LMFDB - Embedded newform 8036.2.a.t.1.14

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:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - 4 x^{19} - 30 x^{18} + 128 x^{17} + 348 x^{16} - 1644 x^{15} - 1934 x^{14} + 10948 x^{13} + \cdots + 9 $ x^20 - 4x^19 - 30x^18 + 128x^17 + 348x^16 - 1644x^15 - 1934x^14 + 10948x^13 + 4748x^12 - 40524x^11 - 220x^10 + 82500x^9 - 21992x^8 - 84720x^7 + 37544x^6 + 34656x^5 - 18823x^4 - 2276x^3 + 1130x^2 + 204*x + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.14
Root			$1.15563$ of defining polynomial
Character	$\chi$	$=$	8036.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.15563 q^{3} +3.94572 q^{5} -1.66452 q^{9} +O(q^{10})$	$q+1.15563 q^{3} +3.94572 q^{5} -1.66452 q^{9} +1.12534 q^{11} +3.54416 q^{13} +4.55979 q^{15} +3.41735 q^{17} -1.28335 q^{19} -3.01618 q^{23} +10.5687 q^{25} -5.39046 q^{27} +7.53562 q^{29} -0.123190 q^{31} +1.30048 q^{33} +0.924137 q^{37} +4.09574 q^{39} -1.00000 q^{41} -3.99122 q^{43} -6.56772 q^{45} +1.47225 q^{47} +3.94919 q^{51} +8.78256 q^{53} +4.44029 q^{55} -1.48308 q^{57} -3.03142 q^{59} +12.2049 q^{61} +13.9842 q^{65} -2.72904 q^{67} -3.48559 q^{69} +5.72921 q^{71} -7.47002 q^{73} +12.2135 q^{75} +10.0679 q^{79} -1.23581 q^{81} -13.5564 q^{83} +13.4839 q^{85} +8.70838 q^{87} -0.778550 q^{89} -0.142362 q^{93} -5.06375 q^{95} -5.90424 q^{97} -1.87316 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q + 4 q^{3} + 8 q^{5} + 16 q^{9}+O(q^{10}) $ 20 * q + 4 * q^3 + 8 * q^5 + 16 * q^9	$ 20 q + 4 q^{3} + 8 q^{5} + 16 q^{9} - 8 q^{11} + 12 q^{13} + 8 q^{15} + 8 q^{17} + 24 q^{19} + 8 q^{23} + 20 q^{25} + 16 q^{27} - 12 q^{29} + 44 q^{33} + 12 q^{37} + 12 q^{39} - 20 q^{41} + 4 q^{43} + 40 q^{45} + 4 q^{47} + 4 q^{51} - 12 q^{53} - 16 q^{55} + 28 q^{57} + 16 q^{59} + 68 q^{61} - 8 q^{65} + 4 q^{67} + 32 q^{69} + 8 q^{71} + 48 q^{73} + 60 q^{75} - 20 q^{79} + 32 q^{81} - 8 q^{83} - 28 q^{85} + 60 q^{89} - 16 q^{93} + 20 q^{95} + 40 q^{97}+O(q^{100}) $ 20 * q + 4 * q^3 + 8 * q^5 + 16 * q^9 - 8 * q^11 + 12 * q^13 + 8 * q^15 + 8 * q^17 + 24 * q^19 + 8 * q^23 + 20 * q^25 + 16 * q^27 - 12 * q^29 + 44 * q^33 + 12 * q^37 + 12 * q^39 - 20 * q^41 + 4 * q^43 + 40 * q^45 + 4 * q^47 + 4 * q^51 - 12 * q^53 - 16 * q^55 + 28 * q^57 + 16 * q^59 + 68 * q^61 - 8 * q^65 + 4 * q^67 + 32 * q^69 + 8 * q^71 + 48 * q^73 + 60 * q^75 - 20 * q^79 + 32 * q^81 - 8 * q^83 - 28 * q^85 + 60 * q^89 - 16 * q^93 + 20 * q^95 + 40 * q^97

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	1.15563	0.667203	0.333602	−	0.942714i	$-0.391736\pi$
$3$	1.15563	0.667203	0.333602	+	0.942714i	$0.391736\pi$
$4$	0	0
$5$	3.94572	1.76458	0.882289	−	0.470708i	$-0.156001\pi$
$5$	3.94572	1.76458	0.882289	+	0.470708i	$0.156001\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−1.66452	−0.554840
$10$	0	0
$11$	1.12534	0.339304	0.169652	−	0.985504i	$-0.445736\pi$
$11$	1.12534	0.339304	0.169652	+	0.985504i	$0.445736\pi$
$12$	0	0
$13$	3.54416	0.982973	0.491486	−	0.870885i	$-0.336454\pi$
$13$	3.54416	0.982973	0.491486	+	0.870885i	$0.336454\pi$
$14$	0	0
$15$	4.55979	1.17733
$16$	0	0
$17$	3.41735	0.828830	0.414415	−	0.910088i	$-0.363986\pi$
$17$	3.41735	0.828830	0.414415	+	0.910088i	$0.363986\pi$
$18$	0	0
$19$	−1.28335	−0.294421	−0.147211	−	0.989105i	$-0.547030\pi$
$19$	−1.28335	−0.294421	−0.147211	+	0.989105i	$0.547030\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−3.01618	−0.628917	−0.314459	−	0.949271i	$-0.601823\pi$
$23$	−3.01618	−0.628917	−0.314459	+	0.949271i	$0.601823\pi$
$24$	0	0
$25$	10.5687	2.11374
$26$	0	0
$27$	−5.39046	−1.03739
$28$	0	0
$29$	7.53562	1.39933	0.699664	−	0.714472i	$-0.253333\pi$
$29$	7.53562	1.39933	0.699664	+	0.714472i	$0.253333\pi$
$30$	0	0
$31$	−0.123190	−0.0221256	−0.0110628	−	0.999939i	$-0.503521\pi$
$31$	−0.123190	−0.0221256	−0.0110628	+	0.999939i	$0.503521\pi$
$32$	0	0
$33$	1.30048	0.226385
$34$	0	0
$35$	0	0
$36$	0	0
$37$	0.924137	0.151927	0.0759636	−	0.997111i	$-0.475797\pi$
$37$	0.924137	0.151927	0.0759636	+	0.997111i	$0.475797\pi$
$38$	0	0
$39$	4.09574	0.655843
$40$	0	0
$41$	−1.00000	−0.156174
$41$	−1.00000	−0.156174
$42$	0	0
$43$	−3.99122	−0.608655	−0.304327	−	0.952567i	$-0.598432\pi$
$43$	−3.99122	−0.608655	−0.304327	+	0.952567i	$0.598432\pi$
$44$	0	0
$45$	−6.56772	−0.979059
$46$	0	0
$47$	1.47225	0.214749	0.107375	−	0.994219i	$-0.465756\pi$
$47$	1.47225	0.214749	0.107375	+	0.994219i	$0.465756\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	3.94919	0.552998
$52$	0	0
$53$	8.78256	1.20638	0.603188	−	0.797599i	$-0.293897\pi$
$53$	8.78256	1.20638	0.603188	+	0.797599i	$0.293897\pi$
$54$	0	0
$55$	4.44029	0.598729
$56$	0	0
$57$	−1.48308	−0.196439
$58$	0	0
$59$	−3.03142	−0.394658	−0.197329	−	0.980337i	$-0.563227\pi$
$59$	−3.03142	−0.394658	−0.197329	+	0.980337i	$0.563227\pi$
$60$	0	0
$61$	12.2049	1.56267	0.781337	−	0.624110i	$-0.214538\pi$
$61$	12.2049	1.56267	0.781337	+	0.624110i	$0.214538\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	13.9842	1.73453
$66$	0	0
$67$	−2.72904	−0.333405	−0.166702	−	0.986007i	$-0.553312\pi$
$67$	−2.72904	−0.333405	−0.166702	+	0.986007i	$0.553312\pi$
$68$	0	0
$69$	−3.48559	−0.419616
$70$	0	0
$71$	5.72921	0.679932	0.339966	−	0.940438i	$-0.389584\pi$
$71$	5.72921	0.679932	0.339966	+	0.940438i	$0.389584\pi$
$72$	0	0
$73$	−7.47002	−0.874300	−0.437150	−	0.899389i	$-0.644012\pi$
$73$	−7.47002	−0.874300	−0.437150	+	0.899389i	$0.644012\pi$
$74$	0	0
$75$	12.2135	1.41029
$76$	0	0
$77$	0	0
$78$	0	0
$79$	10.0679	1.13273	0.566365	−	0.824154i	$-0.308349\pi$
$79$	10.0679	1.13273	0.566365	+	0.824154i	$0.308349\pi$
$80$	0	0
$81$	−1.23581	−0.137313
$82$	0	0
$83$	−13.5564	−1.48801	−0.744004	−	0.668175i	$-0.767076\pi$
$83$	−13.5564	−1.48801	−0.744004	+	0.668175i	$0.767076\pi$
$84$	0	0
$85$	13.4839	1.46253
$86$	0	0
$87$	8.70838	0.933636
$88$	0	0
$89$	−0.778550	−0.0825262	−0.0412631	−	0.999148i	$-0.513138\pi$
$89$	−0.778550	−0.0825262	−0.0412631	+	0.999148i	$0.513138\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−0.142362	−0.0147622
$94$	0	0
$95$	−5.06375	−0.519530
$96$	0	0
$97$	−5.90424	−0.599484	−0.299742	−	0.954020i	$-0.596901\pi$
$97$	−5.90424	−0.599484	−0.299742	+	0.954020i	$0.596901\pi$
$98$	0	0
$99$	−1.87316	−0.188260
$100$	0	0
$101$	0.701602	0.0698120	0.0349060	−	0.999391i	$-0.488887\pi$
$101$	0.701602	0.0698120	0.0349060	+	0.999391i	$0.488887\pi$
$102$	0	0
$103$	−10.1863	−1.00369	−0.501844	−	0.864958i	$-0.667345\pi$
$103$	−10.1863	−1.00369	−0.501844	+	0.864958i	$0.667345\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−9.70206	−0.937934	−0.468967	−	0.883216i	$-0.655374\pi$
$107$	−9.70206	−0.937934	−0.468967	+	0.883216i	$0.655374\pi$
$108$	0	0
$109$	8.43965	0.808371	0.404186	−	0.914677i	$-0.367555\pi$
$109$	8.43965	0.808371	0.404186	+	0.914677i	$0.367555\pi$
$110$	0	0
$111$	1.06796	0.101366
$112$	0	0
$113$	1.43990	0.135455	0.0677274	−	0.997704i	$-0.478425\pi$
$113$	1.43990	0.135455	0.0677274	+	0.997704i	$0.478425\pi$
$114$	0	0
$115$	−11.9010	−1.10977
$116$	0	0
$117$	−5.89932	−0.545393
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−9.73360	−0.884873
$122$	0	0
$123$	−1.15563	−0.104200
$124$	0	0
$125$	21.9724	1.96528
$126$	0	0
$127$	−12.4802	−1.10744	−0.553718	−	0.832704i	$-0.686791\pi$
$127$	−12.4802	−1.10744	−0.553718	+	0.832704i	$0.686791\pi$
$128$	0	0
$129$	−4.61237	−0.406097
$130$	0	0
$131$	20.1838	1.76346	0.881732	−	0.471751i	$-0.156378\pi$
$131$	20.1838	1.76346	0.881732	+	0.471751i	$0.156378\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−21.2692	−1.83056
$136$	0	0
$137$	9.14237	0.781085	0.390542	−	0.920585i	$-0.372287\pi$
$137$	9.14237	0.781085	0.390542	+	0.920585i	$0.372287\pi$
$138$	0	0
$139$	21.1559	1.79442	0.897210	−	0.441604i	$-0.145590\pi$
$139$	21.1559	1.79442	0.897210	+	0.441604i	$0.145590\pi$
$140$	0	0
$141$	1.70137	0.143281
$142$	0	0
$143$	3.98840	0.333527
$144$	0	0
$145$	29.7334	2.46922
$146$	0	0
$147$	0	0
$148$	0	0
$149$	6.12932	0.502134	0.251067	−	0.967970i	$-0.419219\pi$
$149$	6.12932	0.502134	0.251067	+	0.967970i	$0.419219\pi$
$150$	0	0
$151$	7.97172	0.648730	0.324365	−	0.945932i	$-0.394849\pi$
$151$	7.97172	0.648730	0.324365	+	0.945932i	$0.394849\pi$
$152$	0	0
$153$	−5.68825	−0.459868
$154$	0	0
$155$	−0.486073	−0.0390423
$156$	0	0
$157$	−11.2658	−0.899109	−0.449555	−	0.893253i	$-0.648417\pi$
$157$	−11.2658	−0.899109	−0.449555	+	0.893253i	$0.648417\pi$
$158$	0	0
$159$	10.1494	0.804898
$160$	0	0
$161$	0	0
$162$	0	0
$163$	1.74597	0.136755	0.0683775	−	0.997660i	$-0.478218\pi$
$163$	1.74597	0.136755	0.0683775	+	0.997660i	$0.478218\pi$
$164$	0	0
$165$	5.13133	0.399474
$166$	0	0
$167$	−14.2797	−1.10499	−0.552497	−	0.833515i	$-0.686325\pi$
$167$	−14.2797	−1.10499	−0.552497	+	0.833515i	$0.686325\pi$
$168$	0	0
$169$	−0.438935	−0.0337642
$170$	0	0
$171$	2.13617	0.163357
$172$	0	0
$173$	−4.43511	−0.337195	−0.168597	−	0.985685i	$-0.553924\pi$
$173$	−4.43511	−0.337195	−0.168597	+	0.985685i	$0.553924\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−3.50320	−0.263317
$178$	0	0
$179$	1.00948	0.0754519	0.0377260	−	0.999288i	$-0.487989\pi$
$179$	1.00948	0.0754519	0.0377260	+	0.999288i	$0.487989\pi$
$180$	0	0
$181$	14.8421	1.10320	0.551602	−	0.834107i	$-0.314017\pi$
$181$	14.8421	1.10320	0.551602	+	0.834107i	$0.314017\pi$
$182$	0	0
$183$	14.1043	1.04262
$184$	0	0
$185$	3.64638	0.268087
$186$	0	0
$187$	3.84570	0.281225
$188$	0	0
$189$	0	0
$190$	0	0
$191$	4.13720	0.299358	0.149679	−	0.988735i	$-0.452176\pi$
$191$	4.13720	0.299358	0.149679	+	0.988735i	$0.452176\pi$
$192$	0	0
$193$	9.94820	0.716087	0.358044	−	0.933705i	$-0.383444\pi$
$193$	9.94820	0.716087	0.358044	+	0.933705i	$0.383444\pi$
$194$	0	0
$195$	16.1606	1.15729
$196$	0	0
$197$	4.84994	0.345544	0.172772	−	0.984962i	$-0.444728\pi$
$197$	4.84994	0.345544	0.172772	+	0.984962i	$0.444728\pi$
$198$	0	0
$199$	27.3865	1.94137	0.970687	−	0.240345i	$-0.0772607\pi$
$199$	27.3865	1.94137	0.970687	+	0.240345i	$0.0772607\pi$
$200$	0	0
$201$	−3.15375	−0.222449
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−3.94572	−0.275581
$206$	0	0
$207$	5.02049	0.348948
$208$	0	0
$209$	−1.44421	−0.0998984
$210$	0	0
$211$	−20.1342	−1.38610	−0.693048	−	0.720891i	$-0.743733\pi$
$211$	−20.1342	−1.38610	−0.693048	+	0.720891i	$0.743733\pi$
$212$	0	0
$213$	6.62084	0.453653
$214$	0	0
$215$	−15.7482	−1.07402
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−8.63258	−0.583336
$220$	0	0
$221$	12.1116	0.814717
$222$	0	0
$223$	−1.11801	−0.0748676	−0.0374338	−	0.999299i	$-0.511918\pi$
$223$	−1.11801	−0.0748676	−0.0374338	+	0.999299i	$0.511918\pi$
$224$	0	0
$225$	−17.5918	−1.17279
$226$	0	0
$227$	17.8238	1.18301	0.591503	−	0.806303i	$-0.298535\pi$
$227$	17.8238	1.18301	0.591503	+	0.806303i	$0.298535\pi$
$228$	0	0
$229$	−18.6566	−1.23287	−0.616433	−	0.787408i	$-0.711423\pi$
$229$	−18.6566	−1.23287	−0.616433	+	0.787408i	$0.711423\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−5.63485	−0.369151	−0.184575	−	0.982818i	$-0.559091\pi$
$233$	−5.63485	−0.369151	−0.184575	+	0.982818i	$0.559091\pi$
$234$	0	0
$235$	5.80907	0.378942
$236$	0	0
$237$	11.6348	0.755761
$238$	0	0
$239$	−9.13621	−0.590972	−0.295486	−	0.955347i	$-0.595482\pi$
$239$	−9.13621	−0.590972	−0.295486	+	0.955347i	$0.595482\pi$
$240$	0	0
$241$	−23.6516	−1.52353	−0.761766	−	0.647852i	$-0.775668\pi$
$241$	−23.6516	−1.52353	−0.761766	+	0.647852i	$0.775668\pi$
$242$	0	0
$243$	14.7432	0.945779
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−4.54841	−0.289408
$248$	0	0
$249$	−15.6662	−0.992804
$250$	0	0
$251$	−8.96988	−0.566174	−0.283087	−	0.959094i	$-0.591359\pi$
$251$	−8.96988	−0.566174	−0.283087	+	0.959094i	$0.591359\pi$
$252$	0	0
$253$	−3.39424	−0.213394
$254$	0	0
$255$	15.5824	0.975808
$256$	0	0
$257$	−9.23528	−0.576081	−0.288040	−	0.957618i	$-0.593004\pi$
$257$	−9.23528	−0.576081	−0.288040	+	0.957618i	$0.593004\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−12.5432	−0.776403
$262$	0	0
$263$	28.0521	1.72976	0.864882	−	0.501975i	$-0.167393\pi$
$263$	28.0521	1.72976	0.864882	+	0.501975i	$0.167393\pi$
$264$	0	0
$265$	34.6535	2.12875
$266$	0	0
$267$	−0.899716	−0.0550617
$268$	0	0
$269$	7.96236	0.485474	0.242737	−	0.970092i	$-0.421955\pi$
$269$	7.96236	0.485474	0.242737	+	0.970092i	$0.421955\pi$
$270$	0	0
$271$	4.72220	0.286853	0.143426	−	0.989661i	$-0.454188\pi$
$271$	4.72220	0.286853	0.143426	+	0.989661i	$0.454188\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	11.8934	0.717200
$276$	0	0
$277$	5.92448	0.355968	0.177984	−	0.984033i	$-0.443043\pi$
$277$	5.92448	0.355968	0.177984	+	0.984033i	$0.443043\pi$
$278$	0	0
$279$	0.205052	0.0122761
$280$	0	0
$281$	−15.3710	−0.916956	−0.458478	−	0.888706i	$-0.651605\pi$
$281$	−15.3710	−0.916956	−0.458478	+	0.888706i	$0.651605\pi$
$282$	0	0
$283$	−0.0322261	−0.00191564	−0.000957822	−	1.00000i	$-0.500305\pi$
$283$	−0.0322261	−0.00191564	−0.000957822		1.00000i	$0.500305\pi$
$284$	0	0
$285$	−5.85182	−0.346632
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−5.32170	−0.313041
$290$	0	0
$291$	−6.82311	−0.399978
$292$	0	0
$293$	−3.21017	−0.187540	−0.0937700	−	0.995594i	$-0.529892\pi$
$293$	−3.21017	−0.187540	−0.0937700	+	0.995594i	$0.529892\pi$
$294$	0	0
$295$	−11.9611	−0.696404
$296$	0	0
$297$	−6.06612	−0.351992
$298$	0	0
$299$	−10.6898	−0.618209
$300$	0	0
$301$	0	0
$302$	0	0
$303$	0.810792	0.0465788
$304$	0	0
$305$	48.1569	2.75746
$306$	0	0
$307$	−17.0753	−0.974540	−0.487270	−	0.873251i	$-0.662007\pi$
$307$	−17.0753	−0.974540	−0.487270	+	0.873251i	$0.662007\pi$
$308$	0	0
$309$	−11.7716	−0.669663
$310$	0	0
$311$	20.9903	1.19025	0.595126	−	0.803632i	$-0.297102\pi$
$311$	20.9903	1.19025	0.595126	+	0.803632i	$0.297102\pi$
$312$	0	0
$313$	2.67884	0.151417	0.0757085	−	0.997130i	$-0.475878\pi$
$313$	2.67884	0.151417	0.0757085	+	0.997130i	$0.475878\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	21.2252	1.19212	0.596062	−	0.802939i	$-0.296731\pi$
$317$	21.2252	1.19212	0.596062	+	0.802939i	$0.296731\pi$
$318$	0	0
$319$	8.48017	0.474798
$320$	0	0
$321$	−11.2120	−0.625793
$322$	0	0
$323$	−4.38567	−0.244025
$324$	0	0
$325$	37.4571	2.07775
$326$	0	0
$327$	9.75311	0.539348
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−7.04279	−0.387107	−0.193553	−	0.981090i	$-0.562001\pi$
$331$	−7.04279	−0.387107	−0.193553	+	0.981090i	$0.562001\pi$
$332$	0	0
$333$	−1.53824	−0.0842952
$334$	0	0
$335$	−10.7680	−0.588319
$336$	0	0
$337$	−18.8099	−1.02464	−0.512322	−	0.858794i	$-0.671214\pi$
$337$	−18.8099	−1.02464	−0.512322	+	0.858794i	$0.671214\pi$
$338$	0	0
$339$	1.66400	0.0903759
$340$	0	0
$341$	−0.138631	−0.00750730
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−13.7531	−0.740445
$346$	0	0
$347$	−4.68569	−0.251541	−0.125771	−	0.992059i	$-0.540140\pi$
$347$	−4.68569	−0.251541	−0.125771	+	0.992059i	$0.540140\pi$
$348$	0	0
$349$	30.1321	1.61294	0.806468	−	0.591277i	$-0.201376\pi$
$349$	30.1321	1.61294	0.806468	+	0.591277i	$0.201376\pi$
$350$	0	0
$351$	−19.1046	−1.01973
$352$	0	0
$353$	−10.3867	−0.552829	−0.276414	−	0.961039i	$-0.589146\pi$
$353$	−10.3867	−0.552829	−0.276414	+	0.961039i	$0.589146\pi$
$354$	0	0
$355$	22.6058	1.19979
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−17.5176	−0.924542	−0.462271	−	0.886739i	$-0.652965\pi$
$359$	−17.5176	−0.924542	−0.462271	+	0.886739i	$0.652965\pi$
$360$	0	0
$361$	−17.3530	−0.913316
$362$	0	0
$363$	−11.2484	−0.590390
$364$	0	0
$365$	−29.4746	−1.54277
$366$	0	0
$367$	7.27107	0.379547	0.189773	−	0.981828i	$-0.439225\pi$
$367$	7.27107	0.379547	0.189773	+	0.981828i	$0.439225\pi$
$368$	0	0
$369$	1.66452	0.0866514
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−33.3781	−1.72825	−0.864126	−	0.503275i	$-0.832128\pi$
$373$	−33.3781	−1.72825	−0.864126	+	0.503275i	$0.832128\pi$
$374$	0	0
$375$	25.3920	1.31124
$376$	0	0
$377$	26.7074	1.37550
$378$	0	0
$379$	−8.88493	−0.456388	−0.228194	−	0.973616i	$-0.573282\pi$
$379$	−8.88493	−0.456388	−0.228194	+	0.973616i	$0.573282\pi$
$380$	0	0
$381$	−14.4225	−0.738885
$382$	0	0
$383$	−18.8596	−0.963679	−0.481840	−	0.876259i	$-0.660031\pi$
$383$	−18.8596	−0.963679	−0.481840	+	0.876259i	$0.660031\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	6.64346	0.337706
$388$	0	0
$389$	−34.3262	−1.74041	−0.870203	−	0.492693i	$-0.836013\pi$
$389$	−34.3262	−1.74041	−0.870203	+	0.492693i	$0.836013\pi$
$390$	0	0
$391$	−10.3074	−0.521265
$392$	0	0
$393$	23.3250	1.17659
$394$	0	0
$395$	39.7252	1.99879
$396$	0	0
$397$	−10.5501	−0.529495	−0.264747	−	0.964318i	$-0.585289\pi$
$397$	−10.5501	−0.529495	−0.264747	+	0.964318i	$0.585289\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	31.5621	1.57614	0.788068	−	0.615589i	$-0.211082\pi$
$401$	31.5621	1.57614	0.788068	+	0.615589i	$0.211082\pi$
$402$	0	0
$403$	−0.436605	−0.0217488
$404$	0	0
$405$	−4.87617	−0.242299
$406$	0	0
$407$	1.03997	0.0515495
$408$	0	0
$409$	20.1166	0.994703	0.497351	−	0.867549i	$-0.334306\pi$
$409$	20.1166	0.994703	0.497351	+	0.867549i	$0.334306\pi$
$410$	0	0
$411$	10.5652	0.521142
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−53.4897	−2.62571
$416$	0	0
$417$	24.4484	1.19724
$418$	0	0
$419$	−22.6059	−1.10437	−0.552186	−	0.833721i	$-0.686206\pi$
$419$	−22.6059	−1.10437	−0.552186	+	0.833721i	$0.686206\pi$
$420$	0	0
$421$	3.68375	0.179535	0.0897674	−	0.995963i	$-0.471388\pi$
$421$	3.68375	0.179535	0.0897674	+	0.995963i	$0.471388\pi$
$422$	0	0
$423$	−2.45058	−0.119151
$424$	0	0
$425$	36.1169	1.75193
$426$	0	0
$427$	0	0
$428$	0	0
$429$	4.60912	0.222530
$430$	0	0
$431$	40.2662	1.93956	0.969778	−	0.243988i	$-0.0784559\pi$
$431$	40.2662	1.93956	0.969778	+	0.243988i	$0.0784559\pi$
$432$	0	0
$433$	33.0742	1.58945	0.794723	−	0.606972i	$-0.207616\pi$
$433$	33.0742	1.58945	0.794723	+	0.606972i	$0.207616\pi$
$434$	0	0
$435$	34.3608	1.64747
$436$	0	0
$437$	3.87083	0.185167
$438$	0	0
$439$	−10.1150	−0.482761	−0.241381	−	0.970431i	$-0.577600\pi$
$439$	−10.1150	−0.482761	−0.241381	+	0.970431i	$0.577600\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−2.16423	−0.102826	−0.0514128	−	0.998677i	$-0.516372\pi$
$443$	−2.16423	−0.102826	−0.0514128	+	0.998677i	$0.516372\pi$
$444$	0	0
$445$	−3.07194	−0.145624
$446$	0	0
$447$	7.08323	0.335025
$448$	0	0
$449$	−34.2656	−1.61709	−0.808547	−	0.588431i	$-0.799746\pi$
$449$	−34.2656	−1.61709	−0.808547	+	0.588431i	$0.799746\pi$
$450$	0	0
$451$	−1.12534	−0.0529904
$452$	0	0
$453$	9.21236	0.432835
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−16.3160	−0.763230	−0.381615	−	0.924321i	$-0.624632\pi$
$457$	−16.3160	−0.763230	−0.381615	+	0.924321i	$0.624632\pi$
$458$	0	0
$459$	−18.4211	−0.859823
$460$	0	0
$461$	−9.68932	−0.451277	−0.225638	−	0.974211i	$-0.572447\pi$
$461$	−9.68932	−0.451277	−0.225638	+	0.974211i	$0.572447\pi$
$462$	0	0
$463$	23.0970	1.07341	0.536705	−	0.843770i	$-0.319669\pi$
$463$	23.0970	1.07341	0.536705	+	0.843770i	$0.319669\pi$
$464$	0	0
$465$	−0.561720	−0.0260491
$466$	0	0
$467$	−22.0946	−1.02242	−0.511208	−	0.859457i	$-0.670802\pi$
$467$	−22.0946	−1.02242	−0.511208	+	0.859457i	$0.670802\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−13.0191	−0.599889
$472$	0	0
$473$	−4.49150	−0.206519
$474$	0	0
$475$	−13.5634	−0.622329
$476$	0	0
$477$	−14.6187	−0.669346
$478$	0	0
$479$	20.8078	0.950732	0.475366	−	0.879788i	$-0.342316\pi$
$479$	20.8078	0.950732	0.475366	+	0.879788i	$0.342316\pi$
$480$	0	0
$481$	3.27529	0.149340
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−23.2964	−1.05784
$486$	0	0
$487$	−29.1355	−1.32025	−0.660127	−	0.751154i	$-0.729497\pi$
$487$	−29.1355	−1.32025	−0.660127	+	0.751154i	$0.729497\pi$
$488$	0	0
$489$	2.01770	0.0912434
$490$	0	0
$491$	−15.8054	−0.713287	−0.356644	−	0.934240i	$-0.616079\pi$
$491$	−15.8054	−0.713287	−0.356644	+	0.934240i	$0.616079\pi$
$492$	0	0
$493$	25.7519	1.15981
$494$	0	0
$495$	−7.39095	−0.332199
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−39.5339	−1.76978	−0.884890	−	0.465800i	$-0.845767\pi$
$499$	−39.5339	−1.76978	−0.884890	+	0.465800i	$0.845767\pi$
$500$	0	0
$501$	−16.5020	−0.737255
$502$	0	0
$503$	−38.5115	−1.71715	−0.858573	−	0.512692i	$-0.828648\pi$
$503$	−38.5115	−1.71715	−0.858573	+	0.512692i	$0.828648\pi$
$504$	0	0
$505$	2.76832	0.123189
$506$	0	0
$507$	−0.507246	−0.0225276
$508$	0	0
$509$	17.2109	0.762858	0.381429	−	0.924398i	$-0.375432\pi$
$509$	17.2109	0.762858	0.381429	+	0.924398i	$0.375432\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	6.91786	0.305431
$514$	0	0
$515$	−40.1923	−1.77108
$516$	0	0
$517$	1.65678	0.0728653
$518$	0	0
$519$	−5.12534	−0.224977
$520$	0	0
$521$	−7.94448	−0.348054	−0.174027	−	0.984741i	$-0.555678\pi$
$521$	−7.94448	−0.348054	−0.174027	+	0.984741i	$0.555678\pi$
$522$	0	0
$523$	7.98595	0.349201	0.174601	−	0.984639i	$-0.444137\pi$
$523$	7.98595	0.349201	0.174601	+	0.984639i	$0.444137\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−0.420983	−0.0183383
$528$	0	0
$529$	−13.9027	−0.604463
$530$	0	0
$531$	5.04586	0.218972
$532$	0	0
$533$	−3.54416	−0.153515
$534$	0	0
$535$	−38.2816	−1.65506
$536$	0	0
$537$	1.16658	0.0503418
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−0.815853	−0.0350763	−0.0175381	−	0.999846i	$-0.505583\pi$
$541$	−0.815853	−0.0350763	−0.0175381	+	0.999846i	$0.505583\pi$
$542$	0	0
$543$	17.1520	0.736061
$544$	0	0
$545$	33.3005	1.42643
$546$	0	0
$547$	24.0249	1.02723	0.513616	−	0.858020i	$-0.328306\pi$
$547$	24.0249	1.02723	0.513616	+	0.858020i	$0.328306\pi$
$548$	0	0
$549$	−20.3152	−0.867033
$550$	0	0
$551$	−9.67086	−0.411992
$552$	0	0
$553$	0	0
$554$	0	0
$555$	4.21387	0.178869
$556$	0	0
$557$	8.69660	0.368487	0.184243	−	0.982881i	$-0.441017\pi$
$557$	8.69660	0.368487	0.184243	+	0.982881i	$0.441017\pi$
$558$	0	0
$559$	−14.1455	−0.598291
$560$	0	0
$561$	4.44421	0.187634
$562$	0	0
$563$	32.9095	1.38697	0.693486	−	0.720470i	$-0.256074\pi$
$563$	32.9095	1.38697	0.693486	+	0.720470i	$0.256074\pi$
$564$	0	0
$565$	5.68146	0.239021
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−19.8229	−0.831020	−0.415510	−	0.909589i	$-0.636397\pi$
$569$	−19.8229	−0.831020	−0.415510	+	0.909589i	$0.636397\pi$
$570$	0	0
$571$	−5.62422	−0.235366	−0.117683	−	0.993051i	$-0.537547\pi$
$571$	−5.62422	−0.235366	−0.117683	+	0.993051i	$0.537547\pi$
$572$	0	0
$573$	4.78108	0.199732
$574$	0	0
$575$	−31.8771	−1.32937
$576$	0	0
$577$	17.7611	0.739404	0.369702	−	0.929150i	$-0.379460\pi$
$577$	17.7611	0.739404	0.369702	+	0.929150i	$0.379460\pi$
$578$	0	0
$579$	11.4964	0.477776
$580$	0	0
$581$	0	0
$582$	0	0
$583$	9.88340	0.409329
$584$	0	0
$585$	−23.2771	−0.962388
$586$	0	0
$587$	−24.6592	−1.01779	−0.508897	−	0.860827i	$-0.669947\pi$
$587$	−24.6592	−1.01779	−0.508897	+	0.860827i	$0.669947\pi$
$588$	0	0
$589$	0.158096	0.00651424
$590$	0	0
$591$	5.60474	0.230548
$592$	0	0
$593$	33.2007	1.36339	0.681695	−	0.731637i	$-0.261243\pi$
$593$	33.2007	1.36339	0.681695	+	0.731637i	$0.261243\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	31.6486	1.29529
$598$	0	0
$599$	26.3074	1.07489	0.537445	−	0.843299i	$-0.319389\pi$
$599$	26.3074	1.07489	0.537445	+	0.843299i	$0.319389\pi$
$600$	0	0
$601$	−46.3932	−1.89242	−0.946209	−	0.323556i	$-0.895122\pi$
$601$	−46.3932	−1.89242	−0.946209	+	0.323556i	$0.895122\pi$
$602$	0	0
$603$	4.54253	0.184986
$604$	0	0
$605$	−38.4060	−1.56143
$606$	0	0
$607$	−20.5000	−0.832069	−0.416034	−	0.909349i	$-0.636580\pi$
$607$	−20.5000	−0.832069	−0.416034	+	0.909349i	$0.636580\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	5.21788	0.211093
$612$	0	0
$613$	−7.94499	−0.320895	−0.160448	−	0.987044i	$-0.551294\pi$
$613$	−7.94499	−0.320895	−0.160448	+	0.987044i	$0.551294\pi$
$614$	0	0
$615$	−4.55979	−0.183868
$616$	0	0
$617$	1.49848	0.0603265	0.0301632	−	0.999545i	$-0.490397\pi$
$617$	1.49848	0.0603265	0.0301632	+	0.999545i	$0.490397\pi$
$618$	0	0
$619$	−18.4540	−0.741731	−0.370865	−	0.928687i	$-0.620939\pi$
$619$	−18.4540	−0.741731	−0.370865	+	0.928687i	$0.620939\pi$
$620$	0	0
$621$	16.2586	0.652435
$622$	0	0
$623$	0	0
$624$	0	0
$625$	33.8536	1.35415
$626$	0	0
$627$	−1.66898	−0.0666525
$628$	0	0
$629$	3.15810	0.125922
$630$	0	0
$631$	46.7823	1.86237	0.931187	−	0.364543i	$-0.118775\pi$
$631$	46.7823	1.86237	0.931187	+	0.364543i	$0.118775\pi$
$632$	0	0
$633$	−23.2677	−0.924808
$634$	0	0
$635$	−49.2432	−1.95416
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−9.53638	−0.377253
$640$	0	0
$641$	−0.554345	−0.0218953	−0.0109476	−	0.999940i	$-0.503485\pi$
$641$	−0.554345	−0.0218953	−0.0109476	+	0.999940i	$0.503485\pi$
$642$	0	0
$643$	−16.3700	−0.645572	−0.322786	−	0.946472i	$-0.604619\pi$
$643$	−16.3700	−0.645572	−0.322786	+	0.946472i	$0.604619\pi$
$644$	0	0
$645$	−18.1991	−0.716589
$646$	0	0
$647$	6.20653	0.244004	0.122002	−	0.992530i	$-0.461069\pi$
$647$	6.20653	0.244004	0.122002	+	0.992530i	$0.461069\pi$
$648$	0	0
$649$	−3.41140	−0.133909
$650$	0	0
$651$	0	0
$652$	0	0
$653$	1.86445	0.0729616	0.0364808	−	0.999334i	$-0.488385\pi$
$653$	1.86445	0.0729616	0.0364808	+	0.999334i	$0.488385\pi$
$654$	0	0
$655$	79.6394	3.11177
$656$	0	0
$657$	12.4340	0.485097
$658$	0	0
$659$	−35.2076	−1.37149	−0.685747	−	0.727840i	$-0.740524\pi$
$659$	−35.2076	−1.37149	−0.685747	+	0.727840i	$0.740524\pi$
$660$	0	0
$661$	16.1818	0.629397	0.314699	−	0.949192i	$-0.398097\pi$
$661$	16.1818	0.629397	0.314699	+	0.949192i	$0.398097\pi$
$662$	0	0
$663$	13.9966	0.543582
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−22.7288	−0.880062
$668$	0	0
$669$	−1.29201	−0.0499519
$670$	0	0
$671$	13.7347	0.530222
$672$	0	0
$673$	0.0216325	0.000833870 0	0.000416935	−	1.00000i	$-0.499867\pi$
$673$	0.0216325	0.000833870 0	0.000416935		1.00000i	$0.499867\pi$
$674$	0	0
$675$	−56.9700	−2.19278
$676$	0	0
$677$	29.8548	1.14741	0.573706	−	0.819061i	$-0.305505\pi$
$677$	29.8548	1.14741	0.573706	+	0.819061i	$0.305505\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	20.5977	0.789305
$682$	0	0
$683$	−23.2050	−0.887913	−0.443956	−	0.896048i	$-0.646426\pi$
$683$	−23.2050	−0.887913	−0.443956	+	0.896048i	$0.646426\pi$
$684$	0	0
$685$	36.0732	1.37829
$686$	0	0
$687$	−21.5602	−0.822572
$688$	0	0
$689$	31.1268	1.18584
$690$	0	0
$691$	9.06237	0.344749	0.172374	−	0.985032i	$-0.444856\pi$
$691$	9.06237	0.344749	0.172374	+	0.985032i	$0.444856\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	83.4752	3.16639
$696$	0	0
$697$	−3.41735	−0.129441
$698$	0	0
$699$	−6.51179	−0.246299
$700$	0	0
$701$	5.46047	0.206239	0.103120	−	0.994669i	$-0.467118\pi$
$701$	5.46047	0.206239	0.103120	+	0.994669i	$0.467118\pi$
$702$	0	0
$703$	−1.18599	−0.0447306
$704$	0	0
$705$	6.71313	0.252831
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−36.2012	−1.35956	−0.679782	−	0.733414i	$-0.737926\pi$
$709$	−36.2012	−1.35956	−0.679782	+	0.733414i	$0.737926\pi$
$710$	0	0
$711$	−16.7583	−0.628484
$712$	0	0
$713$	0.371563	0.0139152
$714$	0	0
$715$	15.7371	0.588534
$716$	0	0
$717$	−10.5581	−0.394299
$718$	0	0
$719$	42.8296	1.59728	0.798638	−	0.601812i	$-0.205554\pi$
$719$	42.8296	1.59728	0.798638	+	0.601812i	$0.205554\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−27.3325	−1.01651
$724$	0	0
$725$	79.6415	2.95781
$726$	0	0
$727$	−49.0331	−1.81854	−0.909268	−	0.416210i	$-0.863358\pi$
$727$	−49.0331	−1.81854	−0.909268	+	0.416210i	$0.863358\pi$
$728$	0	0
$729$	20.7452	0.768339
$730$	0	0
$731$	−13.6394	−0.504471
$732$	0	0
$733$	12.6149	0.465942	0.232971	−	0.972484i	$-0.425155\pi$
$733$	12.6149	0.465942	0.232971	+	0.972484i	$0.425155\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−3.07111	−0.113126
$738$	0	0
$739$	11.7584	0.432539	0.216270	−	0.976334i	$-0.430611\pi$
$739$	11.7584	0.432539	0.216270	+	0.976334i	$0.430611\pi$
$740$	0	0
$741$	−5.25628	−0.193094
$742$	0	0
$743$	−0.0548021	−0.00201049	−0.00100525	−	0.999999i	$-0.500320\pi$
$743$	−0.0548021	−0.00201049	−0.00100525	+	0.999999i	$0.500320\pi$
$744$	0	0
$745$	24.1846	0.886054
$746$	0	0
$747$	22.5649	0.825606
$748$	0	0
$749$	0	0
$750$	0	0
$751$	39.3802	1.43700	0.718501	−	0.695525i	$-0.244828\pi$
$751$	39.3802	1.43700	0.718501	+	0.695525i	$0.244828\pi$
$752$	0	0
$753$	−10.3659	−0.377753
$754$	0	0
$755$	31.4542	1.14473
$756$	0	0
$757$	3.34906	0.121724	0.0608618	−	0.998146i	$-0.480615\pi$
$757$	3.34906	0.121724	0.0608618	+	0.998146i	$0.480615\pi$
$758$	0	0
$759$	−3.92249	−0.142377
$760$	0	0
$761$	40.4465	1.46618	0.733092	−	0.680130i	$-0.238077\pi$
$761$	40.4465	1.46618	0.733092	+	0.680130i	$0.238077\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−22.4442	−0.811473
$766$	0	0
$767$	−10.7438	−0.387938
$768$	0	0
$769$	13.4548	0.485194	0.242597	−	0.970127i	$-0.422001\pi$
$769$	13.4548	0.485194	0.242597	+	0.970127i	$0.422001\pi$
$770$	0	0
$771$	−10.6726	−0.384363
$772$	0	0
$773$	55.4155	1.99316	0.996579	−	0.0826452i	$-0.0263368\pi$
$773$	55.4155	1.99316	0.996579	+	0.0826452i	$0.0263368\pi$
$774$	0	0
$775$	−1.30196	−0.0467676
$776$	0	0
$777$	0	0
$778$	0	0
$779$	1.28335	0.0459809
$780$	0	0
$781$	6.44733	0.230704
$782$	0	0
$783$	−40.6204	−1.45166
$784$	0	0
$785$	−44.4517	−1.58655
$786$	0	0
$787$	−34.9091	−1.24438	−0.622188	−	0.782868i	$-0.713756\pi$
$787$	−34.9091	−1.24438	−0.622188	+	0.782868i	$0.713756\pi$
$788$	0	0
$789$	32.4178	1.15410
$790$	0	0
$791$	0	0
$792$	0	0
$793$	43.2560	1.53607
$794$	0	0
$795$	40.0466	1.42031
$796$	0	0
$797$	−8.12865	−0.287931	−0.143966	−	0.989583i	$-0.545986\pi$
$797$	−8.12865	−0.287931	−0.143966	+	0.989583i	$0.545986\pi$
$798$	0	0
$799$	5.03118	0.177991
$800$	0	0
$801$	1.29591	0.0457888
$802$	0	0
$803$	−8.40635	−0.296654
$804$	0	0
$805$	0	0
$806$	0	0
$807$	9.20154	0.323910
$808$	0	0
$809$	40.9839	1.44092	0.720458	−	0.693499i	$-0.243932\pi$
$809$	40.9839	1.44092	0.720458	+	0.693499i	$0.243932\pi$
$810$	0	0
$811$	5.69788	0.200080	0.100040	−	0.994983i	$-0.468103\pi$
$811$	5.69788	0.200080	0.100040	+	0.994983i	$0.468103\pi$
$812$	0	0
$813$	5.45711	0.191389
$814$	0	0
$815$	6.88911	0.241315
$816$	0	0
$817$	5.12214	0.179201
$818$	0	0
$819$	0	0
$820$	0	0
$821$	17.3086	0.604076	0.302038	−	0.953296i	$-0.402333\pi$
$821$	17.3086	0.604076	0.302038	+	0.953296i	$0.402333\pi$
$822$	0	0
$823$	−47.3436	−1.65030	−0.825148	−	0.564917i	$-0.808908\pi$
$823$	−47.3436	−1.65030	−0.825148	+	0.564917i	$0.808908\pi$
$824$	0	0
$825$	13.7444	0.478518
$826$	0	0
$827$	50.9924	1.77318	0.886589	−	0.462558i	$-0.153068\pi$
$827$	50.9924	1.77318	0.886589	+	0.462558i	$0.153068\pi$
$828$	0	0
$829$	21.9429	0.762110	0.381055	−	0.924552i	$-0.375561\pi$
$829$	21.9429	0.762110	0.381055	+	0.924552i	$0.375561\pi$
$830$	0	0
$831$	6.84651	0.237503
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−56.3435	−1.94985
$836$	0	0
$837$	0.664050	0.0229529
$838$	0	0
$839$	5.79049	0.199910	0.0999550	−	0.994992i	$-0.468130\pi$
$839$	5.79049	0.199910	0.0999550	+	0.994992i	$0.468130\pi$
$840$	0	0
$841$	27.7855	0.958121
$842$	0	0
$843$	−17.7632	−0.611796
$844$	0	0
$845$	−1.73191	−0.0595796
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−0.0372415	−0.00127812
$850$	0	0
$851$	−2.78736	−0.0955496
$852$	0	0
$853$	−6.63091	−0.227038	−0.113519	−	0.993536i	$-0.536212\pi$
$853$	−6.63091	−0.227038	−0.113519	+	0.993536i	$0.536212\pi$
$854$	0	0
$855$	8.42871	0.288256
$856$	0	0
$857$	3.09276	0.105647	0.0528233	−	0.998604i	$-0.483178\pi$
$857$	3.09276	0.105647	0.0528233	+	0.998604i	$0.483178\pi$
$858$	0	0
$859$	11.7716	0.401640	0.200820	−	0.979628i	$-0.435639\pi$
$859$	11.7716	0.401640	0.200820	+	0.979628i	$0.435639\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	32.7828	1.11594	0.557970	−	0.829861i	$-0.311580\pi$
$863$	32.7828	1.11594	0.557970	+	0.829861i	$0.311580\pi$
$864$	0	0
$865$	−17.4997	−0.595007
$866$	0	0
$867$	−6.14992	−0.208862
$868$	0	0
$869$	11.3299	0.384340
$870$	0	0
$871$	−9.67214	−0.327728
$872$	0	0
$873$	9.82772	0.332618
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−37.6650	−1.27186	−0.635929	−	0.771748i	$-0.719383\pi$
$877$	−37.6650	−1.27186	−0.635929	+	0.771748i	$0.719383\pi$
$878$	0	0
$879$	−3.70976	−0.125127
$880$	0	0
$881$	−57.0582	−1.92234	−0.961170	−	0.275956i	$-0.911006\pi$
$881$	−57.0582	−1.92234	−0.961170	+	0.275956i	$0.911006\pi$
$882$	0	0
$883$	−40.5820	−1.36569	−0.682846	−	0.730562i	$-0.739258\pi$
$883$	−40.5820	−1.36569	−0.682846	+	0.730562i	$0.739258\pi$
$884$	0	0
$885$	−13.8226	−0.464643
$886$	0	0
$887$	−8.35120	−0.280406	−0.140203	−	0.990123i	$-0.544775\pi$
$887$	−8.35120	−0.280406	−0.140203	+	0.990123i	$0.544775\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−1.39072	−0.0465908
$892$	0	0
$893$	−1.88941	−0.0632268
$894$	0	0
$895$	3.98311	0.133141
$896$	0	0
$897$	−12.3535	−0.412471
$898$	0	0
$899$	−0.928312	−0.0309609
$900$	0	0
$901$	30.0131	0.999881
$902$	0	0
$903$	0	0
$904$	0	0
$905$	58.5627	1.94669
$906$	0	0
$907$	−13.9348	−0.462696	−0.231348	−	0.972871i	$-0.574314\pi$
$907$	−13.9348	−0.462696	−0.231348	+	0.972871i	$0.574314\pi$
$908$	0	0
$909$	−1.16783	−0.0387345
$910$	0	0
$911$	−9.88485	−0.327500	−0.163750	−	0.986502i	$-0.552359\pi$
$911$	−9.88485	−0.327500	−0.163750	+	0.986502i	$0.552359\pi$
$912$	0	0
$913$	−15.2556	−0.504887
$914$	0	0
$915$	55.6516	1.83979
$916$	0	0
$917$	0	0
$918$	0	0
$919$	38.6877	1.27619	0.638095	−	0.769958i	$-0.279723\pi$
$919$	38.6877	1.27619	0.638095	+	0.769958i	$0.279723\pi$
$920$	0	0
$921$	−19.7327	−0.650216
$922$	0	0
$923$	20.3052	0.668355
$924$	0	0
$925$	9.76691	0.321134
$926$	0	0
$927$	16.9553	0.556886
$928$	0	0
$929$	−33.3468	−1.09407	−0.547036	−	0.837109i	$-0.684244\pi$
$929$	−33.3468	−1.09407	−0.547036	+	0.837109i	$0.684244\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	24.2571	0.794140
$934$	0	0
$935$	15.1740	0.496244
$936$	0	0
$937$	−25.5278	−0.833957	−0.416979	−	0.908916i	$-0.636911\pi$
$937$	−25.5278	−0.833957	−0.416979	+	0.908916i	$0.636911\pi$
$938$	0	0
$939$	3.09575	0.101026
$940$	0	0
$941$	−16.5550	−0.539679	−0.269839	−	0.962905i	$-0.586971\pi$
$941$	−16.5550	−0.539679	−0.269839	+	0.962905i	$0.586971\pi$
$942$	0	0
$943$	3.01618	0.0982204
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−49.6494	−1.61339	−0.806694	−	0.590970i	$-0.798745\pi$
$947$	−49.6494	−1.61339	−0.806694	+	0.590970i	$0.798745\pi$
$948$	0	0
$949$	−26.4750	−0.859413
$950$	0	0
$951$	24.5284	0.795388
$952$	0	0
$953$	39.4973	1.27944	0.639721	−	0.768607i	$-0.279050\pi$
$953$	39.4973	1.27944	0.639721	+	0.768607i	$0.279050\pi$
$954$	0	0
$955$	16.3242	0.528240
$956$	0	0
$957$	9.79993	0.316787
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−30.9848	−0.999510
$962$	0	0
$963$	16.1493	0.520403
$964$	0	0
$965$	39.2528	1.26359
$966$	0	0
$967$	27.3053	0.878080	0.439040	−	0.898468i	$-0.355319\pi$
$967$	27.3053	0.878080	0.439040	+	0.898468i	$0.355319\pi$
$968$	0	0
$969$	−5.06821	−0.162814
$970$	0	0
$971$	−21.9950	−0.705853	−0.352926	−	0.935651i	$-0.614813\pi$
$971$	−21.9950	−0.705853	−0.352926	+	0.935651i	$0.614813\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	43.2865	1.38628
$976$	0	0
$977$	8.72454	0.279123	0.139561	−	0.990213i	$-0.455431\pi$
$977$	8.72454	0.279123	0.139561	+	0.990213i	$0.455431\pi$
$978$	0	0
$979$	−0.876138	−0.0280015
$980$	0	0
$981$	−14.0480	−0.448517
$982$	0	0
$983$	46.1851	1.47307	0.736537	−	0.676397i	$-0.236460\pi$
$983$	46.1851	1.47307	0.736537	+	0.676397i	$0.236460\pi$
$984$	0	0
$985$	19.1365	0.609739
$986$	0	0
$987$	0	0
$988$	0	0
$989$	12.0382	0.382794
$990$	0	0
$991$	−12.3287	−0.391635	−0.195818	−	0.980640i	$-0.562736\pi$
$991$	−12.3287	−0.391635	−0.195818	+	0.980640i	$0.562736\pi$
$992$	0	0
$993$	−8.13886	−0.258279
$994$	0	0
$995$	108.059	3.42571
$996$	0	0
$997$	39.9390	1.26488	0.632441	−	0.774609i	$-0.282053\pi$
$997$	39.9390	1.26488	0.632441	+	0.774609i	$0.282053\pi$
$998$	0	0
$999$	−4.98152	−0.157608

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8036.2.a.t.1.14	yes	20
7.6	odd	2		8036.2.a.s.1.7	✓	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8036.2.a.s.1.7	✓	20	7.6	odd	2
8036.2.a.t.1.14	yes	20	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)

\(f(q)\)	\(=\)	\(q+1.15563 q^{3} +3.94572 q^{5} -1.66452 q^{9} +O(q^{10})\)	\(q+1.15563 q^{3} +3.94572 q^{5} -1.66452 q^{9} +1.12534 q^{11} +3.54416 q^{13} +4.55979 q^{15} +3.41735 q^{17} -1.28335 q^{19} -3.01618 q^{23} +10.5687 q^{25} -5.39046 q^{27} +7.53562 q^{29} -0.123190 q^{31} +1.30048 q^{33} +0.924137 q^{37} +4.09574 q^{39} -1.00000 q^{41} -3.99122 q^{43} -6.56772 q^{45} +1.47225 q^{47} +3.94919 q^{51} +8.78256 q^{53} +4.44029 q^{55} -1.48308 q^{57} -3.03142 q^{59} +12.2049 q^{61} +13.9842 q^{65} -2.72904 q^{67} -3.48559 q^{69} +5.72921 q^{71} -7.47002 q^{73} +12.2135 q^{75} +10.0679 q^{79} -1.23581 q^{81} -13.5564 q^{83} +13.4839 q^{85} +8.70838 q^{87} -0.778550 q^{89} -0.142362 q^{93} -5.06375 q^{95} -5.90424 q^{97} -1.87316 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 4 q^{3} + 8 q^{5} + 16 q^{9}+O(q^{10}) \) 20 * q + 4 * q^3 + 8 * q^5 + 16 * q^9	\( 20 q + 4 q^{3} + 8 q^{5} + 16 q^{9} - 8 q^{11} + 12 q^{13} + 8 q^{15} + 8 q^{17} + 24 q^{19} + 8 q^{23} + 20 q^{25} + 16 q^{27} - 12 q^{29} + 44 q^{33} + 12 q^{37} + 12 q^{39} - 20 q^{41} + 4 q^{43} + 40 q^{45} + 4 q^{47} + 4 q^{51} - 12 q^{53} - 16 q^{55} + 28 q^{57} + 16 q^{59} + 68 q^{61} - 8 q^{65} + 4 q^{67} + 32 q^{69} + 8 q^{71} + 48 q^{73} + 60 q^{75} - 20 q^{79} + 32 q^{81} - 8 q^{83} - 28 q^{85} + 60 q^{89} - 16 q^{93} + 20 q^{95} + 40 q^{97}+O(q^{100}) \) 20 * q + 4 * q^3 + 8 * q^5 + 16 * q^9 - 8 * q^11 + 12 * q^13 + 8 * q^15 + 8 * q^17 + 24 * q^19 + 8 * q^23 + 20 * q^25 + 16 * q^27 - 12 * q^29 + 44 * q^33 + 12 * q^37 + 12 * q^39 - 20 * q^41 + 4 * q^43 + 40 * q^45 + 4 * q^47 + 4 * q^51 - 12 * q^53 - 16 * q^55 + 28 * q^57 + 16 * q^59 + 68 * q^61 - 8 * q^65 + 4 * q^67 + 32 * q^69 + 8 * q^71 + 48 * q^73 + 60 * q^75 - 20 * q^79 + 32 * q^81 - 8 * q^83 - 28 * q^85 + 60 * q^89 - 16 * q^93 + 20 * q^95 + 40 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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