LMFDB - Embedded newform 736.4.a.h.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [736,4,Mod(1,736)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("736.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(736, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 736 = 2^{5} \cdot 23 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	736.a (trivial)

Newform invariants

comment:select newform

sage:traces = [9,0,0,0,0,0,-42] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$43.4254057642$
Analytic rank:	$1$
Dimension:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - 3x^{8} - 147x^{7} - 97x^{6} + 4561x^{5} + 7383x^{4} - 31427x^{3} - 43981x^{2} + 17596x + 12306 $ x^9 - 3x^8 - 147x^7 - 97x^6 + 4561x^5 + 7383x^4 - 31427x^3 - 43981x^2 + 17596x + 12306
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{10} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$12.7496$ of defining polynomial
Character	$\chi$	$=$	736.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-5.48281 q^{3} +1.44246 q^{5} +5.95524 q^{7} +3.06121 q^{9} -34.3096 q^{11} +76.5477 q^{13} -7.90873 q^{15} -13.1296 q^{17} -45.0212 q^{19} -32.6514 q^{21} -23.0000 q^{23} -122.919 q^{25} +131.252 q^{27} +92.3349 q^{29} +231.013 q^{31} +188.113 q^{33} +8.59019 q^{35} +438.001 q^{37} -419.696 q^{39} +91.0056 q^{41} -415.535 q^{43} +4.41567 q^{45} -476.936 q^{47} -307.535 q^{49} +71.9873 q^{51} -262.366 q^{53} -49.4902 q^{55} +246.843 q^{57} +352.842 q^{59} +685.060 q^{61} +18.2302 q^{63} +110.417 q^{65} -543.860 q^{67} +126.105 q^{69} -840.263 q^{71} +195.510 q^{73} +673.943 q^{75} -204.322 q^{77} +337.507 q^{79} -802.282 q^{81} -1269.79 q^{83} -18.9390 q^{85} -506.255 q^{87} -1112.84 q^{89} +455.860 q^{91} -1266.60 q^{93} -64.9413 q^{95} -605.888 q^{97} -105.029 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q - 42 q^{7} + 71 q^{9} - 66 q^{11} + 48 q^{13} - 90 q^{15} - 32 q^{17} + 6 q^{21} - 207 q^{23} + 135 q^{25} - 88 q^{29} - 556 q^{31} - 230 q^{33} - 368 q^{35} - 368 q^{37} - 468 q^{39} - 216 q^{41} + 552 q^{43}+ \cdots - 552 q^{99}+O(q^{100}) $ 9 * q - 42 * q^7 + 71 * q^9 - 66 * q^11 + 48 * q^13 - 90 * q^15 - 32 * q^17 + 6 * q^21 - 207 * q^23 + 135 * q^25 - 88 * q^29 - 556 * q^31 - 230 * q^33 - 368 * q^35 - 368 * q^37 - 468 * q^39 - 216 * q^41 + 552 * q^43 - 206 * q^45 - 1300 * q^47 + 1041 * q^49 - 1658 * q^51 + 510 * q^53 - 1948 * q^55 - 804 * q^57 - 1444 * q^59 - 882 * q^61 - 3128 * q^63 + 334 * q^65 - 1186 * q^67 - 2440 * q^71 - 368 * q^73 - 1636 * q^75 + 556 * q^77 - 2576 * q^79 + 1069 * q^81 - 1710 * q^83 + 640 * q^85 - 4844 * q^87 - 474 * q^89 - 2790 * q^91 + 1178 * q^93 - 4564 * q^95 - 884 * q^97 - 552 * 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
$3$	−5.48281	−1.05517	−0.527584	−	0.849503i	$-0.676902\pi$
$3$	−5.48281	−1.05517	−0.527584	+	0.849503i	$0.676902\pi$
$4$	0	0
$5$	1.44246	0.129017	0.0645087	−	0.997917i	$-0.479452\pi$
$5$	1.44246	0.129017	0.0645087	+	0.997917i	$0.479452\pi$
$6$	0	0
$7$	5.95524	0.321553	0.160776	−	0.986991i	$-0.448600\pi$
$7$	5.95524	0.321553	0.160776	+	0.986991i	$0.448600\pi$
$8$	0	0
$9$	3.06121	0.113378
$10$	0	0
$11$	−34.3096	−0.940431	−0.470215	−	0.882552i	$-0.655824\pi$
$11$	−34.3096	−0.940431	−0.470215	+	0.882552i	$0.655824\pi$
$12$	0	0
$13$	76.5477	1.63312	0.816558	−	0.577263i	$-0.195879\pi$
$13$	76.5477	1.63312	0.816558	+	0.577263i	$0.195879\pi$
$14$	0	0
$15$	−7.90873	−0.136135
$16$	0	0
$17$	−13.1296	−0.187318	−0.0936590	−	0.995604i	$-0.529856\pi$
$17$	−13.1296	−0.187318	−0.0936590	+	0.995604i	$0.529856\pi$
$18$	0	0
$19$	−45.0212	−0.543609	−0.271805	−	0.962352i	$-0.587620\pi$
$19$	−45.0212	−0.543609	−0.271805	+	0.962352i	$0.587620\pi$
$20$	0	0
$21$	−32.6514	−0.339292
$22$	0	0
$23$	−23.0000	−0.208514
$23$	−23.0000	−0.208514
$24$	0	0
$25$	−122.919	−0.983354
$26$	0	0
$27$	131.252	0.935534
$28$	0	0
$29$	92.3349	0.591247	0.295624	−	0.955305i	$-0.404473\pi$
$29$	92.3349	0.591247	0.295624	+	0.955305i	$0.404473\pi$
$30$	0	0
$31$	231.013	1.33843	0.669213	−	0.743071i	$-0.266631\pi$
$31$	231.013	1.33843	0.669213	+	0.743071i	$0.266631\pi$
$32$	0	0
$33$	188.113	0.992312
$34$	0	0
$35$	8.59019	0.0414859
$36$	0	0
$37$	438.001	1.94613	0.973067	−	0.230524i	$-0.0740439\pi$
$37$	438.001	1.94613	0.973067	+	0.230524i	$0.0740439\pi$
$38$	0	0
$39$	−419.696	−1.72321
$40$	0	0
$41$	91.0056	0.346651	0.173325	−	0.984865i	$-0.444549\pi$
$41$	91.0056	0.346651	0.173325	+	0.984865i	$0.444549\pi$
$42$	0	0
$43$	−415.535	−1.47369	−0.736843	−	0.676064i	$-0.763684\pi$
$43$	−415.535	−1.47369	−0.736843	+	0.676064i	$0.763684\pi$
$44$	0	0
$45$	4.41567	0.0146278
$46$	0	0
$47$	−476.936	−1.48018	−0.740088	−	0.672510i	$-0.765216\pi$
$47$	−476.936	−1.48018	−0.740088	+	0.672510i	$0.765216\pi$
$48$	0	0
$49$	−307.535	−0.896604
$50$	0	0
$51$	71.9873	0.197652
$52$	0	0
$53$	−262.366	−0.679975	−0.339988	−	0.940430i	$-0.610423\pi$
$53$	−262.366	−0.679975	−0.339988	+	0.940430i	$0.610423\pi$
$54$	0	0
$55$	−49.4902	−0.121332
$56$	0	0
$57$	246.843	0.573599
$58$	0	0
$59$	352.842	0.778578	0.389289	−	0.921116i	$-0.372721\pi$
$59$	352.842	0.778578	0.389289	+	0.921116i	$0.372721\pi$
$60$	0	0
$61$	685.060	1.43792	0.718959	−	0.695052i	$-0.244619\pi$
$61$	685.060	1.43792	0.718959	+	0.695052i	$0.244619\pi$
$62$	0	0
$63$	18.2302	0.0364570
$64$	0	0
$65$	110.417	0.210701
$66$	0	0
$67$	−543.860	−0.991688	−0.495844	−	0.868412i	$-0.665141\pi$
$67$	−543.860	−0.991688	−0.495844	+	0.868412i	$0.665141\pi$
$68$	0	0
$69$	126.105	0.220018
$70$	0	0
$71$	−840.263	−1.40452	−0.702259	−	0.711921i	$-0.747825\pi$
$71$	−840.263	−1.40452	−0.702259	+	0.711921i	$0.747825\pi$
$72$	0	0
$73$	195.510	0.313462	0.156731	−	0.987641i	$-0.449904\pi$
$73$	195.510	0.313462	0.156731	+	0.987641i	$0.449904\pi$
$74$	0	0
$75$	673.943	1.03760
$76$	0	0
$77$	−204.322	−0.302398
$78$	0	0
$79$	337.507	0.480664	0.240332	−	0.970691i	$-0.422744\pi$
$79$	337.507	0.480664	0.240332	+	0.970691i	$0.422744\pi$
$80$	0	0
$81$	−802.282	−1.10052
$82$	0	0
$83$	−1269.79	−1.67925	−0.839625	−	0.543166i	$-0.817226\pi$
$83$	−1269.79	−1.67925	−0.839625	+	0.543166i	$0.817226\pi$
$84$	0	0
$85$	−18.9390	−0.0241673
$86$	0	0
$87$	−506.255	−0.623865
$88$	0	0
$89$	−1112.84	−1.32541	−0.662703	−	0.748882i	$-0.730591\pi$
$89$	−1112.84	−1.32541	−0.662703	+	0.748882i	$0.730591\pi$
$90$	0	0
$91$	455.860	0.525133
$92$	0	0
$93$	−1266.60	−1.41226
$94$	0	0
$95$	−64.9413	−0.0701351
$96$	0	0
$97$	−605.888	−0.634212	−0.317106	−	0.948390i	$-0.602711\pi$
$97$	−605.888	−0.634212	−0.317106	+	0.948390i	$0.602711\pi$
$98$	0	0
$99$	−105.029	−0.106624
$100$	0	0
$101$	1546.51	1.52360	0.761800	−	0.647812i	$-0.224316\pi$
$101$	1546.51	1.52360	0.761800	+	0.647812i	$0.224316\pi$
$102$	0	0
$103$	−49.4195	−0.0472762	−0.0236381	−	0.999721i	$-0.507525\pi$
$103$	−49.4195	−0.0472762	−0.0236381	+	0.999721i	$0.507525\pi$
$104$	0	0
$105$	−47.0984	−0.0437746
$106$	0	0
$107$	−1338.48	−1.20930	−0.604652	−	0.796490i	$-0.706688\pi$
$107$	−1338.48	−1.20930	−0.604652	+	0.796490i	$0.706688\pi$
$108$	0	0
$109$	−1421.06	−1.24874	−0.624369	−	0.781130i	$-0.714644\pi$
$109$	−1421.06	−1.24874	−0.624369	+	0.781130i	$0.714644\pi$
$110$	0	0
$111$	−2401.48	−2.05350
$112$	0	0
$113$	−1231.68	−1.02537	−0.512683	−	0.858578i	$-0.671348\pi$
$113$	−1231.68	−1.02537	−0.512683	+	0.858578i	$0.671348\pi$
$114$	0	0
$115$	−33.1766	−0.0269020
$116$	0	0
$117$	234.329	0.185160
$118$	0	0
$119$	−78.1901	−0.0602326
$120$	0	0
$121$	−153.850	−0.115590
$122$	0	0
$123$	−498.966	−0.365775
$124$	0	0
$125$	−357.614	−0.255887
$126$	0	0
$127$	−1314.59	−0.918515	−0.459258	−	0.888303i	$-0.651884\pi$
$127$	−1314.59	−0.918515	−0.459258	+	0.888303i	$0.651884\pi$
$128$	0	0
$129$	2278.30	1.55499
$130$	0	0
$131$	470.698	0.313932	0.156966	−	0.987604i	$-0.449829\pi$
$131$	470.698	0.313932	0.156966	+	0.987604i	$0.449829\pi$
$132$	0	0
$133$	−268.112	−0.174799
$134$	0	0
$135$	189.325	0.120700
$136$	0	0
$137$	−1359.42	−0.847757	−0.423879	−	0.905719i	$-0.639332\pi$
$137$	−1359.42	−0.847757	−0.423879	+	0.905719i	$0.639332\pi$
$138$	0	0
$139$	−972.983	−0.593722	−0.296861	−	0.954921i	$-0.595940\pi$
$139$	−972.983	−0.593722	−0.296861	+	0.954921i	$0.595940\pi$
$140$	0	0
$141$	2614.95	1.56183
$142$	0	0
$143$	−2626.32	−1.53583
$144$	0	0
$145$	133.189	0.0762812
$146$	0	0
$147$	1686.16	0.946067
$148$	0	0
$149$	−2894.03	−1.59119	−0.795597	−	0.605826i	$-0.792843\pi$
$149$	−2894.03	−1.59119	−0.795597	+	0.605826i	$0.792843\pi$
$150$	0	0
$151$	−2235.60	−1.20484	−0.602418	−	0.798181i	$-0.705796\pi$
$151$	−2235.60	−1.20484	−0.602418	+	0.798181i	$0.705796\pi$
$152$	0	0
$153$	−40.1926	−0.0212378
$154$	0	0
$155$	333.227	0.172680
$156$	0	0
$157$	917.699	0.466499	0.233250	−	0.972417i	$-0.425064\pi$
$157$	917.699	0.466499	0.233250	+	0.972417i	$0.425064\pi$
$158$	0	0
$159$	1438.50	0.717488
$160$	0	0
$161$	−136.970	−0.0670484
$162$	0	0
$163$	3160.70	1.51880	0.759401	−	0.650623i	$-0.225492\pi$
$163$	3160.70	1.51880	0.759401	+	0.650623i	$0.225492\pi$
$164$	0	0
$165$	271.346	0.128026
$166$	0	0
$167$	−1475.73	−0.683806	−0.341903	−	0.939735i	$-0.611071\pi$
$167$	−1475.73	−0.683806	−0.341903	+	0.939735i	$0.611071\pi$
$168$	0	0
$169$	3662.55	1.66707
$170$	0	0
$171$	−137.819	−0.0616334
$172$	0	0
$173$	2327.25	1.02276	0.511381	−	0.859354i	$-0.329134\pi$
$173$	2327.25	1.02276	0.511381	+	0.859354i	$0.329134\pi$
$174$	0	0
$175$	−732.014	−0.316200
$176$	0	0
$177$	−1934.57	−0.821531
$178$	0	0
$179$	3465.62	1.44711	0.723555	−	0.690267i	$-0.242507\pi$
$179$	3465.62	1.44711	0.723555	+	0.690267i	$0.242507\pi$
$180$	0	0
$181$	−2340.00	−0.960945	−0.480473	−	0.877010i	$-0.659535\pi$
$181$	−2340.00	−0.960945	−0.480473	+	0.877010i	$0.659535\pi$
$182$	0	0
$183$	−3756.06	−1.51724
$184$	0	0
$185$	631.799	0.251085
$186$	0	0
$187$	450.473	0.176160
$188$	0	0
$189$	781.636	0.300824
$190$	0	0
$191$	−719.948	−0.272741	−0.136371	−	0.990658i	$-0.543544\pi$
$191$	−719.948	−0.272741	−0.136371	+	0.990658i	$0.543544\pi$
$192$	0	0
$193$	−722.665	−0.269526	−0.134763	−	0.990878i	$-0.543027\pi$
$193$	−722.665	−0.269526	−0.134763	+	0.990878i	$0.543027\pi$
$194$	0	0
$195$	−605.395	−0.222324
$196$	0	0
$197$	−4516.25	−1.63335	−0.816673	−	0.577100i	$-0.804184\pi$
$197$	−4516.25	−1.63335	−0.816673	+	0.577100i	$0.804184\pi$
$198$	0	0
$199$	2482.98	0.884493	0.442246	−	0.896894i	$-0.354182\pi$
$199$	2482.98	0.884493	0.442246	+	0.896894i	$0.354182\pi$
$200$	0	0
$201$	2981.88	1.04640
$202$	0	0
$203$	549.877	0.190117
$204$	0	0
$205$	131.272	0.0447240
$206$	0	0
$207$	−70.4078	−0.0236410
$208$	0	0
$209$	1544.66	0.511227
$210$	0	0
$211$	−4832.66	−1.57675	−0.788374	−	0.615196i	$-0.789077\pi$
$211$	−4832.66	−1.57675	−0.788374	+	0.615196i	$0.789077\pi$
$212$	0	0
$213$	4607.00	1.48200
$214$	0	0
$215$	−599.392	−0.190131
$216$	0	0
$217$	1375.74	0.430374
$218$	0	0
$219$	−1071.94	−0.330755
$220$	0	0
$221$	−1005.04	−0.305912
$222$	0	0
$223$	−1278.76	−0.384000	−0.192000	−	0.981395i	$-0.561497\pi$
$223$	−1278.76	−0.384000	−0.192000	+	0.981395i	$0.561497\pi$
$224$	0	0
$225$	−376.282	−0.111491
$226$	0	0
$227$	4662.45	1.36325	0.681625	−	0.731702i	$-0.261274\pi$
$227$	4662.45	1.36325	0.681625	+	0.731702i	$0.261274\pi$
$228$	0	0
$229$	−4162.21	−1.20108	−0.600539	−	0.799595i	$-0.705047\pi$
$229$	−4162.21	−1.20108	−0.600539	+	0.799595i	$0.705047\pi$
$230$	0	0
$231$	1120.26	0.319081
$232$	0	0
$233$	5668.59	1.59383	0.796914	−	0.604093i	$-0.206465\pi$
$233$	5668.59	1.59383	0.796914	+	0.604093i	$0.206465\pi$
$234$	0	0
$235$	−687.961	−0.190969
$236$	0	0
$237$	−1850.49	−0.507181
$238$	0	0
$239$	3530.16	0.955426	0.477713	−	0.878516i	$-0.341466\pi$
$239$	3530.16	0.955426	0.477713	+	0.878516i	$0.341466\pi$
$240$	0	0
$241$	−99.0121	−0.0264644	−0.0132322	−	0.999912i	$-0.504212\pi$
$241$	−99.0121	−0.0264644	−0.0132322	+	0.999912i	$0.504212\pi$
$242$	0	0
$243$	854.959	0.225702
$244$	0	0
$245$	−443.607	−0.115678
$246$	0	0
$247$	−3446.27	−0.887777
$248$	0	0
$249$	6962.03	1.77189
$250$	0	0
$251$	293.478	0.0738016	0.0369008	−	0.999319i	$-0.488251\pi$
$251$	293.478	0.0738016	0.0369008	+	0.999319i	$0.488251\pi$
$252$	0	0
$253$	789.121	0.196093
$254$	0	0
$255$	103.839	0.0255005
$256$	0	0
$257$	−2503.83	−0.607723	−0.303862	−	0.952716i	$-0.598276\pi$
$257$	−2503.83	−0.607723	−0.303862	+	0.952716i	$0.598276\pi$
$258$	0	0
$259$	2608.40	0.625784
$260$	0	0
$261$	282.657	0.0670345
$262$	0	0
$263$	3599.41	0.843914	0.421957	−	0.906616i	$-0.361343\pi$
$263$	3599.41	0.843914	0.421957	+	0.906616i	$0.361343\pi$
$264$	0	0
$265$	−378.452	−0.0877287
$266$	0	0
$267$	6101.51	1.39853
$268$	0	0
$269$	6821.24	1.54609	0.773045	−	0.634351i	$-0.218732\pi$
$269$	6821.24	1.54609	0.773045	+	0.634351i	$0.218732\pi$
$270$	0	0
$271$	2584.33	0.579287	0.289644	−	0.957135i	$-0.406463\pi$
$271$	2584.33	0.579287	0.289644	+	0.957135i	$0.406463\pi$
$272$	0	0
$273$	−2499.39	−0.554103
$274$	0	0
$275$	4217.32	0.924777
$276$	0	0
$277$	2093.40	0.454081	0.227041	−	0.973885i	$-0.427095\pi$
$277$	2093.40	0.454081	0.227041	+	0.973885i	$0.427095\pi$
$278$	0	0
$279$	707.180	0.151748
$280$	0	0
$281$	−3435.43	−0.729325	−0.364663	−	0.931140i	$-0.618816\pi$
$281$	−3435.43	−0.729325	−0.364663	+	0.931140i	$0.618816\pi$
$282$	0	0
$283$	5136.95	1.07901	0.539505	−	0.841982i	$-0.318611\pi$
$283$	5136.95	1.07901	0.539505	+	0.841982i	$0.318611\pi$
$284$	0	0
$285$	356.061	0.0740043
$286$	0	0
$287$	541.960	0.111466
$288$	0	0
$289$	−4740.61	−0.964912
$290$	0	0
$291$	3321.97	0.669200
$292$	0	0
$293$	−505.986	−0.100887	−0.0504437	−	0.998727i	$-0.516064\pi$
$293$	−505.986	−0.100887	−0.0504437	+	0.998727i	$0.516064\pi$
$294$	0	0
$295$	508.960	0.100450
$296$	0	0
$297$	−4503.20	−0.879805
$298$	0	0
$299$	−1760.60	−0.340528
$300$	0	0
$301$	−2474.61	−0.473868
$302$	0	0
$303$	−8479.23	−1.60765
$304$	0	0
$305$	988.172	0.185517
$306$	0	0
$307$	3913.39	0.727522	0.363761	−	0.931492i	$-0.381492\pi$
$307$	3913.39	0.727522	0.363761	+	0.931492i	$0.381492\pi$
$308$	0	0
$309$	270.958	0.0498843
$310$	0	0
$311$	−2376.02	−0.433221	−0.216610	−	0.976258i	$-0.569500\pi$
$311$	−2376.02	−0.433221	−0.216610	+	0.976258i	$0.569500\pi$
$312$	0	0
$313$	1946.78	0.351560	0.175780	−	0.984429i	$-0.443755\pi$
$313$	1946.78	0.351560	0.175780	+	0.984429i	$0.443755\pi$
$314$	0	0
$315$	26.2964	0.00470360
$316$	0	0
$317$	1207.06	0.213865	0.106933	−	0.994266i	$-0.465897\pi$
$317$	1207.06	0.213865	0.106933	+	0.994266i	$0.465897\pi$
$318$	0	0
$319$	−3167.98	−0.556027
$320$	0	0
$321$	7338.62	1.27602
$322$	0	0
$323$	591.112	0.101828
$324$	0	0
$325$	−9409.19	−1.60593
$326$	0	0
$327$	7791.38	1.31763
$328$	0	0
$329$	−2840.27	−0.475955
$330$	0	0
$331$	4311.83	0.716011	0.358005	−	0.933720i	$-0.383457\pi$
$331$	4311.83	0.716011	0.358005	+	0.933720i	$0.383457\pi$
$332$	0	0
$333$	1340.81	0.220649
$334$	0	0
$335$	−784.496	−0.127945
$336$	0	0
$337$	−6342.26	−1.02518	−0.512589	−	0.858634i	$-0.671314\pi$
$337$	−6342.26	−1.02518	−0.512589	+	0.858634i	$0.671314\pi$
$338$	0	0
$339$	6753.05	1.08193
$340$	0	0
$341$	−7925.98	−1.25870
$342$	0	0
$343$	−3874.09	−0.609858
$344$	0	0
$345$	181.901	0.0283861
$346$	0	0
$347$	−6635.25	−1.02651	−0.513255	−	0.858236i	$-0.671561\pi$
$347$	−6635.25	−1.02651	−0.513255	+	0.858236i	$0.671561\pi$
$348$	0	0
$349$	−3695.13	−0.566750	−0.283375	−	0.959009i	$-0.591454\pi$
$349$	−3695.13	−0.566750	−0.283375	+	0.959009i	$0.591454\pi$
$350$	0	0
$351$	10047.0	1.52784
$352$	0	0
$353$	−1734.41	−0.261510	−0.130755	−	0.991415i	$-0.541740\pi$
$353$	−1734.41	−0.261510	−0.130755	+	0.991415i	$0.541740\pi$
$354$	0	0
$355$	−1212.04	−0.181207
$356$	0	0
$357$	428.702	0.0635554
$358$	0	0
$359$	−8420.19	−1.23788	−0.618942	−	0.785437i	$-0.712438\pi$
$359$	−8420.19	−1.23788	−0.618942	+	0.785437i	$0.712438\pi$
$360$	0	0
$361$	−4832.09	−0.704489
$362$	0	0
$363$	843.530	0.121966
$364$	0	0
$365$	282.015	0.0404421
$366$	0	0
$367$	13947.7	1.98382	0.991910	−	0.126947i	$-0.0405178\pi$
$367$	13947.7	1.98382	0.991910	+	0.126947i	$0.0405178\pi$
$368$	0	0
$369$	278.587	0.0393026
$370$	0	0
$371$	−1562.45	−0.218648
$372$	0	0
$373$	2567.77	0.356445	0.178222	−	0.983990i	$-0.442965\pi$
$373$	2567.77	0.356445	0.178222	+	0.983990i	$0.442965\pi$
$374$	0	0
$375$	1960.73	0.270004
$376$	0	0
$377$	7068.03	0.965575
$378$	0	0
$379$	4549.39	0.616587	0.308293	−	0.951291i	$-0.400242\pi$
$379$	4549.39	0.616587	0.308293	+	0.951291i	$0.400242\pi$
$380$	0	0
$381$	7207.67	0.969187
$382$	0	0
$383$	2053.57	0.273976	0.136988	−	0.990573i	$-0.456258\pi$
$383$	2053.57	0.273976	0.136988	+	0.990573i	$0.456258\pi$
$384$	0	0
$385$	−294.726	−0.0390146
$386$	0	0
$387$	−1272.04	−0.167084
$388$	0	0
$389$	−6626.47	−0.863690	−0.431845	−	0.901948i	$-0.642137\pi$
$389$	−6626.47	−0.863690	−0.431845	+	0.901948i	$0.642137\pi$
$390$	0	0
$391$	301.982	0.0390585
$392$	0	0
$393$	−2580.75	−0.331250
$394$	0	0
$395$	486.840	0.0620141
$396$	0	0
$397$	−1245.53	−0.157459	−0.0787296	−	0.996896i	$-0.525086\pi$
$397$	−1245.53	−0.157459	−0.0787296	+	0.996896i	$0.525086\pi$
$398$	0	0
$399$	1470.01	0.184442
$400$	0	0
$401$	−4526.90	−0.563747	−0.281874	−	0.959452i	$-0.590956\pi$
$401$	−4526.90	−0.563747	−0.281874	+	0.959452i	$0.590956\pi$
$402$	0	0
$403$	17683.5	2.18580
$404$	0	0
$405$	−1157.26	−0.141987
$406$	0	0
$407$	−15027.7	−1.83020
$408$	0	0
$409$	−9959.61	−1.20409	−0.602043	−	0.798464i	$-0.705646\pi$
$409$	−9959.61	−1.20409	−0.602043	+	0.798464i	$0.705646\pi$
$410$	0	0
$411$	7453.42	0.894526
$412$	0	0
$413$	2101.26	0.250354
$414$	0	0
$415$	−1831.62	−0.216653
$416$	0	0
$417$	5334.68	0.626476
$418$	0	0
$419$	−6823.75	−0.795613	−0.397807	−	0.917469i	$-0.630229\pi$
$419$	−6823.75	−0.795613	−0.397807	+	0.917469i	$0.630229\pi$
$420$	0	0
$421$	9291.42	1.07562	0.537810	−	0.843066i	$-0.319252\pi$
$421$	9291.42	1.07562	0.537810	+	0.843066i	$0.319252\pi$
$422$	0	0
$423$	−1460.00	−0.167820
$424$	0	0
$425$	1613.89	0.184200
$426$	0	0
$427$	4079.70	0.462366
$428$	0	0
$429$	14399.6	1.62056
$430$	0	0
$431$	15751.8	1.76042	0.880208	−	0.474589i	$-0.157403\pi$
$431$	15751.8	1.76042	0.880208	+	0.474589i	$0.157403\pi$
$432$	0	0
$433$	−11994.0	−1.33116	−0.665582	−	0.746325i	$-0.731817\pi$
$433$	−11994.0	−1.33116	−0.665582	+	0.746325i	$0.731817\pi$
$434$	0	0
$435$	−730.252	−0.0804895
$436$	0	0
$437$	1035.49	0.113350
$438$	0	0
$439$	10701.3	1.16343	0.581714	−	0.813394i	$-0.302382\pi$
$439$	10701.3	1.16343	0.581714	+	0.813394i	$0.302382\pi$
$440$	0	0
$441$	−941.430	−0.101655
$442$	0	0
$443$	−6834.67	−0.733014	−0.366507	−	0.930415i	$-0.619446\pi$
$443$	−6834.67	−0.733014	−0.366507	+	0.930415i	$0.619446\pi$
$444$	0	0
$445$	−1605.23	−0.171001
$446$	0	0
$447$	15867.4	1.67898
$448$	0	0
$449$	−13907.7	−1.46179	−0.730894	−	0.682491i	$-0.760897\pi$
$449$	−13907.7	−1.46179	−0.730894	+	0.682491i	$0.760897\pi$
$450$	0	0
$451$	−3122.37	−0.326001
$452$	0	0
$453$	12257.3	1.27130
$454$	0	0
$455$	657.559	0.0677513
$456$	0	0
$457$	5690.52	0.582476	0.291238	−	0.956651i	$-0.405933\pi$
$457$	5690.52	0.582476	0.291238	+	0.956651i	$0.405933\pi$
$458$	0	0
$459$	−1723.29	−0.175242
$460$	0	0
$461$	−5547.09	−0.560420	−0.280210	−	0.959939i	$-0.590404\pi$
$461$	−5547.09	−0.560420	−0.280210	+	0.959939i	$0.590404\pi$
$462$	0	0
$463$	−8973.41	−0.900712	−0.450356	−	0.892849i	$-0.648703\pi$
$463$	−8973.41	−0.900712	−0.450356	+	0.892849i	$0.648703\pi$
$464$	0	0
$465$	−1827.02	−0.182207
$466$	0	0
$467$	−19200.3	−1.90254	−0.951269	−	0.308363i	$-0.900219\pi$
$467$	−19200.3	−1.90254	−0.951269	+	0.308363i	$0.900219\pi$
$468$	0	0
$469$	−3238.82	−0.318880
$470$	0	0
$471$	−5031.57	−0.492235
$472$	0	0
$473$	14256.8	1.38590
$474$	0	0
$475$	5533.98	0.534561
$476$	0	0
$477$	−803.156	−0.0770944
$478$	0	0
$479$	4974.38	0.474499	0.237250	−	0.971449i	$-0.423754\pi$
$479$	4974.38	0.474499	0.237250	+	0.971449i	$0.423754\pi$
$480$	0	0
$481$	33528.0	3.17826
$482$	0	0
$483$	750.983	0.0707472
$484$	0	0
$485$	−873.969	−0.0818245
$486$	0	0
$487$	1009.47	0.0939294	0.0469647	−	0.998897i	$-0.485045\pi$
$487$	1009.47	0.0939294	0.0469647	+	0.998897i	$0.485045\pi$
$488$	0	0
$489$	−17329.5	−1.60259
$490$	0	0
$491$	−14220.0	−1.30700	−0.653502	−	0.756925i	$-0.726701\pi$
$491$	−14220.0	−1.30700	−0.653502	+	0.756925i	$0.726701\pi$
$492$	0	0
$493$	−1212.32	−0.110751
$494$	0	0
$495$	−151.500	−0.0137564
$496$	0	0
$497$	−5003.96	−0.451627
$498$	0	0
$499$	−4040.61	−0.362490	−0.181245	−	0.983438i	$-0.558013\pi$
$499$	−4040.61	−0.362490	−0.181245	+	0.983438i	$0.558013\pi$
$500$	0	0
$501$	8091.16	0.721530
$502$	0	0
$503$	−3383.69	−0.299943	−0.149971	−	0.988690i	$-0.547918\pi$
$503$	−3383.69	−0.299943	−0.149971	+	0.988690i	$0.547918\pi$
$504$	0	0
$505$	2230.78	0.196571
$506$	0	0
$507$	−20081.1	−1.75904
$508$	0	0
$509$	−10557.0	−0.919310	−0.459655	−	0.888097i	$-0.652027\pi$
$509$	−10557.0	−0.919310	−0.459655	+	0.888097i	$0.652027\pi$
$510$	0	0
$511$	1164.31	0.100795
$512$	0	0
$513$	−5909.12	−0.508565
$514$	0	0
$515$	−71.2856	−0.00609946
$516$	0	0
$517$	16363.5	1.39200
$518$	0	0
$519$	−12759.9	−1.07919
$520$	0	0
$521$	−562.524	−0.0473025	−0.0236513	−	0.999720i	$-0.507529\pi$
$521$	−562.524	−0.0473025	−0.0236513	+	0.999720i	$0.507529\pi$
$522$	0	0
$523$	−9026.26	−0.754667	−0.377333	−	0.926077i	$-0.623159\pi$
$523$	−9026.26	−0.754667	−0.377333	+	0.926077i	$0.623159\pi$
$524$	0	0
$525$	4013.49	0.333644
$526$	0	0
$527$	−3033.12	−0.250711
$528$	0	0
$529$	529.000	0.0434783
$530$	0	0
$531$	1080.12	0.0882738
$532$	0	0
$533$	6966.26	0.566121
$534$	0	0
$535$	−1930.70	−0.156021
$536$	0	0
$537$	−19001.3	−1.52694
$538$	0	0
$539$	10551.4	0.843194
$540$	0	0
$541$	17974.9	1.42847	0.714235	−	0.699906i	$-0.246775\pi$
$541$	17974.9	1.42847	0.714235	+	0.699906i	$0.246775\pi$
$542$	0	0
$543$	12829.8	1.01396
$544$	0	0
$545$	−2049.81	−0.161109
$546$	0	0
$547$	22408.7	1.75160	0.875800	−	0.482673i	$-0.160334\pi$
$547$	22408.7	1.75160	0.875800	+	0.482673i	$0.160334\pi$
$548$	0	0
$549$	2097.11	0.163029
$550$	0	0
$551$	−4157.03	−0.321408
$552$	0	0
$553$	2009.93	0.154559
$554$	0	0
$555$	−3464.03	−0.264937
$556$	0	0
$557$	3607.56	0.274429	0.137215	−	0.990541i	$-0.456185\pi$
$557$	3607.56	0.274429	0.137215	+	0.990541i	$0.456185\pi$
$558$	0	0
$559$	−31808.2	−2.40670
$560$	0	0
$561$	−2469.86	−0.185878
$562$	0	0
$563$	14383.2	1.07669	0.538346	−	0.842724i	$-0.319049\pi$
$563$	14383.2	1.07669	0.538346	+	0.842724i	$0.319049\pi$
$564$	0	0
$565$	−1776.64	−0.132290
$566$	0	0
$567$	−4777.78	−0.353876
$568$	0	0
$569$	20252.0	1.49211	0.746053	−	0.665886i	$-0.231946\pi$
$569$	20252.0	1.49211	0.746053	+	0.665886i	$0.231946\pi$
$570$	0	0
$571$	−24666.8	−1.80784	−0.903919	−	0.427705i	$-0.859322\pi$
$571$	−24666.8	−1.80784	−0.903919	+	0.427705i	$0.859322\pi$
$572$	0	0
$573$	3947.34	0.287788
$574$	0	0
$575$	2827.14	0.205044
$576$	0	0
$577$	13339.4	0.962436	0.481218	−	0.876601i	$-0.340195\pi$
$577$	13339.4	0.962436	0.481218	+	0.876601i	$0.340195\pi$
$578$	0	0
$579$	3962.23	0.284395
$580$	0	0
$581$	−7561.92	−0.539968
$582$	0	0
$583$	9001.66	0.639470
$584$	0	0
$585$	338.009	0.0238888
$586$	0	0
$587$	−9352.70	−0.657628	−0.328814	−	0.944395i	$-0.606649\pi$
$587$	−9352.70	−0.657628	−0.328814	+	0.944395i	$0.606649\pi$
$588$	0	0
$589$	−10400.5	−0.727581
$590$	0	0
$591$	24761.7	1.72345
$592$	0	0
$593$	19319.6	1.33788	0.668939	−	0.743317i	$-0.266749\pi$
$593$	19319.6	1.33788	0.668939	+	0.743317i	$0.266749\pi$
$594$	0	0
$595$	−112.786	−0.00777105
$596$	0	0
$597$	−13613.7	−0.933288
$598$	0	0
$599$	6214.10	0.423875	0.211938	−	0.977283i	$-0.432023\pi$
$599$	6214.10	0.423875	0.211938	+	0.977283i	$0.432023\pi$
$600$	0	0
$601$	−13398.9	−0.909403	−0.454702	−	0.890644i	$-0.650254\pi$
$601$	−13398.9	−0.909403	−0.454702	+	0.890644i	$0.650254\pi$
$602$	0	0
$603$	−1664.87	−0.112436
$604$	0	0
$605$	−221.922	−0.0149131
$606$	0	0
$607$	6729.05	0.449957	0.224978	−	0.974364i	$-0.427769\pi$
$607$	6729.05	0.449957	0.224978	+	0.974364i	$0.427769\pi$
$608$	0	0
$609$	−3014.87	−0.200605
$610$	0	0
$611$	−36508.4	−2.41730
$612$	0	0
$613$	−14280.2	−0.940901	−0.470450	−	0.882426i	$-0.655909\pi$
$613$	−14280.2	−0.940901	−0.470450	+	0.882426i	$0.655909\pi$
$614$	0	0
$615$	−719.739	−0.0471913
$616$	0	0
$617$	13519.7	0.882146	0.441073	−	0.897471i	$-0.354598\pi$
$617$	13519.7	0.882146	0.441073	+	0.897471i	$0.354598\pi$
$618$	0	0
$619$	−25300.2	−1.64281	−0.821407	−	0.570343i	$-0.806810\pi$
$619$	−25300.2	−1.64281	−0.821407	+	0.570343i	$0.806810\pi$
$620$	0	0
$621$	−3018.79	−0.195072
$622$	0	0
$623$	−6627.25	−0.426188
$624$	0	0
$625$	14849.1	0.950341
$626$	0	0
$627$	−8469.09	−0.539430
$628$	0	0
$629$	−5750.80	−0.364546
$630$	0	0
$631$	5030.37	0.317363	0.158681	−	0.987330i	$-0.449276\pi$
$631$	5030.37	0.317363	0.158681	+	0.987330i	$0.449276\pi$
$632$	0	0
$633$	26496.5	1.66373
$634$	0	0
$635$	−1896.25	−0.118505
$636$	0	0
$637$	−23541.1	−1.46426
$638$	0	0
$639$	−2572.22	−0.159242
$640$	0	0
$641$	−3814.93	−0.235071	−0.117536	−	0.993069i	$-0.537499\pi$
$641$	−3814.93	−0.235071	−0.117536	+	0.993069i	$0.537499\pi$
$642$	0	0
$643$	26470.2	1.62346	0.811729	−	0.584035i	$-0.198527\pi$
$643$	26470.2	1.62346	0.811729	+	0.584035i	$0.198527\pi$
$644$	0	0
$645$	3286.35	0.200620
$646$	0	0
$647$	−12390.2	−0.752871	−0.376435	−	0.926443i	$-0.622850\pi$
$647$	−12390.2	−0.752871	−0.376435	+	0.926443i	$0.622850\pi$
$648$	0	0
$649$	−12105.9	−0.732199
$650$	0	0
$651$	−7542.92	−0.454117
$652$	0	0
$653$	−11527.9	−0.690843	−0.345421	−	0.938448i	$-0.612264\pi$
$653$	−11527.9	−0.690843	−0.345421	+	0.938448i	$0.612264\pi$
$654$	0	0
$655$	678.962	0.0405027
$656$	0	0
$657$	598.498	0.0355397
$658$	0	0
$659$	−29611.8	−1.75040	−0.875199	−	0.483763i	$-0.839270\pi$
$659$	−29611.8	−1.75040	−0.875199	+	0.483763i	$0.839270\pi$
$660$	0	0
$661$	2767.89	0.162872	0.0814361	−	0.996679i	$-0.474049\pi$
$661$	2767.89	0.162872	0.0814361	+	0.996679i	$0.474049\pi$
$662$	0	0
$663$	5510.46	0.322788
$664$	0	0
$665$	−386.741	−0.0225521
$666$	0	0
$667$	−2123.70	−0.123284
$668$	0	0
$669$	7011.19	0.405184
$670$	0	0
$671$	−23504.2	−1.35226
$672$	0	0
$673$	−16779.6	−0.961079	−0.480540	−	0.876973i	$-0.659559\pi$
$673$	−16779.6	−0.961079	−0.480540	+	0.876973i	$0.659559\pi$
$674$	0	0
$675$	−16133.4	−0.919962
$676$	0	0
$677$	−1354.04	−0.0768684	−0.0384342	−	0.999261i	$-0.512237\pi$
$677$	−1354.04	−0.0768684	−0.0384342	+	0.999261i	$0.512237\pi$
$678$	0	0
$679$	−3608.21	−0.203933
$680$	0	0
$681$	−25563.3	−1.43846
$682$	0	0
$683$	−34512.3	−1.93349	−0.966746	−	0.255740i	$-0.917681\pi$
$683$	−34512.3	−1.93349	−0.966746	+	0.255740i	$0.917681\pi$
$684$	0	0
$685$	−1960.90	−0.109376
$686$	0	0
$687$	22820.6	1.26734
$688$	0	0
$689$	−20083.5	−1.11048
$690$	0	0
$691$	26091.9	1.43644	0.718222	−	0.695814i	$-0.244956\pi$
$691$	26091.9	1.43644	0.718222	+	0.695814i	$0.244956\pi$
$692$	0	0
$693$	−625.473	−0.0342853
$694$	0	0
$695$	−1403.49	−0.0766005
$696$	0	0
$697$	−1194.87	−0.0649339
$698$	0	0
$699$	−31079.8	−1.68175
$700$	0	0
$701$	−13703.5	−0.738337	−0.369168	−	0.929363i	$-0.620357\pi$
$701$	−13703.5	−0.738337	−0.369168	+	0.929363i	$0.620357\pi$
$702$	0	0
$703$	−19719.4	−1.05794
$704$	0	0
$705$	3771.96	0.201504
$706$	0	0
$707$	9209.84	0.489918
$708$	0	0
$709$	−33264.6	−1.76203	−0.881013	−	0.473091i	$-0.843138\pi$
$709$	−33264.6	−1.76203	−0.881013	+	0.473091i	$0.843138\pi$
$710$	0	0
$711$	1033.18	0.0544968
$712$	0	0
$713$	−5313.30	−0.279081
$714$	0	0
$715$	−3788.36	−0.198149
$716$	0	0
$717$	−19355.2	−1.00813
$718$	0	0
$719$	6519.81	0.338175	0.169088	−	0.985601i	$-0.445918\pi$
$719$	6519.81	0.338175	0.169088	+	0.985601i	$0.445918\pi$
$720$	0	0
$721$	−294.305	−0.0152018
$722$	0	0
$723$	542.864	0.0279244
$724$	0	0
$725$	−11349.7	−0.581406
$726$	0	0
$727$	30762.0	1.56933	0.784663	−	0.619923i	$-0.212836\pi$
$727$	30762.0	1.56933	0.784663	+	0.619923i	$0.212836\pi$
$728$	0	0
$729$	16974.0	0.862370
$730$	0	0
$731$	5455.82	0.276048
$732$	0	0
$733$	2866.80	0.144458	0.0722290	−	0.997388i	$-0.476989\pi$
$733$	2866.80	0.144458	0.0722290	+	0.997388i	$0.476989\pi$
$734$	0	0
$735$	2432.21	0.122059
$736$	0	0
$737$	18659.6	0.932614
$738$	0	0
$739$	−26130.3	−1.30070	−0.650351	−	0.759634i	$-0.725378\pi$
$739$	−26130.3	−1.30070	−0.650351	+	0.759634i	$0.725378\pi$
$740$	0	0
$741$	18895.3	0.936754
$742$	0	0
$743$	−14423.8	−0.712192	−0.356096	−	0.934449i	$-0.615892\pi$
$743$	−14423.8	−0.712192	−0.356096	+	0.934449i	$0.615892\pi$
$744$	0	0
$745$	−4174.52	−0.205292
$746$	0	0
$747$	−3887.10	−0.190390
$748$	0	0
$749$	−7970.95	−0.388855
$750$	0	0
$751$	32892.1	1.59820	0.799100	−	0.601198i	$-0.205310\pi$
$751$	32892.1	1.59820	0.799100	+	0.601198i	$0.205310\pi$
$752$	0	0
$753$	−1609.09	−0.0778730
$754$	0	0
$755$	−3224.76	−0.155445
$756$	0	0
$757$	26854.4	1.28935	0.644676	−	0.764456i	$-0.276992\pi$
$757$	26854.4	1.28935	0.644676	+	0.764456i	$0.276992\pi$
$758$	0	0
$759$	−4326.60	−0.206911
$760$	0	0
$761$	7087.42	0.337607	0.168803	−	0.985650i	$-0.446010\pi$
$761$	7087.42	0.337607	0.168803	+	0.985650i	$0.446010\pi$
$762$	0	0
$763$	−8462.72	−0.401535
$764$	0	0
$765$	−57.9762	−0.00274004
$766$	0	0
$767$	27009.2	1.27151
$768$	0	0
$769$	−8889.76	−0.416870	−0.208435	−	0.978036i	$-0.566837\pi$
$769$	−8889.76	−0.416870	−0.208435	+	0.978036i	$0.566837\pi$
$770$	0	0
$771$	13728.0	0.641250
$772$	0	0
$773$	5361.22	0.249456	0.124728	−	0.992191i	$-0.460194\pi$
$773$	5361.22	0.249456	0.124728	+	0.992191i	$0.460194\pi$
$774$	0	0
$775$	−28396.0	−1.31615
$776$	0	0
$777$	−14301.4	−0.660307
$778$	0	0
$779$	−4097.18	−0.188443
$780$	0	0
$781$	28829.1	1.32085
$782$	0	0
$783$	12119.1	0.553132
$784$	0	0
$785$	1323.74	0.0601866
$786$	0	0
$787$	11774.9	0.533329	0.266665	−	0.963789i	$-0.414078\pi$
$787$	11774.9	0.533329	0.266665	+	0.963789i	$0.414078\pi$
$788$	0	0
$789$	−19734.9	−0.890470
$790$	0	0
$791$	−7334.92	−0.329709
$792$	0	0
$793$	52439.8	2.34829
$794$	0	0
$795$	2074.98	0.0925685
$796$	0	0
$797$	−2534.85	−0.112659	−0.0563293	−	0.998412i	$-0.517940\pi$
$797$	−2534.85	−0.112659	−0.0563293	+	0.998412i	$0.517940\pi$
$798$	0	0
$799$	6262.00	0.277264
$800$	0	0
$801$	−3406.65	−0.150272
$802$	0	0
$803$	−6707.88	−0.294789
$804$	0	0
$805$	−197.574	−0.00865041
$806$	0	0
$807$	−37399.6	−1.63138
$808$	0	0
$809$	−16977.8	−0.737835	−0.368917	−	0.929462i	$-0.620272\pi$
$809$	−16977.8	−0.737835	−0.368917	+	0.929462i	$0.620272\pi$
$810$	0	0
$811$	−11287.0	−0.488704	−0.244352	−	0.969687i	$-0.578575\pi$
$811$	−11287.0	−0.488704	−0.244352	+	0.969687i	$0.578575\pi$
$812$	0	0
$813$	−14169.4	−0.611245
$814$	0	0
$815$	4559.18	0.195952
$816$	0	0
$817$	18707.9	0.801110
$818$	0	0
$819$	1395.48	0.0595386
$820$	0	0
$821$	32434.3	1.37876	0.689382	−	0.724398i	$-0.257882\pi$
$821$	32434.3	1.37876	0.689382	+	0.724398i	$0.257882\pi$
$822$	0	0
$823$	35961.4	1.52313	0.761564	−	0.648090i	$-0.224432\pi$
$823$	35961.4	1.52313	0.761564	+	0.648090i	$0.224432\pi$
$824$	0	0
$825$	−23122.7	−0.975794
$826$	0	0
$827$	18671.7	0.785102	0.392551	−	0.919730i	$-0.371593\pi$
$827$	18671.7	0.785102	0.392551	+	0.919730i	$0.371593\pi$
$828$	0	0
$829$	−35148.2	−1.47256	−0.736278	−	0.676679i	$-0.763418\pi$
$829$	−35148.2	−1.47256	−0.736278	+	0.676679i	$0.763418\pi$
$830$	0	0
$831$	−11477.7	−0.479131
$832$	0	0
$833$	4037.82	0.167950
$834$	0	0
$835$	−2128.68	−0.0882229
$836$	0	0
$837$	30320.9	1.25214
$838$	0	0
$839$	−5885.12	−0.242166	−0.121083	−	0.992642i	$-0.538637\pi$
$839$	−5885.12	−0.242166	−0.121083	+	0.992642i	$0.538637\pi$
$840$	0	0
$841$	−15863.3	−0.650427
$842$	0	0
$843$	18835.8	0.769560
$844$	0	0
$845$	5283.08	0.215081
$846$	0	0
$847$	−916.213	−0.0371682
$848$	0	0
$849$	−28164.9	−1.13854
$850$	0	0
$851$	−10074.0	−0.405797
$852$	0	0
$853$	−320.329	−0.0128580	−0.00642899	−	0.999979i	$-0.502046\pi$
$853$	−320.329	−0.0128580	−0.00642899	+	0.999979i	$0.502046\pi$
$854$	0	0
$855$	−198.799	−0.00795179
$856$	0	0
$857$	23460.1	0.935101	0.467550	−	0.883966i	$-0.345137\pi$
$857$	23460.1	0.935101	0.467550	+	0.883966i	$0.345137\pi$
$858$	0	0
$859$	−2429.53	−0.0965012	−0.0482506	−	0.998835i	$-0.515365\pi$
$859$	−2429.53	−0.0965012	−0.0482506	+	0.998835i	$0.515365\pi$
$860$	0	0
$861$	−2971.46	−0.117616
$862$	0	0
$863$	20239.2	0.798322	0.399161	−	0.916881i	$-0.369302\pi$
$863$	20239.2	0.798322	0.399161	+	0.916881i	$0.369302\pi$
$864$	0	0
$865$	3356.97	0.131954
$866$	0	0
$867$	25991.9	1.01814
$868$	0	0
$869$	−11579.7	−0.452032
$870$	0	0
$871$	−41631.3	−1.61954
$872$	0	0
$873$	−1854.75	−0.0719058
$874$	0	0
$875$	−2129.67	−0.0822813
$876$	0	0
$877$	23209.8	0.893659	0.446830	−	0.894619i	$-0.352553\pi$
$877$	23209.8	0.893659	0.446830	+	0.894619i	$0.352553\pi$
$878$	0	0
$879$	2774.22	0.106453
$880$	0	0
$881$	−16867.3	−0.645033	−0.322517	−	0.946564i	$-0.604529\pi$
$881$	−16867.3	−0.645033	−0.322517	+	0.946564i	$0.604529\pi$
$882$	0	0
$883$	16054.7	0.611874	0.305937	−	0.952052i	$-0.401030\pi$
$883$	16054.7	0.611874	0.305937	+	0.952052i	$0.401030\pi$
$884$	0	0
$885$	−2790.53	−0.105992
$886$	0	0
$887$	−40587.7	−1.53642	−0.768208	−	0.640200i	$-0.778851\pi$
$887$	−40587.7	−1.53642	−0.768208	+	0.640200i	$0.778851\pi$
$888$	0	0
$889$	−7828.72	−0.295351
$890$	0	0
$891$	27526.0	1.03497
$892$	0	0
$893$	21472.3	0.804638
$894$	0	0
$895$	4999.02	0.186702
$896$	0	0
$897$	9653.02	0.359314
$898$	0	0
$899$	21330.6	0.791341
$900$	0	0
$901$	3444.77	0.127372
$902$	0	0
$903$	13567.8	0.500010
$904$	0	0
$905$	−3375.36	−0.123979
$906$	0	0
$907$	1158.51	0.0424118	0.0212059	−	0.999775i	$-0.493249\pi$
$907$	1158.51	0.0424118	0.0212059	+	0.999775i	$0.493249\pi$
$908$	0	0
$909$	4734.20	0.172743
$910$	0	0
$911$	16531.5	0.601221	0.300611	−	0.953747i	$-0.402810\pi$
$911$	16531.5	0.601221	0.300611	+	0.953747i	$0.402810\pi$
$912$	0	0
$913$	43566.1	1.57922
$914$	0	0
$915$	−5417.96	−0.195751
$916$	0	0
$917$	2803.12	0.100946
$918$	0	0
$919$	42868.2	1.53873	0.769365	−	0.638809i	$-0.220573\pi$
$919$	42868.2	1.53873	0.769365	+	0.638809i	$0.220573\pi$
$920$	0	0
$921$	−21456.4	−0.767657
$922$	0	0
$923$	−64320.2	−2.29374
$924$	0	0
$925$	−53838.8	−1.91374
$926$	0	0
$927$	−151.284	−0.00536009
$928$	0	0
$929$	41605.9	1.46937	0.734685	−	0.678409i	$-0.237330\pi$
$929$	41605.9	1.46937	0.734685	+	0.678409i	$0.237330\pi$
$930$	0	0
$931$	13845.6	0.487402
$932$	0	0
$933$	13027.3	0.457121
$934$	0	0
$935$	649.789	0.0227277
$936$	0	0
$937$	−6344.27	−0.221193	−0.110597	−	0.993865i	$-0.535276\pi$
$937$	−6344.27	−0.221193	−0.110597	+	0.993865i	$0.535276\pi$
$938$	0	0
$939$	−10673.8	−0.370955
$940$	0	0
$941$	33596.3	1.16388	0.581938	−	0.813233i	$-0.302295\pi$
$941$	33596.3	1.16388	0.581938	+	0.813233i	$0.302295\pi$
$942$	0	0
$943$	−2093.13	−0.0722817
$944$	0	0
$945$	1127.48	0.0388115
$946$	0	0
$947$	−11090.3	−0.380557	−0.190278	−	0.981730i	$-0.560939\pi$
$947$	−11090.3	−0.380557	−0.190278	+	0.981730i	$0.560939\pi$
$948$	0	0
$949$	14965.8	0.511920
$950$	0	0
$951$	−6618.09	−0.225664
$952$	0	0
$953$	−15187.0	−0.516216	−0.258108	−	0.966116i	$-0.583099\pi$
$953$	−15187.0	−0.516216	−0.258108	+	0.966116i	$0.583099\pi$
$954$	0	0
$955$	−1038.50	−0.0351884
$956$	0	0
$957$	17369.4	0.586702
$958$	0	0
$959$	−8095.65	−0.272599
$960$	0	0
$961$	23576.1	0.791384
$962$	0	0
$963$	−4097.36	−0.137109
$964$	0	0
$965$	−1042.41	−0.0347736
$966$	0	0
$967$	−12052.4	−0.400806	−0.200403	−	0.979714i	$-0.564225\pi$
$967$	−12052.4	−0.400806	−0.200403	+	0.979714i	$0.564225\pi$
$968$	0	0
$969$	−3240.96	−0.107445
$970$	0	0
$971$	2710.79	0.0895915	0.0447958	−	0.998996i	$-0.485736\pi$
$971$	2710.79	0.0895915	0.0447958	+	0.998996i	$0.485736\pi$
$972$	0	0
$973$	−5794.34	−0.190913
$974$	0	0
$975$	51588.8	1.69453
$976$	0	0
$977$	−25037.0	−0.819860	−0.409930	−	0.912117i	$-0.634447\pi$
$977$	−25037.0	−0.819860	−0.409930	+	0.912117i	$0.634447\pi$
$978$	0	0
$979$	38181.2	1.24645
$980$	0	0
$981$	−4350.15	−0.141580
$982$	0	0
$983$	−17600.5	−0.571078	−0.285539	−	0.958367i	$-0.592173\pi$
$983$	−17600.5	−0.571078	−0.285539	+	0.958367i	$0.592173\pi$
$984$	0	0
$985$	−6514.50	−0.210730
$986$	0	0
$987$	15572.7	0.502212
$988$	0	0
$989$	9557.30	0.307285
$990$	0	0
$991$	25965.6	0.832316	0.416158	−	0.909292i	$-0.363376\pi$
$991$	25965.6	0.832316	0.416158	+	0.909292i	$0.363376\pi$
$992$	0	0
$993$	−23640.9	−0.755511
$994$	0	0
$995$	3581.60	0.114115
$996$	0	0
$997$	−15101.7	−0.479716	−0.239858	−	0.970808i	$-0.577101\pi$
$997$	−15101.7	−0.479716	−0.239858	+	0.970808i	$0.577101\pi$
$998$	0	0
$999$	57488.5	1.82067

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	736.4.a.h.1.3	✓	9
4.3	odd	2		736.4.a.i.1.7	yes	9
8.3	odd	2		1472.4.a.bk.1.3		9
8.5	even	2		1472.4.a.bj.1.7		9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
736.4.a.h.1.3	✓	9	1.1	even	1	trivial
736.4.a.i.1.7	yes	9	4.3	odd	2
1472.4.a.bj.1.7		9	8.5	even	2
1472.4.a.bk.1.3		9	8.3	odd	2

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 736 = 2^{5} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	736.a (trivial)

\(f(q)\)	\(=\)	\(q-5.48281 q^{3} +1.44246 q^{5} +5.95524 q^{7} +3.06121 q^{9} -34.3096 q^{11} +76.5477 q^{13} -7.90873 q^{15} -13.1296 q^{17} -45.0212 q^{19} -32.6514 q^{21} -23.0000 q^{23} -122.919 q^{25} +131.252 q^{27} +92.3349 q^{29} +231.013 q^{31} +188.113 q^{33} +8.59019 q^{35} +438.001 q^{37} -419.696 q^{39} +91.0056 q^{41} -415.535 q^{43} +4.41567 q^{45} -476.936 q^{47} -307.535 q^{49} +71.9873 q^{51} -262.366 q^{53} -49.4902 q^{55} +246.843 q^{57} +352.842 q^{59} +685.060 q^{61} +18.2302 q^{63} +110.417 q^{65} -543.860 q^{67} +126.105 q^{69} -840.263 q^{71} +195.510 q^{73} +673.943 q^{75} -204.322 q^{77} +337.507 q^{79} -802.282 q^{81} -1269.79 q^{83} -18.9390 q^{85} -506.255 q^{87} -1112.84 q^{89} +455.860 q^{91} -1266.60 q^{93} -64.9413 q^{95} -605.888 q^{97} -105.029 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q - 42 q^{7} + 71 q^{9} - 66 q^{11} + 48 q^{13} - 90 q^{15} - 32 q^{17} + 6 q^{21} - 207 q^{23} + 135 q^{25} - 88 q^{29} - 556 q^{31} - 230 q^{33} - 368 q^{35} - 368 q^{37} - 468 q^{39} - 216 q^{41} + 552 q^{43}+ \cdots - 552 q^{99}+O(q^{100}) \) 9 * q - 42 * q^7 + 71 * q^9 - 66 * q^11 + 48 * q^13 - 90 * q^15 - 32 * q^17 + 6 * q^21 - 207 * q^23 + 135 * q^25 - 88 * q^29 - 556 * q^31 - 230 * q^33 - 368 * q^35 - 368 * q^37 - 468 * q^39 - 216 * q^41 + 552 * q^43 - 206 * q^45 - 1300 * q^47 + 1041 * q^49 - 1658 * q^51 + 510 * q^53 - 1948 * q^55 - 804 * q^57 - 1444 * q^59 - 882 * q^61 - 3128 * q^63 + 334 * q^65 - 1186 * q^67 - 2440 * q^71 - 368 * q^73 - 1636 * q^75 + 556 * q^77 - 2576 * q^79 + 1069 * q^81 - 1710 * q^83 + 640 * q^85 - 4844 * q^87 - 474 * q^89 - 2790 * q^91 + 1178 * q^93 - 4564 * q^95 - 884 * q^97 - 552 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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