LMFDB - Embedded newform 8036.2.a.r.1.7

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:	$15$
Coefficient field:	$\mathbb{Q}[x]/(x^{15} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{15} - x^{14} - 37 x^{13} + 39 x^{12} + 537 x^{11} - 616 x^{10} - 3853 x^{9} + 4929 x^{8} + \cdots + 3381 $ x^15 - x^14 - 37x^13 + 39x^12 + 537x^11 - 616x^10 - 3853x^9 + 4929x^8 + 13971x^7 - 20311x^6 - 22309x^5 + 38415x^4 + 8429x^3 - 22584x^2 - 399*x + 3381
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 1148)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$-0.449590$ 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)

$f(q)$	$=$	$q-0.449590 q^{3} +0.787803 q^{5} -2.79787 q^{9} +O(q^{10})$	$q-0.449590 q^{3} +0.787803 q^{5} -2.79787 q^{9} +3.10400 q^{11} -5.59137 q^{13} -0.354188 q^{15} -5.38244 q^{17} -4.82385 q^{19} +7.02456 q^{23} -4.37937 q^{25} +2.60666 q^{27} -7.52560 q^{29} -4.90724 q^{31} -1.39553 q^{33} -9.21207 q^{37} +2.51382 q^{39} +1.00000 q^{41} +11.7730 q^{43} -2.20417 q^{45} +5.56287 q^{47} +2.41989 q^{51} +5.94752 q^{53} +2.44534 q^{55} +2.16875 q^{57} +10.2017 q^{59} -4.89403 q^{61} -4.40490 q^{65} +11.7436 q^{67} -3.15817 q^{69} -1.39569 q^{71} +4.28028 q^{73} +1.96892 q^{75} +8.10074 q^{79} +7.22168 q^{81} -2.10452 q^{83} -4.24031 q^{85} +3.38343 q^{87} +16.1614 q^{89} +2.20625 q^{93} -3.80024 q^{95} -2.23832 q^{97} -8.68459 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 15 q + q^{3} - 3 q^{5} + 30 q^{9}+O(q^{10}) $ 15 * q + q^3 - 3 * q^5 + 30 * q^9	$ 15 q + q^{3} - 3 q^{5} + 30 q^{9} + 9 q^{11} - 7 q^{13} + 2 q^{15} - 3 q^{17} - 7 q^{19} - q^{23} + 32 q^{25} - 11 q^{27} + 18 q^{29} - 30 q^{31} + 16 q^{33} + 23 q^{37} + 5 q^{39} + 15 q^{41} + 12 q^{43} + 13 q^{45} + 16 q^{47} + 29 q^{51} + 33 q^{53} - 37 q^{55} + 16 q^{57} + 10 q^{59} - q^{61} + 16 q^{65} + 20 q^{67} - 21 q^{69} + 5 q^{71} + 3 q^{73} + 51 q^{75} + 25 q^{79} + 43 q^{81} - 18 q^{83} + 36 q^{85} + 53 q^{87} + 11 q^{89} + 65 q^{93} - 30 q^{95} - 16 q^{97} - 18 q^{99}+O(q^{100}) $ 15 * q + q^3 - 3 * q^5 + 30 * q^9 + 9 * q^11 - 7 * q^13 + 2 * q^15 - 3 * q^17 - 7 * q^19 - q^23 + 32 * q^25 - 11 * q^27 + 18 * q^29 - 30 * q^31 + 16 * q^33 + 23 * q^37 + 5 * q^39 + 15 * q^41 + 12 * q^43 + 13 * q^45 + 16 * q^47 + 29 * q^51 + 33 * q^53 - 37 * q^55 + 16 * q^57 + 10 * q^59 - q^61 + 16 * q^65 + 20 * q^67 - 21 * q^69 + 5 * q^71 + 3 * q^73 + 51 * q^75 + 25 * q^79 + 43 * q^81 - 18 * q^83 + 36 * q^85 + 53 * q^87 + 11 * q^89 + 65 * q^93 - 30 * q^95 - 16 * 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
$3$	−0.449590	−0.259571	−0.129785	−	0.991542i	$-0.541429\pi$
$3$	−0.449590	−0.259571	−0.129785	+	0.991542i	$0.541429\pi$
$4$	0	0
$5$	0.787803	0.352316	0.176158	−	0.984362i	$-0.443633\pi$
$5$	0.787803	0.352316	0.176158	+	0.984362i	$0.443633\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−2.79787	−0.932623
$10$	0	0
$11$	3.10400	0.935891	0.467946	−	0.883757i	$-0.344994\pi$
$11$	3.10400	0.935891	0.467946	+	0.883757i	$0.344994\pi$
$12$	0	0
$13$	−5.59137	−1.55077	−0.775383	−	0.631491i	$-0.782443\pi$
$13$	−5.59137	−1.55077	−0.775383	+	0.631491i	$0.782443\pi$
$14$	0	0
$15$	−0.354188	−0.0914510
$16$	0	0
$17$	−5.38244	−1.30543	−0.652717	−	0.757602i	$-0.726371\pi$
$17$	−5.38244	−1.30543	−0.652717	+	0.757602i	$0.726371\pi$
$18$	0	0
$19$	−4.82385	−1.10667	−0.553334	−	0.832960i	$-0.686645\pi$
$19$	−4.82385	−1.10667	−0.553334	+	0.832960i	$0.686645\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	7.02456	1.46472	0.732361	−	0.680917i	$-0.238419\pi$
$23$	7.02456	1.46472	0.732361	+	0.680917i	$0.238419\pi$
$24$	0	0
$25$	−4.37937	−0.875873
$26$	0	0
$27$	2.60666	0.501652
$28$	0	0
$29$	−7.52560	−1.39747	−0.698734	−	0.715382i	$-0.746253\pi$
$29$	−7.52560	−1.39747	−0.698734	+	0.715382i	$0.746253\pi$
$30$	0	0
$31$	−4.90724	−0.881366	−0.440683	−	0.897663i	$-0.645264\pi$
$31$	−4.90724	−0.881366	−0.440683	+	0.897663i	$0.645264\pi$
$32$	0	0
$33$	−1.39553	−0.242930
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−9.21207	−1.51446	−0.757228	−	0.653151i	$-0.773447\pi$
$37$	−9.21207	−1.51446	−0.757228	+	0.653151i	$0.773447\pi$
$38$	0	0
$39$	2.51382	0.402534
$40$	0	0
$41$	1.00000	0.156174
$41$	1.00000	0.156174
$42$	0	0
$43$	11.7730	1.79536	0.897680	−	0.440648i	$-0.145251\pi$
$43$	11.7730	1.79536	0.897680	+	0.440648i	$0.145251\pi$
$44$	0	0
$45$	−2.20417	−0.328578
$46$	0	0
$47$	5.56287	0.811428	0.405714	−	0.914000i	$-0.367023\pi$
$47$	5.56287	0.811428	0.405714	+	0.914000i	$0.367023\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	2.41989	0.338853
$52$	0	0
$53$	5.94752	0.816955	0.408477	−	0.912768i	$-0.366060\pi$
$53$	5.94752	0.816955	0.408477	+	0.912768i	$0.366060\pi$
$54$	0	0
$55$	2.44534	0.329730
$56$	0	0
$57$	2.16875	0.287259
$58$	0	0
$59$	10.2017	1.32814	0.664071	−	0.747670i	$-0.268827\pi$
$59$	10.2017	1.32814	0.664071	+	0.747670i	$0.268827\pi$
$60$	0	0
$61$	−4.89403	−0.626616	−0.313308	−	0.949652i	$-0.601437\pi$
$61$	−4.89403	−0.626616	−0.313308	+	0.949652i	$0.601437\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−4.40490	−0.546360
$66$	0	0
$67$	11.7436	1.43472	0.717358	−	0.696705i	$-0.245351\pi$
$67$	11.7436	1.43472	0.717358	+	0.696705i	$0.245351\pi$
$68$	0	0
$69$	−3.15817	−0.380199
$70$	0	0
$71$	−1.39569	−0.165638	−0.0828188	−	0.996565i	$-0.526392\pi$
$71$	−1.39569	−0.165638	−0.0828188	+	0.996565i	$0.526392\pi$
$72$	0	0
$73$	4.28028	0.500968	0.250484	−	0.968121i	$-0.419410\pi$
$73$	4.28028	0.500968	0.250484	+	0.968121i	$0.419410\pi$
$74$	0	0
$75$	1.96892	0.227351
$76$	0	0
$77$	0	0
$78$	0	0
$79$	8.10074	0.911404	0.455702	−	0.890132i	$-0.349388\pi$
$79$	8.10074	0.911404	0.455702	+	0.890132i	$0.349388\pi$
$80$	0	0
$81$	7.22168	0.802409
$82$	0	0
$83$	−2.10452	−0.231001	−0.115500	−	0.993307i	$-0.536847\pi$
$83$	−2.10452	−0.231001	−0.115500	+	0.993307i	$0.536847\pi$
$84$	0	0
$85$	−4.24031	−0.459926
$86$	0	0
$87$	3.38343	0.362742
$88$	0	0
$89$	16.1614	1.71311	0.856553	−	0.516059i	$-0.172602\pi$
$89$	16.1614	1.71311	0.856553	+	0.516059i	$0.172602\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	2.20625	0.228777
$94$	0	0
$95$	−3.80024	−0.389897
$96$	0	0
$97$	−2.23832	−0.227267	−0.113634	−	0.993523i	$-0.536249\pi$
$97$	−2.23832	−0.227267	−0.113634	+	0.993523i	$0.536249\pi$
$98$	0	0
$99$	−8.68459	−0.872834
$100$	0	0
$101$	6.14466	0.611416	0.305708	−	0.952125i	$-0.401107\pi$
$101$	6.14466	0.611416	0.305708	+	0.952125i	$0.401107\pi$
$102$	0	0
$103$	−1.47349	−0.145188	−0.0725938	−	0.997362i	$-0.523128\pi$
$103$	−1.47349	−0.145188	−0.0725938	+	0.997362i	$0.523128\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	8.43556	0.815497	0.407748	−	0.913094i	$-0.366314\pi$
$107$	8.43556	0.815497	0.407748	+	0.913094i	$0.366314\pi$
$108$	0	0
$109$	−10.1612	−0.973266	−0.486633	−	0.873607i	$-0.661775\pi$
$109$	−10.1612	−0.973266	−0.486633	+	0.873607i	$0.661775\pi$
$110$	0	0
$111$	4.14165	0.393108
$112$	0	0
$113$	8.34801	0.785315	0.392657	−	0.919685i	$-0.371556\pi$
$113$	8.34801	0.785315	0.392657	+	0.919685i	$0.371556\pi$
$114$	0	0
$115$	5.53397	0.516045
$116$	0	0
$117$	15.6439	1.44628
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−1.36518	−0.124107
$122$	0	0
$123$	−0.449590	−0.0405381
$124$	0	0
$125$	−7.38909	−0.660901
$126$	0	0
$127$	−15.6031	−1.38455	−0.692274	−	0.721635i	$-0.743391\pi$
$127$	−15.6031	−1.38455	−0.692274	+	0.721635i	$0.743391\pi$
$128$	0	0
$129$	−5.29300	−0.466023
$130$	0	0
$131$	11.7626	1.02771	0.513853	−	0.857878i	$-0.328218\pi$
$131$	11.7626	1.02771	0.513853	+	0.857878i	$0.328218\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	2.05354	0.176740
$136$	0	0
$137$	4.49604	0.384123	0.192061	−	0.981383i	$-0.438483\pi$
$137$	4.49604	0.384123	0.192061	+	0.981383i	$0.438483\pi$
$138$	0	0
$139$	−1.41035	−0.119624	−0.0598121	−	0.998210i	$-0.519050\pi$
$139$	−1.41035	−0.119624	−0.0598121	+	0.998210i	$0.519050\pi$
$140$	0	0
$141$	−2.50101	−0.210623
$142$	0	0
$143$	−17.3556	−1.45135
$144$	0	0
$145$	−5.92869	−0.492351
$146$	0	0
$147$	0	0
$148$	0	0
$149$	8.50284	0.696580	0.348290	−	0.937387i	$-0.386762\pi$
$149$	8.50284	0.696580	0.348290	+	0.937387i	$0.386762\pi$
$150$	0	0
$151$	−2.68539	−0.218534	−0.109267	−	0.994012i	$-0.534850\pi$
$151$	−2.68539	−0.218534	−0.109267	+	0.994012i	$0.534850\pi$
$152$	0	0
$153$	15.0594	1.21748
$154$	0	0
$155$	−3.86594	−0.310520
$156$	0	0
$157$	4.26751	0.340584	0.170292	−	0.985394i	$-0.445529\pi$
$157$	4.26751	0.340584	0.170292	+	0.985394i	$0.445529\pi$
$158$	0	0
$159$	−2.67394	−0.212058
$160$	0	0
$161$	0	0
$162$	0	0
$163$	18.7332	1.46730	0.733649	−	0.679529i	$-0.237816\pi$
$163$	18.7332	1.46730	0.733649	+	0.679529i	$0.237816\pi$
$164$	0	0
$165$	−1.09940	−0.0855882
$166$	0	0
$167$	−11.3399	−0.877510	−0.438755	−	0.898607i	$-0.644580\pi$
$167$	−11.3399	−0.877510	−0.438755	+	0.898607i	$0.644580\pi$
$168$	0	0
$169$	18.2634	1.40488
$170$	0	0
$171$	13.4965	1.03210
$172$	0	0
$173$	12.5023	0.950531	0.475266	−	0.879842i	$-0.342352\pi$
$173$	12.5023	0.950531	0.475266	+	0.879842i	$0.342352\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−4.58656	−0.344747
$178$	0	0
$179$	−6.96188	−0.520355	−0.260178	−	0.965561i	$-0.583781\pi$
$179$	−6.96188	−0.520355	−0.260178	+	0.965561i	$0.583781\pi$
$180$	0	0
$181$	−18.3297	−1.36244	−0.681219	−	0.732080i	$-0.738550\pi$
$181$	−18.3297	−1.36244	−0.681219	+	0.732080i	$0.738550\pi$
$182$	0	0
$183$	2.20031	0.162651
$184$	0	0
$185$	−7.25730	−0.533567
$186$	0	0
$187$	−16.7071	−1.22174
$188$	0	0
$189$	0	0
$190$	0	0
$191$	19.2841	1.39535	0.697675	−	0.716414i	$-0.254218\pi$
$191$	19.2841	1.39535	0.697675	+	0.716414i	$0.254218\pi$
$192$	0	0
$193$	1.39703	0.100561	0.0502803	−	0.998735i	$-0.483989\pi$
$193$	1.39703	0.100561	0.0502803	+	0.998735i	$0.483989\pi$
$194$	0	0
$195$	1.98040	0.141819
$196$	0	0
$197$	12.8161	0.913108	0.456554	−	0.889696i	$-0.349084\pi$
$197$	12.8161	0.913108	0.456554	+	0.889696i	$0.349084\pi$
$198$	0	0
$199$	1.67339	0.118624	0.0593119	−	0.998239i	$-0.481109\pi$
$199$	1.67339	0.118624	0.0593119	+	0.998239i	$0.481109\pi$
$200$	0	0
$201$	−5.27982	−0.372410
$202$	0	0
$203$	0	0
$204$	0	0
$205$	0.787803	0.0550225
$206$	0	0
$207$	−19.6538	−1.36603
$208$	0	0
$209$	−14.9732	−1.03572
$210$	0	0
$211$	−13.1602	−0.905984	−0.452992	−	0.891515i	$-0.649643\pi$
$211$	−13.1602	−0.905984	−0.452992	+	0.891515i	$0.649643\pi$
$212$	0	0
$213$	0.627487	0.0429947
$214$	0	0
$215$	9.27478	0.632535
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−1.92437	−0.130037
$220$	0	0
$221$	30.0952	2.02442
$222$	0	0
$223$	−11.1114	−0.744072	−0.372036	−	0.928218i	$-0.621340\pi$
$223$	−11.1114	−0.744072	−0.372036	+	0.928218i	$0.621340\pi$
$224$	0	0
$225$	12.2529	0.816860
$226$	0	0
$227$	16.0652	1.06628	0.533142	−	0.846026i	$-0.321011\pi$
$227$	16.0652	1.06628	0.533142	+	0.846026i	$0.321011\pi$
$228$	0	0
$229$	25.4126	1.67931	0.839655	−	0.543120i	$-0.182757\pi$
$229$	25.4126	1.67931	0.839655	+	0.543120i	$0.182757\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	20.0557	1.31390	0.656948	−	0.753936i	$-0.271847\pi$
$233$	20.0557	1.31390	0.656948	+	0.753936i	$0.271847\pi$
$234$	0	0
$235$	4.38245	0.285879
$236$	0	0
$237$	−3.64201	−0.236574
$238$	0	0
$239$	6.32352	0.409034	0.204517	−	0.978863i	$-0.434438\pi$
$239$	6.32352	0.409034	0.204517	+	0.978863i	$0.434438\pi$
$240$	0	0
$241$	−25.2122	−1.62406	−0.812030	−	0.583616i	$-0.801637\pi$
$241$	−25.2122	−1.62406	−0.812030	+	0.583616i	$0.801637\pi$
$242$	0	0
$243$	−11.0668	−0.709934
$244$	0	0
$245$	0	0
$246$	0	0
$247$	26.9719	1.71618
$248$	0	0
$249$	0.946169	0.0599610
$250$	0	0
$251$	−15.3360	−0.967998	−0.483999	−	0.875069i	$-0.660816\pi$
$251$	−15.3360	−0.967998	−0.483999	+	0.875069i	$0.660816\pi$
$252$	0	0
$253$	21.8042	1.37082
$254$	0	0
$255$	1.90640	0.119383
$256$	0	0
$257$	−15.2711	−0.952587	−0.476293	−	0.879286i	$-0.658020\pi$
$257$	−15.2711	−0.952587	−0.476293	+	0.879286i	$0.658020\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	21.0556	1.30331
$262$	0	0
$263$	25.2720	1.55834	0.779169	−	0.626814i	$-0.215642\pi$
$263$	25.2720	1.55834	0.779169	+	0.626814i	$0.215642\pi$
$264$	0	0
$265$	4.68547	0.287826
$266$	0	0
$267$	−7.26600	−0.444672
$268$	0	0
$269$	7.10754	0.433355	0.216677	−	0.976243i	$-0.430478\pi$
$269$	7.10754	0.433355	0.216677	+	0.976243i	$0.430478\pi$
$270$	0	0
$271$	−1.01526	−0.0616727	−0.0308363	−	0.999524i	$-0.509817\pi$
$271$	−1.01526	−0.0616727	−0.0308363	+	0.999524i	$0.509817\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−13.5936	−0.819722
$276$	0	0
$277$	−23.0775	−1.38659	−0.693296	−	0.720653i	$-0.743842\pi$
$277$	−23.0775	−1.38659	−0.693296	+	0.720653i	$0.743842\pi$
$278$	0	0
$279$	13.7298	0.821983
$280$	0	0
$281$	−19.3572	−1.15475	−0.577377	−	0.816478i	$-0.695924\pi$
$281$	−19.3572	−1.15475	−0.577377	+	0.816478i	$0.695924\pi$
$282$	0	0
$283$	−0.177467	−0.0105493	−0.00527466	−	0.999986i	$-0.501679\pi$
$283$	−0.177467	−0.0105493	−0.00527466	+	0.999986i	$0.501679\pi$
$284$	0	0
$285$	1.70855	0.101206
$286$	0	0
$287$	0	0
$288$	0	0
$289$	11.9707	0.704159
$290$	0	0
$291$	1.00633	0.0589919
$292$	0	0
$293$	−27.8143	−1.62493	−0.812465	−	0.583010i	$-0.801875\pi$
$293$	−27.8143	−1.62493	−0.812465	+	0.583010i	$0.801875\pi$
$294$	0	0
$295$	8.03689	0.467926
$296$	0	0
$297$	8.09108	0.469492
$298$	0	0
$299$	−39.2769	−2.27144
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−2.76257	−0.158706
$304$	0	0
$305$	−3.85553	−0.220767
$306$	0	0
$307$	−25.7021	−1.46690	−0.733448	−	0.679745i	$-0.762090\pi$
$307$	−25.7021	−1.46690	−0.733448	+	0.679745i	$0.762090\pi$
$308$	0	0
$309$	0.662468	0.0376865
$310$	0	0
$311$	15.5756	0.883213	0.441606	−	0.897209i	$-0.354409\pi$
$311$	15.5756	0.883213	0.441606	+	0.897209i	$0.354409\pi$
$312$	0	0
$313$	−21.6373	−1.22301	−0.611505	−	0.791241i	$-0.709436\pi$
$313$	−21.6373	−1.22301	−0.611505	+	0.791241i	$0.709436\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	23.9737	1.34650	0.673248	−	0.739417i	$-0.264899\pi$
$317$	23.9737	1.34650	0.673248	+	0.739417i	$0.264899\pi$
$318$	0	0
$319$	−23.3595	−1.30788
$320$	0	0
$321$	−3.79254	−0.211679
$322$	0	0
$323$	25.9641	1.44468
$324$	0	0
$325$	24.4867	1.35828
$326$	0	0
$327$	4.56837	0.252631
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−9.64917	−0.530366	−0.265183	−	0.964198i	$-0.585432\pi$
$331$	−9.64917	−0.530366	−0.265183	+	0.964198i	$0.585432\pi$
$332$	0	0
$333$	25.7742	1.41242
$334$	0	0
$335$	9.25168	0.505473
$336$	0	0
$337$	−0.606481	−0.0330371	−0.0165186	−	0.999864i	$-0.505258\pi$
$337$	−0.606481	−0.0330371	−0.0165186	+	0.999864i	$0.505258\pi$
$338$	0	0
$339$	−3.75318	−0.203845
$340$	0	0
$341$	−15.2321	−0.824863
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−2.48801	−0.133950
$346$	0	0
$347$	18.7135	1.00459	0.502296	−	0.864696i	$-0.332489\pi$
$347$	18.7135	1.00459	0.502296	+	0.864696i	$0.332489\pi$
$348$	0	0
$349$	−20.5477	−1.09989	−0.549945	−	0.835201i	$-0.685351\pi$
$349$	−20.5477	−1.09989	−0.549945	+	0.835201i	$0.685351\pi$
$350$	0	0
$351$	−14.5748	−0.777946
$352$	0	0
$353$	3.94997	0.210235	0.105118	−	0.994460i	$-0.466478\pi$
$353$	3.94997	0.210235	0.105118	+	0.994460i	$0.466478\pi$
$354$	0	0
$355$	−1.09953	−0.0583568
$356$	0	0
$357$	0	0
$358$	0	0
$359$	2.04134	0.107738	0.0538690	−	0.998548i	$-0.482845\pi$
$359$	2.04134	0.107738	0.0538690	+	0.998548i	$0.482845\pi$
$360$	0	0
$361$	4.26954	0.224713
$362$	0	0
$363$	0.613771	0.0322146
$364$	0	0
$365$	3.37201	0.176499
$366$	0	0
$367$	16.6087	0.866969	0.433485	−	0.901161i	$-0.357284\pi$
$367$	16.6087	0.866969	0.433485	+	0.901161i	$0.357284\pi$
$368$	0	0
$369$	−2.79787	−0.145651
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−13.8375	−0.716479	−0.358239	−	0.933630i	$-0.616623\pi$
$373$	−13.8375	−0.716479	−0.358239	+	0.933630i	$0.616623\pi$
$374$	0	0
$375$	3.32206	0.171550
$376$	0	0
$377$	42.0784	2.16715
$378$	0	0
$379$	−5.23579	−0.268944	−0.134472	−	0.990917i	$-0.542934\pi$
$379$	−5.23579	−0.268944	−0.134472	+	0.990917i	$0.542934\pi$
$380$	0	0
$381$	7.01498	0.359388
$382$	0	0
$383$	−8.45623	−0.432093	−0.216047	−	0.976383i	$-0.569316\pi$
$383$	−8.45623	−0.432093	−0.216047	+	0.976383i	$0.569316\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−32.9392	−1.67439
$388$	0	0
$389$	−26.0414	−1.32035	−0.660175	−	0.751112i	$-0.729518\pi$
$389$	−26.0414	−1.32035	−0.660175	+	0.751112i	$0.729518\pi$
$390$	0	0
$391$	−37.8093	−1.91210
$392$	0	0
$393$	−5.28836	−0.266762
$394$	0	0
$395$	6.38179	0.321103
$396$	0	0
$397$	3.42795	0.172044	0.0860220	−	0.996293i	$-0.472584\pi$
$397$	3.42795	0.172044	0.0860220	+	0.996293i	$0.472584\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	2.26799	0.113258	0.0566289	−	0.998395i	$-0.481965\pi$
$401$	2.26799	0.113258	0.0566289	+	0.998395i	$0.481965\pi$
$402$	0	0
$403$	27.4382	1.36679
$404$	0	0
$405$	5.68926	0.282702
$406$	0	0
$407$	−28.5943	−1.41737
$408$	0	0
$409$	11.0462	0.546198	0.273099	−	0.961986i	$-0.411951\pi$
$409$	11.0462	0.546198	0.273099	+	0.961986i	$0.411951\pi$
$410$	0	0
$411$	−2.02138	−0.0997071
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−1.65794	−0.0813853
$416$	0	0
$417$	0.634078	0.0310509
$418$	0	0
$419$	29.0178	1.41761	0.708805	−	0.705404i	$-0.249235\pi$
$419$	29.0178	1.41761	0.708805	+	0.705404i	$0.249235\pi$
$420$	0	0
$421$	11.2610	0.548827	0.274414	−	0.961612i	$-0.411516\pi$
$421$	11.2610	0.548827	0.274414	+	0.961612i	$0.411516\pi$
$422$	0	0
$423$	−15.5642	−0.756756
$424$	0	0
$425$	23.5717	1.14340
$426$	0	0
$427$	0	0
$428$	0	0
$429$	7.80291	0.376728
$430$	0	0
$431$	16.5476	0.797071	0.398535	−	0.917153i	$-0.369519\pi$
$431$	16.5476	0.797071	0.398535	+	0.917153i	$0.369519\pi$
$432$	0	0
$433$	4.65533	0.223721	0.111860	−	0.993724i	$-0.464319\pi$
$433$	4.65533	0.223721	0.111860	+	0.993724i	$0.464319\pi$
$434$	0	0
$435$	2.66548	0.127800
$436$	0	0
$437$	−33.8854	−1.62096
$438$	0	0
$439$	22.7846	1.08745	0.543725	−	0.839263i	$-0.317013\pi$
$439$	22.7846	1.08745	0.543725	+	0.839263i	$0.317013\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−29.6057	−1.40661	−0.703305	−	0.710888i	$-0.748293\pi$
$443$	−29.6057	−1.40661	−0.703305	+	0.710888i	$0.748293\pi$
$444$	0	0
$445$	12.7320	0.603555
$446$	0	0
$447$	−3.82279	−0.180812
$448$	0	0
$449$	−2.83513	−0.133798	−0.0668990	−	0.997760i	$-0.521311\pi$
$449$	−2.83513	−0.133798	−0.0668990	+	0.997760i	$0.521311\pi$
$450$	0	0
$451$	3.10400	0.146162
$452$	0	0
$453$	1.20732	0.0567250
$454$	0	0
$455$	0	0
$456$	0	0
$457$	27.6601	1.29389	0.646943	−	0.762538i	$-0.276047\pi$
$457$	27.6601	1.29389	0.646943	+	0.762538i	$0.276047\pi$
$458$	0	0
$459$	−14.0302	−0.654874
$460$	0	0
$461$	−4.80680	−0.223875	−0.111937	−	0.993715i	$-0.535706\pi$
$461$	−4.80680	−0.223875	−0.111937	+	0.993715i	$0.535706\pi$
$462$	0	0
$463$	31.2899	1.45416	0.727082	−	0.686551i	$-0.240876\pi$
$463$	31.2899	1.45416	0.727082	+	0.686551i	$0.240876\pi$
$464$	0	0
$465$	1.73809	0.0806018
$466$	0	0
$467$	−10.8937	−0.504100	−0.252050	−	0.967714i	$-0.581105\pi$
$467$	−10.8937	−0.504100	−0.252050	+	0.967714i	$0.581105\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−1.91863	−0.0884057
$472$	0	0
$473$	36.5433	1.68026
$474$	0	0
$475$	21.1254	0.969300
$476$	0	0
$477$	−16.6404	−0.761911
$478$	0	0
$479$	−14.1384	−0.646000	−0.323000	−	0.946399i	$-0.604691\pi$
$479$	−14.1384	−0.646000	−0.323000	+	0.946399i	$0.604691\pi$
$480$	0	0
$481$	51.5081	2.34857
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−1.76336	−0.0800699
$486$	0	0
$487$	−34.7768	−1.57589	−0.787945	−	0.615746i	$-0.788855\pi$
$487$	−34.7768	−1.57589	−0.787945	+	0.615746i	$0.788855\pi$
$488$	0	0
$489$	−8.42226	−0.380868
$490$	0	0
$491$	31.6187	1.42693	0.713466	−	0.700690i	$-0.247124\pi$
$491$	31.6187	1.42693	0.713466	+	0.700690i	$0.247124\pi$
$492$	0	0
$493$	40.5061	1.82430
$494$	0	0
$495$	−6.84174	−0.307514
$496$	0	0
$497$	0	0
$498$	0	0
$499$	30.8653	1.38172	0.690860	−	0.722989i	$-0.257232\pi$
$499$	30.8653	1.38172	0.690860	+	0.722989i	$0.257232\pi$
$500$	0	0
$501$	5.09832	0.227776
$502$	0	0
$503$	18.3660	0.818900	0.409450	−	0.912333i	$-0.365721\pi$
$503$	18.3660	0.818900	0.409450	+	0.912333i	$0.365721\pi$
$504$	0	0
$505$	4.84078	0.215412
$506$	0	0
$507$	−8.21104	−0.364665
$508$	0	0
$509$	37.0290	1.64128	0.820641	−	0.571444i	$-0.193616\pi$
$509$	37.0290	1.64128	0.820641	+	0.571444i	$0.193616\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−12.5742	−0.555162
$514$	0	0
$515$	−1.16082	−0.0511520
$516$	0	0
$517$	17.2672	0.759408
$518$	0	0
$519$	−5.62090	−0.246730
$520$	0	0
$521$	26.2351	1.14938	0.574690	−	0.818371i	$-0.305122\pi$
$521$	26.2351	1.14938	0.574690	+	0.818371i	$0.305122\pi$
$522$	0	0
$523$	−1.74603	−0.0763485	−0.0381742	−	0.999271i	$-0.512154\pi$
$523$	−1.74603	−0.0763485	−0.0381742	+	0.999271i	$0.512154\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	26.4129	1.15057
$528$	0	0
$529$	26.3444	1.14541
$530$	0	0
$531$	−28.5429	−1.23866
$532$	0	0
$533$	−5.59137	−0.242189
$534$	0	0
$535$	6.64556	0.287313
$536$	0	0
$537$	3.12999	0.135069
$538$	0	0
$539$	0	0
$540$	0	0
$541$	14.3096	0.615219	0.307609	−	0.951513i	$-0.400471\pi$
$541$	14.3096	0.615219	0.307609	+	0.951513i	$0.400471\pi$
$542$	0	0
$543$	8.24086	0.353649
$544$	0	0
$545$	−8.00502	−0.342897
$546$	0	0
$547$	1.40140	0.0599194	0.0299597	−	0.999551i	$-0.490462\pi$
$547$	1.40140	0.0599194	0.0299597	+	0.999551i	$0.490462\pi$
$548$	0	0
$549$	13.6929	0.584397
$550$	0	0
$551$	36.3024	1.54653
$552$	0	0
$553$	0	0
$554$	0	0
$555$	3.26281	0.138498
$556$	0	0
$557$	−13.4128	−0.568318	−0.284159	−	0.958777i	$-0.591714\pi$
$557$	−13.4128	−0.568318	−0.284159	+	0.958777i	$0.591714\pi$
$558$	0	0
$559$	−65.8270	−2.78419
$560$	0	0
$561$	7.51135	0.317129
$562$	0	0
$563$	20.3679	0.858404	0.429202	−	0.903208i	$-0.358795\pi$
$563$	20.3679	0.858404	0.429202	+	0.903208i	$0.358795\pi$
$564$	0	0
$565$	6.57659	0.276679
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−13.9245	−0.583746	−0.291873	−	0.956457i	$-0.594278\pi$
$569$	−13.9245	−0.583746	−0.291873	+	0.956457i	$0.594278\pi$
$570$	0	0
$571$	40.0547	1.67624	0.838118	−	0.545489i	$-0.183656\pi$
$571$	40.0547	1.67624	0.838118	+	0.545489i	$0.183656\pi$
$572$	0	0
$573$	−8.66995	−0.362192
$574$	0	0
$575$	−30.7631	−1.28291
$576$	0	0
$577$	−41.8713	−1.74312	−0.871562	−	0.490285i	$-0.836893\pi$
$577$	−41.8713	−1.74312	−0.871562	+	0.490285i	$0.836893\pi$
$578$	0	0
$579$	−0.628092	−0.0261026
$580$	0	0
$581$	0	0
$582$	0	0
$583$	18.4611	0.764581
$584$	0	0
$585$	12.3243	0.509548
$586$	0	0
$587$	−12.4478	−0.513775	−0.256888	−	0.966441i	$-0.582697\pi$
$587$	−12.4478	−0.513775	−0.256888	+	0.966441i	$0.582697\pi$
$588$	0	0
$589$	23.6718	0.975379
$590$	0	0
$591$	−5.76198	−0.237016
$592$	0	0
$593$	12.6955	0.521342	0.260671	−	0.965428i	$-0.416056\pi$
$593$	12.6955	0.521342	0.260671	+	0.965428i	$0.416056\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−0.752341	−0.0307913
$598$	0	0
$599$	37.9084	1.54890	0.774448	−	0.632637i	$-0.218028\pi$
$599$	37.9084	1.54890	0.774448	+	0.632637i	$0.218028\pi$
$600$	0	0
$601$	−22.3859	−0.913139	−0.456569	−	0.889688i	$-0.650922\pi$
$601$	−22.3859	−0.913139	−0.456569	+	0.889688i	$0.650922\pi$
$602$	0	0
$603$	−32.8572	−1.33805
$604$	0	0
$605$	−1.07549	−0.0437250
$606$	0	0
$607$	−46.7440	−1.89728	−0.948640	−	0.316357i	$-0.897540\pi$
$607$	−46.7440	−1.89728	−0.948640	+	0.316357i	$0.897540\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−31.1041	−1.25834
$612$	0	0
$613$	−0.295420	−0.0119319	−0.00596596	−	0.999982i	$-0.501899\pi$
$613$	−0.295420	−0.0119319	−0.00596596	+	0.999982i	$0.501899\pi$
$614$	0	0
$615$	−0.354188	−0.0142822
$616$	0	0
$617$	−42.2704	−1.70174	−0.850871	−	0.525375i	$-0.823925\pi$
$617$	−42.2704	−1.70174	−0.850871	+	0.525375i	$0.823925\pi$
$618$	0	0
$619$	−31.9134	−1.28271	−0.641354	−	0.767245i	$-0.721627\pi$
$619$	−31.9134	−1.28271	−0.641354	+	0.767245i	$0.721627\pi$
$620$	0	0
$621$	18.3106	0.734781
$622$	0	0
$623$	0	0
$624$	0	0
$625$	16.0757	0.643027
$626$	0	0
$627$	6.73181	0.268843
$628$	0	0
$629$	49.5835	1.97702
$630$	0	0
$631$	−5.19942	−0.206986	−0.103493	−	0.994630i	$-0.533002\pi$
$631$	−5.19942	−0.206986	−0.103493	+	0.994630i	$0.533002\pi$
$632$	0	0
$633$	5.91668	0.235167
$634$	0	0
$635$	−12.2921	−0.487799
$636$	0	0
$637$	0	0
$638$	0	0
$639$	3.90495	0.154478
$640$	0	0
$641$	−8.90534	−0.351740	−0.175870	−	0.984413i	$-0.556274\pi$
$641$	−8.90534	−0.351740	−0.175870	+	0.984413i	$0.556274\pi$
$642$	0	0
$643$	24.4281	0.963350	0.481675	−	0.876350i	$-0.340029\pi$
$643$	24.4281	0.963350	0.481675	+	0.876350i	$0.340029\pi$
$644$	0	0
$645$	−4.16984	−0.164187
$646$	0	0
$647$	24.8284	0.976104	0.488052	−	0.872814i	$-0.337708\pi$
$647$	24.8284	0.976104	0.488052	+	0.872814i	$0.337708\pi$
$648$	0	0
$649$	31.6659	1.24300
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−24.0715	−0.941990	−0.470995	−	0.882136i	$-0.656105\pi$
$653$	−24.0715	−0.941990	−0.470995	+	0.882136i	$0.656105\pi$
$654$	0	0
$655$	9.26663	0.362077
$656$	0	0
$657$	−11.9756	−0.467215
$658$	0	0
$659$	−3.61093	−0.140662	−0.0703310	−	0.997524i	$-0.522406\pi$
$659$	−3.61093	−0.140662	−0.0703310	+	0.997524i	$0.522406\pi$
$660$	0	0
$661$	28.0885	1.09252	0.546258	−	0.837617i	$-0.316052\pi$
$661$	28.0885	1.09252	0.546258	+	0.837617i	$0.316052\pi$
$662$	0	0
$663$	−13.5305	−0.525481
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−52.8640	−2.04690
$668$	0	0
$669$	4.99556	0.193139
$670$	0	0
$671$	−15.1911	−0.586445
$672$	0	0
$673$	34.0798	1.31368	0.656840	−	0.754030i	$-0.271893\pi$
$673$	34.0798	1.31368	0.656840	+	0.754030i	$0.271893\pi$
$674$	0	0
$675$	−11.4155	−0.439384
$676$	0	0
$677$	−9.32910	−0.358546	−0.179273	−	0.983799i	$-0.557375\pi$
$677$	−9.32910	−0.358546	−0.179273	+	0.983799i	$0.557375\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−7.22274	−0.276776
$682$	0	0
$683$	−14.7903	−0.565933	−0.282967	−	0.959130i	$-0.591319\pi$
$683$	−14.7903	−0.565933	−0.282967	+	0.959130i	$0.591319\pi$
$684$	0	0
$685$	3.54200	0.135333
$686$	0	0
$687$	−11.4252	−0.435900
$688$	0	0
$689$	−33.2548	−1.26691
$690$	0	0
$691$	27.7101	1.05414	0.527072	−	0.849821i	$-0.323290\pi$
$691$	27.7101	1.05414	0.527072	+	0.849821i	$0.323290\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−1.11108	−0.0421455
$696$	0	0
$697$	−5.38244	−0.203875
$698$	0	0
$699$	−9.01686	−0.341049
$700$	0	0
$701$	−3.12039	−0.117856	−0.0589278	−	0.998262i	$-0.518768\pi$
$701$	−3.12039	−0.117856	−0.0589278	+	0.998262i	$0.518768\pi$
$702$	0	0
$703$	44.4377	1.67600
$704$	0	0
$705$	−1.97030	−0.0742059
$706$	0	0
$707$	0	0
$708$	0	0
$709$	34.9063	1.31093	0.655467	−	0.755224i	$-0.272472\pi$
$709$	34.9063	1.31093	0.655467	+	0.755224i	$0.272472\pi$
$710$	0	0
$711$	−22.6648	−0.849997
$712$	0	0
$713$	−34.4712	−1.29096
$714$	0	0
$715$	−13.6728	−0.511334
$716$	0	0
$717$	−2.84299	−0.106173
$718$	0	0
$719$	35.8654	1.33755	0.668777	−	0.743463i	$-0.266818\pi$
$719$	35.8654	1.33755	0.668777	+	0.743463i	$0.266818\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	11.3351	0.421558
$724$	0	0
$725$	32.9573	1.22400
$726$	0	0
$727$	8.15799	0.302563	0.151282	−	0.988491i	$-0.451660\pi$
$727$	8.15799	0.302563	0.151282	+	0.988491i	$0.451660\pi$
$728$	0	0
$729$	−16.6895	−0.618131
$730$	0	0
$731$	−63.3673	−2.34373
$732$	0	0
$733$	−37.3000	−1.37771	−0.688854	−	0.724900i	$-0.741886\pi$
$733$	−37.3000	−1.37771	−0.688854	+	0.724900i	$0.741886\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	36.4523	1.34274
$738$	0	0
$739$	0.801245	0.0294743	0.0147371	−	0.999891i	$-0.495309\pi$
$739$	0.801245	0.0294743	0.0147371	+	0.999891i	$0.495309\pi$
$740$	0	0
$741$	−12.1263	−0.445471
$742$	0	0
$743$	16.3460	0.599678	0.299839	−	0.953990i	$-0.403067\pi$
$743$	16.3460	0.599678	0.299839	+	0.953990i	$0.403067\pi$
$744$	0	0
$745$	6.69856	0.245416
$746$	0	0
$747$	5.88816	0.215437
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−39.8775	−1.45515	−0.727575	−	0.686028i	$-0.759353\pi$
$751$	−39.8775	−1.45515	−0.727575	+	0.686028i	$0.759353\pi$
$752$	0	0
$753$	6.89489	0.251264
$754$	0	0
$755$	−2.11556	−0.0769930
$756$	0	0
$757$	24.7946	0.901177	0.450588	−	0.892732i	$-0.351214\pi$
$757$	24.7946	0.901177	0.450588	+	0.892732i	$0.351214\pi$
$758$	0	0
$759$	−9.80296	−0.355825
$760$	0	0
$761$	−12.8427	−0.465549	−0.232774	−	0.972531i	$-0.574780\pi$
$761$	−12.8427	−0.465549	−0.232774	+	0.972531i	$0.574780\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	11.8638	0.428937
$766$	0	0
$767$	−57.0412	−2.05964
$768$	0	0
$769$	43.3315	1.56258	0.781288	−	0.624171i	$-0.214563\pi$
$769$	43.3315	1.56258	0.781288	+	0.624171i	$0.214563\pi$
$770$	0	0
$771$	6.86574	0.247264
$772$	0	0
$773$	−44.3550	−1.59534	−0.797669	−	0.603096i	$-0.793934\pi$
$773$	−44.3550	−1.59534	−0.797669	+	0.603096i	$0.793934\pi$
$774$	0	0
$775$	21.4906	0.771965
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−4.82385	−0.172832
$780$	0	0
$781$	−4.33222	−0.155019
$782$	0	0
$783$	−19.6167	−0.701043
$784$	0	0
$785$	3.36195	0.119993
$786$	0	0
$787$	38.8939	1.38642	0.693208	−	0.720737i	$-0.256197\pi$
$787$	38.8939	1.38642	0.693208	+	0.720737i	$0.256197\pi$
$788$	0	0
$789$	−11.3620	−0.404499
$790$	0	0
$791$	0	0
$792$	0	0
$793$	27.3643	0.971736
$794$	0	0
$795$	−2.10654	−0.0747113
$796$	0	0
$797$	18.0342	0.638803	0.319401	−	0.947619i	$-0.396518\pi$
$797$	18.0342	0.638803	0.319401	+	0.947619i	$0.396518\pi$
$798$	0	0
$799$	−29.9418	−1.05927
$800$	0	0
$801$	−45.2175	−1.59768
$802$	0	0
$803$	13.2860	0.468852
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−3.19548	−0.112486
$808$	0	0
$809$	−49.9425	−1.75589	−0.877943	−	0.478765i	$-0.841085\pi$
$809$	−49.9425	−1.75589	−0.877943	+	0.478765i	$0.841085\pi$
$810$	0	0
$811$	−2.43604	−0.0855409	−0.0427705	−	0.999085i	$-0.513618\pi$
$811$	−2.43604	−0.0855409	−0.0427705	+	0.999085i	$0.513618\pi$
$812$	0	0
$813$	0.456451	0.0160084
$814$	0	0
$815$	14.7581	0.516953
$816$	0	0
$817$	−56.7910	−1.98687
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−10.6180	−0.370572	−0.185286	−	0.982685i	$-0.559321\pi$
$821$	−10.6180	−0.370572	−0.185286	+	0.982685i	$0.559321\pi$
$822$	0	0
$823$	50.4401	1.75823	0.879116	−	0.476608i	$-0.158134\pi$
$823$	50.4401	1.75823	0.879116	+	0.476608i	$0.158134\pi$
$824$	0	0
$825$	6.11152	0.212776
$826$	0	0
$827$	16.6086	0.577538	0.288769	−	0.957399i	$-0.406754\pi$
$827$	16.6086	0.577538	0.288769	+	0.957399i	$0.406754\pi$
$828$	0	0
$829$	15.8505	0.550512	0.275256	−	0.961371i	$-0.411237\pi$
$829$	15.8505	0.550512	0.275256	+	0.961371i	$0.411237\pi$
$830$	0	0
$831$	10.3754	0.359919
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−8.93363	−0.309161
$836$	0	0
$837$	−12.7915	−0.442140
$838$	0	0
$839$	20.4364	0.705544	0.352772	−	0.935709i	$-0.385239\pi$
$839$	20.4364	0.705544	0.352772	+	0.935709i	$0.385239\pi$
$840$	0	0
$841$	27.6346	0.952917
$842$	0	0
$843$	8.70280	0.299740
$844$	0	0
$845$	14.3880	0.494961
$846$	0	0
$847$	0	0
$848$	0	0
$849$	0.0797873	0.00273829
$850$	0	0
$851$	−64.7107	−2.21825
$852$	0	0
$853$	38.5432	1.31969	0.659847	−	0.751400i	$-0.270621\pi$
$853$	38.5432	1.31969	0.659847	+	0.751400i	$0.270621\pi$
$854$	0	0
$855$	10.6326	0.363627
$856$	0	0
$857$	−42.1391	−1.43945	−0.719723	−	0.694261i	$-0.755731\pi$
$857$	−42.1391	−1.43945	−0.719723	+	0.694261i	$0.755731\pi$
$858$	0	0
$859$	14.2451	0.486037	0.243018	−	0.970022i	$-0.421862\pi$
$859$	14.2451	0.486037	0.243018	+	0.970022i	$0.421862\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	10.4037	0.354146	0.177073	−	0.984198i	$-0.443337\pi$
$863$	10.4037	0.354146	0.177073	+	0.984198i	$0.443337\pi$
$864$	0	0
$865$	9.84934	0.334888
$866$	0	0
$867$	−5.38191	−0.182779
$868$	0	0
$869$	25.1447	0.852975
$870$	0	0
$871$	−65.6631	−2.22491
$872$	0	0
$873$	6.26253	0.211955
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−2.38200	−0.0804344	−0.0402172	−	0.999191i	$-0.512805\pi$
$877$	−2.38200	−0.0804344	−0.0402172	+	0.999191i	$0.512805\pi$
$878$	0	0
$879$	12.5050	0.421784
$880$	0	0
$881$	38.7986	1.30716	0.653579	−	0.756858i	$-0.273267\pi$
$881$	38.7986	1.30716	0.653579	+	0.756858i	$0.273267\pi$
$882$	0	0
$883$	−35.8165	−1.20532	−0.602660	−	0.797998i	$-0.705892\pi$
$883$	−35.8165	−1.20532	−0.602660	+	0.797998i	$0.705892\pi$
$884$	0	0
$885$	−3.61330	−0.121460
$886$	0	0
$887$	−34.4998	−1.15839	−0.579196	−	0.815189i	$-0.696633\pi$
$887$	−34.4998	−1.15839	−0.579196	+	0.815189i	$0.696633\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	22.4161	0.750967
$892$	0	0
$893$	−26.8345	−0.897981
$894$	0	0
$895$	−5.48459	−0.183330
$896$	0	0
$897$	17.6585	0.589600
$898$	0	0
$899$	36.9299	1.23168
$900$	0	0
$901$	−32.0122	−1.06648
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−14.4402	−0.480009
$906$	0	0
$907$	2.75723	0.0915523	0.0457761	−	0.998952i	$-0.485424\pi$
$907$	2.75723	0.0915523	0.0457761	+	0.998952i	$0.485424\pi$
$908$	0	0
$909$	−17.1919	−0.570221
$910$	0	0
$911$	4.19693	0.139051	0.0695253	−	0.997580i	$-0.477852\pi$
$911$	4.19693	0.139051	0.0695253	+	0.997580i	$0.477852\pi$
$912$	0	0
$913$	−6.53242	−0.216192
$914$	0	0
$915$	1.73341	0.0573047
$916$	0	0
$917$	0	0
$918$	0	0
$919$	10.8130	0.356689	0.178344	−	0.983968i	$-0.442926\pi$
$919$	10.8130	0.356689	0.178344	+	0.983968i	$0.442926\pi$
$920$	0	0
$921$	11.5554	0.380763
$922$	0	0
$923$	7.80381	0.256865
$924$	0	0
$925$	40.3430	1.32647
$926$	0	0
$927$	4.12264	0.135405
$928$	0	0
$929$	28.3572	0.930370	0.465185	−	0.885214i	$-0.345988\pi$
$929$	28.3572	0.930370	0.465185	+	0.885214i	$0.345988\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−7.00264	−0.229256
$934$	0	0
$935$	−13.1619	−0.430441
$936$	0	0
$937$	11.1289	0.363565	0.181783	−	0.983339i	$-0.441813\pi$
$937$	11.1289	0.363565	0.181783	+	0.983339i	$0.441813\pi$
$938$	0	0
$939$	9.72789	0.317458
$940$	0	0
$941$	43.0876	1.40462	0.702308	−	0.711873i	$-0.252153\pi$
$941$	43.0876	1.40462	0.702308	+	0.711873i	$0.252153\pi$
$942$	0	0
$943$	7.02456	0.228751
$944$	0	0
$945$	0	0
$946$	0	0
$947$	18.7408	0.608996	0.304498	−	0.952513i	$-0.401511\pi$
$947$	18.7408	0.608996	0.304498	+	0.952513i	$0.401511\pi$
$948$	0	0
$949$	−23.9326	−0.776885
$950$	0	0
$951$	−10.7783	−0.349511
$952$	0	0
$953$	−13.7933	−0.446807	−0.223404	−	0.974726i	$-0.571717\pi$
$953$	−13.7933	−0.446807	−0.223404	+	0.974726i	$0.571717\pi$
$954$	0	0
$955$	15.1921	0.491605
$956$	0	0
$957$	10.5022	0.339487
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−6.91899	−0.223193
$962$	0	0
$963$	−23.6016	−0.760551
$964$	0	0
$965$	1.10059	0.0354291
$966$	0	0
$967$	−9.71189	−0.312313	−0.156157	−	0.987732i	$-0.549910\pi$
$967$	−9.71189	−0.312313	−0.156157	+	0.987732i	$0.549910\pi$
$968$	0	0
$969$	−11.6732	−0.374997
$970$	0	0
$971$	58.4091	1.87444	0.937218	−	0.348743i	$-0.113391\pi$
$971$	58.4091	1.87444	0.937218	+	0.348743i	$0.113391\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−11.0089	−0.352569
$976$	0	0
$977$	51.0637	1.63367	0.816837	−	0.576869i	$-0.195726\pi$
$977$	51.0637	1.63367	0.816837	+	0.576869i	$0.195726\pi$
$978$	0	0
$979$	50.1650	1.60328
$980$	0	0
$981$	28.4297	0.907690
$982$	0	0
$983$	61.7375	1.96912	0.984560	−	0.175046i	$-0.0560074\pi$
$983$	61.7375	1.96912	0.984560	+	0.175046i	$0.0560074\pi$
$984$	0	0
$985$	10.0965	0.321703
$986$	0	0
$987$	0	0
$988$	0	0
$989$	82.6998	2.62970
$990$	0	0
$991$	−36.0099	−1.14389	−0.571946	−	0.820292i	$-0.693811\pi$
$991$	−36.0099	−1.14389	−0.571946	+	0.820292i	$0.693811\pi$
$992$	0	0
$993$	4.33817	0.137668
$994$	0	0
$995$	1.31831	0.0417931
$996$	0	0
$997$	15.7614	0.499169	0.249585	−	0.968353i	$-0.419706\pi$
$997$	15.7614	0.499169	0.249585	+	0.968353i	$0.419706\pi$
$998$	0	0
$999$	−24.0128	−0.759730

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8036.2.a.r.1.7		15
7.3	odd	6		1148.2.i.e.821.7	yes	30
7.5	odd	6		1148.2.i.e.165.7	✓	30
7.6	odd	2		8036.2.a.q.1.9		15

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1148.2.i.e.165.7	✓	30	7.5	odd	6
1148.2.i.e.821.7	yes	30	7.3	odd	6
8036.2.a.q.1.9		15	7.6	odd	2
8036.2.a.r.1.7		15	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)

\(f(q)\)	\(=\)	\(q-0.449590 q^{3} +0.787803 q^{5} -2.79787 q^{9} +O(q^{10})\)	\(q-0.449590 q^{3} +0.787803 q^{5} -2.79787 q^{9} +3.10400 q^{11} -5.59137 q^{13} -0.354188 q^{15} -5.38244 q^{17} -4.82385 q^{19} +7.02456 q^{23} -4.37937 q^{25} +2.60666 q^{27} -7.52560 q^{29} -4.90724 q^{31} -1.39553 q^{33} -9.21207 q^{37} +2.51382 q^{39} +1.00000 q^{41} +11.7730 q^{43} -2.20417 q^{45} +5.56287 q^{47} +2.41989 q^{51} +5.94752 q^{53} +2.44534 q^{55} +2.16875 q^{57} +10.2017 q^{59} -4.89403 q^{61} -4.40490 q^{65} +11.7436 q^{67} -3.15817 q^{69} -1.39569 q^{71} +4.28028 q^{73} +1.96892 q^{75} +8.10074 q^{79} +7.22168 q^{81} -2.10452 q^{83} -4.24031 q^{85} +3.38343 q^{87} +16.1614 q^{89} +2.20625 q^{93} -3.80024 q^{95} -2.23832 q^{97} -8.68459 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 15 q + q^{3} - 3 q^{5} + 30 q^{9}+O(q^{10}) \) 15 * q + q^3 - 3 * q^5 + 30 * q^9	\( 15 q + q^{3} - 3 q^{5} + 30 q^{9} + 9 q^{11} - 7 q^{13} + 2 q^{15} - 3 q^{17} - 7 q^{19} - q^{23} + 32 q^{25} - 11 q^{27} + 18 q^{29} - 30 q^{31} + 16 q^{33} + 23 q^{37} + 5 q^{39} + 15 q^{41} + 12 q^{43} + 13 q^{45} + 16 q^{47} + 29 q^{51} + 33 q^{53} - 37 q^{55} + 16 q^{57} + 10 q^{59} - q^{61} + 16 q^{65} + 20 q^{67} - 21 q^{69} + 5 q^{71} + 3 q^{73} + 51 q^{75} + 25 q^{79} + 43 q^{81} - 18 q^{83} + 36 q^{85} + 53 q^{87} + 11 q^{89} + 65 q^{93} - 30 q^{95} - 16 q^{97} - 18 q^{99}+O(q^{100}) \) 15 * q + q^3 - 3 * q^5 + 30 * q^9 + 9 * q^11 - 7 * q^13 + 2 * q^15 - 3 * q^17 - 7 * q^19 - q^23 + 32 * q^25 - 11 * q^27 + 18 * q^29 - 30 * q^31 + 16 * q^33 + 23 * q^37 + 5 * q^39 + 15 * q^41 + 12 * q^43 + 13 * q^45 + 16 * q^47 + 29 * q^51 + 33 * q^53 - 37 * q^55 + 16 * q^57 + 10 * q^59 - q^61 + 16 * q^65 + 20 * q^67 - 21 * q^69 + 5 * q^71 + 3 * q^73 + 51 * q^75 + 25 * q^79 + 43 * q^81 - 18 * q^83 + 36 * q^85 + 53 * q^87 + 11 * q^89 + 65 * q^93 - 30 * q^95 - 16 * q^97 - 18 * q^99

Introduction

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