LMFDB - Embedded newform 8046.2.a.p.1.2

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:	$12$
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			1.2
Root			$-2.78478$ 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)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} -2.78478 q^{5} +2.18432 q^{7} +1.00000 q^{8} +O(q^{10})$	\(q+1.00000 q^{2} +1.00000 q^{4} -2.78478 q^{5} +2.18432 q^{7} +1.00000 q^{8} -2.78478 q^{10} +3.79347 q^{11} +3.92737 q^{13} +2.18432 q^{14} +1.00000 q^{16} +7.46686 q^{17} -5.81644 q^{19} -2.78478 q^{20} +3.79347 q^{22} +3.28490 q^{23} +2.75499 q^{25} +3.92737 q^{26} +2.18432 q^{28} +5.39511 q^{29} +1.61066 q^{31} +1.00000 q^{32} +7.46686 q^{34} -6.08285 q^{35} +4.51065 q^{37} -5.81644 q^{38} -2.78478 q^{40} -2.46063 q^{41} +6.51141 q^{43} +3.79347 q^{44} +3.28490 q^{46} -9.17875 q^{47} -2.22873 q^{49} +2.75499 q^{50} +3.92737 q^{52} -2.04199 q^{53} -10.5640 q^{55} +2.18432 q^{56} +5.39511 q^{58} +2.76719 q^{59} -5.07451 q^{61} +1.61066 q^{62} +1.00000 q^{64} -10.9368 q^{65} -10.5518 q^{67} +7.46686 q^{68} -6.08285 q^{70} -4.21136 q^{71} -15.3730 q^{73} +4.51065 q^{74} -5.81644 q^{76} +8.28617 q^{77} +13.2008 q^{79} -2.78478 q^{80} -2.46063 q^{82} +4.27367 q^{83} -20.7935 q^{85} +6.51141 q^{86} +3.79347 q^{88} -2.59134 q^{89} +8.57864 q^{91} +3.28490 q^{92} -9.17875 q^{94} +16.1975 q^{95} +1.13319 q^{97} -2.22873 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	−2.78478	−1.24539	−0.622695	−	0.782464i	$-0.713962\pi$
$5$	−2.78478	−1.24539	−0.622695	+	0.782464i	$0.713962\pi$
$6$	0	0
$7$	2.18432	0.825596	0.412798	−	0.910823i	$-0.364551\pi$
$7$	2.18432	0.825596	0.412798	+	0.910823i	$0.364551\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	−2.78478	−0.880624
$11$	3.79347	1.14377	0.571887	−	0.820332i	$-0.306211\pi$
$11$	3.79347	1.14377	0.571887	+	0.820332i	$0.306211\pi$
$12$	0	0
$13$	3.92737	1.08926	0.544628	−	0.838678i	$-0.316671\pi$
$13$	3.92737	1.08926	0.544628	+	0.838678i	$0.316671\pi$
$14$	2.18432	0.583785
$15$	0	0
$16$	1.00000	0.250000
$17$	7.46686	1.81098	0.905490	−	0.424368i	$-0.139504\pi$
$17$	7.46686	1.81098	0.905490	+	0.424368i	$0.139504\pi$
$18$	0	0
$19$	−5.81644	−1.33438	−0.667192	−	0.744886i	$-0.732504\pi$
$19$	−5.81644	−1.33438	−0.667192	+	0.744886i	$0.732504\pi$
$20$	−2.78478	−0.622695
$21$	0	0
$22$	3.79347	0.808771
$23$	3.28490	0.684950	0.342475	−	0.939527i	$-0.388735\pi$
$23$	3.28490	0.684950	0.342475	+	0.939527i	$0.388735\pi$
$24$	0	0
$25$	2.75499	0.550997
$26$	3.92737	0.770220
$27$	0	0
$28$	2.18432	0.412798
$29$	5.39511	1.00185	0.500923	−	0.865492i	$-0.332994\pi$
$29$	5.39511	1.00185	0.500923	+	0.865492i	$0.332994\pi$
$30$	0	0
$31$	1.61066	0.289283	0.144641	−	0.989484i	$-0.453797\pi$
$31$	1.61066	0.289283	0.144641	+	0.989484i	$0.453797\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	7.46686	1.28056
$35$	−6.08285	−1.02819
$36$	0	0
$37$	4.51065	0.741546	0.370773	−	0.928724i	$-0.379093\pi$
$37$	4.51065	0.741546	0.370773	+	0.928724i	$0.379093\pi$
$38$	−5.81644	−0.943552
$39$	0	0
$40$	−2.78478	−0.440312
$41$	−2.46063	−0.384286	−0.192143	−	0.981367i	$-0.561544\pi$
$41$	−2.46063	−0.384286	−0.192143	+	0.981367i	$0.561544\pi$
$42$	0	0
$43$	6.51141	0.992981	0.496491	−	0.868042i	$-0.334622\pi$
$43$	6.51141	0.992981	0.496491	+	0.868042i	$0.334622\pi$
$44$	3.79347	0.571887
$45$	0	0
$46$	3.28490	0.484333
$47$	−9.17875	−1.33886	−0.669429	−	0.742876i	$-0.733461\pi$
$47$	−9.17875	−1.33886	−0.669429	+	0.742876i	$0.733461\pi$
$48$	0	0
$49$	−2.22873	−0.318391
$50$	2.75499	0.389614
$51$	0	0
$52$	3.92737	0.544628
$53$	−2.04199	−0.280489	−0.140244	−	0.990117i	$-0.544789\pi$
$53$	−2.04199	−0.280489	−0.140244	+	0.990117i	$0.544789\pi$
$54$	0	0
$55$	−10.5640	−1.42445
$56$	2.18432	0.291892
$57$	0	0
$58$	5.39511	0.708412
$59$	2.76719	0.360258	0.180129	−	0.983643i	$-0.442349\pi$
$59$	2.76719	0.360258	0.180129	+	0.983643i	$0.442349\pi$
$60$	0	0
$61$	−5.07451	−0.649725	−0.324862	−	0.945761i	$-0.605318\pi$
$61$	−5.07451	−0.649725	−0.324862	+	0.945761i	$0.605318\pi$
$62$	1.61066	0.204554
$63$	0	0
$64$	1.00000	0.125000
$65$	−10.9368	−1.35655
$66$	0	0
$67$	−10.5518	−1.28911	−0.644557	−	0.764557i	$-0.722958\pi$
$67$	−10.5518	−1.28911	−0.644557	+	0.764557i	$0.722958\pi$
$68$	7.46686	0.905490
$69$	0	0
$70$	−6.08285	−0.727040
$71$	−4.21136	−0.499796	−0.249898	−	0.968272i	$-0.580397\pi$
$71$	−4.21136	−0.499796	−0.249898	+	0.968272i	$0.580397\pi$
$72$	0	0
$73$	−15.3730	−1.79927	−0.899636	−	0.436640i	$-0.856168\pi$
$73$	−15.3730	−1.79927	−0.899636	+	0.436640i	$0.856168\pi$
$74$	4.51065	0.524352
$75$	0	0
$76$	−5.81644	−0.667192
$77$	8.28617	0.944296
$78$	0	0
$79$	13.2008	1.48520	0.742601	−	0.669734i	$-0.233592\pi$
$79$	13.2008	1.48520	0.742601	+	0.669734i	$0.233592\pi$
$80$	−2.78478	−0.311348
$81$	0	0
$82$	−2.46063	−0.271731
$83$	4.27367	0.469097	0.234548	−	0.972104i	$-0.424639\pi$
$83$	4.27367	0.469097	0.234548	+	0.972104i	$0.424639\pi$
$84$	0	0
$85$	−20.7935	−2.25538
$86$	6.51141	0.702144
$87$	0	0
$88$	3.79347	0.404385
$89$	−2.59134	−0.274681	−0.137341	−	0.990524i	$-0.543855\pi$
$89$	−2.59134	−0.274681	−0.137341	+	0.990524i	$0.543855\pi$
$90$	0	0
$91$	8.57864	0.899285
$92$	3.28490	0.342475
$93$	0	0
$94$	−9.17875	−0.946715
$95$	16.1975	1.66183
$96$	0	0
$97$	1.13319	0.115058	0.0575288	−	0.998344i	$-0.481678\pi$
$97$	1.13319	0.115058	0.0575288	+	0.998344i	$0.481678\pi$
$98$	−2.22873	−0.225136
$99$	0	0
$100$	2.75499	0.275499
$101$	−6.37228	−0.634065	−0.317033	−	0.948415i	$-0.602686\pi$
$101$	−6.37228	−0.634065	−0.317033	+	0.948415i	$0.602686\pi$
$102$	0	0
$103$	1.82993	0.180309	0.0901544	−	0.995928i	$-0.471264\pi$
$103$	1.82993	0.180309	0.0901544	+	0.995928i	$0.471264\pi$
$104$	3.92737	0.385110
$105$	0	0
$106$	−2.04199	−0.198336
$107$	16.9316	1.63683	0.818417	−	0.574624i	$-0.194852\pi$
$107$	16.9316	1.63683	0.818417	+	0.574624i	$0.194852\pi$
$108$	0	0
$109$	−7.04611	−0.674895	−0.337447	−	0.941344i	$-0.609563\pi$
$109$	−7.04611	−0.674895	−0.337447	+	0.941344i	$0.609563\pi$
$110$	−10.5640	−1.00724
$111$	0	0
$112$	2.18432	0.206399
$113$	14.6370	1.37693	0.688467	−	0.725268i	$-0.258284\pi$
$113$	14.6370	1.37693	0.688467	+	0.725268i	$0.258284\pi$
$114$	0	0
$115$	−9.14773	−0.853030
$116$	5.39511	0.500923
$117$	0	0
$118$	2.76719	0.254741
$119$	16.3100	1.49514
$120$	0	0
$121$	3.39043	0.308221
$122$	−5.07451	−0.459425
$123$	0	0
$124$	1.61066	0.144641
$125$	6.25186	0.559184
$126$	0	0
$127$	0.581987	0.0516430	0.0258215	−	0.999667i	$-0.491780\pi$
$127$	0.581987	0.0516430	0.0258215	+	0.999667i	$0.491780\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	−10.9368	−0.959224
$131$	7.96392	0.695811	0.347906	−	0.937530i	$-0.386893\pi$
$131$	7.96392	0.695811	0.347906	+	0.937530i	$0.386893\pi$
$132$	0	0
$133$	−12.7050	−1.10166
$134$	−10.5518	−0.911541
$135$	0	0
$136$	7.46686	0.640278
$137$	19.3760	1.65540	0.827702	−	0.561167i	$-0.189648\pi$
$137$	19.3760	1.65540	0.827702	+	0.561167i	$0.189648\pi$
$138$	0	0
$139$	5.19559	0.440684	0.220342	−	0.975423i	$-0.429283\pi$
$139$	5.19559	0.440684	0.220342	+	0.975423i	$0.429283\pi$
$140$	−6.08285	−0.514095
$141$	0	0
$142$	−4.21136	−0.353409
$143$	14.8984	1.24586
$144$	0	0
$145$	−15.0242	−1.24769
$146$	−15.3730	−1.27228
$147$	0	0
$148$	4.51065	0.370773
$149$	−1.00000	−0.0819232
$149$	−1.00000	−0.0819232
$150$	0	0
$151$	3.67260	0.298872	0.149436	−	0.988771i	$-0.452254\pi$
$151$	3.67260	0.298872	0.149436	+	0.988771i	$0.452254\pi$
$152$	−5.81644	−0.471776
$153$	0	0
$154$	8.28617	0.667718
$155$	−4.48532	−0.360270
$156$	0	0
$157$	20.6000	1.64406	0.822029	−	0.569446i	$-0.192842\pi$
$157$	20.6000	1.64406	0.822029	+	0.569446i	$0.192842\pi$
$158$	13.2008	1.05020
$159$	0	0
$160$	−2.78478	−0.220156
$161$	7.17529	0.565492
$162$	0	0
$163$	−0.781487	−0.0612108	−0.0306054	−	0.999532i	$-0.509744\pi$
$163$	−0.781487	−0.0612108	−0.0306054	+	0.999532i	$0.509744\pi$
$164$	−2.46063	−0.192143
$165$	0	0
$166$	4.27367	0.331701
$167$	−4.43877	−0.343482	−0.171741	−	0.985142i	$-0.554939\pi$
$167$	−4.43877	−0.343482	−0.171741	+	0.985142i	$0.554939\pi$
$168$	0	0
$169$	2.42420	0.186477
$170$	−20.7935	−1.59479
$171$	0	0
$172$	6.51141	0.496491
$173$	11.7888	0.896283	0.448141	−	0.893963i	$-0.352086\pi$
$173$	11.7888	0.896283	0.448141	+	0.893963i	$0.352086\pi$
$174$	0	0
$175$	6.01778	0.454901
$176$	3.79347	0.285944
$177$	0	0
$178$	−2.59134	−0.194229
$179$	−11.4199	−0.853565	−0.426782	−	0.904354i	$-0.640353\pi$
$179$	−11.4199	−0.853565	−0.426782	+	0.904354i	$0.640353\pi$
$180$	0	0
$181$	−19.3924	−1.44142	−0.720712	−	0.693234i	$-0.756185\pi$
$181$	−19.3924	−1.44142	−0.720712	+	0.693234i	$0.756185\pi$
$182$	8.57864	0.635891
$183$	0	0
$184$	3.28490	0.242166
$185$	−12.5611	−0.923514
$186$	0	0
$187$	28.3253	2.07135
$188$	−9.17875	−0.669429
$189$	0	0
$190$	16.1975	1.17509
$191$	13.1005	0.947916	0.473958	−	0.880548i	$-0.342825\pi$
$191$	13.1005	0.947916	0.473958	+	0.880548i	$0.342825\pi$
$192$	0	0
$193$	−4.41114	−0.317521	−0.158760	−	0.987317i	$-0.550750\pi$
$193$	−4.41114	−0.317521	−0.158760	+	0.987317i	$0.550750\pi$
$194$	1.13319	0.0813579
$195$	0	0
$196$	−2.22873	−0.159195
$197$	−7.06752	−0.503540	−0.251770	−	0.967787i	$-0.581013\pi$
$197$	−7.06752	−0.503540	−0.251770	+	0.967787i	$0.581013\pi$
$198$	0	0
$199$	−19.7015	−1.39660	−0.698301	−	0.715804i	$-0.746060\pi$
$199$	−19.7015	−1.39660	−0.698301	+	0.715804i	$0.746060\pi$
$200$	2.75499	0.194807
$201$	0	0
$202$	−6.37228	−0.448352
$203$	11.7847	0.827120
$204$	0	0
$205$	6.85230	0.478586
$206$	1.82993	0.127498
$207$	0	0
$208$	3.92737	0.272314
$209$	−22.0645	−1.52623
$210$	0	0
$211$	13.5788	0.934800	0.467400	−	0.884046i	$-0.345191\pi$
$211$	13.5788	0.934800	0.467400	+	0.884046i	$0.345191\pi$
$212$	−2.04199	−0.140244
$213$	0	0
$214$	16.9316	1.15742
$215$	−18.1328	−1.23665
$216$	0	0
$217$	3.51819	0.238831
$218$	−7.04611	−0.477222
$219$	0	0
$220$	−10.5640	−0.712223
$221$	29.3251	1.97262
$222$	0	0
$223$	−6.54106	−0.438022	−0.219011	−	0.975722i	$-0.570283\pi$
$223$	−6.54106	−0.438022	−0.219011	+	0.975722i	$0.570283\pi$
$224$	2.18432	0.145946
$225$	0	0
$226$	14.6370	0.973639
$227$	−2.84316	−0.188707	−0.0943537	−	0.995539i	$-0.530078\pi$
$227$	−2.84316	−0.188707	−0.0943537	+	0.995539i	$0.530078\pi$
$228$	0	0
$229$	2.85552	0.188698	0.0943490	−	0.995539i	$-0.469923\pi$
$229$	2.85552	0.188698	0.0943490	+	0.995539i	$0.469923\pi$
$230$	−9.14773	−0.603183
$231$	0	0
$232$	5.39511	0.354206
$233$	24.3743	1.59682	0.798408	−	0.602117i	$-0.205676\pi$
$233$	24.3743	1.59682	0.798408	+	0.602117i	$0.205676\pi$
$234$	0	0
$235$	25.5608	1.66740
$236$	2.76719	0.180129
$237$	0	0
$238$	16.3100	1.05722
$239$	19.0073	1.22948	0.614739	−	0.788731i	$-0.289262\pi$
$239$	19.0073	1.22948	0.614739	+	0.788731i	$0.289262\pi$
$240$	0	0
$241$	−27.7788	−1.78939	−0.894695	−	0.446678i	$-0.852607\pi$
$241$	−27.7788	−1.78939	−0.894695	+	0.446678i	$0.852607\pi$
$242$	3.39043	0.217945
$243$	0	0
$244$	−5.07451	−0.324862
$245$	6.20653	0.396521
$246$	0	0
$247$	−22.8433	−1.45348
$248$	1.61066	0.102277
$249$	0	0
$250$	6.25186	0.395403
$251$	−8.03049	−0.506880	−0.253440	−	0.967351i	$-0.581562\pi$
$251$	−8.03049	−0.506880	−0.253440	+	0.967351i	$0.581562\pi$
$252$	0	0
$253$	12.4612	0.783428
$254$	0.581987	0.0365171
$255$	0	0
$256$	1.00000	0.0625000
$257$	23.7384	1.48076	0.740380	−	0.672189i	$-0.234646\pi$
$257$	23.7384	1.48076	0.740380	+	0.672189i	$0.234646\pi$
$258$	0	0
$259$	9.85271	0.612217
$260$	−10.9368	−0.678274
$261$	0	0
$262$	7.96392	0.492013
$263$	−10.4821	−0.646356	−0.323178	−	0.946338i	$-0.604751\pi$
$263$	−10.4821	−0.646356	−0.323178	+	0.946338i	$0.604751\pi$
$264$	0	0
$265$	5.68649	0.349318
$266$	−12.7050	−0.778993
$267$	0	0
$268$	−10.5518	−0.644557
$269$	6.50883	0.396850	0.198425	−	0.980116i	$-0.436417\pi$
$269$	6.50883	0.396850	0.198425	+	0.980116i	$0.436417\pi$
$270$	0	0
$271$	0.237220	0.0144101	0.00720505	−	0.999974i	$-0.497707\pi$
$271$	0.237220	0.0144101	0.00720505	+	0.999974i	$0.497707\pi$
$272$	7.46686	0.452745
$273$	0	0
$274$	19.3760	1.17055
$275$	10.4510	0.630217
$276$	0	0
$277$	−28.0412	−1.68483	−0.842416	−	0.538828i	$-0.818867\pi$
$277$	−28.0412	−1.68483	−0.842416	+	0.538828i	$0.818867\pi$
$278$	5.19559	0.311611
$279$	0	0
$280$	−6.08285	−0.363520
$281$	−6.74939	−0.402635	−0.201317	−	0.979526i	$-0.564522\pi$
$281$	−6.74939	−0.402635	−0.201317	+	0.979526i	$0.564522\pi$
$282$	0	0
$283$	−7.40295	−0.440060	−0.220030	−	0.975493i	$-0.570616\pi$
$283$	−7.40295	−0.440060	−0.220030	+	0.975493i	$0.570616\pi$
$284$	−4.21136	−0.249898
$285$	0	0
$286$	14.8984	0.880958
$287$	−5.37481	−0.317265
$288$	0	0
$289$	38.7540	2.27965
$290$	−15.0242	−0.882250
$291$	0	0
$292$	−15.3730	−0.899636
$293$	20.2061	1.18045	0.590226	−	0.807238i	$-0.299039\pi$
$293$	20.2061	1.18045	0.590226	+	0.807238i	$0.299039\pi$
$294$	0	0
$295$	−7.70601	−0.448661
$296$	4.51065	0.262176
$297$	0	0
$298$	−1.00000	−0.0579284
$299$	12.9010	0.746085
$300$	0	0
$301$	14.2230	0.819802
$302$	3.67260	0.211334
$303$	0	0
$304$	−5.81644	−0.333596
$305$	14.1314	0.809161
$306$	0	0
$307$	9.70048	0.553636	0.276818	−	0.960922i	$-0.410720\pi$
$307$	9.70048	0.553636	0.276818	+	0.960922i	$0.410720\pi$
$308$	8.28617	0.472148
$309$	0	0
$310$	−4.48532	−0.254749
$311$	−15.7395	−0.892503	−0.446252	−	0.894908i	$-0.647241\pi$
$311$	−15.7395	−0.892503	−0.446252	+	0.894908i	$0.647241\pi$
$312$	0	0
$313$	−26.7280	−1.51075	−0.755377	−	0.655291i	$-0.772546\pi$
$313$	−26.7280	−1.51075	−0.755377	+	0.655291i	$0.772546\pi$
$314$	20.6000	1.16252
$315$	0	0
$316$	13.2008	0.742601
$317$	−26.5139	−1.48917	−0.744586	−	0.667527i	$-0.767353\pi$
$317$	−26.5139	−1.48917	−0.744586	+	0.667527i	$0.767353\pi$
$318$	0	0
$319$	20.4662	1.14589
$320$	−2.78478	−0.155674
$321$	0	0
$322$	7.17529	0.399863
$323$	−43.4306	−2.41654
$324$	0	0
$325$	10.8198	0.600177
$326$	−0.781487	−0.0432825
$327$	0	0
$328$	−2.46063	−0.135865
$329$	−20.0493	−1.10536
$330$	0	0
$331$	4.24045	0.233076	0.116538	−	0.993186i	$-0.462820\pi$
$331$	4.24045	0.233076	0.116538	+	0.993186i	$0.462820\pi$
$332$	4.27367	0.234548
$333$	0	0
$334$	−4.43877	−0.242879
$335$	29.3845	1.60545
$336$	0	0
$337$	18.0860	0.985207	0.492604	−	0.870254i	$-0.336045\pi$
$337$	18.0860	0.985207	0.492604	+	0.870254i	$0.336045\pi$
$338$	2.42420	0.131859
$339$	0	0
$340$	−20.7935	−1.12769
$341$	6.10998	0.330874
$342$	0	0
$343$	−20.1585	−1.08846
$344$	6.51141	0.351072
$345$	0	0
$346$	11.7888	0.633768
$347$	7.52017	0.403704	0.201852	−	0.979416i	$-0.435304\pi$
$347$	7.52017	0.403704	0.201852	+	0.979416i	$0.435304\pi$
$348$	0	0
$349$	17.8971	0.958008	0.479004	−	0.877813i	$-0.340998\pi$
$349$	17.8971	0.958008	0.479004	+	0.877813i	$0.340998\pi$
$350$	6.01778	0.321664
$351$	0	0
$352$	3.79347	0.202193
$353$	18.1266	0.964782	0.482391	−	0.875956i	$-0.339769\pi$
$353$	18.1266	0.964782	0.482391	+	0.875956i	$0.339769\pi$
$354$	0	0
$355$	11.7277	0.622441
$356$	−2.59134	−0.137341
$357$	0	0
$358$	−11.4199	−0.603561
$359$	−27.1156	−1.43111	−0.715554	−	0.698557i	$-0.753826\pi$
$359$	−27.1156	−1.43111	−0.715554	+	0.698557i	$0.753826\pi$
$360$	0	0
$361$	14.8310	0.780579
$362$	−19.3924	−1.01924
$363$	0	0
$364$	8.57864	0.449643
$365$	42.8104	2.24080
$366$	0	0
$367$	10.5060	0.548408	0.274204	−	0.961672i	$-0.411586\pi$
$367$	10.5060	0.548408	0.274204	+	0.961672i	$0.411586\pi$
$368$	3.28490	0.171237
$369$	0	0
$370$	−12.5611	−0.653023
$371$	−4.46036	−0.231571
$372$	0	0
$373$	8.41289	0.435603	0.217802	−	0.975993i	$-0.430111\pi$
$373$	8.41289	0.435603	0.217802	+	0.975993i	$0.430111\pi$
$374$	28.3253	1.46467
$375$	0	0
$376$	−9.17875	−0.473358
$377$	21.1886	1.09127
$378$	0	0
$379$	−6.82251	−0.350449	−0.175224	−	0.984529i	$-0.556065\pi$
$379$	−6.82251	−0.350449	−0.175224	+	0.984529i	$0.556065\pi$
$380$	16.1975	0.830914
$381$	0	0
$382$	13.1005	0.670278
$383$	−8.64591	−0.441785	−0.220893	−	0.975298i	$-0.570897\pi$
$383$	−8.64591	−0.441785	−0.220893	+	0.975298i	$0.570897\pi$
$384$	0	0
$385$	−23.0751	−1.17602
$386$	−4.41114	−0.224521
$387$	0	0
$388$	1.13319	0.0575288
$389$	16.0651	0.814532	0.407266	−	0.913310i	$-0.366482\pi$
$389$	16.0651	0.814532	0.407266	+	0.913310i	$0.366482\pi$
$390$	0	0
$391$	24.5279	1.24043
$392$	−2.22873	−0.112568
$393$	0	0
$394$	−7.06752	−0.356056
$395$	−36.7612	−1.84966
$396$	0	0
$397$	20.5577	1.03176	0.515880	−	0.856661i	$-0.327465\pi$
$397$	20.5577	1.03176	0.515880	+	0.856661i	$0.327465\pi$
$398$	−19.7015	−0.987546
$399$	0	0
$400$	2.75499	0.137749
$401$	−13.3660	−0.667468	−0.333734	−	0.942667i	$-0.608309\pi$
$401$	−13.3660	−0.667468	−0.333734	+	0.942667i	$0.608309\pi$
$402$	0	0
$403$	6.32564	0.315103
$404$	−6.37228	−0.317033
$405$	0	0
$406$	11.7847	0.584862
$407$	17.1110	0.848161
$408$	0	0
$409$	31.3584	1.55057	0.775286	−	0.631610i	$-0.217606\pi$
$409$	31.3584	1.55057	0.775286	+	0.631610i	$0.217606\pi$
$410$	6.85230	0.338411
$411$	0	0
$412$	1.82993	0.0901544
$413$	6.04444	0.297427
$414$	0	0
$415$	−11.9012	−0.584208
$416$	3.92737	0.192555
$417$	0	0
$418$	−22.0645	−1.07921
$419$	18.8568	0.921213	0.460607	−	0.887604i	$-0.347632\pi$
$419$	18.8568	0.921213	0.460607	+	0.887604i	$0.347632\pi$
$420$	0	0
$421$	−24.2532	−1.18203	−0.591013	−	0.806662i	$-0.701272\pi$
$421$	−24.2532	−1.18203	−0.591013	+	0.806662i	$0.701272\pi$
$422$	13.5788	0.661004
$423$	0	0
$424$	−2.04199	−0.0991678
$425$	20.5711	0.997845
$426$	0	0
$427$	−11.0844	−0.536411
$428$	16.9316	0.818417
$429$	0	0
$430$	−18.1328	−0.874443
$431$	−3.23977	−0.156054	−0.0780272	−	0.996951i	$-0.524862\pi$
$431$	−3.23977	−0.156054	−0.0780272	+	0.996951i	$0.524862\pi$
$432$	0	0
$433$	36.4730	1.75278	0.876391	−	0.481600i	$-0.159944\pi$
$433$	36.4730	1.75278	0.876391	+	0.481600i	$0.159944\pi$
$434$	3.51819	0.168879
$435$	0	0
$436$	−7.04611	−0.337447
$437$	−19.1065	−0.913985
$438$	0	0
$439$	12.1537	0.580064	0.290032	−	0.957017i	$-0.406334\pi$
$439$	12.1537	0.580064	0.290032	+	0.957017i	$0.406334\pi$
$440$	−10.5640	−0.503618
$441$	0	0
$442$	29.3251	1.39485
$443$	−29.8509	−1.41826	−0.709129	−	0.705078i	$-0.750912\pi$
$443$	−29.8509	−1.41826	−0.709129	+	0.705078i	$0.750912\pi$
$444$	0	0
$445$	7.21630	0.342085
$446$	−6.54106	−0.309728
$447$	0	0
$448$	2.18432	0.103200
$449$	−9.05651	−0.427403	−0.213701	−	0.976899i	$-0.568552\pi$
$449$	−9.05651	−0.427403	−0.213701	+	0.976899i	$0.568552\pi$
$450$	0	0
$451$	−9.33433	−0.439536
$452$	14.6370	0.688467
$453$	0	0
$454$	−2.84316	−0.133436
$455$	−23.8896	−1.11996
$456$	0	0
$457$	−4.58122	−0.214301	−0.107150	−	0.994243i	$-0.534173\pi$
$457$	−4.58122	−0.214301	−0.107150	+	0.994243i	$0.534173\pi$
$458$	2.85552	0.133430
$459$	0	0
$460$	−9.14773	−0.426515
$461$	27.6814	1.28925	0.644626	−	0.764498i	$-0.277013\pi$
$461$	27.6814	1.28925	0.644626	+	0.764498i	$0.277013\pi$
$462$	0	0
$463$	15.9465	0.741095	0.370547	−	0.928814i	$-0.379170\pi$
$463$	15.9465	0.741095	0.370547	+	0.928814i	$0.379170\pi$
$464$	5.39511	0.250461
$465$	0	0
$466$	24.3743	1.12912
$467$	−7.25960	−0.335934	−0.167967	−	0.985793i	$-0.553720\pi$
$467$	−7.25960	−0.335934	−0.167967	+	0.985793i	$0.553720\pi$
$468$	0	0
$469$	−23.0486	−1.06429
$470$	25.5608	1.17903
$471$	0	0
$472$	2.76719	0.127370
$473$	24.7009	1.13575
$474$	0	0
$475$	−16.0242	−0.735242
$476$	16.3100	0.747569
$477$	0	0
$478$	19.0073	0.869372
$479$	−14.5064	−0.662814	−0.331407	−	0.943488i	$-0.607523\pi$
$479$	−14.5064	−0.662814	−0.331407	+	0.943488i	$0.607523\pi$
$480$	0	0
$481$	17.7150	0.807733
$482$	−27.7788	−1.26529
$483$	0	0
$484$	3.39043	0.154111
$485$	−3.15567	−0.143292
$486$	0	0
$487$	10.5737	0.479139	0.239569	−	0.970879i	$-0.422994\pi$
$487$	10.5737	0.479139	0.239569	+	0.970879i	$0.422994\pi$
$488$	−5.07451	−0.229712
$489$	0	0
$490$	6.20653	0.280382
$491$	24.5582	1.10830	0.554148	−	0.832418i	$-0.313044\pi$
$491$	24.5582	1.10830	0.554148	+	0.832418i	$0.313044\pi$
$492$	0	0
$493$	40.2845	1.81432
$494$	−22.8433	−1.02777
$495$	0	0
$496$	1.61066	0.0723206
$497$	−9.19897	−0.412630
$498$	0	0
$499$	−22.1110	−0.989825	−0.494912	−	0.868943i	$-0.664800\pi$
$499$	−22.1110	−0.989825	−0.494912	+	0.868943i	$0.664800\pi$
$500$	6.25186	0.279592
$501$	0	0
$502$	−8.03049	−0.358418
$503$	38.7917	1.72964	0.864819	−	0.502083i	$-0.167433\pi$
$503$	38.7917	1.72964	0.864819	+	0.502083i	$0.167433\pi$
$504$	0	0
$505$	17.7454	0.789659
$506$	12.4612	0.553967
$507$	0	0
$508$	0.581987	0.0258215
$509$	−25.7730	−1.14237	−0.571185	−	0.820821i	$-0.693516\pi$
$509$	−25.7730	−1.14237	−0.571185	+	0.820821i	$0.693516\pi$
$510$	0	0
$511$	−33.5796	−1.48547
$512$	1.00000	0.0441942
$513$	0	0
$514$	23.7384	1.04706
$515$	−5.09596	−0.224555
$516$	0	0
$517$	−34.8193	−1.53135
$518$	9.85271	0.432903
$519$	0	0
$520$	−10.9368	−0.479612
$521$	−27.2126	−1.19221	−0.596103	−	0.802908i	$-0.703285\pi$
$521$	−27.2126	−1.19221	−0.596103	+	0.802908i	$0.703285\pi$
$522$	0	0
$523$	16.8686	0.737614	0.368807	−	0.929506i	$-0.379766\pi$
$523$	16.8686	0.737614	0.368807	+	0.929506i	$0.379766\pi$
$524$	7.96392	0.347906
$525$	0	0
$526$	−10.4821	−0.457043
$527$	12.0266	0.523885
$528$	0	0
$529$	−12.2094	−0.530844
$530$	5.68649	0.247005
$531$	0	0
$532$	−12.7050	−0.550831
$533$	−9.66379	−0.418585
$534$	0	0
$535$	−47.1506	−2.03850
$536$	−10.5518	−0.455770
$537$	0	0
$538$	6.50883	0.280616
$539$	−8.45464	−0.364167
$540$	0	0
$541$	−41.5755	−1.78747	−0.893735	−	0.448595i	$-0.851925\pi$
$541$	−41.5755	−1.78747	−0.893735	+	0.448595i	$0.851925\pi$
$542$	0.237220	0.0101895
$543$	0	0
$544$	7.46686	0.320139
$545$	19.6218	0.840507
$546$	0	0
$547$	11.1070	0.474901	0.237450	−	0.971400i	$-0.423688\pi$
$547$	11.1070	0.474901	0.237450	+	0.971400i	$0.423688\pi$
$548$	19.3760	0.827702
$549$	0	0
$550$	10.4510	0.445631
$551$	−31.3803	−1.33685
$552$	0	0
$553$	28.8347	1.22618
$554$	−28.0412	−1.19136
$555$	0	0
$556$	5.19559	0.220342
$557$	35.2212	1.49237	0.746185	−	0.665739i	$-0.231884\pi$
$557$	35.2212	1.49237	0.746185	+	0.665739i	$0.231884\pi$
$558$	0	0
$559$	25.5727	1.08161
$560$	−6.08285	−0.257047
$561$	0	0
$562$	−6.74939	−0.284706
$563$	5.27331	0.222244	0.111122	−	0.993807i	$-0.464556\pi$
$563$	5.27331	0.222244	0.111122	+	0.993807i	$0.464556\pi$
$564$	0	0
$565$	−40.7608	−1.71482
$566$	−7.40295	−0.311169
$567$	0	0
$568$	−4.21136	−0.176705
$569$	−21.3483	−0.894967	−0.447483	−	0.894292i	$-0.647680\pi$
$569$	−21.3483	−0.894967	−0.447483	+	0.894292i	$0.647680\pi$
$570$	0	0
$571$	−25.1418	−1.05215	−0.526076	−	0.850438i	$-0.676337\pi$
$571$	−25.1418	−1.05215	−0.526076	+	0.850438i	$0.676337\pi$
$572$	14.8984	0.622931
$573$	0	0
$574$	−5.37481	−0.224340
$575$	9.04987	0.377405
$576$	0	0
$577$	−39.5919	−1.64823	−0.824116	−	0.566422i	$-0.808327\pi$
$577$	−39.5919	−1.64823	−0.824116	+	0.566422i	$0.808327\pi$
$578$	38.7540	1.61195
$579$	0	0
$580$	−15.0242	−0.623845
$581$	9.33508	0.387284
$582$	0	0
$583$	−7.74623	−0.320816
$584$	−15.3730	−0.636139
$585$	0	0
$586$	20.2061	0.834705
$587$	−22.4345	−0.925972	−0.462986	−	0.886366i	$-0.653222\pi$
$587$	−22.4345	−0.925972	−0.462986	+	0.886366i	$0.653222\pi$
$588$	0	0
$589$	−9.36829	−0.386014
$590$	−7.70601	−0.317251
$591$	0	0
$592$	4.51065	0.185386
$593$	−17.3689	−0.713256	−0.356628	−	0.934247i	$-0.616074\pi$
$593$	−17.3689	−0.713256	−0.356628	+	0.934247i	$0.616074\pi$
$594$	0	0
$595$	−45.4198	−1.86203
$596$	−1.00000	−0.0409616
$597$	0	0
$598$	12.9010	0.527562
$599$	−17.2697	−0.705620	−0.352810	−	0.935695i	$-0.614774\pi$
$599$	−17.2697	−0.705620	−0.352810	+	0.935695i	$0.614774\pi$
$600$	0	0
$601$	−11.7210	−0.478109	−0.239055	−	0.971006i	$-0.576837\pi$
$601$	−11.7210	−0.478109	−0.239055	+	0.971006i	$0.576837\pi$
$602$	14.2230	0.579687
$603$	0	0
$604$	3.67260	0.149436
$605$	−9.44160	−0.383856
$606$	0	0
$607$	−30.6373	−1.24353	−0.621764	−	0.783204i	$-0.713584\pi$
$607$	−30.6373	−1.24353	−0.621764	+	0.783204i	$0.713584\pi$
$608$	−5.81644	−0.235888
$609$	0	0
$610$	14.1314	0.572163
$611$	−36.0483	−1.45836
$612$	0	0
$613$	48.1817	1.94604	0.973020	−	0.230721i	$-0.0741086\pi$
$613$	48.1817	1.94604	0.973020	+	0.230721i	$0.0741086\pi$
$614$	9.70048	0.391480
$615$	0	0
$616$	8.28617	0.333859
$617$	20.1467	0.811074	0.405537	−	0.914079i	$-0.367085\pi$
$617$	20.1467	0.811074	0.405537	+	0.914079i	$0.367085\pi$
$618$	0	0
$619$	16.1713	0.649979	0.324990	−	0.945718i	$-0.394639\pi$
$619$	16.1713	0.649979	0.324990	+	0.945718i	$0.394639\pi$
$620$	−4.48532	−0.180135
$621$	0	0
$622$	−15.7395	−0.631095
$623$	−5.66032	−0.226776
$624$	0	0
$625$	−31.1850	−1.24740
$626$	−26.7280	−1.06826
$627$	0	0
$628$	20.6000	0.822029
$629$	33.6804	1.34292
$630$	0	0
$631$	28.3158	1.12723	0.563617	−	0.826036i	$-0.309409\pi$
$631$	28.3158	1.12723	0.563617	+	0.826036i	$0.309409\pi$
$632$	13.2008	0.525098
$633$	0	0
$634$	−26.5139	−1.05300
$635$	−1.62070	−0.0643157
$636$	0	0
$637$	−8.75305	−0.346809
$638$	20.4662	0.810264
$639$	0	0
$640$	−2.78478	−0.110078
$641$	−9.37925	−0.370458	−0.185229	−	0.982695i	$-0.559303\pi$
$641$	−9.37925	−0.370458	−0.185229	+	0.982695i	$0.559303\pi$
$642$	0	0
$643$	−29.3434	−1.15719	−0.578595	−	0.815615i	$-0.696399\pi$
$643$	−29.3434	−1.15719	−0.578595	+	0.815615i	$0.696399\pi$
$644$	7.17529	0.282746
$645$	0	0
$646$	−43.4306	−1.70875
$647$	3.62559	0.142537	0.0712684	−	0.997457i	$-0.477295\pi$
$647$	3.62559	0.142537	0.0712684	+	0.997457i	$0.477295\pi$
$648$	0	0
$649$	10.4973	0.412054
$650$	10.8198	0.424389
$651$	0	0
$652$	−0.781487	−0.0306054
$653$	10.1057	0.395466	0.197733	−	0.980256i	$-0.436642\pi$
$653$	10.1057	0.395466	0.197733	+	0.980256i	$0.436642\pi$
$654$	0	0
$655$	−22.1778	−0.866556
$656$	−2.46063	−0.0960714
$657$	0	0
$658$	−20.0493	−0.781605
$659$	8.51959	0.331876	0.165938	−	0.986136i	$-0.446935\pi$
$659$	8.51959	0.331876	0.165938	+	0.986136i	$0.446935\pi$
$660$	0	0
$661$	−44.2789	−1.72225	−0.861125	−	0.508394i	$-0.830239\pi$
$661$	−44.2789	−1.72225	−0.861125	+	0.508394i	$0.830239\pi$
$662$	4.24045	0.164810
$663$	0	0
$664$	4.27367	0.165851
$665$	35.3806	1.37200
$666$	0	0
$667$	17.7224	0.686214
$668$	−4.43877	−0.171741
$669$	0	0
$670$	29.3845	1.13522
$671$	−19.2500	−0.743139
$672$	0	0
$673$	−27.8084	−1.07193	−0.535967	−	0.844239i	$-0.680053\pi$
$673$	−27.8084	−1.07193	−0.535967	+	0.844239i	$0.680053\pi$
$674$	18.0860	0.696647
$675$	0	0
$676$	2.42420	0.0932386
$677$	1.30462	0.0501406	0.0250703	−	0.999686i	$-0.492019\pi$
$677$	1.30462	0.0501406	0.0250703	+	0.999686i	$0.492019\pi$
$678$	0	0
$679$	2.47524	0.0949911
$680$	−20.7935	−0.797396
$681$	0	0
$682$	6.10998	0.233963
$683$	38.5493	1.47505	0.737524	−	0.675321i	$-0.235995\pi$
$683$	38.5493	1.47505	0.737524	+	0.675321i	$0.235995\pi$
$684$	0	0
$685$	−53.9579	−2.06163
$686$	−20.1585	−0.769656
$687$	0	0
$688$	6.51141	0.248245
$689$	−8.01964	−0.305524
$690$	0	0
$691$	12.5778	0.478483	0.239242	−	0.970960i	$-0.423101\pi$
$691$	12.5778	0.478483	0.239242	+	0.970960i	$0.423101\pi$
$692$	11.7888	0.448141
$693$	0	0
$694$	7.52017	0.285462
$695$	−14.4686	−0.548824
$696$	0	0
$697$	−18.3732	−0.695934
$698$	17.8971	0.677414
$699$	0	0
$700$	6.01778	0.227451
$701$	−31.3534	−1.18420	−0.592101	−	0.805863i	$-0.701702\pi$
$701$	−31.3534	−1.18420	−0.592101	+	0.805863i	$0.701702\pi$
$702$	0	0
$703$	−26.2359	−0.989506
$704$	3.79347	0.142972
$705$	0	0
$706$	18.1266	0.682204
$707$	−13.9191	−0.523482
$708$	0	0
$709$	0.0913585	0.00343104	0.00171552	−	0.999999i	$-0.499454\pi$
$709$	0.0913585	0.00343104	0.00171552	+	0.999999i	$0.499454\pi$
$710$	11.7277	0.440133
$711$	0	0
$712$	−2.59134	−0.0971145
$713$	5.29085	0.198144
$714$	0	0
$715$	−41.4886	−1.55159
$716$	−11.4199	−0.426782
$717$	0	0
$718$	−27.1156	−1.01195
$719$	−10.5501	−0.393452	−0.196726	−	0.980458i	$-0.563031\pi$
$719$	−10.5501	−0.393452	−0.196726	+	0.980458i	$0.563031\pi$
$720$	0	0
$721$	3.99717	0.148862
$722$	14.8310	0.551953
$723$	0	0
$724$	−19.3924	−0.720712
$725$	14.8634	0.552014
$726$	0	0
$727$	41.0484	1.52240	0.761200	−	0.648517i	$-0.224610\pi$
$727$	41.0484	1.52240	0.761200	+	0.648517i	$0.224610\pi$
$728$	8.57864	0.317945
$729$	0	0
$730$	42.8104	1.58448
$731$	48.6198	1.79827
$732$	0	0
$733$	12.1993	0.450591	0.225295	−	0.974290i	$-0.427665\pi$
$733$	12.1993	0.450591	0.225295	+	0.974290i	$0.427665\pi$
$734$	10.5060	0.387783
$735$	0	0
$736$	3.28490	0.121083
$737$	−40.0281	−1.47446
$738$	0	0
$739$	−1.43021	−0.0526110	−0.0263055	−	0.999654i	$-0.508374\pi$
$739$	−1.43021	−0.0526110	−0.0263055	+	0.999654i	$0.508374\pi$
$740$	−12.5611	−0.461757
$741$	0	0
$742$	−4.46036	−0.163745
$743$	41.1915	1.51117	0.755585	−	0.655051i	$-0.227353\pi$
$743$	41.1915	1.51117	0.755585	+	0.655051i	$0.227353\pi$
$744$	0	0
$745$	2.78478	0.102026
$746$	8.41289	0.308018
$747$	0	0
$748$	28.3253	1.03568
$749$	36.9840	1.35137
$750$	0	0
$751$	14.3214	0.522596	0.261298	−	0.965258i	$-0.415849\pi$
$751$	14.3214	0.522596	0.261298	+	0.965258i	$0.415849\pi$
$752$	−9.17875	−0.334714
$753$	0	0
$754$	21.1886	0.771642
$755$	−10.2274	−0.372212
$756$	0	0
$757$	29.3350	1.06620	0.533100	−	0.846052i	$-0.321027\pi$
$757$	29.3350	1.06620	0.533100	+	0.846052i	$0.321027\pi$
$758$	−6.82251	−0.247805
$759$	0	0
$760$	16.1975	0.587545
$761$	−47.3418	−1.71614	−0.858070	−	0.513532i	$-0.828337\pi$
$761$	−47.3418	−1.71614	−0.858070	+	0.513532i	$0.828337\pi$
$762$	0	0
$763$	−15.3910	−0.557190
$764$	13.1005	0.473958
$765$	0	0
$766$	−8.64591	−0.312389
$767$	10.8678	0.392412
$768$	0	0
$769$	2.84485	0.102588	0.0512940	−	0.998684i	$-0.483665\pi$
$769$	2.84485	0.102588	0.0512940	+	0.998684i	$0.483665\pi$
$770$	−23.0751	−0.831570
$771$	0	0
$772$	−4.41114	−0.158760
$773$	12.7120	0.457221	0.228610	−	0.973518i	$-0.426582\pi$
$773$	12.7120	0.457221	0.228610	+	0.973518i	$0.426582\pi$
$774$	0	0
$775$	4.43734	0.159394
$776$	1.13319	0.0406790
$777$	0	0
$778$	16.0651	0.575961
$779$	14.3121	0.512784
$780$	0	0
$781$	−15.9757	−0.571654
$782$	24.5279	0.877117
$783$	0	0
$784$	−2.22873	−0.0795976
$785$	−57.3664	−2.04749
$786$	0	0
$787$	12.5949	0.448961	0.224480	−	0.974479i	$-0.427932\pi$
$787$	12.5949	0.448961	0.224480	+	0.974479i	$0.427932\pi$
$788$	−7.06752	−0.251770
$789$	0	0
$790$	−36.7612	−1.30791
$791$	31.9720	1.13679
$792$	0	0
$793$	−19.9295	−0.707716
$794$	20.5577	0.729565
$795$	0	0
$796$	−19.7015	−0.698301
$797$	1.35952	0.0481568	0.0240784	−	0.999710i	$-0.492335\pi$
$797$	1.35952	0.0481568	0.0240784	+	0.999710i	$0.492335\pi$
$798$	0	0
$799$	−68.5364	−2.42464
$800$	2.75499	0.0974035
$801$	0	0
$802$	−13.3660	−0.471971
$803$	−58.3170	−2.05796
$804$	0	0
$805$	−19.9816	−0.704258
$806$	6.32564	0.222811
$807$	0	0
$808$	−6.37228	−0.224176
$809$	49.2666	1.73212	0.866061	−	0.499939i	$-0.166644\pi$
$809$	49.2666	1.73212	0.866061	+	0.499939i	$0.166644\pi$
$810$	0	0
$811$	−27.8956	−0.979545	−0.489773	−	0.871850i	$-0.662920\pi$
$811$	−27.8956	−0.979545	−0.489773	+	0.871850i	$0.662920\pi$
$812$	11.7847	0.413560
$813$	0	0
$814$	17.1110	0.599741
$815$	2.17627	0.0762313
$816$	0	0
$817$	−37.8733	−1.32502
$818$	31.3584	1.09642
$819$	0	0
$820$	6.85230	0.239293
$821$	2.91747	0.101820	0.0509101	−	0.998703i	$-0.483788\pi$
$821$	2.91747	0.101820	0.0509101	+	0.998703i	$0.483788\pi$
$822$	0	0
$823$	−1.32328	−0.0461267	−0.0230634	−	0.999734i	$-0.507342\pi$
$823$	−1.32328	−0.0461267	−0.0230634	+	0.999734i	$0.507342\pi$
$824$	1.82993	0.0637488
$825$	0	0
$826$	6.04444	0.210313
$827$	15.7276	0.546902	0.273451	−	0.961886i	$-0.411835\pi$
$827$	15.7276	0.546902	0.273451	+	0.961886i	$0.411835\pi$
$828$	0	0
$829$	48.2399	1.67544	0.837720	−	0.546100i	$-0.183888\pi$
$829$	48.2399	1.67544	0.837720	+	0.546100i	$0.183888\pi$
$830$	−11.9012	−0.413098
$831$	0	0
$832$	3.92737	0.136157
$833$	−16.6416	−0.576599
$834$	0	0
$835$	12.3610	0.427769
$836$	−22.0645	−0.763117
$837$	0	0
$838$	18.8568	0.651396
$839$	9.01568	0.311256	0.155628	−	0.987816i	$-0.450260\pi$
$839$	9.01568	0.311256	0.155628	+	0.987816i	$0.450260\pi$
$840$	0	0
$841$	0.107167	0.00369541
$842$	−24.2532	−0.835819
$843$	0	0
$844$	13.5788	0.467400
$845$	−6.75087	−0.232237
$846$	0	0
$847$	7.40580	0.254466
$848$	−2.04199	−0.0701222
$849$	0	0
$850$	20.5711	0.705583
$851$	14.8170	0.507922
$852$	0	0
$853$	−23.0418	−0.788937	−0.394468	−	0.918909i	$-0.629071\pi$
$853$	−23.0418	−0.788937	−0.394468	+	0.918909i	$0.629071\pi$
$854$	−11.0844	−0.379300
$855$	0	0
$856$	16.9316	0.578709
$857$	13.6930	0.467745	0.233872	−	0.972267i	$-0.424860\pi$
$857$	13.6930	0.467745	0.233872	+	0.972267i	$0.424860\pi$
$858$	0	0
$859$	20.9896	0.716156	0.358078	−	0.933692i	$-0.383432\pi$
$859$	20.9896	0.716156	0.358078	+	0.933692i	$0.383432\pi$
$860$	−18.1328	−0.618325
$861$	0	0
$862$	−3.23977	−0.110347
$863$	−45.3334	−1.54317	−0.771583	−	0.636128i	$-0.780535\pi$
$863$	−45.3334	−1.54317	−0.771583	+	0.636128i	$0.780535\pi$
$864$	0	0
$865$	−32.8291	−1.11622
$866$	36.4730	1.23940
$867$	0	0
$868$	3.51819	0.119415
$869$	50.0768	1.69874
$870$	0	0
$871$	−41.4410	−1.40417
$872$	−7.04611	−0.238611
$873$	0	0
$874$	−19.1065	−0.646285
$875$	13.6561	0.461660
$876$	0	0
$877$	8.38726	0.283218	0.141609	−	0.989923i	$-0.454773\pi$
$877$	8.38726	0.283218	0.141609	+	0.989923i	$0.454773\pi$
$878$	12.1537	0.410167
$879$	0	0
$880$	−10.5640	−0.356112
$881$	−56.3886	−1.89978	−0.949890	−	0.312584i	$-0.898805\pi$
$881$	−56.3886	−1.89978	−0.949890	+	0.312584i	$0.898805\pi$
$882$	0	0
$883$	21.7284	0.731220	0.365610	−	0.930768i	$-0.380860\pi$
$883$	21.7284	0.731220	0.365610	+	0.930768i	$0.380860\pi$
$884$	29.3251	0.986310
$885$	0	0
$886$	−29.8509	−1.00286
$887$	−48.5951	−1.63166	−0.815831	−	0.578290i	$-0.803720\pi$
$887$	−48.5951	−1.63166	−0.815831	+	0.578290i	$0.803720\pi$
$888$	0	0
$889$	1.27125	0.0426363
$890$	7.21630	0.241891
$891$	0	0
$892$	−6.54106	−0.219011
$893$	53.3876	1.78655
$894$	0	0
$895$	31.8019	1.06302
$896$	2.18432	0.0729731
$897$	0	0
$898$	−9.05651	−0.302220
$899$	8.68966	0.289817
$900$	0	0
$901$	−15.2473	−0.507960
$902$	−9.33433	−0.310799
$903$	0	0
$904$	14.6370	0.486820
$905$	54.0035	1.79514
$906$	0	0
$907$	33.7567	1.12087	0.560437	−	0.828197i	$-0.310633\pi$
$907$	33.7567	1.12087	0.560437	+	0.828197i	$0.310633\pi$
$908$	−2.84316	−0.0943537
$909$	0	0
$910$	−23.8896	−0.791932
$911$	42.8074	1.41827	0.709137	−	0.705071i	$-0.249085\pi$
$911$	42.8074	1.41827	0.709137	+	0.705071i	$0.249085\pi$
$912$	0	0
$913$	16.2121	0.536541
$914$	−4.58122	−0.151533
$915$	0	0
$916$	2.85552	0.0943490
$917$	17.3958	0.574459
$918$	0	0
$919$	−57.6631	−1.90213	−0.951065	−	0.308991i	$-0.900009\pi$
$919$	−57.6631	−1.90213	−0.951065	+	0.308991i	$0.900009\pi$
$920$	−9.14773	−0.301592
$921$	0	0
$922$	27.6814	0.911638
$923$	−16.5395	−0.544406
$924$	0	0
$925$	12.4268	0.408590
$926$	15.9465	0.524033
$927$	0	0
$928$	5.39511	0.177103
$929$	−14.2154	−0.466393	−0.233197	−	0.972430i	$-0.574919\pi$
$929$	−14.2154	−0.466393	−0.233197	+	0.972430i	$0.574919\pi$
$930$	0	0
$931$	12.9633	0.424855
$932$	24.3743	0.798408
$933$	0	0
$934$	−7.25960	−0.237541
$935$	−78.8797	−2.57964
$936$	0	0
$937$	35.9078	1.17306	0.586528	−	0.809929i	$-0.300494\pi$
$937$	35.9078	1.17306	0.586528	+	0.809929i	$0.300494\pi$
$938$	−23.0486	−0.752565
$939$	0	0
$940$	25.5608	0.833700
$941$	−0.858585	−0.0279891	−0.0139945	−	0.999902i	$-0.504455\pi$
$941$	−0.858585	−0.0279891	−0.0139945	+	0.999902i	$0.504455\pi$
$942$	0	0
$943$	−8.08293	−0.263216
$944$	2.76719	0.0900644
$945$	0	0
$946$	24.7009	0.803094
$947$	4.77011	0.155008	0.0775039	−	0.996992i	$-0.475305\pi$
$947$	4.77011	0.155008	0.0775039	+	0.996992i	$0.475305\pi$
$948$	0	0
$949$	−60.3754	−1.95987
$950$	−16.0242	−0.519894
$951$	0	0
$952$	16.3100	0.528611
$953$	−4.14543	−0.134284	−0.0671419	−	0.997743i	$-0.521388\pi$
$953$	−4.14543	−0.134284	−0.0671419	+	0.997743i	$0.521388\pi$
$954$	0	0
$955$	−36.4819	−1.18053
$956$	19.0073	0.614739
$957$	0	0
$958$	−14.5064	−0.468680
$959$	42.3235	1.36670
$960$	0	0
$961$	−28.4058	−0.916316
$962$	17.7150	0.571153
$963$	0	0
$964$	−27.7788	−0.894695
$965$	12.2840	0.395437
$966$	0	0
$967$	−20.4284	−0.656932	−0.328466	−	0.944516i	$-0.606532\pi$
$967$	−20.4284	−0.656932	−0.328466	+	0.944516i	$0.606532\pi$
$968$	3.39043	0.108973
$969$	0	0
$970$	−3.15567	−0.101322
$971$	−51.7496	−1.66072	−0.830362	−	0.557225i	$-0.811866\pi$
$971$	−51.7496	−1.66072	−0.830362	+	0.557225i	$0.811866\pi$
$972$	0	0
$973$	11.3489	0.363827
$974$	10.5737	0.338802
$975$	0	0
$976$	−5.07451	−0.162431
$977$	−35.5583	−1.13761	−0.568805	−	0.822473i	$-0.692594\pi$
$977$	−35.5583	−1.13761	−0.568805	+	0.822473i	$0.692594\pi$
$978$	0	0
$979$	−9.83016	−0.314173
$980$	6.20653	0.198260
$981$	0	0
$982$	24.5582	0.783683
$983$	−26.5476	−0.846738	−0.423369	−	0.905957i	$-0.639153\pi$
$983$	−26.5476	−0.846738	−0.423369	+	0.905957i	$0.639153\pi$
$984$	0	0
$985$	19.6815	0.627104
$986$	40.2845	1.28292
$987$	0	0
$988$	−22.8433	−0.726742
$989$	21.3894	0.680142
$990$	0	0
$991$	−31.7753	−1.00938	−0.504688	−	0.863302i	$-0.668393\pi$
$991$	−31.7753	−1.00938	−0.504688	+	0.863302i	$0.668393\pi$
$992$	1.61066	0.0511384
$993$	0	0
$994$	−9.19897	−0.291773
$995$	54.8643	1.73931
$996$	0	0
$997$	−17.5169	−0.554765	−0.277382	−	0.960760i	$-0.589467\pi$
$997$	−17.5169	−0.554765	−0.277382	+	0.960760i	$0.589467\pi$
$998$	−22.1110	−0.699912
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8046.2.a.p.1.2	yes	12
3.2	odd	2		8046.2.a.i.1.11	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8046.2.a.i.1.11	✓	12	3.2	odd	2
8046.2.a.p.1.2	yes	12	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} -2.78478 q^{5} +2.18432 q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} -2.78478 q^{5} +2.18432 q^{7} +1.00000 q^{8} -2.78478 q^{10} +3.79347 q^{11} +3.92737 q^{13} +2.18432 q^{14} +1.00000 q^{16} +7.46686 q^{17} -5.81644 q^{19} -2.78478 q^{20} +3.79347 q^{22} +3.28490 q^{23} +2.75499 q^{25} +3.92737 q^{26} +2.18432 q^{28} +5.39511 q^{29} +1.61066 q^{31} +1.00000 q^{32} +7.46686 q^{34} -6.08285 q^{35} +4.51065 q^{37} -5.81644 q^{38} -2.78478 q^{40} -2.46063 q^{41} +6.51141 q^{43} +3.79347 q^{44} +3.28490 q^{46} -9.17875 q^{47} -2.22873 q^{49} +2.75499 q^{50} +3.92737 q^{52} -2.04199 q^{53} -10.5640 q^{55} +2.18432 q^{56} +5.39511 q^{58} +2.76719 q^{59} -5.07451 q^{61} +1.61066 q^{62} +1.00000 q^{64} -10.9368 q^{65} -10.5518 q^{67} +7.46686 q^{68} -6.08285 q^{70} -4.21136 q^{71} -15.3730 q^{73} +4.51065 q^{74} -5.81644 q^{76} +8.28617 q^{77} +13.2008 q^{79} -2.78478 q^{80} -2.46063 q^{82} +4.27367 q^{83} -20.7935 q^{85} +6.51141 q^{86} +3.79347 q^{88} -2.59134 q^{89} +8.57864 q^{91} +3.28490 q^{92} -9.17875 q^{94} +16.1975 q^{95} +1.13319 q^{97} -2.22873 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 12 q^{2} + 12 q^{4} + 5 q^{5} + 6 q^{7} + 12 q^{8}+O(q^{10}) \) 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

Modular forms

Varieties

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Representations

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Database

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