LMFDB - Embedded newform 225.6.a.n.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:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{11}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 11 $ x^2 - 11
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 5)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-3.31662$ of defining polynomial
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)

$f(q)$	$=$	$q-6.63325 q^{2} +12.0000 q^{4} -59.6992 q^{7} +132.665 q^{8} +O(q^{10})$	$q-6.63325 q^{2} +12.0000 q^{4} -59.6992 q^{7} +132.665 q^{8} -252.000 q^{11} +119.398 q^{13} +396.000 q^{14} -1264.00 q^{16} +689.858 q^{17} +220.000 q^{19} +1671.58 q^{22} +2434.40 q^{23} -792.000 q^{26} -716.391 q^{28} -6930.00 q^{29} +6752.00 q^{31} +4139.15 q^{32} -4576.00 q^{34} +13969.6 q^{37} -1459.31 q^{38} +198.000 q^{41} +417.895 q^{43} -3024.00 q^{44} -16148.0 q^{46} +10540.2 q^{47} -13243.0 q^{49} +1432.78 q^{52} -5823.99 q^{53} -7920.00 q^{56} +45968.4 q^{58} -24660.0 q^{59} -5698.00 q^{61} -44787.7 q^{62} +12992.0 q^{64} -43640.1 q^{67} +8278.30 q^{68} -53352.0 q^{71} -70922.7 q^{73} -92664.0 q^{74} +2640.00 q^{76} +15044.2 q^{77} -51920.0 q^{79} -1313.38 q^{82} -61841.8 q^{83} -2772.00 q^{86} -33431.6 q^{88} -9990.00 q^{89} -7128.00 q^{91} +29212.8 q^{92} -69916.0 q^{94} -101250. q^{97} +87844.1 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 24 q^{4}+O(q^{10}) $ 2 * q + 24 * q^4	$ 2 q + 24 q^{4} - 504 q^{11} + 792 q^{14} - 2528 q^{16} + 440 q^{19} - 1584 q^{26} - 13860 q^{29} + 13504 q^{31} - 9152 q^{34} + 396 q^{41} - 6048 q^{44} - 32296 q^{46} - 26486 q^{49} - 15840 q^{56} - 49320 q^{59} - 11396 q^{61} + 25984 q^{64} - 106704 q^{71} - 185328 q^{74} + 5280 q^{76} - 103840 q^{79} - 5544 q^{86} - 19980 q^{89} - 14256 q^{91} - 139832 q^{94}+O(q^{100}) $ 2 * q + 24 * q^4 - 504 * q^11 + 792 * q^14 - 2528 * q^16 + 440 * q^19 - 1584 * q^26 - 13860 * q^29 + 13504 * q^31 - 9152 * q^34 + 396 * q^41 - 6048 * q^44 - 32296 * q^46 - 26486 * q^49 - 15840 * q^56 - 49320 * q^59 - 11396 * q^61 + 25984 * q^64 - 106704 * q^71 - 185328 * q^74 + 5280 * q^76 - 103840 * q^79 - 5544 * q^86 - 19980 * q^89 - 14256 * q^91 - 139832 * q^94

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 $

$2$	−6.63325	−1.17260	−0.586302	−	0.810093i	$-0.699417\pi$
$2$	−6.63325	−1.17260	−0.586302	+	0.810093i	$0.699417\pi$
$3$	0	0
$3$	0	0
$4$	12.0000	0.375000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−59.6992	−0.460494	−0.230247	−	0.973132i	$-0.573953\pi$
$7$	−59.6992	−0.460494	−0.230247	+	0.973132i	$0.573953\pi$
$8$	132.665	0.732877
$9$	0	0
$10$	0	0
$11$	−252.000	−0.627941	−0.313970	−	0.949433i	$-0.601659\pi$
$11$	−252.000	−0.627941	−0.313970	+	0.949433i	$0.601659\pi$
$12$	0	0
$13$	119.398	0.195948	0.0979739	−	0.995189i	$-0.468764\pi$
$13$	119.398	0.195948	0.0979739	+	0.995189i	$0.468764\pi$
$14$	396.000	0.539977
$15$	0	0
$16$	−1264.00	−1.23438
$17$	689.858	0.578945	0.289473	−	0.957186i	$-0.406520\pi$
$17$	689.858	0.578945	0.289473	+	0.957186i	$0.406520\pi$
$18$	0	0
$19$	220.000	0.139810	0.0699051	−	0.997554i	$-0.477730\pi$
$19$	220.000	0.139810	0.0699051	+	0.997554i	$0.477730\pi$
$20$	0	0
$21$	0	0
$22$	1671.58	0.736326
$23$	2434.40	0.959561	0.479781	−	0.877388i	$-0.340716\pi$
$23$	2434.40	0.959561	0.479781	+	0.877388i	$0.340716\pi$
$24$	0	0
$25$	0	0
$26$	−792.000	−0.229769
$27$	0	0
$28$	−716.391	−0.172685
$29$	−6930.00	−1.53016	−0.765082	−	0.643932i	$-0.777302\pi$
$29$	−6930.00	−1.53016	−0.765082	+	0.643932i	$0.777302\pi$
$30$	0	0
$31$	6752.00	1.26191	0.630955	−	0.775820i	$-0.282663\pi$
$31$	6752.00	1.26191	0.630955	+	0.775820i	$0.282663\pi$
$32$	4139.15	0.714556
$33$	0	0
$34$	−4576.00	−0.678873
$35$	0	0
$36$	0	0
$37$	13969.6	1.67757	0.838785	−	0.544464i	$-0.183267\pi$
$37$	13969.6	1.67757	0.838785	+	0.544464i	$0.183267\pi$
$38$	−1459.31	−0.163942
$39$	0	0
$40$	0	0
$41$	198.000	0.0183952	0.00919762	−	0.999958i	$-0.497072\pi$
$41$	198.000	0.0183952	0.00919762	+	0.999958i	$0.497072\pi$
$42$	0	0
$43$	417.895	0.0344664	0.0172332	−	0.999851i	$-0.494514\pi$
$43$	417.895	0.0344664	0.0172332	+	0.999851i	$0.494514\pi$
$44$	−3024.00	−0.235478
$45$	0	0
$46$	−16148.0	−1.12519
$47$	10540.2	0.695994	0.347997	−	0.937496i	$-0.386862\pi$
$47$	10540.2	0.695994	0.347997	+	0.937496i	$0.386862\pi$
$48$	0	0
$49$	−13243.0	−0.787945
$50$	0	0
$51$	0	0
$52$	1432.78	0.0734804
$53$	−5823.99	−0.284794	−0.142397	−	0.989810i	$-0.545481\pi$
$53$	−5823.99	−0.284794	−0.142397	+	0.989810i	$0.545481\pi$
$54$	0	0
$55$	0	0
$56$	−7920.00	−0.337485
$57$	0	0
$58$	45968.4	1.79428
$59$	−24660.0	−0.922281	−0.461140	−	0.887327i	$-0.652560\pi$
$59$	−24660.0	−0.922281	−0.461140	+	0.887327i	$0.652560\pi$
$60$	0	0
$61$	−5698.00	−0.196064	−0.0980320	−	0.995183i	$-0.531255\pi$
$61$	−5698.00	−0.196064	−0.0980320	+	0.995183i	$0.531255\pi$
$62$	−44787.7	−1.47972
$63$	0	0
$64$	12992.0	0.396484
$65$	0	0
$66$	0	0
$67$	−43640.1	−1.18768	−0.593840	−	0.804583i	$-0.702389\pi$
$67$	−43640.1	−1.18768	−0.593840	+	0.804583i	$0.702389\pi$
$68$	8278.30	0.217104
$69$	0	0
$70$	0	0
$71$	−53352.0	−1.25604	−0.628022	−	0.778196i	$-0.716135\pi$
$71$	−53352.0	−1.25604	−0.628022	+	0.778196i	$0.716135\pi$
$72$	0	0
$73$	−70922.7	−1.55768	−0.778840	−	0.627223i	$-0.784192\pi$
$73$	−70922.7	−1.55768	−0.778840	+	0.627223i	$0.784192\pi$
$74$	−92664.0	−1.96712
$75$	0	0
$76$	2640.00	0.0524288
$77$	15044.2	0.289163
$78$	0	0
$79$	−51920.0	−0.935981	−0.467990	−	0.883734i	$-0.655022\pi$
$79$	−51920.0	−0.935981	−0.467990	+	0.883734i	$0.655022\pi$
$80$	0	0
$81$	0	0
$82$	−1313.38	−0.0215703
$83$	−61841.8	−0.985342	−0.492671	−	0.870216i	$-0.663979\pi$
$83$	−61841.8	−0.985342	−0.492671	+	0.870216i	$0.663979\pi$
$84$	0	0
$85$	0	0
$86$	−2772.00	−0.0404154
$87$	0	0
$88$	−33431.6	−0.460204
$89$	−9990.00	−0.133687	−0.0668437	−	0.997763i	$-0.521293\pi$
$89$	−9990.00	−0.133687	−0.0668437	+	0.997763i	$0.521293\pi$
$90$	0	0
$91$	−7128.00	−0.0902328
$92$	29212.8	0.359836
$93$	0	0
$94$	−69916.0	−0.816125
$95$	0	0
$96$	0	0
$97$	−101250.	−1.09261	−0.546305	−	0.837586i	$-0.683966\pi$
$97$	−101250.	−1.09261	−0.546305	+	0.837586i	$0.683966\pi$
$98$	87844.1	0.923948
$99$	0	0
$100$	0	0
$101$	109098.	1.06418	0.532088	−	0.846689i	$-0.321408\pi$
$101$	109098.	1.06418	0.532088	+	0.846689i	$0.321408\pi$
$102$	0	0
$103$	−70624.2	−0.655935	−0.327967	−	0.944689i	$-0.606364\pi$
$103$	−70624.2	−0.655935	−0.327967	+	0.944689i	$0.606364\pi$
$104$	15840.0	0.143606
$105$	0	0
$106$	38632.0	0.333951
$107$	−97117.4	−0.820045	−0.410022	−	0.912075i	$-0.634479\pi$
$107$	−97117.4	−0.820045	−0.410022	+	0.912075i	$0.634479\pi$
$108$	0	0
$109$	21010.0	0.169379	0.0846895	−	0.996407i	$-0.473010\pi$
$109$	21010.0	0.169379	0.0846895	+	0.996407i	$0.473010\pi$
$110$	0	0
$111$	0	0
$112$	75459.8	0.568422
$113$	105018.	0.773688	0.386844	−	0.922145i	$-0.373565\pi$
$113$	105018.	0.773688	0.386844	+	0.922145i	$0.373565\pi$
$114$	0	0
$115$	0	0
$116$	−83160.0	−0.573812
$117$	0	0
$118$	163576.	1.08147
$119$	−41184.0	−0.266601
$120$	0	0
$121$	−97547.0	−0.605690
$122$	37796.3	0.229905
$123$	0	0
$124$	81024.0	0.473216
$125$	0	0
$126$	0	0
$127$	−87220.6	−0.479855	−0.239927	−	0.970791i	$-0.577124\pi$
$127$	−87220.6	−0.479855	−0.239927	+	0.970791i	$0.577124\pi$
$128$	−218632.	−1.17947
$129$	0	0
$130$	0	0
$131$	−192852.	−0.981852	−0.490926	−	0.871201i	$-0.663341\pi$
$131$	−192852.	−0.981852	−0.490926	+	0.871201i	$0.663341\pi$
$132$	0	0
$133$	−13133.8	−0.0643817
$134$	289476.	1.39268
$135$	0	0
$136$	91520.0	0.424296
$137$	143570.	0.653525	0.326763	−	0.945106i	$-0.394042\pi$
$137$	143570.	0.653525	0.326763	+	0.945106i	$0.394042\pi$
$138$	0	0
$139$	318340.	1.39751	0.698754	−	0.715362i	$-0.253738\pi$
$139$	318340.	1.39751	0.698754	+	0.715362i	$0.253738\pi$
$140$	0	0
$141$	0	0
$142$	353897.	1.47284
$143$	−30088.4	−0.123044
$144$	0	0
$145$	0	0
$146$	470448.	1.82654
$147$	0	0
$148$	167635.	0.629088
$149$	84150.0	0.310519	0.155260	−	0.987874i	$-0.450379\pi$
$149$	84150.0	0.310519	0.155260	+	0.987874i	$0.450379\pi$
$150$	0	0
$151$	−155848.	−0.556236	−0.278118	−	0.960547i	$-0.589711\pi$
$151$	−155848.	−0.556236	−0.278118	+	0.960547i	$0.589711\pi$
$152$	29186.3	0.102464
$153$	0	0
$154$	−99792.0	−0.339074
$155$	0	0
$156$	0	0
$157$	356643.	1.15474	0.577371	−	0.816482i	$-0.304079\pi$
$157$	356643.	1.15474	0.577371	+	0.816482i	$0.304079\pi$
$158$	344398.	1.09753
$159$	0	0
$160$	0	0
$161$	−145332.	−0.441872
$162$	0	0
$163$	144890.	0.427139	0.213570	−	0.976928i	$-0.431491\pi$
$163$	144890.	0.427139	0.213570	+	0.976928i	$0.431491\pi$
$164$	2376.00	0.00689822
$165$	0	0
$166$	410212.	1.15542
$167$	18102.1	0.0502272	0.0251136	−	0.999685i	$-0.492005\pi$
$167$	18102.1	0.0502272	0.0251136	+	0.999685i	$0.492005\pi$
$168$	0	0
$169$	−357037.	−0.961604
$170$	0	0
$171$	0	0
$172$	5014.74	0.0129249
$173$	−492572.	−1.25128	−0.625640	−	0.780112i	$-0.715162\pi$
$173$	−492572.	−1.25128	−0.625640	+	0.780112i	$0.715162\pi$
$174$	0	0
$175$	0	0
$176$	318528.	0.775115
$177$	0	0
$178$	66266.2	0.156762
$179$	444420.	1.03672	0.518359	−	0.855163i	$-0.326543\pi$
$179$	444420.	1.03672	0.518359	+	0.855163i	$0.326543\pi$
$180$	0	0
$181$	156902.	0.355985	0.177993	−	0.984032i	$-0.443040\pi$
$181$	156902.	0.355985	0.177993	+	0.984032i	$0.443040\pi$
$182$	47281.8	0.105807
$183$	0	0
$184$	322960.	0.703241
$185$	0	0
$186$	0	0
$187$	−173844.	−0.363543
$188$	126483.	0.260998
$189$	0	0
$190$	0	0
$191$	−332352.	−0.659196	−0.329598	−	0.944121i	$-0.606913\pi$
$191$	−332352.	−0.659196	−0.329598	+	0.944121i	$0.606913\pi$
$192$	0	0
$193$	−786120.	−1.51913	−0.759566	−	0.650430i	$-0.774589\pi$
$193$	−786120.	−1.51913	−0.759566	+	0.650430i	$0.774589\pi$
$194$	671616.	1.28120
$195$	0	0
$196$	−158916.	−0.295480
$197$	−59606.4	−0.109428	−0.0547138	−	0.998502i	$-0.517425\pi$
$197$	−59606.4	−0.109428	−0.0547138	+	0.998502i	$0.517425\pi$
$198$	0	0
$199$	395800.	0.708505	0.354253	−	0.935150i	$-0.384735\pi$
$199$	395800.	0.708505	0.354253	+	0.935150i	$0.384735\pi$
$200$	0	0
$201$	0	0
$202$	−723674.	−1.24786
$203$	413716.	0.704631
$204$	0	0
$205$	0	0
$206$	468468.	0.769151
$207$	0	0
$208$	−150920.	−0.241873
$209$	−55440.0	−0.0877925
$210$	0	0
$211$	−251548.	−0.388969	−0.194484	−	0.980906i	$-0.562303\pi$
$211$	−251548.	−0.388969	−0.194484	+	0.980906i	$0.562303\pi$
$212$	−69887.9	−0.106798
$213$	0	0
$214$	644204.	0.961588
$215$	0	0
$216$	0	0
$217$	−403089.	−0.581101
$218$	−139365.	−0.198615
$219$	0	0
$220$	0	0
$221$	82368.0	0.113443
$222$	0	0
$223$	288765.	0.388851	0.194425	−	0.980917i	$-0.437716\pi$
$223$	288765.	0.388851	0.194425	+	0.980917i	$0.437716\pi$
$224$	−247104.	−0.329048
$225$	0	0
$226$	−696608.	−0.907230
$227$	−1.16414e6	−1.49948	−0.749741	−	0.661731i	$-0.769822\pi$
$227$	−1.16414e6	−1.49948	−0.749741	+	0.661731i	$0.769822\pi$
$228$	0	0
$229$	−547670.	−0.690129	−0.345064	−	0.938579i	$-0.612143\pi$
$229$	−547670.	−0.690129	−0.345064	+	0.938579i	$0.612143\pi$
$230$	0	0
$231$	0	0
$232$	−919368.	−1.12142
$233$	48104.3	0.0580489	0.0290245	−	0.999579i	$-0.490760\pi$
$233$	48104.3	0.0580489	0.0290245	+	0.999579i	$0.490760\pi$
$234$	0	0
$235$	0	0
$236$	−295920.	−0.345855
$237$	0	0
$238$	273184.	0.312617
$239$	−1.00584e6	−1.13903	−0.569514	−	0.821982i	$-0.692868\pi$
$239$	−1.00584e6	−1.13903	−0.569514	+	0.821982i	$0.692868\pi$
$240$	0	0
$241$	895202.	0.992838	0.496419	−	0.868083i	$-0.334648\pi$
$241$	895202.	0.992838	0.496419	+	0.868083i	$0.334648\pi$
$242$	647054.	0.710235
$243$	0	0
$244$	−68376.0	−0.0735240
$245$	0	0
$246$	0	0
$247$	26267.7	0.0273955
$248$	895754.	0.924825
$249$	0	0
$250$	0	0
$251$	−558252.	−0.559301	−0.279651	−	0.960102i	$-0.590219\pi$
$251$	−558252.	−0.559301	−0.279651	+	0.960102i	$0.590219\pi$
$252$	0	0
$253$	−613469.	−0.602548
$254$	578556.	0.562680
$255$	0	0
$256$	1.03450e6	0.986572
$257$	787924.	0.744135	0.372067	−	0.928206i	$-0.378649\pi$
$257$	787924.	0.744135	0.372067	+	0.928206i	$0.378649\pi$
$258$	0	0
$259$	−833976.	−0.772510
$260$	0	0
$261$	0	0
$262$	1.27924e6	1.15132
$263$	−1.63173e6	−1.45465	−0.727327	−	0.686291i	$-0.759238\pi$
$263$	−1.63173e6	−1.45465	−0.727327	+	0.686291i	$0.759238\pi$
$264$	0	0
$265$	0	0
$266$	87120.0	0.0754942
$267$	0	0
$268$	−523682.	−0.445380
$269$	−1.73637e6	−1.46306	−0.731529	−	0.681810i	$-0.761193\pi$
$269$	−1.73637e6	−1.46306	−0.731529	+	0.681810i	$0.761193\pi$
$270$	0	0
$271$	−1.72005e6	−1.42271	−0.711357	−	0.702831i	$-0.751919\pi$
$271$	−1.72005e6	−1.42271	−0.711357	+	0.702831i	$0.751919\pi$
$272$	−871980.	−0.714635
$273$	0	0
$274$	−952336.	−0.766326
$275$	0	0
$276$	0	0
$277$	1.27243e6	0.996402	0.498201	−	0.867062i	$-0.333994\pi$
$277$	1.27243e6	0.996402	0.498201	+	0.867062i	$0.333994\pi$
$278$	−2.11163e6	−1.63872
$279$	0	0
$280$	0	0
$281$	−1.46500e6	−1.10681	−0.553404	−	0.832913i	$-0.686671\pi$
$281$	−1.46500e6	−1.10681	−0.553404	+	0.832913i	$0.686671\pi$
$282$	0	0
$283$	1.65051e6	1.22504	0.612521	−	0.790455i	$-0.290156\pi$
$283$	1.65051e6	1.22504	0.612521	+	0.790455i	$0.290156\pi$
$284$	−640224.	−0.471016
$285$	0	0
$286$	199584.	0.144281
$287$	−11820.5	−0.00847089
$288$	0	0
$289$	−943953.	−0.664823
$290$	0	0
$291$	0	0
$292$	−851072.	−0.584130
$293$	2.38772e6	1.62485	0.812426	−	0.583064i	$-0.198146\pi$
$293$	2.38772e6	1.62485	0.812426	+	0.583064i	$0.198146\pi$
$294$	0	0
$295$	0	0
$296$	1.85328e6	1.22945
$297$	0	0
$298$	−558188.	−0.364116
$299$	290664.	0.188024
$300$	0	0
$301$	−24948.0	−0.0158716
$302$	1.03378e6	0.652244
$303$	0	0
$304$	−278080.	−0.172578
$305$	0	0
$306$	0	0
$307$	928264.	0.562115	0.281058	−	0.959691i	$-0.409315\pi$
$307$	928264.	0.562115	0.281058	+	0.959691i	$0.409315\pi$
$308$	180531.	0.108436
$309$	0	0
$310$	0	0
$311$	−568152.	−0.333092	−0.166546	−	0.986034i	$-0.553261\pi$
$311$	−568152.	−0.333092	−0.166546	+	0.986034i	$0.553261\pi$
$312$	0	0
$313$	1.72244e6	0.993766	0.496883	−	0.867818i	$-0.334478\pi$
$313$	1.72244e6	0.993766	0.496883	+	0.867818i	$0.334478\pi$
$314$	−2.36570e6	−1.35405
$315$	0	0
$316$	−623040.	−0.350993
$317$	−131643.	−0.0735785	−0.0367893	−	0.999323i	$-0.511713\pi$
$317$	−131643.	−0.0735785	−0.0367893	+	0.999323i	$0.511713\pi$
$318$	0	0
$319$	1.74636e6	0.960853
$320$	0	0
$321$	0	0
$322$	964023.	0.518141
$323$	151769.	0.0809424
$324$	0	0
$325$	0	0
$326$	−961092.	−0.500865
$327$	0	0
$328$	26267.7	0.0134815
$329$	−629244.	−0.320501
$330$	0	0
$331$	−1.58055e6	−0.792935	−0.396468	−	0.918049i	$-0.629764\pi$
$331$	−1.58055e6	−0.792935	−0.396468	+	0.918049i	$0.629764\pi$
$332$	−742101.	−0.369503
$333$	0	0
$334$	−120076.	−0.0588966
$335$	0	0
$336$	0	0
$337$	1.22885e6	0.589419	0.294709	−	0.955587i	$-0.404777\pi$
$337$	1.22885e6	0.589419	0.294709	+	0.955587i	$0.404777\pi$
$338$	2.36832e6	1.12758
$339$	0	0
$340$	0	0
$341$	−1.70150e6	−0.792405
$342$	0	0
$343$	1.79396e6	0.823338
$344$	55440.0	0.0252596
$345$	0	0
$346$	3.26735e6	1.46726
$347$	3.84224e6	1.71301	0.856506	−	0.516137i	$-0.172630\pi$
$347$	3.84224e6	1.71301	0.856506	+	0.516137i	$0.172630\pi$
$348$	0	0
$349$	1.59445e6	0.700725	0.350362	−	0.936614i	$-0.386058\pi$
$349$	1.59445e6	0.700725	0.350362	+	0.936614i	$0.386058\pi$
$350$	0	0
$351$	0	0
$352$	−1.04307e6	−0.448699
$353$	−295365.	−0.126160	−0.0630802	−	0.998008i	$-0.520092\pi$
$353$	−295365.	−0.126160	−0.0630802	+	0.998008i	$0.520092\pi$
$354$	0	0
$355$	0	0
$356$	−119880.	−0.0501328
$357$	0	0
$358$	−2.94795e6	−1.21566
$359$	1.10484e6	0.452442	0.226221	−	0.974076i	$-0.427363\pi$
$359$	1.10484e6	0.452442	0.226221	+	0.974076i	$0.427363\pi$
$360$	0	0
$361$	−2.42770e6	−0.980453
$362$	−1.04077e6	−0.417430
$363$	0	0
$364$	−85536.0	−0.0338373
$365$	0	0
$366$	0	0
$367$	−1.83760e6	−0.712174	−0.356087	−	0.934453i	$-0.615889\pi$
$367$	−1.83760e6	−0.712174	−0.356087	+	0.934453i	$0.615889\pi$
$368$	−3.07708e6	−1.18446
$369$	0	0
$370$	0	0
$371$	347688.	0.131146
$372$	0	0
$373$	−2.93350e6	−1.09173	−0.545864	−	0.837874i	$-0.683798\pi$
$373$	−2.93350e6	−1.09173	−0.545864	+	0.837874i	$0.683798\pi$
$374$	1.15315e6	0.426292
$375$	0	0
$376$	1.39832e6	0.510078
$377$	−827432.	−0.299832
$378$	0	0
$379$	−5.09342e6	−1.82143	−0.910713	−	0.413040i	$-0.864467\pi$
$379$	−5.09342e6	−1.82143	−0.910713	+	0.413040i	$0.864467\pi$
$380$	0	0
$381$	0	0
$382$	2.20457e6	0.772976
$383$	3.17485e6	1.10593	0.552964	−	0.833205i	$-0.313497\pi$
$383$	3.17485e6	1.10593	0.552964	+	0.833205i	$0.313497\pi$
$384$	0	0
$385$	0	0
$386$	5.21453e6	1.78134
$387$	0	0
$388$	−1.21500e6	−0.409729
$389$	1.79991e6	0.603083	0.301541	−	0.953453i	$-0.402499\pi$
$389$	1.79991e6	0.603083	0.301541	+	0.953453i	$0.402499\pi$
$390$	0	0
$391$	1.67939e6	0.555533
$392$	−1.75688e6	−0.577467
$393$	0	0
$394$	395384.	0.128315
$395$	0	0
$396$	0	0
$397$	−4.90405e6	−1.56163	−0.780817	−	0.624760i	$-0.785197\pi$
$397$	−4.90405e6	−1.56163	−0.780817	+	0.624760i	$0.785197\pi$
$398$	−2.62544e6	−0.830796
$399$	0	0
$400$	0	0
$401$	642798.	0.199624	0.0998122	−	0.995006i	$-0.468176\pi$
$401$	642798.	0.199624	0.0998122	+	0.995006i	$0.468176\pi$
$402$	0	0
$403$	806179.	0.247268
$404$	1.30918e6	0.399066
$405$	0	0
$406$	−2.74428e6	−0.826254
$407$	−3.52035e6	−1.05341
$408$	0	0
$409$	2.05711e6	0.608064	0.304032	−	0.952662i	$-0.401667\pi$
$409$	2.05711e6	0.608064	0.304032	+	0.952662i	$0.401667\pi$
$410$	0	0
$411$	0	0
$412$	−847490.	−0.245975
$413$	1.47218e6	0.424704
$414$	0	0
$415$	0	0
$416$	494208.	0.140016
$417$	0	0
$418$	367747.	0.102946
$419$	−2.93742e6	−0.817393	−0.408697	−	0.912670i	$-0.634017\pi$
$419$	−2.93742e6	−0.817393	−0.408697	+	0.912670i	$0.634017\pi$
$420$	0	0
$421$	2.71770e6	0.747303	0.373651	−	0.927569i	$-0.378106\pi$
$421$	2.71770e6	0.747303	0.373651	+	0.927569i	$0.378106\pi$
$422$	1.66858e6	0.456106
$423$	0	0
$424$	−772640.	−0.208719
$425$	0	0
$426$	0	0
$427$	340166.	0.0902862
$428$	−1.16541e6	−0.307517
$429$	0	0
$430$	0	0
$431$	−4.99435e6	−1.29505	−0.647524	−	0.762045i	$-0.724196\pi$
$431$	−4.99435e6	−1.29505	−0.647524	+	0.762045i	$0.724196\pi$
$432$	0	0
$433$	2.08183e6	0.533612	0.266806	−	0.963750i	$-0.414032\pi$
$433$	2.08183e6	0.533612	0.266806	+	0.963750i	$0.414032\pi$
$434$	2.67379e6	0.681402
$435$	0	0
$436$	252120.	0.0635172
$437$	535569.	0.134156
$438$	0	0
$439$	4.70404e6	1.16496	0.582478	−	0.812846i	$-0.302083\pi$
$439$	4.70404e6	1.16496	0.582478	+	0.812846i	$0.302083\pi$
$440$	0	0
$441$	0	0
$442$	−546368.	−0.133024
$443$	−5.70103e6	−1.38021	−0.690103	−	0.723711i	$-0.742435\pi$
$443$	−5.70103e6	−1.38021	−0.690103	+	0.723711i	$0.742435\pi$
$444$	0	0
$445$	0	0
$446$	−1.91545e6	−0.455968
$447$	0	0
$448$	−775613.	−0.182579
$449$	6.20325e6	1.45212	0.726062	−	0.687630i	$-0.241349\pi$
$449$	6.20325e6	1.45212	0.726062	+	0.687630i	$0.241349\pi$
$450$	0	0
$451$	−49896.0	−0.0115511
$452$	1.26021e6	0.290133
$453$	0	0
$454$	7.72204e6	1.75830
$455$	0	0
$456$	0	0
$457$	−2.15371e6	−0.482388	−0.241194	−	0.970477i	$-0.577539\pi$
$457$	−2.15371e6	−0.482388	−0.241194	+	0.970477i	$0.577539\pi$
$458$	3.63283e6	0.809248
$459$	0	0
$460$	0	0
$461$	3.85130e6	0.844024	0.422012	−	0.906590i	$-0.361324\pi$
$461$	3.85130e6	0.844024	0.422012	+	0.906590i	$0.361324\pi$
$462$	0	0
$463$	2.08213e6	0.451394	0.225697	−	0.974198i	$-0.427534\pi$
$463$	2.08213e6	0.451394	0.225697	+	0.974198i	$0.427534\pi$
$464$	8.75952e6	1.88880
$465$	0	0
$466$	−319088.	−0.0680684
$467$	−1.30822e6	−0.277579	−0.138790	−	0.990322i	$-0.544321\pi$
$467$	−1.30822e6	−0.277579	−0.138790	+	0.990322i	$0.544321\pi$
$468$	0	0
$469$	2.60528e6	0.546919
$470$	0	0
$471$	0	0
$472$	−3.27152e6	−0.675919
$473$	−105309.	−0.0216429
$474$	0	0
$475$	0	0
$476$	−494208.	−0.0999752
$477$	0	0
$478$	6.67199e6	1.33563
$479$	−6.76368e6	−1.34693	−0.673464	−	0.739220i	$-0.735194\pi$
$479$	−6.76368e6	−1.34693	−0.673464	+	0.739220i	$0.735194\pi$
$480$	0	0
$481$	1.66795e6	0.328716
$482$	−5.93810e6	−1.16421
$483$	0	0
$484$	−1.17056e6	−0.227134
$485$	0	0
$486$	0	0
$487$	6.67193e6	1.27476	0.637381	−	0.770549i	$-0.280018\pi$
$487$	6.67193e6	1.27476	0.637381	+	0.770549i	$0.280018\pi$
$488$	−755925.	−0.143691
$489$	0	0
$490$	0	0
$491$	6.87575e6	1.28711	0.643556	−	0.765399i	$-0.277458\pi$
$491$	6.87575e6	1.28711	0.643556	+	0.765399i	$0.277458\pi$
$492$	0	0
$493$	−4.78072e6	−0.885881
$494$	−174240.	−0.0321241
$495$	0	0
$496$	−8.53453e6	−1.55767
$497$	3.18507e6	0.578400
$498$	0	0
$499$	−6.94010e6	−1.24771	−0.623856	−	0.781539i	$-0.714435\pi$
$499$	−6.94010e6	−1.24771	−0.623856	+	0.781539i	$0.714435\pi$
$500$	0	0
$501$	0	0
$502$	3.70302e6	0.655839
$503$	921007.	0.162309	0.0811546	−	0.996702i	$-0.474139\pi$
$503$	921007.	0.162309	0.0811546	+	0.996702i	$0.474139\pi$
$504$	0	0
$505$	0	0
$506$	4.06930e6	0.706550
$507$	0	0
$508$	−1.04665e6	−0.179946
$509$	4.97979e6	0.851955	0.425977	−	0.904734i	$-0.359930\pi$
$509$	4.97979e6	0.851955	0.425977	+	0.904734i	$0.359930\pi$
$510$	0	0
$511$	4.23403e6	0.717302
$512$	134151.	0.0226161
$513$	0	0
$514$	−5.22650e6	−0.872575
$515$	0	0
$516$	0	0
$517$	−2.65614e6	−0.437043
$518$	5.53197e6	0.905848
$519$	0	0
$520$	0	0
$521$	147798.	0.0238547	0.0119274	−	0.999929i	$-0.496203\pi$
$521$	147798.	0.0238547	0.0119274	+	0.999929i	$0.496203\pi$
$522$	0	0
$523$	−1.23884e7	−1.98043	−0.990216	−	0.139543i	$-0.955437\pi$
$523$	−1.23884e7	−1.98043	−0.990216	+	0.139543i	$0.955437\pi$
$524$	−2.31422e6	−0.368194
$525$	0	0
$526$	1.08237e7	1.70573
$527$	4.65792e6	0.730576
$528$	0	0
$529$	−510027.	−0.0792417
$530$	0	0
$531$	0	0
$532$	−157606.	−0.0241431
$533$	23640.9	0.00360451
$534$	0	0
$535$	0	0
$536$	−5.78952e6	−0.870423
$537$	0	0
$538$	1.15178e7	1.71559
$539$	3.33724e6	0.494783
$540$	0	0
$541$	−9.99810e6	−1.46867	−0.734335	−	0.678787i	$-0.762506\pi$
$541$	−9.99810e6	−1.46867	−0.734335	+	0.678787i	$0.762506\pi$
$542$	1.14095e7	1.66828
$543$	0	0
$544$	2.85542e6	0.413688
$545$	0	0
$546$	0	0
$547$	1.18580e7	1.69451	0.847253	−	0.531189i	$-0.178255\pi$
$547$	1.18580e7	1.69451	0.847253	+	0.531189i	$0.178255\pi$
$548$	1.72284e6	0.245072
$549$	0	0
$550$	0	0
$551$	−1.52460e6	−0.213933
$552$	0	0
$553$	3.09958e6	0.431013
$554$	−8.44034e6	−1.16838
$555$	0	0
$556$	3.82008e6	0.524065
$557$	−904550.	−0.123536	−0.0617681	−	0.998091i	$-0.519674\pi$
$557$	−904550.	−0.123536	−0.0617681	+	0.998091i	$0.519674\pi$
$558$	0	0
$559$	49896.0	0.00675361
$560$	0	0
$561$	0	0
$562$	9.71772e6	1.29785
$563$	−8.68719e6	−1.15507	−0.577535	−	0.816366i	$-0.695985\pi$
$563$	−8.68719e6	−1.15507	−0.577535	+	0.816366i	$0.695985\pi$
$564$	0	0
$565$	0	0
$566$	−1.09482e7	−1.43649
$567$	0	0
$568$	−7.07794e6	−0.920526
$569$	−2.27007e6	−0.293940	−0.146970	−	0.989141i	$-0.546952\pi$
$569$	−2.27007e6	−0.293940	−0.146970	+	0.989141i	$0.546952\pi$
$570$	0	0
$571$	1.43807e7	1.84582	0.922908	−	0.385021i	$-0.125806\pi$
$571$	1.43807e7	1.84582	0.922908	+	0.385021i	$0.125806\pi$
$572$	−361061.	−0.0461414
$573$	0	0
$574$	78408.0	0.00993300
$575$	0	0
$576$	0	0
$577$	5.63943e6	0.705173	0.352586	−	0.935779i	$-0.385302\pi$
$577$	5.63943e6	0.705173	0.352586	+	0.935779i	$0.385302\pi$
$578$	6.26148e6	0.779574
$579$	0	0
$580$	0	0
$581$	3.69191e6	0.453744
$582$	0	0
$583$	1.46765e6	0.178834
$584$	−9.40896e6	−1.14159
$585$	0	0
$586$	−1.58383e7	−1.90531
$587$	−1.28473e6	−0.153893	−0.0769464	−	0.997035i	$-0.524517\pi$
$587$	−1.28473e6	−0.153893	−0.0769464	+	0.997035i	$0.524517\pi$
$588$	0	0
$589$	1.48544e6	0.176428
$590$	0	0
$591$	0	0
$592$	−1.76576e7	−2.07075
$593$	7.00943e6	0.818552	0.409276	−	0.912411i	$-0.365781\pi$
$593$	7.00943e6	0.818552	0.409276	+	0.912411i	$0.365781\pi$
$594$	0	0
$595$	0	0
$596$	1.00980e6	0.116445
$597$	0	0
$598$	−1.92805e6	−0.220478
$599$	−8.80020e6	−1.00213	−0.501067	−	0.865409i	$-0.667059\pi$
$599$	−8.80020e6	−1.00213	−0.501067	+	0.865409i	$0.667059\pi$
$600$	0	0
$601$	−1.07670e7	−1.21593	−0.607965	−	0.793964i	$-0.708014\pi$
$601$	−1.07670e7	−1.21593	−0.607965	+	0.793964i	$0.708014\pi$
$602$	165486.	0.0186110
$603$	0	0
$604$	−1.87018e6	−0.208588
$605$	0	0
$606$	0	0
$607$	−1.51219e7	−1.66584	−0.832921	−	0.553391i	$-0.813333\pi$
$607$	−1.51219e7	−1.66584	−0.832921	+	0.553391i	$0.813333\pi$
$608$	910613.	0.0999021
$609$	0	0
$610$	0	0
$611$	1.25849e6	0.136379
$612$	0	0
$613$	8.31622e6	0.893871	0.446936	−	0.894566i	$-0.352515\pi$
$613$	8.31622e6	0.893871	0.446936	+	0.894566i	$0.352515\pi$
$614$	−6.15740e6	−0.659139
$615$	0	0
$616$	1.99584e6	0.211921
$617$	−1.21083e7	−1.28047	−0.640237	−	0.768178i	$-0.721164\pi$
$617$	−1.21083e7	−1.28047	−0.640237	+	0.768178i	$0.721164\pi$
$618$	0	0
$619$	−9.73238e6	−1.02092	−0.510461	−	0.859901i	$-0.670525\pi$
$619$	−9.73238e6	−1.02092	−0.510461	+	0.859901i	$0.670525\pi$
$620$	0	0
$621$	0	0
$622$	3.76869e6	0.390584
$623$	596395.	0.0615622
$624$	0	0
$625$	0	0
$626$	−1.14254e7	−1.16529
$627$	0	0
$628$	4.27972e6	0.433028
$629$	9.63706e6	0.971220
$630$	0	0
$631$	−8.60145e6	−0.859999	−0.430000	−	0.902829i	$-0.641486\pi$
$631$	−8.60145e6	−0.859999	−0.430000	+	0.902829i	$0.641486\pi$
$632$	−6.88797e6	−0.685959
$633$	0	0
$634$	873224.	0.0862785
$635$	0	0
$636$	0	0
$637$	−1.58119e6	−0.154396
$638$	−1.15840e7	−1.12670
$639$	0	0
$640$	0	0
$641$	6.42440e6	0.617572	0.308786	−	0.951132i	$-0.400077\pi$
$641$	6.42440e6	0.617572	0.308786	+	0.951132i	$0.400077\pi$
$642$	0	0
$643$	−3.64721e6	−0.347883	−0.173941	−	0.984756i	$-0.555650\pi$
$643$	−3.64721e6	−0.347883	−0.173941	+	0.984756i	$0.555650\pi$
$644$	−1.74398e6	−0.165702
$645$	0	0
$646$	−1.00672e6	−0.0949134
$647$	−3.78036e6	−0.355036	−0.177518	−	0.984118i	$-0.556807\pi$
$647$	−3.78036e6	−0.355036	−0.177518	+	0.984118i	$0.556807\pi$
$648$	0	0
$649$	6.21432e6	0.579138
$650$	0	0
$651$	0	0
$652$	1.73868e6	0.160177
$653$	−1.66957e7	−1.53223	−0.766113	−	0.642706i	$-0.777812\pi$
$653$	−1.66957e7	−1.53223	−0.766113	+	0.642706i	$0.777812\pi$
$654$	0	0
$655$	0	0
$656$	−250272.	−0.0227066
$657$	0	0
$658$	4.17393e6	0.375821
$659$	−1.22166e6	−0.109581	−0.0547907	−	0.998498i	$-0.517449\pi$
$659$	−1.22166e6	−0.109581	−0.0547907	+	0.998498i	$0.517449\pi$
$660$	0	0
$661$	1.62789e7	1.44918	0.724589	−	0.689182i	$-0.242030\pi$
$661$	1.62789e7	1.44918	0.724589	+	0.689182i	$0.242030\pi$
$662$	1.04842e7	0.929799
$663$	0	0
$664$	−8.20424e6	−0.722135
$665$	0	0
$666$	0	0
$667$	−1.68704e7	−1.46829
$668$	217226.	0.0188352
$669$	0	0
$670$	0	0
$671$	1.43590e6	0.123117
$672$	0	0
$673$	−1.43928e7	−1.22492	−0.612459	−	0.790503i	$-0.709819\pi$
$673$	−1.43928e7	−1.22492	−0.612459	+	0.790503i	$0.709819\pi$
$674$	−8.15126e6	−0.691155
$675$	0	0
$676$	−4.28444e6	−0.360602
$677$	−2.62429e6	−0.220059	−0.110030	−	0.993928i	$-0.535095\pi$
$677$	−2.62429e6	−0.220059	−0.110030	+	0.993928i	$0.535095\pi$
$678$	0	0
$679$	6.04454e6	0.503140
$680$	0	0
$681$	0	0
$682$	1.12865e7	0.929177
$683$	1.03039e7	0.845184	0.422592	−	0.906320i	$-0.361120\pi$
$683$	1.03039e7	0.845184	0.422592	+	0.906320i	$0.361120\pi$
$684$	0	0
$685$	0	0
$686$	−1.18998e7	−0.965449
$687$	0	0
$688$	−528219.	−0.0425444
$689$	−695376.	−0.0558048
$690$	0	0
$691$	4.50285e6	0.358751	0.179375	−	0.983781i	$-0.442592\pi$
$691$	4.50285e6	0.358751	0.179375	+	0.983781i	$0.442592\pi$
$692$	−5.91086e6	−0.469230
$693$	0	0
$694$	−2.54865e7	−2.00868
$695$	0	0
$696$	0	0
$697$	136592.	0.0106498
$698$	−1.05764e7	−0.821672
$699$	0	0
$700$	0	0
$701$	4.88090e6	0.375150	0.187575	−	0.982250i	$-0.439937\pi$
$701$	4.88090e6	0.375150	0.187575	+	0.982250i	$0.439937\pi$
$702$	0	0
$703$	3.07332e6	0.234541
$704$	−3.27398e6	−0.248969
$705$	0	0
$706$	1.95923e6	0.147936
$707$	−6.51307e6	−0.490046
$708$	0	0
$709$	9.96961e6	0.744839	0.372420	−	0.928064i	$-0.378528\pi$
$709$	9.96961e6	0.744839	0.372420	+	0.928064i	$0.378528\pi$
$710$	0	0
$711$	0	0
$712$	−1.32532e6	−0.0979765
$713$	1.64371e7	1.21088
$714$	0	0
$715$	0	0
$716$	5.33304e6	0.388770
$717$	0	0
$718$	−7.32868e6	−0.530536
$719$	−1.19167e7	−0.859675	−0.429838	−	0.902906i	$-0.641429\pi$
$719$	−1.19167e7	−0.859675	−0.429838	+	0.902906i	$0.641429\pi$
$720$	0	0
$721$	4.21621e6	0.302054
$722$	1.61035e7	1.14968
$723$	0	0
$724$	1.88282e6	0.133494
$725$	0	0
$726$	0	0
$727$	−1.38269e6	−0.0970264	−0.0485132	−	0.998823i	$-0.515448\pi$
$727$	−1.38269e6	−0.0970264	−0.0485132	+	0.998823i	$0.515448\pi$
$728$	−945636.	−0.0661296
$729$	0	0
$730$	0	0
$731$	288288.	0.0199541
$732$	0	0
$733$	6.09661e6	0.419110	0.209555	−	0.977797i	$-0.432798\pi$
$733$	6.09661e6	0.419110	0.209555	+	0.977797i	$0.432798\pi$
$734$	1.21893e7	0.835099
$735$	0	0
$736$	1.00764e7	0.685660
$737$	1.09973e7	0.745793
$738$	0	0
$739$	−6.16946e6	−0.415562	−0.207781	−	0.978175i	$-0.566624\pi$
$739$	−6.16946e6	−0.415562	−0.207781	+	0.978175i	$0.566624\pi$
$740$	0	0
$741$	0	0
$742$	−2.30630e6	−0.153782
$743$	−1.57574e7	−1.04716	−0.523578	−	0.851978i	$-0.675403\pi$
$743$	−1.57574e7	−1.04716	−0.523578	+	0.851978i	$0.675403\pi$
$744$	0	0
$745$	0	0
$746$	1.94586e7	1.28016
$747$	0	0
$748$	−2.08613e6	−0.136329
$749$	5.79784e6	0.377626
$750$	0	0
$751$	−1.51816e7	−0.982243	−0.491122	−	0.871091i	$-0.663413\pi$
$751$	−1.51816e7	−0.982243	−0.491122	+	0.871091i	$0.663413\pi$
$752$	−1.33229e7	−0.859118
$753$	0	0
$754$	5.48856e6	0.351585
$755$	0	0
$756$	0	0
$757$	−652274.	−0.0413705	−0.0206852	−	0.999786i	$-0.506585\pi$
$757$	−652274.	−0.0413705	−0.0206852	+	0.999786i	$0.506585\pi$
$758$	3.37859e7	2.13581
$759$	0	0
$760$	0	0
$761$	−4.51420e6	−0.282566	−0.141283	−	0.989969i	$-0.545123\pi$
$761$	−4.51420e6	−0.282566	−0.141283	+	0.989969i	$0.545123\pi$
$762$	0	0
$763$	−1.25428e6	−0.0779980
$764$	−3.98822e6	−0.247199
$765$	0	0
$766$	−2.10596e7	−1.29681
$767$	−2.94437e6	−0.180719
$768$	0	0
$769$	1.20799e7	0.736625	0.368312	−	0.929702i	$-0.379936\pi$
$769$	1.20799e7	0.736625	0.368312	+	0.929702i	$0.379936\pi$
$770$	0	0
$771$	0	0
$772$	−9.43344e6	−0.569674
$773$	−1.04245e7	−0.627492	−0.313746	−	0.949507i	$-0.601584\pi$
$773$	−1.04245e7	−0.627492	−0.313746	+	0.949507i	$0.601584\pi$
$774$	0	0
$775$	0	0
$776$	−1.34323e7	−0.800750
$777$	0	0
$778$	−1.19393e7	−0.707177
$779$	43560.0	0.00257184
$780$	0	0
$781$	1.34447e7	0.788721
$782$	−1.11398e7	−0.651421
$783$	0	0
$784$	1.67392e7	0.972620
$785$	0	0
$786$	0	0
$787$	3.45366e7	1.98766	0.993830	−	0.110913i	$-0.0353776\pi$
$787$	3.45366e7	1.98766	0.993830	+	0.110913i	$0.0353776\pi$
$788$	−715277.	−0.0410354
$789$	0	0
$790$	0	0
$791$	−6.26947e6	−0.356279
$792$	0	0
$793$	−680333.	−0.0384183
$794$	3.25298e7	1.83118
$795$	0	0
$796$	4.74960e6	0.265689
$797$	2.09287e7	1.16707	0.583533	−	0.812089i	$-0.301670\pi$
$797$	2.09287e7	1.16707	0.583533	+	0.812089i	$0.301670\pi$
$798$	0	0
$799$	7.27126e6	0.402942
$800$	0	0
$801$	0	0
$802$	−4.26384e6	−0.234080
$803$	1.78725e7	0.978131
$804$	0	0
$805$	0	0
$806$	−5.34758e6	−0.289948
$807$	0	0
$808$	1.44735e7	0.779910
$809$	2.48797e7	1.33651	0.668257	−	0.743930i	$-0.267041\pi$
$809$	2.48797e7	1.33651	0.668257	+	0.743930i	$0.267041\pi$
$810$	0	0
$811$	−3.95415e6	−0.211106	−0.105553	−	0.994414i	$-0.533661\pi$
$811$	−3.95415e6	−0.211106	−0.105553	+	0.994414i	$0.533661\pi$
$812$	4.96459e6	0.264237
$813$	0	0
$814$	2.33513e7	1.23524
$815$	0	0
$816$	0	0
$817$	91936.8	0.00481875
$818$	−1.36453e7	−0.713018
$819$	0	0
$820$	0	0
$821$	3.43550e6	0.177882	0.0889410	−	0.996037i	$-0.471652\pi$
$821$	3.43550e6	0.177882	0.0889410	+	0.996037i	$0.471652\pi$
$822$	0	0
$823$	3.94833e6	0.203195	0.101598	−	0.994826i	$-0.467605\pi$
$823$	3.94833e6	0.203195	0.101598	+	0.994826i	$0.467605\pi$
$824$	−9.36936e6	−0.480720
$825$	0	0
$826$	−9.76536e6	−0.498010
$827$	3.38176e7	1.71941	0.859705	−	0.510791i	$-0.170647\pi$
$827$	3.38176e7	1.71941	0.859705	+	0.510791i	$0.170647\pi$
$828$	0	0
$829$	−1.52015e7	−0.768244	−0.384122	−	0.923282i	$-0.625496\pi$
$829$	−1.52015e7	−0.768244	−0.384122	+	0.923282i	$0.625496\pi$
$830$	0	0
$831$	0	0
$832$	1.55123e6	0.0776903
$833$	−9.13579e6	−0.456177
$834$	0	0
$835$	0	0
$836$	−665280.	−0.0329222
$837$	0	0
$838$	1.94846e7	0.958478
$839$	2.89012e7	1.41746	0.708729	−	0.705481i	$-0.249269\pi$
$839$	2.89012e7	1.41746	0.708729	+	0.705481i	$0.249269\pi$
$840$	0	0
$841$	2.75138e7	1.34140
$842$	−1.80272e7	−0.876290
$843$	0	0
$844$	−3.01858e6	−0.145863
$845$	0	0
$846$	0	0
$847$	5.82348e6	0.278917
$848$	7.36153e6	0.351543
$849$	0	0
$850$	0	0
$851$	3.40077e7	1.60973
$852$	0	0
$853$	2.02107e7	0.951062	0.475531	−	0.879699i	$-0.342256\pi$
$853$	2.02107e7	0.951062	0.475531	+	0.879699i	$0.342256\pi$
$854$	−2.25641e6	−0.105870
$855$	0	0
$856$	−1.28841e7	−0.600992
$857$	−1.70522e7	−0.793101	−0.396550	−	0.918013i	$-0.629793\pi$
$857$	−1.70522e7	−0.793101	−0.396550	+	0.918013i	$0.629793\pi$
$858$	0	0
$859$	−1.95505e7	−0.904015	−0.452008	−	0.892014i	$-0.649292\pi$
$859$	−1.95505e7	−0.904015	−0.452008	+	0.892014i	$0.649292\pi$
$860$	0	0
$861$	0	0
$862$	3.31288e7	1.51858
$863$	2.70896e7	1.23816	0.619078	−	0.785330i	$-0.287507\pi$
$863$	2.70896e7	1.23816	0.619078	+	0.785330i	$0.287507\pi$
$864$	0	0
$865$	0	0
$866$	−1.38093e7	−0.625716
$867$	0	0
$868$	−4.83707e6	−0.217913
$869$	1.30838e7	0.587741
$870$	0	0
$871$	−5.21057e6	−0.232723
$872$	2.78729e6	0.124134
$873$	0	0
$874$	−3.55256e6	−0.157312
$875$	0	0
$876$	0	0
$877$	1.98285e6	0.0870545	0.0435272	−	0.999052i	$-0.486140\pi$
$877$	1.98285e6	0.0870545	0.0435272	+	0.999052i	$0.486140\pi$
$878$	−3.12031e7	−1.36603
$879$	0	0
$880$	0	0
$881$	4.22840e7	1.83542	0.917712	−	0.397247i	$-0.130034\pi$
$881$	4.22840e7	1.83542	0.917712	+	0.397247i	$0.130034\pi$
$882$	0	0
$883$	−134502.	−0.00580535	−0.00290267	−	0.999996i	$-0.500924\pi$
$883$	−134502.	−0.00580535	−0.00290267	+	0.999996i	$0.500924\pi$
$884$	988416.	0.0425411
$885$	0	0
$886$	3.78164e7	1.61844
$887$	−3.87668e6	−0.165444	−0.0827219	−	0.996573i	$-0.526361\pi$
$887$	−3.87668e6	−0.165444	−0.0827219	+	0.996573i	$0.526361\pi$
$888$	0	0
$889$	5.20700e6	0.220970
$890$	0	0
$891$	0	0
$892$	3.46518e6	0.145819
$893$	2.31885e6	0.0973070
$894$	0	0
$895$	0	0
$896$	1.30522e7	0.543141
$897$	0	0
$898$	−4.11477e7	−1.70277
$899$	−4.67914e7	−1.93093
$900$	0	0
$901$	−4.01773e6	−0.164880
$902$	330973.	0.0135449
$903$	0	0
$904$	1.39322e7	0.567019
$905$	0	0
$906$	0	0
$907$	−2.87363e7	−1.15988	−0.579939	−	0.814660i	$-0.696924\pi$
$907$	−2.87363e7	−1.15988	−0.579939	+	0.814660i	$0.696924\pi$
$908$	−1.39697e7	−0.562306
$909$	0	0
$910$	0	0
$911$	−1.87675e6	−0.0749223	−0.0374611	−	0.999298i	$-0.511927\pi$
$911$	−1.87675e6	−0.0749223	−0.0374611	+	0.999298i	$0.511927\pi$
$912$	0	0
$913$	1.55841e7	0.618736
$914$	1.42861e7	0.565650
$915$	0	0
$916$	−6.57204e6	−0.258798
$917$	1.15131e7	0.452137
$918$	0	0
$919$	6.76852e6	0.264366	0.132183	−	0.991225i	$-0.457801\pi$
$919$	6.76852e6	0.264366	0.132183	+	0.991225i	$0.457801\pi$
$920$	0	0
$921$	0	0
$922$	−2.55466e7	−0.989706
$923$	−6.37015e6	−0.246119
$924$	0	0
$925$	0	0
$926$	−1.38113e7	−0.529306
$927$	0	0
$928$	−2.86843e7	−1.09339
$929$	1.15356e7	0.438530	0.219265	−	0.975665i	$-0.429634\pi$
$929$	1.15356e7	0.438530	0.219265	+	0.975665i	$0.429634\pi$
$930$	0	0
$931$	−2.91346e6	−0.110163
$932$	577252.	0.0217684
$933$	0	0
$934$	8.67772e6	0.325491
$935$	0	0
$936$	0	0
$937$	3.92632e7	1.46096	0.730478	−	0.682936i	$-0.239297\pi$
$937$	3.92632e7	1.46096	0.730478	+	0.682936i	$0.239297\pi$
$938$	−1.72815e7	−0.641319
$939$	0	0
$940$	0	0
$941$	−2.94919e7	−1.08575	−0.542874	−	0.839814i	$-0.682664\pi$
$941$	−2.94919e7	−1.08575	−0.542874	+	0.839814i	$0.682664\pi$
$942$	0	0
$943$	482012.	0.0176514
$944$	3.11702e7	1.13844
$945$	0	0
$946$	698544.	0.0253785
$947$	−2.09628e7	−0.759581	−0.379791	−	0.925072i	$-0.624004\pi$
$947$	−2.09628e7	−0.759581	−0.379791	+	0.925072i	$0.624004\pi$
$948$	0	0
$949$	−8.46806e6	−0.305224
$950$	0	0
$951$	0	0
$952$	−5.46368e6	−0.195386
$953$	−1.64122e7	−0.585375	−0.292687	−	0.956208i	$-0.594549\pi$
$953$	−1.64122e7	−0.585375	−0.292687	+	0.956208i	$0.594549\pi$
$954$	0	0
$955$	0	0
$956$	−1.20701e7	−0.427135
$957$	0	0
$958$	4.48652e7	1.57941
$959$	−8.57102e6	−0.300944
$960$	0	0
$961$	1.69604e7	0.592416
$962$	−1.10639e7	−0.385454
$963$	0	0
$964$	1.07424e7	0.372314
$965$	0	0
$966$	0	0
$967$	−4.71911e7	−1.62291	−0.811454	−	0.584416i	$-0.801324\pi$
$967$	−4.71911e7	−1.62291	−0.811454	+	0.584416i	$0.801324\pi$
$968$	−1.29411e7	−0.443897
$969$	0	0
$970$	0	0
$971$	−3.84771e7	−1.30965	−0.654823	−	0.755783i	$-0.727257\pi$
$971$	−3.84771e7	−1.30965	−0.654823	+	0.755783i	$0.727257\pi$
$972$	0	0
$973$	−1.90047e7	−0.643544
$974$	−4.42566e7	−1.49479
$975$	0	0
$976$	7.20227e6	0.242016
$977$	−2.70184e7	−0.905572	−0.452786	−	0.891619i	$-0.649570\pi$
$977$	−2.70184e7	−0.905572	−0.452786	+	0.891619i	$0.649570\pi$
$978$	0	0
$979$	2.51748e6	0.0839478
$980$	0	0
$981$	0	0
$982$	−4.56086e7	−1.50927
$983$	2.88475e7	0.952192	0.476096	−	0.879393i	$-0.342051\pi$
$983$	2.88475e7	0.952192	0.476096	+	0.879393i	$0.342051\pi$
$984$	0	0
$985$	0	0
$986$	3.17117e7	1.03879
$987$	0	0
$988$	315212.	0.0102733
$989$	1.01732e6	0.0330726
$990$	0	0
$991$	−5.21596e7	−1.68714	−0.843569	−	0.537021i	$-0.819550\pi$
$991$	−5.21596e7	−1.68714	−0.843569	+	0.537021i	$0.819550\pi$
$992$	2.79475e7	0.901704
$993$	0	0
$994$	−2.11274e7	−0.678235
$995$	0	0
$996$	0	0
$997$	−9.78148e6	−0.311650	−0.155825	−	0.987785i	$-0.549804\pi$
$997$	−9.78148e6	−0.311650	−0.155825	+	0.987785i	$0.549804\pi$
$998$	4.60354e7	1.46307
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	225.6.a.n.1.1		2
3.2	odd	2		25.6.a.c.1.2		2
5.2	odd	4		45.6.b.b.19.1		2
5.3	odd	4		45.6.b.b.19.2		2
5.4	even	2	inner	225.6.a.n.1.2		2
12.11	even	2		400.6.a.t.1.1		2
15.2	even	4		5.6.b.a.4.2	yes	2
15.8	even	4		5.6.b.a.4.1	✓	2
15.14	odd	2		25.6.a.c.1.1		2
20.3	even	4		720.6.f.f.289.2		2
20.7	even	4		720.6.f.f.289.1		2
60.23	odd	4		80.6.c.a.49.1		2
60.47	odd	4		80.6.c.a.49.2		2
60.59	even	2		400.6.a.t.1.2		2
105.62	odd	4		245.6.b.a.99.2		2
105.83	odd	4		245.6.b.a.99.1		2
120.53	even	4		320.6.c.f.129.1		2
120.77	even	4		320.6.c.f.129.2		2
120.83	odd	4		320.6.c.g.129.2		2
120.107	odd	4		320.6.c.g.129.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5.6.b.a.4.1	✓	2	15.8	even	4
5.6.b.a.4.2	yes	2	15.2	even	4
25.6.a.c.1.1		2	15.14	odd	2
25.6.a.c.1.2		2	3.2	odd	2
45.6.b.b.19.1		2	5.2	odd	4
45.6.b.b.19.2		2	5.3	odd	4
80.6.c.a.49.1		2	60.23	odd	4
80.6.c.a.49.2		2	60.47	odd	4
225.6.a.n.1.1		2	1.1	even	1	trivial
225.6.a.n.1.2		2	5.4	even	2	inner
245.6.b.a.99.1		2	105.83	odd	4
245.6.b.a.99.2		2	105.62	odd	4
320.6.c.f.129.1		2	120.53	even	4
320.6.c.f.129.2		2	120.77	even	4
320.6.c.g.129.1		2	120.107	odd	4
320.6.c.g.129.2		2	120.83	odd	4
400.6.a.t.1.1		2	12.11	even	2
400.6.a.t.1.2		2	60.59	even	2
720.6.f.f.289.1		2	20.7	even	4
720.6.f.f.289.2		2	20.3	even	4

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)

\(f(q)\)	\(=\)	\(q-6.63325 q^{2} +12.0000 q^{4} -59.6992 q^{7} +132.665 q^{8} +O(q^{10})\)	\(q-6.63325 q^{2} +12.0000 q^{4} -59.6992 q^{7} +132.665 q^{8} -252.000 q^{11} +119.398 q^{13} +396.000 q^{14} -1264.00 q^{16} +689.858 q^{17} +220.000 q^{19} +1671.58 q^{22} +2434.40 q^{23} -792.000 q^{26} -716.391 q^{28} -6930.00 q^{29} +6752.00 q^{31} +4139.15 q^{32} -4576.00 q^{34} +13969.6 q^{37} -1459.31 q^{38} +198.000 q^{41} +417.895 q^{43} -3024.00 q^{44} -16148.0 q^{46} +10540.2 q^{47} -13243.0 q^{49} +1432.78 q^{52} -5823.99 q^{53} -7920.00 q^{56} +45968.4 q^{58} -24660.0 q^{59} -5698.00 q^{61} -44787.7 q^{62} +12992.0 q^{64} -43640.1 q^{67} +8278.30 q^{68} -53352.0 q^{71} -70922.7 q^{73} -92664.0 q^{74} +2640.00 q^{76} +15044.2 q^{77} -51920.0 q^{79} -1313.38 q^{82} -61841.8 q^{83} -2772.00 q^{86} -33431.6 q^{88} -9990.00 q^{89} -7128.00 q^{91} +29212.8 q^{92} -69916.0 q^{94} -101250. q^{97} +87844.1 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 24 q^{4}+O(q^{10}) \) 2 * q + 24 * q^4	\( 2 q + 24 q^{4} - 504 q^{11} + 792 q^{14} - 2528 q^{16} + 440 q^{19} - 1584 q^{26} - 13860 q^{29} + 13504 q^{31} - 9152 q^{34} + 396 q^{41} - 6048 q^{44} - 32296 q^{46} - 26486 q^{49} - 15840 q^{56} - 49320 q^{59} - 11396 q^{61} + 25984 q^{64} - 106704 q^{71} - 185328 q^{74} + 5280 q^{76} - 103840 q^{79} - 5544 q^{86} - 19980 q^{89} - 14256 q^{91} - 139832 q^{94}+O(q^{100}) \) 2 * q + 24 * q^4 - 504 * q^11 + 792 * q^14 - 2528 * q^16 + 440 * q^19 - 1584 * q^26 - 13860 * q^29 + 13504 * q^31 - 9152 * q^34 + 396 * q^41 - 6048 * q^44 - 32296 * q^46 - 26486 * q^49 - 15840 * q^56 - 49320 * q^59 - 11396 * q^61 + 25984 * q^64 - 106704 * q^71 - 185328 * q^74 + 5280 * q^76 - 103840 * q^79 - 5544 * q^86 - 19980 * q^89 - 14256 * q^91 - 139832 * q^94

Introduction

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