LMFDB - Embedded newform 8036.2.a.m.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8036, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("8036.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8036 = 2^{2} \cdot 7^{2} \cdot 41 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8036.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$64.1677830643$
Analytic rank:	$1$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 14x^{6} - 4x^{5} + 60x^{4} + 31x^{3} - 75x^{2} - 60x - 11 $ x^8 - 14x^6 - 4x^5 + 60x^4 + 31x^3 - 75x^2 - 60x - 11
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1148)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.10300$ 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)

$f(q)$	$=$	$q-2.10300 q^{3} -2.93524 q^{5} +1.42260 q^{9} +O(q^{10})$	$q-2.10300 q^{3} -2.93524 q^{5} +1.42260 q^{9} -5.80589 q^{11} -1.53467 q^{13} +6.17281 q^{15} +5.69116 q^{17} +3.30562 q^{19} -2.90777 q^{23} +3.61563 q^{25} +3.31726 q^{27} -1.53671 q^{29} -6.66063 q^{31} +12.2098 q^{33} -11.3433 q^{37} +3.22742 q^{39} +1.00000 q^{41} -2.99712 q^{43} -4.17568 q^{45} +8.97221 q^{47} -11.9685 q^{51} +2.87561 q^{53} +17.0417 q^{55} -6.95172 q^{57} +9.75975 q^{59} +6.56368 q^{61} +4.50464 q^{65} +8.35339 q^{67} +6.11503 q^{69} +12.2707 q^{71} +2.02087 q^{73} -7.60368 q^{75} +10.3165 q^{79} -11.2440 q^{81} -8.68931 q^{83} -16.7049 q^{85} +3.23169 q^{87} -4.80690 q^{89} +14.0073 q^{93} -9.70280 q^{95} -4.61153 q^{97} -8.25948 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 4 q^{9}+O(q^{10}) $ 8 * q + 4 * q^9	$ 8 q + 4 q^{9} - 8 q^{11} - 7 q^{13} - q^{15} - q^{17} + 4 q^{19} - 3 q^{23} - 4 q^{25} - 12 q^{27} - 4 q^{29} + 4 q^{31} + 23 q^{33} - 31 q^{37} + 5 q^{39} + 8 q^{41} - 8 q^{43} + q^{45} + 24 q^{47} - 23 q^{51} - q^{53} + 2 q^{55} - 15 q^{57} + 4 q^{59} - 4 q^{61} - 24 q^{65} - 21 q^{69} + 8 q^{71} + 11 q^{73} - 15 q^{75} + 14 q^{79} - 28 q^{81} - 42 q^{83} - 20 q^{85} + 25 q^{87} - 11 q^{89} - 27 q^{93} - 15 q^{95} - 16 q^{97} - 8 q^{99}+O(q^{100}) $ 8 * q + 4 * q^9 - 8 * q^11 - 7 * q^13 - q^15 - q^17 + 4 * q^19 - 3 * q^23 - 4 * q^25 - 12 * q^27 - 4 * q^29 + 4 * q^31 + 23 * q^33 - 31 * q^37 + 5 * q^39 + 8 * q^41 - 8 * q^43 + q^45 + 24 * q^47 - 23 * q^51 - q^53 + 2 * q^55 - 15 * q^57 + 4 * q^59 - 4 * q^61 - 24 * q^65 - 21 * q^69 + 8 * q^71 + 11 * q^73 - 15 * q^75 + 14 * q^79 - 28 * q^81 - 42 * q^83 - 20 * q^85 + 25 * q^87 - 11 * q^89 - 27 * q^93 - 15 * q^95 - 16 * q^97 - 8 * 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
$3$	−2.10300	−1.21417	−0.607083	−	0.794638i	$-0.707661\pi$
$3$	−2.10300	−1.21417	−0.607083	+	0.794638i	$0.707661\pi$
$4$	0	0
$5$	−2.93524	−1.31268	−0.656340	−	0.754466i	$-0.727896\pi$
$5$	−2.93524	−1.31268	−0.656340	+	0.754466i	$0.727896\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	1.42260	0.474201
$10$	0	0
$11$	−5.80589	−1.75054	−0.875271	−	0.483633i	$-0.839317\pi$
$11$	−5.80589	−1.75054	−0.875271	+	0.483633i	$0.839317\pi$
$12$	0	0
$13$	−1.53467	−0.425642	−0.212821	−	0.977091i	$-0.568265\pi$
$13$	−1.53467	−0.425642	−0.212821	+	0.977091i	$0.568265\pi$
$14$	0	0
$15$	6.17281	1.59381
$16$	0	0
$17$	5.69116	1.38031	0.690154	−	0.723662i	$-0.257543\pi$
$17$	5.69116	1.38031	0.690154	+	0.723662i	$0.257543\pi$
$18$	0	0
$19$	3.30562	0.758362	0.379181	−	0.925323i	$-0.376206\pi$
$19$	3.30562	0.758362	0.379181	+	0.925323i	$0.376206\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−2.90777	−0.606311	−0.303156	−	0.952941i	$-0.598040\pi$
$23$	−2.90777	−0.606311	−0.303156	+	0.952941i	$0.598040\pi$
$24$	0	0
$25$	3.61563	0.723127
$26$	0	0
$27$	3.31726	0.638407
$28$	0	0
$29$	−1.53671	−0.285359	−0.142680	−	0.989769i	$-0.545572\pi$
$29$	−1.53671	−0.285359	−0.142680	+	0.989769i	$0.545572\pi$
$30$	0	0
$31$	−6.66063	−1.19628	−0.598142	−	0.801390i	$-0.704094\pi$
$31$	−6.66063	−1.19628	−0.598142	+	0.801390i	$0.704094\pi$
$32$	0	0
$33$	12.2098	2.12545
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−11.3433	−1.86482	−0.932411	−	0.361399i	$-0.882299\pi$
$37$	−11.3433	−1.86482	−0.932411	+	0.361399i	$0.882299\pi$
$38$	0	0
$39$	3.22742	0.516801
$40$	0	0
$41$	1.00000	0.156174
$41$	1.00000	0.156174
$42$	0	0
$43$	−2.99712	−0.457057	−0.228528	−	0.973537i	$-0.573391\pi$
$43$	−2.99712	−0.457057	−0.228528	+	0.973537i	$0.573391\pi$
$44$	0	0
$45$	−4.17568	−0.622474
$46$	0	0
$47$	8.97221	1.30873	0.654366	−	0.756178i	$-0.272936\pi$
$47$	8.97221	1.30873	0.654366	+	0.756178i	$0.272936\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−11.9685	−1.67593
$52$	0	0
$53$	2.87561	0.394995	0.197498	−	0.980303i	$-0.436719\pi$
$53$	2.87561	0.394995	0.197498	+	0.980303i	$0.436719\pi$
$54$	0	0
$55$	17.0417	2.29790
$56$	0	0
$57$	−6.95172	−0.920778
$58$	0	0
$59$	9.75975	1.27061	0.635306	−	0.772261i	$-0.280874\pi$
$59$	9.75975	1.27061	0.635306	+	0.772261i	$0.280874\pi$
$60$	0	0
$61$	6.56368	0.840393	0.420197	−	0.907433i	$-0.361961\pi$
$61$	6.56368	0.840393	0.420197	+	0.907433i	$0.361961\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	4.50464	0.558732
$66$	0	0
$67$	8.35339	1.02053	0.510265	−	0.860017i	$-0.329547\pi$
$67$	8.35339	1.02053	0.510265	+	0.860017i	$0.329547\pi$
$68$	0	0
$69$	6.11503	0.736163
$70$	0	0
$71$	12.2707	1.45626	0.728131	−	0.685438i	$-0.240389\pi$
$71$	12.2707	1.45626	0.728131	+	0.685438i	$0.240389\pi$
$72$	0	0
$73$	2.02087	0.236525	0.118262	−	0.992982i	$-0.462268\pi$
$73$	2.02087	0.236525	0.118262	+	0.992982i	$0.462268\pi$
$74$	0	0
$75$	−7.60368	−0.877997
$76$	0	0
$77$	0	0
$78$	0	0
$79$	10.3165	1.16069	0.580346	−	0.814370i	$-0.302917\pi$
$79$	10.3165	1.16069	0.580346	+	0.814370i	$0.302917\pi$
$80$	0	0
$81$	−11.2440	−1.24933
$82$	0	0
$83$	−8.68931	−0.953775	−0.476888	−	0.878964i	$-0.658235\pi$
$83$	−8.68931	−0.953775	−0.476888	+	0.878964i	$0.658235\pi$
$84$	0	0
$85$	−16.7049	−1.81190
$86$	0	0
$87$	3.23169	0.346474
$88$	0	0
$89$	−4.80690	−0.509530	−0.254765	−	0.967003i	$-0.581998\pi$
$89$	−4.80690	−0.509530	−0.254765	+	0.967003i	$0.581998\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	14.0073	1.45249
$94$	0	0
$95$	−9.70280	−0.995486
$96$	0	0
$97$	−4.61153	−0.468230	−0.234115	−	0.972209i	$-0.575219\pi$
$97$	−4.61153	−0.468230	−0.234115	+	0.972209i	$0.575219\pi$
$98$	0	0
$99$	−8.25948	−0.830109
$100$	0	0
$101$	5.60656	0.557874	0.278937	−	0.960309i	$-0.410018\pi$
$101$	5.60656	0.557874	0.278937	+	0.960309i	$0.410018\pi$
$102$	0	0
$103$	15.1374	1.49154	0.745768	−	0.666206i	$-0.232083\pi$
$103$	15.1374	1.49154	0.745768	+	0.666206i	$0.232083\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	13.2187	1.27790	0.638949	−	0.769249i	$-0.279370\pi$
$107$	13.2187	1.27790	0.638949	+	0.769249i	$0.279370\pi$
$108$	0	0
$109$	−11.2266	−1.07532	−0.537658	−	0.843163i	$-0.680691\pi$
$109$	−11.2266	−1.07532	−0.537658	+	0.843163i	$0.680691\pi$
$110$	0	0
$111$	23.8549	2.26421
$112$	0	0
$113$	2.08830	0.196451	0.0982255	−	0.995164i	$-0.468683\pi$
$113$	2.08830	0.196451	0.0982255	+	0.995164i	$0.468683\pi$
$114$	0	0
$115$	8.53499	0.795892
$116$	0	0
$117$	−2.18323	−0.201840
$118$	0	0
$119$	0	0
$120$	0	0
$121$	22.7084	2.06440
$122$	0	0
$123$	−2.10300	−0.189621
$124$	0	0
$125$	4.06344	0.363445
$126$	0	0
$127$	−0.678117	−0.0601732	−0.0300866	−	0.999547i	$-0.509578\pi$
$127$	−0.678117	−0.0601732	−0.0300866	+	0.999547i	$0.509578\pi$
$128$	0	0
$129$	6.30294	0.554943
$130$	0	0
$131$	1.82742	0.159663	0.0798314	−	0.996808i	$-0.474562\pi$
$131$	1.82742	0.159663	0.0798314	+	0.996808i	$0.474562\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−9.73696	−0.838024
$136$	0	0
$137$	1.83637	0.156892	0.0784460	−	0.996918i	$-0.475004\pi$
$137$	1.83637	0.156892	0.0784460	+	0.996918i	$0.475004\pi$
$138$	0	0
$139$	−1.74423	−0.147944	−0.0739719	−	0.997260i	$-0.523568\pi$
$139$	−1.74423	−0.147944	−0.0739719	+	0.997260i	$0.523568\pi$
$140$	0	0
$141$	−18.8686	−1.58902
$142$	0	0
$143$	8.91015	0.745105
$144$	0	0
$145$	4.51060	0.374585
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−7.22001	−0.591486	−0.295743	−	0.955268i	$-0.595567\pi$
$149$	−7.22001	−0.591486	−0.295743	+	0.955268i	$0.595567\pi$
$150$	0	0
$151$	4.92765	0.401007	0.200503	−	0.979693i	$-0.435742\pi$
$151$	4.92765	0.401007	0.200503	+	0.979693i	$0.435742\pi$
$152$	0	0
$153$	8.09627	0.654544
$154$	0	0
$155$	19.5505	1.57034
$156$	0	0
$157$	16.3603	1.30569	0.652845	−	0.757491i	$-0.273575\pi$
$157$	16.3603	1.30569	0.652845	+	0.757491i	$0.273575\pi$
$158$	0	0
$159$	−6.04740	−0.479590
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−21.0626	−1.64975	−0.824877	−	0.565313i	$-0.808755\pi$
$163$	−21.0626	−1.64975	−0.824877	+	0.565313i	$0.808755\pi$
$164$	0	0
$165$	−35.8386	−2.79003
$166$	0	0
$167$	−17.3127	−1.33970	−0.669849	−	0.742497i	$-0.733641\pi$
$167$	−17.3127	−1.33970	−0.669849	+	0.742497i	$0.733641\pi$
$168$	0	0
$169$	−10.6448	−0.818829
$170$	0	0
$171$	4.70259	0.359616
$172$	0	0
$173$	−10.0958	−0.767570	−0.383785	−	0.923423i	$-0.625380\pi$
$173$	−10.0958	−0.767570	−0.383785	+	0.923423i	$0.625380\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−20.5247	−1.54273
$178$	0	0
$179$	16.0269	1.19791	0.598955	−	0.800783i	$-0.295583\pi$
$179$	16.0269	1.19791	0.598955	+	0.800783i	$0.295583\pi$
$180$	0	0
$181$	9.31417	0.692317	0.346158	−	0.938176i	$-0.387486\pi$
$181$	9.31417	0.692317	0.346158	+	0.938176i	$0.387486\pi$
$182$	0	0
$183$	−13.8034	−1.02038
$184$	0	0
$185$	33.2952	2.44791
$186$	0	0
$187$	−33.0422	−2.41629
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−4.04040	−0.292353	−0.146176	−	0.989259i	$-0.546697\pi$
$191$	−4.04040	−0.292353	−0.146176	+	0.989259i	$0.546697\pi$
$192$	0	0
$193$	11.8584	0.853584	0.426792	−	0.904350i	$-0.359644\pi$
$193$	11.8584	0.853584	0.426792	+	0.904350i	$0.359644\pi$
$194$	0	0
$195$	−9.47325	−0.678394
$196$	0	0
$197$	−10.4773	−0.746473	−0.373237	−	0.927736i	$-0.621752\pi$
$197$	−10.4773	−0.746473	−0.373237	+	0.927736i	$0.621752\pi$
$198$	0	0
$199$	−10.7571	−0.762548	−0.381274	−	0.924462i	$-0.624515\pi$
$199$	−10.7571	−0.762548	−0.381274	+	0.924462i	$0.624515\pi$
$200$	0	0
$201$	−17.5672	−1.23909
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−2.93524	−0.205006
$206$	0	0
$207$	−4.13660	−0.287514
$208$	0	0
$209$	−19.1921	−1.32754
$210$	0	0
$211$	−23.7031	−1.63179	−0.815896	−	0.578199i	$-0.803756\pi$
$211$	−23.7031	−1.63179	−0.815896	+	0.578199i	$0.803756\pi$
$212$	0	0
$213$	−25.8052	−1.76814
$214$	0	0
$215$	8.79727	0.599969
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−4.24989	−0.287181
$220$	0	0
$221$	−8.73408	−0.587518
$222$	0	0
$223$	−10.6405	−0.712538	−0.356269	−	0.934383i	$-0.615951\pi$
$223$	−10.6405	−0.712538	−0.356269	+	0.934383i	$0.615951\pi$
$224$	0	0
$225$	5.14362	0.342908
$226$	0	0
$227$	−12.5501	−0.832979	−0.416490	−	0.909140i	$-0.636740\pi$
$227$	−12.5501	−0.832979	−0.416490	+	0.909140i	$0.636740\pi$
$228$	0	0
$229$	15.2707	1.00911	0.504557	−	0.863378i	$-0.331656\pi$
$229$	15.2707	1.00911	0.504557	+	0.863378i	$0.331656\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−13.4022	−0.878006	−0.439003	−	0.898486i	$-0.644668\pi$
$233$	−13.4022	−0.878006	−0.439003	+	0.898486i	$0.644668\pi$
$234$	0	0
$235$	−26.3356	−1.71795
$236$	0	0
$237$	−21.6955	−1.40927
$238$	0	0
$239$	30.6159	1.98037	0.990187	−	0.139746i	$-0.0446287\pi$
$239$	30.6159	1.98037	0.990187	+	0.139746i	$0.0446287\pi$
$240$	0	0
$241$	7.16419	0.461486	0.230743	−	0.973015i	$-0.425884\pi$
$241$	7.16419	0.461486	0.230743	+	0.973015i	$0.425884\pi$
$242$	0	0
$243$	13.6944	0.878493
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−5.07306	−0.322791
$248$	0	0
$249$	18.2736	1.15804
$250$	0	0
$251$	−25.6079	−1.61636	−0.808179	−	0.588937i	$-0.799547\pi$
$251$	−25.6079	−1.61636	−0.808179	+	0.588937i	$0.799547\pi$
$252$	0	0
$253$	16.8822	1.06137
$254$	0	0
$255$	35.1304	2.19995
$256$	0	0
$257$	−27.8166	−1.73515	−0.867576	−	0.497304i	$-0.834323\pi$
$257$	−27.8166	−1.73515	−0.867576	+	0.497304i	$0.834323\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−2.18613	−0.135318
$262$	0	0
$263$	24.1217	1.48741	0.743704	−	0.668509i	$-0.233067\pi$
$263$	24.1217	1.48741	0.743704	+	0.668509i	$0.233067\pi$
$264$	0	0
$265$	−8.44060	−0.518502
$266$	0	0
$267$	10.1089	0.618654
$268$	0	0
$269$	7.80055	0.475608	0.237804	−	0.971313i	$-0.423572\pi$
$269$	7.80055	0.475608	0.237804	+	0.971313i	$0.423572\pi$
$270$	0	0
$271$	9.38009	0.569800	0.284900	−	0.958557i	$-0.408040\pi$
$271$	9.38009	0.569800	0.284900	+	0.958557i	$0.408040\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−20.9920	−1.26586
$276$	0	0
$277$	−29.5839	−1.77753	−0.888764	−	0.458366i	$-0.848435\pi$
$277$	−29.5839	−1.77753	−0.888764	+	0.458366i	$0.848435\pi$
$278$	0	0
$279$	−9.47544	−0.567280
$280$	0	0
$281$	25.8917	1.54457	0.772285	−	0.635277i	$-0.219114\pi$
$281$	25.8917	1.54457	0.772285	+	0.635277i	$0.219114\pi$
$282$	0	0
$283$	13.1227	0.780066	0.390033	−	0.920801i	$-0.372464\pi$
$283$	13.1227	0.780066	0.390033	+	0.920801i	$0.372464\pi$
$284$	0	0
$285$	20.4050	1.20869
$286$	0	0
$287$	0	0
$288$	0	0
$289$	15.3893	0.905253
$290$	0	0
$291$	9.69805	0.568509
$292$	0	0
$293$	23.7546	1.38776	0.693880	−	0.720091i	$-0.255900\pi$
$293$	23.7546	1.38776	0.693880	+	0.720091i	$0.255900\pi$
$294$	0	0
$295$	−28.6472	−1.66791
$296$	0	0
$297$	−19.2597	−1.11756
$298$	0	0
$299$	4.46248	0.258072
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−11.7906	−0.677352
$304$	0	0
$305$	−19.2660	−1.10317
$306$	0	0
$307$	−26.9650	−1.53898	−0.769488	−	0.638662i	$-0.779488\pi$
$307$	−26.9650	−1.53898	−0.769488	+	0.638662i	$0.779488\pi$
$308$	0	0
$309$	−31.8340	−1.81097
$310$	0	0
$311$	22.7997	1.29285	0.646426	−	0.762976i	$-0.276263\pi$
$311$	22.7997	1.29285	0.646426	+	0.762976i	$0.276263\pi$
$312$	0	0
$313$	22.6585	1.28073	0.640367	−	0.768069i	$-0.278782\pi$
$313$	22.6585	1.28073	0.640367	+	0.768069i	$0.278782\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−1.15355	−0.0647900	−0.0323950	−	0.999475i	$-0.510313\pi$
$317$	−1.15355	−0.0647900	−0.0323950	+	0.999475i	$0.510313\pi$
$318$	0	0
$319$	8.92195	0.499533
$320$	0	0
$321$	−27.7989	−1.55158
$322$	0	0
$323$	18.8128	1.04677
$324$	0	0
$325$	−5.54882	−0.307793
$326$	0	0
$327$	23.6096	1.30561
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−21.9494	−1.20645	−0.603225	−	0.797571i	$-0.706118\pi$
$331$	−21.9494	−1.20645	−0.603225	+	0.797571i	$0.706118\pi$
$332$	0	0
$333$	−16.1370	−0.884301
$334$	0	0
$335$	−24.5192	−1.33963
$336$	0	0
$337$	−1.27042	−0.0692040	−0.0346020	−	0.999401i	$-0.511016\pi$
$337$	−1.27042	−0.0692040	−0.0346020	+	0.999401i	$0.511016\pi$
$338$	0	0
$339$	−4.39170	−0.238524
$340$	0	0
$341$	38.6709	2.09415
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−17.9491	−0.966346
$346$	0	0
$347$	−8.48096	−0.455282	−0.227641	−	0.973745i	$-0.573101\pi$
$347$	−8.48096	−0.455282	−0.227641	+	0.973745i	$0.573101\pi$
$348$	0	0
$349$	−25.9408	−1.38858	−0.694290	−	0.719695i	$-0.744282\pi$
$349$	−25.9408	−1.38858	−0.694290	+	0.719695i	$0.744282\pi$
$350$	0	0
$351$	−5.09092	−0.271733
$352$	0	0
$353$	−14.5847	−0.776264	−0.388132	−	0.921604i	$-0.626880\pi$
$353$	−14.5847	−0.776264	−0.388132	+	0.921604i	$0.626880\pi$
$354$	0	0
$355$	−36.0174	−1.91160
$356$	0	0
$357$	0	0
$358$	0	0
$359$	22.5018	1.18760	0.593799	−	0.804613i	$-0.297627\pi$
$359$	22.5018	1.18760	0.593799	+	0.804613i	$0.297627\pi$
$360$	0	0
$361$	−8.07286	−0.424887
$362$	0	0
$363$	−47.7557	−2.50652
$364$	0	0
$365$	−5.93174	−0.310481
$366$	0	0
$367$	3.33991	0.174342	0.0871709	−	0.996193i	$-0.472217\pi$
$367$	3.33991	0.174342	0.0871709	+	0.996193i	$0.472217\pi$
$368$	0	0
$369$	1.42260	0.0740578
$370$	0	0
$371$	0	0
$372$	0	0
$373$	27.6430	1.43130	0.715649	−	0.698460i	$-0.246131\pi$
$373$	27.6430	1.43130	0.715649	+	0.698460i	$0.246131\pi$
$374$	0	0
$375$	−8.54542	−0.441283
$376$	0	0
$377$	2.35835	0.121461
$378$	0	0
$379$	−20.8269	−1.06981	−0.534903	−	0.844913i	$-0.679652\pi$
$379$	−20.8269	−1.06981	−0.534903	+	0.844913i	$0.679652\pi$
$380$	0	0
$381$	1.42608	0.0730603
$382$	0	0
$383$	11.3048	0.577648	0.288824	−	0.957382i	$-0.406736\pi$
$383$	11.3048	0.577648	0.288824	+	0.957382i	$0.406736\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−4.26372	−0.216737
$388$	0	0
$389$	6.04606	0.306548	0.153274	−	0.988184i	$-0.451018\pi$
$389$	6.04606	0.306548	0.153274	+	0.988184i	$0.451018\pi$
$390$	0	0
$391$	−16.5486	−0.836897
$392$	0	0
$393$	−3.84307	−0.193857
$394$	0	0
$395$	−30.2813	−1.52362
$396$	0	0
$397$	−14.1774	−0.711545	−0.355772	−	0.934573i	$-0.615782\pi$
$397$	−14.1774	−0.711545	−0.355772	+	0.934573i	$0.615782\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−12.1255	−0.605519	−0.302759	−	0.953067i	$-0.597908\pi$
$401$	−12.1255	−0.605519	−0.302759	+	0.953067i	$0.597908\pi$
$402$	0	0
$403$	10.2219	0.509189
$404$	0	0
$405$	33.0039	1.63998
$406$	0	0
$407$	65.8578	3.26445
$408$	0	0
$409$	−5.48656	−0.271293	−0.135646	−	0.990757i	$-0.543311\pi$
$409$	−5.48656	−0.271293	−0.135646	+	0.990757i	$0.543311\pi$
$410$	0	0
$411$	−3.86189	−0.190493
$412$	0	0
$413$	0	0
$414$	0	0
$415$	25.5052	1.25200
$416$	0	0
$417$	3.66812	0.179629
$418$	0	0
$419$	5.94192	0.290282	0.145141	−	0.989411i	$-0.453636\pi$
$419$	5.94192	0.290282	0.145141	+	0.989411i	$0.453636\pi$
$420$	0	0
$421$	−37.0802	−1.80718	−0.903588	−	0.428402i	$-0.859077\pi$
$421$	−37.0802	−1.80718	−0.903588	+	0.428402i	$0.859077\pi$
$422$	0	0
$423$	12.7639	0.620602
$424$	0	0
$425$	20.5772	0.998139
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−18.7380	−0.904681
$430$	0	0
$431$	−27.5964	−1.32927	−0.664636	−	0.747167i	$-0.731413\pi$
$431$	−27.5964	−1.32927	−0.664636	+	0.747167i	$0.731413\pi$
$432$	0	0
$433$	−19.3644	−0.930595	−0.465297	−	0.885154i	$-0.654053\pi$
$433$	−19.3644	−0.930595	−0.465297	+	0.885154i	$0.654053\pi$
$434$	0	0
$435$	−9.48579	−0.454809
$436$	0	0
$437$	−9.61198	−0.459803
$438$	0	0
$439$	−25.2009	−1.20277	−0.601387	−	0.798958i	$-0.705385\pi$
$439$	−25.2009	−1.20277	−0.601387	+	0.798958i	$0.705385\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	37.2363	1.76915	0.884576	−	0.466396i	$-0.154448\pi$
$443$	37.2363	1.76915	0.884576	+	0.466396i	$0.154448\pi$
$444$	0	0
$445$	14.1094	0.668849
$446$	0	0
$447$	15.1837	0.718163
$448$	0	0
$449$	−17.6022	−0.830698	−0.415349	−	0.909662i	$-0.636341\pi$
$449$	−17.6022	−0.830698	−0.415349	+	0.909662i	$0.636341\pi$
$450$	0	0
$451$	−5.80589	−0.273389
$452$	0	0
$453$	−10.3628	−0.486889
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−27.7729	−1.29916	−0.649581	−	0.760292i	$-0.725056\pi$
$457$	−27.7729	−1.29916	−0.649581	+	0.760292i	$0.725056\pi$
$458$	0	0
$459$	18.8791	0.881199
$460$	0	0
$461$	−5.78612	−0.269486	−0.134743	−	0.990881i	$-0.543021\pi$
$461$	−5.78612	−0.269486	−0.134743	+	0.990881i	$0.543021\pi$
$462$	0	0
$463$	38.3113	1.78048	0.890238	−	0.455496i	$-0.150538\pi$
$463$	38.3113	1.78048	0.890238	+	0.455496i	$0.150538\pi$
$464$	0	0
$465$	−41.1148	−1.90665
$466$	0	0
$467$	−19.6031	−0.907123	−0.453562	−	0.891225i	$-0.649847\pi$
$467$	−19.6031	−0.907123	−0.453562	+	0.891225i	$0.649847\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−34.4056	−1.58533
$472$	0	0
$473$	17.4010	0.800097
$474$	0	0
$475$	11.9519	0.548392
$476$	0	0
$477$	4.09085	0.187307
$478$	0	0
$479$	34.7579	1.58813	0.794065	−	0.607833i	$-0.207961\pi$
$479$	34.7579	1.58813	0.794065	+	0.607833i	$0.207961\pi$
$480$	0	0
$481$	17.4082	0.793747
$482$	0	0
$483$	0	0
$484$	0	0
$485$	13.5360	0.614636
$486$	0	0
$487$	7.94609	0.360072	0.180036	−	0.983660i	$-0.442379\pi$
$487$	7.94609	0.360072	0.180036	+	0.983660i	$0.442379\pi$
$488$	0	0
$489$	44.2947	2.00308
$490$	0	0
$491$	14.6853	0.662736	0.331368	−	0.943502i	$-0.392490\pi$
$491$	14.6853	0.662736	0.331368	+	0.943502i	$0.392490\pi$
$492$	0	0
$493$	−8.74564	−0.393884
$494$	0	0
$495$	24.2436	1.08967
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−0.298874	−0.0133794	−0.00668971	−	0.999978i	$-0.502129\pi$
$499$	−0.298874	−0.0133794	−0.00668971	+	0.999978i	$0.502129\pi$
$500$	0	0
$501$	36.4086	1.62662
$502$	0	0
$503$	17.9191	0.798971	0.399486	−	0.916739i	$-0.369189\pi$
$503$	17.9191	0.798971	0.399486	+	0.916739i	$0.369189\pi$
$504$	0	0
$505$	−16.4566	−0.732309
$506$	0	0
$507$	22.3859	0.994195
$508$	0	0
$509$	−3.92726	−0.174073	−0.0870363	−	0.996205i	$-0.527740\pi$
$509$	−3.92726	−0.174073	−0.0870363	+	0.996205i	$0.527740\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	10.9656	0.484144
$514$	0	0
$515$	−44.4320	−1.95791
$516$	0	0
$517$	−52.0917	−2.29099
$518$	0	0
$519$	21.2315	0.931958
$520$	0	0
$521$	−37.5532	−1.64524	−0.822619	−	0.568593i	$-0.807488\pi$
$521$	−37.5532	−1.64524	−0.822619	+	0.568593i	$0.807488\pi$
$522$	0	0
$523$	1.34686	0.0588941	0.0294470	−	0.999566i	$-0.490625\pi$
$523$	1.34686	0.0588941	0.0294470	+	0.999566i	$0.490625\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−37.9067	−1.65124
$528$	0	0
$529$	−14.5449	−0.632387
$530$	0	0
$531$	13.8843	0.602526
$532$	0	0
$533$	−1.53467	−0.0664741
$534$	0	0
$535$	−38.8000	−1.67747
$536$	0	0
$537$	−33.7046	−1.45446
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−12.4707	−0.536159	−0.268080	−	0.963397i	$-0.586389\pi$
$541$	−12.4707	−0.536159	−0.268080	+	0.963397i	$0.586389\pi$
$542$	0	0
$543$	−19.5877	−0.840588
$544$	0	0
$545$	32.9528	1.41154
$546$	0	0
$547$	−8.05835	−0.344550	−0.172275	−	0.985049i	$-0.555112\pi$
$547$	−8.05835	−0.344550	−0.172275	+	0.985049i	$0.555112\pi$
$548$	0	0
$549$	9.33752	0.398516
$550$	0	0
$551$	−5.07977	−0.216406
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−70.0198	−2.97218
$556$	0	0
$557$	−18.9012	−0.800871	−0.400436	−	0.916325i	$-0.631141\pi$
$557$	−18.9012	−0.800871	−0.400436	+	0.916325i	$0.631141\pi$
$558$	0	0
$559$	4.59961	0.194543
$560$	0	0
$561$	69.4878	2.93378
$562$	0	0
$563$	29.9533	1.26238	0.631190	−	0.775628i	$-0.282567\pi$
$563$	29.9533	1.26238	0.631190	+	0.775628i	$0.282567\pi$
$564$	0	0
$565$	−6.12967	−0.257877
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−7.24713	−0.303816	−0.151908	−	0.988395i	$-0.548542\pi$
$569$	−7.24713	−0.303816	−0.151908	+	0.988395i	$0.548542\pi$
$570$	0	0
$571$	−34.9998	−1.46470	−0.732348	−	0.680930i	$-0.761576\pi$
$571$	−34.9998	−1.46470	−0.732348	+	0.680930i	$0.761576\pi$
$572$	0	0
$573$	8.49695	0.354965
$574$	0	0
$575$	−10.5134	−0.438440
$576$	0	0
$577$	−42.1125	−1.75317	−0.876583	−	0.481251i	$-0.840183\pi$
$577$	−42.1125	−1.75317	−0.876583	+	0.481251i	$0.840183\pi$
$578$	0	0
$579$	−24.9381	−1.03639
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−16.6955	−0.691456
$584$	0	0
$585$	6.40832	0.264951
$586$	0	0
$587$	30.8138	1.27182	0.635910	−	0.771763i	$-0.280625\pi$
$587$	30.8138	1.27182	0.635910	+	0.771763i	$0.280625\pi$
$588$	0	0
$589$	−22.0175	−0.907216
$590$	0	0
$591$	22.0336	0.906343
$592$	0	0
$593$	−30.8266	−1.26590	−0.632949	−	0.774194i	$-0.718156\pi$
$593$	−30.8266	−1.26590	−0.632949	+	0.774194i	$0.718156\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	22.6221	0.925860
$598$	0	0
$599$	−10.7094	−0.437574	−0.218787	−	0.975773i	$-0.570210\pi$
$599$	−10.7094	−0.437574	−0.218787	+	0.975773i	$0.570210\pi$
$600$	0	0
$601$	−9.62891	−0.392771	−0.196386	−	0.980527i	$-0.562920\pi$
$601$	−9.62891	−0.392771	−0.196386	+	0.980527i	$0.562920\pi$
$602$	0	0
$603$	11.8836	0.483936
$604$	0	0
$605$	−66.6545	−2.70989
$606$	0	0
$607$	38.9387	1.58048	0.790238	−	0.612801i	$-0.209957\pi$
$607$	38.9387	1.58048	0.790238	+	0.612801i	$0.209957\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−13.7694	−0.557052
$612$	0	0
$613$	−3.32006	−0.134096	−0.0670481	−	0.997750i	$-0.521358\pi$
$613$	−3.32006	−0.134096	−0.0670481	+	0.997750i	$0.521358\pi$
$614$	0	0
$615$	6.17281	0.248912
$616$	0	0
$617$	−0.562483	−0.0226447	−0.0113224	−	0.999936i	$-0.503604\pi$
$617$	−0.562483	−0.0226447	−0.0113224	+	0.999936i	$0.503604\pi$
$618$	0	0
$619$	−2.20586	−0.0886609	−0.0443304	−	0.999017i	$-0.514115\pi$
$619$	−2.20586	−0.0886609	−0.0443304	+	0.999017i	$0.514115\pi$
$620$	0	0
$621$	−9.64582	−0.387073
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−30.0054	−1.20021
$626$	0	0
$627$	40.3609	1.61186
$628$	0	0
$629$	−64.5564	−2.57403
$630$	0	0
$631$	25.9567	1.03332	0.516660	−	0.856191i	$-0.327175\pi$
$631$	25.9567	1.03332	0.516660	+	0.856191i	$0.327175\pi$
$632$	0	0
$633$	49.8477	1.98127
$634$	0	0
$635$	1.99044	0.0789880
$636$	0	0
$637$	0	0
$638$	0	0
$639$	17.4563	0.690561
$640$	0	0
$641$	−22.7400	−0.898176	−0.449088	−	0.893488i	$-0.648251\pi$
$641$	−22.7400	−0.898176	−0.449088	+	0.893488i	$0.648251\pi$
$642$	0	0
$643$	−8.75563	−0.345288	−0.172644	−	0.984984i	$-0.555231\pi$
$643$	−8.75563	−0.345288	−0.172644	+	0.984984i	$0.555231\pi$
$644$	0	0
$645$	−18.5007	−0.728463
$646$	0	0
$647$	−20.5132	−0.806456	−0.403228	−	0.915100i	$-0.632112\pi$
$647$	−20.5132	−0.806456	−0.403228	+	0.915100i	$0.632112\pi$
$648$	0	0
$649$	−56.6640	−2.22426
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−12.4986	−0.489108	−0.244554	−	0.969636i	$-0.578642\pi$
$653$	−12.4986	−0.489108	−0.244554	+	0.969636i	$0.578642\pi$
$654$	0	0
$655$	−5.36393	−0.209586
$656$	0	0
$657$	2.87490	0.112160
$658$	0	0
$659$	16.8784	0.657490	0.328745	−	0.944419i	$-0.393374\pi$
$659$	16.8784	0.657490	0.328745	+	0.944419i	$0.393374\pi$
$660$	0	0
$661$	9.12923	0.355086	0.177543	−	0.984113i	$-0.443185\pi$
$661$	9.12923	0.355086	0.177543	+	0.984113i	$0.443185\pi$
$662$	0	0
$663$	18.3678	0.713345
$664$	0	0
$665$	0	0
$666$	0	0
$667$	4.46838	0.173017
$668$	0	0
$669$	22.3769	0.865140
$670$	0	0
$671$	−38.1080	−1.47114
$672$	0	0
$673$	3.59435	0.138552	0.0692760	−	0.997598i	$-0.477931\pi$
$673$	3.59435	0.138552	0.0692760	+	0.997598i	$0.477931\pi$
$674$	0	0
$675$	11.9940	0.461650
$676$	0	0
$677$	31.3826	1.20613	0.603066	−	0.797691i	$-0.293946\pi$
$677$	31.3826	1.20613	0.603066	+	0.797691i	$0.293946\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	26.3928	1.01138
$682$	0	0
$683$	−35.0517	−1.34121	−0.670607	−	0.741813i	$-0.733966\pi$
$683$	−35.0517	−1.34121	−0.670607	+	0.741813i	$0.733966\pi$
$684$	0	0
$685$	−5.39020	−0.205949
$686$	0	0
$687$	−32.1142	−1.22523
$688$	0	0
$689$	−4.41312	−0.168127
$690$	0	0
$691$	39.8033	1.51419	0.757095	−	0.653305i	$-0.226618\pi$
$691$	39.8033	1.51419	0.757095	+	0.653305i	$0.226618\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	5.11974	0.194203
$696$	0	0
$697$	5.69116	0.215568
$698$	0	0
$699$	28.1848	1.06605
$700$	0	0
$701$	23.7679	0.897703	0.448852	−	0.893606i	$-0.351833\pi$
$701$	23.7679	0.897703	0.448852	+	0.893606i	$0.351833\pi$
$702$	0	0
$703$	−37.4966	−1.41421
$704$	0	0
$705$	55.3837	2.08587
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−1.82726	−0.0686241	−0.0343120	−	0.999411i	$-0.510924\pi$
$709$	−1.82726	−0.0686241	−0.0343120	+	0.999411i	$0.510924\pi$
$710$	0	0
$711$	14.6762	0.550402
$712$	0	0
$713$	19.3675	0.725320
$714$	0	0
$715$	−26.1534	−0.978083
$716$	0	0
$717$	−64.3851	−2.40451
$718$	0	0
$719$	−22.4992	−0.839078	−0.419539	−	0.907737i	$-0.637808\pi$
$719$	−22.4992	−0.839078	−0.419539	+	0.907737i	$0.637808\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−15.0663	−0.560321
$724$	0	0
$725$	−5.55617	−0.206351
$726$	0	0
$727$	−16.1827	−0.600182	−0.300091	−	0.953911i	$-0.597017\pi$
$727$	−16.1827	−0.600182	−0.300091	+	0.953911i	$0.597017\pi$
$728$	0	0
$729$	4.93282	0.182697
$730$	0	0
$731$	−17.0571	−0.630880
$732$	0	0
$733$	−47.5431	−1.75605	−0.878023	−	0.478618i	$-0.841138\pi$
$733$	−47.5431	−1.75605	−0.878023	+	0.478618i	$0.841138\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−48.4989	−1.78648
$738$	0	0
$739$	−37.2260	−1.36938	−0.684691	−	0.728834i	$-0.740063\pi$
$739$	−37.2260	−1.36938	−0.684691	+	0.728834i	$0.740063\pi$
$740$	0	0
$741$	10.6686	0.391922
$742$	0	0
$743$	−10.8904	−0.399531	−0.199766	−	0.979844i	$-0.564018\pi$
$743$	−10.8904	−0.399531	−0.199766	+	0.979844i	$0.564018\pi$
$744$	0	0
$745$	21.1925	0.776432
$746$	0	0
$747$	−12.3614	−0.452282
$748$	0	0
$749$	0	0
$750$	0	0
$751$	29.5471	1.07819	0.539094	−	0.842245i	$-0.318767\pi$
$751$	29.5471	1.07819	0.539094	+	0.842245i	$0.318767\pi$
$752$	0	0
$753$	53.8535	1.96253
$754$	0	0
$755$	−14.4638	−0.526393
$756$	0	0
$757$	−16.0007	−0.581557	−0.290778	−	0.956790i	$-0.593914\pi$
$757$	−16.0007	−0.581557	−0.290778	+	0.956790i	$0.593914\pi$
$758$	0	0
$759$	−35.5032	−1.28868
$760$	0	0
$761$	36.4027	1.31960	0.659799	−	0.751442i	$-0.270641\pi$
$761$	36.4027	1.31960	0.659799	+	0.751442i	$0.270641\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−23.7645	−0.859207
$766$	0	0
$767$	−14.9780	−0.540826
$768$	0	0
$769$	−22.5697	−0.813886	−0.406943	−	0.913454i	$-0.633405\pi$
$769$	−22.5697	−0.813886	−0.406943	+	0.913454i	$0.633405\pi$
$770$	0	0
$771$	58.4983	2.10677
$772$	0	0
$773$	31.6303	1.13766	0.568832	−	0.822454i	$-0.307396\pi$
$773$	31.6303	1.13766	0.568832	+	0.822454i	$0.307396\pi$
$774$	0	0
$775$	−24.0824	−0.865065
$776$	0	0
$777$	0	0
$778$	0	0
$779$	3.30562	0.118436
$780$	0	0
$781$	−71.2422	−2.54925
$782$	0	0
$783$	−5.09766	−0.182175
$784$	0	0
$785$	−48.0213	−1.71395
$786$	0	0
$787$	35.4344	1.26310	0.631551	−	0.775335i	$-0.282419\pi$
$787$	35.4344	1.26310	0.631551	+	0.775335i	$0.282419\pi$
$788$	0	0
$789$	−50.7279	−1.80596
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−10.0731	−0.357707
$794$	0	0
$795$	17.7506	0.629548
$796$	0	0
$797$	19.9555	0.706861	0.353430	−	0.935461i	$-0.385015\pi$
$797$	19.9555	0.706861	0.353430	+	0.935461i	$0.385015\pi$
$798$	0	0
$799$	51.0623	1.80645
$800$	0	0
$801$	−6.83831	−0.241620
$802$	0	0
$803$	−11.7330	−0.414047
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−16.4045	−0.577467
$808$	0	0
$809$	29.6620	1.04286	0.521431	−	0.853293i	$-0.325398\pi$
$809$	29.6620	1.04286	0.521431	+	0.853293i	$0.325398\pi$
$810$	0	0
$811$	4.22988	0.148531	0.0742655	−	0.997239i	$-0.476339\pi$
$811$	4.22988	0.148531	0.0742655	+	0.997239i	$0.476339\pi$
$812$	0	0
$813$	−19.7263	−0.691832
$814$	0	0
$815$	61.8239	2.16560
$816$	0	0
$817$	−9.90736	−0.346615
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−11.8792	−0.414587	−0.207293	−	0.978279i	$-0.566465\pi$
$821$	−11.8792	−0.414587	−0.207293	+	0.978279i	$0.566465\pi$
$822$	0	0
$823$	−9.92362	−0.345916	−0.172958	−	0.984929i	$-0.555332\pi$
$823$	−9.92362	−0.345916	−0.172958	+	0.984929i	$0.555332\pi$
$824$	0	0
$825$	44.1461	1.53697
$826$	0	0
$827$	34.7169	1.20723	0.603613	−	0.797278i	$-0.293727\pi$
$827$	34.7169	1.20723	0.603613	+	0.797278i	$0.293727\pi$
$828$	0	0
$829$	−5.40968	−0.187886	−0.0939431	−	0.995578i	$-0.529947\pi$
$829$	−5.40968	−0.187886	−0.0939431	+	0.995578i	$0.529947\pi$
$830$	0	0
$831$	62.2150	2.15821
$832$	0	0
$833$	0	0
$834$	0	0
$835$	50.8170	1.75859
$836$	0	0
$837$	−22.0950	−0.763717
$838$	0	0
$839$	8.25395	0.284958	0.142479	−	0.989798i	$-0.454493\pi$
$839$	8.25395	0.284958	0.142479	+	0.989798i	$0.454493\pi$
$840$	0	0
$841$	−26.6385	−0.918570
$842$	0	0
$843$	−54.4502	−1.87536
$844$	0	0
$845$	31.2450	1.07486
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−27.5971	−0.947130
$850$	0	0
$851$	32.9836	1.13066
$852$	0	0
$853$	0.615237	0.0210653	0.0105327	−	0.999945i	$-0.496647\pi$
$853$	0.615237	0.0210653	0.0105327	+	0.999945i	$0.496647\pi$
$854$	0	0
$855$	−13.8032	−0.472061
$856$	0	0
$857$	51.0864	1.74508	0.872539	−	0.488544i	$-0.162472\pi$
$857$	51.0864	1.74508	0.872539	+	0.488544i	$0.162472\pi$
$858$	0	0
$859$	40.2604	1.37367	0.686834	−	0.726815i	$-0.259000\pi$
$859$	40.2604	1.37367	0.686834	+	0.726815i	$0.259000\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−53.0494	−1.80582	−0.902911	−	0.429828i	$-0.858574\pi$
$863$	−53.0494	−1.80582	−0.902911	+	0.429828i	$0.858574\pi$
$864$	0	0
$865$	29.6336	1.00757
$866$	0	0
$867$	−32.3637	−1.09913
$868$	0	0
$869$	−59.8963	−2.03184
$870$	0	0
$871$	−12.8197	−0.434380
$872$	0	0
$873$	−6.56038	−0.222035
$874$	0	0
$875$	0	0
$876$	0	0
$877$	24.2035	0.817296	0.408648	−	0.912692i	$-0.366000\pi$
$877$	24.2035	0.817296	0.408648	+	0.912692i	$0.366000\pi$
$878$	0	0
$879$	−49.9560	−1.68497
$880$	0	0
$881$	38.2014	1.28704	0.643520	−	0.765430i	$-0.277473\pi$
$881$	38.2014	1.28704	0.643520	+	0.765430i	$0.277473\pi$
$882$	0	0
$883$	−7.92322	−0.266637	−0.133319	−	0.991073i	$-0.542563\pi$
$883$	−7.92322	−0.266637	−0.133319	+	0.991073i	$0.542563\pi$
$884$	0	0
$885$	60.2451	2.02512
$886$	0	0
$887$	51.6881	1.73552	0.867758	−	0.496986i	$-0.165560\pi$
$887$	51.6881	1.73552	0.867758	+	0.496986i	$0.165560\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	65.2815	2.18701
$892$	0	0
$893$	29.6588	0.992493
$894$	0	0
$895$	−47.0429	−1.57247
$896$	0	0
$897$	−9.38458	−0.313342
$898$	0	0
$899$	10.2354	0.341371
$900$	0	0
$901$	16.3655	0.545215
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−27.3393	−0.908790
$906$	0	0
$907$	−33.4125	−1.10944	−0.554722	−	0.832036i	$-0.687175\pi$
$907$	−33.4125	−1.10944	−0.554722	+	0.832036i	$0.687175\pi$
$908$	0	0
$909$	7.97592	0.264545
$910$	0	0
$911$	−46.1058	−1.52755	−0.763777	−	0.645480i	$-0.776657\pi$
$911$	−46.1058	−1.52755	−0.763777	+	0.645480i	$0.776657\pi$
$912$	0	0
$913$	50.4492	1.66962
$914$	0	0
$915$	40.5163	1.33943
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−9.07070	−0.299215	−0.149607	−	0.988745i	$-0.547801\pi$
$919$	−9.07070	−0.299215	−0.149607	+	0.988745i	$0.547801\pi$
$920$	0	0
$921$	56.7074	1.86857
$922$	0	0
$923$	−18.8315	−0.619846
$924$	0	0
$925$	−41.0131	−1.34850
$926$	0	0
$927$	21.5346	0.707288
$928$	0	0
$929$	−15.2065	−0.498910	−0.249455	−	0.968386i	$-0.580251\pi$
$929$	−15.2065	−0.498910	−0.249455	+	0.968386i	$0.580251\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−47.9477	−1.56974
$934$	0	0
$935$	96.9869	3.17181
$936$	0	0
$937$	40.5690	1.32533	0.662666	−	0.748915i	$-0.269425\pi$
$937$	40.5690	1.32533	0.662666	+	0.748915i	$0.269425\pi$
$938$	0	0
$939$	−47.6508	−1.55502
$940$	0	0
$941$	−2.09489	−0.0682915	−0.0341457	−	0.999417i	$-0.510871\pi$
$941$	−2.09489	−0.0682915	−0.0341457	+	0.999417i	$0.510871\pi$
$942$	0	0
$943$	−2.90777	−0.0946899
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−33.1986	−1.07881	−0.539405	−	0.842047i	$-0.681351\pi$
$947$	−33.1986	−1.07881	−0.539405	+	0.842047i	$0.681351\pi$
$948$	0	0
$949$	−3.10138	−0.100675
$950$	0	0
$951$	2.42592	0.0786659
$952$	0	0
$953$	26.0205	0.842887	0.421443	−	0.906855i	$-0.361524\pi$
$953$	26.0205	0.842887	0.421443	+	0.906855i	$0.361524\pi$
$954$	0	0
$955$	11.8595	0.383766
$956$	0	0
$957$	−18.7629	−0.606517
$958$	0	0
$959$	0	0
$960$	0	0
$961$	13.3640	0.431096
$962$	0	0
$963$	18.8049	0.605981
$964$	0	0
$965$	−34.8072	−1.12048
$966$	0	0
$967$	41.8805	1.34679	0.673394	−	0.739284i	$-0.264836\pi$
$967$	41.8805	1.34679	0.673394	+	0.739284i	$0.264836\pi$
$968$	0	0
$969$	−39.5634	−1.27096
$970$	0	0
$971$	−46.7046	−1.49882	−0.749411	−	0.662105i	$-0.769663\pi$
$971$	−46.7046	−1.49882	−0.749411	+	0.662105i	$0.769663\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	11.6692	0.373713
$976$	0	0
$977$	19.7169	0.630800	0.315400	−	0.948959i	$-0.397861\pi$
$977$	19.7169	0.630800	0.315400	+	0.948959i	$0.397861\pi$
$978$	0	0
$979$	27.9083	0.891954
$980$	0	0
$981$	−15.9710	−0.509916
$982$	0	0
$983$	−32.2603	−1.02894	−0.514472	−	0.857507i	$-0.672012\pi$
$983$	−32.2603	−1.02894	−0.514472	+	0.857507i	$0.672012\pi$
$984$	0	0
$985$	30.7533	0.979880
$986$	0	0
$987$	0	0
$988$	0	0
$989$	8.71493	0.277119
$990$	0	0
$991$	3.27294	0.103969	0.0519843	−	0.998648i	$-0.483445\pi$
$991$	3.27294	0.103969	0.0519843	+	0.998648i	$0.483445\pi$
$992$	0	0
$993$	46.1596	1.46483
$994$	0	0
$995$	31.5746	1.00098
$996$	0	0
$997$	25.3864	0.803994	0.401997	−	0.915641i	$-0.368316\pi$
$997$	25.3864	0.803994	0.401997	+	0.915641i	$0.368316\pi$
$998$	0	0
$999$	−37.6286	−1.19052

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8036.2.a.m.1.2		8
7.2	even	3		1148.2.i.d.165.7	✓	16
7.4	even	3		1148.2.i.d.821.7	yes	16
7.6	odd	2		8036.2.a.n.1.7		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1148.2.i.d.165.7	✓	16	7.2	even	3
1148.2.i.d.821.7	yes	16	7.4	even	3
8036.2.a.m.1.2		8	1.1	even	1	trivial
8036.2.a.n.1.7		8	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)

\(f(q)\)	\(=\)	\(q-2.10300 q^{3} -2.93524 q^{5} +1.42260 q^{9} +O(q^{10})\)	\(q-2.10300 q^{3} -2.93524 q^{5} +1.42260 q^{9} -5.80589 q^{11} -1.53467 q^{13} +6.17281 q^{15} +5.69116 q^{17} +3.30562 q^{19} -2.90777 q^{23} +3.61563 q^{25} +3.31726 q^{27} -1.53671 q^{29} -6.66063 q^{31} +12.2098 q^{33} -11.3433 q^{37} +3.22742 q^{39} +1.00000 q^{41} -2.99712 q^{43} -4.17568 q^{45} +8.97221 q^{47} -11.9685 q^{51} +2.87561 q^{53} +17.0417 q^{55} -6.95172 q^{57} +9.75975 q^{59} +6.56368 q^{61} +4.50464 q^{65} +8.35339 q^{67} +6.11503 q^{69} +12.2707 q^{71} +2.02087 q^{73} -7.60368 q^{75} +10.3165 q^{79} -11.2440 q^{81} -8.68931 q^{83} -16.7049 q^{85} +3.23169 q^{87} -4.80690 q^{89} +14.0073 q^{93} -9.70280 q^{95} -4.61153 q^{97} -8.25948 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 4 q^{9}+O(q^{10}) \) 8 * q + 4 * q^9	\( 8 q + 4 q^{9} - 8 q^{11} - 7 q^{13} - q^{15} - q^{17} + 4 q^{19} - 3 q^{23} - 4 q^{25} - 12 q^{27} - 4 q^{29} + 4 q^{31} + 23 q^{33} - 31 q^{37} + 5 q^{39} + 8 q^{41} - 8 q^{43} + q^{45} + 24 q^{47} - 23 q^{51} - q^{53} + 2 q^{55} - 15 q^{57} + 4 q^{59} - 4 q^{61} - 24 q^{65} - 21 q^{69} + 8 q^{71} + 11 q^{73} - 15 q^{75} + 14 q^{79} - 28 q^{81} - 42 q^{83} - 20 q^{85} + 25 q^{87} - 11 q^{89} - 27 q^{93} - 15 q^{95} - 16 q^{97} - 8 q^{99}+O(q^{100}) \) 8 * q + 4 * q^9 - 8 * q^11 - 7 * q^13 - q^15 - q^17 + 4 * q^19 - 3 * q^23 - 4 * q^25 - 12 * q^27 - 4 * q^29 + 4 * q^31 + 23 * q^33 - 31 * q^37 + 5 * q^39 + 8 * q^41 - 8 * q^43 + q^45 + 24 * q^47 - 23 * q^51 - q^53 + 2 * q^55 - 15 * q^57 + 4 * q^59 - 4 * q^61 - 24 * q^65 - 21 * q^69 + 8 * q^71 + 11 * q^73 - 15 * q^75 + 14 * q^79 - 28 * q^81 - 42 * q^83 - 20 * q^85 + 25 * q^87 - 11 * q^89 - 27 * q^93 - 15 * q^95 - 16 * q^97 - 8 * q^99

Introduction

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