LMFDB - Embedded newform 4338.2.a.u.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4338,2,Mod(1,4338)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4338, 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("4338.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4338 = 2 \cdot 3^{2} \cdot 241 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4338.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:	$34.6391043968$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.6.131357120.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 10x^{4} + 16x^{3} + 26x^{2} - 30x - 13 $ x^6 - 2x^5 - 10x^4 + 16x^3 + 26x^2 - 30*x - 13
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 482)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$-2.34780$ of defining polynomial
Character	$\chi$	$=$	4338.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} +3.81130 q^{5} +0.407599 q^{7} +1.00000 q^{8} +O(q^{10})$	\(q+1.00000 q^{2} +1.00000 q^{4} +3.81130 q^{5} +0.407599 q^{7} +1.00000 q^{8} +3.81130 q^{10} +4.85999 q^{11} -2.78733 q^{13} +0.407599 q^{14} +1.00000 q^{16} +1.04869 q^{17} -2.71468 q^{19} +3.81130 q^{20} +4.85999 q^{22} -8.67128 q^{23} +9.52598 q^{25} -2.78733 q^{26} +0.407599 q^{28} +4.08819 q^{29} -0.325219 q^{31} +1.00000 q^{32} +1.04869 q^{34} +1.55348 q^{35} +11.4919 q^{37} -2.71468 q^{38} +3.81130 q^{40} -0.439895 q^{41} +9.74430 q^{43} +4.85999 q^{44} -8.67128 q^{46} +4.85474 q^{47} -6.83386 q^{49} +9.52598 q^{50} -2.78733 q^{52} +8.80287 q^{53} +18.5228 q^{55} +0.407599 q^{56} +4.08819 q^{58} -8.47943 q^{59} -6.37079 q^{61} -0.325219 q^{62} +1.00000 q^{64} -10.6234 q^{65} -3.46878 q^{67} +1.04869 q^{68} +1.55348 q^{70} +6.11292 q^{71} +2.77180 q^{73} +11.4919 q^{74} -2.71468 q^{76} +1.98092 q^{77} +8.48606 q^{79} +3.81130 q^{80} -0.439895 q^{82} -15.0594 q^{83} +3.99686 q^{85} +9.74430 q^{86} +4.85999 q^{88} -5.55559 q^{89} -1.13611 q^{91} -8.67128 q^{92} +4.85474 q^{94} -10.3465 q^{95} -12.8685 q^{97} -6.83386 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 6 q^{2} + 6 q^{4} + 5 q^{5} + 10 q^{7} + 6 q^{8}+O(q^{10}) $ 6 * q + 6 * q^2 + 6 * q^4 + 5 * q^5 + 10 * q^7 + 6 * q^8	$ 6 q + 6 q^{2} + 6 q^{4} + 5 q^{5} + 10 q^{7} + 6 q^{8} + 5 q^{10} + 4 q^{11} + 9 q^{13} + 10 q^{14} + 6 q^{16} - q^{17} + 10 q^{19} + 5 q^{20} + 4 q^{22} - 9 q^{23} + 13 q^{25} + 9 q^{26} + 10 q^{28} + q^{29} + 14 q^{31} + 6 q^{32} - q^{34} + 3 q^{35} + 20 q^{37} + 10 q^{38} + 5 q^{40} + 4 q^{41} + 19 q^{43} + 4 q^{44} - 9 q^{46} - q^{47} + 14 q^{49} + 13 q^{50} + 9 q^{52} + 3 q^{53} + 11 q^{55} + 10 q^{56} + q^{58} + q^{59} + 4 q^{61} + 14 q^{62} + 6 q^{64} - 5 q^{65} + 5 q^{67} - q^{68} + 3 q^{70} - 2 q^{71} + 15 q^{73} + 20 q^{74} + 10 q^{76} + 6 q^{77} + 12 q^{79} + 5 q^{80} + 4 q^{82} + 8 q^{83} - 32 q^{85} + 19 q^{86} + 4 q^{88} + 24 q^{89} + q^{91} - 9 q^{92} - q^{94} - 9 q^{95} - 12 q^{97} + 14 q^{98}+O(q^{100}) $ 6 * q + 6 * q^2 + 6 * q^4 + 5 * q^5 + 10 * q^7 + 6 * q^8 + 5 * q^10 + 4 * q^11 + 9 * q^13 + 10 * q^14 + 6 * q^16 - q^17 + 10 * q^19 + 5 * q^20 + 4 * q^22 - 9 * q^23 + 13 * q^25 + 9 * q^26 + 10 * q^28 + q^29 + 14 * q^31 + 6 * q^32 - q^34 + 3 * q^35 + 20 * q^37 + 10 * q^38 + 5 * q^40 + 4 * q^41 + 19 * q^43 + 4 * q^44 - 9 * q^46 - q^47 + 14 * q^49 + 13 * q^50 + 9 * q^52 + 3 * q^53 + 11 * q^55 + 10 * q^56 + q^58 + q^59 + 4 * q^61 + 14 * q^62 + 6 * q^64 - 5 * q^65 + 5 * q^67 - q^68 + 3 * q^70 - 2 * q^71 + 15 * q^73 + 20 * q^74 + 10 * q^76 + 6 * q^77 + 12 * q^79 + 5 * q^80 + 4 * q^82 + 8 * q^83 - 32 * q^85 + 19 * q^86 + 4 * q^88 + 24 * q^89 + q^91 - 9 * q^92 - q^94 - 9 * q^95 - 12 * 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
$5$	3.81130	1.70446	0.852232	−	0.523164i	$-0.175249\pi$
$5$	3.81130	1.70446	0.852232	+	0.523164i	$0.175249\pi$
$6$	0	0
$7$	0.407599	0.154058	0.0770289	−	0.997029i	$-0.475457\pi$
$7$	0.407599	0.154058	0.0770289	+	0.997029i	$0.475457\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	3.81130	1.20524
$11$	4.85999	1.46534	0.732670	−	0.680584i	$-0.238274\pi$
$11$	4.85999	1.46534	0.732670	+	0.680584i	$0.238274\pi$
$12$	0	0
$13$	−2.78733	−0.773067	−0.386534	−	0.922275i	$-0.626328\pi$
$13$	−2.78733	−0.773067	−0.386534	+	0.922275i	$0.626328\pi$
$14$	0.407599	0.108935
$15$	0	0
$16$	1.00000	0.250000
$17$	1.04869	0.254344	0.127172	−	0.991881i	$-0.459410\pi$
$17$	1.04869	0.254344	0.127172	+	0.991881i	$0.459410\pi$
$18$	0	0
$19$	−2.71468	−0.622791	−0.311396	−	0.950280i	$-0.600796\pi$
$19$	−2.71468	−0.622791	−0.311396	+	0.950280i	$0.600796\pi$
$20$	3.81130	0.852232
$21$	0	0
$22$	4.85999	1.03615
$23$	−8.67128	−1.80809	−0.904044	−	0.427440i	$-0.859415\pi$
$23$	−8.67128	−1.80809	−0.904044	+	0.427440i	$0.859415\pi$
$24$	0	0
$25$	9.52598	1.90520
$26$	−2.78733	−0.546641
$27$	0	0
$28$	0.407599	0.0770289
$29$	4.08819	0.759158	0.379579	−	0.925159i	$-0.376069\pi$
$29$	4.08819	0.759158	0.379579	+	0.925159i	$0.376069\pi$
$30$	0	0
$31$	−0.325219	−0.0584110	−0.0292055	−	0.999573i	$-0.509298\pi$
$31$	−0.325219	−0.0584110	−0.0292055	+	0.999573i	$0.509298\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	1.04869	0.179849
$35$	1.55348	0.262586
$36$	0	0
$37$	11.4919	1.88925	0.944627	−	0.328145i	$-0.106423\pi$
$37$	11.4919	1.88925	0.944627	+	0.328145i	$0.106423\pi$
$38$	−2.71468	−0.440380
$39$	0	0
$40$	3.81130	0.602619
$41$	−0.439895	−0.0687000	−0.0343500	−	0.999410i	$-0.510936\pi$
$41$	−0.439895	−0.0687000	−0.0343500	+	0.999410i	$0.510936\pi$
$42$	0	0
$43$	9.74430	1.48599	0.742996	−	0.669296i	$-0.233404\pi$
$43$	9.74430	1.48599	0.742996	+	0.669296i	$0.233404\pi$
$44$	4.85999	0.732670
$45$	0	0
$46$	−8.67128	−1.27851
$47$	4.85474	0.708136	0.354068	−	0.935220i	$-0.384798\pi$
$47$	4.85474	0.708136	0.354068	+	0.935220i	$0.384798\pi$
$48$	0	0
$49$	−6.83386	−0.976266
$50$	9.52598	1.34718
$51$	0	0
$52$	−2.78733	−0.386534
$53$	8.80287	1.20917	0.604584	−	0.796542i	$-0.293339\pi$
$53$	8.80287	1.20917	0.604584	+	0.796542i	$0.293339\pi$
$54$	0	0
$55$	18.5228	2.49762
$56$	0.407599	0.0544677
$57$	0	0
$58$	4.08819	0.536806
$59$	−8.47943	−1.10393	−0.551964	−	0.833868i	$-0.686121\pi$
$59$	−8.47943	−1.10393	−0.551964	+	0.833868i	$0.686121\pi$
$60$	0	0
$61$	−6.37079	−0.815696	−0.407848	−	0.913050i	$-0.633721\pi$
$61$	−6.37079	−0.815696	−0.407848	+	0.913050i	$0.633721\pi$
$62$	−0.325219	−0.0413028
$63$	0	0
$64$	1.00000	0.125000
$65$	−10.6234	−1.31767
$66$	0	0
$67$	−3.46878	−0.423779	−0.211889	−	0.977294i	$-0.567962\pi$
$67$	−3.46878	−0.423779	−0.211889	+	0.977294i	$0.567962\pi$
$68$	1.04869	0.127172
$69$	0	0
$70$	1.55348	0.185676
$71$	6.11292	0.725470	0.362735	−	0.931892i	$-0.381843\pi$
$71$	6.11292	0.725470	0.362735	+	0.931892i	$0.381843\pi$
$72$	0	0
$73$	2.77180	0.324414	0.162207	−	0.986757i	$-0.448139\pi$
$73$	2.77180	0.324414	0.162207	+	0.986757i	$0.448139\pi$
$74$	11.4919	1.33590
$75$	0	0
$76$	−2.71468	−0.311396
$77$	1.98092	0.225747
$78$	0	0
$79$	8.48606	0.954756	0.477378	−	0.878698i	$-0.341587\pi$
$79$	8.48606	0.954756	0.477378	+	0.878698i	$0.341587\pi$
$80$	3.81130	0.426116
$81$	0	0
$82$	−0.439895	−0.0485782
$83$	−15.0594	−1.65298	−0.826489	−	0.562952i	$-0.809665\pi$
$83$	−15.0594	−1.65298	−0.826489	+	0.562952i	$0.809665\pi$
$84$	0	0
$85$	3.99686	0.433521
$86$	9.74430	1.05075
$87$	0	0
$88$	4.85999	0.518076
$89$	−5.55559	−0.588892	−0.294446	−	0.955668i	$-0.595135\pi$
$89$	−5.55559	−0.588892	−0.294446	+	0.955668i	$0.595135\pi$
$90$	0	0
$91$	−1.13611	−0.119097
$92$	−8.67128	−0.904044
$93$	0	0
$94$	4.85474	0.500728
$95$	−10.3465	−1.06152
$96$	0	0
$97$	−12.8685	−1.30660	−0.653301	−	0.757098i	$-0.726616\pi$
$97$	−12.8685	−1.30660	−0.653301	+	0.757098i	$0.726616\pi$
$98$	−6.83386	−0.690324
$99$	0	0
$100$	9.52598	0.952598
$101$	6.52011	0.648776	0.324388	−	0.945924i	$-0.394842\pi$
$101$	6.52011	0.648776	0.324388	+	0.945924i	$0.394842\pi$
$102$	0	0
$103$	−3.90343	−0.384616	−0.192308	−	0.981335i	$-0.561597\pi$
$103$	−3.90343	−0.384616	−0.192308	+	0.981335i	$0.561597\pi$
$104$	−2.78733	−0.273321
$105$	0	0
$106$	8.80287	0.855011
$107$	−2.59961	−0.251313	−0.125657	−	0.992074i	$-0.540104\pi$
$107$	−2.59961	−0.251313	−0.125657	+	0.992074i	$0.540104\pi$
$108$	0	0
$109$	1.89496	0.181505	0.0907524	−	0.995873i	$-0.471073\pi$
$109$	1.89496	0.181505	0.0907524	+	0.995873i	$0.471073\pi$
$110$	18.5228	1.76608
$111$	0	0
$112$	0.407599	0.0385145
$113$	−5.32601	−0.501029	−0.250514	−	0.968113i	$-0.580600\pi$
$113$	−5.32601	−0.501029	−0.250514	+	0.968113i	$0.580600\pi$
$114$	0	0
$115$	−33.0488	−3.08182
$116$	4.08819	0.379579
$117$	0	0
$118$	−8.47943	−0.780595
$119$	0.427444	0.0391837
$120$	0	0
$121$	12.6195	1.14722
$122$	−6.37079	−0.576784
$123$	0	0
$124$	−0.325219	−0.0292055
$125$	17.2499	1.54287
$126$	0	0
$127$	−0.340598	−0.0302232	−0.0151116	−	0.999886i	$-0.504810\pi$
$127$	−0.340598	−0.0302232	−0.0151116	+	0.999886i	$0.504810\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	−10.6234	−0.931730
$131$	−7.55165	−0.659791	−0.329896	−	0.944017i	$-0.607014\pi$
$131$	−7.55165	−0.659791	−0.329896	+	0.944017i	$0.607014\pi$
$132$	0	0
$133$	−1.10650	−0.0959458
$134$	−3.46878	−0.299657
$135$	0	0
$136$	1.04869	0.0899243
$137$	21.5761	1.84337	0.921683	−	0.387943i	$-0.126814\pi$
$137$	21.5761	1.84337	0.921683	+	0.387943i	$0.126814\pi$
$138$	0	0
$139$	−21.1211	−1.79147	−0.895733	−	0.444592i	$-0.853349\pi$
$139$	−21.1211	−1.79147	−0.895733	+	0.444592i	$0.853349\pi$
$140$	1.55348	0.131293
$141$	0	0
$142$	6.11292	0.512985
$143$	−13.5464	−1.13281
$144$	0	0
$145$	15.5813	1.29396
$146$	2.77180	0.229395
$147$	0	0
$148$	11.4919	0.944627
$149$	3.93654	0.322494	0.161247	−	0.986914i	$-0.448448\pi$
$149$	3.93654	0.322494	0.161247	+	0.986914i	$0.448448\pi$
$150$	0	0
$151$	8.17568	0.665327	0.332664	−	0.943046i	$-0.392053\pi$
$151$	8.17568	0.665327	0.332664	+	0.943046i	$0.392053\pi$
$152$	−2.71468	−0.220190
$153$	0	0
$154$	1.98092	0.159627
$155$	−1.23951	−0.0995595
$156$	0	0
$157$	0.0864784	0.00690172	0.00345086	−	0.999994i	$-0.498902\pi$
$157$	0.0864784	0.00690172	0.00345086	+	0.999994i	$0.498902\pi$
$158$	8.48606	0.675115
$159$	0	0
$160$	3.81130	0.301309
$161$	−3.53440	−0.278550
$162$	0	0
$163$	18.9831	1.48687	0.743435	−	0.668809i	$-0.233195\pi$
$163$	18.9831	1.48687	0.743435	+	0.668809i	$0.233195\pi$
$164$	−0.439895	−0.0343500
$165$	0	0
$166$	−15.0594	−1.16883
$167$	−9.43290	−0.729940	−0.364970	−	0.931019i	$-0.618921\pi$
$167$	−9.43290	−0.729940	−0.364970	+	0.931019i	$0.618921\pi$
$168$	0	0
$169$	−5.23077	−0.402367
$170$	3.99686	0.306545
$171$	0	0
$172$	9.74430	0.742996
$173$	−18.2796	−1.38977	−0.694887	−	0.719119i	$-0.744546\pi$
$173$	−18.2796	−1.38977	−0.694887	+	0.719119i	$0.744546\pi$
$174$	0	0
$175$	3.88278	0.293510
$176$	4.85999	0.366335
$177$	0	0
$178$	−5.55559	−0.416409
$179$	−12.8522	−0.960617	−0.480309	−	0.877100i	$-0.659475\pi$
$179$	−12.8522	−0.960617	−0.480309	+	0.877100i	$0.659475\pi$
$180$	0	0
$181$	21.3281	1.58531	0.792654	−	0.609672i	$-0.208699\pi$
$181$	21.3281	1.58531	0.792654	+	0.609672i	$0.208699\pi$
$182$	−1.13611	−0.0842144
$183$	0	0
$184$	−8.67128	−0.639255
$185$	43.7990	3.22017
$186$	0	0
$187$	5.09661	0.372701
$188$	4.85474	0.354068
$189$	0	0
$190$	−10.3465	−0.750611
$191$	−17.7870	−1.28702	−0.643510	−	0.765438i	$-0.722523\pi$
$191$	−17.7870	−1.28702	−0.643510	+	0.765438i	$0.722523\pi$
$192$	0	0
$193$	−14.3394	−1.03217	−0.516087	−	0.856536i	$-0.672612\pi$
$193$	−14.3394	−1.03217	−0.516087	+	0.856536i	$0.672612\pi$
$194$	−12.8685	−0.923907
$195$	0	0
$196$	−6.83386	−0.488133
$197$	−25.1458	−1.79156	−0.895782	−	0.444494i	$-0.853384\pi$
$197$	−25.1458	−1.79156	−0.895782	+	0.444494i	$0.853384\pi$
$198$	0	0
$199$	10.2795	0.728698	0.364349	−	0.931263i	$-0.381292\pi$
$199$	10.2795	0.728698	0.364349	+	0.931263i	$0.381292\pi$
$200$	9.52598	0.673589
$201$	0	0
$202$	6.52011	0.458754
$203$	1.66634	0.116954
$204$	0	0
$205$	−1.67657	−0.117097
$206$	−3.90343	−0.271965
$207$	0	0
$208$	−2.78733	−0.193267
$209$	−13.1933	−0.912601
$210$	0	0
$211$	28.0422	1.93050	0.965252	−	0.261320i	$-0.0841577\pi$
$211$	28.0422	1.93050	0.965252	+	0.261320i	$0.0841577\pi$
$212$	8.80287	0.604584
$213$	0	0
$214$	−2.59961	−0.177705
$215$	37.1384	2.53282
$216$	0	0
$217$	−0.132559	−0.00899868
$218$	1.89496	0.128343
$219$	0	0
$220$	18.5228	1.24881
$221$	−2.92305	−0.196625
$222$	0	0
$223$	−12.9060	−0.864247	−0.432123	−	0.901814i	$-0.642236\pi$
$223$	−12.9060	−0.864247	−0.432123	+	0.901814i	$0.642236\pi$
$224$	0.407599	0.0272338
$225$	0	0
$226$	−5.32601	−0.354281
$227$	24.6376	1.63525	0.817627	−	0.575748i	$-0.195289\pi$
$227$	24.6376	1.63525	0.817627	+	0.575748i	$0.195289\pi$
$228$	0	0
$229$	−8.09344	−0.534830	−0.267415	−	0.963582i	$-0.586169\pi$
$229$	−8.09344	−0.534830	−0.267415	+	0.963582i	$0.586169\pi$
$230$	−33.0488	−2.17917
$231$	0	0
$232$	4.08819	0.268403
$233$	−10.8370	−0.709958	−0.354979	−	0.934874i	$-0.615512\pi$
$233$	−10.8370	−0.709958	−0.354979	+	0.934874i	$0.615512\pi$
$234$	0	0
$235$	18.5028	1.20699
$236$	−8.47943	−0.551964
$237$	0	0
$238$	0.427444	0.0277071
$239$	7.75943	0.501916	0.250958	−	0.967998i	$-0.419254\pi$
$239$	7.75943	0.501916	0.250958	+	0.967998i	$0.419254\pi$
$240$	0	0
$241$	1.00000	0.0644157
$241$	1.00000	0.0644157
$242$	12.6195	0.811209
$243$	0	0
$244$	−6.37079	−0.407848
$245$	−26.0459	−1.66401
$246$	0	0
$247$	7.56673	0.481460
$248$	−0.325219	−0.0206514
$249$	0	0
$250$	17.2499	1.09098
$251$	−22.6364	−1.42880	−0.714399	−	0.699738i	$-0.753300\pi$
$251$	−22.6364	−1.42880	−0.714399	+	0.699738i	$0.753300\pi$
$252$	0	0
$253$	−42.1423	−2.64946
$254$	−0.340598	−0.0213710
$255$	0	0
$256$	1.00000	0.0625000
$257$	−0.167485	−0.0104474	−0.00522370	−	0.999986i	$-0.501663\pi$
$257$	−0.167485	−0.0104474	−0.00522370	+	0.999986i	$0.501663\pi$
$258$	0	0
$259$	4.68408	0.291054
$260$	−10.6234	−0.658833
$261$	0	0
$262$	−7.55165	−0.466543
$263$	11.1147	0.685363	0.342682	−	0.939452i	$-0.388665\pi$
$263$	11.1147	0.685363	0.342682	+	0.939452i	$0.388665\pi$
$264$	0	0
$265$	33.5504	2.06098
$266$	−1.10650	−0.0678440
$267$	0	0
$268$	−3.46878	−0.211889
$269$	9.61340	0.586140	0.293070	−	0.956091i	$-0.405323\pi$
$269$	9.61340	0.586140	0.293070	+	0.956091i	$0.405323\pi$
$270$	0	0
$271$	25.3811	1.54179	0.770895	−	0.636962i	$-0.219809\pi$
$271$	25.3811	1.54179	0.770895	+	0.636962i	$0.219809\pi$
$272$	1.04869	0.0635861
$273$	0	0
$274$	21.5761	1.30346
$275$	46.2961	2.79176
$276$	0	0
$277$	23.2504	1.39698	0.698492	−	0.715618i	$-0.253855\pi$
$277$	23.2504	1.39698	0.698492	+	0.715618i	$0.253855\pi$
$278$	−21.1211	−1.26676
$279$	0	0
$280$	1.55348	0.0928381
$281$	28.3271	1.68985	0.844926	−	0.534883i	$-0.179644\pi$
$281$	28.3271	1.68985	0.844926	+	0.534883i	$0.179644\pi$
$282$	0	0
$283$	−11.6889	−0.694833	−0.347416	−	0.937711i	$-0.612941\pi$
$283$	−11.6889	−0.694833	−0.347416	+	0.937711i	$0.612941\pi$
$284$	6.11292	0.362735
$285$	0	0
$286$	−13.5464	−0.801016
$287$	−0.179300	−0.0105838
$288$	0	0
$289$	−15.9003	−0.935309
$290$	15.5813	0.914966
$291$	0	0
$292$	2.77180	0.162207
$293$	−25.5328	−1.49164	−0.745821	−	0.666146i	$-0.767943\pi$
$293$	−25.5328	−1.49164	−0.745821	+	0.666146i	$0.767943\pi$
$294$	0	0
$295$	−32.3176	−1.88161
$296$	11.4919	0.667952
$297$	0	0
$298$	3.93654	0.228037
$299$	24.1698	1.39777
$300$	0	0
$301$	3.97176	0.228929
$302$	8.17568	0.470457
$303$	0	0
$304$	−2.71468	−0.155698
$305$	−24.2810	−1.39032
$306$	0	0
$307$	−20.6109	−1.17633	−0.588163	−	0.808743i	$-0.700149\pi$
$307$	−20.6109	−1.17633	−0.588163	+	0.808743i	$0.700149\pi$
$308$	1.98092	0.112874
$309$	0	0
$310$	−1.23951	−0.0703992
$311$	−26.3769	−1.49569	−0.747847	−	0.663871i	$-0.768912\pi$
$311$	−26.3769	−1.49569	−0.747847	+	0.663871i	$0.768912\pi$
$312$	0	0
$313$	20.2220	1.14301	0.571507	−	0.820597i	$-0.306359\pi$
$313$	20.2220	1.14301	0.571507	+	0.820597i	$0.306359\pi$
$314$	0.0864784	0.00488026
$315$	0	0
$316$	8.48606	0.477378
$317$	−18.1125	−1.01730	−0.508651	−	0.860973i	$-0.669856\pi$
$317$	−18.1125	−1.01730	−0.508651	+	0.860973i	$0.669856\pi$
$318$	0	0
$319$	19.8685	1.11242
$320$	3.81130	0.213058
$321$	0	0
$322$	−3.53440	−0.196965
$323$	−2.84686	−0.158403
$324$	0	0
$325$	−26.5521	−1.47285
$326$	18.9831	1.05138
$327$	0	0
$328$	−0.439895	−0.0242891
$329$	1.97878	0.109094
$330$	0	0
$331$	−25.9799	−1.42798	−0.713992	−	0.700154i	$-0.753115\pi$
$331$	−25.9799	−1.42798	−0.713992	+	0.700154i	$0.753115\pi$
$332$	−15.0594	−0.826489
$333$	0	0
$334$	−9.43290	−0.516146
$335$	−13.2205	−0.722316
$336$	0	0
$337$	29.6685	1.61615	0.808073	−	0.589082i	$-0.200510\pi$
$337$	29.6685	1.61615	0.808073	+	0.589082i	$0.200510\pi$
$338$	−5.23077	−0.284516
$339$	0	0
$340$	3.99686	0.216760
$341$	−1.58056	−0.0855921
$342$	0	0
$343$	−5.63866	−0.304459
$344$	9.74430	0.525377
$345$	0	0
$346$	−18.2796	−0.982719
$347$	5.43400	0.291713	0.145856	−	0.989306i	$-0.453406\pi$
$347$	5.43400	0.291713	0.145856	+	0.989306i	$0.453406\pi$
$348$	0	0
$349$	−10.7514	−0.575510	−0.287755	−	0.957704i	$-0.592909\pi$
$349$	−10.7514	−0.575510	−0.287755	+	0.957704i	$0.592909\pi$
$350$	3.88278	0.207543
$351$	0	0
$352$	4.85999	0.259038
$353$	16.6773	0.887645	0.443822	−	0.896115i	$-0.353622\pi$
$353$	16.6773	0.887645	0.443822	+	0.896115i	$0.353622\pi$
$354$	0	0
$355$	23.2981	1.23654
$356$	−5.55559	−0.294446
$357$	0	0
$358$	−12.8522	−0.679259
$359$	−15.4699	−0.816469	−0.408234	−	0.912877i	$-0.633855\pi$
$359$	−15.4699	−0.816469	−0.408234	+	0.912877i	$0.633855\pi$
$360$	0	0
$361$	−11.6305	−0.612131
$362$	21.3281	1.12098
$363$	0	0
$364$	−1.13611	−0.0595485
$365$	10.5641	0.552952
$366$	0	0
$367$	−20.6403	−1.07741	−0.538706	−	0.842494i	$-0.681087\pi$
$367$	−20.6403	−1.07741	−0.538706	+	0.842494i	$0.681087\pi$
$368$	−8.67128	−0.452022
$369$	0	0
$370$	43.7990	2.27700
$371$	3.58804	0.186282
$372$	0	0
$373$	13.3245	0.689914	0.344957	−	0.938618i	$-0.387893\pi$
$373$	13.3245	0.689914	0.344957	+	0.938618i	$0.387893\pi$
$374$	5.09661	0.263540
$375$	0	0
$376$	4.85474	0.250364
$377$	−11.3952	−0.586880
$378$	0	0
$379$	−19.3631	−0.994614	−0.497307	−	0.867575i	$-0.665678\pi$
$379$	−19.3631	−0.994614	−0.497307	+	0.867575i	$0.665678\pi$
$380$	−10.3465	−0.530762
$381$	0	0
$382$	−17.7870	−0.910061
$383$	−19.0127	−0.971503	−0.485751	−	0.874097i	$-0.661454\pi$
$383$	−19.0127	−0.971503	−0.485751	+	0.874097i	$0.661454\pi$
$384$	0	0
$385$	7.54989	0.384778
$386$	−14.3394	−0.729857
$387$	0	0
$388$	−12.8685	−0.653301
$389$	−6.76286	−0.342891	−0.171445	−	0.985194i	$-0.554844\pi$
$389$	−6.76286	−0.342891	−0.171445	+	0.985194i	$0.554844\pi$
$390$	0	0
$391$	−9.09348	−0.459877
$392$	−6.83386	−0.345162
$393$	0	0
$394$	−25.1458	−1.26683
$395$	32.3429	1.62735
$396$	0	0
$397$	24.0133	1.20519	0.602596	−	0.798047i	$-0.294133\pi$
$397$	24.0133	1.20519	0.602596	+	0.798047i	$0.294133\pi$
$398$	10.2795	0.515267
$399$	0	0
$400$	9.52598	0.476299
$401$	−2.68925	−0.134295	−0.0671474	−	0.997743i	$-0.521390\pi$
$401$	−2.68925	−0.134295	−0.0671474	+	0.997743i	$0.521390\pi$
$402$	0	0
$403$	0.906494	0.0451557
$404$	6.52011	0.324388
$405$	0	0
$406$	1.66634	0.0826991
$407$	55.8504	2.76840
$408$	0	0
$409$	8.26430	0.408643	0.204322	−	0.978904i	$-0.434501\pi$
$409$	8.26430	0.408643	0.204322	+	0.978904i	$0.434501\pi$
$410$	−1.67657	−0.0827998
$411$	0	0
$412$	−3.90343	−0.192308
$413$	−3.45621	−0.170069
$414$	0	0
$415$	−57.3957	−2.81744
$416$	−2.78733	−0.136660
$417$	0	0
$418$	−13.1933	−0.645307
$419$	16.5813	0.810048	0.405024	−	0.914306i	$-0.367263\pi$
$419$	16.5813	0.810048	0.405024	+	0.914306i	$0.367263\pi$
$420$	0	0
$421$	−18.1121	−0.882731	−0.441366	−	0.897327i	$-0.645506\pi$
$421$	−18.1121	−0.882731	−0.441366	+	0.897327i	$0.645506\pi$
$422$	28.0422	1.36507
$423$	0	0
$424$	8.80287	0.427505
$425$	9.98979	0.484576
$426$	0	0
$427$	−2.59673	−0.125664
$428$	−2.59961	−0.125657
$429$	0	0
$430$	37.1384	1.79097
$431$	9.68282	0.466405	0.233202	−	0.972428i	$-0.425080\pi$
$431$	9.68282	0.466405	0.233202	+	0.972428i	$0.425080\pi$
$432$	0	0
$433$	26.4467	1.27095	0.635473	−	0.772123i	$-0.280805\pi$
$433$	26.4467	1.27095	0.635473	+	0.772123i	$0.280805\pi$
$434$	−0.132559	−0.00636303
$435$	0	0
$436$	1.89496	0.0907524
$437$	23.5398	1.12606
$438$	0	0
$439$	−17.6829	−0.843960	−0.421980	−	0.906605i	$-0.638665\pi$
$439$	−17.6829	−0.843960	−0.421980	+	0.906605i	$0.638665\pi$
$440$	18.5228	0.883042
$441$	0	0
$442$	−2.92305	−0.139035
$443$	8.39543	0.398879	0.199440	−	0.979910i	$-0.436088\pi$
$443$	8.39543	0.398879	0.199440	+	0.979910i	$0.436088\pi$
$444$	0	0
$445$	−21.1740	−1.00374
$446$	−12.9060	−0.611115
$447$	0	0
$448$	0.407599	0.0192572
$449$	−29.1627	−1.37627	−0.688137	−	0.725581i	$-0.741571\pi$
$449$	−29.1627	−1.37627	−0.688137	+	0.725581i	$0.741571\pi$
$450$	0	0
$451$	−2.13788	−0.100669
$452$	−5.32601	−0.250514
$453$	0	0
$454$	24.6376	1.15630
$455$	−4.33007	−0.202997
$456$	0	0
$457$	1.35070	0.0631832	0.0315916	−	0.999501i	$-0.489942\pi$
$457$	1.35070	0.0631832	0.0315916	+	0.999501i	$0.489942\pi$
$458$	−8.09344	−0.378182
$459$	0	0
$460$	−33.0488	−1.54091
$461$	−9.13738	−0.425570	−0.212785	−	0.977099i	$-0.568253\pi$
$461$	−9.13738	−0.425570	−0.212785	+	0.977099i	$0.568253\pi$
$462$	0	0
$463$	−21.4323	−0.996042	−0.498021	−	0.867165i	$-0.665940\pi$
$463$	−21.4323	−0.996042	−0.498021	+	0.867165i	$0.665940\pi$
$464$	4.08819	0.189789
$465$	0	0
$466$	−10.8370	−0.502016
$467$	0.431773	0.0199801	0.00999004	−	0.999950i	$-0.496820\pi$
$467$	0.431773	0.0199801	0.00999004	+	0.999950i	$0.496820\pi$
$468$	0	0
$469$	−1.41387	−0.0652864
$470$	18.5028	0.853472
$471$	0	0
$472$	−8.47943	−0.390298
$473$	47.3571	2.17748
$474$	0	0
$475$	−25.8600	−1.18654
$476$	0.427444	0.0195919
$477$	0	0
$478$	7.75943	0.354908
$479$	−14.3331	−0.654896	−0.327448	−	0.944869i	$-0.606189\pi$
$479$	−14.3331	−0.654896	−0.327448	+	0.944869i	$0.606189\pi$
$480$	0	0
$481$	−32.0317	−1.46052
$482$	1.00000	0.0455488
$483$	0	0
$484$	12.6195	0.573612
$485$	−49.0458	−2.22706
$486$	0	0
$487$	4.52148	0.204888	0.102444	−	0.994739i	$-0.467334\pi$
$487$	4.52148	0.204888	0.102444	+	0.994739i	$0.467334\pi$
$488$	−6.37079	−0.288392
$489$	0	0
$490$	−26.0459	−1.17663
$491$	−26.8044	−1.20967	−0.604833	−	0.796352i	$-0.706760\pi$
$491$	−26.8044	−1.20967	−0.604833	+	0.796352i	$0.706760\pi$
$492$	0	0
$493$	4.28724	0.193088
$494$	7.56673	0.340443
$495$	0	0
$496$	−0.325219	−0.0146028
$497$	2.49162	0.111764
$498$	0	0
$499$	−20.7924	−0.930796	−0.465398	−	0.885102i	$-0.654089\pi$
$499$	−20.7924	−0.930796	−0.465398	+	0.885102i	$0.654089\pi$
$500$	17.2499	0.771437
$501$	0	0
$502$	−22.6364	−1.01031
$503$	6.70010	0.298743	0.149371	−	0.988781i	$-0.452275\pi$
$503$	6.70010	0.298743	0.149371	+	0.988781i	$0.452275\pi$
$504$	0	0
$505$	24.8501	1.10581
$506$	−42.1423	−1.87345
$507$	0	0
$508$	−0.340598	−0.0151116
$509$	24.9632	1.10648	0.553238	−	0.833023i	$-0.313392\pi$
$509$	24.9632	1.10648	0.553238	+	0.833023i	$0.313392\pi$
$510$	0	0
$511$	1.12978	0.0499785
$512$	1.00000	0.0441942
$513$	0	0
$514$	−0.167485	−0.00738743
$515$	−14.8771	−0.655564
$516$	0	0
$517$	23.5939	1.03766
$518$	4.68408	0.205807
$519$	0	0
$520$	−10.6234	−0.465865
$521$	5.71399	0.250335	0.125167	−	0.992136i	$-0.460053\pi$
$521$	5.71399	0.250335	0.125167	+	0.992136i	$0.460053\pi$
$522$	0	0
$523$	−28.2721	−1.23625	−0.618125	−	0.786080i	$-0.712108\pi$
$523$	−28.2721	−1.23625	−0.618125	+	0.786080i	$0.712108\pi$
$524$	−7.55165	−0.329896
$525$	0	0
$526$	11.1147	0.484625
$527$	−0.341053	−0.0148565
$528$	0	0
$529$	52.1911	2.26918
$530$	33.5504	1.45733
$531$	0	0
$532$	−1.10650	−0.0479729
$533$	1.22613	0.0531097
$534$	0	0
$535$	−9.90787	−0.428355
$536$	−3.46878	−0.149828
$537$	0	0
$538$	9.61340	0.414463
$539$	−33.2125	−1.43056
$540$	0	0
$541$	5.25457	0.225912	0.112956	−	0.993600i	$-0.463968\pi$
$541$	5.25457	0.225912	0.112956	+	0.993600i	$0.463968\pi$
$542$	25.3811	1.09021
$543$	0	0
$544$	1.04869	0.0449622
$545$	7.22227	0.309368
$546$	0	0
$547$	31.8626	1.36235	0.681173	−	0.732122i	$-0.261470\pi$
$547$	31.8626	1.36235	0.681173	+	0.732122i	$0.261470\pi$
$548$	21.5761	0.921683
$549$	0	0
$550$	46.2961	1.97407
$551$	−11.0981	−0.472797
$552$	0	0
$553$	3.45891	0.147088
$554$	23.2504	0.987816
$555$	0	0
$556$	−21.1211	−0.895733
$557$	−6.07474	−0.257395	−0.128698	−	0.991684i	$-0.541080\pi$
$557$	−6.07474	−0.257395	−0.128698	+	0.991684i	$0.541080\pi$
$558$	0	0
$559$	−27.1606	−1.14877
$560$	1.55348	0.0656465
$561$	0	0
$562$	28.3271	1.19491
$563$	−14.6444	−0.617189	−0.308595	−	0.951194i	$-0.599859\pi$
$563$	−14.6444	−0.617189	−0.308595	+	0.951194i	$0.599859\pi$
$564$	0	0
$565$	−20.2990	−0.853986
$566$	−11.6889	−0.491321
$567$	0	0
$568$	6.11292	0.256492
$569$	−14.0462	−0.588847	−0.294424	−	0.955675i	$-0.595128\pi$
$569$	−14.0462	−0.588847	−0.294424	+	0.955675i	$0.595128\pi$
$570$	0	0
$571$	40.3221	1.68743	0.843714	−	0.536792i	$-0.180364\pi$
$571$	40.3221	1.68743	0.843714	+	0.536792i	$0.180364\pi$
$572$	−13.5464	−0.566404
$573$	0	0
$574$	−0.179300	−0.00748386
$575$	−82.6025	−3.44476
$576$	0	0
$577$	44.4769	1.85160	0.925799	−	0.378017i	$-0.123394\pi$
$577$	44.4769	1.85160	0.925799	+	0.378017i	$0.123394\pi$
$578$	−15.9003	−0.661363
$579$	0	0
$580$	15.5813	0.646978
$581$	−6.13817	−0.254654
$582$	0	0
$583$	42.7818	1.77184
$584$	2.77180	0.114698
$585$	0	0
$586$	−25.5328	−1.05475
$587$	0.0232592	0.000960012 0	0.000480006	−	1.00000i	$-0.499847\pi$
$587$	0.0232592	0.000960012 0	0.000480006		1.00000i	$0.499847\pi$
$588$	0	0
$589$	0.882867	0.0363779
$590$	−32.3176	−1.33050
$591$	0	0
$592$	11.4919	0.472314
$593$	12.6779	0.520619	0.260309	−	0.965525i	$-0.416175\pi$
$593$	12.6779	0.520619	0.260309	+	0.965525i	$0.416175\pi$
$594$	0	0
$595$	1.62912	0.0667873
$596$	3.93654	0.161247
$597$	0	0
$598$	24.1698	0.988375
$599$	−7.15033	−0.292155	−0.146077	−	0.989273i	$-0.546665\pi$
$599$	−7.15033	−0.292155	−0.146077	+	0.989273i	$0.546665\pi$
$600$	0	0
$601$	−9.90416	−0.403999	−0.202000	−	0.979386i	$-0.564744\pi$
$601$	−9.90416	−0.403999	−0.202000	+	0.979386i	$0.564744\pi$
$602$	3.97176	0.161877
$603$	0	0
$604$	8.17568	0.332664
$605$	48.0965	1.95540
$606$	0	0
$607$	45.2965	1.83853	0.919264	−	0.393641i	$-0.128785\pi$
$607$	45.2965	1.83853	0.919264	+	0.393641i	$0.128785\pi$
$608$	−2.71468	−0.110095
$609$	0	0
$610$	−24.2810	−0.983108
$611$	−13.5318	−0.547437
$612$	0	0
$613$	23.9811	0.968588	0.484294	−	0.874905i	$-0.339077\pi$
$613$	23.9811	0.968588	0.484294	+	0.874905i	$0.339077\pi$
$614$	−20.6109	−0.831788
$615$	0	0
$616$	1.98092	0.0798137
$617$	23.1091	0.930338	0.465169	−	0.885222i	$-0.345994\pi$
$617$	23.1091	0.930338	0.465169	+	0.885222i	$0.345994\pi$
$618$	0	0
$619$	−33.1906	−1.33404	−0.667021	−	0.745039i	$-0.732431\pi$
$619$	−33.1906	−1.33404	−0.667021	+	0.745039i	$0.732431\pi$
$620$	−1.23951	−0.0497797
$621$	0	0
$622$	−26.3769	−1.05762
$623$	−2.26445	−0.0907234
$624$	0	0
$625$	18.1144	0.724576
$626$	20.2220	0.808233
$627$	0	0
$628$	0.0864784	0.00345086
$629$	12.0514	0.480521
$630$	0	0
$631$	47.6969	1.89878	0.949391	−	0.314097i	$-0.101702\pi$
$631$	47.6969	1.89878	0.949391	+	0.314097i	$0.101702\pi$
$632$	8.48606	0.337557
$633$	0	0
$634$	−18.1125	−0.719341
$635$	−1.29812	−0.0515144
$636$	0	0
$637$	19.0483	0.754720
$638$	19.8685	0.786603
$639$	0	0
$640$	3.81130	0.150655
$641$	−14.1477	−0.558802	−0.279401	−	0.960174i	$-0.590136\pi$
$641$	−14.1477	−0.558802	−0.279401	+	0.960174i	$0.590136\pi$
$642$	0	0
$643$	14.6138	0.576310	0.288155	−	0.957584i	$-0.406958\pi$
$643$	14.6138	0.576310	0.288155	+	0.957584i	$0.406958\pi$
$644$	−3.53440	−0.139275
$645$	0	0
$646$	−2.84686	−0.112008
$647$	−10.0049	−0.393333	−0.196667	−	0.980470i	$-0.563012\pi$
$647$	−10.0049	−0.393333	−0.196667	+	0.980470i	$0.563012\pi$
$648$	0	0
$649$	−41.2099	−1.61763
$650$	−26.5521	−1.04146
$651$	0	0
$652$	18.9831	0.743435
$653$	6.43798	0.251938	0.125969	−	0.992034i	$-0.459796\pi$
$653$	6.43798	0.251938	0.125969	+	0.992034i	$0.459796\pi$
$654$	0	0
$655$	−28.7816	−1.12459
$656$	−0.439895	−0.0171750
$657$	0	0
$658$	1.97878	0.0771410
$659$	−14.4481	−0.562818	−0.281409	−	0.959588i	$-0.590802\pi$
$659$	−14.4481	−0.562818	−0.281409	+	0.959588i	$0.590802\pi$
$660$	0	0
$661$	−32.2750	−1.25535	−0.627675	−	0.778475i	$-0.715993\pi$
$661$	−32.2750	−1.25535	−0.627675	+	0.778475i	$0.715993\pi$
$662$	−25.9799	−1.00974
$663$	0	0
$664$	−15.0594	−0.584416
$665$	−4.21720	−0.163536
$666$	0	0
$667$	−35.4498	−1.37262
$668$	−9.43290	−0.364970
$669$	0	0
$670$	−13.2205	−0.510754
$671$	−30.9619	−1.19527
$672$	0	0
$673$	33.1396	1.27744	0.638718	−	0.769441i	$-0.279465\pi$
$673$	33.1396	1.27744	0.638718	+	0.769441i	$0.279465\pi$
$674$	29.6685	1.14279
$675$	0	0
$676$	−5.23077	−0.201183
$677$	28.9944	1.11435	0.557173	−	0.830397i	$-0.311886\pi$
$677$	28.9944	1.11435	0.557173	+	0.830397i	$0.311886\pi$
$678$	0	0
$679$	−5.24520	−0.201292
$680$	3.99686	0.153273
$681$	0	0
$682$	−1.58056	−0.0605227
$683$	−1.80765	−0.0691678	−0.0345839	−	0.999402i	$-0.511011\pi$
$683$	−1.80765	−0.0691678	−0.0345839	+	0.999402i	$0.511011\pi$
$684$	0	0
$685$	82.2327	3.14195
$686$	−5.63866	−0.215285
$687$	0	0
$688$	9.74430	0.371498
$689$	−24.5366	−0.934768
$690$	0	0
$691$	−37.0383	−1.40900	−0.704502	−	0.709702i	$-0.748830\pi$
$691$	−37.0383	−1.40900	−0.704502	+	0.709702i	$0.748830\pi$
$692$	−18.2796	−0.694887
$693$	0	0
$694$	5.43400	0.206272
$695$	−80.4987	−3.05349
$696$	0	0
$697$	−0.461313	−0.0174735
$698$	−10.7514	−0.406947
$699$	0	0
$700$	3.88278	0.146755
$701$	−27.7059	−1.04644	−0.523218	−	0.852199i	$-0.675269\pi$
$701$	−27.7059	−1.04644	−0.523218	+	0.852199i	$0.675269\pi$
$702$	0	0
$703$	−31.1968	−1.17661
$704$	4.85999	0.183168
$705$	0	0
$706$	16.6773	0.627660
$707$	2.65759	0.0999489
$708$	0	0
$709$	−31.3766	−1.17837	−0.589187	−	0.807997i	$-0.700552\pi$
$709$	−31.3766	−1.17837	−0.589187	+	0.807997i	$0.700552\pi$
$710$	23.2981	0.874364
$711$	0	0
$712$	−5.55559	−0.208205
$713$	2.82006	0.105612
$714$	0	0
$715$	−51.6294	−1.93083
$716$	−12.8522	−0.480309
$717$	0	0
$718$	−15.4699	−0.577330
$719$	10.5400	0.393074	0.196537	−	0.980496i	$-0.437030\pi$
$719$	10.5400	0.393074	0.196537	+	0.980496i	$0.437030\pi$
$720$	0	0
$721$	−1.59103	−0.0592531
$722$	−11.6305	−0.432842
$723$	0	0
$724$	21.3281	0.792654
$725$	38.9440	1.44634
$726$	0	0
$727$	−10.4004	−0.385730	−0.192865	−	0.981225i	$-0.561778\pi$
$727$	−10.4004	−0.385730	−0.192865	+	0.981225i	$0.561778\pi$
$728$	−1.13611	−0.0421072
$729$	0	0
$730$	10.5641	0.390996
$731$	10.2187	0.377954
$732$	0	0
$733$	−3.49971	−0.129265	−0.0646323	−	0.997909i	$-0.520587\pi$
$733$	−3.49971	−0.129265	−0.0646323	+	0.997909i	$0.520587\pi$
$734$	−20.6403	−0.761846
$735$	0	0
$736$	−8.67128	−0.319628
$737$	−16.8582	−0.620980
$738$	0	0
$739$	9.90482	0.364355	0.182177	−	0.983266i	$-0.441686\pi$
$739$	9.90482	0.364355	0.182177	+	0.983266i	$0.441686\pi$
$740$	43.7990	1.61008
$741$	0	0
$742$	3.58804	0.131721
$743$	−47.2127	−1.73206	−0.866032	−	0.499988i	$-0.833338\pi$
$743$	−47.2127	−1.73206	−0.866032	+	0.499988i	$0.833338\pi$
$744$	0	0
$745$	15.0033	0.549679
$746$	13.3245	0.487843
$747$	0	0
$748$	5.09661	0.186351
$749$	−1.05960	−0.0387168
$750$	0	0
$751$	13.7635	0.502238	0.251119	−	0.967956i	$-0.419201\pi$
$751$	13.7635	0.502238	0.251119	+	0.967956i	$0.419201\pi$
$752$	4.85474	0.177034
$753$	0	0
$754$	−11.3952	−0.414987
$755$	31.1599	1.13403
$756$	0	0
$757$	−32.1537	−1.16865	−0.584323	−	0.811521i	$-0.698640\pi$
$757$	−32.1537	−1.16865	−0.584323	+	0.811521i	$0.698640\pi$
$758$	−19.3631	−0.703298
$759$	0	0
$760$	−10.3465	−0.375306
$761$	0.885275	0.0320912	0.0160456	−	0.999871i	$-0.494892\pi$
$761$	0.885275	0.0320912	0.0160456	+	0.999871i	$0.494892\pi$
$762$	0	0
$763$	0.772385	0.0279622
$764$	−17.7870	−0.643510
$765$	0	0
$766$	−19.0127	−0.686956
$767$	23.6350	0.853411
$768$	0	0
$769$	−10.2356	−0.369104	−0.184552	−	0.982823i	$-0.559083\pi$
$769$	−10.2356	−0.369104	−0.184552	+	0.982823i	$0.559083\pi$
$770$	7.54989	0.272079
$771$	0	0
$772$	−14.3394	−0.516087
$773$	3.93686	0.141599	0.0707996	−	0.997491i	$-0.477445\pi$
$773$	3.93686	0.141599	0.0707996	+	0.997491i	$0.477445\pi$
$774$	0	0
$775$	−3.09803	−0.111284
$776$	−12.8685	−0.461953
$777$	0	0
$778$	−6.76286	−0.242460
$779$	1.19417	0.0427857
$780$	0	0
$781$	29.7087	1.06306
$782$	−9.09348	−0.325182
$783$	0	0
$784$	−6.83386	−0.244067
$785$	0.329595	0.0117637
$786$	0	0
$787$	−31.8254	−1.13445	−0.567227	−	0.823562i	$-0.691984\pi$
$787$	−31.8254	−1.13445	−0.567227	+	0.823562i	$0.691984\pi$
$788$	−25.1458	−0.895782
$789$	0	0
$790$	32.3429	1.15071
$791$	−2.17087	−0.0771874
$792$	0	0
$793$	17.7575	0.630588
$794$	24.0133	0.852199
$795$	0	0
$796$	10.2795	0.364349
$797$	−18.8466	−0.667581	−0.333790	−	0.942647i	$-0.608328\pi$
$797$	−18.8466	−0.667581	−0.333790	+	0.942647i	$0.608328\pi$
$798$	0	0
$799$	5.09111	0.180110
$800$	9.52598	0.336794
$801$	0	0
$802$	−2.68925	−0.0949608
$803$	13.4709	0.475377
$804$	0	0
$805$	−13.4707	−0.474778
$806$	0.906494	0.0319299
$807$	0	0
$808$	6.52011	0.229377
$809$	47.7190	1.67771	0.838855	−	0.544355i	$-0.183226\pi$
$809$	47.7190	1.67771	0.838855	+	0.544355i	$0.183226\pi$
$810$	0	0
$811$	−10.0133	−0.351616	−0.175808	−	0.984425i	$-0.556254\pi$
$811$	−10.0133	−0.351616	−0.175808	+	0.984425i	$0.556254\pi$
$812$	1.66634	0.0584771
$813$	0	0
$814$	55.8504	1.95756
$815$	72.3501	2.53431
$816$	0	0
$817$	−26.4527	−0.925462
$818$	8.26430	0.288954
$819$	0	0
$820$	−1.67657	−0.0585483
$821$	−32.1295	−1.12133	−0.560663	−	0.828044i	$-0.689454\pi$
$821$	−32.1295	−1.12133	−0.560663	+	0.828044i	$0.689454\pi$
$822$	0	0
$823$	−46.2827	−1.61331	−0.806656	−	0.591021i	$-0.798725\pi$
$823$	−46.2827	−1.61331	−0.806656	+	0.591021i	$0.798725\pi$
$824$	−3.90343	−0.135982
$825$	0	0
$826$	−3.45621	−0.120257
$827$	5.50701	0.191498	0.0957488	−	0.995406i	$-0.469475\pi$
$827$	5.50701	0.191498	0.0957488	+	0.995406i	$0.469475\pi$
$828$	0	0
$829$	33.5088	1.16381	0.581904	−	0.813257i	$-0.302308\pi$
$829$	33.5088	1.16381	0.581904	+	0.813257i	$0.302308\pi$
$830$	−57.3957	−1.99223
$831$	0	0
$832$	−2.78733	−0.0966334
$833$	−7.16660	−0.248308
$834$	0	0
$835$	−35.9516	−1.24416
$836$	−13.1933	−0.456301
$837$	0	0
$838$	16.5813	0.572790
$839$	−40.0484	−1.38262	−0.691312	−	0.722556i	$-0.742967\pi$
$839$	−40.0484	−1.38262	−0.691312	+	0.722556i	$0.742967\pi$
$840$	0	0
$841$	−12.2867	−0.423679
$842$	−18.1121	−0.624185
$843$	0	0
$844$	28.0422	0.965252
$845$	−19.9360	−0.685819
$846$	0	0
$847$	5.14367	0.176739
$848$	8.80287	0.302292
$849$	0	0
$850$	9.98979	0.342647
$851$	−99.6494	−3.41594
$852$	0	0
$853$	48.4427	1.65865	0.829324	−	0.558768i	$-0.188726\pi$
$853$	48.4427	1.65865	0.829324	+	0.558768i	$0.188726\pi$
$854$	−2.59673	−0.0888581
$855$	0	0
$856$	−2.59961	−0.0888527
$857$	−35.5466	−1.21425	−0.607124	−	0.794607i	$-0.707677\pi$
$857$	−35.5466	−1.21425	−0.607124	+	0.794607i	$0.707677\pi$
$858$	0	0
$859$	−36.3859	−1.24147	−0.620735	−	0.784020i	$-0.713166\pi$
$859$	−36.3859	−1.24147	−0.620735	+	0.784020i	$0.713166\pi$
$860$	37.1384	1.26641
$861$	0	0
$862$	9.68282	0.329798
$863$	−36.3696	−1.23803	−0.619017	−	0.785377i	$-0.712469\pi$
$863$	−36.3696	−1.23803	−0.619017	+	0.785377i	$0.712469\pi$
$864$	0	0
$865$	−69.6691	−2.36882
$866$	26.4467	0.898695
$867$	0	0
$868$	−0.132559	−0.00449934
$869$	41.2421	1.39904
$870$	0	0
$871$	9.66865	0.327610
$872$	1.89496	0.0641716
$873$	0	0
$874$	23.5398	0.796245
$875$	7.03102	0.237692
$876$	0	0
$877$	−45.0135	−1.52000	−0.759999	−	0.649924i	$-0.774801\pi$
$877$	−45.0135	−1.52000	−0.759999	+	0.649924i	$0.774801\pi$
$878$	−17.6829	−0.596770
$879$	0	0
$880$	18.5228	0.624405
$881$	−56.3116	−1.89719	−0.948593	−	0.316499i	$-0.897493\pi$
$881$	−56.3116	−1.89719	−0.948593	+	0.316499i	$0.897493\pi$
$882$	0	0
$883$	12.2425	0.411993	0.205997	−	0.978553i	$-0.433956\pi$
$883$	12.2425	0.411993	0.205997	+	0.978553i	$0.433956\pi$
$884$	−2.92305	−0.0983127
$885$	0	0
$886$	8.39543	0.282050
$887$	−19.9539	−0.669987	−0.334994	−	0.942220i	$-0.608734\pi$
$887$	−19.9539	−0.669987	−0.334994	+	0.942220i	$0.608734\pi$
$888$	0	0
$889$	−0.138827	−0.00465612
$890$	−21.1740	−0.709754
$891$	0	0
$892$	−12.9060	−0.432123
$893$	−13.1791	−0.441021
$894$	0	0
$895$	−48.9835	−1.63734
$896$	0.407599	0.0136169
$897$	0	0
$898$	−29.1627	−0.973173
$899$	−1.32956	−0.0443432
$900$	0	0
$901$	9.23148	0.307545
$902$	−2.13788	−0.0711837
$903$	0	0
$904$	−5.32601	−0.177140
$905$	81.2878	2.70210
$906$	0	0
$907$	−45.4569	−1.50937	−0.754685	−	0.656087i	$-0.772210\pi$
$907$	−45.4569	−1.50937	−0.754685	+	0.656087i	$0.772210\pi$
$908$	24.6376	0.817627
$909$	0	0
$910$	−4.33007	−0.143540
$911$	6.85309	0.227053	0.113526	−	0.993535i	$-0.463785\pi$
$911$	6.85309	0.227053	0.113526	+	0.993535i	$0.463785\pi$
$912$	0	0
$913$	−73.1882	−2.42218
$914$	1.35070	0.0446773
$915$	0	0
$916$	−8.09344	−0.267415
$917$	−3.07804	−0.101646
$918$	0	0
$919$	18.9140	0.623915	0.311958	−	0.950096i	$-0.399015\pi$
$919$	18.9140	0.623915	0.311958	+	0.950096i	$0.399015\pi$
$920$	−33.0488	−1.08959
$921$	0	0
$922$	−9.13738	−0.300924
$923$	−17.0387	−0.560837
$924$	0	0
$925$	109.471	3.59940
$926$	−21.4323	−0.704308
$927$	0	0
$928$	4.08819	0.134201
$929$	3.28463	0.107765	0.0538826	−	0.998547i	$-0.482840\pi$
$929$	3.28463	0.107765	0.0538826	+	0.998547i	$0.482840\pi$
$930$	0	0
$931$	18.5518	0.608010
$932$	−10.8370	−0.354979
$933$	0	0
$934$	0.431773	0.0141280
$935$	19.4247	0.635256
$936$	0	0
$937$	2.79976	0.0914643	0.0457322	−	0.998954i	$-0.485438\pi$
$937$	2.79976	0.0914643	0.0457322	+	0.998954i	$0.485438\pi$
$938$	−1.41387	−0.0461645
$939$	0	0
$940$	18.5028	0.603496
$941$	13.0622	0.425816	0.212908	−	0.977072i	$-0.431707\pi$
$941$	13.0622	0.425816	0.212908	+	0.977072i	$0.431707\pi$
$942$	0	0
$943$	3.81445	0.124216
$944$	−8.47943	−0.275982
$945$	0	0
$946$	47.3571	1.53971
$947$	35.5843	1.15633	0.578167	−	0.815918i	$-0.303768\pi$
$947$	35.5843	1.15633	0.578167	+	0.815918i	$0.303768\pi$
$948$	0	0
$949$	−7.72592	−0.250794
$950$	−25.8600	−0.839010
$951$	0	0
$952$	0.427444	0.0138535
$953$	43.4229	1.40661	0.703303	−	0.710890i	$-0.251708\pi$
$953$	43.4229	1.40661	0.703303	+	0.710890i	$0.251708\pi$
$954$	0	0
$955$	−67.7914	−2.19368
$956$	7.75943	0.250958
$957$	0	0
$958$	−14.3331	−0.463081
$959$	8.79437	0.283985
$960$	0	0
$961$	−30.8942	−0.996588
$962$	−32.0317	−1.03274
$963$	0	0
$964$	1.00000	0.0322078
$965$	−54.6518	−1.75930
$966$	0	0
$967$	−56.2588	−1.80916	−0.904581	−	0.426302i	$-0.859816\pi$
$967$	−56.2588	−1.80916	−0.904581	+	0.426302i	$0.859816\pi$
$968$	12.6195	0.405605
$969$	0	0
$970$	−49.0458	−1.57477
$971$	38.1291	1.22362	0.611811	−	0.791004i	$-0.290441\pi$
$971$	38.1291	1.22362	0.611811	+	0.791004i	$0.290441\pi$
$972$	0	0
$973$	−8.60892	−0.275989
$974$	4.52148	0.144878
$975$	0	0
$976$	−6.37079	−0.203924
$977$	20.2896	0.649121	0.324561	−	0.945865i	$-0.394784\pi$
$977$	20.2896	0.649121	0.324561	+	0.945865i	$0.394784\pi$
$978$	0	0
$979$	−27.0001	−0.862927
$980$	−26.0459	−0.832005
$981$	0	0
$982$	−26.8044	−0.855363
$983$	−45.1636	−1.44050	−0.720248	−	0.693717i	$-0.755972\pi$
$983$	−45.1636	−1.44050	−0.720248	+	0.693717i	$0.755972\pi$
$984$	0	0
$985$	−95.8381	−3.05366
$986$	4.28724	0.136534
$987$	0	0
$988$	7.56673	0.240730
$989$	−84.4955	−2.68680
$990$	0	0
$991$	23.7345	0.753953	0.376976	−	0.926223i	$-0.376964\pi$
$991$	23.7345	0.753953	0.376976	+	0.926223i	$0.376964\pi$
$992$	−0.325219	−0.0103257
$993$	0	0
$994$	2.49162	0.0790293
$995$	39.1784	1.24204
$996$	0	0
$997$	−22.1793	−0.702426	−0.351213	−	0.936296i	$-0.614231\pi$
$997$	−22.1793	−0.702426	−0.351213	+	0.936296i	$0.614231\pi$
$998$	−20.7924	−0.658172
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4338.2.a.u.1.6		6
3.2	odd	2		482.2.a.d.1.1	✓	6
12.11	even	2		3856.2.a.h.1.6		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
482.2.a.d.1.1	✓	6	3.2	odd	2
3856.2.a.h.1.6		6	12.11	even	2
4338.2.a.u.1.6		6	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:	\( N \)	\(=\)	\( 4338 = 2 \cdot 3^{2} \cdot 241 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4338.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} +3.81130 q^{5} +0.407599 q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} +3.81130 q^{5} +0.407599 q^{7} +1.00000 q^{8} +3.81130 q^{10} +4.85999 q^{11} -2.78733 q^{13} +0.407599 q^{14} +1.00000 q^{16} +1.04869 q^{17} -2.71468 q^{19} +3.81130 q^{20} +4.85999 q^{22} -8.67128 q^{23} +9.52598 q^{25} -2.78733 q^{26} +0.407599 q^{28} +4.08819 q^{29} -0.325219 q^{31} +1.00000 q^{32} +1.04869 q^{34} +1.55348 q^{35} +11.4919 q^{37} -2.71468 q^{38} +3.81130 q^{40} -0.439895 q^{41} +9.74430 q^{43} +4.85999 q^{44} -8.67128 q^{46} +4.85474 q^{47} -6.83386 q^{49} +9.52598 q^{50} -2.78733 q^{52} +8.80287 q^{53} +18.5228 q^{55} +0.407599 q^{56} +4.08819 q^{58} -8.47943 q^{59} -6.37079 q^{61} -0.325219 q^{62} +1.00000 q^{64} -10.6234 q^{65} -3.46878 q^{67} +1.04869 q^{68} +1.55348 q^{70} +6.11292 q^{71} +2.77180 q^{73} +11.4919 q^{74} -2.71468 q^{76} +1.98092 q^{77} +8.48606 q^{79} +3.81130 q^{80} -0.439895 q^{82} -15.0594 q^{83} +3.99686 q^{85} +9.74430 q^{86} +4.85999 q^{88} -5.55559 q^{89} -1.13611 q^{91} -8.67128 q^{92} +4.85474 q^{94} -10.3465 q^{95} -12.8685 q^{97} -6.83386 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 6 q^{2} + 6 q^{4} + 5 q^{5} + 10 q^{7} + 6 q^{8}+O(q^{10}) \) 6 * q + 6 * q^2 + 6 * q^4 + 5 * q^5 + 10 * q^7 + 6 * q^8	\( 6 q + 6 q^{2} + 6 q^{4} + 5 q^{5} + 10 q^{7} + 6 q^{8} + 5 q^{10} + 4 q^{11} + 9 q^{13} + 10 q^{14} + 6 q^{16} - q^{17} + 10 q^{19} + 5 q^{20} + 4 q^{22} - 9 q^{23} + 13 q^{25} + 9 q^{26} + 10 q^{28} + q^{29} + 14 q^{31} + 6 q^{32} - q^{34} + 3 q^{35} + 20 q^{37} + 10 q^{38} + 5 q^{40} + 4 q^{41} + 19 q^{43} + 4 q^{44} - 9 q^{46} - q^{47} + 14 q^{49} + 13 q^{50} + 9 q^{52} + 3 q^{53} + 11 q^{55} + 10 q^{56} + q^{58} + q^{59} + 4 q^{61} + 14 q^{62} + 6 q^{64} - 5 q^{65} + 5 q^{67} - q^{68} + 3 q^{70} - 2 q^{71} + 15 q^{73} + 20 q^{74} + 10 q^{76} + 6 q^{77} + 12 q^{79} + 5 q^{80} + 4 q^{82} + 8 q^{83} - 32 q^{85} + 19 q^{86} + 4 q^{88} + 24 q^{89} + q^{91} - 9 q^{92} - q^{94} - 9 q^{95} - 12 q^{97} + 14 q^{98}+O(q^{100}) \) 6 * q + 6 * q^2 + 6 * q^4 + 5 * q^5 + 10 * q^7 + 6 * q^8 + 5 * q^10 + 4 * q^11 + 9 * q^13 + 10 * q^14 + 6 * q^16 - q^17 + 10 * q^19 + 5 * q^20 + 4 * q^22 - 9 * q^23 + 13 * q^25 + 9 * q^26 + 10 * q^28 + q^29 + 14 * q^31 + 6 * q^32 - q^34 + 3 * q^35 + 20 * q^37 + 10 * q^38 + 5 * q^40 + 4 * q^41 + 19 * q^43 + 4 * q^44 - 9 * q^46 - q^47 + 14 * q^49 + 13 * q^50 + 9 * q^52 + 3 * q^53 + 11 * q^55 + 10 * q^56 + q^58 + q^59 + 4 * q^61 + 14 * q^62 + 6 * q^64 - 5 * q^65 + 5 * q^67 - q^68 + 3 * q^70 - 2 * q^71 + 15 * q^73 + 20 * q^74 + 10 * q^76 + 6 * q^77 + 12 * q^79 + 5 * q^80 + 4 * q^82 + 8 * q^83 - 32 * q^85 + 19 * q^86 + 4 * q^88 + 24 * q^89 + q^91 - 9 * q^92 - q^94 - 9 * q^95 - 12 * q^97 + 14 * q^98

Introduction

L-functions

Modular forms

Varieties

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Database

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