LMFDB - Embedded newform 7232.2.a.bn.1.16

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [7232,2,Mod(1,7232)] 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("7232.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(7232, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 7232 = 2^{6} \cdot 113 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7232.a (trivial)

Newform invariants

comment:select newform

sage:traces = [18,0,0,0,-10,0,0] 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:	$57.7478107418$
Analytic rank:	$0$
Dimension:	$18$
Coefficient field:	$\mathbb{Q}[x]/(x^{18} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{18} - 43 x^{16} + 760 x^{14} - 7095 x^{12} + 37240 x^{10} - 107142 x^{8} + 149152 x^{6} - 72200 x^{4} + \cdots - 800 $ x^18 - 43x^16 + 760x^14 - 7095x^12 + 37240x^10 - 107142x^8 + 149152x^6 - 72200x^4 + 13504x^2 - 800
Coefficient ring:	$\Z[a_1, \ldots, a_{23}]$
Coefficient ring index:	$ 2^{10}\cdot 5 $
Twist minimal:	no (minimal twist has level 3616)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.16
Root			$2.69830$ of defining polynomial
Character	$\chi$	$=$	7232.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+2.69830 q^{3} -3.42881 q^{5} -2.84717 q^{7} +4.28081 q^{9} +4.71823 q^{11} +6.74669 q^{13} -9.25194 q^{15} -7.43351 q^{17} +4.76127 q^{19} -7.68251 q^{21} +0.359963 q^{23} +6.75672 q^{25} +3.45601 q^{27} +1.26714 q^{29} -3.41913 q^{31} +12.7312 q^{33} +9.76239 q^{35} +4.25370 q^{37} +18.2046 q^{39} +4.67337 q^{41} -10.0952 q^{43} -14.6781 q^{45} +6.65306 q^{47} +1.10637 q^{49} -20.0578 q^{51} -9.13377 q^{53} -16.1779 q^{55} +12.8473 q^{57} -0.225839 q^{59} -12.4007 q^{61} -12.1882 q^{63} -23.1331 q^{65} +10.6330 q^{67} +0.971288 q^{69} +1.60620 q^{71} +4.98656 q^{73} +18.2316 q^{75} -13.4336 q^{77} +11.9260 q^{79} -3.51709 q^{81} +7.11944 q^{83} +25.4881 q^{85} +3.41913 q^{87} +9.32338 q^{89} -19.2090 q^{91} -9.22583 q^{93} -16.3255 q^{95} +1.52558 q^{97} +20.1978 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 18 q - 10 q^{5} + 32 q^{9} + 12 q^{13} + 16 q^{17} - 14 q^{21} + 44 q^{25} - 22 q^{29} + 24 q^{33} + 4 q^{37} + 50 q^{41} - 32 q^{45} + 58 q^{49} - 2 q^{53} + 46 q^{57} - 10 q^{61} + 40 q^{65} - 22 q^{69}+ \cdots + 48 q^{97}+O(q^{100}) $ 18 * q - 10 * q^5 + 32 * q^9 + 12 * q^13 + 16 * q^17 - 14 * q^21 + 44 * q^25 - 22 * q^29 + 24 * q^33 + 4 * q^37 + 50 * q^41 - 32 * q^45 + 58 * q^49 - 2 * q^53 + 46 * q^57 - 10 * q^61 + 40 * q^65 - 22 * q^69 + 62 * q^73 - 14 * q^77 + 46 * q^81 - 12 * q^85 + 52 * q^89 - 16 * q^93 + 48 * q^97

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
$3$	2.69830	1.55786	0.778932	−	0.627109i	$-0.215762\pi$
$3$	2.69830	1.55786	0.778932	+	0.627109i	$0.215762\pi$
$4$	0	0
$5$	−3.42881	−1.53341	−0.766705	−	0.642000i	$-0.778105\pi$
$5$	−3.42881	−1.53341	−0.766705	+	0.642000i	$0.778105\pi$
$6$	0	0
$7$	−2.84717	−1.07613	−0.538064	−	0.842904i	$-0.680844\pi$
$7$	−2.84717	−1.07613	−0.538064	+	0.842904i	$0.680844\pi$
$8$	0	0
$9$	4.28081	1.42694
$10$	0	0
$11$	4.71823	1.42260	0.711299	−	0.702889i	$-0.248107\pi$
$11$	4.71823	1.42260	0.711299	+	0.702889i	$0.248107\pi$
$12$	0	0
$13$	6.74669	1.87119	0.935597	−	0.353070i	$-0.114862\pi$
$13$	6.74669	1.87119	0.935597	+	0.353070i	$0.114862\pi$
$14$	0	0
$15$	−9.25194	−2.38884
$16$	0	0
$17$	−7.43351	−1.80289	−0.901445	−	0.432893i	$-0.857493\pi$
$17$	−7.43351	−1.80289	−0.901445	+	0.432893i	$0.857493\pi$
$18$	0	0
$19$	4.76127	1.09231	0.546156	−	0.837684i	$-0.316091\pi$
$19$	4.76127	1.09231	0.546156	+	0.837684i	$0.316091\pi$
$20$	0	0
$21$	−7.68251	−1.67646
$22$	0	0
$23$	0.359963	0.0750575	0.0375288	−	0.999296i	$-0.488051\pi$
$23$	0.359963	0.0750575	0.0375288	+	0.999296i	$0.488051\pi$
$24$	0	0
$25$	6.75672	1.35134
$26$	0	0
$27$	3.45601	0.665110
$28$	0	0
$29$	1.26714	0.235302	0.117651	−	0.993055i	$-0.462464\pi$
$29$	1.26714	0.235302	0.117651	+	0.993055i	$0.462464\pi$
$30$	0	0
$31$	−3.41913	−0.614094	−0.307047	−	0.951694i	$-0.599341\pi$
$31$	−3.41913	−0.614094	−0.307047	+	0.951694i	$0.599341\pi$
$32$	0	0
$33$	12.7312	2.21621
$34$	0	0
$35$	9.76239	1.65015
$36$	0	0
$37$	4.25370	0.699305	0.349652	−	0.936880i	$-0.386300\pi$
$37$	4.25370	0.699305	0.349652	+	0.936880i	$0.386300\pi$
$38$	0	0
$39$	18.2046	2.91506
$40$	0	0
$41$	4.67337	0.729858	0.364929	−	0.931035i	$-0.381093\pi$
$41$	4.67337	0.729858	0.364929	+	0.931035i	$0.381093\pi$
$42$	0	0
$43$	−10.0952	−1.53950	−0.769749	−	0.638347i	$-0.779619\pi$
$43$	−10.0952	−1.53950	−0.769749	+	0.638347i	$0.779619\pi$
$44$	0	0
$45$	−14.6781	−2.18808
$46$	0	0
$47$	6.65306	0.970449	0.485225	−	0.874390i	$-0.338738\pi$
$47$	6.65306	0.970449	0.485225	+	0.874390i	$0.338738\pi$
$48$	0	0
$49$	1.10637	0.158053
$50$	0	0
$51$	−20.0578	−2.80866
$52$	0	0
$53$	−9.13377	−1.25462	−0.627310	−	0.778770i	$-0.715844\pi$
$53$	−9.13377	−1.25462	−0.627310	+	0.778770i	$0.715844\pi$
$54$	0	0
$55$	−16.1779	−2.18143
$56$	0	0
$57$	12.8473	1.70167
$58$	0	0
$59$	−0.225839	−0.0294018	−0.0147009	−	0.999892i	$-0.504680\pi$
$59$	−0.225839	−0.0294018	−0.0147009	+	0.999892i	$0.504680\pi$
$60$	0	0
$61$	−12.4007	−1.58775	−0.793873	−	0.608083i	$-0.791939\pi$
$61$	−12.4007	−1.58775	−0.793873	+	0.608083i	$0.791939\pi$
$62$	0	0
$63$	−12.1882	−1.53557
$64$	0	0
$65$	−23.1331	−2.86931
$66$	0	0
$67$	10.6330	1.29903	0.649514	−	0.760349i	$-0.274972\pi$
$67$	10.6330	1.29903	0.649514	+	0.760349i	$0.274972\pi$
$68$	0	0
$69$	0.971288	0.116929
$70$	0	0
$71$	1.60620	0.190621	0.0953103	−	0.995448i	$-0.469616\pi$
$71$	1.60620	0.190621	0.0953103	+	0.995448i	$0.469616\pi$
$72$	0	0
$73$	4.98656	0.583633	0.291816	−	0.956474i	$-0.405740\pi$
$73$	4.98656	0.583633	0.291816	+	0.956474i	$0.405740\pi$
$74$	0	0
$75$	18.2316	2.10521
$76$	0	0
$77$	−13.4336	−1.53090
$78$	0	0
$79$	11.9260	1.34178	0.670892	−	0.741555i	$-0.265911\pi$
$79$	11.9260	1.34178	0.670892	+	0.741555i	$0.265911\pi$
$80$	0	0
$81$	−3.51709	−0.390787
$82$	0	0
$83$	7.11944	0.781460	0.390730	−	0.920505i	$-0.372223\pi$
$83$	7.11944	0.781460	0.390730	+	0.920505i	$0.372223\pi$
$84$	0	0
$85$	25.4881	2.76457
$86$	0	0
$87$	3.41913	0.366569
$88$	0	0
$89$	9.32338	0.988276	0.494138	−	0.869383i	$-0.335484\pi$
$89$	9.32338	0.988276	0.494138	+	0.869383i	$0.335484\pi$
$90$	0	0
$91$	−19.2090	−2.01365
$92$	0	0
$93$	−9.22583	−0.956674
$94$	0	0
$95$	−16.3255	−1.67496
$96$	0	0
$97$	1.52558	0.154899	0.0774495	−	0.996996i	$-0.475322\pi$
$97$	1.52558	0.154899	0.0774495	+	0.996996i	$0.475322\pi$
$98$	0	0
$99$	20.1978	2.02996
$100$	0	0
$101$	15.0599	1.49851	0.749256	−	0.662280i	$-0.230411\pi$
$101$	15.0599	1.49851	0.749256	+	0.662280i	$0.230411\pi$
$102$	0	0
$103$	13.8666	1.36632	0.683159	−	0.730269i	$-0.260605\pi$
$103$	13.8666	1.36632	0.683159	+	0.730269i	$0.260605\pi$
$104$	0	0
$105$	26.3418	2.57070
$106$	0	0
$107$	10.4193	1.00727	0.503637	−	0.863916i	$-0.331995\pi$
$107$	10.4193	1.00727	0.503637	+	0.863916i	$0.331995\pi$
$108$	0	0
$109$	9.62687	0.922087	0.461044	−	0.887377i	$-0.347475\pi$
$109$	9.62687	0.922087	0.461044	+	0.887377i	$0.347475\pi$
$110$	0	0
$111$	11.4778	1.08942
$112$	0	0
$113$	−1.00000	−0.0940721
$113$	−1.00000	−0.0940721
$114$	0	0
$115$	−1.23424	−0.115094
$116$	0	0
$117$	28.8813	2.67008
$118$	0	0
$119$	21.1645	1.94014
$120$	0	0
$121$	11.2617	1.02379
$122$	0	0
$123$	12.6102	1.13702
$124$	0	0
$125$	−6.02344	−0.538753
$126$	0	0
$127$	13.2031	1.17159	0.585794	−	0.810460i	$-0.300783\pi$
$127$	13.2031	1.17159	0.585794	+	0.810460i	$0.300783\pi$
$128$	0	0
$129$	−27.2398	−2.39833
$130$	0	0
$131$	−18.1645	−1.58704	−0.793519	−	0.608545i	$-0.791753\pi$
$131$	−18.1645	−1.58704	−0.793519	+	0.608545i	$0.791753\pi$
$132$	0	0
$133$	−13.5562	−1.17547
$134$	0	0
$135$	−11.8500	−1.01989
$136$	0	0
$137$	−19.9747	−1.70655	−0.853276	−	0.521459i	$-0.825388\pi$
$137$	−19.9747	−1.70655	−0.853276	+	0.521459i	$0.825388\pi$
$138$	0	0
$139$	7.81379	0.662757	0.331378	−	0.943498i	$-0.392486\pi$
$139$	7.81379	0.662757	0.331378	+	0.943498i	$0.392486\pi$
$140$	0	0
$141$	17.9519	1.51183
$142$	0	0
$143$	31.8324	2.66196
$144$	0	0
$145$	−4.34479	−0.360815
$146$	0	0
$147$	2.98532	0.246225
$148$	0	0
$149$	−10.8185	−0.886287	−0.443144	−	0.896451i	$-0.646137\pi$
$149$	−10.8185	−0.886287	−0.443144	+	0.896451i	$0.646137\pi$
$150$	0	0
$151$	−7.16369	−0.582973	−0.291487	−	0.956575i	$-0.594150\pi$
$151$	−7.16369	−0.582973	−0.291487	+	0.956575i	$0.594150\pi$
$152$	0	0
$153$	−31.8215	−2.57261
$154$	0	0
$155$	11.7235	0.941657
$156$	0	0
$157$	10.5871	0.844943	0.422471	−	0.906376i	$-0.361163\pi$
$157$	10.5871	0.844943	0.422471	+	0.906376i	$0.361163\pi$
$158$	0	0
$159$	−24.6456	−1.95453
$160$	0	0
$161$	−1.02488	−0.0807715
$162$	0	0
$163$	17.3434	1.35844	0.679220	−	0.733935i	$-0.262318\pi$
$163$	17.3434	1.35844	0.679220	+	0.733935i	$0.262318\pi$
$164$	0	0
$165$	−43.6528	−3.39836
$166$	0	0
$167$	7.37120	0.570401	0.285200	−	0.958468i	$-0.407940\pi$
$167$	7.37120	0.570401	0.285200	+	0.958468i	$0.407940\pi$
$168$	0	0
$169$	32.5178	2.50137
$170$	0	0
$171$	20.3821	1.55866
$172$	0	0
$173$	11.3961	0.866428	0.433214	−	0.901291i	$-0.357379\pi$
$173$	11.3961	0.866428	0.433214	+	0.901291i	$0.357379\pi$
$174$	0	0
$175$	−19.2375	−1.45422
$176$	0	0
$177$	−0.609382	−0.0458040
$178$	0	0
$179$	12.4196	0.928284	0.464142	−	0.885761i	$-0.346363\pi$
$179$	12.4196	0.928284	0.464142	+	0.885761i	$0.346363\pi$
$180$	0	0
$181$	−10.2194	−0.759600	−0.379800	−	0.925069i	$-0.624007\pi$
$181$	−10.2194	−0.759600	−0.379800	+	0.925069i	$0.624007\pi$
$182$	0	0
$183$	−33.4608	−2.47349
$184$	0	0
$185$	−14.5851	−1.07232
$186$	0	0
$187$	−35.0730	−2.56479
$188$	0	0
$189$	−9.83985	−0.715744
$190$	0	0
$191$	8.14408	0.589285	0.294643	−	0.955608i	$-0.404799\pi$
$191$	8.14408	0.589285	0.294643	+	0.955608i	$0.404799\pi$
$192$	0	0
$193$	−10.2754	−0.739639	−0.369819	−	0.929104i	$-0.620580\pi$
$193$	−10.2754	−0.739639	−0.369819	+	0.929104i	$0.620580\pi$
$194$	0	0
$195$	−62.4199	−4.46999
$196$	0	0
$197$	13.9307	0.992519	0.496260	−	0.868174i	$-0.334706\pi$
$197$	13.9307	0.992519	0.496260	+	0.868174i	$0.334706\pi$
$198$	0	0
$199$	20.3218	1.44057	0.720287	−	0.693676i	$-0.244010\pi$
$199$	20.3218	1.44057	0.720287	+	0.693676i	$0.244010\pi$
$200$	0	0
$201$	28.6910	2.02371
$202$	0	0
$203$	−3.60777	−0.253216
$204$	0	0
$205$	−16.0241	−1.11917
$206$	0	0
$207$	1.54093	0.107102
$208$	0	0
$209$	22.4648	1.55392
$210$	0	0
$211$	10.5670	0.727464	0.363732	−	0.931504i	$-0.381502\pi$
$211$	10.5670	0.727464	0.363732	+	0.931504i	$0.381502\pi$
$212$	0	0
$213$	4.33400	0.296961
$214$	0	0
$215$	34.6144	2.36068
$216$	0	0
$217$	9.73484	0.660844
$218$	0	0
$219$	13.4552	0.909220
$220$	0	0
$221$	−50.1515	−3.37356
$222$	0	0
$223$	−8.09983	−0.542404	−0.271202	−	0.962522i	$-0.587421\pi$
$223$	−8.09983	−0.542404	−0.271202	+	0.962522i	$0.587421\pi$
$224$	0	0
$225$	28.9242	1.92828
$226$	0	0
$227$	−20.8655	−1.38489	−0.692445	−	0.721470i	$-0.743467\pi$
$227$	−20.8655	−1.38489	−0.692445	+	0.721470i	$0.743467\pi$
$228$	0	0
$229$	−15.6142	−1.03181	−0.515906	−	0.856645i	$-0.672545\pi$
$229$	−15.6142	−1.03181	−0.515906	+	0.856645i	$0.672545\pi$
$230$	0	0
$231$	−36.2478	−2.38493
$232$	0	0
$233$	−12.6714	−0.830129	−0.415064	−	0.909792i	$-0.636241\pi$
$233$	−12.6714	−0.830129	−0.415064	+	0.909792i	$0.636241\pi$
$234$	0	0
$235$	−22.8121	−1.48810
$236$	0	0
$237$	32.1800	2.09032
$238$	0	0
$239$	−0.568349	−0.0367634	−0.0183817	−	0.999831i	$-0.505851\pi$
$239$	−0.568349	−0.0367634	−0.0183817	+	0.999831i	$0.505851\pi$
$240$	0	0
$241$	14.8124	0.954151	0.477076	−	0.878862i	$-0.341697\pi$
$241$	14.8124	0.954151	0.477076	+	0.878862i	$0.341697\pi$
$242$	0	0
$243$	−19.8582	−1.27390
$244$	0	0
$245$	−3.79354	−0.242360
$246$	0	0
$247$	32.1228	2.04393
$248$	0	0
$249$	19.2104	1.21741
$250$	0	0
$251$	−16.3285	−1.03064	−0.515322	−	0.856997i	$-0.672328\pi$
$251$	−16.3285	−1.03064	−0.515322	+	0.856997i	$0.672328\pi$
$252$	0	0
$253$	1.69839	0.106777
$254$	0	0
$255$	68.7744	4.30682
$256$	0	0
$257$	9.93074	0.619463	0.309731	−	0.950824i	$-0.399761\pi$
$257$	9.93074	0.619463	0.309731	+	0.950824i	$0.399761\pi$
$258$	0	0
$259$	−12.1110	−0.752542
$260$	0	0
$261$	5.42440	0.335762
$262$	0	0
$263$	−15.5531	−0.959043	−0.479521	−	0.877530i	$-0.659190\pi$
$263$	−15.5531	−0.959043	−0.479521	+	0.877530i	$0.659190\pi$
$264$	0	0
$265$	31.3179	1.92385
$266$	0	0
$267$	25.1573	1.53960
$268$	0	0
$269$	4.70625	0.286945	0.143473	−	0.989654i	$-0.454173\pi$
$269$	4.70625	0.286945	0.143473	+	0.989654i	$0.454173\pi$
$270$	0	0
$271$	11.2569	0.683809	0.341905	−	0.939735i	$-0.388928\pi$
$271$	11.2569	0.683809	0.341905	+	0.939735i	$0.388928\pi$
$272$	0	0
$273$	−51.8315	−3.13698
$274$	0	0
$275$	31.8797	1.92242
$276$	0	0
$277$	19.9915	1.20117	0.600586	−	0.799560i	$-0.294934\pi$
$277$	19.9915	1.20117	0.600586	+	0.799560i	$0.294934\pi$
$278$	0	0
$279$	−14.6366	−0.876273
$280$	0	0
$281$	−6.27396	−0.374273	−0.187136	−	0.982334i	$-0.559921\pi$
$281$	−6.27396	−0.374273	−0.187136	+	0.982334i	$0.559921\pi$
$282$	0	0
$283$	25.5139	1.51664	0.758321	−	0.651881i	$-0.226020\pi$
$283$	25.5139	1.51664	0.758321	+	0.651881i	$0.226020\pi$
$284$	0	0
$285$	−44.0510	−2.60936
$286$	0	0
$287$	−13.3059	−0.785421
$288$	0	0
$289$	38.2571	2.25042
$290$	0	0
$291$	4.11646	0.241311
$292$	0	0
$293$	18.4943	1.08045	0.540225	−	0.841521i	$-0.318339\pi$
$293$	18.4943	1.08045	0.540225	+	0.841521i	$0.318339\pi$
$294$	0	0
$295$	0.774360	0.0450850
$296$	0	0
$297$	16.3062	0.946184
$298$	0	0
$299$	2.42856	0.140447
$300$	0	0
$301$	28.7426	1.65670
$302$	0	0
$303$	40.6360	2.33448
$304$	0	0
$305$	42.5196	2.43466
$306$	0	0
$307$	−2.32452	−0.132667	−0.0663337	−	0.997797i	$-0.521130\pi$
$307$	−2.32452	−0.132667	−0.0663337	+	0.997797i	$0.521130\pi$
$308$	0	0
$309$	37.4163	2.12854
$310$	0	0
$311$	−21.9024	−1.24197	−0.620985	−	0.783822i	$-0.713267\pi$
$311$	−21.9024	−1.24197	−0.620985	+	0.783822i	$0.713267\pi$
$312$	0	0
$313$	29.6212	1.67429	0.837145	−	0.546982i	$-0.184223\pi$
$313$	29.6212	1.67429	0.837145	+	0.546982i	$0.184223\pi$
$314$	0	0
$315$	41.7910	2.35465
$316$	0	0
$317$	17.6672	0.992290	0.496145	−	0.868240i	$-0.334748\pi$
$317$	17.6672	0.992290	0.496145	+	0.868240i	$0.334748\pi$
$318$	0	0
$319$	5.97866	0.334741
$320$	0	0
$321$	28.1144	1.56919
$322$	0	0
$323$	−35.3930	−1.96932
$324$	0	0
$325$	45.5854	2.52863
$326$	0	0
$327$	25.9762	1.43649
$328$	0	0
$329$	−18.9424	−1.04433
$330$	0	0
$331$	−10.7008	−0.588170	−0.294085	−	0.955779i	$-0.595015\pi$
$331$	−10.7008	−0.588170	−0.294085	+	0.955779i	$0.595015\pi$
$332$	0	0
$333$	18.2093	0.997864
$334$	0	0
$335$	−36.4585	−1.99194
$336$	0	0
$337$	−2.96901	−0.161732	−0.0808661	−	0.996725i	$-0.525769\pi$
$337$	−2.96901	−0.161732	−0.0808661	+	0.996725i	$0.525769\pi$
$338$	0	0
$339$	−2.69830	−0.146551
$340$	0	0
$341$	−16.1322	−0.873609
$342$	0	0
$343$	16.7802	0.906043
$344$	0	0
$345$	−3.33036	−0.179300
$346$	0	0
$347$	8.56410	0.459745	0.229872	−	0.973221i	$-0.426169\pi$
$347$	8.56410	0.459745	0.229872	+	0.973221i	$0.426169\pi$
$348$	0	0
$349$	4.98871	0.267039	0.133520	−	0.991046i	$-0.457372\pi$
$349$	4.98871	0.267039	0.133520	+	0.991046i	$0.457372\pi$
$350$	0	0
$351$	23.3166	1.24455
$352$	0	0
$353$	17.3096	0.921295	0.460648	−	0.887583i	$-0.347617\pi$
$353$	17.3096	0.921295	0.460648	+	0.887583i	$0.347617\pi$
$354$	0	0
$355$	−5.50734	−0.292299
$356$	0	0
$357$	57.1080	3.02248
$358$	0	0
$359$	−30.8672	−1.62911	−0.814554	−	0.580088i	$-0.803018\pi$
$359$	−30.8672	−1.62911	−0.814554	+	0.580088i	$0.803018\pi$
$360$	0	0
$361$	3.66974	0.193144
$362$	0	0
$363$	30.3873	1.59492
$364$	0	0
$365$	−17.0980	−0.894948
$366$	0	0
$367$	−35.8032	−1.86891	−0.934456	−	0.356078i	$-0.884114\pi$
$367$	−35.8032	−1.86891	−0.934456	+	0.356078i	$0.884114\pi$
$368$	0	0
$369$	20.0058	1.04146
$370$	0	0
$371$	26.0054	1.35013
$372$	0	0
$373$	−11.6236	−0.601850	−0.300925	−	0.953648i	$-0.597295\pi$
$373$	−11.6236	−0.601850	−0.300925	+	0.953648i	$0.597295\pi$
$374$	0	0
$375$	−16.2530	−0.839304
$376$	0	0
$377$	8.54901	0.440297
$378$	0	0
$379$	−27.5715	−1.41625	−0.708126	−	0.706086i	$-0.750459\pi$
$379$	−27.5715	−1.41625	−0.708126	+	0.706086i	$0.750459\pi$
$380$	0	0
$381$	35.6260	1.82517
$382$	0	0
$383$	−20.2709	−1.03580	−0.517898	−	0.855442i	$-0.673285\pi$
$383$	−20.2709	−1.03580	−0.517898	+	0.855442i	$0.673285\pi$
$384$	0	0
$385$	46.0612	2.34749
$386$	0	0
$387$	−43.2155	−2.19677
$388$	0	0
$389$	1.56013	0.0791016	0.0395508	−	0.999218i	$-0.487407\pi$
$389$	1.56013	0.0791016	0.0395508	+	0.999218i	$0.487407\pi$
$390$	0	0
$391$	−2.67579	−0.135320
$392$	0	0
$393$	−49.0132	−2.47239
$394$	0	0
$395$	−40.8921	−2.05750
$396$	0	0
$397$	−9.30840	−0.467175	−0.233588	−	0.972336i	$-0.575047\pi$
$397$	−9.30840	−0.467175	−0.233588	+	0.972336i	$0.575047\pi$
$398$	0	0
$399$	−36.5785	−1.83122
$400$	0	0
$401$	18.6036	0.929019	0.464510	−	0.885568i	$-0.346231\pi$
$401$	18.6036	0.929019	0.464510	+	0.885568i	$0.346231\pi$
$402$	0	0
$403$	−23.0678	−1.14909
$404$	0	0
$405$	12.0594	0.599237
$406$	0	0
$407$	20.0699	0.994830
$408$	0	0
$409$	−4.93856	−0.244196	−0.122098	−	0.992518i	$-0.538962\pi$
$409$	−4.93856	−0.244196	−0.122098	+	0.992518i	$0.538962\pi$
$410$	0	0
$411$	−53.8977	−2.65858
$412$	0	0
$413$	0.643003	0.0316401
$414$	0	0
$415$	−24.4112	−1.19830
$416$	0	0
$417$	21.0839	1.03248
$418$	0	0
$419$	−31.0059	−1.51474	−0.757368	−	0.652988i	$-0.773515\pi$
$419$	−31.0059	−1.51474	−0.757368	+	0.652988i	$0.773515\pi$
$420$	0	0
$421$	21.0530	1.02606	0.513031	−	0.858370i	$-0.328523\pi$
$421$	21.0530	1.02606	0.513031	+	0.858370i	$0.328523\pi$
$422$	0	0
$423$	28.4805	1.38477
$424$	0	0
$425$	−50.2261	−2.43632
$426$	0	0
$427$	35.3069	1.70862
$428$	0	0
$429$	85.8933	4.14697
$430$	0	0
$431$	−22.8825	−1.10221	−0.551106	−	0.834435i	$-0.685794\pi$
$431$	−22.8825	−1.10221	−0.551106	+	0.834435i	$0.685794\pi$
$432$	0	0
$433$	−3.57403	−0.171757	−0.0858786	−	0.996306i	$-0.527370\pi$
$433$	−3.57403	−0.171757	−0.0858786	+	0.996306i	$0.527370\pi$
$434$	0	0
$435$	−11.7235	−0.562100
$436$	0	0
$437$	1.71388	0.0819862
$438$	0	0
$439$	−13.2021	−0.630101	−0.315051	−	0.949075i	$-0.602022\pi$
$439$	−13.2021	−0.630101	−0.315051	+	0.949075i	$0.602022\pi$
$440$	0	0
$441$	4.73617	0.225532
$442$	0	0
$443$	35.9563	1.70833	0.854167	−	0.519998i	$-0.174067\pi$
$443$	35.9563	1.70833	0.854167	+	0.519998i	$0.174067\pi$
$444$	0	0
$445$	−31.9681	−1.51543
$446$	0	0
$447$	−29.1916	−1.38071
$448$	0	0
$449$	2.21274	0.104426	0.0522129	−	0.998636i	$-0.483373\pi$
$449$	2.21274	0.104426	0.0522129	+	0.998636i	$0.483373\pi$
$450$	0	0
$451$	22.0500	1.03830
$452$	0	0
$453$	−19.3298	−0.908192
$454$	0	0
$455$	65.8638	3.08774
$456$	0	0
$457$	4.23134	0.197934	0.0989669	−	0.995091i	$-0.468446\pi$
$457$	4.23134	0.197934	0.0989669	+	0.995091i	$0.468446\pi$
$458$	0	0
$459$	−25.6903	−1.19912
$460$	0	0
$461$	−18.2833	−0.851540	−0.425770	−	0.904831i	$-0.639997\pi$
$461$	−18.2833	−0.851540	−0.425770	+	0.904831i	$0.639997\pi$
$462$	0	0
$463$	28.9269	1.34435	0.672174	−	0.740394i	$-0.265361\pi$
$463$	28.9269	1.34435	0.672174	+	0.740394i	$0.265361\pi$
$464$	0	0
$465$	31.6336	1.46697
$466$	0	0
$467$	10.3400	0.478479	0.239240	−	0.970961i	$-0.423102\pi$
$467$	10.3400	0.478479	0.239240	+	0.970961i	$0.423102\pi$
$468$	0	0
$469$	−30.2740	−1.39792
$470$	0	0
$471$	28.5672	1.31631
$472$	0	0
$473$	−47.6313	−2.19009
$474$	0	0
$475$	32.1706	1.47609
$476$	0	0
$477$	−39.1000	−1.79026
$478$	0	0
$479$	31.8858	1.45690	0.728450	−	0.685099i	$-0.240241\pi$
$479$	31.8858	1.45690	0.728450	+	0.685099i	$0.240241\pi$
$480$	0	0
$481$	28.6984	1.30853
$482$	0	0
$483$	−2.76542	−0.125831
$484$	0	0
$485$	−5.23091	−0.237523
$486$	0	0
$487$	7.32866	0.332093	0.166047	−	0.986118i	$-0.446900\pi$
$487$	7.32866	0.332093	0.166047	+	0.986118i	$0.446900\pi$
$488$	0	0
$489$	46.7977	2.11626
$490$	0	0
$491$	−0.00305169	−0.000137721 0	−6.88605e−5	−	1.00000i	$-0.500022\pi$
$491$	−0.00305169	−0.000137721 0	−6.88605e−5		1.00000i	$0.500022\pi$
$492$	0	0
$493$	−9.41932	−0.424225
$494$	0	0
$495$	−69.2545	−3.11276
$496$	0	0
$497$	−4.57311	−0.205132
$498$	0	0
$499$	15.5176	0.694664	0.347332	−	0.937742i	$-0.387088\pi$
$499$	15.5176	0.694664	0.347332	+	0.937742i	$0.387088\pi$
$500$	0	0
$501$	19.8897	0.888606
$502$	0	0
$503$	−13.0145	−0.580286	−0.290143	−	0.956983i	$-0.593703\pi$
$503$	−13.0145	−0.580286	−0.290143	+	0.956983i	$0.593703\pi$
$504$	0	0
$505$	−51.6374	−2.29783
$506$	0	0
$507$	87.7426	3.89679
$508$	0	0
$509$	−22.6054	−1.00197	−0.500984	−	0.865457i	$-0.667028\pi$
$509$	−22.6054	−1.00197	−0.500984	+	0.865457i	$0.667028\pi$
$510$	0	0
$511$	−14.1976	−0.628064
$512$	0	0
$513$	16.4550	0.726507
$514$	0	0
$515$	−47.5460	−2.09513
$516$	0	0
$517$	31.3907	1.38056
$518$	0	0
$519$	30.7500	1.34978
$520$	0	0
$521$	7.22908	0.316712	0.158356	−	0.987382i	$-0.449381\pi$
$521$	7.22908	0.316712	0.158356	+	0.987382i	$0.449381\pi$
$522$	0	0
$523$	−9.69063	−0.423742	−0.211871	−	0.977298i	$-0.567956\pi$
$523$	−9.69063	−0.423742	−0.211871	+	0.977298i	$0.567956\pi$
$524$	0	0
$525$	−51.9085	−2.26547
$526$	0	0
$527$	25.4161	1.10714
$528$	0	0
$529$	−22.8704	−0.994366
$530$	0	0
$531$	−0.966776	−0.0419545
$532$	0	0
$533$	31.5298	1.36571
$534$	0	0
$535$	−35.7258	−1.54456
$536$	0	0
$537$	33.5118	1.44614
$538$	0	0
$539$	5.22011	0.224846
$540$	0	0
$541$	−1.13780	−0.0489177	−0.0244589	−	0.999701i	$-0.507786\pi$
$541$	−1.13780	−0.0489177	−0.0244589	+	0.999701i	$0.507786\pi$
$542$	0	0
$543$	−27.5749	−1.18335
$544$	0	0
$545$	−33.0087	−1.41394
$546$	0	0
$547$	−26.0203	−1.11255	−0.556275	−	0.830998i	$-0.687770\pi$
$547$	−26.0203	−1.11255	−0.556275	+	0.830998i	$0.687770\pi$
$548$	0	0
$549$	−53.0850	−2.26561
$550$	0	0
$551$	6.03321	0.257024
$552$	0	0
$553$	−33.9554	−1.44393
$554$	0	0
$555$	−39.3550	−1.67053
$556$	0	0
$557$	−23.8048	−1.00864	−0.504320	−	0.863517i	$-0.668257\pi$
$557$	−23.8048	−1.00864	−0.504320	+	0.863517i	$0.668257\pi$
$558$	0	0
$559$	−68.1089	−2.88070
$560$	0	0
$561$	−94.6373	−3.99559
$562$	0	0
$563$	12.5203	0.527667	0.263833	−	0.964568i	$-0.415013\pi$
$563$	12.5203	0.527667	0.263833	+	0.964568i	$0.415013\pi$
$564$	0	0
$565$	3.42881	0.144251
$566$	0	0
$567$	10.0137	0.420538
$568$	0	0
$569$	31.5642	1.32324	0.661619	−	0.749840i	$-0.269870\pi$
$569$	31.5642	1.32324	0.661619	+	0.749840i	$0.269870\pi$
$570$	0	0
$571$	11.9355	0.499484	0.249742	−	0.968312i	$-0.419654\pi$
$571$	11.9355	0.499484	0.249742	+	0.968312i	$0.419654\pi$
$572$	0	0
$573$	21.9752	0.918025
$574$	0	0
$575$	2.43217	0.101428
$576$	0	0
$577$	38.1999	1.59028	0.795142	−	0.606423i	$-0.207396\pi$
$577$	38.1999	1.59028	0.795142	+	0.606423i	$0.207396\pi$
$578$	0	0
$579$	−27.7261	−1.15226
$580$	0	0
$581$	−20.2702	−0.840951
$582$	0	0
$583$	−43.0952	−1.78482
$584$	0	0
$585$	−99.0284	−4.09432
$586$	0	0
$587$	6.63017	0.273657	0.136828	−	0.990595i	$-0.456309\pi$
$587$	6.63017	0.273657	0.136828	+	0.990595i	$0.456309\pi$
$588$	0	0
$589$	−16.2794	−0.670781
$590$	0	0
$591$	37.5891	1.54621
$592$	0	0
$593$	−1.34361	−0.0551756	−0.0275878	−	0.999619i	$-0.508783\pi$
$593$	−1.34361	−0.0551756	−0.0275878	+	0.999619i	$0.508783\pi$
$594$	0	0
$595$	−72.5688	−2.97503
$596$	0	0
$597$	54.8343	2.24422
$598$	0	0
$599$	−5.29237	−0.216240	−0.108120	−	0.994138i	$-0.534483\pi$
$599$	−5.29237	−0.216240	−0.108120	+	0.994138i	$0.534483\pi$
$600$	0	0
$601$	−4.69432	−0.191485	−0.0957427	−	0.995406i	$-0.530523\pi$
$601$	−4.69432	−0.191485	−0.0957427	+	0.995406i	$0.530523\pi$
$602$	0	0
$603$	45.5179	1.85363
$604$	0	0
$605$	−38.6140	−1.56988
$606$	0	0
$607$	−20.2481	−0.821847	−0.410923	−	0.911670i	$-0.634794\pi$
$607$	−20.2481	−0.821847	−0.410923	+	0.911670i	$0.634794\pi$
$608$	0	0
$609$	−9.73484	−0.394475
$610$	0	0
$611$	44.8861	1.81590
$612$	0	0
$613$	19.1313	0.772707	0.386354	−	0.922351i	$-0.373734\pi$
$613$	19.1313	0.772707	0.386354	+	0.922351i	$0.373734\pi$
$614$	0	0
$615$	−43.2378	−1.74352
$616$	0	0
$617$	−12.3171	−0.495867	−0.247934	−	0.968777i	$-0.579751\pi$
$617$	−12.3171	−0.495867	−0.247934	+	0.968777i	$0.579751\pi$
$618$	0	0
$619$	−19.0171	−0.764363	−0.382182	−	0.924087i	$-0.624827\pi$
$619$	−19.0171	−0.764363	−0.382182	+	0.924087i	$0.624827\pi$
$620$	0	0
$621$	1.24404	0.0499215
$622$	0	0
$623$	−26.5452	−1.06351
$624$	0	0
$625$	−13.1304	−0.525214
$626$	0	0
$627$	60.6166	2.42080
$628$	0	0
$629$	−31.6199	−1.26077
$630$	0	0
$631$	0.611512	0.0243439	0.0121720	−	0.999926i	$-0.496125\pi$
$631$	0.611512	0.0243439	0.0121720	+	0.999926i	$0.496125\pi$
$632$	0	0
$633$	28.5130	1.13329
$634$	0	0
$635$	−45.2710	−1.79652
$636$	0	0
$637$	7.46434	0.295748
$638$	0	0
$639$	6.87583	0.272004
$640$	0	0
$641$	40.6063	1.60385	0.801926	−	0.597423i	$-0.203809\pi$
$641$	40.6063	1.60385	0.801926	+	0.597423i	$0.203809\pi$
$642$	0	0
$643$	−39.3007	−1.54987	−0.774933	−	0.632043i	$-0.782217\pi$
$643$	−39.3007	−1.54987	−0.774933	+	0.632043i	$0.782217\pi$
$644$	0	0
$645$	93.3999	3.67762
$646$	0	0
$647$	24.0464	0.945363	0.472681	−	0.881233i	$-0.343286\pi$
$647$	24.0464	0.945363	0.472681	+	0.881233i	$0.343286\pi$
$648$	0	0
$649$	−1.06556	−0.0418270
$650$	0	0
$651$	26.2675	1.02950
$652$	0	0
$653$	43.7991	1.71399	0.856995	−	0.515325i	$-0.172329\pi$
$653$	43.7991	1.71399	0.856995	+	0.515325i	$0.172329\pi$
$654$	0	0
$655$	62.2825	2.43358
$656$	0	0
$657$	21.3465	0.832807
$658$	0	0
$659$	−30.0355	−1.17002	−0.585009	−	0.811027i	$-0.698909\pi$
$659$	−30.0355	−1.17002	−0.585009	+	0.811027i	$0.698909\pi$
$660$	0	0
$661$	7.06842	0.274930	0.137465	−	0.990507i	$-0.456105\pi$
$661$	7.06842	0.274930	0.137465	+	0.990507i	$0.456105\pi$
$662$	0	0
$663$	−135.324	−5.25554
$664$	0	0
$665$	46.4814	1.80247
$666$	0	0
$667$	0.456125	0.0176612
$668$	0	0
$669$	−21.8557	−0.844992
$670$	0	0
$671$	−58.5093	−2.25873
$672$	0	0
$673$	5.52677	0.213041	0.106521	−	0.994310i	$-0.466029\pi$
$673$	5.52677	0.213041	0.106521	+	0.994310i	$0.466029\pi$
$674$	0	0
$675$	23.3513	0.898792
$676$	0	0
$677$	−11.3344	−0.435618	−0.217809	−	0.975991i	$-0.569891\pi$
$677$	−11.3344	−0.435618	−0.217809	+	0.975991i	$0.569891\pi$
$678$	0	0
$679$	−4.34358	−0.166691
$680$	0	0
$681$	−56.3013	−2.15747
$682$	0	0
$683$	−26.7119	−1.02210	−0.511050	−	0.859551i	$-0.670743\pi$
$683$	−26.7119	−1.02210	−0.511050	+	0.859551i	$0.670743\pi$
$684$	0	0
$685$	68.4893	2.61684
$686$	0	0
$687$	−42.1316	−1.60742
$688$	0	0
$689$	−61.6227	−2.34764
$690$	0	0
$691$	42.1885	1.60493	0.802464	−	0.596701i	$-0.203522\pi$
$691$	42.1885	1.60493	0.802464	+	0.596701i	$0.203522\pi$
$692$	0	0
$693$	−57.5067	−2.18450
$694$	0	0
$695$	−26.7920	−1.01628
$696$	0	0
$697$	−34.7396	−1.31585
$698$	0	0
$699$	−34.1911	−1.29323
$700$	0	0
$701$	−25.8789	−0.977433	−0.488716	−	0.872443i	$-0.662535\pi$
$701$	−25.8789	−0.977433	−0.488716	+	0.872443i	$0.662535\pi$
$702$	0	0
$703$	20.2531	0.763858
$704$	0	0
$705$	−61.5538	−2.31825
$706$	0	0
$707$	−42.8780	−1.61259
$708$	0	0
$709$	−31.5871	−1.18628	−0.593139	−	0.805100i	$-0.702112\pi$
$709$	−31.5871	−1.18628	−0.593139	+	0.805100i	$0.702112\pi$
$710$	0	0
$711$	51.0531	1.91464
$712$	0	0
$713$	−1.23076	−0.0460923
$714$	0	0
$715$	−109.147	−4.08187
$716$	0	0
$717$	−1.53357	−0.0572724
$718$	0	0
$719$	−18.2918	−0.682169	−0.341084	−	0.940033i	$-0.610794\pi$
$719$	−18.2918	−0.682169	−0.341084	+	0.940033i	$0.610794\pi$
$720$	0	0
$721$	−39.4806	−1.47033
$722$	0	0
$723$	39.9683	1.48644
$724$	0	0
$725$	8.56172	0.317974
$726$	0	0
$727$	23.8608	0.884949	0.442475	−	0.896781i	$-0.354101\pi$
$727$	23.8608	0.884949	0.442475	+	0.896781i	$0.354101\pi$
$728$	0	0
$729$	−43.0320	−1.59378
$730$	0	0
$731$	75.0425	2.77555
$732$	0	0
$733$	28.0183	1.03488	0.517439	−	0.855720i	$-0.326885\pi$
$733$	28.0183	1.03488	0.517439	+	0.855720i	$0.326885\pi$
$734$	0	0
$735$	−10.2361	−0.377564
$736$	0	0
$737$	50.1689	1.84800
$738$	0	0
$739$	−52.9841	−1.94905	−0.974526	−	0.224277i	$-0.927998\pi$
$739$	−52.9841	−1.94905	−0.974526	+	0.224277i	$0.927998\pi$
$740$	0	0
$741$	86.6769	3.18416
$742$	0	0
$743$	−18.9763	−0.696173	−0.348087	−	0.937462i	$-0.613169\pi$
$743$	−18.9763	−0.696173	−0.348087	+	0.937462i	$0.613169\pi$
$744$	0	0
$745$	37.0946	1.35904
$746$	0	0
$747$	30.4770	1.11509
$748$	0	0
$749$	−29.6656	−1.08396
$750$	0	0
$751$	9.59668	0.350188	0.175094	−	0.984552i	$-0.443977\pi$
$751$	9.59668	0.350188	0.175094	+	0.984552i	$0.443977\pi$
$752$	0	0
$753$	−44.0591	−1.60560
$754$	0	0
$755$	24.5629	0.893936
$756$	0	0
$757$	−7.97049	−0.289692	−0.144846	−	0.989454i	$-0.546269\pi$
$757$	−7.97049	−0.289692	−0.144846	+	0.989454i	$0.546269\pi$
$758$	0	0
$759$	4.58276	0.166343
$760$	0	0
$761$	24.1353	0.874905	0.437453	−	0.899241i	$-0.355881\pi$
$761$	24.1353	0.874905	0.437453	+	0.899241i	$0.355881\pi$
$762$	0	0
$763$	−27.4093	−0.992285
$764$	0	0
$765$	109.110	3.94487
$766$	0	0
$767$	−1.52367	−0.0550165
$768$	0	0
$769$	−53.4244	−1.92653	−0.963266	−	0.268550i	$-0.913456\pi$
$769$	−53.4244	−1.92653	−0.963266	+	0.268550i	$0.913456\pi$
$770$	0	0
$771$	26.7961	0.965038
$772$	0	0
$773$	−42.3927	−1.52476	−0.762380	−	0.647130i	$-0.775969\pi$
$773$	−42.3927	−1.52476	−0.762380	+	0.647130i	$0.775969\pi$
$774$	0	0
$775$	−23.1021	−0.829851
$776$	0	0
$777$	−32.6791	−1.17236
$778$	0	0
$779$	22.2512	0.797232
$780$	0	0
$781$	7.57840	0.271177
$782$	0	0
$783$	4.37926	0.156502
$784$	0	0
$785$	−36.3011	−1.29564
$786$	0	0
$787$	−4.25766	−0.151769	−0.0758845	−	0.997117i	$-0.524178\pi$
$787$	−4.25766	−0.151769	−0.0758845	+	0.997117i	$0.524178\pi$
$788$	0	0
$789$	−41.9668	−1.49406
$790$	0	0
$791$	2.84717	0.101234
$792$	0	0
$793$	−83.6636	−2.97098
$794$	0	0
$795$	84.5051	2.99709
$796$	0	0
$797$	2.48075	0.0878727	0.0439363	−	0.999034i	$-0.486010\pi$
$797$	2.48075	0.0878727	0.0439363	+	0.999034i	$0.486010\pi$
$798$	0	0
$799$	−49.4556	−1.74961
$800$	0	0
$801$	39.9116	1.41021
$802$	0	0
$803$	23.5277	0.830275
$804$	0	0
$805$	3.51410	0.123856
$806$	0	0
$807$	12.6989	0.447021
$808$	0	0
$809$	25.0915	0.882171	0.441085	−	0.897465i	$-0.354594\pi$
$809$	25.0915	0.882171	0.441085	+	0.897465i	$0.354594\pi$
$810$	0	0
$811$	−18.9465	−0.665303	−0.332651	−	0.943050i	$-0.607943\pi$
$811$	−18.9465	−0.665303	−0.332651	+	0.943050i	$0.607943\pi$
$812$	0	0
$813$	30.3745	1.06528
$814$	0	0
$815$	−59.4672	−2.08304
$816$	0	0
$817$	−48.0659	−1.68161
$818$	0	0
$819$	−82.2299	−2.87335
$820$	0	0
$821$	−40.8438	−1.42546	−0.712729	−	0.701440i	$-0.752541\pi$
$821$	−40.8438	−1.42546	−0.712729	+	0.701440i	$0.752541\pi$
$822$	0	0
$823$	−16.4136	−0.572143	−0.286072	−	0.958208i	$-0.592350\pi$
$823$	−16.4136	−0.572143	−0.286072	+	0.958208i	$0.592350\pi$
$824$	0	0
$825$	86.0210	2.99487
$826$	0	0
$827$	−34.5387	−1.20103	−0.600513	−	0.799615i	$-0.705037\pi$
$827$	−34.5387	−1.20103	−0.600513	+	0.799615i	$0.705037\pi$
$828$	0	0
$829$	−28.9537	−1.00560	−0.502802	−	0.864401i	$-0.667698\pi$
$829$	−28.9537	−1.00560	−0.502802	+	0.864401i	$0.667698\pi$
$830$	0	0
$831$	53.9430	1.87126
$832$	0	0
$833$	−8.22423	−0.284953
$834$	0	0
$835$	−25.2744	−0.874658
$836$	0	0
$837$	−11.8165	−0.408440
$838$	0	0
$839$	20.9270	0.722480	0.361240	−	0.932473i	$-0.382353\pi$
$839$	20.9270	0.722480	0.361240	+	0.932473i	$0.382353\pi$
$840$	0	0
$841$	−27.3943	−0.944633
$842$	0	0
$843$	−16.9290	−0.583066
$844$	0	0
$845$	−111.497	−3.83562
$846$	0	0
$847$	−32.0638	−1.10173
$848$	0	0
$849$	68.8440	2.36272
$850$	0	0
$851$	1.53118	0.0524881
$852$	0	0
$853$	44.4090	1.52054	0.760268	−	0.649610i	$-0.225068\pi$
$853$	44.4090	1.52054	0.760268	+	0.649610i	$0.225068\pi$
$854$	0	0
$855$	−69.8864	−2.39006
$856$	0	0
$857$	33.3927	1.14067	0.570337	−	0.821411i	$-0.306813\pi$
$857$	33.3927	1.14067	0.570337	+	0.821411i	$0.306813\pi$
$858$	0	0
$859$	−16.4511	−0.561304	−0.280652	−	0.959810i	$-0.590551\pi$
$859$	−16.4511	−0.561304	−0.280652	+	0.959810i	$0.590551\pi$
$860$	0	0
$861$	−35.9032	−1.22358
$862$	0	0
$863$	−22.5206	−0.766609	−0.383304	−	0.923622i	$-0.625214\pi$
$863$	−22.5206	−0.766609	−0.383304	+	0.923622i	$0.625214\pi$
$864$	0	0
$865$	−39.0750	−1.32859
$866$	0	0
$867$	103.229	3.50584
$868$	0	0
$869$	56.2697	1.90882
$870$	0	0
$871$	71.7376	2.43073
$872$	0	0
$873$	6.53071	0.221031
$874$	0	0
$875$	17.1498	0.579768
$876$	0	0
$877$	−51.2352	−1.73009	−0.865045	−	0.501694i	$-0.832710\pi$
$877$	−51.2352	−1.73009	−0.865045	+	0.501694i	$0.832710\pi$
$878$	0	0
$879$	49.9032	1.68319
$880$	0	0
$881$	−18.9274	−0.637681	−0.318841	−	0.947808i	$-0.603293\pi$
$881$	−18.9274	−0.637681	−0.318841	+	0.947808i	$0.603293\pi$
$882$	0	0
$883$	−32.2404	−1.08498	−0.542488	−	0.840063i	$-0.682518\pi$
$883$	−32.2404	−1.08498	−0.542488	+	0.840063i	$0.682518\pi$
$884$	0	0
$885$	2.08945	0.0702362
$886$	0	0
$887$	55.5605	1.86554	0.932770	−	0.360472i	$-0.117384\pi$
$887$	55.5605	1.86554	0.932770	+	0.360472i	$0.117384\pi$
$888$	0	0
$889$	−37.5915	−1.26078
$890$	0	0
$891$	−16.5944	−0.555934
$892$	0	0
$893$	31.6771	1.06003
$894$	0	0
$895$	−42.5844	−1.42344
$896$	0	0
$897$	6.55297	0.218797
$898$	0	0
$899$	−4.33252	−0.144498
$900$	0	0
$901$	67.8960	2.26194
$902$	0	0
$903$	77.5562	2.58091
$904$	0	0
$905$	35.0402	1.16478
$906$	0	0
$907$	50.3835	1.67296	0.836479	−	0.547999i	$-0.184610\pi$
$907$	50.3835	1.67296	0.836479	+	0.547999i	$0.184610\pi$
$908$	0	0
$909$	64.4684	2.13828
$910$	0	0
$911$	−28.9115	−0.957880	−0.478940	−	0.877848i	$-0.658979\pi$
$911$	−28.9115	−0.957880	−0.478940	+	0.877848i	$0.658979\pi$
$912$	0	0
$913$	33.5911	1.11170
$914$	0	0
$915$	114.731	3.79287
$916$	0	0
$917$	51.7174	1.70786
$918$	0	0
$919$	9.20481	0.303639	0.151819	−	0.988408i	$-0.451487\pi$
$919$	9.20481	0.303639	0.151819	+	0.988408i	$0.451487\pi$
$920$	0	0
$921$	−6.27225	−0.206678
$922$	0	0
$923$	10.8365	0.356688
$924$	0	0
$925$	28.7411	0.945001
$926$	0	0
$927$	59.3604	1.94965
$928$	0	0
$929$	14.9304	0.489850	0.244925	−	0.969542i	$-0.421237\pi$
$929$	14.9304	0.489850	0.244925	+	0.969542i	$0.421237\pi$
$930$	0	0
$931$	5.26774	0.172643
$932$	0	0
$933$	−59.0992	−1.93482
$934$	0	0
$935$	120.258	3.93287
$936$	0	0
$937$	−25.4897	−0.832712	−0.416356	−	0.909202i	$-0.636693\pi$
$937$	−25.4897	−0.832712	−0.416356	+	0.909202i	$0.636693\pi$
$938$	0	0
$939$	79.9268	2.60831
$940$	0	0
$941$	11.8944	0.387747	0.193874	−	0.981027i	$-0.437895\pi$
$941$	11.8944	0.387747	0.193874	+	0.981027i	$0.437895\pi$
$942$	0	0
$943$	1.68224	0.0547813
$944$	0	0
$945$	33.7389	1.09753
$946$	0	0
$947$	−51.7008	−1.68005	−0.840026	−	0.542547i	$-0.817460\pi$
$947$	−51.7008	−1.68005	−0.840026	+	0.542547i	$0.817460\pi$
$948$	0	0
$949$	33.6428	1.09209
$950$	0	0
$951$	47.6714	1.54585
$952$	0	0
$953$	32.9830	1.06842	0.534212	−	0.845351i	$-0.320609\pi$
$953$	32.9830	1.06842	0.534212	+	0.845351i	$0.320609\pi$
$954$	0	0
$955$	−27.9245	−0.903615
$956$	0	0
$957$	16.1322	0.521481
$958$	0	0
$959$	56.8713	1.83647
$960$	0	0
$961$	−19.3096	−0.622889
$962$	0	0
$963$	44.6031	1.43732
$964$	0	0
$965$	35.2323	1.13417
$966$	0	0
$967$	14.7191	0.473335	0.236667	−	0.971591i	$-0.423945\pi$
$967$	14.7191	0.473335	0.236667	+	0.971591i	$0.423945\pi$
$968$	0	0
$969$	−95.5008	−3.06793
$970$	0	0
$971$	−34.5033	−1.10726	−0.553632	−	0.832762i	$-0.686758\pi$
$971$	−34.5033	−1.10726	−0.553632	+	0.832762i	$0.686758\pi$
$972$	0	0
$973$	−22.2472	−0.713212
$974$	0	0
$975$	123.003	3.93925
$976$	0	0
$977$	43.6620	1.39687	0.698436	−	0.715672i	$-0.253880\pi$
$977$	43.6620	1.39687	0.698436	+	0.715672i	$0.253880\pi$
$978$	0	0
$979$	43.9898	1.40592
$980$	0	0
$981$	41.2108	1.31576
$982$	0	0
$983$	−42.8819	−1.36772	−0.683860	−	0.729613i	$-0.739700\pi$
$983$	−42.8819	−1.36772	−0.683860	+	0.729613i	$0.739700\pi$
$984$	0	0
$985$	−47.7656	−1.52194
$986$	0	0
$987$	−51.1122	−1.62692
$988$	0	0
$989$	−3.63389	−0.115551
$990$	0	0
$991$	−29.2724	−0.929868	−0.464934	−	0.885345i	$-0.653922\pi$
$991$	−29.2724	−0.929868	−0.464934	+	0.885345i	$0.653922\pi$
$992$	0	0
$993$	−28.8740	−0.916288
$994$	0	0
$995$	−69.6795	−2.20899
$996$	0	0
$997$	26.4579	0.837930	0.418965	−	0.908002i	$-0.362393\pi$
$997$	26.4579	0.837930	0.418965	+	0.908002i	$0.362393\pi$
$998$	0	0
$999$	14.7008	0.465114

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7232.2.a.bn.1.16		18
4.3	odd	2	inner	7232.2.a.bn.1.3		18
8.3	odd	2		3616.2.a.j.1.16	yes	18
8.5	even	2		3616.2.a.j.1.3	✓	18

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3616.2.a.j.1.3	✓	18	8.5	even	2
3616.2.a.j.1.16	yes	18	8.3	odd	2
7232.2.a.bn.1.3		18	4.3	odd	2	inner
7232.2.a.bn.1.16		18	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 7232 = 2^{6} \cdot 113 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7232.a (trivial)

\(f(q)\)	\(=\)	\(q+2.69830 q^{3} -3.42881 q^{5} -2.84717 q^{7} +4.28081 q^{9} +4.71823 q^{11} +6.74669 q^{13} -9.25194 q^{15} -7.43351 q^{17} +4.76127 q^{19} -7.68251 q^{21} +0.359963 q^{23} +6.75672 q^{25} +3.45601 q^{27} +1.26714 q^{29} -3.41913 q^{31} +12.7312 q^{33} +9.76239 q^{35} +4.25370 q^{37} +18.2046 q^{39} +4.67337 q^{41} -10.0952 q^{43} -14.6781 q^{45} +6.65306 q^{47} +1.10637 q^{49} -20.0578 q^{51} -9.13377 q^{53} -16.1779 q^{55} +12.8473 q^{57} -0.225839 q^{59} -12.4007 q^{61} -12.1882 q^{63} -23.1331 q^{65} +10.6330 q^{67} +0.971288 q^{69} +1.60620 q^{71} +4.98656 q^{73} +18.2316 q^{75} -13.4336 q^{77} +11.9260 q^{79} -3.51709 q^{81} +7.11944 q^{83} +25.4881 q^{85} +3.41913 q^{87} +9.32338 q^{89} -19.2090 q^{91} -9.22583 q^{93} -16.3255 q^{95} +1.52558 q^{97} +20.1978 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q - 10 q^{5} + 32 q^{9} + 12 q^{13} + 16 q^{17} - 14 q^{21} + 44 q^{25} - 22 q^{29} + 24 q^{33} + 4 q^{37} + 50 q^{41} - 32 q^{45} + 58 q^{49} - 2 q^{53} + 46 q^{57} - 10 q^{61} + 40 q^{65} - 22 q^{69}+ \cdots + 48 q^{97}+O(q^{100}) \) 18 * q - 10 * q^5 + 32 * q^9 + 12 * q^13 + 16 * q^17 - 14 * q^21 + 44 * q^25 - 22 * q^29 + 24 * q^33 + 4 * q^37 + 50 * q^41 - 32 * q^45 + 58 * q^49 - 2 * q^53 + 46 * q^57 - 10 * q^61 + 40 * q^65 - 22 * q^69 + 62 * q^73 - 14 * q^77 + 46 * q^81 - 12 * q^85 + 52 * q^89 - 16 * q^93 + 48 * q^97

Introduction

L-functions

Modular forms

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Database

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