LMFDB - Embedded newform 6028.2.a.d.1.26

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6028,2,Mod(1,6028)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6028, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("6028.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6028 = 2^{2} \cdot 11 \cdot 137 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6028.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:	$48.1338223384$
Analytic rank:	$1$
Dimension:	$27$
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.26
Character	$\chi$	$=$	6028.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.94255 q^{3} -2.68803 q^{5} +0.723147 q^{7} +5.65862 q^{9} +O(q^{10})$	$q+2.94255 q^{3} -2.68803 q^{5} +0.723147 q^{7} +5.65862 q^{9} -1.00000 q^{11} -0.951085 q^{13} -7.90967 q^{15} -0.367297 q^{17} -2.91849 q^{19} +2.12790 q^{21} +0.0834206 q^{23} +2.22549 q^{25} +7.82314 q^{27} -8.77384 q^{29} -6.01515 q^{31} -2.94255 q^{33} -1.94384 q^{35} +10.1517 q^{37} -2.79862 q^{39} -5.88427 q^{41} -8.70325 q^{43} -15.2105 q^{45} -8.78796 q^{47} -6.47706 q^{49} -1.08079 q^{51} +5.17335 q^{53} +2.68803 q^{55} -8.58781 q^{57} -3.81515 q^{59} +14.5731 q^{61} +4.09202 q^{63} +2.55654 q^{65} +5.57451 q^{67} +0.245470 q^{69} +2.81508 q^{71} -2.36148 q^{73} +6.54863 q^{75} -0.723147 q^{77} -2.46321 q^{79} +6.04415 q^{81} -15.5427 q^{83} +0.987304 q^{85} -25.8175 q^{87} +16.0343 q^{89} -0.687774 q^{91} -17.6999 q^{93} +7.84497 q^{95} -16.2429 q^{97} -5.65862 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 27 q - 6 q^{3} + q^{5} - 14 q^{7} + 33 q^{9}+O(q^{10}) $ 27 * q - 6 * q^3 + q^5 - 14 * q^7 + 33 * q^9	$ 27 q - 6 q^{3} + q^{5} - 14 q^{7} + 33 q^{9} - 27 q^{11} - 6 q^{15} - 21 q^{17} - 3 q^{19} - 4 q^{21} - 44 q^{23} + 38 q^{25} - 18 q^{27} + q^{29} - 8 q^{31} + 6 q^{33} - 33 q^{35} + 11 q^{37} - 13 q^{39} - 19 q^{41} - 11 q^{43} + 17 q^{45} - 37 q^{47} + 41 q^{49} - 49 q^{51} - 12 q^{53} - q^{55} - 50 q^{57} - 14 q^{59} + 12 q^{61} - 53 q^{63} - 55 q^{65} - 5 q^{67} + 14 q^{69} - 67 q^{71} - 27 q^{73} - 70 q^{75} + 14 q^{77} - 31 q^{79} - 5 q^{81} - 55 q^{83} - 3 q^{85} - 31 q^{87} + 11 q^{89} - 11 q^{91} - 24 q^{93} - 47 q^{95} - q^{97} - 33 q^{99}+O(q^{100}) $ 27 * q - 6 * q^3 + q^5 - 14 * q^7 + 33 * q^9 - 27 * q^11 - 6 * q^15 - 21 * q^17 - 3 * q^19 - 4 * q^21 - 44 * q^23 + 38 * q^25 - 18 * q^27 + q^29 - 8 * q^31 + 6 * q^33 - 33 * q^35 + 11 * q^37 - 13 * q^39 - 19 * q^41 - 11 * q^43 + 17 * q^45 - 37 * q^47 + 41 * q^49 - 49 * q^51 - 12 * q^53 - q^55 - 50 * q^57 - 14 * q^59 + 12 * q^61 - 53 * q^63 - 55 * q^65 - 5 * q^67 + 14 * q^69 - 67 * q^71 - 27 * q^73 - 70 * q^75 + 14 * q^77 - 31 * q^79 - 5 * q^81 - 55 * q^83 - 3 * q^85 - 31 * q^87 + 11 * q^89 - 11 * q^91 - 24 * q^93 - 47 * q^95 - q^97 - 33 * 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
$3$	2.94255	1.69888	0.849442	−	0.527682i	$-0.176939\pi$
$3$	2.94255	1.69888	0.849442	+	0.527682i	$0.176939\pi$
$4$	0	0
$5$	−2.68803	−1.20212	−0.601061	−	0.799203i	$-0.705255\pi$
$5$	−2.68803	−1.20212	−0.601061	+	0.799203i	$0.705255\pi$
$6$	0	0
$7$	0.723147	0.273324	0.136662	−	0.990618i	$-0.456363\pi$
$7$	0.723147	0.273324	0.136662	+	0.990618i	$0.456363\pi$
$8$	0	0
$9$	5.65862	1.88621
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	−0.951085	−0.263783	−0.131892	−	0.991264i	$-0.542105\pi$
$13$	−0.951085	−0.263783	−0.131892	+	0.991264i	$0.542105\pi$
$14$	0	0
$15$	−7.90967	−2.04227
$16$	0	0
$17$	−0.367297	−0.0890826	−0.0445413	−	0.999008i	$-0.514183\pi$
$17$	−0.367297	−0.0890826	−0.0445413	+	0.999008i	$0.514183\pi$
$18$	0	0
$19$	−2.91849	−0.669547	−0.334773	−	0.942299i	$-0.608660\pi$
$19$	−2.91849	−0.669547	−0.334773	+	0.942299i	$0.608660\pi$
$20$	0	0
$21$	2.12790	0.464346
$22$	0	0
$23$	0.0834206	0.0173944	0.00869720	−	0.999962i	$-0.497232\pi$
$23$	0.0834206	0.0173944	0.00869720	+	0.999962i	$0.497232\pi$
$24$	0	0
$25$	2.22549	0.445099
$26$	0	0
$27$	7.82314	1.50556
$28$	0	0
$29$	−8.77384	−1.62926	−0.814631	−	0.579980i	$-0.803061\pi$
$29$	−8.77384	−1.62926	−0.814631	+	0.579980i	$0.803061\pi$
$30$	0	0
$31$	−6.01515	−1.08035	−0.540176	−	0.841552i	$-0.681642\pi$
$31$	−6.01515	−1.08035	−0.540176	+	0.841552i	$0.681642\pi$
$32$	0	0
$33$	−2.94255	−0.512233
$34$	0	0
$35$	−1.94384	−0.328569
$36$	0	0
$37$	10.1517	1.66893	0.834464	−	0.551062i	$-0.185777\pi$
$37$	10.1517	1.66893	0.834464	+	0.551062i	$0.185777\pi$
$38$	0	0
$39$	−2.79862	−0.448138
$40$	0	0
$41$	−5.88427	−0.918969	−0.459484	−	0.888186i	$-0.651966\pi$
$41$	−5.88427	−0.918969	−0.459484	+	0.888186i	$0.651966\pi$
$42$	0	0
$43$	−8.70325	−1.32723	−0.663616	−	0.748073i	$-0.730979\pi$
$43$	−8.70325	−1.32723	−0.663616	+	0.748073i	$0.730979\pi$
$44$	0	0
$45$	−15.2105	−2.26745
$46$	0	0
$47$	−8.78796	−1.28186	−0.640928	−	0.767601i	$-0.721450\pi$
$47$	−8.78796	−1.28186	−0.640928	+	0.767601i	$0.721450\pi$
$48$	0	0
$49$	−6.47706	−0.925294
$50$	0	0
$51$	−1.08079	−0.151341
$52$	0	0
$53$	5.17335	0.710615	0.355307	−	0.934750i	$-0.384376\pi$
$53$	5.17335	0.710615	0.355307	+	0.934750i	$0.384376\pi$
$54$	0	0
$55$	2.68803	0.362454
$56$	0	0
$57$	−8.58781	−1.13748
$58$	0	0
$59$	−3.81515	−0.496690	−0.248345	−	0.968672i	$-0.579887\pi$
$59$	−3.81515	−0.496690	−0.248345	+	0.968672i	$0.579887\pi$
$60$	0	0
$61$	14.5731	1.86590	0.932949	−	0.360008i	$-0.117226\pi$
$61$	14.5731	1.86590	0.932949	+	0.360008i	$0.117226\pi$
$62$	0	0
$63$	4.09202	0.515546
$64$	0	0
$65$	2.55654	0.317100
$66$	0	0
$67$	5.57451	0.681034	0.340517	−	0.940238i	$-0.389398\pi$
$67$	5.57451	0.681034	0.340517	+	0.940238i	$0.389398\pi$
$68$	0	0
$69$	0.245470	0.0295511
$70$	0	0
$71$	2.81508	0.334088	0.167044	−	0.985949i	$-0.446578\pi$
$71$	2.81508	0.334088	0.167044	+	0.985949i	$0.446578\pi$
$72$	0	0
$73$	−2.36148	−0.276390	−0.138195	−	0.990405i	$-0.544130\pi$
$73$	−2.36148	−0.276390	−0.138195	+	0.990405i	$0.544130\pi$
$74$	0	0
$75$	6.54863	0.756171
$76$	0	0
$77$	−0.723147	−0.0824102
$78$	0	0
$79$	−2.46321	−0.277133	−0.138566	−	0.990353i	$-0.544249\pi$
$79$	−2.46321	−0.277133	−0.138566	+	0.990353i	$0.544249\pi$
$80$	0	0
$81$	6.04415	0.671572
$82$	0	0
$83$	−15.5427	−1.70604	−0.853019	−	0.521880i	$-0.825231\pi$
$83$	−15.5427	−1.70604	−0.853019	+	0.521880i	$0.825231\pi$
$84$	0	0
$85$	0.987304	0.107088
$86$	0	0
$87$	−25.8175	−2.76793
$88$	0	0
$89$	16.0343	1.69964	0.849818	−	0.527077i	$-0.176712\pi$
$89$	16.0343	1.69964	0.849818	+	0.527077i	$0.176712\pi$
$90$	0	0
$91$	−0.687774	−0.0720983
$92$	0	0
$93$	−17.6999	−1.83539
$94$	0	0
$95$	7.84497	0.804877
$96$	0	0
$97$	−16.2429	−1.64921	−0.824607	−	0.565707i	$-0.808604\pi$
$97$	−16.2429	−1.64921	−0.824607	+	0.565707i	$0.808604\pi$
$98$	0	0
$99$	−5.65862	−0.568713
$100$	0	0
$101$	−5.52050	−0.549310	−0.274655	−	0.961543i	$-0.588564\pi$
$101$	−5.52050	−0.549310	−0.274655	+	0.961543i	$0.588564\pi$
$102$	0	0
$103$	14.4465	1.42345	0.711726	−	0.702457i	$-0.247914\pi$
$103$	14.4465	1.42345	0.711726	+	0.702457i	$0.247914\pi$
$104$	0	0
$105$	−5.71985	−0.558200
$106$	0	0
$107$	−13.1582	−1.27206	−0.636028	−	0.771666i	$-0.719424\pi$
$107$	−13.1582	−1.27206	−0.636028	+	0.771666i	$0.719424\pi$
$108$	0	0
$109$	10.5953	1.01484	0.507421	−	0.861699i	$-0.330599\pi$
$109$	10.5953	1.01484	0.507421	+	0.861699i	$0.330599\pi$
$110$	0	0
$111$	29.8719	2.83532
$112$	0	0
$113$	−10.7308	−1.00947	−0.504735	−	0.863274i	$-0.668410\pi$
$113$	−10.7308	−1.00947	−0.504735	+	0.863274i	$0.668410\pi$
$114$	0	0
$115$	−0.224237	−0.0209102
$116$	0	0
$117$	−5.38183	−0.497550
$118$	0	0
$119$	−0.265610	−0.0243484
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−17.3148	−1.56122
$124$	0	0
$125$	7.45795	0.667060
$126$	0	0
$127$	−1.77257	−0.157290	−0.0786450	−	0.996903i	$-0.525059\pi$
$127$	−1.77257	−0.157290	−0.0786450	+	0.996903i	$0.525059\pi$
$128$	0	0
$129$	−25.6098	−2.25482
$130$	0	0
$131$	−15.0481	−1.31476	−0.657381	−	0.753559i	$-0.728336\pi$
$131$	−15.0481	−1.31476	−0.657381	+	0.753559i	$0.728336\pi$
$132$	0	0
$133$	−2.11050	−0.183003
$134$	0	0
$135$	−21.0288	−1.80987
$136$	0	0
$137$	1.00000	0.0854358
$137$	1.00000	0.0854358
$138$	0	0
$139$	−10.1569	−0.861500	−0.430750	−	0.902471i	$-0.641751\pi$
$139$	−10.1569	−0.861500	−0.430750	+	0.902471i	$0.641751\pi$
$140$	0	0
$141$	−25.8591	−2.17773
$142$	0	0
$143$	0.951085	0.0795337
$144$	0	0
$145$	23.5843	1.95857
$146$	0	0
$147$	−19.0591	−1.57197
$148$	0	0
$149$	9.99837	0.819098	0.409549	−	0.912288i	$-0.365686\pi$
$149$	9.99837	0.819098	0.409549	+	0.912288i	$0.365686\pi$
$150$	0	0
$151$	10.5364	0.857436	0.428718	−	0.903438i	$-0.358965\pi$
$151$	10.5364	0.857436	0.428718	+	0.903438i	$0.358965\pi$
$152$	0	0
$153$	−2.07839	−0.168028
$154$	0	0
$155$	16.1689	1.29872
$156$	0	0
$157$	−5.33249	−0.425579	−0.212789	−	0.977098i	$-0.568255\pi$
$157$	−5.33249	−0.425579	−0.212789	+	0.977098i	$0.568255\pi$
$158$	0	0
$159$	15.2229	1.20725
$160$	0	0
$161$	0.0603254	0.00475431
$162$	0	0
$163$	−7.07086	−0.553832	−0.276916	−	0.960894i	$-0.589312\pi$
$163$	−7.07086	−0.553832	−0.276916	+	0.960894i	$0.589312\pi$
$164$	0	0
$165$	7.90967	0.615767
$166$	0	0
$167$	4.17024	0.322703	0.161351	−	0.986897i	$-0.448415\pi$
$167$	4.17024	0.322703	0.161351	+	0.986897i	$0.448415\pi$
$168$	0	0
$169$	−12.0954	−0.930418
$170$	0	0
$171$	−16.5146	−1.26290
$172$	0	0
$173$	16.1033	1.22431	0.612155	−	0.790738i	$-0.290303\pi$
$173$	16.1033	1.22431	0.612155	+	0.790738i	$0.290303\pi$
$174$	0	0
$175$	1.60936	0.121656
$176$	0	0
$177$	−11.2263	−0.843819
$178$	0	0
$179$	−4.79903	−0.358697	−0.179348	−	0.983786i	$-0.557399\pi$
$179$	−4.79903	−0.358697	−0.179348	+	0.983786i	$0.557399\pi$
$180$	0	0
$181$	−17.6897	−1.31487	−0.657433	−	0.753513i	$-0.728358\pi$
$181$	−17.6897	−1.31487	−0.657433	+	0.753513i	$0.728358\pi$
$182$	0	0
$183$	42.8822	3.16995
$184$	0	0
$185$	−27.2880	−2.00626
$186$	0	0
$187$	0.367297	0.0268594
$188$	0	0
$189$	5.65728	0.411507
$190$	0	0
$191$	−6.77899	−0.490510	−0.245255	−	0.969459i	$-0.578872\pi$
$191$	−6.77899	−0.490510	−0.245255	+	0.969459i	$0.578872\pi$
$192$	0	0
$193$	9.00187	0.647968	0.323984	−	0.946062i	$-0.394978\pi$
$193$	9.00187	0.647968	0.323984	+	0.946062i	$0.394978\pi$
$194$	0	0
$195$	7.52276	0.538716
$196$	0	0
$197$	24.8096	1.76761	0.883804	−	0.467856i	$-0.154974\pi$
$197$	24.8096	1.76761	0.883804	+	0.467856i	$0.154974\pi$
$198$	0	0
$199$	12.0796	0.856300	0.428150	−	0.903708i	$-0.359165\pi$
$199$	12.0796	0.856300	0.428150	+	0.903708i	$0.359165\pi$
$200$	0	0
$201$	16.4033	1.15700
$202$	0	0
$203$	−6.34478	−0.445316
$204$	0	0
$205$	15.8171	1.10471
$206$	0	0
$207$	0.472046	0.0328095
$208$	0	0
$209$	2.91849	0.201876
$210$	0	0
$211$	−2.66955	−0.183779	−0.0918897	−	0.995769i	$-0.529291\pi$
$211$	−2.66955	−0.183779	−0.0918897	+	0.995769i	$0.529291\pi$
$212$	0	0
$213$	8.28352	0.567578
$214$	0	0
$215$	23.3946	1.59550
$216$	0	0
$217$	−4.34983	−0.295286
$218$	0	0
$219$	−6.94878	−0.469555
$220$	0	0
$221$	0.349330	0.0234985
$222$	0	0
$223$	−0.429317	−0.0287492	−0.0143746	−	0.999897i	$-0.504576\pi$
$223$	−0.429317	−0.0287492	−0.0143746	+	0.999897i	$0.504576\pi$
$224$	0	0
$225$	12.5932	0.839548
$226$	0	0
$227$	−11.0421	−0.732891	−0.366446	−	0.930439i	$-0.619425\pi$
$227$	−11.0421	−0.732891	−0.366446	+	0.930439i	$0.619425\pi$
$228$	0	0
$229$	−3.13946	−0.207461	−0.103731	−	0.994605i	$-0.533078\pi$
$229$	−3.13946	−0.207461	−0.103731	+	0.994605i	$0.533078\pi$
$230$	0	0
$231$	−2.12790	−0.140005
$232$	0	0
$233$	9.14584	0.599164	0.299582	−	0.954071i	$-0.403153\pi$
$233$	9.14584	0.599164	0.299582	+	0.954071i	$0.403153\pi$
$234$	0	0
$235$	23.6223	1.54095
$236$	0	0
$237$	−7.24813	−0.470816
$238$	0	0
$239$	−23.9358	−1.54828	−0.774138	−	0.633017i	$-0.781816\pi$
$239$	−23.9358	−1.54828	−0.774138	+	0.633017i	$0.781816\pi$
$240$	0	0
$241$	1.32935	0.0856307	0.0428154	−	0.999083i	$-0.486367\pi$
$241$	1.32935	0.0856307	0.0428154	+	0.999083i	$0.486367\pi$
$242$	0	0
$243$	−5.68420	−0.364642
$244$	0	0
$245$	17.4105	1.11232
$246$	0	0
$247$	2.77573	0.176615
$248$	0	0
$249$	−45.7354	−2.89836
$250$	0	0
$251$	−4.96934	−0.313662	−0.156831	−	0.987625i	$-0.550128\pi$
$251$	−4.96934	−0.313662	−0.156831	+	0.987625i	$0.550128\pi$
$252$	0	0
$253$	−0.0834206	−0.00524461
$254$	0	0
$255$	2.90520	0.181930
$256$	0	0
$257$	−3.27709	−0.204419	−0.102210	−	0.994763i	$-0.532591\pi$
$257$	−3.27709	−0.204419	−0.102210	+	0.994763i	$0.532591\pi$
$258$	0	0
$259$	7.34117	0.456158
$260$	0	0
$261$	−49.6479	−3.07313
$262$	0	0
$263$	2.07676	0.128058	0.0640291	−	0.997948i	$-0.479605\pi$
$263$	2.07676	0.128058	0.0640291	+	0.997948i	$0.479605\pi$
$264$	0	0
$265$	−13.9061	−0.854246
$266$	0	0
$267$	47.1819	2.88748
$268$	0	0
$269$	−19.7784	−1.20591	−0.602954	−	0.797776i	$-0.706010\pi$
$269$	−19.7784	−1.20591	−0.602954	+	0.797776i	$0.706010\pi$
$270$	0	0
$271$	−12.9967	−0.789492	−0.394746	−	0.918790i	$-0.629167\pi$
$271$	−12.9967	−0.789492	−0.394746	+	0.918790i	$0.629167\pi$
$272$	0	0
$273$	−2.02381	−0.122487
$274$	0	0
$275$	−2.22549	−0.134202
$276$	0	0
$277$	−28.2839	−1.69941	−0.849706	−	0.527257i	$-0.823221\pi$
$277$	−28.2839	−1.69941	−0.849706	+	0.527257i	$0.823221\pi$
$278$	0	0
$279$	−34.0374	−2.03777
$280$	0	0
$281$	−11.9153	−0.710805	−0.355402	−	0.934713i	$-0.615656\pi$
$281$	−11.9153	−0.710805	−0.355402	+	0.934713i	$0.615656\pi$
$282$	0	0
$283$	24.0010	1.42671	0.713356	−	0.700802i	$-0.247174\pi$
$283$	24.0010	1.42671	0.713356	+	0.700802i	$0.247174\pi$
$284$	0	0
$285$	23.0843	1.36739
$286$	0	0
$287$	−4.25519	−0.251176
$288$	0	0
$289$	−16.8651	−0.992064
$290$	0	0
$291$	−47.7955	−2.80182
$292$	0	0
$293$	−29.5493	−1.72629	−0.863144	−	0.504958i	$-0.831508\pi$
$293$	−29.5493	−1.72629	−0.863144	+	0.504958i	$0.831508\pi$
$294$	0	0
$295$	10.2552	0.597082
$296$	0	0
$297$	−7.82314	−0.453945
$298$	0	0
$299$	−0.0793401	−0.00458836
$300$	0	0
$301$	−6.29373	−0.362764
$302$	0	0
$303$	−16.2444	−0.933215
$304$	0	0
$305$	−39.1730	−2.24304
$306$	0	0
$307$	0.228578	0.0130457	0.00652283	−	0.999979i	$-0.497924\pi$
$307$	0.228578	0.0130457	0.00652283	+	0.999979i	$0.497924\pi$
$308$	0	0
$309$	42.5095	2.41828
$310$	0	0
$311$	11.5477	0.654809	0.327404	−	0.944884i	$-0.393826\pi$
$311$	11.5477	0.654809	0.327404	+	0.944884i	$0.393826\pi$
$312$	0	0
$313$	4.76639	0.269412	0.134706	−	0.990886i	$-0.456991\pi$
$313$	4.76639	0.269412	0.134706	+	0.990886i	$0.456991\pi$
$314$	0	0
$315$	−10.9995	−0.619749
$316$	0	0
$317$	−6.38578	−0.358661	−0.179331	−	0.983789i	$-0.557393\pi$
$317$	−6.38578	−0.358661	−0.179331	+	0.983789i	$0.557393\pi$
$318$	0	0
$319$	8.77384	0.491241
$320$	0	0
$321$	−38.7188	−2.16108
$322$	0	0
$323$	1.07195	0.0596450
$324$	0	0
$325$	−2.11663	−0.117410
$326$	0	0
$327$	31.1771	1.72410
$328$	0	0
$329$	−6.35499	−0.350362
$330$	0	0
$331$	26.8425	1.47540	0.737700	−	0.675129i	$-0.235912\pi$
$331$	26.8425	1.47540	0.737700	+	0.675129i	$0.235912\pi$
$332$	0	0
$333$	57.4446	3.14795
$334$	0	0
$335$	−14.9844	−0.818687
$336$	0	0
$337$	−1.10740	−0.0603237	−0.0301618	−	0.999545i	$-0.509602\pi$
$337$	−1.10740	−0.0603237	−0.0301618	+	0.999545i	$0.509602\pi$
$338$	0	0
$339$	−31.5760	−1.71497
$340$	0	0
$341$	6.01515	0.325738
$342$	0	0
$343$	−9.74589	−0.526229
$344$	0	0
$345$	−0.659829	−0.0355240
$346$	0	0
$347$	24.4546	1.31279	0.656395	−	0.754417i	$-0.272080\pi$
$347$	24.4546	1.31279	0.656395	+	0.754417i	$0.272080\pi$
$348$	0	0
$349$	14.1693	0.758468	0.379234	−	0.925301i	$-0.376188\pi$
$349$	14.1693	0.758468	0.379234	+	0.925301i	$0.376188\pi$
$350$	0	0
$351$	−7.44047	−0.397143
$352$	0	0
$353$	19.4925	1.03748	0.518741	−	0.854932i	$-0.326401\pi$
$353$	19.4925	1.03748	0.518741	+	0.854932i	$0.326401\pi$
$354$	0	0
$355$	−7.56701	−0.401615
$356$	0	0
$357$	−0.781571	−0.0413651
$358$	0	0
$359$	21.8229	1.15177	0.575884	−	0.817531i	$-0.304658\pi$
$359$	21.8229	1.15177	0.575884	+	0.817531i	$0.304658\pi$
$360$	0	0
$361$	−10.4824	−0.551707
$362$	0	0
$363$	2.94255	0.154444
$364$	0	0
$365$	6.34772	0.332255
$366$	0	0
$367$	11.1128	0.580083	0.290042	−	0.957014i	$-0.406331\pi$
$367$	11.1128	0.580083	0.290042	+	0.957014i	$0.406331\pi$
$368$	0	0
$369$	−33.2969	−1.73337
$370$	0	0
$371$	3.74109	0.194228
$372$	0	0
$373$	−16.2052	−0.839073	−0.419536	−	0.907739i	$-0.637807\pi$
$373$	−16.2052	−0.839073	−0.419536	+	0.907739i	$0.637807\pi$
$374$	0	0
$375$	21.9454	1.13326
$376$	0	0
$377$	8.34467	0.429772
$378$	0	0
$379$	12.1936	0.626344	0.313172	−	0.949696i	$-0.398608\pi$
$379$	12.1936	0.626344	0.313172	+	0.949696i	$0.398608\pi$
$380$	0	0
$381$	−5.21588	−0.267217
$382$	0	0
$383$	−7.02375	−0.358897	−0.179449	−	0.983767i	$-0.557431\pi$
$383$	−7.02375	−0.358897	−0.179449	+	0.983767i	$0.557431\pi$
$384$	0	0
$385$	1.94384	0.0990672
$386$	0	0
$387$	−49.2484	−2.50344
$388$	0	0
$389$	−19.0234	−0.964523	−0.482262	−	0.876027i	$-0.660185\pi$
$389$	−19.0234	−0.964523	−0.482262	+	0.876027i	$0.660185\pi$
$390$	0	0
$391$	−0.0306401	−0.00154954
$392$	0	0
$393$	−44.2799	−2.23363
$394$	0	0
$395$	6.62118	0.333148
$396$	0	0
$397$	20.2215	1.01489	0.507444	−	0.861685i	$-0.330590\pi$
$397$	20.2215	1.01489	0.507444	+	0.861685i	$0.330590\pi$
$398$	0	0
$399$	−6.21025	−0.310901
$400$	0	0
$401$	5.99258	0.299255	0.149628	−	0.988742i	$-0.452193\pi$
$401$	5.99258	0.299255	0.149628	+	0.988742i	$0.452193\pi$
$402$	0	0
$403$	5.72091	0.284979
$404$	0	0
$405$	−16.2468	−0.807312
$406$	0	0
$407$	−10.1517	−0.503201
$408$	0	0
$409$	−35.4352	−1.75216	−0.876078	−	0.482169i	$-0.839849\pi$
$409$	−35.4352	−1.75216	−0.876078	+	0.482169i	$0.839849\pi$
$410$	0	0
$411$	2.94255	0.145145
$412$	0	0
$413$	−2.75891	−0.135757
$414$	0	0
$415$	41.7793	2.05087
$416$	0	0
$417$	−29.8873	−1.46359
$418$	0	0
$419$	−9.14615	−0.446819	−0.223409	−	0.974725i	$-0.571719\pi$
$419$	−9.14615	−0.446819	−0.223409	+	0.974725i	$0.571719\pi$
$420$	0	0
$421$	−19.5366	−0.952155	−0.476078	−	0.879403i	$-0.657942\pi$
$421$	−19.5366	−0.952155	−0.476078	+	0.879403i	$0.657942\pi$
$422$	0	0
$423$	−49.7278	−2.41785
$424$	0	0
$425$	−0.817417	−0.0396505
$426$	0	0
$427$	10.5385	0.509995
$428$	0	0
$429$	2.79862	0.135119
$430$	0	0
$431$	−24.2888	−1.16995	−0.584975	−	0.811051i	$-0.698896\pi$
$431$	−24.2888	−1.16995	−0.584975	+	0.811051i	$0.698896\pi$
$432$	0	0
$433$	18.8308	0.904951	0.452475	−	0.891777i	$-0.350541\pi$
$433$	18.8308	0.904951	0.452475	+	0.891777i	$0.350541\pi$
$434$	0	0
$435$	69.3982	3.32739
$436$	0	0
$437$	−0.243462	−0.0116464
$438$	0	0
$439$	−30.7843	−1.46926	−0.734628	−	0.678470i	$-0.762643\pi$
$439$	−30.7843	−1.46926	−0.734628	+	0.678470i	$0.762643\pi$
$440$	0	0
$441$	−36.6512	−1.74530
$442$	0	0
$443$	35.6426	1.69343	0.846716	−	0.532045i	$-0.178576\pi$
$443$	35.6426	1.69343	0.846716	+	0.532045i	$0.178576\pi$
$444$	0	0
$445$	−43.1007	−2.04317
$446$	0	0
$447$	29.4207	1.39155
$448$	0	0
$449$	37.4471	1.76724	0.883619	−	0.468206i	$-0.155100\pi$
$449$	37.4471	1.76724	0.883619	+	0.468206i	$0.155100\pi$
$450$	0	0
$451$	5.88427	0.277080
$452$	0	0
$453$	31.0038	1.45669
$454$	0	0
$455$	1.84876	0.0866710
$456$	0	0
$457$	7.13893	0.333945	0.166973	−	0.985962i	$-0.446601\pi$
$457$	7.13893	0.333945	0.166973	+	0.985962i	$0.446601\pi$
$458$	0	0
$459$	−2.87342	−0.134120
$460$	0	0
$461$	−30.1366	−1.40360	−0.701801	−	0.712373i	$-0.747620\pi$
$461$	−30.1366	−1.40360	−0.701801	+	0.712373i	$0.747620\pi$
$462$	0	0
$463$	13.1051	0.609046	0.304523	−	0.952505i	$-0.401503\pi$
$463$	13.1051	0.609046	0.304523	+	0.952505i	$0.401503\pi$
$464$	0	0
$465$	47.5778	2.20637
$466$	0	0
$467$	−19.5254	−0.903527	−0.451763	−	0.892138i	$-0.649205\pi$
$467$	−19.5254	−0.903527	−0.451763	+	0.892138i	$0.649205\pi$
$468$	0	0
$469$	4.03119	0.186143
$470$	0	0
$471$	−15.6911	−0.723009
$472$	0	0
$473$	8.70325	0.400176
$474$	0	0
$475$	−6.49507	−0.298014
$476$	0	0
$477$	29.2740	1.34037
$478$	0	0
$479$	18.9380	0.865297	0.432649	−	0.901563i	$-0.357579\pi$
$479$	18.9380	0.865297	0.432649	+	0.901563i	$0.357579\pi$
$480$	0	0
$481$	−9.65512	−0.440236
$482$	0	0
$483$	0.177511	0.00807702
$484$	0	0
$485$	43.6613	1.98256
$486$	0	0
$487$	24.5189	1.11106	0.555529	−	0.831497i	$-0.312516\pi$
$487$	24.5189	1.11106	0.555529	+	0.831497i	$0.312516\pi$
$488$	0	0
$489$	−20.8064	−0.940897
$490$	0	0
$491$	−4.08435	−0.184324	−0.0921621	−	0.995744i	$-0.529378\pi$
$491$	−4.08435	−0.184324	−0.0921621	+	0.995744i	$0.529378\pi$
$492$	0	0
$493$	3.22261	0.145139
$494$	0	0
$495$	15.2105	0.683663
$496$	0	0
$497$	2.03572	0.0913143
$498$	0	0
$499$	42.2560	1.89164	0.945820	−	0.324692i	$-0.105261\pi$
$499$	42.2560	1.89164	0.945820	+	0.324692i	$0.105261\pi$
$500$	0	0
$501$	12.2712	0.548235
$502$	0	0
$503$	−38.5023	−1.71673	−0.858367	−	0.513036i	$-0.828521\pi$
$503$	−38.5023	−1.71673	−0.858367	+	0.513036i	$0.828521\pi$
$504$	0	0
$505$	14.8393	0.660338
$506$	0	0
$507$	−35.5915	−1.58067
$508$	0	0
$509$	−5.57862	−0.247268	−0.123634	−	0.992328i	$-0.539455\pi$
$509$	−5.57862	−0.247268	−0.123634	+	0.992328i	$0.539455\pi$
$510$	0	0
$511$	−1.70770	−0.0755440
$512$	0	0
$513$	−22.8317	−1.00805
$514$	0	0
$515$	−38.8325	−1.71116
$516$	0	0
$517$	8.78796	0.386494
$518$	0	0
$519$	47.3848	2.07996
$520$	0	0
$521$	27.7554	1.21599	0.607994	−	0.793942i	$-0.291975\pi$
$521$	27.7554	1.21599	0.607994	+	0.793942i	$0.291975\pi$
$522$	0	0
$523$	−3.42179	−0.149624	−0.0748121	−	0.997198i	$-0.523836\pi$
$523$	−3.42179	−0.149624	−0.0748121	+	0.997198i	$0.523836\pi$
$524$	0	0
$525$	4.73562	0.206680
$526$	0	0
$527$	2.20934	0.0962406
$528$	0	0
$529$	−22.9930	−0.999697
$530$	0	0
$531$	−21.5885	−0.936861
$532$	0	0
$533$	5.59644	0.242409
$534$	0	0
$535$	35.3697	1.52917
$536$	0	0
$537$	−14.1214	−0.609384
$538$	0	0
$539$	6.47706	0.278987
$540$	0	0
$541$	35.1972	1.51325	0.756623	−	0.653851i	$-0.226848\pi$
$541$	35.1972	1.51325	0.756623	+	0.653851i	$0.226848\pi$
$542$	0	0
$543$	−52.0529	−2.23380
$544$	0	0
$545$	−28.4803	−1.21996
$546$	0	0
$547$	35.7383	1.52806	0.764031	−	0.645180i	$-0.223218\pi$
$547$	35.7383	1.52806	0.764031	+	0.645180i	$0.223218\pi$
$548$	0	0
$549$	82.4639	3.51947
$550$	0	0
$551$	25.6064	1.09087
$552$	0	0
$553$	−1.78126	−0.0757470
$554$	0	0
$555$	−80.2965	−3.40840
$556$	0	0
$557$	−9.12996	−0.386849	−0.193424	−	0.981115i	$-0.561959\pi$
$557$	−9.12996	−0.386849	−0.193424	+	0.981115i	$0.561959\pi$
$558$	0	0
$559$	8.27753	0.350102
$560$	0	0
$561$	1.08079	0.0456310
$562$	0	0
$563$	−14.7082	−0.619877	−0.309939	−	0.950757i	$-0.600309\pi$
$563$	−14.7082	−0.619877	−0.309939	+	0.950757i	$0.600309\pi$
$564$	0	0
$565$	28.8447	1.21351
$566$	0	0
$567$	4.37081	0.183557
$568$	0	0
$569$	17.5362	0.735155	0.367577	−	0.929993i	$-0.380187\pi$
$569$	17.5362	0.735155	0.367577	+	0.929993i	$0.380187\pi$
$570$	0	0
$571$	3.06644	0.128326	0.0641632	−	0.997939i	$-0.479562\pi$
$571$	3.06644	0.128326	0.0641632	+	0.997939i	$0.479562\pi$
$572$	0	0
$573$	−19.9475	−0.833320
$574$	0	0
$575$	0.185652	0.00774222
$576$	0	0
$577$	1.65422	0.0688659	0.0344329	−	0.999407i	$-0.489037\pi$
$577$	1.65422	0.0688659	0.0344329	+	0.999407i	$0.489037\pi$
$578$	0	0
$579$	26.4885	1.10082
$580$	0	0
$581$	−11.2397	−0.466301
$582$	0	0
$583$	−5.17335	−0.214258
$584$	0	0
$585$	14.4665	0.598116
$586$	0	0
$587$	43.3634	1.78980	0.894900	−	0.446268i	$-0.147247\pi$
$587$	43.3634	1.78980	0.894900	+	0.446268i	$0.147247\pi$
$588$	0	0
$589$	17.5551	0.723346
$590$	0	0
$591$	73.0035	3.00296
$592$	0	0
$593$	8.38531	0.344344	0.172172	−	0.985067i	$-0.444922\pi$
$593$	8.38531	0.344344	0.172172	+	0.985067i	$0.444922\pi$
$594$	0	0
$595$	0.713966	0.0292698
$596$	0	0
$597$	35.5449	1.45476
$598$	0	0
$599$	−11.7124	−0.478555	−0.239277	−	0.970951i	$-0.576911\pi$
$599$	−11.7124	−0.478555	−0.239277	+	0.970951i	$0.576911\pi$
$600$	0	0
$601$	9.48059	0.386721	0.193361	−	0.981128i	$-0.438061\pi$
$601$	9.48059	0.386721	0.193361	+	0.981128i	$0.438061\pi$
$602$	0	0
$603$	31.5440	1.28457
$604$	0	0
$605$	−2.68803	−0.109284
$606$	0	0
$607$	−10.0897	−0.409528	−0.204764	−	0.978811i	$-0.565643\pi$
$607$	−10.0897	−0.409528	−0.204764	+	0.978811i	$0.565643\pi$
$608$	0	0
$609$	−18.6699	−0.756541
$610$	0	0
$611$	8.35810	0.338132
$612$	0	0
$613$	21.7024	0.876553	0.438277	−	0.898840i	$-0.355589\pi$
$613$	21.7024	0.876553	0.438277	+	0.898840i	$0.355589\pi$
$614$	0	0
$615$	46.5426	1.87678
$616$	0	0
$617$	1.71840	0.0691801	0.0345901	−	0.999402i	$-0.488987\pi$
$617$	1.71840	0.0691801	0.0345901	+	0.999402i	$0.488987\pi$
$618$	0	0
$619$	−13.3284	−0.535714	−0.267857	−	0.963459i	$-0.586316\pi$
$619$	−13.3284	−0.535714	−0.267857	+	0.963459i	$0.586316\pi$
$620$	0	0
$621$	0.652612	0.0261884
$622$	0	0
$623$	11.5952	0.464551
$624$	0	0
$625$	−31.1746	−1.24699
$626$	0	0
$627$	8.58781	0.342964
$628$	0	0
$629$	−3.72869	−0.148672
$630$	0	0
$631$	43.5890	1.73525	0.867625	−	0.497219i	$-0.165645\pi$
$631$	43.5890	1.73525	0.867625	+	0.497219i	$0.165645\pi$
$632$	0	0
$633$	−7.85530	−0.312220
$634$	0	0
$635$	4.76471	0.189082
$636$	0	0
$637$	6.16023	0.244077
$638$	0	0
$639$	15.9295	0.630160
$640$	0	0
$641$	23.2000	0.916346	0.458173	−	0.888863i	$-0.348504\pi$
$641$	23.2000	0.916346	0.458173	+	0.888863i	$0.348504\pi$
$642$	0	0
$643$	49.2115	1.94071	0.970356	−	0.241680i	$-0.0776986\pi$
$643$	49.2115	1.94071	0.970356	+	0.241680i	$0.0776986\pi$
$644$	0	0
$645$	68.8398	2.71056
$646$	0	0
$647$	26.5099	1.04221	0.521106	−	0.853492i	$-0.325520\pi$
$647$	26.5099	1.04221	0.521106	+	0.853492i	$0.325520\pi$
$648$	0	0
$649$	3.81515	0.149758
$650$	0	0
$651$	−12.7996	−0.501657
$652$	0	0
$653$	−41.5060	−1.62426	−0.812128	−	0.583480i	$-0.801691\pi$
$653$	−41.5060	−1.62426	−0.812128	+	0.583480i	$0.801691\pi$
$654$	0	0
$655$	40.4498	1.58050
$656$	0	0
$657$	−13.3627	−0.521329
$658$	0	0
$659$	−22.4947	−0.876270	−0.438135	−	0.898909i	$-0.644361\pi$
$659$	−22.4947	−0.876270	−0.438135	+	0.898909i	$0.644361\pi$
$660$	0	0
$661$	45.3991	1.76582	0.882910	−	0.469543i	$-0.155581\pi$
$661$	45.3991	1.76582	0.882910	+	0.469543i	$0.155581\pi$
$662$	0	0
$663$	1.02792	0.0399213
$664$	0	0
$665$	5.67307	0.219992
$666$	0	0
$667$	−0.731920	−0.0283400
$668$	0	0
$669$	−1.26329	−0.0488416
$670$	0	0
$671$	−14.5731	−0.562589
$672$	0	0
$673$	−35.3300	−1.36187	−0.680935	−	0.732344i	$-0.738426\pi$
$673$	−35.3300	−1.36187	−0.680935	+	0.732344i	$0.738426\pi$
$674$	0	0
$675$	17.4103	0.670124
$676$	0	0
$677$	33.5056	1.28773	0.643863	−	0.765141i	$-0.277331\pi$
$677$	33.5056	1.28773	0.643863	+	0.765141i	$0.277331\pi$
$678$	0	0
$679$	−11.7460	−0.450769
$680$	0	0
$681$	−32.4920	−1.24510
$682$	0	0
$683$	10.3575	0.396319	0.198159	−	0.980170i	$-0.436504\pi$
$683$	10.3575	0.396319	0.198159	+	0.980170i	$0.436504\pi$
$684$	0	0
$685$	−2.68803	−0.102704
$686$	0	0
$687$	−9.23803	−0.352453
$688$	0	0
$689$	−4.92030	−0.187448
$690$	0	0
$691$	−30.7223	−1.16873	−0.584366	−	0.811490i	$-0.698657\pi$
$691$	−30.7223	−1.16873	−0.584366	+	0.811490i	$0.698657\pi$
$692$	0	0
$693$	−4.09202	−0.155443
$694$	0	0
$695$	27.3021	1.03563
$696$	0	0
$697$	2.16127	0.0818641
$698$	0	0
$699$	26.9121	1.01791
$700$	0	0
$701$	−34.6940	−1.31037	−0.655187	−	0.755467i	$-0.727410\pi$
$701$	−34.6940	−1.31037	−0.655187	+	0.755467i	$0.727410\pi$
$702$	0	0
$703$	−29.6276	−1.11743
$704$	0	0
$705$	69.5099	2.61789
$706$	0	0
$707$	−3.99213	−0.150140
$708$	0	0
$709$	10.9064	0.409599	0.204800	−	0.978804i	$-0.434346\pi$
$709$	10.9064	0.409599	0.204800	+	0.978804i	$0.434346\pi$
$710$	0	0
$711$	−13.9384	−0.522730
$712$	0	0
$713$	−0.501787	−0.0187921
$714$	0	0
$715$	−2.55654	−0.0956093
$716$	0	0
$717$	−70.4323	−2.63034
$718$	0	0
$719$	−33.7241	−1.25770	−0.628848	−	0.777528i	$-0.716473\pi$
$719$	−33.7241	−1.25770	−0.628848	+	0.777528i	$0.716473\pi$
$720$	0	0
$721$	10.4469	0.389063
$722$	0	0
$723$	3.91167	0.145477
$724$	0	0
$725$	−19.5261	−0.725182
$726$	0	0
$727$	−25.2456	−0.936309	−0.468154	−	0.883647i	$-0.655081\pi$
$727$	−25.2456	−0.936309	−0.468154	+	0.883647i	$0.655081\pi$
$728$	0	0
$729$	−34.8585	−1.29106
$730$	0	0
$731$	3.19668	0.118233
$732$	0	0
$733$	52.2463	1.92976	0.964881	−	0.262686i	$-0.0846083\pi$
$733$	52.2463	1.92976	0.964881	+	0.262686i	$0.0846083\pi$
$734$	0	0
$735$	51.2314	1.88970
$736$	0	0
$737$	−5.57451	−0.205340
$738$	0	0
$739$	35.4002	1.30222	0.651109	−	0.758985i	$-0.274304\pi$
$739$	35.4002	1.30222	0.651109	+	0.758985i	$0.274304\pi$
$740$	0	0
$741$	8.16773	0.300049
$742$	0	0
$743$	−38.0878	−1.39731	−0.698653	−	0.715461i	$-0.746217\pi$
$743$	−38.0878	−1.39731	−0.698653	+	0.715461i	$0.746217\pi$
$744$	0	0
$745$	−26.8759	−0.984657
$746$	0	0
$747$	−87.9505	−3.21794
$748$	0	0
$749$	−9.51535	−0.347683
$750$	0	0
$751$	41.7104	1.52203	0.761017	−	0.648731i	$-0.224700\pi$
$751$	41.7104	1.52203	0.761017	+	0.648731i	$0.224700\pi$
$752$	0	0
$753$	−14.6226	−0.532876
$754$	0	0
$755$	−28.3220	−1.03074
$756$	0	0
$757$	40.5545	1.47398	0.736989	−	0.675905i	$-0.236247\pi$
$757$	40.5545	1.47398	0.736989	+	0.675905i	$0.236247\pi$
$758$	0	0
$759$	−0.245470	−0.00890999
$760$	0	0
$761$	−26.4099	−0.957357	−0.478679	−	0.877990i	$-0.658884\pi$
$761$	−26.4099	−0.957357	−0.478679	+	0.877990i	$0.658884\pi$
$762$	0	0
$763$	7.66192	0.277380
$764$	0	0
$765$	5.58678	0.201991
$766$	0	0
$767$	3.62853	0.131019
$768$	0	0
$769$	36.4706	1.31516	0.657581	−	0.753384i	$-0.271580\pi$
$769$	36.4706	1.31516	0.657581	+	0.753384i	$0.271580\pi$
$770$	0	0
$771$	−9.64301	−0.347285
$772$	0	0
$773$	−15.8555	−0.570282	−0.285141	−	0.958486i	$-0.592040\pi$
$773$	−15.8555	−0.570282	−0.285141	+	0.958486i	$0.592040\pi$
$774$	0	0
$775$	−13.3867	−0.480863
$776$	0	0
$777$	21.6018	0.774960
$778$	0	0
$779$	17.1732	0.615293
$780$	0	0
$781$	−2.81508	−0.100731
$782$	0	0
$783$	−68.6390	−2.45296
$784$	0	0
$785$	14.3339	0.511598
$786$	0	0
$787$	−41.4059	−1.47596	−0.737980	−	0.674823i	$-0.764220\pi$
$787$	−41.4059	−1.47596	−0.737980	+	0.674823i	$0.764220\pi$
$788$	0	0
$789$	6.11097	0.217556
$790$	0	0
$791$	−7.75996	−0.275912
$792$	0	0
$793$	−13.8603	−0.492193
$794$	0	0
$795$	−40.9195	−1.45126
$796$	0	0
$797$	17.1803	0.608558	0.304279	−	0.952583i	$-0.401585\pi$
$797$	17.1803	0.608558	0.304279	+	0.952583i	$0.401585\pi$
$798$	0	0
$799$	3.22779	0.114191
$800$	0	0
$801$	90.7322	3.20587
$802$	0	0
$803$	2.36148	0.0833348
$804$	0	0
$805$	−0.162156	−0.00571526
$806$	0	0
$807$	−58.1989	−2.04870
$808$	0	0
$809$	13.3193	0.468283	0.234142	−	0.972203i	$-0.424772\pi$
$809$	13.3193	0.468283	0.234142	+	0.972203i	$0.424772\pi$
$810$	0	0
$811$	0.383933	0.0134817	0.00674086	−	0.999977i	$-0.497854\pi$
$811$	0.383933	0.0134817	0.00674086	+	0.999977i	$0.497854\pi$
$812$	0	0
$813$	−38.2434	−1.34125
$814$	0	0
$815$	19.0067	0.665774
$816$	0	0
$817$	25.4003	0.888645
$818$	0	0
$819$	−3.89185	−0.135992
$820$	0	0
$821$	15.9756	0.557552	0.278776	−	0.960356i	$-0.410071\pi$
$821$	15.9756	0.557552	0.278776	+	0.960356i	$0.410071\pi$
$822$	0	0
$823$	−36.5930	−1.27555	−0.637776	−	0.770222i	$-0.720146\pi$
$823$	−36.5930	−1.27555	−0.637776	+	0.770222i	$0.720146\pi$
$824$	0	0
$825$	−6.54863	−0.227994
$826$	0	0
$827$	−26.0839	−0.907027	−0.453514	−	0.891249i	$-0.649830\pi$
$827$	−26.0839	−0.907027	−0.453514	+	0.891249i	$0.649830\pi$
$828$	0	0
$829$	8.69460	0.301976	0.150988	−	0.988536i	$-0.451755\pi$
$829$	8.69460	0.301976	0.150988	+	0.988536i	$0.451755\pi$
$830$	0	0
$831$	−83.2268	−2.88710
$832$	0	0
$833$	2.37900	0.0824276
$834$	0	0
$835$	−11.2097	−0.387928
$836$	0	0
$837$	−47.0573	−1.62654
$838$	0	0
$839$	−51.9033	−1.79190	−0.895951	−	0.444153i	$-0.853505\pi$
$839$	−51.9033	−1.79190	−0.895951	+	0.444153i	$0.853505\pi$
$840$	0	0
$841$	47.9804	1.65449
$842$	0	0
$843$	−35.0613	−1.20758
$844$	0	0
$845$	32.5129	1.11848
$846$	0	0
$847$	0.723147	0.0248476
$848$	0	0
$849$	70.6243	2.42382
$850$	0	0
$851$	0.846861	0.0290300
$852$	0	0
$853$	−12.4279	−0.425524	−0.212762	−	0.977104i	$-0.568246\pi$
$853$	−12.4279	−0.425524	−0.212762	+	0.977104i	$0.568246\pi$
$854$	0	0
$855$	44.3918	1.51817
$856$	0	0
$857$	25.8227	0.882088	0.441044	−	0.897486i	$-0.354608\pi$
$857$	25.8227	0.882088	0.441044	+	0.897486i	$0.354608\pi$
$858$	0	0
$859$	24.2176	0.826293	0.413147	−	0.910665i	$-0.364430\pi$
$859$	24.2176	0.826293	0.413147	+	0.910665i	$0.364430\pi$
$860$	0	0
$861$	−12.5211	−0.426719
$862$	0	0
$863$	−53.4396	−1.81911	−0.909553	−	0.415588i	$-0.863576\pi$
$863$	−53.4396	−1.81911	−0.909553	+	0.415588i	$0.863576\pi$
$864$	0	0
$865$	−43.2861	−1.47177
$866$	0	0
$867$	−49.6264	−1.68540
$868$	0	0
$869$	2.46321	0.0835587
$870$	0	0
$871$	−5.30183	−0.179646
$872$	0	0
$873$	−91.9123	−3.11076
$874$	0	0
$875$	5.39320	0.182323
$876$	0	0
$877$	−8.98379	−0.303361	−0.151681	−	0.988430i	$-0.548468\pi$
$877$	−8.98379	−0.303361	−0.151681	+	0.988430i	$0.548468\pi$
$878$	0	0
$879$	−86.9504	−2.93276
$880$	0	0
$881$	−9.30540	−0.313507	−0.156753	−	0.987638i	$-0.550103\pi$
$881$	−9.30540	−0.313507	−0.156753	+	0.987638i	$0.550103\pi$
$882$	0	0
$883$	48.5667	1.63440	0.817200	−	0.576355i	$-0.195525\pi$
$883$	48.5667	1.63440	0.817200	+	0.576355i	$0.195525\pi$
$884$	0	0
$885$	30.1766	1.01437
$886$	0	0
$887$	19.8392	0.666134	0.333067	−	0.942903i	$-0.391917\pi$
$887$	19.8392	0.666134	0.333067	+	0.942903i	$0.391917\pi$
$888$	0	0
$889$	−1.28183	−0.0429911
$890$	0	0
$891$	−6.04415	−0.202487
$892$	0	0
$893$	25.6476	0.858263
$894$	0	0
$895$	12.8999	0.431197
$896$	0	0
$897$	−0.233463	−0.00779509
$898$	0	0
$899$	52.7760	1.76018
$900$	0	0
$901$	−1.90016	−0.0633034
$902$	0	0
$903$	−18.5196	−0.616295
$904$	0	0
$905$	47.5504	1.58063
$906$	0	0
$907$	34.4025	1.14232	0.571158	−	0.820840i	$-0.306494\pi$
$907$	34.4025	1.14232	0.571158	+	0.820840i	$0.306494\pi$
$908$	0	0
$909$	−31.2384	−1.03611
$910$	0	0
$911$	−8.99683	−0.298078	−0.149039	−	0.988831i	$-0.547618\pi$
$911$	−8.99683	−0.298078	−0.149039	+	0.988831i	$0.547618\pi$
$912$	0	0
$913$	15.5427	0.514390
$914$	0	0
$915$	−115.269	−3.81066
$916$	0	0
$917$	−10.8820	−0.359356
$918$	0	0
$919$	−31.3678	−1.03473	−0.517365	−	0.855765i	$-0.673087\pi$
$919$	−31.3678	−1.03473	−0.517365	+	0.855765i	$0.673087\pi$
$920$	0	0
$921$	0.672604	0.0221631
$922$	0	0
$923$	−2.67738	−0.0881270
$924$	0	0
$925$	22.5925	0.742838
$926$	0	0
$927$	81.7471	2.68493
$928$	0	0
$929$	31.9631	1.04868	0.524338	−	0.851510i	$-0.324313\pi$
$929$	31.9631	1.04868	0.524338	+	0.851510i	$0.324313\pi$
$930$	0	0
$931$	18.9032	0.619528
$932$	0	0
$933$	33.9797	1.11244
$934$	0	0
$935$	−0.987304	−0.0322883
$936$	0	0
$937$	−17.4431	−0.569841	−0.284921	−	0.958551i	$-0.591967\pi$
$937$	−17.4431	−0.569841	−0.284921	+	0.958551i	$0.591967\pi$
$938$	0	0
$939$	14.0254	0.457700
$940$	0	0
$941$	12.9481	0.422096	0.211048	−	0.977476i	$-0.432312\pi$
$941$	12.9481	0.422096	0.211048	+	0.977476i	$0.432312\pi$
$942$	0	0
$943$	−0.490870	−0.0159849
$944$	0	0
$945$	−15.2069	−0.494681
$946$	0	0
$947$	−59.1102	−1.92082	−0.960412	−	0.278584i	$-0.910135\pi$
$947$	−59.1102	−1.92082	−0.960412	+	0.278584i	$0.910135\pi$
$948$	0	0
$949$	2.24597	0.0729072
$950$	0	0
$951$	−18.7905	−0.609324
$952$	0	0
$953$	−0.901590	−0.0292054	−0.0146027	−	0.999893i	$-0.504648\pi$
$953$	−0.901590	−0.0292054	−0.0146027	+	0.999893i	$0.504648\pi$
$954$	0	0
$955$	18.2221	0.589653
$956$	0	0
$957$	25.8175	0.834562
$958$	0	0
$959$	0.723147	0.0233516
$960$	0	0
$961$	5.18199	0.167161
$962$	0	0
$963$	−74.4576	−2.39936
$964$	0	0
$965$	−24.1973	−0.778937
$966$	0	0
$967$	10.2863	0.330785	0.165393	−	0.986228i	$-0.447111\pi$
$967$	10.2863	0.330785	0.165393	+	0.986228i	$0.447111\pi$
$968$	0	0
$969$	3.15427	0.101330
$970$	0	0
$971$	26.4423	0.848574	0.424287	−	0.905528i	$-0.360525\pi$
$971$	26.4423	0.848574	0.424287	+	0.905528i	$0.360525\pi$
$972$	0	0
$973$	−7.34496	−0.235469
$974$	0	0
$975$	−6.22830	−0.199465
$976$	0	0
$977$	56.2794	1.80054	0.900269	−	0.435334i	$-0.143370\pi$
$977$	56.2794	1.80054	0.900269	+	0.435334i	$0.143370\pi$
$978$	0	0
$979$	−16.0343	−0.512459
$980$	0	0
$981$	59.9545	1.91420
$982$	0	0
$983$	−55.9510	−1.78456	−0.892279	−	0.451484i	$-0.850895\pi$
$983$	−55.9510	−1.78456	−0.892279	+	0.451484i	$0.850895\pi$
$984$	0	0
$985$	−66.6888	−2.12488
$986$	0	0
$987$	−18.6999	−0.595224
$988$	0	0
$989$	−0.726031	−0.0230864
$990$	0	0
$991$	−18.2996	−0.581307	−0.290653	−	0.956828i	$-0.593873\pi$
$991$	−18.2996	−0.581307	−0.290653	+	0.956828i	$0.593873\pi$
$992$	0	0
$993$	78.9856	2.50653
$994$	0	0
$995$	−32.4703	−1.02938
$996$	0	0
$997$	51.8295	1.64146	0.820728	−	0.571320i	$-0.193568\pi$
$997$	51.8295	1.64146	0.820728	+	0.571320i	$0.193568\pi$
$998$	0	0
$999$	79.4182	2.51268

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6028.2.a.d.1.26	✓	27

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6028.2.a.d.1.26	✓	27	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:	\( N \)	\(=\)	\( 6028 = 2^{2} \cdot 11 \cdot 137 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6028.a (trivial)

\(f(q)\)	\(=\)	\(q+2.94255 q^{3} -2.68803 q^{5} +0.723147 q^{7} +5.65862 q^{9} +O(q^{10})\)	\(q+2.94255 q^{3} -2.68803 q^{5} +0.723147 q^{7} +5.65862 q^{9} -1.00000 q^{11} -0.951085 q^{13} -7.90967 q^{15} -0.367297 q^{17} -2.91849 q^{19} +2.12790 q^{21} +0.0834206 q^{23} +2.22549 q^{25} +7.82314 q^{27} -8.77384 q^{29} -6.01515 q^{31} -2.94255 q^{33} -1.94384 q^{35} +10.1517 q^{37} -2.79862 q^{39} -5.88427 q^{41} -8.70325 q^{43} -15.2105 q^{45} -8.78796 q^{47} -6.47706 q^{49} -1.08079 q^{51} +5.17335 q^{53} +2.68803 q^{55} -8.58781 q^{57} -3.81515 q^{59} +14.5731 q^{61} +4.09202 q^{63} +2.55654 q^{65} +5.57451 q^{67} +0.245470 q^{69} +2.81508 q^{71} -2.36148 q^{73} +6.54863 q^{75} -0.723147 q^{77} -2.46321 q^{79} +6.04415 q^{81} -15.5427 q^{83} +0.987304 q^{85} -25.8175 q^{87} +16.0343 q^{89} -0.687774 q^{91} -17.6999 q^{93} +7.84497 q^{95} -16.2429 q^{97} -5.65862 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 27 q - 6 q^{3} + q^{5} - 14 q^{7} + 33 q^{9}+O(q^{10}) \) 27 * q - 6 * q^3 + q^5 - 14 * q^7 + 33 * q^9	\( 27 q - 6 q^{3} + q^{5} - 14 q^{7} + 33 q^{9} - 27 q^{11} - 6 q^{15} - 21 q^{17} - 3 q^{19} - 4 q^{21} - 44 q^{23} + 38 q^{25} - 18 q^{27} + q^{29} - 8 q^{31} + 6 q^{33} - 33 q^{35} + 11 q^{37} - 13 q^{39} - 19 q^{41} - 11 q^{43} + 17 q^{45} - 37 q^{47} + 41 q^{49} - 49 q^{51} - 12 q^{53} - q^{55} - 50 q^{57} - 14 q^{59} + 12 q^{61} - 53 q^{63} - 55 q^{65} - 5 q^{67} + 14 q^{69} - 67 q^{71} - 27 q^{73} - 70 q^{75} + 14 q^{77} - 31 q^{79} - 5 q^{81} - 55 q^{83} - 3 q^{85} - 31 q^{87} + 11 q^{89} - 11 q^{91} - 24 q^{93} - 47 q^{95} - q^{97} - 33 q^{99}+O(q^{100}) \) 27 * q - 6 * q^3 + q^5 - 14 * q^7 + 33 * q^9 - 27 * q^11 - 6 * q^15 - 21 * q^17 - 3 * q^19 - 4 * q^21 - 44 * q^23 + 38 * q^25 - 18 * q^27 + q^29 - 8 * q^31 + 6 * q^33 - 33 * q^35 + 11 * q^37 - 13 * q^39 - 19 * q^41 - 11 * q^43 + 17 * q^45 - 37 * q^47 + 41 * q^49 - 49 * q^51 - 12 * q^53 - q^55 - 50 * q^57 - 14 * q^59 + 12 * q^61 - 53 * q^63 - 55 * q^65 - 5 * q^67 + 14 * q^69 - 67 * q^71 - 27 * q^73 - 70 * q^75 + 14 * q^77 - 31 * q^79 - 5 * q^81 - 55 * q^83 - 3 * q^85 - 31 * q^87 + 11 * q^89 - 11 * q^91 - 24 * q^93 - 47 * q^95 - q^97 - 33 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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