LMFDB - Embedded newform 6040.2.a.m.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6040,2,Mod(1,6040)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6040, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("6040.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6040 = 2^{3} \cdot 5 \cdot 151 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6040.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:	$48.2296428209$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 3 x^{11} - 18 x^{10} + 54 x^{9} + 110 x^{8} - 335 x^{7} - 258 x^{6} + 825 x^{5} + 168 x^{4} + \cdots - 16 $ x^12 - 3x^11 - 18x^10 + 54x^9 + 110x^8 - 335x^7 - 258x^6 + 825x^5 + 168x^4 - 669x^3 + 39x^2 + 106*x - 16
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.89604$ of defining polynomial
Character	$\chi$	$=$	6040.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.89604 q^{3} -1.00000 q^{5} -1.82001 q^{7} +0.594974 q^{9} +O(q^{10})$	$q-1.89604 q^{3} -1.00000 q^{5} -1.82001 q^{7} +0.594974 q^{9} -1.31797 q^{11} -1.28840 q^{13} +1.89604 q^{15} -4.14326 q^{17} -1.85714 q^{19} +3.45081 q^{21} +1.25766 q^{23} +1.00000 q^{25} +4.56003 q^{27} -0.373811 q^{29} -4.20135 q^{31} +2.49892 q^{33} +1.82001 q^{35} -2.64864 q^{37} +2.44287 q^{39} -10.4606 q^{41} -0.450079 q^{43} -0.594974 q^{45} -4.47477 q^{47} -3.68758 q^{49} +7.85580 q^{51} -7.87985 q^{53} +1.31797 q^{55} +3.52121 q^{57} -14.3721 q^{59} -14.2551 q^{61} -1.08286 q^{63} +1.28840 q^{65} +2.68305 q^{67} -2.38457 q^{69} +6.34437 q^{71} +0.378605 q^{73} -1.89604 q^{75} +2.39871 q^{77} +6.71301 q^{79} -10.4309 q^{81} +5.89660 q^{83} +4.14326 q^{85} +0.708761 q^{87} +5.92773 q^{89} +2.34490 q^{91} +7.96593 q^{93} +1.85714 q^{95} -7.69681 q^{97} -0.784156 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 3 q^{3} - 12 q^{5} + 5 q^{7} + 9 q^{9}+O(q^{10}) $ 12 * q + 3 * q^3 - 12 * q^5 + 5 * q^7 + 9 * q^9	$ 12 q + 3 q^{3} - 12 q^{5} + 5 q^{7} + 9 q^{9} + 10 q^{11} + 11 q^{13} - 3 q^{15} - 4 q^{17} + 5 q^{19} - q^{21} + 18 q^{23} + 12 q^{25} + 9 q^{27} + 16 q^{29} - q^{31} + 8 q^{33} - 5 q^{35} + 2 q^{37} + 6 q^{39} + 4 q^{41} + 7 q^{43} - 9 q^{45} + 3 q^{49} - 4 q^{51} + 39 q^{53} - 10 q^{55} - 15 q^{57} - 4 q^{59} - 32 q^{61} + 3 q^{63} - 11 q^{65} + 4 q^{67} + 12 q^{69} + 24 q^{71} - 10 q^{73} + 3 q^{75} + 38 q^{77} + 32 q^{79} - 8 q^{81} + 9 q^{83} + 4 q^{85} + 3 q^{87} + 15 q^{89} + 18 q^{91} + 36 q^{93} - 5 q^{95} + 15 q^{97} + 28 q^{99}+O(q^{100}) $ 12 * q + 3 * q^3 - 12 * q^5 + 5 * q^7 + 9 * q^9 + 10 * q^11 + 11 * q^13 - 3 * q^15 - 4 * q^17 + 5 * q^19 - q^21 + 18 * q^23 + 12 * q^25 + 9 * q^27 + 16 * q^29 - q^31 + 8 * q^33 - 5 * q^35 + 2 * q^37 + 6 * q^39 + 4 * q^41 + 7 * q^43 - 9 * q^45 + 3 * q^49 - 4 * q^51 + 39 * q^53 - 10 * q^55 - 15 * q^57 - 4 * q^59 - 32 * q^61 + 3 * q^63 - 11 * q^65 + 4 * q^67 + 12 * q^69 + 24 * q^71 - 10 * q^73 + 3 * q^75 + 38 * q^77 + 32 * q^79 - 8 * q^81 + 9 * q^83 + 4 * q^85 + 3 * q^87 + 15 * q^89 + 18 * q^91 + 36 * q^93 - 5 * q^95 + 15 * q^97 + 28 * 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
$3$	−1.89604	−1.09468	−0.547340	−	0.836910i	$-0.684360\pi$
$3$	−1.89604	−1.09468	−0.547340	+	0.836910i	$0.684360\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−1.82001	−0.687897	−0.343949	−	0.938988i	$-0.611765\pi$
$7$	−1.82001	−0.687897	−0.343949	+	0.938988i	$0.611765\pi$
$8$	0	0
$9$	0.594974	0.198325
$10$	0	0
$11$	−1.31797	−0.397382	−0.198691	−	0.980062i	$-0.563669\pi$
$11$	−1.31797	−0.397382	−0.198691	+	0.980062i	$0.563669\pi$
$12$	0	0
$13$	−1.28840	−0.357339	−0.178669	−	0.983909i	$-0.557179\pi$
$13$	−1.28840	−0.357339	−0.178669	+	0.983909i	$0.557179\pi$
$14$	0	0
$15$	1.89604	0.489556
$16$	0	0
$17$	−4.14326	−1.00489	−0.502444	−	0.864610i	$-0.667566\pi$
$17$	−4.14326	−1.00489	−0.502444	+	0.864610i	$0.667566\pi$
$18$	0	0
$19$	−1.85714	−0.426057	−0.213028	−	0.977046i	$-0.568333\pi$
$19$	−1.85714	−0.426057	−0.213028	+	0.977046i	$0.568333\pi$
$20$	0	0
$21$	3.45081	0.753028
$22$	0	0
$23$	1.25766	0.262240	0.131120	−	0.991367i	$-0.458143\pi$
$23$	1.25766	0.262240	0.131120	+	0.991367i	$0.458143\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	4.56003	0.877578
$28$	0	0
$29$	−0.373811	−0.0694150	−0.0347075	−	0.999398i	$-0.511050\pi$
$29$	−0.373811	−0.0694150	−0.0347075	+	0.999398i	$0.511050\pi$
$30$	0	0
$31$	−4.20135	−0.754584	−0.377292	−	0.926094i	$-0.623145\pi$
$31$	−4.20135	−0.754584	−0.377292	+	0.926094i	$0.623145\pi$
$32$	0	0
$33$	2.49892	0.435006
$34$	0	0
$35$	1.82001	0.307637
$36$	0	0
$37$	−2.64864	−0.435434	−0.217717	−	0.976012i	$-0.569861\pi$
$37$	−2.64864	−0.435434	−0.217717	+	0.976012i	$0.569861\pi$
$38$	0	0
$39$	2.44287	0.391171
$40$	0	0
$41$	−10.4606	−1.63367	−0.816836	−	0.576870i	$-0.804274\pi$
$41$	−10.4606	−1.63367	−0.816836	+	0.576870i	$0.804274\pi$
$42$	0	0
$43$	−0.450079	−0.0686364	−0.0343182	−	0.999411i	$-0.510926\pi$
$43$	−0.450079	−0.0686364	−0.0343182	+	0.999411i	$0.510926\pi$
$44$	0	0
$45$	−0.594974	−0.0886935
$46$	0	0
$47$	−4.47477	−0.652712	−0.326356	−	0.945247i	$-0.605821\pi$
$47$	−4.47477	−0.652712	−0.326356	+	0.945247i	$0.605821\pi$
$48$	0	0
$49$	−3.68758	−0.526797
$50$	0	0
$51$	7.85580	1.10003
$52$	0	0
$53$	−7.87985	−1.08238	−0.541191	−	0.840900i	$-0.682026\pi$
$53$	−7.87985	−1.08238	−0.541191	+	0.840900i	$0.682026\pi$
$54$	0	0
$55$	1.31797	0.177715
$56$	0	0
$57$	3.52121	0.466396
$58$	0	0
$59$	−14.3721	−1.87109	−0.935543	−	0.353212i	$-0.885090\pi$
$59$	−14.3721	−1.87109	−0.935543	+	0.353212i	$0.885090\pi$
$60$	0	0
$61$	−14.2551	−1.82518	−0.912590	−	0.408876i	$-0.865921\pi$
$61$	−14.2551	−1.82518	−0.912590	+	0.408876i	$0.865921\pi$
$62$	0	0
$63$	−1.08286	−0.136427
$64$	0	0
$65$	1.28840	0.159807
$66$	0	0
$67$	2.68305	0.327786	0.163893	−	0.986478i	$-0.447595\pi$
$67$	2.68305	0.327786	0.163893	+	0.986478i	$0.447595\pi$
$68$	0	0
$69$	−2.38457	−0.287069
$70$	0	0
$71$	6.34437	0.752939	0.376469	−	0.926429i	$-0.377138\pi$
$71$	6.34437	0.752939	0.376469	+	0.926429i	$0.377138\pi$
$72$	0	0
$73$	0.378605	0.0443124	0.0221562	−	0.999755i	$-0.492947\pi$
$73$	0.378605	0.0443124	0.0221562	+	0.999755i	$0.492947\pi$
$74$	0	0
$75$	−1.89604	−0.218936
$76$	0	0
$77$	2.39871	0.273358
$78$	0	0
$79$	6.71301	0.755272	0.377636	−	0.925954i	$-0.376737\pi$
$79$	6.71301	0.755272	0.377636	+	0.925954i	$0.376737\pi$
$80$	0	0
$81$	−10.4309	−1.15899
$82$	0	0
$83$	5.89660	0.647236	0.323618	−	0.946188i	$-0.395101\pi$
$83$	5.89660	0.647236	0.323618	+	0.946188i	$0.395101\pi$
$84$	0	0
$85$	4.14326	0.449400
$86$	0	0
$87$	0.708761	0.0759872
$88$	0	0
$89$	5.92773	0.628338	0.314169	−	0.949367i	$-0.398274\pi$
$89$	5.92773	0.628338	0.314169	+	0.949367i	$0.398274\pi$
$90$	0	0
$91$	2.34490	0.245812
$92$	0	0
$93$	7.96593	0.826029
$94$	0	0
$95$	1.85714	0.190538
$96$	0	0
$97$	−7.69681	−0.781493	−0.390746	−	0.920498i	$-0.627783\pi$
$97$	−7.69681	−0.781493	−0.390746	+	0.920498i	$0.627783\pi$
$98$	0	0
$99$	−0.784156	−0.0788107
$100$	0	0
$101$	14.8219	1.47483	0.737415	−	0.675440i	$-0.236046\pi$
$101$	14.8219	1.47483	0.737415	+	0.675440i	$0.236046\pi$
$102$	0	0
$103$	−9.39444	−0.925662	−0.462831	−	0.886447i	$-0.653166\pi$
$103$	−9.39444	−0.925662	−0.462831	+	0.886447i	$0.653166\pi$
$104$	0	0
$105$	−3.45081	−0.336764
$106$	0	0
$107$	17.2451	1.66714	0.833571	−	0.552412i	$-0.186293\pi$
$107$	17.2451	1.66714	0.833571	+	0.552412i	$0.186293\pi$
$108$	0	0
$109$	−5.92141	−0.567169	−0.283584	−	0.958947i	$-0.591524\pi$
$109$	−5.92141	−0.567169	−0.283584	+	0.958947i	$0.591524\pi$
$110$	0	0
$111$	5.02193	0.476661
$112$	0	0
$113$	10.2923	0.968218	0.484109	−	0.875008i	$-0.339144\pi$
$113$	10.2923	0.968218	0.484109	+	0.875008i	$0.339144\pi$
$114$	0	0
$115$	−1.25766	−0.117277
$116$	0	0
$117$	−0.766566	−0.0708691
$118$	0	0
$119$	7.54076	0.691260
$120$	0	0
$121$	−9.26296	−0.842087
$122$	0	0
$123$	19.8337	1.78835
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−1.12800	−0.100094	−0.0500471	−	0.998747i	$-0.515937\pi$
$127$	−1.12800	−0.100094	−0.0500471	+	0.998747i	$0.515937\pi$
$128$	0	0
$129$	0.853368	0.0751349
$130$	0	0
$131$	−15.1387	−1.32267	−0.661336	−	0.750090i	$-0.730010\pi$
$131$	−15.1387	−1.32267	−0.661336	+	0.750090i	$0.730010\pi$
$132$	0	0
$133$	3.38000	0.293083
$134$	0	0
$135$	−4.56003	−0.392465
$136$	0	0
$137$	−1.11291	−0.0950823	−0.0475412	−	0.998869i	$-0.515139\pi$
$137$	−1.11291	−0.0950823	−0.0475412	+	0.998869i	$0.515139\pi$
$138$	0	0
$139$	1.62233	0.137604	0.0688021	−	0.997630i	$-0.478082\pi$
$139$	1.62233	0.137604	0.0688021	+	0.997630i	$0.478082\pi$
$140$	0	0
$141$	8.48435	0.714511
$142$	0	0
$143$	1.69807	0.142000
$144$	0	0
$145$	0.373811	0.0310433
$146$	0	0
$147$	6.99180	0.576674
$148$	0	0
$149$	−13.1540	−1.07762	−0.538808	−	0.842429i	$-0.681125\pi$
$149$	−13.1540	−1.07762	−0.538808	+	0.842429i	$0.681125\pi$
$150$	0	0
$151$	1.00000	0.0813788
$151$	1.00000	0.0813788
$152$	0	0
$153$	−2.46513	−0.199294
$154$	0	0
$155$	4.20135	0.337460
$156$	0	0
$157$	15.9516	1.27307	0.636537	−	0.771246i	$-0.280366\pi$
$157$	15.9516	1.27307	0.636537	+	0.771246i	$0.280366\pi$
$158$	0	0
$159$	14.9405	1.18486
$160$	0	0
$161$	−2.28894	−0.180394
$162$	0	0
$163$	1.89656	0.148550	0.0742751	−	0.997238i	$-0.476336\pi$
$163$	1.89656	0.148550	0.0742751	+	0.997238i	$0.476336\pi$
$164$	0	0
$165$	−2.49892	−0.194541
$166$	0	0
$167$	−14.0176	−1.08472	−0.542358	−	0.840148i	$-0.682468\pi$
$167$	−14.0176	−1.08472	−0.542358	+	0.840148i	$0.682468\pi$
$168$	0	0
$169$	−11.3400	−0.872309
$170$	0	0
$171$	−1.10495	−0.0844975
$172$	0	0
$173$	−12.5531	−0.954398	−0.477199	−	0.878795i	$-0.658348\pi$
$173$	−12.5531	−0.954398	−0.477199	+	0.878795i	$0.658348\pi$
$174$	0	0
$175$	−1.82001	−0.137579
$176$	0	0
$177$	27.2501	2.04824
$178$	0	0
$179$	5.33413	0.398691	0.199346	−	0.979929i	$-0.436118\pi$
$179$	5.33413	0.398691	0.199346	+	0.979929i	$0.436118\pi$
$180$	0	0
$181$	−22.3878	−1.66407	−0.832037	−	0.554721i	$-0.812825\pi$
$181$	−22.3878	−1.66407	−0.832037	+	0.554721i	$0.812825\pi$
$182$	0	0
$183$	27.0283	1.99799
$184$	0	0
$185$	2.64864	0.194732
$186$	0	0
$187$	5.46068	0.399325
$188$	0	0
$189$	−8.29928	−0.603684
$190$	0	0
$191$	−13.5961	−0.983776	−0.491888	−	0.870659i	$-0.663693\pi$
$191$	−13.5961	−0.983776	−0.491888	+	0.870659i	$0.663693\pi$
$192$	0	0
$193$	−12.8446	−0.924577	−0.462289	−	0.886729i	$-0.652972\pi$
$193$	−12.8446	−0.924577	−0.462289	+	0.886729i	$0.652972\pi$
$194$	0	0
$195$	−2.44287	−0.174937
$196$	0	0
$197$	17.7278	1.26306	0.631528	−	0.775353i	$-0.282428\pi$
$197$	17.7278	1.26306	0.631528	+	0.775353i	$0.282428\pi$
$198$	0	0
$199$	15.3894	1.09093	0.545463	−	0.838135i	$-0.316354\pi$
$199$	15.3894	1.09093	0.545463	+	0.838135i	$0.316354\pi$
$200$	0	0
$201$	−5.08717	−0.358821
$202$	0	0
$203$	0.680338	0.0477504
$204$	0	0
$205$	10.4606	0.730600
$206$	0	0
$207$	0.748274	0.0520086
$208$	0	0
$209$	2.44765	0.169307
$210$	0	0
$211$	6.20901	0.427446	0.213723	−	0.976894i	$-0.431441\pi$
$211$	6.20901	0.427446	0.213723	+	0.976894i	$0.431441\pi$
$212$	0	0
$213$	−12.0292	−0.824227
$214$	0	0
$215$	0.450079	0.0306951
$216$	0	0
$217$	7.64648	0.519077
$218$	0	0
$219$	−0.717851	−0.0485079
$220$	0	0
$221$	5.33819	0.359086
$222$	0	0
$223$	15.0585	1.00839	0.504196	−	0.863589i	$-0.331789\pi$
$223$	15.0585	1.00839	0.504196	+	0.863589i	$0.331789\pi$
$224$	0	0
$225$	0.594974	0.0396649
$226$	0	0
$227$	2.73213	0.181338	0.0906690	−	0.995881i	$-0.471099\pi$
$227$	2.73213	0.181338	0.0906690	+	0.995881i	$0.471099\pi$
$228$	0	0
$229$	−7.09193	−0.468648	−0.234324	−	0.972159i	$-0.575288\pi$
$229$	−7.09193	−0.468648	−0.234324	+	0.972159i	$0.575288\pi$
$230$	0	0
$231$	−4.54805	−0.299240
$232$	0	0
$233$	5.42647	0.355500	0.177750	−	0.984076i	$-0.443118\pi$
$233$	5.42647	0.355500	0.177750	+	0.984076i	$0.443118\pi$
$234$	0	0
$235$	4.47477	0.291902
$236$	0	0
$237$	−12.7281	−0.826782
$238$	0	0
$239$	−3.30402	−0.213719	−0.106860	−	0.994274i	$-0.534080\pi$
$239$	−3.30402	−0.213719	−0.106860	+	0.994274i	$0.534080\pi$
$240$	0	0
$241$	5.33991	0.343974	0.171987	−	0.985099i	$-0.444981\pi$
$241$	5.33991	0.343974	0.171987	+	0.985099i	$0.444981\pi$
$242$	0	0
$243$	6.09739	0.391147
$244$	0	0
$245$	3.68758	0.235591
$246$	0	0
$247$	2.39274	0.152246
$248$	0	0
$249$	−11.1802	−0.708517
$250$	0	0
$251$	30.0048	1.89388	0.946942	−	0.321406i	$-0.104155\pi$
$251$	30.0048	1.89388	0.946942	+	0.321406i	$0.104155\pi$
$252$	0	0
$253$	−1.65755	−0.104209
$254$	0	0
$255$	−7.85580	−0.491949
$256$	0	0
$257$	2.59275	0.161731	0.0808657	−	0.996725i	$-0.474232\pi$
$257$	2.59275	0.161731	0.0808657	+	0.996725i	$0.474232\pi$
$258$	0	0
$259$	4.82054	0.299534
$260$	0	0
$261$	−0.222408	−0.0137667
$262$	0	0
$263$	−12.1827	−0.751218	−0.375609	−	0.926778i	$-0.622566\pi$
$263$	−12.1827	−0.751218	−0.375609	+	0.926778i	$0.622566\pi$
$264$	0	0
$265$	7.87985	0.484056
$266$	0	0
$267$	−11.2392	−0.687830
$268$	0	0
$269$	−8.25287	−0.503186	−0.251593	−	0.967833i	$-0.580954\pi$
$269$	−8.25287	−0.503186	−0.251593	+	0.967833i	$0.580954\pi$
$270$	0	0
$271$	25.6089	1.55563	0.777816	−	0.628493i	$-0.216328\pi$
$271$	25.6089	1.55563	0.777816	+	0.628493i	$0.216328\pi$
$272$	0	0
$273$	−4.44603	−0.269086
$274$	0	0
$275$	−1.31797	−0.0794764
$276$	0	0
$277$	21.1003	1.26780	0.633898	−	0.773417i	$-0.281454\pi$
$277$	21.1003	1.26780	0.633898	+	0.773417i	$0.281454\pi$
$278$	0	0
$279$	−2.49969	−0.149653
$280$	0	0
$281$	−7.45291	−0.444603	−0.222302	−	0.974978i	$-0.571357\pi$
$281$	−7.45291	−0.444603	−0.222302	+	0.974978i	$0.571357\pi$
$282$	0	0
$283$	11.6329	0.691506	0.345753	−	0.938326i	$-0.387624\pi$
$283$	11.6329	0.691506	0.345753	+	0.938326i	$0.387624\pi$
$284$	0	0
$285$	−3.52121	−0.208578
$286$	0	0
$287$	19.0384	1.12380
$288$	0	0
$289$	0.166623	0.00980136
$290$	0	0
$291$	14.5935	0.855485
$292$	0	0
$293$	−0.600373	−0.0350741	−0.0175371	−	0.999846i	$-0.505583\pi$
$293$	−0.600373	−0.0350741	−0.0175371	+	0.999846i	$0.505583\pi$
$294$	0	0
$295$	14.3721	0.836775
$296$	0	0
$297$	−6.00997	−0.348734
$298$	0	0
$299$	−1.62037	−0.0937084
$300$	0	0
$301$	0.819146	0.0472148
$302$	0	0
$303$	−28.1029	−1.61447
$304$	0	0
$305$	14.2551	0.816245
$306$	0	0
$307$	14.4399	0.824128	0.412064	−	0.911155i	$-0.364808\pi$
$307$	14.4399	0.824128	0.412064	+	0.911155i	$0.364808\pi$
$308$	0	0
$309$	17.8122	1.01330
$310$	0	0
$311$	−14.6949	−0.833269	−0.416635	−	0.909074i	$-0.636791\pi$
$311$	−14.6949	−0.833269	−0.416635	+	0.909074i	$0.636791\pi$
$312$	0	0
$313$	−8.17579	−0.462123	−0.231061	−	0.972939i	$-0.574220\pi$
$313$	−8.17579	−0.462123	−0.231061	+	0.972939i	$0.574220\pi$
$314$	0	0
$315$	1.08286	0.0610120
$316$	0	0
$317$	26.8763	1.50952	0.754762	−	0.655998i	$-0.227752\pi$
$317$	26.8763	1.50952	0.754762	+	0.655998i	$0.227752\pi$
$318$	0	0
$319$	0.492671	0.0275843
$320$	0	0
$321$	−32.6973	−1.82499
$322$	0	0
$323$	7.69461	0.428139
$324$	0	0
$325$	−1.28840	−0.0714677
$326$	0	0
$327$	11.2272	0.620868
$328$	0	0
$329$	8.14410	0.448999
$330$	0	0
$331$	5.96562	0.327900	0.163950	−	0.986469i	$-0.447576\pi$
$331$	5.96562	0.327900	0.163950	+	0.986469i	$0.447576\pi$
$332$	0	0
$333$	−1.57587	−0.0863572
$334$	0	0
$335$	−2.68305	−0.146591
$336$	0	0
$337$	26.9686	1.46907	0.734537	−	0.678569i	$-0.237400\pi$
$337$	26.9686	1.46907	0.734537	+	0.678569i	$0.237400\pi$
$338$	0	0
$339$	−19.5146	−1.05989
$340$	0	0
$341$	5.53724	0.299858
$342$	0	0
$343$	19.4515	1.05028
$344$	0	0
$345$	2.38457	0.128381
$346$	0	0
$347$	22.6377	1.21525	0.607627	−	0.794223i	$-0.292122\pi$
$347$	22.6377	1.21525	0.607627	+	0.794223i	$0.292122\pi$
$348$	0	0
$349$	32.1501	1.72095	0.860477	−	0.509489i	$-0.170166\pi$
$349$	32.1501	1.72095	0.860477	+	0.509489i	$0.170166\pi$
$350$	0	0
$351$	−5.87515	−0.313593
$352$	0	0
$353$	−12.3960	−0.659772	−0.329886	−	0.944021i	$-0.607010\pi$
$353$	−12.3960	−0.659772	−0.329886	+	0.944021i	$0.607010\pi$
$354$	0	0
$355$	−6.34437	−0.336724
$356$	0	0
$357$	−14.2976	−0.756709
$358$	0	0
$359$	−0.360944	−0.0190499	−0.00952495	−	0.999955i	$-0.503032\pi$
$359$	−0.360944	−0.0190499	−0.00952495	+	0.999955i	$0.503032\pi$
$360$	0	0
$361$	−15.5510	−0.818476
$362$	0	0
$363$	17.5630	0.921816
$364$	0	0
$365$	−0.378605	−0.0198171
$366$	0	0
$367$	−23.3396	−1.21832	−0.609159	−	0.793048i	$-0.708493\pi$
$367$	−23.3396	−1.21832	−0.609159	+	0.793048i	$0.708493\pi$
$368$	0	0
$369$	−6.22379	−0.323997
$370$	0	0
$371$	14.3414	0.744567
$372$	0	0
$373$	14.1195	0.731081	0.365540	−	0.930795i	$-0.380884\pi$
$373$	14.1195	0.731081	0.365540	+	0.930795i	$0.380884\pi$
$374$	0	0
$375$	1.89604	0.0979112
$376$	0	0
$377$	0.481619	0.0248047
$378$	0	0
$379$	22.0553	1.13291	0.566453	−	0.824094i	$-0.308315\pi$
$379$	22.0553	1.13291	0.566453	+	0.824094i	$0.308315\pi$
$380$	0	0
$381$	2.13874	0.109571
$382$	0	0
$383$	33.9580	1.73517	0.867586	−	0.497286i	$-0.165670\pi$
$383$	33.9580	1.73517	0.867586	+	0.497286i	$0.165670\pi$
$384$	0	0
$385$	−2.39871	−0.122249
$386$	0	0
$387$	−0.267785	−0.0136123
$388$	0	0
$389$	20.9461	1.06201	0.531005	−	0.847369i	$-0.321815\pi$
$389$	20.9461	1.06201	0.531005	+	0.847369i	$0.321815\pi$
$390$	0	0
$391$	−5.21081	−0.263522
$392$	0	0
$393$	28.7036	1.44790
$394$	0	0
$395$	−6.71301	−0.337768
$396$	0	0
$397$	−13.3377	−0.669401	−0.334700	−	0.942325i	$-0.608635\pi$
$397$	−13.3377	−0.669401	−0.334700	+	0.942325i	$0.608635\pi$
$398$	0	0
$399$	−6.40862	−0.320832
$400$	0	0
$401$	12.4091	0.619680	0.309840	−	0.950789i	$-0.399725\pi$
$401$	12.4091	0.619680	0.309840	+	0.950789i	$0.399725\pi$
$402$	0	0
$403$	5.41303	0.269642
$404$	0	0
$405$	10.4309	0.518317
$406$	0	0
$407$	3.49082	0.173034
$408$	0	0
$409$	−38.8226	−1.91965	−0.959827	−	0.280592i	$-0.909469\pi$
$409$	−38.8226	−1.91965	−0.959827	+	0.280592i	$0.909469\pi$
$410$	0	0
$411$	2.11012	0.104085
$412$	0	0
$413$	26.1573	1.28712
$414$	0	0
$415$	−5.89660	−0.289453
$416$	0	0
$417$	−3.07600	−0.150632
$418$	0	0
$419$	28.2565	1.38042	0.690211	−	0.723608i	$-0.257518\pi$
$419$	28.2565	1.38042	0.690211	+	0.723608i	$0.257518\pi$
$420$	0	0
$421$	23.4395	1.14237	0.571185	−	0.820822i	$-0.306484\pi$
$421$	23.4395	1.14237	0.571185	+	0.820822i	$0.306484\pi$
$422$	0	0
$423$	−2.66237	−0.129449
$424$	0	0
$425$	−4.14326	−0.200978
$426$	0	0
$427$	25.9444	1.25554
$428$	0	0
$429$	−3.21962	−0.155445
$430$	0	0
$431$	−18.8352	−0.907262	−0.453631	−	0.891190i	$-0.649871\pi$
$431$	−18.8352	−0.907262	−0.453631	+	0.891190i	$0.649871\pi$
$432$	0	0
$433$	−31.5219	−1.51485	−0.757424	−	0.652924i	$-0.773542\pi$
$433$	−31.5219	−1.51485	−0.757424	+	0.652924i	$0.773542\pi$
$434$	0	0
$435$	−0.708761	−0.0339825
$436$	0	0
$437$	−2.33564	−0.111729
$438$	0	0
$439$	−9.40334	−0.448797	−0.224398	−	0.974497i	$-0.572042\pi$
$439$	−9.40334	−0.448797	−0.224398	+	0.974497i	$0.572042\pi$
$440$	0	0
$441$	−2.19401	−0.104477
$442$	0	0
$443$	−36.2617	−1.72284	−0.861422	−	0.507890i	$-0.830426\pi$
$443$	−36.2617	−1.72284	−0.861422	+	0.507890i	$0.830426\pi$
$444$	0	0
$445$	−5.92773	−0.281001
$446$	0	0
$447$	24.9405	1.17964
$448$	0	0
$449$	14.1572	0.668118	0.334059	−	0.942552i	$-0.391582\pi$
$449$	14.1572	0.668118	0.334059	+	0.942552i	$0.391582\pi$
$450$	0	0
$451$	13.7867	0.649192
$452$	0	0
$453$	−1.89604	−0.0890838
$454$	0	0
$455$	−2.34490	−0.109931
$456$	0	0
$457$	0.416388	0.0194778	0.00973890	−	0.999953i	$-0.496900\pi$
$457$	0.416388	0.0194778	0.00973890	+	0.999953i	$0.496900\pi$
$458$	0	0
$459$	−18.8934	−0.881868
$460$	0	0
$461$	17.1903	0.800630	0.400315	−	0.916378i	$-0.368901\pi$
$461$	17.1903	0.800630	0.400315	+	0.916378i	$0.368901\pi$
$462$	0	0
$463$	−4.08303	−0.189754	−0.0948772	−	0.995489i	$-0.530246\pi$
$463$	−4.08303	−0.189754	−0.0948772	+	0.995489i	$0.530246\pi$
$464$	0	0
$465$	−7.96593	−0.369411
$466$	0	0
$467$	0.823421	0.0381034	0.0190517	−	0.999818i	$-0.493935\pi$
$467$	0.823421	0.0381034	0.0190517	+	0.999818i	$0.493935\pi$
$468$	0	0
$469$	−4.88316	−0.225483
$470$	0	0
$471$	−30.2449	−1.39361
$472$	0	0
$473$	0.593189	0.0272749
$474$	0	0
$475$	−1.85714	−0.0852113
$476$	0	0
$477$	−4.68831	−0.214663
$478$	0	0
$479$	−18.9329	−0.865065	−0.432532	−	0.901618i	$-0.642380\pi$
$479$	−18.9329	−0.865065	−0.432532	+	0.901618i	$0.642380\pi$
$480$	0	0
$481$	3.41251	0.155597
$482$	0	0
$483$	4.33993	0.197474
$484$	0	0
$485$	7.69681	0.349494
$486$	0	0
$487$	28.9277	1.31084	0.655420	−	0.755265i	$-0.272492\pi$
$487$	28.9277	1.31084	0.655420	+	0.755265i	$0.272492\pi$
$488$	0	0
$489$	−3.59596	−0.162615
$490$	0	0
$491$	1.93657	0.0873962	0.0436981	−	0.999045i	$-0.486086\pi$
$491$	1.93657	0.0873962	0.0436981	+	0.999045i	$0.486086\pi$
$492$	0	0
$493$	1.54880	0.0697543
$494$	0	0
$495$	0.784156	0.0352452
$496$	0	0
$497$	−11.5468	−0.517945
$498$	0	0
$499$	−14.9108	−0.667499	−0.333750	−	0.942662i	$-0.608314\pi$
$499$	−14.9108	−0.667499	−0.333750	+	0.942662i	$0.608314\pi$
$500$	0	0
$501$	26.5780	1.18742
$502$	0	0
$503$	−30.3497	−1.35322	−0.676612	−	0.736339i	$-0.736553\pi$
$503$	−30.3497	−1.35322	−0.676612	+	0.736339i	$0.736553\pi$
$504$	0	0
$505$	−14.8219	−0.659564
$506$	0	0
$507$	21.5011	0.954899
$508$	0	0
$509$	−21.6848	−0.961163	−0.480581	−	0.876950i	$-0.659574\pi$
$509$	−21.6848	−0.961163	−0.480581	+	0.876950i	$0.659574\pi$
$510$	0	0
$511$	−0.689063	−0.0304824
$512$	0	0
$513$	−8.46860	−0.373898
$514$	0	0
$515$	9.39444	0.413968
$516$	0	0
$517$	5.89760	0.259376
$518$	0	0
$519$	23.8013	1.04476
$520$	0	0
$521$	26.3502	1.15442	0.577211	−	0.816595i	$-0.304141\pi$
$521$	26.3502	1.15442	0.577211	+	0.816595i	$0.304141\pi$
$522$	0	0
$523$	−32.7480	−1.43197	−0.715986	−	0.698115i	$-0.754022\pi$
$523$	−32.7480	−1.43197	−0.715986	+	0.698115i	$0.754022\pi$
$524$	0	0
$525$	3.45081	0.150606
$526$	0	0
$527$	17.4073	0.758273
$528$	0	0
$529$	−21.4183	−0.931230
$530$	0	0
$531$	−8.55102	−0.371083
$532$	0	0
$533$	13.4775	0.583774
$534$	0	0
$535$	−17.2451	−0.745569
$536$	0	0
$537$	−10.1137	−0.436439
$538$	0	0
$539$	4.86011	0.209340
$540$	0	0
$541$	−15.0889	−0.648722	−0.324361	−	0.945933i	$-0.605149\pi$
$541$	−15.0889	−0.648722	−0.324361	+	0.945933i	$0.605149\pi$
$542$	0	0
$543$	42.4482	1.82163
$544$	0	0
$545$	5.92141	0.253645
$546$	0	0
$547$	−22.3951	−0.957544	−0.478772	−	0.877939i	$-0.658918\pi$
$547$	−22.3951	−0.957544	−0.478772	+	0.877939i	$0.658918\pi$
$548$	0	0
$549$	−8.48142	−0.361978
$550$	0	0
$551$	0.694219	0.0295747
$552$	0	0
$553$	−12.2177	−0.519550
$554$	0	0
$555$	−5.02193	−0.213169
$556$	0	0
$557$	21.1705	0.897021	0.448510	−	0.893778i	$-0.351955\pi$
$557$	21.1705	0.897021	0.448510	+	0.893778i	$0.351955\pi$
$558$	0	0
$559$	0.579883	0.0245264
$560$	0	0
$561$	−10.3537	−0.437133
$562$	0	0
$563$	27.3006	1.15058	0.575291	−	0.817949i	$-0.304889\pi$
$563$	27.3006	1.15058	0.575291	+	0.817949i	$0.304889\pi$
$564$	0	0
$565$	−10.2923	−0.433000
$566$	0	0
$567$	18.9843	0.797268
$568$	0	0
$569$	36.9446	1.54880	0.774398	−	0.632698i	$-0.218053\pi$
$569$	36.9446	1.54880	0.774398	+	0.632698i	$0.218053\pi$
$570$	0	0
$571$	−30.6451	−1.28246	−0.641230	−	0.767349i	$-0.721575\pi$
$571$	−30.6451	−1.28246	−0.641230	+	0.767349i	$0.721575\pi$
$572$	0	0
$573$	25.7787	1.07692
$574$	0	0
$575$	1.25766	0.0524480
$576$	0	0
$577$	−34.2688	−1.42663	−0.713315	−	0.700844i	$-0.752807\pi$
$577$	−34.2688	−1.42663	−0.713315	+	0.700844i	$0.752807\pi$
$578$	0	0
$579$	24.3540	1.01212
$580$	0	0
$581$	−10.7318	−0.445232
$582$	0	0
$583$	10.3854	0.430119
$584$	0	0
$585$	0.766566	0.0316936
$586$	0	0
$587$	41.4186	1.70953	0.854764	−	0.519016i	$-0.173702\pi$
$587$	41.4186	1.70953	0.854764	+	0.519016i	$0.173702\pi$
$588$	0	0
$589$	7.80248	0.321496
$590$	0	0
$591$	−33.6127	−1.38264
$592$	0	0
$593$	−13.3827	−0.549560	−0.274780	−	0.961507i	$-0.588605\pi$
$593$	−13.3827	−0.549560	−0.274780	+	0.961507i	$0.588605\pi$
$594$	0	0
$595$	−7.54076	−0.309141
$596$	0	0
$597$	−29.1789	−1.19421
$598$	0	0
$599$	39.0965	1.59744	0.798721	−	0.601702i	$-0.205511\pi$
$599$	39.0965	1.59744	0.798721	+	0.601702i	$0.205511\pi$
$600$	0	0
$601$	−15.9008	−0.648609	−0.324304	−	0.945953i	$-0.605130\pi$
$601$	−15.9008	−0.648609	−0.324304	+	0.945953i	$0.605130\pi$
$602$	0	0
$603$	1.59634	0.0650081
$604$	0	0
$605$	9.26296	0.376593
$606$	0	0
$607$	24.1169	0.978873	0.489437	−	0.872039i	$-0.337202\pi$
$607$	24.1169	0.978873	0.489437	+	0.872039i	$0.337202\pi$
$608$	0	0
$609$	−1.28995	−0.0522714
$610$	0	0
$611$	5.76530	0.233239
$612$	0	0
$613$	−17.9784	−0.726141	−0.363070	−	0.931762i	$-0.618272\pi$
$613$	−17.9784	−0.726141	−0.363070	+	0.931762i	$0.618272\pi$
$614$	0	0
$615$	−19.8337	−0.799773
$616$	0	0
$617$	−0.994497	−0.0400369	−0.0200185	−	0.999800i	$-0.506373\pi$
$617$	−0.994497	−0.0400369	−0.0200185	+	0.999800i	$0.506373\pi$
$618$	0	0
$619$	−23.4575	−0.942836	−0.471418	−	0.881910i	$-0.656258\pi$
$619$	−23.4575	−0.942836	−0.471418	+	0.881910i	$0.656258\pi$
$620$	0	0
$621$	5.73496	0.230136
$622$	0	0
$623$	−10.7885	−0.432232
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−4.64084	−0.185337
$628$	0	0
$629$	10.9740	0.437562
$630$	0	0
$631$	4.26028	0.169599	0.0847995	−	0.996398i	$-0.472975\pi$
$631$	4.26028	0.169599	0.0847995	+	0.996398i	$0.472975\pi$
$632$	0	0
$633$	−11.7725	−0.467916
$634$	0	0
$635$	1.12800	0.0447635
$636$	0	0
$637$	4.75109	0.188245
$638$	0	0
$639$	3.77474	0.149326
$640$	0	0
$641$	−26.4778	−1.04581	−0.522905	−	0.852391i	$-0.675152\pi$
$641$	−26.4778	−1.04581	−0.522905	+	0.852391i	$0.675152\pi$
$642$	0	0
$643$	26.9731	1.06371	0.531857	−	0.846834i	$-0.321494\pi$
$643$	26.9731	1.06371	0.531857	+	0.846834i	$0.321494\pi$
$644$	0	0
$645$	−0.853368	−0.0336013
$646$	0	0
$647$	26.4005	1.03791	0.518954	−	0.854802i	$-0.326321\pi$
$647$	26.4005	1.03791	0.518954	+	0.854802i	$0.326321\pi$
$648$	0	0
$649$	18.9419	0.743536
$650$	0	0
$651$	−14.4980	−0.568223
$652$	0	0
$653$	−42.2120	−1.65188	−0.825942	−	0.563755i	$-0.809356\pi$
$653$	−42.2120	−1.65188	−0.825942	+	0.563755i	$0.809356\pi$
$654$	0	0
$655$	15.1387	0.591517
$656$	0	0
$657$	0.225260	0.00878824
$658$	0	0
$659$	−26.6236	−1.03711	−0.518554	−	0.855045i	$-0.673530\pi$
$659$	−26.6236	−1.03711	−0.518554	+	0.855045i	$0.673530\pi$
$660$	0	0
$661$	14.0911	0.548080	0.274040	−	0.961718i	$-0.411640\pi$
$661$	14.0911	0.548080	0.274040	+	0.961718i	$0.411640\pi$
$662$	0	0
$663$	−10.1214	−0.393084
$664$	0	0
$665$	−3.38000	−0.131071
$666$	0	0
$667$	−0.470126	−0.0182034
$668$	0	0
$669$	−28.5515	−1.10387
$670$	0	0
$671$	18.7878	0.725294
$672$	0	0
$673$	−25.6164	−0.987438	−0.493719	−	0.869621i	$-0.664363\pi$
$673$	−25.6164	−0.987438	−0.493719	+	0.869621i	$0.664363\pi$
$674$	0	0
$675$	4.56003	0.175516
$676$	0	0
$677$	−2.90479	−0.111640	−0.0558201	−	0.998441i	$-0.517777\pi$
$677$	−2.90479	−0.111640	−0.0558201	+	0.998441i	$0.517777\pi$
$678$	0	0
$679$	14.0082	0.537587
$680$	0	0
$681$	−5.18023	−0.198507
$682$	0	0
$683$	−2.70316	−0.103434	−0.0517169	−	0.998662i	$-0.516469\pi$
$683$	−2.70316	−0.103434	−0.0517169	+	0.998662i	$0.516469\pi$
$684$	0	0
$685$	1.11291	0.0425221
$686$	0	0
$687$	13.4466	0.513020
$688$	0	0
$689$	10.1524	0.386777
$690$	0	0
$691$	43.9734	1.67283	0.836414	−	0.548099i	$-0.184648\pi$
$691$	43.9734	1.67283	0.836414	+	0.548099i	$0.184648\pi$
$692$	0	0
$693$	1.42717	0.0542137
$694$	0	0
$695$	−1.62233	−0.0615384
$696$	0	0
$697$	43.3410	1.64166
$698$	0	0
$699$	−10.2888	−0.389159
$700$	0	0
$701$	−46.9420	−1.77297	−0.886487	−	0.462753i	$-0.846862\pi$
$701$	−46.9420	−1.77297	−0.886487	+	0.462753i	$0.846862\pi$
$702$	0	0
$703$	4.91889	0.185519
$704$	0	0
$705$	−8.48435	−0.319539
$706$	0	0
$707$	−26.9759	−1.01453
$708$	0	0
$709$	41.9997	1.57733	0.788666	−	0.614822i	$-0.210772\pi$
$709$	41.9997	1.57733	0.788666	+	0.614822i	$0.210772\pi$
$710$	0	0
$711$	3.99406	0.149789
$712$	0	0
$713$	−5.28386	−0.197882
$714$	0	0
$715$	−1.69807	−0.0635043
$716$	0	0
$717$	6.26457	0.233954
$718$	0	0
$719$	−29.7185	−1.10831	−0.554156	−	0.832413i	$-0.686959\pi$
$719$	−29.7185	−1.10831	−0.554156	+	0.832413i	$0.686959\pi$
$720$	0	0
$721$	17.0979	0.636760
$722$	0	0
$723$	−10.1247	−0.376541
$724$	0	0
$725$	−0.373811	−0.0138830
$726$	0	0
$727$	−16.6069	−0.615917	−0.307958	−	0.951400i	$-0.599646\pi$
$727$	−16.6069	−0.615917	−0.307958	+	0.951400i	$0.599646\pi$
$728$	0	0
$729$	19.7319	0.730811
$730$	0	0
$731$	1.86479	0.0689719
$732$	0	0
$733$	4.87916	0.180216	0.0901079	−	0.995932i	$-0.471279\pi$
$733$	4.87916	0.180216	0.0901079	+	0.995932i	$0.471279\pi$
$734$	0	0
$735$	−6.99180	−0.257897
$736$	0	0
$737$	−3.53617	−0.130256
$738$	0	0
$739$	4.82198	0.177380	0.0886898	−	0.996059i	$-0.471732\pi$
$739$	4.82198	0.177380	0.0886898	+	0.996059i	$0.471732\pi$
$740$	0	0
$741$	−4.53674	−0.166661
$742$	0	0
$743$	−8.62996	−0.316603	−0.158301	−	0.987391i	$-0.550602\pi$
$743$	−8.62996	−0.316603	−0.158301	+	0.987391i	$0.550602\pi$
$744$	0	0
$745$	13.1540	0.481924
$746$	0	0
$747$	3.50832	0.128363
$748$	0	0
$749$	−31.3861	−1.14682
$750$	0	0
$751$	−32.1642	−1.17369	−0.586844	−	0.809700i	$-0.699630\pi$
$751$	−32.1642	−1.17369	−0.586844	+	0.809700i	$0.699630\pi$
$752$	0	0
$753$	−56.8903	−2.07320
$754$	0	0
$755$	−1.00000	−0.0363937
$756$	0	0
$757$	40.0557	1.45585	0.727924	−	0.685658i	$-0.240485\pi$
$757$	40.0557	1.45585	0.727924	+	0.685658i	$0.240485\pi$
$758$	0	0
$759$	3.14279	0.114076
$760$	0	0
$761$	0.882057	0.0319746	0.0159873	−	0.999872i	$-0.494911\pi$
$761$	0.882057	0.0319746	0.0159873	+	0.999872i	$0.494911\pi$
$762$	0	0
$763$	10.7770	0.390154
$764$	0	0
$765$	2.46513	0.0891271
$766$	0	0
$767$	18.5170	0.668612
$768$	0	0
$769$	−35.5296	−1.28123	−0.640614	−	0.767863i	$-0.721320\pi$
$769$	−35.5296	−1.28123	−0.640614	+	0.767863i	$0.721320\pi$
$770$	0	0
$771$	−4.91596	−0.177044
$772$	0	0
$773$	−14.8840	−0.535339	−0.267669	−	0.963511i	$-0.586254\pi$
$773$	−14.8840	−0.535339	−0.267669	+	0.963511i	$0.586254\pi$
$774$	0	0
$775$	−4.20135	−0.150917
$776$	0	0
$777$	−9.13994	−0.327894
$778$	0	0
$779$	19.4268	0.696036
$780$	0	0
$781$	−8.36168	−0.299204
$782$	0	0
$783$	−1.70459	−0.0609171
$784$	0	0
$785$	−15.9516	−0.569336
$786$	0	0
$787$	−50.0097	−1.78265	−0.891327	−	0.453360i	$-0.850225\pi$
$787$	−50.0097	−1.78265	−0.891327	+	0.453360i	$0.850225\pi$
$788$	0	0
$789$	23.0989	0.822343
$790$	0	0
$791$	−18.7320	−0.666035
$792$	0	0
$793$	18.3663	0.652207
$794$	0	0
$795$	−14.9405	−0.529886
$796$	0	0
$797$	50.1469	1.77629	0.888147	−	0.459560i	$-0.151993\pi$
$797$	50.1469	1.77629	0.888147	+	0.459560i	$0.151993\pi$
$798$	0	0
$799$	18.5401	0.655903
$800$	0	0
$801$	3.52685	0.124615
$802$	0	0
$803$	−0.498989	−0.0176089
$804$	0	0
$805$	2.28894	0.0806747
$806$	0	0
$807$	15.6478	0.550828
$808$	0	0
$809$	48.5670	1.70753	0.853763	−	0.520662i	$-0.174315\pi$
$809$	48.5670	1.70753	0.853763	+	0.520662i	$0.174315\pi$
$810$	0	0
$811$	−19.3858	−0.680727	−0.340364	−	0.940294i	$-0.610550\pi$
$811$	−19.3858	−0.680727	−0.340364	+	0.940294i	$0.610550\pi$
$812$	0	0
$813$	−48.5556	−1.70292
$814$	0	0
$815$	−1.89656	−0.0664336
$816$	0	0
$817$	0.835858	0.0292430
$818$	0	0
$819$	1.39515	0.0487506
$820$	0	0
$821$	−30.8202	−1.07563	−0.537817	−	0.843062i	$-0.680751\pi$
$821$	−30.8202	−1.07563	−0.537817	+	0.843062i	$0.680751\pi$
$822$	0	0
$823$	−6.15651	−0.214602	−0.107301	−	0.994227i	$-0.534221\pi$
$823$	−6.15651	−0.214602	−0.107301	+	0.994227i	$0.534221\pi$
$824$	0	0
$825$	2.49892	0.0870013
$826$	0	0
$827$	13.0315	0.453151	0.226575	−	0.973994i	$-0.427247\pi$
$827$	13.0315	0.453151	0.226575	+	0.973994i	$0.427247\pi$
$828$	0	0
$829$	13.0808	0.454315	0.227158	−	0.973858i	$-0.427057\pi$
$829$	13.0808	0.454315	0.227158	+	0.973858i	$0.427057\pi$
$830$	0	0
$831$	−40.0071	−1.38783
$832$	0	0
$833$	15.2786	0.529372
$834$	0	0
$835$	14.0176	0.485100
$836$	0	0
$837$	−19.1583	−0.662207
$838$	0	0
$839$	25.9371	0.895447	0.447723	−	0.894172i	$-0.352235\pi$
$839$	25.9371	0.895447	0.447723	+	0.894172i	$0.352235\pi$
$840$	0	0
$841$	−28.8603	−0.995182
$842$	0	0
$843$	14.1310	0.486698
$844$	0	0
$845$	11.3400	0.390108
$846$	0	0
$847$	16.8586	0.579270
$848$	0	0
$849$	−22.0565	−0.756978
$850$	0	0
$851$	−3.33108	−0.114188
$852$	0	0
$853$	−7.92599	−0.271381	−0.135690	−	0.990751i	$-0.543325\pi$
$853$	−7.92599	−0.271381	−0.135690	+	0.990751i	$0.543325\pi$
$854$	0	0
$855$	1.10495	0.0377884
$856$	0	0
$857$	48.6456	1.66170	0.830851	−	0.556495i	$-0.187854\pi$
$857$	48.6456	1.66170	0.830851	+	0.556495i	$0.187854\pi$
$858$	0	0
$859$	−25.9639	−0.885877	−0.442938	−	0.896552i	$-0.646064\pi$
$859$	−25.9639	−0.885877	−0.442938	+	0.896552i	$0.646064\pi$
$860$	0	0
$861$	−36.0975	−1.23020
$862$	0	0
$863$	−4.51134	−0.153568	−0.0767839	−	0.997048i	$-0.524465\pi$
$863$	−4.51134	−0.153568	−0.0767839	+	0.997048i	$0.524465\pi$
$864$	0	0
$865$	12.5531	0.426820
$866$	0	0
$867$	−0.315924	−0.0107294
$868$	0	0
$869$	−8.84752	−0.300132
$870$	0	0
$871$	−3.45685	−0.117131
$872$	0	0
$873$	−4.57940	−0.154989
$874$	0	0
$875$	1.82001	0.0615274
$876$	0	0
$877$	7.98162	0.269520	0.134760	−	0.990878i	$-0.456974\pi$
$877$	7.98162	0.269520	0.134760	+	0.990878i	$0.456974\pi$
$878$	0	0
$879$	1.13833	0.0383950
$880$	0	0
$881$	−52.4173	−1.76598	−0.882992	−	0.469388i	$-0.844475\pi$
$881$	−52.4173	−1.76598	−0.882992	+	0.469388i	$0.844475\pi$
$882$	0	0
$883$	28.3351	0.953552	0.476776	−	0.879025i	$-0.341805\pi$
$883$	28.3351	0.953552	0.476776	+	0.879025i	$0.341805\pi$
$884$	0	0
$885$	−27.2501	−0.916002
$886$	0	0
$887$	8.74703	0.293697	0.146848	−	0.989159i	$-0.453087\pi$
$887$	8.74703	0.293697	0.146848	+	0.989159i	$0.453087\pi$
$888$	0	0
$889$	2.05298	0.0688546
$890$	0	0
$891$	13.7476	0.460563
$892$	0	0
$893$	8.31026	0.278092
$894$	0	0
$895$	−5.33413	−0.178300
$896$	0	0
$897$	3.07229	0.102581
$898$	0	0
$899$	1.57051	0.0523795
$900$	0	0
$901$	32.6483	1.08767
$902$	0	0
$903$	−1.55313	−0.0516851
$904$	0	0
$905$	22.3878	0.744196
$906$	0	0
$907$	20.8731	0.693079	0.346539	−	0.938035i	$-0.387357\pi$
$907$	20.8731	0.693079	0.346539	+	0.938035i	$0.387357\pi$
$908$	0	0
$909$	8.81862	0.292495
$910$	0	0
$911$	8.51944	0.282262	0.141131	−	0.989991i	$-0.454926\pi$
$911$	8.51944	0.282262	0.141131	+	0.989991i	$0.454926\pi$
$912$	0	0
$913$	−7.77153	−0.257200
$914$	0	0
$915$	−27.0283	−0.893528
$916$	0	0
$917$	27.5525	0.909863
$918$	0	0
$919$	−15.0849	−0.497604	−0.248802	−	0.968554i	$-0.580037\pi$
$919$	−15.0849	−0.497604	−0.248802	+	0.968554i	$0.580037\pi$
$920$	0	0
$921$	−27.3786	−0.902157
$922$	0	0
$923$	−8.17411	−0.269054
$924$	0	0
$925$	−2.64864	−0.0870867
$926$	0	0
$927$	−5.58945	−0.183582
$928$	0	0
$929$	5.63160	0.184767	0.0923833	−	0.995724i	$-0.470551\pi$
$929$	5.63160	0.184767	0.0923833	+	0.995724i	$0.470551\pi$
$930$	0	0
$931$	6.84834	0.224445
$932$	0	0
$933$	27.8621	0.912163
$934$	0	0
$935$	−5.46068	−0.178583
$936$	0	0
$937$	32.5649	1.06385	0.531925	−	0.846791i	$-0.321469\pi$
$937$	32.5649	1.06385	0.531925	+	0.846791i	$0.321469\pi$
$938$	0	0
$939$	15.5016	0.505877
$940$	0	0
$941$	−24.7010	−0.805230	−0.402615	−	0.915369i	$-0.631899\pi$
$941$	−24.7010	−0.805230	−0.402615	+	0.915369i	$0.631899\pi$
$942$	0	0
$943$	−13.1559	−0.428414
$944$	0	0
$945$	8.29928	0.269976
$946$	0	0
$947$	−5.14060	−0.167047	−0.0835236	−	0.996506i	$-0.526617\pi$
$947$	−5.14060	−0.167047	−0.0835236	+	0.996506i	$0.526617\pi$
$948$	0	0
$949$	−0.487796	−0.0158345
$950$	0	0
$951$	−50.9586	−1.65245
$952$	0	0
$953$	−45.5080	−1.47415	−0.737074	−	0.675811i	$-0.763793\pi$
$953$	−45.5080	−1.47415	−0.737074	+	0.675811i	$0.763793\pi$
$954$	0	0
$955$	13.5961	0.439958
$956$	0	0
$957$	−0.934124	−0.0301960
$958$	0	0
$959$	2.02550	0.0654069
$960$	0	0
$961$	−13.3487	−0.430602
$962$	0	0
$963$	10.2604	0.330635
$964$	0	0
$965$	12.8446	0.413484
$966$	0	0
$967$	8.08001	0.259836	0.129918	−	0.991525i	$-0.458529\pi$
$967$	8.08001	0.259836	0.129918	+	0.991525i	$0.458529\pi$
$968$	0	0
$969$	−14.5893	−0.468676
$970$	0	0
$971$	−9.72968	−0.312240	−0.156120	−	0.987738i	$-0.549899\pi$
$971$	−9.72968	−0.312240	−0.156120	+	0.987738i	$0.549899\pi$
$972$	0	0
$973$	−2.95265	−0.0946575
$974$	0	0
$975$	2.44287	0.0782343
$976$	0	0
$977$	−47.0914	−1.50659	−0.753293	−	0.657685i	$-0.771536\pi$
$977$	−47.0914	−1.50659	−0.753293	+	0.657685i	$0.771536\pi$
$978$	0	0
$979$	−7.81256	−0.249690
$980$	0	0
$981$	−3.52309	−0.112484
$982$	0	0
$983$	−14.8390	−0.473292	−0.236646	−	0.971596i	$-0.576048\pi$
$983$	−14.8390	−0.473292	−0.236646	+	0.971596i	$0.576048\pi$
$984$	0	0
$985$	−17.7278	−0.564855
$986$	0	0
$987$	−15.4416	−0.491510
$988$	0	0
$989$	−0.566045	−0.0179992
$990$	0	0
$991$	17.3990	0.552698	0.276349	−	0.961057i	$-0.410875\pi$
$991$	17.3990	0.552698	0.276349	+	0.961057i	$0.410875\pi$
$992$	0	0
$993$	−11.3111	−0.358946
$994$	0	0
$995$	−15.3894	−0.487877
$996$	0	0
$997$	−34.7060	−1.09915	−0.549575	−	0.835444i	$-0.685210\pi$
$997$	−34.7060	−1.09915	−0.549575	+	0.835444i	$0.685210\pi$
$998$	0	0
$999$	−12.0779	−0.382127

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6040.2.a.m.1.3	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6040.2.a.m.1.3	✓	12	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:	\( N \)	\(=\)	\( 6040 = 2^{3} \cdot 5 \cdot 151 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6040.a (trivial)

\(f(q)\)	\(=\)	\(q-1.89604 q^{3} -1.00000 q^{5} -1.82001 q^{7} +0.594974 q^{9} +O(q^{10})\)	\(q-1.89604 q^{3} -1.00000 q^{5} -1.82001 q^{7} +0.594974 q^{9} -1.31797 q^{11} -1.28840 q^{13} +1.89604 q^{15} -4.14326 q^{17} -1.85714 q^{19} +3.45081 q^{21} +1.25766 q^{23} +1.00000 q^{25} +4.56003 q^{27} -0.373811 q^{29} -4.20135 q^{31} +2.49892 q^{33} +1.82001 q^{35} -2.64864 q^{37} +2.44287 q^{39} -10.4606 q^{41} -0.450079 q^{43} -0.594974 q^{45} -4.47477 q^{47} -3.68758 q^{49} +7.85580 q^{51} -7.87985 q^{53} +1.31797 q^{55} +3.52121 q^{57} -14.3721 q^{59} -14.2551 q^{61} -1.08286 q^{63} +1.28840 q^{65} +2.68305 q^{67} -2.38457 q^{69} +6.34437 q^{71} +0.378605 q^{73} -1.89604 q^{75} +2.39871 q^{77} +6.71301 q^{79} -10.4309 q^{81} +5.89660 q^{83} +4.14326 q^{85} +0.708761 q^{87} +5.92773 q^{89} +2.34490 q^{91} +7.96593 q^{93} +1.85714 q^{95} -7.69681 q^{97} -0.784156 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 3 q^{3} - 12 q^{5} + 5 q^{7} + 9 q^{9}+O(q^{10}) \) 12 * q + 3 * q^3 - 12 * q^5 + 5 * q^7 + 9 * q^9	\( 12 q + 3 q^{3} - 12 q^{5} + 5 q^{7} + 9 q^{9} + 10 q^{11} + 11 q^{13} - 3 q^{15} - 4 q^{17} + 5 q^{19} - q^{21} + 18 q^{23} + 12 q^{25} + 9 q^{27} + 16 q^{29} - q^{31} + 8 q^{33} - 5 q^{35} + 2 q^{37} + 6 q^{39} + 4 q^{41} + 7 q^{43} - 9 q^{45} + 3 q^{49} - 4 q^{51} + 39 q^{53} - 10 q^{55} - 15 q^{57} - 4 q^{59} - 32 q^{61} + 3 q^{63} - 11 q^{65} + 4 q^{67} + 12 q^{69} + 24 q^{71} - 10 q^{73} + 3 q^{75} + 38 q^{77} + 32 q^{79} - 8 q^{81} + 9 q^{83} + 4 q^{85} + 3 q^{87} + 15 q^{89} + 18 q^{91} + 36 q^{93} - 5 q^{95} + 15 q^{97} + 28 q^{99}+O(q^{100}) \) 12 * q + 3 * q^3 - 12 * q^5 + 5 * q^7 + 9 * q^9 + 10 * q^11 + 11 * q^13 - 3 * q^15 - 4 * q^17 + 5 * q^19 - q^21 + 18 * q^23 + 12 * q^25 + 9 * q^27 + 16 * q^29 - q^31 + 8 * q^33 - 5 * q^35 + 2 * q^37 + 6 * q^39 + 4 * q^41 + 7 * q^43 - 9 * q^45 + 3 * q^49 - 4 * q^51 + 39 * q^53 - 10 * q^55 - 15 * q^57 - 4 * q^59 - 32 * q^61 + 3 * q^63 - 11 * q^65 + 4 * q^67 + 12 * q^69 + 24 * q^71 - 10 * q^73 + 3 * q^75 + 38 * q^77 + 32 * q^79 - 8 * q^81 + 9 * q^83 + 4 * q^85 + 3 * q^87 + 15 * q^89 + 18 * q^91 + 36 * q^93 - 5 * q^95 + 15 * q^97 + 28 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more