LMFDB - Embedded newform 6039.2.a.h.1.9

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:	$13$
Coefficient field:	$\mathbb{Q}[x]/(x^{13} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{13} - 4 x^{12} - 11 x^{11} + 57 x^{10} + 28 x^{9} - 290 x^{8} + 51 x^{7} + 644 x^{6} - 259 x^{5} + \cdots - 35 $ x^13 - 4x^12 - 11x^11 + 57x^10 + 28x^9 - 290x^8 + 51x^7 + 644x^6 - 259x^5 - 640x^4 + 274x^3 + 256x^2 - 74x - 35
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 5 $
Twist minimal:	no (minimal twist has level 2013)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
Root			$-0.638829$ 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)

$f(q)$	$=$	$q+0.638829 q^{2} -1.59190 q^{4} +2.55169 q^{5} -2.98273 q^{7} -2.29461 q^{8} +O(q^{10})$	\(q+0.638829 q^{2} -1.59190 q^{4} +2.55169 q^{5} -2.98273 q^{7} -2.29461 q^{8} +1.63009 q^{10} -1.00000 q^{11} -1.61265 q^{13} -1.90546 q^{14} +1.71793 q^{16} +5.85854 q^{17} +0.125907 q^{19} -4.06203 q^{20} -0.638829 q^{22} +3.09553 q^{23} +1.51112 q^{25} -1.03021 q^{26} +4.74820 q^{28} -0.191996 q^{29} -1.81925 q^{31} +5.68668 q^{32} +3.74261 q^{34} -7.61100 q^{35} -5.11943 q^{37} +0.0804331 q^{38} -5.85513 q^{40} -6.37269 q^{41} +6.32066 q^{43} +1.59190 q^{44} +1.97751 q^{46} +0.116868 q^{47} +1.89669 q^{49} +0.965346 q^{50} +2.56717 q^{52} -8.43820 q^{53} -2.55169 q^{55} +6.84420 q^{56} -0.122652 q^{58} +0.197718 q^{59} +1.00000 q^{61} -1.16219 q^{62} +0.196960 q^{64} -4.11498 q^{65} +2.32585 q^{67} -9.32620 q^{68} -4.86213 q^{70} +4.03481 q^{71} +12.9120 q^{73} -3.27044 q^{74} -0.200431 q^{76} +2.98273 q^{77} +2.21747 q^{79} +4.38362 q^{80} -4.07106 q^{82} -9.40586 q^{83} +14.9492 q^{85} +4.03783 q^{86} +2.29461 q^{88} -4.97987 q^{89} +4.81010 q^{91} -4.92776 q^{92} +0.0746588 q^{94} +0.321275 q^{95} -0.00505976 q^{97} +1.21166 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 13 q - 4 q^{2} + 12 q^{4} - 7 q^{5} + 7 q^{7} - 9 q^{8}+O(q^{10}) $ 13 * q - 4 * q^2 + 12 * q^4 - 7 * q^5 + 7 * q^7 - 9 * q^8	$ 13 q - 4 q^{2} + 12 q^{4} - 7 q^{5} + 7 q^{7} - 9 q^{8} + 2 q^{10} - 13 q^{11} + 9 q^{13} - 7 q^{14} + 2 q^{16} - 19 q^{17} + 14 q^{19} - 19 q^{20} + 4 q^{22} - 5 q^{23} + 2 q^{25} + 4 q^{26} + 7 q^{28} - 10 q^{29} - q^{31} - 7 q^{32} - 2 q^{34} - 16 q^{35} - 8 q^{37} + 10 q^{38} + 14 q^{40} - 21 q^{41} + 11 q^{43} - 12 q^{44} - 8 q^{46} - 22 q^{47} - 19 q^{50} - q^{52} - 16 q^{53} + 7 q^{55} - 13 q^{58} - 19 q^{59} + 13 q^{61} - 3 q^{62} - 13 q^{64} - 13 q^{65} + 12 q^{67} - 36 q^{68} - 20 q^{70} - 5 q^{71} + 18 q^{73} - 6 q^{74} - 5 q^{76} - 7 q^{77} - q^{79} - 6 q^{80} - 22 q^{82} - 48 q^{83} - 2 q^{85} - 26 q^{86} + 9 q^{88} - 15 q^{89} - 11 q^{91} + 24 q^{92} - 23 q^{94} - 17 q^{95} - 17 q^{97} + 15 q^{98}+O(q^{100}) $ 13 * q - 4 * q^2 + 12 * q^4 - 7 * q^5 + 7 * q^7 - 9 * q^8 + 2 * q^10 - 13 * q^11 + 9 * q^13 - 7 * q^14 + 2 * q^16 - 19 * q^17 + 14 * q^19 - 19 * q^20 + 4 * q^22 - 5 * q^23 + 2 * q^25 + 4 * q^26 + 7 * q^28 - 10 * q^29 - q^31 - 7 * q^32 - 2 * q^34 - 16 * q^35 - 8 * q^37 + 10 * q^38 + 14 * q^40 - 21 * q^41 + 11 * q^43 - 12 * q^44 - 8 * q^46 - 22 * q^47 - 19 * q^50 - q^52 - 16 * q^53 + 7 * q^55 - 13 * q^58 - 19 * q^59 + 13 * q^61 - 3 * q^62 - 13 * q^64 - 13 * q^65 + 12 * q^67 - 36 * q^68 - 20 * q^70 - 5 * q^71 + 18 * q^73 - 6 * q^74 - 5 * q^76 - 7 * q^77 - q^79 - 6 * q^80 - 22 * q^82 - 48 * q^83 - 2 * q^85 - 26 * q^86 + 9 * q^88 - 15 * q^89 - 11 * q^91 + 24 * q^92 - 23 * q^94 - 17 * q^95 - 17 * q^97 + 15 * 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 $

$2$	0.638829	0.451721	0.225860	−	0.974160i	$-0.427481\pi$
$2$	0.638829	0.451721	0.225860	+	0.974160i	$0.427481\pi$
$3$	0	0
$3$	0	0
$4$	−1.59190	−0.795949
$5$	2.55169	1.14115	0.570575	−	0.821245i	$-0.306720\pi$
$5$	2.55169	1.14115	0.570575	+	0.821245i	$0.306720\pi$
$6$	0	0
$7$	−2.98273	−1.12737	−0.563683	−	0.825991i	$-0.690616\pi$
$7$	−2.98273	−1.12737	−0.563683	+	0.825991i	$0.690616\pi$
$8$	−2.29461	−0.811267
$9$	0	0
$10$	1.63009	0.515481
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	−1.61265	−0.447269	−0.223634	−	0.974673i	$-0.571792\pi$
$13$	−1.61265	−0.447269	−0.223634	+	0.974673i	$0.571792\pi$
$14$	−1.90546	−0.509255
$15$	0	0
$16$	1.71793	0.429483
$17$	5.85854	1.42091	0.710453	−	0.703745i	$-0.248490\pi$
$17$	5.85854	1.42091	0.710453	+	0.703745i	$0.248490\pi$
$18$	0	0
$19$	0.125907	0.0288850	0.0144425	−	0.999896i	$-0.495403\pi$
$19$	0.125907	0.0288850	0.0144425	+	0.999896i	$0.495403\pi$
$20$	−4.06203	−0.908297
$21$	0	0
$22$	−0.638829	−0.136199
$23$	3.09553	0.645462	0.322731	−	0.946491i	$-0.395399\pi$
$23$	3.09553	0.645462	0.322731	+	0.946491i	$0.395399\pi$
$24$	0	0
$25$	1.51112	0.302223
$26$	−1.03021	−0.202040
$27$	0	0
$28$	4.74820	0.897326
$29$	−0.191996	−0.0356527	−0.0178263	−	0.999841i	$-0.505675\pi$
$29$	−0.191996	−0.0356527	−0.0178263	+	0.999841i	$0.505675\pi$
$30$	0	0
$31$	−1.81925	−0.326747	−0.163374	−	0.986564i	$-0.552238\pi$
$31$	−1.81925	−0.326747	−0.163374	+	0.986564i	$0.552238\pi$
$32$	5.68668	1.00527
$33$	0	0
$34$	3.74261	0.641852
$35$	−7.61100	−1.28649
$36$	0	0
$37$	−5.11943	−0.841629	−0.420814	−	0.907147i	$-0.638256\pi$
$37$	−5.11943	−0.841629	−0.420814	+	0.907147i	$0.638256\pi$
$38$	0.0804331	0.0130480
$39$	0	0
$40$	−5.85513	−0.925777
$41$	−6.37269	−0.995247	−0.497623	−	0.867393i	$-0.665794\pi$
$41$	−6.37269	−0.995247	−0.497623	+	0.867393i	$0.665794\pi$
$42$	0	0
$43$	6.32066	0.963892	0.481946	−	0.876201i	$-0.339930\pi$
$43$	6.32066	0.963892	0.481946	+	0.876201i	$0.339930\pi$
$44$	1.59190	0.239988
$45$	0	0
$46$	1.97751	0.291568
$47$	0.116868	0.0170470	0.00852348	−	0.999964i	$-0.497287\pi$
$47$	0.116868	0.0170470	0.00852348	+	0.999964i	$0.497287\pi$
$48$	0	0
$49$	1.89669	0.270955
$50$	0.965346	0.136521
$51$	0	0
$52$	2.56717	0.356003
$53$	−8.43820	−1.15908	−0.579538	−	0.814945i	$-0.696767\pi$
$53$	−8.43820	−1.15908	−0.579538	+	0.814945i	$0.696767\pi$
$54$	0	0
$55$	−2.55169	−0.344070
$56$	6.84420	0.914595
$57$	0	0
$58$	−0.122652	−0.0161051
$59$	0.197718	0.0257407	0.0128704	−	0.999917i	$-0.495903\pi$
$59$	0.197718	0.0257407	0.0128704	+	0.999917i	$0.495903\pi$
$60$	0	0
$61$	1.00000	0.128037
$61$	1.00000	0.128037
$62$	−1.16219	−0.147598
$63$	0	0
$64$	0.196960	0.0246200
$65$	−4.11498	−0.510401
$66$	0	0
$67$	2.32585	0.284148	0.142074	−	0.989856i	$-0.454623\pi$
$67$	2.32585	0.284148	0.142074	+	0.989856i	$0.454623\pi$
$68$	−9.32620	−1.13097
$69$	0	0
$70$	−4.86213	−0.581136
$71$	4.03481	0.478844	0.239422	−	0.970916i	$-0.423042\pi$
$71$	4.03481	0.478844	0.239422	+	0.970916i	$0.423042\pi$
$72$	0	0
$73$	12.9120	1.51123	0.755617	−	0.655014i	$-0.227337\pi$
$73$	12.9120	1.51123	0.755617	+	0.655014i	$0.227337\pi$
$74$	−3.27044	−0.380181
$75$	0	0
$76$	−0.200431	−0.0229910
$77$	2.98273	0.339914
$78$	0	0
$79$	2.21747	0.249485	0.124743	−	0.992189i	$-0.460189\pi$
$79$	2.21747	0.249485	0.124743	+	0.992189i	$0.460189\pi$
$80$	4.38362	0.490104
$81$	0	0
$82$	−4.07106	−0.449573
$83$	−9.40586	−1.03243	−0.516214	−	0.856460i	$-0.672659\pi$
$83$	−9.40586	−1.03243	−0.516214	+	0.856460i	$0.672659\pi$
$84$	0	0
$85$	14.9492	1.62147
$86$	4.03783	0.435410
$87$	0	0
$88$	2.29461	0.244606
$89$	−4.97987	−0.527865	−0.263933	−	0.964541i	$-0.585020\pi$
$89$	−4.97987	−0.527865	−0.263933	+	0.964541i	$0.585020\pi$
$90$	0	0
$91$	4.81010	0.504236
$92$	−4.92776	−0.513755
$93$	0	0
$94$	0.0746588	0.00770046
$95$	0.321275	0.0329622
$96$	0	0
$97$	−0.00505976	−0.000513741 0	−0.000256871	−	1.00000i	$-0.500082\pi$
$97$	−0.00505976	−0.000513741 0	−0.000256871		1.00000i	$0.500082\pi$
$98$	1.21166	0.122396
$99$	0	0
$100$	−2.40554	−0.240554
$101$	−19.3024	−1.92066	−0.960330	−	0.278867i	$-0.910041\pi$
$101$	−19.3024	−1.92066	−0.960330	+	0.278867i	$0.910041\pi$
$102$	0	0
$103$	2.13746	0.210610	0.105305	−	0.994440i	$-0.466418\pi$
$103$	2.13746	0.210610	0.105305	+	0.994440i	$0.466418\pi$
$104$	3.70040	0.362854
$105$	0	0
$106$	−5.39057	−0.523578
$107$	−10.0210	−0.968764	−0.484382	−	0.874856i	$-0.660956\pi$
$107$	−10.0210	−0.968764	−0.484382	+	0.874856i	$0.660956\pi$
$108$	0	0
$109$	−14.5007	−1.38892	−0.694459	−	0.719533i	$-0.744356\pi$
$109$	−14.5007	−1.38892	−0.694459	+	0.719533i	$0.744356\pi$
$110$	−1.63009	−0.155423
$111$	0	0
$112$	−5.12412	−0.484184
$113$	−19.2021	−1.80639	−0.903193	−	0.429236i	$-0.858783\pi$
$113$	−19.2021	−1.80639	−0.903193	+	0.429236i	$0.858783\pi$
$114$	0	0
$115$	7.89882	0.736569
$116$	0.305637	0.0283777
$117$	0	0
$118$	0.126308	0.0116276
$119$	−17.4745	−1.60188
$120$	0	0
$121$	1.00000	0.0909091
$122$	0.638829	0.0578369
$123$	0	0
$124$	2.89606	0.260074
$125$	−8.90254	−0.796268
$126$	0	0
$127$	6.27538	0.556850	0.278425	−	0.960458i	$-0.410188\pi$
$127$	6.27538	0.556850	0.278425	+	0.960458i	$0.410188\pi$
$128$	−11.2475	−0.994152
$129$	0	0
$130$	−2.62877	−0.230558
$131$	11.8518	1.03550	0.517750	−	0.855532i	$-0.326770\pi$
$131$	11.8518	1.03550	0.517750	+	0.855532i	$0.326770\pi$
$132$	0	0
$133$	−0.375547	−0.0325640
$134$	1.48582	0.128355
$135$	0	0
$136$	−13.4431	−1.15273
$137$	−19.5944	−1.67407	−0.837033	−	0.547152i	$-0.815712\pi$
$137$	−19.5944	−1.67407	−0.837033	+	0.547152i	$0.815712\pi$
$138$	0	0
$139$	−12.3549	−1.04793	−0.523966	−	0.851739i	$-0.675548\pi$
$139$	−12.3549	−1.04793	−0.523966	+	0.851739i	$0.675548\pi$
$140$	12.1159	1.02398
$141$	0	0
$142$	2.57756	0.216304
$143$	1.61265	0.134857
$144$	0	0
$145$	−0.489913	−0.0406851
$146$	8.24856	0.682655
$147$	0	0
$148$	8.14960	0.669893
$149$	11.5552	0.946637	0.473319	−	0.880891i	$-0.343056\pi$
$149$	11.5552	0.946637	0.473319	+	0.880891i	$0.343056\pi$
$150$	0	0
$151$	−13.2277	−1.07645	−0.538226	−	0.842800i	$-0.680905\pi$
$151$	−13.2277	−1.07645	−0.538226	+	0.842800i	$0.680905\pi$
$152$	−0.288907	−0.0234335
$153$	0	0
$154$	1.90546	0.153546
$155$	−4.64216	−0.372868
$156$	0	0
$157$	−17.0282	−1.35900	−0.679499	−	0.733676i	$-0.737803\pi$
$157$	−17.0282	−1.35900	−0.679499	+	0.733676i	$0.737803\pi$
$158$	1.41659	0.112698
$159$	0	0
$160$	14.5106	1.14717
$161$	−9.23313	−0.727672
$162$	0	0
$163$	2.71957	0.213013	0.106506	−	0.994312i	$-0.466034\pi$
$163$	2.71957	0.213013	0.106506	+	0.994312i	$0.466034\pi$
$164$	10.1447	0.792165
$165$	0	0
$166$	−6.00874	−0.466369
$167$	−21.8146	−1.68806	−0.844031	−	0.536294i	$-0.819824\pi$
$167$	−21.8146	−1.68806	−0.844031	+	0.536294i	$0.819824\pi$
$168$	0	0
$169$	−10.3994	−0.799951
$170$	9.54997	0.732450
$171$	0	0
$172$	−10.0618	−0.767209
$173$	9.50168	0.722400	0.361200	−	0.932488i	$-0.382367\pi$
$173$	9.50168	0.722400	0.361200	+	0.932488i	$0.382367\pi$
$174$	0	0
$175$	−4.50726	−0.340717
$176$	−1.71793	−0.129494
$177$	0	0
$178$	−3.18129	−0.238448
$179$	11.6892	0.873692	0.436846	−	0.899536i	$-0.356095\pi$
$179$	11.6892	0.873692	0.436846	+	0.899536i	$0.356095\pi$
$180$	0	0
$181$	−5.10150	−0.379191	−0.189596	−	0.981862i	$-0.560718\pi$
$181$	−5.10150	−0.379191	−0.189596	+	0.981862i	$0.560718\pi$
$182$	3.07283	0.227774
$183$	0	0
$184$	−7.10303	−0.523642
$185$	−13.0632	−0.960425
$186$	0	0
$187$	−5.85854	−0.428419
$188$	−0.186042	−0.0135685
$189$	0	0
$190$	0.205240	0.0148897
$191$	1.98782	0.143834	0.0719169	−	0.997411i	$-0.477088\pi$
$191$	1.98782	0.143834	0.0719169	+	0.997411i	$0.477088\pi$
$192$	0	0
$193$	−7.45469	−0.536600	−0.268300	−	0.963335i	$-0.586462\pi$
$193$	−7.45469	−0.536600	−0.268300	+	0.963335i	$0.586462\pi$
$194$	−0.00323233	−0.000232067 0
$195$	0	0
$196$	−3.01933	−0.215667
$197$	9.63445	0.686426	0.343213	−	0.939258i	$-0.388485\pi$
$197$	9.63445	0.686426	0.343213	+	0.939258i	$0.388485\pi$
$198$	0	0
$199$	11.2101	0.794665	0.397332	−	0.917675i	$-0.369936\pi$
$199$	11.2101	0.794665	0.397332	+	0.917675i	$0.369936\pi$
$200$	−3.46742	−0.245184
$201$	0	0
$202$	−12.3309	−0.867602
$203$	0.572671	0.0401936
$204$	0	0
$205$	−16.2611	−1.13573
$206$	1.36547	0.0951367
$207$	0	0
$208$	−2.77042	−0.192094
$209$	−0.125907	−0.00870917
$210$	0	0
$211$	−10.5409	−0.725666	−0.362833	−	0.931854i	$-0.618190\pi$
$211$	−10.5409	−0.725666	−0.362833	+	0.931854i	$0.618190\pi$
$212$	13.4327	0.922564
$213$	0	0
$214$	−6.40169	−0.437611
$215$	16.1284	1.09995
$216$	0	0
$217$	5.42634	0.368364
$218$	−9.26349	−0.627403
$219$	0	0
$220$	4.06203	0.273862
$221$	−9.44778	−0.635526
$222$	0	0
$223$	2.75074	0.184203	0.0921015	−	0.995750i	$-0.470642\pi$
$223$	2.75074	0.184203	0.0921015	+	0.995750i	$0.470642\pi$
$224$	−16.9618	−1.13331
$225$	0	0
$226$	−12.2669	−0.815981
$227$	−12.4826	−0.828496	−0.414248	−	0.910164i	$-0.635955\pi$
$227$	−12.4826	−0.828496	−0.414248	+	0.910164i	$0.635955\pi$
$228$	0	0
$229$	8.93189	0.590236	0.295118	−	0.955461i	$-0.404641\pi$
$229$	8.93189	0.590236	0.295118	+	0.955461i	$0.404641\pi$
$230$	5.04600	0.332723
$231$	0	0
$232$	0.440555	0.0289238
$233$	−16.0808	−1.05349	−0.526744	−	0.850024i	$-0.676587\pi$
$233$	−16.0808	−1.05349	−0.526744	+	0.850024i	$0.676587\pi$
$234$	0	0
$235$	0.298211	0.0194531
$236$	−0.314747	−0.0204883
$237$	0	0
$238$	−11.1632	−0.723603
$239$	6.20231	0.401194	0.200597	−	0.979674i	$-0.435712\pi$
$239$	6.20231	0.401194	0.200597	+	0.979674i	$0.435712\pi$
$240$	0	0
$241$	1.66908	0.107515	0.0537575	−	0.998554i	$-0.482880\pi$
$241$	1.66908	0.107515	0.0537575	+	0.998554i	$0.482880\pi$
$242$	0.638829	0.0410655
$243$	0	0
$244$	−1.59190	−0.101911
$245$	4.83976	0.309201
$246$	0	0
$247$	−0.203044	−0.0129194
$248$	4.17447	0.265079
$249$	0	0
$250$	−5.68721	−0.359691
$251$	9.43302	0.595407	0.297703	−	0.954658i	$-0.403779\pi$
$251$	9.43302	0.595407	0.297703	+	0.954658i	$0.403779\pi$
$252$	0	0
$253$	−3.09553	−0.194614
$254$	4.00890	0.251541
$255$	0	0
$256$	−7.57918	−0.473699
$257$	15.6496	0.976192	0.488096	−	0.872790i	$-0.337692\pi$
$257$	15.6496	0.976192	0.488096	+	0.872790i	$0.337692\pi$
$258$	0	0
$259$	15.2699	0.948824
$260$	6.55063	0.406253
$261$	0	0
$262$	7.57130	0.467756
$263$	1.18349	0.0729768	0.0364884	−	0.999334i	$-0.488383\pi$
$263$	1.18349	0.0729768	0.0364884	+	0.999334i	$0.488383\pi$
$264$	0	0
$265$	−21.5317	−1.32268
$266$	−0.239910	−0.0147098
$267$	0	0
$268$	−3.70252	−0.226167
$269$	27.5543	1.68002	0.840008	−	0.542574i	$-0.182550\pi$
$269$	27.5543	1.68002	0.840008	+	0.542574i	$0.182550\pi$
$270$	0	0
$271$	7.21580	0.438328	0.219164	−	0.975688i	$-0.429667\pi$
$271$	7.21580	0.438328	0.219164	+	0.975688i	$0.429667\pi$
$272$	10.0646	0.610254
$273$	0	0
$274$	−12.5175	−0.756210
$275$	−1.51112	−0.0911238
$276$	0	0
$277$	15.5916	0.936808	0.468404	−	0.883514i	$-0.344829\pi$
$277$	15.5916	0.936808	0.468404	+	0.883514i	$0.344829\pi$
$278$	−7.89269	−0.473372
$279$	0	0
$280$	17.4643	1.04369
$281$	−9.98524	−0.595670	−0.297835	−	0.954617i	$-0.596264\pi$
$281$	−9.98524	−0.595670	−0.297835	+	0.954617i	$0.596264\pi$
$282$	0	0
$283$	13.8811	0.825145	0.412572	−	0.910925i	$-0.364630\pi$
$283$	13.8811	0.825145	0.412572	+	0.910925i	$0.364630\pi$
$284$	−6.42301	−0.381135
$285$	0	0
$286$	1.03021	0.0609175
$287$	19.0080	1.12201
$288$	0	0
$289$	17.3225	1.01897
$290$	−0.312971	−0.0183783
$291$	0	0
$292$	−20.5546	−1.20286
$293$	−26.7224	−1.56114	−0.780571	−	0.625067i	$-0.785072\pi$
$293$	−26.7224	−1.56114	−0.780571	+	0.625067i	$0.785072\pi$
$294$	0	0
$295$	0.504516	0.0293741
$296$	11.7471	0.682786
$297$	0	0
$298$	7.38179	0.427615
$299$	−4.99200	−0.288695
$300$	0	0
$301$	−18.8528	−1.08666
$302$	−8.45023	−0.486256
$303$	0	0
$304$	0.216299	0.0124056
$305$	2.55169	0.146109
$306$	0	0
$307$	−4.29612	−0.245192	−0.122596	−	0.992457i	$-0.539122\pi$
$307$	−4.29612	−0.245192	−0.122596	+	0.992457i	$0.539122\pi$
$308$	−4.74820	−0.270554
$309$	0	0
$310$	−2.96555	−0.168432
$311$	−7.41621	−0.420535	−0.210267	−	0.977644i	$-0.567433\pi$
$311$	−7.41621	−0.420535	−0.210267	+	0.977644i	$0.567433\pi$
$312$	0	0
$313$	−24.0635	−1.36015	−0.680073	−	0.733144i	$-0.738052\pi$
$313$	−24.0635	−1.36015	−0.680073	+	0.733144i	$0.738052\pi$
$314$	−10.8781	−0.613888
$315$	0	0
$316$	−3.52999	−0.198577
$317$	−14.8151	−0.832101	−0.416051	−	0.909341i	$-0.636586\pi$
$317$	−14.8151	−0.832101	−0.416051	+	0.909341i	$0.636586\pi$
$318$	0	0
$319$	0.191996	0.0107497
$320$	0.502580	0.0280951
$321$	0	0
$322$	−5.89839	−0.328705
$323$	0.737631	0.0410429
$324$	0	0
$325$	−2.43690	−0.135175
$326$	1.73734	0.0962223
$327$	0	0
$328$	14.6228	0.807411
$329$	−0.348586	−0.0192182
$330$	0	0
$331$	4.21655	0.231762	0.115881	−	0.993263i	$-0.463031\pi$
$331$	4.21655	0.231762	0.115881	+	0.993263i	$0.463031\pi$
$332$	14.9732	0.821759
$333$	0	0
$334$	−13.9358	−0.762533
$335$	5.93485	0.324255
$336$	0	0
$337$	19.6842	1.07227	0.536134	−	0.844133i	$-0.319884\pi$
$337$	19.6842	1.07227	0.536134	+	0.844133i	$0.319884\pi$
$338$	−6.64342	−0.361354
$339$	0	0
$340$	−23.7976	−1.29060
$341$	1.81925	0.0985180
$342$	0	0
$343$	15.2218	0.821901
$344$	−14.5035	−0.781974
$345$	0	0
$346$	6.06996	0.326323
$347$	−28.6990	−1.54064	−0.770321	−	0.637656i	$-0.779904\pi$
$347$	−28.6990	−1.54064	−0.770321	+	0.637656i	$0.779904\pi$
$348$	0	0
$349$	16.7748	0.897935	0.448968	−	0.893548i	$-0.351792\pi$
$349$	16.7748	0.897935	0.448968	+	0.893548i	$0.351792\pi$
$350$	−2.87937	−0.153909
$351$	0	0
$352$	−5.68668	−0.303101
$353$	−24.8325	−1.32170	−0.660851	−	0.750517i	$-0.729804\pi$
$353$	−24.8325	−1.32170	−0.660851	+	0.750517i	$0.729804\pi$
$354$	0	0
$355$	10.2956	0.546433
$356$	7.92744	0.420154
$357$	0	0
$358$	7.46740	0.394665
$359$	9.65713	0.509684	0.254842	−	0.966983i	$-0.417977\pi$
$359$	9.65713	0.509684	0.254842	+	0.966983i	$0.417977\pi$
$360$	0	0
$361$	−18.9841	−0.999166
$362$	−3.25899	−0.171288
$363$	0	0
$364$	−7.65719	−0.401346
$365$	32.9474	1.72454
$366$	0	0
$367$	−10.5641	−0.551444	−0.275722	−	0.961237i	$-0.588917\pi$
$367$	−10.5641	−0.551444	−0.275722	+	0.961237i	$0.588917\pi$
$368$	5.31790	0.277215
$369$	0	0
$370$	−8.34515	−0.433844
$371$	25.1689	1.30670
$372$	0	0
$373$	−22.6159	−1.17101	−0.585503	−	0.810670i	$-0.699103\pi$
$373$	−22.6159	−1.17101	−0.585503	+	0.810670i	$0.699103\pi$
$374$	−3.74261	−0.193526
$375$	0	0
$376$	−0.268167	−0.0138296
$377$	0.309622	0.0159463
$378$	0	0
$379$	−31.0913	−1.59705	−0.798526	−	0.601960i	$-0.794387\pi$
$379$	−31.0913	−1.59705	−0.798526	+	0.601960i	$0.794387\pi$
$380$	−0.511437	−0.0262362
$381$	0	0
$382$	1.26988	0.0649727
$383$	−0.266844	−0.0136351	−0.00681755	−	0.999977i	$-0.502170\pi$
$383$	−0.266844	−0.0136351	−0.00681755	+	0.999977i	$0.502170\pi$
$384$	0	0
$385$	7.61100	0.387893
$386$	−4.76227	−0.242393
$387$	0	0
$388$	0.00805462	0.000408912 0
$389$	−16.5671	−0.839988	−0.419994	−	0.907527i	$-0.637968\pi$
$389$	−16.5671	−0.839988	−0.419994	+	0.907527i	$0.637968\pi$
$390$	0	0
$391$	18.1353	0.917140
$392$	−4.35216	−0.219817
$393$	0	0
$394$	6.15477	0.310073
$395$	5.65830	0.284700
$396$	0	0
$397$	12.1572	0.610151	0.305076	−	0.952328i	$-0.401318\pi$
$397$	12.1572	0.610151	0.305076	+	0.952328i	$0.401318\pi$
$398$	7.16136	0.358966
$399$	0	0
$400$	2.59599	0.129800
$401$	9.47284	0.473051	0.236526	−	0.971625i	$-0.423991\pi$
$401$	9.47284	0.473051	0.236526	+	0.971625i	$0.423991\pi$
$402$	0	0
$403$	2.93382	0.146144
$404$	30.7274	1.52875
$405$	0	0
$406$	0.365839	0.0181563
$407$	5.11943	0.253761
$408$	0	0
$409$	−2.91535	−0.144155	−0.0720773	−	0.997399i	$-0.522963\pi$
$409$	−2.91535	−0.144155	−0.0720773	+	0.997399i	$0.522963\pi$
$410$	−10.3881	−0.513031
$411$	0	0
$412$	−3.40261	−0.167634
$413$	−0.589741	−0.0290193
$414$	0	0
$415$	−24.0008	−1.17815
$416$	−9.17063	−0.449627
$417$	0	0
$418$	−0.0804331	−0.00393411
$419$	−33.0816	−1.61614	−0.808071	−	0.589085i	$-0.799488\pi$
$419$	−33.0816	−1.61614	−0.808071	+	0.589085i	$0.799488\pi$
$420$	0	0
$421$	−18.8953	−0.920899	−0.460449	−	0.887686i	$-0.652312\pi$
$421$	−18.8953	−0.920899	−0.460449	+	0.887686i	$0.652312\pi$
$422$	−6.73384	−0.327798
$423$	0	0
$424$	19.3624	0.940319
$425$	8.85294	0.429431
$426$	0	0
$427$	−2.98273	−0.144345
$428$	15.9524	0.771087
$429$	0	0
$430$	10.3033	0.496868
$431$	20.2464	0.975233	0.487616	−	0.873058i	$-0.337867\pi$
$431$	20.2464	0.975233	0.487616	+	0.873058i	$0.337867\pi$
$432$	0	0
$433$	−12.0123	−0.577273	−0.288637	−	0.957439i	$-0.593202\pi$
$433$	−12.0123	−0.577273	−0.288637	+	0.957439i	$0.593202\pi$
$434$	3.46650	0.166398
$435$	0	0
$436$	23.0837	1.10551
$437$	0.389748	0.0186442
$438$	0	0
$439$	28.1149	1.34185	0.670925	−	0.741525i	$-0.265897\pi$
$439$	28.1149	1.34185	0.670925	+	0.741525i	$0.265897\pi$
$440$	5.85513	0.279132
$441$	0	0
$442$	−6.03552	−0.287080
$443$	5.13937	0.244179	0.122089	−	0.992519i	$-0.461041\pi$
$443$	5.13937	0.244179	0.122089	+	0.992519i	$0.461041\pi$
$444$	0	0
$445$	−12.7071	−0.602373
$446$	1.75725	0.0832083
$447$	0	0
$448$	−0.587479	−0.0277558
$449$	−4.47669	−0.211268	−0.105634	−	0.994405i	$-0.533687\pi$
$449$	−4.47669	−0.211268	−0.105634	+	0.994405i	$0.533687\pi$
$450$	0	0
$451$	6.37269	0.300078
$452$	30.5678	1.43779
$453$	0	0
$454$	−7.97422	−0.374249
$455$	12.2739	0.575409
$456$	0	0
$457$	1.13739	0.0532049	0.0266025	−	0.999646i	$-0.491531\pi$
$457$	1.13739	0.0532049	0.0266025	+	0.999646i	$0.491531\pi$
$458$	5.70595	0.266622
$459$	0	0
$460$	−12.5741	−0.586271
$461$	−14.0951	−0.656476	−0.328238	−	0.944595i	$-0.606455\pi$
$461$	−14.0951	−0.656476	−0.328238	+	0.944595i	$0.606455\pi$
$462$	0	0
$463$	19.7373	0.917270	0.458635	−	0.888625i	$-0.348338\pi$
$463$	19.7373	0.917270	0.458635	+	0.888625i	$0.348338\pi$
$464$	−0.329835	−0.0153122
$465$	0	0
$466$	−10.2729	−0.475882
$467$	29.7503	1.37668	0.688341	−	0.725387i	$-0.258339\pi$
$467$	29.7503	1.37668	0.688341	+	0.725387i	$0.258339\pi$
$468$	0	0
$469$	−6.93739	−0.320339
$470$	0.190506	0.00878739
$471$	0	0
$472$	−0.453687	−0.0208826
$473$	−6.32066	−0.290624
$474$	0	0
$475$	0.190260	0.00872974
$476$	27.8175	1.27501
$477$	0	0
$478$	3.96222	0.181228
$479$	−19.4612	−0.889204	−0.444602	−	0.895728i	$-0.646655\pi$
$479$	−19.4612	−0.889204	−0.444602	+	0.895728i	$0.646655\pi$
$480$	0	0
$481$	8.25585	0.376434
$482$	1.06626	0.0485667
$483$	0	0
$484$	−1.59190	−0.0723590
$485$	−0.0129109	−0.000586256 0
$486$	0	0
$487$	−40.5536	−1.83766	−0.918830	−	0.394654i	$-0.870865\pi$
$487$	−40.5536	−1.83766	−0.918830	+	0.394654i	$0.870865\pi$
$488$	−2.29461	−0.103872
$489$	0	0
$490$	3.09178	0.139672
$491$	21.8023	0.983922	0.491961	−	0.870617i	$-0.336280\pi$
$491$	21.8023	0.983922	0.491961	+	0.870617i	$0.336280\pi$
$492$	0	0
$493$	−1.12481	−0.0506591
$494$	−0.129710	−0.00583595
$495$	0	0
$496$	−3.12535	−0.140332
$497$	−12.0348	−0.539833
$498$	0	0
$499$	36.0404	1.61339	0.806695	−	0.590968i	$-0.201254\pi$
$499$	36.0404	1.61339	0.806695	+	0.590968i	$0.201254\pi$
$500$	14.1719	0.633788
$501$	0	0
$502$	6.02609	0.268958
$503$	−15.7865	−0.703883	−0.351942	−	0.936022i	$-0.614478\pi$
$503$	−15.7865	−0.703883	−0.351942	+	0.936022i	$0.614478\pi$
$504$	0	0
$505$	−49.2537	−2.19176
$506$	−1.97751	−0.0879112
$507$	0	0
$508$	−9.98976	−0.443224
$509$	12.0969	0.536188	0.268094	−	0.963393i	$-0.413606\pi$
$509$	12.0969	0.536188	0.268094	+	0.963393i	$0.413606\pi$
$510$	0	0
$511$	−38.5130	−1.70371
$512$	17.6533	0.780172
$513$	0	0
$514$	9.99739	0.440966
$515$	5.45412	0.240337
$516$	0	0
$517$	−0.116868	−0.00513985
$518$	9.75485	0.428603
$519$	0	0
$520$	9.44228	0.414071
$521$	−19.8367	−0.869060	−0.434530	−	0.900657i	$-0.643086\pi$
$521$	−19.8367	−0.869060	−0.434530	+	0.900657i	$0.643086\pi$
$522$	0	0
$523$	−6.02096	−0.263278	−0.131639	−	0.991298i	$-0.542024\pi$
$523$	−6.02096	−0.263278	−0.131639	+	0.991298i	$0.542024\pi$
$524$	−18.8669	−0.824204
$525$	0	0
$526$	0.756045	0.0329651
$527$	−10.6582	−0.464277
$528$	0	0
$529$	−13.4177	−0.583379
$530$	−13.7551	−0.597481
$531$	0	0
$532$	0.597832	0.0259193
$533$	10.2769	0.445143
$534$	0	0
$535$	−25.5704	−1.10551
$536$	−5.33692	−0.230520
$537$	0	0
$538$	17.6025	0.758898
$539$	−1.89669	−0.0816961
$540$	0	0
$541$	29.7325	1.27830	0.639150	−	0.769082i	$-0.279286\pi$
$541$	29.7325	1.27830	0.639150	+	0.769082i	$0.279286\pi$
$542$	4.60966	0.198002
$543$	0	0
$544$	33.3157	1.42840
$545$	−37.0013	−1.58496
$546$	0	0
$547$	25.2902	1.08133	0.540666	−	0.841237i	$-0.318172\pi$
$547$	25.2902	1.08133	0.540666	+	0.841237i	$0.318172\pi$
$548$	31.1923	1.33247
$549$	0	0
$550$	−0.965346	−0.0411625
$551$	−0.0241736	−0.00102983
$552$	0	0
$553$	−6.61413	−0.281261
$554$	9.96037	0.423176
$555$	0	0
$556$	19.6678	0.834100
$557$	6.72947	0.285137	0.142568	−	0.989785i	$-0.454464\pi$
$557$	6.72947	0.285137	0.142568	+	0.989785i	$0.454464\pi$
$558$	0	0
$559$	−10.1930	−0.431119
$560$	−13.0752	−0.552527
$561$	0	0
$562$	−6.37887	−0.269076
$563$	20.8382	0.878224	0.439112	−	0.898432i	$-0.355293\pi$
$563$	20.8382	0.878224	0.439112	+	0.898432i	$0.355293\pi$
$564$	0	0
$565$	−48.9979	−2.06136
$566$	8.86765	0.372735
$567$	0	0
$568$	−9.25832	−0.388471
$569$	−8.15145	−0.341727	−0.170863	−	0.985295i	$-0.554656\pi$
$569$	−8.15145	−0.341727	−0.170863	+	0.985295i	$0.554656\pi$
$570$	0	0
$571$	−31.5196	−1.31906	−0.659528	−	0.751680i	$-0.729244\pi$
$571$	−31.5196	−1.31906	−0.659528	+	0.751680i	$0.729244\pi$
$572$	−2.56717	−0.107339
$573$	0	0
$574$	12.1429	0.506834
$575$	4.67770	0.195074
$576$	0	0
$577$	−42.8089	−1.78216	−0.891079	−	0.453849i	$-0.850051\pi$
$577$	−42.8089	−1.78216	−0.891079	+	0.453849i	$0.850051\pi$
$578$	11.0661	0.460291
$579$	0	0
$580$	0.779891	0.0323832
$581$	28.0551	1.16392
$582$	0	0
$583$	8.43820	0.349474
$584$	−29.6280	−1.22601
$585$	0	0
$586$	−17.0711	−0.705200
$587$	−31.6957	−1.30822	−0.654111	−	0.756398i	$-0.726957\pi$
$587$	−31.6957	−1.30822	−0.654111	+	0.756398i	$0.726957\pi$
$588$	0	0
$589$	−0.229056	−0.00943811
$590$	0.322300	0.0132689
$591$	0	0
$592$	−8.79482	−0.361465
$593$	−11.9630	−0.491263	−0.245632	−	0.969363i	$-0.578995\pi$
$593$	−11.9630	−0.491263	−0.245632	+	0.969363i	$0.578995\pi$
$594$	0	0
$595$	−44.5894	−1.82799
$596$	−18.3947	−0.753474
$597$	0	0
$598$	−3.18904	−0.130409
$599$	28.5018	1.16455	0.582276	−	0.812991i	$-0.302162\pi$
$599$	28.5018	1.16455	0.582276	+	0.812991i	$0.302162\pi$
$600$	0	0
$601$	43.2159	1.76281	0.881407	−	0.472358i	$-0.156597\pi$
$601$	43.2159	1.76281	0.881407	+	0.472358i	$0.156597\pi$
$602$	−12.0438	−0.490867
$603$	0	0
$604$	21.0571	0.856801
$605$	2.55169	0.103741
$606$	0	0
$607$	10.8061	0.438605	0.219302	−	0.975657i	$-0.429622\pi$
$607$	10.8061	0.438605	0.219302	+	0.975657i	$0.429622\pi$
$608$	0.715993	0.0290374
$609$	0	0
$610$	1.63009	0.0660006
$611$	−0.188467	−0.00762457
$612$	0	0
$613$	31.9142	1.28900	0.644500	−	0.764604i	$-0.277065\pi$
$613$	31.9142	1.28900	0.644500	+	0.764604i	$0.277065\pi$
$614$	−2.74449	−0.110758
$615$	0	0
$616$	−6.84420	−0.275761
$617$	13.3066	0.535704	0.267852	−	0.963460i	$-0.413686\pi$
$617$	13.3066	0.535704	0.267852	+	0.963460i	$0.413686\pi$
$618$	0	0
$619$	2.94505	0.118371	0.0591857	−	0.998247i	$-0.481150\pi$
$619$	2.94505	0.118371	0.0591857	+	0.998247i	$0.481150\pi$
$620$	7.38985	0.296783
$621$	0	0
$622$	−4.73769	−0.189964
$623$	14.8536	0.595098
$624$	0	0
$625$	−30.2721	−1.21088
$626$	−15.3724	−0.614406
$627$	0	0
$628$	27.1071	1.08169
$629$	−29.9924	−1.19587
$630$	0	0
$631$	−4.88074	−0.194299	−0.0971496	−	0.995270i	$-0.530973\pi$
$631$	−4.88074	−0.194299	−0.0971496	+	0.995270i	$0.530973\pi$
$632$	−5.08824	−0.202399
$633$	0	0
$634$	−9.46435	−0.375877
$635$	16.0128	0.635449
$636$	0	0
$637$	−3.05869	−0.121190
$638$	0.122652	0.00485586
$639$	0	0
$640$	−28.7002	−1.13448
$641$	−28.6540	−1.13176	−0.565882	−	0.824486i	$-0.691464\pi$
$641$	−28.6540	−1.13176	−0.565882	+	0.824486i	$0.691464\pi$
$642$	0	0
$643$	18.8394	0.742952	0.371476	−	0.928443i	$-0.378852\pi$
$643$	18.8394	0.742952	0.371476	+	0.928443i	$0.378852\pi$
$644$	14.6982	0.579190
$645$	0	0
$646$	0.471221	0.0185399
$647$	−16.1990	−0.636849	−0.318424	−	0.947948i	$-0.603154\pi$
$647$	−16.1990	−0.636849	−0.318424	+	0.947948i	$0.603154\pi$
$648$	0	0
$649$	−0.197718	−0.00776113
$650$	−1.55677	−0.0610614
$651$	0	0
$652$	−4.32927	−0.169547
$653$	−3.97026	−0.155368	−0.0776841	−	0.996978i	$-0.524753\pi$
$653$	−3.97026	−0.155368	−0.0776841	+	0.996978i	$0.524753\pi$
$654$	0	0
$655$	30.2422	1.18166
$656$	−10.9478	−0.427441
$657$	0	0
$658$	−0.222687	−0.00868125
$659$	−1.31317	−0.0511539	−0.0255770	−	0.999673i	$-0.508142\pi$
$659$	−1.31317	−0.0511539	−0.0255770	+	0.999673i	$0.508142\pi$
$660$	0	0
$661$	−27.5377	−1.07109	−0.535546	−	0.844506i	$-0.679894\pi$
$661$	−27.5377	−1.07109	−0.535546	+	0.844506i	$0.679894\pi$
$662$	2.69365	0.104692
$663$	0	0
$664$	21.5828	0.837574
$665$	−0.958279	−0.0371604
$666$	0	0
$667$	−0.594327	−0.0230125
$668$	34.7266	1.34361
$669$	0	0
$670$	3.79136	0.146473
$671$	−1.00000	−0.0386046
$672$	0	0
$673$	−19.8310	−0.764431	−0.382215	−	0.924073i	$-0.624839\pi$
$673$	−19.8310	−0.764431	−0.382215	+	0.924073i	$0.624839\pi$
$674$	12.5749	0.484365
$675$	0	0
$676$	16.5547	0.636720
$677$	12.1366	0.466446	0.233223	−	0.972423i	$-0.425073\pi$
$677$	12.1366	0.466446	0.233223	+	0.972423i	$0.425073\pi$
$678$	0	0
$679$	0.0150919	0.000579175 0
$680$	−34.3025	−1.31544
$681$	0	0
$682$	1.16219	0.0445026
$683$	14.9118	0.570583	0.285291	−	0.958441i	$-0.407910\pi$
$683$	14.9118	0.570583	0.285291	+	0.958441i	$0.407910\pi$
$684$	0	0
$685$	−49.9989	−1.91036
$686$	9.72414	0.371269
$687$	0	0
$688$	10.8585	0.413975
$689$	13.6079	0.518418
$690$	0	0
$691$	−0.934230	−0.0355398	−0.0177699	−	0.999842i	$-0.505657\pi$
$691$	−0.934230	−0.0355398	−0.0177699	+	0.999842i	$0.505657\pi$
$692$	−15.1257	−0.574993
$693$	0	0
$694$	−18.3338	−0.695940
$695$	−31.5259	−1.19585
$696$	0	0
$697$	−37.3347	−1.41415
$698$	10.7162	0.405616
$699$	0	0
$700$	7.17509	0.271193
$701$	22.6242	0.854504	0.427252	−	0.904133i	$-0.359482\pi$
$701$	22.6242	0.854504	0.427252	+	0.904133i	$0.359482\pi$
$702$	0	0
$703$	−0.644572	−0.0243105
$704$	−0.196960	−0.00742321
$705$	0	0
$706$	−15.8637	−0.597040
$707$	57.5739	2.16529
$708$	0	0
$709$	2.42550	0.0910914	0.0455457	−	0.998962i	$-0.485497\pi$
$709$	2.42550	0.0910914	0.0455457	+	0.998962i	$0.485497\pi$
$710$	6.57713	0.246835
$711$	0	0
$712$	11.4269	0.428240
$713$	−5.63154	−0.210903
$714$	0	0
$715$	4.11498	0.153892
$716$	−18.6080	−0.695414
$717$	0	0
$718$	6.16926	0.230235
$719$	−23.6599	−0.882367	−0.441184	−	0.897417i	$-0.645441\pi$
$719$	−23.6599	−0.882367	−0.441184	+	0.897417i	$0.645441\pi$
$720$	0	0
$721$	−6.37545	−0.237434
$722$	−12.1276	−0.451344
$723$	0	0
$724$	8.12106	0.301817
$725$	−0.290128	−0.0107751
$726$	0	0
$727$	13.1377	0.487252	0.243626	−	0.969869i	$-0.421663\pi$
$727$	13.1377	0.487252	0.243626	+	0.969869i	$0.421663\pi$
$728$	−11.0373	−0.409070
$729$	0	0
$730$	21.0478	0.779012
$731$	37.0299	1.36960
$732$	0	0
$733$	−11.9036	−0.439670	−0.219835	−	0.975537i	$-0.570552\pi$
$733$	−11.9036	−0.439670	−0.219835	+	0.975537i	$0.570552\pi$
$734$	−6.74869	−0.249099
$735$	0	0
$736$	17.6033	0.648866
$737$	−2.32585	−0.0856738
$738$	0	0
$739$	40.8485	1.50264	0.751319	−	0.659939i	$-0.229418\pi$
$739$	40.8485	1.50264	0.751319	+	0.659939i	$0.229418\pi$
$740$	20.7953	0.764449
$741$	0	0
$742$	16.0786	0.590265
$743$	−26.4863	−0.971687	−0.485843	−	0.874046i	$-0.661487\pi$
$743$	−26.4863	−0.971687	−0.485843	+	0.874046i	$0.661487\pi$
$744$	0	0
$745$	29.4852	1.08025
$746$	−14.4477	−0.528968
$747$	0	0
$748$	9.32620	0.341000
$749$	29.8899	1.09215
$750$	0	0
$751$	−26.4040	−0.963496	−0.481748	−	0.876310i	$-0.659998\pi$
$751$	−26.4040	−0.963496	−0.481748	+	0.876310i	$0.659998\pi$
$752$	0.200771	0.00732137
$753$	0	0
$754$	0.197795	0.00720328
$755$	−33.7529	−1.22839
$756$	0	0
$757$	−1.21645	−0.0442125	−0.0221063	−	0.999756i	$-0.507037\pi$
$757$	−1.21645	−0.0442125	−0.0221063	+	0.999756i	$0.507037\pi$
$758$	−19.8620	−0.721422
$759$	0	0
$760$	−0.737202	−0.0267411
$761$	10.3188	0.374057	0.187029	−	0.982354i	$-0.440114\pi$
$761$	10.3188	0.374057	0.187029	+	0.982354i	$0.440114\pi$
$762$	0	0
$763$	43.2518	1.56582
$764$	−3.16441	−0.114484
$765$	0	0
$766$	−0.170468	−0.00615926
$767$	−0.318851	−0.0115130
$768$	0	0
$769$	36.5131	1.31670	0.658348	−	0.752714i	$-0.271256\pi$
$769$	36.5131	1.31670	0.658348	+	0.752714i	$0.271256\pi$
$770$	4.86213	0.175219
$771$	0	0
$772$	11.8671	0.427106
$773$	10.2672	0.369285	0.184643	−	0.982806i	$-0.440887\pi$
$773$	10.2672	0.369285	0.184643	+	0.982806i	$0.440887\pi$
$774$	0	0
$775$	−2.74910	−0.0987506
$776$	0.0116102	0.000416781 0
$777$	0	0
$778$	−10.5836	−0.379440
$779$	−0.802366	−0.0287477
$780$	0	0
$781$	−4.03481	−0.144377
$782$	11.5853	0.414291
$783$	0	0
$784$	3.25838	0.116371
$785$	−43.4507	−1.55082
$786$	0	0
$787$	35.9424	1.28121	0.640603	−	0.767872i	$-0.278684\pi$
$787$	35.9424	1.28121	0.640603	+	0.767872i	$0.278684\pi$
$788$	−15.3370	−0.546360
$789$	0	0
$790$	3.61469	0.128605
$791$	57.2748	2.03646
$792$	0	0
$793$	−1.61265	−0.0572669
$794$	7.76636	0.275618
$795$	0	0
$796$	−17.8454	−0.632512
$797$	−8.26717	−0.292838	−0.146419	−	0.989223i	$-0.546775\pi$
$797$	−8.26717	−0.292838	−0.146419	+	0.989223i	$0.546775\pi$
$798$	0	0
$799$	0.684677	0.0242221
$800$	8.59324	0.303817
$801$	0	0
$802$	6.05153	0.213687
$803$	−12.9120	−0.455654
$804$	0	0
$805$	−23.5601	−0.830383
$806$	1.87421	0.0660161
$807$	0	0
$808$	44.2914	1.55817
$809$	25.8669	0.909431	0.454716	−	0.890637i	$-0.349741\pi$
$809$	25.8669	0.909431	0.454716	+	0.890637i	$0.349741\pi$
$810$	0	0
$811$	−5.58015	−0.195945	−0.0979727	−	0.995189i	$-0.531236\pi$
$811$	−5.58015	−0.195945	−0.0979727	+	0.995189i	$0.531236\pi$
$812$	−0.911634	−0.0319921
$813$	0	0
$814$	3.27044	0.114629
$815$	6.93949	0.243080
$816$	0	0
$817$	0.795816	0.0278421
$818$	−1.86241	−0.0651176
$819$	0	0
$820$	25.8860	0.903979
$821$	31.3779	1.09509	0.547547	−	0.836775i	$-0.315562\pi$
$821$	31.3779	1.09509	0.547547	+	0.836775i	$0.315562\pi$
$822$	0	0
$823$	47.2915	1.64848	0.824239	−	0.566242i	$-0.191603\pi$
$823$	47.2915	1.64848	0.824239	+	0.566242i	$0.191603\pi$
$824$	−4.90462	−0.170861
$825$	0	0
$826$	−0.376744	−0.0131086
$827$	−10.2754	−0.357312	−0.178656	−	0.983912i	$-0.557175\pi$
$827$	−10.2754	−0.357312	−0.178656	+	0.983912i	$0.557175\pi$
$828$	0	0
$829$	5.77954	0.200732	0.100366	−	0.994951i	$-0.467999\pi$
$829$	5.77954	0.200732	0.100366	+	0.994951i	$0.467999\pi$
$830$	−15.3324	−0.532196
$831$	0	0
$832$	−0.317627	−0.0110117
$833$	11.1118	0.385002
$834$	0	0
$835$	−55.6640	−1.92633
$836$	0.200431	0.00693205
$837$	0	0
$838$	−21.1335	−0.730045
$839$	9.04271	0.312189	0.156095	−	0.987742i	$-0.450110\pi$
$839$	9.04271	0.312189	0.156095	+	0.987742i	$0.450110\pi$
$840$	0	0
$841$	−28.9631	−0.998729
$842$	−12.0709	−0.415989
$843$	0	0
$844$	16.7800	0.577593
$845$	−26.5359	−0.912864
$846$	0	0
$847$	−2.98273	−0.102488
$848$	−14.4962	−0.497803
$849$	0	0
$850$	5.65552	0.193983
$851$	−15.8473	−0.543239
$852$	0	0
$853$	3.34760	0.114620	0.0573098	−	0.998356i	$-0.481748\pi$
$853$	3.34760	0.114620	0.0573098	+	0.998356i	$0.481748\pi$
$854$	−1.90546	−0.0652034
$855$	0	0
$856$	22.9942	0.785927
$857$	−5.38469	−0.183937	−0.0919687	−	0.995762i	$-0.529316\pi$
$857$	−5.38469	−0.183937	−0.0919687	+	0.995762i	$0.529316\pi$
$858$	0	0
$859$	18.7192	0.638689	0.319345	−	0.947639i	$-0.396537\pi$
$859$	18.7192	0.638689	0.319345	+	0.947639i	$0.396537\pi$
$860$	−25.6747	−0.875500
$861$	0	0
$862$	12.9340	0.440533
$863$	54.0048	1.83834	0.919172	−	0.393855i	$-0.128859\pi$
$863$	54.0048	1.83834	0.919172	+	0.393855i	$0.128859\pi$
$864$	0	0
$865$	24.2453	0.824366
$866$	−7.67380	−0.260766
$867$	0	0
$868$	−8.63817	−0.293199
$869$	−2.21747	−0.0752227
$870$	0	0
$871$	−3.75078	−0.127090
$872$	33.2735	1.12678
$873$	0	0
$874$	0.248983	0.00842197
$875$	26.5539	0.897686
$876$	0	0
$877$	14.3534	0.484678	0.242339	−	0.970192i	$-0.422085\pi$
$877$	14.3534	0.484678	0.242339	+	0.970192i	$0.422085\pi$
$878$	17.9606	0.606141
$879$	0	0
$880$	−4.38362	−0.147772
$881$	−21.9866	−0.740749	−0.370374	−	0.928883i	$-0.620771\pi$
$881$	−21.9866	−0.740749	−0.370374	+	0.928883i	$0.620771\pi$
$882$	0	0
$883$	−1.53489	−0.0516533	−0.0258266	−	0.999666i	$-0.508222\pi$
$883$	−1.53489	−0.0516533	−0.0258266	+	0.999666i	$0.508222\pi$
$884$	15.0399	0.505846
$885$	0	0
$886$	3.28318	0.110301
$887$	2.01163	0.0675439	0.0337720	−	0.999430i	$-0.489248\pi$
$887$	2.01163	0.0675439	0.0337720	+	0.999430i	$0.489248\pi$
$888$	0	0
$889$	−18.7178	−0.627774
$890$	−8.11766	−0.272104
$891$	0	0
$892$	−4.37889	−0.146616
$893$	0.0147145	0.000492402 0
$894$	0	0
$895$	29.8272	0.997013
$896$	33.5484	1.12077
$897$	0	0
$898$	−2.85984	−0.0954342
$899$	0.349288	0.0116494
$900$	0	0
$901$	−49.4355	−1.64694
$902$	4.07106	0.135552
$903$	0	0
$904$	44.0614	1.46546
$905$	−13.0174	−0.432714
$906$	0	0
$907$	−19.3921	−0.643905	−0.321953	−	0.946756i	$-0.604339\pi$
$907$	−19.3921	−0.643905	−0.321953	+	0.946756i	$0.604339\pi$
$908$	19.8709	0.659441
$909$	0	0
$910$	7.84092	0.259924
$911$	−39.0218	−1.29285	−0.646425	−	0.762977i	$-0.723737\pi$
$911$	−39.0218	−1.29285	−0.646425	+	0.762977i	$0.723737\pi$
$912$	0	0
$913$	9.40586	0.311288
$914$	0.726599	0.0240338
$915$	0	0
$916$	−14.2186	−0.469797
$917$	−35.3508	−1.16739
$918$	0	0
$919$	−0.750396	−0.0247533	−0.0123766	−	0.999923i	$-0.503940\pi$
$919$	−0.750396	−0.0247533	−0.0123766	+	0.999923i	$0.503940\pi$
$920$	−18.1247	−0.597554
$921$	0	0
$922$	−9.00438	−0.296544
$923$	−6.50674	−0.214172
$924$	0	0
$925$	−7.73606	−0.254360
$926$	12.6088	0.414350
$927$	0	0
$928$	−1.09182	−0.0358407
$929$	−1.45809	−0.0478385	−0.0239193	−	0.999714i	$-0.507614\pi$
$929$	−1.45809	−0.0478385	−0.0239193	+	0.999714i	$0.507614\pi$
$930$	0	0
$931$	0.238806	0.00782656
$932$	25.5989	0.838521
$933$	0	0
$934$	19.0054	0.621876
$935$	−14.9492	−0.488890
$936$	0	0
$937$	−15.7074	−0.513139	−0.256569	−	0.966526i	$-0.582592\pi$
$937$	−15.7074	−0.513139	−0.256569	+	0.966526i	$0.582592\pi$
$938$	−4.43181	−0.144704
$939$	0	0
$940$	−0.474721	−0.0154837
$941$	−29.3944	−0.958230	−0.479115	−	0.877752i	$-0.659042\pi$
$941$	−29.3944	−0.958230	−0.479115	+	0.877752i	$0.659042\pi$
$942$	0	0
$943$	−19.7268	−0.642394
$944$	0.339666	0.0110552
$945$	0	0
$946$	−4.03783	−0.131281
$947$	−51.5545	−1.67530	−0.837648	−	0.546211i	$-0.816070\pi$
$947$	−51.5545	−1.67530	−0.837648	+	0.546211i	$0.816070\pi$
$948$	0	0
$949$	−20.8225	−0.675927
$950$	0.121544	0.00394340
$951$	0	0
$952$	40.0971	1.29955
$953$	32.4172	1.05010	0.525048	−	0.851073i	$-0.324048\pi$
$953$	32.4172	1.05010	0.525048	+	0.851073i	$0.324048\pi$
$954$	0	0
$955$	5.07231	0.164136
$956$	−9.87343	−0.319330
$957$	0	0
$958$	−12.4324	−0.401672
$959$	58.4450	1.88729
$960$	0	0
$961$	−27.6903	−0.893236
$962$	5.27408	0.170043
$963$	0	0
$964$	−2.65700	−0.0855763
$965$	−19.0220	−0.612341
$966$	0	0
$967$	10.2547	0.329769	0.164884	−	0.986313i	$-0.447275\pi$
$967$	10.2547	0.329769	0.164884	+	0.986313i	$0.447275\pi$
$968$	−2.29461	−0.0737515
$969$	0	0
$970$	−0.00824789	−0.000264824 0
$971$	−42.5331	−1.36495	−0.682475	−	0.730909i	$-0.739097\pi$
$971$	−42.5331	−1.36495	−0.682475	+	0.730909i	$0.739097\pi$
$972$	0	0
$973$	36.8514	1.18140
$974$	−25.9068	−0.830109
$975$	0	0
$976$	1.71793	0.0549896
$977$	50.5368	1.61682	0.808408	−	0.588622i	$-0.200329\pi$
$977$	50.5368	1.61682	0.808408	+	0.588622i	$0.200329\pi$
$978$	0	0
$979$	4.97987	0.159157
$980$	−7.70440	−0.246108
$981$	0	0
$982$	13.9279	0.444458
$983$	5.12871	0.163581	0.0817903	−	0.996650i	$-0.473936\pi$
$983$	5.12871	0.163581	0.0817903	+	0.996650i	$0.473936\pi$
$984$	0	0
$985$	24.5841	0.783315
$986$	−0.718564	−0.0228838
$987$	0	0
$988$	0.323225	0.0102832
$989$	19.5658	0.622156
$990$	0	0
$991$	33.3767	1.06024	0.530122	−	0.847921i	$-0.322146\pi$
$991$	33.3767	1.06024	0.530122	+	0.847921i	$0.322146\pi$
$992$	−10.3455	−0.328470
$993$	0	0
$994$	−7.68816	−0.243854
$995$	28.6048	0.906832
$996$	0	0
$997$	51.8693	1.64272	0.821358	−	0.570413i	$-0.193217\pi$
$997$	51.8693	1.64272	0.821358	+	0.570413i	$0.193217\pi$
$998$	23.0237	0.728801
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6039.2.a.h.1.9		13
3.2	odd	2		2013.2.a.g.1.5	✓	13

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2013.2.a.g.1.5	✓	13	3.2	odd	2
6039.2.a.h.1.9		13	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)

\(f(q)\)	\(=\)	\(q+0.638829 q^{2} -1.59190 q^{4} +2.55169 q^{5} -2.98273 q^{7} -2.29461 q^{8} +O(q^{10})\)	\(q+0.638829 q^{2} -1.59190 q^{4} +2.55169 q^{5} -2.98273 q^{7} -2.29461 q^{8} +1.63009 q^{10} -1.00000 q^{11} -1.61265 q^{13} -1.90546 q^{14} +1.71793 q^{16} +5.85854 q^{17} +0.125907 q^{19} -4.06203 q^{20} -0.638829 q^{22} +3.09553 q^{23} +1.51112 q^{25} -1.03021 q^{26} +4.74820 q^{28} -0.191996 q^{29} -1.81925 q^{31} +5.68668 q^{32} +3.74261 q^{34} -7.61100 q^{35} -5.11943 q^{37} +0.0804331 q^{38} -5.85513 q^{40} -6.37269 q^{41} +6.32066 q^{43} +1.59190 q^{44} +1.97751 q^{46} +0.116868 q^{47} +1.89669 q^{49} +0.965346 q^{50} +2.56717 q^{52} -8.43820 q^{53} -2.55169 q^{55} +6.84420 q^{56} -0.122652 q^{58} +0.197718 q^{59} +1.00000 q^{61} -1.16219 q^{62} +0.196960 q^{64} -4.11498 q^{65} +2.32585 q^{67} -9.32620 q^{68} -4.86213 q^{70} +4.03481 q^{71} +12.9120 q^{73} -3.27044 q^{74} -0.200431 q^{76} +2.98273 q^{77} +2.21747 q^{79} +4.38362 q^{80} -4.07106 q^{82} -9.40586 q^{83} +14.9492 q^{85} +4.03783 q^{86} +2.29461 q^{88} -4.97987 q^{89} +4.81010 q^{91} -4.92776 q^{92} +0.0746588 q^{94} +0.321275 q^{95} -0.00505976 q^{97} +1.21166 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 13 q - 4 q^{2} + 12 q^{4} - 7 q^{5} + 7 q^{7} - 9 q^{8}+O(q^{10}) \) 13 * q - 4 * q^2 + 12 * q^4 - 7 * q^5 + 7 * q^7 - 9 * q^8	\( 13 q - 4 q^{2} + 12 q^{4} - 7 q^{5} + 7 q^{7} - 9 q^{8} + 2 q^{10} - 13 q^{11} + 9 q^{13} - 7 q^{14} + 2 q^{16} - 19 q^{17} + 14 q^{19} - 19 q^{20} + 4 q^{22} - 5 q^{23} + 2 q^{25} + 4 q^{26} + 7 q^{28} - 10 q^{29} - q^{31} - 7 q^{32} - 2 q^{34} - 16 q^{35} - 8 q^{37} + 10 q^{38} + 14 q^{40} - 21 q^{41} + 11 q^{43} - 12 q^{44} - 8 q^{46} - 22 q^{47} - 19 q^{50} - q^{52} - 16 q^{53} + 7 q^{55} - 13 q^{58} - 19 q^{59} + 13 q^{61} - 3 q^{62} - 13 q^{64} - 13 q^{65} + 12 q^{67} - 36 q^{68} - 20 q^{70} - 5 q^{71} + 18 q^{73} - 6 q^{74} - 5 q^{76} - 7 q^{77} - q^{79} - 6 q^{80} - 22 q^{82} - 48 q^{83} - 2 q^{85} - 26 q^{86} + 9 q^{88} - 15 q^{89} - 11 q^{91} + 24 q^{92} - 23 q^{94} - 17 q^{95} - 17 q^{97} + 15 q^{98}+O(q^{100}) \) 13 * q - 4 * q^2 + 12 * q^4 - 7 * q^5 + 7 * q^7 - 9 * q^8 + 2 * q^10 - 13 * q^11 + 9 * q^13 - 7 * q^14 + 2 * q^16 - 19 * q^17 + 14 * q^19 - 19 * q^20 + 4 * q^22 - 5 * q^23 + 2 * q^25 + 4 * q^26 + 7 * q^28 - 10 * q^29 - q^31 - 7 * q^32 - 2 * q^34 - 16 * q^35 - 8 * q^37 + 10 * q^38 + 14 * q^40 - 21 * q^41 + 11 * q^43 - 12 * q^44 - 8 * q^46 - 22 * q^47 - 19 * q^50 - q^52 - 16 * q^53 + 7 * q^55 - 13 * q^58 - 19 * q^59 + 13 * q^61 - 3 * q^62 - 13 * q^64 - 13 * q^65 + 12 * q^67 - 36 * q^68 - 20 * q^70 - 5 * q^71 + 18 * q^73 - 6 * q^74 - 5 * q^76 - 7 * q^77 - q^79 - 6 * q^80 - 22 * q^82 - 48 * q^83 - 2 * q^85 - 26 * q^86 + 9 * q^88 - 15 * q^89 - 11 * q^91 + 24 * q^92 - 23 * q^94 - 17 * q^95 - 17 * q^97 + 15 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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