LMFDB - Embedded newform 8036.2.a.i.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:	$4$
Coefficient field:	4.4.25808.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 10x^{2} - 6x + 9 $ x^4 - 10x^2 - 6x + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 164)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$0.707500$ 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)

$f(q)$	$=$	$q+3.24028 q^{3} -2.56613 q^{5} +7.49944 q^{9} +O(q^{10})$	$q+3.24028 q^{3} -2.56613 q^{5} +7.49944 q^{9} +6.20694 q^{11} -1.41500 q^{13} -8.31498 q^{15} +3.93332 q^{17} -3.82529 q^{19} +3.06557 q^{23} +1.58500 q^{25} +14.5795 q^{27} -8.48057 q^{29} -1.13225 q^{31} +20.1123 q^{33} +8.49944 q^{37} -4.58500 q^{39} +1.00000 q^{41} +5.34832 q^{43} -19.2445 q^{45} +6.65528 q^{47} +12.7451 q^{51} +6.41389 q^{53} -15.9278 q^{55} -12.3950 q^{57} +3.06557 q^{59} -7.41500 q^{61} +3.63107 q^{65} +1.79306 q^{67} +9.93332 q^{69} -3.02422 q^{71} -0.632807 q^{73} +5.13585 q^{75} -14.3059 q^{79} +24.7433 q^{81} +8.76332 q^{83} -10.0934 q^{85} -27.4795 q^{87} +7.89557 q^{89} -3.66882 q^{93} +9.81616 q^{95} -9.86664 q^{97} +46.5486 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 2 q^{3} - 4 q^{5} + 12 q^{9}+O(q^{10}) $ 4 * q - 2 * q^3 - 4 * q^5 + 12 * q^9	$ 4 q - 2 q^{3} - 4 q^{5} + 12 q^{9} + 4 q^{11} - 10 q^{15} + 4 q^{17} - 6 q^{19} - 12 q^{23} + 12 q^{25} + 10 q^{27} - 4 q^{29} + 8 q^{31} + 20 q^{33} + 16 q^{37} - 24 q^{39} + 4 q^{41} + 4 q^{43} - 4 q^{45} + 6 q^{47} - 4 q^{51} - 16 q^{53} + 2 q^{55} + 4 q^{57} - 12 q^{59} - 24 q^{61} + 4 q^{65} + 28 q^{67} + 28 q^{69} - 2 q^{71} - 8 q^{73} - 30 q^{75} - 18 q^{79} + 28 q^{81} + 12 q^{83} + 32 q^{85} - 44 q^{87} - 4 q^{89} - 28 q^{93} + 14 q^{95} - 16 q^{97} + 58 q^{99}+O(q^{100}) $ 4 * q - 2 * q^3 - 4 * q^5 + 12 * q^9 + 4 * q^11 - 10 * q^15 + 4 * q^17 - 6 * q^19 - 12 * q^23 + 12 * q^25 + 10 * q^27 - 4 * q^29 + 8 * q^31 + 20 * q^33 + 16 * q^37 - 24 * q^39 + 4 * q^41 + 4 * q^43 - 4 * q^45 + 6 * q^47 - 4 * q^51 - 16 * q^53 + 2 * q^55 + 4 * q^57 - 12 * q^59 - 24 * q^61 + 4 * q^65 + 28 * q^67 + 28 * q^69 - 2 * q^71 - 8 * q^73 - 30 * q^75 - 18 * q^79 + 28 * q^81 + 12 * q^83 + 32 * q^85 - 44 * q^87 - 4 * q^89 - 28 * q^93 + 14 * q^95 - 16 * q^97 + 58 * 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
$3$	3.24028	1.87078	0.935390	−	0.353619i	$-0.115049\pi$
$3$	3.24028	1.87078	0.935390	+	0.353619i	$0.115049\pi$
$4$	0	0
$5$	−2.56613	−1.14761	−0.573803	−	0.818993i	$-0.694533\pi$
$5$	−2.56613	−1.14761	−0.573803	+	0.818993i	$0.694533\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	7.49944	2.49981
$10$	0	0
$11$	6.20694	1.87146	0.935732	−	0.352712i	$-0.114740\pi$
$11$	6.20694	1.87146	0.935732	+	0.352712i	$0.114740\pi$
$12$	0	0
$13$	−1.41500	−0.392450	−0.196225	−	0.980559i	$-0.562868\pi$
$13$	−1.41500	−0.392450	−0.196225	+	0.980559i	$0.562868\pi$
$14$	0	0
$15$	−8.31498	−2.14692
$16$	0	0
$17$	3.93332	0.953970	0.476985	−	0.878911i	$-0.341730\pi$
$17$	3.93332	0.953970	0.476985	+	0.878911i	$0.341730\pi$
$18$	0	0
$19$	−3.82529	−0.877581	−0.438790	−	0.898589i	$-0.644593\pi$
$19$	−3.82529	−0.877581	−0.438790	+	0.898589i	$0.644593\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	3.06557	0.639215	0.319608	−	0.947550i	$-0.396449\pi$
$23$	3.06557	0.639215	0.319608	+	0.947550i	$0.396449\pi$
$24$	0	0
$25$	1.58500	0.317000
$26$	0	0
$27$	14.5795	2.80582
$28$	0	0
$29$	−8.48057	−1.57480	−0.787401	−	0.616441i	$-0.788574\pi$
$29$	−8.48057	−1.57480	−0.787401	+	0.616441i	$0.788574\pi$
$30$	0	0
$31$	−1.13225	−0.203358	−0.101679	−	0.994817i	$-0.532422\pi$
$31$	−1.13225	−0.203358	−0.101679	+	0.994817i	$0.532422\pi$
$32$	0	0
$33$	20.1123	3.50110
$34$	0	0
$35$	0	0
$36$	0	0
$37$	8.49944	1.39730	0.698650	−	0.715464i	$-0.253784\pi$
$37$	8.49944	1.39730	0.698650	+	0.715464i	$0.253784\pi$
$38$	0	0
$39$	−4.58500	−0.734188
$40$	0	0
$41$	1.00000	0.156174
$41$	1.00000	0.156174
$42$	0	0
$43$	5.34832	0.815611	0.407805	−	0.913069i	$-0.366294\pi$
$43$	5.34832	0.815611	0.407805	+	0.913069i	$0.366294\pi$
$44$	0	0
$45$	−19.2445	−2.86880
$46$	0	0
$47$	6.65528	0.970773	0.485386	−	0.874300i	$-0.338679\pi$
$47$	6.65528	0.970773	0.485386	+	0.874300i	$0.338679\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	12.7451	1.78467
$52$	0	0
$53$	6.41389	0.881015	0.440508	−	0.897749i	$-0.354798\pi$
$53$	6.41389	0.881015	0.440508	+	0.897749i	$0.354798\pi$
$54$	0	0
$55$	−15.9278	−2.14770
$56$	0	0
$57$	−12.3950	−1.64176
$58$	0	0
$59$	3.06557	0.399103	0.199552	−	0.979887i	$-0.436051\pi$
$59$	3.06557	0.399103	0.199552	+	0.979887i	$0.436051\pi$
$60$	0	0
$61$	−7.41500	−0.949393	−0.474697	−	0.880149i	$-0.657442\pi$
$61$	−7.41500	−0.949393	−0.474697	+	0.880149i	$0.657442\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	3.63107	0.450378
$66$	0	0
$67$	1.79306	0.219057	0.109528	−	0.993984i	$-0.465066\pi$
$67$	1.79306	0.219057	0.109528	+	0.993984i	$0.465066\pi$
$68$	0	0
$69$	9.93332	1.19583
$70$	0	0
$71$	−3.02422	−0.358909	−0.179454	−	0.983766i	$-0.557433\pi$
$71$	−3.02422	−0.358909	−0.179454	+	0.983766i	$0.557433\pi$
$72$	0	0
$73$	−0.632807	−0.0740645	−0.0370322	−	0.999314i	$-0.511790\pi$
$73$	−0.632807	−0.0740645	−0.0370322	+	0.999314i	$0.511790\pi$
$74$	0	0
$75$	5.13585	0.593037
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−14.3059	−1.60953	−0.804767	−	0.593591i	$-0.797710\pi$
$79$	−14.3059	−1.60953	−0.804767	+	0.593591i	$0.797710\pi$
$80$	0	0
$81$	24.7433	2.74926
$82$	0	0
$83$	8.76332	0.961899	0.480950	−	0.876748i	$-0.340292\pi$
$83$	8.76332	0.961899	0.480950	+	0.876748i	$0.340292\pi$
$84$	0	0
$85$	−10.0934	−1.09478
$86$	0	0
$87$	−27.4795	−2.94611
$88$	0	0
$89$	7.89557	0.836929	0.418464	−	0.908233i	$-0.362569\pi$
$89$	7.89557	0.836929	0.418464	+	0.908233i	$0.362569\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−3.66882	−0.380439
$94$	0	0
$95$	9.81616	1.00712
$96$	0	0
$97$	−9.86664	−1.00181	−0.500903	−	0.865504i	$-0.666999\pi$
$97$	−9.86664	−1.00181	−0.500903	+	0.865504i	$0.666999\pi$
$98$	0	0
$99$	46.5486	4.67831
$100$	0	0
$101$	4.51832	0.449590	0.224795	−	0.974406i	$-0.427829\pi$
$101$	4.51832	0.449590	0.224795	+	0.974406i	$0.427829\pi$
$102$	0	0
$103$	16.4139	1.61731	0.808654	−	0.588284i	$-0.200196\pi$
$103$	16.4139	1.61731	0.808654	+	0.588284i	$0.200196\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−16.0645	−1.55301	−0.776505	−	0.630111i	$-0.783009\pi$
$107$	−16.0645	−1.55301	−0.776505	+	0.630111i	$0.783009\pi$
$108$	0	0
$109$	−5.61282	−0.537611	−0.268805	−	0.963195i	$-0.586629\pi$
$109$	−5.61282	−0.537611	−0.268805	+	0.963195i	$0.586629\pi$
$110$	0	0
$111$	27.5406	2.61404
$112$	0	0
$113$	5.63170	0.529785	0.264893	−	0.964278i	$-0.414663\pi$
$113$	5.63170	0.529785	0.264893	+	0.964278i	$0.414663\pi$
$114$	0	0
$115$	−7.86664	−0.733568
$116$	0	0
$117$	−10.6117	−0.981053
$118$	0	0
$119$	0	0
$120$	0	0
$121$	27.5262	2.50238
$122$	0	0
$123$	3.24028	0.292167
$124$	0	0
$125$	8.76332	0.783815
$126$	0	0
$127$	1.69775	0.150651	0.0753254	−	0.997159i	$-0.476000\pi$
$127$	1.69775	0.150651	0.0753254	+	0.997159i	$0.476000\pi$
$128$	0	0
$129$	17.3301	1.52583
$130$	0	0
$131$	2.48057	0.216728	0.108364	−	0.994111i	$-0.465439\pi$
$131$	2.48057	0.216728	0.108364	+	0.994111i	$0.465439\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−37.4128	−3.21998
$136$	0	0
$137$	2.76332	0.236086	0.118043	−	0.993008i	$-0.462338\pi$
$137$	2.76332	0.236086	0.118043	+	0.993008i	$0.462338\pi$
$138$	0	0
$139$	−8.99889	−0.763276	−0.381638	−	0.924312i	$-0.624640\pi$
$139$	−8.99889	−0.763276	−0.381638	+	0.924312i	$0.624640\pi$
$140$	0	0
$141$	21.5650	1.81610
$142$	0	0
$143$	−8.78282	−0.734456
$144$	0	0
$145$	21.7622	1.80725
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−8.48057	−0.694755	−0.347378	−	0.937725i	$-0.612928\pi$
$149$	−8.48057	−0.694755	−0.347378	+	0.937725i	$0.612928\pi$
$150$	0	0
$151$	−7.58971	−0.617642	−0.308821	−	0.951120i	$-0.599934\pi$
$151$	−7.58971	−0.617642	−0.308821	+	0.951120i	$0.599934\pi$
$152$	0	0
$153$	29.4977	2.38475
$154$	0	0
$155$	2.90550	0.233375
$156$	0	0
$157$	6.78282	0.541328	0.270664	−	0.962674i	$-0.412757\pi$
$157$	6.78282	0.541328	0.270664	+	0.962674i	$0.412757\pi$
$158$	0	0
$159$	20.7828	1.64818
$160$	0	0
$161$	0	0
$162$	0	0
$163$	15.0278	1.17707	0.588535	−	0.808472i	$-0.299705\pi$
$163$	15.0278	1.17707	0.588535	+	0.808472i	$0.299705\pi$
$164$	0	0
$165$	−51.6106	−4.01788
$166$	0	0
$167$	−5.79306	−0.448280	−0.224140	−	0.974557i	$-0.571957\pi$
$167$	−5.79306	−0.448280	−0.224140	+	0.974557i	$0.571957\pi$
$168$	0	0
$169$	−10.9978	−0.845983
$170$	0	0
$171$	−28.6875	−2.19379
$172$	0	0
$173$	5.43450	0.413178	0.206589	−	0.978428i	$-0.433764\pi$
$173$	5.43450	0.413178	0.206589	+	0.978428i	$0.433764\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	9.93332	0.746634
$178$	0	0
$179$	8.10251	0.605610	0.302805	−	0.953052i	$-0.402077\pi$
$179$	8.10251	0.605610	0.302805	+	0.953052i	$0.402077\pi$
$180$	0	0
$181$	9.24389	0.687093	0.343546	−	0.939136i	$-0.388372\pi$
$181$	9.24389	0.687093	0.343546	+	0.939136i	$0.388372\pi$
$182$	0	0
$183$	−24.0267	−1.77611
$184$	0	0
$185$	−21.8106	−1.60355
$186$	0	0
$187$	24.4139	1.78532
$188$	0	0
$189$	0	0
$190$	0	0
$191$	24.3381	1.76104	0.880521	−	0.474007i	$-0.157193\pi$
$191$	24.3381	1.76104	0.880521	+	0.474007i	$0.157193\pi$
$192$	0	0
$193$	13.2634	0.954720	0.477360	−	0.878708i	$-0.341594\pi$
$193$	13.2634	0.954720	0.477360	+	0.878708i	$0.341594\pi$
$194$	0	0
$195$	11.7657	0.842558
$196$	0	0
$197$	−18.1116	−1.29040	−0.645200	−	0.764013i	$-0.723226\pi$
$197$	−18.1116	−1.29040	−0.645200	+	0.764013i	$0.723226\pi$
$198$	0	0
$199$	−19.7726	−1.40164	−0.700821	−	0.713337i	$-0.747183\pi$
$199$	−19.7726	−1.40164	−0.700821	+	0.713337i	$0.747183\pi$
$200$	0	0
$201$	5.81001	0.409807
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−2.56613	−0.179226
$206$	0	0
$207$	22.9901	1.59792
$208$	0	0
$209$	−23.7433	−1.64236
$210$	0	0
$211$	1.53924	0.105966	0.0529828	−	0.998595i	$-0.483127\pi$
$211$	1.53924	0.105966	0.0529828	+	0.998595i	$0.483127\pi$
$212$	0	0
$213$	−9.79933	−0.671439
$214$	0	0
$215$	−13.7245	−0.936000
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−2.05047	−0.138558
$220$	0	0
$221$	−5.56564	−0.374386
$222$	0	0
$223$	27.6578	1.85210	0.926051	−	0.377399i	$-0.123181\pi$
$223$	27.6578	1.85210	0.926051	+	0.377399i	$0.123181\pi$
$224$	0	0
$225$	11.8866	0.792442
$226$	0	0
$227$	1.79635	0.119228	0.0596141	−	0.998221i	$-0.481013\pi$
$227$	1.79635	0.119228	0.0596141	+	0.998221i	$0.481013\pi$
$228$	0	0
$229$	−21.6128	−1.42822	−0.714108	−	0.700036i	$-0.753167\pi$
$229$	−21.6128	−1.42822	−0.714108	+	0.700036i	$0.753167\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−18.3312	−1.20092	−0.600458	−	0.799656i	$-0.705015\pi$
$233$	−18.3312	−1.20092	−0.600458	+	0.799656i	$0.705015\pi$
$234$	0	0
$235$	−17.0783	−1.11407
$236$	0	0
$237$	−46.3550	−3.01108
$238$	0	0
$239$	11.9586	0.773541	0.386770	−	0.922176i	$-0.373591\pi$
$239$	11.9586	0.773541	0.386770	+	0.922176i	$0.373591\pi$
$240$	0	0
$241$	−7.41500	−0.477642	−0.238821	−	0.971064i	$-0.576761\pi$
$241$	−7.41500	−0.477642	−0.238821	+	0.971064i	$0.576761\pi$
$242$	0	0
$243$	36.4370	2.33743
$244$	0	0
$245$	0	0
$246$	0	0
$247$	5.41278	0.344407
$248$	0	0
$249$	28.3956	1.79950
$250$	0	0
$251$	1.91507	0.120878	0.0604392	−	0.998172i	$-0.480750\pi$
$251$	1.91507	0.120878	0.0604392	+	0.998172i	$0.480750\pi$
$252$	0	0
$253$	19.0278	1.19627
$254$	0	0
$255$	−32.7055	−2.04810
$256$	0	0
$257$	−2.93443	−0.183045	−0.0915224	−	0.995803i	$-0.529173\pi$
$257$	−2.93443	−0.183045	−0.0915224	+	0.995803i	$0.529173\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−63.5996	−3.93671
$262$	0	0
$263$	−3.10692	−0.191581	−0.0957905	−	0.995402i	$-0.530538\pi$
$263$	−3.10692	−0.191581	−0.0957905	+	0.995402i	$0.530538\pi$
$264$	0	0
$265$	−16.4588	−1.01106
$266$	0	0
$267$	25.5839	1.56571
$268$	0	0
$269$	10.2450	0.624649	0.312324	−	0.949976i	$-0.398892\pi$
$269$	10.2450	0.624649	0.312324	+	0.949976i	$0.398892\pi$
$270$	0	0
$271$	11.1989	0.680287	0.340143	−	0.940374i	$-0.389524\pi$
$271$	11.1989	0.680287	0.340143	+	0.940374i	$0.389524\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	9.83801	0.593254
$276$	0	0
$277$	2.52838	0.151915	0.0759577	−	0.997111i	$-0.475799\pi$
$277$	2.52838	0.151915	0.0759577	+	0.997111i	$0.475799\pi$
$278$	0	0
$279$	−8.49125	−0.508358
$280$	0	0
$281$	−23.5656	−1.40581	−0.702904	−	0.711285i	$-0.748114\pi$
$281$	−23.5656	−1.40581	−0.702904	+	0.711285i	$0.748114\pi$
$282$	0	0
$283$	−8.99889	−0.534928	−0.267464	−	0.963568i	$-0.586186\pi$
$283$	−8.99889	−0.534928	−0.267464	+	0.963568i	$0.586186\pi$
$284$	0	0
$285$	31.8072	1.88409
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−1.52900	−0.0899415
$290$	0	0
$291$	−31.9707	−1.87416
$292$	0	0
$293$	−3.17721	−0.185614	−0.0928072	−	0.995684i	$-0.529584\pi$
$293$	−3.17721	−0.185614	−0.0928072	+	0.995684i	$0.529584\pi$
$294$	0	0
$295$	−7.86664	−0.458013
$296$	0	0
$297$	90.4940	5.25100
$298$	0	0
$299$	−4.33778	−0.250860
$300$	0	0
$301$	0	0
$302$	0	0
$303$	14.6406	0.841083
$304$	0	0
$305$	19.0278	1.08953
$306$	0	0
$307$	17.5839	1.00357	0.501783	−	0.864994i	$-0.332678\pi$
$307$	17.5839	1.00357	0.501783	+	0.864994i	$0.332678\pi$
$308$	0	0
$309$	53.1857	3.02563
$310$	0	0
$311$	−1.55749	−0.0883169	−0.0441584	−	0.999025i	$-0.514061\pi$
$311$	−1.55749	−0.0883169	−0.0441584	+	0.999025i	$0.514061\pi$
$312$	0	0
$313$	−20.5255	−1.16017	−0.580086	−	0.814556i	$-0.696981\pi$
$313$	−20.5255	−1.16017	−0.580086	+	0.814556i	$0.696981\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	26.1094	1.46645	0.733225	−	0.679986i	$-0.238014\pi$
$317$	26.1094	1.46645	0.733225	+	0.679986i	$0.238014\pi$
$318$	0	0
$319$	−52.6384	−2.94719
$320$	0	0
$321$	−52.0534	−2.90534
$322$	0	0
$323$	−15.0461	−0.837185
$324$	0	0
$325$	−2.24277	−0.124407
$326$	0	0
$327$	−18.1871	−1.00575
$328$	0	0
$329$	0	0
$330$	0	0
$331$	3.68862	0.202745	0.101373	−	0.994849i	$-0.467677\pi$
$331$	3.68862	0.202745	0.101373	+	0.994849i	$0.467677\pi$
$332$	0	0
$333$	63.7411	3.49299
$334$	0	0
$335$	−4.60121	−0.251391
$336$	0	0
$337$	30.9240	1.68454	0.842269	−	0.539057i	$-0.181219\pi$
$337$	30.9240	1.68454	0.842269	+	0.539057i	$0.181219\pi$
$338$	0	0
$339$	18.2483	0.991111
$340$	0	0
$341$	−7.02782	−0.380578
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−25.4901	−1.37234
$346$	0	0
$347$	27.1264	1.45622	0.728111	−	0.685459i	$-0.240398\pi$
$347$	27.1264	1.45622	0.728111	+	0.685459i	$0.240398\pi$
$348$	0	0
$349$	9.30051	0.497845	0.248922	−	0.968523i	$-0.419924\pi$
$349$	9.30051	0.497845	0.248922	+	0.968523i	$0.419924\pi$
$350$	0	0
$351$	−20.6300	−1.10115
$352$	0	0
$353$	−13.3961	−0.713004	−0.356502	−	0.934295i	$-0.616031\pi$
$353$	−13.3961	−0.713004	−0.356502	+	0.934295i	$0.616031\pi$
$354$	0	0
$355$	7.76052	0.411886
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−28.9611	−1.52851	−0.764255	−	0.644914i	$-0.776893\pi$
$359$	−28.9611	−1.52851	−0.764255	+	0.644914i	$0.776893\pi$
$360$	0	0
$361$	−4.36719	−0.229852
$362$	0	0
$363$	89.1926	4.68140
$364$	0	0
$365$	1.62386	0.0849968
$366$	0	0
$367$	31.4795	1.64321	0.821607	−	0.570054i	$-0.193078\pi$
$367$	31.4795	1.64321	0.821607	+	0.570054i	$0.193078\pi$
$368$	0	0
$369$	7.49944	0.390405
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−15.5266	−0.803939	−0.401969	−	0.915653i	$-0.631674\pi$
$373$	−15.5266	−0.803939	−0.401969	+	0.915653i	$0.631674\pi$
$374$	0	0
$375$	28.3956	1.46634
$376$	0	0
$377$	12.0000	0.618031
$378$	0	0
$379$	1.97107	0.101247	0.0506235	−	0.998718i	$-0.483879\pi$
$379$	1.97107	0.101247	0.0506235	+	0.998718i	$0.483879\pi$
$380$	0	0
$381$	5.50119	0.281834
$382$	0	0
$383$	12.8404	0.656113	0.328056	−	0.944658i	$-0.393606\pi$
$383$	12.8404	0.656113	0.328056	+	0.944658i	$0.393606\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	40.1094	2.03888
$388$	0	0
$389$	1.03886	0.0526724	0.0263362	−	0.999653i	$-0.491616\pi$
$389$	1.03886	0.0526724	0.0263362	+	0.999653i	$0.491616\pi$
$390$	0	0
$391$	12.0579	0.609792
$392$	0	0
$393$	8.03775	0.405451
$394$	0	0
$395$	36.7106	1.84711
$396$	0	0
$397$	24.6406	1.23668	0.618339	−	0.785911i	$-0.287806\pi$
$397$	24.6406	1.23668	0.618339	+	0.785911i	$0.287806\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	0.397236	0.0198370	0.00991851	−	0.999951i	$-0.496843\pi$
$401$	0.397236	0.0198370	0.00991851	+	0.999951i	$0.496843\pi$
$402$	0	0
$403$	1.60213	0.0798080
$404$	0	0
$405$	−63.4945	−3.15507
$406$	0	0
$407$	52.7556	2.61500
$408$	0	0
$409$	2.76395	0.136668	0.0683342	−	0.997662i	$-0.478232\pi$
$409$	2.76395	0.136668	0.0683342	+	0.997662i	$0.478232\pi$
$410$	0	0
$411$	8.95393	0.441665
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−22.4878	−1.10388
$416$	0	0
$417$	−29.1590	−1.42792
$418$	0	0
$419$	−1.48168	−0.0723848	−0.0361924	−	0.999345i	$-0.511523\pi$
$419$	−1.48168	−0.0723848	−0.0361924	+	0.999345i	$0.511523\pi$
$420$	0	0
$421$	−18.9039	−0.921319	−0.460659	−	0.887577i	$-0.652387\pi$
$421$	−18.9039	−0.921319	−0.460659	+	0.887577i	$0.652387\pi$
$422$	0	0
$423$	49.9109	2.42675
$424$	0	0
$425$	6.23431	0.302409
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−28.4588	−1.37401
$430$	0	0
$431$	−7.28164	−0.350744	−0.175372	−	0.984502i	$-0.556113\pi$
$431$	−7.28164	−0.350744	−0.175372	+	0.984502i	$0.556113\pi$
$432$	0	0
$433$	0.397865	0.0191202	0.00956009	−	0.999954i	$-0.496957\pi$
$433$	0.397865	0.0191202	0.00956009	+	0.999954i	$0.496957\pi$
$434$	0	0
$435$	70.5157	3.38097
$436$	0	0
$437$	−11.7267	−0.560963
$438$	0	0
$439$	−33.9798	−1.62177	−0.810885	−	0.585206i	$-0.801014\pi$
$439$	−33.9798	−1.62177	−0.810885	+	0.585206i	$0.801014\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	16.2050	0.769924	0.384962	−	0.922932i	$-0.374215\pi$
$443$	16.2050	0.769924	0.384962	+	0.922932i	$0.374215\pi$
$444$	0	0
$445$	−20.2610	−0.960464
$446$	0	0
$447$	−27.4795	−1.29973
$448$	0	0
$449$	−23.1118	−1.09071	−0.545356	−	0.838204i	$-0.683606\pi$
$449$	−23.1118	−1.09071	−0.545356	+	0.838204i	$0.683606\pi$
$450$	0	0
$451$	6.20694	0.292274
$452$	0	0
$453$	−24.5928	−1.15547
$454$	0	0
$455$	0	0
$456$	0	0
$457$	35.6301	1.66671	0.833353	−	0.552741i	$-0.186418\pi$
$457$	35.6301	1.66671	0.833353	+	0.552741i	$0.186418\pi$
$458$	0	0
$459$	57.3457	2.67667
$460$	0	0
$461$	−6.63058	−0.308817	−0.154409	−	0.988007i	$-0.549347\pi$
$461$	−6.63058	−0.308817	−0.154409	+	0.988007i	$0.549347\pi$
$462$	0	0
$463$	−21.0220	−0.976975	−0.488487	−	0.872571i	$-0.662451\pi$
$463$	−21.0220	−0.976975	−0.488487	+	0.872571i	$0.662451\pi$
$464$	0	0
$465$	9.41464	0.436594
$466$	0	0
$467$	−15.8933	−0.735456	−0.367728	−	0.929933i	$-0.619864\pi$
$467$	−15.8933	−0.735456	−0.367728	+	0.929933i	$0.619864\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	21.9783	1.01271
$472$	0	0
$473$	33.1967	1.52639
$474$	0	0
$475$	−6.06308	−0.278193
$476$	0	0
$477$	48.1006	2.20237
$478$	0	0
$479$	34.8564	1.59263	0.796315	−	0.604882i	$-0.206780\pi$
$479$	34.8564	1.59263	0.796315	+	0.604882i	$0.206780\pi$
$480$	0	0
$481$	−12.0267	−0.548371
$482$	0	0
$483$	0	0
$484$	0	0
$485$	25.3190	1.14968
$486$	0	0
$487$	−9.44393	−0.427945	−0.213973	−	0.976840i	$-0.568640\pi$
$487$	−9.44393	−0.427945	−0.213973	+	0.976840i	$0.568640\pi$
$488$	0	0
$489$	48.6944	2.20204
$490$	0	0
$491$	33.1577	1.49639	0.748193	−	0.663481i	$-0.230922\pi$
$491$	33.1577	1.49639	0.748193	+	0.663481i	$0.230922\pi$
$492$	0	0
$493$	−33.3568	−1.50231
$494$	0	0
$495$	−119.450	−5.36886
$496$	0	0
$497$	0	0
$498$	0	0
$499$	12.7597	0.571203	0.285602	−	0.958348i	$-0.407807\pi$
$499$	12.7597	0.571203	0.285602	+	0.958348i	$0.407807\pi$
$500$	0	0
$501$	−18.7712	−0.838633
$502$	0	0
$503$	17.0025	0.758104	0.379052	−	0.925375i	$-0.376250\pi$
$503$	17.0025	0.758104	0.379052	+	0.925375i	$0.376250\pi$
$504$	0	0
$505$	−11.5946	−0.515952
$506$	0	0
$507$	−35.6359	−1.58265
$508$	0	0
$509$	2.50950	0.111232	0.0556158	−	0.998452i	$-0.482288\pi$
$509$	2.50950	0.111232	0.0556158	+	0.998452i	$0.482288\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−55.7707	−2.46234
$514$	0	0
$515$	−42.1201	−1.85603
$516$	0	0
$517$	41.3090	1.81677
$518$	0	0
$519$	17.6093	0.772964
$520$	0	0
$521$	−13.1300	−0.575237	−0.287618	−	0.957745i	$-0.592864\pi$
$521$	−13.1300	−0.575237	−0.287618	+	0.957745i	$0.592864\pi$
$522$	0	0
$523$	−23.8301	−1.04202	−0.521010	−	0.853551i	$-0.674444\pi$
$523$	−23.8301	−1.04202	−0.521010	+	0.853551i	$0.674444\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−4.45350	−0.193998
$528$	0	0
$529$	−13.6023	−0.591404
$530$	0	0
$531$	22.9901	0.997684
$532$	0	0
$533$	−1.41500	−0.0612904
$534$	0	0
$535$	41.2234	1.78224
$536$	0	0
$537$	26.2544	1.13296
$538$	0	0
$539$	0	0
$540$	0	0
$541$	14.6995	0.631980	0.315990	−	0.948762i	$-0.397663\pi$
$541$	14.6995	0.631980	0.315990	+	0.948762i	$0.397663\pi$
$542$	0	0
$543$	29.9528	1.28540
$544$	0	0
$545$	14.4032	0.616965
$546$	0	0
$547$	20.2209	0.864584	0.432292	−	0.901734i	$-0.357705\pi$
$547$	20.2209	0.864584	0.432292	+	0.901734i	$0.357705\pi$
$548$	0	0
$549$	−55.6084	−2.37331
$550$	0	0
$551$	32.4406	1.38202
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−70.6727	−2.99989
$556$	0	0
$557$	−40.6784	−1.72360	−0.861799	−	0.507249i	$-0.830662\pi$
$557$	−40.6784	−1.72360	−0.861799	+	0.507249i	$0.830662\pi$
$558$	0	0
$559$	−7.56787	−0.320087
$560$	0	0
$561$	79.1079	3.33994
$562$	0	0
$563$	19.4486	0.819661	0.409831	−	0.912162i	$-0.365588\pi$
$563$	19.4486	0.819661	0.409831	+	0.912162i	$0.365588\pi$
$564$	0	0
$565$	−14.4516	−0.607985
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−31.1850	−1.30734	−0.653672	−	0.756778i	$-0.726773\pi$
$569$	−31.1850	−1.30734	−0.653672	+	0.756778i	$0.726773\pi$
$570$	0	0
$571$	−0.190921	−0.00798981	−0.00399491	−	0.999992i	$-0.501272\pi$
$571$	−0.190921	−0.00798981	−0.00399491	+	0.999992i	$0.501272\pi$
$572$	0	0
$573$	78.8623	3.29452
$574$	0	0
$575$	4.85893	0.202631
$576$	0	0
$577$	29.0278	1.20844	0.604222	−	0.796816i	$-0.293484\pi$
$577$	29.0278	1.20844	0.604222	+	0.796816i	$0.293484\pi$
$578$	0	0
$579$	42.9772	1.78607
$580$	0	0
$581$	0	0
$582$	0	0
$583$	39.8106	1.64879
$584$	0	0
$585$	27.2310	1.12586
$586$	0	0
$587$	11.5735	0.477690	0.238845	−	0.971058i	$-0.423231\pi$
$587$	11.5735	0.477690	0.238845	+	0.971058i	$0.423231\pi$
$588$	0	0
$589$	4.33118	0.178463
$590$	0	0
$591$	−58.6869	−2.41405
$592$	0	0
$593$	−34.5557	−1.41903	−0.709517	−	0.704689i	$-0.751087\pi$
$593$	−34.5557	−1.41903	−0.709517	+	0.704689i	$0.751087\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−64.0688	−2.62216
$598$	0	0
$599$	26.9967	1.10305	0.551527	−	0.834157i	$-0.314045\pi$
$599$	26.9967	1.10305	0.551527	+	0.834157i	$0.314045\pi$
$600$	0	0
$601$	−47.8534	−1.95198	−0.975990	−	0.217816i	$-0.930107\pi$
$601$	−47.8534	−1.95198	−0.975990	+	0.217816i	$0.930107\pi$
$602$	0	0
$603$	13.4469	0.547601
$604$	0	0
$605$	−70.6356	−2.87174
$606$	0	0
$607$	−12.3761	−0.502332	−0.251166	−	0.967944i	$-0.580814\pi$
$607$	−12.3761	−0.502332	−0.251166	+	0.967944i	$0.580814\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−9.41722	−0.380980
$612$	0	0
$613$	12.9061	0.521274	0.260637	−	0.965437i	$-0.416067\pi$
$613$	12.9061	0.521274	0.260637	+	0.965437i	$0.416067\pi$
$614$	0	0
$615$	−8.31498	−0.335292
$616$	0	0
$617$	−40.5268	−1.63155	−0.815773	−	0.578372i	$-0.803688\pi$
$617$	−40.5268	−1.63155	−0.815773	+	0.578372i	$0.803688\pi$
$618$	0	0
$619$	2.65889	0.106870	0.0534348	−	0.998571i	$-0.482983\pi$
$619$	2.65889	0.106870	0.0534348	+	0.998571i	$0.482983\pi$
$620$	0	0
$621$	44.6944	1.79353
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−30.4128	−1.21651
$626$	0	0
$627$	−76.9351	−3.07249
$628$	0	0
$629$	33.4310	1.33298
$630$	0	0
$631$	−0.139958	−0.00557165	−0.00278583	−	0.999996i	$-0.500887\pi$
$631$	−0.139958	−0.00557165	−0.00278583	+	0.999996i	$0.500887\pi$
$632$	0	0
$633$	4.98757	0.198238
$634$	0	0
$635$	−4.35663	−0.172888
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−22.6800	−0.897205
$640$	0	0
$641$	9.39675	0.371149	0.185575	−	0.982630i	$-0.440585\pi$
$641$	9.39675	0.371149	0.185575	+	0.982630i	$0.440585\pi$
$642$	0	0
$643$	−6.02863	−0.237746	−0.118873	−	0.992909i	$-0.537928\pi$
$643$	−6.02863	−0.237746	−0.118873	+	0.992909i	$0.537928\pi$
$644$	0	0
$645$	−44.4711	−1.75105
$646$	0	0
$647$	−17.3861	−0.683517	−0.341758	−	0.939788i	$-0.611022\pi$
$647$	−17.3861	−0.683517	−0.341758	+	0.939788i	$0.611022\pi$
$648$	0	0
$649$	19.0278	0.746907
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−8.02832	−0.314173	−0.157086	−	0.987585i	$-0.550210\pi$
$653$	−8.02832	−0.314173	−0.157086	+	0.987585i	$0.550210\pi$
$654$	0	0
$655$	−6.36545	−0.248719
$656$	0	0
$657$	−4.74570	−0.185147
$658$	0	0
$659$	−33.8809	−1.31981	−0.659907	−	0.751348i	$-0.729404\pi$
$659$	−33.8809	−1.31981	−0.659907	+	0.751348i	$0.729404\pi$
$660$	0	0
$661$	−34.3840	−1.33738	−0.668691	−	0.743541i	$-0.733145\pi$
$661$	−34.3840	−1.33738	−0.668691	+	0.743541i	$0.733145\pi$
$662$	0	0
$663$	−18.0343	−0.700393
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−25.9978	−1.00664
$668$	0	0
$669$	89.6191	3.46487
$670$	0	0
$671$	−46.0245	−1.77676
$672$	0	0
$673$	10.3689	0.399693	0.199847	−	0.979827i	$-0.435956\pi$
$673$	10.3689	0.399693	0.199847	+	0.979827i	$0.435956\pi$
$674$	0	0
$675$	23.1085	0.889446
$676$	0	0
$677$	−28.0616	−1.07850	−0.539248	−	0.842147i	$-0.681291\pi$
$677$	−28.0616	−1.07850	−0.539248	+	0.842147i	$0.681291\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	5.82070	0.223050
$682$	0	0
$683$	12.2209	0.467621	0.233810	−	0.972282i	$-0.424881\pi$
$683$	12.2209	0.467621	0.233810	+	0.972282i	$0.424881\pi$
$684$	0	0
$685$	−7.09102	−0.270934
$686$	0	0
$687$	−70.0317	−2.67188
$688$	0	0
$689$	−9.07565	−0.345755
$690$	0	0
$691$	−46.1797	−1.75676	−0.878379	−	0.477964i	$-0.841375\pi$
$691$	−46.1797	−1.75676	−0.878379	+	0.477964i	$0.841375\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	23.0923	0.875940
$696$	0	0
$697$	3.93332	0.148985
$698$	0	0
$699$	−59.3983	−2.24665
$700$	0	0
$701$	−46.8161	−1.76822	−0.884110	−	0.467279i	$-0.845234\pi$
$701$	−46.8161	−1.76822	−0.884110	+	0.467279i	$0.845234\pi$
$702$	0	0
$703$	−32.5128	−1.22624
$704$	0	0
$705$	−55.3385	−2.08417
$706$	0	0
$707$	0	0
$708$	0	0
$709$	42.0840	1.58050	0.790248	−	0.612787i	$-0.209952\pi$
$709$	42.0840	1.58050	0.790248	+	0.612787i	$0.209952\pi$
$710$	0	0
$711$	−107.286	−4.02354
$712$	0	0
$713$	−3.47100	−0.129990
$714$	0	0
$715$	22.5378	0.842867
$716$	0	0
$717$	38.7494	1.44712
$718$	0	0
$719$	−1.77685	−0.0662653	−0.0331327	−	0.999451i	$-0.510548\pi$
$719$	−1.77685	−0.0662653	−0.0331327	+	0.999451i	$0.510548\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−24.0267	−0.893563
$724$	0	0
$725$	−13.4417	−0.499212
$726$	0	0
$727$	17.4346	0.646614	0.323307	−	0.946294i	$-0.395205\pi$
$727$	17.4346	0.646614	0.323307	+	0.946294i	$0.395205\pi$
$728$	0	0
$729$	43.8362	1.62356
$730$	0	0
$731$	21.0366	0.778068
$732$	0	0
$733$	8.31827	0.307242	0.153621	−	0.988130i	$-0.450906\pi$
$733$	8.31827	0.307242	0.153621	+	0.988130i	$0.450906\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	11.1294	0.409957
$738$	0	0
$739$	−0.139958	−0.00514845	−0.00257422	−	0.999997i	$-0.500819\pi$
$739$	−0.139958	−0.00514845	−0.00257422	+	0.999997i	$0.500819\pi$
$740$	0	0
$741$	17.5389	0.644309
$742$	0	0
$743$	−3.92389	−0.143954	−0.0719768	−	0.997406i	$-0.522931\pi$
$743$	−3.92389	−0.143954	−0.0719768	+	0.997406i	$0.522931\pi$
$744$	0	0
$745$	21.7622	0.797306
$746$	0	0
$747$	65.7200	2.40457
$748$	0	0
$749$	0	0
$750$	0	0
$751$	38.2786	1.39681	0.698403	−	0.715705i	$-0.253894\pi$
$751$	38.2786	1.39681	0.698403	+	0.715705i	$0.253894\pi$
$752$	0	0
$753$	6.20538	0.226137
$754$	0	0
$755$	19.4762	0.708810
$756$	0	0
$757$	−31.0640	−1.12904	−0.564519	−	0.825420i	$-0.690938\pi$
$757$	−31.0640	−1.12904	−0.564519	+	0.825420i	$0.690938\pi$
$758$	0	0
$759$	61.6556	2.23795
$760$	0	0
$761$	2.83945	0.102930	0.0514649	−	0.998675i	$-0.483611\pi$
$761$	2.83945	0.102930	0.0514649	+	0.998675i	$0.483611\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−75.6948	−2.73675
$766$	0	0
$767$	−4.33778	−0.156628
$768$	0	0
$769$	−38.7888	−1.39876	−0.699379	−	0.714751i	$-0.746540\pi$
$769$	−38.7888	−1.39876	−0.699379	+	0.714751i	$0.746540\pi$
$770$	0	0
$771$	−9.50839	−0.342436
$772$	0	0
$773$	−25.7150	−0.924905	−0.462453	−	0.886644i	$-0.653030\pi$
$773$	−25.7150	−0.924905	−0.462453	+	0.886644i	$0.653030\pi$
$774$	0	0
$775$	−1.79462	−0.0644646
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−3.82529	−0.137055
$780$	0	0
$781$	−18.7712	−0.671685
$782$	0	0
$783$	−123.642	−4.41861
$784$	0	0
$785$	−17.4056	−0.621232
$786$	0	0
$787$	−41.7817	−1.48936	−0.744679	−	0.667423i	$-0.767397\pi$
$787$	−41.7817	−1.48936	−0.744679	+	0.667423i	$0.767397\pi$
$788$	0	0
$789$	−10.0673	−0.358406
$790$	0	0
$791$	0	0
$792$	0	0
$793$	10.4922	0.372590
$794$	0	0
$795$	−53.3313	−1.89147
$796$	0	0
$797$	−36.2818	−1.28517	−0.642583	−	0.766216i	$-0.722137\pi$
$797$	−36.2818	−1.28517	−0.642583	+	0.766216i	$0.722137\pi$
$798$	0	0
$799$	26.1774	0.926088
$800$	0	0
$801$	59.2124	2.09217
$802$	0	0
$803$	−3.92780	−0.138609
$804$	0	0
$805$	0	0
$806$	0	0
$807$	33.1967	1.16858
$808$	0	0
$809$	−37.8556	−1.33093	−0.665466	−	0.746428i	$-0.731767\pi$
$809$	−37.8556	−1.33093	−0.665466	+	0.746428i	$0.731767\pi$
$810$	0	0
$811$	15.5751	0.546915	0.273457	−	0.961884i	$-0.411833\pi$
$811$	15.5751	0.546915	0.273457	+	0.961884i	$0.411833\pi$
$812$	0	0
$813$	36.2877	1.27267
$814$	0	0
$815$	−38.5633	−1.35081
$816$	0	0
$817$	−20.4588	−0.715764
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−9.42283	−0.328859	−0.164430	−	0.986389i	$-0.552578\pi$
$821$	−9.42283	−0.328859	−0.164430	+	0.986389i	$0.552578\pi$
$822$	0	0
$823$	−13.9636	−0.486741	−0.243371	−	0.969933i	$-0.578253\pi$
$823$	−13.9636	−0.486741	−0.243371	+	0.969933i	$0.578253\pi$
$824$	0	0
$825$	31.8780	1.10985
$826$	0	0
$827$	19.6886	0.684641	0.342320	−	0.939583i	$-0.388787\pi$
$827$	19.6886	0.684641	0.342320	+	0.939583i	$0.388787\pi$
$828$	0	0
$829$	−52.9595	−1.83936	−0.919681	−	0.392668i	$-0.871552\pi$
$829$	−52.9595	−1.83936	−0.919681	+	0.392668i	$0.871552\pi$
$830$	0	0
$831$	8.19266	0.284200
$832$	0	0
$833$	0	0
$834$	0	0
$835$	14.8657	0.514449
$836$	0	0
$837$	−16.5076	−0.570587
$838$	0	0
$839$	17.1392	0.591709	0.295855	−	0.955233i	$-0.404396\pi$
$839$	17.1392	0.591709	0.295855	+	0.955233i	$0.404396\pi$
$840$	0	0
$841$	42.9201	1.48000
$842$	0	0
$843$	−76.3594	−2.62996
$844$	0	0
$845$	28.2217	0.970855
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−29.1590	−1.00073
$850$	0	0
$851$	26.0556	0.893176
$852$	0	0
$853$	24.8256	0.850011	0.425005	−	0.905191i	$-0.360272\pi$
$853$	24.8256	0.850011	0.425005	+	0.905191i	$0.360272\pi$
$854$	0	0
$855$	73.6158	2.51761
$856$	0	0
$857$	−25.6584	−0.876474	−0.438237	−	0.898859i	$-0.644397\pi$
$857$	−25.6584	−0.876474	−0.438237	+	0.898859i	$0.644397\pi$
$858$	0	0
$859$	−11.9818	−0.408812	−0.204406	−	0.978886i	$-0.565526\pi$
$859$	−11.9818	−0.408812	−0.204406	+	0.978886i	$0.565526\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−33.9138	−1.15444	−0.577220	−	0.816589i	$-0.695862\pi$
$863$	−33.9138	−1.15444	−0.577220	+	0.816589i	$0.695862\pi$
$864$	0	0
$865$	−13.9456	−0.474165
$866$	0	0
$867$	−4.95441	−0.168261
$868$	0	0
$869$	−88.7956	−3.01219
$870$	0	0
$871$	−2.53717	−0.0859688
$872$	0	0
$873$	−73.9943	−2.50433
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−14.1871	−0.479066	−0.239533	−	0.970888i	$-0.576994\pi$
$877$	−14.1871	−0.479066	−0.239533	+	0.970888i	$0.576994\pi$
$878$	0	0
$879$	−10.2950	−0.347243
$880$	0	0
$881$	−42.3201	−1.42580	−0.712901	−	0.701265i	$-0.752619\pi$
$881$	−42.3201	−1.42580	−0.712901	+	0.701265i	$0.752619\pi$
$882$	0	0
$883$	51.7258	1.74071	0.870356	−	0.492422i	$-0.163888\pi$
$883$	51.7258	1.74071	0.870356	+	0.492422i	$0.163888\pi$
$884$	0	0
$885$	−25.4901	−0.856842
$886$	0	0
$887$	−30.1270	−1.01157	−0.505783	−	0.862661i	$-0.668796\pi$
$887$	−30.1270	−1.01157	−0.505783	+	0.862661i	$0.668796\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	153.580	5.14514
$892$	0	0
$893$	−25.4584	−0.851932
$894$	0	0
$895$	−20.7921	−0.695002
$896$	0	0
$897$	−14.0556	−0.469304
$898$	0	0
$899$	9.60213	0.320249
$900$	0	0
$901$	25.2279	0.840462
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−23.7210	−0.788512
$906$	0	0
$907$	−4.24848	−0.141068	−0.0705342	−	0.997509i	$-0.522470\pi$
$907$	−4.24848	−0.141068	−0.0705342	+	0.997509i	$0.522470\pi$
$908$	0	0
$909$	33.8849	1.12389
$910$	0	0
$911$	−49.1395	−1.62806	−0.814031	−	0.580821i	$-0.802732\pi$
$911$	−49.1395	−1.62806	−0.814031	+	0.580821i	$0.802732\pi$
$912$	0	0
$913$	54.3934	1.80016
$914$	0	0
$915$	61.6556	2.03827
$916$	0	0
$917$	0	0
$918$	0	0
$919$	6.59412	0.217520	0.108760	−	0.994068i	$-0.465312\pi$
$919$	6.59412	0.217520	0.108760	+	0.994068i	$0.465312\pi$
$920$	0	0
$921$	56.9768	1.87745
$922$	0	0
$923$	4.27927	0.140854
$924$	0	0
$925$	13.4716	0.442944
$926$	0	0
$927$	123.095	4.04297
$928$	0	0
$929$	−54.3833	−1.78426	−0.892130	−	0.451779i	$-0.850789\pi$
$929$	−54.3833	−1.78426	−0.892130	+	0.451779i	$0.850789\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−5.04669	−0.165221
$934$	0	0
$935$	−62.6491	−2.04884
$936$	0	0
$937$	43.0888	1.40765	0.703825	−	0.710374i	$-0.251474\pi$
$937$	43.0888	1.40765	0.703825	+	0.710374i	$0.251474\pi$
$938$	0	0
$939$	−66.5085	−2.17042
$940$	0	0
$941$	−12.6966	−0.413899	−0.206949	−	0.978352i	$-0.566354\pi$
$941$	−12.6966	−0.413899	−0.206949	+	0.978352i	$0.566354\pi$
$942$	0	0
$943$	3.06557	0.0998287
$944$	0	0
$945$	0	0
$946$	0	0
$947$	29.6867	0.964688	0.482344	−	0.875982i	$-0.339785\pi$
$947$	29.6867	0.964688	0.482344	+	0.875982i	$0.339785\pi$
$948$	0	0
$949$	0.895422	0.0290666
$950$	0	0
$951$	84.6019	2.74341
$952$	0	0
$953$	7.62947	0.247143	0.123571	−	0.992336i	$-0.460565\pi$
$953$	7.62947	0.247143	0.123571	+	0.992336i	$0.460565\pi$
$954$	0	0
$955$	−62.4546	−2.02098
$956$	0	0
$957$	−170.563	−5.51353
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−29.7180	−0.958645
$962$	0	0
$963$	−120.475	−3.88224
$964$	0	0
$965$	−34.0355	−1.09564
$966$	0	0
$967$	−37.0187	−1.19044	−0.595221	−	0.803562i	$-0.702935\pi$
$967$	−37.0187	−1.19044	−0.595221	+	0.803562i	$0.702935\pi$
$968$	0	0
$969$	−48.7535	−1.56619
$970$	0	0
$971$	17.2425	0.553338	0.276669	−	0.960965i	$-0.410769\pi$
$971$	17.2425	0.553338	0.276669	+	0.960965i	$0.410769\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−7.26723	−0.232738
$976$	0	0
$977$	−31.6112	−1.01133	−0.505666	−	0.862729i	$-0.668753\pi$
$977$	−31.6112	−1.01133	−0.505666	+	0.862729i	$0.668753\pi$
$978$	0	0
$979$	49.0074	1.56628
$980$	0	0
$981$	−42.0930	−1.34393
$982$	0	0
$983$	−51.8543	−1.65390	−0.826948	−	0.562278i	$-0.809925\pi$
$983$	−51.8543	−1.65390	−0.826948	+	0.562278i	$0.809925\pi$
$984$	0	0
$985$	46.4767	1.48087
$986$	0	0
$987$	0	0
$988$	0	0
$989$	16.3956	0.521351
$990$	0	0
$991$	−37.7209	−1.19824	−0.599121	−	0.800658i	$-0.704483\pi$
$991$	−37.7209	−1.19824	−0.599121	+	0.800658i	$0.704483\pi$
$992$	0	0
$993$	11.9522	0.379291
$994$	0	0
$995$	50.7389	1.60853
$996$	0	0
$997$	19.1882	0.607698	0.303849	−	0.952720i	$-0.401728\pi$
$997$	19.1882	0.607698	0.303849	+	0.952720i	$0.401728\pi$
$998$	0	0
$999$	123.917	3.92058

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8036.2.a.i.1.4		4
7.6	odd	2		164.2.a.a.1.1	✓	4
21.20	even	2		1476.2.a.g.1.2		4
28.27	even	2		656.2.a.i.1.4		4
35.13	even	4		4100.2.d.c.1149.1		8
35.27	even	4		4100.2.d.c.1149.8		8
35.34	odd	2		4100.2.a.c.1.4		4
56.13	odd	2		2624.2.a.v.1.4		4
56.27	even	2		2624.2.a.y.1.1		4
84.83	odd	2		5904.2.a.bp.1.2		4
287.286	odd	2		6724.2.a.c.1.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
164.2.a.a.1.1	✓	4	7.6	odd	2
656.2.a.i.1.4		4	28.27	even	2
1476.2.a.g.1.2		4	21.20	even	2
2624.2.a.v.1.4		4	56.13	odd	2
2624.2.a.y.1.1		4	56.27	even	2
4100.2.a.c.1.4		4	35.34	odd	2
4100.2.d.c.1149.1		8	35.13	even	4
4100.2.d.c.1149.8		8	35.27	even	4
5904.2.a.bp.1.2		4	84.83	odd	2
6724.2.a.c.1.4		4	287.286	odd	2
8036.2.a.i.1.4		4	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)

\(f(q)\)	\(=\)	\(q+3.24028 q^{3} -2.56613 q^{5} +7.49944 q^{9} +O(q^{10})\)	\(q+3.24028 q^{3} -2.56613 q^{5} +7.49944 q^{9} +6.20694 q^{11} -1.41500 q^{13} -8.31498 q^{15} +3.93332 q^{17} -3.82529 q^{19} +3.06557 q^{23} +1.58500 q^{25} +14.5795 q^{27} -8.48057 q^{29} -1.13225 q^{31} +20.1123 q^{33} +8.49944 q^{37} -4.58500 q^{39} +1.00000 q^{41} +5.34832 q^{43} -19.2445 q^{45} +6.65528 q^{47} +12.7451 q^{51} +6.41389 q^{53} -15.9278 q^{55} -12.3950 q^{57} +3.06557 q^{59} -7.41500 q^{61} +3.63107 q^{65} +1.79306 q^{67} +9.93332 q^{69} -3.02422 q^{71} -0.632807 q^{73} +5.13585 q^{75} -14.3059 q^{79} +24.7433 q^{81} +8.76332 q^{83} -10.0934 q^{85} -27.4795 q^{87} +7.89557 q^{89} -3.66882 q^{93} +9.81616 q^{95} -9.86664 q^{97} +46.5486 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 2 q^{3} - 4 q^{5} + 12 q^{9}+O(q^{10}) \) 4 * q - 2 * q^3 - 4 * q^5 + 12 * q^9	\( 4 q - 2 q^{3} - 4 q^{5} + 12 q^{9} + 4 q^{11} - 10 q^{15} + 4 q^{17} - 6 q^{19} - 12 q^{23} + 12 q^{25} + 10 q^{27} - 4 q^{29} + 8 q^{31} + 20 q^{33} + 16 q^{37} - 24 q^{39} + 4 q^{41} + 4 q^{43} - 4 q^{45} + 6 q^{47} - 4 q^{51} - 16 q^{53} + 2 q^{55} + 4 q^{57} - 12 q^{59} - 24 q^{61} + 4 q^{65} + 28 q^{67} + 28 q^{69} - 2 q^{71} - 8 q^{73} - 30 q^{75} - 18 q^{79} + 28 q^{81} + 12 q^{83} + 32 q^{85} - 44 q^{87} - 4 q^{89} - 28 q^{93} + 14 q^{95} - 16 q^{97} + 58 q^{99}+O(q^{100}) \) 4 * q - 2 * q^3 - 4 * q^5 + 12 * q^9 + 4 * q^11 - 10 * q^15 + 4 * q^17 - 6 * q^19 - 12 * q^23 + 12 * q^25 + 10 * q^27 - 4 * q^29 + 8 * q^31 + 20 * q^33 + 16 * q^37 - 24 * q^39 + 4 * q^41 + 4 * q^43 - 4 * q^45 + 6 * q^47 - 4 * q^51 - 16 * q^53 + 2 * q^55 + 4 * q^57 - 12 * q^59 - 24 * q^61 + 4 * q^65 + 28 * q^67 + 28 * q^69 - 2 * q^71 - 8 * q^73 - 30 * q^75 - 18 * q^79 + 28 * q^81 + 12 * q^83 + 32 * q^85 - 44 * q^87 - 4 * q^89 - 28 * q^93 + 14 * q^95 - 16 * q^97 + 58 * q^99

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