LMFDB - Embedded newform 1232.4.a.bb.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1232,4,Mod(1,1232)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1232, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("1232.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1232 = 2^{4} \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1232.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:	$72.6903531271$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} - 47x^{4} + 10x^{3} + 612x^{2} + 240x - 1440 $ x^6 - x^5 - 47x^4 + 10x^3 + 612x^2 + 240x - 1440
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{6}\cdot 5 $
Twist minimal:	no (minimal twist has level 616)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$-2.26862$ of defining polynomial
Character	$\chi$	$=$	1232.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+8.70461 q^{3} -11.2312 q^{5} -7.00000 q^{7} +48.7703 q^{9} +11.0000 q^{11} -70.7822 q^{13} -97.7636 q^{15} -22.6075 q^{17} +150.276 q^{19} -60.9323 q^{21} +39.7508 q^{23} +1.14064 q^{25} +189.502 q^{27} -171.699 q^{29} +48.8411 q^{31} +95.7508 q^{33} +78.6186 q^{35} -253.496 q^{37} -616.132 q^{39} -208.433 q^{41} -346.157 q^{43} -547.751 q^{45} -42.2758 q^{47} +49.0000 q^{49} -196.789 q^{51} -300.206 q^{53} -123.544 q^{55} +1308.10 q^{57} +412.500 q^{59} +781.931 q^{61} -341.392 q^{63} +794.972 q^{65} -359.574 q^{67} +346.016 q^{69} -665.468 q^{71} -1146.92 q^{73} +9.92879 q^{75} -77.0000 q^{77} -1145.57 q^{79} +332.745 q^{81} -606.927 q^{83} +253.910 q^{85} -1494.58 q^{87} -177.318 q^{89} +495.476 q^{91} +425.143 q^{93} -1687.79 q^{95} -463.156 q^{97} +536.473 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 2 q^{5} - 42 q^{7} + 60 q^{9} + 66 q^{11} - 6 q^{13} - 126 q^{15} - 14 q^{17} - 80 q^{19} - 254 q^{23} + 220 q^{25} - 90 q^{27} + 132 q^{29} + 52 q^{31} - 14 q^{35} - 518 q^{37} - 332 q^{39} + 486 q^{41}+ \cdots + 660 q^{99}+O(q^{100}) $ 6 * q + 2 * q^5 - 42 * q^7 + 60 * q^9 + 66 * q^11 - 6 * q^13 - 126 * q^15 - 14 * q^17 - 80 * q^19 - 254 * q^23 + 220 * q^25 - 90 * q^27 + 132 * q^29 + 52 * q^31 - 14 * q^35 - 518 * q^37 - 332 * q^39 + 486 * q^41 - 428 * q^43 - 244 * q^45 - 790 * q^47 + 294 * q^49 - 640 * q^51 - 40 * q^53 + 22 * q^55 + 2276 * q^57 - 436 * q^59 + 1034 * q^61 - 420 * q^63 + 800 * q^65 - 562 * q^67 + 530 * q^69 - 2474 * q^71 - 902 * q^73 - 1986 * q^75 - 462 * q^77 - 1636 * q^79 + 570 * q^81 - 3016 * q^83 + 236 * q^85 - 4340 * q^87 + 1750 * q^89 + 42 * q^91 + 150 * q^93 - 3628 * q^95 - 1250 * q^97 + 660 * q^99

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	0
$2$	0	0
$3$	8.70461	1.67520	0.837602	−	0.546281i	$-0.183957\pi$
$3$	8.70461	1.67520	0.837602	+	0.546281i	$0.183957\pi$
$4$	0	0
$5$	−11.2312	−1.00455	−0.502276	−	0.864707i	$-0.667504\pi$
$5$	−11.2312	−1.00455	−0.502276	+	0.864707i	$0.667504\pi$
$6$	0	0
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	0	0
$9$	48.7703	1.80631
$10$	0	0
$11$	11.0000	0.301511
$11$	11.0000	0.301511
$12$	0	0
$13$	−70.7822	−1.51011	−0.755056	−	0.655660i	$-0.772390\pi$
$13$	−70.7822	−1.51011	−0.755056	+	0.655660i	$0.772390\pi$
$14$	0	0
$15$	−97.7636	−1.68283
$16$	0	0
$17$	−22.6075	−0.322536	−0.161268	−	0.986911i	$-0.551558\pi$
$17$	−22.6075	−0.322536	−0.161268	+	0.986911i	$0.551558\pi$
$18$	0	0
$19$	150.276	1.81451	0.907256	−	0.420578i	$-0.138173\pi$
$19$	150.276	1.81451	0.907256	+	0.420578i	$0.138173\pi$
$20$	0	0
$21$	−60.9323	−0.633168
$22$	0	0
$23$	39.7508	0.360375	0.180187	−	0.983632i	$-0.442330\pi$
$23$	39.7508	0.360375	0.180187	+	0.983632i	$0.442330\pi$
$24$	0	0
$25$	1.14064	0.00912508
$26$	0	0
$27$	189.502	1.35073
$28$	0	0
$29$	−171.699	−1.09944	−0.549721	−	0.835349i	$-0.685266\pi$
$29$	−171.699	−1.09944	−0.549721	+	0.835349i	$0.685266\pi$
$30$	0	0
$31$	48.8411	0.282972	0.141486	−	0.989940i	$-0.454812\pi$
$31$	48.8411	0.282972	0.141486	+	0.989940i	$0.454812\pi$
$32$	0	0
$33$	95.7508	0.505093
$34$	0	0
$35$	78.6186	0.379685
$36$	0	0
$37$	−253.496	−1.12634	−0.563169	−	0.826342i	$-0.690418\pi$
$37$	−253.496	−1.12634	−0.563169	+	0.826342i	$0.690418\pi$
$38$	0	0
$39$	−616.132	−2.52975
$40$	0	0
$41$	−208.433	−0.793946	−0.396973	−	0.917830i	$-0.629939\pi$
$41$	−208.433	−0.793946	−0.396973	+	0.917830i	$0.629939\pi$
$42$	0	0
$43$	−346.157	−1.22764	−0.613819	−	0.789447i	$-0.710367\pi$
$43$	−346.157	−1.22764	−0.613819	+	0.789447i	$0.710367\pi$
$44$	0	0
$45$	−547.751	−1.81453
$46$	0	0
$47$	−42.2758	−0.131203	−0.0656017	−	0.997846i	$-0.520897\pi$
$47$	−42.2758	−0.131203	−0.0656017	+	0.997846i	$0.520897\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	0	0
$51$	−196.789	−0.540314
$52$	0	0
$53$	−300.206	−0.778047	−0.389023	−	0.921228i	$-0.627187\pi$
$53$	−300.206	−0.778047	−0.389023	+	0.921228i	$0.627187\pi$
$54$	0	0
$55$	−123.544	−0.302884
$56$	0	0
$57$	1308.10	3.03968
$58$	0	0
$59$	412.500	0.910220	0.455110	−	0.890435i	$-0.349600\pi$
$59$	412.500	0.910220	0.455110	+	0.890435i	$0.349600\pi$
$60$	0	0
$61$	781.931	1.64125	0.820623	−	0.571470i	$-0.193627\pi$
$61$	781.931	1.64125	0.820623	+	0.571470i	$0.193627\pi$
$62$	0	0
$63$	−341.392	−0.682720
$64$	0	0
$65$	794.972	1.51699
$66$	0	0
$67$	−359.574	−0.655656	−0.327828	−	0.944737i	$-0.606317\pi$
$67$	−359.574	−0.655656	−0.327828	+	0.944737i	$0.606317\pi$
$68$	0	0
$69$	346.016	0.603701
$70$	0	0
$71$	−665.468	−1.11235	−0.556173	−	0.831067i	$-0.687731\pi$
$71$	−665.468	−1.11235	−0.556173	+	0.831067i	$0.687731\pi$
$72$	0	0
$73$	−1146.92	−1.83885	−0.919426	−	0.393262i	$-0.871347\pi$
$73$	−1146.92	−1.83885	−0.919426	+	0.393262i	$0.871347\pi$
$74$	0	0
$75$	9.92879	0.0152864
$76$	0	0
$77$	−77.0000	−0.113961
$78$	0	0
$79$	−1145.57	−1.63148	−0.815741	−	0.578417i	$-0.803671\pi$
$79$	−1145.57	−1.63148	−0.815741	+	0.578417i	$0.803671\pi$
$80$	0	0
$81$	332.745	0.456440
$82$	0	0
$83$	−606.927	−0.802638	−0.401319	−	0.915938i	$-0.631448\pi$
$83$	−606.927	−0.802638	−0.401319	+	0.915938i	$0.631448\pi$
$84$	0	0
$85$	253.910	0.324004
$86$	0	0
$87$	−1494.58	−1.84179
$88$	0	0
$89$	−177.318	−0.211187	−0.105594	−	0.994409i	$-0.533674\pi$
$89$	−177.318	−0.211187	−0.105594	+	0.994409i	$0.533674\pi$
$90$	0	0
$91$	495.476	0.570769
$92$	0	0
$93$	425.143	0.474036
$94$	0	0
$95$	−1687.79	−1.82277
$96$	0	0
$97$	−463.156	−0.484808	−0.242404	−	0.970175i	$-0.577936\pi$
$97$	−463.156	−0.484808	−0.242404	+	0.970175i	$0.577936\pi$
$98$	0	0
$99$	536.473	0.544622
$100$	0	0
$101$	−464.045	−0.457171	−0.228585	−	0.973524i	$-0.573410\pi$
$101$	−464.045	−0.457171	−0.228585	+	0.973524i	$0.573410\pi$
$102$	0	0
$103$	304.396	0.291195	0.145597	−	0.989344i	$-0.453490\pi$
$103$	304.396	0.291195	0.145597	+	0.989344i	$0.453490\pi$
$104$	0	0
$105$	684.345	0.636050
$106$	0	0
$107$	833.013	0.752621	0.376310	−	0.926494i	$-0.377193\pi$
$107$	833.013	0.752621	0.376310	+	0.926494i	$0.377193\pi$
$108$	0	0
$109$	−1496.01	−1.31460	−0.657302	−	0.753627i	$-0.728302\pi$
$109$	−1496.01	−1.31460	−0.657302	+	0.753627i	$0.728302\pi$
$110$	0	0
$111$	−2206.58	−1.88684
$112$	0	0
$113$	247.726	0.206231	0.103115	−	0.994669i	$-0.467119\pi$
$113$	247.726	0.206231	0.103115	+	0.994669i	$0.467119\pi$
$114$	0	0
$115$	−446.451	−0.362015
$116$	0	0
$117$	−3452.07	−2.72773
$118$	0	0
$119$	158.252	0.121907
$120$	0	0
$121$	121.000	0.0909091
$122$	0	0
$123$	−1814.33	−1.33002
$124$	0	0
$125$	1391.09	0.995386
$126$	0	0
$127$	1374.43	0.960322	0.480161	−	0.877180i	$-0.340578\pi$
$127$	1374.43	0.960322	0.480161	+	0.877180i	$0.340578\pi$
$128$	0	0
$129$	−3013.16	−2.05654
$130$	0	0
$131$	−1771.68	−1.18162	−0.590809	−	0.806811i	$-0.701191\pi$
$131$	−1771.68	−1.18162	−0.590809	+	0.806811i	$0.701191\pi$
$132$	0	0
$133$	−1051.93	−0.685821
$134$	0	0
$135$	−2128.34	−1.35688
$136$	0	0
$137$	506.645	0.315953	0.157977	−	0.987443i	$-0.449503\pi$
$137$	506.645	0.315953	0.157977	+	0.987443i	$0.449503\pi$
$138$	0	0
$139$	−1929.96	−1.17768	−0.588839	−	0.808251i	$-0.700415\pi$
$139$	−1929.96	−1.17768	−0.588839	+	0.808251i	$0.700415\pi$
$140$	0	0
$141$	−367.994	−0.219792
$142$	0	0
$143$	−778.605	−0.455316
$144$	0	0
$145$	1928.40	1.10445
$146$	0	0
$147$	426.526	0.239315
$148$	0	0
$149$	3393.29	1.86570	0.932850	−	0.360266i	$-0.117314\pi$
$149$	3393.29	1.86570	0.932850	+	0.360266i	$0.117314\pi$
$150$	0	0
$151$	1574.74	0.848679	0.424339	−	0.905503i	$-0.360506\pi$
$151$	1574.74	0.848679	0.424339	+	0.905503i	$0.360506\pi$
$152$	0	0
$153$	−1102.57	−0.582600
$154$	0	0
$155$	−548.546	−0.284260
$156$	0	0
$157$	2310.54	1.17453	0.587265	−	0.809394i	$-0.300204\pi$
$157$	2310.54	1.17453	0.587265	+	0.809394i	$0.300204\pi$
$158$	0	0
$159$	−2613.18	−1.30339
$160$	0	0
$161$	−278.256	−0.136209
$162$	0	0
$163$	1658.35	0.796883	0.398442	−	0.917194i	$-0.369551\pi$
$163$	1658.35	0.796883	0.398442	+	0.917194i	$0.369551\pi$
$164$	0	0
$165$	−1075.40	−0.507392
$166$	0	0
$167$	−831.741	−0.385402	−0.192701	−	0.981258i	$-0.561725\pi$
$167$	−831.741	−0.385402	−0.192701	+	0.981258i	$0.561725\pi$
$168$	0	0
$169$	2813.13	1.28044
$170$	0	0
$171$	7329.02	3.27757
$172$	0	0
$173$	−1573.66	−0.691578	−0.345789	−	0.938312i	$-0.612389\pi$
$173$	−1573.66	−0.691578	−0.345789	+	0.938312i	$0.612389\pi$
$174$	0	0
$175$	−7.98445	−0.00344896
$176$	0	0
$177$	3590.66	1.52480
$178$	0	0
$179$	−1101.85	−0.460088	−0.230044	−	0.973180i	$-0.573887\pi$
$179$	−1101.85	−0.460088	−0.230044	+	0.973180i	$0.573887\pi$
$180$	0	0
$181$	−3036.38	−1.24692	−0.623459	−	0.781856i	$-0.714273\pi$
$181$	−3036.38	−1.24692	−0.623459	+	0.781856i	$0.714273\pi$
$182$	0	0
$183$	6806.40	2.74942
$184$	0	0
$185$	2847.07	1.13146
$186$	0	0
$187$	−248.682	−0.0972483
$188$	0	0
$189$	−1326.51	−0.510528
$190$	0	0
$191$	−731.342	−0.277058	−0.138529	−	0.990358i	$-0.544237\pi$
$191$	−731.342	−0.277058	−0.138529	+	0.990358i	$0.544237\pi$
$192$	0	0
$193$	−3142.83	−1.17215	−0.586077	−	0.810256i	$-0.699328\pi$
$193$	−3142.83	−1.17215	−0.586077	+	0.810256i	$0.699328\pi$
$194$	0	0
$195$	6919.92	2.54126
$196$	0	0
$197$	501.362	0.181323	0.0906614	−	0.995882i	$-0.471102\pi$
$197$	501.362	0.181323	0.0906614	+	0.995882i	$0.471102\pi$
$198$	0	0
$199$	4570.12	1.62798	0.813988	−	0.580881i	$-0.197292\pi$
$199$	4570.12	1.62798	0.813988	+	0.580881i	$0.197292\pi$
$200$	0	0
$201$	−3129.95	−1.09836
$202$	0	0
$203$	1201.90	0.415550
$204$	0	0
$205$	2340.96	0.797560
$206$	0	0
$207$	1938.66	0.650948
$208$	0	0
$209$	1653.04	0.547096
$210$	0	0
$211$	4396.54	1.43446	0.717228	−	0.696839i	$-0.245411\pi$
$211$	4396.54	1.43446	0.717228	+	0.696839i	$0.245411\pi$
$212$	0	0
$213$	−5792.64	−1.86340
$214$	0	0
$215$	3887.77	1.23323
$216$	0	0
$217$	−341.888	−0.106953
$218$	0	0
$219$	−9983.45	−3.08045
$220$	0	0
$221$	1600.21	0.487066
$222$	0	0
$223$	1681.25	0.504864	0.252432	−	0.967615i	$-0.418770\pi$
$223$	1681.25	0.504864	0.252432	+	0.967615i	$0.418770\pi$
$224$	0	0
$225$	55.6291	0.0164827
$226$	0	0
$227$	−2563.22	−0.749456	−0.374728	−	0.927135i	$-0.622264\pi$
$227$	−2563.22	−0.749456	−0.374728	+	0.927135i	$0.622264\pi$
$228$	0	0
$229$	2637.97	0.761230	0.380615	−	0.924734i	$-0.375712\pi$
$229$	2637.97	0.761230	0.380615	+	0.924734i	$0.375712\pi$
$230$	0	0
$231$	−670.255	−0.190907
$232$	0	0
$233$	4900.60	1.37789	0.688946	−	0.724812i	$-0.258074\pi$
$233$	4900.60	1.37789	0.688946	+	0.724812i	$0.258074\pi$
$234$	0	0
$235$	474.809	0.131801
$236$	0	0
$237$	−9971.78	−2.73307
$238$	0	0
$239$	−4627.72	−1.25248	−0.626239	−	0.779631i	$-0.715407\pi$
$239$	−4627.72	−1.25248	−0.626239	+	0.779631i	$0.715407\pi$
$240$	0	0
$241$	3550.08	0.948884	0.474442	−	0.880287i	$-0.342650\pi$
$241$	3550.08	0.948884	0.474442	+	0.880287i	$0.342650\pi$
$242$	0	0
$243$	−2220.14	−0.586100
$244$	0	0
$245$	−550.331	−0.143507
$246$	0	0
$247$	−10636.9	−2.74012
$248$	0	0
$249$	−5283.07	−1.34458
$250$	0	0
$251$	−3425.23	−0.861350	−0.430675	−	0.902507i	$-0.641724\pi$
$251$	−3425.23	−0.861350	−0.430675	+	0.902507i	$0.641724\pi$
$252$	0	0
$253$	437.259	0.108657
$254$	0	0
$255$	2210.19	0.542774
$256$	0	0
$257$	−5631.21	−1.36679	−0.683396	−	0.730048i	$-0.739498\pi$
$257$	−5631.21	−1.36679	−0.683396	+	0.730048i	$0.739498\pi$
$258$	0	0
$259$	1774.47	0.425716
$260$	0	0
$261$	−8373.84	−1.98593
$262$	0	0
$263$	6642.14	1.55731	0.778654	−	0.627453i	$-0.215903\pi$
$263$	6642.14	1.55731	0.778654	+	0.627453i	$0.215903\pi$
$264$	0	0
$265$	3371.68	0.781588
$266$	0	0
$267$	−1543.48	−0.353782
$268$	0	0
$269$	7906.39	1.79205	0.896025	−	0.444004i	$-0.146442\pi$
$269$	7906.39	1.79205	0.896025	+	0.444004i	$0.146442\pi$
$270$	0	0
$271$	−2843.85	−0.637461	−0.318730	−	0.947845i	$-0.603256\pi$
$271$	−2843.85	−0.637461	−0.318730	+	0.947845i	$0.603256\pi$
$272$	0	0
$273$	4312.92	0.956154
$274$	0	0
$275$	12.5470	0.00275132
$276$	0	0
$277$	−6339.22	−1.37504	−0.687521	−	0.726164i	$-0.741301\pi$
$277$	−6339.22	−1.37504	−0.687521	+	0.726164i	$0.741301\pi$
$278$	0	0
$279$	2382.00	0.511134
$280$	0	0
$281$	7249.42	1.53902	0.769510	−	0.638635i	$-0.220501\pi$
$281$	7249.42	1.53902	0.769510	+	0.638635i	$0.220501\pi$
$282$	0	0
$283$	4074.64	0.855873	0.427937	−	0.903809i	$-0.359241\pi$
$283$	4074.64	0.855873	0.427937	+	0.903809i	$0.359241\pi$
$284$	0	0
$285$	−14691.5	−3.05352
$286$	0	0
$287$	1459.03	0.300083
$288$	0	0
$289$	−4401.90	−0.895970
$290$	0	0
$291$	−4031.59	−0.812152
$292$	0	0
$293$	−9631.88	−1.92048	−0.960239	−	0.279178i	$-0.909938\pi$
$293$	−9631.88	−1.92048	−0.960239	+	0.279178i	$0.909938\pi$
$294$	0	0
$295$	−4632.89	−0.914363
$296$	0	0
$297$	2084.52	0.407260
$298$	0	0
$299$	−2813.65	−0.544207
$300$	0	0
$301$	2423.10	0.464003
$302$	0	0
$303$	−4039.34	−0.765854
$304$	0	0
$305$	−8782.05	−1.64872
$306$	0	0
$307$	−6205.69	−1.15367	−0.576836	−	0.816860i	$-0.695713\pi$
$307$	−6205.69	−1.15367	−0.576836	+	0.816860i	$0.695713\pi$
$308$	0	0
$309$	2649.65	0.487811
$310$	0	0
$311$	5691.60	1.03775	0.518876	−	0.854849i	$-0.326351\pi$
$311$	5691.60	1.03775	0.518876	+	0.854849i	$0.326351\pi$
$312$	0	0
$313$	2423.14	0.437584	0.218792	−	0.975772i	$-0.429788\pi$
$313$	2423.14	0.437584	0.218792	+	0.975772i	$0.429788\pi$
$314$	0	0
$315$	3834.26	0.685828
$316$	0	0
$317$	5217.54	0.924437	0.462218	−	0.886766i	$-0.347054\pi$
$317$	5217.54	0.924437	0.462218	+	0.886766i	$0.347054\pi$
$318$	0	0
$319$	−1888.69	−0.331494
$320$	0	0
$321$	7251.06	1.26079
$322$	0	0
$323$	−3397.37	−0.585246
$324$	0	0
$325$	−80.7367	−0.0137799
$326$	0	0
$327$	−13022.2	−2.20223
$328$	0	0
$329$	295.930	0.0495902
$330$	0	0
$331$	10929.7	1.81496	0.907480	−	0.420095i	$-0.138003\pi$
$331$	10929.7	1.81496	0.907480	+	0.420095i	$0.138003\pi$
$332$	0	0
$333$	−12363.1	−2.03451
$334$	0	0
$335$	4038.46	0.658641
$336$	0	0
$337$	−690.641	−0.111637	−0.0558184	−	0.998441i	$-0.517777\pi$
$337$	−690.641	−0.111637	−0.0558184	+	0.998441i	$0.517777\pi$
$338$	0	0
$339$	2156.36	0.345479
$340$	0	0
$341$	537.253	0.0853192
$342$	0	0
$343$	−343.000	−0.0539949
$344$	0	0
$345$	−3886.18	−0.606450
$346$	0	0
$347$	2451.12	0.379203	0.189601	−	0.981861i	$-0.439280\pi$
$347$	2451.12	0.379203	0.189601	+	0.981861i	$0.439280\pi$
$348$	0	0
$349$	3773.64	0.578792	0.289396	−	0.957209i	$-0.406546\pi$
$349$	3773.64	0.578792	0.289396	+	0.957209i	$0.406546\pi$
$350$	0	0
$351$	−13413.4	−2.03975
$352$	0	0
$353$	−9417.40	−1.41994	−0.709968	−	0.704234i	$-0.751291\pi$
$353$	−9417.40	−1.41994	−0.709968	+	0.704234i	$0.751291\pi$
$354$	0	0
$355$	7474.03	1.11741
$356$	0	0
$357$	1377.52	0.204219
$358$	0	0
$359$	−3802.27	−0.558986	−0.279493	−	0.960148i	$-0.590166\pi$
$359$	−3802.27	−0.558986	−0.279493	+	0.960148i	$0.590166\pi$
$360$	0	0
$361$	15724.0	2.29246
$362$	0	0
$363$	1053.26	0.152291
$364$	0	0
$365$	12881.3	1.84722
$366$	0	0
$367$	−4463.46	−0.634852	−0.317426	−	0.948283i	$-0.602819\pi$
$367$	−4463.46	−0.634852	−0.317426	+	0.948283i	$0.602819\pi$
$368$	0	0
$369$	−10165.3	−1.43411
$370$	0	0
$371$	2101.44	0.294074
$372$	0	0
$373$	−8157.05	−1.13232	−0.566161	−	0.824295i	$-0.691572\pi$
$373$	−8157.05	−1.13232	−0.566161	+	0.824295i	$0.691572\pi$
$374$	0	0
$375$	12108.9	1.66747
$376$	0	0
$377$	12153.3	1.66028
$378$	0	0
$379$	−2071.65	−0.280775	−0.140387	−	0.990097i	$-0.544835\pi$
$379$	−2071.65	−0.280775	−0.140387	+	0.990097i	$0.544835\pi$
$380$	0	0
$381$	11963.9	1.60874
$382$	0	0
$383$	4114.19	0.548891	0.274445	−	0.961603i	$-0.411506\pi$
$383$	4114.19	0.548891	0.274445	+	0.961603i	$0.411506\pi$
$384$	0	0
$385$	864.805	0.114479
$386$	0	0
$387$	−16882.2	−2.21749
$388$	0	0
$389$	5446.31	0.709869	0.354934	−	0.934891i	$-0.384503\pi$
$389$	5446.31	0.709869	0.354934	+	0.934891i	$0.384503\pi$
$390$	0	0
$391$	−898.666	−0.116234
$392$	0	0
$393$	−15421.8	−1.97945
$394$	0	0
$395$	12866.2	1.63891
$396$	0	0
$397$	−10643.5	−1.34554	−0.672772	−	0.739850i	$-0.734897\pi$
$397$	−10643.5	−1.34554	−0.672772	+	0.739850i	$0.734897\pi$
$398$	0	0
$399$	−9156.68	−1.14889
$400$	0	0
$401$	6260.83	0.779678	0.389839	−	0.920883i	$-0.372531\pi$
$401$	6260.83	0.779678	0.389839	+	0.920883i	$0.372531\pi$
$402$	0	0
$403$	−3457.09	−0.427319
$404$	0	0
$405$	−3737.13	−0.458518
$406$	0	0
$407$	−2788.46	−0.339603
$408$	0	0
$409$	−9348.20	−1.13017	−0.565084	−	0.825033i	$-0.691156\pi$
$409$	−9348.20	−1.13017	−0.565084	+	0.825033i	$0.691156\pi$
$410$	0	0
$411$	4410.15	0.529286
$412$	0	0
$413$	−2887.50	−0.344031
$414$	0	0
$415$	6816.54	0.806291
$416$	0	0
$417$	−16799.6	−1.97285
$418$	0	0
$419$	9749.81	1.13678	0.568388	−	0.822760i	$-0.307567\pi$
$419$	9749.81	1.13678	0.568388	+	0.822760i	$0.307567\pi$
$420$	0	0
$421$	−6947.22	−0.804244	−0.402122	−	0.915586i	$-0.631727\pi$
$421$	−6947.22	−0.804244	−0.402122	+	0.915586i	$0.631727\pi$
$422$	0	0
$423$	−2061.80	−0.236994
$424$	0	0
$425$	−25.7869	−0.00294317
$426$	0	0
$427$	−5473.51	−0.620332
$428$	0	0
$429$	−6777.45	−0.762747
$430$	0	0
$431$	−9675.10	−1.08128	−0.540642	−	0.841253i	$-0.681818\pi$
$431$	−9675.10	−1.08128	−0.540642	+	0.841253i	$0.681818\pi$
$432$	0	0
$433$	−9378.86	−1.04092	−0.520461	−	0.853885i	$-0.674240\pi$
$433$	−9378.86	−1.04092	−0.520461	+	0.853885i	$0.674240\pi$
$434$	0	0
$435$	16786.0	1.85017
$436$	0	0
$437$	5973.61	0.653905
$438$	0	0
$439$	−14786.6	−1.60758	−0.803788	−	0.594916i	$-0.797185\pi$
$439$	−14786.6	−1.60758	−0.803788	+	0.594916i	$0.797185\pi$
$440$	0	0
$441$	2389.75	0.258044
$442$	0	0
$443$	1618.90	0.173626	0.0868132	−	0.996225i	$-0.472332\pi$
$443$	1618.90	0.173626	0.0868132	+	0.996225i	$0.472332\pi$
$444$	0	0
$445$	1991.50	0.212149
$446$	0	0
$447$	29537.3	3.12543
$448$	0	0
$449$	14229.7	1.49564	0.747820	−	0.663901i	$-0.231100\pi$
$449$	14229.7	1.49564	0.747820	+	0.663901i	$0.231100\pi$
$450$	0	0
$451$	−2292.76	−0.239384
$452$	0	0
$453$	13707.5	1.42171
$454$	0	0
$455$	−5564.80	−0.573367
$456$	0	0
$457$	−4324.02	−0.442602	−0.221301	−	0.975206i	$-0.571030\pi$
$457$	−4324.02	−0.442602	−0.221301	+	0.975206i	$0.571030\pi$
$458$	0	0
$459$	−4284.16	−0.435659
$460$	0	0
$461$	12979.5	1.31131	0.655657	−	0.755059i	$-0.272392\pi$
$461$	12979.5	1.31131	0.655657	+	0.755059i	$0.272392\pi$
$462$	0	0
$463$	−8306.05	−0.833725	−0.416863	−	0.908970i	$-0.636870\pi$
$463$	−8306.05	−0.833725	−0.416863	+	0.908970i	$0.636870\pi$
$464$	0	0
$465$	−4774.88	−0.476193
$466$	0	0
$467$	−6343.79	−0.628599	−0.314299	−	0.949324i	$-0.601770\pi$
$467$	−6343.79	−0.628599	−0.314299	+	0.949324i	$0.601770\pi$
$468$	0	0
$469$	2517.02	0.247815
$470$	0	0
$471$	20112.4	1.96758
$472$	0	0
$473$	−3807.72	−0.370147
$474$	0	0
$475$	171.410	0.0165576
$476$	0	0
$477$	−14641.1	−1.40539
$478$	0	0
$479$	12501.9	1.19254	0.596272	−	0.802783i	$-0.296648\pi$
$479$	12501.9	1.19254	0.596272	+	0.802783i	$0.296648\pi$
$480$	0	0
$481$	17943.0	1.70090
$482$	0	0
$483$	−2422.11	−0.228178
$484$	0	0
$485$	5201.81	0.487015
$486$	0	0
$487$	3604.74	0.335413	0.167707	−	0.985837i	$-0.446364\pi$
$487$	3604.74	0.335413	0.167707	+	0.985837i	$0.446364\pi$
$488$	0	0
$489$	14435.3	1.33494
$490$	0	0
$491$	20506.8	1.88484	0.942420	−	0.334430i	$-0.108544\pi$
$491$	20506.8	1.88484	0.942420	+	0.334430i	$0.108544\pi$
$492$	0	0
$493$	3881.69	0.354610
$494$	0	0
$495$	−6025.26	−0.547101
$496$	0	0
$497$	4658.27	0.420427
$498$	0	0
$499$	−15075.2	−1.35242	−0.676209	−	0.736710i	$-0.736378\pi$
$499$	−15075.2	−1.35242	−0.676209	+	0.736710i	$0.736378\pi$
$500$	0	0
$501$	−7239.99	−0.645626
$502$	0	0
$503$	−1366.43	−0.121126	−0.0605629	−	0.998164i	$-0.519290\pi$
$503$	−1366.43	−0.121126	−0.0605629	+	0.998164i	$0.519290\pi$
$504$	0	0
$505$	5211.80	0.459252
$506$	0	0
$507$	24487.2	2.14500
$508$	0	0
$509$	12219.6	1.06409	0.532046	−	0.846715i	$-0.321423\pi$
$509$	12219.6	1.06409	0.532046	+	0.846715i	$0.321423\pi$
$510$	0	0
$511$	8028.41	0.695021
$512$	0	0
$513$	28477.7	2.45092
$514$	0	0
$515$	−3418.75	−0.292521
$516$	0	0
$517$	−465.034	−0.0395593
$518$	0	0
$519$	−13698.1	−1.15853
$520$	0	0
$521$	−14153.7	−1.19018	−0.595092	−	0.803658i	$-0.702884\pi$
$521$	−14153.7	−1.19018	−0.595092	+	0.803658i	$0.702884\pi$
$522$	0	0
$523$	13454.2	1.12488	0.562438	−	0.826839i	$-0.309863\pi$
$523$	13454.2	1.12488	0.562438	+	0.826839i	$0.309863\pi$
$524$	0	0
$525$	−69.5015	−0.00577771
$526$	0	0
$527$	−1104.17	−0.0912687
$528$	0	0
$529$	−10586.9	−0.870130
$530$	0	0
$531$	20117.8	1.64414
$532$	0	0
$533$	14753.4	1.19895
$534$	0	0
$535$	−9355.77	−0.756047
$536$	0	0
$537$	−9591.14	−0.770742
$538$	0	0
$539$	539.000	0.0430730
$540$	0	0
$541$	−4252.71	−0.337964	−0.168982	−	0.985619i	$-0.554048\pi$
$541$	−4252.71	−0.337964	−0.168982	+	0.985619i	$0.554048\pi$
$542$	0	0
$543$	−26430.5	−2.08884
$544$	0	0
$545$	16802.1	1.32059
$546$	0	0
$547$	−12313.4	−0.962493	−0.481247	−	0.876585i	$-0.659816\pi$
$547$	−12313.4	−0.962493	−0.481247	+	0.876585i	$0.659816\pi$
$548$	0	0
$549$	38135.0	2.96459
$550$	0	0
$551$	−25802.4	−1.99495
$552$	0	0
$553$	8019.02	0.616643
$554$	0	0
$555$	24782.7	1.89543
$556$	0	0
$557$	−2161.87	−0.164455	−0.0822273	−	0.996614i	$-0.526203\pi$
$557$	−2161.87	−0.164455	−0.0822273	+	0.996614i	$0.526203\pi$
$558$	0	0
$559$	24501.7	1.85387
$560$	0	0
$561$	−2164.68	−0.162911
$562$	0	0
$563$	−8823.05	−0.660475	−0.330237	−	0.943898i	$-0.607129\pi$
$563$	−8823.05	−0.660475	−0.330237	+	0.943898i	$0.607129\pi$
$564$	0	0
$565$	−2782.27	−0.207170
$566$	0	0
$567$	−2329.21	−0.172518
$568$	0	0
$569$	−11914.5	−0.877820	−0.438910	−	0.898531i	$-0.644635\pi$
$569$	−11914.5	−0.877820	−0.438910	+	0.898531i	$0.644635\pi$
$570$	0	0
$571$	2313.34	0.169545	0.0847727	−	0.996400i	$-0.472984\pi$
$571$	2313.34	0.169545	0.0847727	+	0.996400i	$0.472984\pi$
$572$	0	0
$573$	−6366.05	−0.464129
$574$	0	0
$575$	45.3412	0.00328845
$576$	0	0
$577$	9307.61	0.671544	0.335772	−	0.941943i	$-0.391003\pi$
$577$	9307.61	0.671544	0.335772	+	0.941943i	$0.391003\pi$
$578$	0	0
$579$	−27357.1	−1.96360
$580$	0	0
$581$	4248.49	0.303369
$582$	0	0
$583$	−3302.27	−0.234590
$584$	0	0
$585$	38771.0	2.74014
$586$	0	0
$587$	−2306.12	−0.162153	−0.0810764	−	0.996708i	$-0.525836\pi$
$587$	−2306.12	−0.162153	−0.0810764	+	0.996708i	$0.525836\pi$
$588$	0	0
$589$	7339.66	0.513456
$590$	0	0
$591$	4364.17	0.303753
$592$	0	0
$593$	−3588.27	−0.248487	−0.124243	−	0.992252i	$-0.539650\pi$
$593$	−3588.27	−0.248487	−0.124243	+	0.992252i	$0.539650\pi$
$594$	0	0
$595$	−1777.37	−0.122462
$596$	0	0
$597$	39781.1	2.72719
$598$	0	0
$599$	5178.56	0.353239	0.176620	−	0.984279i	$-0.443484\pi$
$599$	5178.56	0.353239	0.176620	+	0.984279i	$0.443484\pi$
$600$	0	0
$601$	−5631.19	−0.382198	−0.191099	−	0.981571i	$-0.561205\pi$
$601$	−5631.19	−0.382198	−0.191099	+	0.981571i	$0.561205\pi$
$602$	0	0
$603$	−17536.5	−1.18432
$604$	0	0
$605$	−1358.98	−0.0913229
$606$	0	0
$607$	−17843.8	−1.19318	−0.596589	−	0.802547i	$-0.703478\pi$
$607$	−17843.8	−1.19318	−0.596589	+	0.802547i	$0.703478\pi$
$608$	0	0
$609$	10462.0	0.696130
$610$	0	0
$611$	2992.37	0.198132
$612$	0	0
$613$	5561.75	0.366455	0.183228	−	0.983071i	$-0.441345\pi$
$613$	5561.75	0.366455	0.183228	+	0.983071i	$0.441345\pi$
$614$	0	0
$615$	20377.2	1.33608
$616$	0	0
$617$	27284.0	1.78025	0.890124	−	0.455718i	$-0.150618\pi$
$617$	27284.0	1.78025	0.890124	+	0.455718i	$0.150618\pi$
$618$	0	0
$619$	−10717.3	−0.695903	−0.347951	−	0.937513i	$-0.613123\pi$
$619$	−10717.3	−0.695903	−0.347951	+	0.937513i	$0.613123\pi$
$620$	0	0
$621$	7532.87	0.486769
$622$	0	0
$623$	1241.23	0.0798213
$624$	0	0
$625$	−15766.3	−1.00904
$626$	0	0
$627$	14389.1	0.916497
$628$	0	0
$629$	5730.90	0.363285
$630$	0	0
$631$	15607.9	0.984690	0.492345	−	0.870400i	$-0.336140\pi$
$631$	15607.9	0.984690	0.492345	+	0.870400i	$0.336140\pi$
$632$	0	0
$633$	38270.1	2.40300
$634$	0	0
$635$	−15436.5	−0.964694
$636$	0	0
$637$	−3468.33	−0.215730
$638$	0	0
$639$	−32455.1	−2.00924
$640$	0	0
$641$	−25322.3	−1.56033	−0.780163	−	0.625576i	$-0.784864\pi$
$641$	−25322.3	−1.56033	−0.780163	+	0.625576i	$0.784864\pi$
$642$	0	0
$643$	26251.3	1.61003	0.805016	−	0.593253i	$-0.202157\pi$
$643$	26251.3	1.61003	0.805016	+	0.593253i	$0.202157\pi$
$644$	0	0
$645$	33841.5	2.06590
$646$	0	0
$647$	−21488.0	−1.30569	−0.652843	−	0.757493i	$-0.726424\pi$
$647$	−21488.0	−1.30569	−0.652843	+	0.757493i	$0.726424\pi$
$648$	0	0
$649$	4537.50	0.274442
$650$	0	0
$651$	−2976.00	−0.179169
$652$	0	0
$653$	17090.3	1.02419	0.512096	−	0.858928i	$-0.328869\pi$
$653$	17090.3	1.02419	0.512096	+	0.858928i	$0.328869\pi$
$654$	0	0
$655$	19898.1	1.18700
$656$	0	0
$657$	−55935.4	−3.32153
$658$	0	0
$659$	−3405.83	−0.201324	−0.100662	−	0.994921i	$-0.532096\pi$
$659$	−3405.83	−0.201324	−0.100662	+	0.994921i	$0.532096\pi$
$660$	0	0
$661$	17772.4	1.04579	0.522895	−	0.852397i	$-0.324852\pi$
$661$	17772.4	1.04579	0.522895	+	0.852397i	$0.324852\pi$
$662$	0	0
$663$	13929.2	0.815935
$664$	0	0
$665$	11814.5	0.688943
$666$	0	0
$667$	−6825.20	−0.396211
$668$	0	0
$669$	14634.6	0.845751
$670$	0	0
$671$	8601.24	0.494854
$672$	0	0
$673$	20376.8	1.16712	0.583558	−	0.812072i	$-0.301660\pi$
$673$	20376.8	1.16712	0.583558	+	0.812072i	$0.301660\pi$
$674$	0	0
$675$	216.153	0.0123255
$676$	0	0
$677$	−3662.84	−0.207938	−0.103969	−	0.994581i	$-0.533154\pi$
$677$	−3662.84	−0.207938	−0.103969	+	0.994581i	$0.533154\pi$
$678$	0	0
$679$	3242.09	0.183240
$680$	0	0
$681$	−22311.8	−1.25549
$682$	0	0
$683$	31485.1	1.76390	0.881951	−	0.471341i	$-0.156230\pi$
$683$	31485.1	1.76390	0.881951	+	0.471341i	$0.156230\pi$
$684$	0	0
$685$	−5690.25	−0.317392
$686$	0	0
$687$	22962.5	1.27522
$688$	0	0
$689$	21249.3	1.17494
$690$	0	0
$691$	15073.2	0.829829	0.414915	−	0.909860i	$-0.363811\pi$
$691$	15073.2	0.829829	0.414915	+	0.909860i	$0.363811\pi$
$692$	0	0
$693$	−3755.31	−0.205848
$694$	0	0
$695$	21675.8	1.18304
$696$	0	0
$697$	4712.14	0.256076
$698$	0	0
$699$	42657.8	2.30825
$700$	0	0
$701$	−20958.4	−1.12922	−0.564612	−	0.825356i	$-0.690974\pi$
$701$	−20958.4	−1.12922	−0.564612	+	0.825356i	$0.690974\pi$
$702$	0	0
$703$	−38094.4	−2.04375
$704$	0	0
$705$	4133.03	0.220793
$706$	0	0
$707$	3248.32	0.172794
$708$	0	0
$709$	11805.1	0.625316	0.312658	−	0.949866i	$-0.398781\pi$
$709$	11805.1	0.625316	0.312658	+	0.949866i	$0.398781\pi$
$710$	0	0
$711$	−55870.0	−2.94696
$712$	0	0
$713$	1941.48	0.101976
$714$	0	0
$715$	8744.69	0.457389
$716$	0	0
$717$	−40282.5	−2.09816
$718$	0	0
$719$	4294.47	0.222749	0.111375	−	0.993778i	$-0.464475\pi$
$719$	4294.47	0.222749	0.111375	+	0.993778i	$0.464475\pi$
$720$	0	0
$721$	−2130.78	−0.110061
$722$	0	0
$723$	30902.1	1.58957
$724$	0	0
$725$	−195.847	−0.0100325
$726$	0	0
$727$	5916.37	0.301824	0.150912	−	0.988547i	$-0.451779\pi$
$727$	5916.37	0.301824	0.150912	+	0.988547i	$0.451779\pi$
$728$	0	0
$729$	−28309.6	−1.43828
$730$	0	0
$731$	7825.73	0.395958
$732$	0	0
$733$	617.535	0.0311176	0.0155588	−	0.999879i	$-0.495047\pi$
$733$	617.535	0.0311176	0.0155588	+	0.999879i	$0.495047\pi$
$734$	0	0
$735$	−4790.41	−0.240404
$736$	0	0
$737$	−3955.31	−0.197688
$738$	0	0
$739$	29262.1	1.45660	0.728298	−	0.685261i	$-0.240312\pi$
$739$	29262.1	1.45660	0.728298	+	0.685261i	$0.240312\pi$
$740$	0	0
$741$	−92590.0	−4.59026
$742$	0	0
$743$	32266.2	1.59318	0.796591	−	0.604519i	$-0.206635\pi$
$743$	32266.2	1.59318	0.796591	+	0.604519i	$0.206635\pi$
$744$	0	0
$745$	−38110.8	−1.87419
$746$	0	0
$747$	−29600.0	−1.44981
$748$	0	0
$749$	−5831.09	−0.284464
$750$	0	0
$751$	25365.3	1.23248	0.616240	−	0.787558i	$-0.288655\pi$
$751$	25365.3	1.23248	0.616240	+	0.787558i	$0.288655\pi$
$752$	0	0
$753$	−29815.3	−1.44294
$754$	0	0
$755$	−17686.3	−0.852542
$756$	0	0
$757$	−32910.6	−1.58012	−0.790062	−	0.613026i	$-0.789952\pi$
$757$	−32910.6	−1.58012	−0.790062	+	0.613026i	$0.789952\pi$
$758$	0	0
$759$	3806.17	0.182023
$760$	0	0
$761$	14386.3	0.685287	0.342644	−	0.939465i	$-0.388678\pi$
$761$	14386.3	0.685287	0.342644	+	0.939465i	$0.388678\pi$
$762$	0	0
$763$	10472.1	0.496874
$764$	0	0
$765$	12383.3	0.585252
$766$	0	0
$767$	−29197.7	−1.37453
$768$	0	0
$769$	−17200.1	−0.806568	−0.403284	−	0.915075i	$-0.632131\pi$
$769$	−17200.1	−0.806568	−0.403284	+	0.915075i	$0.632131\pi$
$770$	0	0
$771$	−49017.5	−2.28965
$772$	0	0
$773$	34246.3	1.59347	0.796735	−	0.604329i	$-0.206559\pi$
$773$	34246.3	1.59347	0.796735	+	0.604329i	$0.206559\pi$
$774$	0	0
$775$	55.7099	0.00258214
$776$	0	0
$777$	15446.1	0.713160
$778$	0	0
$779$	−31322.5	−1.44062
$780$	0	0
$781$	−7320.15	−0.335385
$782$	0	0
$783$	−32537.4	−1.48505
$784$	0	0
$785$	−25950.2	−1.17988
$786$	0	0
$787$	13692.9	0.620203	0.310101	−	0.950703i	$-0.399637\pi$
$787$	13692.9	0.620203	0.310101	+	0.950703i	$0.399637\pi$
$788$	0	0
$789$	57817.3	2.60881
$790$	0	0
$791$	−1734.08	−0.0779479
$792$	0	0
$793$	−55346.8	−2.47846
$794$	0	0
$795$	29349.2	1.30932
$796$	0	0
$797$	6141.98	0.272974	0.136487	−	0.990642i	$-0.456419\pi$
$797$	6141.98	0.272974	0.136487	+	0.990642i	$0.456419\pi$
$798$	0	0
$799$	955.748	0.0423178
$800$	0	0
$801$	−8647.85	−0.381469
$802$	0	0
$803$	−12616.1	−0.554435
$804$	0	0
$805$	3125.16	0.136829
$806$	0	0
$807$	68822.1	3.00205
$808$	0	0
$809$	12690.4	0.551509	0.275754	−	0.961228i	$-0.411072\pi$
$809$	12690.4	0.551509	0.275754	+	0.961228i	$0.411072\pi$
$810$	0	0
$811$	−23397.1	−1.01305	−0.506525	−	0.862225i	$-0.669070\pi$
$811$	−23397.1	−1.01305	−0.506525	+	0.862225i	$0.669070\pi$
$812$	0	0
$813$	−24754.7	−1.06788
$814$	0	0
$815$	−18625.3	−0.800511
$816$	0	0
$817$	−52019.1	−2.22756
$818$	0	0
$819$	24164.5	1.03098
$820$	0	0
$821$	10759.8	0.457395	0.228697	−	0.973498i	$-0.426553\pi$
$821$	10759.8	0.457395	0.228697	+	0.973498i	$0.426553\pi$
$822$	0	0
$823$	9489.09	0.401906	0.200953	−	0.979601i	$-0.435596\pi$
$823$	9489.09	0.401906	0.200953	+	0.979601i	$0.435596\pi$
$824$	0	0
$825$	109.217	0.00460902
$826$	0	0
$827$	25988.4	1.09275	0.546375	−	0.837540i	$-0.316007\pi$
$827$	25988.4	1.09275	0.546375	+	0.837540i	$0.316007\pi$
$828$	0	0
$829$	1723.25	0.0721965	0.0360982	−	0.999348i	$-0.488507\pi$
$829$	1723.25	0.0721965	0.0360982	+	0.999348i	$0.488507\pi$
$830$	0	0
$831$	−55180.4	−2.30348
$832$	0	0
$833$	−1107.77	−0.0460766
$834$	0	0
$835$	9341.48	0.387156
$836$	0	0
$837$	9255.50	0.382219
$838$	0	0
$839$	−8950.36	−0.368296	−0.184148	−	0.982898i	$-0.558953\pi$
$839$	−8950.36	−0.368296	−0.184148	+	0.982898i	$0.558953\pi$
$840$	0	0
$841$	5091.71	0.208771
$842$	0	0
$843$	63103.4	2.57817
$844$	0	0
$845$	−31594.9	−1.28627
$846$	0	0
$847$	−847.000	−0.0343604
$848$	0	0
$849$	35468.2	1.43376
$850$	0	0
$851$	−10076.7	−0.405904
$852$	0	0
$853$	−5985.68	−0.240265	−0.120132	−	0.992758i	$-0.538332\pi$
$853$	−5985.68	−0.240265	−0.120132	+	0.992758i	$0.538332\pi$
$854$	0	0
$855$	−82313.9	−3.29249
$856$	0	0
$857$	−18380.8	−0.732645	−0.366323	−	0.930488i	$-0.619383\pi$
$857$	−18380.8	−0.732645	−0.366323	+	0.930488i	$0.619383\pi$
$858$	0	0
$859$	24625.6	0.978133	0.489066	−	0.872247i	$-0.337338\pi$
$859$	24625.6	0.978133	0.489066	+	0.872247i	$0.337338\pi$
$860$	0	0
$861$	12700.3	0.502701
$862$	0	0
$863$	−22569.6	−0.890243	−0.445121	−	0.895470i	$-0.646839\pi$
$863$	−22569.6	−0.890243	−0.445121	+	0.895470i	$0.646839\pi$
$864$	0	0
$865$	17674.1	0.694727
$866$	0	0
$867$	−38316.9	−1.50093
$868$	0	0
$869$	−12601.3	−0.491911
$870$	0	0
$871$	25451.4	0.990114
$872$	0	0
$873$	−22588.3	−0.875712
$874$	0	0
$875$	−9737.66	−0.376220
$876$	0	0
$877$	22763.4	0.876473	0.438237	−	0.898860i	$-0.355603\pi$
$877$	22763.4	0.876473	0.438237	+	0.898860i	$0.355603\pi$
$878$	0	0
$879$	−83841.8	−3.21719
$880$	0	0
$881$	−29753.1	−1.13781	−0.568903	−	0.822405i	$-0.692632\pi$
$881$	−29753.1	−1.13781	−0.568903	+	0.822405i	$0.692632\pi$
$882$	0	0
$883$	−17862.0	−0.680752	−0.340376	−	0.940289i	$-0.610554\pi$
$883$	−17862.0	−0.680752	−0.340376	+	0.940289i	$0.610554\pi$
$884$	0	0
$885$	−40327.5	−1.53174
$886$	0	0
$887$	25953.9	0.982464	0.491232	−	0.871029i	$-0.336547\pi$
$887$	25953.9	0.982464	0.491232	+	0.871029i	$0.336547\pi$
$888$	0	0
$889$	−9621.01	−0.362968
$890$	0	0
$891$	3660.19	0.137622
$892$	0	0
$893$	−6353.05	−0.238070
$894$	0	0
$895$	12375.1	0.462183
$896$	0	0
$897$	−24491.8	−0.911657
$898$	0	0
$899$	−8386.00	−0.311111
$900$	0	0
$901$	6786.90	0.250948
$902$	0	0
$903$	21092.1	0.777300
$904$	0	0
$905$	34102.2	1.25259
$906$	0	0
$907$	46572.1	1.70496	0.852481	−	0.522758i	$-0.175097\pi$
$907$	46572.1	1.70496	0.852481	+	0.522758i	$0.175097\pi$
$908$	0	0
$909$	−22631.6	−0.825791
$910$	0	0
$911$	−13240.4	−0.481532	−0.240766	−	0.970583i	$-0.577399\pi$
$911$	−13240.4	−0.481532	−0.240766	+	0.970583i	$0.577399\pi$
$912$	0	0
$913$	−6676.20	−0.242004
$914$	0	0
$915$	−76444.3	−2.76194
$916$	0	0
$917$	12401.7	0.446610
$918$	0	0
$919$	5210.60	0.187031	0.0935156	−	0.995618i	$-0.470189\pi$
$919$	5210.60	0.187031	0.0935156	+	0.995618i	$0.470189\pi$
$920$	0	0
$921$	−54018.1	−1.93264
$922$	0	0
$923$	47103.3	1.67977
$924$	0	0
$925$	−289.146	−0.0102779
$926$	0	0
$927$	14845.5	0.525988
$928$	0	0
$929$	−41924.2	−1.48061	−0.740305	−	0.672271i	$-0.765319\pi$
$929$	−41924.2	−1.48061	−0.740305	+	0.672271i	$0.765319\pi$
$930$	0	0
$931$	7363.54	0.259216
$932$	0	0
$933$	49543.2	1.73845
$934$	0	0
$935$	2793.01	0.0976910
$936$	0	0
$937$	−48975.6	−1.70754	−0.853768	−	0.520653i	$-0.825688\pi$
$937$	−48975.6	−1.70754	−0.853768	+	0.520653i	$0.825688\pi$
$938$	0	0
$939$	21092.5	0.733042
$940$	0	0
$941$	−18223.0	−0.631298	−0.315649	−	0.948876i	$-0.602222\pi$
$941$	−18223.0	−0.631298	−0.315649	+	0.948876i	$0.602222\pi$
$942$	0	0
$943$	−8285.39	−0.286118
$944$	0	0
$945$	14898.4	0.512852
$946$	0	0
$947$	33873.6	1.16235	0.581174	−	0.813779i	$-0.302594\pi$
$947$	33873.6	1.16235	0.581174	+	0.813779i	$0.302594\pi$
$948$	0	0
$949$	81181.2	2.77687
$950$	0	0
$951$	45416.7	1.54862
$952$	0	0
$953$	−41893.1	−1.42398	−0.711990	−	0.702190i	$-0.752206\pi$
$953$	−41893.1	−1.42398	−0.711990	+	0.702190i	$0.752206\pi$
$954$	0	0
$955$	8213.88	0.278319
$956$	0	0
$957$	−16440.4	−0.555320
$958$	0	0
$959$	−3546.51	−0.119419
$960$	0	0
$961$	−27405.5	−0.919927
$962$	0	0
$963$	40626.3	1.35947
$964$	0	0
$965$	35297.8	1.17749
$966$	0	0
$967$	−45599.6	−1.51643	−0.758213	−	0.652007i	$-0.773927\pi$
$967$	−45599.6	−1.51643	−0.758213	+	0.652007i	$0.773927\pi$
$968$	0	0
$969$	−29572.8	−0.980406
$970$	0	0
$971$	11781.9	0.389392	0.194696	−	0.980864i	$-0.437628\pi$
$971$	11781.9	0.389392	0.194696	+	0.980864i	$0.437628\pi$
$972$	0	0
$973$	13509.7	0.445120
$974$	0	0
$975$	−702.782	−0.0230841
$976$	0	0
$977$	33583.6	1.09973	0.549864	−	0.835254i	$-0.314680\pi$
$977$	33583.6	1.09973	0.549864	+	0.835254i	$0.314680\pi$
$978$	0	0
$979$	−1950.50	−0.0636753
$980$	0	0
$981$	−72960.9	−2.37458
$982$	0	0
$983$	−2709.05	−0.0878995	−0.0439498	−	0.999034i	$-0.513994\pi$
$983$	−2709.05	−0.0878995	−0.0439498	+	0.999034i	$0.513994\pi$
$984$	0	0
$985$	−5630.92	−0.182148
$986$	0	0
$987$	2575.96	0.0830737
$988$	0	0
$989$	−13760.0	−0.442410
$990$	0	0
$991$	22183.3	0.711077	0.355538	−	0.934662i	$-0.384297\pi$
$991$	22183.3	0.711077	0.355538	+	0.934662i	$0.384297\pi$
$992$	0	0
$993$	95139.0	3.04043
$994$	0	0
$995$	−51328.1	−1.63539
$996$	0	0
$997$	25840.5	0.820838	0.410419	−	0.911897i	$-0.365382\pi$
$997$	25840.5	0.820838	0.410419	+	0.911897i	$0.365382\pi$
$998$	0	0
$999$	−48038.0	−1.52138

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1232.4.a.bb.1.6		6
4.3	odd	2		616.4.a.i.1.1	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
616.4.a.i.1.1	✓	6	4.3	odd	2
1232.4.a.bb.1.6		6	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1232 = 2^{4} \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1232.a (trivial)

\(f(q)\)	\(=\)	\(q+8.70461 q^{3} -11.2312 q^{5} -7.00000 q^{7} +48.7703 q^{9} +11.0000 q^{11} -70.7822 q^{13} -97.7636 q^{15} -22.6075 q^{17} +150.276 q^{19} -60.9323 q^{21} +39.7508 q^{23} +1.14064 q^{25} +189.502 q^{27} -171.699 q^{29} +48.8411 q^{31} +95.7508 q^{33} +78.6186 q^{35} -253.496 q^{37} -616.132 q^{39} -208.433 q^{41} -346.157 q^{43} -547.751 q^{45} -42.2758 q^{47} +49.0000 q^{49} -196.789 q^{51} -300.206 q^{53} -123.544 q^{55} +1308.10 q^{57} +412.500 q^{59} +781.931 q^{61} -341.392 q^{63} +794.972 q^{65} -359.574 q^{67} +346.016 q^{69} -665.468 q^{71} -1146.92 q^{73} +9.92879 q^{75} -77.0000 q^{77} -1145.57 q^{79} +332.745 q^{81} -606.927 q^{83} +253.910 q^{85} -1494.58 q^{87} -177.318 q^{89} +495.476 q^{91} +425.143 q^{93} -1687.79 q^{95} -463.156 q^{97} +536.473 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 2 q^{5} - 42 q^{7} + 60 q^{9} + 66 q^{11} - 6 q^{13} - 126 q^{15} - 14 q^{17} - 80 q^{19} - 254 q^{23} + 220 q^{25} - 90 q^{27} + 132 q^{29} + 52 q^{31} - 14 q^{35} - 518 q^{37} - 332 q^{39} + 486 q^{41}+ \cdots + 660 q^{99}+O(q^{100}) \) 6 * q + 2 * q^5 - 42 * q^7 + 60 * q^9 + 66 * q^11 - 6 * q^13 - 126 * q^15 - 14 * q^17 - 80 * q^19 - 254 * q^23 + 220 * q^25 - 90 * q^27 + 132 * q^29 + 52 * q^31 - 14 * q^35 - 518 * q^37 - 332 * q^39 + 486 * q^41 - 428 * q^43 - 244 * q^45 - 790 * q^47 + 294 * q^49 - 640 * q^51 - 40 * q^53 + 22 * q^55 + 2276 * q^57 - 436 * q^59 + 1034 * q^61 - 420 * q^63 + 800 * q^65 - 562 * q^67 + 530 * q^69 - 2474 * q^71 - 902 * q^73 - 1986 * q^75 - 462 * q^77 - 1636 * q^79 + 570 * q^81 - 3016 * q^83 + 236 * q^85 - 4340 * q^87 + 1750 * q^89 + 42 * q^91 + 150 * q^93 - 3628 * q^95 - 1250 * q^97 + 660 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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