LMFDB - Embedded newform 8036.2.a.s.1.13

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:	$1$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - 4 x^{19} - 30 x^{18} + 128 x^{17} + 348 x^{16} - 1644 x^{15} - 1934 x^{14} + 10948 x^{13} + \cdots + 9 $ x^20 - 4x^19 - 30x^18 + 128x^17 + 348x^16 - 1644x^15 - 1934x^14 + 10948x^13 + 4748x^12 - 40524x^11 - 220x^10 + 82500x^9 - 21992x^8 - 84720x^7 + 37544x^6 + 34656x^5 - 18823x^4 - 2276x^3 + 1130x^2 + 204*x + 9
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			1.13
Root			$-0.146508$ 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)

$f(q)$	$=$	$q+0.146508 q^{3} -3.56882 q^{5} -2.97854 q^{9} +O(q^{10})$	$q+0.146508 q^{3} -3.56882 q^{5} -2.97854 q^{9} -5.58095 q^{11} +6.25954 q^{13} -0.522862 q^{15} +6.24549 q^{17} -7.23364 q^{19} +5.27797 q^{23} +7.73647 q^{25} -0.875905 q^{27} -4.42408 q^{29} -1.06333 q^{31} -0.817655 q^{33} +2.40613 q^{37} +0.917075 q^{39} +1.00000 q^{41} +12.2876 q^{43} +10.6299 q^{45} +4.35831 q^{47} +0.915016 q^{51} +9.39440 q^{53} +19.9174 q^{55} -1.05979 q^{57} +13.2911 q^{59} -14.3104 q^{61} -22.3392 q^{65} -4.58700 q^{67} +0.773267 q^{69} -0.0228021 q^{71} -12.7838 q^{73} +1.13346 q^{75} +1.46033 q^{79} +8.80728 q^{81} -4.21437 q^{83} -22.2890 q^{85} -0.648164 q^{87} +3.56397 q^{89} -0.155787 q^{93} +25.8156 q^{95} -10.0736 q^{97} +16.6230 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q - 4 q^{3} - 8 q^{5} + 16 q^{9}+O(q^{10}) $ 20 * q - 4 * q^3 - 8 * q^5 + 16 * q^9	$ 20 q - 4 q^{3} - 8 q^{5} + 16 q^{9} - 8 q^{11} - 12 q^{13} + 8 q^{15} - 8 q^{17} - 24 q^{19} + 8 q^{23} + 20 q^{25} - 16 q^{27} - 12 q^{29} - 44 q^{33} + 12 q^{37} + 12 q^{39} + 20 q^{41} + 4 q^{43} - 40 q^{45} - 4 q^{47} + 4 q^{51} - 12 q^{53} + 16 q^{55} + 28 q^{57} - 16 q^{59} - 68 q^{61} - 8 q^{65} + 4 q^{67} - 32 q^{69} + 8 q^{71} - 48 q^{73} - 60 q^{75} - 20 q^{79} + 32 q^{81} + 8 q^{83} - 28 q^{85} - 60 q^{89} - 16 q^{93} + 20 q^{95} - 40 q^{97}+O(q^{100}) $ 20 * q - 4 * q^3 - 8 * q^5 + 16 * q^9 - 8 * q^11 - 12 * q^13 + 8 * q^15 - 8 * q^17 - 24 * q^19 + 8 * q^23 + 20 * q^25 - 16 * q^27 - 12 * q^29 - 44 * q^33 + 12 * q^37 + 12 * q^39 + 20 * q^41 + 4 * q^43 - 40 * q^45 - 4 * q^47 + 4 * q^51 - 12 * q^53 + 16 * q^55 + 28 * q^57 - 16 * q^59 - 68 * q^61 - 8 * q^65 + 4 * q^67 - 32 * q^69 + 8 * q^71 - 48 * q^73 - 60 * q^75 - 20 * q^79 + 32 * q^81 + 8 * q^83 - 28 * q^85 - 60 * q^89 - 16 * q^93 + 20 * q^95 - 40 * q^97

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
$3$	0.146508	0.0845866	0.0422933	−	0.999105i	$-0.486534\pi$
$3$	0.146508	0.0845866	0.0422933	+	0.999105i	$0.486534\pi$
$4$	0	0
$5$	−3.56882	−1.59602	−0.798012	−	0.602641i	$-0.794115\pi$
$5$	−3.56882	−1.59602	−0.798012	+	0.602641i	$0.794115\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−2.97854	−0.992845
$10$	0	0
$11$	−5.58095	−1.68272	−0.841359	−	0.540476i	$-0.818244\pi$
$11$	−5.58095	−1.68272	−0.841359	+	0.540476i	$0.818244\pi$
$12$	0	0
$13$	6.25954	1.73608	0.868042	−	0.496490i	$-0.165378\pi$
$13$	6.25954	1.73608	0.868042	+	0.496490i	$0.165378\pi$
$14$	0	0
$15$	−0.522862	−0.135002
$16$	0	0
$17$	6.24549	1.51475	0.757377	−	0.652978i	$-0.226481\pi$
$17$	6.24549	1.51475	0.757377	+	0.652978i	$0.226481\pi$
$18$	0	0
$19$	−7.23364	−1.65951	−0.829756	−	0.558127i	$-0.811520\pi$
$19$	−7.23364	−1.65951	−0.829756	+	0.558127i	$0.811520\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	5.27797	1.10053	0.550266	−	0.834989i	$-0.314526\pi$
$23$	5.27797	1.10053	0.550266	+	0.834989i	$0.314526\pi$
$24$	0	0
$25$	7.73647	1.54729
$26$	0	0
$27$	−0.875905	−0.168568
$28$	0	0
$29$	−4.42408	−0.821531	−0.410765	−	0.911741i	$-0.634738\pi$
$29$	−4.42408	−0.821531	−0.410765	+	0.911741i	$0.634738\pi$
$30$	0	0
$31$	−1.06333	−0.190981	−0.0954903	−	0.995430i	$-0.530442\pi$
$31$	−1.06333	−0.190981	−0.0954903	+	0.995430i	$0.530442\pi$
$32$	0	0
$33$	−0.817655	−0.142336
$34$	0	0
$35$	0	0
$36$	0	0
$37$	2.40613	0.395566	0.197783	−	0.980246i	$-0.436626\pi$
$37$	2.40613	0.395566	0.197783	+	0.980246i	$0.436626\pi$
$38$	0	0
$39$	0.917075	0.146850
$40$	0	0
$41$	1.00000	0.156174
$41$	1.00000	0.156174
$42$	0	0
$43$	12.2876	1.87384	0.936920	−	0.349543i	$-0.113663\pi$
$43$	12.2876	1.87384	0.936920	+	0.349543i	$0.113663\pi$
$44$	0	0
$45$	10.6299	1.58460
$46$	0	0
$47$	4.35831	0.635725	0.317862	−	0.948137i	$-0.397035\pi$
$47$	4.35831	0.635725	0.317862	+	0.948137i	$0.397035\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	0.915016	0.128128
$52$	0	0
$53$	9.39440	1.29042	0.645210	−	0.764005i	$-0.276770\pi$
$53$	9.39440	1.29042	0.645210	+	0.764005i	$0.276770\pi$
$54$	0	0
$55$	19.9174	2.68566
$56$	0	0
$57$	−1.05979	−0.140372
$58$	0	0
$59$	13.2911	1.73035	0.865177	−	0.501466i	$-0.167206\pi$
$59$	13.2911	1.73035	0.865177	+	0.501466i	$0.167206\pi$
$60$	0	0
$61$	−14.3104	−1.83226	−0.916130	−	0.400881i	$-0.868704\pi$
$61$	−14.3104	−1.83226	−0.916130	+	0.400881i	$0.868704\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−22.3392	−2.77083
$66$	0	0
$67$	−4.58700	−0.560392	−0.280196	−	0.959943i	$-0.590399\pi$
$67$	−4.58700	−0.560392	−0.280196	+	0.959943i	$0.590399\pi$
$68$	0	0
$69$	0.773267	0.0930904
$70$	0	0
$71$	−0.0228021	−0.00270612	−0.00135306	−	0.999999i	$-0.500431\pi$
$71$	−0.0228021	−0.00270612	−0.00135306	+	0.999999i	$0.500431\pi$
$72$	0	0
$73$	−12.7838	−1.49623	−0.748115	−	0.663569i	$-0.769041\pi$
$73$	−12.7838	−1.49623	−0.748115	+	0.663569i	$0.769041\pi$
$74$	0	0
$75$	1.13346	0.130880
$76$	0	0
$77$	0	0
$78$	0	0
$79$	1.46033	0.164300	0.0821499	−	0.996620i	$-0.473821\pi$
$79$	1.46033	0.164300	0.0821499	+	0.996620i	$0.473821\pi$
$80$	0	0
$81$	8.80728	0.978586
$82$	0	0
$83$	−4.21437	−0.462588	−0.231294	−	0.972884i	$-0.574296\pi$
$83$	−4.21437	−0.462588	−0.231294	+	0.972884i	$0.574296\pi$
$84$	0	0
$85$	−22.2890	−2.41758
$86$	0	0
$87$	−0.648164	−0.0694905
$88$	0	0
$89$	3.56397	0.377780	0.188890	−	0.981998i	$-0.439511\pi$
$89$	3.56397	0.377780	0.188890	+	0.981998i	$0.439511\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−0.155787	−0.0161544
$94$	0	0
$95$	25.8156	2.64862
$96$	0	0
$97$	−10.0736	−1.02282	−0.511409	−	0.859337i	$-0.670876\pi$
$97$	−10.0736	−1.02282	−0.511409	+	0.859337i	$0.670876\pi$
$98$	0	0
$99$	16.6230	1.67068
$100$	0	0
$101$	−6.78163	−0.674797	−0.337398	−	0.941362i	$-0.609547\pi$
$101$	−6.78163	−0.674797	−0.337398	+	0.941362i	$0.609547\pi$
$102$	0	0
$103$	−14.8830	−1.46647	−0.733235	−	0.679976i	$-0.761990\pi$
$103$	−14.8830	−1.46647	−0.733235	+	0.679976i	$0.761990\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	3.23983	0.313207	0.156603	−	0.987662i	$-0.449946\pi$
$107$	3.23983	0.313207	0.156603	+	0.987662i	$0.449946\pi$
$108$	0	0
$109$	−12.0580	−1.15495	−0.577473	−	0.816410i	$-0.695961\pi$
$109$	−12.0580	−1.15495	−0.577473	+	0.816410i	$0.695961\pi$
$110$	0	0
$111$	0.352519	0.0334596
$112$	0	0
$113$	2.15189	0.202433	0.101216	−	0.994864i	$-0.467727\pi$
$113$	2.15189	0.202433	0.101216	+	0.994864i	$0.467727\pi$
$114$	0	0
$115$	−18.8361	−1.75648
$116$	0	0
$117$	−18.6443	−1.72366
$118$	0	0
$119$	0	0
$120$	0	0
$121$	20.1470	1.83154
$122$	0	0
$123$	0.146508	0.0132102
$124$	0	0
$125$	−9.76596	−0.873494
$126$	0	0
$127$	10.4780	0.929769	0.464885	−	0.885371i	$-0.346096\pi$
$127$	10.4780	0.929769	0.464885	+	0.885371i	$0.346096\pi$
$128$	0	0
$129$	1.80024	0.158502
$130$	0	0
$131$	15.1943	1.32753	0.663765	−	0.747941i	$-0.268957\pi$
$131$	15.1943	1.32753	0.663765	+	0.747941i	$0.268957\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	3.12595	0.269039
$136$	0	0
$137$	12.2648	1.04785	0.523927	−	0.851763i	$-0.324467\pi$
$137$	12.2648	1.04785	0.523927	+	0.851763i	$0.324467\pi$
$138$	0	0
$139$	−5.87366	−0.498197	−0.249099	−	0.968478i	$-0.580134\pi$
$139$	−5.87366	−0.498197	−0.249099	+	0.968478i	$0.580134\pi$
$140$	0	0
$141$	0.638529	0.0537738
$142$	0	0
$143$	−34.9342	−2.92134
$144$	0	0
$145$	15.7887	1.31118
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−3.43183	−0.281147	−0.140573	−	0.990070i	$-0.544895\pi$
$149$	−3.43183	−0.281147	−0.140573	+	0.990070i	$0.544895\pi$
$150$	0	0
$151$	−1.00796	−0.0820267	−0.0410134	−	0.999159i	$-0.513059\pi$
$151$	−1.00796	−0.0820267	−0.0410134	+	0.999159i	$0.513059\pi$
$152$	0	0
$153$	−18.6024	−1.50392
$154$	0	0
$155$	3.79485	0.304810
$156$	0	0
$157$	−10.1710	−0.811731	−0.405866	−	0.913933i	$-0.633030\pi$
$157$	−10.1710	−0.811731	−0.405866	+	0.913933i	$0.633030\pi$
$158$	0	0
$159$	1.37636	0.109152
$160$	0	0
$161$	0	0
$162$	0	0
$163$	12.8358	1.00538	0.502690	−	0.864467i	$-0.332344\pi$
$163$	12.8358	1.00538	0.502690	+	0.864467i	$0.332344\pi$
$164$	0	0
$165$	2.91806	0.227171
$166$	0	0
$167$	19.6228	1.51846	0.759230	−	0.650822i	$-0.225576\pi$
$167$	19.6228	1.51846	0.759230	+	0.650822i	$0.225576\pi$
$168$	0	0
$169$	26.1819	2.01399
$170$	0	0
$171$	21.5457	1.64764
$172$	0	0
$173$	2.34884	0.178579	0.0892895	−	0.996006i	$-0.471540\pi$
$173$	2.34884	0.178579	0.0892895	+	0.996006i	$0.471540\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	1.94726	0.146365
$178$	0	0
$179$	−1.34712	−0.100689	−0.0503443	−	0.998732i	$-0.516032\pi$
$179$	−1.34712	−0.100689	−0.0503443	+	0.998732i	$0.516032\pi$
$180$	0	0
$181$	−25.3273	−1.88256	−0.941281	−	0.337623i	$-0.890377\pi$
$181$	−25.3273	−1.88256	−0.941281	+	0.337623i	$0.890377\pi$
$182$	0	0
$183$	−2.09660	−0.154985
$184$	0	0
$185$	−8.58705	−0.631333
$186$	0	0
$187$	−34.8557	−2.54890
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−13.7103	−0.992042	−0.496021	−	0.868311i	$-0.665206\pi$
$191$	−13.7103	−0.992042	−0.496021	+	0.868311i	$0.665206\pi$
$192$	0	0
$193$	−14.2959	−1.02904	−0.514520	−	0.857478i	$-0.672030\pi$
$193$	−14.2959	−1.02904	−0.514520	+	0.857478i	$0.672030\pi$
$194$	0	0
$195$	−3.27288	−0.234375
$196$	0	0
$197$	−8.46578	−0.603162	−0.301581	−	0.953441i	$-0.597514\pi$
$197$	−8.46578	−0.603162	−0.301581	+	0.953441i	$0.597514\pi$
$198$	0	0
$199$	2.27139	0.161014	0.0805072	−	0.996754i	$-0.474346\pi$
$199$	2.27139	0.161014	0.0805072	+	0.996754i	$0.474346\pi$
$200$	0	0
$201$	−0.672034	−0.0474017
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−3.56882	−0.249257
$206$	0	0
$207$	−15.7206	−1.09266
$208$	0	0
$209$	40.3706	2.79249
$210$	0	0
$211$	−3.57292	−0.245970	−0.122985	−	0.992409i	$-0.539247\pi$
$211$	−3.57292	−0.245970	−0.122985	+	0.992409i	$0.539247\pi$
$212$	0	0
$213$	−0.00334070	−0.000228901 0
$214$	0	0
$215$	−43.8522	−2.99070
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−1.87293	−0.126561
$220$	0	0
$221$	39.0939	2.62974
$222$	0	0
$223$	−5.99708	−0.401594	−0.200797	−	0.979633i	$-0.564353\pi$
$223$	−5.99708	−0.401594	−0.200797	+	0.979633i	$0.564353\pi$
$224$	0	0
$225$	−23.0433	−1.53622
$226$	0	0
$227$	−8.33020	−0.552894	−0.276447	−	0.961029i	$-0.589157\pi$
$227$	−8.33020	−0.552894	−0.276447	+	0.961029i	$0.589157\pi$
$228$	0	0
$229$	−21.5041	−1.42103	−0.710515	−	0.703682i	$-0.751538\pi$
$229$	−21.5041	−1.42103	−0.710515	+	0.703682i	$0.751538\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	6.16205	0.403689	0.201845	−	0.979418i	$-0.435306\pi$
$233$	6.16205	0.403689	0.201845	+	0.979418i	$0.435306\pi$
$234$	0	0
$235$	−15.5540	−1.01463
$236$	0	0
$237$	0.213950	0.0138976
$238$	0	0
$239$	−3.12242	−0.201973	−0.100986	−	0.994888i	$-0.532200\pi$
$239$	−3.12242	−0.201973	−0.100986	+	0.994888i	$0.532200\pi$
$240$	0	0
$241$	−29.8712	−1.92417	−0.962086	−	0.272746i	$-0.912068\pi$
$241$	−29.8712	−1.92417	−0.962086	+	0.272746i	$0.912068\pi$
$242$	0	0
$243$	3.91806	0.251343
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−45.2793	−2.88105
$248$	0	0
$249$	−0.617441	−0.0391287
$250$	0	0
$251$	−6.93673	−0.437843	−0.218921	−	0.975743i	$-0.570254\pi$
$251$	−6.93673	−0.437843	−0.218921	+	0.975743i	$0.570254\pi$
$252$	0	0
$253$	−29.4561	−1.85189
$254$	0	0
$255$	−3.26553	−0.204495
$256$	0	0
$257$	−10.8457	−0.676535	−0.338267	−	0.941050i	$-0.609841\pi$
$257$	−10.8457	−0.676535	−0.338267	+	0.941050i	$0.609841\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	13.1773	0.815653
$262$	0	0
$263$	−21.5532	−1.32903	−0.664515	−	0.747275i	$-0.731362\pi$
$263$	−21.5532	−1.32903	−0.664515	+	0.747275i	$0.731362\pi$
$264$	0	0
$265$	−33.5269	−2.05954
$266$	0	0
$267$	0.522151	0.0319551
$268$	0	0
$269$	8.56276	0.522081	0.261040	−	0.965328i	$-0.415934\pi$
$269$	8.56276	0.522081	0.261040	+	0.965328i	$0.415934\pi$
$270$	0	0
$271$	13.5941	0.825784	0.412892	−	0.910780i	$-0.364519\pi$
$271$	13.5941	0.825784	0.412892	+	0.910780i	$0.364519\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−43.1768	−2.60366
$276$	0	0
$277$	−0.322861	−0.0193988	−0.00969941	−	0.999953i	$-0.503087\pi$
$277$	−0.322861	−0.0193988	−0.00969941	+	0.999953i	$0.503087\pi$
$278$	0	0
$279$	3.16718	0.189614
$280$	0	0
$281$	−2.28975	−0.136595	−0.0682975	−	0.997665i	$-0.521757\pi$
$281$	−2.28975	−0.136595	−0.0682975	+	0.997665i	$0.521757\pi$
$282$	0	0
$283$	−15.2741	−0.907950	−0.453975	−	0.891014i	$-0.649995\pi$
$283$	−15.2741	−0.907950	−0.453975	+	0.891014i	$0.649995\pi$
$284$	0	0
$285$	3.78220	0.224038
$286$	0	0
$287$	0	0
$288$	0	0
$289$	22.0061	1.29448
$290$	0	0
$291$	−1.47587	−0.0865168
$292$	0	0
$293$	−18.3619	−1.07272	−0.536358	−	0.843991i	$-0.680200\pi$
$293$	−18.3619	−1.07272	−0.536358	+	0.843991i	$0.680200\pi$
$294$	0	0
$295$	−47.4336	−2.76169
$296$	0	0
$297$	4.88838	0.283653
$298$	0	0
$299$	33.0377	1.91062
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−0.993565	−0.0570788
$304$	0	0
$305$	51.0713	2.92433
$306$	0	0
$307$	27.4007	1.56384	0.781920	−	0.623379i	$-0.214241\pi$
$307$	27.4007	1.56384	0.781920	+	0.623379i	$0.214241\pi$
$308$	0	0
$309$	−2.18049	−0.124044
$310$	0	0
$311$	−9.83010	−0.557414	−0.278707	−	0.960376i	$-0.589906\pi$
$311$	−9.83010	−0.557414	−0.278707	+	0.960376i	$0.589906\pi$
$312$	0	0
$313$	−0.0229956	−0.00129979	−0.000649895	−	1.00000i	$-0.500207\pi$
$313$	−0.0229956	−0.00129979	−0.000649895		1.00000i	$0.500207\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−8.32439	−0.467544	−0.233772	−	0.972291i	$-0.575107\pi$
$317$	−8.32439	−0.467544	−0.233772	+	0.972291i	$0.575107\pi$
$318$	0	0
$319$	24.6905	1.38240
$320$	0	0
$321$	0.474663	0.0264931
$322$	0	0
$323$	−45.1776	−2.51375
$324$	0	0
$325$	48.4267	2.68623
$326$	0	0
$327$	−1.76660	−0.0976930
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−18.9999	−1.04433	−0.522166	−	0.852844i	$-0.674876\pi$
$331$	−18.9999	−1.04433	−0.522166	+	0.852844i	$0.674876\pi$
$332$	0	0
$333$	−7.16675	−0.392736
$334$	0	0
$335$	16.3702	0.894399
$336$	0	0
$337$	−15.8051	−0.860956	−0.430478	−	0.902601i	$-0.641655\pi$
$337$	−15.8051	−0.860956	−0.430478	+	0.902601i	$0.641655\pi$
$338$	0	0
$339$	0.315270	0.0171231
$340$	0	0
$341$	5.93441	0.321367
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−2.75965	−0.148575
$346$	0	0
$347$	16.6774	0.895289	0.447645	−	0.894212i	$-0.352263\pi$
$347$	16.6774	0.895289	0.447645	+	0.894212i	$0.352263\pi$
$348$	0	0
$349$	−4.89892	−0.262233	−0.131116	−	0.991367i	$-0.541856\pi$
$349$	−4.89892	−0.262233	−0.131116	+	0.991367i	$0.541856\pi$
$350$	0	0
$351$	−5.48277	−0.292648
$352$	0	0
$353$	6.05193	0.322112	0.161056	−	0.986945i	$-0.448510\pi$
$353$	6.05193	0.322112	0.161056	+	0.986945i	$0.448510\pi$
$354$	0	0
$355$	0.0813767	0.00431903
$356$	0	0
$357$	0	0
$358$	0	0
$359$	18.5703	0.980103	0.490052	−	0.871693i	$-0.336978\pi$
$359$	18.5703	0.980103	0.490052	+	0.871693i	$0.336978\pi$
$360$	0	0
$361$	33.3256	1.75398
$362$	0	0
$363$	2.95170	0.154924
$364$	0	0
$365$	45.6230	2.38802
$366$	0	0
$367$	26.4297	1.37962	0.689810	−	0.723990i	$-0.257694\pi$
$367$	26.4297	1.37962	0.689810	+	0.723990i	$0.257694\pi$
$368$	0	0
$369$	−2.97854	−0.155056
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−30.4746	−1.57792	−0.788958	−	0.614447i	$-0.789379\pi$
$373$	−30.4746	−1.57792	−0.788958	+	0.614447i	$0.789379\pi$
$374$	0	0
$375$	−1.43080	−0.0738860
$376$	0	0
$377$	−27.6927	−1.42625
$378$	0	0
$379$	−33.0447	−1.69739	−0.848697	−	0.528880i	$-0.822612\pi$
$379$	−33.0447	−1.69739	−0.848697	+	0.528880i	$0.822612\pi$
$380$	0	0
$381$	1.53511	0.0786461
$382$	0	0
$383$	5.81336	0.297049	0.148524	−	0.988909i	$-0.452548\pi$
$383$	5.81336	0.297049	0.148524	+	0.988909i	$0.452548\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−36.5990	−1.86043
$388$	0	0
$389$	9.76527	0.495119	0.247559	−	0.968873i	$-0.420371\pi$
$389$	9.76527	0.495119	0.247559	+	0.968873i	$0.420371\pi$
$390$	0	0
$391$	32.9635	1.66704
$392$	0	0
$393$	2.22609	0.112291
$394$	0	0
$395$	−5.21165	−0.262227
$396$	0	0
$397$	4.72934	0.237359	0.118679	−	0.992933i	$-0.462134\pi$
$397$	4.72934	0.237359	0.118679	+	0.992933i	$0.462134\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−23.6521	−1.18113	−0.590566	−	0.806989i	$-0.701095\pi$
$401$	−23.6521	−1.18113	−0.590566	+	0.806989i	$0.701095\pi$
$402$	0	0
$403$	−6.65599	−0.331558
$404$	0	0
$405$	−31.4316	−1.56185
$406$	0	0
$407$	−13.4285	−0.665626
$408$	0	0
$409$	22.6946	1.12217	0.561087	−	0.827757i	$-0.310383\pi$
$409$	22.6946	1.12217	0.561087	+	0.827757i	$0.310383\pi$
$410$	0	0
$411$	1.79690	0.0886344
$412$	0	0
$413$	0	0
$414$	0	0
$415$	15.0403	0.738301
$416$	0	0
$417$	−0.860540	−0.0421408
$418$	0	0
$419$	−16.3286	−0.797706	−0.398853	−	0.917015i	$-0.630592\pi$
$419$	−16.3286	−0.797706	−0.398853	+	0.917015i	$0.630592\pi$
$420$	0	0
$421$	13.9893	0.681794	0.340897	−	0.940101i	$-0.389269\pi$
$421$	13.9893	0.681794	0.340897	+	0.940101i	$0.389269\pi$
$422$	0	0
$423$	−12.9814	−0.631176
$424$	0	0
$425$	48.3180	2.34377
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−5.11815	−0.247106
$430$	0	0
$431$	13.1555	0.633680	0.316840	−	0.948479i	$-0.397378\pi$
$431$	13.1555	0.633680	0.316840	+	0.948479i	$0.397378\pi$
$432$	0	0
$433$	23.1641	1.11319	0.556597	−	0.830782i	$-0.312107\pi$
$433$	23.1641	1.11319	0.556597	+	0.830782i	$0.312107\pi$
$434$	0	0
$435$	2.31318	0.110909
$436$	0	0
$437$	−38.1789	−1.82635
$438$	0	0
$439$	3.29543	0.157282	0.0786412	−	0.996903i	$-0.474942\pi$
$439$	3.29543	0.157282	0.0786412	+	0.996903i	$0.474942\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−9.60201	−0.456205	−0.228103	−	0.973637i	$-0.573252\pi$
$443$	−9.60201	−0.456205	−0.228103	+	0.973637i	$0.573252\pi$
$444$	0	0
$445$	−12.7192	−0.602946
$446$	0	0
$447$	−0.502792	−0.0237813
$448$	0	0
$449$	5.38167	0.253977	0.126988	−	0.991904i	$-0.459469\pi$
$449$	5.38167	0.253977	0.126988	+	0.991904i	$0.459469\pi$
$450$	0	0
$451$	−5.58095	−0.262796
$452$	0	0
$453$	−0.147675	−0.00693837
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−17.5108	−0.819122	−0.409561	−	0.912283i	$-0.634318\pi$
$457$	−17.5108	−0.819122	−0.409561	+	0.912283i	$0.634318\pi$
$458$	0	0
$459$	−5.47046	−0.255339
$460$	0	0
$461$	−16.5516	−0.770887	−0.385443	−	0.922731i	$-0.625952\pi$
$461$	−16.5516	−0.770887	−0.385443	+	0.922731i	$0.625952\pi$
$462$	0	0
$463$	20.4013	0.948131	0.474066	−	0.880490i	$-0.342786\pi$
$463$	20.4013	0.948131	0.474066	+	0.880490i	$0.342786\pi$
$464$	0	0
$465$	0.555977	0.0257828
$466$	0	0
$467$	22.4563	1.03915	0.519576	−	0.854424i	$-0.326090\pi$
$467$	22.4563	1.03915	0.519576	+	0.854424i	$0.326090\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−1.49013	−0.0686616
$472$	0	0
$473$	−68.5764	−3.15315
$474$	0	0
$475$	−55.9628	−2.56775
$476$	0	0
$477$	−27.9815	−1.28119
$478$	0	0
$479$	−24.7303	−1.12996	−0.564978	−	0.825106i	$-0.691115\pi$
$479$	−24.7303	−1.12996	−0.564978	+	0.825106i	$0.691115\pi$
$480$	0	0
$481$	15.0613	0.686736
$482$	0	0
$483$	0	0
$484$	0	0
$485$	35.9508	1.63244
$486$	0	0
$487$	42.3397	1.91859	0.959297	−	0.282400i	$-0.0911304\pi$
$487$	42.3397	1.91859	0.959297	+	0.282400i	$0.0911304\pi$
$488$	0	0
$489$	1.88056	0.0850417
$490$	0	0
$491$	−36.9495	−1.66751	−0.833754	−	0.552136i	$-0.813813\pi$
$491$	−36.9495	−1.66751	−0.833754	+	0.552136i	$0.813813\pi$
$492$	0	0
$493$	−27.6305	−1.24442
$494$	0	0
$495$	−59.3246	−2.66644
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−6.34747	−0.284152	−0.142076	−	0.989856i	$-0.545378\pi$
$499$	−6.34747	−0.284152	−0.142076	+	0.989856i	$0.545378\pi$
$500$	0	0
$501$	2.87491	0.128442
$502$	0	0
$503$	9.57194	0.426792	0.213396	−	0.976966i	$-0.431548\pi$
$503$	9.57194	0.426792	0.213396	+	0.976966i	$0.431548\pi$
$504$	0	0
$505$	24.2024	1.07699
$506$	0	0
$507$	3.83586	0.170357
$508$	0	0
$509$	−12.4150	−0.550284	−0.275142	−	0.961404i	$-0.588725\pi$
$509$	−12.4150	−0.550284	−0.275142	+	0.961404i	$0.588725\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	6.33599	0.279741
$514$	0	0
$515$	53.1149	2.34052
$516$	0	0
$517$	−24.3235	−1.06975
$518$	0	0
$519$	0.344125	0.0151054
$520$	0	0
$521$	−21.3737	−0.936397	−0.468199	−	0.883623i	$-0.655097\pi$
$521$	−21.3737	−0.936397	−0.468199	+	0.883623i	$0.655097\pi$
$522$	0	0
$523$	−19.4095	−0.848718	−0.424359	−	0.905494i	$-0.639501\pi$
$523$	−19.4095	−0.848718	−0.424359	+	0.905494i	$0.639501\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−6.64104	−0.289288
$528$	0	0
$529$	4.85697	0.211173
$530$	0	0
$531$	−39.5880	−1.71797
$532$	0	0
$533$	6.25954	0.271131
$534$	0	0
$535$	−11.5624	−0.499885
$536$	0	0
$537$	−0.197365	−0.00851692
$538$	0	0
$539$	0	0
$540$	0	0
$541$	18.6391	0.801359	0.400680	−	0.916218i	$-0.368774\pi$
$541$	18.6391	0.801359	0.400680	+	0.916218i	$0.368774\pi$
$542$	0	0
$543$	−3.71066	−0.159240
$544$	0	0
$545$	43.0328	1.84332
$546$	0	0
$547$	−25.5848	−1.09393	−0.546963	−	0.837157i	$-0.684216\pi$
$547$	−25.5848	−1.09393	−0.546963	+	0.837157i	$0.684216\pi$
$548$	0	0
$549$	42.6241	1.81915
$550$	0	0
$551$	32.0022	1.36334
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−1.25808	−0.0534023
$556$	0	0
$557$	43.8379	1.85747	0.928735	−	0.370743i	$-0.120897\pi$
$557$	43.8379	1.85747	0.928735	+	0.370743i	$0.120897\pi$
$558$	0	0
$559$	76.9147	3.25315
$560$	0	0
$561$	−5.10666	−0.215603
$562$	0	0
$563$	−0.377258	−0.0158995	−0.00794977	−	0.999968i	$-0.502531\pi$
$563$	−0.377258	−0.0158995	−0.00794977	+	0.999968i	$0.502531\pi$
$564$	0	0
$565$	−7.67971	−0.323088
$566$	0	0
$567$	0	0
$568$	0	0
$569$	32.1648	1.34842	0.674210	−	0.738539i	$-0.264484\pi$
$569$	32.1648	1.34842	0.674210	+	0.738539i	$0.264484\pi$
$570$	0	0
$571$	23.9046	1.00038	0.500189	−	0.865917i	$-0.333264\pi$
$571$	23.9046	1.00038	0.500189	+	0.865917i	$0.333264\pi$
$572$	0	0
$573$	−2.00867	−0.0839135
$574$	0	0
$575$	40.8329	1.70285
$576$	0	0
$577$	−28.0194	−1.16646	−0.583231	−	0.812306i	$-0.698212\pi$
$577$	−28.0194	−1.16646	−0.583231	+	0.812306i	$0.698212\pi$
$578$	0	0
$579$	−2.09447	−0.0870430
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−52.4296	−2.17141
$584$	0	0
$585$	66.5380	2.75101
$586$	0	0
$587$	−30.3058	−1.25085	−0.625427	−	0.780282i	$-0.715075\pi$
$587$	−30.3058	−1.25085	−0.625427	+	0.780282i	$0.715075\pi$
$588$	0	0
$589$	7.69178	0.316934
$590$	0	0
$591$	−1.24031	−0.0510194
$592$	0	0
$593$	−31.3996	−1.28943	−0.644713	−	0.764425i	$-0.723023\pi$
$593$	−31.3996	−1.28943	−0.644713	+	0.764425i	$0.723023\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0.332777	0.0136197
$598$	0	0
$599$	17.7301	0.724431	0.362215	−	0.932094i	$-0.382020\pi$
$599$	17.7301	0.724431	0.362215	+	0.932094i	$0.382020\pi$
$600$	0	0
$601$	21.8509	0.891319	0.445659	−	0.895203i	$-0.352969\pi$
$601$	21.8509	0.891319	0.445659	+	0.895203i	$0.352969\pi$
$602$	0	0
$603$	13.6626	0.556382
$604$	0	0
$605$	−71.9008	−2.92318
$606$	0	0
$607$	0.438480	0.0177974	0.00889868	−	0.999960i	$-0.497167\pi$
$607$	0.438480	0.0177974	0.00889868	+	0.999960i	$0.497167\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	27.2810	1.10367
$612$	0	0
$613$	−8.33445	−0.336625	−0.168313	−	0.985734i	$-0.553832\pi$
$613$	−8.33445	−0.336625	−0.168313	+	0.985734i	$0.553832\pi$
$614$	0	0
$615$	−0.522862	−0.0210838
$616$	0	0
$617$	41.6709	1.67761	0.838803	−	0.544434i	$-0.183256\pi$
$617$	41.6709	1.67761	0.838803	+	0.544434i	$0.183256\pi$
$618$	0	0
$619$	−22.0300	−0.885461	−0.442731	−	0.896655i	$-0.645990\pi$
$619$	−22.0300	−0.885461	−0.442731	+	0.896655i	$0.645990\pi$
$620$	0	0
$621$	−4.62300	−0.185515
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−3.82939	−0.153176
$626$	0	0
$627$	5.91463	0.236207
$628$	0	0
$629$	15.0275	0.599185
$630$	0	0
$631$	7.37811	0.293718	0.146859	−	0.989157i	$-0.453084\pi$
$631$	7.37811	0.293718	0.146859	+	0.989157i	$0.453084\pi$
$632$	0	0
$633$	−0.523463	−0.0208058
$634$	0	0
$635$	−37.3940	−1.48393
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0.0679170	0.00268675
$640$	0	0
$641$	−40.7267	−1.60861	−0.804304	−	0.594219i	$-0.797461\pi$
$641$	−40.7267	−1.60861	−0.804304	+	0.594219i	$0.797461\pi$
$642$	0	0
$643$	6.53076	0.257548	0.128774	−	0.991674i	$-0.458896\pi$
$643$	6.53076	0.257548	0.128774	+	0.991674i	$0.458896\pi$
$644$	0	0
$645$	−6.42472	−0.252973
$646$	0	0
$647$	16.7724	0.659391	0.329695	−	0.944087i	$-0.393054\pi$
$647$	16.7724	0.659391	0.329695	+	0.944087i	$0.393054\pi$
$648$	0	0
$649$	−74.1770	−2.91170
$650$	0	0
$651$	0	0
$652$	0	0
$653$	29.4507	1.15249	0.576247	−	0.817275i	$-0.304516\pi$
$653$	29.4507	1.15249	0.576247	+	0.817275i	$0.304516\pi$
$654$	0	0
$655$	−54.2256	−2.11877
$656$	0	0
$657$	38.0770	1.48552
$658$	0	0
$659$	9.00387	0.350741	0.175370	−	0.984503i	$-0.443888\pi$
$659$	9.00387	0.350741	0.175370	+	0.984503i	$0.443888\pi$
$660$	0	0
$661$	−10.7229	−0.417074	−0.208537	−	0.978014i	$-0.566870\pi$
$661$	−10.7229	−0.417074	−0.208537	+	0.978014i	$0.566870\pi$
$662$	0	0
$663$	5.72758	0.222441
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−23.3502	−0.904121
$668$	0	0
$669$	−0.878622	−0.0339695
$670$	0	0
$671$	79.8656	3.08318
$672$	0	0
$673$	−19.7406	−0.760944	−0.380472	−	0.924792i	$-0.624238\pi$
$673$	−19.7406	−0.760944	−0.380472	+	0.924792i	$0.624238\pi$
$674$	0	0
$675$	−6.77642	−0.260824
$676$	0	0
$677$	−2.56119	−0.0984345	−0.0492173	−	0.998788i	$-0.515673\pi$
$677$	−2.56119	−0.0984345	−0.0492173	+	0.998788i	$0.515673\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−1.22044	−0.0467675
$682$	0	0
$683$	−39.3304	−1.50494	−0.752469	−	0.658628i	$-0.771137\pi$
$683$	−39.3304	−1.50494	−0.752469	+	0.658628i	$0.771137\pi$
$684$	0	0
$685$	−43.7709	−1.67240
$686$	0	0
$687$	−3.15053	−0.120200
$688$	0	0
$689$	58.8046	2.24028
$690$	0	0
$691$	−28.4369	−1.08179	−0.540895	−	0.841090i	$-0.681915\pi$
$691$	−28.4369	−1.08179	−0.540895	+	0.841090i	$0.681915\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	20.9620	0.795135
$696$	0	0
$697$	6.24549	0.236565
$698$	0	0
$699$	0.902792	0.0341467
$700$	0	0
$701$	−1.90674	−0.0720165	−0.0360083	−	0.999351i	$-0.511464\pi$
$701$	−1.90674	−0.0720165	−0.0360083	+	0.999351i	$0.511464\pi$
$702$	0	0
$703$	−17.4051	−0.656446
$704$	0	0
$705$	−2.27879	−0.0858243
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−1.66673	−0.0625954	−0.0312977	−	0.999510i	$-0.509964\pi$
$709$	−1.66673	−0.0625954	−0.0312977	+	0.999510i	$0.509964\pi$
$710$	0	0
$711$	−4.34964	−0.163124
$712$	0	0
$713$	−5.61225	−0.210180
$714$	0	0
$715$	124.674	4.66253
$716$	0	0
$717$	−0.457461	−0.0170842
$718$	0	0
$719$	−5.77861	−0.215506	−0.107753	−	0.994178i	$-0.534366\pi$
$719$	−5.77861	−0.215506	−0.107753	+	0.994178i	$0.534366\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−4.37638	−0.162759
$724$	0	0
$725$	−34.2267	−1.27115
$726$	0	0
$727$	−35.6801	−1.32330	−0.661651	−	0.749812i	$-0.730144\pi$
$727$	−35.6801	−1.32330	−0.661651	+	0.749812i	$0.730144\pi$
$728$	0	0
$729$	−25.8478	−0.957326
$730$	0	0
$731$	76.7420	2.83841
$732$	0	0
$733$	3.72876	0.137725	0.0688624	−	0.997626i	$-0.478063\pi$
$733$	3.72876	0.137725	0.0688624	+	0.997626i	$0.478063\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	25.5998	0.942981
$738$	0	0
$739$	−47.8076	−1.75863	−0.879316	−	0.476239i	$-0.842000\pi$
$739$	−47.8076	−1.75863	−0.879316	+	0.476239i	$0.842000\pi$
$740$	0	0
$741$	−6.63379	−0.243698
$742$	0	0
$743$	−23.1744	−0.850186	−0.425093	−	0.905150i	$-0.639759\pi$
$743$	−23.1744	−0.850186	−0.425093	+	0.905150i	$0.639759\pi$
$744$	0	0
$745$	12.2476	0.448717
$746$	0	0
$747$	12.5527	0.459278
$748$	0	0
$749$	0	0
$750$	0	0
$751$	36.3210	1.32537	0.662685	−	0.748898i	$-0.269417\pi$
$751$	36.3210	1.32537	0.662685	+	0.748898i	$0.269417\pi$
$752$	0	0
$753$	−1.01629	−0.0370356
$754$	0	0
$755$	3.59723	0.130917
$756$	0	0
$757$	−21.0931	−0.766643	−0.383321	−	0.923615i	$-0.625220\pi$
$757$	−21.0931	−0.766643	−0.383321	+	0.923615i	$0.625220\pi$
$758$	0	0
$759$	−4.31556	−0.156645
$760$	0	0
$761$	26.6886	0.967459	0.483730	−	0.875217i	$-0.339282\pi$
$761$	26.6886	0.967459	0.483730	+	0.875217i	$0.339282\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	66.3886	2.40029
$766$	0	0
$767$	83.1962	3.00404
$768$	0	0
$769$	−35.4811	−1.27948	−0.639740	−	0.768591i	$-0.720958\pi$
$769$	−35.4811	−1.27948	−0.639740	+	0.768591i	$0.720958\pi$
$770$	0	0
$771$	−1.58898	−0.0572258
$772$	0	0
$773$	5.67891	0.204256	0.102128	−	0.994771i	$-0.467435\pi$
$773$	5.67891	0.204256	0.102128	+	0.994771i	$0.467435\pi$
$774$	0	0
$775$	−8.22646	−0.295503
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−7.23364	−0.259172
$780$	0	0
$781$	0.127257	0.00455363
$782$	0	0
$783$	3.87507	0.138484
$784$	0	0
$785$	36.2983	1.29554
$786$	0	0
$787$	22.3367	0.796216	0.398108	−	0.917338i	$-0.369667\pi$
$787$	22.3367	0.796216	0.398108	+	0.917338i	$0.369667\pi$
$788$	0	0
$789$	−3.15773	−0.112418
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−89.5766	−3.18096
$794$	0	0
$795$	−4.91197	−0.174210
$796$	0	0
$797$	10.6460	0.377101	0.188550	−	0.982064i	$-0.439621\pi$
$797$	10.6460	0.377101	0.188550	+	0.982064i	$0.439621\pi$
$798$	0	0
$799$	27.2198	0.962966
$800$	0	0
$801$	−10.6154	−0.375077
$802$	0	0
$803$	71.3456	2.51773
$804$	0	0
$805$	0	0
$806$	0	0
$807$	1.25452	0.0441611
$808$	0	0
$809$	−29.8892	−1.05085	−0.525425	−	0.850840i	$-0.676094\pi$
$809$	−29.8892	−1.05085	−0.525425	+	0.850840i	$0.676094\pi$
$810$	0	0
$811$	30.5838	1.07394	0.536971	−	0.843601i	$-0.319568\pi$
$811$	30.5838	1.07394	0.536971	+	0.843601i	$0.319568\pi$
$812$	0	0
$813$	1.99165	0.0698503
$814$	0	0
$815$	−45.8088	−1.60461
$816$	0	0
$817$	−88.8841	−3.10966
$818$	0	0
$819$	0	0
$820$	0	0
$821$	51.0330	1.78106	0.890532	−	0.454921i	$-0.150332\pi$
$821$	51.0330	1.78106	0.890532	+	0.454921i	$0.150332\pi$
$822$	0	0
$823$	−7.29326	−0.254227	−0.127114	−	0.991888i	$-0.540571\pi$
$823$	−7.29326	−0.254227	−0.127114	+	0.991888i	$0.540571\pi$
$824$	0	0
$825$	−6.32576	−0.220235
$826$	0	0
$827$	20.8265	0.724209	0.362105	−	0.932137i	$-0.382058\pi$
$827$	20.8265	0.724209	0.362105	+	0.932137i	$0.382058\pi$
$828$	0	0
$829$	4.70821	0.163523	0.0817614	−	0.996652i	$-0.473945\pi$
$829$	4.70821	0.163523	0.0817614	+	0.996652i	$0.473945\pi$
$830$	0	0
$831$	−0.0473018	−0.00164088
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−70.0304	−2.42350
$836$	0	0
$837$	0.931381	0.0321932
$838$	0	0
$839$	−41.1009	−1.41896	−0.709480	−	0.704725i	$-0.751070\pi$
$839$	−41.1009	−1.41896	−0.709480	+	0.704725i	$0.751070\pi$
$840$	0	0
$841$	−9.42754	−0.325088
$842$	0	0
$843$	−0.335467	−0.0115541
$844$	0	0
$845$	−93.4383	−3.21438
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−2.23778	−0.0768005
$850$	0	0
$851$	12.6995	0.435333
$852$	0	0
$853$	−33.8597	−1.15934	−0.579668	−	0.814853i	$-0.696818\pi$
$853$	−33.8597	−1.15934	−0.579668	+	0.814853i	$0.696818\pi$
$854$	0	0
$855$	−76.8926	−2.62967
$856$	0	0
$857$	−18.6073	−0.635613	−0.317806	−	0.948156i	$-0.602946\pi$
$857$	−18.6073	−0.635613	−0.317806	+	0.948156i	$0.602946\pi$
$858$	0	0
$859$	19.1801	0.654415	0.327207	−	0.944953i	$-0.393892\pi$
$859$	19.1801	0.654415	0.327207	+	0.944953i	$0.393892\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−32.3298	−1.10052	−0.550259	−	0.834994i	$-0.685471\pi$
$863$	−32.3298	−1.10052	−0.550259	+	0.834994i	$0.685471\pi$
$864$	0	0
$865$	−8.38258	−0.285016
$866$	0	0
$867$	3.22408	0.109495
$868$	0	0
$869$	−8.15002	−0.276470
$870$	0	0
$871$	−28.7125	−0.972887
$872$	0	0
$873$	30.0046	1.01550
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−11.3251	−0.382422	−0.191211	−	0.981549i	$-0.561241\pi$
$877$	−11.3251	−0.382422	−0.191211	+	0.981549i	$0.561241\pi$
$878$	0	0
$879$	−2.69018	−0.0907375
$880$	0	0
$881$	5.02177	0.169188	0.0845939	−	0.996416i	$-0.473041\pi$
$881$	5.02177	0.169188	0.0845939	+	0.996416i	$0.473041\pi$
$882$	0	0
$883$	−0.651629	−0.0219291	−0.0109645	−	0.999940i	$-0.503490\pi$
$883$	−0.651629	−0.0219291	−0.0109645	+	0.999940i	$0.503490\pi$
$884$	0	0
$885$	−6.94941	−0.233602
$886$	0	0
$887$	−15.4506	−0.518779	−0.259389	−	0.965773i	$-0.583521\pi$
$887$	−15.4506	−0.518779	−0.259389	+	0.965773i	$0.583521\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−49.1529	−1.64669
$892$	0	0
$893$	−31.5264	−1.05499
$894$	0	0
$895$	4.80764	0.160702
$896$	0	0
$897$	4.84030	0.161613
$898$	0	0
$899$	4.70428	0.156896
$900$	0	0
$901$	58.6726	1.95467
$902$	0	0
$903$	0	0
$904$	0	0
$905$	90.3885	3.00462
$906$	0	0
$907$	12.7773	0.424263	0.212132	−	0.977241i	$-0.431959\pi$
$907$	12.7773	0.424263	0.212132	+	0.977241i	$0.431959\pi$
$908$	0	0
$909$	20.1993	0.669969
$910$	0	0
$911$	−18.8263	−0.623744	−0.311872	−	0.950124i	$-0.600956\pi$
$911$	−18.8263	−0.623744	−0.311872	+	0.950124i	$0.600956\pi$
$912$	0	0
$913$	23.5202	0.778405
$914$	0	0
$915$	7.48237	0.247359
$916$	0	0
$917$	0	0
$918$	0	0
$919$	47.2712	1.55933	0.779667	−	0.626194i	$-0.215388\pi$
$919$	47.2712	1.55933	0.779667	+	0.626194i	$0.215388\pi$
$920$	0	0
$921$	4.01443	0.132280
$922$	0	0
$923$	−0.142731	−0.00469804
$924$	0	0
$925$	18.6150	0.612056
$926$	0	0
$927$	44.3297	1.45598
$928$	0	0
$929$	−25.1270	−0.824390	−0.412195	−	0.911096i	$-0.635238\pi$
$929$	−25.1270	−0.824390	−0.412195	+	0.911096i	$0.635238\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−1.44019	−0.0471498
$934$	0	0
$935$	124.394	4.06811
$936$	0	0
$937$	28.3047	0.924674	0.462337	−	0.886704i	$-0.347011\pi$
$937$	28.3047	0.924674	0.462337	+	0.886704i	$0.347011\pi$
$938$	0	0
$939$	−0.00336905	−0.000109945 0
$940$	0	0
$941$	−2.63050	−0.0857519	−0.0428759	−	0.999080i	$-0.513652\pi$
$941$	−2.63050	−0.0857519	−0.0428759	+	0.999080i	$0.513652\pi$
$942$	0	0
$943$	5.27797	0.171874
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−37.3928	−1.21510	−0.607552	−	0.794280i	$-0.707848\pi$
$947$	−37.3928	−1.21510	−0.607552	+	0.794280i	$0.707848\pi$
$948$	0	0
$949$	−80.0207	−2.59758
$950$	0	0
$951$	−1.21959	−0.0395480
$952$	0	0
$953$	15.4868	0.501667	0.250833	−	0.968030i	$-0.419295\pi$
$953$	15.4868	0.501667	0.250833	+	0.968030i	$0.419295\pi$
$954$	0	0
$955$	48.9295	1.58332
$956$	0	0
$957$	3.61737	0.116933
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−29.8693	−0.963526
$962$	0	0
$963$	−9.64996	−0.310966
$964$	0	0
$965$	51.0194	1.64237
$966$	0	0
$967$	37.4311	1.20370	0.601852	−	0.798608i	$-0.294430\pi$
$967$	37.4311	1.20370	0.601852	+	0.798608i	$0.294430\pi$
$968$	0	0
$969$	−6.61890	−0.212630
$970$	0	0
$971$	−24.2651	−0.778704	−0.389352	−	0.921089i	$-0.627301\pi$
$971$	−24.2651	−0.778704	−0.389352	+	0.921089i	$0.627301\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	7.09492	0.227219
$976$	0	0
$977$	−16.1282	−0.515986	−0.257993	−	0.966147i	$-0.583061\pi$
$977$	−16.1282	−0.515986	−0.257993	+	0.966147i	$0.583061\pi$
$978$	0	0
$979$	−19.8903	−0.635697
$980$	0	0
$981$	35.9152	1.14668
$982$	0	0
$983$	−12.9169	−0.411986	−0.205993	−	0.978553i	$-0.566042\pi$
$983$	−12.9169	−0.411986	−0.205993	+	0.978553i	$0.566042\pi$
$984$	0	0
$985$	30.2128	0.962661
$986$	0	0
$987$	0	0
$988$	0	0
$989$	64.8536	2.06222
$990$	0	0
$991$	−19.8559	−0.630743	−0.315372	−	0.948968i	$-0.602129\pi$
$991$	−19.8559	−0.630743	−0.315372	+	0.948968i	$0.602129\pi$
$992$	0	0
$993$	−2.78365	−0.0883365
$994$	0	0
$995$	−8.10617	−0.256983
$996$	0	0
$997$	38.6874	1.22524	0.612621	−	0.790377i	$-0.290115\pi$
$997$	38.6874	1.22524	0.612621	+	0.790377i	$0.290115\pi$
$998$	0	0
$999$	−2.10754	−0.0666798

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8036.2.a.s.1.13	✓	20
7.6	odd	2		8036.2.a.t.1.8	yes	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8036.2.a.s.1.13	✓	20	1.1	even	1	trivial
8036.2.a.t.1.8	yes	20	7.6	odd	2

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)

\(f(q)\)	\(=\)	\(q+0.146508 q^{3} -3.56882 q^{5} -2.97854 q^{9} +O(q^{10})\)	\(q+0.146508 q^{3} -3.56882 q^{5} -2.97854 q^{9} -5.58095 q^{11} +6.25954 q^{13} -0.522862 q^{15} +6.24549 q^{17} -7.23364 q^{19} +5.27797 q^{23} +7.73647 q^{25} -0.875905 q^{27} -4.42408 q^{29} -1.06333 q^{31} -0.817655 q^{33} +2.40613 q^{37} +0.917075 q^{39} +1.00000 q^{41} +12.2876 q^{43} +10.6299 q^{45} +4.35831 q^{47} +0.915016 q^{51} +9.39440 q^{53} +19.9174 q^{55} -1.05979 q^{57} +13.2911 q^{59} -14.3104 q^{61} -22.3392 q^{65} -4.58700 q^{67} +0.773267 q^{69} -0.0228021 q^{71} -12.7838 q^{73} +1.13346 q^{75} +1.46033 q^{79} +8.80728 q^{81} -4.21437 q^{83} -22.2890 q^{85} -0.648164 q^{87} +3.56397 q^{89} -0.155787 q^{93} +25.8156 q^{95} -10.0736 q^{97} +16.6230 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 4 q^{3} - 8 q^{5} + 16 q^{9}+O(q^{10}) \) 20 * q - 4 * q^3 - 8 * q^5 + 16 * q^9	\( 20 q - 4 q^{3} - 8 q^{5} + 16 q^{9} - 8 q^{11} - 12 q^{13} + 8 q^{15} - 8 q^{17} - 24 q^{19} + 8 q^{23} + 20 q^{25} - 16 q^{27} - 12 q^{29} - 44 q^{33} + 12 q^{37} + 12 q^{39} + 20 q^{41} + 4 q^{43} - 40 q^{45} - 4 q^{47} + 4 q^{51} - 12 q^{53} + 16 q^{55} + 28 q^{57} - 16 q^{59} - 68 q^{61} - 8 q^{65} + 4 q^{67} - 32 q^{69} + 8 q^{71} - 48 q^{73} - 60 q^{75} - 20 q^{79} + 32 q^{81} + 8 q^{83} - 28 q^{85} - 60 q^{89} - 16 q^{93} + 20 q^{95} - 40 q^{97}+O(q^{100}) \) 20 * q - 4 * q^3 - 8 * q^5 + 16 * q^9 - 8 * q^11 - 12 * q^13 + 8 * q^15 - 8 * q^17 - 24 * q^19 + 8 * q^23 + 20 * q^25 - 16 * q^27 - 12 * q^29 - 44 * q^33 + 12 * q^37 + 12 * q^39 + 20 * q^41 + 4 * q^43 - 40 * q^45 - 4 * q^47 + 4 * q^51 - 12 * q^53 + 16 * q^55 + 28 * q^57 - 16 * q^59 - 68 * q^61 - 8 * q^65 + 4 * q^67 - 32 * q^69 + 8 * q^71 - 48 * q^73 - 60 * q^75 - 20 * q^79 + 32 * q^81 + 8 * q^83 - 28 * q^85 - 60 * q^89 - 16 * q^93 + 20 * q^95 - 40 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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