LMFDB - Embedded newform 6040.2.a.l.1.3

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			1.3
Root			$-1.10984$ of defining polynomial
Character	$\chi$	$=$	6040.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.57672 q^{3} +1.00000 q^{5} -0.208807 q^{7} -0.513952 q^{9} +O(q^{10})$	$q-1.57672 q^{3} +1.00000 q^{5} -0.208807 q^{7} -0.513952 q^{9} -0.616299 q^{11} -3.24227 q^{13} -1.57672 q^{15} +1.94776 q^{17} +1.81681 q^{19} +0.329230 q^{21} -1.42747 q^{23} +1.00000 q^{25} +5.54052 q^{27} -4.26518 q^{29} +6.21450 q^{31} +0.971732 q^{33} -0.208807 q^{35} +4.31510 q^{37} +5.11215 q^{39} +1.53452 q^{41} -1.84534 q^{43} -0.513952 q^{45} -4.09784 q^{47} -6.95640 q^{49} -3.07107 q^{51} +10.8297 q^{53} -0.616299 q^{55} -2.86460 q^{57} +8.06057 q^{59} -11.1832 q^{61} +0.107317 q^{63} -3.24227 q^{65} +5.24700 q^{67} +2.25071 q^{69} +8.10268 q^{71} -1.11838 q^{73} -1.57672 q^{75} +0.128687 q^{77} -7.47039 q^{79} -7.19400 q^{81} -13.6712 q^{83} +1.94776 q^{85} +6.72499 q^{87} -13.1975 q^{89} +0.677008 q^{91} -9.79853 q^{93} +1.81681 q^{95} +9.40168 q^{97} +0.316748 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q + 9 q^{5} - 2 q^{7} - 3 q^{9}+O(q^{10}) $ 9 * q + 9 * q^5 - 2 * q^7 - 3 * q^9	$ 9 q + 9 q^{5} - 2 q^{7} - 3 q^{9} - 6 q^{11} - 9 q^{13} - 2 q^{17} - 10 q^{19} - 9 q^{21} - 6 q^{23} + 9 q^{25} + 12 q^{27} - 6 q^{29} + 9 q^{31} - 11 q^{33} - 2 q^{35} - 12 q^{37} - 3 q^{39} - 20 q^{41} + q^{43} - 3 q^{45} + 22 q^{47} - 29 q^{49} + 2 q^{51} - 35 q^{53} - 6 q^{55} - 20 q^{57} + 14 q^{59} - 22 q^{61} - 12 q^{63} - 9 q^{65} + 4 q^{67} + 5 q^{69} - 22 q^{71} - 34 q^{73} - 5 q^{77} + 8 q^{79} - 31 q^{81} - 3 q^{83} - 2 q^{85} - 5 q^{89} - 7 q^{91} - 21 q^{93} - 10 q^{95} - 33 q^{97} - 15 q^{99}+O(q^{100}) $ 9 * q + 9 * q^5 - 2 * q^7 - 3 * q^9 - 6 * q^11 - 9 * q^13 - 2 * q^17 - 10 * q^19 - 9 * q^21 - 6 * q^23 + 9 * q^25 + 12 * q^27 - 6 * q^29 + 9 * q^31 - 11 * q^33 - 2 * q^35 - 12 * q^37 - 3 * q^39 - 20 * q^41 + q^43 - 3 * q^45 + 22 * q^47 - 29 * q^49 + 2 * q^51 - 35 * q^53 - 6 * q^55 - 20 * q^57 + 14 * q^59 - 22 * q^61 - 12 * q^63 - 9 * q^65 + 4 * q^67 + 5 * q^69 - 22 * q^71 - 34 * q^73 - 5 * q^77 + 8 * q^79 - 31 * q^81 - 3 * q^83 - 2 * q^85 - 5 * q^89 - 7 * q^91 - 21 * q^93 - 10 * q^95 - 33 * q^97 - 15 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	−1.57672	−0.910320	−0.455160	−	0.890410i	$-0.650418\pi$
$3$	−1.57672	−0.910320	−0.455160	+	0.890410i	$0.650418\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−0.208807	−0.0789215	−0.0394608	−	0.999221i	$-0.512564\pi$
$7$	−0.208807	−0.0789215	−0.0394608	+	0.999221i	$0.512564\pi$
$8$	0	0
$9$	−0.513952	−0.171317
$10$	0	0
$11$	−0.616299	−0.185821	−0.0929106	−	0.995674i	$-0.529617\pi$
$11$	−0.616299	−0.185821	−0.0929106	+	0.995674i	$0.529617\pi$
$12$	0	0
$13$	−3.24227	−0.899244	−0.449622	−	0.893219i	$-0.648441\pi$
$13$	−3.24227	−0.899244	−0.449622	+	0.893219i	$0.648441\pi$
$14$	0	0
$15$	−1.57672	−0.407107
$16$	0	0
$17$	1.94776	0.472401	0.236201	−	0.971704i	$-0.424098\pi$
$17$	1.94776	0.472401	0.236201	+	0.971704i	$0.424098\pi$
$18$	0	0
$19$	1.81681	0.416805	0.208402	−	0.978043i	$-0.433174\pi$
$19$	1.81681	0.416805	0.208402	+	0.978043i	$0.433174\pi$
$20$	0	0
$21$	0.329230	0.0718438
$22$	0	0
$23$	−1.42747	−0.297647	−0.148824	−	0.988864i	$-0.547549\pi$
$23$	−1.42747	−0.297647	−0.148824	+	0.988864i	$0.547549\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	5.54052	1.06627
$28$	0	0
$29$	−4.26518	−0.792023	−0.396012	−	0.918245i	$-0.629606\pi$
$29$	−4.26518	−0.792023	−0.396012	+	0.918245i	$0.629606\pi$
$30$	0	0
$31$	6.21450	1.11616	0.558079	−	0.829788i	$-0.311539\pi$
$31$	6.21450	1.11616	0.558079	+	0.829788i	$0.311539\pi$
$32$	0	0
$33$	0.971732	0.169157
$34$	0	0
$35$	−0.208807	−0.0352948
$36$	0	0
$37$	4.31510	0.709397	0.354699	−	0.934981i	$-0.384583\pi$
$37$	4.31510	0.709397	0.354699	+	0.934981i	$0.384583\pi$
$38$	0	0
$39$	5.11215	0.818600
$40$	0	0
$41$	1.53452	0.239652	0.119826	−	0.992795i	$-0.461766\pi$
$41$	1.53452	0.239652	0.119826	+	0.992795i	$0.461766\pi$
$42$	0	0
$43$	−1.84534	−0.281412	−0.140706	−	0.990051i	$-0.544937\pi$
$43$	−1.84534	−0.281412	−0.140706	+	0.990051i	$0.544937\pi$
$44$	0	0
$45$	−0.513952	−0.0766155
$46$	0	0
$47$	−4.09784	−0.597732	−0.298866	−	0.954295i	$-0.596608\pi$
$47$	−4.09784	−0.597732	−0.298866	+	0.954295i	$0.596608\pi$
$48$	0	0
$49$	−6.95640	−0.993771
$50$	0	0
$51$	−3.07107	−0.430036
$52$	0	0
$53$	10.8297	1.48757	0.743785	−	0.668419i	$-0.233029\pi$
$53$	10.8297	1.48757	0.743785	+	0.668419i	$0.233029\pi$
$54$	0	0
$55$	−0.616299	−0.0831018
$56$	0	0
$57$	−2.86460	−0.379426
$58$	0	0
$59$	8.06057	1.04940	0.524699	−	0.851288i	$-0.324178\pi$
$59$	8.06057	1.04940	0.524699	+	0.851288i	$0.324178\pi$
$60$	0	0
$61$	−11.1832	−1.43187	−0.715933	−	0.698169i	$-0.753998\pi$
$61$	−11.1832	−1.43187	−0.715933	+	0.698169i	$0.753998\pi$
$62$	0	0
$63$	0.107317	0.0135206
$64$	0	0
$65$	−3.24227	−0.402154
$66$	0	0
$67$	5.24700	0.641023	0.320511	−	0.947245i	$-0.396145\pi$
$67$	5.24700	0.641023	0.320511	+	0.947245i	$0.396145\pi$
$68$	0	0
$69$	2.25071	0.270954
$70$	0	0
$71$	8.10268	0.961611	0.480805	−	0.876827i	$-0.340344\pi$
$71$	8.10268	0.961611	0.480805	+	0.876827i	$0.340344\pi$
$72$	0	0
$73$	−1.11838	−0.130896	−0.0654480	−	0.997856i	$-0.520848\pi$
$73$	−1.11838	−0.130896	−0.0654480	+	0.997856i	$0.520848\pi$
$74$	0	0
$75$	−1.57672	−0.182064
$76$	0	0
$77$	0.128687	0.0146653
$78$	0	0
$79$	−7.47039	−0.840484	−0.420242	−	0.907412i	$-0.638055\pi$
$79$	−7.47039	−0.840484	−0.420242	+	0.907412i	$0.638055\pi$
$80$	0	0
$81$	−7.19400	−0.799333
$82$	0	0
$83$	−13.6712	−1.50061	−0.750305	−	0.661092i	$-0.770093\pi$
$83$	−13.6712	−1.50061	−0.750305	+	0.661092i	$0.770093\pi$
$84$	0	0
$85$	1.94776	0.211264
$86$	0	0
$87$	6.72499	0.720995
$88$	0	0
$89$	−13.1975	−1.39893	−0.699467	−	0.714665i	$-0.746579\pi$
$89$	−13.1975	−1.39893	−0.699467	+	0.714665i	$0.746579\pi$
$90$	0	0
$91$	0.677008	0.0709697
$92$	0	0
$93$	−9.79853	−1.01606
$94$	0	0
$95$	1.81681	0.186401
$96$	0	0
$97$	9.40168	0.954596	0.477298	−	0.878741i	$-0.341616\pi$
$97$	9.40168	0.954596	0.477298	+	0.878741i	$0.341616\pi$
$98$	0	0
$99$	0.316748	0.0318344
$100$	0	0
$101$	−13.2614	−1.31956	−0.659781	−	0.751458i	$-0.729351\pi$
$101$	−13.2614	−1.31956	−0.659781	+	0.751458i	$0.729351\pi$
$102$	0	0
$103$	1.44604	0.142483	0.0712415	−	0.997459i	$-0.477304\pi$
$103$	1.44604	0.142483	0.0712415	+	0.997459i	$0.477304\pi$
$104$	0	0
$105$	0.329230	0.0321295
$106$	0	0
$107$	10.7757	1.04173	0.520865	−	0.853639i	$-0.325610\pi$
$107$	10.7757	1.04173	0.520865	+	0.853639i	$0.325610\pi$
$108$	0	0
$109$	5.75641	0.551364	0.275682	−	0.961249i	$-0.411096\pi$
$109$	5.75641	0.551364	0.275682	+	0.961249i	$0.411096\pi$
$110$	0	0
$111$	−6.80370	−0.645779
$112$	0	0
$113$	−5.46809	−0.514394	−0.257197	−	0.966359i	$-0.582799\pi$
$113$	−5.46809	−0.514394	−0.257197	+	0.966359i	$0.582799\pi$
$114$	0	0
$115$	−1.42747	−0.133112
$116$	0	0
$117$	1.66637	0.154056
$118$	0	0
$119$	−0.406705	−0.0372826
$120$	0	0
$121$	−10.6202	−0.965470
$122$	0	0
$123$	−2.41951	−0.218160
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−7.43210	−0.659492	−0.329746	−	0.944070i	$-0.606963\pi$
$127$	−7.43210	−0.659492	−0.329746	+	0.944070i	$0.606963\pi$
$128$	0	0
$129$	2.90958	0.256175
$130$	0	0
$131$	7.40484	0.646964	0.323482	−	0.946234i	$-0.395146\pi$
$131$	7.40484	0.646964	0.323482	+	0.946234i	$0.395146\pi$
$132$	0	0
$133$	−0.379362	−0.0328949
$134$	0	0
$135$	5.54052	0.476852
$136$	0	0
$137$	−17.9623	−1.53462	−0.767312	−	0.641273i	$-0.778406\pi$
$137$	−17.9623	−1.53462	−0.767312	+	0.641273i	$0.778406\pi$
$138$	0	0
$139$	−2.74094	−0.232483	−0.116242	−	0.993221i	$-0.537085\pi$
$139$	−2.74094	−0.232483	−0.116242	+	0.993221i	$0.537085\pi$
$140$	0	0
$141$	6.46115	0.544127
$142$	0	0
$143$	1.99821	0.167099
$144$	0	0
$145$	−4.26518	−0.354204
$146$	0	0
$147$	10.9683	0.904650
$148$	0	0
$149$	−17.6431	−1.44538	−0.722689	−	0.691174i	$-0.757094\pi$
$149$	−17.6431	−1.44538	−0.722689	+	0.691174i	$0.757094\pi$
$150$	0	0
$151$	−1.00000	−0.0813788
$151$	−1.00000	−0.0813788
$152$	0	0
$153$	−1.00106	−0.0809305
$154$	0	0
$155$	6.21450	0.499161
$156$	0	0
$157$	−3.25706	−0.259942	−0.129971	−	0.991518i	$-0.541488\pi$
$157$	−3.25706	−0.259942	−0.129971	+	0.991518i	$0.541488\pi$
$158$	0	0
$159$	−17.0754	−1.35416
$160$	0	0
$161$	0.298064	0.0234908
$162$	0	0
$163$	−7.69472	−0.602697	−0.301348	−	0.953514i	$-0.597437\pi$
$163$	−7.69472	−0.602697	−0.301348	+	0.953514i	$0.597437\pi$
$164$	0	0
$165$	0.971732	0.0756492
$166$	0	0
$167$	20.9172	1.61862	0.809311	−	0.587381i	$-0.199841\pi$
$167$	20.9172	1.61862	0.809311	+	0.587381i	$0.199841\pi$
$168$	0	0
$169$	−2.48769	−0.191361
$170$	0	0
$171$	−0.933754	−0.0714060
$172$	0	0
$173$	−14.4578	−1.09921	−0.549605	−	0.835425i	$-0.685222\pi$
$173$	−14.4578	−1.09921	−0.549605	+	0.835425i	$0.685222\pi$
$174$	0	0
$175$	−0.208807	−0.0157843
$176$	0	0
$177$	−12.7093	−0.955287
$178$	0	0
$179$	−8.11436	−0.606496	−0.303248	−	0.952912i	$-0.598071\pi$
$179$	−8.11436	−0.606496	−0.303248	+	0.952912i	$0.598071\pi$
$180$	0	0
$181$	22.6472	1.68335	0.841676	−	0.539983i	$-0.181569\pi$
$181$	22.6472	1.68335	0.841676	+	0.539983i	$0.181569\pi$
$182$	0	0
$183$	17.6328	1.30346
$184$	0	0
$185$	4.31510	0.317252
$186$	0	0
$187$	−1.20040	−0.0877822
$188$	0	0
$189$	−1.15690	−0.0841519
$190$	0	0
$191$	3.36141	0.243223	0.121611	−	0.992578i	$-0.461194\pi$
$191$	3.36141	0.243223	0.121611	+	0.992578i	$0.461194\pi$
$192$	0	0
$193$	3.99457	0.287535	0.143768	−	0.989611i	$-0.454078\pi$
$193$	3.99457	0.287535	0.143768	+	0.989611i	$0.454078\pi$
$194$	0	0
$195$	5.11215	0.366089
$196$	0	0
$197$	15.4145	1.09823	0.549117	−	0.835745i	$-0.314964\pi$
$197$	15.4145	1.09823	0.549117	+	0.835745i	$0.314964\pi$
$198$	0	0
$199$	4.61808	0.327367	0.163683	−	0.986513i	$-0.447662\pi$
$199$	4.61808	0.327367	0.163683	+	0.986513i	$0.447662\pi$
$200$	0	0
$201$	−8.27305	−0.583536
$202$	0	0
$203$	0.890597	0.0625077
$204$	0	0
$205$	1.53452	0.107175
$206$	0	0
$207$	0.733649	0.0509921
$208$	0	0
$209$	−1.11970	−0.0774512
$210$	0	0
$211$	−0.713130	−0.0490939	−0.0245470	−	0.999699i	$-0.507814\pi$
$211$	−0.713130	−0.0490939	−0.0245470	+	0.999699i	$0.507814\pi$
$212$	0	0
$213$	−12.7757	−0.875374
$214$	0	0
$215$	−1.84534	−0.125851
$216$	0	0
$217$	−1.29763	−0.0880888
$218$	0	0
$219$	1.76337	0.119157
$220$	0	0
$221$	−6.31516	−0.424804
$222$	0	0
$223$	3.64969	0.244401	0.122201	−	0.992505i	$-0.461005\pi$
$223$	3.64969	0.244401	0.122201	+	0.992505i	$0.461005\pi$
$224$	0	0
$225$	−0.513952	−0.0342635
$226$	0	0
$227$	5.20840	0.345694	0.172847	−	0.984949i	$-0.444703\pi$
$227$	5.20840	0.345694	0.172847	+	0.984949i	$0.444703\pi$
$228$	0	0
$229$	5.13516	0.339341	0.169670	−	0.985501i	$-0.445730\pi$
$229$	5.13516	0.339341	0.169670	+	0.985501i	$0.445730\pi$
$230$	0	0
$231$	−0.202904	−0.0133501
$232$	0	0
$233$	2.78606	0.182521	0.0912605	−	0.995827i	$-0.470910\pi$
$233$	2.78606	0.182521	0.0912605	+	0.995827i	$0.470910\pi$
$234$	0	0
$235$	−4.09784	−0.267314
$236$	0	0
$237$	11.7787	0.765110
$238$	0	0
$239$	−13.6225	−0.881167	−0.440584	−	0.897712i	$-0.645229\pi$
$239$	−13.6225	−0.881167	−0.440584	+	0.897712i	$0.645229\pi$
$240$	0	0
$241$	−28.6786	−1.84735	−0.923677	−	0.383173i	$-0.874831\pi$
$241$	−28.6786	−1.84735	−0.923677	+	0.383173i	$0.874831\pi$
$242$	0	0
$243$	−5.27864	−0.338625
$244$	0	0
$245$	−6.95640	−0.444428
$246$	0	0
$247$	−5.89059	−0.374809
$248$	0	0
$249$	21.5557	1.36604
$250$	0	0
$251$	−6.32430	−0.399186	−0.199593	−	0.979879i	$-0.563962\pi$
$251$	−6.32430	−0.399186	−0.199593	+	0.979879i	$0.563962\pi$
$252$	0	0
$253$	0.879746	0.0553092
$254$	0	0
$255$	−3.07107	−0.192318
$256$	0	0
$257$	−22.1344	−1.38071	−0.690354	−	0.723472i	$-0.742545\pi$
$257$	−22.1344	−1.38071	−0.690354	+	0.723472i	$0.742545\pi$
$258$	0	0
$259$	−0.901021	−0.0559867
$260$	0	0
$261$	2.19210	0.135687
$262$	0	0
$263$	−9.72503	−0.599671	−0.299836	−	0.953991i	$-0.596932\pi$
$263$	−9.72503	−0.599671	−0.299836	+	0.953991i	$0.596932\pi$
$264$	0	0
$265$	10.8297	0.665261
$266$	0	0
$267$	20.8088	1.27348
$268$	0	0
$269$	−0.848837	−0.0517545	−0.0258773	−	0.999665i	$-0.508238\pi$
$269$	−0.848837	−0.0517545	−0.0258773	+	0.999665i	$0.508238\pi$
$270$	0	0
$271$	−3.31253	−0.201222	−0.100611	−	0.994926i	$-0.532080\pi$
$271$	−3.31253	−0.201222	−0.100611	+	0.994926i	$0.532080\pi$
$272$	0	0
$273$	−1.06745	−0.0646051
$274$	0	0
$275$	−0.616299	−0.0371643
$276$	0	0
$277$	−21.1085	−1.26829	−0.634144	−	0.773215i	$-0.718647\pi$
$277$	−21.1085	−1.26829	−0.634144	+	0.773215i	$0.718647\pi$
$278$	0	0
$279$	−3.19396	−0.191217
$280$	0	0
$281$	−6.31579	−0.376768	−0.188384	−	0.982095i	$-0.560325\pi$
$281$	−6.31579	−0.376768	−0.188384	+	0.982095i	$0.560325\pi$
$282$	0	0
$283$	13.7968	0.820134	0.410067	−	0.912055i	$-0.365505\pi$
$283$	13.7968	0.820134	0.410067	+	0.912055i	$0.365505\pi$
$284$	0	0
$285$	−2.86460	−0.169684
$286$	0	0
$287$	−0.320418	−0.0189137
$288$	0	0
$289$	−13.2062	−0.776837
$290$	0	0
$291$	−14.8238	−0.868988
$292$	0	0
$293$	−0.0503915	−0.00294390	−0.00147195	−	0.999999i	$-0.500469\pi$
$293$	−0.0503915	−0.00294390	−0.00147195	+	0.999999i	$0.500469\pi$
$294$	0	0
$295$	8.06057	0.469305
$296$	0	0
$297$	−3.41462	−0.198136
$298$	0	0
$299$	4.62823	0.267657
$300$	0	0
$301$	0.385319	0.0222094
$302$	0	0
$303$	20.9096	1.20122
$304$	0	0
$305$	−11.1832	−0.640350
$306$	0	0
$307$	−29.5064	−1.68402	−0.842008	−	0.539464i	$-0.818627\pi$
$307$	−29.5064	−1.68402	−0.842008	+	0.539464i	$0.818627\pi$
$308$	0	0
$309$	−2.28001	−0.129705
$310$	0	0
$311$	−15.6955	−0.890012	−0.445006	−	0.895528i	$-0.646798\pi$
$311$	−15.6955	−0.890012	−0.445006	+	0.895528i	$0.646798\pi$
$312$	0	0
$313$	19.2491	1.08803	0.544013	−	0.839077i	$-0.316904\pi$
$313$	19.2491	1.08803	0.544013	+	0.839077i	$0.316904\pi$
$314$	0	0
$315$	0.107317	0.00604661
$316$	0	0
$317$	−8.23495	−0.462521	−0.231260	−	0.972892i	$-0.574285\pi$
$317$	−8.23495	−0.462521	−0.231260	+	0.972892i	$0.574285\pi$
$318$	0	0
$319$	2.62863	0.147175
$320$	0	0
$321$	−16.9903	−0.948307
$322$	0	0
$323$	3.53871	0.196899
$324$	0	0
$325$	−3.24227	−0.179849
$326$	0	0
$327$	−9.07625	−0.501918
$328$	0	0
$329$	0.855657	0.0471739
$330$	0	0
$331$	8.87968	0.488072	0.244036	−	0.969766i	$-0.421529\pi$
$331$	8.87968	0.488072	0.244036	+	0.969766i	$0.421529\pi$
$332$	0	0
$333$	−2.21775	−0.121532
$334$	0	0
$335$	5.24700	0.286674
$336$	0	0
$337$	16.6349	0.906159	0.453080	−	0.891470i	$-0.350325\pi$
$337$	16.6349	0.906159	0.453080	+	0.891470i	$0.350325\pi$
$338$	0	0
$339$	8.62165	0.468264
$340$	0	0
$341$	−3.82999	−0.207406
$342$	0	0
$343$	2.91419	0.157351
$344$	0	0
$345$	2.25071	0.121174
$346$	0	0
$347$	24.2974	1.30435	0.652176	−	0.758068i	$-0.273856\pi$
$347$	24.2974	1.30435	0.652176	+	0.758068i	$0.273856\pi$
$348$	0	0
$349$	−12.5363	−0.671051	−0.335525	−	0.942031i	$-0.608914\pi$
$349$	−12.5363	−0.671051	−0.335525	+	0.942031i	$0.608914\pi$
$350$	0	0
$351$	−17.9639	−0.958840
$352$	0	0
$353$	−18.4445	−0.981703	−0.490852	−	0.871243i	$-0.663314\pi$
$353$	−18.4445	−0.981703	−0.490852	+	0.871243i	$0.663314\pi$
$354$	0	0
$355$	8.10268	0.430045
$356$	0	0
$357$	0.641260	0.0339391
$358$	0	0
$359$	12.0751	0.637300	0.318650	−	0.947873i	$-0.396771\pi$
$359$	12.0751	0.637300	0.318650	+	0.947873i	$0.396771\pi$
$360$	0	0
$361$	−15.6992	−0.826274
$362$	0	0
$363$	16.7450	0.878887
$364$	0	0
$365$	−1.11838	−0.0585385
$366$	0	0
$367$	−11.7726	−0.614523	−0.307262	−	0.951625i	$-0.599413\pi$
$367$	−11.7726	−0.614523	−0.307262	+	0.951625i	$0.599413\pi$
$368$	0	0
$369$	−0.788669	−0.0410565
$370$	0	0
$371$	−2.26131	−0.117401
$372$	0	0
$373$	−1.70761	−0.0884168	−0.0442084	−	0.999022i	$-0.514077\pi$
$373$	−1.70761	−0.0884168	−0.0442084	+	0.999022i	$0.514077\pi$
$374$	0	0
$375$	−1.57672	−0.0814215
$376$	0	0
$377$	13.8289	0.712222
$378$	0	0
$379$	7.29129	0.374528	0.187264	−	0.982310i	$-0.440038\pi$
$379$	7.29129	0.374528	0.187264	+	0.982310i	$0.440038\pi$
$380$	0	0
$381$	11.7183	0.600349
$382$	0	0
$383$	13.6210	0.696002	0.348001	−	0.937494i	$-0.386861\pi$
$383$	13.6210	0.696002	0.348001	+	0.937494i	$0.386861\pi$
$384$	0	0
$385$	0.128687	0.00655852
$386$	0	0
$387$	0.948416	0.0482107
$388$	0	0
$389$	−22.8664	−1.15937	−0.579685	−	0.814841i	$-0.696824\pi$
$389$	−22.8664	−1.15937	−0.579685	+	0.814841i	$0.696824\pi$
$390$	0	0
$391$	−2.78036	−0.140609
$392$	0	0
$393$	−11.6754	−0.588944
$394$	0	0
$395$	−7.47039	−0.375876
$396$	0	0
$397$	−3.92578	−0.197029	−0.0985146	−	0.995136i	$-0.531409\pi$
$397$	−3.92578	−0.197029	−0.0985146	+	0.995136i	$0.531409\pi$
$398$	0	0
$399$	0.598148	0.0299449
$400$	0	0
$401$	23.5785	1.17745	0.588727	−	0.808332i	$-0.299629\pi$
$401$	23.5785	1.17745	0.588727	+	0.808332i	$0.299629\pi$
$402$	0	0
$403$	−20.1491	−1.00370
$404$	0	0
$405$	−7.19400	−0.357473
$406$	0	0
$407$	−2.65939	−0.131821
$408$	0	0
$409$	−31.4830	−1.55673	−0.778367	−	0.627810i	$-0.783952\pi$
$409$	−31.4830	−1.55673	−0.778367	+	0.627810i	$0.783952\pi$
$410$	0	0
$411$	28.3216	1.39700
$412$	0	0
$413$	−1.68310	−0.0828200
$414$	0	0
$415$	−13.6712	−0.671093
$416$	0	0
$417$	4.32169	0.211634
$418$	0	0
$419$	4.50972	0.220314	0.110157	−	0.993914i	$-0.464865\pi$
$419$	4.50972	0.220314	0.110157	+	0.993914i	$0.464865\pi$
$420$	0	0
$421$	−27.7864	−1.35423	−0.677113	−	0.735879i	$-0.736769\pi$
$421$	−27.7864	−1.35423	−0.677113	+	0.735879i	$0.736769\pi$
$422$	0	0
$423$	2.10609	0.102402
$424$	0	0
$425$	1.94776	0.0944802
$426$	0	0
$427$	2.33513	0.113005
$428$	0	0
$429$	−3.15062	−0.152113
$430$	0	0
$431$	−27.0231	−1.30166	−0.650829	−	0.759225i	$-0.725578\pi$
$431$	−27.0231	−1.30166	−0.650829	+	0.759225i	$0.725578\pi$
$432$	0	0
$433$	−30.9410	−1.48693	−0.743464	−	0.668775i	$-0.766819\pi$
$433$	−30.9410	−1.48693	−0.743464	+	0.668775i	$0.766819\pi$
$434$	0	0
$435$	6.72499	0.322439
$436$	0	0
$437$	−2.59343	−0.124061
$438$	0	0
$439$	−36.2431	−1.72979	−0.864895	−	0.501952i	$-0.832615\pi$
$439$	−36.2431	−1.72979	−0.864895	+	0.501952i	$0.832615\pi$
$440$	0	0
$441$	3.57526	0.170250
$442$	0	0
$443$	−25.3711	−1.20542	−0.602709	−	0.797961i	$-0.705912\pi$
$443$	−25.3711	−1.20542	−0.602709	+	0.797961i	$0.705912\pi$
$444$	0	0
$445$	−13.1975	−0.625622
$446$	0	0
$447$	27.8182	1.31576
$448$	0	0
$449$	18.9558	0.894580	0.447290	−	0.894389i	$-0.352389\pi$
$449$	18.9558	0.894580	0.447290	+	0.894389i	$0.352389\pi$
$450$	0	0
$451$	−0.945723	−0.0445323
$452$	0	0
$453$	1.57672	0.0740808
$454$	0	0
$455$	0.677008	0.0317386
$456$	0	0
$457$	−14.2217	−0.665265	−0.332632	−	0.943057i	$-0.607937\pi$
$457$	−14.2217	−0.665265	−0.332632	+	0.943057i	$0.607937\pi$
$458$	0	0
$459$	10.7916	0.503709
$460$	0	0
$461$	−24.0031	−1.11794	−0.558969	−	0.829189i	$-0.688803\pi$
$461$	−24.0031	−1.11794	−0.558969	+	0.829189i	$0.688803\pi$
$462$	0	0
$463$	−11.8075	−0.548743	−0.274372	−	0.961624i	$-0.588470\pi$
$463$	−11.8075	−0.548743	−0.274372	+	0.961624i	$0.588470\pi$
$464$	0	0
$465$	−9.79853	−0.454396
$466$	0	0
$467$	−4.03966	−0.186933	−0.0934667	−	0.995622i	$-0.529795\pi$
$467$	−4.03966	−0.186933	−0.0934667	+	0.995622i	$0.529795\pi$
$468$	0	0
$469$	−1.09561	−0.0505905
$470$	0	0
$471$	5.13547	0.236630
$472$	0	0
$473$	1.13728	0.0522923
$474$	0	0
$475$	1.81681	0.0833610
$476$	0	0
$477$	−5.56593	−0.254847
$478$	0	0
$479$	−24.1312	−1.10258	−0.551291	−	0.834313i	$-0.685864\pi$
$479$	−24.1312	−1.10258	−0.551291	+	0.834313i	$0.685864\pi$
$480$	0	0
$481$	−13.9907	−0.637921
$482$	0	0
$483$	−0.469964	−0.0213841
$484$	0	0
$485$	9.40168	0.426908
$486$	0	0
$487$	18.0591	0.818334	0.409167	−	0.912460i	$-0.365819\pi$
$487$	18.0591	0.818334	0.409167	+	0.912460i	$0.365819\pi$
$488$	0	0
$489$	12.1324	0.548647
$490$	0	0
$491$	−19.3288	−0.872295	−0.436148	−	0.899875i	$-0.643657\pi$
$491$	−19.3288	−0.872295	−0.436148	+	0.899875i	$0.643657\pi$
$492$	0	0
$493$	−8.30754	−0.374153
$494$	0	0
$495$	0.316748	0.0142368
$496$	0	0
$497$	−1.69189	−0.0758918
$498$	0	0
$499$	5.06467	0.226726	0.113363	−	0.993554i	$-0.463838\pi$
$499$	5.06467	0.226726	0.113363	+	0.993554i	$0.463838\pi$
$500$	0	0
$501$	−32.9806	−1.47346
$502$	0	0
$503$	−28.2555	−1.25985	−0.629925	−	0.776656i	$-0.716915\pi$
$503$	−28.2555	−1.25985	−0.629925	+	0.776656i	$0.716915\pi$
$504$	0	0
$505$	−13.2614	−0.590126
$506$	0	0
$507$	3.92239	0.174199
$508$	0	0
$509$	31.2474	1.38502	0.692508	−	0.721410i	$-0.256506\pi$
$509$	31.2474	1.38502	0.692508	+	0.721410i	$0.256506\pi$
$510$	0	0
$511$	0.233524	0.0103305
$512$	0	0
$513$	10.0661	0.444428
$514$	0	0
$515$	1.44604	0.0637203
$516$	0	0
$517$	2.52550	0.111071
$518$	0	0
$519$	22.7960	1.00063
$520$	0	0
$521$	16.0452	0.702954	0.351477	−	0.936196i	$-0.385679\pi$
$521$	16.0452	0.702954	0.351477	+	0.936196i	$0.385679\pi$
$522$	0	0
$523$	−3.44740	−0.150744	−0.0753722	−	0.997155i	$-0.524015\pi$
$523$	−3.44740	−0.150744	−0.0753722	+	0.997155i	$0.524015\pi$
$524$	0	0
$525$	0.329230	0.0143688
$526$	0	0
$527$	12.1044	0.527274
$528$	0	0
$529$	−20.9623	−0.911406
$530$	0	0
$531$	−4.14275	−0.179780
$532$	0	0
$533$	−4.97532	−0.215505
$534$	0	0
$535$	10.7757	0.465876
$536$	0	0
$537$	12.7941	0.552106
$538$	0	0
$539$	4.28723	0.184664
$540$	0	0
$541$	−21.3423	−0.917578	−0.458789	−	0.888545i	$-0.651717\pi$
$541$	−21.3423	−0.917578	−0.458789	+	0.888545i	$0.651717\pi$
$542$	0	0
$543$	−35.7083	−1.53239
$544$	0	0
$545$	5.75641	0.246578
$546$	0	0
$547$	31.9664	1.36679	0.683393	−	0.730050i	$-0.260503\pi$
$547$	31.9664	1.36679	0.683393	+	0.730050i	$0.260503\pi$
$548$	0	0
$549$	5.74764	0.245303
$550$	0	0
$551$	−7.74902	−0.330119
$552$	0	0
$553$	1.55987	0.0663323
$554$	0	0
$555$	−6.80370	−0.288801
$556$	0	0
$557$	29.5091	1.25034	0.625171	−	0.780488i	$-0.285029\pi$
$557$	29.5091	1.25034	0.625171	+	0.780488i	$0.285029\pi$
$558$	0	0
$559$	5.98309	0.253058
$560$	0	0
$561$	1.89270	0.0799099
$562$	0	0
$563$	−47.0845	−1.98438	−0.992188	−	0.124748i	$-0.960188\pi$
$563$	−47.0845	−1.98438	−0.992188	+	0.124748i	$0.960188\pi$
$564$	0	0
$565$	−5.46809	−0.230044
$566$	0	0
$567$	1.50215	0.0630846
$568$	0	0
$569$	−12.8848	−0.540158	−0.270079	−	0.962838i	$-0.587050\pi$
$569$	−12.8848	−0.540158	−0.270079	+	0.962838i	$0.587050\pi$
$570$	0	0
$571$	−40.7747	−1.70637	−0.853183	−	0.521611i	$-0.825331\pi$
$571$	−40.7747	−1.70637	−0.853183	+	0.521611i	$0.825331\pi$
$572$	0	0
$573$	−5.30000	−0.221411
$574$	0	0
$575$	−1.42747	−0.0595294
$576$	0	0
$577$	44.8273	1.86618	0.933091	−	0.359639i	$-0.117100\pi$
$577$	44.8273	1.86618	0.933091	+	0.359639i	$0.117100\pi$
$578$	0	0
$579$	−6.29832	−0.261749
$580$	0	0
$581$	2.85464	0.118430
$582$	0	0
$583$	−6.67432	−0.276422
$584$	0	0
$585$	1.66637	0.0688960
$586$	0	0
$587$	19.2724	0.795455	0.397728	−	0.917504i	$-0.369799\pi$
$587$	19.2724	0.795455	0.397728	+	0.917504i	$0.369799\pi$
$588$	0	0
$589$	11.2906	0.465220
$590$	0	0
$591$	−24.3043	−0.999745
$592$	0	0
$593$	14.6610	0.602055	0.301027	−	0.953616i	$-0.402670\pi$
$593$	14.6610	0.602055	0.301027	+	0.953616i	$0.402670\pi$
$594$	0	0
$595$	−0.406705	−0.0166733
$596$	0	0
$597$	−7.28142	−0.298009
$598$	0	0
$599$	12.0961	0.494232	0.247116	−	0.968986i	$-0.420517\pi$
$599$	12.0961	0.494232	0.247116	+	0.968986i	$0.420517\pi$
$600$	0	0
$601$	12.0154	0.490117	0.245059	−	0.969508i	$-0.421193\pi$
$601$	12.0154	0.490117	0.245059	+	0.969508i	$0.421193\pi$
$602$	0	0
$603$	−2.69671	−0.109818
$604$	0	0
$605$	−10.6202	−0.431772
$606$	0	0
$607$	−22.7414	−0.923047	−0.461523	−	0.887128i	$-0.652697\pi$
$607$	−22.7414	−0.923047	−0.461523	+	0.887128i	$0.652697\pi$
$608$	0	0
$609$	−1.40422	−0.0569020
$610$	0	0
$611$	13.2863	0.537506
$612$	0	0
$613$	17.0718	0.689522	0.344761	−	0.938690i	$-0.387960\pi$
$613$	17.0718	0.689522	0.344761	+	0.938690i	$0.387960\pi$
$614$	0	0
$615$	−2.41951	−0.0975639
$616$	0	0
$617$	−34.6057	−1.39317	−0.696587	−	0.717473i	$-0.745299\pi$
$617$	−34.6057	−1.39317	−0.696587	+	0.717473i	$0.745299\pi$
$618$	0	0
$619$	−30.1899	−1.21344	−0.606718	−	0.794917i	$-0.707514\pi$
$619$	−30.1899	−1.21344	−0.606718	+	0.794917i	$0.707514\pi$
$620$	0	0
$621$	−7.90890	−0.317373
$622$	0	0
$623$	2.75573	0.110406
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	1.76545	0.0705054
$628$	0	0
$629$	8.40477	0.335120
$630$	0	0
$631$	−44.2735	−1.76250	−0.881251	−	0.472649i	$-0.843298\pi$
$631$	−44.2735	−1.76250	−0.881251	+	0.472649i	$0.843298\pi$
$632$	0	0
$633$	1.12441	0.0446912
$634$	0	0
$635$	−7.43210	−0.294934
$636$	0	0
$637$	22.5545	0.893643
$638$	0	0
$639$	−4.16439	−0.164741
$640$	0	0
$641$	32.3085	1.27611	0.638055	−	0.769991i	$-0.279739\pi$
$641$	32.3085	1.27611	0.638055	+	0.769991i	$0.279739\pi$
$642$	0	0
$643$	−0.498956	−0.0196769	−0.00983845	−	0.999952i	$-0.503132\pi$
$643$	−0.498956	−0.0196769	−0.00983845	+	0.999952i	$0.503132\pi$
$644$	0	0
$645$	2.90958	0.114565
$646$	0	0
$647$	39.2668	1.54374	0.771869	−	0.635781i	$-0.219322\pi$
$647$	39.2668	1.54374	0.771869	+	0.635781i	$0.219322\pi$
$648$	0	0
$649$	−4.96773	−0.195000
$650$	0	0
$651$	2.04600	0.0801890
$652$	0	0
$653$	23.0748	0.902987	0.451493	−	0.892274i	$-0.350891\pi$
$653$	23.0748	0.902987	0.451493	+	0.892274i	$0.350891\pi$
$654$	0	0
$655$	7.40484	0.289331
$656$	0	0
$657$	0.574792	0.0224248
$658$	0	0
$659$	36.2100	1.41054	0.705271	−	0.708938i	$-0.250825\pi$
$659$	36.2100	1.41054	0.705271	+	0.708938i	$0.250825\pi$
$660$	0	0
$661$	39.4735	1.53534	0.767670	−	0.640846i	$-0.221416\pi$
$661$	39.4735	1.53534	0.767670	+	0.640846i	$0.221416\pi$
$662$	0	0
$663$	9.95725	0.386707
$664$	0	0
$665$	−0.379362	−0.0147110
$666$	0	0
$667$	6.08839	0.235743
$668$	0	0
$669$	−5.75454	−0.222483
$670$	0	0
$671$	6.89222	0.266071
$672$	0	0
$673$	13.5574	0.522598	0.261299	−	0.965258i	$-0.415849\pi$
$673$	13.5574	0.522598	0.261299	+	0.965258i	$0.415849\pi$
$674$	0	0
$675$	5.54052	0.213255
$676$	0	0
$677$	−41.1325	−1.58085	−0.790426	−	0.612558i	$-0.790141\pi$
$677$	−41.1325	−1.58085	−0.790426	+	0.612558i	$0.790141\pi$
$678$	0	0
$679$	−1.96313	−0.0753382
$680$	0	0
$681$	−8.21219	−0.314692
$682$	0	0
$683$	19.1400	0.732371	0.366186	−	0.930542i	$-0.380664\pi$
$683$	19.1400	0.732371	0.366186	+	0.930542i	$0.380664\pi$
$684$	0	0
$685$	−17.9623	−0.686305
$686$	0	0
$687$	−8.09671	−0.308909
$688$	0	0
$689$	−35.1127	−1.33769
$690$	0	0
$691$	7.34560	0.279440	0.139720	−	0.990191i	$-0.455380\pi$
$691$	7.34560	0.279440	0.139720	+	0.990191i	$0.455380\pi$
$692$	0	0
$693$	−0.0661392	−0.00251242
$694$	0	0
$695$	−2.74094	−0.103970
$696$	0	0
$697$	2.98887	0.113212
$698$	0	0
$699$	−4.39284	−0.166153
$700$	0	0
$701$	−27.2689	−1.02993	−0.514966	−	0.857210i	$-0.672196\pi$
$701$	−27.2689	−1.02993	−0.514966	+	0.857210i	$0.672196\pi$
$702$	0	0
$703$	7.83971	0.295680
$704$	0	0
$705$	6.46115	0.243341
$706$	0	0
$707$	2.76908	0.104142
$708$	0	0
$709$	8.89231	0.333958	0.166979	−	0.985960i	$-0.446599\pi$
$709$	8.89231	0.333958	0.166979	+	0.985960i	$0.446599\pi$
$710$	0	0
$711$	3.83942	0.143990
$712$	0	0
$713$	−8.87098	−0.332221
$714$	0	0
$715$	1.99821	0.0747288
$716$	0	0
$717$	21.4789	0.802144
$718$	0	0
$719$	−21.7533	−0.811260	−0.405630	−	0.914037i	$-0.632948\pi$
$719$	−21.7533	−0.811260	−0.405630	+	0.914037i	$0.632948\pi$
$720$	0	0
$721$	−0.301944	−0.0112450
$722$	0	0
$723$	45.2182	1.68168
$724$	0	0
$725$	−4.26518	−0.158405
$726$	0	0
$727$	9.99766	0.370793	0.185396	−	0.982664i	$-0.440643\pi$
$727$	9.99766	0.370793	0.185396	+	0.982664i	$0.440643\pi$
$728$	0	0
$729$	29.9049	1.10759
$730$	0	0
$731$	−3.59428	−0.132939
$732$	0	0
$733$	26.6425	0.984061	0.492031	−	0.870578i	$-0.336255\pi$
$733$	26.6425	0.984061	0.492031	+	0.870578i	$0.336255\pi$
$734$	0	0
$735$	10.9683	0.404572
$736$	0	0
$737$	−3.23372	−0.119116
$738$	0	0
$739$	47.9446	1.76367	0.881835	−	0.471558i	$-0.156308\pi$
$739$	47.9446	1.76367	0.881835	+	0.471558i	$0.156308\pi$
$740$	0	0
$741$	9.28782	0.341196
$742$	0	0
$743$	−20.8792	−0.765983	−0.382992	−	0.923752i	$-0.625106\pi$
$743$	−20.8792	−0.765983	−0.382992	+	0.923752i	$0.625106\pi$
$744$	0	0
$745$	−17.6431	−0.646392
$746$	0	0
$747$	7.02635	0.257081
$748$	0	0
$749$	−2.25004	−0.0822148
$750$	0	0
$751$	3.44860	0.125841	0.0629207	−	0.998019i	$-0.479958\pi$
$751$	3.44860	0.125841	0.0629207	+	0.998019i	$0.479958\pi$
$752$	0	0
$753$	9.97166	0.363387
$754$	0	0
$755$	−1.00000	−0.0363937
$756$	0	0
$757$	2.99315	0.108788	0.0543940	−	0.998520i	$-0.482677\pi$
$757$	2.99315	0.108788	0.0543940	+	0.998520i	$0.482677\pi$
$758$	0	0
$759$	−1.38711	−0.0503490
$760$	0	0
$761$	−24.5750	−0.890844	−0.445422	−	0.895321i	$-0.646946\pi$
$761$	−24.5750	−0.890844	−0.445422	+	0.895321i	$0.646946\pi$
$762$	0	0
$763$	−1.20198	−0.0435145
$764$	0	0
$765$	−1.00106	−0.0361932
$766$	0	0
$767$	−26.1345	−0.943664
$768$	0	0
$769$	−37.2537	−1.34340	−0.671702	−	0.740822i	$-0.734436\pi$
$769$	−37.2537	−1.34340	−0.671702	+	0.740822i	$0.734436\pi$
$770$	0	0
$771$	34.8998	1.25689
$772$	0	0
$773$	21.3887	0.769299	0.384649	−	0.923063i	$-0.374322\pi$
$773$	21.3887	0.769299	0.384649	+	0.923063i	$0.374322\pi$
$774$	0	0
$775$	6.21450	0.223231
$776$	0	0
$777$	1.42066	0.0509658
$778$	0	0
$779$	2.78793	0.0998879
$780$	0	0
$781$	−4.99368	−0.178688
$782$	0	0
$783$	−23.6313	−0.844514
$784$	0	0
$785$	−3.25706	−0.116249
$786$	0	0
$787$	30.1181	1.07359	0.536797	−	0.843711i	$-0.319634\pi$
$787$	30.1181	1.07359	0.536797	+	0.843711i	$0.319634\pi$
$788$	0	0
$789$	15.3337	0.545893
$790$	0	0
$791$	1.14177	0.0405968
$792$	0	0
$793$	36.2590	1.28760
$794$	0	0
$795$	−17.0754	−0.605601
$796$	0	0
$797$	−17.1673	−0.608099	−0.304049	−	0.952656i	$-0.598339\pi$
$797$	−17.1673	−0.608099	−0.304049	+	0.952656i	$0.598339\pi$
$798$	0	0
$799$	−7.98161	−0.282369
$800$	0	0
$801$	6.78289	0.239662
$802$	0	0
$803$	0.689255	0.0243233
$804$	0	0
$805$	0.298064	0.0105054
$806$	0	0
$807$	1.33838	0.0471132
$808$	0	0
$809$	−14.1584	−0.497784	−0.248892	−	0.968531i	$-0.580066\pi$
$809$	−14.1584	−0.497784	−0.248892	+	0.968531i	$0.580066\pi$
$810$	0	0
$811$	21.0139	0.737897	0.368948	−	0.929450i	$-0.379718\pi$
$811$	21.0139	0.737897	0.368948	+	0.929450i	$0.379718\pi$
$812$	0	0
$813$	5.22294	0.183176
$814$	0	0
$815$	−7.69472	−0.269534
$816$	0	0
$817$	−3.35263	−0.117294
$818$	0	0
$819$	−0.347950	−0.0121583
$820$	0	0
$821$	−34.3762	−1.19974	−0.599869	−	0.800098i	$-0.704781\pi$
$821$	−34.3762	−1.19974	−0.599869	+	0.800098i	$0.704781\pi$
$822$	0	0
$823$	28.5373	0.994749	0.497375	−	0.867536i	$-0.334297\pi$
$823$	28.5373	0.994749	0.497375	+	0.867536i	$0.334297\pi$
$824$	0	0
$825$	0.971732	0.0338314
$826$	0	0
$827$	−31.2311	−1.08601	−0.543005	−	0.839729i	$-0.682714\pi$
$827$	−31.2311	−1.08601	−0.543005	+	0.839729i	$0.682714\pi$
$828$	0	0
$829$	13.1885	0.458056	0.229028	−	0.973420i	$-0.426445\pi$
$829$	13.1885	0.458056	0.229028	+	0.973420i	$0.426445\pi$
$830$	0	0
$831$	33.2822	1.15455
$832$	0	0
$833$	−13.5494	−0.469459
$834$	0	0
$835$	20.9172	0.723870
$836$	0	0
$837$	34.4316	1.19013
$838$	0	0
$839$	7.03792	0.242976	0.121488	−	0.992593i	$-0.461233\pi$
$839$	7.03792	0.242976	0.121488	+	0.992593i	$0.461233\pi$
$840$	0	0
$841$	−10.8083	−0.372699
$842$	0	0
$843$	9.95823	0.342980
$844$	0	0
$845$	−2.48769	−0.0855791
$846$	0	0
$847$	2.21756	0.0761964
$848$	0	0
$849$	−21.7537	−0.746584
$850$	0	0
$851$	−6.15965	−0.211150
$852$	0	0
$853$	2.21589	0.0758705	0.0379353	−	0.999280i	$-0.487922\pi$
$853$	2.21589	0.0758705	0.0379353	+	0.999280i	$0.487922\pi$
$854$	0	0
$855$	−0.933754	−0.0319337
$856$	0	0
$857$	−0.678188	−0.0231665	−0.0115832	−	0.999933i	$-0.503687\pi$
$857$	−0.678188	−0.0231665	−0.0115832	+	0.999933i	$0.503687\pi$
$858$	0	0
$859$	12.2532	0.418073	0.209036	−	0.977908i	$-0.432967\pi$
$859$	12.2532	0.418073	0.209036	+	0.977908i	$0.432967\pi$
$860$	0	0
$861$	0.505209	0.0172175
$862$	0	0
$863$	−46.7632	−1.59184	−0.795920	−	0.605402i	$-0.793012\pi$
$863$	−46.7632	−1.59184	−0.795920	+	0.605402i	$0.793012\pi$
$864$	0	0
$865$	−14.4578	−0.491582
$866$	0	0
$867$	20.8225	0.707170
$868$	0	0
$869$	4.60400	0.156180
$870$	0	0
$871$	−17.0122	−0.576436
$872$	0	0
$873$	−4.83202	−0.163539
$874$	0	0
$875$	−0.208807	−0.00705895
$876$	0	0
$877$	21.3470	0.720838	0.360419	−	0.932791i	$-0.382634\pi$
$877$	21.3470	0.720838	0.360419	+	0.932791i	$0.382634\pi$
$878$	0	0
$879$	0.0794533	0.00267989
$880$	0	0
$881$	−40.7670	−1.37348	−0.686738	−	0.726905i	$-0.740958\pi$
$881$	−40.7670	−1.37348	−0.686738	+	0.726905i	$0.740958\pi$
$882$	0	0
$883$	34.0744	1.14669	0.573347	−	0.819313i	$-0.305645\pi$
$883$	34.0744	1.14669	0.573347	+	0.819313i	$0.305645\pi$
$884$	0	0
$885$	−12.7093	−0.427217
$886$	0	0
$887$	−33.3868	−1.12102	−0.560510	−	0.828148i	$-0.689395\pi$
$887$	−33.3868	−1.12102	−0.560510	+	0.828148i	$0.689395\pi$
$888$	0	0
$889$	1.55187	0.0520481
$890$	0	0
$891$	4.43366	0.148533
$892$	0	0
$893$	−7.44500	−0.249137
$894$	0	0
$895$	−8.11436	−0.271233
$896$	0	0
$897$	−7.29742	−0.243654
$898$	0	0
$899$	−26.5059	−0.884023
$900$	0	0
$901$	21.0936	0.702729
$902$	0	0
$903$	−0.607541	−0.0202177
$904$	0	0
$905$	22.6472	0.752818
$906$	0	0
$907$	30.3554	1.00794	0.503968	−	0.863723i	$-0.331873\pi$
$907$	30.3554	1.00794	0.503968	+	0.863723i	$0.331873\pi$
$908$	0	0
$909$	6.81574	0.226064
$910$	0	0
$911$	47.3403	1.56845	0.784227	−	0.620474i	$-0.213060\pi$
$911$	47.3403	1.56845	0.784227	+	0.620474i	$0.213060\pi$
$912$	0	0
$913$	8.42556	0.278845
$914$	0	0
$915$	17.6328	0.582923
$916$	0	0
$917$	−1.54618	−0.0510594
$918$	0	0
$919$	6.15031	0.202880	0.101440	−	0.994842i	$-0.467655\pi$
$919$	6.15031	0.202880	0.101440	+	0.994842i	$0.467655\pi$
$920$	0	0
$921$	46.5233	1.53299
$922$	0	0
$923$	−26.2711	−0.864723
$924$	0	0
$925$	4.31510	0.141879
$926$	0	0
$927$	−0.743198	−0.0244098
$928$	0	0
$929$	35.7758	1.17377	0.586883	−	0.809672i	$-0.300355\pi$
$929$	35.7758	1.17377	0.586883	+	0.809672i	$0.300355\pi$
$930$	0	0
$931$	−12.6385	−0.414209
$932$	0	0
$933$	24.7475	0.810196
$934$	0	0
$935$	−1.20040	−0.0392574
$936$	0	0
$937$	6.49148	0.212067	0.106034	−	0.994363i	$-0.466185\pi$
$937$	6.49148	0.212067	0.106034	+	0.994363i	$0.466185\pi$
$938$	0	0
$939$	−30.3505	−0.990452
$940$	0	0
$941$	48.9031	1.59420	0.797098	−	0.603850i	$-0.206367\pi$
$941$	48.9031	1.59420	0.797098	+	0.603850i	$0.206367\pi$
$942$	0	0
$943$	−2.19047	−0.0713316
$944$	0	0
$945$	−1.15690	−0.0376339
$946$	0	0
$947$	23.8981	0.776585	0.388292	−	0.921536i	$-0.373065\pi$
$947$	23.8981	0.776585	0.388292	+	0.921536i	$0.373065\pi$
$948$	0	0
$949$	3.62608	0.117707
$950$	0	0
$951$	12.9842	0.421042
$952$	0	0
$953$	−44.3409	−1.43634	−0.718172	−	0.695866i	$-0.755021\pi$
$953$	−44.3409	−1.43634	−0.718172	+	0.695866i	$0.755021\pi$
$954$	0	0
$955$	3.36141	0.108773
$956$	0	0
$957$	−4.14461	−0.133976
$958$	0	0
$959$	3.75065	0.121115
$960$	0	0
$961$	7.62001	0.245807
$962$	0	0
$963$	−5.53821	−0.178466
$964$	0	0
$965$	3.99457	0.128590
$966$	0	0
$967$	43.2516	1.39088	0.695440	−	0.718585i	$-0.255210\pi$
$967$	43.2516	1.39088	0.695440	+	0.718585i	$0.255210\pi$
$968$	0	0
$969$	−5.57956	−0.179241
$970$	0	0
$971$	−48.3956	−1.55309	−0.776544	−	0.630062i	$-0.783029\pi$
$971$	−48.3956	−1.55309	−0.776544	+	0.630062i	$0.783029\pi$
$972$	0	0
$973$	0.572326	0.0183479
$974$	0	0
$975$	5.11215	0.163720
$976$	0	0
$977$	−4.81703	−0.154110	−0.0770552	−	0.997027i	$-0.524552\pi$
$977$	−4.81703	−0.154110	−0.0770552	+	0.997027i	$0.524552\pi$
$978$	0	0
$979$	8.13362	0.259952
$980$	0	0
$981$	−2.95852	−0.0944583
$982$	0	0
$983$	−0.128226	−0.00408979	−0.00204490	−	0.999998i	$-0.500651\pi$
$983$	−0.128226	−0.00408979	−0.00204490	+	0.999998i	$0.500651\pi$
$984$	0	0
$985$	15.4145	0.491146
$986$	0	0
$987$	−1.34913	−0.0429433
$988$	0	0
$989$	2.63416	0.0837614
$990$	0	0
$991$	10.3265	0.328033	0.164016	−	0.986458i	$-0.447555\pi$
$991$	10.3265	0.328033	0.164016	+	0.986458i	$0.447555\pi$
$992$	0	0
$993$	−14.0008	−0.444301
$994$	0	0
$995$	4.61808	0.146403
$996$	0	0
$997$	49.1178	1.55558	0.777789	−	0.628526i	$-0.216341\pi$
$997$	49.1178	1.55558	0.777789	+	0.628526i	$0.216341\pi$
$998$	0	0
$999$	23.9079	0.756412

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6040.2.a.l.1.3	✓	9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6040.2.a.l.1.3	✓	9	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 6040 = 2^{3} \cdot 5 \cdot 151 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6040.a (trivial)

\(f(q)\)	\(=\)	\(q-1.57672 q^{3} +1.00000 q^{5} -0.208807 q^{7} -0.513952 q^{9} +O(q^{10})\)	\(q-1.57672 q^{3} +1.00000 q^{5} -0.208807 q^{7} -0.513952 q^{9} -0.616299 q^{11} -3.24227 q^{13} -1.57672 q^{15} +1.94776 q^{17} +1.81681 q^{19} +0.329230 q^{21} -1.42747 q^{23} +1.00000 q^{25} +5.54052 q^{27} -4.26518 q^{29} +6.21450 q^{31} +0.971732 q^{33} -0.208807 q^{35} +4.31510 q^{37} +5.11215 q^{39} +1.53452 q^{41} -1.84534 q^{43} -0.513952 q^{45} -4.09784 q^{47} -6.95640 q^{49} -3.07107 q^{51} +10.8297 q^{53} -0.616299 q^{55} -2.86460 q^{57} +8.06057 q^{59} -11.1832 q^{61} +0.107317 q^{63} -3.24227 q^{65} +5.24700 q^{67} +2.25071 q^{69} +8.10268 q^{71} -1.11838 q^{73} -1.57672 q^{75} +0.128687 q^{77} -7.47039 q^{79} -7.19400 q^{81} -13.6712 q^{83} +1.94776 q^{85} +6.72499 q^{87} -13.1975 q^{89} +0.677008 q^{91} -9.79853 q^{93} +1.81681 q^{95} +9.40168 q^{97} +0.316748 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q + 9 q^{5} - 2 q^{7} - 3 q^{9}+O(q^{10}) \) 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

Related objects

Downloads

Learn more