LMFDB - Embedded newform 6024.2.a.k.1.7

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:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{7} - 10x^{6} + 25x^{5} + 5x^{4} - 36x^{3} + 11x^{2} + 3x - 1 $ x^8 - 2x^7 - 10x^6 + 25x^5 + 5x^4 - 36x^3 + 11x^2 + 3*x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$1.53651$ 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)

$f(q)$	$=$	$q+1.00000 q^{3} +1.27524 q^{5} +0.298981 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +1.27524 q^{5} +0.298981 q^{7} +1.00000 q^{9} -2.92606 q^{11} +1.03806 q^{13} +1.27524 q^{15} -3.09343 q^{17} +3.33558 q^{19} +0.298981 q^{21} -5.64365 q^{23} -3.37377 q^{25} +1.00000 q^{27} +0.183331 q^{29} -10.8391 q^{31} -2.92606 q^{33} +0.381271 q^{35} -6.37909 q^{37} +1.03806 q^{39} -2.88666 q^{41} +2.09280 q^{43} +1.27524 q^{45} +6.91639 q^{47} -6.91061 q^{49} -3.09343 q^{51} -9.47327 q^{53} -3.73142 q^{55} +3.33558 q^{57} +13.0446 q^{59} -11.2767 q^{61} +0.298981 q^{63} +1.32377 q^{65} -13.7265 q^{67} -5.64365 q^{69} +5.81983 q^{71} +4.30272 q^{73} -3.37377 q^{75} -0.874836 q^{77} +2.95063 q^{79} +1.00000 q^{81} -15.5508 q^{83} -3.94486 q^{85} +0.183331 q^{87} +4.27736 q^{89} +0.310360 q^{91} -10.8391 q^{93} +4.25365 q^{95} +2.81946 q^{97} -2.92606 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 8 q^{3} - 5 q^{5} + q^{7} + 8 q^{9}+O(q^{10}) $ 8 * q + 8 * q^3 - 5 * q^5 + q^7 + 8 * q^9	$ 8 q + 8 q^{3} - 5 q^{5} + q^{7} + 8 q^{9} - 6 q^{11} - 4 q^{13} - 5 q^{15} - 12 q^{17} - 10 q^{19} + q^{21} - 6 q^{23} - 7 q^{25} + 8 q^{27} - 2 q^{29} - 3 q^{31} - 6 q^{33} + 4 q^{35} - 16 q^{37} - 4 q^{39} + 2 q^{41} - 2 q^{43} - 5 q^{45} - 15 q^{47} - 21 q^{49} - 12 q^{51} + 13 q^{53} - 33 q^{55} - 10 q^{57} + 8 q^{59} - 38 q^{61} + q^{63} + 16 q^{65} - 31 q^{67} - 6 q^{69} + 5 q^{71} - 26 q^{73} - 7 q^{75} - 24 q^{77} - 25 q^{79} + 8 q^{81} - 7 q^{83} - 18 q^{85} - 2 q^{87} - 14 q^{89} - 6 q^{91} - 3 q^{93} - q^{95} - 51 q^{97} - 6 q^{99}+O(q^{100}) $ 8 * q + 8 * q^3 - 5 * q^5 + q^7 + 8 * q^9 - 6 * q^11 - 4 * q^13 - 5 * q^15 - 12 * q^17 - 10 * q^19 + q^21 - 6 * q^23 - 7 * q^25 + 8 * q^27 - 2 * q^29 - 3 * q^31 - 6 * q^33 + 4 * q^35 - 16 * q^37 - 4 * q^39 + 2 * q^41 - 2 * q^43 - 5 * q^45 - 15 * q^47 - 21 * q^49 - 12 * q^51 + 13 * q^53 - 33 * q^55 - 10 * q^57 + 8 * q^59 - 38 * q^61 + q^63 + 16 * q^65 - 31 * q^67 - 6 * q^69 + 5 * q^71 - 26 * q^73 - 7 * q^75 - 24 * q^77 - 25 * q^79 + 8 * q^81 - 7 * q^83 - 18 * q^85 - 2 * q^87 - 14 * q^89 - 6 * q^91 - 3 * q^93 - q^95 - 51 * q^97 - 6 * 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
$5$	1.27524	0.570303	0.285152	−	0.958482i	$-0.407956\pi$
$5$	1.27524	0.570303	0.285152	+	0.958482i	$0.407956\pi$
$6$	0	0
$7$	0.298981	0.113004	0.0565020	−	0.998402i	$-0.482005\pi$
$7$	0.298981	0.113004	0.0565020	+	0.998402i	$0.482005\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−2.92606	−0.882241	−0.441121	−	0.897448i	$-0.645419\pi$
$11$	−2.92606	−0.882241	−0.441121	+	0.897448i	$0.645419\pi$
$12$	0	0
$13$	1.03806	0.287906	0.143953	−	0.989585i	$-0.454019\pi$
$13$	1.03806	0.287906	0.143953	+	0.989585i	$0.454019\pi$
$14$	0	0
$15$	1.27524	0.329265
$16$	0	0
$17$	−3.09343	−0.750267	−0.375134	−	0.926971i	$-0.622403\pi$
$17$	−3.09343	−0.750267	−0.375134	+	0.926971i	$0.622403\pi$
$18$	0	0
$19$	3.33558	0.765233	0.382617	−	0.923907i	$-0.375023\pi$
$19$	3.33558	0.765233	0.382617	+	0.923907i	$0.375023\pi$
$20$	0	0
$21$	0.298981	0.0652429
$22$	0	0
$23$	−5.64365	−1.17678	−0.588391	−	0.808577i	$-0.700238\pi$
$23$	−5.64365	−1.17678	−0.588391	+	0.808577i	$0.700238\pi$
$24$	0	0
$25$	−3.37377	−0.674754
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	0.183331	0.0340436	0.0170218	−	0.999855i	$-0.494582\pi$
$29$	0.183331	0.0340436	0.0170218	+	0.999855i	$0.494582\pi$
$30$	0	0
$31$	−10.8391	−1.94675	−0.973376	−	0.229213i	$-0.926385\pi$
$31$	−10.8391	−1.94675	−0.973376	+	0.229213i	$0.926385\pi$
$32$	0	0
$33$	−2.92606	−0.509362
$34$	0	0
$35$	0.381271	0.0644466
$36$	0	0
$37$	−6.37909	−1.04872	−0.524358	−	0.851498i	$-0.675695\pi$
$37$	−6.37909	−1.04872	−0.524358	+	0.851498i	$0.675695\pi$
$38$	0	0
$39$	1.03806	0.166223
$40$	0	0
$41$	−2.88666	−0.450820	−0.225410	−	0.974264i	$-0.572372\pi$
$41$	−2.88666	−0.450820	−0.225410	+	0.974264i	$0.572372\pi$
$42$	0	0
$43$	2.09280	0.319149	0.159574	−	0.987186i	$-0.448988\pi$
$43$	2.09280	0.319149	0.159574	+	0.987186i	$0.448988\pi$
$44$	0	0
$45$	1.27524	0.190101
$46$	0	0
$47$	6.91639	1.00886	0.504429	−	0.863453i	$-0.331703\pi$
$47$	6.91639	1.00886	0.504429	+	0.863453i	$0.331703\pi$
$48$	0	0
$49$	−6.91061	−0.987230
$50$	0	0
$51$	−3.09343	−0.433167
$52$	0	0
$53$	−9.47327	−1.30125	−0.650627	−	0.759398i	$-0.725494\pi$
$53$	−9.47327	−1.30125	−0.650627	+	0.759398i	$0.725494\pi$
$54$	0	0
$55$	−3.73142	−0.503145
$56$	0	0
$57$	3.33558	0.441808
$58$	0	0
$59$	13.0446	1.69826	0.849128	−	0.528187i	$-0.177128\pi$
$59$	13.0446	1.69826	0.849128	+	0.528187i	$0.177128\pi$
$60$	0	0
$61$	−11.2767	−1.44384	−0.721919	−	0.691978i	$-0.756739\pi$
$61$	−11.2767	−1.44384	−0.721919	+	0.691978i	$0.756739\pi$
$62$	0	0
$63$	0.298981	0.0376680
$64$	0	0
$65$	1.32377	0.164194
$66$	0	0
$67$	−13.7265	−1.67695	−0.838477	−	0.544937i	$-0.816553\pi$
$67$	−13.7265	−1.67695	−0.838477	+	0.544937i	$0.816553\pi$
$68$	0	0
$69$	−5.64365	−0.679415
$70$	0	0
$71$	5.81983	0.690687	0.345344	−	0.938476i	$-0.387762\pi$
$71$	5.81983	0.690687	0.345344	+	0.938476i	$0.387762\pi$
$72$	0	0
$73$	4.30272	0.503596	0.251798	−	0.967780i	$-0.418978\pi$
$73$	4.30272	0.503596	0.251798	+	0.967780i	$0.418978\pi$
$74$	0	0
$75$	−3.37377	−0.389570
$76$	0	0
$77$	−0.874836	−0.0996969
$78$	0	0
$79$	2.95063	0.331971	0.165986	−	0.986128i	$-0.446919\pi$
$79$	2.95063	0.331971	0.165986	+	0.986128i	$0.446919\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−15.5508	−1.70692	−0.853458	−	0.521161i	$-0.825499\pi$
$83$	−15.5508	−1.70692	−0.853458	+	0.521161i	$0.825499\pi$
$84$	0	0
$85$	−3.94486	−0.427880
$86$	0	0
$87$	0.183331	0.0196551
$88$	0	0
$89$	4.27736	0.453399	0.226700	−	0.973965i	$-0.427206\pi$
$89$	4.27736	0.453399	0.226700	+	0.973965i	$0.427206\pi$
$90$	0	0
$91$	0.310360	0.0325345
$92$	0	0
$93$	−10.8391	−1.12396
$94$	0	0
$95$	4.25365	0.436415
$96$	0	0
$97$	2.81946	0.286273	0.143137	−	0.989703i	$-0.454281\pi$
$97$	2.81946	0.286273	0.143137	+	0.989703i	$0.454281\pi$
$98$	0	0
$99$	−2.92606	−0.294080
$100$	0	0
$101$	9.87311	0.982412	0.491206	−	0.871044i	$-0.336556\pi$
$101$	9.87311	0.982412	0.491206	+	0.871044i	$0.336556\pi$
$102$	0	0
$103$	4.94339	0.487086	0.243543	−	0.969890i	$-0.421690\pi$
$103$	4.94339	0.487086	0.243543	+	0.969890i	$0.421690\pi$
$104$	0	0
$105$	0.381271	0.0372082
$106$	0	0
$107$	4.88726	0.472470	0.236235	−	0.971696i	$-0.424087\pi$
$107$	4.88726	0.472470	0.236235	+	0.971696i	$0.424087\pi$
$108$	0	0
$109$	−7.58959	−0.726951	−0.363476	−	0.931604i	$-0.618410\pi$
$109$	−7.58959	−0.726951	−0.363476	+	0.931604i	$0.618410\pi$
$110$	0	0
$111$	−6.37909	−0.605476
$112$	0	0
$113$	5.07412	0.477333	0.238666	−	0.971102i	$-0.423290\pi$
$113$	5.07412	0.477333	0.238666	+	0.971102i	$0.423290\pi$
$114$	0	0
$115$	−7.19698	−0.671122
$116$	0	0
$117$	1.03806	0.0959687
$118$	0	0
$119$	−0.924876	−0.0847832
$120$	0	0
$121$	−2.43815	−0.221650
$122$	0	0
$123$	−2.88666	−0.260281
$124$	0	0
$125$	−10.6785	−0.955118
$126$	0	0
$127$	19.6435	1.74308	0.871538	−	0.490328i	$-0.163123\pi$
$127$	19.6435	1.74308	0.871538	+	0.490328i	$0.163123\pi$
$128$	0	0
$129$	2.09280	0.184261
$130$	0	0
$131$	11.3222	0.989222	0.494611	−	0.869114i	$-0.335311\pi$
$131$	11.3222	0.989222	0.494611	+	0.869114i	$0.335311\pi$
$132$	0	0
$133$	0.997272	0.0864745
$134$	0	0
$135$	1.27524	0.109755
$136$	0	0
$137$	2.90570	0.248251	0.124125	−	0.992267i	$-0.460387\pi$
$137$	2.90570	0.248251	0.124125	+	0.992267i	$0.460387\pi$
$138$	0	0
$139$	−20.2270	−1.71564	−0.857818	−	0.513954i	$-0.828180\pi$
$139$	−20.2270	−1.71564	−0.857818	+	0.513954i	$0.828180\pi$
$140$	0	0
$141$	6.91639	0.582465
$142$	0	0
$143$	−3.03743	−0.254003
$144$	0	0
$145$	0.233790	0.0194152
$146$	0	0
$147$	−6.91061	−0.569978
$148$	0	0
$149$	−22.9048	−1.87643	−0.938216	−	0.346049i	$-0.887523\pi$
$149$	−22.9048	−1.87643	−0.938216	+	0.346049i	$0.887523\pi$
$150$	0	0
$151$	−0.836346	−0.0680609	−0.0340304	−	0.999421i	$-0.510834\pi$
$151$	−0.836346	−0.0680609	−0.0340304	+	0.999421i	$0.510834\pi$
$152$	0	0
$153$	−3.09343	−0.250089
$154$	0	0
$155$	−13.8224	−1.11024
$156$	0	0
$157$	19.3353	1.54313	0.771564	−	0.636152i	$-0.219475\pi$
$157$	19.3353	1.54313	0.771564	+	0.636152i	$0.219475\pi$
$158$	0	0
$159$	−9.47327	−0.751279
$160$	0	0
$161$	−1.68734	−0.132981
$162$	0	0
$163$	−7.91910	−0.620272	−0.310136	−	0.950692i	$-0.600375\pi$
$163$	−7.91910	−0.620272	−0.310136	+	0.950692i	$0.600375\pi$
$164$	0	0
$165$	−3.73142	−0.290491
$166$	0	0
$167$	0.264256	0.0204488	0.0102244	−	0.999948i	$-0.496745\pi$
$167$	0.264256	0.0204488	0.0102244	+	0.999948i	$0.496745\pi$
$168$	0	0
$169$	−11.9224	−0.917110
$170$	0	0
$171$	3.33558	0.255078
$172$	0	0
$173$	−8.57352	−0.651833	−0.325916	−	0.945399i	$-0.605673\pi$
$173$	−8.57352	−0.651833	−0.325916	+	0.945399i	$0.605673\pi$
$174$	0	0
$175$	−1.00869	−0.0762500
$176$	0	0
$177$	13.0446	0.980489
$178$	0	0
$179$	11.4646	0.856906	0.428453	−	0.903564i	$-0.359059\pi$
$179$	11.4646	0.856906	0.428453	+	0.903564i	$0.359059\pi$
$180$	0	0
$181$	4.52183	0.336105	0.168053	−	0.985778i	$-0.446252\pi$
$181$	4.52183	0.336105	0.168053	+	0.985778i	$0.446252\pi$
$182$	0	0
$183$	−11.2767	−0.833600
$184$	0	0
$185$	−8.13484	−0.598086
$186$	0	0
$187$	9.05158	0.661917
$188$	0	0
$189$	0.298981	0.0217476
$190$	0	0
$191$	9.41650	0.681354	0.340677	−	0.940180i	$-0.389344\pi$
$191$	9.41650	0.681354	0.340677	+	0.940180i	$0.389344\pi$
$192$	0	0
$193$	−5.47258	−0.393925	−0.196962	−	0.980411i	$-0.563108\pi$
$193$	−5.47258	−0.393925	−0.196962	+	0.980411i	$0.563108\pi$
$194$	0	0
$195$	1.32377	0.0947973
$196$	0	0
$197$	−13.9532	−0.994125	−0.497062	−	0.867715i	$-0.665588\pi$
$197$	−13.9532	−0.994125	−0.497062	+	0.867715i	$0.665588\pi$
$198$	0	0
$199$	−5.46291	−0.387256	−0.193628	−	0.981075i	$-0.562025\pi$
$199$	−5.46291	−0.387256	−0.193628	+	0.981075i	$0.562025\pi$
$200$	0	0
$201$	−13.7265	−0.968190
$202$	0	0
$203$	0.0548123	0.00384707
$204$	0	0
$205$	−3.68117	−0.257104
$206$	0	0
$207$	−5.64365	−0.392261
$208$	0	0
$209$	−9.76011	−0.675121
$210$	0	0
$211$	−5.35714	−0.368801	−0.184400	−	0.982851i	$-0.559034\pi$
$211$	−5.35714	−0.368801	−0.184400	+	0.982851i	$0.559034\pi$
$212$	0	0
$213$	5.81983	0.398769
$214$	0	0
$215$	2.66881	0.182012
$216$	0	0
$217$	−3.24067	−0.219991
$218$	0	0
$219$	4.30272	0.290751
$220$	0	0
$221$	−3.21117	−0.216006
$222$	0	0
$223$	25.8029	1.72789	0.863944	−	0.503588i	$-0.167987\pi$
$223$	25.8029	1.72789	0.863944	+	0.503588i	$0.167987\pi$
$224$	0	0
$225$	−3.37377	−0.224918
$226$	0	0
$227$	4.81244	0.319413	0.159707	−	0.987165i	$-0.448945\pi$
$227$	4.81244	0.319413	0.159707	+	0.987165i	$0.448945\pi$
$228$	0	0
$229$	2.22169	0.146813	0.0734066	−	0.997302i	$-0.476613\pi$
$229$	2.22169	0.146813	0.0734066	+	0.997302i	$0.476613\pi$
$230$	0	0
$231$	−0.874836	−0.0575600
$232$	0	0
$233$	−17.1040	−1.12052	−0.560260	−	0.828317i	$-0.689299\pi$
$233$	−17.1040	−1.12052	−0.560260	+	0.828317i	$0.689299\pi$
$234$	0	0
$235$	8.82003	0.575355
$236$	0	0
$237$	2.95063	0.191664
$238$	0	0
$239$	−2.26452	−0.146480	−0.0732400	−	0.997314i	$-0.523334\pi$
$239$	−2.26452	−0.146480	−0.0732400	+	0.997314i	$0.523334\pi$
$240$	0	0
$241$	−4.75567	−0.306340	−0.153170	−	0.988200i	$-0.548948\pi$
$241$	−4.75567	−0.306340	−0.153170	+	0.988200i	$0.548948\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−8.81266	−0.563020
$246$	0	0
$247$	3.46253	0.220315
$248$	0	0
$249$	−15.5508	−0.985489
$250$	0	0
$251$	1.00000	0.0631194
$251$	1.00000	0.0631194
$252$	0	0
$253$	16.5137	1.03821
$254$	0	0
$255$	−3.94486	−0.247036
$256$	0	0
$257$	11.0711	0.690599	0.345299	−	0.938493i	$-0.387777\pi$
$257$	11.0711	0.690599	0.345299	+	0.938493i	$0.387777\pi$
$258$	0	0
$259$	−1.90722	−0.118509
$260$	0	0
$261$	0.183331	0.0113479
$262$	0	0
$263$	−21.8548	−1.34763	−0.673814	−	0.738901i	$-0.735345\pi$
$263$	−21.8548	−1.34763	−0.673814	+	0.738901i	$0.735345\pi$
$264$	0	0
$265$	−12.0807	−0.742109
$266$	0	0
$267$	4.27736	0.261770
$268$	0	0
$269$	19.2686	1.17483	0.587414	−	0.809286i	$-0.300146\pi$
$269$	19.2686	1.17483	0.587414	+	0.809286i	$0.300146\pi$
$270$	0	0
$271$	−2.71190	−0.164736	−0.0823682	−	0.996602i	$-0.526248\pi$
$271$	−2.71190	−0.164736	−0.0823682	+	0.996602i	$0.526248\pi$
$272$	0	0
$273$	0.310360	0.0187838
$274$	0	0
$275$	9.87187	0.595296
$276$	0	0
$277$	−3.34489	−0.200975	−0.100487	−	0.994938i	$-0.532040\pi$
$277$	−3.34489	−0.200975	−0.100487	+	0.994938i	$0.532040\pi$
$278$	0	0
$279$	−10.8391	−0.648917
$280$	0	0
$281$	31.2770	1.86583	0.932916	−	0.360095i	$-0.117256\pi$
$281$	31.2770	1.86583	0.932916	+	0.360095i	$0.117256\pi$
$282$	0	0
$283$	−9.97848	−0.593159	−0.296580	−	0.955008i	$-0.595846\pi$
$283$	−9.97848	−0.593159	−0.296580	+	0.955008i	$0.595846\pi$
$284$	0	0
$285$	4.25365	0.251964
$286$	0	0
$287$	−0.863054	−0.0509445
$288$	0	0
$289$	−7.43068	−0.437099
$290$	0	0
$291$	2.81946	0.165280
$292$	0	0
$293$	1.90441	0.111257	0.0556284	−	0.998452i	$-0.482284\pi$
$293$	1.90441	0.111257	0.0556284	+	0.998452i	$0.482284\pi$
$294$	0	0
$295$	16.6349	0.968521
$296$	0	0
$297$	−2.92606	−0.169787
$298$	0	0
$299$	−5.85844	−0.338802
$300$	0	0
$301$	0.625706	0.0360651
$302$	0	0
$303$	9.87311	0.567196
$304$	0	0
$305$	−14.3805	−0.823425
$306$	0	0
$307$	28.8790	1.64821	0.824106	−	0.566436i	$-0.191678\pi$
$307$	28.8790	1.64821	0.824106	+	0.566436i	$0.191678\pi$
$308$	0	0
$309$	4.94339	0.281219
$310$	0	0
$311$	−11.1302	−0.631137	−0.315569	−	0.948903i	$-0.602195\pi$
$311$	−11.1302	−0.631137	−0.315569	+	0.948903i	$0.602195\pi$
$312$	0	0
$313$	13.3513	0.754658	0.377329	−	0.926079i	$-0.376843\pi$
$313$	13.3513	0.754658	0.377329	+	0.926079i	$0.376843\pi$
$314$	0	0
$315$	0.381271	0.0214822
$316$	0	0
$317$	21.6239	1.21452	0.607260	−	0.794503i	$-0.292269\pi$
$317$	21.6239	1.21452	0.607260	+	0.794503i	$0.292269\pi$
$318$	0	0
$319$	−0.536437	−0.0300347
$320$	0	0
$321$	4.88726	0.272781
$322$	0	0
$323$	−10.3184	−0.574130
$324$	0	0
$325$	−3.50218	−0.194266
$326$	0	0
$327$	−7.58959	−0.419705
$328$	0	0
$329$	2.06787	0.114005
$330$	0	0
$331$	−27.4060	−1.50637	−0.753184	−	0.657810i	$-0.771483\pi$
$331$	−27.4060	−1.50637	−0.753184	+	0.657810i	$0.771483\pi$
$332$	0	0
$333$	−6.37909	−0.349572
$334$	0	0
$335$	−17.5045	−0.956372
$336$	0	0
$337$	2.28065	0.124235	0.0621174	−	0.998069i	$-0.480215\pi$
$337$	2.28065	0.124235	0.0621174	+	0.998069i	$0.480215\pi$
$338$	0	0
$339$	5.07412	0.275588
$340$	0	0
$341$	31.7158	1.71751
$342$	0	0
$343$	−4.15900	−0.224565
$344$	0	0
$345$	−7.19698	−0.387473
$346$	0	0
$347$	12.4230	0.666904	0.333452	−	0.942767i	$-0.391786\pi$
$347$	12.4230	0.666904	0.333452	+	0.942767i	$0.391786\pi$
$348$	0	0
$349$	−6.28600	−0.336482	−0.168241	−	0.985746i	$-0.553809\pi$
$349$	−6.28600	−0.336482	−0.168241	+	0.985746i	$0.553809\pi$
$350$	0	0
$351$	1.03806	0.0554075
$352$	0	0
$353$	−22.8962	−1.21864	−0.609321	−	0.792924i	$-0.708558\pi$
$353$	−22.8962	−1.21864	−0.609321	+	0.792924i	$0.708558\pi$
$354$	0	0
$355$	7.42167	0.393901
$356$	0	0
$357$	−0.924876	−0.0489496
$358$	0	0
$359$	17.0884	0.901890	0.450945	−	0.892552i	$-0.351087\pi$
$359$	17.0884	0.901890	0.450945	+	0.892552i	$0.351087\pi$
$360$	0	0
$361$	−7.87394	−0.414418
$362$	0	0
$363$	−2.43815	−0.127970
$364$	0	0
$365$	5.48699	0.287202
$366$	0	0
$367$	−19.5874	−1.02245	−0.511227	−	0.859446i	$-0.670809\pi$
$367$	−19.5874	−1.02245	−0.511227	+	0.859446i	$0.670809\pi$
$368$	0	0
$369$	−2.88666	−0.150273
$370$	0	0
$371$	−2.83232	−0.147047
$372$	0	0
$373$	−23.7508	−1.22977	−0.614884	−	0.788618i	$-0.710797\pi$
$373$	−23.7508	−1.22977	−0.614884	+	0.788618i	$0.710797\pi$
$374$	0	0
$375$	−10.6785	−0.551437
$376$	0	0
$377$	0.190308	0.00980137
$378$	0	0
$379$	19.0668	0.979397	0.489699	−	0.871892i	$-0.337107\pi$
$379$	19.0668	0.979397	0.489699	+	0.871892i	$0.337107\pi$
$380$	0	0
$381$	19.6435	1.00637
$382$	0	0
$383$	−20.2231	−1.03335	−0.516677	−	0.856181i	$-0.672831\pi$
$383$	−20.2231	−1.03335	−0.516677	+	0.856181i	$0.672831\pi$
$384$	0	0
$385$	−1.11562	−0.0568574
$386$	0	0
$387$	2.09280	0.106383
$388$	0	0
$389$	−10.3333	−0.523921	−0.261960	−	0.965079i	$-0.584369\pi$
$389$	−10.3333	−0.523921	−0.261960	+	0.965079i	$0.584369\pi$
$390$	0	0
$391$	17.4582	0.882901
$392$	0	0
$393$	11.3222	0.571128
$394$	0	0
$395$	3.76275	0.189324
$396$	0	0
$397$	−28.7931	−1.44508	−0.722541	−	0.691328i	$-0.757026\pi$
$397$	−28.7931	−1.44508	−0.722541	+	0.691328i	$0.757026\pi$
$398$	0	0
$399$	0.997272	0.0499261
$400$	0	0
$401$	35.0670	1.75116	0.875581	−	0.483072i	$-0.160479\pi$
$401$	35.0670	1.75116	0.875581	+	0.483072i	$0.160479\pi$
$402$	0	0
$403$	−11.2516	−0.560482
$404$	0	0
$405$	1.27524	0.0633670
$406$	0	0
$407$	18.6656	0.925220
$408$	0	0
$409$	−0.828608	−0.0409720	−0.0204860	−	0.999790i	$-0.506521\pi$
$409$	−0.828608	−0.0409720	−0.0204860	+	0.999790i	$0.506521\pi$
$410$	0	0
$411$	2.90570	0.143328
$412$	0	0
$413$	3.90007	0.191910
$414$	0	0
$415$	−19.8309	−0.973460
$416$	0	0
$417$	−20.2270	−0.990523
$418$	0	0
$419$	−8.52861	−0.416650	−0.208325	−	0.978060i	$-0.566801\pi$
$419$	−8.52861	−0.416650	−0.208325	+	0.978060i	$0.566801\pi$
$420$	0	0
$421$	31.3246	1.52667	0.763335	−	0.646003i	$-0.223561\pi$
$421$	31.3246	1.52667	0.763335	+	0.646003i	$0.223561\pi$
$422$	0	0
$423$	6.91639	0.336286
$424$	0	0
$425$	10.4365	0.506246
$426$	0	0
$427$	−3.37152	−0.163159
$428$	0	0
$429$	−3.03743	−0.146648
$430$	0	0
$431$	−36.9785	−1.78119	−0.890597	−	0.454794i	$-0.849713\pi$
$431$	−36.9785	−1.78119	−0.890597	+	0.454794i	$0.849713\pi$
$432$	0	0
$433$	−16.0026	−0.769037	−0.384518	−	0.923117i	$-0.625632\pi$
$433$	−16.0026	−0.769037	−0.384518	+	0.923117i	$0.625632\pi$
$434$	0	0
$435$	0.233790	0.0112094
$436$	0	0
$437$	−18.8248	−0.900513
$438$	0	0
$439$	18.3350	0.875081	0.437541	−	0.899199i	$-0.355850\pi$
$439$	18.3350	0.875081	0.437541	+	0.899199i	$0.355850\pi$
$440$	0	0
$441$	−6.91061	−0.329077
$442$	0	0
$443$	31.0293	1.47425	0.737124	−	0.675758i	$-0.236184\pi$
$443$	31.0293	1.47425	0.737124	+	0.675758i	$0.236184\pi$
$444$	0	0
$445$	5.45464	0.258575
$446$	0	0
$447$	−22.9048	−1.08336
$448$	0	0
$449$	−31.2183	−1.47328	−0.736640	−	0.676285i	$-0.763589\pi$
$449$	−31.2183	−1.47328	−0.736640	+	0.676285i	$0.763589\pi$
$450$	0	0
$451$	8.44654	0.397732
$452$	0	0
$453$	−0.836346	−0.0392950
$454$	0	0
$455$	0.395782	0.0185546
$456$	0	0
$457$	−14.5909	−0.682535	−0.341267	−	0.939966i	$-0.610856\pi$
$457$	−14.5909	−0.682535	−0.341267	+	0.939966i	$0.610856\pi$
$458$	0	0
$459$	−3.09343	−0.144389
$460$	0	0
$461$	3.28276	0.152893	0.0764467	−	0.997074i	$-0.475642\pi$
$461$	3.28276	0.152893	0.0764467	+	0.997074i	$0.475642\pi$
$462$	0	0
$463$	−24.2200	−1.12560	−0.562800	−	0.826593i	$-0.690276\pi$
$463$	−24.2200	−1.12560	−0.562800	+	0.826593i	$0.690276\pi$
$464$	0	0
$465$	−13.8224	−0.640997
$466$	0	0
$467$	22.4726	1.03991	0.519954	−	0.854194i	$-0.325949\pi$
$467$	22.4726	1.03991	0.519954	+	0.854194i	$0.325949\pi$
$468$	0	0
$469$	−4.10394	−0.189503
$470$	0	0
$471$	19.3353	0.890925
$472$	0	0
$473$	−6.12367	−0.281566
$474$	0	0
$475$	−11.2535	−0.516345
$476$	0	0
$477$	−9.47327	−0.433751
$478$	0	0
$479$	−17.5384	−0.801351	−0.400675	−	0.916220i	$-0.631224\pi$
$479$	−17.5384	−0.801351	−0.400675	+	0.916220i	$0.631224\pi$
$480$	0	0
$481$	−6.62187	−0.301931
$482$	0	0
$483$	−1.68734	−0.0767767
$484$	0	0
$485$	3.59548	0.163262
$486$	0	0
$487$	−16.2864	−0.738007	−0.369004	−	0.929428i	$-0.620301\pi$
$487$	−16.2864	−0.738007	−0.369004	+	0.929428i	$0.620301\pi$
$488$	0	0
$489$	−7.91910	−0.358114
$490$	0	0
$491$	−39.9124	−1.80122	−0.900611	−	0.434627i	$-0.856880\pi$
$491$	−39.9124	−1.80122	−0.900611	+	0.434627i	$0.856880\pi$
$492$	0	0
$493$	−0.567121	−0.0255418
$494$	0	0
$495$	−3.73142	−0.167715
$496$	0	0
$497$	1.74002	0.0780505
$498$	0	0
$499$	30.1635	1.35030	0.675152	−	0.737678i	$-0.264078\pi$
$499$	30.1635	1.35030	0.675152	+	0.737678i	$0.264078\pi$
$500$	0	0
$501$	0.264256	0.0118061
$502$	0	0
$503$	39.1920	1.74748	0.873741	−	0.486391i	$-0.161687\pi$
$503$	39.1920	1.74748	0.873741	+	0.486391i	$0.161687\pi$
$504$	0	0
$505$	12.5906	0.560272
$506$	0	0
$507$	−11.9224	−0.529494
$508$	0	0
$509$	3.24622	0.143886	0.0719431	−	0.997409i	$-0.477080\pi$
$509$	3.24622	0.143886	0.0719431	+	0.997409i	$0.477080\pi$
$510$	0	0
$511$	1.28643	0.0569083
$512$	0	0
$513$	3.33558	0.147269
$514$	0	0
$515$	6.30399	0.277787
$516$	0	0
$517$	−20.2378	−0.890057
$518$	0	0
$519$	−8.57352	−0.376336
$520$	0	0
$521$	5.25042	0.230025	0.115012	−	0.993364i	$-0.463309\pi$
$521$	5.25042	0.230025	0.115012	+	0.993364i	$0.463309\pi$
$522$	0	0
$523$	40.1655	1.75632	0.878158	−	0.478371i	$-0.158772\pi$
$523$	40.1655	1.75632	0.878158	+	0.478371i	$0.158772\pi$
$524$	0	0
$525$	−1.00869	−0.0440229
$526$	0	0
$527$	33.5299	1.46058
$528$	0	0
$529$	8.85074	0.384815
$530$	0	0
$531$	13.0446	0.566086
$532$	0	0
$533$	−2.99652	−0.129794
$534$	0	0
$535$	6.23242	0.269451
$536$	0	0
$537$	11.4646	0.494735
$538$	0	0
$539$	20.2209	0.870975
$540$	0	0
$541$	−34.5102	−1.48371	−0.741855	−	0.670560i	$-0.766054\pi$
$541$	−34.5102	−1.48371	−0.741855	+	0.670560i	$0.766054\pi$
$542$	0	0
$543$	4.52183	0.194050
$544$	0	0
$545$	−9.67853	−0.414582
$546$	0	0
$547$	−32.4126	−1.38586	−0.692931	−	0.721004i	$-0.743681\pi$
$547$	−32.4126	−1.38586	−0.692931	+	0.721004i	$0.743681\pi$
$548$	0	0
$549$	−11.2767	−0.481279
$550$	0	0
$551$	0.611513	0.0260513
$552$	0	0
$553$	0.882180	0.0375141
$554$	0	0
$555$	−8.13484	−0.345305
$556$	0	0
$557$	2.95729	0.125304	0.0626522	−	0.998035i	$-0.480044\pi$
$557$	2.95729	0.125304	0.0626522	+	0.998035i	$0.480044\pi$
$558$	0	0
$559$	2.17245	0.0918849
$560$	0	0
$561$	9.05158	0.382158
$562$	0	0
$563$	40.8931	1.72344	0.861719	−	0.507386i	$-0.169388\pi$
$563$	40.8931	1.72344	0.861719	+	0.507386i	$0.169388\pi$
$564$	0	0
$565$	6.47070	0.272224
$566$	0	0
$567$	0.298981	0.0125560
$568$	0	0
$569$	27.3853	1.14805	0.574025	−	0.818837i	$-0.305381\pi$
$569$	27.3853	1.14805	0.574025	+	0.818837i	$0.305381\pi$
$570$	0	0
$571$	−30.2051	−1.26405	−0.632023	−	0.774950i	$-0.717775\pi$
$571$	−30.2051	−1.26405	−0.632023	+	0.774950i	$0.717775\pi$
$572$	0	0
$573$	9.41650	0.393380
$574$	0	0
$575$	19.0404	0.794038
$576$	0	0
$577$	−22.1862	−0.923624	−0.461812	−	0.886978i	$-0.652800\pi$
$577$	−22.1862	−0.923624	−0.461812	+	0.886978i	$0.652800\pi$
$578$	0	0
$579$	−5.47258	−0.227433
$580$	0	0
$581$	−4.64937	−0.192889
$582$	0	0
$583$	27.7194	1.14802
$584$	0	0
$585$	1.32377	0.0547312
$586$	0	0
$587$	19.4765	0.803882	0.401941	−	0.915665i	$-0.368336\pi$
$587$	19.4765	0.803882	0.401941	+	0.915665i	$0.368336\pi$
$588$	0	0
$589$	−36.1545	−1.48972
$590$	0	0
$591$	−13.9532	−0.573958
$592$	0	0
$593$	−2.07357	−0.0851515	−0.0425758	−	0.999093i	$-0.513556\pi$
$593$	−2.07357	−0.0851515	−0.0425758	+	0.999093i	$0.513556\pi$
$594$	0	0
$595$	−1.17944	−0.0483521
$596$	0	0
$597$	−5.46291	−0.223582
$598$	0	0
$599$	6.28469	0.256785	0.128393	−	0.991723i	$-0.459018\pi$
$599$	6.28469	0.256785	0.128393	+	0.991723i	$0.459018\pi$
$600$	0	0
$601$	−39.4147	−1.60776	−0.803880	−	0.594791i	$-0.797235\pi$
$601$	−39.4147	−1.60776	−0.803880	+	0.594791i	$0.797235\pi$
$602$	0	0
$603$	−13.7265	−0.558984
$604$	0	0
$605$	−3.10922	−0.126408
$606$	0	0
$607$	−17.0388	−0.691584	−0.345792	−	0.938311i	$-0.612390\pi$
$607$	−17.0388	−0.691584	−0.345792	+	0.938311i	$0.612390\pi$
$608$	0	0
$609$	0.0548123	0.00222111
$610$	0	0
$611$	7.17962	0.290456
$612$	0	0
$613$	−22.3939	−0.904480	−0.452240	−	0.891896i	$-0.649375\pi$
$613$	−22.3939	−0.904480	−0.452240	+	0.891896i	$0.649375\pi$
$614$	0	0
$615$	−3.68117	−0.148439
$616$	0	0
$617$	17.5827	0.707855	0.353927	−	0.935273i	$-0.384846\pi$
$617$	17.5827	0.707855	0.353927	+	0.935273i	$0.384846\pi$
$618$	0	0
$619$	−7.89289	−0.317242	−0.158621	−	0.987340i	$-0.550705\pi$
$619$	−7.89289	−0.317242	−0.158621	+	0.987340i	$0.550705\pi$
$620$	0	0
$621$	−5.64365	−0.226472
$622$	0	0
$623$	1.27885	0.0512359
$624$	0	0
$625$	3.25119	0.130048
$626$	0	0
$627$	−9.76011	−0.389781
$628$	0	0
$629$	19.7333	0.786817
$630$	0	0
$631$	−42.0884	−1.67551	−0.837756	−	0.546044i	$-0.816133\pi$
$631$	−42.0884	−1.67551	−0.837756	+	0.546044i	$0.816133\pi$
$632$	0	0
$633$	−5.35714	−0.212927
$634$	0	0
$635$	25.0501	0.994082
$636$	0	0
$637$	−7.17363	−0.284229
$638$	0	0
$639$	5.81983	0.230229
$640$	0	0
$641$	−18.5948	−0.734451	−0.367225	−	0.930132i	$-0.619692\pi$
$641$	−18.5948	−0.734451	−0.367225	+	0.930132i	$0.619692\pi$
$642$	0	0
$643$	34.8920	1.37600	0.688002	−	0.725708i	$-0.258488\pi$
$643$	34.8920	1.37600	0.688002	+	0.725708i	$0.258488\pi$
$644$	0	0
$645$	2.66881	0.105084
$646$	0	0
$647$	13.2068	0.519214	0.259607	−	0.965714i	$-0.416407\pi$
$647$	13.2068	0.519214	0.259607	+	0.965714i	$0.416407\pi$
$648$	0	0
$649$	−38.1692	−1.49827
$650$	0	0
$651$	−3.24067	−0.127012
$652$	0	0
$653$	−35.5486	−1.39112	−0.695561	−	0.718467i	$-0.744844\pi$
$653$	−35.5486	−1.39112	−0.695561	+	0.718467i	$0.744844\pi$
$654$	0	0
$655$	14.4384	0.564156
$656$	0	0
$657$	4.30272	0.167865
$658$	0	0
$659$	48.4611	1.88778	0.943889	−	0.330263i	$-0.107137\pi$
$659$	48.4611	1.88778	0.943889	+	0.330263i	$0.107137\pi$
$660$	0	0
$661$	−8.14683	−0.316875	−0.158437	−	0.987369i	$-0.550646\pi$
$661$	−8.14683	−0.316875	−0.158437	+	0.987369i	$0.550646\pi$
$662$	0	0
$663$	−3.21117	−0.124711
$664$	0	0
$665$	1.27176	0.0493167
$666$	0	0
$667$	−1.03465	−0.0400619
$668$	0	0
$669$	25.8029	0.997597
$670$	0	0
$671$	32.9964	1.27381
$672$	0	0
$673$	−2.81869	−0.108653	−0.0543263	−	0.998523i	$-0.517301\pi$
$673$	−2.81869	−0.108653	−0.0543263	+	0.998523i	$0.517301\pi$
$674$	0	0
$675$	−3.37377	−0.129857
$676$	0	0
$677$	−11.0834	−0.425968	−0.212984	−	0.977056i	$-0.568318\pi$
$677$	−11.0834	−0.425968	−0.212984	+	0.977056i	$0.568318\pi$
$678$	0	0
$679$	0.842965	0.0323500
$680$	0	0
$681$	4.81244	0.184413
$682$	0	0
$683$	−46.3032	−1.77174	−0.885872	−	0.463930i	$-0.846439\pi$
$683$	−46.3032	−1.77174	−0.885872	+	0.463930i	$0.846439\pi$
$684$	0	0
$685$	3.70546	0.141578
$686$	0	0
$687$	2.22169	0.0847627
$688$	0	0
$689$	−9.83382	−0.374639
$690$	0	0
$691$	45.7545	1.74058	0.870291	−	0.492538i	$-0.163931\pi$
$691$	45.7545	1.74058	0.870291	+	0.492538i	$0.163931\pi$
$692$	0	0
$693$	−0.874836	−0.0332323
$694$	0	0
$695$	−25.7943	−0.978432
$696$	0	0
$697$	8.92967	0.338235
$698$	0	0
$699$	−17.1040	−0.646932
$700$	0	0
$701$	−27.1537	−1.02558	−0.512790	−	0.858514i	$-0.671388\pi$
$701$	−27.1537	−1.02558	−0.512790	+	0.858514i	$0.671388\pi$
$702$	0	0
$703$	−21.2779	−0.802512
$704$	0	0
$705$	8.82003	0.332182
$706$	0	0
$707$	2.95187	0.111016
$708$	0	0
$709$	39.5850	1.48665	0.743323	−	0.668933i	$-0.233249\pi$
$709$	39.5850	1.48665	0.743323	+	0.668933i	$0.233249\pi$
$710$	0	0
$711$	2.95063	0.110657
$712$	0	0
$713$	61.1718	2.29090
$714$	0	0
$715$	−3.87344	−0.144858
$716$	0	0
$717$	−2.26452	−0.0845702
$718$	0	0
$719$	−9.28754	−0.346367	−0.173183	−	0.984890i	$-0.555405\pi$
$719$	−9.28754	−0.346367	−0.173183	+	0.984890i	$0.555405\pi$
$720$	0	0
$721$	1.47798	0.0550427
$722$	0	0
$723$	−4.75567	−0.176865
$724$	0	0
$725$	−0.618516	−0.0229711
$726$	0	0
$727$	11.1000	0.411675	0.205838	−	0.978586i	$-0.434008\pi$
$727$	11.1000	0.411675	0.205838	+	0.978586i	$0.434008\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−6.47393	−0.239447
$732$	0	0
$733$	−20.5892	−0.760479	−0.380240	−	0.924888i	$-0.624158\pi$
$733$	−20.5892	−0.760479	−0.380240	+	0.924888i	$0.624158\pi$
$734$	0	0
$735$	−8.81266	−0.325060
$736$	0	0
$737$	40.1645	1.47948
$738$	0	0
$739$	2.22583	0.0818786	0.0409393	−	0.999162i	$-0.486965\pi$
$739$	2.22583	0.0818786	0.0409393	+	0.999162i	$0.486965\pi$
$740$	0	0
$741$	3.46253	0.127199
$742$	0	0
$743$	34.6655	1.27175	0.635876	−	0.771791i	$-0.280639\pi$
$743$	34.6655	1.27175	0.635876	+	0.771791i	$0.280639\pi$
$744$	0	0
$745$	−29.2090	−1.07014
$746$	0	0
$747$	−15.5508	−0.568972
$748$	0	0
$749$	1.46120	0.0533910
$750$	0	0
$751$	2.94616	0.107507	0.0537535	−	0.998554i	$-0.482881\pi$
$751$	2.94616	0.107507	0.0537535	+	0.998554i	$0.482881\pi$
$752$	0	0
$753$	1.00000	0.0364420
$754$	0	0
$755$	−1.06654	−0.0388153
$756$	0	0
$757$	−36.0822	−1.31143	−0.655715	−	0.755008i	$-0.727633\pi$
$757$	−36.0822	−1.31143	−0.655715	+	0.755008i	$0.727633\pi$
$758$	0	0
$759$	16.5137	0.599408
$760$	0	0
$761$	19.1228	0.693201	0.346600	−	0.938013i	$-0.387336\pi$
$761$	19.1228	0.693201	0.346600	+	0.938013i	$0.387336\pi$
$762$	0	0
$763$	−2.26914	−0.0821484
$764$	0	0
$765$	−3.94486	−0.142627
$766$	0	0
$767$	13.5410	0.488938
$768$	0	0
$769$	6.89746	0.248729	0.124364	−	0.992237i	$-0.460311\pi$
$769$	6.89746	0.248729	0.124364	+	0.992237i	$0.460311\pi$
$770$	0	0
$771$	11.0711	0.398717
$772$	0	0
$773$	−42.9135	−1.54349	−0.771747	−	0.635930i	$-0.780617\pi$
$773$	−42.9135	−1.54349	−0.771747	+	0.635930i	$0.780617\pi$
$774$	0	0
$775$	36.5685	1.31358
$776$	0	0
$777$	−1.90722	−0.0684212
$778$	0	0
$779$	−9.62866	−0.344983
$780$	0	0
$781$	−17.0292	−0.609353
$782$	0	0
$783$	0.183331	0.00655170
$784$	0	0
$785$	24.6571	0.880050
$786$	0	0
$787$	36.8804	1.31464	0.657322	−	0.753610i	$-0.271689\pi$
$787$	36.8804	1.31464	0.657322	+	0.753610i	$0.271689\pi$
$788$	0	0
$789$	−21.8548	−0.778053
$790$	0	0
$791$	1.51706	0.0539405
$792$	0	0
$793$	−11.7059	−0.415689
$794$	0	0
$795$	−12.0807	−0.428457
$796$	0	0
$797$	−48.4419	−1.71590	−0.857949	−	0.513734i	$-0.828262\pi$
$797$	−48.4419	−1.71590	−0.857949	+	0.513734i	$0.828262\pi$
$798$	0	0
$799$	−21.3954	−0.756914
$800$	0	0
$801$	4.27736	0.151133
$802$	0	0
$803$	−12.5900	−0.444293
$804$	0	0
$805$	−2.15176	−0.0758395
$806$	0	0
$807$	19.2686	0.678288
$808$	0	0
$809$	−32.3955	−1.13897	−0.569483	−	0.822003i	$-0.692857\pi$
$809$	−32.3955	−1.13897	−0.569483	+	0.822003i	$0.692857\pi$
$810$	0	0
$811$	49.3730	1.73372	0.866861	−	0.498551i	$-0.166134\pi$
$811$	49.3730	1.73372	0.866861	+	0.498551i	$0.166134\pi$
$812$	0	0
$813$	−2.71190	−0.0951106
$814$	0	0
$815$	−10.0987	−0.353743
$816$	0	0
$817$	6.98069	0.244223
$818$	0	0
$819$	0.310360	0.0108448
$820$	0	0
$821$	48.2870	1.68523	0.842615	−	0.538517i	$-0.181015\pi$
$821$	48.2870	1.68523	0.842615	+	0.538517i	$0.181015\pi$
$822$	0	0
$823$	20.9099	0.728872	0.364436	−	0.931228i	$-0.381262\pi$
$823$	20.9099	0.728872	0.364436	+	0.931228i	$0.381262\pi$
$824$	0	0
$825$	9.87187	0.343694
$826$	0	0
$827$	−13.5220	−0.470205	−0.235103	−	0.971971i	$-0.575543\pi$
$827$	−13.5220	−0.470205	−0.235103	+	0.971971i	$0.575543\pi$
$828$	0	0
$829$	26.2797	0.912732	0.456366	−	0.889792i	$-0.349151\pi$
$829$	26.2797	0.912732	0.456366	+	0.889792i	$0.349151\pi$
$830$	0	0
$831$	−3.34489	−0.116033
$832$	0	0
$833$	21.3775	0.740686
$834$	0	0
$835$	0.336989	0.0116620
$836$	0	0
$837$	−10.8391	−0.374653
$838$	0	0
$839$	26.1515	0.902851	0.451425	−	0.892309i	$-0.350916\pi$
$839$	26.1515	0.902851	0.451425	+	0.892309i	$0.350916\pi$
$840$	0	0
$841$	−28.9664	−0.998841
$842$	0	0
$843$	31.2770	1.07724
$844$	0	0
$845$	−15.2039	−0.523031
$846$	0	0
$847$	−0.728960	−0.0250473
$848$	0	0
$849$	−9.97848	−0.342461
$850$	0	0
$851$	36.0013	1.23411
$852$	0	0
$853$	30.8991	1.05796	0.528982	−	0.848633i	$-0.322574\pi$
$853$	30.8991	1.05796	0.528982	+	0.848633i	$0.322574\pi$
$854$	0	0
$855$	4.25365	0.145472
$856$	0	0
$857$	21.9124	0.748514	0.374257	−	0.927325i	$-0.377898\pi$
$857$	21.9124	0.748514	0.374257	+	0.927325i	$0.377898\pi$
$858$	0	0
$859$	−41.1066	−1.40254	−0.701269	−	0.712897i	$-0.747383\pi$
$859$	−41.1066	−1.40254	−0.701269	+	0.712897i	$0.747383\pi$
$860$	0	0
$861$	−0.863054	−0.0294128
$862$	0	0
$863$	−23.0569	−0.784865	−0.392433	−	0.919781i	$-0.628366\pi$
$863$	−23.0569	−0.784865	−0.392433	+	0.919781i	$0.628366\pi$
$864$	0	0
$865$	−10.9333	−0.371742
$866$	0	0
$867$	−7.43068	−0.252359
$868$	0	0
$869$	−8.63372	−0.292879
$870$	0	0
$871$	−14.2489	−0.482805
$872$	0	0
$873$	2.81946	0.0954244
$874$	0	0
$875$	−3.19268	−0.107932
$876$	0	0
$877$	−46.9816	−1.58645	−0.793227	−	0.608925i	$-0.791601\pi$
$877$	−46.9816	−1.58645	−0.793227	+	0.608925i	$0.791601\pi$
$878$	0	0
$879$	1.90441	0.0642342
$880$	0	0
$881$	15.0857	0.508249	0.254124	−	0.967172i	$-0.418213\pi$
$881$	15.0857	0.508249	0.254124	+	0.967172i	$0.418213\pi$
$882$	0	0
$883$	38.7642	1.30452	0.652259	−	0.757996i	$-0.273821\pi$
$883$	38.7642	1.30452	0.652259	+	0.757996i	$0.273821\pi$
$884$	0	0
$885$	16.6349	0.559176
$886$	0	0
$887$	−55.0150	−1.84722	−0.923612	−	0.383328i	$-0.874778\pi$
$887$	−55.0150	−1.84722	−0.923612	+	0.383328i	$0.874778\pi$
$888$	0	0
$889$	5.87302	0.196975
$890$	0	0
$891$	−2.92606	−0.0980268
$892$	0	0
$893$	23.0701	0.772012
$894$	0	0
$895$	14.6201	0.488696
$896$	0	0
$897$	−5.85844	−0.195608
$898$	0	0
$899$	−1.98713	−0.0662745
$900$	0	0
$901$	29.3049	0.976288
$902$	0	0
$903$	0.625706	0.0208222
$904$	0	0
$905$	5.76641	0.191682
$906$	0	0
$907$	41.0971	1.36461	0.682303	−	0.731070i	$-0.260979\pi$
$907$	41.0971	1.36461	0.682303	+	0.731070i	$0.260979\pi$
$908$	0	0
$909$	9.87311	0.327471
$910$	0	0
$911$	−57.8959	−1.91818	−0.959089	−	0.283106i	$-0.908635\pi$
$911$	−57.8959	−1.91818	−0.959089	+	0.283106i	$0.908635\pi$
$912$	0	0
$913$	45.5025	1.50591
$914$	0	0
$915$	−14.3805	−0.475405
$916$	0	0
$917$	3.38511	0.111786
$918$	0	0
$919$	−28.9907	−0.956314	−0.478157	−	0.878275i	$-0.658695\pi$
$919$	−28.9907	−0.956314	−0.478157	+	0.878275i	$0.658695\pi$
$920$	0	0
$921$	28.8790	0.951595
$922$	0	0
$923$	6.04134	0.198853
$924$	0	0
$925$	21.5216	0.707625
$926$	0	0
$927$	4.94339	0.162362
$928$	0	0
$929$	−10.2354	−0.335814	−0.167907	−	0.985803i	$-0.553701\pi$
$929$	−10.2354	−0.335814	−0.167907	+	0.985803i	$0.553701\pi$
$930$	0	0
$931$	−23.0509	−0.755461
$932$	0	0
$933$	−11.1302	−0.364387
$934$	0	0
$935$	11.5429	0.377493
$936$	0	0
$937$	12.2006	0.398576	0.199288	−	0.979941i	$-0.436137\pi$
$937$	12.2006	0.398576	0.199288	+	0.979941i	$0.436137\pi$
$938$	0	0
$939$	13.3513	0.435702
$940$	0	0
$941$	26.5066	0.864090	0.432045	−	0.901852i	$-0.357792\pi$
$941$	26.5066	0.864090	0.432045	+	0.901852i	$0.357792\pi$
$942$	0	0
$943$	16.2913	0.530517
$944$	0	0
$945$	0.381271	0.0124027
$946$	0	0
$947$	16.9787	0.551735	0.275867	−	0.961196i	$-0.411035\pi$
$947$	16.9787	0.551735	0.275867	+	0.961196i	$0.411035\pi$
$948$	0	0
$949$	4.46648	0.144988
$950$	0	0
$951$	21.6239	0.701203
$952$	0	0
$953$	−2.73607	−0.0886300	−0.0443150	−	0.999018i	$-0.514111\pi$
$953$	−2.73607	−0.0886300	−0.0443150	+	0.999018i	$0.514111\pi$
$954$	0	0
$955$	12.0083	0.388578
$956$	0	0
$957$	−0.536437	−0.0173406
$958$	0	0
$959$	0.868749	0.0280534
$960$	0	0
$961$	86.4852	2.78984
$962$	0	0
$963$	4.88726	0.157490
$964$	0	0
$965$	−6.97883	−0.224657
$966$	0	0
$967$	31.1551	1.00188	0.500940	−	0.865482i	$-0.332988\pi$
$967$	31.1551	1.00188	0.500940	+	0.865482i	$0.332988\pi$
$968$	0	0
$969$	−10.3184	−0.331474
$970$	0	0
$971$	27.1851	0.872411	0.436205	−	0.899847i	$-0.356322\pi$
$971$	27.1851	0.872411	0.436205	+	0.899847i	$0.356322\pi$
$972$	0	0
$973$	−6.04749	−0.193874
$974$	0	0
$975$	−3.50218	−0.112159
$976$	0	0
$977$	13.4258	0.429530	0.214765	−	0.976666i	$-0.431101\pi$
$977$	13.4258	0.429530	0.214765	+	0.976666i	$0.431101\pi$
$978$	0	0
$979$	−12.5158	−0.400007
$980$	0	0
$981$	−7.58959	−0.242317
$982$	0	0
$983$	10.0422	0.320297	0.160149	−	0.987093i	$-0.448803\pi$
$983$	10.0422	0.320297	0.160149	+	0.987093i	$0.448803\pi$
$984$	0	0
$985$	−17.7936	−0.566953
$986$	0	0
$987$	2.06787	0.0658209
$988$	0	0
$989$	−11.8110	−0.375569
$990$	0	0
$991$	−40.7160	−1.29339	−0.646694	−	0.762750i	$-0.723849\pi$
$991$	−40.7160	−1.29339	−0.646694	+	0.762750i	$0.723849\pi$
$992$	0	0
$993$	−27.4060	−0.869702
$994$	0	0
$995$	−6.96650	−0.220853
$996$	0	0
$997$	−20.3246	−0.643687	−0.321844	−	0.946793i	$-0.604303\pi$
$997$	−20.3246	−0.643687	−0.321844	+	0.946793i	$0.604303\pi$
$998$	0	0
$999$	−6.37909	−0.201825

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6024.2.a.k.1.7	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6024.2.a.k.1.7	✓	8	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +1.27524 q^{5} +0.298981 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +1.27524 q^{5} +0.298981 q^{7} +1.00000 q^{9} -2.92606 q^{11} +1.03806 q^{13} +1.27524 q^{15} -3.09343 q^{17} +3.33558 q^{19} +0.298981 q^{21} -5.64365 q^{23} -3.37377 q^{25} +1.00000 q^{27} +0.183331 q^{29} -10.8391 q^{31} -2.92606 q^{33} +0.381271 q^{35} -6.37909 q^{37} +1.03806 q^{39} -2.88666 q^{41} +2.09280 q^{43} +1.27524 q^{45} +6.91639 q^{47} -6.91061 q^{49} -3.09343 q^{51} -9.47327 q^{53} -3.73142 q^{55} +3.33558 q^{57} +13.0446 q^{59} -11.2767 q^{61} +0.298981 q^{63} +1.32377 q^{65} -13.7265 q^{67} -5.64365 q^{69} +5.81983 q^{71} +4.30272 q^{73} -3.37377 q^{75} -0.874836 q^{77} +2.95063 q^{79} +1.00000 q^{81} -15.5508 q^{83} -3.94486 q^{85} +0.183331 q^{87} +4.27736 q^{89} +0.310360 q^{91} -10.8391 q^{93} +4.25365 q^{95} +2.81946 q^{97} -2.92606 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{3} - 5 q^{5} + q^{7} + 8 q^{9}+O(q^{10}) \) 8 * q + 8 * q^3 - 5 * q^5 + q^7 + 8 * q^9	\( 8 q + 8 q^{3} - 5 q^{5} + q^{7} + 8 q^{9} - 6 q^{11} - 4 q^{13} - 5 q^{15} - 12 q^{17} - 10 q^{19} + q^{21} - 6 q^{23} - 7 q^{25} + 8 q^{27} - 2 q^{29} - 3 q^{31} - 6 q^{33} + 4 q^{35} - 16 q^{37} - 4 q^{39} + 2 q^{41} - 2 q^{43} - 5 q^{45} - 15 q^{47} - 21 q^{49} - 12 q^{51} + 13 q^{53} - 33 q^{55} - 10 q^{57} + 8 q^{59} - 38 q^{61} + q^{63} + 16 q^{65} - 31 q^{67} - 6 q^{69} + 5 q^{71} - 26 q^{73} - 7 q^{75} - 24 q^{77} - 25 q^{79} + 8 q^{81} - 7 q^{83} - 18 q^{85} - 2 q^{87} - 14 q^{89} - 6 q^{91} - 3 q^{93} - q^{95} - 51 q^{97} - 6 q^{99}+O(q^{100}) \) 8 * q + 8 * q^3 - 5 * q^5 + q^7 + 8 * q^9 - 6 * q^11 - 4 * q^13 - 5 * q^15 - 12 * q^17 - 10 * q^19 + q^21 - 6 * q^23 - 7 * q^25 + 8 * q^27 - 2 * q^29 - 3 * q^31 - 6 * q^33 + 4 * q^35 - 16 * q^37 - 4 * q^39 + 2 * q^41 - 2 * q^43 - 5 * q^45 - 15 * q^47 - 21 * q^49 - 12 * q^51 + 13 * q^53 - 33 * q^55 - 10 * q^57 + 8 * q^59 - 38 * q^61 + q^63 + 16 * q^65 - 31 * q^67 - 6 * q^69 + 5 * q^71 - 26 * q^73 - 7 * q^75 - 24 * q^77 - 25 * q^79 + 8 * q^81 - 7 * q^83 - 18 * q^85 - 2 * q^87 - 14 * q^89 - 6 * q^91 - 3 * q^93 - q^95 - 51 * q^97 - 6 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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