LMFDB - Embedded newform 8008.2.a.t.1.7

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:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - x^{9} - 17x^{8} + 13x^{7} + 96x^{6} - 49x^{5} - 207x^{4} + 58x^{3} + 169x^{2} - 12x - 32 $ x^10 - x^9 - 17x^8 + 13x^7 + 96x^6 - 49x^5 - 207x^4 + 58x^3 + 169x^2 - 12x - 32
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$1.37400$ 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)

$f(q)$	$=$	$q+1.37400 q^{3} -0.949904 q^{5} +1.00000 q^{7} -1.11212 q^{9} +O(q^{10})$	$q+1.37400 q^{3} -0.949904 q^{5} +1.00000 q^{7} -1.11212 q^{9} -1.00000 q^{11} +1.00000 q^{13} -1.30517 q^{15} -5.27209 q^{17} -3.79428 q^{19} +1.37400 q^{21} +2.13174 q^{23} -4.09768 q^{25} -5.65006 q^{27} +2.06862 q^{29} -0.698043 q^{31} -1.37400 q^{33} -0.949904 q^{35} +5.10641 q^{37} +1.37400 q^{39} -1.72658 q^{41} +0.353503 q^{43} +1.05641 q^{45} +12.2124 q^{47} +1.00000 q^{49} -7.24386 q^{51} +12.3677 q^{53} +0.949904 q^{55} -5.21334 q^{57} -0.154213 q^{59} +5.42802 q^{61} -1.11212 q^{63} -0.949904 q^{65} +1.39660 q^{67} +2.92901 q^{69} +3.90040 q^{71} +11.5476 q^{73} -5.63022 q^{75} -1.00000 q^{77} +13.0818 q^{79} -4.42683 q^{81} +12.1100 q^{83} +5.00798 q^{85} +2.84228 q^{87} +6.64156 q^{89} +1.00000 q^{91} -0.959112 q^{93} +3.60420 q^{95} +9.20030 q^{97} +1.11212 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + q^{3} + 3 q^{5} + 10 q^{7} + 5 q^{9}+O(q^{10}) $ 10 * q + q^3 + 3 * q^5 + 10 * q^7 + 5 * q^9	$ 10 q + q^{3} + 3 q^{5} + 10 q^{7} + 5 q^{9} - 10 q^{11} + 10 q^{13} + 9 q^{15} + 7 q^{17} + 17 q^{19} + q^{21} + 15 q^{23} + 13 q^{25} + 7 q^{27} - q^{29} - 6 q^{31} - q^{33} + 3 q^{35} - 12 q^{37} + q^{39} + 4 q^{41} + 17 q^{43} - 15 q^{45} + 21 q^{47} + 10 q^{49} + 3 q^{51} + 20 q^{53} - 3 q^{55} + 12 q^{57} + 9 q^{59} + 6 q^{61} + 5 q^{63} + 3 q^{65} + 26 q^{67} + 30 q^{69} + 18 q^{71} - 4 q^{73} + 48 q^{75} - 10 q^{77} + 19 q^{79} - 14 q^{81} + 40 q^{83} - 25 q^{85} - 25 q^{87} - 19 q^{89} + 10 q^{91} + q^{93} + 11 q^{95} - 22 q^{97} - 5 q^{99}+O(q^{100}) $ 10 * q + q^3 + 3 * q^5 + 10 * q^7 + 5 * q^9 - 10 * q^11 + 10 * q^13 + 9 * q^15 + 7 * q^17 + 17 * q^19 + q^21 + 15 * q^23 + 13 * q^25 + 7 * q^27 - q^29 - 6 * q^31 - q^33 + 3 * q^35 - 12 * q^37 + q^39 + 4 * q^41 + 17 * q^43 - 15 * q^45 + 21 * q^47 + 10 * q^49 + 3 * q^51 + 20 * q^53 - 3 * q^55 + 12 * q^57 + 9 * q^59 + 6 * q^61 + 5 * q^63 + 3 * q^65 + 26 * q^67 + 30 * q^69 + 18 * q^71 - 4 * q^73 + 48 * q^75 - 10 * q^77 + 19 * q^79 - 14 * q^81 + 40 * q^83 - 25 * q^85 - 25 * q^87 - 19 * q^89 + 10 * q^91 + q^93 + 11 * q^95 - 22 * q^97 - 5 * 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
$3$	1.37400	0.793280	0.396640	−	0.917974i	$-0.370176\pi$
$3$	1.37400	0.793280	0.396640	+	0.917974i	$0.370176\pi$
$4$	0	0
$5$	−0.949904	−0.424810	−0.212405	−	0.977182i	$-0.568130\pi$
$5$	−0.949904	−0.424810	−0.212405	+	0.977182i	$0.568130\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	−1.11212	−0.370706
$10$	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
$15$	−1.30517	−0.336993
$16$	0	0
$17$	−5.27209	−1.27867	−0.639335	−	0.768928i	$-0.720790\pi$
$17$	−5.27209	−1.27867	−0.639335	+	0.768928i	$0.720790\pi$
$18$	0	0
$19$	−3.79428	−0.870466	−0.435233	−	0.900318i	$-0.643334\pi$
$19$	−3.79428	−0.870466	−0.435233	+	0.900318i	$0.643334\pi$
$20$	0	0
$21$	1.37400	0.299832
$22$	0	0
$23$	2.13174	0.444498	0.222249	−	0.974990i	$-0.428660\pi$
$23$	2.13174	0.444498	0.222249	+	0.974990i	$0.428660\pi$
$24$	0	0
$25$	−4.09768	−0.819536
$26$	0	0
$27$	−5.65006	−1.08735
$28$	0	0
$29$	2.06862	0.384132	0.192066	−	0.981382i	$-0.438481\pi$
$29$	2.06862	0.384132	0.192066	+	0.981382i	$0.438481\pi$
$30$	0	0
$31$	−0.698043	−0.125372	−0.0626861	−	0.998033i	$-0.519967\pi$
$31$	−0.698043	−0.125372	−0.0626861	+	0.998033i	$0.519967\pi$
$32$	0	0
$33$	−1.37400	−0.239183
$34$	0	0
$35$	−0.949904	−0.160563
$36$	0	0
$37$	5.10641	0.839489	0.419745	−	0.907642i	$-0.362120\pi$
$37$	5.10641	0.839489	0.419745	+	0.907642i	$0.362120\pi$
$38$	0	0
$39$	1.37400	0.220016
$40$	0	0
$41$	−1.72658	−0.269646	−0.134823	−	0.990870i	$-0.543047\pi$
$41$	−1.72658	−0.269646	−0.134823	+	0.990870i	$0.543047\pi$
$42$	0	0
$43$	0.353503	0.0539087	0.0269544	−	0.999637i	$-0.491419\pi$
$43$	0.353503	0.0539087	0.0269544	+	0.999637i	$0.491419\pi$
$44$	0	0
$45$	1.05641	0.157480
$46$	0	0
$47$	12.2124	1.78136	0.890679	−	0.454633i	$-0.150230\pi$
$47$	12.2124	1.78136	0.890679	+	0.454633i	$0.150230\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−7.24386	−1.01434
$52$	0	0
$53$	12.3677	1.69884	0.849418	−	0.527720i	$-0.176953\pi$
$53$	12.3677	1.69884	0.849418	+	0.527720i	$0.176953\pi$
$54$	0	0
$55$	0.949904	0.128085
$56$	0	0
$57$	−5.21334	−0.690524
$58$	0	0
$59$	−0.154213	−0.0200769	−0.0100384	−	0.999950i	$-0.503195\pi$
$59$	−0.154213	−0.0200769	−0.0100384	+	0.999950i	$0.503195\pi$
$60$	0	0
$61$	5.42802	0.694987	0.347494	−	0.937682i	$-0.387033\pi$
$61$	5.42802	0.694987	0.347494	+	0.937682i	$0.387033\pi$
$62$	0	0
$63$	−1.11212	−0.140114
$64$	0	0
$65$	−0.949904	−0.117821
$66$	0	0
$67$	1.39660	0.170622	0.0853110	−	0.996354i	$-0.472812\pi$
$67$	1.39660	0.170622	0.0853110	+	0.996354i	$0.472812\pi$
$68$	0	0
$69$	2.92901	0.352612
$70$	0	0
$71$	3.90040	0.462892	0.231446	−	0.972848i	$-0.425654\pi$
$71$	3.90040	0.462892	0.231446	+	0.972848i	$0.425654\pi$
$72$	0	0
$73$	11.5476	1.35155	0.675774	−	0.737109i	$-0.263810\pi$
$73$	11.5476	1.35155	0.675774	+	0.737109i	$0.263810\pi$
$74$	0	0
$75$	−5.63022	−0.650122
$76$	0	0
$77$	−1.00000	−0.113961
$78$	0	0
$79$	13.0818	1.47182	0.735908	−	0.677082i	$-0.236756\pi$
$79$	13.0818	1.47182	0.735908	+	0.677082i	$0.236756\pi$
$80$	0	0
$81$	−4.42683	−0.491871
$82$	0	0
$83$	12.1100	1.32924	0.664621	−	0.747181i	$-0.268593\pi$
$83$	12.1100	1.32924	0.664621	+	0.747181i	$0.268593\pi$
$84$	0	0
$85$	5.00798	0.543192
$86$	0	0
$87$	2.84228	0.304725
$88$	0	0
$89$	6.64156	0.704004	0.352002	−	0.935999i	$-0.385501\pi$
$89$	6.64156	0.704004	0.352002	+	0.935999i	$0.385501\pi$
$90$	0	0
$91$	1.00000	0.104828
$92$	0	0
$93$	−0.959112	−0.0994553
$94$	0	0
$95$	3.60420	0.369783
$96$	0	0
$97$	9.20030	0.934149	0.467074	−	0.884218i	$-0.345308\pi$
$97$	9.20030	0.934149	0.467074	+	0.884218i	$0.345308\pi$
$98$	0	0
$99$	1.11212	0.111772
$100$	0	0
$101$	14.0154	1.39458	0.697292	−	0.716787i	$-0.254388\pi$
$101$	14.0154	1.39458	0.697292	+	0.716787i	$0.254388\pi$
$102$	0	0
$103$	−0.726493	−0.0715835	−0.0357918	−	0.999359i	$-0.511395\pi$
$103$	−0.726493	−0.0715835	−0.0357918	+	0.999359i	$0.511395\pi$
$104$	0	0
$105$	−1.30517	−0.127372
$106$	0	0
$107$	−18.0592	−1.74585	−0.872924	−	0.487856i	$-0.837779\pi$
$107$	−18.0592	−1.74585	−0.872924	+	0.487856i	$0.837779\pi$
$108$	0	0
$109$	−10.0605	−0.963622	−0.481811	−	0.876275i	$-0.660021\pi$
$109$	−10.0605	−0.963622	−0.481811	+	0.876275i	$0.660021\pi$
$110$	0	0
$111$	7.01622	0.665950
$112$	0	0
$113$	−10.4434	−0.982430	−0.491215	−	0.871038i	$-0.663447\pi$
$113$	−10.4434	−0.982430	−0.491215	+	0.871038i	$0.663447\pi$
$114$	0	0
$115$	−2.02495	−0.188827
$116$	0	0
$117$	−1.11212	−0.102815
$118$	0	0
$119$	−5.27209	−0.483292
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−2.37232	−0.213905
$124$	0	0
$125$	8.64192	0.772957
$126$	0	0
$127$	0.0624536	0.00554186	0.00277093	−	0.999996i	$-0.499118\pi$
$127$	0.0624536	0.00554186	0.00277093	+	0.999996i	$0.499118\pi$
$128$	0	0
$129$	0.485714	0.0427647
$130$	0	0
$131$	−8.36118	−0.730520	−0.365260	−	0.930906i	$-0.619020\pi$
$131$	−8.36118	−0.730520	−0.365260	+	0.930906i	$0.619020\pi$
$132$	0	0
$133$	−3.79428	−0.329005
$134$	0	0
$135$	5.36701	0.461919
$136$	0	0
$137$	19.9385	1.70346	0.851730	−	0.523981i	$-0.175554\pi$
$137$	19.9385	1.70346	0.851730	+	0.523981i	$0.175554\pi$
$138$	0	0
$139$	19.7369	1.67406	0.837031	−	0.547155i	$-0.184289\pi$
$139$	19.7369	1.67406	0.837031	+	0.547155i	$0.184289\pi$
$140$	0	0
$141$	16.7798	1.41312
$142$	0	0
$143$	−1.00000	−0.0836242
$144$	0	0
$145$	−1.96499	−0.163183
$146$	0	0
$147$	1.37400	0.113326
$148$	0	0
$149$	10.0793	0.825730	0.412865	−	0.910792i	$-0.364528\pi$
$149$	10.0793	0.825730	0.412865	+	0.910792i	$0.364528\pi$
$150$	0	0
$151$	−6.14659	−0.500203	−0.250101	−	0.968220i	$-0.580464\pi$
$151$	−6.14659	−0.500203	−0.250101	+	0.968220i	$0.580464\pi$
$152$	0	0
$153$	5.86319	0.474011
$154$	0	0
$155$	0.663073	0.0532593
$156$	0	0
$157$	−8.42941	−0.672740	−0.336370	−	0.941730i	$-0.609199\pi$
$157$	−8.42941	−0.672740	−0.336370	+	0.941730i	$0.609199\pi$
$158$	0	0
$159$	16.9933	1.34765
$160$	0	0
$161$	2.13174	0.168005
$162$	0	0
$163$	6.42209	0.503017	0.251508	−	0.967855i	$-0.419073\pi$
$163$	6.42209	0.503017	0.251508	+	0.967855i	$0.419073\pi$
$164$	0	0
$165$	1.30517	0.101607
$166$	0	0
$167$	−9.45765	−0.731855	−0.365928	−	0.930643i	$-0.619248\pi$
$167$	−9.45765	−0.731855	−0.365928	+	0.930643i	$0.619248\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	4.21969	0.322687
$172$	0	0
$173$	15.4030	1.17107	0.585535	−	0.810647i	$-0.300885\pi$
$173$	15.4030	1.17107	0.585535	+	0.810647i	$0.300885\pi$
$174$	0	0
$175$	−4.09768	−0.309756
$176$	0	0
$177$	−0.211889	−0.0159266
$178$	0	0
$179$	−10.5983	−0.792156	−0.396078	−	0.918217i	$-0.629629\pi$
$179$	−10.5983	−0.792156	−0.396078	+	0.918217i	$0.629629\pi$
$180$	0	0
$181$	−16.6526	−1.23778	−0.618890	−	0.785477i	$-0.712418\pi$
$181$	−16.6526	−1.23778	−0.618890	+	0.785477i	$0.712418\pi$
$182$	0	0
$183$	7.45811	0.551320
$184$	0	0
$185$	−4.85060	−0.356623
$186$	0	0
$187$	5.27209	0.385533
$188$	0	0
$189$	−5.65006	−0.410981
$190$	0	0
$191$	9.89456	0.715945	0.357973	−	0.933732i	$-0.383468\pi$
$191$	9.89456	0.715945	0.357973	+	0.933732i	$0.383468\pi$
$192$	0	0
$193$	−24.2809	−1.74777	−0.873887	−	0.486128i	$-0.838409\pi$
$193$	−24.2809	−1.74777	−0.873887	+	0.486128i	$0.838409\pi$
$194$	0	0
$195$	−1.30517	−0.0934651
$196$	0	0
$197$	−27.3170	−1.94625	−0.973126	−	0.230273i	$-0.926038\pi$
$197$	−27.3170	−1.94625	−0.973126	+	0.230273i	$0.926038\pi$
$198$	0	0
$199$	1.18040	0.0836760	0.0418380	−	0.999124i	$-0.486679\pi$
$199$	1.18040	0.0836760	0.0418380	+	0.999124i	$0.486679\pi$
$200$	0	0
$201$	1.91893	0.135351
$202$	0	0
$203$	2.06862	0.145188
$204$	0	0
$205$	1.64008	0.114548
$206$	0	0
$207$	−2.37075	−0.164778
$208$	0	0
$209$	3.79428	0.262456
$210$	0	0
$211$	10.6430	0.732695	0.366347	−	0.930478i	$-0.380608\pi$
$211$	10.6430	0.732695	0.366347	+	0.930478i	$0.380608\pi$
$212$	0	0
$213$	5.35915	0.367203
$214$	0	0
$215$	−0.335794	−0.0229010
$216$	0	0
$217$	−0.698043	−0.0473862
$218$	0	0
$219$	15.8665	1.07216
$220$	0	0
$221$	−5.27209	−0.354639
$222$	0	0
$223$	−14.5060	−0.971396	−0.485698	−	0.874127i	$-0.661435\pi$
$223$	−14.5060	−0.971396	−0.485698	+	0.874127i	$0.661435\pi$
$224$	0	0
$225$	4.55711	0.303807
$226$	0	0
$227$	−21.4430	−1.42322	−0.711612	−	0.702573i	$-0.752034\pi$
$227$	−21.4430	−1.42322	−0.711612	+	0.702573i	$0.752034\pi$
$228$	0	0
$229$	−15.9833	−1.05621	−0.528104	−	0.849180i	$-0.677097\pi$
$229$	−15.9833	−1.05621	−0.528104	+	0.849180i	$0.677097\pi$
$230$	0	0
$231$	−1.37400	−0.0904027
$232$	0	0
$233$	20.0663	1.31458	0.657292	−	0.753636i	$-0.271702\pi$
$233$	20.0663	1.31458	0.657292	+	0.753636i	$0.271702\pi$
$234$	0	0
$235$	−11.6006	−0.756738
$236$	0	0
$237$	17.9744	1.16756
$238$	0	0
$239$	−21.1041	−1.36511	−0.682556	−	0.730833i	$-0.739132\pi$
$239$	−21.1041	−1.36511	−0.682556	+	0.730833i	$0.739132\pi$
$240$	0	0
$241$	25.2423	1.62600	0.812999	−	0.582265i	$-0.197834\pi$
$241$	25.2423	1.62600	0.812999	+	0.582265i	$0.197834\pi$
$242$	0	0
$243$	10.8677	0.697163
$244$	0	0
$245$	−0.949904	−0.0606871
$246$	0	0
$247$	−3.79428	−0.241424
$248$	0	0
$249$	16.6391	1.05446
$250$	0	0
$251$	4.42203	0.279116	0.139558	−	0.990214i	$-0.455432\pi$
$251$	4.42203	0.279116	0.139558	+	0.990214i	$0.455432\pi$
$252$	0	0
$253$	−2.13174	−0.134021
$254$	0	0
$255$	6.88097	0.430903
$256$	0	0
$257$	11.0743	0.690797	0.345399	−	0.938456i	$-0.387744\pi$
$257$	11.0743	0.690797	0.345399	+	0.938456i	$0.387744\pi$
$258$	0	0
$259$	5.10641	0.317297
$260$	0	0
$261$	−2.30055	−0.142400
$262$	0	0
$263$	15.0378	0.927267	0.463634	−	0.886027i	$-0.346545\pi$
$263$	15.0378	0.927267	0.463634	+	0.886027i	$0.346545\pi$
$264$	0	0
$265$	−11.7481	−0.721683
$266$	0	0
$267$	9.12551	0.558472
$268$	0	0
$269$	26.1564	1.59479	0.797393	−	0.603460i	$-0.206212\pi$
$269$	26.1564	1.59479	0.797393	+	0.603460i	$0.206212\pi$
$270$	0	0
$271$	12.9400	0.786049	0.393024	−	0.919528i	$-0.371429\pi$
$271$	12.9400	0.786049	0.393024	+	0.919528i	$0.371429\pi$
$272$	0	0
$273$	1.37400	0.0831584
$274$	0	0
$275$	4.09768	0.247100
$276$	0	0
$277$	12.5223	0.752395	0.376197	−	0.926540i	$-0.377232\pi$
$277$	12.5223	0.752395	0.376197	+	0.926540i	$0.377232\pi$
$278$	0	0
$279$	0.776306	0.0464762
$280$	0	0
$281$	−10.4363	−0.622580	−0.311290	−	0.950315i	$-0.600761\pi$
$281$	−10.4363	−0.622580	−0.311290	+	0.950315i	$0.600761\pi$
$282$	0	0
$283$	−3.13451	−0.186328	−0.0931638	−	0.995651i	$-0.529698\pi$
$283$	−3.13451	−0.186328	−0.0931638	+	0.995651i	$0.529698\pi$
$284$	0	0
$285$	4.95217	0.293341
$286$	0	0
$287$	−1.72658	−0.101917
$288$	0	0
$289$	10.7949	0.634996
$290$	0	0
$291$	12.6412	0.741042
$292$	0	0
$293$	−18.1624	−1.06106	−0.530529	−	0.847667i	$-0.678007\pi$
$293$	−18.1624	−1.06106	−0.530529	+	0.847667i	$0.678007\pi$
$294$	0	0
$295$	0.146488	0.00852885
$296$	0	0
$297$	5.65006	0.327850
$298$	0	0
$299$	2.13174	0.123282
$300$	0	0
$301$	0.353503	0.0203756
$302$	0	0
$303$	19.2572	1.10630
$304$	0	0
$305$	−5.15610	−0.295237
$306$	0	0
$307$	5.11621	0.291998	0.145999	−	0.989285i	$-0.453360\pi$
$307$	5.11621	0.291998	0.145999	+	0.989285i	$0.453360\pi$
$308$	0	0
$309$	−0.998203	−0.0567858
$310$	0	0
$311$	−0.137870	−0.00781789	−0.00390894	−	0.999992i	$-0.501244\pi$
$311$	−0.137870	−0.00781789	−0.00390894	+	0.999992i	$0.501244\pi$
$312$	0	0
$313$	−15.4547	−0.873549	−0.436775	−	0.899571i	$-0.643879\pi$
$313$	−15.4547	−0.873549	−0.436775	+	0.899571i	$0.643879\pi$
$314$	0	0
$315$	1.05641	0.0595217
$316$	0	0
$317$	25.8929	1.45429	0.727145	−	0.686484i	$-0.240847\pi$
$317$	25.8929	1.45429	0.727145	+	0.686484i	$0.240847\pi$
$318$	0	0
$319$	−2.06862	−0.115820
$320$	0	0
$321$	−24.8134	−1.38495
$322$	0	0
$323$	20.0038	1.11304
$324$	0	0
$325$	−4.09768	−0.227299
$326$	0	0
$327$	−13.8232	−0.764422
$328$	0	0
$329$	12.2124	0.673290
$330$	0	0
$331$	−1.08534	−0.0596558	−0.0298279	−	0.999555i	$-0.509496\pi$
$331$	−1.08534	−0.0596558	−0.0298279	+	0.999555i	$0.509496\pi$
$332$	0	0
$333$	−5.67894	−0.311204
$334$	0	0
$335$	−1.32664	−0.0724819
$336$	0	0
$337$	−14.3888	−0.783806	−0.391903	−	0.920007i	$-0.628183\pi$
$337$	−14.3888	−0.783806	−0.391903	+	0.920007i	$0.628183\pi$
$338$	0	0
$339$	−14.3492	−0.779343
$340$	0	0
$341$	0.698043	0.0378011
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	−2.78228	−0.149793
$346$	0	0
$347$	−1.73580	−0.0931829	−0.0465914	−	0.998914i	$-0.514836\pi$
$347$	−1.73580	−0.0931829	−0.0465914	+	0.998914i	$0.514836\pi$
$348$	0	0
$349$	−25.4668	−1.36321	−0.681604	−	0.731721i	$-0.738717\pi$
$349$	−25.4668	−1.36321	−0.681604	+	0.731721i	$0.738717\pi$
$350$	0	0
$351$	−5.65006	−0.301578
$352$	0	0
$353$	10.4265	0.554948	0.277474	−	0.960733i	$-0.410503\pi$
$353$	10.4265	0.554948	0.277474	+	0.960733i	$0.410503\pi$
$354$	0	0
$355$	−3.70500	−0.196641
$356$	0	0
$357$	−7.24386	−0.383386
$358$	0	0
$359$	24.5847	1.29753	0.648766	−	0.760988i	$-0.275285\pi$
$359$	24.5847	1.29753	0.648766	+	0.760988i	$0.275285\pi$
$360$	0	0
$361$	−4.60347	−0.242288
$362$	0	0
$363$	1.37400	0.0721164
$364$	0	0
$365$	−10.9691	−0.574151
$366$	0	0
$367$	−4.37136	−0.228183	−0.114092	−	0.993470i	$-0.536396\pi$
$367$	−4.37136	−0.228183	−0.114092	+	0.993470i	$0.536396\pi$
$368$	0	0
$369$	1.92016	0.0999595
$370$	0	0
$371$	12.3677	0.642100
$372$	0	0
$373$	7.40644	0.383491	0.191746	−	0.981445i	$-0.438585\pi$
$373$	7.40644	0.383491	0.191746	+	0.981445i	$0.438585\pi$
$374$	0	0
$375$	11.8740	0.613172
$376$	0	0
$377$	2.06862	0.106539
$378$	0	0
$379$	18.5383	0.952247	0.476123	−	0.879378i	$-0.342042\pi$
$379$	18.5383	0.952247	0.476123	+	0.879378i	$0.342042\pi$
$380$	0	0
$381$	0.0858114	0.00439625
$382$	0	0
$383$	−12.0534	−0.615901	−0.307951	−	0.951402i	$-0.599643\pi$
$383$	−12.0534	−0.615901	−0.307951	+	0.951402i	$0.599643\pi$
$384$	0	0
$385$	0.949904	0.0484116
$386$	0	0
$387$	−0.393138	−0.0199843
$388$	0	0
$389$	−1.02869	−0.0521567	−0.0260783	−	0.999660i	$-0.508302\pi$
$389$	−1.02869	−0.0521567	−0.0260783	+	0.999660i	$0.508302\pi$
$390$	0	0
$391$	−11.2387	−0.568366
$392$	0	0
$393$	−11.4883	−0.579507
$394$	0	0
$395$	−12.4264	−0.625242
$396$	0	0
$397$	−16.9336	−0.849874	−0.424937	−	0.905223i	$-0.639704\pi$
$397$	−16.9336	−0.849874	−0.424937	+	0.905223i	$0.639704\pi$
$398$	0	0
$399$	−5.21334	−0.260994
$400$	0	0
$401$	−9.91840	−0.495301	−0.247651	−	0.968849i	$-0.579659\pi$
$401$	−9.91840	−0.495301	−0.247651	+	0.968849i	$0.579659\pi$
$402$	0	0
$403$	−0.698043	−0.0347720
$404$	0	0
$405$	4.20507	0.208951
$406$	0	0
$407$	−5.10641	−0.253115
$408$	0	0
$409$	32.0379	1.58417	0.792085	−	0.610410i	$-0.208995\pi$
$409$	32.0379	1.58417	0.792085	+	0.610410i	$0.208995\pi$
$410$	0	0
$411$	27.3955	1.35132
$412$	0	0
$413$	−0.154213	−0.00758834
$414$	0	0
$415$	−11.5033	−0.564675
$416$	0	0
$417$	27.1185	1.32800
$418$	0	0
$419$	29.3105	1.43191	0.715956	−	0.698146i	$-0.245991\pi$
$419$	29.3105	1.43191	0.715956	+	0.698146i	$0.245991\pi$
$420$	0	0
$421$	19.9891	0.974209	0.487104	−	0.873344i	$-0.338053\pi$
$421$	19.9891	0.974209	0.487104	+	0.873344i	$0.338053\pi$
$422$	0	0
$423$	−13.5816	−0.660360
$424$	0	0
$425$	21.6034	1.04792
$426$	0	0
$427$	5.42802	0.262680
$428$	0	0
$429$	−1.37400	−0.0663374
$430$	0	0
$431$	18.9607	0.913305	0.456652	−	0.889645i	$-0.349048\pi$
$431$	18.9607	0.913305	0.456652	+	0.889645i	$0.349048\pi$
$432$	0	0
$433$	−4.12173	−0.198078	−0.0990388	−	0.995084i	$-0.531577\pi$
$433$	−4.12173	−0.198078	−0.0990388	+	0.995084i	$0.531577\pi$
$434$	0	0
$435$	−2.69990	−0.129450
$436$	0	0
$437$	−8.08840	−0.386921
$438$	0	0
$439$	−26.5264	−1.26604	−0.633019	−	0.774136i	$-0.718184\pi$
$439$	−26.5264	−1.26604	−0.633019	+	0.774136i	$0.718184\pi$
$440$	0	0
$441$	−1.11212	−0.0529580
$442$	0	0
$443$	29.2519	1.38980	0.694901	−	0.719106i	$-0.255448\pi$
$443$	29.2519	1.38980	0.694901	+	0.719106i	$0.255448\pi$
$444$	0	0
$445$	−6.30884	−0.299068
$446$	0	0
$447$	13.8490	0.655036
$448$	0	0
$449$	8.41086	0.396933	0.198467	−	0.980108i	$-0.436404\pi$
$449$	8.41086	0.396933	0.198467	+	0.980108i	$0.436404\pi$
$450$	0	0
$451$	1.72658	0.0813013
$452$	0	0
$453$	−8.44543	−0.396801
$454$	0	0
$455$	−0.949904	−0.0445322
$456$	0	0
$457$	−16.1021	−0.753223	−0.376612	−	0.926371i	$-0.622911\pi$
$457$	−16.1021	−0.753223	−0.376612	+	0.926371i	$0.622911\pi$
$458$	0	0
$459$	29.7876	1.39037
$460$	0	0
$461$	−4.42732	−0.206201	−0.103100	−	0.994671i	$-0.532876\pi$
$461$	−4.42732	−0.206201	−0.103100	+	0.994671i	$0.532876\pi$
$462$	0	0
$463$	−34.6176	−1.60882	−0.804408	−	0.594077i	$-0.797517\pi$
$463$	−34.6176	−1.60882	−0.804408	+	0.594077i	$0.797517\pi$
$464$	0	0
$465$	0.911064	0.0422496
$466$	0	0
$467$	−10.2277	−0.473280	−0.236640	−	0.971597i	$-0.576046\pi$
$467$	−10.2277	−0.473280	−0.236640	+	0.971597i	$0.576046\pi$
$468$	0	0
$469$	1.39660	0.0644890
$470$	0	0
$471$	−11.5820	−0.533672
$472$	0	0
$473$	−0.353503	−0.0162541
$474$	0	0
$475$	15.5477	0.713379
$476$	0	0
$477$	−13.7544	−0.629769
$478$	0	0
$479$	9.63553	0.440259	0.220129	−	0.975471i	$-0.429352\pi$
$479$	9.63553	0.440259	0.220129	+	0.975471i	$0.429352\pi$
$480$	0	0
$481$	5.10641	0.232832
$482$	0	0
$483$	2.92901	0.133275
$484$	0	0
$485$	−8.73940	−0.396836
$486$	0	0
$487$	2.89915	0.131373	0.0656866	−	0.997840i	$-0.479076\pi$
$487$	2.89915	0.131373	0.0656866	+	0.997840i	$0.479076\pi$
$488$	0	0
$489$	8.82396	0.399033
$490$	0	0
$491$	11.1058	0.501196	0.250598	−	0.968091i	$-0.419373\pi$
$491$	11.1058	0.501196	0.250598	+	0.968091i	$0.419373\pi$
$492$	0	0
$493$	−10.9059	−0.491178
$494$	0	0
$495$	−1.05641	−0.0474819
$496$	0	0
$497$	3.90040	0.174957
$498$	0	0
$499$	−7.70845	−0.345078	−0.172539	−	0.985003i	$-0.555197\pi$
$499$	−7.70845	−0.345078	−0.172539	+	0.985003i	$0.555197\pi$
$500$	0	0
$501$	−12.9948	−0.580566
$502$	0	0
$503$	24.5492	1.09459	0.547297	−	0.836939i	$-0.315657\pi$
$503$	24.5492	1.09459	0.547297	+	0.836939i	$0.315657\pi$
$504$	0	0
$505$	−13.3133	−0.592434
$506$	0	0
$507$	1.37400	0.0610216
$508$	0	0
$509$	9.90091	0.438850	0.219425	−	0.975629i	$-0.429582\pi$
$509$	9.90091	0.438850	0.219425	+	0.975629i	$0.429582\pi$
$510$	0	0
$511$	11.5476	0.510837
$512$	0	0
$513$	21.4379	0.946505
$514$	0	0
$515$	0.690099	0.0304094
$516$	0	0
$517$	−12.2124	−0.537099
$518$	0	0
$519$	21.1638	0.928987
$520$	0	0
$521$	−34.1144	−1.49458	−0.747289	−	0.664499i	$-0.768645\pi$
$521$	−34.1144	−1.49458	−0.747289	+	0.664499i	$0.768645\pi$
$522$	0	0
$523$	28.0037	1.22452	0.612259	−	0.790657i	$-0.290261\pi$
$523$	28.0037	1.22452	0.612259	+	0.790657i	$0.290261\pi$
$524$	0	0
$525$	−5.63022	−0.245723
$526$	0	0
$527$	3.68014	0.160310
$528$	0	0
$529$	−18.4557	−0.802421
$530$	0	0
$531$	0.171503	0.00744262
$532$	0	0
$533$	−1.72658	−0.0747863
$534$	0	0
$535$	17.1545	0.741654
$536$	0	0
$537$	−14.5621	−0.628402
$538$	0	0
$539$	−1.00000	−0.0430730
$540$	0	0
$541$	−18.6191	−0.800498	−0.400249	−	0.916406i	$-0.631076\pi$
$541$	−18.6191	−0.800498	−0.400249	+	0.916406i	$0.631076\pi$
$542$	0	0
$543$	−22.8808	−0.981907
$544$	0	0
$545$	9.55652	0.409356
$546$	0	0
$547$	−19.9922	−0.854806	−0.427403	−	0.904061i	$-0.640571\pi$
$547$	−19.9922	−0.854806	−0.427403	+	0.904061i	$0.640571\pi$
$548$	0	0
$549$	−6.03661	−0.257636
$550$	0	0
$551$	−7.84890	−0.334374
$552$	0	0
$553$	13.0818	0.556294
$554$	0	0
$555$	−6.66474	−0.282902
$556$	0	0
$557$	39.9707	1.69361	0.846807	−	0.531901i	$-0.178522\pi$
$557$	39.9707	1.69361	0.846807	+	0.531901i	$0.178522\pi$
$558$	0	0
$559$	0.353503	0.0149516
$560$	0	0
$561$	7.24386	0.305836
$562$	0	0
$563$	12.9189	0.544468	0.272234	−	0.962231i	$-0.412237\pi$
$563$	12.9189	0.544468	0.272234	+	0.962231i	$0.412237\pi$
$564$	0	0
$565$	9.92021	0.417346
$566$	0	0
$567$	−4.42683	−0.185910
$568$	0	0
$569$	−23.2751	−0.975744	−0.487872	−	0.872915i	$-0.662227\pi$
$569$	−23.2751	−0.975744	−0.487872	+	0.872915i	$0.662227\pi$
$570$	0	0
$571$	4.82039	0.201727	0.100863	−	0.994900i	$-0.467839\pi$
$571$	4.82039	0.201727	0.100863	+	0.994900i	$0.467839\pi$
$572$	0	0
$573$	13.5951	0.567945
$574$	0	0
$575$	−8.73519	−0.364283
$576$	0	0
$577$	19.8058	0.824528	0.412264	−	0.911064i	$-0.364738\pi$
$577$	19.8058	0.824528	0.412264	+	0.911064i	$0.364738\pi$
$578$	0	0
$579$	−33.3620	−1.38648
$580$	0	0
$581$	12.1100	0.502406
$582$	0	0
$583$	−12.3677	−0.512218
$584$	0	0
$585$	1.05641	0.0436770
$586$	0	0
$587$	−21.0529	−0.868947	−0.434474	−	0.900685i	$-0.643066\pi$
$587$	−21.0529	−0.868947	−0.434474	+	0.900685i	$0.643066\pi$
$588$	0	0
$589$	2.64857	0.109132
$590$	0	0
$591$	−37.5335	−1.54392
$592$	0	0
$593$	27.6718	1.13634	0.568172	−	0.822910i	$-0.307651\pi$
$593$	27.6718	1.13634	0.568172	+	0.822910i	$0.307651\pi$
$594$	0	0
$595$	5.00798	0.205307
$596$	0	0
$597$	1.62187	0.0663786
$598$	0	0
$599$	47.4357	1.93817	0.969086	−	0.246725i	$-0.0793543\pi$
$599$	47.4357	1.93817	0.969086	+	0.246725i	$0.0793543\pi$
$600$	0	0
$601$	−34.9190	−1.42437	−0.712187	−	0.701990i	$-0.752295\pi$
$601$	−34.9190	−1.42437	−0.712187	+	0.701990i	$0.752295\pi$
$602$	0	0
$603$	−1.55319	−0.0632506
$604$	0	0
$605$	−0.949904	−0.0386191
$606$	0	0
$607$	35.5429	1.44264	0.721321	−	0.692601i	$-0.243535\pi$
$607$	35.5429	1.44264	0.721321	+	0.692601i	$0.243535\pi$
$608$	0	0
$609$	2.84228	0.115175
$610$	0	0
$611$	12.2124	0.494060
$612$	0	0
$613$	22.9735	0.927891	0.463945	−	0.885864i	$-0.346433\pi$
$613$	22.9735	0.927891	0.463945	+	0.885864i	$0.346433\pi$
$614$	0	0
$615$	2.25348	0.0908689
$616$	0	0
$617$	−34.8994	−1.40500	−0.702499	−	0.711685i	$-0.747932\pi$
$617$	−34.8994	−1.40500	−0.702499	+	0.711685i	$0.747932\pi$
$618$	0	0
$619$	−8.87944	−0.356895	−0.178447	−	0.983949i	$-0.557107\pi$
$619$	−8.87944	−0.356895	−0.178447	+	0.983949i	$0.557107\pi$
$620$	0	0
$621$	−12.0445	−0.483327
$622$	0	0
$623$	6.64156	0.266088
$624$	0	0
$625$	12.2794	0.491177
$626$	0	0
$627$	5.21334	0.208201
$628$	0	0
$629$	−26.9215	−1.07343
$630$	0	0
$631$	−11.7508	−0.467790	−0.233895	−	0.972262i	$-0.575147\pi$
$631$	−11.7508	−0.467790	−0.233895	+	0.972262i	$0.575147\pi$
$632$	0	0
$633$	14.6235	0.581232
$634$	0	0
$635$	−0.0593249	−0.00235424
$636$	0	0
$637$	1.00000	0.0396214
$638$	0	0
$639$	−4.33771	−0.171597
$640$	0	0
$641$	20.3627	0.804277	0.402139	−	0.915579i	$-0.368267\pi$
$641$	20.3627	0.804277	0.402139	+	0.915579i	$0.368267\pi$
$642$	0	0
$643$	−21.7778	−0.858831	−0.429416	−	0.903107i	$-0.641280\pi$
$643$	−21.7778	−0.858831	−0.429416	+	0.903107i	$0.641280\pi$
$644$	0	0
$645$	−0.461382	−0.0181669
$646$	0	0
$647$	22.2145	0.873340	0.436670	−	0.899622i	$-0.356158\pi$
$647$	22.2145	0.873340	0.436670	+	0.899622i	$0.356158\pi$
$648$	0	0
$649$	0.154213	0.00605340
$650$	0	0
$651$	−0.959112	−0.0375906
$652$	0	0
$653$	−23.9938	−0.938951	−0.469475	−	0.882946i	$-0.655557\pi$
$653$	−23.9938	−0.938951	−0.469475	+	0.882946i	$0.655557\pi$
$654$	0	0
$655$	7.94232	0.310332
$656$	0	0
$657$	−12.8423	−0.501027
$658$	0	0
$659$	−1.31003	−0.0510316	−0.0255158	−	0.999674i	$-0.508123\pi$
$659$	−1.31003	−0.0510316	−0.0255158	+	0.999674i	$0.508123\pi$
$660$	0	0
$661$	−7.68908	−0.299070	−0.149535	−	0.988756i	$-0.547778\pi$
$661$	−7.68908	−0.299070	−0.149535	+	0.988756i	$0.547778\pi$
$662$	0	0
$663$	−7.24386	−0.281328
$664$	0	0
$665$	3.60420	0.139765
$666$	0	0
$667$	4.40975	0.170746
$668$	0	0
$669$	−19.9313	−0.770589
$670$	0	0
$671$	−5.42802	−0.209547
$672$	0	0
$673$	2.92503	0.112752	0.0563758	−	0.998410i	$-0.482046\pi$
$673$	2.92503	0.112752	0.0563758	+	0.998410i	$0.482046\pi$
$674$	0	0
$675$	23.1521	0.891127
$676$	0	0
$677$	3.89797	0.149811	0.0749055	−	0.997191i	$-0.476134\pi$
$677$	3.89797	0.149811	0.0749055	+	0.997191i	$0.476134\pi$
$678$	0	0
$679$	9.20030	0.353075
$680$	0	0
$681$	−29.4628	−1.12902
$682$	0	0
$683$	6.83939	0.261702	0.130851	−	0.991402i	$-0.458229\pi$
$683$	6.83939	0.261702	0.130851	+	0.991402i	$0.458229\pi$
$684$	0	0
$685$	−18.9396	−0.723647
$686$	0	0
$687$	−21.9611	−0.837869
$688$	0	0
$689$	12.3677	0.471172
$690$	0	0
$691$	20.3394	0.773746	0.386873	−	0.922133i	$-0.373555\pi$
$691$	20.3394	0.773746	0.386873	+	0.922133i	$0.373555\pi$
$692$	0	0
$693$	1.11212	0.0422459
$694$	0	0
$695$	−18.7482	−0.711158
$696$	0	0
$697$	9.10267	0.344788
$698$	0	0
$699$	27.5711	1.04283
$700$	0	0
$701$	−31.1580	−1.17682	−0.588412	−	0.808562i	$-0.700247\pi$
$701$	−31.1580	−1.17682	−0.588412	+	0.808562i	$0.700247\pi$
$702$	0	0
$703$	−19.3751	−0.730747
$704$	0	0
$705$	−15.9392	−0.600306
$706$	0	0
$707$	14.0154	0.527104
$708$	0	0
$709$	17.2132	0.646455	0.323228	−	0.946321i	$-0.395232\pi$
$709$	17.2132	0.646455	0.323228	+	0.946321i	$0.395232\pi$
$710$	0	0
$711$	−14.5485	−0.545611
$712$	0	0
$713$	−1.48804	−0.0557277
$714$	0	0
$715$	0.949904	0.0355244
$716$	0	0
$717$	−28.9971	−1.08292
$718$	0	0
$719$	0.686781	0.0256126	0.0128063	−	0.999918i	$-0.495924\pi$
$719$	0.686781	0.0256126	0.0128063	+	0.999918i	$0.495924\pi$
$720$	0	0
$721$	−0.726493	−0.0270560
$722$	0	0
$723$	34.6829	1.28987
$724$	0	0
$725$	−8.47653	−0.314811
$726$	0	0
$727$	27.9120	1.03520	0.517600	−	0.855623i	$-0.326825\pi$
$727$	27.9120	1.03520	0.517600	+	0.855623i	$0.326825\pi$
$728$	0	0
$729$	28.2127	1.04492
$730$	0	0
$731$	−1.86370	−0.0689315
$732$	0	0
$733$	−29.7389	−1.09843	−0.549216	−	0.835681i	$-0.685073\pi$
$733$	−29.7389	−1.09843	−0.549216	+	0.835681i	$0.685073\pi$
$734$	0	0
$735$	−1.30517	−0.0481419
$736$	0	0
$737$	−1.39660	−0.0514444
$738$	0	0
$739$	3.94687	0.145188	0.0725940	−	0.997362i	$-0.476872\pi$
$739$	3.94687	0.145188	0.0725940	+	0.997362i	$0.476872\pi$
$740$	0	0
$741$	−5.21334	−0.191517
$742$	0	0
$743$	−0.570963	−0.0209466	−0.0104733	−	0.999945i	$-0.503334\pi$
$743$	−0.570963	−0.0209466	−0.0104733	+	0.999945i	$0.503334\pi$
$744$	0	0
$745$	−9.57439	−0.350778
$746$	0	0
$747$	−13.4677	−0.492758
$748$	0	0
$749$	−18.0592	−0.659869
$750$	0	0
$751$	28.0021	1.02181	0.510905	−	0.859637i	$-0.329310\pi$
$751$	28.0021	1.02181	0.510905	+	0.859637i	$0.329310\pi$
$752$	0	0
$753$	6.07587	0.221417
$754$	0	0
$755$	5.83867	0.212491
$756$	0	0
$757$	−44.9109	−1.63231	−0.816157	−	0.577830i	$-0.803900\pi$
$757$	−44.9109	−1.63231	−0.816157	+	0.577830i	$0.803900\pi$
$758$	0	0
$759$	−2.92901	−0.106316
$760$	0	0
$761$	−46.7107	−1.69326	−0.846630	−	0.532181i	$-0.821372\pi$
$761$	−46.7107	−1.69326	−0.846630	+	0.532181i	$0.821372\pi$
$762$	0	0
$763$	−10.0605	−0.364215
$764$	0	0
$765$	−5.56947	−0.201365
$766$	0	0
$767$	−0.154213	−0.00556832
$768$	0	0
$769$	−44.3840	−1.60053	−0.800264	−	0.599647i	$-0.795308\pi$
$769$	−44.3840	−1.60053	−0.800264	+	0.599647i	$0.795308\pi$
$770$	0	0
$771$	15.2161	0.547996
$772$	0	0
$773$	−19.6968	−0.708444	−0.354222	−	0.935161i	$-0.615254\pi$
$773$	−19.6968	−0.708444	−0.354222	+	0.935161i	$0.615254\pi$
$774$	0	0
$775$	2.86036	0.102747
$776$	0	0
$777$	7.01622	0.251706
$778$	0	0
$779$	6.55111	0.234718
$780$	0	0
$781$	−3.90040	−0.139567
$782$	0	0
$783$	−11.6878	−0.417688
$784$	0	0
$785$	8.00713	0.285787
$786$	0	0
$787$	28.9062	1.03039	0.515197	−	0.857072i	$-0.327719\pi$
$787$	28.9062	1.03039	0.515197	+	0.857072i	$0.327719\pi$
$788$	0	0
$789$	20.6619	0.735583
$790$	0	0
$791$	−10.4434	−0.371324
$792$	0	0
$793$	5.42802	0.192755
$794$	0	0
$795$	−16.1420	−0.572497
$796$	0	0
$797$	41.1744	1.45847	0.729235	−	0.684263i	$-0.239876\pi$
$797$	41.1744	1.45847	0.729235	+	0.684263i	$0.239876\pi$
$798$	0	0
$799$	−64.3847	−2.27777
$800$	0	0
$801$	−7.38620	−0.260979
$802$	0	0
$803$	−11.5476	−0.407507
$804$	0	0
$805$	−2.02495	−0.0713700
$806$	0	0
$807$	35.9390	1.26511
$808$	0	0
$809$	−19.1716	−0.674036	−0.337018	−	0.941498i	$-0.609418\pi$
$809$	−19.1716	−0.674036	−0.337018	+	0.941498i	$0.609418\pi$
$810$	0	0
$811$	−1.71804	−0.0603285	−0.0301643	−	0.999545i	$-0.509603\pi$
$811$	−1.71804	−0.0603285	−0.0301643	+	0.999545i	$0.509603\pi$
$812$	0	0
$813$	17.7796	0.623557
$814$	0	0
$815$	−6.10036	−0.213686
$816$	0	0
$817$	−1.34129	−0.0469257
$818$	0	0
$819$	−1.11212	−0.0388606
$820$	0	0
$821$	21.3872	0.746418	0.373209	−	0.927747i	$-0.378257\pi$
$821$	21.3872	0.746418	0.373209	+	0.927747i	$0.378257\pi$
$822$	0	0
$823$	11.2905	0.393561	0.196781	−	0.980448i	$-0.436951\pi$
$823$	11.2905	0.393561	0.196781	+	0.980448i	$0.436951\pi$
$824$	0	0
$825$	5.63022	0.196019
$826$	0	0
$827$	27.1375	0.943663	0.471831	−	0.881689i	$-0.343593\pi$
$827$	27.1375	0.943663	0.471831	+	0.881689i	$0.343593\pi$
$828$	0	0
$829$	4.85754	0.168709	0.0843547	−	0.996436i	$-0.473117\pi$
$829$	4.85754	0.168709	0.0843547	+	0.996436i	$0.473117\pi$
$830$	0	0
$831$	17.2057	0.596860
$832$	0	0
$833$	−5.27209	−0.182667
$834$	0	0
$835$	8.98386	0.310899
$836$	0	0
$837$	3.94398	0.136324
$838$	0	0
$839$	36.2106	1.25013	0.625065	−	0.780573i	$-0.285072\pi$
$839$	36.2106	1.25013	0.625065	+	0.780573i	$0.285072\pi$
$840$	0	0
$841$	−24.7208	−0.852442
$842$	0	0
$843$	−14.3395	−0.493880
$844$	0	0
$845$	−0.949904	−0.0326777
$846$	0	0
$847$	1.00000	0.0343604
$848$	0	0
$849$	−4.30683	−0.147810
$850$	0	0
$851$	10.8855	0.373151
$852$	0	0
$853$	47.3328	1.62064	0.810322	−	0.585985i	$-0.199292\pi$
$853$	47.3328	1.62064	0.810322	+	0.585985i	$0.199292\pi$
$854$	0	0
$855$	−4.00830	−0.137081
$856$	0	0
$857$	27.8950	0.952876	0.476438	−	0.879208i	$-0.341928\pi$
$857$	27.8950	0.952876	0.476438	+	0.879208i	$0.341928\pi$
$858$	0	0
$859$	34.0691	1.16242	0.581211	−	0.813753i	$-0.302579\pi$
$859$	34.0691	1.16242	0.581211	+	0.813753i	$0.302579\pi$
$860$	0	0
$861$	−2.37232	−0.0808484
$862$	0	0
$863$	9.64716	0.328393	0.164197	−	0.986428i	$-0.447497\pi$
$863$	9.64716	0.328393	0.164197	+	0.986428i	$0.447497\pi$
$864$	0	0
$865$	−14.6314	−0.497482
$866$	0	0
$867$	14.8323	0.503730
$868$	0	0
$869$	−13.0818	−0.443769
$870$	0	0
$871$	1.39660	0.0473220
$872$	0	0
$873$	−10.2318	−0.346295
$874$	0	0
$875$	8.64192	0.292150
$876$	0	0
$877$	4.31740	0.145788	0.0728942	−	0.997340i	$-0.476776\pi$
$877$	4.31740	0.145788	0.0728942	+	0.997340i	$0.476776\pi$
$878$	0	0
$879$	−24.9552	−0.841716
$880$	0	0
$881$	45.7580	1.54163	0.770814	−	0.637061i	$-0.219850\pi$
$881$	45.7580	1.54163	0.770814	+	0.637061i	$0.219850\pi$
$882$	0	0
$883$	0.219394	0.00738320	0.00369160	−	0.999993i	$-0.498825\pi$
$883$	0.219394	0.00738320	0.00369160	+	0.999993i	$0.498825\pi$
$884$	0	0
$885$	0.201274	0.00676577
$886$	0	0
$887$	−57.7428	−1.93881	−0.969406	−	0.245462i	$-0.921060\pi$
$887$	−57.7428	−1.93881	−0.969406	+	0.245462i	$0.921060\pi$
$888$	0	0
$889$	0.0624536	0.00209463
$890$	0	0
$891$	4.42683	0.148305
$892$	0	0
$893$	−46.3371	−1.55061
$894$	0	0
$895$	10.0674	0.336516
$896$	0	0
$897$	2.92901	0.0977969
$898$	0	0
$899$	−1.44398	−0.0481595
$900$	0	0
$901$	−65.2037	−2.17225
$902$	0	0
$903$	0.485714	0.0161636
$904$	0	0
$905$	15.8184	0.525822
$906$	0	0
$907$	24.2794	0.806183	0.403091	−	0.915160i	$-0.367936\pi$
$907$	24.2794	0.806183	0.403091	+	0.915160i	$0.367936\pi$
$908$	0	0
$909$	−15.5868	−0.516981
$910$	0	0
$911$	−16.6734	−0.552415	−0.276208	−	0.961098i	$-0.589078\pi$
$911$	−16.6734	−0.552415	−0.276208	+	0.961098i	$0.589078\pi$
$912$	0	0
$913$	−12.1100	−0.400781
$914$	0	0
$915$	−7.08449	−0.234206
$916$	0	0
$917$	−8.36118	−0.276111
$918$	0	0
$919$	20.0245	0.660547	0.330273	−	0.943885i	$-0.392859\pi$
$919$	20.0245	0.660547	0.330273	+	0.943885i	$0.392859\pi$
$920$	0	0
$921$	7.02969	0.231636
$922$	0	0
$923$	3.90040	0.128383
$924$	0	0
$925$	−20.9245	−0.687992
$926$	0	0
$927$	0.807947	0.0265365
$928$	0	0
$929$	−35.2374	−1.15610	−0.578051	−	0.816001i	$-0.696186\pi$
$929$	−35.2374	−1.15610	−0.578051	+	0.816001i	$0.696186\pi$
$930$	0	0
$931$	−3.79428	−0.124352
$932$	0	0
$933$	−0.189434	−0.00620178
$934$	0	0
$935$	−5.00798	−0.163778
$936$	0	0
$937$	−31.3305	−1.02352	−0.511761	−	0.859128i	$-0.671007\pi$
$937$	−31.3305	−1.02352	−0.511761	+	0.859128i	$0.671007\pi$
$938$	0	0
$939$	−21.2347	−0.692969
$940$	0	0
$941$	−10.0001	−0.325994	−0.162997	−	0.986627i	$-0.552116\pi$
$941$	−10.0001	−0.325994	−0.162997	+	0.986627i	$0.552116\pi$
$942$	0	0
$943$	−3.68061	−0.119857
$944$	0	0
$945$	5.36701	0.174589
$946$	0	0
$947$	−14.3627	−0.466724	−0.233362	−	0.972390i	$-0.574973\pi$
$947$	−14.3627	−0.466724	−0.233362	+	0.972390i	$0.574973\pi$
$948$	0	0
$949$	11.5476	0.374852
$950$	0	0
$951$	35.5769	1.15366
$952$	0	0
$953$	−9.15005	−0.296399	−0.148200	−	0.988957i	$-0.547348\pi$
$953$	−9.15005	−0.296399	−0.148200	+	0.988957i	$0.547348\pi$
$954$	0	0
$955$	−9.39888	−0.304141
$956$	0	0
$957$	−2.84228	−0.0918780
$958$	0	0
$959$	19.9385	0.643847
$960$	0	0
$961$	−30.5127	−0.984282
$962$	0	0
$963$	20.0840	0.647197
$964$	0	0
$965$	23.0645	0.742472
$966$	0	0
$967$	−34.7522	−1.11756	−0.558778	−	0.829317i	$-0.688730\pi$
$967$	−34.7522	−1.11756	−0.558778	+	0.829317i	$0.688730\pi$
$968$	0	0
$969$	27.4852	0.882952
$970$	0	0
$971$	27.7594	0.890840	0.445420	−	0.895322i	$-0.353054\pi$
$971$	27.7594	0.890840	0.445420	+	0.895322i	$0.353054\pi$
$972$	0	0
$973$	19.7369	0.632736
$974$	0	0
$975$	−5.63022	−0.180311
$976$	0	0
$977$	4.60923	0.147462	0.0737312	−	0.997278i	$-0.476509\pi$
$977$	4.60923	0.147462	0.0737312	+	0.997278i	$0.476509\pi$
$978$	0	0
$979$	−6.64156	−0.212265
$980$	0	0
$981$	11.1885	0.357221
$982$	0	0
$983$	−19.3032	−0.615676	−0.307838	−	0.951439i	$-0.599605\pi$
$983$	−19.3032	−0.615676	−0.307838	+	0.951439i	$0.599605\pi$
$984$	0	0
$985$	25.9485	0.826787
$986$	0	0
$987$	16.7798	0.534108
$988$	0	0
$989$	0.753577	0.0239623
$990$	0	0
$991$	−2.36790	−0.0752188	−0.0376094	−	0.999293i	$-0.511974\pi$
$991$	−2.36790	−0.0752188	−0.0376094	+	0.999293i	$0.511974\pi$
$992$	0	0
$993$	−1.49126	−0.0473238
$994$	0	0
$995$	−1.12126	−0.0355464
$996$	0	0
$997$	−5.37453	−0.170213	−0.0851065	−	0.996372i	$-0.527123\pi$
$997$	−5.37453	−0.170213	−0.0851065	+	0.996372i	$0.527123\pi$
$998$	0	0
$999$	−28.8515	−0.912822

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8008.2.a.t.1.7	✓	10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8008.2.a.t.1.7	✓	10	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)

\(f(q)\)	\(=\)	\(q+1.37400 q^{3} -0.949904 q^{5} +1.00000 q^{7} -1.11212 q^{9} +O(q^{10})\)	\(q+1.37400 q^{3} -0.949904 q^{5} +1.00000 q^{7} -1.11212 q^{9} -1.00000 q^{11} +1.00000 q^{13} -1.30517 q^{15} -5.27209 q^{17} -3.79428 q^{19} +1.37400 q^{21} +2.13174 q^{23} -4.09768 q^{25} -5.65006 q^{27} +2.06862 q^{29} -0.698043 q^{31} -1.37400 q^{33} -0.949904 q^{35} +5.10641 q^{37} +1.37400 q^{39} -1.72658 q^{41} +0.353503 q^{43} +1.05641 q^{45} +12.2124 q^{47} +1.00000 q^{49} -7.24386 q^{51} +12.3677 q^{53} +0.949904 q^{55} -5.21334 q^{57} -0.154213 q^{59} +5.42802 q^{61} -1.11212 q^{63} -0.949904 q^{65} +1.39660 q^{67} +2.92901 q^{69} +3.90040 q^{71} +11.5476 q^{73} -5.63022 q^{75} -1.00000 q^{77} +13.0818 q^{79} -4.42683 q^{81} +12.1100 q^{83} +5.00798 q^{85} +2.84228 q^{87} +6.64156 q^{89} +1.00000 q^{91} -0.959112 q^{93} +3.60420 q^{95} +9.20030 q^{97} +1.11212 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + q^{3} + 3 q^{5} + 10 q^{7} + 5 q^{9}+O(q^{10}) \) 10 * q + q^3 + 3 * q^5 + 10 * q^7 + 5 * q^9	\( 10 q + q^{3} + 3 q^{5} + 10 q^{7} + 5 q^{9} - 10 q^{11} + 10 q^{13} + 9 q^{15} + 7 q^{17} + 17 q^{19} + q^{21} + 15 q^{23} + 13 q^{25} + 7 q^{27} - q^{29} - 6 q^{31} - q^{33} + 3 q^{35} - 12 q^{37} + q^{39} + 4 q^{41} + 17 q^{43} - 15 q^{45} + 21 q^{47} + 10 q^{49} + 3 q^{51} + 20 q^{53} - 3 q^{55} + 12 q^{57} + 9 q^{59} + 6 q^{61} + 5 q^{63} + 3 q^{65} + 26 q^{67} + 30 q^{69} + 18 q^{71} - 4 q^{73} + 48 q^{75} - 10 q^{77} + 19 q^{79} - 14 q^{81} + 40 q^{83} - 25 q^{85} - 25 q^{87} - 19 q^{89} + 10 q^{91} + q^{93} + 11 q^{95} - 22 q^{97} - 5 q^{99}+O(q^{100}) \) 10 * q + q^3 + 3 * q^5 + 10 * q^7 + 5 * q^9 - 10 * q^11 + 10 * q^13 + 9 * q^15 + 7 * q^17 + 17 * q^19 + q^21 + 15 * q^23 + 13 * q^25 + 7 * q^27 - q^29 - 6 * q^31 - q^33 + 3 * q^35 - 12 * q^37 + q^39 + 4 * q^41 + 17 * q^43 - 15 * q^45 + 21 * q^47 + 10 * q^49 + 3 * q^51 + 20 * q^53 - 3 * q^55 + 12 * q^57 + 9 * q^59 + 6 * q^61 + 5 * q^63 + 3 * q^65 + 26 * q^67 + 30 * q^69 + 18 * q^71 - 4 * q^73 + 48 * q^75 - 10 * q^77 + 19 * q^79 - 14 * q^81 + 40 * q^83 - 25 * q^85 - 25 * q^87 - 19 * q^89 + 10 * q^91 + q^93 + 11 * q^95 - 22 * q^97 - 5 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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