LMFDB - Embedded newform 6028.2.a.c.1.6

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:	$25$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
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)

$f(q)$	$=$	$q-2.30272 q^{3} -2.25624 q^{5} -2.39335 q^{7} +2.30252 q^{9} +O(q^{10})$	$q-2.30272 q^{3} -2.25624 q^{5} -2.39335 q^{7} +2.30252 q^{9} +1.00000 q^{11} -2.79999 q^{13} +5.19549 q^{15} +1.45960 q^{17} -6.96162 q^{19} +5.51122 q^{21} +2.28409 q^{23} +0.0906293 q^{25} +1.60611 q^{27} +4.69777 q^{29} -2.86141 q^{31} -2.30272 q^{33} +5.39998 q^{35} +5.35930 q^{37} +6.44759 q^{39} +1.80817 q^{41} +5.80973 q^{43} -5.19503 q^{45} +5.45976 q^{47} -1.27187 q^{49} -3.36104 q^{51} +3.77771 q^{53} -2.25624 q^{55} +16.0307 q^{57} +12.7354 q^{59} -0.632895 q^{61} -5.51073 q^{63} +6.31745 q^{65} -8.66914 q^{67} -5.25963 q^{69} -11.6583 q^{71} -6.97494 q^{73} -0.208694 q^{75} -2.39335 q^{77} +4.41537 q^{79} -10.6060 q^{81} -12.5797 q^{83} -3.29320 q^{85} -10.8176 q^{87} +17.5607 q^{89} +6.70135 q^{91} +6.58902 q^{93} +15.7071 q^{95} -0.855925 q^{97} +2.30252 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 25 q - 11 q^{3} - 2 q^{5} - 9 q^{7} + 18 q^{9}+O(q^{10}) $ 25 * q - 11 * q^3 - 2 * q^5 - 9 * q^7 + 18 * q^9	$ 25 q - 11 q^{3} - 2 q^{5} - 9 q^{7} + 18 q^{9} + 25 q^{11} - 4 q^{13} - 10 q^{15} - 19 q^{17} - 12 q^{19} + 8 q^{21} - 31 q^{23} - q^{25} - 44 q^{27} - q^{29} - 8 q^{31} - 11 q^{33} - 16 q^{35} - 14 q^{37} - 18 q^{39} - 5 q^{41} - 15 q^{43} - 15 q^{45} - 41 q^{47} + 2 q^{49} + 10 q^{51} + 4 q^{53} - 2 q^{55} - 3 q^{57} - 35 q^{59} - 4 q^{61} - 45 q^{63} - 28 q^{65} - 30 q^{67} - 3 q^{69} + 4 q^{71} - 7 q^{73} - 18 q^{75} - 9 q^{77} - 9 q^{79} + 29 q^{81} - 72 q^{83} - 33 q^{87} - 30 q^{89} - 10 q^{91} + 7 q^{93} + 9 q^{95} - 37 q^{97} + 18 q^{99}+O(q^{100}) $ 25 * q - 11 * q^3 - 2 * q^5 - 9 * q^7 + 18 * q^9 + 25 * q^11 - 4 * q^13 - 10 * q^15 - 19 * q^17 - 12 * q^19 + 8 * q^21 - 31 * q^23 - q^25 - 44 * q^27 - q^29 - 8 * q^31 - 11 * q^33 - 16 * q^35 - 14 * q^37 - 18 * q^39 - 5 * q^41 - 15 * q^43 - 15 * q^45 - 41 * q^47 + 2 * q^49 + 10 * q^51 + 4 * q^53 - 2 * q^55 - 3 * q^57 - 35 * q^59 - 4 * q^61 - 45 * q^63 - 28 * q^65 - 30 * q^67 - 3 * q^69 + 4 * q^71 - 7 * q^73 - 18 * q^75 - 9 * q^77 - 9 * q^79 + 29 * q^81 - 72 * q^83 - 33 * q^87 - 30 * q^89 - 10 * q^91 + 7 * q^93 + 9 * q^95 - 37 * q^97 + 18 * 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
$3$	−2.30272	−1.32948	−0.664738	−	0.747077i	$-0.731457\pi$
$3$	−2.30272	−1.32948	−0.664738	+	0.747077i	$0.731457\pi$
$4$	0	0
$5$	−2.25624	−1.00902	−0.504511	−	0.863405i	$-0.668327\pi$
$5$	−2.25624	−1.00902	−0.504511	+	0.863405i	$0.668327\pi$
$6$	0	0
$7$	−2.39335	−0.904602	−0.452301	−	0.891865i	$-0.649397\pi$
$7$	−2.39335	−0.904602	−0.452301	+	0.891865i	$0.649397\pi$
$8$	0	0
$9$	2.30252	0.767505
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	−2.79999	−0.776577	−0.388288	−	0.921538i	$-0.626934\pi$
$13$	−2.79999	−0.776577	−0.388288	+	0.921538i	$0.626934\pi$
$14$	0	0
$15$	5.19549	1.34147
$16$	0	0
$17$	1.45960	0.354004	0.177002	−	0.984210i	$-0.443360\pi$
$17$	1.45960	0.354004	0.177002	+	0.984210i	$0.443360\pi$
$18$	0	0
$19$	−6.96162	−1.59710	−0.798552	−	0.601925i	$-0.794400\pi$
$19$	−6.96162	−1.59710	−0.798552	+	0.601925i	$0.794400\pi$
$20$	0	0
$21$	5.51122	1.20265
$22$	0	0
$23$	2.28409	0.476266	0.238133	−	0.971233i	$-0.423465\pi$
$23$	2.28409	0.476266	0.238133	+	0.971233i	$0.423465\pi$
$24$	0	0
$25$	0.0906293	0.0181259
$26$	0	0
$27$	1.60611	0.309096
$28$	0	0
$29$	4.69777	0.872354	0.436177	−	0.899861i	$-0.356332\pi$
$29$	4.69777	0.872354	0.436177	+	0.899861i	$0.356332\pi$
$30$	0	0
$31$	−2.86141	−0.513924	−0.256962	−	0.966422i	$-0.582722\pi$
$31$	−2.86141	−0.513924	−0.256962	+	0.966422i	$0.582722\pi$
$32$	0	0
$33$	−2.30272	−0.400852
$34$	0	0
$35$	5.39998	0.912763
$36$	0	0
$37$	5.35930	0.881064	0.440532	−	0.897737i	$-0.354790\pi$
$37$	5.35930	0.881064	0.440532	+	0.897737i	$0.354790\pi$
$38$	0	0
$39$	6.44759	1.03244
$40$	0	0
$41$	1.80817	0.282389	0.141194	−	0.989982i	$-0.454906\pi$
$41$	1.80817	0.282389	0.141194	+	0.989982i	$0.454906\pi$
$42$	0	0
$43$	5.80973	0.885975	0.442987	−	0.896528i	$-0.353919\pi$
$43$	5.80973	0.885975	0.442987	+	0.896528i	$0.353919\pi$
$44$	0	0
$45$	−5.19503	−0.774430
$46$	0	0
$47$	5.45976	0.796388	0.398194	−	0.917301i	$-0.369637\pi$
$47$	5.45976	0.796388	0.398194	+	0.917301i	$0.369637\pi$
$48$	0	0
$49$	−1.27187	−0.181696
$50$	0	0
$51$	−3.36104	−0.470640
$52$	0	0
$53$	3.77771	0.518908	0.259454	−	0.965755i	$-0.416457\pi$
$53$	3.77771	0.518908	0.259454	+	0.965755i	$0.416457\pi$
$54$	0	0
$55$	−2.25624	−0.304232
$56$	0	0
$57$	16.0307	2.12331
$58$	0	0
$59$	12.7354	1.65801	0.829007	−	0.559239i	$-0.188906\pi$
$59$	12.7354	1.65801	0.829007	+	0.559239i	$0.188906\pi$
$60$	0	0
$61$	−0.632895	−0.0810339	−0.0405169	−	0.999179i	$-0.512900\pi$
$61$	−0.632895	−0.0810339	−0.0405169	+	0.999179i	$0.512900\pi$
$62$	0	0
$63$	−5.51073	−0.694287
$64$	0	0
$65$	6.31745	0.783583
$66$	0	0
$67$	−8.66914	−1.05910	−0.529552	−	0.848277i	$-0.677640\pi$
$67$	−8.66914	−1.05910	−0.529552	+	0.848277i	$0.677640\pi$
$68$	0	0
$69$	−5.25963	−0.633185
$70$	0	0
$71$	−11.6583	−1.38359	−0.691793	−	0.722096i	$-0.743179\pi$
$71$	−11.6583	−1.38359	−0.691793	+	0.722096i	$0.743179\pi$
$72$	0	0
$73$	−6.97494	−0.816355	−0.408177	−	0.912903i	$-0.633836\pi$
$73$	−6.97494	−0.816355	−0.408177	+	0.912903i	$0.633836\pi$
$74$	0	0
$75$	−0.208694	−0.0240979
$76$	0	0
$77$	−2.39335	−0.272748
$78$	0	0
$79$	4.41537	0.496768	0.248384	−	0.968662i	$-0.420101\pi$
$79$	4.41537	0.496768	0.248384	+	0.968662i	$0.420101\pi$
$80$	0	0
$81$	−10.6060	−1.17844
$82$	0	0
$83$	−12.5797	−1.38080	−0.690402	−	0.723426i	$-0.742567\pi$
$83$	−12.5797	−1.38080	−0.690402	+	0.723426i	$0.742567\pi$
$84$	0	0
$85$	−3.29320	−0.357198
$86$	0	0
$87$	−10.8176	−1.15977
$88$	0	0
$89$	17.5607	1.86143	0.930714	−	0.365747i	$-0.119186\pi$
$89$	17.5607	1.86143	0.930714	+	0.365747i	$0.119186\pi$
$90$	0	0
$91$	6.70135	0.702493
$92$	0	0
$93$	6.58902	0.683249
$94$	0	0
$95$	15.7071	1.61151
$96$	0	0
$97$	−0.855925	−0.0869060	−0.0434530	−	0.999055i	$-0.513836\pi$
$97$	−0.855925	−0.0869060	−0.0434530	+	0.999055i	$0.513836\pi$
$98$	0	0
$99$	2.30252	0.231412
$100$	0	0
$101$	4.75780	0.473419	0.236709	−	0.971581i	$-0.423931\pi$
$101$	4.75780	0.473419	0.236709	+	0.971581i	$0.423931\pi$
$102$	0	0
$103$	9.41374	0.927563	0.463782	−	0.885950i	$-0.346492\pi$
$103$	9.41374	0.927563	0.463782	+	0.885950i	$0.346492\pi$
$104$	0	0
$105$	−12.4346	−1.21350
$106$	0	0
$107$	8.39158	0.811245	0.405622	−	0.914041i	$-0.367055\pi$
$107$	8.39158	0.811245	0.405622	+	0.914041i	$0.367055\pi$
$108$	0	0
$109$	14.3684	1.37624	0.688120	−	0.725597i	$-0.258436\pi$
$109$	14.3684	1.37624	0.688120	+	0.725597i	$0.258436\pi$
$110$	0	0
$111$	−12.3410	−1.17135
$112$	0	0
$113$	10.0743	0.947710	0.473855	−	0.880603i	$-0.342862\pi$
$113$	10.0743	0.947710	0.473855	+	0.880603i	$0.342862\pi$
$114$	0	0
$115$	−5.15347	−0.480563
$116$	0	0
$117$	−6.44702	−0.596027
$118$	0	0
$119$	−3.49332	−0.320233
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−4.16370	−0.375429
$124$	0	0
$125$	11.0767	0.990733
$126$	0	0
$127$	−11.8311	−1.04984	−0.524922	−	0.851151i	$-0.675905\pi$
$127$	−11.8311	−1.04984	−0.524922	+	0.851151i	$0.675905\pi$
$128$	0	0
$129$	−13.3782	−1.17788
$130$	0	0
$131$	−3.18513	−0.278286	−0.139143	−	0.990272i	$-0.544435\pi$
$131$	−3.18513	−0.278286	−0.139143	+	0.990272i	$0.544435\pi$
$132$	0	0
$133$	16.6616	1.44474
$134$	0	0
$135$	−3.62377	−0.311885
$136$	0	0
$137$	−1.00000	−0.0854358
$137$	−1.00000	−0.0854358
$138$	0	0
$139$	18.4921	1.56848	0.784240	−	0.620458i	$-0.213053\pi$
$139$	18.4921	1.56848	0.784240	+	0.620458i	$0.213053\pi$
$140$	0	0
$141$	−12.5723	−1.05878
$142$	0	0
$143$	−2.79999	−0.234147
$144$	0	0
$145$	−10.5993	−0.880224
$146$	0	0
$147$	2.92876	0.241560
$148$	0	0
$149$	23.5381	1.92832	0.964159	−	0.265324i	$-0.0854789\pi$
$149$	23.5381	1.92832	0.964159	+	0.265324i	$0.0854789\pi$
$150$	0	0
$151$	2.48187	0.201972	0.100986	−	0.994888i	$-0.467800\pi$
$151$	2.48187	0.201972	0.100986	+	0.994888i	$0.467800\pi$
$152$	0	0
$153$	3.36074	0.271700
$154$	0	0
$155$	6.45603	0.518561
$156$	0	0
$157$	3.80632	0.303778	0.151889	−	0.988398i	$-0.451464\pi$
$157$	3.80632	0.303778	0.151889	+	0.988398i	$0.451464\pi$
$158$	0	0
$159$	−8.69900	−0.689876
$160$	0	0
$161$	−5.46664	−0.430831
$162$	0	0
$163$	−10.7155	−0.839300	−0.419650	−	0.907686i	$-0.637847\pi$
$163$	−10.7155	−0.839300	−0.419650	+	0.907686i	$0.637847\pi$
$164$	0	0
$165$	5.19549	0.404469
$166$	0	0
$167$	−11.6625	−0.902473	−0.451236	−	0.892404i	$-0.649017\pi$
$167$	−11.6625	−0.902473	−0.451236	+	0.892404i	$0.649017\pi$
$168$	0	0
$169$	−5.16007	−0.396928
$170$	0	0
$171$	−16.0292	−1.22579
$172$	0	0
$173$	−13.1546	−1.00013	−0.500063	−	0.865989i	$-0.666690\pi$
$173$	−13.1546	−1.00013	−0.500063	+	0.865989i	$0.666690\pi$
$174$	0	0
$175$	−0.216908	−0.0163967
$176$	0	0
$177$	−29.3261	−2.20429
$178$	0	0
$179$	−3.48131	−0.260205	−0.130103	−	0.991501i	$-0.541531\pi$
$179$	−3.48131	−0.260205	−0.130103	+	0.991501i	$0.541531\pi$
$180$	0	0
$181$	12.1185	0.900761	0.450381	−	0.892837i	$-0.351288\pi$
$181$	12.1185	0.900761	0.450381	+	0.892837i	$0.351288\pi$
$182$	0	0
$183$	1.45738	0.107733
$184$	0	0
$185$	−12.0919	−0.889013
$186$	0	0
$187$	1.45960	0.106736
$188$	0	0
$189$	−3.84398	−0.279609
$190$	0	0
$191$	−11.4103	−0.825618	−0.412809	−	0.910818i	$-0.635452\pi$
$191$	−11.4103	−0.825618	−0.412809	+	0.910818i	$0.635452\pi$
$192$	0	0
$193$	−4.04775	−0.291363	−0.145682	−	0.989332i	$-0.546538\pi$
$193$	−4.04775	−0.291363	−0.145682	+	0.989332i	$0.546538\pi$
$194$	0	0
$195$	−14.5473	−1.04176
$196$	0	0
$197$	−13.0065	−0.926676	−0.463338	−	0.886182i	$-0.653348\pi$
$197$	−13.0065	−0.926676	−0.463338	+	0.886182i	$0.653348\pi$
$198$	0	0
$199$	−26.0268	−1.84499	−0.922497	−	0.386005i	$-0.873855\pi$
$199$	−26.0268	−1.84499	−0.922497	+	0.386005i	$0.873855\pi$
$200$	0	0
$201$	19.9626	1.40805
$202$	0	0
$203$	−11.2434	−0.789133
$204$	0	0
$205$	−4.07967	−0.284936
$206$	0	0
$207$	5.25916	0.365537
$208$	0	0
$209$	−6.96162	−0.481545
$210$	0	0
$211$	15.1393	1.04223	0.521115	−	0.853486i	$-0.325516\pi$
$211$	15.1393	1.04223	0.521115	+	0.853486i	$0.325516\pi$
$212$	0	0
$213$	26.8458	1.83944
$214$	0	0
$215$	−13.1081	−0.893968
$216$	0	0
$217$	6.84835	0.464896
$218$	0	0
$219$	16.0613	1.08532
$220$	0	0
$221$	−4.08685	−0.274911
$222$	0	0
$223$	−25.4912	−1.70702	−0.853509	−	0.521078i	$-0.825530\pi$
$223$	−25.4912	−1.70702	−0.853509	+	0.521078i	$0.825530\pi$
$224$	0	0
$225$	0.208675	0.0139117
$226$	0	0
$227$	13.8713	0.920669	0.460334	−	0.887746i	$-0.347730\pi$
$227$	13.8713	0.920669	0.460334	+	0.887746i	$0.347730\pi$
$228$	0	0
$229$	−24.1039	−1.59283	−0.796414	−	0.604752i	$-0.793272\pi$
$229$	−24.1039	−1.59283	−0.796414	+	0.604752i	$0.793272\pi$
$230$	0	0
$231$	5.51122	0.362611
$232$	0	0
$233$	2.02046	0.132365	0.0661823	−	0.997808i	$-0.478918\pi$
$233$	2.02046	0.132365	0.0661823	+	0.997808i	$0.478918\pi$
$234$	0	0
$235$	−12.3186	−0.803574
$236$	0	0
$237$	−10.1674	−0.660441
$238$	0	0
$239$	−11.3043	−0.731213	−0.365606	−	0.930770i	$-0.619138\pi$
$239$	−11.3043	−0.731213	−0.365606	+	0.930770i	$0.619138\pi$
$240$	0	0
$241$	6.53535	0.420979	0.210490	−	0.977596i	$-0.432494\pi$
$241$	6.53535	0.420979	0.210490	+	0.977596i	$0.432494\pi$
$242$	0	0
$243$	19.6042	1.25761
$244$	0	0
$245$	2.86965	0.183335
$246$	0	0
$247$	19.4924	1.24027
$248$	0	0
$249$	28.9676	1.83575
$250$	0	0
$251$	−10.8612	−0.685552	−0.342776	−	0.939417i	$-0.611367\pi$
$251$	−10.8612	−0.685552	−0.342776	+	0.939417i	$0.611367\pi$
$252$	0	0
$253$	2.28409	0.143600
$254$	0	0
$255$	7.58332	0.474886
$256$	0	0
$257$	−16.7706	−1.04612	−0.523061	−	0.852295i	$-0.675210\pi$
$257$	−16.7706	−1.04612	−0.523061	+	0.852295i	$0.675210\pi$
$258$	0	0
$259$	−12.8267	−0.797012
$260$	0	0
$261$	10.8167	0.669536
$262$	0	0
$263$	15.6252	0.963489	0.481744	−	0.876312i	$-0.340003\pi$
$263$	15.6252	0.963489	0.481744	+	0.876312i	$0.340003\pi$
$264$	0	0
$265$	−8.52343	−0.523590
$266$	0	0
$267$	−40.4373	−2.47472
$268$	0	0
$269$	−17.2380	−1.05102	−0.525511	−	0.850787i	$-0.676126\pi$
$269$	−17.2380	−1.05102	−0.525511	+	0.850787i	$0.676126\pi$
$270$	0	0
$271$	1.25991	0.0765343	0.0382672	−	0.999268i	$-0.487816\pi$
$271$	1.25991	0.0765343	0.0382672	+	0.999268i	$0.487816\pi$
$272$	0	0
$273$	−15.4313	−0.933947
$274$	0	0
$275$	0.0906293	0.00546515
$276$	0	0
$277$	−1.84051	−0.110585	−0.0552927	−	0.998470i	$-0.517609\pi$
$277$	−1.84051	−0.110585	−0.0552927	+	0.998470i	$0.517609\pi$
$278$	0	0
$279$	−6.58844	−0.394439
$280$	0	0
$281$	13.7119	0.817981	0.408991	−	0.912539i	$-0.365881\pi$
$281$	13.7119	0.817981	0.408991	+	0.912539i	$0.365881\pi$
$282$	0	0
$283$	13.1492	0.781639	0.390819	−	0.920467i	$-0.372192\pi$
$283$	13.1492	0.781639	0.390819	+	0.920467i	$0.372192\pi$
$284$	0	0
$285$	−36.1690	−2.14247
$286$	0	0
$287$	−4.32758	−0.255449
$288$	0	0
$289$	−14.8696	−0.874681
$290$	0	0
$291$	1.97095	0.115539
$292$	0	0
$293$	−0.192125	−0.0112241	−0.00561204	−	0.999984i	$-0.501786\pi$
$293$	−0.192125	−0.0112241	−0.00561204	+	0.999984i	$0.501786\pi$
$294$	0	0
$295$	−28.7342	−1.67297
$296$	0	0
$297$	1.60611	0.0931959
$298$	0	0
$299$	−6.39543	−0.369857
$300$	0	0
$301$	−13.9047	−0.801454
$302$	0	0
$303$	−10.9559	−0.629398
$304$	0	0
$305$	1.42796	0.0817650
$306$	0	0
$307$	−24.6117	−1.40466	−0.702332	−	0.711850i	$-0.747858\pi$
$307$	−24.6117	−1.40466	−0.702332	+	0.711850i	$0.747858\pi$
$308$	0	0
$309$	−21.6772	−1.23317
$310$	0	0
$311$	−28.3101	−1.60532	−0.802658	−	0.596439i	$-0.796582\pi$
$311$	−28.3101	−1.60532	−0.802658	+	0.596439i	$0.796582\pi$
$312$	0	0
$313$	−20.8914	−1.18085	−0.590425	−	0.807092i	$-0.701040\pi$
$313$	−20.8914	−1.18085	−0.590425	+	0.807092i	$0.701040\pi$
$314$	0	0
$315$	12.4335	0.700551
$316$	0	0
$317$	24.9864	1.40337	0.701687	−	0.712485i	$-0.252431\pi$
$317$	24.9864	1.40337	0.701687	+	0.712485i	$0.252431\pi$
$318$	0	0
$319$	4.69777	0.263025
$320$	0	0
$321$	−19.3235	−1.07853
$322$	0	0
$323$	−10.1611	−0.565381
$324$	0	0
$325$	−0.253761	−0.0140761
$326$	0	0
$327$	−33.0863	−1.82968
$328$	0	0
$329$	−13.0671	−0.720414
$330$	0	0
$331$	9.13441	0.502073	0.251036	−	0.967978i	$-0.419229\pi$
$331$	9.13441	0.502073	0.251036	+	0.967978i	$0.419229\pi$
$332$	0	0
$333$	12.3399	0.676221
$334$	0	0
$335$	19.5597	1.06866
$336$	0	0
$337$	−7.42834	−0.404648	−0.202324	−	0.979319i	$-0.564849\pi$
$337$	−7.42834	−0.404648	−0.202324	+	0.979319i	$0.564849\pi$
$338$	0	0
$339$	−23.1983	−1.25996
$340$	0	0
$341$	−2.86141	−0.154954
$342$	0	0
$343$	19.7975	1.06896
$344$	0	0
$345$	11.8670	0.638897
$346$	0	0
$347$	−22.2327	−1.19351	−0.596756	−	0.802422i	$-0.703544\pi$
$347$	−22.2327	−1.19351	−0.596756	+	0.802422i	$0.703544\pi$
$348$	0	0
$349$	−24.5191	−1.31248	−0.656239	−	0.754553i	$-0.727854\pi$
$349$	−24.5191	−1.31248	−0.656239	+	0.754553i	$0.727854\pi$
$350$	0	0
$351$	−4.49709	−0.240037
$352$	0	0
$353$	30.8393	1.64141	0.820706	−	0.571351i	$-0.193581\pi$
$353$	30.8393	1.64141	0.820706	+	0.571351i	$0.193581\pi$
$354$	0	0
$355$	26.3040	1.39607
$356$	0	0
$357$	8.04415	0.425741
$358$	0	0
$359$	19.4059	1.02420	0.512102	−	0.858925i	$-0.328867\pi$
$359$	19.4059	1.02420	0.512102	+	0.858925i	$0.328867\pi$
$360$	0	0
$361$	29.4641	1.55074
$362$	0	0
$363$	−2.30272	−0.120861
$364$	0	0
$365$	15.7372	0.823720
$366$	0	0
$367$	6.31559	0.329671	0.164836	−	0.986321i	$-0.447291\pi$
$367$	6.31559	0.329671	0.164836	+	0.986321i	$0.447291\pi$
$368$	0	0
$369$	4.16334	0.216735
$370$	0	0
$371$	−9.04138	−0.469405
$372$	0	0
$373$	6.27036	0.324667	0.162334	−	0.986736i	$-0.448098\pi$
$373$	6.27036	0.324667	0.162334	+	0.986736i	$0.448098\pi$
$374$	0	0
$375$	−25.5066	−1.31716
$376$	0	0
$377$	−13.1537	−0.677450
$378$	0	0
$379$	9.67380	0.496910	0.248455	−	0.968643i	$-0.420077\pi$
$379$	9.67380	0.496910	0.248455	+	0.968643i	$0.420077\pi$
$380$	0	0
$381$	27.2438	1.39574
$382$	0	0
$383$	−29.3085	−1.49759	−0.748797	−	0.662800i	$-0.769368\pi$
$383$	−29.3085	−1.49759	−0.748797	+	0.662800i	$0.769368\pi$
$384$	0	0
$385$	5.39998	0.275208
$386$	0	0
$387$	13.3770	0.679990
$388$	0	0
$389$	−22.8725	−1.15968	−0.579841	−	0.814730i	$-0.696885\pi$
$389$	−22.8725	−1.15968	−0.579841	+	0.814730i	$0.696885\pi$
$390$	0	0
$391$	3.33385	0.168600
$392$	0	0
$393$	7.33445	0.369974
$394$	0	0
$395$	−9.96214	−0.501250
$396$	0	0
$397$	−7.62423	−0.382649	−0.191325	−	0.981527i	$-0.561278\pi$
$397$	−7.62423	−0.382649	−0.191325	+	0.981527i	$0.561278\pi$
$398$	0	0
$399$	−38.3670	−1.92075
$400$	0	0
$401$	−30.7077	−1.53347	−0.766734	−	0.641965i	$-0.778119\pi$
$401$	−30.7077	−1.53347	−0.766734	+	0.641965i	$0.778119\pi$
$402$	0	0
$403$	8.01191	0.399101
$404$	0	0
$405$	23.9296	1.18907
$406$	0	0
$407$	5.35930	0.265651
$408$	0	0
$409$	24.5804	1.21542	0.607711	−	0.794158i	$-0.292088\pi$
$409$	24.5804	1.21542	0.607711	+	0.794158i	$0.292088\pi$
$410$	0	0
$411$	2.30272	0.113585
$412$	0	0
$413$	−30.4804	−1.49984
$414$	0	0
$415$	28.3829	1.39326
$416$	0	0
$417$	−42.5821	−2.08525
$418$	0	0
$419$	11.0785	0.541218	0.270609	−	0.962689i	$-0.412775\pi$
$419$	11.0785	0.541218	0.270609	+	0.962689i	$0.412775\pi$
$420$	0	0
$421$	31.0310	1.51236	0.756178	−	0.654365i	$-0.227064\pi$
$421$	31.0310	1.51236	0.756178	+	0.654365i	$0.227064\pi$
$422$	0	0
$423$	12.5712	0.611232
$424$	0	0
$425$	0.132282	0.00641663
$426$	0	0
$427$	1.51474	0.0733034
$428$	0	0
$429$	6.44759	0.311292
$430$	0	0
$431$	18.4182	0.887175	0.443588	−	0.896231i	$-0.353706\pi$
$431$	18.4182	0.887175	0.443588	+	0.896231i	$0.353706\pi$
$432$	0	0
$433$	4.40646	0.211761	0.105880	−	0.994379i	$-0.466234\pi$
$433$	4.40646	0.211761	0.105880	+	0.994379i	$0.466234\pi$
$434$	0	0
$435$	24.4072	1.17024
$436$	0	0
$437$	−15.9010	−0.760647
$438$	0	0
$439$	−19.9681	−0.953024	−0.476512	−	0.879168i	$-0.658099\pi$
$439$	−19.9681	−0.953024	−0.476512	+	0.879168i	$0.658099\pi$
$440$	0	0
$441$	−2.92850	−0.139453
$442$	0	0
$443$	−21.3742	−1.01552	−0.507760	−	0.861499i	$-0.669526\pi$
$443$	−21.3742	−1.01552	−0.507760	+	0.861499i	$0.669526\pi$
$444$	0	0
$445$	−39.6212	−1.87822
$446$	0	0
$447$	−54.2017	−2.56365
$448$	0	0
$449$	35.4654	1.67372	0.836859	−	0.547419i	$-0.184389\pi$
$449$	35.4654	1.67372	0.836859	+	0.547419i	$0.184389\pi$
$450$	0	0
$451$	1.80817	0.0851433
$452$	0	0
$453$	−5.71505	−0.268516
$454$	0	0
$455$	−15.1199	−0.708831
$456$	0	0
$457$	−21.2510	−0.994082	−0.497041	−	0.867727i	$-0.665580\pi$
$457$	−21.2510	−0.994082	−0.497041	+	0.867727i	$0.665580\pi$
$458$	0	0
$459$	2.34427	0.109421
$460$	0	0
$461$	2.47696	0.115364	0.0576818	−	0.998335i	$-0.481629\pi$
$461$	2.47696	0.115364	0.0576818	+	0.998335i	$0.481629\pi$
$462$	0	0
$463$	15.2984	0.710976	0.355488	−	0.934681i	$-0.384315\pi$
$463$	15.2984	0.710976	0.355488	+	0.934681i	$0.384315\pi$
$464$	0	0
$465$	−14.8664	−0.689414
$466$	0	0
$467$	−33.3289	−1.54228	−0.771139	−	0.636667i	$-0.780312\pi$
$467$	−33.3289	−1.54228	−0.771139	+	0.636667i	$0.780312\pi$
$468$	0	0
$469$	20.7483	0.958068
$470$	0	0
$471$	−8.76489	−0.403865
$472$	0	0
$473$	5.80973	0.267131
$474$	0	0
$475$	−0.630927	−0.0289489
$476$	0	0
$477$	8.69824	0.398265
$478$	0	0
$479$	−24.3502	−1.11259	−0.556295	−	0.830985i	$-0.687778\pi$
$479$	−24.3502	−1.11259	−0.556295	+	0.830985i	$0.687778\pi$
$480$	0	0
$481$	−15.0060	−0.684214
$482$	0	0
$483$	12.5881	0.572780
$484$	0	0
$485$	1.93117	0.0876901
$486$	0	0
$487$	3.56295	0.161453	0.0807264	−	0.996736i	$-0.474276\pi$
$487$	3.56295	0.161453	0.0807264	+	0.996736i	$0.474276\pi$
$488$	0	0
$489$	24.6747	1.11583
$490$	0	0
$491$	−5.42514	−0.244833	−0.122417	−	0.992479i	$-0.539064\pi$
$491$	−5.42514	−0.244833	−0.122417	+	0.992479i	$0.539064\pi$
$492$	0	0
$493$	6.85684	0.308817
$494$	0	0
$495$	−5.19503	−0.233499
$496$	0	0
$497$	27.9024	1.25159
$498$	0	0
$499$	−38.0661	−1.70407	−0.852036	−	0.523484i	$-0.824632\pi$
$499$	−38.0661	−1.70407	−0.852036	+	0.523484i	$0.824632\pi$
$500$	0	0
$501$	26.8555	1.19982
$502$	0	0
$503$	−40.9393	−1.82539	−0.912697	−	0.408637i	$-0.866004\pi$
$503$	−40.9393	−1.82539	−0.912697	+	0.408637i	$0.866004\pi$
$504$	0	0
$505$	−10.7347	−0.477690
$506$	0	0
$507$	11.8822	0.527706
$508$	0	0
$509$	24.0329	1.06524	0.532619	−	0.846355i	$-0.321208\pi$
$509$	24.0329	1.06524	0.532619	+	0.846355i	$0.321208\pi$
$510$	0	0
$511$	16.6935	0.738476
$512$	0	0
$513$	−11.1811	−0.493658
$514$	0	0
$515$	−21.2397	−0.935932
$516$	0	0
$517$	5.45976	0.240120
$518$	0	0
$519$	30.2914	1.32964
$520$	0	0
$521$	33.8429	1.48269	0.741343	−	0.671126i	$-0.234189\pi$
$521$	33.8429	1.48269	0.741343	+	0.671126i	$0.234189\pi$
$522$	0	0
$523$	−22.7458	−0.994605	−0.497303	−	0.867577i	$-0.665676\pi$
$523$	−22.7458	−0.994605	−0.497303	+	0.867577i	$0.665676\pi$
$524$	0	0
$525$	0.499478	0.0217990
$526$	0	0
$527$	−4.17650	−0.181931
$528$	0	0
$529$	−17.7829	−0.773170
$530$	0	0
$531$	29.3236	1.27253
$532$	0	0
$533$	−5.06285	−0.219296
$534$	0	0
$535$	−18.9334	−0.818564
$536$	0	0
$537$	8.01647	0.345936
$538$	0	0
$539$	−1.27187	−0.0547833
$540$	0	0
$541$	35.9256	1.54456	0.772282	−	0.635280i	$-0.219115\pi$
$541$	35.9256	1.54456	0.772282	+	0.635280i	$0.219115\pi$
$542$	0	0
$543$	−27.9055	−1.19754
$544$	0	0
$545$	−32.4185	−1.38866
$546$	0	0
$547$	−24.2993	−1.03896	−0.519482	−	0.854482i	$-0.673875\pi$
$547$	−24.2993	−1.03896	−0.519482	+	0.854482i	$0.673875\pi$
$548$	0	0
$549$	−1.45725	−0.0621939
$550$	0	0
$551$	−32.7041	−1.39324
$552$	0	0
$553$	−10.5675	−0.449377
$554$	0	0
$555$	27.8442	1.18192
$556$	0	0
$557$	28.2614	1.19747	0.598737	−	0.800946i	$-0.295670\pi$
$557$	28.2614	1.19747	0.598737	+	0.800946i	$0.295670\pi$
$558$	0	0
$559$	−16.2672	−0.688028
$560$	0	0
$561$	−3.36104	−0.141903
$562$	0	0
$563$	−8.20462	−0.345784	−0.172892	−	0.984941i	$-0.555311\pi$
$563$	−8.20462	−0.345784	−0.172892	+	0.984941i	$0.555311\pi$
$564$	0	0
$565$	−22.7301	−0.956260
$566$	0	0
$567$	25.3838	1.06602
$568$	0	0
$569$	2.34214	0.0981874	0.0490937	−	0.998794i	$-0.484367\pi$
$569$	2.34214	0.0981874	0.0490937	+	0.998794i	$0.484367\pi$
$570$	0	0
$571$	27.4790	1.14996	0.574979	−	0.818168i	$-0.305010\pi$
$571$	27.4790	1.14996	0.574979	+	0.818168i	$0.305010\pi$
$572$	0	0
$573$	26.2746	1.09764
$574$	0	0
$575$	0.207006	0.00863274
$576$	0	0
$577$	−3.20156	−0.133283	−0.0666413	−	0.997777i	$-0.521228\pi$
$577$	−3.20156	−0.133283	−0.0666413	+	0.997777i	$0.521228\pi$
$578$	0	0
$579$	9.32083	0.387360
$580$	0	0
$581$	30.1077	1.24908
$582$	0	0
$583$	3.77771	0.156457
$584$	0	0
$585$	14.5460	0.601404
$586$	0	0
$587$	2.18366	0.0901295	0.0450647	−	0.998984i	$-0.485651\pi$
$587$	2.18366	0.0901295	0.0450647	+	0.998984i	$0.485651\pi$
$588$	0	0
$589$	19.9200	0.820790
$590$	0	0
$591$	29.9503	1.23199
$592$	0	0
$593$	−0.503657	−0.0206827	−0.0103414	−	0.999947i	$-0.503292\pi$
$593$	−0.503657	−0.0206827	−0.0103414	+	0.999947i	$0.503292\pi$
$594$	0	0
$595$	7.88179	0.323122
$596$	0	0
$597$	59.9325	2.45287
$598$	0	0
$599$	−12.7952	−0.522797	−0.261398	−	0.965231i	$-0.584184\pi$
$599$	−12.7952	−0.522797	−0.261398	+	0.965231i	$0.584184\pi$
$600$	0	0
$601$	−19.3320	−0.788569	−0.394285	−	0.918988i	$-0.629008\pi$
$601$	−19.3320	−0.788569	−0.394285	+	0.918988i	$0.629008\pi$
$602$	0	0
$603$	−19.9608	−0.812868
$604$	0	0
$605$	−2.25624	−0.0917293
$606$	0	0
$607$	11.6232	0.471769	0.235885	−	0.971781i	$-0.424201\pi$
$607$	11.6232	0.471769	0.235885	+	0.971781i	$0.424201\pi$
$608$	0	0
$609$	25.8904	1.04913
$610$	0	0
$611$	−15.2873	−0.618457
$612$	0	0
$613$	38.1895	1.54246	0.771231	−	0.636556i	$-0.219642\pi$
$613$	38.1895	1.54246	0.771231	+	0.636556i	$0.219642\pi$
$614$	0	0
$615$	9.39433	0.378816
$616$	0	0
$617$	−2.96484	−0.119360	−0.0596799	−	0.998218i	$-0.519008\pi$
$617$	−2.96484	−0.119360	−0.0596799	+	0.998218i	$0.519008\pi$
$618$	0	0
$619$	5.86037	0.235548	0.117774	−	0.993040i	$-0.462424\pi$
$619$	5.86037	0.235548	0.117774	+	0.993040i	$0.462424\pi$
$620$	0	0
$621$	3.66850	0.147212
$622$	0	0
$623$	−42.0289	−1.68385
$624$	0	0
$625$	−25.4449	−1.01780
$626$	0	0
$627$	16.0307	0.640203
$628$	0	0
$629$	7.82241	0.311900
$630$	0	0
$631$	32.5288	1.29495	0.647475	−	0.762087i	$-0.275825\pi$
$631$	32.5288	1.29495	0.647475	+	0.762087i	$0.275825\pi$
$632$	0	0
$633$	−34.8615	−1.38562
$634$	0	0
$635$	26.6939	1.05931
$636$	0	0
$637$	3.56122	0.141101
$638$	0	0
$639$	−26.8434	−1.06191
$640$	0	0
$641$	−19.2238	−0.759294	−0.379647	−	0.925131i	$-0.623954\pi$
$641$	−19.2238	−0.759294	−0.379647	+	0.925131i	$0.623954\pi$
$642$	0	0
$643$	−12.2482	−0.483022	−0.241511	−	0.970398i	$-0.577643\pi$
$643$	−12.2482	−0.483022	−0.241511	+	0.970398i	$0.577643\pi$
$644$	0	0
$645$	30.1844	1.18851
$646$	0	0
$647$	−15.9103	−0.625498	−0.312749	−	0.949836i	$-0.601250\pi$
$647$	−15.9103	−0.625498	−0.312749	+	0.949836i	$0.601250\pi$
$648$	0	0
$649$	12.7354	0.499910
$650$	0	0
$651$	−15.7698	−0.618069
$652$	0	0
$653$	33.5501	1.31292	0.656459	−	0.754361i	$-0.272053\pi$
$653$	33.5501	1.31292	0.656459	+	0.754361i	$0.272053\pi$
$654$	0	0
$655$	7.18642	0.280797
$656$	0	0
$657$	−16.0599	−0.626557
$658$	0	0
$659$	−27.7159	−1.07966	−0.539829	−	0.841775i	$-0.681511\pi$
$659$	−27.7159	−1.07966	−0.539829	+	0.841775i	$0.681511\pi$
$660$	0	0
$661$	19.3385	0.752181	0.376090	−	0.926583i	$-0.377268\pi$
$661$	19.3385	0.752181	0.376090	+	0.926583i	$0.377268\pi$
$662$	0	0
$663$	9.41087	0.365488
$664$	0	0
$665$	−37.5926	−1.45778
$666$	0	0
$667$	10.7301	0.415473
$668$	0	0
$669$	58.6991	2.26944
$670$	0	0
$671$	−0.632895	−0.0244326
$672$	0	0
$673$	30.5074	1.17597	0.587986	−	0.808871i	$-0.299921\pi$
$673$	30.5074	1.17597	0.587986	+	0.808871i	$0.299921\pi$
$674$	0	0
$675$	0.145561	0.00560263
$676$	0	0
$677$	−47.7064	−1.83351	−0.916753	−	0.399454i	$-0.869200\pi$
$677$	−47.7064	−1.83351	−0.916753	+	0.399454i	$0.869200\pi$
$678$	0	0
$679$	2.04853	0.0786153
$680$	0	0
$681$	−31.9417	−1.22401
$682$	0	0
$683$	23.3563	0.893705	0.446853	−	0.894608i	$-0.352545\pi$
$683$	23.3563	0.893705	0.446853	+	0.894608i	$0.352545\pi$
$684$	0	0
$685$	2.25624	0.0862066
$686$	0	0
$687$	55.5044	2.11763
$688$	0	0
$689$	−10.5775	−0.402972
$690$	0	0
$691$	−33.9284	−1.29070	−0.645348	−	0.763889i	$-0.723288\pi$
$691$	−33.9284	−1.29070	−0.645348	+	0.763889i	$0.723288\pi$
$692$	0	0
$693$	−5.51073	−0.209335
$694$	0	0
$695$	−41.7226	−1.58263
$696$	0	0
$697$	2.63919	0.0999666
$698$	0	0
$699$	−4.65255	−0.175975
$700$	0	0
$701$	−0.00478035	−0.000180551 0	−9.02756e−5	−	1.00000i	$-0.500029\pi$
$701$	−0.00478035	−0.000180551 0	−9.02756e−5		1.00000i	$0.500029\pi$
$702$	0	0
$703$	−37.3094	−1.40715
$704$	0	0
$705$	28.3662	1.06833
$706$	0	0
$707$	−11.3871	−0.428255
$708$	0	0
$709$	36.4883	1.37035	0.685173	−	0.728380i	$-0.259727\pi$
$709$	36.4883	1.37035	0.685173	+	0.728380i	$0.259727\pi$
$710$	0	0
$711$	10.1665	0.381272
$712$	0	0
$713$	−6.53572	−0.244765
$714$	0	0
$715$	6.31745	0.236259
$716$	0	0
$717$	26.0306	0.972129
$718$	0	0
$719$	5.91631	0.220641	0.110321	−	0.993896i	$-0.464812\pi$
$719$	5.91631	0.220641	0.110321	+	0.993896i	$0.464812\pi$
$720$	0	0
$721$	−22.5304	−0.839075
$722$	0	0
$723$	−15.0491	−0.559681
$724$	0	0
$725$	0.425756	0.0158122
$726$	0	0
$727$	27.0469	1.00311	0.501557	−	0.865125i	$-0.332761\pi$
$727$	27.0469	1.00311	0.501557	+	0.865125i	$0.332761\pi$
$728$	0	0
$729$	−13.3252	−0.493524
$730$	0	0
$731$	8.47985	0.313639
$732$	0	0
$733$	24.7172	0.912952	0.456476	−	0.889736i	$-0.349112\pi$
$733$	24.7172	0.912952	0.456476	+	0.889736i	$0.349112\pi$
$734$	0	0
$735$	−6.60799	−0.243740
$736$	0	0
$737$	−8.66914	−0.319332
$738$	0	0
$739$	34.9129	1.28429	0.642145	−	0.766583i	$-0.278045\pi$
$739$	34.9129	1.28429	0.642145	+	0.766583i	$0.278045\pi$
$740$	0	0
$741$	−44.8856	−1.64891
$742$	0	0
$743$	27.0918	0.993901	0.496951	−	0.867779i	$-0.334453\pi$
$743$	27.0918	0.993901	0.496951	+	0.867779i	$0.334453\pi$
$744$	0	0
$745$	−53.1077	−1.94572
$746$	0	0
$747$	−28.9650	−1.05978
$748$	0	0
$749$	−20.0840	−0.733853
$750$	0	0
$751$	39.6413	1.44653	0.723265	−	0.690570i	$-0.242640\pi$
$751$	39.6413	1.44653	0.723265	+	0.690570i	$0.242640\pi$
$752$	0	0
$753$	25.0102	0.911424
$754$	0	0
$755$	−5.59970	−0.203794
$756$	0	0
$757$	−52.3000	−1.90087	−0.950437	−	0.310916i	$-0.899364\pi$
$757$	−52.3000	−1.90087	−0.950437	+	0.310916i	$0.899364\pi$
$758$	0	0
$759$	−5.25963	−0.190912
$760$	0	0
$761$	31.9728	1.15901	0.579506	−	0.814968i	$-0.303245\pi$
$761$	31.9728	1.15901	0.579506	+	0.814968i	$0.303245\pi$
$762$	0	0
$763$	−34.3886	−1.24495
$764$	0	0
$765$	−7.58265	−0.274151
$766$	0	0
$767$	−35.6591	−1.28757
$768$	0	0
$769$	−28.3822	−1.02349	−0.511745	−	0.859137i	$-0.671001\pi$
$769$	−28.3822	−1.02349	−0.511745	+	0.859137i	$0.671001\pi$
$770$	0	0
$771$	38.6180	1.39079
$772$	0	0
$773$	2.47086	0.0888708	0.0444354	−	0.999012i	$-0.485851\pi$
$773$	2.47086	0.0888708	0.0444354	+	0.999012i	$0.485851\pi$
$774$	0	0
$775$	−0.259327	−0.00931532
$776$	0	0
$777$	29.5363	1.05961
$778$	0	0
$779$	−12.5878	−0.451004
$780$	0	0
$781$	−11.6583	−0.417167
$782$	0	0
$783$	7.54513	0.269641
$784$	0	0
$785$	−8.58799	−0.306518
$786$	0	0
$787$	−2.50324	−0.0892308	−0.0446154	−	0.999004i	$-0.514206\pi$
$787$	−2.50324	−0.0892308	−0.0446154	+	0.999004i	$0.514206\pi$
$788$	0	0
$789$	−35.9804	−1.28093
$790$	0	0
$791$	−24.1113	−0.857300
$792$	0	0
$793$	1.77210	0.0629290
$794$	0	0
$795$	19.6271	0.696100
$796$	0	0
$797$	−7.33726	−0.259899	−0.129950	−	0.991521i	$-0.541482\pi$
$797$	−7.33726	−0.259899	−0.129950	+	0.991521i	$0.541482\pi$
$798$	0	0
$799$	7.96905	0.281925
$800$	0	0
$801$	40.4338	1.42866
$802$	0	0
$803$	−6.97494	−0.246140
$804$	0	0
$805$	12.3341	0.434718
$806$	0	0
$807$	39.6944	1.39731
$808$	0	0
$809$	7.77207	0.273251	0.136626	−	0.990623i	$-0.456374\pi$
$809$	7.77207	0.273251	0.136626	+	0.990623i	$0.456374\pi$
$810$	0	0
$811$	26.9338	0.945775	0.472887	−	0.881123i	$-0.343212\pi$
$811$	26.9338	0.945775	0.472887	+	0.881123i	$0.343212\pi$
$812$	0	0
$813$	−2.90123	−0.101751
$814$	0	0
$815$	24.1767	0.846872
$816$	0	0
$817$	−40.4451	−1.41499
$818$	0	0
$819$	15.4300	0.539167
$820$	0	0
$821$	−14.1866	−0.495117	−0.247559	−	0.968873i	$-0.579628\pi$
$821$	−14.1866	−0.495117	−0.247559	+	0.968873i	$0.579628\pi$
$822$	0	0
$823$	−19.7720	−0.689208	−0.344604	−	0.938748i	$-0.611987\pi$
$823$	−19.7720	−0.689208	−0.344604	+	0.938748i	$0.611987\pi$
$824$	0	0
$825$	−0.208694	−0.00726579
$826$	0	0
$827$	−8.62567	−0.299944	−0.149972	−	0.988690i	$-0.547918\pi$
$827$	−8.62567	−0.299944	−0.149972	+	0.988690i	$0.547918\pi$
$828$	0	0
$829$	52.5859	1.82639	0.913193	−	0.407528i	$-0.133609\pi$
$829$	52.5859	1.82639	0.913193	+	0.407528i	$0.133609\pi$
$830$	0	0
$831$	4.23818	0.147021
$832$	0	0
$833$	−1.85642	−0.0643210
$834$	0	0
$835$	26.3135	0.910615
$836$	0	0
$837$	−4.59573	−0.158852
$838$	0	0
$839$	25.1992	0.869973	0.434986	−	0.900437i	$-0.356753\pi$
$839$	25.1992	0.869973	0.434986	+	0.900437i	$0.356753\pi$
$840$	0	0
$841$	−6.93097	−0.238999
$842$	0	0
$843$	−31.5746	−1.08749
$844$	0	0
$845$	11.6424	0.400509
$846$	0	0
$847$	−2.39335	−0.0822365
$848$	0	0
$849$	−30.2789	−1.03917
$850$	0	0
$851$	12.2411	0.419621
$852$	0	0
$853$	28.8887	0.989129	0.494564	−	0.869141i	$-0.335328\pi$
$853$	28.8887	0.989129	0.494564	+	0.869141i	$0.335328\pi$
$854$	0	0
$855$	36.1658	1.23685
$856$	0	0
$857$	−46.3734	−1.58409	−0.792043	−	0.610465i	$-0.790983\pi$
$857$	−46.3734	−1.58409	−0.792043	+	0.610465i	$0.790983\pi$
$858$	0	0
$859$	5.65215	0.192849	0.0964245	−	0.995340i	$-0.469259\pi$
$859$	5.65215	0.192849	0.0964245	+	0.995340i	$0.469259\pi$
$860$	0	0
$861$	9.96521	0.339613
$862$	0	0
$863$	−35.9833	−1.22489	−0.612443	−	0.790515i	$-0.709813\pi$
$863$	−35.9833	−1.22489	−0.612443	+	0.790515i	$0.709813\pi$
$864$	0	0
$865$	29.6800	1.00915
$866$	0	0
$867$	34.2405	1.16287
$868$	0	0
$869$	4.41537	0.149781
$870$	0	0
$871$	24.2735	0.822476
$872$	0	0
$873$	−1.97078	−0.0667008
$874$	0	0
$875$	−26.5105	−0.896219
$876$	0	0
$877$	−12.3264	−0.416232	−0.208116	−	0.978104i	$-0.566733\pi$
$877$	−12.3264	−0.416232	−0.208116	+	0.978104i	$0.566733\pi$
$878$	0	0
$879$	0.442411	0.0149222
$880$	0	0
$881$	27.2899	0.919420	0.459710	−	0.888069i	$-0.347953\pi$
$881$	27.2899	0.919420	0.459710	+	0.888069i	$0.347953\pi$
$882$	0	0
$883$	−42.3434	−1.42497	−0.712485	−	0.701687i	$-0.752430\pi$
$883$	−42.3434	−1.42497	−0.712485	+	0.701687i	$0.752430\pi$
$884$	0	0
$885$	66.1669	2.22418
$886$	0	0
$887$	−16.3252	−0.548146	−0.274073	−	0.961709i	$-0.588371\pi$
$887$	−16.3252	−0.548146	−0.274073	+	0.961709i	$0.588371\pi$
$888$	0	0
$889$	28.3160	0.949690
$890$	0	0
$891$	−10.6060	−0.355313
$892$	0	0
$893$	−38.0088	−1.27192
$894$	0	0
$895$	7.85467	0.262553
$896$	0	0
$897$	14.7269	0.491717
$898$	0	0
$899$	−13.4422	−0.448323
$900$	0	0
$901$	5.51393	0.183696
$902$	0	0
$903$	32.0186	1.06551
$904$	0	0
$905$	−27.3423	−0.908888
$906$	0	0
$907$	6.07610	0.201754	0.100877	−	0.994899i	$-0.467835\pi$
$907$	6.07610	0.201754	0.100877	+	0.994899i	$0.467835\pi$
$908$	0	0
$909$	10.9549	0.363351
$910$	0	0
$911$	24.7948	0.821490	0.410745	−	0.911750i	$-0.365269\pi$
$911$	24.7948	0.821490	0.410745	+	0.911750i	$0.365269\pi$
$912$	0	0
$913$	−12.5797	−0.416328
$914$	0	0
$915$	−3.28820	−0.108705
$916$	0	0
$917$	7.62313	0.251738
$918$	0	0
$919$	25.1971	0.831175	0.415588	−	0.909553i	$-0.363576\pi$
$919$	25.1971	0.831175	0.415588	+	0.909553i	$0.363576\pi$
$920$	0	0
$921$	56.6738	1.86747
$922$	0	0
$923$	32.6431	1.07446
$924$	0	0
$925$	0.485710	0.0159700
$926$	0	0
$927$	21.6753	0.711910
$928$	0	0
$929$	−43.7704	−1.43606	−0.718030	−	0.696012i	$-0.754956\pi$
$929$	−43.7704	−1.43606	−0.718030	+	0.696012i	$0.754956\pi$
$930$	0	0
$931$	8.85428	0.290187
$932$	0	0
$933$	65.1901	2.13423
$934$	0	0
$935$	−3.29320	−0.107699
$936$	0	0
$937$	−52.8168	−1.72545	−0.862724	−	0.505674i	$-0.831244\pi$
$937$	−52.8168	−1.72545	−0.862724	+	0.505674i	$0.831244\pi$
$938$	0	0
$939$	48.1070	1.56991
$940$	0	0
$941$	36.2253	1.18091	0.590455	−	0.807071i	$-0.298948\pi$
$941$	36.2253	1.18091	0.590455	+	0.807071i	$0.298948\pi$
$942$	0	0
$943$	4.13003	0.134492
$944$	0	0
$945$	8.67296	0.282131
$946$	0	0
$947$	−41.3321	−1.34311	−0.671557	−	0.740953i	$-0.734374\pi$
$947$	−41.3321	−1.34311	−0.671557	+	0.740953i	$0.734374\pi$
$948$	0	0
$949$	19.5297	0.633962
$950$	0	0
$951$	−57.5366	−1.86575
$952$	0	0
$953$	−27.9446	−0.905214	−0.452607	−	0.891710i	$-0.649506\pi$
$953$	−27.9446	−0.905214	−0.452607	+	0.891710i	$0.649506\pi$
$954$	0	0
$955$	25.7443	0.833067
$956$	0	0
$957$	−10.8176	−0.349685
$958$	0	0
$959$	2.39335	0.0772853
$960$	0	0
$961$	−22.8123	−0.735882
$962$	0	0
$963$	19.3217	0.622635
$964$	0	0
$965$	9.13270	0.293992
$966$	0	0
$967$	−45.4023	−1.46004	−0.730019	−	0.683427i	$-0.760489\pi$
$967$	−45.4023	−1.46004	−0.730019	+	0.683427i	$0.760489\pi$
$968$	0	0
$969$	23.3983	0.751661
$970$	0	0
$971$	−22.8229	−0.732422	−0.366211	−	0.930532i	$-0.619345\pi$
$971$	−22.8229	−0.732422	−0.366211	+	0.930532i	$0.619345\pi$
$972$	0	0
$973$	−44.2581	−1.41885
$974$	0	0
$975$	0.584340	0.0187139
$976$	0	0
$977$	−5.63004	−0.180121	−0.0900605	−	0.995936i	$-0.528706\pi$
$977$	−5.63004	−0.180121	−0.0900605	+	0.995936i	$0.528706\pi$
$978$	0	0
$979$	17.5607	0.561242
$980$	0	0
$981$	33.0834	1.05627
$982$	0	0
$983$	33.2235	1.05967	0.529833	−	0.848102i	$-0.322255\pi$
$983$	33.2235	1.05967	0.529833	+	0.848102i	$0.322255\pi$
$984$	0	0
$985$	29.3458	0.935036
$986$	0	0
$987$	30.0899	0.957773
$988$	0	0
$989$	13.2700	0.421960
$990$	0	0
$991$	18.6523	0.592509	0.296254	−	0.955109i	$-0.404262\pi$
$991$	18.6523	0.592509	0.296254	+	0.955109i	$0.404262\pi$
$992$	0	0
$993$	−21.0340	−0.667493
$994$	0	0
$995$	58.7228	1.86164
$996$	0	0
$997$	43.2770	1.37060	0.685298	−	0.728262i	$-0.259672\pi$
$997$	43.2770	1.37060	0.685298	+	0.728262i	$0.259672\pi$
$998$	0	0
$999$	8.60762	0.272333

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6028.2.a.c.1.6	✓	25

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6028.2.a.c.1.6	✓	25	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)

\(f(q)\)	\(=\)	\(q-2.30272 q^{3} -2.25624 q^{5} -2.39335 q^{7} +2.30252 q^{9} +O(q^{10})\)	\(q-2.30272 q^{3} -2.25624 q^{5} -2.39335 q^{7} +2.30252 q^{9} +1.00000 q^{11} -2.79999 q^{13} +5.19549 q^{15} +1.45960 q^{17} -6.96162 q^{19} +5.51122 q^{21} +2.28409 q^{23} +0.0906293 q^{25} +1.60611 q^{27} +4.69777 q^{29} -2.86141 q^{31} -2.30272 q^{33} +5.39998 q^{35} +5.35930 q^{37} +6.44759 q^{39} +1.80817 q^{41} +5.80973 q^{43} -5.19503 q^{45} +5.45976 q^{47} -1.27187 q^{49} -3.36104 q^{51} +3.77771 q^{53} -2.25624 q^{55} +16.0307 q^{57} +12.7354 q^{59} -0.632895 q^{61} -5.51073 q^{63} +6.31745 q^{65} -8.66914 q^{67} -5.25963 q^{69} -11.6583 q^{71} -6.97494 q^{73} -0.208694 q^{75} -2.39335 q^{77} +4.41537 q^{79} -10.6060 q^{81} -12.5797 q^{83} -3.29320 q^{85} -10.8176 q^{87} +17.5607 q^{89} +6.70135 q^{91} +6.58902 q^{93} +15.7071 q^{95} -0.855925 q^{97} +2.30252 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 25 q - 11 q^{3} - 2 q^{5} - 9 q^{7} + 18 q^{9}+O(q^{10}) \) 25 * q - 11 * q^3 - 2 * q^5 - 9 * q^7 + 18 * q^9	\( 25 q - 11 q^{3} - 2 q^{5} - 9 q^{7} + 18 q^{9} + 25 q^{11} - 4 q^{13} - 10 q^{15} - 19 q^{17} - 12 q^{19} + 8 q^{21} - 31 q^{23} - q^{25} - 44 q^{27} - q^{29} - 8 q^{31} - 11 q^{33} - 16 q^{35} - 14 q^{37} - 18 q^{39} - 5 q^{41} - 15 q^{43} - 15 q^{45} - 41 q^{47} + 2 q^{49} + 10 q^{51} + 4 q^{53} - 2 q^{55} - 3 q^{57} - 35 q^{59} - 4 q^{61} - 45 q^{63} - 28 q^{65} - 30 q^{67} - 3 q^{69} + 4 q^{71} - 7 q^{73} - 18 q^{75} - 9 q^{77} - 9 q^{79} + 29 q^{81} - 72 q^{83} - 33 q^{87} - 30 q^{89} - 10 q^{91} + 7 q^{93} + 9 q^{95} - 37 q^{97} + 18 q^{99}+O(q^{100}) \) 25 * q - 11 * q^3 - 2 * q^5 - 9 * q^7 + 18 * q^9 + 25 * q^11 - 4 * q^13 - 10 * q^15 - 19 * q^17 - 12 * q^19 + 8 * q^21 - 31 * q^23 - q^25 - 44 * q^27 - q^29 - 8 * q^31 - 11 * q^33 - 16 * q^35 - 14 * q^37 - 18 * q^39 - 5 * q^41 - 15 * q^43 - 15 * q^45 - 41 * q^47 + 2 * q^49 + 10 * q^51 + 4 * q^53 - 2 * q^55 - 3 * q^57 - 35 * q^59 - 4 * q^61 - 45 * q^63 - 28 * q^65 - 30 * q^67 - 3 * q^69 + 4 * q^71 - 7 * q^73 - 18 * q^75 - 9 * q^77 - 9 * q^79 + 29 * q^81 - 72 * q^83 - 33 * q^87 - 30 * q^89 - 10 * q^91 + 7 * q^93 + 9 * q^95 - 37 * q^97 + 18 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more