LMFDB - Embedded newform 8008.2.a.j.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

chi = DirichletCharacter(H, H._module([0, 0, 0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("8008.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8008 = 2^{3} \cdot 7 \cdot 11 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8008.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$63.9442019386$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	5.5.668973.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 9x^{3} - x^{2} + 7x - 2 $ x^5 - x^4 - 9x^3 - x^2 + 7x - 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 3 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-1.26787$ 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)

$f(q)$	$=$	$q+2.43421 q^{3} +1.30958 q^{5} +1.00000 q^{7} +2.92540 q^{9} +O(q^{10})$	$q+2.43421 q^{3} +1.30958 q^{5} +1.00000 q^{7} +2.92540 q^{9} -1.00000 q^{11} -1.00000 q^{13} +3.18779 q^{15} +5.30958 q^{17} -1.44400 q^{19} +2.43421 q^{21} +8.93159 q^{23} -3.28500 q^{25} -0.181603 q^{27} -5.58176 q^{29} -6.13083 q^{31} -2.43421 q^{33} +1.30958 q^{35} +8.97877 q^{37} -2.43421 q^{39} -3.64943 q^{41} +6.67538 q^{43} +3.83103 q^{45} -2.40679 q^{47} +1.00000 q^{49} +12.9246 q^{51} +4.61582 q^{53} -1.30958 q^{55} -3.51501 q^{57} +14.3055 q^{59} +11.6220 q^{61} +2.92540 q^{63} -1.30958 q^{65} +0.118445 q^{67} +21.7414 q^{69} +10.5248 q^{71} +13.9976 q^{73} -7.99640 q^{75} -1.00000 q^{77} -11.0968 q^{79} -9.21825 q^{81} +14.3412 q^{83} +6.95331 q^{85} -13.5872 q^{87} -12.3124 q^{89} -1.00000 q^{91} -14.9237 q^{93} -1.89103 q^{95} +17.7254 q^{97} -2.92540 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + q^{3} - q^{5} + 5 q^{7} + 6 q^{9}+O(q^{10}) $ 5 * q + q^3 - q^5 + 5 * q^7 + 6 * q^9	$ 5 q + q^{3} - q^{5} + 5 q^{7} + 6 q^{9} - 5 q^{11} - 5 q^{13} - 5 q^{15} + 19 q^{17} - 5 q^{19} + q^{21} + 5 q^{23} + 12 q^{25} - 11 q^{27} - 17 q^{29} + 4 q^{31} - q^{33} - q^{35} + 10 q^{37} - q^{39} + 10 q^{41} - 25 q^{43} + q^{45} + 3 q^{47} + 5 q^{49} - q^{51} + 22 q^{53} + q^{55} + 16 q^{57} + 21 q^{59} + 26 q^{61} + 6 q^{63} + q^{65} + 28 q^{67} + 16 q^{69} + 28 q^{71} - 4 q^{73} - 5 q^{77} - 11 q^{79} + 5 q^{81} + 33 q^{85} + 31 q^{87} - 37 q^{89} - 5 q^{91} - 49 q^{93} + 29 q^{95} + 18 q^{97} - 6 q^{99}+O(q^{100}) $ 5 * q + q^3 - q^5 + 5 * q^7 + 6 * q^9 - 5 * q^11 - 5 * q^13 - 5 * q^15 + 19 * q^17 - 5 * q^19 + q^21 + 5 * q^23 + 12 * q^25 - 11 * q^27 - 17 * q^29 + 4 * q^31 - q^33 - q^35 + 10 * q^37 - q^39 + 10 * q^41 - 25 * q^43 + q^45 + 3 * q^47 + 5 * q^49 - q^51 + 22 * q^53 + q^55 + 16 * q^57 + 21 * q^59 + 26 * q^61 + 6 * q^63 + q^65 + 28 * q^67 + 16 * q^69 + 28 * q^71 - 4 * q^73 - 5 * q^77 - 11 * q^79 + 5 * q^81 + 33 * q^85 + 31 * q^87 - 37 * q^89 - 5 * q^91 - 49 * q^93 + 29 * q^95 + 18 * q^97 - 6 * 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
$3$	2.43421	1.40539	0.702697	−	0.711489i	$-0.251979\pi$
$3$	2.43421	1.40539	0.702697	+	0.711489i	$0.251979\pi$
$4$	0	0
$5$	1.30958	0.585661	0.292831	−	0.956164i	$-0.405403\pi$
$5$	1.30958	0.585661	0.292831	+	0.956164i	$0.405403\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	2.92540	0.975132
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	3.18779	0.823085
$16$	0	0
$17$	5.30958	1.28776	0.643881	−	0.765126i	$-0.277323\pi$
$17$	5.30958	1.28776	0.643881	+	0.765126i	$0.277323\pi$
$18$	0	0
$19$	−1.44400	−0.331277	−0.165638	−	0.986187i	$-0.552968\pi$
$19$	−1.44400	−0.331277	−0.165638	+	0.986187i	$0.552968\pi$
$20$	0	0
$21$	2.43421	0.531189
$22$	0	0
$23$	8.93159	1.86236	0.931182	−	0.364554i	$-0.118779\pi$
$23$	8.93159	1.86236	0.931182	+	0.364554i	$0.118779\pi$
$24$	0	0
$25$	−3.28500	−0.657001
$26$	0	0
$27$	−0.181603	−0.0349496
$28$	0	0
$29$	−5.58176	−1.03651	−0.518254	−	0.855227i	$-0.673418\pi$
$29$	−5.58176	−1.03651	−0.518254	+	0.855227i	$0.673418\pi$
$30$	0	0
$31$	−6.13083	−1.10113	−0.550564	−	0.834793i	$-0.685587\pi$
$31$	−6.13083	−1.10113	−0.550564	+	0.834793i	$0.685587\pi$
$32$	0	0
$33$	−2.43421	−0.423742
$34$	0	0
$35$	1.30958	0.221359
$36$	0	0
$37$	8.97877	1.47610	0.738050	−	0.674746i	$-0.235747\pi$
$37$	8.97877	1.47610	0.738050	+	0.674746i	$0.235747\pi$
$38$	0	0
$39$	−2.43421	−0.389786
$40$	0	0
$41$	−3.64943	−0.569945	−0.284973	−	0.958536i	$-0.591985\pi$
$41$	−3.64943	−0.569945	−0.284973	+	0.958536i	$0.591985\pi$
$42$	0	0
$43$	6.67538	1.01799	0.508993	−	0.860771i	$-0.330018\pi$
$43$	6.67538	1.01799	0.508993	+	0.860771i	$0.330018\pi$
$44$	0	0
$45$	3.83103	0.571097
$46$	0	0
$47$	−2.40679	−0.351066	−0.175533	−	0.984474i	$-0.556165\pi$
$47$	−2.40679	−0.351066	−0.175533	+	0.984474i	$0.556165\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	12.9246	1.80981
$52$	0	0
$53$	4.61582	0.634031	0.317016	−	0.948420i	$-0.397319\pi$
$53$	4.61582	0.634031	0.317016	+	0.948420i	$0.397319\pi$
$54$	0	0
$55$	−1.30958	−0.176584
$56$	0	0
$57$	−3.51501	−0.465574
$58$	0	0
$59$	14.3055	1.86242	0.931208	−	0.364488i	$-0.118756\pi$
$59$	14.3055	1.86242	0.931208	+	0.364488i	$0.118756\pi$
$60$	0	0
$61$	11.6220	1.48805	0.744023	−	0.668154i	$-0.232915\pi$
$61$	11.6220	1.48805	0.744023	+	0.668154i	$0.232915\pi$
$62$	0	0
$63$	2.92540	0.368565
$64$	0	0
$65$	−1.30958	−0.162433
$66$	0	0
$67$	0.118445	0.0144704	0.00723518	−	0.999974i	$-0.497697\pi$
$67$	0.118445	0.0144704	0.00723518	+	0.999974i	$0.497697\pi$
$68$	0	0
$69$	21.7414	2.61736
$70$	0	0
$71$	10.5248	1.24906	0.624532	−	0.780999i	$-0.285290\pi$
$71$	10.5248	1.24906	0.624532	+	0.780999i	$0.285290\pi$
$72$	0	0
$73$	13.9976	1.63829	0.819147	−	0.573583i	$-0.194447\pi$
$73$	13.9976	1.63829	0.819147	+	0.573583i	$0.194447\pi$
$74$	0	0
$75$	−7.99640	−0.923345
$76$	0	0
$77$	−1.00000	−0.113961
$78$	0	0
$79$	−11.0968	−1.24848	−0.624242	−	0.781231i	$-0.714592\pi$
$79$	−11.0968	−1.24848	−0.624242	+	0.781231i	$0.714592\pi$
$80$	0	0
$81$	−9.21825	−1.02425
$82$	0	0
$83$	14.3412	1.57415	0.787077	−	0.616855i	$-0.211593\pi$
$83$	14.3412	1.57415	0.787077	+	0.616855i	$0.211593\pi$
$84$	0	0
$85$	6.95331	0.754192
$86$	0	0
$87$	−13.5872	−1.45670
$88$	0	0
$89$	−12.3124	−1.30511	−0.652557	−	0.757739i	$-0.726304\pi$
$89$	−12.3124	−1.30511	−0.652557	+	0.757739i	$0.726304\pi$
$90$	0	0
$91$	−1.00000	−0.104828
$92$	0	0
$93$	−14.9237	−1.54752
$94$	0	0
$95$	−1.89103	−0.194016
$96$	0	0
$97$	17.7254	1.79974	0.899871	−	0.436156i	$-0.143660\pi$
$97$	17.7254	1.79974	0.899871	+	0.436156i	$0.143660\pi$
$98$	0	0
$99$	−2.92540	−0.294013
$100$	0	0
$101$	−5.64158	−0.561359	−0.280679	−	0.959802i	$-0.590560\pi$
$101$	−5.64158	−0.561359	−0.280679	+	0.959802i	$0.590560\pi$
$102$	0	0
$103$	−2.83512	−0.279353	−0.139676	−	0.990197i	$-0.544606\pi$
$103$	−2.83512	−0.279353	−0.139676	+	0.990197i	$0.544606\pi$
$104$	0	0
$105$	3.18779	0.311097
$106$	0	0
$107$	−3.21900	−0.311192	−0.155596	−	0.987821i	$-0.549730\pi$
$107$	−3.21900	−0.311192	−0.155596	+	0.987821i	$0.549730\pi$
$108$	0	0
$109$	16.5812	1.58819	0.794095	−	0.607793i	$-0.207945\pi$
$109$	16.5812	1.58819	0.794095	+	0.607793i	$0.207945\pi$
$110$	0	0
$111$	21.8562	2.07450
$112$	0	0
$113$	0.535512	0.0503767	0.0251884	−	0.999683i	$-0.491981\pi$
$113$	0.535512	0.0503767	0.0251884	+	0.999683i	$0.491981\pi$
$114$	0	0
$115$	11.6966	1.09071
$116$	0	0
$117$	−2.92540	−0.270453
$118$	0	0
$119$	5.30958	0.486728
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−8.88349	−0.800998
$124$	0	0
$125$	−10.8499	−0.970441
$126$	0	0
$127$	−14.1678	−1.25718	−0.628592	−	0.777735i	$-0.716369\pi$
$127$	−14.1678	−1.25718	−0.628592	+	0.777735i	$0.716369\pi$
$128$	0	0
$129$	16.2493	1.43067
$130$	0	0
$131$	−20.0250	−1.74959	−0.874797	−	0.484490i	$-0.839005\pi$
$131$	−20.0250	−1.74959	−0.874797	+	0.484490i	$0.839005\pi$
$132$	0	0
$133$	−1.44400	−0.125211
$134$	0	0
$135$	−0.237824	−0.0204686
$136$	0	0
$137$	−9.24445	−0.789807	−0.394903	−	0.918723i	$-0.629222\pi$
$137$	−9.24445	−0.789807	−0.394903	+	0.918723i	$0.629222\pi$
$138$	0	0
$139$	7.52314	0.638104	0.319052	−	0.947737i	$-0.396635\pi$
$139$	7.52314	0.638104	0.319052	+	0.947737i	$0.396635\pi$
$140$	0	0
$141$	−5.85864	−0.493386
$142$	0	0
$143$	1.00000	0.0836242
$144$	0	0
$145$	−7.30976	−0.607042
$146$	0	0
$147$	2.43421	0.200771
$148$	0	0
$149$	−16.7948	−1.37588	−0.687940	−	0.725768i	$-0.741485\pi$
$149$	−16.7948	−1.37588	−0.687940	+	0.725768i	$0.741485\pi$
$150$	0	0
$151$	−2.11016	−0.171722	−0.0858611	−	0.996307i	$-0.527364\pi$
$151$	−2.11016	−0.171722	−0.0858611	+	0.996307i	$0.527364\pi$
$152$	0	0
$153$	15.5326	1.25574
$154$	0	0
$155$	−8.02880	−0.644888
$156$	0	0
$157$	−6.71425	−0.535855	−0.267928	−	0.963439i	$-0.586339\pi$
$157$	−6.71425	−0.535855	−0.267928	+	0.963439i	$0.586339\pi$
$158$	0	0
$159$	11.2359	0.891064
$160$	0	0
$161$	8.93159	0.703908
$162$	0	0
$163$	−3.84582	−0.301228	−0.150614	−	0.988593i	$-0.548125\pi$
$163$	−3.84582	−0.301228	−0.150614	+	0.988593i	$0.548125\pi$
$164$	0	0
$165$	−3.18779	−0.248169
$166$	0	0
$167$	0.452598	0.0350231	0.0175115	−	0.999847i	$-0.494426\pi$
$167$	0.452598	0.0350231	0.0175115	+	0.999847i	$0.494426\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−4.22428	−0.323038
$172$	0	0
$173$	−7.14061	−0.542891	−0.271445	−	0.962454i	$-0.587502\pi$
$173$	−7.14061	−0.542891	−0.271445	+	0.962454i	$0.587502\pi$
$174$	0	0
$175$	−3.28500	−0.248323
$176$	0	0
$177$	34.8226	2.61743
$178$	0	0
$179$	−7.42940	−0.555299	−0.277650	−	0.960682i	$-0.589555\pi$
$179$	−7.42940	−0.555299	−0.277650	+	0.960682i	$0.589555\pi$
$180$	0	0
$181$	12.5572	0.933369	0.466684	−	0.884424i	$-0.345448\pi$
$181$	12.5572	0.933369	0.466684	+	0.884424i	$0.345448\pi$
$182$	0	0
$183$	28.2904	2.09129
$184$	0	0
$185$	11.7584	0.864495
$186$	0	0
$187$	−5.30958	−0.388275
$188$	0	0
$189$	−0.181603	−0.0132097
$190$	0	0
$191$	23.7540	1.71878	0.859390	−	0.511320i	$-0.170843\pi$
$191$	23.7540	1.71878	0.859390	+	0.511320i	$0.170843\pi$
$192$	0	0
$193$	−3.35777	−0.241698	−0.120849	−	0.992671i	$-0.538562\pi$
$193$	−3.35777	−0.241698	−0.120849	+	0.992671i	$0.538562\pi$
$194$	0	0
$195$	−3.18779	−0.228283
$196$	0	0
$197$	4.76640	0.339592	0.169796	−	0.985479i	$-0.445689\pi$
$197$	4.76640	0.339592	0.169796	+	0.985479i	$0.445689\pi$
$198$	0	0
$199$	1.11182	0.0788146	0.0394073	−	0.999223i	$-0.487453\pi$
$199$	1.11182	0.0788146	0.0394073	+	0.999223i	$0.487453\pi$
$200$	0	0
$201$	0.288321	0.0203366
$202$	0	0
$203$	−5.58176	−0.391763
$204$	0	0
$205$	−4.77922	−0.333795
$206$	0	0
$207$	26.1284	1.81605
$208$	0	0
$209$	1.44400	0.0998837
$210$	0	0
$211$	19.4015	1.33566	0.667828	−	0.744315i	$-0.267224\pi$
$211$	19.4015	1.33566	0.667828	+	0.744315i	$0.267224\pi$
$212$	0	0
$213$	25.6196	1.75543
$214$	0	0
$215$	8.74193	0.596195
$216$	0	0
$217$	−6.13083	−0.416188
$218$	0	0
$219$	34.0731	2.30245
$220$	0	0
$221$	−5.30958	−0.357161
$222$	0	0
$223$	26.6327	1.78346	0.891729	−	0.452570i	$-0.149493\pi$
$223$	26.6327	1.78346	0.891729	+	0.452570i	$0.149493\pi$
$224$	0	0
$225$	−9.60994	−0.640662
$226$	0	0
$227$	−4.88683	−0.324351	−0.162175	−	0.986762i	$-0.551851\pi$
$227$	−4.88683	−0.324351	−0.162175	+	0.986762i	$0.551851\pi$
$228$	0	0
$229$	19.7445	1.30476	0.652378	−	0.757894i	$-0.273771\pi$
$229$	19.7445	1.30476	0.652378	+	0.757894i	$0.273771\pi$
$230$	0	0
$231$	−2.43421	−0.160159
$232$	0	0
$233$	−2.27887	−0.149293	−0.0746467	−	0.997210i	$-0.523783\pi$
$233$	−2.27887	−0.149293	−0.0746467	+	0.997210i	$0.523783\pi$
$234$	0	0
$235$	−3.15188	−0.205606
$236$	0	0
$237$	−27.0119	−1.75461
$238$	0	0
$239$	−2.91921	−0.188828	−0.0944139	−	0.995533i	$-0.530098\pi$
$239$	−2.91921	−0.188828	−0.0944139	+	0.995533i	$0.530098\pi$
$240$	0	0
$241$	−18.2628	−1.17641	−0.588207	−	0.808711i	$-0.700166\pi$
$241$	−18.2628	−1.17641	−0.588207	+	0.808711i	$0.700166\pi$
$242$	0	0
$243$	−21.8944	−1.40452
$244$	0	0
$245$	1.30958	0.0836659
$246$	0	0
$247$	1.44400	0.0918796
$248$	0	0
$249$	34.9096	2.21231
$250$	0	0
$251$	−22.7636	−1.43682	−0.718411	−	0.695618i	$-0.755130\pi$
$251$	−22.7636	−1.43682	−0.718411	+	0.695618i	$0.755130\pi$
$252$	0	0
$253$	−8.93159	−0.561524
$254$	0	0
$255$	16.9258	1.05994
$256$	0	0
$257$	25.5863	1.59603	0.798015	−	0.602638i	$-0.205884\pi$
$257$	25.5863	1.59603	0.798015	+	0.602638i	$0.205884\pi$
$258$	0	0
$259$	8.97877	0.557913
$260$	0	0
$261$	−16.3289	−1.01073
$262$	0	0
$263$	−9.11200	−0.561870	−0.280935	−	0.959727i	$-0.590645\pi$
$263$	−9.11200	−0.561870	−0.280935	+	0.959727i	$0.590645\pi$
$264$	0	0
$265$	6.04477	0.371328
$266$	0	0
$267$	−29.9711	−1.83420
$268$	0	0
$269$	−1.12797	−0.0687738	−0.0343869	−	0.999409i	$-0.510948\pi$
$269$	−1.12797	−0.0687738	−0.0343869	+	0.999409i	$0.510948\pi$
$270$	0	0
$271$	8.43287	0.512260	0.256130	−	0.966642i	$-0.417552\pi$
$271$	8.43287	0.512260	0.256130	+	0.966642i	$0.417552\pi$
$272$	0	0
$273$	−2.43421	−0.147325
$274$	0	0
$275$	3.28500	0.198093
$276$	0	0
$277$	−9.78523	−0.587937	−0.293969	−	0.955815i	$-0.594976\pi$
$277$	−9.78523	−0.587937	−0.293969	+	0.955815i	$0.594976\pi$
$278$	0	0
$279$	−17.9351	−1.07375
$280$	0	0
$281$	5.05889	0.301788	0.150894	−	0.988550i	$-0.451785\pi$
$281$	5.05889	0.301788	0.150894	+	0.988550i	$0.451785\pi$
$282$	0	0
$283$	−28.2306	−1.67814	−0.839068	−	0.544026i	$-0.816899\pi$
$283$	−28.2306	−1.67814	−0.839068	+	0.544026i	$0.816899\pi$
$284$	0	0
$285$	−4.60318	−0.272669
$286$	0	0
$287$	−3.64943	−0.215419
$288$	0	0
$289$	11.1916	0.658331
$290$	0	0
$291$	43.1474	2.52935
$292$	0	0
$293$	−16.0621	−0.938358	−0.469179	−	0.883103i	$-0.655450\pi$
$293$	−16.0621	−0.938358	−0.469179	+	0.883103i	$0.655450\pi$
$294$	0	0
$295$	18.7342	1.09075
$296$	0	0
$297$	0.181603	0.0105377
$298$	0	0
$299$	−8.93159	−0.516527
$300$	0	0
$301$	6.67538	0.384762
$302$	0	0
$303$	−13.7328	−0.788930
$304$	0	0
$305$	15.2199	0.871491
$306$	0	0
$307$	−31.4794	−1.79663	−0.898313	−	0.439356i	$-0.855207\pi$
$307$	−31.4794	−1.79663	−0.898313	+	0.439356i	$0.855207\pi$
$308$	0	0
$309$	−6.90129	−0.392601
$310$	0	0
$311$	−8.27672	−0.469330	−0.234665	−	0.972076i	$-0.575399\pi$
$311$	−8.27672	−0.469330	−0.234665	+	0.972076i	$0.575399\pi$
$312$	0	0
$313$	−31.1040	−1.75810	−0.879052	−	0.476726i	$-0.841823\pi$
$313$	−31.1040	−1.75810	−0.879052	+	0.476726i	$0.841823\pi$
$314$	0	0
$315$	3.83103	0.215854
$316$	0	0
$317$	1.45211	0.0815585	0.0407793	−	0.999168i	$-0.487016\pi$
$317$	1.45211	0.0815585	0.0407793	+	0.999168i	$0.487016\pi$
$318$	0	0
$319$	5.58176	0.312519
$320$	0	0
$321$	−7.83572	−0.437347
$322$	0	0
$323$	−7.66704	−0.426605
$324$	0	0
$325$	3.28500	0.182219
$326$	0	0
$327$	40.3622	2.23203
$328$	0	0
$329$	−2.40679	−0.132691
$330$	0	0
$331$	6.13794	0.337372	0.168686	−	0.985670i	$-0.446048\pi$
$331$	6.13794	0.337372	0.168686	+	0.985670i	$0.446048\pi$
$332$	0	0
$333$	26.2664	1.43939
$334$	0	0
$335$	0.155113	0.00847473
$336$	0	0
$337$	−25.7355	−1.40190	−0.700951	−	0.713209i	$-0.747241\pi$
$337$	−25.7355	−1.40190	−0.700951	+	0.713209i	$0.747241\pi$
$338$	0	0
$339$	1.30355	0.0707992
$340$	0	0
$341$	6.13083	0.332003
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	28.4720	1.53288
$346$	0	0
$347$	8.13009	0.436446	0.218223	−	0.975899i	$-0.429974\pi$
$347$	8.13009	0.436446	0.218223	+	0.975899i	$0.429974\pi$
$348$	0	0
$349$	17.5158	0.937597	0.468799	−	0.883305i	$-0.344687\pi$
$349$	17.5158	0.937597	0.468799	+	0.883305i	$0.344687\pi$
$350$	0	0
$351$	0.181603	0.00969328
$352$	0	0
$353$	−5.48758	−0.292075	−0.146037	−	0.989279i	$-0.546652\pi$
$353$	−5.48758	−0.292075	−0.146037	+	0.989279i	$0.546652\pi$
$354$	0	0
$355$	13.7830	0.731528
$356$	0	0
$357$	12.9246	0.684045
$358$	0	0
$359$	−15.1990	−0.802173	−0.401086	−	0.916040i	$-0.631367\pi$
$359$	−15.1990	−0.802173	−0.401086	+	0.916040i	$0.631367\pi$
$360$	0	0
$361$	−16.9149	−0.890256
$362$	0	0
$363$	2.43421	0.127763
$364$	0	0
$365$	18.3309	0.959486
$366$	0	0
$367$	0.0566587	0.00295756	0.00147878	−	0.999999i	$-0.499529\pi$
$367$	0.0566587	0.00295756	0.00147878	+	0.999999i	$0.499529\pi$
$368$	0	0
$369$	−10.6760	−0.555772
$370$	0	0
$371$	4.61582	0.239641
$372$	0	0
$373$	−8.30927	−0.430238	−0.215119	−	0.976588i	$-0.569014\pi$
$373$	−8.30927	−0.430238	−0.215119	+	0.976588i	$0.569014\pi$
$374$	0	0
$375$	−26.4109	−1.36385
$376$	0	0
$377$	5.58176	0.287476
$378$	0	0
$379$	−30.8510	−1.58471	−0.792354	−	0.610061i	$-0.791145\pi$
$379$	−30.8510	−1.58471	−0.792354	+	0.610061i	$0.791145\pi$
$380$	0	0
$381$	−34.4873	−1.76684
$382$	0	0
$383$	32.8915	1.68068	0.840338	−	0.542063i	$-0.182357\pi$
$383$	32.8915	1.68068	0.840338	+	0.542063i	$0.182357\pi$
$384$	0	0
$385$	−1.30958	−0.0667423
$386$	0	0
$387$	19.5281	0.992670
$388$	0	0
$389$	1.09936	0.0557397	0.0278698	−	0.999612i	$-0.491128\pi$
$389$	1.09936	0.0557397	0.0278698	+	0.999612i	$0.491128\pi$
$390$	0	0
$391$	47.4230	2.39828
$392$	0	0
$393$	−48.7452	−2.45887
$394$	0	0
$395$	−14.5321	−0.731189
$396$	0	0
$397$	−27.4618	−1.37827	−0.689135	−	0.724633i	$-0.742009\pi$
$397$	−27.4618	−1.37827	−0.689135	+	0.724633i	$0.742009\pi$
$398$	0	0
$399$	−3.51501	−0.175971
$400$	0	0
$401$	23.1990	1.15850	0.579251	−	0.815149i	$-0.303345\pi$
$401$	23.1990	1.15850	0.579251	+	0.815149i	$0.303345\pi$
$402$	0	0
$403$	6.13083	0.305398
$404$	0	0
$405$	−12.0720	−0.599863
$406$	0	0
$407$	−8.97877	−0.445061
$408$	0	0
$409$	5.56815	0.275327	0.137664	−	0.990479i	$-0.456041\pi$
$409$	5.56815	0.275327	0.137664	+	0.990479i	$0.456041\pi$
$410$	0	0
$411$	−22.5030	−1.10999
$412$	0	0
$413$	14.3055	0.703927
$414$	0	0
$415$	18.7810	0.921921
$416$	0	0
$417$	18.3129	0.896788
$418$	0	0
$419$	3.09822	0.151358	0.0756790	−	0.997132i	$-0.475888\pi$
$419$	3.09822	0.151358	0.0756790	+	0.997132i	$0.475888\pi$
$420$	0	0
$421$	34.7521	1.69371	0.846857	−	0.531821i	$-0.178492\pi$
$421$	34.7521	1.69371	0.846857	+	0.531821i	$0.178492\pi$
$422$	0	0
$423$	−7.04081	−0.342336
$424$	0	0
$425$	−17.4420	−0.846061
$426$	0	0
$427$	11.6220	0.562428
$428$	0	0
$429$	2.43421	0.117525
$430$	0	0
$431$	−36.5946	−1.76270	−0.881350	−	0.472465i	$-0.843364\pi$
$431$	−36.5946	−1.76270	−0.881350	+	0.472465i	$0.843364\pi$
$432$	0	0
$433$	−27.9134	−1.34143	−0.670716	−	0.741714i	$-0.734013\pi$
$433$	−27.9134	−1.34143	−0.670716	+	0.741714i	$0.734013\pi$
$434$	0	0
$435$	−17.7935	−0.853134
$436$	0	0
$437$	−12.8972	−0.616958
$438$	0	0
$439$	−37.7039	−1.79951	−0.899754	−	0.436398i	$-0.856254\pi$
$439$	−37.7039	−1.79951	−0.899754	+	0.436398i	$0.856254\pi$
$440$	0	0
$441$	2.92540	0.139305
$442$	0	0
$443$	3.03045	0.143981	0.0719906	−	0.997405i	$-0.477065\pi$
$443$	3.03045	0.143981	0.0719906	+	0.997405i	$0.477065\pi$
$444$	0	0
$445$	−16.1241	−0.764355
$446$	0	0
$447$	−40.8820	−1.93365
$448$	0	0
$449$	6.47823	0.305727	0.152863	−	0.988247i	$-0.451151\pi$
$449$	6.47823	0.305727	0.152863	+	0.988247i	$0.451151\pi$
$450$	0	0
$451$	3.64943	0.171845
$452$	0	0
$453$	−5.13658	−0.241337
$454$	0	0
$455$	−1.30958	−0.0613940
$456$	0	0
$457$	4.03690	0.188838	0.0944192	−	0.995533i	$-0.469901\pi$
$457$	4.03690	0.188838	0.0944192	+	0.995533i	$0.469901\pi$
$458$	0	0
$459$	−0.964238	−0.0450068
$460$	0	0
$461$	−12.3352	−0.574509	−0.287254	−	0.957854i	$-0.592742\pi$
$461$	−12.3352	−0.574509	−0.287254	+	0.957854i	$0.592742\pi$
$462$	0	0
$463$	−14.9993	−0.697074	−0.348537	−	0.937295i	$-0.613321\pi$
$463$	−14.9993	−0.697074	−0.348537	+	0.937295i	$0.613321\pi$
$464$	0	0
$465$	−19.5438	−0.906322
$466$	0	0
$467$	−7.14272	−0.330525	−0.165263	−	0.986250i	$-0.552847\pi$
$467$	−7.14272	−0.330525	−0.165263	+	0.986250i	$0.552847\pi$
$468$	0	0
$469$	0.118445	0.00546929
$470$	0	0
$471$	−16.3439	−0.753088
$472$	0	0
$473$	−6.67538	−0.306934
$474$	0	0
$475$	4.74355	0.217649
$476$	0	0
$477$	13.5031	0.618264
$478$	0	0
$479$	16.7466	0.765174	0.382587	−	0.923920i	$-0.375033\pi$
$479$	16.7466	0.765174	0.382587	+	0.923920i	$0.375033\pi$
$480$	0	0
$481$	−8.97877	−0.409396
$482$	0	0
$483$	21.7414	0.989267
$484$	0	0
$485$	23.2128	1.05404
$486$	0	0
$487$	30.3938	1.37728	0.688638	−	0.725105i	$-0.258209\pi$
$487$	30.3938	1.37728	0.688638	+	0.725105i	$0.258209\pi$
$488$	0	0
$489$	−9.36155	−0.423344
$490$	0	0
$491$	7.26242	0.327749	0.163874	−	0.986481i	$-0.447601\pi$
$491$	7.26242	0.327749	0.163874	+	0.986481i	$0.447601\pi$
$492$	0	0
$493$	−29.6368	−1.33478
$494$	0	0
$495$	−3.83103	−0.172192
$496$	0	0
$497$	10.5248	0.472102
$498$	0	0
$499$	−33.4915	−1.49929	−0.749643	−	0.661843i	$-0.769775\pi$
$499$	−33.4915	−1.49929	−0.749643	+	0.661843i	$0.769775\pi$
$500$	0	0
$501$	1.10172	0.0492212
$502$	0	0
$503$	26.2830	1.17190	0.585951	−	0.810346i	$-0.300721\pi$
$503$	26.2830	1.17190	0.585951	+	0.810346i	$0.300721\pi$
$504$	0	0
$505$	−7.38810	−0.328766
$506$	0	0
$507$	2.43421	0.108107
$508$	0	0
$509$	26.6308	1.18039	0.590194	−	0.807262i	$-0.299051\pi$
$509$	26.6308	1.18039	0.590194	+	0.807262i	$0.299051\pi$
$510$	0	0
$511$	13.9976	0.619217
$512$	0	0
$513$	0.262236	0.0115780
$514$	0	0
$515$	−3.71282	−0.163606
$516$	0	0
$517$	2.40679	0.105850
$518$	0	0
$519$	−17.3818	−0.762975
$520$	0	0
$521$	15.8034	0.692357	0.346179	−	0.938169i	$-0.387479\pi$
$521$	15.8034	0.692357	0.346179	+	0.938169i	$0.387479\pi$
$522$	0	0
$523$	30.1601	1.31881	0.659405	−	0.751788i	$-0.270808\pi$
$523$	30.1601	1.31881	0.659405	+	0.751788i	$0.270808\pi$
$524$	0	0
$525$	−7.99640	−0.348992
$526$	0	0
$527$	−32.5521	−1.41799
$528$	0	0
$529$	56.7732	2.46840
$530$	0	0
$531$	41.8492	1.81610
$532$	0	0
$533$	3.64943	0.158074
$534$	0	0
$535$	−4.21553	−0.182253
$536$	0	0
$537$	−18.0847	−0.780414
$538$	0	0
$539$	−1.00000	−0.0430730
$540$	0	0
$541$	37.4540	1.61027	0.805136	−	0.593090i	$-0.202092\pi$
$541$	37.4540	1.61027	0.805136	+	0.593090i	$0.202092\pi$
$542$	0	0
$543$	30.5669	1.31175
$544$	0	0
$545$	21.7144	0.930142
$546$	0	0
$547$	17.3336	0.741132	0.370566	−	0.928806i	$-0.379164\pi$
$547$	17.3336	0.741132	0.370566	+	0.928806i	$0.379164\pi$
$548$	0	0
$549$	33.9990	1.45104
$550$	0	0
$551$	8.06008	0.343371
$552$	0	0
$553$	−11.0968	−0.471883
$554$	0	0
$555$	28.6225	1.21496
$556$	0	0
$557$	−4.80454	−0.203575	−0.101787	−	0.994806i	$-0.532456\pi$
$557$	−4.80454	−0.203575	−0.101787	+	0.994806i	$0.532456\pi$
$558$	0	0
$559$	−6.67538	−0.282338
$560$	0	0
$561$	−12.9246	−0.545679
$562$	0	0
$563$	8.15734	0.343791	0.171895	−	0.985115i	$-0.445011\pi$
$563$	8.15734	0.343791	0.171895	+	0.985115i	$0.445011\pi$
$564$	0	0
$565$	0.701295	0.0295037
$566$	0	0
$567$	−9.21825	−0.387130
$568$	0	0
$569$	−36.6773	−1.53759	−0.768797	−	0.639493i	$-0.779144\pi$
$569$	−36.6773	−1.53759	−0.768797	+	0.639493i	$0.779144\pi$
$570$	0	0
$571$	5.57563	0.233333	0.116666	−	0.993171i	$-0.462779\pi$
$571$	5.57563	0.233333	0.116666	+	0.993171i	$0.462779\pi$
$572$	0	0
$573$	57.8224	2.41556
$574$	0	0
$575$	−29.3403	−1.22357
$576$	0	0
$577$	−43.0825	−1.79355	−0.896774	−	0.442490i	$-0.854095\pi$
$577$	−43.0825	−1.79355	−0.896774	+	0.442490i	$0.854095\pi$
$578$	0	0
$579$	−8.17353	−0.339680
$580$	0	0
$581$	14.3412	0.594974
$582$	0	0
$583$	−4.61582	−0.191168
$584$	0	0
$585$	−3.83103	−0.158394
$586$	0	0
$587$	−23.3648	−0.964368	−0.482184	−	0.876070i	$-0.660156\pi$
$587$	−23.3648	−0.964368	−0.482184	+	0.876070i	$0.660156\pi$
$588$	0	0
$589$	8.85292	0.364778
$590$	0	0
$591$	11.6024	0.477260
$592$	0	0
$593$	−28.2788	−1.16127	−0.580636	−	0.814163i	$-0.697196\pi$
$593$	−28.2788	−1.16127	−0.580636	+	0.814163i	$0.697196\pi$
$594$	0	0
$595$	6.95331	0.285058
$596$	0	0
$597$	2.70640	0.110766
$598$	0	0
$599$	26.9244	1.10010	0.550051	−	0.835131i	$-0.314608\pi$
$599$	26.9244	1.10010	0.550051	+	0.835131i	$0.314608\pi$
$600$	0	0
$601$	20.6556	0.842561	0.421281	−	0.906930i	$-0.361581\pi$
$601$	20.6556	0.842561	0.421281	+	0.906930i	$0.361581\pi$
$602$	0	0
$603$	0.346499	0.0141105
$604$	0	0
$605$	1.30958	0.0532419
$606$	0	0
$607$	−17.1738	−0.697061	−0.348531	−	0.937297i	$-0.613319\pi$
$607$	−17.1738	−0.697061	−0.348531	+	0.937297i	$0.613319\pi$
$608$	0	0
$609$	−13.5872	−0.550581
$610$	0	0
$611$	2.40679	0.0973683
$612$	0	0
$613$	8.19637	0.331048	0.165524	−	0.986206i	$-0.447068\pi$
$613$	8.19637	0.331048	0.165524	+	0.986206i	$0.447068\pi$
$614$	0	0
$615$	−11.6336	−0.469113
$616$	0	0
$617$	19.2026	0.773067	0.386533	−	0.922275i	$-0.373672\pi$
$617$	19.2026	0.773067	0.386533	+	0.922275i	$0.373672\pi$
$618$	0	0
$619$	31.0776	1.24911	0.624557	−	0.780979i	$-0.285280\pi$
$619$	31.0776	1.24911	0.624557	+	0.780979i	$0.285280\pi$
$620$	0	0
$621$	−1.62201	−0.0650889
$622$	0	0
$623$	−12.3124	−0.493287
$624$	0	0
$625$	2.21628	0.0886510
$626$	0	0
$627$	3.51501	0.140376
$628$	0	0
$629$	47.6735	1.90087
$630$	0	0
$631$	−42.8626	−1.70633	−0.853166	−	0.521640i	$-0.825320\pi$
$631$	−42.8626	−1.70633	−0.853166	+	0.521640i	$0.825320\pi$
$632$	0	0
$633$	47.2275	1.87712
$634$	0	0
$635$	−18.5538	−0.736284
$636$	0	0
$637$	−1.00000	−0.0396214
$638$	0	0
$639$	30.7892	1.21800
$640$	0	0
$641$	31.8047	1.25621	0.628106	−	0.778128i	$-0.283831\pi$
$641$	31.8047	1.25621	0.628106	+	0.778128i	$0.283831\pi$
$642$	0	0
$643$	−46.9897	−1.85309	−0.926545	−	0.376183i	$-0.877236\pi$
$643$	−46.9897	−1.85309	−0.926545	+	0.376183i	$0.877236\pi$
$644$	0	0
$645$	21.2797	0.837888
$646$	0	0
$647$	−7.71409	−0.303272	−0.151636	−	0.988436i	$-0.548454\pi$
$647$	−7.71409	−0.303272	−0.151636	+	0.988436i	$0.548454\pi$
$648$	0	0
$649$	−14.3055	−0.561540
$650$	0	0
$651$	−14.9237	−0.584907
$652$	0	0
$653$	−28.6944	−1.12290	−0.561449	−	0.827511i	$-0.689756\pi$
$653$	−28.6944	−1.12290	−0.561449	+	0.827511i	$0.689756\pi$
$654$	0	0
$655$	−26.2243	−1.02467
$656$	0	0
$657$	40.9485	1.59755
$658$	0	0
$659$	−37.2716	−1.45189	−0.725947	−	0.687750i	$-0.758598\pi$
$659$	−37.2716	−1.45189	−0.725947	+	0.687750i	$0.758598\pi$
$660$	0	0
$661$	21.1393	0.822223	0.411112	−	0.911585i	$-0.365141\pi$
$661$	21.1393	0.822223	0.411112	+	0.911585i	$0.365141\pi$
$662$	0	0
$663$	−12.9246	−0.501952
$664$	0	0
$665$	−1.89103	−0.0733311
$666$	0	0
$667$	−49.8540	−1.93035
$668$	0	0
$669$	64.8297	2.50646
$670$	0	0
$671$	−11.6220	−0.448663
$672$	0	0
$673$	15.6947	0.604985	0.302492	−	0.953152i	$-0.402181\pi$
$673$	15.6947	0.604985	0.302492	+	0.953152i	$0.402181\pi$
$674$	0	0
$675$	0.596568	0.0229619
$676$	0	0
$677$	2.89450	0.111245	0.0556224	−	0.998452i	$-0.482286\pi$
$677$	2.89450	0.111245	0.0556224	+	0.998452i	$0.482286\pi$
$678$	0	0
$679$	17.7254	0.680239
$680$	0	0
$681$	−11.8956	−0.455840
$682$	0	0
$683$	−29.7398	−1.13796	−0.568980	−	0.822351i	$-0.692662\pi$
$683$	−29.7398	−1.13796	−0.568980	+	0.822351i	$0.692662\pi$
$684$	0	0
$685$	−12.1063	−0.462559
$686$	0	0
$687$	48.0624	1.83370
$688$	0	0
$689$	−4.61582	−0.175849
$690$	0	0
$691$	−25.6734	−0.976663	−0.488331	−	0.872658i	$-0.662394\pi$
$691$	−25.6734	−0.976663	−0.488331	+	0.872658i	$0.662394\pi$
$692$	0	0
$693$	−2.92540	−0.111127
$694$	0	0
$695$	9.85214	0.373713
$696$	0	0
$697$	−19.3769	−0.733954
$698$	0	0
$699$	−5.54725	−0.209816
$700$	0	0
$701$	2.85211	0.107723	0.0538614	−	0.998548i	$-0.482847\pi$
$701$	2.85211	0.107723	0.0538614	+	0.998548i	$0.482847\pi$
$702$	0	0
$703$	−12.9654	−0.488998
$704$	0	0
$705$	−7.67235	−0.288957
$706$	0	0
$707$	−5.64158	−0.212174
$708$	0	0
$709$	14.6803	0.551331	0.275666	−	0.961254i	$-0.411102\pi$
$709$	14.6803	0.551331	0.275666	+	0.961254i	$0.411102\pi$
$710$	0	0
$711$	−32.4624	−1.21744
$712$	0	0
$713$	−54.7580	−2.05070
$714$	0	0
$715$	1.30958	0.0489755
$716$	0	0
$717$	−7.10597	−0.265377
$718$	0	0
$719$	26.4770	0.987427	0.493714	−	0.869625i	$-0.335639\pi$
$719$	26.4770	0.987427	0.493714	+	0.869625i	$0.335639\pi$
$720$	0	0
$721$	−2.83512	−0.105585
$722$	0	0
$723$	−44.4557	−1.65332
$724$	0	0
$725$	18.3361	0.680986
$726$	0	0
$727$	9.34346	0.346530	0.173265	−	0.984875i	$-0.444568\pi$
$727$	9.34346	0.346530	0.173265	+	0.984875i	$0.444568\pi$
$728$	0	0
$729$	−25.6408	−0.949661
$730$	0	0
$731$	35.4434	1.31092
$732$	0	0
$733$	−14.3676	−0.530678	−0.265339	−	0.964155i	$-0.585484\pi$
$733$	−14.3676	−0.530678	−0.265339	+	0.964155i	$0.585484\pi$
$734$	0	0
$735$	3.18779	0.117584
$736$	0	0
$737$	−0.118445	−0.00436298
$738$	0	0
$739$	−14.5720	−0.536038	−0.268019	−	0.963414i	$-0.586369\pi$
$739$	−14.5720	−0.536038	−0.268019	+	0.963414i	$0.586369\pi$
$740$	0	0
$741$	3.51501	0.129127
$742$	0	0
$743$	−9.45395	−0.346832	−0.173416	−	0.984849i	$-0.555480\pi$
$743$	−9.45395	−0.346832	−0.173416	+	0.984849i	$0.555480\pi$
$744$	0	0
$745$	−21.9941	−0.805800
$746$	0	0
$747$	41.9538	1.53501
$748$	0	0
$749$	−3.21900	−0.117620
$750$	0	0
$751$	23.0879	0.842491	0.421245	−	0.906947i	$-0.361593\pi$
$751$	23.0879	0.842491	0.421245	+	0.906947i	$0.361593\pi$
$752$	0	0
$753$	−55.4114	−2.01930
$754$	0	0
$755$	−2.76342	−0.100571
$756$	0	0
$757$	−1.63004	−0.0592446	−0.0296223	−	0.999561i	$-0.509430\pi$
$757$	−1.63004	−0.0592446	−0.0296223	+	0.999561i	$0.509430\pi$
$758$	0	0
$759$	−21.7414	−0.789162
$760$	0	0
$761$	21.3870	0.775280	0.387640	−	0.921811i	$-0.373290\pi$
$761$	21.3870	0.775280	0.387640	+	0.921811i	$0.373290\pi$
$762$	0	0
$763$	16.5812	0.600280
$764$	0	0
$765$	20.3412	0.735437
$766$	0	0
$767$	−14.3055	−0.516541
$768$	0	0
$769$	20.2435	0.730000	0.365000	−	0.931007i	$-0.381069\pi$
$769$	20.2435	0.730000	0.365000	+	0.931007i	$0.381069\pi$
$770$	0	0
$771$	62.2825	2.24305
$772$	0	0
$773$	35.3131	1.27013	0.635063	−	0.772461i	$-0.280974\pi$
$773$	35.3131	1.27013	0.635063	+	0.772461i	$0.280974\pi$
$774$	0	0
$775$	20.1398	0.723443
$776$	0	0
$777$	21.8562	0.784088
$778$	0	0
$779$	5.26978	0.188810
$780$	0	0
$781$	−10.5248	−0.376607
$782$	0	0
$783$	1.01367	0.0362255
$784$	0	0
$785$	−8.79283	−0.313830
$786$	0	0
$787$	−18.0870	−0.644731	−0.322365	−	0.946615i	$-0.604478\pi$
$787$	−18.0870	−0.644731	−0.322365	+	0.946615i	$0.604478\pi$
$788$	0	0
$789$	−22.1805	−0.789648
$790$	0	0
$791$	0.535512	0.0190406
$792$	0	0
$793$	−11.6220	−0.412710
$794$	0	0
$795$	14.7143	0.521861
$796$	0	0
$797$	32.5035	1.15133	0.575666	−	0.817685i	$-0.304743\pi$
$797$	32.5035	1.15133	0.575666	+	0.817685i	$0.304743\pi$
$798$	0	0
$799$	−12.7790	−0.452090
$800$	0	0
$801$	−36.0187	−1.27266
$802$	0	0
$803$	−13.9976	−0.493964
$804$	0	0
$805$	11.6966	0.412251
$806$	0	0
$807$	−2.74573	−0.0966543
$808$	0	0
$809$	40.2892	1.41649	0.708247	−	0.705965i	$-0.249486\pi$
$809$	40.2892	1.41649	0.708247	+	0.705965i	$0.249486\pi$
$810$	0	0
$811$	−53.3286	−1.87262	−0.936310	−	0.351175i	$-0.885782\pi$
$811$	−53.3286	−1.87262	−0.936310	+	0.351175i	$0.885782\pi$
$812$	0	0
$813$	20.5274	0.719927
$814$	0	0
$815$	−5.03640	−0.176418
$816$	0	0
$817$	−9.63926	−0.337235
$818$	0	0
$819$	−2.92540	−0.102222
$820$	0	0
$821$	−5.40557	−0.188656	−0.0943278	−	0.995541i	$-0.530070\pi$
$821$	−5.40557	−0.188656	−0.0943278	+	0.995541i	$0.530070\pi$
$822$	0	0
$823$	−55.1841	−1.92360	−0.961798	−	0.273759i	$-0.911733\pi$
$823$	−55.1841	−1.92360	−0.961798	+	0.273759i	$0.911733\pi$
$824$	0	0
$825$	7.99640	0.278399
$826$	0	0
$827$	−40.4077	−1.40511	−0.702557	−	0.711628i	$-0.747958\pi$
$827$	−40.4077	−1.40511	−0.702557	+	0.711628i	$0.747958\pi$
$828$	0	0
$829$	−55.2329	−1.91832	−0.959159	−	0.282867i	$-0.908714\pi$
$829$	−55.2329	−1.91832	−0.959159	+	0.282867i	$0.908714\pi$
$830$	0	0
$831$	−23.8193	−0.826283
$832$	0	0
$833$	5.30958	0.183966
$834$	0	0
$835$	0.592712	0.0205116
$836$	0	0
$837$	1.11338	0.0384840
$838$	0	0
$839$	−3.06303	−0.105747	−0.0528737	−	0.998601i	$-0.516838\pi$
$839$	−3.06303	−0.105747	−0.0528737	+	0.998601i	$0.516838\pi$
$840$	0	0
$841$	2.15610	0.0743481
$842$	0	0
$843$	12.3144	0.424131
$844$	0	0
$845$	1.30958	0.0450509
$846$	0	0
$847$	1.00000	0.0343604
$848$	0	0
$849$	−68.7194	−2.35844
$850$	0	0
$851$	80.1946	2.74904
$852$	0	0
$853$	19.0017	0.650606	0.325303	−	0.945610i	$-0.394534\pi$
$853$	19.0017	0.650606	0.325303	+	0.945610i	$0.394534\pi$
$854$	0	0
$855$	−5.53202	−0.189191
$856$	0	0
$857$	−33.1025	−1.13076	−0.565380	−	0.824830i	$-0.691271\pi$
$857$	−33.1025	−1.13076	−0.565380	+	0.824830i	$0.691271\pi$
$858$	0	0
$859$	−14.0693	−0.480038	−0.240019	−	0.970768i	$-0.577154\pi$
$859$	−14.0693	−0.480038	−0.240019	+	0.970768i	$0.577154\pi$
$860$	0	0
$861$	−8.88349	−0.302749
$862$	0	0
$863$	52.3893	1.78335	0.891677	−	0.452672i	$-0.149529\pi$
$863$	52.3893	1.78335	0.891677	+	0.452672i	$0.149529\pi$
$864$	0	0
$865$	−9.35119	−0.317950
$866$	0	0
$867$	27.2428	0.925214
$868$	0	0
$869$	11.0968	0.376432
$870$	0	0
$871$	−0.118445	−0.00401336
$872$	0	0
$873$	51.8538	1.75499
$874$	0	0
$875$	−10.8499	−0.366792
$876$	0	0
$877$	−1.86423	−0.0629505	−0.0314753	−	0.999505i	$-0.510021\pi$
$877$	−1.86423	−0.0629505	−0.0314753	+	0.999505i	$0.510021\pi$
$878$	0	0
$879$	−39.0986	−1.31876
$880$	0	0
$881$	−22.4824	−0.757451	−0.378726	−	0.925509i	$-0.623638\pi$
$881$	−22.4824	−0.757451	−0.378726	+	0.925509i	$0.623638\pi$
$882$	0	0
$883$	−36.4502	−1.22665	−0.613323	−	0.789832i	$-0.710168\pi$
$883$	−36.4502	−1.22665	−0.613323	+	0.789832i	$0.710168\pi$
$884$	0	0
$885$	45.6030	1.53293
$886$	0	0
$887$	19.8258	0.665684	0.332842	−	0.942983i	$-0.391992\pi$
$887$	19.8258	0.665684	0.332842	+	0.942983i	$0.391992\pi$
$888$	0	0
$889$	−14.1678	−0.475171
$890$	0	0
$891$	9.21825	0.308823
$892$	0	0
$893$	3.47541	0.116300
$894$	0	0
$895$	−9.72938	−0.325217
$896$	0	0
$897$	−21.7414	−0.725924
$898$	0	0
$899$	34.2208	1.14133
$900$	0	0
$901$	24.5080	0.816481
$902$	0	0
$903$	16.2493	0.540743
$904$	0	0
$905$	16.4446	0.546638
$906$	0	0
$907$	−8.72419	−0.289682	−0.144841	−	0.989455i	$-0.546267\pi$
$907$	−8.72419	−0.289682	−0.144841	+	0.989455i	$0.546267\pi$
$908$	0	0
$909$	−16.5039	−0.547399
$910$	0	0
$911$	−46.6394	−1.54523	−0.772616	−	0.634874i	$-0.781052\pi$
$911$	−46.6394	−1.54523	−0.772616	+	0.634874i	$0.781052\pi$
$912$	0	0
$913$	−14.3412	−0.474625
$914$	0	0
$915$	37.0486	1.22479
$916$	0	0
$917$	−20.0250	−0.661284
$918$	0	0
$919$	−43.1913	−1.42475	−0.712375	−	0.701799i	$-0.752380\pi$
$919$	−43.1913	−1.42475	−0.712375	+	0.701799i	$0.752380\pi$
$920$	0	0
$921$	−76.6277	−2.52497
$922$	0	0
$923$	−10.5248	−0.346428
$924$	0	0
$925$	−29.4953	−0.969799
$926$	0	0
$927$	−8.29385	−0.272406
$928$	0	0
$929$	38.9013	1.27631	0.638155	−	0.769908i	$-0.279698\pi$
$929$	38.9013	1.27631	0.638155	+	0.769908i	$0.279698\pi$
$930$	0	0
$931$	−1.44400	−0.0473252
$932$	0	0
$933$	−20.1473	−0.659593
$934$	0	0
$935$	−6.95331	−0.227398
$936$	0	0
$937$	27.9816	0.914120	0.457060	−	0.889436i	$-0.348903\pi$
$937$	27.9816	0.914120	0.457060	+	0.889436i	$0.348903\pi$
$938$	0	0
$939$	−75.7138	−2.47083
$940$	0	0
$941$	17.9977	0.586707	0.293354	−	0.956004i	$-0.405229\pi$
$941$	17.9977	0.586707	0.293354	+	0.956004i	$0.405229\pi$
$942$	0	0
$943$	−32.5952	−1.06145
$944$	0	0
$945$	−0.237824	−0.00773642
$946$	0	0
$947$	13.5523	0.440390	0.220195	−	0.975456i	$-0.429331\pi$
$947$	13.5523	0.440390	0.220195	+	0.975456i	$0.429331\pi$
$948$	0	0
$949$	−13.9976	−0.454381
$950$	0	0
$951$	3.53474	0.114622
$952$	0	0
$953$	−5.56431	−0.180246	−0.0901229	−	0.995931i	$-0.528726\pi$
$953$	−5.56431	−0.180246	−0.0901229	+	0.995931i	$0.528726\pi$
$954$	0	0
$955$	31.1078	1.00662
$956$	0	0
$957$	13.5872	0.439212
$958$	0	0
$959$	−9.24445	−0.298519
$960$	0	0
$961$	6.58702	0.212484
$962$	0	0
$963$	−9.41684	−0.303453
$964$	0	0
$965$	−4.39726	−0.141553
$966$	0	0
$967$	19.6187	0.630895	0.315447	−	0.948943i	$-0.397845\pi$
$967$	19.6187	0.630895	0.315447	+	0.948943i	$0.397845\pi$
$968$	0	0
$969$	−18.6632	−0.599549
$970$	0	0
$971$	30.1798	0.968516	0.484258	−	0.874925i	$-0.339090\pi$
$971$	30.1798	0.968516	0.484258	+	0.874925i	$0.339090\pi$
$972$	0	0
$973$	7.52314	0.241181
$974$	0	0
$975$	7.99640	0.256090
$976$	0	0
$977$	11.1444	0.356541	0.178270	−	0.983982i	$-0.442950\pi$
$977$	11.1444	0.356541	0.178270	+	0.983982i	$0.442950\pi$
$978$	0	0
$979$	12.3124	0.393507
$980$	0	0
$981$	48.5066	1.54870
$982$	0	0
$983$	40.0228	1.27653	0.638264	−	0.769817i	$-0.279653\pi$
$983$	40.0228	1.27653	0.638264	+	0.769817i	$0.279653\pi$
$984$	0	0
$985$	6.24197	0.198886
$986$	0	0
$987$	−5.85864	−0.186483
$988$	0	0
$989$	59.6217	1.89586
$990$	0	0
$991$	−18.5061	−0.587866	−0.293933	−	0.955826i	$-0.594964\pi$
$991$	−18.5061	−0.587866	−0.293933	+	0.955826i	$0.594964\pi$
$992$	0	0
$993$	14.9411	0.474140
$994$	0	0
$995$	1.45601	0.0461587
$996$	0	0
$997$	−50.3183	−1.59360	−0.796798	−	0.604245i	$-0.793475\pi$
$997$	−50.3183	−1.59360	−0.796798	+	0.604245i	$0.793475\pi$
$998$	0	0
$999$	−1.63058	−0.0515891

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8008.2.a.j.1.5	✓	5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8008.2.a.j.1.5	✓	5	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)

\(f(q)\)	\(=\)	\(q+2.43421 q^{3} +1.30958 q^{5} +1.00000 q^{7} +2.92540 q^{9} +O(q^{10})\)	\(q+2.43421 q^{3} +1.30958 q^{5} +1.00000 q^{7} +2.92540 q^{9} -1.00000 q^{11} -1.00000 q^{13} +3.18779 q^{15} +5.30958 q^{17} -1.44400 q^{19} +2.43421 q^{21} +8.93159 q^{23} -3.28500 q^{25} -0.181603 q^{27} -5.58176 q^{29} -6.13083 q^{31} -2.43421 q^{33} +1.30958 q^{35} +8.97877 q^{37} -2.43421 q^{39} -3.64943 q^{41} +6.67538 q^{43} +3.83103 q^{45} -2.40679 q^{47} +1.00000 q^{49} +12.9246 q^{51} +4.61582 q^{53} -1.30958 q^{55} -3.51501 q^{57} +14.3055 q^{59} +11.6220 q^{61} +2.92540 q^{63} -1.30958 q^{65} +0.118445 q^{67} +21.7414 q^{69} +10.5248 q^{71} +13.9976 q^{73} -7.99640 q^{75} -1.00000 q^{77} -11.0968 q^{79} -9.21825 q^{81} +14.3412 q^{83} +6.95331 q^{85} -13.5872 q^{87} -12.3124 q^{89} -1.00000 q^{91} -14.9237 q^{93} -1.89103 q^{95} +17.7254 q^{97} -2.92540 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + q^{3} - q^{5} + 5 q^{7} + 6 q^{9}+O(q^{10}) \) 5 * q + q^3 - q^5 + 5 * q^7 + 6 * q^9	\( 5 q + q^{3} - q^{5} + 5 q^{7} + 6 q^{9} - 5 q^{11} - 5 q^{13} - 5 q^{15} + 19 q^{17} - 5 q^{19} + q^{21} + 5 q^{23} + 12 q^{25} - 11 q^{27} - 17 q^{29} + 4 q^{31} - q^{33} - q^{35} + 10 q^{37} - q^{39} + 10 q^{41} - 25 q^{43} + q^{45} + 3 q^{47} + 5 q^{49} - q^{51} + 22 q^{53} + q^{55} + 16 q^{57} + 21 q^{59} + 26 q^{61} + 6 q^{63} + q^{65} + 28 q^{67} + 16 q^{69} + 28 q^{71} - 4 q^{73} - 5 q^{77} - 11 q^{79} + 5 q^{81} + 33 q^{85} + 31 q^{87} - 37 q^{89} - 5 q^{91} - 49 q^{93} + 29 q^{95} + 18 q^{97} - 6 q^{99}+O(q^{100}) \) 5 * q + q^3 - q^5 + 5 * q^7 + 6 * q^9 - 5 * q^11 - 5 * q^13 - 5 * q^15 + 19 * q^17 - 5 * q^19 + q^21 + 5 * q^23 + 12 * q^25 - 11 * q^27 - 17 * q^29 + 4 * q^31 - q^33 - q^35 + 10 * q^37 - q^39 + 10 * q^41 - 25 * q^43 + q^45 + 3 * q^47 + 5 * q^49 - q^51 + 22 * q^53 + q^55 + 16 * q^57 + 21 * q^59 + 26 * q^61 + 6 * q^63 + q^65 + 28 * q^67 + 16 * q^69 + 28 * q^71 - 4 * q^73 - 5 * q^77 - 11 * q^79 + 5 * q^81 + 33 * q^85 + 31 * q^87 - 37 * q^89 - 5 * q^91 - 49 * q^93 + 29 * q^95 + 18 * q^97 - 6 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more