LMFDB - Embedded newform 8036.2.a.r.1.5

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			1.5
Root			$-2.11769$ 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)

$f(q)$	$=$	$q-2.11769 q^{3} -3.93656 q^{5} +1.48461 q^{9} +O(q^{10})$	$q-2.11769 q^{3} -3.93656 q^{5} +1.48461 q^{9} +4.13020 q^{11} -1.64438 q^{13} +8.33642 q^{15} +5.20325 q^{17} +6.58422 q^{19} +7.17263 q^{23} +10.4965 q^{25} +3.20913 q^{27} -9.38888 q^{29} -6.46362 q^{31} -8.74649 q^{33} +11.3246 q^{37} +3.48230 q^{39} +1.00000 q^{41} -0.742453 q^{43} -5.84425 q^{45} -0.264604 q^{47} -11.0189 q^{51} -0.838544 q^{53} -16.2588 q^{55} -13.9433 q^{57} +5.28576 q^{59} -2.56825 q^{61} +6.47323 q^{65} +12.9381 q^{67} -15.1894 q^{69} +5.81754 q^{71} -12.4835 q^{73} -22.2284 q^{75} +2.51028 q^{79} -11.2498 q^{81} +8.61393 q^{83} -20.4829 q^{85} +19.8827 q^{87} -1.18149 q^{89} +13.6879 q^{93} -25.9192 q^{95} -6.32575 q^{97} +6.13173 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−2.11769	−1.22265	−0.611324	−	0.791380i	$-0.709363\pi$
$3$	−2.11769	−1.22265	−0.611324	+	0.791380i	$0.709363\pi$
$4$	0	0
$5$	−3.93656	−1.76049	−0.880243	−	0.474524i	$-0.842620\pi$
$5$	−3.93656	−1.76049	−0.880243	+	0.474524i	$0.842620\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	1.48461	0.494869
$10$	0	0
$11$	4.13020	1.24530	0.622652	−	0.782499i	$-0.286055\pi$
$11$	4.13020	1.24530	0.622652	+	0.782499i	$0.286055\pi$
$12$	0	0
$13$	−1.64438	−0.456070	−0.228035	−	0.973653i	$-0.573230\pi$
$13$	−1.64438	−0.456070	−0.228035	+	0.973653i	$0.573230\pi$
$14$	0	0
$15$	8.33642	2.15245
$16$	0	0
$17$	5.20325	1.26197	0.630987	−	0.775794i	$-0.282650\pi$
$17$	5.20325	1.26197	0.630987	+	0.775794i	$0.282650\pi$
$18$	0	0
$19$	6.58422	1.51052	0.755262	−	0.655423i	$-0.227510\pi$
$19$	6.58422	1.51052	0.755262	+	0.655423i	$0.227510\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	7.17263	1.49560	0.747798	−	0.663926i	$-0.231111\pi$
$23$	7.17263	1.49560	0.747798	+	0.663926i	$0.231111\pi$
$24$	0	0
$25$	10.4965	2.09931
$26$	0	0
$27$	3.20913	0.617598
$28$	0	0
$29$	−9.38888	−1.74347	−0.871736	−	0.489976i	$-0.837006\pi$
$29$	−9.38888	−1.74347	−0.871736	+	0.489976i	$0.837006\pi$
$30$	0	0
$31$	−6.46362	−1.16090	−0.580450	−	0.814296i	$-0.697123\pi$
$31$	−6.46362	−1.16090	−0.580450	+	0.814296i	$0.697123\pi$
$32$	0	0
$33$	−8.74649	−1.52257
$34$	0	0
$35$	0	0
$36$	0	0
$37$	11.3246	1.86175	0.930873	−	0.365343i	$-0.119048\pi$
$37$	11.3246	1.86175	0.930873	+	0.365343i	$0.119048\pi$
$38$	0	0
$39$	3.48230	0.557614
$40$	0	0
$41$	1.00000	0.156174
$41$	1.00000	0.156174
$42$	0	0
$43$	−0.742453	−0.113223	−0.0566115	−	0.998396i	$-0.518030\pi$
$43$	−0.742453	−0.113223	−0.0566115	+	0.998396i	$0.518030\pi$
$44$	0	0
$45$	−5.84425	−0.871209
$46$	0	0
$47$	−0.264604	−0.0385964	−0.0192982	−	0.999814i	$-0.506143\pi$
$47$	−0.264604	−0.0385964	−0.0192982	+	0.999814i	$0.506143\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−11.0189	−1.54295
$52$	0	0
$53$	−0.838544	−0.115183	−0.0575914	−	0.998340i	$-0.518342\pi$
$53$	−0.838544	−0.115183	−0.0575914	+	0.998340i	$0.518342\pi$
$54$	0	0
$55$	−16.2588	−2.19234
$56$	0	0
$57$	−13.9433	−1.84684
$58$	0	0
$59$	5.28576	0.688148	0.344074	−	0.938943i	$-0.388193\pi$
$59$	5.28576	0.688148	0.344074	+	0.938943i	$0.388193\pi$
$60$	0	0
$61$	−2.56825	−0.328831	−0.164415	−	0.986391i	$-0.552574\pi$
$61$	−2.56825	−0.328831	−0.164415	+	0.986391i	$0.552574\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	6.47323	0.802905
$66$	0	0
$67$	12.9381	1.58064	0.790318	−	0.612697i	$-0.209915\pi$
$67$	12.9381	1.58064	0.790318	+	0.612697i	$0.209915\pi$
$68$	0	0
$69$	−15.1894	−1.82859
$70$	0	0
$71$	5.81754	0.690415	0.345207	−	0.938526i	$-0.387809\pi$
$71$	5.81754	0.690415	0.345207	+	0.938526i	$0.387809\pi$
$72$	0	0
$73$	−12.4835	−1.46109	−0.730544	−	0.682866i	$-0.760733\pi$
$73$	−12.4835	−1.46109	−0.730544	+	0.682866i	$0.760733\pi$
$74$	0	0
$75$	−22.2284	−2.56672
$76$	0	0
$77$	0	0
$78$	0	0
$79$	2.51028	0.282429	0.141214	−	0.989979i	$-0.454899\pi$
$79$	2.51028	0.282429	0.141214	+	0.989979i	$0.454899\pi$
$80$	0	0
$81$	−11.2498	−1.24997
$82$	0	0
$83$	8.61393	0.945501	0.472751	−	0.881196i	$-0.343261\pi$
$83$	8.61393	0.945501	0.472751	+	0.881196i	$0.343261\pi$
$84$	0	0
$85$	−20.4829	−2.22169
$86$	0	0
$87$	19.8827	2.13165
$88$	0	0
$89$	−1.18149	−0.125238	−0.0626190	−	0.998038i	$-0.519945\pi$
$89$	−1.18149	−0.125238	−0.0626190	+	0.998038i	$0.519945\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	13.6879	1.41937
$94$	0	0
$95$	−25.9192	−2.65926
$96$	0	0
$97$	−6.32575	−0.642283	−0.321141	−	0.947031i	$-0.604066\pi$
$97$	−6.32575	−0.642283	−0.321141	+	0.947031i	$0.604066\pi$
$98$	0	0
$99$	6.13173	0.616262
$100$	0	0
$101$	−13.1843	−1.31189	−0.655946	−	0.754808i	$-0.727730\pi$
$101$	−13.1843	−1.31189	−0.655946	+	0.754808i	$0.727730\pi$
$102$	0	0
$103$	−15.8507	−1.56182	−0.780909	−	0.624645i	$-0.785244\pi$
$103$	−15.8507	−1.56182	−0.780909	+	0.624645i	$0.785244\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	7.98270	0.771717	0.385858	−	0.922558i	$-0.373905\pi$
$107$	7.98270	0.771717	0.385858	+	0.922558i	$0.373905\pi$
$108$	0	0
$109$	12.0240	1.15169	0.575847	−	0.817557i	$-0.304672\pi$
$109$	12.0240	1.15169	0.575847	+	0.817557i	$0.304672\pi$
$110$	0	0
$111$	−23.9819	−2.27626
$112$	0	0
$113$	−18.6408	−1.75358	−0.876788	−	0.480876i	$-0.840319\pi$
$113$	−18.6408	−1.75358	−0.876788	+	0.480876i	$0.840319\pi$
$114$	0	0
$115$	−28.2355	−2.63298
$116$	0	0
$117$	−2.44126	−0.225695
$118$	0	0
$119$	0	0
$120$	0	0
$121$	6.05859	0.550781
$122$	0	0
$123$	−2.11769	−0.190946
$124$	0	0
$125$	−21.6375	−1.93532
$126$	0	0
$127$	−4.15387	−0.368596	−0.184298	−	0.982870i	$-0.559001\pi$
$127$	−4.15387	−0.368596	−0.184298	+	0.982870i	$0.559001\pi$
$128$	0	0
$129$	1.57228	0.138432
$130$	0	0
$131$	10.5045	0.917786	0.458893	−	0.888492i	$-0.348246\pi$
$131$	10.5045	0.917786	0.458893	+	0.888492i	$0.348246\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−12.6330	−1.08727
$136$	0	0
$137$	1.41888	0.121223	0.0606117	−	0.998161i	$-0.480695\pi$
$137$	1.41888	0.121223	0.0606117	+	0.998161i	$0.480695\pi$
$138$	0	0
$139$	10.3332	0.876448	0.438224	−	0.898866i	$-0.355608\pi$
$139$	10.3332	0.876448	0.438224	+	0.898866i	$0.355608\pi$
$140$	0	0
$141$	0.560348	0.0471898
$142$	0	0
$143$	−6.79165	−0.567946
$144$	0	0
$145$	36.9599	3.06936
$146$	0	0
$147$	0	0
$148$	0	0
$149$	1.28402	0.105191	0.0525954	−	0.998616i	$-0.483251\pi$
$149$	1.28402	0.105191	0.0525954	+	0.998616i	$0.483251\pi$
$150$	0	0
$151$	6.45601	0.525382	0.262691	−	0.964880i	$-0.415390\pi$
$151$	6.45601	0.525382	0.262691	+	0.964880i	$0.415390\pi$
$152$	0	0
$153$	7.72478	0.624511
$154$	0	0
$155$	25.4444	2.04375
$156$	0	0
$157$	16.2843	1.29963	0.649815	−	0.760093i	$-0.274846\pi$
$157$	16.2843	1.29963	0.649815	+	0.760093i	$0.274846\pi$
$158$	0	0
$159$	1.77577	0.140828
$160$	0	0
$161$	0	0
$162$	0	0
$163$	1.33662	0.104692	0.0523459	−	0.998629i	$-0.483330\pi$
$163$	1.33662	0.104692	0.0523459	+	0.998629i	$0.483330\pi$
$164$	0	0
$165$	34.4311	2.68046
$166$	0	0
$167$	−9.64888	−0.746653	−0.373326	−	0.927700i	$-0.621783\pi$
$167$	−9.64888	−0.746653	−0.373326	+	0.927700i	$0.621783\pi$
$168$	0	0
$169$	−10.2960	−0.792000
$170$	0	0
$171$	9.77498	0.747511
$172$	0	0
$173$	−0.394838	−0.0300190	−0.0150095	−	0.999887i	$-0.504778\pi$
$173$	−0.394838	−0.0300190	−0.0150095	+	0.999887i	$0.504778\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−11.1936	−0.841362
$178$	0	0
$179$	−7.95229	−0.594382	−0.297191	−	0.954818i	$-0.596050\pi$
$179$	−7.95229	−0.594382	−0.297191	+	0.954818i	$0.596050\pi$
$180$	0	0
$181$	−2.47729	−0.184135	−0.0920677	−	0.995753i	$-0.529348\pi$
$181$	−2.47729	−0.184135	−0.0920677	+	0.995753i	$0.529348\pi$
$182$	0	0
$183$	5.43875	0.402044
$184$	0	0
$185$	−44.5799	−3.27758
$186$	0	0
$187$	21.4905	1.57154
$188$	0	0
$189$	0	0
$190$	0	0
$191$	8.03672	0.581517	0.290758	−	0.956797i	$-0.406092\pi$
$191$	8.03672	0.581517	0.290758	+	0.956797i	$0.406092\pi$
$192$	0	0
$193$	−18.7700	−1.35109	−0.675547	−	0.737317i	$-0.736092\pi$
$193$	−18.7700	−1.35109	−0.675547	+	0.737317i	$0.736092\pi$
$194$	0	0
$195$	−13.7083	−0.981670
$196$	0	0
$197$	10.9468	0.779924	0.389962	−	0.920831i	$-0.372488\pi$
$197$	10.9468	0.779924	0.389962	+	0.920831i	$0.372488\pi$
$198$	0	0
$199$	20.8786	1.48004	0.740022	−	0.672583i	$-0.234815\pi$
$199$	20.8786	1.48004	0.740022	+	0.672583i	$0.234815\pi$
$200$	0	0
$201$	−27.3988	−1.93256
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−3.93656	−0.274942
$206$	0	0
$207$	10.6485	0.740124
$208$	0	0
$209$	27.1942	1.88106
$210$	0	0
$211$	4.44265	0.305845	0.152922	−	0.988238i	$-0.451132\pi$
$211$	4.44265	0.305845	0.152922	+	0.988238i	$0.451132\pi$
$212$	0	0
$213$	−12.3197	−0.844134
$214$	0	0
$215$	2.92271	0.199327
$216$	0	0
$217$	0	0
$218$	0	0
$219$	26.4362	1.78640
$220$	0	0
$221$	−8.55614	−0.575549
$222$	0	0
$223$	−9.49744	−0.635995	−0.317998	−	0.948091i	$-0.603010\pi$
$223$	−9.49744	−0.635995	−0.317998	+	0.948091i	$0.603010\pi$
$224$	0	0
$225$	15.5832	1.03888
$226$	0	0
$227$	12.6954	0.842621	0.421311	−	0.906916i	$-0.361570\pi$
$227$	12.6954	0.842621	0.421311	+	0.906916i	$0.361570\pi$
$228$	0	0
$229$	13.2558	0.875966	0.437983	−	0.898983i	$-0.355693\pi$
$229$	13.2558	0.875966	0.437983	+	0.898983i	$0.355693\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	11.8763	0.778042	0.389021	−	0.921229i	$-0.372813\pi$
$233$	11.8763	0.778042	0.389021	+	0.921229i	$0.372813\pi$
$234$	0	0
$235$	1.04163	0.0679484
$236$	0	0
$237$	−5.31600	−0.345311
$238$	0	0
$239$	−22.4170	−1.45004	−0.725018	−	0.688730i	$-0.758169\pi$
$239$	−22.4170	−1.45004	−0.725018	+	0.688730i	$0.758169\pi$
$240$	0	0
$241$	25.0278	1.61218	0.806092	−	0.591791i	$-0.201579\pi$
$241$	25.0278	1.61218	0.806092	+	0.591791i	$0.201579\pi$
$242$	0	0
$243$	14.1961	0.910680
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−10.8270	−0.688905
$248$	0	0
$249$	−18.2416	−1.15602
$250$	0	0
$251$	2.08571	0.131649	0.0658244	−	0.997831i	$-0.479032\pi$
$251$	2.08571	0.131649	0.0658244	+	0.997831i	$0.479032\pi$
$252$	0	0
$253$	29.6244	1.86247
$254$	0	0
$255$	43.3765	2.71634
$256$	0	0
$257$	18.9277	1.18068	0.590339	−	0.807156i	$-0.298994\pi$
$257$	18.9277	1.18068	0.590339	+	0.807156i	$0.298994\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−13.9388	−0.862790
$262$	0	0
$263$	−5.16279	−0.318351	−0.159175	−	0.987250i	$-0.550884\pi$
$263$	−5.16279	−0.318351	−0.159175	+	0.987250i	$0.550884\pi$
$264$	0	0
$265$	3.30098	0.202778
$266$	0	0
$267$	2.50204	0.153122
$268$	0	0
$269$	−6.40064	−0.390254	−0.195127	−	0.980778i	$-0.562512\pi$
$269$	−6.40064	−0.390254	−0.195127	+	0.980778i	$0.562512\pi$
$270$	0	0
$271$	31.9665	1.94183	0.970915	−	0.239425i	$-0.0769590\pi$
$271$	31.9665	1.94183	0.970915	+	0.239425i	$0.0769590\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	43.3529	2.61428
$276$	0	0
$277$	24.0243	1.44348	0.721740	−	0.692165i	$-0.243343\pi$
$277$	24.0243	1.44348	0.721740	+	0.692165i	$0.243343\pi$
$278$	0	0
$279$	−9.59593	−0.574493
$280$	0	0
$281$	−13.6948	−0.816965	−0.408483	−	0.912766i	$-0.633942\pi$
$281$	−13.6948	−0.816965	−0.408483	+	0.912766i	$0.633942\pi$
$282$	0	0
$283$	14.1506	0.841165	0.420582	−	0.907254i	$-0.361826\pi$
$283$	14.1506	0.841165	0.420582	+	0.907254i	$0.361826\pi$
$284$	0	0
$285$	54.8888	3.25133
$286$	0	0
$287$	0	0
$288$	0	0
$289$	10.0738	0.592577
$290$	0	0
$291$	13.3960	0.785286
$292$	0	0
$293$	−4.12691	−0.241097	−0.120548	−	0.992707i	$-0.538465\pi$
$293$	−4.12691	−0.241097	−0.120548	+	0.992707i	$0.538465\pi$
$294$	0	0
$295$	−20.8077	−1.21147
$296$	0	0
$297$	13.2544	0.769097
$298$	0	0
$299$	−11.7946	−0.682097
$300$	0	0
$301$	0	0
$302$	0	0
$303$	27.9203	1.60398
$304$	0	0
$305$	10.1101	0.578901
$306$	0	0
$307$	−5.97471	−0.340995	−0.170497	−	0.985358i	$-0.554537\pi$
$307$	−5.97471	−0.340995	−0.170497	+	0.985358i	$0.554537\pi$
$308$	0	0
$309$	33.5669	1.90955
$310$	0	0
$311$	−16.1431	−0.915390	−0.457695	−	0.889109i	$-0.651325\pi$
$311$	−16.1431	−0.915390	−0.457695	+	0.889109i	$0.651325\pi$
$312$	0	0
$313$	7.74303	0.437662	0.218831	−	0.975763i	$-0.429776\pi$
$313$	7.74303	0.437662	0.218831	+	0.975763i	$0.429776\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−24.3917	−1.36997	−0.684987	−	0.728556i	$-0.740192\pi$
$317$	−24.3917	−1.36997	−0.684987	+	0.728556i	$0.740192\pi$
$318$	0	0
$319$	−38.7780	−2.17115
$320$	0	0
$321$	−16.9049	−0.943538
$322$	0	0
$323$	34.2594	1.90624
$324$	0	0
$325$	−17.2604	−0.957432
$326$	0	0
$327$	−25.4632	−1.40812
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−24.4302	−1.34281	−0.671403	−	0.741092i	$-0.734308\pi$
$331$	−24.4302	−1.34281	−0.671403	+	0.741092i	$0.734308\pi$
$332$	0	0
$333$	16.8125	0.921320
$334$	0	0
$335$	−50.9315	−2.78269
$336$	0	0
$337$	10.3432	0.563428	0.281714	−	0.959498i	$-0.409097\pi$
$337$	10.3432	0.563428	0.281714	+	0.959498i	$0.409097\pi$
$338$	0	0
$339$	39.4754	2.14401
$340$	0	0
$341$	−26.6961	−1.44567
$342$	0	0
$343$	0	0
$344$	0	0
$345$	59.7940	3.21920
$346$	0	0
$347$	−23.4882	−1.26091	−0.630457	−	0.776224i	$-0.717133\pi$
$347$	−23.4882	−1.26091	−0.630457	+	0.776224i	$0.717133\pi$
$348$	0	0
$349$	−5.54452	−0.296791	−0.148396	−	0.988928i	$-0.547411\pi$
$349$	−5.54452	−0.296791	−0.148396	+	0.988928i	$0.547411\pi$
$350$	0	0
$351$	−5.27705	−0.281668
$352$	0	0
$353$	3.44192	0.183195	0.0915974	−	0.995796i	$-0.470803\pi$
$353$	3.44192	0.183195	0.0915974	+	0.995796i	$0.470803\pi$
$354$	0	0
$355$	−22.9011	−1.21546
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−7.62296	−0.402324	−0.201162	−	0.979558i	$-0.564472\pi$
$359$	−7.62296	−0.402324	−0.201162	+	0.979558i	$0.564472\pi$
$360$	0	0
$361$	24.3520	1.28168
$362$	0	0
$363$	−12.8302	−0.673411
$364$	0	0
$365$	49.1422	2.57222
$366$	0	0
$367$	−12.7933	−0.667806	−0.333903	−	0.942607i	$-0.608366\pi$
$367$	−12.7933	−0.667806	−0.333903	+	0.942607i	$0.608366\pi$
$368$	0	0
$369$	1.48461	0.0772855
$370$	0	0
$371$	0	0
$372$	0	0
$373$	37.4785	1.94056	0.970281	−	0.241979i	$-0.0777965\pi$
$373$	37.4785	1.94056	0.970281	+	0.241979i	$0.0777965\pi$
$374$	0	0
$375$	45.8215	2.36621
$376$	0	0
$377$	15.4389	0.795146
$378$	0	0
$379$	−9.08847	−0.466844	−0.233422	−	0.972376i	$-0.574992\pi$
$379$	−9.08847	−0.466844	−0.233422	+	0.972376i	$0.574992\pi$
$380$	0	0
$381$	8.79660	0.450663
$382$	0	0
$383$	−14.3554	−0.733524	−0.366762	−	0.930315i	$-0.619534\pi$
$383$	−14.3554	−0.733524	−0.366762	+	0.930315i	$0.619534\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−1.10225	−0.0560305
$388$	0	0
$389$	−26.6068	−1.34902	−0.674510	−	0.738265i	$-0.735645\pi$
$389$	−26.6068	−1.34902	−0.674510	+	0.738265i	$0.735645\pi$
$390$	0	0
$391$	37.3210	1.88740
$392$	0	0
$393$	−22.2453	−1.12213
$394$	0	0
$395$	−9.88189	−0.497212
$396$	0	0
$397$	−30.0384	−1.50758	−0.753792	−	0.657113i	$-0.771777\pi$
$397$	−30.0384	−1.50758	−0.753792	+	0.657113i	$0.771777\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	14.8185	0.739999	0.369999	−	0.929032i	$-0.379358\pi$
$401$	14.8185	0.739999	0.369999	+	0.929032i	$0.379358\pi$
$402$	0	0
$403$	10.6287	0.529452
$404$	0	0
$405$	44.2854	2.20056
$406$	0	0
$407$	46.7727	2.31844
$408$	0	0
$409$	18.6040	0.919908	0.459954	−	0.887943i	$-0.347866\pi$
$409$	18.6040	0.919908	0.459954	+	0.887943i	$0.347866\pi$
$410$	0	0
$411$	−3.00475	−0.148214
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−33.9093	−1.66454
$416$	0	0
$417$	−21.8824	−1.07159
$418$	0	0
$419$	1.89906	0.0927751	0.0463875	−	0.998924i	$-0.485229\pi$
$419$	1.89906	0.0927751	0.0463875	+	0.998924i	$0.485229\pi$
$420$	0	0
$421$	5.78259	0.281826	0.140913	−	0.990022i	$-0.454996\pi$
$421$	5.78259	0.281826	0.140913	+	0.990022i	$0.454996\pi$
$422$	0	0
$423$	−0.392832	−0.0191002
$424$	0	0
$425$	54.6161	2.64927
$426$	0	0
$427$	0	0
$428$	0	0
$429$	14.3826	0.694398
$430$	0	0
$431$	−27.9485	−1.34623	−0.673117	−	0.739536i	$-0.735045\pi$
$431$	−27.9485	−1.34623	−0.673117	+	0.739536i	$0.735045\pi$
$432$	0	0
$433$	−12.1725	−0.584975	−0.292487	−	0.956269i	$-0.594483\pi$
$433$	−12.1725	−0.584975	−0.292487	+	0.956269i	$0.594483\pi$
$434$	0	0
$435$	−78.2697	−3.75274
$436$	0	0
$437$	47.2262	2.25913
$438$	0	0
$439$	10.3773	0.495284	0.247642	−	0.968852i	$-0.420344\pi$
$439$	10.3773	0.495284	0.247642	+	0.968852i	$0.420344\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−38.9198	−1.84913	−0.924567	−	0.381020i	$-0.875573\pi$
$443$	−38.9198	−1.84913	−0.924567	+	0.381020i	$0.875573\pi$
$444$	0	0
$445$	4.65103	0.220480
$446$	0	0
$447$	−2.71915	−0.128611
$448$	0	0
$449$	−10.8233	−0.510783	−0.255391	−	0.966838i	$-0.582204\pi$
$449$	−10.8233	−0.510783	−0.255391	+	0.966838i	$0.582204\pi$
$450$	0	0
$451$	4.13020	0.194484
$452$	0	0
$453$	−13.6718	−0.642358
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−10.2417	−0.479086	−0.239543	−	0.970886i	$-0.576998\pi$
$457$	−10.2417	−0.479086	−0.239543	+	0.970886i	$0.576998\pi$
$458$	0	0
$459$	16.6979	0.779392
$460$	0	0
$461$	24.4368	1.13814	0.569068	−	0.822290i	$-0.307304\pi$
$461$	24.4368	1.13814	0.569068	+	0.822290i	$0.307304\pi$
$462$	0	0
$463$	−21.0456	−0.978072	−0.489036	−	0.872264i	$-0.662651\pi$
$463$	−21.0456	−0.978072	−0.489036	+	0.872264i	$0.662651\pi$
$464$	0	0
$465$	−53.8834	−2.49878
$466$	0	0
$467$	12.2601	0.567329	0.283664	−	0.958924i	$-0.408450\pi$
$467$	12.2601	0.567329	0.283664	+	0.958924i	$0.408450\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−34.4851	−1.58899
$472$	0	0
$473$	−3.06648	−0.140997
$474$	0	0
$475$	69.1116	3.17106
$476$	0	0
$477$	−1.24491	−0.0570004
$478$	0	0
$479$	22.7134	1.03780	0.518902	−	0.854834i	$-0.326341\pi$
$479$	22.7134	1.03780	0.518902	+	0.854834i	$0.326341\pi$
$480$	0	0
$481$	−18.6219	−0.849087
$482$	0	0
$483$	0	0
$484$	0	0
$485$	24.9017	1.13073
$486$	0	0
$487$	−28.5514	−1.29379	−0.646893	−	0.762581i	$-0.723932\pi$
$487$	−28.5514	−1.29379	−0.646893	+	0.762581i	$0.723932\pi$
$488$	0	0
$489$	−2.83054	−0.128001
$490$	0	0
$491$	8.49643	0.383438	0.191719	−	0.981450i	$-0.438594\pi$
$491$	8.49643	0.383438	0.191719	+	0.981450i	$0.438594\pi$
$492$	0	0
$493$	−48.8527	−2.20021
$494$	0	0
$495$	−24.1379	−1.08492
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−8.05158	−0.360438	−0.180219	−	0.983627i	$-0.557681\pi$
$499$	−8.05158	−0.360438	−0.180219	+	0.983627i	$0.557681\pi$
$500$	0	0
$501$	20.4333	0.912893
$502$	0	0
$503$	31.3484	1.39776	0.698879	−	0.715240i	$-0.253683\pi$
$503$	31.3484	1.39776	0.698879	+	0.715240i	$0.253683\pi$
$504$	0	0
$505$	51.9010	2.30957
$506$	0	0
$507$	21.8037	0.968337
$508$	0	0
$509$	−3.98621	−0.176686	−0.0883429	−	0.996090i	$-0.528157\pi$
$509$	−3.98621	−0.176686	−0.0883429	+	0.996090i	$0.528157\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	21.1296	0.932897
$514$	0	0
$515$	62.3974	2.74956
$516$	0	0
$517$	−1.09287	−0.0480642
$518$	0	0
$519$	0.836144	0.0367026
$520$	0	0
$521$	42.5988	1.86629	0.933144	−	0.359503i	$-0.117054\pi$
$521$	42.5988	1.86629	0.933144	+	0.359503i	$0.117054\pi$
$522$	0	0
$523$	13.4853	0.589673	0.294836	−	0.955548i	$-0.404735\pi$
$523$	13.4853	0.589673	0.294836	+	0.955548i	$0.404735\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−33.6318	−1.46502
$528$	0	0
$529$	28.4466	1.23681
$530$	0	0
$531$	7.84728	0.340543
$532$	0	0
$533$	−1.64438	−0.0712262
$534$	0	0
$535$	−31.4244	−1.35860
$536$	0	0
$537$	16.8405	0.726720
$538$	0	0
$539$	0	0
$540$	0	0
$541$	31.8373	1.36879	0.684395	−	0.729111i	$-0.260066\pi$
$541$	31.8373	1.36879	0.684395	+	0.729111i	$0.260066\pi$
$542$	0	0
$543$	5.24613	0.225133
$544$	0	0
$545$	−47.3334	−2.02754
$546$	0	0
$547$	−29.2689	−1.25145	−0.625723	−	0.780045i	$-0.715196\pi$
$547$	−29.2689	−1.25145	−0.625723	+	0.780045i	$0.715196\pi$
$548$	0	0
$549$	−3.81284	−0.162728
$550$	0	0
$551$	−61.8185	−2.63356
$552$	0	0
$553$	0	0
$554$	0	0
$555$	94.4063	4.00732
$556$	0	0
$557$	7.63655	0.323571	0.161785	−	0.986826i	$-0.448275\pi$
$557$	7.63655	0.323571	0.161785	+	0.986826i	$0.448275\pi$
$558$	0	0
$559$	1.22088	0.0516377
$560$	0	0
$561$	−45.5102	−1.92144
$562$	0	0
$563$	−1.77861	−0.0749593	−0.0374797	−	0.999297i	$-0.511933\pi$
$563$	−1.77861	−0.0749593	−0.0374797	+	0.999297i	$0.511933\pi$
$564$	0	0
$565$	73.3806	3.08715
$566$	0	0
$567$	0	0
$568$	0	0
$569$	8.83946	0.370570	0.185285	−	0.982685i	$-0.440679\pi$
$569$	8.83946	0.370570	0.185285	+	0.982685i	$0.440679\pi$
$570$	0	0
$571$	6.50457	0.272208	0.136104	−	0.990695i	$-0.456542\pi$
$571$	6.50457	0.272208	0.136104	+	0.990695i	$0.456542\pi$
$572$	0	0
$573$	−17.0193	−0.710990
$574$	0	0
$575$	75.2878	3.13972
$576$	0	0
$577$	25.7240	1.07090	0.535452	−	0.844566i	$-0.320141\pi$
$577$	25.7240	1.07090	0.535452	+	0.844566i	$0.320141\pi$
$578$	0	0
$579$	39.7490	1.65191
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−3.46336	−0.143438
$584$	0	0
$585$	9.61019	0.397333
$586$	0	0
$587$	27.1254	1.11959	0.559794	−	0.828632i	$-0.310880\pi$
$587$	27.1254	1.11959	0.559794	+	0.828632i	$0.310880\pi$
$588$	0	0
$589$	−42.5579	−1.75357
$590$	0	0
$591$	−23.1818	−0.953573
$592$	0	0
$593$	−25.0630	−1.02921	−0.514607	−	0.857426i	$-0.672062\pi$
$593$	−25.0630	−1.02921	−0.514607	+	0.857426i	$0.672062\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−44.2143	−1.80957
$598$	0	0
$599$	19.8368	0.810510	0.405255	−	0.914204i	$-0.367183\pi$
$599$	19.8368	0.810510	0.405255	+	0.914204i	$0.367183\pi$
$600$	0	0
$601$	19.3130	0.787792	0.393896	−	0.919155i	$-0.371127\pi$
$601$	19.3130	0.787792	0.393896	+	0.919155i	$0.371127\pi$
$602$	0	0
$603$	19.2079	0.782208
$604$	0	0
$605$	−23.8500	−0.969641
$606$	0	0
$607$	14.9873	0.608318	0.304159	−	0.952621i	$-0.401625\pi$
$607$	14.9873	0.608318	0.304159	+	0.952621i	$0.401625\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	0.435110	0.0176027
$612$	0	0
$613$	11.2933	0.456131	0.228066	−	0.973646i	$-0.426760\pi$
$613$	11.2933	0.456131	0.228066	+	0.973646i	$0.426760\pi$
$614$	0	0
$615$	8.33642	0.336157
$616$	0	0
$617$	46.4782	1.87114	0.935572	−	0.353137i	$-0.114885\pi$
$617$	46.4782	1.87114	0.935572	+	0.353137i	$0.114885\pi$
$618$	0	0
$619$	36.4425	1.46475	0.732374	−	0.680902i	$-0.238412\pi$
$619$	36.4425	1.46475	0.732374	+	0.680902i	$0.238412\pi$
$620$	0	0
$621$	23.0179	0.923677
$622$	0	0
$623$	0	0
$624$	0	0
$625$	32.6946	1.30779
$626$	0	0
$627$	−57.5888	−2.29988
$628$	0	0
$629$	58.9245	2.34947
$630$	0	0
$631$	−15.2500	−0.607095	−0.303547	−	0.952816i	$-0.598171\pi$
$631$	−15.2500	−0.607095	−0.303547	+	0.952816i	$0.598171\pi$
$632$	0	0
$633$	−9.40816	−0.373941
$634$	0	0
$635$	16.3520	0.648908
$636$	0	0
$637$	0	0
$638$	0	0
$639$	8.63675	0.341665
$640$	0	0
$641$	21.4291	0.846398	0.423199	−	0.906037i	$-0.360907\pi$
$641$	21.4291	0.846398	0.423199	+	0.906037i	$0.360907\pi$
$642$	0	0
$643$	−26.2975	−1.03707	−0.518536	−	0.855056i	$-0.673523\pi$
$643$	−26.2975	−1.03707	−0.518536	+	0.855056i	$0.673523\pi$
$644$	0	0
$645$	−6.18940	−0.243707
$646$	0	0
$647$	−16.6289	−0.653749	−0.326875	−	0.945068i	$-0.605995\pi$
$647$	−16.6289	−0.653749	−0.326875	+	0.945068i	$0.605995\pi$
$648$	0	0
$649$	21.8313	0.856953
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−7.05664	−0.276148	−0.138074	−	0.990422i	$-0.544091\pi$
$653$	−7.05664	−0.276148	−0.138074	+	0.990422i	$0.544091\pi$
$654$	0	0
$655$	−41.3518	−1.61575
$656$	0	0
$657$	−18.5331	−0.723046
$658$	0	0
$659$	45.2945	1.76442	0.882211	−	0.470854i	$-0.156054\pi$
$659$	45.2945	1.76442	0.882211	+	0.470854i	$0.156054\pi$
$660$	0	0
$661$	40.1854	1.56303	0.781516	−	0.623886i	$-0.214447\pi$
$661$	40.1854	1.56303	0.781516	+	0.623886i	$0.214447\pi$
$662$	0	0
$663$	18.1193	0.703693
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−67.3430	−2.60753
$668$	0	0
$669$	20.1126	0.777599
$670$	0	0
$671$	−10.6074	−0.409494
$672$	0	0
$673$	15.5616	0.599856	0.299928	−	0.953962i	$-0.403037\pi$
$673$	15.5616	0.599856	0.299928	+	0.953962i	$0.403037\pi$
$674$	0	0
$675$	33.6848	1.29653
$676$	0	0
$677$	−26.4420	−1.01625	−0.508124	−	0.861284i	$-0.669661\pi$
$677$	−26.4420	−1.01625	−0.508124	+	0.861284i	$0.669661\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−26.8848	−1.03023
$682$	0	0
$683$	−1.07364	−0.0410817	−0.0205408	−	0.999789i	$-0.506539\pi$
$683$	−1.07364	−0.0410817	−0.0205408	+	0.999789i	$0.506539\pi$
$684$	0	0
$685$	−5.58553	−0.213412
$686$	0	0
$687$	−28.0716	−1.07100
$688$	0	0
$689$	1.37889	0.0525315
$690$	0	0
$691$	−8.62945	−0.328280	−0.164140	−	0.986437i	$-0.552485\pi$
$691$	−8.62945	−0.328280	−0.164140	+	0.986437i	$0.552485\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−40.6772	−1.54297
$696$	0	0
$697$	5.20325	0.197087
$698$	0	0
$699$	−25.1503	−0.951272
$700$	0	0
$701$	−44.1873	−1.66893	−0.834465	−	0.551061i	$-0.814223\pi$
$701$	−44.1873	−1.66893	−0.834465	+	0.551061i	$0.814223\pi$
$702$	0	0
$703$	74.5634	2.81221
$704$	0	0
$705$	−2.20585	−0.0830770
$706$	0	0
$707$	0	0
$708$	0	0
$709$	16.9496	0.636554	0.318277	−	0.947998i	$-0.396896\pi$
$709$	16.9496	0.636554	0.318277	+	0.947998i	$0.396896\pi$
$710$	0	0
$711$	3.72678	0.139765
$712$	0	0
$713$	−46.3611	−1.73624
$714$	0	0
$715$	26.7358	0.999860
$716$	0	0
$717$	47.4723	1.77288
$718$	0	0
$719$	−20.8173	−0.776355	−0.388177	−	0.921585i	$-0.626895\pi$
$719$	−20.8173	−0.776355	−0.388177	+	0.921585i	$0.626895\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−53.0011	−1.97113
$724$	0	0
$725$	−98.5508	−3.66008
$726$	0	0
$727$	−7.51361	−0.278664	−0.139332	−	0.990246i	$-0.544496\pi$
$727$	−7.51361	−0.278664	−0.139332	+	0.990246i	$0.544496\pi$
$728$	0	0
$729$	3.68636	0.136532
$730$	0	0
$731$	−3.86317	−0.142884
$732$	0	0
$733$	−24.7527	−0.914261	−0.457131	−	0.889400i	$-0.651123\pi$
$733$	−24.7527	−0.914261	−0.457131	+	0.889400i	$0.651123\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	53.4369	1.96837
$738$	0	0
$739$	36.3640	1.33767	0.668836	−	0.743410i	$-0.266793\pi$
$739$	36.3640	1.33767	0.668836	+	0.743410i	$0.266793\pi$
$740$	0	0
$741$	22.9282	0.842289
$742$	0	0
$743$	−11.7748	−0.431976	−0.215988	−	0.976396i	$-0.569297\pi$
$743$	−11.7748	−0.431976	−0.215988	+	0.976396i	$0.569297\pi$
$744$	0	0
$745$	−5.05462	−0.185187
$746$	0	0
$747$	12.7883	0.467899
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−2.08643	−0.0761350	−0.0380675	−	0.999275i	$-0.512120\pi$
$751$	−2.08643	−0.0761350	−0.0380675	+	0.999275i	$0.512120\pi$
$752$	0	0
$753$	−4.41688	−0.160960
$754$	0	0
$755$	−25.4145	−0.924928
$756$	0	0
$757$	−40.7156	−1.47983	−0.739917	−	0.672698i	$-0.765135\pi$
$757$	−40.7156	−1.47983	−0.739917	+	0.672698i	$0.765135\pi$
$758$	0	0
$759$	−62.7353	−2.27715
$760$	0	0
$761$	19.7790	0.716989	0.358495	−	0.933532i	$-0.383290\pi$
$761$	19.7790	0.716989	0.358495	+	0.933532i	$0.383290\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−30.4091	−1.09944
$766$	0	0
$767$	−8.69183	−0.313844
$768$	0	0
$769$	−24.9578	−0.900003	−0.450001	−	0.893028i	$-0.648577\pi$
$769$	−24.9578	−0.900003	−0.450001	+	0.893028i	$0.648577\pi$
$770$	0	0
$771$	−40.0830	−1.44355
$772$	0	0
$773$	−13.6827	−0.492133	−0.246067	−	0.969253i	$-0.579138\pi$
$773$	−13.6827	−0.492133	−0.246067	+	0.969253i	$0.579138\pi$
$774$	0	0
$775$	−67.8456	−2.43709
$776$	0	0
$777$	0	0
$778$	0	0
$779$	6.58422	0.235904
$780$	0	0
$781$	24.0276	0.859776
$782$	0	0
$783$	−30.1302	−1.07676
$784$	0	0
$785$	−64.1042	−2.28798
$786$	0	0
$787$	−10.7902	−0.384630	−0.192315	−	0.981333i	$-0.561599\pi$
$787$	−10.7902	−0.384630	−0.192315	+	0.981333i	$0.561599\pi$
$788$	0	0
$789$	10.9332	0.389231
$790$	0	0
$791$	0	0
$792$	0	0
$793$	4.22319	0.149970
$794$	0	0
$795$	−6.99045	−0.247926
$796$	0	0
$797$	3.70563	0.131260	0.0656301	−	0.997844i	$-0.479094\pi$
$797$	3.70563	0.131260	0.0656301	+	0.997844i	$0.479094\pi$
$798$	0	0
$799$	−1.37680	−0.0487076
$800$	0	0
$801$	−1.75405	−0.0619764
$802$	0	0
$803$	−51.5595	−1.81950
$804$	0	0
$805$	0	0
$806$	0	0
$807$	13.5546	0.477144
$808$	0	0
$809$	21.1550	0.743769	0.371884	−	0.928279i	$-0.378712\pi$
$809$	21.1550	0.743769	0.371884	+	0.928279i	$0.378712\pi$
$810$	0	0
$811$	−12.1506	−0.426667	−0.213333	−	0.976979i	$-0.568432\pi$
$811$	−12.1506	−0.426667	−0.213333	+	0.976979i	$0.568432\pi$
$812$	0	0
$813$	−67.6952	−2.37417
$814$	0	0
$815$	−5.26167	−0.184308
$816$	0	0
$817$	−4.88848	−0.171026
$818$	0	0
$819$	0	0
$820$	0	0
$821$	15.2915	0.533675	0.266838	−	0.963741i	$-0.414021\pi$
$821$	15.2915	0.533675	0.266838	+	0.963741i	$0.414021\pi$
$822$	0	0
$823$	57.1962	1.99373	0.996867	−	0.0791017i	$-0.0252052\pi$
$823$	57.1962	1.99373	0.996867	+	0.0791017i	$0.0252052\pi$
$824$	0	0
$825$	−91.8079	−3.19634
$826$	0	0
$827$	−29.1296	−1.01294	−0.506469	−	0.862258i	$-0.669049\pi$
$827$	−29.1296	−1.01294	−0.506469	+	0.862258i	$0.669049\pi$
$828$	0	0
$829$	21.4260	0.744155	0.372077	−	0.928202i	$-0.378646\pi$
$829$	21.4260	0.744155	0.372077	+	0.928202i	$0.378646\pi$
$830$	0	0
$831$	−50.8759	−1.76487
$832$	0	0
$833$	0	0
$834$	0	0
$835$	37.9834	1.31447
$836$	0	0
$837$	−20.7426	−0.716969
$838$	0	0
$839$	13.9582	0.481891	0.240945	−	0.970539i	$-0.422543\pi$
$839$	13.9582	0.481891	0.240945	+	0.970539i	$0.422543\pi$
$840$	0	0
$841$	59.1511	2.03969
$842$	0	0
$843$	29.0014	0.998861
$844$	0	0
$845$	40.5309	1.39430
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−29.9665	−1.02845
$850$	0	0
$851$	81.2269	2.78442
$852$	0	0
$853$	−10.9836	−0.376070	−0.188035	−	0.982162i	$-0.560212\pi$
$853$	−10.9836	−0.376070	−0.188035	+	0.982162i	$0.560212\pi$
$854$	0	0
$855$	−38.4798	−1.31598
$856$	0	0
$857$	−34.9320	−1.19325	−0.596627	−	0.802519i	$-0.703493\pi$
$857$	−34.9320	−1.19325	−0.596627	+	0.802519i	$0.703493\pi$
$858$	0	0
$859$	−4.09973	−0.139881	−0.0699405	−	0.997551i	$-0.522281\pi$
$859$	−4.09973	−0.139881	−0.0699405	+	0.997551i	$0.522281\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	11.6514	0.396618	0.198309	−	0.980140i	$-0.436455\pi$
$863$	11.6514	0.396618	0.198309	+	0.980140i	$0.436455\pi$
$864$	0	0
$865$	1.55431	0.0528480
$866$	0	0
$867$	−21.3332	−0.724513
$868$	0	0
$869$	10.3680	0.351710
$870$	0	0
$871$	−21.2752	−0.720881
$872$	0	0
$873$	−9.39125	−0.317846
$874$	0	0
$875$	0	0
$876$	0	0
$877$	19.8783	0.671243	0.335621	−	0.941997i	$-0.391054\pi$
$877$	19.8783	0.671243	0.335621	+	0.941997i	$0.391054\pi$
$878$	0	0
$879$	8.73952	0.294777
$880$	0	0
$881$	−34.3176	−1.15619	−0.578095	−	0.815970i	$-0.696204\pi$
$881$	−34.3176	−1.15619	−0.578095	+	0.815970i	$0.696204\pi$
$882$	0	0
$883$	−18.0828	−0.608535	−0.304267	−	0.952587i	$-0.598412\pi$
$883$	−18.0828	−0.608535	−0.304267	+	0.952587i	$0.598412\pi$
$884$	0	0
$885$	44.0643	1.48121
$886$	0	0
$887$	−18.8490	−0.632888	−0.316444	−	0.948611i	$-0.602489\pi$
$887$	−18.8490	−0.632888	−0.316444	+	0.948611i	$0.602489\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−46.4638	−1.55660
$892$	0	0
$893$	−1.74221	−0.0583008
$894$	0	0
$895$	31.3047	1.04640
$896$	0	0
$897$	24.9772	0.833965
$898$	0	0
$899$	60.6862	2.02400
$900$	0	0
$901$	−4.36315	−0.145358
$902$	0	0
$903$	0	0
$904$	0	0
$905$	9.75201	0.324168
$906$	0	0
$907$	33.7374	1.12023	0.560117	−	0.828414i	$-0.310756\pi$
$907$	33.7374	1.12023	0.560117	+	0.828414i	$0.310756\pi$
$908$	0	0
$909$	−19.5736	−0.649214
$910$	0	0
$911$	42.0812	1.39421	0.697106	−	0.716968i	$-0.254471\pi$
$911$	42.0812	1.39421	0.697106	+	0.716968i	$0.254471\pi$
$912$	0	0
$913$	35.5773	1.17744
$914$	0	0
$915$	−21.4100	−0.707793
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−22.8624	−0.754159	−0.377080	−	0.926181i	$-0.623072\pi$
$919$	−22.8624	−0.754159	−0.377080	+	0.926181i	$0.623072\pi$
$920$	0	0
$921$	12.6526	0.416917
$922$	0	0
$923$	−9.56627	−0.314878
$924$	0	0
$925$	118.869	3.90838
$926$	0	0
$927$	−23.5321	−0.772895
$928$	0	0
$929$	−31.1426	−1.02176	−0.510878	−	0.859653i	$-0.670680\pi$
$929$	−31.1426	−1.02176	−0.510878	+	0.859653i	$0.670680\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	34.1860	1.11920
$934$	0	0
$935$	−84.5987	−2.76667
$936$	0	0
$937$	6.94013	0.226724	0.113362	−	0.993554i	$-0.463838\pi$
$937$	6.94013	0.226724	0.113362	+	0.993554i	$0.463838\pi$
$938$	0	0
$939$	−16.3973	−0.535106
$940$	0	0
$941$	14.4228	0.470171	0.235086	−	0.971975i	$-0.424463\pi$
$941$	14.4228	0.470171	0.235086	+	0.971975i	$0.424463\pi$
$942$	0	0
$943$	7.17263	0.233573
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−29.9397	−0.972909	−0.486455	−	0.873706i	$-0.661710\pi$
$947$	−29.9397	−0.972909	−0.486455	+	0.873706i	$0.661710\pi$
$948$	0	0
$949$	20.5277	0.666358
$950$	0	0
$951$	51.6540	1.67500
$952$	0	0
$953$	−0.521703	−0.0168996	−0.00844981	−	0.999964i	$-0.502690\pi$
$953$	−0.521703	−0.0168996	−0.00844981	+	0.999964i	$0.502690\pi$
$954$	0	0
$955$	−31.6371	−1.02375
$956$	0	0
$957$	82.1198	2.65455
$958$	0	0
$959$	0	0
$960$	0	0
$961$	10.7784	0.347689
$962$	0	0
$963$	11.8512	0.381898
$964$	0	0
$965$	73.8893	2.37858
$966$	0	0
$967$	29.9003	0.961529	0.480765	−	0.876850i	$-0.340359\pi$
$967$	29.9003	0.961529	0.480765	+	0.876850i	$0.340359\pi$
$968$	0	0
$969$	−72.5507	−2.33066
$970$	0	0
$971$	−16.7863	−0.538699	−0.269349	−	0.963043i	$-0.586809\pi$
$971$	−16.7863	−0.538699	−0.269349	+	0.963043i	$0.586809\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	36.5521	1.17060
$976$	0	0
$977$	−57.9457	−1.85385	−0.926923	−	0.375251i	$-0.877557\pi$
$977$	−57.9457	−1.85385	−0.926923	+	0.375251i	$0.877557\pi$
$978$	0	0
$979$	−4.87981	−0.155959
$980$	0	0
$981$	17.8510	0.569938
$982$	0	0
$983$	−53.6430	−1.71095	−0.855473	−	0.517847i	$-0.826734\pi$
$983$	−53.6430	−1.71095	−0.855473	+	0.517847i	$0.826734\pi$
$984$	0	0
$985$	−43.0926	−1.37304
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−5.32534	−0.169336
$990$	0	0
$991$	14.3369	0.455428	0.227714	−	0.973728i	$-0.426875\pi$
$991$	14.3369	0.455428	0.227714	+	0.973728i	$0.426875\pi$
$992$	0	0
$993$	51.7356	1.64178
$994$	0	0
$995$	−82.1899	−2.60559
$996$	0	0
$997$	−25.8404	−0.818373	−0.409186	−	0.912451i	$-0.634187\pi$
$997$	−25.8404	−0.818373	−0.409186	+	0.912451i	$0.634187\pi$
$998$	0	0
$999$	36.3420	1.14981

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8036.2.a.r.1.5		15
7.3	odd	6		1148.2.i.e.821.5	yes	30
7.5	odd	6		1148.2.i.e.165.5	✓	30
7.6	odd	2		8036.2.a.q.1.11		15

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1148.2.i.e.165.5	✓	30	7.5	odd	6
1148.2.i.e.821.5	yes	30	7.3	odd	6
8036.2.a.q.1.11		15	7.6	odd	2
8036.2.a.r.1.5		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)

\(f(q)\)	\(=\)	\(q-2.11769 q^{3} -3.93656 q^{5} +1.48461 q^{9} +O(q^{10})\)	\(q-2.11769 q^{3} -3.93656 q^{5} +1.48461 q^{9} +4.13020 q^{11} -1.64438 q^{13} +8.33642 q^{15} +5.20325 q^{17} +6.58422 q^{19} +7.17263 q^{23} +10.4965 q^{25} +3.20913 q^{27} -9.38888 q^{29} -6.46362 q^{31} -8.74649 q^{33} +11.3246 q^{37} +3.48230 q^{39} +1.00000 q^{41} -0.742453 q^{43} -5.84425 q^{45} -0.264604 q^{47} -11.0189 q^{51} -0.838544 q^{53} -16.2588 q^{55} -13.9433 q^{57} +5.28576 q^{59} -2.56825 q^{61} +6.47323 q^{65} +12.9381 q^{67} -15.1894 q^{69} +5.81754 q^{71} -12.4835 q^{73} -22.2284 q^{75} +2.51028 q^{79} -11.2498 q^{81} +8.61393 q^{83} -20.4829 q^{85} +19.8827 q^{87} -1.18149 q^{89} +13.6879 q^{93} -25.9192 q^{95} -6.32575 q^{97} +6.13173 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 15 q + q^{3} - 3 q^{5} + 30 q^{9}+O(q^{10}) \) 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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