LMFDB - Embedded newform 6040.2.a.m.1.10

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:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 3 x^{11} - 18 x^{10} + 54 x^{9} + 110 x^{8} - 335 x^{7} - 258 x^{6} + 825 x^{5} + 168 x^{4} + \cdots - 16 $ x^12 - 3x^11 - 18x^10 + 54x^9 + 110x^8 - 335x^7 - 258x^6 + 825x^5 + 168x^4 - 669x^3 + 39x^2 + 106*x - 16
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.10
Root			$2.24403$ 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)

$f(q)$	$=$	$q+2.24403 q^{3} -1.00000 q^{5} +3.06714 q^{7} +2.03565 q^{9} +O(q^{10})$	$q+2.24403 q^{3} -1.00000 q^{5} +3.06714 q^{7} +2.03565 q^{9} +1.64079 q^{11} +4.87967 q^{13} -2.24403 q^{15} +3.34022 q^{17} +2.83756 q^{19} +6.88274 q^{21} +8.66064 q^{23} +1.00000 q^{25} -2.16402 q^{27} -3.80401 q^{29} -2.88911 q^{31} +3.68197 q^{33} -3.06714 q^{35} -6.42015 q^{37} +10.9501 q^{39} +3.28477 q^{41} -3.64381 q^{43} -2.03565 q^{45} +0.752377 q^{47} +2.40733 q^{49} +7.49555 q^{51} +1.41475 q^{53} -1.64079 q^{55} +6.36755 q^{57} +5.30059 q^{59} -0.454366 q^{61} +6.24362 q^{63} -4.87967 q^{65} +1.24860 q^{67} +19.4347 q^{69} -14.5473 q^{71} -5.49692 q^{73} +2.24403 q^{75} +5.03252 q^{77} -4.55114 q^{79} -10.9631 q^{81} -5.47254 q^{83} -3.34022 q^{85} -8.53630 q^{87} -7.79092 q^{89} +14.9666 q^{91} -6.48323 q^{93} -2.83756 q^{95} +4.71652 q^{97} +3.34007 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 3 q^{3} - 12 q^{5} + 5 q^{7} + 9 q^{9}+O(q^{10}) $ 12 * q + 3 * q^3 - 12 * q^5 + 5 * q^7 + 9 * q^9	$ 12 q + 3 q^{3} - 12 q^{5} + 5 q^{7} + 9 q^{9} + 10 q^{11} + 11 q^{13} - 3 q^{15} - 4 q^{17} + 5 q^{19} - q^{21} + 18 q^{23} + 12 q^{25} + 9 q^{27} + 16 q^{29} - q^{31} + 8 q^{33} - 5 q^{35} + 2 q^{37} + 6 q^{39} + 4 q^{41} + 7 q^{43} - 9 q^{45} + 3 q^{49} - 4 q^{51} + 39 q^{53} - 10 q^{55} - 15 q^{57} - 4 q^{59} - 32 q^{61} + 3 q^{63} - 11 q^{65} + 4 q^{67} + 12 q^{69} + 24 q^{71} - 10 q^{73} + 3 q^{75} + 38 q^{77} + 32 q^{79} - 8 q^{81} + 9 q^{83} + 4 q^{85} + 3 q^{87} + 15 q^{89} + 18 q^{91} + 36 q^{93} - 5 q^{95} + 15 q^{97} + 28 q^{99}+O(q^{100}) $ 12 * q + 3 * q^3 - 12 * q^5 + 5 * q^7 + 9 * q^9 + 10 * q^11 + 11 * q^13 - 3 * q^15 - 4 * q^17 + 5 * q^19 - q^21 + 18 * q^23 + 12 * q^25 + 9 * q^27 + 16 * q^29 - q^31 + 8 * q^33 - 5 * q^35 + 2 * q^37 + 6 * q^39 + 4 * q^41 + 7 * q^43 - 9 * q^45 + 3 * q^49 - 4 * q^51 + 39 * q^53 - 10 * q^55 - 15 * q^57 - 4 * q^59 - 32 * q^61 + 3 * q^63 - 11 * q^65 + 4 * q^67 + 12 * q^69 + 24 * q^71 - 10 * q^73 + 3 * q^75 + 38 * q^77 + 32 * q^79 - 8 * q^81 + 9 * q^83 + 4 * q^85 + 3 * q^87 + 15 * q^89 + 18 * q^91 + 36 * q^93 - 5 * q^95 + 15 * q^97 + 28 * 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
$3$	2.24403	1.29559	0.647794	−	0.761815i	$-0.275692\pi$
$3$	2.24403	1.29559	0.647794	+	0.761815i	$0.275692\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	3.06714	1.15927	0.579634	−	0.814877i	$-0.303195\pi$
$7$	3.06714	1.15927	0.579634	+	0.814877i	$0.303195\pi$
$8$	0	0
$9$	2.03565	0.678550
$10$	0	0
$11$	1.64079	0.494716	0.247358	−	0.968924i	$-0.420438\pi$
$11$	1.64079	0.494716	0.247358	+	0.968924i	$0.420438\pi$
$12$	0	0
$13$	4.87967	1.35338	0.676688	−	0.736270i	$-0.263415\pi$
$13$	4.87967	1.35338	0.676688	+	0.736270i	$0.263415\pi$
$14$	0	0
$15$	−2.24403	−0.579405
$16$	0	0
$17$	3.34022	0.810123	0.405062	−	0.914289i	$-0.367250\pi$
$17$	3.34022	0.810123	0.405062	+	0.914289i	$0.367250\pi$
$18$	0	0
$19$	2.83756	0.650981	0.325490	−	0.945545i	$-0.394471\pi$
$19$	2.83756	0.650981	0.325490	+	0.945545i	$0.394471\pi$
$20$	0	0
$21$	6.88274	1.50194
$22$	0	0
$23$	8.66064	1.80587	0.902935	−	0.429778i	$-0.141408\pi$
$23$	8.66064	1.80587	0.902935	+	0.429778i	$0.141408\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−2.16402	−0.416466
$28$	0	0
$29$	−3.80401	−0.706387	−0.353193	−	0.935550i	$-0.614904\pi$
$29$	−3.80401	−0.706387	−0.353193	+	0.935550i	$0.614904\pi$
$30$	0	0
$31$	−2.88911	−0.518899	−0.259450	−	0.965757i	$-0.583541\pi$
$31$	−2.88911	−0.518899	−0.259450	+	0.965757i	$0.583541\pi$
$32$	0	0
$33$	3.68197	0.640948
$34$	0	0
$35$	−3.06714	−0.518441
$36$	0	0
$37$	−6.42015	−1.05547	−0.527733	−	0.849410i	$-0.676958\pi$
$37$	−6.42015	−1.05547	−0.527733	+	0.849410i	$0.676958\pi$
$38$	0	0
$39$	10.9501	1.75342
$40$	0	0
$41$	3.28477	0.512995	0.256498	−	0.966545i	$-0.417431\pi$
$41$	3.28477	0.512995	0.256498	+	0.966545i	$0.417431\pi$
$42$	0	0
$43$	−3.64381	−0.555676	−0.277838	−	0.960628i	$-0.589618\pi$
$43$	−3.64381	−0.555676	−0.277838	+	0.960628i	$0.589618\pi$
$44$	0	0
$45$	−2.03565	−0.303457
$46$	0	0
$47$	0.752377	0.109746	0.0548728	−	0.998493i	$-0.482525\pi$
$47$	0.752377	0.109746	0.0548728	+	0.998493i	$0.482525\pi$
$48$	0	0
$49$	2.40733	0.343905
$50$	0	0
$51$	7.49555	1.04959
$52$	0	0
$53$	1.41475	0.194331	0.0971657	−	0.995268i	$-0.469022\pi$
$53$	1.41475	0.194331	0.0971657	+	0.995268i	$0.469022\pi$
$54$	0	0
$55$	−1.64079	−0.221244
$56$	0	0
$57$	6.36755	0.843403
$58$	0	0
$59$	5.30059	0.690077	0.345039	−	0.938588i	$-0.387866\pi$
$59$	5.30059	0.690077	0.345039	+	0.938588i	$0.387866\pi$
$60$	0	0
$61$	−0.454366	−0.0581756	−0.0290878	−	0.999577i	$-0.509260\pi$
$61$	−0.454366	−0.0581756	−0.0290878	+	0.999577i	$0.509260\pi$
$62$	0	0
$63$	6.24362	0.786622
$64$	0	0
$65$	−4.87967	−0.605248
$66$	0	0
$67$	1.24860	0.152541	0.0762703	−	0.997087i	$-0.475699\pi$
$67$	1.24860	0.152541	0.0762703	+	0.997087i	$0.475699\pi$
$68$	0	0
$69$	19.4347	2.33966
$70$	0	0
$71$	−14.5473	−1.72645	−0.863224	−	0.504820i	$-0.831559\pi$
$71$	−14.5473	−1.72645	−0.863224	+	0.504820i	$0.831559\pi$
$72$	0	0
$73$	−5.49692	−0.643365	−0.321683	−	0.946848i	$-0.604248\pi$
$73$	−5.49692	−0.643365	−0.321683	+	0.946848i	$0.604248\pi$
$74$	0	0
$75$	2.24403	0.259118
$76$	0	0
$77$	5.03252	0.573508
$78$	0	0
$79$	−4.55114	−0.512043	−0.256022	−	0.966671i	$-0.582412\pi$
$79$	−4.55114	−0.512043	−0.256022	+	0.966671i	$0.582412\pi$
$80$	0	0
$81$	−10.9631	−1.21812
$82$	0	0
$83$	−5.47254	−0.600690	−0.300345	−	0.953831i	$-0.597102\pi$
$83$	−5.47254	−0.600690	−0.300345	+	0.953831i	$0.597102\pi$
$84$	0	0
$85$	−3.34022	−0.362298
$86$	0	0
$87$	−8.53630	−0.915187
$88$	0	0
$89$	−7.79092	−0.825836	−0.412918	−	0.910768i	$-0.635490\pi$
$89$	−7.79092	−0.825836	−0.412918	+	0.910768i	$0.635490\pi$
$90$	0	0
$91$	14.9666	1.56893
$92$	0	0
$93$	−6.48323	−0.672280
$94$	0	0
$95$	−2.83756	−0.291127
$96$	0	0
$97$	4.71652	0.478891	0.239445	−	0.970910i	$-0.423034\pi$
$97$	4.71652	0.478891	0.239445	+	0.970910i	$0.423034\pi$
$98$	0	0
$99$	3.34007	0.335689
$100$	0	0
$101$	12.0737	1.20138	0.600691	−	0.799482i	$-0.294892\pi$
$101$	12.0737	1.20138	0.600691	+	0.799482i	$0.294892\pi$
$102$	0	0
$103$	2.66466	0.262557	0.131279	−	0.991346i	$-0.458092\pi$
$103$	2.66466	0.262557	0.131279	+	0.991346i	$0.458092\pi$
$104$	0	0
$105$	−6.88274	−0.671686
$106$	0	0
$107$	−12.7395	−1.23157	−0.615787	−	0.787913i	$-0.711162\pi$
$107$	−12.7395	−1.23157	−0.615787	+	0.787913i	$0.711162\pi$
$108$	0	0
$109$	12.9096	1.23652	0.618259	−	0.785975i	$-0.287838\pi$
$109$	12.9096	1.23652	0.618259	+	0.785975i	$0.287838\pi$
$110$	0	0
$111$	−14.4070	−1.36745
$112$	0	0
$113$	9.95239	0.936242	0.468121	−	0.883664i	$-0.344931\pi$
$113$	9.95239	0.936242	0.468121	+	0.883664i	$0.344931\pi$
$114$	0	0
$115$	−8.66064	−0.807609
$116$	0	0
$117$	9.93330	0.918334
$118$	0	0
$119$	10.2449	0.939151
$120$	0	0
$121$	−8.30782	−0.755257
$122$	0	0
$123$	7.37111	0.664631
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−14.0398	−1.24583	−0.622915	−	0.782289i	$-0.714052\pi$
$127$	−14.0398	−1.24583	−0.622915	+	0.782289i	$0.714052\pi$
$128$	0	0
$129$	−8.17681	−0.719928
$130$	0	0
$131$	13.3232	1.16405	0.582027	−	0.813169i	$-0.302260\pi$
$131$	13.3232	1.16405	0.582027	+	0.813169i	$0.302260\pi$
$132$	0	0
$133$	8.70318	0.754662
$134$	0	0
$135$	2.16402	0.186249
$136$	0	0
$137$	−5.86127	−0.500762	−0.250381	−	0.968147i	$-0.580556\pi$
$137$	−5.86127	−0.500762	−0.250381	+	0.968147i	$0.580556\pi$
$138$	0	0
$139$	7.20672	0.611266	0.305633	−	0.952149i	$-0.401132\pi$
$139$	7.20672	0.611266	0.305633	+	0.952149i	$0.401132\pi$
$140$	0	0
$141$	1.68835	0.142185
$142$	0	0
$143$	8.00649	0.669536
$144$	0	0
$145$	3.80401	0.315906
$146$	0	0
$147$	5.40212	0.445559
$148$	0	0
$149$	21.6115	1.77048	0.885240	−	0.465134i	$-0.153994\pi$
$149$	21.6115	1.77048	0.885240	+	0.465134i	$0.153994\pi$
$150$	0	0
$151$	1.00000	0.0813788
$151$	1.00000	0.0813788
$152$	0	0
$153$	6.79953	0.549710
$154$	0	0
$155$	2.88911	0.232059
$156$	0	0
$157$	12.6535	1.00986	0.504928	−	0.863161i	$-0.331519\pi$
$157$	12.6535	1.00986	0.504928	+	0.863161i	$0.331519\pi$
$158$	0	0
$159$	3.17475	0.251774
$160$	0	0
$161$	26.5634	2.09349
$162$	0	0
$163$	−17.3172	−1.35638	−0.678192	−	0.734885i	$-0.737236\pi$
$163$	−17.3172	−1.35638	−0.678192	+	0.734885i	$0.737236\pi$
$164$	0	0
$165$	−3.68197	−0.286641
$166$	0	0
$167$	−15.6039	−1.20747	−0.603733	−	0.797186i	$-0.706321\pi$
$167$	−15.6039	−1.20747	−0.603733	+	0.797186i	$0.706321\pi$
$168$	0	0
$169$	10.8111	0.831626
$170$	0	0
$171$	5.77628	0.441723
$172$	0	0
$173$	−2.96064	−0.225094	−0.112547	−	0.993646i	$-0.535901\pi$
$173$	−2.96064	−0.225094	−0.112547	+	0.993646i	$0.535901\pi$
$174$	0	0
$175$	3.06714	0.231854
$176$	0	0
$177$	11.8946	0.894057
$178$	0	0
$179$	1.32991	0.0994022	0.0497011	−	0.998764i	$-0.484173\pi$
$179$	1.32991	0.0994022	0.0497011	+	0.998764i	$0.484173\pi$
$180$	0	0
$181$	−8.52607	−0.633738	−0.316869	−	0.948469i	$-0.602631\pi$
$181$	−8.52607	−0.633738	−0.316869	+	0.948469i	$0.602631\pi$
$182$	0	0
$183$	−1.01961	−0.0753716
$184$	0	0
$185$	6.42015	0.472019
$186$	0	0
$187$	5.48059	0.400781
$188$	0	0
$189$	−6.63736	−0.482797
$190$	0	0
$191$	3.92040	0.283670	0.141835	−	0.989890i	$-0.454700\pi$
$191$	3.92040	0.283670	0.141835	+	0.989890i	$0.454700\pi$
$192$	0	0
$193$	10.3308	0.743630	0.371815	−	0.928307i	$-0.378736\pi$
$193$	10.3308	0.743630	0.371815	+	0.928307i	$0.378736\pi$
$194$	0	0
$195$	−10.9501	−0.784153
$196$	0	0
$197$	16.2677	1.15902	0.579512	−	0.814963i	$-0.303243\pi$
$197$	16.2677	1.15902	0.579512	+	0.814963i	$0.303243\pi$
$198$	0	0
$199$	14.7638	1.04658	0.523290	−	0.852155i	$-0.324704\pi$
$199$	14.7638	1.04658	0.523290	+	0.852155i	$0.324704\pi$
$200$	0	0
$201$	2.80189	0.197630
$202$	0	0
$203$	−11.6674	−0.818892
$204$	0	0
$205$	−3.28477	−0.229418
$206$	0	0
$207$	17.6301	1.22537
$208$	0	0
$209$	4.65583	0.322050
$210$	0	0
$211$	−3.50955	−0.241608	−0.120804	−	0.992676i	$-0.538547\pi$
$211$	−3.50955	−0.241608	−0.120804	+	0.992676i	$0.538547\pi$
$212$	0	0
$213$	−32.6446	−2.23677
$214$	0	0
$215$	3.64381	0.248506
$216$	0	0
$217$	−8.86129	−0.601544
$218$	0	0
$219$	−12.3352	−0.833537
$220$	0	0
$221$	16.2992	1.09640
$222$	0	0
$223$	19.7047	1.31952	0.659761	−	0.751476i	$-0.270658\pi$
$223$	19.7047	1.31952	0.659761	+	0.751476i	$0.270658\pi$
$224$	0	0
$225$	2.03565	0.135710
$226$	0	0
$227$	6.38150	0.423555	0.211778	−	0.977318i	$-0.432075\pi$
$227$	6.38150	0.423555	0.211778	+	0.977318i	$0.432075\pi$
$228$	0	0
$229$	−10.5577	−0.697675	−0.348837	−	0.937183i	$-0.613423\pi$
$229$	−10.5577	−0.697675	−0.348837	+	0.937183i	$0.613423\pi$
$230$	0	0
$231$	11.2931	0.743031
$232$	0	0
$233$	−24.2694	−1.58994	−0.794970	−	0.606649i	$-0.792513\pi$
$233$	−24.2694	−1.58994	−0.794970	+	0.606649i	$0.792513\pi$
$234$	0	0
$235$	−0.752377	−0.0490797
$236$	0	0
$237$	−10.2129	−0.663398
$238$	0	0
$239$	−5.20718	−0.336824	−0.168412	−	0.985717i	$-0.553864\pi$
$239$	−5.20718	−0.336824	−0.168412	+	0.985717i	$0.553864\pi$
$240$	0	0
$241$	1.48878	0.0959010	0.0479505	−	0.998850i	$-0.484731\pi$
$241$	1.48878	0.0959010	0.0479505	+	0.998850i	$0.484731\pi$
$242$	0	0
$243$	−18.1094	−1.16172
$244$	0	0
$245$	−2.40733	−0.153799
$246$	0	0
$247$	13.8463	0.881021
$248$	0	0
$249$	−12.2805	−0.778247
$250$	0	0
$251$	−3.59954	−0.227201	−0.113600	−	0.993527i	$-0.536238\pi$
$251$	−3.59954	−0.227201	−0.113600	+	0.993527i	$0.536238\pi$
$252$	0	0
$253$	14.2103	0.893392
$254$	0	0
$255$	−7.49555	−0.469390
$256$	0	0
$257$	18.9195	1.18017	0.590084	−	0.807342i	$-0.299094\pi$
$257$	18.9195	1.18017	0.590084	+	0.807342i	$0.299094\pi$
$258$	0	0
$259$	−19.6915	−1.22357
$260$	0	0
$261$	−7.74364	−0.479319
$262$	0	0
$263$	−11.3455	−0.699592	−0.349796	−	0.936826i	$-0.613749\pi$
$263$	−11.3455	−0.699592	−0.349796	+	0.936826i	$0.613749\pi$
$264$	0	0
$265$	−1.41475	−0.0869077
$266$	0	0
$267$	−17.4830	−1.06994
$268$	0	0
$269$	−7.51136	−0.457976	−0.228988	−	0.973429i	$-0.573542\pi$
$269$	−7.51136	−0.457976	−0.228988	+	0.973429i	$0.573542\pi$
$270$	0	0
$271$	3.94252	0.239491	0.119746	−	0.992805i	$-0.461792\pi$
$271$	3.94252	0.239491	0.119746	+	0.992805i	$0.461792\pi$
$272$	0	0
$273$	33.5854	2.03268
$274$	0	0
$275$	1.64079	0.0989431
$276$	0	0
$277$	−7.90727	−0.475102	−0.237551	−	0.971375i	$-0.576345\pi$
$277$	−7.90727	−0.475102	−0.237551	+	0.971375i	$0.576345\pi$
$278$	0	0
$279$	−5.88122	−0.352099
$280$	0	0
$281$	23.0021	1.37219	0.686094	−	0.727513i	$-0.259324\pi$
$281$	23.0021	1.37219	0.686094	+	0.727513i	$0.259324\pi$
$282$	0	0
$283$	4.36492	0.259468	0.129734	−	0.991549i	$-0.458588\pi$
$283$	4.36492	0.259468	0.129734	+	0.991549i	$0.458588\pi$
$284$	0	0
$285$	−6.36755	−0.377181
$286$	0	0
$287$	10.0748	0.594699
$288$	0	0
$289$	−5.84290	−0.343700
$290$	0	0
$291$	10.5840	0.620445
$292$	0	0
$293$	14.2719	0.833771	0.416886	−	0.908959i	$-0.363122\pi$
$293$	14.2719	0.833771	0.416886	+	0.908959i	$0.363122\pi$
$294$	0	0
$295$	−5.30059	−0.308612
$296$	0	0
$297$	−3.55070	−0.206032
$298$	0	0
$299$	42.2611	2.44402
$300$	0	0
$301$	−11.1761	−0.644178
$302$	0	0
$303$	27.0938	1.55650
$304$	0	0
$305$	0.454366	0.0260169
$306$	0	0
$307$	−12.6144	−0.719941	−0.359970	−	0.932964i	$-0.617213\pi$
$307$	−12.6144	−0.719941	−0.359970	+	0.932964i	$0.617213\pi$
$308$	0	0
$309$	5.97957	0.340166
$310$	0	0
$311$	14.0957	0.799291	0.399646	−	0.916670i	$-0.369133\pi$
$311$	14.0957	0.799291	0.399646	+	0.916670i	$0.369133\pi$
$312$	0	0
$313$	−6.97121	−0.394036	−0.197018	−	0.980400i	$-0.563126\pi$
$313$	−6.97121	−0.394036	−0.197018	+	0.980400i	$0.563126\pi$
$314$	0	0
$315$	−6.24362	−0.351788
$316$	0	0
$317$	−15.5648	−0.874204	−0.437102	−	0.899412i	$-0.643995\pi$
$317$	−15.5648	−0.874204	−0.437102	+	0.899412i	$0.643995\pi$
$318$	0	0
$319$	−6.24157	−0.349461
$320$	0	0
$321$	−28.5877	−1.59561
$322$	0	0
$323$	9.47808	0.527375
$324$	0	0
$325$	4.87967	0.270675
$326$	0	0
$327$	28.9695	1.60202
$328$	0	0
$329$	2.30764	0.127225
$330$	0	0
$331$	−7.27313	−0.399767	−0.199884	−	0.979820i	$-0.564056\pi$
$331$	−7.27313	−0.399767	−0.199884	+	0.979820i	$0.564056\pi$
$332$	0	0
$333$	−13.0692	−0.716187
$334$	0	0
$335$	−1.24860	−0.0682182
$336$	0	0
$337$	−26.3985	−1.43802	−0.719010	−	0.695000i	$-0.755404\pi$
$337$	−26.3985	−1.43802	−0.719010	+	0.695000i	$0.755404\pi$
$338$	0	0
$339$	22.3334	1.21298
$340$	0	0
$341$	−4.74041	−0.256707
$342$	0	0
$343$	−14.0863	−0.760591
$344$	0	0
$345$	−19.4347	−1.04633
$346$	0	0
$347$	19.0057	1.02028	0.510141	−	0.860091i	$-0.329593\pi$
$347$	19.0057	1.02028	0.510141	+	0.860091i	$0.329593\pi$
$348$	0	0
$349$	−7.13098	−0.381712	−0.190856	−	0.981618i	$-0.561126\pi$
$349$	−7.13098	−0.381712	−0.190856	+	0.981618i	$0.561126\pi$
$350$	0	0
$351$	−10.5597	−0.563636
$352$	0	0
$353$	27.6101	1.46954	0.734769	−	0.678317i	$-0.237290\pi$
$353$	27.6101	1.46954	0.734769	+	0.678317i	$0.237290\pi$
$354$	0	0
$355$	14.5473	0.772091
$356$	0	0
$357$	22.9899	1.21675
$358$	0	0
$359$	8.08496	0.426708	0.213354	−	0.976975i	$-0.431561\pi$
$359$	8.08496	0.426708	0.213354	+	0.976975i	$0.431561\pi$
$360$	0	0
$361$	−10.9483	−0.576224
$362$	0	0
$363$	−18.6430	−0.978502
$364$	0	0
$365$	5.49692	0.287722
$366$	0	0
$367$	−10.0266	−0.523387	−0.261693	−	0.965151i	$-0.584281\pi$
$367$	−10.0266	−0.523387	−0.261693	+	0.965151i	$0.584281\pi$
$368$	0	0
$369$	6.68665	0.348093
$370$	0	0
$371$	4.33925	0.225282
$372$	0	0
$373$	−10.1214	−0.524066	−0.262033	−	0.965059i	$-0.584393\pi$
$373$	−10.1214	−0.524066	−0.262033	+	0.965059i	$0.584393\pi$
$374$	0	0
$375$	−2.24403	−0.115881
$376$	0	0
$377$	−18.5623	−0.956007
$378$	0	0
$379$	−0.229974	−0.0118130	−0.00590648	−	0.999983i	$-0.501880\pi$
$379$	−0.229974	−0.0118130	−0.00590648	+	0.999983i	$0.501880\pi$
$380$	0	0
$381$	−31.5057	−1.61408
$382$	0	0
$383$	23.7878	1.21550	0.607750	−	0.794128i	$-0.292072\pi$
$383$	23.7878	1.21550	0.607750	+	0.794128i	$0.292072\pi$
$384$	0	0
$385$	−5.03252	−0.256481
$386$	0	0
$387$	−7.41753	−0.377054
$388$	0	0
$389$	11.8771	0.602191	0.301096	−	0.953594i	$-0.402648\pi$
$389$	11.8771	0.602191	0.301096	+	0.953594i	$0.402648\pi$
$390$	0	0
$391$	28.9285	1.46298
$392$	0	0
$393$	29.8976	1.50814
$394$	0	0
$395$	4.55114	0.228993
$396$	0	0
$397$	11.5074	0.577539	0.288769	−	0.957399i	$-0.406754\pi$
$397$	11.5074	0.577539	0.288769	+	0.957399i	$0.406754\pi$
$398$	0	0
$399$	19.5302	0.977731
$400$	0	0
$401$	−4.08731	−0.204110	−0.102055	−	0.994779i	$-0.532542\pi$
$401$	−4.08731	−0.204110	−0.102055	+	0.994779i	$0.532542\pi$
$402$	0	0
$403$	−14.0979	−0.702265
$404$	0	0
$405$	10.9631	0.544760
$406$	0	0
$407$	−10.5341	−0.522156
$408$	0	0
$409$	−39.7168	−1.96387	−0.981935	−	0.189216i	$-0.939405\pi$
$409$	−39.7168	−1.96387	−0.981935	+	0.189216i	$0.939405\pi$
$410$	0	0
$411$	−13.1528	−0.648782
$412$	0	0
$413$	16.2576	0.799985
$414$	0	0
$415$	5.47254	0.268637
$416$	0	0
$417$	16.1721	0.791949
$418$	0	0
$419$	16.3797	0.800199	0.400099	−	0.916472i	$-0.368976\pi$
$419$	16.3797	0.800199	0.400099	+	0.916472i	$0.368976\pi$
$420$	0	0
$421$	1.36103	0.0663325	0.0331663	−	0.999450i	$-0.489441\pi$
$421$	1.36103	0.0663325	0.0331663	+	0.999450i	$0.489441\pi$
$422$	0	0
$423$	1.53158	0.0744679
$424$	0	0
$425$	3.34022	0.162025
$426$	0	0
$427$	−1.39360	−0.0674411
$428$	0	0
$429$	17.9668	0.867443
$430$	0	0
$431$	3.25649	0.156859	0.0784297	−	0.996920i	$-0.475009\pi$
$431$	3.25649	0.156859	0.0784297	+	0.996920i	$0.475009\pi$
$432$	0	0
$433$	8.84720	0.425169	0.212585	−	0.977143i	$-0.431812\pi$
$433$	8.84720	0.425169	0.212585	+	0.977143i	$0.431812\pi$
$434$	0	0
$435$	8.53630	0.409284
$436$	0	0
$437$	24.5751	1.17559
$438$	0	0
$439$	1.56726	0.0748011	0.0374005	−	0.999300i	$-0.488092\pi$
$439$	1.56726	0.0748011	0.0374005	+	0.999300i	$0.488092\pi$
$440$	0	0
$441$	4.90049	0.233357
$442$	0	0
$443$	−34.6750	−1.64746	−0.823730	−	0.566982i	$-0.808111\pi$
$443$	−34.6750	−1.64746	−0.823730	+	0.566982i	$0.808111\pi$
$444$	0	0
$445$	7.79092	0.369325
$446$	0	0
$447$	48.4967	2.29381
$448$	0	0
$449$	4.31292	0.203539	0.101770	−	0.994808i	$-0.467550\pi$
$449$	4.31292	0.203539	0.101770	+	0.994808i	$0.467550\pi$
$450$	0	0
$451$	5.38961	0.253787
$452$	0	0
$453$	2.24403	0.105434
$454$	0	0
$455$	−14.9666	−0.701645
$456$	0	0
$457$	−39.6528	−1.85488	−0.927441	−	0.373971i	$-0.877996\pi$
$457$	−39.6528	−1.85488	−0.927441	+	0.373971i	$0.877996\pi$
$458$	0	0
$459$	−7.22832	−0.337389
$460$	0	0
$461$	10.1292	0.471763	0.235882	−	0.971782i	$-0.424202\pi$
$461$	10.1292	0.471763	0.235882	+	0.971782i	$0.424202\pi$
$462$	0	0
$463$	11.3689	0.528356	0.264178	−	0.964474i	$-0.414899\pi$
$463$	11.3689	0.528356	0.264178	+	0.964474i	$0.414899\pi$
$464$	0	0
$465$	6.48323	0.300653
$466$	0	0
$467$	−23.8500	−1.10365	−0.551824	−	0.833961i	$-0.686068\pi$
$467$	−23.8500	−1.10365	−0.551824	+	0.833961i	$0.686068\pi$
$468$	0	0
$469$	3.82962	0.176836
$470$	0	0
$471$	28.3947	1.30836
$472$	0	0
$473$	−5.97872	−0.274902
$474$	0	0
$475$	2.83756	0.130196
$476$	0	0
$477$	2.87995	0.131864
$478$	0	0
$479$	36.5586	1.67040	0.835202	−	0.549944i	$-0.185351\pi$
$479$	36.5586	1.67040	0.835202	+	0.549944i	$0.185351\pi$
$480$	0	0
$481$	−31.3282	−1.42844
$482$	0	0
$483$	59.6089	2.71230
$484$	0	0
$485$	−4.71652	−0.214166
$486$	0	0
$487$	19.3569	0.877146	0.438573	−	0.898696i	$-0.355484\pi$
$487$	19.3569	0.877146	0.438573	+	0.898696i	$0.355484\pi$
$488$	0	0
$489$	−38.8601	−1.75732
$490$	0	0
$491$	−7.87637	−0.355455	−0.177728	−	0.984080i	$-0.556875\pi$
$491$	−7.87637	−0.355455	−0.177728	+	0.984080i	$0.556875\pi$
$492$	0	0
$493$	−12.7062	−0.572261
$494$	0	0
$495$	−3.34007	−0.150125
$496$	0	0
$497$	−44.6186	−2.00142
$498$	0	0
$499$	8.34329	0.373497	0.186748	−	0.982408i	$-0.440205\pi$
$499$	8.34329	0.373497	0.186748	+	0.982408i	$0.440205\pi$
$500$	0	0
$501$	−35.0156	−1.56438
$502$	0	0
$503$	−14.0768	−0.627653	−0.313826	−	0.949480i	$-0.601611\pi$
$503$	−14.0768	−0.627653	−0.313826	+	0.949480i	$0.601611\pi$
$504$	0	0
$505$	−12.0737	−0.537274
$506$	0	0
$507$	24.2605	1.07745
$508$	0	0
$509$	−10.7138	−0.474882	−0.237441	−	0.971402i	$-0.576309\pi$
$509$	−10.7138	−0.474882	−0.237441	+	0.971402i	$0.576309\pi$
$510$	0	0
$511$	−16.8598	−0.745834
$512$	0	0
$513$	−6.14054	−0.271112
$514$	0	0
$515$	−2.66466	−0.117419
$516$	0	0
$517$	1.23449	0.0542928
$518$	0	0
$519$	−6.64376	−0.291629
$520$	0	0
$521$	−36.2935	−1.59005	−0.795023	−	0.606579i	$-0.792541\pi$
$521$	−36.2935	−1.59005	−0.795023	+	0.606579i	$0.792541\pi$
$522$	0	0
$523$	−19.2087	−0.839937	−0.419968	−	0.907539i	$-0.637959\pi$
$523$	−19.2087	−0.839937	−0.419968	+	0.907539i	$0.637959\pi$
$524$	0	0
$525$	6.88274	0.300387
$526$	0	0
$527$	−9.65027	−0.420372
$528$	0	0
$529$	52.0068	2.26116
$530$	0	0
$531$	10.7901	0.468252
$532$	0	0
$533$	16.0286	0.694275
$534$	0	0
$535$	12.7395	0.550776
$536$	0	0
$537$	2.98436	0.128784
$538$	0	0
$539$	3.94992	0.170135
$540$	0	0
$541$	−43.3655	−1.86443	−0.932214	−	0.361907i	$-0.882126\pi$
$541$	−43.3655	−1.86443	−0.932214	+	0.361907i	$0.882126\pi$
$542$	0	0
$543$	−19.1327	−0.821063
$544$	0	0
$545$	−12.9096	−0.552987
$546$	0	0
$547$	9.63815	0.412097	0.206049	−	0.978542i	$-0.433939\pi$
$547$	9.63815	0.412097	0.206049	+	0.978542i	$0.433939\pi$
$548$	0	0
$549$	−0.924930	−0.0394751
$550$	0	0
$551$	−10.7941	−0.459844
$552$	0	0
$553$	−13.9590	−0.593596
$554$	0	0
$555$	14.4070	0.611542
$556$	0	0
$557$	−11.0277	−0.467259	−0.233629	−	0.972326i	$-0.575060\pi$
$557$	−11.0277	−0.467259	−0.233629	+	0.972326i	$0.575060\pi$
$558$	0	0
$559$	−17.7806	−0.752039
$560$	0	0
$561$	12.2986	0.519247
$562$	0	0
$563$	13.9596	0.588327	0.294163	−	0.955755i	$-0.404959\pi$
$563$	13.9596	0.588327	0.294163	+	0.955755i	$0.404959\pi$
$564$	0	0
$565$	−9.95239	−0.418700
$566$	0	0
$567$	−33.6253	−1.41213
$568$	0	0
$569$	−0.883735	−0.0370481	−0.0185240	−	0.999828i	$-0.505897\pi$
$569$	−0.883735	−0.0370481	−0.0185240	+	0.999828i	$0.505897\pi$
$570$	0	0
$571$	−8.08749	−0.338451	−0.169226	−	0.985577i	$-0.554127\pi$
$571$	−8.08749	−0.338451	−0.169226	+	0.985577i	$0.554127\pi$
$572$	0	0
$573$	8.79748	0.367520
$574$	0	0
$575$	8.66064	0.361174
$576$	0	0
$577$	−27.7491	−1.15521	−0.577604	−	0.816317i	$-0.696012\pi$
$577$	−27.7491	−1.15521	−0.577604	+	0.816317i	$0.696012\pi$
$578$	0	0
$579$	23.1827	0.963439
$580$	0	0
$581$	−16.7850	−0.696361
$582$	0	0
$583$	2.32131	0.0961388
$584$	0	0
$585$	−9.93330	−0.410691
$586$	0	0
$587$	−24.1773	−0.997902	−0.498951	−	0.866630i	$-0.666281\pi$
$587$	−24.1773	−0.997902	−0.498951	+	0.866630i	$0.666281\pi$
$588$	0	0
$589$	−8.19801	−0.337793
$590$	0	0
$591$	36.5051	1.50162
$592$	0	0
$593$	−32.0484	−1.31607	−0.658034	−	0.752988i	$-0.728612\pi$
$593$	−32.0484	−1.31607	−0.658034	+	0.752988i	$0.728612\pi$
$594$	0	0
$595$	−10.2449	−0.420001
$596$	0	0
$597$	33.1304	1.35594
$598$	0	0
$599$	6.01048	0.245582	0.122791	−	0.992433i	$-0.460816\pi$
$599$	6.01048	0.245582	0.122791	+	0.992433i	$0.460816\pi$
$600$	0	0
$601$	−18.0523	−0.736367	−0.368184	−	0.929753i	$-0.620020\pi$
$601$	−18.0523	−0.736367	−0.368184	+	0.929753i	$0.620020\pi$
$602$	0	0
$603$	2.54171	0.103506
$604$	0	0
$605$	8.30782	0.337761
$606$	0	0
$607$	4.45170	0.180689	0.0903445	−	0.995911i	$-0.471203\pi$
$607$	4.45170	0.180689	0.0903445	+	0.995911i	$0.471203\pi$
$608$	0	0
$609$	−26.1820	−1.06095
$610$	0	0
$611$	3.67135	0.148527
$612$	0	0
$613$	−24.0290	−0.970524	−0.485262	−	0.874369i	$-0.661276\pi$
$613$	−24.0290	−0.970524	−0.485262	+	0.874369i	$0.661276\pi$
$614$	0	0
$615$	−7.37111	−0.297232
$616$	0	0
$617$	−42.3091	−1.70330	−0.851651	−	0.524110i	$-0.824398\pi$
$617$	−42.3091	−1.70330	−0.851651	+	0.524110i	$0.824398\pi$
$618$	0	0
$619$	27.4328	1.10262	0.551309	−	0.834301i	$-0.314129\pi$
$619$	27.4328	1.10262	0.551309	+	0.834301i	$0.314129\pi$
$620$	0	0
$621$	−18.7418	−0.752084
$622$	0	0
$623$	−23.8958	−0.957366
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	10.4478	0.417245
$628$	0	0
$629$	−21.4447	−0.855058
$630$	0	0
$631$	−6.85553	−0.272914	−0.136457	−	0.990646i	$-0.543572\pi$
$631$	−6.85553	−0.272914	−0.136457	+	0.990646i	$0.543572\pi$
$632$	0	0
$633$	−7.87552	−0.313024
$634$	0	0
$635$	14.0398	0.557153
$636$	0	0
$637$	11.7470	0.465432
$638$	0	0
$639$	−29.6133	−1.17148
$640$	0	0
$641$	0.656138	0.0259159	0.0129580	−	0.999916i	$-0.495875\pi$
$641$	0.656138	0.0259159	0.0129580	+	0.999916i	$0.495875\pi$
$642$	0	0
$643$	−39.7733	−1.56851	−0.784253	−	0.620441i	$-0.786954\pi$
$643$	−39.7733	−1.56851	−0.784253	+	0.620441i	$0.786954\pi$
$644$	0	0
$645$	8.17681	0.321962
$646$	0	0
$647$	−42.6018	−1.67485	−0.837426	−	0.546551i	$-0.815940\pi$
$647$	−42.6018	−1.67485	−0.837426	+	0.546551i	$0.815940\pi$
$648$	0	0
$649$	8.69712	0.341392
$650$	0	0
$651$	−19.8850	−0.779353
$652$	0	0
$653$	40.7612	1.59511	0.797554	−	0.603247i	$-0.206127\pi$
$653$	40.7612	1.59511	0.797554	+	0.603247i	$0.206127\pi$
$654$	0	0
$655$	−13.3232	−0.520581
$656$	0	0
$657$	−11.1898	−0.436556
$658$	0	0
$659$	5.14229	0.200315	0.100158	−	0.994972i	$-0.468065\pi$
$659$	5.14229	0.200315	0.100158	+	0.994972i	$0.468065\pi$
$660$	0	0
$661$	5.60965	0.218190	0.109095	−	0.994031i	$-0.465205\pi$
$661$	5.60965	0.218190	0.109095	+	0.994031i	$0.465205\pi$
$662$	0	0
$663$	36.5758	1.42049
$664$	0	0
$665$	−8.70318	−0.337495
$666$	0	0
$667$	−32.9452	−1.27564
$668$	0	0
$669$	44.2178	1.70956
$670$	0	0
$671$	−0.745517	−0.0287804
$672$	0	0
$673$	3.88120	0.149609	0.0748046	−	0.997198i	$-0.476167\pi$
$673$	3.88120	0.149609	0.0748046	+	0.997198i	$0.476167\pi$
$674$	0	0
$675$	−2.16402	−0.0832933
$676$	0	0
$677$	23.4932	0.902919	0.451459	−	0.892292i	$-0.350904\pi$
$677$	23.4932	0.902919	0.451459	+	0.892292i	$0.350904\pi$
$678$	0	0
$679$	14.4662	0.555163
$680$	0	0
$681$	14.3203	0.548754
$682$	0	0
$683$	40.8344	1.56249	0.781243	−	0.624228i	$-0.214586\pi$
$683$	40.8344	1.56249	0.781243	+	0.624228i	$0.214586\pi$
$684$	0	0
$685$	5.86127	0.223948
$686$	0	0
$687$	−23.6918	−0.903900
$688$	0	0
$689$	6.90353	0.263004
$690$	0	0
$691$	−7.15895	−0.272339	−0.136170	−	0.990686i	$-0.543479\pi$
$691$	−7.15895	−0.272339	−0.136170	+	0.990686i	$0.543479\pi$
$692$	0	0
$693$	10.2444	0.389154
$694$	0	0
$695$	−7.20672	−0.273366
$696$	0	0
$697$	10.9719	0.415589
$698$	0	0
$699$	−54.4611	−2.05991
$700$	0	0
$701$	−6.35669	−0.240089	−0.120044	−	0.992769i	$-0.538304\pi$
$701$	−6.35669	−0.240089	−0.120044	+	0.992769i	$0.538304\pi$
$702$	0	0
$703$	−18.2176	−0.687088
$704$	0	0
$705$	−1.68835	−0.0635871
$706$	0	0
$707$	37.0318	1.39272
$708$	0	0
$709$	25.5682	0.960234	0.480117	−	0.877204i	$-0.340594\pi$
$709$	25.5682	0.960234	0.480117	+	0.877204i	$0.340594\pi$
$710$	0	0
$711$	−9.26454	−0.347447
$712$	0	0
$713$	−25.0215	−0.937064
$714$	0	0
$715$	−8.00649	−0.299426
$716$	0	0
$717$	−11.6850	−0.436386
$718$	0	0
$719$	14.1275	0.526868	0.263434	−	0.964677i	$-0.415145\pi$
$719$	14.1275	0.526868	0.263434	+	0.964677i	$0.415145\pi$
$720$	0	0
$721$	8.17289	0.304374
$722$	0	0
$723$	3.34087	0.124248
$724$	0	0
$725$	−3.80401	−0.141277
$726$	0	0
$727$	−4.14414	−0.153698	−0.0768488	−	0.997043i	$-0.524486\pi$
$727$	−4.14414	−0.153698	−0.0768488	+	0.997043i	$0.524486\pi$
$728$	0	0
$729$	−7.74863	−0.286986
$730$	0	0
$731$	−12.1712	−0.450166
$732$	0	0
$733$	−32.7266	−1.20879	−0.604393	−	0.796686i	$-0.706584\pi$
$733$	−32.7266	−1.20879	−0.604393	+	0.796686i	$0.706584\pi$
$734$	0	0
$735$	−5.40212	−0.199260
$736$	0	0
$737$	2.04868	0.0754642
$738$	0	0
$739$	38.3372	1.41026	0.705129	−	0.709079i	$-0.250889\pi$
$739$	38.3372	1.41026	0.705129	+	0.709079i	$0.250889\pi$
$740$	0	0
$741$	31.0715	1.14144
$742$	0	0
$743$	2.30637	0.0846125	0.0423062	−	0.999105i	$-0.486529\pi$
$743$	2.30637	0.0846125	0.0423062	+	0.999105i	$0.486529\pi$
$744$	0	0
$745$	−21.6115	−0.791783
$746$	0	0
$747$	−11.1402	−0.407598
$748$	0	0
$749$	−39.0738	−1.42772
$750$	0	0
$751$	−19.0473	−0.695045	−0.347522	−	0.937672i	$-0.612977\pi$
$751$	−19.0473	−0.695045	−0.347522	+	0.937672i	$0.612977\pi$
$752$	0	0
$753$	−8.07746	−0.294359
$754$	0	0
$755$	−1.00000	−0.0363937
$756$	0	0
$757$	35.8151	1.30172	0.650861	−	0.759197i	$-0.274408\pi$
$757$	35.8151	1.30172	0.650861	+	0.759197i	$0.274408\pi$
$758$	0	0
$759$	31.8882	1.15747
$760$	0	0
$761$	−14.8422	−0.538028	−0.269014	−	0.963136i	$-0.586698\pi$
$761$	−14.8422	−0.538028	−0.269014	+	0.963136i	$0.586698\pi$
$762$	0	0
$763$	39.5956	1.43346
$764$	0	0
$765$	−6.79953	−0.245838
$766$	0	0
$767$	25.8651	0.933934
$768$	0	0
$769$	48.0867	1.73405	0.867025	−	0.498264i	$-0.166029\pi$
$769$	48.0867	1.73405	0.867025	+	0.498264i	$0.166029\pi$
$770$	0	0
$771$	42.4560	1.52901
$772$	0	0
$773$	42.4906	1.52828	0.764141	−	0.645050i	$-0.223163\pi$
$773$	42.4906	1.52828	0.764141	+	0.645050i	$0.223163\pi$
$774$	0	0
$775$	−2.88911	−0.103780
$776$	0	0
$777$	−44.1882	−1.58524
$778$	0	0
$779$	9.32073	0.333950
$780$	0	0
$781$	−23.8690	−0.854101
$782$	0	0
$783$	8.23197	0.294186
$784$	0	0
$785$	−12.6535	−0.451621
$786$	0	0
$787$	−47.9982	−1.71095	−0.855475	−	0.517844i	$-0.826735\pi$
$787$	−47.9982	−1.71095	−0.855475	+	0.517844i	$0.826735\pi$
$788$	0	0
$789$	−25.4595	−0.906383
$790$	0	0
$791$	30.5254	1.08536
$792$	0	0
$793$	−2.21715	−0.0787334
$794$	0	0
$795$	−3.17475	−0.112597
$796$	0	0
$797$	7.41273	0.262572	0.131286	−	0.991345i	$-0.458089\pi$
$797$	7.41273	0.262572	0.131286	+	0.991345i	$0.458089\pi$
$798$	0	0
$799$	2.51311	0.0889074
$800$	0	0
$801$	−15.8596	−0.560371
$802$	0	0
$803$	−9.01926	−0.318283
$804$	0	0
$805$	−26.5634	−0.936236
$806$	0	0
$807$	−16.8557	−0.593348
$808$	0	0
$809$	−41.0886	−1.44460	−0.722298	−	0.691582i	$-0.756914\pi$
$809$	−41.0886	−1.44460	−0.722298	+	0.691582i	$0.756914\pi$
$810$	0	0
$811$	−14.1499	−0.496868	−0.248434	−	0.968649i	$-0.579916\pi$
$811$	−14.1499	−0.496868	−0.248434	+	0.968649i	$0.579916\pi$
$812$	0	0
$813$	8.84712	0.310282
$814$	0	0
$815$	17.3172	0.606593
$816$	0	0
$817$	−10.3395	−0.361734
$818$	0	0
$819$	30.4668	1.06460
$820$	0	0
$821$	28.0909	0.980379	0.490190	−	0.871616i	$-0.336928\pi$
$821$	28.0909	0.980379	0.490190	+	0.871616i	$0.336928\pi$
$822$	0	0
$823$	9.99994	0.348576	0.174288	−	0.984695i	$-0.444238\pi$
$823$	9.99994	0.348576	0.174288	+	0.984695i	$0.444238\pi$
$824$	0	0
$825$	3.68197	0.128190
$826$	0	0
$827$	42.7739	1.48739	0.743697	−	0.668517i	$-0.233071\pi$
$827$	42.7739	1.48739	0.743697	+	0.668517i	$0.233071\pi$
$828$	0	0
$829$	49.6921	1.72588	0.862939	−	0.505308i	$-0.168621\pi$
$829$	49.6921	1.72588	0.862939	+	0.505308i	$0.168621\pi$
$830$	0	0
$831$	−17.7441	−0.615537
$832$	0	0
$833$	8.04103	0.278605
$834$	0	0
$835$	15.6039	0.539996
$836$	0	0
$837$	6.25210	0.216104
$838$	0	0
$839$	41.1290	1.41993	0.709965	−	0.704237i	$-0.248711\pi$
$839$	41.1290	1.41993	0.709965	+	0.704237i	$0.248711\pi$
$840$	0	0
$841$	−14.5295	−0.501018
$842$	0	0
$843$	51.6172	1.77779
$844$	0	0
$845$	−10.8111	−0.371914
$846$	0	0
$847$	−25.4812	−0.875546
$848$	0	0
$849$	9.79500	0.336164
$850$	0	0
$851$	−55.6027	−1.90603
$852$	0	0
$853$	−20.3226	−0.695834	−0.347917	−	0.937525i	$-0.613111\pi$
$853$	−20.3226	−0.695834	−0.347917	+	0.937525i	$0.613111\pi$
$854$	0	0
$855$	−5.77628	−0.197545
$856$	0	0
$857$	21.1567	0.722700	0.361350	−	0.932430i	$-0.382316\pi$
$857$	21.1567	0.722700	0.361350	+	0.932430i	$0.382316\pi$
$858$	0	0
$859$	30.2455	1.03196	0.515982	−	0.856599i	$-0.327427\pi$
$859$	30.2455	1.03196	0.515982	+	0.856599i	$0.327427\pi$
$860$	0	0
$861$	22.6082	0.770486
$862$	0	0
$863$	21.2448	0.723183	0.361591	−	0.932337i	$-0.382234\pi$
$863$	21.2448	0.723183	0.361591	+	0.932337i	$0.382234\pi$
$864$	0	0
$865$	2.96064	0.100665
$866$	0	0
$867$	−13.1116	−0.445294
$868$	0	0
$869$	−7.46745	−0.253316
$870$	0	0
$871$	6.09274	0.206445
$872$	0	0
$873$	9.60120	0.324951
$874$	0	0
$875$	−3.06714	−0.103688
$876$	0	0
$877$	26.0298	0.878963	0.439481	−	0.898252i	$-0.355162\pi$
$877$	26.0298	0.878963	0.439481	+	0.898252i	$0.355162\pi$
$878$	0	0
$879$	32.0264	1.08022
$880$	0	0
$881$	−55.6674	−1.87548	−0.937740	−	0.347337i	$-0.887086\pi$
$881$	−55.6674	−1.87548	−0.937740	+	0.347337i	$0.887086\pi$
$882$	0	0
$883$	−35.1135	−1.18166	−0.590832	−	0.806794i	$-0.701201\pi$
$883$	−35.1135	−1.18166	−0.590832	+	0.806794i	$0.701201\pi$
$884$	0	0
$885$	−11.8946	−0.399834
$886$	0	0
$887$	34.5975	1.16167	0.580835	−	0.814022i	$-0.302726\pi$
$887$	34.5975	1.16167	0.580835	+	0.814022i	$0.302726\pi$
$888$	0	0
$889$	−43.0620	−1.44425
$890$	0	0
$891$	−17.9881	−0.602623
$892$	0	0
$893$	2.13491	0.0714422
$894$	0	0
$895$	−1.32991	−0.0444540
$896$	0	0
$897$	94.8349	3.16644
$898$	0	0
$899$	10.9902	0.366543
$900$	0	0
$901$	4.72560	0.157433
$902$	0	0
$903$	−25.0794	−0.834590
$904$	0	0
$905$	8.52607	0.283416
$906$	0	0
$907$	36.7016	1.21866	0.609329	−	0.792917i	$-0.291439\pi$
$907$	36.7016	1.21866	0.609329	+	0.792917i	$0.291439\pi$
$908$	0	0
$909$	24.5779	0.815198
$910$	0	0
$911$	1.70249	0.0564061	0.0282031	−	0.999602i	$-0.491021\pi$
$911$	1.70249	0.0564061	0.0282031	+	0.999602i	$0.491021\pi$
$912$	0	0
$913$	−8.97927	−0.297170
$914$	0	0
$915$	1.01961	0.0337072
$916$	0	0
$917$	40.8641	1.34945
$918$	0	0
$919$	−21.0944	−0.695842	−0.347921	−	0.937524i	$-0.613112\pi$
$919$	−21.0944	−0.695842	−0.347921	+	0.937524i	$0.613112\pi$
$920$	0	0
$921$	−28.3070	−0.932747
$922$	0	0
$923$	−70.9860	−2.33653
$924$	0	0
$925$	−6.42015	−0.211093
$926$	0	0
$927$	5.42432	0.178158
$928$	0	0
$929$	16.7461	0.549421	0.274710	−	0.961527i	$-0.411418\pi$
$929$	16.7461	0.549421	0.274710	+	0.961527i	$0.411418\pi$
$930$	0	0
$931$	6.83095	0.223875
$932$	0	0
$933$	31.6310	1.03555
$934$	0	0
$935$	−5.48059	−0.179235
$936$	0	0
$937$	38.5595	1.25968	0.629842	−	0.776723i	$-0.283119\pi$
$937$	38.5595	1.25968	0.629842	+	0.776723i	$0.283119\pi$
$938$	0	0
$939$	−15.6436	−0.510508
$940$	0	0
$941$	−14.6165	−0.476483	−0.238241	−	0.971206i	$-0.576571\pi$
$941$	−14.6165	−0.476483	−0.238241	+	0.971206i	$0.576571\pi$
$942$	0	0
$943$	28.4482	0.926402
$944$	0	0
$945$	6.63736	0.215913
$946$	0	0
$947$	−21.5372	−0.699865	−0.349932	−	0.936775i	$-0.613795\pi$
$947$	−21.5372	−0.699865	−0.349932	+	0.936775i	$0.613795\pi$
$948$	0	0
$949$	−26.8231	−0.870715
$950$	0	0
$951$	−34.9277	−1.13261
$952$	0	0
$953$	−2.04608	−0.0662791	−0.0331395	−	0.999451i	$-0.510551\pi$
$953$	−2.04608	−0.0662791	−0.0331395	+	0.999451i	$0.510551\pi$
$954$	0	0
$955$	−3.92040	−0.126861
$956$	0	0
$957$	−14.0062	−0.452757
$958$	0	0
$959$	−17.9773	−0.580518
$960$	0	0
$961$	−22.6531	−0.730744
$962$	0	0
$963$	−25.9332	−0.835684
$964$	0	0
$965$	−10.3308	−0.332562
$966$	0	0
$967$	−2.07088	−0.0665950	−0.0332975	−	0.999445i	$-0.510601\pi$
$967$	−2.07088	−0.0665950	−0.0332975	+	0.999445i	$0.510601\pi$
$968$	0	0
$969$	21.2691	0.683261
$970$	0	0
$971$	−40.0916	−1.28660	−0.643301	−	0.765614i	$-0.722435\pi$
$971$	−40.0916	−1.28660	−0.643301	+	0.765614i	$0.722435\pi$
$972$	0	0
$973$	22.1040	0.708622
$974$	0	0
$975$	10.9501	0.350684
$976$	0	0
$977$	5.24815	0.167903	0.0839517	−	0.996470i	$-0.473246\pi$
$977$	5.24815	0.167903	0.0839517	+	0.996470i	$0.473246\pi$
$978$	0	0
$979$	−12.7832	−0.408554
$980$	0	0
$981$	26.2795	0.839039
$982$	0	0
$983$	21.5523	0.687411	0.343706	−	0.939077i	$-0.388318\pi$
$983$	21.5523	0.687411	0.343706	+	0.939077i	$0.388318\pi$
$984$	0	0
$985$	−16.2677	−0.518332
$986$	0	0
$987$	5.17841	0.164831
$988$	0	0
$989$	−31.5578	−1.00348
$990$	0	0
$991$	−39.4467	−1.25306	−0.626532	−	0.779395i	$-0.715526\pi$
$991$	−39.4467	−1.25306	−0.626532	+	0.779395i	$0.715526\pi$
$992$	0	0
$993$	−16.3211	−0.517934
$994$	0	0
$995$	−14.7638	−0.468045
$996$	0	0
$997$	7.20203	0.228090	0.114045	−	0.993476i	$-0.463619\pi$
$997$	7.20203	0.228090	0.114045	+	0.993476i	$0.463619\pi$
$998$	0	0
$999$	13.8934	0.439566

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6040.2.a.m.1.10	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6040.2.a.m.1.10	✓	12	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)

\(f(q)\)	\(=\)	\(q+2.24403 q^{3} -1.00000 q^{5} +3.06714 q^{7} +2.03565 q^{9} +O(q^{10})\)	\(q+2.24403 q^{3} -1.00000 q^{5} +3.06714 q^{7} +2.03565 q^{9} +1.64079 q^{11} +4.87967 q^{13} -2.24403 q^{15} +3.34022 q^{17} +2.83756 q^{19} +6.88274 q^{21} +8.66064 q^{23} +1.00000 q^{25} -2.16402 q^{27} -3.80401 q^{29} -2.88911 q^{31} +3.68197 q^{33} -3.06714 q^{35} -6.42015 q^{37} +10.9501 q^{39} +3.28477 q^{41} -3.64381 q^{43} -2.03565 q^{45} +0.752377 q^{47} +2.40733 q^{49} +7.49555 q^{51} +1.41475 q^{53} -1.64079 q^{55} +6.36755 q^{57} +5.30059 q^{59} -0.454366 q^{61} +6.24362 q^{63} -4.87967 q^{65} +1.24860 q^{67} +19.4347 q^{69} -14.5473 q^{71} -5.49692 q^{73} +2.24403 q^{75} +5.03252 q^{77} -4.55114 q^{79} -10.9631 q^{81} -5.47254 q^{83} -3.34022 q^{85} -8.53630 q^{87} -7.79092 q^{89} +14.9666 q^{91} -6.48323 q^{93} -2.83756 q^{95} +4.71652 q^{97} +3.34007 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 3 q^{3} - 12 q^{5} + 5 q^{7} + 9 q^{9}+O(q^{10}) \) 12 * q + 3 * q^3 - 12 * q^5 + 5 * q^7 + 9 * q^9	\( 12 q + 3 q^{3} - 12 q^{5} + 5 q^{7} + 9 q^{9} + 10 q^{11} + 11 q^{13} - 3 q^{15} - 4 q^{17} + 5 q^{19} - q^{21} + 18 q^{23} + 12 q^{25} + 9 q^{27} + 16 q^{29} - q^{31} + 8 q^{33} - 5 q^{35} + 2 q^{37} + 6 q^{39} + 4 q^{41} + 7 q^{43} - 9 q^{45} + 3 q^{49} - 4 q^{51} + 39 q^{53} - 10 q^{55} - 15 q^{57} - 4 q^{59} - 32 q^{61} + 3 q^{63} - 11 q^{65} + 4 q^{67} + 12 q^{69} + 24 q^{71} - 10 q^{73} + 3 q^{75} + 38 q^{77} + 32 q^{79} - 8 q^{81} + 9 q^{83} + 4 q^{85} + 3 q^{87} + 15 q^{89} + 18 q^{91} + 36 q^{93} - 5 q^{95} + 15 q^{97} + 28 q^{99}+O(q^{100}) \) 12 * q + 3 * q^3 - 12 * q^5 + 5 * q^7 + 9 * q^9 + 10 * q^11 + 11 * q^13 - 3 * q^15 - 4 * q^17 + 5 * q^19 - q^21 + 18 * q^23 + 12 * q^25 + 9 * q^27 + 16 * q^29 - q^31 + 8 * q^33 - 5 * q^35 + 2 * q^37 + 6 * q^39 + 4 * q^41 + 7 * q^43 - 9 * q^45 + 3 * q^49 - 4 * q^51 + 39 * q^53 - 10 * q^55 - 15 * q^57 - 4 * q^59 - 32 * q^61 + 3 * q^63 - 11 * q^65 + 4 * q^67 + 12 * q^69 + 24 * q^71 - 10 * q^73 + 3 * q^75 + 38 * q^77 + 32 * q^79 - 8 * q^81 + 9 * q^83 + 4 * q^85 + 3 * q^87 + 15 * q^89 + 18 * q^91 + 36 * q^93 - 5 * q^95 + 15 * q^97 + 28 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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