LMFDB - Embedded newform 4012.2.a.i.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4012,2,Mod(1,4012)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4012, 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("4012.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4012 = 2^{2} \cdot 17 \cdot 59 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4012.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:	$32.0359812909$
Analytic rank:	$0$
Dimension:	$18$
Coefficient field:	$\mathbb{Q}[x]/(x^{18} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{18} - 8 x^{17} - 4 x^{16} + 178 x^{15} - 265 x^{14} - 1405 x^{13} + 3503 x^{12} + 4295 x^{11} + \cdots - 16 $ x^18 - 8x^17 - 4x^16 + 178x^15 - 265x^14 - 1405x^13 + 3503x^12 + 4295x^11 - 17374x^10 - 893x^9 + 38112x^8 - 18700x^7 - 32137x^6 + 26381x^5 + 3964x^4 - 5788x^3 - 108x^2 + 232*x - 16
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-2.15409$ of defining polynomial
Character	$\chi$	$=$	4012.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.15409 q^{3} -0.228433 q^{5} +2.56317 q^{7} +1.64008 q^{9} +O(q^{10})$	$q-2.15409 q^{3} -0.228433 q^{5} +2.56317 q^{7} +1.64008 q^{9} -0.376697 q^{11} +2.42602 q^{13} +0.492063 q^{15} -1.00000 q^{17} -3.73570 q^{19} -5.52129 q^{21} -1.19475 q^{23} -4.94782 q^{25} +2.92938 q^{27} +7.25625 q^{29} +8.58188 q^{31} +0.811437 q^{33} -0.585512 q^{35} -5.61593 q^{37} -5.22586 q^{39} +6.42624 q^{41} +8.49857 q^{43} -0.374648 q^{45} -0.519671 q^{47} -0.430154 q^{49} +2.15409 q^{51} +3.21643 q^{53} +0.0860499 q^{55} +8.04702 q^{57} -1.00000 q^{59} -13.3844 q^{61} +4.20381 q^{63} -0.554182 q^{65} -7.65363 q^{67} +2.57358 q^{69} -3.61624 q^{71} +4.36323 q^{73} +10.6580 q^{75} -0.965539 q^{77} +3.27862 q^{79} -11.2304 q^{81} -17.8899 q^{83} +0.228433 q^{85} -15.6306 q^{87} +7.74376 q^{89} +6.21831 q^{91} -18.4861 q^{93} +0.853356 q^{95} -1.60594 q^{97} -0.617814 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 18 q + 8 q^{3} + 4 q^{5} + 2 q^{7} + 18 q^{9}+O(q^{10}) $ 18 * q + 8 * q^3 + 4 * q^5 + 2 * q^7 + 18 * q^9	$ 18 q + 8 q^{3} + 4 q^{5} + 2 q^{7} + 18 q^{9} + 12 q^{11} + 2 q^{13} - 18 q^{17} + 5 q^{19} - 3 q^{21} + 21 q^{23} + 16 q^{25} + 26 q^{27} + 14 q^{29} + 15 q^{31} + 19 q^{33} + 20 q^{35} + 2 q^{37} - 14 q^{39} + 34 q^{41} + 21 q^{43} + 49 q^{45} + 69 q^{47} + 28 q^{49} - 8 q^{51} - 4 q^{53} + 18 q^{55} + 5 q^{57} - 18 q^{59} + 11 q^{61} + 35 q^{63} + 27 q^{65} + 34 q^{67} - 4 q^{69} + 37 q^{71} + 18 q^{73} + 72 q^{75} + 11 q^{77} + 11 q^{79} + 30 q^{81} + 28 q^{83} - 4 q^{85} + 7 q^{87} + 44 q^{89} - 23 q^{91} - 3 q^{93} - 11 q^{95} + 11 q^{97} + 56 q^{99}+O(q^{100}) $ 18 * q + 8 * q^3 + 4 * q^5 + 2 * q^7 + 18 * q^9 + 12 * q^11 + 2 * q^13 - 18 * q^17 + 5 * q^19 - 3 * q^21 + 21 * q^23 + 16 * q^25 + 26 * q^27 + 14 * q^29 + 15 * q^31 + 19 * q^33 + 20 * q^35 + 2 * q^37 - 14 * q^39 + 34 * q^41 + 21 * q^43 + 49 * q^45 + 69 * q^47 + 28 * q^49 - 8 * q^51 - 4 * q^53 + 18 * q^55 + 5 * q^57 - 18 * q^59 + 11 * q^61 + 35 * q^63 + 27 * q^65 + 34 * q^67 - 4 * q^69 + 37 * q^71 + 18 * q^73 + 72 * q^75 + 11 * q^77 + 11 * q^79 + 30 * q^81 + 28 * q^83 - 4 * q^85 + 7 * q^87 + 44 * q^89 - 23 * q^91 - 3 * q^93 - 11 * q^95 + 11 * q^97 + 56 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	−2.15409	−1.24366	−0.621831	−	0.783152i	$-0.713611\pi$
$3$	−2.15409	−1.24366	−0.621831	+	0.783152i	$0.713611\pi$
$4$	0	0
$5$	−0.228433	−0.102158	−0.0510791	−	0.998695i	$-0.516266\pi$
$5$	−0.228433	−0.102158	−0.0510791	+	0.998695i	$0.516266\pi$
$6$	0	0
$7$	2.56317	0.968788	0.484394	−	0.874850i	$-0.339040\pi$
$7$	2.56317	0.968788	0.484394	+	0.874850i	$0.339040\pi$
$8$	0	0
$9$	1.64008	0.546694
$10$	0	0
$11$	−0.376697	−0.113578	−0.0567892	−	0.998386i	$-0.518086\pi$
$11$	−0.376697	−0.113578	−0.0567892	+	0.998386i	$0.518086\pi$
$12$	0	0
$13$	2.42602	0.672857	0.336429	−	0.941709i	$-0.390781\pi$
$13$	2.42602	0.672857	0.336429	+	0.941709i	$0.390781\pi$
$14$	0	0
$15$	0.492063	0.127050
$16$	0	0
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	0	0
$19$	−3.73570	−0.857028	−0.428514	−	0.903535i	$-0.640963\pi$
$19$	−3.73570	−0.857028	−0.428514	+	0.903535i	$0.640963\pi$
$20$	0	0
$21$	−5.52129	−1.20484
$22$	0	0
$23$	−1.19475	−0.249122	−0.124561	−	0.992212i	$-0.539752\pi$
$23$	−1.19475	−0.249122	−0.124561	+	0.992212i	$0.539752\pi$
$24$	0	0
$25$	−4.94782	−0.989564
$26$	0	0
$27$	2.92938	0.563759
$28$	0	0
$29$	7.25625	1.34745	0.673726	−	0.738982i	$-0.264693\pi$
$29$	7.25625	1.34745	0.673726	+	0.738982i	$0.264693\pi$
$30$	0	0
$31$	8.58188	1.54135	0.770676	−	0.637228i	$-0.219919\pi$
$31$	8.58188	1.54135	0.770676	+	0.637228i	$0.219919\pi$
$32$	0	0
$33$	0.811437	0.141253
$34$	0	0
$35$	−0.585512	−0.0989696
$36$	0	0
$37$	−5.61593	−0.923253	−0.461626	−	0.887074i	$-0.652734\pi$
$37$	−5.61593	−0.923253	−0.461626	+	0.887074i	$0.652734\pi$
$38$	0	0
$39$	−5.22586	−0.836807
$40$	0	0
$41$	6.42624	1.00361	0.501805	−	0.864981i	$-0.332669\pi$
$41$	6.42624	1.00361	0.501805	+	0.864981i	$0.332669\pi$
$42$	0	0
$43$	8.49857	1.29602	0.648010	−	0.761632i	$-0.275602\pi$
$43$	8.49857	1.29602	0.648010	+	0.761632i	$0.275602\pi$
$44$	0	0
$45$	−0.374648	−0.0558493
$46$	0	0
$47$	−0.519671	−0.0758017	−0.0379009	−	0.999282i	$-0.512067\pi$
$47$	−0.519671	−0.0758017	−0.0379009	+	0.999282i	$0.512067\pi$
$48$	0	0
$49$	−0.430154	−0.0614505
$50$	0	0
$51$	2.15409	0.301632
$52$	0	0
$53$	3.21643	0.441811	0.220906	−	0.975295i	$-0.429099\pi$
$53$	3.21643	0.441811	0.220906	+	0.975295i	$0.429099\pi$
$54$	0	0
$55$	0.0860499	0.0116030
$56$	0	0
$57$	8.04702	1.06585
$58$	0	0
$59$	−1.00000	−0.130189
$59$	−1.00000	−0.130189
$60$	0	0
$61$	−13.3844	−1.71370	−0.856850	−	0.515566i	$-0.827582\pi$
$61$	−13.3844	−1.71370	−0.856850	+	0.515566i	$0.827582\pi$
$62$	0	0
$63$	4.20381	0.529631
$64$	0	0
$65$	−0.554182	−0.0687378
$66$	0	0
$67$	−7.65363	−0.935040	−0.467520	−	0.883983i	$-0.654852\pi$
$67$	−7.65363	−0.935040	−0.467520	+	0.883983i	$0.654852\pi$
$68$	0	0
$69$	2.57358	0.309823
$70$	0	0
$71$	−3.61624	−0.429169	−0.214585	−	0.976705i	$-0.568840\pi$
$71$	−3.61624	−0.429169	−0.214585	+	0.976705i	$0.568840\pi$
$72$	0	0
$73$	4.36323	0.510678	0.255339	−	0.966852i	$-0.417813\pi$
$73$	4.36323	0.510678	0.255339	+	0.966852i	$0.417813\pi$
$74$	0	0
$75$	10.6580	1.23068
$76$	0	0
$77$	−0.965539	−0.110033
$78$	0	0
$79$	3.27862	0.368873	0.184437	−	0.982844i	$-0.440954\pi$
$79$	3.27862	0.368873	0.184437	+	0.982844i	$0.440954\pi$
$80$	0	0
$81$	−11.2304	−1.24782
$82$	0	0
$83$	−17.8899	−1.96368	−0.981838	−	0.189722i	$-0.939241\pi$
$83$	−17.8899	−1.96368	−0.981838	+	0.189722i	$0.939241\pi$
$84$	0	0
$85$	0.228433	0.0247770
$86$	0	0
$87$	−15.6306	−1.67577
$88$	0	0
$89$	7.74376	0.820837	0.410418	−	0.911897i	$-0.365383\pi$
$89$	7.74376	0.820837	0.410418	+	0.911897i	$0.365383\pi$
$90$	0	0
$91$	6.21831	0.651856
$92$	0	0
$93$	−18.4861	−1.91692
$94$	0	0
$95$	0.853356	0.0875524
$96$	0	0
$97$	−1.60594	−0.163058	−0.0815291	−	0.996671i	$-0.525980\pi$
$97$	−1.60594	−0.163058	−0.0815291	+	0.996671i	$0.525980\pi$
$98$	0	0
$99$	−0.617814	−0.0620927
$100$	0	0
$101$	−4.51135	−0.448896	−0.224448	−	0.974486i	$-0.572058\pi$
$101$	−4.51135	−0.448896	−0.224448	+	0.974486i	$0.572058\pi$
$102$	0	0
$103$	14.0957	1.38889	0.694447	−	0.719544i	$-0.255649\pi$
$103$	14.0957	1.38889	0.694447	+	0.719544i	$0.255649\pi$
$104$	0	0
$105$	1.26124	0.123085
$106$	0	0
$107$	10.6384	1.02846	0.514228	−	0.857653i	$-0.328078\pi$
$107$	10.6384	1.02846	0.514228	+	0.857653i	$0.328078\pi$
$108$	0	0
$109$	−7.59238	−0.727218	−0.363609	−	0.931552i	$-0.618456\pi$
$109$	−7.59238	−0.727218	−0.363609	+	0.931552i	$0.618456\pi$
$110$	0	0
$111$	12.0972	1.14821
$112$	0	0
$113$	−1.66828	−0.156939	−0.0784695	−	0.996917i	$-0.525003\pi$
$113$	−1.66828	−0.156939	−0.0784695	+	0.996917i	$0.525003\pi$
$114$	0	0
$115$	0.272919	0.0254498
$116$	0	0
$117$	3.97887	0.367847
$118$	0	0
$119$	−2.56317	−0.234966
$120$	0	0
$121$	−10.8581	−0.987100
$122$	0	0
$123$	−13.8427	−1.24815
$124$	0	0
$125$	2.27241	0.203250
$126$	0	0
$127$	11.7284	1.04073	0.520366	−	0.853944i	$-0.325796\pi$
$127$	11.7284	1.04073	0.520366	+	0.853944i	$0.325796\pi$
$128$	0	0
$129$	−18.3066	−1.61181
$130$	0	0
$131$	−4.05334	−0.354142	−0.177071	−	0.984198i	$-0.556662\pi$
$131$	−4.05334	−0.354142	−0.177071	+	0.984198i	$0.556662\pi$
$132$	0	0
$133$	−9.57524	−0.830278
$134$	0	0
$135$	−0.669165	−0.0575926
$136$	0	0
$137$	11.5673	0.988265	0.494133	−	0.869387i	$-0.335486\pi$
$137$	11.5673	0.988265	0.494133	+	0.869387i	$0.335486\pi$
$138$	0	0
$139$	11.2737	0.956224	0.478112	−	0.878299i	$-0.341321\pi$
$139$	11.2737	0.956224	0.478112	+	0.878299i	$0.341321\pi$
$140$	0	0
$141$	1.11941	0.0942717
$142$	0	0
$143$	−0.913875	−0.0764220
$144$	0	0
$145$	−1.65756	−0.137653
$146$	0	0
$147$	0.926588	0.0764237
$148$	0	0
$149$	22.0634	1.80751	0.903754	−	0.428052i	$-0.140800\pi$
$149$	22.0634	1.80751	0.903754	+	0.428052i	$0.140800\pi$
$150$	0	0
$151$	21.6789	1.76421	0.882103	−	0.471057i	$-0.156127\pi$
$151$	21.6789	1.76421	0.882103	+	0.471057i	$0.156127\pi$
$152$	0	0
$153$	−1.64008	−0.132593
$154$	0	0
$155$	−1.96038	−0.157462
$156$	0	0
$157$	−7.74123	−0.617818	−0.308909	−	0.951092i	$-0.599964\pi$
$157$	−7.74123	−0.617818	−0.308909	+	0.951092i	$0.599964\pi$
$158$	0	0
$159$	−6.92847	−0.549464
$160$	0	0
$161$	−3.06234	−0.241346
$162$	0	0
$163$	−3.84005	−0.300776	−0.150388	−	0.988627i	$-0.548052\pi$
$163$	−3.84005	−0.300776	−0.150388	+	0.988627i	$0.548052\pi$
$164$	0	0
$165$	−0.185359	−0.0144302
$166$	0	0
$167$	19.5715	1.51449	0.757246	−	0.653130i	$-0.226545\pi$
$167$	19.5715	1.51449	0.757246	+	0.653130i	$0.226545\pi$
$168$	0	0
$169$	−7.11442	−0.547263
$170$	0	0
$171$	−6.12686	−0.468532
$172$	0	0
$173$	20.4440	1.55433	0.777163	−	0.629300i	$-0.216658\pi$
$173$	20.4440	1.55433	0.777163	+	0.629300i	$0.216658\pi$
$174$	0	0
$175$	−12.6821	−0.958677
$176$	0	0
$177$	2.15409	0.161911
$178$	0	0
$179$	25.3559	1.89519	0.947594	−	0.319476i	$-0.103507\pi$
$179$	25.3559	1.89519	0.947594	+	0.319476i	$0.103507\pi$
$180$	0	0
$181$	19.0934	1.41920	0.709599	−	0.704606i	$-0.248876\pi$
$181$	19.0934	1.41920	0.709599	+	0.704606i	$0.248876\pi$
$182$	0	0
$183$	28.8312	2.13126
$184$	0	0
$185$	1.28286	0.0943178
$186$	0	0
$187$	0.376697	0.0275468
$188$	0	0
$189$	7.50850	0.546163
$190$	0	0
$191$	24.3146	1.75935	0.879673	−	0.475580i	$-0.157762\pi$
$191$	24.3146	1.75935	0.879673	+	0.475580i	$0.157762\pi$
$192$	0	0
$193$	−15.3869	−1.10757	−0.553787	−	0.832658i	$-0.686818\pi$
$193$	−15.3869	−1.10757	−0.553787	+	0.832658i	$0.686818\pi$
$194$	0	0
$195$	1.19376	0.0854866
$196$	0	0
$197$	5.42822	0.386745	0.193372	−	0.981125i	$-0.438057\pi$
$197$	5.42822	0.386745	0.193372	+	0.981125i	$0.438057\pi$
$198$	0	0
$199$	−20.7649	−1.47198	−0.735992	−	0.676991i	$-0.763284\pi$
$199$	−20.7649	−1.47198	−0.735992	+	0.676991i	$0.763284\pi$
$200$	0	0
$201$	16.4866	1.16287
$202$	0	0
$203$	18.5990	1.30539
$204$	0	0
$205$	−1.46796	−0.102527
$206$	0	0
$207$	−1.95948	−0.136193
$208$	0	0
$209$	1.40723	0.0973399
$210$	0	0
$211$	18.2544	1.25669	0.628344	−	0.777936i	$-0.283733\pi$
$211$	18.2544	1.25669	0.628344	+	0.777936i	$0.283733\pi$
$212$	0	0
$213$	7.78970	0.533741
$214$	0	0
$215$	−1.94135	−0.132399
$216$	0	0
$217$	21.9968	1.49324
$218$	0	0
$219$	−9.39878	−0.635111
$220$	0	0
$221$	−2.42602	−0.163192
$222$	0	0
$223$	6.37952	0.427204	0.213602	−	0.976921i	$-0.431480\pi$
$223$	6.37952	0.427204	0.213602	+	0.976921i	$0.431480\pi$
$224$	0	0
$225$	−8.11483	−0.540989
$226$	0	0
$227$	−2.90289	−0.192671	−0.0963357	−	0.995349i	$-0.530712\pi$
$227$	−2.90289	−0.192671	−0.0963357	+	0.995349i	$0.530712\pi$
$228$	0	0
$229$	6.20513	0.410047	0.205023	−	0.978757i	$-0.434273\pi$
$229$	6.20513	0.410047	0.205023	+	0.978757i	$0.434273\pi$
$230$	0	0
$231$	2.07985	0.136844
$232$	0	0
$233$	2.36620	0.155015	0.0775074	−	0.996992i	$-0.475304\pi$
$233$	2.36620	0.155015	0.0775074	+	0.996992i	$0.475304\pi$
$234$	0	0
$235$	0.118710	0.00774377
$236$	0	0
$237$	−7.06242	−0.458753
$238$	0	0
$239$	14.5856	0.943461	0.471731	−	0.881743i	$-0.343629\pi$
$239$	14.5856	0.943461	0.471731	+	0.881743i	$0.343629\pi$
$240$	0	0
$241$	−22.5181	−1.45052	−0.725260	−	0.688475i	$-0.758280\pi$
$241$	−22.5181	−1.45052	−0.725260	+	0.688475i	$0.758280\pi$
$242$	0	0
$243$	15.4031	0.988106
$244$	0	0
$245$	0.0982611	0.00627767
$246$	0	0
$247$	−9.06289	−0.576658
$248$	0	0
$249$	38.5364	2.44215
$250$	0	0
$251$	−21.1834	−1.33708	−0.668542	−	0.743674i	$-0.733081\pi$
$251$	−21.1834	−1.33708	−0.668542	+	0.743674i	$0.733081\pi$
$252$	0	0
$253$	0.450057	0.0282948
$254$	0	0
$255$	−0.492063	−0.0308142
$256$	0	0
$257$	−0.935145	−0.0583327	−0.0291664	−	0.999575i	$-0.509285\pi$
$257$	−0.935145	−0.0583327	−0.0291664	+	0.999575i	$0.509285\pi$
$258$	0	0
$259$	−14.3946	−0.894436
$260$	0	0
$261$	11.9008	0.736644
$262$	0	0
$263$	23.9263	1.47536	0.737679	−	0.675152i	$-0.235922\pi$
$263$	23.9263	1.47536	0.737679	+	0.675152i	$0.235922\pi$
$264$	0	0
$265$	−0.734738	−0.0451346
$266$	0	0
$267$	−16.6807	−1.02084
$268$	0	0
$269$	−6.06263	−0.369645	−0.184823	−	0.982772i	$-0.559171\pi$
$269$	−6.06263	−0.369645	−0.184823	+	0.982772i	$0.559171\pi$
$270$	0	0
$271$	28.6012	1.73740	0.868700	−	0.495339i	$-0.164956\pi$
$271$	28.6012	1.73740	0.868700	+	0.495339i	$0.164956\pi$
$272$	0	0
$273$	−13.3948	−0.810688
$274$	0	0
$275$	1.86383	0.112393
$276$	0	0
$277$	6.67364	0.400980	0.200490	−	0.979696i	$-0.435747\pi$
$277$	6.67364	0.400980	0.200490	+	0.979696i	$0.435747\pi$
$278$	0	0
$279$	14.0750	0.842648
$280$	0	0
$281$	3.44815	0.205699	0.102850	−	0.994697i	$-0.467204\pi$
$281$	3.44815	0.205699	0.102850	+	0.994697i	$0.467204\pi$
$282$	0	0
$283$	−14.8164	−0.880746	−0.440373	−	0.897815i	$-0.645154\pi$
$283$	−14.8164	−0.880746	−0.440373	+	0.897815i	$0.645154\pi$
$284$	0	0
$285$	−1.83820	−0.108886
$286$	0	0
$287$	16.4716	0.972285
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	3.45932	0.202789
$292$	0	0
$293$	−5.04351	−0.294645	−0.147323	−	0.989088i	$-0.547066\pi$
$293$	−5.04351	−0.294645	−0.147323	+	0.989088i	$0.547066\pi$
$294$	0	0
$295$	0.228433	0.0132999
$296$	0	0
$297$	−1.10349	−0.0640308
$298$	0	0
$299$	−2.89848	−0.167623
$300$	0	0
$301$	21.7833	1.25557
$302$	0	0
$303$	9.71782	0.558274
$304$	0	0
$305$	3.05744	0.175068
$306$	0	0
$307$	5.25326	0.299820	0.149910	−	0.988700i	$-0.452102\pi$
$307$	5.25326	0.299820	0.149910	+	0.988700i	$0.452102\pi$
$308$	0	0
$309$	−30.3634	−1.72731
$310$	0	0
$311$	3.46248	0.196339	0.0981696	−	0.995170i	$-0.468701\pi$
$311$	3.46248	0.196339	0.0981696	+	0.995170i	$0.468701\pi$
$312$	0	0
$313$	1.96228	0.110915	0.0554573	−	0.998461i	$-0.482338\pi$
$313$	1.96228	0.110915	0.0554573	+	0.998461i	$0.482338\pi$
$314$	0	0
$315$	−0.960288	−0.0541061
$316$	0	0
$317$	−19.5325	−1.09705	−0.548527	−	0.836133i	$-0.684811\pi$
$317$	−19.5325	−1.09705	−0.548527	+	0.836133i	$0.684811\pi$
$318$	0	0
$319$	−2.73341	−0.153041
$320$	0	0
$321$	−22.9161	−1.27905
$322$	0	0
$323$	3.73570	0.207860
$324$	0	0
$325$	−12.0035	−0.665835
$326$	0	0
$327$	16.3546	0.904413
$328$	0	0
$329$	−1.33200	−0.0734358
$330$	0	0
$331$	−12.2855	−0.675274	−0.337637	−	0.941276i	$-0.609628\pi$
$331$	−12.2855	−0.675274	−0.337637	+	0.941276i	$0.609628\pi$
$332$	0	0
$333$	−9.21058	−0.504737
$334$	0	0
$335$	1.74834	0.0955219
$336$	0	0
$337$	−18.1205	−0.987086	−0.493543	−	0.869721i	$-0.664298\pi$
$337$	−18.1205	−0.987086	−0.493543	+	0.869721i	$0.664298\pi$
$338$	0	0
$339$	3.59363	0.195179
$340$	0	0
$341$	−3.23277	−0.175064
$342$	0	0
$343$	−19.0448	−1.02832
$344$	0	0
$345$	−0.587890	−0.0316509
$346$	0	0
$347$	−29.3658	−1.57644	−0.788219	−	0.615394i	$-0.788997\pi$
$347$	−29.3658	−1.57644	−0.788219	+	0.615394i	$0.788997\pi$
$348$	0	0
$349$	0.805425	0.0431134	0.0215567	−	0.999768i	$-0.493138\pi$
$349$	0.805425	0.0431134	0.0215567	+	0.999768i	$0.493138\pi$
$350$	0	0
$351$	7.10673	0.379329
$352$	0	0
$353$	29.3891	1.56423	0.782113	−	0.623137i	$-0.214142\pi$
$353$	29.3891	1.56423	0.782113	+	0.623137i	$0.214142\pi$
$354$	0	0
$355$	0.826068	0.0438431
$356$	0	0
$357$	5.52129	0.292218
$358$	0	0
$359$	0.452667	0.0238908	0.0119454	−	0.999929i	$-0.496198\pi$
$359$	0.452667	0.0238908	0.0119454	+	0.999929i	$0.496198\pi$
$360$	0	0
$361$	−5.04455	−0.265502
$362$	0	0
$363$	23.3893	1.22762
$364$	0	0
$365$	−0.996705	−0.0521699
$366$	0	0
$367$	14.6840	0.766498	0.383249	−	0.923645i	$-0.374805\pi$
$367$	14.6840	0.766498	0.383249	+	0.923645i	$0.374805\pi$
$368$	0	0
$369$	10.5396	0.548668
$370$	0	0
$371$	8.24427	0.428021
$372$	0	0
$373$	31.6781	1.64023	0.820116	−	0.572198i	$-0.193909\pi$
$373$	31.6781	1.64023	0.820116	+	0.572198i	$0.193909\pi$
$374$	0	0
$375$	−4.89496	−0.252774
$376$	0	0
$377$	17.6038	0.906642
$378$	0	0
$379$	21.7499	1.11722	0.558609	−	0.829431i	$-0.311335\pi$
$379$	21.7499	1.11722	0.558609	+	0.829431i	$0.311335\pi$
$380$	0	0
$381$	−25.2641	−1.29432
$382$	0	0
$383$	21.8670	1.11735	0.558677	−	0.829386i	$-0.311309\pi$
$383$	21.8670	1.11735	0.558677	+	0.829386i	$0.311309\pi$
$384$	0	0
$385$	0.220560	0.0112408
$386$	0	0
$387$	13.9384	0.708526
$388$	0	0
$389$	−19.0231	−0.964511	−0.482256	−	0.876031i	$-0.660182\pi$
$389$	−19.0231	−0.964511	−0.482256	+	0.876031i	$0.660182\pi$
$390$	0	0
$391$	1.19475	0.0604209
$392$	0	0
$393$	8.73125	0.440433
$394$	0	0
$395$	−0.748942	−0.0376834
$396$	0	0
$397$	38.3467	1.92457	0.962283	−	0.272050i	$-0.0877016\pi$
$397$	38.3467	1.92457	0.962283	+	0.272050i	$0.0877016\pi$
$398$	0	0
$399$	20.6259	1.03259
$400$	0	0
$401$	25.0302	1.24995	0.624973	−	0.780646i	$-0.285110\pi$
$401$	25.0302	1.24995	0.624973	+	0.780646i	$0.285110\pi$
$402$	0	0
$403$	20.8198	1.03711
$404$	0	0
$405$	2.56538	0.127475
$406$	0	0
$407$	2.11550	0.104862
$408$	0	0
$409$	33.9806	1.68024	0.840118	−	0.542404i	$-0.182486\pi$
$409$	33.9806	1.68024	0.840118	+	0.542404i	$0.182486\pi$
$410$	0	0
$411$	−24.9170	−1.22907
$412$	0	0
$413$	−2.56317	−0.126125
$414$	0	0
$415$	4.08664	0.200605
$416$	0	0
$417$	−24.2845	−1.18922
$418$	0	0
$419$	20.2808	0.990784	0.495392	−	0.868669i	$-0.335024\pi$
$419$	20.2808	0.990784	0.495392	+	0.868669i	$0.335024\pi$
$420$	0	0
$421$	−27.6691	−1.34851	−0.674255	−	0.738499i	$-0.735535\pi$
$421$	−27.6691	−1.34851	−0.674255	+	0.738499i	$0.735535\pi$
$422$	0	0
$423$	−0.852303	−0.0414404
$424$	0	0
$425$	4.94782	0.240004
$426$	0	0
$427$	−34.3066	−1.66021
$428$	0	0
$429$	1.96856	0.0950431
$430$	0	0
$431$	38.4832	1.85367	0.926836	−	0.375465i	$-0.122517\pi$
$431$	38.4832	1.85367	0.926836	+	0.375465i	$0.122517\pi$
$432$	0	0
$433$	17.1505	0.824198	0.412099	−	0.911139i	$-0.364796\pi$
$433$	17.1505	0.824198	0.412099	+	0.911139i	$0.364796\pi$
$434$	0	0
$435$	3.57053	0.171194
$436$	0	0
$437$	4.46321	0.213504
$438$	0	0
$439$	−15.1093	−0.721126	−0.360563	−	0.932735i	$-0.617415\pi$
$439$	−15.1093	−0.721126	−0.360563	+	0.932735i	$0.617415\pi$
$440$	0	0
$441$	−0.705487	−0.0335946
$442$	0	0
$443$	4.18381	0.198779	0.0993894	−	0.995049i	$-0.468311\pi$
$443$	4.18381	0.198779	0.0993894	+	0.995049i	$0.468311\pi$
$444$	0	0
$445$	−1.76893	−0.0838552
$446$	0	0
$447$	−47.5265	−2.24793
$448$	0	0
$449$	20.0729	0.947299	0.473649	−	0.880714i	$-0.342936\pi$
$449$	20.0729	0.947299	0.473649	+	0.880714i	$0.342936\pi$
$450$	0	0
$451$	−2.42074	−0.113988
$452$	0	0
$453$	−46.6982	−2.19407
$454$	0	0
$455$	−1.42046	−0.0665924
$456$	0	0
$457$	−34.7944	−1.62762	−0.813808	−	0.581134i	$-0.802609\pi$
$457$	−34.7944	−1.62762	−0.813808	+	0.581134i	$0.802609\pi$
$458$	0	0
$459$	−2.92938	−0.136732
$460$	0	0
$461$	−14.8857	−0.693296	−0.346648	−	0.937995i	$-0.612680\pi$
$461$	−14.8857	−0.693296	−0.346648	+	0.937995i	$0.612680\pi$
$462$	0	0
$463$	3.56149	0.165517	0.0827583	−	0.996570i	$-0.473627\pi$
$463$	3.56149	0.165517	0.0827583	+	0.996570i	$0.473627\pi$
$464$	0	0
$465$	4.22283	0.195829
$466$	0	0
$467$	−26.3684	−1.22018	−0.610092	−	0.792330i	$-0.708868\pi$
$467$	−26.3684	−1.22018	−0.610092	+	0.792330i	$0.708868\pi$
$468$	0	0
$469$	−19.6176	−0.905855
$470$	0	0
$471$	16.6753	0.768356
$472$	0	0
$473$	−3.20138	−0.147200
$474$	0	0
$475$	18.4836	0.848084
$476$	0	0
$477$	5.27522	0.241536
$478$	0	0
$479$	19.4615	0.889217	0.444608	−	0.895725i	$-0.353343\pi$
$479$	19.4615	0.889217	0.444608	+	0.895725i	$0.353343\pi$
$480$	0	0
$481$	−13.6244	−0.621217
$482$	0	0
$483$	6.59653	0.300153
$484$	0	0
$485$	0.366848	0.0166577
$486$	0	0
$487$	−17.9309	−0.812525	−0.406263	−	0.913756i	$-0.633168\pi$
$487$	−17.9309	−0.812525	−0.406263	+	0.913756i	$0.633168\pi$
$488$	0	0
$489$	8.27180	0.374064
$490$	0	0
$491$	−14.5764	−0.657823	−0.328912	−	0.944361i	$-0.606682\pi$
$491$	−14.5764	−0.657823	−0.328912	+	0.944361i	$0.606682\pi$
$492$	0	0
$493$	−7.25625	−0.326805
$494$	0	0
$495$	0.141129	0.00634327
$496$	0	0
$497$	−9.26905	−0.415774
$498$	0	0
$499$	−34.3884	−1.53944	−0.769718	−	0.638384i	$-0.779603\pi$
$499$	−34.3884	−1.53944	−0.769718	+	0.638384i	$0.779603\pi$
$500$	0	0
$501$	−42.1588	−1.88351
$502$	0	0
$503$	36.0150	1.60583	0.802914	−	0.596094i	$-0.203282\pi$
$503$	36.0150	1.60583	0.802914	+	0.596094i	$0.203282\pi$
$504$	0	0
$505$	1.03054	0.0458584
$506$	0	0
$507$	15.3251	0.680610
$508$	0	0
$509$	19.4004	0.859909	0.429955	−	0.902851i	$-0.358530\pi$
$509$	19.4004	0.859909	0.429955	+	0.902851i	$0.358530\pi$
$510$	0	0
$511$	11.1837	0.494738
$512$	0	0
$513$	−10.9433	−0.483157
$514$	0	0
$515$	−3.21992	−0.141887
$516$	0	0
$517$	0.195758	0.00860944
$518$	0	0
$519$	−44.0380	−1.93305
$520$	0	0
$521$	−38.8115	−1.70036	−0.850182	−	0.526489i	$-0.823508\pi$
$521$	−38.8115	−1.70036	−0.850182	+	0.526489i	$0.823508\pi$
$522$	0	0
$523$	35.4731	1.55113	0.775565	−	0.631267i	$-0.217465\pi$
$523$	35.4731	1.55113	0.775565	+	0.631267i	$0.217465\pi$
$524$	0	0
$525$	27.3183	1.19227
$526$	0	0
$527$	−8.58188	−0.373833
$528$	0	0
$529$	−21.5726	−0.937938
$530$	0	0
$531$	−1.64008	−0.0711735
$532$	0	0
$533$	15.5902	0.675286
$534$	0	0
$535$	−2.43016	−0.105065
$536$	0	0
$537$	−54.6188	−2.35697
$538$	0	0
$539$	0.162038	0.00697945
$540$	0	0
$541$	−29.8367	−1.28278	−0.641390	−	0.767215i	$-0.721642\pi$
$541$	−29.8367	−1.28278	−0.641390	+	0.767215i	$0.721642\pi$
$542$	0	0
$543$	−41.1287	−1.76500
$544$	0	0
$545$	1.73435	0.0742913
$546$	0	0
$547$	20.8022	0.889436	0.444718	−	0.895671i	$-0.353304\pi$
$547$	20.8022	0.889436	0.444718	+	0.895671i	$0.353304\pi$
$548$	0	0
$549$	−21.9516	−0.936870
$550$	0	0
$551$	−27.1072	−1.15480
$552$	0	0
$553$	8.40365	0.357360
$554$	0	0
$555$	−2.76339	−0.117299
$556$	0	0
$557$	−27.4074	−1.16129	−0.580644	−	0.814157i	$-0.697199\pi$
$557$	−27.4074	−1.16129	−0.580644	+	0.814157i	$0.697199\pi$
$558$	0	0
$559$	20.6177	0.872036
$560$	0	0
$561$	−0.811437	−0.0342589
$562$	0	0
$563$	−9.68873	−0.408331	−0.204166	−	0.978936i	$-0.565448\pi$
$563$	−9.68873	−0.408331	−0.204166	+	0.978936i	$0.565448\pi$
$564$	0	0
$565$	0.381091	0.0160326
$566$	0	0
$567$	−28.7854	−1.20887
$568$	0	0
$569$	15.5283	0.650979	0.325490	−	0.945546i	$-0.394471\pi$
$569$	15.5283	0.650979	0.325490	+	0.945546i	$0.394471\pi$
$570$	0	0
$571$	−16.7305	−0.700150	−0.350075	−	0.936722i	$-0.613844\pi$
$571$	−16.7305	−0.700150	−0.350075	+	0.936722i	$0.613844\pi$
$572$	0	0
$573$	−52.3758	−2.18803
$574$	0	0
$575$	5.91138	0.246522
$576$	0	0
$577$	24.4870	1.01941	0.509704	−	0.860350i	$-0.329755\pi$
$577$	24.4870	1.01941	0.509704	+	0.860350i	$0.329755\pi$
$578$	0	0
$579$	33.1447	1.37745
$580$	0	0
$581$	−45.8550	−1.90238
$582$	0	0
$583$	−1.21162	−0.0501802
$584$	0	0
$585$	−0.908904	−0.0375786
$586$	0	0
$587$	14.4127	0.594877	0.297438	−	0.954741i	$-0.403868\pi$
$587$	14.4127	0.594877	0.297438	+	0.954741i	$0.403868\pi$
$588$	0	0
$589$	−32.0593	−1.32098
$590$	0	0
$591$	−11.6928	−0.480980
$592$	0	0
$593$	6.57865	0.270153	0.135076	−	0.990835i	$-0.456872\pi$
$593$	6.57865	0.270153	0.135076	+	0.990835i	$0.456872\pi$
$594$	0	0
$595$	0.585512	0.0240036
$596$	0	0
$597$	44.7293	1.83065
$598$	0	0
$599$	21.7762	0.889751	0.444875	−	0.895592i	$-0.353248\pi$
$599$	21.7762	0.889751	0.444875	+	0.895592i	$0.353248\pi$
$600$	0	0
$601$	7.99677	0.326195	0.163098	−	0.986610i	$-0.447851\pi$
$601$	7.99677	0.326195	0.163098	+	0.986610i	$0.447851\pi$
$602$	0	0
$603$	−12.5526	−0.511181
$604$	0	0
$605$	2.48034	0.100840
$606$	0	0
$607$	18.7519	0.761117	0.380559	−	0.924757i	$-0.375732\pi$
$607$	18.7519	0.761117	0.380559	+	0.924757i	$0.375732\pi$
$608$	0	0
$609$	−40.0638	−1.62347
$610$	0	0
$611$	−1.26073	−0.0510037
$612$	0	0
$613$	−3.52184	−0.142246	−0.0711228	−	0.997468i	$-0.522658\pi$
$613$	−3.52184	−0.142246	−0.0711228	+	0.997468i	$0.522658\pi$
$614$	0	0
$615$	3.16212	0.127509
$616$	0	0
$617$	−34.7482	−1.39891	−0.699454	−	0.714678i	$-0.746573\pi$
$617$	−34.7482	−1.39891	−0.699454	+	0.714678i	$0.746573\pi$
$618$	0	0
$619$	−36.5857	−1.47050	−0.735252	−	0.677794i	$-0.762936\pi$
$619$	−36.5857	−1.47050	−0.735252	+	0.677794i	$0.762936\pi$
$620$	0	0
$621$	−3.49986	−0.140445
$622$	0	0
$623$	19.8486	0.795217
$624$	0	0
$625$	24.2200	0.968800
$626$	0	0
$627$	−3.03129	−0.121058
$628$	0	0
$629$	5.61593	0.223922
$630$	0	0
$631$	36.1710	1.43994	0.719972	−	0.694003i	$-0.244155\pi$
$631$	36.1710	1.43994	0.719972	+	0.694003i	$0.244155\pi$
$632$	0	0
$633$	−39.3216	−1.56289
$634$	0	0
$635$	−2.67916	−0.106319
$636$	0	0
$637$	−1.04356	−0.0413474
$638$	0	0
$639$	−5.93094	−0.234624
$640$	0	0
$641$	9.74334	0.384839	0.192419	−	0.981313i	$-0.438367\pi$
$641$	9.74334	0.384839	0.192419	+	0.981313i	$0.438367\pi$
$642$	0	0
$643$	17.6461	0.695894	0.347947	−	0.937514i	$-0.386879\pi$
$643$	17.6461	0.695894	0.347947	+	0.937514i	$0.386879\pi$
$644$	0	0
$645$	4.18183	0.164660
$646$	0	0
$647$	11.5012	0.452160	0.226080	−	0.974109i	$-0.427409\pi$
$647$	11.5012	0.452160	0.226080	+	0.974109i	$0.427409\pi$
$648$	0	0
$649$	0.376697	0.0147866
$650$	0	0
$651$	−47.3830	−1.85709
$652$	0	0
$653$	−19.4450	−0.760942	−0.380471	−	0.924793i	$-0.624238\pi$
$653$	−19.4450	−0.760942	−0.380471	+	0.924793i	$0.624238\pi$
$654$	0	0
$655$	0.925916	0.0361785
$656$	0	0
$657$	7.15606	0.279185
$658$	0	0
$659$	3.37833	0.131601	0.0658004	−	0.997833i	$-0.479040\pi$
$659$	3.37833	0.131601	0.0658004	+	0.997833i	$0.479040\pi$
$660$	0	0
$661$	−22.1603	−0.861937	−0.430968	−	0.902367i	$-0.641828\pi$
$661$	−22.1603	−0.861937	−0.430968	+	0.902367i	$0.641828\pi$
$662$	0	0
$663$	5.22586	0.202955
$664$	0	0
$665$	2.18730	0.0848197
$666$	0	0
$667$	−8.66937	−0.335679
$668$	0	0
$669$	−13.7420	−0.531297
$670$	0	0
$671$	5.04187	0.194639
$672$	0	0
$673$	−45.8262	−1.76647	−0.883235	−	0.468931i	$-0.844639\pi$
$673$	−45.8262	−1.76647	−0.883235	+	0.468931i	$0.844639\pi$
$674$	0	0
$675$	−14.4940	−0.557875
$676$	0	0
$677$	41.4430	1.59278	0.796392	−	0.604781i	$-0.206739\pi$
$677$	41.4430	1.59278	0.796392	+	0.604781i	$0.206739\pi$
$678$	0	0
$679$	−4.11629	−0.157969
$680$	0	0
$681$	6.25307	0.239618
$682$	0	0
$683$	11.0456	0.422648	0.211324	−	0.977416i	$-0.432222\pi$
$683$	11.0456	0.422648	0.211324	+	0.977416i	$0.432222\pi$
$684$	0	0
$685$	−2.64236	−0.100959
$686$	0	0
$687$	−13.3664	−0.509959
$688$	0	0
$689$	7.80314	0.297276
$690$	0	0
$691$	−35.7319	−1.35931	−0.679654	−	0.733533i	$-0.737870\pi$
$691$	−35.7319	−1.35931	−0.679654	+	0.733533i	$0.737870\pi$
$692$	0	0
$693$	−1.58356	−0.0601546
$694$	0	0
$695$	−2.57528	−0.0976861
$696$	0	0
$697$	−6.42624	−0.243411
$698$	0	0
$699$	−5.09699	−0.192786
$700$	0	0
$701$	10.3589	0.391250	0.195625	−	0.980679i	$-0.437326\pi$
$701$	10.3589	0.391250	0.195625	+	0.980679i	$0.437326\pi$
$702$	0	0
$703$	20.9794	0.791254
$704$	0	0
$705$	−0.255711	−0.00963062
$706$	0	0
$707$	−11.5634	−0.434885
$708$	0	0
$709$	25.6831	0.964548	0.482274	−	0.876020i	$-0.339811\pi$
$709$	25.6831	0.964548	0.482274	+	0.876020i	$0.339811\pi$
$710$	0	0
$711$	5.37720	0.201661
$712$	0	0
$713$	−10.2532	−0.383984
$714$	0	0
$715$	0.208759	0.00780713
$716$	0	0
$717$	−31.4185	−1.17335
$718$	0	0
$719$	39.3460	1.46736	0.733680	−	0.679495i	$-0.237801\pi$
$719$	39.3460	1.46736	0.733680	+	0.679495i	$0.237801\pi$
$720$	0	0
$721$	36.1298	1.34554
$722$	0	0
$723$	48.5059	1.80396
$724$	0	0
$725$	−35.9026	−1.33339
$726$	0	0
$727$	48.8649	1.81230	0.906150	−	0.422957i	$-0.139008\pi$
$727$	48.8649	1.81230	0.906150	+	0.422957i	$0.139008\pi$
$728$	0	0
$729$	0.511642	0.0189497
$730$	0	0
$731$	−8.49857	−0.314331
$732$	0	0
$733$	−3.88165	−0.143372	−0.0716860	−	0.997427i	$-0.522838\pi$
$733$	−3.88165	−0.143372	−0.0716860	+	0.997427i	$0.522838\pi$
$734$	0	0
$735$	−0.211663	−0.00780730
$736$	0	0
$737$	2.88310	0.106200
$738$	0	0
$739$	−30.0021	−1.10365	−0.551823	−	0.833961i	$-0.686068\pi$
$739$	−30.0021	−1.10365	−0.551823	+	0.833961i	$0.686068\pi$
$740$	0	0
$741$	19.5222	0.717167
$742$	0	0
$743$	−9.81131	−0.359942	−0.179971	−	0.983672i	$-0.557600\pi$
$743$	−9.81131	−0.359942	−0.179971	+	0.983672i	$0.557600\pi$
$744$	0	0
$745$	−5.04001	−0.184652
$746$	0	0
$747$	−29.3410	−1.07353
$748$	0	0
$749$	27.2681	0.996356
$750$	0	0
$751$	−34.5965	−1.26244	−0.631222	−	0.775602i	$-0.717446\pi$
$751$	−34.5965	−1.26244	−0.631222	+	0.775602i	$0.717446\pi$
$752$	0	0
$753$	45.6309	1.66288
$754$	0	0
$755$	−4.95217	−0.180228
$756$	0	0
$757$	−4.53160	−0.164704	−0.0823519	−	0.996603i	$-0.526243\pi$
$757$	−4.53160	−0.164704	−0.0823519	+	0.996603i	$0.526243\pi$
$758$	0	0
$759$	−0.969461	−0.0351892
$760$	0	0
$761$	9.07918	0.329120	0.164560	−	0.986367i	$-0.447380\pi$
$761$	9.07918	0.329120	0.164560	+	0.986367i	$0.447380\pi$
$762$	0	0
$763$	−19.4606	−0.704520
$764$	0	0
$765$	0.374648	0.0135454
$766$	0	0
$767$	−2.42602	−0.0875985
$768$	0	0
$769$	16.2084	0.584491	0.292246	−	0.956343i	$-0.405598\pi$
$769$	16.2084	0.584491	0.292246	+	0.956343i	$0.405598\pi$
$770$	0	0
$771$	2.01438	0.0725462
$772$	0	0
$773$	−3.10428	−0.111653	−0.0558267	−	0.998440i	$-0.517779\pi$
$773$	−3.10428	−0.111653	−0.0558267	+	0.998440i	$0.517779\pi$
$774$	0	0
$775$	−42.4616	−1.52527
$776$	0	0
$777$	31.0072	1.11238
$778$	0	0
$779$	−24.0065	−0.860122
$780$	0	0
$781$	1.36223	0.0487444
$782$	0	0
$783$	21.2563	0.759638
$784$	0	0
$785$	1.76835	0.0631151
$786$	0	0
$787$	38.2791	1.36450	0.682251	−	0.731118i	$-0.261001\pi$
$787$	38.2791	1.36450	0.682251	+	0.731118i	$0.261001\pi$
$788$	0	0
$789$	−51.5392	−1.83485
$790$	0	0
$791$	−4.27610	−0.152041
$792$	0	0
$793$	−32.4709	−1.15308
$794$	0	0
$795$	1.58269	0.0561322
$796$	0	0
$797$	−50.8193	−1.80011	−0.900055	−	0.435776i	$-0.856474\pi$
$797$	−50.8193	−1.80011	−0.900055	+	0.435776i	$0.856474\pi$
$798$	0	0
$799$	0.519671	0.0183846
$800$	0	0
$801$	12.7004	0.448747
$802$	0	0
$803$	−1.64362	−0.0580020
$804$	0	0
$805$	0.699538	0.0246555
$806$	0	0
$807$	13.0594	0.459713
$808$	0	0
$809$	−32.0959	−1.12843	−0.564216	−	0.825627i	$-0.690822\pi$
$809$	−32.0959	−1.12843	−0.564216	+	0.825627i	$0.690822\pi$
$810$	0	0
$811$	9.99552	0.350990	0.175495	−	0.984480i	$-0.443847\pi$
$811$	9.99552	0.350990	0.175495	+	0.984480i	$0.443847\pi$
$812$	0	0
$813$	−61.6094	−2.16074
$814$	0	0
$815$	0.877193	0.0307267
$816$	0	0
$817$	−31.7481	−1.11073
$818$	0	0
$819$	10.1985	0.356366
$820$	0	0
$821$	−26.8493	−0.937046	−0.468523	−	0.883451i	$-0.655214\pi$
$821$	−26.8493	−0.937046	−0.468523	+	0.883451i	$0.655214\pi$
$822$	0	0
$823$	−21.9088	−0.763692	−0.381846	−	0.924226i	$-0.624712\pi$
$823$	−21.9088	−0.763692	−0.381846	+	0.924226i	$0.624712\pi$
$824$	0	0
$825$	−4.01484	−0.139779
$826$	0	0
$827$	−5.22666	−0.181749	−0.0908744	−	0.995862i	$-0.528966\pi$
$827$	−5.22666	−0.181749	−0.0908744	+	0.995862i	$0.528966\pi$
$828$	0	0
$829$	8.63153	0.299785	0.149893	−	0.988702i	$-0.452107\pi$
$829$	8.63153	0.299785	0.149893	+	0.988702i	$0.452107\pi$
$830$	0	0
$831$	−14.3756	−0.498684
$832$	0	0
$833$	0.430154	0.0149039
$834$	0	0
$835$	−4.47078	−0.154718
$836$	0	0
$837$	25.1396	0.868951
$838$	0	0
$839$	−14.6457	−0.505625	−0.252813	−	0.967515i	$-0.581356\pi$
$839$	−14.6457	−0.505625	−0.252813	+	0.967515i	$0.581356\pi$
$840$	0	0
$841$	23.6531	0.815625
$842$	0	0
$843$	−7.42760	−0.255820
$844$	0	0
$845$	1.62517	0.0559074
$846$	0	0
$847$	−27.8312	−0.956290
$848$	0	0
$849$	31.9159	1.09535
$850$	0	0
$851$	6.70960	0.230002
$852$	0	0
$853$	23.0298	0.788525	0.394262	−	0.918998i	$-0.371000\pi$
$853$	23.0298	0.788525	0.394262	+	0.918998i	$0.371000\pi$
$854$	0	0
$855$	1.39957	0.0478644
$856$	0	0
$857$	4.92393	0.168198	0.0840991	−	0.996457i	$-0.473199\pi$
$857$	4.92393	0.168198	0.0840991	+	0.996457i	$0.473199\pi$
$858$	0	0
$859$	−38.7118	−1.32083	−0.660415	−	0.750901i	$-0.729620\pi$
$859$	−38.7118	−1.32083	−0.660415	+	0.750901i	$0.729620\pi$
$860$	0	0
$861$	−35.4811	−1.20919
$862$	0	0
$863$	−2.97344	−0.101217	−0.0506085	−	0.998719i	$-0.516116\pi$
$863$	−2.97344	−0.101217	−0.0506085	+	0.998719i	$0.516116\pi$
$864$	0	0
$865$	−4.67007	−0.158787
$866$	0	0
$867$	−2.15409	−0.0731566
$868$	0	0
$869$	−1.23504	−0.0418960
$870$	0	0
$871$	−18.5679	−0.629148
$872$	0	0
$873$	−2.63387	−0.0891429
$874$	0	0
$875$	5.82456	0.196906
$876$	0	0
$877$	−28.6008	−0.965782	−0.482891	−	0.875680i	$-0.660413\pi$
$877$	−28.6008	−0.965782	−0.482891	+	0.875680i	$0.660413\pi$
$878$	0	0
$879$	10.8642	0.366439
$880$	0	0
$881$	−23.5580	−0.793690	−0.396845	−	0.917886i	$-0.629895\pi$
$881$	−23.5580	−0.793690	−0.396845	+	0.917886i	$0.629895\pi$
$882$	0	0
$883$	47.0950	1.58487	0.792437	−	0.609954i	$-0.208812\pi$
$883$	47.0950	1.58487	0.792437	+	0.609954i	$0.208812\pi$
$884$	0	0
$885$	−0.492063	−0.0165405
$886$	0	0
$887$	−57.6105	−1.93437	−0.967186	−	0.254069i	$-0.918231\pi$
$887$	−57.6105	−1.93437	−0.967186	+	0.254069i	$0.918231\pi$
$888$	0	0
$889$	30.0620	1.00825
$890$	0	0
$891$	4.23045	0.141725
$892$	0	0
$893$	1.94133	0.0649642
$894$	0	0
$895$	−5.79211	−0.193609
$896$	0	0
$897$	6.24357	0.208467
$898$	0	0
$899$	62.2723	2.07690
$900$	0	0
$901$	−3.21643	−0.107155
$902$	0	0
$903$	−46.9231	−1.56150
$904$	0	0
$905$	−4.36154	−0.144983
$906$	0	0
$907$	9.62127	0.319469	0.159734	−	0.987160i	$-0.448936\pi$
$907$	9.62127	0.319469	0.159734	+	0.987160i	$0.448936\pi$
$908$	0	0
$909$	−7.39898	−0.245409
$910$	0	0
$911$	0.170367	0.00564451	0.00282226	−	0.999996i	$-0.499102\pi$
$911$	0.170367	0.00564451	0.00282226	+	0.999996i	$0.499102\pi$
$912$	0	0
$913$	6.73908	0.223031
$914$	0	0
$915$	−6.58598	−0.217726
$916$	0	0
$917$	−10.3894	−0.343089
$918$	0	0
$919$	−31.5463	−1.04062	−0.520308	−	0.853979i	$-0.674183\pi$
$919$	−31.5463	−1.04062	−0.520308	+	0.853979i	$0.674183\pi$
$920$	0	0
$921$	−11.3160	−0.372874
$922$	0	0
$923$	−8.77308	−0.288770
$924$	0	0
$925$	27.7866	0.913617
$926$	0	0
$927$	23.1182	0.759300
$928$	0	0
$929$	−34.5620	−1.13394	−0.566971	−	0.823738i	$-0.691885\pi$
$929$	−34.5620	−1.13394	−0.566971	+	0.823738i	$0.691885\pi$
$930$	0	0
$931$	1.60692	0.0526648
$932$	0	0
$933$	−7.45848	−0.244180
$934$	0	0
$935$	−0.0860499	−0.00281413
$936$	0	0
$937$	−23.2990	−0.761145	−0.380572	−	0.924751i	$-0.624273\pi$
$937$	−23.2990	−0.761145	−0.380572	+	0.924751i	$0.624273\pi$
$938$	0	0
$939$	−4.22692	−0.137940
$940$	0	0
$941$	54.6280	1.78082	0.890410	−	0.455159i	$-0.150418\pi$
$941$	54.6280	1.78082	0.890410	+	0.455159i	$0.150418\pi$
$942$	0	0
$943$	−7.67772	−0.250021
$944$	0	0
$945$	−1.71519	−0.0557950
$946$	0	0
$947$	−51.4480	−1.67184	−0.835918	−	0.548855i	$-0.815064\pi$
$947$	−51.4480	−1.67184	−0.835918	+	0.548855i	$0.815064\pi$
$948$	0	0
$949$	10.5853	0.343613
$950$	0	0
$951$	42.0746	1.36436
$952$	0	0
$953$	48.9404	1.58534	0.792668	−	0.609653i	$-0.208691\pi$
$953$	48.9404	1.58534	0.792668	+	0.609653i	$0.208691\pi$
$954$	0	0
$955$	−5.55425	−0.179731
$956$	0	0
$957$	5.88799	0.190332
$958$	0	0
$959$	29.6491	0.957419
$960$	0	0
$961$	42.6487	1.37576
$962$	0	0
$963$	17.4479	0.562251
$964$	0	0
$965$	3.51487	0.113148
$966$	0	0
$967$	47.5082	1.52776	0.763880	−	0.645358i	$-0.223292\pi$
$967$	47.5082	1.52776	0.763880	+	0.645358i	$0.223292\pi$
$968$	0	0
$969$	−8.04702	−0.258507
$970$	0	0
$971$	−27.4869	−0.882095	−0.441048	−	0.897484i	$-0.645393\pi$
$971$	−27.4869	−0.882095	−0.441048	+	0.897484i	$0.645393\pi$
$972$	0	0
$973$	28.8965	0.926378
$974$	0	0
$975$	25.8566	0.828073
$976$	0	0
$977$	−52.5993	−1.68280	−0.841400	−	0.540413i	$-0.818268\pi$
$977$	−52.5993	−1.68280	−0.841400	+	0.540413i	$0.818268\pi$
$978$	0	0
$979$	−2.91705	−0.0932293
$980$	0	0
$981$	−12.4521	−0.397566
$982$	0	0
$983$	47.4925	1.51477	0.757387	−	0.652966i	$-0.226476\pi$
$983$	47.4925	1.51477	0.757387	+	0.652966i	$0.226476\pi$
$984$	0	0
$985$	−1.23998	−0.0395091
$986$	0	0
$987$	2.86925	0.0913293
$988$	0	0
$989$	−10.1536	−0.322867
$990$	0	0
$991$	−14.9337	−0.474385	−0.237192	−	0.971463i	$-0.576227\pi$
$991$	−14.9337	−0.474385	−0.237192	+	0.971463i	$0.576227\pi$
$992$	0	0
$993$	26.4641	0.839812
$994$	0	0
$995$	4.74337	0.150375
$996$	0	0
$997$	37.0690	1.17399	0.586994	−	0.809591i	$-0.300311\pi$
$997$	37.0690	1.17399	0.586994	+	0.809591i	$0.300311\pi$
$998$	0	0
$999$	−16.4512	−0.520492

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4012.2.a.i.1.3	✓	18

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4012.2.a.i.1.3	✓	18	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:	\( N \)	\(=\)	\( 4012 = 2^{2} \cdot 17 \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4012.a (trivial)

\(f(q)\)	\(=\)	\(q-2.15409 q^{3} -0.228433 q^{5} +2.56317 q^{7} +1.64008 q^{9} +O(q^{10})\)	\(q-2.15409 q^{3} -0.228433 q^{5} +2.56317 q^{7} +1.64008 q^{9} -0.376697 q^{11} +2.42602 q^{13} +0.492063 q^{15} -1.00000 q^{17} -3.73570 q^{19} -5.52129 q^{21} -1.19475 q^{23} -4.94782 q^{25} +2.92938 q^{27} +7.25625 q^{29} +8.58188 q^{31} +0.811437 q^{33} -0.585512 q^{35} -5.61593 q^{37} -5.22586 q^{39} +6.42624 q^{41} +8.49857 q^{43} -0.374648 q^{45} -0.519671 q^{47} -0.430154 q^{49} +2.15409 q^{51} +3.21643 q^{53} +0.0860499 q^{55} +8.04702 q^{57} -1.00000 q^{59} -13.3844 q^{61} +4.20381 q^{63} -0.554182 q^{65} -7.65363 q^{67} +2.57358 q^{69} -3.61624 q^{71} +4.36323 q^{73} +10.6580 q^{75} -0.965539 q^{77} +3.27862 q^{79} -11.2304 q^{81} -17.8899 q^{83} +0.228433 q^{85} -15.6306 q^{87} +7.74376 q^{89} +6.21831 q^{91} -18.4861 q^{93} +0.853356 q^{95} -1.60594 q^{97} -0.617814 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q + 8 q^{3} + 4 q^{5} + 2 q^{7} + 18 q^{9}+O(q^{10}) \) 18 * q + 8 * q^3 + 4 * q^5 + 2 * q^7 + 18 * q^9	\( 18 q + 8 q^{3} + 4 q^{5} + 2 q^{7} + 18 q^{9} + 12 q^{11} + 2 q^{13} - 18 q^{17} + 5 q^{19} - 3 q^{21} + 21 q^{23} + 16 q^{25} + 26 q^{27} + 14 q^{29} + 15 q^{31} + 19 q^{33} + 20 q^{35} + 2 q^{37} - 14 q^{39} + 34 q^{41} + 21 q^{43} + 49 q^{45} + 69 q^{47} + 28 q^{49} - 8 q^{51} - 4 q^{53} + 18 q^{55} + 5 q^{57} - 18 q^{59} + 11 q^{61} + 35 q^{63} + 27 q^{65} + 34 q^{67} - 4 q^{69} + 37 q^{71} + 18 q^{73} + 72 q^{75} + 11 q^{77} + 11 q^{79} + 30 q^{81} + 28 q^{83} - 4 q^{85} + 7 q^{87} + 44 q^{89} - 23 q^{91} - 3 q^{93} - 11 q^{95} + 11 q^{97} + 56 q^{99}+O(q^{100}) \) 18 * q + 8 * q^3 + 4 * q^5 + 2 * q^7 + 18 * q^9 + 12 * q^11 + 2 * q^13 - 18 * q^17 + 5 * q^19 - 3 * q^21 + 21 * q^23 + 16 * q^25 + 26 * q^27 + 14 * q^29 + 15 * q^31 + 19 * q^33 + 20 * q^35 + 2 * q^37 - 14 * q^39 + 34 * q^41 + 21 * q^43 + 49 * q^45 + 69 * q^47 + 28 * q^49 - 8 * q^51 - 4 * q^53 + 18 * q^55 + 5 * q^57 - 18 * q^59 + 11 * q^61 + 35 * q^63 + 27 * q^65 + 34 * q^67 - 4 * q^69 + 37 * q^71 + 18 * q^73 + 72 * q^75 + 11 * q^77 + 11 * q^79 + 30 * q^81 + 28 * q^83 - 4 * q^85 + 7 * q^87 + 44 * q^89 - 23 * q^91 - 3 * q^93 - 11 * q^95 + 11 * q^97 + 56 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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