213
$590$	0	0
$591$	0	0
$592$	3919.71	0.272127
$593$	21486.0	1.48790	0.743950	−	0.668236i	$-0.232950\pi$
$593$	21486.0	1.48790	0.743950	+	0.668236i	$0.232950\pi$
$594$	0	0
$595$	0	0
$596$	−5434.14	−0.373475
$597$	0	0
$598$	18707.7	1.27929
$599$	25923.5	1.76829	0.884145	−	0.467212i	$-0.154742\pi$
$599$	25923.5	1.76829	0.884145	+	0.467212i	$0.154742\pi$
$600$	0	0
$601$	−250.000	−0.0169679	−0.00848395	−	0.999964i	$-0.502701\pi$
$601$	−250.000	−0.0169679	−0.00848395	+	0.999964i	$0.502701\pi$
$602$	−15872.0	−1.07458
$603$	0	0
$604$	−5504.00	−0.370786
$605$	0	0
$606$	0	0
$607$	−13340.4	−0.892041	−0.446020	−	0.895023i	$-0.647159\pi$
$607$	−13340.4	−0.892041	−0.446020	+	0.895023i	$0.647159\pi$
$608$	−2688.00	−0.179297
$609$	0	0
$610$	0	0
$611$	−6681.32	−0.442385
$612$	0	0
$613$	−14721.2	−0.969955	−0.484977	−	0.874527i	$-0.661172\pi$
$613$	−14721.2	−0.969955	−0.484977	+	0.874527i	$0.661172\pi$
$614$	534.505	0.0351317
$615$	0	0
$616$	3968.00	0.259538
$617$	14410.0	0.940235	0.470117	−	0.882604i	$-0.344212\pi$
$617$	14410.0	0.940235	0.470117	+	0.882604i	$0.344212\pi$
$618$	0	0
$619$	−21580.0	−1.40125	−0.700625	−	0.713530i	$-0.747095\pi$
$619$	−21580.0	−1.40125	−0.700625	+	0.713530i	$0.747095\pi$
$620$	0	0
$621$	0	0
$622$	7215.82	0.465158
$623$	15872.0	1.02070
$624$	0	0
$625$	0	0
$626$	−15589.7	−0.995354
$627$	0	0
$628$	−7393.99	−0.469829
$629$	−15188.9	−0.962829
$630$	0	0
$631$	−15160.0	−0.956434	−0.478217	−	0.878242i	$-0.658717\pi$
$631$	−15160.0	−0.956434	−0.478217	+	0.878242i	$0.658717\pi$
$632$	7488.00	0.471292
$633$	0	0
$634$	−18668.0	−1.16940
$635$	0	0
$636$	0	0
$637$	10222.4	0.635835
$638$	8928.00	0.554017
$639$	0	0
$640$	0	0
$641$	6102.27	0.376014	0.188007	−	0.982168i	$-0.439797\pi$
$641$	6102.27	0.376014	0.188007	+	0.982168i	$0.439797\pi$
$642$	0	0
$643$	−14298.0	−0.876919	−0.438459	−	0.898751i	$-0.644476\pi$
$643$	−14298.0	−0.876919	−0.438459	+	0.898751i	$0.644476\pi$
$644$	12471.8	0.763133
$645$	0	0
$646$	10416.0	0.634384
$647$	−11916.0	−0.724059	−0.362030	−	0.932167i	$-0.617916\pi$
$647$	−11916.0	−0.724059	−0.362030	+	0.932167i	$0.617916\pi$
$648$	0	0
$649$	14384.0	0.869987
$650$	0	0
$651$	0	0
$652$	890.842	0.0535093
$653$	24646.0	1.47699	0.738493	−	0.674261i	$-0.235538\pi$
$653$	24646.0	1.47699	0.738493	+	0.674261i	$0.235538\pi$
$654$	0	0
$655$	0	0
$656$	3563.37	0.212083
$657$	0	0
$658$	−4454.21	−0.263896
$659$	−17928.2	−1.05976	−0.529881	−	0.848072i	$-0.677764\pi$
$659$	−17928.2	−1.05976	−0.529881	+	0.848072i	$0.677764\pi$
$660$	0	0
$661$	−14638.0	−0.861350	−0.430675	−	0.902507i	$-0.641724\pi$
$661$	−14638.0	−0.861350	−0.430675	+	0.902507i	$0.641724\pi$
$662$	−3272.00	−0.192100
$663$	0	0
$664$	10432.0	0.609699
$665$	0	0
$666$	0	0
$667$	28061.5	1.62901
$668$	8688.00	0.503217
$669$	0	0
$670$	0	0
$671$	−7973.04	−0.458712
$672$	0	0
$673$	−31134.9	−1.78330	−0.891652	−	0.452721i	$-0.850453\pi$
$673$	−31134.9	−1.78330	−0.891652	+	0.452721i	$0.850453\pi$
$674$	−356.337	−0.0203644
$675$	0	0
$676$	9068.00	0.515931
$677$	6382.00	0.362305	0.181152	−	0.983455i	$-0.442017\pi$
$677$	6382.00	0.362305	0.181152	+	0.983455i	$0.442017\pi$
$678$	0	0
$679$	−16864.0	−0.953138
$680$	0	0
$681$	0	0
$682$	−712.674	−0.0400142
$683$	−4544.00	−0.254570	−0.127285	−	0.991866i	$-0.540626\pi$
$683$	−4544.00	−0.254570	−0.127285	+	0.991866i	$0.540626\pi$
$684$	0	0
$685$	0	0
$686$	−8463.00	−0.471019
$687$	0	0
$688$	−5701.39	−0.315935
$689$	−49308.1	−2.72640
$690$	0	0
$691$	13564.0	0.746742	0.373371	−	0.927682i	$-0.378202\pi$
$691$	13564.0	0.746742	0.373371	+	0.927682i	$0.378202\pi$
$692$	13736.0	0.754573
$693$	0	0
$694$	7680.00	0.420070
$695$	0	0
$696$	0	0
$697$	−13808.1	−0.750384
$698$	−13364.0	−0.724692
$699$	0	0
$700$	0	0
$701$	1492.16	0.0803968	0.0401984	−	0.999192i	$-0.487201\pi$
$701$	1492.16	0.0803968	0.0401984	+	0.999192i	$0.487201\pi$
$702$	0	0
$703$	20578.5	1.10403
$704$	1425.35	0.0763066
$705$	0	0
$706$	4220.00	0.224960
$707$	−14384.0	−0.765157
$708$	0	0
$709$	−17826.0	−0.944245	−0.472122	−	0.881533i	$-0.656512\pi$
$709$	−17826.0	−0.944245	−0.472122	+	0.881533i	$0.656512\pi$
$710$	0	0
$711$	0	0
$712$	5701.39	0.300096
$713$	−2240.00	−0.117656
$714$	0	0
$715$	0	0
$716$	−13451.7	−0.702115
$717$	0	0
$718$	3741.54	0.194475
$719$	18796.8	0.974967	0.487484	−	0.873132i	$-0.337915\pi$
$719$	18796.8	0.974967	0.487484	+	0.873132i	$0.337915\pi$
$720$	0	0
$721$	−15376.0	−0.794219
$722$	−394.000	−0.0203091
$723$	0	0
$724$	−6712.00	−0.344544
$725$	0	0
$726$	0	0
$727$	33072.5	1.68720	0.843598	−	0.536975i	$-0.180433\pi$
$727$	33072.5	1.68720	0.843598	+	0.536975i	$0.180433\pi$
$728$	11904.0	0.606032
$729$	0	0
$730$	0	0
$731$	22092.9	1.11783
$732$	0	0
$733$	34765.1	1.75181	0.875907	−	0.482481i	$-0.160264\pi$
$733$	34765.1	1.75181	0.875907	+	0.482481i	$0.160264\pi$
$734$	19286.7	0.969872
$735$	0	0
$736$	4480.00	0.224368
$737$	−18848.0	−0.942028
$738$	0	0
$739$	−22940.0	−1.14190	−0.570948	−	0.820986i	$-0.693424\pi$
$739$	−22940.0	−1.14190	−0.570948	+	0.820986i	$0.693424\pi$
$740$	0	0
$741$	0	0
$742$	−32872.1	−1.62638
$743$	26460.0	1.30649	0.653246	−	0.757146i	$-0.273407\pi$
$743$	26460.0	1.30649	0.653246	+	0.757146i	$0.273407\pi$
$744$	0	0
$745$	0	0
$746$	23117.4	1.13457
$747$	0	0
$748$	−5523.22	−0.269985
$749$	15500.7	0.756184
$750$	0	0
$751$	−30160.0	−1.46545	−0.732726	−	0.680524i	$-0.761752\pi$
$751$	−30160.0	−1.46545	−0.732726	+	0.680524i	$0.761752\pi$
$752$	−1600.00	−0.0775878
$753$	0	0
$754$	26784.0	1.29365
$755$	0	0
$756$	0	0
$757$	−18596.3	−0.892860	−0.446430	−	0.894818i	$-0.647305\pi$
$757$	−18596.3	−0.892860	−0.446430	+	0.894818i	$0.647305\pi$
$758$	12440.0	0.596097
$759$	0	0
$760$	0	0
$761$	5478.68	0.260975	0.130488	−	0.991450i	$-0.458346\pi$
$761$	5478.68	0.260975	0.130488	+	0.991450i	$0.458346\pi$
$762$	0	0
$763$	25433.5	1.20676
$764$	−6948.57	−0.329045
$765$	0	0
$766$	27800.0	1.31130
$767$	43152.0	2.03146
$768$	0	0
$769$	29406.0	1.37894	0.689472	−	0.724313i	$-0.257843\pi$
$769$	29406.0	1.37894	0.689472	+	0.724313i	$0.257843\pi$
$770$	0	0
$771$	0	0
$772$	9799.27	0.456844
$773$	−31122.0	−1.44810	−0.724050	−	0.689748i	$-0.757721\pi$
$773$	−31122.0	−1.44810	−0.724050	+	0.689748i	$0.757721\pi$
$774$	0	0
$775$	0	0
$776$	−6057.73	−0.280232
$777$	0	0
$778$	−7260.36	−0.334572
$779$	18707.7	0.860427
$780$	0	0
$781$	−20832.0	−0.954453
$782$	−17360.0	−0.793852
$783$	0	0
$784$	2448.00	0.111516
$785$	0	0
$786$	0	0
$787$	−34431.1	−1.55951	−0.779755	−	0.626085i	$-0.784656\pi$
$787$	−34431.1	−1.55951	−0.779755	+	0.626085i	$0.784656\pi$
$788$	−2104.00	−0.0951166
$789$	0	0
$790$	0	0
$791$	−1737.14	−0.0780856
$792$	0	0
$793$	−23919.1	−1.07111
$794$	−1024.47	−0.0457897
$795$	0	0
$796$	−2976.00	−0.132514
$797$	13066.0	0.580704	0.290352	−	0.956920i	$-0.406228\pi$
$797$	13066.0	0.580704	0.290352	+	0.956920i	$0.406228\pi$
$798$	0	0
$799$	6200.00	0.274518
$800$	0	0
$801$	0	0
$802$	−5968.64	−0.262793
$803$	9920.00	0.435952
$804$	0	0
$805$	0	0
$806$	−2138.02	−0.0934350
$807$	0	0
$808$	−5166.89	−0.224963
$809$	39954.3	1.73636	0.868181	−	0.496247i	$-0.165289\pi$
$809$	39954.3	1.73636	0.868181	+	0.496247i	$0.165289\pi$
$810$	0	0
$811$	14428.0	0.624705	0.312352	−	0.949966i	$-0.398883\pi$
$811$	14428.0	0.624705	0.312352	+	0.949966i	$0.398883\pi$
$812$	17856.0	0.771703
$813$	0	0
$814$	−10912.0	−0.469859
$815$	0	0
$816$	0	0
$817$	−29932.3	−1.28176
$818$	−29644.0	−1.26709
$819$	0	0
$820$	0	0
$821$	−16591.9	−0.705314	−0.352657	−	0.935753i	$-0.614722\pi$
$821$	−16591.9	−0.705314	−0.352657	+	0.935753i	$0.614722\pi$
$822$	0	0
$823$	25322.2	1.07251	0.536255	−	0.844056i	$-0.319838\pi$
$823$	25322.2	1.07251	0.536255	+	0.844056i	$0.319838\pi$
$824$	−5523.22	−0.233508
$825$	0	0
$826$	28768.0	1.21182
$827$	−16592.0	−0.697655	−0.348827	−	0.937187i	$-0.613420\pi$
$827$	−16592.0	−0.697655	−0.348827	+	0.937187i	$0.613420\pi$
$828$	0	0
$829$	18234.0	0.763924	0.381962	−	0.924178i	$-0.375249\pi$
$829$	18234.0	0.763924	0.381962	+	0.924178i	$0.375249\pi$
$830$	0	0
$831$	0	0
$832$	4276.04	0.178179
$833$	−9486.00	−0.394562
$834$	0	0
$835$	0	0
$836$	7483.08	0.309578
$837$	0	0
$838$	17326.9	0.714257
$839$	−31357.6	−1.29033	−0.645165	−	0.764044i	$-0.723211\pi$
$839$	−31357.6	−1.29033	−0.645165	+	0.764044i	$0.723211\pi$
$840$	0	0
$841$	15787.0	0.647300
$842$	−9788.00	−0.400614
$843$	0	0
$844$	5904.00	0.240787
$845$	0	0
$846$	0	0
$847$	18596.3	0.754401
$848$	−11808.0	−0.478170
$849$	0	0
$850$	0	0
$851$	−34297.4	−1.38155
$852$	0	0
$853$	−2249.38	−0.0902898	−0.0451449	−	0.998980i	$-0.514375\pi$
$853$	−2249.38	−0.0902898	−0.0451449	+	0.998980i	$0.514375\pi$
$854$	−15946.1	−0.638950
$855$	0	0
$856$	5568.00	0.222325
$857$	−25206.0	−1.00469	−0.502346	−	0.864667i	$-0.667530\pi$
$857$	−25206.0	−1.00469	−0.502346	+	0.864667i	$0.667530\pi$
$858$	0	0
$859$	17540.0	0.696690	0.348345	−	0.937366i	$-0.386744\pi$
$859$	17540.0	0.696690	0.348345	+	0.937366i	$0.386744\pi$
$860$	0	0
$861$	0	0
$862$	19598.5	0.774395
$863$	−33108.0	−1.30592	−0.652960	−	0.757392i	$-0.726473\pi$
$863$	−33108.0	−1.30592	−0.652960	+	0.757392i	$0.726473\pi$
$864$	0	0
$865$	0	0
$866$	7393.99	0.290136
$867$	0	0
$868$	−1425.35	−0.0557367
$869$	−20845.7	−0.813743
$870$	0	0
$871$	−56544.0	−2.19968
$872$	9136.00	0.354798
$873$	0	0
$874$	23520.0	0.910270
$875$	0	0
$876$	0	0
$877$	29152.8	1.12249	0.561243	−	0.827651i	$-0.310323\pi$
$877$	29152.8	1.12249	0.561243	+	0.827651i	$0.310323\pi$
$878$	18576.0	0.714020
$879$	0	0
$880$	0	0
$881$	−8819.34	−0.337266	−0.168633	−	0.985679i	$-0.553935\pi$
$881$	−8819.34	−0.337266	−0.168633	+	0.985679i	$0.553935\pi$
$882$	0	0
$883$	−19643.1	−0.748632	−0.374316	−	0.927301i	$-0.622122\pi$
$883$	−19643.1	−0.748632	−0.374316	+	0.927301i	$0.622122\pi$
$884$	−16569.7	−0.630428
$885$	0	0
$886$	−14736.0	−0.558765
$887$	−8004.00	−0.302985	−0.151493	−	0.988458i	$-0.548408\pi$
$887$	−8004.00	−0.302985	−0.151493	+	0.988458i	$0.548408\pi$
$888$	0	0
$889$	−29264.0	−1.10403
$890$	0	0
$891$	0	0
$892$	−3296.12	−0.123724
$893$	−8400.00	−0.314776
$894$	0	0
$895$	0	0
$896$	2850.70	0.106289
$897$	0	0
$898$	801.758	0.0297940
$899$	−3207.03	−0.118977
$900$	0	0
$901$	45756.0	1.69185
$902$	−9920.00	−0.366186
$903$	0	0
$904$	−624.000	−0.0229579
$905$	0	0
$906$	0	0
$907$	−1336.26	−0.0489194	−0.0244597	−	0.999701i	$-0.507787\pi$
$907$	−1336.26	−0.0489194	−0.0244597	+	0.999701i	$0.507787\pi$
$908$	8800.00	0.321628
$909$	0	0
$910$	0	0
$911$	32070.3	1.16634	0.583171	−	0.812350i	$-0.301812\pi$
$911$	32070.3	1.16634	0.583171	+	0.812350i	$0.301812\pi$
$912$	0	0
$913$	−29041.5	−1.05272
$914$	−29486.9	−1.06711
$915$	0	0
$916$	20984.0	0.756911
$917$	10416.0	0.375100
$918$	0	0
$919$	2832.00	0.101653	0.0508265	−	0.998707i	$-0.483814\pi$
$919$	2832.00	0.101653	0.0508265	+	0.998707i	$0.483814\pi$
$920$	0	0
$921$	0	0
$922$	9487.47	0.338886
$923$	−62496.0	−2.22869
$924$	0	0
$925$	0	0
$926$	14565.3	0.516895
$927$	0	0
$928$	6414.06	0.226888
$929$	31936.7	1.12789	0.563945	−	0.825813i	$-0.309283\pi$
$929$	31936.7	1.12789	0.563945	+	0.825813i	$0.309283\pi$
$930$	0	0
$931$	12852.0	0.452425
$932$	3368.00	0.118372
$933$	0	0
$934$	11328.0	0.396856
$935$	0	0
$936$	0	0
$937$	25745.3	0.897613	0.448807	−	0.893629i	$-0.351849\pi$
$937$	25745.3	0.897613	0.448807	+	0.893629i	$0.351849\pi$
$938$	−37696.0	−1.31217
$939$	0	0
$940$	0	0
$941$	−7059.93	−0.244577	−0.122289	−	0.992495i	$-0.539023\pi$
$941$	−7059.93	−0.244577	−0.122289	+	0.992495i	$0.539023\pi$
$942$	0	0
$943$	−31179.5	−1.07672
$944$	10333.8	0.356288
$945$	0	0
$946$	15872.0	0.545500
$947$	−1752.00	−0.0601186	−0.0300593	−	0.999548i	$-0.509570\pi$
$947$	−1752.00	−0.0601186	−0.0300593	+	0.999548i	$0.509570\pi$
$948$	0	0
$949$	29760.0	1.01797
$950$	0	0
$951$	0	0
$952$	−11046.4	−0.376069
$953$	−3850.00	−0.130864	−0.0654322	−	0.997857i	$-0.520843\pi$
$953$	−3850.00	−0.130864	−0.0654322	+	0.997857i	$0.520843\pi$
$954$	0	0
$955$	0	0
$956$	23340.1	0.789615
$957$	0	0
$958$	−20222.1	−0.681991
$959$	−23651.9	−0.796411
$960$	0	0
$961$	−29535.0	−0.991407
$962$	−32736.0	−1.09714
$963$	0	0
$964$	10120.0	0.338115
$965$	0	0
$966$	0	0
$967$	24475.9	0.813952	0.406976	−	0.913439i	$-0.366583\pi$
$967$	24475.9	0.813952	0.406976	+	0.913439i	$0.366583\pi$
$968$	6680.00	0.221801
$969$	0	0
$970$	0	0
$971$	2204.83	0.0728697	0.0364349	−	0.999336i	$-0.488400\pi$
$971$	2204.83	0.0728697	0.0364349	+	0.999336i	$0.488400\pi$
$972$	0	0
$973$	19153.1	0.631059
$974$	23117.4	0.760501
$975$	0	0
$976$	−5728.00	−0.187857
$977$	−29982.0	−0.981790	−0.490895	−	0.871219i	$-0.663330\pi$
$977$	−29982.0	−0.981790	−0.490895	+	0.871219i	$0.663330\pi$
$978$	0	0
$979$	−15872.0	−0.518153
$980$	0	0
$981$	0	0
$982$	935.384	0.0303965
$983$	−35284.0	−1.14485	−0.572424	−	0.819958i	$-0.693997\pi$
$983$	−35284.0	−1.14485	−0.572424	+	0.819958i	$0.693997\pi$
$984$	0	0
$985$	0	0
$986$	−24854.5	−0.802767
$987$	0	0
$988$	22449.2	0.722880
$989$	49887.2	1.60396
$990$	0	0
$991$	−11528.0	−0.369525	−0.184762	−	0.982783i	$-0.559152\pi$
$991$	−11528.0	−0.369525	−0.184762	+	0.982783i	$0.559152\pi$
$992$	−512.000	−0.0163871
$993$	0	0
$994$	−41664.0	−1.32948
$995$	0	0
$996$	0	0
$997$	−9955.16	−0.316232	−0.158116	−	0.987421i	$-0.550542\pi$
$997$	−9955.16	−0.316232	−0.158116	+	0.987421i	$0.550542\pi$
$998$	−18408.0	−0.583863
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	450.4.a.u.1.1		2
3.2	odd	2		450.4.a.v.1.1		2
5.2	odd	4		90.4.c.c.19.2	yes	4
5.3	odd	4		90.4.c.c.19.4	yes	4
5.4	even	2		450.4.a.v.1.2		2
15.2	even	4		90.4.c.c.19.3	yes	4
15.8	even	4		90.4.c.c.19.1	✓	4
15.14	odd	2	inner	450.4.a.u.1.2		2
20.3	even	4		720.4.f.l.289.3		4
20.7	even	4		720.4.f.l.289.4		4
60.23	odd	4		720.4.f.l.289.2		4
60.47	odd	4		720.4.f.l.289.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
90.4.c.c.19.1	✓	4	15.8	even	4
90.4.c.c.19.2	yes	4	5.2	odd	4
90.4.c.c.19.3	yes	4	15.2	even	4
90.4.c.c.19.4	yes	4	5.3	odd	4
450.4.a.u.1.1		2	1.1	even	1	trivial
450.4.a.u.1.2		2	15.14	odd	2	inner
450.4.a.v.1.1		2	3.2	odd	2
450.4.a.v.1.2		2	5.4	even	2
720.4.f.l.289.1		4	60.47	odd	4
720.4.f.l.289.2		4	60.23	odd	4
720.4.f.l.289.3		4	20.3	even	4
720.4.f.l.289.4		4	20.7	even	4

Overview	Random
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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:	\( N \)	\(=\)	\( 450 = 2 \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	450.a (trivial)

\(f(q)\)	\(=\)	\(q-2.00000 q^{2} +4.00000 q^{4} -22.2711 q^{7} -8.00000 q^{8} +O(q^{10})\)	\(q-2.00000 q^{2} +4.00000 q^{4} -22.2711 q^{7} -8.00000 q^{8} +22.2711 q^{11} +66.8132 q^{13} +44.5421 q^{14} +16.0000 q^{16} -62.0000 q^{17} +84.0000 q^{19} -44.5421 q^{22} -140.000 q^{23} -133.626 q^{26} -89.0842 q^{28} -200.440 q^{29} +16.0000 q^{31} -32.0000 q^{32} +124.000 q^{34} +244.982 q^{37} -168.000 q^{38} +222.711 q^{41} -356.337 q^{43} +89.0842 q^{44} +280.000 q^{46} -100.000 q^{47} +153.000 q^{49} +267.253 q^{52} -738.000 q^{53} +178.168 q^{56} +400.879 q^{58} +645.861 q^{59} -358.000 q^{61} -32.0000 q^{62} +64.0000 q^{64} -846.300 q^{67} -248.000 q^{68} -935.384 q^{71} +445.421 q^{73} -489.963 q^{74} +336.000 q^{76} -496.000 q^{77} -936.000 q^{79} -445.421 q^{82} -1304.00 q^{83} +712.674 q^{86} -178.168 q^{88} -712.674 q^{89} -1488.00 q^{91} -560.000 q^{92} +200.000 q^{94} +757.216 q^{97} -306.000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 4 q^{2} + 8 q^{4} - 16 q^{8}+O(q^{10}) \) 2 * q - 4 * q^2 + 8 * q^4 - 16 * q^8	\( 2 q - 4 q^{2} + 8 q^{4} - 16 q^{8} + 32 q^{16} - 124 q^{17} + 168 q^{19} - 280 q^{23} + 32 q^{31} - 64 q^{32} + 248 q^{34} - 336 q^{38} + 560 q^{46} - 200 q^{47} + 306 q^{49} - 1476 q^{53} - 716 q^{61} - 64 q^{62} + 128 q^{64} - 496 q^{68} + 672 q^{76} - 992 q^{77} - 1872 q^{79} - 2608 q^{83} - 2976 q^{91} - 1120 q^{92} + 400 q^{94} - 612 q^{98}+O(q^{100}) \) 2 * q - 4 * q^2 + 8 * q^4 - 16 * q^8 + 32 * q^16 - 124 * q^17 + 168 * q^19 - 280 * q^23 + 32 * q^31 - 64 * q^32 + 248 * q^34 - 336 * q^38 + 560 * q^46 - 200 * q^47 + 306 * q^49 - 1476 * q^53 - 716 * q^61 - 64 * q^62 + 128 * q^64 - 496 * q^68 + 672 * q^76 - 992 * q^77 - 1872 * q^79 - 2608 * q^83 - 2976 * q^91 - 1120 * q^92 + 400 * q^94 - 612 * q^98

Introduction

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