LMFDB - Embedded newform 6039.2.a.f.1.4

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:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 5x^{11} - 5x^{10} + 48x^{9} - 173x^{7} + 29x^{6} + 281x^{5} - 41x^{4} - 201x^{3} + 8x^{2} + 49x + 8 $ x^12 - 5x^11 - 5x^10 + 48x^9 - 173x^7 + 29x^6 + 281x^5 - 41x^4 - 201x^3 + 8x^2 + 49x + 8
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 2013)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$1.23554$ of defining polynomial
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)

$f(q)$	$=$	$q-0.235537 q^{2} -1.94452 q^{4} -2.04419 q^{5} -0.723848 q^{7} +0.929081 q^{8} +O(q^{10})$	\(q-0.235537 q^{2} -1.94452 q^{4} -2.04419 q^{5} -0.723848 q^{7} +0.929081 q^{8} +0.481482 q^{10} -1.00000 q^{11} -6.69193 q^{13} +0.170493 q^{14} +3.67021 q^{16} +4.95425 q^{17} -3.92369 q^{19} +3.97497 q^{20} +0.235537 q^{22} -5.80424 q^{23} -0.821284 q^{25} +1.57620 q^{26} +1.40754 q^{28} +2.22988 q^{29} +1.52276 q^{31} -2.72263 q^{32} -1.16691 q^{34} +1.47968 q^{35} -9.58753 q^{37} +0.924173 q^{38} -1.89922 q^{40} +6.49981 q^{41} -9.19853 q^{43} +1.94452 q^{44} +1.36711 q^{46} -11.2181 q^{47} -6.47604 q^{49} +0.193443 q^{50} +13.0126 q^{52} +0.677460 q^{53} +2.04419 q^{55} -0.672513 q^{56} -0.525219 q^{58} +0.691323 q^{59} -1.00000 q^{61} -0.358666 q^{62} -6.69914 q^{64} +13.6796 q^{65} -1.07551 q^{67} -9.63366 q^{68} -0.348520 q^{70} -9.28792 q^{71} +8.26502 q^{73} +2.25822 q^{74} +7.62969 q^{76} +0.723848 q^{77} -12.7876 q^{79} -7.50261 q^{80} -1.53095 q^{82} -12.1560 q^{83} -10.1274 q^{85} +2.16659 q^{86} -0.929081 q^{88} -1.81839 q^{89} +4.84394 q^{91} +11.2865 q^{92} +2.64228 q^{94} +8.02076 q^{95} +14.4696 q^{97} +1.52535 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 7 q^{2} + 13 q^{4} + 7 q^{5} - 15 q^{7} + 18 q^{8}+O(q^{10}) $ 12 * q + 7 * q^2 + 13 * q^4 + 7 * q^5 - 15 * q^7 + 18 * q^8	$ 12 q + 7 q^{2} + 13 q^{4} + 7 q^{5} - 15 q^{7} + 18 q^{8} - 6 q^{10} - 12 q^{11} - 11 q^{13} - 3 q^{14} + 19 q^{16} + 33 q^{17} - 24 q^{19} + 11 q^{20} - 7 q^{22} + 9 q^{23} + 11 q^{25} + 16 q^{26} - 41 q^{28} + 16 q^{29} + q^{31} + 28 q^{32} + 32 q^{34} + 22 q^{35} - 6 q^{37} - 12 q^{38} + 26 q^{40} + 21 q^{41} - 39 q^{43} - 13 q^{44} + 18 q^{47} + 31 q^{49} + 44 q^{50} + 3 q^{52} + 14 q^{53} - 7 q^{55} - 16 q^{56} + 33 q^{58} + 23 q^{59} - 12 q^{61} + 25 q^{62} + 12 q^{64} + 29 q^{65} + 96 q^{68} + 44 q^{70} + 19 q^{71} - 42 q^{73} - 38 q^{74} + 11 q^{76} + 15 q^{77} - 11 q^{79} + 44 q^{80} - 14 q^{82} + 56 q^{83} + 16 q^{85} + 18 q^{86} - 18 q^{88} + 55 q^{89} + 11 q^{91} + 4 q^{92} - 5 q^{94} - 15 q^{95} - 7 q^{97} - 6 q^{98}+O(q^{100}) $ 12 * q + 7 * q^2 + 13 * q^4 + 7 * q^5 - 15 * q^7 + 18 * q^8 - 6 * q^10 - 12 * q^11 - 11 * q^13 - 3 * q^14 + 19 * q^16 + 33 * q^17 - 24 * q^19 + 11 * q^20 - 7 * q^22 + 9 * q^23 + 11 * q^25 + 16 * q^26 - 41 * q^28 + 16 * q^29 + q^31 + 28 * q^32 + 32 * q^34 + 22 * q^35 - 6 * q^37 - 12 * q^38 + 26 * q^40 + 21 * q^41 - 39 * q^43 - 13 * q^44 + 18 * q^47 + 31 * q^49 + 44 * q^50 + 3 * q^52 + 14 * q^53 - 7 * q^55 - 16 * q^56 + 33 * q^58 + 23 * q^59 - 12 * q^61 + 25 * q^62 + 12 * q^64 + 29 * q^65 + 96 * q^68 + 44 * q^70 + 19 * q^71 - 42 * q^73 - 38 * q^74 + 11 * q^76 + 15 * q^77 - 11 * q^79 + 44 * q^80 - 14 * q^82 + 56 * q^83 + 16 * q^85 + 18 * q^86 - 18 * q^88 + 55 * q^89 + 11 * q^91 + 4 * q^92 - 5 * q^94 - 15 * q^95 - 7 * q^97 - 6 * 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 $

$2$	−0.235537	−0.166550	−0.0832749	−	0.996527i	$-0.526538\pi$
$2$	−0.235537	−0.166550	−0.0832749	+	0.996527i	$0.526538\pi$
$3$	0	0
$3$	0	0
$4$	−1.94452	−0.972261
$5$	−2.04419	−0.914190	−0.457095	−	0.889418i	$-0.651110\pi$
$5$	−2.04419	−0.914190	−0.457095	+	0.889418i	$0.651110\pi$
$6$	0	0
$7$	−0.723848	−0.273589	−0.136794	−	0.990599i	$-0.543680\pi$
$7$	−0.723848	−0.273589	−0.136794	+	0.990599i	$0.543680\pi$
$8$	0.929081	0.328480
$9$	0	0
$10$	0.481482	0.152258
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	−6.69193	−1.85601	−0.928004	−	0.372569i	$-0.878477\pi$
$13$	−6.69193	−1.85601	−0.928004	+	0.372569i	$0.878477\pi$
$14$	0.170493	0.0455661
$15$	0	0
$16$	3.67021	0.917553
$17$	4.95425	1.20158	0.600792	−	0.799406i	$-0.294852\pi$
$17$	4.95425	1.20158	0.600792	+	0.799406i	$0.294852\pi$
$18$	0	0
$19$	−3.92369	−0.900155	−0.450078	−	0.892989i	$-0.648604\pi$
$19$	−3.92369	−0.900155	−0.450078	+	0.892989i	$0.648604\pi$
$20$	3.97497	0.888831
$21$	0	0
$22$	0.235537	0.0502167
$23$	−5.80424	−1.21027	−0.605134	−	0.796124i	$-0.706880\pi$
$23$	−5.80424	−1.21027	−0.605134	+	0.796124i	$0.706880\pi$
$24$	0	0
$25$	−0.821284	−0.164257
$26$	1.57620	0.309118
$27$	0	0
$28$	1.40754	0.266000
$29$	2.22988	0.414078	0.207039	−	0.978333i	$-0.433617\pi$
$29$	2.22988	0.414078	0.207039	+	0.978333i	$0.433617\pi$
$30$	0	0
$31$	1.52276	0.273495	0.136748	−	0.990606i	$-0.456335\pi$
$31$	1.52276	0.273495	0.136748	+	0.990606i	$0.456335\pi$
$32$	−2.72263	−0.481298
$33$	0	0
$34$	−1.16691	−0.200123
$35$	1.47968	0.250112
$36$	0	0
$37$	−9.58753	−1.57618	−0.788090	−	0.615560i	$-0.788930\pi$
$37$	−9.58753	−1.57618	−0.788090	+	0.615560i	$0.788930\pi$
$38$	0.924173	0.149921
$39$	0	0
$40$	−1.89922	−0.300293
$41$	6.49981	1.01510	0.507550	−	0.861622i	$-0.330551\pi$
$41$	6.49981	1.01510	0.507550	+	0.861622i	$0.330551\pi$
$42$	0	0
$43$	−9.19853	−1.40276	−0.701382	−	0.712786i	$-0.747433\pi$
$43$	−9.19853	−1.40276	−0.701382	+	0.712786i	$0.747433\pi$
$44$	1.94452	0.293148
$45$	0	0
$46$	1.36711	0.201570
$47$	−11.2181	−1.63633	−0.818165	−	0.574983i	$-0.805009\pi$
$47$	−11.2181	−1.63633	−0.818165	+	0.574983i	$0.805009\pi$
$48$	0	0
$49$	−6.47604	−0.925149
$50$	0.193443	0.0273569
$51$	0	0
$52$	13.0126	1.80453
$53$	0.677460	0.0930563	0.0465281	−	0.998917i	$-0.485184\pi$
$53$	0.677460	0.0930563	0.0465281	+	0.998917i	$0.485184\pi$
$54$	0	0
$55$	2.04419	0.275639
$56$	−0.672513	−0.0898683
$57$	0	0
$58$	−0.525219	−0.0689647
$59$	0.691323	0.0900026	0.0450013	−	0.998987i	$-0.485671\pi$
$59$	0.691323	0.0900026	0.0450013	+	0.998987i	$0.485671\pi$
$60$	0	0
$61$	−1.00000	−0.128037
$61$	−1.00000	−0.128037
$62$	−0.358666	−0.0455506
$63$	0	0
$64$	−6.69914	−0.837393
$65$	13.6796	1.69674
$66$	0	0
$67$	−1.07551	−0.131394	−0.0656972	−	0.997840i	$-0.520927\pi$
$67$	−1.07551	−0.131394	−0.0656972	+	0.997840i	$0.520927\pi$
$68$	−9.63366	−1.16825
$69$	0	0
$70$	−0.348520	−0.0416561
$71$	−9.28792	−1.10227	−0.551137	−	0.834415i	$-0.685806\pi$
$71$	−9.28792	−1.10227	−0.551137	+	0.834415i	$0.685806\pi$
$72$	0	0
$73$	8.26502	0.967347	0.483674	−	0.875248i	$-0.339302\pi$
$73$	8.26502	0.967347	0.483674	+	0.875248i	$0.339302\pi$
$74$	2.25822	0.262512
$75$	0	0
$76$	7.62969	0.875186
$77$	0.723848	0.0824901
$78$	0	0
$79$	−12.7876	−1.43872	−0.719360	−	0.694637i	$-0.755565\pi$
$79$	−12.7876	−1.43872	−0.719360	+	0.694637i	$0.755565\pi$
$80$	−7.50261	−0.838818
$81$	0	0
$82$	−1.53095	−0.169065
$83$	−12.1560	−1.33429	−0.667146	−	0.744927i	$-0.732484\pi$
$83$	−12.1560	−1.33429	−0.667146	+	0.744927i	$0.732484\pi$
$84$	0	0
$85$	−10.1274	−1.09848
$86$	2.16659	0.233630
$87$	0	0
$88$	−0.929081	−0.0990403
$89$	−1.81839	−0.192749	−0.0963745	−	0.995345i	$-0.530725\pi$
$89$	−1.81839	−0.192749	−0.0963745	+	0.995345i	$0.530725\pi$
$90$	0	0
$91$	4.84394	0.507783
$92$	11.2865	1.17670
$93$	0	0
$94$	2.64228	0.272531
$95$	8.02076	0.822913
$96$	0	0
$97$	14.4696	1.46917	0.734583	−	0.678519i	$-0.237378\pi$
$97$	14.4696	1.46917	0.734583	+	0.678519i	$0.237378\pi$
$98$	1.52535	0.154083
$99$	0	0
$100$	1.59701	0.159701
$101$	17.2171	1.71317	0.856583	−	0.516010i	$-0.172583\pi$
$101$	17.2171	1.71317	0.856583	+	0.516010i	$0.172583\pi$
$102$	0	0
$103$	−13.3611	−1.31651	−0.658255	−	0.752795i	$-0.728705\pi$
$103$	−13.3611	−1.31651	−0.658255	+	0.752795i	$0.728705\pi$
$104$	−6.21735	−0.609661
$105$	0	0
$106$	−0.159567	−0.0154985
$107$	11.2568	1.08824	0.544118	−	0.839008i	$-0.316864\pi$
$107$	11.2568	1.08824	0.544118	+	0.839008i	$0.316864\pi$
$108$	0	0
$109$	−7.14756	−0.684612	−0.342306	−	0.939588i	$-0.611208\pi$
$109$	−7.14756	−0.684612	−0.342306	+	0.939588i	$0.611208\pi$
$110$	−0.481482	−0.0459076
$111$	0	0
$112$	−2.65667	−0.251032
$113$	−7.60648	−0.715557	−0.357779	−	0.933806i	$-0.616466\pi$
$113$	−7.60648	−0.715557	−0.357779	+	0.933806i	$0.616466\pi$
$114$	0	0
$115$	11.8650	1.10641
$116$	−4.33605	−0.402592
$117$	0	0
$118$	−0.162832	−0.0149899
$119$	−3.58613	−0.328740
$120$	0	0
$121$	1.00000	0.0909091
$122$	0.235537	0.0213245
$123$	0	0
$124$	−2.96104	−0.265909
$125$	11.8998	1.06435
$126$	0	0
$127$	−14.9395	−1.32567	−0.662833	−	0.748767i	$-0.730646\pi$
$127$	−14.9395	−1.32567	−0.662833	+	0.748767i	$0.730646\pi$
$128$	7.02316	0.620766
$129$	0	0
$130$	−3.22205	−0.282592
$131$	−3.43654	−0.300252	−0.150126	−	0.988667i	$-0.547968\pi$
$131$	−3.43654	−0.300252	−0.150126	+	0.988667i	$0.547968\pi$
$132$	0	0
$133$	2.84015	0.246272
$134$	0.253322	0.0218837
$135$	0	0
$136$	4.60290	0.394696
$137$	7.41928	0.633872	0.316936	−	0.948447i	$-0.397346\pi$
$137$	7.41928	0.633872	0.316936	+	0.948447i	$0.397346\pi$
$138$	0	0
$139$	6.57357	0.557563	0.278781	−	0.960355i	$-0.410070\pi$
$139$	6.57357	0.557563	0.278781	+	0.960355i	$0.410070\pi$
$140$	−2.87728	−0.243174
$141$	0	0
$142$	2.18765	0.183583
$143$	6.69193	0.559608
$144$	0	0
$145$	−4.55830	−0.378546
$146$	−1.94672	−0.161111
$147$	0	0
$148$	18.6432	1.53246
$149$	−1.66917	−0.136744	−0.0683721	−	0.997660i	$-0.521780\pi$
$149$	−1.66917	−0.136744	−0.0683721	+	0.997660i	$0.521780\pi$
$150$	0	0
$151$	23.5099	1.91321	0.956603	−	0.291395i	$-0.0941192\pi$
$151$	23.5099	1.91321	0.956603	+	0.291395i	$0.0941192\pi$
$152$	−3.64542	−0.295683
$153$	0	0
$154$	−0.170493	−0.0137387
$155$	−3.11281	−0.250027
$156$	0	0
$157$	−14.7167	−1.17452	−0.587259	−	0.809399i	$-0.699793\pi$
$157$	−14.7167	−1.17452	−0.587259	+	0.809399i	$0.699793\pi$
$158$	3.01196	0.239619
$159$	0	0
$160$	5.56558	0.439998
$161$	4.20139	0.331116
$162$	0	0
$163$	−15.4056	−1.20666	−0.603329	−	0.797492i	$-0.706159\pi$
$163$	−15.4056	−1.20666	−0.603329	+	0.797492i	$0.706159\pi$
$164$	−12.6390	−0.986942
$165$	0	0
$166$	2.86318	0.222226
$167$	−0.495246	−0.0383233	−0.0191617	−	0.999816i	$-0.506100\pi$
$167$	−0.495246	−0.0383233	−0.0191617	+	0.999816i	$0.506100\pi$
$168$	0	0
$169$	31.7820	2.44477
$170$	2.38539	0.182951
$171$	0	0
$172$	17.8868	1.36385
$173$	2.80094	0.212952	0.106476	−	0.994315i	$-0.466043\pi$
$173$	2.80094	0.212952	0.106476	+	0.994315i	$0.466043\pi$
$174$	0	0
$175$	0.594485	0.0449388
$176$	−3.67021	−0.276653
$177$	0	0
$178$	0.428298	0.0321023
$179$	17.7330	1.32543	0.662713	−	0.748873i	$-0.269405\pi$
$179$	17.7330	1.32543	0.662713	+	0.748873i	$0.269405\pi$
$180$	0	0
$181$	2.37734	0.176706	0.0883532	−	0.996089i	$-0.471840\pi$
$181$	2.37734	0.176706	0.0883532	+	0.996089i	$0.471840\pi$
$182$	−1.14093	−0.0845711
$183$	0	0
$184$	−5.39261	−0.397548
$185$	19.5987	1.44093
$186$	0	0
$187$	−4.95425	−0.362291
$188$	21.8139	1.59094
$189$	0	0
$190$	−1.88919	−0.137056
$191$	13.3357	0.964937	0.482468	−	0.875913i	$-0.339740\pi$
$191$	13.3357	0.964937	0.482468	+	0.875913i	$0.339740\pi$
$192$	0	0
$193$	5.21165	0.375143	0.187572	−	0.982251i	$-0.439938\pi$
$193$	5.21165	0.375143	0.187572	+	0.982251i	$0.439938\pi$
$194$	−3.40813	−0.244689
$195$	0	0
$196$	12.5928	0.899487
$197$	−18.1294	−1.29167	−0.645834	−	0.763478i	$-0.723490\pi$
$197$	−18.1294	−1.29167	−0.645834	+	0.763478i	$0.723490\pi$
$198$	0	0
$199$	3.73408	0.264702	0.132351	−	0.991203i	$-0.457747\pi$
$199$	3.73408	0.264702	0.132351	+	0.991203i	$0.457747\pi$
$200$	−0.763040	−0.0539550
$201$	0	0
$202$	−4.05526	−0.285327
$203$	−1.61409	−0.113287
$204$	0	0
$205$	−13.2869	−0.927994
$206$	3.14704	0.219264
$207$	0	0
$208$	−24.5608	−1.70299
$209$	3.92369	0.271407
$210$	0	0
$211$	0.901886	0.0620884	0.0310442	−	0.999518i	$-0.490117\pi$
$211$	0.901886	0.0620884	0.0310442	+	0.999518i	$0.490117\pi$
$212$	−1.31734	−0.0904750
$213$	0	0
$214$	−2.65139	−0.181246
$215$	18.8036	1.28239
$216$	0	0
$217$	−1.10224	−0.0748252
$218$	1.68352	0.114022
$219$	0	0
$220$	−3.97497	−0.267993
$221$	−33.1535	−2.23015
$222$	0	0
$223$	−8.19380	−0.548697	−0.274349	−	0.961630i	$-0.588462\pi$
$223$	−8.19380	−0.548697	−0.274349	+	0.961630i	$0.588462\pi$
$224$	1.97077	0.131678
$225$	0	0
$226$	1.79161	0.119176
$227$	1.21582	0.0806971	0.0403486	−	0.999186i	$-0.487153\pi$
$227$	1.21582	0.0806971	0.0403486	+	0.999186i	$0.487153\pi$
$228$	0	0
$229$	−18.9103	−1.24963	−0.624813	−	0.780774i	$-0.714825\pi$
$229$	−18.9103	−1.24963	−0.624813	+	0.780774i	$0.714825\pi$
$230$	−2.79464	−0.184273
$231$	0	0
$232$	2.07174	0.136016
$233$	−13.3678	−0.875757	−0.437878	−	0.899034i	$-0.644270\pi$
$233$	−13.3678	−0.875757	−0.437878	+	0.899034i	$0.644270\pi$
$234$	0	0
$235$	22.9320	1.49592
$236$	−1.34429	−0.0875061
$237$	0	0
$238$	0.844665	0.0547515
$239$	−20.1278	−1.30196	−0.650979	−	0.759095i	$-0.725642\pi$
$239$	−20.1278	−1.30196	−0.650979	+	0.759095i	$0.725642\pi$
$240$	0	0
$241$	26.3031	1.69433	0.847165	−	0.531330i	$-0.178308\pi$
$241$	26.3031	1.69433	0.847165	+	0.531330i	$0.178308\pi$
$242$	−0.235537	−0.0151409
$243$	0	0
$244$	1.94452	0.124485
$245$	13.2383	0.845762
$246$	0	0
$247$	26.2570	1.67070
$248$	1.41477	0.0898377
$249$	0	0
$250$	−2.80285	−0.177268
$251$	25.7154	1.62314	0.811571	−	0.584253i	$-0.198613\pi$
$251$	25.7154	1.62314	0.811571	+	0.584253i	$0.198613\pi$
$252$	0	0
$253$	5.80424	0.364909
$254$	3.51881	0.220790
$255$	0	0
$256$	11.7441	0.734005
$257$	−12.0034	−0.748751	−0.374376	−	0.927277i	$-0.622143\pi$
$257$	−12.0034	−0.748751	−0.374376	+	0.927277i	$0.622143\pi$
$258$	0	0
$259$	6.93991	0.431225
$260$	−26.6003	−1.64968
$261$	0	0
$262$	0.809433	0.0500069
$263$	17.8845	1.10280	0.551402	−	0.834240i	$-0.314093\pi$
$263$	17.8845	1.10280	0.551402	+	0.834240i	$0.314093\pi$
$264$	0	0
$265$	−1.38486	−0.0850711
$266$	−0.668960	−0.0410166
$267$	0	0
$268$	2.09135	0.127750
$269$	−17.4003	−1.06092	−0.530459	−	0.847711i	$-0.677980\pi$
$269$	−17.4003	−1.06092	−0.530459	+	0.847711i	$0.677980\pi$
$270$	0	0
$271$	26.4891	1.60910	0.804548	−	0.593887i	$-0.202407\pi$
$271$	26.4891	1.60910	0.804548	+	0.593887i	$0.202407\pi$
$272$	18.1832	1.10252
$273$	0	0
$274$	−1.74752	−0.105571
$275$	0.821284	0.0495253
$276$	0	0
$277$	−6.58293	−0.395530	−0.197765	−	0.980249i	$-0.563368\pi$
$277$	−6.58293	−0.395530	−0.197765	+	0.980249i	$0.563368\pi$
$278$	−1.54832	−0.0928619
$279$	0	0
$280$	1.37474	0.0821567
$281$	−25.5501	−1.52419	−0.762097	−	0.647463i	$-0.775830\pi$
$281$	−25.5501	−1.52419	−0.762097	+	0.647463i	$0.775830\pi$
$282$	0	0
$283$	−12.2727	−0.729535	−0.364767	−	0.931099i	$-0.618851\pi$
$283$	−12.2727	−0.729535	−0.364767	+	0.931099i	$0.618851\pi$
$284$	18.0606	1.07170
$285$	0	0
$286$	−1.57620	−0.0932025
$287$	−4.70487	−0.277720
$288$	0	0
$289$	7.54463	0.443802
$290$	1.07365	0.0630468
$291$	0	0
$292$	−16.0715	−0.940514
$293$	0.877691	0.0512753	0.0256376	−	0.999671i	$-0.491838\pi$
$293$	0.877691	0.0512753	0.0256376	+	0.999671i	$0.491838\pi$
$294$	0	0
$295$	−1.41320	−0.0822795
$296$	−8.90759	−0.517743
$297$	0	0
$298$	0.393152	0.0227747
$299$	38.8416	2.24627
$300$	0	0
$301$	6.65834	0.383780
$302$	−5.53744	−0.318644
$303$	0	0
$304$	−14.4008	−0.825940
$305$	2.04419	0.117050
$306$	0	0
$307$	13.9557	0.796495	0.398247	−	0.917278i	$-0.369619\pi$
$307$	13.9557	0.796495	0.398247	+	0.917278i	$0.369619\pi$
$308$	−1.40754	−0.0802019
$309$	0	0
$310$	0.733181	0.0416419
$311$	2.93986	0.166704	0.0833522	−	0.996520i	$-0.473437\pi$
$311$	2.93986	0.166704	0.0833522	+	0.996520i	$0.473437\pi$
$312$	0	0
$313$	23.2085	1.31182	0.655910	−	0.754839i	$-0.272285\pi$
$313$	23.2085	1.31182	0.655910	+	0.754839i	$0.272285\pi$
$314$	3.46632	0.195616
$315$	0	0
$316$	24.8658	1.39881
$317$	27.8500	1.56421	0.782107	−	0.623145i	$-0.214145\pi$
$317$	27.8500	1.56421	0.782107	+	0.623145i	$0.214145\pi$
$318$	0	0
$319$	−2.22988	−0.124849
$320$	13.6943	0.765536
$321$	0	0
$322$	−0.989582	−0.0551472
$323$	−19.4389	−1.08161
$324$	0	0
$325$	5.49598	0.304862
$326$	3.62858	0.200969
$327$	0	0
$328$	6.03885	0.333440
$329$	8.12021	0.447682
$330$	0	0
$331$	1.64974	0.0906780	0.0453390	−	0.998972i	$-0.485563\pi$
$331$	1.64974	0.0906780	0.0453390	+	0.998972i	$0.485563\pi$
$332$	23.6376	1.29728
$333$	0	0
$334$	0.116649	0.00638274
$335$	2.19855	0.120119
$336$	0	0
$337$	22.6128	1.23180	0.615899	−	0.787825i	$-0.288793\pi$
$337$	22.6128	1.23180	0.615899	+	0.787825i	$0.288793\pi$
$338$	−7.48583	−0.407176
$339$	0	0
$340$	19.6930	1.06800
$341$	−1.52276	−0.0824620
$342$	0	0
$343$	9.75460	0.526699
$344$	−8.54618	−0.460779
$345$	0	0
$346$	−0.659726	−0.0354671
$347$	−25.1103	−1.34799	−0.673996	−	0.738735i	$-0.735423\pi$
$347$	−25.1103	−1.34799	−0.673996	+	0.738735i	$0.735423\pi$
$348$	0	0
$349$	34.0980	1.82522	0.912611	−	0.408828i	$-0.134063\pi$
$349$	34.0980	1.82522	0.912611	+	0.408828i	$0.134063\pi$
$350$	−0.140023	−0.00748455
$351$	0	0
$352$	2.72263	0.145117
$353$	7.12957	0.379469	0.189734	−	0.981835i	$-0.439237\pi$
$353$	7.12957	0.379469	0.189734	+	0.981835i	$0.439237\pi$
$354$	0	0
$355$	18.9863	1.00769
$356$	3.53590	0.187402
$357$	0	0
$358$	−4.17678	−0.220749
$359$	−22.2802	−1.17590	−0.587951	−	0.808896i	$-0.700065\pi$
$359$	−22.2802	−1.17590	−0.587951	+	0.808896i	$0.700065\pi$
$360$	0	0
$361$	−3.60469	−0.189721
$362$	−0.559952	−0.0294304
$363$	0	0
$364$	−9.41915	−0.493698
$365$	−16.8953	−0.884339
$366$	0	0
$367$	−28.8535	−1.50614	−0.753070	−	0.657940i	$-0.771428\pi$
$367$	−28.8535	−1.50614	−0.753070	+	0.657940i	$0.771428\pi$
$368$	−21.3028	−1.11048
$369$	0	0
$370$	−4.61623	−0.239986
$371$	−0.490378	−0.0254592
$372$	0	0
$373$	0.759702	0.0393359	0.0196679	−	0.999807i	$-0.493739\pi$
$373$	0.759702	0.0393359	0.0196679	+	0.999807i	$0.493739\pi$
$374$	1.16691	0.0603395
$375$	0	0
$376$	−10.4225	−0.537501
$377$	−14.9222	−0.768533
$378$	0	0
$379$	2.21021	0.113531	0.0567654	−	0.998388i	$-0.481921\pi$
$379$	2.21021	0.113531	0.0567654	+	0.998388i	$0.481921\pi$
$380$	−15.5965	−0.800086
$381$	0	0
$382$	−3.14105	−0.160710
$383$	−2.71839	−0.138903	−0.0694516	−	0.997585i	$-0.522125\pi$
$383$	−2.71839	−0.138903	−0.0694516	+	0.997585i	$0.522125\pi$
$384$	0	0
$385$	−1.47968	−0.0754116
$386$	−1.22754	−0.0624800
$387$	0	0
$388$	−28.1365	−1.42841
$389$	23.6314	1.19816	0.599079	−	0.800690i	$-0.295534\pi$
$389$	23.6314	1.19816	0.599079	+	0.800690i	$0.295534\pi$
$390$	0	0
$391$	−28.7557	−1.45424
$392$	−6.01677	−0.303893
$393$	0	0
$394$	4.27015	0.215127
$395$	26.1403	1.31526
$396$	0	0
$397$	6.09075	0.305686	0.152843	−	0.988250i	$-0.451157\pi$
$397$	6.09075	0.305686	0.152843	+	0.988250i	$0.451157\pi$
$398$	−0.879513	−0.0440860
$399$	0	0
$400$	−3.01429	−0.150714
$401$	−1.55568	−0.0776869	−0.0388434	−	0.999245i	$-0.512367\pi$
$401$	−1.55568	−0.0776869	−0.0388434	+	0.999245i	$0.512367\pi$
$402$	0	0
$403$	−10.1902	−0.507610
$404$	−33.4790	−1.66564
$405$	0	0
$406$	0.380179	0.0188680
$407$	9.58753	0.475236
$408$	0	0
$409$	26.1040	1.29076	0.645380	−	0.763861i	$-0.276699\pi$
$409$	26.1040	1.29076	0.645380	+	0.763861i	$0.276699\pi$
$410$	3.12955	0.154557
$411$	0	0
$412$	25.9810	1.27999
$413$	−0.500413	−0.0246237
$414$	0	0
$415$	24.8491	1.21980
$416$	18.2197	0.893293
$417$	0	0
$418$	−0.924173	−0.0452028
$419$	2.57443	0.125769	0.0628846	−	0.998021i	$-0.479970\pi$
$419$	2.57443	0.125769	0.0628846	+	0.998021i	$0.479970\pi$
$420$	0	0
$421$	21.6153	1.05347	0.526733	−	0.850031i	$-0.323417\pi$
$421$	21.6153	1.05347	0.526733	+	0.850031i	$0.323417\pi$
$422$	−0.212427	−0.0103408
$423$	0	0
$424$	0.629415	0.0305671
$425$	−4.06885	−0.197368
$426$	0	0
$427$	0.723848	0.0350294
$428$	−21.8891	−1.05805
$429$	0	0
$430$	−4.42893	−0.213582
$431$	−3.39836	−0.163693	−0.0818467	−	0.996645i	$-0.526082\pi$
$431$	−3.39836	−0.163693	−0.0818467	+	0.996645i	$0.526082\pi$
$432$	0	0
$433$	5.46595	0.262677	0.131338	−	0.991338i	$-0.458073\pi$
$433$	5.46595	0.262677	0.131338	+	0.991338i	$0.458073\pi$
$434$	0.259619	0.0124621
$435$	0	0
$436$	13.8986	0.665622
$437$	22.7740	1.08943
$438$	0	0
$439$	−4.20007	−0.200459	−0.100229	−	0.994964i	$-0.531958\pi$
$439$	−4.20007	−0.200459	−0.100229	+	0.994964i	$0.531958\pi$
$440$	1.89922	0.0905417
$441$	0	0
$442$	7.80888	0.371431
$443$	3.16172	0.150218	0.0751089	−	0.997175i	$-0.476070\pi$
$443$	3.16172	0.150218	0.0751089	+	0.997175i	$0.476070\pi$
$444$	0	0
$445$	3.71714	0.176209
$446$	1.92994	0.0913854
$447$	0	0
$448$	4.84916	0.229101
$449$	32.2853	1.52364	0.761818	−	0.647791i	$-0.224307\pi$
$449$	32.2853	1.52364	0.761818	+	0.647791i	$0.224307\pi$
$450$	0	0
$451$	−6.49981	−0.306064
$452$	14.7910	0.695708
$453$	0	0
$454$	−0.286372	−0.0134401
$455$	−9.90194	−0.464210
$456$	0	0
$457$	−33.1974	−1.55291	−0.776456	−	0.630172i	$-0.782984\pi$
$457$	−33.1974	−1.55291	−0.776456	+	0.630172i	$0.782984\pi$
$458$	4.45407	0.208125
$459$	0	0
$460$	−23.0717	−1.07572
$461$	−14.2044	−0.661566	−0.330783	−	0.943707i	$-0.607313\pi$
$461$	−14.2044	−0.661566	−0.330783	+	0.943707i	$0.607313\pi$
$462$	0	0
$463$	24.0580	1.11807	0.559034	−	0.829145i	$-0.311172\pi$
$463$	24.0580	1.11807	0.559034	+	0.829145i	$0.311172\pi$
$464$	8.18413	0.379939
$465$	0	0
$466$	3.14862	0.145857
$467$	23.5408	1.08934	0.544668	−	0.838652i	$-0.316656\pi$
$467$	23.5408	1.08934	0.544668	+	0.838652i	$0.316656\pi$
$468$	0	0
$469$	0.778505	0.0359480
$470$	−5.40133	−0.249145
$471$	0	0
$472$	0.642295	0.0295640
$473$	9.19853	0.422949
$474$	0	0
$475$	3.22246	0.147857
$476$	6.97330	0.319621
$477$	0	0
$478$	4.74084	0.216841
$479$	35.3983	1.61739	0.808696	−	0.588227i	$-0.200174\pi$
$479$	35.3983	1.61739	0.808696	+	0.588227i	$0.200174\pi$
$480$	0	0
$481$	64.1591	2.92540
$482$	−6.19535	−0.282190
$483$	0	0
$484$	−1.94452	−0.0883874
$485$	−29.5786	−1.34310
$486$	0	0
$487$	−1.35675	−0.0614802	−0.0307401	−	0.999527i	$-0.509786\pi$
$487$	−1.35675	−0.0614802	−0.0307401	+	0.999527i	$0.509786\pi$
$488$	−0.929081	−0.0420575
$489$	0	0
$490$	−3.11810	−0.140861
$491$	−1.46644	−0.0661794	−0.0330897	−	0.999452i	$-0.510535\pi$
$491$	−1.46644	−0.0661794	−0.0330897	+	0.999452i	$0.510535\pi$
$492$	0	0
$493$	11.0474	0.497550
$494$	−6.18450	−0.278254
$495$	0	0
$496$	5.58884	0.250946
$497$	6.72304	0.301570
$498$	0	0
$499$	−10.0901	−0.451696	−0.225848	−	0.974163i	$-0.572515\pi$
$499$	−10.0901	−0.451696	−0.225848	+	0.974163i	$0.572515\pi$
$500$	−23.1395	−1.03483
$501$	0	0
$502$	−6.05693	−0.270334
$503$	24.1016	1.07464	0.537319	−	0.843379i	$-0.319437\pi$
$503$	24.1016	1.07464	0.537319	+	0.843379i	$0.319437\pi$
$504$	0	0
$505$	−35.1950	−1.56616
$506$	−1.36711	−0.0607756
$507$	0	0
$508$	29.0502	1.28889
$509$	−18.4344	−0.817091	−0.408545	−	0.912738i	$-0.633964\pi$
$509$	−18.4344	−0.817091	−0.408545	+	0.912738i	$0.633964\pi$
$510$	0	0
$511$	−5.98261	−0.264655
$512$	−16.8125	−0.743014
$513$	0	0
$514$	2.82724	0.124704
$515$	27.3127	1.20354
$516$	0	0
$517$	11.2181	0.493372
$518$	−1.63461	−0.0718204
$519$	0	0
$520$	12.7094	0.557346
$521$	−43.4513	−1.90364	−0.951818	−	0.306665i	$-0.900787\pi$
$521$	−43.4513	−1.90364	−0.951818	+	0.306665i	$0.900787\pi$
$522$	0	0
$523$	−13.4676	−0.588895	−0.294448	−	0.955668i	$-0.595136\pi$
$523$	−13.4676	−0.588895	−0.294448	+	0.955668i	$0.595136\pi$
$524$	6.68244	0.291924
$525$	0	0
$526$	−4.21245	−0.183672
$527$	7.54413	0.328627
$528$	0	0
$529$	10.6892	0.464748
$530$	0.326185	0.0141686
$531$	0	0
$532$	−5.52274	−0.239441
$533$	−43.4963	−1.88403
$534$	0	0
$535$	−23.0111	−0.994855
$536$	−0.999236	−0.0431604
$537$	0	0
$538$	4.09842	0.176696
$539$	6.47604	0.278943
$540$	0	0
$541$	8.21000	0.352975	0.176488	−	0.984303i	$-0.443526\pi$
$541$	8.21000	0.352975	0.176488	+	0.984303i	$0.443526\pi$
$542$	−6.23916	−0.267995
$543$	0	0
$544$	−13.4886	−0.578319
$545$	14.6110	0.625866
$546$	0	0
$547$	−4.03523	−0.172534	−0.0862671	−	0.996272i	$-0.527494\pi$
$547$	−4.03523	−0.172534	−0.0862671	+	0.996272i	$0.527494\pi$
$548$	−14.4270	−0.616289
$549$	0	0
$550$	−0.193443	−0.00824843
$551$	−8.74935	−0.372735
$552$	0	0
$553$	9.25629	0.393618
$554$	1.55052	0.0658755
$555$	0	0
$556$	−12.7824	−0.542096
$557$	−13.0822	−0.554310	−0.277155	−	0.960825i	$-0.589392\pi$
$557$	−13.0822	−0.554310	−0.277155	+	0.960825i	$0.589392\pi$
$558$	0	0
$559$	61.5560	2.60354
$560$	5.43075	0.229491
$561$	0	0
$562$	6.01800	0.253854
$563$	21.0959	0.889087	0.444544	−	0.895757i	$-0.353366\pi$
$563$	21.0959	0.889087	0.444544	+	0.895757i	$0.353366\pi$
$564$	0	0
$565$	15.5491	0.654155
$566$	2.89067	0.121504
$567$	0	0
$568$	−8.62923	−0.362075
$569$	13.6484	0.572171	0.286085	−	0.958204i	$-0.407646\pi$
$569$	13.6484	0.572171	0.286085	+	0.958204i	$0.407646\pi$
$570$	0	0
$571$	21.6085	0.904287	0.452144	−	0.891945i	$-0.350659\pi$
$571$	21.6085	0.904287	0.452144	+	0.891945i	$0.350659\pi$
$572$	−13.0126	−0.544085
$573$	0	0
$574$	1.10817	0.0462542
$575$	4.76693	0.198795
$576$	0	0
$577$	−15.3266	−0.638056	−0.319028	−	0.947745i	$-0.603356\pi$
$577$	−15.3266	−0.638056	−0.319028	+	0.947745i	$0.603356\pi$
$578$	−1.77704	−0.0739151
$579$	0	0
$580$	8.86372	0.368046
$581$	8.79908	0.365047
$582$	0	0
$583$	−0.677460	−0.0280575
$584$	7.67887	0.317754
$585$	0	0
$586$	−0.206729	−0.00853989
$587$	2.51692	0.103884	0.0519422	−	0.998650i	$-0.483459\pi$
$587$	2.51692	0.103884	0.0519422	+	0.998650i	$0.483459\pi$
$588$	0	0
$589$	−5.97482	−0.246188
$590$	0.332860	0.0137036
$591$	0	0
$592$	−35.1883	−1.44623
$593$	−3.42712	−0.140735	−0.0703674	−	0.997521i	$-0.522417\pi$
$593$	−3.42712	−0.140735	−0.0703674	+	0.997521i	$0.522417\pi$
$594$	0	0
$595$	7.33072	0.300530
$596$	3.24575	0.132951
$597$	0	0
$598$	−9.14863	−0.374115
$599$	−45.9062	−1.87568	−0.937838	−	0.347075i	$-0.887175\pi$
$599$	−45.9062	−1.87568	−0.937838	+	0.347075i	$0.887175\pi$
$600$	0	0
$601$	−38.3645	−1.56492	−0.782460	−	0.622701i	$-0.786035\pi$
$601$	−38.3645	−1.56492	−0.782460	+	0.622701i	$0.786035\pi$
$602$	−1.56828	−0.0639185
$603$	0	0
$604$	−45.7155	−1.86014
$605$	−2.04419	−0.0831082
$606$	0	0
$607$	4.04915	0.164350	0.0821750	−	0.996618i	$-0.473813\pi$
$607$	4.04915	0.164350	0.0821750	+	0.996618i	$0.473813\pi$
$608$	10.6828	0.433243
$609$	0	0
$610$	−0.481482	−0.0194947
$611$	75.0709	3.03704
$612$	0	0
$613$	8.11035	0.327574	0.163787	−	0.986496i	$-0.447629\pi$
$613$	8.11035	0.327574	0.163787	+	0.986496i	$0.447629\pi$
$614$	−3.28709	−0.132656
$615$	0	0
$616$	0.672513	0.0270963
$617$	27.6683	1.11388	0.556941	−	0.830552i	$-0.311975\pi$
$617$	27.6683	1.11388	0.556941	+	0.830552i	$0.311975\pi$
$618$	0	0
$619$	−21.5573	−0.866459	−0.433230	−	0.901284i	$-0.642626\pi$
$619$	−21.5573	−0.866459	−0.433230	+	0.901284i	$0.642626\pi$
$620$	6.05292	0.243091
$621$	0	0
$622$	−0.692447	−0.0277646
$623$	1.31624	0.0527339
$624$	0	0
$625$	−20.2191	−0.808763
$626$	−5.46645	−0.218483
$627$	0	0
$628$	28.6169	1.14194
$629$	−47.4990	−1.89391
$630$	0	0
$631$	−11.4131	−0.454347	−0.227174	−	0.973854i	$-0.572949\pi$
$631$	−11.4131	−0.454347	−0.227174	+	0.973854i	$0.572949\pi$
$632$	−11.8807	−0.472590
$633$	0	0
$634$	−6.55971	−0.260519
$635$	30.5392	1.21191
$636$	0	0
$637$	43.3373	1.71708
$638$	0.525219	0.0207936
$639$	0	0
$640$	−14.3567	−0.567498
$641$	28.1026	1.10999	0.554994	−	0.831855i	$-0.312721\pi$
$641$	28.1026	1.10999	0.554994	+	0.831855i	$0.312721\pi$
$642$	0	0
$643$	−44.2173	−1.74376	−0.871881	−	0.489718i	$-0.837100\pi$
$643$	−44.2173	−1.74376	−0.871881	+	0.489718i	$0.837100\pi$
$644$	−8.16969	−0.321931
$645$	0	0
$646$	4.57859	0.180142
$647$	−7.80996	−0.307041	−0.153521	−	0.988145i	$-0.549061\pi$
$647$	−7.80996	−0.307041	−0.153521	+	0.988145i	$0.549061\pi$
$648$	0	0
$649$	−0.691323	−0.0271368
$650$	−1.29451	−0.0507747
$651$	0	0
$652$	29.9565	1.17319
$653$	0.776841	0.0304002	0.0152001	−	0.999884i	$-0.495161\pi$
$653$	0.776841	0.0304002	0.0152001	+	0.999884i	$0.495161\pi$
$654$	0	0
$655$	7.02495	0.274488
$656$	23.8557	0.931408
$657$	0	0
$658$	−1.91261	−0.0745613
$659$	1.79547	0.0699415	0.0349708	−	0.999388i	$-0.488866\pi$
$659$	1.79547	0.0699415	0.0349708	+	0.999388i	$0.488866\pi$
$660$	0	0
$661$	−19.1867	−0.746277	−0.373138	−	0.927776i	$-0.621718\pi$
$661$	−19.1867	−0.746277	−0.373138	+	0.927776i	$0.621718\pi$
$662$	−0.388575	−0.0151024
$663$	0	0
$664$	−11.2939	−0.438288
$665$	−5.80581	−0.225140
$666$	0	0
$667$	−12.9428	−0.501146
$668$	0.963017	0.0372603
$669$	0	0
$670$	−0.517839	−0.0200059
$671$	1.00000	0.0386046
$672$	0	0
$673$	3.89545	0.150159	0.0750793	−	0.997178i	$-0.476079\pi$
$673$	3.89545	0.150159	0.0750793	+	0.997178i	$0.476079\pi$
$674$	−5.32615	−0.205156
$675$	0	0
$676$	−61.8008	−2.37695
$677$	28.7331	1.10430	0.552151	−	0.833744i	$-0.313807\pi$
$677$	28.7331	1.10430	0.552151	+	0.833744i	$0.313807\pi$
$678$	0	0
$679$	−10.4738	−0.401947
$680$	−9.40921	−0.360827
$681$	0	0
$682$	0.358666	0.0137340
$683$	−36.2785	−1.38816	−0.694079	−	0.719898i	$-0.744188\pi$
$683$	−36.2785	−1.38816	−0.694079	+	0.719898i	$0.744188\pi$
$684$	0	0
$685$	−15.1664	−0.579479
$686$	−2.29757	−0.0877216
$687$	0	0
$688$	−33.7606	−1.28711
$689$	−4.53352	−0.172713
$690$	0	0
$691$	−2.46840	−0.0939023	−0.0469512	−	0.998897i	$-0.514951\pi$
$691$	−2.46840	−0.0939023	−0.0469512	+	0.998897i	$0.514951\pi$
$692$	−5.44650	−0.207045
$693$	0	0
$694$	5.91440	0.224508
$695$	−13.4376	−0.509718
$696$	0	0
$697$	32.2017	1.21973
$698$	−8.03133	−0.303990
$699$	0	0
$700$	−1.15599	−0.0436923
$701$	−18.9603	−0.716122	−0.358061	−	0.933698i	$-0.616562\pi$
$701$	−18.9603	−0.716122	−0.358061	+	0.933698i	$0.616562\pi$
$702$	0	0
$703$	37.6184	1.41881
$704$	6.69914	0.252483
$705$	0	0
$706$	−1.67928	−0.0632004
$707$	−12.4626	−0.468703
$708$	0	0
$709$	−22.4904	−0.844645	−0.422323	−	0.906446i	$-0.638785\pi$
$709$	−22.4904	−0.844645	−0.422323	+	0.906446i	$0.638785\pi$
$710$	−4.47197	−0.167830
$711$	0	0
$712$	−1.68943	−0.0633141
$713$	−8.83845	−0.331003
$714$	0	0
$715$	−13.6796	−0.511588
$716$	−34.4822	−1.28866
$717$	0	0
$718$	5.24780	0.195846
$719$	−39.3485	−1.46745	−0.733725	−	0.679446i	$-0.762220\pi$
$719$	−39.3485	−1.46745	−0.733725	+	0.679446i	$0.762220\pi$
$720$	0	0
$721$	9.67141	0.360182
$722$	0.849038	0.0315979
$723$	0	0
$724$	−4.62280	−0.171805
$725$	−1.83137	−0.0680152
$726$	0	0
$727$	−12.6859	−0.470495	−0.235247	−	0.971936i	$-0.575590\pi$
$727$	−12.6859	−0.470495	−0.235247	+	0.971936i	$0.575590\pi$
$728$	4.50041	0.166796
$729$	0	0
$730$	3.97946	0.147286
$731$	−45.5719	−1.68554
$732$	0	0
$733$	−10.6832	−0.394593	−0.197297	−	0.980344i	$-0.563216\pi$
$733$	−10.6832	−0.394593	−0.197297	+	0.980344i	$0.563216\pi$
$734$	6.79607	0.250847
$735$	0	0
$736$	15.8028	0.582499
$737$	1.07551	0.0396169
$738$	0	0
$739$	−17.8007	−0.654808	−0.327404	−	0.944885i	$-0.606174\pi$
$739$	−17.8007	−0.654808	−0.327404	+	0.944885i	$0.606174\pi$
$740$	−38.1102	−1.40096
$741$	0	0
$742$	0.115502	0.00424022
$743$	−3.71765	−0.136387	−0.0681936	−	0.997672i	$-0.521724\pi$
$743$	−3.71765	−0.136387	−0.0681936	+	0.997672i	$0.521724\pi$
$744$	0	0
$745$	3.41211	0.125010
$746$	−0.178938	−0.00655138
$747$	0	0
$748$	9.63366	0.352241
$749$	−8.14821	−0.297729
$750$	0	0
$751$	49.1587	1.79383	0.896914	−	0.442206i	$-0.145804\pi$
$751$	49.1587	1.79383	0.896914	+	0.442206i	$0.145804\pi$
$752$	−41.1729	−1.50142
$753$	0	0
$754$	3.51473	0.127999
$755$	−48.0587	−1.74903
$756$	0	0
$757$	0.318867	0.0115894	0.00579471	−	0.999983i	$-0.498155\pi$
$757$	0.318867	0.0115894	0.00579471	+	0.999983i	$0.498155\pi$
$758$	−0.520586	−0.0189085
$759$	0	0
$760$	7.45194	0.270310
$761$	39.6622	1.43775	0.718877	−	0.695138i	$-0.244657\pi$
$761$	39.6622	1.43775	0.718877	+	0.695138i	$0.244657\pi$
$762$	0	0
$763$	5.17375	0.187302
$764$	−25.9316	−0.938171
$765$	0	0
$766$	0.640281	0.0231343
$767$	−4.62629	−0.167046
$768$	0	0
$769$	−46.7484	−1.68579	−0.842895	−	0.538078i	$-0.819151\pi$
$769$	−46.7484	−1.68579	−0.842895	+	0.538078i	$0.819151\pi$
$770$	0.348520	0.0125598
$771$	0	0
$772$	−10.1342	−0.364737
$773$	−18.4478	−0.663521	−0.331760	−	0.943364i	$-0.607643\pi$
$773$	−18.4478	−0.663521	−0.331760	+	0.943364i	$0.607643\pi$
$774$	0	0
$775$	−1.25062	−0.0449235
$776$	13.4434	0.482591
$777$	0	0
$778$	−5.56606	−0.199553
$779$	−25.5032	−0.913748
$780$	0	0
$781$	9.28792	0.332348
$782$	6.77303	0.242203
$783$	0	0
$784$	−23.7685	−0.848873
$785$	30.0837	1.07373
$786$	0	0
$787$	−26.9722	−0.961455	−0.480727	−	0.876870i	$-0.659627\pi$
$787$	−26.9722	−0.961455	−0.480727	+	0.876870i	$0.659627\pi$
$788$	35.2531	1.25584
$789$	0	0
$790$	−6.15702	−0.219057
$791$	5.50593	0.195768
$792$	0	0
$793$	6.69193	0.237638
$794$	−1.43460	−0.0509119
$795$	0	0
$796$	−7.26100	−0.257359
$797$	−46.7418	−1.65568	−0.827839	−	0.560965i	$-0.810430\pi$
$797$	−46.7418	−1.65568	−0.827839	+	0.560965i	$0.810430\pi$
$798$	0	0
$799$	−55.5774	−1.96619
$800$	2.23606	0.0790565
$801$	0	0
$802$	0.366420	0.0129387
$803$	−8.26502	−0.291666
$804$	0	0
$805$	−8.58843	−0.302702
$806$	2.40017	0.0845423
$807$	0	0
$808$	15.9961	0.562740
$809$	22.8417	0.803070	0.401535	−	0.915844i	$-0.368477\pi$
$809$	22.8417	0.803070	0.401535	+	0.915844i	$0.368477\pi$
$810$	0	0
$811$	−16.9355	−0.594686	−0.297343	−	0.954771i	$-0.596100\pi$
$811$	−16.9355	−0.594686	−0.297343	+	0.954771i	$0.596100\pi$
$812$	3.13864	0.110145
$813$	0	0
$814$	−2.25822	−0.0791505
$815$	31.4919	1.10311
$816$	0	0
$817$	36.0921	1.26270
$818$	−6.14846	−0.214976
$819$	0	0
$820$	25.8366	0.902253
$821$	−54.5444	−1.90361	−0.951806	−	0.306702i	$-0.900775\pi$
$821$	−54.5444	−1.90361	−0.951806	+	0.306702i	$0.900775\pi$
$822$	0	0
$823$	22.8598	0.796843	0.398422	−	0.917202i	$-0.369558\pi$
$823$	22.8598	0.796843	0.398422	+	0.917202i	$0.369558\pi$
$824$	−12.4136	−0.432447
$825$	0	0
$826$	0.117866	0.00410107
$827$	−25.2383	−0.877621	−0.438811	−	0.898580i	$-0.644600\pi$
$827$	−25.2383	−0.877621	−0.438811	+	0.898580i	$0.644600\pi$
$828$	0	0
$829$	24.4323	0.848569	0.424284	−	0.905529i	$-0.360526\pi$
$829$	24.4323	0.848569	0.424284	+	0.905529i	$0.360526\pi$
$830$	−5.85289	−0.203157
$831$	0	0
$832$	44.8302	1.55421
$833$	−32.0840	−1.11164
$834$	0	0
$835$	1.01238	0.0350348
$836$	−7.62969	−0.263878
$837$	0	0
$838$	−0.606374	−0.0209468
$839$	−8.27376	−0.285642	−0.142821	−	0.989749i	$-0.545617\pi$
$839$	−8.27376	−0.285642	−0.142821	+	0.989749i	$0.545617\pi$
$840$	0	0
$841$	−24.0276	−0.828539
$842$	−5.09120	−0.175454
$843$	0	0
$844$	−1.75374	−0.0603661
$845$	−64.9684	−2.23498
$846$	0	0
$847$	−0.723848	−0.0248717
$848$	2.48642	0.0853841
$849$	0	0
$850$	0.958365	0.0328716
$851$	55.6483	1.90760
$852$	0	0
$853$	−29.5424	−1.01151	−0.505756	−	0.862677i	$-0.668786\pi$
$853$	−29.5424	−1.01151	−0.505756	+	0.862677i	$0.668786\pi$
$854$	−0.170493	−0.00583415
$855$	0	0
$856$	10.4585	0.357464
$857$	7.09758	0.242449	0.121224	−	0.992625i	$-0.461318\pi$
$857$	7.09758	0.242449	0.121224	+	0.992625i	$0.461318\pi$
$858$	0	0
$859$	−9.65252	−0.329340	−0.164670	−	0.986349i	$-0.552656\pi$
$859$	−9.65252	−0.329340	−0.164670	+	0.986349i	$0.552656\pi$
$860$	−36.5639	−1.24682
$861$	0	0
$862$	0.800440	0.0272631
$863$	10.9290	0.372028	0.186014	−	0.982547i	$-0.440443\pi$
$863$	10.9290	0.372028	0.186014	+	0.982547i	$0.440443\pi$
$864$	0	0
$865$	−5.72566	−0.194678
$866$	−1.28743	−0.0437487
$867$	0	0
$868$	2.14334	0.0727497
$869$	12.7876	0.433790
$870$	0	0
$871$	7.19724	0.243869
$872$	−6.64066	−0.224881
$873$	0	0
$874$	−5.36412	−0.181444
$875$	−8.61365	−0.291195
$876$	0	0
$877$	30.6324	1.03438	0.517191	−	0.855870i	$-0.326978\pi$
$877$	30.6324	1.03438	0.517191	+	0.855870i	$0.326978\pi$
$878$	0.989273	0.0333863
$879$	0	0
$880$	7.50261	0.252913
$881$	54.9924	1.85274	0.926371	−	0.376613i	$-0.122911\pi$
$881$	54.9924	1.85274	0.926371	+	0.376613i	$0.122911\pi$
$882$	0	0
$883$	−49.5978	−1.66910	−0.834550	−	0.550932i	$-0.814272\pi$
$883$	−49.5978	−1.66910	−0.834550	+	0.550932i	$0.814272\pi$
$884$	64.4678	2.16829
$885$	0	0
$886$	−0.744702	−0.0250187
$887$	18.8950	0.634433	0.317217	−	0.948353i	$-0.397252\pi$
$887$	18.8950	0.634433	0.317217	+	0.948353i	$0.397252\pi$
$888$	0	0
$889$	10.8139	0.362687
$890$	−0.875523	−0.0293476
$891$	0	0
$892$	15.9330	0.533477
$893$	44.0164	1.47295
$894$	0	0
$895$	−36.2496	−1.21169
$896$	−5.08370	−0.169834
$897$	0	0
$898$	−7.60437	−0.253761
$899$	3.39557	0.113249
$900$	0	0
$901$	3.35631	0.111815
$902$	1.53095	0.0509749
$903$	0	0
$904$	−7.06703	−0.235046
$905$	−4.85974	−0.161543
$906$	0	0
$907$	−47.4305	−1.57490	−0.787452	−	0.616376i	$-0.788600\pi$
$907$	−47.4305	−1.57490	−0.787452	+	0.616376i	$0.788600\pi$
$908$	−2.36420	−0.0784587
$909$	0	0
$910$	2.33227	0.0773141
$911$	−47.6999	−1.58037	−0.790184	−	0.612869i	$-0.790015\pi$
$911$	−47.6999	−1.58037	−0.790184	+	0.612869i	$0.790015\pi$
$912$	0	0
$913$	12.1560	0.402304
$914$	7.81923	0.258637
$915$	0	0
$916$	36.7715	1.21496
$917$	2.48753	0.0821456
$918$	0	0
$919$	2.67395	0.0882056	0.0441028	−	0.999027i	$-0.485957\pi$
$919$	2.67395	0.0882056	0.0441028	+	0.999027i	$0.485957\pi$
$920$	11.0235	0.363435
$921$	0	0
$922$	3.34567	0.110184
$923$	62.1542	2.04583
$924$	0	0
$925$	7.87409	0.258898
$926$	−5.66654	−0.186214
$927$	0	0
$928$	−6.07115	−0.199295
$929$	25.3166	0.830609	0.415305	−	0.909682i	$-0.363675\pi$
$929$	25.3166	0.830609	0.415305	+	0.909682i	$0.363675\pi$
$930$	0	0
$931$	25.4100	0.832778
$932$	25.9941	0.851464
$933$	0	0
$934$	−5.54472	−0.181429
$935$	10.1274	0.331203
$936$	0	0
$937$	5.84409	0.190918	0.0954591	−	0.995433i	$-0.469568\pi$
$937$	5.84409	0.190918	0.0954591	+	0.995433i	$0.469568\pi$
$938$	−0.183367	−0.00598714
$939$	0	0
$940$	−44.5917	−1.45442
$941$	−0.401984	−0.0131043	−0.00655215	−	0.999979i	$-0.502086\pi$
$941$	−0.401984	−0.0131043	−0.00655215	+	0.999979i	$0.502086\pi$
$942$	0	0
$943$	−37.7265	−1.22854
$944$	2.53730	0.0825822
$945$	0	0
$946$	−2.16659	−0.0704421
$947$	55.3298	1.79798	0.898989	−	0.437972i	$-0.144303\pi$
$947$	55.3298	1.79798	0.898989	+	0.437972i	$0.144303\pi$
$948$	0	0
$949$	−55.3090	−1.79541
$950$	−0.759009	−0.0246255
$951$	0	0
$952$	−3.33180	−0.107984
$953$	7.06683	0.228917	0.114458	−	0.993428i	$-0.463487\pi$
$953$	7.06683	0.228917	0.114458	+	0.993428i	$0.463487\pi$
$954$	0	0
$955$	−27.2607	−0.882135
$956$	39.1389	1.26584
$957$	0	0
$958$	−8.33762	−0.269376
$959$	−5.37043	−0.173420
$960$	0	0
$961$	−28.6812	−0.925200
$962$	−15.1118	−0.487225
$963$	0	0
$964$	−51.1469	−1.64733
$965$	−10.6536	−0.342952
$966$	0	0
$967$	43.5548	1.40063	0.700314	−	0.713835i	$-0.253043\pi$
$967$	43.5548	1.40063	0.700314	+	0.713835i	$0.253043\pi$
$968$	0.929081	0.0298618
$969$	0	0
$970$	6.96686	0.223693
$971$	−54.5295	−1.74993	−0.874967	−	0.484183i	$-0.839117\pi$
$971$	−54.5295	−1.74993	−0.874967	+	0.484183i	$0.839117\pi$
$972$	0	0
$973$	−4.75826	−0.152543
$974$	0.319565	0.0102395
$975$	0	0
$976$	−3.67021	−0.117481
$977$	45.8335	1.46634	0.733171	−	0.680044i	$-0.238039\pi$
$977$	45.8335	1.46634	0.733171	+	0.680044i	$0.238039\pi$
$978$	0	0
$979$	1.81839	0.0581160
$980$	−25.7421	−0.822302
$981$	0	0
$982$	0.345400	0.0110222
$983$	38.0224	1.21272	0.606362	−	0.795188i	$-0.292628\pi$
$983$	38.0224	1.21272	0.606362	+	0.795188i	$0.292628\pi$
$984$	0	0
$985$	37.0600	1.18083
$986$	−2.60207	−0.0828668
$987$	0	0
$988$	−51.0574	−1.62435
$989$	53.3905	1.69772
$990$	0	0
$991$	47.8525	1.52009	0.760043	−	0.649873i	$-0.225178\pi$
$991$	47.8525	1.52009	0.760043	+	0.649873i	$0.225178\pi$
$992$	−4.14591	−0.131633
$993$	0	0
$994$	−1.58352	−0.0502264
$995$	−7.63317	−0.241988
$996$	0	0
$997$	−58.2323	−1.84424	−0.922118	−	0.386908i	$-0.873543\pi$
$997$	−58.2323	−1.84424	−0.922118	+	0.386908i	$0.873543\pi$
$998$	2.37660	0.0752298
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6039.2.a.f.1.4		12
3.2	odd	2		2013.2.a.c.1.9	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2013.2.a.c.1.9	✓	12	3.2	odd	2
6039.2.a.f.1.4		12	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-0.235537 q^{2} -1.94452 q^{4} -2.04419 q^{5} -0.723848 q^{7} +0.929081 q^{8} +O(q^{10})\)	\(q-0.235537 q^{2} -1.94452 q^{4} -2.04419 q^{5} -0.723848 q^{7} +0.929081 q^{8} +0.481482 q^{10} -1.00000 q^{11} -6.69193 q^{13} +0.170493 q^{14} +3.67021 q^{16} +4.95425 q^{17} -3.92369 q^{19} +3.97497 q^{20} +0.235537 q^{22} -5.80424 q^{23} -0.821284 q^{25} +1.57620 q^{26} +1.40754 q^{28} +2.22988 q^{29} +1.52276 q^{31} -2.72263 q^{32} -1.16691 q^{34} +1.47968 q^{35} -9.58753 q^{37} +0.924173 q^{38} -1.89922 q^{40} +6.49981 q^{41} -9.19853 q^{43} +1.94452 q^{44} +1.36711 q^{46} -11.2181 q^{47} -6.47604 q^{49} +0.193443 q^{50} +13.0126 q^{52} +0.677460 q^{53} +2.04419 q^{55} -0.672513 q^{56} -0.525219 q^{58} +0.691323 q^{59} -1.00000 q^{61} -0.358666 q^{62} -6.69914 q^{64} +13.6796 q^{65} -1.07551 q^{67} -9.63366 q^{68} -0.348520 q^{70} -9.28792 q^{71} +8.26502 q^{73} +2.25822 q^{74} +7.62969 q^{76} +0.723848 q^{77} -12.7876 q^{79} -7.50261 q^{80} -1.53095 q^{82} -12.1560 q^{83} -10.1274 q^{85} +2.16659 q^{86} -0.929081 q^{88} -1.81839 q^{89} +4.84394 q^{91} +11.2865 q^{92} +2.64228 q^{94} +8.02076 q^{95} +14.4696 q^{97} +1.52535 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 7 q^{2} + 13 q^{4} + 7 q^{5} - 15 q^{7} + 18 q^{8}+O(q^{10}) \) 12 * q + 7 * q^2 + 13 * q^4 + 7 * q^5 - 15 * q^7 + 18 * q^8	\( 12 q + 7 q^{2} + 13 q^{4} + 7 q^{5} - 15 q^{7} + 18 q^{8} - 6 q^{10} - 12 q^{11} - 11 q^{13} - 3 q^{14} + 19 q^{16} + 33 q^{17} - 24 q^{19} + 11 q^{20} - 7 q^{22} + 9 q^{23} + 11 q^{25} + 16 q^{26} - 41 q^{28} + 16 q^{29} + q^{31} + 28 q^{32} + 32 q^{34} + 22 q^{35} - 6 q^{37} - 12 q^{38} + 26 q^{40} + 21 q^{41} - 39 q^{43} - 13 q^{44} + 18 q^{47} + 31 q^{49} + 44 q^{50} + 3 q^{52} + 14 q^{53} - 7 q^{55} - 16 q^{56} + 33 q^{58} + 23 q^{59} - 12 q^{61} + 25 q^{62} + 12 q^{64} + 29 q^{65} + 96 q^{68} + 44 q^{70} + 19 q^{71} - 42 q^{73} - 38 q^{74} + 11 q^{76} + 15 q^{77} - 11 q^{79} + 44 q^{80} - 14 q^{82} + 56 q^{83} + 16 q^{85} + 18 q^{86} - 18 q^{88} + 55 q^{89} + 11 q^{91} + 4 q^{92} - 5 q^{94} - 15 q^{95} - 7 q^{97} - 6 q^{98}+O(q^{100}) \) 12 * q + 7 * q^2 + 13 * q^4 + 7 * q^5 - 15 * q^7 + 18 * q^8 - 6 * q^10 - 12 * q^11 - 11 * q^13 - 3 * q^14 + 19 * q^16 + 33 * q^17 - 24 * q^19 + 11 * q^20 - 7 * q^22 + 9 * q^23 + 11 * q^25 + 16 * q^26 - 41 * q^28 + 16 * q^29 + q^31 + 28 * q^32 + 32 * q^34 + 22 * q^35 - 6 * q^37 - 12 * q^38 + 26 * q^40 + 21 * q^41 - 39 * q^43 - 13 * q^44 + 18 * q^47 + 31 * q^49 + 44 * q^50 + 3 * q^52 + 14 * q^53 - 7 * q^55 - 16 * q^56 + 33 * q^58 + 23 * q^59 - 12 * q^61 + 25 * q^62 + 12 * q^64 + 29 * q^65 + 96 * q^68 + 44 * q^70 + 19 * q^71 - 42 * q^73 - 38 * q^74 + 11 * q^76 + 15 * q^77 - 11 * q^79 + 44 * q^80 - 14 * q^82 + 56 * q^83 + 16 * q^85 + 18 * q^86 - 18 * q^88 + 55 * q^89 + 11 * q^91 + 4 * q^92 - 5 * q^94 - 15 * q^95 - 7 * q^97 - 6 * q^98

Introduction

L-functions

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