LMFDB - Embedded newform 6039.2.a.p.1.18

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:	$0$
Dimension:	$25$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.18
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	$-0.337260\pi$
$2$	1.38389	0.978558	0.489279	+	0.872127i	$0.337260\pi$
$3$	0	0
$3$	0	0
$4$	−0.0848502	−0.0424251
$5$	4.17750	1.86824	0.934118	−	0.356964i	$-0.116188\pi$
$5$	4.17750	1.86824	0.934118	+	0.356964i	$0.116188\pi$
$6$	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	$-0.349378\pi$
$17$	3.75806	0.911463	0.455731	+	0.890117i	$0.349378\pi$
$18$	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	$-0.464281\pi$
$23$	1.07408	0.223961	0.111981	+	0.993710i	$0.464281\pi$
$24$	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	$-0.348636\pi$
$29$	4.93072	0.915611	0.457805	+	0.889052i	$0.348636\pi$
$30$	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	$-0.553047\pi$
$41$	−2.12431	−0.331761	−0.165881	+	0.986146i	$0.553047\pi$
$42$	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	$-0.631230\pi$
$47$	−5.49399	−0.801380	−0.400690	+	0.916214i	$0.631230\pi$
$48$	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	$-0.525420\pi$
$53$	−1.16155	−0.159551	−0.0797753	+	0.996813i	$0.525420\pi$
$54$	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	$-0.305317\pi$
$59$	8.82087	1.14838	0.574189	+	0.818722i	$0.305317\pi$
$60$	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	$-0.586133\pi$
$71$	−4.50469	−0.534608	−0.267304	+	0.963612i	$0.586133\pi$
$72$	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	$-0.901957\pi$
$83$	−17.3634	−1.90588	−0.952938	+	0.303164i	$0.901957\pi$
$84$	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	$-0.325858\pi$
$89$	9.81507	1.04040	0.520198	+	0.854046i	$0.325858\pi$
$90$	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	$-0.421671\pi$
$101$	4.89636	0.487206	0.243603	+	0.969875i	$0.421671\pi$
$102$	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	$-0.641513\pi$
$107$	−8.89746	−0.860150	−0.430075	+	0.902793i	$0.641513\pi$
$108$	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	$-0.325567\pi$
$113$	11.0762	1.04196	0.520980	+	0.853569i	$0.325567\pi$
$114$	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	$-0.479534\pi$
$131$	1.47079	0.128504	0.0642518	+	0.997934i	$0.479534\pi$
$132$	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	$-0.614424\pi$
$137$	−8.23496	−0.703560	−0.351780	+	0.936083i	$0.614424\pi$
$138$	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	$-0.511238\pi$
$149$	−0.861759	−0.0705980	−0.0352990	+	0.999377i	$0.511238\pi$
$150$	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	$-0.649692\pi$
$167$	−11.7114	−0.906258	−0.453129	+	0.891445i	$0.649692\pi$
$168$	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	$-0.713477\pi$
$173$	−16.3491	−1.24300	−0.621501	+	0.783414i	$0.713477\pi$
$174$	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	$-0.373261\pi$
$179$	10.3748	0.775447	0.387724	+	0.921776i	$0.373261\pi$
$180$	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	$-0.542507\pi$
$191$	−3.68014	−0.266285	−0.133143	+	0.991097i	$0.542507\pi$
$192$	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	$-0.616508\pi$
$197$	−10.0468	−0.715804	−0.357902	+	0.933759i	$0.616508\pi$
$198$	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	$-0.486288\pi$
$227$	1.29762	0.0861264	0.0430632	+	0.999072i	$0.486288\pi$
$228$	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	$-0.0888945\pi$
$233$	29.3459	1.92251	0.961257	+	0.275654i	$0.0888945\pi$
$234$	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	$-0.414283\pi$
$239$	8.22595	0.532093	0.266046	+	0.963960i	$0.414283\pi$
$240$	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	$-0.374623\pi$
$251$	12.1603	0.767553	0.383777	+	0.923426i	$0.374623\pi$
$252$	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	$-0.451551\pi$
$257$	4.86126	0.303237	0.151619	+	0.988439i	$0.451551\pi$
$258$	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	$-0.330843\pi$
$263$	16.4365	1.01352	0.506760	+	0.862087i	$0.330843\pi$
$264$	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	$-0.507695\pi$
$269$	−0.792880	−0.0483428	−0.0241714	+	0.999708i	$0.507695\pi$
$270$	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	$-0.410041\pi$
$281$	9.34938	0.557737	0.278869	+	0.960329i	$0.410041\pi$
$282$	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	$-0.846366\pi$
$293$	−30.3237	−1.77153	−0.885765	+	0.464134i	$0.846366\pi$
$294$	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	$-0.400049\pi$
$311$	10.8939	0.617739	0.308869	+	0.951104i	$0.400049\pi$
$312$	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	$-0.550538\pi$
$317$	−5.62995	−0.316209	−0.158105	+	0.987422i	$0.550538\pi$
$318$	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	$-0.288960\pi$
$347$	22.9304	1.23097	0.615485	+	0.788148i	$0.288960\pi$
$348$	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	$-0.287402\pi$
$353$	23.2725	1.23867	0.619336	+	0.785126i	$0.287402\pi$
$354$	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	$-0.575230\pi$
$359$	−8.87297	−0.468298	−0.234149	+	0.972201i	$0.575230\pi$
$360$	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	$-0.670090\pi$
$383$	−19.9338	−1.01857	−0.509286	+	0.860597i	$0.670090\pi$
$384$	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	$-0.536020\pi$
$389$	−4.45423	−0.225839	−0.112919	+	0.993604i	$0.536020\pi$
$390$	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	$-0.599770\pi$
$401$	−12.3485	−0.616657	−0.308329	+	0.951280i	$0.599770\pi$
$402$	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	$-0.461582\pi$
$419$	4.92913	0.240804	0.120402	+	0.992725i	$0.461582\pi$
$420$	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	$-0.183084\pi$
$431$	34.8402	1.67820	0.839098	+	0.543981i	$0.183084\pi$
$432$	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	$-0.367856\pi$
$443$	16.9778	0.806642	0.403321	+	0.915059i	$0.367856\pi$
$444$	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	$-0.379400\pi$
$449$	15.6750	0.739751	0.369875	+	0.929081i	$0.379400\pi$
$450$	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	$-0.597550\pi$
$461$	−12.9551	−0.603377	−0.301688	+	0.953407i	$0.597550\pi$
$462$	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	$-0.505867\pi$
$467$	−0.796623	−0.0368633	−0.0184317	+	0.999830i	$0.505867\pi$
$468$	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	$-0.616941\pi$
$479$	−15.7217	−0.718342	−0.359171	+	0.933272i	$0.616941\pi$
$480$	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	$-0.544217\pi$
$491$	−6.13635	−0.276929	−0.138465	+	0.990367i	$0.544217\pi$
$492$	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	$-0.709064\pi$
$503$	−27.3878	−1.22116	−0.610581	+	0.791954i	$0.709064\pi$
$504$	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	$-0.384935\pi$
$509$	15.9582	0.707335	0.353667	+	0.935371i	$0.384935\pi$
$510$	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	$-0.441271\pi$
$521$	8.37497	0.366914	0.183457	+	0.983028i	$0.441271\pi$
$522$	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	$-0.366879\pi$
$557$	19.1700	0.812257	0.406128	+	0.913816i	$0.366879\pi$
$558$	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	$-0.371144\pi$
$563$	18.6901	0.787694	0.393847	+	0.919176i	$0.371144\pi$
$564$	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	$-0.354195\pi$
$569$	21.0967	0.884420	0.442210	+	0.896911i	$0.354195\pi$
$570$	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	$-0.332741\pi$
$587$	24.3061	1.00322	0.501610	+	0.865094i	$0.332741\pi$
$588$	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	$-0.849364\pi$
$593$	−43.3506	−1.78020	−0.890098	+	0.455770i	$0.849364\pi$
$594$	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	$-0.431127\pi$
$599$	10.5087	0.429372	0.214686	+	0.976683i	$0.431127\pi$
$600$	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	$-0.569204\pi$
$617$	−10.7158	−0.431401	−0.215701	+	0.976460i	$0.569204\pi$
$618$	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	$-0.521180\pi$
$641$	−3.36670	−0.132977	−0.0664883	+	0.997787i	$0.521180\pi$
$642$	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	$-0.473450\pi$
$647$	4.23834	0.166626	0.0833131	+	0.996523i	$0.473450\pi$
$648$	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	$-0.705562\pi$
$653$	−30.7582	−1.20366	−0.601830	+	0.798624i	$0.705562\pi$
$654$	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	$-0.804883\pi$
$659$	−41.9946	−1.63588	−0.817939	+	0.575306i	$0.804883\pi$
$660$	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	$-0.395544\pi$
$677$	16.7719	0.644598	0.322299	+	0.946638i	$0.395544\pi$
$678$	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	$-0.369220\pi$
$683$	20.8759	0.798792	0.399396	+	0.916778i	$0.369220\pi$
$684$	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	$-0.866488\pi$
$701$	−48.3627	−1.82663	−0.913317	+	0.407250i	$0.866488\pi$
$702$	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	$-0.266965\pi$
$719$	35.8470	1.33687	0.668434	+	0.743772i	$0.266965\pi$
$720$	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	$-0.370933\pi$
$743$	21.5042	0.788912	0.394456	+	0.918915i	$0.370933\pi$
$744$	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	$-0.570207\pi$
$761$	−12.0705	−0.437554	−0.218777	+	0.975775i	$0.570207\pi$
$762$	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	$-0.760066\pi$
$773$	−40.5428	−1.45822	−0.729111	+	0.684395i	$0.760066\pi$
$774$	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	$-0.275638\pi$
$797$	36.5833	1.29585	0.647924	+	0.761705i	$0.275638\pi$
$798$	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	$-0.367124\pi$
$809$	23.0629	0.810848	0.405424	+	0.914129i	$0.367124\pi$
$810$	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	$-0.599805\pi$
$821$	−17.6752	−0.616868	−0.308434	+	0.951246i	$0.599805\pi$
$822$	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	$-0.277220\pi$
$827$	37.0472	1.28826	0.644129	+	0.764917i	$0.277220\pi$
$828$	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	$-0.833892\pi$
$839$	−50.2205	−1.73380	−0.866902	+	0.498478i	$0.833892\pi$
$840$	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	$-0.205106\pi$
$857$	46.8091	1.59897	0.799484	+	0.600687i	$0.205106\pi$
$858$	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	$-0.820726\pi$
$863$	−49.6791	−1.69110	−0.845548	+	0.533900i	$0.820726\pi$
$864$	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	$-0.544221\pi$
$881$	−8.22046	−0.276954	−0.138477	+	0.990366i	$0.544221\pi$
$882$	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	$-0.688859\pi$
$887$	−33.3037	−1.11823	−0.559115	+	0.829090i	$0.688859\pi$
$888$	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	$-0.180252\pi$
$911$	50.9427	1.68781	0.843904	+	0.536494i	$0.180252\pi$
$912$	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	$-0.191999\pi$
$929$	50.2019	1.64707	0.823535	+	0.567265i	$0.191999\pi$
$930$	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	$-0.240497\pi$
$941$	44.6577	1.45580	0.727899	+	0.685684i	$0.240497\pi$
$942$	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	$-0.724663\pi$
$947$	−39.9218	−1.29728	−0.648641	+	0.761094i	$0.724663\pi$
$948$	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	$-0.844268\pi$
$953$	−54.4983	−1.76537	−0.882687	+	0.469962i	$0.844268\pi$
$954$	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	$-0.374430\pi$
$971$	23.9525	0.768673	0.384337	+	0.923193i	$0.374430\pi$
$972$	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	$-0.125391\pi$
$977$	57.7260	1.84682	0.923409	+	0.383818i	$0.125391\pi$
$978$	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	$-0.456990\pi$
$983$	8.44703	0.269418	0.134709	+	0.990885i	$0.456990\pi$
$984$	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.p.1.18	yes	25
3.2	odd	2		6039.2.a.m.1.8	✓	25

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6039.2.a.m.1.8	✓	25	3.2	odd	2
6039.2.a.p.1.18	yes	25	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:	\( N \)	\(=\)	\( 6039 = 3^{2} \cdot 11 \cdot 61 \)
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

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Database

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