LMFDB - Embedded newform 1764.4.k.q.1549.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1764,4,Mod(361,1764)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1764, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 4]))

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

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

chi := DirichletCharacter("1764.361");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1764.k (of order $3$, degree $2$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$104.079369250$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{3})$
Coefficient field:	$\Q(\sqrt{-3}, \sqrt{193})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} + 49x^{2} + 48x + 2304 $ x^4 - x^3 + 49x^2 + 48x + 2304
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 84)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			1549.2
Root			$3.72311 - 6.44862i$ of defining polynomial
Character	$\chi$	$=$	1764.1549
Dual form			1764.4.k.q.361.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.723111 - 1.25246i) q^{5} +O(q^{10})$	$q+(0.723111 - 1.25246i) q^{5} +(23.0618 + 39.9442i) q^{11} +32.2311 q^{13} +(-38.8924 - 67.3637i) q^{17} +(6.33067 - 10.9650i) q^{19} +(-50.4622 + 87.4031i) q^{23} +(61.4542 + 106.442i) q^{25} -213.908 q^{29} +(21.0378 + 36.4385i) q^{31} +(-155.040 + 268.537i) q^{37} +44.0320 q^{41} +381.339 q^{43} +(179.032 - 310.093i) q^{47} +(92.4920 + 160.201i) q^{53} +66.7049 q^{55} +(-227.325 - 393.738i) q^{59} +(5.92444 - 10.2614i) q^{61} +(23.3067 - 40.3683i) q^{65} +(-295.180 - 511.266i) q^{67} -494.366 q^{71} +(487.825 + 844.937i) q^{73} +(-149.667 + 259.231i) q^{79} -1406.07 q^{83} -112.494 q^{85} +(347.629 - 602.112i) q^{89} +(-9.15555 - 15.8579i) q^{95} -481.940 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 11 q^{5}+O(q^{10}) $ 4 * q - 11 * q^5	$ 4 q - 11 q^{5} - 5 q^{11} - 10 q^{13} - 100 q^{17} + 67 q^{19} + 76 q^{23} + 93 q^{25} - 550 q^{29} + 362 q^{31} + 5 q^{37} - 324 q^{41} + 1442 q^{43} + 216 q^{47} + 495 q^{53} + 1406 q^{55} - 173 q^{59} - 532 q^{61} + 510 q^{65} - 111 q^{67} - 3200 q^{71} + 1215 q^{73} - 1460 q^{79} - 2818 q^{83} + 328 q^{85} + 1974 q^{89} + 658 q^{95} - 1122 q^{97}+O(q^{100}) $ 4 * q - 11 * q^5 - 5 * q^11 - 10 * q^13 - 100 * q^17 + 67 * q^19 + 76 * q^23 + 93 * q^25 - 550 * q^29 + 362 * q^31 + 5 * q^37 - 324 * q^41 + 1442 * q^43 + 216 * q^47 + 495 * q^53 + 1406 * q^55 - 173 * q^59 - 532 * q^61 + 510 * q^65 - 111 * q^67 - 3200 * q^71 + 1215 * q^73 - 1460 * q^79 - 2818 * q^83 + 328 * q^85 + 1974 * q^89 + 658 * q^95 - 1122 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1764\mathbb{Z}\right)^\times$.

$n$	$785$	$883$	$1081$
$\chi(n)$	$1$	$1$	$e\left(\frac{1}{3}\right)$

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
$3$		0			0
$3$		0			0
$4$		0			0
$5$	0.723111	−	1.25246i	0.0646770	−	0.112024i	−0.831874	−	0.554965i	$-0.812732\pi$
$5$	0.723111	−	1.25246i	0.0646770	−	0.112024i	0.896551	+	0.442941i	$0.146065\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$		0			0
$9$		0			0
$10$		0			0
$11$	23.0618	+	39.9442i	0.632126	+	1.09487i	0.987116	+	0.160005i	$0.0511509\pi$
$11$	23.0618	+	39.9442i	0.632126	+	1.09487i	−0.354990	+	0.934870i	$0.615516\pi$
$12$		0			0
$13$	32.2311			0.687639			0.343819	−	0.939036i	$-0.388279\pi$
$13$	32.2311			0.687639			0.343819	+	0.939036i	$0.388279\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−38.8924	−	67.3637i	−0.554871	−	0.961064i	−0.997914	−	0.0645639i	$-0.979434\pi$
$17$	−38.8924	−	67.3637i	−0.554871	−	0.961064i	0.443043	−	0.896500i	$-0.353899\pi$
$18$		0			0
$19$	6.33067	−	10.9650i	0.0764397	−	0.132397i	−0.825272	−	0.564736i	$-0.808978\pi$
$19$	6.33067	−	10.9650i	0.0764397	−	0.132397i	0.901711	+	0.432338i	$0.142311\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−50.4622	+	87.4031i	−0.457483	+	0.792383i	−0.998827	−	0.0484177i	$-0.984582\pi$
$23$	−50.4622	+	87.4031i	−0.457483	+	0.792383i	0.541345	+	0.840801i	$0.317915\pi$
$24$		0			0
$25$	61.4542	+	106.442i	0.491634	+	0.851535i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−213.908			−1.36972			−0.684859	−	0.728676i	$-0.740136\pi$
$29$	−213.908			−1.36972			−0.684859	+	0.728676i	$0.740136\pi$
$30$		0			0
$31$	21.0378	+	36.4385i	0.121887	+	0.211114i	0.920512	−	0.390715i	$-0.127772\pi$
$31$	21.0378	+	36.4385i	0.121887	+	0.211114i	−0.798625	+	0.601829i	$0.794439\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−155.040	+	268.537i	−0.688876	+	1.19317i	0.283326	+	0.959024i	$0.408562\pi$
$37$	−155.040	+	268.537i	−0.688876	+	1.19317i	−0.972202	+	0.234145i	$0.924771\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	44.0320			0.167723			0.0838615	−	0.996477i	$-0.473275\pi$
$41$	44.0320			0.167723			0.0838615	+	0.996477i	$0.473275\pi$
$42$		0			0
$43$	381.339			1.35241			0.676205	−	0.736714i	$-0.263624\pi$
$43$	381.339			1.35241			0.676205	+	0.736714i	$0.263624\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	179.032	−	310.093i	0.555628	−	0.962375i	−0.442227	−	0.896903i	$-0.645811\pi$
$47$	179.032	−	310.093i	0.555628	−	0.962375i	0.997854	−	0.0654721i	$-0.0208553\pi$
$48$		0			0
$49$		0			0
$50$		0			0
$51$		0			0
$52$		0			0
$53$	92.4920	+	160.201i	0.239712	+	0.415194i	0.960632	−	0.277825i	$-0.0896135\pi$
$53$	92.4920	+	160.201i	0.239712	+	0.415194i	−0.720919	+	0.693019i	$0.756280\pi$
$54$		0			0
$55$	66.7049			0.163536
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−227.325	−	393.738i	−0.501613	−	0.868820i	−0.999998	−	0.00186377i	$-0.999407\pi$
$59$	−227.325	−	393.738i	−0.501613	−	0.868820i	0.498385	−	0.866956i	$-0.333927\pi$
$60$		0			0
$61$	5.92444	−	10.2614i	0.0124352	−	0.0215384i	−0.859741	−	0.510731i	$-0.829375\pi$
$61$	5.92444	−	10.2614i	0.0124352	−	0.0215384i	0.872176	+	0.489192i	$0.162708\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	23.3067	−	40.3683i	0.0444744	−	0.0770319i
$66$		0			0
$67$	−295.180	−	511.266i	−0.538238	−	0.932255i	−0.998999	−	0.0447309i	$-0.985757\pi$
$67$	−295.180	−	511.266i	−0.538238	−	0.932255i	0.460761	−	0.887524i	$-0.347576\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−494.366			−0.826345			−0.413172	−	0.910653i	$-0.635579\pi$
$71$	−494.366			−0.826345			−0.413172	+	0.910653i	$0.635579\pi$
$72$		0			0
$73$	487.825	+	844.937i	0.782131	+	1.35469i	0.930698	+	0.365789i	$0.119201\pi$
$73$	487.825	+	844.937i	0.782131	+	1.35469i	−0.148567	+	0.988902i	$0.547466\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−149.667	+	259.231i	−0.213150	+	0.369187i	−0.952699	−	0.303916i	$-0.901706\pi$
$79$	−149.667	+	259.231i	−0.213150	+	0.369187i	0.739549	+	0.673103i	$0.235039\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−1406.07			−1.85947			−0.929735	−	0.368229i	$-0.879964\pi$
$83$	−1406.07			−1.85947			−0.929735	+	0.368229i	$0.879964\pi$
$84$		0			0
$85$	−112.494			−0.143550
$86$		0			0
$87$		0			0
$88$		0			0
$89$	347.629	−	602.112i	0.414030	−	0.717120i	−0.581296	−	0.813692i	$-0.697454\pi$
$89$	347.629	−	602.112i	0.414030	−	0.717120i	0.995326	+	0.0965715i	$0.0307877\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−9.15555	−	15.8579i	−0.00988779	−	0.0171261i
$96$		0			0
$97$	−481.940			−0.504470			−0.252235	−	0.967666i	$-0.581166\pi$
$97$	−481.940			−0.504470			−0.252235	+	0.967666i	$0.581166\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	592.052	+	1025.46i	0.583281	+	1.01027i	0.995087	+	0.0990014i	$0.0315648\pi$
$101$	592.052	+	1025.46i	0.583281	+	1.01027i	−0.411806	+	0.911272i	$0.635102\pi$
$102$		0			0
$103$	−641.765	+	1111.57i	−0.613932	+	1.06336i	0.376639	+	0.926360i	$0.377080\pi$
$103$	−641.765	+	1111.57i	−0.613932	+	1.06336i	−0.990571	+	0.137001i	$0.956253\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	806.684	−	1397.22i	0.728833	−	1.26238i	−0.228544	−	0.973533i	$-0.573397\pi$
$107$	806.684	−	1397.22i	0.728833	−	1.26238i	0.957377	−	0.288842i	$-0.0932701\pi$
$108$		0			0
$109$	76.9164	+	133.223i	0.0675895	+	0.117069i	0.897840	−	0.440322i	$-0.145136\pi$
$109$	76.9164	+	133.223i	0.0675895	+	0.117069i	−0.830250	+	0.557391i	$0.811803\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−1581.08			−1.31625			−0.658123	−	0.752910i	$-0.728650\pi$
$113$	−1581.08			−1.31625			−0.658123	+	0.752910i	$0.728650\pi$
$114$		0			0
$115$	72.9796	+	126.404i	0.0591772	+	0.102498i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−398.191	+	689.687i	−0.299167	+	0.518172i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	358.531			0.256544
$126$		0			0
$127$	1916.30			1.33893			0.669465	−	0.742844i	$-0.266523\pi$
$127$	1916.30			1.33893			0.669465	+	0.742844i	$0.266523\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−1250.24	+	2165.48i	−0.833849	+	1.44427i	0.0611158	+	0.998131i	$0.480534\pi$
$131$	−1250.24	+	2165.48i	−0.833849	+	1.44427i	−0.894964	+	0.446137i	$0.852799\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$	145.446	+	251.920i	0.0907030	+	0.157102i	0.907807	−	0.419388i	$-0.137755\pi$
$137$	145.446	+	251.920i	0.0907030	+	0.157102i	−0.817104	+	0.576490i	$0.804422\pi$
$138$		0			0
$139$	1348.77			0.823028			0.411514	−	0.911403i	$-0.365000\pi$
$139$	1348.77			0.823028			0.411514	+	0.911403i	$0.365000\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	743.307	+	1287.44i	0.434674	+	0.752878i
$144$		0			0
$145$	−154.680	+	267.913i	−0.0885892	+	0.153441i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−1424.40	+	2467.14i	−0.783165	+	1.35648i	0.146924	+	0.989148i	$0.453063\pi$
$149$	−1424.40	+	2467.14i	−0.783165	+	1.35648i	−0.930089	+	0.367334i	$0.880271\pi$
$150$		0			0
$151$	744.656	+	1289.78i	0.401320	+	0.695106i	0.993885	−	0.110416i	$-0.0352183\pi$
$151$	744.656	+	1289.78i	0.401320	+	0.695106i	−0.592566	+	0.805522i	$0.701885\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	60.8506			0.0315331
$156$		0			0
$157$	1821.69	+	3155.26i	0.926030	+	1.60393i	0.789896	+	0.613241i	$0.210135\pi$
$157$	1821.69	+	3155.26i	0.926030	+	1.60393i	0.136134	+	0.990690i	$0.456532\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−384.581	+	666.114i	−0.184802	+	0.320087i	−0.943510	−	0.331345i	$-0.892498\pi$
$163$	−384.581	+	666.114i	−0.184802	+	0.320087i	0.758708	+	0.651431i	$0.225831\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−2399.78			−1.11198			−0.555991	−	0.831188i	$-0.687661\pi$
$167$	−2399.78			−1.11198			−0.555991	+	0.831188i	$0.687661\pi$
$168$		0			0
$169$	−1158.16			−0.527153
$170$		0			0
$171$		0			0
$172$		0			0
$173$	1668.32	−	2889.62i	0.733181	−	1.26991i	−0.222335	−	0.974970i	$-0.571368\pi$
$173$	1668.32	−	2889.62i	0.733181	−	1.26991i	0.955517	−	0.294937i	$-0.0952986\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$	1230.89	+	2131.96i	0.513970	+	0.890223i	0.999869	+	0.0162074i	$0.00515919\pi$
$179$	1230.89	+	2131.96i	0.513970	+	0.890223i	−0.485898	+	0.874015i	$0.661507\pi$
$180$		0			0
$181$	−1316.74			−0.540732			−0.270366	−	0.962758i	$-0.587145\pi$
$181$	−1316.74			−0.540732			−0.270366	+	0.962758i	$0.587145\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	224.222	+	388.364i	0.0891089	+	0.154341i
$186$		0			0
$187$	1793.86	−	3107.05i	0.701497	−	1.21503i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−1633.98	+	2830.14i	−0.619010	+	1.07216i	0.370657	+	0.928770i	$0.379133\pi$
$191$	−1633.98	+	2830.14i	−0.619010	+	1.07216i	−0.989667	+	0.143387i	$0.954201\pi$
$192$		0			0
$193$	−116.836	−	202.366i	−0.0435753	−	0.0754747i	0.843415	−	0.537262i	$-0.180542\pi$
$193$	−116.836	−	202.366i	−0.0435753	−	0.0754747i	−0.886990	+	0.461788i	$0.847208\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	31.8632			0.0115236			0.00576182	−	0.999983i	$-0.498166\pi$
$197$	31.8632			0.0115236			0.00576182	+	0.999983i	$0.498166\pi$
$198$		0			0
$199$	−739.451	−	1280.77i	−0.263408	−	0.456237i	0.703737	−	0.710461i	$-0.251513\pi$
$199$	−739.451	−	1280.77i	−0.263408	−	0.456237i	−0.967145	+	0.254224i	$0.918180\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$	31.8400	−	55.1485i	0.0108478	−	0.0187890i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	583.986			0.193278
$210$		0			0
$211$	−4498.67			−1.46778			−0.733889	−	0.679269i	$-0.762297\pi$
$211$	−4498.67			−1.46778			−0.733889	+	0.679269i	$0.762297\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	275.750	−	477.613i	0.0874698	−	0.151502i
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−1253.55	−	2171.21i	−0.381551	−	0.660865i
$222$		0			0
$223$	−5382.75			−1.61639			−0.808196	−	0.588913i	$-0.799556\pi$
$223$	−5382.75			−1.61639			−0.808196	+	0.588913i	$0.799556\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	2712.78	+	4698.68i	0.793188	+	1.37384i	0.923983	+	0.382433i	$0.124914\pi$
$227$	2712.78	+	4698.68i	0.793188	+	1.37384i	−0.130795	+	0.991409i	$0.541753\pi$
$228$		0			0
$229$	994.873	−	1723.17i	0.287088	−	0.497250i	−0.686026	−	0.727577i	$-0.740646\pi$
$229$	994.873	−	1723.17i	0.287088	−	0.497250i	0.973113	+	0.230327i	$0.0739796\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−3148.83	+	5453.93i	−0.885351	+	1.53347i	−0.0400396	+	0.999198i	$0.512748\pi$
$233$	−3148.83	+	5453.93i	−0.885351	+	1.53347i	−0.845311	+	0.534274i	$0.820585\pi$
$234$		0			0
$235$	−258.920	−	448.463i	−0.0718727	−	0.124487i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−3395.77			−0.919054			−0.459527	−	0.888164i	$-0.651981\pi$
$239$	−3395.77			−0.919054			−0.459527	+	0.888164i	$0.651981\pi$
$240$		0			0
$241$	−3186.97	−	5519.99i	−0.851829	−	1.47541i	−0.879556	−	0.475795i	$-0.842160\pi$
$241$	−3186.97	−	5519.99i	−0.851829	−	1.47541i	0.0277273	−	0.999616i	$-0.491173\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$	204.044	−	353.415i	0.0525629	−	0.0910416i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−2650.91			−0.666630			−0.333315	−	0.942815i	$-0.608167\pi$
$251$	−2650.91			−0.666630			−0.333315	+	0.942815i	$0.608167\pi$
$252$		0			0
$253$	−4654.99			−1.15675
$254$		0			0
$255$		0			0
$256$		0			0
$257$	936.035	−	1621.26i	0.227192	−	0.393507i	−0.729783	−	0.683679i	$-0.760379\pi$
$257$	936.035	−	1621.26i	0.227192	−	0.393507i	0.956975	+	0.290171i	$0.0937123\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$	3063.05	+	5305.35i	0.718158	+	1.24389i	0.961729	+	0.274003i	$0.0883480\pi$
$263$	3063.05	+	5305.35i	0.718158	+	1.24389i	−0.243570	+	0.969883i	$0.578319\pi$
$264$		0			0
$265$	267.528			0.0620155
$266$		0			0
$267$		0			0
$268$		0			0
$269$	4252.06	+	7364.78i	0.963764	+	1.66929i	0.712899	+	0.701267i	$0.247382\pi$
$269$	4252.06	+	7364.78i	0.963764	+	1.66929i	0.250866	+	0.968022i	$0.419285\pi$
$270$		0			0
$271$	1060.77	−	1837.31i	0.237776	−	0.411840i	−0.722300	−	0.691580i	$-0.756915\pi$
$271$	1060.77	−	1837.31i	0.237776	−	0.411840i	0.960076	+	0.279740i	$0.0902483\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−2834.49	+	4909.48i	−0.621549	+	1.07655i
$276$		0			0
$277$	4010.68	+	6946.71i	0.869959	+	1.50681i	0.862038	+	0.506844i	$0.169188\pi$
$277$	4010.68	+	6946.71i	0.869959	+	1.50681i	0.00792096	+	0.999969i	$0.497479\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	8244.17			1.75020			0.875100	−	0.483943i	$-0.160796\pi$
$281$	8244.17			1.75020			0.875100	+	0.483943i	$0.160796\pi$
$282$		0			0
$283$	3050.64	+	5283.86i	0.640784	+	1.10987i	0.985258	+	0.171074i	$0.0547238\pi$
$283$	3050.64	+	5283.86i	0.640784	+	1.10987i	−0.344475	+	0.938796i	$0.611943\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−568.744	+	985.094i	−0.115763	+	0.200508i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−1965.98			−0.391993			−0.195996	−	0.980605i	$-0.562794\pi$
$293$	−1965.98			−0.391993			−0.195996	+	0.980605i	$0.562794\pi$
$294$		0			0
$295$	−657.524			−0.129771
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−1626.45	+	2817.10i	−0.314583	+	0.544873i
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−8.56806	−	14.8403i	−0.00160854	−	0.00278608i
$306$		0			0
$307$	−997.810			−0.185498			−0.0927492	−	0.995690i	$-0.529565\pi$
$307$	−997.810			−0.185498			−0.0927492	+	0.995690i	$0.529565\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	3450.72	+	5976.82i	0.629171	+	1.08976i	0.987718	+	0.156245i	$0.0499388\pi$
$311$	3450.72	+	5976.82i	0.629171	+	1.08976i	−0.358547	+	0.933512i	$0.616728\pi$
$312$		0			0
$313$	−3170.86	+	5492.08i	−0.572612	+	0.991792i	0.423685	+	0.905810i	$0.360736\pi$
$313$	−3170.86	+	5492.08i	−0.572612	+	0.991792i	−0.996297	+	0.0859827i	$0.972597\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	612.487	−	1060.86i	0.108519	−	0.187961i	−0.806651	−	0.591028i	$-0.798722\pi$
$317$	612.487	−	1060.86i	0.108519	−	0.187961i	0.915171	+	0.403067i	$0.132056\pi$
$318$		0			0
$319$	−4933.11	−	8544.40i	−0.865834	−	1.49967i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−984.860			−0.169657
$324$		0			0
$325$	1980.74	+	3430.74i	0.338066	+	0.585548i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−1868.15	+	3235.73i	−0.310220	+	0.537317i	−0.978410	−	0.206674i	$-0.933736\pi$
$331$	−1868.15	+	3235.73i	−0.310220	+	0.537317i	0.668190	+	0.743991i	$0.267069\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−853.790			−0.139246
$336$		0			0
$337$	−3928.18			−0.634960			−0.317480	−	0.948265i	$-0.602837\pi$
$337$	−3928.18			−0.634960			−0.317480	+	0.948265i	$0.602837\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−970.337	+	1680.67i	−0.154096	+	0.266902i
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$	271.003	+	469.390i	0.0419256	+	0.0726173i	0.886227	−	0.463252i	$-0.153317\pi$
$347$	271.003	+	469.390i	0.0419256	+	0.0726173i	−0.844301	+	0.535869i	$0.819984\pi$
$348$		0			0
$349$	−2331.24			−0.357561			−0.178780	−	0.983889i	$-0.557215\pi$
$349$	−2331.24			−0.357561			−0.178780	+	0.983889i	$0.557215\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	1280.15	+	2217.28i	0.193018	+	0.334318i	0.946249	−	0.323439i	$-0.104839\pi$
$353$	1280.15	+	2217.28i	0.193018	+	0.334318i	−0.753231	+	0.657756i	$0.771506\pi$
$354$		0			0
$355$	−357.482	+	619.176i	−0.0534455	+	0.0925703i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−2284.27	+	3956.46i	−0.335819	+	0.581655i	−0.983642	−	0.180136i	$-0.942346\pi$
$359$	−2284.27	+	3956.46i	−0.335819	+	0.581655i	0.647823	+	0.761791i	$0.275680\pi$
$360$		0			0
$361$	3349.35	+	5801.24i	0.488314	+	0.845785i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	1411.01			0.202344
$366$		0			0
$367$	−3190.91	−	5526.82i	−0.453854	−	0.786098i	0.544768	−	0.838587i	$-0.316618\pi$
$367$	−3190.91	−	5526.82i	−0.453854	−	0.786098i	−0.998621	+	0.0524893i	$0.983284\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	3186.01	−	5518.33i	0.442266	−	0.766027i	−0.555591	−	0.831456i	$-0.687508\pi$
$373$	3186.01	−	5518.33i	0.442266	−	0.766027i	0.997857	+	0.0654285i	$0.0208414\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−6894.51			−0.941870
$378$		0			0
$379$	1494.59			0.202564			0.101282	−	0.994858i	$-0.467706\pi$
$379$	1494.59			0.202564			0.101282	+	0.994858i	$0.467706\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−4410.13	+	7638.57i	−0.588374	+	1.01909i	0.406072	+	0.913841i	$0.366898\pi$
$383$	−4410.13	+	7638.57i	−0.588374	+	1.01909i	−0.994446	+	0.105252i	$0.966435\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−4939.24	−	8555.01i	−0.643777	−	1.11505i	−0.984583	−	0.174921i	$-0.944033\pi$
$389$	−4939.24	−	8555.01i	−0.643777	−	1.11505i	0.340806	−	0.940134i	$-0.389300\pi$
$390$		0			0
$391$	7850.40			1.01537
$392$		0			0
$393$		0			0
$394$		0			0
$395$	216.452	+	374.906i	0.0275718	+	0.0477558i
$396$		0			0
$397$	970.898	−	1681.64i	0.122740	−	0.212593i	−0.798107	−	0.602516i	$-0.794165\pi$
$397$	970.898	−	1681.64i	0.122740	−	0.212593i	0.920847	+	0.389923i	$0.127498\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	3792.48	−	6568.77i	0.472288	−	0.818026i	−0.527209	−	0.849735i	$-0.676762\pi$
$401$	3792.48	−	6568.77i	0.472288	−	0.818026i	0.999497	+	0.0317090i	$0.0100950\pi$
$402$		0			0
$403$	678.071	+	1174.45i	0.0838142	+	0.145170i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−14302.0			−1.74183
$408$		0			0
$409$	4353.90	+	7541.18i	0.526373	+	0.911705i	0.999528	+	0.0307253i	$0.00978172\pi$
$409$	4353.90	+	7541.18i	0.526373	+	0.911705i	−0.473155	+	0.880979i	$0.656885\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$	−1016.74	+	1761.05i	−0.120265	+	0.208305i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	6647.96			0.775117			0.387558	−	0.921845i	$-0.373319\pi$
$419$	6647.96			0.775117			0.387558	+	0.921845i	$0.373319\pi$
$420$		0			0
$421$	11670.6			1.35105			0.675524	−	0.737338i	$-0.263917\pi$
$421$	11670.6			1.35105			0.675524	+	0.737338i	$0.263917\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	4780.21	−	8279.57i	0.545586	−	0.944983i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−782.821	−	1355.89i	−0.0874876	−	0.151533i	0.818961	−	0.573849i	$-0.194550\pi$
$431$	−782.821	−	1355.89i	−0.0874876	−	0.151533i	−0.906449	+	0.422316i	$0.861217\pi$
$432$		0			0
$433$	−15446.4			−1.71434			−0.857168	−	0.515037i	$-0.827778\pi$
$433$	−15446.4			−1.71434			−0.857168	+	0.515037i	$0.827778\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	638.919	+	1106.64i	0.0699397	+	0.121139i
$438$		0			0
$439$	5348.70	−	9264.21i	0.581502	−	1.00719i	−0.413800	−	0.910368i	$-0.635799\pi$
$439$	5348.70	−	9264.21i	0.581502	−	1.00719i	0.995302	−	0.0968229i	$-0.0308681\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−526.981	+	912.758i	−0.0565183	+	0.0978926i	−0.892900	−	0.450254i	$-0.851333\pi$
$443$	−526.981	+	912.758i	−0.0565183	+	0.0978926i	0.836382	+	0.548147i	$0.184667\pi$
$444$		0			0
$445$	−502.749	−	870.787i	−0.0535564	−	0.0927624i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−1139.33			−0.119752			−0.0598759	−	0.998206i	$-0.519070\pi$
$449$	−1139.33			−0.119752			−0.0598759	+	0.998206i	$0.519070\pi$
$450$		0			0
$451$	1015.46	+	1758.82i	0.106022	+	0.183636i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−2852.08	+	4939.95i	−0.291936	+	0.505648i	−0.974267	−	0.225395i	$-0.927633\pi$
$457$	−2852.08	+	4939.95i	−0.291936	+	0.505648i	0.682332	+	0.731043i	$0.260966\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−6476.39			−0.654307			−0.327154	−	0.944971i	$-0.606089\pi$
$461$	−6476.39			−0.654307			−0.327154	+	0.944971i	$0.606089\pi$
$462$		0			0
$463$	−232.366			−0.0233239			−0.0116619	−	0.999932i	$-0.503712\pi$
$463$	−232.366			−0.0233239			−0.0116619	+	0.999932i	$0.503712\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	518.583	−	898.212i	0.0513858	−	0.0890028i	−0.839188	−	0.543841i	$-0.816970\pi$
$467$	518.583	−	898.212i	0.0513858	−	0.0890028i	0.890574	+	0.454838i	$0.150303\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$	8794.35	+	15232.3i	0.854893	+	1.48072i
$474$		0			0
$475$	1556.18			0.150321
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−5142.32	−	8906.77i	−0.490519	−	0.849605i	0.509421	−	0.860517i	$-0.329860\pi$
$479$	−5142.32	−	8906.77i	−0.490519	−	0.849605i	−0.999940	+	0.0109129i	$0.996526\pi$
$480$		0			0
$481$	−4997.11	+	8655.25i	−0.473698	+	0.820469i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−348.496	+	603.614i	−0.0326276	+	0.0565127i
$486$		0			0
$487$	−5422.92	−	9392.77i	−0.504591	−	0.873977i	−0.999986	−	0.00530928i	$-0.998310\pi$
$487$	−5422.92	−	9392.77i	−0.504591	−	0.873977i	0.495395	−	0.868668i	$-0.335023\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−8442.11			−0.775941			−0.387971	−	0.921672i	$-0.626824\pi$
$491$	−8442.11			−0.775941			−0.387971	+	0.921672i	$0.626824\pi$
$492$		0			0
$493$	8319.42	+	14409.7i	0.760016	+	1.31639i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	−4905.42	+	8496.45i	−0.440074	+	0.762231i	−0.997694	−	0.0678654i	$-0.978381\pi$
$499$	−4905.42	+	8496.45i	−0.440074	+	0.762231i	0.557620	+	0.830096i	$0.311714\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−6433.96			−0.570330			−0.285165	−	0.958478i	$-0.592048\pi$
$503$	−6433.96			−0.570330			−0.285165	+	0.958478i	$0.592048\pi$
$504$		0			0
$505$	1712.48			0.150900
$506$		0			0
$507$		0			0
$508$		0			0
$509$	10280.3	−	17806.0i	0.895220	−	1.55057i	0.0616885	−	0.998095i	$-0.480351\pi$
$509$	10280.3	−	17806.0i	0.895220	−	1.55057i	0.833532	−	0.552471i	$-0.186315\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$	928.135	+	1607.58i	0.0794146	+	0.137550i
$516$		0			0
$517$	16515.2			1.40491
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−9563.06	−	16563.7i	−0.804156	−	1.39284i	−0.916859	−	0.399211i	$-0.869284\pi$
$521$	−9563.06	−	16563.7i	−0.804156	−	1.39284i	0.112703	−	0.993629i	$-0.464049\pi$
$522$		0			0
$523$	−1522.33	+	2636.75i	−0.127279	+	0.220454i	−0.922621	−	0.385707i	$-0.873958\pi$
$523$	−1522.33	+	2636.75i	−0.127279	+	0.220454i	0.795343	+	0.606160i	$0.207291\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	1636.42	−	2834.36i	0.135263	−	0.234282i
$528$		0			0
$529$	990.629	+	1715.82i	0.0814193	+	0.141022i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	1419.20			0.115333
$534$		0			0
$535$	−1166.64	−	2020.69i	−0.0942774	−	0.163293i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	2947.55	−	5105.30i	0.234242	−	0.405719i	−0.724810	−	0.688949i	$-0.758072\pi$
$541$	2947.55	−	5105.30i	0.234242	−	0.405719i	0.959052	+	0.283230i	$0.0914058\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	222.476			0.0174860
$546$		0			0
$547$	11151.1			0.871641			0.435821	−	0.900034i	$-0.356458\pi$
$547$	11151.1			0.871641			0.435821	+	0.900034i	$0.356458\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−1354.18	+	2345.51i	−0.104701	+	0.181347i
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$	872.173	+	1510.65i	0.0663468	+	0.114916i	0.897291	−	0.441440i	$-0.145532\pi$
$557$	872.173	+	1510.65i	0.0663468	+	0.114916i	−0.830944	+	0.556356i	$0.812199\pi$
$558$		0			0
$559$	12291.0			0.929969
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−5762.19	−	9980.41i	−0.431345	−	0.747112i	0.565644	−	0.824650i	$-0.308628\pi$
$563$	−5762.19	−	9980.41i	−0.431345	−	0.747112i	−0.996989	+	0.0775374i	$0.975294\pi$
$564$		0			0
$565$	−1143.30	+	1980.25i	−0.0851309	+	0.147451i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	1411.38	−	2444.58i	0.103986	−	0.180110i	−0.809337	−	0.587344i	$-0.800174\pi$
$569$	1411.38	−	2444.58i	0.103986	−	0.180110i	0.913324	+	0.407235i	$0.133507\pi$
$570$		0			0
$571$	2583.82	+	4475.31i	0.189369	+	0.327996i	0.945040	−	0.326955i	$-0.106023\pi$
$571$	2583.82	+	4475.31i	0.189369	+	0.327996i	−0.755671	+	0.654951i	$0.772689\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−12404.5			−0.899656
$576$		0			0
$577$	−7357.61	−	12743.7i	−0.530851	−	0.919461i	−0.999352	−	0.0359981i	$-0.988539\pi$
$577$	−7357.61	−	12743.7i	−0.530851	−	0.919461i	0.468501	−	0.883463i	$-0.344794\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	−4266.06	+	7389.03i	−0.303057	+	0.524910i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	9981.64			0.701851			0.350925	−	0.936403i	$-0.385867\pi$
$587$	9981.64			0.701851			0.350925	+	0.936403i	$0.385867\pi$
$588$		0			0
$589$	532.733			0.0372680
$590$		0			0
$591$		0			0
$592$		0			0
$593$	837.605	−	1450.78i	0.0580039	−	0.100466i	−0.835565	−	0.549391i	$-0.814860\pi$
$593$	837.605	−	1450.78i	0.0580039	−	0.100466i	0.893569	+	0.448925i	$0.148193\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	519.955	+	900.588i	0.0354671	+	0.0614308i	0.883214	−	0.468970i	$-0.155375\pi$
$599$	519.955	+	900.588i	0.0354671	+	0.0614308i	−0.847747	+	0.530401i	$0.822041\pi$
$600$		0			0
$601$	−4472.61			−0.303563			−0.151782	−	0.988414i	$-0.548501\pi$
$601$	−4472.61			−0.303563			−0.151782	+	0.988414i	$0.548501\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	575.873	+	997.441i	0.0386984	+	0.0670277i
$606$		0			0
$607$	−8895.38	+	15407.3i	−0.594814	+	1.03025i	0.398759	+	0.917056i	$0.369441\pi$
$607$	−8895.38	+	15407.3i	−0.594814	+	1.03025i	−0.993573	+	0.113193i	$0.963892\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	5770.40	−	9994.63i	0.382071	−	0.661766i
$612$		0			0
$613$	2163.37	+	3747.06i	0.142541	+	0.246888i	0.928453	−	0.371450i	$-0.121139\pi$
$613$	2163.37	+	3747.06i	0.142541	+	0.246888i	−0.785912	+	0.618339i	$0.787806\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	18866.2			1.23100			0.615498	−	0.788139i	$-0.288955\pi$
$617$	18866.2			1.23100			0.615498	+	0.788139i	$0.288955\pi$
$618$		0			0
$619$	−5089.69	−	8815.60i	−0.330488	−	0.572422i	0.652120	−	0.758116i	$-0.273880\pi$
$619$	−5089.69	−	8815.60i	−0.330488	−	0.572422i	−0.982608	+	0.185694i	$0.940547\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−7422.52	+	12856.2i	−0.475041	+	0.822796i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	24119.5			1.52895
$630$		0			0
$631$	−1661.72			−0.104837			−0.0524184	−	0.998625i	$-0.516693\pi$
$631$	−1661.72			−0.104837			−0.0524184	+	0.998625i	$0.516693\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	1385.70	−	2400.10i	0.0865980	−	0.149992i
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−1138.82	−	1972.50i	−0.0701728	−	0.121543i	0.828804	−	0.559539i	$-0.189022\pi$
$641$	−1138.82	−	1972.50i	−0.0701728	−	0.121543i	−0.898977	+	0.437996i	$0.855688\pi$
$642$		0			0
$643$	−1217.38			−0.0746638			−0.0373319	−	0.999303i	$-0.511886\pi$
$643$	−1217.38			−0.0746638			−0.0373319	+	0.999303i	$0.511886\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−5365.90	−	9294.01i	−0.326052	−	0.564738i	0.655673	−	0.755045i	$-0.272385\pi$
$647$	−5365.90	−	9294.01i	−0.326052	−	0.564738i	−0.981725	+	0.190307i	$0.939052\pi$
$648$		0			0
$649$	10485.0	−	18160.6i	0.634166	−	1.09841i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	7614.03	−	13187.9i	0.456294	−	0.790324i	−0.542468	−	0.840077i	$-0.682510\pi$
$653$	7614.03	−	13187.9i	0.456294	−	0.790324i	0.998762	+	0.0497526i	$0.0158433\pi$
$654$		0			0
$655$	1808.13	+	3131.77i	0.107862	+	0.186822i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−10590.8			−0.626038			−0.313019	−	0.949747i	$-0.601340\pi$
$659$	−10590.8			−0.626038			−0.313019	+	0.949747i	$0.601340\pi$
$660$		0			0
$661$	−1934.15	−	3350.04i	−0.113812	−	0.197128i	0.803492	−	0.595315i	$-0.202973\pi$
$661$	−1934.15	−	3350.04i	−0.113812	−	0.197128i	−0.917304	+	0.398187i	$0.869639\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	10794.3	−	18696.3i	0.626622	−	1.08534i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	546.512			0.0314424
$672$		0			0
$673$	11028.7			0.631687			0.315843	−	0.948811i	$-0.397713\pi$
$673$	11028.7			0.631687			0.315843	+	0.948811i	$0.397713\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−9396.24	+	16274.8i	−0.533422	+	0.923914i	0.465816	+	0.884882i	$0.345761\pi$
$677$	−9396.24	+	16274.8i	−0.533422	+	0.923914i	−0.999238	+	0.0390325i	$0.987572\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−10121.9	−	17531.6i	−0.567062	−	0.982181i	−0.996855	−	0.0792531i	$-0.974746\pi$
$683$	−10121.9	−	17531.6i	−0.567062	−	0.982181i	0.429792	−	0.902928i	$-0.358587\pi$
$684$		0			0
$685$	420.695			0.0234656
$686$		0			0
$687$		0			0
$688$		0			0
$689$	2981.12	+	5163.45i	0.164835	+	0.285503i
$690$		0			0
$691$	−9601.45	+	16630.2i	−0.528591	+	0.915546i	0.470854	+	0.882211i	$0.343946\pi$
$691$	−9601.45	+	16630.2i	−0.528591	+	0.915546i	−0.999444	+	0.0333346i	$0.989387\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	975.308	−	1689.28i	0.0532310	−	0.0921988i
$696$		0			0
$697$	−1712.51	−	2966.16i	−0.0930646	−	0.161193i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	22156.9			1.19380			0.596900	−	0.802316i	$-0.296399\pi$
$701$	22156.9			1.19380			0.596900	+	0.802316i	$0.296399\pi$
$702$		0			0
$703$	1963.01	+	3400.04i	0.105315	+	0.182411i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	13604.5	−	23563.8i	0.720634	−	1.24817i	−0.240112	−	0.970745i	$-0.577184\pi$
$709$	13604.5	−	23563.8i	0.720634	−	1.24817i	0.960746	−	0.277429i	$-0.0894823\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−4246.45			−0.223045
$714$		0			0
$715$	2149.97			0.112454
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−10776.2	+	18664.9i	−0.558947	+	0.968125i	0.438638	+	0.898664i	$0.355461\pi$
$719$	−10776.2	+	18664.9i	−0.558947	+	0.968125i	−0.997585	+	0.0694605i	$0.977872\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−13145.6	−	22768.8i	−0.673399	−	1.16636i
$726$		0			0
$727$	20599.1			1.05086			0.525431	−	0.850836i	$-0.323904\pi$
$727$	20599.1			1.05086			0.525431	+	0.850836i	$0.323904\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−14831.2	−	25688.4i	−0.750412	−	1.29975i
$732$		0			0
$733$	9880.58	−	17113.7i	0.497882	−	0.862357i	−0.502115	−	0.864801i	$-0.667445\pi$
$733$	9880.58	−	17113.7i	0.497882	−	0.862357i	0.999997	+	0.00244415i	$0.000777998\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	13614.7	−	23581.4i	0.680468	−	1.17861i
$738$		0			0
$739$	−16704.4	−	28933.0i	−0.831506	−	1.44021i	−0.896844	−	0.442348i	$-0.854146\pi$
$739$	−16704.4	−	28933.0i	−0.831506	−	1.44021i	0.0653376	−	0.997863i	$-0.479188\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	36225.4			1.78867			0.894335	−	0.447398i	$-0.147649\pi$
$743$	36225.4			1.78867			0.894335	+	0.447398i	$0.147649\pi$
$744$		0			0
$745$	2060.00	+	3568.03i	0.101306	+	0.175466i
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	13660.2	−	23660.2i	0.663740	−	1.14963i	−0.315885	−	0.948797i	$-0.602302\pi$
$751$	13660.2	−	23660.2i	0.663740	−	1.14963i	0.979625	−	0.200834i	$-0.0643651\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	2153.88			0.103825
$756$		0			0
$757$	−9918.43			−0.476211			−0.238105	−	0.971239i	$-0.576526\pi$
$757$	−9918.43			−0.476211			−0.238105	+	0.971239i	$0.576526\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	2817.52	−	4880.08i	0.134211	−	0.232461i	−0.791084	−	0.611707i	$-0.790483\pi$
$761$	2817.52	−	4880.08i	0.134211	−	0.232461i	0.925296	+	0.379246i	$0.123817\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−7326.93	−	12690.6i	−0.344929	−	0.597434i
$768$		0			0
$769$	−12089.3			−0.566907			−0.283453	−	0.958986i	$-0.591480\pi$
$769$	−12089.3			−0.566907			−0.283453	+	0.958986i	$0.591480\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	7085.68	+	12272.8i	0.329695	+	0.571049i	0.982451	−	0.186519i	$-0.0597207\pi$
$773$	7085.68	+	12272.8i	0.329695	+	0.571049i	−0.652756	+	0.757568i	$0.726387\pi$
$774$		0			0
$775$	−2585.72	+	4478.60i	−0.119848	+	0.207582i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	278.752	−	482.812i	0.0128207	−	0.0222061i
$780$		0			0
$781$	−11401.0	−	19747.0i	−0.522354	−	0.904744i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	5269.14			0.239571
$786$		0			0
$787$	−12853.8	−	22263.5i	−0.582197	−	1.00840i	−0.995218	−	0.0976740i	$-0.968860\pi$
$787$	−12853.8	−	22263.5i	−0.582197	−	1.00840i	0.413021	−	0.910721i	$-0.364474\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	190.951	−	330.737i	0.00855092	−	0.0148106i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−6194.68			−0.275316			−0.137658	−	0.990480i	$-0.543957\pi$
$797$	−6194.68			−0.275316			−0.137658	+	0.990480i	$0.543957\pi$
$798$		0			0
$799$	−27852.0			−1.23321
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−22500.2	+	38971.5i	−0.988811	+	1.71267i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	15407.5	+	26686.7i	0.669593	+	1.15977i	0.978018	+	0.208520i	$0.0668646\pi$
$809$	15407.5	+	26686.7i	0.669593	+	1.15977i	−0.308425	+	0.951249i	$0.599802\pi$
$810$		0			0
$811$	−43024.1			−1.86286			−0.931431	−	0.363917i	$-0.881439\pi$
$811$	−43024.1			−1.86286			−0.931431	+	0.363917i	$0.881439\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	556.190	+	963.349i	0.0239049	+	0.0414045i
$816$		0			0
$817$	2414.13	−	4181.39i	0.103378	−	0.179056i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−5576.12	+	9658.12i	−0.237038	+	0.410561i	−0.959863	−	0.280470i	$-0.909510\pi$
$821$	−5576.12	+	9658.12i	−0.237038	+	0.410561i	0.722825	+	0.691031i	$0.242843\pi$
$822$		0			0
$823$	22456.4	+	38895.6i	0.951130	+	1.64741i	0.742985	+	0.669308i	$0.233409\pi$
$823$	22456.4	+	38895.6i	0.951130	+	1.64741i	0.208145	+	0.978098i	$0.433257\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	3213.42			0.135117			0.0675584	−	0.997715i	$-0.478479\pi$
$827$	3213.42			0.135117			0.0675584	+	0.997715i	$0.478479\pi$
$828$		0			0
$829$	−4397.45	−	7616.61i	−0.184234	−	0.319102i	0.759084	−	0.650992i	$-0.225647\pi$
$829$	−4397.45	−	7616.61i	−0.184234	−	0.319102i	−0.943318	+	0.331890i	$0.892314\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$	−1735.31	+	3005.65i	−0.0719197	+	0.124568i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	45817.0			1.88532			0.942658	−	0.333761i	$-0.108318\pi$
$839$	45817.0			1.88532			0.942658	+	0.333761i	$0.108318\pi$
$840$		0			0
$841$	21367.8			0.876125
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−837.475	+	1450.55i	−0.0340947	+	0.0590537i
$846$		0			0
$847$		0			0
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−15647.3	−	27102.0i	−0.630298	−	1.09171i
$852$		0			0
$853$	16373.6			0.657234			0.328617	−	0.944463i	$-0.393417\pi$
$853$	16373.6			0.657234			0.328617	+	0.944463i	$0.393417\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	2215.10	+	3836.67i	0.0882922	+	0.152927i	0.906789	−	0.421584i	$-0.138526\pi$
$857$	2215.10	+	3836.67i	0.0882922	+	0.152927i	−0.818497	+	0.574511i	$0.805192\pi$
$858$		0			0
$859$	2472.19	−	4281.97i	0.0981958	−	0.170080i	−0.812742	−	0.582624i	$-0.802026\pi$
$859$	2472.19	−	4281.97i	0.0981958	−	0.170080i	0.910938	+	0.412544i	$0.135360\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	6589.31	−	11413.0i	0.259911	−	0.450178i	−0.706307	−	0.707905i	$-0.749640\pi$
$863$	6589.31	−	11413.0i	0.259911	−	0.450178i	0.966218	+	0.257727i	$0.0829736\pi$
$864$		0			0
$865$	−2412.77	−	4179.04i	−0.0948399	−	0.164268i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−13806.4			−0.538951
$870$		0			0
$871$	−9513.96	−	16478.7i	−0.370113	−	0.641054i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	6492.95	−	11246.1i	0.250001	−	0.433015i	−0.713524	−	0.700630i	$-0.752902\pi$
$877$	6492.95	−	11246.1i	0.250001	−	0.433015i	0.963526	+	0.267615i	$0.0862356\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	36877.4			1.41025			0.705126	−	0.709082i	$-0.250890\pi$
$881$	36877.4			1.41025			0.705126	+	0.709082i	$0.250890\pi$
$882$		0			0
$883$	−24874.5			−0.948012			−0.474006	−	0.880522i	$-0.657192\pi$
$883$	−24874.5			−0.948012			−0.474006	+	0.880522i	$0.657192\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−2205.32	+	3819.72i	−0.0834806	+	0.144593i	−0.904743	−	0.425958i	$-0.859937\pi$
$887$	−2205.32	+	3819.72i	−0.0834806	+	0.144593i	0.821262	+	0.570551i	$0.193270\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−2266.78	−	3926.18i	−0.0849440	−	0.147127i
$894$		0			0
$895$	3560.27			0.132968
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−4500.16	−	7794.50i	−0.166951	−	0.289167i
$900$		0			0
$901$	7194.48	−	12461.2i	0.266019	−	0.460758i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−952.149	+	1649.17i	−0.0349729	+	0.0605749i
$906$		0			0
$907$	3637.00	+	6299.46i	0.133147	+	0.230618i	0.924888	−	0.380239i	$-0.124158\pi$
$907$	3637.00	+	6299.46i	0.133147	+	0.230618i	−0.791741	+	0.610857i	$0.790825\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	49491.9			1.79993			0.899967	−	0.435957i	$-0.143590\pi$
$911$	49491.9			1.79993			0.899967	+	0.435957i	$0.143590\pi$
$912$		0			0
$913$	−32426.4	−	56164.2i	−1.17542	−	2.03589i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	2507.95	−	4343.89i	0.0900213	−	0.155922i	−0.817499	−	0.575931i	$-0.804640\pi$
$919$	2507.95	−	4343.89i	0.0900213	−	0.155922i	0.907520	+	0.420009i	$0.137973\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−15934.0			−0.568227
$924$		0			0
$925$	−38111.4			−1.35470
$926$		0			0
$927$		0			0
$928$		0			0
$929$	26079.1	−	45170.4i	0.921021	−	1.59525i	0.123181	−	0.992384i	$-0.460690\pi$
$929$	26079.1	−	45170.4i	0.921021	−	1.59525i	0.797839	−	0.602870i	$-0.205976\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−2594.32	−	4493.49i	−0.0907414	−	0.157169i
$936$		0			0
$937$	14821.9			0.516766			0.258383	−	0.966042i	$-0.416810\pi$
$937$	14821.9			0.516766			0.258383	+	0.966042i	$0.416810\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−13246.1	−	22943.0i	−0.458886	−	0.794814i	0.540016	−	0.841655i	$-0.318418\pi$
$941$	−13246.1	−	22943.0i	−0.458886	−	0.794814i	−0.998902	+	0.0468405i	$0.985085\pi$
$942$		0			0
$943$	−2221.95	+	3848.53i	−0.0767304	+	0.132901i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	3205.12	−	5551.44i	0.109982	−	0.190494i	−0.805781	−	0.592214i	$-0.798254\pi$
$947$	3205.12	−	5551.44i	0.109982	−	0.190494i	0.915763	+	0.401720i	$0.131587\pi$
$948$		0			0
$949$	15723.1	+	27233.3i	0.537824	+	0.931538i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	25108.5			0.853458			0.426729	−	0.904380i	$-0.359666\pi$
$953$	25108.5			0.853458			0.426729	+	0.904380i	$0.359666\pi$
$954$		0			0
$955$	2363.10	+	4093.02i	0.0800715	+	0.138688i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	14010.3	−	24266.6i	0.470287	−	0.814561i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−337.941			−0.0112733
$966$		0			0
$967$	32928.1			1.09503			0.547516	−	0.836795i	$-0.315574\pi$
$967$	32928.1			1.09503			0.547516	+	0.836795i	$0.315574\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	15771.6	−	27317.2i	0.521251	−	0.902834i	−0.478443	−	0.878119i	$-0.658799\pi$
$971$	15771.6	−	27317.2i	0.521251	−	0.902834i	0.999695	−	0.0247155i	$-0.00786798\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	26940.9	+	46663.0i	0.882206	+	1.52803i	0.848883	+	0.528581i	$0.177276\pi$
$977$	26940.9	+	46663.0i	0.882206	+	1.52803i	0.0333230	+	0.999445i	$0.489391\pi$
$978$		0			0
$979$	32067.8			1.04688
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−17844.3	−	30907.3i	−0.578988	−	1.00284i	−0.995596	−	0.0937503i	$-0.970114\pi$
$983$	−17844.3	−	30907.3i	−0.578988	−	1.00284i	0.416608	−	0.909086i	$-0.363219\pi$
$984$		0			0
$985$	23.0406	−	39.9075i	0.000745314	−	0.00129092i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−19243.2	+	33330.2i	−0.618704	+	1.07163i
$990$		0			0
$991$	−7419.86	−	12851.6i	−0.237840	−	0.411951i	0.722254	−	0.691628i	$-0.243106\pi$
$991$	−7419.86	−	12851.6i	−0.237840	−	0.411951i	−0.960094	+	0.279676i	$0.909773\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−2138.82			−0.0681459
$996$		0			0
$997$	8686.03	+	15044.7i	0.275917	+	0.477903i	0.970366	−	0.241639i	$-0.0776851\pi$
$997$	8686.03	+	15044.7i	0.275917	+	0.477903i	−0.694449	+	0.719542i	$0.744352\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1764.4.k.q.1549.2		4
3.2	odd	2		588.4.i.j.373.1		4
7.2	even	3		1764.4.a.y.1.1		2
7.3	odd	6		252.4.k.f.109.1		4
7.4	even	3	inner	1764.4.k.q.361.2		4
7.5	odd	6		1764.4.a.o.1.2		2
7.6	odd	2		252.4.k.f.37.1		4
21.2	odd	6		588.4.a.f.1.2		2
21.5	even	6		588.4.a.i.1.1		2
21.11	odd	6		588.4.i.j.361.1		4
21.17	even	6		84.4.i.a.25.2	✓	4
21.20	even	2		84.4.i.a.37.2	yes	4
84.23	even	6		2352.4.a.bx.1.2		2
84.47	odd	6		2352.4.a.bt.1.1		2
84.59	odd	6		336.4.q.i.193.2		4
84.83	odd	2		336.4.q.i.289.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
84.4.i.a.25.2	✓	4	21.17	even	6
84.4.i.a.37.2	yes	4	21.20	even	2
252.4.k.f.37.1		4	7.6	odd	2
252.4.k.f.109.1		4	7.3	odd	6
336.4.q.i.193.2		4	84.59	odd	6
336.4.q.i.289.2		4	84.83	odd	2
588.4.a.f.1.2		2	21.2	odd	6
588.4.a.i.1.1		2	21.5	even	6
588.4.i.j.361.1		4	21.11	odd	6
588.4.i.j.373.1		4	3.2	odd	2
1764.4.a.o.1.2		2	7.5	odd	6
1764.4.a.y.1.1		2	7.2	even	3
1764.4.k.q.361.2		4	7.4	even	3	inner
1764.4.k.q.1549.2		4	1.1	even	1	trivial
2352.4.a.bt.1.1		2	84.47	odd	6
2352.4.a.bx.1.2		2	84.23	even	6

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1764.k (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(0.723111 - 1.25246i) q^{5} +O(q^{10})\)	\(q+(0.723111 - 1.25246i) q^{5} +(23.0618 + 39.9442i) q^{11} +32.2311 q^{13} +(-38.8924 - 67.3637i) q^{17} +(6.33067 - 10.9650i) q^{19} +(-50.4622 + 87.4031i) q^{23} +(61.4542 + 106.442i) q^{25} -213.908 q^{29} +(21.0378 + 36.4385i) q^{31} +(-155.040 + 268.537i) q^{37} +44.0320 q^{41} +381.339 q^{43} +(179.032 - 310.093i) q^{47} +(92.4920 + 160.201i) q^{53} +66.7049 q^{55} +(-227.325 - 393.738i) q^{59} +(5.92444 - 10.2614i) q^{61} +(23.3067 - 40.3683i) q^{65} +(-295.180 - 511.266i) q^{67} -494.366 q^{71} +(487.825 + 844.937i) q^{73} +(-149.667 + 259.231i) q^{79} -1406.07 q^{83} -112.494 q^{85} +(347.629 - 602.112i) q^{89} +(-9.15555 - 15.8579i) q^{95} -481.940 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 11 q^{5}+O(q^{10}) \) 4 * q - 11 * q^5	\( 4 q - 11 q^{5} - 5 q^{11} - 10 q^{13} - 100 q^{17} + 67 q^{19} + 76 q^{23} + 93 q^{25} - 550 q^{29} + 362 q^{31} + 5 q^{37} - 324 q^{41} + 1442 q^{43} + 216 q^{47} + 495 q^{53} + 1406 q^{55} - 173 q^{59} - 532 q^{61} + 510 q^{65} - 111 q^{67} - 3200 q^{71} + 1215 q^{73} - 1460 q^{79} - 2818 q^{83} + 328 q^{85} + 1974 q^{89} + 658 q^{95} - 1122 q^{97}+O(q^{100}) \) 4 * q - 11 * q^5 - 5 * q^11 - 10 * q^13 - 100 * q^17 + 67 * q^19 + 76 * q^23 + 93 * q^25 - 550 * q^29 + 362 * q^31 + 5 * q^37 - 324 * q^41 + 1442 * q^43 + 216 * q^47 + 495 * q^53 + 1406 * q^55 - 173 * q^59 - 532 * q^61 + 510 * q^65 - 111 * q^67 - 3200 * q^71 + 1215 * q^73 - 1460 * q^79 - 2818 * q^83 + 328 * q^85 + 1974 * q^89 + 658 * q^95 - 1122 * q^97

Introduction

L-functions

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Varieties

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Database

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