LMFDB - Embedded newform 8008.2.a.n.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8008,2,Mod(1,8008)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8008, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("8008.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8008 = 2^{3} \cdot 7 \cdot 11 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8008.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:	$63.9442019386$
Analytic rank:	$0$
Dimension:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - 3x^{8} - 15x^{7} + 45x^{6} + 64x^{5} - 201x^{4} - 63x^{3} + 282x^{2} + 3x - 116 $ x^9 - 3x^8 - 15x^7 + 45x^6 + 64x^5 - 201x^4 - 63x^3 + 282x^2 + 3x - 116
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$1.21838$ of defining polynomial
Character	$\chi$	$=$	8008.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.21838 q^{3} +2.90030 q^{5} -1.00000 q^{7} -1.51556 q^{9} +O(q^{10})$	$q-1.21838 q^{3} +2.90030 q^{5} -1.00000 q^{7} -1.51556 q^{9} +1.00000 q^{11} -1.00000 q^{13} -3.53365 q^{15} +3.20996 q^{17} -0.338048 q^{19} +1.21838 q^{21} +7.17564 q^{23} +3.41172 q^{25} +5.50165 q^{27} +10.5557 q^{29} +1.34712 q^{31} -1.21838 q^{33} -2.90030 q^{35} +2.97901 q^{37} +1.21838 q^{39} -3.83082 q^{41} -2.40992 q^{43} -4.39557 q^{45} -12.1040 q^{47} +1.00000 q^{49} -3.91094 q^{51} +1.77133 q^{53} +2.90030 q^{55} +0.411870 q^{57} -6.34524 q^{59} -6.55153 q^{61} +1.51556 q^{63} -2.90030 q^{65} -0.479975 q^{67} -8.74263 q^{69} -7.64425 q^{71} -5.07758 q^{73} -4.15676 q^{75} -1.00000 q^{77} +1.11643 q^{79} -2.15641 q^{81} +7.65360 q^{83} +9.30983 q^{85} -12.8609 q^{87} +15.7332 q^{89} +1.00000 q^{91} -1.64130 q^{93} -0.980440 q^{95} +16.9715 q^{97} -1.51556 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q - 3 q^{3} + q^{5} - 9 q^{7} + 12 q^{9}+O(q^{10}) $ 9 * q - 3 * q^3 + q^5 - 9 * q^7 + 12 * q^9	$ 9 q - 3 q^{3} + q^{5} - 9 q^{7} + 12 q^{9} + 9 q^{11} - 9 q^{13} + 11 q^{15} + 7 q^{17} - 17 q^{19} + 3 q^{21} + 11 q^{23} + 18 q^{25} - 9 q^{27} + 9 q^{29} - 8 q^{31} - 3 q^{33} - q^{35} + 2 q^{37} + 3 q^{39} + 18 q^{41} + 7 q^{43} + 5 q^{45} + 15 q^{47} + 9 q^{49} - 7 q^{51} - 4 q^{53} + q^{55} + 22 q^{57} - 23 q^{59} + 12 q^{61} - 12 q^{63} - q^{65} - 16 q^{67} - 32 q^{69} - 6 q^{71} + 4 q^{73} - 14 q^{75} - 9 q^{77} + 21 q^{79} + 5 q^{81} - 16 q^{83} + 53 q^{85} + 41 q^{87} + 5 q^{89} + 9 q^{91} + 29 q^{93} + 19 q^{95} + 18 q^{97} + 12 q^{99}+O(q^{100}) $ 9 * q - 3 * q^3 + q^5 - 9 * q^7 + 12 * q^9 + 9 * q^11 - 9 * q^13 + 11 * q^15 + 7 * q^17 - 17 * q^19 + 3 * q^21 + 11 * q^23 + 18 * q^25 - 9 * q^27 + 9 * q^29 - 8 * q^31 - 3 * q^33 - q^35 + 2 * q^37 + 3 * q^39 + 18 * q^41 + 7 * q^43 + 5 * q^45 + 15 * q^47 + 9 * q^49 - 7 * q^51 - 4 * q^53 + q^55 + 22 * q^57 - 23 * q^59 + 12 * q^61 - 12 * q^63 - q^65 - 16 * q^67 - 32 * q^69 - 6 * q^71 + 4 * q^73 - 14 * q^75 - 9 * q^77 + 21 * q^79 + 5 * q^81 - 16 * q^83 + 53 * q^85 + 41 * q^87 + 5 * q^89 + 9 * q^91 + 29 * q^93 + 19 * q^95 + 18 * q^97 + 12 * 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
$3$	−1.21838	−0.703430	−0.351715	−	0.936107i	$-0.614401\pi$
$3$	−1.21838	−0.703430	−0.351715	+	0.936107i	$0.614401\pi$
$4$	0	0
$5$	2.90030	1.29705	0.648526	−	0.761193i	$-0.275386\pi$
$5$	2.90030	1.29705	0.648526	+	0.761193i	$0.275386\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	−1.51556	−0.505186
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	−3.53365	−0.912385
$16$	0	0
$17$	3.20996	0.778529	0.389265	−	0.921126i	$-0.372729\pi$
$17$	3.20996	0.778529	0.389265	+	0.921126i	$0.372729\pi$
$18$	0	0
$19$	−0.338048	−0.0775536	−0.0387768	−	0.999248i	$-0.512346\pi$
$19$	−0.338048	−0.0775536	−0.0387768	+	0.999248i	$0.512346\pi$
$20$	0	0
$21$	1.21838	0.265872
$22$	0	0
$23$	7.17564	1.49622	0.748112	−	0.663573i	$-0.230961\pi$
$23$	7.17564	1.49622	0.748112	+	0.663573i	$0.230961\pi$
$24$	0	0
$25$	3.41172	0.682344
$26$	0	0
$27$	5.50165	1.05879
$28$	0	0
$29$	10.5557	1.96015	0.980075	−	0.198626i	$-0.0636479\pi$
$29$	10.5557	1.96015	0.980075	+	0.198626i	$0.0636479\pi$
$30$	0	0
$31$	1.34712	0.241950	0.120975	−	0.992656i	$-0.461398\pi$
$31$	1.34712	0.241950	0.120975	+	0.992656i	$0.461398\pi$
$32$	0	0
$33$	−1.21838	−0.212092
$34$	0	0
$35$	−2.90030	−0.490240
$36$	0	0
$37$	2.97901	0.489746	0.244873	−	0.969555i	$-0.421254\pi$
$37$	2.97901	0.489746	0.244873	+	0.969555i	$0.421254\pi$
$38$	0	0
$39$	1.21838	0.195096
$40$	0	0
$41$	−3.83082	−0.598273	−0.299137	−	0.954210i	$-0.596699\pi$
$41$	−3.83082	−0.598273	−0.299137	+	0.954210i	$0.596699\pi$
$42$	0	0
$43$	−2.40992	−0.367510	−0.183755	−	0.982972i	$-0.558825\pi$
$43$	−2.40992	−0.367510	−0.183755	+	0.982972i	$0.558825\pi$
$44$	0	0
$45$	−4.39557	−0.655253
$46$	0	0
$47$	−12.1040	−1.76555	−0.882777	−	0.469793i	$-0.844329\pi$
$47$	−12.1040	−1.76555	−0.882777	+	0.469793i	$0.844329\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−3.91094	−0.547641
$52$	0	0
$53$	1.77133	0.243311	0.121656	−	0.992572i	$-0.461180\pi$
$53$	1.77133	0.243311	0.121656	+	0.992572i	$0.461180\pi$
$54$	0	0
$55$	2.90030	0.391076
$56$	0	0
$57$	0.411870	0.0545535
$58$	0	0
$59$	−6.34524	−0.826080	−0.413040	−	0.910713i	$-0.635533\pi$
$59$	−6.34524	−0.826080	−0.413040	+	0.910713i	$0.635533\pi$
$60$	0	0
$61$	−6.55153	−0.838837	−0.419419	−	0.907793i	$-0.637766\pi$
$61$	−6.55153	−0.838837	−0.419419	+	0.907793i	$0.637766\pi$
$62$	0	0
$63$	1.51556	0.190942
$64$	0	0
$65$	−2.90030	−0.359737
$66$	0	0
$67$	−0.479975	−0.0586383	−0.0293192	−	0.999570i	$-0.509334\pi$
$67$	−0.479975	−0.0586383	−0.0293192	+	0.999570i	$0.509334\pi$
$68$	0	0
$69$	−8.74263	−1.05249
$70$	0	0
$71$	−7.64425	−0.907205	−0.453603	−	0.891204i	$-0.649861\pi$
$71$	−7.64425	−0.907205	−0.453603	+	0.891204i	$0.649861\pi$
$72$	0	0
$73$	−5.07758	−0.594286	−0.297143	−	0.954833i	$-0.596034\pi$
$73$	−5.07758	−0.594286	−0.297143	+	0.954833i	$0.596034\pi$
$74$	0	0
$75$	−4.15676	−0.479981
$76$	0	0
$77$	−1.00000	−0.113961
$78$	0	0
$79$	1.11643	0.125608	0.0628038	−	0.998026i	$-0.479996\pi$
$79$	1.11643	0.125608	0.0628038	+	0.998026i	$0.479996\pi$
$80$	0	0
$81$	−2.15641	−0.239601
$82$	0	0
$83$	7.65360	0.840092	0.420046	−	0.907503i	$-0.362014\pi$
$83$	7.65360	0.840092	0.420046	+	0.907503i	$0.362014\pi$
$84$	0	0
$85$	9.30983	1.00979
$86$	0	0
$87$	−12.8609	−1.37883
$88$	0	0
$89$	15.7332	1.66771	0.833856	−	0.551983i	$-0.186129\pi$
$89$	15.7332	1.66771	0.833856	+	0.551983i	$0.186129\pi$
$90$	0	0
$91$	1.00000	0.104828
$92$	0	0
$93$	−1.64130	−0.170195
$94$	0	0
$95$	−0.980440	−0.100591
$96$	0	0
$97$	16.9715	1.72320	0.861598	−	0.507591i	$-0.169464\pi$
$97$	16.9715	1.72320	0.861598	+	0.507591i	$0.169464\pi$
$98$	0	0
$99$	−1.51556	−0.152319
$100$	0	0
$101$	8.90578	0.886159	0.443079	−	0.896482i	$-0.353886\pi$
$101$	8.90578	0.886159	0.443079	+	0.896482i	$0.353886\pi$
$102$	0	0
$103$	−12.8118	−1.26238	−0.631191	−	0.775628i	$-0.717434\pi$
$103$	−12.8118	−1.26238	−0.631191	+	0.775628i	$0.717434\pi$
$104$	0	0
$105$	3.53365	0.344849
$106$	0	0
$107$	−9.75077	−0.942642	−0.471321	−	0.881962i	$-0.656223\pi$
$107$	−9.75077	−0.942642	−0.471321	+	0.881962i	$0.656223\pi$
$108$	0	0
$109$	4.90638	0.469946	0.234973	−	0.972002i	$-0.424500\pi$
$109$	4.90638	0.469946	0.234973	+	0.972002i	$0.424500\pi$
$110$	0	0
$111$	−3.62955	−0.344502
$112$	0	0
$113$	−13.2893	−1.25015	−0.625077	−	0.780563i	$-0.714933\pi$
$113$	−13.2893	−1.25015	−0.625077	+	0.780563i	$0.714933\pi$
$114$	0	0
$115$	20.8115	1.94068
$116$	0	0
$117$	1.51556	0.140113
$118$	0	0
$119$	−3.20996	−0.294256
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	4.66738	0.420844
$124$	0	0
$125$	−4.60649	−0.412017
$126$	0	0
$127$	6.66573	0.591488	0.295744	−	0.955267i	$-0.404432\pi$
$127$	6.66573	0.591488	0.295744	+	0.955267i	$0.404432\pi$
$128$	0	0
$129$	2.93619	0.258517
$130$	0	0
$131$	8.77634	0.766792	0.383396	−	0.923584i	$-0.374754\pi$
$131$	8.77634	0.766792	0.383396	+	0.923584i	$0.374754\pi$
$132$	0	0
$133$	0.338048	0.0293125
$134$	0	0
$135$	15.9564	1.37331
$136$	0	0
$137$	9.29231	0.793896	0.396948	−	0.917841i	$-0.370069\pi$
$137$	9.29231	0.793896	0.396948	+	0.917841i	$0.370069\pi$
$138$	0	0
$139$	−5.54313	−0.470162	−0.235081	−	0.971976i	$-0.575536\pi$
$139$	−5.54313	−0.470162	−0.235081	+	0.971976i	$0.575536\pi$
$140$	0	0
$141$	14.7473	1.24194
$142$	0	0
$143$	−1.00000	−0.0836242
$144$	0	0
$145$	30.6148	2.54242
$146$	0	0
$147$	−1.21838	−0.100490
$148$	0	0
$149$	−3.44869	−0.282528	−0.141264	−	0.989972i	$-0.545117\pi$
$149$	−3.44869	−0.282528	−0.141264	+	0.989972i	$0.545117\pi$
$150$	0	0
$151$	20.3094	1.65275	0.826377	−	0.563117i	$-0.190398\pi$
$151$	20.3094	1.65275	0.826377	+	0.563117i	$0.190398\pi$
$152$	0	0
$153$	−4.86488	−0.393302
$154$	0	0
$155$	3.90704	0.313821
$156$	0	0
$157$	14.0052	1.11774	0.558869	−	0.829256i	$-0.311236\pi$
$157$	14.0052	1.11774	0.558869	+	0.829256i	$0.311236\pi$
$158$	0	0
$159$	−2.15815	−0.171152
$160$	0	0
$161$	−7.17564	−0.565519
$162$	0	0
$163$	23.2055	1.81760	0.908799	−	0.417235i	$-0.137001\pi$
$163$	23.2055	1.81760	0.908799	+	0.417235i	$0.137001\pi$
$164$	0	0
$165$	−3.53365	−0.275095
$166$	0	0
$167$	−19.4507	−1.50514	−0.752569	−	0.658514i	$-0.771185\pi$
$167$	−19.4507	−1.50514	−0.752569	+	0.658514i	$0.771185\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	0.512332	0.0391790
$172$	0	0
$173$	13.3217	1.01283	0.506416	−	0.862289i	$-0.330970\pi$
$173$	13.3217	1.01283	0.506416	+	0.862289i	$0.330970\pi$
$174$	0	0
$175$	−3.41172	−0.257902
$176$	0	0
$177$	7.73090	0.581090
$178$	0	0
$179$	−15.7801	−1.17946	−0.589731	−	0.807600i	$-0.700766\pi$
$179$	−15.7801	−1.17946	−0.589731	+	0.807600i	$0.700766\pi$
$180$	0	0
$181$	−2.04562	−0.152050	−0.0760251	−	0.997106i	$-0.524223\pi$
$181$	−2.04562	−0.152050	−0.0760251	+	0.997106i	$0.524223\pi$
$182$	0	0
$183$	7.98223	0.590063
$184$	0	0
$185$	8.64000	0.635226
$186$	0	0
$187$	3.20996	0.234735
$188$	0	0
$189$	−5.50165	−0.400186
$190$	0	0
$191$	19.8934	1.43944	0.719718	−	0.694267i	$-0.244271\pi$
$191$	19.8934	1.43944	0.719718	+	0.694267i	$0.244271\pi$
$192$	0	0
$193$	2.68351	0.193163	0.0965817	−	0.995325i	$-0.469209\pi$
$193$	2.68351	0.193163	0.0965817	+	0.995325i	$0.469209\pi$
$194$	0	0
$195$	3.53365	0.253050
$196$	0	0
$197$	11.8597	0.844968	0.422484	−	0.906370i	$-0.361158\pi$
$197$	11.8597	0.844968	0.422484	+	0.906370i	$0.361158\pi$
$198$	0	0
$199$	−16.0890	−1.14052	−0.570261	−	0.821463i	$-0.693158\pi$
$199$	−16.0890	−1.14052	−0.570261	+	0.821463i	$0.693158\pi$
$200$	0	0
$201$	0.584791	0.0412480
$202$	0	0
$203$	−10.5557	−0.740867
$204$	0	0
$205$	−11.1105	−0.775992
$206$	0	0
$207$	−10.8751	−0.755872
$208$	0	0
$209$	−0.338048	−0.0233833
$210$	0	0
$211$	−8.95441	−0.616447	−0.308224	−	0.951314i	$-0.599734\pi$
$211$	−8.95441	−0.616447	−0.308224	+	0.951314i	$0.599734\pi$
$212$	0	0
$213$	9.31357	0.638155
$214$	0	0
$215$	−6.98949	−0.476679
$216$	0	0
$217$	−1.34712	−0.0914483
$218$	0	0
$219$	6.18641	0.418039
$220$	0	0
$221$	−3.20996	−0.215925
$222$	0	0
$223$	8.41486	0.563501	0.281750	−	0.959488i	$-0.409085\pi$
$223$	8.41486	0.563501	0.281750	+	0.959488i	$0.409085\pi$
$224$	0	0
$225$	−5.17066	−0.344711
$226$	0	0
$227$	10.9637	0.727687	0.363844	−	0.931460i	$-0.381464\pi$
$227$	10.9637	0.727687	0.363844	+	0.931460i	$0.381464\pi$
$228$	0	0
$229$	13.4275	0.887315	0.443658	−	0.896196i	$-0.353681\pi$
$229$	13.4275	0.887315	0.443658	+	0.896196i	$0.353681\pi$
$230$	0	0
$231$	1.21838	0.0801633
$232$	0	0
$233$	18.8452	1.23459	0.617294	−	0.786732i	$-0.288229\pi$
$233$	18.8452	1.23459	0.617294	+	0.786732i	$0.288229\pi$
$234$	0	0
$235$	−35.1053	−2.29001
$236$	0	0
$237$	−1.36023	−0.0883562
$238$	0	0
$239$	−26.9165	−1.74109	−0.870543	−	0.492093i	$-0.836232\pi$
$239$	−26.9165	−1.74109	−0.870543	+	0.492093i	$0.836232\pi$
$240$	0	0
$241$	17.9783	1.15809	0.579043	−	0.815297i	$-0.303426\pi$
$241$	17.9783	1.15809	0.579043	+	0.815297i	$0.303426\pi$
$242$	0	0
$243$	−13.8776	−0.890251
$244$	0	0
$245$	2.90030	0.185293
$246$	0	0
$247$	0.338048	0.0215095
$248$	0	0
$249$	−9.32497	−0.590946
$250$	0	0
$251$	17.4480	1.10131	0.550653	−	0.834734i	$-0.314379\pi$
$251$	17.4480	1.10131	0.550653	+	0.834734i	$0.314379\pi$
$252$	0	0
$253$	7.17564	0.451128
$254$	0	0
$255$	−11.3429	−0.710318
$256$	0	0
$257$	−16.0870	−1.00348	−0.501740	−	0.865019i	$-0.667307\pi$
$257$	−16.0870	−1.00348	−0.501740	+	0.865019i	$0.667307\pi$
$258$	0	0
$259$	−2.97901	−0.185107
$260$	0	0
$261$	−15.9978	−0.990241
$262$	0	0
$263$	−8.61980	−0.531520	−0.265760	−	0.964039i	$-0.585623\pi$
$263$	−8.61980	−0.531520	−0.265760	+	0.964039i	$0.585623\pi$
$264$	0	0
$265$	5.13738	0.315587
$266$	0	0
$267$	−19.1689	−1.17312
$268$	0	0
$269$	7.93841	0.484013	0.242007	−	0.970275i	$-0.422194\pi$
$269$	7.93841	0.484013	0.242007	+	0.970275i	$0.422194\pi$
$270$	0	0
$271$	27.9301	1.69663	0.848315	−	0.529492i	$-0.177617\pi$
$271$	27.9301	1.69663	0.848315	+	0.529492i	$0.177617\pi$
$272$	0	0
$273$	−1.21838	−0.0737395
$274$	0	0
$275$	3.41172	0.205734
$276$	0	0
$277$	18.6275	1.11922	0.559609	−	0.828757i	$-0.310951\pi$
$277$	18.6275	1.11922	0.559609	+	0.828757i	$0.310951\pi$
$278$	0	0
$279$	−2.04164	−0.122230
$280$	0	0
$281$	15.9773	0.953124	0.476562	−	0.879141i	$-0.341883\pi$
$281$	15.9773	0.953124	0.476562	+	0.879141i	$0.341883\pi$
$282$	0	0
$283$	28.0250	1.66592	0.832958	−	0.553336i	$-0.186646\pi$
$283$	28.0250	1.66592	0.832958	+	0.553336i	$0.186646\pi$
$284$	0	0
$285$	1.19454	0.0707587
$286$	0	0
$287$	3.83082	0.226126
$288$	0	0
$289$	−6.69617	−0.393893
$290$	0	0
$291$	−20.6777	−1.21215
$292$	0	0
$293$	1.21281	0.0708530	0.0354265	−	0.999372i	$-0.488721\pi$
$293$	1.21281	0.0708530	0.0354265	+	0.999372i	$0.488721\pi$
$294$	0	0
$295$	−18.4031	−1.07147
$296$	0	0
$297$	5.50165	0.319238
$298$	0	0
$299$	−7.17564	−0.414978
$300$	0	0
$301$	2.40992	0.138906
$302$	0	0
$303$	−10.8506	−0.623351
$304$	0	0
$305$	−19.0014	−1.08802
$306$	0	0
$307$	−15.8553	−0.904910	−0.452455	−	0.891787i	$-0.649452\pi$
$307$	−15.8553	−0.904910	−0.452455	+	0.891787i	$0.649452\pi$
$308$	0	0
$309$	15.6096	0.887997
$310$	0	0
$311$	−23.4291	−1.32854	−0.664271	−	0.747492i	$-0.731258\pi$
$311$	−23.4291	−1.32854	−0.664271	+	0.747492i	$0.731258\pi$
$312$	0	0
$313$	−29.3280	−1.65772	−0.828859	−	0.559458i	$-0.811009\pi$
$313$	−29.3280	−1.65772	−0.828859	+	0.559458i	$0.811009\pi$
$314$	0	0
$315$	4.39557	0.247662
$316$	0	0
$317$	−7.88166	−0.442678	−0.221339	−	0.975197i	$-0.571043\pi$
$317$	−7.88166	−0.442678	−0.221339	+	0.975197i	$0.571043\pi$
$318$	0	0
$319$	10.5557	0.591008
$320$	0	0
$321$	11.8801	0.663083
$322$	0	0
$323$	−1.08512	−0.0603777
$324$	0	0
$325$	−3.41172	−0.189248
$326$	0	0
$327$	−5.97782	−0.330574
$328$	0	0
$329$	12.1040	0.667316
$330$	0	0
$331$	10.0779	0.553930	0.276965	−	0.960880i	$-0.410671\pi$
$331$	10.0779	0.553930	0.276965	+	0.960880i	$0.410671\pi$
$332$	0	0
$333$	−4.51486	−0.247413
$334$	0	0
$335$	−1.39207	−0.0760570
$336$	0	0
$337$	−18.8753	−1.02820	−0.514102	−	0.857729i	$-0.671875\pi$
$337$	−18.8753	−1.02820	−0.514102	+	0.857729i	$0.671875\pi$
$338$	0	0
$339$	16.1914	0.879396
$340$	0	0
$341$	1.34712	0.0729505
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	−25.3562	−1.36513
$346$	0	0
$347$	18.3631	0.985784	0.492892	−	0.870091i	$-0.335940\pi$
$347$	18.3631	0.985784	0.492892	+	0.870091i	$0.335940\pi$
$348$	0	0
$349$	−2.22415	−0.119056	−0.0595281	−	0.998227i	$-0.518960\pi$
$349$	−2.22415	−0.119056	−0.0595281	+	0.998227i	$0.518960\pi$
$350$	0	0
$351$	−5.50165	−0.293656
$352$	0	0
$353$	5.40051	0.287440	0.143720	−	0.989618i	$-0.454093\pi$
$353$	5.40051	0.287440	0.143720	+	0.989618i	$0.454093\pi$
$354$	0	0
$355$	−22.1706	−1.17669
$356$	0	0
$357$	3.91094	0.206989
$358$	0	0
$359$	−24.5262	−1.29444	−0.647222	−	0.762302i	$-0.724069\pi$
$359$	−24.5262	−1.29444	−0.647222	+	0.762302i	$0.724069\pi$
$360$	0	0
$361$	−18.8857	−0.993985
$362$	0	0
$363$	−1.21838	−0.0639482
$364$	0	0
$365$	−14.7265	−0.770820
$366$	0	0
$367$	29.7524	1.55306	0.776532	−	0.630077i	$-0.216977\pi$
$367$	29.7524	1.55306	0.776532	+	0.630077i	$0.216977\pi$
$368$	0	0
$369$	5.80583	0.302240
$370$	0	0
$371$	−1.77133	−0.0919629
$372$	0	0
$373$	25.3278	1.31142	0.655711	−	0.755012i	$-0.272369\pi$
$373$	25.3278	1.31142	0.655711	+	0.755012i	$0.272369\pi$
$374$	0	0
$375$	5.61244	0.289825
$376$	0	0
$377$	−10.5557	−0.543648
$378$	0	0
$379$	−12.3986	−0.636872	−0.318436	−	0.947944i	$-0.603158\pi$
$379$	−12.3986	−0.636872	−0.318436	+	0.947944i	$0.603158\pi$
$380$	0	0
$381$	−8.12137	−0.416070
$382$	0	0
$383$	19.6711	1.00515	0.502574	−	0.864534i	$-0.332387\pi$
$383$	19.6711	1.00515	0.502574	+	0.864534i	$0.332387\pi$
$384$	0	0
$385$	−2.90030	−0.147813
$386$	0	0
$387$	3.65238	0.185661
$388$	0	0
$389$	9.16212	0.464538	0.232269	−	0.972652i	$-0.425385\pi$
$389$	9.16212	0.464538	0.232269	+	0.972652i	$0.425385\pi$
$390$	0	0
$391$	23.0335	1.16485
$392$	0	0
$393$	−10.6929	−0.539385
$394$	0	0
$395$	3.23796	0.162920
$396$	0	0
$397$	11.0924	0.556713	0.278357	−	0.960478i	$-0.410210\pi$
$397$	11.0924	0.556713	0.278357	+	0.960478i	$0.410210\pi$
$398$	0	0
$399$	−0.411870	−0.0206193
$400$	0	0
$401$	21.1617	1.05676	0.528382	−	0.849006i	$-0.322799\pi$
$401$	21.1617	1.05676	0.528382	+	0.849006i	$0.322799\pi$
$402$	0	0
$403$	−1.34712	−0.0671047
$404$	0	0
$405$	−6.25422	−0.310775
$406$	0	0
$407$	2.97901	0.147664
$408$	0	0
$409$	39.4751	1.95192	0.975959	−	0.217955i	$-0.0699385\pi$
$409$	39.4751	1.95192	0.975959	+	0.217955i	$0.0699385\pi$
$410$	0	0
$411$	−11.3215	−0.558450
$412$	0	0
$413$	6.34524	0.312229
$414$	0	0
$415$	22.1977	1.08964
$416$	0	0
$417$	6.75362	0.330726
$418$	0	0
$419$	27.8391	1.36003	0.680016	−	0.733198i	$-0.261973\pi$
$419$	27.8391	1.36003	0.680016	+	0.733198i	$0.261973\pi$
$420$	0	0
$421$	19.0248	0.927211	0.463606	−	0.886042i	$-0.346555\pi$
$421$	19.0248	0.927211	0.463606	+	0.886042i	$0.346555\pi$
$422$	0	0
$423$	18.3444	0.891933
$424$	0	0
$425$	10.9515	0.531224
$426$	0	0
$427$	6.55153	0.317051
$428$	0	0
$429$	1.21838	0.0588238
$430$	0	0
$431$	−5.54985	−0.267327	−0.133664	−	0.991027i	$-0.542674\pi$
$431$	−5.54985	−0.267327	−0.133664	+	0.991027i	$0.542674\pi$
$432$	0	0
$433$	15.7316	0.756010	0.378005	−	0.925803i	$-0.376610\pi$
$433$	15.7316	0.756010	0.378005	+	0.925803i	$0.376610\pi$
$434$	0	0
$435$	−37.3003	−1.78841
$436$	0	0
$437$	−2.42571	−0.116037
$438$	0	0
$439$	8.98296	0.428733	0.214367	−	0.976753i	$-0.431231\pi$
$439$	8.98296	0.428733	0.214367	+	0.976753i	$0.431231\pi$
$440$	0	0
$441$	−1.51556	−0.0721695
$442$	0	0
$443$	−4.35431	−0.206879	−0.103440	−	0.994636i	$-0.532985\pi$
$443$	−4.35431	−0.206879	−0.103440	+	0.994636i	$0.532985\pi$
$444$	0	0
$445$	45.6308	2.16311
$446$	0	0
$447$	4.20181	0.198739
$448$	0	0
$449$	24.4540	1.15406	0.577028	−	0.816724i	$-0.304212\pi$
$449$	24.4540	1.15406	0.577028	+	0.816724i	$0.304212\pi$
$450$	0	0
$451$	−3.83082	−0.180386
$452$	0	0
$453$	−24.7445	−1.16260
$454$	0	0
$455$	2.90030	0.135968
$456$	0	0
$457$	23.9544	1.12054	0.560269	−	0.828311i	$-0.310698\pi$
$457$	23.9544	1.12054	0.560269	+	0.828311i	$0.310698\pi$
$458$	0	0
$459$	17.6601	0.824301
$460$	0	0
$461$	−40.4206	−1.88257	−0.941287	−	0.337608i	$-0.890382\pi$
$461$	−40.4206	−1.88257	−0.941287	+	0.337608i	$0.890382\pi$
$462$	0	0
$463$	13.9033	0.646141	0.323071	−	0.946375i	$-0.395285\pi$
$463$	13.9033	0.646141	0.323071	+	0.946375i	$0.395285\pi$
$464$	0	0
$465$	−4.76025	−0.220751
$466$	0	0
$467$	3.41389	0.157976	0.0789880	−	0.996876i	$-0.474831\pi$
$467$	3.41389	0.157976	0.0789880	+	0.996876i	$0.474831\pi$
$468$	0	0
$469$	0.479975	0.0221632
$470$	0	0
$471$	−17.0636	−0.786250
$472$	0	0
$473$	−2.40992	−0.110808
$474$	0	0
$475$	−1.15332	−0.0529182
$476$	0	0
$477$	−2.68456	−0.122917
$478$	0	0
$479$	29.0810	1.32874	0.664371	−	0.747403i	$-0.268699\pi$
$479$	29.0810	1.32874	0.664371	+	0.747403i	$0.268699\pi$
$480$	0	0
$481$	−2.97901	−0.135831
$482$	0	0
$483$	8.74263	0.397803
$484$	0	0
$485$	49.2224	2.23507
$486$	0	0
$487$	−40.9117	−1.85389	−0.926944	−	0.375201i	$-0.877574\pi$
$487$	−40.9117	−1.85389	−0.926944	+	0.375201i	$0.877574\pi$
$488$	0	0
$489$	−28.2731	−1.27855
$490$	0	0
$491$	21.8139	0.984447	0.492223	−	0.870469i	$-0.336184\pi$
$491$	21.8139	0.984447	0.492223	+	0.870469i	$0.336184\pi$
$492$	0	0
$493$	33.8835	1.52603
$494$	0	0
$495$	−4.39557	−0.197566
$496$	0	0
$497$	7.64425	0.342891
$498$	0	0
$499$	3.60399	0.161337	0.0806684	−	0.996741i	$-0.474295\pi$
$499$	3.60399	0.161337	0.0806684	+	0.996741i	$0.474295\pi$
$500$	0	0
$501$	23.6982	1.05876
$502$	0	0
$503$	−40.9551	−1.82610	−0.913049	−	0.407850i	$-0.866279\pi$
$503$	−40.9551	−1.82610	−0.913049	+	0.407850i	$0.866279\pi$
$504$	0	0
$505$	25.8294	1.14939
$506$	0	0
$507$	−1.21838	−0.0541100
$508$	0	0
$509$	−25.9845	−1.15174	−0.575872	−	0.817540i	$-0.695337\pi$
$509$	−25.9845	−1.15174	−0.575872	+	0.817540i	$0.695337\pi$
$510$	0	0
$511$	5.07758	0.224619
$512$	0	0
$513$	−1.85982	−0.0821132
$514$	0	0
$515$	−37.1579	−1.63737
$516$	0	0
$517$	−12.1040	−0.532334
$518$	0	0
$519$	−16.2309	−0.712456
$520$	0	0
$521$	4.37871	0.191835	0.0959173	−	0.995389i	$-0.469422\pi$
$521$	4.37871	0.191835	0.0959173	+	0.995389i	$0.469422\pi$
$522$	0	0
$523$	−39.8178	−1.74111	−0.870556	−	0.492070i	$-0.836240\pi$
$523$	−39.8178	−1.74111	−0.870556	+	0.492070i	$0.836240\pi$
$524$	0	0
$525$	4.15676	0.181416
$526$	0	0
$527$	4.32419	0.188365
$528$	0	0
$529$	28.4898	1.23869
$530$	0	0
$531$	9.61659	0.417324
$532$	0	0
$533$	3.83082	0.165931
$534$	0	0
$535$	−28.2801	−1.22266
$536$	0	0
$537$	19.2261	0.829669
$538$	0	0
$539$	1.00000	0.0430730
$540$	0	0
$541$	10.7361	0.461581	0.230790	−	0.973003i	$-0.425869\pi$
$541$	10.7361	0.461581	0.230790	+	0.973003i	$0.425869\pi$
$542$	0	0
$543$	2.49234	0.106957
$544$	0	0
$545$	14.2299	0.609544
$546$	0	0
$547$	32.5328	1.39100	0.695500	−	0.718526i	$-0.255183\pi$
$547$	32.5328	1.39100	0.695500	+	0.718526i	$0.255183\pi$
$548$	0	0
$549$	9.92923	0.423769
$550$	0	0
$551$	−3.56835	−0.152017
$552$	0	0
$553$	−1.11643	−0.0474752
$554$	0	0
$555$	−10.5268	−0.446837
$556$	0	0
$557$	−14.5348	−0.615859	−0.307930	−	0.951409i	$-0.599636\pi$
$557$	−14.5348	−0.615859	−0.307930	+	0.951409i	$0.599636\pi$
$558$	0	0
$559$	2.40992	0.101929
$560$	0	0
$561$	−3.91094	−0.165120
$562$	0	0
$563$	10.6786	0.450050	0.225025	−	0.974353i	$-0.427754\pi$
$563$	10.6786	0.450050	0.225025	+	0.974353i	$0.427754\pi$
$564$	0	0
$565$	−38.5430	−1.62152
$566$	0	0
$567$	2.15641	0.0905605
$568$	0	0
$569$	10.7695	0.451481	0.225740	−	0.974187i	$-0.427520\pi$
$569$	10.7695	0.451481	0.225740	+	0.974187i	$0.427520\pi$
$570$	0	0
$571$	−18.0492	−0.755334	−0.377667	−	0.925941i	$-0.623274\pi$
$571$	−18.0492	−0.755334	−0.377667	+	0.925941i	$0.623274\pi$
$572$	0	0
$573$	−24.2376	−1.01254
$574$	0	0
$575$	24.4813	1.02094
$576$	0	0
$577$	−17.9289	−0.746388	−0.373194	−	0.927753i	$-0.621737\pi$
$577$	−17.9289	−0.746388	−0.373194	+	0.927753i	$0.621737\pi$
$578$	0	0
$579$	−3.26953	−0.135877
$580$	0	0
$581$	−7.65360	−0.317525
$582$	0	0
$583$	1.77133	0.0733610
$584$	0	0
$585$	4.39557	0.181734
$586$	0	0
$587$	−20.0383	−0.827069	−0.413534	−	0.910488i	$-0.635706\pi$
$587$	−20.0383	−0.827069	−0.413534	+	0.910488i	$0.635706\pi$
$588$	0	0
$589$	−0.455391	−0.0187640
$590$	0	0
$591$	−14.4496	−0.594376
$592$	0	0
$593$	−24.5002	−1.00610	−0.503050	−	0.864257i	$-0.667789\pi$
$593$	−24.5002	−1.00610	−0.503050	+	0.864257i	$0.667789\pi$
$594$	0	0
$595$	−9.30983	−0.381666
$596$	0	0
$597$	19.6025	0.802278
$598$	0	0
$599$	12.6370	0.516335	0.258168	−	0.966100i	$-0.416881\pi$
$599$	12.6370	0.516335	0.258168	+	0.966100i	$0.416881\pi$
$600$	0	0
$601$	1.87669	0.0765517	0.0382759	−	0.999267i	$-0.487813\pi$
$601$	1.87669	0.0765517	0.0382759	+	0.999267i	$0.487813\pi$
$602$	0	0
$603$	0.727431	0.0296233
$604$	0	0
$605$	2.90030	0.117914
$606$	0	0
$607$	25.9699	1.05408	0.527042	−	0.849839i	$-0.323301\pi$
$607$	25.9699	1.05408	0.527042	+	0.849839i	$0.323301\pi$
$608$	0	0
$609$	12.8609	0.521148
$610$	0	0
$611$	12.1040	0.489676
$612$	0	0
$613$	−23.2858	−0.940505	−0.470252	−	0.882532i	$-0.655837\pi$
$613$	−23.2858	−0.940505	−0.470252	+	0.882532i	$0.655837\pi$
$614$	0	0
$615$	13.5368	0.545856
$616$	0	0
$617$	14.2406	0.573306	0.286653	−	0.958034i	$-0.407457\pi$
$617$	14.2406	0.573306	0.286653	+	0.958034i	$0.407457\pi$
$618$	0	0
$619$	30.9387	1.24353	0.621766	−	0.783203i	$-0.286416\pi$
$619$	30.9387	1.24353	0.621766	+	0.783203i	$0.286416\pi$
$620$	0	0
$621$	39.4778	1.58419
$622$	0	0
$623$	−15.7332	−0.630336
$624$	0	0
$625$	−30.4188	−1.21675
$626$	0	0
$627$	0.411870	0.0164485
$628$	0	0
$629$	9.56249	0.381281
$630$	0	0
$631$	−2.77785	−0.110585	−0.0552923	−	0.998470i	$-0.517609\pi$
$631$	−2.77785	−0.110585	−0.0552923	+	0.998470i	$0.517609\pi$
$632$	0	0
$633$	10.9098	0.433627
$634$	0	0
$635$	19.3326	0.767191
$636$	0	0
$637$	−1.00000	−0.0396214
$638$	0	0
$639$	11.5853	0.458308
$640$	0	0
$641$	−27.1749	−1.07335	−0.536673	−	0.843790i	$-0.680319\pi$
$641$	−27.1749	−1.07335	−0.536673	+	0.843790i	$0.680319\pi$
$642$	0	0
$643$	4.95387	0.195362	0.0976808	−	0.995218i	$-0.468858\pi$
$643$	4.95387	0.195362	0.0976808	+	0.995218i	$0.468858\pi$
$644$	0	0
$645$	8.51583	0.335311
$646$	0	0
$647$	32.6204	1.28244	0.641220	−	0.767357i	$-0.278429\pi$
$647$	32.6204	1.28244	0.641220	+	0.767357i	$0.278429\pi$
$648$	0	0
$649$	−6.34524	−0.249073
$650$	0	0
$651$	1.64130	0.0643275
$652$	0	0
$653$	−2.88950	−0.113075	−0.0565374	−	0.998400i	$-0.518006\pi$
$653$	−2.88950	−0.113075	−0.0565374	+	0.998400i	$0.518006\pi$
$654$	0	0
$655$	25.4540	0.994569
$656$	0	0
$657$	7.69537	0.300225
$658$	0	0
$659$	−2.94885	−0.114871	−0.0574355	−	0.998349i	$-0.518292\pi$
$659$	−2.94885	−0.114871	−0.0574355	+	0.998349i	$0.518292\pi$
$660$	0	0
$661$	−28.9628	−1.12652	−0.563261	−	0.826279i	$-0.690454\pi$
$661$	−28.9628	−1.12652	−0.563261	+	0.826279i	$0.690454\pi$
$662$	0	0
$663$	3.91094	0.151888
$664$	0	0
$665$	0.980440	0.0380198
$666$	0	0
$667$	75.7441	2.93282
$668$	0	0
$669$	−10.2525	−0.396383
$670$	0	0
$671$	−6.55153	−0.252919
$672$	0	0
$673$	4.52456	0.174409	0.0872045	−	0.996190i	$-0.472207\pi$
$673$	4.52456	0.174409	0.0872045	+	0.996190i	$0.472207\pi$
$674$	0	0
$675$	18.7701	0.722461
$676$	0	0
$677$	−9.94320	−0.382148	−0.191074	−	0.981576i	$-0.561197\pi$
$677$	−9.94320	−0.382148	−0.191074	+	0.981576i	$0.561197\pi$
$678$	0	0
$679$	−16.9715	−0.651307
$680$	0	0
$681$	−13.3579	−0.511877
$682$	0	0
$683$	36.2544	1.38723	0.693617	−	0.720344i	$-0.256016\pi$
$683$	36.2544	1.38723	0.693617	+	0.720344i	$0.256016\pi$
$684$	0	0
$685$	26.9504	1.02972
$686$	0	0
$687$	−16.3598	−0.624164
$688$	0	0
$689$	−1.77133	−0.0674823
$690$	0	0
$691$	−45.4198	−1.72785	−0.863926	−	0.503619i	$-0.832002\pi$
$691$	−45.4198	−1.72785	−0.863926	+	0.503619i	$0.832002\pi$
$692$	0	0
$693$	1.51556	0.0575713
$694$	0	0
$695$	−16.0767	−0.609825
$696$	0	0
$697$	−12.2968	−0.465773
$698$	0	0
$699$	−22.9605	−0.868447
$700$	0	0
$701$	−25.0214	−0.945045	−0.472523	−	0.881319i	$-0.656657\pi$
$701$	−25.0214	−0.945045	−0.472523	+	0.881319i	$0.656657\pi$
$702$	0	0
$703$	−1.00705	−0.0379815
$704$	0	0
$705$	42.7714	1.61086
$706$	0	0
$707$	−8.90578	−0.334936
$708$	0	0
$709$	−49.3904	−1.85490	−0.927448	−	0.373952i	$-0.878003\pi$
$709$	−49.3904	−1.85490	−0.927448	+	0.373952i	$0.878003\pi$
$710$	0	0
$711$	−1.69201	−0.0634552
$712$	0	0
$713$	9.66643	0.362011
$714$	0	0
$715$	−2.90030	−0.108465
$716$	0	0
$717$	32.7945	1.22473
$718$	0	0
$719$	0.951868	0.0354987	0.0177493	−	0.999842i	$-0.494350\pi$
$719$	0.951868	0.0354987	0.0177493	+	0.999842i	$0.494350\pi$
$720$	0	0
$721$	12.8118	0.477135
$722$	0	0
$723$	−21.9044	−0.814633
$724$	0	0
$725$	36.0132	1.33750
$726$	0	0
$727$	−22.3584	−0.829229	−0.414614	−	0.909997i	$-0.636083\pi$
$727$	−22.3584	−0.829229	−0.414614	+	0.909997i	$0.636083\pi$
$728$	0	0
$729$	23.3774	0.865830
$730$	0	0
$731$	−7.73575	−0.286117
$732$	0	0
$733$	−33.9992	−1.25579	−0.627895	−	0.778298i	$-0.716083\pi$
$733$	−33.9992	−1.25579	−0.627895	+	0.778298i	$0.716083\pi$
$734$	0	0
$735$	−3.53365	−0.130341
$736$	0	0
$737$	−0.479975	−0.0176801
$738$	0	0
$739$	33.6754	1.23877	0.619384	−	0.785088i	$-0.287382\pi$
$739$	33.6754	1.23877	0.619384	+	0.785088i	$0.287382\pi$
$740$	0	0
$741$	−0.411870	−0.0151304
$742$	0	0
$743$	21.7180	0.796755	0.398378	−	0.917221i	$-0.369573\pi$
$743$	21.7180	0.796755	0.398378	+	0.917221i	$0.369573\pi$
$744$	0	0
$745$	−10.0022	−0.366453
$746$	0	0
$747$	−11.5995	−0.424403
$748$	0	0
$749$	9.75077	0.356285
$750$	0	0
$751$	1.64775	0.0601272	0.0300636	−	0.999548i	$-0.490429\pi$
$751$	1.64775	0.0601272	0.0300636	+	0.999548i	$0.490429\pi$
$752$	0	0
$753$	−21.2582	−0.774692
$754$	0	0
$755$	58.9033	2.14371
$756$	0	0
$757$	−7.58858	−0.275812	−0.137906	−	0.990445i	$-0.544037\pi$
$757$	−7.58858	−0.275812	−0.137906	+	0.990445i	$0.544037\pi$
$758$	0	0
$759$	−8.74263	−0.317337
$760$	0	0
$761$	−0.206389	−0.00748159	−0.00374079	−	0.999993i	$-0.501191\pi$
$761$	−0.206389	−0.00748159	−0.00374079	+	0.999993i	$0.501191\pi$
$762$	0	0
$763$	−4.90638	−0.177623
$764$	0	0
$765$	−14.1096	−0.510133
$766$	0	0
$767$	6.34524	0.229113
$768$	0	0
$769$	12.1132	0.436812	0.218406	−	0.975858i	$-0.429914\pi$
$769$	12.1132	0.436812	0.218406	+	0.975858i	$0.429914\pi$
$770$	0	0
$771$	19.6000	0.705878
$772$	0	0
$773$	44.7549	1.60972	0.804862	−	0.593462i	$-0.202239\pi$
$773$	44.7549	1.60972	0.804862	+	0.593462i	$0.202239\pi$
$774$	0	0
$775$	4.59599	0.165093
$776$	0	0
$777$	3.62955	0.130209
$778$	0	0
$779$	1.29500	0.0463982
$780$	0	0
$781$	−7.64425	−0.273533
$782$	0	0
$783$	58.0740	2.07539
$784$	0	0
$785$	40.6193	1.44976
$786$	0	0
$787$	27.7427	0.988919	0.494460	−	0.869201i	$-0.335366\pi$
$787$	27.7427	0.988919	0.494460	+	0.869201i	$0.335366\pi$
$788$	0	0
$789$	10.5022	0.373887
$790$	0	0
$791$	13.2893	0.472514
$792$	0	0
$793$	6.55153	0.232652
$794$	0	0
$795$	−6.25927	−0.221993
$796$	0	0
$797$	−31.1866	−1.10469	−0.552343	−	0.833617i	$-0.686266\pi$
$797$	−31.1866	−1.10469	−0.552343	+	0.833617i	$0.686266\pi$
$798$	0	0
$799$	−38.8534	−1.37453
$800$	0	0
$801$	−23.8445	−0.842505
$802$	0	0
$803$	−5.07758	−0.179184
$804$	0	0
$805$	−20.8115	−0.733508
$806$	0	0
$807$	−9.67197	−0.340469
$808$	0	0
$809$	47.8068	1.68080	0.840399	−	0.541969i	$-0.182321\pi$
$809$	47.8068	1.68080	0.840399	+	0.541969i	$0.182321\pi$
$810$	0	0
$811$	−40.5500	−1.42390	−0.711951	−	0.702229i	$-0.752188\pi$
$811$	−40.5500	−1.42390	−0.711951	+	0.702229i	$0.752188\pi$
$812$	0	0
$813$	−34.0293	−1.19346
$814$	0	0
$815$	67.3029	2.35752
$816$	0	0
$817$	0.814670	0.0285017
$818$	0	0
$819$	−1.51556	−0.0529579
$820$	0	0
$821$	20.5251	0.716329	0.358165	−	0.933658i	$-0.383403\pi$
$821$	20.5251	0.716329	0.358165	+	0.933658i	$0.383403\pi$
$822$	0	0
$823$	−47.5837	−1.65866	−0.829332	−	0.558757i	$-0.811279\pi$
$823$	−47.5837	−1.65866	−0.829332	+	0.558757i	$0.811279\pi$
$824$	0	0
$825$	−4.15676	−0.144720
$826$	0	0
$827$	10.2056	0.354882	0.177441	−	0.984131i	$-0.443218\pi$
$827$	10.2056	0.354882	0.177441	+	0.984131i	$0.443218\pi$
$828$	0	0
$829$	−36.1405	−1.25521	−0.627606	−	0.778531i	$-0.715965\pi$
$829$	−36.1405	−1.25521	−0.627606	+	0.778531i	$0.715965\pi$
$830$	0	0
$831$	−22.6953	−0.787291
$832$	0	0
$833$	3.20996	0.111218
$834$	0	0
$835$	−56.4127	−1.95224
$836$	0	0
$837$	7.41137	0.256175
$838$	0	0
$839$	10.5312	0.363576	0.181788	−	0.983338i	$-0.441812\pi$
$839$	10.5312	0.363576	0.181788	+	0.983338i	$0.441812\pi$
$840$	0	0
$841$	82.4235	2.84219
$842$	0	0
$843$	−19.4663	−0.670456
$844$	0	0
$845$	2.90030	0.0997732
$846$	0	0
$847$	−1.00000	−0.0343604
$848$	0	0
$849$	−34.1451	−1.17186
$850$	0	0
$851$	21.3763	0.732769
$852$	0	0
$853$	−33.2746	−1.13930	−0.569651	−	0.821887i	$-0.692922\pi$
$853$	−33.2746	−1.13930	−0.569651	+	0.821887i	$0.692922\pi$
$854$	0	0
$855$	1.48591	0.0508172
$856$	0	0
$857$	−32.9235	−1.12465	−0.562323	−	0.826917i	$-0.690092\pi$
$857$	−32.9235	−1.12465	−0.562323	+	0.826917i	$0.690092\pi$
$858$	0	0
$859$	−33.8803	−1.15598	−0.577991	−	0.816043i	$-0.696163\pi$
$859$	−33.8803	−1.15598	−0.577991	+	0.816043i	$0.696163\pi$
$860$	0	0
$861$	−4.66738	−0.159064
$862$	0	0
$863$	−11.6252	−0.395727	−0.197864	−	0.980230i	$-0.563400\pi$
$863$	−11.6252	−0.395727	−0.197864	+	0.980230i	$0.563400\pi$
$864$	0	0
$865$	38.6369	1.31370
$866$	0	0
$867$	8.15846	0.277076
$868$	0	0
$869$	1.11643	0.0378721
$870$	0	0
$871$	0.479975	0.0162633
$872$	0	0
$873$	−25.7213	−0.870535
$874$	0	0
$875$	4.60649	0.155728
$876$	0	0
$877$	50.0458	1.68993	0.844964	−	0.534823i	$-0.179622\pi$
$877$	50.0458	1.68993	0.844964	+	0.534823i	$0.179622\pi$
$878$	0	0
$879$	−1.47766	−0.0498402
$880$	0	0
$881$	−28.0109	−0.943710	−0.471855	−	0.881676i	$-0.656415\pi$
$881$	−28.0109	−0.943710	−0.471855	+	0.881676i	$0.656415\pi$
$882$	0	0
$883$	13.6504	0.459373	0.229686	−	0.973265i	$-0.426230\pi$
$883$	13.6504	0.459373	0.229686	+	0.973265i	$0.426230\pi$
$884$	0	0
$885$	22.4219	0.753704
$886$	0	0
$887$	−10.9354	−0.367173	−0.183587	−	0.983004i	$-0.558771\pi$
$887$	−10.9354	−0.367173	−0.183587	+	0.983004i	$0.558771\pi$
$888$	0	0
$889$	−6.66573	−0.223561
$890$	0	0
$891$	−2.15641	−0.0722423
$892$	0	0
$893$	4.09174	0.136925
$894$	0	0
$895$	−45.7670	−1.52982
$896$	0	0
$897$	8.74263	0.291908
$898$	0	0
$899$	14.2198	0.474258
$900$	0	0
$901$	5.68590	0.189425
$902$	0	0
$903$	−2.93619	−0.0977104
$904$	0	0
$905$	−5.93292	−0.197217
$906$	0	0
$907$	25.9696	0.862305	0.431152	−	0.902279i	$-0.358107\pi$
$907$	25.9696	0.862305	0.431152	+	0.902279i	$0.358107\pi$
$908$	0	0
$909$	−13.4972	−0.447675
$910$	0	0
$911$	−22.7039	−0.752213	−0.376106	−	0.926576i	$-0.622737\pi$
$911$	−22.7039	−0.752213	−0.376106	+	0.926576i	$0.622737\pi$
$912$	0	0
$913$	7.65360	0.253297
$914$	0	0
$915$	23.1508	0.765343
$916$	0	0
$917$	−8.77634	−0.289820
$918$	0	0
$919$	35.0952	1.15768	0.578841	−	0.815440i	$-0.303505\pi$
$919$	35.0952	1.15768	0.578841	+	0.815440i	$0.303505\pi$
$920$	0	0
$921$	19.3177	0.636541
$922$	0	0
$923$	7.64425	0.251613
$924$	0	0
$925$	10.1635	0.334175
$926$	0	0
$927$	19.4170	0.637738
$928$	0	0
$929$	−49.6592	−1.62927	−0.814633	−	0.579977i	$-0.803062\pi$
$929$	−49.6592	−1.62927	−0.814633	+	0.579977i	$0.803062\pi$
$930$	0	0
$931$	−0.338048	−0.0110791
$932$	0	0
$933$	28.5455	0.934537
$934$	0	0
$935$	9.30983	0.304464
$936$	0	0
$937$	−10.6784	−0.348847	−0.174423	−	0.984671i	$-0.555806\pi$
$937$	−10.6784	−0.348847	−0.174423	+	0.984671i	$0.555806\pi$
$938$	0	0
$939$	35.7326	1.16609
$940$	0	0
$941$	46.1623	1.50485	0.752424	−	0.658679i	$-0.228885\pi$
$941$	46.1623	1.50485	0.752424	+	0.658679i	$0.228885\pi$
$942$	0	0
$943$	−27.4886	−0.895151
$944$	0	0
$945$	−15.9564	−0.519062
$946$	0	0
$947$	15.5959	0.506799	0.253400	−	0.967362i	$-0.418451\pi$
$947$	15.5959	0.506799	0.253400	+	0.967362i	$0.418451\pi$
$948$	0	0
$949$	5.07758	0.164825
$950$	0	0
$951$	9.60283	0.311393
$952$	0	0
$953$	−13.3604	−0.432786	−0.216393	−	0.976306i	$-0.569429\pi$
$953$	−13.3604	−0.432786	−0.216393	+	0.976306i	$0.569429\pi$
$954$	0	0
$955$	57.6967	1.86702
$956$	0	0
$957$	−12.8609	−0.415733
$958$	0	0
$959$	−9.29231	−0.300064
$960$	0	0
$961$	−29.1853	−0.941460
$962$	0	0
$963$	14.7779	0.476210
$964$	0	0
$965$	7.78298	0.250543
$966$	0	0
$967$	23.8066	0.765569	0.382785	−	0.923838i	$-0.374965\pi$
$967$	23.8066	0.765569	0.382785	+	0.923838i	$0.374965\pi$
$968$	0	0
$969$	1.32208	0.0424715
$970$	0	0
$971$	−6.72845	−0.215926	−0.107963	−	0.994155i	$-0.534433\pi$
$971$	−6.72845	−0.215926	−0.107963	+	0.994155i	$0.534433\pi$
$972$	0	0
$973$	5.54313	0.177705
$974$	0	0
$975$	4.15676	0.133123
$976$	0	0
$977$	39.2128	1.25453	0.627265	−	0.778806i	$-0.284174\pi$
$977$	39.2128	1.25453	0.627265	+	0.778806i	$0.284174\pi$
$978$	0	0
$979$	15.7332	0.502834
$980$	0	0
$981$	−7.43590	−0.237410
$982$	0	0
$983$	18.3795	0.586214	0.293107	−	0.956080i	$-0.405311\pi$
$983$	18.3795	0.586214	0.293107	+	0.956080i	$0.405311\pi$
$984$	0	0
$985$	34.3966	1.09597
$986$	0	0
$987$	−14.7473	−0.469410
$988$	0	0
$989$	−17.2927	−0.549877
$990$	0	0
$991$	27.3735	0.869549	0.434775	−	0.900539i	$-0.356828\pi$
$991$	27.3735	0.869549	0.434775	+	0.900539i	$0.356828\pi$
$992$	0	0
$993$	−12.2787	−0.389651
$994$	0	0
$995$	−46.6630	−1.47932
$996$	0	0
$997$	56.7976	1.79880	0.899399	−	0.437129i	$-0.144005\pi$
$997$	56.7976	1.79880	0.899399	+	0.437129i	$0.144005\pi$
$998$	0	0
$999$	16.3895	0.518540

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8008.2.a.n.1.4	✓	9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8008.2.a.n.1.4	✓	9	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:	\( N \)	\(=\)	\( 8008 = 2^{3} \cdot 7 \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8008.a (trivial)

\(f(q)\)	\(=\)	\(q-1.21838 q^{3} +2.90030 q^{5} -1.00000 q^{7} -1.51556 q^{9} +O(q^{10})\)	\(q-1.21838 q^{3} +2.90030 q^{5} -1.00000 q^{7} -1.51556 q^{9} +1.00000 q^{11} -1.00000 q^{13} -3.53365 q^{15} +3.20996 q^{17} -0.338048 q^{19} +1.21838 q^{21} +7.17564 q^{23} +3.41172 q^{25} +5.50165 q^{27} +10.5557 q^{29} +1.34712 q^{31} -1.21838 q^{33} -2.90030 q^{35} +2.97901 q^{37} +1.21838 q^{39} -3.83082 q^{41} -2.40992 q^{43} -4.39557 q^{45} -12.1040 q^{47} +1.00000 q^{49} -3.91094 q^{51} +1.77133 q^{53} +2.90030 q^{55} +0.411870 q^{57} -6.34524 q^{59} -6.55153 q^{61} +1.51556 q^{63} -2.90030 q^{65} -0.479975 q^{67} -8.74263 q^{69} -7.64425 q^{71} -5.07758 q^{73} -4.15676 q^{75} -1.00000 q^{77} +1.11643 q^{79} -2.15641 q^{81} +7.65360 q^{83} +9.30983 q^{85} -12.8609 q^{87} +15.7332 q^{89} +1.00000 q^{91} -1.64130 q^{93} -0.980440 q^{95} +16.9715 q^{97} -1.51556 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q - 3 q^{3} + q^{5} - 9 q^{7} + 12 q^{9}+O(q^{10}) \) 9 * q - 3 * q^3 + q^5 - 9 * q^7 + 12 * q^9	\( 9 q - 3 q^{3} + q^{5} - 9 q^{7} + 12 q^{9} + 9 q^{11} - 9 q^{13} + 11 q^{15} + 7 q^{17} - 17 q^{19} + 3 q^{21} + 11 q^{23} + 18 q^{25} - 9 q^{27} + 9 q^{29} - 8 q^{31} - 3 q^{33} - q^{35} + 2 q^{37} + 3 q^{39} + 18 q^{41} + 7 q^{43} + 5 q^{45} + 15 q^{47} + 9 q^{49} - 7 q^{51} - 4 q^{53} + q^{55} + 22 q^{57} - 23 q^{59} + 12 q^{61} - 12 q^{63} - q^{65} - 16 q^{67} - 32 q^{69} - 6 q^{71} + 4 q^{73} - 14 q^{75} - 9 q^{77} + 21 q^{79} + 5 q^{81} - 16 q^{83} + 53 q^{85} + 41 q^{87} + 5 q^{89} + 9 q^{91} + 29 q^{93} + 19 q^{95} + 18 q^{97} + 12 q^{99}+O(q^{100}) \) 9 * q - 3 * q^3 + q^5 - 9 * q^7 + 12 * q^9 + 9 * q^11 - 9 * q^13 + 11 * q^15 + 7 * q^17 - 17 * q^19 + 3 * q^21 + 11 * q^23 + 18 * q^25 - 9 * q^27 + 9 * q^29 - 8 * q^31 - 3 * q^33 - q^35 + 2 * q^37 + 3 * q^39 + 18 * q^41 + 7 * q^43 + 5 * q^45 + 15 * q^47 + 9 * q^49 - 7 * q^51 - 4 * q^53 + q^55 + 22 * q^57 - 23 * q^59 + 12 * q^61 - 12 * q^63 - q^65 - 16 * q^67 - 32 * q^69 - 6 * q^71 + 4 * q^73 - 14 * q^75 - 9 * q^77 + 21 * q^79 + 5 * q^81 - 16 * q^83 + 53 * q^85 + 41 * q^87 + 5 * q^89 + 9 * q^91 + 29 * q^93 + 19 * q^95 + 18 * q^97 + 12 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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