LMFDB - Embedded newform 6040.2.a.l.1.8

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			1.8
Root			$1.10659$ 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)

$f(q)$	$=$	$q+2.35518 q^{3} +1.00000 q^{5} +0.202920 q^{7} +2.54689 q^{9} +O(q^{10})$	$q+2.35518 q^{3} +1.00000 q^{5} +0.202920 q^{7} +2.54689 q^{9} -4.49118 q^{11} -0.979565 q^{13} +2.35518 q^{15} +0.563306 q^{17} -3.51749 q^{19} +0.477913 q^{21} -8.46590 q^{23} +1.00000 q^{25} -1.06715 q^{27} -3.31208 q^{29} +5.73102 q^{31} -10.5776 q^{33} +0.202920 q^{35} -8.46526 q^{37} -2.30706 q^{39} -6.59632 q^{41} -0.444501 q^{43} +2.54689 q^{45} +10.8979 q^{47} -6.95882 q^{49} +1.32669 q^{51} -6.63714 q^{53} -4.49118 q^{55} -8.28435 q^{57} +1.09121 q^{59} +0.633746 q^{61} +0.516815 q^{63} -0.979565 q^{65} +4.51926 q^{67} -19.9388 q^{69} -14.7218 q^{71} +0.361789 q^{73} +2.35518 q^{75} -0.911349 q^{77} +2.63388 q^{79} -10.1540 q^{81} +11.1266 q^{83} +0.563306 q^{85} -7.80055 q^{87} +6.68105 q^{89} -0.198773 q^{91} +13.4976 q^{93} -3.51749 q^{95} +10.1080 q^{97} -11.4386 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	2.35518	1.35977	0.679883	−	0.733320i	$-0.262030\pi$
$3$	2.35518	1.35977	0.679883	+	0.733320i	$0.262030\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	0.202920	0.0766964	0.0383482	−	0.999264i	$-0.487790\pi$
$7$	0.202920	0.0766964	0.0383482	+	0.999264i	$0.487790\pi$
$8$	0	0
$9$	2.54689	0.848965
$10$	0	0
$11$	−4.49118	−1.35414	−0.677071	−	0.735918i	$-0.736751\pi$
$11$	−4.49118	−1.35414	−0.677071	+	0.735918i	$0.736751\pi$
$12$	0	0
$13$	−0.979565	−0.271682	−0.135841	−	0.990731i	$-0.543374\pi$
$13$	−0.979565	−0.271682	−0.135841	+	0.990731i	$0.543374\pi$
$14$	0	0
$15$	2.35518	0.608106
$16$	0	0
$17$	0.563306	0.136622	0.0683109	−	0.997664i	$-0.478239\pi$
$17$	0.563306	0.136622	0.0683109	+	0.997664i	$0.478239\pi$
$18$	0	0
$19$	−3.51749	−0.806968	−0.403484	−	0.914987i	$-0.632201\pi$
$19$	−3.51749	−0.806968	−0.403484	+	0.914987i	$0.632201\pi$
$20$	0	0
$21$	0.477913	0.104289
$22$	0	0
$23$	−8.46590	−1.76526	−0.882631	−	0.470067i	$-0.844230\pi$
$23$	−8.46590	−1.76526	−0.882631	+	0.470067i	$0.844230\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−1.06715	−0.205373
$28$	0	0
$29$	−3.31208	−0.615037	−0.307519	−	0.951542i	$-0.599499\pi$
$29$	−3.31208	−0.615037	−0.307519	+	0.951542i	$0.599499\pi$
$30$	0	0
$31$	5.73102	1.02932	0.514660	−	0.857394i	$-0.327918\pi$
$31$	5.73102	1.02932	0.514660	+	0.857394i	$0.327918\pi$
$32$	0	0
$33$	−10.5776	−1.84132
$34$	0	0
$35$	0.202920	0.0342997
$36$	0	0
$37$	−8.46526	−1.39168	−0.695840	−	0.718197i	$-0.744968\pi$
$37$	−8.46526	−1.39168	−0.695840	+	0.718197i	$0.744968\pi$
$38$	0	0
$39$	−2.30706	−0.369425
$40$	0	0
$41$	−6.59632	−1.03017	−0.515086	−	0.857138i	$-0.672240\pi$
$41$	−6.59632	−1.03017	−0.515086	+	0.857138i	$0.672240\pi$
$42$	0	0
$43$	−0.444501	−0.0677858	−0.0338929	−	0.999425i	$-0.510791\pi$
$43$	−0.444501	−0.0677858	−0.0338929	+	0.999425i	$0.510791\pi$
$44$	0	0
$45$	2.54689	0.379668
$46$	0	0
$47$	10.8979	1.58962	0.794811	−	0.606857i	$-0.207570\pi$
$47$	10.8979	1.58962	0.794811	+	0.606857i	$0.207570\pi$
$48$	0	0
$49$	−6.95882	−0.994118
$50$	0	0
$51$	1.32669	0.185774
$52$	0	0
$53$	−6.63714	−0.911682	−0.455841	−	0.890061i	$-0.650661\pi$
$53$	−6.63714	−0.911682	−0.455841	+	0.890061i	$0.650661\pi$
$54$	0	0
$55$	−4.49118	−0.605590
$56$	0	0
$57$	−8.28435	−1.09729
$58$	0	0
$59$	1.09121	0.142063	0.0710315	−	0.997474i	$-0.477371\pi$
$59$	1.09121	0.142063	0.0710315	+	0.997474i	$0.477371\pi$
$60$	0	0
$61$	0.633746	0.0811429	0.0405714	−	0.999177i	$-0.487082\pi$
$61$	0.633746	0.0811429	0.0405714	+	0.999177i	$0.487082\pi$
$62$	0	0
$63$	0.516815	0.0651126
$64$	0	0
$65$	−0.979565	−0.121500
$66$	0	0
$67$	4.51926	0.552115	0.276057	−	0.961141i	$-0.410972\pi$
$67$	4.51926	0.552115	0.276057	+	0.961141i	$0.410972\pi$
$68$	0	0
$69$	−19.9388	−2.40034
$70$	0	0
$71$	−14.7218	−1.74715	−0.873577	−	0.486687i	$-0.838205\pi$
$71$	−14.7218	−1.74715	−0.873577	+	0.486687i	$0.838205\pi$
$72$	0	0
$73$	0.361789	0.0423442	0.0211721	−	0.999776i	$-0.493260\pi$
$73$	0.361789	0.0423442	0.0211721	+	0.999776i	$0.493260\pi$
$74$	0	0
$75$	2.35518	0.271953
$76$	0	0
$77$	−0.911349	−0.103858
$78$	0	0
$79$	2.63388	0.296334	0.148167	−	0.988962i	$-0.452663\pi$
$79$	2.63388	0.296334	0.148167	+	0.988962i	$0.452663\pi$
$80$	0	0
$81$	−10.1540	−1.12822
$82$	0	0
$83$	11.1266	1.22130	0.610650	−	0.791901i	$-0.290908\pi$
$83$	11.1266	1.22130	0.610650	+	0.791901i	$0.290908\pi$
$84$	0	0
$85$	0.563306	0.0610991
$86$	0	0
$87$	−7.80055	−0.836307
$88$	0	0
$89$	6.68105	0.708190	0.354095	−	0.935209i	$-0.384789\pi$
$89$	6.68105	0.708190	0.354095	+	0.935209i	$0.384789\pi$
$90$	0	0
$91$	−0.198773	−0.0208371
$92$	0	0
$93$	13.4976	1.39964
$94$	0	0
$95$	−3.51749	−0.360887
$96$	0	0
$97$	10.1080	1.02631	0.513154	−	0.858296i	$-0.328477\pi$
$97$	10.1080	1.02631	0.513154	+	0.858296i	$0.328477\pi$
$98$	0	0
$99$	−11.4386	−1.14962
$100$	0	0
$101$	−9.21387	−0.916814	−0.458407	−	0.888742i	$-0.651580\pi$
$101$	−9.21387	−0.916814	−0.458407	+	0.888742i	$0.651580\pi$
$102$	0	0
$103$	−4.96729	−0.489441	−0.244721	−	0.969594i	$-0.578696\pi$
$103$	−4.96729	−0.489441	−0.244721	+	0.969594i	$0.578696\pi$
$104$	0	0
$105$	0.477913	0.0466396
$106$	0	0
$107$	−7.94662	−0.768229	−0.384114	−	0.923285i	$-0.625493\pi$
$107$	−7.94662	−0.768229	−0.384114	+	0.923285i	$0.625493\pi$
$108$	0	0
$109$	12.1664	1.16533	0.582666	−	0.812711i	$-0.302009\pi$
$109$	12.1664	1.16533	0.582666	+	0.812711i	$0.302009\pi$
$110$	0	0
$111$	−19.9372	−1.89236
$112$	0	0
$113$	−14.9767	−1.40889	−0.704445	−	0.709758i	$-0.748804\pi$
$113$	−14.9767	−1.40889	−0.704445	+	0.709758i	$0.748804\pi$
$114$	0	0
$115$	−8.46590	−0.789449
$116$	0	0
$117$	−2.49485	−0.230649
$118$	0	0
$119$	0.114306	0.0104784
$120$	0	0
$121$	9.17068	0.833698
$122$	0	0
$123$	−15.5356	−1.40079
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	5.31664	0.471776	0.235888	−	0.971780i	$-0.424200\pi$
$127$	5.31664	0.471776	0.235888	+	0.971780i	$0.424200\pi$
$128$	0	0
$129$	−1.04688	−0.0921729
$130$	0	0
$131$	17.3314	1.51425	0.757127	−	0.653267i	$-0.226602\pi$
$131$	17.3314	1.51425	0.757127	+	0.653267i	$0.226602\pi$
$132$	0	0
$133$	−0.713769	−0.0618916
$134$	0	0
$135$	−1.06715	−0.0918456
$136$	0	0
$137$	−3.42352	−0.292491	−0.146246	−	0.989248i	$-0.546719\pi$
$137$	−3.42352	−0.292491	−0.146246	+	0.989248i	$0.546719\pi$
$138$	0	0
$139$	−2.58726	−0.219448	−0.109724	−	0.993962i	$-0.534997\pi$
$139$	−2.58726	−0.219448	−0.109724	+	0.993962i	$0.534997\pi$
$140$	0	0
$141$	25.6666	2.16152
$142$	0	0
$143$	4.39940	0.367896
$144$	0	0
$145$	−3.31208	−0.275053
$146$	0	0
$147$	−16.3893	−1.35177
$148$	0	0
$149$	7.04914	0.577488	0.288744	−	0.957406i	$-0.406762\pi$
$149$	7.04914	0.577488	0.288744	+	0.957406i	$0.406762\pi$
$150$	0	0
$151$	−1.00000	−0.0813788
$151$	−1.00000	−0.0813788
$152$	0	0
$153$	1.43468	0.115987
$154$	0	0
$155$	5.73102	0.460326
$156$	0	0
$157$	−23.2092	−1.85229	−0.926147	−	0.377162i	$-0.876900\pi$
$157$	−23.2092	−1.85229	−0.926147	+	0.377162i	$0.876900\pi$
$158$	0	0
$159$	−15.6317	−1.23967
$160$	0	0
$161$	−1.71790	−0.135389
$162$	0	0
$163$	21.8289	1.70977	0.854884	−	0.518819i	$-0.173628\pi$
$163$	21.8289	1.70977	0.854884	+	0.518819i	$0.173628\pi$
$164$	0	0
$165$	−10.5776	−0.823461
$166$	0	0
$167$	−9.28307	−0.718346	−0.359173	−	0.933271i	$-0.616941\pi$
$167$	−9.28307	−0.718346	−0.359173	+	0.933271i	$0.616941\pi$
$168$	0	0
$169$	−12.0405	−0.926189
$170$	0	0
$171$	−8.95868	−0.685088
$172$	0	0
$173$	19.6824	1.49642	0.748211	−	0.663460i	$-0.230913\pi$
$173$	19.6824	1.49642	0.748211	+	0.663460i	$0.230913\pi$
$174$	0	0
$175$	0.202920	0.0153393
$176$	0	0
$177$	2.56999	0.193172
$178$	0	0
$179$	−3.71951	−0.278009	−0.139005	−	0.990292i	$-0.544390\pi$
$179$	−3.71951	−0.278009	−0.139005	+	0.990292i	$0.544390\pi$
$180$	0	0
$181$	14.4230	1.07205	0.536027	−	0.844201i	$-0.319924\pi$
$181$	14.4230	1.07205	0.536027	+	0.844201i	$0.319924\pi$
$182$	0	0
$183$	1.49259	0.110335
$184$	0	0
$185$	−8.46526	−0.622378
$186$	0	0
$187$	−2.52991	−0.185005
$188$	0	0
$189$	−0.216546	−0.0157514
$190$	0	0
$191$	13.1268	0.949820	0.474910	−	0.880034i	$-0.342481\pi$
$191$	13.1268	0.949820	0.474910	+	0.880034i	$0.342481\pi$
$192$	0	0
$193$	−2.80478	−0.201893	−0.100946	−	0.994892i	$-0.532187\pi$
$193$	−2.80478	−0.201893	−0.100946	+	0.994892i	$0.532187\pi$
$194$	0	0
$195$	−2.30706	−0.165212
$196$	0	0
$197$	−24.0126	−1.71083	−0.855414	−	0.517945i	$-0.826697\pi$
$197$	−24.0126	−1.71083	−0.855414	+	0.517945i	$0.826697\pi$
$198$	0	0
$199$	−25.4102	−1.80128	−0.900641	−	0.434564i	$-0.856902\pi$
$199$	−25.4102	−1.80128	−0.900641	+	0.434564i	$0.856902\pi$
$200$	0	0
$201$	10.6437	0.750747
$202$	0	0
$203$	−0.672086	−0.0471712
$204$	0	0
$205$	−6.59632	−0.460707
$206$	0	0
$207$	−21.5617	−1.49864
$208$	0	0
$209$	15.7977	1.09275
$210$	0	0
$211$	7.42509	0.511164	0.255582	−	0.966787i	$-0.417733\pi$
$211$	7.42509	0.511164	0.255582	+	0.966787i	$0.417733\pi$
$212$	0	0
$213$	−34.6725	−2.37572
$214$	0	0
$215$	−0.444501	−0.0303147
$216$	0	0
$217$	1.16294	0.0789452
$218$	0	0
$219$	0.852079	0.0575782
$220$	0	0
$221$	−0.551795	−0.0371177
$222$	0	0
$223$	10.6305	0.711874	0.355937	−	0.934510i	$-0.384162\pi$
$223$	10.6305	0.711874	0.355937	+	0.934510i	$0.384162\pi$
$224$	0	0
$225$	2.54689	0.169793
$226$	0	0
$227$	−21.4230	−1.42190	−0.710949	−	0.703244i	$-0.751734\pi$
$227$	−21.4230	−1.42190	−0.710949	+	0.703244i	$0.751734\pi$
$228$	0	0
$229$	26.5539	1.75473	0.877365	−	0.479824i	$-0.159299\pi$
$229$	26.5539	1.75473	0.877365	+	0.479824i	$0.159299\pi$
$230$	0	0
$231$	−2.14639	−0.141222
$232$	0	0
$233$	−14.9502	−0.979419	−0.489709	−	0.871886i	$-0.662897\pi$
$233$	−14.9502	−0.979419	−0.489709	+	0.871886i	$0.662897\pi$
$234$	0	0
$235$	10.8979	0.710901
$236$	0	0
$237$	6.20327	0.402946
$238$	0	0
$239$	29.9489	1.93723	0.968617	−	0.248559i	$-0.0799569\pi$
$239$	29.9489	1.93723	0.968617	+	0.248559i	$0.0799569\pi$
$240$	0	0
$241$	−11.1974	−0.721285	−0.360643	−	0.932704i	$-0.617443\pi$
$241$	−11.1974	−0.721285	−0.360643	+	0.932704i	$0.617443\pi$
$242$	0	0
$243$	−20.7131	−1.32875
$244$	0	0
$245$	−6.95882	−0.444583
$246$	0	0
$247$	3.44561	0.219239
$248$	0	0
$249$	26.2051	1.66068
$250$	0	0
$251$	−14.2145	−0.897210	−0.448605	−	0.893730i	$-0.648079\pi$
$251$	−14.2145	−0.897210	−0.448605	+	0.893730i	$0.648079\pi$
$252$	0	0
$253$	38.0219	2.39041
$254$	0	0
$255$	1.32669	0.0830805
$256$	0	0
$257$	−31.4537	−1.96203	−0.981015	−	0.193932i	$-0.937876\pi$
$257$	−31.4537	−1.96203	−0.981015	+	0.193932i	$0.937876\pi$
$258$	0	0
$259$	−1.71777	−0.106737
$260$	0	0
$261$	−8.43551	−0.522145
$262$	0	0
$263$	24.3949	1.50426	0.752128	−	0.659018i	$-0.229028\pi$
$263$	24.3949	1.50426	0.752128	+	0.659018i	$0.229028\pi$
$264$	0	0
$265$	−6.63714	−0.407716
$266$	0	0
$267$	15.7351	0.962973
$268$	0	0
$269$	25.5613	1.55850	0.779250	−	0.626713i	$-0.215600\pi$
$269$	25.5613	1.55850	0.779250	+	0.626713i	$0.215600\pi$
$270$	0	0
$271$	−16.2039	−0.984319	−0.492160	−	0.870505i	$-0.663792\pi$
$271$	−16.2039	−0.984319	−0.492160	+	0.870505i	$0.663792\pi$
$272$	0	0
$273$	−0.468147	−0.0283336
$274$	0	0
$275$	−4.49118	−0.270828
$276$	0	0
$277$	21.5615	1.29550	0.647752	−	0.761851i	$-0.275709\pi$
$277$	21.5615	1.29550	0.647752	+	0.761851i	$0.275709\pi$
$278$	0	0
$279$	14.5963	0.873857
$280$	0	0
$281$	0.415755	0.0248019	0.0124009	−	0.999923i	$-0.496053\pi$
$281$	0.415755	0.0248019	0.0124009	+	0.999923i	$0.496053\pi$
$282$	0	0
$283$	15.8910	0.944620	0.472310	−	0.881433i	$-0.343420\pi$
$283$	15.8910	0.944620	0.472310	+	0.881433i	$0.343420\pi$
$284$	0	0
$285$	−8.28435	−0.490722
$286$	0	0
$287$	−1.33852	−0.0790106
$288$	0	0
$289$	−16.6827	−0.981334
$290$	0	0
$291$	23.8061	1.39554
$292$	0	0
$293$	12.7581	0.745335	0.372668	−	0.927965i	$-0.378443\pi$
$293$	12.7581	0.745335	0.372668	+	0.927965i	$0.378443\pi$
$294$	0	0
$295$	1.09121	0.0635325
$296$	0	0
$297$	4.79276	0.278104
$298$	0	0
$299$	8.29290	0.479591
$300$	0	0
$301$	−0.0901981	−0.00519893
$302$	0	0
$303$	−21.7004	−1.24665
$304$	0	0
$305$	0.633746	0.0362882
$306$	0	0
$307$	−15.6385	−0.892539	−0.446269	−	0.894899i	$-0.647248\pi$
$307$	−15.6385	−0.892539	−0.446269	+	0.894899i	$0.647248\pi$
$308$	0	0
$309$	−11.6989	−0.665526
$310$	0	0
$311$	−9.81184	−0.556378	−0.278189	−	0.960526i	$-0.589734\pi$
$311$	−9.81184	−0.556378	−0.278189	+	0.960526i	$0.589734\pi$
$312$	0	0
$313$	14.7719	0.834956	0.417478	−	0.908687i	$-0.362914\pi$
$313$	14.7719	0.834956	0.417478	+	0.908687i	$0.362914\pi$
$314$	0	0
$315$	0.516815	0.0291192
$316$	0	0
$317$	−33.8917	−1.90355	−0.951773	−	0.306802i	$-0.900741\pi$
$317$	−33.8917	−1.90355	−0.951773	+	0.306802i	$0.900741\pi$
$318$	0	0
$319$	14.8751	0.832847
$320$	0	0
$321$	−18.7158	−1.04461
$322$	0	0
$323$	−1.98143	−0.110249
$324$	0	0
$325$	−0.979565	−0.0543365
$326$	0	0
$327$	28.6542	1.58458
$328$	0	0
$329$	2.21140	0.121918
$330$	0	0
$331$	16.3916	0.900962	0.450481	−	0.892786i	$-0.351253\pi$
$331$	16.3916	0.900962	0.450481	+	0.892786i	$0.351253\pi$
$332$	0	0
$333$	−21.5601	−1.18149
$334$	0	0
$335$	4.51926	0.246913
$336$	0	0
$337$	7.16366	0.390229	0.195115	−	0.980780i	$-0.437492\pi$
$337$	7.16366	0.390229	0.195115	+	0.980780i	$0.437492\pi$
$338$	0	0
$339$	−35.2729	−1.91576
$340$	0	0
$341$	−25.7390	−1.39385
$342$	0	0
$343$	−2.83252	−0.152942
$344$	0	0
$345$	−19.9388	−1.07347
$346$	0	0
$347$	−3.68889	−0.198030	−0.0990150	−	0.995086i	$-0.531569\pi$
$347$	−3.68889	−0.198030	−0.0990150	+	0.995086i	$0.531569\pi$
$348$	0	0
$349$	18.3785	0.983777	0.491888	−	0.870658i	$-0.336307\pi$
$349$	18.3785	0.983777	0.491888	+	0.870658i	$0.336307\pi$
$350$	0	0
$351$	1.04534	0.0557962
$352$	0	0
$353$	−23.9540	−1.27494	−0.637472	−	0.770473i	$-0.720020\pi$
$353$	−23.9540	−1.27494	−0.637472	+	0.770473i	$0.720020\pi$
$354$	0	0
$355$	−14.7218	−0.781351
$356$	0	0
$357$	0.269212	0.0142482
$358$	0	0
$359$	−1.35465	−0.0714959	−0.0357479	−	0.999361i	$-0.511381\pi$
$359$	−1.35465	−0.0714959	−0.0357479	+	0.999361i	$0.511381\pi$
$360$	0	0
$361$	−6.62724	−0.348802
$362$	0	0
$363$	21.5986	1.13363
$364$	0	0
$365$	0.361789	0.0189369
$366$	0	0
$367$	13.2011	0.689092	0.344546	−	0.938769i	$-0.388033\pi$
$367$	13.2011	0.689092	0.344546	+	0.938769i	$0.388033\pi$
$368$	0	0
$369$	−16.8001	−0.874580
$370$	0	0
$371$	−1.34681	−0.0699227
$372$	0	0
$373$	−6.09681	−0.315681	−0.157840	−	0.987465i	$-0.550453\pi$
$373$	−6.09681	−0.315681	−0.157840	+	0.987465i	$0.550453\pi$
$374$	0	0
$375$	2.35518	0.121621
$376$	0	0
$377$	3.24439	0.167095
$378$	0	0
$379$	−17.9581	−0.922447	−0.461224	−	0.887284i	$-0.652589\pi$
$379$	−17.9581	−0.922447	−0.461224	+	0.887284i	$0.652589\pi$
$380$	0	0
$381$	12.5217	0.641505
$382$	0	0
$383$	19.4959	0.996196	0.498098	−	0.867121i	$-0.334032\pi$
$383$	19.4959	0.996196	0.498098	+	0.867121i	$0.334032\pi$
$384$	0	0
$385$	−0.911349	−0.0464466
$386$	0	0
$387$	−1.13210	−0.0575477
$388$	0	0
$389$	31.7748	1.61105	0.805523	−	0.592564i	$-0.201884\pi$
$389$	31.7748	1.61105	0.805523	+	0.592564i	$0.201884\pi$
$390$	0	0
$391$	−4.76889	−0.241173
$392$	0	0
$393$	40.8187	2.05903
$394$	0	0
$395$	2.63388	0.132525
$396$	0	0
$397$	27.9452	1.40253	0.701265	−	0.712900i	$-0.252619\pi$
$397$	27.9452	1.40253	0.701265	+	0.712900i	$0.252619\pi$
$398$	0	0
$399$	−1.68106	−0.0841581
$400$	0	0
$401$	−1.23547	−0.0616963	−0.0308482	−	0.999524i	$-0.509821\pi$
$401$	−1.23547	−0.0616963	−0.0308482	+	0.999524i	$0.509821\pi$
$402$	0	0
$403$	−5.61390	−0.279648
$404$	0	0
$405$	−10.1540	−0.504557
$406$	0	0
$407$	38.0190	1.88453
$408$	0	0
$409$	−25.5433	−1.26304	−0.631518	−	0.775361i	$-0.717568\pi$
$409$	−25.5433	−1.26304	−0.631518	+	0.775361i	$0.717568\pi$
$410$	0	0
$411$	−8.06302	−0.397719
$412$	0	0
$413$	0.221427	0.0108957
$414$	0	0
$415$	11.1266	0.546182
$416$	0	0
$417$	−6.09347	−0.298399
$418$	0	0
$419$	−1.30024	−0.0635208	−0.0317604	−	0.999496i	$-0.510111\pi$
$419$	−1.30024	−0.0635208	−0.0317604	+	0.999496i	$0.510111\pi$
$420$	0	0
$421$	9.81643	0.478423	0.239212	−	0.970967i	$-0.423111\pi$
$421$	9.81643	0.478423	0.239212	+	0.970967i	$0.423111\pi$
$422$	0	0
$423$	27.7558	1.34953
$424$	0	0
$425$	0.563306	0.0273244
$426$	0	0
$427$	0.128600	0.00622337
$428$	0	0
$429$	10.3614	0.500253
$430$	0	0
$431$	−7.06097	−0.340115	−0.170057	−	0.985434i	$-0.554395\pi$
$431$	−7.06097	−0.340115	−0.170057	+	0.985434i	$0.554395\pi$
$432$	0	0
$433$	−21.1737	−1.01754	−0.508771	−	0.860902i	$-0.669900\pi$
$433$	−21.1737	−1.01754	−0.508771	+	0.860902i	$0.669900\pi$
$434$	0	0
$435$	−7.80055	−0.374008
$436$	0	0
$437$	29.7787	1.42451
$438$	0	0
$439$	−26.3068	−1.25555	−0.627777	−	0.778393i	$-0.716035\pi$
$439$	−26.3068	−1.25555	−0.627777	+	0.778393i	$0.716035\pi$
$440$	0	0
$441$	−17.7234	−0.843971
$442$	0	0
$443$	−13.0573	−0.620371	−0.310186	−	0.950676i	$-0.600391\pi$
$443$	−13.0573	−0.620371	−0.310186	+	0.950676i	$0.600391\pi$
$444$	0	0
$445$	6.68105	0.316712
$446$	0	0
$447$	16.6020	0.785249
$448$	0	0
$449$	30.3665	1.43308	0.716541	−	0.697545i	$-0.245724\pi$
$449$	30.3665	1.43308	0.716541	+	0.697545i	$0.245724\pi$
$450$	0	0
$451$	29.6253	1.39500
$452$	0	0
$453$	−2.35518	−0.110656
$454$	0	0
$455$	−0.198773	−0.00931863
$456$	0	0
$457$	−25.5804	−1.19660	−0.598301	−	0.801271i	$-0.704157\pi$
$457$	−25.5804	−1.19660	−0.598301	+	0.801271i	$0.704157\pi$
$458$	0	0
$459$	−0.601132	−0.0280584
$460$	0	0
$461$	−36.9012	−1.71866	−0.859330	−	0.511422i	$-0.829119\pi$
$461$	−36.9012	−1.71866	−0.859330	+	0.511422i	$0.829119\pi$
$462$	0	0
$463$	−0.538716	−0.0250362	−0.0125181	−	0.999922i	$-0.503985\pi$
$463$	−0.538716	−0.0250362	−0.0125181	+	0.999922i	$0.503985\pi$
$464$	0	0
$465$	13.4976	0.625936
$466$	0	0
$467$	−3.12127	−0.144435	−0.0722176	−	0.997389i	$-0.523008\pi$
$467$	−3.12127	−0.144435	−0.0722176	+	0.997389i	$0.523008\pi$
$468$	0	0
$469$	0.917046	0.0423453
$470$	0	0
$471$	−54.6619	−2.51869
$472$	0	0
$473$	1.99633	0.0917915
$474$	0	0
$475$	−3.51749	−0.161394
$476$	0	0
$477$	−16.9041	−0.773985
$478$	0	0
$479$	−28.6507	−1.30908	−0.654542	−	0.756026i	$-0.727139\pi$
$479$	−28.6507	−1.30908	−0.654542	+	0.756026i	$0.727139\pi$
$480$	0	0
$481$	8.29227	0.378095
$482$	0	0
$483$	−4.04597	−0.184098
$484$	0	0
$485$	10.1080	0.458979
$486$	0	0
$487$	1.67099	0.0757196	0.0378598	−	0.999283i	$-0.487946\pi$
$487$	1.67099	0.0757196	0.0378598	+	0.999283i	$0.487946\pi$
$488$	0	0
$489$	51.4110	2.32489
$490$	0	0
$491$	9.05502	0.408647	0.204324	−	0.978903i	$-0.434501\pi$
$491$	9.05502	0.408647	0.204324	+	0.978903i	$0.434501\pi$
$492$	0	0
$493$	−1.86571	−0.0840275
$494$	0	0
$495$	−11.4386	−0.514125
$496$	0	0
$497$	−2.98734	−0.134000
$498$	0	0
$499$	5.04777	0.225969	0.112985	−	0.993597i	$-0.463959\pi$
$499$	5.04777	0.225969	0.112985	+	0.993597i	$0.463959\pi$
$500$	0	0
$501$	−21.8633	−0.976782
$502$	0	0
$503$	28.4586	1.26891	0.634453	−	0.772961i	$-0.281225\pi$
$503$	28.4586	1.26891	0.634453	+	0.772961i	$0.281225\pi$
$504$	0	0
$505$	−9.21387	−0.410012
$506$	0	0
$507$	−28.3575	−1.25940
$508$	0	0
$509$	14.6592	0.649756	0.324878	−	0.945756i	$-0.394677\pi$
$509$	14.6592	0.649756	0.324878	+	0.945756i	$0.394677\pi$
$510$	0	0
$511$	0.0734141	0.00324765
$512$	0	0
$513$	3.75369	0.165729
$514$	0	0
$515$	−4.96729	−0.218885
$516$	0	0
$517$	−48.9444	−2.15257
$518$	0	0
$519$	46.3556	2.03479
$520$	0	0
$521$	−5.01177	−0.219570	−0.109785	−	0.993955i	$-0.535016\pi$
$521$	−5.01177	−0.219570	−0.109785	+	0.993955i	$0.535016\pi$
$522$	0	0
$523$	−39.1308	−1.71107	−0.855535	−	0.517744i	$-0.826772\pi$
$523$	−39.1308	−1.71107	−0.855535	+	0.517744i	$0.826772\pi$
$524$	0	0
$525$	0.477913	0.0208579
$526$	0	0
$527$	3.22832	0.140628
$528$	0	0
$529$	48.6714	2.11615
$530$	0	0
$531$	2.77919	0.120606
$532$	0	0
$533$	6.46153	0.279880
$534$	0	0
$535$	−7.94662	−0.343562
$536$	0	0
$537$	−8.76014	−0.378028
$538$	0	0
$539$	31.2533	1.34618
$540$	0	0
$541$	6.96641	0.299509	0.149755	−	0.988723i	$-0.452152\pi$
$541$	6.96641	0.299509	0.149755	+	0.988723i	$0.452152\pi$
$542$	0	0
$543$	33.9689	1.45774
$544$	0	0
$545$	12.1664	0.521153
$546$	0	0
$547$	38.9850	1.66688	0.833439	−	0.552611i	$-0.186368\pi$
$547$	38.9850	1.66688	0.833439	+	0.552611i	$0.186368\pi$
$548$	0	0
$549$	1.61408	0.0688874
$550$	0	0
$551$	11.6502	0.496316
$552$	0	0
$553$	0.534466	0.0227278
$554$	0	0
$555$	−19.9372	−0.846289
$556$	0	0
$557$	−26.8368	−1.13711	−0.568557	−	0.822644i	$-0.692498\pi$
$557$	−26.8368	−1.13711	−0.568557	+	0.822644i	$0.692498\pi$
$558$	0	0
$559$	0.435418	0.0184162
$560$	0	0
$561$	−5.95840	−0.251564
$562$	0	0
$563$	−37.1806	−1.56697	−0.783487	−	0.621408i	$-0.786561\pi$
$563$	−37.1806	−1.56697	−0.783487	+	0.621408i	$0.786561\pi$
$564$	0	0
$565$	−14.9767	−0.630075
$566$	0	0
$567$	−2.06045	−0.0865308
$568$	0	0
$569$	−20.6274	−0.864745	−0.432372	−	0.901695i	$-0.642323\pi$
$569$	−20.6274	−0.864745	−0.432372	+	0.901695i	$0.642323\pi$
$570$	0	0
$571$	1.64544	0.0688594	0.0344297	−	0.999407i	$-0.489039\pi$
$571$	1.64544	0.0688594	0.0344297	+	0.999407i	$0.489039\pi$
$572$	0	0
$573$	30.9160	1.29153
$574$	0	0
$575$	−8.46590	−0.353052
$576$	0	0
$577$	6.62015	0.275601	0.137800	−	0.990460i	$-0.455997\pi$
$577$	6.62015	0.275601	0.137800	+	0.990460i	$0.455997\pi$
$578$	0	0
$579$	−6.60578	−0.274527
$580$	0	0
$581$	2.25780	0.0936693
$582$	0	0
$583$	29.8086	1.23455
$584$	0	0
$585$	−2.49485	−0.103149
$586$	0	0
$587$	43.5390	1.79705	0.898523	−	0.438927i	$-0.144641\pi$
$587$	43.5390	1.79705	0.898523	+	0.438927i	$0.144641\pi$
$588$	0	0
$589$	−20.1588	−0.830629
$590$	0	0
$591$	−56.5541	−2.32633
$592$	0	0
$593$	−7.30485	−0.299974	−0.149987	−	0.988688i	$-0.547923\pi$
$593$	−7.30485	−0.299974	−0.149987	+	0.988688i	$0.547923\pi$
$594$	0	0
$595$	0.114306	0.00468609
$596$	0	0
$597$	−59.8457	−2.44932
$598$	0	0
$599$	27.2901	1.11504	0.557521	−	0.830163i	$-0.311753\pi$
$599$	27.2901	1.11504	0.557521	+	0.830163i	$0.311753\pi$
$600$	0	0
$601$	−27.6305	−1.12707	−0.563535	−	0.826092i	$-0.690559\pi$
$601$	−27.6305	−1.12707	−0.563535	+	0.826092i	$0.690559\pi$
$602$	0	0
$603$	11.5101	0.468726
$604$	0	0
$605$	9.17068	0.372841
$606$	0	0
$607$	−33.9348	−1.37737	−0.688685	−	0.725060i	$-0.741812\pi$
$607$	−33.9348	−1.37737	−0.688685	+	0.725060i	$0.741812\pi$
$608$	0	0
$609$	−1.58289	−0.0641418
$610$	0	0
$611$	−10.6752	−0.431873
$612$	0	0
$613$	−44.6722	−1.80429	−0.902147	−	0.431428i	$-0.858010\pi$
$613$	−44.6722	−1.80429	−0.902147	+	0.431428i	$0.858010\pi$
$614$	0	0
$615$	−15.5356	−0.626454
$616$	0	0
$617$	7.08617	0.285279	0.142639	−	0.989775i	$-0.454441\pi$
$617$	7.08617	0.285279	0.142639	+	0.989775i	$0.454441\pi$
$618$	0	0
$619$	32.3033	1.29838	0.649190	−	0.760626i	$-0.275108\pi$
$619$	32.3033	1.29838	0.649190	+	0.760626i	$0.275108\pi$
$620$	0	0
$621$	9.03438	0.362537
$622$	0	0
$623$	1.35572	0.0543157
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	37.2065	1.48588
$628$	0	0
$629$	−4.76853	−0.190134
$630$	0	0
$631$	−7.24903	−0.288579	−0.144290	−	0.989536i	$-0.546090\pi$
$631$	−7.24903	−0.288579	−0.144290	+	0.989536i	$0.546090\pi$
$632$	0	0
$633$	17.4875	0.695064
$634$	0	0
$635$	5.31664	0.210985
$636$	0	0
$637$	6.81662	0.270084
$638$	0	0
$639$	−37.4948	−1.48327
$640$	0	0
$641$	−3.98697	−0.157476	−0.0787379	−	0.996895i	$-0.525089\pi$
$641$	−3.98697	−0.157476	−0.0787379	+	0.996895i	$0.525089\pi$
$642$	0	0
$643$	5.46730	0.215609	0.107805	−	0.994172i	$-0.465618\pi$
$643$	5.46730	0.215609	0.107805	+	0.994172i	$0.465618\pi$
$644$	0	0
$645$	−1.04688	−0.0412210
$646$	0	0
$647$	2.48685	0.0977683	0.0488842	−	0.998804i	$-0.484433\pi$
$647$	2.48685	0.0977683	0.0488842	+	0.998804i	$0.484433\pi$
$648$	0	0
$649$	−4.90080	−0.192373
$650$	0	0
$651$	2.73893	0.107347
$652$	0	0
$653$	5.39300	0.211045	0.105522	−	0.994417i	$-0.466349\pi$
$653$	5.39300	0.211045	0.105522	+	0.994417i	$0.466349\pi$
$654$	0	0
$655$	17.3314	0.677195
$656$	0	0
$657$	0.921438	0.0359487
$658$	0	0
$659$	−6.08375	−0.236989	−0.118495	−	0.992955i	$-0.537807\pi$
$659$	−6.08375	−0.236989	−0.118495	+	0.992955i	$0.537807\pi$
$660$	0	0
$661$	−37.6924	−1.46606	−0.733032	−	0.680194i	$-0.761896\pi$
$661$	−37.6924	−1.46606	−0.733032	+	0.680194i	$0.761896\pi$
$662$	0	0
$663$	−1.29958	−0.0504715
$664$	0	0
$665$	−0.713769	−0.0276788
$666$	0	0
$667$	28.0397	1.08570
$668$	0	0
$669$	25.0369	0.967983
$670$	0	0
$671$	−2.84627	−0.109879
$672$	0	0
$673$	−31.3561	−1.20869	−0.604343	−	0.796724i	$-0.706565\pi$
$673$	−31.3561	−1.20869	−0.604343	+	0.796724i	$0.706565\pi$
$674$	0	0
$675$	−1.06715	−0.0410746
$676$	0	0
$677$	17.0170	0.654016	0.327008	−	0.945021i	$-0.393960\pi$
$677$	17.0170	0.654016	0.327008	+	0.945021i	$0.393960\pi$
$678$	0	0
$679$	2.05111	0.0787142
$680$	0	0
$681$	−50.4552	−1.93345
$682$	0	0
$683$	19.7605	0.756116	0.378058	−	0.925782i	$-0.376592\pi$
$683$	19.7605	0.756116	0.378058	+	0.925782i	$0.376592\pi$
$684$	0	0
$685$	−3.42352	−0.130806
$686$	0	0
$687$	62.5393	2.38602
$688$	0	0
$689$	6.50151	0.247688
$690$	0	0
$691$	−3.84967	−0.146448	−0.0732242	−	0.997316i	$-0.523329\pi$
$691$	−3.84967	−0.146448	−0.0732242	+	0.997316i	$0.523329\pi$
$692$	0	0
$693$	−2.32111	−0.0881716
$694$	0	0
$695$	−2.58726	−0.0981403
$696$	0	0
$697$	−3.71575	−0.140744
$698$	0	0
$699$	−35.2104	−1.33178
$700$	0	0
$701$	−39.6526	−1.49766	−0.748829	−	0.662763i	$-0.769384\pi$
$701$	−39.6526	−1.49766	−0.748829	+	0.662763i	$0.769384\pi$
$702$	0	0
$703$	29.7765	1.12304
$704$	0	0
$705$	25.6666	0.966659
$706$	0	0
$707$	−1.86968	−0.0703164
$708$	0	0
$709$	−8.24413	−0.309615	−0.154807	−	0.987945i	$-0.549476\pi$
$709$	−8.24413	−0.309615	−0.154807	+	0.987945i	$0.549476\pi$
$710$	0	0
$711$	6.70821	0.251577
$712$	0	0
$713$	−48.5182	−1.81702
$714$	0	0
$715$	4.39940	0.164528
$716$	0	0
$717$	70.5352	2.63419
$718$	0	0
$719$	−2.88559	−0.107614	−0.0538071	−	0.998551i	$-0.517136\pi$
$719$	−2.88559	−0.107614	−0.0538071	+	0.998551i	$0.517136\pi$
$720$	0	0
$721$	−1.00796	−0.0375384
$722$	0	0
$723$	−26.3718	−0.980779
$724$	0	0
$725$	−3.31208	−0.123007
$726$	0	0
$727$	31.2477	1.15891	0.579457	−	0.815003i	$-0.303265\pi$
$727$	31.2477	1.15891	0.579457	+	0.815003i	$0.303265\pi$
$728$	0	0
$729$	−18.3212	−0.678563
$730$	0	0
$731$	−0.250390	−0.00926102
$732$	0	0
$733$	−41.4275	−1.53016	−0.765079	−	0.643936i	$-0.777300\pi$
$733$	−41.4275	−1.53016	−0.765079	+	0.643936i	$0.777300\pi$
$734$	0	0
$735$	−16.3893	−0.604529
$736$	0	0
$737$	−20.2968	−0.747642
$738$	0	0
$739$	0.263160	0.00968051	0.00484025	−	0.999988i	$-0.498459\pi$
$739$	0.263160	0.00968051	0.00484025	+	0.999988i	$0.498459\pi$
$740$	0	0
$741$	8.11506	0.298114
$742$	0	0
$743$	3.12566	0.114669	0.0573347	−	0.998355i	$-0.481740\pi$
$743$	3.12566	0.114669	0.0573347	+	0.998355i	$0.481740\pi$
$744$	0	0
$745$	7.04914	0.258260
$746$	0	0
$747$	28.3382	1.03684
$748$	0	0
$749$	−1.61253	−0.0589204
$750$	0	0
$751$	17.9728	0.655835	0.327918	−	0.944706i	$-0.393653\pi$
$751$	17.9728	0.655835	0.327918	+	0.944706i	$0.393653\pi$
$752$	0	0
$753$	−33.4777	−1.22000
$754$	0	0
$755$	−1.00000	−0.0363937
$756$	0	0
$757$	36.5849	1.32970	0.664850	−	0.746977i	$-0.268495\pi$
$757$	36.5849	1.32970	0.664850	+	0.746977i	$0.268495\pi$
$758$	0	0
$759$	89.5485	3.25040
$760$	0	0
$761$	−23.3244	−0.845509	−0.422755	−	0.906244i	$-0.638937\pi$
$761$	−23.3244	−0.845509	−0.422755	+	0.906244i	$0.638937\pi$
$762$	0	0
$763$	2.46881	0.0893769
$764$	0	0
$765$	1.43468	0.0518710
$766$	0	0
$767$	−1.06891	−0.0385960
$768$	0	0
$769$	42.3109	1.52577	0.762886	−	0.646533i	$-0.223782\pi$
$769$	42.3109	1.52577	0.762886	+	0.646533i	$0.223782\pi$
$770$	0	0
$771$	−74.0793	−2.66790
$772$	0	0
$773$	39.9119	1.43553	0.717766	−	0.696284i	$-0.245165\pi$
$773$	39.9119	1.43553	0.717766	+	0.696284i	$0.245165\pi$
$774$	0	0
$775$	5.73102	0.205864
$776$	0	0
$777$	−4.04566	−0.145137
$778$	0	0
$779$	23.2025	0.831317
$780$	0	0
$781$	66.1181	2.36589
$782$	0	0
$783$	3.53448	0.126312
$784$	0	0
$785$	−23.2092	−0.828371
$786$	0	0
$787$	7.41388	0.264276	0.132138	−	0.991231i	$-0.457816\pi$
$787$	7.41388	0.264276	0.132138	+	0.991231i	$0.457816\pi$
$788$	0	0
$789$	57.4545	2.04544
$790$	0	0
$791$	−3.03907	−0.108057
$792$	0	0
$793$	−0.620795	−0.0220451
$794$	0	0
$795$	−15.6317	−0.554399
$796$	0	0
$797$	−37.4922	−1.32804	−0.664022	−	0.747713i	$-0.731152\pi$
$797$	−37.4922	−1.32804	−0.664022	+	0.747713i	$0.731152\pi$
$798$	0	0
$799$	6.13886	0.217177
$800$	0	0
$801$	17.0159	0.601228
$802$	0	0
$803$	−1.62486	−0.0573400
$804$	0	0
$805$	−1.71790	−0.0605479
$806$	0	0
$807$	60.2016	2.11920
$808$	0	0
$809$	−14.2543	−0.501155	−0.250578	−	0.968097i	$-0.580621\pi$
$809$	−14.2543	−0.501155	−0.250578	+	0.968097i	$0.580621\pi$
$810$	0	0
$811$	15.1211	0.530972	0.265486	−	0.964115i	$-0.414468\pi$
$811$	15.1211	0.530972	0.265486	+	0.964115i	$0.414468\pi$
$812$	0	0
$813$	−38.1633	−1.33844
$814$	0	0
$815$	21.8289	0.764632
$816$	0	0
$817$	1.56353	0.0547010
$818$	0	0
$819$	−0.506254	−0.0176899
$820$	0	0
$821$	−43.2687	−1.51009	−0.755043	−	0.655675i	$-0.772384\pi$
$821$	−43.2687	−1.51009	−0.755043	+	0.655675i	$0.772384\pi$
$822$	0	0
$823$	55.4572	1.93312	0.966558	−	0.256450i	$-0.0825528\pi$
$823$	55.4572	1.93312	0.966558	+	0.256450i	$0.0825528\pi$
$824$	0	0
$825$	−10.5776	−0.368263
$826$	0	0
$827$	−14.8934	−0.517896	−0.258948	−	0.965891i	$-0.583376\pi$
$827$	−14.8934	−0.517896	−0.258948	+	0.965891i	$0.583376\pi$
$828$	0	0
$829$	−17.0618	−0.592581	−0.296290	−	0.955098i	$-0.595750\pi$
$829$	−17.0618	−0.592581	−0.296290	+	0.955098i	$0.595750\pi$
$830$	0	0
$831$	50.7813	1.76158
$832$	0	0
$833$	−3.91995	−0.135818
$834$	0	0
$835$	−9.28307	−0.321254
$836$	0	0
$837$	−6.11585	−0.211395
$838$	0	0
$839$	29.1530	1.00647	0.503236	−	0.864149i	$-0.332143\pi$
$839$	29.1530	1.00647	0.503236	+	0.864149i	$0.332143\pi$
$840$	0	0
$841$	−18.0302	−0.621729
$842$	0	0
$843$	0.979179	0.0337247
$844$	0	0
$845$	−12.0405	−0.414204
$846$	0	0
$847$	1.86091	0.0639417
$848$	0	0
$849$	37.4261	1.28446
$850$	0	0
$851$	71.6660	2.45668
$852$	0	0
$853$	−11.1847	−0.382957	−0.191478	−	0.981497i	$-0.561328\pi$
$853$	−11.1847	−0.382957	−0.191478	+	0.981497i	$0.561328\pi$
$854$	0	0
$855$	−8.95868	−0.306380
$856$	0	0
$857$	7.86905	0.268802	0.134401	−	0.990927i	$-0.457089\pi$
$857$	7.86905	0.268802	0.134401	+	0.990927i	$0.457089\pi$
$858$	0	0
$859$	−38.1597	−1.30199	−0.650996	−	0.759082i	$-0.725648\pi$
$859$	−38.1597	−1.30199	−0.650996	+	0.759082i	$0.725648\pi$
$860$	0	0
$861$	−3.15247	−0.107436
$862$	0	0
$863$	39.0500	1.32928	0.664639	−	0.747164i	$-0.268585\pi$
$863$	39.0500	1.32928	0.664639	+	0.747164i	$0.268585\pi$
$864$	0	0
$865$	19.6824	0.669221
$866$	0	0
$867$	−39.2908	−1.33439
$868$	0	0
$869$	−11.8292	−0.401279
$870$	0	0
$871$	−4.42690	−0.150000
$872$	0	0
$873$	25.7439	0.871300
$874$	0	0
$875$	0.202920	0.00685994
$876$	0	0
$877$	−24.6150	−0.831190	−0.415595	−	0.909550i	$-0.636427\pi$
$877$	−24.6150	−0.831190	−0.415595	+	0.909550i	$0.636427\pi$
$878$	0	0
$879$	30.0476	1.01348
$880$	0	0
$881$	28.1672	0.948977	0.474489	−	0.880262i	$-0.342633\pi$
$881$	28.1672	0.948977	0.474489	+	0.880262i	$0.342633\pi$
$882$	0	0
$883$	34.7308	1.16878	0.584392	−	0.811471i	$-0.301333\pi$
$883$	34.7308	1.16878	0.584392	+	0.811471i	$0.301333\pi$
$884$	0	0
$885$	2.56999	0.0863893
$886$	0	0
$887$	−8.95032	−0.300522	−0.150261	−	0.988646i	$-0.548011\pi$
$887$	−8.95032	−0.300522	−0.150261	+	0.988646i	$0.548011\pi$
$888$	0	0
$889$	1.07885	0.0361835
$890$	0	0
$891$	45.6035	1.52777
$892$	0	0
$893$	−38.3333	−1.28278
$894$	0	0
$895$	−3.71951	−0.124330
$896$	0	0
$897$	19.5313	0.652131
$898$	0	0
$899$	−18.9816	−0.633070
$900$	0	0
$901$	−3.73874	−0.124556
$902$	0	0
$903$	−0.212433	−0.00706933
$904$	0	0
$905$	14.4230	0.479437
$906$	0	0
$907$	−49.6056	−1.64713	−0.823563	−	0.567225i	$-0.808017\pi$
$907$	−49.6056	−1.64713	−0.823563	+	0.567225i	$0.808017\pi$
$908$	0	0
$909$	−23.4667	−0.778343
$910$	0	0
$911$	9.01846	0.298795	0.149398	−	0.988777i	$-0.452267\pi$
$911$	9.01846	0.298795	0.149398	+	0.988777i	$0.452267\pi$
$912$	0	0
$913$	−49.9714	−1.65381
$914$	0	0
$915$	1.49259	0.0493435
$916$	0	0
$917$	3.51689	0.116138
$918$	0	0
$919$	4.94530	0.163130	0.0815652	−	0.996668i	$-0.474008\pi$
$919$	4.94530	0.163130	0.0815652	+	0.996668i	$0.474008\pi$
$920$	0	0
$921$	−36.8317	−1.21364
$922$	0	0
$923$	14.4209	0.474671
$924$	0	0
$925$	−8.46526	−0.278336
$926$	0	0
$927$	−12.6511	−0.415518
$928$	0	0
$929$	12.0661	0.395875	0.197938	−	0.980215i	$-0.436576\pi$
$929$	12.0661	0.395875	0.197938	+	0.980215i	$0.436576\pi$
$930$	0	0
$931$	24.4776	0.802222
$932$	0	0
$933$	−23.1087	−0.756544
$934$	0	0
$935$	−2.52991	−0.0827368
$936$	0	0
$937$	−0.272066	−0.00888800	−0.00444400	−	0.999990i	$-0.501415\pi$
$937$	−0.272066	−0.00888800	−0.00444400	+	0.999990i	$0.501415\pi$
$938$	0	0
$939$	34.7905	1.13534
$940$	0	0
$941$	−19.8673	−0.647656	−0.323828	−	0.946116i	$-0.604970\pi$
$941$	−19.8673	−0.647656	−0.323828	+	0.946116i	$0.604970\pi$
$942$	0	0
$943$	55.8438	1.81852
$944$	0	0
$945$	−0.216546	−0.00704423
$946$	0	0
$947$	−36.7169	−1.19314	−0.596569	−	0.802562i	$-0.703470\pi$
$947$	−36.7169	−1.19314	−0.596569	+	0.802562i	$0.703470\pi$
$948$	0	0
$949$	−0.354396	−0.0115042
$950$	0	0
$951$	−79.8212	−2.58838
$952$	0	0
$953$	−41.4553	−1.34287	−0.671434	−	0.741065i	$-0.734321\pi$
$953$	−41.4553	−1.34287	−0.671434	+	0.741065i	$0.734321\pi$
$954$	0	0
$955$	13.1268	0.424773
$956$	0	0
$957$	35.0337	1.13248
$958$	0	0
$959$	−0.694700	−0.0224330
$960$	0	0
$961$	1.84454	0.0595012
$962$	0	0
$963$	−20.2392	−0.652199
$964$	0	0
$965$	−2.80478	−0.0902891
$966$	0	0
$967$	−33.6334	−1.08158	−0.540788	−	0.841159i	$-0.681874\pi$
$967$	−33.6334	−1.08158	−0.540788	+	0.841159i	$0.681874\pi$
$968$	0	0
$969$	−4.66662	−0.149914
$970$	0	0
$971$	−17.7577	−0.569871	−0.284935	−	0.958547i	$-0.591972\pi$
$971$	−17.7577	−0.569871	−0.284935	+	0.958547i	$0.591972\pi$
$972$	0	0
$973$	−0.525006	−0.0168309
$974$	0	0
$975$	−2.30706	−0.0738849
$976$	0	0
$977$	−15.0637	−0.481930	−0.240965	−	0.970534i	$-0.577464\pi$
$977$	−15.0637	−0.481930	−0.240965	+	0.970534i	$0.577464\pi$
$978$	0	0
$979$	−30.0058	−0.958989
$980$	0	0
$981$	30.9866	0.989326
$982$	0	0
$983$	−37.5910	−1.19897	−0.599484	−	0.800387i	$-0.704627\pi$
$983$	−37.5910	−1.19897	−0.599484	+	0.800387i	$0.704627\pi$
$984$	0	0
$985$	−24.0126	−0.765105
$986$	0	0
$987$	5.20825	0.165781
$988$	0	0
$989$	3.76310	0.119660
$990$	0	0
$991$	36.0602	1.14549	0.572745	−	0.819734i	$-0.305879\pi$
$991$	36.0602	1.14549	0.572745	+	0.819734i	$0.305879\pi$
$992$	0	0
$993$	38.6052	1.22510
$994$	0	0
$995$	−25.4102	−0.805558
$996$	0	0
$997$	13.8167	0.437578	0.218789	−	0.975772i	$-0.429789\pi$
$997$	13.8167	0.437578	0.218789	+	0.975772i	$0.429789\pi$
$998$	0	0
$999$	9.03370	0.285813

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6040.2.a.l.1.8	✓	9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6040.2.a.l.1.8	✓	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)

\(f(q)\)	\(=\)	\(q+2.35518 q^{3} +1.00000 q^{5} +0.202920 q^{7} +2.54689 q^{9} +O(q^{10})\)	\(q+2.35518 q^{3} +1.00000 q^{5} +0.202920 q^{7} +2.54689 q^{9} -4.49118 q^{11} -0.979565 q^{13} +2.35518 q^{15} +0.563306 q^{17} -3.51749 q^{19} +0.477913 q^{21} -8.46590 q^{23} +1.00000 q^{25} -1.06715 q^{27} -3.31208 q^{29} +5.73102 q^{31} -10.5776 q^{33} +0.202920 q^{35} -8.46526 q^{37} -2.30706 q^{39} -6.59632 q^{41} -0.444501 q^{43} +2.54689 q^{45} +10.8979 q^{47} -6.95882 q^{49} +1.32669 q^{51} -6.63714 q^{53} -4.49118 q^{55} -8.28435 q^{57} +1.09121 q^{59} +0.633746 q^{61} +0.516815 q^{63} -0.979565 q^{65} +4.51926 q^{67} -19.9388 q^{69} -14.7218 q^{71} +0.361789 q^{73} +2.35518 q^{75} -0.911349 q^{77} +2.63388 q^{79} -10.1540 q^{81} +11.1266 q^{83} +0.563306 q^{85} -7.80055 q^{87} +6.68105 q^{89} -0.198773 q^{91} +13.4976 q^{93} -3.51749 q^{95} +10.1080 q^{97} -11.4386 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q + 9 q^{5} - 2 q^{7} - 3 q^{9}+O(q^{10}) \) 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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