LMFDB - Embedded newform 8008.2.a.y.1.10

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:	$14$
Coefficient field:	$\mathbb{Q}[x]/(x^{14} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{14} - 3 x^{13} - 27 x^{12} + 78 x^{11} + 273 x^{10} - 750 x^{9} - 1306 x^{8} + 3378 x^{7} + \cdots - 128 $ x^14 - 3x^13 - 27x^12 + 78x^11 + 273x^10 - 750x^9 - 1306x^8 + 3378x^7 + 2996x^6 - 7275x^5 - 2804x^4 + 6417x^3 + 538x^2 - 1032*x - 128
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{5} $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.10
Root			$-1.44419$ 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)

$f(q)$	$=$	$q+1.44419 q^{3} -3.20437 q^{5} +1.00000 q^{7} -0.914318 q^{9} +O(q^{10})$	$q+1.44419 q^{3} -3.20437 q^{5} +1.00000 q^{7} -0.914318 q^{9} -1.00000 q^{11} -1.00000 q^{13} -4.62772 q^{15} -2.31952 q^{17} +6.29372 q^{19} +1.44419 q^{21} -3.44251 q^{23} +5.26801 q^{25} -5.65302 q^{27} +7.75096 q^{29} +7.59336 q^{31} -1.44419 q^{33} -3.20437 q^{35} +4.91871 q^{37} -1.44419 q^{39} -6.73003 q^{41} +9.09518 q^{43} +2.92982 q^{45} +1.34640 q^{47} +1.00000 q^{49} -3.34982 q^{51} -7.22970 q^{53} +3.20437 q^{55} +9.08933 q^{57} -4.77419 q^{59} -1.83923 q^{61} -0.914318 q^{63} +3.20437 q^{65} +10.8247 q^{67} -4.97163 q^{69} -3.88452 q^{71} -11.1855 q^{73} +7.60800 q^{75} -1.00000 q^{77} -0.193203 q^{79} -5.42107 q^{81} -0.916052 q^{83} +7.43259 q^{85} +11.1939 q^{87} -9.66381 q^{89} -1.00000 q^{91} +10.9663 q^{93} -20.1674 q^{95} -15.9476 q^{97} +0.914318 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 14 q - 3 q^{3} - 6 q^{5} + 14 q^{7} + 21 q^{9}+O(q^{10}) $ 14 * q - 3 * q^3 - 6 * q^5 + 14 * q^7 + 21 * q^9	$ 14 q - 3 q^{3} - 6 q^{5} + 14 q^{7} + 21 q^{9} - 14 q^{11} - 14 q^{13} - 6 q^{15} - 6 q^{17} - 13 q^{19} - 3 q^{21} - 9 q^{23} + 22 q^{25} - 18 q^{27} + 2 q^{29} - 2 q^{31} + 3 q^{33} - 6 q^{35} - q^{37} + 3 q^{39} - 16 q^{41} - 15 q^{43} - 44 q^{45} - 8 q^{47} + 14 q^{49} - 14 q^{51} - 6 q^{53} + 6 q^{55} - 10 q^{57} - 36 q^{59} - 19 q^{61} + 21 q^{63} + 6 q^{65} - 34 q^{67} - q^{69} - 10 q^{71} + 9 q^{73} - 44 q^{75} - 14 q^{77} - q^{79} + 42 q^{81} - 56 q^{83} + 21 q^{85} - 5 q^{87} - 14 q^{89} - 14 q^{91} - 20 q^{93} + q^{95} - 14 q^{97} - 21 q^{99}+O(q^{100}) $ 14 * q - 3 * q^3 - 6 * q^5 + 14 * q^7 + 21 * q^9 - 14 * q^11 - 14 * q^13 - 6 * q^15 - 6 * q^17 - 13 * q^19 - 3 * q^21 - 9 * q^23 + 22 * q^25 - 18 * q^27 + 2 * q^29 - 2 * q^31 + 3 * q^33 - 6 * q^35 - q^37 + 3 * q^39 - 16 * q^41 - 15 * q^43 - 44 * q^45 - 8 * q^47 + 14 * q^49 - 14 * q^51 - 6 * q^53 + 6 * q^55 - 10 * q^57 - 36 * q^59 - 19 * q^61 + 21 * q^63 + 6 * q^65 - 34 * q^67 - q^69 - 10 * q^71 + 9 * q^73 - 44 * q^75 - 14 * q^77 - q^79 + 42 * q^81 - 56 * q^83 + 21 * q^85 - 5 * q^87 - 14 * q^89 - 14 * q^91 - 20 * q^93 + q^95 - 14 * q^97 - 21 * 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
$3$	1.44419	0.833803	0.416901	−	0.908952i	$-0.363116\pi$
$3$	1.44419	0.833803	0.416901	+	0.908952i	$0.363116\pi$
$4$	0	0
$5$	−3.20437	−1.43304	−0.716520	−	0.697567i	$-0.754266\pi$
$5$	−3.20437	−1.43304	−0.716520	+	0.697567i	$0.754266\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	−0.914318	−0.304773
$10$	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
$15$	−4.62772	−1.19487
$16$	0	0
$17$	−2.31952	−0.562565	−0.281283	−	0.959625i	$-0.590760\pi$
$17$	−2.31952	−0.562565	−0.281283	+	0.959625i	$0.590760\pi$
$18$	0	0
$19$	6.29372	1.44388	0.721940	−	0.691956i	$-0.243251\pi$
$19$	6.29372	1.44388	0.721940	+	0.691956i	$0.243251\pi$
$20$	0	0
$21$	1.44419	0.315148
$22$	0	0
$23$	−3.44251	−0.717812	−0.358906	−	0.933374i	$-0.616850\pi$
$23$	−3.44251	−0.717812	−0.358906	+	0.933374i	$0.616850\pi$
$24$	0	0
$25$	5.26801	1.05360
$26$	0	0
$27$	−5.65302	−1.08792
$28$	0	0
$29$	7.75096	1.43932	0.719659	−	0.694328i	$-0.244298\pi$
$29$	7.75096	1.43932	0.719659	+	0.694328i	$0.244298\pi$
$30$	0	0
$31$	7.59336	1.36381	0.681904	−	0.731441i	$-0.261152\pi$
$31$	7.59336	1.36381	0.681904	+	0.731441i	$0.261152\pi$
$32$	0	0
$33$	−1.44419	−0.251401
$34$	0	0
$35$	−3.20437	−0.541638
$36$	0	0
$37$	4.91871	0.808632	0.404316	−	0.914619i	$-0.367510\pi$
$37$	4.91871	0.808632	0.404316	+	0.914619i	$0.367510\pi$
$38$	0	0
$39$	−1.44419	−0.231255
$40$	0	0
$41$	−6.73003	−1.05105	−0.525527	−	0.850777i	$-0.676132\pi$
$41$	−6.73003	−1.05105	−0.525527	+	0.850777i	$0.676132\pi$
$42$	0	0
$43$	9.09518	1.38700	0.693501	−	0.720456i	$-0.256067\pi$
$43$	9.09518	1.38700	0.693501	+	0.720456i	$0.256067\pi$
$44$	0	0
$45$	2.92982	0.436751
$46$	0	0
$47$	1.34640	0.196392	0.0981961	−	0.995167i	$-0.468693\pi$
$47$	1.34640	0.196392	0.0981961	+	0.995167i	$0.468693\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−3.34982	−0.469068
$52$	0	0
$53$	−7.22970	−0.993076	−0.496538	−	0.868015i	$-0.665396\pi$
$53$	−7.22970	−0.993076	−0.496538	+	0.868015i	$0.665396\pi$
$54$	0	0
$55$	3.20437	0.432078
$56$	0	0
$57$	9.08933	1.20391
$58$	0	0
$59$	−4.77419	−0.621547	−0.310773	−	0.950484i	$-0.600588\pi$
$59$	−4.77419	−0.621547	−0.310773	+	0.950484i	$0.600588\pi$
$60$	0	0
$61$	−1.83923	−0.235490	−0.117745	−	0.993044i	$-0.537566\pi$
$61$	−1.83923	−0.235490	−0.117745	+	0.993044i	$0.537566\pi$
$62$	0	0
$63$	−0.914318	−0.115193
$64$	0	0
$65$	3.20437	0.397454
$66$	0	0
$67$	10.8247	1.32245	0.661226	−	0.750187i	$-0.270036\pi$
$67$	10.8247	1.32245	0.661226	+	0.750187i	$0.270036\pi$
$68$	0	0
$69$	−4.97163	−0.598514
$70$	0	0
$71$	−3.88452	−0.461008	−0.230504	−	0.973071i	$-0.574038\pi$
$71$	−3.88452	−0.461008	−0.230504	+	0.973071i	$0.574038\pi$
$72$	0	0
$73$	−11.1855	−1.30916	−0.654579	−	0.755993i	$-0.727154\pi$
$73$	−11.1855	−1.30916	−0.654579	+	0.755993i	$0.727154\pi$
$74$	0	0
$75$	7.60800	0.878496
$76$	0	0
$77$	−1.00000	−0.113961
$78$	0	0
$79$	−0.193203	−0.0217370	−0.0108685	−	0.999941i	$-0.503460\pi$
$79$	−0.193203	−0.0217370	−0.0108685	+	0.999941i	$0.503460\pi$
$80$	0	0
$81$	−5.42107	−0.602341
$82$	0	0
$83$	−0.916052	−0.100550	−0.0502749	−	0.998735i	$-0.516010\pi$
$83$	−0.916052	−0.100550	−0.0502749	+	0.998735i	$0.516010\pi$
$84$	0	0
$85$	7.43259	0.806178
$86$	0	0
$87$	11.1939	1.20011
$88$	0	0
$89$	−9.66381	−1.02436	−0.512181	−	0.858878i	$-0.671162\pi$
$89$	−9.66381	−1.02436	−0.512181	+	0.858878i	$0.671162\pi$
$90$	0	0
$91$	−1.00000	−0.104828
$92$	0	0
$93$	10.9663	1.13715
$94$	0	0
$95$	−20.1674	−2.06914
$96$	0	0
$97$	−15.9476	−1.61924	−0.809618	−	0.586957i	$-0.800326\pi$
$97$	−15.9476	−1.61924	−0.809618	+	0.586957i	$0.800326\pi$
$98$	0	0
$99$	0.914318	0.0918924
$100$	0	0
$101$	−19.0368	−1.89423	−0.947117	−	0.320890i	$-0.896018\pi$
$101$	−19.0368	−1.89423	−0.947117	+	0.320890i	$0.896018\pi$
$102$	0	0
$103$	8.07416	0.795570	0.397785	−	0.917479i	$-0.369779\pi$
$103$	8.07416	0.795570	0.397785	+	0.917479i	$0.369779\pi$
$104$	0	0
$105$	−4.62772	−0.451619
$106$	0	0
$107$	−11.5277	−1.11443	−0.557214	−	0.830369i	$-0.688130\pi$
$107$	−11.5277	−1.11443	−0.557214	+	0.830369i	$0.688130\pi$
$108$	0	0
$109$	−12.6152	−1.20831	−0.604157	−	0.796865i	$-0.706490\pi$
$109$	−12.6152	−1.20831	−0.604157	+	0.796865i	$0.706490\pi$
$110$	0	0
$111$	7.10355	0.674239
$112$	0	0
$113$	1.36869	0.128756	0.0643778	−	0.997926i	$-0.479494\pi$
$113$	1.36869	0.128756	0.0643778	+	0.997926i	$0.479494\pi$
$114$	0	0
$115$	11.0311	1.02865
$116$	0	0
$117$	0.914318	0.0845287
$118$	0	0
$119$	−2.31952	−0.212630
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−9.71944	−0.876372
$124$	0	0
$125$	−0.858794	−0.0768129
$126$	0	0
$127$	−5.79620	−0.514330	−0.257165	−	0.966368i	$-0.582788\pi$
$127$	−5.79620	−0.514330	−0.257165	+	0.966368i	$0.582788\pi$
$128$	0	0
$129$	13.1352	1.15649
$130$	0	0
$131$	−5.40661	−0.472378	−0.236189	−	0.971707i	$-0.575898\pi$
$131$	−5.40661	−0.472378	−0.236189	+	0.971707i	$0.575898\pi$
$132$	0	0
$133$	6.29372	0.545735
$134$	0	0
$135$	18.1144	1.55904
$136$	0	0
$137$	−3.40955	−0.291298	−0.145649	−	0.989336i	$-0.546527\pi$
$137$	−3.40955	−0.291298	−0.145649	+	0.989336i	$0.546527\pi$
$138$	0	0
$139$	14.4035	1.22169	0.610843	−	0.791752i	$-0.290831\pi$
$139$	14.4035	1.22169	0.610843	+	0.791752i	$0.290831\pi$
$140$	0	0
$141$	1.94445	0.163752
$142$	0	0
$143$	1.00000	0.0836242
$144$	0	0
$145$	−24.8370	−2.06260
$146$	0	0
$147$	1.44419	0.119115
$148$	0	0
$149$	−20.7281	−1.69811	−0.849057	−	0.528301i	$-0.822829\pi$
$149$	−20.7281	−1.69811	−0.849057	+	0.528301i	$0.822829\pi$
$150$	0	0
$151$	−3.57387	−0.290838	−0.145419	−	0.989370i	$-0.546453\pi$
$151$	−3.57387	−0.290838	−0.145419	+	0.989370i	$0.546453\pi$
$152$	0	0
$153$	2.12077	0.171454
$154$	0	0
$155$	−24.3320	−1.95439
$156$	0	0
$157$	−24.9355	−1.99007	−0.995035	−	0.0995229i	$-0.968268\pi$
$157$	−24.9355	−1.99007	−0.995035	+	0.0995229i	$0.968268\pi$
$158$	0	0
$159$	−10.4411	−0.828030
$160$	0	0
$161$	−3.44251	−0.271307
$162$	0	0
$163$	−8.63735	−0.676529	−0.338265	−	0.941051i	$-0.609840\pi$
$163$	−8.63735	−0.676529	−0.338265	+	0.941051i	$0.609840\pi$
$164$	0	0
$165$	4.62772	0.360268
$166$	0	0
$167$	4.69294	0.363150	0.181575	−	0.983377i	$-0.441880\pi$
$167$	4.69294	0.363150	0.181575	+	0.983377i	$0.441880\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−5.75447	−0.440055
$172$	0	0
$173$	5.64279	0.429013	0.214507	−	0.976723i	$-0.431186\pi$
$173$	5.64279	0.429013	0.214507	+	0.976723i	$0.431186\pi$
$174$	0	0
$175$	5.26801	0.398224
$176$	0	0
$177$	−6.89484	−0.518248
$178$	0	0
$179$	15.9075	1.18898	0.594491	−	0.804103i	$-0.297354\pi$
$179$	15.9075	1.18898	0.594491	+	0.804103i	$0.297354\pi$
$180$	0	0
$181$	−3.03636	−0.225691	−0.112845	−	0.993613i	$-0.535996\pi$
$181$	−3.03636	−0.225691	−0.112845	+	0.993613i	$0.535996\pi$
$182$	0	0
$183$	−2.65620	−0.196352
$184$	0	0
$185$	−15.7614	−1.15880
$186$	0	0
$187$	2.31952	0.169620
$188$	0	0
$189$	−5.65302	−0.411196
$190$	0	0
$191$	2.02901	0.146814	0.0734071	−	0.997302i	$-0.476613\pi$
$191$	2.02901	0.146814	0.0734071	+	0.997302i	$0.476613\pi$
$192$	0	0
$193$	−18.6853	−1.34500	−0.672499	−	0.740098i	$-0.734779\pi$
$193$	−18.6853	−1.34500	−0.672499	+	0.740098i	$0.734779\pi$
$194$	0	0
$195$	4.62772	0.331398
$196$	0	0
$197$	−2.29290	−0.163362	−0.0816811	−	0.996659i	$-0.526029\pi$
$197$	−2.29290	−0.163362	−0.0816811	+	0.996659i	$0.526029\pi$
$198$	0	0
$199$	−19.8927	−1.41015	−0.705077	−	0.709131i	$-0.749088\pi$
$199$	−19.8927	−1.41015	−0.705077	+	0.709131i	$0.749088\pi$
$200$	0	0
$201$	15.6330	1.10266
$202$	0	0
$203$	7.75096	0.544011
$204$	0	0
$205$	21.5655	1.50620
$206$	0	0
$207$	3.14754	0.218769
$208$	0	0
$209$	−6.29372	−0.435346
$210$	0	0
$211$	−12.4066	−0.854106	−0.427053	−	0.904227i	$-0.640448\pi$
$211$	−12.4066	−0.854106	−0.427053	+	0.904227i	$0.640448\pi$
$212$	0	0
$213$	−5.60999	−0.384390
$214$	0	0
$215$	−29.1443	−1.98763
$216$	0	0
$217$	7.59336	0.515471
$218$	0	0
$219$	−16.1539	−1.09158
$220$	0	0
$221$	2.31952	0.156027
$222$	0	0
$223$	3.84264	0.257322	0.128661	−	0.991689i	$-0.458932\pi$
$223$	3.84264	0.257322	0.128661	+	0.991689i	$0.458932\pi$
$224$	0	0
$225$	−4.81663	−0.321109
$226$	0	0
$227$	−14.2631	−0.946675	−0.473337	−	0.880881i	$-0.656951\pi$
$227$	−14.2631	−0.946675	−0.473337	+	0.880881i	$0.656951\pi$
$228$	0	0
$229$	21.2055	1.40130	0.700648	−	0.713507i	$-0.252894\pi$
$229$	21.2055	1.40130	0.700648	+	0.713507i	$0.252894\pi$
$230$	0	0
$231$	−1.44419	−0.0950207
$232$	0	0
$233$	−15.2038	−0.996037	−0.498019	−	0.867166i	$-0.665939\pi$
$233$	−15.2038	−0.996037	−0.498019	+	0.867166i	$0.665939\pi$
$234$	0	0
$235$	−4.31436	−0.281438
$236$	0	0
$237$	−0.279021	−0.0181244
$238$	0	0
$239$	28.1442	1.82050	0.910248	−	0.414063i	$-0.135891\pi$
$239$	28.1442	1.82050	0.910248	+	0.414063i	$0.135891\pi$
$240$	0	0
$241$	−3.48559	−0.224527	−0.112263	−	0.993678i	$-0.535810\pi$
$241$	−3.48559	−0.224527	−0.112263	+	0.993678i	$0.535810\pi$
$242$	0	0
$243$	9.13000	0.585690
$244$	0	0
$245$	−3.20437	−0.204720
$246$	0	0
$247$	−6.29372	−0.400460
$248$	0	0
$249$	−1.32295	−0.0838387
$250$	0	0
$251$	3.01109	0.190058	0.0950291	−	0.995474i	$-0.469706\pi$
$251$	3.01109	0.190058	0.0950291	+	0.995474i	$0.469706\pi$
$252$	0	0
$253$	3.44251	0.216428
$254$	0	0
$255$	10.7341	0.672193
$256$	0	0
$257$	24.4961	1.52803	0.764014	−	0.645200i	$-0.223226\pi$
$257$	24.4961	1.52803	0.764014	+	0.645200i	$0.223226\pi$
$258$	0	0
$259$	4.91871	0.305634
$260$	0	0
$261$	−7.08685	−0.438665
$262$	0	0
$263$	18.3192	1.12961	0.564804	−	0.825225i	$-0.308952\pi$
$263$	18.3192	1.12961	0.564804	+	0.825225i	$0.308952\pi$
$264$	0	0
$265$	23.1667	1.42312
$266$	0	0
$267$	−13.9564	−0.854115
$268$	0	0
$269$	13.8621	0.845189	0.422594	−	0.906319i	$-0.361119\pi$
$269$	13.8621	0.845189	0.422594	+	0.906319i	$0.361119\pi$
$270$	0	0
$271$	21.2214	1.28911	0.644555	−	0.764558i	$-0.277042\pi$
$271$	21.2214	1.28911	0.644555	+	0.764558i	$0.277042\pi$
$272$	0	0
$273$	−1.44419	−0.0874063
$274$	0	0
$275$	−5.26801	−0.317673
$276$	0	0
$277$	−10.3438	−0.621497	−0.310749	−	0.950492i	$-0.600580\pi$
$277$	−10.3438	−0.621497	−0.310749	+	0.950492i	$0.600580\pi$
$278$	0	0
$279$	−6.94275	−0.415652
$280$	0	0
$281$	−30.5867	−1.82465	−0.912324	−	0.409469i	$-0.865714\pi$
$281$	−30.5867	−1.82465	−0.912324	+	0.409469i	$0.865714\pi$
$282$	0	0
$283$	−5.97881	−0.355404	−0.177702	−	0.984084i	$-0.556866\pi$
$283$	−5.97881	−0.355404	−0.177702	+	0.984084i	$0.556866\pi$
$284$	0	0
$285$	−29.1256	−1.72525
$286$	0	0
$287$	−6.73003	−0.397261
$288$	0	0
$289$	−11.6198	−0.683520
$290$	0	0
$291$	−23.0314	−1.35012
$292$	0	0
$293$	−1.92545	−0.112486	−0.0562430	−	0.998417i	$-0.517912\pi$
$293$	−1.92545	−0.112486	−0.0562430	+	0.998417i	$0.517912\pi$
$294$	0	0
$295$	15.2983	0.890701
$296$	0	0
$297$	5.65302	0.328021
$298$	0	0
$299$	3.44251	0.199085
$300$	0	0
$301$	9.09518	0.524237
$302$	0	0
$303$	−27.4927	−1.57942
$304$	0	0
$305$	5.89359	0.337466
$306$	0	0
$307$	−2.64519	−0.150969	−0.0754845	−	0.997147i	$-0.524050\pi$
$307$	−2.64519	−0.150969	−0.0754845	+	0.997147i	$0.524050\pi$
$308$	0	0
$309$	11.6606	0.663349
$310$	0	0
$311$	19.4111	1.10070	0.550352	−	0.834932i	$-0.314493\pi$
$311$	19.4111	1.10070	0.550352	+	0.834932i	$0.314493\pi$
$312$	0	0
$313$	3.05781	0.172838	0.0864188	−	0.996259i	$-0.472458\pi$
$313$	3.05781	0.172838	0.0864188	+	0.996259i	$0.472458\pi$
$314$	0	0
$315$	2.92982	0.165076
$316$	0	0
$317$	−13.2197	−0.742491	−0.371245	−	0.928535i	$-0.621069\pi$
$317$	−13.2197	−0.742491	−0.371245	+	0.928535i	$0.621069\pi$
$318$	0	0
$319$	−7.75096	−0.433971
$320$	0	0
$321$	−16.6482	−0.929214
$322$	0	0
$323$	−14.5984	−0.812276
$324$	0	0
$325$	−5.26801	−0.292216
$326$	0	0
$327$	−18.2187	−1.00750
$328$	0	0
$329$	1.34640	0.0742293
$330$	0	0
$331$	−34.6991	−1.90723	−0.953617	−	0.301021i	$-0.902673\pi$
$331$	−34.6991	−1.90723	−0.953617	+	0.301021i	$0.902673\pi$
$332$	0	0
$333$	−4.49727	−0.246449
$334$	0	0
$335$	−34.6865	−1.89513
$336$	0	0
$337$	1.19120	0.0648890	0.0324445	−	0.999474i	$-0.489671\pi$
$337$	1.19120	0.0648890	0.0324445	+	0.999474i	$0.489671\pi$
$338$	0	0
$339$	1.97665	0.107357
$340$	0	0
$341$	−7.59336	−0.411204
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	15.9310	0.857694
$346$	0	0
$347$	−1.36302	−0.0731705	−0.0365852	−	0.999331i	$-0.511648\pi$
$347$	−1.36302	−0.0731705	−0.0365852	+	0.999331i	$0.511648\pi$
$348$	0	0
$349$	29.7802	1.59410	0.797050	−	0.603913i	$-0.206393\pi$
$349$	29.7802	1.59410	0.797050	+	0.603913i	$0.206393\pi$
$350$	0	0
$351$	5.65302	0.301736
$352$	0	0
$353$	−25.1209	−1.33705	−0.668526	−	0.743689i	$-0.733074\pi$
$353$	−25.1209	−1.33705	−0.668526	+	0.743689i	$0.733074\pi$
$354$	0	0
$355$	12.4475	0.660643
$356$	0	0
$357$	−3.34982	−0.177291
$358$	0	0
$359$	26.6694	1.40756	0.703779	−	0.710419i	$-0.251495\pi$
$359$	26.6694	1.40756	0.703779	+	0.710419i	$0.251495\pi$
$360$	0	0
$361$	20.6110	1.08479
$362$	0	0
$363$	1.44419	0.0758003
$364$	0	0
$365$	35.8424	1.87608
$366$	0	0
$367$	7.86807	0.410710	0.205355	−	0.978688i	$-0.434165\pi$
$367$	7.86807	0.410710	0.205355	+	0.978688i	$0.434165\pi$
$368$	0	0
$369$	6.15339	0.320333
$370$	0	0
$371$	−7.22970	−0.375348
$372$	0	0
$373$	21.9264	1.13530	0.567652	−	0.823269i	$-0.307852\pi$
$373$	21.9264	1.13530	0.567652	+	0.823269i	$0.307852\pi$
$374$	0	0
$375$	−1.24026	−0.0640468
$376$	0	0
$377$	−7.75096	−0.399195
$378$	0	0
$379$	−15.4705	−0.794667	−0.397333	−	0.917674i	$-0.630064\pi$
$379$	−15.4705	−0.794667	−0.397333	+	0.917674i	$0.630064\pi$
$380$	0	0
$381$	−8.37081	−0.428850
$382$	0	0
$383$	−19.2393	−0.983080	−0.491540	−	0.870855i	$-0.663566\pi$
$383$	−19.2393	−0.983080	−0.491540	+	0.870855i	$0.663566\pi$
$384$	0	0
$385$	3.20437	0.163310
$386$	0	0
$387$	−8.31588	−0.422720
$388$	0	0
$389$	2.09352	0.106145	0.0530727	−	0.998591i	$-0.483098\pi$
$389$	2.09352	0.106145	0.0530727	+	0.998591i	$0.483098\pi$
$390$	0	0
$391$	7.98494	0.403816
$392$	0	0
$393$	−7.80817	−0.393870
$394$	0	0
$395$	0.619093	0.0311500
$396$	0	0
$397$	−2.42903	−0.121910	−0.0609549	−	0.998141i	$-0.519415\pi$
$397$	−2.42903	−0.121910	−0.0609549	+	0.998141i	$0.519415\pi$
$398$	0	0
$399$	9.08933	0.455036
$400$	0	0
$401$	37.1875	1.85706	0.928528	−	0.371262i	$-0.121075\pi$
$401$	37.1875	1.85706	0.928528	+	0.371262i	$0.121075\pi$
$402$	0	0
$403$	−7.59336	−0.378252
$404$	0	0
$405$	17.3711	0.863178
$406$	0	0
$407$	−4.91871	−0.243812
$408$	0	0
$409$	−14.9890	−0.741159	−0.370579	−	0.928801i	$-0.620841\pi$
$409$	−14.9890	−0.741159	−0.370579	+	0.928801i	$0.620841\pi$
$410$	0	0
$411$	−4.92404	−0.242885
$412$	0	0
$413$	−4.77419	−0.234923
$414$	0	0
$415$	2.93537	0.144092
$416$	0	0
$417$	20.8013	1.01865
$418$	0	0
$419$	11.0522	0.539933	0.269967	−	0.962870i	$-0.412987\pi$
$419$	11.0522	0.539933	0.269967	+	0.962870i	$0.412987\pi$
$420$	0	0
$421$	−16.3196	−0.795368	−0.397684	−	0.917522i	$-0.630186\pi$
$421$	−16.3196	−0.795368	−0.397684	+	0.917522i	$0.630186\pi$
$422$	0	0
$423$	−1.23104	−0.0598550
$424$	0	0
$425$	−12.2192	−0.592719
$426$	0	0
$427$	−1.83923	−0.0890067
$428$	0	0
$429$	1.44419	0.0697261
$430$	0	0
$431$	17.8849	0.861485	0.430742	−	0.902475i	$-0.358252\pi$
$431$	17.8849	0.861485	0.430742	+	0.902475i	$0.358252\pi$
$432$	0	0
$433$	−17.6237	−0.846939	−0.423469	−	0.905910i	$-0.639188\pi$
$433$	−17.6237	−0.846939	−0.423469	+	0.905910i	$0.639188\pi$
$434$	0	0
$435$	−35.8693	−1.71980
$436$	0	0
$437$	−21.6662	−1.03643
$438$	0	0
$439$	−31.0473	−1.48181	−0.740903	−	0.671612i	$-0.765602\pi$
$439$	−31.0473	−1.48181	−0.740903	+	0.671612i	$0.765602\pi$
$440$	0	0
$441$	−0.914318	−0.0435390
$442$	0	0
$443$	−19.1022	−0.907575	−0.453787	−	0.891110i	$-0.649927\pi$
$443$	−19.1022	−0.907575	−0.453787	+	0.891110i	$0.649927\pi$
$444$	0	0
$445$	30.9664	1.46795
$446$	0	0
$447$	−29.9353	−1.41589
$448$	0	0
$449$	−10.0218	−0.472960	−0.236480	−	0.971636i	$-0.575994\pi$
$449$	−10.0218	−0.472960	−0.236480	+	0.971636i	$0.575994\pi$
$450$	0	0
$451$	6.73003	0.316905
$452$	0	0
$453$	−5.16135	−0.242501
$454$	0	0
$455$	3.20437	0.150223
$456$	0	0
$457$	−29.7473	−1.39152	−0.695760	−	0.718274i	$-0.744932\pi$
$457$	−29.7473	−1.39152	−0.695760	+	0.718274i	$0.744932\pi$
$458$	0	0
$459$	13.1123	0.612028
$460$	0	0
$461$	−34.5306	−1.60825	−0.804125	−	0.594460i	$-0.797366\pi$
$461$	−34.5306	−1.60825	−0.804125	+	0.594460i	$0.797366\pi$
$462$	0	0
$463$	20.2120	0.939332	0.469666	−	0.882844i	$-0.344374\pi$
$463$	20.2120	0.939332	0.469666	+	0.882844i	$0.344374\pi$
$464$	0	0
$465$	−35.1400	−1.62958
$466$	0	0
$467$	−21.6012	−0.999585	−0.499792	−	0.866145i	$-0.666590\pi$
$467$	−21.6012	−0.999585	−0.499792	+	0.866145i	$0.666590\pi$
$468$	0	0
$469$	10.8247	0.499840
$470$	0	0
$471$	−36.0116	−1.65933
$472$	0	0
$473$	−9.09518	−0.418197
$474$	0	0
$475$	33.1554	1.52127
$476$	0	0
$477$	6.61025	0.302663
$478$	0	0
$479$	−3.51296	−0.160511	−0.0802556	−	0.996774i	$-0.525574\pi$
$479$	−3.51296	−0.160511	−0.0802556	+	0.996774i	$0.525574\pi$
$480$	0	0
$481$	−4.91871	−0.224274
$482$	0	0
$483$	−4.97163	−0.226217
$484$	0	0
$485$	51.1022	2.32043
$486$	0	0
$487$	12.5414	0.568307	0.284154	−	0.958779i	$-0.408287\pi$
$487$	12.5414	0.568307	0.284154	+	0.958779i	$0.408287\pi$
$488$	0	0
$489$	−12.4740	−0.564092
$490$	0	0
$491$	3.99406	0.180250	0.0901248	−	0.995930i	$-0.471273\pi$
$491$	3.99406	0.180250	0.0901248	+	0.995930i	$0.471273\pi$
$492$	0	0
$493$	−17.9785	−0.809710
$494$	0	0
$495$	−2.92982	−0.131685
$496$	0	0
$497$	−3.88452	−0.174245
$498$	0	0
$499$	−33.4341	−1.49672	−0.748358	−	0.663295i	$-0.769157\pi$
$499$	−33.4341	−1.49672	−0.748358	+	0.663295i	$0.769157\pi$
$500$	0	0
$501$	6.77749	0.302796
$502$	0	0
$503$	30.2391	1.34830	0.674148	−	0.738596i	$-0.264511\pi$
$503$	30.2391	1.34830	0.674148	+	0.738596i	$0.264511\pi$
$504$	0	0
$505$	61.0010	2.71451
$506$	0	0
$507$	1.44419	0.0641387
$508$	0	0
$509$	−8.11821	−0.359833	−0.179917	−	0.983682i	$-0.557583\pi$
$509$	−8.11821	−0.359833	−0.179917	+	0.983682i	$0.557583\pi$
$510$	0	0
$511$	−11.1855	−0.494816
$512$	0	0
$513$	−35.5785	−1.57083
$514$	0	0
$515$	−25.8726	−1.14008
$516$	0	0
$517$	−1.34640	−0.0592145
$518$	0	0
$519$	8.14925	0.357713
$520$	0	0
$521$	−1.43863	−0.0630277	−0.0315139	−	0.999503i	$-0.510033\pi$
$521$	−1.43863	−0.0630277	−0.0315139	+	0.999503i	$0.510033\pi$
$522$	0	0
$523$	−29.7573	−1.30119	−0.650597	−	0.759423i	$-0.725481\pi$
$523$	−29.7573	−1.30119	−0.650597	+	0.759423i	$0.725481\pi$
$524$	0	0
$525$	7.60800	0.332040
$526$	0	0
$527$	−17.6129	−0.767231
$528$	0	0
$529$	−11.1492	−0.484746
$530$	0	0
$531$	4.36513	0.189431
$532$	0	0
$533$	6.73003	0.291510
$534$	0	0
$535$	36.9392	1.59702
$536$	0	0
$537$	22.9734	0.991376
$538$	0	0
$539$	−1.00000	−0.0430730
$540$	0	0
$541$	20.4480	0.879128	0.439564	−	0.898211i	$-0.355133\pi$
$541$	20.4480	0.879128	0.439564	+	0.898211i	$0.355133\pi$
$542$	0	0
$543$	−4.38508	−0.188182
$544$	0	0
$545$	40.4237	1.73156
$546$	0	0
$547$	33.2759	1.42278	0.711388	−	0.702799i	$-0.248067\pi$
$547$	33.2759	1.42278	0.711388	+	0.702799i	$0.248067\pi$
$548$	0	0
$549$	1.68164	0.0717708
$550$	0	0
$551$	48.7824	2.07820
$552$	0	0
$553$	−0.193203	−0.00821581
$554$	0	0
$555$	−22.7624	−0.966211
$556$	0	0
$557$	3.87173	0.164050	0.0820252	−	0.996630i	$-0.473861\pi$
$557$	3.87173	0.164050	0.0820252	+	0.996630i	$0.473861\pi$
$558$	0	0
$559$	−9.09518	−0.384685
$560$	0	0
$561$	3.34982	0.141429
$562$	0	0
$563$	−14.8940	−0.627708	−0.313854	−	0.949471i	$-0.601620\pi$
$563$	−14.8940	−0.627708	−0.313854	+	0.949471i	$0.601620\pi$
$564$	0	0
$565$	−4.38580	−0.184512
$566$	0	0
$567$	−5.42107	−0.227663
$568$	0	0
$569$	−29.9316	−1.25480	−0.627399	−	0.778698i	$-0.715880\pi$
$569$	−29.9316	−1.25480	−0.627399	+	0.778698i	$0.715880\pi$
$570$	0	0
$571$	−11.1058	−0.464762	−0.232381	−	0.972625i	$-0.574652\pi$
$571$	−11.1058	−0.464762	−0.232381	+	0.972625i	$0.574652\pi$
$572$	0	0
$573$	2.93028	0.122414
$574$	0	0
$575$	−18.1351	−0.756288
$576$	0	0
$577$	−33.5890	−1.39833	−0.699165	−	0.714961i	$-0.746445\pi$
$577$	−33.5890	−1.39833	−0.699165	+	0.714961i	$0.746445\pi$
$578$	0	0
$579$	−26.9851	−1.12146
$580$	0	0
$581$	−0.916052	−0.0380042
$582$	0	0
$583$	7.22970	0.299424
$584$	0	0
$585$	−2.92982	−0.121133
$586$	0	0
$587$	7.92290	0.327013	0.163507	−	0.986542i	$-0.447719\pi$
$587$	7.92290	0.327013	0.163507	+	0.986542i	$0.447719\pi$
$588$	0	0
$589$	47.7905	1.96917
$590$	0	0
$591$	−3.31138	−0.136212
$592$	0	0
$593$	18.3912	0.755235	0.377618	−	0.925962i	$-0.376743\pi$
$593$	18.3912	0.755235	0.377618	+	0.925962i	$0.376743\pi$
$594$	0	0
$595$	7.43259	0.304707
$596$	0	0
$597$	−28.7288	−1.17579
$598$	0	0
$599$	29.5765	1.20846	0.604232	−	0.796808i	$-0.293480\pi$
$599$	29.5765	1.20846	0.604232	+	0.796808i	$0.293480\pi$
$600$	0	0
$601$	12.3041	0.501895	0.250948	−	0.968001i	$-0.419258\pi$
$601$	12.3041	0.501895	0.250948	+	0.968001i	$0.419258\pi$
$602$	0	0
$603$	−9.89725	−0.403047
$604$	0	0
$605$	−3.20437	−0.130276
$606$	0	0
$607$	−31.0101	−1.25866	−0.629331	−	0.777137i	$-0.716671\pi$
$607$	−31.0101	−1.25866	−0.629331	+	0.777137i	$0.716671\pi$
$608$	0	0
$609$	11.1939	0.453598
$610$	0	0
$611$	−1.34640	−0.0544694
$612$	0	0
$613$	−44.2716	−1.78811	−0.894056	−	0.447955i	$-0.852152\pi$
$613$	−44.2716	−1.78811	−0.894056	+	0.447955i	$0.852152\pi$
$614$	0	0
$615$	31.1447	1.25588
$616$	0	0
$617$	41.0672	1.65330	0.826651	−	0.562715i	$-0.190243\pi$
$617$	41.0672	1.65330	0.826651	+	0.562715i	$0.190243\pi$
$618$	0	0
$619$	−25.9283	−1.04214	−0.521072	−	0.853513i	$-0.674468\pi$
$619$	−25.9283	−1.04214	−0.521072	+	0.853513i	$0.674468\pi$
$620$	0	0
$621$	19.4605	0.780924
$622$	0	0
$623$	−9.66381	−0.387172
$624$	0	0
$625$	−23.5881	−0.943525
$626$	0	0
$627$	−9.08933	−0.362993
$628$	0	0
$629$	−11.4090	−0.454908
$630$	0	0
$631$	35.3034	1.40541	0.702704	−	0.711483i	$-0.251976\pi$
$631$	35.3034	1.40541	0.702704	+	0.711483i	$0.251976\pi$
$632$	0	0
$633$	−17.9175	−0.712156
$634$	0	0
$635$	18.5732	0.737055
$636$	0	0
$637$	−1.00000	−0.0396214
$638$	0	0
$639$	3.55169	0.140503
$640$	0	0
$641$	44.0440	1.73963	0.869816	−	0.493376i	$-0.164237\pi$
$641$	44.0440	1.73963	0.869816	+	0.493376i	$0.164237\pi$
$642$	0	0
$643$	−17.3413	−0.683875	−0.341937	−	0.939723i	$-0.611083\pi$
$643$	−17.3413	−0.683875	−0.341937	+	0.939723i	$0.611083\pi$
$644$	0	0
$645$	−42.0899	−1.65729
$646$	0	0
$647$	−8.57704	−0.337198	−0.168599	−	0.985685i	$-0.553924\pi$
$647$	−8.57704	−0.337198	−0.168599	+	0.985685i	$0.553924\pi$
$648$	0	0
$649$	4.77419	0.187403
$650$	0	0
$651$	10.9663	0.429801
$652$	0	0
$653$	4.46090	0.174569	0.0872843	−	0.996183i	$-0.472181\pi$
$653$	4.46090	0.174569	0.0872843	+	0.996183i	$0.472181\pi$
$654$	0	0
$655$	17.3248	0.676936
$656$	0	0
$657$	10.2271	0.398996
$658$	0	0
$659$	−12.4968	−0.486805	−0.243402	−	0.969925i	$-0.578264\pi$
$659$	−12.4968	−0.486805	−0.243402	+	0.969925i	$0.578264\pi$
$660$	0	0
$661$	−7.91620	−0.307905	−0.153952	−	0.988078i	$-0.549200\pi$
$661$	−7.91620	−0.307905	−0.153952	+	0.988078i	$0.549200\pi$
$662$	0	0
$663$	3.34982	0.130096
$664$	0	0
$665$	−20.1674	−0.782060
$666$	0	0
$667$	−26.6827	−1.03316
$668$	0	0
$669$	5.54949	0.214556
$670$	0	0
$671$	1.83923	0.0710028
$672$	0	0
$673$	49.8822	1.92282	0.961408	−	0.275126i	$-0.0887195\pi$
$673$	49.8822	1.92282	0.961408	+	0.275126i	$0.0887195\pi$
$674$	0	0
$675$	−29.7801	−1.14624
$676$	0	0
$677$	13.7608	0.528870	0.264435	−	0.964404i	$-0.414815\pi$
$677$	13.7608	0.528870	0.264435	+	0.964404i	$0.414815\pi$
$678$	0	0
$679$	−15.9476	−0.612014
$680$	0	0
$681$	−20.5986	−0.789340
$682$	0	0
$683$	−9.42960	−0.360814	−0.180407	−	0.983592i	$-0.557741\pi$
$683$	−9.42960	−0.360814	−0.180407	+	0.983592i	$0.557741\pi$
$684$	0	0
$685$	10.9255	0.417441
$686$	0	0
$687$	30.6247	1.16841
$688$	0	0
$689$	7.22970	0.275430
$690$	0	0
$691$	27.3993	1.04232	0.521159	−	0.853460i	$-0.325500\pi$
$691$	27.3993	1.04232	0.521159	+	0.853460i	$0.325500\pi$
$692$	0	0
$693$	0.914318	0.0347321
$694$	0	0
$695$	−46.1541	−1.75072
$696$	0	0
$697$	15.6104	0.591286
$698$	0	0
$699$	−21.9572	−0.830499
$700$	0	0
$701$	−20.4288	−0.771587	−0.385793	−	0.922585i	$-0.626072\pi$
$701$	−20.4288	−0.771587	−0.385793	+	0.922585i	$0.626072\pi$
$702$	0	0
$703$	30.9570	1.16757
$704$	0	0
$705$	−6.23075	−0.234664
$706$	0	0
$707$	−19.0368	−0.715953
$708$	0	0
$709$	35.7380	1.34217	0.671085	−	0.741380i	$-0.265828\pi$
$709$	35.7380	1.34217	0.671085	+	0.741380i	$0.265828\pi$
$710$	0	0
$711$	0.176649	0.00662484
$712$	0	0
$713$	−26.1402	−0.978958
$714$	0	0
$715$	−3.20437	−0.119837
$716$	0	0
$717$	40.6455	1.51793
$718$	0	0
$719$	36.2923	1.35347	0.676737	−	0.736225i	$-0.263393\pi$
$719$	36.2923	1.35347	0.676737	+	0.736225i	$0.263393\pi$
$720$	0	0
$721$	8.07416	0.300697
$722$	0	0
$723$	−5.03385	−0.187211
$724$	0	0
$725$	40.8321	1.51647
$726$	0	0
$727$	−12.4117	−0.460326	−0.230163	−	0.973152i	$-0.573926\pi$
$727$	−12.4117	−0.460326	−0.230163	+	0.973152i	$0.573926\pi$
$728$	0	0
$729$	29.4486	1.09069
$730$	0	0
$731$	−21.0964	−0.780279
$732$	0	0
$733$	−53.4904	−1.97571	−0.987856	−	0.155372i	$-0.950342\pi$
$733$	−53.4904	−1.97571	−0.987856	+	0.155372i	$0.950342\pi$
$734$	0	0
$735$	−4.62772	−0.170696
$736$	0	0
$737$	−10.8247	−0.398734
$738$	0	0
$739$	−18.7385	−0.689306	−0.344653	−	0.938730i	$-0.612003\pi$
$739$	−18.7385	−0.689306	−0.344653	+	0.938730i	$0.612003\pi$
$740$	0	0
$741$	−9.08933	−0.333905
$742$	0	0
$743$	−12.1809	−0.446875	−0.223438	−	0.974718i	$-0.571728\pi$
$743$	−12.1809	−0.446875	−0.223438	+	0.974718i	$0.571728\pi$
$744$	0	0
$745$	66.4207	2.43346
$746$	0	0
$747$	0.837563	0.0306448
$748$	0	0
$749$	−11.5277	−0.421215
$750$	0	0
$751$	−1.39702	−0.0509781	−0.0254891	−	0.999675i	$-0.508114\pi$
$751$	−1.39702	−0.0509781	−0.0254891	+	0.999675i	$0.508114\pi$
$752$	0	0
$753$	4.34858	0.158471
$754$	0	0
$755$	11.4520	0.416782
$756$	0	0
$757$	28.5775	1.03867	0.519333	−	0.854572i	$-0.326180\pi$
$757$	28.5775	1.03867	0.519333	+	0.854572i	$0.326180\pi$
$758$	0	0
$759$	4.97163	0.180459
$760$	0	0
$761$	−7.46884	−0.270745	−0.135373	−	0.990795i	$-0.543223\pi$
$761$	−7.46884	−0.270745	−0.135373	+	0.990795i	$0.543223\pi$
$762$	0	0
$763$	−12.6152	−0.456700
$764$	0	0
$765$	−6.79575	−0.245701
$766$	0	0
$767$	4.77419	0.172386
$768$	0	0
$769$	−6.26487	−0.225917	−0.112958	−	0.993600i	$-0.536033\pi$
$769$	−6.26487	−0.225917	−0.112958	+	0.993600i	$0.536033\pi$
$770$	0	0
$771$	35.3771	1.27407
$772$	0	0
$773$	0.328063	0.0117996	0.00589980	−	0.999983i	$-0.498122\pi$
$773$	0.328063	0.0117996	0.00589980	+	0.999983i	$0.498122\pi$
$774$	0	0
$775$	40.0019	1.43691
$776$	0	0
$777$	7.10355	0.254839
$778$	0	0
$779$	−42.3570	−1.51760
$780$	0	0
$781$	3.88452	0.138999
$782$	0	0
$783$	−43.8163	−1.56587
$784$	0	0
$785$	79.9027	2.85185
$786$	0	0
$787$	−39.7572	−1.41719	−0.708596	−	0.705614i	$-0.750671\pi$
$787$	−39.7572	−1.41719	−0.708596	+	0.705614i	$0.750671\pi$
$788$	0	0
$789$	26.4563	0.941871
$790$	0	0
$791$	1.36869	0.0486650
$792$	0	0
$793$	1.83923	0.0653131
$794$	0	0
$795$	33.4571	1.18660
$796$	0	0
$797$	−4.10621	−0.145450	−0.0727248	−	0.997352i	$-0.523169\pi$
$797$	−4.10621	−0.145450	−0.0727248	+	0.997352i	$0.523169\pi$
$798$	0	0
$799$	−3.12299	−0.110483
$800$	0	0
$801$	8.83579	0.312197
$802$	0	0
$803$	11.1855	0.394726
$804$	0	0
$805$	11.0311	0.388794
$806$	0	0
$807$	20.0195	0.704721
$808$	0	0
$809$	36.6278	1.28776	0.643882	−	0.765125i	$-0.277323\pi$
$809$	36.6278	1.28776	0.643882	+	0.765125i	$0.277323\pi$
$810$	0	0
$811$	−1.01553	−0.0356602	−0.0178301	−	0.999841i	$-0.505676\pi$
$811$	−1.01553	−0.0356602	−0.0178301	+	0.999841i	$0.505676\pi$
$812$	0	0
$813$	30.6478	1.07486
$814$	0	0
$815$	27.6773	0.969493
$816$	0	0
$817$	57.2425	2.00266
$818$	0	0
$819$	0.914318	0.0319489
$820$	0	0
$821$	−27.9550	−0.975636	−0.487818	−	0.872945i	$-0.662207\pi$
$821$	−27.9550	−0.975636	−0.487818	+	0.872945i	$0.662207\pi$
$822$	0	0
$823$	15.8816	0.553597	0.276798	−	0.960928i	$-0.410727\pi$
$823$	15.8816	0.553597	0.276798	+	0.960928i	$0.410727\pi$
$824$	0	0
$825$	−7.60800	−0.264876
$826$	0	0
$827$	23.4062	0.813915	0.406957	−	0.913447i	$-0.366590\pi$
$827$	23.4062	0.813915	0.406957	+	0.913447i	$0.366590\pi$
$828$	0	0
$829$	−2.62109	−0.0910341	−0.0455171	−	0.998964i	$-0.514494\pi$
$829$	−2.62109	−0.0910341	−0.0455171	+	0.998964i	$0.514494\pi$
$830$	0	0
$831$	−14.9384	−0.518206
$832$	0	0
$833$	−2.31952	−0.0803664
$834$	0	0
$835$	−15.0379	−0.520409
$836$	0	0
$837$	−42.9254	−1.48372
$838$	0	0
$839$	−34.9194	−1.20555	−0.602776	−	0.797910i	$-0.705939\pi$
$839$	−34.9194	−1.20555	−0.602776	+	0.797910i	$0.705939\pi$
$840$	0	0
$841$	31.0775	1.07164
$842$	0	0
$843$	−44.1729	−1.52140
$844$	0	0
$845$	−3.20437	−0.110234
$846$	0	0
$847$	1.00000	0.0343604
$848$	0	0
$849$	−8.63454	−0.296336
$850$	0	0
$851$	−16.9327	−0.580445
$852$	0	0
$853$	−18.9813	−0.649907	−0.324954	−	0.945730i	$-0.605349\pi$
$853$	−18.9813	−0.649907	−0.324954	+	0.945730i	$0.605349\pi$
$854$	0	0
$855$	18.4395	0.630616
$856$	0	0
$857$	−31.9430	−1.09115	−0.545577	−	0.838061i	$-0.683689\pi$
$857$	−31.9430	−1.09115	−0.545577	+	0.838061i	$0.683689\pi$
$858$	0	0
$859$	−41.8276	−1.42714	−0.713570	−	0.700584i	$-0.752923\pi$
$859$	−41.8276	−1.42714	−0.713570	+	0.700584i	$0.752923\pi$
$860$	0	0
$861$	−9.71944	−0.331237
$862$	0	0
$863$	−45.1376	−1.53650	−0.768250	−	0.640149i	$-0.778872\pi$
$863$	−45.1376	−1.53650	−0.768250	+	0.640149i	$0.778872\pi$
$864$	0	0
$865$	−18.0816	−0.614793
$866$	0	0
$867$	−16.7813	−0.569921
$868$	0	0
$869$	0.193203	0.00655395
$870$	0	0
$871$	−10.8247	−0.366782
$872$	0	0
$873$	14.5812	0.493499
$874$	0	0
$875$	−0.858794	−0.0290325
$876$	0	0
$877$	57.8694	1.95411	0.977055	−	0.212985i	$-0.0683186\pi$
$877$	57.8694	1.95411	0.977055	+	0.212985i	$0.0683186\pi$
$878$	0	0
$879$	−2.78071	−0.0937912
$880$	0	0
$881$	41.5819	1.40093	0.700465	−	0.713687i	$-0.252976\pi$
$881$	41.5819	1.40093	0.700465	+	0.713687i	$0.252976\pi$
$882$	0	0
$883$	21.5021	0.723604	0.361802	−	0.932255i	$-0.382162\pi$
$883$	21.5021	0.723604	0.361802	+	0.932255i	$0.382162\pi$
$884$	0	0
$885$	22.0936	0.742669
$886$	0	0
$887$	−11.9618	−0.401638	−0.200819	−	0.979628i	$-0.564360\pi$
$887$	−11.9618	−0.401638	−0.200819	+	0.979628i	$0.564360\pi$
$888$	0	0
$889$	−5.79620	−0.194398
$890$	0	0
$891$	5.42107	0.181613
$892$	0	0
$893$	8.47385	0.283567
$894$	0	0
$895$	−50.9735	−1.70386
$896$	0	0
$897$	4.97163	0.165998
$898$	0	0
$899$	58.8559	1.96295
$900$	0	0
$901$	16.7694	0.558670
$902$	0	0
$903$	13.1352	0.437111
$904$	0	0
$905$	9.72962	0.323424
$906$	0	0
$907$	14.1790	0.470805	0.235402	−	0.971898i	$-0.424359\pi$
$907$	14.1790	0.470805	0.235402	+	0.971898i	$0.424359\pi$
$908$	0	0
$909$	17.4057	0.577311
$910$	0	0
$911$	12.5401	0.415472	0.207736	−	0.978185i	$-0.433390\pi$
$911$	12.5401	0.415472	0.207736	+	0.978185i	$0.433390\pi$
$912$	0	0
$913$	0.916052	0.0303169
$914$	0	0
$915$	8.51145	0.281380
$916$	0	0
$917$	−5.40661	−0.178542
$918$	0	0
$919$	44.3897	1.46428	0.732141	−	0.681153i	$-0.238521\pi$
$919$	44.3897	1.46428	0.732141	+	0.681153i	$0.238521\pi$
$920$	0	0
$921$	−3.82015	−0.125878
$922$	0	0
$923$	3.88452	0.127861
$924$	0	0
$925$	25.9118	0.851975
$926$	0	0
$927$	−7.38235	−0.242468
$928$	0	0
$929$	−13.8764	−0.455269	−0.227634	−	0.973747i	$-0.573099\pi$
$929$	−13.8764	−0.455269	−0.227634	+	0.973747i	$0.573099\pi$
$930$	0	0
$931$	6.29372	0.206268
$932$	0	0
$933$	28.0334	0.917771
$934$	0	0
$935$	−7.43259	−0.243072
$936$	0	0
$937$	51.7319	1.69001	0.845003	−	0.534761i	$-0.179598\pi$
$937$	51.7319	1.69001	0.845003	+	0.534761i	$0.179598\pi$
$938$	0	0
$939$	4.41605	0.144112
$940$	0	0
$941$	36.8613	1.20164	0.600821	−	0.799383i	$-0.294840\pi$
$941$	36.8613	1.20164	0.600821	+	0.799383i	$0.294840\pi$
$942$	0	0
$943$	23.1682	0.754459
$944$	0	0
$945$	18.1144	0.589260
$946$	0	0
$947$	17.5413	0.570014	0.285007	−	0.958525i	$-0.408004\pi$
$947$	17.5413	0.570014	0.285007	+	0.958525i	$0.408004\pi$
$948$	0	0
$949$	11.1855	0.363095
$950$	0	0
$951$	−19.0917	−0.619091
$952$	0	0
$953$	−21.5672	−0.698630	−0.349315	−	0.937005i	$-0.613586\pi$
$953$	−21.5672	−0.698630	−0.349315	+	0.937005i	$0.613586\pi$
$954$	0	0
$955$	−6.50172	−0.210391
$956$	0	0
$957$	−11.1939	−0.361846
$958$	0	0
$959$	−3.40955	−0.110100
$960$	0	0
$961$	26.6592	0.859973
$962$	0	0
$963$	10.5400	0.339648
$964$	0	0
$965$	59.8747	1.92743
$966$	0	0
$967$	−29.8799	−0.960874	−0.480437	−	0.877029i	$-0.659522\pi$
$967$	−29.8799	−0.960874	−0.480437	+	0.877029i	$0.659522\pi$
$968$	0	0
$969$	−21.0828	−0.677278
$970$	0	0
$971$	−16.3515	−0.524744	−0.262372	−	0.964967i	$-0.584505\pi$
$971$	−16.3515	−0.524744	−0.262372	+	0.964967i	$0.584505\pi$
$972$	0	0
$973$	14.4035	0.461754
$974$	0	0
$975$	−7.60800	−0.243651
$976$	0	0
$977$	−33.9649	−1.08663	−0.543317	−	0.839527i	$-0.682832\pi$
$977$	−33.9649	−1.08663	−0.543317	+	0.839527i	$0.682832\pi$
$978$	0	0
$979$	9.66381	0.308857
$980$	0	0
$981$	11.5343	0.368261
$982$	0	0
$983$	−44.2905	−1.41265	−0.706324	−	0.707889i	$-0.749648\pi$
$983$	−44.2905	−1.41265	−0.706324	+	0.707889i	$0.749648\pi$
$984$	0	0
$985$	7.34730	0.234104
$986$	0	0
$987$	1.94445	0.0618926
$988$	0	0
$989$	−31.3102	−0.995606
$990$	0	0
$991$	3.59295	0.114134	0.0570669	−	0.998370i	$-0.481825\pi$
$991$	3.59295	0.114134	0.0570669	+	0.998370i	$0.481825\pi$
$992$	0	0
$993$	−50.1121	−1.59026
$994$	0	0
$995$	63.7435	2.02081
$996$	0	0
$997$	5.16274	0.163506	0.0817528	−	0.996653i	$-0.473948\pi$
$997$	5.16274	0.163506	0.0817528	+	0.996653i	$0.473948\pi$
$998$	0	0
$999$	−27.8056	−0.879729

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8008.2.a.y.1.10	✓	14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8008.2.a.y.1.10	✓	14	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)

\(f(q)\)	\(=\)	\(q+1.44419 q^{3} -3.20437 q^{5} +1.00000 q^{7} -0.914318 q^{9} +O(q^{10})\)	\(q+1.44419 q^{3} -3.20437 q^{5} +1.00000 q^{7} -0.914318 q^{9} -1.00000 q^{11} -1.00000 q^{13} -4.62772 q^{15} -2.31952 q^{17} +6.29372 q^{19} +1.44419 q^{21} -3.44251 q^{23} +5.26801 q^{25} -5.65302 q^{27} +7.75096 q^{29} +7.59336 q^{31} -1.44419 q^{33} -3.20437 q^{35} +4.91871 q^{37} -1.44419 q^{39} -6.73003 q^{41} +9.09518 q^{43} +2.92982 q^{45} +1.34640 q^{47} +1.00000 q^{49} -3.34982 q^{51} -7.22970 q^{53} +3.20437 q^{55} +9.08933 q^{57} -4.77419 q^{59} -1.83923 q^{61} -0.914318 q^{63} +3.20437 q^{65} +10.8247 q^{67} -4.97163 q^{69} -3.88452 q^{71} -11.1855 q^{73} +7.60800 q^{75} -1.00000 q^{77} -0.193203 q^{79} -5.42107 q^{81} -0.916052 q^{83} +7.43259 q^{85} +11.1939 q^{87} -9.66381 q^{89} -1.00000 q^{91} +10.9663 q^{93} -20.1674 q^{95} -15.9476 q^{97} +0.914318 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q - 3 q^{3} - 6 q^{5} + 14 q^{7} + 21 q^{9}+O(q^{10}) \) 14 * q - 3 * q^3 - 6 * q^5 + 14 * q^7 + 21 * q^9	\( 14 q - 3 q^{3} - 6 q^{5} + 14 q^{7} + 21 q^{9} - 14 q^{11} - 14 q^{13} - 6 q^{15} - 6 q^{17} - 13 q^{19} - 3 q^{21} - 9 q^{23} + 22 q^{25} - 18 q^{27} + 2 q^{29} - 2 q^{31} + 3 q^{33} - 6 q^{35} - q^{37} + 3 q^{39} - 16 q^{41} - 15 q^{43} - 44 q^{45} - 8 q^{47} + 14 q^{49} - 14 q^{51} - 6 q^{53} + 6 q^{55} - 10 q^{57} - 36 q^{59} - 19 q^{61} + 21 q^{63} + 6 q^{65} - 34 q^{67} - q^{69} - 10 q^{71} + 9 q^{73} - 44 q^{75} - 14 q^{77} - q^{79} + 42 q^{81} - 56 q^{83} + 21 q^{85} - 5 q^{87} - 14 q^{89} - 14 q^{91} - 20 q^{93} + q^{95} - 14 q^{97} - 21 q^{99}+O(q^{100}) \) 14 * q - 3 * q^3 - 6 * q^5 + 14 * q^7 + 21 * q^9 - 14 * q^11 - 14 * q^13 - 6 * q^15 - 6 * q^17 - 13 * q^19 - 3 * q^21 - 9 * q^23 + 22 * q^25 - 18 * q^27 + 2 * q^29 - 2 * q^31 + 3 * q^33 - 6 * q^35 - q^37 + 3 * q^39 - 16 * q^41 - 15 * q^43 - 44 * q^45 - 8 * q^47 + 14 * q^49 - 14 * q^51 - 6 * q^53 + 6 * q^55 - 10 * q^57 - 36 * q^59 - 19 * q^61 + 21 * q^63 + 6 * q^65 - 34 * q^67 - q^69 - 10 * q^71 + 9 * q^73 - 44 * q^75 - 14 * q^77 - q^79 + 42 * q^81 - 56 * q^83 + 21 * q^85 - 5 * q^87 - 14 * q^89 - 14 * q^91 - 20 * q^93 + q^95 - 14 * q^97 - 21 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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