LMFDB - Embedded newform 8008.2.a.l.1.8

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:	$1$
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} - 9x^{6} + 12x^{5} + 24x^{4} - 10x^{3} - 18x^{2} - 2x + 1 $ x^8 - 2x^7 - 9x^6 + 12x^5 + 24x^4 - 10x^3 - 18x^2 - 2*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Root			$2.26876$ 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)

$f(q)$	$=$	$q+2.32056 q^{3} -4.00673 q^{5} +1.00000 q^{7} +2.38501 q^{9} +O(q^{10})$	$q+2.32056 q^{3} -4.00673 q^{5} +1.00000 q^{7} +2.38501 q^{9} +1.00000 q^{11} +1.00000 q^{13} -9.29787 q^{15} -2.56348 q^{17} -0.392314 q^{19} +2.32056 q^{21} +4.91549 q^{23} +11.0539 q^{25} -1.42712 q^{27} -10.5359 q^{29} +3.48242 q^{31} +2.32056 q^{33} -4.00673 q^{35} +11.2827 q^{37} +2.32056 q^{39} -9.26332 q^{41} -10.8987 q^{43} -9.55610 q^{45} -4.98149 q^{47} +1.00000 q^{49} -5.94871 q^{51} +3.32229 q^{53} -4.00673 q^{55} -0.910389 q^{57} -2.18420 q^{59} -3.81145 q^{61} +2.38501 q^{63} -4.00673 q^{65} -1.98464 q^{67} +11.4067 q^{69} +8.42487 q^{71} -15.3928 q^{73} +25.6512 q^{75} +1.00000 q^{77} +7.20901 q^{79} -10.4668 q^{81} +15.2313 q^{83} +10.2712 q^{85} -24.4491 q^{87} -6.35307 q^{89} +1.00000 q^{91} +8.08118 q^{93} +1.57189 q^{95} +9.91123 q^{97} +2.38501 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 5 q^{3} - 7 q^{5} + 8 q^{7} + 7 q^{9}+O(q^{10}) $ 8 * q - 5 * q^3 - 7 * q^5 + 8 * q^7 + 7 * q^9	$ 8 q - 5 q^{3} - 7 q^{5} + 8 q^{7} + 7 q^{9} + 8 q^{11} + 8 q^{13} - 5 q^{15} + 3 q^{17} - 19 q^{19} - 5 q^{21} + q^{23} + 15 q^{25} - 11 q^{27} - 21 q^{29} + 2 q^{31} - 5 q^{33} - 7 q^{35} - 12 q^{37} - 5 q^{39} - 6 q^{41} - 19 q^{43} - 17 q^{45} - 7 q^{47} + 8 q^{49} - 19 q^{51} - 26 q^{53} - 7 q^{55} + 2 q^{57} - 17 q^{59} + 7 q^{63} - 7 q^{65} - 24 q^{67} + 20 q^{69} + 2 q^{71} + 2 q^{73} + 18 q^{75} + 8 q^{77} + 7 q^{79} - 4 q^{81} - 6 q^{83} + 19 q^{85} - 13 q^{87} + 13 q^{89} + 8 q^{91} + 13 q^{93} + 21 q^{95} + 18 q^{97} + 7 q^{99}+O(q^{100}) $ 8 * q - 5 * q^3 - 7 * q^5 + 8 * q^7 + 7 * q^9 + 8 * q^11 + 8 * q^13 - 5 * q^15 + 3 * q^17 - 19 * q^19 - 5 * q^21 + q^23 + 15 * q^25 - 11 * q^27 - 21 * q^29 + 2 * q^31 - 5 * q^33 - 7 * q^35 - 12 * q^37 - 5 * q^39 - 6 * q^41 - 19 * q^43 - 17 * q^45 - 7 * q^47 + 8 * q^49 - 19 * q^51 - 26 * q^53 - 7 * q^55 + 2 * q^57 - 17 * q^59 + 7 * q^63 - 7 * q^65 - 24 * q^67 + 20 * q^69 + 2 * q^71 + 2 * q^73 + 18 * q^75 + 8 * q^77 + 7 * q^79 - 4 * q^81 - 6 * q^83 + 19 * q^85 - 13 * q^87 + 13 * q^89 + 8 * q^91 + 13 * q^93 + 21 * q^95 + 18 * q^97 + 7 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	2.32056	1.33978	0.669889	−	0.742461i	$-0.266342\pi$
$3$	2.32056	1.33978	0.669889	+	0.742461i	$0.266342\pi$
$4$	0	0
$5$	−4.00673	−1.79186	−0.895932	−	0.444192i	$-0.853491\pi$
$5$	−4.00673	−1.79186	−0.895932	+	0.444192i	$0.853491\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	2.38501	0.795004
$10$	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
$15$	−9.29787	−2.40070
$16$	0	0
$17$	−2.56348	−0.621734	−0.310867	−	0.950453i	$-0.600619\pi$
$17$	−2.56348	−0.621734	−0.310867	+	0.950453i	$0.600619\pi$
$18$	0	0
$19$	−0.392314	−0.0900029	−0.0450015	−	0.998987i	$-0.514329\pi$
$19$	−0.392314	−0.0900029	−0.0450015	+	0.998987i	$0.514329\pi$
$20$	0	0
$21$	2.32056	0.506388
$22$	0	0
$23$	4.91549	1.02495	0.512475	−	0.858702i	$-0.328729\pi$
$23$	4.91549	1.02495	0.512475	+	0.858702i	$0.328729\pi$
$24$	0	0
$25$	11.0539	2.21078
$26$	0	0
$27$	−1.42712	−0.274649
$28$	0	0
$29$	−10.5359	−1.95646	−0.978230	−	0.207524i	$-0.933459\pi$
$29$	−10.5359	−1.95646	−0.978230	+	0.207524i	$0.933459\pi$
$30$	0	0
$31$	3.48242	0.625461	0.312731	−	0.949842i	$-0.398756\pi$
$31$	3.48242	0.625461	0.312731	+	0.949842i	$0.398756\pi$
$32$	0	0
$33$	2.32056	0.403958
$34$	0	0
$35$	−4.00673	−0.677261
$36$	0	0
$37$	11.2827	1.85487	0.927435	−	0.373985i	$-0.122009\pi$
$37$	11.2827	1.85487	0.927435	+	0.373985i	$0.122009\pi$
$38$	0	0
$39$	2.32056	0.371587
$40$	0	0
$41$	−9.26332	−1.44669	−0.723344	−	0.690488i	$-0.757396\pi$
$41$	−9.26332	−1.44669	−0.723344	+	0.690488i	$0.757396\pi$
$42$	0	0
$43$	−10.8987	−1.66204	−0.831022	−	0.556240i	$-0.812243\pi$
$43$	−10.8987	−1.66204	−0.831022	+	0.556240i	$0.812243\pi$
$44$	0	0
$45$	−9.55610	−1.42454
$46$	0	0
$47$	−4.98149	−0.726624	−0.363312	−	0.931667i	$-0.618354\pi$
$47$	−4.98149	−0.726624	−0.363312	+	0.931667i	$0.618354\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−5.94871	−0.832986
$52$	0	0
$53$	3.32229	0.456351	0.228176	−	0.973620i	$-0.426724\pi$
$53$	3.32229	0.456351	0.228176	+	0.973620i	$0.426724\pi$
$54$	0	0
$55$	−4.00673	−0.540267
$56$	0	0
$57$	−0.910389	−0.120584
$58$	0	0
$59$	−2.18420	−0.284359	−0.142180	−	0.989841i	$-0.545411\pi$
$59$	−2.18420	−0.284359	−0.142180	+	0.989841i	$0.545411\pi$
$60$	0	0
$61$	−3.81145	−0.488007	−0.244003	−	0.969774i	$-0.578461\pi$
$61$	−3.81145	−0.488007	−0.244003	+	0.969774i	$0.578461\pi$
$62$	0	0
$63$	2.38501	0.300483
$64$	0	0
$65$	−4.00673	−0.496974
$66$	0	0
$67$	−1.98464	−0.242462	−0.121231	−	0.992624i	$-0.538684\pi$
$67$	−1.98464	−0.242462	−0.121231	+	0.992624i	$0.538684\pi$
$68$	0	0
$69$	11.4067	1.37321
$70$	0	0
$71$	8.42487	0.999849	0.499924	−	0.866069i	$-0.333361\pi$
$71$	8.42487	0.999849	0.499924	+	0.866069i	$0.333361\pi$
$72$	0	0
$73$	−15.3928	−1.80159	−0.900797	−	0.434241i	$-0.857016\pi$
$73$	−15.3928	−1.80159	−0.900797	+	0.434241i	$0.857016\pi$
$74$	0	0
$75$	25.6512	2.96195
$76$	0	0
$77$	1.00000	0.113961
$78$	0	0
$79$	7.20901	0.811076	0.405538	−	0.914078i	$-0.367084\pi$
$79$	7.20901	0.811076	0.405538	+	0.914078i	$0.367084\pi$
$80$	0	0
$81$	−10.4668	−1.16297
$82$	0	0
$83$	15.2313	1.67185	0.835926	−	0.548842i	$-0.184931\pi$
$83$	15.2313	1.67185	0.835926	+	0.548842i	$0.184931\pi$
$84$	0	0
$85$	10.2712	1.11406
$86$	0	0
$87$	−24.4491	−2.62122
$88$	0	0
$89$	−6.35307	−0.673424	−0.336712	−	0.941608i	$-0.609315\pi$
$89$	−6.35307	−0.673424	−0.336712	+	0.941608i	$0.609315\pi$
$90$	0	0
$91$	1.00000	0.104828
$92$	0	0
$93$	8.08118	0.837979
$94$	0	0
$95$	1.57189	0.161273
$96$	0	0
$97$	9.91123	1.00633	0.503166	−	0.864190i	$-0.332168\pi$
$97$	9.91123	1.00633	0.503166	+	0.864190i	$0.332168\pi$
$98$	0	0
$99$	2.38501	0.239703
$100$	0	0
$101$	−4.95092	−0.492634	−0.246317	−	0.969189i	$-0.579221\pi$
$101$	−4.95092	−0.492634	−0.246317	+	0.969189i	$0.579221\pi$
$102$	0	0
$103$	5.15461	0.507898	0.253949	−	0.967218i	$-0.418270\pi$
$103$	5.15461	0.507898	0.253949	+	0.967218i	$0.418270\pi$
$104$	0	0
$105$	−9.29787	−0.907379
$106$	0	0
$107$	12.9709	1.25395	0.626973	−	0.779041i	$-0.284294\pi$
$107$	12.9709	1.25395	0.626973	+	0.779041i	$0.284294\pi$
$108$	0	0
$109$	−10.4820	−1.00400	−0.501998	−	0.864869i	$-0.667402\pi$
$109$	−10.4820	−1.00400	−0.501998	+	0.864869i	$0.667402\pi$
$110$	0	0
$111$	26.1823	2.48511
$112$	0	0
$113$	−6.87596	−0.646836	−0.323418	−	0.946256i	$-0.604832\pi$
$113$	−6.87596	−0.646836	−0.323418	+	0.946256i	$0.604832\pi$
$114$	0	0
$115$	−19.6950	−1.83657
$116$	0	0
$117$	2.38501	0.220494
$118$	0	0
$119$	−2.56348	−0.234994
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−21.4961	−1.93824
$124$	0	0
$125$	−24.2562	−2.16954
$126$	0	0
$127$	−0.488074	−0.0433095	−0.0216548	−	0.999766i	$-0.506893\pi$
$127$	−0.488074	−0.0433095	−0.0216548	+	0.999766i	$0.506893\pi$
$128$	0	0
$129$	−25.2912	−2.22677
$130$	0	0
$131$	−20.1539	−1.76085	−0.880426	−	0.474184i	$-0.842743\pi$
$131$	−20.1539	−1.76085	−0.880426	+	0.474184i	$0.842743\pi$
$132$	0	0
$133$	−0.392314	−0.0340179
$134$	0	0
$135$	5.71807	0.492133
$136$	0	0
$137$	−4.42902	−0.378397	−0.189198	−	0.981939i	$-0.560589\pi$
$137$	−4.42902	−0.378397	−0.189198	+	0.981939i	$0.560589\pi$
$138$	0	0
$139$	−12.4510	−1.05608	−0.528038	−	0.849221i	$-0.677072\pi$
$139$	−12.4510	−1.05608	−0.528038	+	0.849221i	$0.677072\pi$
$140$	0	0
$141$	−11.5599	−0.973515
$142$	0	0
$143$	1.00000	0.0836242
$144$	0	0
$145$	42.2143	3.50571
$146$	0	0
$147$	2.32056	0.191397
$148$	0	0
$149$	−14.4315	−1.18228	−0.591138	−	0.806570i	$-0.701321\pi$
$149$	−14.4315	−1.18228	−0.591138	+	0.806570i	$0.701321\pi$
$150$	0	0
$151$	−1.22337	−0.0995561	−0.0497781	−	0.998760i	$-0.515851\pi$
$151$	−1.22337	−0.0995561	−0.0497781	+	0.998760i	$0.515851\pi$
$152$	0	0
$153$	−6.11392	−0.494281
$154$	0	0
$155$	−13.9531	−1.12074
$156$	0	0
$157$	−16.8606	−1.34562	−0.672810	−	0.739815i	$-0.734913\pi$
$157$	−16.8606	−1.34562	−0.672810	+	0.739815i	$0.734913\pi$
$158$	0	0
$159$	7.70958	0.611410
$160$	0	0
$161$	4.91549	0.387395
$162$	0	0
$163$	3.81980	0.299190	0.149595	−	0.988747i	$-0.452203\pi$
$163$	3.81980	0.299190	0.149595	+	0.988747i	$0.452203\pi$
$164$	0	0
$165$	−9.29787	−0.723838
$166$	0	0
$167$	−0.590145	−0.0456668	−0.0228334	−	0.999739i	$-0.507269\pi$
$167$	−0.590145	−0.0456668	−0.0228334	+	0.999739i	$0.507269\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−0.935673	−0.0715527
$172$	0	0
$173$	−19.9134	−1.51399	−0.756994	−	0.653421i	$-0.773333\pi$
$173$	−19.9134	−1.51399	−0.756994	+	0.653421i	$0.773333\pi$
$174$	0	0
$175$	11.0539	0.835595
$176$	0	0
$177$	−5.06858	−0.380978
$178$	0	0
$179$	−4.97040	−0.371505	−0.185753	−	0.982597i	$-0.559472\pi$
$179$	−4.97040	−0.371505	−0.185753	+	0.982597i	$0.559472\pi$
$180$	0	0
$181$	−6.37724	−0.474016	−0.237008	−	0.971508i	$-0.576167\pi$
$181$	−6.37724	−0.474016	−0.237008	+	0.971508i	$0.576167\pi$
$182$	0	0
$183$	−8.84472	−0.653821
$184$	0	0
$185$	−45.2068	−3.32367
$186$	0	0
$187$	−2.56348	−0.187460
$188$	0	0
$189$	−1.42712	−0.103808
$190$	0	0
$191$	−15.1091	−1.09326	−0.546628	−	0.837375i	$-0.684089\pi$
$191$	−15.1091	−1.09326	−0.546628	+	0.837375i	$0.684089\pi$
$192$	0	0
$193$	5.31135	0.382319	0.191160	−	0.981559i	$-0.438775\pi$
$193$	5.31135	0.382319	0.191160	+	0.981559i	$0.438775\pi$
$194$	0	0
$195$	−9.29787	−0.665834
$196$	0	0
$197$	12.1044	0.862403	0.431201	−	0.902256i	$-0.358090\pi$
$197$	12.1044	0.862403	0.431201	+	0.902256i	$0.358090\pi$
$198$	0	0
$199$	23.7910	1.68650	0.843248	−	0.537524i	$-0.180640\pi$
$199$	23.7910	1.68650	0.843248	+	0.537524i	$0.180640\pi$
$200$	0	0
$201$	−4.60547	−0.324845
$202$	0	0
$203$	−10.5359	−0.739472
$204$	0	0
$205$	37.1156	2.59227
$206$	0	0
$207$	11.7235	0.814840
$208$	0	0
$209$	−0.392314	−0.0271369
$210$	0	0
$211$	−3.19055	−0.219647	−0.109823	−	0.993951i	$-0.535029\pi$
$211$	−3.19055	−0.219647	−0.109823	+	0.993951i	$0.535029\pi$
$212$	0	0
$213$	19.5505	1.33957
$214$	0	0
$215$	43.6683	2.97815
$216$	0	0
$217$	3.48242	0.236402
$218$	0	0
$219$	−35.7200	−2.41373
$220$	0	0
$221$	−2.56348	−0.172438
$222$	0	0
$223$	0.772334	0.0517193	0.0258597	−	0.999666i	$-0.491768\pi$
$223$	0.772334	0.0517193	0.0258597	+	0.999666i	$0.491768\pi$
$224$	0	0
$225$	26.3636	1.75758
$226$	0	0
$227$	−4.56119	−0.302737	−0.151368	−	0.988477i	$-0.548368\pi$
$227$	−4.56119	−0.302737	−0.151368	+	0.988477i	$0.548368\pi$
$228$	0	0
$229$	−4.07816	−0.269493	−0.134746	−	0.990880i	$-0.543022\pi$
$229$	−4.07816	−0.269493	−0.134746	+	0.990880i	$0.543022\pi$
$230$	0	0
$231$	2.32056	0.152682
$232$	0	0
$233$	−26.6878	−1.74838	−0.874189	−	0.485585i	$-0.838607\pi$
$233$	−26.6878	−1.74838	−0.874189	+	0.485585i	$0.838607\pi$
$234$	0	0
$235$	19.9595	1.30201
$236$	0	0
$237$	16.7290	1.08666
$238$	0	0
$239$	−10.1738	−0.658088	−0.329044	−	0.944315i	$-0.606727\pi$
$239$	−10.1738	−0.658088	−0.329044	+	0.944315i	$0.606727\pi$
$240$	0	0
$241$	−21.0742	−1.35751	−0.678755	−	0.734364i	$-0.737480\pi$
$241$	−21.0742	−1.35751	−0.678755	+	0.734364i	$0.737480\pi$
$242$	0	0
$243$	−20.0074	−1.28348
$244$	0	0
$245$	−4.00673	−0.255981
$246$	0	0
$247$	−0.392314	−0.0249623
$248$	0	0
$249$	35.3452	2.23991
$250$	0	0
$251$	8.70093	0.549198	0.274599	−	0.961559i	$-0.411455\pi$
$251$	8.70093	0.549198	0.274599	+	0.961559i	$0.411455\pi$
$252$	0	0
$253$	4.91549	0.309034
$254$	0	0
$255$	23.8349	1.49260
$256$	0	0
$257$	−24.8427	−1.54965	−0.774823	−	0.632178i	$-0.782161\pi$
$257$	−24.8427	−1.54965	−0.774823	+	0.632178i	$0.782161\pi$
$258$	0	0
$259$	11.2827	0.701075
$260$	0	0
$261$	−25.1281	−1.55539
$262$	0	0
$263$	−5.82154	−0.358971	−0.179486	−	0.983761i	$-0.557443\pi$
$263$	−5.82154	−0.358971	−0.179486	+	0.983761i	$0.557443\pi$
$264$	0	0
$265$	−13.3115	−0.817720
$266$	0	0
$267$	−14.7427	−0.902238
$268$	0	0
$269$	−27.2781	−1.66318	−0.831589	−	0.555392i	$-0.812568\pi$
$269$	−27.2781	−1.66318	−0.831589	+	0.555392i	$0.812568\pi$
$270$	0	0
$271$	21.9511	1.33343	0.666717	−	0.745311i	$-0.267699\pi$
$271$	21.9511	1.33343	0.666717	+	0.745311i	$0.267699\pi$
$272$	0	0
$273$	2.32056	0.140447
$274$	0	0
$275$	11.0539	0.666574
$276$	0	0
$277$	−5.98284	−0.359474	−0.179737	−	0.983715i	$-0.557525\pi$
$277$	−5.98284	−0.359474	−0.179737	+	0.983715i	$0.557525\pi$
$278$	0	0
$279$	8.30562	0.497244
$280$	0	0
$281$	−21.4694	−1.28076	−0.640379	−	0.768059i	$-0.721223\pi$
$281$	−21.4694	−1.28076	−0.640379	+	0.768059i	$0.721223\pi$
$282$	0	0
$283$	3.10223	0.184408	0.0922042	−	0.995740i	$-0.470609\pi$
$283$	3.10223	0.184408	0.0922042	+	0.995740i	$0.470609\pi$
$284$	0	0
$285$	3.64768	0.216070
$286$	0	0
$287$	−9.26332	−0.546797
$288$	0	0
$289$	−10.4286	−0.613446
$290$	0	0
$291$	22.9996	1.34826
$292$	0	0
$293$	9.18764	0.536748	0.268374	−	0.963315i	$-0.413514\pi$
$293$	9.18764	0.536748	0.268374	+	0.963315i	$0.413514\pi$
$294$	0	0
$295$	8.75151	0.509533
$296$	0	0
$297$	−1.42712	−0.0828098
$298$	0	0
$299$	4.91549	0.284270
$300$	0	0
$301$	−10.8987	−0.628193
$302$	0	0
$303$	−11.4889	−0.660021
$304$	0	0
$305$	15.2715	0.874442
$306$	0	0
$307$	5.30868	0.302982	0.151491	−	0.988459i	$-0.451592\pi$
$307$	5.30868	0.302982	0.151491	+	0.988459i	$0.451592\pi$
$308$	0	0
$309$	11.9616	0.680471
$310$	0	0
$311$	31.8669	1.80700	0.903502	−	0.428583i	$-0.140987\pi$
$311$	31.8669	1.80700	0.903502	+	0.428583i	$0.140987\pi$
$312$	0	0
$313$	−16.9461	−0.957848	−0.478924	−	0.877856i	$-0.658973\pi$
$313$	−16.9461	−0.957848	−0.478924	+	0.877856i	$0.658973\pi$
$314$	0	0
$315$	−9.55610	−0.538425
$316$	0	0
$317$	6.90319	0.387722	0.193861	−	0.981029i	$-0.437899\pi$
$317$	6.90319	0.387722	0.193861	+	0.981029i	$0.437899\pi$
$318$	0	0
$319$	−10.5359	−0.589895
$320$	0	0
$321$	30.0998	1.68001
$322$	0	0
$323$	1.00569	0.0559579
$324$	0	0
$325$	11.0539	0.613159
$326$	0	0
$327$	−24.3242	−1.34513
$328$	0	0
$329$	−4.98149	−0.274638
$330$	0	0
$331$	−10.6016	−0.582715	−0.291358	−	0.956614i	$-0.594107\pi$
$331$	−10.6016	−0.582715	−0.291358	+	0.956614i	$0.594107\pi$
$332$	0	0
$333$	26.9094	1.47463
$334$	0	0
$335$	7.95190	0.434458
$336$	0	0
$337$	9.69190	0.527951	0.263976	−	0.964529i	$-0.414966\pi$
$337$	9.69190	0.527951	0.263976	+	0.964529i	$0.414966\pi$
$338$	0	0
$339$	−15.9561	−0.866617
$340$	0	0
$341$	3.48242	0.188584
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	−45.7036	−2.46060
$346$	0	0
$347$	19.6823	1.05660	0.528300	−	0.849058i	$-0.322830\pi$
$347$	19.6823	1.05660	0.528300	+	0.849058i	$0.322830\pi$
$348$	0	0
$349$	−34.3241	−1.83733	−0.918664	−	0.395041i	$-0.870730\pi$
$349$	−34.3241	−1.83733	−0.918664	+	0.395041i	$0.870730\pi$
$350$	0	0
$351$	−1.42712	−0.0761739
$352$	0	0
$353$	−14.3423	−0.763362	−0.381681	−	0.924294i	$-0.624655\pi$
$353$	−14.3423	−0.763362	−0.381681	+	0.924294i	$0.624655\pi$
$354$	0	0
$355$	−33.7562	−1.79159
$356$	0	0
$357$	−5.94871	−0.314839
$358$	0	0
$359$	−6.24537	−0.329618	−0.164809	−	0.986326i	$-0.552701\pi$
$359$	−6.24537	−0.329618	−0.164809	+	0.986326i	$0.552701\pi$
$360$	0	0
$361$	−18.8461	−0.991899
$362$	0	0
$363$	2.32056	0.121798
$364$	0	0
$365$	61.6748	3.22821
$366$	0	0
$367$	27.4456	1.43265	0.716326	−	0.697766i	$-0.245823\pi$
$367$	27.4456	1.43265	0.716326	+	0.697766i	$0.245823\pi$
$368$	0	0
$369$	−22.0931	−1.15012
$370$	0	0
$371$	3.32229	0.172485
$372$	0	0
$373$	−3.54890	−0.183755	−0.0918775	−	0.995770i	$-0.529287\pi$
$373$	−3.54890	−0.183755	−0.0918775	+	0.995770i	$0.529287\pi$
$374$	0	0
$375$	−56.2881	−2.90671
$376$	0	0
$377$	−10.5359	−0.542624
$378$	0	0
$379$	−21.5265	−1.10574	−0.552871	−	0.833267i	$-0.686468\pi$
$379$	−21.5265	−1.10574	−0.552871	+	0.833267i	$0.686468\pi$
$380$	0	0
$381$	−1.13261	−0.0580252
$382$	0	0
$383$	2.70270	0.138101	0.0690506	−	0.997613i	$-0.478003\pi$
$383$	2.70270	0.138101	0.0690506	+	0.997613i	$0.478003\pi$
$384$	0	0
$385$	−4.00673	−0.204202
$386$	0	0
$387$	−25.9936	−1.32133
$388$	0	0
$389$	−2.97539	−0.150858	−0.0754292	−	0.997151i	$-0.524033\pi$
$389$	−2.97539	−0.150858	−0.0754292	+	0.997151i	$0.524033\pi$
$390$	0	0
$391$	−12.6007	−0.637247
$392$	0	0
$393$	−46.7683	−2.35915
$394$	0	0
$395$	−28.8845	−1.45334
$396$	0	0
$397$	3.88351	0.194908	0.0974540	−	0.995240i	$-0.468930\pi$
$397$	3.88351	0.194908	0.0974540	+	0.995240i	$0.468930\pi$
$398$	0	0
$399$	−0.910389	−0.0455764
$400$	0	0
$401$	−27.6753	−1.38204	−0.691020	−	0.722836i	$-0.742838\pi$
$401$	−27.6753	−1.38204	−0.691020	+	0.722836i	$0.742838\pi$
$402$	0	0
$403$	3.48242	0.173472
$404$	0	0
$405$	41.9374	2.08389
$406$	0	0
$407$	11.2827	0.559264
$408$	0	0
$409$	32.2370	1.59402	0.797008	−	0.603969i	$-0.206415\pi$
$409$	32.2370	1.59402	0.797008	+	0.603969i	$0.206415\pi$
$410$	0	0
$411$	−10.2778	−0.506967
$412$	0	0
$413$	−2.18420	−0.107478
$414$	0	0
$415$	−61.0277	−2.99573
$416$	0	0
$417$	−28.8932	−1.41491
$418$	0	0
$419$	−28.1083	−1.37318	−0.686589	−	0.727046i	$-0.740893\pi$
$419$	−28.1083	−1.37318	−0.686589	+	0.727046i	$0.740893\pi$
$420$	0	0
$421$	14.0614	0.685313	0.342656	−	0.939461i	$-0.388673\pi$
$421$	14.0614	0.685313	0.342656	+	0.939461i	$0.388673\pi$
$422$	0	0
$423$	−11.8809	−0.577669
$424$	0	0
$425$	−28.3364	−1.37452
$426$	0	0
$427$	−3.81145	−0.184449
$428$	0	0
$429$	2.32056	0.112038
$430$	0	0
$431$	34.2180	1.64822	0.824111	−	0.566428i	$-0.191675\pi$
$431$	34.2180	1.64822	0.824111	+	0.566428i	$0.191675\pi$
$432$	0	0
$433$	−8.12488	−0.390457	−0.195228	−	0.980758i	$-0.562545\pi$
$433$	−8.12488	−0.390457	−0.195228	+	0.980758i	$0.562545\pi$
$434$	0	0
$435$	97.9610	4.69687
$436$	0	0
$437$	−1.92841	−0.0922486
$438$	0	0
$439$	−20.7647	−0.991045	−0.495522	−	0.868595i	$-0.665023\pi$
$439$	−20.7647	−0.991045	−0.495522	+	0.868595i	$0.665023\pi$
$440$	0	0
$441$	2.38501	0.113572
$442$	0	0
$443$	11.7064	0.556186	0.278093	−	0.960554i	$-0.410298\pi$
$443$	11.7064	0.556186	0.278093	+	0.960554i	$0.410298\pi$
$444$	0	0
$445$	25.4550	1.20668
$446$	0	0
$447$	−33.4893	−1.58399
$448$	0	0
$449$	16.6895	0.787628	0.393814	−	0.919190i	$-0.371155\pi$
$449$	16.6895	0.787628	0.393814	+	0.919190i	$0.371155\pi$
$450$	0	0
$451$	−9.26332	−0.436193
$452$	0	0
$453$	−2.83890	−0.133383
$454$	0	0
$455$	−4.00673	−0.187838
$456$	0	0
$457$	41.6170	1.94676	0.973381	−	0.229195i	$-0.0736093\pi$
$457$	41.6170	1.94676	0.973381	+	0.229195i	$0.0736093\pi$
$458$	0	0
$459$	3.65838	0.170759
$460$	0	0
$461$	3.55766	0.165697	0.0828484	−	0.996562i	$-0.473598\pi$
$461$	3.55766	0.165697	0.0828484	+	0.996562i	$0.473598\pi$
$462$	0	0
$463$	31.1321	1.44683	0.723416	−	0.690412i	$-0.242571\pi$
$463$	31.1321	1.44683	0.723416	+	0.690412i	$0.242571\pi$
$464$	0	0
$465$	−32.3791	−1.50154
$466$	0	0
$467$	25.3692	1.17395	0.586973	−	0.809606i	$-0.300319\pi$
$467$	25.3692	1.17395	0.586973	+	0.809606i	$0.300319\pi$
$468$	0	0
$469$	−1.98464	−0.0916419
$470$	0	0
$471$	−39.1260	−1.80283
$472$	0	0
$473$	−10.8987	−0.501125
$474$	0	0
$475$	−4.33659	−0.198976
$476$	0	0
$477$	7.92370	0.362801
$478$	0	0
$479$	16.2786	0.743790	0.371895	−	0.928275i	$-0.378708\pi$
$479$	16.2786	0.743790	0.371895	+	0.928275i	$0.378708\pi$
$480$	0	0
$481$	11.2827	0.514448
$482$	0	0
$483$	11.4067	0.519023
$484$	0	0
$485$	−39.7116	−1.80321
$486$	0	0
$487$	−36.0323	−1.63278	−0.816390	−	0.577501i	$-0.804028\pi$
$487$	−36.0323	−1.63278	−0.816390	+	0.577501i	$0.804028\pi$
$488$	0	0
$489$	8.86409	0.400848
$490$	0	0
$491$	−36.4962	−1.64705	−0.823525	−	0.567280i	$-0.807996\pi$
$491$	−36.4962	−1.64705	−0.823525	+	0.567280i	$0.807996\pi$
$492$	0	0
$493$	27.0084	1.21640
$494$	0	0
$495$	−9.55610	−0.429515
$496$	0	0
$497$	8.42487	0.377907
$498$	0	0
$499$	16.1932	0.724906	0.362453	−	0.932002i	$-0.381939\pi$
$499$	16.1932	0.724906	0.362453	+	0.932002i	$0.381939\pi$
$500$	0	0
$501$	−1.36947	−0.0611834
$502$	0	0
$503$	−21.8276	−0.973245	−0.486623	−	0.873612i	$-0.661771\pi$
$503$	−21.8276	−0.973245	−0.486623	+	0.873612i	$0.661771\pi$
$504$	0	0
$505$	19.8370	0.882734
$506$	0	0
$507$	2.32056	0.103060
$508$	0	0
$509$	27.2036	1.20578	0.602889	−	0.797825i	$-0.294016\pi$
$509$	27.2036	1.20578	0.602889	+	0.797825i	$0.294016\pi$
$510$	0	0
$511$	−15.3928	−0.680938
$512$	0	0
$513$	0.559878	0.0247192
$514$	0	0
$515$	−20.6531	−0.910085
$516$	0	0
$517$	−4.98149	−0.219086
$518$	0	0
$519$	−46.2103	−2.02841
$520$	0	0
$521$	9.23194	0.404459	0.202229	−	0.979338i	$-0.435181\pi$
$521$	9.23194	0.404459	0.202229	+	0.979338i	$0.435181\pi$
$522$	0	0
$523$	−17.9921	−0.786741	−0.393371	−	0.919380i	$-0.628691\pi$
$523$	−17.9921	−0.786741	−0.393371	+	0.919380i	$0.628691\pi$
$524$	0	0
$525$	25.6512	1.11951
$526$	0	0
$527$	−8.92710	−0.388871
$528$	0	0
$529$	1.16205	0.0505241
$530$	0	0
$531$	−5.20935	−0.226067
$532$	0	0
$533$	−9.26332	−0.401239
$534$	0	0
$535$	−51.9709	−2.24690
$536$	0	0
$537$	−11.5341	−0.497734
$538$	0	0
$539$	1.00000	0.0430730
$540$	0	0
$541$	5.47114	0.235223	0.117611	−	0.993060i	$-0.462476\pi$
$541$	5.47114	0.235223	0.117611	+	0.993060i	$0.462476\pi$
$542$	0	0
$543$	−14.7988	−0.635077
$544$	0	0
$545$	41.9987	1.79902
$546$	0	0
$547$	10.0450	0.429495	0.214748	−	0.976670i	$-0.431107\pi$
$547$	10.0450	0.429495	0.214748	+	0.976670i	$0.431107\pi$
$548$	0	0
$549$	−9.09037	−0.387967
$550$	0	0
$551$	4.13336	0.176087
$552$	0	0
$553$	7.20901	0.306558
$554$	0	0
$555$	−104.905	−4.45298
$556$	0	0
$557$	−2.10000	−0.0889798	−0.0444899	−	0.999010i	$-0.514166\pi$
$557$	−2.10000	−0.0889798	−0.0444899	+	0.999010i	$0.514166\pi$
$558$	0	0
$559$	−10.8987	−0.460968
$560$	0	0
$561$	−5.94871	−0.251155
$562$	0	0
$563$	−12.9870	−0.547336	−0.273668	−	0.961824i	$-0.588237\pi$
$563$	−12.9870	−0.547336	−0.273668	+	0.961824i	$0.588237\pi$
$564$	0	0
$565$	27.5501	1.15904
$566$	0	0
$567$	−10.4668	−0.439562
$568$	0	0
$569$	15.7698	0.661104	0.330552	−	0.943788i	$-0.392765\pi$
$569$	15.7698	0.661104	0.330552	+	0.943788i	$0.392765\pi$
$570$	0	0
$571$	−30.1155	−1.26029	−0.630147	−	0.776476i	$-0.717006\pi$
$571$	−30.1155	−1.26029	−0.630147	+	0.776476i	$0.717006\pi$
$572$	0	0
$573$	−35.0616	−1.46472
$574$	0	0
$575$	54.3352	2.26594
$576$	0	0
$577$	−22.0334	−0.917261	−0.458630	−	0.888627i	$-0.651660\pi$
$577$	−22.0334	−0.917261	−0.458630	+	0.888627i	$0.651660\pi$
$578$	0	0
$579$	12.3253	0.512223
$580$	0	0
$581$	15.2313	0.631901
$582$	0	0
$583$	3.32229	0.137595
$584$	0	0
$585$	−9.55610	−0.395096
$586$	0	0
$587$	−36.0185	−1.48664	−0.743322	−	0.668934i	$-0.766751\pi$
$587$	−36.0185	−1.48664	−0.743322	+	0.668934i	$0.766751\pi$
$588$	0	0
$589$	−1.36620	−0.0562933
$590$	0	0
$591$	28.0890	1.15543
$592$	0	0
$593$	26.9159	1.10530	0.552652	−	0.833412i	$-0.313616\pi$
$593$	26.9159	1.10530	0.552652	+	0.833412i	$0.313616\pi$
$594$	0	0
$595$	10.2712	0.421076
$596$	0	0
$597$	55.2084	2.25953
$598$	0	0
$599$	5.47314	0.223627	0.111813	−	0.993729i	$-0.464334\pi$
$599$	5.47314	0.223627	0.111813	+	0.993729i	$0.464334\pi$
$600$	0	0
$601$	−28.2099	−1.15071	−0.575353	−	0.817905i	$-0.695135\pi$
$601$	−28.2099	−1.15071	−0.575353	+	0.817905i	$0.695135\pi$
$602$	0	0
$603$	−4.73338	−0.192758
$604$	0	0
$605$	−4.00673	−0.162897
$606$	0	0
$607$	11.8051	0.479155	0.239577	−	0.970877i	$-0.422991\pi$
$607$	11.8051	0.479155	0.239577	+	0.970877i	$0.422991\pi$
$608$	0	0
$609$	−24.4491	−0.990728
$610$	0	0
$611$	−4.98149	−0.201529
$612$	0	0
$613$	−26.6356	−1.07580	−0.537900	−	0.843009i	$-0.680782\pi$
$613$	−26.6356	−1.07580	−0.537900	+	0.843009i	$0.680782\pi$
$614$	0	0
$615$	86.1291	3.47306
$616$	0	0
$617$	29.8659	1.20235	0.601177	−	0.799116i	$-0.294698\pi$
$617$	29.8659	1.20235	0.601177	+	0.799116i	$0.294698\pi$
$618$	0	0
$619$	−41.7789	−1.67924	−0.839618	−	0.543177i	$-0.817221\pi$
$619$	−41.7789	−1.67924	−0.839618	+	0.543177i	$0.817221\pi$
$620$	0	0
$621$	−7.01499	−0.281502
$622$	0	0
$623$	−6.35307	−0.254530
$624$	0	0
$625$	41.9188	1.67675
$626$	0	0
$627$	−0.910389	−0.0363574
$628$	0	0
$629$	−28.9230	−1.15324
$630$	0	0
$631$	11.8623	0.472230	0.236115	−	0.971725i	$-0.424126\pi$
$631$	11.8623	0.472230	0.236115	+	0.971725i	$0.424126\pi$
$632$	0	0
$633$	−7.40388	−0.294278
$634$	0	0
$635$	1.95558	0.0776048
$636$	0	0
$637$	1.00000	0.0396214
$638$	0	0
$639$	20.0934	0.794884
$640$	0	0
$641$	3.96782	0.156720	0.0783598	−	0.996925i	$-0.475032\pi$
$641$	3.96782	0.156720	0.0783598	+	0.996925i	$0.475032\pi$
$642$	0	0
$643$	40.7858	1.60843	0.804217	−	0.594336i	$-0.202585\pi$
$643$	40.7858	1.60843	0.804217	+	0.594336i	$0.202585\pi$
$644$	0	0
$645$	101.335	3.99007
$646$	0	0
$647$	14.4704	0.568891	0.284446	−	0.958692i	$-0.408191\pi$
$647$	14.4704	0.568891	0.284446	+	0.958692i	$0.408191\pi$
$648$	0	0
$649$	−2.18420	−0.0857375
$650$	0	0
$651$	8.08118	0.316726
$652$	0	0
$653$	12.3302	0.482520	0.241260	−	0.970461i	$-0.422439\pi$
$653$	12.3302	0.482520	0.241260	+	0.970461i	$0.422439\pi$
$654$	0	0
$655$	80.7511	3.15520
$656$	0	0
$657$	−36.7121	−1.43227
$658$	0	0
$659$	11.0320	0.429746	0.214873	−	0.976642i	$-0.431066\pi$
$659$	11.0320	0.429746	0.214873	+	0.976642i	$0.431066\pi$
$660$	0	0
$661$	−6.76898	−0.263283	−0.131641	−	0.991297i	$-0.542025\pi$
$661$	−6.76898	−0.263283	−0.131641	+	0.991297i	$0.542025\pi$
$662$	0	0
$663$	−5.94871	−0.231029
$664$	0	0
$665$	1.57189	0.0609555
$666$	0	0
$667$	−51.7889	−2.00527
$668$	0	0
$669$	1.79225	0.0692924
$670$	0	0
$671$	−3.81145	−0.147140
$672$	0	0
$673$	29.6495	1.14291	0.571453	−	0.820635i	$-0.306380\pi$
$673$	29.6495	1.14291	0.571453	+	0.820635i	$0.306380\pi$
$674$	0	0
$675$	−15.7752	−0.607187
$676$	0	0
$677$	5.72755	0.220128	0.110064	−	0.993925i	$-0.464894\pi$
$677$	5.72755	0.220128	0.110064	+	0.993925i	$0.464894\pi$
$678$	0	0
$679$	9.91123	0.380358
$680$	0	0
$681$	−10.5845	−0.405600
$682$	0	0
$683$	27.6536	1.05814	0.529068	−	0.848579i	$-0.322542\pi$
$683$	27.6536	1.05814	0.529068	+	0.848579i	$0.322542\pi$
$684$	0	0
$685$	17.7459	0.678035
$686$	0	0
$687$	−9.46364	−0.361060
$688$	0	0
$689$	3.32229	0.126569
$690$	0	0
$691$	−10.7215	−0.407865	−0.203932	−	0.978985i	$-0.565372\pi$
$691$	−10.7215	−0.407865	−0.203932	+	0.978985i	$0.565372\pi$
$692$	0	0
$693$	2.38501	0.0905991
$694$	0	0
$695$	49.8876	1.89234
$696$	0	0
$697$	23.7463	0.899455
$698$	0	0
$699$	−61.9308	−2.34244
$700$	0	0
$701$	47.1552	1.78103	0.890513	−	0.454958i	$-0.150346\pi$
$701$	47.1552	1.78103	0.890513	+	0.454958i	$0.150346\pi$
$702$	0	0
$703$	−4.42637	−0.166944
$704$	0	0
$705$	46.3172	1.74441
$706$	0	0
$707$	−4.95092	−0.186198
$708$	0	0
$709$	−29.9938	−1.12644	−0.563221	−	0.826307i	$-0.690438\pi$
$709$	−29.9938	−1.12644	−0.563221	+	0.826307i	$0.690438\pi$
$710$	0	0
$711$	17.1936	0.644809
$712$	0	0
$713$	17.1178	0.641067
$714$	0	0
$715$	−4.00673	−0.149843
$716$	0	0
$717$	−23.6089	−0.881692
$718$	0	0
$719$	−41.5460	−1.54940	−0.774701	−	0.632327i	$-0.782100\pi$
$719$	−41.5460	−1.54940	−0.774701	+	0.632327i	$0.782100\pi$
$720$	0	0
$721$	5.15461	0.191968
$722$	0	0
$723$	−48.9041	−1.81876
$724$	0	0
$725$	−116.462	−4.32529
$726$	0	0
$727$	44.1621	1.63788	0.818941	−	0.573878i	$-0.194562\pi$
$727$	44.1621	1.63788	0.818941	+	0.573878i	$0.194562\pi$
$728$	0	0
$729$	−15.0282	−0.556599
$730$	0	0
$731$	27.9387	1.03335
$732$	0	0
$733$	45.3133	1.67368	0.836842	−	0.547444i	$-0.184399\pi$
$733$	45.3133	1.67368	0.836842	+	0.547444i	$0.184399\pi$
$734$	0	0
$735$	−9.29787	−0.342957
$736$	0	0
$737$	−1.98464	−0.0731050
$738$	0	0
$739$	47.4693	1.74619	0.873094	−	0.487552i	$-0.162110\pi$
$739$	47.4693	1.74619	0.873094	+	0.487552i	$0.162110\pi$
$740$	0	0
$741$	−0.910389	−0.0334440
$742$	0	0
$743$	−4.23893	−0.155511	−0.0777557	−	0.996972i	$-0.524775\pi$
$743$	−4.23893	−0.155511	−0.0777557	+	0.996972i	$0.524775\pi$
$744$	0	0
$745$	57.8232	2.11848
$746$	0	0
$747$	36.3268	1.32913
$748$	0	0
$749$	12.9709	0.473947
$750$	0	0
$751$	3.11227	0.113569	0.0567843	−	0.998386i	$-0.481915\pi$
$751$	3.11227	0.113569	0.0567843	+	0.998386i	$0.481915\pi$
$752$	0	0
$753$	20.1911	0.735803
$754$	0	0
$755$	4.90170	0.178391
$756$	0	0
$757$	33.9244	1.23300	0.616502	−	0.787353i	$-0.288549\pi$
$757$	33.9244	1.23300	0.616502	+	0.787353i	$0.288549\pi$
$758$	0	0
$759$	11.4067	0.414037
$760$	0	0
$761$	53.4394	1.93718	0.968588	−	0.248672i	$-0.0799941\pi$
$761$	53.4394	1.93718	0.968588	+	0.248672i	$0.0799941\pi$
$762$	0	0
$763$	−10.4820	−0.379475
$764$	0	0
$765$	24.4968	0.885685
$766$	0	0
$767$	−2.18420	−0.0788670
$768$	0	0
$769$	14.9032	0.537425	0.268712	−	0.963220i	$-0.413402\pi$
$769$	14.9032	0.537425	0.268712	+	0.963220i	$0.413402\pi$
$770$	0	0
$771$	−57.6491	−2.07618
$772$	0	0
$773$	−35.4916	−1.27654	−0.638272	−	0.769811i	$-0.720351\pi$
$773$	−35.4916	−1.27654	−0.638272	+	0.769811i	$0.720351\pi$
$774$	0	0
$775$	38.4942	1.38275
$776$	0	0
$777$	26.1823	0.939284
$778$	0	0
$779$	3.63413	0.130206
$780$	0	0
$781$	8.42487	0.301466
$782$	0	0
$783$	15.0359	0.537340
$784$	0	0
$785$	67.5557	2.41117
$786$	0	0
$787$	36.5186	1.30175	0.650873	−	0.759186i	$-0.274403\pi$
$787$	36.5186	1.30175	0.650873	+	0.759186i	$0.274403\pi$
$788$	0	0
$789$	−13.5092	−0.480942
$790$	0	0
$791$	−6.87596	−0.244481
$792$	0	0
$793$	−3.81145	−0.135349
$794$	0	0
$795$	−30.8902	−1.09556
$796$	0	0
$797$	−0.496911	−0.0176015	−0.00880074	−	0.999961i	$-0.502801\pi$
$797$	−0.496911	−0.0176015	−0.00880074	+	0.999961i	$0.502801\pi$
$798$	0	0
$799$	12.7699	0.451767
$800$	0	0
$801$	−15.1521	−0.535375
$802$	0	0
$803$	−15.3928	−0.543201
$804$	0	0
$805$	−19.6950	−0.694159
$806$	0	0
$807$	−63.3006	−2.22829
$808$	0	0
$809$	8.08253	0.284167	0.142083	−	0.989855i	$-0.454620\pi$
$809$	8.08253	0.284167	0.142083	+	0.989855i	$0.454620\pi$
$810$	0	0
$811$	34.7835	1.22141	0.610707	−	0.791857i	$-0.290885\pi$
$811$	34.7835	1.22141	0.610707	+	0.791857i	$0.290885\pi$
$812$	0	0
$813$	50.9389	1.78651
$814$	0	0
$815$	−15.3049	−0.536108
$816$	0	0
$817$	4.27573	0.149589
$818$	0	0
$819$	2.38501	0.0833391
$820$	0	0
$821$	36.9545	1.28972	0.644860	−	0.764301i	$-0.276916\pi$
$821$	36.9545	1.28972	0.644860	+	0.764301i	$0.276916\pi$
$822$	0	0
$823$	40.5159	1.41230	0.706148	−	0.708064i	$-0.250431\pi$
$823$	40.5159	1.41230	0.706148	+	0.708064i	$0.250431\pi$
$824$	0	0
$825$	25.6512	0.893061
$826$	0	0
$827$	−26.5951	−0.924801	−0.462400	−	0.886671i	$-0.653012\pi$
$827$	−26.5951	−0.924801	−0.462400	+	0.886671i	$0.653012\pi$
$828$	0	0
$829$	−51.9434	−1.80407	−0.902035	−	0.431663i	$-0.857927\pi$
$829$	−51.9434	−1.80407	−0.902035	+	0.431663i	$0.857927\pi$
$830$	0	0
$831$	−13.8835	−0.481615
$832$	0	0
$833$	−2.56348	−0.0888192
$834$	0	0
$835$	2.36455	0.0818287
$836$	0	0
$837$	−4.96982	−0.171782
$838$	0	0
$839$	−23.3308	−0.805468	−0.402734	−	0.915317i	$-0.631940\pi$
$839$	−23.3308	−0.805468	−0.402734	+	0.915317i	$0.631940\pi$
$840$	0	0
$841$	82.0043	2.82773
$842$	0	0
$843$	−49.8212	−1.71593
$844$	0	0
$845$	−4.00673	−0.137836
$846$	0	0
$847$	1.00000	0.0343604
$848$	0	0
$849$	7.19892	0.247066
$850$	0	0
$851$	55.4602	1.90115
$852$	0	0
$853$	−5.64778	−0.193376	−0.0966882	−	0.995315i	$-0.530825\pi$
$853$	−5.64778	−0.193376	−0.0966882	+	0.995315i	$0.530825\pi$
$854$	0	0
$855$	3.74899	0.128213
$856$	0	0
$857$	39.6579	1.35469	0.677345	−	0.735666i	$-0.263131\pi$
$857$	39.6579	1.35469	0.677345	+	0.735666i	$0.263131\pi$
$858$	0	0
$859$	34.1399	1.16484	0.582419	−	0.812889i	$-0.302106\pi$
$859$	34.1399	1.16484	0.582419	+	0.812889i	$0.302106\pi$
$860$	0	0
$861$	−21.4961	−0.732586
$862$	0	0
$863$	17.2222	0.586252	0.293126	−	0.956074i	$-0.405304\pi$
$863$	17.2222	0.586252	0.293126	+	0.956074i	$0.405304\pi$
$864$	0	0
$865$	79.7876	2.71286
$866$	0	0
$867$	−24.2002	−0.821882
$868$	0	0
$869$	7.20901	0.244549
$870$	0	0
$871$	−1.98464	−0.0672468
$872$	0	0
$873$	23.6384	0.800039
$874$	0	0
$875$	−24.2562	−0.820011
$876$	0	0
$877$	6.44954	0.217786	0.108893	−	0.994054i	$-0.465269\pi$
$877$	6.44954	0.217786	0.108893	+	0.994054i	$0.465269\pi$
$878$	0	0
$879$	21.3205	0.719122
$880$	0	0
$881$	−2.79727	−0.0942423	−0.0471212	−	0.998889i	$-0.515005\pi$
$881$	−2.79727	−0.0942423	−0.0471212	+	0.998889i	$0.515005\pi$
$882$	0	0
$883$	9.63033	0.324086	0.162043	−	0.986784i	$-0.448192\pi$
$883$	9.63033	0.324086	0.162043	+	0.986784i	$0.448192\pi$
$884$	0	0
$885$	20.3084	0.682661
$886$	0	0
$887$	18.8977	0.634523	0.317261	−	0.948338i	$-0.397237\pi$
$887$	18.8977	0.634523	0.317261	+	0.948338i	$0.397237\pi$
$888$	0	0
$889$	−0.488074	−0.0163695
$890$	0	0
$891$	−10.4668	−0.350649
$892$	0	0
$893$	1.95431	0.0653983
$894$	0	0
$895$	19.9151	0.665687
$896$	0	0
$897$	11.4067	0.380859
$898$	0	0
$899$	−36.6903	−1.22369
$900$	0	0
$901$	−8.51661	−0.283729
$902$	0	0
$903$	−25.2912	−0.841639
$904$	0	0
$905$	25.5519	0.849373
$906$	0	0
$907$	−21.1641	−0.702742	−0.351371	−	0.936236i	$-0.614284\pi$
$907$	−21.1641	−0.702742	−0.351371	+	0.936236i	$0.614284\pi$
$908$	0	0
$909$	−11.8080	−0.391646
$910$	0	0
$911$	6.69583	0.221843	0.110921	−	0.993829i	$-0.464620\pi$
$911$	6.69583	0.221843	0.110921	+	0.993829i	$0.464620\pi$
$912$	0	0
$913$	15.2313	0.504082
$914$	0	0
$915$	35.4384	1.17156
$916$	0	0
$917$	−20.1539	−0.665539
$918$	0	0
$919$	−51.5838	−1.70159	−0.850796	−	0.525495i	$-0.823880\pi$
$919$	−51.5838	−1.70159	−0.850796	+	0.525495i	$0.823880\pi$
$920$	0	0
$921$	12.3191	0.405929
$922$	0	0
$923$	8.42487	0.277308
$924$	0	0
$925$	124.718	4.10070
$926$	0	0
$927$	12.2938	0.403781
$928$	0	0
$929$	−15.9466	−0.523191	−0.261596	−	0.965178i	$-0.584249\pi$
$929$	−15.9466	−0.523191	−0.261596	+	0.965178i	$0.584249\pi$
$930$	0	0
$931$	−0.392314	−0.0128576
$932$	0	0
$933$	73.9491	2.42098
$934$	0	0
$935$	10.2712	0.335903
$936$	0	0
$937$	2.86325	0.0935384	0.0467692	−	0.998906i	$-0.485107\pi$
$937$	2.86325	0.0935384	0.0467692	+	0.998906i	$0.485107\pi$
$938$	0	0
$939$	−39.3244	−1.28330
$940$	0	0
$941$	54.6952	1.78301	0.891506	−	0.453008i	$-0.149649\pi$
$941$	54.6952	1.78301	0.891506	+	0.453008i	$0.149649\pi$
$942$	0	0
$943$	−45.5338	−1.48278
$944$	0	0
$945$	5.71807	0.186009
$946$	0	0
$947$	36.0849	1.17260	0.586301	−	0.810093i	$-0.300584\pi$
$947$	36.0849	1.17260	0.586301	+	0.810093i	$0.300584\pi$
$948$	0	0
$949$	−15.3928	−0.499672
$950$	0	0
$951$	16.0193	0.519461
$952$	0	0
$953$	−38.9282	−1.26101	−0.630505	−	0.776185i	$-0.717152\pi$
$953$	−38.9282	−1.26101	−0.630505	+	0.776185i	$0.717152\pi$
$954$	0	0
$955$	60.5381	1.95897
$956$	0	0
$957$	−24.4491	−0.790328
$958$	0	0
$959$	−4.42902	−0.143020
$960$	0	0
$961$	−18.8727	−0.608798
$962$	0	0
$963$	30.9358	0.996892
$964$	0	0
$965$	−21.2811	−0.685064
$966$	0	0
$967$	30.7855	0.989995	0.494997	−	0.868894i	$-0.335169\pi$
$967$	30.7855	0.989995	0.494997	+	0.868894i	$0.335169\pi$
$968$	0	0
$969$	2.33376	0.0749712
$970$	0	0
$971$	8.64293	0.277365	0.138682	−	0.990337i	$-0.455713\pi$
$971$	8.64293	0.277365	0.138682	+	0.990337i	$0.455713\pi$
$972$	0	0
$973$	−12.4510	−0.399159
$974$	0	0
$975$	25.6512	0.821496
$976$	0	0
$977$	−8.40726	−0.268972	−0.134486	−	0.990915i	$-0.542938\pi$
$977$	−8.40726	−0.268972	−0.134486	+	0.990915i	$0.542938\pi$
$978$	0	0
$979$	−6.35307	−0.203045
$980$	0	0
$981$	−24.9998	−0.798181
$982$	0	0
$983$	2.33725	0.0745467	0.0372734	−	0.999305i	$-0.488133\pi$
$983$	2.33725	0.0745467	0.0372734	+	0.999305i	$0.488133\pi$
$984$	0	0
$985$	−48.4990	−1.54531
$986$	0	0
$987$	−11.5599	−0.367954
$988$	0	0
$989$	−53.5727	−1.70351
$990$	0	0
$991$	−11.1374	−0.353793	−0.176896	−	0.984229i	$-0.556606\pi$
$991$	−11.1374	−0.353793	−0.176896	+	0.984229i	$0.556606\pi$
$992$	0	0
$993$	−24.6016	−0.780709
$994$	0	0
$995$	−95.3240	−3.02197
$996$	0	0
$997$	−50.2917	−1.59275	−0.796377	−	0.604800i	$-0.793253\pi$
$997$	−50.2917	−1.59275	−0.796377	+	0.604800i	$0.793253\pi$
$998$	0	0
$999$	−16.1018	−0.509438

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8008.2.a.l.1.8	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8008.2.a.l.1.8	✓	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:	\( N \)	\(=\)	\( 8008 = 2^{3} \cdot 7 \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8008.a (trivial)

\(f(q)\)	\(=\)	\(q+2.32056 q^{3} -4.00673 q^{5} +1.00000 q^{7} +2.38501 q^{9} +O(q^{10})\)	\(q+2.32056 q^{3} -4.00673 q^{5} +1.00000 q^{7} +2.38501 q^{9} +1.00000 q^{11} +1.00000 q^{13} -9.29787 q^{15} -2.56348 q^{17} -0.392314 q^{19} +2.32056 q^{21} +4.91549 q^{23} +11.0539 q^{25} -1.42712 q^{27} -10.5359 q^{29} +3.48242 q^{31} +2.32056 q^{33} -4.00673 q^{35} +11.2827 q^{37} +2.32056 q^{39} -9.26332 q^{41} -10.8987 q^{43} -9.55610 q^{45} -4.98149 q^{47} +1.00000 q^{49} -5.94871 q^{51} +3.32229 q^{53} -4.00673 q^{55} -0.910389 q^{57} -2.18420 q^{59} -3.81145 q^{61} +2.38501 q^{63} -4.00673 q^{65} -1.98464 q^{67} +11.4067 q^{69} +8.42487 q^{71} -15.3928 q^{73} +25.6512 q^{75} +1.00000 q^{77} +7.20901 q^{79} -10.4668 q^{81} +15.2313 q^{83} +10.2712 q^{85} -24.4491 q^{87} -6.35307 q^{89} +1.00000 q^{91} +8.08118 q^{93} +1.57189 q^{95} +9.91123 q^{97} +2.38501 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 5 q^{3} - 7 q^{5} + 8 q^{7} + 7 q^{9}+O(q^{10}) \) 8 * q - 5 * q^3 - 7 * q^5 + 8 * q^7 + 7 * q^9	\( 8 q - 5 q^{3} - 7 q^{5} + 8 q^{7} + 7 q^{9} + 8 q^{11} + 8 q^{13} - 5 q^{15} + 3 q^{17} - 19 q^{19} - 5 q^{21} + q^{23} + 15 q^{25} - 11 q^{27} - 21 q^{29} + 2 q^{31} - 5 q^{33} - 7 q^{35} - 12 q^{37} - 5 q^{39} - 6 q^{41} - 19 q^{43} - 17 q^{45} - 7 q^{47} + 8 q^{49} - 19 q^{51} - 26 q^{53} - 7 q^{55} + 2 q^{57} - 17 q^{59} + 7 q^{63} - 7 q^{65} - 24 q^{67} + 20 q^{69} + 2 q^{71} + 2 q^{73} + 18 q^{75} + 8 q^{77} + 7 q^{79} - 4 q^{81} - 6 q^{83} + 19 q^{85} - 13 q^{87} + 13 q^{89} + 8 q^{91} + 13 q^{93} + 21 q^{95} + 18 q^{97} + 7 q^{99}+O(q^{100}) \) 8 * q - 5 * q^3 - 7 * q^5 + 8 * q^7 + 7 * q^9 + 8 * q^11 + 8 * q^13 - 5 * q^15 + 3 * q^17 - 19 * q^19 - 5 * q^21 + q^23 + 15 * q^25 - 11 * q^27 - 21 * q^29 + 2 * q^31 - 5 * q^33 - 7 * q^35 - 12 * q^37 - 5 * q^39 - 6 * q^41 - 19 * q^43 - 17 * q^45 - 7 * q^47 + 8 * q^49 - 19 * q^51 - 26 * q^53 - 7 * q^55 + 2 * q^57 - 17 * q^59 + 7 * q^63 - 7 * q^65 - 24 * q^67 + 20 * q^69 + 2 * q^71 + 2 * q^73 + 18 * q^75 + 8 * q^77 + 7 * q^79 - 4 * q^81 - 6 * q^83 + 19 * q^85 - 13 * q^87 + 13 * q^89 + 8 * q^91 + 13 * q^93 + 21 * q^95 + 18 * q^97 + 7 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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