LMFDB - Embedded newform 8036.2.a.s.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:	$1$
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			$-0.915292$ 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)

$f(q)$	$=$	$q+0.915292 q^{3} +1.31004 q^{5} -2.16224 q^{9} +O(q^{10})$	$q+0.915292 q^{3} +1.31004 q^{5} -2.16224 q^{9} +2.72778 q^{11} -6.20323 q^{13} +1.19907 q^{15} +6.76678 q^{17} -5.15682 q^{19} +6.76186 q^{23} -3.28379 q^{25} -4.72496 q^{27} -6.37080 q^{29} -2.28883 q^{31} +2.49672 q^{33} +9.35867 q^{37} -5.67776 q^{39} +1.00000 q^{41} -6.74692 q^{43} -2.83262 q^{45} -8.15236 q^{47} +6.19357 q^{51} -12.5686 q^{53} +3.57351 q^{55} -4.72000 q^{57} -1.48363 q^{59} +2.36836 q^{61} -8.12648 q^{65} +2.61180 q^{67} +6.18908 q^{69} -1.28105 q^{71} -3.81398 q^{73} -3.00563 q^{75} +4.28357 q^{79} +2.16201 q^{81} +2.35962 q^{83} +8.86476 q^{85} -5.83114 q^{87} -3.98293 q^{89} -2.09495 q^{93} -6.75565 q^{95} +7.03488 q^{97} -5.89813 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	0.915292	0.528444	0.264222	−	0.964462i	$-0.414885\pi$
$3$	0.915292	0.528444	0.264222	+	0.964462i	$0.414885\pi$
$4$	0	0
$5$	1.31004	0.585868	0.292934	−	0.956133i	$-0.405368\pi$
$5$	1.31004	0.585868	0.292934	+	0.956133i	$0.405368\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−2.16224	−0.720747
$10$	0	0
$11$	2.72778	0.822458	0.411229	−	0.911532i	$-0.365100\pi$
$11$	2.72778	0.822458	0.411229	+	0.911532i	$0.365100\pi$
$12$	0	0
$13$	−6.20323	−1.72047	−0.860233	−	0.509902i	$-0.829682\pi$
$13$	−6.20323	−1.72047	−0.860233	+	0.509902i	$0.829682\pi$
$14$	0	0
$15$	1.19907	0.309598
$16$	0	0
$17$	6.76678	1.64118	0.820592	−	0.571514i	$-0.193644\pi$
$17$	6.76678	1.64118	0.820592	+	0.571514i	$0.193644\pi$
$18$	0	0
$19$	−5.15682	−1.18306	−0.591528	−	0.806284i	$-0.701475\pi$
$19$	−5.15682	−1.18306	−0.591528	+	0.806284i	$0.701475\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	6.76186	1.40995	0.704973	−	0.709234i	$-0.250959\pi$
$23$	6.76186	1.40995	0.704973	+	0.709234i	$0.250959\pi$
$24$	0	0
$25$	−3.28379	−0.656758
$26$	0	0
$27$	−4.72496	−0.909318
$28$	0	0
$29$	−6.37080	−1.18303	−0.591514	−	0.806295i	$-0.701470\pi$
$29$	−6.37080	−1.18303	−0.591514	+	0.806295i	$0.701470\pi$
$30$	0	0
$31$	−2.28883	−0.411086	−0.205543	−	0.978648i	$-0.565896\pi$
$31$	−2.28883	−0.411086	−0.205543	+	0.978648i	$0.565896\pi$
$32$	0	0
$33$	2.49672	0.434623
$34$	0	0
$35$	0	0
$36$	0	0
$37$	9.35867	1.53856	0.769278	−	0.638915i	$-0.220616\pi$
$37$	9.35867	1.53856	0.769278	+	0.638915i	$0.220616\pi$
$38$	0	0
$39$	−5.67776	−0.909169
$40$	0	0
$41$	1.00000	0.156174
$41$	1.00000	0.156174
$42$	0	0
$43$	−6.74692	−1.02890	−0.514448	−	0.857521i	$-0.672003\pi$
$43$	−6.74692	−1.02890	−0.514448	+	0.857521i	$0.672003\pi$
$44$	0	0
$45$	−2.83262	−0.422263
$46$	0	0
$47$	−8.15236	−1.18914	−0.594572	−	0.804042i	$-0.702678\pi$
$47$	−8.15236	−1.18914	−0.594572	+	0.804042i	$0.702678\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	6.19357	0.867274
$52$	0	0
$53$	−12.5686	−1.72643	−0.863215	−	0.504837i	$-0.831552\pi$
$53$	−12.5686	−1.72643	−0.863215	+	0.504837i	$0.831552\pi$
$54$	0	0
$55$	3.57351	0.481852
$56$	0	0
$57$	−4.72000	−0.625179
$58$	0	0
$59$	−1.48363	−0.193152	−0.0965759	−	0.995326i	$-0.530789\pi$
$59$	−1.48363	−0.193152	−0.0965759	+	0.995326i	$0.530789\pi$
$60$	0	0
$61$	2.36836	0.303237	0.151619	−	0.988439i	$-0.451551\pi$
$61$	2.36836	0.303237	0.151619	+	0.988439i	$0.451551\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−8.12648	−1.00797
$66$	0	0
$67$	2.61180	0.319083	0.159541	−	0.987191i	$-0.448998\pi$
$67$	2.61180	0.319083	0.159541	+	0.987191i	$0.448998\pi$
$68$	0	0
$69$	6.18908	0.745077
$70$	0	0
$71$	−1.28105	−0.152033	−0.0760164	−	0.997107i	$-0.524220\pi$
$71$	−1.28105	−0.152033	−0.0760164	+	0.997107i	$0.524220\pi$
$72$	0	0
$73$	−3.81398	−0.446393	−0.223196	−	0.974774i	$-0.571649\pi$
$73$	−3.81398	−0.446393	−0.223196	+	0.974774i	$0.571649\pi$
$74$	0	0
$75$	−3.00563	−0.347060
$76$	0	0
$77$	0	0
$78$	0	0
$79$	4.28357	0.481940	0.240970	−	0.970533i	$-0.422535\pi$
$79$	4.28357	0.481940	0.240970	+	0.970533i	$0.422535\pi$
$80$	0	0
$81$	2.16201	0.240223
$82$	0	0
$83$	2.35962	0.259002	0.129501	−	0.991579i	$-0.458662\pi$
$83$	2.35962	0.259002	0.129501	+	0.991579i	$0.458662\pi$
$84$	0	0
$85$	8.86476	0.961518
$86$	0	0
$87$	−5.83114	−0.625164
$88$	0	0
$89$	−3.98293	−0.422190	−0.211095	−	0.977466i	$-0.567703\pi$
$89$	−3.98293	−0.422190	−0.211095	+	0.977466i	$0.567703\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−2.09495	−0.217236
$94$	0	0
$95$	−6.75565	−0.693115
$96$	0	0
$97$	7.03488	0.714284	0.357142	−	0.934050i	$-0.383751\pi$
$97$	7.03488	0.714284	0.357142	+	0.934050i	$0.383751\pi$
$98$	0	0
$99$	−5.89813	−0.592784
$100$	0	0
$101$	6.24828	0.621727	0.310864	−	0.950455i	$-0.399382\pi$
$101$	6.24828	0.621727	0.310864	+	0.950455i	$0.399382\pi$
$102$	0	0
$103$	−7.53553	−0.742498	−0.371249	−	0.928533i	$-0.621070\pi$
$103$	−7.53553	−0.742498	−0.371249	+	0.928533i	$0.621070\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−17.8145	−1.72220	−0.861098	−	0.508439i	$-0.830223\pi$
$107$	−17.8145	−1.72220	−0.861098	+	0.508439i	$0.830223\pi$
$108$	0	0
$109$	−7.83067	−0.750042	−0.375021	−	0.927016i	$-0.622365\pi$
$109$	−7.83067	−0.750042	−0.375021	+	0.927016i	$0.622365\pi$
$110$	0	0
$111$	8.56591	0.813040
$112$	0	0
$113$	5.31934	0.500401	0.250201	−	0.968194i	$-0.419503\pi$
$113$	5.31934	0.500401	0.250201	+	0.968194i	$0.419503\pi$
$114$	0	0
$115$	8.85832	0.826042
$116$	0	0
$117$	13.4129	1.24002
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−3.55920	−0.323563
$122$	0	0
$123$	0.915292	0.0825291
$124$	0	0
$125$	−10.8521	−0.970642
$126$	0	0
$127$	9.66189	0.857354	0.428677	−	0.903458i	$-0.358980\pi$
$127$	9.66189	0.857354	0.428677	+	0.903458i	$0.358980\pi$
$128$	0	0
$129$	−6.17540	−0.543714
$130$	0	0
$131$	−17.2464	−1.50683	−0.753413	−	0.657547i	$-0.771594\pi$
$131$	−17.2464	−1.50683	−0.753413	+	0.657547i	$0.771594\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−6.18989	−0.532741
$136$	0	0
$137$	−8.10990	−0.692875	−0.346438	−	0.938073i	$-0.612609\pi$
$137$	−8.10990	−0.692875	−0.346438	+	0.938073i	$0.612609\pi$
$138$	0	0
$139$	2.72783	0.231372	0.115686	−	0.993286i	$-0.463093\pi$
$139$	2.72783	0.231372	0.115686	+	0.993286i	$0.463093\pi$
$140$	0	0
$141$	−7.46179	−0.628396
$142$	0	0
$143$	−16.9211	−1.41501
$144$	0	0
$145$	−8.34601	−0.693099
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−4.82784	−0.395512	−0.197756	−	0.980251i	$-0.563365\pi$
$149$	−4.82784	−0.395512	−0.197756	+	0.980251i	$0.563365\pi$
$150$	0	0
$151$	2.35596	0.191726	0.0958628	−	0.995395i	$-0.469439\pi$
$151$	2.35596	0.191726	0.0958628	+	0.995395i	$0.469439\pi$
$152$	0	0
$153$	−14.6314	−1.18288
$154$	0	0
$155$	−2.99846	−0.240842
$156$	0	0
$157$	−12.6689	−1.01109	−0.505543	−	0.862801i	$-0.668708\pi$
$157$	−12.6689	−1.01109	−0.505543	+	0.862801i	$0.668708\pi$
$158$	0	0
$159$	−11.5039	−0.912321
$160$	0	0
$161$	0	0
$162$	0	0
$163$	8.78527	0.688115	0.344058	−	0.938949i	$-0.388198\pi$
$163$	8.78527	0.688115	0.344058	+	0.938949i	$0.388198\pi$
$164$	0	0
$165$	3.27080	0.254632
$166$	0	0
$167$	3.24661	0.251230	0.125615	−	0.992079i	$-0.459910\pi$
$167$	3.24661	0.251230	0.125615	+	0.992079i	$0.459910\pi$
$168$	0	0
$169$	25.4800	1.96000
$170$	0	0
$171$	11.1503	0.852684
$172$	0	0
$173$	0.370585	0.0281751	0.0140875	−	0.999901i	$-0.495516\pi$
$173$	0.370585	0.0281751	0.0140875	+	0.999901i	$0.495516\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−1.35795	−0.102070
$178$	0	0
$179$	−24.0644	−1.79866	−0.899330	−	0.437271i	$-0.855945\pi$
$179$	−24.0644	−1.79866	−0.899330	+	0.437271i	$0.855945\pi$
$180$	0	0
$181$	1.03031	0.0765824	0.0382912	−	0.999267i	$-0.487809\pi$
$181$	1.03031	0.0765824	0.0382912	+	0.999267i	$0.487809\pi$
$182$	0	0
$183$	2.16774	0.160244
$184$	0	0
$185$	12.2602	0.901391
$186$	0	0
$187$	18.4583	1.34980
$188$	0	0
$189$	0	0
$190$	0	0
$191$	8.25690	0.597448	0.298724	−	0.954340i	$-0.403439\pi$
$191$	8.25690	0.597448	0.298724	+	0.954340i	$0.403439\pi$
$192$	0	0
$193$	9.77019	0.703274	0.351637	−	0.936137i	$-0.385625\pi$
$193$	9.77019	0.703274	0.351637	+	0.936137i	$0.385625\pi$
$194$	0	0
$195$	−7.43810	−0.532653
$196$	0	0
$197$	2.63912	0.188030	0.0940149	−	0.995571i	$-0.470030\pi$
$197$	2.63912	0.188030	0.0940149	+	0.995571i	$0.470030\pi$
$198$	0	0
$199$	−7.19404	−0.509972	−0.254986	−	0.966945i	$-0.582071\pi$
$199$	−7.19404	−0.509972	−0.254986	+	0.966945i	$0.582071\pi$
$200$	0	0
$201$	2.39056	0.168617
$202$	0	0
$203$	0	0
$204$	0	0
$205$	1.31004	0.0914972
$206$	0	0
$207$	−14.6208	−1.01621
$208$	0	0
$209$	−14.0667	−0.973014
$210$	0	0
$211$	−19.7516	−1.35976	−0.679879	−	0.733324i	$-0.737968\pi$
$211$	−19.7516	−1.35976	−0.679879	+	0.733324i	$0.737968\pi$
$212$	0	0
$213$	−1.17254	−0.0803409
$214$	0	0
$215$	−8.83875	−0.602798
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−3.49090	−0.235893
$220$	0	0
$221$	−41.9758	−2.82360
$222$	0	0
$223$	−19.8914	−1.33203	−0.666014	−	0.745939i	$-0.732001\pi$
$223$	−19.8914	−1.33203	−0.666014	+	0.745939i	$0.732001\pi$
$224$	0	0
$225$	7.10035	0.473357
$226$	0	0
$227$	−16.0483	−1.06516	−0.532581	−	0.846379i	$-0.678778\pi$
$227$	−16.0483	−1.06516	−0.532581	+	0.846379i	$0.678778\pi$
$228$	0	0
$229$	−20.4940	−1.35428	−0.677142	−	0.735852i	$-0.736782\pi$
$229$	−20.4940	−1.35428	−0.677142	+	0.735852i	$0.736782\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−15.2859	−1.00141	−0.500705	−	0.865618i	$-0.666926\pi$
$233$	−15.2859	−1.00141	−0.500705	+	0.865618i	$0.666926\pi$
$234$	0	0
$235$	−10.6799	−0.696682
$236$	0	0
$237$	3.92072	0.254678
$238$	0	0
$239$	27.3135	1.76676	0.883380	−	0.468657i	$-0.155262\pi$
$239$	27.3135	1.76676	0.883380	+	0.468657i	$0.155262\pi$
$240$	0	0
$241$	15.7932	1.01733	0.508663	−	0.860965i	$-0.330140\pi$
$241$	15.7932	1.01733	0.508663	+	0.860965i	$0.330140\pi$
$242$	0	0
$243$	16.1537	1.03626
$244$	0	0
$245$	0	0
$246$	0	0
$247$	31.9889	2.03541
$248$	0	0
$249$	2.15974	0.136868
$250$	0	0
$251$	−15.7718	−0.995505	−0.497753	−	0.867319i	$-0.665841\pi$
$251$	−15.7718	−0.995505	−0.497753	+	0.867319i	$0.665841\pi$
$252$	0	0
$253$	18.4449	1.15962
$254$	0	0
$255$	8.11384	0.508108
$256$	0	0
$257$	7.38691	0.460783	0.230391	−	0.973098i	$-0.425999\pi$
$257$	7.38691	0.460783	0.230391	+	0.973098i	$0.425999\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	13.7752	0.852664
$262$	0	0
$263$	−7.79849	−0.480875	−0.240438	−	0.970665i	$-0.577291\pi$
$263$	−7.79849	−0.480875	−0.240438	+	0.970665i	$0.577291\pi$
$264$	0	0
$265$	−16.4654	−1.01146
$266$	0	0
$267$	−3.64555	−0.223104
$268$	0	0
$269$	−11.3794	−0.693815	−0.346907	−	0.937899i	$-0.612768\pi$
$269$	−11.3794	−0.693815	−0.346907	+	0.937899i	$0.612768\pi$
$270$	0	0
$271$	20.2606	1.23075	0.615373	−	0.788236i	$-0.289005\pi$
$271$	20.2606	1.23075	0.615373	+	0.788236i	$0.289005\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−8.95747	−0.540156
$276$	0	0
$277$	22.4498	1.34888	0.674440	−	0.738330i	$-0.264385\pi$
$277$	22.4498	1.34888	0.674440	+	0.738330i	$0.264385\pi$
$278$	0	0
$279$	4.94900	0.296289
$280$	0	0
$281$	28.7285	1.71380	0.856898	−	0.515485i	$-0.172388\pi$
$281$	28.7285	1.71380	0.856898	+	0.515485i	$0.172388\pi$
$282$	0	0
$283$	−25.3840	−1.50892	−0.754462	−	0.656344i	$-0.772102\pi$
$283$	−25.3840	−1.50892	−0.754462	+	0.656344i	$0.772102\pi$
$284$	0	0
$285$	−6.18339	−0.366272
$286$	0	0
$287$	0	0
$288$	0	0
$289$	28.7893	1.69349
$290$	0	0
$291$	6.43897	0.377459
$292$	0	0
$293$	−26.3359	−1.53856	−0.769280	−	0.638912i	$-0.779385\pi$
$293$	−26.3359	−1.53856	−0.769280	+	0.638912i	$0.779385\pi$
$294$	0	0
$295$	−1.94361	−0.113161
$296$	0	0
$297$	−12.8887	−0.747876
$298$	0	0
$299$	−41.9454	−2.42576
$300$	0	0
$301$	0	0
$302$	0	0
$303$	5.71900	0.328548
$304$	0	0
$305$	3.10265	0.177657
$306$	0	0
$307$	29.8539	1.70385	0.851926	−	0.523663i	$-0.175435\pi$
$307$	29.8539	1.70385	0.851926	+	0.523663i	$0.175435\pi$
$308$	0	0
$309$	−6.89721	−0.392369
$310$	0	0
$311$	−1.09275	−0.0619642	−0.0309821	−	0.999520i	$-0.509863\pi$
$311$	−1.09275	−0.0619642	−0.0309821	+	0.999520i	$0.509863\pi$
$312$	0	0
$313$	−25.8202	−1.45945	−0.729723	−	0.683743i	$-0.760351\pi$
$313$	−25.8202	−1.45945	−0.729723	+	0.683743i	$0.760351\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	6.09034	0.342067	0.171034	−	0.985265i	$-0.445289\pi$
$317$	6.09034	0.342067	0.171034	+	0.985265i	$0.445289\pi$
$318$	0	0
$319$	−17.3782	−0.972991
$320$	0	0
$321$	−16.3055	−0.910084
$322$	0	0
$323$	−34.8951	−1.94161
$324$	0	0
$325$	20.3701	1.12993
$326$	0	0
$327$	−7.16735	−0.396355
$328$	0	0
$329$	0	0
$330$	0	0
$331$	13.7596	0.756298	0.378149	−	0.925745i	$-0.376561\pi$
$331$	13.7596	0.756298	0.378149	+	0.925745i	$0.376561\pi$
$332$	0	0
$333$	−20.2357	−1.10891
$334$	0	0
$335$	3.42157	0.186940
$336$	0	0
$337$	−28.3125	−1.54228	−0.771141	−	0.636664i	$-0.780314\pi$
$337$	−28.3125	−1.54228	−0.771141	+	0.636664i	$0.780314\pi$
$338$	0	0
$339$	4.86875	0.264434
$340$	0	0
$341$	−6.24343	−0.338101
$342$	0	0
$343$	0	0
$344$	0	0
$345$	8.10794	0.436517
$346$	0	0
$347$	−14.3979	−0.772920	−0.386460	−	0.922306i	$-0.626302\pi$
$347$	−14.3979	−0.772920	−0.386460	+	0.922306i	$0.626302\pi$
$348$	0	0
$349$	−15.5569	−0.832741	−0.416371	−	0.909195i	$-0.636698\pi$
$349$	−15.5569	−0.832741	−0.416371	+	0.909195i	$0.636698\pi$
$350$	0	0
$351$	29.3100	1.56445
$352$	0	0
$353$	−4.63356	−0.246620	−0.123310	−	0.992368i	$-0.539351\pi$
$353$	−4.63356	−0.246620	−0.123310	+	0.992368i	$0.539351\pi$
$354$	0	0
$355$	−1.67823	−0.0890712
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−6.24826	−0.329771	−0.164885	−	0.986313i	$-0.552725\pi$
$359$	−6.24826	−0.329771	−0.164885	+	0.986313i	$0.552725\pi$
$360$	0	0
$361$	7.59281	0.399622
$362$	0	0
$363$	−3.25770	−0.170985
$364$	0	0
$365$	−4.99647	−0.261527
$366$	0	0
$367$	−35.2894	−1.84209	−0.921047	−	0.389452i	$-0.872664\pi$
$367$	−35.2894	−1.84209	−0.921047	+	0.389452i	$0.872664\pi$
$368$	0	0
$369$	−2.16224	−0.112562
$370$	0	0
$371$	0	0
$372$	0	0
$373$	29.5655	1.53084	0.765421	−	0.643530i	$-0.222531\pi$
$373$	29.5655	1.53084	0.765421	+	0.643530i	$0.222531\pi$
$374$	0	0
$375$	−9.93284	−0.512930
$376$	0	0
$377$	39.5195	2.03536
$378$	0	0
$379$	−9.02380	−0.463522	−0.231761	−	0.972773i	$-0.574449\pi$
$379$	−9.02380	−0.463522	−0.231761	+	0.972773i	$0.574449\pi$
$380$	0	0
$381$	8.84344	0.453063
$382$	0	0
$383$	20.0681	1.02543	0.512716	−	0.858559i	$-0.328640\pi$
$383$	20.0681	1.02543	0.512716	+	0.858559i	$0.328640\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	14.5885	0.741574
$388$	0	0
$389$	35.4162	1.79567	0.897836	−	0.440330i	$-0.145139\pi$
$389$	35.4162	1.79567	0.897836	+	0.440330i	$0.145139\pi$
$390$	0	0
$391$	45.7560	2.31398
$392$	0	0
$393$	−15.7855	−0.796273
$394$	0	0
$395$	5.61166	0.282353
$396$	0	0
$397$	−20.8612	−1.04699	−0.523496	−	0.852028i	$-0.675372\pi$
$397$	−20.8612	−1.04699	−0.523496	+	0.852028i	$0.675372\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	9.68818	0.483805	0.241902	−	0.970301i	$-0.422229\pi$
$401$	9.68818	0.483805	0.241902	+	0.970301i	$0.422229\pi$
$402$	0	0
$403$	14.1981	0.707259
$404$	0	0
$405$	2.83232	0.140739
$406$	0	0
$407$	25.5284	1.26540
$408$	0	0
$409$	14.5208	0.718009	0.359004	−	0.933336i	$-0.383116\pi$
$409$	14.5208	0.718009	0.359004	+	0.933336i	$0.383116\pi$
$410$	0	0
$411$	−7.42292	−0.366146
$412$	0	0
$413$	0	0
$414$	0	0
$415$	3.09120	0.151741
$416$	0	0
$417$	2.49676	0.122267
$418$	0	0
$419$	7.75224	0.378721	0.189361	−	0.981908i	$-0.439358\pi$
$419$	7.75224	0.378721	0.189361	+	0.981908i	$0.439358\pi$
$420$	0	0
$421$	6.10905	0.297737	0.148869	−	0.988857i	$-0.452437\pi$
$421$	6.10905	0.297737	0.148869	+	0.988857i	$0.452437\pi$
$422$	0	0
$423$	17.6274	0.857072
$424$	0	0
$425$	−22.2207	−1.07786
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−15.4877	−0.747753
$430$	0	0
$431$	20.0724	0.966853	0.483427	−	0.875385i	$-0.339392\pi$
$431$	20.0724	0.966853	0.483427	+	0.875385i	$0.339392\pi$
$432$	0	0
$433$	−18.0613	−0.867973	−0.433986	−	0.900919i	$-0.642893\pi$
$433$	−18.0613	−0.867973	−0.433986	+	0.900919i	$0.642893\pi$
$434$	0	0
$435$	−7.63904	−0.366264
$436$	0	0
$437$	−34.8697	−1.66804
$438$	0	0
$439$	4.24242	0.202480	0.101240	−	0.994862i	$-0.467719\pi$
$439$	4.24242	0.202480	0.101240	+	0.994862i	$0.467719\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	16.3712	0.777819	0.388909	−	0.921276i	$-0.372852\pi$
$443$	16.3712	0.777819	0.388909	+	0.921276i	$0.372852\pi$
$444$	0	0
$445$	−5.21781	−0.247348
$446$	0	0
$447$	−4.41888	−0.209006
$448$	0	0
$449$	−14.4030	−0.679721	−0.339860	−	0.940476i	$-0.610380\pi$
$449$	−14.4030	−0.679721	−0.339860	+	0.940476i	$0.610380\pi$
$450$	0	0
$451$	2.72778	0.128446
$452$	0	0
$453$	2.15639	0.101316
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−12.7736	−0.597523	−0.298761	−	0.954328i	$-0.596573\pi$
$457$	−12.7736	−0.597523	−0.298761	+	0.954328i	$0.596573\pi$
$458$	0	0
$459$	−31.9727	−1.49236
$460$	0	0
$461$	3.83939	0.178818	0.0894092	−	0.995995i	$-0.471502\pi$
$461$	3.83939	0.178818	0.0894092	+	0.995995i	$0.471502\pi$
$462$	0	0
$463$	42.3474	1.96805	0.984025	−	0.178030i	$-0.0569723\pi$
$463$	42.3474	1.96805	0.984025	+	0.178030i	$0.0569723\pi$
$464$	0	0
$465$	−2.74447	−0.127272
$466$	0	0
$467$	27.2929	1.26296	0.631482	−	0.775390i	$-0.282447\pi$
$467$	27.2929	1.26296	0.631482	+	0.775390i	$0.282447\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−11.5957	−0.534302
$472$	0	0
$473$	−18.4041	−0.846224
$474$	0	0
$475$	16.9339	0.776982
$476$	0	0
$477$	27.1763	1.24432
$478$	0	0
$479$	−24.9162	−1.13845	−0.569225	−	0.822182i	$-0.692757\pi$
$479$	−24.9162	−1.13845	−0.569225	+	0.822182i	$0.692757\pi$
$480$	0	0
$481$	−58.0539	−2.64703
$482$	0	0
$483$	0	0
$484$	0	0
$485$	9.21598	0.418476
$486$	0	0
$487$	17.1545	0.777346	0.388673	−	0.921376i	$-0.372934\pi$
$487$	17.1545	0.777346	0.388673	+	0.921376i	$0.372934\pi$
$488$	0	0
$489$	8.04108	0.363630
$490$	0	0
$491$	20.8028	0.938815	0.469408	−	0.882982i	$-0.344468\pi$
$491$	20.8028	0.938815	0.469408	+	0.882982i	$0.344468\pi$
$492$	0	0
$493$	−43.1098	−1.94157
$494$	0	0
$495$	−7.72679	−0.347293
$496$	0	0
$497$	0	0
$498$	0	0
$499$	14.8858	0.666380	0.333190	−	0.942860i	$-0.391875\pi$
$499$	14.8858	0.666380	0.333190	+	0.942860i	$0.391875\pi$
$500$	0	0
$501$	2.97159	0.132761
$502$	0	0
$503$	4.55717	0.203194	0.101597	−	0.994826i	$-0.467605\pi$
$503$	4.55717	0.203194	0.101597	+	0.994826i	$0.467605\pi$
$504$	0	0
$505$	8.18550	0.364250
$506$	0	0
$507$	23.3216	1.03575
$508$	0	0
$509$	−5.37084	−0.238058	−0.119029	−	0.992891i	$-0.537978\pi$
$509$	−5.37084	−0.238058	−0.119029	+	0.992891i	$0.537978\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	24.3658	1.07577
$514$	0	0
$515$	−9.87186	−0.435006
$516$	0	0
$517$	−22.2379	−0.978021
$518$	0	0
$519$	0.339193	0.0148889
$520$	0	0
$521$	14.7276	0.645228	0.322614	−	0.946531i	$-0.395438\pi$
$521$	14.7276	0.645228	0.322614	+	0.946531i	$0.395438\pi$
$522$	0	0
$523$	39.0551	1.70776	0.853880	−	0.520470i	$-0.174243\pi$
$523$	39.0551	1.70776	0.853880	+	0.520470i	$0.174243\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−15.4880	−0.674668
$528$	0	0
$529$	22.7228	0.987947
$530$	0	0
$531$	3.20796	0.139214
$532$	0	0
$533$	−6.20323	−0.268692
$534$	0	0
$535$	−23.3378	−1.00898
$536$	0	0
$537$	−22.0260	−0.950490
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−11.7979	−0.507230	−0.253615	−	0.967305i	$-0.581620\pi$
$541$	−11.7979	−0.507230	−0.253615	+	0.967305i	$0.581620\pi$
$542$	0	0
$543$	0.943034	0.0404695
$544$	0	0
$545$	−10.2585	−0.439426
$546$	0	0
$547$	−16.4776	−0.704530	−0.352265	−	0.935900i	$-0.614588\pi$
$547$	−16.4776	−0.704530	−0.352265	+	0.935900i	$0.614588\pi$
$548$	0	0
$549$	−5.12097	−0.218557
$550$	0	0
$551$	32.8531	1.39959
$552$	0	0
$553$	0	0
$554$	0	0
$555$	11.2217	0.476334
$556$	0	0
$557$	−29.7784	−1.26175	−0.630876	−	0.775884i	$-0.717304\pi$
$557$	−29.7784	−1.26175	−0.630876	+	0.775884i	$0.717304\pi$
$558$	0	0
$559$	41.8527	1.77018
$560$	0	0
$561$	16.8947	0.713296
$562$	0	0
$563$	−27.9737	−1.17895	−0.589476	−	0.807786i	$-0.700666\pi$
$563$	−27.9737	−1.17895	−0.589476	+	0.807786i	$0.700666\pi$
$564$	0	0
$565$	6.96856	0.293169
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−39.1698	−1.64208	−0.821041	−	0.570870i	$-0.806606\pi$
$569$	−39.1698	−1.64208	−0.821041	+	0.570870i	$0.806606\pi$
$570$	0	0
$571$	−9.58472	−0.401108	−0.200554	−	0.979683i	$-0.564274\pi$
$571$	−9.58472	−0.401108	−0.200554	+	0.979683i	$0.564274\pi$
$572$	0	0
$573$	7.55747	0.315718
$574$	0	0
$575$	−22.2045	−0.925994
$576$	0	0
$577$	−17.4043	−0.724549	−0.362275	−	0.932071i	$-0.618000\pi$
$577$	−17.4043	−0.724549	−0.362275	+	0.932071i	$0.618000\pi$
$578$	0	0
$579$	8.94257	0.371641
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−34.2844	−1.41991
$584$	0	0
$585$	17.5714	0.726488
$586$	0	0
$587$	−26.8005	−1.10617	−0.553087	−	0.833123i	$-0.686550\pi$
$587$	−26.8005	−1.10617	−0.553087	+	0.833123i	$0.686550\pi$
$588$	0	0
$589$	11.8031	0.486338
$590$	0	0
$591$	2.41557	0.0993632
$592$	0	0
$593$	−4.07921	−0.167513	−0.0837564	−	0.996486i	$-0.526692\pi$
$593$	−4.07921	−0.167513	−0.0837564	+	0.996486i	$0.526692\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−6.58465	−0.269492
$598$	0	0
$599$	0.348239	0.0142287	0.00711434	−	0.999975i	$-0.497735\pi$
$599$	0.348239	0.0142287	0.00711434	+	0.999975i	$0.497735\pi$
$600$	0	0
$601$	−12.9913	−0.529924	−0.264962	−	0.964259i	$-0.585359\pi$
$601$	−12.9913	−0.529924	−0.264962	+	0.964259i	$0.585359\pi$
$602$	0	0
$603$	−5.64735	−0.229978
$604$	0	0
$605$	−4.66269	−0.189565
$606$	0	0
$607$	32.8925	1.33506	0.667532	−	0.744581i	$-0.267351\pi$
$607$	32.8925	1.33506	0.667532	+	0.744581i	$0.267351\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	50.5709	2.04588
$612$	0	0
$613$	−47.5587	−1.92088	−0.960439	−	0.278490i	$-0.910166\pi$
$613$	−47.5587	−1.92088	−0.960439	+	0.278490i	$0.910166\pi$
$614$	0	0
$615$	1.19907	0.0483512
$616$	0	0
$617$	14.3219	0.576578	0.288289	−	0.957544i	$-0.406914\pi$
$617$	14.3219	0.576578	0.288289	+	0.957544i	$0.406914\pi$
$618$	0	0
$619$	−0.567473	−0.0228087	−0.0114043	−	0.999935i	$-0.503630\pi$
$619$	−0.567473	−0.0228087	−0.0114043	+	0.999935i	$0.503630\pi$
$620$	0	0
$621$	−31.9495	−1.28209
$622$	0	0
$623$	0	0
$624$	0	0
$625$	2.20225	0.0880900
$626$	0	0
$627$	−12.8751	−0.514183
$628$	0	0
$629$	63.3280	2.52505
$630$	0	0
$631$	19.5021	0.776366	0.388183	−	0.921582i	$-0.373103\pi$
$631$	19.5021	0.776366	0.388183	+	0.921582i	$0.373103\pi$
$632$	0	0
$633$	−18.0785	−0.718556
$634$	0	0
$635$	12.6575	0.502296
$636$	0	0
$637$	0	0
$638$	0	0
$639$	2.76994	0.109577
$640$	0	0
$641$	−19.0255	−0.751461	−0.375731	−	0.926729i	$-0.622608\pi$
$641$	−19.0255	−0.751461	−0.375731	+	0.926729i	$0.622608\pi$
$642$	0	0
$643$	−44.8947	−1.77047	−0.885237	−	0.465141i	$-0.846004\pi$
$643$	−44.8947	−1.77047	−0.885237	+	0.465141i	$0.846004\pi$
$644$	0	0
$645$	−8.09003	−0.318545
$646$	0	0
$647$	−5.67864	−0.223250	−0.111625	−	0.993750i	$-0.535606\pi$
$647$	−5.67864	−0.223250	−0.111625	+	0.993750i	$0.535606\pi$
$648$	0	0
$649$	−4.04701	−0.158859
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−4.31189	−0.168737	−0.0843686	−	0.996435i	$-0.526887\pi$
$653$	−4.31189	−0.168737	−0.0843686	+	0.996435i	$0.526887\pi$
$654$	0	0
$655$	−22.5935	−0.882802
$656$	0	0
$657$	8.24674	0.321736
$658$	0	0
$659$	42.9372	1.67260	0.836298	−	0.548275i	$-0.184715\pi$
$659$	42.9372	1.67260	0.836298	+	0.548275i	$0.184715\pi$
$660$	0	0
$661$	−14.6588	−0.570160	−0.285080	−	0.958504i	$-0.592020\pi$
$661$	−14.6588	−0.570160	−0.285080	+	0.958504i	$0.592020\pi$
$662$	0	0
$663$	−38.4201	−1.49211
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−43.0785	−1.66801
$668$	0	0
$669$	−18.2065	−0.703902
$670$	0	0
$671$	6.46037	0.249400
$672$	0	0
$673$	−26.0296	−1.00337	−0.501684	−	0.865051i	$-0.667286\pi$
$673$	−26.0296	−1.00337	−0.501684	+	0.865051i	$0.667286\pi$
$674$	0	0
$675$	15.5158	0.597202
$676$	0	0
$677$	36.3578	1.39735	0.698673	−	0.715441i	$-0.253774\pi$
$677$	36.3578	1.39735	0.698673	+	0.715441i	$0.253774\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−14.6889	−0.562878
$682$	0	0
$683$	17.5708	0.672328	0.336164	−	0.941803i	$-0.390870\pi$
$683$	17.5708	0.672328	0.336164	+	0.941803i	$0.390870\pi$
$684$	0	0
$685$	−10.6243	−0.405934
$686$	0	0
$687$	−18.7580	−0.715663
$688$	0	0
$689$	77.9658	2.97026
$690$	0	0
$691$	−25.1288	−0.955944	−0.477972	−	0.878375i	$-0.658628\pi$
$691$	−25.1288	−0.955944	−0.477972	+	0.878375i	$0.658628\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	3.57357	0.135553
$696$	0	0
$697$	6.76678	0.256310
$698$	0	0
$699$	−13.9910	−0.529189
$700$	0	0
$701$	−12.8610	−0.485753	−0.242877	−	0.970057i	$-0.578091\pi$
$701$	−12.8610	−0.485753	−0.242877	+	0.970057i	$0.578091\pi$
$702$	0	0
$703$	−48.2610	−1.82020
$704$	0	0
$705$	−9.77525	−0.368157
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−23.9369	−0.898969	−0.449484	−	0.893288i	$-0.648392\pi$
$709$	−23.9369	−0.898969	−0.449484	+	0.893288i	$0.648392\pi$
$710$	0	0
$711$	−9.26212	−0.347357
$712$	0	0
$713$	−15.4767	−0.579609
$714$	0	0
$715$	−22.1673	−0.829009
$716$	0	0
$717$	24.9998	0.933634
$718$	0	0
$719$	−38.7742	−1.44603	−0.723017	−	0.690830i	$-0.757245\pi$
$719$	−38.7742	−1.44603	−0.723017	+	0.690830i	$0.757245\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	14.4553	0.537600
$724$	0	0
$725$	20.9204	0.776964
$726$	0	0
$727$	−19.6322	−0.728117	−0.364059	−	0.931376i	$-0.618609\pi$
$727$	−19.6322	−0.728117	−0.364059	+	0.931376i	$0.618609\pi$
$728$	0	0
$729$	8.29935	0.307383
$730$	0	0
$731$	−45.6549	−1.68861
$732$	0	0
$733$	−35.7663	−1.32106	−0.660529	−	0.750801i	$-0.729668\pi$
$733$	−35.7663	−1.32106	−0.660529	+	0.750801i	$0.729668\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	7.12444	0.262432
$738$	0	0
$739$	−22.6233	−0.832212	−0.416106	−	0.909316i	$-0.636606\pi$
$739$	−22.6233	−0.832212	−0.416106	+	0.909316i	$0.636606\pi$
$740$	0	0
$741$	29.2792	1.07560
$742$	0	0
$743$	16.9685	0.622515	0.311257	−	0.950326i	$-0.399250\pi$
$743$	16.9685	0.622515	0.311257	+	0.950326i	$0.399250\pi$
$744$	0	0
$745$	−6.32467	−0.231718
$746$	0	0
$747$	−5.10207	−0.186675
$748$	0	0
$749$	0	0
$750$	0	0
$751$	38.1262	1.39124	0.695622	−	0.718408i	$-0.255129\pi$
$751$	38.1262	1.39124	0.695622	+	0.718408i	$0.255129\pi$
$752$	0	0
$753$	−14.4358	−0.526069
$754$	0	0
$755$	3.08641	0.112326
$756$	0	0
$757$	15.1230	0.549656	0.274828	−	0.961493i	$-0.411379\pi$
$757$	15.1230	0.549656	0.274828	+	0.961493i	$0.411379\pi$
$758$	0	0
$759$	16.8825	0.612794
$760$	0	0
$761$	43.5322	1.57804	0.789020	−	0.614367i	$-0.210589\pi$
$761$	43.5322	1.57804	0.789020	+	0.614367i	$0.210589\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−19.1677	−0.693011
$766$	0	0
$767$	9.20327	0.332311
$768$	0	0
$769$	−5.22236	−0.188323	−0.0941616	−	0.995557i	$-0.530017\pi$
$769$	−5.22236	−0.188323	−0.0941616	+	0.995557i	$0.530017\pi$
$770$	0	0
$771$	6.76118	0.243498
$772$	0	0
$773$	5.83438	0.209848	0.104924	−	0.994480i	$-0.466540\pi$
$773$	5.83438	0.209848	0.104924	+	0.994480i	$0.466540\pi$
$774$	0	0
$775$	7.51604	0.269984
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−5.15682	−0.184762
$780$	0	0
$781$	−3.49443	−0.125041
$782$	0	0
$783$	30.1018	1.07575
$784$	0	0
$785$	−16.5967	−0.592363
$786$	0	0
$787$	54.2719	1.93459	0.967293	−	0.253662i	$-0.0816352\pi$
$787$	54.2719	1.93459	0.967293	+	0.253662i	$0.0816352\pi$
$788$	0	0
$789$	−7.13789	−0.254116
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−14.6915	−0.521709
$794$	0	0
$795$	−15.0706	−0.534500
$796$	0	0
$797$	−20.7610	−0.735393	−0.367697	−	0.929946i	$-0.619854\pi$
$797$	−20.7610	−0.735393	−0.367697	+	0.929946i	$0.619854\pi$
$798$	0	0
$799$	−55.1652	−1.95160
$800$	0	0
$801$	8.61206	0.304292
$802$	0	0
$803$	−10.4037	−0.367139
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−10.4155	−0.366642
$808$	0	0
$809$	2.74903	0.0966509	0.0483254	−	0.998832i	$-0.484612\pi$
$809$	2.74903	0.0966509	0.0483254	+	0.998832i	$0.484612\pi$
$810$	0	0
$811$	38.6188	1.35609	0.678045	−	0.735020i	$-0.262827\pi$
$811$	38.6188	1.35609	0.678045	+	0.735020i	$0.262827\pi$
$812$	0	0
$813$	18.5444	0.650380
$814$	0	0
$815$	11.5091	0.403145
$816$	0	0
$817$	34.7927	1.21724
$818$	0	0
$819$	0	0
$820$	0	0
$821$	32.0655	1.11910	0.559548	−	0.828798i	$-0.310975\pi$
$821$	32.0655	1.11910	0.559548	+	0.828798i	$0.310975\pi$
$822$	0	0
$823$	24.9174	0.868566	0.434283	−	0.900776i	$-0.357002\pi$
$823$	24.9174	0.868566	0.434283	+	0.900776i	$0.357002\pi$
$824$	0	0
$825$	−8.19870	−0.285442
$826$	0	0
$827$	16.0445	0.557924	0.278962	−	0.960302i	$-0.410010\pi$
$827$	16.0445	0.557924	0.278962	+	0.960302i	$0.410010\pi$
$828$	0	0
$829$	23.2341	0.806955	0.403477	−	0.914990i	$-0.367801\pi$
$829$	23.2341	0.806955	0.403477	+	0.914990i	$0.367801\pi$
$830$	0	0
$831$	20.5481	0.712807
$832$	0	0
$833$	0	0
$834$	0	0
$835$	4.25319	0.147188
$836$	0	0
$837$	10.8146	0.373808
$838$	0	0
$839$	9.97240	0.344286	0.172143	−	0.985072i	$-0.444931\pi$
$839$	9.97240	0.344286	0.172143	+	0.985072i	$0.444931\pi$
$840$	0	0
$841$	11.5871	0.399556
$842$	0	0
$843$	26.2949	0.905645
$844$	0	0
$845$	33.3799	1.14830
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−23.2338	−0.797382
$850$	0	0
$851$	63.2820	2.16928
$852$	0	0
$853$	−20.8869	−0.715155	−0.357577	−	0.933883i	$-0.616397\pi$
$853$	−20.8869	−0.715155	−0.357577	+	0.933883i	$0.616397\pi$
$854$	0	0
$855$	14.6073	0.499561
$856$	0	0
$857$	−25.0377	−0.855272	−0.427636	−	0.903951i	$-0.640653\pi$
$857$	−25.0377	−0.855272	−0.427636	+	0.903951i	$0.640653\pi$
$858$	0	0
$859$	−33.8628	−1.15538	−0.577692	−	0.816255i	$-0.696047\pi$
$859$	−33.8628	−1.15538	−0.577692	+	0.816255i	$0.696047\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	47.2687	1.60905	0.804523	−	0.593922i	$-0.202421\pi$
$863$	47.2687	1.60905	0.804523	+	0.593922i	$0.202421\pi$
$864$	0	0
$865$	0.485482	0.0165069
$866$	0	0
$867$	26.3506	0.894912
$868$	0	0
$869$	11.6847	0.396375
$870$	0	0
$871$	−16.2016	−0.548971
$872$	0	0
$873$	−15.2111	−0.514818
$874$	0	0
$875$	0	0
$876$	0	0
$877$	24.3287	0.821522	0.410761	−	0.911743i	$-0.365263\pi$
$877$	24.3287	0.821522	0.410761	+	0.911743i	$0.365263\pi$
$878$	0	0
$879$	−24.1050	−0.813042
$880$	0	0
$881$	−46.4242	−1.56407	−0.782035	−	0.623235i	$-0.785818\pi$
$881$	−46.4242	−1.56407	−0.782035	+	0.623235i	$0.785818\pi$
$882$	0	0
$883$	−4.22330	−0.142125	−0.0710626	−	0.997472i	$-0.522639\pi$
$883$	−4.22330	−0.142125	−0.0710626	+	0.997472i	$0.522639\pi$
$884$	0	0
$885$	−1.77897	−0.0597995
$886$	0	0
$887$	5.72536	0.192239	0.0961194	−	0.995370i	$-0.469357\pi$
$887$	5.72536	0.192239	0.0961194	+	0.995370i	$0.469357\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	5.89750	0.197574
$892$	0	0
$893$	42.0403	1.40682
$894$	0	0
$895$	−31.5254	−1.05378
$896$	0	0
$897$	−38.3922	−1.28188
$898$	0	0
$899$	14.5817	0.486326
$900$	0	0
$901$	−85.0489	−2.83339
$902$	0	0
$903$	0	0
$904$	0	0
$905$	1.34975	0.0448672
$906$	0	0
$907$	−18.2656	−0.606501	−0.303250	−	0.952911i	$-0.598072\pi$
$907$	−18.2656	−0.606501	−0.303250	+	0.952911i	$0.598072\pi$
$908$	0	0
$909$	−13.5103	−0.448108
$910$	0	0
$911$	−22.8893	−0.758356	−0.379178	−	0.925324i	$-0.623793\pi$
$911$	−22.8893	−0.758356	−0.379178	+	0.925324i	$0.623793\pi$
$912$	0	0
$913$	6.43654	0.213018
$914$	0	0
$915$	2.83983	0.0938819
$916$	0	0
$917$	0	0
$918$	0	0
$919$	4.67612	0.154251	0.0771254	−	0.997021i	$-0.475426\pi$
$919$	4.67612	0.154251	0.0771254	+	0.997021i	$0.475426\pi$
$920$	0	0
$921$	27.3250	0.900390
$922$	0	0
$923$	7.94665	0.261567
$924$	0	0
$925$	−30.7319	−1.01046
$926$	0	0
$927$	16.2936	0.535153
$928$	0	0
$929$	−33.1752	−1.08844	−0.544222	−	0.838941i	$-0.683175\pi$
$929$	−33.1752	−1.08844	−0.544222	+	0.838941i	$0.683175\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−1.00019	−0.0327446
$934$	0	0
$935$	24.1811	0.790808
$936$	0	0
$937$	54.1119	1.76776	0.883879	−	0.467715i	$-0.154923\pi$
$937$	54.1119	1.76776	0.883879	+	0.467715i	$0.154923\pi$
$938$	0	0
$939$	−23.6330	−0.771235
$940$	0	0
$941$	19.1721	0.624993	0.312497	−	0.949919i	$-0.398835\pi$
$941$	19.1721	0.624993	0.312497	+	0.949919i	$0.398835\pi$
$942$	0	0
$943$	6.76186	0.220197
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−1.20600	−0.0391898	−0.0195949	−	0.999808i	$-0.506238\pi$
$947$	−1.20600	−0.0391898	−0.0195949	+	0.999808i	$0.506238\pi$
$948$	0	0
$949$	23.6590	0.768003
$950$	0	0
$951$	5.57443	0.180763
$952$	0	0
$953$	35.0276	1.13466	0.567328	−	0.823492i	$-0.307977\pi$
$953$	35.0276	1.13466	0.567328	+	0.823492i	$0.307977\pi$
$954$	0	0
$955$	10.8169	0.350026
$956$	0	0
$957$	−15.9061	−0.514171
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−25.7613	−0.831008
$962$	0	0
$963$	38.5193	1.24127
$964$	0	0
$965$	12.7993	0.412026
$966$	0	0
$967$	−36.8246	−1.18420	−0.592099	−	0.805865i	$-0.701701\pi$
$967$	−36.8246	−1.18420	−0.592099	+	0.805865i	$0.701701\pi$
$968$	0	0
$969$	−31.9392	−1.02603
$970$	0	0
$971$	3.28648	0.105468	0.0527341	−	0.998609i	$-0.483206\pi$
$971$	3.28648	0.105468	0.0527341	+	0.998609i	$0.483206\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	18.6446	0.597105
$976$	0	0
$977$	−9.64478	−0.308564	−0.154282	−	0.988027i	$-0.549306\pi$
$977$	−9.64478	−0.308564	−0.154282	+	0.988027i	$0.549306\pi$
$978$	0	0
$979$	−10.8646	−0.347234
$980$	0	0
$981$	16.9318	0.540591
$982$	0	0
$983$	29.4596	0.939615	0.469808	−	0.882769i	$-0.344323\pi$
$983$	29.4596	0.939615	0.469808	+	0.882769i	$0.344323\pi$
$984$	0	0
$985$	3.45736	0.110161
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−45.6218	−1.45069
$990$	0	0
$991$	−18.9223	−0.601086	−0.300543	−	0.953768i	$-0.597168\pi$
$991$	−18.9223	−0.601086	−0.300543	+	0.953768i	$0.597168\pi$
$992$	0	0
$993$	12.5941	0.399661
$994$	0	0
$995$	−9.42449	−0.298777
$996$	0	0
$997$	50.4615	1.59813	0.799066	−	0.601243i	$-0.205328\pi$
$997$	50.4615	1.59813	0.799066	+	0.601243i	$0.205328\pi$
$998$	0	0
$999$	−44.2193	−1.39904

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8036.2.a.s.1.14	✓	20	1.1	even	1	trivial
8036.2.a.t.1.7	yes	20	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)

\(f(q)\)	\(=\)	\(q+0.915292 q^{3} +1.31004 q^{5} -2.16224 q^{9} +O(q^{10})\)	\(q+0.915292 q^{3} +1.31004 q^{5} -2.16224 q^{9} +2.72778 q^{11} -6.20323 q^{13} +1.19907 q^{15} +6.76678 q^{17} -5.15682 q^{19} +6.76186 q^{23} -3.28379 q^{25} -4.72496 q^{27} -6.37080 q^{29} -2.28883 q^{31} +2.49672 q^{33} +9.35867 q^{37} -5.67776 q^{39} +1.00000 q^{41} -6.74692 q^{43} -2.83262 q^{45} -8.15236 q^{47} +6.19357 q^{51} -12.5686 q^{53} +3.57351 q^{55} -4.72000 q^{57} -1.48363 q^{59} +2.36836 q^{61} -8.12648 q^{65} +2.61180 q^{67} +6.18908 q^{69} -1.28105 q^{71} -3.81398 q^{73} -3.00563 q^{75} +4.28357 q^{79} +2.16201 q^{81} +2.35962 q^{83} +8.86476 q^{85} -5.83114 q^{87} -3.98293 q^{89} -2.09495 q^{93} -6.75565 q^{95} +7.03488 q^{97} -5.89813 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 4 q^{3} - 8 q^{5} + 16 q^{9}+O(q^{10}) \) 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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