LMFDB - Embedded newform 8046.2.a.i.1.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8046,2,Mod(1,8046)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8046, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("8046.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8046 = 2 \cdot 3^{3} \cdot 149 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8046.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:	$64.2476334663$
Analytic rank:	$1$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 5 x^{11} - 23 x^{10} + 142 x^{9} + 104 x^{8} - 1302 x^{7} + 607 x^{6} + 4323 x^{5} - 4461 x^{4} + \cdots - 553 $ x^12 - 5x^11 - 23x^10 + 142x^9 + 104x^8 - 1302x^7 + 607x^6 + 4323x^5 - 4461x^4 - 3333x^3 + 4805x^2 - 224*x - 553
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			1.12
Root			$-3.72799$ of defining polynomial
Character	$\chi$	$=$	8046.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} +3.72799 q^{5} -2.68331 q^{7} -1.00000 q^{8} +O(q^{10})$	\(q-1.00000 q^{2} +1.00000 q^{4} +3.72799 q^{5} -2.68331 q^{7} -1.00000 q^{8} -3.72799 q^{10} +3.03613 q^{11} -1.51763 q^{13} +2.68331 q^{14} +1.00000 q^{16} +3.16608 q^{17} -7.34640 q^{19} +3.72799 q^{20} -3.03613 q^{22} +4.36415 q^{23} +8.89789 q^{25} +1.51763 q^{26} -2.68331 q^{28} -3.00007 q^{29} -1.76116 q^{31} -1.00000 q^{32} -3.16608 q^{34} -10.0033 q^{35} -11.0054 q^{37} +7.34640 q^{38} -3.72799 q^{40} -7.36102 q^{41} -4.31638 q^{43} +3.03613 q^{44} -4.36415 q^{46} +4.55389 q^{47} +0.200143 q^{49} -8.89789 q^{50} -1.51763 q^{52} -10.4838 q^{53} +11.3186 q^{55} +2.68331 q^{56} +3.00007 q^{58} -3.61539 q^{59} -1.93968 q^{61} +1.76116 q^{62} +1.00000 q^{64} -5.65770 q^{65} +0.348084 q^{67} +3.16608 q^{68} +10.0033 q^{70} -1.77899 q^{71} -9.05505 q^{73} +11.0054 q^{74} -7.34640 q^{76} -8.14687 q^{77} +8.59998 q^{79} +3.72799 q^{80} +7.36102 q^{82} +0.134921 q^{83} +11.8031 q^{85} +4.31638 q^{86} -3.03613 q^{88} +13.2087 q^{89} +4.07226 q^{91} +4.36415 q^{92} -4.55389 q^{94} -27.3873 q^{95} -10.9196 q^{97} -0.200143 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 12 q^{2} + 12 q^{4} - 5 q^{5} + 6 q^{7} - 12 q^{8}+O(q^{10}) $ 12 * q - 12 * q^2 + 12 * q^4 - 5 * q^5 + 6 * q^7 - 12 * q^8	$ 12 q - 12 q^{2} + 12 q^{4} - 5 q^{5} + 6 q^{7} - 12 q^{8} + 5 q^{10} - 6 q^{11} + 3 q^{13} - 6 q^{14} + 12 q^{16} - 6 q^{17} + 8 q^{19} - 5 q^{20} + 6 q^{22} - 11 q^{23} + 11 q^{25} - 3 q^{26} + 6 q^{28} - 29 q^{29} + 2 q^{31} - 12 q^{32} + 6 q^{34} - 4 q^{35} + 5 q^{37} - 8 q^{38} + 5 q^{40} - 22 q^{41} + 9 q^{43} - 6 q^{44} + 11 q^{46} - 15 q^{47} + 14 q^{49} - 11 q^{50} + 3 q^{52} - 12 q^{53} + 13 q^{55} - 6 q^{56} + 29 q^{58} - 34 q^{59} - 4 q^{61} - 2 q^{62} + 12 q^{64} - 12 q^{65} + q^{67} - 6 q^{68} + 4 q^{70} - 21 q^{71} - 2 q^{73} - 5 q^{74} + 8 q^{76} - 34 q^{77} + 9 q^{79} - 5 q^{80} + 22 q^{82} - 10 q^{83} + 5 q^{85} - 9 q^{86} + 6 q^{88} + 2 q^{89} + 17 q^{91} - 11 q^{92} + 15 q^{94} - 69 q^{95} - 13 q^{97} - 14 q^{98}+O(q^{100}) $ 12 * q - 12 * q^2 + 12 * q^4 - 5 * q^5 + 6 * q^7 - 12 * q^8 + 5 * q^10 - 6 * q^11 + 3 * q^13 - 6 * q^14 + 12 * q^16 - 6 * q^17 + 8 * q^19 - 5 * q^20 + 6 * q^22 - 11 * q^23 + 11 * q^25 - 3 * q^26 + 6 * q^28 - 29 * q^29 + 2 * q^31 - 12 * q^32 + 6 * q^34 - 4 * q^35 + 5 * q^37 - 8 * q^38 + 5 * q^40 - 22 * q^41 + 9 * q^43 - 6 * q^44 + 11 * q^46 - 15 * q^47 + 14 * q^49 - 11 * q^50 + 3 * q^52 - 12 * q^53 + 13 * q^55 - 6 * q^56 + 29 * q^58 - 34 * q^59 - 4 * q^61 - 2 * q^62 + 12 * q^64 - 12 * q^65 + q^67 - 6 * q^68 + 4 * q^70 - 21 * q^71 - 2 * q^73 - 5 * q^74 + 8 * q^76 - 34 * q^77 + 9 * q^79 - 5 * q^80 + 22 * q^82 - 10 * q^83 + 5 * q^85 - 9 * q^86 + 6 * q^88 + 2 * q^89 + 17 * q^91 - 11 * q^92 + 15 * q^94 - 69 * q^95 - 13 * q^97 - 14 * q^98

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 $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	3.72799	1.66721	0.833603	−	0.552364i	$-0.186274\pi$
$5$	3.72799	1.66721	0.833603	+	0.552364i	$0.186274\pi$
$6$	0	0
$7$	−2.68331	−1.01420	−0.507098	−	0.861889i	$-0.669282\pi$
$7$	−2.68331	−1.01420	−0.507098	+	0.861889i	$0.669282\pi$
$8$	−1.00000	−0.353553
$9$	0	0
$10$	−3.72799	−1.17889
$11$	3.03613	0.915427	0.457714	−	0.889100i	$-0.348669\pi$
$11$	3.03613	0.915427	0.457714	+	0.889100i	$0.348669\pi$
$12$	0	0
$13$	−1.51763	−0.420914	−0.210457	−	0.977603i	$-0.567495\pi$
$13$	−1.51763	−0.420914	−0.210457	+	0.977603i	$0.567495\pi$
$14$	2.68331	0.717144
$15$	0	0
$16$	1.00000	0.250000
$17$	3.16608	0.767886	0.383943	−	0.923357i	$-0.374566\pi$
$17$	3.16608	0.767886	0.383943	+	0.923357i	$0.374566\pi$
$18$	0	0
$19$	−7.34640	−1.68538	−0.842690	−	0.538400i	$-0.819029\pi$
$19$	−7.34640	−1.68538	−0.842690	+	0.538400i	$0.819029\pi$
$20$	3.72799	0.833603
$21$	0	0
$22$	−3.03613	−0.647305
$23$	4.36415	0.909989	0.454995	−	0.890494i	$-0.349641\pi$
$23$	4.36415	0.909989	0.454995	+	0.890494i	$0.349641\pi$
$24$	0	0
$25$	8.89789	1.77958
$26$	1.51763	0.297631
$27$	0	0
$28$	−2.68331	−0.507098
$29$	−3.00007	−0.557100	−0.278550	−	0.960422i	$-0.589854\pi$
$29$	−3.00007	−0.557100	−0.278550	+	0.960422i	$0.589854\pi$
$30$	0	0
$31$	−1.76116	−0.316313	−0.158157	−	0.987414i	$-0.550555\pi$
$31$	−1.76116	−0.316313	−0.158157	+	0.987414i	$0.550555\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	−3.16608	−0.542978
$35$	−10.0033	−1.69087
$36$	0	0
$37$	−11.0054	−1.80928	−0.904639	−	0.426178i	$-0.859860\pi$
$37$	−11.0054	−1.80928	−0.904639	+	0.426178i	$0.859860\pi$
$38$	7.34640	1.19174
$39$	0	0
$40$	−3.72799	−0.589446
$41$	−7.36102	−1.14960	−0.574799	−	0.818295i	$-0.694920\pi$
$41$	−7.36102	−1.14960	−0.574799	+	0.818295i	$0.694920\pi$
$42$	0	0
$43$	−4.31638	−0.658243	−0.329121	−	0.944288i	$-0.606752\pi$
$43$	−4.31638	−0.658243	−0.329121	+	0.944288i	$0.606752\pi$
$44$	3.03613	0.457714
$45$	0	0
$46$	−4.36415	−0.643459
$47$	4.55389	0.664254	0.332127	−	0.943235i	$-0.392234\pi$
$47$	4.55389	0.664254	0.332127	+	0.943235i	$0.392234\pi$
$48$	0	0
$49$	0.200143	0.0285918
$50$	−8.89789	−1.25835
$51$	0	0
$52$	−1.51763	−0.210457
$53$	−10.4838	−1.44006	−0.720031	−	0.693942i	$-0.755872\pi$
$53$	−10.4838	−1.44006	−0.720031	+	0.693942i	$0.755872\pi$
$54$	0	0
$55$	11.3186	1.52621
$56$	2.68331	0.358572
$57$	0	0
$58$	3.00007	0.393929
$59$	−3.61539	−0.470684	−0.235342	−	0.971913i	$-0.575621\pi$
$59$	−3.61539	−0.470684	−0.235342	+	0.971913i	$0.575621\pi$
$60$	0	0
$61$	−1.93968	−0.248351	−0.124176	−	0.992260i	$-0.539629\pi$
$61$	−1.93968	−0.248351	−0.124176	+	0.992260i	$0.539629\pi$
$62$	1.76116	0.223667
$63$	0	0
$64$	1.00000	0.125000
$65$	−5.65770	−0.701751
$66$	0	0
$67$	0.348084	0.0425252	0.0212626	−	0.999774i	$-0.493231\pi$
$67$	0.348084	0.0425252	0.0212626	+	0.999774i	$0.493231\pi$
$68$	3.16608	0.383943
$69$	0	0
$70$	10.0033	1.19563
$71$	−1.77899	−0.211127	−0.105564	−	0.994413i	$-0.533665\pi$
$71$	−1.77899	−0.211127	−0.105564	+	0.994413i	$0.533665\pi$
$72$	0	0
$73$	−9.05505	−1.05981	−0.529907	−	0.848056i	$-0.677773\pi$
$73$	−9.05505	−1.05981	−0.529907	+	0.848056i	$0.677773\pi$
$74$	11.0054	1.27935
$75$	0	0
$76$	−7.34640	−0.842690
$77$	−8.14687	−0.928422
$78$	0	0
$79$	8.59998	0.967573	0.483786	−	0.875186i	$-0.339261\pi$
$79$	8.59998	0.967573	0.483786	+	0.875186i	$0.339261\pi$
$80$	3.72799	0.416802
$81$	0	0
$82$	7.36102	0.812889
$83$	0.134921	0.0148095	0.00740474	−	0.999973i	$-0.497643\pi$
$83$	0.134921	0.0148095	0.00740474	+	0.999973i	$0.497643\pi$
$84$	0	0
$85$	11.8031	1.28022
$86$	4.31638	0.465448
$87$	0	0
$88$	−3.03613	−0.323652
$89$	13.2087	1.40011	0.700057	−	0.714087i	$-0.253158\pi$
$89$	13.2087	1.40011	0.700057	+	0.714087i	$0.253158\pi$
$90$	0	0
$91$	4.07226	0.426889
$92$	4.36415	0.454995
$93$	0	0
$94$	−4.55389	−0.469698
$95$	−27.3873	−2.80988
$96$	0	0
$97$	−10.9196	−1.10871	−0.554357	−	0.832279i	$-0.687036\pi$
$97$	−10.9196	−1.10871	−0.554357	+	0.832279i	$0.687036\pi$
$98$	−0.200143	−0.0202175
$99$	0	0
$100$	8.89789	0.889789
$101$	−7.12802	−0.709265	−0.354632	−	0.935006i	$-0.615394\pi$
$101$	−7.12802	−0.709265	−0.354632	+	0.935006i	$0.615394\pi$
$102$	0	0
$103$	12.1181	1.19403	0.597015	−	0.802230i	$-0.296353\pi$
$103$	12.1181	1.19403	0.597015	+	0.802230i	$0.296353\pi$
$104$	1.51763	0.148816
$105$	0	0
$106$	10.4838	1.01828
$107$	−2.99883	−0.289908	−0.144954	−	0.989438i	$-0.546303\pi$
$107$	−2.99883	−0.289908	−0.144954	+	0.989438i	$0.546303\pi$
$108$	0	0
$109$	−10.6605	−1.02109	−0.510543	−	0.859852i	$-0.670556\pi$
$109$	−10.6605	−1.02109	−0.510543	+	0.859852i	$0.670556\pi$
$110$	−11.3186	−1.07919
$111$	0	0
$112$	−2.68331	−0.253549
$113$	−4.55326	−0.428335	−0.214168	−	0.976797i	$-0.568704\pi$
$113$	−4.55326	−0.428335	−0.214168	+	0.976797i	$0.568704\pi$
$114$	0	0
$115$	16.2695	1.51714
$116$	−3.00007	−0.278550
$117$	0	0
$118$	3.61539	0.332824
$119$	−8.49556	−0.778787
$120$	0	0
$121$	−1.78193	−0.161993
$122$	1.93968	0.175611
$123$	0	0
$124$	−1.76116	−0.158157
$125$	14.5313	1.29972
$126$	0	0
$127$	9.85022	0.874066	0.437033	−	0.899446i	$-0.356029\pi$
$127$	9.85022	0.874066	0.437033	+	0.899446i	$0.356029\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	5.65770	0.496213
$131$	−9.08179	−0.793479	−0.396740	−	0.917931i	$-0.629858\pi$
$131$	−9.08179	−0.793479	−0.396740	+	0.917931i	$0.629858\pi$
$132$	0	0
$133$	19.7127	1.70930
$134$	−0.348084	−0.0300699
$135$	0	0
$136$	−3.16608	−0.271489
$137$	12.2378	1.04555	0.522775	−	0.852471i	$-0.324897\pi$
$137$	12.2378	1.04555	0.522775	+	0.852471i	$0.324897\pi$
$138$	0	0
$139$	7.30626	0.619709	0.309855	−	0.950784i	$-0.399720\pi$
$139$	7.30626	0.619709	0.309855	+	0.950784i	$0.399720\pi$
$140$	−10.0033	−0.845436
$141$	0	0
$142$	1.77899	0.149289
$143$	−4.60771	−0.385316
$144$	0	0
$145$	−11.1842	−0.928800
$146$	9.05505	0.749401
$147$	0	0
$148$	−11.0054	−0.904639
$149$	1.00000	0.0819232
$149$	1.00000	0.0819232
$150$	0	0
$151$	−20.8449	−1.69634	−0.848168	−	0.529728i	$-0.822294\pi$
$151$	−20.8449	−1.69634	−0.848168	+	0.529728i	$0.822294\pi$
$152$	7.34640	0.595872
$153$	0	0
$154$	8.14687	0.656493
$155$	−6.56558	−0.527360
$156$	0	0
$157$	−14.4371	−1.15220	−0.576101	−	0.817378i	$-0.695427\pi$
$157$	−14.4371	−1.15220	−0.576101	+	0.817378i	$0.695427\pi$
$158$	−8.59998	−0.684177
$159$	0	0
$160$	−3.72799	−0.294723
$161$	−11.7104	−0.922906
$162$	0	0
$163$	22.9863	1.80043	0.900214	−	0.435448i	$-0.143410\pi$
$163$	22.9863	1.80043	0.900214	+	0.435448i	$0.143410\pi$
$164$	−7.36102	−0.574799
$165$	0	0
$166$	−0.134921	−0.0104719
$167$	−11.1508	−0.862874	−0.431437	−	0.902143i	$-0.641993\pi$
$167$	−11.1508	−0.862874	−0.431437	+	0.902143i	$0.641993\pi$
$168$	0	0
$169$	−10.6968	−0.822831
$170$	−11.8031	−0.905256
$171$	0	0
$172$	−4.31638	−0.329121
$173$	−9.79349	−0.744585	−0.372293	−	0.928115i	$-0.621428\pi$
$173$	−9.79349	−0.744585	−0.372293	+	0.928115i	$0.621428\pi$
$174$	0	0
$175$	−23.8758	−1.80484
$176$	3.03613	0.228857
$177$	0	0
$178$	−13.2087	−0.990031
$179$	−18.3388	−1.37071	−0.685354	−	0.728210i	$-0.740353\pi$
$179$	−18.3388	−1.37071	−0.685354	+	0.728210i	$0.740353\pi$
$180$	0	0
$181$	11.4340	0.849885	0.424943	−	0.905220i	$-0.360294\pi$
$181$	11.4340	0.849885	0.424943	+	0.905220i	$0.360294\pi$
$182$	−4.07226	−0.301856
$183$	0	0
$184$	−4.36415	−0.321730
$185$	−41.0280	−3.01644
$186$	0	0
$187$	9.61261	0.702944
$188$	4.55389	0.332127
$189$	0	0
$190$	27.3873	1.98688
$191$	1.34890	0.0976029	0.0488015	−	0.998808i	$-0.484460\pi$
$191$	1.34890	0.0976029	0.0488015	+	0.998808i	$0.484460\pi$
$192$	0	0
$193$	−7.26527	−0.522965	−0.261483	−	0.965208i	$-0.584211\pi$
$193$	−7.26527	−0.522965	−0.261483	+	0.965208i	$0.584211\pi$
$194$	10.9196	0.783979
$195$	0	0
$196$	0.200143	0.0142959
$197$	4.47541	0.318859	0.159430	−	0.987209i	$-0.449034\pi$
$197$	4.47541	0.318859	0.159430	+	0.987209i	$0.449034\pi$
$198$	0	0
$199$	18.6122	1.31939	0.659693	−	0.751536i	$-0.270687\pi$
$199$	18.6122	1.31939	0.659693	+	0.751536i	$0.270687\pi$
$200$	−8.89789	−0.629176
$201$	0	0
$202$	7.12802	0.501526
$203$	8.05012	0.565008
$204$	0	0
$205$	−27.4418	−1.91662
$206$	−12.1181	−0.844306
$207$	0	0
$208$	−1.51763	−0.105229
$209$	−22.3046	−1.54284
$210$	0	0
$211$	4.66630	0.321241	0.160621	−	0.987016i	$-0.448650\pi$
$211$	4.66630	0.321241	0.160621	+	0.987016i	$0.448650\pi$
$212$	−10.4838	−0.720031
$213$	0	0
$214$	2.99883	0.204996
$215$	−16.0914	−1.09743
$216$	0	0
$217$	4.72573	0.320804
$218$	10.6605	0.722017
$219$	0	0
$220$	11.3186	0.763103
$221$	−4.80493	−0.323214
$222$	0	0
$223$	−7.01396	−0.469690	−0.234845	−	0.972033i	$-0.575458\pi$
$223$	−7.01396	−0.469690	−0.234845	+	0.972033i	$0.575458\pi$
$224$	2.68331	0.179286
$225$	0	0
$226$	4.55326	0.302879
$227$	28.9841	1.92374	0.961871	−	0.273504i	$-0.0881825\pi$
$227$	28.9841	1.92374	0.961871	+	0.273504i	$0.0881825\pi$
$228$	0	0
$229$	−1.49235	−0.0986172	−0.0493086	−	0.998784i	$-0.515702\pi$
$229$	−1.49235	−0.0986172	−0.0493086	+	0.998784i	$0.515702\pi$
$230$	−16.2695	−1.07278
$231$	0	0
$232$	3.00007	0.196964
$233$	−16.5298	−1.08291	−0.541453	−	0.840731i	$-0.682126\pi$
$233$	−16.5298	−1.08291	−0.541453	+	0.840731i	$0.682126\pi$
$234$	0	0
$235$	16.9769	1.10745
$236$	−3.61539	−0.235342
$237$	0	0
$238$	8.49556	0.550685
$239$	6.51060	0.421136	0.210568	−	0.977579i	$-0.432469\pi$
$239$	6.51060	0.421136	0.210568	+	0.977579i	$0.432469\pi$
$240$	0	0
$241$	−3.15693	−0.203356	−0.101678	−	0.994817i	$-0.532421\pi$
$241$	−3.15693	−0.203356	−0.101678	+	0.994817i	$0.532421\pi$
$242$	1.78193	0.114547
$243$	0	0
$244$	−1.93968	−0.124176
$245$	0.746129	0.0476685
$246$	0	0
$247$	11.1491	0.709400
$248$	1.76116	0.111834
$249$	0	0
$250$	−14.5313	−0.919038
$251$	20.0049	1.26269	0.631347	−	0.775500i	$-0.282502\pi$
$251$	20.0049	1.26269	0.631347	+	0.775500i	$0.282502\pi$
$252$	0	0
$253$	13.2501	0.833029
$254$	−9.85022	−0.618058
$255$	0	0
$256$	1.00000	0.0625000
$257$	−6.28291	−0.391917	−0.195959	−	0.980612i	$-0.562782\pi$
$257$	−6.28291	−0.391917	−0.195959	+	0.980612i	$0.562782\pi$
$258$	0	0
$259$	29.5309	1.83496
$260$	−5.65770	−0.350875
$261$	0	0
$262$	9.08179	0.561075
$263$	22.9242	1.41357	0.706785	−	0.707429i	$-0.250145\pi$
$263$	22.9242	1.41357	0.706785	+	0.707429i	$0.250145\pi$
$264$	0	0
$265$	−39.0835	−2.40088
$266$	−19.7127	−1.20866
$267$	0	0
$268$	0.348084	0.0212626
$269$	−5.77491	−0.352102	−0.176051	−	0.984381i	$-0.556332\pi$
$269$	−5.77491	−0.352102	−0.176051	+	0.984381i	$0.556332\pi$
$270$	0	0
$271$	−32.1746	−1.95447	−0.977233	−	0.212171i	$-0.931947\pi$
$271$	−32.1746	−1.95447	−0.977233	+	0.212171i	$0.931947\pi$
$272$	3.16608	0.191972
$273$	0	0
$274$	−12.2378	−0.739315
$275$	27.0151	1.62907
$276$	0	0
$277$	12.3691	0.743185	0.371593	−	0.928396i	$-0.378812\pi$
$277$	12.3691	0.743185	0.371593	+	0.928396i	$0.378812\pi$
$278$	−7.30626	−0.438201
$279$	0	0
$280$	10.0033	0.597814
$281$	−7.67646	−0.457939	−0.228970	−	0.973434i	$-0.573536\pi$
$281$	−7.67646	−0.457939	−0.228970	+	0.973434i	$0.573536\pi$
$282$	0	0
$283$	−4.14480	−0.246383	−0.123191	−	0.992383i	$-0.539313\pi$
$283$	−4.14480	−0.246383	−0.123191	+	0.992383i	$0.539313\pi$
$284$	−1.77899	−0.105564
$285$	0	0
$286$	4.60771	0.272460
$287$	19.7519	1.16592
$288$	0	0
$289$	−6.97596	−0.410351
$290$	11.1842	0.656761
$291$	0	0
$292$	−9.05505	−0.529907
$293$	−12.7783	−0.746517	−0.373259	−	0.927727i	$-0.621760\pi$
$293$	−12.7783	−0.746517	−0.373259	+	0.927727i	$0.621760\pi$
$294$	0	0
$295$	−13.4781	−0.784728
$296$	11.0054	0.639676
$297$	0	0
$298$	−1.00000	−0.0579284
$299$	−6.62316	−0.383027
$300$	0	0
$301$	11.5822	0.667586
$302$	20.8449	1.19949
$303$	0	0
$304$	−7.34640	−0.421345
$305$	−7.23112	−0.414053
$306$	0	0
$307$	17.6013	1.00456	0.502281	−	0.864705i	$-0.332494\pi$
$307$	17.6013	1.00456	0.502281	+	0.864705i	$0.332494\pi$
$308$	−8.14687	−0.464211
$309$	0	0
$310$	6.56558	0.372900
$311$	−10.6147	−0.601903	−0.300951	−	0.953639i	$-0.597304\pi$
$311$	−10.6147	−0.601903	−0.300951	+	0.953639i	$0.597304\pi$
$312$	0	0
$313$	−17.2722	−0.976284	−0.488142	−	0.872764i	$-0.662325\pi$
$313$	−17.2722	−0.976284	−0.488142	+	0.872764i	$0.662325\pi$
$314$	14.4371	0.814731
$315$	0	0
$316$	8.59998	0.483786
$317$	7.28092	0.408937	0.204469	−	0.978873i	$-0.434453\pi$
$317$	7.28092	0.408937	0.204469	+	0.978873i	$0.434453\pi$
$318$	0	0
$319$	−9.10861	−0.509984
$320$	3.72799	0.208401
$321$	0	0
$322$	11.7104	0.652593
$323$	−23.2593	−1.29418
$324$	0	0
$325$	−13.5037	−0.749049
$326$	−22.9863	−1.27309
$327$	0	0
$328$	7.36102	0.406444
$329$	−12.2195	−0.673683
$330$	0	0
$331$	−7.05357	−0.387699	−0.193850	−	0.981031i	$-0.562097\pi$
$331$	−7.05357	−0.387699	−0.193850	+	0.981031i	$0.562097\pi$
$332$	0.134921	0.00740474
$333$	0	0
$334$	11.1508	0.610144
$335$	1.29765	0.0708984
$336$	0	0
$337$	−2.20317	−0.120015	−0.0600073	−	0.998198i	$-0.519112\pi$
$337$	−2.20317	−0.120015	−0.0600073	+	0.998198i	$0.519112\pi$
$338$	10.6968	0.581829
$339$	0	0
$340$	11.8031	0.640112
$341$	−5.34710	−0.289562
$342$	0	0
$343$	18.2461	0.985197
$344$	4.31638	0.232724
$345$	0	0
$346$	9.79349	0.526501
$347$	−8.47386	−0.454901	−0.227450	−	0.973790i	$-0.573039\pi$
$347$	−8.47386	−0.454901	−0.227450	+	0.973790i	$0.573039\pi$
$348$	0	0
$349$	30.7803	1.64763	0.823816	−	0.566857i	$-0.191841\pi$
$349$	30.7803	1.64763	0.823816	+	0.566857i	$0.191841\pi$
$350$	23.8758	1.27621
$351$	0	0
$352$	−3.03613	−0.161826
$353$	25.4586	1.35502	0.677511	−	0.735513i	$-0.263059\pi$
$353$	25.4586	1.35502	0.677511	+	0.735513i	$0.263059\pi$
$354$	0	0
$355$	−6.63205	−0.351993
$356$	13.2087	0.700057
$357$	0	0
$358$	18.3388	0.969237
$359$	0.739243	0.0390157	0.0195079	−	0.999810i	$-0.493790\pi$
$359$	0.739243	0.0390157	0.0195079	+	0.999810i	$0.493790\pi$
$360$	0	0
$361$	34.9696	1.84050
$362$	−11.4340	−0.600960
$363$	0	0
$364$	4.07226	0.213445
$365$	−33.7571	−1.76693
$366$	0	0
$367$	−24.3186	−1.26942	−0.634711	−	0.772750i	$-0.718881\pi$
$367$	−24.3186	−1.26942	−0.634711	+	0.772750i	$0.718881\pi$
$368$	4.36415	0.227497
$369$	0	0
$370$	41.0280	2.13295
$371$	28.1313	1.46050
$372$	0	0
$373$	2.50837	0.129879	0.0649393	−	0.997889i	$-0.479315\pi$
$373$	2.50837	0.129879	0.0649393	+	0.997889i	$0.479315\pi$
$374$	−9.61261	−0.497056
$375$	0	0
$376$	−4.55389	−0.234849
$377$	4.55300	0.234491
$378$	0	0
$379$	32.3360	1.66099	0.830494	−	0.557028i	$-0.188059\pi$
$379$	32.3360	1.66099	0.830494	+	0.557028i	$0.188059\pi$
$380$	−27.3873	−1.40494
$381$	0	0
$382$	−1.34890	−0.0690157
$383$	6.93966	0.354600	0.177300	−	0.984157i	$-0.443264\pi$
$383$	6.93966	0.354600	0.177300	+	0.984157i	$0.443264\pi$
$384$	0	0
$385$	−30.3714	−1.54787
$386$	7.26527	0.369792
$387$	0	0
$388$	−10.9196	−0.554357
$389$	23.9613	1.21489	0.607444	−	0.794362i	$-0.292195\pi$
$389$	23.9613	1.21489	0.607444	+	0.794362i	$0.292195\pi$
$390$	0	0
$391$	13.8172	0.698768
$392$	−0.200143	−0.0101087
$393$	0	0
$394$	−4.47541	−0.225468
$395$	32.0606	1.61314
$396$	0	0
$397$	−18.7734	−0.942209	−0.471105	−	0.882077i	$-0.656145\pi$
$397$	−18.7734	−0.942209	−0.471105	+	0.882077i	$0.656145\pi$
$398$	−18.6122	−0.932946
$399$	0	0
$400$	8.89789	0.444894
$401$	−37.4767	−1.87150	−0.935748	−	0.352670i	$-0.885274\pi$
$401$	−37.4767	−1.87150	−0.935748	+	0.352670i	$0.885274\pi$
$402$	0	0
$403$	2.67278	0.133141
$404$	−7.12802	−0.354632
$405$	0	0
$406$	−8.05012	−0.399521
$407$	−33.4138	−1.65626
$408$	0	0
$409$	7.66375	0.378948	0.189474	−	0.981886i	$-0.439322\pi$
$409$	7.66375	0.378948	0.189474	+	0.981886i	$0.439322\pi$
$410$	27.4418	1.35525
$411$	0	0
$412$	12.1181	0.597015
$413$	9.70122	0.477366
$414$	0	0
$415$	0.502983	0.0246905
$416$	1.51763	0.0744078
$417$	0	0
$418$	22.3046	1.09095
$419$	−35.3284	−1.72591	−0.862953	−	0.505285i	$-0.831387\pi$
$419$	−35.3284	−1.72591	−0.862953	+	0.505285i	$0.831387\pi$
$420$	0	0
$421$	22.0077	1.07259	0.536296	−	0.844030i	$-0.319823\pi$
$421$	22.0077	1.07259	0.536296	+	0.844030i	$0.319823\pi$
$422$	−4.66630	−0.227152
$423$	0	0
$424$	10.4838	0.509139
$425$	28.1714	1.36651
$426$	0	0
$427$	5.20477	0.251876
$428$	−2.99883	−0.144954
$429$	0	0
$430$	16.0914	0.775997
$431$	−33.2706	−1.60259	−0.801295	−	0.598269i	$-0.795855\pi$
$431$	−33.2706	−1.60259	−0.801295	+	0.598269i	$0.795855\pi$
$432$	0	0
$433$	−33.4569	−1.60783	−0.803917	−	0.594741i	$-0.797254\pi$
$433$	−33.4569	−1.60783	−0.803917	+	0.594741i	$0.797254\pi$
$434$	−4.72573	−0.226842
$435$	0	0
$436$	−10.6605	−0.510543
$437$	−32.0608	−1.53368
$438$	0	0
$439$	−14.8073	−0.706713	−0.353356	−	0.935489i	$-0.614960\pi$
$439$	−14.8073	−0.706713	−0.353356	+	0.935489i	$0.614960\pi$
$440$	−11.3186	−0.539595
$441$	0	0
$442$	4.80493	0.228547
$443$	−34.0503	−1.61778	−0.808889	−	0.587962i	$-0.799930\pi$
$443$	−34.0503	−1.61778	−0.808889	+	0.587962i	$0.799930\pi$
$444$	0	0
$445$	49.2417	2.33428
$446$	7.01396	0.332121
$447$	0	0
$448$	−2.68331	−0.126774
$449$	12.2168	0.576546	0.288273	−	0.957548i	$-0.406919\pi$
$449$	12.2168	0.576546	0.288273	+	0.957548i	$0.406919\pi$
$450$	0	0
$451$	−22.3490	−1.05237
$452$	−4.55326	−0.214168
$453$	0	0
$454$	−28.9841	−1.36029
$455$	15.1813	0.711712
$456$	0	0
$457$	−5.68028	−0.265712	−0.132856	−	0.991135i	$-0.542415\pi$
$457$	−5.68028	−0.265712	−0.132856	+	0.991135i	$0.542415\pi$
$458$	1.49235	0.0697329
$459$	0	0
$460$	16.2695	0.758570
$461$	2.69247	0.125401	0.0627004	−	0.998032i	$-0.480029\pi$
$461$	2.69247	0.125401	0.0627004	+	0.998032i	$0.480029\pi$
$462$	0	0
$463$	7.31923	0.340153	0.170077	−	0.985431i	$-0.445598\pi$
$463$	7.31923	0.340153	0.170077	+	0.985431i	$0.445598\pi$
$464$	−3.00007	−0.139275
$465$	0	0
$466$	16.5298	0.765730
$467$	41.8088	1.93468	0.967341	−	0.253479i	$-0.0815748\pi$
$467$	41.8088	1.93468	0.967341	+	0.253479i	$0.0815748\pi$
$468$	0	0
$469$	−0.934017	−0.0431289
$470$	−16.9769	−0.783084
$471$	0	0
$472$	3.61539	0.166412
$473$	−13.1051	−0.602573
$474$	0	0
$475$	−65.3674	−2.99926
$476$	−8.49556	−0.389393
$477$	0	0
$478$	−6.51060	−0.297788
$479$	−24.8975	−1.13759	−0.568797	−	0.822478i	$-0.692591\pi$
$479$	−24.8975	−1.13759	−0.568797	+	0.822478i	$0.692591\pi$
$480$	0	0
$481$	16.7021	0.761551
$482$	3.15693	0.143794
$483$	0	0
$484$	−1.78193	−0.0809967
$485$	−40.7080	−1.84845
$486$	0	0
$487$	33.5846	1.52186	0.760932	−	0.648832i	$-0.224742\pi$
$487$	33.5846	1.52186	0.760932	+	0.648832i	$0.224742\pi$
$488$	1.93968	0.0878054
$489$	0	0
$490$	−0.746129	−0.0337067
$491$	−4.55572	−0.205597	−0.102798	−	0.994702i	$-0.532780\pi$
$491$	−4.55572	−0.205597	−0.102798	+	0.994702i	$0.532780\pi$
$492$	0	0
$493$	−9.49846	−0.427789
$494$	−11.1491	−0.501622
$495$	0	0
$496$	−1.76116	−0.0790784
$497$	4.77358	0.214124
$498$	0	0
$499$	8.32927	0.372869	0.186435	−	0.982467i	$-0.440307\pi$
$499$	8.32927	0.372869	0.186435	+	0.982467i	$0.440307\pi$
$500$	14.5313	0.649858
$501$	0	0
$502$	−20.0049	−0.892860
$503$	−10.0572	−0.448429	−0.224214	−	0.974540i	$-0.571982\pi$
$503$	−10.0572	−0.448429	−0.224214	+	0.974540i	$0.571982\pi$
$504$	0	0
$505$	−26.5732	−1.18249
$506$	−13.2501	−0.589040
$507$	0	0
$508$	9.85022	0.437033
$509$	32.6070	1.44528	0.722640	−	0.691224i	$-0.242928\pi$
$509$	32.6070	1.44528	0.722640	+	0.691224i	$0.242928\pi$
$510$	0	0
$511$	24.2975	1.07486
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	6.28291	0.277127
$515$	45.1760	1.99069
$516$	0	0
$517$	13.8262	0.608076
$518$	−29.5309	−1.29751
$519$	0	0
$520$	5.65770	0.248106
$521$	32.9185	1.44219	0.721094	−	0.692838i	$-0.243640\pi$
$521$	32.9185	1.44219	0.721094	+	0.692838i	$0.243640\pi$
$522$	0	0
$523$	−43.1958	−1.88882	−0.944409	−	0.328772i	$-0.893365\pi$
$523$	−43.1958	−1.88882	−0.944409	+	0.328772i	$0.893365\pi$
$524$	−9.08179	−0.396740
$525$	0	0
$526$	−22.9242	−0.999545
$527$	−5.57596	−0.242893
$528$	0	0
$529$	−3.95416	−0.171920
$530$	39.0835	1.69768
$531$	0	0
$532$	19.7127	0.854652
$533$	11.1713	0.483882
$534$	0	0
$535$	−11.1796	−0.483336
$536$	−0.348084	−0.0150349
$537$	0	0
$538$	5.77491	0.248974
$539$	0.607659	0.0261737
$540$	0	0
$541$	−17.8775	−0.768612	−0.384306	−	0.923206i	$-0.625559\pi$
$541$	−17.8775	−0.768612	−0.384306	+	0.923206i	$0.625559\pi$
$542$	32.1746	1.38202
$543$	0	0
$544$	−3.16608	−0.135744
$545$	−39.7420	−1.70236
$546$	0	0
$547$	37.2678	1.59346	0.796728	−	0.604338i	$-0.206563\pi$
$547$	37.2678	1.59346	0.796728	+	0.604338i	$0.206563\pi$
$548$	12.2378	0.522775
$549$	0	0
$550$	−27.0151	−1.15193
$551$	22.0397	0.938924
$552$	0	0
$553$	−23.0764	−0.981308
$554$	−12.3691	−0.525511
$555$	0	0
$556$	7.30626	0.309855
$557$	8.96823	0.379996	0.189998	−	0.981784i	$-0.439152\pi$
$557$	8.96823	0.379996	0.189998	+	0.981784i	$0.439152\pi$
$558$	0	0
$559$	6.55067	0.277064
$560$	−10.0033	−0.422718
$561$	0	0
$562$	7.67646	0.323812
$563$	−6.43248	−0.271097	−0.135548	−	0.990771i	$-0.543280\pi$
$563$	−6.43248	−0.271097	−0.135548	+	0.990771i	$0.543280\pi$
$564$	0	0
$565$	−16.9745	−0.714123
$566$	4.14480	0.174219
$567$	0	0
$568$	1.77899	0.0746447
$569$	0.866264	0.0363157	0.0181578	−	0.999835i	$-0.494220\pi$
$569$	0.866264	0.0363157	0.0181578	+	0.999835i	$0.494220\pi$
$570$	0	0
$571$	38.7345	1.62099	0.810493	−	0.585748i	$-0.199199\pi$
$571$	38.7345	1.62099	0.810493	+	0.585748i	$0.199199\pi$
$572$	−4.60771	−0.192658
$573$	0	0
$574$	−19.7519	−0.824428
$575$	38.8317	1.61940
$576$	0	0
$577$	9.58882	0.399188	0.199594	−	0.979879i	$-0.436038\pi$
$577$	9.58882	0.399188	0.199594	+	0.979879i	$0.436038\pi$
$578$	6.97596	0.290162
$579$	0	0
$580$	−11.1842	−0.464400
$581$	−0.362034	−0.0150197
$582$	0	0
$583$	−31.8302	−1.31827
$584$	9.05505	0.374701
$585$	0	0
$586$	12.7783	0.527867
$587$	4.71487	0.194604	0.0973018	−	0.995255i	$-0.468979\pi$
$587$	4.71487	0.194604	0.0973018	+	0.995255i	$0.468979\pi$
$588$	0	0
$589$	12.9382	0.533108
$590$	13.4781	0.554886
$591$	0	0
$592$	−11.0054	−0.452320
$593$	47.8498	1.96496	0.982478	−	0.186379i	$-0.0596754\pi$
$593$	47.8498	1.96496	0.982478	+	0.186379i	$0.0596754\pi$
$594$	0	0
$595$	−31.6713	−1.29840
$596$	1.00000	0.0409616
$597$	0	0
$598$	6.62316	0.270841
$599$	−12.1155	−0.495027	−0.247513	−	0.968884i	$-0.579613\pi$
$599$	−12.1155	−0.495027	−0.247513	+	0.968884i	$0.579613\pi$
$600$	0	0
$601$	−17.9338	−0.731536	−0.365768	−	0.930706i	$-0.619194\pi$
$601$	−17.9338	−0.731536	−0.365768	+	0.930706i	$0.619194\pi$
$602$	−11.5822	−0.472055
$603$	0	0
$604$	−20.8449	−0.848168
$605$	−6.64300	−0.270076
$606$	0	0
$607$	−31.8665	−1.29342	−0.646710	−	0.762736i	$-0.723856\pi$
$607$	−31.8665	−1.29342	−0.646710	+	0.762736i	$0.723856\pi$
$608$	7.34640	0.297936
$609$	0	0
$610$	7.23112	0.292779
$611$	−6.91112	−0.279594
$612$	0	0
$613$	−9.75657	−0.394064	−0.197032	−	0.980397i	$-0.563130\pi$
$613$	−9.75657	−0.394064	−0.197032	+	0.980397i	$0.563130\pi$
$614$	−17.6013	−0.710332
$615$	0	0
$616$	8.14687	0.328247
$617$	17.3527	0.698595	0.349298	−	0.937012i	$-0.386420\pi$
$617$	17.3527	0.698595	0.349298	+	0.937012i	$0.386420\pi$
$618$	0	0
$619$	6.61009	0.265682	0.132841	−	0.991137i	$-0.457590\pi$
$619$	6.61009	0.265682	0.132841	+	0.991137i	$0.457590\pi$
$620$	−6.56558	−0.263680
$621$	0	0
$622$	10.6147	0.425610
$623$	−35.4429	−1.41999
$624$	0	0
$625$	9.68294	0.387318
$626$	17.2722	0.690337
$627$	0	0
$628$	−14.4371	−0.576101
$629$	−34.8440	−1.38932
$630$	0	0
$631$	5.83702	0.232368	0.116184	−	0.993228i	$-0.462934\pi$
$631$	5.83702	0.232368	0.116184	+	0.993228i	$0.462934\pi$
$632$	−8.59998	−0.342089
$633$	0	0
$634$	−7.28092	−0.289162
$635$	36.7215	1.45725
$636$	0	0
$637$	−0.303742	−0.0120347
$638$	9.10861	0.360613
$639$	0	0
$640$	−3.72799	−0.147362
$641$	−8.97358	−0.354435	−0.177218	−	0.984172i	$-0.556710\pi$
$641$	−8.97358	−0.354435	−0.177218	+	0.984172i	$0.556710\pi$
$642$	0	0
$643$	12.5976	0.496799	0.248400	−	0.968658i	$-0.420095\pi$
$643$	12.5976	0.496799	0.248400	+	0.968658i	$0.420095\pi$
$644$	−11.7104	−0.461453
$645$	0	0
$646$	23.2593	0.915123
$647$	30.1280	1.18445	0.592226	−	0.805772i	$-0.298249\pi$
$647$	30.1280	1.18445	0.592226	+	0.805772i	$0.298249\pi$
$648$	0	0
$649$	−10.9768	−0.430877
$650$	13.5037	0.529658
$651$	0	0
$652$	22.9863	0.900214
$653$	−24.2342	−0.948356	−0.474178	−	0.880429i	$-0.657255\pi$
$653$	−24.2342	−0.948356	−0.474178	+	0.880429i	$0.657255\pi$
$654$	0	0
$655$	−33.8568	−1.32289
$656$	−7.36102	−0.287399
$657$	0	0
$658$	12.2195	0.476366
$659$	−0.635814	−0.0247678	−0.0123839	−	0.999923i	$-0.503942\pi$
$659$	−0.635814	−0.0247678	−0.0123839	+	0.999923i	$0.503942\pi$
$660$	0	0
$661$	23.3171	0.906929	0.453464	−	0.891274i	$-0.350188\pi$
$661$	23.3171	0.906929	0.453464	+	0.891274i	$0.350188\pi$
$662$	7.05357	0.274145
$663$	0	0
$664$	−0.134921	−0.00523594
$665$	73.4885	2.84976
$666$	0	0
$667$	−13.0928	−0.506955
$668$	−11.1508	−0.431437
$669$	0	0
$670$	−1.29765	−0.0501327
$671$	−5.88913	−0.227347
$672$	0	0
$673$	−43.9800	−1.69530	−0.847652	−	0.530553i	$-0.821984\pi$
$673$	−43.9800	−1.69530	−0.847652	+	0.530553i	$0.821984\pi$
$674$	2.20317	0.0848631
$675$	0	0
$676$	−10.6968	−0.411416
$677$	29.1623	1.12080	0.560399	−	0.828223i	$-0.310648\pi$
$677$	29.1623	1.12080	0.560399	+	0.828223i	$0.310648\pi$
$678$	0	0
$679$	29.3006	1.12445
$680$	−11.8031	−0.452628
$681$	0	0
$682$	5.34710	0.204751
$683$	9.47599	0.362589	0.181294	−	0.983429i	$-0.441971\pi$
$683$	9.47599	0.362589	0.181294	+	0.983429i	$0.441971\pi$
$684$	0	0
$685$	45.6225	1.74315
$686$	−18.2461	−0.696640
$687$	0	0
$688$	−4.31638	−0.164561
$689$	15.9105	0.606143
$690$	0	0
$691$	−44.8611	−1.70660	−0.853299	−	0.521422i	$-0.825402\pi$
$691$	−44.8611	−1.70660	−0.853299	+	0.521422i	$0.825402\pi$
$692$	−9.79349	−0.372293
$693$	0	0
$694$	8.47386	0.321663
$695$	27.2377	1.03318
$696$	0	0
$697$	−23.3055	−0.882760
$698$	−30.7803	−1.16505
$699$	0	0
$700$	−23.8758	−0.902419
$701$	−23.7948	−0.898716	−0.449358	−	0.893352i	$-0.648347\pi$
$701$	−23.7948	−0.898716	−0.449358	+	0.893352i	$0.648347\pi$
$702$	0	0
$703$	80.8501	3.04932
$704$	3.03613	0.114428
$705$	0	0
$706$	−25.4586	−0.958145
$707$	19.1267	0.719333
$708$	0	0
$709$	−29.0140	−1.08964	−0.544822	−	0.838552i	$-0.683403\pi$
$709$	−29.0140	−1.08964	−0.544822	+	0.838552i	$0.683403\pi$
$710$	6.63205	0.248896
$711$	0	0
$712$	−13.2087	−0.495015
$713$	−7.68597	−0.287842
$714$	0	0
$715$	−17.1775	−0.642402
$716$	−18.3388	−0.685354
$717$	0	0
$718$	−0.739243	−0.0275883
$719$	−13.9547	−0.520423	−0.260211	−	0.965552i	$-0.583792\pi$
$719$	−13.9547	−0.520423	−0.260211	+	0.965552i	$0.583792\pi$
$720$	0	0
$721$	−32.5165	−1.21098
$722$	−34.9696	−1.30143
$723$	0	0
$724$	11.4340	0.424943
$725$	−26.6943	−0.991402
$726$	0	0
$727$	3.70803	0.137523	0.0687616	−	0.997633i	$-0.478095\pi$
$727$	3.70803	0.137523	0.0687616	+	0.997633i	$0.478095\pi$
$728$	−4.07226	−0.150928
$729$	0	0
$730$	33.7571	1.24941
$731$	−13.6660	−0.505455
$732$	0	0
$733$	15.1030	0.557842	0.278921	−	0.960314i	$-0.410023\pi$
$733$	15.1030	0.557842	0.278921	+	0.960314i	$0.410023\pi$
$734$	24.3186	0.897617
$735$	0	0
$736$	−4.36415	−0.160865
$737$	1.05683	0.0389288
$738$	0	0
$739$	−46.4878	−1.71008	−0.855041	−	0.518561i	$-0.826468\pi$
$739$	−46.4878	−1.71008	−0.855041	+	0.518561i	$0.826468\pi$
$740$	−41.0280	−1.50822
$741$	0	0
$742$	−28.1313	−1.03273
$743$	34.4107	1.26240	0.631202	−	0.775618i	$-0.282562\pi$
$743$	34.4107	1.26240	0.631202	+	0.775618i	$0.282562\pi$
$744$	0	0
$745$	3.72799	0.136583
$746$	−2.50837	−0.0918381
$747$	0	0
$748$	9.61261	0.351472
$749$	8.04678	0.294023
$750$	0	0
$751$	−18.1491	−0.662269	−0.331135	−	0.943584i	$-0.607431\pi$
$751$	−18.1491	−0.662269	−0.331135	+	0.943584i	$0.607431\pi$
$752$	4.55389	0.166063
$753$	0	0
$754$	−4.55300	−0.165810
$755$	−77.7096	−2.82814
$756$	0	0
$757$	19.0081	0.690861	0.345430	−	0.938444i	$-0.387733\pi$
$757$	19.0081	0.690861	0.345430	+	0.938444i	$0.387733\pi$
$758$	−32.3360	−1.17450
$759$	0	0
$760$	27.3873	0.993441
$761$	0.514324	0.0186442	0.00932212	−	0.999957i	$-0.497033\pi$
$761$	0.514324	0.0186442	0.00932212	+	0.999957i	$0.497033\pi$
$762$	0	0
$763$	28.6053	1.03558
$764$	1.34890	0.0488015
$765$	0	0
$766$	−6.93966	−0.250740
$767$	5.48682	0.198118
$768$	0	0
$769$	−22.5648	−0.813706	−0.406853	−	0.913494i	$-0.633374\pi$
$769$	−22.5648	−0.813706	−0.406853	+	0.913494i	$0.633374\pi$
$770$	30.3714	1.09451
$771$	0	0
$772$	−7.26527	−0.261483
$773$	−22.3684	−0.804534	−0.402267	−	0.915522i	$-0.631778\pi$
$773$	−22.3684	−0.804534	−0.402267	+	0.915522i	$0.631778\pi$
$774$	0	0
$775$	−15.6706	−0.562904
$776$	10.9196	0.391989
$777$	0	0
$778$	−23.9613	−0.859056
$779$	54.0770	1.93751
$780$	0	0
$781$	−5.40124	−0.193272
$782$	−13.8172	−0.494104
$783$	0	0
$784$	0.200143	0.00714795
$785$	−53.8212	−1.92096
$786$	0	0
$787$	−6.81781	−0.243029	−0.121514	−	0.992590i	$-0.538775\pi$
$787$	−6.81781	−0.243029	−0.121514	+	0.992590i	$0.538775\pi$
$788$	4.47541	0.159430
$789$	0	0
$790$	−32.0606	−1.14066
$791$	12.2178	0.434415
$792$	0	0
$793$	2.94372	0.104535
$794$	18.7734	0.666242
$795$	0	0
$796$	18.6122	0.659693
$797$	−18.6309	−0.659941	−0.329970	−	0.943991i	$-0.607039\pi$
$797$	−18.6309	−0.659941	−0.329970	+	0.943991i	$0.607039\pi$
$798$	0	0
$799$	14.4180	0.510071
$800$	−8.89789	−0.314588
$801$	0	0
$802$	37.4767	1.32335
$803$	−27.4923	−0.970182
$804$	0	0
$805$	−43.6561	−1.53868
$806$	−2.67278	−0.0941448
$807$	0	0
$808$	7.12802	0.250763
$809$	34.1985	1.20236	0.601178	−	0.799115i	$-0.294698\pi$
$809$	34.1985	1.20236	0.601178	+	0.799115i	$0.294698\pi$
$810$	0	0
$811$	−12.8426	−0.450965	−0.225482	−	0.974247i	$-0.572396\pi$
$811$	−12.8426	−0.450965	−0.225482	+	0.974247i	$0.572396\pi$
$812$	8.05012	0.282504
$813$	0	0
$814$	33.4138	1.17115
$815$	85.6927	3.00168
$816$	0	0
$817$	31.7099	1.10939
$818$	−7.66375	−0.267957
$819$	0	0
$820$	−27.4418	−0.958309
$821$	−28.4301	−0.992217	−0.496108	−	0.868261i	$-0.665238\pi$
$821$	−28.4301	−0.992217	−0.496108	+	0.868261i	$0.665238\pi$
$822$	0	0
$823$	−12.4052	−0.432418	−0.216209	−	0.976347i	$-0.569369\pi$
$823$	−12.4052	−0.432418	−0.216209	+	0.976347i	$0.569369\pi$
$824$	−12.1181	−0.422153
$825$	0	0
$826$	−9.70122	−0.337549
$827$	30.7034	1.06766	0.533831	−	0.845591i	$-0.320752\pi$
$827$	30.7034	1.06766	0.533831	+	0.845591i	$0.320752\pi$
$828$	0	0
$829$	18.3653	0.637854	0.318927	−	0.947779i	$-0.396678\pi$
$829$	18.3653	0.637854	0.318927	+	0.947779i	$0.396678\pi$
$830$	−0.502983	−0.0174588
$831$	0	0
$832$	−1.51763	−0.0526143
$833$	0.633667	0.0219553
$834$	0	0
$835$	−41.5700	−1.43859
$836$	−22.3046	−0.771421
$837$	0	0
$838$	35.3284	1.22040
$839$	−36.6263	−1.26448	−0.632239	−	0.774773i	$-0.717864\pi$
$839$	−36.6263	−1.26448	−0.632239	+	0.774773i	$0.717864\pi$
$840$	0	0
$841$	−19.9996	−0.689640
$842$	−22.0077	−0.758437
$843$	0	0
$844$	4.66630	0.160621
$845$	−39.8775	−1.37183
$846$	0	0
$847$	4.78146	0.164293
$848$	−10.4838	−0.360016
$849$	0	0
$850$	−28.1714	−0.966270
$851$	−48.0293	−1.64642
$852$	0	0
$853$	16.5200	0.565633	0.282816	−	0.959174i	$-0.408731\pi$
$853$	16.5200	0.565633	0.282816	+	0.959174i	$0.408731\pi$
$854$	−5.20477	−0.178104
$855$	0	0
$856$	2.99883	0.102498
$857$	38.7649	1.32418	0.662092	−	0.749423i	$-0.269669\pi$
$857$	38.7649	1.32418	0.662092	+	0.749423i	$0.269669\pi$
$858$	0	0
$859$	−31.3944	−1.07116	−0.535581	−	0.844484i	$-0.679907\pi$
$859$	−31.3944	−1.07116	−0.535581	+	0.844484i	$0.679907\pi$
$860$	−16.0914	−0.548713
$861$	0	0
$862$	33.2706	1.13320
$863$	−13.2569	−0.451269	−0.225635	−	0.974212i	$-0.572446\pi$
$863$	−13.2569	−0.451269	−0.225635	+	0.974212i	$0.572446\pi$
$864$	0	0
$865$	−36.5100	−1.24138
$866$	33.4569	1.13691
$867$	0	0
$868$	4.72573	0.160402
$869$	26.1106	0.885742
$870$	0	0
$871$	−0.528262	−0.0178995
$872$	10.6605	0.361009
$873$	0	0
$874$	32.0608	1.08447
$875$	−38.9919	−1.31817
$876$	0	0
$877$	28.3016	0.955679	0.477839	−	0.878447i	$-0.341420\pi$
$877$	28.3016	0.955679	0.477839	+	0.878447i	$0.341420\pi$
$878$	14.8073	0.499721
$879$	0	0
$880$	11.3186	0.381551
$881$	−41.8860	−1.41118	−0.705588	−	0.708622i	$-0.749317\pi$
$881$	−41.8860	−1.41118	−0.705588	+	0.708622i	$0.749317\pi$
$882$	0	0
$883$	−42.1014	−1.41682	−0.708412	−	0.705799i	$-0.750588\pi$
$883$	−42.1014	−1.41682	−0.708412	+	0.705799i	$0.750588\pi$
$884$	−4.80493	−0.161607
$885$	0	0
$886$	34.0503	1.14394
$887$	−33.1791	−1.11405	−0.557023	−	0.830497i	$-0.688056\pi$
$887$	−33.1791	−1.11405	−0.557023	+	0.830497i	$0.688056\pi$
$888$	0	0
$889$	−26.4312	−0.886473
$890$	−49.2417	−1.65059
$891$	0	0
$892$	−7.01396	−0.234845
$893$	−33.4547	−1.11952
$894$	0	0
$895$	−68.3669	−2.28525
$896$	2.68331	0.0896430
$897$	0	0
$898$	−12.2168	−0.407680
$899$	5.28361	0.176218
$900$	0	0
$901$	−33.1925	−1.10580
$902$	22.3490	0.744140
$903$	0	0
$904$	4.55326	0.151439
$905$	42.6259	1.41693
$906$	0	0
$907$	−34.3574	−1.14082	−0.570410	−	0.821360i	$-0.693216\pi$
$907$	−34.3574	−1.14082	−0.570410	+	0.821360i	$0.693216\pi$
$908$	28.9841	0.961871
$909$	0	0
$910$	−15.1813	−0.503257
$911$	−37.7454	−1.25056	−0.625280	−	0.780400i	$-0.715015\pi$
$911$	−37.7454	−1.25056	−0.625280	+	0.780400i	$0.715015\pi$
$912$	0	0
$913$	0.409637	0.0135570
$914$	5.68028	0.187887
$915$	0	0
$916$	−1.49235	−0.0493086
$917$	24.3692	0.804743
$918$	0	0
$919$	−49.5954	−1.63600	−0.818000	−	0.575218i	$-0.804917\pi$
$919$	−49.5954	−1.63600	−0.818000	+	0.575218i	$0.804917\pi$
$920$	−16.2695	−0.536390
$921$	0	0
$922$	−2.69247	−0.0886717
$923$	2.69984	0.0888665
$924$	0	0
$925$	−97.9249	−3.21975
$926$	−7.31923	−0.240525
$927$	0	0
$928$	3.00007	0.0984822
$929$	−6.32218	−0.207424	−0.103712	−	0.994607i	$-0.533072\pi$
$929$	−6.32218	−0.207424	−0.103712	+	0.994607i	$0.533072\pi$
$930$	0	0
$931$	−1.47033	−0.0481881
$932$	−16.5298	−0.541453
$933$	0	0
$934$	−41.8088	−1.36803
$935$	35.8357	1.17195
$936$	0	0
$937$	−28.2690	−0.923506	−0.461753	−	0.887008i	$-0.652779\pi$
$937$	−28.2690	−0.923506	−0.461753	+	0.887008i	$0.652779\pi$
$938$	0.934017	0.0304967
$939$	0	0
$940$	16.9769	0.553724
$941$	−4.80946	−0.156784	−0.0783920	−	0.996923i	$-0.524979\pi$
$941$	−4.80946	−0.156784	−0.0783920	+	0.996923i	$0.524979\pi$
$942$	0	0
$943$	−32.1246	−1.04612
$944$	−3.61539	−0.117671
$945$	0	0
$946$	13.1051	0.426083
$947$	21.3831	0.694856	0.347428	−	0.937707i	$-0.387055\pi$
$947$	21.3831	0.694856	0.347428	+	0.937707i	$0.387055\pi$
$948$	0	0
$949$	13.7422	0.446091
$950$	65.3674	2.12080
$951$	0	0
$952$	8.49556	0.275343
$953$	−47.7187	−1.54576	−0.772881	−	0.634551i	$-0.781185\pi$
$953$	−47.7187	−1.54576	−0.772881	+	0.634551i	$0.781185\pi$
$954$	0	0
$955$	5.02868	0.162724
$956$	6.51060	0.210568
$957$	0	0
$958$	24.8975	0.804400
$959$	−32.8379	−1.06039
$960$	0	0
$961$	−27.8983	−0.899946
$962$	−16.7021	−0.538498
$963$	0	0
$964$	−3.15693	−0.101678
$965$	−27.0848	−0.871891
$966$	0	0
$967$	7.82754	0.251717	0.125858	−	0.992048i	$-0.459832\pi$
$967$	7.82754	0.251717	0.125858	+	0.992048i	$0.459832\pi$
$968$	1.78193	0.0572733
$969$	0	0
$970$	40.7080	1.30705
$971$	1.90224	0.0610456	0.0305228	−	0.999534i	$-0.490283\pi$
$971$	1.90224	0.0610456	0.0305228	+	0.999534i	$0.490283\pi$
$972$	0	0
$973$	−19.6050	−0.628506
$974$	−33.5846	−1.07612
$975$	0	0
$976$	−1.93968	−0.0620878
$977$	−35.6734	−1.14129	−0.570646	−	0.821196i	$-0.693307\pi$
$977$	−35.6734	−1.14129	−0.570646	+	0.821196i	$0.693307\pi$
$978$	0	0
$979$	40.1032	1.28170
$980$	0.746129	0.0238342
$981$	0	0
$982$	4.55572	0.145379
$983$	−9.95422	−0.317490	−0.158745	−	0.987320i	$-0.550745\pi$
$983$	−9.95422	−0.317490	−0.158745	+	0.987320i	$0.550745\pi$
$984$	0	0
$985$	16.6843	0.531605
$986$	9.49846	0.302493
$987$	0	0
$988$	11.1491	0.354700
$989$	−18.8374	−0.598993
$990$	0	0
$991$	−10.3371	−0.328370	−0.164185	−	0.986430i	$-0.552499\pi$
$991$	−10.3371	−0.328370	−0.164185	+	0.986430i	$0.552499\pi$
$992$	1.76116	0.0559168
$993$	0	0
$994$	−4.77358	−0.151409
$995$	69.3861	2.19969
$996$	0	0
$997$	−54.6167	−1.72973	−0.864863	−	0.502007i	$-0.832595\pi$
$997$	−54.6167	−1.72973	−0.864863	+	0.502007i	$0.832595\pi$
$998$	−8.32927	−0.263658
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8046.2.a.i.1.12	✓	12
3.2	odd	2		8046.2.a.p.1.1	yes	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8046.2.a.i.1.12	✓	12	1.1	even	1	trivial
8046.2.a.p.1.1	yes	12	3.2	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:	\( N \)	\(=\)	\( 8046 = 2 \cdot 3^{3} \cdot 149 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8046.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} +3.72799 q^{5} -2.68331 q^{7} -1.00000 q^{8} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{4} +3.72799 q^{5} -2.68331 q^{7} -1.00000 q^{8} -3.72799 q^{10} +3.03613 q^{11} -1.51763 q^{13} +2.68331 q^{14} +1.00000 q^{16} +3.16608 q^{17} -7.34640 q^{19} +3.72799 q^{20} -3.03613 q^{22} +4.36415 q^{23} +8.89789 q^{25} +1.51763 q^{26} -2.68331 q^{28} -3.00007 q^{29} -1.76116 q^{31} -1.00000 q^{32} -3.16608 q^{34} -10.0033 q^{35} -11.0054 q^{37} +7.34640 q^{38} -3.72799 q^{40} -7.36102 q^{41} -4.31638 q^{43} +3.03613 q^{44} -4.36415 q^{46} +4.55389 q^{47} +0.200143 q^{49} -8.89789 q^{50} -1.51763 q^{52} -10.4838 q^{53} +11.3186 q^{55} +2.68331 q^{56} +3.00007 q^{58} -3.61539 q^{59} -1.93968 q^{61} +1.76116 q^{62} +1.00000 q^{64} -5.65770 q^{65} +0.348084 q^{67} +3.16608 q^{68} +10.0033 q^{70} -1.77899 q^{71} -9.05505 q^{73} +11.0054 q^{74} -7.34640 q^{76} -8.14687 q^{77} +8.59998 q^{79} +3.72799 q^{80} +7.36102 q^{82} +0.134921 q^{83} +11.8031 q^{85} +4.31638 q^{86} -3.03613 q^{88} +13.2087 q^{89} +4.07226 q^{91} +4.36415 q^{92} -4.55389 q^{94} -27.3873 q^{95} -10.9196 q^{97} -0.200143 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 12 q^{2} + 12 q^{4} - 5 q^{5} + 6 q^{7} - 12 q^{8}+O(q^{10}) \) 12 * q - 12 * q^2 + 12 * q^4 - 5 * q^5 + 6 * q^7 - 12 * q^8	\( 12 q - 12 q^{2} + 12 q^{4} - 5 q^{5} + 6 q^{7} - 12 q^{8} + 5 q^{10} - 6 q^{11} + 3 q^{13} - 6 q^{14} + 12 q^{16} - 6 q^{17} + 8 q^{19} - 5 q^{20} + 6 q^{22} - 11 q^{23} + 11 q^{25} - 3 q^{26} + 6 q^{28} - 29 q^{29} + 2 q^{31} - 12 q^{32} + 6 q^{34} - 4 q^{35} + 5 q^{37} - 8 q^{38} + 5 q^{40} - 22 q^{41} + 9 q^{43} - 6 q^{44} + 11 q^{46} - 15 q^{47} + 14 q^{49} - 11 q^{50} + 3 q^{52} - 12 q^{53} + 13 q^{55} - 6 q^{56} + 29 q^{58} - 34 q^{59} - 4 q^{61} - 2 q^{62} + 12 q^{64} - 12 q^{65} + q^{67} - 6 q^{68} + 4 q^{70} - 21 q^{71} - 2 q^{73} - 5 q^{74} + 8 q^{76} - 34 q^{77} + 9 q^{79} - 5 q^{80} + 22 q^{82} - 10 q^{83} + 5 q^{85} - 9 q^{86} + 6 q^{88} + 2 q^{89} + 17 q^{91} - 11 q^{92} + 15 q^{94} - 69 q^{95} - 13 q^{97} - 14 q^{98}+O(q^{100}) \) 12 * q - 12 * q^2 + 12 * q^4 - 5 * q^5 + 6 * q^7 - 12 * q^8 + 5 * q^10 - 6 * q^11 + 3 * q^13 - 6 * q^14 + 12 * q^16 - 6 * q^17 + 8 * q^19 - 5 * q^20 + 6 * q^22 - 11 * q^23 + 11 * q^25 - 3 * q^26 + 6 * q^28 - 29 * q^29 + 2 * q^31 - 12 * q^32 + 6 * q^34 - 4 * q^35 + 5 * q^37 - 8 * q^38 + 5 * q^40 - 22 * q^41 + 9 * q^43 - 6 * q^44 + 11 * q^46 - 15 * q^47 + 14 * q^49 - 11 * q^50 + 3 * q^52 - 12 * q^53 + 13 * q^55 - 6 * q^56 + 29 * q^58 - 34 * q^59 - 4 * q^61 - 2 * q^62 + 12 * q^64 - 12 * q^65 + q^67 - 6 * q^68 + 4 * q^70 - 21 * q^71 - 2 * q^73 - 5 * q^74 + 8 * q^76 - 34 * q^77 + 9 * q^79 - 5 * q^80 + 22 * q^82 - 10 * q^83 + 5 * q^85 - 9 * q^86 + 6 * q^88 + 2 * q^89 + 17 * q^91 - 11 * q^92 + 15 * q^94 - 69 * q^95 - 13 * q^97 - 14 * q^98

Introduction

L-functions

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Database

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