LMFDB - Embedded newform 225.6.a.f.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [225,6,Mod(1,225)]

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([0, 0]))

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("225.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 225 = 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	225.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:	$36.0863594579$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 5)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	225.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+2.00000 q^{2} -28.0000 q^{4} -192.000 q^{7} -120.000 q^{8} +148.000 q^{11} -286.000 q^{13} -384.000 q^{14} +656.000 q^{16} -1678.00 q^{17} +1060.00 q^{19} +296.000 q^{22} +2976.00 q^{23} -572.000 q^{26} +5376.00 q^{28} +3410.00 q^{29} -2448.00 q^{31} +5152.00 q^{32} -3356.00 q^{34} -182.000 q^{37} +2120.00 q^{38} +9398.00 q^{41} +1244.00 q^{43} -4144.00 q^{44} +5952.00 q^{46} -12088.0 q^{47} +20057.0 q^{49} +8008.00 q^{52} +23846.0 q^{53} +23040.0 q^{56} +6820.00 q^{58} +20020.0 q^{59} +32302.0 q^{61} -4896.00 q^{62} -10688.0 q^{64} -60972.0 q^{67} +46984.0 q^{68} +32648.0 q^{71} +38774.0 q^{73} -364.000 q^{74} -29680.0 q^{76} -28416.0 q^{77} -33360.0 q^{79} +18796.0 q^{82} +16716.0 q^{83} +2488.00 q^{86} -17760.0 q^{88} -101370. q^{89} +54912.0 q^{91} -83328.0 q^{92} -24176.0 q^{94} +119038. q^{97} +40114.0 q^{98} +O(q^{100})$

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	0.353553	0.176777	−	0.984251i	$-0.443433\pi$
$2$	2.00000	0.353553	0.176777	+	0.984251i	$0.443433\pi$
$3$	0	0
$3$	0	0
$4$	−28.0000	−0.875000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−192.000	−1.48100	−0.740502	−	0.672054i	$-0.765412\pi$
$7$	−192.000	−1.48100	−0.740502	+	0.672054i	$0.765412\pi$
$8$	−120.000	−0.662913
$9$	0	0
$10$	0	0
$11$	148.000	0.368791	0.184395	−	0.982852i	$-0.440967\pi$
$11$	148.000	0.368791	0.184395	+	0.982852i	$0.440967\pi$
$12$	0	0
$13$	−286.000	−0.469362	−0.234681	−	0.972072i	$-0.575405\pi$
$13$	−286.000	−0.469362	−0.234681	+	0.972072i	$0.575405\pi$
$14$	−384.000	−0.523614
$15$	0	0
$16$	656.000	0.640625
$17$	−1678.00	−1.40822	−0.704109	−	0.710092i	$-0.748653\pi$
$17$	−1678.00	−1.40822	−0.704109	+	0.710092i	$0.748653\pi$
$18$	0	0
$19$	1060.00	0.673631	0.336815	−	0.941571i	$-0.390650\pi$
$19$	1060.00	0.673631	0.336815	+	0.941571i	$0.390650\pi$
$20$	0	0
$21$	0	0
$22$	296.000	0.130387
$23$	2976.00	1.17304	0.586521	−	0.809934i	$-0.300497\pi$
$23$	2976.00	1.17304	0.586521	+	0.809934i	$0.300497\pi$
$24$	0	0
$25$	0	0
$26$	−572.000	−0.165944
$27$	0	0
$28$	5376.00	1.29588
$29$	3410.00	0.752938	0.376469	−	0.926429i	$-0.377138\pi$
$29$	3410.00	0.752938	0.376469	+	0.926429i	$0.377138\pi$
$30$	0	0
$31$	−2448.00	−0.457517	−0.228758	−	0.973483i	$-0.573467\pi$
$31$	−2448.00	−0.457517	−0.228758	+	0.973483i	$0.573467\pi$
$32$	5152.00	0.889408
$33$	0	0
$34$	−3356.00	−0.497880
$35$	0	0
$36$	0	0
$37$	−182.000	−0.0218558	−0.0109279	−	0.999940i	$-0.503479\pi$
$37$	−182.000	−0.0218558	−0.0109279	+	0.999940i	$0.503479\pi$
$38$	2120.00	0.238164
$39$	0	0
$40$	0	0
$41$	9398.00	0.873124	0.436562	−	0.899674i	$-0.356196\pi$
$41$	9398.00	0.873124	0.436562	+	0.899674i	$0.356196\pi$
$42$	0	0
$43$	1244.00	0.102600	0.0513002	−	0.998683i	$-0.483663\pi$
$43$	1244.00	0.102600	0.0513002	+	0.998683i	$0.483663\pi$
$44$	−4144.00	−0.322692
$45$	0	0
$46$	5952.00	0.414733
$47$	−12088.0	−0.798196	−0.399098	−	0.916908i	$-0.630677\pi$
$47$	−12088.0	−0.798196	−0.399098	+	0.916908i	$0.630677\pi$
$48$	0	0
$49$	20057.0	1.19337
$50$	0	0
$51$	0	0
$52$	8008.00	0.410691
$53$	23846.0	1.16607	0.583037	−	0.812446i	$-0.301864\pi$
$53$	23846.0	1.16607	0.583037	+	0.812446i	$0.301864\pi$
$54$	0	0
$55$	0	0
$56$	23040.0	0.981776
$57$	0	0
$58$	6820.00	0.266204
$59$	20020.0	0.748745	0.374373	−	0.927278i	$-0.377858\pi$
$59$	20020.0	0.748745	0.374373	+	0.927278i	$0.377858\pi$
$60$	0	0
$61$	32302.0	1.11149	0.555744	−	0.831353i	$-0.312433\pi$
$61$	32302.0	1.11149	0.555744	+	0.831353i	$0.312433\pi$
$62$	−4896.00	−0.161757
$63$	0	0
$64$	−10688.0	−0.326172
$65$	0	0
$66$	0	0
$67$	−60972.0	−1.65937	−0.829685	−	0.558231i	$-0.811480\pi$
$67$	−60972.0	−1.65937	−0.829685	+	0.558231i	$0.811480\pi$
$68$	46984.0	1.23219
$69$	0	0
$70$	0	0
$71$	32648.0	0.768618	0.384309	−	0.923204i	$-0.374440\pi$
$71$	32648.0	0.768618	0.384309	+	0.923204i	$0.374440\pi$
$72$	0	0
$73$	38774.0	0.851596	0.425798	−	0.904818i	$-0.359993\pi$
$73$	38774.0	0.851596	0.425798	+	0.904818i	$0.359993\pi$
$74$	−364.000	−0.00772720
$75$	0	0
$76$	−29680.0	−0.589427
$77$	−28416.0	−0.546180
$78$	0	0
$79$	−33360.0	−0.601393	−0.300696	−	0.953720i	$-0.597219\pi$
$79$	−33360.0	−0.601393	−0.300696	+	0.953720i	$0.597219\pi$
$80$	0	0
$81$	0	0
$82$	18796.0	0.308696
$83$	16716.0	0.266340	0.133170	−	0.991093i	$-0.457484\pi$
$83$	16716.0	0.266340	0.133170	+	0.991093i	$0.457484\pi$
$84$	0	0
$85$	0	0
$86$	2488.00	0.0362747
$87$	0	0
$88$	−17760.0	−0.244476
$89$	−101370.	−1.35655	−0.678273	−	0.734810i	$-0.737271\pi$
$89$	−101370.	−1.35655	−0.678273	+	0.734810i	$0.737271\pi$
$90$	0	0
$91$	54912.0	0.695126
$92$	−83328.0	−1.02641
$93$	0	0
$94$	−24176.0	−0.282205
$95$	0	0
$96$	0	0
$97$	119038.	1.28457	0.642283	−	0.766468i	$-0.277987\pi$
$97$	119038.	1.28457	0.642283	+	0.766468i	$0.277987\pi$
$98$	40114.0	0.421921
$99$	0	0
$100$	0	0
$101$	89898.0	0.876893	0.438446	−	0.898757i	$-0.355529\pi$
$101$	89898.0	0.876893	0.438446	+	0.898757i	$0.355529\pi$
$102$	0	0
$103$	19504.0	0.181147	0.0905734	−	0.995890i	$-0.471130\pi$
$103$	19504.0	0.181147	0.0905734	+	0.995890i	$0.471130\pi$
$104$	34320.0	0.311146
$105$	0	0
$106$	47692.0	0.412269
$107$	158292.	1.33659	0.668297	−	0.743895i	$-0.267024\pi$
$107$	158292.	1.33659	0.668297	+	0.743895i	$0.267024\pi$
$108$	0	0
$109$	36830.0	0.296917	0.148459	−	0.988919i	$-0.452569\pi$
$109$	36830.0	0.296917	0.148459	+	0.988919i	$0.452569\pi$
$110$	0	0
$111$	0	0
$112$	−125952.	−0.948768
$113$	11186.0	0.0824098	0.0412049	−	0.999151i	$-0.486880\pi$
$113$	11186.0	0.0824098	0.0412049	+	0.999151i	$0.486880\pi$
$114$	0	0
$115$	0	0
$116$	−95480.0	−0.658821
$117$	0	0
$118$	40040.0	0.264721
$119$	322176.	2.08557
$120$	0	0
$121$	−139147.	−0.863993
$122$	64604.0	0.392970
$123$	0	0
$124$	68544.0	0.400327
$125$	0	0
$126$	0	0
$127$	−70552.0	−0.388150	−0.194075	−	0.980987i	$-0.562171\pi$
$127$	−70552.0	−0.388150	−0.194075	+	0.980987i	$0.562171\pi$
$128$	−186240.	−1.00473
$129$	0	0
$130$	0	0
$131$	−76452.0	−0.389234	−0.194617	−	0.980879i	$-0.562346\pi$
$131$	−76452.0	−0.389234	−0.194617	+	0.980879i	$0.562346\pi$
$132$	0	0
$133$	−203520.	−0.997650
$134$	−121944.	−0.586676
$135$	0	0
$136$	201360.	0.933525
$137$	−144918.	−0.659661	−0.329831	−	0.944040i	$-0.606992\pi$
$137$	−144918.	−0.659661	−0.329831	+	0.944040i	$0.606992\pi$
$138$	0	0
$139$	112220.	0.492644	0.246322	−	0.969188i	$-0.420778\pi$
$139$	112220.	0.492644	0.246322	+	0.969188i	$0.420778\pi$
$140$	0	0
$141$	0	0
$142$	65296.0	0.271748
$143$	−42328.0	−0.173096
$144$	0	0
$145$	0	0
$146$	77548.0	0.301085
$147$	0	0
$148$	5096.00	0.0191238
$149$	−403750.	−1.48986	−0.744932	−	0.667140i	$-0.767518\pi$
$149$	−403750.	−1.48986	−0.744932	+	0.667140i	$0.767518\pi$
$150$	0	0
$151$	−446648.	−1.59413	−0.797064	−	0.603895i	$-0.793615\pi$
$151$	−446648.	−1.59413	−0.797064	+	0.603895i	$0.793615\pi$
$152$	−127200.	−0.446558
$153$	0	0
$154$	−56832.0	−0.193104
$155$	0	0
$156$	0	0
$157$	262258.	0.849141	0.424570	−	0.905395i	$-0.360425\pi$
$157$	262258.	0.849141	0.424570	+	0.905395i	$0.360425\pi$
$158$	−66720.0	−0.212625
$159$	0	0
$160$	0	0
$161$	−571392.	−1.73728
$162$	0	0
$163$	154564.	0.455658	0.227829	−	0.973701i	$-0.426837\pi$
$163$	154564.	0.455658	0.227829	+	0.973701i	$0.426837\pi$
$164$	−263144.	−0.763983
$165$	0	0
$166$	33432.0	0.0941656
$167$	396672.	1.10063	0.550314	−	0.834958i	$-0.314508\pi$
$167$	396672.	1.10063	0.550314	+	0.834958i	$0.314508\pi$
$168$	0	0
$169$	−289497.	−0.779700
$170$	0	0
$171$	0	0
$172$	−34832.0	−0.0897754
$173$	−573474.	−1.45680	−0.728398	−	0.685155i	$-0.759735\pi$
$173$	−573474.	−1.45680	−0.728398	+	0.685155i	$0.759735\pi$
$174$	0	0
$175$	0	0
$176$	97088.0	0.236257
$177$	0	0
$178$	−202740.	−0.479611
$179$	594460.	1.38672	0.693362	−	0.720589i	$-0.256129\pi$
$179$	594460.	1.38672	0.693362	+	0.720589i	$0.256129\pi$
$180$	0	0
$181$	−107098.	−0.242988	−0.121494	−	0.992592i	$-0.538769\pi$
$181$	−107098.	−0.242988	−0.121494	+	0.992592i	$0.538769\pi$
$182$	109824.	0.245764
$183$	0	0
$184$	−357120.	−0.777624
$185$	0	0
$186$	0	0
$187$	−248344.	−0.519337
$188$	338464.	0.698422
$189$	0	0
$190$	0	0
$191$	−469552.	−0.931323	−0.465661	−	0.884963i	$-0.654184\pi$
$191$	−469552.	−0.931323	−0.465661	+	0.884963i	$0.654184\pi$
$192$	0	0
$193$	−52706.0	−0.101851	−0.0509257	−	0.998702i	$-0.516217\pi$
$193$	−52706.0	−0.101851	−0.0509257	+	0.998702i	$0.516217\pi$
$194$	238076.	0.454163
$195$	0	0
$196$	−561596.	−1.04420
$197$	455862.	0.836889	0.418444	−	0.908242i	$-0.362575\pi$
$197$	455862.	0.836889	0.418444	+	0.908242i	$0.362575\pi$
$198$	0	0
$199$	865000.	1.54840	0.774200	−	0.632940i	$-0.218152\pi$
$199$	865000.	1.54840	0.774200	+	0.632940i	$0.218152\pi$
$200$	0	0
$201$	0	0
$202$	179796.	0.310028
$203$	−654720.	−1.11510
$204$	0	0
$205$	0	0
$206$	39008.0	0.0640451
$207$	0	0
$208$	−187616.	−0.300685
$209$	156880.	0.248429
$210$	0	0
$211$	1.10565e6	1.70967	0.854835	−	0.518900i	$-0.173658\pi$
$211$	1.10565e6	1.70967	0.854835	+	0.518900i	$0.173658\pi$
$212$	−667688.	−1.02031
$213$	0	0
$214$	316584.	0.472557
$215$	0	0
$216$	0	0
$217$	470016.	0.677584
$218$	73660.0	0.104976
$219$	0	0
$220$	0	0
$221$	479908.	0.660963
$222$	0	0
$223$	−1.12158e6	−1.51031	−0.755156	−	0.655545i	$-0.772439\pi$
$223$	−1.12158e6	−1.51031	−0.755156	+	0.655545i	$0.772439\pi$
$224$	−989184.	−1.31722
$225$	0	0
$226$	22372.0	0.0291363
$227$	−23348.0	−0.0300736	−0.0150368	−	0.999887i	$-0.504787\pi$
$227$	−23348.0	−0.0300736	−0.0150368	+	0.999887i	$0.504787\pi$
$228$	0	0
$229$	−596010.	−0.751043	−0.375522	−	0.926814i	$-0.622536\pi$
$229$	−596010.	−0.751043	−0.375522	+	0.926814i	$0.622536\pi$
$230$	0	0
$231$	0	0
$232$	−409200.	−0.499132
$233$	−485334.	−0.585667	−0.292834	−	0.956163i	$-0.594598\pi$
$233$	−485334.	−0.585667	−0.292834	+	0.956163i	$0.594598\pi$
$234$	0	0
$235$	0	0
$236$	−560560.	−0.655152
$237$	0	0
$238$	644352.	0.737362
$239$	48880.0	0.0553524	0.0276762	−	0.999617i	$-0.491189\pi$
$239$	48880.0	0.0553524	0.0276762	+	0.999617i	$0.491189\pi$
$240$	0	0
$241$	−110798.	−0.122882	−0.0614411	−	0.998111i	$-0.519570\pi$
$241$	−110798.	−0.122882	−0.0614411	+	0.998111i	$0.519570\pi$
$242$	−278294.	−0.305468
$243$	0	0
$244$	−904456.	−0.972552
$245$	0	0
$246$	0	0
$247$	−303160.	−0.316176
$248$	293760.	0.303294
$249$	0	0
$250$	0	0
$251$	1.64375e6	1.64684	0.823419	−	0.567434i	$-0.192064\pi$
$251$	1.64375e6	1.64684	0.823419	+	0.567434i	$0.192064\pi$
$252$	0	0
$253$	440448.	0.432607
$254$	−141104.	−0.137232
$255$	0	0
$256$	−30464.0	−0.0290527
$257$	1.30624e6	1.23365	0.616823	−	0.787102i	$-0.288419\pi$
$257$	1.30624e6	1.23365	0.616823	+	0.787102i	$0.288419\pi$
$258$	0	0
$259$	34944.0	0.0323685
$260$	0	0
$261$	0	0
$262$	−152904.	−0.137615
$263$	2.12834e6	1.89736	0.948682	−	0.316231i	$-0.102417\pi$
$263$	2.12834e6	1.89736	0.948682	+	0.316231i	$0.102417\pi$
$264$	0	0
$265$	0	0
$266$	−407040.	−0.352722
$267$	0	0
$268$	1.70722e6	1.45195
$269$	1.44109e6	1.21426	0.607128	−	0.794604i	$-0.292321\pi$
$269$	1.44109e6	1.21426	0.607128	+	0.794604i	$0.292321\pi$
$270$	0	0
$271$	−93248.0	−0.0771288	−0.0385644	−	0.999256i	$-0.512278\pi$
$271$	−93248.0	−0.0771288	−0.0385644	+	0.999256i	$0.512278\pi$
$272$	−1.10077e6	−0.902139
$273$	0	0
$274$	−289836.	−0.233225
$275$	0	0
$276$	0	0
$277$	110298.	0.0863711	0.0431855	−	0.999067i	$-0.486249\pi$
$277$	110298.	0.0863711	0.0431855	+	0.999067i	$0.486249\pi$
$278$	224440.	0.174176
$279$	0	0
$280$	0	0
$281$	192198.	0.145205	0.0726027	−	0.997361i	$-0.476869\pi$
$281$	192198.	0.145205	0.0726027	+	0.997361i	$0.476869\pi$
$282$	0	0
$283$	331884.	0.246332	0.123166	−	0.992386i	$-0.460695\pi$
$283$	331884.	0.246332	0.123166	+	0.992386i	$0.460695\pi$
$284$	−914144.	−0.672541
$285$	0	0
$286$	−84656.0	−0.0611988
$287$	−1.80442e6	−1.29310
$288$	0	0
$289$	1.39583e6	0.983076
$290$	0	0
$291$	0	0
$292$	−1.08567e6	−0.745146
$293$	2.19481e6	1.49358	0.746788	−	0.665063i	$-0.231595\pi$
$293$	2.19481e6	1.49358	0.746788	+	0.665063i	$0.231595\pi$
$294$	0	0
$295$	0	0
$296$	21840.0	0.0144885
$297$	0	0
$298$	−807500.	−0.526747
$299$	−851136.	−0.550581
$300$	0	0
$301$	−238848.	−0.151952
$302$	−893296.	−0.563609
$303$	0	0
$304$	695360.	0.431545
$305$	0	0
$306$	0	0
$307$	2.37751e6	1.43971	0.719857	−	0.694123i	$-0.244207\pi$
$307$	2.37751e6	1.43971	0.719857	+	0.694123i	$0.244207\pi$
$308$	795648.	0.477908
$309$	0	0
$310$	0	0
$311$	2.37305e6	1.39125	0.695626	−	0.718405i	$-0.255127\pi$
$311$	2.37305e6	1.39125	0.695626	+	0.718405i	$0.255127\pi$
$312$	0	0
$313$	1.42941e6	0.824702	0.412351	−	0.911025i	$-0.364708\pi$
$313$	1.42941e6	0.824702	0.412351	+	0.911025i	$0.364708\pi$
$314$	524516.	0.300217
$315$	0	0
$316$	934080.	0.526219
$317$	2.12462e6	1.18750	0.593750	−	0.804650i	$-0.297647\pi$
$317$	2.12462e6	1.18750	0.593750	+	0.804650i	$0.297647\pi$
$318$	0	0
$319$	504680.	0.277677
$320$	0	0
$321$	0	0
$322$	−1.14278e6	−0.614221
$323$	−1.77868e6	−0.948618
$324$	0	0
$325$	0	0
$326$	309128.	0.161100
$327$	0	0
$328$	−1.12776e6	−0.578805
$329$	2.32090e6	1.18213
$330$	0	0
$331$	3.09985e6	1.55515	0.777573	−	0.628793i	$-0.216451\pi$
$331$	3.09985e6	1.55515	0.777573	+	0.628793i	$0.216451\pi$
$332$	−468048.	−0.233048
$333$	0	0
$334$	793344.	0.389131
$335$	0	0
$336$	0	0
$337$	−2.40008e6	−1.15120	−0.575601	−	0.817731i	$-0.695232\pi$
$337$	−2.40008e6	−1.15120	−0.575601	+	0.817731i	$0.695232\pi$
$338$	−578994.	−0.275665
$339$	0	0
$340$	0	0
$341$	−362304.	−0.168728
$342$	0	0
$343$	−624000.	−0.286384
$344$	−149280.	−0.0680151
$345$	0	0
$346$	−1.14695e6	−0.515055
$347$	1.77741e6	0.792436	0.396218	−	0.918156i	$-0.370322\pi$
$347$	1.77741e6	0.792436	0.396218	+	0.918156i	$0.370322\pi$
$348$	0	0
$349$	−2.14805e6	−0.944019	−0.472010	−	0.881593i	$-0.656471\pi$
$349$	−2.14805e6	−0.944019	−0.472010	+	0.881593i	$0.656471\pi$
$350$	0	0
$351$	0	0
$352$	762496.	0.328005
$353$	−661854.	−0.282700	−0.141350	−	0.989960i	$-0.545144\pi$
$353$	−661854.	−0.282700	−0.141350	+	0.989960i	$0.545144\pi$
$354$	0	0
$355$	0	0
$356$	2.83836e6	1.18698
$357$	0	0
$358$	1.18892e6	0.490281
$359$	259320.	0.106194	0.0530970	−	0.998589i	$-0.483091\pi$
$359$	259320.	0.106194	0.0530970	+	0.998589i	$0.483091\pi$
$360$	0	0
$361$	−1.35250e6	−0.546222
$362$	−214196.	−0.0859093
$363$	0	0
$364$	−1.53754e6	−0.608236
$365$	0	0
$366$	0	0
$367$	1.49993e6	0.581307	0.290653	−	0.956828i	$-0.406127\pi$
$367$	1.49993e6	0.581307	0.290653	+	0.956828i	$0.406127\pi$
$368$	1.95226e6	0.751480
$369$	0	0
$370$	0	0
$371$	−4.57843e6	−1.72696
$372$	0	0
$373$	2.23807e6	0.832918	0.416459	−	0.909154i	$-0.363271\pi$
$373$	2.23807e6	0.832918	0.416459	+	0.909154i	$0.363271\pi$
$374$	−496688.	−0.183614
$375$	0	0
$376$	1.45056e6	0.529135
$377$	−975260.	−0.353400
$378$	0	0
$379$	3.15934e6	1.12979	0.564896	−	0.825162i	$-0.308916\pi$
$379$	3.15934e6	1.12979	0.564896	+	0.825162i	$0.308916\pi$
$380$	0	0
$381$	0	0
$382$	−939104.	−0.329272
$383$	342216.	0.119207	0.0596037	−	0.998222i	$-0.481016\pi$
$383$	342216.	0.119207	0.0596037	+	0.998222i	$0.481016\pi$
$384$	0	0
$385$	0	0
$386$	−105412.	−0.0360099
$387$	0	0
$388$	−3.33306e6	−1.12399
$389$	−88470.0	−0.0296430	−0.0148215	−	0.999890i	$-0.504718\pi$
$389$	−88470.0	−0.0296430	−0.0148215	+	0.999890i	$0.504718\pi$
$390$	0	0
$391$	−4.99373e6	−1.65190
$392$	−2.40684e6	−0.791101
$393$	0	0
$394$	911724.	0.295885
$395$	0	0
$396$	0	0
$397$	5.45674e6	1.73763	0.868814	−	0.495138i	$-0.164883\pi$
$397$	5.45674e6	1.73763	0.868814	+	0.495138i	$0.164883\pi$
$398$	1.73000e6	0.547442
$399$	0	0
$400$	0	0
$401$	−4.04680e6	−1.25676	−0.628378	−	0.777908i	$-0.716281\pi$
$401$	−4.04680e6	−1.25676	−0.628378	+	0.777908i	$0.716281\pi$
$402$	0	0
$403$	700128.	0.214741
$404$	−2.51714e6	−0.767281
$405$	0	0
$406$	−1.30944e6	−0.394249
$407$	−26936.0	−0.00806022
$408$	0	0
$409$	−2.71207e6	−0.801664	−0.400832	−	0.916151i	$-0.631279\pi$
$409$	−2.71207e6	−0.801664	−0.400832	+	0.916151i	$0.631279\pi$
$410$	0	0
$411$	0	0
$412$	−546112.	−0.158503
$413$	−3.84384e6	−1.10889
$414$	0	0
$415$	0	0
$416$	−1.47347e6	−0.417454
$417$	0	0
$418$	313760.	0.0878328
$419$	−3.71746e6	−1.03445	−0.517227	−	0.855848i	$-0.673036\pi$
$419$	−3.71746e6	−1.03445	−0.517227	+	0.855848i	$0.673036\pi$
$420$	0	0
$421$	3.55250e6	0.976853	0.488426	−	0.872605i	$-0.337571\pi$
$421$	3.55250e6	0.976853	0.488426	+	0.872605i	$0.337571\pi$
$422$	2.21130e6	0.604460
$423$	0	0
$424$	−2.86152e6	−0.773005
$425$	0	0
$426$	0	0
$427$	−6.20198e6	−1.64612
$428$	−4.43218e6	−1.16952
$429$	0	0
$430$	0	0
$431$	4.06205e6	1.05330	0.526650	−	0.850082i	$-0.323448\pi$
$431$	4.06205e6	1.05330	0.526650	+	0.850082i	$0.323448\pi$
$432$	0	0
$433$	−7.26287e6	−1.86161	−0.930804	−	0.365518i	$-0.880892\pi$
$433$	−7.26287e6	−1.86161	−0.930804	+	0.365518i	$0.880892\pi$
$434$	940032.	0.239562
$435$	0	0
$436$	−1.03124e6	−0.259803
$437$	3.15456e6	0.790197
$438$	0	0
$439$	−5.41028e6	−1.33986	−0.669928	−	0.742426i	$-0.733675\pi$
$439$	−5.41028e6	−1.33986	−0.669928	+	0.742426i	$0.733675\pi$
$440$	0	0
$441$	0	0
$442$	959816.	0.233686
$443$	−6.51524e6	−1.57733	−0.788663	−	0.614826i	$-0.789226\pi$
$443$	−6.51524e6	−1.57733	−0.788663	+	0.614826i	$0.789226\pi$
$444$	0	0
$445$	0	0
$446$	−2.24315e6	−0.533976
$447$	0	0
$448$	2.05210e6	0.483062
$449$	509950.	0.119375	0.0596873	−	0.998217i	$-0.480990\pi$
$449$	509950.	0.119375	0.0596873	+	0.998217i	$0.480990\pi$
$450$	0	0
$451$	1.39090e6	0.322000
$452$	−313208.	−0.0721085
$453$	0	0
$454$	−46696.0	−0.0106326
$455$	0	0
$456$	0	0
$457$	−1.22084e6	−0.273444	−0.136722	−	0.990609i	$-0.543657\pi$
$457$	−1.22084e6	−0.273444	−0.136722	+	0.990609i	$0.543657\pi$
$458$	−1.19202e6	−0.265534
$459$	0	0
$460$	0	0
$461$	4.07210e6	0.892413	0.446207	−	0.894930i	$-0.352775\pi$
$461$	4.07210e6	0.892413	0.446207	+	0.894930i	$0.352775\pi$
$462$	0	0
$463$	−2.02294e6	−0.438561	−0.219280	−	0.975662i	$-0.570371\pi$
$463$	−2.02294e6	−0.438561	−0.219280	+	0.975662i	$0.570371\pi$
$464$	2.23696e6	0.482351
$465$	0	0
$466$	−970668.	−0.207065
$467$	3.25097e6	0.689797	0.344898	−	0.938640i	$-0.387913\pi$
$467$	3.25097e6	0.689797	0.344898	+	0.938640i	$0.387913\pi$
$468$	0	0
$469$	1.17066e7	2.45753
$470$	0	0
$471$	0	0
$472$	−2.40240e6	−0.496353
$473$	184112.	0.0378381
$474$	0	0
$475$	0	0
$476$	−9.02093e6	−1.82488
$477$	0	0
$478$	97760.0	0.0195700
$479$	3.27936e6	0.653056	0.326528	−	0.945188i	$-0.394121\pi$
$479$	3.27936e6	0.653056	0.326528	+	0.945188i	$0.394121\pi$
$480$	0	0
$481$	52052.0	0.0102583
$482$	−221596.	−0.0434455
$483$	0	0
$484$	3.89612e6	0.755994
$485$	0	0
$486$	0	0
$487$	8.53197e6	1.63015	0.815074	−	0.579357i	$-0.196696\pi$
$487$	8.53197e6	1.63015	0.815074	+	0.579357i	$0.196696\pi$
$488$	−3.87624e6	−0.736819
$489$	0	0
$490$	0	0
$491$	−1.51265e6	−0.283162	−0.141581	−	0.989927i	$-0.545219\pi$
$491$	−1.51265e6	−0.283162	−0.141581	+	0.989927i	$0.545219\pi$
$492$	0	0
$493$	−5.72198e6	−1.06030
$494$	−606320.	−0.111785
$495$	0	0
$496$	−1.60589e6	−0.293097
$497$	−6.26842e6	−1.13833
$498$	0	0
$499$	−6.49190e6	−1.16713	−0.583567	−	0.812065i	$-0.698343\pi$
$499$	−6.49190e6	−1.16713	−0.583567	+	0.812065i	$0.698343\pi$
$500$	0	0
$501$	0	0
$502$	3.28750e6	0.582245
$503$	8.61770e6	1.51870	0.759349	−	0.650684i	$-0.225518\pi$
$503$	8.61770e6	1.51870	0.759349	+	0.650684i	$0.225518\pi$
$504$	0	0
$505$	0	0
$506$	880896.	0.152950
$507$	0	0
$508$	1.97546e6	0.339632
$509$	−2.67323e6	−0.457343	−0.228671	−	0.973504i	$-0.573438\pi$
$509$	−2.67323e6	−0.457343	−0.228671	+	0.973504i	$0.573438\pi$
$510$	0	0
$511$	−7.44461e6	−1.26122
$512$	5.89875e6	0.994455
$513$	0	0
$514$	2.61248e6	0.436160
$515$	0	0
$516$	0	0
$517$	−1.78902e6	−0.294367
$518$	69888.0	0.0114440
$519$	0	0
$520$	0	0
$521$	−6.18500e6	−0.998264	−0.499132	−	0.866526i	$-0.666348\pi$
$521$	−6.18500e6	−0.998264	−0.499132	+	0.866526i	$0.666348\pi$
$522$	0	0
$523$	6.89452e6	1.10217	0.551087	−	0.834448i	$-0.314213\pi$
$523$	6.89452e6	1.10217	0.551087	+	0.834448i	$0.314213\pi$
$524$	2.14066e6	0.340580
$525$	0	0
$526$	4.25667e6	0.670820
$527$	4.10774e6	0.644283
$528$	0	0
$529$	2.42023e6	0.376026
$530$	0	0
$531$	0	0
$532$	5.69856e6	0.872943
$533$	−2.68783e6	−0.409811
$534$	0	0
$535$	0	0
$536$	7.31664e6	1.10002
$537$	0	0
$538$	2.88218e6	0.429304
$539$	2.96844e6	0.440104
$540$	0	0
$541$	155502.	0.0228425	0.0114212	−	0.999935i	$-0.496364\pi$
$541$	155502.	0.0228425	0.0114212	+	0.999935i	$0.496364\pi$
$542$	−186496.	−0.0272691
$543$	0	0
$544$	−8.64506e6	−1.25248
$545$	0	0
$546$	0	0
$547$	−1.26544e7	−1.80831	−0.904157	−	0.427201i	$-0.859500\pi$
$547$	−1.26544e7	−1.80831	−0.904157	+	0.427201i	$0.859500\pi$
$548$	4.05770e6	0.577204
$549$	0	0
$550$	0	0
$551$	3.61460e6	0.507202
$552$	0	0
$553$	6.40512e6	0.890665
$554$	220596.	0.0305368
$555$	0	0
$556$	−3.14216e6	−0.431064
$557$	−7.07786e6	−0.966638	−0.483319	−	0.875444i	$-0.660569\pi$
$557$	−7.07786e6	−0.966638	−0.483319	+	0.875444i	$0.660569\pi$
$558$	0	0
$559$	−355784.	−0.0481567
$560$	0	0
$561$	0	0
$562$	384396.	0.0513379
$563$	846636.	0.112571	0.0562854	−	0.998415i	$-0.482074\pi$
$563$	846636.	0.112571	0.0562854	+	0.998415i	$0.482074\pi$
$564$	0	0
$565$	0	0
$566$	663768.	0.0870914
$567$	0	0
$568$	−3.91776e6	−0.509527
$569$	−4.96041e6	−0.642299	−0.321149	−	0.947029i	$-0.604069\pi$
$569$	−4.96041e6	−0.642299	−0.321149	+	0.947029i	$0.604069\pi$
$570$	0	0
$571$	8.96505e6	1.15070	0.575351	−	0.817907i	$-0.304866\pi$
$571$	8.96505e6	1.15070	0.575351	+	0.817907i	$0.304866\pi$
$572$	1.18518e6	0.151459
$573$	0	0
$574$	−3.60883e6	−0.457180
$575$	0	0
$576$	0	0
$577$	2.86080e6	0.357724	0.178862	−	0.983874i	$-0.442758\pi$
$577$	2.86080e6	0.357724	0.178862	+	0.983874i	$0.442758\pi$
$578$	2.79165e6	0.347570
$579$	0	0
$580$	0	0
$581$	−3.20947e6	−0.394451
$582$	0	0
$583$	3.52921e6	0.430037
$584$	−4.65288e6	−0.564534
$585$	0	0
$586$	4.38961e6	0.528059
$587$	−6.74027e6	−0.807387	−0.403694	−	0.914894i	$-0.632274\pi$
$587$	−6.74027e6	−0.807387	−0.403694	+	0.914894i	$0.632274\pi$
$588$	0	0
$589$	−2.59488e6	−0.308197
$590$	0	0
$591$	0	0
$592$	−119392.	−0.0140014
$593$	−1.78609e6	−0.208578	−0.104289	−	0.994547i	$-0.533257\pi$
$593$	−1.78609e6	−0.208578	−0.104289	+	0.994547i	$0.533257\pi$
$594$	0	0
$595$	0	0
$596$	1.13050e7	1.30363
$597$	0	0
$598$	−1.70227e6	−0.194660
$599$	−4.94620e6	−0.563254	−0.281627	−	0.959524i	$-0.590874\pi$
$599$	−4.94620e6	−0.563254	−0.281627	+	0.959524i	$0.590874\pi$
$600$	0	0
$601$	−4.58100e6	−0.517337	−0.258669	−	0.965966i	$-0.583284\pi$
$601$	−4.58100e6	−0.517337	−0.258669	+	0.965966i	$0.583284\pi$
$602$	−477696.	−0.0537230
$603$	0	0
$604$	1.25061e7	1.39486
$605$	0	0
$606$	0	0
$607$	−7.07999e6	−0.779940	−0.389970	−	0.920828i	$-0.627515\pi$
$607$	−7.07999e6	−0.779940	−0.389970	+	0.920828i	$0.627515\pi$
$608$	5.46112e6	0.599132
$609$	0	0
$610$	0	0
$611$	3.45717e6	0.374643
$612$	0	0
$613$	−5.09609e6	−0.547754	−0.273877	−	0.961765i	$-0.588306\pi$
$613$	−5.09609e6	−0.547754	−0.273877	+	0.961765i	$0.588306\pi$
$614$	4.75502e6	0.509016
$615$	0	0
$616$	3.40992e6	0.362070
$617$	−1.30003e7	−1.37480	−0.687400	−	0.726279i	$-0.741248\pi$
$617$	−1.30003e7	−1.37480	−0.687400	+	0.726279i	$0.741248\pi$
$618$	0	0
$619$	4.84406e6	0.508139	0.254070	−	0.967186i	$-0.418231\pi$
$619$	4.84406e6	0.508139	0.254070	+	0.967186i	$0.418231\pi$
$620$	0	0
$621$	0	0
$622$	4.74610e6	0.491882
$623$	1.94630e7	2.00905
$624$	0	0
$625$	0	0
$626$	2.85883e6	0.291576
$627$	0	0
$628$	−7.34322e6	−0.742998
$629$	305396.	0.0307777
$630$	0	0
$631$	6.22775e6	0.622670	0.311335	−	0.950300i	$-0.399224\pi$
$631$	6.22775e6	0.622670	0.311335	+	0.950300i	$0.399224\pi$
$632$	4.00320e6	0.398671
$633$	0	0
$634$	4.24924e6	0.419845
$635$	0	0
$636$	0	0
$637$	−5.73630e6	−0.560123
$638$	1.00936e6	0.0981735
$639$	0	0
$640$	0	0
$641$	−1.53280e6	−0.147347	−0.0736734	−	0.997282i	$-0.523472\pi$
$641$	−1.53280e6	−0.147347	−0.0736734	+	0.997282i	$0.523472\pi$
$642$	0	0
$643$	1.74382e7	1.66332	0.831659	−	0.555287i	$-0.187391\pi$
$643$	1.74382e7	1.66332	0.831659	+	0.555287i	$0.187391\pi$
$644$	1.59990e7	1.52012
$645$	0	0
$646$	−3.55736e6	−0.335387
$647$	−4.25469e6	−0.399583	−0.199792	−	0.979838i	$-0.564026\pi$
$647$	−4.25469e6	−0.399583	−0.199792	+	0.979838i	$0.564026\pi$
$648$	0	0
$649$	2.96296e6	0.276130
$650$	0	0
$651$	0	0
$652$	−4.32779e6	−0.398701
$653$	3.01085e6	0.276316	0.138158	−	0.990410i	$-0.455882\pi$
$653$	3.01085e6	0.276316	0.138158	+	0.990410i	$0.455882\pi$
$654$	0	0
$655$	0	0
$656$	6.16509e6	0.559345
$657$	0	0
$658$	4.64179e6	0.417947
$659$	8.11462e6	0.727871	0.363936	−	0.931424i	$-0.381433\pi$
$659$	8.11462e6	0.727871	0.363936	+	0.931424i	$0.381433\pi$
$660$	0	0
$661$	2.47370e6	0.220213	0.110107	−	0.993920i	$-0.464881\pi$
$661$	2.47370e6	0.220213	0.110107	+	0.993920i	$0.464881\pi$
$662$	6.19970e6	0.549827
$663$	0	0
$664$	−2.00592e6	−0.176560
$665$	0	0
$666$	0	0
$667$	1.01482e7	0.883228
$668$	−1.11068e7	−0.963049
$669$	0	0
$670$	0	0
$671$	4.78070e6	0.409907
$672$	0	0
$673$	−5.77063e6	−0.491117	−0.245559	−	0.969382i	$-0.578971\pi$
$673$	−5.77063e6	−0.491117	−0.245559	+	0.969382i	$0.578971\pi$
$674$	−4.80016e6	−0.407011
$675$	0	0
$676$	8.10592e6	0.682237
$677$	1.67197e7	1.40203	0.701014	−	0.713147i	$-0.252731\pi$
$677$	1.67197e7	1.40203	0.701014	+	0.713147i	$0.252731\pi$
$678$	0	0
$679$	−2.28553e7	−1.90245
$680$	0	0
$681$	0	0
$682$	−724608.	−0.0596544
$683$	7.14532e6	0.586097	0.293049	−	0.956098i	$-0.405330\pi$
$683$	7.14532e6	0.586097	0.293049	+	0.956098i	$0.405330\pi$
$684$	0	0
$685$	0	0
$686$	−1.24800e6	−0.101252
$687$	0	0
$688$	816064.	0.0657284
$689$	−6.81996e6	−0.547310
$690$	0	0
$691$	−8.78395e6	−0.699833	−0.349917	−	0.936781i	$-0.613790\pi$
$691$	−8.78395e6	−0.699833	−0.349917	+	0.936781i	$0.613790\pi$
$692$	1.60573e7	1.27470
$693$	0	0
$694$	3.55482e6	0.280169
$695$	0	0
$696$	0	0
$697$	−1.57698e7	−1.22955
$698$	−4.29610e6	−0.333761
$699$	0	0
$700$	0	0
$701$	1.60141e7	1.23086	0.615428	−	0.788193i	$-0.288983\pi$
$701$	1.60141e7	1.23086	0.615428	+	0.788193i	$0.288983\pi$
$702$	0	0
$703$	−192920.	−0.0147228
$704$	−1.58182e6	−0.120289
$705$	0	0
$706$	−1.32371e6	−0.0999495
$707$	−1.72604e7	−1.29868
$708$	0	0
$709$	−1.91354e7	−1.42962	−0.714811	−	0.699318i	$-0.753487\pi$
$709$	−1.91354e7	−1.42962	−0.714811	+	0.699318i	$0.753487\pi$
$710$	0	0
$711$	0	0
$712$	1.21644e7	0.899271
$713$	−7.28525e6	−0.536686
$714$	0	0
$715$	0	0
$716$	−1.66449e7	−1.21338
$717$	0	0
$718$	518640.	0.0375452
$719$	−1.02934e7	−0.742566	−0.371283	−	0.928520i	$-0.621082\pi$
$719$	−1.02934e7	−0.742566	−0.371283	+	0.928520i	$0.621082\pi$
$720$	0	0
$721$	−3.74477e6	−0.268279
$722$	−2.70500e6	−0.193119
$723$	0	0
$724$	2.99874e6	0.212615
$725$	0	0
$726$	0	0
$727$	1.93264e7	1.35618	0.678088	−	0.734981i	$-0.262809\pi$
$727$	1.93264e7	1.35618	0.678088	+	0.734981i	$0.262809\pi$
$728$	−6.58944e6	−0.460808
$729$	0	0
$730$	0	0
$731$	−2.08743e6	−0.144484
$732$	0	0
$733$	−5.26197e6	−0.361733	−0.180866	−	0.983508i	$-0.557890\pi$
$733$	−5.26197e6	−0.361733	−0.180866	+	0.983508i	$0.557890\pi$
$734$	2.99986e6	0.205523
$735$	0	0
$736$	1.53324e7	1.04331
$737$	−9.02386e6	−0.611961
$738$	0	0
$739$	2.82944e7	1.90585	0.952927	−	0.303199i	$-0.0980548\pi$
$739$	2.82944e7	1.90585	0.952927	+	0.303199i	$0.0980548\pi$
$740$	0	0
$741$	0	0
$742$	−9.15686e6	−0.610572
$743$	2.09863e7	1.39464	0.697321	−	0.716759i	$-0.254375\pi$
$743$	2.09863e7	1.39464	0.697321	+	0.716759i	$0.254375\pi$
$744$	0	0
$745$	0	0
$746$	4.47615e6	0.294481
$747$	0	0
$748$	6.95363e6	0.454420
$749$	−3.03921e7	−1.97950
$750$	0	0
$751$	−1.89668e7	−1.22714	−0.613572	−	0.789639i	$-0.710268\pi$
$751$	−1.89668e7	−1.22714	−0.613572	+	0.789639i	$0.710268\pi$
$752$	−7.92973e6	−0.511345
$753$	0	0
$754$	−1.95052e6	−0.124946
$755$	0	0
$756$	0	0
$757$	1.08257e7	0.686617	0.343309	−	0.939223i	$-0.388452\pi$
$757$	1.08257e7	0.686617	0.343309	+	0.939223i	$0.388452\pi$
$758$	6.31868e6	0.399442
$759$	0	0
$760$	0	0
$761$	−1.90534e7	−1.19264	−0.596322	−	0.802745i	$-0.703372\pi$
$761$	−1.90534e7	−1.19264	−0.596322	+	0.802745i	$0.703372\pi$
$762$	0	0
$763$	−7.07136e6	−0.439736
$764$	1.31475e7	0.814908
$765$	0	0
$766$	684432.	0.0421462
$767$	−5.72572e6	−0.351432
$768$	0	0
$769$	−1.57826e7	−0.962415	−0.481208	−	0.876607i	$-0.659802\pi$
$769$	−1.57826e7	−0.962415	−0.481208	+	0.876607i	$0.659802\pi$
$770$	0	0
$771$	0	0
$772$	1.47577e6	0.0891199
$773$	−2.44049e7	−1.46902	−0.734510	−	0.678598i	$-0.762588\pi$
$773$	−2.44049e7	−1.46902	−0.734510	+	0.678598i	$0.762588\pi$
$774$	0	0
$775$	0	0
$776$	−1.42846e7	−0.851555
$777$	0	0
$778$	−176940.	−0.0104804
$779$	9.96188e6	0.588163
$780$	0	0
$781$	4.83190e6	0.283459
$782$	−9.98746e6	−0.584034
$783$	0	0
$784$	1.31574e7	0.764504
$785$	0	0
$786$	0	0
$787$	−3.37607e7	−1.94301	−0.971505	−	0.237019i	$-0.923830\pi$
$787$	−3.37607e7	−1.94301	−0.971505	+	0.237019i	$0.923830\pi$
$788$	−1.27641e7	−0.732278
$789$	0	0
$790$	0	0
$791$	−2.14771e6	−0.122049
$792$	0	0
$793$	−9.23837e6	−0.521690
$794$	1.09135e7	0.614344
$795$	0	0
$796$	−2.42200e7	−1.35485
$797$	2.19885e7	1.22617	0.613083	−	0.790019i	$-0.289929\pi$
$797$	2.19885e7	1.22617	0.613083	+	0.790019i	$0.289929\pi$
$798$	0	0
$799$	2.02837e7	1.12403
$800$	0	0
$801$	0	0
$802$	−8.09360e6	−0.444330
$803$	5.73855e6	0.314061
$804$	0	0
$805$	0	0
$806$	1.40026e6	0.0759224
$807$	0	0
$808$	−1.07878e7	−0.581303
$809$	2.93597e7	1.57717	0.788587	−	0.614923i	$-0.210813\pi$
$809$	2.93597e7	1.57717	0.788587	+	0.614923i	$0.210813\pi$
$810$	0	0
$811$	3.17703e7	1.69617	0.848083	−	0.529863i	$-0.177757\pi$
$811$	3.17703e7	1.69617	0.848083	+	0.529863i	$0.177757\pi$
$812$	1.83322e7	0.975716
$813$	0	0
$814$	−53872.0	−0.00284972
$815$	0	0
$816$	0	0
$817$	1.31864e6	0.0691148
$818$	−5.42414e6	−0.283431
$819$	0	0
$820$	0	0
$821$	2.71430e6	0.140540	0.0702699	−	0.997528i	$-0.477614\pi$
$821$	2.71430e6	0.140540	0.0702699	+	0.997528i	$0.477614\pi$
$822$	0	0
$823$	1.25866e7	0.647753	0.323877	−	0.946099i	$-0.395014\pi$
$823$	1.25866e7	0.647753	0.323877	+	0.946099i	$0.395014\pi$
$824$	−2.34048e6	−0.120084
$825$	0	0
$826$	−7.68768e6	−0.392053
$827$	−8.72355e6	−0.443537	−0.221768	−	0.975099i	$-0.571183\pi$
$827$	−8.72355e6	−0.443537	−0.221768	+	0.975099i	$0.571183\pi$
$828$	0	0
$829$	−1.06178e7	−0.536597	−0.268299	−	0.963336i	$-0.586461\pi$
$829$	−1.06178e7	−0.536597	−0.268299	+	0.963336i	$0.586461\pi$
$830$	0	0
$831$	0	0
$832$	3.05677e6	0.153093
$833$	−3.36556e7	−1.68053
$834$	0	0
$835$	0	0
$836$	−4.39264e6	−0.217375
$837$	0	0
$838$	−7.43492e6	−0.365735
$839$	−1.67765e7	−0.822805	−0.411403	−	0.911454i	$-0.634961\pi$
$839$	−1.67765e7	−0.822805	−0.411403	+	0.911454i	$0.634961\pi$
$840$	0	0
$841$	−8.88305e6	−0.433084
$842$	7.10500e6	0.345370
$843$	0	0
$844$	−3.09583e7	−1.49596
$845$	0	0
$846$	0	0
$847$	2.67162e7	1.27958
$848$	1.56430e7	0.747016
$849$	0	0
$850$	0	0
$851$	−541632.	−0.0256378
$852$	0	0
$853$	2.20186e7	1.03613	0.518067	−	0.855340i	$-0.326652\pi$
$853$	2.20186e7	1.03613	0.518067	+	0.855340i	$0.326652\pi$
$854$	−1.24040e7	−0.581991
$855$	0	0
$856$	−1.89950e7	−0.886045
$857$	3.16676e7	1.47287	0.736434	−	0.676510i	$-0.236508\pi$
$857$	3.16676e7	1.47287	0.736434	+	0.676510i	$0.236508\pi$
$858$	0	0
$859$	1.58064e7	0.730886	0.365443	−	0.930834i	$-0.380918\pi$
$859$	1.58064e7	0.730886	0.365443	+	0.930834i	$0.380918\pi$
$860$	0	0
$861$	0	0
$862$	8.12410e6	0.372398
$863$	−1.44287e7	−0.659476	−0.329738	−	0.944072i	$-0.606960\pi$
$863$	−1.44287e7	−0.659476	−0.329738	+	0.944072i	$0.606960\pi$
$864$	0	0
$865$	0	0
$866$	−1.45257e7	−0.658178
$867$	0	0
$868$	−1.31604e7	−0.592886
$869$	−4.93728e6	−0.221788
$870$	0	0
$871$	1.74380e7	0.778845
$872$	−4.41960e6	−0.196830
$873$	0	0
$874$	6.30912e6	0.279377
$875$	0	0
$876$	0	0
$877$	−247902.	−0.0108838	−0.00544191	−	0.999985i	$-0.501732\pi$
$877$	−247902.	−0.0108838	−0.00544191	+	0.999985i	$0.501732\pi$
$878$	−1.08206e7	−0.473711
$879$	0	0
$880$	0	0
$881$	−4.10268e7	−1.78085	−0.890426	−	0.455128i	$-0.849594\pi$
$881$	−4.10268e7	−1.78085	−0.890426	+	0.455128i	$0.849594\pi$
$882$	0	0
$883$	−4.18015e7	−1.80422	−0.902112	−	0.431503i	$-0.857984\pi$
$883$	−4.18015e7	−1.80422	−0.902112	+	0.431503i	$0.857984\pi$
$884$	−1.34374e7	−0.578343
$885$	0	0
$886$	−1.30305e7	−0.557669
$887$	−2.10476e7	−0.898241	−0.449120	−	0.893471i	$-0.648263\pi$
$887$	−2.10476e7	−0.898241	−0.449120	+	0.893471i	$0.648263\pi$
$888$	0	0
$889$	1.35460e7	0.574852
$890$	0	0
$891$	0	0
$892$	3.14041e7	1.32152
$893$	−1.28133e7	−0.537690
$894$	0	0
$895$	0	0
$896$	3.57581e7	1.48800
$897$	0	0
$898$	1.01990e6	0.0422053
$899$	−8.34768e6	−0.344482
$900$	0	0
$901$	−4.00136e7	−1.64208
$902$	2.78181e6	0.113844
$903$	0	0
$904$	−1.34232e6	−0.0546305
$905$	0	0
$906$	0	0
$907$	−7.48309e6	−0.302039	−0.151019	−	0.988531i	$-0.548256\pi$
$907$	−7.48309e6	−0.302039	−0.151019	+	0.988531i	$0.548256\pi$
$908$	653744.	0.0263144
$909$	0	0
$910$	0	0
$911$	6.63165e6	0.264744	0.132372	−	0.991200i	$-0.457741\pi$
$911$	6.63165e6	0.264744	0.132372	+	0.991200i	$0.457741\pi$
$912$	0	0
$913$	2.47397e6	0.0982239
$914$	−2.44168e6	−0.0966772
$915$	0	0
$916$	1.66883e7	0.657163
$917$	1.46788e7	0.576457
$918$	0	0
$919$	−1.68976e7	−0.659990	−0.329995	−	0.943983i	$-0.607047\pi$
$919$	−1.68976e7	−0.659990	−0.329995	+	0.943983i	$0.607047\pi$
$920$	0	0
$921$	0	0
$922$	8.14420e6	0.315516
$923$	−9.33733e6	−0.360760
$924$	0	0
$925$	0	0
$926$	−4.04587e6	−0.155055
$927$	0	0
$928$	1.75683e7	0.669669
$929$	1.28653e7	0.489081	0.244541	−	0.969639i	$-0.421363\pi$
$929$	1.28653e7	0.489081	0.244541	+	0.969639i	$0.421363\pi$
$930$	0	0
$931$	2.12604e7	0.803892
$932$	1.35894e7	0.512459
$933$	0	0
$934$	6.50194e6	0.243880
$935$	0	0
$936$	0	0
$937$	−1.06887e7	−0.397718	−0.198859	−	0.980028i	$-0.563724\pi$
$937$	−1.06887e7	−0.397718	−0.198859	+	0.980028i	$0.563724\pi$
$938$	2.34132e7	0.868870
$939$	0	0
$940$	0	0
$941$	−2.82455e7	−1.03986	−0.519930	−	0.854209i	$-0.674042\pi$
$941$	−2.82455e7	−1.03986	−0.519930	+	0.854209i	$0.674042\pi$
$942$	0	0
$943$	2.79684e7	1.02421
$944$	1.31331e7	0.479665
$945$	0	0
$946$	368224.	0.0133778
$947$	−1.70892e7	−0.619222	−0.309611	−	0.950863i	$-0.600199\pi$
$947$	−1.70892e7	−0.619222	−0.309611	+	0.950863i	$0.600199\pi$
$948$	0	0
$949$	−1.10894e7	−0.399706
$950$	0	0
$951$	0	0
$952$	−3.86611e7	−1.38255
$953$	2.22259e7	0.792735	0.396367	−	0.918092i	$-0.370271\pi$
$953$	2.22259e7	0.792735	0.396367	+	0.918092i	$0.370271\pi$
$954$	0	0
$955$	0	0
$956$	−1.36864e6	−0.0484333
$957$	0	0
$958$	6.55872e6	0.230890
$959$	2.78243e7	0.976961
$960$	0	0
$961$	−2.26364e7	−0.790678
$962$	104104.	0.00362685
$963$	0	0
$964$	3.10234e6	0.107522
$965$	0	0
$966$	0	0
$967$	−2.41551e7	−0.830696	−0.415348	−	0.909663i	$-0.636340\pi$
$967$	−2.41551e7	−0.830696	−0.415348	+	0.909663i	$0.636340\pi$
$968$	1.66976e7	0.572752
$969$	0	0
$970$	0	0
$971$	5.48313e7	1.86630	0.933149	−	0.359491i	$-0.117050\pi$
$971$	5.48313e7	1.86630	0.933149	+	0.359491i	$0.117050\pi$
$972$	0	0
$973$	−2.15462e7	−0.729608
$974$	1.70639e7	0.576344
$975$	0	0
$976$	2.11901e7	0.712047
$977$	−1.56612e7	−0.524915	−0.262457	−	0.964944i	$-0.584533\pi$
$977$	−1.56612e7	−0.524915	−0.262457	+	0.964944i	$0.584533\pi$
$978$	0	0
$979$	−1.50028e7	−0.500281
$980$	0	0
$981$	0	0
$982$	−3.02530e6	−0.100113
$983$	−1.63420e7	−0.539412	−0.269706	−	0.962943i	$-0.586927\pi$
$983$	−1.63420e7	−0.539412	−0.269706	+	0.962943i	$0.586927\pi$
$984$	0	0
$985$	0	0
$986$	−1.14440e7	−0.374873
$987$	0	0
$988$	8.48848e6	0.276654
$989$	3.70214e6	0.120355
$990$	0	0
$991$	1.37576e7	0.444997	0.222498	−	0.974933i	$-0.428579\pi$
$991$	1.37576e7	0.444997	0.222498	+	0.974933i	$0.428579\pi$
$992$	−1.26121e7	−0.406919
$993$	0	0
$994$	−1.25368e7	−0.402459
$995$	0	0
$996$	0	0
$997$	1.29097e7	0.411320	0.205660	−	0.978624i	$-0.434066\pi$
$997$	1.29097e7	0.411320	0.205660	+	0.978624i	$0.434066\pi$
$998$	−1.29838e7	−0.412644
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	225.6.a.f.1.1		1
3.2	odd	2		25.6.a.a.1.1		1
5.2	odd	4		225.6.b.e.199.2		2
5.3	odd	4		225.6.b.e.199.1		2
5.4	even	2		45.6.a.b.1.1		1
12.11	even	2		400.6.a.g.1.1		1
15.2	even	4		25.6.b.a.24.1		2
15.8	even	4		25.6.b.a.24.2		2
15.14	odd	2		5.6.a.a.1.1	✓	1
20.19	odd	2		720.6.a.a.1.1		1
60.23	odd	4		400.6.c.j.49.1		2
60.47	odd	4		400.6.c.j.49.2		2
60.59	even	2		80.6.a.e.1.1		1
105.104	even	2		245.6.a.b.1.1		1
120.29	odd	2		320.6.a.j.1.1		1
120.59	even	2		320.6.a.g.1.1		1
165.164	even	2		605.6.a.a.1.1		1
195.194	odd	2		845.6.a.b.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5.6.a.a.1.1	✓	1	15.14	odd	2
25.6.a.a.1.1		1	3.2	odd	2
25.6.b.a.24.1		2	15.2	even	4
25.6.b.a.24.2		2	15.8	even	4
45.6.a.b.1.1		1	5.4	even	2
80.6.a.e.1.1		1	60.59	even	2
225.6.a.f.1.1		1	1.1	even	1	trivial
225.6.b.e.199.1		2	5.3	odd	4
225.6.b.e.199.2		2	5.2	odd	4
245.6.a.b.1.1		1	105.104	even	2
320.6.a.g.1.1		1	120.59	even	2
320.6.a.j.1.1		1	120.29	odd	2
400.6.a.g.1.1		1	12.11	even	2
400.6.c.j.49.1		2	60.23	odd	4
400.6.c.j.49.2		2	60.47	odd	4
605.6.a.a.1.1		1	165.164	even	2
720.6.a.a.1.1		1	20.19	odd	2
845.6.a.b.1.1		1	195.194	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:	\( N \)	\(=\)	\( 225 = 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	225.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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