LMFDB - Embedded newform 6040.2.a.n.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:	$13$
Coefficient field:	$\mathbb{Q}[x]/(x^{13} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{13} - 4 x^{12} - 14 x^{11} + 70 x^{10} + 41 x^{9} - 403 x^{8} + 109 x^{7} + 870 x^{6} - 444 x^{5} + \cdots + 11 $ x^13 - 4x^12 - 14x^11 + 70x^10 + 41x^9 - 403x^8 + 109x^7 + 870x^6 - 444x^5 - 708x^4 + 373x^3 + 108x^2 - 82x + 11
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.79963$ 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)

$f(q)$	$=$	$q-2.79963 q^{3} -1.00000 q^{5} +2.90399 q^{7} +4.83794 q^{9} +O(q^{10})$	$q-2.79963 q^{3} -1.00000 q^{5} +2.90399 q^{7} +4.83794 q^{9} +2.35658 q^{11} +3.77013 q^{13} +2.79963 q^{15} -1.37967 q^{17} +2.68463 q^{19} -8.13009 q^{21} -5.78013 q^{23} +1.00000 q^{25} -5.14556 q^{27} -8.01030 q^{29} -7.99345 q^{31} -6.59754 q^{33} -2.90399 q^{35} -3.34612 q^{37} -10.5550 q^{39} -2.12262 q^{41} +3.22975 q^{43} -4.83794 q^{45} +0.895162 q^{47} +1.43313 q^{49} +3.86258 q^{51} -0.533712 q^{53} -2.35658 q^{55} -7.51597 q^{57} +8.55197 q^{59} -0.172407 q^{61} +14.0493 q^{63} -3.77013 q^{65} -3.98670 q^{67} +16.1822 q^{69} +4.50214 q^{71} -9.79377 q^{73} -2.79963 q^{75} +6.84346 q^{77} +4.26466 q^{79} -0.108154 q^{81} -12.5596 q^{83} +1.37967 q^{85} +22.4259 q^{87} +1.84280 q^{89} +10.9484 q^{91} +22.3787 q^{93} -2.68463 q^{95} -9.02980 q^{97} +11.4010 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 13 q - 4 q^{3} - 13 q^{5} + 5 q^{9}+O(q^{10}) $ 13 * q - 4 * q^3 - 13 * q^5 + 5 * q^9	$ 13 q - 4 q^{3} - 13 q^{5} + 5 q^{9} - 14 q^{11} + 5 q^{13} + 4 q^{15} - 8 q^{17} + 16 q^{19} - 5 q^{21} - 4 q^{23} + 13 q^{25} + 2 q^{27} - 6 q^{29} + 11 q^{31} - 19 q^{33} + 6 q^{37} + 7 q^{39} - 18 q^{41} + 7 q^{43} - 5 q^{45} - 22 q^{47} - q^{49} + 12 q^{51} - 17 q^{53} + 14 q^{55} - 16 q^{57} - 6 q^{59} + 10 q^{61} - 5 q^{65} + 12 q^{67} + 13 q^{69} - 16 q^{71} - 24 q^{73} - 4 q^{75} - 11 q^{77} + 36 q^{79} - 19 q^{81} + q^{83} + 8 q^{85} - 8 q^{87} - 53 q^{89} + 23 q^{91} - 9 q^{93} - 16 q^{95} - 21 q^{97} - 9 q^{99}+O(q^{100}) $ 13 * q - 4 * q^3 - 13 * q^5 + 5 * q^9 - 14 * q^11 + 5 * q^13 + 4 * q^15 - 8 * q^17 + 16 * q^19 - 5 * q^21 - 4 * q^23 + 13 * q^25 + 2 * q^27 - 6 * q^29 + 11 * q^31 - 19 * q^33 + 6 * q^37 + 7 * q^39 - 18 * q^41 + 7 * q^43 - 5 * q^45 - 22 * q^47 - q^49 + 12 * q^51 - 17 * q^53 + 14 * q^55 - 16 * q^57 - 6 * q^59 + 10 * q^61 - 5 * q^65 + 12 * q^67 + 13 * q^69 - 16 * q^71 - 24 * q^73 - 4 * q^75 - 11 * q^77 + 36 * q^79 - 19 * q^81 + q^83 + 8 * q^85 - 8 * q^87 - 53 * q^89 + 23 * q^91 - 9 * q^93 - 16 * q^95 - 21 * q^97 - 9 * 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
$3$	−2.79963	−1.61637	−0.808184	−	0.588930i	$-0.799549\pi$
$3$	−2.79963	−1.61637	−0.808184	+	0.588930i	$0.799549\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	2.90399	1.09760	0.548802	−	0.835953i	$-0.315084\pi$
$7$	2.90399	1.09760	0.548802	+	0.835953i	$0.315084\pi$
$8$	0	0
$9$	4.83794	1.61265
$10$	0	0
$11$	2.35658	0.710534	0.355267	−	0.934765i	$-0.384390\pi$
$11$	2.35658	0.710534	0.355267	+	0.934765i	$0.384390\pi$
$12$	0	0
$13$	3.77013	1.04565	0.522823	−	0.852441i	$-0.324879\pi$
$13$	3.77013	1.04565	0.522823	+	0.852441i	$0.324879\pi$
$14$	0	0
$15$	2.79963	0.722862
$16$	0	0
$17$	−1.37967	−0.334620	−0.167310	−	0.985904i	$-0.553508\pi$
$17$	−1.37967	−0.334620	−0.167310	+	0.985904i	$0.553508\pi$
$18$	0	0
$19$	2.68463	0.615896	0.307948	−	0.951403i	$-0.400358\pi$
$19$	2.68463	0.615896	0.307948	+	0.951403i	$0.400358\pi$
$20$	0	0
$21$	−8.13009	−1.77413
$22$	0	0
$23$	−5.78013	−1.20524	−0.602620	−	0.798028i	$-0.705876\pi$
$23$	−5.78013	−1.20524	−0.602620	+	0.798028i	$0.705876\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−5.14556	−0.990263
$28$	0	0
$29$	−8.01030	−1.48748	−0.743738	−	0.668471i	$-0.766949\pi$
$29$	−8.01030	−1.48748	−0.743738	+	0.668471i	$0.766949\pi$
$30$	0	0
$31$	−7.99345	−1.43567	−0.717833	−	0.696215i	$-0.754866\pi$
$31$	−7.99345	−1.43567	−0.717833	+	0.696215i	$0.754866\pi$
$32$	0	0
$33$	−6.59754	−1.14849
$34$	0	0
$35$	−2.90399	−0.490863
$36$	0	0
$37$	−3.34612	−0.550099	−0.275050	−	0.961430i	$-0.588694\pi$
$37$	−3.34612	−0.550099	−0.275050	+	0.961430i	$0.588694\pi$
$38$	0	0
$39$	−10.5550	−1.69015
$40$	0	0
$41$	−2.12262	−0.331497	−0.165749	−	0.986168i	$-0.553004\pi$
$41$	−2.12262	−0.331497	−0.165749	+	0.986168i	$0.553004\pi$
$42$	0	0
$43$	3.22975	0.492533	0.246266	−	0.969202i	$-0.420796\pi$
$43$	3.22975	0.492533	0.246266	+	0.969202i	$0.420796\pi$
$44$	0	0
$45$	−4.83794	−0.721198
$46$	0	0
$47$	0.895162	0.130573	0.0652864	−	0.997867i	$-0.479204\pi$
$47$	0.895162	0.130573	0.0652864	+	0.997867i	$0.479204\pi$
$48$	0	0
$49$	1.43313	0.204733
$50$	0	0
$51$	3.86258	0.540870
$52$	0	0
$53$	−0.533712	−0.0733109	−0.0366555	−	0.999328i	$-0.511670\pi$
$53$	−0.533712	−0.0733109	−0.0366555	+	0.999328i	$0.511670\pi$
$54$	0	0
$55$	−2.35658	−0.317761
$56$	0	0
$57$	−7.51597	−0.995515
$58$	0	0
$59$	8.55197	1.11337	0.556686	−	0.830723i	$-0.312073\pi$
$59$	8.55197	1.11337	0.556686	+	0.830723i	$0.312073\pi$
$60$	0	0
$61$	−0.172407	−0.0220744	−0.0110372	−	0.999939i	$-0.503513\pi$
$61$	−0.172407	−0.0220744	−0.0110372	+	0.999939i	$0.503513\pi$
$62$	0	0
$63$	14.0493	1.77005
$64$	0	0
$65$	−3.77013	−0.467628
$66$	0	0
$67$	−3.98670	−0.487053	−0.243526	−	0.969894i	$-0.578304\pi$
$67$	−3.98670	−0.487053	−0.243526	+	0.969894i	$0.578304\pi$
$68$	0	0
$69$	16.1822	1.94811
$70$	0	0
$71$	4.50214	0.534305	0.267153	−	0.963654i	$-0.413917\pi$
$71$	4.50214	0.534305	0.267153	+	0.963654i	$0.413917\pi$
$72$	0	0
$73$	−9.79377	−1.14627	−0.573137	−	0.819459i	$-0.694274\pi$
$73$	−9.79377	−1.14627	−0.573137	+	0.819459i	$0.694274\pi$
$74$	0	0
$75$	−2.79963	−0.323274
$76$	0	0
$77$	6.84346	0.779885
$78$	0	0
$79$	4.26466	0.479811	0.239906	−	0.970796i	$-0.422883\pi$
$79$	4.26466	0.479811	0.239906	+	0.970796i	$0.422883\pi$
$80$	0	0
$81$	−0.108154	−0.0120171
$82$	0	0
$83$	−12.5596	−1.37860	−0.689299	−	0.724477i	$-0.742081\pi$
$83$	−12.5596	−1.37860	−0.689299	+	0.724477i	$0.742081\pi$
$84$	0	0
$85$	1.37967	0.149647
$86$	0	0
$87$	22.4259	2.40431
$88$	0	0
$89$	1.84280	0.195337	0.0976685	−	0.995219i	$-0.468862\pi$
$89$	1.84280	0.195337	0.0976685	+	0.995219i	$0.468862\pi$
$90$	0	0
$91$	10.9484	1.14771
$92$	0	0
$93$	22.3787	2.32056
$94$	0	0
$95$	−2.68463	−0.275437
$96$	0	0
$97$	−9.02980	−0.916837	−0.458419	−	0.888736i	$-0.651584\pi$
$97$	−9.02980	−0.916837	−0.458419	+	0.888736i	$0.651584\pi$
$98$	0	0
$99$	11.4010	1.14584
$100$	0	0
$101$	0.822161	0.0818080	0.0409040	−	0.999163i	$-0.486976\pi$
$101$	0.822161	0.0818080	0.0409040	+	0.999163i	$0.486976\pi$
$102$	0	0
$103$	15.3261	1.51013	0.755064	−	0.655652i	$-0.227606\pi$
$103$	15.3261	1.51013	0.755064	+	0.655652i	$0.227606\pi$
$104$	0	0
$105$	8.13009	0.793416
$106$	0	0
$107$	−7.68258	−0.742703	−0.371352	−	0.928492i	$-0.621106\pi$
$107$	−7.68258	−0.742703	−0.371352	+	0.928492i	$0.621106\pi$
$108$	0	0
$109$	17.0763	1.63561	0.817806	−	0.575494i	$-0.195190\pi$
$109$	17.0763	1.63561	0.817806	+	0.575494i	$0.195190\pi$
$110$	0	0
$111$	9.36792	0.889163
$112$	0	0
$113$	−6.86534	−0.645837	−0.322918	−	0.946427i	$-0.604664\pi$
$113$	−6.86534	−0.645837	−0.322918	+	0.946427i	$0.604664\pi$
$114$	0	0
$115$	5.78013	0.539000
$116$	0	0
$117$	18.2397	1.68626
$118$	0	0
$119$	−4.00656	−0.367280
$120$	0	0
$121$	−5.44655	−0.495141
$122$	0	0
$123$	5.94255	0.535822
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	5.83291	0.517587	0.258794	−	0.965933i	$-0.416675\pi$
$127$	5.83291	0.517587	0.258794	+	0.965933i	$0.416675\pi$
$128$	0	0
$129$	−9.04212	−0.796114
$130$	0	0
$131$	−1.74896	−0.152807	−0.0764037	−	0.997077i	$-0.524344\pi$
$131$	−1.74896	−0.152807	−0.0764037	+	0.997077i	$0.524344\pi$
$132$	0	0
$133$	7.79612	0.676009
$134$	0	0
$135$	5.14556	0.442859
$136$	0	0
$137$	−8.25783	−0.705514	−0.352757	−	0.935715i	$-0.614756\pi$
$137$	−8.25783	−0.705514	−0.352757	+	0.935715i	$0.614756\pi$
$138$	0	0
$139$	1.92368	0.163165	0.0815823	−	0.996667i	$-0.474003\pi$
$139$	1.92368	0.163165	0.0815823	+	0.996667i	$0.474003\pi$
$140$	0	0
$141$	−2.50612	−0.211054
$142$	0	0
$143$	8.88461	0.742968
$144$	0	0
$145$	8.01030	0.665220
$146$	0	0
$147$	−4.01224	−0.330924
$148$	0	0
$149$	11.6380	0.953423	0.476711	−	0.879060i	$-0.341829\pi$
$149$	11.6380	0.953423	0.476711	+	0.879060i	$0.341829\pi$
$150$	0	0
$151$	1.00000	0.0813788
$151$	1.00000	0.0813788
$152$	0	0
$153$	−6.67478	−0.539624
$154$	0	0
$155$	7.99345	0.642049
$156$	0	0
$157$	−15.5462	−1.24072	−0.620361	−	0.784316i	$-0.713014\pi$
$157$	−15.5462	−1.24072	−0.620361	+	0.784316i	$0.713014\pi$
$158$	0	0
$159$	1.49420	0.118498
$160$	0	0
$161$	−16.7854	−1.32288
$162$	0	0
$163$	8.17142	0.640035	0.320017	−	0.947412i	$-0.396311\pi$
$163$	8.17142	0.640035	0.320017	+	0.947412i	$0.396311\pi$
$164$	0	0
$165$	6.59754	0.513618
$166$	0	0
$167$	−12.5589	−0.971835	−0.485918	−	0.874005i	$-0.661515\pi$
$167$	−12.5589	−0.971835	−0.485918	+	0.874005i	$0.661515\pi$
$168$	0	0
$169$	1.21391	0.0933776
$170$	0	0
$171$	12.9881	0.993222
$172$	0	0
$173$	−19.6473	−1.49376	−0.746878	−	0.664961i	$-0.768448\pi$
$173$	−19.6473	−1.49376	−0.746878	+	0.664961i	$0.768448\pi$
$174$	0	0
$175$	2.90399	0.219521
$176$	0	0
$177$	−23.9424	−1.79962
$178$	0	0
$179$	10.9981	0.822039	0.411020	−	0.911627i	$-0.365173\pi$
$179$	10.9981	0.822039	0.411020	+	0.911627i	$0.365173\pi$
$180$	0	0
$181$	−14.6564	−1.08940	−0.544701	−	0.838631i	$-0.683357\pi$
$181$	−14.6564	−1.08940	−0.544701	+	0.838631i	$0.683357\pi$
$182$	0	0
$183$	0.482676	0.0356804
$184$	0	0
$185$	3.34612	0.246012
$186$	0	0
$187$	−3.25131	−0.237759
$188$	0	0
$189$	−14.9426	−1.08692
$190$	0	0
$191$	−10.1983	−0.737924	−0.368962	−	0.929444i	$-0.620287\pi$
$191$	−10.1983	−0.737924	−0.368962	+	0.929444i	$0.620287\pi$
$192$	0	0
$193$	−7.99179	−0.575262	−0.287631	−	0.957741i	$-0.592868\pi$
$193$	−7.99179	−0.575262	−0.287631	+	0.957741i	$0.592868\pi$
$194$	0	0
$195$	10.5550	0.755858
$196$	0	0
$197$	−5.00020	−0.356249	−0.178125	−	0.984008i	$-0.557003\pi$
$197$	−5.00020	−0.356249	−0.178125	+	0.984008i	$0.557003\pi$
$198$	0	0
$199$	−13.2276	−0.937677	−0.468839	−	0.883284i	$-0.655327\pi$
$199$	−13.2276	−0.937677	−0.468839	+	0.883284i	$0.655327\pi$
$200$	0	0
$201$	11.1613	0.787256
$202$	0	0
$203$	−23.2618	−1.63266
$204$	0	0
$205$	2.12262	0.148250
$206$	0	0
$207$	−27.9639	−1.94363
$208$	0	0
$209$	6.32653	0.437615
$210$	0	0
$211$	11.6680	0.803257	0.401629	−	0.915803i	$-0.368444\pi$
$211$	11.6680	0.803257	0.401629	+	0.915803i	$0.368444\pi$
$212$	0	0
$213$	−12.6043	−0.863634
$214$	0	0
$215$	−3.22975	−0.220267
$216$	0	0
$217$	−23.2129	−1.57579
$218$	0	0
$219$	27.4190	1.85280
$220$	0	0
$221$	−5.20156	−0.349895
$222$	0	0
$223$	13.0222	0.872033	0.436016	−	0.899939i	$-0.356389\pi$
$223$	13.0222	0.872033	0.436016	+	0.899939i	$0.356389\pi$
$224$	0	0
$225$	4.83794	0.322529
$226$	0	0
$227$	0.513201	0.0340623	0.0170312	−	0.999855i	$-0.494579\pi$
$227$	0.513201	0.0340623	0.0170312	+	0.999855i	$0.494579\pi$
$228$	0	0
$229$	15.8790	1.04931	0.524655	−	0.851315i	$-0.324194\pi$
$229$	15.8790	1.04931	0.524655	+	0.851315i	$0.324194\pi$
$230$	0	0
$231$	−19.1592	−1.26058
$232$	0	0
$233$	−14.7405	−0.965679	−0.482840	−	0.875709i	$-0.660395\pi$
$233$	−14.7405	−0.965679	−0.482840	+	0.875709i	$0.660395\pi$
$234$	0	0
$235$	−0.895162	−0.0583939
$236$	0	0
$237$	−11.9395	−0.775552
$238$	0	0
$239$	−17.8677	−1.15577	−0.577884	−	0.816119i	$-0.696122\pi$
$239$	−17.8677	−1.15577	−0.577884	+	0.816119i	$0.696122\pi$
$240$	0	0
$241$	15.0559	0.969836	0.484918	−	0.874560i	$-0.338849\pi$
$241$	15.0559	0.969836	0.484918	+	0.874560i	$0.338849\pi$
$242$	0	0
$243$	15.7395	1.00969
$244$	0	0
$245$	−1.43313	−0.0915595
$246$	0	0
$247$	10.1214	0.644010
$248$	0	0
$249$	35.1623	2.22832
$250$	0	0
$251$	6.39841	0.403864	0.201932	−	0.979400i	$-0.435278\pi$
$251$	6.39841	0.403864	0.201932	+	0.979400i	$0.435278\pi$
$252$	0	0
$253$	−13.6213	−0.856364
$254$	0	0
$255$	−3.86258	−0.241884
$256$	0	0
$257$	−21.4363	−1.33716	−0.668581	−	0.743639i	$-0.733098\pi$
$257$	−21.4363	−1.33716	−0.668581	+	0.743639i	$0.733098\pi$
$258$	0	0
$259$	−9.71710	−0.603791
$260$	0	0
$261$	−38.7534	−2.39877
$262$	0	0
$263$	−22.4994	−1.38737	−0.693685	−	0.720278i	$-0.744014\pi$
$263$	−22.4994	−1.38737	−0.693685	+	0.720278i	$0.744014\pi$
$264$	0	0
$265$	0.533712	0.0327857
$266$	0	0
$267$	−5.15918	−0.315736
$268$	0	0
$269$	10.0265	0.611324	0.305662	−	0.952140i	$-0.401122\pi$
$269$	10.0265	0.611324	0.305662	+	0.952140i	$0.401122\pi$
$270$	0	0
$271$	24.0132	1.45870	0.729348	−	0.684143i	$-0.239824\pi$
$271$	24.0132	1.45870	0.729348	+	0.684143i	$0.239824\pi$
$272$	0	0
$273$	−30.6515	−1.85512
$274$	0	0
$275$	2.35658	0.142107
$276$	0	0
$277$	−17.9230	−1.07689	−0.538445	−	0.842661i	$-0.680988\pi$
$277$	−17.9230	−1.07689	−0.538445	+	0.842661i	$0.680988\pi$
$278$	0	0
$279$	−38.6718	−2.31522
$280$	0	0
$281$	−26.9181	−1.60580	−0.802901	−	0.596112i	$-0.796711\pi$
$281$	−26.9181	−1.60580	−0.802901	+	0.596112i	$0.796711\pi$
$282$	0	0
$283$	16.9093	1.00516	0.502578	−	0.864532i	$-0.332385\pi$
$283$	16.9093	1.00516	0.502578	+	0.864532i	$0.332385\pi$
$284$	0	0
$285$	7.51597	0.445208
$286$	0	0
$287$	−6.16405	−0.363853
$288$	0	0
$289$	−15.0965	−0.888029
$290$	0	0
$291$	25.2801	1.48195
$292$	0	0
$293$	10.6135	0.620048	0.310024	−	0.950729i	$-0.399663\pi$
$293$	10.6135	0.620048	0.310024	+	0.950729i	$0.399663\pi$
$294$	0	0
$295$	−8.55197	−0.497915
$296$	0	0
$297$	−12.1259	−0.703616
$298$	0	0
$299$	−21.7918	−1.26026
$300$	0	0
$301$	9.37915	0.540605
$302$	0	0
$303$	−2.30175	−0.132232
$304$	0	0
$305$	0.172407	0.00987199
$306$	0	0
$307$	3.17084	0.180970	0.0904848	−	0.995898i	$-0.471158\pi$
$307$	3.17084	0.180970	0.0904848	+	0.995898i	$0.471158\pi$
$308$	0	0
$309$	−42.9075	−2.44092
$310$	0	0
$311$	2.76377	0.156719	0.0783594	−	0.996925i	$-0.475032\pi$
$311$	2.76377	0.156719	0.0783594	+	0.996925i	$0.475032\pi$
$312$	0	0
$313$	30.8216	1.74214	0.871070	−	0.491159i	$-0.163426\pi$
$313$	30.8216	1.74214	0.871070	+	0.491159i	$0.163426\pi$
$314$	0	0
$315$	−14.0493	−0.791589
$316$	0	0
$317$	12.0255	0.675420	0.337710	−	0.941250i	$-0.390348\pi$
$317$	12.0255	0.675420	0.337710	+	0.941250i	$0.390348\pi$
$318$	0	0
$319$	−18.8769	−1.05690
$320$	0	0
$321$	21.5084	1.20048
$322$	0	0
$323$	−3.70391	−0.206091
$324$	0	0
$325$	3.77013	0.209129
$326$	0	0
$327$	−47.8073	−2.64375
$328$	0	0
$329$	2.59954	0.143317
$330$	0	0
$331$	8.79420	0.483373	0.241687	−	0.970354i	$-0.422299\pi$
$331$	8.79420	0.483373	0.241687	+	0.970354i	$0.422299\pi$
$332$	0	0
$333$	−16.1883	−0.887116
$334$	0	0
$335$	3.98670	0.217817
$336$	0	0
$337$	−10.5046	−0.572224	−0.286112	−	0.958196i	$-0.592363\pi$
$337$	−10.5046	−0.572224	−0.286112	+	0.958196i	$0.592363\pi$
$338$	0	0
$339$	19.2204	1.04391
$340$	0	0
$341$	−18.8372	−1.02009
$342$	0	0
$343$	−16.1661	−0.872888
$344$	0	0
$345$	−16.1822	−0.871222
$346$	0	0
$347$	−15.1901	−0.815447	−0.407723	−	0.913106i	$-0.633677\pi$
$347$	−15.1901	−0.815447	−0.407723	+	0.913106i	$0.633677\pi$
$348$	0	0
$349$	11.1907	0.599023	0.299511	−	0.954093i	$-0.403176\pi$
$349$	11.1907	0.599023	0.299511	+	0.954093i	$0.403176\pi$
$350$	0	0
$351$	−19.3994	−1.03547
$352$	0	0
$353$	4.21110	0.224134	0.112067	−	0.993701i	$-0.464253\pi$
$353$	4.21110	0.224134	0.112067	+	0.993701i	$0.464253\pi$
$354$	0	0
$355$	−4.50214	−0.238949
$356$	0	0
$357$	11.2169	0.593660
$358$	0	0
$359$	−4.33215	−0.228642	−0.114321	−	0.993444i	$-0.536469\pi$
$359$	−4.33215	−0.228642	−0.114321	+	0.993444i	$0.536469\pi$
$360$	0	0
$361$	−11.7928	−0.620672
$362$	0	0
$363$	15.2483	0.800330
$364$	0	0
$365$	9.79377	0.512629
$366$	0	0
$367$	2.25393	0.117654	0.0588272	−	0.998268i	$-0.481264\pi$
$367$	2.25393	0.117654	0.0588272	+	0.998268i	$0.481264\pi$
$368$	0	0
$369$	−10.2691	−0.534588
$370$	0	0
$371$	−1.54989	−0.0804663
$372$	0	0
$373$	1.54369	0.0799291	0.0399646	−	0.999201i	$-0.487275\pi$
$373$	1.54369	0.0799291	0.0399646	+	0.999201i	$0.487275\pi$
$374$	0	0
$375$	2.79963	0.144572
$376$	0	0
$377$	−30.1999	−1.55537
$378$	0	0
$379$	−5.79665	−0.297754	−0.148877	−	0.988856i	$-0.547566\pi$
$379$	−5.79665	−0.297754	−0.148877	+	0.988856i	$0.547566\pi$
$380$	0	0
$381$	−16.3300	−0.836611
$382$	0	0
$383$	14.2075	0.725972	0.362986	−	0.931795i	$-0.381757\pi$
$383$	14.2075	0.725972	0.362986	+	0.931795i	$0.381757\pi$
$384$	0	0
$385$	−6.84346	−0.348775
$386$	0	0
$387$	15.6253	0.794281
$388$	0	0
$389$	21.3101	1.08047	0.540234	−	0.841515i	$-0.318336\pi$
$389$	21.3101	1.08047	0.540234	+	0.841515i	$0.318336\pi$
$390$	0	0
$391$	7.97469	0.403298
$392$	0	0
$393$	4.89645	0.246993
$394$	0	0
$395$	−4.26466	−0.214578
$396$	0	0
$397$	−15.5468	−0.780270	−0.390135	−	0.920758i	$-0.627572\pi$
$397$	−15.5468	−0.780270	−0.390135	+	0.920758i	$0.627572\pi$
$398$	0	0
$399$	−21.8263	−1.09268
$400$	0	0
$401$	−14.0819	−0.703214	−0.351607	−	0.936148i	$-0.614365\pi$
$401$	−14.0819	−0.703214	−0.351607	+	0.936148i	$0.614365\pi$
$402$	0	0
$403$	−30.1364	−1.50120
$404$	0	0
$405$	0.108154	0.00537423
$406$	0	0
$407$	−7.88539	−0.390864
$408$	0	0
$409$	5.20995	0.257615	0.128808	−	0.991670i	$-0.458885\pi$
$409$	5.20995	0.257615	0.128808	+	0.991670i	$0.458885\pi$
$410$	0	0
$411$	23.1189	1.14037
$412$	0	0
$413$	24.8348	1.22204
$414$	0	0
$415$	12.5596	0.616527
$416$	0	0
$417$	−5.38560	−0.263734
$418$	0	0
$419$	18.6951	0.913315	0.456657	−	0.889643i	$-0.349047\pi$
$419$	18.6951	0.913315	0.456657	+	0.889643i	$0.349047\pi$
$420$	0	0
$421$	−32.0249	−1.56080	−0.780399	−	0.625282i	$-0.784984\pi$
$421$	−32.0249	−1.56080	−0.780399	+	0.625282i	$0.784984\pi$
$422$	0	0
$423$	4.33074	0.210568
$424$	0	0
$425$	−1.37967	−0.0669241
$426$	0	0
$427$	−0.500667	−0.0242290
$428$	0	0
$429$	−24.8736	−1.20091
$430$	0	0
$431$	28.9990	1.39683	0.698416	−	0.715692i	$-0.253889\pi$
$431$	28.9990	1.39683	0.698416	+	0.715692i	$0.253889\pi$
$432$	0	0
$433$	−21.8468	−1.04989	−0.524945	−	0.851136i	$-0.675914\pi$
$433$	−21.8468	−1.04989	−0.524945	+	0.851136i	$0.675914\pi$
$434$	0	0
$435$	−22.4259	−1.07524
$436$	0	0
$437$	−15.5175	−0.742302
$438$	0	0
$439$	−33.9837	−1.62195	−0.810976	−	0.585079i	$-0.801063\pi$
$439$	−33.9837	−1.62195	−0.810976	+	0.585079i	$0.801063\pi$
$440$	0	0
$441$	6.93341	0.330162
$442$	0	0
$443$	12.9008	0.612937	0.306469	−	0.951881i	$-0.400853\pi$
$443$	12.9008	0.612937	0.306469	+	0.951881i	$0.400853\pi$
$444$	0	0
$445$	−1.84280	−0.0873573
$446$	0	0
$447$	−32.5821	−1.54108
$448$	0	0
$449$	3.56145	0.168075	0.0840376	−	0.996463i	$-0.473218\pi$
$449$	3.56145	0.168075	0.0840376	+	0.996463i	$0.473218\pi$
$450$	0	0
$451$	−5.00211	−0.235540
$452$	0	0
$453$	−2.79963	−0.131538
$454$	0	0
$455$	−10.9484	−0.513270
$456$	0	0
$457$	−12.9124	−0.604017	−0.302008	−	0.953305i	$-0.597657\pi$
$457$	−12.9124	−0.604017	−0.302008	+	0.953305i	$0.597657\pi$
$458$	0	0
$459$	7.09919	0.331362
$460$	0	0
$461$	−8.04545	−0.374714	−0.187357	−	0.982292i	$-0.559992\pi$
$461$	−8.04545	−0.374714	−0.187357	+	0.982292i	$0.559992\pi$
$462$	0	0
$463$	36.9499	1.71721	0.858605	−	0.512638i	$-0.171332\pi$
$463$	36.9499	1.71721	0.858605	+	0.512638i	$0.171332\pi$
$464$	0	0
$465$	−22.3787	−1.03779
$466$	0	0
$467$	−28.2102	−1.30541	−0.652706	−	0.757612i	$-0.726366\pi$
$467$	−28.2102	−1.30541	−0.652706	+	0.757612i	$0.726366\pi$
$468$	0	0
$469$	−11.5773	−0.534590
$470$	0	0
$471$	43.5236	2.00546
$472$	0	0
$473$	7.61115	0.349961
$474$	0	0
$475$	2.68463	0.123179
$476$	0	0
$477$	−2.58207	−0.118225
$478$	0	0
$479$	−7.93091	−0.362372	−0.181186	−	0.983449i	$-0.557994\pi$
$479$	−7.93091	−0.362372	−0.181186	+	0.983449i	$0.557994\pi$
$480$	0	0
$481$	−12.6153	−0.575210
$482$	0	0
$483$	46.9930	2.13825
$484$	0	0
$485$	9.02980	0.410022
$486$	0	0
$487$	−21.6830	−0.982549	−0.491274	−	0.871005i	$-0.663469\pi$
$487$	−21.6830	−0.982549	−0.491274	+	0.871005i	$0.663469\pi$
$488$	0	0
$489$	−22.8770	−1.03453
$490$	0	0
$491$	26.5587	1.19858	0.599288	−	0.800533i	$-0.295450\pi$
$491$	26.5587	1.19858	0.599288	+	0.800533i	$0.295450\pi$
$492$	0	0
$493$	11.0516	0.497740
$494$	0	0
$495$	−11.4010	−0.512436
$496$	0	0
$497$	13.0741	0.586455
$498$	0	0
$499$	−6.72575	−0.301086	−0.150543	−	0.988603i	$-0.548102\pi$
$499$	−6.72575	−0.301086	−0.150543	+	0.988603i	$0.548102\pi$
$500$	0	0
$501$	35.1602	1.57084
$502$	0	0
$503$	−40.3870	−1.80077	−0.900383	−	0.435097i	$-0.856714\pi$
$503$	−40.3870	−1.80077	−0.900383	+	0.435097i	$0.856714\pi$
$504$	0	0
$505$	−0.822161	−0.0365857
$506$	0	0
$507$	−3.39850	−0.150933
$508$	0	0
$509$	−15.0218	−0.665831	−0.332916	−	0.942957i	$-0.608032\pi$
$509$	−15.0218	−0.665831	−0.332916	+	0.942957i	$0.608032\pi$
$510$	0	0
$511$	−28.4410	−1.25815
$512$	0	0
$513$	−13.8139	−0.609899
$514$	0	0
$515$	−15.3261	−0.675349
$516$	0	0
$517$	2.10952	0.0927764
$518$	0	0
$519$	55.0052	2.41446
$520$	0	0
$521$	0.818847	0.0358743	0.0179372	−	0.999839i	$-0.494290\pi$
$521$	0.818847	0.0358743	0.0179372	+	0.999839i	$0.494290\pi$
$522$	0	0
$523$	8.48983	0.371235	0.185617	−	0.982622i	$-0.440572\pi$
$523$	8.48983	0.371235	0.185617	+	0.982622i	$0.440572\pi$
$524$	0	0
$525$	−8.13009	−0.354826
$526$	0	0
$527$	11.0284	0.480403
$528$	0	0
$529$	10.4099	0.452603
$530$	0	0
$531$	41.3739	1.79547
$532$	0	0
$533$	−8.00256	−0.346629
$534$	0	0
$535$	7.68258	0.332147
$536$	0	0
$537$	−30.7907	−1.32872
$538$	0	0
$539$	3.37729	0.145470
$540$	0	0
$541$	−9.25728	−0.398002	−0.199001	−	0.979999i	$-0.563770\pi$
$541$	−9.25728	−0.398002	−0.199001	+	0.979999i	$0.563770\pi$
$542$	0	0
$543$	41.0325	1.76087
$544$	0	0
$545$	−17.0763	−0.731468
$546$	0	0
$547$	−12.9869	−0.555278	−0.277639	−	0.960685i	$-0.589552\pi$
$547$	−12.9869	−0.555278	−0.277639	+	0.960685i	$0.589552\pi$
$548$	0	0
$549$	−0.834094	−0.0355983
$550$	0	0
$551$	−21.5047	−0.916130
$552$	0	0
$553$	12.3845	0.526642
$554$	0	0
$555$	−9.36792	−0.397646
$556$	0	0
$557$	2.02586	0.0858386	0.0429193	−	0.999079i	$-0.486334\pi$
$557$	2.02586	0.0858386	0.0429193	+	0.999079i	$0.486334\pi$
$558$	0	0
$559$	12.1766	0.515015
$560$	0	0
$561$	9.10247	0.384306
$562$	0	0
$563$	10.6811	0.450154	0.225077	−	0.974341i	$-0.427737\pi$
$563$	10.6811	0.450154	0.225077	+	0.974341i	$0.427737\pi$
$564$	0	0
$565$	6.86534	0.288827
$566$	0	0
$567$	−0.314078	−0.0131900
$568$	0	0
$569$	−1.71383	−0.0718476	−0.0359238	−	0.999355i	$-0.511437\pi$
$569$	−1.71383	−0.0718476	−0.0359238	+	0.999355i	$0.511437\pi$
$570$	0	0
$571$	−39.7981	−1.66550	−0.832749	−	0.553650i	$-0.813234\pi$
$571$	−39.7981	−1.66550	−0.832749	+	0.553650i	$0.813234\pi$
$572$	0	0
$573$	28.5515	1.19276
$574$	0	0
$575$	−5.78013	−0.241048
$576$	0	0
$577$	−21.4710	−0.893848	−0.446924	−	0.894572i	$-0.647481\pi$
$577$	−21.4710	−0.893848	−0.446924	+	0.894572i	$0.647481\pi$
$578$	0	0
$579$	22.3741	0.929835
$580$	0	0
$581$	−36.4730	−1.51315
$582$	0	0
$583$	−1.25773	−0.0520899
$584$	0	0
$585$	−18.2397	−0.754118
$586$	0	0
$587$	8.73408	0.360494	0.180247	−	0.983621i	$-0.442310\pi$
$587$	8.73408	0.360494	0.180247	+	0.983621i	$0.442310\pi$
$588$	0	0
$589$	−21.4594	−0.884221
$590$	0	0
$591$	13.9987	0.575830
$592$	0	0
$593$	8.65959	0.355607	0.177803	−	0.984066i	$-0.443101\pi$
$593$	8.65959	0.355607	0.177803	+	0.984066i	$0.443101\pi$
$594$	0	0
$595$	4.00656	0.164253
$596$	0	0
$597$	37.0323	1.51563
$598$	0	0
$599$	3.75266	0.153330	0.0766649	−	0.997057i	$-0.475573\pi$
$599$	3.75266	0.153330	0.0766649	+	0.997057i	$0.475573\pi$
$600$	0	0
$601$	4.26631	0.174026	0.0870132	−	0.996207i	$-0.472268\pi$
$601$	4.26631	0.174026	0.0870132	+	0.996207i	$0.472268\pi$
$602$	0	0
$603$	−19.2874	−0.785444
$604$	0	0
$605$	5.44655	0.221434
$606$	0	0
$607$	−29.6146	−1.20202	−0.601010	−	0.799241i	$-0.705235\pi$
$607$	−29.6146	−1.20202	−0.601010	+	0.799241i	$0.705235\pi$
$608$	0	0
$609$	65.1245	2.63898
$610$	0	0
$611$	3.37488	0.136533
$612$	0	0
$613$	13.2794	0.536348	0.268174	−	0.963370i	$-0.413580\pi$
$613$	13.2794	0.536348	0.268174	+	0.963370i	$0.413580\pi$
$614$	0	0
$615$	−5.94255	−0.239627
$616$	0	0
$617$	−13.8090	−0.555930	−0.277965	−	0.960591i	$-0.589660\pi$
$617$	−13.8090	−0.555930	−0.277965	+	0.960591i	$0.589660\pi$
$618$	0	0
$619$	38.0829	1.53068	0.765341	−	0.643625i	$-0.222570\pi$
$619$	38.0829	1.53068	0.765341	+	0.643625i	$0.222570\pi$
$620$	0	0
$621$	29.7420	1.19350
$622$	0	0
$623$	5.35148	0.214402
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−17.7120	−0.707347
$628$	0	0
$629$	4.61656	0.184074
$630$	0	0
$631$	−11.6671	−0.464459	−0.232230	−	0.972661i	$-0.574602\pi$
$631$	−11.6671	−0.464459	−0.232230	+	0.972661i	$0.574602\pi$
$632$	0	0
$633$	−32.6661	−1.29836
$634$	0	0
$635$	−5.83291	−0.231472
$636$	0	0
$637$	5.40310	0.214079
$638$	0	0
$639$	21.7811	0.861646
$640$	0	0
$641$	−28.8797	−1.14068	−0.570341	−	0.821408i	$-0.693189\pi$
$641$	−28.8797	−1.14068	−0.570341	+	0.821408i	$0.693189\pi$
$642$	0	0
$643$	−36.8751	−1.45421	−0.727107	−	0.686525i	$-0.759135\pi$
$643$	−36.8751	−1.45421	−0.727107	+	0.686525i	$0.759135\pi$
$644$	0	0
$645$	9.04212	0.356033
$646$	0	0
$647$	17.8587	0.702098	0.351049	−	0.936357i	$-0.385825\pi$
$647$	17.8587	0.702098	0.351049	+	0.936357i	$0.385825\pi$
$648$	0	0
$649$	20.1534	0.791088
$650$	0	0
$651$	64.9875	2.54706
$652$	0	0
$653$	−30.1185	−1.17863	−0.589314	−	0.807904i	$-0.700602\pi$
$653$	−30.1185	−1.17863	−0.589314	+	0.807904i	$0.700602\pi$
$654$	0	0
$655$	1.74896	0.0683376
$656$	0	0
$657$	−47.3817	−1.84854
$658$	0	0
$659$	−11.6781	−0.454915	−0.227458	−	0.973788i	$-0.573041\pi$
$659$	−11.6781	−0.454915	−0.227458	+	0.973788i	$0.573041\pi$
$660$	0	0
$661$	−19.0320	−0.740259	−0.370130	−	0.928980i	$-0.620687\pi$
$661$	−19.0320	−0.740259	−0.370130	+	0.928980i	$0.620687\pi$
$662$	0	0
$663$	14.5624	0.565559
$664$	0	0
$665$	−7.79612	−0.302321
$666$	0	0
$667$	46.3006	1.79277
$668$	0	0
$669$	−36.4574	−1.40953
$670$	0	0
$671$	−0.406290	−0.0156846
$672$	0	0
$673$	6.38881	0.246271	0.123135	−	0.992390i	$-0.460705\pi$
$673$	6.38881	0.246271	0.123135	+	0.992390i	$0.460705\pi$
$674$	0	0
$675$	−5.14556	−0.198053
$676$	0	0
$677$	−15.0951	−0.580150	−0.290075	−	0.957004i	$-0.593680\pi$
$677$	−15.0951	−0.580150	−0.290075	+	0.957004i	$0.593680\pi$
$678$	0	0
$679$	−26.2224	−1.00632
$680$	0	0
$681$	−1.43677	−0.0550573
$682$	0	0
$683$	−14.9778	−0.573109	−0.286555	−	0.958064i	$-0.592510\pi$
$683$	−14.9778	−0.573109	−0.286555	+	0.958064i	$0.592510\pi$
$684$	0	0
$685$	8.25783	0.315515
$686$	0	0
$687$	−44.4552	−1.69607
$688$	0	0
$689$	−2.01216	−0.0766574
$690$	0	0
$691$	0.0102231	0.000388904 0	0.000194452	−	1.00000i	$-0.499938\pi$
$691$	0.0102231	0.000388904 0	0.000194452		1.00000i	$0.499938\pi$
$692$	0	0
$693$	33.1083	1.25768
$694$	0	0
$695$	−1.92368	−0.0729694
$696$	0	0
$697$	2.92852	0.110926
$698$	0	0
$699$	41.2678	1.56089
$700$	0	0
$701$	6.31131	0.238375	0.119188	−	0.992872i	$-0.461971\pi$
$701$	6.31131	0.238375	0.119188	+	0.992872i	$0.461971\pi$
$702$	0	0
$703$	−8.98310	−0.338804
$704$	0	0
$705$	2.50612	0.0943861
$706$	0	0
$707$	2.38754	0.0897928
$708$	0	0
$709$	31.1207	1.16876	0.584382	−	0.811479i	$-0.301337\pi$
$709$	31.1207	1.16876	0.584382	+	0.811479i	$0.301337\pi$
$710$	0	0
$711$	20.6322	0.773766
$712$	0	0
$713$	46.2031	1.73032
$714$	0	0
$715$	−8.88461	−0.332265
$716$	0	0
$717$	50.0231	1.86815
$718$	0	0
$719$	18.7783	0.700314	0.350157	−	0.936691i	$-0.386128\pi$
$719$	18.7783	0.700314	0.350157	+	0.936691i	$0.386128\pi$
$720$	0	0
$721$	44.5068	1.65752
$722$	0	0
$723$	−42.1510	−1.56761
$724$	0	0
$725$	−8.01030	−0.297495
$726$	0	0
$727$	−7.99920	−0.296674	−0.148337	−	0.988937i	$-0.547392\pi$
$727$	−7.99920	−0.296674	−0.148337	+	0.988937i	$0.547392\pi$
$728$	0	0
$729$	−43.7402	−1.62001
$730$	0	0
$731$	−4.45601	−0.164811
$732$	0	0
$733$	30.5673	1.12903	0.564515	−	0.825423i	$-0.309063\pi$
$733$	30.5673	1.12903	0.564515	+	0.825423i	$0.309063\pi$
$734$	0	0
$735$	4.01224	0.147994
$736$	0	0
$737$	−9.39495	−0.346068
$738$	0	0
$739$	−47.4560	−1.74570	−0.872849	−	0.487990i	$-0.837730\pi$
$739$	−47.4560	−1.74570	−0.872849	+	0.487990i	$0.837730\pi$
$740$	0	0
$741$	−28.3362	−1.04096
$742$	0	0
$743$	−41.1810	−1.51078	−0.755392	−	0.655273i	$-0.772553\pi$
$743$	−41.1810	−1.51078	−0.755392	+	0.655273i	$0.772553\pi$
$744$	0	0
$745$	−11.6380	−0.426384
$746$	0	0
$747$	−60.7627	−2.22319
$748$	0	0
$749$	−22.3101	−0.815194
$750$	0	0
$751$	53.7895	1.96281	0.981404	−	0.191956i	$-0.0614830\pi$
$751$	53.7895	1.96281	0.981404	+	0.191956i	$0.0614830\pi$
$752$	0	0
$753$	−17.9132	−0.652793
$754$	0	0
$755$	−1.00000	−0.0363937
$756$	0	0
$757$	38.0578	1.38324	0.691618	−	0.722263i	$-0.256898\pi$
$757$	38.0578	1.38324	0.691618	+	0.722263i	$0.256898\pi$
$758$	0	0
$759$	38.1346	1.38420
$760$	0	0
$761$	−5.81652	−0.210849	−0.105424	−	0.994427i	$-0.533620\pi$
$761$	−5.81652	−0.210849	−0.105424	+	0.994427i	$0.533620\pi$
$762$	0	0
$763$	49.5893	1.79525
$764$	0	0
$765$	6.67478	0.241327
$766$	0	0
$767$	32.2421	1.16419
$768$	0	0
$769$	−49.8355	−1.79712	−0.898558	−	0.438855i	$-0.855384\pi$
$769$	−49.8355	−1.79712	−0.898558	+	0.438855i	$0.855384\pi$
$770$	0	0
$771$	60.0139	2.16135
$772$	0	0
$773$	−8.17538	−0.294048	−0.147024	−	0.989133i	$-0.546969\pi$
$773$	−8.17538	−0.294048	−0.147024	+	0.989133i	$0.546969\pi$
$774$	0	0
$775$	−7.99345	−0.287133
$776$	0	0
$777$	27.2043	0.975949
$778$	0	0
$779$	−5.69844	−0.204168
$780$	0	0
$781$	10.6096	0.379642
$782$	0	0
$783$	41.2175	1.47299
$784$	0	0
$785$	15.5462	0.554868
$786$	0	0
$787$	43.4086	1.54735	0.773675	−	0.633583i	$-0.218416\pi$
$787$	43.4086	1.54735	0.773675	+	0.633583i	$0.218416\pi$
$788$	0	0
$789$	62.9900	2.24250
$790$	0	0
$791$	−19.9368	−0.708873
$792$	0	0
$793$	−0.649997	−0.0230821
$794$	0	0
$795$	−1.49420	−0.0529937
$796$	0	0
$797$	−3.75458	−0.132994	−0.0664971	−	0.997787i	$-0.521182\pi$
$797$	−3.75458	−0.132994	−0.0664971	+	0.997787i	$0.521182\pi$
$798$	0	0
$799$	−1.23503	−0.0436923
$800$	0	0
$801$	8.91538	0.315009
$802$	0	0
$803$	−23.0798	−0.814467
$804$	0	0
$805$	16.7854	0.591608
$806$	0	0
$807$	−28.0704	−0.988124
$808$	0	0
$809$	17.7008	0.622328	0.311164	−	0.950356i	$-0.399281\pi$
$809$	17.7008	0.622328	0.311164	+	0.950356i	$0.399281\pi$
$810$	0	0
$811$	−47.4312	−1.66554	−0.832768	−	0.553622i	$-0.813245\pi$
$811$	−47.4312	−1.66554	−0.832768	+	0.553622i	$0.813245\pi$
$812$	0	0
$813$	−67.2280	−2.35779
$814$	0	0
$815$	−8.17142	−0.286232
$816$	0	0
$817$	8.67068	0.303349
$818$	0	0
$819$	52.9678	1.85084
$820$	0	0
$821$	48.7570	1.70163	0.850815	−	0.525466i	$-0.176109\pi$
$821$	48.7570	1.70163	0.850815	+	0.525466i	$0.176109\pi$
$822$	0	0
$823$	34.2445	1.19369	0.596844	−	0.802357i	$-0.296421\pi$
$823$	34.2445	1.19369	0.596844	+	0.802357i	$0.296421\pi$
$824$	0	0
$825$	−6.59754	−0.229697
$826$	0	0
$827$	−32.4276	−1.12762	−0.563809	−	0.825905i	$-0.690665\pi$
$827$	−32.4276	−1.12762	−0.563809	+	0.825905i	$0.690665\pi$
$828$	0	0
$829$	22.5128	0.781901	0.390950	−	0.920412i	$-0.372146\pi$
$829$	22.5128	0.781901	0.390950	+	0.920412i	$0.372146\pi$
$830$	0	0
$831$	50.1778	1.74065
$832$	0	0
$833$	−1.97726	−0.0685079
$834$	0	0
$835$	12.5589	0.434618
$836$	0	0
$837$	41.1307	1.42169
$838$	0	0
$839$	6.43559	0.222181	0.111091	−	0.993810i	$-0.464566\pi$
$839$	6.43559	0.222181	0.111091	+	0.993810i	$0.464566\pi$
$840$	0	0
$841$	35.1650	1.21258
$842$	0	0
$843$	75.3609	2.59557
$844$	0	0
$845$	−1.21391	−0.0417597
$846$	0	0
$847$	−15.8167	−0.543468
$848$	0	0
$849$	−47.3399	−1.62470
$850$	0	0
$851$	19.3410	0.663002
$852$	0	0
$853$	−13.3131	−0.455831	−0.227916	−	0.973681i	$-0.573191\pi$
$853$	−13.3131	−0.455831	−0.227916	+	0.973681i	$0.573191\pi$
$854$	0	0
$855$	−12.9881	−0.444183
$856$	0	0
$857$	11.5294	0.393835	0.196918	−	0.980420i	$-0.436907\pi$
$857$	11.5294	0.393835	0.196918	+	0.980420i	$0.436907\pi$
$858$	0	0
$859$	30.0237	1.02439	0.512197	−	0.858868i	$-0.328832\pi$
$859$	30.0237	1.02439	0.512197	+	0.858868i	$0.328832\pi$
$860$	0	0
$861$	17.2571	0.588120
$862$	0	0
$863$	−33.8520	−1.15234	−0.576168	−	0.817332i	$-0.695452\pi$
$863$	−33.8520	−1.15234	−0.576168	+	0.817332i	$0.695452\pi$
$864$	0	0
$865$	19.6473	0.668028
$866$	0	0
$867$	42.2646	1.43538
$868$	0	0
$869$	10.0500	0.340922
$870$	0	0
$871$	−15.0304	−0.509285
$872$	0	0
$873$	−43.6856	−1.47853
$874$	0	0
$875$	−2.90399	−0.0981726
$876$	0	0
$877$	−35.3388	−1.19331	−0.596654	−	0.802499i	$-0.703503\pi$
$877$	−35.3388	−1.19331	−0.596654	+	0.802499i	$0.703503\pi$
$878$	0	0
$879$	−29.7139	−1.00223
$880$	0	0
$881$	1.00914	0.0339986	0.0169993	−	0.999856i	$-0.494589\pi$
$881$	1.00914	0.0339986	0.0169993	+	0.999856i	$0.494589\pi$
$882$	0	0
$883$	18.9154	0.636554	0.318277	−	0.947998i	$-0.396896\pi$
$883$	18.9154	0.636554	0.318277	+	0.947998i	$0.396896\pi$
$884$	0	0
$885$	23.9424	0.804814
$886$	0	0
$887$	−37.7793	−1.26850	−0.634252	−	0.773127i	$-0.718692\pi$
$887$	−37.7793	−1.26850	−0.634252	+	0.773127i	$0.718692\pi$
$888$	0	0
$889$	16.9387	0.568105
$890$	0	0
$891$	−0.254874	−0.00853859
$892$	0	0
$893$	2.40318	0.0804192
$894$	0	0
$895$	−10.9981	−0.367627
$896$	0	0
$897$	61.0092	2.03704
$898$	0	0
$899$	64.0299	2.13552
$900$	0	0
$901$	0.736349	0.0245313
$902$	0	0
$903$	−26.2582	−0.873817
$904$	0	0
$905$	14.6564	0.487195
$906$	0	0
$907$	−11.3409	−0.376568	−0.188284	−	0.982115i	$-0.560293\pi$
$907$	−11.3409	−0.376568	−0.188284	+	0.982115i	$0.560293\pi$
$908$	0	0
$909$	3.97756	0.131927
$910$	0	0
$911$	−25.9553	−0.859936	−0.429968	−	0.902844i	$-0.641475\pi$
$911$	−25.9553	−0.859936	−0.429968	+	0.902844i	$0.641475\pi$
$912$	0	0
$913$	−29.5977	−0.979541
$914$	0	0
$915$	−0.482676	−0.0159568
$916$	0	0
$917$	−5.07896	−0.167722
$918$	0	0
$919$	−41.6533	−1.37401	−0.687007	−	0.726651i	$-0.741076\pi$
$919$	−41.6533	−1.37401	−0.687007	+	0.726651i	$0.741076\pi$
$920$	0	0
$921$	−8.87719	−0.292513
$922$	0	0
$923$	16.9737	0.558695
$924$	0	0
$925$	−3.34612	−0.110020
$926$	0	0
$927$	74.1468	2.43530
$928$	0	0
$929$	48.4493	1.58957	0.794786	−	0.606890i	$-0.207583\pi$
$929$	48.4493	1.58957	0.794786	+	0.606890i	$0.207583\pi$
$930$	0	0
$931$	3.84743	0.126094
$932$	0	0
$933$	−7.73753	−0.253315
$934$	0	0
$935$	3.25131	0.106329
$936$	0	0
$937$	45.6294	1.49065	0.745324	−	0.666702i	$-0.232295\pi$
$937$	45.6294	1.49065	0.745324	+	0.666702i	$0.232295\pi$
$938$	0	0
$939$	−86.2892	−2.81594
$940$	0	0
$941$	49.7829	1.62287	0.811437	−	0.584439i	$-0.198685\pi$
$941$	49.7829	1.62287	0.811437	+	0.584439i	$0.198685\pi$
$942$	0	0
$943$	12.2690	0.399534
$944$	0	0
$945$	14.9426	0.486084
$946$	0	0
$947$	−52.1778	−1.69555	−0.847776	−	0.530355i	$-0.822059\pi$
$947$	−52.1778	−1.69555	−0.847776	+	0.530355i	$0.822059\pi$
$948$	0	0
$949$	−36.9238	−1.19860
$950$	0	0
$951$	−33.6670	−1.09173
$952$	0	0
$953$	−42.6244	−1.38074	−0.690370	−	0.723456i	$-0.742552\pi$
$953$	−42.6244	−1.38074	−0.690370	+	0.723456i	$0.742552\pi$
$954$	0	0
$955$	10.1983	0.330010
$956$	0	0
$957$	52.8483	1.70834
$958$	0	0
$959$	−23.9806	−0.774374
$960$	0	0
$961$	32.8952	1.06114
$962$	0	0
$963$	−37.1679	−1.19772
$964$	0	0
$965$	7.99179	0.257265
$966$	0	0
$967$	−37.3888	−1.20234	−0.601171	−	0.799120i	$-0.705299\pi$
$967$	−37.3888	−1.20234	−0.601171	+	0.799120i	$0.705299\pi$
$968$	0	0
$969$	10.3696	0.333119
$970$	0	0
$971$	−20.0818	−0.644457	−0.322228	−	0.946662i	$-0.604432\pi$
$971$	−20.0818	−0.644457	−0.322228	+	0.946662i	$0.604432\pi$
$972$	0	0
$973$	5.58634	0.179090
$974$	0	0
$975$	−10.5550	−0.338030
$976$	0	0
$977$	0.951341	0.0304361	0.0152180	−	0.999884i	$-0.495156\pi$
$977$	0.951341	0.0304361	0.0152180	+	0.999884i	$0.495156\pi$
$978$	0	0
$979$	4.34271	0.138794
$980$	0	0
$981$	82.6141	2.63766
$982$	0	0
$983$	−30.6318	−0.977002	−0.488501	−	0.872563i	$-0.662456\pi$
$983$	−30.6318	−0.977002	−0.488501	+	0.872563i	$0.662456\pi$
$984$	0	0
$985$	5.00020	0.159320
$986$	0	0
$987$	−7.27775	−0.231653
$988$	0	0
$989$	−18.6684	−0.593620
$990$	0	0
$991$	−21.0850	−0.669788	−0.334894	−	0.942256i	$-0.608701\pi$
$991$	−21.0850	−0.669788	−0.334894	+	0.942256i	$0.608701\pi$
$992$	0	0
$993$	−24.6205	−0.781309
$994$	0	0
$995$	13.2276	0.419342
$996$	0	0
$997$	−1.96143	−0.0621190	−0.0310595	−	0.999518i	$-0.509888\pi$
$997$	−1.96143	−0.0621190	−0.0310595	+	0.999518i	$0.509888\pi$
$998$	0	0
$999$	17.2177	0.544743

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6040.2.a.n.1.1	✓	13

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6040.2.a.n.1.1	✓	13	1.1	even	1	trivial

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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)

\(f(q)\)	\(=\)	\(q-2.79963 q^{3} -1.00000 q^{5} +2.90399 q^{7} +4.83794 q^{9} +O(q^{10})\)	\(q-2.79963 q^{3} -1.00000 q^{5} +2.90399 q^{7} +4.83794 q^{9} +2.35658 q^{11} +3.77013 q^{13} +2.79963 q^{15} -1.37967 q^{17} +2.68463 q^{19} -8.13009 q^{21} -5.78013 q^{23} +1.00000 q^{25} -5.14556 q^{27} -8.01030 q^{29} -7.99345 q^{31} -6.59754 q^{33} -2.90399 q^{35} -3.34612 q^{37} -10.5550 q^{39} -2.12262 q^{41} +3.22975 q^{43} -4.83794 q^{45} +0.895162 q^{47} +1.43313 q^{49} +3.86258 q^{51} -0.533712 q^{53} -2.35658 q^{55} -7.51597 q^{57} +8.55197 q^{59} -0.172407 q^{61} +14.0493 q^{63} -3.77013 q^{65} -3.98670 q^{67} +16.1822 q^{69} +4.50214 q^{71} -9.79377 q^{73} -2.79963 q^{75} +6.84346 q^{77} +4.26466 q^{79} -0.108154 q^{81} -12.5596 q^{83} +1.37967 q^{85} +22.4259 q^{87} +1.84280 q^{89} +10.9484 q^{91} +22.3787 q^{93} -2.68463 q^{95} -9.02980 q^{97} +11.4010 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 13 q - 4 q^{3} - 13 q^{5} + 5 q^{9}+O(q^{10}) \) 13 * q - 4 * q^3 - 13 * q^5 + 5 * q^9	\( 13 q - 4 q^{3} - 13 q^{5} + 5 q^{9} - 14 q^{11} + 5 q^{13} + 4 q^{15} - 8 q^{17} + 16 q^{19} - 5 q^{21} - 4 q^{23} + 13 q^{25} + 2 q^{27} - 6 q^{29} + 11 q^{31} - 19 q^{33} + 6 q^{37} + 7 q^{39} - 18 q^{41} + 7 q^{43} - 5 q^{45} - 22 q^{47} - q^{49} + 12 q^{51} - 17 q^{53} + 14 q^{55} - 16 q^{57} - 6 q^{59} + 10 q^{61} - 5 q^{65} + 12 q^{67} + 13 q^{69} - 16 q^{71} - 24 q^{73} - 4 q^{75} - 11 q^{77} + 36 q^{79} - 19 q^{81} + q^{83} + 8 q^{85} - 8 q^{87} - 53 q^{89} + 23 q^{91} - 9 q^{93} - 16 q^{95} - 21 q^{97} - 9 q^{99}+O(q^{100}) \) 13 * q - 4 * q^3 - 13 * q^5 + 5 * q^9 - 14 * q^11 + 5 * q^13 + 4 * q^15 - 8 * q^17 + 16 * q^19 - 5 * q^21 - 4 * q^23 + 13 * q^25 + 2 * q^27 - 6 * q^29 + 11 * q^31 - 19 * q^33 + 6 * q^37 + 7 * q^39 - 18 * q^41 + 7 * q^43 - 5 * q^45 - 22 * q^47 - q^49 + 12 * q^51 - 17 * q^53 + 14 * q^55 - 16 * q^57 - 6 * q^59 + 10 * q^61 - 5 * q^65 + 12 * q^67 + 13 * q^69 - 16 * q^71 - 24 * q^73 - 4 * q^75 - 11 * q^77 + 36 * q^79 - 19 * q^81 + q^83 + 8 * q^85 - 8 * q^87 - 53 * q^89 + 23 * q^91 - 9 * q^93 - 16 * q^95 - 21 * q^97 - 9 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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