LMFDB - Embedded newform 6024.2.a.q.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6024,2,Mod(1,6024)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6024, 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("6024.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6024 = 2^{3} \cdot 3 \cdot 251 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6024.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:	$48.1018821776$
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} - x^{17} - 57 x^{16} + 51 x^{15} + 1328 x^{14} - 1116 x^{13} - 16275 x^{12} + 13699 x^{11} + \cdots + 44032 $ x^18 - x^17 - 57x^16 + 51x^15 + 1328x^14 - 1116x^13 - 16275x^12 + 13699x^11 + 112394x^10 - 101250x^9 - 432956x^8 + 439806x^7 + 844117x^6 - 1010785x^5 - 593552x^4 + 965980x^3 - 88040x^2 - 184992x + 44032
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-3.36568$ of defining polynomial
Character	$\chi$	$=$	6024.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} -3.36568 q^{5} -2.47862 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -3.36568 q^{5} -2.47862 q^{7} +1.00000 q^{9} -6.62075 q^{11} -1.10011 q^{13} -3.36568 q^{15} -4.07636 q^{17} -0.635651 q^{19} -2.47862 q^{21} -8.21898 q^{23} +6.32781 q^{25} +1.00000 q^{27} -3.47455 q^{29} +8.19684 q^{31} -6.62075 q^{33} +8.34225 q^{35} +4.16843 q^{37} -1.10011 q^{39} -7.94459 q^{41} +3.69096 q^{43} -3.36568 q^{45} -9.19615 q^{47} -0.856428 q^{49} -4.07636 q^{51} -9.69155 q^{53} +22.2833 q^{55} -0.635651 q^{57} +2.89156 q^{59} -1.30308 q^{61} -2.47862 q^{63} +3.70263 q^{65} +0.588636 q^{67} -8.21898 q^{69} +11.7563 q^{71} -10.5132 q^{73} +6.32781 q^{75} +16.4103 q^{77} +2.94961 q^{79} +1.00000 q^{81} +5.25007 q^{83} +13.7197 q^{85} -3.47455 q^{87} +8.28016 q^{89} +2.72676 q^{91} +8.19684 q^{93} +2.13940 q^{95} -4.92699 q^{97} -6.62075 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 18 q + 18 q^{3} + q^{5} + 7 q^{7} + 18 q^{9}+O(q^{10}) $ 18 * q + 18 * q^3 + q^5 + 7 * q^7 + 18 * q^9	$ 18 q + 18 q^{3} + q^{5} + 7 q^{7} + 18 q^{9} + 8 q^{11} + 2 q^{13} + q^{15} + 2 q^{17} + 21 q^{19} + 7 q^{21} - 2 q^{23} + 25 q^{25} + 18 q^{27} + 5 q^{29} + 23 q^{31} + 8 q^{33} + 17 q^{35} + 15 q^{37} + 2 q^{39} + 20 q^{41} + 37 q^{43} + q^{45} - q^{47} + 33 q^{49} + 2 q^{51} + 2 q^{53} + 26 q^{55} + 21 q^{57} + 24 q^{59} + 20 q^{61} + 7 q^{63} + 26 q^{65} + 49 q^{67} - 2 q^{69} + 15 q^{71} + 15 q^{73} + 25 q^{75} + 6 q^{77} + 23 q^{79} + 18 q^{81} + 38 q^{83} + 21 q^{85} + 5 q^{87} + 31 q^{89} + 48 q^{91} + 23 q^{93} + 13 q^{95} + 25 q^{97} + 8 q^{99}+O(q^{100}) $ 18 * q + 18 * q^3 + q^5 + 7 * q^7 + 18 * q^9 + 8 * q^11 + 2 * q^13 + q^15 + 2 * q^17 + 21 * q^19 + 7 * q^21 - 2 * q^23 + 25 * q^25 + 18 * q^27 + 5 * q^29 + 23 * q^31 + 8 * q^33 + 17 * q^35 + 15 * q^37 + 2 * q^39 + 20 * q^41 + 37 * q^43 + q^45 - q^47 + 33 * q^49 + 2 * q^51 + 2 * q^53 + 26 * q^55 + 21 * q^57 + 24 * q^59 + 20 * q^61 + 7 * q^63 + 26 * q^65 + 49 * q^67 - 2 * q^69 + 15 * q^71 + 15 * q^73 + 25 * q^75 + 6 * q^77 + 23 * q^79 + 18 * q^81 + 38 * q^83 + 21 * q^85 + 5 * q^87 + 31 * q^89 + 48 * q^91 + 23 * q^93 + 13 * q^95 + 25 * q^97 + 8 * 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
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	−3.36568	−1.50518	−0.752589	−	0.658490i	$-0.771195\pi$
$5$	−3.36568	−1.50518	−0.752589	+	0.658490i	$0.771195\pi$
$6$	0	0
$7$	−2.47862	−0.936831	−0.468416	−	0.883508i	$-0.655175\pi$
$7$	−2.47862	−0.936831	−0.468416	+	0.883508i	$0.655175\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−6.62075	−1.99623	−0.998115	−	0.0613641i	$-0.980455\pi$
$11$	−6.62075	−1.99623	−0.998115	+	0.0613641i	$0.980455\pi$
$12$	0	0
$13$	−1.10011	−0.305116	−0.152558	−	0.988295i	$-0.548751\pi$
$13$	−1.10011	−0.305116	−0.152558	+	0.988295i	$0.548751\pi$
$14$	0	0
$15$	−3.36568	−0.869015
$16$	0	0
$17$	−4.07636	−0.988661	−0.494331	−	0.869274i	$-0.664587\pi$
$17$	−4.07636	−0.988661	−0.494331	+	0.869274i	$0.664587\pi$
$18$	0	0
$19$	−0.635651	−0.145828	−0.0729141	−	0.997338i	$-0.523230\pi$
$19$	−0.635651	−0.145828	−0.0729141	+	0.997338i	$0.523230\pi$
$20$	0	0
$21$	−2.47862	−0.540880
$22$	0	0
$23$	−8.21898	−1.71378	−0.856888	−	0.515502i	$-0.827605\pi$
$23$	−8.21898	−1.71378	−0.856888	+	0.515502i	$0.827605\pi$
$24$	0	0
$25$	6.32781	1.26556
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−3.47455	−0.645208	−0.322604	−	0.946534i	$-0.604558\pi$
$29$	−3.47455	−0.645208	−0.322604	+	0.946534i	$0.604558\pi$
$30$	0	0
$31$	8.19684	1.47220	0.736098	−	0.676875i	$-0.236666\pi$
$31$	8.19684	1.47220	0.736098	+	0.676875i	$0.236666\pi$
$32$	0	0
$33$	−6.62075	−1.15252
$34$	0	0
$35$	8.34225	1.41010
$36$	0	0
$37$	4.16843	0.685285	0.342643	−	0.939466i	$-0.388678\pi$
$37$	4.16843	0.685285	0.342643	+	0.939466i	$0.388678\pi$
$38$	0	0
$39$	−1.10011	−0.176159
$40$	0	0
$41$	−7.94459	−1.24074	−0.620368	−	0.784311i	$-0.713017\pi$
$41$	−7.94459	−1.24074	−0.620368	+	0.784311i	$0.713017\pi$
$42$	0	0
$43$	3.69096	0.562867	0.281433	−	0.959581i	$-0.409190\pi$
$43$	3.69096	0.562867	0.281433	+	0.959581i	$0.409190\pi$
$44$	0	0
$45$	−3.36568	−0.501726
$46$	0	0
$47$	−9.19615	−1.34140	−0.670698	−	0.741730i	$-0.734005\pi$
$47$	−9.19615	−1.34140	−0.670698	+	0.741730i	$0.734005\pi$
$48$	0	0
$49$	−0.856428	−0.122347
$50$	0	0
$51$	−4.07636	−0.570804
$52$	0	0
$53$	−9.69155	−1.33124	−0.665619	−	0.746292i	$-0.731832\pi$
$53$	−9.69155	−1.33124	−0.665619	+	0.746292i	$0.731832\pi$
$54$	0	0
$55$	22.2833	3.00468
$56$	0	0
$57$	−0.635651	−0.0841940
$58$	0	0
$59$	2.89156	0.376448	0.188224	−	0.982126i	$-0.439727\pi$
$59$	2.89156	0.376448	0.188224	+	0.982126i	$0.439727\pi$
$60$	0	0
$61$	−1.30308	−0.166843	−0.0834214	−	0.996514i	$-0.526585\pi$
$61$	−1.30308	−0.166843	−0.0834214	+	0.996514i	$0.526585\pi$
$62$	0	0
$63$	−2.47862	−0.312277
$64$	0	0
$65$	3.70263	0.459254
$66$	0	0
$67$	0.588636	0.0719133	0.0359566	−	0.999353i	$-0.488552\pi$
$67$	0.588636	0.0719133	0.0359566	+	0.999353i	$0.488552\pi$
$68$	0	0
$69$	−8.21898	−0.989449
$70$	0	0
$71$	11.7563	1.39521	0.697607	−	0.716480i	$-0.254248\pi$
$71$	11.7563	1.39521	0.697607	+	0.716480i	$0.254248\pi$
$72$	0	0
$73$	−10.5132	−1.23047	−0.615236	−	0.788343i	$-0.710939\pi$
$73$	−10.5132	−1.23047	−0.615236	+	0.788343i	$0.710939\pi$
$74$	0	0
$75$	6.32781	0.730672
$76$	0	0
$77$	16.4103	1.87013
$78$	0	0
$79$	2.94961	0.331857	0.165928	−	0.986138i	$-0.446938\pi$
$79$	2.94961	0.331857	0.165928	+	0.986138i	$0.446938\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	5.25007	0.576270	0.288135	−	0.957590i	$-0.406965\pi$
$83$	5.25007	0.576270	0.288135	+	0.957590i	$0.406965\pi$
$84$	0	0
$85$	13.7197	1.48811
$86$	0	0
$87$	−3.47455	−0.372511
$88$	0	0
$89$	8.28016	0.877695	0.438848	−	0.898562i	$-0.355387\pi$
$89$	8.28016	0.877695	0.438848	+	0.898562i	$0.355387\pi$
$90$	0	0
$91$	2.72676	0.285842
$92$	0	0
$93$	8.19684	0.849973
$94$	0	0
$95$	2.13940	0.219498
$96$	0	0
$97$	−4.92699	−0.500260	−0.250130	−	0.968212i	$-0.580473\pi$
$97$	−4.92699	−0.500260	−0.250130	+	0.968212i	$0.580473\pi$
$98$	0	0
$99$	−6.62075	−0.665410
$100$	0	0
$101$	9.46322	0.941626	0.470813	−	0.882233i	$-0.343961\pi$
$101$	9.46322	0.941626	0.470813	+	0.882233i	$0.343961\pi$
$102$	0	0
$103$	−19.7258	−1.94364	−0.971821	−	0.235720i	$-0.924255\pi$
$103$	−19.7258	−1.94364	−0.971821	+	0.235720i	$0.924255\pi$
$104$	0	0
$105$	8.34225	0.814121
$106$	0	0
$107$	−9.80405	−0.947794	−0.473897	−	0.880580i	$-0.657153\pi$
$107$	−9.80405	−0.947794	−0.473897	+	0.880580i	$0.657153\pi$
$108$	0	0
$109$	−5.53171	−0.529841	−0.264921	−	0.964270i	$-0.585346\pi$
$109$	−5.53171	−0.529841	−0.264921	+	0.964270i	$0.585346\pi$
$110$	0	0
$111$	4.16843	0.395650
$112$	0	0
$113$	15.9980	1.50497	0.752483	−	0.658611i	$-0.228856\pi$
$113$	15.9980	1.50497	0.752483	+	0.658611i	$0.228856\pi$
$114$	0	0
$115$	27.6625	2.57954
$116$	0	0
$117$	−1.10011	−0.101705
$118$	0	0
$119$	10.1037	0.926209
$120$	0	0
$121$	32.8343	2.98494
$122$	0	0
$123$	−7.94459	−0.716340
$124$	0	0
$125$	−4.46898	−0.399718
$126$	0	0
$127$	−7.17561	−0.636733	−0.318366	−	0.947968i	$-0.603134\pi$
$127$	−7.17561	−0.636733	−0.318366	+	0.947968i	$0.603134\pi$
$128$	0	0
$129$	3.69096	0.324971
$130$	0	0
$131$	16.5685	1.44760	0.723798	−	0.690012i	$-0.242395\pi$
$131$	16.5685	1.44760	0.723798	+	0.690012i	$0.242395\pi$
$132$	0	0
$133$	1.57554	0.136616
$134$	0	0
$135$	−3.36568	−0.289672
$136$	0	0
$137$	1.24096	0.106023	0.0530113	−	0.998594i	$-0.483118\pi$
$137$	1.24096	0.106023	0.0530113	+	0.998594i	$0.483118\pi$
$138$	0	0
$139$	9.69595	0.822400	0.411200	−	0.911545i	$-0.365110\pi$
$139$	9.69595	0.822400	0.411200	+	0.911545i	$0.365110\pi$
$140$	0	0
$141$	−9.19615	−0.774456
$142$	0	0
$143$	7.28357	0.609082
$144$	0	0
$145$	11.6942	0.971152
$146$	0	0
$147$	−0.856428	−0.0706370
$148$	0	0
$149$	7.12641	0.583818	0.291909	−	0.956446i	$-0.405709\pi$
$149$	7.12641	0.583818	0.291909	+	0.956446i	$0.405709\pi$
$150$	0	0
$151$	−7.10792	−0.578435	−0.289217	−	0.957263i	$-0.593395\pi$
$151$	−7.10792	−0.578435	−0.289217	+	0.957263i	$0.593395\pi$
$152$	0	0
$153$	−4.07636	−0.329554
$154$	0	0
$155$	−27.5880	−2.21592
$156$	0	0
$157$	−14.9908	−1.19640	−0.598200	−	0.801347i	$-0.704117\pi$
$157$	−14.9908	−1.19640	−0.598200	+	0.801347i	$0.704117\pi$
$158$	0	0
$159$	−9.69155	−0.768590
$160$	0	0
$161$	20.3718	1.60552
$162$	0	0
$163$	16.4106	1.28538	0.642688	−	0.766128i	$-0.277819\pi$
$163$	16.4106	1.28538	0.642688	+	0.766128i	$0.277819\pi$
$164$	0	0
$165$	22.2833	1.73475
$166$	0	0
$167$	3.79495	0.293662	0.146831	−	0.989162i	$-0.453093\pi$
$167$	3.79495	0.293662	0.146831	+	0.989162i	$0.453093\pi$
$168$	0	0
$169$	−11.7898	−0.906904
$170$	0	0
$171$	−0.635651	−0.0486094
$172$	0	0
$173$	1.77724	0.135121	0.0675605	−	0.997715i	$-0.478478\pi$
$173$	1.77724	0.135121	0.0675605	+	0.997715i	$0.478478\pi$
$174$	0	0
$175$	−15.6843	−1.18562
$176$	0	0
$177$	2.89156	0.217343
$178$	0	0
$179$	−5.24524	−0.392048	−0.196024	−	0.980599i	$-0.562803\pi$
$179$	−5.24524	−0.392048	−0.196024	+	0.980599i	$0.562803\pi$
$180$	0	0
$181$	−17.9914	−1.33729	−0.668646	−	0.743581i	$-0.733126\pi$
$181$	−17.9914	−1.33729	−0.668646	+	0.743581i	$0.733126\pi$
$182$	0	0
$183$	−1.30308	−0.0963267
$184$	0	0
$185$	−14.0296	−1.03148
$186$	0	0
$187$	26.9885	1.97360
$188$	0	0
$189$	−2.47862	−0.180293
$190$	0	0
$191$	−18.3162	−1.32532	−0.662658	−	0.748923i	$-0.730571\pi$
$191$	−18.3162	−1.32532	−0.662658	+	0.748923i	$0.730571\pi$
$192$	0	0
$193$	16.2969	1.17308	0.586538	−	0.809922i	$-0.300490\pi$
$193$	16.2969	1.17308	0.586538	+	0.809922i	$0.300490\pi$
$194$	0	0
$195$	3.70263	0.265151
$196$	0	0
$197$	−24.6509	−1.75630	−0.878152	−	0.478382i	$-0.841223\pi$
$197$	−24.6509	−1.75630	−0.878152	+	0.478382i	$0.841223\pi$
$198$	0	0
$199$	1.65197	0.117105	0.0585524	−	0.998284i	$-0.481352\pi$
$199$	1.65197	0.117105	0.0585524	+	0.998284i	$0.481352\pi$
$200$	0	0
$201$	0.588636	0.0415192
$202$	0	0
$203$	8.61210	0.604451
$204$	0	0
$205$	26.7390	1.86753
$206$	0	0
$207$	−8.21898	−0.571259
$208$	0	0
$209$	4.20848	0.291107
$210$	0	0
$211$	19.5981	1.34919	0.674593	−	0.738190i	$-0.264319\pi$
$211$	19.5981	1.34919	0.674593	+	0.738190i	$0.264319\pi$
$212$	0	0
$213$	11.7563	0.805528
$214$	0	0
$215$	−12.4226	−0.847215
$216$	0	0
$217$	−20.3169	−1.37920
$218$	0	0
$219$	−10.5132	−0.710414
$220$	0	0
$221$	4.48445	0.301657
$222$	0	0
$223$	−16.7285	−1.12022	−0.560112	−	0.828417i	$-0.689242\pi$
$223$	−16.7285	−1.12022	−0.560112	+	0.828417i	$0.689242\pi$
$224$	0	0
$225$	6.32781	0.421854
$226$	0	0
$227$	17.5465	1.16460	0.582302	−	0.812973i	$-0.302152\pi$
$227$	17.5465	1.16460	0.582302	+	0.812973i	$0.302152\pi$
$228$	0	0
$229$	0.171282	0.0113186	0.00565932	−	0.999984i	$-0.498199\pi$
$229$	0.171282	0.0113186	0.00565932	+	0.999984i	$0.498199\pi$
$230$	0	0
$231$	16.4103	1.07972
$232$	0	0
$233$	−8.57696	−0.561895	−0.280948	−	0.959723i	$-0.590649\pi$
$233$	−8.57696	−0.561895	−0.280948	+	0.959723i	$0.590649\pi$
$234$	0	0
$235$	30.9513	2.01904
$236$	0	0
$237$	2.94961	0.191598
$238$	0	0
$239$	7.13488	0.461517	0.230759	−	0.973011i	$-0.425879\pi$
$239$	7.13488	0.461517	0.230759	+	0.973011i	$0.425879\pi$
$240$	0	0
$241$	−11.9005	−0.766576	−0.383288	−	0.923629i	$-0.625208\pi$
$241$	−11.9005	−0.766576	−0.383288	+	0.923629i	$0.625208\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	2.88246	0.184154
$246$	0	0
$247$	0.699287	0.0444946
$248$	0	0
$249$	5.25007	0.332710
$250$	0	0
$251$	1.00000	0.0631194
$251$	1.00000	0.0631194
$252$	0	0
$253$	54.4158	3.42109
$254$	0	0
$255$	13.7197	0.859162
$256$	0	0
$257$	6.29182	0.392473	0.196237	−	0.980557i	$-0.437128\pi$
$257$	6.29182	0.392473	0.196237	+	0.980557i	$0.437128\pi$
$258$	0	0
$259$	−10.3320	−0.641997
$260$	0	0
$261$	−3.47455	−0.215069
$262$	0	0
$263$	−15.1549	−0.934490	−0.467245	−	0.884128i	$-0.654753\pi$
$263$	−15.1549	−0.934490	−0.467245	+	0.884128i	$0.654753\pi$
$264$	0	0
$265$	32.6187	2.00375
$266$	0	0
$267$	8.28016	0.506738
$268$	0	0
$269$	16.4276	1.00161	0.500804	−	0.865561i	$-0.333038\pi$
$269$	16.4276	1.00161	0.500804	+	0.865561i	$0.333038\pi$
$270$	0	0
$271$	−12.2432	−0.743719	−0.371859	−	0.928289i	$-0.621280\pi$
$271$	−12.2432	−0.743719	−0.371859	+	0.928289i	$0.621280\pi$
$272$	0	0
$273$	2.72676	0.165031
$274$	0	0
$275$	−41.8948	−2.52635
$276$	0	0
$277$	−27.2694	−1.63846	−0.819228	−	0.573467i	$-0.805598\pi$
$277$	−27.2694	−1.63846	−0.819228	+	0.573467i	$0.805598\pi$
$278$	0	0
$279$	8.19684	0.490732
$280$	0	0
$281$	−10.7498	−0.641276	−0.320638	−	0.947202i	$-0.603897\pi$
$281$	−10.7498	−0.641276	−0.320638	+	0.947202i	$0.603897\pi$
$282$	0	0
$283$	8.90259	0.529204	0.264602	−	0.964358i	$-0.414759\pi$
$283$	8.90259	0.529204	0.264602	+	0.964358i	$0.414759\pi$
$284$	0	0
$285$	2.13940	0.126727
$286$	0	0
$287$	19.6916	1.16236
$288$	0	0
$289$	−0.383324	−0.0225485
$290$	0	0
$291$	−4.92699	−0.288825
$292$	0	0
$293$	25.1757	1.47078	0.735391	−	0.677643i	$-0.236999\pi$
$293$	25.1757	1.47078	0.735391	+	0.677643i	$0.236999\pi$
$294$	0	0
$295$	−9.73205	−0.566622
$296$	0	0
$297$	−6.62075	−0.384175
$298$	0	0
$299$	9.04181	0.522901
$300$	0	0
$301$	−9.14851	−0.527311
$302$	0	0
$303$	9.46322	0.543648
$304$	0	0
$305$	4.38576	0.251128
$306$	0	0
$307$	−12.4813	−0.712345	−0.356172	−	0.934420i	$-0.615918\pi$
$307$	−12.4813	−0.712345	−0.356172	+	0.934420i	$0.615918\pi$
$308$	0	0
$309$	−19.7258	−1.12216
$310$	0	0
$311$	23.5101	1.33314	0.666568	−	0.745444i	$-0.267762\pi$
$311$	23.5101	1.33314	0.666568	+	0.745444i	$0.267762\pi$
$312$	0	0
$313$	1.41010	0.0797034	0.0398517	−	0.999206i	$-0.487311\pi$
$313$	1.41010	0.0797034	0.0398517	+	0.999206i	$0.487311\pi$
$314$	0	0
$315$	8.34225	0.470033
$316$	0	0
$317$	3.71969	0.208918	0.104459	−	0.994529i	$-0.466689\pi$
$317$	3.71969	0.208918	0.104459	+	0.994529i	$0.466689\pi$
$318$	0	0
$319$	23.0041	1.28798
$320$	0	0
$321$	−9.80405	−0.547209
$322$	0	0
$323$	2.59114	0.144175
$324$	0	0
$325$	−6.96130	−0.386143
$326$	0	0
$327$	−5.53171	−0.305904
$328$	0	0
$329$	22.7938	1.25666
$330$	0	0
$331$	−8.99902	−0.494631	−0.247315	−	0.968935i	$-0.579548\pi$
$331$	−8.99902	−0.494631	−0.247315	+	0.968935i	$0.579548\pi$
$332$	0	0
$333$	4.16843	0.228428
$334$	0	0
$335$	−1.98116	−0.108242
$336$	0	0
$337$	4.88705	0.266215	0.133107	−	0.991102i	$-0.457505\pi$
$337$	4.88705	0.266215	0.133107	+	0.991102i	$0.457505\pi$
$338$	0	0
$339$	15.9980	0.868893
$340$	0	0
$341$	−54.2692	−2.93884
$342$	0	0
$343$	19.4731	1.05145
$344$	0	0
$345$	27.6625	1.48930
$346$	0	0
$347$	18.6438	1.00085	0.500427	−	0.865779i	$-0.333177\pi$
$347$	18.6438	1.00085	0.500427	+	0.865779i	$0.333177\pi$
$348$	0	0
$349$	10.6833	0.571865	0.285932	−	0.958250i	$-0.407697\pi$
$349$	10.6833	0.571865	0.285932	+	0.958250i	$0.407697\pi$
$350$	0	0
$351$	−1.10011	−0.0587196
$352$	0	0
$353$	−12.7411	−0.678139	−0.339070	−	0.940761i	$-0.610112\pi$
$353$	−12.7411	−0.678139	−0.339070	+	0.940761i	$0.610112\pi$
$354$	0	0
$355$	−39.5679	−2.10005
$356$	0	0
$357$	10.1037	0.534747
$358$	0	0
$359$	−9.37882	−0.494995	−0.247498	−	0.968888i	$-0.579608\pi$
$359$	−9.37882	−0.494995	−0.247498	+	0.968888i	$0.579608\pi$
$360$	0	0
$361$	−18.5959	−0.978734
$362$	0	0
$363$	32.8343	1.72335
$364$	0	0
$365$	35.3839	1.85208
$366$	0	0
$367$	28.7692	1.50174	0.750871	−	0.660449i	$-0.229634\pi$
$367$	28.7692	1.50174	0.750871	+	0.660449i	$0.229634\pi$
$368$	0	0
$369$	−7.94459	−0.413579
$370$	0	0
$371$	24.0217	1.24714
$372$	0	0
$373$	−10.0108	−0.518341	−0.259171	−	0.965832i	$-0.583449\pi$
$373$	−10.0108	−0.518341	−0.259171	+	0.965832i	$0.583449\pi$
$374$	0	0
$375$	−4.46898	−0.230777
$376$	0	0
$377$	3.82239	0.196863
$378$	0	0
$379$	30.2890	1.55584	0.777922	−	0.628361i	$-0.216274\pi$
$379$	30.2890	1.55584	0.777922	+	0.628361i	$0.216274\pi$
$380$	0	0
$381$	−7.17561	−0.367618
$382$	0	0
$383$	−18.5300	−0.946839	−0.473420	−	0.880837i	$-0.656981\pi$
$383$	−18.5300	−0.946839	−0.473420	+	0.880837i	$0.656981\pi$
$384$	0	0
$385$	−55.2320	−2.81488
$386$	0	0
$387$	3.69096	0.187622
$388$	0	0
$389$	−26.8869	−1.36322	−0.681609	−	0.731716i	$-0.738720\pi$
$389$	−26.8869	−1.36322	−0.681609	+	0.731716i	$0.738720\pi$
$390$	0	0
$391$	33.5035	1.69435
$392$	0	0
$393$	16.5685	0.835770
$394$	0	0
$395$	−9.92744	−0.499503
$396$	0	0
$397$	−19.3435	−0.970824	−0.485412	−	0.874286i	$-0.661330\pi$
$397$	−19.3435	−0.970824	−0.485412	+	0.874286i	$0.661330\pi$
$398$	0	0
$399$	1.57554	0.0788756
$400$	0	0
$401$	−1.97215	−0.0984843	−0.0492421	−	0.998787i	$-0.515681\pi$
$401$	−1.97215	−0.0984843	−0.0492421	+	0.998787i	$0.515681\pi$
$402$	0	0
$403$	−9.01745	−0.449191
$404$	0	0
$405$	−3.36568	−0.167242
$406$	0	0
$407$	−27.5981	−1.36799
$408$	0	0
$409$	−10.6856	−0.528368	−0.264184	−	0.964472i	$-0.585103\pi$
$409$	−10.6856	−0.528368	−0.264184	+	0.964472i	$0.585103\pi$
$410$	0	0
$411$	1.24096	0.0612122
$412$	0	0
$413$	−7.16708	−0.352669
$414$	0	0
$415$	−17.6701	−0.867390
$416$	0	0
$417$	9.69595	0.474813
$418$	0	0
$419$	0.992049	0.0484648	0.0242324	−	0.999706i	$-0.492286\pi$
$419$	0.992049	0.0484648	0.0242324	+	0.999706i	$0.492286\pi$
$420$	0	0
$421$	34.8429	1.69814	0.849070	−	0.528280i	$-0.177163\pi$
$421$	34.8429	1.69814	0.849070	+	0.528280i	$0.177163\pi$
$422$	0	0
$423$	−9.19615	−0.447132
$424$	0	0
$425$	−25.7944	−1.25121
$426$	0	0
$427$	3.22985	0.156304
$428$	0	0
$429$	7.28357	0.351654
$430$	0	0
$431$	−41.2830	−1.98853	−0.994265	−	0.106941i	$-0.965894\pi$
$431$	−41.2830	−1.98853	−0.994265	+	0.106941i	$0.965894\pi$
$432$	0	0
$433$	−13.8439	−0.665296	−0.332648	−	0.943051i	$-0.607942\pi$
$433$	−13.8439	−0.665296	−0.332648	+	0.943051i	$0.607942\pi$
$434$	0	0
$435$	11.6942	0.560695
$436$	0	0
$437$	5.22440	0.249917
$438$	0	0
$439$	−4.77871	−0.228076	−0.114038	−	0.993476i	$-0.536379\pi$
$439$	−4.77871	−0.228076	−0.114038	+	0.993476i	$0.536379\pi$
$440$	0	0
$441$	−0.856428	−0.0407823
$442$	0	0
$443$	−11.3938	−0.541336	−0.270668	−	0.962673i	$-0.587245\pi$
$443$	−11.3938	−0.541336	−0.270668	+	0.962673i	$0.587245\pi$
$444$	0	0
$445$	−27.8684	−1.32109
$446$	0	0
$447$	7.12641	0.337068
$448$	0	0
$449$	15.7482	0.743204	0.371602	−	0.928392i	$-0.378809\pi$
$449$	15.7482	0.743204	0.371602	+	0.928392i	$0.378809\pi$
$450$	0	0
$451$	52.5991	2.47680
$452$	0	0
$453$	−7.10792	−0.333959
$454$	0	0
$455$	−9.17742	−0.430244
$456$	0	0
$457$	8.58475	0.401578	0.200789	−	0.979635i	$-0.435649\pi$
$457$	8.58475	0.401578	0.200789	+	0.979635i	$0.435649\pi$
$458$	0	0
$459$	−4.07636	−0.190268
$460$	0	0
$461$	−8.56401	−0.398866	−0.199433	−	0.979911i	$-0.563910\pi$
$461$	−8.56401	−0.398866	−0.199433	+	0.979911i	$0.563910\pi$
$462$	0	0
$463$	−6.45796	−0.300127	−0.150064	−	0.988676i	$-0.547948\pi$
$463$	−6.45796	−0.300127	−0.150064	+	0.988676i	$0.547948\pi$
$464$	0	0
$465$	−27.5880	−1.27936
$466$	0	0
$467$	−5.10312	−0.236144	−0.118072	−	0.993005i	$-0.537671\pi$
$467$	−5.10312	−0.236144	−0.118072	+	0.993005i	$0.537671\pi$
$468$	0	0
$469$	−1.45901	−0.0673706
$470$	0	0
$471$	−14.9908	−0.690741
$472$	0	0
$473$	−24.4369	−1.12361
$474$	0	0
$475$	−4.02228	−0.184555
$476$	0	0
$477$	−9.69155	−0.443746
$478$	0	0
$479$	−15.4383	−0.705393	−0.352696	−	0.935738i	$-0.614735\pi$
$479$	−15.4383	−0.705393	−0.352696	+	0.935738i	$0.614735\pi$
$480$	0	0
$481$	−4.58574	−0.209092
$482$	0	0
$483$	20.3718	0.926947
$484$	0	0
$485$	16.5827	0.752981
$486$	0	0
$487$	18.8823	0.855639	0.427819	−	0.903864i	$-0.359282\pi$
$487$	18.8823	0.855639	0.427819	+	0.903864i	$0.359282\pi$
$488$	0	0
$489$	16.4106	0.742112
$490$	0	0
$491$	−24.0262	−1.08429	−0.542143	−	0.840287i	$-0.682387\pi$
$491$	−24.0262	−1.08429	−0.542143	+	0.840287i	$0.682387\pi$
$492$	0	0
$493$	14.1635	0.637892
$494$	0	0
$495$	22.2833	1.00156
$496$	0	0
$497$	−29.1394	−1.30708
$498$	0	0
$499$	−4.37244	−0.195737	−0.0978686	−	0.995199i	$-0.531202\pi$
$499$	−4.37244	−0.195737	−0.0978686	+	0.995199i	$0.531202\pi$
$500$	0	0
$501$	3.79495	0.169546
$502$	0	0
$503$	−24.0811	−1.07372	−0.536862	−	0.843670i	$-0.680391\pi$
$503$	−24.0811	−1.07372	−0.536862	+	0.843670i	$0.680391\pi$
$504$	0	0
$505$	−31.8502	−1.41731
$506$	0	0
$507$	−11.7898	−0.523601
$508$	0	0
$509$	24.7694	1.09788	0.548941	−	0.835861i	$-0.315031\pi$
$509$	24.7694	1.09788	0.548941	+	0.835861i	$0.315031\pi$
$510$	0	0
$511$	26.0582	1.15275
$512$	0	0
$513$	−0.635651	−0.0280647
$514$	0	0
$515$	66.3908	2.92553
$516$	0	0
$517$	60.8854	2.67774
$518$	0	0
$519$	1.77724	0.0780121
$520$	0	0
$521$	8.54634	0.374422	0.187211	−	0.982320i	$-0.440055\pi$
$521$	8.54634	0.374422	0.187211	+	0.982320i	$0.440055\pi$
$522$	0	0
$523$	32.7320	1.43127	0.715635	−	0.698474i	$-0.246137\pi$
$523$	32.7320	1.43127	0.715635	+	0.698474i	$0.246137\pi$
$524$	0	0
$525$	−15.6843	−0.684517
$526$	0	0
$527$	−33.4132	−1.45550
$528$	0	0
$529$	44.5517	1.93703
$530$	0	0
$531$	2.89156	0.125483
$532$	0	0
$533$	8.73994	0.378569
$534$	0	0
$535$	32.9973	1.42660
$536$	0	0
$537$	−5.24524	−0.226349
$538$	0	0
$539$	5.67020	0.244233
$540$	0	0
$541$	−26.7811	−1.15141	−0.575705	−	0.817658i	$-0.695272\pi$
$541$	−26.7811	−1.15141	−0.575705	+	0.817658i	$0.695272\pi$
$542$	0	0
$543$	−17.9914	−0.772086
$544$	0	0
$545$	18.6180	0.797506
$546$	0	0
$547$	23.4469	1.00252	0.501258	−	0.865298i	$-0.332871\pi$
$547$	23.4469	1.00252	0.501258	+	0.865298i	$0.332871\pi$
$548$	0	0
$549$	−1.30308	−0.0556142
$550$	0	0
$551$	2.20860	0.0940895
$552$	0	0
$553$	−7.31096	−0.310894
$554$	0	0
$555$	−14.0296	−0.595523
$556$	0	0
$557$	24.2750	1.02857	0.514283	−	0.857621i	$-0.328058\pi$
$557$	24.2750	1.02857	0.514283	+	0.857621i	$0.328058\pi$
$558$	0	0
$559$	−4.06047	−0.171740
$560$	0	0
$561$	26.9885	1.13946
$562$	0	0
$563$	0.112238	0.00473027	0.00236513	−	0.999997i	$-0.499247\pi$
$563$	0.112238	0.00473027	0.00236513	+	0.999997i	$0.499247\pi$
$564$	0	0
$565$	−53.8442	−2.26524
$566$	0	0
$567$	−2.47862	−0.104092
$568$	0	0
$569$	5.55159	0.232735	0.116367	−	0.993206i	$-0.462875\pi$
$569$	5.55159	0.232735	0.116367	+	0.993206i	$0.462875\pi$
$570$	0	0
$571$	−32.4313	−1.35721	−0.678605	−	0.734504i	$-0.737415\pi$
$571$	−32.4313	−1.35721	−0.678605	+	0.734504i	$0.737415\pi$
$572$	0	0
$573$	−18.3162	−0.765171
$574$	0	0
$575$	−52.0082	−2.16889
$576$	0	0
$577$	−44.0739	−1.83482	−0.917411	−	0.397942i	$-0.869725\pi$
$577$	−44.0739	−1.83482	−0.917411	+	0.397942i	$0.869725\pi$
$578$	0	0
$579$	16.2969	0.677276
$580$	0	0
$581$	−13.0130	−0.539868
$582$	0	0
$583$	64.1653	2.65746
$584$	0	0
$585$	3.70263	0.153085
$586$	0	0
$587$	34.2514	1.41371	0.706853	−	0.707361i	$-0.250114\pi$
$587$	34.2514	1.41371	0.706853	+	0.707361i	$0.250114\pi$
$588$	0	0
$589$	−5.21033	−0.214688
$590$	0	0
$591$	−24.6509	−1.01400
$592$	0	0
$593$	13.8748	0.569769	0.284884	−	0.958562i	$-0.408045\pi$
$593$	13.8748	0.569769	0.284884	+	0.958562i	$0.408045\pi$
$594$	0	0
$595$	−34.0060	−1.39411
$596$	0	0
$597$	1.65197	0.0676104
$598$	0	0
$599$	9.53105	0.389428	0.194714	−	0.980860i	$-0.437622\pi$
$599$	9.53105	0.389428	0.194714	+	0.980860i	$0.437622\pi$
$600$	0	0
$601$	−22.1262	−0.902546	−0.451273	−	0.892386i	$-0.649030\pi$
$601$	−22.1262	−0.902546	−0.451273	+	0.892386i	$0.649030\pi$
$602$	0	0
$603$	0.588636	0.0239711
$604$	0	0
$605$	−110.510	−4.49286
$606$	0	0
$607$	−31.7456	−1.28852	−0.644258	−	0.764809i	$-0.722834\pi$
$607$	−31.7456	−1.28852	−0.644258	+	0.764809i	$0.722834\pi$
$608$	0	0
$609$	8.61210	0.348980
$610$	0	0
$611$	10.1168	0.409282
$612$	0	0
$613$	−19.6666	−0.794326	−0.397163	−	0.917748i	$-0.630005\pi$
$613$	−19.6666	−0.794326	−0.397163	+	0.917748i	$0.630005\pi$
$614$	0	0
$615$	26.7390	1.07822
$616$	0	0
$617$	−11.1500	−0.448884	−0.224442	−	0.974488i	$-0.572056\pi$
$617$	−11.1500	−0.448884	−0.224442	+	0.974488i	$0.572056\pi$
$618$	0	0
$619$	11.7162	0.470913	0.235456	−	0.971885i	$-0.424342\pi$
$619$	11.7162	0.470913	0.235456	+	0.971885i	$0.424342\pi$
$620$	0	0
$621$	−8.21898	−0.329816
$622$	0	0
$623$	−20.5234	−0.822252
$624$	0	0
$625$	−16.5979	−0.663915
$626$	0	0
$627$	4.20848	0.168071
$628$	0	0
$629$	−16.9920	−0.677515
$630$	0	0
$631$	34.7686	1.38412	0.692058	−	0.721842i	$-0.256704\pi$
$631$	34.7686	1.38412	0.692058	+	0.721842i	$0.256704\pi$
$632$	0	0
$633$	19.5981	0.778953
$634$	0	0
$635$	24.1508	0.958396
$636$	0	0
$637$	0.942167	0.0373300
$638$	0	0
$639$	11.7563	0.465072
$640$	0	0
$641$	3.70414	0.146305	0.0731523	−	0.997321i	$-0.476694\pi$
$641$	3.70414	0.146305	0.0731523	+	0.997321i	$0.476694\pi$
$642$	0	0
$643$	−6.94456	−0.273867	−0.136933	−	0.990580i	$-0.543725\pi$
$643$	−6.94456	−0.273867	−0.136933	+	0.990580i	$0.543725\pi$
$644$	0	0
$645$	−12.4226	−0.489140
$646$	0	0
$647$	−14.6740	−0.576896	−0.288448	−	0.957496i	$-0.593139\pi$
$647$	−14.6740	−0.576896	−0.288448	+	0.957496i	$0.593139\pi$
$648$	0	0
$649$	−19.1443	−0.751478
$650$	0	0
$651$	−20.3169	−0.796281
$652$	0	0
$653$	−29.6167	−1.15899	−0.579495	−	0.814976i	$-0.696750\pi$
$653$	−29.6167	−1.15899	−0.579495	+	0.814976i	$0.696750\pi$
$654$	0	0
$655$	−55.7642	−2.17889
$656$	0	0
$657$	−10.5132	−0.410157
$658$	0	0
$659$	43.8761	1.70917	0.854586	−	0.519310i	$-0.173811\pi$
$659$	43.8761	1.70917	0.854586	+	0.519310i	$0.173811\pi$
$660$	0	0
$661$	8.44928	0.328639	0.164319	−	0.986407i	$-0.447457\pi$
$661$	8.44928	0.328639	0.164319	+	0.986407i	$0.447457\pi$
$662$	0	0
$663$	4.48445	0.174162
$664$	0	0
$665$	−5.30276	−0.205632
$666$	0	0
$667$	28.5573	1.10574
$668$	0	0
$669$	−16.7285	−0.646762
$670$	0	0
$671$	8.62739	0.333057
$672$	0	0
$673$	−38.0912	−1.46831	−0.734153	−	0.678984i	$-0.762421\pi$
$673$	−38.0912	−1.46831	−0.734153	+	0.678984i	$0.762421\pi$
$674$	0	0
$675$	6.32781	0.243557
$676$	0	0
$677$	−10.5876	−0.406913	−0.203456	−	0.979084i	$-0.565218\pi$
$677$	−10.5876	−0.406913	−0.203456	+	0.979084i	$0.565218\pi$
$678$	0	0
$679$	12.2122	0.468660
$680$	0	0
$681$	17.5465	0.672384
$682$	0	0
$683$	−5.65643	−0.216437	−0.108219	−	0.994127i	$-0.534515\pi$
$683$	−5.65643	−0.216437	−0.108219	+	0.994127i	$0.534515\pi$
$684$	0	0
$685$	−4.17668	−0.159583
$686$	0	0
$687$	0.171282	0.00653482
$688$	0	0
$689$	10.6618	0.406182
$690$	0	0
$691$	6.37695	0.242591	0.121295	−	0.992616i	$-0.461295\pi$
$691$	6.37695	0.242591	0.121295	+	0.992616i	$0.461295\pi$
$692$	0	0
$693$	16.4103	0.623377
$694$	0	0
$695$	−32.6335	−1.23786
$696$	0	0
$697$	32.3850	1.22667
$698$	0	0
$699$	−8.57696	−0.324410
$700$	0	0
$701$	24.1137	0.910762	0.455381	−	0.890297i	$-0.349503\pi$
$701$	24.1137	0.910762	0.455381	+	0.890297i	$0.349503\pi$
$702$	0	0
$703$	−2.64966	−0.0999340
$704$	0	0
$705$	30.9513	1.16569
$706$	0	0
$707$	−23.4558	−0.882144
$708$	0	0
$709$	−47.3946	−1.77994	−0.889970	−	0.456019i	$-0.849275\pi$
$709$	−47.3946	−1.77994	−0.889970	+	0.456019i	$0.849275\pi$
$710$	0	0
$711$	2.94961	0.110619
$712$	0	0
$713$	−67.3697	−2.52302
$714$	0	0
$715$	−24.5142	−0.916778
$716$	0	0
$717$	7.13488	0.266457
$718$	0	0
$719$	9.53252	0.355503	0.177752	−	0.984075i	$-0.443118\pi$
$719$	9.53252	0.355503	0.177752	+	0.984075i	$0.443118\pi$
$720$	0	0
$721$	48.8928	1.82086
$722$	0	0
$723$	−11.9005	−0.442583
$724$	0	0
$725$	−21.9863	−0.816550
$726$	0	0
$727$	39.1378	1.45154	0.725771	−	0.687937i	$-0.241483\pi$
$727$	39.1378	1.45154	0.725771	+	0.687937i	$0.241483\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−15.0457	−0.556485
$732$	0	0
$733$	−22.9341	−0.847090	−0.423545	−	0.905875i	$-0.639214\pi$
$733$	−22.9341	−0.847090	−0.423545	+	0.905875i	$0.639214\pi$
$734$	0	0
$735$	2.88246	0.106321
$736$	0	0
$737$	−3.89721	−0.143556
$738$	0	0
$739$	−3.53558	−0.130059	−0.0650293	−	0.997883i	$-0.520714\pi$
$739$	−3.53558	−0.130059	−0.0650293	+	0.997883i	$0.520714\pi$
$740$	0	0
$741$	0.699287	0.0256889
$742$	0	0
$743$	31.7266	1.16394	0.581968	−	0.813212i	$-0.302283\pi$
$743$	31.7266	1.16394	0.581968	+	0.813212i	$0.302283\pi$
$744$	0	0
$745$	−23.9852	−0.878751
$746$	0	0
$747$	5.25007	0.192090
$748$	0	0
$749$	24.3005	0.887923
$750$	0	0
$751$	−0.779435	−0.0284420	−0.0142210	−	0.999899i	$-0.504527\pi$
$751$	−0.779435	−0.0284420	−0.0142210	+	0.999899i	$0.504527\pi$
$752$	0	0
$753$	1.00000	0.0364420
$754$	0	0
$755$	23.9230	0.870647
$756$	0	0
$757$	−31.0054	−1.12691	−0.563455	−	0.826147i	$-0.690528\pi$
$757$	−31.0054	−1.12691	−0.563455	+	0.826147i	$0.690528\pi$
$758$	0	0
$759$	54.4158	1.97517
$760$	0	0
$761$	−14.3398	−0.519817	−0.259908	−	0.965633i	$-0.583692\pi$
$761$	−14.3398	−0.519817	−0.259908	+	0.965633i	$0.583692\pi$
$762$	0	0
$763$	13.7110	0.496372
$764$	0	0
$765$	13.7197	0.496037
$766$	0	0
$767$	−3.18104	−0.114861
$768$	0	0
$769$	−26.3452	−0.950032	−0.475016	−	0.879977i	$-0.657558\pi$
$769$	−26.3452	−0.950032	−0.475016	+	0.879977i	$0.657558\pi$
$770$	0	0
$771$	6.29182	0.226594
$772$	0	0
$773$	8.27449	0.297613	0.148806	−	0.988866i	$-0.452457\pi$
$773$	8.27449	0.297613	0.148806	+	0.988866i	$0.452457\pi$
$774$	0	0
$775$	51.8681	1.86316
$776$	0	0
$777$	−10.3320	−0.370657
$778$	0	0
$779$	5.04998	0.180934
$780$	0	0
$781$	−77.8354	−2.78517
$782$	0	0
$783$	−3.47455	−0.124170
$784$	0	0
$785$	50.4544	1.80079
$786$	0	0
$787$	5.87670	0.209482	0.104741	−	0.994500i	$-0.466599\pi$
$787$	5.87670	0.209482	0.104741	+	0.994500i	$0.466599\pi$
$788$	0	0
$789$	−15.1549	−0.539528
$790$	0	0
$791$	−39.6530	−1.40990
$792$	0	0
$793$	1.43354	0.0509064
$794$	0	0
$795$	32.6187	1.15687
$796$	0	0
$797$	−36.8702	−1.30601	−0.653004	−	0.757354i	$-0.726492\pi$
$797$	−36.8702	−1.30601	−0.653004	+	0.757354i	$0.726492\pi$
$798$	0	0
$799$	37.4868	1.32619
$800$	0	0
$801$	8.28016	0.292565
$802$	0	0
$803$	69.6050	2.45631
$804$	0	0
$805$	−68.5649	−2.41659
$806$	0	0
$807$	16.4276	0.578278
$808$	0	0
$809$	33.8664	1.19068	0.595339	−	0.803475i	$-0.297018\pi$
$809$	33.8664	1.19068	0.595339	+	0.803475i	$0.297018\pi$
$810$	0	0
$811$	−2.00005	−0.0702314	−0.0351157	−	0.999383i	$-0.511180\pi$
$811$	−2.00005	−0.0702314	−0.0351157	+	0.999383i	$0.511180\pi$
$812$	0	0
$813$	−12.2432	−0.429386
$814$	0	0
$815$	−55.2328	−1.93472
$816$	0	0
$817$	−2.34616	−0.0820819
$818$	0	0
$819$	2.72676	0.0952808
$820$	0	0
$821$	0.968378	0.0337966	0.0168983	−	0.999857i	$-0.494621\pi$
$821$	0.968378	0.0337966	0.0168983	+	0.999857i	$0.494621\pi$
$822$	0	0
$823$	41.0634	1.43138	0.715690	−	0.698418i	$-0.246112\pi$
$823$	41.0634	1.43138	0.715690	+	0.698418i	$0.246112\pi$
$824$	0	0
$825$	−41.8948	−1.45859
$826$	0	0
$827$	47.2937	1.64456	0.822281	−	0.569081i	$-0.192701\pi$
$827$	47.2937	1.64456	0.822281	+	0.569081i	$0.192701\pi$
$828$	0	0
$829$	32.9161	1.14322	0.571611	−	0.820524i	$-0.306319\pi$
$829$	32.9161	1.14322	0.571611	+	0.820524i	$0.306319\pi$
$830$	0	0
$831$	−27.2694	−0.945963
$832$	0	0
$833$	3.49111	0.120960
$834$	0	0
$835$	−12.7726	−0.442014
$836$	0	0
$837$	8.19684	0.283324
$838$	0	0
$839$	4.94949	0.170875	0.0854376	−	0.996344i	$-0.472771\pi$
$839$	4.94949	0.170875	0.0854376	+	0.996344i	$0.472771\pi$
$840$	0	0
$841$	−16.9275	−0.583707
$842$	0	0
$843$	−10.7498	−0.370241
$844$	0	0
$845$	39.6805	1.36505
$846$	0	0
$847$	−81.3839	−2.79638
$848$	0	0
$849$	8.90259	0.305536
$850$	0	0
$851$	−34.2602	−1.17443
$852$	0	0
$853$	−13.7065	−0.469303	−0.234652	−	0.972080i	$-0.575395\pi$
$853$	−13.7065	−0.469303	−0.234652	+	0.972080i	$0.575395\pi$
$854$	0	0
$855$	2.13940	0.0731658
$856$	0	0
$857$	−25.9652	−0.886953	−0.443476	−	0.896286i	$-0.646255\pi$
$857$	−25.9652	−0.886953	−0.443476	+	0.896286i	$0.646255\pi$
$858$	0	0
$859$	−31.2599	−1.06657	−0.533287	−	0.845935i	$-0.679043\pi$
$859$	−31.2599	−1.06657	−0.533287	+	0.845935i	$0.679043\pi$
$860$	0	0
$861$	19.6916	0.671090
$862$	0	0
$863$	50.3492	1.71391	0.856954	−	0.515394i	$-0.172354\pi$
$863$	50.3492	1.71391	0.856954	+	0.515394i	$0.172354\pi$
$864$	0	0
$865$	−5.98162	−0.203381
$866$	0	0
$867$	−0.383324	−0.0130184
$868$	0	0
$869$	−19.5286	−0.662463
$870$	0	0
$871$	−0.647565	−0.0219419
$872$	0	0
$873$	−4.92699	−0.166753
$874$	0	0
$875$	11.0769	0.374468
$876$	0	0
$877$	21.1624	0.714603	0.357301	−	0.933989i	$-0.383697\pi$
$877$	21.1624	0.714603	0.357301	+	0.933989i	$0.383697\pi$
$878$	0	0
$879$	25.1757	0.849156
$880$	0	0
$881$	−52.7057	−1.77570	−0.887850	−	0.460133i	$-0.847802\pi$
$881$	−52.7057	−1.77570	−0.887850	+	0.460133i	$0.847802\pi$
$882$	0	0
$883$	−40.9508	−1.37810	−0.689051	−	0.724712i	$-0.741973\pi$
$883$	−40.9508	−1.37810	−0.689051	+	0.724712i	$0.741973\pi$
$884$	0	0
$885$	−9.73205	−0.327139
$886$	0	0
$887$	9.60600	0.322538	0.161269	−	0.986910i	$-0.448441\pi$
$887$	9.60600	0.322538	0.161269	+	0.986910i	$0.448441\pi$
$888$	0	0
$889$	17.7856	0.596511
$890$	0	0
$891$	−6.62075	−0.221803
$892$	0	0
$893$	5.84554	0.195614
$894$	0	0
$895$	17.6538	0.590102
$896$	0	0
$897$	9.04181	0.301897
$898$	0	0
$899$	−28.4803	−0.949872
$900$	0	0
$901$	39.5062	1.31614
$902$	0	0
$903$	−9.14851	−0.304443
$904$	0	0
$905$	60.5534	2.01286
$906$	0	0
$907$	−3.11366	−0.103388	−0.0516938	−	0.998663i	$-0.516462\pi$
$907$	−3.11366	−0.103388	−0.0516938	+	0.998663i	$0.516462\pi$
$908$	0	0
$909$	9.46322	0.313875
$910$	0	0
$911$	−29.0106	−0.961165	−0.480583	−	0.876949i	$-0.659575\pi$
$911$	−29.0106	−0.961165	−0.480583	+	0.876949i	$0.659575\pi$
$912$	0	0
$913$	−34.7594	−1.15037
$914$	0	0
$915$	4.38576	0.144989
$916$	0	0
$917$	−41.0670	−1.35615
$918$	0	0
$919$	−5.02133	−0.165638	−0.0828192	−	0.996565i	$-0.526392\pi$
$919$	−5.02133	−0.165638	−0.0828192	+	0.996565i	$0.526392\pi$
$920$	0	0
$921$	−12.4813	−0.411273
$922$	0	0
$923$	−12.9332	−0.425703
$924$	0	0
$925$	26.3770	0.867271
$926$	0	0
$927$	−19.7258	−0.647881
$928$	0	0
$929$	36.7732	1.20649	0.603245	−	0.797556i	$-0.293874\pi$
$929$	36.7732	1.20649	0.603245	+	0.797556i	$0.293874\pi$
$930$	0	0
$931$	0.544389	0.0178416
$932$	0	0
$933$	23.5101	0.769687
$934$	0	0
$935$	−90.8348	−2.97061
$936$	0	0
$937$	24.0687	0.786290	0.393145	−	0.919476i	$-0.371387\pi$
$937$	24.0687	0.786290	0.393145	+	0.919476i	$0.371387\pi$
$938$	0	0
$939$	1.41010	0.0460168
$940$	0	0
$941$	21.9684	0.716148	0.358074	−	0.933693i	$-0.383433\pi$
$941$	21.9684	0.716148	0.358074	+	0.933693i	$0.383433\pi$
$942$	0	0
$943$	65.2965	2.12635
$944$	0	0
$945$	8.34225	0.271374
$946$	0	0
$947$	25.5873	0.831475	0.415738	−	0.909485i	$-0.363523\pi$
$947$	25.5873	0.831475	0.415738	+	0.909485i	$0.363523\pi$
$948$	0	0
$949$	11.5657	0.375437
$950$	0	0
$951$	3.71969	0.120619
$952$	0	0
$953$	−43.7377	−1.41680	−0.708402	−	0.705809i	$-0.750584\pi$
$953$	−43.7377	−1.41680	−0.708402	+	0.705809i	$0.750584\pi$
$954$	0	0
$955$	61.6466	1.99484
$956$	0	0
$957$	23.0041	0.743618
$958$	0	0
$959$	−3.07588	−0.0993253
$960$	0	0
$961$	36.1882	1.16736
$962$	0	0
$963$	−9.80405	−0.315931
$964$	0	0
$965$	−54.8501	−1.76569
$966$	0	0
$967$	32.5252	1.04594	0.522970	−	0.852351i	$-0.324824\pi$
$967$	32.5252	1.04594	0.522970	+	0.852351i	$0.324824\pi$
$968$	0	0
$969$	2.59114	0.0832393
$970$	0	0
$971$	−56.4363	−1.81113	−0.905563	−	0.424212i	$-0.860551\pi$
$971$	−56.4363	−1.81113	−0.905563	+	0.424212i	$0.860551\pi$
$972$	0	0
$973$	−24.0326	−0.770450
$974$	0	0
$975$	−6.96130	−0.222940
$976$	0	0
$977$	−22.3609	−0.715389	−0.357695	−	0.933839i	$-0.616437\pi$
$977$	−22.3609	−0.715389	−0.357695	+	0.933839i	$0.616437\pi$
$978$	0	0
$979$	−54.8209	−1.75208
$980$	0	0
$981$	−5.53171	−0.176614
$982$	0	0
$983$	37.7700	1.20468	0.602338	−	0.798241i	$-0.294236\pi$
$983$	37.7700	1.20468	0.602338	+	0.798241i	$0.294236\pi$
$984$	0	0
$985$	82.9671	2.64355
$986$	0	0
$987$	22.7938	0.725534
$988$	0	0
$989$	−30.3360	−0.964628
$990$	0	0
$991$	60.9006	1.93457	0.967285	−	0.253693i	$-0.0816452\pi$
$991$	60.9006	1.93457	0.967285	+	0.253693i	$0.0816452\pi$
$992$	0	0
$993$	−8.99902	−0.285575
$994$	0	0
$995$	−5.55999	−0.176263
$996$	0	0
$997$	10.2905	0.325904	0.162952	−	0.986634i	$-0.447898\pi$
$997$	10.2905	0.325904	0.162952	+	0.986634i	$0.447898\pi$
$998$	0	0
$999$	4.16843	0.131883

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6024.2.a.q.1.2	✓	18

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6024.2.a.q.1.2	✓	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:	\( N \)	\(=\)	\( 6024 = 2^{3} \cdot 3 \cdot 251 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6024.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -3.36568 q^{5} -2.47862 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -3.36568 q^{5} -2.47862 q^{7} +1.00000 q^{9} -6.62075 q^{11} -1.10011 q^{13} -3.36568 q^{15} -4.07636 q^{17} -0.635651 q^{19} -2.47862 q^{21} -8.21898 q^{23} +6.32781 q^{25} +1.00000 q^{27} -3.47455 q^{29} +8.19684 q^{31} -6.62075 q^{33} +8.34225 q^{35} +4.16843 q^{37} -1.10011 q^{39} -7.94459 q^{41} +3.69096 q^{43} -3.36568 q^{45} -9.19615 q^{47} -0.856428 q^{49} -4.07636 q^{51} -9.69155 q^{53} +22.2833 q^{55} -0.635651 q^{57} +2.89156 q^{59} -1.30308 q^{61} -2.47862 q^{63} +3.70263 q^{65} +0.588636 q^{67} -8.21898 q^{69} +11.7563 q^{71} -10.5132 q^{73} +6.32781 q^{75} +16.4103 q^{77} +2.94961 q^{79} +1.00000 q^{81} +5.25007 q^{83} +13.7197 q^{85} -3.47455 q^{87} +8.28016 q^{89} +2.72676 q^{91} +8.19684 q^{93} +2.13940 q^{95} -4.92699 q^{97} -6.62075 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q + 18 q^{3} + q^{5} + 7 q^{7} + 18 q^{9}+O(q^{10}) \) 18 * q + 18 * q^3 + q^5 + 7 * q^7 + 18 * q^9	\( 18 q + 18 q^{3} + q^{5} + 7 q^{7} + 18 q^{9} + 8 q^{11} + 2 q^{13} + q^{15} + 2 q^{17} + 21 q^{19} + 7 q^{21} - 2 q^{23} + 25 q^{25} + 18 q^{27} + 5 q^{29} + 23 q^{31} + 8 q^{33} + 17 q^{35} + 15 q^{37} + 2 q^{39} + 20 q^{41} + 37 q^{43} + q^{45} - q^{47} + 33 q^{49} + 2 q^{51} + 2 q^{53} + 26 q^{55} + 21 q^{57} + 24 q^{59} + 20 q^{61} + 7 q^{63} + 26 q^{65} + 49 q^{67} - 2 q^{69} + 15 q^{71} + 15 q^{73} + 25 q^{75} + 6 q^{77} + 23 q^{79} + 18 q^{81} + 38 q^{83} + 21 q^{85} + 5 q^{87} + 31 q^{89} + 48 q^{91} + 23 q^{93} + 13 q^{95} + 25 q^{97} + 8 q^{99}+O(q^{100}) \) 18 * q + 18 * q^3 + q^5 + 7 * q^7 + 18 * q^9 + 8 * q^11 + 2 * q^13 + q^15 + 2 * q^17 + 21 * q^19 + 7 * q^21 - 2 * q^23 + 25 * q^25 + 18 * q^27 + 5 * q^29 + 23 * q^31 + 8 * q^33 + 17 * q^35 + 15 * q^37 + 2 * q^39 + 20 * q^41 + 37 * q^43 + q^45 - q^47 + 33 * q^49 + 2 * q^51 + 2 * q^53 + 26 * q^55 + 21 * q^57 + 24 * q^59 + 20 * q^61 + 7 * q^63 + 26 * q^65 + 49 * q^67 - 2 * q^69 + 15 * q^71 + 15 * q^73 + 25 * q^75 + 6 * q^77 + 23 * q^79 + 18 * q^81 + 38 * q^83 + 21 * q^85 + 5 * q^87 + 31 * q^89 + 48 * q^91 + 23 * q^93 + 13 * q^95 + 25 * q^97 + 8 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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