LMFDB - Embedded newform 6024.2.a.n.1.12

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			1.12
Root			$2.93768$ 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)

$f(q)$	$=$	$q-1.00000 q^{3} +1.93768 q^{5} +2.74501 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +1.93768 q^{5} +2.74501 q^{7} +1.00000 q^{9} -0.761191 q^{11} -0.823866 q^{13} -1.93768 q^{15} -3.77869 q^{17} +2.01103 q^{19} -2.74501 q^{21} -7.45981 q^{23} -1.24540 q^{25} -1.00000 q^{27} -5.07546 q^{29} +10.0268 q^{31} +0.761191 q^{33} +5.31895 q^{35} +9.29400 q^{37} +0.823866 q^{39} -9.81220 q^{41} -11.3748 q^{43} +1.93768 q^{45} -12.1550 q^{47} +0.535089 q^{49} +3.77869 q^{51} +0.223843 q^{53} -1.47494 q^{55} -2.01103 q^{57} -6.47898 q^{59} -11.4458 q^{61} +2.74501 q^{63} -1.59639 q^{65} -5.53107 q^{67} +7.45981 q^{69} +1.82740 q^{71} -5.35370 q^{73} +1.24540 q^{75} -2.08948 q^{77} +4.27265 q^{79} +1.00000 q^{81} -1.53959 q^{83} -7.32189 q^{85} +5.07546 q^{87} -0.918278 q^{89} -2.26152 q^{91} -10.0268 q^{93} +3.89673 q^{95} -6.91613 q^{97} -0.761191 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	1.93768	0.866557	0.433278	−	0.901260i	$-0.357357\pi$
$5$	1.93768	0.866557	0.433278	+	0.901260i	$0.357357\pi$
$6$	0	0
$7$	2.74501	1.03752	0.518758	−	0.854921i	$-0.326394\pi$
$7$	2.74501	1.03752	0.518758	+	0.854921i	$0.326394\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−0.761191	−0.229508	−0.114754	−	0.993394i	$-0.536608\pi$
$11$	−0.761191	−0.229508	−0.114754	+	0.993394i	$0.536608\pi$
$12$	0	0
$13$	−0.823866	−0.228499	−0.114250	−	0.993452i	$-0.536446\pi$
$13$	−0.823866	−0.228499	−0.114250	+	0.993452i	$0.536446\pi$
$14$	0	0
$15$	−1.93768	−0.500307
$16$	0	0
$17$	−3.77869	−0.916467	−0.458233	−	0.888832i	$-0.651518\pi$
$17$	−3.77869	−0.916467	−0.458233	+	0.888832i	$0.651518\pi$
$18$	0	0
$19$	2.01103	0.461361	0.230681	−	0.973030i	$-0.425905\pi$
$19$	2.01103	0.461361	0.230681	+	0.973030i	$0.425905\pi$
$20$	0	0
$21$	−2.74501	−0.599011
$22$	0	0
$23$	−7.45981	−1.55548	−0.777739	−	0.628587i	$-0.783634\pi$
$23$	−7.45981	−1.55548	−0.777739	+	0.628587i	$0.783634\pi$
$24$	0	0
$25$	−1.24540	−0.249079
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−5.07546	−0.942489	−0.471245	−	0.882003i	$-0.656195\pi$
$29$	−5.07546	−0.942489	−0.471245	+	0.882003i	$0.656195\pi$
$30$	0	0
$31$	10.0268	1.80086	0.900429	−	0.435003i	$-0.143253\pi$
$31$	10.0268	1.80086	0.900429	+	0.435003i	$0.143253\pi$
$32$	0	0
$33$	0.761191	0.132506
$34$	0	0
$35$	5.31895	0.899067
$36$	0	0
$37$	9.29400	1.52792	0.763962	−	0.645261i	$-0.223252\pi$
$37$	9.29400	1.52792	0.763962	+	0.645261i	$0.223252\pi$
$38$	0	0
$39$	0.823866	0.131924
$40$	0	0
$41$	−9.81220	−1.53241	−0.766204	−	0.642597i	$-0.777857\pi$
$41$	−9.81220	−1.53241	−0.766204	+	0.642597i	$0.777857\pi$
$42$	0	0
$43$	−11.3748	−1.73465	−0.867324	−	0.497744i	$-0.834162\pi$
$43$	−11.3748	−1.73465	−0.867324	+	0.497744i	$0.834162\pi$
$44$	0	0
$45$	1.93768	0.288852
$46$	0	0
$47$	−12.1550	−1.77299	−0.886494	−	0.462741i	$-0.846866\pi$
$47$	−12.1550	−1.77299	−0.886494	+	0.462741i	$0.846866\pi$
$48$	0	0
$49$	0.535089	0.0764413
$50$	0	0
$51$	3.77869	0.529122
$52$	0	0
$53$	0.223843	0.0307472	0.0153736	−	0.999882i	$-0.495106\pi$
$53$	0.223843	0.0307472	0.0153736	+	0.999882i	$0.495106\pi$
$54$	0	0
$55$	−1.47494	−0.198881
$56$	0	0
$57$	−2.01103	−0.266367
$58$	0	0
$59$	−6.47898	−0.843491	−0.421746	−	0.906714i	$-0.638582\pi$
$59$	−6.47898	−0.843491	−0.421746	+	0.906714i	$0.638582\pi$
$60$	0	0
$61$	−11.4458	−1.46549	−0.732745	−	0.680504i	$-0.761761\pi$
$61$	−11.4458	−1.46549	−0.732745	+	0.680504i	$0.761761\pi$
$62$	0	0
$63$	2.74501	0.345839
$64$	0	0
$65$	−1.59639	−0.198008
$66$	0	0
$67$	−5.53107	−0.675727	−0.337864	−	0.941195i	$-0.609704\pi$
$67$	−5.53107	−0.675727	−0.337864	+	0.941195i	$0.609704\pi$
$68$	0	0
$69$	7.45981	0.898056
$70$	0	0
$71$	1.82740	0.216872	0.108436	−	0.994103i	$-0.465416\pi$
$71$	1.82740	0.216872	0.108436	+	0.994103i	$0.465416\pi$
$72$	0	0
$73$	−5.35370	−0.626604	−0.313302	−	0.949654i	$-0.601435\pi$
$73$	−5.35370	−0.626604	−0.313302	+	0.949654i	$0.601435\pi$
$74$	0	0
$75$	1.24540	0.143806
$76$	0	0
$77$	−2.08948	−0.238118
$78$	0	0
$79$	4.27265	0.480710	0.240355	−	0.970685i	$-0.422736\pi$
$79$	4.27265	0.480710	0.240355	+	0.970685i	$0.422736\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−1.53959	−0.168992	−0.0844958	−	0.996424i	$-0.526928\pi$
$83$	−1.53959	−0.168992	−0.0844958	+	0.996424i	$0.526928\pi$
$84$	0	0
$85$	−7.32189	−0.794170
$86$	0	0
$87$	5.07546	0.544146
$88$	0	0
$89$	−0.918278	−0.0973373	−0.0486687	−	0.998815i	$-0.515498\pi$
$89$	−0.918278	−0.0973373	−0.0486687	+	0.998815i	$0.515498\pi$
$90$	0	0
$91$	−2.26152	−0.237072
$92$	0	0
$93$	−10.0268	−1.03973
$94$	0	0
$95$	3.89673	0.399796
$96$	0	0
$97$	−6.91613	−0.702227	−0.351113	−	0.936333i	$-0.614197\pi$
$97$	−6.91613	−0.702227	−0.351113	+	0.936333i	$0.614197\pi$
$98$	0	0
$99$	−0.761191	−0.0765025
$100$	0	0
$101$	−11.4839	−1.14269	−0.571345	−	0.820710i	$-0.693578\pi$
$101$	−11.4839	−1.14269	−0.571345	+	0.820710i	$0.693578\pi$
$102$	0	0
$103$	12.7495	1.25625	0.628123	−	0.778114i	$-0.283824\pi$
$103$	12.7495	1.25625	0.628123	+	0.778114i	$0.283824\pi$
$104$	0	0
$105$	−5.31895	−0.519077
$106$	0	0
$107$	2.95987	0.286141	0.143071	−	0.989712i	$-0.454302\pi$
$107$	2.95987	0.286141	0.143071	+	0.989712i	$0.454302\pi$
$108$	0	0
$109$	−10.2671	−0.983410	−0.491705	−	0.870762i	$-0.663626\pi$
$109$	−10.2671	−0.983410	−0.491705	+	0.870762i	$0.663626\pi$
$110$	0	0
$111$	−9.29400	−0.882147
$112$	0	0
$113$	12.6039	1.18568	0.592838	−	0.805322i	$-0.298007\pi$
$113$	12.6039	1.18568	0.592838	+	0.805322i	$0.298007\pi$
$114$	0	0
$115$	−14.4547	−1.34791
$116$	0	0
$117$	−0.823866	−0.0761664
$118$	0	0
$119$	−10.3725	−0.950850
$120$	0	0
$121$	−10.4206	−0.947326
$122$	0	0
$123$	9.81220	0.884737
$124$	0	0
$125$	−12.1016	−1.08240
$126$	0	0
$127$	9.31183	0.826291	0.413145	−	0.910665i	$-0.364430\pi$
$127$	9.31183	0.826291	0.413145	+	0.910665i	$0.364430\pi$
$128$	0	0
$129$	11.3748	1.00150
$130$	0	0
$131$	13.9160	1.21584	0.607921	−	0.793997i	$-0.292004\pi$
$131$	13.9160	1.21584	0.607921	+	0.793997i	$0.292004\pi$
$132$	0	0
$133$	5.52029	0.478670
$134$	0	0
$135$	−1.93768	−0.166769
$136$	0	0
$137$	7.02077	0.599825	0.299912	−	0.953967i	$-0.403043\pi$
$137$	7.02077	0.599825	0.299912	+	0.953967i	$0.403043\pi$
$138$	0	0
$139$	−1.57657	−0.133723	−0.0668617	−	0.997762i	$-0.521299\pi$
$139$	−1.57657	−0.133723	−0.0668617	+	0.997762i	$0.521299\pi$
$140$	0	0
$141$	12.1550	1.02363
$142$	0	0
$143$	0.627119	0.0524423
$144$	0	0
$145$	−9.83462	−0.816721
$146$	0	0
$147$	−0.535089	−0.0441334
$148$	0	0
$149$	6.41574	0.525598	0.262799	−	0.964851i	$-0.415354\pi$
$149$	6.41574	0.525598	0.262799	+	0.964851i	$0.415354\pi$
$150$	0	0
$151$	14.0580	1.14402	0.572012	−	0.820245i	$-0.306163\pi$
$151$	14.0580	1.14402	0.572012	+	0.820245i	$0.306163\pi$
$152$	0	0
$153$	−3.77869	−0.305489
$154$	0	0
$155$	19.4286	1.56055
$156$	0	0
$157$	−3.69589	−0.294964	−0.147482	−	0.989065i	$-0.547117\pi$
$157$	−3.69589	−0.294964	−0.147482	+	0.989065i	$0.547117\pi$
$158$	0	0
$159$	−0.223843	−0.0177519
$160$	0	0
$161$	−20.4773	−1.61384
$162$	0	0
$163$	15.0540	1.17912	0.589560	−	0.807725i	$-0.299301\pi$
$163$	15.0540	1.17912	0.589560	+	0.807725i	$0.299301\pi$
$164$	0	0
$165$	1.47494	0.114824
$166$	0	0
$167$	24.5463	1.89945	0.949726	−	0.313082i	$-0.101361\pi$
$167$	24.5463	1.89945	0.949726	+	0.313082i	$0.101361\pi$
$168$	0	0
$169$	−12.3212	−0.947788
$170$	0	0
$171$	2.01103	0.153787
$172$	0	0
$173$	0.0112732	0.000857084 0	0.000428542	−	1.00000i	$-0.499864\pi$
$173$	0.0112732	0.000857084 0	0.000428542		1.00000i	$0.499864\pi$
$174$	0	0
$175$	−3.41862	−0.258424
$176$	0	0
$177$	6.47898	0.486990
$178$	0	0
$179$	3.67671	0.274810	0.137405	−	0.990515i	$-0.456124\pi$
$179$	3.67671	0.274810	0.137405	+	0.990515i	$0.456124\pi$
$180$	0	0
$181$	11.0588	0.821992	0.410996	−	0.911637i	$-0.365181\pi$
$181$	11.0588	0.821992	0.410996	+	0.911637i	$0.365181\pi$
$182$	0	0
$183$	11.4458	0.846101
$184$	0	0
$185$	18.0088	1.32403
$186$	0	0
$187$	2.87630	0.210336
$188$	0	0
$189$	−2.74501	−0.199670
$190$	0	0
$191$	−19.5045	−1.41130	−0.705649	−	0.708561i	$-0.749345\pi$
$191$	−19.5045	−1.41130	−0.705649	+	0.708561i	$0.749345\pi$
$192$	0	0
$193$	7.25182	0.521997	0.260999	−	0.965339i	$-0.415948\pi$
$193$	7.25182	0.521997	0.260999	+	0.965339i	$0.415948\pi$
$194$	0	0
$195$	1.59639	0.114320
$196$	0	0
$197$	26.8394	1.91223	0.956114	−	0.292995i	$-0.0946518\pi$
$197$	26.8394	1.91223	0.956114	+	0.292995i	$0.0946518\pi$
$198$	0	0
$199$	3.75849	0.266432	0.133216	−	0.991087i	$-0.457470\pi$
$199$	3.75849	0.266432	0.133216	+	0.991087i	$0.457470\pi$
$200$	0	0
$201$	5.53107	0.390131
$202$	0	0
$203$	−13.9322	−0.977849
$204$	0	0
$205$	−19.0129	−1.32792
$206$	0	0
$207$	−7.45981	−0.518493
$208$	0	0
$209$	−1.53077	−0.105886
$210$	0	0
$211$	4.32004	0.297404	0.148702	−	0.988882i	$-0.452491\pi$
$211$	4.32004	0.297404	0.148702	+	0.988882i	$0.452491\pi$
$212$	0	0
$213$	−1.82740	−0.125211
$214$	0	0
$215$	−22.0408	−1.50317
$216$	0	0
$217$	27.5236	1.86842
$218$	0	0
$219$	5.35370	0.361770
$220$	0	0
$221$	3.11313	0.209412
$222$	0	0
$223$	4.19839	0.281145	0.140573	−	0.990070i	$-0.455106\pi$
$223$	4.19839	0.281145	0.140573	+	0.990070i	$0.455106\pi$
$224$	0	0
$225$	−1.24540	−0.0830264
$226$	0	0
$227$	−9.41114	−0.624639	−0.312320	−	0.949977i	$-0.601106\pi$
$227$	−9.41114	−0.624639	−0.312320	+	0.949977i	$0.601106\pi$
$228$	0	0
$229$	−18.0274	−1.19128	−0.595641	−	0.803251i	$-0.703102\pi$
$229$	−18.0274	−1.19128	−0.595641	+	0.803251i	$0.703102\pi$
$230$	0	0
$231$	2.08948	0.137477
$232$	0	0
$233$	−30.0229	−1.96687	−0.983434	−	0.181268i	$-0.941980\pi$
$233$	−30.0229	−1.96687	−0.983434	+	0.181268i	$0.941980\pi$
$234$	0	0
$235$	−23.5525	−1.53639
$236$	0	0
$237$	−4.27265	−0.277538
$238$	0	0
$239$	−12.7757	−0.826390	−0.413195	−	0.910643i	$-0.635587\pi$
$239$	−12.7757	−0.826390	−0.413195	+	0.910643i	$0.635587\pi$
$240$	0	0
$241$	−22.4241	−1.44446	−0.722232	−	0.691651i	$-0.756884\pi$
$241$	−22.4241	−1.44446	−0.722232	+	0.691651i	$0.756884\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	1.03683	0.0662407
$246$	0	0
$247$	−1.65682	−0.105421
$248$	0	0
$249$	1.53959	0.0975674
$250$	0	0
$251$	−1.00000	−0.0631194
$251$	−1.00000	−0.0631194
$252$	0	0
$253$	5.67834	0.356994
$254$	0	0
$255$	7.32189	0.458515
$256$	0	0
$257$	−7.52799	−0.469583	−0.234791	−	0.972046i	$-0.575441\pi$
$257$	−7.52799	−0.469583	−0.234791	+	0.972046i	$0.575441\pi$
$258$	0	0
$259$	25.5121	1.58525
$260$	0	0
$261$	−5.07546	−0.314163
$262$	0	0
$263$	−5.58652	−0.344479	−0.172240	−	0.985055i	$-0.555100\pi$
$263$	−5.58652	−0.344479	−0.172240	+	0.985055i	$0.555100\pi$
$264$	0	0
$265$	0.433736	0.0266442
$266$	0	0
$267$	0.918278	0.0561977
$268$	0	0
$269$	17.4250	1.06242	0.531210	−	0.847240i	$-0.321737\pi$
$269$	17.4250	1.06242	0.531210	+	0.847240i	$0.321737\pi$
$270$	0	0
$271$	−14.1258	−0.858081	−0.429040	−	0.903285i	$-0.641148\pi$
$271$	−14.1258	−0.858081	−0.429040	+	0.903285i	$0.641148\pi$
$272$	0	0
$273$	2.26152	0.136874
$274$	0	0
$275$	0.947983	0.0571655
$276$	0	0
$277$	0.155792	0.00936062	0.00468031	−	0.999989i	$-0.498510\pi$
$277$	0.155792	0.00936062	0.00468031	+	0.999989i	$0.498510\pi$
$278$	0	0
$279$	10.0268	0.600286
$280$	0	0
$281$	−32.7374	−1.95295	−0.976473	−	0.215638i	$-0.930817\pi$
$281$	−32.7374	−1.95295	−0.976473	+	0.215638i	$0.930817\pi$
$282$	0	0
$283$	23.7938	1.41440	0.707198	−	0.707016i	$-0.249959\pi$
$283$	23.7938	1.41440	0.707198	+	0.707016i	$0.249959\pi$
$284$	0	0
$285$	−3.89673	−0.230822
$286$	0	0
$287$	−26.9346	−1.58990
$288$	0	0
$289$	−2.72151	−0.160089
$290$	0	0
$291$	6.91613	0.405431
$292$	0	0
$293$	−6.41298	−0.374650	−0.187325	−	0.982298i	$-0.559982\pi$
$293$	−6.41298	−0.374650	−0.187325	+	0.982298i	$0.559982\pi$
$294$	0	0
$295$	−12.5542	−0.730933
$296$	0	0
$297$	0.761191	0.0441688
$298$	0	0
$299$	6.14589	0.355426
$300$	0	0
$301$	−31.2241	−1.79973
$302$	0	0
$303$	11.4839	0.659732
$304$	0	0
$305$	−22.1784	−1.26993
$306$	0	0
$307$	5.78320	0.330065	0.165032	−	0.986288i	$-0.447227\pi$
$307$	5.78320	0.330065	0.165032	+	0.986288i	$0.447227\pi$
$308$	0	0
$309$	−12.7495	−0.725294
$310$	0	0
$311$	10.8706	0.616414	0.308207	−	0.951319i	$-0.400271\pi$
$311$	10.8706	0.616414	0.308207	+	0.951319i	$0.400271\pi$
$312$	0	0
$313$	−19.5999	−1.10785	−0.553926	−	0.832566i	$-0.686871\pi$
$313$	−19.5999	−1.10785	−0.553926	+	0.832566i	$0.686871\pi$
$314$	0	0
$315$	5.31895	0.299689
$316$	0	0
$317$	9.06650	0.509226	0.254613	−	0.967043i	$-0.418052\pi$
$317$	9.06650	0.509226	0.254613	+	0.967043i	$0.418052\pi$
$318$	0	0
$319$	3.86339	0.216308
$320$	0	0
$321$	−2.95987	−0.165204
$322$	0	0
$323$	−7.59904	−0.422822
$324$	0	0
$325$	1.02604	0.0569144
$326$	0	0
$327$	10.2671	0.567772
$328$	0	0
$329$	−33.3656	−1.83950
$330$	0	0
$331$	−25.8957	−1.42335	−0.711677	−	0.702507i	$-0.752064\pi$
$331$	−25.8957	−1.42335	−0.711677	+	0.702507i	$0.752064\pi$
$332$	0	0
$333$	9.29400	0.509308
$334$	0	0
$335$	−10.7174	−0.585556
$336$	0	0
$337$	21.1923	1.15442	0.577210	−	0.816596i	$-0.304142\pi$
$337$	21.1923	1.15442	0.577210	+	0.816596i	$0.304142\pi$
$338$	0	0
$339$	−12.6039	−0.684550
$340$	0	0
$341$	−7.63227	−0.413311
$342$	0	0
$343$	−17.7463	−0.958208
$344$	0	0
$345$	14.4547	0.778217
$346$	0	0
$347$	16.3214	0.876181	0.438090	−	0.898931i	$-0.355655\pi$
$347$	16.3214	0.876181	0.438090	+	0.898931i	$0.355655\pi$
$348$	0	0
$349$	33.5248	1.79454	0.897270	−	0.441482i	$-0.145547\pi$
$349$	33.5248	1.79454	0.897270	+	0.441482i	$0.145547\pi$
$350$	0	0
$351$	0.823866	0.0439747
$352$	0	0
$353$	−12.4661	−0.663505	−0.331753	−	0.943366i	$-0.607640\pi$
$353$	−12.4661	−0.663505	−0.331753	+	0.943366i	$0.607640\pi$
$354$	0	0
$355$	3.54092	0.187932
$356$	0	0
$357$	10.3725	0.548973
$358$	0	0
$359$	3.85227	0.203315	0.101658	−	0.994819i	$-0.467585\pi$
$359$	3.85227	0.203315	0.101658	+	0.994819i	$0.467585\pi$
$360$	0	0
$361$	−14.9558	−0.787146
$362$	0	0
$363$	10.4206	0.546939
$364$	0	0
$365$	−10.3738	−0.542988
$366$	0	0
$367$	−12.7362	−0.664824	−0.332412	−	0.943134i	$-0.607863\pi$
$367$	−12.7362	−0.664824	−0.332412	+	0.943134i	$0.607863\pi$
$368$	0	0
$369$	−9.81220	−0.510803
$370$	0	0
$371$	0.614452	0.0319007
$372$	0	0
$373$	12.5013	0.647291	0.323645	−	0.946178i	$-0.395092\pi$
$373$	12.5013	0.647291	0.323645	+	0.946178i	$0.395092\pi$
$374$	0	0
$375$	12.1016	0.624923
$376$	0	0
$377$	4.18150	0.215358
$378$	0	0
$379$	12.5586	0.645089	0.322545	−	0.946554i	$-0.395462\pi$
$379$	12.5586	0.645089	0.322545	+	0.946554i	$0.395462\pi$
$380$	0	0
$381$	−9.31183	−0.477059
$382$	0	0
$383$	−5.53250	−0.282697	−0.141349	−	0.989960i	$-0.545144\pi$
$383$	−5.53250	−0.282697	−0.141349	+	0.989960i	$0.545144\pi$
$384$	0	0
$385$	−4.04874	−0.206343
$386$	0	0
$387$	−11.3748	−0.578216
$388$	0	0
$389$	26.5870	1.34802	0.674008	−	0.738724i	$-0.264571\pi$
$389$	26.5870	1.34802	0.674008	+	0.738724i	$0.264571\pi$
$390$	0	0
$391$	28.1883	1.42554
$392$	0	0
$393$	−13.9160	−0.701967
$394$	0	0
$395$	8.27903	0.416563
$396$	0	0
$397$	−0.907800	−0.0455612	−0.0227806	−	0.999740i	$-0.507252\pi$
$397$	−0.907800	−0.0455612	−0.0227806	+	0.999740i	$0.507252\pi$
$398$	0	0
$399$	−5.52029	−0.276360
$400$	0	0
$401$	−0.922773	−0.0460811	−0.0230405	−	0.999735i	$-0.507335\pi$
$401$	−0.922773	−0.0460811	−0.0230405	+	0.999735i	$0.507335\pi$
$402$	0	0
$403$	−8.26070	−0.411495
$404$	0	0
$405$	1.93768	0.0962841
$406$	0	0
$407$	−7.07450	−0.350670
$408$	0	0
$409$	−8.80249	−0.435255	−0.217628	−	0.976032i	$-0.569832\pi$
$409$	−8.80249	−0.435255	−0.217628	+	0.976032i	$0.569832\pi$
$410$	0	0
$411$	−7.02077	−0.346309
$412$	0	0
$413$	−17.7849	−0.875136
$414$	0	0
$415$	−2.98323	−0.146441
$416$	0	0
$417$	1.57657	0.0772052
$418$	0	0
$419$	4.85607	0.237235	0.118617	−	0.992940i	$-0.462154\pi$
$419$	4.85607	0.237235	0.118617	+	0.992940i	$0.462154\pi$
$420$	0	0
$421$	−5.45618	−0.265918	−0.132959	−	0.991122i	$-0.542448\pi$
$421$	−5.45618	−0.265918	−0.132959	+	0.991122i	$0.542448\pi$
$422$	0	0
$423$	−12.1550	−0.590996
$424$	0	0
$425$	4.70596	0.228273
$426$	0	0
$427$	−31.4190	−1.52047
$428$	0	0
$429$	−0.627119	−0.0302776
$430$	0	0
$431$	38.3652	1.84798	0.923992	−	0.382411i	$-0.124906\pi$
$431$	38.3652	1.84798	0.923992	+	0.382411i	$0.124906\pi$
$432$	0	0
$433$	24.3802	1.17164	0.585818	−	0.810443i	$-0.300773\pi$
$433$	24.3802	1.17164	0.585818	+	0.810443i	$0.300773\pi$
$434$	0	0
$435$	9.83462	0.471534
$436$	0	0
$437$	−15.0019	−0.717638
$438$	0	0
$439$	35.0581	1.67323	0.836616	−	0.547790i	$-0.184531\pi$
$439$	35.0581	1.67323	0.836616	+	0.547790i	$0.184531\pi$
$440$	0	0
$441$	0.535089	0.0254804
$442$	0	0
$443$	−34.5000	−1.63914	−0.819572	−	0.572976i	$-0.805789\pi$
$443$	−34.5000	−1.63914	−0.819572	+	0.572976i	$0.805789\pi$
$444$	0	0
$445$	−1.77933	−0.0843483
$446$	0	0
$447$	−6.41574	−0.303454
$448$	0	0
$449$	−36.3975	−1.71770	−0.858852	−	0.512224i	$-0.828822\pi$
$449$	−36.3975	−1.71770	−0.858852	+	0.512224i	$0.828822\pi$
$450$	0	0
$451$	7.46896	0.351699
$452$	0	0
$453$	−14.0580	−0.660503
$454$	0	0
$455$	−4.38211	−0.205436
$456$	0	0
$457$	−21.4952	−1.00550	−0.502751	−	0.864431i	$-0.667679\pi$
$457$	−21.4952	−1.00550	−0.502751	+	0.864431i	$0.667679\pi$
$458$	0	0
$459$	3.77869	0.176374
$460$	0	0
$461$	−11.2402	−0.523510	−0.261755	−	0.965134i	$-0.584301\pi$
$461$	−11.2402	−0.523510	−0.261755	+	0.965134i	$0.584301\pi$
$462$	0	0
$463$	29.4779	1.36996	0.684978	−	0.728564i	$-0.259812\pi$
$463$	29.4779	1.36996	0.684978	+	0.728564i	$0.259812\pi$
$464$	0	0
$465$	−19.4286	−0.900982
$466$	0	0
$467$	11.3973	0.527406	0.263703	−	0.964604i	$-0.415056\pi$
$467$	11.3973	0.527406	0.263703	+	0.964604i	$0.415056\pi$
$468$	0	0
$469$	−15.1828	−0.701078
$470$	0	0
$471$	3.69589	0.170298
$472$	0	0
$473$	8.65843	0.398115
$474$	0	0
$475$	−2.50452	−0.114915
$476$	0	0
$477$	0.223843	0.0102491
$478$	0	0
$479$	12.7141	0.580922	0.290461	−	0.956887i	$-0.406191\pi$
$479$	12.7141	0.580922	0.290461	+	0.956887i	$0.406191\pi$
$480$	0	0
$481$	−7.65701	−0.349130
$482$	0	0
$483$	20.4773	0.931748
$484$	0	0
$485$	−13.4013	−0.608519
$486$	0	0
$487$	−37.9478	−1.71958	−0.859789	−	0.510650i	$-0.829405\pi$
$487$	−37.9478	−1.71958	−0.859789	+	0.510650i	$0.829405\pi$
$488$	0	0
$489$	−15.0540	−0.680765
$490$	0	0
$491$	−33.4825	−1.51104	−0.755522	−	0.655123i	$-0.772617\pi$
$491$	−33.4825	−1.51104	−0.755522	+	0.655123i	$0.772617\pi$
$492$	0	0
$493$	19.1786	0.863760
$494$	0	0
$495$	−1.47494	−0.0662938
$496$	0	0
$497$	5.01623	0.225009
$498$	0	0
$499$	−40.4884	−1.81251	−0.906255	−	0.422732i	$-0.861071\pi$
$499$	−40.4884	−1.81251	−0.906255	+	0.422732i	$0.861071\pi$
$500$	0	0
$501$	−24.5463	−1.09665
$502$	0	0
$503$	2.91165	0.129824	0.0649120	−	0.997891i	$-0.479323\pi$
$503$	2.91165	0.129824	0.0649120	+	0.997891i	$0.479323\pi$
$504$	0	0
$505$	−22.2521	−0.990206
$506$	0	0
$507$	12.3212	0.547206
$508$	0	0
$509$	−2.64246	−0.117125	−0.0585624	−	0.998284i	$-0.518652\pi$
$509$	−2.64246	−0.117125	−0.0585624	+	0.998284i	$0.518652\pi$
$510$	0	0
$511$	−14.6960	−0.650112
$512$	0	0
$513$	−2.01103	−0.0887890
$514$	0	0
$515$	24.7044	1.08861
$516$	0	0
$517$	9.25226	0.406914
$518$	0	0
$519$	−0.0112732	−0.000494838 0
$520$	0	0
$521$	−10.1089	−0.442881	−0.221441	−	0.975174i	$-0.571076\pi$
$521$	−10.1089	−0.442881	−0.221441	+	0.975174i	$0.571076\pi$
$522$	0	0
$523$	−29.2299	−1.27813	−0.639066	−	0.769152i	$-0.720679\pi$
$523$	−29.2299	−1.27813	−0.639066	+	0.769152i	$0.720679\pi$
$524$	0	0
$525$	3.41862	0.149201
$526$	0	0
$527$	−37.8880	−1.65043
$528$	0	0
$529$	32.6488	1.41951
$530$	0	0
$531$	−6.47898	−0.281164
$532$	0	0
$533$	8.08394	0.350154
$534$	0	0
$535$	5.73527	0.247958
$536$	0	0
$537$	−3.67671	−0.158662
$538$	0	0
$539$	−0.407305	−0.0175439
$540$	0	0
$541$	4.99758	0.214863	0.107431	−	0.994212i	$-0.465737\pi$
$541$	4.99758	0.214863	0.107431	+	0.994212i	$0.465737\pi$
$542$	0	0
$543$	−11.0588	−0.474577
$544$	0	0
$545$	−19.8944	−0.852181
$546$	0	0
$547$	−12.2466	−0.523627	−0.261814	−	0.965118i	$-0.584321\pi$
$547$	−12.2466	−0.523627	−0.261814	+	0.965118i	$0.584321\pi$
$548$	0	0
$549$	−11.4458	−0.488496
$550$	0	0
$551$	−10.2069	−0.434828
$552$	0	0
$553$	11.7285	0.498745
$554$	0	0
$555$	−18.0088	−0.764431
$556$	0	0
$557$	−4.89718	−0.207500	−0.103750	−	0.994603i	$-0.533084\pi$
$557$	−4.89718	−0.207500	−0.103750	+	0.994603i	$0.533084\pi$
$558$	0	0
$559$	9.37135	0.396366
$560$	0	0
$561$	−2.87630	−0.121438
$562$	0	0
$563$	−40.0750	−1.68896	−0.844481	−	0.535586i	$-0.820091\pi$
$563$	−40.0750	−1.68896	−0.844481	+	0.535586i	$0.820091\pi$
$564$	0	0
$565$	24.4223	1.02746
$566$	0	0
$567$	2.74501	0.115280
$568$	0	0
$569$	3.18419	0.133488	0.0667442	−	0.997770i	$-0.478739\pi$
$569$	3.18419	0.133488	0.0667442	+	0.997770i	$0.478739\pi$
$570$	0	0
$571$	9.22818	0.386187	0.193094	−	0.981180i	$-0.438148\pi$
$571$	9.22818	0.386187	0.193094	+	0.981180i	$0.438148\pi$
$572$	0	0
$573$	19.5045	0.814814
$574$	0	0
$575$	9.29042	0.387437
$576$	0	0
$577$	−3.49139	−0.145348	−0.0726742	−	0.997356i	$-0.523153\pi$
$577$	−3.49139	−0.145348	−0.0726742	+	0.997356i	$0.523153\pi$
$578$	0	0
$579$	−7.25182	−0.301375
$580$	0	0
$581$	−4.22619	−0.175332
$582$	0	0
$583$	−0.170387	−0.00705672
$584$	0	0
$585$	−1.59639	−0.0660026
$586$	0	0
$587$	−6.76526	−0.279232	−0.139616	−	0.990206i	$-0.544587\pi$
$587$	−6.76526	−0.279232	−0.139616	+	0.990206i	$0.544587\pi$
$588$	0	0
$589$	20.1641	0.830846
$590$	0	0
$591$	−26.8394	−1.10403
$592$	0	0
$593$	40.0027	1.64271	0.821357	−	0.570414i	$-0.193217\pi$
$593$	40.0027	1.64271	0.821357	+	0.570414i	$0.193217\pi$
$594$	0	0
$595$	−20.0987	−0.823965
$596$	0	0
$597$	−3.75849	−0.153825
$598$	0	0
$599$	−21.4936	−0.878204	−0.439102	−	0.898437i	$-0.644703\pi$
$599$	−21.4936	−0.878204	−0.439102	+	0.898437i	$0.644703\pi$
$600$	0	0
$601$	−23.5125	−0.959095	−0.479548	−	0.877516i	$-0.659199\pi$
$601$	−23.5125	−0.959095	−0.479548	+	0.877516i	$0.659199\pi$
$602$	0	0
$603$	−5.53107	−0.225242
$604$	0	0
$605$	−20.1918	−0.820912
$606$	0	0
$607$	−16.4602	−0.668098	−0.334049	−	0.942556i	$-0.608415\pi$
$607$	−16.4602	−0.668098	−0.334049	+	0.942556i	$0.608415\pi$
$608$	0	0
$609$	13.9322	0.564561
$610$	0	0
$611$	10.0141	0.405126
$612$	0	0
$613$	−23.6589	−0.955574	−0.477787	−	0.878476i	$-0.658561\pi$
$613$	−23.6589	−0.955574	−0.477787	+	0.878476i	$0.658561\pi$
$614$	0	0
$615$	19.0129	0.766675
$616$	0	0
$617$	−26.5259	−1.06789	−0.533946	−	0.845519i	$-0.679291\pi$
$617$	−26.5259	−1.06789	−0.533946	+	0.845519i	$0.679291\pi$
$618$	0	0
$619$	43.5361	1.74986	0.874931	−	0.484247i	$-0.160906\pi$
$619$	43.5361	1.74986	0.874931	+	0.484247i	$0.160906\pi$
$620$	0	0
$621$	7.45981	0.299352
$622$	0	0
$623$	−2.52068	−0.100989
$624$	0	0
$625$	−17.2220	−0.688881
$626$	0	0
$627$	1.53077	0.0611333
$628$	0	0
$629$	−35.1191	−1.40029
$630$	0	0
$631$	−32.1314	−1.27913	−0.639566	−	0.768737i	$-0.720886\pi$
$631$	−32.1314	−1.27913	−0.639566	+	0.768737i	$0.720886\pi$
$632$	0	0
$633$	−4.32004	−0.171706
$634$	0	0
$635$	18.0433	0.716028
$636$	0	0
$637$	−0.440842	−0.0174668
$638$	0	0
$639$	1.82740	0.0722908
$640$	0	0
$641$	−7.76756	−0.306800	−0.153400	−	0.988164i	$-0.549022\pi$
$641$	−7.76756	−0.306800	−0.153400	+	0.988164i	$0.549022\pi$
$642$	0	0
$643$	37.9938	1.49833	0.749164	−	0.662385i	$-0.230456\pi$
$643$	37.9938	1.49833	0.749164	+	0.662385i	$0.230456\pi$
$644$	0	0
$645$	22.0408	0.867856
$646$	0	0
$647$	26.7043	1.04985	0.524926	−	0.851148i	$-0.324093\pi$
$647$	26.7043	1.04985	0.524926	+	0.851148i	$0.324093\pi$
$648$	0	0
$649$	4.93174	0.193588
$650$	0	0
$651$	−27.5236	−1.07873
$652$	0	0
$653$	−7.77682	−0.304330	−0.152165	−	0.988355i	$-0.548625\pi$
$653$	−7.77682	−0.304330	−0.152165	+	0.988355i	$0.548625\pi$
$654$	0	0
$655$	26.9647	1.05360
$656$	0	0
$657$	−5.35370	−0.208868
$658$	0	0
$659$	33.0555	1.28766	0.643831	−	0.765168i	$-0.277344\pi$
$659$	33.0555	1.28766	0.643831	+	0.765168i	$0.277344\pi$
$660$	0	0
$661$	35.0089	1.36169	0.680845	−	0.732427i	$-0.261613\pi$
$661$	35.0089	1.36169	0.680845	+	0.732427i	$0.261613\pi$
$662$	0	0
$663$	−3.11313	−0.120904
$664$	0	0
$665$	10.6966	0.414795
$666$	0	0
$667$	37.8620	1.46602
$668$	0	0
$669$	−4.19839	−0.162319
$670$	0	0
$671$	8.71246	0.336341
$672$	0	0
$673$	−7.39029	−0.284875	−0.142437	−	0.989804i	$-0.545494\pi$
$673$	−7.39029	−0.284875	−0.142437	+	0.989804i	$0.545494\pi$
$674$	0	0
$675$	1.24540	0.0479353
$676$	0	0
$677$	50.1414	1.92709	0.963545	−	0.267545i	$-0.0862123\pi$
$677$	50.1414	1.92709	0.963545	+	0.267545i	$0.0862123\pi$
$678$	0	0
$679$	−18.9849	−0.728572
$680$	0	0
$681$	9.41114	0.360636
$682$	0	0
$683$	−3.55053	−0.135857	−0.0679285	−	0.997690i	$-0.521639\pi$
$683$	−3.55053	−0.135857	−0.0679285	+	0.997690i	$0.521639\pi$
$684$	0	0
$685$	13.6040	0.519782
$686$	0	0
$687$	18.0274	0.687787
$688$	0	0
$689$	−0.184417	−0.00702572
$690$	0	0
$691$	4.03339	0.153437	0.0767187	−	0.997053i	$-0.475556\pi$
$691$	4.03339	0.153437	0.0767187	+	0.997053i	$0.475556\pi$
$692$	0	0
$693$	−2.08948	−0.0793727
$694$	0	0
$695$	−3.05490	−0.115879
$696$	0	0
$697$	37.0773	1.40440
$698$	0	0
$699$	30.0229	1.13557
$700$	0	0
$701$	39.0627	1.47538	0.737689	−	0.675141i	$-0.235917\pi$
$701$	39.0627	1.47538	0.737689	+	0.675141i	$0.235917\pi$
$702$	0	0
$703$	18.6905	0.704925
$704$	0	0
$705$	23.5525	0.887038
$706$	0	0
$707$	−31.5234	−1.18556
$708$	0	0
$709$	12.1178	0.455095	0.227547	−	0.973767i	$-0.426929\pi$
$709$	12.1178	0.455095	0.227547	+	0.973767i	$0.426929\pi$
$710$	0	0
$711$	4.27265	0.160237
$712$	0	0
$713$	−74.7977	−2.80120
$714$	0	0
$715$	1.21516	0.0454443
$716$	0	0
$717$	12.7757	0.477116
$718$	0	0
$719$	9.68067	0.361028	0.180514	−	0.983572i	$-0.442224\pi$
$719$	9.68067	0.361028	0.180514	+	0.983572i	$0.442224\pi$
$720$	0	0
$721$	34.9975	1.30338
$722$	0	0
$723$	22.4241	0.833962
$724$	0	0
$725$	6.32095	0.234754
$726$	0	0
$727$	13.0397	0.483617	0.241809	−	0.970324i	$-0.422259\pi$
$727$	13.0397	0.483617	0.241809	+	0.970324i	$0.422259\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	42.9820	1.58975
$732$	0	0
$733$	−1.30279	−0.0481197	−0.0240599	−	0.999711i	$-0.507659\pi$
$733$	−1.30279	−0.0481197	−0.0240599	+	0.999711i	$0.507659\pi$
$734$	0	0
$735$	−1.03683	−0.0382441
$736$	0	0
$737$	4.21019	0.155085
$738$	0	0
$739$	−5.49031	−0.201964	−0.100982	−	0.994888i	$-0.532198\pi$
$739$	−5.49031	−0.201964	−0.100982	+	0.994888i	$0.532198\pi$
$740$	0	0
$741$	1.65682	0.0608647
$742$	0	0
$743$	−18.8493	−0.691512	−0.345756	−	0.938324i	$-0.612378\pi$
$743$	−18.8493	−0.691512	−0.345756	+	0.938324i	$0.612378\pi$
$744$	0	0
$745$	12.4317	0.455461
$746$	0	0
$747$	−1.53959	−0.0563306
$748$	0	0
$749$	8.12487	0.296876
$750$	0	0
$751$	−1.45765	−0.0531902	−0.0265951	−	0.999646i	$-0.508466\pi$
$751$	−1.45765	−0.0531902	−0.0265951	+	0.999646i	$0.508466\pi$
$752$	0	0
$753$	1.00000	0.0364420
$754$	0	0
$755$	27.2399	0.991362
$756$	0	0
$757$	−9.01661	−0.327714	−0.163857	−	0.986484i	$-0.552394\pi$
$757$	−9.01661	−0.327714	−0.163857	+	0.986484i	$0.552394\pi$
$758$	0	0
$759$	−5.67834	−0.206111
$760$	0	0
$761$	20.4447	0.741121	0.370560	−	0.928808i	$-0.379166\pi$
$761$	20.4447	0.741121	0.370560	+	0.928808i	$0.379166\pi$
$762$	0	0
$763$	−28.1833	−1.02030
$764$	0	0
$765$	−7.32189	−0.264723
$766$	0	0
$767$	5.33781	0.192737
$768$	0	0
$769$	−23.5093	−0.847768	−0.423884	−	0.905716i	$-0.639334\pi$
$769$	−23.5093	−0.847768	−0.423884	+	0.905716i	$0.639334\pi$
$770$	0	0
$771$	7.52799	0.271114
$772$	0	0
$773$	−14.6500	−0.526923	−0.263462	−	0.964670i	$-0.584864\pi$
$773$	−14.6500	−0.526923	−0.263462	+	0.964670i	$0.584864\pi$
$774$	0	0
$775$	−12.4873	−0.448556
$776$	0	0
$777$	−25.5121	−0.915242
$778$	0	0
$779$	−19.7326	−0.706994
$780$	0	0
$781$	−1.39100	−0.0497739
$782$	0	0
$783$	5.07546	0.181382
$784$	0	0
$785$	−7.16145	−0.255603
$786$	0	0
$787$	10.2002	0.363599	0.181800	−	0.983336i	$-0.441808\pi$
$787$	10.2002	0.363599	0.181800	+	0.983336i	$0.441808\pi$
$788$	0	0
$789$	5.58652	0.198885
$790$	0	0
$791$	34.5979	1.23016
$792$	0	0
$793$	9.42984	0.334863
$794$	0	0
$795$	−0.433736	−0.0153830
$796$	0	0
$797$	−10.3971	−0.368283	−0.184141	−	0.982900i	$-0.558950\pi$
$797$	−10.3971	−0.368283	−0.184141	+	0.982900i	$0.558950\pi$
$798$	0	0
$799$	45.9299	1.62488
$800$	0	0
$801$	−0.918278	−0.0324458
$802$	0	0
$803$	4.07519	0.143810
$804$	0	0
$805$	−39.6784	−1.39848
$806$	0	0
$807$	−17.4250	−0.613389
$808$	0	0
$809$	−36.9675	−1.29971	−0.649853	−	0.760060i	$-0.725170\pi$
$809$	−36.9675	−1.29971	−0.649853	+	0.760060i	$0.725170\pi$
$810$	0	0
$811$	46.8325	1.64451	0.822256	−	0.569118i	$-0.192715\pi$
$811$	46.8325	1.64451	0.822256	+	0.569118i	$0.192715\pi$
$812$	0	0
$813$	14.1258	0.495413
$814$	0	0
$815$	29.1698	1.02177
$816$	0	0
$817$	−22.8751	−0.800299
$818$	0	0
$819$	−2.26152	−0.0790240
$820$	0	0
$821$	29.7611	1.03867	0.519335	−	0.854571i	$-0.326180\pi$
$821$	29.7611	1.03867	0.519335	+	0.854571i	$0.326180\pi$
$822$	0	0
$823$	−52.9609	−1.84610	−0.923051	−	0.384677i	$-0.874313\pi$
$823$	−52.9609	−1.84610	−0.923051	+	0.384677i	$0.874313\pi$
$824$	0	0
$825$	−0.947983	−0.0330045
$826$	0	0
$827$	35.8906	1.24804	0.624020	−	0.781409i	$-0.285499\pi$
$827$	35.8906	1.24804	0.624020	+	0.781409i	$0.285499\pi$
$828$	0	0
$829$	−30.2396	−1.05026	−0.525132	−	0.851021i	$-0.675984\pi$
$829$	−30.2396	−1.05026	−0.525132	+	0.851021i	$0.675984\pi$
$830$	0	0
$831$	−0.155792	−0.00540436
$832$	0	0
$833$	−2.02193	−0.0700559
$834$	0	0
$835$	47.5629	1.64598
$836$	0	0
$837$	−10.0268	−0.346575
$838$	0	0
$839$	−21.5076	−0.742525	−0.371262	−	0.928528i	$-0.621075\pi$
$839$	−21.5076	−0.742525	−0.371262	+	0.928528i	$0.621075\pi$
$840$	0	0
$841$	−3.23970	−0.111714
$842$	0	0
$843$	32.7374	1.12753
$844$	0	0
$845$	−23.8746	−0.821312
$846$	0	0
$847$	−28.6046	−0.982867
$848$	0	0
$849$	−23.7938	−0.816602
$850$	0	0
$851$	−69.3315	−2.37665
$852$	0	0
$853$	32.4924	1.11252	0.556260	−	0.831009i	$-0.312236\pi$
$853$	32.4924	1.11252	0.556260	+	0.831009i	$0.312236\pi$
$854$	0	0
$855$	3.89673	0.133265
$856$	0	0
$857$	49.7922	1.70087	0.850435	−	0.526081i	$-0.176339\pi$
$857$	49.7922	1.70087	0.850435	+	0.526081i	$0.176339\pi$
$858$	0	0
$859$	−4.73705	−0.161626	−0.0808131	−	0.996729i	$-0.525752\pi$
$859$	−4.73705	−0.161626	−0.0808131	+	0.996729i	$0.525752\pi$
$860$	0	0
$861$	26.9346	0.917929
$862$	0	0
$863$	3.18886	0.108550	0.0542751	−	0.998526i	$-0.482715\pi$
$863$	3.18886	0.108550	0.0542751	+	0.998526i	$0.482715\pi$
$864$	0	0
$865$	0.0218438	0.000742712 0
$866$	0	0
$867$	2.72151	0.0924274
$868$	0	0
$869$	−3.25230	−0.110327
$870$	0	0
$871$	4.55686	0.154403
$872$	0	0
$873$	−6.91613	−0.234076
$874$	0	0
$875$	−33.2190	−1.12301
$876$	0	0
$877$	10.3745	0.350321	0.175161	−	0.984540i	$-0.443956\pi$
$877$	10.3745	0.350321	0.175161	+	0.984540i	$0.443956\pi$
$878$	0	0
$879$	6.41298	0.216304
$880$	0	0
$881$	−40.7768	−1.37381	−0.686903	−	0.726749i	$-0.741030\pi$
$881$	−40.7768	−1.37381	−0.686903	+	0.726749i	$0.741030\pi$
$882$	0	0
$883$	−52.7711	−1.77589	−0.887945	−	0.459949i	$-0.847868\pi$
$883$	−52.7711	−1.77589	−0.887945	+	0.459949i	$0.847868\pi$
$884$	0	0
$885$	12.5542	0.422004
$886$	0	0
$887$	−50.4748	−1.69478	−0.847389	−	0.530972i	$-0.821827\pi$
$887$	−50.4748	−1.69478	−0.847389	+	0.530972i	$0.821827\pi$
$888$	0	0
$889$	25.5611	0.857291
$890$	0	0
$891$	−0.761191	−0.0255008
$892$	0	0
$893$	−24.4440	−0.817987
$894$	0	0
$895$	7.12429	0.238139
$896$	0	0
$897$	−6.14589	−0.205205
$898$	0	0
$899$	−50.8904	−1.69729
$900$	0	0
$901$	−0.845833	−0.0281788
$902$	0	0
$903$	31.2241	1.03907
$904$	0	0
$905$	21.4284	0.712302
$906$	0	0
$907$	48.6703	1.61607	0.808036	−	0.589133i	$-0.200531\pi$
$907$	48.6703	1.61607	0.808036	+	0.589133i	$0.200531\pi$
$908$	0	0
$909$	−11.4839	−0.380897
$910$	0	0
$911$	−0.410551	−0.0136022	−0.00680109	−	0.999977i	$-0.502165\pi$
$911$	−0.410551	−0.0136022	−0.00680109	+	0.999977i	$0.502165\pi$
$912$	0	0
$913$	1.17192	0.0387849
$914$	0	0
$915$	22.1784	0.733194
$916$	0	0
$917$	38.1995	1.26146
$918$	0	0
$919$	16.5743	0.546736	0.273368	−	0.961910i	$-0.411862\pi$
$919$	16.5743	0.546736	0.273368	+	0.961910i	$0.411862\pi$
$920$	0	0
$921$	−5.78320	−0.190563
$922$	0	0
$923$	−1.50553	−0.0495552
$924$	0	0
$925$	−11.5747	−0.380574
$926$	0	0
$927$	12.7495	0.418748
$928$	0	0
$929$	−44.8409	−1.47118	−0.735590	−	0.677427i	$-0.763095\pi$
$929$	−44.8409	−1.47118	−0.735590	+	0.677427i	$0.763095\pi$
$930$	0	0
$931$	1.07608	0.0352670
$932$	0	0
$933$	−10.8706	−0.355887
$934$	0	0
$935$	5.57335	0.182268
$936$	0	0
$937$	−10.3194	−0.337119	−0.168560	−	0.985691i	$-0.553912\pi$
$937$	−10.3194	−0.337119	−0.168560	+	0.985691i	$0.553912\pi$
$938$	0	0
$939$	19.5999	0.639618
$940$	0	0
$941$	10.7859	0.351611	0.175806	−	0.984425i	$-0.443747\pi$
$941$	10.7859	0.351611	0.175806	+	0.984425i	$0.443747\pi$
$942$	0	0
$943$	73.1972	2.38363
$944$	0	0
$945$	−5.31895	−0.173026
$946$	0	0
$947$	6.49472	0.211050	0.105525	−	0.994417i	$-0.466348\pi$
$947$	6.49472	0.211050	0.105525	+	0.994417i	$0.466348\pi$
$948$	0	0
$949$	4.41074	0.143179
$950$	0	0
$951$	−9.06650	−0.294001
$952$	0	0
$953$	7.31257	0.236877	0.118439	−	0.992961i	$-0.462211\pi$
$953$	7.31257	0.236877	0.118439	+	0.992961i	$0.462211\pi$
$954$	0	0
$955$	−37.7936	−1.22297
$956$	0	0
$957$	−3.86339	−0.124886
$958$	0	0
$959$	19.2721	0.622328
$960$	0	0
$961$	69.5358	2.24309
$962$	0	0
$963$	2.95987	0.0953804
$964$	0	0
$965$	14.0517	0.452340
$966$	0	0
$967$	−21.9348	−0.705377	−0.352688	−	0.935741i	$-0.614732\pi$
$967$	−21.9348	−0.705377	−0.352688	+	0.935741i	$0.614732\pi$
$968$	0	0
$969$	7.59904	0.244116
$970$	0	0
$971$	−30.9564	−0.993439	−0.496719	−	0.867911i	$-0.665462\pi$
$971$	−30.9564	−0.993439	−0.496719	+	0.867911i	$0.665462\pi$
$972$	0	0
$973$	−4.32772	−0.138740
$974$	0	0
$975$	−1.02604	−0.0328595
$976$	0	0
$977$	15.3239	0.490253	0.245127	−	0.969491i	$-0.421170\pi$
$977$	15.3239	0.490253	0.245127	+	0.969491i	$0.421170\pi$
$978$	0	0
$979$	0.698985	0.0223397
$980$	0	0
$981$	−10.2671	−0.327803
$982$	0	0
$983$	21.2665	0.678295	0.339147	−	0.940733i	$-0.389862\pi$
$983$	21.2665	0.678295	0.339147	+	0.940733i	$0.389862\pi$
$984$	0	0
$985$	52.0062	1.65705
$986$	0	0
$987$	33.3656	1.06204
$988$	0	0
$989$	84.8542	2.69821
$990$	0	0
$991$	−47.5312	−1.50988	−0.754939	−	0.655795i	$-0.772334\pi$
$991$	−47.5312	−1.50988	−0.754939	+	0.655795i	$0.772334\pi$
$992$	0	0
$993$	25.8957	0.821774
$994$	0	0
$995$	7.28275	0.230879
$996$	0	0
$997$	−15.5995	−0.494041	−0.247021	−	0.969010i	$-0.579452\pi$
$997$	−15.5995	−0.494041	−0.247021	+	0.969010i	$0.579452\pi$
$998$	0	0
$999$	−9.29400	−0.294049

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6024.2.a.n.1.12	✓	14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6024.2.a.n.1.12	✓	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +1.93768 q^{5} +2.74501 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +1.93768 q^{5} +2.74501 q^{7} +1.00000 q^{9} -0.761191 q^{11} -0.823866 q^{13} -1.93768 q^{15} -3.77869 q^{17} +2.01103 q^{19} -2.74501 q^{21} -7.45981 q^{23} -1.24540 q^{25} -1.00000 q^{27} -5.07546 q^{29} +10.0268 q^{31} +0.761191 q^{33} +5.31895 q^{35} +9.29400 q^{37} +0.823866 q^{39} -9.81220 q^{41} -11.3748 q^{43} +1.93768 q^{45} -12.1550 q^{47} +0.535089 q^{49} +3.77869 q^{51} +0.223843 q^{53} -1.47494 q^{55} -2.01103 q^{57} -6.47898 q^{59} -11.4458 q^{61} +2.74501 q^{63} -1.59639 q^{65} -5.53107 q^{67} +7.45981 q^{69} +1.82740 q^{71} -5.35370 q^{73} +1.24540 q^{75} -2.08948 q^{77} +4.27265 q^{79} +1.00000 q^{81} -1.53959 q^{83} -7.32189 q^{85} +5.07546 q^{87} -0.918278 q^{89} -2.26152 q^{91} -10.0268 q^{93} +3.89673 q^{95} -6.91613 q^{97} -0.761191 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q - 14 q^{3} - 7 q^{5} + q^{7} + 14 q^{9}+O(q^{10}) \) 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

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Representations

Groups

Database

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