LMFDB - Embedded newform 1620.3.b.b.809.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1620,3,Mod(809,1620)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1620, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 1]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("1620.809");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1620 = 2^{2} \cdot 3^{4} \cdot 5 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1620.b (of order $2$, degree $1$, 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:	$44.1418028264$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	no (minimal twist has level 180)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			809.3
Character	$\chi$	$=$	1620.809
Dual form			1620.3.b.b.809.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-4.80229 - 1.39212i) q^{5} -12.3801i q^{7} +O(q^{10})$	$q+(-4.80229 - 1.39212i) q^{5} -12.3801i q^{7} +4.62899i q^{11} +1.26353i q^{13} +22.9122 q^{17} +32.8729 q^{19} +0.797117 q^{23} +(21.1240 + 13.3707i) q^{25} +39.6411i q^{29} +1.46043 q^{31} +(-17.2345 + 59.4527i) q^{35} -44.9884i q^{37} -39.1346i q^{41} -36.3817i q^{43} +61.0408 q^{47} -104.266 q^{49} -13.9872 q^{53} +(6.44409 - 22.2297i) q^{55} -58.0138i q^{59} -55.0506 q^{61} +(1.75898 - 6.06783i) q^{65} -66.9919i q^{67} -70.6560i q^{71} -23.1498i q^{73} +57.3071 q^{77} -29.1451 q^{79} -130.433 q^{83} +(-110.031 - 31.8965i) q^{85} +48.0873i q^{89} +15.6425 q^{91} +(-157.865 - 45.7629i) q^{95} -19.3761i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q+O(q^{10}) $ 24 * q	$ 24 q - 6 q^{25} - 60 q^{31} - 216 q^{49} + 42 q^{55} - 96 q^{61} - 228 q^{79} - 96 q^{85} - 84 q^{91}+O(q^{100}) $ 24 * q - 6 * q^25 - 60 * q^31 - 216 * q^49 + 42 * q^55 - 96 * q^61 - 228 * q^79 - 96 * q^85 - 84 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1620\mathbb{Z}\right)^\times$.

$n$	$811$	$1297$	$1541$
$\chi(n)$	$1$	$-1$	$-1$

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
$5$	−4.80229	−	1.39212i	−0.960458	−	0.278424i
$5$	−4.80229	−	1.39212i	−0.960458	−	0.278424i
$6$		0			0
$7$		−	12.3801i		−	1.76858i	−0.466938	−	0.884290i	$-0.654643\pi$
$7$		−	12.3801i		−	1.76858i	0.466938	−	0.884290i	$-0.345357\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$			4.62899i			0.420817i	0.977614	+	0.210408i	$0.0674794\pi$
$11$			4.62899i			0.420817i	−0.977614	+	0.210408i	$0.932521\pi$
$12$		0			0
$13$			1.26353i			0.0971944i	0.998818	+	0.0485972i	$0.0154751\pi$
$13$			1.26353i			0.0971944i	−0.998818	+	0.0485972i	$0.984525\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	22.9122			1.34778			0.673889	−	0.738833i	$-0.264623\pi$
$17$	22.9122			1.34778			0.673889	+	0.738833i	$0.264623\pi$
$18$		0			0
$19$	32.8729			1.73015			0.865075	−	0.501642i	$-0.167270\pi$
$19$	32.8729			1.73015			0.865075	+	0.501642i	$0.167270\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	0.797117			0.0346573			0.0173286	−	0.999850i	$-0.494484\pi$
$23$	0.797117			0.0346573			0.0173286	+	0.999850i	$0.494484\pi$
$24$		0			0
$25$	21.1240	+	13.3707i	0.844961	+	0.534828i
$26$		0			0
$27$		0			0
$28$		0			0
$29$			39.6411i			1.36694i	0.729980	+	0.683468i	$0.239529\pi$
$29$			39.6411i			1.36694i	−0.729980	+	0.683468i	$0.760471\pi$
$30$		0			0
$31$	1.46043			0.0471107			0.0235553	−	0.999723i	$-0.492501\pi$
$31$	1.46043			0.0471107			0.0235553	+	0.999723i	$0.492501\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−17.2345	+	59.4527i	−0.492414	+	1.69865i
$36$		0			0
$37$		−	44.9884i		−	1.21590i	−0.793974	−	0.607952i	$-0.791991\pi$
$37$		−	44.9884i		−	1.21590i	0.793974	−	0.607952i	$-0.208009\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	39.1346i		−	0.954502i	−0.878767	−	0.477251i	$-0.841633\pi$
$41$		−	39.1346i		−	0.954502i	0.878767	−	0.477251i	$-0.158367\pi$
$42$		0			0
$43$		−	36.3817i		−	0.846085i	−0.906110	−	0.423043i	$-0.860962\pi$
$43$		−	36.3817i		−	0.846085i	0.906110	−	0.423043i	$-0.139038\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	61.0408			1.29874			0.649370	−	0.760473i	$-0.275033\pi$
$47$	61.0408			1.29874			0.649370	+	0.760473i	$0.275033\pi$
$48$		0			0
$49$	−104.266			−2.12788
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−13.9872			−0.263909			−0.131954	−	0.991256i	$-0.542125\pi$
$53$	−13.9872			−0.263909			−0.131954	+	0.991256i	$0.542125\pi$
$54$		0			0
$55$	6.44409	−	22.2297i	0.117165	−	0.404177i
$56$		0			0
$57$		0			0
$58$		0			0
$59$		−	58.0138i		−	0.983285i	−0.870797	−	0.491643i	$-0.836397\pi$
$59$		−	58.0138i		−	0.983285i	0.870797	−	0.491643i	$-0.163603\pi$
$60$		0			0
$61$	−55.0506			−0.902469			−0.451234	−	0.892405i	$-0.649016\pi$
$61$	−55.0506			−0.902469			−0.451234	+	0.892405i	$0.649016\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	1.75898	−	6.06783i	0.0270612	−	0.0933512i
$66$		0			0
$67$		−	66.9919i		−	0.999880i	−0.866060	−	0.499940i	$-0.833355\pi$
$67$		−	66.9919i		−	0.999880i	0.866060	−	0.499940i	$-0.166645\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	70.6560i		−	0.995156i	−0.867419	−	0.497578i	$-0.834223\pi$
$71$		−	70.6560i		−	0.995156i	0.867419	−	0.497578i	$-0.165777\pi$
$72$		0			0
$73$		−	23.1498i		−	0.317121i	−0.987349	−	0.158561i	$-0.949315\pi$
$73$		−	23.1498i		−	0.317121i	0.987349	−	0.158561i	$-0.0506853\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	57.3071			0.744248
$78$		0			0
$79$	−29.1451			−0.368925			−0.184462	−	0.982840i	$-0.559054\pi$
$79$	−29.1451			−0.368925			−0.184462	+	0.982840i	$0.559054\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−130.433			−1.57148			−0.785741	−	0.618556i	$-0.787718\pi$
$83$	−130.433			−1.57148			−0.785741	+	0.618556i	$0.787718\pi$
$84$		0			0
$85$	−110.031	−	31.8965i	−1.29448	−	0.375253i
$86$		0			0
$87$		0			0
$88$		0			0
$89$			48.0873i			0.540307i	0.962817	+	0.270153i	$0.0870744\pi$
$89$			48.0873i			0.540307i	−0.962817	+	0.270153i	$0.912926\pi$
$90$		0			0
$91$	15.6425			0.171896
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−157.865	−	45.7629i	−1.66174	−	0.481715i
$96$		0			0
$97$		−	19.3761i		−	0.199753i	−0.995000	−	0.0998766i	$-0.968155\pi$
$97$		−	19.3761i		−	0.199753i	0.995000	−	0.0998766i	$-0.0318448\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$			129.502i			1.28219i	0.767460	+	0.641097i	$0.221520\pi$
$101$			129.502i			1.28219i	−0.767460	+	0.641097i	$0.778480\pi$
$102$		0			0
$103$		−	8.40246i		−	0.0815772i	−0.999168	−	0.0407886i	$-0.987013\pi$
$103$		−	8.40246i		−	0.0815772i	0.999168	−	0.0407886i	$-0.0129870\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	61.1389			0.571391			0.285696	−	0.958320i	$-0.407775\pi$
$107$	61.1389			0.571391			0.285696	+	0.958320i	$0.407775\pi$
$108$		0			0
$109$	−161.771			−1.48414			−0.742068	−	0.670324i	$-0.766155\pi$
$109$	−161.771			−1.48414			−0.742068	+	0.670324i	$0.766155\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	34.2475			0.303075			0.151538	−	0.988451i	$-0.451578\pi$
$113$	34.2475			0.303075			0.151538	+	0.988451i	$0.451578\pi$
$114$		0			0
$115$	−3.82799	−	1.10968i	−0.0332869	−	0.00964940i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		−	283.655i		−	2.38365i
$120$		0			0
$121$	99.5725			0.822913
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−82.8301	−	93.6172i	−0.662641	−	0.748937i
$126$		0			0
$127$		−	104.320i		−	0.821421i	−0.911766	−	0.410710i	$-0.865281\pi$
$127$		−	104.320i		−	0.821421i	0.911766	−	0.410710i	$-0.134719\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$			219.634i			1.67660i	0.545211	+	0.838299i	$0.316449\pi$
$131$			219.634i			1.67660i	−0.545211	+	0.838299i	$0.683551\pi$
$132$		0			0
$133$		−	406.968i		−	3.05991i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	189.475			1.38303			0.691514	−	0.722363i	$-0.256944\pi$
$137$	189.475			1.38303			0.691514	+	0.722363i	$0.256944\pi$
$138$		0			0
$139$	220.040			1.58302			0.791511	−	0.611155i	$-0.209295\pi$
$139$	220.040			1.58302			0.791511	+	0.611155i	$0.209295\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−5.84885			−0.0409010
$144$		0			0
$145$	55.1851	−	190.368i	0.380587	−	1.31289i
$146$		0			0
$147$		0			0
$148$		0			0
$149$		−	10.3969i		−	0.0697782i	−0.999391	−	0.0348891i	$-0.988892\pi$
$149$		−	10.3969i		−	0.0697782i	0.999391	−	0.0348891i	$-0.0111078\pi$
$150$		0			0
$151$	−81.7621			−0.541471			−0.270736	−	0.962654i	$-0.587267\pi$
$151$	−81.7621			−0.541471			−0.270736	+	0.962654i	$0.587267\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−7.01341	−	2.03309i	−0.0452478	−	0.0131167i
$156$		0			0
$157$		−	132.312i		−	0.842749i	−0.906887	−	0.421375i	$-0.861548\pi$
$157$		−	132.312i		−	0.842749i	0.906887	−	0.421375i	$-0.138452\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		−	9.86836i		−	0.0612942i
$162$		0			0
$163$			81.7721i			0.501669i	0.968030	+	0.250835i	$0.0807051\pi$
$163$			81.7721i			0.501669i	−0.968030	+	0.250835i	$0.919295\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	27.7023			0.165882			0.0829411	−	0.996554i	$-0.473569\pi$
$167$	27.7023			0.165882			0.0829411	+	0.996554i	$0.473569\pi$
$168$		0			0
$169$	167.403			0.990553
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−278.615			−1.61049			−0.805246	−	0.592940i	$-0.797967\pi$
$173$	−278.615			−1.61049			−0.805246	+	0.592940i	$0.797967\pi$
$174$		0			0
$175$	165.530	−	261.517i	0.945887	−	1.49438i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		−	175.928i		−	0.982839i	−0.870923	−	0.491419i	$-0.836478\pi$
$179$		−	175.928i		−	0.982839i	0.870923	−	0.491419i	$-0.163522\pi$
$180$		0			0
$181$	−180.887			−0.999377			−0.499689	−	0.866205i	$-0.666552\pi$
$181$	−180.887			−0.999377			−0.499689	+	0.866205i	$0.666552\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−62.6292	+	216.048i	−0.338536	+	1.16782i
$186$		0			0
$187$			106.060i			0.567167i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		−	138.769i		−	0.726540i	−0.931684	−	0.363270i	$-0.881660\pi$
$191$		−	138.769i		−	0.726540i	0.931684	−	0.363270i	$-0.118340\pi$
$192$		0			0
$193$		−	178.975i		−	0.927332i	−0.886010	−	0.463666i	$-0.846534\pi$
$193$		−	178.975i		−	0.927332i	0.886010	−	0.463666i	$-0.153466\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	136.877			0.694809			0.347405	−	0.937715i	$-0.387063\pi$
$197$	136.877			0.694809			0.347405	+	0.937715i	$0.387063\pi$
$198$		0			0
$199$	11.4634			0.0576050			0.0288025	−	0.999585i	$-0.490831\pi$
$199$	11.4634			0.0576050			0.0288025	+	0.999585i	$0.490831\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	490.760			2.41754
$204$		0			0
$205$	−54.4799	+	187.936i	−0.265756	+	0.916760i
$206$		0			0
$207$		0			0
$208$		0			0
$209$			152.168i			0.728077i
$210$		0			0
$211$	−77.4456			−0.367041			−0.183520	−	0.983016i	$-0.558749\pi$
$211$	−77.4456			−0.367041			−0.183520	+	0.983016i	$0.558749\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−50.6476	+	174.715i	−0.235570	+	0.812630i
$216$		0			0
$217$		−	18.0802i		−	0.0833190i
$218$		0			0
$219$		0			0
$220$		0			0
$221$			28.9502i			0.130996i
$222$		0			0
$223$		−	144.235i		−	0.646794i	−0.946263	−	0.323397i	$-0.895175\pi$
$223$		−	144.235i		−	0.646794i	0.946263	−	0.323397i	$-0.104825\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−49.1034			−0.216314			−0.108157	−	0.994134i	$-0.534495\pi$
$227$	−49.1034			−0.216314			−0.108157	+	0.994134i	$0.534495\pi$
$228$		0			0
$229$	−100.980			−0.440962			−0.220481	−	0.975391i	$-0.570763\pi$
$229$	−100.980			−0.440962			−0.220481	+	0.975391i	$0.570763\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−120.466			−0.517021			−0.258510	−	0.966009i	$-0.583232\pi$
$233$	−120.466			−0.517021			−0.258510	+	0.966009i	$0.583232\pi$
$234$		0			0
$235$	−293.136	−	84.9759i	−1.24739	−	0.361600i
$236$		0			0
$237$		0			0
$238$		0			0
$239$			181.396i			0.758978i	0.925196	+	0.379489i	$0.123900\pi$
$239$			181.396i			0.758978i	−0.925196	+	0.379489i	$0.876100\pi$
$240$		0			0
$241$	162.308			0.673475			0.336738	−	0.941599i	$-0.390676\pi$
$241$	162.308			0.673475			0.336738	+	0.941599i	$0.390676\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	500.716	+	145.150i	2.04374	+	0.592451i
$246$		0			0
$247$			41.5358i			0.168161i
$248$		0			0
$249$		0			0
$250$		0			0
$251$			445.457i			1.77473i	0.461069	+	0.887364i	$0.347466\pi$
$251$			445.457i			1.77473i	−0.461069	+	0.887364i	$0.652534\pi$
$252$		0			0
$253$			3.68984i			0.0145844i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−247.417			−0.962713			−0.481357	−	0.876525i	$-0.659856\pi$
$257$	−247.417			−0.962713			−0.481357	+	0.876525i	$0.659856\pi$
$258$		0			0
$259$	−556.960			−2.15042
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−86.0242			−0.327088			−0.163544	−	0.986536i	$-0.552293\pi$
$263$	−86.0242			−0.327088			−0.163544	+	0.986536i	$0.552293\pi$
$264$		0			0
$265$	67.1704	+	19.4718i	0.253473	+	0.0734784i
$266$		0			0
$267$		0			0
$268$		0			0
$269$		−	368.618i		−	1.37033i	−0.728389	−	0.685164i	$-0.759731\pi$
$269$		−	368.618i		−	1.37033i	0.728389	−	0.685164i	$-0.240269\pi$
$270$		0			0
$271$	−130.740			−0.482434			−0.241217	−	0.970471i	$-0.577546\pi$
$271$	−130.740			−0.482434			−0.241217	+	0.970471i	$0.577546\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−61.8928	+	97.7828i	−0.225065	+	0.355574i
$276$		0			0
$277$		−	217.270i		−	0.784370i	−0.919886	−	0.392185i	$-0.871719\pi$
$277$		−	217.270i		−	0.784370i	0.919886	−	0.392185i	$-0.128281\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	172.327i		−	0.613264i	−0.951828	−	0.306632i	$-0.900798\pi$
$281$		−	172.327i		−	0.613264i	0.951828	−	0.306632i	$-0.0992021\pi$
$282$		0			0
$283$			436.158i			1.54119i	0.637324	+	0.770596i	$0.280041\pi$
$283$			436.158i			1.54119i	−0.637324	+	0.770596i	$0.719959\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−484.489			−1.68811
$288$		0			0
$289$	235.970			0.816504
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−29.0406			−0.0991148			−0.0495574	−	0.998771i	$-0.515781\pi$
$293$	−29.0406			−0.0991148			−0.0495574	+	0.998771i	$0.515781\pi$
$294$		0			0
$295$	−80.7621	+	278.599i	−0.273770	+	0.944405i
$296$		0			0
$297$		0			0
$298$		0			0
$299$			1.00718i			0.00336849i
$300$		0			0
$301$	−450.407			−1.49637
$302$		0			0
$303$		0			0
$304$		0			0
$305$	264.369	+	76.6369i	0.866784	+	0.251269i
$306$		0			0
$307$		−	306.163i		−	0.997273i	−0.866811	−	0.498637i	$-0.833834\pi$
$307$		−	306.163i		−	0.997273i	0.866811	−	0.498637i	$-0.166166\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		−	397.221i		−	1.27724i	−0.769524	−	0.638618i	$-0.779506\pi$
$311$		−	397.221i		−	1.27724i	0.769524	−	0.638618i	$-0.220494\pi$
$312$		0			0
$313$		−	391.762i		−	1.25164i	−0.779969	−	0.625818i	$-0.784765\pi$
$313$		−	391.762i		−	1.25164i	0.779969	−	0.625818i	$-0.215235\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−161.788			−0.510372			−0.255186	−	0.966892i	$-0.582137\pi$
$317$	−161.788			−0.510372			−0.255186	+	0.966892i	$0.582137\pi$
$318$		0			0
$319$	−183.498			−0.575230
$320$		0			0
$321$		0			0
$322$		0			0
$323$	753.190			2.33186
$324$		0			0
$325$	−16.8943	+	26.6908i	−0.0519823	+	0.0821255i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		−	755.689i		−	2.29693i
$330$		0			0
$331$	237.189			0.716582			0.358291	−	0.933610i	$-0.383360\pi$
$331$	237.189			0.716582			0.358291	+	0.933610i	$0.383360\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−93.2607	+	321.715i	−0.278390	+	0.960343i
$336$		0			0
$337$		−	422.049i		−	1.25237i	−0.779674	−	0.626185i	$-0.784615\pi$
$337$		−	422.049i		−	1.25237i	0.779674	−	0.626185i	$-0.215385\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$			6.76031i			0.0198250i
$342$		0			0
$343$			684.196i			1.99474i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−552.495			−1.59220			−0.796102	−	0.605163i	$-0.793108\pi$
$347$	−552.495			−1.59220			−0.796102	+	0.605163i	$0.793108\pi$
$348$		0			0
$349$	397.177			1.13804			0.569022	−	0.822322i	$-0.307322\pi$
$349$	397.177			1.13804			0.569022	+	0.822322i	$0.307322\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−308.675			−0.874433			−0.437216	−	0.899356i	$-0.644036\pi$
$353$	−308.675			−0.874433			−0.437216	+	0.899356i	$0.644036\pi$
$354$		0			0
$355$	−98.3615	+	339.311i	−0.277075	+	0.955806i
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	278.225i		−	0.775000i	−0.921870	−	0.387500i	$-0.873339\pi$
$359$		−	278.225i		−	0.775000i	0.921870	−	0.387500i	$-0.126661\pi$
$360$		0			0
$361$	719.626			1.99342
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−32.2273	+	111.172i	−0.0882940	+	0.304582i
$366$		0			0
$367$			608.243i			1.65734i	0.559738	+	0.828669i	$0.310902\pi$
$367$			608.243i			1.65734i	−0.559738	+	0.828669i	$0.689098\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$			173.162i			0.466744i
$372$		0			0
$373$			358.878i			0.962140i	0.876682	+	0.481070i	$0.159752\pi$
$373$			358.878i			0.962140i	−0.876682	+	0.481070i	$0.840248\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−50.0877			−0.132859
$378$		0			0
$379$	353.208			0.931948			0.465974	−	0.884798i	$-0.345704\pi$
$379$	353.208			0.931948			0.465974	+	0.884798i	$0.345704\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	271.388			0.708586			0.354293	−	0.935135i	$-0.384722\pi$
$383$	271.388			0.708586			0.354293	+	0.935135i	$0.384722\pi$
$384$		0			0
$385$	−275.206	−	79.7783i	−0.714820	−	0.207216i
$386$		0			0
$387$		0			0
$388$		0			0
$389$		−	262.697i		−	0.675314i	−0.941269	−	0.337657i	$-0.890366\pi$
$389$		−	262.697i		−	0.675314i	0.941269	−	0.337657i	$-0.109634\pi$
$390$		0			0
$391$	18.2637			0.0467103
$392$		0			0
$393$		0			0
$394$		0			0
$395$	139.963	+	40.5733i	0.354337	+	0.102717i
$396$		0			0
$397$			53.4516i			0.134639i	0.997731	+	0.0673194i	$0.0214447\pi$
$397$			53.4516i			0.134639i	−0.997731	+	0.0673194i	$0.978555\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		−	184.917i		−	0.461139i	−0.973056	−	0.230570i	$-0.925941\pi$
$401$		−	184.917i		−	0.461139i	0.973056	−	0.230570i	$-0.0740590\pi$
$402$		0			0
$403$			1.84529i			0.00457889i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	208.251			0.511673
$408$		0			0
$409$	232.105			0.567493			0.283746	−	0.958899i	$-0.408423\pi$
$409$	232.105			0.567493			0.283746	+	0.958899i	$0.408423\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−718.215			−1.73902
$414$		0			0
$415$	626.377	+	181.578i	1.50934	+	0.437537i
$416$		0			0
$417$		0			0
$418$		0			0
$419$		−	388.092i		−	0.926235i	−0.886297	−	0.463117i	$-0.846731\pi$
$419$		−	388.092i		−	0.926235i	0.886297	−	0.463117i	$-0.153269\pi$
$420$		0			0
$421$	154.003			0.365803			0.182902	−	0.983131i	$-0.441451\pi$
$421$	154.003			0.365803			0.182902	+	0.983131i	$0.441451\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	483.998	+	306.353i	1.13882	+	0.720830i
$426$		0			0
$427$			681.530i			1.59609i
$428$		0			0
$429$		0			0
$430$		0			0
$431$			238.147i			0.552544i	0.961079	+	0.276272i	$0.0890991\pi$
$431$			238.147i			0.552544i	−0.961079	+	0.276272i	$0.910901\pi$
$432$		0			0
$433$		−	523.956i		−	1.21006i	−0.796203	−	0.605030i	$-0.793161\pi$
$433$		−	523.956i		−	1.21006i	0.796203	−	0.605030i	$-0.206839\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	26.2035			0.0599623
$438$		0			0
$439$	456.345			1.03951			0.519755	−	0.854316i	$-0.326023\pi$
$439$	456.345			1.03951			0.519755	+	0.854316i	$0.326023\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	271.780			0.613499			0.306749	−	0.951790i	$-0.400759\pi$
$443$	271.780			0.613499			0.306749	+	0.951790i	$0.400759\pi$
$444$		0			0
$445$	66.9432	−	230.929i	0.150434	−	0.518942i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			386.192i			0.860116i	0.902801	+	0.430058i	$0.141507\pi$
$449$			386.192i			0.860116i	−0.902801	+	0.430058i	$0.858493\pi$
$450$		0			0
$451$	181.153			0.401671
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−75.1201	−	21.7763i	−0.165099	−	0.0478599i
$456$		0			0
$457$			4.48832i			0.00982126i	0.999988	+	0.00491063i	$0.00156311\pi$
$457$			4.48832i			0.00982126i	−0.999988	+	0.00491063i	$0.998437\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$			327.360i			0.710108i	0.934846	+	0.355054i	$0.115538\pi$
$461$			327.360i			0.710108i	−0.934846	+	0.355054i	$0.884462\pi$
$462$		0			0
$463$		−	212.141i		−	0.458188i	−0.973404	−	0.229094i	$-0.926424\pi$
$463$		−	212.141i		−	0.458188i	0.973404	−	0.229094i	$-0.0735763\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−471.246			−1.00909			−0.504546	−	0.863385i	$-0.668340\pi$
$467$	−471.246			−1.00909			−0.504546	+	0.863385i	$0.668340\pi$
$468$		0			0
$469$	−829.364			−1.76837
$470$		0			0
$471$		0			0
$472$		0			0
$473$	168.410			0.356047
$474$		0			0
$475$	694.407	+	439.534i	1.46191	+	0.925334i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		−	880.071i		−	1.83731i	−0.395062	−	0.918654i	$-0.629277\pi$
$479$		−	880.071i		−	1.83731i	0.395062	−	0.918654i	$-0.370723\pi$
$480$		0			0
$481$	56.8441			0.118179
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−26.9737	+	93.0495i	−0.0556160	+	0.191855i
$486$		0			0
$487$			606.421i			1.24522i	0.782533	+	0.622609i	$0.213927\pi$
$487$			606.421i			1.24522i	−0.782533	+	0.622609i	$0.786073\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$			563.512i			1.14768i	0.818967	+	0.573841i	$0.194547\pi$
$491$			563.512i			1.14768i	−0.818967	+	0.573841i	$0.805453\pi$
$492$		0			0
$493$			908.266i			1.84233i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−874.726			−1.76001
$498$		0			0
$499$	71.8813			0.144051			0.0720254	−	0.997403i	$-0.477054\pi$
$499$	71.8813			0.144051			0.0720254	+	0.997403i	$0.477054\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	124.538			0.247591			0.123795	−	0.992308i	$-0.460493\pi$
$503$	124.538			0.247591			0.123795	+	0.992308i	$0.460493\pi$
$504$		0			0
$505$	180.281	−	621.904i	0.356993	−	1.23149i
$506$		0			0
$507$		0			0
$508$		0			0
$509$		−	538.157i		−	1.05728i	−0.848845	−	0.528642i	$-0.822702\pi$
$509$		−	538.157i		−	1.05728i	0.848845	−	0.528642i	$-0.177298\pi$
$510$		0			0
$511$	−286.596			−0.560854
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−11.6972	+	40.3510i	−0.0227130	+	0.0783515i
$516$		0			0
$517$			282.557i			0.546532i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	147.767i		−	0.283622i	−0.989894	−	0.141811i	$-0.954707\pi$
$521$		−	147.767i		−	0.283622i	0.989894	−	0.141811i	$-0.0452925\pi$
$522$		0			0
$523$			701.166i			1.34066i	0.742063	+	0.670331i	$0.233848\pi$
$523$			701.166i			1.34066i	−0.742063	+	0.670331i	$0.766152\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	33.4617			0.0634947
$528$		0			0
$529$	−528.365			−0.998799
$530$		0			0
$531$		0			0
$532$		0			0
$533$	49.4476			0.0927723
$534$		0			0
$535$	−293.607	−	85.1125i	−0.548798	−	0.159089i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		−	482.646i		−	0.895446i
$540$		0			0
$541$	−573.411			−1.05991			−0.529955	−	0.848026i	$-0.677791\pi$
$541$	−573.411			−1.05991			−0.529955	+	0.848026i	$0.677791\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	776.871	+	225.204i	1.42545	+	0.413218i
$546$		0			0
$547$			305.020i			0.557624i	0.960346	+	0.278812i	$0.0899406\pi$
$547$			305.020i			0.557624i	−0.960346	+	0.278812i	$0.910059\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$			1303.12i			2.36501i
$552$		0			0
$553$			360.818i			0.652473i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	686.875			1.23317			0.616585	−	0.787288i	$-0.288516\pi$
$557$	686.875			1.23317			0.616585	+	0.787288i	$0.288516\pi$
$558$		0			0
$559$	45.9692			0.0822348
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−465.062			−0.826043			−0.413021	−	0.910721i	$-0.635527\pi$
$563$	−465.062			−0.826043			−0.413021	+	0.910721i	$0.635527\pi$
$564$		0			0
$565$	−164.467	−	47.6765i	−0.291091	−	0.0843833i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	1101.68i		−	1.93618i	−0.250611	−	0.968088i	$-0.580631\pi$
$569$		−	1101.68i		−	1.93618i	0.250611	−	0.968088i	$-0.419369\pi$
$570$		0			0
$571$	762.054			1.33460			0.667298	−	0.744791i	$-0.267451\pi$
$571$	762.054			1.33460			0.667298	+	0.744791i	$0.267451\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	16.8383	+	10.6580i	0.0292840	+	0.0185357i
$576$		0			0
$577$			673.918i			1.16797i	0.811765	+	0.583985i	$0.198507\pi$
$577$			673.918i			1.16797i	−0.811765	+	0.583985i	$0.801493\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$			1614.77i			2.77929i
$582$		0			0
$583$		−	64.7464i		−	0.111057i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	389.360			0.663305			0.331653	−	0.943402i	$-0.392394\pi$
$587$	389.360			0.663305			0.331653	+	0.943402i	$0.392394\pi$
$588$		0			0
$589$	48.0085			0.0815085
$590$		0			0
$591$		0			0
$592$		0			0
$593$	640.797			1.08060			0.540301	−	0.841472i	$-0.318310\pi$
$593$	640.797			1.08060			0.540301	+	0.841472i	$0.318310\pi$
$594$		0			0
$595$	−394.881	+	1362.19i	−0.663665	+	2.28940i
$596$		0			0
$597$		0			0
$598$		0			0
$599$			466.425i			0.778673i	0.921096	+	0.389336i	$0.127296\pi$
$599$			466.425i			0.778673i	−0.921096	+	0.389336i	$0.872704\pi$
$600$		0			0
$601$	−986.430			−1.64131			−0.820657	−	0.571421i	$-0.806392\pi$
$601$	−986.430			−1.64131			−0.820657	+	0.571421i	$0.806392\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−478.176	−	138.617i	−0.790374	−	0.229118i
$606$		0			0
$607$			869.964i			1.43322i	0.697474	+	0.716610i	$0.254307\pi$
$607$			869.964i			1.43322i	−0.697474	+	0.716610i	$0.745693\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			77.1267i			0.126230i
$612$		0			0
$613$			376.400i			0.614029i	0.951705	+	0.307014i	$0.0993300\pi$
$613$			376.400i			0.614029i	−0.951705	+	0.307014i	$0.900670\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−116.125			−0.188208			−0.0941041	−	0.995562i	$-0.529999\pi$
$617$	−116.125			−0.188208			−0.0941041	+	0.995562i	$0.529999\pi$
$618$		0			0
$619$	239.530			0.386963			0.193482	−	0.981104i	$-0.438022\pi$
$619$	239.530			0.386963			0.193482	+	0.981104i	$0.438022\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	595.324			0.955576
$624$		0			0
$625$	267.448	+	564.886i	0.427917	+	0.903818i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		−	1030.78i		−	1.63877i
$630$		0			0
$631$	−590.863			−0.936392			−0.468196	−	0.883625i	$-0.655096\pi$
$631$	−590.863			−0.936392			−0.468196	+	0.883625i	$0.655096\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−145.226	+	500.977i	−0.228703	+	0.788941i
$636$		0			0
$637$		−	131.743i		−	0.206818i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		−	766.027i		−	1.19505i	−0.801850	−	0.597525i	$-0.796151\pi$
$641$		−	766.027i		−	1.19505i	0.801850	−	0.597525i	$-0.203849\pi$
$642$		0			0
$643$			76.0526i			0.118278i	0.998250	+	0.0591389i	$0.0188355\pi$
$643$			76.0526i			0.118278i	−0.998250	+	0.0591389i	$0.981165\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	287.130			0.443786			0.221893	−	0.975071i	$-0.428776\pi$
$647$	287.130			0.443786			0.221893	+	0.975071i	$0.428776\pi$
$648$		0			0
$649$	268.545			0.413783
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−224.432			−0.343694			−0.171847	−	0.985124i	$-0.554973\pi$
$653$	−224.432			−0.343694			−0.171847	+	0.985124i	$0.554973\pi$
$654$		0			0
$655$	305.757	−	1054.75i	0.466804	−	1.61030i
$656$		0			0
$657$		0			0
$658$		0			0
$659$			1088.80i			1.65220i	0.563520	+	0.826102i	$0.309447\pi$
$659$			1088.80i			1.65220i	−0.563520	+	0.826102i	$0.690553\pi$
$660$		0			0
$661$	878.752			1.32943			0.664714	−	0.747098i	$-0.268553\pi$
$661$	878.752			1.32943			0.664714	+	0.747098i	$0.268553\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−566.548	+	1954.38i	−0.851951	+	2.93892i
$666$		0			0
$667$			31.5986i			0.0473743i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		−	254.828i		−	0.379774i
$672$		0			0
$673$			556.285i			0.826575i	0.910601	+	0.413287i	$0.135619\pi$
$673$			556.285i			0.826575i	−0.910601	+	0.413287i	$0.864381\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	562.016			0.830157			0.415078	−	0.909786i	$-0.363754\pi$
$677$	562.016			0.830157			0.415078	+	0.909786i	$0.363754\pi$
$678$		0			0
$679$	−239.877			−0.353280
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−500.931			−0.733427			−0.366714	−	0.930334i	$-0.619517\pi$
$683$	−500.931			−0.733427			−0.366714	+	0.930334i	$0.619517\pi$
$684$		0			0
$685$	−909.913	−	263.771i	−1.32834	−	0.385067i
$686$		0			0
$687$		0			0
$688$		0			0
$689$		−	17.6732i		−	0.0256504i
$690$		0			0
$691$	722.500			1.04559			0.522793	−	0.852460i	$-0.324890\pi$
$691$	722.500			1.04559			0.522793	+	0.852460i	$0.324890\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−1056.70	−	306.322i	−1.52043	−	0.440751i
$696$		0			0
$697$		−	896.660i		−	1.28646i
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	788.759i		−	1.12519i	−0.826732	−	0.562595i	$-0.809803\pi$
$701$		−	788.759i		−	1.12519i	0.826732	−	0.562595i	$-0.190197\pi$
$702$		0			0
$703$		−	1478.90i		−	2.10370i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	1603.24			2.26766
$708$		0			0
$709$	−399.286			−0.563168			−0.281584	−	0.959537i	$-0.590860\pi$
$709$	−399.286			−0.563168			−0.281584	+	0.959537i	$0.590860\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	1.16413			0.00163273
$714$		0			0
$715$	28.0879	+	8.14229i	0.0392838	+	0.0113878i
$716$		0			0
$717$		0			0
$718$		0			0
$719$		−	844.389i		−	1.17439i	−0.809444	−	0.587197i	$-0.800232\pi$
$719$		−	844.389i		−	1.17439i	0.809444	−	0.587197i	$-0.199768\pi$
$720$		0			0
$721$	−104.023			−0.144276
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−530.030	+	837.380i	−0.731076	+	1.15501i
$726$		0			0
$727$			405.715i			0.558067i	0.960281	+	0.279033i	$0.0900141\pi$
$727$			405.715i			0.558067i	−0.960281	+	0.279033i	$0.909986\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		−	833.585i		−	1.14033i
$732$		0			0
$733$		−	1204.99i		−	1.64392i	−0.569548	−	0.821958i	$-0.692882\pi$
$733$		−	1204.99i		−	1.64392i	0.569548	−	0.821958i	$-0.307118\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	310.105			0.420766
$738$		0			0
$739$	539.145			0.729561			0.364780	−	0.931094i	$-0.381144\pi$
$739$	539.145			0.729561			0.364780	+	0.931094i	$0.381144\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−311.004			−0.418579			−0.209289	−	0.977854i	$-0.567115\pi$
$743$	−311.004			−0.418579			−0.209289	+	0.977854i	$0.567115\pi$
$744$		0			0
$745$	−14.4738	+	49.9292i	−0.0194279	+	0.0670190i
$746$		0			0
$747$		0			0
$748$		0			0
$749$		−	756.903i		−	1.01055i
$750$		0			0
$751$	926.577			1.23379			0.616895	−	0.787045i	$-0.288390\pi$
$751$	926.577			1.23379			0.616895	+	0.787045i	$0.288390\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	392.646	+	113.822i	0.520060	+	0.150758i
$756$		0			0
$757$			450.364i			0.594933i	0.954732	+	0.297467i	$0.0961417\pi$
$757$			450.364i			0.594933i	−0.954732	+	0.297467i	$0.903858\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	671.249i		−	0.882062i	−0.897492	−	0.441031i	$-0.854613\pi$
$761$		−	671.249i		−	0.882062i	0.897492	−	0.441031i	$-0.145387\pi$
$762$		0			0
$763$			2002.73i			2.62481i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	73.3021			0.0955698
$768$		0			0
$769$	−74.7039			−0.0971443			−0.0485721	−	0.998820i	$-0.515467\pi$
$769$	−74.7039			−0.0971443			−0.0485721	+	0.998820i	$0.515467\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−1226.34			−1.58647			−0.793234	−	0.608917i	$-0.791604\pi$
$773$	−1226.34			−1.58647			−0.793234	+	0.608917i	$0.791604\pi$
$774$		0			0
$775$	30.8502	+	19.5270i	0.0398067	+	0.0251961i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		−	1286.47i		−	1.65143i
$780$		0			0
$781$	327.066			0.418778
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−184.193	+	635.399i	−0.234641	+	0.809426i
$786$		0			0
$787$			259.191i			0.329341i	0.986349	+	0.164670i	$0.0526561\pi$
$787$			259.191i			0.329341i	−0.986349	+	0.164670i	$0.947344\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	423.986i		−	0.536013i
$792$		0			0
$793$		−	69.5579i		−	0.0877149i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	1485.11			1.86337			0.931686	−	0.363265i	$-0.118338\pi$
$797$	1485.11			1.86337			0.931686	+	0.363265i	$0.118338\pi$
$798$		0			0
$799$	1398.58			1.75041
$800$		0			0
$801$		0			0
$802$		0			0
$803$	107.160			0.133450
$804$		0			0
$805$	−13.7379	+	47.3908i	−0.0170657	+	0.0588705i
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	10.2026i		−	0.0126114i	−0.999980	−	0.00630569i	$-0.997993\pi$
$809$		−	10.2026i		−	0.0126114i	0.999980	−	0.00630569i	$-0.00200718\pi$
$810$		0			0
$811$	1113.80			1.37337			0.686685	−	0.726955i	$-0.259065\pi$
$811$	1113.80			1.37337			0.686685	+	0.726955i	$0.259065\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	113.836	−	392.694i	0.139677	−	0.481833i
$816$		0			0
$817$		−	1195.97i		−	1.46386i
$818$		0			0
$819$		0			0
$820$		0			0
$821$			307.485i			0.374525i	0.982310	+	0.187262i	$0.0599614\pi$
$821$			307.485i			0.374525i	−0.982310	+	0.187262i	$0.940039\pi$
$822$		0			0
$823$			595.499i			0.723570i	0.932261	+	0.361785i	$0.117833\pi$
$823$			595.499i			0.723570i	−0.932261	+	0.361785i	$0.882167\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−1110.81			−1.34318			−0.671592	−	0.740921i	$-0.734389\pi$
$827$	−1110.81			−1.34318			−0.671592	+	0.740921i	$0.734389\pi$
$828$		0			0
$829$	390.454			0.470994			0.235497	−	0.971875i	$-0.424328\pi$
$829$	390.454			0.470994			0.235497	+	0.971875i	$0.424328\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−2388.96			−2.86790
$834$		0			0
$835$	−133.035	−	38.5649i	−0.159323	−	0.0461855i
$836$		0			0
$837$		0			0
$838$		0			0
$839$		−	1072.65i		−	1.27849i	−0.769003	−	0.639246i	$-0.779247\pi$
$839$		−	1072.65i		−	1.27849i	0.769003	−	0.639246i	$-0.220753\pi$
$840$		0			0
$841$	−730.420			−0.868514
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−803.920	−	233.045i	−0.951385	−	0.275793i
$846$		0			0
$847$		−	1232.71i		−	1.45539i
$848$		0			0
$849$		0			0
$850$		0			0
$851$		−	35.8611i		−	0.0421399i
$852$		0			0
$853$		−	909.447i		−	1.06617i	−0.846060	−	0.533087i	$-0.821032\pi$
$853$		−	909.447i		−	1.06617i	0.846060	−	0.533087i	$-0.178968\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−651.464			−0.760169			−0.380084	−	0.924952i	$-0.624105\pi$
$857$	−651.464			−0.760169			−0.380084	+	0.924952i	$0.624105\pi$
$858$		0			0
$859$	460.478			0.536063			0.268032	−	0.963410i	$-0.413627\pi$
$859$	460.478			0.536063			0.268032	+	0.963410i	$0.413627\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	359.492			0.416560			0.208280	−	0.978069i	$-0.433213\pi$
$863$	359.492			0.416560			0.208280	+	0.978069i	$0.433213\pi$
$864$		0			0
$865$	1337.99	+	387.865i	1.54681	+	0.448399i
$866$		0			0
$867$		0			0
$868$		0			0
$869$		−	134.912i		−	0.155250i
$870$		0			0
$871$	84.6462			0.0971827
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−1158.99	+	1025.44i	−1.32456	+	1.17193i
$876$		0			0
$877$		−	1139.04i		−	1.29879i	−0.760451	−	0.649395i	$-0.775022\pi$
$877$		−	1139.04i		−	1.29879i	0.760451	−	0.649395i	$-0.224978\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			1612.41i			1.83020i	0.403227	+	0.915100i	$0.367888\pi$
$881$			1612.41i			1.83020i	−0.403227	+	0.915100i	$0.632112\pi$
$882$		0			0
$883$			688.542i			0.779775i	0.920862	+	0.389888i	$0.127486\pi$
$883$			688.542i			0.779775i	−0.920862	+	0.389888i	$0.872514\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−1125.59			−1.26899			−0.634494	−	0.772927i	$-0.718792\pi$
$887$	−1125.59			−1.26899			−0.634494	+	0.772927i	$0.718792\pi$
$888$		0			0
$889$	−1291.49			−1.45275
$890$		0			0
$891$		0			0
$892$		0			0
$893$	2006.59			2.24702
$894$		0			0
$895$	−244.913	+	844.858i	−0.273645	+	0.943976i
$896$		0			0
$897$		0			0
$898$		0			0
$899$			57.8931i			0.0643972i
$900$		0			0
$901$	−320.477			−0.355690
$902$		0			0
$903$		0			0
$904$		0			0
$905$	868.674	+	251.816i	0.959860	+	0.278250i
$906$		0			0
$907$		−	623.987i		−	0.687968i	−0.938976	−	0.343984i	$-0.888223\pi$
$907$		−	623.987i		−	0.687968i	0.938976	−	0.343984i	$-0.111777\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			1525.40i			1.67442i	0.546879	+	0.837212i	$0.315816\pi$
$911$			1525.40i			1.67442i	−0.546879	+	0.837212i	$0.684184\pi$
$912$		0			0
$913$		−	603.772i		−	0.661306i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	2719.09			2.96520
$918$		0			0
$919$	−875.498			−0.952663			−0.476332	−	0.879266i	$-0.658034\pi$
$919$	−875.498			−0.952663			−0.476332	+	0.879266i	$0.658034\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	89.2759			0.0967236
$924$		0			0
$925$	601.527	−	950.337i	0.650300	−	1.02739i
$926$		0			0
$927$		0			0
$928$		0			0
$929$		−	394.172i		−	0.424297i	−0.977237	−	0.212148i	$-0.931954\pi$
$929$		−	394.172i		−	0.424297i	0.977237	−	0.212148i	$-0.0680460\pi$
$930$		0			0
$931$	−3427.52			−3.68155
$932$		0			0
$933$		0			0
$934$		0			0
$935$	147.648	−	509.333i	0.157913	−	0.544741i
$936$		0			0
$937$		−	600.533i		−	0.640911i	−0.947264	−	0.320455i	$-0.896164\pi$
$937$		−	600.533i		−	0.640911i	0.947264	−	0.320455i	$-0.103836\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		−	660.729i		−	0.702156i	−0.936346	−	0.351078i	$-0.885815\pi$
$941$		−	660.729i		−	0.702156i	0.936346	−	0.351078i	$-0.114185\pi$
$942$		0			0
$943$		−	31.1949i		−	0.0330804i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	1265.59			1.33642			0.668210	−	0.743973i	$-0.267061\pi$
$947$	1265.59			1.33642			0.668210	+	0.743973i	$0.267061\pi$
$948$		0			0
$949$	29.2505			0.0308224
$950$		0			0
$951$		0			0
$952$		0			0
$953$	428.958			0.450113			0.225057	−	0.974346i	$-0.427743\pi$
$953$	428.958			0.450113			0.225057	+	0.974346i	$0.427743\pi$
$954$		0			0
$955$	−193.183	+	666.410i	−0.202286	+	0.697812i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		−	2345.71i		−	2.44599i
$960$		0			0
$961$	−958.867			−0.997781
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−249.154	+	859.491i	−0.258191	+	0.890664i
$966$		0			0
$967$			443.284i			0.458412i	0.973378	+	0.229206i	$0.0736129\pi$
$967$			443.284i			0.458412i	−0.973378	+	0.229206i	$0.926387\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$			444.042i			0.457304i	0.973508	+	0.228652i	$0.0734317\pi$
$971$			444.042i			0.457304i	−0.973508	+	0.228652i	$0.926568\pi$
$972$		0			0
$973$		−	2724.11i		−	2.79970i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−230.950			−0.236387			−0.118194	−	0.992991i	$-0.537710\pi$
$977$	−230.950			−0.236387			−0.118194	+	0.992991i	$0.537710\pi$
$978$		0			0
$979$	−222.595			−0.227370
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−1121.24			−1.14063			−0.570316	−	0.821426i	$-0.693179\pi$
$983$	−1121.24			−1.14063			−0.570316	+	0.821426i	$0.693179\pi$
$984$		0			0
$985$	−657.325	−	190.549i	−0.667335	−	0.193451i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		−	29.0005i		−	0.0293230i
$990$		0			0
$991$	182.717			0.184377			0.0921884	−	0.995742i	$-0.470614\pi$
$991$	182.717			0.184377			0.0921884	+	0.995742i	$0.470614\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−55.0506	−	15.9584i	−0.0553272	−	0.0160386i
$996$		0			0
$997$		−	1233.24i		−	1.23695i	−0.785806	−	0.618473i	$-0.787751\pi$
$997$		−	1233.24i		−	1.23695i	0.785806	−	0.618473i	$-0.212249\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1620.3.b.b.809.3		24
3.2	odd	2	inner	1620.3.b.b.809.22		24
5.4	even	2	inner	1620.3.b.b.809.21		24
9.2	odd	6		180.3.t.a.149.7	yes	24
9.4	even	3		180.3.t.a.29.6	✓	24
9.5	odd	6		540.3.t.a.89.5		24
9.7	even	3		540.3.t.a.449.10		24
15.14	odd	2	inner	1620.3.b.b.809.4		24
45.2	even	12		900.3.p.f.401.1		24
45.4	even	6		180.3.t.a.29.7	yes	24
45.7	odd	12		2700.3.p.f.2501.1		24
45.13	odd	12		900.3.p.f.101.12		24
45.14	odd	6		540.3.t.a.89.10		24
45.22	odd	12		900.3.p.f.101.1		24
45.23	even	12		2700.3.p.f.1601.12		24
45.29	odd	6		180.3.t.a.149.6	yes	24
45.32	even	12		2700.3.p.f.1601.1		24
45.34	even	6		540.3.t.a.449.5		24
45.38	even	12		900.3.p.f.401.12		24
45.43	odd	12		2700.3.p.f.2501.12		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
180.3.t.a.29.6	✓	24	9.4	even	3
180.3.t.a.29.7	yes	24	45.4	even	6
180.3.t.a.149.6	yes	24	45.29	odd	6
180.3.t.a.149.7	yes	24	9.2	odd	6
540.3.t.a.89.5		24	9.5	odd	6
540.3.t.a.89.10		24	45.14	odd	6
540.3.t.a.449.5		24	45.34	even	6
540.3.t.a.449.10		24	9.7	even	3
900.3.p.f.101.1		24	45.22	odd	12
900.3.p.f.101.12		24	45.13	odd	12
900.3.p.f.401.1		24	45.2	even	12
900.3.p.f.401.12		24	45.38	even	12
1620.3.b.b.809.3		24	1.1	even	1	trivial
1620.3.b.b.809.4		24	15.14	odd	2	inner
1620.3.b.b.809.21		24	5.4	even	2	inner
1620.3.b.b.809.22		24	3.2	odd	2	inner
2700.3.p.f.1601.1		24	45.32	even	12
2700.3.p.f.1601.12		24	45.23	even	12
2700.3.p.f.2501.1		24	45.7	odd	12
2700.3.p.f.2501.12		24	45.43	odd	12

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:	\( N \)	\(=\)	\( 1620 = 2^{2} \cdot 3^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1620.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-4.80229 - 1.39212i) q^{5} -12.3801i q^{7} +O(q^{10})\)	\(q+(-4.80229 - 1.39212i) q^{5} -12.3801i q^{7} +4.62899i q^{11} +1.26353i q^{13} +22.9122 q^{17} +32.8729 q^{19} +0.797117 q^{23} +(21.1240 + 13.3707i) q^{25} +39.6411i q^{29} +1.46043 q^{31} +(-17.2345 + 59.4527i) q^{35} -44.9884i q^{37} -39.1346i q^{41} -36.3817i q^{43} +61.0408 q^{47} -104.266 q^{49} -13.9872 q^{53} +(6.44409 - 22.2297i) q^{55} -58.0138i q^{59} -55.0506 q^{61} +(1.75898 - 6.06783i) q^{65} -66.9919i q^{67} -70.6560i q^{71} -23.1498i q^{73} +57.3071 q^{77} -29.1451 q^{79} -130.433 q^{83} +(-110.031 - 31.8965i) q^{85} +48.0873i q^{89} +15.6425 q^{91} +(-157.865 - 45.7629i) q^{95} -19.3761i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q+O(q^{10}) \) 24 * q	\( 24 q - 6 q^{25} - 60 q^{31} - 216 q^{49} + 42 q^{55} - 96 q^{61} - 228 q^{79} - 96 q^{85} - 84 q^{91}+O(q^{100}) \) 24 * q - 6 * q^25 - 60 * q^31 - 216 * q^49 + 42 * q^55 - 96 * q^61 - 228 * q^79 - 96 * q^85 - 84 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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