LMFDB - Embedded newform 4008.2.a.j.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4008,2,Mod(1,4008)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4008, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("4008.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4008 = 2^{3} \cdot 3 \cdot 167 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4008.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:	$32.0040411301$
Analytic rank:	$1$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 26x^{8} - 3x^{7} + 220x^{6} + 42x^{5} - 675x^{4} - 67x^{3} + 628x^{2} - 48x - 2 $ x^10 - 26x^8 - 3x^7 + 220x^6 + 42x^5 - 675x^4 - 67x^3 + 628x^2 - 48x - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$3.44741$ of defining polynomial
Character	$\chi$	$=$	4008.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} -4.44741 q^{5} +2.42708 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} -4.44741 q^{5} +2.42708 q^{7} +1.00000 q^{9} -0.762494 q^{11} +0.148977 q^{13} +4.44741 q^{15} -0.334532 q^{17} -4.24502 q^{19} -2.42708 q^{21} +4.31827 q^{23} +14.7795 q^{25} -1.00000 q^{27} -7.41711 q^{29} -2.55344 q^{31} +0.762494 q^{33} -10.7942 q^{35} +10.9302 q^{37} -0.148977 q^{39} +2.59245 q^{41} +4.61890 q^{43} -4.44741 q^{45} -3.82949 q^{47} -1.10930 q^{49} +0.334532 q^{51} +10.1388 q^{53} +3.39113 q^{55} +4.24502 q^{57} +6.52551 q^{59} +0.628633 q^{61} +2.42708 q^{63} -0.662562 q^{65} -6.47849 q^{67} -4.31827 q^{69} -0.134220 q^{71} +3.02269 q^{73} -14.7795 q^{75} -1.85063 q^{77} -1.69105 q^{79} +1.00000 q^{81} -4.64029 q^{83} +1.48780 q^{85} +7.41711 q^{87} -9.98708 q^{89} +0.361578 q^{91} +2.55344 q^{93} +18.8794 q^{95} +9.08991 q^{97} -0.762494 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 10 q^{3} - 10 q^{5} + q^{7} + 10 q^{9}+O(q^{10}) $ 10 * q - 10 * q^3 - 10 * q^5 + q^7 + 10 * q^9	$ 10 q - 10 q^{3} - 10 q^{5} + q^{7} + 10 q^{9} - q^{11} - 6 q^{13} + 10 q^{15} - 9 q^{17} + 2 q^{19} - q^{21} + 7 q^{23} + 12 q^{25} - 10 q^{27} - 13 q^{29} + 23 q^{31} + q^{33} + q^{35} - 6 q^{37} + 6 q^{39} - 12 q^{41} - 10 q^{45} + 10 q^{47} + 7 q^{49} + 9 q^{51} - 26 q^{53} + 11 q^{55} - 2 q^{57} - 10 q^{59} - 10 q^{61} + q^{63} - 22 q^{65} - 5 q^{67} - 7 q^{69} + 25 q^{71} - 8 q^{73} - 12 q^{75} - 46 q^{77} + 26 q^{79} + 10 q^{81} - 14 q^{83} + 9 q^{85} + 13 q^{87} - 31 q^{89} - 3 q^{91} - 23 q^{93} - 5 q^{95} - 32 q^{97} - q^{99}+O(q^{100}) $ 10 * q - 10 * q^3 - 10 * q^5 + q^7 + 10 * q^9 - q^11 - 6 * q^13 + 10 * q^15 - 9 * q^17 + 2 * q^19 - q^21 + 7 * q^23 + 12 * q^25 - 10 * q^27 - 13 * q^29 + 23 * q^31 + q^33 + q^35 - 6 * q^37 + 6 * q^39 - 12 * q^41 - 10 * q^45 + 10 * q^47 + 7 * q^49 + 9 * q^51 - 26 * q^53 + 11 * q^55 - 2 * q^57 - 10 * q^59 - 10 * q^61 + q^63 - 22 * q^65 - 5 * q^67 - 7 * q^69 + 25 * q^71 - 8 * q^73 - 12 * q^75 - 46 * q^77 + 26 * q^79 + 10 * q^81 - 14 * q^83 + 9 * q^85 + 13 * q^87 - 31 * q^89 - 3 * q^91 - 23 * q^93 - 5 * q^95 - 32 * q^97 - 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
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	−4.44741	−1.98894	−0.994472	−	0.105003i	$-0.966515\pi$
$5$	−4.44741	−1.98894	−0.994472	+	0.105003i	$0.966515\pi$
$6$	0	0
$7$	2.42708	0.917349	0.458674	−	0.888604i	$-0.348324\pi$
$7$	2.42708	0.917349	0.458674	+	0.888604i	$0.348324\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−0.762494	−0.229901	−0.114950	−	0.993371i	$-0.536671\pi$
$11$	−0.762494	−0.229901	−0.114950	+	0.993371i	$0.536671\pi$
$12$	0	0
$13$	0.148977	0.0413188	0.0206594	−	0.999787i	$-0.493423\pi$
$13$	0.148977	0.0413188	0.0206594	+	0.999787i	$0.493423\pi$
$14$	0	0
$15$	4.44741	1.14832
$16$	0	0
$17$	−0.334532	−0.0811360	−0.0405680	−	0.999177i	$-0.512917\pi$
$17$	−0.334532	−0.0811360	−0.0405680	+	0.999177i	$0.512917\pi$
$18$	0	0
$19$	−4.24502	−0.973875	−0.486937	−	0.873437i	$-0.661886\pi$
$19$	−4.24502	−0.973875	−0.486937	+	0.873437i	$0.661886\pi$
$20$	0	0
$21$	−2.42708	−0.529632
$22$	0	0
$23$	4.31827	0.900421	0.450211	−	0.892922i	$-0.351349\pi$
$23$	4.31827	0.900421	0.450211	+	0.892922i	$0.351349\pi$
$24$	0	0
$25$	14.7795	2.95590
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−7.41711	−1.37732	−0.688661	−	0.725083i	$-0.741801\pi$
$29$	−7.41711	−1.37732	−0.688661	+	0.725083i	$0.741801\pi$
$30$	0	0
$31$	−2.55344	−0.458612	−0.229306	−	0.973354i	$-0.573646\pi$
$31$	−2.55344	−0.458612	−0.229306	+	0.973354i	$0.573646\pi$
$32$	0	0
$33$	0.762494	0.132733
$34$	0	0
$35$	−10.7942	−1.82456
$36$	0	0
$37$	10.9302	1.79692	0.898458	−	0.439060i	$-0.144689\pi$
$37$	10.9302	1.79692	0.898458	+	0.439060i	$0.144689\pi$
$38$	0	0
$39$	−0.148977	−0.0238554
$40$	0	0
$41$	2.59245	0.404873	0.202437	−	0.979295i	$-0.435114\pi$
$41$	2.59245	0.404873	0.202437	+	0.979295i	$0.435114\pi$
$42$	0	0
$43$	4.61890	0.704376	0.352188	−	0.935929i	$-0.385438\pi$
$43$	4.61890	0.704376	0.352188	+	0.935929i	$0.385438\pi$
$44$	0	0
$45$	−4.44741	−0.662981
$46$	0	0
$47$	−3.82949	−0.558588	−0.279294	−	0.960206i	$-0.590100\pi$
$47$	−3.82949	−0.558588	−0.279294	+	0.960206i	$0.590100\pi$
$48$	0	0
$49$	−1.10930	−0.158471
$50$	0	0
$51$	0.334532	0.0468439
$52$	0	0
$53$	10.1388	1.39268	0.696339	−	0.717713i	$-0.254811\pi$
$53$	10.1388	1.39268	0.696339	+	0.717713i	$0.254811\pi$
$54$	0	0
$55$	3.39113	0.457259
$56$	0	0
$57$	4.24502	0.562267
$58$	0	0
$59$	6.52551	0.849548	0.424774	−	0.905299i	$-0.360353\pi$
$59$	6.52551	0.849548	0.424774	+	0.905299i	$0.360353\pi$
$60$	0	0
$61$	0.628633	0.0804882	0.0402441	−	0.999190i	$-0.487186\pi$
$61$	0.628633	0.0804882	0.0402441	+	0.999190i	$0.487186\pi$
$62$	0	0
$63$	2.42708	0.305783
$64$	0	0
$65$	−0.662562	−0.0821807
$66$	0	0
$67$	−6.47849	−0.791474	−0.395737	−	0.918364i	$-0.629511\pi$
$67$	−6.47849	−0.791474	−0.395737	+	0.918364i	$0.629511\pi$
$68$	0	0
$69$	−4.31827	−0.519858
$70$	0	0
$71$	−0.134220	−0.0159290	−0.00796448	−	0.999968i	$-0.502535\pi$
$71$	−0.134220	−0.0159290	−0.00796448	+	0.999968i	$0.502535\pi$
$72$	0	0
$73$	3.02269	0.353779	0.176889	−	0.984231i	$-0.443396\pi$
$73$	3.02269	0.353779	0.176889	+	0.984231i	$0.443396\pi$
$74$	0	0
$75$	−14.7795	−1.70659
$76$	0	0
$77$	−1.85063	−0.210899
$78$	0	0
$79$	−1.69105	−0.190258	−0.0951291	−	0.995465i	$-0.530326\pi$
$79$	−1.69105	−0.190258	−0.0951291	+	0.995465i	$0.530326\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−4.64029	−0.509337	−0.254669	−	0.967028i	$-0.581966\pi$
$83$	−4.64029	−0.509337	−0.254669	+	0.967028i	$0.581966\pi$
$84$	0	0
$85$	1.48780	0.161375
$86$	0	0
$87$	7.41711	0.795198
$88$	0	0
$89$	−9.98708	−1.05863	−0.529314	−	0.848426i	$-0.677551\pi$
$89$	−9.98708	−1.05863	−0.529314	+	0.848426i	$0.677551\pi$
$90$	0	0
$91$	0.361578	0.0379037
$92$	0	0
$93$	2.55344	0.264780
$94$	0	0
$95$	18.8794	1.93698
$96$	0	0
$97$	9.08991	0.922940	0.461470	−	0.887156i	$-0.347322\pi$
$97$	9.08991	0.922940	0.461470	+	0.887156i	$0.347322\pi$
$98$	0	0
$99$	−0.762494	−0.0766336
$100$	0	0
$101$	−14.7768	−1.47035	−0.735175	−	0.677878i	$-0.762900\pi$
$101$	−14.7768	−1.47035	−0.735175	+	0.677878i	$0.762900\pi$
$102$	0	0
$103$	5.90650	0.581985	0.290992	−	0.956725i	$-0.406015\pi$
$103$	5.90650	0.581985	0.290992	+	0.956725i	$0.406015\pi$
$104$	0	0
$105$	10.7942	1.05341
$106$	0	0
$107$	1.76179	0.170319	0.0851593	−	0.996367i	$-0.472860\pi$
$107$	1.76179	0.170319	0.0851593	+	0.996367i	$0.472860\pi$
$108$	0	0
$109$	−6.32940	−0.606246	−0.303123	−	0.952951i	$-0.598029\pi$
$109$	−6.32940	−0.606246	−0.303123	+	0.952951i	$0.598029\pi$
$110$	0	0
$111$	−10.9302	−1.03745
$112$	0	0
$113$	−7.19118	−0.676490	−0.338245	−	0.941058i	$-0.609833\pi$
$113$	−7.19118	−0.676490	−0.338245	+	0.941058i	$0.609833\pi$
$114$	0	0
$115$	−19.2051	−1.79089
$116$	0	0
$117$	0.148977	0.0137729
$118$	0	0
$119$	−0.811936	−0.0744300
$120$	0	0
$121$	−10.4186	−0.947146
$122$	0	0
$123$	−2.59245	−0.233754
$124$	0	0
$125$	−43.4934	−3.89017
$126$	0	0
$127$	9.54803	0.847250	0.423625	−	0.905838i	$-0.360757\pi$
$127$	9.54803	0.847250	0.423625	+	0.905838i	$0.360757\pi$
$128$	0	0
$129$	−4.61890	−0.406672
$130$	0	0
$131$	−15.2977	−1.33657	−0.668283	−	0.743907i	$-0.732971\pi$
$131$	−15.2977	−1.33657	−0.668283	+	0.743907i	$0.732971\pi$
$132$	0	0
$133$	−10.3030	−0.893383
$134$	0	0
$135$	4.44741	0.382772
$136$	0	0
$137$	−2.35962	−0.201596	−0.100798	−	0.994907i	$-0.532140\pi$
$137$	−2.35962	−0.201596	−0.100798	+	0.994907i	$0.532140\pi$
$138$	0	0
$139$	−10.9696	−0.930427	−0.465214	−	0.885198i	$-0.654022\pi$
$139$	−10.9696	−0.930427	−0.465214	+	0.885198i	$0.654022\pi$
$140$	0	0
$141$	3.82949	0.322501
$142$	0	0
$143$	−0.113594	−0.00949921
$144$	0	0
$145$	32.9870	2.73942
$146$	0	0
$147$	1.10930	0.0914934
$148$	0	0
$149$	13.0034	1.06528	0.532642	−	0.846341i	$-0.321199\pi$
$149$	13.0034	1.06528	0.532642	+	0.846341i	$0.321199\pi$
$150$	0	0
$151$	19.5157	1.58816	0.794082	−	0.607811i	$-0.207952\pi$
$151$	19.5157	1.58816	0.794082	+	0.607811i	$0.207952\pi$
$152$	0	0
$153$	−0.334532	−0.0270453
$154$	0	0
$155$	11.3562	0.912153
$156$	0	0
$157$	−21.0683	−1.68143	−0.840715	−	0.541478i	$-0.817865\pi$
$157$	−21.0683	−1.68143	−0.840715	+	0.541478i	$0.817865\pi$
$158$	0	0
$159$	−10.1388	−0.804063
$160$	0	0
$161$	10.4808	0.826000
$162$	0	0
$163$	19.2272	1.50599	0.752994	−	0.658028i	$-0.228609\pi$
$163$	19.2272	1.50599	0.752994	+	0.658028i	$0.228609\pi$
$164$	0	0
$165$	−3.39113	−0.263999
$166$	0	0
$167$	−1.00000	−0.0773823
$167$	−1.00000	−0.0773823
$168$	0	0
$169$	−12.9778	−0.998293
$170$	0	0
$171$	−4.24502	−0.324625
$172$	0	0
$173$	−15.5563	−1.18272	−0.591362	−	0.806406i	$-0.701410\pi$
$173$	−15.5563	−1.18272	−0.591362	+	0.806406i	$0.701410\pi$
$174$	0	0
$175$	35.8710	2.71159
$176$	0	0
$177$	−6.52551	−0.490487
$178$	0	0
$179$	−12.3545	−0.923416	−0.461708	−	0.887032i	$-0.652763\pi$
$179$	−12.3545	−0.923416	−0.461708	+	0.887032i	$0.652763\pi$
$180$	0	0
$181$	13.9570	1.03741	0.518707	−	0.854952i	$-0.326414\pi$
$181$	13.9570	1.03741	0.518707	+	0.854952i	$0.326414\pi$
$182$	0	0
$183$	−0.628633	−0.0464699
$184$	0	0
$185$	−48.6112	−3.57396
$186$	0	0
$187$	0.255079	0.0186532
$188$	0	0
$189$	−2.42708	−0.176544
$190$	0	0
$191$	−21.9171	−1.58586	−0.792932	−	0.609310i	$-0.791446\pi$
$191$	−21.9171	−1.58586	−0.792932	+	0.609310i	$0.791446\pi$
$192$	0	0
$193$	−10.5270	−0.757753	−0.378876	−	0.925447i	$-0.623689\pi$
$193$	−10.5270	−0.757753	−0.378876	+	0.925447i	$0.623689\pi$
$194$	0	0
$195$	0.662562	0.0474471
$196$	0	0
$197$	−17.5510	−1.25045	−0.625227	−	0.780443i	$-0.714994\pi$
$197$	−17.5510	−1.25045	−0.625227	+	0.780443i	$0.714994\pi$
$198$	0	0
$199$	0.498014	0.0353033	0.0176516	−	0.999844i	$-0.494381\pi$
$199$	0.498014	0.0353033	0.0176516	+	0.999844i	$0.494381\pi$
$200$	0	0
$201$	6.47849	0.456958
$202$	0	0
$203$	−18.0019	−1.26349
$204$	0	0
$205$	−11.5297	−0.805270
$206$	0	0
$207$	4.31827	0.300140
$208$	0	0
$209$	3.23680	0.223894
$210$	0	0
$211$	−25.6923	−1.76873	−0.884366	−	0.466794i	$-0.845409\pi$
$211$	−25.6923	−1.76873	−0.884366	+	0.466794i	$0.845409\pi$
$212$	0	0
$213$	0.134220	0.00919658
$214$	0	0
$215$	−20.5422	−1.40096
$216$	0	0
$217$	−6.19740	−0.420707
$218$	0	0
$219$	−3.02269	−0.204254
$220$	0	0
$221$	−0.0498376	−0.00335244
$222$	0	0
$223$	−16.1201	−1.07948	−0.539742	−	0.841831i	$-0.681478\pi$
$223$	−16.1201	−1.07948	−0.539742	+	0.841831i	$0.681478\pi$
$224$	0	0
$225$	14.7795	0.985299
$226$	0	0
$227$	−5.05834	−0.335734	−0.167867	−	0.985810i	$-0.553688\pi$
$227$	−5.05834	−0.335734	−0.167867	+	0.985810i	$0.553688\pi$
$228$	0	0
$229$	−15.6317	−1.03297	−0.516484	−	0.856297i	$-0.672760\pi$
$229$	−15.6317	−1.03297	−0.516484	+	0.856297i	$0.672760\pi$
$230$	0	0
$231$	1.85063	0.121763
$232$	0	0
$233$	1.22752	0.0804172	0.0402086	−	0.999191i	$-0.487198\pi$
$233$	1.22752	0.0804172	0.0402086	+	0.999191i	$0.487198\pi$
$234$	0	0
$235$	17.0313	1.11100
$236$	0	0
$237$	1.69105	0.109846
$238$	0	0
$239$	−16.6299	−1.07570	−0.537849	−	0.843041i	$-0.680763\pi$
$239$	−16.6299	−1.07570	−0.537849	+	0.843041i	$0.680763\pi$
$240$	0	0
$241$	4.07079	0.262222	0.131111	−	0.991368i	$-0.458145\pi$
$241$	4.07079	0.262222	0.131111	+	0.991368i	$0.458145\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	4.93351	0.315190
$246$	0	0
$247$	−0.632410	−0.0402393
$248$	0	0
$249$	4.64029	0.294066
$250$	0	0
$251$	9.81152	0.619298	0.309649	−	0.950851i	$-0.399789\pi$
$251$	9.81152	0.619298	0.309649	+	0.950851i	$0.399789\pi$
$252$	0	0
$253$	−3.29265	−0.207007
$254$	0	0
$255$	−1.48780	−0.0931699
$256$	0	0
$257$	−10.4322	−0.650746	−0.325373	−	0.945586i	$-0.605490\pi$
$257$	−10.4322	−0.650746	−0.325373	+	0.945586i	$0.605490\pi$
$258$	0	0
$259$	26.5285	1.64840
$260$	0	0
$261$	−7.41711	−0.459108
$262$	0	0
$263$	23.0223	1.41962	0.709808	−	0.704395i	$-0.248782\pi$
$263$	23.0223	1.41962	0.709808	+	0.704395i	$0.248782\pi$
$264$	0	0
$265$	−45.0916	−2.76996
$266$	0	0
$267$	9.98708	0.611199
$268$	0	0
$269$	4.51311	0.275169	0.137585	−	0.990490i	$-0.456066\pi$
$269$	4.51311	0.275169	0.137585	+	0.990490i	$0.456066\pi$
$270$	0	0
$271$	−8.88576	−0.539771	−0.269886	−	0.962892i	$-0.586986\pi$
$271$	−8.88576	−0.539771	−0.269886	+	0.962892i	$0.586986\pi$
$272$	0	0
$273$	−0.361578	−0.0218837
$274$	0	0
$275$	−11.2693	−0.679563
$276$	0	0
$277$	−12.4338	−0.747073	−0.373536	−	0.927616i	$-0.621855\pi$
$277$	−12.4338	−0.747073	−0.373536	+	0.927616i	$0.621855\pi$
$278$	0	0
$279$	−2.55344	−0.152871
$280$	0	0
$281$	−9.49299	−0.566304	−0.283152	−	0.959075i	$-0.591380\pi$
$281$	−9.49299	−0.566304	−0.283152	+	0.959075i	$0.591380\pi$
$282$	0	0
$283$	21.3564	1.26950	0.634752	−	0.772716i	$-0.281102\pi$
$283$	21.3564	1.26950	0.634752	+	0.772716i	$0.281102\pi$
$284$	0	0
$285$	−18.8794	−1.11832
$286$	0	0
$287$	6.29208	0.371410
$288$	0	0
$289$	−16.8881	−0.993417
$290$	0	0
$291$	−9.08991	−0.532860
$292$	0	0
$293$	−5.03154	−0.293946	−0.146973	−	0.989141i	$-0.546953\pi$
$293$	−5.03154	−0.293946	−0.146973	+	0.989141i	$0.546953\pi$
$294$	0	0
$295$	−29.0216	−1.68970
$296$	0	0
$297$	0.762494	0.0442444
$298$	0	0
$299$	0.643322	0.0372043
$300$	0	0
$301$	11.2104	0.646159
$302$	0	0
$303$	14.7768	0.848907
$304$	0	0
$305$	−2.79579	−0.160087
$306$	0	0
$307$	−15.2440	−0.870018	−0.435009	−	0.900426i	$-0.643255\pi$
$307$	−15.2440	−0.870018	−0.435009	+	0.900426i	$0.643255\pi$
$308$	0	0
$309$	−5.90650	−0.336009
$310$	0	0
$311$	26.2210	1.48686	0.743428	−	0.668816i	$-0.233198\pi$
$311$	26.2210	1.48686	0.743428	+	0.668816i	$0.233198\pi$
$312$	0	0
$313$	17.1744	0.970753	0.485376	−	0.874305i	$-0.338683\pi$
$313$	17.1744	0.970753	0.485376	+	0.874305i	$0.338683\pi$
$314$	0	0
$315$	−10.7942	−0.608185
$316$	0	0
$317$	2.28356	0.128258	0.0641288	−	0.997942i	$-0.479573\pi$
$317$	2.28356	0.128258	0.0641288	+	0.997942i	$0.479573\pi$
$318$	0	0
$319$	5.65550	0.316647
$320$	0	0
$321$	−1.76179	−0.0983335
$322$	0	0
$323$	1.42010	0.0790163
$324$	0	0
$325$	2.20180	0.122134
$326$	0	0
$327$	6.32940	0.350016
$328$	0	0
$329$	−9.29446	−0.512420
$330$	0	0
$331$	5.20651	0.286175	0.143088	−	0.989710i	$-0.454297\pi$
$331$	5.20651	0.286175	0.143088	+	0.989710i	$0.454297\pi$
$332$	0	0
$333$	10.9302	0.598972
$334$	0	0
$335$	28.8125	1.57420
$336$	0	0
$337$	−19.2700	−1.04970	−0.524852	−	0.851193i	$-0.675879\pi$
$337$	−19.2700	−1.04970	−0.524852	+	0.851193i	$0.675879\pi$
$338$	0	0
$339$	7.19118	0.390571
$340$	0	0
$341$	1.94698	0.105435
$342$	0	0
$343$	−19.6819	−1.06272
$344$	0	0
$345$	19.2051	1.03397
$346$	0	0
$347$	6.75451	0.362601	0.181300	−	0.983428i	$-0.441969\pi$
$347$	6.75451	0.362601	0.181300	+	0.983428i	$0.441969\pi$
$348$	0	0
$349$	10.4141	0.557453	0.278727	−	0.960371i	$-0.410088\pi$
$349$	10.4141	0.557453	0.278727	+	0.960371i	$0.410088\pi$
$350$	0	0
$351$	−0.148977	−0.00795180
$352$	0	0
$353$	−32.0933	−1.70815	−0.854077	−	0.520147i	$-0.825877\pi$
$353$	−32.0933	−1.70815	−0.854077	+	0.520147i	$0.825877\pi$
$354$	0	0
$355$	0.596931	0.0316818
$356$	0	0
$357$	0.811936	0.0429722
$358$	0	0
$359$	31.8286	1.67985	0.839924	−	0.542705i	$-0.182600\pi$
$359$	31.8286	1.67985	0.839924	+	0.542705i	$0.182600\pi$
$360$	0	0
$361$	−0.979799	−0.0515684
$362$	0	0
$363$	10.4186	0.546835
$364$	0	0
$365$	−13.4431	−0.703646
$366$	0	0
$367$	−11.9252	−0.622490	−0.311245	−	0.950330i	$-0.600746\pi$
$367$	−11.9252	−0.622490	−0.311245	+	0.950330i	$0.600746\pi$
$368$	0	0
$369$	2.59245	0.134958
$370$	0	0
$371$	24.6077	1.27757
$372$	0	0
$373$	−14.5989	−0.755900	−0.377950	−	0.925826i	$-0.623371\pi$
$373$	−14.5989	−0.755900	−0.377950	+	0.925826i	$0.623371\pi$
$374$	0	0
$375$	43.4934	2.24599
$376$	0	0
$377$	−1.10498	−0.0569093
$378$	0	0
$379$	−24.2409	−1.24517	−0.622585	−	0.782552i	$-0.713918\pi$
$379$	−24.2409	−1.24517	−0.622585	+	0.782552i	$0.713918\pi$
$380$	0	0
$381$	−9.54803	−0.489160
$382$	0	0
$383$	24.9787	1.27635	0.638177	−	0.769890i	$-0.279689\pi$
$383$	24.9787	1.27635	0.638177	+	0.769890i	$0.279689\pi$
$384$	0	0
$385$	8.23053	0.419466
$386$	0	0
$387$	4.61890	0.234792
$388$	0	0
$389$	35.5821	1.80408	0.902041	−	0.431650i	$-0.142069\pi$
$389$	35.5821	1.80408	0.902041	+	0.431650i	$0.142069\pi$
$390$	0	0
$391$	−1.44460	−0.0730566
$392$	0	0
$393$	15.2977	0.771667
$394$	0	0
$395$	7.52081	0.378413
$396$	0	0
$397$	9.66644	0.485145	0.242572	−	0.970133i	$-0.422009\pi$
$397$	9.66644	0.485145	0.242572	+	0.970133i	$0.422009\pi$
$398$	0	0
$399$	10.3030	0.515795
$400$	0	0
$401$	−37.9050	−1.89288	−0.946441	−	0.322875i	$-0.895350\pi$
$401$	−37.9050	−1.89288	−0.946441	+	0.322875i	$0.895350\pi$
$402$	0	0
$403$	−0.380404	−0.0189493
$404$	0	0
$405$	−4.44741	−0.220994
$406$	0	0
$407$	−8.33422	−0.413112
$408$	0	0
$409$	−33.9866	−1.68053	−0.840264	−	0.542178i	$-0.817600\pi$
$409$	−33.9866	−1.68053	−0.840264	+	0.542178i	$0.817600\pi$
$410$	0	0
$411$	2.35962	0.116392
$412$	0	0
$413$	15.8379	0.779332
$414$	0	0
$415$	20.6373	1.01304
$416$	0	0
$417$	10.9696	0.537182
$418$	0	0
$419$	18.8987	0.923260	0.461630	−	0.887073i	$-0.347265\pi$
$419$	18.8987	0.923260	0.461630	+	0.887073i	$0.347265\pi$
$420$	0	0
$421$	8.63780	0.420981	0.210490	−	0.977596i	$-0.432494\pi$
$421$	8.63780	0.420981	0.210490	+	0.977596i	$0.432494\pi$
$422$	0	0
$423$	−3.82949	−0.186196
$424$	0	0
$425$	−4.94422	−0.239830
$426$	0	0
$427$	1.52574	0.0738358
$428$	0	0
$429$	0.113594	0.00548437
$430$	0	0
$431$	4.66917	0.224906	0.112453	−	0.993657i	$-0.464129\pi$
$431$	4.66917	0.224906	0.112453	+	0.993657i	$0.464129\pi$
$432$	0	0
$433$	33.9632	1.63217	0.816083	−	0.577935i	$-0.196141\pi$
$433$	33.9632	1.63217	0.816083	+	0.577935i	$0.196141\pi$
$434$	0	0
$435$	−32.9870	−1.58160
$436$	0	0
$437$	−18.3311	−0.876897
$438$	0	0
$439$	9.42573	0.449866	0.224933	−	0.974374i	$-0.427784\pi$
$439$	9.42573	0.449866	0.224933	+	0.974374i	$0.427784\pi$
$440$	0	0
$441$	−1.10930	−0.0528237
$442$	0	0
$443$	5.33593	0.253518	0.126759	−	0.991934i	$-0.459543\pi$
$443$	5.33593	0.253518	0.126759	+	0.991934i	$0.459543\pi$
$444$	0	0
$445$	44.4167	2.10555
$446$	0	0
$447$	−13.0034	−0.615042
$448$	0	0
$449$	27.2621	1.28658	0.643288	−	0.765624i	$-0.277570\pi$
$449$	27.2621	1.28658	0.643288	+	0.765624i	$0.277570\pi$
$450$	0	0
$451$	−1.97673	−0.0930806
$452$	0	0
$453$	−19.5157	−0.916927
$454$	0	0
$455$	−1.60809	−0.0753884
$456$	0	0
$457$	1.22588	0.0573443	0.0286722	−	0.999589i	$-0.490872\pi$
$457$	1.22588	0.0573443	0.0286722	+	0.999589i	$0.490872\pi$
$458$	0	0
$459$	0.334532	0.0156146
$460$	0	0
$461$	−1.84380	−0.0858742	−0.0429371	−	0.999078i	$-0.513672\pi$
$461$	−1.84380	−0.0858742	−0.0429371	+	0.999078i	$0.513672\pi$
$462$	0	0
$463$	19.6812	0.914662	0.457331	−	0.889297i	$-0.348806\pi$
$463$	19.6812	0.914662	0.457331	+	0.889297i	$0.348806\pi$
$464$	0	0
$465$	−11.3562	−0.526632
$466$	0	0
$467$	12.9824	0.600753	0.300376	−	0.953821i	$-0.402888\pi$
$467$	12.9824	0.600753	0.300376	+	0.953821i	$0.402888\pi$
$468$	0	0
$469$	−15.7238	−0.726057
$470$	0	0
$471$	21.0683	0.970774
$472$	0	0
$473$	−3.52189	−0.161937
$474$	0	0
$475$	−62.7392	−2.87867
$476$	0	0
$477$	10.1388	0.464226
$478$	0	0
$479$	2.16954	0.0991287	0.0495644	−	0.998771i	$-0.484217\pi$
$479$	2.16954	0.0991287	0.0495644	+	0.998771i	$0.484217\pi$
$480$	0	0
$481$	1.62835	0.0742463
$482$	0	0
$483$	−10.4808	−0.476891
$484$	0	0
$485$	−40.4266	−1.83568
$486$	0	0
$487$	−18.4390	−0.835550	−0.417775	−	0.908550i	$-0.637190\pi$
$487$	−18.4390	−0.835550	−0.417775	+	0.908550i	$0.637190\pi$
$488$	0	0
$489$	−19.2272	−0.869482
$490$	0	0
$491$	−16.2614	−0.733867	−0.366934	−	0.930247i	$-0.619592\pi$
$491$	−16.2614	−0.733867	−0.366934	+	0.930247i	$0.619592\pi$
$492$	0	0
$493$	2.48126	0.111750
$494$	0	0
$495$	3.39113	0.152420
$496$	0	0
$497$	−0.325762	−0.0146124
$498$	0	0
$499$	26.1176	1.16918	0.584592	−	0.811328i	$-0.301255\pi$
$499$	26.1176	1.16918	0.584592	+	0.811328i	$0.301255\pi$
$500$	0	0
$501$	1.00000	0.0446767
$502$	0	0
$503$	−3.28163	−0.146321	−0.0731603	−	0.997320i	$-0.523308\pi$
$503$	−3.28163	−0.146321	−0.0731603	+	0.997320i	$0.523308\pi$
$504$	0	0
$505$	65.7187	2.92444
$506$	0	0
$507$	12.9778	0.576365
$508$	0	0
$509$	−14.2656	−0.632312	−0.316156	−	0.948707i	$-0.602392\pi$
$509$	−14.2656	−0.632312	−0.316156	+	0.948707i	$0.602392\pi$
$510$	0	0
$511$	7.33630	0.324539
$512$	0	0
$513$	4.24502	0.187422
$514$	0	0
$515$	−26.2687	−1.15754
$516$	0	0
$517$	2.91996	0.128420
$518$	0	0
$519$	15.5563	0.682846
$520$	0	0
$521$	−39.7935	−1.74338	−0.871692	−	0.490054i	$-0.836977\pi$
$521$	−39.7935	−1.74338	−0.871692	+	0.490054i	$0.836977\pi$
$522$	0	0
$523$	18.2924	0.799872	0.399936	−	0.916543i	$-0.369032\pi$
$523$	18.2924	0.799872	0.399936	+	0.916543i	$0.369032\pi$
$524$	0	0
$525$	−35.8710	−1.56554
$526$	0	0
$527$	0.854209	0.0372099
$528$	0	0
$529$	−4.35256	−0.189242
$530$	0	0
$531$	6.52551	0.283183
$532$	0	0
$533$	0.386216	0.0167289
$534$	0	0
$535$	−7.83541	−0.338754
$536$	0	0
$537$	12.3545	0.533134
$538$	0	0
$539$	0.845834	0.0364326
$540$	0	0
$541$	−12.5735	−0.540577	−0.270289	−	0.962779i	$-0.587119\pi$
$541$	−12.5735	−0.540577	−0.270289	+	0.962779i	$0.587119\pi$
$542$	0	0
$543$	−13.9570	−0.598951
$544$	0	0
$545$	28.1494	1.20579
$546$	0	0
$547$	−31.9078	−1.36428	−0.682140	−	0.731221i	$-0.738951\pi$
$547$	−31.9078	−1.36428	−0.682140	+	0.731221i	$0.738951\pi$
$548$	0	0
$549$	0.628633	0.0268294
$550$	0	0
$551$	31.4858	1.34134
$552$	0	0
$553$	−4.10431	−0.174533
$554$	0	0
$555$	48.6112	2.06343
$556$	0	0
$557$	23.2218	0.983941	0.491971	−	0.870612i	$-0.336277\pi$
$557$	23.2218	0.983941	0.491971	+	0.870612i	$0.336277\pi$
$558$	0	0
$559$	0.688110	0.0291040
$560$	0	0
$561$	−0.255079	−0.0107694
$562$	0	0
$563$	21.6036	0.910484	0.455242	−	0.890368i	$-0.349553\pi$
$563$	21.6036	0.910484	0.455242	+	0.890368i	$0.349553\pi$
$564$	0	0
$565$	31.9822	1.34550
$566$	0	0
$567$	2.42708	0.101928
$568$	0	0
$569$	0.607174	0.0254541	0.0127270	−	0.999919i	$-0.495949\pi$
$569$	0.607174	0.0254541	0.0127270	+	0.999919i	$0.495949\pi$
$570$	0	0
$571$	−36.0464	−1.50849	−0.754246	−	0.656591i	$-0.771998\pi$
$571$	−36.0464	−1.50849	−0.754246	+	0.656591i	$0.771998\pi$
$572$	0	0
$573$	21.9171	0.915599
$574$	0	0
$575$	63.8218	2.66155
$576$	0	0
$577$	1.00596	0.0418787	0.0209394	−	0.999781i	$-0.493334\pi$
$577$	1.00596	0.0418787	0.0209394	+	0.999781i	$0.493334\pi$
$578$	0	0
$579$	10.5270	0.437489
$580$	0	0
$581$	−11.2623	−0.467240
$582$	0	0
$583$	−7.73081	−0.320177
$584$	0	0
$585$	−0.662562	−0.0273936
$586$	0	0
$587$	28.1197	1.16063	0.580313	−	0.814394i	$-0.302930\pi$
$587$	28.1197	1.16063	0.580313	+	0.814394i	$0.302930\pi$
$588$	0	0
$589$	10.8394	0.446630
$590$	0	0
$591$	17.5510	0.721950
$592$	0	0
$593$	−16.1400	−0.662790	−0.331395	−	0.943492i	$-0.607519\pi$
$593$	−16.1400	−0.662790	−0.331395	+	0.943492i	$0.607519\pi$
$594$	0	0
$595$	3.61101	0.148037
$596$	0	0
$597$	−0.498014	−0.0203824
$598$	0	0
$599$	15.2866	0.624593	0.312297	−	0.949985i	$-0.398902\pi$
$599$	15.2866	0.624593	0.312297	+	0.949985i	$0.398902\pi$
$600$	0	0
$601$	−16.1023	−0.656828	−0.328414	−	0.944534i	$-0.606514\pi$
$601$	−16.1023	−0.656828	−0.328414	+	0.944534i	$0.606514\pi$
$602$	0	0
$603$	−6.47849	−0.263825
$604$	0	0
$605$	46.3358	1.88382
$606$	0	0
$607$	−26.8559	−1.09005	−0.545024	−	0.838421i	$-0.683479\pi$
$607$	−26.8559	−1.09005	−0.545024	+	0.838421i	$0.683479\pi$
$608$	0	0
$609$	18.0019	0.729474
$610$	0	0
$611$	−0.570505	−0.0230802
$612$	0	0
$613$	15.7868	0.637625	0.318812	−	0.947818i	$-0.396716\pi$
$613$	15.7868	0.637625	0.318812	+	0.947818i	$0.396716\pi$
$614$	0	0
$615$	11.5297	0.464923
$616$	0	0
$617$	−28.9926	−1.16720	−0.583599	−	0.812042i	$-0.698356\pi$
$617$	−28.9926	−1.16720	−0.583599	+	0.812042i	$0.698356\pi$
$618$	0	0
$619$	3.78489	0.152128	0.0760639	−	0.997103i	$-0.475765\pi$
$619$	3.78489	0.152128	0.0760639	+	0.997103i	$0.475765\pi$
$620$	0	0
$621$	−4.31827	−0.173286
$622$	0	0
$623$	−24.2394	−0.971131
$624$	0	0
$625$	119.536	4.78143
$626$	0	0
$627$	−3.23680	−0.129265
$628$	0	0
$629$	−3.65651	−0.145795
$630$	0	0
$631$	27.7314	1.10397	0.551984	−	0.833854i	$-0.313871\pi$
$631$	27.7314	1.10397	0.551984	+	0.833854i	$0.313871\pi$
$632$	0	0
$633$	25.6923	1.02118
$634$	0	0
$635$	−42.4640	−1.68513
$636$	0	0
$637$	−0.165260	−0.00654784
$638$	0	0
$639$	−0.134220	−0.00530965
$640$	0	0
$641$	1.98963	0.0785855	0.0392928	−	0.999228i	$-0.487490\pi$
$641$	1.98963	0.0785855	0.0392928	+	0.999228i	$0.487490\pi$
$642$	0	0
$643$	18.7552	0.739631	0.369816	−	0.929105i	$-0.379421\pi$
$643$	18.7552	0.739631	0.369816	+	0.929105i	$0.379421\pi$
$644$	0	0
$645$	20.5422	0.808847
$646$	0	0
$647$	−6.22188	−0.244607	−0.122304	−	0.992493i	$-0.539028\pi$
$647$	−6.22188	−0.244607	−0.122304	+	0.992493i	$0.539028\pi$
$648$	0	0
$649$	−4.97566	−0.195312
$650$	0	0
$651$	6.19740	0.242895
$652$	0	0
$653$	29.4085	1.15084	0.575422	−	0.817857i	$-0.304838\pi$
$653$	29.4085	1.15084	0.575422	+	0.817857i	$0.304838\pi$
$654$	0	0
$655$	68.0352	2.65836
$656$	0	0
$657$	3.02269	0.117926
$658$	0	0
$659$	−43.6285	−1.69953	−0.849763	−	0.527164i	$-0.823255\pi$
$659$	−43.6285	−1.69953	−0.849763	+	0.527164i	$0.823255\pi$
$660$	0	0
$661$	−27.7637	−1.07988	−0.539942	−	0.841702i	$-0.681554\pi$
$661$	−27.7637	−1.07988	−0.539942	+	0.841702i	$0.681554\pi$
$662$	0	0
$663$	0.0498376	0.00193553
$664$	0	0
$665$	45.8217	1.77689
$666$	0	0
$667$	−32.0291	−1.24017
$668$	0	0
$669$	16.1201	0.623240
$670$	0	0
$671$	−0.479329	−0.0185043
$672$	0	0
$673$	6.78369	0.261492	0.130746	−	0.991416i	$-0.458263\pi$
$673$	6.78369	0.261492	0.130746	+	0.991416i	$0.458263\pi$
$674$	0	0
$675$	−14.7795	−0.568863
$676$	0	0
$677$	−1.62466	−0.0624407	−0.0312203	−	0.999513i	$-0.509939\pi$
$677$	−1.62466	−0.0624407	−0.0312203	+	0.999513i	$0.509939\pi$
$678$	0	0
$679$	22.0619	0.846658
$680$	0	0
$681$	5.05834	0.193836
$682$	0	0
$683$	−48.0581	−1.83889	−0.919447	−	0.393215i	$-0.871363\pi$
$683$	−48.0581	−1.83889	−0.919447	+	0.393215i	$0.871363\pi$
$684$	0	0
$685$	10.4942	0.400963
$686$	0	0
$687$	15.6317	0.596385
$688$	0	0
$689$	1.51045	0.0575437
$690$	0	0
$691$	5.89940	0.224424	0.112212	−	0.993684i	$-0.464206\pi$
$691$	5.89940	0.224424	0.112212	+	0.993684i	$0.464206\pi$
$692$	0	0
$693$	−1.85063	−0.0702997
$694$	0	0
$695$	48.7862	1.85057
$696$	0	0
$697$	−0.867259	−0.0328498
$698$	0	0
$699$	−1.22752	−0.0464289
$700$	0	0
$701$	9.99283	0.377424	0.188712	−	0.982032i	$-0.439569\pi$
$701$	9.99283	0.377424	0.188712	+	0.982032i	$0.439569\pi$
$702$	0	0
$703$	−46.3990	−1.74997
$704$	0	0
$705$	−17.0313	−0.641437
$706$	0	0
$707$	−35.8645	−1.34882
$708$	0	0
$709$	23.6535	0.888326	0.444163	−	0.895946i	$-0.353501\pi$
$709$	23.6535	0.888326	0.444163	+	0.895946i	$0.353501\pi$
$710$	0	0
$711$	−1.69105	−0.0634194
$712$	0	0
$713$	−11.0264	−0.412944
$714$	0	0
$715$	0.505200	0.0188934
$716$	0	0
$717$	16.6299	0.621055
$718$	0	0
$719$	13.4324	0.500943	0.250472	−	0.968124i	$-0.419414\pi$
$719$	13.4324	0.500943	0.250472	+	0.968124i	$0.419414\pi$
$720$	0	0
$721$	14.3355	0.533883
$722$	0	0
$723$	−4.07079	−0.151394
$724$	0	0
$725$	−109.621	−4.07122
$726$	0	0
$727$	−30.2167	−1.12067	−0.560337	−	0.828265i	$-0.689328\pi$
$727$	−30.2167	−1.12067	−0.560337	+	0.828265i	$0.689328\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−1.54517	−0.0571503
$732$	0	0
$733$	−0.159014	−0.00587331	−0.00293665	−	0.999996i	$-0.500935\pi$
$733$	−0.159014	−0.00587331	−0.00293665	+	0.999996i	$0.500935\pi$
$734$	0	0
$735$	−4.93351	−0.181975
$736$	0	0
$737$	4.93981	0.181960
$738$	0	0
$739$	−3.35588	−0.123448	−0.0617240	−	0.998093i	$-0.519660\pi$
$739$	−3.35588	−0.123448	−0.0617240	+	0.998093i	$0.519660\pi$
$740$	0	0
$741$	0.632410	0.0232322
$742$	0	0
$743$	1.49736	0.0549330	0.0274665	−	0.999623i	$-0.491256\pi$
$743$	1.49736	0.0549330	0.0274665	+	0.999623i	$0.491256\pi$
$744$	0	0
$745$	−57.8317	−2.11879
$746$	0	0
$747$	−4.64029	−0.169779
$748$	0	0
$749$	4.27600	0.156242
$750$	0	0
$751$	16.2496	0.592957	0.296479	−	0.955039i	$-0.404188\pi$
$751$	16.2496	0.592957	0.296479	+	0.955039i	$0.404188\pi$
$752$	0	0
$753$	−9.81152	−0.357552
$754$	0	0
$755$	−86.7943	−3.15877
$756$	0	0
$757$	−31.0135	−1.12720	−0.563602	−	0.826047i	$-0.690585\pi$
$757$	−31.0135	−1.12720	−0.563602	+	0.826047i	$0.690585\pi$
$758$	0	0
$759$	3.29265	0.119516
$760$	0	0
$761$	−31.0952	−1.12720	−0.563600	−	0.826048i	$-0.690584\pi$
$761$	−31.0952	−1.12720	−0.563600	+	0.826048i	$0.690584\pi$
$762$	0	0
$763$	−15.3619	−0.556139
$764$	0	0
$765$	1.48780	0.0537917
$766$	0	0
$767$	0.972150	0.0351023
$768$	0	0
$769$	−24.4653	−0.882241	−0.441120	−	0.897448i	$-0.645419\pi$
$769$	−24.4653	−0.882241	−0.441120	+	0.897448i	$0.645419\pi$
$770$	0	0
$771$	10.4322	0.375708
$772$	0	0
$773$	−33.6931	−1.21186	−0.605929	−	0.795519i	$-0.707198\pi$
$773$	−33.6931	−1.21186	−0.605929	+	0.795519i	$0.707198\pi$
$774$	0	0
$775$	−37.7386	−1.35561
$776$	0	0
$777$	−26.5285	−0.951703
$778$	0	0
$779$	−11.0050	−0.394296
$780$	0	0
$781$	0.102342	0.00366208
$782$	0	0
$783$	7.41711	0.265066
$784$	0	0
$785$	93.6993	3.34427
$786$	0	0
$787$	−8.54950	−0.304757	−0.152378	−	0.988322i	$-0.548693\pi$
$787$	−8.54950	−0.304757	−0.152378	+	0.988322i	$0.548693\pi$
$788$	0	0
$789$	−23.0223	−0.819616
$790$	0	0
$791$	−17.4536	−0.620577
$792$	0	0
$793$	0.0936518	0.00332567
$794$	0	0
$795$	45.0916	1.59924
$796$	0	0
$797$	23.1646	0.820533	0.410266	−	0.911966i	$-0.365436\pi$
$797$	23.1646	0.820533	0.410266	+	0.911966i	$0.365436\pi$
$798$	0	0
$799$	1.28109	0.0453216
$800$	0	0
$801$	−9.98708	−0.352876
$802$	0	0
$803$	−2.30478	−0.0813340
$804$	0	0
$805$	−46.6123	−1.64287
$806$	0	0
$807$	−4.51311	−0.158869
$808$	0	0
$809$	19.6040	0.689239	0.344620	−	0.938742i	$-0.388008\pi$
$809$	19.6040	0.689239	0.344620	+	0.938742i	$0.388008\pi$
$810$	0	0
$811$	8.01820	0.281557	0.140779	−	0.990041i	$-0.455039\pi$
$811$	8.01820	0.281557	0.140779	+	0.990041i	$0.455039\pi$
$812$	0	0
$813$	8.88576	0.311637
$814$	0	0
$815$	−85.5111	−2.99532
$816$	0	0
$817$	−19.6073	−0.685974
$818$	0	0
$819$	0.361578	0.0126346
$820$	0	0
$821$	32.8430	1.14623	0.573114	−	0.819475i	$-0.305735\pi$
$821$	32.8430	1.14623	0.573114	+	0.819475i	$0.305735\pi$
$822$	0	0
$823$	−40.6516	−1.41702	−0.708512	−	0.705699i	$-0.750633\pi$
$823$	−40.6516	−1.41702	−0.708512	+	0.705699i	$0.750633\pi$
$824$	0	0
$825$	11.2693	0.392346
$826$	0	0
$827$	−41.6511	−1.44835	−0.724174	−	0.689617i	$-0.757779\pi$
$827$	−41.6511	−1.44835	−0.724174	+	0.689617i	$0.757779\pi$
$828$	0	0
$829$	10.4807	0.364009	0.182005	−	0.983298i	$-0.441741\pi$
$829$	10.4807	0.364009	0.182005	+	0.983298i	$0.441741\pi$
$830$	0	0
$831$	12.4338	0.431323
$832$	0	0
$833$	0.371096	0.0128577
$834$	0	0
$835$	4.44741	0.153909
$836$	0	0
$837$	2.55344	0.0882599
$838$	0	0
$839$	26.4224	0.912201	0.456100	−	0.889928i	$-0.349246\pi$
$839$	26.4224	0.912201	0.456100	+	0.889928i	$0.349246\pi$
$840$	0	0
$841$	26.0135	0.897017
$842$	0	0
$843$	9.49299	0.326956
$844$	0	0
$845$	57.7177	1.98555
$846$	0	0
$847$	−25.2867	−0.868863
$848$	0	0
$849$	−21.3564	−0.732949
$850$	0	0
$851$	47.1996	1.61798
$852$	0	0
$853$	−9.86959	−0.337928	−0.168964	−	0.985622i	$-0.554042\pi$
$853$	−9.86959	−0.337928	−0.168964	+	0.985622i	$0.554042\pi$
$854$	0	0
$855$	18.8794	0.645661
$856$	0	0
$857$	−11.7976	−0.402997	−0.201498	−	0.979489i	$-0.564581\pi$
$857$	−11.7976	−0.402997	−0.201498	+	0.979489i	$0.564581\pi$
$858$	0	0
$859$	−27.0195	−0.921893	−0.460947	−	0.887428i	$-0.652490\pi$
$859$	−27.0195	−0.921893	−0.460947	+	0.887428i	$0.652490\pi$
$860$	0	0
$861$	−6.29208	−0.214434
$862$	0	0
$863$	−18.0670	−0.615008	−0.307504	−	0.951547i	$-0.599494\pi$
$863$	−18.0670	−0.615008	−0.307504	+	0.951547i	$0.599494\pi$
$864$	0	0
$865$	69.1853	2.35237
$866$	0	0
$867$	16.8881	0.573550
$868$	0	0
$869$	1.28942	0.0437405
$870$	0	0
$871$	−0.965146	−0.0327027
$872$	0	0
$873$	9.08991	0.307647
$874$	0	0
$875$	−105.562	−3.56864
$876$	0	0
$877$	19.8497	0.670278	0.335139	−	0.942169i	$-0.391217\pi$
$877$	19.8497	0.670278	0.335139	+	0.942169i	$0.391217\pi$
$878$	0	0
$879$	5.03154	0.169710
$880$	0	0
$881$	−18.4618	−0.621994	−0.310997	−	0.950411i	$-0.600663\pi$
$881$	−18.4618	−0.621994	−0.310997	+	0.950411i	$0.600663\pi$
$882$	0	0
$883$	−44.2092	−1.48776	−0.743879	−	0.668314i	$-0.767016\pi$
$883$	−44.2092	−1.48776	−0.743879	+	0.668314i	$0.767016\pi$
$884$	0	0
$885$	29.0216	0.975551
$886$	0	0
$887$	−44.8445	−1.50573	−0.752865	−	0.658175i	$-0.771329\pi$
$887$	−44.8445	−1.50573	−0.752865	+	0.658175i	$0.771329\pi$
$888$	0	0
$889$	23.1738	0.777224
$890$	0	0
$891$	−0.762494	−0.0255445
$892$	0	0
$893$	16.2563	0.543995
$894$	0	0
$895$	54.9454	1.83662
$896$	0	0
$897$	−0.643322	−0.0214799
$898$	0	0
$899$	18.9392	0.631656
$900$	0	0
$901$	−3.39177	−0.112996
$902$	0	0
$903$	−11.2104	−0.373060
$904$	0	0
$905$	−62.0724	−2.06336
$906$	0	0
$907$	32.1371	1.06709	0.533547	−	0.845770i	$-0.320859\pi$
$907$	32.1371	1.06709	0.533547	+	0.845770i	$0.320859\pi$
$908$	0	0
$909$	−14.7768	−0.490116
$910$	0	0
$911$	−18.7175	−0.620138	−0.310069	−	0.950714i	$-0.600352\pi$
$911$	−18.7175	−0.620138	−0.310069	+	0.950714i	$0.600352\pi$
$912$	0	0
$913$	3.53819	0.117097
$914$	0	0
$915$	2.79579	0.0924260
$916$	0	0
$917$	−37.1287	−1.22610
$918$	0	0
$919$	−46.9203	−1.54776	−0.773879	−	0.633333i	$-0.781686\pi$
$919$	−46.9203	−1.54776	−0.773879	+	0.633333i	$0.781686\pi$
$920$	0	0
$921$	15.2440	0.502305
$922$	0	0
$923$	−0.0199956	−0.000658165 0
$924$	0	0
$925$	161.543	5.31150
$926$	0	0
$927$	5.90650	0.193995
$928$	0	0
$929$	57.4967	1.88640	0.943202	−	0.332220i	$-0.107798\pi$
$929$	57.4967	1.88640	0.943202	+	0.332220i	$0.107798\pi$
$930$	0	0
$931$	4.70900	0.154331
$932$	0	0
$933$	−26.2210	−0.858437
$934$	0	0
$935$	−1.13444	−0.0371002
$936$	0	0
$937$	−30.3608	−0.991843	−0.495921	−	0.868367i	$-0.665170\pi$
$937$	−30.3608	−0.991843	−0.495921	+	0.868367i	$0.665170\pi$
$938$	0	0
$939$	−17.1744	−0.560464
$940$	0	0
$941$	43.3467	1.41306	0.706531	−	0.707682i	$-0.250259\pi$
$941$	43.3467	1.41306	0.706531	+	0.707682i	$0.250259\pi$
$942$	0	0
$943$	11.1949	0.364556
$944$	0	0
$945$	10.7942	0.351136
$946$	0	0
$947$	43.9177	1.42713	0.713567	−	0.700587i	$-0.247078\pi$
$947$	43.9177	1.42713	0.713567	+	0.700587i	$0.247078\pi$
$948$	0	0
$949$	0.450311	0.0146177
$950$	0	0
$951$	−2.28356	−0.0740496
$952$	0	0
$953$	2.89208	0.0936837	0.0468419	−	0.998902i	$-0.485084\pi$
$953$	2.89208	0.0936837	0.0468419	+	0.998902i	$0.485084\pi$
$954$	0	0
$955$	97.4743	3.15419
$956$	0	0
$957$	−5.65550	−0.182816
$958$	0	0
$959$	−5.72698	−0.184934
$960$	0	0
$961$	−24.4799	−0.789675
$962$	0	0
$963$	1.76179	0.0567729
$964$	0	0
$965$	46.8181	1.50713
$966$	0	0
$967$	−0.347511	−0.0111752	−0.00558760	−	0.999984i	$-0.501779\pi$
$967$	−0.347511	−0.0111752	−0.00558760	+	0.999984i	$0.501779\pi$
$968$	0	0
$969$	−1.42010	−0.0456201
$970$	0	0
$971$	52.0146	1.66923	0.834614	−	0.550835i	$-0.185691\pi$
$971$	52.0146	1.66923	0.834614	+	0.550835i	$0.185691\pi$
$972$	0	0
$973$	−26.6240	−0.853526
$974$	0	0
$975$	−2.20180	−0.0705141
$976$	0	0
$977$	−32.1282	−1.02787	−0.513936	−	0.857828i	$-0.671813\pi$
$977$	−32.1282	−1.02787	−0.513936	+	0.857828i	$0.671813\pi$
$978$	0	0
$979$	7.61509	0.243379
$980$	0	0
$981$	−6.32940	−0.202082
$982$	0	0
$983$	42.2106	1.34631	0.673155	−	0.739501i	$-0.264939\pi$
$983$	42.2106	1.34631	0.673155	+	0.739501i	$0.264939\pi$
$984$	0	0
$985$	78.0564	2.48708
$986$	0	0
$987$	9.29446	0.295846
$988$	0	0
$989$	19.9457	0.634235
$990$	0	0
$991$	44.4264	1.41125	0.705625	−	0.708585i	$-0.250666\pi$
$991$	44.4264	1.41125	0.705625	+	0.708585i	$0.250666\pi$
$992$	0	0
$993$	−5.20651	−0.165223
$994$	0	0
$995$	−2.21488	−0.0702163
$996$	0	0
$997$	−35.7351	−1.13174	−0.565871	−	0.824493i	$-0.691460\pi$
$997$	−35.7351	−1.13174	−0.565871	+	0.824493i	$0.691460\pi$
$998$	0	0
$999$	−10.9302	−0.345816

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4008.2.a.j.1.1	✓	10
4.3	odd	2		8016.2.a.bd.1.1		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4008.2.a.j.1.1	✓	10	1.1	even	1	trivial
8016.2.a.bd.1.1		10	4.3	odd	2

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:	\( N \)	\(=\)	\( 4008 = 2^{3} \cdot 3 \cdot 167 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4008.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -4.44741 q^{5} +2.42708 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -4.44741 q^{5} +2.42708 q^{7} +1.00000 q^{9} -0.762494 q^{11} +0.148977 q^{13} +4.44741 q^{15} -0.334532 q^{17} -4.24502 q^{19} -2.42708 q^{21} +4.31827 q^{23} +14.7795 q^{25} -1.00000 q^{27} -7.41711 q^{29} -2.55344 q^{31} +0.762494 q^{33} -10.7942 q^{35} +10.9302 q^{37} -0.148977 q^{39} +2.59245 q^{41} +4.61890 q^{43} -4.44741 q^{45} -3.82949 q^{47} -1.10930 q^{49} +0.334532 q^{51} +10.1388 q^{53} +3.39113 q^{55} +4.24502 q^{57} +6.52551 q^{59} +0.628633 q^{61} +2.42708 q^{63} -0.662562 q^{65} -6.47849 q^{67} -4.31827 q^{69} -0.134220 q^{71} +3.02269 q^{73} -14.7795 q^{75} -1.85063 q^{77} -1.69105 q^{79} +1.00000 q^{81} -4.64029 q^{83} +1.48780 q^{85} +7.41711 q^{87} -9.98708 q^{89} +0.361578 q^{91} +2.55344 q^{93} +18.8794 q^{95} +9.08991 q^{97} -0.762494 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 10 q^{3} - 10 q^{5} + q^{7} + 10 q^{9}+O(q^{10}) \) 10 * q - 10 * q^3 - 10 * q^5 + q^7 + 10 * q^9	\( 10 q - 10 q^{3} - 10 q^{5} + q^{7} + 10 q^{9} - q^{11} - 6 q^{13} + 10 q^{15} - 9 q^{17} + 2 q^{19} - q^{21} + 7 q^{23} + 12 q^{25} - 10 q^{27} - 13 q^{29} + 23 q^{31} + q^{33} + q^{35} - 6 q^{37} + 6 q^{39} - 12 q^{41} - 10 q^{45} + 10 q^{47} + 7 q^{49} + 9 q^{51} - 26 q^{53} + 11 q^{55} - 2 q^{57} - 10 q^{59} - 10 q^{61} + q^{63} - 22 q^{65} - 5 q^{67} - 7 q^{69} + 25 q^{71} - 8 q^{73} - 12 q^{75} - 46 q^{77} + 26 q^{79} + 10 q^{81} - 14 q^{83} + 9 q^{85} + 13 q^{87} - 31 q^{89} - 3 q^{91} - 23 q^{93} - 5 q^{95} - 32 q^{97} - q^{99}+O(q^{100}) \) 10 * q - 10 * q^3 - 10 * q^5 + q^7 + 10 * q^9 - q^11 - 6 * q^13 + 10 * q^15 - 9 * q^17 + 2 * q^19 - q^21 + 7 * q^23 + 12 * q^25 - 10 * q^27 - 13 * q^29 + 23 * q^31 + q^33 + q^35 - 6 * q^37 + 6 * q^39 - 12 * q^41 - 10 * q^45 + 10 * q^47 + 7 * q^49 + 9 * q^51 - 26 * q^53 + 11 * q^55 - 2 * q^57 - 10 * q^59 - 10 * q^61 + q^63 - 22 * q^65 - 5 * q^67 - 7 * q^69 + 25 * q^71 - 8 * q^73 - 12 * q^75 - 46 * q^77 + 26 * q^79 + 10 * q^81 - 14 * q^83 + 9 * q^85 + 13 * q^87 - 31 * q^89 - 3 * q^91 - 23 * q^93 - 5 * q^95 - 32 * q^97 - q^99

Introduction

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