LMFDB - Embedded newform 8024.2.a.z.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8024,2,Mod(1,8024)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8024, 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("8024.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8024 = 2^{3} \cdot 17 \cdot 59 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8024.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:	$64.0719625819$
Analytic rank:	$1$
Dimension:	$24$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Character	$\chi$	$=$	8024.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.58401 q^{3} +1.28987 q^{5} +3.31857 q^{7} +3.67713 q^{9} +O(q^{10})$	$q-2.58401 q^{3} +1.28987 q^{5} +3.31857 q^{7} +3.67713 q^{9} -6.36578 q^{11} -0.280650 q^{13} -3.33304 q^{15} +1.00000 q^{17} +0.344773 q^{19} -8.57524 q^{21} +7.63670 q^{23} -3.33624 q^{25} -1.74972 q^{27} -1.34090 q^{29} +0.0693813 q^{31} +16.4493 q^{33} +4.28052 q^{35} -11.3813 q^{37} +0.725204 q^{39} +8.03569 q^{41} -4.25544 q^{43} +4.74302 q^{45} -6.79756 q^{47} +4.01291 q^{49} -2.58401 q^{51} +13.7134 q^{53} -8.21102 q^{55} -0.890898 q^{57} +1.00000 q^{59} +11.1539 q^{61} +12.2028 q^{63} -0.362001 q^{65} -4.52081 q^{67} -19.7333 q^{69} -8.19571 q^{71} +11.5031 q^{73} +8.62089 q^{75} -21.1253 q^{77} -5.01208 q^{79} -6.51009 q^{81} -16.9476 q^{83} +1.28987 q^{85} +3.46491 q^{87} -4.22267 q^{89} -0.931356 q^{91} -0.179282 q^{93} +0.444711 q^{95} +0.200891 q^{97} -23.4078 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q - 5 q^{3} - 7 q^{5} - 12 q^{7} + 33 q^{9}+O(q^{10}) $ 24 * q - 5 * q^3 - 7 * q^5 - 12 * q^7 + 33 * q^9	$ 24 q - 5 q^{3} - 7 q^{5} - 12 q^{7} + 33 q^{9} - 15 q^{11} - 17 q^{13} - 7 q^{15} + 24 q^{17} - q^{19} + 10 q^{21} - 21 q^{23} + 19 q^{25} - 26 q^{27} - 18 q^{29} - 23 q^{31} + 10 q^{33} - 5 q^{35} - 29 q^{37} - 44 q^{39} + 14 q^{41} - 23 q^{43} - 24 q^{45} - 17 q^{47} + 46 q^{49} - 5 q^{51} + 7 q^{53} - 33 q^{55} - 2 q^{57} + 24 q^{59} - 9 q^{61} - 52 q^{63} - 15 q^{65} + 2 q^{67} + 2 q^{69} - 51 q^{71} - 18 q^{73} + 7 q^{75} + 14 q^{77} - 36 q^{79} + 12 q^{81} - 39 q^{83} - 7 q^{85} - 5 q^{87} + 46 q^{89} - 52 q^{91} - 20 q^{93} - 52 q^{95} + 50 q^{97} - 53 q^{99}+O(q^{100}) $ 24 * q - 5 * q^3 - 7 * q^5 - 12 * q^7 + 33 * q^9 - 15 * q^11 - 17 * q^13 - 7 * q^15 + 24 * q^17 - q^19 + 10 * q^21 - 21 * q^23 + 19 * q^25 - 26 * q^27 - 18 * q^29 - 23 * q^31 + 10 * q^33 - 5 * q^35 - 29 * q^37 - 44 * q^39 + 14 * q^41 - 23 * q^43 - 24 * q^45 - 17 * q^47 + 46 * q^49 - 5 * q^51 + 7 * q^53 - 33 * q^55 - 2 * q^57 + 24 * q^59 - 9 * q^61 - 52 * q^63 - 15 * q^65 + 2 * q^67 + 2 * q^69 - 51 * q^71 - 18 * q^73 + 7 * q^75 + 14 * q^77 - 36 * q^79 + 12 * q^81 - 39 * q^83 - 7 * q^85 - 5 * q^87 + 46 * q^89 - 52 * q^91 - 20 * q^93 - 52 * q^95 + 50 * q^97 - 53 * 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
$3$	−2.58401	−1.49188	−0.745941	−	0.666012i	$-0.768000\pi$
$3$	−2.58401	−1.49188	−0.745941	+	0.666012i	$0.768000\pi$
$4$	0	0
$5$	1.28987	0.576847	0.288423	−	0.957503i	$-0.406869\pi$
$5$	1.28987	0.576847	0.288423	+	0.957503i	$0.406869\pi$
$6$	0	0
$7$	3.31857	1.25430	0.627151	−	0.778898i	$-0.284221\pi$
$7$	3.31857	1.25430	0.627151	+	0.778898i	$0.284221\pi$
$8$	0	0
$9$	3.67713	1.22571
$10$	0	0
$11$	−6.36578	−1.91936	−0.959678	−	0.281101i	$-0.909300\pi$
$11$	−6.36578	−1.91936	−0.959678	+	0.281101i	$0.909300\pi$
$12$	0	0
$13$	−0.280650	−0.0778383	−0.0389191	−	0.999242i	$-0.512391\pi$
$13$	−0.280650	−0.0778383	−0.0389191	+	0.999242i	$0.512391\pi$
$14$	0	0
$15$	−3.33304	−0.860587
$16$	0	0
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	0	0
$19$	0.344773	0.0790963	0.0395481	−	0.999218i	$-0.487408\pi$
$19$	0.344773	0.0790963	0.0395481	+	0.999218i	$0.487408\pi$
$20$	0	0
$21$	−8.57524	−1.87127
$22$	0	0
$23$	7.63670	1.59236	0.796181	−	0.605058i	$-0.206850\pi$
$23$	7.63670	1.59236	0.796181	+	0.605058i	$0.206850\pi$
$24$	0	0
$25$	−3.33624	−0.667248
$26$	0	0
$27$	−1.74972	−0.336734
$28$	0	0
$29$	−1.34090	−0.248999	−0.124500	−	0.992220i	$-0.539733\pi$
$29$	−1.34090	−0.248999	−0.124500	+	0.992220i	$0.539733\pi$
$30$	0	0
$31$	0.0693813	0.0124612	0.00623062	−	0.999981i	$-0.498017\pi$
$31$	0.0693813	0.0124612	0.00623062	+	0.999981i	$0.498017\pi$
$32$	0	0
$33$	16.4493	2.86345
$34$	0	0
$35$	4.28052	0.723540
$36$	0	0
$37$	−11.3813	−1.87107	−0.935533	−	0.353238i	$-0.885081\pi$
$37$	−11.3813	−1.87107	−0.935533	+	0.353238i	$0.885081\pi$
$38$	0	0
$39$	0.725204	0.116125
$40$	0	0
$41$	8.03569	1.25496	0.627482	−	0.778631i	$-0.284085\pi$
$41$	8.03569	1.25496	0.627482	+	0.778631i	$0.284085\pi$
$42$	0	0
$43$	−4.25544	−0.648948	−0.324474	−	0.945895i	$-0.605187\pi$
$43$	−4.25544	−0.648948	−0.324474	+	0.945895i	$0.605187\pi$
$44$	0	0
$45$	4.74302	0.707047
$46$	0	0
$47$	−6.79756	−0.991526	−0.495763	−	0.868458i	$-0.665112\pi$
$47$	−6.79756	−0.991526	−0.495763	+	0.868458i	$0.665112\pi$
$48$	0	0
$49$	4.01291	0.573273
$50$	0	0
$51$	−2.58401	−0.361834
$52$	0	0
$53$	13.7134	1.88368	0.941842	−	0.336057i	$-0.109093\pi$
$53$	13.7134	1.88368	0.941842	+	0.336057i	$0.109093\pi$
$54$	0	0
$55$	−8.21102	−1.10717
$56$	0	0
$57$	−0.890898	−0.118002
$58$	0	0
$59$	1.00000	0.130189
$59$	1.00000	0.130189
$60$	0	0
$61$	11.1539	1.42811	0.714055	−	0.700089i	$-0.246857\pi$
$61$	11.1539	1.42811	0.714055	+	0.700089i	$0.246857\pi$
$62$	0	0
$63$	12.2028	1.53741
$64$	0	0
$65$	−0.362001	−0.0449007
$66$	0	0
$67$	−4.52081	−0.552304	−0.276152	−	0.961114i	$-0.589059\pi$
$67$	−4.52081	−0.552304	−0.276152	+	0.961114i	$0.589059\pi$
$68$	0	0
$69$	−19.7333	−2.37562
$70$	0	0
$71$	−8.19571	−0.972652	−0.486326	−	0.873778i	$-0.661663\pi$
$71$	−8.19571	−0.972652	−0.486326	+	0.873778i	$0.661663\pi$
$72$	0	0
$73$	11.5031	1.34634	0.673170	−	0.739487i	$-0.264932\pi$
$73$	11.5031	1.34634	0.673170	+	0.739487i	$0.264932\pi$
$74$	0	0
$75$	8.62089	0.995455
$76$	0	0
$77$	−21.1253	−2.40745
$78$	0	0
$79$	−5.01208	−0.563903	−0.281951	−	0.959429i	$-0.590982\pi$
$79$	−5.01208	−0.563903	−0.281951	+	0.959429i	$0.590982\pi$
$80$	0	0
$81$	−6.51009	−0.723343
$82$	0	0
$83$	−16.9476	−1.86024	−0.930121	−	0.367254i	$-0.880298\pi$
$83$	−16.9476	−1.86024	−0.930121	+	0.367254i	$0.880298\pi$
$84$	0	0
$85$	1.28987	0.139906
$86$	0	0
$87$	3.46491	0.371477
$88$	0	0
$89$	−4.22267	−0.447603	−0.223801	−	0.974635i	$-0.571847\pi$
$89$	−4.22267	−0.447603	−0.223801	+	0.974635i	$0.571847\pi$
$90$	0	0
$91$	−0.931356	−0.0976327
$92$	0	0
$93$	−0.179282	−0.0185907
$94$	0	0
$95$	0.444711	0.0456264
$96$	0	0
$97$	0.200891	0.0203974	0.0101987	−	0.999948i	$-0.496754\pi$
$97$	0.200891	0.0203974	0.0101987	+	0.999948i	$0.496754\pi$
$98$	0	0
$99$	−23.4078	−2.35258
$100$	0	0
$101$	−9.80961	−0.976093	−0.488046	−	0.872818i	$-0.662290\pi$
$101$	−9.80961	−0.976093	−0.488046	+	0.872818i	$0.662290\pi$
$102$	0	0
$103$	−1.40051	−0.137996	−0.0689980	−	0.997617i	$-0.521980\pi$
$103$	−1.40051	−0.137996	−0.0689980	+	0.997617i	$0.521980\pi$
$104$	0	0
$105$	−11.0609	−1.07944
$106$	0	0
$107$	−19.0097	−1.83774	−0.918870	−	0.394559i	$-0.870897\pi$
$107$	−19.0097	−1.83774	−0.918870	+	0.394559i	$0.870897\pi$
$108$	0	0
$109$	−5.07393	−0.485995	−0.242997	−	0.970027i	$-0.578131\pi$
$109$	−5.07393	−0.485995	−0.242997	+	0.970027i	$0.578131\pi$
$110$	0	0
$111$	29.4093	2.79141
$112$	0	0
$113$	8.50650	0.800225	0.400112	−	0.916466i	$-0.368971\pi$
$113$	8.50650	0.800225	0.400112	+	0.916466i	$0.368971\pi$
$114$	0	0
$115$	9.85034	0.918549
$116$	0	0
$117$	−1.03199	−0.0954072
$118$	0	0
$119$	3.31857	0.304213
$120$	0	0
$121$	29.5232	2.68393
$122$	0	0
$123$	−20.7644	−1.87226
$124$	0	0
$125$	−10.7527	−0.961746
$126$	0	0
$127$	−7.30442	−0.648163	−0.324081	−	0.946029i	$-0.605055\pi$
$127$	−7.30442	−0.648163	−0.324081	+	0.946029i	$0.605055\pi$
$128$	0	0
$129$	10.9961	0.968154
$130$	0	0
$131$	10.2427	0.894908	0.447454	−	0.894307i	$-0.352331\pi$
$131$	10.2427	0.894908	0.447454	+	0.894307i	$0.352331\pi$
$132$	0	0
$133$	1.14415	0.0992106
$134$	0	0
$135$	−2.25691	−0.194244
$136$	0	0
$137$	12.0710	1.03130	0.515648	−	0.856801i	$-0.327551\pi$
$137$	12.0710	1.03130	0.515648	+	0.856801i	$0.327551\pi$
$138$	0	0
$139$	−8.90272	−0.755119	−0.377560	−	0.925985i	$-0.623237\pi$
$139$	−8.90272	−0.755119	−0.377560	+	0.925985i	$0.623237\pi$
$140$	0	0
$141$	17.5650	1.47924
$142$	0	0
$143$	1.78656	0.149399
$144$	0	0
$145$	−1.72959	−0.143634
$146$	0	0
$147$	−10.3694	−0.855255
$148$	0	0
$149$	14.1340	1.15790	0.578952	−	0.815362i	$-0.303462\pi$
$149$	14.1340	1.15790	0.578952	+	0.815362i	$0.303462\pi$
$150$	0	0
$151$	−7.86847	−0.640327	−0.320163	−	0.947362i	$-0.603738\pi$
$151$	−7.86847	−0.640327	−0.320163	+	0.947362i	$0.603738\pi$
$152$	0	0
$153$	3.67713	0.297279
$154$	0	0
$155$	0.0894927	0.00718823
$156$	0	0
$157$	15.3325	1.22367	0.611833	−	0.790987i	$-0.290432\pi$
$157$	15.3325	1.22367	0.611833	+	0.790987i	$0.290432\pi$
$158$	0	0
$159$	−35.4357	−2.81023
$160$	0	0
$161$	25.3429	1.99730
$162$	0	0
$163$	4.79459	0.375541	0.187771	−	0.982213i	$-0.439874\pi$
$163$	4.79459	0.375541	0.187771	+	0.982213i	$0.439874\pi$
$164$	0	0
$165$	21.2174	1.65177
$166$	0	0
$167$	−11.6187	−0.899081	−0.449540	−	0.893260i	$-0.648412\pi$
$167$	−11.6187	−0.899081	−0.449540	+	0.893260i	$0.648412\pi$
$168$	0	0
$169$	−12.9212	−0.993941
$170$	0	0
$171$	1.26777	0.0969492
$172$	0	0
$173$	1.46519	0.111396	0.0556982	−	0.998448i	$-0.482262\pi$
$173$	1.46519	0.111396	0.0556982	+	0.998448i	$0.482262\pi$
$174$	0	0
$175$	−11.0715	−0.836930
$176$	0	0
$177$	−2.58401	−0.194226
$178$	0	0
$179$	11.5968	0.866782	0.433391	−	0.901206i	$-0.357317\pi$
$179$	11.5968	0.866782	0.433391	+	0.901206i	$0.357317\pi$
$180$	0	0
$181$	6.57197	0.488491	0.244245	−	0.969713i	$-0.421460\pi$
$181$	6.57197	0.488491	0.244245	+	0.969713i	$0.421460\pi$
$182$	0	0
$183$	−28.8218	−2.13057
$184$	0	0
$185$	−14.6803	−1.07932
$186$	0	0
$187$	−6.36578	−0.465512
$188$	0	0
$189$	−5.80658	−0.422366
$190$	0	0
$191$	−4.24437	−0.307112	−0.153556	−	0.988140i	$-0.549072\pi$
$191$	−4.24437	−0.307112	−0.153556	+	0.988140i	$0.549072\pi$
$192$	0	0
$193$	−1.36039	−0.0979230	−0.0489615	−	0.998801i	$-0.515591\pi$
$193$	−1.36039	−0.0979230	−0.0489615	+	0.998801i	$0.515591\pi$
$194$	0	0
$195$	0.935417	0.0669866
$196$	0	0
$197$	−18.3727	−1.30900	−0.654500	−	0.756062i	$-0.727121\pi$
$197$	−18.3727	−1.30900	−0.654500	+	0.756062i	$0.727121\pi$
$198$	0	0
$199$	10.7685	0.763360	0.381680	−	0.924294i	$-0.375346\pi$
$199$	10.7685	0.763360	0.381680	+	0.924294i	$0.375346\pi$
$200$	0	0
$201$	11.6818	0.823973
$202$	0	0
$203$	−4.44988	−0.312320
$204$	0	0
$205$	10.3650	0.723922
$206$	0	0
$207$	28.0812	1.95178
$208$	0	0
$209$	−2.19475	−0.151814
$210$	0	0
$211$	−3.02037	−0.207931	−0.103965	−	0.994581i	$-0.533153\pi$
$211$	−3.02037	−0.207931	−0.103965	+	0.994581i	$0.533153\pi$
$212$	0	0
$213$	21.1778	1.45108
$214$	0	0
$215$	−5.48895	−0.374343
$216$	0	0
$217$	0.230247	0.0156302
$218$	0	0
$219$	−29.7243	−2.00858
$220$	0	0
$221$	−0.280650	−0.0188786
$222$	0	0
$223$	12.9413	0.866612	0.433306	−	0.901247i	$-0.357347\pi$
$223$	12.9413	0.866612	0.433306	+	0.901247i	$0.357347\pi$
$224$	0	0
$225$	−12.2678	−0.817853
$226$	0	0
$227$	1.90305	0.126310	0.0631549	−	0.998004i	$-0.479884\pi$
$227$	1.90305	0.126310	0.0631549	+	0.998004i	$0.479884\pi$
$228$	0	0
$229$	26.8200	1.77232	0.886158	−	0.463384i	$-0.153365\pi$
$229$	26.8200	1.77232	0.886158	+	0.463384i	$0.153365\pi$
$230$	0	0
$231$	54.5881	3.59163
$232$	0	0
$233$	−12.2216	−0.800665	−0.400333	−	0.916370i	$-0.631105\pi$
$233$	−12.2216	−0.800665	−0.400333	+	0.916370i	$0.631105\pi$
$234$	0	0
$235$	−8.76796	−0.571958
$236$	0	0
$237$	12.9513	0.841276
$238$	0	0
$239$	10.2811	0.665030	0.332515	−	0.943098i	$-0.392103\pi$
$239$	10.2811	0.665030	0.332515	+	0.943098i	$0.392103\pi$
$240$	0	0
$241$	−3.92390	−0.252761	−0.126380	−	0.991982i	$-0.540336\pi$
$241$	−3.92390	−0.252761	−0.126380	+	0.991982i	$0.540336\pi$
$242$	0	0
$243$	22.0713	1.41588
$244$	0	0
$245$	5.17612	0.330690
$246$	0	0
$247$	−0.0967604	−0.00615672
$248$	0	0
$249$	43.7929	2.77526
$250$	0	0
$251$	−2.71642	−0.171459	−0.0857294	−	0.996318i	$-0.527322\pi$
$251$	−2.71642	−0.171459	−0.0857294	+	0.996318i	$0.527322\pi$
$252$	0	0
$253$	−48.6136	−3.05631
$254$	0	0
$255$	−3.33304	−0.208723
$256$	0	0
$257$	0.296773	0.0185122	0.00925611	−	0.999957i	$-0.497054\pi$
$257$	0.296773	0.0185122	0.00925611	+	0.999957i	$0.497054\pi$
$258$	0	0
$259$	−37.7695	−2.34688
$260$	0	0
$261$	−4.93067	−0.305201
$262$	0	0
$263$	−17.7231	−1.09285	−0.546425	−	0.837508i	$-0.684012\pi$
$263$	−17.7231	−1.09285	−0.546425	+	0.837508i	$0.684012\pi$
$264$	0	0
$265$	17.6885	1.08660
$266$	0	0
$267$	10.9115	0.667770
$268$	0	0
$269$	−3.08054	−0.187824	−0.0939120	−	0.995581i	$-0.529937\pi$
$269$	−3.08054	−0.187824	−0.0939120	+	0.995581i	$0.529937\pi$
$270$	0	0
$271$	20.7484	1.26038	0.630189	−	0.776442i	$-0.282978\pi$
$271$	20.7484	1.26038	0.630189	+	0.776442i	$0.282978\pi$
$272$	0	0
$273$	2.40664	0.145656
$274$	0	0
$275$	21.2378	1.28069
$276$	0	0
$277$	−24.9233	−1.49750	−0.748749	−	0.662853i	$-0.769345\pi$
$277$	−24.9233	−1.49750	−0.748749	+	0.662853i	$0.769345\pi$
$278$	0	0
$279$	0.255124	0.0152739
$280$	0	0
$281$	−32.0445	−1.91161	−0.955806	−	0.293997i	$-0.905014\pi$
$281$	−32.0445	−1.91161	−0.955806	+	0.293997i	$0.905014\pi$
$282$	0	0
$283$	−20.3329	−1.20867	−0.604334	−	0.796731i	$-0.706561\pi$
$283$	−20.3329	−1.20867	−0.604334	+	0.796731i	$0.706561\pi$
$284$	0	0
$285$	−1.14914	−0.0680692
$286$	0	0
$287$	26.6670	1.57410
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	−0.519105	−0.0304305
$292$	0	0
$293$	−2.19565	−0.128271	−0.0641356	−	0.997941i	$-0.520429\pi$
$293$	−2.19565	−0.128271	−0.0641356	+	0.997941i	$0.520429\pi$
$294$	0	0
$295$	1.28987	0.0750990
$296$	0	0
$297$	11.1384	0.646313
$298$	0	0
$299$	−2.14324	−0.123947
$300$	0	0
$301$	−14.1220	−0.813977
$302$	0	0
$303$	25.3482	1.45621
$304$	0	0
$305$	14.3871	0.823801
$306$	0	0
$307$	−6.31997	−0.360700	−0.180350	−	0.983603i	$-0.557723\pi$
$307$	−6.31997	−0.360700	−0.180350	+	0.983603i	$0.557723\pi$
$308$	0	0
$309$	3.61893	0.205874
$310$	0	0
$311$	16.6502	0.944147	0.472073	−	0.881559i	$-0.343506\pi$
$311$	16.6502	0.944147	0.472073	+	0.881559i	$0.343506\pi$
$312$	0	0
$313$	21.1383	1.19481	0.597404	−	0.801940i	$-0.296199\pi$
$313$	21.1383	1.19481	0.597404	+	0.801940i	$0.296199\pi$
$314$	0	0
$315$	15.7400	0.886851
$316$	0	0
$317$	11.9251	0.669779	0.334889	−	0.942257i	$-0.391301\pi$
$317$	11.9251	0.669779	0.334889	+	0.942257i	$0.391301\pi$
$318$	0	0
$319$	8.53589	0.477918
$320$	0	0
$321$	49.1215	2.74169
$322$	0	0
$323$	0.344773	0.0191837
$324$	0	0
$325$	0.936315	0.0519374
$326$	0	0
$327$	13.1111	0.725047
$328$	0	0
$329$	−22.5582	−1.24367
$330$	0	0
$331$	−9.99513	−0.549382	−0.274691	−	0.961533i	$-0.588576\pi$
$331$	−9.99513	−0.549382	−0.274691	+	0.961533i	$0.588576\pi$
$332$	0	0
$333$	−41.8504	−2.29339
$334$	0	0
$335$	−5.83124	−0.318595
$336$	0	0
$337$	−5.86839	−0.319672	−0.159836	−	0.987144i	$-0.551097\pi$
$337$	−5.86839	−0.319672	−0.159836	+	0.987144i	$0.551097\pi$
$338$	0	0
$339$	−21.9809	−1.19384
$340$	0	0
$341$	−0.441666	−0.0239176
$342$	0	0
$343$	−9.91287	−0.535245
$344$	0	0
$345$	−25.4534	−1.37037
$346$	0	0
$347$	−21.5803	−1.15849	−0.579246	−	0.815153i	$-0.696653\pi$
$347$	−21.5803	−1.15849	−0.579246	+	0.815153i	$0.696653\pi$
$348$	0	0
$349$	−9.03894	−0.483844	−0.241922	−	0.970296i	$-0.577778\pi$
$349$	−9.03894	−0.483844	−0.241922	+	0.970296i	$0.577778\pi$
$350$	0	0
$351$	0.491059	0.0262108
$352$	0	0
$353$	−22.1530	−1.17909	−0.589543	−	0.807737i	$-0.700692\pi$
$353$	−22.1530	−1.17909	−0.589543	+	0.807737i	$0.700692\pi$
$354$	0	0
$355$	−10.5714	−0.561071
$356$	0	0
$357$	−8.57524	−0.453850
$358$	0	0
$359$	11.5102	0.607484	0.303742	−	0.952754i	$-0.401764\pi$
$359$	11.5102	0.607484	0.303742	+	0.952754i	$0.401764\pi$
$360$	0	0
$361$	−18.8811	−0.993744
$362$	0	0
$363$	−76.2884	−4.00410
$364$	0	0
$365$	14.8375	0.776632
$366$	0	0
$367$	−1.58475	−0.0827230	−0.0413615	−	0.999144i	$-0.513170\pi$
$367$	−1.58475	−0.0827230	−0.0413615	+	0.999144i	$0.513170\pi$
$368$	0	0
$369$	29.5483	1.53822
$370$	0	0
$371$	45.5090	2.36271
$372$	0	0
$373$	−12.1452	−0.628854	−0.314427	−	0.949282i	$-0.601812\pi$
$373$	−12.1452	−0.628854	−0.314427	+	0.949282i	$0.601812\pi$
$374$	0	0
$375$	27.7850	1.43481
$376$	0	0
$377$	0.376324	0.0193817
$378$	0	0
$379$	−34.0290	−1.74795	−0.873976	−	0.485969i	$-0.838467\pi$
$379$	−34.0290	−1.74795	−0.873976	+	0.485969i	$0.838467\pi$
$380$	0	0
$381$	18.8747	0.966982
$382$	0	0
$383$	12.2367	0.625267	0.312633	−	0.949874i	$-0.398789\pi$
$383$	12.2367	0.625267	0.312633	+	0.949874i	$0.398789\pi$
$384$	0	0
$385$	−27.2489	−1.38873
$386$	0	0
$387$	−15.6478	−0.795423
$388$	0	0
$389$	11.2736	0.571594	0.285797	−	0.958290i	$-0.407742\pi$
$389$	11.2736	0.571594	0.285797	+	0.958290i	$0.407742\pi$
$390$	0	0
$391$	7.63670	0.386205
$392$	0	0
$393$	−26.4673	−1.33510
$394$	0	0
$395$	−6.46492	−0.325285
$396$	0	0
$397$	−29.6136	−1.48627	−0.743133	−	0.669144i	$-0.766661\pi$
$397$	−29.6136	−1.48627	−0.743133	+	0.669144i	$0.766661\pi$
$398$	0	0
$399$	−2.95651	−0.148010
$400$	0	0
$401$	−19.9132	−0.994416	−0.497208	−	0.867631i	$-0.665641\pi$
$401$	−19.9132	−0.994416	−0.497208	+	0.867631i	$0.665641\pi$
$402$	0	0
$403$	−0.0194718	−0.000969962 0
$404$	0	0
$405$	−8.39716	−0.417258
$406$	0	0
$407$	72.4506	3.59124
$408$	0	0
$409$	−29.9779	−1.48231	−0.741155	−	0.671334i	$-0.765722\pi$
$409$	−29.9779	−1.48231	−0.741155	+	0.671334i	$0.765722\pi$
$410$	0	0
$411$	−31.1917	−1.53857
$412$	0	0
$413$	3.31857	0.163296
$414$	0	0
$415$	−21.8602	−1.07307
$416$	0	0
$417$	23.0048	1.12655
$418$	0	0
$419$	19.2044	0.938198	0.469099	−	0.883145i	$-0.344579\pi$
$419$	19.2044	0.938198	0.469099	+	0.883145i	$0.344579\pi$
$420$	0	0
$421$	9.03632	0.440403	0.220202	−	0.975454i	$-0.429328\pi$
$421$	9.03632	0.440403	0.220202	+	0.975454i	$0.429328\pi$
$422$	0	0
$423$	−24.9955	−1.21532
$424$	0	0
$425$	−3.33624	−0.161831
$426$	0	0
$427$	37.0150	1.79128
$428$	0	0
$429$	−4.61649	−0.222886
$430$	0	0
$431$	14.3681	0.692087	0.346044	−	0.938218i	$-0.387525\pi$
$431$	14.3681	0.692087	0.346044	+	0.938218i	$0.387525\pi$
$432$	0	0
$433$	−20.7794	−0.998595	−0.499298	−	0.866431i	$-0.666409\pi$
$433$	−20.7794	−0.998595	−0.499298	+	0.866431i	$0.666409\pi$
$434$	0	0
$435$	4.46928	0.214285
$436$	0	0
$437$	2.63293	0.125950
$438$	0	0
$439$	−25.1748	−1.20153	−0.600763	−	0.799427i	$-0.705137\pi$
$439$	−25.1748	−1.20153	−0.600763	+	0.799427i	$0.705137\pi$
$440$	0	0
$441$	14.7560	0.702667
$442$	0	0
$443$	18.3120	0.870028	0.435014	−	0.900424i	$-0.356743\pi$
$443$	18.3120	0.870028	0.435014	+	0.900424i	$0.356743\pi$
$444$	0	0
$445$	−5.44669	−0.258198
$446$	0	0
$447$	−36.5225	−1.72746
$448$	0	0
$449$	26.0917	1.23134	0.615672	−	0.788003i	$-0.288885\pi$
$449$	26.0917	1.23134	0.615672	+	0.788003i	$0.288885\pi$
$450$	0	0
$451$	−51.1535	−2.40872
$452$	0	0
$453$	20.3322	0.955292
$454$	0	0
$455$	−1.20133	−0.0563191
$456$	0	0
$457$	−31.9702	−1.49550	−0.747751	−	0.663979i	$-0.768866\pi$
$457$	−31.9702	−1.49550	−0.747751	+	0.663979i	$0.768866\pi$
$458$	0	0
$459$	−1.74972	−0.0816700
$460$	0	0
$461$	−12.4052	−0.577768	−0.288884	−	0.957364i	$-0.593284\pi$
$461$	−12.4052	−0.577768	−0.288884	+	0.957364i	$0.593284\pi$
$462$	0	0
$463$	−13.3369	−0.619816	−0.309908	−	0.950766i	$-0.600298\pi$
$463$	−13.3369	−0.619816	−0.309908	+	0.950766i	$0.600298\pi$
$464$	0	0
$465$	−0.231250	−0.0107240
$466$	0	0
$467$	−13.8022	−0.638691	−0.319345	−	0.947638i	$-0.603463\pi$
$467$	−13.8022	−0.638691	−0.319345	+	0.947638i	$0.603463\pi$
$468$	0	0
$469$	−15.0026	−0.692756
$470$	0	0
$471$	−39.6194	−1.82556
$472$	0	0
$473$	27.0892	1.24556
$474$	0	0
$475$	−1.15024	−0.0527768
$476$	0	0
$477$	50.4261	2.30885
$478$	0	0
$479$	−20.6717	−0.944512	−0.472256	−	0.881461i	$-0.656560\pi$
$479$	−20.6717	−0.944512	−0.472256	+	0.881461i	$0.656560\pi$
$480$	0	0
$481$	3.19415	0.145641
$482$	0	0
$483$	−65.4865	−2.97974
$484$	0	0
$485$	0.259123	0.0117662
$486$	0	0
$487$	−1.23905	−0.0561467	−0.0280734	−	0.999606i	$-0.508937\pi$
$487$	−1.23905	−0.0561467	−0.0280734	+	0.999606i	$0.508937\pi$
$488$	0	0
$489$	−12.3893	−0.560263
$490$	0	0
$491$	14.2312	0.642247	0.321123	−	0.947037i	$-0.395940\pi$
$491$	14.2312	0.642247	0.321123	+	0.947037i	$0.395940\pi$
$492$	0	0
$493$	−1.34090	−0.0603912
$494$	0	0
$495$	−30.1930	−1.35708
$496$	0	0
$497$	−27.1980	−1.22000
$498$	0	0
$499$	−21.1795	−0.948126	−0.474063	−	0.880491i	$-0.657213\pi$
$499$	−21.1795	−0.948126	−0.474063	+	0.880491i	$0.657213\pi$
$500$	0	0
$501$	30.0228	1.34132
$502$	0	0
$503$	−9.93762	−0.443097	−0.221548	−	0.975149i	$-0.571111\pi$
$503$	−9.93762	−0.443097	−0.221548	+	0.975149i	$0.571111\pi$
$504$	0	0
$505$	−12.6531	−0.563056
$506$	0	0
$507$	33.3887	1.48284
$508$	0	0
$509$	−6.29045	−0.278819	−0.139410	−	0.990235i	$-0.544520\pi$
$509$	−6.29045	−0.278819	−0.139410	+	0.990235i	$0.544520\pi$
$510$	0	0
$511$	38.1740	1.68872
$512$	0	0
$513$	−0.603256	−0.0266344
$514$	0	0
$515$	−1.80647	−0.0796026
$516$	0	0
$517$	43.2718	1.90309
$518$	0	0
$519$	−3.78608	−0.166190
$520$	0	0
$521$	−16.6965	−0.731489	−0.365744	−	0.930715i	$-0.619186\pi$
$521$	−16.6965	−0.731489	−0.365744	+	0.930715i	$0.619186\pi$
$522$	0	0
$523$	−37.2715	−1.62977	−0.814885	−	0.579622i	$-0.803200\pi$
$523$	−37.2715	−1.62977	−0.814885	+	0.579622i	$0.803200\pi$
$524$	0	0
$525$	28.6090	1.24860
$526$	0	0
$527$	0.0693813	0.00302230
$528$	0	0
$529$	35.3192	1.53562
$530$	0	0
$531$	3.67713	0.159574
$532$	0	0
$533$	−2.25522	−0.0976843
$534$	0	0
$535$	−24.5201	−1.06009
$536$	0	0
$537$	−29.9662	−1.29314
$538$	0	0
$539$	−25.5453	−1.10031
$540$	0	0
$541$	−45.4959	−1.95602	−0.978011	−	0.208553i	$-0.933125\pi$
$541$	−45.4959	−1.95602	−0.978011	+	0.208553i	$0.933125\pi$
$542$	0	0
$543$	−16.9821	−0.728770
$544$	0	0
$545$	−6.54471	−0.280344
$546$	0	0
$547$	−8.95524	−0.382899	−0.191449	−	0.981503i	$-0.561319\pi$
$547$	−8.95524	−0.382899	−0.191449	+	0.981503i	$0.561319\pi$
$548$	0	0
$549$	41.0144	1.75045
$550$	0	0
$551$	−0.462306	−0.0196949
$552$	0	0
$553$	−16.6329	−0.707304
$554$	0	0
$555$	37.9342	1.61022
$556$	0	0
$557$	40.7808	1.72794	0.863970	−	0.503543i	$-0.167971\pi$
$557$	40.7808	1.72794	0.863970	+	0.503543i	$0.167971\pi$
$558$	0	0
$559$	1.19429	0.0505130
$560$	0	0
$561$	16.4493	0.694489
$562$	0	0
$563$	30.9681	1.30515	0.652574	−	0.757725i	$-0.273689\pi$
$563$	30.9681	1.30515	0.652574	+	0.757725i	$0.273689\pi$
$564$	0	0
$565$	10.9723	0.461607
$566$	0	0
$567$	−21.6042	−0.907291
$568$	0	0
$569$	−25.1954	−1.05625	−0.528123	−	0.849168i	$-0.677104\pi$
$569$	−25.1954	−1.05625	−0.528123	+	0.849168i	$0.677104\pi$
$570$	0	0
$571$	−42.4776	−1.77763	−0.888816	−	0.458264i	$-0.848471\pi$
$571$	−42.4776	−1.77763	−0.888816	+	0.458264i	$0.848471\pi$
$572$	0	0
$573$	10.9675	0.458174
$574$	0	0
$575$	−25.4779	−1.06250
$576$	0	0
$577$	−6.57146	−0.273573	−0.136787	−	0.990601i	$-0.543677\pi$
$577$	−6.57146	−0.273573	−0.136787	+	0.990601i	$0.543677\pi$
$578$	0	0
$579$	3.51527	0.146090
$580$	0	0
$581$	−56.2418	−2.33330
$582$	0	0
$583$	−87.2967	−3.61546
$584$	0	0
$585$	−1.33113	−0.0550353
$586$	0	0
$587$	−7.00931	−0.289305	−0.144653	−	0.989483i	$-0.546206\pi$
$587$	−7.00931	−0.289305	−0.144653	+	0.989483i	$0.546206\pi$
$588$	0	0
$589$	0.0239208	0.000985638 0
$590$	0	0
$591$	47.4753	1.95287
$592$	0	0
$593$	27.5634	1.13189	0.565947	−	0.824442i	$-0.308511\pi$
$593$	27.5634	1.13189	0.565947	+	0.824442i	$0.308511\pi$
$594$	0	0
$595$	4.28052	0.175484
$596$	0	0
$597$	−27.8260	−1.13884
$598$	0	0
$599$	13.3832	0.546824	0.273412	−	0.961897i	$-0.411848\pi$
$599$	13.3832	0.546824	0.273412	+	0.961897i	$0.411848\pi$
$600$	0	0
$601$	−26.2477	−1.07067	−0.535334	−	0.844641i	$-0.679814\pi$
$601$	−26.2477	−1.07067	−0.535334	+	0.844641i	$0.679814\pi$
$602$	0	0
$603$	−16.6236	−0.676965
$604$	0	0
$605$	38.0811	1.54822
$606$	0	0
$607$	16.9355	0.687393	0.343696	−	0.939081i	$-0.388321\pi$
$607$	16.9355	0.687393	0.343696	+	0.939081i	$0.388321\pi$
$608$	0	0
$609$	11.4985	0.465945
$610$	0	0
$611$	1.90773	0.0771787
$612$	0	0
$613$	−3.68492	−0.148832	−0.0744162	−	0.997227i	$-0.523709\pi$
$613$	−3.68492	−0.148832	−0.0744162	+	0.997227i	$0.523709\pi$
$614$	0	0
$615$	−26.7833	−1.08001
$616$	0	0
$617$	−20.9823	−0.844714	−0.422357	−	0.906429i	$-0.638797\pi$
$617$	−20.9823	−0.844714	−0.422357	+	0.906429i	$0.638797\pi$
$618$	0	0
$619$	−13.0704	−0.525343	−0.262672	−	0.964885i	$-0.584604\pi$
$619$	−13.0704	−0.525343	−0.262672	+	0.964885i	$0.584604\pi$
$620$	0	0
$621$	−13.3621	−0.536203
$622$	0	0
$623$	−14.0132	−0.561429
$624$	0	0
$625$	2.81170	0.112468
$626$	0	0
$627$	5.67126	0.226488
$628$	0	0
$629$	−11.3813	−0.453800
$630$	0	0
$631$	−1.26100	−0.0501995	−0.0250998	−	0.999685i	$-0.507990\pi$
$631$	−1.26100	−0.0501995	−0.0250998	+	0.999685i	$0.507990\pi$
$632$	0	0
$633$	7.80468	0.310208
$634$	0	0
$635$	−9.42174	−0.373890
$636$	0	0
$637$	−1.12622	−0.0446226
$638$	0	0
$639$	−30.1367	−1.19219
$640$	0	0
$641$	−15.5591	−0.614547	−0.307274	−	0.951621i	$-0.599417\pi$
$641$	−15.5591	−0.614547	−0.307274	+	0.951621i	$0.599417\pi$
$642$	0	0
$643$	5.91007	0.233070	0.116535	−	0.993187i	$-0.462821\pi$
$643$	5.91007	0.233070	0.116535	+	0.993187i	$0.462821\pi$
$644$	0	0
$645$	14.1835	0.558476
$646$	0	0
$647$	1.88839	0.0742403	0.0371202	−	0.999311i	$-0.488182\pi$
$647$	1.88839	0.0742403	0.0371202	+	0.999311i	$0.488182\pi$
$648$	0	0
$649$	−6.36578	−0.249879
$650$	0	0
$651$	−0.594961	−0.0233183
$652$	0	0
$653$	9.08077	0.355358	0.177679	−	0.984088i	$-0.443141\pi$
$653$	9.08077	0.355358	0.177679	+	0.984088i	$0.443141\pi$
$654$	0	0
$655$	13.2117	0.516224
$656$	0	0
$657$	42.2986	1.65022
$658$	0	0
$659$	23.0239	0.896886	0.448443	−	0.893812i	$-0.351979\pi$
$659$	23.0239	0.896886	0.448443	+	0.893812i	$0.351979\pi$
$660$	0	0
$661$	0.619347	0.0240898	0.0120449	−	0.999927i	$-0.496166\pi$
$661$	0.619347	0.0240898	0.0120449	+	0.999927i	$0.496166\pi$
$662$	0	0
$663$	0.725204	0.0281646
$664$	0	0
$665$	1.47581	0.0572293
$666$	0	0
$667$	−10.2401	−0.396497
$668$	0	0
$669$	−33.4405	−1.29288
$670$	0	0
$671$	−71.0033	−2.74105
$672$	0	0
$673$	−41.6920	−1.60711	−0.803554	−	0.595232i	$-0.797060\pi$
$673$	−41.6920	−1.60711	−0.803554	+	0.595232i	$0.797060\pi$
$674$	0	0
$675$	5.83749	0.224685
$676$	0	0
$677$	−22.9649	−0.882611	−0.441306	−	0.897357i	$-0.645485\pi$
$677$	−22.9649	−0.882611	−0.441306	+	0.897357i	$0.645485\pi$
$678$	0	0
$679$	0.666670	0.0255845
$680$	0	0
$681$	−4.91751	−0.188439
$682$	0	0
$683$	−20.9646	−0.802187	−0.401093	−	0.916037i	$-0.631370\pi$
$683$	−20.9646	−0.802187	−0.401093	+	0.916037i	$0.631370\pi$
$684$	0	0
$685$	15.5700	0.594899
$686$	0	0
$687$	−69.3033	−2.64409
$688$	0	0
$689$	−3.84867	−0.146623
$690$	0	0
$691$	−15.0468	−0.572406	−0.286203	−	0.958169i	$-0.592393\pi$
$691$	−15.0468	−0.572406	−0.286203	+	0.958169i	$0.592393\pi$
$692$	0	0
$693$	−77.6806	−2.95084
$694$	0	0
$695$	−11.4833	−0.435588
$696$	0	0
$697$	8.03569	0.304374
$698$	0	0
$699$	31.5809	1.19450
$700$	0	0
$701$	−26.6821	−1.00777	−0.503884	−	0.863771i	$-0.668096\pi$
$701$	−26.6821	−1.00777	−0.503884	+	0.863771i	$0.668096\pi$
$702$	0	0
$703$	−3.92395	−0.147994
$704$	0	0
$705$	22.6565	0.853294
$706$	0	0
$707$	−32.5539	−1.22431
$708$	0	0
$709$	−1.89180	−0.0710479	−0.0355240	−	0.999369i	$-0.511310\pi$
$709$	−1.89180	−0.0710479	−0.0355240	+	0.999369i	$0.511310\pi$
$710$	0	0
$711$	−18.4301	−0.691182
$712$	0	0
$713$	0.529844	0.0198428
$714$	0	0
$715$	2.30442	0.0861805
$716$	0	0
$717$	−26.5666	−0.992146
$718$	0	0
$719$	18.9641	0.707241	0.353620	−	0.935389i	$-0.384950\pi$
$719$	18.9641	0.707241	0.353620	+	0.935389i	$0.384950\pi$
$720$	0	0
$721$	−4.64768	−0.173089
$722$	0	0
$723$	10.1394	0.377089
$724$	0	0
$725$	4.47357	0.166144
$726$	0	0
$727$	−1.13546	−0.0421119	−0.0210560	−	0.999778i	$-0.506703\pi$
$727$	−1.13546	−0.0421119	−0.0210560	+	0.999778i	$0.506703\pi$
$728$	0	0
$729$	−37.5024	−1.38898
$730$	0	0
$731$	−4.25544	−0.157393
$732$	0	0
$733$	6.74291	0.249055	0.124528	−	0.992216i	$-0.460258\pi$
$733$	6.74291	0.249055	0.124528	+	0.992216i	$0.460258\pi$
$734$	0	0
$735$	−13.3752	−0.493351
$736$	0	0
$737$	28.7785	1.06007
$738$	0	0
$739$	22.6316	0.832515	0.416258	−	0.909247i	$-0.363341\pi$
$739$	22.6316	0.832515	0.416258	+	0.909247i	$0.363341\pi$
$740$	0	0
$741$	0.250030	0.00918509
$742$	0	0
$743$	−29.4582	−1.08072	−0.540359	−	0.841434i	$-0.681712\pi$
$743$	−29.4582	−1.08072	−0.540359	+	0.841434i	$0.681712\pi$
$744$	0	0
$745$	18.2310	0.667933
$746$	0	0
$747$	−62.3186	−2.28012
$748$	0	0
$749$	−63.0852	−2.30508
$750$	0	0
$751$	−19.4343	−0.709167	−0.354583	−	0.935024i	$-0.615377\pi$
$751$	−19.4343	−0.709167	−0.354583	+	0.935024i	$0.615377\pi$
$752$	0	0
$753$	7.01927	0.255796
$754$	0	0
$755$	−10.1493	−0.369370
$756$	0	0
$757$	0.932315	0.0338856	0.0169428	−	0.999856i	$-0.494607\pi$
$757$	0.932315	0.0338856	0.0169428	+	0.999856i	$0.494607\pi$
$758$	0	0
$759$	125.618	4.55965
$760$	0	0
$761$	−20.9971	−0.761145	−0.380572	−	0.924751i	$-0.624273\pi$
$761$	−20.9971	−0.761145	−0.380572	+	0.924751i	$0.624273\pi$
$762$	0	0
$763$	−16.8382	−0.609584
$764$	0	0
$765$	4.74302	0.171484
$766$	0	0
$767$	−0.280650	−0.0101337
$768$	0	0
$769$	8.97271	0.323564	0.161782	−	0.986827i	$-0.448276\pi$
$769$	8.97271	0.323564	0.161782	+	0.986827i	$0.448276\pi$
$770$	0	0
$771$	−0.766867	−0.0276180
$772$	0	0
$773$	29.2026	1.05034	0.525172	−	0.850996i	$-0.324001\pi$
$773$	29.2026	1.05034	0.525172	+	0.850996i	$0.324001\pi$
$774$	0	0
$775$	−0.231473	−0.00831474
$776$	0	0
$777$	97.5969	3.50127
$778$	0	0
$779$	2.77049	0.0992630
$780$	0	0
$781$	52.1721	1.86687
$782$	0	0
$783$	2.34621	0.0838465
$784$	0	0
$785$	19.7769	0.705868
$786$	0	0
$787$	5.90308	0.210422	0.105211	−	0.994450i	$-0.466448\pi$
$787$	5.90308	0.210422	0.105211	+	0.994450i	$0.466448\pi$
$788$	0	0
$789$	45.7966	1.63040
$790$	0	0
$791$	28.2294	1.00372
$792$	0	0
$793$	−3.13034	−0.111162
$794$	0	0
$795$	−45.7074	−1.62107
$796$	0	0
$797$	42.0068	1.48796	0.743978	−	0.668204i	$-0.232937\pi$
$797$	42.0068	1.48796	0.743978	+	0.668204i	$0.232937\pi$
$798$	0	0
$799$	−6.79756	−0.240480
$800$	0	0
$801$	−15.5273	−0.548632
$802$	0	0
$803$	−73.2265	−2.58411
$804$	0	0
$805$	32.6890	1.15214
$806$	0	0
$807$	7.96017	0.280211
$808$	0	0
$809$	33.7235	1.18566	0.592828	−	0.805329i	$-0.298011\pi$
$809$	33.7235	1.18566	0.592828	+	0.805329i	$0.298011\pi$
$810$	0	0
$811$	−46.6241	−1.63720	−0.818598	−	0.574367i	$-0.805248\pi$
$811$	−46.6241	−1.63720	−0.818598	+	0.574367i	$0.805248\pi$
$812$	0	0
$813$	−53.6142	−1.88033
$814$	0	0
$815$	6.18439	0.216630
$816$	0	0
$817$	−1.46716	−0.0513294
$818$	0	0
$819$	−3.42472	−0.119669
$820$	0	0
$821$	−16.0855	−0.561389	−0.280695	−	0.959797i	$-0.590565\pi$
$821$	−16.0855	−0.561389	−0.280695	+	0.959797i	$0.590565\pi$
$822$	0	0
$823$	5.43151	0.189331	0.0946653	−	0.995509i	$-0.469822\pi$
$823$	5.43151	0.189331	0.0946653	+	0.995509i	$0.469822\pi$
$824$	0	0
$825$	−54.8788	−1.91063
$826$	0	0
$827$	27.0332	0.940035	0.470017	−	0.882657i	$-0.344248\pi$
$827$	27.0332	0.940035	0.470017	+	0.882657i	$0.344248\pi$
$828$	0	0
$829$	19.1113	0.663763	0.331881	−	0.943321i	$-0.392317\pi$
$829$	19.1113	0.663763	0.331881	+	0.943321i	$0.392317\pi$
$830$	0	0
$831$	64.4023	2.23409
$832$	0	0
$833$	4.01291	0.139039
$834$	0	0
$835$	−14.9866	−0.518632
$836$	0	0
$837$	−0.121398	−0.00419613
$838$	0	0
$839$	−7.45405	−0.257343	−0.128671	−	0.991687i	$-0.541071\pi$
$839$	−7.45405	−0.257343	−0.128671	+	0.991687i	$0.541071\pi$
$840$	0	0
$841$	−27.2020	−0.937999
$842$	0	0
$843$	82.8034	2.85190
$844$	0	0
$845$	−16.6667	−0.573352
$846$	0	0
$847$	97.9749	3.36646
$848$	0	0
$849$	52.5406	1.80319
$850$	0	0
$851$	−86.9152	−2.97942
$852$	0	0
$853$	51.3335	1.75763	0.878813	−	0.477167i	$-0.158336\pi$
$853$	51.3335	1.75763	0.878813	+	0.477167i	$0.158336\pi$
$854$	0	0
$855$	1.63526	0.0559248
$856$	0	0
$857$	−34.1770	−1.16746	−0.583732	−	0.811947i	$-0.698408\pi$
$857$	−34.1770	−1.16746	−0.583732	+	0.811947i	$0.698408\pi$
$858$	0	0
$859$	−43.3496	−1.47907	−0.739534	−	0.673119i	$-0.764954\pi$
$859$	−43.3496	−1.47907	−0.739534	+	0.673119i	$0.764954\pi$
$860$	0	0
$861$	−68.9080	−2.34838
$862$	0	0
$863$	−11.4309	−0.389113	−0.194556	−	0.980891i	$-0.562327\pi$
$863$	−11.4309	−0.389113	−0.194556	+	0.980891i	$0.562327\pi$
$864$	0	0
$865$	1.88990	0.0642586
$866$	0	0
$867$	−2.58401	−0.0877577
$868$	0	0
$869$	31.9058	1.08233
$870$	0	0
$871$	1.26876	0.0429904
$872$	0	0
$873$	0.738702	0.0250013
$874$	0	0
$875$	−35.6834	−1.20632
$876$	0	0
$877$	35.2238	1.18942	0.594712	−	0.803939i	$-0.297266\pi$
$877$	35.2238	1.18942	0.594712	+	0.803939i	$0.297266\pi$
$878$	0	0
$879$	5.67359	0.191365
$880$	0	0
$881$	8.20717	0.276507	0.138253	−	0.990397i	$-0.455851\pi$
$881$	8.20717	0.276507	0.138253	+	0.990397i	$0.455851\pi$
$882$	0	0
$883$	23.8262	0.801815	0.400908	−	0.916118i	$-0.368695\pi$
$883$	23.8262	0.801815	0.400908	+	0.916118i	$0.368695\pi$
$884$	0	0
$885$	−3.33304	−0.112039
$886$	0	0
$887$	47.4008	1.59156	0.795782	−	0.605584i	$-0.207060\pi$
$887$	47.4008	1.59156	0.795782	+	0.605584i	$0.207060\pi$
$888$	0	0
$889$	−24.2402	−0.812992
$890$	0	0
$891$	41.4418	1.38835
$892$	0	0
$893$	−2.34361	−0.0784260
$894$	0	0
$895$	14.9583	0.500000
$896$	0	0
$897$	5.53816	0.184914
$898$	0	0
$899$	−0.0930335	−0.00310284
$900$	0	0
$901$	13.7134	0.456860
$902$	0	0
$903$	36.4914	1.21436
$904$	0	0
$905$	8.47697	0.281784
$906$	0	0
$907$	20.9796	0.696618	0.348309	−	0.937380i	$-0.386756\pi$
$907$	20.9796	0.696618	0.348309	+	0.937380i	$0.386756\pi$
$908$	0	0
$909$	−36.0712	−1.19641
$910$	0	0
$911$	−49.7359	−1.64782	−0.823912	−	0.566717i	$-0.808213\pi$
$911$	−49.7359	−1.64782	−0.823912	+	0.566717i	$0.808213\pi$
$912$	0	0
$913$	107.885	3.57047
$914$	0	0
$915$	−37.1764	−1.22901
$916$	0	0
$917$	33.9911	1.12248
$918$	0	0
$919$	−34.7507	−1.14632	−0.573160	−	0.819443i	$-0.694283\pi$
$919$	−34.7507	−1.14632	−0.573160	+	0.819443i	$0.694283\pi$
$920$	0	0
$921$	16.3309	0.538121
$922$	0	0
$923$	2.30012	0.0757095
$924$	0	0
$925$	37.9706	1.24847
$926$	0	0
$927$	−5.14985	−0.169143
$928$	0	0
$929$	4.23419	0.138919	0.0694596	−	0.997585i	$-0.477873\pi$
$929$	4.23419	0.138919	0.0694596	+	0.997585i	$0.477873\pi$
$930$	0	0
$931$	1.38354	0.0453437
$932$	0	0
$933$	−43.0244	−1.40856
$934$	0	0
$935$	−8.21102	−0.268529
$936$	0	0
$937$	−15.2515	−0.498246	−0.249123	−	0.968472i	$-0.580142\pi$
$937$	−15.2515	−0.498246	−0.249123	+	0.968472i	$0.580142\pi$
$938$	0	0
$939$	−54.6217	−1.78251
$940$	0	0
$941$	4.28129	0.139566	0.0697830	−	0.997562i	$-0.477769\pi$
$941$	4.28129	0.139566	0.0697830	+	0.997562i	$0.477769\pi$
$942$	0	0
$943$	61.3662	1.99836
$944$	0	0
$945$	−7.48972	−0.243641
$946$	0	0
$947$	43.4918	1.41329	0.706646	−	0.707567i	$-0.250207\pi$
$947$	43.4918	1.41329	0.706646	+	0.707567i	$0.250207\pi$
$948$	0	0
$949$	−3.22835	−0.104797
$950$	0	0
$951$	−30.8146	−0.999231
$952$	0	0
$953$	51.0671	1.65423	0.827113	−	0.562035i	$-0.189981\pi$
$953$	51.0671	1.65423	0.827113	+	0.562035i	$0.189981\pi$
$954$	0	0
$955$	−5.47467	−0.177156
$956$	0	0
$957$	−22.0569	−0.712997
$958$	0	0
$959$	40.0585	1.29356
$960$	0	0
$961$	−30.9952	−0.999845
$962$	0	0
$963$	−69.9013	−2.25254
$964$	0	0
$965$	−1.75472	−0.0564865
$966$	0	0
$967$	40.5544	1.30414	0.652071	−	0.758158i	$-0.273900\pi$
$967$	40.5544	1.30414	0.652071	+	0.758158i	$0.273900\pi$
$968$	0	0
$969$	−0.890898	−0.0286198
$970$	0	0
$971$	−12.5197	−0.401775	−0.200887	−	0.979614i	$-0.564383\pi$
$971$	−12.5197	−0.401775	−0.200887	+	0.979614i	$0.564383\pi$
$972$	0	0
$973$	−29.5443	−0.947147
$974$	0	0
$975$	−2.41945	−0.0774845
$976$	0	0
$977$	−19.0153	−0.608352	−0.304176	−	0.952616i	$-0.598381\pi$
$977$	−19.0153	−0.608352	−0.304176	+	0.952616i	$0.598381\pi$
$978$	0	0
$979$	26.8806	0.859109
$980$	0	0
$981$	−18.6575	−0.595689
$982$	0	0
$983$	20.9625	0.668600	0.334300	−	0.942467i	$-0.391500\pi$
$983$	20.9625	0.668600	0.334300	+	0.942467i	$0.391500\pi$
$984$	0	0
$985$	−23.6983	−0.755092
$986$	0	0
$987$	58.2907	1.85541
$988$	0	0
$989$	−32.4975	−1.03336
$990$	0	0
$991$	−52.9748	−1.68280	−0.841400	−	0.540413i	$-0.818268\pi$
$991$	−52.9748	−1.68280	−0.841400	+	0.540413i	$0.818268\pi$
$992$	0	0
$993$	25.8276	0.819613
$994$	0	0
$995$	13.8900	0.440342
$996$	0	0
$997$	−10.7404	−0.340151	−0.170076	−	0.985431i	$-0.554401\pi$
$997$	−10.7404	−0.340151	−0.170076	+	0.985431i	$0.554401\pi$
$998$	0	0
$999$	19.9140	0.630052

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8024.2.a.z.1.4	✓	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8024.2.a.z.1.4	✓	24	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:	\( N \)	\(=\)	\( 8024 = 2^{3} \cdot 17 \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8024.a (trivial)

\(f(q)\)	\(=\)	\(q-2.58401 q^{3} +1.28987 q^{5} +3.31857 q^{7} +3.67713 q^{9} +O(q^{10})\)	\(q-2.58401 q^{3} +1.28987 q^{5} +3.31857 q^{7} +3.67713 q^{9} -6.36578 q^{11} -0.280650 q^{13} -3.33304 q^{15} +1.00000 q^{17} +0.344773 q^{19} -8.57524 q^{21} +7.63670 q^{23} -3.33624 q^{25} -1.74972 q^{27} -1.34090 q^{29} +0.0693813 q^{31} +16.4493 q^{33} +4.28052 q^{35} -11.3813 q^{37} +0.725204 q^{39} +8.03569 q^{41} -4.25544 q^{43} +4.74302 q^{45} -6.79756 q^{47} +4.01291 q^{49} -2.58401 q^{51} +13.7134 q^{53} -8.21102 q^{55} -0.890898 q^{57} +1.00000 q^{59} +11.1539 q^{61} +12.2028 q^{63} -0.362001 q^{65} -4.52081 q^{67} -19.7333 q^{69} -8.19571 q^{71} +11.5031 q^{73} +8.62089 q^{75} -21.1253 q^{77} -5.01208 q^{79} -6.51009 q^{81} -16.9476 q^{83} +1.28987 q^{85} +3.46491 q^{87} -4.22267 q^{89} -0.931356 q^{91} -0.179282 q^{93} +0.444711 q^{95} +0.200891 q^{97} -23.4078 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 5 q^{3} - 7 q^{5} - 12 q^{7} + 33 q^{9}+O(q^{10}) \) 24 * q - 5 * q^3 - 7 * q^5 - 12 * q^7 + 33 * q^9	\( 24 q - 5 q^{3} - 7 q^{5} - 12 q^{7} + 33 q^{9} - 15 q^{11} - 17 q^{13} - 7 q^{15} + 24 q^{17} - q^{19} + 10 q^{21} - 21 q^{23} + 19 q^{25} - 26 q^{27} - 18 q^{29} - 23 q^{31} + 10 q^{33} - 5 q^{35} - 29 q^{37} - 44 q^{39} + 14 q^{41} - 23 q^{43} - 24 q^{45} - 17 q^{47} + 46 q^{49} - 5 q^{51} + 7 q^{53} - 33 q^{55} - 2 q^{57} + 24 q^{59} - 9 q^{61} - 52 q^{63} - 15 q^{65} + 2 q^{67} + 2 q^{69} - 51 q^{71} - 18 q^{73} + 7 q^{75} + 14 q^{77} - 36 q^{79} + 12 q^{81} - 39 q^{83} - 7 q^{85} - 5 q^{87} + 46 q^{89} - 52 q^{91} - 20 q^{93} - 52 q^{95} + 50 q^{97} - 53 q^{99}+O(q^{100}) \) 24 * q - 5 * q^3 - 7 * q^5 - 12 * q^7 + 33 * q^9 - 15 * q^11 - 17 * q^13 - 7 * q^15 + 24 * q^17 - q^19 + 10 * q^21 - 21 * q^23 + 19 * q^25 - 26 * q^27 - 18 * q^29 - 23 * q^31 + 10 * q^33 - 5 * q^35 - 29 * q^37 - 44 * q^39 + 14 * q^41 - 23 * q^43 - 24 * q^45 - 17 * q^47 + 46 * q^49 - 5 * q^51 + 7 * q^53 - 33 * q^55 - 2 * q^57 + 24 * q^59 - 9 * q^61 - 52 * q^63 - 15 * q^65 + 2 * q^67 + 2 * q^69 - 51 * q^71 - 18 * q^73 + 7 * q^75 + 14 * q^77 - 36 * q^79 + 12 * q^81 - 39 * q^83 - 7 * q^85 - 5 * q^87 + 46 * q^89 - 52 * q^91 - 20 * q^93 - 52 * q^95 + 50 * q^97 - 53 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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