LMFDB - Embedded newform 2240.4.a.ca.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2240,4,Mod(1,2240)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2240, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2240.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2240 = 2^{6} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2240.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:	$132.164278413$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	$\mathbb{Q}[x]/(x^{4} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 68x^{2} - 168x - 99 $ x^4 - 2x^3 - 68x^2 - 168*x - 99
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 1120)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.914579$ of defining polynomial
Character	$\chi$	$=$	2240.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-4.75423 q^{3} -5.00000 q^{5} -7.00000 q^{7} -4.39727 q^{9} +O(q^{10})$	$q-4.75423 q^{3} -5.00000 q^{5} -7.00000 q^{7} -4.39727 q^{9} -47.7308 q^{11} -48.7494 q^{13} +23.7712 q^{15} +78.2159 q^{17} -23.0153 q^{19} +33.2796 q^{21} +32.9847 q^{23} +25.0000 q^{25} +149.270 q^{27} +158.595 q^{29} +55.9277 q^{31} +226.923 q^{33} +35.0000 q^{35} +165.083 q^{37} +231.766 q^{39} +69.2477 q^{41} +221.017 q^{43} +21.9863 q^{45} -83.8679 q^{47} +49.0000 q^{49} -371.857 q^{51} -63.4374 q^{53} +238.654 q^{55} +109.420 q^{57} -244.152 q^{59} -461.583 q^{61} +30.7809 q^{63} +243.747 q^{65} -620.922 q^{67} -156.817 q^{69} +293.354 q^{71} +993.671 q^{73} -118.856 q^{75} +334.116 q^{77} +345.852 q^{79} -590.938 q^{81} -313.818 q^{83} -391.079 q^{85} -753.995 q^{87} +529.928 q^{89} +341.246 q^{91} -265.893 q^{93} +115.076 q^{95} +121.755 q^{97} +209.885 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 9 q^{3} - 20 q^{5} - 28 q^{7} + 55 q^{9}+O(q^{10}) $ 4 * q - 9 * q^3 - 20 * q^5 - 28 * q^7 + 55 * q^9	$ 4 q - 9 q^{3} - 20 q^{5} - 28 q^{7} + 55 q^{9} - 33 q^{11} + 83 q^{13} + 45 q^{15} - 13 q^{17} - 132 q^{19} + 63 q^{21} + 92 q^{23} + 100 q^{25} - 495 q^{27} + 113 q^{29} + 94 q^{31} + 11 q^{33} + 140 q^{35} + 54 q^{37} + 145 q^{39} + 428 q^{41} - 604 q^{43} - 275 q^{45} + 709 q^{47} + 196 q^{49} + 13 q^{51} - 554 q^{53} + 165 q^{55} + 1044 q^{57} + 100 q^{59} + 588 q^{61} - 385 q^{63} - 415 q^{65} - 484 q^{67} + 540 q^{69} - 128 q^{71} + 1508 q^{73} - 225 q^{75} + 231 q^{77} + 587 q^{79} + 2548 q^{81} - 220 q^{83} + 65 q^{85} + 2887 q^{87} - 1392 q^{89} - 581 q^{91} - 2074 q^{93} + 660 q^{95} + 643 q^{97} - 690 q^{99}+O(q^{100}) $ 4 * q - 9 * q^3 - 20 * q^5 - 28 * q^7 + 55 * q^9 - 33 * q^11 + 83 * q^13 + 45 * q^15 - 13 * q^17 - 132 * q^19 + 63 * q^21 + 92 * q^23 + 100 * q^25 - 495 * q^27 + 113 * q^29 + 94 * q^31 + 11 * q^33 + 140 * q^35 + 54 * q^37 + 145 * q^39 + 428 * q^41 - 604 * q^43 - 275 * q^45 + 709 * q^47 + 196 * q^49 + 13 * q^51 - 554 * q^53 + 165 * q^55 + 1044 * q^57 + 100 * q^59 + 588 * q^61 - 385 * q^63 - 415 * q^65 - 484 * q^67 + 540 * q^69 - 128 * q^71 + 1508 * q^73 - 225 * q^75 + 231 * q^77 + 587 * q^79 + 2548 * q^81 - 220 * q^83 + 65 * q^85 + 2887 * q^87 - 1392 * q^89 - 581 * q^91 - 2074 * q^93 + 660 * q^95 + 643 * q^97 - 690 * 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
$3$	−4.75423	−0.914953	−0.457476	−	0.889222i	$-0.651247\pi$
$3$	−4.75423	−0.914953	−0.457476	+	0.889222i	$0.651247\pi$
$4$	0	0
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	0	0
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	0	0
$9$	−4.39727	−0.162862
$10$	0	0
$11$	−47.7308	−1.30831	−0.654154	−	0.756361i	$-0.726975\pi$
$11$	−47.7308	−1.30831	−0.654154	+	0.756361i	$0.726975\pi$
$12$	0	0
$13$	−48.7494	−1.04005	−0.520025	−	0.854151i	$-0.674077\pi$
$13$	−48.7494	−1.04005	−0.520025	+	0.854151i	$0.674077\pi$
$14$	0	0
$15$	23.7712	0.409179
$16$	0	0
$17$	78.2159	1.11589	0.557945	−	0.829878i	$-0.311590\pi$
$17$	78.2159	1.11589	0.557945	+	0.829878i	$0.311590\pi$
$18$	0	0
$19$	−23.0153	−0.277898	−0.138949	−	0.990300i	$-0.544372\pi$
$19$	−23.0153	−0.277898	−0.138949	+	0.990300i	$0.544372\pi$
$20$	0	0
$21$	33.2796	0.345820
$22$	0	0
$23$	32.9847	0.299034	0.149517	−	0.988759i	$-0.452228\pi$
$23$	32.9847	0.299034	0.149517	+	0.988759i	$0.452228\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	149.270	1.06396
$28$	0	0
$29$	158.595	1.01553	0.507763	−	0.861497i	$-0.330472\pi$
$29$	158.595	1.01553	0.507763	+	0.861497i	$0.330472\pi$
$30$	0	0
$31$	55.9277	0.324029	0.162015	−	0.986788i	$-0.448201\pi$
$31$	55.9277	0.324029	0.162015	+	0.986788i	$0.448201\pi$
$32$	0	0
$33$	226.923	1.19704
$34$	0	0
$35$	35.0000	0.169031
$36$	0	0
$37$	165.083	0.733499	0.366750	−	0.930320i	$-0.380471\pi$
$37$	165.083	0.733499	0.366750	+	0.930320i	$0.380471\pi$
$38$	0	0
$39$	231.766	0.951597
$40$	0	0
$41$	69.2477	0.263772	0.131886	−	0.991265i	$-0.457897\pi$
$41$	69.2477	0.263772	0.131886	+	0.991265i	$0.457897\pi$
$42$	0	0
$43$	221.017	0.783832	0.391916	−	0.920001i	$-0.371812\pi$
$43$	221.017	0.783832	0.391916	+	0.920001i	$0.371812\pi$
$44$	0	0
$45$	21.9863	0.0728340
$46$	0	0
$47$	−83.8679	−0.260285	−0.130142	−	0.991495i	$-0.541543\pi$
$47$	−83.8679	−0.260285	−0.130142	+	0.991495i	$0.541543\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	0	0
$51$	−371.857	−1.02099
$52$	0	0
$53$	−63.4374	−0.164411	−0.0822057	−	0.996615i	$-0.526196\pi$
$53$	−63.4374	−0.164411	−0.0822057	+	0.996615i	$0.526196\pi$
$54$	0	0
$55$	238.654	0.585093
$56$	0	0
$57$	109.420	0.254264
$58$	0	0
$59$	−244.152	−0.538743	−0.269371	−	0.963036i	$-0.586816\pi$
$59$	−244.152	−0.538743	−0.269371	+	0.963036i	$0.586816\pi$
$60$	0	0
$61$	−461.583	−0.968846	−0.484423	−	0.874834i	$-0.660970\pi$
$61$	−461.583	−0.968846	−0.484423	+	0.874834i	$0.660970\pi$
$62$	0	0
$63$	30.7809	0.0615560
$64$	0	0
$65$	243.747	0.465125
$66$	0	0
$67$	−620.922	−1.13220	−0.566102	−	0.824335i	$-0.691549\pi$
$67$	−620.922	−1.13220	−0.566102	+	0.824335i	$0.691549\pi$
$68$	0	0
$69$	−156.817	−0.273602
$70$	0	0
$71$	293.354	0.490348	0.245174	−	0.969479i	$-0.421155\pi$
$71$	293.354	0.490348	0.245174	+	0.969479i	$0.421155\pi$
$72$	0	0
$73$	993.671	1.59316	0.796578	−	0.604536i	$-0.206641\pi$
$73$	993.671	1.59316	0.796578	+	0.604536i	$0.206641\pi$
$74$	0	0
$75$	−118.856	−0.182991
$76$	0	0
$77$	334.116	0.494494
$78$	0	0
$79$	345.852	0.492549	0.246275	−	0.969200i	$-0.420794\pi$
$79$	345.852	0.492549	0.246275	+	0.969200i	$0.420794\pi$
$80$	0	0
$81$	−590.938	−0.810614
$82$	0	0
$83$	−313.818	−0.415013	−0.207506	−	0.978234i	$-0.566535\pi$
$83$	−313.818	−0.415013	−0.207506	+	0.978234i	$0.566535\pi$
$84$	0	0
$85$	−391.079	−0.499041
$86$	0	0
$87$	−753.995	−0.929159
$88$	0	0
$89$	529.928	0.631148	0.315574	−	0.948901i	$-0.397803\pi$
$89$	529.928	0.631148	0.315574	+	0.948901i	$0.397803\pi$
$90$	0	0
$91$	341.246	0.393102
$92$	0	0
$93$	−265.893	−0.296471
$94$	0	0
$95$	115.076	0.124280
$96$	0	0
$97$	121.755	0.127447	0.0637234	−	0.997968i	$-0.479702\pi$
$97$	121.755	0.127447	0.0637234	+	0.997968i	$0.479702\pi$
$98$	0	0
$99$	209.885	0.213073
$100$	0	0
$101$	357.751	0.352451	0.176225	−	0.984350i	$-0.443611\pi$
$101$	357.751	0.352451	0.176225	+	0.984350i	$0.443611\pi$
$102$	0	0
$103$	768.986	0.735635	0.367818	−	0.929898i	$-0.380105\pi$
$103$	768.986	0.735635	0.367818	+	0.929898i	$0.380105\pi$
$104$	0	0
$105$	−166.398	−0.154655
$106$	0	0
$107$	−1110.71	−1.00352	−0.501761	−	0.865007i	$-0.667314\pi$
$107$	−1110.71	−1.00352	−0.501761	+	0.865007i	$0.667314\pi$
$108$	0	0
$109$	441.419	0.387892	0.193946	−	0.981012i	$-0.437871\pi$
$109$	441.419	0.387892	0.193946	+	0.981012i	$0.437871\pi$
$110$	0	0
$111$	−784.843	−0.671117
$112$	0	0
$113$	−106.282	−0.0884796	−0.0442398	−	0.999021i	$-0.514087\pi$
$113$	−106.282	−0.0884796	−0.0442398	+	0.999021i	$0.514087\pi$
$114$	0	0
$115$	−164.924	−0.133732
$116$	0	0
$117$	214.364	0.169384
$118$	0	0
$119$	−547.511	−0.421767
$120$	0	0
$121$	947.232	0.711670
$122$	0	0
$123$	−329.219	−0.241339
$124$	0	0
$125$	−125.000	−0.0894427
$126$	0	0
$127$	160.779	0.112337	0.0561686	−	0.998421i	$-0.482112\pi$
$127$	160.779	0.112337	0.0561686	+	0.998421i	$0.482112\pi$
$128$	0	0
$129$	−1050.77	−0.717169
$130$	0	0
$131$	−2515.29	−1.67757	−0.838787	−	0.544459i	$-0.816735\pi$
$131$	−2515.29	−1.67757	−0.838787	+	0.544459i	$0.816735\pi$
$132$	0	0
$133$	161.107	0.105036
$134$	0	0
$135$	−746.350	−0.475819
$136$	0	0
$137$	−989.039	−0.616783	−0.308392	−	0.951259i	$-0.599791\pi$
$137$	−989.039	−0.616783	−0.308392	+	0.951259i	$0.599791\pi$
$138$	0	0
$139$	−405.360	−0.247354	−0.123677	−	0.992323i	$-0.539469\pi$
$139$	−405.360	−0.247354	−0.123677	+	0.992323i	$0.539469\pi$
$140$	0	0
$141$	398.727	0.238148
$142$	0	0
$143$	2326.85	1.36071
$144$	0	0
$145$	−792.973	−0.454157
$146$	0	0
$147$	−232.957	−0.130708
$148$	0	0
$149$	3444.19	1.89368	0.946842	−	0.321699i	$-0.104254\pi$
$149$	3444.19	1.89368	0.946842	+	0.321699i	$0.104254\pi$
$150$	0	0
$151$	1647.98	0.888153	0.444076	−	0.895989i	$-0.353532\pi$
$151$	1647.98	0.888153	0.444076	+	0.895989i	$0.353532\pi$
$152$	0	0
$153$	−343.936	−0.181736
$154$	0	0
$155$	−279.638	−0.144910
$156$	0	0
$157$	2832.70	1.43996	0.719982	−	0.693992i	$-0.244150\pi$
$157$	2832.70	1.43996	0.719982	+	0.693992i	$0.244150\pi$
$158$	0	0
$159$	301.596	0.150429
$160$	0	0
$161$	−230.893	−0.113024
$162$	0	0
$163$	−2464.41	−1.18422	−0.592109	−	0.805858i	$-0.701705\pi$
$163$	−2464.41	−1.18422	−0.592109	+	0.805858i	$0.701705\pi$
$164$	0	0
$165$	−1134.62	−0.535332
$166$	0	0
$167$	−799.488	−0.370457	−0.185228	−	0.982696i	$-0.559302\pi$
$167$	−799.488	−0.370457	−0.185228	+	0.982696i	$0.559302\pi$
$168$	0	0
$169$	179.505	0.0817048
$170$	0	0
$171$	101.204	0.0452590
$172$	0	0
$173$	2338.71	1.02780	0.513899	−	0.857851i	$-0.328200\pi$
$173$	2338.71	1.02780	0.513899	+	0.857851i	$0.328200\pi$
$174$	0	0
$175$	−175.000	−0.0755929
$176$	0	0
$177$	1160.75	0.492924
$178$	0	0
$179$	−1231.72	−0.514319	−0.257159	−	0.966369i	$-0.582787\pi$
$179$	−1231.72	−0.514319	−0.257159	+	0.966369i	$0.582787\pi$
$180$	0	0
$181$	−3429.69	−1.40844	−0.704218	−	0.709984i	$-0.748702\pi$
$181$	−3429.69	−1.40844	−0.704218	+	0.709984i	$0.748702\pi$
$182$	0	0
$183$	2194.47	0.886448
$184$	0	0
$185$	−825.415	−0.328031
$186$	0	0
$187$	−3733.31	−1.45993
$188$	0	0
$189$	−1044.89	−0.402140
$190$	0	0
$191$	−707.531	−0.268038	−0.134019	−	0.990979i	$-0.542788\pi$
$191$	−707.531	−0.268038	−0.134019	+	0.990979i	$0.542788\pi$
$192$	0	0
$193$	−768.906	−0.286773	−0.143386	−	0.989667i	$-0.545799\pi$
$193$	−768.906	−0.286773	−0.143386	+	0.989667i	$0.545799\pi$
$194$	0	0
$195$	−1158.83	−0.425567
$196$	0	0
$197$	3274.74	1.18434	0.592172	−	0.805811i	$-0.298271\pi$
$197$	3274.74	1.18434	0.592172	+	0.805811i	$0.298271\pi$
$198$	0	0
$199$	2942.01	1.04801	0.524004	−	0.851716i	$-0.324438\pi$
$199$	2942.01	1.04801	0.524004	+	0.851716i	$0.324438\pi$
$200$	0	0
$201$	2952.01	1.03591
$202$	0	0
$203$	−1110.16	−0.383833
$204$	0	0
$205$	−346.238	−0.117963
$206$	0	0
$207$	−145.043	−0.0487013
$208$	0	0
$209$	1098.54	0.363577
$210$	0	0
$211$	3340.88	1.09003	0.545014	−	0.838427i	$-0.316524\pi$
$211$	3340.88	1.09003	0.545014	+	0.838427i	$0.316524\pi$
$212$	0	0
$213$	−1394.67	−0.448645
$214$	0	0
$215$	−1105.08	−0.350540
$216$	0	0
$217$	−391.494	−0.122472
$218$	0	0
$219$	−4724.14	−1.45766
$220$	0	0
$221$	−3812.98	−1.16058
$222$	0	0
$223$	−919.003	−0.275969	−0.137984	−	0.990434i	$-0.544062\pi$
$223$	−919.003	−0.275969	−0.137984	+	0.990434i	$0.544062\pi$
$224$	0	0
$225$	−109.932	−0.0325724
$226$	0	0
$227$	−3716.79	−1.08675	−0.543374	−	0.839491i	$-0.682853\pi$
$227$	−3716.79	−1.08675	−0.543374	+	0.839491i	$0.682853\pi$
$228$	0	0
$229$	5523.57	1.59392	0.796960	−	0.604032i	$-0.206440\pi$
$229$	5523.57	1.59392	0.796960	+	0.604032i	$0.206440\pi$
$230$	0	0
$231$	−1588.46	−0.452438
$232$	0	0
$233$	−2943.43	−0.827599	−0.413800	−	0.910368i	$-0.635799\pi$
$233$	−2943.43	−0.827599	−0.413800	+	0.910368i	$0.635799\pi$
$234$	0	0
$235$	419.339	0.116403
$236$	0	0
$237$	−1644.26	−0.450659
$238$	0	0
$239$	−3642.42	−0.985809	−0.492905	−	0.870083i	$-0.664065\pi$
$239$	−3642.42	−0.985809	−0.492905	+	0.870083i	$0.664065\pi$
$240$	0	0
$241$	3293.89	0.880407	0.440204	−	0.897898i	$-0.354906\pi$
$241$	3293.89	0.880407	0.440204	+	0.897898i	$0.354906\pi$
$242$	0	0
$243$	−1220.83	−0.322290
$244$	0	0
$245$	−245.000	−0.0638877
$246$	0	0
$247$	1121.98	0.289028
$248$	0	0
$249$	1491.97	0.379717
$250$	0	0
$251$	−3165.71	−0.796087	−0.398043	−	0.917367i	$-0.630311\pi$
$251$	−3165.71	−0.796087	−0.398043	+	0.917367i	$0.630311\pi$
$252$	0	0
$253$	−1574.39	−0.391229
$254$	0	0
$255$	1859.28	0.456599
$256$	0	0
$257$	−1392.66	−0.338022	−0.169011	−	0.985614i	$-0.554057\pi$
$257$	−1392.66	−0.338022	−0.169011	+	0.985614i	$0.554057\pi$
$258$	0	0
$259$	−1155.58	−0.277237
$260$	0	0
$261$	−697.383	−0.165390
$262$	0	0
$263$	−6122.99	−1.43559	−0.717794	−	0.696256i	$-0.754848\pi$
$263$	−6122.99	−1.43559	−0.717794	+	0.696256i	$0.754848\pi$
$264$	0	0
$265$	317.187	0.0735270
$266$	0	0
$267$	−2519.40	−0.577471
$268$	0	0
$269$	−970.403	−0.219950	−0.109975	−	0.993934i	$-0.535077\pi$
$269$	−970.403	−0.219950	−0.109975	+	0.993934i	$0.535077\pi$
$270$	0	0
$271$	−7522.49	−1.68619	−0.843097	−	0.537761i	$-0.819270\pi$
$271$	−7522.49	−1.68619	−0.843097	+	0.537761i	$0.819270\pi$
$272$	0	0
$273$	−1622.36	−0.359670
$274$	0	0
$275$	−1193.27	−0.261662
$276$	0	0
$277$	−3194.79	−0.692982	−0.346491	−	0.938053i	$-0.612627\pi$
$277$	−3194.79	−0.692982	−0.346491	+	0.938053i	$0.612627\pi$
$278$	0	0
$279$	−245.929	−0.0527720
$280$	0	0
$281$	−720.884	−0.153040	−0.0765201	−	0.997068i	$-0.524381\pi$
$281$	−720.884	−0.153040	−0.0765201	+	0.997068i	$0.524381\pi$
$282$	0	0
$283$	−7815.32	−1.64160	−0.820799	−	0.571217i	$-0.806472\pi$
$283$	−7815.32	−1.64160	−0.820799	+	0.571217i	$0.806472\pi$
$284$	0	0
$285$	−547.100	−0.113710
$286$	0	0
$287$	−484.734	−0.0996966
$288$	0	0
$289$	1204.73	0.245212
$290$	0	0
$291$	−578.852	−0.116608
$292$	0	0
$293$	4817.55	0.960561	0.480280	−	0.877115i	$-0.340535\pi$
$293$	4817.55	0.960561	0.480280	+	0.877115i	$0.340535\pi$
$294$	0	0
$295$	1220.76	0.240933
$296$	0	0
$297$	−7124.78	−1.39199
$298$	0	0
$299$	−1607.99	−0.311011
$300$	0	0
$301$	−1547.12	−0.296261
$302$	0	0
$303$	−1700.83	−0.322476
$304$	0	0
$305$	2307.91	0.433281
$306$	0	0
$307$	5574.91	1.03641	0.518204	−	0.855257i	$-0.326601\pi$
$307$	5574.91	1.03641	0.518204	+	0.855257i	$0.326601\pi$
$308$	0	0
$309$	−3655.94	−0.673071
$310$	0	0
$311$	7028.07	1.28143	0.640716	−	0.767778i	$-0.278638\pi$
$311$	7028.07	1.28143	0.640716	+	0.767778i	$0.278638\pi$
$312$	0	0
$313$	−8279.29	−1.49512	−0.747561	−	0.664193i	$-0.768775\pi$
$313$	−8279.29	−1.49512	−0.747561	+	0.664193i	$0.768775\pi$
$314$	0	0
$315$	−153.904	−0.0275287
$316$	0	0
$317$	−9385.60	−1.66293	−0.831463	−	0.555580i	$-0.812496\pi$
$317$	−9385.60	−1.66293	−0.831463	+	0.555580i	$0.812496\pi$
$318$	0	0
$319$	−7569.85	−1.32862
$320$	0	0
$321$	5280.59	0.918174
$322$	0	0
$323$	−1800.16	−0.310104
$324$	0	0
$325$	−1218.74	−0.208010
$326$	0	0
$327$	−2098.61	−0.354903
$328$	0	0
$329$	587.075	0.0983784
$330$	0	0
$331$	4561.41	0.757455	0.378728	−	0.925508i	$-0.376362\pi$
$331$	4561.41	0.757455	0.378728	+	0.925508i	$0.376362\pi$
$332$	0	0
$333$	−725.914	−0.119459
$334$	0	0
$335$	3104.61	0.506337
$336$	0	0
$337$	−5388.05	−0.870936	−0.435468	−	0.900204i	$-0.643417\pi$
$337$	−5388.05	−0.870936	−0.435468	+	0.900204i	$0.643417\pi$
$338$	0	0
$339$	505.291	0.0809547
$340$	0	0
$341$	−2669.47	−0.423930
$342$	0	0
$343$	−343.000	−0.0539949
$344$	0	0
$345$	784.085	0.122359
$346$	0	0
$347$	12282.9	1.90024	0.950119	−	0.311886i	$-0.100961\pi$
$347$	12282.9	1.90024	0.950119	+	0.311886i	$0.100961\pi$
$348$	0	0
$349$	−5073.63	−0.778181	−0.389090	−	0.921200i	$-0.627211\pi$
$349$	−5073.63	−0.778181	−0.389090	+	0.921200i	$0.627211\pi$
$350$	0	0
$351$	−7276.82	−1.10658
$352$	0	0
$353$	5728.69	0.863760	0.431880	−	0.901931i	$-0.357850\pi$
$353$	5728.69	0.863760	0.431880	+	0.901931i	$0.357850\pi$
$354$	0	0
$355$	−1466.77	−0.219290
$356$	0	0
$357$	2603.00	0.385897
$358$	0	0
$359$	−3277.22	−0.481797	−0.240899	−	0.970550i	$-0.577442\pi$
$359$	−3277.22	−0.481797	−0.240899	+	0.970550i	$0.577442\pi$
$360$	0	0
$361$	−6329.30	−0.922773
$362$	0	0
$363$	−4503.36	−0.651144
$364$	0	0
$365$	−4968.35	−0.712481
$366$	0	0
$367$	−619.243	−0.0880769	−0.0440385	−	0.999030i	$-0.514022\pi$
$367$	−619.243	−0.0880769	−0.0440385	+	0.999030i	$0.514022\pi$
$368$	0	0
$369$	−304.500	−0.0429584
$370$	0	0
$371$	444.062	0.0621416
$372$	0	0
$373$	−4043.42	−0.561288	−0.280644	−	0.959812i	$-0.590548\pi$
$373$	−4043.42	−0.561288	−0.280644	+	0.959812i	$0.590548\pi$
$374$	0	0
$375$	594.279	0.0818358
$376$	0	0
$377$	−7731.39	−1.05620
$378$	0	0
$379$	−5706.73	−0.773444	−0.386722	−	0.922196i	$-0.626393\pi$
$379$	−5706.73	−0.773444	−0.386722	+	0.922196i	$0.626393\pi$
$380$	0	0
$381$	−764.381	−0.102783
$382$	0	0
$383$	2632.05	0.351152	0.175576	−	0.984466i	$-0.443821\pi$
$383$	2632.05	0.351152	0.175576	+	0.984466i	$0.443821\pi$
$384$	0	0
$385$	−1670.58	−0.221144
$386$	0	0
$387$	−971.871	−0.127656
$388$	0	0
$389$	11822.0	1.54088	0.770438	−	0.637515i	$-0.220038\pi$
$389$	11822.0	1.54088	0.770438	+	0.637515i	$0.220038\pi$
$390$	0	0
$391$	2579.93	0.333690
$392$	0	0
$393$	11958.3	1.53490
$394$	0	0
$395$	−1729.26	−0.220275
$396$	0	0
$397$	−965.178	−0.122017	−0.0610087	−	0.998137i	$-0.519432\pi$
$397$	−965.178	−0.122017	−0.0610087	+	0.998137i	$0.519432\pi$
$398$	0	0
$399$	−765.940	−0.0961027
$400$	0	0
$401$	14784.6	1.84116	0.920582	−	0.390550i	$-0.127715\pi$
$401$	14784.6	1.84116	0.920582	+	0.390550i	$0.127715\pi$
$402$	0	0
$403$	−2726.44	−0.337007
$404$	0	0
$405$	2954.69	0.362518
$406$	0	0
$407$	−7879.55	−0.959643
$408$	0	0
$409$	−587.656	−0.0710457	−0.0355229	−	0.999369i	$-0.511310\pi$
$409$	−587.656	−0.0710457	−0.0355229	+	0.999369i	$0.511310\pi$
$410$	0	0
$411$	4702.12	0.564327
$412$	0	0
$413$	1709.06	0.203626
$414$	0	0
$415$	1569.09	0.185599
$416$	0	0
$417$	1927.17	0.226317
$418$	0	0
$419$	9406.12	1.09670	0.548352	−	0.836248i	$-0.315255\pi$
$419$	9406.12	1.09670	0.548352	+	0.836248i	$0.315255\pi$
$420$	0	0
$421$	10083.1	1.16727	0.583633	−	0.812017i	$-0.301631\pi$
$421$	10083.1	1.16727	0.583633	+	0.812017i	$0.301631\pi$
$422$	0	0
$423$	368.790	0.0423904
$424$	0	0
$425$	1955.40	0.223178
$426$	0	0
$427$	3231.08	0.366189
$428$	0	0
$429$	−11062.4	−1.24498
$430$	0	0
$431$	−254.372	−0.0284284	−0.0142142	−	0.999899i	$-0.504525\pi$
$431$	−254.372	−0.0284284	−0.0142142	+	0.999899i	$0.504525\pi$
$432$	0	0
$433$	14730.1	1.63484	0.817418	−	0.576046i	$-0.195405\pi$
$433$	14730.1	1.63484	0.817418	+	0.576046i	$0.195405\pi$
$434$	0	0
$435$	3769.98	0.415532
$436$	0	0
$437$	−759.153	−0.0831011
$438$	0	0
$439$	388.166	0.0422008	0.0211004	−	0.999777i	$-0.493283\pi$
$439$	388.166	0.0422008	0.0211004	+	0.999777i	$0.493283\pi$
$440$	0	0
$441$	−215.466	−0.0232660
$442$	0	0
$443$	−10172.9	−1.09104	−0.545518	−	0.838099i	$-0.683667\pi$
$443$	−10172.9	−1.09104	−0.545518	+	0.838099i	$0.683667\pi$
$444$	0	0
$445$	−2649.64	−0.282258
$446$	0	0
$447$	−16374.5	−1.73263
$448$	0	0
$449$	−6441.31	−0.677025	−0.338513	−	0.940962i	$-0.609924\pi$
$449$	−6441.31	−0.677025	−0.338513	+	0.940962i	$0.609924\pi$
$450$	0	0
$451$	−3305.25	−0.345095
$452$	0	0
$453$	−7834.90	−0.812618
$454$	0	0
$455$	−1706.23	−0.175801
$456$	0	0
$457$	−17815.4	−1.82357	−0.911785	−	0.410668i	$-0.865296\pi$
$457$	−17815.4	−1.82357	−0.911785	+	0.410668i	$0.865296\pi$
$458$	0	0
$459$	11675.3	1.18727
$460$	0	0
$461$	2212.14	0.223491	0.111746	−	0.993737i	$-0.464356\pi$
$461$	2212.14	0.223491	0.111746	+	0.993737i	$0.464356\pi$
$462$	0	0
$463$	−9159.44	−0.919385	−0.459693	−	0.888078i	$-0.652040\pi$
$463$	−9159.44	−0.919385	−0.459693	+	0.888078i	$0.652040\pi$
$464$	0	0
$465$	1329.47	0.132586
$466$	0	0
$467$	−6387.57	−0.632937	−0.316468	−	0.948603i	$-0.602497\pi$
$467$	−6387.57	−0.632937	−0.316468	+	0.948603i	$0.602497\pi$
$468$	0	0
$469$	4346.45	0.427933
$470$	0	0
$471$	−13467.3	−1.31750
$472$	0	0
$473$	−10549.3	−1.02549
$474$	0	0
$475$	−575.382	−0.0555797
$476$	0	0
$477$	278.951	0.0267763
$478$	0	0
$479$	−10712.9	−1.02189	−0.510943	−	0.859615i	$-0.670704\pi$
$479$	−10712.9	−1.02189	−0.510943	+	0.859615i	$0.670704\pi$
$480$	0	0
$481$	−8047.70	−0.762876
$482$	0	0
$483$	1097.72	0.103412
$484$	0	0
$485$	−608.775	−0.0569960
$486$	0	0
$487$	−11230.7	−1.04499	−0.522495	−	0.852643i	$-0.674999\pi$
$487$	−11230.7	−1.04499	−0.522495	+	0.852643i	$0.674999\pi$
$488$	0	0
$489$	11716.4	1.08350
$490$	0	0
$491$	19995.2	1.83782	0.918909	−	0.394470i	$-0.129072\pi$
$491$	19995.2	1.83782	0.918909	+	0.394470i	$0.129072\pi$
$492$	0	0
$493$	12404.6	1.13322
$494$	0	0
$495$	−1049.43	−0.0952893
$496$	0	0
$497$	−2053.48	−0.185334
$498$	0	0
$499$	−13856.6	−1.24310	−0.621548	−	0.783376i	$-0.713496\pi$
$499$	−13856.6	−1.24310	−0.621548	+	0.783376i	$0.713496\pi$
$500$	0	0
$501$	3800.95	0.338950
$502$	0	0
$503$	13761.4	1.21986	0.609931	−	0.792455i	$-0.291197\pi$
$503$	13761.4	1.21986	0.609931	+	0.792455i	$0.291197\pi$
$504$	0	0
$505$	−1788.75	−0.157621
$506$	0	0
$507$	−853.411	−0.0747560
$508$	0	0
$509$	14019.8	1.22086	0.610428	−	0.792072i	$-0.290998\pi$
$509$	14019.8	1.22086	0.610428	+	0.792072i	$0.290998\pi$
$510$	0	0
$511$	−6955.70	−0.602156
$512$	0	0
$513$	−3435.49	−0.295674
$514$	0	0
$515$	−3844.93	−0.328986
$516$	0	0
$517$	4003.08	0.340533
$518$	0	0
$519$	−11118.8	−0.940387
$520$	0	0
$521$	−15380.3	−1.29332	−0.646661	−	0.762777i	$-0.723835\pi$
$521$	−15380.3	−1.29332	−0.646661	+	0.762777i	$0.723835\pi$
$522$	0	0
$523$	−3737.35	−0.312472	−0.156236	−	0.987720i	$-0.549936\pi$
$523$	−3737.35	−0.312472	−0.156236	+	0.987720i	$0.549936\pi$
$524$	0	0
$525$	831.991	0.0691639
$526$	0	0
$527$	4374.43	0.361581
$528$	0	0
$529$	−11079.0	−0.910578
$530$	0	0
$531$	1073.60	0.0877406
$532$	0	0
$533$	−3375.78	−0.274337
$534$	0	0
$535$	5553.57	0.448788
$536$	0	0
$537$	5855.88	0.470577
$538$	0	0
$539$	−2338.81	−0.186901
$540$	0	0
$541$	5552.39	0.441249	0.220624	−	0.975359i	$-0.429190\pi$
$541$	5552.39	0.441249	0.220624	+	0.975359i	$0.429190\pi$
$542$	0	0
$543$	16305.6	1.28865
$544$	0	0
$545$	−2207.09	−0.173471
$546$	0	0
$547$	−25025.1	−1.95612	−0.978058	−	0.208335i	$-0.933196\pi$
$547$	−25025.1	−1.95612	−0.978058	+	0.208335i	$0.933196\pi$
$548$	0	0
$549$	2029.70	0.157788
$550$	0	0
$551$	−3650.10	−0.282213
$552$	0	0
$553$	−2420.96	−0.186166
$554$	0	0
$555$	3924.21	0.300133
$556$	0	0
$557$	−14957.0	−1.13779	−0.568896	−	0.822409i	$-0.692629\pi$
$557$	−14957.0	−1.13779	−0.568896	+	0.822409i	$0.692629\pi$
$558$	0	0
$559$	−10774.4	−0.815225
$560$	0	0
$561$	17749.0	1.33577
$562$	0	0
$563$	−8173.82	−0.611875	−0.305937	−	0.952052i	$-0.598970\pi$
$563$	−8173.82	−0.611875	−0.305937	+	0.952052i	$0.598970\pi$
$564$	0	0
$565$	531.411	0.0395693
$566$	0	0
$567$	4136.56	0.306383
$568$	0	0
$569$	7344.79	0.541141	0.270571	−	0.962700i	$-0.412788\pi$
$569$	7344.79	0.541141	0.270571	+	0.962700i	$0.412788\pi$
$570$	0	0
$571$	22283.4	1.63315	0.816576	−	0.577238i	$-0.195869\pi$
$571$	22283.4	1.63315	0.816576	+	0.577238i	$0.195869\pi$
$572$	0	0
$573$	3363.77	0.245242
$574$	0	0
$575$	824.618	0.0598069
$576$	0	0
$577$	−9774.34	−0.705218	−0.352609	−	0.935771i	$-0.614706\pi$
$577$	−9774.34	−0.705218	−0.352609	+	0.935771i	$0.614706\pi$
$578$	0	0
$579$	3655.56	0.262383
$580$	0	0
$581$	2196.73	0.156860
$582$	0	0
$583$	3027.92	0.215101
$584$	0	0
$585$	−1071.82	−0.0757510
$586$	0	0
$587$	26794.3	1.88402	0.942011	−	0.335582i	$-0.108933\pi$
$587$	26794.3	1.88402	0.942011	+	0.335582i	$0.108933\pi$
$588$	0	0
$589$	−1287.19	−0.0900471
$590$	0	0
$591$	−15568.9	−1.08362
$592$	0	0
$593$	−3865.32	−0.267672	−0.133836	−	0.991003i	$-0.542730\pi$
$593$	−3865.32	−0.267672	−0.133836	+	0.991003i	$0.542730\pi$
$594$	0	0
$595$	2737.56	0.188620
$596$	0	0
$597$	−13987.0	−0.958878
$598$	0	0
$599$	−509.159	−0.0347306	−0.0173653	−	0.999849i	$-0.505528\pi$
$599$	−509.159	−0.0347306	−0.0173653	+	0.999849i	$0.505528\pi$
$600$	0	0
$601$	11322.1	0.768450	0.384225	−	0.923240i	$-0.374469\pi$
$601$	11322.1	0.768450	0.384225	+	0.923240i	$0.374469\pi$
$602$	0	0
$603$	2730.36	0.184393
$604$	0	0
$605$	−4736.16	−0.318268
$606$	0	0
$607$	−24447.3	−1.63474	−0.817368	−	0.576116i	$-0.804568\pi$
$607$	−24447.3	−1.63474	−0.817368	+	0.576116i	$0.804568\pi$
$608$	0	0
$609$	5277.97	0.351189
$610$	0	0
$611$	4088.51	0.270709
$612$	0	0
$613$	13210.2	0.870400	0.435200	−	0.900334i	$-0.356678\pi$
$613$	13210.2	0.870400	0.435200	+	0.900334i	$0.356678\pi$
$614$	0	0
$615$	1646.10	0.107930
$616$	0	0
$617$	2283.50	0.148996	0.0744979	−	0.997221i	$-0.476265\pi$
$617$	2283.50	0.148996	0.0744979	+	0.997221i	$0.476265\pi$
$618$	0	0
$619$	−4226.49	−0.274438	−0.137219	−	0.990541i	$-0.543816\pi$
$619$	−4226.49	−0.274438	−0.137219	+	0.990541i	$0.543816\pi$
$620$	0	0
$621$	4923.63	0.318162
$622$	0	0
$623$	−3709.49	−0.238552
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	−5222.71	−0.332655
$628$	0	0
$629$	12912.1	0.818505
$630$	0	0
$631$	−18142.7	−1.14461	−0.572305	−	0.820041i	$-0.693951\pi$
$631$	−18142.7	−1.14461	−0.572305	+	0.820041i	$0.693951\pi$
$632$	0	0
$633$	−15883.3	−0.997324
$634$	0	0
$635$	−803.895	−0.0502388
$636$	0	0
$637$	−2388.72	−0.148579
$638$	0	0
$639$	−1289.96	−0.0798590
$640$	0	0
$641$	−14059.8	−0.866350	−0.433175	−	0.901310i	$-0.642607\pi$
$641$	−14059.8	−0.866350	−0.433175	+	0.901310i	$0.642607\pi$
$642$	0	0
$643$	29418.6	1.80429	0.902144	−	0.431436i	$-0.141993\pi$
$643$	29418.6	1.80429	0.902144	+	0.431436i	$0.141993\pi$
$644$	0	0
$645$	5253.83	0.320728
$646$	0	0
$647$	4740.48	0.288049	0.144024	−	0.989574i	$-0.453996\pi$
$647$	4740.48	0.288049	0.144024	+	0.989574i	$0.453996\pi$
$648$	0	0
$649$	11653.6	0.704842
$650$	0	0
$651$	1861.25	0.112056
$652$	0	0
$653$	−14043.1	−0.841573	−0.420787	−	0.907160i	$-0.638246\pi$
$653$	−14043.1	−0.841573	−0.420787	+	0.907160i	$0.638246\pi$
$654$	0	0
$655$	12576.5	0.750234
$656$	0	0
$657$	−4369.44	−0.259464
$658$	0	0
$659$	3803.31	0.224819	0.112410	−	0.993662i	$-0.464143\pi$
$659$	3803.31	0.224819	0.112410	+	0.993662i	$0.464143\pi$
$660$	0	0
$661$	13101.4	0.770929	0.385464	−	0.922723i	$-0.374041\pi$
$661$	13101.4	0.770929	0.385464	+	0.922723i	$0.374041\pi$
$662$	0	0
$663$	18127.8	1.06188
$664$	0	0
$665$	−805.535	−0.0469734
$666$	0	0
$667$	5231.20	0.303677
$668$	0	0
$669$	4369.16	0.252498
$670$	0	0
$671$	22031.7	1.26755
$672$	0	0
$673$	30039.1	1.72054	0.860268	−	0.509842i	$-0.170296\pi$
$673$	30039.1	1.72054	0.860268	+	0.509842i	$0.170296\pi$
$674$	0	0
$675$	3731.75	0.212793
$676$	0	0
$677$	−7794.32	−0.442482	−0.221241	−	0.975219i	$-0.571011\pi$
$677$	−7794.32	−0.442482	−0.221241	+	0.975219i	$0.571011\pi$
$678$	0	0
$679$	−852.285	−0.0481704
$680$	0	0
$681$	17670.5	0.994323
$682$	0	0
$683$	−20258.0	−1.13492	−0.567461	−	0.823401i	$-0.692074\pi$
$683$	−20258.0	−1.13492	−0.567461	+	0.823401i	$0.692074\pi$
$684$	0	0
$685$	4945.20	0.275834
$686$	0	0
$687$	−26260.3	−1.45836
$688$	0	0
$689$	3092.54	0.170996
$690$	0	0
$691$	22210.0	1.22273	0.611367	−	0.791347i	$-0.290620\pi$
$691$	22210.0	1.22273	0.611367	+	0.791347i	$0.290620\pi$
$692$	0	0
$693$	−1469.20	−0.0805342
$694$	0	0
$695$	2026.80	0.110620
$696$	0	0
$697$	5416.27	0.294341
$698$	0	0
$699$	13993.8	0.757214
$700$	0	0
$701$	3033.60	0.163449	0.0817244	−	0.996655i	$-0.473957\pi$
$701$	3033.60	0.163449	0.0817244	+	0.996655i	$0.473957\pi$
$702$	0	0
$703$	−3799.43	−0.203838
$704$	0	0
$705$	−1993.64	−0.106503
$706$	0	0
$707$	−2504.26	−0.133214
$708$	0	0
$709$	−16465.4	−0.872171	−0.436086	−	0.899905i	$-0.643636\pi$
$709$	−16465.4	−0.872171	−0.436086	+	0.899905i	$0.643636\pi$
$710$	0	0
$711$	−1520.80	−0.0802174
$712$	0	0
$713$	1844.76	0.0968958
$714$	0	0
$715$	−11634.3	−0.608526
$716$	0	0
$717$	17316.9	0.901969
$718$	0	0
$719$	32975.7	1.71041	0.855206	−	0.518288i	$-0.173430\pi$
$719$	32975.7	1.71041	0.855206	+	0.518288i	$0.173430\pi$
$720$	0	0
$721$	−5382.90	−0.278044
$722$	0	0
$723$	−15659.9	−0.805531
$724$	0	0
$725$	3964.86	0.203105
$726$	0	0
$727$	−8657.41	−0.441658	−0.220829	−	0.975313i	$-0.570876\pi$
$727$	−8657.41	−0.441658	−0.220829	+	0.975313i	$0.570876\pi$
$728$	0	0
$729$	21759.4	1.10549
$730$	0	0
$731$	17287.0	0.874671
$732$	0	0
$733$	22964.1	1.15716	0.578580	−	0.815625i	$-0.303607\pi$
$733$	22964.1	1.15716	0.578580	+	0.815625i	$0.303607\pi$
$734$	0	0
$735$	1164.79	0.0584542
$736$	0	0
$737$	29637.1	1.48127
$738$	0	0
$739$	−19362.2	−0.963803	−0.481901	−	0.876226i	$-0.660054\pi$
$739$	−19362.2	−0.963803	−0.481901	+	0.876226i	$0.660054\pi$
$740$	0	0
$741$	−5334.16	−0.264447
$742$	0	0
$743$	14763.9	0.728986	0.364493	−	0.931206i	$-0.381242\pi$
$743$	14763.9	0.728986	0.364493	+	0.931206i	$0.381242\pi$
$744$	0	0
$745$	−17220.9	−0.846881
$746$	0	0
$747$	1379.94	0.0675897
$748$	0	0
$749$	7775.00	0.379295
$750$	0	0
$751$	−28089.6	−1.36485	−0.682425	−	0.730956i	$-0.739075\pi$
$751$	−28089.6	−1.36485	−0.682425	+	0.730956i	$0.739075\pi$
$752$	0	0
$753$	15050.5	0.728382
$754$	0	0
$755$	−8239.92	−0.397194
$756$	0	0
$757$	−19596.3	−0.940871	−0.470436	−	0.882434i	$-0.655903\pi$
$757$	−19596.3	−0.940871	−0.470436	+	0.882434i	$0.655903\pi$
$758$	0	0
$759$	7485.01	0.357956
$760$	0	0
$761$	−12978.5	−0.618225	−0.309113	−	0.951025i	$-0.600032\pi$
$761$	−12978.5	−0.618225	−0.309113	+	0.951025i	$0.600032\pi$
$762$	0	0
$763$	−3089.93	−0.146610
$764$	0	0
$765$	1719.68	0.0812748
$766$	0	0
$767$	11902.2	0.560320
$768$	0	0
$769$	14638.2	0.686433	0.343217	−	0.939256i	$-0.388484\pi$
$769$	14638.2	0.686433	0.343217	+	0.939256i	$0.388484\pi$
$770$	0	0
$771$	6621.03	0.309275
$772$	0	0
$773$	16852.6	0.784147	0.392074	−	0.919934i	$-0.371758\pi$
$773$	16852.6	0.784147	0.392074	+	0.919934i	$0.371758\pi$
$774$	0	0
$775$	1398.19	0.0648058
$776$	0	0
$777$	5493.90	0.253658
$778$	0	0
$779$	−1593.75	−0.0733019
$780$	0	0
$781$	−14002.0	−0.641526
$782$	0	0
$783$	23673.4	1.08048
$784$	0	0
$785$	−14163.5	−0.643972
$786$	0	0
$787$	13934.1	0.631126	0.315563	−	0.948905i	$-0.397807\pi$
$787$	13934.1	0.631126	0.315563	+	0.948905i	$0.397807\pi$
$788$	0	0
$789$	29110.1	1.31349
$790$	0	0
$791$	743.976	0.0334422
$792$	0	0
$793$	22501.9	1.00765
$794$	0	0
$795$	−1507.98	−0.0672737
$796$	0	0
$797$	−36608.0	−1.62701	−0.813503	−	0.581561i	$-0.802442\pi$
$797$	−36608.0	−1.62701	−0.813503	+	0.581561i	$0.802442\pi$
$798$	0	0
$799$	−6559.80	−0.290449
$800$	0	0
$801$	−2330.23	−0.102790
$802$	0	0
$803$	−47428.7	−2.08434
$804$	0	0
$805$	1154.47	0.0505460
$806$	0	0
$807$	4613.52	0.201244
$808$	0	0
$809$	−20466.8	−0.889464	−0.444732	−	0.895664i	$-0.646701\pi$
$809$	−20466.8	−0.889464	−0.444732	+	0.895664i	$0.646701\pi$
$810$	0	0
$811$	17371.6	0.752157	0.376078	−	0.926588i	$-0.377272\pi$
$811$	17371.6	0.752157	0.376078	+	0.926588i	$0.377272\pi$
$812$	0	0
$813$	35763.7	1.54279
$814$	0	0
$815$	12322.1	0.529599
$816$	0	0
$817$	−5086.77	−0.217826
$818$	0	0
$819$	−1500.55	−0.0640213
$820$	0	0
$821$	−23358.9	−0.992972	−0.496486	−	0.868045i	$-0.665377\pi$
$821$	−23358.9	−0.992972	−0.496486	+	0.868045i	$0.665377\pi$
$822$	0	0
$823$	−191.031	−0.00809104	−0.00404552	−	0.999992i	$-0.501288\pi$
$823$	−191.031	−0.00809104	−0.00404552	+	0.999992i	$0.501288\pi$
$824$	0	0
$825$	5673.09	0.239408
$826$	0	0
$827$	−42864.5	−1.80235	−0.901177	−	0.433452i	$-0.857295\pi$
$827$	−42864.5	−1.80235	−0.901177	+	0.433452i	$0.857295\pi$
$828$	0	0
$829$	8271.80	0.346552	0.173276	−	0.984873i	$-0.444565\pi$
$829$	8271.80	0.346552	0.173276	+	0.984873i	$0.444565\pi$
$830$	0	0
$831$	15188.8	0.634046
$832$	0	0
$833$	3832.58	0.159413
$834$	0	0
$835$	3997.44	0.165673
$836$	0	0
$837$	8348.32	0.344755
$838$	0	0
$839$	42316.8	1.74129	0.870643	−	0.491915i	$-0.163703\pi$
$839$	42316.8	1.74129	0.870643	+	0.491915i	$0.163703\pi$
$840$	0	0
$841$	763.229	0.0312940
$842$	0	0
$843$	3427.25	0.140025
$844$	0	0
$845$	−897.527	−0.0365395
$846$	0	0
$847$	−6630.62	−0.268986
$848$	0	0
$849$	37155.8	1.50198
$850$	0	0
$851$	5445.21	0.219341
$852$	0	0
$853$	−33770.7	−1.35555	−0.677776	−	0.735268i	$-0.737056\pi$
$853$	−33770.7	−1.35555	−0.677776	+	0.735268i	$0.737056\pi$
$854$	0	0
$855$	−506.022	−0.0202404
$856$	0	0
$857$	13518.3	0.538830	0.269415	−	0.963024i	$-0.413170\pi$
$857$	13518.3	0.538830	0.269415	+	0.963024i	$0.413170\pi$
$858$	0	0
$859$	27385.8	1.08777	0.543883	−	0.839161i	$-0.316954\pi$
$859$	27385.8	1.08777	0.543883	+	0.839161i	$0.316954\pi$
$860$	0	0
$861$	2304.54	0.0912176
$862$	0	0
$863$	34154.1	1.34718	0.673592	−	0.739103i	$-0.264751\pi$
$863$	34154.1	1.34718	0.673592	+	0.739103i	$0.264751\pi$
$864$	0	0
$865$	−11693.6	−0.459646
$866$	0	0
$867$	−5727.55	−0.224357
$868$	0	0
$869$	−16507.8	−0.644406
$870$	0	0
$871$	30269.6	1.17755
$872$	0	0
$873$	−535.389	−0.0207562
$874$	0	0
$875$	875.000	0.0338062
$876$	0	0
$877$	−43254.1	−1.66544	−0.832719	−	0.553697i	$-0.813217\pi$
$877$	−43254.1	−1.66544	−0.832719	+	0.553697i	$0.813217\pi$
$878$	0	0
$879$	−22903.8	−0.878868
$880$	0	0
$881$	−32955.0	−1.26025	−0.630125	−	0.776493i	$-0.716997\pi$
$881$	−32955.0	−1.26025	−0.630125	+	0.776493i	$0.716997\pi$
$882$	0	0
$883$	−37636.3	−1.43439	−0.717193	−	0.696875i	$-0.754573\pi$
$883$	−37636.3	−1.43439	−0.717193	+	0.696875i	$0.754573\pi$
$884$	0	0
$885$	−5803.77	−0.220442
$886$	0	0
$887$	−30670.1	−1.16099	−0.580496	−	0.814263i	$-0.697141\pi$
$887$	−30670.1	−1.16099	−0.580496	+	0.814263i	$0.697141\pi$
$888$	0	0
$889$	−1125.45	−0.0424595
$890$	0	0
$891$	28206.0	1.06053
$892$	0	0
$893$	1930.24	0.0723327
$894$	0	0
$895$	6158.60	0.230010
$896$	0	0
$897$	7644.74	0.284560
$898$	0	0
$899$	8869.82	0.329060
$900$	0	0
$901$	−4961.82	−0.183465
$902$	0	0
$903$	7355.36	0.271064
$904$	0	0
$905$	17148.5	0.629872
$906$	0	0
$907$	24717.3	0.904877	0.452439	−	0.891795i	$-0.350554\pi$
$907$	24717.3	0.904877	0.452439	+	0.891795i	$0.350554\pi$
$908$	0	0
$909$	−1573.13	−0.0574008
$910$	0	0
$911$	25484.3	0.926821	0.463411	−	0.886144i	$-0.346626\pi$
$911$	25484.3	0.926821	0.463411	+	0.886144i	$0.346626\pi$
$912$	0	0
$913$	14978.8	0.542964
$914$	0	0
$915$	−10972.4	−0.396432
$916$	0	0
$917$	17607.1	0.634064
$918$	0	0
$919$	−28510.6	−1.02337	−0.511686	−	0.859172i	$-0.670979\pi$
$919$	−28510.6	−1.02337	−0.511686	+	0.859172i	$0.670979\pi$
$920$	0	0
$921$	−26504.4	−0.948264
$922$	0	0
$923$	−14300.8	−0.509987
$924$	0	0
$925$	4127.07	0.146700
$926$	0	0
$927$	−3381.44	−0.119807
$928$	0	0
$929$	−23929.6	−0.845106	−0.422553	−	0.906338i	$-0.638866\pi$
$929$	−23929.6	−0.845106	−0.422553	+	0.906338i	$0.638866\pi$
$930$	0	0
$931$	−1127.75	−0.0396998
$932$	0	0
$933$	−33413.1	−1.17245
$934$	0	0
$935$	18666.5	0.652900
$936$	0	0
$937$	24744.6	0.862721	0.431360	−	0.902180i	$-0.358034\pi$
$937$	24744.6	0.862721	0.431360	+	0.902180i	$0.358034\pi$
$938$	0	0
$939$	39361.7	1.36797
$940$	0	0
$941$	−16805.7	−0.582199	−0.291100	−	0.956693i	$-0.594021\pi$
$941$	−16805.7	−0.582199	−0.291100	+	0.956693i	$0.594021\pi$
$942$	0	0
$943$	2284.11	0.0788770
$944$	0	0
$945$	5224.45	0.179843
$946$	0	0
$947$	5212.41	0.178860	0.0894301	−	0.995993i	$-0.471495\pi$
$947$	5212.41	0.178860	0.0894301	+	0.995993i	$0.471495\pi$
$948$	0	0
$949$	−48440.9	−1.65696
$950$	0	0
$951$	44621.3	1.52150
$952$	0	0
$953$	−28456.5	−0.967257	−0.483628	−	0.875273i	$-0.660681\pi$
$953$	−28456.5	−0.967257	−0.483628	+	0.875273i	$0.660681\pi$
$954$	0	0
$955$	3537.66	0.119870
$956$	0	0
$957$	35988.8	1.21563
$958$	0	0
$959$	6923.27	0.233122
$960$	0	0
$961$	−26663.1	−0.895005
$962$	0	0
$963$	4884.11	0.163435
$964$	0	0
$965$	3844.53	0.128249
$966$	0	0
$967$	50857.6	1.69128	0.845640	−	0.533753i	$-0.179219\pi$
$967$	50857.6	1.69128	0.845640	+	0.533753i	$0.179219\pi$
$968$	0	0
$969$	8558.38	0.283731
$970$	0	0
$971$	33639.8	1.11180	0.555898	−	0.831251i	$-0.312375\pi$
$971$	33639.8	1.11180	0.555898	+	0.831251i	$0.312375\pi$
$972$	0	0
$973$	2837.52	0.0934909
$974$	0	0
$975$	5794.15	0.190319
$976$	0	0
$977$	−36600.6	−1.19852	−0.599262	−	0.800553i	$-0.704539\pi$
$977$	−36600.6	−1.19852	−0.599262	+	0.800553i	$0.704539\pi$
$978$	0	0
$979$	−25293.9	−0.825736
$980$	0	0
$981$	−1941.04	−0.0631728
$982$	0	0
$983$	39139.4	1.26994	0.634971	−	0.772536i	$-0.281012\pi$
$983$	39139.4	1.26994	0.634971	+	0.772536i	$0.281012\pi$
$984$	0	0
$985$	−16373.7	−0.529655
$986$	0	0
$987$	−2791.09	−0.0900116
$988$	0	0
$989$	7290.18	0.234393
$990$	0	0
$991$	30319.0	0.971862	0.485931	−	0.873997i	$-0.338481\pi$
$991$	30319.0	0.971862	0.485931	+	0.873997i	$0.338481\pi$
$992$	0	0
$993$	−21686.0	−0.693036
$994$	0	0
$995$	−14710.1	−0.468684
$996$	0	0
$997$	40669.9	1.29191	0.645953	−	0.763378i	$-0.276460\pi$
$997$	40669.9	1.29191	0.645953	+	0.763378i	$0.276460\pi$
$998$	0	0
$999$	24641.9	0.780416

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2240.4.a.ca.1.2		4
4.3	odd	2		2240.4.a.cl.1.3		4
8.3	odd	2		1120.4.a.e.1.2	✓	4
8.5	even	2		1120.4.a.p.1.3	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1120.4.a.e.1.2	✓	4	8.3	odd	2
1120.4.a.p.1.3	yes	4	8.5	even	2
2240.4.a.ca.1.2		4	1.1	even	1	trivial
2240.4.a.cl.1.3		4	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 2240 = 2^{6} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2240.a (trivial)

\(f(q)\)	\(=\)	\(q-4.75423 q^{3} -5.00000 q^{5} -7.00000 q^{7} -4.39727 q^{9} +O(q^{10})\)	\(q-4.75423 q^{3} -5.00000 q^{5} -7.00000 q^{7} -4.39727 q^{9} -47.7308 q^{11} -48.7494 q^{13} +23.7712 q^{15} +78.2159 q^{17} -23.0153 q^{19} +33.2796 q^{21} +32.9847 q^{23} +25.0000 q^{25} +149.270 q^{27} +158.595 q^{29} +55.9277 q^{31} +226.923 q^{33} +35.0000 q^{35} +165.083 q^{37} +231.766 q^{39} +69.2477 q^{41} +221.017 q^{43} +21.9863 q^{45} -83.8679 q^{47} +49.0000 q^{49} -371.857 q^{51} -63.4374 q^{53} +238.654 q^{55} +109.420 q^{57} -244.152 q^{59} -461.583 q^{61} +30.7809 q^{63} +243.747 q^{65} -620.922 q^{67} -156.817 q^{69} +293.354 q^{71} +993.671 q^{73} -118.856 q^{75} +334.116 q^{77} +345.852 q^{79} -590.938 q^{81} -313.818 q^{83} -391.079 q^{85} -753.995 q^{87} +529.928 q^{89} +341.246 q^{91} -265.893 q^{93} +115.076 q^{95} +121.755 q^{97} +209.885 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 9 q^{3} - 20 q^{5} - 28 q^{7} + 55 q^{9}+O(q^{10}) \) 4 * q - 9 * q^3 - 20 * q^5 - 28 * q^7 + 55 * q^9	\( 4 q - 9 q^{3} - 20 q^{5} - 28 q^{7} + 55 q^{9} - 33 q^{11} + 83 q^{13} + 45 q^{15} - 13 q^{17} - 132 q^{19} + 63 q^{21} + 92 q^{23} + 100 q^{25} - 495 q^{27} + 113 q^{29} + 94 q^{31} + 11 q^{33} + 140 q^{35} + 54 q^{37} + 145 q^{39} + 428 q^{41} - 604 q^{43} - 275 q^{45} + 709 q^{47} + 196 q^{49} + 13 q^{51} - 554 q^{53} + 165 q^{55} + 1044 q^{57} + 100 q^{59} + 588 q^{61} - 385 q^{63} - 415 q^{65} - 484 q^{67} + 540 q^{69} - 128 q^{71} + 1508 q^{73} - 225 q^{75} + 231 q^{77} + 587 q^{79} + 2548 q^{81} - 220 q^{83} + 65 q^{85} + 2887 q^{87} - 1392 q^{89} - 581 q^{91} - 2074 q^{93} + 660 q^{95} + 643 q^{97} - 690 q^{99}+O(q^{100}) \) 4 * q - 9 * q^3 - 20 * q^5 - 28 * q^7 + 55 * q^9 - 33 * q^11 + 83 * q^13 + 45 * q^15 - 13 * q^17 - 132 * q^19 + 63 * q^21 + 92 * q^23 + 100 * q^25 - 495 * q^27 + 113 * q^29 + 94 * q^31 + 11 * q^33 + 140 * q^35 + 54 * q^37 + 145 * q^39 + 428 * q^41 - 604 * q^43 - 275 * q^45 + 709 * q^47 + 196 * q^49 + 13 * q^51 - 554 * q^53 + 165 * q^55 + 1044 * q^57 + 100 * q^59 + 588 * q^61 - 385 * q^63 - 415 * q^65 - 484 * q^67 + 540 * q^69 - 128 * q^71 + 1508 * q^73 - 225 * q^75 + 231 * q^77 + 587 * q^79 + 2548 * q^81 - 220 * q^83 + 65 * q^85 + 2887 * q^87 - 1392 * q^89 - 581 * q^91 - 2074 * q^93 + 660 * q^95 + 643 * q^97 - 690 * q^99

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