LMFDB - Embedded newform 1104.6.a.y.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1104,6,Mod(1,1104)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1104, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 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("1104.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 1104 = 2^{4} \cdot 3 \cdot 23 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	1104.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:	$177.063737074$
Analytic rank:	$0$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - x^{6} - 2429x^{5} - 54929x^{4} - 436970x^{3} - 1590048x^{2} - 2711880x - 1760400 $ x^7 - x^6 - 2429x^5 - 54929x^4 - 436970x^3 - 1590048x^2 - 2711880*x - 1760400
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{14}\cdot 3 $
Twist minimal:	no (minimal twist has level 552)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-2.10756$ of defining polynomial
Character	$\chi$	$=$	1104.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+9.00000 q^{3} -32.8770 q^{5} +11.9330 q^{7} +81.0000 q^{9} +O(q^{10})$	$q+9.00000 q^{3} -32.8770 q^{5} +11.9330 q^{7} +81.0000 q^{9} -21.1543 q^{11} -812.097 q^{13} -295.893 q^{15} -800.289 q^{17} -1299.69 q^{19} +107.397 q^{21} -529.000 q^{23} -2044.10 q^{25} +729.000 q^{27} -4237.13 q^{29} +3124.27 q^{31} -190.389 q^{33} -392.320 q^{35} +8807.57 q^{37} -7308.87 q^{39} +10339.0 q^{41} +10479.1 q^{43} -2663.03 q^{45} +22231.1 q^{47} -16664.6 q^{49} -7202.60 q^{51} -18474.8 q^{53} +695.490 q^{55} -11697.2 q^{57} +26355.9 q^{59} +18250.6 q^{61} +966.572 q^{63} +26699.3 q^{65} -15327.2 q^{67} -4761.00 q^{69} -59488.7 q^{71} +56758.6 q^{73} -18396.9 q^{75} -252.434 q^{77} +99756.1 q^{79} +6561.00 q^{81} -79359.7 q^{83} +26311.1 q^{85} -38134.1 q^{87} +5492.24 q^{89} -9690.74 q^{91} +28118.4 q^{93} +42729.8 q^{95} -65412.8 q^{97} -1713.50 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q + 63 q^{3} + 80 q^{5} - 46 q^{7} + 567 q^{9}+O(q^{10}) $ 7 * q + 63 * q^3 + 80 * q^5 - 46 * q^7 + 567 * q^9	$ 7 q + 63 q^{3} + 80 q^{5} - 46 q^{7} + 567 q^{9} - 124 q^{11} + 966 q^{13} + 720 q^{15} + 652 q^{17} + 542 q^{19} - 414 q^{21} - 3703 q^{23} + 5165 q^{25} + 5103 q^{27} + 5222 q^{29} + 2660 q^{31} - 1116 q^{33} + 17088 q^{35} - 202 q^{37} + 8694 q^{39} - 706 q^{41} + 7086 q^{43} + 6480 q^{45} + 37696 q^{47} - 34469 q^{49} + 5868 q^{51} + 30180 q^{53} + 51104 q^{55} + 4878 q^{57} + 12284 q^{59} + 53534 q^{61} - 3726 q^{63} + 46920 q^{65} - 47750 q^{67} - 33327 q^{69} + 2552 q^{71} - 19474 q^{73} + 46485 q^{75} + 36264 q^{77} + 4510 q^{79} + 45927 q^{81} + 47580 q^{83} + 224944 q^{85} + 46998 q^{87} + 150664 q^{89} - 57460 q^{91} + 23940 q^{93} + 21648 q^{95} + 392426 q^{97} - 10044 q^{99}+O(q^{100}) $ 7 * q + 63 * q^3 + 80 * q^5 - 46 * q^7 + 567 * q^9 - 124 * q^11 + 966 * q^13 + 720 * q^15 + 652 * q^17 + 542 * q^19 - 414 * q^21 - 3703 * q^23 + 5165 * q^25 + 5103 * q^27 + 5222 * q^29 + 2660 * q^31 - 1116 * q^33 + 17088 * q^35 - 202 * q^37 + 8694 * q^39 - 706 * q^41 + 7086 * q^43 + 6480 * q^45 + 37696 * q^47 - 34469 * q^49 + 5868 * q^51 + 30180 * q^53 + 51104 * q^55 + 4878 * q^57 + 12284 * q^59 + 53534 * q^61 - 3726 * q^63 + 46920 * q^65 - 47750 * q^67 - 33327 * q^69 + 2552 * q^71 - 19474 * q^73 + 46485 * q^75 + 36264 * q^77 + 4510 * q^79 + 45927 * q^81 + 47580 * q^83 + 224944 * q^85 + 46998 * q^87 + 150664 * q^89 - 57460 * q^91 + 23940 * q^93 + 21648 * q^95 + 392426 * q^97 - 10044 * q^99

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 $

$2$	0	0
$2$	0	0
$3$	9.00000	0.577350
$3$	9.00000	0.577350
$4$	0	0
$5$	−32.8770	−0.588121	−0.294061	−	0.955787i	$-0.595007\pi$
$5$	−32.8770	−0.588121	−0.294061	+	0.955787i	$0.595007\pi$
$6$	0	0
$7$	11.9330	0.0920458	0.0460229	−	0.998940i	$-0.485345\pi$
$7$	11.9330	0.0920458	0.0460229	+	0.998940i	$0.485345\pi$
$8$	0	0
$9$	81.0000	0.333333
$10$	0	0
$11$	−21.1543	−0.0527130	−0.0263565	−	0.999653i	$-0.508391\pi$
$11$	−21.1543	−0.0527130	−0.0263565	+	0.999653i	$0.508391\pi$
$12$	0	0
$13$	−812.097	−1.33275	−0.666376	−	0.745616i	$-0.732155\pi$
$13$	−812.097	−1.33275	−0.666376	+	0.745616i	$0.732155\pi$
$14$	0	0
$15$	−295.893	−0.339552
$16$	0	0
$17$	−800.289	−0.671621	−0.335811	−	0.941930i	$-0.609010\pi$
$17$	−800.289	−0.671621	−0.335811	+	0.941930i	$0.609010\pi$
$18$	0	0
$19$	−1299.69	−0.825952	−0.412976	−	0.910742i	$-0.635511\pi$
$19$	−1299.69	−0.825952	−0.412976	+	0.910742i	$0.635511\pi$
$20$	0	0
$21$	107.397	0.0531427
$22$	0	0
$23$	−529.000	−0.208514
$23$	−529.000	−0.208514
$24$	0	0
$25$	−2044.10	−0.654113
$26$	0	0
$27$	729.000	0.192450
$28$	0	0
$29$	−4237.13	−0.935570	−0.467785	−	0.883842i	$-0.654948\pi$
$29$	−4237.13	−0.935570	−0.467785	+	0.883842i	$0.654948\pi$
$30$	0	0
$31$	3124.27	0.583908	0.291954	−	0.956432i	$-0.405695\pi$
$31$	3124.27	0.583908	0.291954	+	0.956432i	$0.405695\pi$
$32$	0	0
$33$	−190.389	−0.0304338
$34$	0	0
$35$	−392.320	−0.0541341
$36$	0	0
$37$	8807.57	1.05767	0.528837	−	0.848723i	$-0.322628\pi$
$37$	8807.57	1.05767	0.528837	+	0.848723i	$0.322628\pi$
$38$	0	0
$39$	−7308.87	−0.769465
$40$	0	0
$41$	10339.0	0.960552	0.480276	−	0.877117i	$-0.340537\pi$
$41$	10339.0	0.960552	0.480276	+	0.877117i	$0.340537\pi$
$42$	0	0
$43$	10479.1	0.864274	0.432137	−	0.901808i	$-0.357760\pi$
$43$	10479.1	0.864274	0.432137	+	0.901808i	$0.357760\pi$
$44$	0	0
$45$	−2663.03	−0.196040
$46$	0	0
$47$	22231.1	1.46797	0.733985	−	0.679166i	$-0.237658\pi$
$47$	22231.1	1.46797	0.733985	+	0.679166i	$0.237658\pi$
$48$	0	0
$49$	−16664.6	−0.991528
$50$	0	0
$51$	−7202.60	−0.387761
$52$	0	0
$53$	−18474.8	−0.903421	−0.451710	−	0.892165i	$-0.649186\pi$
$53$	−18474.8	−0.903421	−0.451710	+	0.892165i	$0.649186\pi$
$54$	0	0
$55$	695.490	0.0310016
$56$	0	0
$57$	−11697.2	−0.476863
$58$	0	0
$59$	26355.9	0.985706	0.492853	−	0.870112i	$-0.335954\pi$
$59$	26355.9	0.985706	0.492853	+	0.870112i	$0.335954\pi$
$60$	0	0
$61$	18250.6	0.627989	0.313995	−	0.949425i	$-0.398333\pi$
$61$	18250.6	0.627989	0.313995	+	0.949425i	$0.398333\pi$
$62$	0	0
$63$	966.572	0.0306819
$64$	0	0
$65$	26699.3	0.783820
$66$	0	0
$67$	−15327.2	−0.417134	−0.208567	−	0.978008i	$-0.566880\pi$
$67$	−15327.2	−0.417134	−0.208567	+	0.978008i	$0.566880\pi$
$68$	0	0
$69$	−4761.00	−0.120386
$70$	0	0
$71$	−59488.7	−1.40052	−0.700259	−	0.713889i	$-0.746932\pi$
$71$	−59488.7	−1.40052	−0.700259	+	0.713889i	$0.746932\pi$
$72$	0	0
$73$	56758.6	1.24659	0.623296	−	0.781986i	$-0.285793\pi$
$73$	56758.6	1.24659	0.623296	+	0.781986i	$0.285793\pi$
$74$	0	0
$75$	−18396.9	−0.377653
$76$	0	0
$77$	−252.434	−0.00485201
$78$	0	0
$79$	99756.1	1.79834	0.899170	−	0.437600i	$-0.144171\pi$
$79$	99756.1	1.79834	0.899170	+	0.437600i	$0.144171\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	0	0
$83$	−79359.7	−1.26446	−0.632229	−	0.774781i	$-0.717860\pi$
$83$	−79359.7	−1.26446	−0.632229	+	0.774781i	$0.717860\pi$
$84$	0	0
$85$	26311.1	0.394995
$86$	0	0
$87$	−38134.1	−0.540152
$88$	0	0
$89$	5492.24	0.0734978	0.0367489	−	0.999325i	$-0.488300\pi$
$89$	5492.24	0.0734978	0.0367489	+	0.999325i	$0.488300\pi$
$90$	0	0
$91$	−9690.74	−0.122674
$92$	0	0
$93$	28118.4	0.337119
$94$	0	0
$95$	42729.8	0.485760
$96$	0	0
$97$	−65412.8	−0.705884	−0.352942	−	0.935645i	$-0.614819\pi$
$97$	−65412.8	−0.705884	−0.352942	+	0.935645i	$0.614819\pi$
$98$	0	0
$99$	−1713.50	−0.0175710
$100$	0	0
$101$	−872.760	−0.00851317	−0.00425658	−	0.999991i	$-0.501355\pi$
$101$	−872.760	−0.00851317	−0.00425658	+	0.999991i	$0.501355\pi$
$102$	0	0
$103$	−138480.	−1.28615	−0.643076	−	0.765802i	$-0.722342\pi$
$103$	−138480.	−1.28615	−0.643076	+	0.765802i	$0.722342\pi$
$104$	0	0
$105$	−3530.88	−0.0312543
$106$	0	0
$107$	−151896.	−1.28258	−0.641292	−	0.767297i	$-0.721601\pi$
$107$	−151896.	−1.28258	−0.641292	+	0.767297i	$0.721601\pi$
$108$	0	0
$109$	89799.1	0.723945	0.361973	−	0.932189i	$-0.382103\pi$
$109$	89799.1	0.723945	0.361973	+	0.932189i	$0.382103\pi$
$110$	0	0
$111$	79268.2	0.610649
$112$	0	0
$113$	173015.	1.27464	0.637320	−	0.770599i	$-0.280043\pi$
$113$	173015.	1.27464	0.637320	+	0.770599i	$0.280043\pi$
$114$	0	0
$115$	17391.9	0.122632
$116$	0	0
$117$	−65779.8	−0.444251
$118$	0	0
$119$	−9549.84	−0.0618199
$120$	0	0
$121$	−160603.	−0.997221
$122$	0	0
$123$	93051.4	0.554575
$124$	0	0
$125$	169945.	0.972819
$126$	0	0
$127$	114845.	0.631834	0.315917	−	0.948787i	$-0.397688\pi$
$127$	114845.	0.631834	0.315917	+	0.948787i	$0.397688\pi$
$128$	0	0
$129$	94311.6	0.498989
$130$	0	0
$131$	137092.	0.697966	0.348983	−	0.937129i	$-0.386527\pi$
$131$	137092.	0.697966	0.348983	+	0.937129i	$0.386527\pi$
$132$	0	0
$133$	−15509.1	−0.0760254
$134$	0	0
$135$	−23967.3	−0.113184
$136$	0	0
$137$	413311.	1.88138	0.940688	−	0.339273i	$-0.110181\pi$
$137$	413311.	1.88138	0.940688	+	0.339273i	$0.110181\pi$
$138$	0	0
$139$	−276160.	−1.21234	−0.606168	−	0.795336i	$-0.707294\pi$
$139$	−276160.	−1.21234	−0.606168	+	0.795336i	$0.707294\pi$
$140$	0	0
$141$	200080.	0.847533
$142$	0	0
$143$	17179.4	0.0702533
$144$	0	0
$145$	139304.	0.550229
$146$	0	0
$147$	−149981.	−0.572459
$148$	0	0
$149$	38975.4	0.143822	0.0719109	−	0.997411i	$-0.477090\pi$
$149$	38975.4	0.143822	0.0719109	+	0.997411i	$0.477090\pi$
$150$	0	0
$151$	233457.	0.833228	0.416614	−	0.909084i	$-0.363217\pi$
$151$	233457.	0.833228	0.416614	+	0.909084i	$0.363217\pi$
$152$	0	0
$153$	−64823.4	−0.223874
$154$	0	0
$155$	−102717.	−0.343409
$156$	0	0
$157$	250939.	0.812492	0.406246	−	0.913764i	$-0.366838\pi$
$157$	250939.	0.812492	0.406246	+	0.913764i	$0.366838\pi$
$158$	0	0
$159$	−166273.	−0.521590
$160$	0	0
$161$	−6312.55	−0.0191929
$162$	0	0
$163$	−115615.	−0.340835	−0.170417	−	0.985372i	$-0.554512\pi$
$163$	−115615.	−0.340835	−0.170417	+	0.985372i	$0.554512\pi$
$164$	0	0
$165$	6259.41	0.0178988
$166$	0	0
$167$	418379.	1.16086	0.580428	−	0.814312i	$-0.302885\pi$
$167$	418379.	1.16086	0.580428	+	0.814312i	$0.302885\pi$
$168$	0	0
$169$	288208.	0.776228
$170$	0	0
$171$	−105275.	−0.275317
$172$	0	0
$173$	447361.	1.13643	0.568215	−	0.822880i	$-0.307634\pi$
$173$	447361.	1.13643	0.568215	+	0.822880i	$0.307634\pi$
$174$	0	0
$175$	−24392.3	−0.0602084
$176$	0	0
$177$	237203.	0.569098
$178$	0	0
$179$	−192583.	−0.449247	−0.224623	−	0.974446i	$-0.572115\pi$
$179$	−192583.	−0.449247	−0.224623	+	0.974446i	$0.572115\pi$
$180$	0	0
$181$	458555.	1.04039	0.520193	−	0.854049i	$-0.325860\pi$
$181$	458555.	1.04039	0.520193	+	0.854049i	$0.325860\pi$
$182$	0	0
$183$	164255.	0.362570
$184$	0	0
$185$	−289566.	−0.622041
$186$	0	0
$187$	16929.6	0.0354032
$188$	0	0
$189$	8699.15	0.0177142
$190$	0	0
$191$	370636.	0.735130	0.367565	−	0.929998i	$-0.380192\pi$
$191$	370636.	0.735130	0.367565	+	0.929998i	$0.380192\pi$
$192$	0	0
$193$	428687.	0.828412	0.414206	−	0.910183i	$-0.364059\pi$
$193$	428687.	0.828412	0.414206	+	0.910183i	$0.364059\pi$
$194$	0	0
$195$	240294.	0.452539
$196$	0	0
$197$	717981.	1.31810	0.659049	−	0.752100i	$-0.270959\pi$
$197$	717981.	1.31810	0.659049	+	0.752100i	$0.270959\pi$
$198$	0	0
$199$	−639377.	−1.14452	−0.572261	−	0.820071i	$-0.693934\pi$
$199$	−639377.	−1.14452	−0.572261	+	0.820071i	$0.693934\pi$
$200$	0	0
$201$	−137945.	−0.240833
$202$	0	0
$203$	−50561.6	−0.0861153
$204$	0	0
$205$	−339917.	−0.564921
$206$	0	0
$207$	−42849.0	−0.0695048
$208$	0	0
$209$	27494.0	0.0435384
$210$	0	0
$211$	−557509.	−0.862076	−0.431038	−	0.902334i	$-0.641853\pi$
$211$	−557509.	−0.862076	−0.431038	+	0.902334i	$0.641853\pi$
$212$	0	0
$213$	−535399.	−0.808590
$214$	0	0
$215$	−344520.	−0.508298
$216$	0	0
$217$	37281.9	0.0537463
$218$	0	0
$219$	510827.	0.719720
$220$	0	0
$221$	649912.	0.895105
$222$	0	0
$223$	877977.	1.18228	0.591141	−	0.806568i	$-0.298678\pi$
$223$	877977.	1.18228	0.591141	+	0.806568i	$0.298678\pi$
$224$	0	0
$225$	−165572.	−0.218038
$226$	0	0
$227$	91962.5	0.118453	0.0592265	−	0.998245i	$-0.481137\pi$
$227$	91962.5	0.118453	0.0592265	+	0.998245i	$0.481137\pi$
$228$	0	0
$229$	−1.15774e6	−1.45889	−0.729443	−	0.684042i	$-0.760220\pi$
$229$	−1.15774e6	−1.45889	−0.729443	+	0.684042i	$0.760220\pi$
$230$	0	0
$231$	−2271.91	−0.00280131
$232$	0	0
$233$	−131720.	−0.158950	−0.0794750	−	0.996837i	$-0.525324\pi$
$233$	−131720.	−0.158950	−0.0794750	+	0.996837i	$0.525324\pi$
$234$	0	0
$235$	−730893.	−0.863344
$236$	0	0
$237$	897805.	1.03827
$238$	0	0
$239$	−1.52695e6	−1.72913	−0.864567	−	0.502517i	$-0.832407\pi$
$239$	−1.52695e6	−1.72913	−0.864567	+	0.502517i	$0.832407\pi$
$240$	0	0
$241$	912477.	1.01200	0.505999	−	0.862534i	$-0.331124\pi$
$241$	912477.	1.01200	0.505999	+	0.862534i	$0.331124\pi$
$242$	0	0
$243$	59049.0	0.0641500
$244$	0	0
$245$	547882.	0.583138
$246$	0	0
$247$	1.05547e6	1.10079
$248$	0	0
$249$	−714237.	−0.730035
$250$	0	0
$251$	1.42166e6	1.42434	0.712169	−	0.702008i	$-0.247713\pi$
$251$	1.42166e6	1.42434	0.712169	+	0.702008i	$0.247713\pi$
$252$	0	0
$253$	11190.6	0.0109914
$254$	0	0
$255$	236800.	0.228050
$256$	0	0
$257$	1.45875e6	1.37768	0.688838	−	0.724915i	$-0.258121\pi$
$257$	1.45875e6	1.37768	0.688838	+	0.724915i	$0.258121\pi$
$258$	0	0
$259$	105101.	0.0973545
$260$	0	0
$261$	−343207.	−0.311857
$262$	0	0
$263$	−1.30328e6	−1.16184	−0.580921	−	0.813960i	$-0.697308\pi$
$263$	−1.30328e6	−1.16184	−0.580921	+	0.813960i	$0.697308\pi$
$264$	0	0
$265$	607395.	0.531321
$266$	0	0
$267$	49430.1	0.0424340
$268$	0	0
$269$	−254511.	−0.214450	−0.107225	−	0.994235i	$-0.534196\pi$
$269$	−254511.	−0.214450	−0.107225	+	0.994235i	$0.534196\pi$
$270$	0	0
$271$	826554.	0.683672	0.341836	−	0.939760i	$-0.388951\pi$
$271$	826554.	0.683672	0.341836	+	0.939760i	$0.388951\pi$
$272$	0	0
$273$	−87216.6	−0.0708260
$274$	0	0
$275$	43241.7	0.0344803
$276$	0	0
$277$	−1.71774e6	−1.34511	−0.672556	−	0.740047i	$-0.734803\pi$
$277$	−1.71774e6	−1.34511	−0.672556	+	0.740047i	$0.734803\pi$
$278$	0	0
$279$	253066.	0.194636
$280$	0	0
$281$	−663317.	−0.501136	−0.250568	−	0.968099i	$-0.580617\pi$
$281$	−663317.	−0.501136	−0.250568	+	0.968099i	$0.580617\pi$
$282$	0	0
$283$	−1.96474e6	−1.45827	−0.729135	−	0.684370i	$-0.760077\pi$
$283$	−1.96474e6	−1.45827	−0.729135	+	0.684370i	$0.760077\pi$
$284$	0	0
$285$	384568.	0.280453
$286$	0	0
$287$	123376.	0.0884148
$288$	0	0
$289$	−779395.	−0.548925
$290$	0	0
$291$	−588715.	−0.407542
$292$	0	0
$293$	1.75300e6	1.19293	0.596463	−	0.802641i	$-0.296572\pi$
$293$	1.75300e6	1.19293	0.596463	+	0.802641i	$0.296572\pi$
$294$	0	0
$295$	−866502.	−0.579715
$296$	0	0
$297$	−15421.5	−0.0101446
$298$	0	0
$299$	429599.	0.277898
$300$	0	0
$301$	125046.	0.0795528
$302$	0	0
$303$	−7854.84	−0.00491508
$304$	0	0
$305$	−600024.	−0.369334
$306$	0	0
$307$	2.49999e6	1.51388	0.756940	−	0.653484i	$-0.226693\pi$
$307$	2.49999e6	1.51388	0.756940	+	0.653484i	$0.226693\pi$
$308$	0	0
$309$	−1.24632e6	−0.742561
$310$	0	0
$311$	2.87886e6	1.68779	0.843896	−	0.536507i	$-0.180256\pi$
$311$	2.87886e6	1.68779	0.843896	+	0.536507i	$0.180256\pi$
$312$	0	0
$313$	−1.02097e6	−0.589050	−0.294525	−	0.955644i	$-0.595161\pi$
$313$	−1.02097e6	−0.589050	−0.294525	+	0.955644i	$0.595161\pi$
$314$	0	0
$315$	−31778.0	−0.0180447
$316$	0	0
$317$	1.83047e6	1.02309	0.511545	−	0.859256i	$-0.329073\pi$
$317$	1.83047e6	1.02309	0.511545	+	0.859256i	$0.329073\pi$
$318$	0	0
$319$	89633.5	0.0493167
$320$	0	0
$321$	−1.36706e6	−0.740501
$322$	0	0
$323$	1.04012e6	0.554727
$324$	0	0
$325$	1.66001e6	0.871771
$326$	0	0
$327$	808192.	0.417970
$328$	0	0
$329$	265284.	0.135120
$330$	0	0
$331$	2.89470e6	1.45222	0.726112	−	0.687576i	$-0.241325\pi$
$331$	2.89470e6	1.45222	0.726112	+	0.687576i	$0.241325\pi$
$332$	0	0
$333$	713414.	0.352558
$334$	0	0
$335$	503912.	0.245326
$336$	0	0
$337$	2.07903e6	0.997208	0.498604	−	0.866830i	$-0.333846\pi$
$337$	2.07903e6	0.997208	0.498604	+	0.866830i	$0.333846\pi$
$338$	0	0
$339$	1.55714e6	0.735914
$340$	0	0
$341$	−66091.8	−0.0307795
$342$	0	0
$343$	−399416.	−0.183312
$344$	0	0
$345$	156527.	0.0708015
$346$	0	0
$347$	−3.48174e6	−1.55229	−0.776144	−	0.630555i	$-0.782827\pi$
$347$	−3.48174e6	−1.55229	−0.776144	+	0.630555i	$0.782827\pi$
$348$	0	0
$349$	520577.	0.228782	0.114391	−	0.993436i	$-0.463508\pi$
$349$	520577.	0.228782	0.114391	+	0.993436i	$0.463508\pi$
$350$	0	0
$351$	−592019.	−0.256488
$352$	0	0
$353$	−3.14717e6	−1.34426	−0.672131	−	0.740432i	$-0.734621\pi$
$353$	−3.14717e6	−1.34426	−0.672131	+	0.740432i	$0.734621\pi$
$354$	0	0
$355$	1.95581e6	0.823675
$356$	0	0
$357$	−85948.5	−0.0356918
$358$	0	0
$359$	754168.	0.308839	0.154419	−	0.988005i	$-0.450649\pi$
$359$	754168.	0.308839	0.154419	+	0.988005i	$0.450649\pi$
$360$	0	0
$361$	−786914.	−0.317804
$362$	0	0
$363$	−1.44543e6	−0.575746
$364$	0	0
$365$	−1.86605e6	−0.733147
$366$	0	0
$367$	−610274.	−0.236516	−0.118258	−	0.992983i	$-0.537731\pi$
$367$	−610274.	−0.236516	−0.118258	+	0.992983i	$0.537731\pi$
$368$	0	0
$369$	837463.	0.320184
$370$	0	0
$371$	−220459.	−0.0831561
$372$	0	0
$373$	3.28956e6	1.22424	0.612119	−	0.790766i	$-0.290317\pi$
$373$	3.28956e6	1.22424	0.612119	+	0.790766i	$0.290317\pi$
$374$	0	0
$375$	1.52950e6	0.561657
$376$	0	0
$377$	3.44096e6	1.24688
$378$	0	0
$379$	4.08482e6	1.46075	0.730373	−	0.683048i	$-0.239346\pi$
$379$	4.08482e6	1.46075	0.730373	+	0.683048i	$0.239346\pi$
$380$	0	0
$381$	1.03361e6	0.364789
$382$	0	0
$383$	1.70924e6	0.595395	0.297698	−	0.954660i	$-0.403781\pi$
$383$	1.70924e6	0.595395	0.297698	+	0.954660i	$0.403781\pi$
$384$	0	0
$385$	8299.27	0.00285357
$386$	0	0
$387$	848804.	0.288091
$388$	0	0
$389$	2.81728e6	0.943964	0.471982	−	0.881608i	$-0.343539\pi$
$389$	2.81728e6	0.943964	0.471982	+	0.881608i	$0.343539\pi$
$390$	0	0
$391$	423353.	0.140043
$392$	0	0
$393$	1.23383e6	0.402971
$394$	0	0
$395$	−3.27968e6	−1.05764
$396$	0	0
$397$	−3.59156e6	−1.14368	−0.571842	−	0.820364i	$-0.693771\pi$
$397$	−3.59156e6	−1.14368	−0.571842	+	0.820364i	$0.693771\pi$
$398$	0	0
$399$	−139582.	−0.0438933
$400$	0	0
$401$	1.40697e6	0.436943	0.218472	−	0.975843i	$-0.429893\pi$
$401$	1.40697e6	0.436943	0.218472	+	0.975843i	$0.429893\pi$
$402$	0	0
$403$	−2.53721e6	−0.778204
$404$	0	0
$405$	−215706.	−0.0653468
$406$	0	0
$407$	−186318.	−0.0557532
$408$	0	0
$409$	−516097.	−0.152554	−0.0762769	−	0.997087i	$-0.524303\pi$
$409$	−516097.	−0.152554	−0.0762769	+	0.997087i	$0.524303\pi$
$410$	0	0
$411$	3.71980e6	1.08621
$412$	0	0
$413$	314504.	0.0907301
$414$	0	0
$415$	2.60911e6	0.743655
$416$	0	0
$417$	−2.48544e6	−0.699943
$418$	0	0
$419$	−1.10273e6	−0.306857	−0.153428	−	0.988160i	$-0.549031\pi$
$419$	−1.10273e6	−0.306857	−0.153428	+	0.988160i	$0.549031\pi$
$420$	0	0
$421$	−5.84763e6	−1.60796	−0.803978	−	0.594659i	$-0.797287\pi$
$421$	−5.84763e6	−1.60796	−0.803978	+	0.594659i	$0.797287\pi$
$422$	0	0
$423$	1.80072e6	0.489323
$424$	0	0
$425$	1.63587e6	0.439317
$426$	0	0
$427$	217784.	0.0578038
$428$	0	0
$429$	154614.	0.0405608
$430$	0	0
$431$	4.33571e6	1.12426	0.562130	−	0.827049i	$-0.309982\pi$
$431$	4.33571e6	1.12426	0.562130	+	0.827049i	$0.309982\pi$
$432$	0	0
$433$	1.70149e6	0.436124	0.218062	−	0.975935i	$-0.430027\pi$
$433$	1.70149e6	0.436124	0.218062	+	0.975935i	$0.430027\pi$
$434$	0	0
$435$	1.25373e6	0.317675
$436$	0	0
$437$	687534.	0.172223
$438$	0	0
$439$	7.33064e6	1.81543	0.907717	−	0.419582i	$-0.137823\pi$
$439$	7.33064e6	1.81543	0.907717	+	0.419582i	$0.137823\pi$
$440$	0	0
$441$	−1.34983e6	−0.330509
$442$	0	0
$443$	−3.73122e6	−0.903319	−0.451659	−	0.892190i	$-0.649168\pi$
$443$	−3.73122e6	−0.903319	−0.451659	+	0.892190i	$0.649168\pi$
$444$	0	0
$445$	−180568.	−0.0432256
$446$	0	0
$447$	350779.	0.0830356
$448$	0	0
$449$	1.21769e6	0.285050	0.142525	−	0.989791i	$-0.454478\pi$
$449$	1.21769e6	0.285050	0.142525	+	0.989791i	$0.454478\pi$
$450$	0	0
$451$	−218716.	−0.0506335
$452$	0	0
$453$	2.10111e6	0.481064
$454$	0	0
$455$	318602.	0.0721473
$456$	0	0
$457$	−5.46320e6	−1.22365	−0.611825	−	0.790993i	$-0.709564\pi$
$457$	−5.46320e6	−1.22365	−0.611825	+	0.790993i	$0.709564\pi$
$458$	0	0
$459$	−583411.	−0.129254
$460$	0	0
$461$	7.55170e6	1.65498	0.827490	−	0.561481i	$-0.189768\pi$
$461$	7.55170e6	1.65498	0.827490	+	0.561481i	$0.189768\pi$
$462$	0	0
$463$	4.53918e6	0.984068	0.492034	−	0.870576i	$-0.336254\pi$
$463$	4.53918e6	0.984068	0.492034	+	0.870576i	$0.336254\pi$
$464$	0	0
$465$	−924449.	−0.198267
$466$	0	0
$467$	3.16848e6	0.672292	0.336146	−	0.941810i	$-0.390876\pi$
$467$	3.16848e6	0.672292	0.336146	+	0.941810i	$0.390876\pi$
$468$	0	0
$469$	−182899.	−0.0383955
$470$	0	0
$471$	2.25845e6	0.469093
$472$	0	0
$473$	−221677.	−0.0455584
$474$	0	0
$475$	2.65669e6	0.540266
$476$	0	0
$477$	−1.49646e6	−0.301140
$478$	0	0
$479$	524949.	0.104539	0.0522695	−	0.998633i	$-0.483355\pi$
$479$	524949.	0.104539	0.0522695	+	0.998633i	$0.483355\pi$
$480$	0	0
$481$	−7.15260e6	−1.40962
$482$	0	0
$483$	−56812.9	−0.0110810
$484$	0	0
$485$	2.15057e6	0.415145
$486$	0	0
$487$	1.57524e6	0.300971	0.150486	−	0.988612i	$-0.451916\pi$
$487$	1.57524e6	0.300971	0.150486	+	0.988612i	$0.451916\pi$
$488$	0	0
$489$	−1.04053e6	−0.196781
$490$	0	0
$491$	−1.13576e6	−0.212609	−0.106304	−	0.994334i	$-0.533902\pi$
$491$	−1.13576e6	−0.212609	−0.106304	+	0.994334i	$0.533902\pi$
$492$	0	0
$493$	3.39093e6	0.628349
$494$	0	0
$495$	56334.7	0.0103339
$496$	0	0
$497$	−709878.	−0.128912
$498$	0	0
$499$	−646939.	−0.116309	−0.0581543	−	0.998308i	$-0.518522\pi$
$499$	−646939.	−0.116309	−0.0581543	+	0.998308i	$0.518522\pi$
$500$	0	0
$501$	3.76541e6	0.670220
$502$	0	0
$503$	−3.74759e6	−0.660438	−0.330219	−	0.943904i	$-0.607123\pi$
$503$	−3.74759e6	−0.660438	−0.330219	+	0.943904i	$0.607123\pi$
$504$	0	0
$505$	28693.7	0.00500678
$506$	0	0
$507$	2.59387e6	0.448156
$508$	0	0
$509$	6.98821e6	1.19556	0.597780	−	0.801660i	$-0.296050\pi$
$509$	6.98821e6	1.19556	0.597780	+	0.801660i	$0.296050\pi$
$510$	0	0
$511$	677299.	0.114744
$512$	0	0
$513$	−947471.	−0.158954
$514$	0	0
$515$	4.55279e6	0.756414
$516$	0	0
$517$	−470285.	−0.0773810
$518$	0	0
$519$	4.02624e6	0.656118
$520$	0	0
$521$	4.50959e6	0.727851	0.363926	−	0.931428i	$-0.381436\pi$
$521$	4.50959e6	0.727851	0.363926	+	0.931428i	$0.381436\pi$
$522$	0	0
$523$	2.19875e6	0.351497	0.175749	−	0.984435i	$-0.443765\pi$
$523$	2.19875e6	0.351497	0.175749	+	0.984435i	$0.443765\pi$
$524$	0	0
$525$	−219530.	−0.0347613
$526$	0	0
$527$	−2.50032e6	−0.392165
$528$	0	0
$529$	279841.	0.0434783
$530$	0	0
$531$	2.13483e6	0.328569
$532$	0	0
$533$	−8.39631e6	−1.28018
$534$	0	0
$535$	4.99387e6	0.754315
$536$	0	0
$537$	−1.73325e6	−0.259373
$538$	0	0
$539$	352528.	0.0522664
$540$	0	0
$541$	4.04994e6	0.594916	0.297458	−	0.954735i	$-0.403861\pi$
$541$	4.04994e6	0.594916	0.297458	+	0.954735i	$0.403861\pi$
$542$	0	0
$543$	4.12699e6	0.600667
$544$	0	0
$545$	−2.95232e6	−0.425767
$546$	0	0
$547$	−4.92426e6	−0.703676	−0.351838	−	0.936061i	$-0.614443\pi$
$547$	−4.92426e6	−0.703676	−0.351838	+	0.936061i	$0.614443\pi$
$548$	0	0
$549$	1.47830e6	0.209330
$550$	0	0
$551$	5.50694e6	0.772736
$552$	0	0
$553$	1.19039e6	0.165530
$554$	0	0
$555$	−2.60610e6	−0.359135
$556$	0	0
$557$	−2.12786e6	−0.290606	−0.145303	−	0.989387i	$-0.546416\pi$
$557$	−2.12786e6	−0.290606	−0.145303	+	0.989387i	$0.546416\pi$
$558$	0	0
$559$	−8.51001e6	−1.15186
$560$	0	0
$561$	152366.	0.0204400
$562$	0	0
$563$	4.11948e6	0.547736	0.273868	−	0.961767i	$-0.411697\pi$
$563$	4.11948e6	0.547736	0.273868	+	0.961767i	$0.411697\pi$
$564$	0	0
$565$	−5.68821e6	−0.749643
$566$	0	0
$567$	78292.3	0.0102273
$568$	0	0
$569$	4.63578e6	0.600263	0.300132	−	0.953898i	$-0.402969\pi$
$569$	4.63578e6	0.600263	0.300132	+	0.953898i	$0.402969\pi$
$570$	0	0
$571$	−4.43528e6	−0.569287	−0.284643	−	0.958633i	$-0.591875\pi$
$571$	−4.43528e6	−0.569287	−0.284643	+	0.958633i	$0.591875\pi$
$572$	0	0
$573$	3.33572e6	0.424427
$574$	0	0
$575$	1.08133e6	0.136392
$576$	0	0
$577$	−1.33423e7	−1.66836	−0.834182	−	0.551489i	$-0.814060\pi$
$577$	−1.33423e7	−1.66836	−0.834182	+	0.551489i	$0.814060\pi$
$578$	0	0
$579$	3.85818e6	0.478284
$580$	0	0
$581$	−946998.	−0.116388
$582$	0	0
$583$	390822.	0.0476220
$584$	0	0
$585$	2.16264e6	0.261273
$586$	0	0
$587$	6.80053e6	0.814605	0.407303	−	0.913293i	$-0.366469\pi$
$587$	6.80053e6	0.814605	0.407303	+	0.913293i	$0.366469\pi$
$588$	0	0
$589$	−4.06057e6	−0.482280
$590$	0	0
$591$	6.46183e6	0.761004
$592$	0	0
$593$	760654.	0.0888281	0.0444140	−	0.999013i	$-0.485858\pi$
$593$	760654.	0.0888281	0.0444140	+	0.999013i	$0.485858\pi$
$594$	0	0
$595$	313970.	0.0363576
$596$	0	0
$597$	−5.75440e6	−0.660791
$598$	0	0
$599$	−1.33143e7	−1.51619	−0.758093	−	0.652147i	$-0.773869\pi$
$599$	−1.33143e7	−1.51619	−0.758093	+	0.652147i	$0.773869\pi$
$600$	0	0
$601$	−8.81550e6	−0.995545	−0.497772	−	0.867308i	$-0.665849\pi$
$601$	−8.81550e6	−0.995545	−0.497772	+	0.867308i	$0.665849\pi$
$602$	0	0
$603$	−1.24150e6	−0.139045
$604$	0	0
$605$	5.28016e6	0.586487
$606$	0	0
$607$	−4.97779e6	−0.548359	−0.274179	−	0.961679i	$-0.588406\pi$
$607$	−4.97779e6	−0.548359	−0.274179	+	0.961679i	$0.588406\pi$
$608$	0	0
$609$	−455054.	−0.0497187
$610$	0	0
$611$	−1.80538e7	−1.95644
$612$	0	0
$613$	−3.81041e6	−0.409562	−0.204781	−	0.978808i	$-0.565648\pi$
$613$	−3.81041e6	−0.409562	−0.204781	+	0.978808i	$0.565648\pi$
$614$	0	0
$615$	−3.05925e6	−0.326157
$616$	0	0
$617$	1.11430e7	1.17839	0.589196	−	0.807990i	$-0.299444\pi$
$617$	1.11430e7	1.17839	0.589196	+	0.807990i	$0.299444\pi$
$618$	0	0
$619$	7.29111e6	0.764834	0.382417	−	0.923990i	$-0.375092\pi$
$619$	7.29111e6	0.764834	0.382417	+	0.923990i	$0.375092\pi$
$620$	0	0
$621$	−385641.	−0.0401286
$622$	0	0
$623$	65538.8	0.00676517
$624$	0	0
$625$	800566.	0.0819779
$626$	0	0
$627$	247446.	0.0251369
$628$	0	0
$629$	−7.04860e6	−0.710357
$630$	0	0
$631$	−1.35430e7	−1.35407	−0.677035	−	0.735951i	$-0.736735\pi$
$631$	−1.35430e7	−1.35407	−0.677035	+	0.735951i	$0.736735\pi$
$632$	0	0
$633$	−5.01758e6	−0.497720
$634$	0	0
$635$	−3.77576e6	−0.371595
$636$	0	0
$637$	1.35333e7	1.32146
$638$	0	0
$639$	−4.81859e6	−0.466839
$640$	0	0
$641$	1.36938e7	1.31637	0.658186	−	0.752855i	$-0.271324\pi$
$641$	1.36938e7	1.31637	0.658186	+	0.752855i	$0.271324\pi$
$642$	0	0
$643$	−2.00054e7	−1.90818	−0.954090	−	0.299520i	$-0.903174\pi$
$643$	−2.00054e7	−1.90818	−0.954090	+	0.299520i	$0.903174\pi$
$644$	0	0
$645$	−3.10068e6	−0.293466
$646$	0	0
$647$	−1.96156e7	−1.84222	−0.921110	−	0.389304i	$-0.872716\pi$
$647$	−1.96156e7	−1.84222	−0.921110	+	0.389304i	$0.872716\pi$
$648$	0	0
$649$	−557541.	−0.0519595
$650$	0	0
$651$	335537.	0.0310304
$652$	0	0
$653$	4.14890e6	0.380759	0.190380	−	0.981711i	$-0.439028\pi$
$653$	4.14890e6	0.380759	0.190380	+	0.981711i	$0.439028\pi$
$654$	0	0
$655$	−4.50717e6	−0.410489
$656$	0	0
$657$	4.59745e6	0.415531
$658$	0	0
$659$	−4.76733e6	−0.427623	−0.213812	−	0.976875i	$-0.568588\pi$
$659$	−4.76733e6	−0.427623	−0.213812	+	0.976875i	$0.568588\pi$
$660$	0	0
$661$	8.68651e6	0.773289	0.386644	−	0.922229i	$-0.373634\pi$
$661$	8.68651e6	0.773289	0.386644	+	0.922229i	$0.373634\pi$
$662$	0	0
$663$	5.84921e6	0.516789
$664$	0	0
$665$	509894.	0.0447121
$666$	0	0
$667$	2.24144e6	0.195080
$668$	0	0
$669$	7.90180e6	0.682591
$670$	0	0
$671$	−386079.	−0.0331032
$672$	0	0
$673$	1.41827e7	1.20704	0.603519	−	0.797349i	$-0.293765\pi$
$673$	1.41827e7	1.20704	0.603519	+	0.797349i	$0.293765\pi$
$674$	0	0
$675$	−1.49015e6	−0.125884
$676$	0	0
$677$	−1.72377e7	−1.44546	−0.722731	−	0.691129i	$-0.757114\pi$
$677$	−1.72377e7	−1.44546	−0.722731	+	0.691129i	$0.757114\pi$
$678$	0	0
$679$	−780570.	−0.0649736
$680$	0	0
$681$	827662.	0.0683888
$682$	0	0
$683$	−7.08632e6	−0.581258	−0.290629	−	0.956836i	$-0.593865\pi$
$683$	−7.08632e6	−0.581258	−0.290629	+	0.956836i	$0.593865\pi$
$684$	0	0
$685$	−1.35884e7	−1.10648
$686$	0	0
$687$	−1.04196e7	−0.842288
$688$	0	0
$689$	1.50033e7	1.20404
$690$	0	0
$691$	−1.97744e7	−1.57546	−0.787732	−	0.616018i	$-0.788745\pi$
$691$	−1.97744e7	−1.57546	−0.787732	+	0.616018i	$0.788745\pi$
$692$	0	0
$693$	−20447.2	−0.00161734
$694$	0	0
$695$	9.07929e6	0.713001
$696$	0	0
$697$	−8.27422e6	−0.645127
$698$	0	0
$699$	−1.18548e6	−0.0917698
$700$	0	0
$701$	−6.07708e6	−0.467089	−0.233545	−	0.972346i	$-0.575033\pi$
$701$	−6.07708e6	−0.467089	−0.233545	+	0.972346i	$0.575033\pi$
$702$	0	0
$703$	−1.14471e7	−0.873588
$704$	0	0
$705$	−6.57803e6	−0.498452
$706$	0	0
$707$	−10414.6	−0.000783602 0
$708$	0	0
$709$	1.86123e7	1.39054	0.695272	−	0.718747i	$-0.255284\pi$
$709$	1.86123e7	1.39054	0.695272	+	0.718747i	$0.255284\pi$
$710$	0	0
$711$	8.08024e6	0.599447
$712$	0	0
$713$	−1.65274e6	−0.121753
$714$	0	0
$715$	−564805.	−0.0413175
$716$	0	0
$717$	−1.37425e7	−0.998316
$718$	0	0
$719$	−3.37138e6	−0.243212	−0.121606	−	0.992578i	$-0.538805\pi$
$719$	−3.37138e6	−0.243212	−0.121606	+	0.992578i	$0.538805\pi$
$720$	0	0
$721$	−1.65247e6	−0.118385
$722$	0	0
$723$	8.21230e6	0.584277
$724$	0	0
$725$	8.66113e6	0.611969
$726$	0	0
$727$	5.07291e6	0.355976	0.177988	−	0.984033i	$-0.443041\pi$
$727$	5.07291e6	0.355976	0.177988	+	0.984033i	$0.443041\pi$
$728$	0	0
$729$	531441.	0.0370370
$730$	0	0
$731$	−8.38628e6	−0.580465
$732$	0	0
$733$	−2.47947e7	−1.70451	−0.852255	−	0.523126i	$-0.824766\pi$
$733$	−2.47947e7	−1.70451	−0.852255	+	0.523126i	$0.824766\pi$
$734$	0	0
$735$	4.93094e6	0.336675
$736$	0	0
$737$	324237.	0.0219884
$738$	0	0
$739$	6.98541e6	0.470523	0.235262	−	0.971932i	$-0.424405\pi$
$739$	6.98541e6	0.470523	0.235262	+	0.971932i	$0.424405\pi$
$740$	0	0
$741$	9.49924e6	0.635541
$742$	0	0
$743$	632349.	0.0420228	0.0210114	−	0.999779i	$-0.493311\pi$
$743$	632349.	0.0420228	0.0210114	+	0.999779i	$0.493311\pi$
$744$	0	0
$745$	−1.28139e6	−0.0845847
$746$	0	0
$747$	−6.42813e6	−0.421486
$748$	0	0
$749$	−1.81257e6	−0.118057
$750$	0	0
$751$	1.19559e7	0.773541	0.386771	−	0.922176i	$-0.373591\pi$
$751$	1.19559e7	0.773541	0.386771	+	0.922176i	$0.373591\pi$
$752$	0	0
$753$	1.27950e7	0.822342
$754$	0	0
$755$	−7.67534e6	−0.490039
$756$	0	0
$757$	2.13022e7	1.35109	0.675546	−	0.737318i	$-0.263908\pi$
$757$	2.13022e7	1.35109	0.675546	+	0.737318i	$0.263908\pi$
$758$	0	0
$759$	100716.	0.00634590
$760$	0	0
$761$	595985.	0.0373056	0.0186528	−	0.999826i	$-0.494062\pi$
$761$	595985.	0.0373056	0.0186528	+	0.999826i	$0.494062\pi$
$762$	0	0
$763$	1.07157e6	0.0666361
$764$	0	0
$765$	2.13120e6	0.131665
$766$	0	0
$767$	−2.14035e7	−1.31370
$768$	0	0
$769$	−2.76974e7	−1.68897	−0.844487	−	0.535575i	$-0.820095\pi$
$769$	−2.76974e7	−1.68897	−0.844487	+	0.535575i	$0.820095\pi$
$770$	0	0
$771$	1.31287e7	0.795402
$772$	0	0
$773$	684547.	0.0412054	0.0206027	−	0.999788i	$-0.493441\pi$
$773$	684547.	0.0412054	0.0206027	+	0.999788i	$0.493441\pi$
$774$	0	0
$775$	−6.38633e6	−0.381942
$776$	0	0
$777$	945906.	0.0562076
$778$	0	0
$779$	−1.34375e7	−0.793369
$780$	0	0
$781$	1.25844e6	0.0738255
$782$	0	0
$783$	−3.08887e6	−0.180051
$784$	0	0
$785$	−8.25012e6	−0.477844
$786$	0	0
$787$	−1.63797e7	−0.942693	−0.471346	−	0.881948i	$-0.656232\pi$
$787$	−1.63797e7	−0.942693	−0.471346	+	0.881948i	$0.656232\pi$
$788$	0	0
$789$	−1.17295e7	−0.670790
$790$	0	0
$791$	2.06459e6	0.117325
$792$	0	0
$793$	−1.48212e7	−0.836954
$794$	0	0
$795$	5.46656e6	0.306758
$796$	0	0
$797$	1.22309e7	0.682042	0.341021	−	0.940056i	$-0.389227\pi$
$797$	1.22309e7	0.682042	0.341021	+	0.940056i	$0.389227\pi$
$798$	0	0
$799$	−1.77913e7	−0.985920
$800$	0	0
$801$	444871.	0.0244993
$802$	0	0
$803$	−1.20069e6	−0.0657116
$804$	0	0
$805$	207538.	0.0112877
$806$	0	0
$807$	−2.29059e6	−0.123812
$808$	0	0
$809$	2.97550e6	0.159841	0.0799205	−	0.996801i	$-0.474533\pi$
$809$	2.97550e6	0.159841	0.0799205	+	0.996801i	$0.474533\pi$
$810$	0	0
$811$	−1.13521e7	−0.606072	−0.303036	−	0.952979i	$-0.598000\pi$
$811$	−1.13521e7	−0.606072	−0.303036	+	0.952979i	$0.598000\pi$
$812$	0	0
$813$	7.43898e6	0.394718
$814$	0	0
$815$	3.80106e6	0.200452
$816$	0	0
$817$	−1.36195e7	−0.713848
$818$	0	0
$819$	−784950.	−0.0408914
$820$	0	0
$821$	3.72721e7	1.92986	0.964930	−	0.262507i	$-0.0845494\pi$
$821$	3.72721e7	1.92986	0.964930	+	0.262507i	$0.0845494\pi$
$822$	0	0
$823$	−7.12997e6	−0.366934	−0.183467	−	0.983026i	$-0.558732\pi$
$823$	−7.12997e6	−0.366934	−0.183467	+	0.983026i	$0.558732\pi$
$824$	0	0
$825$	389175.	0.0199072
$826$	0	0
$827$	1.97960e6	0.100650	0.0503250	−	0.998733i	$-0.483974\pi$
$827$	1.97960e6	0.100650	0.0503250	+	0.998733i	$0.483974\pi$
$828$	0	0
$829$	−2.74155e7	−1.38551	−0.692755	−	0.721173i	$-0.743603\pi$
$829$	−2.74155e7	−1.38551	−0.692755	+	0.721173i	$0.743603\pi$
$830$	0	0
$831$	−1.54597e7	−0.776600
$832$	0	0
$833$	1.33365e7	0.665931
$834$	0	0
$835$	−1.37550e7	−0.682724
$836$	0	0
$837$	2.27759e6	0.112373
$838$	0	0
$839$	3.53095e7	1.73176	0.865878	−	0.500254i	$-0.166760\pi$
$839$	3.53095e7	1.73176	0.865878	+	0.500254i	$0.166760\pi$
$840$	0	0
$841$	−2.55791e6	−0.124708
$842$	0	0
$843$	−5.96985e6	−0.289331
$844$	0	0
$845$	−9.47541e6	−0.456516
$846$	0	0
$847$	−1.91648e6	−0.0917900
$848$	0	0
$849$	−1.76826e7	−0.841933
$850$	0	0
$851$	−4.65921e6	−0.220540
$852$	0	0
$853$	−9.71493e6	−0.457159	−0.228579	−	0.973525i	$-0.573408\pi$
$853$	−9.71493e6	−0.457159	−0.228579	+	0.973525i	$0.573408\pi$
$854$	0	0
$855$	3.46111e6	0.161920
$856$	0	0
$857$	1.53251e6	0.0712774	0.0356387	−	0.999365i	$-0.488653\pi$
$857$	1.53251e6	0.0712774	0.0356387	+	0.999365i	$0.488653\pi$
$858$	0	0
$859$	−1.42418e7	−0.658539	−0.329269	−	0.944236i	$-0.606802\pi$
$859$	−1.42418e7	−0.658539	−0.329269	+	0.944236i	$0.606802\pi$
$860$	0	0
$861$	1.11038e6	0.0510463
$862$	0	0
$863$	6.25885e6	0.286067	0.143034	−	0.989718i	$-0.454314\pi$
$863$	6.25885e6	0.286067	0.143034	+	0.989718i	$0.454314\pi$
$864$	0	0
$865$	−1.47079e7	−0.668358
$866$	0	0
$867$	−7.01455e6	−0.316922
$868$	0	0
$869$	−2.11027e6	−0.0947958
$870$	0	0
$871$	1.24472e7	0.555937
$872$	0	0
$873$	−5.29843e6	−0.235295
$874$	0	0
$875$	2.02795e6	0.0895439
$876$	0	0
$877$	7.40611e6	0.325155	0.162578	−	0.986696i	$-0.448019\pi$
$877$	7.40611e6	0.325155	0.162578	+	0.986696i	$0.448019\pi$
$878$	0	0
$879$	1.57770e7	0.688736
$880$	0	0
$881$	−2.28474e7	−0.991739	−0.495869	−	0.868397i	$-0.665150\pi$
$881$	−2.28474e7	−0.991739	−0.495869	+	0.868397i	$0.665150\pi$
$882$	0	0
$883$	−1.73858e7	−0.750399	−0.375199	−	0.926944i	$-0.622426\pi$
$883$	−1.73858e7	−0.750399	−0.375199	+	0.926944i	$0.622426\pi$
$884$	0	0
$885$	−7.79852e6	−0.334699
$886$	0	0
$887$	−900330.	−0.0384231	−0.0192116	−	0.999815i	$-0.506116\pi$
$887$	−900330.	−0.0384231	−0.0192116	+	0.999815i	$0.506116\pi$
$888$	0	0
$889$	1.37044e6	0.0581577
$890$	0	0
$891$	−138794.	−0.00585700
$892$	0	0
$893$	−2.88935e7	−1.21247
$894$	0	0
$895$	6.33154e6	0.264212
$896$	0	0
$897$	3.86639e6	0.160444
$898$	0	0
$899$	−1.32379e7	−0.546287
$900$	0	0
$901$	1.47852e7	0.606757
$902$	0	0
$903$	1.12542e6	0.0459298
$904$	0	0
$905$	−1.50759e7	−0.611873
$906$	0	0
$907$	−7.49317e6	−0.302446	−0.151223	−	0.988500i	$-0.548321\pi$
$907$	−7.49317e6	−0.302446	−0.151223	+	0.988500i	$0.548321\pi$
$908$	0	0
$909$	−70693.5	−0.00283772
$910$	0	0
$911$	2.39623e7	0.956604	0.478302	−	0.878195i	$-0.341252\pi$
$911$	2.39623e7	0.956604	0.478302	+	0.878195i	$0.341252\pi$
$912$	0	0
$913$	1.67880e6	0.0666534
$914$	0	0
$915$	−5.40022e6	−0.213235
$916$	0	0
$917$	1.63592e6	0.0642449
$918$	0	0
$919$	−3.84668e7	−1.50244	−0.751220	−	0.660052i	$-0.770534\pi$
$919$	−3.84668e7	−1.50244	−0.751220	+	0.660052i	$0.770534\pi$
$920$	0	0
$921$	2.24999e7	0.874040
$922$	0	0
$923$	4.83106e7	1.86654
$924$	0	0
$925$	−1.80036e7	−0.691839
$926$	0	0
$927$	−1.12168e7	−0.428718
$928$	0	0
$929$	4.25371e7	1.61707	0.808535	−	0.588448i	$-0.200261\pi$
$929$	4.25371e7	1.61707	0.808535	+	0.588448i	$0.200261\pi$
$930$	0	0
$931$	2.16588e7	0.818954
$932$	0	0
$933$	2.59097e7	0.974447
$934$	0	0
$935$	−556593.	−0.0208213
$936$	0	0
$937$	1.41125e7	0.525114	0.262557	−	0.964916i	$-0.415434\pi$
$937$	1.41125e7	0.525114	0.262557	+	0.964916i	$0.415434\pi$
$938$	0	0
$939$	−9.18872e6	−0.340088
$940$	0	0
$941$	−3.41583e7	−1.25754	−0.628770	−	0.777591i	$-0.716441\pi$
$941$	−3.41583e7	−1.25754	−0.628770	+	0.777591i	$0.716441\pi$
$942$	0	0
$943$	−5.46936e6	−0.200289
$944$	0	0
$945$	−286002.	−0.0104181
$946$	0	0
$947$	7.55224e6	0.273654	0.136827	−	0.990595i	$-0.456310\pi$
$947$	7.55224e6	0.273654	0.136827	+	0.990595i	$0.456310\pi$
$948$	0	0
$949$	−4.60935e7	−1.66140
$950$	0	0
$951$	1.64742e7	0.590682
$952$	0	0
$953$	1.22889e7	0.438310	0.219155	−	0.975690i	$-0.429670\pi$
$953$	1.22889e7	0.438310	0.219155	+	0.975690i	$0.429670\pi$
$954$	0	0
$955$	−1.21854e7	−0.432345
$956$	0	0
$957$	806702.	0.0284730
$958$	0	0
$959$	4.93203e6	0.173173
$960$	0	0
$961$	−1.88681e7	−0.659052
$962$	0	0
$963$	−1.23036e7	−0.427528
$964$	0	0
$965$	−1.40939e7	−0.487207
$966$	0	0
$967$	2.74334e7	0.943439	0.471720	−	0.881749i	$-0.343633\pi$
$967$	2.74334e7	0.943439	0.471720	+	0.881749i	$0.343633\pi$
$968$	0	0
$969$	9.36112e6	0.320272
$970$	0	0
$971$	−3.99172e7	−1.35866	−0.679332	−	0.733831i	$-0.737730\pi$
$971$	−3.99172e7	−1.35866	−0.679332	+	0.733831i	$0.737730\pi$
$972$	0	0
$973$	−3.29541e6	−0.111591
$974$	0	0
$975$	1.49401e7	0.503317
$976$	0	0
$977$	−1.00752e7	−0.337689	−0.168844	−	0.985643i	$-0.554004\pi$
$977$	−1.00752e7	−0.337689	−0.168844	+	0.985643i	$0.554004\pi$
$978$	0	0
$979$	−116185.	−0.00387429
$980$	0	0
$981$	7.27372e6	0.241315
$982$	0	0
$983$	1.32878e6	0.0438601	0.0219300	−	0.999760i	$-0.493019\pi$
$983$	1.32878e6	0.0438601	0.0219300	+	0.999760i	$0.493019\pi$
$984$	0	0
$985$	−2.36051e7	−0.775201
$986$	0	0
$987$	2.38755e6	0.0780118
$988$	0	0
$989$	−5.54342e6	−0.180214
$990$	0	0
$991$	1.43544e7	0.464302	0.232151	−	0.972680i	$-0.425424\pi$
$991$	1.43544e7	0.464302	0.232151	+	0.972680i	$0.425424\pi$
$992$	0	0
$993$	2.60523e7	0.838442
$994$	0	0
$995$	2.10208e7	0.673118
$996$	0	0
$997$	−2.69771e7	−0.859523	−0.429761	−	0.902942i	$-0.641402\pi$
$997$	−2.69771e7	−0.859523	−0.429761	+	0.902942i	$0.641402\pi$
$998$	0	0
$999$	6.42072e6	0.203550

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1104.6.a.y.1.3		7
4.3	odd	2		552.6.a.f.1.3	✓	7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
552.6.a.f.1.3	✓	7	4.3	odd	2
1104.6.a.y.1.3		7	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 1104 = 2^{4} \cdot 3 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	1104.a (trivial)

\(f(q)\)	\(=\)	\(q+9.00000 q^{3} -32.8770 q^{5} +11.9330 q^{7} +81.0000 q^{9} +O(q^{10})\)	\(q+9.00000 q^{3} -32.8770 q^{5} +11.9330 q^{7} +81.0000 q^{9} -21.1543 q^{11} -812.097 q^{13} -295.893 q^{15} -800.289 q^{17} -1299.69 q^{19} +107.397 q^{21} -529.000 q^{23} -2044.10 q^{25} +729.000 q^{27} -4237.13 q^{29} +3124.27 q^{31} -190.389 q^{33} -392.320 q^{35} +8807.57 q^{37} -7308.87 q^{39} +10339.0 q^{41} +10479.1 q^{43} -2663.03 q^{45} +22231.1 q^{47} -16664.6 q^{49} -7202.60 q^{51} -18474.8 q^{53} +695.490 q^{55} -11697.2 q^{57} +26355.9 q^{59} +18250.6 q^{61} +966.572 q^{63} +26699.3 q^{65} -15327.2 q^{67} -4761.00 q^{69} -59488.7 q^{71} +56758.6 q^{73} -18396.9 q^{75} -252.434 q^{77} +99756.1 q^{79} +6561.00 q^{81} -79359.7 q^{83} +26311.1 q^{85} -38134.1 q^{87} +5492.24 q^{89} -9690.74 q^{91} +28118.4 q^{93} +42729.8 q^{95} -65412.8 q^{97} -1713.50 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + 63 q^{3} + 80 q^{5} - 46 q^{7} + 567 q^{9}+O(q^{10}) \) 7 * q + 63 * q^3 + 80 * q^5 - 46 * q^7 + 567 * q^9	\( 7 q + 63 q^{3} + 80 q^{5} - 46 q^{7} + 567 q^{9} - 124 q^{11} + 966 q^{13} + 720 q^{15} + 652 q^{17} + 542 q^{19} - 414 q^{21} - 3703 q^{23} + 5165 q^{25} + 5103 q^{27} + 5222 q^{29} + 2660 q^{31} - 1116 q^{33} + 17088 q^{35} - 202 q^{37} + 8694 q^{39} - 706 q^{41} + 7086 q^{43} + 6480 q^{45} + 37696 q^{47} - 34469 q^{49} + 5868 q^{51} + 30180 q^{53} + 51104 q^{55} + 4878 q^{57} + 12284 q^{59} + 53534 q^{61} - 3726 q^{63} + 46920 q^{65} - 47750 q^{67} - 33327 q^{69} + 2552 q^{71} - 19474 q^{73} + 46485 q^{75} + 36264 q^{77} + 4510 q^{79} + 45927 q^{81} + 47580 q^{83} + 224944 q^{85} + 46998 q^{87} + 150664 q^{89} - 57460 q^{91} + 23940 q^{93} + 21648 q^{95} + 392426 q^{97} - 10044 q^{99}+O(q^{100}) \) 7 * q + 63 * q^3 + 80 * q^5 - 46 * q^7 + 567 * q^9 - 124 * q^11 + 966 * q^13 + 720 * q^15 + 652 * q^17 + 542 * q^19 - 414 * q^21 - 3703 * q^23 + 5165 * q^25 + 5103 * q^27 + 5222 * q^29 + 2660 * q^31 - 1116 * q^33 + 17088 * q^35 - 202 * q^37 + 8694 * q^39 - 706 * q^41 + 7086 * q^43 + 6480 * q^45 + 37696 * q^47 - 34469 * q^49 + 5868 * q^51 + 30180 * q^53 + 51104 * q^55 + 4878 * q^57 + 12284 * q^59 + 53534 * q^61 - 3726 * q^63 + 46920 * q^65 - 47750 * q^67 - 33327 * q^69 + 2552 * q^71 - 19474 * q^73 + 46485 * q^75 + 36264 * q^77 + 4510 * q^79 + 45927 * q^81 + 47580 * q^83 + 224944 * q^85 + 46998 * q^87 + 150664 * q^89 - 57460 * q^91 + 23940 * q^93 + 21648 * q^95 + 392426 * q^97 - 10044 * q^99

Introduction

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Modular forms

Varieties

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