LMFDB - Embedded newform 6024.2.a.p.1.4

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:	$14$
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} - 6 x^{13} - 29 x^{12} + 210 x^{11} + 280 x^{10} - 2796 x^{9} - 863 x^{8} + 17652 x^{7} + \cdots - 3364 $ x^14 - 6x^13 - 29x^12 + 210x^11 + 280x^10 - 2796x^9 - 863x^8 + 17652x^7 - 1300x^6 - 53458x^5 + 10892x^4 + 67298x^3 - 13761x^2 - 22140*x - 3364
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-2.15802$ 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)

$f(q)$	$=$	$q-1.00000 q^{3} -2.15802 q^{5} -0.0890340 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} -2.15802 q^{5} -0.0890340 q^{7} +1.00000 q^{9} +6.37243 q^{11} +3.74097 q^{13} +2.15802 q^{15} +4.37258 q^{17} +5.14883 q^{19} +0.0890340 q^{21} +1.37077 q^{23} -0.342943 q^{25} -1.00000 q^{27} +3.68963 q^{29} +10.8286 q^{31} -6.37243 q^{33} +0.192137 q^{35} +2.87385 q^{37} -3.74097 q^{39} +2.86978 q^{41} -8.06219 q^{43} -2.15802 q^{45} -7.11981 q^{47} -6.99207 q^{49} -4.37258 q^{51} +1.35322 q^{53} -13.7519 q^{55} -5.14883 q^{57} -1.49740 q^{59} +1.63235 q^{61} -0.0890340 q^{63} -8.07310 q^{65} +6.57177 q^{67} -1.37077 q^{69} +5.51440 q^{71} -11.7255 q^{73} +0.342943 q^{75} -0.567363 q^{77} -0.145483 q^{79} +1.00000 q^{81} +10.3541 q^{83} -9.43612 q^{85} -3.68963 q^{87} -3.70733 q^{89} -0.333073 q^{91} -10.8286 q^{93} -11.1113 q^{95} +8.43969 q^{97} +6.37243 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 14 q - 14 q^{3} + 6 q^{5} + 3 q^{7} + 14 q^{9}+O(q^{10}) $ 14 * q - 14 * q^3 + 6 * q^5 + 3 * q^7 + 14 * q^9	$ 14 q - 14 q^{3} + 6 q^{5} + 3 q^{7} + 14 q^{9} + 7 q^{11} + 5 q^{13} - 6 q^{15} + 28 q^{17} - q^{19} - 3 q^{21} - 3 q^{23} + 24 q^{25} - 14 q^{27} + 15 q^{29} + 9 q^{31} - 7 q^{33} - 14 q^{35} + 4 q^{37} - 5 q^{39} + 32 q^{41} - 3 q^{43} + 6 q^{45} + 15 q^{47} + 27 q^{49} - 28 q^{51} + 8 q^{53} + q^{57} - 11 q^{59} + 25 q^{61} + 3 q^{63} + 18 q^{65} - 20 q^{67} + 3 q^{69} + 33 q^{71} + 8 q^{73} - 24 q^{75} + 14 q^{77} - 17 q^{79} + 14 q^{81} - 4 q^{83} + 16 q^{85} - 15 q^{87} + 14 q^{89} - 28 q^{91} - 9 q^{93} + 3 q^{95} + 9 q^{97} + 7 q^{99}+O(q^{100}) $ 14 * q - 14 * q^3 + 6 * q^5 + 3 * q^7 + 14 * q^9 + 7 * q^11 + 5 * q^13 - 6 * q^15 + 28 * q^17 - q^19 - 3 * q^21 - 3 * q^23 + 24 * q^25 - 14 * q^27 + 15 * q^29 + 9 * q^31 - 7 * q^33 - 14 * q^35 + 4 * q^37 - 5 * q^39 + 32 * q^41 - 3 * q^43 + 6 * q^45 + 15 * q^47 + 27 * q^49 - 28 * q^51 + 8 * q^53 + q^57 - 11 * q^59 + 25 * q^61 + 3 * q^63 + 18 * q^65 - 20 * q^67 + 3 * q^69 + 33 * q^71 + 8 * q^73 - 24 * q^75 + 14 * q^77 - 17 * q^79 + 14 * q^81 - 4 * q^83 + 16 * q^85 - 15 * q^87 + 14 * q^89 - 28 * q^91 - 9 * q^93 + 3 * q^95 + 9 * q^97 + 7 * 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
$5$	−2.15802	−0.965097	−0.482548	−	0.875869i	$-0.660289\pi$
$5$	−2.15802	−0.965097	−0.482548	+	0.875869i	$0.660289\pi$
$6$	0	0
$7$	−0.0890340	−0.0336517	−0.0168258	−	0.999858i	$-0.505356\pi$
$7$	−0.0890340	−0.0336517	−0.0168258	+	0.999858i	$0.505356\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	6.37243	1.92136	0.960681	−	0.277656i	$-0.0895575\pi$
$11$	6.37243	1.92136	0.960681	+	0.277656i	$0.0895575\pi$
$12$	0	0
$13$	3.74097	1.03756	0.518779	−	0.854908i	$-0.326387\pi$
$13$	3.74097	1.03756	0.518779	+	0.854908i	$0.326387\pi$
$14$	0	0
$15$	2.15802	0.557199
$16$	0	0
$17$	4.37258	1.06051	0.530253	−	0.847840i	$-0.322097\pi$
$17$	4.37258	1.06051	0.530253	+	0.847840i	$0.322097\pi$
$18$	0	0
$19$	5.14883	1.18122	0.590611	−	0.806956i	$-0.298887\pi$
$19$	5.14883	1.18122	0.590611	+	0.806956i	$0.298887\pi$
$20$	0	0
$21$	0.0890340	0.0194288
$22$	0	0
$23$	1.37077	0.285824	0.142912	−	0.989735i	$-0.454353\pi$
$23$	1.37077	0.285824	0.142912	+	0.989735i	$0.454353\pi$
$24$	0	0
$25$	−0.342943	−0.0685886
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	3.68963	0.685147	0.342574	−	0.939491i	$-0.388701\pi$
$29$	3.68963	0.685147	0.342574	+	0.939491i	$0.388701\pi$
$30$	0	0
$31$	10.8286	1.94487	0.972436	−	0.233171i	$-0.0749102\pi$
$31$	10.8286	1.94487	0.972436	+	0.233171i	$0.0749102\pi$
$32$	0	0
$33$	−6.37243	−1.10930
$34$	0	0
$35$	0.192137	0.0324771
$36$	0	0
$37$	2.87385	0.472457	0.236229	−	0.971697i	$-0.424089\pi$
$37$	2.87385	0.472457	0.236229	+	0.971697i	$0.424089\pi$
$38$	0	0
$39$	−3.74097	−0.599035
$40$	0	0
$41$	2.86978	0.448185	0.224093	−	0.974568i	$-0.428058\pi$
$41$	2.86978	0.448185	0.224093	+	0.974568i	$0.428058\pi$
$42$	0	0
$43$	−8.06219	−1.22947	−0.614736	−	0.788733i	$-0.710737\pi$
$43$	−8.06219	−1.22947	−0.614736	+	0.788733i	$0.710737\pi$
$44$	0	0
$45$	−2.15802	−0.321699
$46$	0	0
$47$	−7.11981	−1.03853	−0.519266	−	0.854613i	$-0.673794\pi$
$47$	−7.11981	−1.03853	−0.519266	+	0.854613i	$0.673794\pi$
$48$	0	0
$49$	−6.99207	−0.998868
$50$	0	0
$51$	−4.37258	−0.612283
$52$	0	0
$53$	1.35322	0.185879	0.0929396	−	0.995672i	$-0.470374\pi$
$53$	1.35322	0.185879	0.0929396	+	0.995672i	$0.470374\pi$
$54$	0	0
$55$	−13.7519	−1.85430
$56$	0	0
$57$	−5.14883	−0.681979
$58$	0	0
$59$	−1.49740	−0.194945	−0.0974724	−	0.995238i	$-0.531076\pi$
$59$	−1.49740	−0.194945	−0.0974724	+	0.995238i	$0.531076\pi$
$60$	0	0
$61$	1.63235	0.209001	0.104500	−	0.994525i	$-0.466676\pi$
$61$	1.63235	0.209001	0.104500	+	0.994525i	$0.466676\pi$
$62$	0	0
$63$	−0.0890340	−0.0112172
$64$	0	0
$65$	−8.07310	−1.00134
$66$	0	0
$67$	6.57177	0.802870	0.401435	−	0.915888i	$-0.368512\pi$
$67$	6.57177	0.802870	0.401435	+	0.915888i	$0.368512\pi$
$68$	0	0
$69$	−1.37077	−0.165021
$70$	0	0
$71$	5.51440	0.654439	0.327220	−	0.944948i	$-0.393888\pi$
$71$	5.51440	0.654439	0.327220	+	0.944948i	$0.393888\pi$
$72$	0	0
$73$	−11.7255	−1.37237	−0.686183	−	0.727429i	$-0.740715\pi$
$73$	−11.7255	−1.37237	−0.686183	+	0.727429i	$0.740715\pi$
$74$	0	0
$75$	0.342943	0.0395996
$76$	0	0
$77$	−0.567363	−0.0646570
$78$	0	0
$79$	−0.145483	−0.0163682	−0.00818409	−	0.999967i	$-0.502605\pi$
$79$	−0.145483	−0.0163682	−0.00818409	+	0.999967i	$0.502605\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	10.3541	1.13651	0.568255	−	0.822852i	$-0.307619\pi$
$83$	10.3541	1.13651	0.568255	+	0.822852i	$0.307619\pi$
$84$	0	0
$85$	−9.43612	−1.02349
$86$	0	0
$87$	−3.68963	−0.395570
$88$	0	0
$89$	−3.70733	−0.392976	−0.196488	−	0.980506i	$-0.562954\pi$
$89$	−3.70733	−0.392976	−0.196488	+	0.980506i	$0.562954\pi$
$90$	0	0
$91$	−0.333073	−0.0349156
$92$	0	0
$93$	−10.8286	−1.12287
$94$	0	0
$95$	−11.1113	−1.13999
$96$	0	0
$97$	8.43969	0.856921	0.428461	−	0.903561i	$-0.359056\pi$
$97$	8.43969	0.856921	0.428461	+	0.903561i	$0.359056\pi$
$98$	0	0
$99$	6.37243	0.640454
$100$	0	0
$101$	2.88449	0.287018	0.143509	−	0.989649i	$-0.454161\pi$
$101$	2.88449	0.287018	0.143509	+	0.989649i	$0.454161\pi$
$102$	0	0
$103$	−8.06957	−0.795118	−0.397559	−	0.917577i	$-0.630143\pi$
$103$	−8.06957	−0.795118	−0.397559	+	0.917577i	$0.630143\pi$
$104$	0	0
$105$	−0.192137	−0.0187507
$106$	0	0
$107$	−13.5425	−1.30920	−0.654601	−	0.755975i	$-0.727163\pi$
$107$	−13.5425	−1.30920	−0.654601	+	0.755975i	$0.727163\pi$
$108$	0	0
$109$	−10.4611	−1.00200	−0.500998	−	0.865448i	$-0.667034\pi$
$109$	−10.4611	−1.00200	−0.500998	+	0.865448i	$0.667034\pi$
$110$	0	0
$111$	−2.87385	−0.272773
$112$	0	0
$113$	8.48407	0.798114	0.399057	−	0.916926i	$-0.369338\pi$
$113$	8.48407	0.798114	0.399057	+	0.916926i	$0.369338\pi$
$114$	0	0
$115$	−2.95814	−0.275848
$116$	0	0
$117$	3.74097	0.345853
$118$	0	0
$119$	−0.389308	−0.0356878
$120$	0	0
$121$	29.6079	2.69163
$122$	0	0
$123$	−2.86978	−0.258760
$124$	0	0
$125$	11.5302	1.03129
$126$	0	0
$127$	12.7775	1.13382	0.566911	−	0.823779i	$-0.308138\pi$
$127$	12.7775	1.13382	0.566911	+	0.823779i	$0.308138\pi$
$128$	0	0
$129$	8.06219	0.709836
$130$	0	0
$131$	−8.81334	−0.770025	−0.385013	−	0.922911i	$-0.625803\pi$
$131$	−8.81334	−0.770025	−0.385013	+	0.922911i	$0.625803\pi$
$132$	0	0
$133$	−0.458420	−0.0397501
$134$	0	0
$135$	2.15802	0.185733
$136$	0	0
$137$	1.75616	0.150039	0.0750196	−	0.997182i	$-0.476098\pi$
$137$	1.75616	0.150039	0.0750196	+	0.997182i	$0.476098\pi$
$138$	0	0
$139$	−1.35600	−0.115015	−0.0575073	−	0.998345i	$-0.518315\pi$
$139$	−1.35600	−0.115015	−0.0575073	+	0.998345i	$0.518315\pi$
$140$	0	0
$141$	7.11981	0.599596
$142$	0	0
$143$	23.8391	1.99352
$144$	0	0
$145$	−7.96230	−0.661233
$146$	0	0
$147$	6.99207	0.576696
$148$	0	0
$149$	17.3341	1.42007	0.710033	−	0.704169i	$-0.248680\pi$
$149$	17.3341	1.42007	0.710033	+	0.704169i	$0.248680\pi$
$150$	0	0
$151$	−5.06723	−0.412365	−0.206182	−	0.978514i	$-0.566104\pi$
$151$	−5.06723	−0.412365	−0.206182	+	0.978514i	$0.566104\pi$
$152$	0	0
$153$	4.37258	0.353502
$154$	0	0
$155$	−23.3683	−1.87699
$156$	0	0
$157$	1.44040	0.114957	0.0574784	−	0.998347i	$-0.481694\pi$
$157$	1.44040	0.114957	0.0574784	+	0.998347i	$0.481694\pi$
$158$	0	0
$159$	−1.35322	−0.107317
$160$	0	0
$161$	−0.122045	−0.00961847
$162$	0	0
$163$	−21.5536	−1.68821	−0.844103	−	0.536181i	$-0.819867\pi$
$163$	−21.5536	−1.68821	−0.844103	+	0.536181i	$0.819867\pi$
$164$	0	0
$165$	13.7519	1.07058
$166$	0	0
$167$	−15.3689	−1.18928	−0.594642	−	0.803991i	$-0.702706\pi$
$167$	−15.3689	−1.18928	−0.594642	+	0.803991i	$0.702706\pi$
$168$	0	0
$169$	0.994864	0.0765280
$170$	0	0
$171$	5.14883	0.393741
$172$	0	0
$173$	−10.0170	−0.761576	−0.380788	−	0.924662i	$-0.624347\pi$
$173$	−10.0170	−0.761576	−0.380788	+	0.924662i	$0.624347\pi$
$174$	0	0
$175$	0.0305336	0.00230812
$176$	0	0
$177$	1.49740	0.112551
$178$	0	0
$179$	−23.4977	−1.75630	−0.878149	−	0.478387i	$-0.841222\pi$
$179$	−23.4977	−1.75630	−0.878149	+	0.478387i	$0.841222\pi$
$180$	0	0
$181$	−4.18660	−0.311188	−0.155594	−	0.987821i	$-0.549729\pi$
$181$	−4.18660	−0.311188	−0.155594	+	0.987821i	$0.549729\pi$
$182$	0	0
$183$	−1.63235	−0.120667
$184$	0	0
$185$	−6.20182	−0.455967
$186$	0	0
$187$	27.8640	2.03761
$188$	0	0
$189$	0.0890340	0.00647627
$190$	0	0
$191$	24.9789	1.80741	0.903704	−	0.428157i	$-0.140837\pi$
$191$	24.9789	1.80741	0.903704	+	0.428157i	$0.140837\pi$
$192$	0	0
$193$	−20.6393	−1.48565	−0.742825	−	0.669486i	$-0.766515\pi$
$193$	−20.6393	−1.48565	−0.742825	+	0.669486i	$0.766515\pi$
$194$	0	0
$195$	8.07310	0.578126
$196$	0	0
$197$	18.7699	1.33730	0.668652	−	0.743576i	$-0.266872\pi$
$197$	18.7699	1.33730	0.668652	+	0.743576i	$0.266872\pi$
$198$	0	0
$199$	4.24748	0.301096	0.150548	−	0.988603i	$-0.451896\pi$
$199$	4.24748	0.301096	0.150548	+	0.988603i	$0.451896\pi$
$200$	0	0
$201$	−6.57177	−0.463537
$202$	0	0
$203$	−0.328502	−0.0230563
$204$	0	0
$205$	−6.19306	−0.432542
$206$	0	0
$207$	1.37077	0.0952748
$208$	0	0
$209$	32.8106	2.26955
$210$	0	0
$211$	8.87678	0.611103	0.305551	−	0.952176i	$-0.401159\pi$
$211$	8.87678	0.611103	0.305551	+	0.952176i	$0.401159\pi$
$212$	0	0
$213$	−5.51440	−0.377841
$214$	0	0
$215$	17.3984	1.18656
$216$	0	0
$217$	−0.964112	−0.0654482
$218$	0	0
$219$	11.7255	0.792335
$220$	0	0
$221$	16.3577	1.10034
$222$	0	0
$223$	6.34611	0.424967	0.212483	−	0.977165i	$-0.431845\pi$
$223$	6.34611	0.424967	0.212483	+	0.977165i	$0.431845\pi$
$224$	0	0
$225$	−0.342943	−0.0228629
$226$	0	0
$227$	−23.1820	−1.53864	−0.769320	−	0.638863i	$-0.779405\pi$
$227$	−23.1820	−1.53864	−0.769320	+	0.638863i	$0.779405\pi$
$228$	0	0
$229$	−3.99134	−0.263755	−0.131878	−	0.991266i	$-0.542101\pi$
$229$	−3.99134	−0.263755	−0.131878	+	0.991266i	$0.542101\pi$
$230$	0	0
$231$	0.567363	0.0373297
$232$	0	0
$233$	15.2900	1.00168	0.500841	−	0.865539i	$-0.333024\pi$
$233$	15.2900	1.00168	0.500841	+	0.865539i	$0.333024\pi$
$234$	0	0
$235$	15.3647	1.00228
$236$	0	0
$237$	0.145483	0.00945017
$238$	0	0
$239$	25.2450	1.63297	0.816483	−	0.577370i	$-0.195921\pi$
$239$	25.2450	1.63297	0.816483	+	0.577370i	$0.195921\pi$
$240$	0	0
$241$	25.4758	1.64104	0.820519	−	0.571619i	$-0.193685\pi$
$241$	25.4758	1.64104	0.820519	+	0.571619i	$0.193685\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	15.0890	0.964004
$246$	0	0
$247$	19.2616	1.22559
$248$	0	0
$249$	−10.3541	−0.656165
$250$	0	0
$251$	1.00000	0.0631194
$251$	1.00000	0.0631194
$252$	0	0
$253$	8.73512	0.549172
$254$	0	0
$255$	9.43612	0.590913
$256$	0	0
$257$	11.7511	0.733014	0.366507	−	0.930415i	$-0.380554\pi$
$257$	11.7511	0.733014	0.366507	+	0.930415i	$0.380554\pi$
$258$	0	0
$259$	−0.255870	−0.0158990
$260$	0	0
$261$	3.68963	0.228382
$262$	0	0
$263$	25.1327	1.54975	0.774873	−	0.632116i	$-0.217814\pi$
$263$	25.1327	1.54975	0.774873	+	0.632116i	$0.217814\pi$
$264$	0	0
$265$	−2.92028	−0.179391
$266$	0	0
$267$	3.70733	0.226885
$268$	0	0
$269$	13.5598	0.826755	0.413378	−	0.910560i	$-0.364349\pi$
$269$	13.5598	0.826755	0.413378	+	0.910560i	$0.364349\pi$
$270$	0	0
$271$	−11.7518	−0.713870	−0.356935	−	0.934129i	$-0.616178\pi$
$271$	−11.7518	−0.713870	−0.356935	+	0.934129i	$0.616178\pi$
$272$	0	0
$273$	0.333073	0.0201585
$274$	0	0
$275$	−2.18538	−0.131783
$276$	0	0
$277$	−10.0425	−0.603398	−0.301699	−	0.953403i	$-0.597554\pi$
$277$	−10.0425	−0.603398	−0.301699	+	0.953403i	$0.597554\pi$
$278$	0	0
$279$	10.8286	0.648290
$280$	0	0
$281$	−6.94249	−0.414154	−0.207077	−	0.978325i	$-0.566395\pi$
$281$	−6.94249	−0.414154	−0.207077	+	0.978325i	$0.566395\pi$
$282$	0	0
$283$	−26.4147	−1.57019	−0.785096	−	0.619374i	$-0.787386\pi$
$283$	−26.4147	−1.57019	−0.785096	+	0.619374i	$0.787386\pi$
$284$	0	0
$285$	11.1113	0.658175
$286$	0	0
$287$	−0.255508	−0.0150822
$288$	0	0
$289$	2.11943	0.124673
$290$	0	0
$291$	−8.43969	−0.494744
$292$	0	0
$293$	26.1155	1.52568	0.762841	−	0.646587i	$-0.223804\pi$
$293$	26.1155	1.52568	0.762841	+	0.646587i	$0.223804\pi$
$294$	0	0
$295$	3.23142	0.188141
$296$	0	0
$297$	−6.37243	−0.369766
$298$	0	0
$299$	5.12800	0.296560
$300$	0	0
$301$	0.717808	0.0413738
$302$	0	0
$303$	−2.88449	−0.165710
$304$	0	0
$305$	−3.52264	−0.201706
$306$	0	0
$307$	−7.39437	−0.422019	−0.211010	−	0.977484i	$-0.567675\pi$
$307$	−7.39437	−0.422019	−0.211010	+	0.977484i	$0.567675\pi$
$308$	0	0
$309$	8.06957	0.459062
$310$	0	0
$311$	−7.68152	−0.435579	−0.217790	−	0.975996i	$-0.569885\pi$
$311$	−7.68152	−0.435579	−0.217790	+	0.975996i	$0.569885\pi$
$312$	0	0
$313$	0.206449	0.0116692	0.00583459	−	0.999983i	$-0.498143\pi$
$313$	0.206449	0.0116692	0.00583459	+	0.999983i	$0.498143\pi$
$314$	0	0
$315$	0.192137	0.0108257
$316$	0	0
$317$	15.1957	0.853473	0.426737	−	0.904376i	$-0.359663\pi$
$317$	15.1957	0.853473	0.426737	+	0.904376i	$0.359663\pi$
$318$	0	0
$319$	23.5119	1.31641
$320$	0	0
$321$	13.5425	0.755868
$322$	0	0
$323$	22.5136	1.25269
$324$	0	0
$325$	−1.28294	−0.0711647
$326$	0	0
$327$	10.4611	0.578503
$328$	0	0
$329$	0.633905	0.0349483
$330$	0	0
$331$	−10.3329	−0.567946	−0.283973	−	0.958832i	$-0.591653\pi$
$331$	−10.3329	−0.567946	−0.283973	+	0.958832i	$0.591653\pi$
$332$	0	0
$333$	2.87385	0.157486
$334$	0	0
$335$	−14.1820	−0.774847
$336$	0	0
$337$	−13.2545	−0.722018	−0.361009	−	0.932562i	$-0.617568\pi$
$337$	−13.2545	−0.722018	−0.361009	+	0.932562i	$0.617568\pi$
$338$	0	0
$339$	−8.48407	−0.460791
$340$	0	0
$341$	69.0045	3.73680
$342$	0	0
$343$	1.24577	0.0672652
$344$	0	0
$345$	2.95814	0.159261
$346$	0	0
$347$	−27.8973	−1.49761	−0.748804	−	0.662792i	$-0.769371\pi$
$347$	−27.8973	−1.49761	−0.748804	+	0.662792i	$0.769371\pi$
$348$	0	0
$349$	26.2385	1.40451	0.702257	−	0.711923i	$-0.252176\pi$
$349$	26.2385	1.40451	0.702257	+	0.711923i	$0.252176\pi$
$350$	0	0
$351$	−3.74097	−0.199678
$352$	0	0
$353$	−19.2709	−1.02568	−0.512842	−	0.858483i	$-0.671407\pi$
$353$	−19.2709	−1.02568	−0.512842	+	0.858483i	$0.671407\pi$
$354$	0	0
$355$	−11.9002	−0.631597
$356$	0	0
$357$	0.389308	0.0206044
$358$	0	0
$359$	−33.6004	−1.77336	−0.886680	−	0.462384i	$-0.846994\pi$
$359$	−33.6004	−1.77336	−0.886680	+	0.462384i	$0.846994\pi$
$360$	0	0
$361$	7.51042	0.395285
$362$	0	0
$363$	−29.6079	−1.55401
$364$	0	0
$365$	25.3039	1.32446
$366$	0	0
$367$	4.85381	0.253367	0.126683	−	0.991943i	$-0.459567\pi$
$367$	4.85381	0.253367	0.126683	+	0.991943i	$0.459567\pi$
$368$	0	0
$369$	2.86978	0.149395
$370$	0	0
$371$	−0.120483	−0.00625514
$372$	0	0
$373$	−10.1297	−0.524497	−0.262248	−	0.965000i	$-0.584464\pi$
$373$	−10.1297	−0.524497	−0.262248	+	0.965000i	$0.584464\pi$
$374$	0	0
$375$	−11.5302	−0.595416
$376$	0	0
$377$	13.8028	0.710880
$378$	0	0
$379$	−20.3255	−1.04405	−0.522026	−	0.852930i	$-0.674824\pi$
$379$	−20.3255	−1.04405	−0.522026	+	0.852930i	$0.674824\pi$
$380$	0	0
$381$	−12.7775	−0.654613
$382$	0	0
$383$	−16.7879	−0.857820	−0.428910	−	0.903347i	$-0.641102\pi$
$383$	−16.7879	−0.857820	−0.428910	+	0.903347i	$0.641102\pi$
$384$	0	0
$385$	1.22438	0.0624003
$386$	0	0
$387$	−8.06219	−0.409824
$388$	0	0
$389$	12.9377	0.655966	0.327983	−	0.944684i	$-0.393631\pi$
$389$	12.9377	0.655966	0.327983	+	0.944684i	$0.393631\pi$
$390$	0	0
$391$	5.99378	0.303118
$392$	0	0
$393$	8.81334	0.444574
$394$	0	0
$395$	0.313956	0.0157969
$396$	0	0
$397$	−32.9278	−1.65260	−0.826300	−	0.563230i	$-0.809559\pi$
$397$	−32.9278	−1.65260	−0.826300	+	0.563230i	$0.809559\pi$
$398$	0	0
$399$	0.458420	0.0229497
$400$	0	0
$401$	21.3606	1.06670	0.533348	−	0.845896i	$-0.320934\pi$
$401$	21.3606	1.06670	0.533348	+	0.845896i	$0.320934\pi$
$402$	0	0
$403$	40.5094	2.01792
$404$	0	0
$405$	−2.15802	−0.107233
$406$	0	0
$407$	18.3134	0.907761
$408$	0	0
$409$	−20.8694	−1.03193	−0.515963	−	0.856611i	$-0.672566\pi$
$409$	−20.8694	−1.03193	−0.515963	+	0.856611i	$0.672566\pi$
$410$	0	0
$411$	−1.75616	−0.0866252
$412$	0	0
$413$	0.133319	0.00656022
$414$	0	0
$415$	−22.3444	−1.09684
$416$	0	0
$417$	1.35600	0.0664037
$418$	0	0
$419$	24.0812	1.17644	0.588221	−	0.808700i	$-0.299828\pi$
$419$	24.0812	1.17644	0.588221	+	0.808700i	$0.299828\pi$
$420$	0	0
$421$	−4.41763	−0.215302	−0.107651	−	0.994189i	$-0.534333\pi$
$421$	−4.41763	−0.215302	−0.107651	+	0.994189i	$0.534333\pi$
$422$	0	0
$423$	−7.11981	−0.346177
$424$	0	0
$425$	−1.49954	−0.0727386
$426$	0	0
$427$	−0.145334	−0.00703322
$428$	0	0
$429$	−23.8391	−1.15096
$430$	0	0
$431$	−6.52361	−0.314231	−0.157116	−	0.987580i	$-0.550220\pi$
$431$	−6.52361	−0.314231	−0.157116	+	0.987580i	$0.550220\pi$
$432$	0	0
$433$	23.1144	1.11081	0.555405	−	0.831580i	$-0.312563\pi$
$433$	23.1144	1.11081	0.555405	+	0.831580i	$0.312563\pi$
$434$	0	0
$435$	7.96230	0.381763
$436$	0	0
$437$	7.05784	0.337622
$438$	0	0
$439$	−3.55975	−0.169897	−0.0849487	−	0.996385i	$-0.527073\pi$
$439$	−3.55975	−0.169897	−0.0849487	+	0.996385i	$0.527073\pi$
$440$	0	0
$441$	−6.99207	−0.332956
$442$	0	0
$443$	−2.87946	−0.136807	−0.0684037	−	0.997658i	$-0.521791\pi$
$443$	−2.87946	−0.136807	−0.0684037	+	0.997658i	$0.521791\pi$
$444$	0	0
$445$	8.00050	0.379260
$446$	0	0
$447$	−17.3341	−0.819875
$448$	0	0
$449$	23.0192	1.08634	0.543171	−	0.839622i	$-0.317224\pi$
$449$	23.0192	1.08634	0.543171	+	0.839622i	$0.317224\pi$
$450$	0	0
$451$	18.2875	0.861125
$452$	0	0
$453$	5.06723	0.238079
$454$	0	0
$455$	0.718780	0.0336969
$456$	0	0
$457$	1.40771	0.0658501	0.0329250	−	0.999458i	$-0.489518\pi$
$457$	1.40771	0.0658501	0.0329250	+	0.999458i	$0.489518\pi$
$458$	0	0
$459$	−4.37258	−0.204094
$460$	0	0
$461$	−32.0510	−1.49277	−0.746383	−	0.665517i	$-0.768211\pi$
$461$	−32.0510	−1.49277	−0.746383	+	0.665517i	$0.768211\pi$
$462$	0	0
$463$	−19.4968	−0.906092	−0.453046	−	0.891487i	$-0.649663\pi$
$463$	−19.4968	−0.906092	−0.453046	+	0.891487i	$0.649663\pi$
$464$	0	0
$465$	23.3683	1.08368
$466$	0	0
$467$	36.4337	1.68595	0.842975	−	0.537952i	$-0.180802\pi$
$467$	36.4337	1.68595	0.842975	+	0.537952i	$0.180802\pi$
$468$	0	0
$469$	−0.585111	−0.0270179
$470$	0	0
$471$	−1.44040	−0.0663703
$472$	0	0
$473$	−51.3758	−2.36226
$474$	0	0
$475$	−1.76575	−0.0810183
$476$	0	0
$477$	1.35322	0.0619597
$478$	0	0
$479$	−22.3614	−1.02172	−0.510858	−	0.859665i	$-0.670672\pi$
$479$	−22.3614	−1.02172	−0.510858	+	0.859665i	$0.670672\pi$
$480$	0	0
$481$	10.7510	0.490202
$482$	0	0
$483$	0.122045	0.00555323
$484$	0	0
$485$	−18.2130	−0.827012
$486$	0	0
$487$	−7.94757	−0.360139	−0.180069	−	0.983654i	$-0.557632\pi$
$487$	−7.94757	−0.360139	−0.180069	+	0.983654i	$0.557632\pi$
$488$	0	0
$489$	21.5536	0.974687
$490$	0	0
$491$	−5.18003	−0.233771	−0.116886	−	0.993145i	$-0.537291\pi$
$491$	−5.18003	−0.233771	−0.116886	+	0.993145i	$0.537291\pi$
$492$	0	0
$493$	16.1332	0.726602
$494$	0	0
$495$	−13.7519	−0.618100
$496$	0	0
$497$	−0.490969	−0.0220230
$498$	0	0
$499$	−5.04036	−0.225637	−0.112819	−	0.993616i	$-0.535988\pi$
$499$	−5.04036	−0.225637	−0.112819	+	0.993616i	$0.535988\pi$
$500$	0	0
$501$	15.3689	0.686633
$502$	0	0
$503$	28.9710	1.29175	0.645877	−	0.763441i	$-0.276492\pi$
$503$	28.9710	1.29175	0.645877	+	0.763441i	$0.276492\pi$
$504$	0	0
$505$	−6.22479	−0.277000
$506$	0	0
$507$	−0.994864	−0.0441835
$508$	0	0
$509$	−5.72353	−0.253691	−0.126846	−	0.991922i	$-0.540485\pi$
$509$	−5.72353	−0.253691	−0.126846	+	0.991922i	$0.540485\pi$
$510$	0	0
$511$	1.04397	0.0461824
$512$	0	0
$513$	−5.14883	−0.227326
$514$	0	0
$515$	17.4143	0.767366
$516$	0	0
$517$	−45.3705	−1.99539
$518$	0	0
$519$	10.0170	0.439696
$520$	0	0
$521$	31.6846	1.38813	0.694065	−	0.719913i	$-0.255818\pi$
$521$	31.6846	1.38813	0.694065	+	0.719913i	$0.255818\pi$
$522$	0	0
$523$	−19.6406	−0.858825	−0.429413	−	0.903108i	$-0.641279\pi$
$523$	−19.6406	−0.858825	−0.429413	+	0.903108i	$0.641279\pi$
$524$	0	0
$525$	−0.0305336	−0.00133259
$526$	0	0
$527$	47.3488	2.06255
$528$	0	0
$529$	−21.1210	−0.918304
$530$	0	0
$531$	−1.49740	−0.0649816
$532$	0	0
$533$	10.7358	0.465018
$534$	0	0
$535$	29.2250	1.26351
$536$	0	0
$537$	23.4977	1.01400
$538$	0	0
$539$	−44.5565	−1.91919
$540$	0	0
$541$	19.7211	0.847875	0.423937	−	0.905691i	$-0.360648\pi$
$541$	19.7211	0.847875	0.423937	+	0.905691i	$0.360648\pi$
$542$	0	0
$543$	4.18660	0.179664
$544$	0	0
$545$	22.5754	0.967023
$546$	0	0
$547$	−25.9101	−1.10783	−0.553917	−	0.832572i	$-0.686867\pi$
$547$	−25.9101	−1.10783	−0.553917	+	0.832572i	$0.686867\pi$
$548$	0	0
$549$	1.63235	0.0696669
$550$	0	0
$551$	18.9973	0.809311
$552$	0	0
$553$	0.0129530	0.000550816 0
$554$	0	0
$555$	6.20182	0.263253
$556$	0	0
$557$	−1.09032	−0.0461982	−0.0230991	−	0.999733i	$-0.507353\pi$
$557$	−1.09032	−0.0461982	−0.0230991	+	0.999733i	$0.507353\pi$
$558$	0	0
$559$	−30.1604	−1.27565
$560$	0	0
$561$	−27.8640	−1.17642
$562$	0	0
$563$	−31.5964	−1.33163	−0.665815	−	0.746117i	$-0.731916\pi$
$563$	−31.5964	−1.33163	−0.665815	+	0.746117i	$0.731916\pi$
$564$	0	0
$565$	−18.3088	−0.770257
$566$	0	0
$567$	−0.0890340	−0.00373907
$568$	0	0
$569$	−8.02734	−0.336524	−0.168262	−	0.985742i	$-0.553815\pi$
$569$	−8.02734	−0.336524	−0.168262	+	0.985742i	$0.553815\pi$
$570$	0	0
$571$	−36.6865	−1.53528	−0.767640	−	0.640881i	$-0.778569\pi$
$571$	−36.6865	−1.53528	−0.767640	+	0.640881i	$0.778569\pi$
$572$	0	0
$573$	−24.9789	−1.04351
$574$	0	0
$575$	−0.470095	−0.0196043
$576$	0	0
$577$	40.6433	1.69200	0.846001	−	0.533181i	$-0.179004\pi$
$577$	40.6433	1.69200	0.846001	+	0.533181i	$0.179004\pi$
$578$	0	0
$579$	20.6393	0.857740
$580$	0	0
$581$	−0.921867	−0.0382455
$582$	0	0
$583$	8.62331	0.357141
$584$	0	0
$585$	−8.07310	−0.333781
$586$	0	0
$587$	−10.3231	−0.426082	−0.213041	−	0.977043i	$-0.568337\pi$
$587$	−10.3231	−0.426082	−0.213041	+	0.977043i	$0.568337\pi$
$588$	0	0
$589$	55.7545	2.29732
$590$	0	0
$591$	−18.7699	−0.772092
$592$	0	0
$593$	19.5553	0.803041	0.401521	−	0.915850i	$-0.368482\pi$
$593$	19.5553	0.803041	0.401521	+	0.915850i	$0.368482\pi$
$594$	0	0
$595$	0.840135	0.0344422
$596$	0	0
$597$	−4.24748	−0.173838
$598$	0	0
$599$	−13.0892	−0.534809	−0.267404	−	0.963584i	$-0.586166\pi$
$599$	−13.0892	−0.534809	−0.267404	+	0.963584i	$0.586166\pi$
$600$	0	0
$601$	44.9038	1.83166	0.915832	−	0.401561i	$-0.131532\pi$
$601$	44.9038	1.83166	0.915832	+	0.401561i	$0.131532\pi$
$602$	0	0
$603$	6.57177	0.267623
$604$	0	0
$605$	−63.8945	−2.59768
$606$	0	0
$607$	37.2506	1.51195	0.755977	−	0.654599i	$-0.227162\pi$
$607$	37.2506	1.51195	0.755977	+	0.654599i	$0.227162\pi$
$608$	0	0
$609$	0.328502	0.0133116
$610$	0	0
$611$	−26.6350	−1.07754
$612$	0	0
$613$	−13.1864	−0.532595	−0.266297	−	0.963891i	$-0.585800\pi$
$613$	−13.1864	−0.532595	−0.266297	+	0.963891i	$0.585800\pi$
$614$	0	0
$615$	6.19306	0.249728
$616$	0	0
$617$	−32.9143	−1.32508	−0.662540	−	0.749026i	$-0.730522\pi$
$617$	−32.9143	−1.32508	−0.662540	+	0.749026i	$0.730522\pi$
$618$	0	0
$619$	29.8087	1.19811	0.599057	−	0.800706i	$-0.295542\pi$
$619$	29.8087	1.19811	0.599057	+	0.800706i	$0.295542\pi$
$620$	0	0
$621$	−1.37077	−0.0550069
$622$	0	0
$623$	0.330078	0.0132243
$624$	0	0
$625$	−23.1677	−0.926707
$626$	0	0
$627$	−32.8106	−1.31033
$628$	0	0
$629$	12.5661	0.501044
$630$	0	0
$631$	−4.70129	−0.187155	−0.0935777	−	0.995612i	$-0.529830\pi$
$631$	−4.70129	−0.187155	−0.0935777	+	0.995612i	$0.529830\pi$
$632$	0	0
$633$	−8.87678	−0.352820
$634$	0	0
$635$	−27.5742	−1.09425
$636$	0	0
$637$	−26.1571	−1.03638
$638$	0	0
$639$	5.51440	0.218146
$640$	0	0
$641$	27.0174	1.06712	0.533562	−	0.845761i	$-0.320853\pi$
$641$	27.0174	1.06712	0.533562	+	0.845761i	$0.320853\pi$
$642$	0	0
$643$	21.7656	0.858351	0.429176	−	0.903221i	$-0.358804\pi$
$643$	21.7656	0.858351	0.429176	+	0.903221i	$0.358804\pi$
$644$	0	0
$645$	−17.3984	−0.685060
$646$	0	0
$647$	32.4831	1.27704	0.638521	−	0.769604i	$-0.279546\pi$
$647$	32.4831	1.27704	0.638521	+	0.769604i	$0.279546\pi$
$648$	0	0
$649$	−9.54208	−0.374559
$650$	0	0
$651$	0.964112	0.0377865
$652$	0	0
$653$	−19.1314	−0.748671	−0.374335	−	0.927293i	$-0.622129\pi$
$653$	−19.1314	−0.748671	−0.374335	+	0.927293i	$0.622129\pi$
$654$	0	0
$655$	19.0194	0.743149
$656$	0	0
$657$	−11.7255	−0.457455
$658$	0	0
$659$	23.3204	0.908433	0.454216	−	0.890891i	$-0.349919\pi$
$659$	23.3204	0.908433	0.454216	+	0.890891i	$0.349919\pi$
$660$	0	0
$661$	−20.2098	−0.786072	−0.393036	−	0.919523i	$-0.628575\pi$
$661$	−20.2098	−0.786072	−0.393036	+	0.919523i	$0.628575\pi$
$662$	0	0
$663$	−16.3577	−0.635280
$664$	0	0
$665$	0.989281	0.0383627
$666$	0	0
$667$	5.05762	0.195832
$668$	0	0
$669$	−6.34611	−0.245355
$670$	0	0
$671$	10.4020	0.401566
$672$	0	0
$673$	13.8534	0.534010	0.267005	−	0.963695i	$-0.413966\pi$
$673$	13.8534	0.534010	0.267005	+	0.963695i	$0.413966\pi$
$674$	0	0
$675$	0.342943	0.0131999
$676$	0	0
$677$	3.40348	0.130806	0.0654032	−	0.997859i	$-0.479167\pi$
$677$	3.40348	0.130806	0.0654032	+	0.997859i	$0.479167\pi$
$678$	0	0
$679$	−0.751419	−0.0288368
$680$	0	0
$681$	23.1820	0.888335
$682$	0	0
$683$	−0.256424	−0.00981180	−0.00490590	−	0.999988i	$-0.501562\pi$
$683$	−0.256424	−0.00981180	−0.00490590	+	0.999988i	$0.501562\pi$
$684$	0	0
$685$	−3.78984	−0.144802
$686$	0	0
$687$	3.99134	0.152279
$688$	0	0
$689$	5.06236	0.192860
$690$	0	0
$691$	9.29790	0.353709	0.176854	−	0.984237i	$-0.443408\pi$
$691$	9.29790	0.353709	0.176854	+	0.984237i	$0.443408\pi$
$692$	0	0
$693$	−0.567363	−0.0215523
$694$	0	0
$695$	2.92628	0.111000
$696$	0	0
$697$	12.5484	0.475303
$698$	0	0
$699$	−15.2900	−0.578322
$700$	0	0
$701$	−10.0041	−0.377851	−0.188925	−	0.981991i	$-0.560500\pi$
$701$	−10.0041	−0.377851	−0.188925	+	0.981991i	$0.560500\pi$
$702$	0	0
$703$	14.7969	0.558077
$704$	0	0
$705$	−15.3647	−0.578668
$706$	0	0
$707$	−0.256818	−0.00965862
$708$	0	0
$709$	40.5260	1.52198	0.760992	−	0.648761i	$-0.224713\pi$
$709$	40.5260	1.52198	0.760992	+	0.648761i	$0.224713\pi$
$710$	0	0
$711$	−0.145483	−0.00545606
$712$	0	0
$713$	14.8435	0.555892
$714$	0	0
$715$	−51.4453	−1.92394
$716$	0	0
$717$	−25.2450	−0.942793
$718$	0	0
$719$	38.9426	1.45231	0.726157	−	0.687529i	$-0.241304\pi$
$719$	38.9426	1.45231	0.726157	+	0.687529i	$0.241304\pi$
$720$	0	0
$721$	0.718465	0.0267570
$722$	0	0
$723$	−25.4758	−0.947454
$724$	0	0
$725$	−1.26533	−0.0469933
$726$	0	0
$727$	−32.3198	−1.19867	−0.599337	−	0.800497i	$-0.704569\pi$
$727$	−32.3198	−1.19867	−0.599337	+	0.800497i	$0.704569\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−35.2525	−1.30386
$732$	0	0
$733$	27.8874	1.03005	0.515023	−	0.857176i	$-0.327783\pi$
$733$	27.8874	1.03005	0.515023	+	0.857176i	$0.327783\pi$
$734$	0	0
$735$	−15.0890	−0.556568
$736$	0	0
$737$	41.8782	1.54260
$738$	0	0
$739$	−9.62125	−0.353923	−0.176962	−	0.984218i	$-0.556627\pi$
$739$	−9.62125	−0.353923	−0.176962	+	0.984218i	$0.556627\pi$
$740$	0	0
$741$	−19.2616	−0.707593
$742$	0	0
$743$	−10.6721	−0.391521	−0.195760	−	0.980652i	$-0.562717\pi$
$743$	−10.6721	−0.391521	−0.195760	+	0.980652i	$0.562717\pi$
$744$	0	0
$745$	−37.4074	−1.37050
$746$	0	0
$747$	10.3541	0.378837
$748$	0	0
$749$	1.20574	0.0440568
$750$	0	0
$751$	−45.7984	−1.67121	−0.835604	−	0.549332i	$-0.814882\pi$
$751$	−45.7984	−1.67121	−0.835604	+	0.549332i	$0.814882\pi$
$752$	0	0
$753$	−1.00000	−0.0364420
$754$	0	0
$755$	10.9352	0.397972
$756$	0	0
$757$	18.1109	0.658250	0.329125	−	0.944286i	$-0.393246\pi$
$757$	18.1109	0.658250	0.329125	+	0.944286i	$0.393246\pi$
$758$	0	0
$759$	−8.73512	−0.317065
$760$	0	0
$761$	27.2075	0.986270	0.493135	−	0.869953i	$-0.335851\pi$
$761$	27.2075	0.986270	0.493135	+	0.869953i	$0.335851\pi$
$762$	0	0
$763$	0.931397	0.0337188
$764$	0	0
$765$	−9.43612	−0.341164
$766$	0	0
$767$	−5.60173	−0.202267
$768$	0	0
$769$	−1.74702	−0.0629993	−0.0314996	−	0.999504i	$-0.510028\pi$
$769$	−1.74702	−0.0629993	−0.0314996	+	0.999504i	$0.510028\pi$
$770$	0	0
$771$	−11.7511	−0.423206
$772$	0	0
$773$	22.8714	0.822629	0.411314	−	0.911494i	$-0.365070\pi$
$773$	22.8714	0.822629	0.411314	+	0.911494i	$0.365070\pi$
$774$	0	0
$775$	−3.71359	−0.133396
$776$	0	0
$777$	0.255870	0.00917928
$778$	0	0
$779$	14.7760	0.529406
$780$	0	0
$781$	35.1402	1.25741
$782$	0	0
$783$	−3.68963	−0.131857
$784$	0	0
$785$	−3.10842	−0.110944
$786$	0	0
$787$	21.3962	0.762693	0.381346	−	0.924432i	$-0.375461\pi$
$787$	21.3962	0.762693	0.381346	+	0.924432i	$0.375461\pi$
$788$	0	0
$789$	−25.1327	−0.894747
$790$	0	0
$791$	−0.755370	−0.0268579
$792$	0	0
$793$	6.10657	0.216850
$794$	0	0
$795$	2.92028	0.103572
$796$	0	0
$797$	16.3928	0.580663	0.290331	−	0.956926i	$-0.406234\pi$
$797$	16.3928	0.580663	0.290331	+	0.956926i	$0.406234\pi$
$798$	0	0
$799$	−31.1319	−1.10137
$800$	0	0
$801$	−3.70733	−0.130992
$802$	0	0
$803$	−74.7199	−2.63681
$804$	0	0
$805$	0.263375	0.00928275
$806$	0	0
$807$	−13.5598	−0.477327
$808$	0	0
$809$	−24.8549	−0.873852	−0.436926	−	0.899498i	$-0.643933\pi$
$809$	−24.8549	−0.873852	−0.436926	+	0.899498i	$0.643933\pi$
$810$	0	0
$811$	−45.3587	−1.59276	−0.796380	−	0.604797i	$-0.793254\pi$
$811$	−45.3587	−1.59276	−0.796380	+	0.604797i	$0.793254\pi$
$812$	0	0
$813$	11.7518	0.412153
$814$	0	0
$815$	46.5131	1.62928
$816$	0	0
$817$	−41.5108	−1.45228
$818$	0	0
$819$	−0.333073	−0.0116385
$820$	0	0
$821$	−0.489606	−0.0170874	−0.00854369	−	0.999964i	$-0.502720\pi$
$821$	−0.489606	−0.0170874	−0.00854369	+	0.999964i	$0.502720\pi$
$822$	0	0
$823$	16.1702	0.563657	0.281829	−	0.959465i	$-0.409059\pi$
$823$	16.1702	0.563657	0.281829	+	0.959465i	$0.409059\pi$
$824$	0	0
$825$	2.18538	0.0760852
$826$	0	0
$827$	46.2064	1.60675	0.803377	−	0.595471i	$-0.203035\pi$
$827$	46.2064	1.60675	0.803377	+	0.595471i	$0.203035\pi$
$828$	0	0
$829$	2.79186	0.0969653	0.0484827	−	0.998824i	$-0.484561\pi$
$829$	2.79186	0.0969653	0.0484827	+	0.998824i	$0.484561\pi$
$830$	0	0
$831$	10.0425	0.348372
$832$	0	0
$833$	−30.5734	−1.05930
$834$	0	0
$835$	33.1665	1.14777
$836$	0	0
$837$	−10.8286	−0.374291
$838$	0	0
$839$	32.7217	1.12968	0.564839	−	0.825201i	$-0.308938\pi$
$839$	32.7217	1.12968	0.564839	+	0.825201i	$0.308938\pi$
$840$	0	0
$841$	−15.3866	−0.530574
$842$	0	0
$843$	6.94249	0.239112
$844$	0	0
$845$	−2.14694	−0.0738569
$846$	0	0
$847$	−2.63611	−0.0905778
$848$	0	0
$849$	26.4147	0.906551
$850$	0	0
$851$	3.93937	0.135040
$852$	0	0
$853$	−51.5390	−1.76466	−0.882330	−	0.470631i	$-0.844026\pi$
$853$	−51.5390	−1.76466	−0.882330	+	0.470631i	$0.844026\pi$
$854$	0	0
$855$	−11.1113	−0.379998
$856$	0	0
$857$	5.97823	0.204212	0.102106	−	0.994773i	$-0.467442\pi$
$857$	5.97823	0.204212	0.102106	+	0.994773i	$0.467442\pi$
$858$	0	0
$859$	−22.1363	−0.755279	−0.377640	−	0.925953i	$-0.623264\pi$
$859$	−22.1363	−0.755279	−0.377640	+	0.925953i	$0.623264\pi$
$860$	0	0
$861$	0.255508	0.00870770
$862$	0	0
$863$	9.95947	0.339024	0.169512	−	0.985528i	$-0.445781\pi$
$863$	9.95947	0.339024	0.169512	+	0.985528i	$0.445781\pi$
$864$	0	0
$865$	21.6168	0.734994
$866$	0	0
$867$	−2.11943	−0.0719798
$868$	0	0
$869$	−0.927084	−0.0314492
$870$	0	0
$871$	24.5848	0.833024
$872$	0	0
$873$	8.43969	0.285640
$874$	0	0
$875$	−1.02658	−0.0347047
$876$	0	0
$877$	−33.6494	−1.13626	−0.568130	−	0.822939i	$-0.692333\pi$
$877$	−33.6494	−1.13626	−0.568130	+	0.822939i	$0.692333\pi$
$878$	0	0
$879$	−26.1155	−0.880852
$880$	0	0
$881$	26.2159	0.883236	0.441618	−	0.897203i	$-0.354405\pi$
$881$	26.2159	0.883236	0.441618	+	0.897203i	$0.354405\pi$
$882$	0	0
$883$	−16.4504	−0.553601	−0.276800	−	0.960927i	$-0.589274\pi$
$883$	−16.4504	−0.553601	−0.276800	+	0.960927i	$0.589274\pi$
$884$	0	0
$885$	−3.23142	−0.108623
$886$	0	0
$887$	−35.1032	−1.17865	−0.589325	−	0.807896i	$-0.700606\pi$
$887$	−35.1032	−1.17865	−0.589325	+	0.807896i	$0.700606\pi$
$888$	0	0
$889$	−1.13763	−0.0381550
$890$	0	0
$891$	6.37243	0.213485
$892$	0	0
$893$	−36.6587	−1.22674
$894$	0	0
$895$	50.7085	1.69500
$896$	0	0
$897$	−5.12800	−0.171219
$898$	0	0
$899$	39.9535	1.33252
$900$	0	0
$901$	5.91706	0.197126
$902$	0	0
$903$	−0.717808	−0.0238872
$904$	0	0
$905$	9.03478	0.300326
$906$	0	0
$907$	37.7541	1.25360	0.626802	−	0.779179i	$-0.284364\pi$
$907$	37.7541	1.25360	0.626802	+	0.779179i	$0.284364\pi$
$908$	0	0
$909$	2.88449	0.0956725
$910$	0	0
$911$	−31.1250	−1.03122	−0.515609	−	0.856824i	$-0.672434\pi$
$911$	−31.1250	−1.03122	−0.515609	+	0.856824i	$0.672434\pi$
$912$	0	0
$913$	65.9809	2.18365
$914$	0	0
$915$	3.52264	0.116455
$916$	0	0
$917$	0.784686	0.0259126
$918$	0	0
$919$	42.9497	1.41678	0.708390	−	0.705821i	$-0.249422\pi$
$919$	42.9497	1.41678	0.708390	+	0.705821i	$0.249422\pi$
$920$	0	0
$921$	7.39437	0.243653
$922$	0	0
$923$	20.6292	0.679019
$924$	0	0
$925$	−0.985565	−0.0324052
$926$	0	0
$927$	−8.06957	−0.265039
$928$	0	0
$929$	−45.7279	−1.50028	−0.750142	−	0.661277i	$-0.770015\pi$
$929$	−45.7279	−1.50028	−0.750142	+	0.661277i	$0.770015\pi$
$930$	0	0
$931$	−36.0010	−1.17988
$932$	0	0
$933$	7.68152	0.251482
$934$	0	0
$935$	−60.1310	−1.96650
$936$	0	0
$937$	41.7370	1.36349	0.681745	−	0.731590i	$-0.261221\pi$
$937$	41.7370	1.36349	0.681745	+	0.731590i	$0.261221\pi$
$938$	0	0
$939$	−0.206449	−0.00673721
$940$	0	0
$941$	−45.1687	−1.47246	−0.736228	−	0.676733i	$-0.763395\pi$
$941$	−45.1687	−1.47246	−0.736228	+	0.676733i	$0.763395\pi$
$942$	0	0
$943$	3.93380	0.128102
$944$	0	0
$945$	−0.192137	−0.00625022
$946$	0	0
$947$	5.59158	0.181702	0.0908509	−	0.995865i	$-0.471041\pi$
$947$	5.59158	0.181702	0.0908509	+	0.995865i	$0.471041\pi$
$948$	0	0
$949$	−43.8647	−1.42391
$950$	0	0
$951$	−15.1957	−0.492753
$952$	0	0
$953$	22.6328	0.733150	0.366575	−	0.930389i	$-0.380530\pi$
$953$	22.6328	0.733150	0.366575	+	0.930389i	$0.380530\pi$
$954$	0	0
$955$	−53.9050	−1.74432
$956$	0	0
$957$	−23.5119	−0.760033
$958$	0	0
$959$	−0.156358	−0.00504907
$960$	0	0
$961$	86.2583	2.78252
$962$	0	0
$963$	−13.5425	−0.436401
$964$	0	0
$965$	44.5401	1.43380
$966$	0	0
$967$	40.8442	1.31346	0.656731	−	0.754125i	$-0.271939\pi$
$967$	40.8442	1.31346	0.656731	+	0.754125i	$0.271939\pi$
$968$	0	0
$969$	−22.5136	−0.723242
$970$	0	0
$971$	11.8000	0.378681	0.189341	−	0.981911i	$-0.439365\pi$
$971$	11.8000	0.378681	0.189341	+	0.981911i	$0.439365\pi$
$972$	0	0
$973$	0.120730	0.00387043
$974$	0	0
$975$	1.28294	0.0410870
$976$	0	0
$977$	−17.6054	−0.563247	−0.281623	−	0.959525i	$-0.590873\pi$
$977$	−17.6054	−0.563247	−0.281623	+	0.959525i	$0.590873\pi$
$978$	0	0
$979$	−23.6247	−0.755049
$980$	0	0
$981$	−10.4611	−0.333999
$982$	0	0
$983$	39.9900	1.27548	0.637741	−	0.770251i	$-0.279869\pi$
$983$	39.9900	1.27548	0.637741	+	0.770251i	$0.279869\pi$
$984$	0	0
$985$	−40.5059	−1.29063
$986$	0	0
$987$	−0.633905	−0.0201774
$988$	0	0
$989$	−11.0514	−0.351413
$990$	0	0
$991$	−2.07342	−0.0658644	−0.0329322	−	0.999458i	$-0.510485\pi$
$991$	−2.07342	−0.0658644	−0.0329322	+	0.999458i	$0.510485\pi$
$992$	0	0
$993$	10.3329	0.327904
$994$	0	0
$995$	−9.16614	−0.290586
$996$	0	0
$997$	−21.3236	−0.675324	−0.337662	−	0.941267i	$-0.609636\pi$
$997$	−21.3236	−0.675324	−0.337662	+	0.941267i	$0.609636\pi$
$998$	0	0
$999$	−2.87385	−0.0909245

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6024.2.a.p.1.4	✓	14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6024.2.a.p.1.4	✓	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -2.15802 q^{5} -0.0890340 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -2.15802 q^{5} -0.0890340 q^{7} +1.00000 q^{9} +6.37243 q^{11} +3.74097 q^{13} +2.15802 q^{15} +4.37258 q^{17} +5.14883 q^{19} +0.0890340 q^{21} +1.37077 q^{23} -0.342943 q^{25} -1.00000 q^{27} +3.68963 q^{29} +10.8286 q^{31} -6.37243 q^{33} +0.192137 q^{35} +2.87385 q^{37} -3.74097 q^{39} +2.86978 q^{41} -8.06219 q^{43} -2.15802 q^{45} -7.11981 q^{47} -6.99207 q^{49} -4.37258 q^{51} +1.35322 q^{53} -13.7519 q^{55} -5.14883 q^{57} -1.49740 q^{59} +1.63235 q^{61} -0.0890340 q^{63} -8.07310 q^{65} +6.57177 q^{67} -1.37077 q^{69} +5.51440 q^{71} -11.7255 q^{73} +0.342943 q^{75} -0.567363 q^{77} -0.145483 q^{79} +1.00000 q^{81} +10.3541 q^{83} -9.43612 q^{85} -3.68963 q^{87} -3.70733 q^{89} -0.333073 q^{91} -10.8286 q^{93} -11.1113 q^{95} +8.43969 q^{97} +6.37243 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q - 14 q^{3} + 6 q^{5} + 3 q^{7} + 14 q^{9}+O(q^{10}) \) 14 * q - 14 * q^3 + 6 * q^5 + 3 * q^7 + 14 * q^9	\( 14 q - 14 q^{3} + 6 q^{5} + 3 q^{7} + 14 q^{9} + 7 q^{11} + 5 q^{13} - 6 q^{15} + 28 q^{17} - q^{19} - 3 q^{21} - 3 q^{23} + 24 q^{25} - 14 q^{27} + 15 q^{29} + 9 q^{31} - 7 q^{33} - 14 q^{35} + 4 q^{37} - 5 q^{39} + 32 q^{41} - 3 q^{43} + 6 q^{45} + 15 q^{47} + 27 q^{49} - 28 q^{51} + 8 q^{53} + q^{57} - 11 q^{59} + 25 q^{61} + 3 q^{63} + 18 q^{65} - 20 q^{67} + 3 q^{69} + 33 q^{71} + 8 q^{73} - 24 q^{75} + 14 q^{77} - 17 q^{79} + 14 q^{81} - 4 q^{83} + 16 q^{85} - 15 q^{87} + 14 q^{89} - 28 q^{91} - 9 q^{93} + 3 q^{95} + 9 q^{97} + 7 q^{99}+O(q^{100}) \) 14 * q - 14 * q^3 + 6 * q^5 + 3 * q^7 + 14 * q^9 + 7 * q^11 + 5 * q^13 - 6 * q^15 + 28 * q^17 - q^19 - 3 * q^21 - 3 * q^23 + 24 * q^25 - 14 * q^27 + 15 * q^29 + 9 * q^31 - 7 * q^33 - 14 * q^35 + 4 * q^37 - 5 * q^39 + 32 * q^41 - 3 * q^43 + 6 * q^45 + 15 * q^47 + 27 * q^49 - 28 * q^51 + 8 * q^53 + q^57 - 11 * q^59 + 25 * q^61 + 3 * q^63 + 18 * q^65 - 20 * q^67 + 3 * q^69 + 33 * q^71 + 8 * q^73 - 24 * q^75 + 14 * q^77 - 17 * q^79 + 14 * q^81 - 4 * q^83 + 16 * q^85 - 15 * q^87 + 14 * q^89 - 28 * q^91 - 9 * q^93 + 3 * q^95 + 9 * q^97 + 7 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more