LMFDB - Embedded newform 8036.2.a.p.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8036,2,Mod(1,8036)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8036, 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("8036.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8036 = 2^{2} \cdot 7^{2} \cdot 41 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8036.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:	$64.1677830643$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 2x^{9} - 18x^{8} + 32x^{7} + 110x^{6} - 154x^{5} - 282x^{4} + 256x^{3} + 253x^{2} - 126x - 49 $ x^10 - 2x^9 - 18x^8 + 32x^7 + 110x^6 - 154x^5 - 282x^4 + 256x^3 + 253x^2 - 126*x - 49
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.91097$ of defining polynomial
Character	$\chi$	$=$	8036.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.91097 q^{3} +1.95825 q^{5} +0.651806 q^{9} +O(q^{10})$	$q-1.91097 q^{3} +1.95825 q^{5} +0.651806 q^{9} +0.643473 q^{11} -3.67505 q^{13} -3.74216 q^{15} -4.14775 q^{17} -0.715949 q^{19} -6.03491 q^{23} -1.16525 q^{25} +4.48733 q^{27} -8.58210 q^{29} +6.28884 q^{31} -1.22966 q^{33} +5.47767 q^{37} +7.02290 q^{39} +1.00000 q^{41} -10.5823 q^{43} +1.27640 q^{45} -9.33410 q^{47} +7.92622 q^{51} -0.928777 q^{53} +1.26008 q^{55} +1.36816 q^{57} +3.27970 q^{59} +3.37016 q^{61} -7.19667 q^{65} +13.9906 q^{67} +11.5325 q^{69} -0.555038 q^{71} +13.7022 q^{73} +2.22675 q^{75} +3.09679 q^{79} -10.5306 q^{81} -6.57829 q^{83} -8.12234 q^{85} +16.4001 q^{87} +16.3958 q^{89} -12.0178 q^{93} -1.40201 q^{95} +18.8641 q^{97} +0.419420 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + 2 q^{3} + 4 q^{5} + 10 q^{9}+O(q^{10}) $ 10 * q + 2 * q^3 + 4 * q^5 + 10 * q^9	$ 10 q + 2 q^{3} + 4 q^{5} + 10 q^{9} + 6 q^{13} - 4 q^{15} + 12 q^{17} + 8 q^{19} + 4 q^{23} + 16 q^{25} + 8 q^{27} + 2 q^{29} + 8 q^{31} - 6 q^{33} - 2 q^{37} - 2 q^{39} + 10 q^{41} - 2 q^{43} + 44 q^{45} - 14 q^{47} + 14 q^{51} + 8 q^{53} + 8 q^{55} - 10 q^{57} + 24 q^{59} + 14 q^{61} + 2 q^{65} - 8 q^{67} + 16 q^{69} + 10 q^{71} + 44 q^{73} - 50 q^{75} + 10 q^{79} - 14 q^{81} + 20 q^{83} + 8 q^{85} + 20 q^{87} + 6 q^{89} + 8 q^{93} + 4 q^{95} + 46 q^{97} + 2 q^{99}+O(q^{100}) $ 10 * q + 2 * q^3 + 4 * q^5 + 10 * q^9 + 6 * q^13 - 4 * q^15 + 12 * q^17 + 8 * q^19 + 4 * q^23 + 16 * q^25 + 8 * q^27 + 2 * q^29 + 8 * q^31 - 6 * q^33 - 2 * q^37 - 2 * q^39 + 10 * q^41 - 2 * q^43 + 44 * q^45 - 14 * q^47 + 14 * q^51 + 8 * q^53 + 8 * q^55 - 10 * q^57 + 24 * q^59 + 14 * q^61 + 2 * q^65 - 8 * q^67 + 16 * q^69 + 10 * q^71 + 44 * q^73 - 50 * q^75 + 10 * q^79 - 14 * q^81 + 20 * q^83 + 8 * q^85 + 20 * q^87 + 6 * q^89 + 8 * q^93 + 4 * q^95 + 46 * q^97 + 2 * 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
$3$	−1.91097	−1.10330	−0.551649	−	0.834076i	$-0.686001\pi$
$3$	−1.91097	−1.10330	−0.551649	+	0.834076i	$0.686001\pi$
$4$	0	0
$5$	1.95825	0.875757	0.437879	−	0.899034i	$-0.355730\pi$
$5$	1.95825	0.875757	0.437879	+	0.899034i	$0.355730\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	0.651806	0.217269
$10$	0	0
$11$	0.643473	0.194015	0.0970073	−	0.995284i	$-0.469073\pi$
$11$	0.643473	0.194015	0.0970073	+	0.995284i	$0.469073\pi$
$12$	0	0
$13$	−3.67505	−1.01927	−0.509637	−	0.860389i	$-0.670220\pi$
$13$	−3.67505	−1.01927	−0.509637	+	0.860389i	$0.670220\pi$
$14$	0	0
$15$	−3.74216	−0.966222
$16$	0	0
$17$	−4.14775	−1.00598	−0.502988	−	0.864293i	$-0.667766\pi$
$17$	−4.14775	−1.00598	−0.502988	+	0.864293i	$0.667766\pi$
$18$	0	0
$19$	−0.715949	−0.164250	−0.0821250	−	0.996622i	$-0.526171\pi$
$19$	−0.715949	−0.164250	−0.0821250	+	0.996622i	$0.526171\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−6.03491	−1.25836	−0.629182	−	0.777258i	$-0.716610\pi$
$23$	−6.03491	−1.25836	−0.629182	+	0.777258i	$0.716610\pi$
$24$	0	0
$25$	−1.16525	−0.233049
$26$	0	0
$27$	4.48733	0.863587
$28$	0	0
$29$	−8.58210	−1.59366	−0.796828	−	0.604206i	$-0.793490\pi$
$29$	−8.58210	−1.59366	−0.796828	+	0.604206i	$0.793490\pi$
$30$	0	0
$31$	6.28884	1.12951	0.564754	−	0.825259i	$-0.308971\pi$
$31$	6.28884	1.12951	0.564754	+	0.825259i	$0.308971\pi$
$32$	0	0
$33$	−1.22966	−0.214056
$34$	0	0
$35$	0	0
$36$	0	0
$37$	5.47767	0.900523	0.450261	−	0.892897i	$-0.351331\pi$
$37$	5.47767	0.900523	0.450261	+	0.892897i	$0.351331\pi$
$38$	0	0
$39$	7.02290	1.12456
$40$	0	0
$41$	1.00000	0.156174
$41$	1.00000	0.156174
$42$	0	0
$43$	−10.5823	−1.61379	−0.806893	−	0.590698i	$-0.798853\pi$
$43$	−10.5823	−1.61379	−0.806893	+	0.590698i	$0.798853\pi$
$44$	0	0
$45$	1.27640	0.190275
$46$	0	0
$47$	−9.33410	−1.36152	−0.680760	−	0.732507i	$-0.738350\pi$
$47$	−9.33410	−1.36152	−0.680760	+	0.732507i	$0.738350\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	7.92622	1.10989
$52$	0	0
$53$	−0.928777	−0.127577	−0.0637887	−	0.997963i	$-0.520318\pi$
$53$	−0.928777	−0.127577	−0.0637887	+	0.997963i	$0.520318\pi$
$54$	0	0
$55$	1.26008	0.169910
$56$	0	0
$57$	1.36816	0.181217
$58$	0	0
$59$	3.27970	0.426980	0.213490	−	0.976945i	$-0.431517\pi$
$59$	3.27970	0.426980	0.213490	+	0.976945i	$0.431517\pi$
$60$	0	0
$61$	3.37016	0.431504	0.215752	−	0.976448i	$-0.430780\pi$
$61$	3.37016	0.431504	0.215752	+	0.976448i	$0.430780\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−7.19667	−0.892637
$66$	0	0
$67$	13.9906	1.70923	0.854614	−	0.519264i	$-0.173794\pi$
$67$	13.9906	1.70923	0.854614	+	0.519264i	$0.173794\pi$
$68$	0	0
$69$	11.5325	1.38835
$70$	0	0
$71$	−0.555038	−0.0658709	−0.0329355	−	0.999457i	$-0.510486\pi$
$71$	−0.555038	−0.0658709	−0.0329355	+	0.999457i	$0.510486\pi$
$72$	0	0
$73$	13.7022	1.60373	0.801863	−	0.597508i	$-0.203842\pi$
$73$	13.7022	1.60373	0.801863	+	0.597508i	$0.203842\pi$
$74$	0	0
$75$	2.22675	0.257123
$76$	0	0
$77$	0	0
$78$	0	0
$79$	3.09679	0.348416	0.174208	−	0.984709i	$-0.444264\pi$
$79$	3.09679	0.348416	0.174208	+	0.984709i	$0.444264\pi$
$80$	0	0
$81$	−10.5306	−1.17006
$82$	0	0
$83$	−6.57829	−0.722061	−0.361030	−	0.932554i	$-0.617575\pi$
$83$	−6.57829	−0.722061	−0.361030	+	0.932554i	$0.617575\pi$
$84$	0	0
$85$	−8.12234	−0.880992
$86$	0	0
$87$	16.4001	1.75828
$88$	0	0
$89$	16.3958	1.73795	0.868974	−	0.494858i	$-0.164780\pi$
$89$	16.3958	1.73795	0.868974	+	0.494858i	$0.164780\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−12.0178	−1.24619
$94$	0	0
$95$	−1.40201	−0.143843
$96$	0	0
$97$	18.8641	1.91536	0.957678	−	0.287840i	$-0.0929372\pi$
$97$	18.8641	1.91536	0.957678	+	0.287840i	$0.0929372\pi$
$98$	0	0
$99$	0.419420	0.0421533
$100$	0	0
$101$	−6.25842	−0.622736	−0.311368	−	0.950289i	$-0.600787\pi$
$101$	−6.25842	−0.622736	−0.311368	+	0.950289i	$0.600787\pi$
$102$	0	0
$103$	−4.69885	−0.462991	−0.231496	−	0.972836i	$-0.574362\pi$
$103$	−4.69885	−0.462991	−0.231496	+	0.972836i	$0.574362\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	3.02610	0.292544	0.146272	−	0.989244i	$-0.453273\pi$
$107$	3.02610	0.292544	0.146272	+	0.989244i	$0.453273\pi$
$108$	0	0
$109$	13.5508	1.29793	0.648967	−	0.760817i	$-0.275201\pi$
$109$	13.5508	1.29793	0.648967	+	0.760817i	$0.275201\pi$
$110$	0	0
$111$	−10.4677	−0.993546
$112$	0	0
$113$	7.33757	0.690260	0.345130	−	0.938555i	$-0.387835\pi$
$113$	7.33757	0.690260	0.345130	+	0.938555i	$0.387835\pi$
$114$	0	0
$115$	−11.8179	−1.10202
$116$	0	0
$117$	−2.39542	−0.221456
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−10.5859	−0.962358
$122$	0	0
$123$	−1.91097	−0.172306
$124$	0	0
$125$	−12.0731	−1.07985
$126$	0	0
$127$	−19.8301	−1.75963	−0.879817	−	0.475312i	$-0.842335\pi$
$127$	−19.8301	−1.75963	−0.879817	+	0.475312i	$0.842335\pi$
$128$	0	0
$129$	20.2225	1.78049
$130$	0	0
$131$	−13.2327	−1.15614	−0.578072	−	0.815986i	$-0.696195\pi$
$131$	−13.2327	−1.15614	−0.578072	+	0.815986i	$0.696195\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	8.78732	0.756292
$136$	0	0
$137$	13.4024	1.14505	0.572523	−	0.819889i	$-0.305965\pi$
$137$	13.4024	1.14505	0.572523	+	0.819889i	$0.305965\pi$
$138$	0	0
$139$	22.4485	1.90406	0.952030	−	0.306003i	$-0.0989918\pi$
$139$	22.4485	1.90406	0.952030	+	0.306003i	$0.0989918\pi$
$140$	0	0
$141$	17.8372	1.50216
$142$	0	0
$143$	−2.36480	−0.197754
$144$	0	0
$145$	−16.8059	−1.39566
$146$	0	0
$147$	0	0
$148$	0	0
$149$	19.9192	1.63185	0.815924	−	0.578159i	$-0.196229\pi$
$149$	19.9192	1.63185	0.815924	+	0.578159i	$0.196229\pi$
$150$	0	0
$151$	19.3734	1.57659	0.788294	−	0.615299i	$-0.210965\pi$
$151$	19.3734	1.57659	0.788294	+	0.615299i	$0.210965\pi$
$152$	0	0
$153$	−2.70353	−0.218567
$154$	0	0
$155$	12.3151	0.989175
$156$	0	0
$157$	5.70389	0.455220	0.227610	−	0.973752i	$-0.426909\pi$
$157$	5.70389	0.455220	0.227610	+	0.973752i	$0.426909\pi$
$158$	0	0
$159$	1.77486	0.140756
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−11.3263	−0.887146	−0.443573	−	0.896238i	$-0.646289\pi$
$163$	−11.3263	−0.887146	−0.443573	+	0.896238i	$0.646289\pi$
$164$	0	0
$165$	−2.40798	−0.187461
$166$	0	0
$167$	6.12840	0.474230	0.237115	−	0.971482i	$-0.423798\pi$
$167$	6.12840	0.474230	0.237115	+	0.971482i	$0.423798\pi$
$168$	0	0
$169$	0.505968	0.0389206
$170$	0	0
$171$	−0.466660	−0.0356864
$172$	0	0
$173$	−0.598647	−0.0455143	−0.0227571	−	0.999741i	$-0.507244\pi$
$173$	−0.598647	−0.0455143	−0.0227571	+	0.999741i	$0.507244\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−6.26741	−0.471087
$178$	0	0
$179$	−10.8022	−0.807397	−0.403699	−	0.914892i	$-0.632276\pi$
$179$	−10.8022	−0.807397	−0.403699	+	0.914892i	$0.632276\pi$
$180$	0	0
$181$	13.8512	1.02955	0.514777	−	0.857324i	$-0.327875\pi$
$181$	13.8512	1.02955	0.514777	+	0.857324i	$0.327875\pi$
$182$	0	0
$183$	−6.44027	−0.476078
$184$	0	0
$185$	10.7267	0.788639
$186$	0	0
$187$	−2.66897	−0.195174
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−20.2949	−1.46849	−0.734243	−	0.678886i	$-0.762463\pi$
$191$	−20.2949	−1.46849	−0.734243	+	0.678886i	$0.762463\pi$
$192$	0	0
$193$	2.74477	0.197573	0.0987866	−	0.995109i	$-0.468504\pi$
$193$	2.74477	0.197573	0.0987866	+	0.995109i	$0.468504\pi$
$194$	0	0
$195$	13.7526	0.984846
$196$	0	0
$197$	−14.7453	−1.05056	−0.525280	−	0.850929i	$-0.676039\pi$
$197$	−14.7453	−1.05056	−0.525280	+	0.850929i	$0.676039\pi$
$198$	0	0
$199$	−9.53698	−0.676059	−0.338029	−	0.941136i	$-0.609760\pi$
$199$	−9.53698	−0.676059	−0.338029	+	0.941136i	$0.609760\pi$
$200$	0	0
$201$	−26.7357	−1.88579
$202$	0	0
$203$	0	0
$204$	0	0
$205$	1.95825	0.136770
$206$	0	0
$207$	−3.93359	−0.273403
$208$	0	0
$209$	−0.460694	−0.0318669
$210$	0	0
$211$	−17.4826	−1.20355	−0.601775	−	0.798666i	$-0.705540\pi$
$211$	−17.4826	−1.20355	−0.601775	+	0.798666i	$0.705540\pi$
$212$	0	0
$213$	1.06066	0.0726754
$214$	0	0
$215$	−20.7228	−1.41328
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−26.1846	−1.76939
$220$	0	0
$221$	15.2432	1.02537
$222$	0	0
$223$	1.69394	0.113435	0.0567174	−	0.998390i	$-0.481937\pi$
$223$	1.69394	0.113435	0.0567174	+	0.998390i	$0.481937\pi$
$224$	0	0
$225$	−0.759515	−0.0506343
$226$	0	0
$227$	20.5800	1.36594	0.682970	−	0.730446i	$-0.260688\pi$
$227$	20.5800	1.36594	0.682970	+	0.730446i	$0.260688\pi$
$228$	0	0
$229$	13.1024	0.865830	0.432915	−	0.901435i	$-0.357485\pi$
$229$	13.1024	0.865830	0.432915	+	0.901435i	$0.357485\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−6.60096	−0.432443	−0.216222	−	0.976344i	$-0.569373\pi$
$233$	−6.60096	−0.432443	−0.216222	+	0.976344i	$0.569373\pi$
$234$	0	0
$235$	−18.2785	−1.19236
$236$	0	0
$237$	−5.91787	−0.384407
$238$	0	0
$239$	−8.80511	−0.569555	−0.284778	−	0.958594i	$-0.591920\pi$
$239$	−8.80511	−0.569555	−0.284778	+	0.958594i	$0.591920\pi$
$240$	0	0
$241$	3.09739	0.199520	0.0997602	−	0.995012i	$-0.468192\pi$
$241$	3.09739	0.199520	0.0997602	+	0.995012i	$0.468192\pi$
$242$	0	0
$243$	6.66161	0.427343
$244$	0	0
$245$	0	0
$246$	0	0
$247$	2.63115	0.167416
$248$	0	0
$249$	12.5709	0.796649
$250$	0	0
$251$	19.4178	1.22564	0.612819	−	0.790223i	$-0.290035\pi$
$251$	19.4178	1.22564	0.612819	+	0.790223i	$0.290035\pi$
$252$	0	0
$253$	−3.88330	−0.244141
$254$	0	0
$255$	15.5215	0.971997
$256$	0	0
$257$	2.62884	0.163983	0.0819913	−	0.996633i	$-0.473872\pi$
$257$	2.62884	0.163983	0.0819913	+	0.996633i	$0.473872\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−5.59386	−0.346252
$262$	0	0
$263$	−3.36712	−0.207625	−0.103813	−	0.994597i	$-0.533104\pi$
$263$	−3.36712	−0.207625	−0.103813	+	0.994597i	$0.533104\pi$
$264$	0	0
$265$	−1.81878	−0.111727
$266$	0	0
$267$	−31.3318	−1.91748
$268$	0	0
$269$	11.7979	0.719332	0.359666	−	0.933081i	$-0.382891\pi$
$269$	11.7979	0.719332	0.359666	+	0.933081i	$0.382891\pi$
$270$	0	0
$271$	−6.86899	−0.417261	−0.208631	−	0.977994i	$-0.566901\pi$
$271$	−6.86899	−0.417261	−0.208631	+	0.977994i	$0.566901\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−0.749805	−0.0452150
$276$	0	0
$277$	−25.4503	−1.52916	−0.764579	−	0.644530i	$-0.777053\pi$
$277$	−25.4503	−1.52916	−0.764579	+	0.644530i	$0.777053\pi$
$278$	0	0
$279$	4.09910	0.245407
$280$	0	0
$281$	30.1862	1.80076	0.900379	−	0.435106i	$-0.143289\pi$
$281$	30.1862	1.80076	0.900379	+	0.435106i	$0.143289\pi$
$282$	0	0
$283$	0.426937	0.0253788	0.0126894	−	0.999919i	$-0.495961\pi$
$283$	0.426937	0.0253788	0.0126894	+	0.999919i	$0.495961\pi$
$284$	0	0
$285$	2.67920	0.158702
$286$	0	0
$287$	0	0
$288$	0	0
$289$	0.203821	0.0119895
$290$	0	0
$291$	−36.0487	−2.11321
$292$	0	0
$293$	−12.6014	−0.736184	−0.368092	−	0.929789i	$-0.619989\pi$
$293$	−12.6014	−0.736184	−0.368092	+	0.929789i	$0.619989\pi$
$294$	0	0
$295$	6.42248	0.373931
$296$	0	0
$297$	2.88748	0.167548
$298$	0	0
$299$	22.1786	1.28262
$300$	0	0
$301$	0	0
$302$	0	0
$303$	11.9596	0.687064
$304$	0	0
$305$	6.59962	0.377893
$306$	0	0
$307$	−8.09076	−0.461764	−0.230882	−	0.972982i	$-0.574161\pi$
$307$	−8.09076	−0.461764	−0.230882	+	0.972982i	$0.574161\pi$
$308$	0	0
$309$	8.97936	0.510818
$310$	0	0
$311$	0.357500	0.0202720	0.0101360	−	0.999949i	$-0.496774\pi$
$311$	0.357500	0.0202720	0.0101360	+	0.999949i	$0.496774\pi$
$312$	0	0
$313$	26.5989	1.50346	0.751729	−	0.659472i	$-0.229220\pi$
$313$	26.5989	1.50346	0.751729	+	0.659472i	$0.229220\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−25.5545	−1.43528	−0.717642	−	0.696413i	$-0.754778\pi$
$317$	−25.5545	−1.43528	−0.717642	+	0.696413i	$0.754778\pi$
$318$	0	0
$319$	−5.52235	−0.309192
$320$	0	0
$321$	−5.78278	−0.322764
$322$	0	0
$323$	2.96958	0.165232
$324$	0	0
$325$	4.28234	0.237541
$326$	0	0
$327$	−25.8952	−1.43201
$328$	0	0
$329$	0	0
$330$	0	0
$331$	16.0465	0.881998	0.440999	−	0.897508i	$-0.354624\pi$
$331$	16.0465	0.881998	0.440999	+	0.897508i	$0.354624\pi$
$332$	0	0
$333$	3.57038	0.195655
$334$	0	0
$335$	27.3972	1.49687
$336$	0	0
$337$	−6.34485	−0.345626	−0.172813	−	0.984955i	$-0.555286\pi$
$337$	−6.34485	−0.345626	−0.172813	+	0.984955i	$0.555286\pi$
$338$	0	0
$339$	−14.0219	−0.761564
$340$	0	0
$341$	4.04670	0.219141
$342$	0	0
$343$	0	0
$344$	0	0
$345$	22.5836	1.21586
$346$	0	0
$347$	−25.4136	−1.36427	−0.682136	−	0.731225i	$-0.738949\pi$
$347$	−25.4136	−1.36427	−0.682136	+	0.731225i	$0.738949\pi$
$348$	0	0
$349$	5.87318	0.314384	0.157192	−	0.987568i	$-0.449756\pi$
$349$	5.87318	0.314384	0.157192	+	0.987568i	$0.449756\pi$
$350$	0	0
$351$	−16.4911	−0.880232
$352$	0	0
$353$	18.3167	0.974901	0.487451	−	0.873151i	$-0.337927\pi$
$353$	18.3167	0.974901	0.487451	+	0.873151i	$0.337927\pi$
$354$	0	0
$355$	−1.08691	−0.0576870
$356$	0	0
$357$	0	0
$358$	0	0
$359$	11.1404	0.587967	0.293983	−	0.955810i	$-0.405019\pi$
$359$	11.1404	0.587967	0.293983	+	0.955810i	$0.405019\pi$
$360$	0	0
$361$	−18.4874	−0.973022
$362$	0	0
$363$	20.2294	1.06177
$364$	0	0
$365$	26.8324	1.40447
$366$	0	0
$367$	−5.92956	−0.309520	−0.154760	−	0.987952i	$-0.549460\pi$
$367$	−5.92956	−0.309520	−0.154760	+	0.987952i	$0.549460\pi$
$368$	0	0
$369$	0.651806	0.0339317
$370$	0	0
$371$	0	0
$372$	0	0
$373$	3.33117	0.172481	0.0862406	−	0.996274i	$-0.472515\pi$
$373$	3.33117	0.172481	0.0862406	+	0.996274i	$0.472515\pi$
$374$	0	0
$375$	23.0714	1.19140
$376$	0	0
$377$	31.5396	1.62437
$378$	0	0
$379$	−5.80640	−0.298255	−0.149127	−	0.988818i	$-0.547646\pi$
$379$	−5.80640	−0.298255	−0.149127	+	0.988818i	$0.547646\pi$
$380$	0	0
$381$	37.8947	1.94140
$382$	0	0
$383$	0.396175	0.0202436	0.0101218	−	0.999949i	$-0.496778\pi$
$383$	0.396175	0.0202436	0.0101218	+	0.999949i	$0.496778\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−6.89761	−0.350625
$388$	0	0
$389$	37.1505	1.88361	0.941803	−	0.336166i	$-0.109130\pi$
$389$	37.1505	1.88361	0.941803	+	0.336166i	$0.109130\pi$
$390$	0	0
$391$	25.0313	1.26589
$392$	0	0
$393$	25.2872	1.27557
$394$	0	0
$395$	6.06429	0.305128
$396$	0	0
$397$	11.1069	0.557441	0.278720	−	0.960372i	$-0.410090\pi$
$397$	11.1069	0.557441	0.278720	+	0.960372i	$0.410090\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−14.3534	−0.716774	−0.358387	−	0.933573i	$-0.616673\pi$
$401$	−14.3534	−0.716774	−0.358387	+	0.933573i	$0.616673\pi$
$402$	0	0
$403$	−23.1118	−1.15128
$404$	0	0
$405$	−20.6215	−1.02469
$406$	0	0
$407$	3.52473	0.174715
$408$	0	0
$409$	21.8460	1.08021	0.540107	−	0.841597i	$-0.318384\pi$
$409$	21.8460	1.08021	0.540107	+	0.841597i	$0.318384\pi$
$410$	0	0
$411$	−25.6116	−1.26333
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−12.8819	−0.632350
$416$	0	0
$417$	−42.8985	−2.10075
$418$	0	0
$419$	−11.2265	−0.548450	−0.274225	−	0.961666i	$-0.588421\pi$
$419$	−11.2265	−0.548450	−0.274225	+	0.961666i	$0.588421\pi$
$420$	0	0
$421$	22.0291	1.07363	0.536817	−	0.843699i	$-0.319627\pi$
$421$	22.0291	1.07363	0.536817	+	0.843699i	$0.319627\pi$
$422$	0	0
$423$	−6.08403	−0.295815
$424$	0	0
$425$	4.83315	0.234442
$426$	0	0
$427$	0	0
$428$	0	0
$429$	4.51905	0.218182
$430$	0	0
$431$	20.5384	0.989300	0.494650	−	0.869092i	$-0.335296\pi$
$431$	20.5384	0.989300	0.494650	+	0.869092i	$0.335296\pi$
$432$	0	0
$433$	20.9277	1.00572	0.502862	−	0.864367i	$-0.332281\pi$
$433$	20.9277	1.00572	0.502862	+	0.864367i	$0.332281\pi$
$434$	0	0
$435$	32.1156	1.53983
$436$	0	0
$437$	4.32068	0.206686
$438$	0	0
$439$	8.02256	0.382896	0.191448	−	0.981503i	$-0.438682\pi$
$439$	8.02256	0.382896	0.191448	+	0.981503i	$0.438682\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	23.3599	1.10986	0.554931	−	0.831897i	$-0.312745\pi$
$443$	23.3599	1.10986	0.554931	+	0.831897i	$0.312745\pi$
$444$	0	0
$445$	32.1070	1.52202
$446$	0	0
$447$	−38.0651	−1.80042
$448$	0	0
$449$	26.8838	1.26872	0.634362	−	0.773036i	$-0.281263\pi$
$449$	26.8838	1.26872	0.634362	+	0.773036i	$0.281263\pi$
$450$	0	0
$451$	0.643473	0.0303000
$452$	0	0
$453$	−37.0221	−1.73945
$454$	0	0
$455$	0	0
$456$	0	0
$457$	36.1981	1.69327	0.846637	−	0.532170i	$-0.178623\pi$
$457$	36.1981	1.69327	0.846637	+	0.532170i	$0.178623\pi$
$458$	0	0
$459$	−18.6123	−0.868748
$460$	0	0
$461$	−10.0948	−0.470163	−0.235081	−	0.971976i	$-0.575536\pi$
$461$	−10.0948	−0.470163	−0.235081	+	0.971976i	$0.575536\pi$
$462$	0	0
$463$	5.51652	0.256374	0.128187	−	0.991750i	$-0.459084\pi$
$463$	5.51652	0.256374	0.128187	+	0.991750i	$0.459084\pi$
$464$	0	0
$465$	−23.5338	−1.09136
$466$	0	0
$467$	−16.7658	−0.775830	−0.387915	−	0.921695i	$-0.626805\pi$
$467$	−16.7658	−0.775830	−0.387915	+	0.921695i	$0.626805\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−10.9000	−0.502243
$472$	0	0
$473$	−6.80943	−0.313098
$474$	0	0
$475$	0.834257	0.0382783
$476$	0	0
$477$	−0.605382	−0.0277186
$478$	0	0
$479$	29.4210	1.34428	0.672141	−	0.740423i	$-0.265375\pi$
$479$	29.4210	1.34428	0.672141	+	0.740423i	$0.265375\pi$
$480$	0	0
$481$	−20.1307	−0.917880
$482$	0	0
$483$	0	0
$484$	0	0
$485$	36.9406	1.67739
$486$	0	0
$487$	−30.2436	−1.37047	−0.685234	−	0.728323i	$-0.740300\pi$
$487$	−30.2436	−1.37047	−0.685234	+	0.728323i	$0.740300\pi$
$488$	0	0
$489$	21.6443	0.978787
$490$	0	0
$491$	1.38354	0.0624383	0.0312192	−	0.999513i	$-0.490061\pi$
$491$	1.38354	0.0624383	0.0312192	+	0.999513i	$0.490061\pi$
$492$	0	0
$493$	35.5964	1.60318
$494$	0	0
$495$	0.821330	0.0369160
$496$	0	0
$497$	0	0
$498$	0	0
$499$	5.03472	0.225385	0.112693	−	0.993630i	$-0.464052\pi$
$499$	5.03472	0.225385	0.112693	+	0.993630i	$0.464052\pi$
$500$	0	0
$501$	−11.7112	−0.523218
$502$	0	0
$503$	−15.9805	−0.712533	−0.356267	−	0.934384i	$-0.615951\pi$
$503$	−15.9805	−0.712533	−0.356267	+	0.934384i	$0.615951\pi$
$504$	0	0
$505$	−12.2556	−0.545365
$506$	0	0
$507$	−0.966889	−0.0429411
$508$	0	0
$509$	−27.0090	−1.19715	−0.598576	−	0.801066i	$-0.704267\pi$
$509$	−27.0090	−1.19715	−0.598576	+	0.801066i	$0.704267\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−3.21270	−0.141844
$514$	0	0
$515$	−9.20153	−0.405468
$516$	0	0
$517$	−6.00625	−0.264155
$518$	0	0
$519$	1.14400	0.0502159
$520$	0	0
$521$	−15.9602	−0.699229	−0.349615	−	0.936894i	$-0.613687\pi$
$521$	−15.9602	−0.699229	−0.349615	+	0.936894i	$0.613687\pi$
$522$	0	0
$523$	1.19280	0.0521575	0.0260787	−	0.999660i	$-0.491698\pi$
$523$	1.19280	0.0521575	0.0260787	+	0.999660i	$0.491698\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−26.0845	−1.13626
$528$	0	0
$529$	13.4201	0.583482
$530$	0	0
$531$	2.13773	0.0927695
$532$	0	0
$533$	−3.67505	−0.159184
$534$	0	0
$535$	5.92587	0.256198
$536$	0	0
$537$	20.6428	0.890800
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−24.4332	−1.05047	−0.525233	−	0.850959i	$-0.676022\pi$
$541$	−24.4332	−1.05047	−0.525233	+	0.850959i	$0.676022\pi$
$542$	0	0
$543$	−26.4693	−1.13591
$544$	0	0
$545$	26.5359	1.13667
$546$	0	0
$547$	36.7175	1.56993	0.784964	−	0.619541i	$-0.212681\pi$
$547$	36.7175	1.56993	0.784964	+	0.619541i	$0.212681\pi$
$548$	0	0
$549$	2.19669	0.0937523
$550$	0	0
$551$	6.14435	0.261758
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−20.4983	−0.870105
$556$	0	0
$557$	17.3858	0.736660	0.368330	−	0.929695i	$-0.379930\pi$
$557$	17.3858	0.736660	0.368330	+	0.929695i	$0.379930\pi$
$558$	0	0
$559$	38.8904	1.64489
$560$	0	0
$561$	5.10031	0.215335
$562$	0	0
$563$	36.0940	1.52118	0.760590	−	0.649232i	$-0.224910\pi$
$563$	36.0940	1.52118	0.760590	+	0.649232i	$0.224910\pi$
$564$	0	0
$565$	14.3688	0.604501
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−10.0482	−0.421245	−0.210622	−	0.977568i	$-0.567549\pi$
$569$	−10.0482	−0.421245	−0.210622	+	0.977568i	$0.567549\pi$
$570$	0	0
$571$	33.7668	1.41309	0.706547	−	0.707666i	$-0.250252\pi$
$571$	33.7668	1.41309	0.706547	+	0.707666i	$0.250252\pi$
$572$	0	0
$573$	38.7829	1.62018
$574$	0	0
$575$	7.03215	0.293261
$576$	0	0
$577$	−38.3567	−1.59681	−0.798405	−	0.602120i	$-0.794323\pi$
$577$	−38.3567	−1.59681	−0.798405	+	0.602120i	$0.794323\pi$
$578$	0	0
$579$	−5.24518	−0.217982
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−0.597643	−0.0247519
$584$	0	0
$585$	−4.69083	−0.193942
$586$	0	0
$587$	46.0630	1.90122	0.950612	−	0.310382i	$-0.100457\pi$
$587$	46.0630	1.90122	0.950612	+	0.310382i	$0.100457\pi$
$588$	0	0
$589$	−4.50248	−0.185522
$590$	0	0
$591$	28.1778	1.15908
$592$	0	0
$593$	18.4036	0.755743	0.377872	−	0.925858i	$-0.376656\pi$
$593$	18.4036	0.755743	0.377872	+	0.925858i	$0.376656\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	18.2249	0.745895
$598$	0	0
$599$	41.2588	1.68579	0.842894	−	0.538080i	$-0.180850\pi$
$599$	41.2588	1.68579	0.842894	+	0.538080i	$0.180850\pi$
$600$	0	0
$601$	4.38098	0.178704	0.0893519	−	0.996000i	$-0.471520\pi$
$601$	4.38098	0.178704	0.0893519	+	0.996000i	$0.471520\pi$
$602$	0	0
$603$	9.11918	0.371362
$604$	0	0
$605$	−20.7299	−0.842792
$606$	0	0
$607$	10.0526	0.408024	0.204012	−	0.978968i	$-0.434602\pi$
$607$	10.0526	0.408024	0.204012	+	0.978968i	$0.434602\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	34.3033	1.38776
$612$	0	0
$613$	25.0095	1.01013	0.505063	−	0.863083i	$-0.331469\pi$
$613$	25.0095	1.01013	0.505063	+	0.863083i	$0.331469\pi$
$614$	0	0
$615$	−3.74216	−0.150899
$616$	0	0
$617$	13.4390	0.541032	0.270516	−	0.962716i	$-0.412806\pi$
$617$	13.4390	0.541032	0.270516	+	0.962716i	$0.412806\pi$
$618$	0	0
$619$	31.2843	1.25742	0.628710	−	0.777639i	$-0.283583\pi$
$619$	31.2843	1.25742	0.628710	+	0.777639i	$0.283583\pi$
$620$	0	0
$621$	−27.0806	−1.08671
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−17.8160	−0.712639
$626$	0	0
$627$	0.880373	0.0351587
$628$	0	0
$629$	−22.7200	−0.905905
$630$	0	0
$631$	−12.7490	−0.507530	−0.253765	−	0.967266i	$-0.581669\pi$
$631$	−12.7490	−0.507530	−0.253765	+	0.967266i	$0.581669\pi$
$632$	0	0
$633$	33.4087	1.32788
$634$	0	0
$635$	−38.8323	−1.54101
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−0.361777	−0.0143117
$640$	0	0
$641$	12.9352	0.510909	0.255455	−	0.966821i	$-0.417775\pi$
$641$	12.9352	0.510909	0.255455	+	0.966821i	$0.417775\pi$
$642$	0	0
$643$	−31.2875	−1.23386	−0.616929	−	0.787019i	$-0.711624\pi$
$643$	−31.2875	−1.23386	−0.616929	+	0.787019i	$0.711624\pi$
$644$	0	0
$645$	39.6007	1.55928
$646$	0	0
$647$	−50.4938	−1.98512	−0.992558	−	0.121771i	$-0.961143\pi$
$647$	−50.4938	−1.98512	−0.992558	+	0.121771i	$0.961143\pi$
$648$	0	0
$649$	2.11040	0.0828404
$650$	0	0
$651$	0	0
$652$	0	0
$653$	37.3278	1.46075	0.730374	−	0.683047i	$-0.239346\pi$
$653$	37.3278	1.46075	0.730374	+	0.683047i	$0.239346\pi$
$654$	0	0
$655$	−25.9129	−1.01250
$656$	0	0
$657$	8.93120	0.348439
$658$	0	0
$659$	3.19788	0.124572	0.0622859	−	0.998058i	$-0.480161\pi$
$659$	3.19788	0.124572	0.0622859	+	0.998058i	$0.480161\pi$
$660$	0	0
$661$	−13.6505	−0.530942	−0.265471	−	0.964119i	$-0.585527\pi$
$661$	−13.6505	−0.530942	−0.265471	+	0.964119i	$0.585527\pi$
$662$	0	0
$663$	−29.1292	−1.13129
$664$	0	0
$665$	0	0
$666$	0	0
$667$	51.7922	2.00540
$668$	0	0
$669$	−3.23707	−0.125152
$670$	0	0
$671$	2.16861	0.0837181
$672$	0	0
$673$	−27.9374	−1.07691	−0.538454	−	0.842655i	$-0.680991\pi$
$673$	−27.9374	−1.07691	−0.538454	+	0.842655i	$0.680991\pi$
$674$	0	0
$675$	−5.22884	−0.201258
$676$	0	0
$677$	1.03395	0.0397380	0.0198690	−	0.999803i	$-0.493675\pi$
$677$	1.03395	0.0397380	0.0198690	+	0.999803i	$0.493675\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−39.3277	−1.50704
$682$	0	0
$683$	−23.3348	−0.892880	−0.446440	−	0.894814i	$-0.647308\pi$
$683$	−23.3348	−0.892880	−0.446440	+	0.894814i	$0.647308\pi$
$684$	0	0
$685$	26.2453	1.00278
$686$	0	0
$687$	−25.0383	−0.955269
$688$	0	0
$689$	3.41330	0.130036
$690$	0	0
$691$	42.5815	1.61988	0.809939	−	0.586514i	$-0.199500\pi$
$691$	42.5815	1.61988	0.809939	+	0.586514i	$0.199500\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	43.9599	1.66750
$696$	0	0
$697$	−4.14775	−0.157107
$698$	0	0
$699$	12.6142	0.477114
$700$	0	0
$701$	−36.9932	−1.39721	−0.698606	−	0.715506i	$-0.746196\pi$
$701$	−36.9932	−1.39721	−0.698606	+	0.715506i	$0.746196\pi$
$702$	0	0
$703$	−3.92173	−0.147911
$704$	0	0
$705$	34.9297	1.31553
$706$	0	0
$707$	0	0
$708$	0	0
$709$	23.1663	0.870030	0.435015	−	0.900423i	$-0.356743\pi$
$709$	23.1663	0.870030	0.435015	+	0.900423i	$0.356743\pi$
$710$	0	0
$711$	2.01850	0.0756998
$712$	0	0
$713$	−37.9525	−1.42133
$714$	0	0
$715$	−4.63087	−0.173185
$716$	0	0
$717$	16.8263	0.628390
$718$	0	0
$719$	−10.5992	−0.395283	−0.197642	−	0.980274i	$-0.563328\pi$
$719$	−10.5992	−0.395283	−0.197642	+	0.980274i	$0.563328\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−5.91902	−0.220131
$724$	0	0
$725$	10.0003	0.371400
$726$	0	0
$727$	−52.8233	−1.95911	−0.979554	−	0.201182i	$-0.935522\pi$
$727$	−52.8233	−1.95911	−0.979554	+	0.201182i	$0.935522\pi$
$728$	0	0
$729$	18.8616	0.698576
$730$	0	0
$731$	43.8927	1.62343
$732$	0	0
$733$	−13.0832	−0.483237	−0.241619	−	0.970371i	$-0.577678\pi$
$733$	−13.0832	−0.483237	−0.241619	+	0.970371i	$0.577678\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	9.00260	0.331615
$738$	0	0
$739$	8.43697	0.310359	0.155180	−	0.987886i	$-0.450404\pi$
$739$	8.43697	0.310359	0.155180	+	0.987886i	$0.450404\pi$
$740$	0	0
$741$	−5.02804	−0.184710
$742$	0	0
$743$	−29.0216	−1.06470	−0.532350	−	0.846524i	$-0.678691\pi$
$743$	−29.0216	−1.06470	−0.532350	+	0.846524i	$0.678691\pi$
$744$	0	0
$745$	39.0069	1.42910
$746$	0	0
$747$	−4.28777	−0.156881
$748$	0	0
$749$	0	0
$750$	0	0
$751$	12.4098	0.452839	0.226419	−	0.974030i	$-0.427298\pi$
$751$	12.4098	0.452839	0.226419	+	0.974030i	$0.427298\pi$
$752$	0	0
$753$	−37.1068	−1.35225
$754$	0	0
$755$	37.9381	1.38071
$756$	0	0
$757$	−1.47607	−0.0536486	−0.0268243	−	0.999640i	$-0.508539\pi$
$757$	−1.47607	−0.0536486	−0.0268243	+	0.999640i	$0.508539\pi$
$758$	0	0
$759$	7.42087	0.269361
$760$	0	0
$761$	28.3360	1.02718	0.513589	−	0.858036i	$-0.328316\pi$
$761$	28.3360	1.02718	0.513589	+	0.858036i	$0.328316\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−5.29419	−0.191412
$766$	0	0
$767$	−12.0530	−0.435210
$768$	0	0
$769$	−45.3967	−1.63705	−0.818523	−	0.574473i	$-0.805207\pi$
$769$	−45.3967	−1.63705	−0.818523	+	0.574473i	$0.805207\pi$
$770$	0	0
$771$	−5.02364	−0.180922
$772$	0	0
$773$	36.9475	1.32891	0.664455	−	0.747328i	$-0.268664\pi$
$773$	36.9475	1.32891	0.664455	+	0.747328i	$0.268664\pi$
$774$	0	0
$775$	−7.32804	−0.263231
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−0.715949	−0.0256515
$780$	0	0
$781$	−0.357153	−0.0127799
$782$	0	0
$783$	−38.5107	−1.37626
$784$	0	0
$785$	11.1696	0.398662
$786$	0	0
$787$	−43.0165	−1.53337	−0.766686	−	0.642023i	$-0.778096\pi$
$787$	−43.0165	−1.53337	−0.766686	+	0.642023i	$0.778096\pi$
$788$	0	0
$789$	6.43446	0.229073
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−12.3855	−0.439821
$794$	0	0
$795$	3.47563	0.123268
$796$	0	0
$797$	5.89036	0.208647	0.104324	−	0.994543i	$-0.466732\pi$
$797$	5.89036	0.208647	0.104324	+	0.994543i	$0.466732\pi$
$798$	0	0
$799$	38.7155	1.36966
$800$	0	0
$801$	10.6869	0.377601
$802$	0	0
$803$	8.81703	0.311146
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−22.5455	−0.793639
$808$	0	0
$809$	40.7584	1.43299	0.716495	−	0.697592i	$-0.245745\pi$
$809$	40.7584	1.43299	0.716495	+	0.697592i	$0.245745\pi$
$810$	0	0
$811$	17.1874	0.603530	0.301765	−	0.953382i	$-0.402424\pi$
$811$	17.1874	0.603530	0.301765	+	0.953382i	$0.402424\pi$
$812$	0	0
$813$	13.1264	0.460364
$814$	0	0
$815$	−22.1798	−0.776924
$816$	0	0
$817$	7.57639	0.265064
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−17.3992	−0.607236	−0.303618	−	0.952794i	$-0.598195\pi$
$821$	−17.3992	−0.607236	−0.303618	+	0.952794i	$0.598195\pi$
$822$	0	0
$823$	−10.6879	−0.372555	−0.186278	−	0.982497i	$-0.559642\pi$
$823$	−10.6879	−0.372555	−0.186278	+	0.982497i	$0.559642\pi$
$824$	0	0
$825$	1.43286	0.0498856
$826$	0	0
$827$	2.70437	0.0940401	0.0470200	−	0.998894i	$-0.485028\pi$
$827$	2.70437	0.0940401	0.0470200	+	0.998894i	$0.485028\pi$
$828$	0	0
$829$	−31.6178	−1.09813	−0.549067	−	0.835779i	$-0.685017\pi$
$829$	−31.6178	−1.09813	−0.549067	+	0.835779i	$0.685017\pi$
$830$	0	0
$831$	48.6347	1.68712
$832$	0	0
$833$	0	0
$834$	0	0
$835$	12.0010	0.415310
$836$	0	0
$837$	28.2201	0.975428
$838$	0	0
$839$	6.93740	0.239506	0.119753	−	0.992804i	$-0.461790\pi$
$839$	6.93740	0.239506	0.119753	+	0.992804i	$0.461790\pi$
$840$	0	0
$841$	44.6525	1.53974
$842$	0	0
$843$	−57.6849	−1.98678
$844$	0	0
$845$	0.990813	0.0340850
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−0.815864	−0.0280004
$850$	0	0
$851$	−33.0572	−1.13319
$852$	0	0
$853$	−12.4961	−0.427859	−0.213929	−	0.976849i	$-0.568626\pi$
$853$	−12.4961	−0.427859	−0.213929	+	0.976849i	$0.568626\pi$
$854$	0	0
$855$	−0.913838	−0.0312526
$856$	0	0
$857$	13.9355	0.476029	0.238014	−	0.971262i	$-0.423504\pi$
$857$	13.9355	0.476029	0.238014	+	0.971262i	$0.423504\pi$
$858$	0	0
$859$	−48.7496	−1.66332	−0.831658	−	0.555289i	$-0.812608\pi$
$859$	−48.7496	−1.66332	−0.831658	+	0.555289i	$0.812608\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	17.4213	0.593029	0.296514	−	0.955028i	$-0.404176\pi$
$863$	17.4213	0.593029	0.296514	+	0.955028i	$0.404176\pi$
$864$	0	0
$865$	−1.17230	−0.0398595
$866$	0	0
$867$	−0.389496	−0.0132280
$868$	0	0
$869$	1.99270	0.0675977
$870$	0	0
$871$	−51.4162	−1.74217
$872$	0	0
$873$	12.2957	0.416147
$874$	0	0
$875$	0	0
$876$	0	0
$877$	23.2166	0.783967	0.391984	−	0.919972i	$-0.371789\pi$
$877$	23.2166	0.783967	0.391984	+	0.919972i	$0.371789\pi$
$878$	0	0
$879$	24.0810	0.812231
$880$	0	0
$881$	−55.2811	−1.86247	−0.931233	−	0.364424i	$-0.881266\pi$
$881$	−55.2811	−1.86247	−0.931233	+	0.364424i	$0.881266\pi$
$882$	0	0
$883$	1.01055	0.0340078	0.0170039	−	0.999855i	$-0.494587\pi$
$883$	1.01055	0.0340078	0.0170039	+	0.999855i	$0.494587\pi$
$884$	0	0
$885$	−12.2732	−0.412558
$886$	0	0
$887$	−39.1254	−1.31370	−0.656852	−	0.754020i	$-0.728112\pi$
$887$	−39.1254	−1.31370	−0.656852	+	0.754020i	$0.728112\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−6.77614	−0.227009
$892$	0	0
$893$	6.68274	0.223629
$894$	0	0
$895$	−21.1535	−0.707084
$896$	0	0
$897$	−42.3826	−1.41511
$898$	0	0
$899$	−53.9714	−1.80005
$900$	0	0
$901$	3.85233	0.128340
$902$	0	0
$903$	0	0
$904$	0	0
$905$	27.1242	0.901639
$906$	0	0
$907$	14.6946	0.487926	0.243963	−	0.969785i	$-0.421553\pi$
$907$	14.6946	0.487926	0.243963	+	0.969785i	$0.421553\pi$
$908$	0	0
$909$	−4.07927	−0.135301
$910$	0	0
$911$	11.2682	0.373331	0.186665	−	0.982424i	$-0.440232\pi$
$911$	11.2682	0.373331	0.186665	+	0.982424i	$0.440232\pi$
$912$	0	0
$913$	−4.23295	−0.140090
$914$	0	0
$915$	−12.6117	−0.416929
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−11.6108	−0.383004	−0.191502	−	0.981492i	$-0.561336\pi$
$919$	−11.6108	−0.383004	−0.191502	+	0.981492i	$0.561336\pi$
$920$	0	0
$921$	15.4612	0.509464
$922$	0	0
$923$	2.03979	0.0671406
$924$	0	0
$925$	−6.38283	−0.209866
$926$	0	0
$927$	−3.06274	−0.100594
$928$	0	0
$929$	23.8126	0.781265	0.390632	−	0.920547i	$-0.372256\pi$
$929$	23.8126	0.781265	0.390632	+	0.920547i	$0.372256\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−0.683172	−0.0223661
$934$	0	0
$935$	−5.22651	−0.170925
$936$	0	0
$937$	50.1180	1.63728	0.818641	−	0.574305i	$-0.194728\pi$
$937$	50.1180	1.63728	0.818641	+	0.574305i	$0.194728\pi$
$938$	0	0
$939$	−50.8297	−1.65876
$940$	0	0
$941$	−0.313120	−0.0102074	−0.00510371	−	0.999987i	$-0.501625\pi$
$941$	−0.313120	−0.0102074	−0.00510371	+	0.999987i	$0.501625\pi$
$942$	0	0
$943$	−6.03491	−0.196524
$944$	0	0
$945$	0	0
$946$	0	0
$947$	37.3034	1.21220	0.606100	−	0.795389i	$-0.292733\pi$
$947$	37.3034	1.21220	0.606100	+	0.795389i	$0.292733\pi$
$948$	0	0
$949$	−50.3564	−1.63464
$950$	0	0
$951$	48.8339	1.58355
$952$	0	0
$953$	−45.1980	−1.46411	−0.732053	−	0.681248i	$-0.761438\pi$
$953$	−45.1980	−1.46411	−0.732053	+	0.681248i	$0.761438\pi$
$954$	0	0
$955$	−39.7425	−1.28604
$956$	0	0
$957$	10.5531	0.341132
$958$	0	0
$959$	0	0
$960$	0	0
$961$	8.54945	0.275789
$962$	0	0
$963$	1.97243	0.0635606
$964$	0	0
$965$	5.37496	0.173026
$966$	0	0
$967$	42.3517	1.36194	0.680969	−	0.732312i	$-0.261559\pi$
$967$	42.3517	1.36194	0.680969	+	0.732312i	$0.261559\pi$
$968$	0	0
$969$	−5.67477	−0.182300
$970$	0	0
$971$	−27.1336	−0.870758	−0.435379	−	0.900247i	$-0.643386\pi$
$971$	−27.1336	−0.870758	−0.435379	+	0.900247i	$0.643386\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−8.18341	−0.262079
$976$	0	0
$977$	34.4694	1.10278	0.551388	−	0.834249i	$-0.314099\pi$
$977$	34.4694	1.10278	0.551388	+	0.834249i	$0.314099\pi$
$978$	0	0
$979$	10.5502	0.337187
$980$	0	0
$981$	8.83251	0.282000
$982$	0	0
$983$	13.3302	0.425168	0.212584	−	0.977143i	$-0.431812\pi$
$983$	13.3302	0.425168	0.212584	+	0.977143i	$0.431812\pi$
$984$	0	0
$985$	−28.8750	−0.920035
$986$	0	0
$987$	0	0
$988$	0	0
$989$	63.8632	2.03073
$990$	0	0
$991$	36.2557	1.15170	0.575850	−	0.817555i	$-0.304671\pi$
$991$	36.2557	1.15170	0.575850	+	0.817555i	$0.304671\pi$
$992$	0	0
$993$	−30.6645	−0.973107
$994$	0	0
$995$	−18.6758	−0.592063
$996$	0	0
$997$	9.06746	0.287169	0.143585	−	0.989638i	$-0.454137\pi$
$997$	9.06746	0.287169	0.143585	+	0.989638i	$0.454137\pi$
$998$	0	0
$999$	24.5801	0.777679

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8036.2.a.p.1.2	yes	10
7.6	odd	2		8036.2.a.o.1.9	✓	10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8036.2.a.o.1.9	✓	10	7.6	odd	2
8036.2.a.p.1.2	yes	10	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:	\( N \)	\(=\)	\( 8036 = 2^{2} \cdot 7^{2} \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8036.a (trivial)

\(f(q)\)	\(=\)	\(q-1.91097 q^{3} +1.95825 q^{5} +0.651806 q^{9} +O(q^{10})\)	\(q-1.91097 q^{3} +1.95825 q^{5} +0.651806 q^{9} +0.643473 q^{11} -3.67505 q^{13} -3.74216 q^{15} -4.14775 q^{17} -0.715949 q^{19} -6.03491 q^{23} -1.16525 q^{25} +4.48733 q^{27} -8.58210 q^{29} +6.28884 q^{31} -1.22966 q^{33} +5.47767 q^{37} +7.02290 q^{39} +1.00000 q^{41} -10.5823 q^{43} +1.27640 q^{45} -9.33410 q^{47} +7.92622 q^{51} -0.928777 q^{53} +1.26008 q^{55} +1.36816 q^{57} +3.27970 q^{59} +3.37016 q^{61} -7.19667 q^{65} +13.9906 q^{67} +11.5325 q^{69} -0.555038 q^{71} +13.7022 q^{73} +2.22675 q^{75} +3.09679 q^{79} -10.5306 q^{81} -6.57829 q^{83} -8.12234 q^{85} +16.4001 q^{87} +16.3958 q^{89} -12.0178 q^{93} -1.40201 q^{95} +18.8641 q^{97} +0.419420 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 2 q^{3} + 4 q^{5} + 10 q^{9}+O(q^{10}) \) 10 * q + 2 * q^3 + 4 * q^5 + 10 * q^9	\( 10 q + 2 q^{3} + 4 q^{5} + 10 q^{9} + 6 q^{13} - 4 q^{15} + 12 q^{17} + 8 q^{19} + 4 q^{23} + 16 q^{25} + 8 q^{27} + 2 q^{29} + 8 q^{31} - 6 q^{33} - 2 q^{37} - 2 q^{39} + 10 q^{41} - 2 q^{43} + 44 q^{45} - 14 q^{47} + 14 q^{51} + 8 q^{53} + 8 q^{55} - 10 q^{57} + 24 q^{59} + 14 q^{61} + 2 q^{65} - 8 q^{67} + 16 q^{69} + 10 q^{71} + 44 q^{73} - 50 q^{75} + 10 q^{79} - 14 q^{81} + 20 q^{83} + 8 q^{85} + 20 q^{87} + 6 q^{89} + 8 q^{93} + 4 q^{95} + 46 q^{97} + 2 q^{99}+O(q^{100}) \) 10 * q + 2 * q^3 + 4 * q^5 + 10 * q^9 + 6 * q^13 - 4 * q^15 + 12 * q^17 + 8 * q^19 + 4 * q^23 + 16 * q^25 + 8 * q^27 + 2 * q^29 + 8 * q^31 - 6 * q^33 - 2 * q^37 - 2 * q^39 + 10 * q^41 - 2 * q^43 + 44 * q^45 - 14 * q^47 + 14 * q^51 + 8 * q^53 + 8 * q^55 - 10 * q^57 + 24 * q^59 + 14 * q^61 + 2 * q^65 - 8 * q^67 + 16 * q^69 + 10 * q^71 + 44 * q^73 - 50 * q^75 + 10 * q^79 - 14 * q^81 + 20 * q^83 + 8 * q^85 + 20 * q^87 + 6 * q^89 + 8 * q^93 + 4 * q^95 + 46 * q^97 + 2 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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