LMFDB - Embedded newform 6040.2.a.l.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6040,2,Mod(1,6040)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6040, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("6040.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6040 = 2^{3} \cdot 5 \cdot 151 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6040.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:	$48.2296428209$
Analytic rank:	$1$
Dimension:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - x^{8} - 11x^{7} + 9x^{6} + 32x^{5} - 17x^{4} - 27x^{3} + 10x^{2} + 3x - 1 $ x^9 - x^8 - 11x^7 + 9x^6 + 32x^5 - 17x^4 - 27x^3 + 10x^2 + 3*x - 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$0.390645$ of defining polynomial
Character	$\chi$	$=$	6040.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.96251 q^{3} +1.00000 q^{5} -2.16922 q^{7} +0.851433 q^{9} +O(q^{10})$	$q-1.96251 q^{3} +1.00000 q^{5} -2.16922 q^{7} +0.851433 q^{9} +3.01214 q^{11} -2.71469 q^{13} -1.96251 q^{15} +0.748145 q^{17} -4.29171 q^{19} +4.25712 q^{21} +1.71882 q^{23} +1.00000 q^{25} +4.21658 q^{27} -1.74832 q^{29} +4.16689 q^{31} -5.91134 q^{33} -2.16922 q^{35} +4.69387 q^{37} +5.32759 q^{39} -9.89262 q^{41} +8.49792 q^{43} +0.851433 q^{45} +12.8181 q^{47} -2.29447 q^{49} -1.46824 q^{51} -7.88590 q^{53} +3.01214 q^{55} +8.42251 q^{57} +6.20478 q^{59} -9.79583 q^{61} -1.84695 q^{63} -2.71469 q^{65} +2.54968 q^{67} -3.37319 q^{69} -6.64355 q^{71} -6.53691 q^{73} -1.96251 q^{75} -6.53400 q^{77} -5.44102 q^{79} -10.8294 q^{81} +4.01722 q^{83} +0.748145 q^{85} +3.43108 q^{87} +15.0573 q^{89} +5.88877 q^{91} -8.17755 q^{93} -4.29171 q^{95} -2.91029 q^{97} +2.56463 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q + 9 q^{5} - 2 q^{7} - 3 q^{9}+O(q^{10}) $ 9 * q + 9 * q^5 - 2 * q^7 - 3 * q^9	$ 9 q + 9 q^{5} - 2 q^{7} - 3 q^{9} - 6 q^{11} - 9 q^{13} - 2 q^{17} - 10 q^{19} - 9 q^{21} - 6 q^{23} + 9 q^{25} + 12 q^{27} - 6 q^{29} + 9 q^{31} - 11 q^{33} - 2 q^{35} - 12 q^{37} - 3 q^{39} - 20 q^{41} + q^{43} - 3 q^{45} + 22 q^{47} - 29 q^{49} + 2 q^{51} - 35 q^{53} - 6 q^{55} - 20 q^{57} + 14 q^{59} - 22 q^{61} - 12 q^{63} - 9 q^{65} + 4 q^{67} + 5 q^{69} - 22 q^{71} - 34 q^{73} - 5 q^{77} + 8 q^{79} - 31 q^{81} - 3 q^{83} - 2 q^{85} - 5 q^{89} - 7 q^{91} - 21 q^{93} - 10 q^{95} - 33 q^{97} - 15 q^{99}+O(q^{100}) $ 9 * q + 9 * q^5 - 2 * q^7 - 3 * q^9 - 6 * q^11 - 9 * q^13 - 2 * q^17 - 10 * q^19 - 9 * q^21 - 6 * q^23 + 9 * q^25 + 12 * q^27 - 6 * q^29 + 9 * q^31 - 11 * q^33 - 2 * q^35 - 12 * q^37 - 3 * q^39 - 20 * q^41 + q^43 - 3 * q^45 + 22 * q^47 - 29 * q^49 + 2 * q^51 - 35 * q^53 - 6 * q^55 - 20 * q^57 + 14 * q^59 - 22 * q^61 - 12 * q^63 - 9 * q^65 + 4 * q^67 + 5 * q^69 - 22 * q^71 - 34 * q^73 - 5 * q^77 + 8 * q^79 - 31 * q^81 - 3 * q^83 - 2 * q^85 - 5 * q^89 - 7 * q^91 - 21 * q^93 - 10 * q^95 - 33 * q^97 - 15 * 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
$3$	−1.96251	−1.13305	−0.566527	−	0.824043i	$-0.691713\pi$
$3$	−1.96251	−1.13305	−0.566527	+	0.824043i	$0.691713\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−2.16922	−0.819890	−0.409945	−	0.912110i	$-0.634452\pi$
$7$	−2.16922	−0.819890	−0.409945	+	0.912110i	$0.634452\pi$
$8$	0	0
$9$	0.851433	0.283811
$10$	0	0
$11$	3.01214	0.908193	0.454097	−	0.890952i	$-0.349962\pi$
$11$	3.01214	0.908193	0.454097	+	0.890952i	$0.349962\pi$
$12$	0	0
$13$	−2.71469	−0.752919	−0.376460	−	0.926433i	$-0.622859\pi$
$13$	−2.71469	−0.752919	−0.376460	+	0.926433i	$0.622859\pi$
$14$	0	0
$15$	−1.96251	−0.506717
$16$	0	0
$17$	0.748145	0.181452	0.0907259	−	0.995876i	$-0.471081\pi$
$17$	0.748145	0.181452	0.0907259	+	0.995876i	$0.471081\pi$
$18$	0	0
$19$	−4.29171	−0.984585	−0.492293	−	0.870430i	$-0.663841\pi$
$19$	−4.29171	−0.984585	−0.492293	+	0.870430i	$0.663841\pi$
$20$	0	0
$21$	4.25712	0.928979
$22$	0	0
$23$	1.71882	0.358398	0.179199	−	0.983813i	$-0.442649\pi$
$23$	1.71882	0.358398	0.179199	+	0.983813i	$0.442649\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	4.21658	0.811481
$28$	0	0
$29$	−1.74832	−0.324654	−0.162327	−	0.986737i	$-0.551900\pi$
$29$	−1.74832	−0.324654	−0.162327	+	0.986737i	$0.551900\pi$
$30$	0	0
$31$	4.16689	0.748396	0.374198	−	0.927349i	$-0.377918\pi$
$31$	4.16689	0.748396	0.374198	+	0.927349i	$0.377918\pi$
$32$	0	0
$33$	−5.91134	−1.02903
$34$	0	0
$35$	−2.16922	−0.366666
$36$	0	0
$37$	4.69387	0.771668	0.385834	−	0.922568i	$-0.373914\pi$
$37$	4.69387	0.771668	0.385834	+	0.922568i	$0.373914\pi$
$38$	0	0
$39$	5.32759	0.853098
$40$	0	0
$41$	−9.89262	−1.54497	−0.772484	−	0.635034i	$-0.780986\pi$
$41$	−9.89262	−1.54497	−0.772484	+	0.635034i	$0.780986\pi$
$42$	0	0
$43$	8.49792	1.29592	0.647960	−	0.761674i	$-0.275622\pi$
$43$	8.49792	1.29592	0.647960	+	0.761674i	$0.275622\pi$
$44$	0	0
$45$	0.851433	0.126924
$46$	0	0
$47$	12.8181	1.86971	0.934856	−	0.355028i	$-0.115529\pi$
$47$	12.8181	1.86971	0.934856	+	0.355028i	$0.115529\pi$
$48$	0	0
$49$	−2.29447	−0.327781
$50$	0	0
$51$	−1.46824	−0.205595
$52$	0	0
$53$	−7.88590	−1.08321	−0.541606	−	0.840633i	$-0.682183\pi$
$53$	−7.88590	−1.08321	−0.541606	+	0.840633i	$0.682183\pi$
$54$	0	0
$55$	3.01214	0.406156
$56$	0	0
$57$	8.42251	1.11559
$58$	0	0
$59$	6.20478	0.807793	0.403897	−	0.914805i	$-0.367656\pi$
$59$	6.20478	0.807793	0.403897	+	0.914805i	$0.367656\pi$
$60$	0	0
$61$	−9.79583	−1.25423	−0.627114	−	0.778927i	$-0.715764\pi$
$61$	−9.79583	−1.25423	−0.627114	+	0.778927i	$0.715764\pi$
$62$	0	0
$63$	−1.84695	−0.232694
$64$	0	0
$65$	−2.71469	−0.336716
$66$	0	0
$67$	2.54968	0.311493	0.155746	−	0.987797i	$-0.450222\pi$
$67$	2.54968	0.311493	0.155746	+	0.987797i	$0.450222\pi$
$68$	0	0
$69$	−3.37319	−0.406084
$70$	0	0
$71$	−6.64355	−0.788445	−0.394222	−	0.919015i	$-0.628986\pi$
$71$	−6.64355	−0.788445	−0.394222	+	0.919015i	$0.628986\pi$
$72$	0	0
$73$	−6.53691	−0.765088	−0.382544	−	0.923937i	$-0.624952\pi$
$73$	−6.53691	−0.765088	−0.382544	+	0.923937i	$0.624952\pi$
$74$	0	0
$75$	−1.96251	−0.226611
$76$	0	0
$77$	−6.53400	−0.744618
$78$	0	0
$79$	−5.44102	−0.612163	−0.306081	−	0.952005i	$-0.599018\pi$
$79$	−5.44102	−0.612163	−0.306081	+	0.952005i	$0.599018\pi$
$80$	0	0
$81$	−10.8294	−1.20326
$82$	0	0
$83$	4.01722	0.440947	0.220473	−	0.975393i	$-0.429240\pi$
$83$	4.01722	0.440947	0.220473	+	0.975393i	$0.429240\pi$
$84$	0	0
$85$	0.748145	0.0811477
$86$	0	0
$87$	3.43108	0.367850
$88$	0	0
$89$	15.0573	1.59607	0.798036	−	0.602610i	$-0.205873\pi$
$89$	15.0573	1.59607	0.798036	+	0.602610i	$0.205873\pi$
$90$	0	0
$91$	5.88877	0.617311
$92$	0	0
$93$	−8.17755	−0.847973
$94$	0	0
$95$	−4.29171	−0.440320
$96$	0	0
$97$	−2.91029	−0.295495	−0.147748	−	0.989025i	$-0.547202\pi$
$97$	−2.91029	−0.295495	−0.147748	+	0.989025i	$0.547202\pi$
$98$	0	0
$99$	2.56463	0.257755
$100$	0	0
$101$	2.73382	0.272025	0.136013	−	0.990707i	$-0.456571\pi$
$101$	2.73382	0.272025	0.136013	+	0.990707i	$0.456571\pi$
$102$	0	0
$103$	3.62541	0.357222	0.178611	−	0.983920i	$-0.442840\pi$
$103$	3.62541	0.357222	0.178611	+	0.983920i	$0.442840\pi$
$104$	0	0
$105$	4.25712	0.415452
$106$	0	0
$107$	−3.20902	−0.310227	−0.155114	−	0.987897i	$-0.549574\pi$
$107$	−3.20902	−0.310227	−0.155114	+	0.987897i	$0.549574\pi$
$108$	0	0
$109$	−4.88474	−0.467873	−0.233936	−	0.972252i	$-0.575161\pi$
$109$	−4.88474	−0.467873	−0.233936	+	0.972252i	$0.575161\pi$
$110$	0	0
$111$	−9.21176	−0.874341
$112$	0	0
$113$	8.30803	0.781554	0.390777	−	0.920485i	$-0.372206\pi$
$113$	8.30803	0.781554	0.390777	+	0.920485i	$0.372206\pi$
$114$	0	0
$115$	1.71882	0.160281
$116$	0	0
$117$	−2.31138	−0.213687
$118$	0	0
$119$	−1.62289	−0.148770
$120$	0	0
$121$	−1.92703	−0.175185
$122$	0	0
$123$	19.4143	1.75053
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	2.07348	0.183992	0.0919958	−	0.995759i	$-0.470675\pi$
$127$	2.07348	0.183992	0.0919958	+	0.995759i	$0.470675\pi$
$128$	0	0
$129$	−16.6772	−1.46835
$130$	0	0
$131$	−13.3055	−1.16251	−0.581253	−	0.813723i	$-0.697437\pi$
$131$	−13.3055	−1.16251	−0.581253	+	0.813723i	$0.697437\pi$
$132$	0	0
$133$	9.30968	0.807251
$134$	0	0
$135$	4.21658	0.362905
$136$	0	0
$137$	18.3490	1.56766	0.783832	−	0.620973i	$-0.213262\pi$
$137$	18.3490	1.56766	0.783832	+	0.620973i	$0.213262\pi$
$138$	0	0
$139$	−0.0634977	−0.00538581	−0.00269290	−	0.999996i	$-0.500857\pi$
$139$	−0.0634977	−0.00538581	−0.00269290	+	0.999996i	$0.500857\pi$
$140$	0	0
$141$	−25.1556	−2.11848
$142$	0	0
$143$	−8.17701	−0.683796
$144$	0	0
$145$	−1.74832	−0.145190
$146$	0	0
$147$	4.50291	0.371393
$148$	0	0
$149$	11.4828	0.940711	0.470356	−	0.882477i	$-0.344126\pi$
$149$	11.4828	0.940711	0.470356	+	0.882477i	$0.344126\pi$
$150$	0	0
$151$	−1.00000	−0.0813788
$151$	−1.00000	−0.0813788
$152$	0	0
$153$	0.636995	0.0514980
$154$	0	0
$155$	4.16689	0.334693
$156$	0	0
$157$	2.26410	0.180695	0.0903475	−	0.995910i	$-0.471202\pi$
$157$	2.26410	0.180695	0.0903475	+	0.995910i	$0.471202\pi$
$158$	0	0
$159$	15.4761	1.22734
$160$	0	0
$161$	−3.72850	−0.293847
$162$	0	0
$163$	−18.5884	−1.45595	−0.727976	−	0.685602i	$-0.759539\pi$
$163$	−18.5884	−1.45595	−0.727976	+	0.685602i	$0.759539\pi$
$164$	0	0
$165$	−5.91134	−0.460197
$166$	0	0
$167$	1.75344	0.135685	0.0678426	−	0.997696i	$-0.478388\pi$
$167$	1.75344	0.135685	0.0678426	+	0.997696i	$0.478388\pi$
$168$	0	0
$169$	−5.63047	−0.433113
$170$	0	0
$171$	−3.65410	−0.279436
$172$	0	0
$173$	17.5676	1.33564	0.667819	−	0.744324i	$-0.267228\pi$
$173$	17.5676	1.33564	0.667819	+	0.744324i	$0.267228\pi$
$174$	0	0
$175$	−2.16922	−0.163978
$176$	0	0
$177$	−12.1769	−0.915273
$178$	0	0
$179$	−16.1306	−1.20566	−0.602829	−	0.797870i	$-0.705960\pi$
$179$	−16.1306	−1.20566	−0.602829	+	0.797870i	$0.705960\pi$
$180$	0	0
$181$	−16.9945	−1.26319	−0.631596	−	0.775298i	$-0.717600\pi$
$181$	−16.9945	−1.26319	−0.631596	+	0.775298i	$0.717600\pi$
$182$	0	0
$183$	19.2244	1.42111
$184$	0	0
$185$	4.69387	0.345100
$186$	0	0
$187$	2.25352	0.164793
$188$	0	0
$189$	−9.14670	−0.665325
$190$	0	0
$191$	−8.52383	−0.616763	−0.308381	−	0.951263i	$-0.599787\pi$
$191$	−8.52383	−0.616763	−0.308381	+	0.951263i	$0.599787\pi$
$192$	0	0
$193$	−21.7917	−1.56860	−0.784300	−	0.620382i	$-0.786977\pi$
$193$	−21.7917	−1.56860	−0.784300	+	0.620382i	$0.786977\pi$
$194$	0	0
$195$	5.32759	0.381517
$196$	0	0
$197$	11.6273	0.828408	0.414204	−	0.910184i	$-0.364060\pi$
$197$	11.6273	0.828408	0.414204	+	0.910184i	$0.364060\pi$
$198$	0	0
$199$	−20.9477	−1.48495	−0.742473	−	0.669876i	$-0.766347\pi$
$199$	−20.9477	−1.48495	−0.742473	+	0.669876i	$0.766347\pi$
$200$	0	0
$201$	−5.00376	−0.352938
$202$	0	0
$203$	3.79249	0.266180
$204$	0	0
$205$	−9.89262	−0.690931
$206$	0	0
$207$	1.46346	0.101717
$208$	0	0
$209$	−12.9272	−0.894194
$210$	0	0
$211$	−9.21057	−0.634082	−0.317041	−	0.948412i	$-0.602689\pi$
$211$	−9.21057	−0.634082	−0.317041	+	0.948412i	$0.602689\pi$
$212$	0	0
$213$	13.0380	0.893350
$214$	0	0
$215$	8.49792	0.579553
$216$	0	0
$217$	−9.03892	−0.613602
$218$	0	0
$219$	12.8287	0.866886
$220$	0	0
$221$	−2.03098	−0.136619
$222$	0	0
$223$	13.4874	0.903185	0.451592	−	0.892224i	$-0.350856\pi$
$223$	13.4874	0.903185	0.451592	+	0.892224i	$0.350856\pi$
$224$	0	0
$225$	0.851433	0.0567622
$226$	0	0
$227$	10.3020	0.683765	0.341882	−	0.939743i	$-0.388936\pi$
$227$	10.3020	0.683765	0.341882	+	0.939743i	$0.388936\pi$
$228$	0	0
$229$	−14.3497	−0.948258	−0.474129	−	0.880455i	$-0.657237\pi$
$229$	−14.3497	−0.948258	−0.474129	+	0.880455i	$0.657237\pi$
$230$	0	0
$231$	12.8230	0.843693
$232$	0	0
$233$	−20.7413	−1.35881	−0.679404	−	0.733764i	$-0.737762\pi$
$233$	−20.7413	−1.35881	−0.679404	+	0.733764i	$0.737762\pi$
$234$	0	0
$235$	12.8181	0.836160
$236$	0	0
$237$	10.6780	0.693613
$238$	0	0
$239$	6.25664	0.404708	0.202354	−	0.979312i	$-0.435141\pi$
$239$	6.25664	0.404708	0.202354	+	0.979312i	$0.435141\pi$
$240$	0	0
$241$	20.1542	1.29825	0.649125	−	0.760682i	$-0.275135\pi$
$241$	20.1542	1.29825	0.649125	+	0.760682i	$0.275135\pi$
$242$	0	0
$243$	8.60296	0.551880
$244$	0	0
$245$	−2.29447	−0.146588
$246$	0	0
$247$	11.6506	0.741313
$248$	0	0
$249$	−7.88381	−0.499616
$250$	0	0
$251$	4.67467	0.295063	0.147531	−	0.989057i	$-0.452867\pi$
$251$	4.67467	0.295063	0.147531	+	0.989057i	$0.452867\pi$
$252$	0	0
$253$	5.17731	0.325495
$254$	0	0
$255$	−1.46824	−0.0919447
$256$	0	0
$257$	−4.57646	−0.285472	−0.142736	−	0.989761i	$-0.545590\pi$
$257$	−4.57646	−0.285472	−0.142736	+	0.989761i	$0.545590\pi$
$258$	0	0
$259$	−10.1821	−0.632683
$260$	0	0
$261$	−1.48857	−0.0921404
$262$	0	0
$263$	−16.0834	−0.991745	−0.495873	−	0.868395i	$-0.665152\pi$
$263$	−16.0834	−0.991745	−0.495873	+	0.868395i	$0.665152\pi$
$264$	0	0
$265$	−7.88590	−0.484427
$266$	0	0
$267$	−29.5501	−1.80843
$268$	0	0
$269$	−1.32579	−0.0808347	−0.0404174	−	0.999183i	$-0.512869\pi$
$269$	−1.32579	−0.0808347	−0.0404174	+	0.999183i	$0.512869\pi$
$270$	0	0
$271$	6.50832	0.395353	0.197676	−	0.980267i	$-0.436661\pi$
$271$	6.50832	0.395353	0.197676	+	0.980267i	$0.436661\pi$
$272$	0	0
$273$	−11.5567	−0.699446
$274$	0	0
$275$	3.01214	0.181639
$276$	0	0
$277$	−14.4607	−0.868859	−0.434430	−	0.900706i	$-0.643050\pi$
$277$	−14.4607	−0.868859	−0.434430	+	0.900706i	$0.643050\pi$
$278$	0	0
$279$	3.54783	0.212403
$280$	0	0
$281$	−6.05087	−0.360964	−0.180482	−	0.983578i	$-0.557766\pi$
$281$	−6.05087	−0.360964	−0.180482	+	0.983578i	$0.557766\pi$
$282$	0	0
$283$	−20.6207	−1.22577	−0.612887	−	0.790170i	$-0.709992\pi$
$283$	−20.6207	−1.22577	−0.612887	+	0.790170i	$0.709992\pi$
$284$	0	0
$285$	8.42251	0.498906
$286$	0	0
$287$	21.4593	1.26670
$288$	0	0
$289$	−16.4403	−0.967075
$290$	0	0
$291$	5.71147	0.334812
$292$	0	0
$293$	16.7712	0.979786	0.489893	−	0.871783i	$-0.337036\pi$
$293$	16.7712	0.979786	0.489893	+	0.871783i	$0.337036\pi$
$294$	0	0
$295$	6.20478	0.361256
$296$	0	0
$297$	12.7009	0.736981
$298$	0	0
$299$	−4.66605	−0.269845
$300$	0	0
$301$	−18.4339	−1.06251
$302$	0	0
$303$	−5.36514	−0.308219
$304$	0	0
$305$	−9.79583	−0.560908
$306$	0	0
$307$	10.7421	0.613081	0.306541	−	0.951858i	$-0.400828\pi$
$307$	10.7421	0.613081	0.306541	+	0.951858i	$0.400828\pi$
$308$	0	0
$309$	−7.11489	−0.404752
$310$	0	0
$311$	2.82492	0.160186	0.0800932	−	0.996787i	$-0.474478\pi$
$311$	2.82492	0.160186	0.0800932	+	0.996787i	$0.474478\pi$
$312$	0	0
$313$	−28.7090	−1.62273	−0.811365	−	0.584540i	$-0.801275\pi$
$313$	−28.7090	−1.62273	−0.811365	+	0.584540i	$0.801275\pi$
$314$	0	0
$315$	−1.84695	−0.104064
$316$	0	0
$317$	−9.72265	−0.546078	−0.273039	−	0.962003i	$-0.588029\pi$
$317$	−9.72265	−0.546078	−0.273039	+	0.962003i	$0.588029\pi$
$318$	0	0
$319$	−5.26616	−0.294849
$320$	0	0
$321$	6.29772	0.351504
$322$	0	0
$323$	−3.21082	−0.178655
$324$	0	0
$325$	−2.71469	−0.150584
$326$	0	0
$327$	9.58633	0.530125
$328$	0	0
$329$	−27.8053	−1.53296
$330$	0	0
$331$	0.830070	0.0456248	0.0228124	−	0.999740i	$-0.492738\pi$
$331$	0.830070	0.0456248	0.0228124	+	0.999740i	$0.492738\pi$
$332$	0	0
$333$	3.99652	0.219008
$334$	0	0
$335$	2.54968	0.139304
$336$	0	0
$337$	−26.9322	−1.46709	−0.733544	−	0.679642i	$-0.762135\pi$
$337$	−26.9322	−1.46709	−0.733544	+	0.679642i	$0.762135\pi$
$338$	0	0
$339$	−16.3046	−0.885542
$340$	0	0
$341$	12.5512	0.679688
$342$	0	0
$343$	20.1618	1.08863
$344$	0	0
$345$	−3.37319	−0.181606
$346$	0	0
$347$	−26.5136	−1.42332	−0.711662	−	0.702522i	$-0.752057\pi$
$347$	−26.5136	−1.42332	−0.711662	+	0.702522i	$0.752057\pi$
$348$	0	0
$349$	−16.3698	−0.876254	−0.438127	−	0.898913i	$-0.644358\pi$
$349$	−16.3698	−0.876254	−0.438127	+	0.898913i	$0.644358\pi$
$350$	0	0
$351$	−11.4467	−0.610979
$352$	0	0
$353$	−8.02380	−0.427064	−0.213532	−	0.976936i	$-0.568497\pi$
$353$	−8.02380	−0.427064	−0.213532	+	0.976936i	$0.568497\pi$
$354$	0	0
$355$	−6.64355	−0.352603
$356$	0	0
$357$	3.18494	0.168565
$358$	0	0
$359$	4.18033	0.220629	0.110315	−	0.993897i	$-0.464814\pi$
$359$	4.18033	0.220629	0.110315	+	0.993897i	$0.464814\pi$
$360$	0	0
$361$	−0.581244	−0.0305918
$362$	0	0
$363$	3.78181	0.198494
$364$	0	0
$365$	−6.53691	−0.342158
$366$	0	0
$367$	−20.8757	−1.08970	−0.544851	−	0.838533i	$-0.683414\pi$
$367$	−20.8757	−1.08970	−0.544851	+	0.838533i	$0.683414\pi$
$368$	0	0
$369$	−8.42290	−0.438479
$370$	0	0
$371$	17.1063	0.888114
$372$	0	0
$373$	−31.6271	−1.63759	−0.818796	−	0.574085i	$-0.805358\pi$
$373$	−31.6271	−1.63759	−0.818796	+	0.574085i	$0.805358\pi$
$374$	0	0
$375$	−1.96251	−0.101343
$376$	0	0
$377$	4.74613	0.244438
$378$	0	0
$379$	−7.38975	−0.379586	−0.189793	−	0.981824i	$-0.560782\pi$
$379$	−7.38975	−0.379586	−0.189793	+	0.981824i	$0.560782\pi$
$380$	0	0
$381$	−4.06922	−0.208472
$382$	0	0
$383$	−30.4978	−1.55836	−0.779182	−	0.626798i	$-0.784365\pi$
$383$	−30.4978	−1.55836	−0.779182	+	0.626798i	$0.784365\pi$
$384$	0	0
$385$	−6.53400	−0.333003
$386$	0	0
$387$	7.23541	0.367796
$388$	0	0
$389$	27.2328	1.38076	0.690379	−	0.723448i	$-0.257444\pi$
$389$	27.2328	1.38076	0.690379	+	0.723448i	$0.257444\pi$
$390$	0	0
$391$	1.28592	0.0650320
$392$	0	0
$393$	26.1121	1.31718
$394$	0	0
$395$	−5.44102	−0.273768
$396$	0	0
$397$	13.2555	0.665273	0.332637	−	0.943055i	$-0.392062\pi$
$397$	13.2555	0.665273	0.332637	+	0.943055i	$0.392062\pi$
$398$	0	0
$399$	−18.2703	−0.914659
$400$	0	0
$401$	−20.4575	−1.02160	−0.510800	−	0.859699i	$-0.670651\pi$
$401$	−20.4575	−1.02160	−0.510800	+	0.859699i	$0.670651\pi$
$402$	0	0
$403$	−11.3118	−0.563481
$404$	0	0
$405$	−10.8294	−0.538115
$406$	0	0
$407$	14.1386	0.700824
$408$	0	0
$409$	21.9027	1.08302	0.541509	−	0.840695i	$-0.317853\pi$
$409$	21.9027	1.08302	0.541509	+	0.840695i	$0.317853\pi$
$410$	0	0
$411$	−36.0101	−1.77625
$412$	0	0
$413$	−13.4596	−0.662301
$414$	0	0
$415$	4.01722	0.197197
$416$	0	0
$417$	0.124615	0.00610241
$418$	0	0
$419$	33.9905	1.66055	0.830273	−	0.557357i	$-0.188185\pi$
$419$	33.9905	1.66055	0.830273	+	0.557357i	$0.188185\pi$
$420$	0	0
$421$	4.66631	0.227422	0.113711	−	0.993514i	$-0.463726\pi$
$421$	4.66631	0.227422	0.113711	+	0.993514i	$0.463726\pi$
$422$	0	0
$423$	10.9137	0.530645
$424$	0	0
$425$	0.748145	0.0362904
$426$	0	0
$427$	21.2494	1.02833
$428$	0	0
$429$	16.0474	0.774778
$430$	0	0
$431$	−11.8922	−0.572829	−0.286414	−	0.958106i	$-0.592463\pi$
$431$	−11.8922	−0.572829	−0.286414	+	0.958106i	$0.592463\pi$
$432$	0	0
$433$	2.07028	0.0994911	0.0497455	−	0.998762i	$-0.484159\pi$
$433$	2.07028	0.0994911	0.0497455	+	0.998762i	$0.484159\pi$
$434$	0	0
$435$	3.43108	0.164508
$436$	0	0
$437$	−7.37666	−0.352874
$438$	0	0
$439$	−19.5142	−0.931361	−0.465680	−	0.884953i	$-0.654190\pi$
$439$	−19.5142	−0.931361	−0.465680	+	0.884953i	$0.654190\pi$
$440$	0	0
$441$	−1.95358	−0.0930278
$442$	0	0
$443$	26.5032	1.25920	0.629601	−	0.776918i	$-0.283218\pi$
$443$	26.5032	1.25920	0.629601	+	0.776918i	$0.283218\pi$
$444$	0	0
$445$	15.0573	0.713785
$446$	0	0
$447$	−22.5352	−1.06588
$448$	0	0
$449$	−37.4184	−1.76588	−0.882941	−	0.469484i	$-0.844440\pi$
$449$	−37.4184	−1.76588	−0.882941	+	0.469484i	$0.844440\pi$
$450$	0	0
$451$	−29.7979	−1.40313
$452$	0	0
$453$	1.96251	0.0922066
$454$	0	0
$455$	5.88877	0.276070
$456$	0	0
$457$	−5.56466	−0.260304	−0.130152	−	0.991494i	$-0.541547\pi$
$457$	−5.56466	−0.260304	−0.130152	+	0.991494i	$0.541547\pi$
$458$	0	0
$459$	3.15461	0.147245
$460$	0	0
$461$	0.103086	0.00480120	0.00240060	−	0.999997i	$-0.499236\pi$
$461$	0.103086	0.00480120	0.00240060	+	0.999997i	$0.499236\pi$
$462$	0	0
$463$	−25.5155	−1.18581	−0.592904	−	0.805273i	$-0.702019\pi$
$463$	−25.5155	−1.18581	−0.592904	+	0.805273i	$0.702019\pi$
$464$	0	0
$465$	−8.17755	−0.379225
$466$	0	0
$467$	4.88782	0.226181	0.113091	−	0.993585i	$-0.463925\pi$
$467$	4.88782	0.226181	0.113091	+	0.993585i	$0.463925\pi$
$468$	0	0
$469$	−5.53083	−0.255390
$470$	0	0
$471$	−4.44331	−0.204737
$472$	0	0
$473$	25.5969	1.17695
$474$	0	0
$475$	−4.29171	−0.196917
$476$	0	0
$477$	−6.71432	−0.307427
$478$	0	0
$479$	7.80514	0.356626	0.178313	−	0.983974i	$-0.442936\pi$
$479$	7.80514	0.356626	0.178313	+	0.983974i	$0.442936\pi$
$480$	0	0
$481$	−12.7424	−0.581004
$482$	0	0
$483$	7.31721	0.332944
$484$	0	0
$485$	−2.91029	−0.132150
$486$	0	0
$487$	−11.3311	−0.513462	−0.256731	−	0.966483i	$-0.582645\pi$
$487$	−11.3311	−0.513462	−0.256731	+	0.966483i	$0.582645\pi$
$488$	0	0
$489$	36.4798	1.64967
$490$	0	0
$491$	−0.687619	−0.0310318	−0.0155159	−	0.999880i	$-0.504939\pi$
$491$	−0.687619	−0.0310318	−0.0155159	+	0.999880i	$0.504939\pi$
$492$	0	0
$493$	−1.30799	−0.0589091
$494$	0	0
$495$	2.56463	0.115272
$496$	0	0
$497$	14.4114	0.646438
$498$	0	0
$499$	18.6479	0.834793	0.417396	−	0.908725i	$-0.362943\pi$
$499$	18.6479	0.834793	0.417396	+	0.908725i	$0.362943\pi$
$500$	0	0
$501$	−3.44114	−0.153739
$502$	0	0
$503$	−24.8823	−1.10945	−0.554724	−	0.832035i	$-0.687176\pi$
$503$	−24.8823	−1.10945	−0.554724	+	0.832035i	$0.687176\pi$
$504$	0	0
$505$	2.73382	0.121653
$506$	0	0
$507$	11.0498	0.490740
$508$	0	0
$509$	−18.0862	−0.801658	−0.400829	−	0.916153i	$-0.631278\pi$
$509$	−18.0862	−0.801658	−0.400829	+	0.916153i	$0.631278\pi$
$510$	0	0
$511$	14.1800	0.627288
$512$	0	0
$513$	−18.0963	−0.798972
$514$	0	0
$515$	3.62541	0.159755
$516$	0	0
$517$	38.6099	1.69806
$518$	0	0
$519$	−34.4765	−1.51335
$520$	0	0
$521$	−3.52627	−0.154489	−0.0772444	−	0.997012i	$-0.524612\pi$
$521$	−3.52627	−0.154489	−0.0772444	+	0.997012i	$0.524612\pi$
$522$	0	0
$523$	−1.83625	−0.0802934	−0.0401467	−	0.999194i	$-0.512783\pi$
$523$	−1.83625	−0.0802934	−0.0401467	+	0.999194i	$0.512783\pi$
$524$	0	0
$525$	4.25712	0.185796
$526$	0	0
$527$	3.11744	0.135798
$528$	0	0
$529$	−20.0457	−0.871551
$530$	0	0
$531$	5.28295	0.229261
$532$	0	0
$533$	26.8554	1.16324
$534$	0	0
$535$	−3.20902	−0.138738
$536$	0	0
$537$	31.6564	1.36608
$538$	0	0
$539$	−6.91125	−0.297688
$540$	0	0
$541$	5.43440	0.233643	0.116822	−	0.993153i	$-0.462729\pi$
$541$	5.43440	0.233643	0.116822	+	0.993153i	$0.462729\pi$
$542$	0	0
$543$	33.3518	1.43126
$544$	0	0
$545$	−4.88474	−0.209239
$546$	0	0
$547$	−13.7390	−0.587436	−0.293718	−	0.955892i	$-0.594893\pi$
$547$	−13.7390	−0.587436	−0.293718	+	0.955892i	$0.594893\pi$
$548$	0	0
$549$	−8.34050	−0.355964
$550$	0	0
$551$	7.50326	0.319650
$552$	0	0
$553$	11.8028	0.501906
$554$	0	0
$555$	−9.21176	−0.391017
$556$	0	0
$557$	−30.8848	−1.30863	−0.654315	−	0.756222i	$-0.727043\pi$
$557$	−30.8848	−1.30863	−0.654315	+	0.756222i	$0.727043\pi$
$558$	0	0
$559$	−23.0692	−0.975723
$560$	0	0
$561$	−4.42254	−0.186720
$562$	0	0
$563$	−34.0179	−1.43368	−0.716842	−	0.697235i	$-0.754413\pi$
$563$	−34.0179	−1.43368	−0.716842	+	0.697235i	$0.754413\pi$
$564$	0	0
$565$	8.30803	0.349521
$566$	0	0
$567$	23.4913	0.986542
$568$	0	0
$569$	−16.6839	−0.699425	−0.349712	−	0.936857i	$-0.613721\pi$
$569$	−16.6839	−0.699425	−0.349712	+	0.936857i	$0.613721\pi$
$570$	0	0
$571$	13.6448	0.571016	0.285508	−	0.958376i	$-0.407838\pi$
$571$	13.6448	0.571016	0.285508	+	0.958376i	$0.407838\pi$
$572$	0	0
$573$	16.7281	0.698825
$574$	0	0
$575$	1.71882	0.0716796
$576$	0	0
$577$	34.1896	1.42333	0.711666	−	0.702518i	$-0.247941\pi$
$577$	34.1896	1.42333	0.711666	+	0.702518i	$0.247941\pi$
$578$	0	0
$579$	42.7663	1.77731
$580$	0	0
$581$	−8.71424	−0.361528
$582$	0	0
$583$	−23.7534	−0.983766
$584$	0	0
$585$	−2.31138	−0.0955636
$586$	0	0
$587$	9.60515	0.396447	0.198223	−	0.980157i	$-0.436483\pi$
$587$	9.60515	0.396447	0.198223	+	0.980157i	$0.436483\pi$
$588$	0	0
$589$	−17.8831	−0.736859
$590$	0	0
$591$	−22.8186	−0.938631
$592$	0	0
$593$	11.4144	0.468733	0.234366	−	0.972148i	$-0.424698\pi$
$593$	11.4144	0.468733	0.234366	+	0.972148i	$0.424698\pi$
$594$	0	0
$595$	−1.62289	−0.0665322
$596$	0	0
$597$	41.1101	1.68252
$598$	0	0
$599$	23.1359	0.945307	0.472653	−	0.881248i	$-0.343296\pi$
$599$	23.1359	0.945307	0.472653	+	0.881248i	$0.343296\pi$
$600$	0	0
$601$	−31.2599	−1.27512	−0.637559	−	0.770402i	$-0.720056\pi$
$601$	−31.2599	−1.27512	−0.637559	+	0.770402i	$0.720056\pi$
$602$	0	0
$603$	2.17088	0.0884051
$604$	0	0
$605$	−1.92703	−0.0783450
$606$	0	0
$607$	26.2273	1.06453	0.532267	−	0.846577i	$-0.321340\pi$
$607$	26.2273	1.06453	0.532267	+	0.846577i	$0.321340\pi$
$608$	0	0
$609$	−7.44278	−0.301597
$610$	0	0
$611$	−34.7971	−1.40774
$612$	0	0
$613$	24.3200	0.982275	0.491138	−	0.871082i	$-0.336581\pi$
$613$	24.3200	0.982275	0.491138	+	0.871082i	$0.336581\pi$
$614$	0	0
$615$	19.4143	0.782862
$616$	0	0
$617$	6.43402	0.259024	0.129512	−	0.991578i	$-0.458659\pi$
$617$	6.43402	0.259024	0.129512	+	0.991578i	$0.458659\pi$
$618$	0	0
$619$	42.7419	1.71794	0.858971	−	0.512024i	$-0.171104\pi$
$619$	42.7419	1.71794	0.858971	+	0.512024i	$0.171104\pi$
$620$	0	0
$621$	7.24753	0.290833
$622$	0	0
$623$	−32.6627	−1.30860
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	25.3697	1.01317
$628$	0	0
$629$	3.51170	0.140021
$630$	0	0
$631$	23.1177	0.920303	0.460151	−	0.887840i	$-0.347795\pi$
$631$	23.1177	0.920303	0.460151	+	0.887840i	$0.347795\pi$
$632$	0	0
$633$	18.0758	0.718449
$634$	0	0
$635$	2.07348	0.0822836
$636$	0	0
$637$	6.22876	0.246793
$638$	0	0
$639$	−5.65654	−0.223769
$640$	0	0
$641$	4.45325	0.175893	0.0879464	−	0.996125i	$-0.471970\pi$
$641$	4.45325	0.175893	0.0879464	+	0.996125i	$0.471970\pi$
$642$	0	0
$643$	0.00677368	0.000267128 0	0.000133564	−	1.00000i	$-0.499957\pi$
$643$	0.00677368	0.000267128 0	0.000133564		1.00000i	$0.499957\pi$
$644$	0	0
$645$	−16.6772	−0.656665
$646$	0	0
$647$	−45.8824	−1.80383	−0.901913	−	0.431919i	$-0.857837\pi$
$647$	−45.8824	−1.80383	−0.901913	+	0.431919i	$0.857837\pi$
$648$	0	0
$649$	18.6896	0.733632
$650$	0	0
$651$	17.7389	0.695244
$652$	0	0
$653$	−43.3779	−1.69751	−0.848754	−	0.528788i	$-0.822647\pi$
$653$	−43.3779	−1.69751	−0.848754	+	0.528788i	$0.822647\pi$
$654$	0	0
$655$	−13.3055	−0.519889
$656$	0	0
$657$	−5.56574	−0.217140
$658$	0	0
$659$	−30.8265	−1.20083	−0.600414	−	0.799689i	$-0.704998\pi$
$659$	−30.8265	−1.20083	−0.600414	+	0.799689i	$0.704998\pi$
$660$	0	0
$661$	−3.59334	−0.139765	−0.0698824	−	0.997555i	$-0.522262\pi$
$661$	−3.59334	−0.139765	−0.0698824	+	0.997555i	$0.522262\pi$
$662$	0	0
$663$	3.98581	0.154796
$664$	0	0
$665$	9.30968	0.361014
$666$	0	0
$667$	−3.00503	−0.116355
$668$	0	0
$669$	−26.4692	−1.02336
$670$	0	0
$671$	−29.5064	−1.13908
$672$	0	0
$673$	17.8727	0.688942	0.344471	−	0.938797i	$-0.388058\pi$
$673$	17.8727	0.688942	0.344471	+	0.938797i	$0.388058\pi$
$674$	0	0
$675$	4.21658	0.162296
$676$	0	0
$677$	−50.1714	−1.92824	−0.964122	−	0.265460i	$-0.914476\pi$
$677$	−50.1714	−1.92824	−0.964122	+	0.265460i	$0.914476\pi$
$678$	0	0
$679$	6.31308	0.242274
$680$	0	0
$681$	−20.2177	−0.774742
$682$	0	0
$683$	−43.2419	−1.65461	−0.827303	−	0.561756i	$-0.810126\pi$
$683$	−43.2419	−1.65461	−0.827303	+	0.561756i	$0.810126\pi$
$684$	0	0
$685$	18.3490	0.701081
$686$	0	0
$687$	28.1615	1.07443
$688$	0	0
$689$	21.4078	0.815571
$690$	0	0
$691$	−11.3083	−0.430188	−0.215094	−	0.976593i	$-0.569006\pi$
$691$	−11.3083	−0.430188	−0.215094	+	0.976593i	$0.569006\pi$
$692$	0	0
$693$	−5.56326	−0.211331
$694$	0	0
$695$	−0.0634977	−0.00240861
$696$	0	0
$697$	−7.40112	−0.280337
$698$	0	0
$699$	40.7050	1.53960
$700$	0	0
$701$	16.5642	0.625620	0.312810	−	0.949816i	$-0.398730\pi$
$701$	16.5642	0.625620	0.312810	+	0.949816i	$0.398730\pi$
$702$	0	0
$703$	−20.1447	−0.759773
$704$	0	0
$705$	−25.1556	−0.947415
$706$	0	0
$707$	−5.93027	−0.223031
$708$	0	0
$709$	46.7990	1.75757	0.878786	−	0.477216i	$-0.158354\pi$
$709$	46.7990	1.75757	0.878786	+	0.477216i	$0.158354\pi$
$710$	0	0
$711$	−4.63267	−0.173739
$712$	0	0
$713$	7.16212	0.268224
$714$	0	0
$715$	−8.17701	−0.305803
$716$	0	0
$717$	−12.2787	−0.458556
$718$	0	0
$719$	−31.6279	−1.17952	−0.589760	−	0.807578i	$-0.700778\pi$
$719$	−31.6279	−1.17952	−0.589760	+	0.807578i	$0.700778\pi$
$720$	0	0
$721$	−7.86432	−0.292883
$722$	0	0
$723$	−39.5528	−1.47099
$724$	0	0
$725$	−1.74832	−0.0649308
$726$	0	0
$727$	−16.5897	−0.615277	−0.307639	−	0.951503i	$-0.599539\pi$
$727$	−16.5897	−0.615277	−0.307639	+	0.951503i	$0.599539\pi$
$728$	0	0
$729$	15.6047	0.577952
$730$	0	0
$731$	6.35768	0.235147
$732$	0	0
$733$	−29.1792	−1.07776	−0.538880	−	0.842383i	$-0.681152\pi$
$733$	−29.1792	−1.07776	−0.538880	+	0.842383i	$0.681152\pi$
$734$	0	0
$735$	4.50291	0.166092
$736$	0	0
$737$	7.67998	0.282896
$738$	0	0
$739$	−0.503787	−0.0185321	−0.00926604	−	0.999957i	$-0.502950\pi$
$739$	−0.503787	−0.0185321	−0.00926604	+	0.999957i	$0.502950\pi$
$740$	0	0
$741$	−22.8645	−0.839948
$742$	0	0
$743$	−26.4912	−0.971868	−0.485934	−	0.873995i	$-0.661521\pi$
$743$	−26.4912	−0.971868	−0.485934	+	0.873995i	$0.661521\pi$
$744$	0	0
$745$	11.4828	0.420699
$746$	0	0
$747$	3.42039	0.125146
$748$	0	0
$749$	6.96108	0.254352
$750$	0	0
$751$	48.4483	1.76790	0.883952	−	0.467577i	$-0.154873\pi$
$751$	48.4483	1.76790	0.883952	+	0.467577i	$0.154873\pi$
$752$	0	0
$753$	−9.17408	−0.334322
$754$	0	0
$755$	−1.00000	−0.0363937
$756$	0	0
$757$	6.76709	0.245954	0.122977	−	0.992410i	$-0.460756\pi$
$757$	6.76709	0.245954	0.122977	+	0.992410i	$0.460756\pi$
$758$	0	0
$759$	−10.1605	−0.368803
$760$	0	0
$761$	8.93154	0.323768	0.161884	−	0.986810i	$-0.448243\pi$
$761$	8.93154	0.323768	0.161884	+	0.986810i	$0.448243\pi$
$762$	0	0
$763$	10.5961	0.383604
$764$	0	0
$765$	0.636995	0.0230306
$766$	0	0
$767$	−16.8440	−0.608203
$768$	0	0
$769$	−15.1159	−0.545095	−0.272548	−	0.962142i	$-0.587866\pi$
$769$	−15.1159	−0.545095	−0.272548	+	0.962142i	$0.587866\pi$
$770$	0	0
$771$	8.98134	0.323455
$772$	0	0
$773$	0.404826	0.0145606	0.00728028	−	0.999973i	$-0.497683\pi$
$773$	0.404826	0.0145606	0.00728028	+	0.999973i	$0.497683\pi$
$774$	0	0
$775$	4.16689	0.149679
$776$	0	0
$777$	19.9824	0.716863
$778$	0	0
$779$	42.4562	1.52115
$780$	0	0
$781$	−20.0113	−0.716060
$782$	0	0
$783$	−7.37191	−0.263450
$784$	0	0
$785$	2.26410	0.0808092
$786$	0	0
$787$	42.2257	1.50519	0.752593	−	0.658486i	$-0.228803\pi$
$787$	42.2257	1.50519	0.752593	+	0.658486i	$0.228803\pi$
$788$	0	0
$789$	31.5638	1.12370
$790$	0	0
$791$	−18.0220	−0.640788
$792$	0	0
$793$	26.5926	0.944332
$794$	0	0
$795$	15.4761	0.548882
$796$	0	0
$797$	−0.488145	−0.0172910	−0.00864549	−	0.999963i	$-0.502752\pi$
$797$	−0.488145	−0.0172910	−0.00864549	+	0.999963i	$0.502752\pi$
$798$	0	0
$799$	9.58980	0.339263
$800$	0	0
$801$	12.8203	0.452983
$802$	0	0
$803$	−19.6901	−0.694848
$804$	0	0
$805$	−3.72850	−0.131412
$806$	0	0
$807$	2.60187	0.0915901
$808$	0	0
$809$	−21.1286	−0.742843	−0.371422	−	0.928464i	$-0.621130\pi$
$809$	−21.1286	−0.742843	−0.371422	+	0.928464i	$0.621130\pi$
$810$	0	0
$811$	−20.8571	−0.732392	−0.366196	−	0.930538i	$-0.619340\pi$
$811$	−20.8571	−0.732392	−0.366196	+	0.930538i	$0.619340\pi$
$812$	0	0
$813$	−12.7726	−0.447956
$814$	0	0
$815$	−18.5884	−0.651122
$816$	0	0
$817$	−36.4706	−1.27594
$818$	0	0
$819$	5.01389	0.175200
$820$	0	0
$821$	−30.9000	−1.07842	−0.539209	−	0.842172i	$-0.681277\pi$
$821$	−30.9000	−1.07842	−0.539209	+	0.842172i	$0.681277\pi$
$822$	0	0
$823$	3.04961	0.106303	0.0531513	−	0.998586i	$-0.483073\pi$
$823$	3.04961	0.106303	0.0531513	+	0.998586i	$0.483073\pi$
$824$	0	0
$825$	−5.91134	−0.205806
$826$	0	0
$827$	41.4450	1.44118	0.720591	−	0.693360i	$-0.243870\pi$
$827$	41.4450	1.44118	0.720591	+	0.693360i	$0.243870\pi$
$828$	0	0
$829$	−3.61104	−0.125417	−0.0627083	−	0.998032i	$-0.519974\pi$
$829$	−3.61104	−0.125417	−0.0627083	+	0.998032i	$0.519974\pi$
$830$	0	0
$831$	28.3792	0.984464
$832$	0	0
$833$	−1.71659	−0.0594764
$834$	0	0
$835$	1.75344	0.0606803
$836$	0	0
$837$	17.5700	0.607309
$838$	0	0
$839$	7.85452	0.271168	0.135584	−	0.990766i	$-0.456709\pi$
$839$	7.85452	0.271168	0.135584	+	0.990766i	$0.456709\pi$
$840$	0	0
$841$	−25.9434	−0.894600
$842$	0	0
$843$	11.8749	0.408992
$844$	0	0
$845$	−5.63047	−0.193694
$846$	0	0
$847$	4.18017	0.143632
$848$	0	0
$849$	40.4683	1.38887
$850$	0	0
$851$	8.06791	0.276564
$852$	0	0
$853$	33.5124	1.14744	0.573721	−	0.819051i	$-0.305500\pi$
$853$	33.5124	1.14744	0.573721	+	0.819051i	$0.305500\pi$
$854$	0	0
$855$	−3.65410	−0.124968
$856$	0	0
$857$	−24.7458	−0.845299	−0.422649	−	0.906293i	$-0.638900\pi$
$857$	−24.7458	−0.845299	−0.422649	+	0.906293i	$0.638900\pi$
$858$	0	0
$859$	20.9899	0.716166	0.358083	−	0.933690i	$-0.383430\pi$
$859$	20.9899	0.716166	0.358083	+	0.933690i	$0.383430\pi$
$860$	0	0
$861$	−42.1140	−1.43524
$862$	0	0
$863$	28.6868	0.976510	0.488255	−	0.872701i	$-0.337634\pi$
$863$	28.6868	0.976510	0.488255	+	0.872701i	$0.337634\pi$
$864$	0	0
$865$	17.5676	0.597315
$866$	0	0
$867$	32.2642	1.09575
$868$	0	0
$869$	−16.3891	−0.555962
$870$	0	0
$871$	−6.92159	−0.234529
$872$	0	0
$873$	−2.47792	−0.0838649
$874$	0	0
$875$	−2.16922	−0.0733332
$876$	0	0
$877$	38.5479	1.30167	0.650835	−	0.759219i	$-0.274419\pi$
$877$	38.5479	1.30167	0.650835	+	0.759219i	$0.274419\pi$
$878$	0	0
$879$	−32.9137	−1.11015
$880$	0	0
$881$	53.0568	1.78753	0.893765	−	0.448535i	$-0.148054\pi$
$881$	53.0568	1.78753	0.893765	+	0.448535i	$0.148054\pi$
$882$	0	0
$883$	28.7759	0.968386	0.484193	−	0.874961i	$-0.339113\pi$
$883$	28.7759	0.968386	0.484193	+	0.874961i	$0.339113\pi$
$884$	0	0
$885$	−12.1769	−0.409323
$886$	0	0
$887$	42.2665	1.41917	0.709586	−	0.704619i	$-0.248882\pi$
$887$	42.2665	1.41917	0.709586	+	0.704619i	$0.248882\pi$
$888$	0	0
$889$	−4.49784	−0.150853
$890$	0	0
$891$	−32.6195	−1.09279
$892$	0	0
$893$	−55.0115	−1.84089
$894$	0	0
$895$	−16.1306	−0.539187
$896$	0	0
$897$	9.15716	0.305749
$898$	0	0
$899$	−7.28504	−0.242970
$900$	0	0
$901$	−5.89980	−0.196551
$902$	0	0
$903$	36.1766	1.20388
$904$	0	0
$905$	−16.9945	−0.564916
$906$	0	0
$907$	7.95833	0.264252	0.132126	−	0.991233i	$-0.457820\pi$
$907$	7.95833	0.264252	0.132126	+	0.991233i	$0.457820\pi$
$908$	0	0
$909$	2.32767	0.0772038
$910$	0	0
$911$	−22.4319	−0.743201	−0.371600	−	0.928393i	$-0.621191\pi$
$911$	−22.4319	−0.743201	−0.371600	+	0.928393i	$0.621191\pi$
$912$	0	0
$913$	12.1004	0.400465
$914$	0	0
$915$	19.2244	0.635539
$916$	0	0
$917$	28.8626	0.953127
$918$	0	0
$919$	−47.4862	−1.56642	−0.783212	−	0.621755i	$-0.786420\pi$
$919$	−47.4862	−1.56642	−0.783212	+	0.621755i	$0.786420\pi$
$920$	0	0
$921$	−21.0813	−0.694654
$922$	0	0
$923$	18.0352	0.593635
$924$	0	0
$925$	4.69387	0.154334
$926$	0	0
$927$	3.08679	0.101384
$928$	0	0
$929$	13.1937	0.432871	0.216436	−	0.976297i	$-0.430557\pi$
$929$	13.1937	0.432871	0.216436	+	0.976297i	$0.430557\pi$
$930$	0	0
$931$	9.84718	0.322728
$932$	0	0
$933$	−5.54392	−0.181500
$934$	0	0
$935$	2.25352	0.0736978
$936$	0	0
$937$	−36.0411	−1.17741	−0.588706	−	0.808347i	$-0.700363\pi$
$937$	−36.0411	−1.17741	−0.588706	+	0.808347i	$0.700363\pi$
$938$	0	0
$939$	56.3416	1.83864
$940$	0	0
$941$	41.8765	1.36514	0.682568	−	0.730822i	$-0.260863\pi$
$941$	41.8765	1.36514	0.682568	+	0.730822i	$0.260863\pi$
$942$	0	0
$943$	−17.0036	−0.553714
$944$	0	0
$945$	−9.14670	−0.297542
$946$	0	0
$947$	−27.3169	−0.887679	−0.443840	−	0.896106i	$-0.646384\pi$
$947$	−27.3169	−0.887679	−0.443840	+	0.896106i	$0.646384\pi$
$948$	0	0
$949$	17.7457	0.576049
$950$	0	0
$951$	19.0808	0.618736
$952$	0	0
$953$	−14.3197	−0.463861	−0.231931	−	0.972732i	$-0.574504\pi$
$953$	−14.3197	−0.463861	−0.231931	+	0.972732i	$0.574504\pi$
$954$	0	0
$955$	−8.52383	−0.275825
$956$	0	0
$957$	10.3349	0.334079
$958$	0	0
$959$	−39.8032	−1.28531
$960$	0	0
$961$	−13.6370	−0.439904
$962$	0	0
$963$	−2.73226	−0.0880459
$964$	0	0
$965$	−21.7917	−0.701499
$966$	0	0
$967$	40.2007	1.29277	0.646384	−	0.763013i	$-0.276281\pi$
$967$	40.2007	1.29277	0.646384	+	0.763013i	$0.276281\pi$
$968$	0	0
$969$	6.30126	0.202426
$970$	0	0
$971$	−40.0498	−1.28526	−0.642630	−	0.766177i	$-0.722157\pi$
$971$	−40.0498	−1.28526	−0.642630	+	0.766177i	$0.722157\pi$
$972$	0	0
$973$	0.137741	0.00441577
$974$	0	0
$975$	5.32759	0.170620
$976$	0	0
$977$	−48.8020	−1.56131	−0.780657	−	0.624959i	$-0.785116\pi$
$977$	−48.8020	−1.56131	−0.780657	+	0.624959i	$0.785116\pi$
$978$	0	0
$979$	45.3547	1.44954
$980$	0	0
$981$	−4.15903	−0.132787
$982$	0	0
$983$	20.1288	0.642009	0.321004	−	0.947078i	$-0.395980\pi$
$983$	20.1288	0.642009	0.321004	+	0.947078i	$0.395980\pi$
$984$	0	0
$985$	11.6273	0.370475
$986$	0	0
$987$	54.5681	1.73692
$988$	0	0
$989$	14.6064	0.464456
$990$	0	0
$991$	−21.8226	−0.693219	−0.346609	−	0.938010i	$-0.612667\pi$
$991$	−21.8226	−0.693219	−0.346609	+	0.938010i	$0.612667\pi$
$992$	0	0
$993$	−1.62902	−0.0516953
$994$	0	0
$995$	−20.9477	−0.664088
$996$	0	0
$997$	27.6835	0.876745	0.438373	−	0.898793i	$-0.355555\pi$
$997$	27.6835	0.876745	0.438373	+	0.898793i	$0.355555\pi$
$998$	0	0
$999$	19.7921	0.626194

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6040.2.a.l.1.1	✓	9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6040.2.a.l.1.1	✓	9	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:	\( N \)	\(=\)	\( 6040 = 2^{3} \cdot 5 \cdot 151 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6040.a (trivial)

\(f(q)\)	\(=\)	\(q-1.96251 q^{3} +1.00000 q^{5} -2.16922 q^{7} +0.851433 q^{9} +O(q^{10})\)	\(q-1.96251 q^{3} +1.00000 q^{5} -2.16922 q^{7} +0.851433 q^{9} +3.01214 q^{11} -2.71469 q^{13} -1.96251 q^{15} +0.748145 q^{17} -4.29171 q^{19} +4.25712 q^{21} +1.71882 q^{23} +1.00000 q^{25} +4.21658 q^{27} -1.74832 q^{29} +4.16689 q^{31} -5.91134 q^{33} -2.16922 q^{35} +4.69387 q^{37} +5.32759 q^{39} -9.89262 q^{41} +8.49792 q^{43} +0.851433 q^{45} +12.8181 q^{47} -2.29447 q^{49} -1.46824 q^{51} -7.88590 q^{53} +3.01214 q^{55} +8.42251 q^{57} +6.20478 q^{59} -9.79583 q^{61} -1.84695 q^{63} -2.71469 q^{65} +2.54968 q^{67} -3.37319 q^{69} -6.64355 q^{71} -6.53691 q^{73} -1.96251 q^{75} -6.53400 q^{77} -5.44102 q^{79} -10.8294 q^{81} +4.01722 q^{83} +0.748145 q^{85} +3.43108 q^{87} +15.0573 q^{89} +5.88877 q^{91} -8.17755 q^{93} -4.29171 q^{95} -2.91029 q^{97} +2.56463 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q + 9 q^{5} - 2 q^{7} - 3 q^{9}+O(q^{10}) \) 9 * q + 9 * q^5 - 2 * q^7 - 3 * q^9	\( 9 q + 9 q^{5} - 2 q^{7} - 3 q^{9} - 6 q^{11} - 9 q^{13} - 2 q^{17} - 10 q^{19} - 9 q^{21} - 6 q^{23} + 9 q^{25} + 12 q^{27} - 6 q^{29} + 9 q^{31} - 11 q^{33} - 2 q^{35} - 12 q^{37} - 3 q^{39} - 20 q^{41} + q^{43} - 3 q^{45} + 22 q^{47} - 29 q^{49} + 2 q^{51} - 35 q^{53} - 6 q^{55} - 20 q^{57} + 14 q^{59} - 22 q^{61} - 12 q^{63} - 9 q^{65} + 4 q^{67} + 5 q^{69} - 22 q^{71} - 34 q^{73} - 5 q^{77} + 8 q^{79} - 31 q^{81} - 3 q^{83} - 2 q^{85} - 5 q^{89} - 7 q^{91} - 21 q^{93} - 10 q^{95} - 33 q^{97} - 15 q^{99}+O(q^{100}) \) 9 * q + 9 * q^5 - 2 * q^7 - 3 * q^9 - 6 * q^11 - 9 * q^13 - 2 * q^17 - 10 * q^19 - 9 * q^21 - 6 * q^23 + 9 * q^25 + 12 * q^27 - 6 * q^29 + 9 * q^31 - 11 * q^33 - 2 * q^35 - 12 * q^37 - 3 * q^39 - 20 * q^41 + q^43 - 3 * q^45 + 22 * q^47 - 29 * q^49 + 2 * q^51 - 35 * q^53 - 6 * q^55 - 20 * q^57 + 14 * q^59 - 22 * q^61 - 12 * q^63 - 9 * q^65 + 4 * q^67 + 5 * q^69 - 22 * q^71 - 34 * q^73 - 5 * q^77 + 8 * q^79 - 31 * q^81 - 3 * q^83 - 2 * q^85 - 5 * q^89 - 7 * q^91 - 21 * q^93 - 10 * q^95 - 33 * q^97 - 15 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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