LMFDB - Embedded newform 8008.2.a.v.1.2

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:	$11$
Coefficient field:	$\mathbb{Q}[x]/(x^{11} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{11} - 2 x^{10} - 19 x^{9} + 33 x^{8} + 120 x^{7} - 178 x^{6} - 296 x^{5} + 380 x^{4} + 280 x^{3} + \cdots + 64 $ x^11 - 2x^10 - 19x^9 + 33x^8 + 120x^7 - 178x^6 - 296x^5 + 380x^4 + 280x^3 - 295x^2 - 87x + 64
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			1.2
Root			$2.73758$ 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)

$f(q)$	$=$	$q-2.73758 q^{3} +4.10495 q^{5} -1.00000 q^{7} +4.49435 q^{9} +O(q^{10})$	$q-2.73758 q^{3} +4.10495 q^{5} -1.00000 q^{7} +4.49435 q^{9} +1.00000 q^{11} -1.00000 q^{13} -11.2376 q^{15} -4.75014 q^{17} -6.50940 q^{19} +2.73758 q^{21} -1.11828 q^{23} +11.8506 q^{25} -4.09091 q^{27} +5.69879 q^{29} +9.59749 q^{31} -2.73758 q^{33} -4.10495 q^{35} +0.925635 q^{37} +2.73758 q^{39} -5.67379 q^{41} -5.94812 q^{43} +18.4491 q^{45} -1.11828 q^{47} +1.00000 q^{49} +13.0039 q^{51} -10.7803 q^{53} +4.10495 q^{55} +17.8200 q^{57} +7.08159 q^{59} +7.14995 q^{61} -4.49435 q^{63} -4.10495 q^{65} -11.8256 q^{67} +3.06139 q^{69} -7.59293 q^{71} +13.5277 q^{73} -32.4421 q^{75} -1.00000 q^{77} -16.4510 q^{79} -2.28386 q^{81} +0.448733 q^{83} -19.4991 q^{85} -15.6009 q^{87} -7.30086 q^{89} +1.00000 q^{91} -26.2739 q^{93} -26.7208 q^{95} -4.67291 q^{97} +4.49435 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 11 q - 2 q^{3} + 2 q^{5} - 11 q^{7} + 9 q^{9}+O(q^{10}) $ 11 * q - 2 * q^3 + 2 * q^5 - 11 * q^7 + 9 * q^9	$ 11 q - 2 q^{3} + 2 q^{5} - 11 q^{7} + 9 q^{9} + 11 q^{11} - 11 q^{13} - 7 q^{15} - 4 q^{17} - 16 q^{19} + 2 q^{21} - 3 q^{23} + 11 q^{25} - 11 q^{27} + q^{29} + 14 q^{31} - 2 q^{33} - 2 q^{35} - 8 q^{37} + 2 q^{39} + 4 q^{41} - 30 q^{43} + 13 q^{45} - 3 q^{47} + 11 q^{49} - 14 q^{51} - 5 q^{53} + 2 q^{55} - 22 q^{57} + 11 q^{59} + 15 q^{61} - 9 q^{63} - 2 q^{65} - 41 q^{67} + 12 q^{69} + q^{71} - 8 q^{73} - 24 q^{75} - 11 q^{77} - 26 q^{79} + 19 q^{81} - 31 q^{83} - 27 q^{85} - 25 q^{87} + 2 q^{89} + 11 q^{91} - 37 q^{93} - 6 q^{97} + 9 q^{99}+O(q^{100}) $ 11 * q - 2 * q^3 + 2 * q^5 - 11 * q^7 + 9 * q^9 + 11 * q^11 - 11 * q^13 - 7 * q^15 - 4 * q^17 - 16 * q^19 + 2 * q^21 - 3 * q^23 + 11 * q^25 - 11 * q^27 + q^29 + 14 * q^31 - 2 * q^33 - 2 * q^35 - 8 * q^37 + 2 * q^39 + 4 * q^41 - 30 * q^43 + 13 * q^45 - 3 * q^47 + 11 * q^49 - 14 * q^51 - 5 * q^53 + 2 * q^55 - 22 * q^57 + 11 * q^59 + 15 * q^61 - 9 * q^63 - 2 * q^65 - 41 * q^67 + 12 * q^69 + q^71 - 8 * q^73 - 24 * q^75 - 11 * q^77 - 26 * q^79 + 19 * q^81 - 31 * q^83 - 27 * q^85 - 25 * q^87 + 2 * q^89 + 11 * q^91 - 37 * q^93 - 6 * q^97 + 9 * 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
$3$	−2.73758	−1.58054	−0.790272	−	0.612757i	$-0.790061\pi$
$3$	−2.73758	−1.58054	−0.790272	+	0.612757i	$0.790061\pi$
$4$	0	0
$5$	4.10495	1.83579	0.917895	−	0.396823i	$-0.129887\pi$
$5$	4.10495	1.83579	0.917895	+	0.396823i	$0.129887\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	4.49435	1.49812
$10$	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
$15$	−11.2376	−2.90155
$16$	0	0
$17$	−4.75014	−1.15208	−0.576040	−	0.817422i	$-0.695403\pi$
$17$	−4.75014	−1.15208	−0.576040	+	0.817422i	$0.695403\pi$
$18$	0	0
$19$	−6.50940	−1.49336	−0.746680	−	0.665184i	$-0.768353\pi$
$19$	−6.50940	−1.49336	−0.746680	+	0.665184i	$0.768353\pi$
$20$	0	0
$21$	2.73758	0.597389
$22$	0	0
$23$	−1.11828	−0.233178	−0.116589	−	0.993180i	$-0.537196\pi$
$23$	−1.11828	−0.233178	−0.116589	+	0.993180i	$0.537196\pi$
$24$	0	0
$25$	11.8506	2.37013
$26$	0	0
$27$	−4.09091	−0.787296
$28$	0	0
$29$	5.69879	1.05824	0.529120	−	0.848547i	$-0.322522\pi$
$29$	5.69879	1.05824	0.529120	+	0.848547i	$0.322522\pi$
$30$	0	0
$31$	9.59749	1.72376	0.861880	−	0.507112i	$-0.169287\pi$
$31$	9.59749	1.72376	0.861880	+	0.507112i	$0.169287\pi$
$32$	0	0
$33$	−2.73758	−0.476552
$34$	0	0
$35$	−4.10495	−0.693864
$36$	0	0
$37$	0.925635	0.152174	0.0760868	−	0.997101i	$-0.475757\pi$
$37$	0.925635	0.152174	0.0760868	+	0.997101i	$0.475757\pi$
$38$	0	0
$39$	2.73758	0.438364
$40$	0	0
$41$	−5.67379	−0.886097	−0.443049	−	0.896498i	$-0.646103\pi$
$41$	−5.67379	−0.886097	−0.443049	+	0.896498i	$0.646103\pi$
$42$	0	0
$43$	−5.94812	−0.907080	−0.453540	−	0.891236i	$-0.649839\pi$
$43$	−5.94812	−0.907080	−0.453540	+	0.891236i	$0.649839\pi$
$44$	0	0
$45$	18.4491	2.75023
$46$	0	0
$47$	−1.11828	−0.163118	−0.0815591	−	0.996669i	$-0.525990\pi$
$47$	−1.11828	−0.163118	−0.0815591	+	0.996669i	$0.525990\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	13.0039	1.82091
$52$	0	0
$53$	−10.7803	−1.48079	−0.740396	−	0.672171i	$-0.765362\pi$
$53$	−10.7803	−1.48079	−0.740396	+	0.672171i	$0.765362\pi$
$54$	0	0
$55$	4.10495	0.553512
$56$	0	0
$57$	17.8200	2.36032
$58$	0	0
$59$	7.08159	0.921945	0.460972	−	0.887415i	$-0.347501\pi$
$59$	7.08159	0.921945	0.460972	+	0.887415i	$0.347501\pi$
$60$	0	0
$61$	7.14995	0.915457	0.457729	−	0.889092i	$-0.348663\pi$
$61$	7.14995	0.915457	0.457729	+	0.889092i	$0.348663\pi$
$62$	0	0
$63$	−4.49435	−0.566235
$64$	0	0
$65$	−4.10495	−0.509157
$66$	0	0
$67$	−11.8256	−1.44473	−0.722366	−	0.691511i	$-0.756946\pi$
$67$	−11.8256	−1.44473	−0.722366	+	0.691511i	$0.756946\pi$
$68$	0	0
$69$	3.06139	0.368548
$70$	0	0
$71$	−7.59293	−0.901115	−0.450557	−	0.892748i	$-0.648775\pi$
$71$	−7.59293	−0.901115	−0.450557	+	0.892748i	$0.648775\pi$
$72$	0	0
$73$	13.5277	1.58330	0.791651	−	0.610973i	$-0.209222\pi$
$73$	13.5277	1.58330	0.791651	+	0.610973i	$0.209222\pi$
$74$	0	0
$75$	−32.4421	−3.74609
$76$	0	0
$77$	−1.00000	−0.113961
$78$	0	0
$79$	−16.4510	−1.85089	−0.925443	−	0.378887i	$-0.876307\pi$
$79$	−16.4510	−1.85089	−0.925443	+	0.378887i	$0.876307\pi$
$80$	0	0
$81$	−2.28386	−0.253762
$82$	0	0
$83$	0.448733	0.0492548	0.0246274	−	0.999697i	$-0.492160\pi$
$83$	0.448733	0.0492548	0.0246274	+	0.999697i	$0.492160\pi$
$84$	0	0
$85$	−19.4991	−2.11498
$86$	0	0
$87$	−15.6009	−1.67259
$88$	0	0
$89$	−7.30086	−0.773890	−0.386945	−	0.922103i	$-0.626470\pi$
$89$	−7.30086	−0.773890	−0.386945	+	0.922103i	$0.626470\pi$
$90$	0	0
$91$	1.00000	0.104828
$92$	0	0
$93$	−26.2739	−2.72448
$94$	0	0
$95$	−26.7208	−2.74150
$96$	0	0
$97$	−4.67291	−0.474462	−0.237231	−	0.971453i	$-0.576240\pi$
$97$	−4.67291	−0.474462	−0.237231	+	0.971453i	$0.576240\pi$
$98$	0	0
$99$	4.49435	0.451699
$100$	0	0
$101$	−2.43553	−0.242344	−0.121172	−	0.992632i	$-0.538665\pi$
$101$	−2.43553	−0.242344	−0.121172	+	0.992632i	$0.538665\pi$
$102$	0	0
$103$	−3.10548	−0.305992	−0.152996	−	0.988227i	$-0.548892\pi$
$103$	−3.10548	−0.305992	−0.152996	+	0.988227i	$0.548892\pi$
$104$	0	0
$105$	11.2376	1.09668
$106$	0	0
$107$	−14.7739	−1.42824	−0.714121	−	0.700022i	$-0.753173\pi$
$107$	−14.7739	−1.42824	−0.714121	+	0.700022i	$0.753173\pi$
$108$	0	0
$109$	17.3006	1.65710	0.828549	−	0.559916i	$-0.189167\pi$
$109$	17.3006	1.65710	0.828549	+	0.559916i	$0.189167\pi$
$110$	0	0
$111$	−2.53400	−0.240517
$112$	0	0
$113$	11.0346	1.03805	0.519024	−	0.854760i	$-0.326296\pi$
$113$	11.0346	1.03805	0.519024	+	0.854760i	$0.326296\pi$
$114$	0	0
$115$	−4.59050	−0.428066
$116$	0	0
$117$	−4.49435	−0.415503
$118$	0	0
$119$	4.75014	0.435445
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	15.5325	1.40052
$124$	0	0
$125$	28.1215	2.51527
$126$	0	0
$127$	0.0460941	0.00409019	0.00204510	−	0.999998i	$-0.499349\pi$
$127$	0.0460941	0.00409019	0.00204510	+	0.999998i	$0.499349\pi$
$128$	0	0
$129$	16.2835	1.43368
$130$	0	0
$131$	−4.33688	−0.378915	−0.189457	−	0.981889i	$-0.560673\pi$
$131$	−4.33688	−0.378915	−0.189457	+	0.981889i	$0.560673\pi$
$132$	0	0
$133$	6.50940	0.564437
$134$	0	0
$135$	−16.7930	−1.44531
$136$	0	0
$137$	−17.5442	−1.49890	−0.749451	−	0.662060i	$-0.769682\pi$
$137$	−17.5442	−1.49890	−0.749451	+	0.662060i	$0.769682\pi$
$138$	0	0
$139$	1.29871	0.110155	0.0550776	−	0.998482i	$-0.482459\pi$
$139$	1.29871	0.110155	0.0550776	+	0.998482i	$0.482459\pi$
$140$	0	0
$141$	3.06139	0.257815
$142$	0	0
$143$	−1.00000	−0.0836242
$144$	0	0
$145$	23.3933	1.94271
$146$	0	0
$147$	−2.73758	−0.225792
$148$	0	0
$149$	−14.0494	−1.15097	−0.575485	−	0.817813i	$-0.695187\pi$
$149$	−14.0494	−1.15097	−0.575485	+	0.817813i	$0.695187\pi$
$150$	0	0
$151$	4.02435	0.327497	0.163749	−	0.986502i	$-0.447641\pi$
$151$	4.02435	0.327497	0.163749	+	0.986502i	$0.447641\pi$
$152$	0	0
$153$	−21.3488	−1.72595
$154$	0	0
$155$	39.3972	3.16446
$156$	0	0
$157$	5.42576	0.433023	0.216511	−	0.976280i	$-0.430532\pi$
$157$	5.42576	0.433023	0.216511	+	0.976280i	$0.430532\pi$
$158$	0	0
$159$	29.5120	2.34046
$160$	0	0
$161$	1.11828	0.0881330
$162$	0	0
$163$	−5.58145	−0.437173	−0.218587	−	0.975818i	$-0.570145\pi$
$163$	−5.58145	−0.437173	−0.218587	+	0.975818i	$0.570145\pi$
$164$	0	0
$165$	−11.2376	−0.874849
$166$	0	0
$167$	14.6129	1.13078	0.565389	−	0.824825i	$-0.308726\pi$
$167$	14.6129	1.13078	0.565389	+	0.824825i	$0.308726\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−29.2556	−2.23723
$172$	0	0
$173$	0.495744	0.0376907	0.0188454	−	0.999822i	$-0.494001\pi$
$173$	0.495744	0.0376907	0.0188454	+	0.999822i	$0.494001\pi$
$174$	0	0
$175$	−11.8506	−0.895824
$176$	0	0
$177$	−19.3864	−1.45717
$178$	0	0
$179$	21.8525	1.63333	0.816666	−	0.577111i	$-0.195820\pi$
$179$	21.8525	1.63333	0.816666	+	0.577111i	$0.195820\pi$
$180$	0	0
$181$	−19.5070	−1.44994	−0.724971	−	0.688779i	$-0.758147\pi$
$181$	−19.5070	−1.44994	−0.724971	+	0.688779i	$0.758147\pi$
$182$	0	0
$183$	−19.5736	−1.44692
$184$	0	0
$185$	3.79969	0.279359
$186$	0	0
$187$	−4.75014	−0.347365
$188$	0	0
$189$	4.09091	0.297570
$190$	0	0
$191$	12.3802	0.895798	0.447899	−	0.894084i	$-0.352172\pi$
$191$	12.3802	0.895798	0.447899	+	0.894084i	$0.352172\pi$
$192$	0	0
$193$	−1.29492	−0.0932101	−0.0466050	−	0.998913i	$-0.514840\pi$
$193$	−1.29492	−0.0932101	−0.0466050	+	0.998913i	$0.514840\pi$
$194$	0	0
$195$	11.2376	0.804744
$196$	0	0
$197$	2.83608	0.202062	0.101031	−	0.994883i	$-0.467786\pi$
$197$	2.83608	0.202062	0.101031	+	0.994883i	$0.467786\pi$
$198$	0	0
$199$	−22.6556	−1.60602	−0.803008	−	0.595968i	$-0.796768\pi$
$199$	−22.6556	−1.60602	−0.803008	+	0.595968i	$0.796768\pi$
$200$	0	0
$201$	32.3737	2.28346
$202$	0	0
$203$	−5.69879	−0.399977
$204$	0	0
$205$	−23.2906	−1.62669
$206$	0	0
$207$	−5.02595	−0.349328
$208$	0	0
$209$	−6.50940	−0.450265
$210$	0	0
$211$	14.8913	1.02516	0.512581	−	0.858639i	$-0.328689\pi$
$211$	14.8913	1.02516	0.512581	+	0.858639i	$0.328689\pi$
$212$	0	0
$213$	20.7863	1.42425
$214$	0	0
$215$	−24.4168	−1.66521
$216$	0	0
$217$	−9.59749	−0.651520
$218$	0	0
$219$	−37.0333	−2.50248
$220$	0	0
$221$	4.75014	0.319529
$222$	0	0
$223$	2.26934	0.151966	0.0759831	−	0.997109i	$-0.475790\pi$
$223$	2.26934	0.151966	0.0759831	+	0.997109i	$0.475790\pi$
$224$	0	0
$225$	53.2609	3.55073
$226$	0	0
$227$	−25.4287	−1.68776	−0.843882	−	0.536528i	$-0.819735\pi$
$227$	−25.4287	−1.68776	−0.843882	+	0.536528i	$0.819735\pi$
$228$	0	0
$229$	9.51218	0.628582	0.314291	−	0.949327i	$-0.398233\pi$
$229$	9.51218	0.628582	0.314291	+	0.949327i	$0.398233\pi$
$230$	0	0
$231$	2.73758	0.180120
$232$	0	0
$233$	−12.2947	−0.805453	−0.402726	−	0.915320i	$-0.631937\pi$
$233$	−12.2947	−0.805453	−0.402726	+	0.915320i	$0.631937\pi$
$234$	0	0
$235$	−4.59050	−0.299451
$236$	0	0
$237$	45.0361	2.92541
$238$	0	0
$239$	−3.80412	−0.246068	−0.123034	−	0.992402i	$-0.539262\pi$
$239$	−3.80412	−0.246068	−0.123034	+	0.992402i	$0.539262\pi$
$240$	0	0
$241$	−3.24913	−0.209295	−0.104648	−	0.994509i	$-0.533371\pi$
$241$	−3.24913	−0.209295	−0.104648	+	0.994509i	$0.533371\pi$
$242$	0	0
$243$	18.5250	1.18838
$244$	0	0
$245$	4.10495	0.262256
$246$	0	0
$247$	6.50940	0.414183
$248$	0	0
$249$	−1.22844	−0.0778493
$250$	0	0
$251$	19.7788	1.24843	0.624213	−	0.781254i	$-0.285420\pi$
$251$	19.7788	1.24843	0.624213	+	0.781254i	$0.285420\pi$
$252$	0	0
$253$	−1.11828	−0.0703058
$254$	0	0
$255$	53.3804	3.34281
$256$	0	0
$257$	23.6628	1.47605	0.738023	−	0.674776i	$-0.235760\pi$
$257$	23.6628	1.47605	0.738023	+	0.674776i	$0.235760\pi$
$258$	0	0
$259$	−0.925635	−0.0575162
$260$	0	0
$261$	25.6124	1.58537
$262$	0	0
$263$	−17.0474	−1.05119	−0.525594	−	0.850736i	$-0.676157\pi$
$263$	−17.0474	−1.05119	−0.525594	+	0.850736i	$0.676157\pi$
$264$	0	0
$265$	−44.2527	−2.71842
$266$	0	0
$267$	19.9867	1.22317
$268$	0	0
$269$	−15.5996	−0.951127	−0.475563	−	0.879681i	$-0.657756\pi$
$269$	−15.5996	−0.951127	−0.475563	+	0.879681i	$0.657756\pi$
$270$	0	0
$271$	1.42104	0.0863222	0.0431611	−	0.999068i	$-0.486257\pi$
$271$	1.42104	0.0863222	0.0431611	+	0.999068i	$0.486257\pi$
$272$	0	0
$273$	−2.73758	−0.165686
$274$	0	0
$275$	11.8506	0.714620
$276$	0	0
$277$	−17.9636	−1.07933	−0.539664	−	0.841880i	$-0.681449\pi$
$277$	−17.9636	−1.07933	−0.539664	+	0.841880i	$0.681449\pi$
$278$	0	0
$279$	43.1345	2.58239
$280$	0	0
$281$	−25.4760	−1.51977	−0.759886	−	0.650057i	$-0.774745\pi$
$281$	−25.4760	−1.51977	−0.759886	+	0.650057i	$0.774745\pi$
$282$	0	0
$283$	7.02772	0.417754	0.208877	−	0.977942i	$-0.433019\pi$
$283$	7.02772	0.417754	0.208877	+	0.977942i	$0.433019\pi$
$284$	0	0
$285$	73.1504	4.33305
$286$	0	0
$287$	5.67379	0.334913
$288$	0	0
$289$	5.56387	0.327286
$290$	0	0
$291$	12.7925	0.749908
$292$	0	0
$293$	−13.3207	−0.778202	−0.389101	−	0.921195i	$-0.627214\pi$
$293$	−13.3207	−0.778202	−0.389101	+	0.921195i	$0.627214\pi$
$294$	0	0
$295$	29.0696	1.69250
$296$	0	0
$297$	−4.09091	−0.237379
$298$	0	0
$299$	1.11828	0.0646719
$300$	0	0
$301$	5.94812	0.342844
$302$	0	0
$303$	6.66746	0.383035
$304$	0	0
$305$	29.3502	1.68059
$306$	0	0
$307$	−31.0142	−1.77007	−0.885036	−	0.465522i	$-0.845867\pi$
$307$	−31.0142	−1.77007	−0.885036	+	0.465522i	$0.845867\pi$
$308$	0	0
$309$	8.50149	0.483633
$310$	0	0
$311$	8.50915	0.482510	0.241255	−	0.970462i	$-0.422441\pi$
$311$	8.50915	0.482510	0.241255	+	0.970462i	$0.422441\pi$
$312$	0	0
$313$	−0.907015	−0.0512675	−0.0256338	−	0.999671i	$-0.508160\pi$
$313$	−0.907015	−0.0512675	−0.0256338	+	0.999671i	$0.508160\pi$
$314$	0	0
$315$	−18.4491	−1.03949
$316$	0	0
$317$	3.15391	0.177141	0.0885706	−	0.996070i	$-0.471770\pi$
$317$	3.15391	0.177141	0.0885706	+	0.996070i	$0.471770\pi$
$318$	0	0
$319$	5.69879	0.319071
$320$	0	0
$321$	40.4446	2.25740
$322$	0	0
$323$	30.9206	1.72047
$324$	0	0
$325$	−11.8506	−0.657355
$326$	0	0
$327$	−47.3619	−2.61912
$328$	0	0
$329$	1.11828	0.0616529
$330$	0	0
$331$	−12.9482	−0.711700	−0.355850	−	0.934543i	$-0.615809\pi$
$331$	−12.9482	−0.711700	−0.355850	+	0.934543i	$0.615809\pi$
$332$	0	0
$333$	4.16013	0.227974
$334$	0	0
$335$	−48.5437	−2.65223
$336$	0	0
$337$	−5.81871	−0.316965	−0.158483	−	0.987362i	$-0.550660\pi$
$337$	−5.81871	−0.316965	−0.158483	+	0.987362i	$0.550660\pi$
$338$	0	0
$339$	−30.2081	−1.64068
$340$	0	0
$341$	9.59749	0.519733
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	12.5669	0.676577
$346$	0	0
$347$	−17.7976	−0.955427	−0.477713	−	0.878516i	$-0.658534\pi$
$347$	−17.7976	−0.955427	−0.477713	+	0.878516i	$0.658534\pi$
$348$	0	0
$349$	6.89931	0.369312	0.184656	−	0.982803i	$-0.440883\pi$
$349$	6.89931	0.369312	0.184656	+	0.982803i	$0.440883\pi$
$350$	0	0
$351$	4.09091	0.218357
$352$	0	0
$353$	−11.2510	−0.598832	−0.299416	−	0.954123i	$-0.596792\pi$
$353$	−11.2510	−0.598832	−0.299416	+	0.954123i	$0.596792\pi$
$354$	0	0
$355$	−31.1686	−1.65426
$356$	0	0
$357$	−13.0039	−0.688240
$358$	0	0
$359$	15.8649	0.837319	0.418659	−	0.908143i	$-0.362500\pi$
$359$	15.8649	0.837319	0.418659	+	0.908143i	$0.362500\pi$
$360$	0	0
$361$	23.3723	1.23012
$362$	0	0
$363$	−2.73758	−0.143686
$364$	0	0
$365$	55.5307	2.90661
$366$	0	0
$367$	−30.3570	−1.58462	−0.792312	−	0.610116i	$-0.791123\pi$
$367$	−30.3570	−1.58462	−0.792312	+	0.610116i	$0.791123\pi$
$368$	0	0
$369$	−25.5000	−1.32748
$370$	0	0
$371$	10.7803	0.559687
$372$	0	0
$373$	−18.3516	−0.950209	−0.475104	−	0.879930i	$-0.657590\pi$
$373$	−18.3516	−0.950209	−0.475104	+	0.879930i	$0.657590\pi$
$374$	0	0
$375$	−76.9850	−3.97549
$376$	0	0
$377$	−5.69879	−0.293503
$378$	0	0
$379$	−2.57415	−0.132225	−0.0661126	−	0.997812i	$-0.521060\pi$
$379$	−2.57415	−0.132225	−0.0661126	+	0.997812i	$0.521060\pi$
$380$	0	0
$381$	−0.126186	−0.00646473
$382$	0	0
$383$	16.7836	0.857601	0.428800	−	0.903399i	$-0.358936\pi$
$383$	16.7836	0.857601	0.428800	+	0.903399i	$0.358936\pi$
$384$	0	0
$385$	−4.10495	−0.209208
$386$	0	0
$387$	−26.7330	−1.35891
$388$	0	0
$389$	−27.7088	−1.40489	−0.702446	−	0.711737i	$-0.747909\pi$
$389$	−27.7088	−1.40489	−0.702446	+	0.711737i	$0.747909\pi$
$390$	0	0
$391$	5.31200	0.268640
$392$	0	0
$393$	11.8726	0.598892
$394$	0	0
$395$	−67.5307	−3.39784
$396$	0	0
$397$	16.0873	0.807398	0.403699	−	0.914892i	$-0.367724\pi$
$397$	16.0873	0.807398	0.403699	+	0.914892i	$0.367724\pi$
$398$	0	0
$399$	−17.8200	−0.892117
$400$	0	0
$401$	−3.51703	−0.175632	−0.0878162	−	0.996137i	$-0.527989\pi$
$401$	−3.51703	−0.175632	−0.0878162	+	0.996137i	$0.527989\pi$
$402$	0	0
$403$	−9.59749	−0.478085
$404$	0	0
$405$	−9.37513	−0.465854
$406$	0	0
$407$	0.925635	0.0458820
$408$	0	0
$409$	2.07105	0.102407	0.0512034	−	0.998688i	$-0.483694\pi$
$409$	2.07105	0.102407	0.0512034	+	0.998688i	$0.483694\pi$
$410$	0	0
$411$	48.0286	2.36908
$412$	0	0
$413$	−7.08159	−0.348462
$414$	0	0
$415$	1.84203	0.0904215
$416$	0	0
$417$	−3.55533	−0.174105
$418$	0	0
$419$	11.4498	0.559360	0.279680	−	0.960093i	$-0.409772\pi$
$419$	11.4498	0.559360	0.279680	+	0.960093i	$0.409772\pi$
$420$	0	0
$421$	−26.0663	−1.27039	−0.635196	−	0.772351i	$-0.719081\pi$
$421$	−26.0663	−1.27039	−0.635196	+	0.772351i	$0.719081\pi$
$422$	0	0
$423$	−5.02595	−0.244370
$424$	0	0
$425$	−56.2922	−2.73057
$426$	0	0
$427$	−7.14995	−0.346010
$428$	0	0
$429$	2.73758	0.132172
$430$	0	0
$431$	−32.5791	−1.56928	−0.784640	−	0.619951i	$-0.787152\pi$
$431$	−32.5791	−1.56928	−0.784640	+	0.619951i	$0.787152\pi$
$432$	0	0
$433$	11.0807	0.532506	0.266253	−	0.963903i	$-0.414214\pi$
$433$	11.0807	0.532506	0.266253	+	0.963903i	$0.414214\pi$
$434$	0	0
$435$	−64.0410	−3.07053
$436$	0	0
$437$	7.27935	0.348219
$438$	0	0
$439$	−0.444568	−0.0212181	−0.0106090	−	0.999944i	$-0.503377\pi$
$439$	−0.444568	−0.0212181	−0.0106090	+	0.999944i	$0.503377\pi$
$440$	0	0
$441$	4.49435	0.214017
$442$	0	0
$443$	35.9096	1.70611	0.853057	−	0.521817i	$-0.174746\pi$
$443$	35.9096	1.70611	0.853057	+	0.521817i	$0.174746\pi$
$444$	0	0
$445$	−29.9697	−1.42070
$446$	0	0
$447$	38.4613	1.81916
$448$	0	0
$449$	−11.0626	−0.522074	−0.261037	−	0.965329i	$-0.584064\pi$
$449$	−11.0626	−0.522074	−0.261037	+	0.965329i	$0.584064\pi$
$450$	0	0
$451$	−5.67379	−0.267168
$452$	0	0
$453$	−11.0170	−0.517624
$454$	0	0
$455$	4.10495	0.192443
$456$	0	0
$457$	−4.64123	−0.217108	−0.108554	−	0.994091i	$-0.534622\pi$
$457$	−4.64123	−0.217108	−0.108554	+	0.994091i	$0.534622\pi$
$458$	0	0
$459$	19.4324	0.907027
$460$	0	0
$461$	12.2326	0.569727	0.284864	−	0.958568i	$-0.408052\pi$
$461$	12.2326	0.569727	0.284864	+	0.958568i	$0.408052\pi$
$462$	0	0
$463$	23.2408	1.08009	0.540047	−	0.841635i	$-0.318407\pi$
$463$	23.2408	1.08009	0.540047	+	0.841635i	$0.318407\pi$
$464$	0	0
$465$	−107.853	−5.00157
$466$	0	0
$467$	−21.8663	−1.01185	−0.505927	−	0.862576i	$-0.668849\pi$
$467$	−21.8663	−1.01185	−0.505927	+	0.862576i	$0.668849\pi$
$468$	0	0
$469$	11.8256	0.546058
$470$	0	0
$471$	−14.8535	−0.684411
$472$	0	0
$473$	−5.94812	−0.273495
$474$	0	0
$475$	−77.1406	−3.53945
$476$	0	0
$477$	−48.4506	−2.21840
$478$	0	0
$479$	37.6424	1.71992	0.859962	−	0.510359i	$-0.170487\pi$
$479$	37.6424	1.71992	0.859962	+	0.510359i	$0.170487\pi$
$480$	0	0
$481$	−0.925635	−0.0422053
$482$	0	0
$483$	−3.06139	−0.139298
$484$	0	0
$485$	−19.1821	−0.871013
$486$	0	0
$487$	1.63042	0.0738815	0.0369408	−	0.999317i	$-0.488239\pi$
$487$	1.63042	0.0738815	0.0369408	+	0.999317i	$0.488239\pi$
$488$	0	0
$489$	15.2797	0.690971
$490$	0	0
$491$	12.6170	0.569395	0.284698	−	0.958617i	$-0.408107\pi$
$491$	12.6170	0.569395	0.284698	+	0.958617i	$0.408107\pi$
$492$	0	0
$493$	−27.0701	−1.21918
$494$	0	0
$495$	18.4491	0.829225
$496$	0	0
$497$	7.59293	0.340589
$498$	0	0
$499$	−33.2460	−1.48830	−0.744148	−	0.668014i	$-0.767144\pi$
$499$	−33.2460	−1.48830	−0.744148	+	0.668014i	$0.767144\pi$
$500$	0	0
$501$	−40.0039	−1.78724
$502$	0	0
$503$	26.7280	1.19174	0.595870	−	0.803081i	$-0.296807\pi$
$503$	26.7280	1.19174	0.595870	+	0.803081i	$0.296807\pi$
$504$	0	0
$505$	−9.99773	−0.444893
$506$	0	0
$507$	−2.73758	−0.121580
$508$	0	0
$509$	12.1032	0.536464	0.268232	−	0.963354i	$-0.413561\pi$
$509$	12.1032	0.536464	0.268232	+	0.963354i	$0.413561\pi$
$510$	0	0
$511$	−13.5277	−0.598432
$512$	0	0
$513$	26.6294	1.17572
$514$	0	0
$515$	−12.7478	−0.561737
$516$	0	0
$517$	−1.11828	−0.0491820
$518$	0	0
$519$	−1.35714	−0.0595718
$520$	0	0
$521$	−43.8908	−1.92289	−0.961445	−	0.274998i	$-0.911323\pi$
$521$	−43.8908	−1.92289	−0.961445	+	0.274998i	$0.911323\pi$
$522$	0	0
$523$	−37.8996	−1.65723	−0.828617	−	0.559815i	$-0.810872\pi$
$523$	−37.8996	−1.65723	−0.828617	+	0.559815i	$0.810872\pi$
$524$	0	0
$525$	32.4421	1.41589
$526$	0	0
$527$	−45.5895	−1.98591
$528$	0	0
$529$	−21.7494	−0.945628
$530$	0	0
$531$	31.8272	1.38118
$532$	0	0
$533$	5.67379	0.245759
$534$	0	0
$535$	−60.6460	−2.62195
$536$	0	0
$537$	−59.8230	−2.58155
$538$	0	0
$539$	1.00000	0.0430730
$540$	0	0
$541$	−17.9675	−0.772482	−0.386241	−	0.922398i	$-0.626227\pi$
$541$	−17.9675	−0.772482	−0.386241	+	0.922398i	$0.626227\pi$
$542$	0	0
$543$	53.4019	2.29170
$544$	0	0
$545$	71.0182	3.04209
$546$	0	0
$547$	44.8092	1.91590	0.957951	−	0.286931i	$-0.0926352\pi$
$547$	44.8092	1.91590	0.957951	+	0.286931i	$0.0926352\pi$
$548$	0	0
$549$	32.1344	1.37146
$550$	0	0
$551$	−37.0957	−1.58033
$552$	0	0
$553$	16.4510	0.699569
$554$	0	0
$555$	−10.4020	−0.441539
$556$	0	0
$557$	−29.5051	−1.25017	−0.625085	−	0.780556i	$-0.714936\pi$
$557$	−29.5051	−1.25017	−0.625085	+	0.780556i	$0.714936\pi$
$558$	0	0
$559$	5.94812	0.251579
$560$	0	0
$561$	13.0039	0.549025
$562$	0	0
$563$	−21.0327	−0.886423	−0.443211	−	0.896417i	$-0.646161\pi$
$563$	−21.0327	−0.886423	−0.443211	+	0.896417i	$0.646161\pi$
$564$	0	0
$565$	45.2965	1.90564
$566$	0	0
$567$	2.28386	0.0959131
$568$	0	0
$569$	−14.1213	−0.591998	−0.295999	−	0.955188i	$-0.595652\pi$
$569$	−14.1213	−0.591998	−0.295999	+	0.955188i	$0.595652\pi$
$570$	0	0
$571$	−1.79404	−0.0750784	−0.0375392	−	0.999295i	$-0.511952\pi$
$571$	−1.79404	−0.0750784	−0.0375392	+	0.999295i	$0.511952\pi$
$572$	0	0
$573$	−33.8917	−1.41585
$574$	0	0
$575$	−13.2524	−0.552662
$576$	0	0
$577$	−23.8800	−0.994137	−0.497069	−	0.867711i	$-0.665590\pi$
$577$	−23.8800	−0.994137	−0.497069	+	0.867711i	$0.665590\pi$
$578$	0	0
$579$	3.54494	0.147323
$580$	0	0
$581$	−0.448733	−0.0186166
$582$	0	0
$583$	−10.7803	−0.446476
$584$	0	0
$585$	−18.4491	−0.762776
$586$	0	0
$587$	32.4688	1.34013	0.670066	−	0.742302i	$-0.266266\pi$
$587$	32.4688	1.34013	0.670066	+	0.742302i	$0.266266\pi$
$588$	0	0
$589$	−62.4739	−2.57419
$590$	0	0
$591$	−7.76400	−0.319368
$592$	0	0
$593$	−13.0750	−0.536926	−0.268463	−	0.963290i	$-0.586516\pi$
$593$	−13.0750	−0.536926	−0.268463	+	0.963290i	$0.586516\pi$
$594$	0	0
$595$	19.4991	0.799386
$596$	0	0
$597$	62.0217	2.53838
$598$	0	0
$599$	23.0615	0.942268	0.471134	−	0.882062i	$-0.343845\pi$
$599$	23.0615	0.942268	0.471134	+	0.882062i	$0.343845\pi$
$600$	0	0
$601$	−15.4448	−0.630008	−0.315004	−	0.949090i	$-0.602006\pi$
$601$	−15.4448	−0.630008	−0.315004	+	0.949090i	$0.602006\pi$
$602$	0	0
$603$	−53.1486	−2.16438
$604$	0	0
$605$	4.10495	0.166890
$606$	0	0
$607$	−21.1164	−0.857087	−0.428544	−	0.903521i	$-0.640973\pi$
$607$	−21.1164	−0.857087	−0.428544	+	0.903521i	$0.640973\pi$
$608$	0	0
$609$	15.6009	0.632181
$610$	0	0
$611$	1.11828	0.0452409
$612$	0	0
$613$	−30.6429	−1.23766	−0.618828	−	0.785526i	$-0.712392\pi$
$613$	−30.6429	−1.23766	−0.618828	+	0.785526i	$0.712392\pi$
$614$	0	0
$615$	63.7600	2.57105
$616$	0	0
$617$	29.9055	1.20395	0.601975	−	0.798515i	$-0.294381\pi$
$617$	29.9055	1.20395	0.601975	+	0.798515i	$0.294381\pi$
$618$	0	0
$619$	43.2643	1.73894	0.869470	−	0.493985i	$-0.164460\pi$
$619$	43.2643	1.73894	0.869470	+	0.493985i	$0.164460\pi$
$620$	0	0
$621$	4.57479	0.183580
$622$	0	0
$623$	7.30086	0.292503
$624$	0	0
$625$	56.1844	2.24738
$626$	0	0
$627$	17.8200	0.711663
$628$	0	0
$629$	−4.39690	−0.175316
$630$	0	0
$631$	18.2390	0.726083	0.363041	−	0.931773i	$-0.381738\pi$
$631$	18.2390	0.726083	0.363041	+	0.931773i	$0.381738\pi$
$632$	0	0
$633$	−40.7663	−1.62031
$634$	0	0
$635$	0.189214	0.00750874
$636$	0	0
$637$	−1.00000	−0.0396214
$638$	0	0
$639$	−34.1253	−1.34998
$640$	0	0
$641$	34.6487	1.36854	0.684270	−	0.729228i	$-0.260121\pi$
$641$	34.6487	1.36854	0.684270	+	0.729228i	$0.260121\pi$
$642$	0	0
$643$	−3.19624	−0.126047	−0.0630236	−	0.998012i	$-0.520074\pi$
$643$	−3.19624	−0.126047	−0.0630236	+	0.998012i	$0.520074\pi$
$644$	0	0
$645$	66.8429	2.63194
$646$	0	0
$647$	−26.5155	−1.04243	−0.521216	−	0.853425i	$-0.674522\pi$
$647$	−26.5155	−1.04243	−0.521216	+	0.853425i	$0.674522\pi$
$648$	0	0
$649$	7.08159	0.277977
$650$	0	0
$651$	26.2739	1.02976
$652$	0	0
$653$	−23.2349	−0.909253	−0.454627	−	0.890682i	$-0.650227\pi$
$653$	−23.2349	−0.909253	−0.454627	+	0.890682i	$0.650227\pi$
$654$	0	0
$655$	−17.8027	−0.695609
$656$	0	0
$657$	60.7984	2.37197
$658$	0	0
$659$	29.3533	1.14344	0.571721	−	0.820448i	$-0.306276\pi$
$659$	29.3533	1.14344	0.571721	+	0.820448i	$0.306276\pi$
$660$	0	0
$661$	−37.7097	−1.46674	−0.733369	−	0.679831i	$-0.762053\pi$
$661$	−37.7097	−1.46674	−0.733369	+	0.679831i	$0.762053\pi$
$662$	0	0
$663$	−13.0039	−0.505030
$664$	0	0
$665$	26.7208	1.03619
$666$	0	0
$667$	−6.37286	−0.246758
$668$	0	0
$669$	−6.21250	−0.240189
$670$	0	0
$671$	7.14995	0.276021
$672$	0	0
$673$	−7.03880	−0.271326	−0.135663	−	0.990755i	$-0.543316\pi$
$673$	−7.03880	−0.271326	−0.135663	+	0.990755i	$0.543316\pi$
$674$	0	0
$675$	−48.4799	−1.86599
$676$	0	0
$677$	−32.8697	−1.26328	−0.631642	−	0.775260i	$-0.717619\pi$
$677$	−32.8697	−1.26328	−0.631642	+	0.775260i	$0.717619\pi$
$678$	0	0
$679$	4.67291	0.179330
$680$	0	0
$681$	69.6132	2.66758
$682$	0	0
$683$	34.5845	1.32334	0.661670	−	0.749795i	$-0.269848\pi$
$683$	34.5845	1.32334	0.661670	+	0.749795i	$0.269848\pi$
$684$	0	0
$685$	−72.0181	−2.75167
$686$	0	0
$687$	−26.0404	−0.993502
$688$	0	0
$689$	10.7803	0.410698
$690$	0	0
$691$	12.1223	0.461155	0.230577	−	0.973054i	$-0.425939\pi$
$691$	12.1223	0.461155	0.230577	+	0.973054i	$0.425939\pi$
$692$	0	0
$693$	−4.49435	−0.170726
$694$	0	0
$695$	5.33115	0.202222
$696$	0	0
$697$	26.9513	1.02085
$698$	0	0
$699$	33.6577	1.27305
$700$	0	0
$701$	−24.9265	−0.941461	−0.470731	−	0.882277i	$-0.656010\pi$
$701$	−24.9265	−0.941461	−0.470731	+	0.882277i	$0.656010\pi$
$702$	0	0
$703$	−6.02534	−0.227250
$704$	0	0
$705$	12.5669	0.473295
$706$	0	0
$707$	2.43553	0.0915975
$708$	0	0
$709$	−26.8247	−1.00742	−0.503711	−	0.863872i	$-0.668032\pi$
$709$	−26.8247	−1.00742	−0.503711	+	0.863872i	$0.668032\pi$
$710$	0	0
$711$	−73.9367	−2.77284
$712$	0	0
$713$	−10.7327	−0.401943
$714$	0	0
$715$	−4.10495	−0.153517
$716$	0	0
$717$	10.4141	0.388921
$718$	0	0
$719$	−27.0067	−1.00718	−0.503590	−	0.863943i	$-0.667988\pi$
$719$	−27.0067	−1.00718	−0.503590	+	0.863943i	$0.667988\pi$
$720$	0	0
$721$	3.10548	0.115654
$722$	0	0
$723$	8.89477	0.330800
$724$	0	0
$725$	67.5343	2.50816
$726$	0	0
$727$	−7.47024	−0.277056	−0.138528	−	0.990359i	$-0.544237\pi$
$727$	−7.47024	−0.277056	−0.138528	+	0.990359i	$0.544237\pi$
$728$	0	0
$729$	−43.8621	−1.62452
$730$	0	0
$731$	28.2544	1.04503
$732$	0	0
$733$	21.3873	0.789957	0.394979	−	0.918690i	$-0.370752\pi$
$733$	21.3873	0.789957	0.394979	+	0.918690i	$0.370752\pi$
$734$	0	0
$735$	−11.2376	−0.414507
$736$	0	0
$737$	−11.8256	−0.435603
$738$	0	0
$739$	−42.8605	−1.57665	−0.788325	−	0.615259i	$-0.789051\pi$
$739$	−42.8605	−1.57665	−0.788325	+	0.615259i	$0.789051\pi$
$740$	0	0
$741$	−17.8200	−0.654635
$742$	0	0
$743$	−30.8790	−1.13284	−0.566421	−	0.824116i	$-0.691672\pi$
$743$	−30.8790	−1.13284	−0.566421	+	0.824116i	$0.691672\pi$
$744$	0	0
$745$	−57.6720	−2.11294
$746$	0	0
$747$	2.01676	0.0737895
$748$	0	0
$749$	14.7739	0.539825
$750$	0	0
$751$	51.6924	1.88628	0.943142	−	0.332391i	$-0.107855\pi$
$751$	51.6924	1.88628	0.943142	+	0.332391i	$0.107855\pi$
$752$	0	0
$753$	−54.1460	−1.97319
$754$	0	0
$755$	16.5198	0.601216
$756$	0	0
$757$	−2.25558	−0.0819805	−0.0409902	−	0.999160i	$-0.513051\pi$
$757$	−2.25558	−0.0819805	−0.0409902	+	0.999160i	$0.513051\pi$
$758$	0	0
$759$	3.06139	0.111121
$760$	0	0
$761$	30.5461	1.10729	0.553647	−	0.832751i	$-0.313236\pi$
$761$	30.5461	1.10729	0.553647	+	0.832751i	$0.313236\pi$
$762$	0	0
$763$	−17.3006	−0.626324
$764$	0	0
$765$	−87.6359	−3.16848
$766$	0	0
$767$	−7.08159	−0.255701
$768$	0	0
$769$	44.1057	1.59049	0.795246	−	0.606287i	$-0.207342\pi$
$769$	44.1057	1.59049	0.795246	+	0.606287i	$0.207342\pi$
$770$	0	0
$771$	−64.7789	−2.33295
$772$	0	0
$773$	−36.4084	−1.30952	−0.654760	−	0.755837i	$-0.727230\pi$
$773$	−36.4084	−1.30952	−0.654760	+	0.755837i	$0.727230\pi$
$774$	0	0
$775$	113.736	4.08553
$776$	0	0
$777$	2.53400	0.0909068
$778$	0	0
$779$	36.9330	1.32326
$780$	0	0
$781$	−7.59293	−0.271696
$782$	0	0
$783$	−23.3132	−0.833147
$784$	0	0
$785$	22.2725	0.794939
$786$	0	0
$787$	5.67746	0.202379	0.101190	−	0.994867i	$-0.467735\pi$
$787$	5.67746	0.202379	0.101190	+	0.994867i	$0.467735\pi$
$788$	0	0
$789$	46.6686	1.66145
$790$	0	0
$791$	−11.0346	−0.392345
$792$	0	0
$793$	−7.14995	−0.253902
$794$	0	0
$795$	121.145	4.29659
$796$	0	0
$797$	43.9545	1.55695	0.778475	−	0.627676i	$-0.215994\pi$
$797$	43.9545	1.55695	0.778475	+	0.627676i	$0.215994\pi$
$798$	0	0
$799$	5.31200	0.187925
$800$	0	0
$801$	−32.8126	−1.15938
$802$	0	0
$803$	13.5277	0.477384
$804$	0	0
$805$	4.59050	0.161794
$806$	0	0
$807$	42.7053	1.50330
$808$	0	0
$809$	2.11004	0.0741850	0.0370925	−	0.999312i	$-0.488190\pi$
$809$	2.11004	0.0741850	0.0370925	+	0.999312i	$0.488190\pi$
$810$	0	0
$811$	15.5301	0.545336	0.272668	−	0.962108i	$-0.412094\pi$
$811$	15.5301	0.545336	0.272668	+	0.962108i	$0.412094\pi$
$812$	0	0
$813$	−3.89022	−0.136436
$814$	0	0
$815$	−22.9116	−0.802559
$816$	0	0
$817$	38.7187	1.35460
$818$	0	0
$819$	4.49435	0.157045
$820$	0	0
$821$	−35.5799	−1.24175	−0.620873	−	0.783911i	$-0.713222\pi$
$821$	−35.5799	−1.24175	−0.620873	+	0.783911i	$0.713222\pi$
$822$	0	0
$823$	10.8156	0.377007	0.188503	−	0.982073i	$-0.439636\pi$
$823$	10.8156	0.377007	0.188503	+	0.982073i	$0.439636\pi$
$824$	0	0
$825$	−32.4421	−1.12949
$826$	0	0
$827$	−23.0872	−0.802821	−0.401410	−	0.915898i	$-0.631480\pi$
$827$	−23.0872	−0.802821	−0.401410	+	0.915898i	$0.631480\pi$
$828$	0	0
$829$	38.5237	1.33798	0.668991	−	0.743270i	$-0.266726\pi$
$829$	38.5237	1.33798	0.668991	+	0.743270i	$0.266726\pi$
$830$	0	0
$831$	49.1768	1.70592
$832$	0	0
$833$	−4.75014	−0.164583
$834$	0	0
$835$	59.9851	2.07587
$836$	0	0
$837$	−39.2625	−1.35711
$838$	0	0
$839$	−35.0629	−1.21050	−0.605252	−	0.796034i	$-0.706928\pi$
$839$	−35.0629	−1.21050	−0.605252	+	0.796034i	$0.706928\pi$
$840$	0	0
$841$	3.47624	0.119871
$842$	0	0
$843$	69.7426	2.40206
$844$	0	0
$845$	4.10495	0.141215
$846$	0	0
$847$	−1.00000	−0.0343604
$848$	0	0
$849$	−19.2389	−0.660279
$850$	0	0
$851$	−1.03512	−0.0354835
$852$	0	0
$853$	−19.7858	−0.677451	−0.338726	−	0.940885i	$-0.609996\pi$
$853$	−19.7858	−0.677451	−0.338726	+	0.940885i	$0.609996\pi$
$854$	0	0
$855$	−120.093	−4.10708
$856$	0	0
$857$	43.4926	1.48568	0.742840	−	0.669469i	$-0.233478\pi$
$857$	43.4926	1.48568	0.742840	+	0.669469i	$0.233478\pi$
$858$	0	0
$859$	−1.88762	−0.0644049	−0.0322024	−	0.999481i	$-0.510252\pi$
$859$	−1.88762	−0.0644049	−0.0322024	+	0.999481i	$0.510252\pi$
$860$	0	0
$861$	−15.5325	−0.529345
$862$	0	0
$863$	−47.5393	−1.61826	−0.809129	−	0.587631i	$-0.800061\pi$
$863$	−47.5393	−1.61826	−0.809129	+	0.587631i	$0.800061\pi$
$864$	0	0
$865$	2.03501	0.0691923
$866$	0	0
$867$	−15.2315	−0.517290
$868$	0	0
$869$	−16.4510	−0.558063
$870$	0	0
$871$	11.8256	0.400697
$872$	0	0
$873$	−21.0017	−0.710800
$874$	0	0
$875$	−28.1215	−0.950682
$876$	0	0
$877$	−54.0558	−1.82533	−0.912667	−	0.408704i	$-0.865981\pi$
$877$	−54.0558	−1.82533	−0.912667	+	0.408704i	$0.865981\pi$
$878$	0	0
$879$	36.4664	1.22998
$880$	0	0
$881$	−17.1489	−0.577761	−0.288881	−	0.957365i	$-0.593283\pi$
$881$	−17.1489	−0.577761	−0.288881	+	0.957365i	$0.593283\pi$
$882$	0	0
$883$	16.2158	0.545707	0.272853	−	0.962056i	$-0.412033\pi$
$883$	16.2158	0.545707	0.272853	+	0.962056i	$0.412033\pi$
$884$	0	0
$885$	−79.5804	−2.67506
$886$	0	0
$887$	13.9302	0.467731	0.233866	−	0.972269i	$-0.424862\pi$
$887$	13.9302	0.467731	0.233866	+	0.972269i	$0.424862\pi$
$888$	0	0
$889$	−0.0460941	−0.00154595
$890$	0	0
$891$	−2.28386	−0.0765121
$892$	0	0
$893$	7.27935	0.243594
$894$	0	0
$895$	89.7034	2.99846
$896$	0	0
$897$	−3.06139	−0.102217
$898$	0	0
$899$	54.6941	1.82415
$900$	0	0
$901$	51.2081	1.70599
$902$	0	0
$903$	−16.2835	−0.541880
$904$	0	0
$905$	−80.0752	−2.66179
$906$	0	0
$907$	−48.2268	−1.60135	−0.800673	−	0.599102i	$-0.795524\pi$
$907$	−48.2268	−1.60135	−0.800673	+	0.599102i	$0.795524\pi$
$908$	0	0
$909$	−10.9461	−0.363060
$910$	0	0
$911$	6.92426	0.229411	0.114706	−	0.993400i	$-0.463408\pi$
$911$	6.92426	0.229411	0.114706	+	0.993400i	$0.463408\pi$
$912$	0	0
$913$	0.448733	0.0148509
$914$	0	0
$915$	−80.3486	−2.65624
$916$	0	0
$917$	4.33688	0.143216
$918$	0	0
$919$	−45.8371	−1.51203	−0.756014	−	0.654556i	$-0.772856\pi$
$919$	−45.8371	−1.51203	−0.756014	+	0.654556i	$0.772856\pi$
$920$	0	0
$921$	84.9038	2.79768
$922$	0	0
$923$	7.59293	0.249924
$924$	0	0
$925$	10.9694	0.360671
$926$	0	0
$927$	−13.9571	−0.458411
$928$	0	0
$929$	42.6822	1.40036	0.700179	−	0.713968i	$-0.253104\pi$
$929$	42.6822	1.40036	0.700179	+	0.713968i	$0.253104\pi$
$930$	0	0
$931$	−6.50940	−0.213337
$932$	0	0
$933$	−23.2945	−0.762628
$934$	0	0
$935$	−19.4991	−0.637689
$936$	0	0
$937$	−10.8195	−0.353458	−0.176729	−	0.984260i	$-0.556552\pi$
$937$	−10.8195	−0.353458	−0.176729	+	0.984260i	$0.556552\pi$
$938$	0	0
$939$	2.48303	0.0810305
$940$	0	0
$941$	−29.5369	−0.962875	−0.481438	−	0.876480i	$-0.659885\pi$
$941$	−29.5369	−0.962875	−0.481438	+	0.876480i	$0.659885\pi$
$942$	0	0
$943$	6.34490	0.206618
$944$	0	0
$945$	16.7930	0.546276
$946$	0	0
$947$	−41.3390	−1.34334	−0.671668	−	0.740852i	$-0.734422\pi$
$947$	−41.3390	−1.34334	−0.671668	+	0.740852i	$0.734422\pi$
$948$	0	0
$949$	−13.5277	−0.439129
$950$	0	0
$951$	−8.63408	−0.279979
$952$	0	0
$953$	−1.36665	−0.0442700	−0.0221350	−	0.999755i	$-0.507046\pi$
$953$	−1.36665	−0.0442700	−0.0221350	+	0.999755i	$0.507046\pi$
$954$	0	0
$955$	50.8200	1.64450
$956$	0	0
$957$	−15.6009	−0.504306
$958$	0	0
$959$	17.5442	0.566531
$960$	0	0
$961$	61.1118	1.97135
$962$	0	0
$963$	−66.3989	−2.13967
$964$	0	0
$965$	−5.31557	−0.171114
$966$	0	0
$967$	50.5079	1.62423	0.812113	−	0.583501i	$-0.198317\pi$
$967$	50.5079	1.62423	0.812113	+	0.583501i	$0.198317\pi$
$968$	0	0
$969$	−84.6477	−2.71928
$970$	0	0
$971$	−15.1043	−0.484720	−0.242360	−	0.970186i	$-0.577921\pi$
$971$	−15.1043	−0.484720	−0.242360	+	0.970186i	$0.577921\pi$
$972$	0	0
$973$	−1.29871	−0.0416348
$974$	0	0
$975$	32.4421	1.03898
$976$	0	0
$977$	27.6707	0.885265	0.442633	−	0.896703i	$-0.354045\pi$
$977$	27.6707	0.885265	0.442633	+	0.896703i	$0.354045\pi$
$978$	0	0
$979$	−7.30086	−0.233337
$980$	0	0
$981$	77.7551	2.48253
$982$	0	0
$983$	19.6661	0.627250	0.313625	−	0.949547i	$-0.398457\pi$
$983$	19.6661	0.627250	0.313625	+	0.949547i	$0.398457\pi$
$984$	0	0
$985$	11.6420	0.370944
$986$	0	0
$987$	−3.06139	−0.0974451
$988$	0	0
$989$	6.65168	0.211511
$990$	0	0
$991$	−41.1064	−1.30579	−0.652893	−	0.757450i	$-0.726445\pi$
$991$	−41.1064	−1.30579	−0.652893	+	0.757450i	$0.726445\pi$
$992$	0	0
$993$	35.4469	1.12487
$994$	0	0
$995$	−93.0003	−2.94831
$996$	0	0
$997$	−60.9967	−1.93178	−0.965892	−	0.258944i	$-0.916625\pi$
$997$	−60.9967	−1.93178	−0.965892	+	0.258944i	$0.916625\pi$
$998$	0	0
$999$	−3.78669	−0.119806

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8008.2.a.v.1.2	✓	11

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8008.2.a.v.1.2	✓	11	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)

\(f(q)\)	\(=\)	\(q-2.73758 q^{3} +4.10495 q^{5} -1.00000 q^{7} +4.49435 q^{9} +O(q^{10})\)	\(q-2.73758 q^{3} +4.10495 q^{5} -1.00000 q^{7} +4.49435 q^{9} +1.00000 q^{11} -1.00000 q^{13} -11.2376 q^{15} -4.75014 q^{17} -6.50940 q^{19} +2.73758 q^{21} -1.11828 q^{23} +11.8506 q^{25} -4.09091 q^{27} +5.69879 q^{29} +9.59749 q^{31} -2.73758 q^{33} -4.10495 q^{35} +0.925635 q^{37} +2.73758 q^{39} -5.67379 q^{41} -5.94812 q^{43} +18.4491 q^{45} -1.11828 q^{47} +1.00000 q^{49} +13.0039 q^{51} -10.7803 q^{53} +4.10495 q^{55} +17.8200 q^{57} +7.08159 q^{59} +7.14995 q^{61} -4.49435 q^{63} -4.10495 q^{65} -11.8256 q^{67} +3.06139 q^{69} -7.59293 q^{71} +13.5277 q^{73} -32.4421 q^{75} -1.00000 q^{77} -16.4510 q^{79} -2.28386 q^{81} +0.448733 q^{83} -19.4991 q^{85} -15.6009 q^{87} -7.30086 q^{89} +1.00000 q^{91} -26.2739 q^{93} -26.7208 q^{95} -4.67291 q^{97} +4.49435 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 11 q - 2 q^{3} + 2 q^{5} - 11 q^{7} + 9 q^{9}+O(q^{10}) \) 11 * q - 2 * q^3 + 2 * q^5 - 11 * q^7 + 9 * q^9	\( 11 q - 2 q^{3} + 2 q^{5} - 11 q^{7} + 9 q^{9} + 11 q^{11} - 11 q^{13} - 7 q^{15} - 4 q^{17} - 16 q^{19} + 2 q^{21} - 3 q^{23} + 11 q^{25} - 11 q^{27} + q^{29} + 14 q^{31} - 2 q^{33} - 2 q^{35} - 8 q^{37} + 2 q^{39} + 4 q^{41} - 30 q^{43} + 13 q^{45} - 3 q^{47} + 11 q^{49} - 14 q^{51} - 5 q^{53} + 2 q^{55} - 22 q^{57} + 11 q^{59} + 15 q^{61} - 9 q^{63} - 2 q^{65} - 41 q^{67} + 12 q^{69} + q^{71} - 8 q^{73} - 24 q^{75} - 11 q^{77} - 26 q^{79} + 19 q^{81} - 31 q^{83} - 27 q^{85} - 25 q^{87} + 2 q^{89} + 11 q^{91} - 37 q^{93} - 6 q^{97} + 9 q^{99}+O(q^{100}) \) 11 * q - 2 * q^3 + 2 * q^5 - 11 * q^7 + 9 * q^9 + 11 * q^11 - 11 * q^13 - 7 * q^15 - 4 * q^17 - 16 * q^19 + 2 * q^21 - 3 * q^23 + 11 * q^25 - 11 * q^27 + q^29 + 14 * q^31 - 2 * q^33 - 2 * q^35 - 8 * q^37 + 2 * q^39 + 4 * q^41 - 30 * q^43 + 13 * q^45 - 3 * q^47 + 11 * q^49 - 14 * q^51 - 5 * q^53 + 2 * q^55 - 22 * q^57 + 11 * q^59 + 15 * q^61 - 9 * q^63 - 2 * q^65 - 41 * q^67 + 12 * q^69 + q^71 - 8 * q^73 - 24 * q^75 - 11 * q^77 - 26 * q^79 + 19 * q^81 - 31 * q^83 - 27 * q^85 - 25 * q^87 + 2 * q^89 + 11 * q^91 - 37 * q^93 - 6 * q^97 + 9 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more