LMFDB - Embedded newform 1232.4.a.p.1.2

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:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{137})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{2} - x - 34$ x^2 - x - 34
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 154)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-5.35235$ 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)

$f(q)$	$=$	$q+8.35235 q^{3} +2.35235 q^{5} -7.00000 q^{7} +42.7617 q^{9} +11.0000 q^{11} +81.5235 q^{13} +19.6477 q^{15} -20.5906 q^{17} +104.933 q^{19} -58.4664 q^{21} +145.285 q^{23} -119.466 q^{25} +131.648 q^{27} -92.3624 q^{29} +50.9430 q^{31} +91.8758 q^{33} -16.4664 q^{35} +19.7617 q^{37} +680.913 q^{39} -102.953 q^{41} -142.819 q^{43} +100.591 q^{45} -504.456 q^{47} +49.0000 q^{49} -171.980 q^{51} -521.047 q^{53} +25.8758 q^{55} +876.436 q^{57} +433.017 q^{59} -429.235 q^{61} -299.332 q^{63} +191.772 q^{65} +1045.91 q^{67} +1213.47 q^{69} +800.044 q^{71} +862.188 q^{73} -997.826 q^{75} -77.0000 q^{77} +218.114 q^{79} -55.0000 q^{81} +1044.95 q^{83} -48.4363 q^{85} -771.443 q^{87} +494.601 q^{89} -570.664 q^{91} +425.493 q^{93} +246.839 q^{95} +1413.78 q^{97} +470.379 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$2 q + 5 q^{3} - 7 q^{5} - 14 q^{7} + 27 q^{9} + 22 q^{11} + 46 q^{13} + 51 q^{15} - 88 q^{17} + 46 q^{19} - 35 q^{21} + 115 q^{23} - 157 q^{25} + 275 q^{27} - 372 q^{29} + 137 q^{31} + 55 q^{33} + 49 q^{35}+ \cdots + 297 q^{99}+O(q^{100})$ 2 * q + 5 * q^3 - 7 * q^5 - 14 * q^7 + 27 * q^9 + 22 * q^11 + 46 * q^13 + 51 * q^15 - 88 * q^17 + 46 * q^19 - 35 * q^21 + 115 * q^23 - 157 * q^25 + 275 * q^27 - 372 * q^29 + 137 * q^31 + 55 * q^33 + 49 * q^35 - 19 * q^37 + 800 * q^39 - 440 * q^41 - 192 * q^43 + 248 * q^45 - 728 * q^47 + 98 * q^49 + 54 * q^51 - 808 * q^53 - 77 * q^55 + 1074 * q^57 + 35 * q^59 + 312 * q^61 - 189 * q^63 + 524 * q^65 + 301 * q^67 + 1315 * q^69 + 137 * q^71 + 788 * q^73 - 872 * q^75 - 154 * q^77 + 366 * q^79 - 110 * q^81 + 2324 * q^83 + 582 * q^85 + 166 * q^87 + 1235 * q^89 - 322 * q^91 + 137 * q^93 + 798 * q^95 + 709 * q^97 + 297 * 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
$3$	8.35235	1.60741	0.803705	−	0.595028i	$-0.202859\pi$
$3$	8.35235	1.60741	0.803705	+	0.595028i	$0.202859\pi$
$4$	0	0
$5$	2.35235	0.210401	0.105200	−	0.994451i	$-0.466452\pi$
$5$	2.35235	0.210401	0.105200	+	0.994451i	$0.466452\pi$
$6$	0	0
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	0	0
$9$	42.7617	1.58377
$10$	0	0
$11$	11.0000	0.301511
$11$	11.0000	0.301511
$12$	0	0
$13$	81.5235	1.73927	0.869637	−	0.493692i	$-0.164353\pi$
$13$	81.5235	1.73927	0.869637	+	0.493692i	$0.164353\pi$
$14$	0	0
$15$	19.6477	0.338200
$16$	0	0
$17$	−20.5906	−0.293762	−0.146881	−	0.989154i	$-0.546923\pi$
$17$	−20.5906	−0.293762	−0.146881	+	0.989154i	$0.546923\pi$
$18$	0	0
$19$	104.933	1.26701	0.633507	−	0.773737i	$-0.281615\pi$
$19$	104.933	1.26701	0.633507	+	0.773737i	$0.281615\pi$
$20$	0	0
$21$	−58.4664	−0.607544
$22$	0	0
$23$	145.285	1.31713	0.658567	−	0.752522i	$-0.271163\pi$
$23$	145.285	1.31713	0.658567	+	0.752522i	$0.271163\pi$
$24$	0	0
$25$	−119.466	−0.955732
$26$	0	0
$27$	131.648	0.938356
$28$	0	0
$29$	−92.3624	−0.591423	−0.295712	−	0.955277i	$-0.595557\pi$
$29$	−92.3624	−0.591423	−0.295712	+	0.955277i	$0.595557\pi$
$30$	0	0
$31$	50.9430	0.295149	0.147575	−	0.989051i	$-0.452853\pi$
$31$	50.9430	0.295149	0.147575	+	0.989051i	$0.452853\pi$
$32$	0	0
$33$	91.8758	0.484653
$34$	0	0
$35$	−16.4664	−0.0795239
$36$	0	0
$37$	19.7617	0.0878057	0.0439029	−	0.999036i	$-0.486021\pi$
$37$	19.7617	0.0878057	0.0439029	+	0.999036i	$0.486021\pi$
$38$	0	0
$39$	680.913	2.79573
$40$	0	0
$41$	−102.953	−0.392160	−0.196080	−	0.980588i	$-0.562821\pi$
$41$	−102.953	−0.392160	−0.196080	+	0.980588i	$0.562821\pi$
$42$	0	0
$43$	−142.819	−0.506504	−0.253252	−	0.967400i	$-0.581500\pi$
$43$	−142.819	−0.506504	−0.253252	+	0.967400i	$0.581500\pi$
$44$	0	0
$45$	100.591	0.333226
$46$	0	0
$47$	−504.456	−1.56559	−0.782793	−	0.622282i	$-0.786206\pi$
$47$	−504.456	−1.56559	−0.782793	+	0.622282i	$0.786206\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	0	0
$51$	−171.980	−0.472196
$52$	0	0
$53$	−521.047	−1.35040	−0.675201	−	0.737634i	$-0.735943\pi$
$53$	−521.047	−1.35040	−0.675201	+	0.737634i	$0.735943\pi$
$54$	0	0
$55$	25.8758	0.0634382
$56$	0	0
$57$	876.436	2.03661
$58$	0	0
$59$	433.017	0.955491	0.477746	−	0.878498i	$-0.341454\pi$
$59$	433.017	0.955491	0.477746	+	0.878498i	$0.341454\pi$
$60$	0	0
$61$	−429.235	−0.900949	−0.450475	−	0.892789i	$-0.648745\pi$
$61$	−429.235	−0.900949	−0.450475	+	0.892789i	$0.648745\pi$
$62$	0	0
$63$	−299.332	−0.598608
$64$	0	0
$65$	191.772	0.365944
$66$	0	0
$67$	1045.91	1.90714	0.953569	−	0.301176i	$-0.0973792\pi$
$67$	1045.91	1.90714	0.953569	+	0.301176i	$0.0973792\pi$
$68$	0	0
$69$	1213.47	2.11717
$70$	0	0
$71$	800.044	1.33729	0.668646	−	0.743581i	$-0.266874\pi$
$71$	800.044	1.33729	0.668646	+	0.743581i	$0.266874\pi$
$72$	0	0
$73$	862.188	1.38235	0.691174	−	0.722688i	$-0.257094\pi$
$73$	862.188	1.38235	0.691174	+	0.722688i	$0.257094\pi$
$74$	0	0
$75$	−997.826	−1.53625
$76$	0	0
$77$	−77.0000	−0.113961
$78$	0	0
$79$	218.114	0.310630	0.155315	−	0.987865i	$-0.450361\pi$
$79$	218.114	0.310630	0.155315	+	0.987865i	$0.450361\pi$
$80$	0	0
$81$	−55.0000	−0.0754458
$82$	0	0
$83$	1044.95	1.38191	0.690955	−	0.722898i	$-0.257190\pi$
$83$	1044.95	1.38191	0.690955	+	0.722898i	$0.257190\pi$
$84$	0	0
$85$	−48.4363	−0.0618077
$86$	0	0
$87$	−771.443	−0.950660
$88$	0	0
$89$	494.601	0.589074	0.294537	−	0.955640i	$-0.404835\pi$
$89$	494.601	0.589074	0.294537	+	0.955640i	$0.404835\pi$
$90$	0	0
$91$	−570.664	−0.657383
$92$	0	0
$93$	425.493	0.474426
$94$	0	0
$95$	246.839	0.266580
$96$	0	0
$97$	1413.78	1.47987	0.739934	−	0.672680i	$-0.234857\pi$
$97$	1413.78	1.47987	0.739934	+	0.672680i	$0.234857\pi$
$98$	0	0
$99$	470.379	0.477524
$100$	0	0
$101$	−1364.99	−1.34476	−0.672382	−	0.740204i	$-0.734729\pi$
$101$	−1364.99	−1.34476	−0.672382	+	0.740204i	$0.734729\pi$
$102$	0	0
$103$	−408.497	−0.390780	−0.195390	−	0.980726i	$-0.562597\pi$
$103$	−408.497	−0.390780	−0.195390	+	0.980726i	$0.562597\pi$
$104$	0	0
$105$	−137.534	−0.127828
$106$	0	0
$107$	1049.39	0.948115	0.474057	−	0.880494i	$-0.342789\pi$
$107$	1049.39	0.948115	0.474057	+	0.880494i	$0.342789\pi$
$108$	0	0
$109$	863.081	0.758423	0.379212	−	0.925310i	$-0.376195\pi$
$109$	863.081	0.758423	0.379212	+	0.925310i	$0.376195\pi$
$110$	0	0
$111$	165.057	0.141140
$112$	0	0
$113$	314.909	0.262161	0.131080	−	0.991372i	$-0.458155\pi$
$113$	314.909	0.262161	0.131080	+	0.991372i	$0.458155\pi$
$114$	0	0
$115$	341.762	0.277126
$116$	0	0
$117$	3486.09	2.75461
$118$	0	0
$119$	144.134	0.111032
$120$	0	0
$121$	121.000	0.0909091
$122$	0	0
$123$	−859.899	−0.630362
$124$	0	0
$125$	−575.071	−0.411487
$126$	0	0
$127$	100.973	0.0705505	0.0352753	−	0.999378i	$-0.488769\pi$
$127$	100.973	0.0705505	0.0352753	+	0.999378i	$0.488769\pi$
$128$	0	0
$129$	−1192.87	−0.814160
$130$	0	0
$131$	1540.70	1.02757	0.513786	−	0.857918i	$-0.328242\pi$
$131$	1540.70	1.02757	0.513786	+	0.857918i	$0.328242\pi$
$132$	0	0
$133$	−734.530	−0.478886
$134$	0	0
$135$	309.681	0.197431
$136$	0	0
$137$	121.299	0.0756442	0.0378221	−	0.999284i	$-0.487958\pi$
$137$	121.299	0.0756442	0.0378221	+	0.999284i	$0.487958\pi$
$138$	0	0
$139$	1439.94	0.878663	0.439331	−	0.898325i	$-0.355215\pi$
$139$	1439.94	0.878663	0.439331	+	0.898325i	$0.355215\pi$
$140$	0	0
$141$	−4213.40	−2.51654
$142$	0	0
$143$	896.758	0.524411
$144$	0	0
$145$	−217.269	−0.124436
$146$	0	0
$147$	409.265	0.229630
$148$	0	0
$149$	−960.711	−0.528218	−0.264109	−	0.964493i	$-0.585078\pi$
$149$	−960.711	−0.528218	−0.264109	+	0.964493i	$0.585078\pi$
$150$	0	0
$151$	2752.05	1.48317	0.741584	−	0.670860i	$-0.234075\pi$
$151$	2752.05	1.48317	0.741584	+	0.670860i	$0.234075\pi$
$152$	0	0
$153$	−880.490	−0.465251
$154$	0	0
$155$	119.836	0.0620996
$156$	0	0
$157$	−1548.04	−0.786922	−0.393461	−	0.919341i	$-0.628722\pi$
$157$	−1548.04	−0.786922	−0.393461	+	0.919341i	$0.628722\pi$
$158$	0	0
$159$	−4351.97	−2.17065
$160$	0	0
$161$	−1017.00	−0.497830
$162$	0	0
$163$	−3644.22	−1.75115	−0.875574	−	0.483085i	$-0.839516\pi$
$163$	−3644.22	−1.75115	−0.875574	+	0.483085i	$0.839516\pi$
$164$	0	0
$165$	216.124	0.101971
$166$	0	0
$167$	−2199.26	−1.01906	−0.509531	−	0.860452i	$-0.670181\pi$
$167$	−2199.26	−1.01906	−0.509531	+	0.860452i	$0.670181\pi$
$168$	0	0
$169$	4449.08	2.02507
$170$	0	0
$171$	4487.11	2.00666
$172$	0	0
$173$	1506.10	0.661888	0.330944	−	0.943650i	$-0.392633\pi$
$173$	1506.10	0.661888	0.330944	+	0.943650i	$0.392633\pi$
$174$	0	0
$175$	836.265	0.361233
$176$	0	0
$177$	3616.71	1.53587
$178$	0	0
$179$	−3114.10	−1.30033	−0.650165	−	0.759793i	$-0.725300\pi$
$179$	−3114.10	−1.30033	−0.650165	+	0.759793i	$0.725300\pi$
$180$	0	0
$181$	−1491.37	−0.612447	−0.306223	−	0.951960i	$-0.599065\pi$
$181$	−1491.37	−0.612447	−0.306223	+	0.951960i	$0.599065\pi$
$182$	0	0
$183$	−3585.12	−1.44820
$184$	0	0
$185$	46.4866	0.0184744
$186$	0	0
$187$	−226.497	−0.0885726
$188$	0	0
$189$	−921.534	−0.354665
$190$	0	0
$191$	−1553.58	−0.588551	−0.294275	−	0.955721i	$-0.595078\pi$
$191$	−1553.58	−0.588551	−0.294275	+	0.955721i	$0.595078\pi$
$192$	0	0
$193$	42.3760	0.0158046	0.00790231	−	0.999969i	$-0.497485\pi$
$193$	42.3760	0.0158046	0.00790231	+	0.999969i	$0.497485\pi$
$194$	0	0
$195$	1601.75	0.588222
$196$	0	0
$197$	−4120.51	−1.49022	−0.745112	−	0.666939i	$-0.767604\pi$
$197$	−4120.51	−1.49022	−0.745112	+	0.666939i	$0.767604\pi$
$198$	0	0
$199$	−3601.21	−1.28283	−0.641414	−	0.767195i	$-0.721652\pi$
$199$	−3601.21	−1.28283	−0.641414	+	0.767195i	$0.721652\pi$
$200$	0	0
$201$	8735.80	3.06555
$202$	0	0
$203$	646.537	0.223537
$204$	0	0
$205$	−242.181	−0.0825107
$206$	0	0
$207$	6212.65	2.08603
$208$	0	0
$209$	1154.26	0.382019
$210$	0	0
$211$	−2741.91	−0.894602	−0.447301	−	0.894383i	$-0.647615\pi$
$211$	−2741.91	−0.894602	−0.447301	+	0.894383i	$0.647615\pi$
$212$	0	0
$213$	6682.25	2.14958
$214$	0	0
$215$	−335.960	−0.106569
$216$	0	0
$217$	−356.601	−0.111556
$218$	0	0
$219$	7201.30	2.22200
$220$	0	0
$221$	−1678.62	−0.510932
$222$	0	0
$223$	−3251.53	−0.976405	−0.488203	−	0.872730i	$-0.662347\pi$
$223$	−3251.53	−0.976405	−0.488203	+	0.872730i	$0.662347\pi$
$224$	0	0
$225$	−5108.59	−1.51366
$226$	0	0
$227$	−3058.09	−0.894153	−0.447077	−	0.894496i	$-0.647535\pi$
$227$	−3058.09	−0.894153	−0.447077	+	0.894496i	$0.647535\pi$
$228$	0	0
$229$	−3767.60	−1.08721	−0.543603	−	0.839343i	$-0.682940\pi$
$229$	−3767.60	−1.08721	−0.543603	+	0.839343i	$0.682940\pi$
$230$	0	0
$231$	−643.131	−0.183181
$232$	0	0
$233$	−4462.05	−1.25459	−0.627294	−	0.778783i	$-0.715837\pi$
$233$	−4462.05	−1.25459	−0.627294	+	0.778783i	$0.715837\pi$
$234$	0	0
$235$	−1186.66	−0.329400
$236$	0	0
$237$	1821.77	0.499310
$238$	0	0
$239$	1840.24	0.498056	0.249028	−	0.968496i	$-0.419889\pi$
$239$	1840.24	0.498056	0.249028	+	0.968496i	$0.419889\pi$
$240$	0	0
$241$	459.993	0.122949	0.0614747	−	0.998109i	$-0.480420\pi$
$241$	459.993	0.122949	0.0614747	+	0.998109i	$0.480420\pi$
$242$	0	0
$243$	−4013.87	−1.05963
$244$	0	0
$245$	115.265	0.0300572
$246$	0	0
$247$	8554.50	2.20368
$248$	0	0
$249$	8727.81	2.22130
$250$	0	0
$251$	−4432.49	−1.11465	−0.557323	−	0.830296i	$-0.688171\pi$
$251$	−4432.49	−1.11465	−0.557323	+	0.830296i	$0.688171\pi$
$252$	0	0
$253$	1598.14	0.397131
$254$	0	0
$255$	−404.557	−0.0993503
$256$	0	0
$257$	−894.161	−0.217028	−0.108514	−	0.994095i	$-0.534609\pi$
$257$	−894.161	−0.217028	−0.108514	+	0.994095i	$0.534609\pi$
$258$	0	0
$259$	−138.332	−0.0331874
$260$	0	0
$261$	−3949.58	−0.936677
$262$	0	0
$263$	7533.82	1.76637	0.883185	−	0.469025i	$-0.155395\pi$
$263$	7533.82	1.76637	0.883185	+	0.469025i	$0.155395\pi$
$264$	0	0
$265$	−1225.68	−0.284125
$266$	0	0
$267$	4131.08	0.946883
$268$	0	0
$269$	3850.40	0.872726	0.436363	−	0.899771i	$-0.356266\pi$
$269$	3850.40	0.872726	0.436363	+	0.899771i	$0.356266\pi$
$270$	0	0
$271$	7395.67	1.65777	0.828883	−	0.559422i	$-0.188977\pi$
$271$	7395.67	1.65777	0.828883	+	0.559422i	$0.188977\pi$
$272$	0	0
$273$	−4766.39	−1.05669
$274$	0	0
$275$	−1314.13	−0.288164
$276$	0	0
$277$	−1024.48	−0.222220	−0.111110	−	0.993808i	$-0.535441\pi$
$277$	−1024.48	−0.222220	−0.111110	+	0.993808i	$0.535441\pi$
$278$	0	0
$279$	2178.41	0.467448
$280$	0	0
$281$	7361.56	1.56283	0.781413	−	0.624014i	$-0.214499\pi$
$281$	7361.56	1.56283	0.781413	+	0.624014i	$0.214499\pi$
$282$	0	0
$283$	−8245.30	−1.73192	−0.865958	−	0.500116i	$-0.833291\pi$
$283$	−8245.30	−1.73192	−0.865958	+	0.500116i	$0.833291\pi$
$284$	0	0
$285$	2061.68	0.428504
$286$	0	0
$287$	720.671	0.148223
$288$	0	0
$289$	−4489.03	−0.913704
$290$	0	0
$291$	11808.3	2.37875
$292$	0	0
$293$	−3900.22	−0.777655	−0.388828	−	0.921310i	$-0.627120\pi$
$293$	−3900.22	−0.777655	−0.388828	+	0.921310i	$0.627120\pi$
$294$	0	0
$295$	1018.61	0.201036
$296$	0	0
$297$	1448.12	0.282925
$298$	0	0
$299$	11844.2	2.29085
$300$	0	0
$301$	999.732	0.191440
$302$	0	0
$303$	−11400.8	−2.16159
$304$	0	0
$305$	−1009.71	−0.189560
$306$	0	0
$307$	1180.90	0.219536	0.109768	−	0.993957i	$-0.464989\pi$
$307$	1180.90	0.219536	0.109768	+	0.993957i	$0.464989\pi$
$308$	0	0
$309$	−3411.91	−0.628144
$310$	0	0
$311$	−8462.39	−1.54295	−0.771476	−	0.636258i	$-0.780481\pi$
$311$	−8462.39	−1.54295	−0.771476	+	0.636258i	$0.780481\pi$
$312$	0	0
$313$	−1070.58	−0.193332	−0.0966658	−	0.995317i	$-0.530818\pi$
$313$	−1070.58	−0.193332	−0.0966658	+	0.995317i	$0.530818\pi$
$314$	0	0
$315$	−704.134	−0.125948
$316$	0	0
$317$	4188.78	0.742161	0.371081	−	0.928601i	$-0.378987\pi$
$317$	4188.78	0.742161	0.371081	+	0.928601i	$0.378987\pi$
$318$	0	0
$319$	−1015.99	−0.178321
$320$	0	0
$321$	8764.87	1.52401
$322$	0	0
$323$	−2160.63	−0.372200
$324$	0	0
$325$	−9739.32	−1.66228
$326$	0	0
$327$	7208.75	1.21910
$328$	0	0
$329$	3531.19	0.591736
$330$	0	0
$331$	−1413.92	−0.234792	−0.117396	−	0.993085i	$-0.537455\pi$
$331$	−1413.92	−0.234792	−0.117396	+	0.993085i	$0.537455\pi$
$332$	0	0
$333$	845.047	0.139064
$334$	0	0
$335$	2460.35	0.401263
$336$	0	0
$337$	3332.50	0.538673	0.269336	−	0.963046i	$-0.413196\pi$
$337$	3332.50	0.538673	0.269336	+	0.963046i	$0.413196\pi$
$338$	0	0
$339$	2630.23	0.421400
$340$	0	0
$341$	560.372	0.0889908
$342$	0	0
$343$	−343.000	−0.0539949
$344$	0	0
$345$	2854.51	0.445455
$346$	0	0
$347$	11550.7	1.78696	0.893480	−	0.449103i	$-0.148256\pi$
$347$	11550.7	1.78696	0.893480	+	0.449103i	$0.148256\pi$
$348$	0	0
$349$	−12512.1	−1.91907	−0.959534	−	0.281592i	$-0.909138\pi$
$349$	−12512.1	−1.91907	−0.959534	+	0.281592i	$0.909138\pi$
$350$	0	0
$351$	10732.4	1.63206
$352$	0	0
$353$	−335.950	−0.0506538	−0.0253269	−	0.999679i	$-0.508063\pi$
$353$	−335.950	−0.0506538	−0.0253269	+	0.999679i	$0.508063\pi$
$354$	0	0
$355$	1881.98	0.281367
$356$	0	0
$357$	1203.86	0.178473
$358$	0	0
$359$	−1767.91	−0.259907	−0.129953	−	0.991520i	$-0.541483\pi$
$359$	−1767.91	−0.259907	−0.129953	+	0.991520i	$0.541483\pi$
$360$	0	0
$361$	4151.91	0.605323
$362$	0	0
$363$	1010.63	0.146128
$364$	0	0
$365$	2028.17	0.290847
$366$	0	0
$367$	1373.56	0.195366	0.0976830	−	0.995218i	$-0.468857\pi$
$367$	1373.56	0.195366	0.0976830	+	0.995218i	$0.468857\pi$
$368$	0	0
$369$	−4402.45	−0.621091
$370$	0	0
$371$	3647.33	0.510404
$372$	0	0
$373$	10297.9	1.42951	0.714753	−	0.699377i	$-0.246539\pi$
$373$	10297.9	1.42951	0.714753	+	0.699377i	$0.246539\pi$
$374$	0	0
$375$	−4803.19	−0.661429
$376$	0	0
$377$	−7529.71	−1.02865
$378$	0	0
$379$	−6853.35	−0.928846	−0.464423	−	0.885613i	$-0.653738\pi$
$379$	−6853.35	−0.928846	−0.464423	+	0.885613i	$0.653738\pi$
$380$	0	0
$381$	843.363	0.113404
$382$	0	0
$383$	2300.66	0.306941	0.153470	−	0.988153i	$-0.450955\pi$
$383$	2300.66	0.306941	0.153470	+	0.988153i	$0.450955\pi$
$384$	0	0
$385$	−181.131	−0.0239774
$386$	0	0
$387$	−6107.18	−0.802185
$388$	0	0
$389$	2759.84	0.359715	0.179858	−	0.983693i	$-0.442436\pi$
$389$	2759.84	0.359715	0.179858	+	0.983693i	$0.442436\pi$
$390$	0	0
$391$	−2991.51	−0.386924
$392$	0	0
$393$	12868.5	1.65173
$394$	0	0
$395$	513.081	0.0653567
$396$	0	0
$397$	−5496.27	−0.694836	−0.347418	−	0.937710i	$-0.612941\pi$
$397$	−5496.27	−0.694836	−0.347418	+	0.937710i	$0.612941\pi$
$398$	0	0
$399$	−6135.05	−0.769767
$400$	0	0
$401$	−12144.8	−1.51243	−0.756215	−	0.654323i	$-0.772954\pi$
$401$	−12144.8	−1.51243	−0.756215	+	0.654323i	$0.772954\pi$
$402$	0	0
$403$	4153.05	0.513345
$404$	0	0
$405$	−129.379	−0.0158738
$406$	0	0
$407$	217.379	0.0264744
$408$	0	0
$409$	587.946	0.0710809	0.0355404	−	0.999368i	$-0.488685\pi$
$409$	587.946	0.0710809	0.0355404	+	0.999368i	$0.488685\pi$
$410$	0	0
$411$	1013.13	0.121591
$412$	0	0
$413$	−3031.12	−0.361142
$414$	0	0
$415$	2458.10	0.290755
$416$	0	0
$417$	12026.9	1.41237
$418$	0	0
$419$	−8368.74	−0.975751	−0.487875	−	0.872913i	$-0.662228\pi$
$419$	−8368.74	−0.975751	−0.487875	+	0.872913i	$0.662228\pi$
$420$	0	0
$421$	−12879.5	−1.49100	−0.745499	−	0.666507i	$-0.767789\pi$
$421$	−12879.5	−1.49100	−0.745499	+	0.666507i	$0.767789\pi$
$422$	0	0
$423$	−21571.4	−2.47953
$424$	0	0
$425$	2459.89	0.280758
$426$	0	0
$427$	3004.64	0.340527
$428$	0	0
$429$	7490.04	0.842943
$430$	0	0
$431$	−16473.6	−1.84108	−0.920541	−	0.390645i	$-0.872252\pi$
$431$	−16473.6	−1.84108	−0.920541	+	0.390645i	$0.872252\pi$
$432$	0	0
$433$	−16301.9	−1.80928	−0.904642	−	0.426172i	$-0.859862\pi$
$433$	−16301.9	−1.80928	−0.904642	+	0.426172i	$0.859862\pi$
$434$	0	0
$435$	−1814.70	−0.200019
$436$	0	0
$437$	15245.2	1.66883
$438$	0	0
$439$	11035.6	1.19978	0.599889	−	0.800083i	$-0.295211\pi$
$439$	11035.6	1.19978	0.599889	+	0.800083i	$0.295211\pi$
$440$	0	0
$441$	2095.33	0.226253
$442$	0	0
$443$	−3713.24	−0.398243	−0.199121	−	0.979975i	$-0.563809\pi$
$443$	−3713.24	−0.398243	−0.199121	+	0.979975i	$0.563809\pi$
$444$	0	0
$445$	1163.47	0.123941
$446$	0	0
$447$	−8024.20	−0.849064
$448$	0	0
$449$	11014.2	1.15767	0.578834	−	0.815445i	$-0.303508\pi$
$449$	11014.2	1.15767	0.578834	+	0.815445i	$0.303508\pi$
$450$	0	0
$451$	−1132.48	−0.118241
$452$	0	0
$453$	22986.1	2.38406
$454$	0	0
$455$	−1342.40	−0.138314
$456$	0	0
$457$	11077.3	1.13386	0.566930	−	0.823766i	$-0.308131\pi$
$457$	11077.3	1.13386	0.566930	+	0.823766i	$0.308131\pi$
$458$	0	0
$459$	−2710.70	−0.275653
$460$	0	0
$461$	5182.57	0.523592	0.261796	−	0.965123i	$-0.415685\pi$
$461$	5182.57	0.523592	0.261796	+	0.965123i	$0.415685\pi$
$462$	0	0
$463$	−9439.70	−0.947517	−0.473758	−	0.880655i	$-0.657103\pi$
$463$	−9439.70	−0.947517	−0.473758	+	0.880655i	$0.657103\pi$
$464$	0	0
$465$	1000.91	0.0998195
$466$	0	0
$467$	−490.043	−0.0485578	−0.0242789	−	0.999705i	$-0.507729\pi$
$467$	−490.043	−0.0485578	−0.0242789	+	0.999705i	$0.507729\pi$
$468$	0	0
$469$	−7321.37	−0.720830
$470$	0	0
$471$	−12929.7	−1.26491
$472$	0	0
$473$	−1571.01	−0.152717
$474$	0	0
$475$	−12536.0	−1.21092
$476$	0	0
$477$	−22280.9	−2.13872
$478$	0	0
$479$	−15375.6	−1.46666	−0.733329	−	0.679874i	$-0.762034\pi$
$479$	−15375.6	−1.46666	−0.733329	+	0.679874i	$0.762034\pi$
$480$	0	0
$481$	1611.05	0.152718
$482$	0	0
$483$	−8494.31	−0.800217
$484$	0	0
$485$	3325.69	0.311365
$486$	0	0
$487$	−11235.8	−1.04547	−0.522733	−	0.852496i	$-0.675088\pi$
$487$	−11235.8	−1.04547	−0.522733	+	0.852496i	$0.675088\pi$
$488$	0	0
$489$	−30437.8	−2.81481
$490$	0	0
$491$	1877.83	0.172598	0.0862988	−	0.996269i	$-0.472496\pi$
$491$	1877.83	0.172598	0.0862988	+	0.996269i	$0.472496\pi$
$492$	0	0
$493$	1901.80	0.173738
$494$	0	0
$495$	1106.50	0.100471
$496$	0	0
$497$	−5600.31	−0.505449
$498$	0	0
$499$	21395.3	1.91941	0.959704	−	0.281014i	$-0.0906708\pi$
$499$	21395.3	1.91941	0.959704	+	0.281014i	$0.0906708\pi$
$500$	0	0
$501$	−18368.9	−1.63805
$502$	0	0
$503$	−101.167	−0.00896782	−0.00448391	−	0.999990i	$-0.501427\pi$
$503$	−101.167	−0.00896782	−0.00448391	+	0.999990i	$0.501427\pi$
$504$	0	0
$505$	−3210.93	−0.282939
$506$	0	0
$507$	37160.3	3.25512
$508$	0	0
$509$	17744.1	1.54517	0.772587	−	0.634909i	$-0.218962\pi$
$509$	17744.1	1.54517	0.772587	+	0.634909i	$0.218962\pi$
$510$	0	0
$511$	−6035.32	−0.522479
$512$	0	0
$513$	13814.2	1.18891
$514$	0	0
$515$	−960.927	−0.0822204
$516$	0	0
$517$	−5549.02	−0.472042
$518$	0	0
$519$	12579.5	1.06393
$520$	0	0
$521$	−12669.9	−1.06541	−0.532707	−	0.846300i	$-0.678825\pi$
$521$	−12669.9	−1.06541	−0.532707	+	0.846300i	$0.678825\pi$
$522$	0	0
$523$	8960.78	0.749192	0.374596	−	0.927188i	$-0.377781\pi$
$523$	8960.78	0.749192	0.374596	+	0.927188i	$0.377781\pi$
$524$	0	0
$525$	6984.78	0.580649
$526$	0	0
$527$	−1048.95	−0.0867036
$528$	0	0
$529$	8940.80	0.734840
$530$	0	0
$531$	18516.6	1.51328
$532$	0	0
$533$	−8393.09	−0.682073
$534$	0	0
$535$	2468.53	0.199484
$536$	0	0
$537$	−26010.1	−2.09016
$538$	0	0
$539$	539.000	0.0430730
$540$	0	0
$541$	16692.5	1.32656	0.663279	−	0.748373i	$-0.269164\pi$
$541$	16692.5	1.32656	0.663279	+	0.748373i	$0.269164\pi$
$542$	0	0
$543$	−12456.5	−0.984453
$544$	0	0
$545$	2030.27	0.159573
$546$	0	0
$547$	20444.7	1.59808	0.799041	−	0.601277i	$-0.205341\pi$
$547$	20444.7	1.59808	0.799041	+	0.601277i	$0.205341\pi$
$548$	0	0
$549$	−18354.8	−1.42690
$550$	0	0
$551$	−9691.85	−0.749341
$552$	0	0
$553$	−1526.80	−0.117407
$554$	0	0
$555$	388.272	0.0296959
$556$	0	0
$557$	−7842.15	−0.596557	−0.298279	−	0.954479i	$-0.596412\pi$
$557$	−7842.15	−0.596557	−0.298279	+	0.954479i	$0.596412\pi$
$558$	0	0
$559$	−11643.1	−0.880948
$560$	0	0
$561$	−1891.78	−0.142372
$562$	0	0
$563$	−18045.9	−1.35088	−0.675438	−	0.737417i	$-0.736045\pi$
$563$	−18045.9	−1.35088	−0.675438	+	0.737417i	$0.736045\pi$
$564$	0	0
$565$	740.777	0.0551588
$566$	0	0
$567$	385.000	0.0285158
$568$	0	0
$569$	−5416.93	−0.399103	−0.199551	−	0.979887i	$-0.563948\pi$
$569$	−5416.93	−0.399103	−0.199551	+	0.979887i	$0.563948\pi$
$570$	0	0
$571$	9301.85	0.681735	0.340867	−	0.940111i	$-0.389279\pi$
$571$	9301.85	0.681735	0.340867	+	0.940111i	$0.389279\pi$
$572$	0	0
$573$	−12976.1	−0.946042
$574$	0	0
$575$	−17356.7	−1.25883
$576$	0	0
$577$	−20255.6	−1.46144	−0.730720	−	0.682677i	$-0.760816\pi$
$577$	−20255.6	−1.46144	−0.730720	+	0.682677i	$0.760816\pi$
$578$	0	0
$579$	353.939	0.0254045
$580$	0	0
$581$	−7314.67	−0.522313
$582$	0	0
$583$	−5731.52	−0.407162
$584$	0	0
$585$	8200.50	0.579571
$586$	0	0
$587$	24500.9	1.72276	0.861381	−	0.507960i	$-0.169600\pi$
$587$	24500.9	1.72276	0.861381	+	0.507960i	$0.169600\pi$
$588$	0	0
$589$	5345.59	0.373958
$590$	0	0
$591$	−34415.9	−2.39540
$592$	0	0
$593$	3130.19	0.216765	0.108382	−	0.994109i	$-0.465433\pi$
$593$	3130.19	0.216765	0.108382	+	0.994109i	$0.465433\pi$
$594$	0	0
$595$	339.054	0.0233611
$596$	0	0
$597$	−30078.6	−2.06203
$598$	0	0
$599$	−13783.7	−0.940208	−0.470104	−	0.882611i	$-0.655784\pi$
$599$	−13783.7	−0.940208	−0.470104	+	0.882611i	$0.655784\pi$
$600$	0	0
$601$	−15509.3	−1.05264	−0.526320	−	0.850287i	$-0.676429\pi$
$601$	−15509.3	−1.05264	−0.526320	+	0.850287i	$0.676429\pi$
$602$	0	0
$603$	44724.9	3.02046
$604$	0	0
$605$	284.634	0.0191273
$606$	0	0
$607$	10643.7	0.711722	0.355861	−	0.934539i	$-0.384188\pi$
$607$	10643.7	0.711722	0.355861	+	0.934539i	$0.384188\pi$
$608$	0	0
$609$	5400.10	0.359316
$610$	0	0
$611$	−41125.1	−2.72298
$612$	0	0
$613$	13187.6	0.868910	0.434455	−	0.900694i	$-0.356941\pi$
$613$	13187.6	0.868910	0.434455	+	0.900694i	$0.356941\pi$
$614$	0	0
$615$	−2022.78	−0.132629
$616$	0	0
$617$	−13314.5	−0.868757	−0.434379	−	0.900730i	$-0.643032\pi$
$617$	−13314.5	−0.868757	−0.434379	+	0.900730i	$0.643032\pi$
$618$	0	0
$619$	−28386.1	−1.84319	−0.921593	−	0.388158i	$-0.873111\pi$
$619$	−28386.1	−1.84319	−0.921593	+	0.388158i	$0.873111\pi$
$620$	0	0
$621$	19126.5	1.23594
$622$	0	0
$623$	−3462.20	−0.222649
$624$	0	0
$625$	13580.5	0.869154
$626$	0	0
$627$	9640.80	0.614061
$628$	0	0
$629$	−406.906	−0.0257940
$630$	0	0
$631$	4782.25	0.301709	0.150855	−	0.988556i	$-0.451797\pi$
$631$	4782.25	0.301709	0.150855	+	0.988556i	$0.451797\pi$
$632$	0	0
$633$	−22901.4	−1.43799
$634$	0	0
$635$	237.524	0.0148439
$636$	0	0
$637$	3994.65	0.248468
$638$	0	0
$639$	34211.3	2.11796
$640$	0	0
$641$	−1892.25	−0.116598	−0.0582991	−	0.998299i	$-0.518568\pi$
$641$	−1892.25	−0.116598	−0.0582991	+	0.998299i	$0.518568\pi$
$642$	0	0
$643$	−14182.1	−0.869811	−0.434906	−	0.900476i	$-0.643218\pi$
$643$	−14182.1	−0.869811	−0.434906	+	0.900476i	$0.643218\pi$
$644$	0	0
$645$	−2806.05	−0.171300
$646$	0	0
$647$	6763.25	0.410959	0.205480	−	0.978661i	$-0.434125\pi$
$647$	6763.25	0.410959	0.205480	+	0.978661i	$0.434125\pi$
$648$	0	0
$649$	4763.19	0.288091
$650$	0	0
$651$	−2978.45	−0.179316
$652$	0	0
$653$	4924.05	0.295088	0.147544	−	0.989055i	$-0.452863\pi$
$653$	4924.05	0.295088	0.147544	+	0.989055i	$0.452863\pi$
$654$	0	0
$655$	3624.28	0.216202
$656$	0	0
$657$	36868.7	2.18932
$658$	0	0
$659$	−19546.2	−1.15540	−0.577702	−	0.816248i	$-0.696050\pi$
$659$	−19546.2	−1.15540	−0.577702	+	0.816248i	$0.696050\pi$
$660$	0	0
$661$	−16873.1	−0.992874	−0.496437	−	0.868073i	$-0.665359\pi$
$661$	−16873.1	−0.992874	−0.496437	+	0.868073i	$0.665359\pi$
$662$	0	0
$663$	−14020.4	−0.821278
$664$	0	0
$665$	−1727.87	−0.100758
$666$	0	0
$667$	−13418.9	−0.778983
$668$	0	0
$669$	−27157.9	−1.56948
$670$	0	0
$671$	−4721.58	−0.271646
$672$	0	0
$673$	−5675.73	−0.325086	−0.162543	−	0.986701i	$-0.551970\pi$
$673$	−5675.73	−0.325086	−0.162543	+	0.986701i	$0.551970\pi$
$674$	0	0
$675$	−15727.5	−0.896816
$676$	0	0
$677$	−26954.3	−1.53019	−0.765094	−	0.643919i	$-0.777308\pi$
$677$	−26954.3	−1.53019	−0.765094	+	0.643919i	$0.777308\pi$
$678$	0	0
$679$	−9896.43	−0.559337
$680$	0	0
$681$	−25542.3	−1.43727
$682$	0	0
$683$	−21935.9	−1.22892	−0.614460	−	0.788948i	$-0.710626\pi$
$683$	−21935.9	−1.22892	−0.614460	+	0.788948i	$0.710626\pi$
$684$	0	0
$685$	285.337	0.0159156
$686$	0	0
$687$	−31468.3	−1.74759
$688$	0	0
$689$	−42477.6	−2.34872
$690$	0	0
$691$	14307.0	0.787649	0.393825	−	0.919186i	$-0.371152\pi$
$691$	14307.0	0.787649	0.393825	+	0.919186i	$0.371152\pi$
$692$	0	0
$693$	−3292.65	−0.180487
$694$	0	0
$695$	3387.24	0.184871
$696$	0	0
$697$	2119.86	0.115202
$698$	0	0
$699$	−37268.6	−2.01664
$700$	0	0
$701$	−18083.0	−0.974304	−0.487152	−	0.873317i	$-0.661964\pi$
$701$	−18083.0	−0.974304	−0.487152	+	0.873317i	$0.661964\pi$
$702$	0	0
$703$	2073.66	0.111251
$704$	0	0
$705$	−9911.38	−0.529481
$706$	0	0
$707$	9554.91	0.508273
$708$	0	0
$709$	11094.7	0.587689	0.293844	−	0.955853i	$-0.405065\pi$
$709$	11094.7	0.587689	0.293844	+	0.955853i	$0.405065\pi$
$710$	0	0
$711$	9326.94	0.491966
$712$	0	0
$713$	7401.26	0.388751
$714$	0	0
$715$	2109.49	0.110336
$716$	0	0
$717$	15370.3	0.800580
$718$	0	0
$719$	−5377.04	−0.278901	−0.139451	−	0.990229i	$-0.544534\pi$
$719$	−5377.04	−0.278901	−0.139451	+	0.990229i	$0.544534\pi$
$720$	0	0
$721$	2859.48	0.147701
$722$	0	0
$723$	3842.03	0.197630
$724$	0	0
$725$	11034.2	0.565242
$726$	0	0
$727$	22454.6	1.14552	0.572762	−	0.819722i	$-0.305872\pi$
$727$	22454.6	1.14552	0.572762	+	0.819722i	$0.305872\pi$
$728$	0	0
$729$	−32040.2	−1.62781
$730$	0	0
$731$	2940.72	0.148792
$732$	0	0
$733$	−19535.4	−0.984386	−0.492193	−	0.870486i	$-0.663805\pi$
$733$	−19535.4	−0.984386	−0.492193	+	0.870486i	$0.663805\pi$
$734$	0	0
$735$	962.735	0.0483143
$736$	0	0
$737$	11505.0	0.575023
$738$	0	0
$739$	17528.7	0.872533	0.436267	−	0.899817i	$-0.356300\pi$
$739$	17528.7	0.872533	0.436267	+	0.899817i	$0.356300\pi$
$740$	0	0
$741$	71450.2	3.54222
$742$	0	0
$743$	−25311.7	−1.24979	−0.624895	−	0.780709i	$-0.714858\pi$
$743$	−25311.7	−1.24979	−0.624895	+	0.780709i	$0.714858\pi$
$744$	0	0
$745$	−2259.93	−0.111137
$746$	0	0
$747$	44684.0	2.18862
$748$	0	0
$749$	−7345.73	−0.358354
$750$	0	0
$751$	11666.5	0.566865	0.283432	−	0.958992i	$-0.408527\pi$
$751$	11666.5	0.566865	0.283432	+	0.958992i	$0.408527\pi$
$752$	0	0
$753$	−37021.7	−1.79169
$754$	0	0
$755$	6473.78	0.312059
$756$	0	0
$757$	12942.5	0.621402	0.310701	−	0.950508i	$-0.399436\pi$
$757$	12942.5	0.621402	0.310701	+	0.950508i	$0.399436\pi$
$758$	0	0
$759$	13348.2	0.638352
$760$	0	0
$761$	2447.96	0.116608	0.0583040	−	0.998299i	$-0.481431\pi$
$761$	2447.96	0.116608	0.0583040	+	0.998299i	$0.481431\pi$
$762$	0	0
$763$	−6041.56	−0.286657
$764$	0	0
$765$	−2071.22	−0.0978891
$766$	0	0
$767$	35301.0	1.66186
$768$	0	0
$769$	30557.1	1.43292	0.716461	−	0.697627i	$-0.245761\pi$
$769$	30557.1	1.43292	0.716461	+	0.697627i	$0.245761\pi$
$770$	0	0
$771$	−7468.34	−0.348853
$772$	0	0
$773$	25089.5	1.16741	0.583704	−	0.811966i	$-0.301603\pi$
$773$	25089.5	1.16741	0.583704	+	0.811966i	$0.301603\pi$
$774$	0	0
$775$	−6085.97	−0.282083
$776$	0	0
$777$	−1155.40	−0.0533458
$778$	0	0
$779$	−10803.2	−0.496872
$780$	0	0
$781$	8800.48	0.403209
$782$	0	0
$783$	−12159.3	−0.554965
$784$	0	0
$785$	−3641.53	−0.165569
$786$	0	0
$787$	−929.983	−0.0421224	−0.0210612	−	0.999778i	$-0.506704\pi$
$787$	−929.983	−0.0421224	−0.0210612	+	0.999778i	$0.506704\pi$
$788$	0	0
$789$	62925.1	2.83928
$790$	0	0
$791$	−2204.36	−0.0990875
$792$	0	0
$793$	−34992.7	−1.56700
$794$	0	0
$795$	−10237.3	−0.456706
$796$	0	0
$797$	13740.7	0.610692	0.305346	−	0.952242i	$-0.401228\pi$
$797$	13740.7	0.610692	0.305346	+	0.952242i	$0.401228\pi$
$798$	0	0
$799$	10387.1	0.459910
$800$	0	0
$801$	21150.0	0.932956
$802$	0	0
$803$	9484.07	0.416794
$804$	0	0
$805$	−2392.33	−0.104744
$806$	0	0
$807$	32159.9	1.40283
$808$	0	0
$809$	16504.8	0.717277	0.358639	−	0.933477i	$-0.383241\pi$
$809$	16504.8	0.717277	0.358639	+	0.933477i	$0.383241\pi$
$810$	0	0
$811$	11540.3	0.499673	0.249836	−	0.968288i	$-0.419623\pi$
$811$	11540.3	0.499673	0.249836	+	0.968288i	$0.419623\pi$
$812$	0	0
$813$	61771.2	2.66471
$814$	0	0
$815$	−8572.47	−0.368442
$816$	0	0
$817$	−14986.4	−0.641747
$818$	0	0
$819$	−24402.6	−1.04114
$820$	0	0
$821$	1510.96	0.0642298	0.0321149	−	0.999484i	$-0.489776\pi$
$821$	1510.96	0.0642298	0.0321149	+	0.999484i	$0.489776\pi$
$822$	0	0
$823$	−3331.48	−0.141103	−0.0705516	−	0.997508i	$-0.522476\pi$
$823$	−3331.48	−0.141103	−0.0705516	+	0.997508i	$0.522476\pi$
$824$	0	0
$825$	−10976.1	−0.463198
$826$	0	0
$827$	−13979.7	−0.587813	−0.293907	−	0.955834i	$-0.594955\pi$
$827$	−13979.7	−0.587813	−0.293907	+	0.955834i	$0.594955\pi$
$828$	0	0
$829$	40518.7	1.69755	0.848777	−	0.528752i	$-0.177340\pi$
$829$	40518.7	1.69755	0.848777	+	0.528752i	$0.177340\pi$
$830$	0	0
$831$	−8556.79	−0.357198
$832$	0	0
$833$	−1008.94	−0.0419660
$834$	0	0
$835$	−5173.42	−0.214411
$836$	0	0
$837$	6706.52	0.276955
$838$	0	0
$839$	27143.3	1.11691	0.558457	−	0.829534i	$-0.311394\pi$
$839$	27143.3	1.11691	0.558457	+	0.829534i	$0.311394\pi$
$840$	0	0
$841$	−15858.2	−0.650219
$842$	0	0
$843$	61486.4	2.51210
$844$	0	0
$845$	10465.8	0.426076
$846$	0	0
$847$	−847.000	−0.0343604
$848$	0	0
$849$	−68867.7	−2.78390
$850$	0	0
$851$	2871.09	0.115652
$852$	0	0
$853$	20197.7	0.810733	0.405367	−	0.914154i	$-0.367144\pi$
$853$	20197.7	0.810733	0.405367	+	0.914154i	$0.367144\pi$
$854$	0	0
$855$	10555.3	0.422202
$856$	0	0
$857$	13913.2	0.554569	0.277285	−	0.960788i	$-0.410565\pi$
$857$	13913.2	0.554569	0.277285	+	0.960788i	$0.410565\pi$
$858$	0	0
$859$	43528.1	1.72894	0.864471	−	0.502683i	$-0.167654\pi$
$859$	43528.1	1.72894	0.864471	+	0.502683i	$0.167654\pi$
$860$	0	0
$861$	6019.30	0.238254
$862$	0	0
$863$	−28016.6	−1.10509	−0.552547	−	0.833482i	$-0.686344\pi$
$863$	−28016.6	−1.10509	−0.552547	+	0.833482i	$0.686344\pi$
$864$	0	0
$865$	3542.88	0.139262
$866$	0	0
$867$	−37493.9	−1.46870
$868$	0	0
$869$	2399.26	0.0936584
$870$	0	0
$871$	85266.2	3.31703
$872$	0	0
$873$	60455.5	2.34377
$874$	0	0
$875$	4025.49	0.155527
$876$	0	0
$877$	4700.43	0.180983	0.0904916	−	0.995897i	$-0.471156\pi$
$877$	4700.43	0.180983	0.0904916	+	0.995897i	$0.471156\pi$
$878$	0	0
$879$	−32576.0	−1.25001
$880$	0	0
$881$	23698.4	0.906267	0.453133	−	0.891443i	$-0.350306\pi$
$881$	23698.4	0.906267	0.453133	+	0.891443i	$0.350306\pi$
$882$	0	0
$883$	−24496.4	−0.933602	−0.466801	−	0.884362i	$-0.654594\pi$
$883$	−24496.4	−0.933602	−0.466801	+	0.884362i	$0.654594\pi$
$884$	0	0
$885$	8507.76	0.323147
$886$	0	0
$887$	−32811.7	−1.24206	−0.621031	−	0.783786i	$-0.713286\pi$
$887$	−32811.7	−1.24206	−0.621031	+	0.783786i	$0.713286\pi$
$888$	0	0
$889$	−706.812	−0.0266656
$890$	0	0
$891$	−605.000	−0.0227478
$892$	0	0
$893$	−52934.1	−1.98362
$894$	0	0
$895$	−7325.46	−0.273590
$896$	0	0
$897$	98926.6	3.68234
$898$	0	0
$899$	−4705.21	−0.174558
$900$	0	0
$901$	10728.7	0.396697
$902$	0	0
$903$	8350.11	0.307723
$904$	0	0
$905$	−3508.23	−0.128859
$906$	0	0
$907$	16608.9	0.608037	0.304018	−	0.952666i	$-0.401672\pi$
$907$	16608.9	0.608037	0.304018	+	0.952666i	$0.401672\pi$
$908$	0	0
$909$	−58369.2	−2.12980
$910$	0	0
$911$	35779.4	1.30123	0.650617	−	0.759406i	$-0.274510\pi$
$911$	35779.4	1.30123	0.650617	+	0.759406i	$0.274510\pi$
$912$	0	0
$913$	11494.5	0.416661
$914$	0	0
$915$	−8433.46	−0.304701
$916$	0	0
$917$	−10784.9	−0.388386
$918$	0	0
$919$	31726.8	1.13881	0.569406	−	0.822056i	$-0.307173\pi$
$919$	31726.8	1.13881	0.569406	+	0.822056i	$0.307173\pi$
$920$	0	0
$921$	9863.28	0.352884
$922$	0	0
$923$	65222.4	2.32592
$924$	0	0
$925$	−2360.87	−0.0839187
$926$	0	0
$927$	−17468.0	−0.618905
$928$	0	0
$929$	30258.0	1.06860	0.534302	−	0.845294i	$-0.320575\pi$
$929$	30258.0	1.06860	0.534302	+	0.845294i	$0.320575\pi$
$930$	0	0
$931$	5141.71	0.181002
$932$	0	0
$933$	−70680.8	−2.48016
$934$	0	0
$935$	−532.799	−0.0186357
$936$	0	0
$937$	20434.2	0.712438	0.356219	−	0.934402i	$-0.384066\pi$
$937$	20434.2	0.712438	0.356219	+	0.934402i	$0.384066\pi$
$938$	0	0
$939$	−8941.86	−0.310763
$940$	0	0
$941$	−29128.2	−1.00909	−0.504545	−	0.863385i	$-0.668340\pi$
$941$	−29128.2	−1.00909	−0.504545	+	0.863385i	$0.668340\pi$
$942$	0	0
$943$	−14957.6	−0.516527
$944$	0	0
$945$	−2167.77	−0.0746217
$946$	0	0
$947$	−7701.21	−0.264262	−0.132131	−	0.991232i	$-0.542182\pi$
$947$	−7701.21	−0.264262	−0.132131	+	0.991232i	$0.542182\pi$
$948$	0	0
$949$	70288.6	2.40428
$950$	0	0
$951$	34986.1	1.19296
$952$	0	0
$953$	4283.51	0.145600	0.0727998	−	0.997347i	$-0.476807\pi$
$953$	4283.51	0.145600	0.0727998	+	0.997347i	$0.476807\pi$
$954$	0	0
$955$	−3654.57	−0.123831
$956$	0	0
$957$	−8485.87	−0.286635
$958$	0	0
$959$	−849.092	−0.0285908
$960$	0	0
$961$	−27195.8	−0.912887
$962$	0	0
$963$	44873.7	1.50159
$964$	0	0
$965$	99.6832	0.00332530
$966$	0	0
$967$	−38091.6	−1.26675	−0.633373	−	0.773847i	$-0.718330\pi$
$967$	−38091.6	−1.26675	−0.633373	+	0.773847i	$0.718330\pi$
$968$	0	0
$969$	−18046.3	−0.598279
$970$	0	0
$971$	4194.45	0.138626	0.0693132	−	0.997595i	$-0.477919\pi$
$971$	4194.45	0.138626	0.0693132	+	0.997595i	$0.477919\pi$
$972$	0	0
$973$	−10079.6	−0.332103
$974$	0	0
$975$	−81346.2	−2.67196
$976$	0	0
$977$	−25613.1	−0.838725	−0.419362	−	0.907819i	$-0.637746\pi$
$977$	−25613.1	−0.838725	−0.419362	+	0.907819i	$0.637746\pi$
$978$	0	0
$979$	5440.61	0.177612
$980$	0	0
$981$	36906.8	1.20117
$982$	0	0
$983$	23674.3	0.768152	0.384076	−	0.923301i	$-0.374520\pi$
$983$	23674.3	0.768152	0.384076	+	0.923301i	$0.374520\pi$
$984$	0	0
$985$	−9692.88	−0.313544
$986$	0	0
$987$	29493.8	0.951162
$988$	0	0
$989$	−20749.5	−0.667133
$990$	0	0
$991$	−47328.6	−1.51710	−0.758548	−	0.651617i	$-0.774091\pi$
$991$	−47328.6	−1.51710	−0.758548	+	0.651617i	$0.774091\pi$
$992$	0	0
$993$	−11809.6	−0.377408
$994$	0	0
$995$	−8471.30	−0.269908
$996$	0	0
$997$	−36318.1	−1.15367	−0.576834	−	0.816862i	$-0.695712\pi$
$997$	−36318.1	−1.15367	−0.576834	+	0.816862i	$0.695712\pi$
$998$	0	0
$999$	2601.59	0.0823930

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1232.4.a.p.1.2		2
4.3	odd	2		154.4.a.f.1.1	✓	2
12.11	even	2		1386.4.a.ba.1.1		2
28.27	even	2		1078.4.a.j.1.2		2
44.43	even	2		1694.4.a.l.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
154.4.a.f.1.1	✓	2	4.3	odd	2
1078.4.a.j.1.2		2	28.27	even	2
1232.4.a.p.1.2		2	1.1	even	1	trivial
1386.4.a.ba.1.1		2	12.11	even	2
1694.4.a.l.1.1		2	44.43	even	2

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Classical	Maass
Hilbert	Bianchi

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

Introduction

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Modular forms

Varieties

Fields

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Groups

Database

Properties

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Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	1232.4.a.p.1.2

Level	$1232$
Weight	$4$
Character	1232.1
Self dual	yes
Analytic conductor	$72.690$
Analytic rank	$0$
Dimension	$2$
CM	no
Inner twists	$1$