LMFDB - Embedded newform 25.6.a.c.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [25,6,Mod(1,25)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(25, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("25.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 25 = 5^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	25.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$4.00959549532$
Analytic rank:	$0$
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			1.2
Root			$3.31662$ of defining polynomial
Character	$\chi$	$=$	25.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+6.63325 q^{2} +19.8997 q^{3} +12.0000 q^{4} +132.000 q^{6} -59.6992 q^{7} -132.665 q^{8} +153.000 q^{9} +252.000 q^{11} +238.797 q^{12} +119.398 q^{13} -396.000 q^{14} -1264.00 q^{16} -689.858 q^{17} +1014.89 q^{18} +220.000 q^{19} -1188.00 q^{21} +1671.58 q^{22} -2434.40 q^{23} -2640.00 q^{24} +792.000 q^{26} -1790.98 q^{27} -716.391 q^{28} +6930.00 q^{29} +6752.00 q^{31} -4139.15 q^{32} +5014.74 q^{33} -4576.00 q^{34} +1836.00 q^{36} +13969.6 q^{37} +1459.31 q^{38} +2376.00 q^{39} -198.000 q^{41} -7880.30 q^{42} +417.895 q^{43} +3024.00 q^{44} -16148.0 q^{46} -10540.2 q^{47} -25153.3 q^{48} -13243.0 q^{49} -13728.0 q^{51} +1432.78 q^{52} +5823.99 q^{53} -11880.0 q^{54} +7920.00 q^{56} +4377.94 q^{57} +45968.4 q^{58} +24660.0 q^{59} -5698.00 q^{61} +44787.7 q^{62} -9133.98 q^{63} +12992.0 q^{64} +33264.0 q^{66} -43640.1 q^{67} -8278.30 q^{68} -48444.0 q^{69} +53352.0 q^{71} -20297.7 q^{72} -70922.7 q^{73} +92664.0 q^{74} +2640.00 q^{76} -15044.2 q^{77} +15760.6 q^{78} -51920.0 q^{79} -72819.0 q^{81} -1313.38 q^{82} +61841.8 q^{83} -14256.0 q^{84} +2772.00 q^{86} +137905. q^{87} -33431.6 q^{88} +9990.00 q^{89} -7128.00 q^{91} -29212.8 q^{92} +134363. q^{93} -69916.0 q^{94} -82368.0 q^{96} -101250. q^{97} -87844.1 q^{98} +38556.0 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 24 q^{4} + 264 q^{6} + 306 q^{9} + 504 q^{11} - 792 q^{14} - 2528 q^{16} + 440 q^{19} - 2376 q^{21} - 5280 q^{24} + 1584 q^{26} + 13860 q^{29} + 13504 q^{31} - 9152 q^{34} + 3672 q^{36} + 4752 q^{39}+ \cdots + 77112 q^{99}+O(q^{100}) $ 2 * q + 24 * q^4 + 264 * q^6 + 306 * q^9 + 504 * q^11 - 792 * q^14 - 2528 * q^16 + 440 * q^19 - 2376 * q^21 - 5280 * q^24 + 1584 * q^26 + 13860 * q^29 + 13504 * q^31 - 9152 * q^34 + 3672 * q^36 + 4752 * q^39 - 396 * q^41 + 6048 * q^44 - 32296 * q^46 - 26486 * q^49 - 27456 * q^51 - 23760 * q^54 + 15840 * q^56 + 49320 * q^59 - 11396 * q^61 + 25984 * q^64 + 66528 * q^66 - 96888 * q^69 + 106704 * q^71 + 185328 * q^74 + 5280 * q^76 - 103840 * q^79 - 145638 * q^81 - 28512 * q^84 + 5544 * q^86 + 19980 * q^89 - 14256 * q^91 - 139832 * q^94 - 164736 * q^96 + 77112 * q^99

Coefficient data

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

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

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

$2$	6.63325	1.17260	0.586302	−	0.810093i	$-0.300583\pi$
$2$	6.63325	1.17260	0.586302	+	0.810093i	$0.300583\pi$
$3$	19.8997	1.27657	0.638285	−	0.769800i	$-0.279644\pi$
$3$	19.8997	1.27657	0.638285	+	0.769800i	$0.279644\pi$
$4$	12.0000	0.375000
$5$	0	0
$5$	0	0
$6$	132.000	1.49691
$7$	−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
$9$	153.000	0.629630
$10$	0	0
$11$	252.000	0.627941	0.313970	−	0.949433i	$-0.398341\pi$
$11$	252.000	0.627941	0.313970	+	0.949433i	$0.398341\pi$
$12$	238.797	0.478714
$13$	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	$-0.593480\pi$
$17$	−689.858	−0.578945	−0.289473	+	0.957186i	$0.593480\pi$
$18$	1014.89	0.738306
$19$	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
$21$	−1188.00	−0.587852
$22$	1671.58	0.736326
$23$	−2434.40	−0.959561	−0.479781	−	0.877388i	$-0.659284\pi$
$23$	−2434.40	−0.959561	−0.479781	+	0.877388i	$0.659284\pi$
$24$	−2640.00	−0.935569
$25$	0	0
$26$	792.000	0.229769
$27$	−1790.98	−0.472804
$28$	−716.391	−0.172685
$29$	6930.00	1.53016	0.765082	−	0.643932i	$-0.222698\pi$
$29$	6930.00	1.53016	0.765082	+	0.643932i	$0.222698\pi$
$30$	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
$33$	5014.74	0.801610
$34$	−4576.00	−0.678873
$35$	0	0
$36$	1836.00	0.236111
$37$	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
$39$	2376.00	0.250141
$40$	0	0
$41$	−198.000	−0.0183952	−0.00919762	−	0.999958i	$-0.502928\pi$
$41$	−198.000	−0.0183952	−0.00919762	+	0.999958i	$0.502928\pi$
$42$	−7880.30	−0.689318
$43$	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	$-0.613138\pi$
$47$	−10540.2	−0.695994	−0.347997	+	0.937496i	$0.613138\pi$
$48$	−25153.3	−1.57577
$49$	−13243.0	−0.787945
$50$	0	0
$51$	−13728.0	−0.739064
$52$	1432.78	0.0734804
$53$	5823.99	0.284794	0.142397	−	0.989810i	$-0.454519\pi$
$53$	5823.99	0.284794	0.142397	+	0.989810i	$0.454519\pi$
$54$	−11880.0	−0.554411
$55$	0	0
$56$	7920.00	0.337485
$57$	4377.94	0.178477
$58$	45968.4	1.79428
$59$	24660.0	0.922281	0.461140	−	0.887327i	$-0.347440\pi$
$59$	24660.0	0.922281	0.461140	+	0.887327i	$0.347440\pi$
$60$	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
$63$	−9133.98	−0.289941
$64$	12992.0	0.396484
$65$	0	0
$66$	33264.0	0.939971
$67$	−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
$69$	−48444.0	−1.22495
$70$	0	0
$71$	53352.0	1.25604	0.628022	−	0.778196i	$-0.283865\pi$
$71$	53352.0	1.25604	0.628022	+	0.778196i	$0.283865\pi$
$72$	−20297.7	−0.461441
$73$	−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
$78$	15760.6	0.293316
$79$	−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
$81$	−72819.0	−1.23320
$82$	−1313.38	−0.0215703
$83$	61841.8	0.985342	0.492671	−	0.870216i	$-0.336021\pi$
$83$	61841.8	0.985342	0.492671	+	0.870216i	$0.336021\pi$
$84$	−14256.0	−0.220445
$85$	0	0
$86$	2772.00	0.0404154
$87$	137905.	1.95336
$88$	−33431.6	−0.460204
$89$	9990.00	0.133687	0.0668437	−	0.997763i	$-0.478707\pi$
$89$	9990.00	0.133687	0.0668437	+	0.997763i	$0.478707\pi$
$90$	0	0
$91$	−7128.00	−0.0902328
$92$	−29212.8	−0.359836
$93$	134363.	1.61092
$94$	−69916.0	−0.816125
$95$	0	0
$96$	−82368.0	−0.912180
$97$	−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
$99$	38556.0	0.395370
$100$	0	0
$101$	−109098.	−1.06418	−0.532088	−	0.846689i	$-0.678592\pi$
$101$	−109098.	−1.06418	−0.532088	+	0.846689i	$0.678592\pi$
$102$	−91061.3	−0.866629
$103$	−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	$-0.365521\pi$
$107$	97117.4	0.820045	0.410022	+	0.912075i	$0.365521\pi$
$108$	−21491.7	−0.177301
$109$	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
$111$	277992.	2.14153
$112$	75459.8	0.568422
$113$	−105018.	−0.773688	−0.386844	−	0.922145i	$-0.626435\pi$
$113$	−105018.	−0.773688	−0.386844	+	0.922145i	$0.626435\pi$
$114$	29040.0	0.209283
$115$	0	0
$116$	83160.0	0.573812
$117$	18268.0	0.123375
$118$	163576.	1.08147
$119$	41184.0	0.266601
$120$	0	0
$121$	−97547.0	−0.605690
$122$	−37796.3	−0.229905
$123$	−3940.15	−0.0234828
$124$	81024.0	0.473216
$125$	0	0
$126$	−60588.0	−0.339985
$127$	−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
$129$	8316.00	0.0439987
$130$	0	0
$131$	192852.	0.981852	0.490926	−	0.871201i	$-0.336659\pi$
$131$	192852.	0.981852	0.490926	+	0.871201i	$0.336659\pi$
$132$	60176.8	0.300604
$133$	−13133.8	−0.0643817
$134$	−289476.	−1.39268
$135$	0	0
$136$	91520.0	0.424296
$137$	−143570.	−0.653525	−0.326763	−	0.945106i	$-0.605958\pi$
$137$	−143570.	−0.653525	−0.326763	+	0.945106i	$0.605958\pi$
$138$	−321341.	−1.43638
$139$	318340.	1.39751	0.698754	−	0.715362i	$-0.253738\pi$
$139$	318340.	1.39751	0.698754	+	0.715362i	$0.253738\pi$
$140$	0	0
$141$	−209748.	−0.888485
$142$	353897.	1.47284
$143$	30088.4	0.123044
$144$	−193392.	−0.777199
$145$	0	0
$146$	−470448.	−1.82654
$147$	−263532.	−1.00587
$148$	167635.	0.629088
$149$	−84150.0	−0.310519	−0.155260	−	0.987874i	$-0.549621\pi$
$149$	−84150.0	−0.310519	−0.155260	+	0.987874i	$0.549621\pi$
$150$	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
$153$	−105548.	−0.364521
$154$	−99792.0	−0.339074
$155$	0	0
$156$	28512.0	0.0938029
$157$	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
$159$	115896.	0.363560
$160$	0	0
$161$	145332.	0.441872
$162$	−483027.	−1.44605
$163$	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	$-0.507995\pi$
$167$	−18102.1	−0.0502272	−0.0251136	+	0.999685i	$0.507995\pi$
$168$	157606.	0.430824
$169$	−357037.	−0.961604
$170$	0	0
$171$	33660.0	0.0880286
$172$	5014.74	0.0129249
$173$	492572.	1.25128	0.625640	−	0.780112i	$-0.284838\pi$
$173$	492572.	1.25128	0.625640	+	0.780112i	$0.284838\pi$
$174$	914760.	2.29052
$175$	0	0
$176$	−318528.	−0.775115
$177$	490728.	1.17736
$178$	66266.2	0.156762
$179$	−444420.	−1.03672	−0.518359	−	0.855163i	$-0.673457\pi$
$179$	−444420.	−1.03672	−0.518359	+	0.855163i	$0.673457\pi$
$180$	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
$183$	−113389.	−0.250289
$184$	322960.	0.703241
$185$	0	0
$186$	891264.	1.88897
$187$	−173844.	−0.363543
$188$	−126483.	−0.260998
$189$	106920.	0.217723
$190$	0	0
$191$	332352.	0.659196	0.329598	−	0.944121i	$-0.393087\pi$
$191$	332352.	0.659196	0.329598	+	0.944121i	$0.393087\pi$
$192$	258538.	0.506140
$193$	−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	$-0.482575\pi$
$197$	59606.4	0.109428	0.0547138	+	0.998502i	$0.482575\pi$
$198$	255752.	0.463613
$199$	395800.	0.708505	0.354253	−	0.935150i	$-0.384735\pi$
$199$	395800.	0.708505	0.354253	+	0.935150i	$0.384735\pi$
$200$	0	0
$201$	−868428.	−1.51616
$202$	−723674.	−1.24786
$203$	−413716.	−0.704631
$204$	−164736.	−0.277149
$205$	0	0
$206$	−468468.	−0.769151
$207$	−372464.	−0.604168
$208$	−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
$213$	1.06169e6	1.60343
$214$	644204.	0.961588
$215$	0	0
$216$	237600.	0.346507
$217$	−403089.	−0.581101
$218$	139365.	0.198615
$219$	−1.41134e6	−1.98849
$220$	0	0
$221$	−82368.0	−0.113443
$222$	1.84399e6	2.51117
$223$	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	$-0.230178\pi$
$227$	1.16414e6	1.49948	0.749741	+	0.661731i	$0.230178\pi$
$228$	52535.3	0.0669290
$229$	−547670.	−0.690129	−0.345064	−	0.938579i	$-0.612143\pi$
$229$	−547670.	−0.690129	−0.345064	+	0.938579i	$0.612143\pi$
$230$	0	0
$231$	−299376.	−0.369137
$232$	−919368.	−1.12142
$233$	−48104.3	−0.0580489	−0.0290245	−	0.999579i	$-0.509240\pi$
$233$	−48104.3	−0.0580489	−0.0290245	+	0.999579i	$0.509240\pi$
$234$	121176.	0.144669
$235$	0	0
$236$	295920.	0.345855
$237$	−1.03319e6	−1.19484
$238$	273184.	0.312617
$239$	1.00584e6	1.13903	0.569514	−	0.821982i	$-0.307132\pi$
$239$	1.00584e6	1.13903	0.569514	+	0.821982i	$0.307132\pi$
$240$	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
$243$	−1.01387e6	−1.10146
$244$	−68376.0	−0.0735240
$245$	0	0
$246$	−26136.0	−0.0275360
$247$	26267.7	0.0273955
$248$	−895754.	−0.924825
$249$	1.23064e6	1.25786
$250$	0	0
$251$	558252.	0.559301	0.279651	−	0.960102i	$-0.409781\pi$
$251$	558252.	0.559301	0.279651	+	0.960102i	$0.409781\pi$
$252$	−109608.	−0.108728
$253$	−613469.	−0.602548
$254$	−578556.	−0.562680
$255$	0	0
$256$	1.03450e6	0.986572
$257$	−787924.	−0.744135	−0.372067	−	0.928206i	$-0.621351\pi$
$257$	−787924.	−0.744135	−0.372067	+	0.928206i	$0.621351\pi$
$258$	55162.1	0.0515931
$259$	−833976.	−0.772510
$260$	0	0
$261$	1.06029e6	0.963437
$262$	1.27924e6	1.15132
$263$	1.63173e6	1.45465	0.727327	−	0.686291i	$-0.240762\pi$
$263$	1.63173e6	1.45465	0.727327	+	0.686291i	$0.240762\pi$
$264$	−665280.	−0.587482
$265$	0	0
$266$	−87120.0	−0.0754942
$267$	198798.	0.170661
$268$	−523682.	−0.445380
$269$	1.73637e6	1.46306	0.731529	−	0.681810i	$-0.238807\pi$
$269$	1.73637e6	1.46306	0.731529	+	0.681810i	$0.238807\pi$
$270$	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
$273$	−141845.	−0.115188
$274$	−952336.	−0.766326
$275$	0	0
$276$	−581328.	−0.459355
$277$	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
$279$	1.03306e6	0.794536
$280$	0	0
$281$	1.46500e6	1.10681	0.553404	−	0.832913i	$-0.313329\pi$
$281$	1.46500e6	1.10681	0.553404	+	0.832913i	$0.313329\pi$
$282$	−1.39131e6	−1.04184
$283$	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
$288$	−633290.	−0.449905
$289$	−943953.	−0.664823
$290$	0	0
$291$	−2.01485e6	−1.39479
$292$	−851072.	−0.584130
$293$	−2.38772e6	−1.62485	−0.812426	−	0.583064i	$-0.801854\pi$
$293$	−2.38772e6	−1.62485	−0.812426	+	0.583064i	$0.801854\pi$
$294$	−1.74808e6	−1.17948
$295$	0	0
$296$	−1.85328e6	−1.22945
$297$	−451326.	−0.296893
$298$	−558188.	−0.364116
$299$	−290664.	−0.188024
$300$	0	0
$301$	−24948.0	−0.0158716
$302$	−1.03378e6	−0.652244
$303$	−2.17102e6	−1.35849
$304$	−278080.	−0.172578
$305$	0	0
$306$	−700128.	−0.427439
$307$	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
$309$	−1.40540e6	−0.837346
$310$	0	0
$311$	568152.	0.333092	0.166546	−	0.986034i	$-0.446739\pi$
$311$	568152.	0.333092	0.166546	+	0.986034i	$0.446739\pi$
$312$	−315212.	−0.183323
$313$	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	$-0.488287\pi$
$317$	131643.	0.0735785	0.0367893	+	0.999323i	$0.488287\pi$
$318$	768767.	0.426311
$319$	1.74636e6	0.960853
$320$	0	0
$321$	1.93261e6	1.04684
$322$	964023.	0.518141
$323$	−151769.	−0.0809424
$324$	−873828.	−0.462449
$325$	0	0
$326$	961092.	0.500865
$327$	418094.	0.216224
$328$	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
$333$	2.13735e6	1.05625
$334$	−120076.	−0.0588966
$335$	0	0
$336$	1.50163e6	0.725630
$337$	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
$339$	−2.08982e6	−0.987667
$340$	0	0
$341$	1.70150e6	0.792405
$342$	223275.	0.103223
$343$	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	$-0.827370\pi$
$347$	−3.84224e6	−1.71301	−0.856506	+	0.516137i	$0.827370\pi$
$348$	1.65486e6	0.732511
$349$	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
$351$	−213840.	−0.0926448
$352$	−1.04307e6	−0.448699
$353$	295365.	0.126160	0.0630802	−	0.998008i	$-0.479908\pi$
$353$	295365.	0.126160	0.0630802	+	0.998008i	$0.479908\pi$
$354$	3.25512e6	1.38057
$355$	0	0
$356$	119880.	0.0501328
$357$	819551.	0.340334
$358$	−2.94795e6	−1.21566
$359$	−1.10484e6	−0.452442	−0.226221	−	0.974076i	$-0.572637\pi$
$359$	−1.10484e6	−0.452442	−0.226221	+	0.974076i	$0.572637\pi$
$360$	0	0
$361$	−2.42770e6	−0.980453
$362$	1.04077e6	0.417430
$363$	−1.94116e6	−0.773206
$364$	−85536.0	−0.0338373
$365$	0	0
$366$	−752136.	−0.293490
$367$	−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
$369$	−30294.0	−0.0115822
$370$	0	0
$371$	−347688.	−0.131146
$372$	1.61236e6	0.604093
$373$	−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
$378$	709227.	0.255303
$379$	−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
$381$	−1.73567e6	−0.612568
$382$	2.20457e6	0.772976
$383$	−3.17485e6	−1.10593	−0.552964	−	0.833205i	$-0.686503\pi$
$383$	−3.17485e6	−1.10593	−0.552964	+	0.833205i	$0.686503\pi$
$384$	4.35072e6	1.50568
$385$	0	0
$386$	−5.21453e6	−1.78134
$387$	63937.9	0.0217011
$388$	−1.21500e6	−0.409729
$389$	−1.79991e6	−0.603083	−0.301541	−	0.953453i	$-0.597501\pi$
$389$	−1.79991e6	−0.603083	−0.301541	+	0.953453i	$0.597501\pi$
$390$	0	0
$391$	1.67939e6	0.555533
$392$	1.75688e6	0.577467
$393$	3.83771e6	1.25340
$394$	395384.	0.128315
$395$	0	0
$396$	462672.	0.148264
$397$	−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
$399$	−261360.	−0.0821877
$400$	0	0
$401$	−642798.	−0.199624	−0.0998122	−	0.995006i	$-0.531824\pi$
$401$	−642798.	−0.199624	−0.0998122	+	0.995006i	$0.531824\pi$
$402$	−5.76050e6	−1.77785
$403$	806179.	0.247268
$404$	−1.30918e6	−0.399066
$405$	0	0
$406$	−2.74428e6	−0.826254
$407$	3.52035e6	1.05341
$408$	1.82123e6	0.541643
$409$	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
$411$	−2.85701e6	−0.834271
$412$	−847490.	−0.245975
$413$	−1.47218e6	−0.424704
$414$	−2.47064e6	−0.708450
$415$	0	0
$416$	−494208.	−0.140016
$417$	6.33489e6	1.78402
$418$	367747.	0.102946
$419$	2.93742e6	0.817393	0.408697	−	0.912670i	$-0.365983\pi$
$419$	2.93742e6	0.817393	0.408697	+	0.912670i	$0.365983\pi$
$420$	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
$423$	−1.61266e6	−0.438219
$424$	−772640.	−0.208719
$425$	0	0
$426$	7.04246e6	1.88019
$427$	340166.	0.0902862
$428$	1.16541e6	0.307517
$429$	598752.	0.157074
$430$	0	0
$431$	4.99435e6	1.29505	0.647524	−	0.762045i	$-0.275804\pi$
$431$	4.99435e6	1.29505	0.647524	+	0.762045i	$0.275804\pi$
$432$	2.26380e6	0.583617
$433$	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
$438$	−9.36180e6	−2.33171
$439$	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
$441$	−2.02618e6	−0.496114
$442$	−546368.	−0.133024
$443$	5.70103e6	1.38021	0.690103	−	0.723711i	$-0.257565\pi$
$443$	5.70103e6	1.38021	0.690103	+	0.723711i	$0.257565\pi$
$444$	3.33590e6	0.803075
$445$	0	0
$446$	1.91545e6	0.455968
$447$	−1.67456e6	−0.396399
$448$	−775613.	−0.182579
$449$	−6.20325e6	−1.45212	−0.726062	−	0.687630i	$-0.758651\pi$
$449$	−6.20325e6	−1.45212	−0.726062	+	0.687630i	$0.758651\pi$
$450$	0	0
$451$	−49896.0	−0.0115511
$452$	−1.26021e6	−0.290133
$453$	−3.10134e6	−0.710074
$454$	7.72204e6	1.75830
$455$	0	0
$456$	−580800.	−0.130802
$457$	−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
$459$	1.23552e6	0.273727
$460$	0	0
$461$	−3.85130e6	−0.844024	−0.422012	−	0.906590i	$-0.638676\pi$
$461$	−3.85130e6	−0.844024	−0.422012	+	0.906590i	$0.638676\pi$
$462$	−1.98584e6	−0.432851
$463$	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	$-0.455679\pi$
$467$	1.30822e6	0.277579	0.138790	+	0.990322i	$0.455679\pi$
$468$	219216.	0.0462655
$469$	2.60528e6	0.546919
$470$	0	0
$471$	7.09711e6	1.47411
$472$	−3.27152e6	−0.675919
$473$	105309.	0.0216429
$474$	−6.85344e6	−1.40108
$475$	0	0
$476$	494208.	0.0999752
$477$	891071.	0.179315
$478$	6.67199e6	1.33563
$479$	6.76368e6	1.34693	0.673464	−	0.739220i	$-0.264806\pi$
$479$	6.76368e6	1.34693	0.673464	+	0.739220i	$0.264806\pi$
$480$	0	0
$481$	1.66795e6	0.328716
$482$	5.93810e6	1.16421
$483$	2.89207e6	0.564080
$484$	−1.17056e6	−0.227134
$485$	0	0
$486$	−6.72527e6	−1.29157
$487$	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
$489$	2.88328e6	0.545273
$490$	0	0
$491$	−6.87575e6	−1.28711	−0.643556	−	0.765399i	$-0.722542\pi$
$491$	−6.87575e6	−1.28711	−0.643556	+	0.765399i	$0.722542\pi$
$492$	−47281.8	−0.00880605
$493$	−4.78072e6	−0.885881
$494$	174240.	0.0321241
$495$	0	0
$496$	−8.53453e6	−1.55767
$497$	−3.18507e6	−0.578400
$498$	8.16312e6	1.47497
$499$	−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
$501$	−360228.	−0.0641185
$502$	3.70302e6	0.655839
$503$	−921007.	−0.162309	−0.0811546	−	0.996702i	$-0.525861\pi$
$503$	−921007.	−0.162309	−0.0811546	+	0.996702i	$0.525861\pi$
$504$	1.21176e6	0.212491
$505$	0	0
$506$	−4.06930e6	−0.706550
$507$	−7.10495e6	−1.22755
$508$	−1.04665e6	−0.179946
$509$	−4.97979e6	−0.851955	−0.425977	−	0.904734i	$-0.640070\pi$
$509$	−4.97979e6	−0.851955	−0.425977	+	0.904734i	$0.640070\pi$
$510$	0	0
$511$	4.23403e6	0.717302
$512$	−134151.	−0.0226161
$513$	−394015.	−0.0661027
$514$	−5.22650e6	−0.872575
$515$	0	0
$516$	99792.0	0.0164995
$517$	−2.65614e6	−0.437043
$518$	−5.53197e6	−0.905848
$519$	9.80206e6	1.59735
$520$	0	0
$521$	−147798.	−0.0238547	−0.0119274	−	0.999929i	$-0.503797\pi$
$521$	−147798.	−0.0238547	−0.0119274	+	0.999929i	$0.503797\pi$
$522$	7.03317e6	1.12973
$523$	−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
$528$	−6.33863e6	−0.989488
$529$	−510027.	−0.0792417
$530$	0	0
$531$	3.77298e6	0.580695
$532$	−157606.	−0.0241431
$533$	−23640.9	−0.00360451
$534$	1.31868e6	0.200118
$535$	0	0
$536$	5.78952e6	0.870423
$537$	−8.84385e6	−1.32344
$538$	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
$543$	3.12231e6	0.454440
$544$	2.85542e6	0.413688
$545$	0	0
$546$	−940896.	−0.135070
$547$	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
$549$	−871794.	−0.123448
$550$	0	0
$551$	1.52460e6	0.213933
$552$	6.42682e6	0.897736
$553$	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	$-0.480326\pi$
$557$	904550.	0.123536	0.0617681	+	0.998091i	$0.480326\pi$
$558$	6.85252e6	0.931676
$559$	49896.0	0.00675361
$560$	0	0
$561$	−3.45946e6	−0.464088
$562$	9.71772e6	1.29785
$563$	8.68719e6	1.15507	0.577535	−	0.816366i	$-0.304015\pi$
$563$	8.68719e6	1.15507	0.577535	+	0.816366i	$0.304015\pi$
$564$	−2.51698e6	−0.333182
$565$	0	0
$566$	1.09482e7	1.43649
$567$	4.34724e6	0.567879
$568$	−7.07794e6	−0.920526
$569$	2.27007e6	0.293940	0.146970	−	0.989141i	$-0.453048\pi$
$569$	2.27007e6	0.293940	0.146970	+	0.989141i	$0.453048\pi$
$570$	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
$573$	6.61372e6	0.841510
$574$	78408.0	0.00993300
$575$	0	0
$576$	1.98778e6	0.249638
$577$	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
$579$	−1.56436e7	−1.93928
$580$	0	0
$581$	−3.69191e6	−0.453744
$582$	−1.33650e7	−1.63554
$583$	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	$-0.475483\pi$
$587$	1.28473e6	0.153893	0.0769464	+	0.997035i	$0.475483\pi$
$588$	−3.16239e6	−0.377200
$589$	1.48544e6	0.176428
$590$	0	0
$591$	1.18615e6	0.139692
$592$	−1.76576e7	−2.07075
$593$	−7.00943e6	−0.818552	−0.409276	−	0.912411i	$-0.634219\pi$
$593$	−7.00943e6	−0.818552	−0.409276	+	0.912411i	$0.634219\pi$
$594$	−2.99376e6	−0.348138
$595$	0	0
$596$	−1.00980e6	−0.116445
$597$	7.87632e6	0.904456
$598$	−1.92805e6	−0.220478
$599$	8.80020e6	1.00213	0.501067	−	0.865409i	$-0.332941\pi$
$599$	8.80020e6	1.00213	0.501067	+	0.865409i	$0.332941\pi$
$600$	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
$603$	−6.67694e6	−0.747798
$604$	−1.87018e6	−0.208588
$605$	0	0
$606$	−1.44009e7	−1.59298
$607$	−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
$609$	−8.23284e6	−0.899511
$610$	0	0
$611$	−1.25849e6	−0.136379
$612$	−1.26658e6	−0.136695
$613$	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	$-0.278836\pi$
$617$	1.21083e7	1.28047	0.640237	+	0.768178i	$0.278836\pi$
$618$	−9.32240e6	−0.981875
$619$	−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
$621$	4.35996e6	0.453684
$622$	3.76869e6	0.390584
$623$	−596395.	−0.0615622
$624$	−3.00326e6	−0.308768
$625$	0	0
$626$	1.14254e7	1.16529
$627$	1.10324e6	0.112073
$628$	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
$633$	−5.00574e6	−0.496546
$634$	873224.	0.0862785
$635$	0	0
$636$	1.39075e6	0.136335
$637$	−1.58119e6	−0.154396
$638$	1.15840e7	1.12670
$639$	8.16286e6	0.790842
$640$	0	0
$641$	−6.42440e6	−0.617572	−0.308786	−	0.951132i	$-0.599923\pi$
$641$	−6.42440e6	−0.617572	−0.308786	+	0.951132i	$0.599923\pi$
$642$	1.28195e7	1.22753
$643$	−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	$-0.443193\pi$
$647$	3.78036e6	0.355036	0.177518	+	0.984118i	$0.443193\pi$
$648$	9.66053e6	0.903782
$649$	6.21432e6	0.579138
$650$	0	0
$651$	−8.02138e6	−0.741816
$652$	1.73868e6	0.160177
$653$	1.66957e7	1.53223	0.766113	−	0.642706i	$-0.222188\pi$
$653$	1.66957e7	1.53223	0.766113	+	0.642706i	$0.222188\pi$
$654$	2.77332e6	0.253545
$655$	0	0
$656$	250272.	0.0227066
$657$	−1.08512e7	−0.980761
$658$	4.17393e6	0.375821
$659$	1.22166e6	0.109581	0.0547907	−	0.998498i	$-0.482551\pi$
$659$	1.22166e6	0.109581	0.0547907	+	0.998498i	$0.482551\pi$
$660$	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
$663$	−1.63910e6	−0.144818
$664$	−8.20424e6	−0.722135
$665$	0	0
$666$	1.41776e7	1.23856
$667$	−1.68704e7	−1.46829
$668$	−217226.	−0.0188352
$669$	5.74636e6	0.496395
$670$	0	0
$671$	−1.43590e6	−0.123117
$672$	4.91731e6	0.420053
$673$	−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	$-0.464905\pi$
$677$	2.62429e6	0.220059	0.110030	+	0.993928i	$0.464905\pi$
$678$	−1.38623e7	−1.15814
$679$	6.04454e6	0.503140
$680$	0	0
$681$	2.31661e7	1.91419
$682$	1.12865e7	0.929177
$683$	−1.03039e7	−0.845184	−0.422592	−	0.906320i	$-0.638880\pi$
$683$	−1.03039e7	−0.845184	−0.422592	+	0.906320i	$0.638880\pi$
$684$	403920.	0.0330107
$685$	0	0
$686$	1.18998e7	0.965449
$687$	−1.08985e7	−0.880998
$688$	−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
$693$	−2.30176e6	−0.182066
$694$	−2.54865e7	−2.00868
$695$	0	0
$696$	−1.82952e7	−1.43157
$697$	136592.	0.0106498
$698$	1.05764e7	0.821672
$699$	−957264.	−0.0741035
$700$	0	0
$701$	−4.88090e6	−0.375150	−0.187575	−	0.982250i	$-0.560063\pi$
$701$	−4.88090e6	−0.375150	−0.187575	+	0.982250i	$0.560063\pi$
$702$	−1.41845e6	−0.108636
$703$	3.07332e6	0.234541
$704$	3.27398e6	0.248969
$705$	0	0
$706$	1.95923e6	0.147936
$707$	6.51307e6	0.490046
$708$	5.88873e6	0.441508
$709$	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
$711$	−7.94376e6	−0.589321
$712$	−1.32532e6	−0.0979765
$713$	−1.64371e7	−1.21088
$714$	5.43629e6	0.399077
$715$	0	0
$716$	−5.33304e6	−0.388770
$717$	2.00160e7	1.45405
$718$	−7.32868e6	−0.530536
$719$	1.19167e7	0.859675	0.429838	−	0.902906i	$-0.358571\pi$
$719$	1.19167e7	0.859675	0.429838	+	0.902906i	$0.358571\pi$
$720$	0	0
$721$	4.21621e6	0.302054
$722$	−1.61035e7	−1.14968
$723$	1.78143e7	1.26743
$724$	1.88282e6	0.133494
$725$	0	0
$726$	−1.28762e7	−0.906664
$727$	−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
$729$	−2.48079e6	−0.172890
$730$	0	0
$731$	−288288.	−0.0199541
$732$	−1.36067e6	−0.0938585
$733$	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
$738$	−200948.	−0.0135813
$739$	−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
$741$	522720.	0.0349723
$742$	−2.30630e6	−0.153782
$743$	1.57574e7	1.04716	0.523578	−	0.851978i	$-0.324597\pi$
$743$	1.57574e7	1.04716	0.523578	+	0.851978i	$0.324597\pi$
$744$	−1.78253e7	−1.18060
$745$	0	0
$746$	−1.94586e7	−1.28016
$747$	9.46179e6	0.620400
$748$	−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
$753$	1.11091e7	0.713987
$754$	5.48856e6	0.351585
$755$	0	0
$756$	1.28304e6	0.0816461
$757$	−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
$759$	−1.22079e7	−0.769194
$760$	0	0
$761$	4.51420e6	0.282566	0.141283	−	0.989969i	$-0.454877\pi$
$761$	4.51420e6	0.282566	0.141283	+	0.989969i	$0.454877\pi$
$762$	−1.15131e7	−0.718300
$763$	−1.25428e6	−0.0779980
$764$	3.98822e6	0.247199
$765$	0	0
$766$	−2.10596e7	−1.29681
$767$	2.94437e6	0.180719
$768$	2.05862e7	1.25943
$769$	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
$771$	−1.56795e7	−0.949939
$772$	−9.43344e6	−0.569674
$773$	1.04245e7	0.627492	0.313746	−	0.949507i	$-0.398416\pi$
$773$	1.04245e7	0.627492	0.313746	+	0.949507i	$0.398416\pi$
$774$	424116.	0.0254467
$775$	0	0
$776$	1.34323e7	0.800750
$777$	−1.65959e7	−0.986163
$778$	−1.19393e7	−0.707177
$779$	−43560.0	−0.00257184
$780$	0	0
$781$	1.34447e7	0.788721
$782$	1.11398e7	0.651421
$783$	−1.24115e7	−0.723467
$784$	1.67392e7	0.972620
$785$	0	0
$786$	2.54565e7	1.46974
$787$	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
$789$	3.24711e7	1.85697
$790$	0	0
$791$	6.26947e6	0.356279
$792$	−5.11503e6	−0.289758
$793$	−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	$-0.698330\pi$
$797$	−2.09287e7	−1.16707	−0.583533	+	0.812089i	$0.698330\pi$
$798$	−1.73367e6	−0.0963736
$799$	7.27126e6	0.402942
$800$	0	0
$801$	1.52847e6	0.0841735
$802$	−4.26384e6	−0.234080
$803$	−1.78725e7	−0.978131
$804$	−1.04211e7	−0.568558
$805$	0	0
$806$	5.34758e6	0.289948
$807$	3.45533e7	1.86770
$808$	1.44735e7	0.779910
$809$	−2.48797e7	−1.33651	−0.668257	−	0.743930i	$-0.732959\pi$
$809$	−2.48797e7	−1.33651	−0.668257	+	0.743930i	$0.732959\pi$
$810$	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
$813$	−3.42285e7	−1.81619
$814$	2.33513e7	1.23524
$815$	0	0
$816$	1.73522e7	0.912282
$817$	91936.8	0.00481875
$818$	1.36453e7	0.713018
$819$	−1.09058e6	−0.0568132
$820$	0	0
$821$	−3.43550e6	−0.177882	−0.0889410	−	0.996037i	$-0.528348\pi$
$821$	−3.43550e6	−0.177882	−0.0889410	+	0.996037i	$0.528348\pi$
$822$	−1.89512e7	−0.978269
$823$	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	$-0.829353\pi$
$827$	−3.38176e7	−1.71941	−0.859705	+	0.510791i	$0.829353\pi$
$828$	−4.46956e6	−0.226563
$829$	−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
$831$	2.53210e7	1.27198
$832$	1.55123e6	0.0776903
$833$	9.13579e6	0.456177
$834$	4.20209e7	2.09194
$835$	0	0
$836$	665280.	0.0329222
$837$	−1.20927e7	−0.596635
$838$	1.94846e7	0.958478
$839$	−2.89012e7	−1.41746	−0.708729	−	0.705481i	$-0.750731\pi$
$839$	−2.89012e7	−1.41746	−0.708729	+	0.705481i	$0.750731\pi$
$840$	0	0
$841$	2.75138e7	1.34140
$842$	1.80272e7	0.876290
$843$	2.91532e7	1.41292
$844$	−3.01858e6	−0.145863
$845$	0	0
$846$	−1.06971e7	−0.513857
$847$	5.82348e6	0.278917
$848$	−7.36153e6	−0.351543
$849$	3.28446e7	1.56385
$850$	0	0
$851$	−3.40077e7	−1.60973
$852$	1.27403e7	0.601285
$853$	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	$-0.370207\pi$
$857$	1.70522e7	0.793101	0.396550	+	0.918013i	$0.370207\pi$
$858$	3.97167e6	0.184185
$859$	−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
$861$	235224.	0.0108137
$862$	3.31288e7	1.51858
$863$	−2.70896e7	−1.23816	−0.619078	−	0.785330i	$-0.712493\pi$
$863$	−2.70896e7	−1.23816	−0.619078	+	0.785330i	$0.712493\pi$
$864$	7.41312e6	0.337844
$865$	0	0
$866$	1.38093e7	0.625716
$867$	−1.87844e7	−0.848692
$868$	−4.83707e6	−0.217913
$869$	−1.30838e7	−0.587741
$870$	0	0
$871$	−5.21057e6	−0.232723
$872$	−2.78729e6	−0.124134
$873$	−1.54912e7	−0.687940
$874$	−3.55256e6	−0.157312
$875$	0	0
$876$	−1.69361e7	−0.745682
$877$	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
$879$	−4.75150e7	−2.07424
$880$	0	0
$881$	−4.22840e7	−1.83542	−0.917712	−	0.397247i	$-0.869966\pi$
$881$	−4.22840e7	−1.83542	−0.917712	+	0.397247i	$0.869966\pi$
$882$	−1.34402e7	−0.581745
$883$	−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	$-0.473639\pi$
$887$	3.87668e6	0.165444	0.0827219	+	0.996573i	$0.473639\pi$
$888$	−3.68798e7	−1.56948
$889$	5.20700e6	0.220970
$890$	0	0
$891$	−1.83504e7	−0.774374
$892$	3.46518e6	0.145819
$893$	−2.31885e6	−0.0973070
$894$	−1.11078e7	−0.464819
$895$	0	0
$896$	−1.30522e7	−0.543141
$897$	−5.78414e6	−0.240026
$898$	−4.11477e7	−1.70277
$899$	4.67914e7	1.93093
$900$	0	0
$901$	−4.01773e6	−0.164880
$902$	−330973.	−0.0135449
$903$	−496459.	−0.0202611
$904$	1.39322e7	0.567019
$905$	0	0
$906$	−2.05719e7	−0.832635
$907$	−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
$909$	−1.66920e7	−0.670037
$910$	0	0
$911$	1.87675e6	0.0749223	0.0374611	−	0.999298i	$-0.488073\pi$
$911$	1.87675e6	0.0749223	0.0374611	+	0.999298i	$0.488073\pi$
$912$	−5.53372e6	−0.220308
$913$	1.55841e7	0.618736
$914$	−1.42861e7	−0.565650
$915$	0	0
$916$	−6.57204e6	−0.258798
$917$	−1.15131e7	−0.452137
$918$	8.19551e6	0.320974
$919$	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
$921$	1.84722e7	0.717579
$922$	−2.55466e7	−0.989706
$923$	6.37015e6	0.246119
$924$	−3.59251e6	−0.138426
$925$	0	0
$926$	1.38113e7	0.529306
$927$	−1.08055e7	−0.412996
$928$	−2.86843e7	−1.09339
$929$	−1.15356e7	−0.438530	−0.219265	−	0.975665i	$-0.570366\pi$
$929$	−1.15356e7	−0.438530	−0.219265	+	0.975665i	$0.570366\pi$
$930$	0	0
$931$	−2.91346e6	−0.110163
$932$	−577252.	−0.0217684
$933$	1.13061e7	0.425214
$934$	8.67772e6	0.325491
$935$	0	0
$936$	−2.42352e6	−0.0904184
$937$	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
$939$	3.42762e7	1.26861
$940$	0	0
$941$	2.94919e7	1.08575	0.542874	−	0.839814i	$-0.317336\pi$
$941$	2.94919e7	1.08575	0.542874	+	0.839814i	$0.317336\pi$
$942$	4.70769e7	1.72855
$943$	482012.	0.0176514
$944$	−3.11702e7	−1.13844
$945$	0	0
$946$	698544.	0.0253785
$947$	2.09628e7	0.759581	0.379791	−	0.925072i	$-0.375996\pi$
$947$	2.09628e7	0.759581	0.379791	+	0.925072i	$0.375996\pi$
$948$	−1.23983e7	−0.448067
$949$	−8.46806e6	−0.305224
$950$	0	0
$951$	2.61967e6	0.0939281
$952$	−5.46368e6	−0.195386
$953$	1.64122e7	0.585375	0.292687	−	0.956208i	$-0.405451\pi$
$953$	1.64122e7	0.585375	0.292687	+	0.956208i	$0.405451\pi$
$954$	5.91070e6	0.210265
$955$	0	0
$956$	1.20701e7	0.427135
$957$	3.47521e7	1.22660
$958$	4.48652e7	1.57941
$959$	8.57102e6	0.300944
$960$	0	0
$961$	1.69604e7	0.592416
$962$	1.10639e7	0.385454
$963$	1.48590e7	0.516325
$964$	1.07424e7	0.372314
$965$	0	0
$966$	1.91838e7	0.661443
$967$	−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
$969$	−3.02016e6	−0.103329
$970$	0	0
$971$	3.84771e7	1.30965	0.654823	−	0.755783i	$-0.272743\pi$
$971$	3.84771e7	1.30965	0.654823	+	0.755783i	$0.272743\pi$
$972$	−1.21665e7	−0.413046
$973$	−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	$-0.350430\pi$
$977$	2.70184e7	0.905572	0.452786	+	0.891619i	$0.350430\pi$
$978$	1.91255e7	0.639389
$979$	2.51748e6	0.0839478
$980$	0	0
$981$	3.21453e6	0.106646
$982$	−4.56086e7	−1.50927
$983$	−2.88475e7	−0.952192	−0.476096	−	0.879393i	$-0.657949\pi$
$983$	−2.88475e7	−0.952192	−0.476096	+	0.879393i	$0.657949\pi$
$984$	522720.	0.0172100
$985$	0	0
$986$	−3.17117e7	−1.03879
$987$	1.25218e7	0.409142
$988$	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
$993$	−3.14525e7	−1.01224
$994$	−2.11274e7	−0.678235
$995$	0	0
$996$	1.47676e7	0.471696
$997$	−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
$999$	−2.50193e7	−0.793161

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	25.6.a.c.1.2		2
3.2	odd	2		225.6.a.n.1.1		2
4.3	odd	2		400.6.a.t.1.1		2
5.2	odd	4		5.6.b.a.4.2	yes	2
5.3	odd	4		5.6.b.a.4.1	✓	2
5.4	even	2	inner	25.6.a.c.1.1		2
15.2	even	4		45.6.b.b.19.1		2
15.8	even	4		45.6.b.b.19.2		2
15.14	odd	2		225.6.a.n.1.2		2
20.3	even	4		80.6.c.a.49.1		2
20.7	even	4		80.6.c.a.49.2		2
20.19	odd	2		400.6.a.t.1.2		2
35.13	even	4		245.6.b.a.99.1		2
35.27	even	4		245.6.b.a.99.2		2
40.3	even	4		320.6.c.g.129.2		2
40.13	odd	4		320.6.c.f.129.1		2
40.27	even	4		320.6.c.g.129.1		2
40.37	odd	4		320.6.c.f.129.2		2
60.23	odd	4		720.6.f.f.289.2		2
60.47	odd	4		720.6.f.f.289.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5.6.b.a.4.1	✓	2	5.3	odd	4
5.6.b.a.4.2	yes	2	5.2	odd	4
25.6.a.c.1.1		2	5.4	even	2	inner
25.6.a.c.1.2		2	1.1	even	1	trivial
45.6.b.b.19.1		2	15.2	even	4
45.6.b.b.19.2		2	15.8	even	4
80.6.c.a.49.1		2	20.3	even	4
80.6.c.a.49.2		2	20.7	even	4
225.6.a.n.1.1		2	3.2	odd	2
225.6.a.n.1.2		2	15.14	odd	2
245.6.b.a.99.1		2	35.13	even	4
245.6.b.a.99.2		2	35.27	even	4
320.6.c.f.129.1		2	40.13	odd	4
320.6.c.f.129.2		2	40.37	odd	4
320.6.c.g.129.1		2	40.27	even	4
320.6.c.g.129.2		2	40.3	even	4
400.6.a.t.1.1		2	4.3	odd	2
400.6.a.t.1.2		2	20.19	odd	2
720.6.f.f.289.1		2	60.47	odd	4
720.6.f.f.289.2		2	60.23	odd	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:	\( N \)	\(=\)	\( 25 = 5^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	25.a (trivial)

\(f(q)\)	\(=\)	\(q+6.63325 q^{2} +19.8997 q^{3} +12.0000 q^{4} +132.000 q^{6} -59.6992 q^{7} -132.665 q^{8} +153.000 q^{9} +252.000 q^{11} +238.797 q^{12} +119.398 q^{13} -396.000 q^{14} -1264.00 q^{16} -689.858 q^{17} +1014.89 q^{18} +220.000 q^{19} -1188.00 q^{21} +1671.58 q^{22} -2434.40 q^{23} -2640.00 q^{24} +792.000 q^{26} -1790.98 q^{27} -716.391 q^{28} +6930.00 q^{29} +6752.00 q^{31} -4139.15 q^{32} +5014.74 q^{33} -4576.00 q^{34} +1836.00 q^{36} +13969.6 q^{37} +1459.31 q^{38} +2376.00 q^{39} -198.000 q^{41} -7880.30 q^{42} +417.895 q^{43} +3024.00 q^{44} -16148.0 q^{46} -10540.2 q^{47} -25153.3 q^{48} -13243.0 q^{49} -13728.0 q^{51} +1432.78 q^{52} +5823.99 q^{53} -11880.0 q^{54} +7920.00 q^{56} +4377.94 q^{57} +45968.4 q^{58} +24660.0 q^{59} -5698.00 q^{61} +44787.7 q^{62} -9133.98 q^{63} +12992.0 q^{64} +33264.0 q^{66} -43640.1 q^{67} -8278.30 q^{68} -48444.0 q^{69} +53352.0 q^{71} -20297.7 q^{72} -70922.7 q^{73} +92664.0 q^{74} +2640.00 q^{76} -15044.2 q^{77} +15760.6 q^{78} -51920.0 q^{79} -72819.0 q^{81} -1313.38 q^{82} +61841.8 q^{83} -14256.0 q^{84} +2772.00 q^{86} +137905. q^{87} -33431.6 q^{88} +9990.00 q^{89} -7128.00 q^{91} -29212.8 q^{92} +134363. q^{93} -69916.0 q^{94} -82368.0 q^{96} -101250. q^{97} -87844.1 q^{98} +38556.0 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 24 q^{4} + 264 q^{6} + 306 q^{9} + 504 q^{11} - 792 q^{14} - 2528 q^{16} + 440 q^{19} - 2376 q^{21} - 5280 q^{24} + 1584 q^{26} + 13860 q^{29} + 13504 q^{31} - 9152 q^{34} + 3672 q^{36} + 4752 q^{39}+ \cdots + 77112 q^{99}+O(q^{100}) \) 2 * q + 24 * q^4 + 264 * q^6 + 306 * q^9 + 504 * q^11 - 792 * q^14 - 2528 * q^16 + 440 * q^19 - 2376 * q^21 - 5280 * q^24 + 1584 * q^26 + 13860 * q^29 + 13504 * q^31 - 9152 * q^34 + 3672 * q^36 + 4752 * q^39 - 396 * q^41 + 6048 * q^44 - 32296 * q^46 - 26486 * q^49 - 27456 * q^51 - 23760 * q^54 + 15840 * q^56 + 49320 * q^59 - 11396 * q^61 + 25984 * q^64 + 66528 * q^66 - 96888 * q^69 + 106704 * q^71 + 185328 * q^74 + 5280 * q^76 - 103840 * q^79 - 145638 * q^81 - 28512 * q^84 + 5544 * q^86 + 19980 * q^89 - 14256 * q^91 - 139832 * q^94 - 164736 * q^96 + 77112 * q^99

Introduction

L-functions

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Database

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