LMFDB - Embedded newform 1323.4.a.bk.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1323,4,Mod(1,1323)] 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("1323.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1323, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 1323 = 3^{3} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1323.a (trivial)

Newform invariants

comment:select newform

sage:traces = [7,1,0,31,1,0,0,84,0,-12,98,0,124] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

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

Self dual:	yes
Analytic conductor:	$78.0595269376$
Analytic rank:	$0$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - x^{6} - 43x^{5} + 10x^{4} + 513x^{3} + 258x^{2} - 936x - 504 $ x^7 - x^6 - 43x^5 + 10x^4 + 513x^3 + 258x^2 - 936*x - 504
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 3^{3} $
Twist minimal:	no (minimal twist has level 189)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.63185$ of defining polynomial
Character	$\chi$	$=$	1323.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.63185 q^{2} -5.33705 q^{4} -12.3790 q^{5} +21.7641 q^{8} +20.2007 q^{10} -29.0051 q^{11} +52.9884 q^{13} +7.18051 q^{16} +122.047 q^{17} -141.105 q^{19} +66.0672 q^{20} +47.3321 q^{22} +60.2992 q^{23} +28.2389 q^{25} -86.4693 q^{26} -126.997 q^{29} -150.849 q^{31} -185.831 q^{32} -199.163 q^{34} -341.581 q^{37} +230.262 q^{38} -269.418 q^{40} -292.082 q^{41} +290.696 q^{43} +154.802 q^{44} -98.3995 q^{46} -284.082 q^{47} -46.0818 q^{50} -282.802 q^{52} +387.993 q^{53} +359.053 q^{55} +207.241 q^{58} -269.312 q^{59} +239.606 q^{61} +246.163 q^{62} +245.804 q^{64} -655.941 q^{65} -712.250 q^{67} -651.370 q^{68} +270.507 q^{71} -146.083 q^{73} +557.411 q^{74} +753.083 q^{76} -652.792 q^{79} -88.8874 q^{80} +476.636 q^{82} +35.0239 q^{83} -1510.81 q^{85} -474.373 q^{86} -631.271 q^{88} +1394.56 q^{89} -321.820 q^{92} +463.581 q^{94} +1746.73 q^{95} -805.822 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q + q^{2} + 31 q^{4} + q^{5} + 84 q^{8} - 12 q^{10} + 98 q^{11} + 124 q^{13} + 139 q^{16} - 30 q^{17} - 182 q^{19} + 110 q^{20} + 276 q^{22} - 6 q^{23} + 388 q^{25} + 245 q^{26} + 323 q^{29} - 26 q^{31}+ \cdots + 3730 q^{97}+O(q^{100}) $ 7 * q + q^2 + 31 * q^4 + q^5 + 84 * q^8 - 12 * q^10 + 98 * q^11 + 124 * q^13 + 139 * q^16 - 30 * q^17 - 182 * q^19 + 110 * q^20 + 276 * q^22 - 6 * q^23 + 388 * q^25 + 245 * q^26 + 323 * q^29 - 26 * q^31 + 398 * q^32 - 114 * q^34 - 112 * q^37 - 1015 * q^38 + 147 * q^40 + 524 * q^41 + 8 * q^43 + 937 * q^44 - 339 * q^46 - 288 * q^47 + 2576 * q^50 + 1075 * q^52 + 1353 * q^53 - 156 * q^55 - 81 * q^58 - 165 * q^59 - 56 * q^61 + 1215 * q^62 - 1706 * q^64 + 1694 * q^65 - 988 * q^67 - 2625 * q^68 + 792 * q^71 - 1487 * q^73 + 2736 * q^74 - 1952 * q^76 - 1273 * q^79 + 2501 * q^80 + 2049 * q^82 + 1170 * q^83 + 216 * q^85 - 160 * q^86 + 9 * q^88 - 1058 * q^89 + 3834 * q^92 - 1653 * q^94 + 3260 * q^95 + 3730 * 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 $

$2$	−1.63185	−0.576948	−0.288474	−	0.957488i	$-0.593148\pi$
$2$	−1.63185	−0.576948	−0.288474	+	0.957488i	$0.593148\pi$
$3$	0	0
$3$	0	0
$4$	−5.33705	−0.667131
$5$	−12.3790	−1.10721	−0.553604	−	0.832780i	$-0.686748\pi$
$5$	−12.3790	−1.10721	−0.553604	+	0.832780i	$0.686748\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	21.7641	0.961848
$9$	0	0
$10$	20.2007	0.638802
$11$	−29.0051	−0.795033	−0.397517	−	0.917595i	$-0.630128\pi$
$11$	−29.0051	−0.795033	−0.397517	+	0.917595i	$0.630128\pi$
$12$	0	0
$13$	52.9884	1.13049	0.565243	−	0.824924i	$-0.308782\pi$
$13$	52.9884	1.13049	0.565243	+	0.824924i	$0.308782\pi$
$14$	0	0
$15$	0	0
$16$	7.18051	0.112196
$17$	122.047	1.74122	0.870609	−	0.491975i	$-0.163725\pi$
$17$	122.047	1.74122	0.870609	+	0.491975i	$0.163725\pi$
$18$	0	0
$19$	−141.105	−1.70377	−0.851885	−	0.523728i	$-0.824541\pi$
$19$	−141.105	−1.70377	−0.851885	+	0.523728i	$0.824541\pi$
$20$	66.0672	0.738654
$21$	0	0
$22$	47.3321	0.458693
$23$	60.2992	0.546663	0.273332	−	0.961920i	$-0.411874\pi$
$23$	60.2992	0.546663	0.273332	+	0.961920i	$0.411874\pi$
$24$	0	0
$25$	28.2389	0.225912
$26$	−86.4693	−0.652232
$27$	0	0
$28$	0	0
$29$	−126.997	−0.813201	−0.406601	−	0.913606i	$-0.633286\pi$
$29$	−126.997	−0.813201	−0.406601	+	0.913606i	$0.633286\pi$
$30$	0	0
$31$	−150.849	−0.873975	−0.436988	−	0.899468i	$-0.643955\pi$
$31$	−150.849	−0.873975	−0.436988	+	0.899468i	$0.643955\pi$
$32$	−185.831	−1.02658
$33$	0	0
$34$	−199.163	−1.00459
$35$	0	0
$36$	0	0
$37$	−341.581	−1.51772	−0.758860	−	0.651254i	$-0.774243\pi$
$37$	−341.581	−1.51772	−0.758860	+	0.651254i	$0.774243\pi$
$38$	230.262	0.982987
$39$	0	0
$40$	−269.418	−1.06497
$41$	−292.082	−1.11258	−0.556288	−	0.830990i	$-0.687775\pi$
$41$	−292.082	−1.11258	−0.556288	+	0.830990i	$0.687775\pi$
$42$	0	0
$43$	290.696	1.03095	0.515473	−	0.856906i	$-0.327616\pi$
$43$	290.696	1.03095	0.515473	+	0.856906i	$0.327616\pi$
$44$	154.802	0.530392
$45$	0	0
$46$	−98.3995	−0.315396
$47$	−284.082	−0.881652	−0.440826	−	0.897593i	$-0.645314\pi$
$47$	−284.082	−0.881652	−0.440826	+	0.897593i	$0.645314\pi$
$48$	0	0
$49$	0	0
$50$	−46.0818	−0.130339
$51$	0	0
$52$	−282.802	−0.754183
$53$	387.993	1.00556	0.502782	−	0.864413i	$-0.332310\pi$
$53$	387.993	1.00556	0.502782	+	0.864413i	$0.332310\pi$
$54$	0	0
$55$	359.053	0.880268
$56$	0	0
$57$	0	0
$58$	207.241	0.469175
$59$	−269.312	−0.594261	−0.297131	−	0.954837i	$-0.596030\pi$
$59$	−269.312	−0.594261	−0.297131	+	0.954837i	$0.596030\pi$
$60$	0	0
$61$	239.606	0.502925	0.251462	−	0.967867i	$-0.419089\pi$
$61$	239.606	0.502925	0.251462	+	0.967867i	$0.419089\pi$
$62$	246.163	0.504238
$63$	0	0
$64$	245.804	0.480087
$65$	−655.941	−1.25168
$66$	0	0
$67$	−712.250	−1.29873	−0.649367	−	0.760475i	$-0.724966\pi$
$67$	−712.250	−1.29873	−0.649367	+	0.760475i	$0.724966\pi$
$68$	−651.370	−1.16162
$69$	0	0
$70$	0	0
$71$	270.507	0.452160	0.226080	−	0.974109i	$-0.427409\pi$
$71$	270.507	0.452160	0.226080	+	0.974109i	$0.427409\pi$
$72$	0	0
$73$	−146.083	−0.234215	−0.117108	−	0.993119i	$-0.537362\pi$
$73$	−146.083	−0.234215	−0.117108	+	0.993119i	$0.537362\pi$
$74$	557.411	0.875645
$75$	0	0
$76$	753.083	1.13664
$77$	0	0
$78$	0	0
$79$	−652.792	−0.929681	−0.464841	−	0.885394i	$-0.653888\pi$
$79$	−652.792	−0.929681	−0.464841	+	0.885394i	$0.653888\pi$
$80$	−88.8874	−0.124224
$81$	0	0
$82$	476.636	0.641898
$83$	35.0239	0.0463178	0.0231589	−	0.999732i	$-0.492628\pi$
$83$	35.0239	0.0463178	0.0231589	+	0.999732i	$0.492628\pi$
$84$	0	0
$85$	−1510.81	−1.92789
$86$	−474.373	−0.594802
$87$	0	0
$88$	−631.271	−0.764701
$89$	1394.56	1.66093	0.830465	−	0.557071i	$-0.188075\pi$
$89$	1394.56	1.66093	0.830465	+	0.557071i	$0.188075\pi$
$90$	0	0
$91$	0	0
$92$	−321.820	−0.364696
$93$	0	0
$94$	463.581	0.508667
$95$	1746.73	1.88643
$96$	0	0
$97$	−805.822	−0.843493	−0.421746	−	0.906714i	$-0.638583\pi$
$97$	−805.822	−0.843493	−0.421746	+	0.906714i	$0.638583\pi$
$98$	0	0
$99$	0	0
$100$	−150.713	−0.150713
$101$	1014.81	0.999778	0.499889	−	0.866089i	$-0.333374\pi$
$101$	1014.81	0.999778	0.499889	+	0.866089i	$0.333374\pi$
$102$	0	0
$103$	−1039.39	−0.994310	−0.497155	−	0.867662i	$-0.665622\pi$
$103$	−1039.39	−0.994310	−0.497155	+	0.867662i	$0.665622\pi$
$104$	1153.25	1.08736
$105$	0	0
$106$	−633.147	−0.580158
$107$	608.338	0.549628	0.274814	−	0.961497i	$-0.411384\pi$
$107$	608.338	0.549628	0.274814	+	0.961497i	$0.411384\pi$
$108$	0	0
$109$	1077.71	0.947029	0.473514	−	0.880786i	$-0.342985\pi$
$109$	1077.71	0.947029	0.473514	+	0.880786i	$0.342985\pi$
$110$	−585.923	−0.507869
$111$	0	0
$112$	0	0
$113$	2064.78	1.71893	0.859463	−	0.511199i	$-0.170798\pi$
$113$	2064.78	1.71893	0.859463	+	0.511199i	$0.170798\pi$
$114$	0	0
$115$	−746.442	−0.605270
$116$	677.792	0.542512
$117$	0	0
$118$	439.478	0.342858
$119$	0	0
$120$	0	0
$121$	−489.704	−0.367922
$122$	−391.002	−0.290161
$123$	0	0
$124$	805.087	0.583056
$125$	1197.80	0.857078
$126$	0	0
$127$	−130.430	−0.0911320	−0.0455660	−	0.998961i	$-0.514509\pi$
$127$	−130.430	−0.0911320	−0.0455660	+	0.998961i	$0.514509\pi$
$128$	1085.53	0.749594
$129$	0	0
$130$	1070.40	0.722157
$131$	−1905.07	−1.27058	−0.635292	−	0.772272i	$-0.719120\pi$
$131$	−1905.07	−1.27058	−0.635292	+	0.772272i	$0.719120\pi$
$132$	0	0
$133$	0	0
$134$	1162.29	0.749302
$135$	0	0
$136$	2656.24	1.67479
$137$	−559.298	−0.348789	−0.174394	−	0.984676i	$-0.555797\pi$
$137$	−559.298	−0.348789	−0.174394	+	0.984676i	$0.555797\pi$
$138$	0	0
$139$	699.079	0.426583	0.213292	−	0.976989i	$-0.431582\pi$
$139$	699.079	0.426583	0.213292	+	0.976989i	$0.431582\pi$
$140$	0	0
$141$	0	0
$142$	−441.429	−0.260872
$143$	−1536.93	−0.898775
$144$	0	0
$145$	1572.10	0.900384
$146$	238.386	0.135130
$147$	0	0
$148$	1823.04	1.01252
$149$	1614.90	0.887904	0.443952	−	0.896050i	$-0.353576\pi$
$149$	1614.90	0.887904	0.443952	+	0.896050i	$0.353576\pi$
$150$	0	0
$151$	−3.59271	−0.00193623	−0.000968116	−	1.00000i	$-0.500308\pi$
$151$	−3.59271	−0.00193623	−0.000968116		1.00000i	$0.500308\pi$
$152$	−3071.02	−1.63877
$153$	0	0
$154$	0	0
$155$	1867.35	0.967673
$156$	0	0
$157$	−697.746	−0.354689	−0.177345	−	0.984149i	$-0.556751\pi$
$157$	−697.746	−0.354689	−0.177345	+	0.984149i	$0.556751\pi$
$158$	1065.26	0.536377
$159$	0	0
$160$	2300.39	1.13664
$161$	0	0
$162$	0	0
$163$	145.908	0.0701130	0.0350565	−	0.999385i	$-0.488839\pi$
$163$	145.908	0.0701130	0.0350565	+	0.999385i	$0.488839\pi$
$164$	1558.86	0.742234
$165$	0	0
$166$	−57.1539	−0.0267229
$167$	−1816.18	−0.841556	−0.420778	−	0.907164i	$-0.638243\pi$
$167$	−1816.18	−0.841556	−0.420778	+	0.907164i	$0.638243\pi$
$168$	0	0
$169$	610.766	0.278000
$170$	2465.43	1.11229
$171$	0	0
$172$	−1551.46	−0.687776
$173$	−2424.68	−1.06558	−0.532790	−	0.846247i	$-0.678856\pi$
$173$	−2424.68	−1.06558	−0.532790	+	0.846247i	$0.678856\pi$
$174$	0	0
$175$	0	0
$176$	−208.272	−0.0891992
$177$	0	0
$178$	−2275.71	−0.958270
$179$	2835.36	1.18394	0.591969	−	0.805961i	$-0.298351\pi$
$179$	2835.36	1.18394	0.591969	+	0.805961i	$0.298351\pi$
$180$	0	0
$181$	219.212	0.0900214	0.0450107	−	0.998987i	$-0.485668\pi$
$181$	219.212	0.0900214	0.0450107	+	0.998987i	$0.485668\pi$
$182$	0	0
$183$	0	0
$184$	1312.36	0.525807
$185$	4228.43	1.68043
$186$	0	0
$187$	−3539.98	−1.38433
$188$	1516.16	0.588178
$189$	0	0
$190$	−2850.41	−1.08837
$191$	223.424	0.0846410	0.0423205	−	0.999104i	$-0.486525\pi$
$191$	223.424	0.0846410	0.0423205	+	0.999104i	$0.486525\pi$
$192$	0	0
$193$	−352.158	−0.131342	−0.0656708	−	0.997841i	$-0.520919\pi$
$193$	−352.158	−0.131342	−0.0656708	+	0.997841i	$0.520919\pi$
$194$	1314.98	0.486651
$195$	0	0
$196$	0	0
$197$	5347.45	1.93396	0.966981	−	0.254850i	$-0.0820261\pi$
$197$	5347.45	1.93396	0.966981	+	0.254850i	$0.0820261\pi$
$198$	0	0
$199$	−582.228	−0.207402	−0.103701	−	0.994609i	$-0.533069\pi$
$199$	−582.228	−0.207402	−0.103701	+	0.994609i	$0.533069\pi$
$200$	614.596	0.217292
$201$	0	0
$202$	−1656.03	−0.576820
$203$	0	0
$204$	0	0
$205$	3615.68	1.23185
$206$	1696.13	0.573665
$207$	0	0
$208$	380.484	0.126836
$209$	4092.76	1.35455
$210$	0	0
$211$	5467.96	1.78403	0.892014	−	0.452008i	$-0.149292\pi$
$211$	5467.96	1.78403	0.892014	+	0.452008i	$0.149292\pi$
$212$	−2070.74	−0.670843
$213$	0	0
$214$	−992.719	−0.317107
$215$	−3598.51	−1.14147
$216$	0	0
$217$	0	0
$218$	−1758.67	−0.546386
$219$	0	0
$220$	−1916.29	−0.587254
$221$	6467.06	1.96842
$222$	0	0
$223$	−4368.05	−1.31169	−0.655844	−	0.754897i	$-0.727687\pi$
$223$	−4368.05	−1.31169	−0.655844	+	0.754897i	$0.727687\pi$
$224$	0	0
$225$	0	0
$226$	−3369.43	−0.991730
$227$	−636.952	−0.186238	−0.0931189	−	0.995655i	$-0.529684\pi$
$227$	−636.952	−0.186238	−0.0931189	+	0.995655i	$0.529684\pi$
$228$	0	0
$229$	−983.495	−0.283804	−0.141902	−	0.989881i	$-0.545322\pi$
$229$	−983.495	−0.283804	−0.141902	+	0.989881i	$0.545322\pi$
$230$	1218.09	0.349209
$231$	0	0
$232$	−2763.99	−0.782176
$233$	−1677.37	−0.471623	−0.235811	−	0.971799i	$-0.575775\pi$
$233$	−1677.37	−0.471623	−0.235811	+	0.971799i	$0.575775\pi$
$234$	0	0
$235$	3516.64	0.976173
$236$	1437.33	0.396450
$237$	0	0
$238$	0	0
$239$	−2705.18	−0.732148	−0.366074	−	0.930586i	$-0.619298\pi$
$239$	−2705.18	−0.732148	−0.366074	+	0.930586i	$0.619298\pi$
$240$	0	0
$241$	2140.73	0.572185	0.286092	−	0.958202i	$-0.407644\pi$
$241$	2140.73	0.572185	0.286092	+	0.958202i	$0.407644\pi$
$242$	799.126	0.212272
$243$	0	0
$244$	−1278.79	−0.335517
$245$	0	0
$246$	0	0
$247$	−7476.91	−1.92609
$248$	−3283.09	−0.840631
$249$	0	0
$250$	−1954.64	−0.494489
$251$	−918.484	−0.230973	−0.115486	−	0.993309i	$-0.536843\pi$
$251$	−918.484	−0.230973	−0.115486	+	0.993309i	$0.536843\pi$
$252$	0	0
$253$	−1748.98	−0.434616
$254$	212.842	0.0525784
$255$	0	0
$256$	−3737.86	−0.912563
$257$	−4232.18	−1.02722	−0.513611	−	0.858023i	$-0.671692\pi$
$257$	−4232.18	−1.02722	−0.513611	+	0.858023i	$0.671692\pi$
$258$	0	0
$259$	0	0
$260$	3500.79	0.835038
$261$	0	0
$262$	3108.79	0.733060
$263$	5825.05	1.36573	0.682867	−	0.730542i	$-0.260733\pi$
$263$	5825.05	1.36573	0.682867	+	0.730542i	$0.260733\pi$
$264$	0	0
$265$	−4802.95	−1.11337
$266$	0	0
$267$	0	0
$268$	3801.31	0.866426
$269$	2592.23	0.587550	0.293775	−	0.955875i	$-0.405088\pi$
$269$	2592.23	0.587550	0.293775	+	0.955875i	$0.405088\pi$
$270$	0	0
$271$	3709.89	0.831586	0.415793	−	0.909459i	$-0.363504\pi$
$271$	3709.89	0.831586	0.415793	+	0.909459i	$0.363504\pi$
$272$	876.359	0.195357
$273$	0	0
$274$	912.694	0.201233
$275$	−819.073	−0.179607
$276$	0	0
$277$	−323.380	−0.0701445	−0.0350722	−	0.999385i	$-0.511166\pi$
$277$	−323.380	−0.0701445	−0.0350722	+	0.999385i	$0.511166\pi$
$278$	−1140.79	−0.246116
$279$	0	0
$280$	0	0
$281$	4990.53	1.05947	0.529733	−	0.848165i	$-0.322292\pi$
$281$	4990.53	1.05947	0.529733	+	0.848165i	$0.322292\pi$
$282$	0	0
$283$	−2458.67	−0.516441	−0.258221	−	0.966086i	$-0.583136\pi$
$283$	−2458.67	−0.516441	−0.258221	+	0.966086i	$0.583136\pi$
$284$	−1443.71	−0.301650
$285$	0	0
$286$	2508.05	0.518546
$287$	0	0
$288$	0	0
$289$	9982.43	2.03184
$290$	−2565.44	−0.519474
$291$	0	0
$292$	779.651	0.156252
$293$	−3028.06	−0.603758	−0.301879	−	0.953346i	$-0.597614\pi$
$293$	−3028.06	−0.603758	−0.301879	+	0.953346i	$0.597614\pi$
$294$	0	0
$295$	3333.80	0.657972
$296$	−7434.22	−1.45982
$297$	0	0
$298$	−2635.28	−0.512274
$299$	3195.16	0.617995
$300$	0	0
$301$	0	0
$302$	5.86279	0.00111710
$303$	0	0
$304$	−1013.20	−0.191155
$305$	−2966.08	−0.556843
$306$	0	0
$307$	8277.86	1.53890	0.769450	−	0.638707i	$-0.220530\pi$
$307$	8277.86	1.53890	0.769450	+	0.638707i	$0.220530\pi$
$308$	0	0
$309$	0	0
$310$	−3047.25	−0.558297
$311$	−824.071	−0.150253	−0.0751267	−	0.997174i	$-0.523936\pi$
$311$	−824.071	−0.150253	−0.0751267	+	0.997174i	$0.523936\pi$
$312$	0	0
$313$	2427.80	0.438426	0.219213	−	0.975677i	$-0.429651\pi$
$313$	2427.80	0.438426	0.219213	+	0.975677i	$0.429651\pi$
$314$	1138.62	0.204637
$315$	0	0
$316$	3483.98	0.620219
$317$	7970.35	1.41218	0.706088	−	0.708125i	$-0.250458\pi$
$317$	7970.35	1.41218	0.706088	+	0.708125i	$0.250458\pi$
$318$	0	0
$319$	3683.58	0.646522
$320$	−3042.81	−0.531556
$321$	0	0
$322$	0	0
$323$	−17221.4	−2.96664
$324$	0	0
$325$	1496.34	0.255390
$326$	−238.101	−0.0404515
$327$	0	0
$328$	−6356.92	−1.07013
$329$	0	0
$330$	0	0
$331$	2273.66	0.377558	0.188779	−	0.982020i	$-0.439547\pi$
$331$	2273.66	0.377558	0.188779	+	0.982020i	$0.439547\pi$
$332$	−186.924	−0.0309000
$333$	0	0
$334$	2963.73	0.485534
$335$	8816.92	1.43797
$336$	0	0
$337$	3142.92	0.508030	0.254015	−	0.967200i	$-0.418249\pi$
$337$	3142.92	0.508030	0.254015	+	0.967200i	$0.418249\pi$
$338$	−996.681	−0.160391
$339$	0	0
$340$	8063.29	1.28616
$341$	4375.38	0.694839
$342$	0	0
$343$	0	0
$344$	6326.74	0.991613
$345$	0	0
$346$	3956.73	0.614784
$347$	8966.74	1.38720	0.693602	−	0.720358i	$-0.256023\pi$
$347$	8966.74	1.38720	0.693602	+	0.720358i	$0.256023\pi$
$348$	0	0
$349$	140.981	0.0216234	0.0108117	−	0.999942i	$-0.496558\pi$
$349$	140.981	0.0216234	0.0108117	+	0.999942i	$0.496558\pi$
$350$	0	0
$351$	0	0
$352$	5390.03	0.816164
$353$	8208.98	1.23773	0.618867	−	0.785496i	$-0.287592\pi$
$353$	8208.98	1.23773	0.618867	+	0.785496i	$0.287592\pi$
$354$	0	0
$355$	−3348.60	−0.500635
$356$	−7442.82	−1.10806
$357$	0	0
$358$	−4626.90	−0.683070
$359$	8749.05	1.28623	0.643116	−	0.765769i	$-0.277641\pi$
$359$	8749.05	1.28623	0.643116	+	0.765769i	$0.277641\pi$
$360$	0	0
$361$	13051.5	1.90284
$362$	−357.722	−0.0519376
$363$	0	0
$364$	0	0
$365$	1808.36	0.259325
$366$	0	0
$367$	5917.89	0.841721	0.420860	−	0.907125i	$-0.361728\pi$
$367$	5917.89	0.841721	0.420860	+	0.907125i	$0.361728\pi$
$368$	432.979	0.0613332
$369$	0	0
$370$	−6900.18	−0.969522
$371$	0	0
$372$	0	0
$373$	11660.7	1.61868	0.809342	−	0.587338i	$-0.199824\pi$
$373$	11660.7	1.61868	0.809342	+	0.587338i	$0.199824\pi$
$374$	5776.73	0.798684
$375$	0	0
$376$	−6182.80	−0.848015
$377$	−6729.39	−0.919313
$378$	0	0
$379$	−14306.7	−1.93902	−0.969508	−	0.245060i	$-0.921192\pi$
$379$	−14306.7	−1.93902	−0.969508	+	0.245060i	$0.921192\pi$
$380$	−9322.39	−1.25850
$381$	0	0
$382$	−364.596	−0.0488334
$383$	−10840.5	−1.44627	−0.723136	−	0.690705i	$-0.757300\pi$
$383$	−10840.5	−1.44627	−0.723136	+	0.690705i	$0.757300\pi$
$384$	0	0
$385$	0	0
$386$	574.671	0.0757772
$387$	0	0
$388$	4300.71	0.562721
$389$	−11175.5	−1.45660	−0.728301	−	0.685257i	$-0.759690\pi$
$389$	−11175.5	−1.45660	−0.728301	+	0.685257i	$0.759690\pi$
$390$	0	0
$391$	7359.33	0.951860
$392$	0	0
$393$	0	0
$394$	−8726.27	−1.11579
$395$	8080.89	1.02935
$396$	0	0
$397$	13077.9	1.65330	0.826650	−	0.562716i	$-0.190244\pi$
$397$	13077.9	1.65330	0.826650	+	0.562716i	$0.190244\pi$
$398$	950.111	0.119660
$399$	0	0
$400$	202.770	0.0253463
$401$	2733.63	0.340427	0.170213	−	0.985407i	$-0.445554\pi$
$401$	2733.63	0.340427	0.170213	+	0.985407i	$0.445554\pi$
$402$	0	0
$403$	−7993.22	−0.988017
$404$	−5416.10	−0.666983
$405$	0	0
$406$	0	0
$407$	9907.60	1.20664
$408$	0	0
$409$	−11521.1	−1.39286	−0.696431	−	0.717623i	$-0.745230\pi$
$409$	−11521.1	−1.39286	−0.696431	+	0.717623i	$0.745230\pi$
$410$	−5900.26	−0.710715
$411$	0	0
$412$	5547.27	0.663336
$413$	0	0
$414$	0	0
$415$	−433.560	−0.0512834
$416$	−9846.86	−1.16053
$417$	0	0
$418$	−6678.78	−0.781507
$419$	2426.62	0.282931	0.141465	−	0.989943i	$-0.454819\pi$
$419$	2426.62	0.282931	0.141465	+	0.989943i	$0.454819\pi$
$420$	0	0
$421$	10188.2	1.17943	0.589716	−	0.807611i	$-0.299240\pi$
$421$	10188.2	1.17943	0.589716	+	0.807611i	$0.299240\pi$
$422$	−8922.91	−1.02929
$423$	0	0
$424$	8444.32	0.967199
$425$	3446.47	0.393361
$426$	0	0
$427$	0	0
$428$	−3246.73	−0.366674
$429$	0	0
$430$	5872.25	0.658570
$431$	−6780.40	−0.757773	−0.378887	−	0.925443i	$-0.623693\pi$
$431$	−6780.40	−0.757773	−0.378887	+	0.925443i	$0.623693\pi$
$432$	0	0
$433$	−6122.85	−0.679550	−0.339775	−	0.940507i	$-0.610351\pi$
$433$	−6122.85	−0.679550	−0.339775	+	0.940507i	$0.610351\pi$
$434$	0	0
$435$	0	0
$436$	−5751.81	−0.631792
$437$	−8508.50	−0.931389
$438$	0	0
$439$	15747.5	1.71204	0.856022	−	0.516939i	$-0.172929\pi$
$439$	15747.5	1.71204	0.856022	+	0.516939i	$0.172929\pi$
$440$	7814.48	0.846684
$441$	0	0
$442$	−10553.3	−1.13568
$443$	15063.8	1.61558	0.807789	−	0.589472i	$-0.200664\pi$
$443$	15063.8	1.61558	0.807789	+	0.589472i	$0.200664\pi$
$444$	0	0
$445$	−17263.2	−1.83900
$446$	7128.02	0.756775
$447$	0	0
$448$	0	0
$449$	−175.064	−0.0184004	−0.00920020	−	0.999958i	$-0.502929\pi$
$449$	−175.064	−0.0184004	−0.00920020	+	0.999958i	$0.502929\pi$
$450$	0	0
$451$	8471.88	0.884535
$452$	−11019.9	−1.14675
$453$	0	0
$454$	1039.41	0.107450
$455$	0	0
$456$	0	0
$457$	−12593.5	−1.28905	−0.644527	−	0.764582i	$-0.722946\pi$
$457$	−12593.5	−1.28905	−0.644527	+	0.764582i	$0.722946\pi$
$458$	1604.92	0.163740
$459$	0	0
$460$	3983.80	0.403795
$461$	−13781.7	−1.39236	−0.696182	−	0.717865i	$-0.745119\pi$
$461$	−13781.7	−1.39236	−0.696182	+	0.717865i	$0.745119\pi$
$462$	0	0
$463$	8867.45	0.890076	0.445038	−	0.895512i	$-0.353190\pi$
$463$	8867.45	0.890076	0.445038	+	0.895512i	$0.353190\pi$
$464$	−911.907	−0.0912376
$465$	0	0
$466$	2737.22	0.272102
$467$	12839.0	1.27221	0.636103	−	0.771604i	$-0.280545\pi$
$467$	12839.0	1.27221	0.636103	+	0.771604i	$0.280545\pi$
$468$	0	0
$469$	0	0
$470$	−5738.65	−0.563201
$471$	0	0
$472$	−5861.34	−0.571589
$473$	−8431.66	−0.819636
$474$	0	0
$475$	−3984.65	−0.384902
$476$	0	0
$477$	0	0
$478$	4414.46	0.422411
$479$	−9955.21	−0.949614	−0.474807	−	0.880090i	$-0.657482\pi$
$479$	−9955.21	−0.949614	−0.474807	+	0.880090i	$0.657482\pi$
$480$	0	0
$481$	−18099.8	−1.71576
$482$	−3493.36	−0.330121
$483$	0	0
$484$	2613.58	0.245452
$485$	9975.25	0.933923
$486$	0	0
$487$	−16716.0	−1.55539	−0.777693	−	0.628644i	$-0.783610\pi$
$487$	−16716.0	−1.55539	−0.777693	+	0.628644i	$0.783610\pi$
$488$	5214.82	0.483737
$489$	0	0
$490$	0	0
$491$	−2981.39	−0.274029	−0.137014	−	0.990569i	$-0.543751\pi$
$491$	−2981.39	−0.274029	−0.137014	+	0.990569i	$0.543751\pi$
$492$	0	0
$493$	−15499.6	−1.41596
$494$	12201.2	1.11125
$495$	0	0
$496$	−1083.17	−0.0980561
$497$	0	0
$498$	0	0
$499$	8169.47	0.732897	0.366449	−	0.930438i	$-0.380574\pi$
$499$	8169.47	0.732897	0.366449	+	0.930438i	$0.380574\pi$
$500$	−6392.73	−0.571783
$501$	0	0
$502$	1498.83	0.133259
$503$	1858.71	0.164763	0.0823814	−	0.996601i	$-0.473747\pi$
$503$	1858.71	0.164763	0.0823814	+	0.996601i	$0.473747\pi$
$504$	0	0
$505$	−12562.3	−1.10696
$506$	2854.09	0.250750
$507$	0	0
$508$	696.110	0.0607970
$509$	14749.0	1.28436	0.642179	−	0.766554i	$-0.278030\pi$
$509$	14749.0	1.28436	0.642179	+	0.766554i	$0.278030\pi$
$510$	0	0
$511$	0	0
$512$	−2584.58	−0.223093
$513$	0	0
$514$	6906.30	0.592653
$515$	12866.6	1.10091
$516$	0	0
$517$	8239.83	0.700943
$518$	0	0
$519$	0	0
$520$	−14276.0	−1.20393
$521$	−13232.1	−1.11268	−0.556342	−	0.830953i	$-0.687796\pi$
$521$	−13232.1	−1.11268	−0.556342	+	0.830953i	$0.687796\pi$
$522$	0	0
$523$	16100.9	1.34617	0.673083	−	0.739567i	$-0.264969\pi$
$523$	16100.9	1.34617	0.673083	+	0.739567i	$0.264969\pi$
$524$	10167.4	0.847646
$525$	0	0
$526$	−9505.64	−0.787958
$527$	−18410.6	−1.52178
$528$	0	0
$529$	−8531.01	−0.701159
$530$	7837.71	0.642356
$531$	0	0
$532$	0	0
$533$	−15477.0	−1.25775
$534$	0	0
$535$	−7530.60	−0.608553
$536$	−15501.5	−1.24918
$537$	0	0
$538$	−4230.14	−0.338986
$539$	0	0
$540$	0	0
$541$	3622.62	0.287890	0.143945	−	0.989586i	$-0.454021\pi$
$541$	3622.62	0.287890	0.143945	+	0.989586i	$0.454021\pi$
$542$	−6054.00	−0.479782
$543$	0	0
$544$	−22680.0	−1.78750
$545$	−13341.0	−1.04856
$546$	0	0
$547$	14543.8	1.13683	0.568417	−	0.822741i	$-0.307556\pi$
$547$	14543.8	1.13683	0.568417	+	0.822741i	$0.307556\pi$
$548$	2985.00	0.232688
$549$	0	0
$550$	1336.61	0.103624
$551$	17919.9	1.38551
$552$	0	0
$553$	0	0
$554$	527.709	0.0404697
$555$	0	0
$556$	−3731.02	−0.284587
$557$	−11533.3	−0.877346	−0.438673	−	0.898647i	$-0.644551\pi$
$557$	−11533.3	−0.877346	−0.438673	+	0.898647i	$0.644551\pi$
$558$	0	0
$559$	15403.5	1.16547
$560$	0	0
$561$	0	0
$562$	−8143.81	−0.611256
$563$	276.313	0.0206842	0.0103421	−	0.999947i	$-0.496708\pi$
$563$	276.313	0.0206842	0.0103421	+	0.999947i	$0.496708\pi$
$564$	0	0
$565$	−25559.9	−1.90321
$566$	4012.20	0.297960
$567$	0	0
$568$	5887.36	0.434909
$569$	−15729.5	−1.15890	−0.579452	−	0.815007i	$-0.696733\pi$
$569$	−15729.5	−1.15890	−0.579452	+	0.815007i	$0.696733\pi$
$570$	0	0
$571$	7718.30	0.565676	0.282838	−	0.959168i	$-0.408724\pi$
$571$	7718.30	0.565676	0.282838	+	0.959168i	$0.408724\pi$
$572$	8202.69	0.599601
$573$	0	0
$574$	0	0
$575$	1702.79	0.123498
$576$	0	0
$577$	−15638.4	−1.12831	−0.564156	−	0.825668i	$-0.690798\pi$
$577$	−15638.4	−1.12831	−0.564156	+	0.825668i	$0.690798\pi$
$578$	−16289.9	−1.17227
$579$	0	0
$580$	−8390.37	−0.600674
$581$	0	0
$582$	0	0
$583$	−11253.8	−0.799457
$584$	−3179.37	−0.225279
$585$	0	0
$586$	4941.35	0.348337
$587$	−14520.1	−1.02097	−0.510484	−	0.859888i	$-0.670534\pi$
$587$	−14520.1	−1.02097	−0.510484	+	0.859888i	$0.670534\pi$
$588$	0	0
$589$	21285.5	1.48905
$590$	−5440.28	−0.379615
$591$	0	0
$592$	−2452.73	−0.170281
$593$	5938.70	0.411253	0.205627	−	0.978631i	$-0.434077\pi$
$593$	5938.70	0.411253	0.205627	+	0.978631i	$0.434077\pi$
$594$	0	0
$595$	0	0
$596$	−8618.80	−0.592349
$597$	0	0
$598$	−5214.03	−0.356551
$599$	7264.82	0.495547	0.247774	−	0.968818i	$-0.420301\pi$
$599$	7264.82	0.495547	0.247774	+	0.968818i	$0.420301\pi$
$600$	0	0
$601$	1691.27	0.114789	0.0573945	−	0.998352i	$-0.481721\pi$
$601$	1691.27	0.114789	0.0573945	+	0.998352i	$0.481721\pi$
$602$	0	0
$603$	0	0
$604$	19.1745	0.00129172
$605$	6062.03	0.407366
$606$	0	0
$607$	−17410.4	−1.16420	−0.582099	−	0.813118i	$-0.697768\pi$
$607$	−17410.4	−1.16420	−0.582099	+	0.813118i	$0.697768\pi$
$608$	26221.6	1.74905
$609$	0	0
$610$	4840.21	0.321269
$611$	−15053.0	−0.996696
$612$	0	0
$613$	10205.4	0.672418	0.336209	−	0.941787i	$-0.390855\pi$
$613$	10205.4	0.672418	0.336209	+	0.941787i	$0.390855\pi$
$614$	−13508.3	−0.887865
$615$	0	0
$616$	0	0
$617$	8936.08	0.583068	0.291534	−	0.956560i	$-0.405834\pi$
$617$	8936.08	0.583068	0.291534	+	0.956560i	$0.405834\pi$
$618$	0	0
$619$	18560.2	1.20517	0.602584	−	0.798056i	$-0.294138\pi$
$619$	18560.2	1.20517	0.602584	+	0.798056i	$0.294138\pi$
$620$	−9966.15	−0.645565
$621$	0	0
$622$	1344.76	0.0866883
$623$	0	0
$624$	0	0
$625$	−18357.4	−1.17488
$626$	−3961.82	−0.252949
$627$	0	0
$628$	3723.91	0.236624
$629$	−41688.9	−2.64268
$630$	0	0
$631$	−6191.13	−0.390594	−0.195297	−	0.980744i	$-0.562567\pi$
$631$	−6191.13	−0.390594	−0.195297	+	0.980744i	$0.562567\pi$
$632$	−14207.4	−0.894212
$633$	0	0
$634$	−13006.5	−0.814751
$635$	1614.59	0.100902
$636$	0	0
$637$	0	0
$638$	−6011.06	−0.373010
$639$	0	0
$640$	−13437.7	−0.829957
$641$	1592.53	0.0981298	0.0490649	−	0.998796i	$-0.484376\pi$
$641$	1592.53	0.0981298	0.0490649	+	0.998796i	$0.484376\pi$
$642$	0	0
$643$	20101.4	1.23285	0.616426	−	0.787413i	$-0.288580\pi$
$643$	20101.4	1.23285	0.616426	+	0.787413i	$0.288580\pi$
$644$	0	0
$645$	0	0
$646$	28102.8	1.71159
$647$	−13953.3	−0.847855	−0.423927	−	0.905696i	$-0.639349\pi$
$647$	−13953.3	−0.847855	−0.423927	+	0.905696i	$0.639349\pi$
$648$	0	0
$649$	7811.42	0.472458
$650$	−2441.80	−0.147347
$651$	0	0
$652$	−778.720	−0.0467746
$653$	32040.0	1.92010	0.960049	−	0.279832i	$-0.0902788\pi$
$653$	32040.0	1.92010	0.960049	+	0.279832i	$0.0902788\pi$
$654$	0	0
$655$	23582.8	1.40680
$656$	−2097.30	−0.124826
$657$	0	0
$658$	0	0
$659$	−22240.2	−1.31465	−0.657325	−	0.753607i	$-0.728312\pi$
$659$	−22240.2	−1.31465	−0.657325	+	0.753607i	$0.728312\pi$
$660$	0	0
$661$	−913.832	−0.0537730	−0.0268865	−	0.999638i	$-0.508559\pi$
$661$	−913.832	−0.0537730	−0.0268865	+	0.999638i	$0.508559\pi$
$662$	−3710.28	−0.217831
$663$	0	0
$664$	762.265	0.0445506
$665$	0	0
$666$	0	0
$667$	−7657.85	−0.444547
$668$	9693.02	0.561429
$669$	0	0
$670$	−14387.9	−0.829634
$671$	−6949.80	−0.399842
$672$	0	0
$673$	−7951.84	−0.455454	−0.227727	−	0.973725i	$-0.573129\pi$
$673$	−7951.84	−0.455454	−0.227727	+	0.973725i	$0.573129\pi$
$674$	−5128.79	−0.293106
$675$	0	0
$676$	−3259.69	−0.185462
$677$	2633.40	0.149498	0.0747488	−	0.997202i	$-0.476184\pi$
$677$	2633.40	0.149498	0.0747488	+	0.997202i	$0.476184\pi$
$678$	0	0
$679$	0	0
$680$	−32881.6	−1.85434
$681$	0	0
$682$	−7139.99	−0.400886
$683$	−31241.6	−1.75026	−0.875129	−	0.483890i	$-0.839224\pi$
$683$	−31241.6	−1.75026	−0.875129	+	0.483890i	$0.839224\pi$
$684$	0	0
$685$	6923.54	0.386182
$686$	0	0
$687$	0	0
$688$	2087.34	0.115668
$689$	20559.1	1.13678
$690$	0	0
$691$	5773.50	0.317850	0.158925	−	0.987291i	$-0.449197\pi$
$691$	5773.50	0.317850	0.158925	+	0.987291i	$0.449197\pi$
$692$	12940.7	0.710882
$693$	0	0
$694$	−14632.4	−0.800344
$695$	−8653.87	−0.472317
$696$	0	0
$697$	−35647.7	−1.93724
$698$	−230.061	−0.0124756
$699$	0	0
$700$	0	0
$701$	18643.2	1.00449	0.502243	−	0.864726i	$-0.332508\pi$
$701$	18643.2	1.00449	0.502243	+	0.864726i	$0.332508\pi$
$702$	0	0
$703$	48198.8	2.58585
$704$	−7129.58	−0.381685
$705$	0	0
$706$	−13395.9	−0.714107
$707$	0	0
$708$	0	0
$709$	10420.4	0.551970	0.275985	−	0.961162i	$-0.410996\pi$
$709$	10420.4	0.551970	0.275985	+	0.961162i	$0.410996\pi$
$710$	5464.43	0.288840
$711$	0	0
$712$	30351.3	1.59756
$713$	−9096.06	−0.477770
$714$	0	0
$715$	19025.6	0.995131
$716$	−15132.5	−0.789842
$717$	0	0
$718$	−14277.2	−0.742088
$719$	8269.85	0.428947	0.214474	−	0.976730i	$-0.431196\pi$
$719$	8269.85	0.428947	0.214474	+	0.976730i	$0.431196\pi$
$720$	0	0
$721$	0	0
$722$	−21298.2	−1.09784
$723$	0	0
$724$	−1169.94	−0.0600561
$725$	−3586.27	−0.183712
$726$	0	0
$727$	4112.34	0.209791	0.104896	−	0.994483i	$-0.466549\pi$
$727$	4112.34	0.209791	0.104896	+	0.994483i	$0.466549\pi$
$728$	0	0
$729$	0	0
$730$	−2950.97	−0.149617
$731$	35478.5	1.79510
$732$	0	0
$733$	28562.7	1.43927	0.719636	−	0.694352i	$-0.244309\pi$
$733$	28562.7	1.43927	0.719636	+	0.694352i	$0.244309\pi$
$734$	−9657.14	−0.485629
$735$	0	0
$736$	−11205.4	−0.561193
$737$	20658.9	1.03254
$738$	0	0
$739$	23183.9	1.15404	0.577019	−	0.816731i	$-0.304216\pi$
$739$	23183.9	1.15404	0.577019	+	0.816731i	$0.304216\pi$
$740$	−22567.3	−1.12107
$741$	0	0
$742$	0	0
$743$	−7212.11	−0.356106	−0.178053	−	0.984021i	$-0.556980\pi$
$743$	−7212.11	−0.356106	−0.178053	+	0.984021i	$0.556980\pi$
$744$	0	0
$745$	−19990.8	−0.983096
$746$	−19028.6	−0.933896
$747$	0	0
$748$	18893.1	0.923528
$749$	0	0
$750$	0	0
$751$	−21762.2	−1.05741	−0.528703	−	0.848807i	$-0.677322\pi$
$751$	−21762.2	−1.05741	−0.528703	+	0.848807i	$0.677322\pi$
$752$	−2039.86	−0.0989174
$753$	0	0
$754$	10981.4	0.530396
$755$	44.4741	0.00214381
$756$	0	0
$757$	−29737.8	−1.42779	−0.713896	−	0.700252i	$-0.753071\pi$
$757$	−29737.8	−1.42779	−0.713896	+	0.700252i	$0.753071\pi$
$758$	23346.5	1.11871
$759$	0	0
$760$	38016.1	1.81446
$761$	30539.9	1.45476	0.727378	−	0.686237i	$-0.240739\pi$
$761$	30539.9	1.45476	0.727378	+	0.686237i	$0.240739\pi$
$762$	0	0
$763$	0	0
$764$	−1192.43	−0.0564667
$765$	0	0
$766$	17690.1	0.834424
$767$	−14270.4	−0.671805
$768$	0	0
$769$	−39801.3	−1.86642	−0.933208	−	0.359338i	$-0.883003\pi$
$769$	−39801.3	−1.86642	−0.933208	+	0.359338i	$0.883003\pi$
$770$	0	0
$771$	0	0
$772$	1879.49	0.0876220
$773$	−15163.9	−0.705574	−0.352787	−	0.935704i	$-0.614766\pi$
$773$	−15163.9	−0.705574	−0.352787	+	0.935704i	$0.614766\pi$
$774$	0	0
$775$	−4259.81	−0.197441
$776$	−17538.0	−0.811312
$777$	0	0
$778$	18236.7	0.840383
$779$	41214.2	1.89557
$780$	0	0
$781$	−7846.10	−0.359482
$782$	−12009.4	−0.549173
$783$	0	0
$784$	0	0
$785$	8637.38	0.392715
$786$	0	0
$787$	−21917.3	−0.992717	−0.496359	−	0.868118i	$-0.665330\pi$
$787$	−21917.3	−0.992717	−0.496359	+	0.868118i	$0.665330\pi$
$788$	−28539.6	−1.29021
$789$	0	0
$790$	−13186.8	−0.593882
$791$	0	0
$792$	0	0
$793$	12696.3	0.568550
$794$	−21341.2	−0.953868
$795$	0	0
$796$	3107.38	0.138364
$797$	3205.98	0.142486	0.0712431	−	0.997459i	$-0.477303\pi$
$797$	3205.98	0.142486	0.0712431	+	0.997459i	$0.477303\pi$
$798$	0	0
$799$	−34671.3	−1.53515
$800$	−5247.66	−0.231916
$801$	0	0
$802$	−4460.89	−0.196408
$803$	4237.15	0.186209
$804$	0	0
$805$	0	0
$806$	13043.8	0.570034
$807$	0	0
$808$	22086.5	0.961634
$809$	39211.8	1.70410	0.852048	−	0.523464i	$-0.175360\pi$
$809$	39211.8	1.70410	0.852048	+	0.523464i	$0.175360\pi$
$810$	0	0
$811$	−14711.3	−0.636973	−0.318486	−	0.947927i	$-0.603175\pi$
$811$	−14711.3	−0.636973	−0.318486	+	0.947927i	$0.603175\pi$
$812$	0	0
$813$	0	0
$814$	−16167.8	−0.696167
$815$	−1806.19	−0.0776297
$816$	0	0
$817$	−41018.5	−1.75650
$818$	18800.7	0.803609
$819$	0	0
$820$	−19297.1	−0.821808
$821$	−42916.2	−1.82434	−0.912171	−	0.409810i	$-0.865595\pi$
$821$	−42916.2	−1.82434	−0.912171	+	0.409810i	$0.865595\pi$
$822$	0	0
$823$	6882.13	0.291490	0.145745	−	0.989322i	$-0.453442\pi$
$823$	6882.13	0.291490	0.145745	+	0.989322i	$0.453442\pi$
$824$	−22621.4	−0.956375
$825$	0	0
$826$	0	0
$827$	−6598.74	−0.277461	−0.138731	−	0.990330i	$-0.544302\pi$
$827$	−6598.74	−0.277461	−0.138731	+	0.990330i	$0.544302\pi$
$828$	0	0
$829$	23585.6	0.988134	0.494067	−	0.869424i	$-0.335510\pi$
$829$	23585.6	0.988134	0.494067	+	0.869424i	$0.335510\pi$
$830$	707.507	0.0295879
$831$	0	0
$832$	13024.8	0.542731
$833$	0	0
$834$	0	0
$835$	22482.4	0.931779
$836$	−21843.3	−0.903666
$837$	0	0
$838$	−3959.88	−0.163236
$839$	−656.729	−0.0270236	−0.0135118	−	0.999909i	$-0.504301\pi$
$839$	−656.729	−0.0270236	−0.0135118	+	0.999909i	$0.504301\pi$
$840$	0	0
$841$	−8260.64	−0.338703
$842$	−16625.6	−0.680470
$843$	0	0
$844$	−29182.8	−1.19018
$845$	−7560.65	−0.307804
$846$	0	0
$847$	0	0
$848$	2785.99	0.112820
$849$	0	0
$850$	−5624.14	−0.226949
$851$	−20597.1	−0.829682
$852$	0	0
$853$	26201.7	1.05173	0.525867	−	0.850567i	$-0.323741\pi$
$853$	26201.7	1.05173	0.525867	+	0.850567i	$0.323741\pi$
$854$	0	0
$855$	0	0
$856$	13239.9	0.528659
$857$	23208.5	0.925072	0.462536	−	0.886601i	$-0.346940\pi$
$857$	23208.5	0.925072	0.462536	+	0.886601i	$0.346940\pi$
$858$	0	0
$859$	26175.4	1.03969	0.519845	−	0.854260i	$-0.325990\pi$
$859$	26175.4	1.03969	0.519845	+	0.854260i	$0.325990\pi$
$860$	19205.4	0.761512
$861$	0	0
$862$	11064.6	0.437196
$863$	5651.84	0.222933	0.111466	−	0.993768i	$-0.464445\pi$
$863$	5651.84	0.222933	0.111466	+	0.993768i	$0.464445\pi$
$864$	0	0
$865$	30015.1	1.17982
$866$	9991.60	0.392065
$867$	0	0
$868$	0	0
$869$	18934.3	0.739128
$870$	0	0
$871$	−37741.0	−1.46820
$872$	23455.5	0.910897
$873$	0	0
$874$	13884.6	0.537363
$875$	0	0
$876$	0	0
$877$	37510.2	1.44428	0.722138	−	0.691749i	$-0.243160\pi$
$877$	37510.2	1.44428	0.722138	+	0.691749i	$0.243160\pi$
$878$	−25697.6	−0.987760
$879$	0	0
$880$	2578.19	0.0987622
$881$	22133.8	0.846434	0.423217	−	0.906028i	$-0.360901\pi$
$881$	22133.8	0.846434	0.423217	+	0.906028i	$0.360901\pi$
$882$	0	0
$883$	20678.1	0.788080	0.394040	−	0.919093i	$-0.371077\pi$
$883$	20678.1	0.788080	0.394040	+	0.919093i	$0.371077\pi$
$884$	−34515.0	−1.31320
$885$	0	0
$886$	−24581.9	−0.932104
$887$	37469.8	1.41839	0.709196	−	0.705012i	$-0.249058\pi$
$887$	37469.8	1.41839	0.709196	+	0.705012i	$0.249058\pi$
$888$	0	0
$889$	0	0
$890$	28171.0	1.06100
$891$	0	0
$892$	23312.5	0.875068
$893$	40085.3	1.50213
$894$	0	0
$895$	−35098.9	−1.31087
$896$	0	0
$897$	0	0
$898$	285.679	0.0106161
$899$	19157.4	0.710718
$900$	0	0
$901$	47353.3	1.75091
$902$	−13824.9	−0.510330
$903$	0	0
$904$	44938.2	1.65334
$905$	−2713.61	−0.0996725
$906$	0	0
$907$	20503.9	0.750630	0.375315	−	0.926897i	$-0.377535\pi$
$907$	20503.9	0.750630	0.375315	+	0.926897i	$0.377535\pi$
$908$	3399.45	0.124245
$909$	0	0
$910$	0	0
$911$	3533.81	0.128519	0.0642593	−	0.997933i	$-0.479532\pi$
$911$	3533.81	0.128519	0.0642593	+	0.997933i	$0.479532\pi$
$912$	0	0
$913$	−1015.87	−0.0368242
$914$	20550.7	0.743717
$915$	0	0
$916$	5248.96	0.189335
$917$	0	0
$918$	0	0
$919$	29803.8	1.06979	0.534895	−	0.844919i	$-0.320351\pi$
$919$	29803.8	1.06979	0.534895	+	0.844919i	$0.320351\pi$
$920$	−16245.7	−0.582178
$921$	0	0
$922$	22489.8	0.803321
$923$	14333.7	0.511160
$924$	0	0
$925$	−9645.90	−0.342870
$926$	−14470.4	−0.513528
$927$	0	0
$928$	23600.0	0.834815
$929$	−14446.4	−0.510194	−0.255097	−	0.966915i	$-0.582107\pi$
$929$	−14446.4	−0.510194	−0.255097	+	0.966915i	$0.582107\pi$
$930$	0	0
$931$	0	0
$932$	8952.20	0.314634
$933$	0	0
$934$	−20951.5	−0.733996
$935$	43821.3	1.53274
$936$	0	0
$937$	−29384.8	−1.02450	−0.512251	−	0.858836i	$-0.671188\pi$
$937$	−29384.8	−1.02450	−0.512251	+	0.858836i	$0.671188\pi$
$938$	0	0
$939$	0	0
$940$	−18768.5	−0.651235
$941$	−4659.97	−0.161435	−0.0807176	−	0.996737i	$-0.525721\pi$
$941$	−4659.97	−0.161435	−0.0807176	+	0.996737i	$0.525721\pi$
$942$	0	0
$943$	−17612.3	−0.608204
$944$	−1933.80	−0.0666735
$945$	0	0
$946$	13759.2	0.472887
$947$	54321.9	1.86402	0.932008	−	0.362437i	$-0.118055\pi$
$947$	54321.9	1.86402	0.932008	+	0.362437i	$0.118055\pi$
$948$	0	0
$949$	−7740.69	−0.264777
$950$	6502.37	0.222068
$951$	0	0
$952$	0	0
$953$	−23606.5	−0.802404	−0.401202	−	0.915990i	$-0.631407\pi$
$953$	−23606.5	−0.802404	−0.401202	+	0.915990i	$0.631407\pi$
$954$	0	0
$955$	−2765.77	−0.0937153
$956$	14437.7	0.488439
$957$	0	0
$958$	16245.5	0.547878
$959$	0	0
$960$	0	0
$961$	−7035.67	−0.236168
$962$	29536.3	0.989905
$963$	0	0
$964$	−11425.2	−0.381722
$965$	4359.36	0.145422
$966$	0	0
$967$	7021.95	0.233517	0.116758	−	0.993160i	$-0.462750\pi$
$967$	7021.95	0.233517	0.116758	+	0.993160i	$0.462750\pi$
$968$	−10658.0	−0.353885
$969$	0	0
$970$	−16278.2	−0.538825
$971$	1483.48	0.0490290	0.0245145	−	0.999699i	$-0.492196\pi$
$971$	1483.48	0.0490290	0.0245145	+	0.999699i	$0.492196\pi$
$972$	0	0
$973$	0	0
$974$	27278.1	0.897377
$975$	0	0
$976$	1720.49	0.0564259
$977$	26152.1	0.856375	0.428187	−	0.903690i	$-0.359152\pi$
$977$	26152.1	0.856375	0.428187	+	0.903690i	$0.359152\pi$
$978$	0	0
$979$	−40449.3	−1.32049
$980$	0	0
$981$	0	0
$982$	4865.19	0.158100
$983$	−39185.3	−1.27143	−0.635716	−	0.771923i	$-0.719295\pi$
$983$	−39185.3	−1.27143	−0.635716	+	0.771923i	$0.719295\pi$
$984$	0	0
$985$	−66196.0	−2.14130
$986$	25293.2	0.816936
$987$	0	0
$988$	39904.6	1.28496
$989$	17528.7	0.563580
$990$	0	0
$991$	−23009.7	−0.737564	−0.368782	−	0.929516i	$-0.620225\pi$
$991$	−23009.7	−0.737564	−0.368782	+	0.929516i	$0.620225\pi$
$992$	28032.3	0.897204
$993$	0	0
$994$	0	0
$995$	7207.38	0.229638
$996$	0	0
$997$	11154.2	0.354320	0.177160	−	0.984182i	$-0.443309\pi$
$997$	11154.2	0.354320	0.177160	+	0.984182i	$0.443309\pi$
$998$	−13331.4	−0.422843
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1323.4.a.bk.1.3		7
3.2	odd	2		1323.4.a.bh.1.5		7
7.3	odd	6		189.4.e.f.163.5	yes	14
7.5	odd	6		189.4.e.f.109.5	✓	14
7.6	odd	2		1323.4.a.bj.1.3		7
21.5	even	6		189.4.e.g.109.3	yes	14
21.17	even	6		189.4.e.g.163.3	yes	14
21.20	even	2		1323.4.a.bi.1.5		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
189.4.e.f.109.5	✓	14	7.5	odd	6
189.4.e.f.163.5	yes	14	7.3	odd	6
189.4.e.g.109.3	yes	14	21.5	even	6
189.4.e.g.163.3	yes	14	21.17	even	6
1323.4.a.bh.1.5		7	3.2	odd	2
1323.4.a.bi.1.5		7	21.20	even	2
1323.4.a.bj.1.3		7	7.6	odd	2
1323.4.a.bk.1.3		7	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:	\( N \)	\(=\)	\( 1323 = 3^{3} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1323.a (trivial)

\(f(q)\)	\(=\)	\(q-1.63185 q^{2} -5.33705 q^{4} -12.3790 q^{5} +21.7641 q^{8} +20.2007 q^{10} -29.0051 q^{11} +52.9884 q^{13} +7.18051 q^{16} +122.047 q^{17} -141.105 q^{19} +66.0672 q^{20} +47.3321 q^{22} +60.2992 q^{23} +28.2389 q^{25} -86.4693 q^{26} -126.997 q^{29} -150.849 q^{31} -185.831 q^{32} -199.163 q^{34} -341.581 q^{37} +230.262 q^{38} -269.418 q^{40} -292.082 q^{41} +290.696 q^{43} +154.802 q^{44} -98.3995 q^{46} -284.082 q^{47} -46.0818 q^{50} -282.802 q^{52} +387.993 q^{53} +359.053 q^{55} +207.241 q^{58} -269.312 q^{59} +239.606 q^{61} +246.163 q^{62} +245.804 q^{64} -655.941 q^{65} -712.250 q^{67} -651.370 q^{68} +270.507 q^{71} -146.083 q^{73} +557.411 q^{74} +753.083 q^{76} -652.792 q^{79} -88.8874 q^{80} +476.636 q^{82} +35.0239 q^{83} -1510.81 q^{85} -474.373 q^{86} -631.271 q^{88} +1394.56 q^{89} -321.820 q^{92} +463.581 q^{94} +1746.73 q^{95} -805.822 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + q^{2} + 31 q^{4} + q^{5} + 84 q^{8} - 12 q^{10} + 98 q^{11} + 124 q^{13} + 139 q^{16} - 30 q^{17} - 182 q^{19} + 110 q^{20} + 276 q^{22} - 6 q^{23} + 388 q^{25} + 245 q^{26} + 323 q^{29} - 26 q^{31}+ \cdots + 3730 q^{97}+O(q^{100}) \) 7 * q + q^2 + 31 * q^4 + q^5 + 84 * q^8 - 12 * q^10 + 98 * q^11 + 124 * q^13 + 139 * q^16 - 30 * q^17 - 182 * q^19 + 110 * q^20 + 276 * q^22 - 6 * q^23 + 388 * q^25 + 245 * q^26 + 323 * q^29 - 26 * q^31 + 398 * q^32 - 114 * q^34 - 112 * q^37 - 1015 * q^38 + 147 * q^40 + 524 * q^41 + 8 * q^43 + 937 * q^44 - 339 * q^46 - 288 * q^47 + 2576 * q^50 + 1075 * q^52 + 1353 * q^53 - 156 * q^55 - 81 * q^58 - 165 * q^59 - 56 * q^61 + 1215 * q^62 - 1706 * q^64 + 1694 * q^65 - 988 * q^67 - 2625 * q^68 + 792 * q^71 - 1487 * q^73 + 2736 * q^74 - 1952 * q^76 - 1273 * q^79 + 2501 * q^80 + 2049 * q^82 + 1170 * q^83 + 216 * q^85 - 160 * q^86 + 9 * q^88 - 1058 * q^89 + 3834 * q^92 - 1653 * q^94 + 3260 * q^95 + 3730 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more