LMFDB - Embedded newform 8046.2.a.t.1.6

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:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 4 x^{15} - 46 x^{14} + 192 x^{13} + 752 x^{12} - 3378 x^{11} - 5277 x^{10} + 27132 x^{9} + \cdots - 4260 $ x^16 - 4x^15 - 46x^14 + 192x^13 + 752x^12 - 3378x^11 - 5277x^10 + 27132x^9 + 15173x^8 - 102466x^7 - 16612x^6 + 186120x^5 + 117x^4 - 156722x^3 + 10545x^2 + 48580*x - 4260
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$-1.16198$ 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)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} -1.16198 q^{5} +1.20200 q^{7} +1.00000 q^{8} +O(q^{10})$	\(q+1.00000 q^{2} +1.00000 q^{4} -1.16198 q^{5} +1.20200 q^{7} +1.00000 q^{8} -1.16198 q^{10} +6.54441 q^{11} -6.22516 q^{13} +1.20200 q^{14} +1.00000 q^{16} +5.41568 q^{17} +6.55699 q^{19} -1.16198 q^{20} +6.54441 q^{22} -5.20334 q^{23} -3.64980 q^{25} -6.22516 q^{26} +1.20200 q^{28} +7.94963 q^{29} +9.69843 q^{31} +1.00000 q^{32} +5.41568 q^{34} -1.39670 q^{35} -0.452266 q^{37} +6.55699 q^{38} -1.16198 q^{40} -3.44983 q^{41} -4.03734 q^{43} +6.54441 q^{44} -5.20334 q^{46} +7.01004 q^{47} -5.55520 q^{49} -3.64980 q^{50} -6.22516 q^{52} -7.59192 q^{53} -7.60448 q^{55} +1.20200 q^{56} +7.94963 q^{58} -2.60856 q^{59} +10.4221 q^{61} +9.69843 q^{62} +1.00000 q^{64} +7.23352 q^{65} -13.1512 q^{67} +5.41568 q^{68} -1.39670 q^{70} -6.49535 q^{71} +0.0143359 q^{73} -0.452266 q^{74} +6.55699 q^{76} +7.86635 q^{77} +6.36611 q^{79} -1.16198 q^{80} -3.44983 q^{82} -4.94655 q^{83} -6.29293 q^{85} -4.03734 q^{86} +6.54441 q^{88} +0.749144 q^{89} -7.48262 q^{91} -5.20334 q^{92} +7.01004 q^{94} -7.61910 q^{95} -4.71701 q^{97} -5.55520 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 16 q^{2} + 16 q^{4} + 4 q^{5} + 6 q^{7} + 16 q^{8}+O(q^{10}) $ 16 * q + 16 * q^2 + 16 * q^4 + 4 * q^5 + 6 * q^7 + 16 * q^8	$ 16 q + 16 q^{2} + 16 q^{4} + 4 q^{5} + 6 q^{7} + 16 q^{8} + 4 q^{10} + 6 q^{11} + 6 q^{13} + 6 q^{14} + 16 q^{16} + q^{17} + 10 q^{19} + 4 q^{20} + 6 q^{22} + 10 q^{23} + 28 q^{25} + 6 q^{26} + 6 q^{28} + 6 q^{29} + 21 q^{31} + 16 q^{32} + q^{34} + 16 q^{35} + 17 q^{37} + 10 q^{38} + 4 q^{40} - 4 q^{41} + 16 q^{43} + 6 q^{44} + 10 q^{46} + 25 q^{47} + 36 q^{49} + 28 q^{50} + 6 q^{52} + 14 q^{53} + 19 q^{55} + 6 q^{56} + 6 q^{58} + 6 q^{59} + 23 q^{61} + 21 q^{62} + 16 q^{64} + 20 q^{65} + 22 q^{67} + q^{68} + 16 q^{70} + 10 q^{71} + 16 q^{73} + 17 q^{74} + 10 q^{76} - 2 q^{77} + 37 q^{79} + 4 q^{80} - 4 q^{82} + 33 q^{83} + 43 q^{85} + 16 q^{86} + 6 q^{88} - 3 q^{89} + 28 q^{91} + 10 q^{92} + 25 q^{94} + 14 q^{95} - 3 q^{97} + 36 q^{98}+O(q^{100}) $ 16 * q + 16 * q^2 + 16 * q^4 + 4 * q^5 + 6 * q^7 + 16 * q^8 + 4 * q^10 + 6 * q^11 + 6 * q^13 + 6 * q^14 + 16 * q^16 + q^17 + 10 * q^19 + 4 * q^20 + 6 * q^22 + 10 * q^23 + 28 * q^25 + 6 * q^26 + 6 * q^28 + 6 * q^29 + 21 * q^31 + 16 * q^32 + q^34 + 16 * q^35 + 17 * q^37 + 10 * q^38 + 4 * q^40 - 4 * q^41 + 16 * q^43 + 6 * q^44 + 10 * q^46 + 25 * q^47 + 36 * q^49 + 28 * q^50 + 6 * q^52 + 14 * q^53 + 19 * q^55 + 6 * q^56 + 6 * q^58 + 6 * q^59 + 23 * q^61 + 21 * q^62 + 16 * q^64 + 20 * q^65 + 22 * q^67 + q^68 + 16 * q^70 + 10 * q^71 + 16 * q^73 + 17 * q^74 + 10 * q^76 - 2 * q^77 + 37 * q^79 + 4 * q^80 - 4 * q^82 + 33 * q^83 + 43 * q^85 + 16 * q^86 + 6 * q^88 - 3 * q^89 + 28 * q^91 + 10 * q^92 + 25 * q^94 + 14 * q^95 - 3 * q^97 + 36 * 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
$5$	−1.16198	−0.519654	−0.259827	−	0.965655i	$-0.583666\pi$
$5$	−1.16198	−0.519654	−0.259827	+	0.965655i	$0.583666\pi$
$6$	0	0
$7$	1.20200	0.454312	0.227156	−	0.973858i	$-0.427057\pi$
$7$	1.20200	0.454312	0.227156	+	0.973858i	$0.427057\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	−1.16198	−0.367451
$11$	6.54441	1.97321	0.986606	−	0.163120i	$-0.0521559\pi$
$11$	6.54441	1.97321	0.986606	+	0.163120i	$0.0521559\pi$
$12$	0	0
$13$	−6.22516	−1.72655	−0.863274	−	0.504736i	$-0.831590\pi$
$13$	−6.22516	−1.72655	−0.863274	+	0.504736i	$0.831590\pi$
$14$	1.20200	0.321247
$15$	0	0
$16$	1.00000	0.250000
$17$	5.41568	1.31350	0.656748	−	0.754110i	$-0.271931\pi$
$17$	5.41568	1.31350	0.656748	+	0.754110i	$0.271931\pi$
$18$	0	0
$19$	6.55699	1.50428	0.752138	−	0.659005i	$-0.229023\pi$
$19$	6.55699	1.50428	0.752138	+	0.659005i	$0.229023\pi$
$20$	−1.16198	−0.259827
$21$	0	0
$22$	6.54441	1.39527
$23$	−5.20334	−1.08497	−0.542486	−	0.840065i	$-0.682517\pi$
$23$	−5.20334	−1.08497	−0.542486	+	0.840065i	$0.682517\pi$
$24$	0	0
$25$	−3.64980	−0.729959
$26$	−6.22516	−1.22085
$27$	0	0
$28$	1.20200	0.227156
$29$	7.94963	1.47621	0.738104	−	0.674687i	$-0.235721\pi$
$29$	7.94963	1.47621	0.738104	+	0.674687i	$0.235721\pi$
$30$	0	0
$31$	9.69843	1.74189	0.870945	−	0.491380i	$-0.163507\pi$
$31$	9.69843	1.74189	0.870945	+	0.491380i	$0.163507\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	5.41568	0.928782
$35$	−1.39670	−0.236085
$36$	0	0
$37$	−0.452266	−0.0743520	−0.0371760	−	0.999309i	$-0.511836\pi$
$37$	−0.452266	−0.0743520	−0.0371760	+	0.999309i	$0.511836\pi$
$38$	6.55699	1.06368
$39$	0	0
$40$	−1.16198	−0.183726
$41$	−3.44983	−0.538772	−0.269386	−	0.963032i	$-0.586821\pi$
$41$	−3.44983	−0.538772	−0.269386	+	0.963032i	$0.586821\pi$
$42$	0	0
$43$	−4.03734	−0.615688	−0.307844	−	0.951437i	$-0.599608\pi$
$43$	−4.03734	−0.615688	−0.307844	+	0.951437i	$0.599608\pi$
$44$	6.54441	0.986606
$45$	0	0
$46$	−5.20334	−0.767191
$47$	7.01004	1.02252	0.511259	−	0.859426i	$-0.329179\pi$
$47$	7.01004	1.02252	0.511259	+	0.859426i	$0.329179\pi$
$48$	0	0
$49$	−5.55520	−0.793601
$50$	−3.64980	−0.516159
$51$	0	0
$52$	−6.22516	−0.863274
$53$	−7.59192	−1.04283	−0.521415	−	0.853303i	$-0.674596\pi$
$53$	−7.59192	−1.04283	−0.521415	+	0.853303i	$0.674596\pi$
$54$	0	0
$55$	−7.60448	−1.02539
$56$	1.20200	0.160624
$57$	0	0
$58$	7.94963	1.04384
$59$	−2.60856	−0.339606	−0.169803	−	0.985478i	$-0.554313\pi$
$59$	−2.60856	−0.339606	−0.169803	+	0.985478i	$0.554313\pi$
$60$	0	0
$61$	10.4221	1.33442	0.667210	−	0.744870i	$-0.267489\pi$
$61$	10.4221	1.33442	0.667210	+	0.744870i	$0.267489\pi$
$62$	9.69843	1.23170
$63$	0	0
$64$	1.00000	0.125000
$65$	7.23352	0.897208
$66$	0	0
$67$	−13.1512	−1.60668	−0.803338	−	0.595524i	$-0.796945\pi$
$67$	−13.1512	−1.60668	−0.803338	+	0.595524i	$0.796945\pi$
$68$	5.41568	0.656748
$69$	0	0
$70$	−1.39670	−0.166937
$71$	−6.49535	−0.770856	−0.385428	−	0.922738i	$-0.625946\pi$
$71$	−6.49535	−0.770856	−0.385428	+	0.922738i	$0.625946\pi$
$72$	0	0
$73$	0.0143359	0.00167789	0.000838945	−	1.00000i	$-0.499733\pi$
$73$	0.0143359	0.00167789	0.000838945		1.00000i	$0.499733\pi$
$74$	−0.452266	−0.0525748
$75$	0	0
$76$	6.55699	0.752138
$77$	7.86635	0.896454
$78$	0	0
$79$	6.36611	0.716244	0.358122	−	0.933675i	$-0.383417\pi$
$79$	6.36611	0.716244	0.358122	+	0.933675i	$0.383417\pi$
$80$	−1.16198	−0.129914
$81$	0	0
$82$	−3.44983	−0.380970
$83$	−4.94655	−0.542954	−0.271477	−	0.962445i	$-0.587512\pi$
$83$	−4.94655	−0.542954	−0.271477	+	0.962445i	$0.587512\pi$
$84$	0	0
$85$	−6.29293	−0.682564
$86$	−4.03734	−0.435357
$87$	0	0
$88$	6.54441	0.697636
$89$	0.749144	0.0794091	0.0397046	−	0.999211i	$-0.487358\pi$
$89$	0.749144	0.0794091	0.0397046	+	0.999211i	$0.487358\pi$
$90$	0	0
$91$	−7.48262	−0.784391
$92$	−5.20334	−0.542486
$93$	0	0
$94$	7.01004	0.723030
$95$	−7.61910	−0.781704
$96$	0	0
$97$	−4.71701	−0.478940	−0.239470	−	0.970904i	$-0.576974\pi$
$97$	−4.71701	−0.478940	−0.239470	+	0.970904i	$0.576974\pi$
$98$	−5.55520	−0.561160
$99$	0	0
$100$	−3.64980	−0.364980
$101$	11.1377	1.10824	0.554121	−	0.832436i	$-0.313054\pi$
$101$	11.1377	1.10824	0.554121	+	0.832436i	$0.313054\pi$
$102$	0	0
$103$	6.56735	0.647100	0.323550	−	0.946211i	$-0.395124\pi$
$103$	6.56735	0.647100	0.323550	+	0.946211i	$0.395124\pi$
$104$	−6.22516	−0.610427
$105$	0	0
$106$	−7.59192	−0.737393
$107$	4.86501	0.470318	0.235159	−	0.971957i	$-0.424439\pi$
$107$	4.86501	0.470318	0.235159	+	0.971957i	$0.424439\pi$
$108$	0	0
$109$	13.8402	1.32565	0.662826	−	0.748774i	$-0.269357\pi$
$109$	13.8402	1.32565	0.662826	+	0.748774i	$0.269357\pi$
$110$	−7.60448	−0.725059
$111$	0	0
$112$	1.20200	0.113578
$113$	7.75371	0.729408	0.364704	−	0.931124i	$-0.381170\pi$
$113$	7.75371	0.729408	0.364704	+	0.931124i	$0.381170\pi$
$114$	0	0
$115$	6.04619	0.563810
$116$	7.94963	0.738104
$117$	0	0
$118$	−2.60856	−0.240137
$119$	6.50963	0.596737
$120$	0	0
$121$	31.8292	2.89357
$122$	10.4221	0.943577
$123$	0	0
$124$	9.69843	0.870945
$125$	10.0509	0.898981
$126$	0	0
$127$	8.28473	0.735151	0.367575	−	0.929994i	$-0.380188\pi$
$127$	8.28473	0.735151	0.367575	+	0.929994i	$0.380188\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	7.23352	0.634422
$131$	−20.2814	−1.77200	−0.885998	−	0.463689i	$-0.846525\pi$
$131$	−20.2814	−1.77200	−0.885998	+	0.463689i	$0.846525\pi$
$132$	0	0
$133$	7.88148	0.683411
$134$	−13.1512	−1.13609
$135$	0	0
$136$	5.41568	0.464391
$137$	−4.25353	−0.363403	−0.181702	−	0.983354i	$-0.558160\pi$
$137$	−4.25353	−0.363403	−0.181702	+	0.983354i	$0.558160\pi$
$138$	0	0
$139$	−20.9154	−1.77402	−0.887012	−	0.461747i	$-0.847223\pi$
$139$	−20.9154	−1.77402	−0.887012	+	0.461747i	$0.847223\pi$
$140$	−1.39670	−0.118043
$141$	0	0
$142$	−6.49535	−0.545078
$143$	−40.7400	−3.40685
$144$	0	0
$145$	−9.23733	−0.767118
$146$	0.0143359	0.00118645
$147$	0	0
$148$	−0.452266	−0.0371760
$149$	−1.00000	−0.0819232
$149$	−1.00000	−0.0819232
$150$	0	0
$151$	23.0985	1.87973	0.939865	−	0.341545i	$-0.110950\pi$
$151$	23.0985	1.87973	0.939865	+	0.341545i	$0.110950\pi$
$152$	6.55699	0.531842
$153$	0	0
$154$	7.86635	0.633889
$155$	−11.2694	−0.905181
$156$	0	0
$157$	9.88220	0.788685	0.394343	−	0.918963i	$-0.370972\pi$
$157$	9.88220	0.788685	0.394343	+	0.918963i	$0.370972\pi$
$158$	6.36611	0.506461
$159$	0	0
$160$	−1.16198	−0.0918628
$161$	−6.25440	−0.492916
$162$	0	0
$163$	24.9483	1.95410	0.977052	−	0.213003i	$-0.0683245\pi$
$163$	24.9483	1.95410	0.977052	+	0.213003i	$0.0683245\pi$
$164$	−3.44983	−0.269386
$165$	0	0
$166$	−4.94655	−0.383927
$167$	−0.483805	−0.0374379	−0.0187190	−	0.999825i	$-0.505959\pi$
$167$	−0.483805	−0.0374379	−0.0187190	+	0.999825i	$0.505959\pi$
$168$	0	0
$169$	25.7526	1.98097
$170$	−6.29293	−0.482646
$171$	0	0
$172$	−4.03734	−0.307844
$173$	13.8608	1.05382	0.526908	−	0.849922i	$-0.323351\pi$
$173$	13.8608	1.05382	0.526908	+	0.849922i	$0.323351\pi$
$174$	0	0
$175$	−4.38704	−0.331629
$176$	6.54441	0.493303
$177$	0	0
$178$	0.749144	0.0561507
$179$	12.5615	0.938888	0.469444	−	0.882962i	$-0.344454\pi$
$179$	12.5615	0.938888	0.469444	+	0.882962i	$0.344454\pi$
$180$	0	0
$181$	8.32741	0.618972	0.309486	−	0.950904i	$-0.399843\pi$
$181$	8.32741	0.618972	0.309486	+	0.950904i	$0.399843\pi$
$182$	−7.48262	−0.554648
$183$	0	0
$184$	−5.20334	−0.383595
$185$	0.525525	0.0386374
$186$	0	0
$187$	35.4424	2.59181
$188$	7.01004	0.511259
$189$	0	0
$190$	−7.61910	−0.552748
$191$	−23.8583	−1.72633	−0.863164	−	0.504924i	$-0.831520\pi$
$191$	−23.8583	−1.72633	−0.863164	+	0.504924i	$0.831520\pi$
$192$	0	0
$193$	−1.94641	−0.140105	−0.0700527	−	0.997543i	$-0.522317\pi$
$193$	−1.94641	−0.140105	−0.0700527	+	0.997543i	$0.522317\pi$
$194$	−4.71701	−0.338662
$195$	0	0
$196$	−5.55520	−0.396800
$197$	−6.79778	−0.484322	−0.242161	−	0.970236i	$-0.577856\pi$
$197$	−6.79778	−0.484322	−0.242161	+	0.970236i	$0.577856\pi$
$198$	0	0
$199$	−5.86635	−0.415855	−0.207927	−	0.978144i	$-0.566672\pi$
$199$	−5.86635	−0.415855	−0.207927	+	0.978144i	$0.566672\pi$
$200$	−3.64980	−0.258080
$201$	0	0
$202$	11.1377	0.783646
$203$	9.55542	0.670659
$204$	0	0
$205$	4.00864	0.279975
$206$	6.56735	0.457569
$207$	0	0
$208$	−6.22516	−0.431637
$209$	42.9116	2.96826
$210$	0	0
$211$	17.0082	1.17090	0.585448	−	0.810710i	$-0.300919\pi$
$211$	17.0082	1.17090	0.585448	+	0.810710i	$0.300919\pi$
$212$	−7.59192	−0.521415
$213$	0	0
$214$	4.86501	0.332565
$215$	4.69132	0.319945
$216$	0	0
$217$	11.6575	0.791362
$218$	13.8402	0.937377
$219$	0	0
$220$	−7.60448	−0.512694
$221$	−33.7135	−2.26781
$222$	0	0
$223$	−11.0866	−0.742410	−0.371205	−	0.928551i	$-0.621055\pi$
$223$	−11.0866	−0.742410	−0.371205	+	0.928551i	$0.621055\pi$
$224$	1.20200	0.0803118
$225$	0	0
$226$	7.75371	0.515769
$227$	−18.9421	−1.25723	−0.628616	−	0.777716i	$-0.716378\pi$
$227$	−18.9421	−1.25723	−0.628616	+	0.777716i	$0.716378\pi$
$228$	0	0
$229$	−2.84801	−0.188202	−0.0941008	−	0.995563i	$-0.529998\pi$
$229$	−2.84801	−0.188202	−0.0941008	+	0.995563i	$0.529998\pi$
$230$	6.04619	0.398674
$231$	0	0
$232$	7.94963	0.521919
$233$	−9.74322	−0.638300	−0.319150	−	0.947704i	$-0.603397\pi$
$233$	−9.74322	−0.638300	−0.319150	+	0.947704i	$0.603397\pi$
$234$	0	0
$235$	−8.14554	−0.531356
$236$	−2.60856	−0.169803
$237$	0	0
$238$	6.50963	0.421957
$239$	−6.42430	−0.415554	−0.207777	−	0.978176i	$-0.566623\pi$
$239$	−6.42430	−0.415554	−0.207777	+	0.978176i	$0.566623\pi$
$240$	0	0
$241$	−15.0728	−0.970927	−0.485463	−	0.874257i	$-0.661349\pi$
$241$	−15.0728	−0.970927	−0.485463	+	0.874257i	$0.661349\pi$
$242$	31.8292	2.04606
$243$	0	0
$244$	10.4221	0.667210
$245$	6.45505	0.412398
$246$	0	0
$247$	−40.8183	−2.59721
$248$	9.69843	0.615851
$249$	0	0
$250$	10.0509	0.635675
$251$	13.7711	0.869222	0.434611	−	0.900618i	$-0.356886\pi$
$251$	13.7711	0.869222	0.434611	+	0.900618i	$0.356886\pi$
$252$	0	0
$253$	−34.0528	−2.14088
$254$	8.28473	0.519830
$255$	0	0
$256$	1.00000	0.0625000
$257$	20.6170	1.28605	0.643026	−	0.765844i	$-0.277679\pi$
$257$	20.6170	1.28605	0.643026	+	0.765844i	$0.277679\pi$
$258$	0	0
$259$	−0.543622	−0.0337790
$260$	7.23352	0.448604
$261$	0	0
$262$	−20.2814	−1.25299
$263$	−8.91070	−0.549457	−0.274729	−	0.961522i	$-0.588588\pi$
$263$	−8.91070	−0.549457	−0.274729	+	0.961522i	$0.588588\pi$
$264$	0	0
$265$	8.82168	0.541911
$266$	7.88148	0.483244
$267$	0	0
$268$	−13.1512	−0.803338
$269$	−13.6612	−0.832936	−0.416468	−	0.909150i	$-0.636732\pi$
$269$	−13.6612	−0.832936	−0.416468	+	0.909150i	$0.636732\pi$
$270$	0	0
$271$	25.6514	1.55821	0.779104	−	0.626894i	$-0.215674\pi$
$271$	25.6514	1.55821	0.779104	+	0.626894i	$0.215674\pi$
$272$	5.41568	0.328374
$273$	0	0
$274$	−4.25353	−0.256965
$275$	−23.8858	−1.44036
$276$	0	0
$277$	7.54881	0.453564	0.226782	−	0.973946i	$-0.427179\pi$
$277$	7.54881	0.453564	0.226782	+	0.973946i	$0.427179\pi$
$278$	−20.9154	−1.25442
$279$	0	0
$280$	−1.39670	−0.0834687
$281$	7.26882	0.433621	0.216811	−	0.976214i	$-0.430435\pi$
$281$	7.26882	0.433621	0.216811	+	0.976214i	$0.430435\pi$
$282$	0	0
$283$	4.41231	0.262285	0.131142	−	0.991364i	$-0.458135\pi$
$283$	4.41231	0.262285	0.131142	+	0.991364i	$0.458135\pi$
$284$	−6.49535	−0.385428
$285$	0	0
$286$	−40.7400	−2.40900
$287$	−4.14668	−0.244771
$288$	0	0
$289$	12.3296	0.725272
$290$	−9.23733	−0.542434
$291$	0	0
$292$	0.0143359	0.000838945 0
$293$	−29.8166	−1.74190	−0.870952	−	0.491368i	$-0.836497\pi$
$293$	−29.8166	−1.74190	−0.870952	+	0.491368i	$0.836497\pi$
$294$	0	0
$295$	3.03110	0.176478
$296$	−0.452266	−0.0262874
$297$	0	0
$298$	−1.00000	−0.0579284
$299$	32.3916	1.87326
$300$	0	0
$301$	−4.85287	−0.279715
$302$	23.0985	1.32917
$303$	0	0
$304$	6.55699	0.376069
$305$	−12.1103	−0.693437
$306$	0	0
$307$	−2.43966	−0.139239	−0.0696194	−	0.997574i	$-0.522178\pi$
$307$	−2.43966	−0.139239	−0.0696194	+	0.997574i	$0.522178\pi$
$308$	7.86635	0.448227
$309$	0	0
$310$	−11.2694	−0.640059
$311$	4.86640	0.275948	0.137974	−	0.990436i	$-0.455941\pi$
$311$	4.86640	0.275948	0.137974	+	0.990436i	$0.455941\pi$
$312$	0	0
$313$	−27.9292	−1.57865	−0.789325	−	0.613976i	$-0.789569\pi$
$313$	−27.9292	−1.57865	−0.789325	+	0.613976i	$0.789569\pi$
$314$	9.88220	0.557685
$315$	0	0
$316$	6.36611	0.358122
$317$	34.3555	1.92960	0.964799	−	0.262989i	$-0.0847082\pi$
$317$	34.3555	1.92960	0.964799	+	0.262989i	$0.0847082\pi$
$318$	0	0
$319$	52.0256	2.91287
$320$	−1.16198	−0.0649568
$321$	0	0
$322$	−6.25440	−0.348544
$323$	35.5106	1.97586
$324$	0	0
$325$	22.7206	1.26031
$326$	24.9483	1.38176
$327$	0	0
$328$	−3.44983	−0.190485
$329$	8.42604	0.464543
$330$	0	0
$331$	−35.3008	−1.94031	−0.970155	−	0.242487i	$-0.922037\pi$
$331$	−35.3008	−1.94031	−0.970155	+	0.242487i	$0.922037\pi$
$332$	−4.94655	−0.271477
$333$	0	0
$334$	−0.483805	−0.0264726
$335$	15.2815	0.834916
$336$	0	0
$337$	17.2600	0.940211	0.470105	−	0.882610i	$-0.344216\pi$
$337$	17.2600	0.940211	0.470105	+	0.882610i	$0.344216\pi$
$338$	25.7526	1.40076
$339$	0	0
$340$	−6.29293	−0.341282
$341$	63.4705	3.43712
$342$	0	0
$343$	−15.0913	−0.814854
$344$	−4.03734	−0.217679
$345$	0	0
$346$	13.8608	0.745160
$347$	16.3930	0.880021	0.440011	−	0.897993i	$-0.354975\pi$
$347$	16.3930	0.880021	0.440011	+	0.897993i	$0.354975\pi$
$348$	0	0
$349$	11.1877	0.598865	0.299433	−	0.954117i	$-0.403203\pi$
$349$	11.1877	0.598865	0.299433	+	0.954117i	$0.403203\pi$
$350$	−4.38704	−0.234497
$351$	0	0
$352$	6.54441	0.348818
$353$	2.97652	0.158424	0.0792121	−	0.996858i	$-0.474760\pi$
$353$	2.97652	0.158424	0.0792121	+	0.996858i	$0.474760\pi$
$354$	0	0
$355$	7.54748	0.400579
$356$	0.749144	0.0397046
$357$	0	0
$358$	12.5615	0.663894
$359$	12.4611	0.657670	0.328835	−	0.944387i	$-0.393344\pi$
$359$	12.4611	0.657670	0.328835	+	0.944387i	$0.393344\pi$
$360$	0	0
$361$	23.9941	1.26285
$362$	8.32741	0.437679
$363$	0	0
$364$	−7.48262	−0.392196
$365$	−0.0166581	−0.000871923 0
$366$	0	0
$367$	17.7698	0.927576	0.463788	−	0.885946i	$-0.346490\pi$
$367$	17.7698	0.927576	0.463788	+	0.885946i	$0.346490\pi$
$368$	−5.20334	−0.271243
$369$	0	0
$370$	0.525525	0.0273207
$371$	−9.12546	−0.473770
$372$	0	0
$373$	8.84727	0.458094	0.229047	−	0.973415i	$-0.426439\pi$
$373$	8.84727	0.458094	0.229047	+	0.973415i	$0.426439\pi$
$374$	35.4424	1.83268
$375$	0	0
$376$	7.01004	0.361515
$377$	−49.4877	−2.54875
$378$	0	0
$379$	1.97758	0.101582	0.0507908	−	0.998709i	$-0.483826\pi$
$379$	1.97758	0.101582	0.0507908	+	0.998709i	$0.483826\pi$
$380$	−7.61910	−0.390852
$381$	0	0
$382$	−23.8583	−1.22070
$383$	−26.7331	−1.36600	−0.683000	−	0.730418i	$-0.739325\pi$
$383$	−26.7331	−1.36600	−0.683000	+	0.730418i	$0.739325\pi$
$384$	0	0
$385$	−9.14056	−0.465846
$386$	−1.94641	−0.0990695
$387$	0	0
$388$	−4.71701	−0.239470
$389$	16.3202	0.827465	0.413732	−	0.910399i	$-0.364225\pi$
$389$	16.3202	0.827465	0.413732	+	0.910399i	$0.364225\pi$
$390$	0	0
$391$	−28.1796	−1.42511
$392$	−5.55520	−0.280580
$393$	0	0
$394$	−6.79778	−0.342467
$395$	−7.39731	−0.372199
$396$	0	0
$397$	18.2921	0.918054	0.459027	−	0.888422i	$-0.348198\pi$
$397$	18.2921	0.918054	0.459027	+	0.888422i	$0.348198\pi$
$398$	−5.86635	−0.294054
$399$	0	0
$400$	−3.64980	−0.182490
$401$	−5.80074	−0.289675	−0.144837	−	0.989455i	$-0.546266\pi$
$401$	−5.80074	−0.289675	−0.144837	+	0.989455i	$0.546266\pi$
$402$	0	0
$403$	−60.3743	−3.00746
$404$	11.1377	0.554121
$405$	0	0
$406$	9.55542	0.474228
$407$	−2.95981	−0.146712
$408$	0	0
$409$	21.2740	1.05193	0.525967	−	0.850505i	$-0.323704\pi$
$409$	21.2740	1.05193	0.525967	+	0.850505i	$0.323704\pi$
$410$	4.00864	0.197973
$411$	0	0
$412$	6.56735	0.323550
$413$	−3.13548	−0.154287
$414$	0	0
$415$	5.74780	0.282148
$416$	−6.22516	−0.305213
$417$	0	0
$418$	42.9116	2.09887
$419$	−26.7929	−1.30892	−0.654460	−	0.756097i	$-0.727104\pi$
$419$	−26.7929	−1.30892	−0.654460	+	0.756097i	$0.727104\pi$
$420$	0	0
$421$	−21.6344	−1.05440	−0.527198	−	0.849742i	$-0.676757\pi$
$421$	−21.6344	−1.05440	−0.527198	+	0.849742i	$0.676757\pi$
$422$	17.0082	0.827948
$423$	0	0
$424$	−7.59192	−0.368696
$425$	−19.7661	−0.958799
$426$	0	0
$427$	12.5274	0.606243
$428$	4.86501	0.235159
$429$	0	0
$430$	4.69132	0.226235
$431$	13.1356	0.632722	0.316361	−	0.948639i	$-0.397539\pi$
$431$	13.1356	0.632722	0.316361	+	0.948639i	$0.397539\pi$
$432$	0	0
$433$	−18.1794	−0.873645	−0.436823	−	0.899548i	$-0.643896\pi$
$433$	−18.1794	−0.873645	−0.436823	+	0.899548i	$0.643896\pi$
$434$	11.6575	0.559577
$435$	0	0
$436$	13.8402	0.662826
$437$	−34.1182	−1.63210
$438$	0	0
$439$	20.8456	0.994905	0.497452	−	0.867491i	$-0.334269\pi$
$439$	20.8456	0.994905	0.497452	+	0.867491i	$0.334269\pi$
$440$	−7.60448	−0.362530
$441$	0	0
$442$	−33.7135	−1.60359
$443$	−5.09801	−0.242214	−0.121107	−	0.992639i	$-0.538644\pi$
$443$	−5.09801	−0.242214	−0.121107	+	0.992639i	$0.538644\pi$
$444$	0	0
$445$	−0.870492	−0.0412653
$446$	−11.0866	−0.524963
$447$	0	0
$448$	1.20200	0.0567890
$449$	8.13634	0.383978	0.191989	−	0.981397i	$-0.438506\pi$
$449$	8.13634	0.383978	0.191989	+	0.981397i	$0.438506\pi$
$450$	0	0
$451$	−22.5771	−1.06311
$452$	7.75371	0.364704
$453$	0	0
$454$	−18.9421	−0.888997
$455$	8.69467	0.407612
$456$	0	0
$457$	9.54401	0.446450	0.223225	−	0.974767i	$-0.428342\pi$
$457$	9.54401	0.446450	0.223225	+	0.974767i	$0.428342\pi$
$458$	−2.84801	−0.133079
$459$	0	0
$460$	6.04619	0.281905
$461$	−39.9662	−1.86141	−0.930706	−	0.365767i	$-0.880807\pi$
$461$	−39.9662	−1.86141	−0.930706	+	0.365767i	$0.880807\pi$
$462$	0	0
$463$	−16.8056	−0.781021	−0.390510	−	0.920598i	$-0.627701\pi$
$463$	−16.8056	−0.781021	−0.390510	+	0.920598i	$0.627701\pi$
$464$	7.94963	0.369052
$465$	0	0
$466$	−9.74322	−0.451346
$467$	39.1289	1.81067	0.905334	−	0.424700i	$-0.139620\pi$
$467$	39.1289	1.81067	0.905334	+	0.424700i	$0.139620\pi$
$468$	0	0
$469$	−15.8077	−0.729932
$470$	−8.14554	−0.375726
$471$	0	0
$472$	−2.60856	−0.120069
$473$	−26.4220	−1.21488
$474$	0	0
$475$	−23.9317	−1.09806
$476$	6.50963	0.298368
$477$	0	0
$478$	−6.42430	−0.293841
$479$	−32.3479	−1.47801	−0.739006	−	0.673698i	$-0.764705\pi$
$479$	−32.3479	−1.47801	−0.739006	+	0.673698i	$0.764705\pi$
$480$	0	0
$481$	2.81543	0.128372
$482$	−15.0728	−0.686549
$483$	0	0
$484$	31.8292	1.44678
$485$	5.48108	0.248883
$486$	0	0
$487$	12.4954	0.566222	0.283111	−	0.959087i	$-0.408634\pi$
$487$	12.4954	0.566222	0.283111	+	0.959087i	$0.408634\pi$
$488$	10.4221	0.471788
$489$	0	0
$490$	6.45505	0.291609
$491$	28.5946	1.29045	0.645227	−	0.763990i	$-0.276763\pi$
$491$	28.5946	1.29045	0.645227	+	0.763990i	$0.276763\pi$
$492$	0	0
$493$	43.0527	1.93899
$494$	−40.8183	−1.83650
$495$	0	0
$496$	9.69843	0.435473
$497$	−7.80739	−0.350209
$498$	0	0
$499$	22.1731	0.992602	0.496301	−	0.868150i	$-0.334691\pi$
$499$	22.1731	0.992602	0.496301	+	0.868150i	$0.334691\pi$
$500$	10.0509	0.449490
$501$	0	0
$502$	13.7711	0.614633
$503$	2.91304	0.129886	0.0649430	−	0.997889i	$-0.479313\pi$
$503$	2.91304	0.129886	0.0649430	+	0.997889i	$0.479313\pi$
$504$	0	0
$505$	−12.9418	−0.575903
$506$	−34.0528	−1.51383
$507$	0	0
$508$	8.28473	0.367575
$509$	−6.45412	−0.286074	−0.143037	−	0.989717i	$-0.545687\pi$
$509$	−6.45412	−0.286074	−0.143037	+	0.989717i	$0.545687\pi$
$510$	0	0
$511$	0.0172317	0.000762286 0
$512$	1.00000	0.0441942
$513$	0	0
$514$	20.6170	0.909377
$515$	−7.63115	−0.336269
$516$	0	0
$517$	45.8765	2.01765
$518$	−0.543622	−0.0238854
$519$	0	0
$520$	7.23352	0.317211
$521$	19.7036	0.863232	0.431616	−	0.902058i	$-0.357944\pi$
$521$	19.7036	0.863232	0.431616	+	0.902058i	$0.357944\pi$
$522$	0	0
$523$	10.5604	0.461775	0.230888	−	0.972980i	$-0.425837\pi$
$523$	10.5604	0.461775	0.230888	+	0.972980i	$0.425837\pi$
$524$	−20.2814	−0.885998
$525$	0	0
$526$	−8.91070	−0.388525
$527$	52.5237	2.28797
$528$	0	0
$529$	4.07476	0.177163
$530$	8.82168	0.383189
$531$	0	0
$532$	7.88148	0.341705
$533$	21.4757	0.930216
$534$	0	0
$535$	−5.65305	−0.244403
$536$	−13.1512	−0.568046
$537$	0	0
$538$	−13.6612	−0.588975
$539$	−36.3555	−1.56594
$540$	0	0
$541$	−4.89515	−0.210459	−0.105230	−	0.994448i	$-0.533558\pi$
$541$	−4.89515	−0.210459	−0.105230	+	0.994448i	$0.533558\pi$
$542$	25.6514	1.10182
$543$	0	0
$544$	5.41568	0.232196
$545$	−16.0821	−0.688880
$546$	0	0
$547$	−25.6608	−1.09718	−0.548589	−	0.836092i	$-0.684835\pi$
$547$	−25.6608	−1.09718	−0.548589	+	0.836092i	$0.684835\pi$
$548$	−4.25353	−0.181702
$549$	0	0
$550$	−23.8858	−1.01849
$551$	52.1256	2.22063
$552$	0	0
$553$	7.65205	0.325398
$554$	7.54881	0.320718
$555$	0	0
$556$	−20.9154	−0.887012
$557$	−9.48044	−0.401699	−0.200850	−	0.979622i	$-0.564370\pi$
$557$	−9.48044	−0.401699	−0.200850	+	0.979622i	$0.564370\pi$
$558$	0	0
$559$	25.1331	1.06302
$560$	−1.39670	−0.0590213
$561$	0	0
$562$	7.26882	0.306616
$563$	−29.0665	−1.22501	−0.612504	−	0.790468i	$-0.709838\pi$
$563$	−29.0665	−1.22501	−0.612504	+	0.790468i	$0.709838\pi$
$564$	0	0
$565$	−9.00967	−0.379040
$566$	4.41231	0.185463
$567$	0	0
$568$	−6.49535	−0.272539
$569$	−10.0108	−0.419673	−0.209837	−	0.977736i	$-0.567293\pi$
$569$	−10.0108	−0.419673	−0.209837	+	0.977736i	$0.567293\pi$
$570$	0	0
$571$	−6.37514	−0.266791	−0.133396	−	0.991063i	$-0.542588\pi$
$571$	−6.37514	−0.266791	−0.133396	+	0.991063i	$0.542588\pi$
$572$	−40.7400	−1.70342
$573$	0	0
$574$	−4.14668	−0.173079
$575$	18.9911	0.791985
$576$	0	0
$577$	−3.07946	−0.128200	−0.0640998	−	0.997943i	$-0.520418\pi$
$577$	−3.07946	−0.128200	−0.0640998	+	0.997943i	$0.520418\pi$
$578$	12.3296	0.512845
$579$	0	0
$580$	−9.23733	−0.383559
$581$	−5.94573	−0.246671
$582$	0	0
$583$	−49.6846	−2.05773
$584$	0.0143359	0.000593224 0
$585$	0	0
$586$	−29.8166	−1.23171
$587$	−4.37886	−0.180735	−0.0903675	−	0.995908i	$-0.528804\pi$
$587$	−4.37886	−0.180735	−0.0903675	+	0.995908i	$0.528804\pi$
$588$	0	0
$589$	63.5925	2.62028
$590$	3.03110	0.124788
$591$	0	0
$592$	−0.452266	−0.0185880
$593$	−43.8585	−1.80105	−0.900526	−	0.434803i	$-0.856818\pi$
$593$	−43.8585	−1.80105	−0.900526	+	0.434803i	$0.856818\pi$
$594$	0	0
$595$	−7.56408	−0.310097
$596$	−1.00000	−0.0409616
$597$	0	0
$598$	32.3916	1.32459
$599$	18.3984	0.751738	0.375869	−	0.926673i	$-0.377344\pi$
$599$	18.3984	0.751738	0.375869	+	0.926673i	$0.377344\pi$
$600$	0	0
$601$	−12.4263	−0.506878	−0.253439	−	0.967351i	$-0.581562\pi$
$601$	−12.4263	−0.506878	−0.253439	+	0.967351i	$0.581562\pi$
$602$	−4.85287	−0.197788
$603$	0	0
$604$	23.0985	0.939865
$605$	−36.9850	−1.50365
$606$	0	0
$607$	21.6727	0.879667	0.439833	−	0.898079i	$-0.355038\pi$
$607$	21.6727	0.879667	0.439833	+	0.898079i	$0.355038\pi$
$608$	6.55699	0.265921
$609$	0	0
$610$	−12.1103	−0.490334
$611$	−43.6386	−1.76543
$612$	0	0
$613$	5.21402	0.210592	0.105296	−	0.994441i	$-0.466421\pi$
$613$	5.21402	0.210592	0.105296	+	0.994441i	$0.466421\pi$
$614$	−2.43966	−0.0984567
$615$	0	0
$616$	7.86635	0.316944
$617$	2.69475	0.108487	0.0542433	−	0.998528i	$-0.482725\pi$
$617$	2.69475	0.108487	0.0542433	+	0.998528i	$0.482725\pi$
$618$	0	0
$619$	22.7411	0.914042	0.457021	−	0.889456i	$-0.348916\pi$
$619$	22.7411	0.914042	0.457021	+	0.889456i	$0.348916\pi$
$620$	−11.2694	−0.452590
$621$	0	0
$622$	4.86640	0.195125
$623$	0.900469	0.0360765
$624$	0	0
$625$	6.57000	0.262800
$626$	−27.9292	−1.11627
$627$	0	0
$628$	9.88220	0.394343
$629$	−2.44933	−0.0976611
$630$	0	0
$631$	7.09179	0.282320	0.141160	−	0.989987i	$-0.454917\pi$
$631$	7.09179	0.282320	0.141160	+	0.989987i	$0.454917\pi$
$632$	6.36611	0.253230
$633$	0	0
$634$	34.3555	1.36443
$635$	−9.62671	−0.382024
$636$	0	0
$637$	34.5820	1.37019
$638$	52.0256	2.05971
$639$	0	0
$640$	−1.16198	−0.0459314
$641$	−36.9150	−1.45805	−0.729027	−	0.684485i	$-0.760027\pi$
$641$	−36.9150	−1.45805	−0.729027	+	0.684485i	$0.760027\pi$
$642$	0	0
$643$	−20.1802	−0.795830	−0.397915	−	0.917422i	$-0.630266\pi$
$643$	−20.1802	−0.795830	−0.397915	+	0.917422i	$0.630266\pi$
$644$	−6.25440	−0.246458
$645$	0	0
$646$	35.5106	1.39714
$647$	−46.3337	−1.82157	−0.910783	−	0.412885i	$-0.864521\pi$
$647$	−46.3337	−1.82157	−0.910783	+	0.412885i	$0.864521\pi$
$648$	0	0
$649$	−17.0715	−0.670114
$650$	22.7206	0.891174
$651$	0	0
$652$	24.9483	0.977052
$653$	20.0494	0.784593	0.392296	−	0.919839i	$-0.371681\pi$
$653$	20.0494	0.784593	0.392296	+	0.919839i	$0.371681\pi$
$654$	0	0
$655$	23.5666	0.920825
$656$	−3.44983	−0.134693
$657$	0	0
$658$	8.42604	0.328481
$659$	−1.65080	−0.0643060	−0.0321530	−	0.999483i	$-0.510236\pi$
$659$	−1.65080	−0.0643060	−0.0321530	+	0.999483i	$0.510236\pi$
$660$	0	0
$661$	21.6939	0.843796	0.421898	−	0.906643i	$-0.361364\pi$
$661$	21.6939	0.843796	0.421898	+	0.906643i	$0.361364\pi$
$662$	−35.3008	−1.37201
$663$	0	0
$664$	−4.94655	−0.191963
$665$	−9.15814	−0.355137
$666$	0	0
$667$	−41.3646	−1.60164
$668$	−0.483805	−0.0187190
$669$	0	0
$670$	15.2815	0.590375
$671$	68.2067	2.63309
$672$	0	0
$673$	21.9200	0.844954	0.422477	−	0.906374i	$-0.361161\pi$
$673$	21.9200	0.844954	0.422477	+	0.906374i	$0.361161\pi$
$674$	17.2600	0.664829
$675$	0	0
$676$	25.7526	0.990484
$677$	−16.8178	−0.646360	−0.323180	−	0.946338i	$-0.604752\pi$
$677$	−16.8178	−0.646360	−0.323180	+	0.946338i	$0.604752\pi$
$678$	0	0
$679$	−5.66983	−0.217588
$680$	−6.29293	−0.241323
$681$	0	0
$682$	63.4705	2.43041
$683$	5.16341	0.197572	0.0987862	−	0.995109i	$-0.468504\pi$
$683$	5.16341	0.197572	0.0987862	+	0.995109i	$0.468504\pi$
$684$	0	0
$685$	4.94252	0.188844
$686$	−15.0913	−0.576189
$687$	0	0
$688$	−4.03734	−0.153922
$689$	47.2609	1.80050
$690$	0	0
$691$	−5.13851	−0.195478	−0.0977391	−	0.995212i	$-0.531161\pi$
$691$	−5.13851	−0.195478	−0.0977391	+	0.995212i	$0.531161\pi$
$692$	13.8608	0.526908
$693$	0	0
$694$	16.3930	0.622269
$695$	24.3034	0.921879
$696$	0	0
$697$	−18.6832	−0.707676
$698$	11.1877	0.423462
$699$	0	0
$700$	−4.38704	−0.165815
$701$	0.572198	0.0216116	0.0108058	−	0.999942i	$-0.496560\pi$
$701$	0.572198	0.0216116	0.0108058	+	0.999942i	$0.496560\pi$
$702$	0	0
$703$	−2.96550	−0.111846
$704$	6.54441	0.246652
$705$	0	0
$706$	2.97652	0.112023
$707$	13.3875	0.503488
$708$	0	0
$709$	21.5520	0.809401	0.404700	−	0.914449i	$-0.367376\pi$
$709$	21.5520	0.809401	0.404700	+	0.914449i	$0.367376\pi$
$710$	7.54748	0.283252
$711$	0	0
$712$	0.749144	0.0280754
$713$	−50.4643	−1.88990
$714$	0	0
$715$	47.3391	1.77038
$716$	12.5615	0.469444
$717$	0	0
$718$	12.4611	0.465043
$719$	20.4402	0.762290	0.381145	−	0.924515i	$-0.375530\pi$
$719$	20.4402	0.762290	0.381145	+	0.924515i	$0.375530\pi$
$720$	0	0
$721$	7.89393	0.293985
$722$	23.9941	0.892968
$723$	0	0
$724$	8.32741	0.309486
$725$	−29.0145	−1.07757
$726$	0	0
$727$	27.2348	1.01008	0.505041	−	0.863095i	$-0.331477\pi$
$727$	27.2348	1.01008	0.505041	+	0.863095i	$0.331477\pi$
$728$	−7.48262	−0.277324
$729$	0	0
$730$	−0.0166581	−0.000616543 0
$731$	−21.8649	−0.808704
$732$	0	0
$733$	51.5419	1.90374	0.951872	−	0.306496i	$-0.0991567\pi$
$733$	51.5419	1.90374	0.951872	+	0.306496i	$0.0991567\pi$
$734$	17.7698	0.655896
$735$	0	0
$736$	−5.20334	−0.191798
$737$	−86.0668	−3.17031
$738$	0	0
$739$	−31.6187	−1.16311	−0.581556	−	0.813506i	$-0.697556\pi$
$739$	−31.6187	−1.16311	−0.581556	+	0.813506i	$0.697556\pi$
$740$	0.525525	0.0193187
$741$	0	0
$742$	−9.12546	−0.335006
$743$	−14.9299	−0.547726	−0.273863	−	0.961769i	$-0.588301\pi$
$743$	−14.9299	−0.547726	−0.273863	+	0.961769i	$0.588301\pi$
$744$	0	0
$745$	1.16198	0.0425717
$746$	8.84727	0.323921
$747$	0	0
$748$	35.4424	1.29590
$749$	5.84772	0.213671
$750$	0	0
$751$	−48.9190	−1.78508	−0.892540	−	0.450969i	$-0.851078\pi$
$751$	−48.9190	−1.78508	−0.892540	+	0.450969i	$0.851078\pi$
$752$	7.01004	0.255630
$753$	0	0
$754$	−49.4877	−1.80223
$755$	−26.8401	−0.976810
$756$	0	0
$757$	−13.9746	−0.507916	−0.253958	−	0.967215i	$-0.581732\pi$
$757$	−13.9746	−0.507916	−0.253958	+	0.967215i	$0.581732\pi$
$758$	1.97758	0.0718290
$759$	0	0
$760$	−7.61910	−0.276374
$761$	17.7366	0.642950	0.321475	−	0.946918i	$-0.395821\pi$
$761$	17.7366	0.642950	0.321475	+	0.946918i	$0.395821\pi$
$762$	0	0
$763$	16.6359	0.602259
$764$	−23.8583	−0.863164
$765$	0	0
$766$	−26.7331	−0.965908
$767$	16.2387	0.586345
$768$	0	0
$769$	36.4891	1.31583	0.657916	−	0.753091i	$-0.271438\pi$
$769$	36.4891	1.31583	0.657916	+	0.753091i	$0.271438\pi$
$770$	−9.14056	−0.329403
$771$	0	0
$772$	−1.94641	−0.0700527
$773$	0.753106	0.0270873	0.0135437	−	0.999908i	$-0.495689\pi$
$773$	0.753106	0.0270873	0.0135437	+	0.999908i	$0.495689\pi$
$774$	0	0
$775$	−35.3973	−1.27151
$776$	−4.71701	−0.169331
$777$	0	0
$778$	16.3202	0.585106
$779$	−22.6205	−0.810463
$780$	0	0
$781$	−42.5082	−1.52106
$782$	−28.1796	−1.00770
$783$	0	0
$784$	−5.55520	−0.198400
$785$	−11.4829	−0.409844
$786$	0	0
$787$	−17.0619	−0.608191	−0.304095	−	0.952642i	$-0.598354\pi$
$787$	−17.0619	−0.608191	−0.304095	+	0.952642i	$0.598354\pi$
$788$	−6.79778	−0.242161
$789$	0	0
$790$	−7.39731	−0.263185
$791$	9.31993	0.331379
$792$	0	0
$793$	−64.8795	−2.30394
$794$	18.2921	0.649162
$795$	0	0
$796$	−5.86635	−0.207927
$797$	−5.43382	−0.192476	−0.0962379	−	0.995358i	$-0.530681\pi$
$797$	−5.43382	−0.192476	−0.0962379	+	0.995358i	$0.530681\pi$
$798$	0	0
$799$	37.9641	1.34307
$800$	−3.64980	−0.129040
$801$	0	0
$802$	−5.80074	−0.204831
$803$	0.0938199	0.00331083
$804$	0	0
$805$	7.26750	0.256146
$806$	−60.3743	−2.12659
$807$	0	0
$808$	11.1377	0.391823
$809$	−50.4104	−1.77234	−0.886168	−	0.463364i	$-0.846642\pi$
$809$	−50.4104	−1.77234	−0.886168	+	0.463364i	$0.846642\pi$
$810$	0	0
$811$	−40.9799	−1.43900	−0.719500	−	0.694492i	$-0.755629\pi$
$811$	−40.9799	−1.43900	−0.719500	+	0.694492i	$0.755629\pi$
$812$	9.55542	0.335330
$813$	0	0
$814$	−2.95981	−0.103741
$815$	−28.9895	−1.01546
$816$	0	0
$817$	−26.4728	−0.926165
$818$	21.2740	0.743829
$819$	0	0
$820$	4.00864	0.139988
$821$	13.7785	0.480872	0.240436	−	0.970665i	$-0.422710\pi$
$821$	13.7785	0.480872	0.240436	+	0.970665i	$0.422710\pi$
$822$	0	0
$823$	−32.3601	−1.12800	−0.564001	−	0.825774i	$-0.690739\pi$
$823$	−32.3601	−1.12800	−0.564001	+	0.825774i	$0.690739\pi$
$824$	6.56735	0.228785
$825$	0	0
$826$	−3.13548	−0.109097
$827$	32.3551	1.12510	0.562549	−	0.826764i	$-0.309821\pi$
$827$	32.3551	1.12510	0.562549	+	0.826764i	$0.309821\pi$
$828$	0	0
$829$	−16.5876	−0.576112	−0.288056	−	0.957614i	$-0.593009\pi$
$829$	−16.5876	−0.576112	−0.288056	+	0.957614i	$0.593009\pi$
$830$	5.74780	0.199509
$831$	0	0
$832$	−6.22516	−0.215819
$833$	−30.0852	−1.04239
$834$	0	0
$835$	0.562173	0.0194548
$836$	42.9116	1.48413
$837$	0	0
$838$	−26.7929	−0.925546
$839$	39.6490	1.36884	0.684418	−	0.729090i	$-0.260056\pi$
$839$	39.6490	1.36884	0.684418	+	0.729090i	$0.260056\pi$
$840$	0	0
$841$	34.1966	1.17919
$842$	−21.6344	−0.745571
$843$	0	0
$844$	17.0082	0.585448
$845$	−29.9240	−1.02942
$846$	0	0
$847$	38.2586	1.31458
$848$	−7.59192	−0.260708
$849$	0	0
$850$	−19.7661	−0.677973
$851$	2.35329	0.0806698
$852$	0	0
$853$	−46.4302	−1.58974	−0.794870	−	0.606779i	$-0.792461\pi$
$853$	−46.4302	−1.58974	−0.794870	+	0.606779i	$0.792461\pi$
$854$	12.5274	0.428678
$855$	0	0
$856$	4.86501	0.166282
$857$	−32.6257	−1.11447	−0.557236	−	0.830354i	$-0.688138\pi$
$857$	−32.6257	−1.11447	−0.557236	+	0.830354i	$0.688138\pi$
$858$	0	0
$859$	−40.8344	−1.39325	−0.696626	−	0.717435i	$-0.745316\pi$
$859$	−40.8344	−1.39325	−0.696626	+	0.717435i	$0.745316\pi$
$860$	4.69132	0.159973
$861$	0	0
$862$	13.1356	0.447402
$863$	23.5119	0.800356	0.400178	−	0.916437i	$-0.368948\pi$
$863$	23.5119	0.800356	0.400178	+	0.916437i	$0.368948\pi$
$864$	0	0
$865$	−16.1060	−0.547620
$866$	−18.1794	−0.617761
$867$	0	0
$868$	11.6575	0.395681
$869$	41.6624	1.41330
$870$	0	0
$871$	81.8683	2.77400
$872$	13.8402	0.468688
$873$	0	0
$874$	−34.1182	−1.15407
$875$	12.0812	0.408418
$876$	0	0
$877$	15.3090	0.516947	0.258474	−	0.966018i	$-0.416781\pi$
$877$	15.3090	0.516947	0.258474	+	0.966018i	$0.416781\pi$
$878$	20.8456	0.703504
$879$	0	0
$880$	−7.60448	−0.256347
$881$	46.0410	1.55116	0.775580	−	0.631250i	$-0.217458\pi$
$881$	46.0410	1.55116	0.775580	+	0.631250i	$0.217458\pi$
$882$	0	0
$883$	−50.1193	−1.68665	−0.843325	−	0.537404i	$-0.819405\pi$
$883$	−50.1193	−1.68665	−0.843325	+	0.537404i	$0.819405\pi$
$884$	−33.7135	−1.13391
$885$	0	0
$886$	−5.09801	−0.171271
$887$	−30.3218	−1.01811	−0.509053	−	0.860735i	$-0.670004\pi$
$887$	−30.3218	−1.01811	−0.509053	+	0.860735i	$0.670004\pi$
$888$	0	0
$889$	9.95821	0.333988
$890$	−0.870492	−0.0291790
$891$	0	0
$892$	−11.0866	−0.371205
$893$	45.9647	1.53815
$894$	0	0
$895$	−14.5962	−0.487897
$896$	1.20200	0.0401559
$897$	0	0
$898$	8.13634	0.271513
$899$	77.0989	2.57139
$900$	0	0
$901$	−41.1154	−1.36975
$902$	−22.5771	−0.751734
$903$	0	0
$904$	7.75371	0.257885
$905$	−9.67630	−0.321651
$906$	0	0
$907$	−50.3208	−1.67087	−0.835437	−	0.549587i	$-0.814785\pi$
$907$	−50.3208	−1.67087	−0.835437	+	0.549587i	$0.814785\pi$
$908$	−18.9421	−0.628616
$909$	0	0
$910$	8.69467	0.288225
$911$	19.5452	0.647561	0.323781	−	0.946132i	$-0.395046\pi$
$911$	19.5452	0.647561	0.323781	+	0.946132i	$0.395046\pi$
$912$	0	0
$913$	−32.3722	−1.07136
$914$	9.54401	0.315688
$915$	0	0
$916$	−2.84801	−0.0941008
$917$	−24.3782	−0.805039
$918$	0	0
$919$	−2.83794	−0.0936149	−0.0468074	−	0.998904i	$-0.514905\pi$
$919$	−2.83794	−0.0936149	−0.0468074	+	0.998904i	$0.514905\pi$
$920$	6.04619	0.199337
$921$	0	0
$922$	−39.9662	−1.31622
$923$	40.4346	1.33092
$924$	0	0
$925$	1.65068	0.0542740
$926$	−16.8056	−0.552265
$927$	0	0
$928$	7.94963	0.260959
$929$	−32.7811	−1.07551	−0.537757	−	0.843100i	$-0.680728\pi$
$929$	−32.7811	−1.07551	−0.537757	+	0.843100i	$0.680728\pi$
$930$	0	0
$931$	−36.4254	−1.19379
$932$	−9.74322	−0.319150
$933$	0	0
$934$	39.1289	1.28034
$935$	−41.1835	−1.34684
$936$	0	0
$937$	−10.5231	−0.343775	−0.171887	−	0.985117i	$-0.554986\pi$
$937$	−10.5231	−0.343775	−0.171887	+	0.985117i	$0.554986\pi$
$938$	−15.8077	−0.516140
$939$	0	0
$940$	−8.14554	−0.265678
$941$	−34.4915	−1.12439	−0.562196	−	0.827004i	$-0.690043\pi$
$941$	−34.4915	−1.12439	−0.562196	+	0.827004i	$0.690043\pi$
$942$	0	0
$943$	17.9506	0.584553
$944$	−2.60856	−0.0849014
$945$	0	0
$946$	−26.4220	−0.859053
$947$	−13.1454	−0.427167	−0.213583	−	0.976925i	$-0.568513\pi$
$947$	−13.1454	−0.427167	−0.213583	+	0.976925i	$0.568513\pi$
$948$	0	0
$949$	−0.0892432	−0.00289696
$950$	−23.9317	−0.776446
$951$	0	0
$952$	6.50963	0.210978
$953$	−3.04290	−0.0985693	−0.0492847	−	0.998785i	$-0.515694\pi$
$953$	−3.04290	−0.0985693	−0.0492847	+	0.998785i	$0.515694\pi$
$954$	0	0
$955$	27.7229	0.897093
$956$	−6.42430	−0.207777
$957$	0	0
$958$	−32.3479	−1.04511
$959$	−5.11272	−0.165098
$960$	0	0
$961$	63.0596	2.03418
$962$	2.81543	0.0907730
$963$	0	0
$964$	−15.0728	−0.485463
$965$	2.26169	0.0728064
$966$	0	0
$967$	12.2359	0.393479	0.196740	−	0.980456i	$-0.436965\pi$
$967$	12.2359	0.393479	0.196740	+	0.980456i	$0.436965\pi$
$968$	31.8292	1.02303
$969$	0	0
$970$	5.48108	0.175987
$971$	−58.0780	−1.86381	−0.931906	−	0.362701i	$-0.881855\pi$
$971$	−58.0780	−1.86381	−0.931906	+	0.362701i	$0.881855\pi$
$972$	0	0
$973$	−25.1403	−0.805960
$974$	12.4954	0.400379
$975$	0	0
$976$	10.4221	0.333605
$977$	−11.1619	−0.357101	−0.178551	−	0.983931i	$-0.557141\pi$
$977$	−11.1619	−0.357101	−0.178551	+	0.983931i	$0.557141\pi$
$978$	0	0
$979$	4.90270	0.156691
$980$	6.45505	0.206199
$981$	0	0
$982$	28.5946	0.912489
$983$	−52.6756	−1.68009	−0.840046	−	0.542516i	$-0.817472\pi$
$983$	−52.6756	−1.68009	−0.840046	+	0.542516i	$0.817472\pi$
$984$	0	0
$985$	7.89890	0.251680
$986$	43.0527	1.37108
$987$	0	0
$988$	−40.8183	−1.29860
$989$	21.0076	0.668004
$990$	0	0
$991$	−0.155139	−0.00492816	−0.00246408	−	0.999997i	$-0.500784\pi$
$991$	−0.155139	−0.00492816	−0.00246408	+	0.999997i	$0.500784\pi$
$992$	9.69843	0.307926
$993$	0	0
$994$	−7.80739	−0.247635
$995$	6.81660	0.216101
$996$	0	0
$997$	−14.4510	−0.457666	−0.228833	−	0.973466i	$-0.573491\pi$
$997$	−14.4510	−0.457666	−0.228833	+	0.973466i	$0.573491\pi$
$998$	22.1731	0.701876
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8046.2.a.t.1.6	yes	16
3.2	odd	2		8046.2.a.s.1.11	✓	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8046.2.a.s.1.11	✓	16	3.2	odd	2
8046.2.a.t.1.6	yes	16	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:	\( N \)	\(=\)	\( 8046 = 2 \cdot 3^{3} \cdot 149 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8046.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} -1.16198 q^{5} +1.20200 q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} -1.16198 q^{5} +1.20200 q^{7} +1.00000 q^{8} -1.16198 q^{10} +6.54441 q^{11} -6.22516 q^{13} +1.20200 q^{14} +1.00000 q^{16} +5.41568 q^{17} +6.55699 q^{19} -1.16198 q^{20} +6.54441 q^{22} -5.20334 q^{23} -3.64980 q^{25} -6.22516 q^{26} +1.20200 q^{28} +7.94963 q^{29} +9.69843 q^{31} +1.00000 q^{32} +5.41568 q^{34} -1.39670 q^{35} -0.452266 q^{37} +6.55699 q^{38} -1.16198 q^{40} -3.44983 q^{41} -4.03734 q^{43} +6.54441 q^{44} -5.20334 q^{46} +7.01004 q^{47} -5.55520 q^{49} -3.64980 q^{50} -6.22516 q^{52} -7.59192 q^{53} -7.60448 q^{55} +1.20200 q^{56} +7.94963 q^{58} -2.60856 q^{59} +10.4221 q^{61} +9.69843 q^{62} +1.00000 q^{64} +7.23352 q^{65} -13.1512 q^{67} +5.41568 q^{68} -1.39670 q^{70} -6.49535 q^{71} +0.0143359 q^{73} -0.452266 q^{74} +6.55699 q^{76} +7.86635 q^{77} +6.36611 q^{79} -1.16198 q^{80} -3.44983 q^{82} -4.94655 q^{83} -6.29293 q^{85} -4.03734 q^{86} +6.54441 q^{88} +0.749144 q^{89} -7.48262 q^{91} -5.20334 q^{92} +7.01004 q^{94} -7.61910 q^{95} -4.71701 q^{97} -5.55520 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 16 q^{2} + 16 q^{4} + 4 q^{5} + 6 q^{7} + 16 q^{8}+O(q^{10}) \) 16 * q + 16 * q^2 + 16 * q^4 + 4 * q^5 + 6 * q^7 + 16 * q^8	\( 16 q + 16 q^{2} + 16 q^{4} + 4 q^{5} + 6 q^{7} + 16 q^{8} + 4 q^{10} + 6 q^{11} + 6 q^{13} + 6 q^{14} + 16 q^{16} + q^{17} + 10 q^{19} + 4 q^{20} + 6 q^{22} + 10 q^{23} + 28 q^{25} + 6 q^{26} + 6 q^{28} + 6 q^{29} + 21 q^{31} + 16 q^{32} + q^{34} + 16 q^{35} + 17 q^{37} + 10 q^{38} + 4 q^{40} - 4 q^{41} + 16 q^{43} + 6 q^{44} + 10 q^{46} + 25 q^{47} + 36 q^{49} + 28 q^{50} + 6 q^{52} + 14 q^{53} + 19 q^{55} + 6 q^{56} + 6 q^{58} + 6 q^{59} + 23 q^{61} + 21 q^{62} + 16 q^{64} + 20 q^{65} + 22 q^{67} + q^{68} + 16 q^{70} + 10 q^{71} + 16 q^{73} + 17 q^{74} + 10 q^{76} - 2 q^{77} + 37 q^{79} + 4 q^{80} - 4 q^{82} + 33 q^{83} + 43 q^{85} + 16 q^{86} + 6 q^{88} - 3 q^{89} + 28 q^{91} + 10 q^{92} + 25 q^{94} + 14 q^{95} - 3 q^{97} + 36 q^{98}+O(q^{100}) \) 16 * q + 16 * q^2 + 16 * q^4 + 4 * q^5 + 6 * q^7 + 16 * q^8 + 4 * q^10 + 6 * q^11 + 6 * q^13 + 6 * q^14 + 16 * q^16 + q^17 + 10 * q^19 + 4 * q^20 + 6 * q^22 + 10 * q^23 + 28 * q^25 + 6 * q^26 + 6 * q^28 + 6 * q^29 + 21 * q^31 + 16 * q^32 + q^34 + 16 * q^35 + 17 * q^37 + 10 * q^38 + 4 * q^40 - 4 * q^41 + 16 * q^43 + 6 * q^44 + 10 * q^46 + 25 * q^47 + 36 * q^49 + 28 * q^50 + 6 * q^52 + 14 * q^53 + 19 * q^55 + 6 * q^56 + 6 * q^58 + 6 * q^59 + 23 * q^61 + 21 * q^62 + 16 * q^64 + 20 * q^65 + 22 * q^67 + q^68 + 16 * q^70 + 10 * q^71 + 16 * q^73 + 17 * q^74 + 10 * q^76 - 2 * q^77 + 37 * q^79 + 4 * q^80 - 4 * q^82 + 33 * q^83 + 43 * q^85 + 16 * q^86 + 6 * q^88 - 3 * q^89 + 28 * q^91 + 10 * q^92 + 25 * q^94 + 14 * q^95 - 3 * q^97 + 36 * q^98

Introduction

L-functions

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