LMFDB - Embedded newform 6024.2.a.n.1.14

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:	$1$
Dimension:	$14$
Coefficient field:	$\mathbb{Q}[x]/(x^{14} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{14} - 7 x^{13} - 22 x^{12} + 214 x^{11} + 91 x^{10} - 2481 x^{9} + 1285 x^{8} + 13253 x^{7} + \cdots - 5832 $ x^14 - 7x^13 - 22x^12 + 214x^11 + 91x^10 - 2481x^9 + 1285x^8 + 13253x^7 - 14287x^6 - 29907x^5 + 49572x^4 + 10044x^3 - 53784x^2 + 32076*x - 5832
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			1.14
Root			$5.38033$ 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)

$f(q)$	$=$	$q-1.00000 q^{3} +4.38033 q^{5} +3.18300 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +4.38033 q^{5} +3.18300 q^{7} +1.00000 q^{9} -2.48542 q^{11} -5.09405 q^{13} -4.38033 q^{15} -7.14597 q^{17} -3.01153 q^{19} -3.18300 q^{21} +5.68695 q^{23} +14.1873 q^{25} -1.00000 q^{27} -7.79309 q^{29} -9.08829 q^{31} +2.48542 q^{33} +13.9426 q^{35} -4.53681 q^{37} +5.09405 q^{39} -3.01216 q^{41} +5.11251 q^{43} +4.38033 q^{45} -7.32621 q^{47} +3.13147 q^{49} +7.14597 q^{51} -13.7925 q^{53} -10.8870 q^{55} +3.01153 q^{57} +7.09125 q^{59} -4.19509 q^{61} +3.18300 q^{63} -22.3136 q^{65} +0.195217 q^{67} -5.68695 q^{69} -9.61479 q^{71} -10.8768 q^{73} -14.1873 q^{75} -7.91108 q^{77} -12.1514 q^{79} +1.00000 q^{81} -0.215339 q^{83} -31.3017 q^{85} +7.79309 q^{87} -9.49827 q^{89} -16.2143 q^{91} +9.08829 q^{93} -13.1915 q^{95} +5.25570 q^{97} -2.48542 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 14 q - 14 q^{3} - 7 q^{5} + q^{7} + 14 q^{9}+O(q^{10}) $ 14 * q - 14 * q^3 - 7 * q^5 + q^7 + 14 * q^9	$ 14 q - 14 q^{3} - 7 q^{5} + q^{7} + 14 q^{9} + 10 q^{11} - 9 q^{13} + 7 q^{15} - 22 q^{17} - 6 q^{19} - q^{21} + 3 q^{23} + 23 q^{25} - 14 q^{27} - 12 q^{29} - 13 q^{31} - 10 q^{33} + 23 q^{35} + 5 q^{37} + 9 q^{39} - 52 q^{41} + 16 q^{43} - 7 q^{45} + q^{47} + 9 q^{49} + 22 q^{51} - 13 q^{53} - 12 q^{55} + 6 q^{57} + 12 q^{59} - 20 q^{61} + q^{63} - 40 q^{65} + 21 q^{67} - 3 q^{69} - 5 q^{71} - 14 q^{73} - 23 q^{75} - 14 q^{77} - 23 q^{79} + 14 q^{81} + 25 q^{83} - 11 q^{85} + 12 q^{87} - 79 q^{89} + 6 q^{91} + 13 q^{93} + 3 q^{95} - 17 q^{97} + 10 q^{99}+O(q^{100}) $ 14 * q - 14 * q^3 - 7 * q^5 + q^7 + 14 * q^9 + 10 * q^11 - 9 * q^13 + 7 * q^15 - 22 * q^17 - 6 * q^19 - q^21 + 3 * q^23 + 23 * q^25 - 14 * q^27 - 12 * q^29 - 13 * q^31 - 10 * q^33 + 23 * q^35 + 5 * q^37 + 9 * q^39 - 52 * q^41 + 16 * q^43 - 7 * q^45 + q^47 + 9 * q^49 + 22 * q^51 - 13 * q^53 - 12 * q^55 + 6 * q^57 + 12 * q^59 - 20 * q^61 + q^63 - 40 * q^65 + 21 * q^67 - 3 * q^69 - 5 * q^71 - 14 * q^73 - 23 * q^75 - 14 * q^77 - 23 * q^79 + 14 * q^81 + 25 * q^83 - 11 * q^85 + 12 * q^87 - 79 * q^89 + 6 * q^91 + 13 * q^93 + 3 * q^95 - 17 * q^97 + 10 * 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
$5$	4.38033	1.95894	0.979471	−	0.201585i	$-0.0646092\pi$
$5$	4.38033	1.95894	0.979471	+	0.201585i	$0.0646092\pi$
$6$	0	0
$7$	3.18300	1.20306	0.601530	−	0.798850i	$-0.294558\pi$
$7$	3.18300	1.20306	0.601530	+	0.798850i	$0.294558\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−2.48542	−0.749382	−0.374691	−	0.927150i	$-0.622251\pi$
$11$	−2.48542	−0.749382	−0.374691	+	0.927150i	$0.622251\pi$
$12$	0	0
$13$	−5.09405	−1.41283	−0.706417	−	0.707795i	$-0.749690\pi$
$13$	−5.09405	−1.41283	−0.706417	+	0.707795i	$0.749690\pi$
$14$	0	0
$15$	−4.38033	−1.13100
$16$	0	0
$17$	−7.14597	−1.73315	−0.866576	−	0.499045i	$-0.833684\pi$
$17$	−7.14597	−1.73315	−0.866576	+	0.499045i	$0.833684\pi$
$18$	0	0
$19$	−3.01153	−0.690891	−0.345446	−	0.938439i	$-0.612272\pi$
$19$	−3.01153	−0.690891	−0.345446	+	0.938439i	$0.612272\pi$
$20$	0	0
$21$	−3.18300	−0.694587
$22$	0	0
$23$	5.68695	1.18581	0.592905	−	0.805272i	$-0.297981\pi$
$23$	5.68695	1.18581	0.592905	+	0.805272i	$0.297981\pi$
$24$	0	0
$25$	14.1873	2.83745
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−7.79309	−1.44714	−0.723570	−	0.690251i	$-0.757500\pi$
$29$	−7.79309	−1.44714	−0.723570	+	0.690251i	$0.757500\pi$
$30$	0	0
$31$	−9.08829	−1.63230	−0.816152	−	0.577837i	$-0.803897\pi$
$31$	−9.08829	−1.63230	−0.816152	+	0.577837i	$0.803897\pi$
$32$	0	0
$33$	2.48542	0.432656
$34$	0	0
$35$	13.9426	2.35672
$36$	0	0
$37$	−4.53681	−0.745846	−0.372923	−	0.927862i	$-0.621644\pi$
$37$	−4.53681	−0.745846	−0.372923	+	0.927862i	$0.621644\pi$
$38$	0	0
$39$	5.09405	0.815701
$40$	0	0
$41$	−3.01216	−0.470421	−0.235210	−	0.971944i	$-0.575578\pi$
$41$	−3.01216	−0.470421	−0.235210	+	0.971944i	$0.575578\pi$
$42$	0	0
$43$	5.11251	0.779650	0.389825	−	0.920889i	$-0.372535\pi$
$43$	5.11251	0.779650	0.389825	+	0.920889i	$0.372535\pi$
$44$	0	0
$45$	4.38033	0.652981
$46$	0	0
$47$	−7.32621	−1.06864	−0.534319	−	0.845283i	$-0.679432\pi$
$47$	−7.32621	−1.06864	−0.534319	+	0.845283i	$0.679432\pi$
$48$	0	0
$49$	3.13147	0.447352
$50$	0	0
$51$	7.14597	1.00064
$52$	0	0
$53$	−13.7925	−1.89454	−0.947272	−	0.320430i	$-0.896173\pi$
$53$	−13.7925	−1.89454	−0.947272	+	0.320430i	$0.896173\pi$
$54$	0	0
$55$	−10.8870	−1.46800
$56$	0	0
$57$	3.01153	0.398886
$58$	0	0
$59$	7.09125	0.923202	0.461601	−	0.887088i	$-0.347275\pi$
$59$	7.09125	0.923202	0.461601	+	0.887088i	$0.347275\pi$
$60$	0	0
$61$	−4.19509	−0.537126	−0.268563	−	0.963262i	$-0.586549\pi$
$61$	−4.19509	−0.537126	−0.268563	+	0.963262i	$0.586549\pi$
$62$	0	0
$63$	3.18300	0.401020
$64$	0	0
$65$	−22.3136	−2.76766
$66$	0	0
$67$	0.195217	0.0238496	0.0119248	−	0.999929i	$-0.496204\pi$
$67$	0.195217	0.0238496	0.0119248	+	0.999929i	$0.496204\pi$
$68$	0	0
$69$	−5.68695	−0.684628
$70$	0	0
$71$	−9.61479	−1.14107	−0.570533	−	0.821275i	$-0.693263\pi$
$71$	−9.61479	−1.14107	−0.570533	+	0.821275i	$0.693263\pi$
$72$	0	0
$73$	−10.8768	−1.27304	−0.636518	−	0.771262i	$-0.719626\pi$
$73$	−10.8768	−1.27304	−0.636518	+	0.771262i	$0.719626\pi$
$74$	0	0
$75$	−14.1873	−1.63820
$76$	0	0
$77$	−7.91108	−0.901552
$78$	0	0
$79$	−12.1514	−1.36714	−0.683572	−	0.729883i	$-0.739574\pi$
$79$	−12.1514	−1.36714	−0.683572	+	0.729883i	$0.739574\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−0.215339	−0.0236365	−0.0118183	−	0.999930i	$-0.503762\pi$
$83$	−0.215339	−0.0236365	−0.0118183	+	0.999930i	$0.503762\pi$
$84$	0	0
$85$	−31.3017	−3.39514
$86$	0	0
$87$	7.79309	0.835507
$88$	0	0
$89$	−9.49827	−1.00681	−0.503407	−	0.864049i	$-0.667920\pi$
$89$	−9.49827	−1.00681	−0.503407	+	0.864049i	$0.667920\pi$
$90$	0	0
$91$	−16.2143	−1.69972
$92$	0	0
$93$	9.08829	0.942411
$94$	0	0
$95$	−13.1915	−1.35342
$96$	0	0
$97$	5.25570	0.533636	0.266818	−	0.963747i	$-0.414028\pi$
$97$	5.25570	0.533636	0.266818	+	0.963747i	$0.414028\pi$
$98$	0	0
$99$	−2.48542	−0.249794
$100$	0	0
$101$	18.4245	1.83330	0.916651	−	0.399688i	$-0.130882\pi$
$101$	18.4245	1.83330	0.916651	+	0.399688i	$0.130882\pi$
$102$	0	0
$103$	−2.98651	−0.294270	−0.147135	−	0.989116i	$-0.547005\pi$
$103$	−2.98651	−0.294270	−0.147135	+	0.989116i	$0.547005\pi$
$104$	0	0
$105$	−13.9426	−1.36066
$106$	0	0
$107$	18.4911	1.78760	0.893799	−	0.448467i	$-0.148030\pi$
$107$	18.4911	1.78760	0.893799	+	0.448467i	$0.148030\pi$
$108$	0	0
$109$	14.1175	1.35221	0.676106	−	0.736805i	$-0.263666\pi$
$109$	14.1175	1.35221	0.676106	+	0.736805i	$0.263666\pi$
$110$	0	0
$111$	4.53681	0.430615
$112$	0	0
$113$	−9.38864	−0.883209	−0.441604	−	0.897210i	$-0.645591\pi$
$113$	−9.38864	−0.883209	−0.441604	+	0.897210i	$0.645591\pi$
$114$	0	0
$115$	24.9107	2.32293
$116$	0	0
$117$	−5.09405	−0.470945
$118$	0	0
$119$	−22.7456	−2.08509
$120$	0	0
$121$	−4.82269	−0.438426
$122$	0	0
$123$	3.01216	0.271598
$124$	0	0
$125$	40.2433	3.59947
$126$	0	0
$127$	4.32695	0.383955	0.191977	−	0.981399i	$-0.438510\pi$
$127$	4.32695	0.383955	0.191977	+	0.981399i	$0.438510\pi$
$128$	0	0
$129$	−5.11251	−0.450131
$130$	0	0
$131$	5.68193	0.496433	0.248216	−	0.968705i	$-0.420156\pi$
$131$	5.68193	0.496433	0.248216	+	0.968705i	$0.420156\pi$
$132$	0	0
$133$	−9.58568	−0.831184
$134$	0	0
$135$	−4.38033	−0.376999
$136$	0	0
$137$	−2.38093	−0.203416	−0.101708	−	0.994814i	$-0.532431\pi$
$137$	−2.38093	−0.203416	−0.101708	+	0.994814i	$0.532431\pi$
$138$	0	0
$139$	18.6400	1.58103	0.790514	−	0.612444i	$-0.209814\pi$
$139$	18.6400	1.58103	0.790514	+	0.612444i	$0.209814\pi$
$140$	0	0
$141$	7.32621	0.616978
$142$	0	0
$143$	12.6609	1.05875
$144$	0	0
$145$	−34.1363	−2.83486
$146$	0	0
$147$	−3.13147	−0.258279
$148$	0	0
$149$	−6.42269	−0.526168	−0.263084	−	0.964773i	$-0.584740\pi$
$149$	−6.42269	−0.526168	−0.263084	+	0.964773i	$0.584740\pi$
$150$	0	0
$151$	14.8567	1.20902	0.604512	−	0.796596i	$-0.293368\pi$
$151$	14.8567	1.20902	0.604512	+	0.796596i	$0.293368\pi$
$152$	0	0
$153$	−7.14597	−0.577717
$154$	0	0
$155$	−39.8097	−3.19759
$156$	0	0
$157$	−16.2330	−1.29553	−0.647765	−	0.761840i	$-0.724296\pi$
$157$	−16.2330	−1.29553	−0.647765	+	0.761840i	$0.724296\pi$
$158$	0	0
$159$	13.7925	1.09382
$160$	0	0
$161$	18.1015	1.42660
$162$	0	0
$163$	−7.05138	−0.552307	−0.276154	−	0.961114i	$-0.589060\pi$
$163$	−7.05138	−0.552307	−0.276154	+	0.961114i	$0.589060\pi$
$164$	0	0
$165$	10.8870	0.847548
$166$	0	0
$167$	−11.0413	−0.854402	−0.427201	−	0.904157i	$-0.640500\pi$
$167$	−11.0413	−0.854402	−0.427201	+	0.904157i	$0.640500\pi$
$168$	0	0
$169$	12.9493	0.996102
$170$	0	0
$171$	−3.01153	−0.230297
$172$	0	0
$173$	−4.36063	−0.331533	−0.165766	−	0.986165i	$-0.553010\pi$
$173$	−4.36063	−0.331533	−0.165766	+	0.986165i	$0.553010\pi$
$174$	0	0
$175$	45.1580	3.41363
$176$	0	0
$177$	−7.09125	−0.533011
$178$	0	0
$179$	7.05542	0.527347	0.263673	−	0.964612i	$-0.415066\pi$
$179$	7.05542	0.527347	0.263673	+	0.964612i	$0.415066\pi$
$180$	0	0
$181$	−5.25308	−0.390459	−0.195229	−	0.980758i	$-0.562545\pi$
$181$	−5.25308	−0.390459	−0.195229	+	0.980758i	$0.562545\pi$
$182$	0	0
$183$	4.19509	0.310110
$184$	0	0
$185$	−19.8727	−1.46107
$186$	0	0
$187$	17.7607	1.29879
$188$	0	0
$189$	−3.18300	−0.231529
$190$	0	0
$191$	18.3724	1.32938	0.664691	−	0.747118i	$-0.268563\pi$
$191$	18.3724	1.32938	0.664691	+	0.747118i	$0.268563\pi$
$192$	0	0
$193$	26.2709	1.89102	0.945511	−	0.325589i	$-0.105562\pi$
$193$	26.2709	1.89102	0.945511	+	0.325589i	$0.105562\pi$
$194$	0	0
$195$	22.3136	1.59791
$196$	0	0
$197$	5.11262	0.364259	0.182129	−	0.983275i	$-0.441701\pi$
$197$	5.11262	0.364259	0.182129	+	0.983275i	$0.441701\pi$
$198$	0	0
$199$	−11.3643	−0.805594	−0.402797	−	0.915289i	$-0.631962\pi$
$199$	−11.3643	−0.805594	−0.402797	+	0.915289i	$0.631962\pi$
$200$	0	0
$201$	−0.195217	−0.0137696
$202$	0	0
$203$	−24.8054	−1.74100
$204$	0	0
$205$	−13.1943	−0.921527
$206$	0	0
$207$	5.68695	0.395270
$208$	0	0
$209$	7.48491	0.517742
$210$	0	0
$211$	19.8807	1.36864	0.684322	−	0.729180i	$-0.260098\pi$
$211$	19.8807	1.36864	0.684322	+	0.729180i	$0.260098\pi$
$212$	0	0
$213$	9.61479	0.658795
$214$	0	0
$215$	22.3945	1.52729
$216$	0	0
$217$	−28.9280	−1.96376
$218$	0	0
$219$	10.8768	0.734988
$220$	0	0
$221$	36.4019	2.44866
$222$	0	0
$223$	−7.00735	−0.469247	−0.234623	−	0.972086i	$-0.575386\pi$
$223$	−7.00735	−0.469247	−0.234623	+	0.972086i	$0.575386\pi$
$224$	0	0
$225$	14.1873	0.945818
$226$	0	0
$227$	25.2863	1.67831	0.839156	−	0.543891i	$-0.183050\pi$
$227$	25.2863	1.67831	0.839156	+	0.543891i	$0.183050\pi$
$228$	0	0
$229$	12.8250	0.847500	0.423750	−	0.905779i	$-0.360713\pi$
$229$	12.8250	0.847500	0.423750	+	0.905779i	$0.360713\pi$
$230$	0	0
$231$	7.91108	0.520511
$232$	0	0
$233$	−14.4063	−0.943787	−0.471893	−	0.881656i	$-0.656429\pi$
$233$	−14.4063	−0.943787	−0.471893	+	0.881656i	$0.656429\pi$
$234$	0	0
$235$	−32.0912	−2.09340
$236$	0	0
$237$	12.1514	0.789321
$238$	0	0
$239$	1.17595	0.0760656	0.0380328	−	0.999276i	$-0.487891\pi$
$239$	1.17595	0.0760656	0.0380328	+	0.999276i	$0.487891\pi$
$240$	0	0
$241$	1.08662	0.0699954	0.0349977	−	0.999387i	$-0.488858\pi$
$241$	1.08662	0.0699954	0.0349977	+	0.999387i	$0.488858\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	13.7169	0.876338
$246$	0	0
$247$	15.3409	0.976115
$248$	0	0
$249$	0.215339	0.0136466
$250$	0	0
$251$	−1.00000	−0.0631194
$251$	−1.00000	−0.0631194
$252$	0	0
$253$	−14.1345	−0.888626
$254$	0	0
$255$	31.3017	1.96019
$256$	0	0
$257$	−1.40163	−0.0874315	−0.0437157	−	0.999044i	$-0.513920\pi$
$257$	−1.40163	−0.0874315	−0.0437157	+	0.999044i	$0.513920\pi$
$258$	0	0
$259$	−14.4406	−0.897298
$260$	0	0
$261$	−7.79309	−0.482380
$262$	0	0
$263$	18.0922	1.11561	0.557805	−	0.829972i	$-0.311644\pi$
$263$	18.0922	1.11561	0.557805	+	0.829972i	$0.311644\pi$
$264$	0	0
$265$	−60.4156	−3.71130
$266$	0	0
$267$	9.49827	0.581285
$268$	0	0
$269$	−11.2989	−0.688908	−0.344454	−	0.938803i	$-0.611936\pi$
$269$	−11.2989	−0.688908	−0.344454	+	0.938803i	$0.611936\pi$
$270$	0	0
$271$	12.5073	0.759762	0.379881	−	0.925035i	$-0.375965\pi$
$271$	12.5073	0.759762	0.379881	+	0.925035i	$0.375965\pi$
$272$	0	0
$273$	16.2143	0.981336
$274$	0	0
$275$	−35.2613	−2.12634
$276$	0	0
$277$	8.60347	0.516932	0.258466	−	0.966020i	$-0.416783\pi$
$277$	8.60347	0.516932	0.258466	+	0.966020i	$0.416783\pi$
$278$	0	0
$279$	−9.08829	−0.544101
$280$	0	0
$281$	−1.99365	−0.118931	−0.0594657	−	0.998230i	$-0.518940\pi$
$281$	−1.99365	−0.118931	−0.0594657	+	0.998230i	$0.518940\pi$
$282$	0	0
$283$	−15.4527	−0.918569	−0.459285	−	0.888289i	$-0.651894\pi$
$283$	−15.4527	−0.918569	−0.459285	+	0.888289i	$0.651894\pi$
$284$	0	0
$285$	13.1915	0.781395
$286$	0	0
$287$	−9.58770	−0.565944
$288$	0	0
$289$	34.0649	2.00382
$290$	0	0
$291$	−5.25570	−0.308095
$292$	0	0
$293$	0.216096	0.0126245	0.00631224	−	0.999980i	$-0.497991\pi$
$293$	0.216096	0.0126245	0.00631224	+	0.999980i	$0.497991\pi$
$294$	0	0
$295$	31.0620	1.80850
$296$	0	0
$297$	2.48542	0.144219
$298$	0	0
$299$	−28.9696	−1.67535
$300$	0	0
$301$	16.2731	0.937966
$302$	0	0
$303$	−18.4245	−1.05846
$304$	0	0
$305$	−18.3759	−1.05220
$306$	0	0
$307$	24.4940	1.39794	0.698972	−	0.715149i	$-0.253641\pi$
$307$	24.4940	1.39794	0.698972	+	0.715149i	$0.253641\pi$
$308$	0	0
$309$	2.98651	0.169897
$310$	0	0
$311$	−26.9909	−1.53051	−0.765255	−	0.643727i	$-0.777387\pi$
$311$	−26.9909	−1.53051	−0.765255	+	0.643727i	$0.777387\pi$
$312$	0	0
$313$	2.62866	0.148581	0.0742903	−	0.997237i	$-0.476331\pi$
$313$	2.62866	0.148581	0.0742903	+	0.997237i	$0.476331\pi$
$314$	0	0
$315$	13.9426	0.785575
$316$	0	0
$317$	18.4658	1.03714	0.518571	−	0.855035i	$-0.326464\pi$
$317$	18.4658	1.03714	0.518571	+	0.855035i	$0.326464\pi$
$318$	0	0
$319$	19.3691	1.08446
$320$	0	0
$321$	−18.4911	−1.03207
$322$	0	0
$323$	21.5203	1.19742
$324$	0	0
$325$	−72.2706	−4.00885
$326$	0	0
$327$	−14.1175	−0.780700
$328$	0	0
$329$	−23.3193	−1.28564
$330$	0	0
$331$	9.00349	0.494877	0.247438	−	0.968904i	$-0.420411\pi$
$331$	9.00349	0.494877	0.247438	+	0.968904i	$0.420411\pi$
$332$	0	0
$333$	−4.53681	−0.248615
$334$	0	0
$335$	0.855116	0.0467200
$336$	0	0
$337$	−22.4839	−1.22478	−0.612389	−	0.790556i	$-0.709792\pi$
$337$	−22.4839	−1.22478	−0.612389	+	0.790556i	$0.709792\pi$
$338$	0	0
$339$	9.38864	0.509921
$340$	0	0
$341$	22.5882	1.22322
$342$	0	0
$343$	−12.3135	−0.664868
$344$	0	0
$345$	−24.9107	−1.34115
$346$	0	0
$347$	−9.87566	−0.530153	−0.265077	−	0.964227i	$-0.585397\pi$
$347$	−9.87566	−0.530153	−0.265077	+	0.964227i	$0.585397\pi$
$348$	0	0
$349$	−15.2989	−0.818932	−0.409466	−	0.912325i	$-0.634285\pi$
$349$	−15.2989	−0.818932	−0.409466	+	0.912325i	$0.634285\pi$
$350$	0	0
$351$	5.09405	0.271900
$352$	0	0
$353$	18.2570	0.971724	0.485862	−	0.874035i	$-0.338506\pi$
$353$	18.2570	0.971724	0.485862	+	0.874035i	$0.338506\pi$
$354$	0	0
$355$	−42.1160	−2.23528
$356$	0	0
$357$	22.7456	1.20382
$358$	0	0
$359$	11.0506	0.583228	0.291614	−	0.956536i	$-0.405808\pi$
$359$	11.0506	0.583228	0.291614	+	0.956536i	$0.405808\pi$
$360$	0	0
$361$	−9.93071	−0.522669
$362$	0	0
$363$	4.82269	0.253125
$364$	0	0
$365$	−47.6440	−2.49380
$366$	0	0
$367$	15.5273	0.810517	0.405258	−	0.914202i	$-0.367182\pi$
$367$	15.5273	0.810517	0.405258	+	0.914202i	$0.367182\pi$
$368$	0	0
$369$	−3.01216	−0.156807
$370$	0	0
$371$	−43.9015	−2.27925
$372$	0	0
$373$	−4.47032	−0.231464	−0.115732	−	0.993280i	$-0.536921\pi$
$373$	−4.47032	−0.231464	−0.115732	+	0.993280i	$0.536921\pi$
$374$	0	0
$375$	−40.2433	−2.07815
$376$	0	0
$377$	39.6984	2.04457
$378$	0	0
$379$	−7.88379	−0.404963	−0.202481	−	0.979286i	$-0.564901\pi$
$379$	−7.88379	−0.404963	−0.202481	+	0.979286i	$0.564901\pi$
$380$	0	0
$381$	−4.32695	−0.221676
$382$	0	0
$383$	11.5549	0.590427	0.295213	−	0.955431i	$-0.404609\pi$
$383$	11.5549	0.590427	0.295213	+	0.955431i	$0.404609\pi$
$384$	0	0
$385$	−34.6531	−1.76609
$386$	0	0
$387$	5.11251	0.259883
$388$	0	0
$389$	26.7050	1.35400	0.676998	−	0.735985i	$-0.263281\pi$
$389$	26.7050	1.35400	0.676998	+	0.735985i	$0.263281\pi$
$390$	0	0
$391$	−40.6388	−2.05519
$392$	0	0
$393$	−5.68193	−0.286616
$394$	0	0
$395$	−53.2273	−2.67816
$396$	0	0
$397$	−11.9967	−0.602099	−0.301049	−	0.953609i	$-0.597337\pi$
$397$	−11.9967	−0.602099	−0.301049	+	0.953609i	$0.597337\pi$
$398$	0	0
$399$	9.58568	0.479884
$400$	0	0
$401$	−8.45352	−0.422149	−0.211074	−	0.977470i	$-0.567696\pi$
$401$	−8.45352	−0.422149	−0.211074	+	0.977470i	$0.567696\pi$
$402$	0	0
$403$	46.2962	2.30618
$404$	0	0
$405$	4.38033	0.217660
$406$	0	0
$407$	11.2759	0.558924
$408$	0	0
$409$	−2.76452	−0.136697	−0.0683483	−	0.997662i	$-0.521773\pi$
$409$	−2.76452	−0.136697	−0.0683483	+	0.997662i	$0.521773\pi$
$410$	0	0
$411$	2.38093	0.117442
$412$	0	0
$413$	22.5714	1.11067
$414$	0	0
$415$	−0.943256	−0.0463026
$416$	0	0
$417$	−18.6400	−0.912807
$418$	0	0
$419$	8.72103	0.426050	0.213025	−	0.977047i	$-0.431668\pi$
$419$	8.72103	0.426050	0.213025	+	0.977047i	$0.431668\pi$
$420$	0	0
$421$	33.0669	1.61158	0.805791	−	0.592200i	$-0.201740\pi$
$421$	33.0669	1.61158	0.805791	+	0.592200i	$0.201740\pi$
$422$	0	0
$423$	−7.32621	−0.356213
$424$	0	0
$425$	−101.382	−4.91774
$426$	0	0
$427$	−13.3530	−0.646195
$428$	0	0
$429$	−12.6609	−0.611272
$430$	0	0
$431$	−20.0249	−0.964565	−0.482283	−	0.876016i	$-0.660192\pi$
$431$	−20.0249	−0.964565	−0.482283	+	0.876016i	$0.660192\pi$
$432$	0	0
$433$	38.1315	1.83248	0.916242	−	0.400626i	$-0.131207\pi$
$433$	38.1315	1.83248	0.916242	+	0.400626i	$0.131207\pi$
$434$	0	0
$435$	34.1363	1.63671
$436$	0	0
$437$	−17.1264	−0.819266
$438$	0	0
$439$	−16.9590	−0.809409	−0.404704	−	0.914448i	$-0.632626\pi$
$439$	−16.9590	−0.809409	−0.404704	+	0.914448i	$0.632626\pi$
$440$	0	0
$441$	3.13147	0.149117
$442$	0	0
$443$	−19.2186	−0.913105	−0.456552	−	0.889697i	$-0.650916\pi$
$443$	−19.2186	−0.913105	−0.456552	+	0.889697i	$0.650916\pi$
$444$	0	0
$445$	−41.6055	−1.97229
$446$	0	0
$447$	6.42269	0.303783
$448$	0	0
$449$	−19.5240	−0.921393	−0.460696	−	0.887558i	$-0.652400\pi$
$449$	−19.5240	−0.921393	−0.460696	+	0.887558i	$0.652400\pi$
$450$	0	0
$451$	7.48649	0.352525
$452$	0	0
$453$	−14.8567	−0.698030
$454$	0	0
$455$	−71.0241	−3.32966
$456$	0	0
$457$	−16.2307	−0.759239	−0.379619	−	0.925143i	$-0.623945\pi$
$457$	−16.2307	−0.759239	−0.379619	+	0.925143i	$0.623945\pi$
$458$	0	0
$459$	7.14597	0.333545
$460$	0	0
$461$	−3.49231	−0.162653	−0.0813265	−	0.996688i	$-0.525916\pi$
$461$	−3.49231	−0.162653	−0.0813265	+	0.996688i	$0.525916\pi$
$462$	0	0
$463$	18.5241	0.860886	0.430443	−	0.902618i	$-0.358357\pi$
$463$	18.5241	0.860886	0.430443	+	0.902618i	$0.358357\pi$
$464$	0	0
$465$	39.8097	1.84613
$466$	0	0
$467$	−23.6047	−1.09229	−0.546147	−	0.837689i	$-0.683906\pi$
$467$	−23.6047	−1.09229	−0.546147	+	0.837689i	$0.683906\pi$
$468$	0	0
$469$	0.621376	0.0286925
$470$	0	0
$471$	16.2330	0.747975
$472$	0	0
$473$	−12.7067	−0.584256
$474$	0	0
$475$	−42.7253	−1.96037
$476$	0	0
$477$	−13.7925	−0.631515
$478$	0	0
$479$	−25.4706	−1.16378	−0.581891	−	0.813267i	$-0.697687\pi$
$479$	−25.4706	−1.16378	−0.581891	+	0.813267i	$0.697687\pi$
$480$	0	0
$481$	23.1107	1.05376
$482$	0	0
$483$	−18.1015	−0.823648
$484$	0	0
$485$	23.0217	1.04536
$486$	0	0
$487$	−11.2632	−0.510383	−0.255191	−	0.966891i	$-0.582138\pi$
$487$	−11.2632	−0.510383	−0.255191	+	0.966891i	$0.582138\pi$
$488$	0	0
$489$	7.05138	0.318875
$490$	0	0
$491$	−16.1412	−0.728440	−0.364220	−	0.931313i	$-0.618664\pi$
$491$	−16.1412	−0.728440	−0.364220	+	0.931313i	$0.618664\pi$
$492$	0	0
$493$	55.6892	2.50811
$494$	0	0
$495$	−10.8870	−0.489332
$496$	0	0
$497$	−30.6039	−1.37277
$498$	0	0
$499$	1.82639	0.0817605	0.0408802	−	0.999164i	$-0.486984\pi$
$499$	1.82639	0.0817605	0.0408802	+	0.999164i	$0.486984\pi$
$500$	0	0
$501$	11.0413	0.493289
$502$	0	0
$503$	−18.0792	−0.806113	−0.403057	−	0.915175i	$-0.632052\pi$
$503$	−18.0792	−0.806113	−0.403057	+	0.915175i	$0.632052\pi$
$504$	0	0
$505$	80.7052	3.59133
$506$	0	0
$507$	−12.9493	−0.575100
$508$	0	0
$509$	−26.9612	−1.19503	−0.597517	−	0.801856i	$-0.703846\pi$
$509$	−26.9612	−1.19503	−0.597517	+	0.801856i	$0.703846\pi$
$510$	0	0
$511$	−34.6209	−1.53154
$512$	0	0
$513$	3.01153	0.132962
$514$	0	0
$515$	−13.0819	−0.576457
$516$	0	0
$517$	18.2087	0.800819
$518$	0	0
$519$	4.36063	0.191411
$520$	0	0
$521$	0.585740	0.0256617	0.0128309	−	0.999918i	$-0.495916\pi$
$521$	0.585740	0.0256617	0.0128309	+	0.999918i	$0.495916\pi$
$522$	0	0
$523$	−12.5308	−0.547934	−0.273967	−	0.961739i	$-0.588336\pi$
$523$	−12.5308	−0.547934	−0.273967	+	0.961739i	$0.588336\pi$
$524$	0	0
$525$	−45.1580	−1.97086
$526$	0	0
$527$	64.9446	2.82903
$528$	0	0
$529$	9.34138	0.406147
$530$	0	0
$531$	7.09125	0.307734
$532$	0	0
$533$	15.3441	0.664627
$534$	0	0
$535$	80.9969	3.50180
$536$	0	0
$537$	−7.05542	−0.304464
$538$	0	0
$539$	−7.78301	−0.335238
$540$	0	0
$541$	−5.16294	−0.221972	−0.110986	−	0.993822i	$-0.535401\pi$
$541$	−5.16294	−0.221972	−0.110986	+	0.993822i	$0.535401\pi$
$542$	0	0
$543$	5.25308	0.225431
$544$	0	0
$545$	61.8393	2.64890
$546$	0	0
$547$	28.0314	1.19854	0.599269	−	0.800548i	$-0.295458\pi$
$547$	28.0314	1.19854	0.599269	+	0.800548i	$0.295458\pi$
$548$	0	0
$549$	−4.19509	−0.179042
$550$	0	0
$551$	23.4691	0.999817
$552$	0	0
$553$	−38.6780	−1.64476
$554$	0	0
$555$	19.8727	0.843549
$556$	0	0
$557$	−10.9148	−0.462477	−0.231238	−	0.972897i	$-0.574278\pi$
$557$	−10.9148	−0.462477	−0.231238	+	0.972897i	$0.574278\pi$
$558$	0	0
$559$	−26.0434	−1.10152
$560$	0	0
$561$	−17.7607	−0.749859
$562$	0	0
$563$	6.67442	0.281293	0.140647	−	0.990060i	$-0.455082\pi$
$563$	6.67442	0.281293	0.140647	+	0.990060i	$0.455082\pi$
$564$	0	0
$565$	−41.1253	−1.73015
$566$	0	0
$567$	3.18300	0.133673
$568$	0	0
$569$	−22.7926	−0.955517	−0.477758	−	0.878491i	$-0.658551\pi$
$569$	−22.7926	−0.955517	−0.477758	+	0.878491i	$0.658551\pi$
$570$	0	0
$571$	6.21930	0.260270	0.130135	−	0.991496i	$-0.458459\pi$
$571$	6.21930	0.260270	0.130135	+	0.991496i	$0.458459\pi$
$572$	0	0
$573$	−18.3724	−0.767519
$574$	0	0
$575$	80.6823	3.36468
$576$	0	0
$577$	14.6705	0.610741	0.305371	−	0.952234i	$-0.401220\pi$
$577$	14.6705	0.610741	0.305371	+	0.952234i	$0.401220\pi$
$578$	0	0
$579$	−26.2709	−1.09178
$580$	0	0
$581$	−0.685423	−0.0284362
$582$	0	0
$583$	34.2801	1.41974
$584$	0	0
$585$	−22.3136	−0.922554
$586$	0	0
$587$	−5.63636	−0.232637	−0.116319	−	0.993212i	$-0.537109\pi$
$587$	−5.63636	−0.232637	−0.116319	+	0.993212i	$0.537109\pi$
$588$	0	0
$589$	27.3696	1.12775
$590$	0	0
$591$	−5.11262	−0.210305
$592$	0	0
$593$	−2.21981	−0.0911568	−0.0455784	−	0.998961i	$-0.514513\pi$
$593$	−2.21981	−0.0911568	−0.0455784	+	0.998961i	$0.514513\pi$
$594$	0	0
$595$	−99.6332	−4.08456
$596$	0	0
$597$	11.3643	0.465110
$598$	0	0
$599$	−2.84849	−0.116386	−0.0581930	−	0.998305i	$-0.518534\pi$
$599$	−2.84849	−0.116386	−0.0581930	+	0.998305i	$0.518534\pi$
$600$	0	0
$601$	34.3516	1.40123	0.700615	−	0.713539i	$-0.252909\pi$
$601$	34.3516	1.40123	0.700615	+	0.713539i	$0.252909\pi$
$602$	0	0
$603$	0.195217	0.00794986
$604$	0	0
$605$	−21.1249	−0.858851
$606$	0	0
$607$	−32.9241	−1.33635	−0.668175	−	0.744004i	$-0.732924\pi$
$607$	−32.9241	−1.33635	−0.668175	+	0.744004i	$0.732924\pi$
$608$	0	0
$609$	24.8054	1.00516
$610$	0	0
$611$	37.3201	1.50981
$612$	0	0
$613$	4.69628	0.189681	0.0948404	−	0.995492i	$-0.469766\pi$
$613$	4.69628	0.189681	0.0948404	+	0.995492i	$0.469766\pi$
$614$	0	0
$615$	13.1943	0.532044
$616$	0	0
$617$	18.6289	0.749971	0.374986	−	0.927031i	$-0.377648\pi$
$617$	18.6289	0.749971	0.374986	+	0.927031i	$0.377648\pi$
$618$	0	0
$619$	30.4652	1.22450	0.612249	−	0.790665i	$-0.290265\pi$
$619$	30.4652	1.22450	0.612249	+	0.790665i	$0.290265\pi$
$620$	0	0
$621$	−5.68695	−0.228209
$622$	0	0
$623$	−30.2330	−1.21126
$624$	0	0
$625$	105.342	4.21369
$626$	0	0
$627$	−7.48491	−0.298918
$628$	0	0
$629$	32.4199	1.29267
$630$	0	0
$631$	9.76394	0.388696	0.194348	−	0.980933i	$-0.437741\pi$
$631$	9.76394	0.388696	0.194348	+	0.980933i	$0.437741\pi$
$632$	0	0
$633$	−19.8807	−0.790187
$634$	0	0
$635$	18.9535	0.752145
$636$	0	0
$637$	−15.9518	−0.632035
$638$	0	0
$639$	−9.61479	−0.380355
$640$	0	0
$641$	−41.6157	−1.64372	−0.821860	−	0.569690i	$-0.807063\pi$
$641$	−41.6157	−1.64372	−0.821860	+	0.569690i	$0.807063\pi$
$642$	0	0
$643$	−19.0405	−0.750885	−0.375443	−	0.926846i	$-0.622509\pi$
$643$	−19.0405	−0.750885	−0.375443	+	0.926846i	$0.622509\pi$
$644$	0	0
$645$	−22.3945	−0.881781
$646$	0	0
$647$	−19.4237	−0.763624	−0.381812	−	0.924240i	$-0.624700\pi$
$647$	−19.4237	−0.763624	−0.381812	+	0.924240i	$0.624700\pi$
$648$	0	0
$649$	−17.6247	−0.691832
$650$	0	0
$651$	28.9280	1.13378
$652$	0	0
$653$	−26.5694	−1.03974	−0.519871	−	0.854245i	$-0.674020\pi$
$653$	−26.5694	−1.03974	−0.519871	+	0.854245i	$0.674020\pi$
$654$	0	0
$655$	24.8887	0.972483
$656$	0	0
$657$	−10.8768	−0.424345
$658$	0	0
$659$	18.7323	0.729708	0.364854	−	0.931065i	$-0.381119\pi$
$659$	18.7323	0.729708	0.364854	+	0.931065i	$0.381119\pi$
$660$	0	0
$661$	−42.1969	−1.64127	−0.820634	−	0.571454i	$-0.806380\pi$
$661$	−42.1969	−1.64127	−0.820634	+	0.571454i	$0.806380\pi$
$662$	0	0
$663$	−36.4019	−1.41373
$664$	0	0
$665$	−41.9884	−1.62824
$666$	0	0
$667$	−44.3189	−1.71603
$668$	0	0
$669$	7.00735	0.270920
$670$	0	0
$671$	10.4266	0.402513
$672$	0	0
$673$	34.3445	1.32388	0.661941	−	0.749556i	$-0.269733\pi$
$673$	34.3445	1.32388	0.661941	+	0.749556i	$0.269733\pi$
$674$	0	0
$675$	−14.1873	−0.546068
$676$	0	0
$677$	44.9324	1.72689	0.863447	−	0.504440i	$-0.168301\pi$
$677$	44.9324	1.72689	0.863447	+	0.504440i	$0.168301\pi$
$678$	0	0
$679$	16.7289	0.641996
$680$	0	0
$681$	−25.2863	−0.968974
$682$	0	0
$683$	−29.6314	−1.13381	−0.566907	−	0.823782i	$-0.691860\pi$
$683$	−29.6314	−1.13381	−0.566907	+	0.823782i	$0.691860\pi$
$684$	0	0
$685$	−10.4292	−0.398480
$686$	0	0
$687$	−12.8250	−0.489304
$688$	0	0
$689$	70.2596	2.67668
$690$	0	0
$691$	−18.3462	−0.697922	−0.348961	−	0.937137i	$-0.613465\pi$
$691$	−18.3462	−0.697922	−0.348961	+	0.937137i	$0.613465\pi$
$692$	0	0
$693$	−7.91108	−0.300517
$694$	0	0
$695$	81.6495	3.09714
$696$	0	0
$697$	21.5248	0.815311
$698$	0	0
$699$	14.4063	0.544896
$700$	0	0
$701$	3.62093	0.136761	0.0683803	−	0.997659i	$-0.478217\pi$
$701$	3.62093	0.136761	0.0683803	+	0.997659i	$0.478217\pi$
$702$	0	0
$703$	13.6627	0.515299
$704$	0	0
$705$	32.0912	1.20863
$706$	0	0
$707$	58.6450	2.20557
$708$	0	0
$709$	−24.0779	−0.904265	−0.452133	−	0.891951i	$-0.649337\pi$
$709$	−24.0779	−0.904265	−0.452133	+	0.891951i	$0.649337\pi$
$710$	0	0
$711$	−12.1514	−0.455715
$712$	0	0
$713$	−51.6846	−1.93560
$714$	0	0
$715$	55.4587	2.07404
$716$	0	0
$717$	−1.17595	−0.0439165
$718$	0	0
$719$	−15.5426	−0.579643	−0.289821	−	0.957081i	$-0.593596\pi$
$719$	−15.5426	−0.579643	−0.289821	+	0.957081i	$0.593596\pi$
$720$	0	0
$721$	−9.50605	−0.354024
$722$	0	0
$723$	−1.08662	−0.0404119
$724$	0	0
$725$	−110.563	−4.10619
$726$	0	0
$727$	0.674572	0.0250185	0.0125093	−	0.999922i	$-0.496018\pi$
$727$	0.674572	0.0250185	0.0125093	+	0.999922i	$0.496018\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−36.5338	−1.35125
$732$	0	0
$733$	−27.7027	−1.02322	−0.511611	−	0.859217i	$-0.670951\pi$
$733$	−27.7027	−1.02322	−0.511611	+	0.859217i	$0.670951\pi$
$734$	0	0
$735$	−13.7169	−0.505954
$736$	0	0
$737$	−0.485197	−0.0178725
$738$	0	0
$739$	14.4129	0.530185	0.265093	−	0.964223i	$-0.414597\pi$
$739$	14.4129	0.530185	0.265093	+	0.964223i	$0.414597\pi$
$740$	0	0
$741$	−15.3409	−0.563561
$742$	0	0
$743$	−9.47164	−0.347481	−0.173740	−	0.984791i	$-0.555585\pi$
$743$	−9.47164	−0.347481	−0.173740	+	0.984791i	$0.555585\pi$
$744$	0	0
$745$	−28.1335	−1.03073
$746$	0	0
$747$	−0.215339	−0.00787884
$748$	0	0
$749$	58.8570	2.15059
$750$	0	0
$751$	−29.8611	−1.08965	−0.544823	−	0.838551i	$-0.683403\pi$
$751$	−29.8611	−1.08965	−0.544823	+	0.838551i	$0.683403\pi$
$752$	0	0
$753$	1.00000	0.0364420
$754$	0	0
$755$	65.0774	2.36841
$756$	0	0
$757$	9.89821	0.359757	0.179878	−	0.983689i	$-0.442430\pi$
$757$	9.89821	0.359757	0.179878	+	0.983689i	$0.442430\pi$
$758$	0	0
$759$	14.1345	0.513048
$760$	0	0
$761$	−50.1047	−1.81630	−0.908148	−	0.418650i	$-0.862504\pi$
$761$	−50.1047	−1.81630	−0.908148	+	0.418650i	$0.862504\pi$
$762$	0	0
$763$	44.9360	1.62679
$764$	0	0
$765$	−31.3017	−1.13171
$766$	0	0
$767$	−36.1232	−1.30433
$768$	0	0
$769$	13.4957	0.486668	0.243334	−	0.969943i	$-0.421759\pi$
$769$	13.4957	0.486668	0.243334	+	0.969943i	$0.421759\pi$
$770$	0	0
$771$	1.40163	0.0504786
$772$	0	0
$773$	−45.0969	−1.62202	−0.811012	−	0.585029i	$-0.801083\pi$
$773$	−45.0969	−1.62202	−0.811012	+	0.585029i	$0.801083\pi$
$774$	0	0
$775$	−128.938	−4.63159
$776$	0	0
$777$	14.4406	0.518055
$778$	0	0
$779$	9.07120	0.325010
$780$	0	0
$781$	23.8968	0.855095
$782$	0	0
$783$	7.79309	0.278502
$784$	0	0
$785$	−71.1056	−2.53787
$786$	0	0
$787$	−10.6532	−0.379745	−0.189872	−	0.981809i	$-0.560807\pi$
$787$	−10.6532	−0.379745	−0.189872	+	0.981809i	$0.560807\pi$
$788$	0	0
$789$	−18.0922	−0.644098
$790$	0	0
$791$	−29.8840	−1.06255
$792$	0	0
$793$	21.3700	0.758871
$794$	0	0
$795$	60.4156	2.14272
$796$	0	0
$797$	−9.43745	−0.334292	−0.167146	−	0.985932i	$-0.553455\pi$
$797$	−9.43745	−0.334292	−0.167146	+	0.985932i	$0.553455\pi$
$798$	0	0
$799$	52.3529	1.85211
$800$	0	0
$801$	−9.49827	−0.335605
$802$	0	0
$803$	27.0335	0.953991
$804$	0	0
$805$	79.2907	2.79463
$806$	0	0
$807$	11.2989	0.397741
$808$	0	0
$809$	−17.2530	−0.606584	−0.303292	−	0.952898i	$-0.598086\pi$
$809$	−17.2530	−0.606584	−0.303292	+	0.952898i	$0.598086\pi$
$810$	0	0
$811$	−51.9499	−1.82421	−0.912104	−	0.409958i	$-0.865543\pi$
$811$	−51.9499	−1.82421	−0.912104	+	0.409958i	$0.865543\pi$
$812$	0	0
$813$	−12.5073	−0.438649
$814$	0	0
$815$	−30.8874	−1.08194
$816$	0	0
$817$	−15.3965	−0.538654
$818$	0	0
$819$	−16.2143	−0.566575
$820$	0	0
$821$	20.7910	0.725611	0.362806	−	0.931865i	$-0.381819\pi$
$821$	20.7910	0.725611	0.362806	+	0.931865i	$0.381819\pi$
$822$	0	0
$823$	49.5210	1.72619	0.863097	−	0.505039i	$-0.168522\pi$
$823$	49.5210	1.72619	0.863097	+	0.505039i	$0.168522\pi$
$824$	0	0
$825$	35.2613	1.22764
$826$	0	0
$827$	−15.3343	−0.533224	−0.266612	−	0.963804i	$-0.585904\pi$
$827$	−15.3343	−0.533224	−0.266612	+	0.963804i	$0.585904\pi$
$828$	0	0
$829$	34.3694	1.19370	0.596850	−	0.802353i	$-0.296419\pi$
$829$	34.3694	1.19370	0.596850	+	0.802353i	$0.296419\pi$
$830$	0	0
$831$	−8.60347	−0.298451
$832$	0	0
$833$	−22.3774	−0.775330
$834$	0	0
$835$	−48.3645	−1.67372
$836$	0	0
$837$	9.08829	0.314137
$838$	0	0
$839$	−34.7586	−1.20000	−0.600000	−	0.800000i	$-0.704833\pi$
$839$	−34.7586	−1.20000	−0.600000	+	0.800000i	$0.704833\pi$
$840$	0	0
$841$	31.7322	1.09421
$842$	0	0
$843$	1.99365	0.0686650
$844$	0	0
$845$	56.7223	1.95131
$846$	0	0
$847$	−15.3506	−0.527453
$848$	0	0
$849$	15.4527	0.530336
$850$	0	0
$851$	−25.8006	−0.884432
$852$	0	0
$853$	28.7898	0.985745	0.492873	−	0.870101i	$-0.335947\pi$
$853$	28.7898	0.985745	0.492873	+	0.870101i	$0.335947\pi$
$854$	0	0
$855$	−13.1915	−0.451139
$856$	0	0
$857$	26.4264	0.902710	0.451355	−	0.892345i	$-0.350941\pi$
$857$	26.4264	0.902710	0.451355	+	0.892345i	$0.350941\pi$
$858$	0	0
$859$	10.1193	0.345266	0.172633	−	0.984986i	$-0.444773\pi$
$859$	10.1193	0.345266	0.172633	+	0.984986i	$0.444773\pi$
$860$	0	0
$861$	9.58770	0.326748
$862$	0	0
$863$	15.5095	0.527950	0.263975	−	0.964530i	$-0.414966\pi$
$863$	15.5095	0.527950	0.263975	+	0.964530i	$0.414966\pi$
$864$	0	0
$865$	−19.1010	−0.649454
$866$	0	0
$867$	−34.0649	−1.15690
$868$	0	0
$869$	30.2014	1.02451
$870$	0	0
$871$	−0.994446	−0.0336955
$872$	0	0
$873$	5.25570	0.177879
$874$	0	0
$875$	128.094	4.33037
$876$	0	0
$877$	33.8803	1.14406	0.572029	−	0.820234i	$-0.306157\pi$
$877$	33.8803	1.14406	0.572029	+	0.820234i	$0.306157\pi$
$878$	0	0
$879$	−0.216096	−0.00728874
$880$	0	0
$881$	−49.5058	−1.66789	−0.833946	−	0.551847i	$-0.813923\pi$
$881$	−49.5058	−1.66789	−0.833946	+	0.551847i	$0.813923\pi$
$882$	0	0
$883$	33.8632	1.13959	0.569793	−	0.821788i	$-0.307023\pi$
$883$	33.8632	1.13959	0.569793	+	0.821788i	$0.307023\pi$
$884$	0	0
$885$	−31.0620	−1.04414
$886$	0	0
$887$	−55.4572	−1.86207	−0.931035	−	0.364929i	$-0.881093\pi$
$887$	−55.4572	−1.86207	−0.931035	+	0.364929i	$0.881093\pi$
$888$	0	0
$889$	13.7727	0.461921
$890$	0	0
$891$	−2.48542	−0.0832647
$892$	0	0
$893$	22.0631	0.738313
$894$	0	0
$895$	30.9050	1.03304
$896$	0	0
$897$	28.9696	0.967266
$898$	0	0
$899$	70.8258	2.36217
$900$	0	0
$901$	98.5607	3.28353
$902$	0	0
$903$	−16.2731	−0.541535
$904$	0	0
$905$	−23.0102	−0.764886
$906$	0	0
$907$	−47.3583	−1.57251	−0.786254	−	0.617904i	$-0.787982\pi$
$907$	−47.3583	−1.57251	−0.786254	+	0.617904i	$0.787982\pi$
$908$	0	0
$909$	18.4245	0.611101
$910$	0	0
$911$	1.50706	0.0499312	0.0249656	−	0.999688i	$-0.492052\pi$
$911$	1.50706	0.0499312	0.0249656	+	0.999688i	$0.492052\pi$
$912$	0	0
$913$	0.535208	0.0177128
$914$	0	0
$915$	18.3759	0.607487
$916$	0	0
$917$	18.0856	0.597238
$918$	0	0
$919$	−9.28240	−0.306198	−0.153099	−	0.988211i	$-0.548925\pi$
$919$	−9.28240	−0.306198	−0.153099	+	0.988211i	$0.548925\pi$
$920$	0	0
$921$	−24.4940	−0.807103
$922$	0	0
$923$	48.9782	1.61214
$924$	0	0
$925$	−64.3649	−2.11630
$926$	0	0
$927$	−2.98651	−0.0980898
$928$	0	0
$929$	−37.4027	−1.22714	−0.613571	−	0.789640i	$-0.710268\pi$
$929$	−37.4027	−1.22714	−0.613571	+	0.789640i	$0.710268\pi$
$930$	0	0
$931$	−9.43049	−0.309072
$932$	0	0
$933$	26.9909	0.883641
$934$	0	0
$935$	77.7979	2.54426
$936$	0	0
$937$	28.8469	0.942388	0.471194	−	0.882030i	$-0.343823\pi$
$937$	28.8469	0.942388	0.471194	+	0.882030i	$0.343823\pi$
$938$	0	0
$939$	−2.62866	−0.0857830
$940$	0	0
$941$	22.8540	0.745019	0.372510	−	0.928028i	$-0.378497\pi$
$941$	22.8540	0.745019	0.372510	+	0.928028i	$0.378497\pi$
$942$	0	0
$943$	−17.1300	−0.557830
$944$	0	0
$945$	−13.9426	−0.453552
$946$	0	0
$947$	−10.6677	−0.346652	−0.173326	−	0.984864i	$-0.555451\pi$
$947$	−10.6677	−0.346652	−0.173326	+	0.984864i	$0.555451\pi$
$948$	0	0
$949$	55.4071	1.79859
$950$	0	0
$951$	−18.4658	−0.598794
$952$	0	0
$953$	−35.7502	−1.15806	−0.579032	−	0.815305i	$-0.696569\pi$
$953$	−35.7502	−1.15806	−0.579032	+	0.815305i	$0.696569\pi$
$954$	0	0
$955$	80.4773	2.60418
$956$	0	0
$957$	−19.3691	−0.626114
$958$	0	0
$959$	−7.57848	−0.244722
$960$	0	0
$961$	51.5969	1.66442
$962$	0	0
$963$	18.4911	0.595866
$964$	0	0
$965$	115.075	3.70440
$966$	0	0
$967$	16.0959	0.517610	0.258805	−	0.965930i	$-0.416671\pi$
$967$	16.0959	0.517610	0.258805	+	0.965930i	$0.416671\pi$
$968$	0	0
$969$	−21.5203	−0.691331
$970$	0	0
$971$	−28.0600	−0.900489	−0.450245	−	0.892905i	$-0.648663\pi$
$971$	−28.0600	−0.900489	−0.450245	+	0.892905i	$0.648663\pi$
$972$	0	0
$973$	59.3312	1.90207
$974$	0	0
$975$	72.2706	2.31451
$976$	0	0
$977$	−22.5462	−0.721317	−0.360658	−	0.932698i	$-0.617448\pi$
$977$	−22.5462	−0.721317	−0.360658	+	0.932698i	$0.617448\pi$
$978$	0	0
$979$	23.6072	0.754489
$980$	0	0
$981$	14.1175	0.450737
$982$	0	0
$983$	−50.4541	−1.60924	−0.804618	−	0.593793i	$-0.797630\pi$
$983$	−50.4541	−1.60924	−0.804618	+	0.593793i	$0.797630\pi$
$984$	0	0
$985$	22.3949	0.713562
$986$	0	0
$987$	23.3193	0.742262
$988$	0	0
$989$	29.0746	0.924518
$990$	0	0
$991$	−42.7766	−1.35884	−0.679421	−	0.733748i	$-0.737769\pi$
$991$	−42.7766	−1.35884	−0.679421	+	0.733748i	$0.737769\pi$
$992$	0	0
$993$	−9.00349	−0.285717
$994$	0	0
$995$	−49.7794	−1.57811
$996$	0	0
$997$	−22.2866	−0.705823	−0.352911	−	0.935657i	$-0.614808\pi$
$997$	−22.2866	−0.705823	−0.352911	+	0.935657i	$0.614808\pi$
$998$	0	0
$999$	4.53681	0.143538

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6024.2.a.n.1.14	✓	14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6024.2.a.n.1.14	✓	14	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +4.38033 q^{5} +3.18300 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +4.38033 q^{5} +3.18300 q^{7} +1.00000 q^{9} -2.48542 q^{11} -5.09405 q^{13} -4.38033 q^{15} -7.14597 q^{17} -3.01153 q^{19} -3.18300 q^{21} +5.68695 q^{23} +14.1873 q^{25} -1.00000 q^{27} -7.79309 q^{29} -9.08829 q^{31} +2.48542 q^{33} +13.9426 q^{35} -4.53681 q^{37} +5.09405 q^{39} -3.01216 q^{41} +5.11251 q^{43} +4.38033 q^{45} -7.32621 q^{47} +3.13147 q^{49} +7.14597 q^{51} -13.7925 q^{53} -10.8870 q^{55} +3.01153 q^{57} +7.09125 q^{59} -4.19509 q^{61} +3.18300 q^{63} -22.3136 q^{65} +0.195217 q^{67} -5.68695 q^{69} -9.61479 q^{71} -10.8768 q^{73} -14.1873 q^{75} -7.91108 q^{77} -12.1514 q^{79} +1.00000 q^{81} -0.215339 q^{83} -31.3017 q^{85} +7.79309 q^{87} -9.49827 q^{89} -16.2143 q^{91} +9.08829 q^{93} -13.1915 q^{95} +5.25570 q^{97} -2.48542 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q - 14 q^{3} - 7 q^{5} + q^{7} + 14 q^{9}+O(q^{10}) \) 14 * q - 14 * q^3 - 7 * q^5 + q^7 + 14 * q^9	\( 14 q - 14 q^{3} - 7 q^{5} + q^{7} + 14 q^{9} + 10 q^{11} - 9 q^{13} + 7 q^{15} - 22 q^{17} - 6 q^{19} - q^{21} + 3 q^{23} + 23 q^{25} - 14 q^{27} - 12 q^{29} - 13 q^{31} - 10 q^{33} + 23 q^{35} + 5 q^{37} + 9 q^{39} - 52 q^{41} + 16 q^{43} - 7 q^{45} + q^{47} + 9 q^{49} + 22 q^{51} - 13 q^{53} - 12 q^{55} + 6 q^{57} + 12 q^{59} - 20 q^{61} + q^{63} - 40 q^{65} + 21 q^{67} - 3 q^{69} - 5 q^{71} - 14 q^{73} - 23 q^{75} - 14 q^{77} - 23 q^{79} + 14 q^{81} + 25 q^{83} - 11 q^{85} + 12 q^{87} - 79 q^{89} + 6 q^{91} + 13 q^{93} + 3 q^{95} - 17 q^{97} + 10 q^{99}+O(q^{100}) \) 14 * q - 14 * q^3 - 7 * q^5 + q^7 + 14 * q^9 + 10 * q^11 - 9 * q^13 + 7 * q^15 - 22 * q^17 - 6 * q^19 - q^21 + 3 * q^23 + 23 * q^25 - 14 * q^27 - 12 * q^29 - 13 * q^31 - 10 * q^33 + 23 * q^35 + 5 * q^37 + 9 * q^39 - 52 * q^41 + 16 * q^43 - 7 * q^45 + q^47 + 9 * q^49 + 22 * q^51 - 13 * q^53 - 12 * q^55 + 6 * q^57 + 12 * q^59 - 20 * q^61 + q^63 - 40 * q^65 + 21 * q^67 - 3 * q^69 - 5 * q^71 - 14 * q^73 - 23 * q^75 - 14 * q^77 - 23 * q^79 + 14 * q^81 + 25 * q^83 - 11 * q^85 + 12 * q^87 - 79 * q^89 + 6 * q^91 + 13 * q^93 + 3 * q^95 - 17 * q^97 + 10 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more