LMFDB - Embedded newform 8036.2.a.q.1.4

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			1.4
Root			$1.93225$ 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)

$f(q)$	$=$	$q-1.93225 q^{3} -0.475465 q^{5} +0.733587 q^{9} +O(q^{10})$	$q-1.93225 q^{3} -0.475465 q^{5} +0.733587 q^{9} +0.608408 q^{11} +0.258249 q^{13} +0.918718 q^{15} +6.67585 q^{17} -8.55857 q^{19} -0.540802 q^{23} -4.77393 q^{25} +4.37927 q^{27} +9.17655 q^{29} -3.19512 q^{31} -1.17560 q^{33} +2.11667 q^{37} -0.499002 q^{39} -1.00000 q^{41} -9.45997 q^{43} -0.348795 q^{45} -5.21836 q^{47} -12.8994 q^{51} +11.1314 q^{53} -0.289277 q^{55} +16.5373 q^{57} -6.80041 q^{59} +1.40208 q^{61} -0.122789 q^{65} +15.9998 q^{67} +1.04496 q^{69} +3.72154 q^{71} -9.31128 q^{73} +9.22443 q^{75} +5.46345 q^{79} -10.6626 q^{81} +0.971073 q^{83} -3.17414 q^{85} -17.7314 q^{87} +9.78989 q^{89} +6.17377 q^{93} +4.06930 q^{95} +3.26994 q^{97} +0.446320 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−1.93225	−1.11558	−0.557792	−	0.829981i	$-0.688351\pi$
$3$	−1.93225	−1.11558	−0.557792	+	0.829981i	$0.688351\pi$
$4$	0	0
$5$	−0.475465	−0.212635	−0.106317	−	0.994332i	$-0.533906\pi$
$5$	−0.475465	−0.212635	−0.106317	+	0.994332i	$0.533906\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	0.733587	0.244529
$10$	0	0
$11$	0.608408	0.183442	0.0917210	−	0.995785i	$-0.470763\pi$
$11$	0.608408	0.183442	0.0917210	+	0.995785i	$0.470763\pi$
$12$	0	0
$13$	0.258249	0.0716255	0.0358127	−	0.999359i	$-0.488598\pi$
$13$	0.258249	0.0716255	0.0358127	+	0.999359i	$0.488598\pi$
$14$	0	0
$15$	0.918718	0.237212
$16$	0	0
$17$	6.67585	1.61913	0.809566	−	0.587029i	$-0.199703\pi$
$17$	6.67585	1.61913	0.809566	+	0.587029i	$0.199703\pi$
$18$	0	0
$19$	−8.55857	−1.96347	−0.981735	−	0.190252i	$-0.939070\pi$
$19$	−8.55857	−1.96347	−0.981735	+	0.190252i	$0.939070\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−0.540802	−0.112765	−0.0563825	−	0.998409i	$-0.517957\pi$
$23$	−0.540802	−0.112765	−0.0563825	+	0.998409i	$0.517957\pi$
$24$	0	0
$25$	−4.77393	−0.954787
$26$	0	0
$27$	4.37927	0.842792
$28$	0	0
$29$	9.17655	1.70404	0.852021	−	0.523508i	$-0.175377\pi$
$29$	9.17655	1.70404	0.852021	+	0.523508i	$0.175377\pi$
$30$	0	0
$31$	−3.19512	−0.573861	−0.286930	−	0.957951i	$-0.592635\pi$
$31$	−3.19512	−0.573861	−0.286930	+	0.957951i	$0.592635\pi$
$32$	0	0
$33$	−1.17560	−0.204645
$34$	0	0
$35$	0	0
$36$	0	0
$37$	2.11667	0.347979	0.173989	−	0.984748i	$-0.444334\pi$
$37$	2.11667	0.347979	0.173989	+	0.984748i	$0.444334\pi$
$38$	0	0
$39$	−0.499002	−0.0799043
$40$	0	0
$41$	−1.00000	−0.156174
$41$	−1.00000	−0.156174
$42$	0	0
$43$	−9.45997	−1.44263	−0.721316	−	0.692606i	$-0.756462\pi$
$43$	−9.45997	−1.44263	−0.721316	+	0.692606i	$0.756462\pi$
$44$	0	0
$45$	−0.348795	−0.0519953
$46$	0	0
$47$	−5.21836	−0.761176	−0.380588	−	0.924745i	$-0.624278\pi$
$47$	−5.21836	−0.761176	−0.380588	+	0.924745i	$0.624278\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−12.8994	−1.80628
$52$	0	0
$53$	11.1314	1.52901	0.764507	−	0.644615i	$-0.222982\pi$
$53$	11.1314	1.52901	0.764507	+	0.644615i	$0.222982\pi$
$54$	0	0
$55$	−0.289277	−0.0390061
$56$	0	0
$57$	16.5373	2.19042
$58$	0	0
$59$	−6.80041	−0.885338	−0.442669	−	0.896685i	$-0.645968\pi$
$59$	−6.80041	−0.885338	−0.442669	+	0.896685i	$0.645968\pi$
$60$	0	0
$61$	1.40208	0.179517	0.0897587	−	0.995964i	$-0.471390\pi$
$61$	1.40208	0.179517	0.0897587	+	0.995964i	$0.471390\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−0.122789	−0.0152301
$66$	0	0
$67$	15.9998	1.95468	0.977340	−	0.211674i	$-0.0678915\pi$
$67$	15.9998	1.95468	0.977340	+	0.211674i	$0.0678915\pi$
$68$	0	0
$69$	1.04496	0.125799
$70$	0	0
$71$	3.72154	0.441665	0.220833	−	0.975312i	$-0.429123\pi$
$71$	3.72154	0.441665	0.220833	+	0.975312i	$0.429123\pi$
$72$	0	0
$73$	−9.31128	−1.08980	−0.544902	−	0.838500i	$-0.683433\pi$
$73$	−9.31128	−1.08980	−0.544902	+	0.838500i	$0.683433\pi$
$74$	0	0
$75$	9.22443	1.06515
$76$	0	0
$77$	0	0
$78$	0	0
$79$	5.46345	0.614686	0.307343	−	0.951599i	$-0.400560\pi$
$79$	5.46345	0.614686	0.307343	+	0.951599i	$0.400560\pi$
$80$	0	0
$81$	−10.6626	−1.18473
$82$	0	0
$83$	0.971073	0.106589	0.0532945	−	0.998579i	$-0.483028\pi$
$83$	0.971073	0.106589	0.0532945	+	0.998579i	$0.483028\pi$
$84$	0	0
$85$	−3.17414	−0.344284
$86$	0	0
$87$	−17.7314	−1.90100
$88$	0	0
$89$	9.78989	1.03773	0.518863	−	0.854857i	$-0.326355\pi$
$89$	9.78989	1.03773	0.518863	+	0.854857i	$0.326355\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	6.17377	0.640190
$94$	0	0
$95$	4.06930	0.417502
$96$	0	0
$97$	3.26994	0.332012	0.166006	−	0.986125i	$-0.446913\pi$
$97$	3.26994	0.332012	0.166006	+	0.986125i	$0.446913\pi$
$98$	0	0
$99$	0.446320	0.0448569
$100$	0	0
$101$	−15.3557	−1.52795	−0.763973	−	0.645248i	$-0.776754\pi$
$101$	−15.3557	−1.52795	−0.763973	+	0.645248i	$0.776754\pi$
$102$	0	0
$103$	−15.9051	−1.56717	−0.783586	−	0.621283i	$-0.786612\pi$
$103$	−15.9051	−1.56717	−0.783586	+	0.621283i	$0.786612\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−9.33023	−0.901988	−0.450994	−	0.892527i	$-0.648930\pi$
$107$	−9.33023	−0.901988	−0.450994	+	0.892527i	$0.648930\pi$
$108$	0	0
$109$	−13.1284	−1.25747	−0.628736	−	0.777619i	$-0.716428\pi$
$109$	−13.1284	−1.25747	−0.628736	+	0.777619i	$0.716428\pi$
$110$	0	0
$111$	−4.08994	−0.388200
$112$	0	0
$113$	−8.66144	−0.814800	−0.407400	−	0.913250i	$-0.633565\pi$
$113$	−8.66144	−0.814800	−0.407400	+	0.913250i	$0.633565\pi$
$114$	0	0
$115$	0.257133	0.0239778
$116$	0	0
$117$	0.189448	0.0175145
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−10.6298	−0.966349
$122$	0	0
$123$	1.93225	0.174225
$124$	0	0
$125$	4.64717	0.415655
$126$	0	0
$127$	15.3874	1.36541	0.682705	−	0.730694i	$-0.260803\pi$
$127$	15.3874	1.36541	0.682705	+	0.730694i	$0.260803\pi$
$128$	0	0
$129$	18.2790	1.60938
$130$	0	0
$131$	−7.68476	−0.671421	−0.335710	−	0.941965i	$-0.608976\pi$
$131$	−7.68476	−0.671421	−0.335710	+	0.941965i	$0.608976\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−2.08219	−0.179207
$136$	0	0
$137$	19.5241	1.66805	0.834026	−	0.551725i	$-0.186030\pi$
$137$	19.5241	1.66805	0.834026	+	0.551725i	$0.186030\pi$
$138$	0	0
$139$	−12.3761	−1.04972	−0.524862	−	0.851188i	$-0.675883\pi$
$139$	−12.3761	−1.04972	−0.524862	+	0.851188i	$0.675883\pi$
$140$	0	0
$141$	10.0832	0.849157
$142$	0	0
$143$	0.157121	0.0131391
$144$	0	0
$145$	−4.36313	−0.362338
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−5.81485	−0.476371	−0.238186	−	0.971220i	$-0.576553\pi$
$149$	−5.81485	−0.476371	−0.238186	+	0.971220i	$0.576553\pi$
$150$	0	0
$151$	5.96305	0.485266	0.242633	−	0.970118i	$-0.421989\pi$
$151$	5.96305	0.485266	0.242633	+	0.970118i	$0.421989\pi$
$152$	0	0
$153$	4.89732	0.395925
$154$	0	0
$155$	1.51917	0.122023
$156$	0	0
$157$	−15.8012	−1.26108	−0.630538	−	0.776159i	$-0.717166\pi$
$157$	−15.8012	−1.26108	−0.630538	+	0.776159i	$0.717166\pi$
$158$	0	0
$159$	−21.5086	−1.70575
$160$	0	0
$161$	0	0
$162$	0	0
$163$	17.1423	1.34269	0.671345	−	0.741145i	$-0.265717\pi$
$163$	17.1423	1.34269	0.671345	+	0.741145i	$0.265717\pi$
$164$	0	0
$165$	0.558956	0.0435146
$166$	0	0
$167$	14.3673	1.11178	0.555888	−	0.831257i	$-0.312378\pi$
$167$	14.3673	1.11178	0.555888	+	0.831257i	$0.312378\pi$
$168$	0	0
$169$	−12.9333	−0.994870
$170$	0	0
$171$	−6.27845	−0.480125
$172$	0	0
$173$	25.2690	1.92116	0.960582	−	0.277996i	$-0.0896701\pi$
$173$	25.2690	1.92116	0.960582	+	0.277996i	$0.0896701\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	13.1401	0.987670
$178$	0	0
$179$	20.2626	1.51450	0.757250	−	0.653125i	$-0.226543\pi$
$179$	20.2626	1.51450	0.757250	+	0.653125i	$0.226543\pi$
$180$	0	0
$181$	8.26666	0.614456	0.307228	−	0.951636i	$-0.400599\pi$
$181$	8.26666	0.614456	0.307228	+	0.951636i	$0.400599\pi$
$182$	0	0
$183$	−2.70916	−0.200267
$184$	0	0
$185$	−1.00640	−0.0739923
$186$	0	0
$187$	4.06164	0.297017
$188$	0	0
$189$	0	0
$190$	0	0
$191$	21.2773	1.53957	0.769786	−	0.638302i	$-0.220363\pi$
$191$	21.2773	1.53957	0.769786	+	0.638302i	$0.220363\pi$
$192$	0	0
$193$	−10.1778	−0.732615	−0.366307	−	0.930494i	$-0.619378\pi$
$193$	−10.1778	−0.732615	−0.366307	+	0.930494i	$0.619378\pi$
$194$	0	0
$195$	0.237258	0.0169904
$196$	0	0
$197$	15.0683	1.07357	0.536787	−	0.843718i	$-0.319638\pi$
$197$	15.0683	1.07357	0.536787	+	0.843718i	$0.319638\pi$
$198$	0	0
$199$	4.57936	0.324622	0.162311	−	0.986740i	$-0.448105\pi$
$199$	4.57936	0.324622	0.162311	+	0.986740i	$0.448105\pi$
$200$	0	0
$201$	−30.9155	−2.18061
$202$	0	0
$203$	0	0
$204$	0	0
$205$	0.475465	0.0332080
$206$	0	0
$207$	−0.396725	−0.0275743
$208$	0	0
$209$	−5.20711	−0.360183
$210$	0	0
$211$	−25.2177	−1.73606	−0.868028	−	0.496515i	$-0.834613\pi$
$211$	−25.2177	−1.73606	−0.868028	+	0.496515i	$0.834613\pi$
$212$	0	0
$213$	−7.19094	−0.492715
$214$	0	0
$215$	4.49789	0.306753
$216$	0	0
$217$	0	0
$218$	0	0
$219$	17.9917	1.21577
$220$	0	0
$221$	1.72403	0.115971
$222$	0	0
$223$	−9.02063	−0.604066	−0.302033	−	0.953298i	$-0.597665\pi$
$223$	−9.02063	−0.604066	−0.302033	+	0.953298i	$0.597665\pi$
$224$	0	0
$225$	−3.50209	−0.233473
$226$	0	0
$227$	17.7610	1.17884	0.589421	−	0.807826i	$-0.299356\pi$
$227$	17.7610	1.17884	0.589421	+	0.807826i	$0.299356\pi$
$228$	0	0
$229$	21.5866	1.42648	0.713242	−	0.700918i	$-0.247226\pi$
$229$	21.5866	1.42648	0.713242	+	0.700918i	$0.247226\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−6.03999	−0.395693	−0.197846	−	0.980233i	$-0.563395\pi$
$233$	−6.03999	−0.395693	−0.197846	+	0.980233i	$0.563395\pi$
$234$	0	0
$235$	2.48115	0.161852
$236$	0	0
$237$	−10.5567	−0.685734
$238$	0	0
$239$	−7.84860	−0.507683	−0.253842	−	0.967246i	$-0.581694\pi$
$239$	−7.84860	−0.507683	−0.253842	+	0.967246i	$0.581694\pi$
$240$	0	0
$241$	−26.3385	−1.69661	−0.848306	−	0.529506i	$-0.822377\pi$
$241$	−26.3385	−1.69661	−0.848306	+	0.529506i	$0.822377\pi$
$242$	0	0
$243$	7.46500	0.478880
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−2.21024	−0.140634
$248$	0	0
$249$	−1.87635	−0.118909
$250$	0	0
$251$	−11.7507	−0.741699	−0.370849	−	0.928693i	$-0.620933\pi$
$251$	−11.7507	−0.741699	−0.370849	+	0.928693i	$0.620933\pi$
$252$	0	0
$253$	−0.329029	−0.0206858
$254$	0	0
$255$	6.13322	0.384077
$256$	0	0
$257$	−19.0458	−1.18805	−0.594023	−	0.804448i	$-0.702461\pi$
$257$	−19.0458	−1.18805	−0.594023	+	0.804448i	$0.702461\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	6.73179	0.416688
$262$	0	0
$263$	−11.7032	−0.721651	−0.360825	−	0.932633i	$-0.617505\pi$
$263$	−11.7032	−0.721651	−0.360825	+	0.932633i	$0.617505\pi$
$264$	0	0
$265$	−5.29259	−0.325121
$266$	0	0
$267$	−18.9165	−1.15767
$268$	0	0
$269$	25.1797	1.53524	0.767618	−	0.640908i	$-0.221442\pi$
$269$	25.1797	1.53524	0.767618	+	0.640908i	$0.221442\pi$
$270$	0	0
$271$	13.2398	0.804262	0.402131	−	0.915582i	$-0.368270\pi$
$271$	13.2398	0.804262	0.402131	+	0.915582i	$0.368270\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−2.90450	−0.175148
$276$	0	0
$277$	2.96026	0.177865	0.0889323	−	0.996038i	$-0.471655\pi$
$277$	2.96026	0.177865	0.0889323	+	0.996038i	$0.471655\pi$
$278$	0	0
$279$	−2.34390	−0.140326
$280$	0	0
$281$	11.8626	0.707666	0.353833	−	0.935309i	$-0.384878\pi$
$281$	11.8626	0.707666	0.353833	+	0.935309i	$0.384878\pi$
$282$	0	0
$283$	−24.4471	−1.45323	−0.726615	−	0.687044i	$-0.758908\pi$
$283$	−24.4471	−1.45323	−0.726615	+	0.687044i	$0.758908\pi$
$284$	0	0
$285$	−7.86291	−0.465759
$286$	0	0
$287$	0	0
$288$	0	0
$289$	27.5670	1.62159
$290$	0	0
$291$	−6.31834	−0.370388
$292$	0	0
$293$	31.9974	1.86931	0.934653	−	0.355561i	$-0.115710\pi$
$293$	31.9974	1.86931	0.934653	+	0.355561i	$0.115710\pi$
$294$	0	0
$295$	3.23336	0.188254
$296$	0	0
$297$	2.66439	0.154603
$298$	0	0
$299$	−0.139662	−0.00807685
$300$	0	0
$301$	0	0
$302$	0	0
$303$	29.6710	1.70455
$304$	0	0
$305$	−0.666639	−0.0381716
$306$	0	0
$307$	−1.37931	−0.0787211	−0.0393606	−	0.999225i	$-0.512532\pi$
$307$	−1.37931	−0.0787211	−0.0393606	+	0.999225i	$0.512532\pi$
$308$	0	0
$309$	30.7326	1.74831
$310$	0	0
$311$	6.57209	0.372669	0.186335	−	0.982486i	$-0.440339\pi$
$311$	6.57209	0.372669	0.186335	+	0.982486i	$0.440339\pi$
$312$	0	0
$313$	10.2077	0.576976	0.288488	−	0.957484i	$-0.406847\pi$
$313$	10.2077	0.576976	0.288488	+	0.957484i	$0.406847\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−11.3693	−0.638565	−0.319282	−	0.947660i	$-0.603442\pi$
$317$	−11.3693	−0.638565	−0.319282	+	0.947660i	$0.603442\pi$
$318$	0	0
$319$	5.58309	0.312593
$320$	0	0
$321$	18.0283	1.00624
$322$	0	0
$323$	−57.1357	−3.17912
$324$	0	0
$325$	−1.23286	−0.0683870
$326$	0	0
$327$	25.3673	1.40282
$328$	0	0
$329$	0	0
$330$	0	0
$331$	19.3419	1.06313	0.531564	−	0.847018i	$-0.321605\pi$
$331$	19.3419	1.06313	0.531564	+	0.847018i	$0.321605\pi$
$332$	0	0
$333$	1.55276	0.0850909
$334$	0	0
$335$	−7.60733	−0.415633
$336$	0	0
$337$	13.0087	0.708629	0.354315	−	0.935126i	$-0.384714\pi$
$337$	13.0087	0.708629	0.354315	+	0.935126i	$0.384714\pi$
$338$	0	0
$339$	16.7361	0.908978
$340$	0	0
$341$	−1.94394	−0.105270
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−0.496845	−0.0267492
$346$	0	0
$347$	9.25837	0.497015	0.248508	−	0.968630i	$-0.420060\pi$
$347$	9.25837	0.497015	0.248508	+	0.968630i	$0.420060\pi$
$348$	0	0
$349$	3.45465	0.184923	0.0924615	−	0.995716i	$-0.470527\pi$
$349$	3.45465	0.184923	0.0924615	+	0.995716i	$0.470527\pi$
$350$	0	0
$351$	1.13094	0.0603653
$352$	0	0
$353$	7.57050	0.402937	0.201468	−	0.979495i	$-0.435429\pi$
$353$	7.57050	0.402937	0.201468	+	0.979495i	$0.435429\pi$
$354$	0	0
$355$	−1.76946	−0.0939133
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−15.2984	−0.807419	−0.403710	−	0.914887i	$-0.632279\pi$
$359$	−15.2984	−0.807419	−0.403710	+	0.914887i	$0.632279\pi$
$360$	0	0
$361$	54.2491	2.85522
$362$	0	0
$363$	20.5395	1.07804
$364$	0	0
$365$	4.42719	0.231730
$366$	0	0
$367$	−11.1193	−0.580421	−0.290211	−	0.956963i	$-0.593725\pi$
$367$	−11.1193	−0.580421	−0.290211	+	0.956963i	$0.593725\pi$
$368$	0	0
$369$	−0.733587	−0.0381890
$370$	0	0
$371$	0	0
$372$	0	0
$373$	30.7948	1.59449	0.797247	−	0.603653i	$-0.206289\pi$
$373$	30.7948	1.59449	0.797247	+	0.603653i	$0.206289\pi$
$374$	0	0
$375$	−8.97949	−0.463699
$376$	0	0
$377$	2.36984	0.122053
$378$	0	0
$379$	15.7083	0.806881	0.403441	−	0.915006i	$-0.367814\pi$
$379$	15.7083	0.806881	0.403441	+	0.915006i	$0.367814\pi$
$380$	0	0
$381$	−29.7323	−1.52323
$382$	0	0
$383$	0.295229	0.0150855	0.00754274	−	0.999972i	$-0.497599\pi$
$383$	0.295229	0.0150855	0.00754274	+	0.999972i	$0.497599\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−6.93971	−0.352765
$388$	0	0
$389$	6.05446	0.306974	0.153487	−	0.988151i	$-0.450950\pi$
$389$	6.05446	0.306974	0.153487	+	0.988151i	$0.450950\pi$
$390$	0	0
$391$	−3.61031	−0.182581
$392$	0	0
$393$	14.8489	0.749027
$394$	0	0
$395$	−2.59768	−0.130703
$396$	0	0
$397$	23.1683	1.16278	0.581392	−	0.813623i	$-0.302508\pi$
$397$	23.1683	1.16278	0.581392	+	0.813623i	$0.302508\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	35.4197	1.76878	0.884388	−	0.466752i	$-0.154576\pi$
$401$	35.4197	1.76878	0.884388	+	0.466752i	$0.154576\pi$
$402$	0	0
$403$	−0.825137	−0.0411030
$404$	0	0
$405$	5.06970	0.251916
$406$	0	0
$407$	1.28780	0.0638339
$408$	0	0
$409$	−20.5932	−1.01827	−0.509135	−	0.860687i	$-0.670035\pi$
$409$	−20.5932	−1.01827	−0.509135	+	0.860687i	$0.670035\pi$
$410$	0	0
$411$	−37.7253	−1.86085
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−0.461712	−0.0226645
$416$	0	0
$417$	23.9136	1.17105
$418$	0	0
$419$	37.3854	1.82640	0.913198	−	0.407517i	$-0.133605\pi$
$419$	37.3854	1.82640	0.913198	+	0.407517i	$0.133605\pi$
$420$	0	0
$421$	9.03447	0.440313	0.220156	−	0.975465i	$-0.429343\pi$
$421$	9.03447	0.440313	0.220156	+	0.975465i	$0.429343\pi$
$422$	0	0
$423$	−3.82812	−0.186130
$424$	0	0
$425$	−31.8701	−1.54593
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−0.303597	−0.0146578
$430$	0	0
$431$	5.86857	0.282679	0.141340	−	0.989961i	$-0.454859\pi$
$431$	5.86857	0.282679	0.141340	+	0.989961i	$0.454859\pi$
$432$	0	0
$433$	27.3406	1.31391	0.656953	−	0.753931i	$-0.271845\pi$
$433$	27.3406	1.31391	0.656953	+	0.753931i	$0.271845\pi$
$434$	0	0
$435$	8.43066	0.404219
$436$	0	0
$437$	4.62849	0.221411
$438$	0	0
$439$	30.0278	1.43315	0.716574	−	0.697512i	$-0.245709\pi$
$439$	30.0278	1.43315	0.716574	+	0.697512i	$0.245709\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−7.60397	−0.361276	−0.180638	−	0.983550i	$-0.557816\pi$
$443$	−7.60397	−0.361276	−0.180638	+	0.983550i	$0.557816\pi$
$444$	0	0
$445$	−4.65476	−0.220657
$446$	0	0
$447$	11.2357	0.531432
$448$	0	0
$449$	0.589677	0.0278286	0.0139143	−	0.999903i	$-0.495571\pi$
$449$	0.589677	0.0278286	0.0139143	+	0.999903i	$0.495571\pi$
$450$	0	0
$451$	−0.608408	−0.0286488
$452$	0	0
$453$	−11.5221	−0.541356
$454$	0	0
$455$	0	0
$456$	0	0
$457$	1.29570	0.0606101	0.0303051	−	0.999541i	$-0.490352\pi$
$457$	1.29570	0.0606101	0.0303051	+	0.999541i	$0.490352\pi$
$458$	0	0
$459$	29.2354	1.36459
$460$	0	0
$461$	22.0668	1.02775	0.513876	−	0.857864i	$-0.328209\pi$
$461$	22.0668	1.02775	0.513876	+	0.857864i	$0.328209\pi$
$462$	0	0
$463$	−20.1079	−0.934493	−0.467246	−	0.884127i	$-0.654754\pi$
$463$	−20.1079	−0.934493	−0.467246	+	0.884127i	$0.654754\pi$
$464$	0	0
$465$	−2.93541	−0.136127
$466$	0	0
$467$	29.8752	1.38246	0.691230	−	0.722635i	$-0.257069\pi$
$467$	29.8752	1.38246	0.691230	+	0.722635i	$0.257069\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	30.5319	1.40684
$472$	0	0
$473$	−5.75552	−0.264639
$474$	0	0
$475$	40.8580	1.87470
$476$	0	0
$477$	8.16585	0.373888
$478$	0	0
$479$	−38.5445	−1.76115	−0.880573	−	0.473911i	$-0.842842\pi$
$479$	−38.5445	−1.76115	−0.880573	+	0.473911i	$0.842842\pi$
$480$	0	0
$481$	0.546629	0.0249241
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−1.55474	−0.0705973
$486$	0	0
$487$	−18.7629	−0.850226	−0.425113	−	0.905140i	$-0.639766\pi$
$487$	−18.7629	−0.850226	−0.425113	+	0.905140i	$0.639766\pi$
$488$	0	0
$489$	−33.1232	−1.49788
$490$	0	0
$491$	−1.61032	−0.0726725	−0.0363363	−	0.999340i	$-0.511569\pi$
$491$	−1.61032	−0.0726725	−0.0363363	+	0.999340i	$0.511569\pi$
$492$	0	0
$493$	61.2613	2.75907
$494$	0	0
$495$	−0.212210	−0.00953813
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−4.49401	−0.201179	−0.100590	−	0.994928i	$-0.532073\pi$
$499$	−4.49401	−0.201179	−0.100590	+	0.994928i	$0.532073\pi$
$500$	0	0
$501$	−27.7613	−1.24028
$502$	0	0
$503$	−0.671350	−0.0299340	−0.0149670	−	0.999888i	$-0.504764\pi$
$503$	−0.671350	−0.0299340	−0.0149670	+	0.999888i	$0.504764\pi$
$504$	0	0
$505$	7.30109	0.324894
$506$	0	0
$507$	24.9904	1.10986
$508$	0	0
$509$	22.5947	1.00149	0.500745	−	0.865595i	$-0.333059\pi$
$509$	22.5947	1.00149	0.500745	+	0.865595i	$0.333059\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−37.4803	−1.65480
$514$	0	0
$515$	7.56231	0.333235
$516$	0	0
$517$	−3.17490	−0.139632
$518$	0	0
$519$	−48.8260	−2.14322
$520$	0	0
$521$	−1.35329	−0.0592888	−0.0296444	−	0.999561i	$-0.509437\pi$
$521$	−1.35329	−0.0592888	−0.0296444	+	0.999561i	$0.509437\pi$
$522$	0	0
$523$	−16.5312	−0.722858	−0.361429	−	0.932400i	$-0.617711\pi$
$523$	−16.5312	−0.722858	−0.361429	+	0.932400i	$0.617711\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−21.3301	−0.929156
$528$	0	0
$529$	−22.7075	−0.987284
$530$	0	0
$531$	−4.98869	−0.216491
$532$	0	0
$533$	−0.258249	−0.0111860
$534$	0	0
$535$	4.43620	0.191794
$536$	0	0
$537$	−39.1524	−1.68955
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−6.33057	−0.272172	−0.136086	−	0.990697i	$-0.543452\pi$
$541$	−6.33057	−0.272172	−0.136086	+	0.990697i	$0.543452\pi$
$542$	0	0
$543$	−15.9732	−0.685477
$544$	0	0
$545$	6.24210	0.267382
$546$	0	0
$547$	28.5272	1.21973	0.609867	−	0.792503i	$-0.291223\pi$
$547$	28.5272	1.21973	0.609867	+	0.792503i	$0.291223\pi$
$548$	0	0
$549$	1.02854	0.0438972
$550$	0	0
$551$	−78.5381	−3.34584
$552$	0	0
$553$	0	0
$554$	0	0
$555$	1.94462	0.0825447
$556$	0	0
$557$	−39.1109	−1.65718	−0.828591	−	0.559855i	$-0.810857\pi$
$557$	−39.1109	−1.65718	−0.828591	+	0.559855i	$0.810857\pi$
$558$	0	0
$559$	−2.44303	−0.103329
$560$	0	0
$561$	−7.84811	−0.331347
$562$	0	0
$563$	29.1984	1.23057	0.615284	−	0.788306i	$-0.289042\pi$
$563$	29.1984	1.23057	0.615284	+	0.788306i	$0.289042\pi$
$564$	0	0
$565$	4.11822	0.173255
$566$	0	0
$567$	0	0
$568$	0	0
$569$	13.3981	0.561679	0.280840	−	0.959755i	$-0.409387\pi$
$569$	13.3981	0.561679	0.280840	+	0.959755i	$0.409387\pi$
$570$	0	0
$571$	−2.05488	−0.0859941	−0.0429970	−	0.999075i	$-0.513691\pi$
$571$	−2.05488	−0.0859941	−0.0429970	+	0.999075i	$0.513691\pi$
$572$	0	0
$573$	−41.1131	−1.71752
$574$	0	0
$575$	2.58175	0.107667
$576$	0	0
$577$	20.0423	0.834372	0.417186	−	0.908821i	$-0.363016\pi$
$577$	20.0423	0.834372	0.417186	+	0.908821i	$0.363016\pi$
$578$	0	0
$579$	19.6661	0.817294
$580$	0	0
$581$	0	0
$582$	0	0
$583$	6.77243	0.280486
$584$	0	0
$585$	−0.0900761	−0.00372419
$586$	0	0
$587$	17.2770	0.713098	0.356549	−	0.934277i	$-0.383953\pi$
$587$	17.2770	0.713098	0.356549	+	0.934277i	$0.383953\pi$
$588$	0	0
$589$	27.3457	1.12676
$590$	0	0
$591$	−29.1158	−1.19766
$592$	0	0
$593$	−4.43785	−0.182241	−0.0911203	−	0.995840i	$-0.529045\pi$
$593$	−4.43785	−0.182241	−0.0911203	+	0.995840i	$0.529045\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−8.84846	−0.362144
$598$	0	0
$599$	36.5564	1.49366	0.746828	−	0.665017i	$-0.231576\pi$
$599$	36.5564	1.49366	0.746828	+	0.665017i	$0.231576\pi$
$600$	0	0
$601$	12.7836	0.521453	0.260726	−	0.965413i	$-0.416038\pi$
$601$	12.7836	0.521453	0.260726	+	0.965413i	$0.416038\pi$
$602$	0	0
$603$	11.7372	0.477976
$604$	0	0
$605$	5.05412	0.205479
$606$	0	0
$607$	−15.7778	−0.640401	−0.320200	−	0.947350i	$-0.603750\pi$
$607$	−15.7778	−0.640401	−0.320200	+	0.947350i	$0.603750\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−1.34764	−0.0545196
$612$	0	0
$613$	42.3571	1.71079	0.855394	−	0.517978i	$-0.173315\pi$
$613$	42.3571	1.71079	0.855394	+	0.517978i	$0.173315\pi$
$614$	0	0
$615$	−0.918718	−0.0370463
$616$	0	0
$617$	−6.74340	−0.271479	−0.135739	−	0.990745i	$-0.543341\pi$
$617$	−6.74340	−0.271479	−0.135739	+	0.990745i	$0.543341\pi$
$618$	0	0
$619$	2.13188	0.0856874	0.0428437	−	0.999082i	$-0.486358\pi$
$619$	2.13188	0.0856874	0.0428437	+	0.999082i	$0.486358\pi$
$620$	0	0
$621$	−2.36832	−0.0950374
$622$	0	0
$623$	0	0
$624$	0	0
$625$	21.6601	0.866404
$626$	0	0
$627$	10.0614	0.401815
$628$	0	0
$629$	14.1306	0.563424
$630$	0	0
$631$	28.8631	1.14902	0.574511	−	0.818497i	$-0.305192\pi$
$631$	28.8631	1.14902	0.574511	+	0.818497i	$0.305192\pi$
$632$	0	0
$633$	48.7268	1.93672
$634$	0	0
$635$	−7.31617	−0.290333
$636$	0	0
$637$	0	0
$638$	0	0
$639$	2.73007	0.108000
$640$	0	0
$641$	−19.1831	−0.757685	−0.378843	−	0.925461i	$-0.623678\pi$
$641$	−19.1831	−0.757685	−0.378843	+	0.925461i	$0.623678\pi$
$642$	0	0
$643$	32.3924	1.27743	0.638716	−	0.769442i	$-0.279466\pi$
$643$	32.3924	1.27743	0.638716	+	0.769442i	$0.279466\pi$
$644$	0	0
$645$	−8.69104	−0.342209
$646$	0	0
$647$	35.4947	1.39544	0.697719	−	0.716371i	$-0.254198\pi$
$647$	35.4947	1.39544	0.697719	+	0.716371i	$0.254198\pi$
$648$	0	0
$649$	−4.13743	−0.162408
$650$	0	0
$651$	0	0
$652$	0	0
$653$	39.0785	1.52926	0.764630	−	0.644469i	$-0.222922\pi$
$653$	39.0785	1.52926	0.764630	+	0.644469i	$0.222922\pi$
$654$	0	0
$655$	3.65384	0.142767
$656$	0	0
$657$	−6.83063	−0.266488
$658$	0	0
$659$	19.6445	0.765242	0.382621	−	0.923905i	$-0.375022\pi$
$659$	19.6445	0.765242	0.382621	+	0.923905i	$0.375022\pi$
$660$	0	0
$661$	11.5191	0.448041	0.224020	−	0.974584i	$-0.428082\pi$
$661$	11.5191	0.448041	0.224020	+	0.974584i	$0.428082\pi$
$662$	0	0
$663$	−3.33126	−0.129376
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−4.96269	−0.192156
$668$	0	0
$669$	17.4301	0.673887
$670$	0	0
$671$	0.853035	0.0329310
$672$	0	0
$673$	16.9330	0.652719	0.326359	−	0.945246i	$-0.394178\pi$
$673$	16.9330	0.652719	0.326359	+	0.945246i	$0.394178\pi$
$674$	0	0
$675$	−20.9064	−0.804686
$676$	0	0
$677$	−26.3909	−1.01429	−0.507143	−	0.861862i	$-0.669298\pi$
$677$	−26.3909	−1.01429	−0.507143	+	0.861862i	$0.669298\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−34.3188	−1.31510
$682$	0	0
$683$	−26.8163	−1.02610	−0.513049	−	0.858359i	$-0.671484\pi$
$683$	−26.8163	−1.02610	−0.513049	+	0.858359i	$0.671484\pi$
$684$	0	0
$685$	−9.28301	−0.354686
$686$	0	0
$687$	−41.7107	−1.59136
$688$	0	0
$689$	2.87467	0.109516
$690$	0	0
$691$	30.1588	1.14730	0.573648	−	0.819102i	$-0.305528\pi$
$691$	30.1588	1.14730	0.573648	+	0.819102i	$0.305528\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	5.88439	0.223207
$696$	0	0
$697$	−6.67585	−0.252866
$698$	0	0
$699$	11.6708	0.441429
$700$	0	0
$701$	−18.1146	−0.684178	−0.342089	−	0.939668i	$-0.611134\pi$
$701$	−18.1146	−0.684178	−0.342089	+	0.939668i	$0.611134\pi$
$702$	0	0
$703$	−18.1157	−0.683246
$704$	0	0
$705$	−4.79420	−0.180560
$706$	0	0
$707$	0	0
$708$	0	0
$709$	37.7306	1.41700	0.708501	−	0.705709i	$-0.249372\pi$
$709$	37.7306	1.41700	0.708501	+	0.705709i	$0.249372\pi$
$710$	0	0
$711$	4.00791	0.150308
$712$	0	0
$713$	1.72793	0.0647114
$714$	0	0
$715$	−0.0747056	−0.00279383
$716$	0	0
$717$	15.1654	0.566364
$718$	0	0
$719$	−27.6396	−1.03078	−0.515391	−	0.856955i	$-0.672353\pi$
$719$	−27.6396	−1.03078	−0.515391	+	0.856955i	$0.672353\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	50.8926	1.89271
$724$	0	0
$725$	−43.8082	−1.62700
$726$	0	0
$727$	36.3955	1.34983	0.674917	−	0.737894i	$-0.264180\pi$
$727$	36.3955	1.34983	0.674917	+	0.737894i	$0.264180\pi$
$728$	0	0
$729$	17.5636	0.650504
$730$	0	0
$731$	−63.1533	−2.33581
$732$	0	0
$733$	18.5815	0.686323	0.343161	−	0.939276i	$-0.388502\pi$
$733$	18.5815	0.686323	0.343161	+	0.939276i	$0.388502\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	9.73438	0.358571
$738$	0	0
$739$	−19.9347	−0.733309	−0.366654	−	0.930357i	$-0.619497\pi$
$739$	−19.9347	−0.733309	−0.366654	+	0.930357i	$0.619497\pi$
$740$	0	0
$741$	4.27074	0.156890
$742$	0	0
$743$	−6.00507	−0.220305	−0.110152	−	0.993915i	$-0.535134\pi$
$743$	−6.00507	−0.220305	−0.110152	+	0.993915i	$0.535134\pi$
$744$	0	0
$745$	2.76476	0.101293
$746$	0	0
$747$	0.712366	0.0260641
$748$	0	0
$749$	0	0
$750$	0	0
$751$	15.2815	0.557632	0.278816	−	0.960345i	$-0.410058\pi$
$751$	15.2815	0.557632	0.278816	+	0.960345i	$0.410058\pi$
$752$	0	0
$753$	22.7053	0.827428
$754$	0	0
$755$	−2.83523	−0.103184
$756$	0	0
$757$	−34.8279	−1.26584	−0.632921	−	0.774217i	$-0.718144\pi$
$757$	−34.8279	−1.26584	−0.632921	+	0.774217i	$0.718144\pi$
$758$	0	0
$759$	0.635765	0.0230768
$760$	0	0
$761$	−24.2028	−0.877350	−0.438675	−	0.898646i	$-0.644552\pi$
$761$	−24.2028	−0.877350	−0.438675	+	0.898646i	$0.644552\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−2.32851	−0.0841873
$766$	0	0
$767$	−1.75620	−0.0634128
$768$	0	0
$769$	−8.20909	−0.296027	−0.148014	−	0.988985i	$-0.547288\pi$
$769$	−8.20909	−0.296027	−0.148014	+	0.988985i	$0.547288\pi$
$770$	0	0
$771$	36.8013	1.32537
$772$	0	0
$773$	−6.17621	−0.222143	−0.111071	−	0.993812i	$-0.535428\pi$
$773$	−6.17621	−0.222143	−0.111071	+	0.993812i	$0.535428\pi$
$774$	0	0
$775$	15.2533	0.547914
$776$	0	0
$777$	0	0
$778$	0	0
$779$	8.55857	0.306643
$780$	0	0
$781$	2.26421	0.0810199
$782$	0	0
$783$	40.1866	1.43615
$784$	0	0
$785$	7.51294	0.268148
$786$	0	0
$787$	−22.0855	−0.787262	−0.393631	−	0.919269i	$-0.628781\pi$
$787$	−22.0855	−0.787262	−0.393631	+	0.919269i	$0.628781\pi$
$788$	0	0
$789$	22.6135	0.805062
$790$	0	0
$791$	0	0
$792$	0	0
$793$	0.362085	0.0128580
$794$	0	0
$795$	10.2266	0.362701
$796$	0	0
$797$	27.3116	0.967425	0.483713	−	0.875227i	$-0.339288\pi$
$797$	27.3116	0.967425	0.483713	+	0.875227i	$0.339288\pi$
$798$	0	0
$799$	−34.8370	−1.23245
$800$	0	0
$801$	7.18174	0.253754
$802$	0	0
$803$	−5.66506	−0.199916
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−48.6535	−1.71269
$808$	0	0
$809$	11.5846	0.407294	0.203647	−	0.979044i	$-0.434721\pi$
$809$	11.5846	0.407294	0.203647	+	0.979044i	$0.434721\pi$
$810$	0	0
$811$	−34.9836	−1.22844	−0.614221	−	0.789134i	$-0.710530\pi$
$811$	−34.9836	−1.22844	−0.614221	+	0.789134i	$0.710530\pi$
$812$	0	0
$813$	−25.5826	−0.897222
$814$	0	0
$815$	−8.15058	−0.285502
$816$	0	0
$817$	80.9638	2.83256
$818$	0	0
$819$	0	0
$820$	0	0
$821$	23.3594	0.815249	0.407625	−	0.913150i	$-0.366357\pi$
$821$	23.3594	0.815249	0.407625	+	0.913150i	$0.366357\pi$
$822$	0	0
$823$	5.52046	0.192431	0.0962156	−	0.995361i	$-0.469326\pi$
$823$	5.52046	0.192431	0.0962156	+	0.995361i	$0.469326\pi$
$824$	0	0
$825$	5.61222	0.195392
$826$	0	0
$827$	−33.6872	−1.17142	−0.585710	−	0.810521i	$-0.699184\pi$
$827$	−33.6872	−1.17142	−0.585710	+	0.810521i	$0.699184\pi$
$828$	0	0
$829$	20.9791	0.728635	0.364318	−	0.931275i	$-0.381302\pi$
$829$	20.9791	0.728635	0.364318	+	0.931275i	$0.381302\pi$
$830$	0	0
$831$	−5.71995	−0.198423
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−6.83117	−0.236402
$836$	0	0
$837$	−13.9923	−0.483645
$838$	0	0
$839$	42.6347	1.47191	0.735957	−	0.677029i	$-0.236733\pi$
$839$	42.6347	1.47191	0.735957	+	0.677029i	$0.236733\pi$
$840$	0	0
$841$	55.2090	1.90376
$842$	0	0
$843$	−22.9216	−0.789461
$844$	0	0
$845$	6.14934	0.211544
$846$	0	0
$847$	0	0
$848$	0	0
$849$	47.2379	1.62120
$850$	0	0
$851$	−1.14470	−0.0392398
$852$	0	0
$853$	−5.48928	−0.187949	−0.0939746	−	0.995575i	$-0.529957\pi$
$853$	−5.48928	−0.187949	−0.0939746	+	0.995575i	$0.529957\pi$
$854$	0	0
$855$	2.98519	0.102091
$856$	0	0
$857$	30.7723	1.05116	0.525581	−	0.850743i	$-0.323848\pi$
$857$	30.7723	1.05116	0.525581	+	0.850743i	$0.323848\pi$
$858$	0	0
$859$	12.1640	0.415030	0.207515	−	0.978232i	$-0.433462\pi$
$859$	12.1640	0.415030	0.207515	+	0.978232i	$0.433462\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−0.631149	−0.0214846	−0.0107423	−	0.999942i	$-0.503419\pi$
$863$	−0.631149	−0.0214846	−0.0107423	+	0.999942i	$0.503419\pi$
$864$	0	0
$865$	−12.0145	−0.408506
$866$	0	0
$867$	−53.2663	−1.80902
$868$	0	0
$869$	3.32401	0.112759
$870$	0	0
$871$	4.13192	0.140005
$872$	0	0
$873$	2.39879	0.0811866
$874$	0	0
$875$	0	0
$876$	0	0
$877$	6.79128	0.229325	0.114663	−	0.993404i	$-0.463421\pi$
$877$	6.79128	0.229325	0.114663	+	0.993404i	$0.463421\pi$
$878$	0	0
$879$	−61.8269	−2.08537
$880$	0	0
$881$	−37.5322	−1.26449	−0.632247	−	0.774767i	$-0.717867\pi$
$881$	−37.5322	−1.26449	−0.632247	+	0.774767i	$0.717867\pi$
$882$	0	0
$883$	20.0969	0.676313	0.338157	−	0.941090i	$-0.390197\pi$
$883$	20.0969	0.676313	0.338157	+	0.941090i	$0.390197\pi$
$884$	0	0
$885$	−6.24766	−0.210013
$886$	0	0
$887$	−23.9581	−0.804433	−0.402217	−	0.915544i	$-0.631760\pi$
$887$	−23.9581	−0.804433	−0.402217	+	0.915544i	$0.631760\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−6.48722	−0.217330
$892$	0	0
$893$	44.6617	1.49455
$894$	0	0
$895$	−9.63418	−0.322035
$896$	0	0
$897$	0.269861	0.00901041
$898$	0	0
$899$	−29.3202	−0.977882
$900$	0	0
$901$	74.3115	2.47568
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−3.93051	−0.130655
$906$	0	0
$907$	13.8339	0.459346	0.229673	−	0.973268i	$-0.426234\pi$
$907$	13.8339	0.459346	0.229673	+	0.973268i	$0.426234\pi$
$908$	0	0
$909$	−11.2647	−0.373627
$910$	0	0
$911$	11.1585	0.369697	0.184849	−	0.982767i	$-0.440820\pi$
$911$	11.1585	0.369697	0.184849	+	0.982767i	$0.440820\pi$
$912$	0	0
$913$	0.590809	0.0195529
$914$	0	0
$915$	1.28811	0.0425837
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−31.9623	−1.05434	−0.527169	−	0.849761i	$-0.676746\pi$
$919$	−31.9623	−1.05434	−0.527169	+	0.849761i	$0.676746\pi$
$920$	0	0
$921$	2.66516	0.0878201
$922$	0	0
$923$	0.961084	0.0316345
$924$	0	0
$925$	−10.1049	−0.332245
$926$	0	0
$927$	−11.6677	−0.383219
$928$	0	0
$929$	−21.0007	−0.689011	−0.344505	−	0.938784i	$-0.611953\pi$
$929$	−21.0007	−0.689011	−0.344505	+	0.938784i	$0.611953\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−12.6989	−0.415744
$934$	0	0
$935$	−1.93117	−0.0631561
$936$	0	0
$937$	14.6936	0.480019	0.240009	−	0.970771i	$-0.422850\pi$
$937$	14.6936	0.480019	0.240009	+	0.970771i	$0.422850\pi$
$938$	0	0
$939$	−19.7239	−0.643666
$940$	0	0
$941$	15.2667	0.497680	0.248840	−	0.968545i	$-0.419951\pi$
$941$	15.2667	0.497680	0.248840	+	0.968545i	$0.419951\pi$
$942$	0	0
$943$	0.540802	0.0176109
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−21.8411	−0.709740	−0.354870	−	0.934916i	$-0.615475\pi$
$947$	−21.8411	−0.709740	−0.354870	+	0.934916i	$0.615475\pi$
$948$	0	0
$949$	−2.40463	−0.0780576
$950$	0	0
$951$	21.9684	0.712373
$952$	0	0
$953$	−48.7302	−1.57853	−0.789264	−	0.614055i	$-0.789537\pi$
$953$	−48.7302	−1.57853	−0.789264	+	0.614055i	$0.789537\pi$
$954$	0	0
$955$	−10.1166	−0.327366
$956$	0	0
$957$	−10.7879	−0.348724
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−20.7912	−0.670684
$962$	0	0
$963$	−6.84454	−0.220562
$964$	0	0
$965$	4.83920	0.155779
$966$	0	0
$967$	−25.1717	−0.809467	−0.404734	−	0.914435i	$-0.632636\pi$
$967$	−25.1717	−0.809467	−0.404734	+	0.914435i	$0.632636\pi$
$968$	0	0
$969$	110.401	3.54658
$970$	0	0
$971$	46.2440	1.48404	0.742020	−	0.670378i	$-0.233868\pi$
$971$	46.2440	1.48404	0.742020	+	0.670378i	$0.233868\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	2.38220	0.0762915
$976$	0	0
$977$	11.1753	0.357529	0.178765	−	0.983892i	$-0.442790\pi$
$977$	11.1753	0.357529	0.178765	+	0.983892i	$0.442790\pi$
$978$	0	0
$979$	5.95625	0.190363
$980$	0	0
$981$	−9.63082	−0.307489
$982$	0	0
$983$	−17.8140	−0.568179	−0.284090	−	0.958798i	$-0.591691\pi$
$983$	−17.8140	−0.568179	−0.284090	+	0.958798i	$0.591691\pi$
$984$	0	0
$985$	−7.16447	−0.228279
$986$	0	0
$987$	0	0
$988$	0	0
$989$	5.11597	0.162678
$990$	0	0
$991$	−4.45574	−0.141541	−0.0707706	−	0.997493i	$-0.522546\pi$
$991$	−4.45574	−0.141541	−0.0707706	+	0.997493i	$0.522546\pi$
$992$	0	0
$993$	−37.3734	−1.18601
$994$	0	0
$995$	−2.17733	−0.0690259
$996$	0	0
$997$	−34.9717	−1.10756	−0.553782	−	0.832662i	$-0.686816\pi$
$997$	−34.9717	−1.10756	−0.553782	+	0.832662i	$0.686816\pi$
$998$	0	0
$999$	9.26949	0.293274

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8036.2.a.q.1.4		15
7.2	even	3		1148.2.i.e.165.12	✓	30
7.4	even	3		1148.2.i.e.821.12	yes	30
7.6	odd	2		8036.2.a.r.1.12		15

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1148.2.i.e.165.12	✓	30	7.2	even	3
1148.2.i.e.821.12	yes	30	7.4	even	3
8036.2.a.q.1.4		15	1.1	even	1	trivial
8036.2.a.r.1.12		15	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)

\(f(q)\)	\(=\)	\(q-1.93225 q^{3} -0.475465 q^{5} +0.733587 q^{9} +O(q^{10})\)	\(q-1.93225 q^{3} -0.475465 q^{5} +0.733587 q^{9} +0.608408 q^{11} +0.258249 q^{13} +0.918718 q^{15} +6.67585 q^{17} -8.55857 q^{19} -0.540802 q^{23} -4.77393 q^{25} +4.37927 q^{27} +9.17655 q^{29} -3.19512 q^{31} -1.17560 q^{33} +2.11667 q^{37} -0.499002 q^{39} -1.00000 q^{41} -9.45997 q^{43} -0.348795 q^{45} -5.21836 q^{47} -12.8994 q^{51} +11.1314 q^{53} -0.289277 q^{55} +16.5373 q^{57} -6.80041 q^{59} +1.40208 q^{61} -0.122789 q^{65} +15.9998 q^{67} +1.04496 q^{69} +3.72154 q^{71} -9.31128 q^{73} +9.22443 q^{75} +5.46345 q^{79} -10.6626 q^{81} +0.971073 q^{83} -3.17414 q^{85} -17.7314 q^{87} +9.78989 q^{89} +6.17377 q^{93} +4.06930 q^{95} +3.26994 q^{97} +0.446320 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 15 q - q^{3} + 3 q^{5} + 30 q^{9}+O(q^{10}) \) 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

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