LMFDB - Embedded newform 6039.2.a.m.1.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6039,2,Mod(1,6039)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6039, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("6039.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6039 = 3^{2} \cdot 11 \cdot 61 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6039.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:	$48.2216577807$
Analytic rank:	$1$
Dimension:	$25$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Character	$\chi$	$=$	6039.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.38389 q^{2} -0.0848502 q^{4} -4.17750 q^{5} +3.01863 q^{7} +2.88520 q^{8} +O(q^{10})$	\(q-1.38389 q^{2} -0.0848502 q^{4} -4.17750 q^{5} +3.01863 q^{7} +2.88520 q^{8} +5.78120 q^{10} +1.00000 q^{11} +0.616430 q^{13} -4.17745 q^{14} -3.82310 q^{16} -3.75806 q^{17} +5.19562 q^{19} +0.354462 q^{20} -1.38389 q^{22} -1.07408 q^{23} +12.4515 q^{25} -0.853071 q^{26} -0.256132 q^{28} -4.93072 q^{29} -5.32555 q^{31} -0.479657 q^{32} +5.20074 q^{34} -12.6104 q^{35} -3.81337 q^{37} -7.19016 q^{38} -12.0529 q^{40} +2.12431 q^{41} -7.97863 q^{43} -0.0848502 q^{44} +1.48641 q^{46} +5.49399 q^{47} +2.11215 q^{49} -17.2315 q^{50} -0.0523042 q^{52} +1.16155 q^{53} -4.17750 q^{55} +8.70937 q^{56} +6.82356 q^{58} -8.82087 q^{59} +1.00000 q^{61} +7.36998 q^{62} +8.30999 q^{64} -2.57514 q^{65} +7.66581 q^{67} +0.318872 q^{68} +17.4513 q^{70} +4.50469 q^{71} +4.44227 q^{73} +5.27728 q^{74} -0.440849 q^{76} +3.01863 q^{77} +13.8498 q^{79} +15.9710 q^{80} -2.93981 q^{82} +17.3634 q^{83} +15.6993 q^{85} +11.0415 q^{86} +2.88520 q^{88} -9.81507 q^{89} +1.86078 q^{91} +0.0911360 q^{92} -7.60307 q^{94} -21.7047 q^{95} -12.6854 q^{97} -2.92298 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 25 q - 5 q^{2} + 25 q^{4} - 12 q^{5} - 4 q^{7} - 15 q^{8}+O(q^{10}) $ 25 * q - 5 * q^2 + 25 * q^4 - 12 * q^5 - 4 * q^7 - 15 * q^8	$ 25 q - 5 q^{2} + 25 q^{4} - 12 q^{5} - 4 q^{7} - 15 q^{8} - 12 q^{10} + 25 q^{11} - 4 q^{13} - 14 q^{14} + 21 q^{16} - 16 q^{17} - 18 q^{19} - 28 q^{20} - 5 q^{22} - 8 q^{23} + 29 q^{25} - 16 q^{26} + 18 q^{28} - 28 q^{29} - 8 q^{31} - 35 q^{32} + 6 q^{34} - 22 q^{35} + 4 q^{37} + 4 q^{38} - 12 q^{40} - 58 q^{41} - 26 q^{43} + 25 q^{44} + 8 q^{46} - 20 q^{47} + 23 q^{49} - 27 q^{50} - 2 q^{52} - 36 q^{53} - 12 q^{55} - 70 q^{56} + 12 q^{58} - 18 q^{59} + 25 q^{61} - 42 q^{62} + 35 q^{64} - 76 q^{65} - 8 q^{67} - 28 q^{68} + 76 q^{70} - 24 q^{71} + 2 q^{73} - 40 q^{74} - 64 q^{76} - 4 q^{77} - 22 q^{79} - 36 q^{80} + 30 q^{82} - 14 q^{83} - 70 q^{86} - 15 q^{88} - 82 q^{89} - 6 q^{91} - 48 q^{92} - 16 q^{94} - 34 q^{95} + 16 q^{97} - 35 q^{98}+O(q^{100}) $ 25 * q - 5 * q^2 + 25 * q^4 - 12 * q^5 - 4 * q^7 - 15 * q^8 - 12 * q^10 + 25 * q^11 - 4 * q^13 - 14 * q^14 + 21 * q^16 - 16 * q^17 - 18 * q^19 - 28 * q^20 - 5 * q^22 - 8 * q^23 + 29 * q^25 - 16 * q^26 + 18 * q^28 - 28 * q^29 - 8 * q^31 - 35 * q^32 + 6 * q^34 - 22 * q^35 + 4 * q^37 + 4 * q^38 - 12 * q^40 - 58 * q^41 - 26 * q^43 + 25 * q^44 + 8 * q^46 - 20 * q^47 + 23 * q^49 - 27 * q^50 - 2 * q^52 - 36 * q^53 - 12 * q^55 - 70 * q^56 + 12 * q^58 - 18 * q^59 + 25 * q^61 - 42 * q^62 + 35 * q^64 - 76 * q^65 - 8 * q^67 - 28 * q^68 + 76 * q^70 - 24 * q^71 + 2 * q^73 - 40 * q^74 - 64 * q^76 - 4 * q^77 - 22 * q^79 - 36 * q^80 + 30 * q^82 - 14 * q^83 - 70 * q^86 - 15 * q^88 - 82 * q^89 - 6 * q^91 - 48 * q^92 - 16 * q^94 - 34 * q^95 + 16 * q^97 - 35 * 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 $

$2$	−1.38389	−0.978558	−0.489279	−	0.872127i	$-0.662740\pi$
$2$	−1.38389	−0.978558	−0.489279	+	0.872127i	$0.662740\pi$
$3$	0	0
$3$	0	0
$4$	−0.0848502	−0.0424251
$5$	−4.17750	−1.86824	−0.934118	−	0.356964i	$-0.883812\pi$
$5$	−4.17750	−1.86824	−0.934118	+	0.356964i	$0.883812\pi$
$6$	0	0
$7$	3.01863	1.14094	0.570468	−	0.821320i	$-0.306762\pi$
$7$	3.01863	1.14094	0.570468	+	0.821320i	$0.306762\pi$
$8$	2.88520	1.02007
$9$	0	0
$10$	5.78120	1.82818
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	0.616430	0.170967	0.0854834	−	0.996340i	$-0.472757\pi$
$13$	0.616430	0.170967	0.0854834	+	0.996340i	$0.472757\pi$
$14$	−4.17745	−1.11647
$15$	0	0
$16$	−3.82310	−0.955775
$17$	−3.75806	−0.911463	−0.455731	−	0.890117i	$-0.650622\pi$
$17$	−3.75806	−0.911463	−0.455731	+	0.890117i	$0.650622\pi$
$18$	0	0
$19$	5.19562	1.19196	0.595978	−	0.803000i	$-0.296764\pi$
$19$	5.19562	1.19196	0.595978	+	0.803000i	$0.296764\pi$
$20$	0.354462	0.0792601
$21$	0	0
$22$	−1.38389	−0.295046
$23$	−1.07408	−0.223961	−0.111981	−	0.993710i	$-0.535719\pi$
$23$	−1.07408	−0.223961	−0.111981	+	0.993710i	$0.535719\pi$
$24$	0	0
$25$	12.4515	2.49031
$26$	−0.853071	−0.167301
$27$	0	0
$28$	−0.256132	−0.0484043
$29$	−4.93072	−0.915611	−0.457805	−	0.889052i	$-0.651364\pi$
$29$	−4.93072	−0.915611	−0.457805	+	0.889052i	$0.651364\pi$
$30$	0	0
$31$	−5.32555	−0.956498	−0.478249	−	0.878224i	$-0.658728\pi$
$31$	−5.32555	−0.956498	−0.478249	+	0.878224i	$0.658728\pi$
$32$	−0.479657	−0.0847921
$33$	0	0
$34$	5.20074	0.891919
$35$	−12.6104	−2.13154
$36$	0	0
$37$	−3.81337	−0.626913	−0.313457	−	0.949602i	$-0.601487\pi$
$37$	−3.81337	−0.626913	−0.313457	+	0.949602i	$0.601487\pi$
$38$	−7.19016	−1.16640
$39$	0	0
$40$	−12.0529	−1.90574
$41$	2.12431	0.331761	0.165881	−	0.986146i	$-0.446953\pi$
$41$	2.12431	0.331761	0.165881	+	0.986146i	$0.446953\pi$
$42$	0	0
$43$	−7.97863	−1.21673	−0.608365	−	0.793658i	$-0.708174\pi$
$43$	−7.97863	−1.21673	−0.608365	+	0.793658i	$0.708174\pi$
$44$	−0.0848502	−0.0127917
$45$	0	0
$46$	1.48641	0.219159
$47$	5.49399	0.801380	0.400690	−	0.916214i	$-0.368770\pi$
$47$	5.49399	0.801380	0.400690	+	0.916214i	$0.368770\pi$
$48$	0	0
$49$	2.11215	0.301735
$50$	−17.2315	−2.43691
$51$	0	0
$52$	−0.0523042	−0.00725329
$53$	1.16155	0.159551	0.0797753	−	0.996813i	$-0.474580\pi$
$53$	1.16155	0.159551	0.0797753	+	0.996813i	$0.474580\pi$
$54$	0	0
$55$	−4.17750	−0.563294
$56$	8.70937	1.16384
$57$	0	0
$58$	6.82356	0.895978
$59$	−8.82087	−1.14838	−0.574189	−	0.818722i	$-0.694683\pi$
$59$	−8.82087	−1.14838	−0.574189	+	0.818722i	$0.694683\pi$
$60$	0	0
$61$	1.00000	0.128037
$61$	1.00000	0.128037
$62$	7.36998	0.935988
$63$	0	0
$64$	8.30999	1.03875
$65$	−2.57514	−0.319407
$66$	0	0
$67$	7.66581	0.936528	0.468264	−	0.883589i	$-0.344880\pi$
$67$	7.66581	0.936528	0.468264	+	0.883589i	$0.344880\pi$
$68$	0.318872	0.0386689
$69$	0	0
$70$	17.4513	2.08583
$71$	4.50469	0.534608	0.267304	−	0.963612i	$-0.413867\pi$
$71$	4.50469	0.534608	0.267304	+	0.963612i	$0.413867\pi$
$72$	0	0
$73$	4.44227	0.519928	0.259964	−	0.965618i	$-0.416289\pi$
$73$	4.44227	0.519928	0.259964	+	0.965618i	$0.416289\pi$
$74$	5.27728	0.613471
$75$	0	0
$76$	−0.440849	−0.0505689
$77$	3.01863	0.344005
$78$	0	0
$79$	13.8498	1.55822	0.779110	−	0.626887i	$-0.215671\pi$
$79$	13.8498	1.55822	0.779110	+	0.626887i	$0.215671\pi$
$80$	15.9710	1.78561
$81$	0	0
$82$	−2.93981	−0.324648
$83$	17.3634	1.90588	0.952938	−	0.303164i	$-0.0980432\pi$
$83$	17.3634	1.90588	0.952938	+	0.303164i	$0.0980432\pi$
$84$	0	0
$85$	15.6993	1.70283
$86$	11.0415	1.19064
$87$	0	0
$88$	2.88520	0.307564
$89$	−9.81507	−1.04040	−0.520198	−	0.854046i	$-0.674142\pi$
$89$	−9.81507	−1.04040	−0.520198	+	0.854046i	$0.674142\pi$
$90$	0	0
$91$	1.86078	0.195062
$92$	0.0911360	0.00950159
$93$	0	0
$94$	−7.60307	−0.784197
$95$	−21.7047	−2.22686
$96$	0	0
$97$	−12.6854	−1.28801	−0.644005	−	0.765021i	$-0.722728\pi$
$97$	−12.6854	−1.28801	−0.644005	+	0.765021i	$0.722728\pi$
$98$	−2.92298	−0.295265
$99$	0	0
$100$	−1.05652	−0.105652
$101$	−4.89636	−0.487206	−0.243603	−	0.969875i	$-0.578329\pi$
$101$	−4.89636	−0.487206	−0.243603	+	0.969875i	$0.578329\pi$
$102$	0	0
$103$	2.60455	0.256634	0.128317	−	0.991733i	$-0.459042\pi$
$103$	2.60455	0.256634	0.128317	+	0.991733i	$0.459042\pi$
$104$	1.77852	0.174399
$105$	0	0
$106$	−1.60745	−0.156130
$107$	8.89746	0.860150	0.430075	−	0.902793i	$-0.358487\pi$
$107$	8.89746	0.860150	0.430075	+	0.902793i	$0.358487\pi$
$108$	0	0
$109$	7.99722	0.765995	0.382997	−	0.923749i	$-0.374892\pi$
$109$	7.99722	0.765995	0.382997	+	0.923749i	$0.374892\pi$
$110$	5.78120	0.551216
$111$	0	0
$112$	−11.5405	−1.09048
$113$	−11.0762	−1.04196	−0.520980	−	0.853569i	$-0.674433\pi$
$113$	−11.0762	−1.04196	−0.520980	+	0.853569i	$0.674433\pi$
$114$	0	0
$115$	4.48698	0.418413
$116$	0.418372	0.0388449
$117$	0	0
$118$	12.2071	1.12375
$119$	−11.3442	−1.03992
$120$	0	0
$121$	1.00000	0.0909091
$122$	−1.38389	−0.125291
$123$	0	0
$124$	0.451875	0.0405795
$125$	−31.1288	−2.78425
$126$	0	0
$127$	−17.5801	−1.55998	−0.779990	−	0.625792i	$-0.784776\pi$
$127$	−17.5801	−1.55998	−0.779990	+	0.625792i	$0.784776\pi$
$128$	−10.5408	−0.931684
$129$	0	0
$130$	3.56371	0.312558
$131$	−1.47079	−0.128504	−0.0642518	−	0.997934i	$-0.520466\pi$
$131$	−1.47079	−0.128504	−0.0642518	+	0.997934i	$0.520466\pi$
$132$	0	0
$133$	15.6837	1.35995
$134$	−10.6086	−0.916446
$135$	0	0
$136$	−10.8428	−0.929759
$137$	8.23496	0.703560	0.351780	−	0.936083i	$-0.385576\pi$
$137$	8.23496	0.703560	0.351780	+	0.936083i	$0.385576\pi$
$138$	0	0
$139$	8.06242	0.683845	0.341923	−	0.939728i	$-0.388922\pi$
$139$	8.06242	0.683845	0.341923	+	0.939728i	$0.388922\pi$
$140$	1.06999	0.0904308
$141$	0	0
$142$	−6.23399	−0.523145
$143$	0.616430	0.0515485
$144$	0	0
$145$	20.5981	1.71058
$146$	−6.14761	−0.508779
$147$	0	0
$148$	0.323565	0.0265969
$149$	0.861759	0.0705980	0.0352990	−	0.999377i	$-0.488762\pi$
$149$	0.861759	0.0705980	0.0352990	+	0.999377i	$0.488762\pi$
$150$	0	0
$151$	14.8034	1.20469	0.602343	−	0.798237i	$-0.294234\pi$
$151$	14.8034	1.20469	0.602343	+	0.798237i	$0.294234\pi$
$152$	14.9904	1.21588
$153$	0	0
$154$	−4.17745	−0.336629
$155$	22.2475	1.78696
$156$	0	0
$157$	−9.62609	−0.768245	−0.384123	−	0.923282i	$-0.625496\pi$
$157$	−9.62609	−0.768245	−0.384123	+	0.923282i	$0.625496\pi$
$158$	−19.1666	−1.52481
$159$	0	0
$160$	2.00377	0.158412
$161$	−3.24226	−0.255526
$162$	0	0
$163$	4.48782	0.351513	0.175756	−	0.984434i	$-0.443763\pi$
$163$	4.48782	0.351513	0.175756	+	0.984434i	$0.443763\pi$
$164$	−0.180248	−0.0140750
$165$	0	0
$166$	−24.0290	−1.86501
$167$	11.7114	0.906258	0.453129	−	0.891445i	$-0.350308\pi$
$167$	11.7114	0.906258	0.453129	+	0.891445i	$0.350308\pi$
$168$	0	0
$169$	−12.6200	−0.970770
$170$	−21.7261	−1.66632
$171$	0	0
$172$	0.676988	0.0516199
$173$	16.3491	1.24300	0.621501	−	0.783414i	$-0.286523\pi$
$173$	16.3491	1.24300	0.621501	+	0.783414i	$0.286523\pi$
$174$	0	0
$175$	37.5866	2.84128
$176$	−3.82310	−0.288177
$177$	0	0
$178$	13.5830	1.01809
$179$	−10.3748	−0.775447	−0.387724	−	0.921776i	$-0.626739\pi$
$179$	−10.3748	−0.775447	−0.387724	+	0.921776i	$0.626739\pi$
$180$	0	0
$181$	9.42898	0.700851	0.350425	−	0.936591i	$-0.386037\pi$
$181$	9.42898	0.700851	0.350425	+	0.936591i	$0.386037\pi$
$182$	−2.57511	−0.190880
$183$	0	0
$184$	−3.09894	−0.228457
$185$	15.9303	1.17122
$186$	0	0
$187$	−3.75806	−0.274816
$188$	−0.466166	−0.0339986
$189$	0	0
$190$	30.0369	2.17911
$191$	3.68014	0.266285	0.133143	−	0.991097i	$-0.457493\pi$
$191$	3.68014	0.266285	0.133143	+	0.991097i	$0.457493\pi$
$192$	0	0
$193$	−10.7515	−0.773908	−0.386954	−	0.922099i	$-0.626473\pi$
$193$	−10.7515	−0.773908	−0.386954	+	0.922099i	$0.626473\pi$
$194$	17.5552	1.26039
$195$	0	0
$196$	−0.179216	−0.0128012
$197$	10.0468	0.715804	0.357902	−	0.933759i	$-0.383492\pi$
$197$	10.0468	0.715804	0.357902	+	0.933759i	$0.383492\pi$
$198$	0	0
$199$	−21.1409	−1.49864	−0.749318	−	0.662211i	$-0.769618\pi$
$199$	−21.1409	−1.49864	−0.749318	+	0.662211i	$0.769618\pi$
$200$	35.9252	2.54030
$201$	0	0
$202$	6.77602	0.476759
$203$	−14.8840	−1.04465
$204$	0	0
$205$	−8.87431	−0.619809
$206$	−3.60441	−0.251131
$207$	0	0
$208$	−2.35667	−0.163406
$209$	5.19562	0.359388
$210$	0	0
$211$	−0.0825164	−0.00568067	−0.00284033	−	0.999996i	$-0.500904\pi$
$211$	−0.0825164	−0.00568067	−0.00284033	+	0.999996i	$0.500904\pi$
$212$	−0.0985575	−0.00676896
$213$	0	0
$214$	−12.3131	−0.841706
$215$	33.3307	2.27314
$216$	0	0
$217$	−16.0759	−1.09130
$218$	−11.0673	−0.749570
$219$	0	0
$220$	0.354462	0.0238978
$221$	−2.31658	−0.155830
$222$	0	0
$223$	−23.7367	−1.58952	−0.794762	−	0.606921i	$-0.792404\pi$
$223$	−23.7367	−1.58952	−0.794762	+	0.606921i	$0.792404\pi$
$224$	−1.44791	−0.0967424
$225$	0	0
$226$	15.3282	1.01962
$227$	−1.29762	−0.0861264	−0.0430632	−	0.999072i	$-0.513712\pi$
$227$	−1.29762	−0.0861264	−0.0430632	+	0.999072i	$0.513712\pi$
$228$	0	0
$229$	−2.40227	−0.158747	−0.0793734	−	0.996845i	$-0.525292\pi$
$229$	−2.40227	−0.158747	−0.0793734	+	0.996845i	$0.525292\pi$
$230$	−6.20948	−0.409441
$231$	0	0
$232$	−14.2261	−0.933990
$233$	−29.3459	−1.92251	−0.961257	−	0.275654i	$-0.911106\pi$
$233$	−29.3459	−1.92251	−0.961257	+	0.275654i	$0.911106\pi$
$234$	0	0
$235$	−22.9511	−1.49717
$236$	0.748452	0.0487201
$237$	0	0
$238$	15.6991	1.01762
$239$	−8.22595	−0.532093	−0.266046	−	0.963960i	$-0.585717\pi$
$239$	−8.22595	−0.532093	−0.266046	+	0.963960i	$0.585717\pi$
$240$	0	0
$241$	28.3849	1.82843	0.914216	−	0.405228i	$-0.132808\pi$
$241$	28.3849	1.82843	0.914216	+	0.405228i	$0.132808\pi$
$242$	−1.38389	−0.0889598
$243$	0	0
$244$	−0.0848502	−0.00543198
$245$	−8.82350	−0.563713
$246$	0	0
$247$	3.20273	0.203785
$248$	−15.3653	−0.975698
$249$	0	0
$250$	43.0788	2.72455
$251$	−12.1603	−0.767553	−0.383777	−	0.923426i	$-0.625377\pi$
$251$	−12.1603	−0.767553	−0.383777	+	0.923426i	$0.625377\pi$
$252$	0	0
$253$	−1.07408	−0.0675269
$254$	24.3289	1.52653
$255$	0	0
$256$	−2.03269	−0.127043
$257$	−4.86126	−0.303237	−0.151619	−	0.988439i	$-0.548449\pi$
$257$	−4.86126	−0.303237	−0.151619	+	0.988439i	$0.548449\pi$
$258$	0	0
$259$	−11.5112	−0.715268
$260$	0.218501	0.0135509
$261$	0	0
$262$	2.03541	0.125748
$263$	−16.4365	−1.01352	−0.506760	−	0.862087i	$-0.669157\pi$
$263$	−16.4365	−1.01352	−0.506760	+	0.862087i	$0.669157\pi$
$264$	0	0
$265$	−4.85236	−0.298078
$266$	−21.7045	−1.33079
$267$	0	0
$268$	−0.650446	−0.0397323
$269$	0.792880	0.0483428	0.0241714	−	0.999708i	$-0.492305\pi$
$269$	0.792880	0.0483428	0.0241714	+	0.999708i	$0.492305\pi$
$270$	0	0
$271$	−15.0667	−0.915238	−0.457619	−	0.889148i	$-0.651298\pi$
$271$	−15.0667	−0.915238	−0.457619	+	0.889148i	$0.651298\pi$
$272$	14.3674	0.871153
$273$	0	0
$274$	−11.3963	−0.688474
$275$	12.4515	0.750856
$276$	0	0
$277$	28.6056	1.71874	0.859371	−	0.511352i	$-0.170855\pi$
$277$	28.6056	1.71874	0.859371	+	0.511352i	$0.170855\pi$
$278$	−11.1575	−0.669182
$279$	0	0
$280$	−36.3834	−2.17432
$281$	−9.34938	−0.557737	−0.278869	−	0.960329i	$-0.589959\pi$
$281$	−9.34938	−0.557737	−0.278869	+	0.960329i	$0.589959\pi$
$282$	0	0
$283$	−10.9419	−0.650428	−0.325214	−	0.945640i	$-0.605436\pi$
$283$	−10.9419	−0.650428	−0.325214	+	0.945640i	$0.605436\pi$
$284$	−0.382224	−0.0226808
$285$	0	0
$286$	−0.853071	−0.0504431
$287$	6.41251	0.378518
$288$	0	0
$289$	−2.87701	−0.169236
$290$	−28.5055	−1.67390
$291$	0	0
$292$	−0.376927	−0.0220580
$293$	30.3237	1.77153	0.885765	−	0.464134i	$-0.153634\pi$
$293$	30.3237	1.77153	0.885765	+	0.464134i	$0.153634\pi$
$294$	0	0
$295$	36.8492	2.14544
$296$	−11.0023	−0.639497
$297$	0	0
$298$	−1.19258	−0.0690842
$299$	−0.662095	−0.0382900
$300$	0	0
$301$	−24.0846	−1.38821
$302$	−20.4863	−1.17885
$303$	0	0
$304$	−19.8634	−1.13924
$305$	−4.17750	−0.239203
$306$	0	0
$307$	3.65208	0.208435	0.104218	−	0.994555i	$-0.466766\pi$
$307$	3.65208	0.208435	0.104218	+	0.994555i	$0.466766\pi$
$308$	−0.256132	−0.0145945
$309$	0	0
$310$	−30.7881	−1.74865
$311$	−10.8939	−0.617739	−0.308869	−	0.951104i	$-0.599951\pi$
$311$	−10.8939	−0.617739	−0.308869	+	0.951104i	$0.599951\pi$
$312$	0	0
$313$	17.5728	0.993273	0.496636	−	0.867959i	$-0.334568\pi$
$313$	17.5728	0.993273	0.496636	+	0.867959i	$0.334568\pi$
$314$	13.3214	0.751772
$315$	0	0
$316$	−1.17516	−0.0661077
$317$	5.62995	0.316209	0.158105	−	0.987422i	$-0.449462\pi$
$317$	5.62995	0.316209	0.158105	+	0.987422i	$0.449462\pi$
$318$	0	0
$319$	−4.93072	−0.276067
$320$	−34.7150	−1.94063
$321$	0	0
$322$	4.48692	0.250046
$323$	−19.5254	−1.08642
$324$	0	0
$325$	7.67550	0.425760
$326$	−6.21064	−0.343976
$327$	0	0
$328$	6.12906	0.338421
$329$	16.5843	0.914324
$330$	0	0
$331$	−17.3316	−0.952630	−0.476315	−	0.879275i	$-0.658028\pi$
$331$	−17.3316	−0.952630	−0.476315	+	0.879275i	$0.658028\pi$
$332$	−1.47329	−0.0808570
$333$	0	0
$334$	−16.2073	−0.886826
$335$	−32.0239	−1.74965
$336$	0	0
$337$	−9.56047	−0.520792	−0.260396	−	0.965502i	$-0.583853\pi$
$337$	−9.56047	−0.520792	−0.260396	+	0.965502i	$0.583853\pi$
$338$	17.4647	0.949955
$339$	0	0
$340$	−1.33209	−0.0722427
$341$	−5.32555	−0.288395
$342$	0	0
$343$	−14.7546	−0.796675
$344$	−23.0200	−1.24115
$345$	0	0
$346$	−22.6254	−1.21635
$347$	−22.9304	−1.23097	−0.615485	−	0.788148i	$-0.711040\pi$
$347$	−22.9304	−1.23097	−0.615485	+	0.788148i	$0.711040\pi$
$348$	0	0
$349$	35.5378	1.90230	0.951148	−	0.308735i	$-0.0999057\pi$
$349$	35.5378	1.90230	0.951148	+	0.308735i	$0.0999057\pi$
$350$	−52.0157	−2.78036
$351$	0	0
$352$	−0.479657	−0.0255658
$353$	−23.2725	−1.23867	−0.619336	−	0.785126i	$-0.712598\pi$
$353$	−23.2725	−1.23867	−0.619336	+	0.785126i	$0.712598\pi$
$354$	0	0
$355$	−18.8184	−0.998774
$356$	0.832811	0.0441389
$357$	0	0
$358$	14.3575	0.758820
$359$	8.87297	0.468298	0.234149	−	0.972201i	$-0.424770\pi$
$359$	8.87297	0.468298	0.234149	+	0.972201i	$0.424770\pi$
$360$	0	0
$361$	7.99446	0.420761
$362$	−13.0487	−0.685823
$363$	0	0
$364$	−0.157887	−0.00827554
$365$	−18.5576	−0.971348
$366$	0	0
$367$	−27.1883	−1.41922	−0.709609	−	0.704595i	$-0.751128\pi$
$367$	−27.1883	−1.41922	−0.709609	+	0.704595i	$0.751128\pi$
$368$	4.10632	0.214057
$369$	0	0
$370$	−22.0458	−1.14611
$371$	3.50628	0.182037
$372$	0	0
$373$	−32.8792	−1.70242	−0.851209	−	0.524827i	$-0.824130\pi$
$373$	−32.8792	−1.70242	−0.851209	+	0.524827i	$0.824130\pi$
$374$	5.20074	0.268924
$375$	0	0
$376$	15.8513	0.817466
$377$	−3.03944	−0.156539
$378$	0	0
$379$	−2.48630	−0.127712	−0.0638562	−	0.997959i	$-0.520340\pi$
$379$	−2.48630	−0.127712	−0.0638562	+	0.997959i	$0.520340\pi$
$380$	1.84165	0.0944747
$381$	0	0
$382$	−5.09290	−0.260576
$383$	19.9338	1.01857	0.509286	−	0.860597i	$-0.329910\pi$
$383$	19.9338	1.01857	0.509286	+	0.860597i	$0.329910\pi$
$384$	0	0
$385$	−12.6104	−0.642683
$386$	14.8789	0.757314
$387$	0	0
$388$	1.07636	0.0546440
$389$	4.45423	0.225839	0.112919	−	0.993604i	$-0.463980\pi$
$389$	4.45423	0.225839	0.112919	+	0.993604i	$0.463980\pi$
$390$	0	0
$391$	4.03646	0.204132
$392$	6.09397	0.307792
$393$	0	0
$394$	−13.9036	−0.700455
$395$	−57.8575	−2.91113
$396$	0	0
$397$	13.5888	0.682005	0.341002	−	0.940062i	$-0.389234\pi$
$397$	13.5888	0.682005	0.341002	+	0.940062i	$0.389234\pi$
$398$	29.2566	1.46650
$399$	0	0
$400$	−47.6035	−2.38017
$401$	12.3485	0.616657	0.308329	−	0.951280i	$-0.400230\pi$
$401$	12.3485	0.616657	0.308329	+	0.951280i	$0.400230\pi$
$402$	0	0
$403$	−3.28283	−0.163529
$404$	0.415457	0.0206698
$405$	0	0
$406$	20.5978	1.02225
$407$	−3.81337	−0.189021
$408$	0	0
$409$	0.988963	0.0489011	0.0244505	−	0.999701i	$-0.492216\pi$
$409$	0.988963	0.0489011	0.0244505	+	0.999701i	$0.492216\pi$
$410$	12.2811	0.606518
$411$	0	0
$412$	−0.220997	−0.0108877
$413$	−26.6270	−1.31023
$414$	0	0
$415$	−72.5355	−3.56063
$416$	−0.295675	−0.0144966
$417$	0	0
$418$	−7.19016	−0.351682
$419$	−4.92913	−0.240804	−0.120402	−	0.992725i	$-0.538418\pi$
$419$	−4.92913	−0.240804	−0.120402	+	0.992725i	$0.538418\pi$
$420$	0	0
$421$	−30.4846	−1.48573	−0.742863	−	0.669443i	$-0.766533\pi$
$421$	−30.4846	−1.48573	−0.742863	+	0.669443i	$0.766533\pi$
$422$	0.114194	0.00555886
$423$	0	0
$424$	3.35130	0.162753
$425$	−46.7936	−2.26982
$426$	0	0
$427$	3.01863	0.146082
$428$	−0.754951	−0.0364920
$429$	0	0
$430$	−46.1261	−2.22440
$431$	−34.8402	−1.67820	−0.839098	−	0.543981i	$-0.816916\pi$
$431$	−34.8402	−1.67820	−0.839098	+	0.543981i	$0.816916\pi$
$432$	0	0
$433$	−18.1325	−0.871394	−0.435697	−	0.900093i	$-0.643498\pi$
$433$	−18.1325	−0.871394	−0.435697	+	0.900093i	$0.643498\pi$
$434$	22.2473	1.06790
$435$	0	0
$436$	−0.678566	−0.0324974
$437$	−5.58052	−0.266952
$438$	0	0
$439$	26.5629	1.26778	0.633889	−	0.773424i	$-0.281458\pi$
$439$	26.5629	1.26778	0.633889	+	0.773424i	$0.281458\pi$
$440$	−12.0529	−0.574601
$441$	0	0
$442$	3.20589	0.152489
$443$	−16.9778	−0.806642	−0.403321	−	0.915059i	$-0.632144\pi$
$443$	−16.9778	−0.806642	−0.403321	+	0.915059i	$0.632144\pi$
$444$	0	0
$445$	41.0025	1.94371
$446$	32.8489	1.55544
$447$	0	0
$448$	25.0848	1.18515
$449$	−15.6750	−0.739751	−0.369875	−	0.929081i	$-0.620600\pi$
$449$	−15.6750	−0.739751	−0.369875	+	0.929081i	$0.620600\pi$
$450$	0	0
$451$	2.12431	0.100030
$452$	0.939817	0.0442053
$453$	0	0
$454$	1.79577	0.0842796
$455$	−7.77340	−0.364422
$456$	0	0
$457$	23.3260	1.09115	0.545573	−	0.838063i	$-0.316312\pi$
$457$	23.3260	1.09115	0.545573	+	0.838063i	$0.316312\pi$
$458$	3.32448	0.155343
$459$	0	0
$460$	−0.380721	−0.0177512
$461$	12.9551	0.603377	0.301688	−	0.953407i	$-0.402450\pi$
$461$	12.9551	0.603377	0.301688	+	0.953407i	$0.402450\pi$
$462$	0	0
$463$	−6.48668	−0.301461	−0.150731	−	0.988575i	$-0.548163\pi$
$463$	−6.48668	−0.301461	−0.150731	+	0.988575i	$0.548163\pi$
$464$	18.8506	0.875118
$465$	0	0
$466$	40.6115	1.88129
$467$	0.796623	0.0368633	0.0184317	−	0.999830i	$-0.494133\pi$
$467$	0.796623	0.0368633	0.0184317	+	0.999830i	$0.494133\pi$
$468$	0	0
$469$	23.1403	1.06852
$470$	31.7618	1.46506
$471$	0	0
$472$	−25.4500	−1.17143
$473$	−7.97863	−0.366858
$474$	0	0
$475$	64.6934	2.96834
$476$	0.962558	0.0441188
$477$	0	0
$478$	11.3838	0.520683
$479$	15.7217	0.718342	0.359171	−	0.933272i	$-0.383059\pi$
$479$	15.7217	0.718342	0.359171	+	0.933272i	$0.383059\pi$
$480$	0	0
$481$	−2.35067	−0.107181
$482$	−39.2815	−1.78923
$483$	0	0
$484$	−0.0848502	−0.00385683
$485$	52.9934	2.40631
$486$	0	0
$487$	−33.5473	−1.52017	−0.760086	−	0.649822i	$-0.774843\pi$
$487$	−33.5473	−1.52017	−0.760086	+	0.649822i	$0.774843\pi$
$488$	2.88520	0.130607
$489$	0	0
$490$	12.2108	0.551625
$491$	6.13635	0.276929	0.138465	−	0.990367i	$-0.455783\pi$
$491$	6.13635	0.276929	0.138465	+	0.990367i	$0.455783\pi$
$492$	0	0
$493$	18.5299	0.834545
$494$	−4.43223	−0.199415
$495$	0	0
$496$	20.3601	0.914197
$497$	13.5980	0.609954
$498$	0	0
$499$	−31.0927	−1.39190	−0.695950	−	0.718091i	$-0.745016\pi$
$499$	−31.0927	−1.39190	−0.695950	+	0.718091i	$0.745016\pi$
$500$	2.64129	0.118122
$501$	0	0
$502$	16.8286	0.751095
$503$	27.3878	1.22116	0.610581	−	0.791954i	$-0.290936\pi$
$503$	27.3878	1.22116	0.610581	+	0.791954i	$0.290936\pi$
$504$	0	0
$505$	20.4546	0.910216
$506$	1.48641	0.0660789
$507$	0	0
$508$	1.49167	0.0661823
$509$	−15.9582	−0.707335	−0.353667	−	0.935371i	$-0.615065\pi$
$509$	−15.9582	−0.707335	−0.353667	+	0.935371i	$0.615065\pi$
$510$	0	0
$511$	13.4096	0.593205
$512$	23.8946	1.05600
$513$	0	0
$514$	6.72745	0.296735
$515$	−10.8805	−0.479453
$516$	0	0
$517$	5.49399	0.241625
$518$	15.9302	0.699931
$519$	0	0
$520$	−7.42979	−0.325818
$521$	−8.37497	−0.366914	−0.183457	−	0.983028i	$-0.558729\pi$
$521$	−8.37497	−0.366914	−0.183457	+	0.983028i	$0.558729\pi$
$522$	0	0
$523$	0.277965	0.0121546	0.00607729	−	0.999982i	$-0.498066\pi$
$523$	0.277965	0.0121546	0.00607729	+	0.999982i	$0.498066\pi$
$524$	0.124797	0.00545178
$525$	0	0
$526$	22.7464	0.991788
$527$	20.0137	0.871812
$528$	0	0
$529$	−21.8464	−0.949841
$530$	6.71514	0.291687
$531$	0	0
$532$	−1.33076	−0.0576959
$533$	1.30949	0.0567202
$534$	0	0
$535$	−37.1692	−1.60696
$536$	22.1174	0.955326
$537$	0	0
$538$	−1.09726	−0.0473062
$539$	2.11215	0.0909766
$540$	0	0
$541$	−4.88770	−0.210139	−0.105069	−	0.994465i	$-0.533506\pi$
$541$	−4.88770	−0.210139	−0.105069	+	0.994465i	$0.533506\pi$
$542$	20.8507	0.895613
$543$	0	0
$544$	1.80258	0.0772849
$545$	−33.4084	−1.43106
$546$	0	0
$547$	−30.6918	−1.31229	−0.656143	−	0.754637i	$-0.727813\pi$
$547$	−30.6918	−1.31229	−0.656143	+	0.754637i	$0.727813\pi$
$548$	−0.698738	−0.0298486
$549$	0	0
$550$	−17.2315	−0.734756
$551$	−25.6181	−1.09137
$552$	0	0
$553$	41.8074	1.77783
$554$	−39.5870	−1.68189
$555$	0	0
$556$	−0.684098	−0.0290122
$557$	−19.1700	−0.812257	−0.406128	−	0.913816i	$-0.633121\pi$
$557$	−19.1700	−0.812257	−0.406128	+	0.913816i	$0.633121\pi$
$558$	0	0
$559$	−4.91826	−0.208020
$560$	48.2106	2.03727
$561$	0	0
$562$	12.9385	0.545778
$563$	−18.6901	−0.787694	−0.393847	−	0.919176i	$-0.628856\pi$
$563$	−18.6901	−0.787694	−0.393847	+	0.919176i	$0.628856\pi$
$564$	0	0
$565$	46.2708	1.94663
$566$	15.1424	0.636481
$567$	0	0
$568$	12.9969	0.545339
$569$	−21.0967	−0.884420	−0.442210	−	0.896911i	$-0.645805\pi$
$569$	−21.0967	−0.884420	−0.442210	+	0.896911i	$0.645805\pi$
$570$	0	0
$571$	−1.82922	−0.0765506	−0.0382753	−	0.999267i	$-0.512186\pi$
$571$	−1.82922	−0.0765506	−0.0382753	+	0.999267i	$0.512186\pi$
$572$	−0.0523042	−0.00218695
$573$	0	0
$574$	−8.87420	−0.370402
$575$	−13.3740	−0.557733
$576$	0	0
$577$	−2.06740	−0.0860670	−0.0430335	−	0.999074i	$-0.513702\pi$
$577$	−2.06740	−0.0860670	−0.0430335	+	0.999074i	$0.513702\pi$
$578$	3.98146	0.165607
$579$	0	0
$580$	−1.74775	−0.0725714
$581$	52.4136	2.17448
$582$	0	0
$583$	1.16155	0.0481063
$584$	12.8168	0.530364
$585$	0	0
$586$	−41.9647	−1.73354
$587$	−24.3061	−1.00322	−0.501610	−	0.865094i	$-0.667259\pi$
$587$	−24.3061	−1.00322	−0.501610	+	0.865094i	$0.667259\pi$
$588$	0	0
$589$	−27.6696	−1.14010
$590$	−50.9952	−2.09944
$591$	0	0
$592$	14.5789	0.599188
$593$	43.3506	1.78020	0.890098	−	0.455770i	$-0.150636\pi$
$593$	43.3506	1.78020	0.890098	+	0.455770i	$0.150636\pi$
$594$	0	0
$595$	47.3904	1.94282
$596$	−0.0731204	−0.00299513
$597$	0	0
$598$	0.916267	0.0374689
$599$	−10.5087	−0.429372	−0.214686	−	0.976683i	$-0.568873\pi$
$599$	−10.5087	−0.429372	−0.214686	+	0.976683i	$0.568873\pi$
$600$	0	0
$601$	1.02909	0.0419775	0.0209887	−	0.999780i	$-0.493319\pi$
$601$	1.02909	0.0419775	0.0209887	+	0.999780i	$0.493319\pi$
$602$	33.3304	1.35844
$603$	0	0
$604$	−1.25607	−0.0511090
$605$	−4.17750	−0.169840
$606$	0	0
$607$	−38.1349	−1.54785	−0.773924	−	0.633279i	$-0.781709\pi$
$607$	−38.1349	−1.54785	−0.773924	+	0.633279i	$0.781709\pi$
$608$	−2.49211	−0.101069
$609$	0	0
$610$	5.78120	0.234074
$611$	3.38666	0.137009
$612$	0	0
$613$	−8.81410	−0.355998	−0.177999	−	0.984031i	$-0.556962\pi$
$613$	−8.81410	−0.355998	−0.177999	+	0.984031i	$0.556962\pi$
$614$	−5.05407	−0.203966
$615$	0	0
$616$	8.70937	0.350910
$617$	10.7158	0.431401	0.215701	−	0.976460i	$-0.430796\pi$
$617$	10.7158	0.431401	0.215701	+	0.976460i	$0.430796\pi$
$618$	0	0
$619$	−10.5467	−0.423910	−0.211955	−	0.977279i	$-0.567983\pi$
$619$	−10.5467	−0.423910	−0.211955	+	0.977279i	$0.567983\pi$
$620$	−1.88771	−0.0758122
$621$	0	0
$622$	15.0760	0.604493
$623$	−29.6281	−1.18703
$624$	0	0
$625$	67.7831	2.71132
$626$	−24.3188	−0.971975
$627$	0	0
$628$	0.816776	0.0325929
$629$	14.3308	0.571408
$630$	0	0
$631$	5.20291	0.207125	0.103562	−	0.994623i	$-0.466976\pi$
$631$	5.20291	0.207125	0.103562	+	0.994623i	$0.466976\pi$
$632$	39.9594	1.58950
$633$	0	0
$634$	−7.79122	−0.309429
$635$	73.4408	2.91441
$636$	0	0
$637$	1.30199	0.0515867
$638$	6.82356	0.270148
$639$	0	0
$640$	44.0342	1.74061
$641$	3.36670	0.132977	0.0664883	−	0.997787i	$-0.478820\pi$
$641$	3.36670	0.132977	0.0664883	+	0.997787i	$0.478820\pi$
$642$	0	0
$643$	21.5138	0.848421	0.424210	−	0.905564i	$-0.360552\pi$
$643$	21.5138	0.848421	0.424210	+	0.905564i	$0.360552\pi$
$644$	0.275106	0.0108407
$645$	0	0
$646$	27.0210	1.06313
$647$	−4.23834	−0.166626	−0.0833131	−	0.996523i	$-0.526550\pi$
$647$	−4.23834	−0.166626	−0.0833131	+	0.996523i	$0.526550\pi$
$648$	0	0
$649$	−8.82087	−0.346249
$650$	−10.6220	−0.416631
$651$	0	0
$652$	−0.380792	−0.0149130
$653$	30.7582	1.20366	0.601830	−	0.798624i	$-0.294438\pi$
$653$	30.7582	1.20366	0.601830	+	0.798624i	$0.294438\pi$
$654$	0	0
$655$	6.14423	0.240075
$656$	−8.12145	−0.317089
$657$	0	0
$658$	−22.9509	−0.894718
$659$	41.9946	1.63588	0.817939	−	0.575306i	$-0.195117\pi$
$659$	41.9946	1.63588	0.817939	+	0.575306i	$0.195117\pi$
$660$	0	0
$661$	27.7342	1.07874	0.539368	−	0.842070i	$-0.318663\pi$
$661$	27.7342	1.07874	0.539368	+	0.842070i	$0.318663\pi$
$662$	23.9850	0.932203
$663$	0	0
$664$	50.0968	1.94413
$665$	−65.5186	−2.54070
$666$	0	0
$667$	5.29599	0.205061
$668$	−0.993718	−0.0384481
$669$	0	0
$670$	44.3176	1.71214
$671$	1.00000	0.0386046
$672$	0	0
$673$	−32.8714	−1.26710	−0.633550	−	0.773702i	$-0.718403\pi$
$673$	−32.8714	−1.26710	−0.633550	+	0.773702i	$0.718403\pi$
$674$	13.2306	0.509625
$675$	0	0
$676$	1.07081	0.0411850
$677$	−16.7719	−0.644598	−0.322299	−	0.946638i	$-0.604456\pi$
$677$	−16.7719	−0.644598	−0.322299	+	0.946638i	$0.604456\pi$
$678$	0	0
$679$	−38.2926	−1.46954
$680$	45.2956	1.73701
$681$	0	0
$682$	7.36998	0.282211
$683$	−20.8759	−0.798792	−0.399396	−	0.916778i	$-0.630780\pi$
$683$	−20.8759	−0.798792	−0.399396	+	0.916778i	$0.630780\pi$
$684$	0	0
$685$	−34.4016	−1.31442
$686$	20.4188	0.779593
$687$	0	0
$688$	30.5031	1.16292
$689$	0.716012	0.0272779
$690$	0	0
$691$	−6.28344	−0.239033	−0.119517	−	0.992832i	$-0.538134\pi$
$691$	−6.28344	−0.239033	−0.119517	+	0.992832i	$0.538134\pi$
$692$	−1.38723	−0.0527345
$693$	0	0
$694$	31.7332	1.20458
$695$	−33.6808	−1.27758
$696$	0	0
$697$	−7.98327	−0.302388
$698$	−49.1804	−1.86151
$699$	0	0
$700$	−3.18923	−0.120542
$701$	48.3627	1.82663	0.913317	−	0.407250i	$-0.133512\pi$
$701$	48.3627	1.82663	0.913317	+	0.407250i	$0.133512\pi$
$702$	0	0
$703$	−19.8128	−0.747254
$704$	8.30999	0.313195
$705$	0	0
$706$	32.2066	1.21211
$707$	−14.7803	−0.555871
$708$	0	0
$709$	−36.7254	−1.37925	−0.689625	−	0.724167i	$-0.742225\pi$
$709$	−36.7254	−1.37925	−0.689625	+	0.724167i	$0.742225\pi$
$710$	26.0425	0.977358
$711$	0	0
$712$	−28.3185	−1.06128
$713$	5.72008	0.214219
$714$	0	0
$715$	−2.57514	−0.0963047
$716$	0.880302	0.0328984
$717$	0	0
$718$	−12.2792	−0.458256
$719$	−35.8470	−1.33687	−0.668434	−	0.743772i	$-0.733035\pi$
$719$	−35.8470	−1.33687	−0.668434	+	0.743772i	$0.733035\pi$
$720$	0	0
$721$	7.86219	0.292803
$722$	−11.0634	−0.411739
$723$	0	0
$724$	−0.800051	−0.0297337
$725$	−61.3950	−2.28015
$726$	0	0
$727$	−47.3533	−1.75624	−0.878118	−	0.478443i	$-0.841201\pi$
$727$	−47.3533	−1.75624	−0.878118	+	0.478443i	$0.841201\pi$
$728$	5.36871	0.198978
$729$	0	0
$730$	25.6816	0.950520
$731$	29.9841	1.10900
$732$	0	0
$733$	−9.34736	−0.345253	−0.172626	−	0.984987i	$-0.555225\pi$
$733$	−9.34736	−0.345253	−0.172626	+	0.984987i	$0.555225\pi$
$734$	37.6256	1.38879
$735$	0	0
$736$	0.515190	0.0189902
$737$	7.66581	0.282374
$738$	0	0
$739$	−20.2360	−0.744394	−0.372197	−	0.928154i	$-0.621395\pi$
$739$	−20.2360	−0.744394	−0.372197	+	0.928154i	$0.621395\pi$
$740$	−1.35169	−0.0496892
$741$	0	0
$742$	−4.85231	−0.178134
$743$	−21.5042	−0.788912	−0.394456	−	0.918915i	$-0.629067\pi$
$743$	−21.5042	−0.788912	−0.394456	+	0.918915i	$0.629067\pi$
$744$	0	0
$745$	−3.60000	−0.131894
$746$	45.5011	1.66591
$747$	0	0
$748$	0.318872	0.0116591
$749$	26.8582	0.981376
$750$	0	0
$751$	19.6460	0.716891	0.358446	−	0.933551i	$-0.383307\pi$
$751$	19.6460	0.716891	0.358446	+	0.933551i	$0.383307\pi$
$752$	−21.0041	−0.765939
$753$	0	0
$754$	4.20625	0.153183
$755$	−61.8414	−2.25064
$756$	0	0
$757$	10.3041	0.374509	0.187255	−	0.982311i	$-0.440041\pi$
$757$	10.3041	0.374509	0.187255	+	0.982311i	$0.440041\pi$
$758$	3.44076	0.124974
$759$	0	0
$760$	−62.6225	−2.27156
$761$	12.0705	0.437554	0.218777	−	0.975775i	$-0.429793\pi$
$761$	12.0705	0.437554	0.218777	+	0.975775i	$0.429793\pi$
$762$	0	0
$763$	24.1407	0.873951
$764$	−0.312261	−0.0112972
$765$	0	0
$766$	−27.5862	−0.996731
$767$	−5.43744	−0.196335
$768$	0	0
$769$	−19.7134	−0.710884	−0.355442	−	0.934698i	$-0.615670\pi$
$769$	−19.7134	−0.710884	−0.355442	+	0.934698i	$0.615670\pi$
$770$	17.4513	0.628902
$771$	0	0
$772$	0.912265	0.0328331
$773$	40.5428	1.45822	0.729111	−	0.684395i	$-0.239934\pi$
$773$	40.5428	1.45822	0.729111	+	0.684395i	$0.239934\pi$
$774$	0	0
$775$	−66.3113	−2.38197
$776$	−36.6000	−1.31386
$777$	0	0
$778$	−6.16417	−0.220996
$779$	11.0371	0.395445
$780$	0	0
$781$	4.50469	0.161190
$782$	−5.58601	−0.199755
$783$	0	0
$784$	−8.07495	−0.288391
$785$	40.2130	1.43526
$786$	0	0
$787$	−15.6568	−0.558105	−0.279053	−	0.960276i	$-0.590020\pi$
$787$	−15.6568	−0.558105	−0.279053	+	0.960276i	$0.590020\pi$
$788$	−0.852472	−0.0303681
$789$	0	0
$790$	80.0683	2.84870
$791$	−33.4350	−1.18881
$792$	0	0
$793$	0.616430	0.0218901
$794$	−18.8055	−0.667381
$795$	0	0
$796$	1.79381	0.0635798
$797$	−36.5833	−1.29585	−0.647924	−	0.761705i	$-0.724362\pi$
$797$	−36.5833	−1.29585	−0.647924	+	0.761705i	$0.724362\pi$
$798$	0	0
$799$	−20.6467	−0.730428
$800$	−5.97246	−0.211158
$801$	0	0
$802$	−17.0890	−0.603434
$803$	4.44227	0.156764
$804$	0	0
$805$	13.5445	0.477382
$806$	4.54307	0.160023
$807$	0	0
$808$	−14.1270	−0.496986
$809$	−23.0629	−0.810848	−0.405424	−	0.914129i	$-0.632876\pi$
$809$	−23.0629	−0.810848	−0.405424	+	0.914129i	$0.632876\pi$
$810$	0	0
$811$	−14.9831	−0.526126	−0.263063	−	0.964779i	$-0.584733\pi$
$811$	−14.9831	−0.526126	−0.263063	+	0.964779i	$0.584733\pi$
$812$	1.26291	0.0443195
$813$	0	0
$814$	5.27728	0.184968
$815$	−18.7479	−0.656709
$816$	0	0
$817$	−41.4539	−1.45029
$818$	−1.36861	−0.0478525
$819$	0	0
$820$	0.752987	0.0262954
$821$	17.6752	0.616868	0.308434	−	0.951246i	$-0.400195\pi$
$821$	17.6752	0.616868	0.308434	+	0.951246i	$0.400195\pi$
$822$	0	0
$823$	21.8275	0.760857	0.380429	−	0.924810i	$-0.375776\pi$
$823$	21.8275	0.760857	0.380429	+	0.924810i	$0.375776\pi$
$824$	7.51466	0.261786
$825$	0	0
$826$	36.8488	1.28213
$827$	−37.0472	−1.28826	−0.644129	−	0.764917i	$-0.722780\pi$
$827$	−37.0472	−1.28826	−0.644129	+	0.764917i	$0.722780\pi$
$828$	0	0
$829$	18.0240	0.625999	0.312999	−	0.949753i	$-0.398666\pi$
$829$	18.0240	0.625999	0.312999	+	0.949753i	$0.398666\pi$
$830$	100.381	3.48428
$831$	0	0
$832$	5.12253	0.177592
$833$	−7.93757	−0.275020
$834$	0	0
$835$	−48.9246	−1.69310
$836$	−0.440849	−0.0152471
$837$	0	0
$838$	6.82137	0.235640
$839$	50.2205	1.73380	0.866902	−	0.498478i	$-0.166108\pi$
$839$	50.2205	1.73380	0.866902	+	0.498478i	$0.166108\pi$
$840$	0	0
$841$	−4.68804	−0.161657
$842$	42.1872	1.45387
$843$	0	0
$844$	0.00700154	0.000241003 0
$845$	52.7202	1.81363
$846$	0	0
$847$	3.01863	0.103721
$848$	−4.44071	−0.152495
$849$	0	0
$850$	64.7571	2.22115
$851$	4.09586	0.140404
$852$	0	0
$853$	−39.6192	−1.35654	−0.678268	−	0.734815i	$-0.737269\pi$
$853$	−39.6192	−1.35654	−0.678268	+	0.734815i	$0.737269\pi$
$854$	−4.17745	−0.142950
$855$	0	0
$856$	25.6710	0.877416
$857$	−46.8091	−1.59897	−0.799484	−	0.600687i	$-0.794894\pi$
$857$	−46.8091	−1.59897	−0.799484	+	0.600687i	$0.794894\pi$
$858$	0	0
$859$	41.3263	1.41003	0.705017	−	0.709190i	$-0.250939\pi$
$859$	41.3263	1.41003	0.705017	+	0.709190i	$0.250939\pi$
$860$	−2.82812	−0.0964381
$861$	0	0
$862$	48.2150	1.64221
$863$	49.6791	1.69110	0.845548	−	0.533900i	$-0.179274\pi$
$863$	49.6791	1.69110	0.845548	+	0.533900i	$0.179274\pi$
$864$	0	0
$865$	−68.2985	−2.32222
$866$	25.0934	0.852710
$867$	0	0
$868$	1.36404	0.0462987
$869$	13.8498	0.469821
$870$	0	0
$871$	4.72543	0.160115
$872$	23.0736	0.781370
$873$	0	0
$874$	7.72282	0.261228
$875$	−93.9665	−3.17665
$876$	0	0
$877$	26.5377	0.896116	0.448058	−	0.894005i	$-0.352116\pi$
$877$	26.5377	0.896116	0.448058	+	0.894005i	$0.352116\pi$
$878$	−36.7601	−1.24059
$879$	0	0
$880$	15.9710	0.538383
$881$	8.22046	0.276954	0.138477	−	0.990366i	$-0.455779\pi$
$881$	8.22046	0.276954	0.138477	+	0.990366i	$0.455779\pi$
$882$	0	0
$883$	0.259727	0.00874051	0.00437026	−	0.999990i	$-0.498609\pi$
$883$	0.259727	0.00874051	0.00437026	+	0.999990i	$0.498609\pi$
$884$	0.196562	0.00661110
$885$	0	0
$886$	23.4955	0.789345
$887$	33.3037	1.11823	0.559115	−	0.829090i	$-0.311141\pi$
$887$	33.3037	1.11823	0.559115	+	0.829090i	$0.311141\pi$
$888$	0	0
$889$	−53.0678	−1.77984
$890$	−56.7429	−1.90203
$891$	0	0
$892$	2.01406	0.0674357
$893$	28.5447	0.955211
$894$	0	0
$895$	43.3407	1.44872
$896$	−31.8188	−1.06299
$897$	0	0
$898$	21.6925	0.723889
$899$	26.2588	0.875780
$900$	0	0
$901$	−4.36516	−0.145424
$902$	−2.93981	−0.0978849
$903$	0	0
$904$	−31.9571	−1.06288
$905$	−39.3896	−1.30935
$906$	0	0
$907$	−26.9334	−0.894308	−0.447154	−	0.894457i	$-0.647562\pi$
$907$	−26.9334	−0.894308	−0.447154	+	0.894457i	$0.647562\pi$
$908$	0.110104	0.00365392
$909$	0	0
$910$	10.7575	0.356608
$911$	−50.9427	−1.68781	−0.843904	−	0.536494i	$-0.819748\pi$
$911$	−50.9427	−1.68781	−0.843904	+	0.536494i	$0.819748\pi$
$912$	0	0
$913$	17.3634	0.574643
$914$	−32.2807	−1.06775
$915$	0	0
$916$	0.203833	0.00673485
$917$	−4.43978	−0.146614
$918$	0	0
$919$	−5.94386	−0.196070	−0.0980350	−	0.995183i	$-0.531256\pi$
$919$	−5.94386	−0.196070	−0.0980350	+	0.995183i	$0.531256\pi$
$920$	12.9458	0.426812
$921$	0	0
$922$	−17.9284	−0.590439
$923$	2.77682	0.0914003
$924$	0	0
$925$	−47.4823	−1.56121
$926$	8.97684	0.294997
$927$	0	0
$928$	2.36505	0.0776366
$929$	−50.2019	−1.64707	−0.823535	−	0.567265i	$-0.808001\pi$
$929$	−50.2019	−1.64707	−0.823535	+	0.567265i	$0.808001\pi$
$930$	0	0
$931$	10.9739	0.359655
$932$	2.49001	0.0815629
$933$	0	0
$934$	−1.10244	−0.0360729
$935$	15.6993	0.513422
$936$	0	0
$937$	38.1542	1.24644	0.623222	−	0.782045i	$-0.285823\pi$
$937$	38.1542	1.24644	0.623222	+	0.782045i	$0.285823\pi$
$938$	−32.0236	−1.04561
$939$	0	0
$940$	1.94741	0.0635175
$941$	−44.6577	−1.45580	−0.727899	−	0.685684i	$-0.759503\pi$
$941$	−44.6577	−1.45580	−0.727899	+	0.685684i	$0.759503\pi$
$942$	0	0
$943$	−2.28168	−0.0743017
$944$	33.7231	1.09759
$945$	0	0
$946$	11.0415	0.358991
$947$	39.9218	1.29728	0.648641	−	0.761094i	$-0.275337\pi$
$947$	39.9218	1.29728	0.648641	+	0.761094i	$0.275337\pi$
$948$	0	0
$949$	2.73835	0.0888905
$950$	−89.5286	−2.90469
$951$	0	0
$952$	−32.7303	−1.06080
$953$	54.4983	1.76537	0.882687	−	0.469962i	$-0.155732\pi$
$953$	54.4983	1.76537	0.882687	+	0.469962i	$0.155732\pi$
$954$	0	0
$955$	−15.3738	−0.497484
$956$	0.697974	0.0225741
$957$	0	0
$958$	−21.7571	−0.702939
$959$	24.8583	0.802717
$960$	0	0
$961$	−2.63847	−0.0851119
$962$	3.25307	0.104883
$963$	0	0
$964$	−2.40846	−0.0775714
$965$	44.9143	1.44584
$966$	0	0
$967$	−21.4053	−0.688349	−0.344175	−	0.938906i	$-0.611841\pi$
$967$	−21.4053	−0.688349	−0.344175	+	0.938906i	$0.611841\pi$
$968$	2.88520	0.0927339
$969$	0	0
$970$	−73.3370	−2.35471
$971$	−23.9525	−0.768673	−0.384337	−	0.923193i	$-0.625570\pi$
$971$	−23.9525	−0.768673	−0.384337	+	0.923193i	$0.625570\pi$
$972$	0	0
$973$	24.3375	0.780224
$974$	46.4257	1.48758
$975$	0	0
$976$	−3.82310	−0.122374
$977$	−57.7260	−1.84682	−0.923409	−	0.383818i	$-0.874609\pi$
$977$	−57.7260	−1.84682	−0.923409	+	0.383818i	$0.874609\pi$
$978$	0	0
$979$	−9.81507	−0.313691
$980$	0.748676	0.0239156
$981$	0	0
$982$	−8.49202	−0.270991
$983$	−8.44703	−0.269418	−0.134709	−	0.990885i	$-0.543010\pi$
$983$	−8.44703	−0.269418	−0.134709	+	0.990885i	$0.543010\pi$
$984$	0	0
$985$	−41.9705	−1.33729
$986$	−25.6433	−0.816651
$987$	0	0
$988$	−0.271753	−0.00864561
$989$	8.56969	0.272500
$990$	0	0
$991$	−58.9931	−1.87398	−0.936989	−	0.349359i	$-0.886399\pi$
$991$	−58.9931	−1.87398	−0.936989	+	0.349359i	$0.886399\pi$
$992$	2.55444	0.0811035
$993$	0	0
$994$	−18.8181	−0.596875
$995$	88.3160	2.79981
$996$	0	0
$997$	−24.9714	−0.790852	−0.395426	−	0.918498i	$-0.629403\pi$
$997$	−24.9714	−0.790852	−0.395426	+	0.918498i	$0.629403\pi$
$998$	43.0288	1.36205
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6039.2.a.m.1.8	✓	25
3.2	odd	2		6039.2.a.p.1.18	yes	25

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6039.2.a.m.1.8	✓	25	1.1	even	1	trivial
6039.2.a.p.1.18	yes	25	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 6039 = 3^{2} \cdot 11 \cdot 61 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6039.a (trivial)

\(f(q)\)	\(=\)	\(q-1.38389 q^{2} -0.0848502 q^{4} -4.17750 q^{5} +3.01863 q^{7} +2.88520 q^{8} +O(q^{10})\)	\(q-1.38389 q^{2} -0.0848502 q^{4} -4.17750 q^{5} +3.01863 q^{7} +2.88520 q^{8} +5.78120 q^{10} +1.00000 q^{11} +0.616430 q^{13} -4.17745 q^{14} -3.82310 q^{16} -3.75806 q^{17} +5.19562 q^{19} +0.354462 q^{20} -1.38389 q^{22} -1.07408 q^{23} +12.4515 q^{25} -0.853071 q^{26} -0.256132 q^{28} -4.93072 q^{29} -5.32555 q^{31} -0.479657 q^{32} +5.20074 q^{34} -12.6104 q^{35} -3.81337 q^{37} -7.19016 q^{38} -12.0529 q^{40} +2.12431 q^{41} -7.97863 q^{43} -0.0848502 q^{44} +1.48641 q^{46} +5.49399 q^{47} +2.11215 q^{49} -17.2315 q^{50} -0.0523042 q^{52} +1.16155 q^{53} -4.17750 q^{55} +8.70937 q^{56} +6.82356 q^{58} -8.82087 q^{59} +1.00000 q^{61} +7.36998 q^{62} +8.30999 q^{64} -2.57514 q^{65} +7.66581 q^{67} +0.318872 q^{68} +17.4513 q^{70} +4.50469 q^{71} +4.44227 q^{73} +5.27728 q^{74} -0.440849 q^{76} +3.01863 q^{77} +13.8498 q^{79} +15.9710 q^{80} -2.93981 q^{82} +17.3634 q^{83} +15.6993 q^{85} +11.0415 q^{86} +2.88520 q^{88} -9.81507 q^{89} +1.86078 q^{91} +0.0911360 q^{92} -7.60307 q^{94} -21.7047 q^{95} -12.6854 q^{97} -2.92298 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 25 q - 5 q^{2} + 25 q^{4} - 12 q^{5} - 4 q^{7} - 15 q^{8}+O(q^{10}) \) 25 * q - 5 * q^2 + 25 * q^4 - 12 * q^5 - 4 * q^7 - 15 * q^8	\( 25 q - 5 q^{2} + 25 q^{4} - 12 q^{5} - 4 q^{7} - 15 q^{8} - 12 q^{10} + 25 q^{11} - 4 q^{13} - 14 q^{14} + 21 q^{16} - 16 q^{17} - 18 q^{19} - 28 q^{20} - 5 q^{22} - 8 q^{23} + 29 q^{25} - 16 q^{26} + 18 q^{28} - 28 q^{29} - 8 q^{31} - 35 q^{32} + 6 q^{34} - 22 q^{35} + 4 q^{37} + 4 q^{38} - 12 q^{40} - 58 q^{41} - 26 q^{43} + 25 q^{44} + 8 q^{46} - 20 q^{47} + 23 q^{49} - 27 q^{50} - 2 q^{52} - 36 q^{53} - 12 q^{55} - 70 q^{56} + 12 q^{58} - 18 q^{59} + 25 q^{61} - 42 q^{62} + 35 q^{64} - 76 q^{65} - 8 q^{67} - 28 q^{68} + 76 q^{70} - 24 q^{71} + 2 q^{73} - 40 q^{74} - 64 q^{76} - 4 q^{77} - 22 q^{79} - 36 q^{80} + 30 q^{82} - 14 q^{83} - 70 q^{86} - 15 q^{88} - 82 q^{89} - 6 q^{91} - 48 q^{92} - 16 q^{94} - 34 q^{95} + 16 q^{97} - 35 q^{98}+O(q^{100}) \) 25 * q - 5 * q^2 + 25 * q^4 - 12 * q^5 - 4 * q^7 - 15 * q^8 - 12 * q^10 + 25 * q^11 - 4 * q^13 - 14 * q^14 + 21 * q^16 - 16 * q^17 - 18 * q^19 - 28 * q^20 - 5 * q^22 - 8 * q^23 + 29 * q^25 - 16 * q^26 + 18 * q^28 - 28 * q^29 - 8 * q^31 - 35 * q^32 + 6 * q^34 - 22 * q^35 + 4 * q^37 + 4 * q^38 - 12 * q^40 - 58 * q^41 - 26 * q^43 + 25 * q^44 + 8 * q^46 - 20 * q^47 + 23 * q^49 - 27 * q^50 - 2 * q^52 - 36 * q^53 - 12 * q^55 - 70 * q^56 + 12 * q^58 - 18 * q^59 + 25 * q^61 - 42 * q^62 + 35 * q^64 - 76 * q^65 - 8 * q^67 - 28 * q^68 + 76 * q^70 - 24 * q^71 + 2 * q^73 - 40 * q^74 - 64 * q^76 - 4 * q^77 - 22 * q^79 - 36 * q^80 + 30 * q^82 - 14 * q^83 - 70 * q^86 - 15 * q^88 - 82 * q^89 - 6 * q^91 - 48 * q^92 - 16 * q^94 - 34 * q^95 + 16 * q^97 - 35 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more