LMFDB - Embedded newform 252.6.k.d.37.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [252,6,Mod(37,252)] 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("252.37"); S:= CuspForms(chi, 6); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(252, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 2])) N = Newforms(chi, 6, names="a")

Level:	$ N $	$=$	$ 252 = 2^{2} \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	252.k (of order $3$, degree $2$, minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,42] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	no
Analytic conductor:	$40.4167225929$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{3})$
Coefficient field:	$\Q(\sqrt{-3}, \sqrt{109})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} + 28x^{2} + 27x + 729 $ x^4 - x^3 + 28x^2 + 27x + 729
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{2}\cdot 3 $
Twist minimal:	no (minimal twist has level 28)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			37.1
Root			$-2.36008 + 4.08777i$ of defining polynomial
Character	$\chi$	$=$	252.37
Dual form			252.6.k.d.109.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(-20.8209 + 36.0629i) q^{5} +(28.0000 - 126.582i) q^{7} +(55.3768 + 95.9154i) q^{11} +179.135 q^{13} +(-177.784 - 307.930i) q^{17} +(-977.981 + 1693.91i) q^{19} +(773.363 - 1339.50i) q^{23} +(695.479 + 1204.60i) q^{25} -6273.94 q^{29} +(-3002.09 - 5199.77i) q^{31} +(3981.93 + 3645.31i) q^{35} +(4844.23 - 8390.44i) q^{37} -10577.3 q^{41} +6716.00 q^{43} +(13620.4 - 23591.2i) q^{47} +(-15239.0 - 7088.59i) q^{49} +(-16339.7 - 28301.2i) q^{53} -4611.98 q^{55} +(-246.332 - 426.659i) q^{59} +(-20276.1 + 35119.3i) q^{61} +(-3729.75 + 6460.12i) q^{65} +(3844.07 + 6658.13i) q^{67} -77879.3 q^{71} +(-36958.0 - 64013.1i) q^{73} +(13691.7 - 4324.07i) q^{77} +(21966.6 - 38047.2i) q^{79} -41194.2 q^{83} +14806.5 q^{85} +(32862.3 - 56919.1i) q^{89} +(5015.77 - 22675.2i) q^{91} +(-40724.9 - 70537.7i) q^{95} -68534.5 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 42 q^{5} + 112 q^{7} + 660 q^{11} - 1288 q^{13} - 210 q^{17} - 3724 q^{19} + 24 q^{23} - 2480 q^{25} - 11064 q^{29} - 2800 q^{31} + 28644 q^{35} + 13238 q^{37} - 8232 q^{41} + 26864 q^{43} + 8064 q^{47}+ \cdots + 16520 q^{97}+O(q^{100}) $ 4 * q + 42 * q^5 + 112 * q^7 + 660 * q^11 - 1288 * q^13 - 210 * q^17 - 3724 * q^19 + 24 * q^23 - 2480 * q^25 - 11064 * q^29 - 2800 * q^31 + 28644 * q^35 + 13238 * q^37 - 8232 * q^41 + 26864 * q^43 + 8064 * q^47 - 60956 * q^49 - 53958 * q^53 + 82656 * q^55 + 36036 * q^59 - 83986 * q^61 - 76308 * q^65 + 2660 * q^67 - 143136 * q^71 - 31318 * q^73 - 77658 * q^77 - 51136 * q^79 - 12432 * q^83 + 53964 * q^85 + 166278 * q^89 - 36064 * q^91 + 66432 * q^95 + 16520 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/252\mathbb{Z}\right)^\times$.

$n$	$29$	$73$	$127$
$\chi(n)$	$1$	$e\left(\frac{1}{3}\right)$	$1$

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
$5$	−20.8209	+	36.0629i	−0.372456	+	0.645113i	−0.989943	−	0.141469i	$-0.954818\pi$
$5$	−20.8209	+	36.0629i	−0.372456	+	0.645113i	0.617487	+	0.786581i	$0.288151\pi$
$6$		0			0
$7$	28.0000	−	126.582i	0.215980	−	0.976398i
$7$	28.0000	−	126.582i	0.215980	−	0.976398i
$8$		0			0
$9$		0			0
$10$		0			0
$11$	55.3768	+	95.9154i	0.137989	+	0.239005i	0.926735	−	0.375714i	$-0.122603\pi$
$11$	55.3768	+	95.9154i	0.137989	+	0.239005i	−0.788746	+	0.614719i	$0.789269\pi$
$12$		0			0
$13$	179.135			0.293982			0.146991	−	0.989138i	$-0.453041\pi$
$13$	179.135			0.293982			0.146991	+	0.989138i	$0.453041\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−177.784	−	307.930i	−0.149200	−	0.258422i	0.781732	−	0.623615i	$-0.214337\pi$
$17$	−177.784	−	307.930i	−0.149200	−	0.258422i	−0.930932	+	0.365192i	$0.881003\pi$
$18$		0			0
$19$	−977.981	+	1693.91i	−0.621508	+	1.07648i	0.367697	+	0.929946i	$0.380146\pi$
$19$	−977.981	+	1693.91i	−0.621508	+	1.07648i	−0.989205	+	0.146538i	$0.953187\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	773.363	−	1339.50i	0.304834	−	0.527988i	−0.672390	−	0.740197i	$-0.734732\pi$
$23$	773.363	−	1339.50i	0.304834	−	0.527988i	0.977224	+	0.212209i	$0.0680657\pi$
$24$		0			0
$25$	695.479	+	1204.60i	0.222553	+	0.385473i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−6273.94			−1.38531			−0.692653	−	0.721271i	$-0.743558\pi$
$29$	−6273.94			−1.38531			−0.692653	+	0.721271i	$0.743558\pi$
$30$		0			0
$31$	−3002.09	−	5199.77i	−0.561073	−	0.971806i	−0.997403	−	0.0720199i	$-0.977055\pi$
$31$	−3002.09	−	5199.77i	−0.561073	−	0.971806i	0.436331	−	0.899786i	$-0.356278\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	3981.93	+	3645.31i	0.549444	+	0.502996i
$36$		0			0
$37$	4844.23	−	8390.44i	0.581728	−	1.00758i	−0.413547	−	0.910483i	$-0.635710\pi$
$37$	4844.23	−	8390.44i	0.581728	−	1.00758i	0.995275	−	0.0970996i	$-0.0309565\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−10577.3			−0.982686			−0.491343	−	0.870966i	$-0.663494\pi$
$41$	−10577.3			−0.982686			−0.491343	+	0.870966i	$0.663494\pi$
$42$		0			0
$43$	6716.00			0.553910			0.276955	−	0.960883i	$-0.410675\pi$
$43$	6716.00			0.553910			0.276955	+	0.960883i	$0.410675\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	13620.4	−	23591.2i	0.899384	−	1.55778i	0.0711012	−	0.997469i	$-0.477349\pi$
$47$	13620.4	−	23591.2i	0.899384	−	1.55778i	0.828283	−	0.560310i	$-0.189318\pi$
$48$		0			0
$49$	−15239.0	−	7088.59i	−0.906706	−	0.421764i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−16339.7	−	28301.2i	−0.799014	−	1.38393i	−0.920259	−	0.391311i	$-0.872022\pi$
$53$	−16339.7	−	28301.2i	−0.799014	−	1.38393i	0.121244	−	0.992623i	$-0.461311\pi$
$54$		0			0
$55$	−4611.98			−0.205580
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−246.332	−	426.659i	−0.00921277	−	0.0159570i	0.861382	−	0.507957i	$-0.169599\pi$
$59$	−246.332	−	426.659i	−0.00921277	−	0.0159570i	−0.870595	+	0.492000i	$0.836266\pi$
$60$		0			0
$61$	−20276.1	+	35119.3i	−0.697686	+	1.20843i	0.271580	+	0.962416i	$0.412454\pi$
$61$	−20276.1	+	35119.3i	−0.697686	+	1.20843i	−0.969267	+	0.246012i	$0.920880\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−3729.75	+	6460.12i	−0.109495	+	0.189652i
$66$		0			0
$67$	3844.07	+	6658.13i	0.104618	+	0.181203i	0.913582	−	0.406655i	$-0.133305\pi$
$67$	3844.07	+	6658.13i	0.104618	+	0.181203i	−0.808964	+	0.587858i	$0.799971\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−77879.3			−1.83348			−0.916740	−	0.399484i	$-0.869189\pi$
$71$	−77879.3			−1.83348			−0.916740	+	0.399484i	$0.869189\pi$
$72$		0			0
$73$	−36958.0	−	64013.1i	−0.811710	−	1.40592i	−0.911667	−	0.410931i	$-0.865204\pi$
$73$	−36958.0	−	64013.1i	−0.811710	−	1.40592i	0.0999567	−	0.994992i	$-0.468130\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	13691.7	−	4324.07i	0.263167	−	0.0831125i
$78$		0			0
$79$	21966.6	−	38047.2i	0.395999	−	0.685891i	−0.597229	−	0.802071i	$-0.703732\pi$
$79$	21966.6	−	38047.2i	0.395999	−	0.685891i	0.993228	+	0.116180i	$0.0370650\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−41194.2			−0.656359			−0.328179	−	0.944615i	$-0.606435\pi$
$83$	−41194.2			−0.656359			−0.328179	+	0.944615i	$0.606435\pi$
$84$		0			0
$85$	14806.5			0.222282
$86$		0			0
$87$		0			0
$88$		0			0
$89$	32862.3	−	56919.1i	0.439767	−	0.761699i	−0.557904	−	0.829905i	$-0.688394\pi$
$89$	32862.3	−	56919.1i	0.439767	−	0.761699i	0.997671	+	0.0682064i	$0.0217277\pi$
$90$		0			0
$91$	5015.77	−	22675.2i	0.0634942	−	0.287044i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−40724.9	−	70537.7i	−0.462969	−	0.801885i
$96$		0			0
$97$	−68534.5			−0.739571			−0.369786	−	0.929117i	$-0.620569\pi$
$97$	−68534.5			−0.739571			−0.369786	+	0.929117i	$0.620569\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−58679.2	−	101635.i	−0.572375	−	0.991383i	−0.996321	−	0.0856959i	$-0.972689\pi$
$101$	−58679.2	−	101635.i	−0.572375	−	0.991383i	0.423946	−	0.905688i	$-0.360645\pi$
$102$		0			0
$103$	5409.51	−	9369.55i	0.0502418	−	0.0870213i	−0.839811	−	0.542879i	$-0.817334\pi$
$103$	5409.51	−	9369.55i	0.0502418	−	0.0870213i	0.890053	+	0.455858i	$0.150667\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	55200.1	−	95609.3i	0.466101	−	0.807311i	−0.533149	−	0.846021i	$-0.678992\pi$
$107$	55200.1	−	95609.3i	0.466101	−	0.807311i	0.999250	+	0.0387102i	$0.0123249\pi$
$108$		0			0
$109$	−6130.59	−	10618.5i	−0.0494238	−	0.0856045i	0.840255	−	0.542191i	$-0.182405\pi$
$109$	−6130.59	−	10618.5i	−0.0494238	−	0.0856045i	−0.889679	+	0.456587i	$0.849072\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	122314.			0.901117			0.450558	−	0.892747i	$-0.351225\pi$
$113$	122314.			0.901117			0.450558	+	0.892747i	$0.351225\pi$
$114$		0			0
$115$	32204.2	+	55779.4i	0.227075	+	0.393305i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−43956.4	+	13882.2i	−0.284547	+	0.0898648i
$120$		0			0
$121$	74392.3	−	128851.i	0.461918	−	0.800065i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−188053.			−1.07648
$126$		0			0
$127$	−169073.			−0.930174			−0.465087	−	0.885265i	$-0.653977\pi$
$127$	−169073.			−0.930174			−0.465087	+	0.885265i	$0.653977\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−188725.	+	326882.i	−0.960842	+	1.66423i	−0.240449	+	0.970662i	$0.577295\pi$
$131$	−188725.	+	326882.i	−0.960842	+	1.66423i	−0.720393	+	0.693566i	$0.756039\pi$
$132$		0			0
$133$	187035.	+	171224.i	0.916843	+	0.839337i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	158315.	+	274209.i	0.720643	+	1.24819i	0.960742	+	0.277442i	$0.0894867\pi$
$137$	158315.	+	274209.i	0.720643	+	1.24819i	−0.240099	+	0.970748i	$0.577180\pi$
$138$		0			0
$139$	−229447.			−1.00727			−0.503634	−	0.863917i	$-0.668004\pi$
$139$	−229447.			−1.00727			−0.503634	+	0.863917i	$0.668004\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	9919.90	+	17181.8i	0.0405665	+	0.0702632i
$144$		0			0
$145$	130629.	−	226257.i	0.515965	−	0.893678i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	78750.8	−	136400.i	0.290596	−	0.503326i	−0.683355	−	0.730086i	$-0.739480\pi$
$149$	78750.8	−	136400.i	0.290596	−	0.503326i	0.973951	+	0.226760i	$0.0728132\pi$
$150$		0			0
$151$	73392.3	+	127119.i	0.261944	+	0.453700i	0.966758	−	0.255691i	$-0.0823031\pi$
$151$	73392.3	+	127119.i	0.261944	+	0.453700i	−0.704815	+	0.709392i	$0.748970\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	250025.			0.835899
$156$		0			0
$157$	163742.	+	283610.i	0.530166	+	0.918275i	0.999381	+	0.0351904i	$0.0112038\pi$
$157$	163742.	+	283610.i	0.530166	+	0.918275i	−0.469215	+	0.883084i	$0.655463\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−147903.	−	135400.i	−0.449689	−	0.411674i
$162$		0			0
$163$	−174093.	+	301538.i	−0.513231	+	0.888942i	0.486651	+	0.873596i	$0.338218\pi$
$163$	−174093.	+	301538.i	−0.513231	+	0.888942i	−0.999882	+	0.0153459i	$0.995115\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	547771.			1.51987			0.759937	−	0.649996i	$-0.225230\pi$
$167$	547771.			1.51987			0.759937	+	0.649996i	$0.225230\pi$
$168$		0			0
$169$	−339204.			−0.913574
$170$		0			0
$171$		0			0
$172$		0			0
$173$	91270.5	−	158085.i	0.231854	−	0.401583i	−0.726500	−	0.687167i	$-0.758854\pi$
$173$	91270.5	−	158085.i	0.231854	−	0.401583i	0.958354	+	0.285584i	$0.0921874\pi$
$174$		0			0
$175$	171955.	−	54306.1i	0.424442	−	0.134046i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	68061.5	+	117886.i	0.158770	+	0.274998i	0.934425	−	0.356159i	$-0.115914\pi$
$179$	68061.5	+	117886.i	0.158770	+	0.274998i	−0.775655	+	0.631157i	$0.782580\pi$
$180$		0			0
$181$	591381.			1.34175			0.670874	−	0.741571i	$-0.265919\pi$
$181$	591381.			1.34175			0.670874	+	0.741571i	$0.265919\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	201722.	+	349394.i	0.433336	+	0.750560i
$186$		0			0
$187$	19690.2	−	34104.4i	0.0411761	−	0.0713192i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−220596.	+	382083.i	−0.437536	+	0.757834i	−0.997499	−	0.0706837i	$-0.977482\pi$
$191$	−220596.	+	382083.i	−0.437536	+	0.757834i	0.559963	+	0.828517i	$0.310815\pi$
$192$		0			0
$193$	−224188.	−	388304.i	−0.433230	−	0.750376i	0.563920	−	0.825830i	$-0.309293\pi$
$193$	−224188.	−	388304.i	−0.433230	−	0.750376i	−0.997149	+	0.0754539i	$0.975959\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	542019.			0.995059			0.497530	−	0.867447i	$-0.334241\pi$
$197$	542019.			0.995059			0.497530	+	0.867447i	$0.334241\pi$
$198$		0			0
$199$	98979.3	+	171437.i	0.177179	+	0.306883i	0.940913	−	0.338648i	$-0.109970\pi$
$199$	98979.3	+	171437.i	0.177179	+	0.306883i	−0.763734	+	0.645531i	$0.776636\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−175670.	+	794168.i	−0.299198	+	1.35261i
$204$		0			0
$205$	220229.	−	381448.i	0.366007	−	0.633943i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−216630.			−0.343046
$210$		0			0
$211$	1.13658e6			1.75749			0.878745	−	0.477292i	$-0.158382\pi$
$211$	1.13658e6			1.75749			0.878745	+	0.477292i	$0.158382\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−139833.	+	242198.i	−0.206307	+	0.357335i
$216$		0			0
$217$	−742255.	+	234417.i	−1.07005	+	0.337940i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−31847.2	−	55161.0i	−0.0438623	−	0.0759717i
$222$		0			0
$223$	−497078.			−0.669364			−0.334682	−	0.942331i	$-0.608629\pi$
$223$	−497078.			−0.669364			−0.334682	+	0.942331i	$0.608629\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	467270.	+	809335.i	0.601870	+	1.04247i	0.992538	+	0.121938i	$0.0389109\pi$
$227$	467270.	+	809335.i	0.601870	+	1.04247i	−0.390668	+	0.920532i	$0.627756\pi$
$228$		0			0
$229$	−526030.	+	911110.i	−0.662860	+	1.14811i	0.317001	+	0.948425i	$0.397324\pi$
$229$	−526030.	+	911110.i	−0.662860	+	1.14811i	−0.979861	+	0.199682i	$0.936009\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	207291.	−	359039.i	0.250145	−	0.433263i	−0.713421	−	0.700736i	$-0.752855\pi$
$233$	207291.	−	359039.i	0.250145	−	0.433263i	0.963565	+	0.267473i	$0.0861885\pi$
$234$		0			0
$235$	567179.	+	982382.i	0.669962	+	1.16041i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−481799.			−0.545596			−0.272798	−	0.962071i	$-0.587949\pi$
$239$	−481799.			−0.545596			−0.272798	+	0.962071i	$0.587949\pi$
$240$		0			0
$241$	−14480.5	−	25081.0i	−0.0160598	−	0.0278165i	0.857884	−	0.513844i	$-0.171779\pi$
$241$	−14480.5	−	25081.0i	−0.0160598	−	0.0278165i	−0.873944	+	0.486027i	$0.838446\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	572925.	−	401971.i	0.609793	−	0.427839i
$246$		0			0
$247$	−175190.	+	303439.i	−0.182712	+	0.316467i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−553855.			−0.554896			−0.277448	−	0.960741i	$-0.589489\pi$
$251$	−553855.			−0.554896			−0.277448	+	0.960741i	$0.589489\pi$
$252$		0			0
$253$	171305.			0.168256
$254$		0			0
$255$		0			0
$256$		0			0
$257$	939844.	−	1.62786e6i	0.887612	−	1.53739i	0.0449211	−	0.998991i	$-0.485696\pi$
$257$	939844.	−	1.62786e6i	0.887612	−	1.53739i	0.842691	−	0.538398i	$-0.180970\pi$
$258$		0			0
$259$	−926441.	−	848124.i	−0.858160	−	0.785615i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−541685.	−	938226.i	−0.482900	−	0.836408i	0.516907	−	0.856042i	$-0.327083\pi$
$263$	−541685.	−	938226.i	−0.482900	−	0.836408i	−0.999807	+	0.0196337i	$0.993750\pi$
$264$		0			0
$265$	1.36083e6			1.19039
$266$		0			0
$267$		0			0
$268$		0			0
$269$	554068.	+	959673.i	0.466855	+	0.808617i	0.999283	−	0.0378586i	$-0.0120537\pi$
$269$	554068.	+	959673.i	0.466855	+	0.808617i	−0.532428	+	0.846475i	$0.678720\pi$
$270$		0			0
$271$	161004.	−	278867.i	0.133172	−	0.230660i	−0.791726	−	0.610877i	$-0.790817\pi$
$271$	161004.	−	278867.i	0.133172	−	0.230660i	0.924898	+	0.380216i	$0.124150\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−77026.7	+	133414.i	−0.0614200	+	0.106383i
$276$		0			0
$277$	−728732.	−	1.26220e6i	−0.570648	−	0.988391i	−0.996500	−	0.0835984i	$-0.973359\pi$
$277$	−728732.	−	1.26220e6i	−0.570648	−	0.988391i	0.425851	−	0.904793i	$-0.359975\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−1.03123e6			−0.779092			−0.389546	−	0.921007i	$-0.627368\pi$
$281$	−1.03123e6			−0.779092			−0.389546	+	0.921007i	$0.627368\pi$
$282$		0			0
$283$	−39832.4	−	68991.8i	−0.0295645	−	0.0512072i	0.850865	−	0.525385i	$-0.176079\pi$
$283$	−39832.4	−	68991.8i	−0.0295645	−	0.0512072i	−0.880429	+	0.474178i	$0.842745\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−296164.	+	1.33889e6i	−0.212240	+	0.959493i
$288$		0			0
$289$	646714.	−	1.12014e6i	0.455479	−	0.788912i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−1.44146e6			−0.980920			−0.490460	−	0.871464i	$-0.663171\pi$
$293$	−1.44146e6			−0.980920			−0.490460	+	0.871464i	$0.663171\pi$
$294$		0			0
$295$	20515.4			0.0137254
$296$		0			0
$297$		0			0
$298$		0			0
$299$	138536.	−	239952.i	0.0896159	−	0.155219i
$300$		0			0
$301$	188048.	−	850125.i	0.119633	−	0.540837i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−844335.	−	1.46243e6i	−0.519715	−	0.900173i
$306$		0			0
$307$	−1.71962e6			−1.04132			−0.520662	−	0.853763i	$-0.674315\pi$
$307$	−1.71962e6			−1.04132			−0.520662	+	0.853763i	$0.674315\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−79804.7	−	138226.i	−0.0467873	−	0.0810379i	0.841683	−	0.539971i	$-0.181565\pi$
$311$	−79804.7	−	138226.i	−0.0467873	−	0.0810379i	−0.888471	+	0.458933i	$0.848232\pi$
$312$		0			0
$313$	−226308.	+	391978.i	−0.130569	+	0.226152i	−0.923896	−	0.382644i	$-0.875014\pi$
$313$	−226308.	+	391978.i	−0.130569	+	0.226152i	0.793327	+	0.608796i	$0.208347\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	913154.	−	1.58163e6i	0.510382	−	0.884008i	−0.489545	−	0.871978i	$-0.662837\pi$
$317$	913154.	−	1.58163e6i	0.510382	−	0.884008i	0.999928	−	0.0120304i	$-0.00382947\pi$
$318$		0			0
$319$	−347431.	−	601768.i	−0.191158	−	0.331095i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	695477.			0.370917
$324$		0			0
$325$	124584.	+	215786.i	0.0654267	+	0.113322i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−2.60485e6	−	2.38465e6i	−1.32676	−	1.21461i
$330$		0			0
$331$	−1.04106e6	+	1.80317e6i	−0.522283	+	0.904620i	0.477381	+	0.878696i	$0.341586\pi$
$331$	−1.04106e6	+	1.80317e6i	−0.522283	+	0.904620i	−0.999664	+	0.0259240i	$0.991747\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−320149.			−0.155862
$336$		0			0
$337$	329493.			0.158041			0.0790207	−	0.996873i	$-0.474821\pi$
$337$	329493.			0.158041			0.0790207	+	0.996873i	$0.474821\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	332492.	−	575893.i	0.154844	−	0.268198i
$342$		0			0
$343$	−1.32398e6	+	1.73050e6i	−0.607640	+	0.794213i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−621590.	−	1.07663e6i	−0.277128	−	0.480000i	0.693542	−	0.720416i	$-0.256049\pi$
$347$	−621590.	−	1.07663e6i	−0.277128	−	0.480000i	−0.970670	+	0.240417i	$0.922716\pi$
$348$		0			0
$349$	−1.07921e6			−0.474287			−0.237143	−	0.971475i	$-0.576211\pi$
$349$	−1.07921e6			−0.474287			−0.237143	+	0.971475i	$0.576211\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−263261.	−	455981.i	−0.112447	−	0.194764i	0.804309	−	0.594211i	$-0.202536\pi$
$353$	−263261.	−	455981.i	−0.112447	−	0.194764i	−0.916756	+	0.399447i	$0.869202\pi$
$354$		0			0
$355$	1.62152e6	−	2.80855e6i	0.682891	−	1.18280i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	713788.	−	1.23632e6i	0.292303	−	0.506283i	−0.682051	−	0.731305i	$-0.738912\pi$
$359$	713788.	−	1.23632e6i	0.292303	−	0.506283i	0.974354	+	0.225021i	$0.0722452\pi$
$360$		0			0
$361$	−674846.	−	1.16887e6i	−0.272544	−	0.472060i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	3.07799e6			1.20930
$366$		0			0
$367$	985646.	+	1.70719e6i	0.381993	+	0.661632i	0.991347	−	0.131267i	$-0.0419045\pi$
$367$	985646.	+	1.70719e6i	0.381993	+	0.661632i	−0.609354	+	0.792898i	$0.708571\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−4.03993e6	+	1.27588e6i	−1.52384	+	0.481254i
$372$		0			0
$373$	1.32258e6	−	2.29078e6i	0.492211	−	0.852535i	−0.507749	−	0.861505i	$-0.669522\pi$
$373$	1.32258e6	−	2.29078e6i	0.492211	−	0.852535i	0.999960	+	0.00897066i	$0.00285549\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−1.12388e6			−0.407255
$378$		0			0
$379$	−1.17483e6			−0.420123			−0.210062	−	0.977688i	$-0.567366\pi$
$379$	−1.17483e6			−0.420123			−0.210062	+	0.977688i	$0.567366\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−1.95438e6	+	3.38509e6i	−0.680788	+	1.17916i	0.293952	+	0.955820i	$0.405029\pi$
$383$	−1.95438e6	+	3.38509e6i	−0.680788	+	1.17916i	−0.974741	+	0.223340i	$0.928304\pi$
$384$		0			0
$385$	−129135.	+	583794.i	−0.0444011	+	0.200728i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	97739.7	+	169290.i	0.0327489	+	0.0567228i	0.881935	−	0.471370i	$-0.156240\pi$
$389$	97739.7	+	169290.i	0.0327489	+	0.0567228i	−0.849186	+	0.528093i	$0.822907\pi$
$390$		0			0
$391$	−549965.			−0.181925
$392$		0			0
$393$		0			0
$394$		0			0
$395$	914728.	+	1.58436e6i	0.294985	+	0.510928i
$396$		0			0
$397$	1.65213e6	−	2.86157e6i	0.526099	−	0.911229i	−0.473439	−	0.880827i	$-0.656988\pi$
$397$	1.65213e6	−	2.86157e6i	0.526099	−	0.911229i	0.999538	−	0.0304029i	$-0.00967905\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	2.19319e6	−	3.79872e6i	0.681107	−	1.17971i	−0.293536	−	0.955948i	$-0.594832\pi$
$401$	2.19319e6	−	3.79872e6i	0.681107	−	1.17971i	0.974643	−	0.223764i	$-0.0718346\pi$
$402$		0			0
$403$	−537778.	−	931459.i	−0.164946	−	0.285694i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	1.07303e6			0.321089
$408$		0			0
$409$	1.60722e6	+	2.78379e6i	0.475081	+	0.822865i	0.999593	−	0.0285384i	$-0.00908528\pi$
$409$	1.60722e6	+	2.78379e6i	0.475081	+	0.822865i	−0.524511	+	0.851403i	$0.675752\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−60904.6	+	19234.7i	−0.0175701	+	0.00554894i
$414$		0			0
$415$	857702.	−	1.48558e6i	0.244465	−	0.423425i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−4.48815e6			−1.24891			−0.624457	−	0.781059i	$-0.714680\pi$
$419$	−4.48815e6			−1.24891			−0.624457	+	0.781059i	$0.714680\pi$
$420$		0			0
$421$	−4.05150e6			−1.11406			−0.557032	−	0.830491i	$-0.688060\pi$
$421$	−4.05150e6			−1.11406			−0.557032	+	0.830491i	$0.688060\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	247289.	−	428318.i	0.0664100	−	0.115025i
$426$		0			0
$427$	3.87774e6	+	3.54993e6i	1.02922	+	0.942215i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	1.41541e6	+	2.45156e6i	0.367019	+	0.635696i	0.989098	−	0.147259i	$-0.0470449\pi$
$431$	1.41541e6	+	2.45156e6i	0.367019	+	0.635696i	−0.622079	+	0.782955i	$0.713712\pi$
$432$		0			0
$433$	−3.39735e6			−0.870804			−0.435402	−	0.900236i	$-0.643394\pi$
$433$	−3.39735e6			−0.870804			−0.435402	+	0.900236i	$0.643394\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	1.51267e6	+	2.62002e6i	0.378914	+	0.656297i
$438$		0			0
$439$	−1.69631e6	+	2.93809e6i	−0.420091	+	0.727618i	−0.995948	−	0.0899319i	$-0.971335\pi$
$439$	−1.69631e6	+	2.93809e6i	−0.420091	+	0.727618i	0.575857	+	0.817550i	$0.304668\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	2.33478e6	−	4.04397e6i	0.565246	−	0.979035i	−0.431781	−	0.901979i	$-0.642115\pi$
$443$	2.33478e6	−	4.04397e6i	0.565246	−	0.979035i	0.997027	−	0.0770561i	$-0.0245521\pi$
$444$		0			0
$445$	1.36845e6	+	2.37022e6i	0.327588	+	0.567399i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	782382.			0.183148			0.0915742	−	0.995798i	$-0.470810\pi$
$449$	782382.			0.183148			0.0915742	+	0.995798i	$0.470810\pi$
$450$		0			0
$451$	−585736.	−	1.01453e6i	−0.135600	−	0.234867i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	713301.	+	653002.i	0.161527	+	0.147872i
$456$		0			0
$457$	265708.	−	460219.i	0.0595132	−	0.103080i	−0.834734	−	0.550654i	$-0.814378\pi$
$457$	265708.	−	460219.i	0.0595132	−	0.103080i	0.894247	+	0.447574i	$0.147712\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	1.34721e6			0.295245			0.147623	−	0.989044i	$-0.452838\pi$
$461$	1.34721e6			0.295245			0.147623	+	0.989044i	$0.452838\pi$
$462$		0			0
$463$	6.81207e6			1.47682			0.738409	−	0.674354i	$-0.235578\pi$
$463$	6.81207e6			1.47682			0.738409	+	0.674354i	$0.235578\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−510769.	+	884678.i	−0.108376	+	0.187712i	−0.915112	−	0.403199i	$-0.867898\pi$
$467$	−510769.	+	884678.i	−0.108376	+	0.187712i	0.806737	+	0.590911i	$0.201232\pi$
$468$		0			0
$469$	950433.	−	300163.i	0.199521	−	0.0630122i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	371910.	+	644168.i	0.0764338	+	0.132387i
$474$		0			0
$475$	−2.72066e6			−0.553274
$476$		0			0
$477$		0			0
$478$		0			0
$479$	4.15913e6	+	7.20382e6i	0.828254	+	1.43458i	0.899407	+	0.437113i	$0.143999\pi$
$479$	4.15913e6	+	7.20382e6i	0.828254	+	1.43458i	−0.0711525	+	0.997465i	$0.522668\pi$
$480$		0			0
$481$	867769.	−	1.50302e6i	0.171018	−	0.296212i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	1.42695e6	−	2.47155e6i	0.275458	−	0.477107i
$486$		0			0
$487$	−301362.	−	521975.i	−0.0575793	−	0.0997303i	0.835799	−	0.549036i	$-0.185005\pi$
$487$	−301362.	−	521975.i	−0.0575793	−	0.0997303i	−0.893378	+	0.449305i	$0.851672\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	5.27617e6			0.987678			0.493839	−	0.869553i	$-0.335593\pi$
$491$	5.27617e6			0.987678			0.493839	+	0.869553i	$0.335593\pi$
$492$		0			0
$493$	1.11540e6	+	1.93194e6i	0.206688	+	0.357994i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−2.18062e6	+	9.85812e6i	−0.395995	+	1.79021i
$498$		0			0
$499$	4.09788e6	−	7.09774e6i	0.736730	−	1.27605i	−0.217230	−	0.976120i	$-0.569702\pi$
$499$	4.09788e6	−	7.09774e6i	0.736730	−	1.27605i	0.953960	−	0.299933i	$-0.0969644\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−3.81947e6			−0.673106			−0.336553	−	0.941665i	$-0.609261\pi$
$503$	−3.81947e6			−0.673106			−0.336553	+	0.941665i	$0.609261\pi$
$504$		0			0
$505$	4.88702e6			0.852739
$506$		0			0
$507$		0			0
$508$		0			0
$509$	1.00020e6	−	1.73239e6i	0.171116	−	0.296382i	−0.767694	−	0.640816i	$-0.778596\pi$
$509$	1.00020e6	−	1.73239e6i	0.171116	−	0.296382i	0.938810	+	0.344434i	$0.111929\pi$
$510$		0			0
$511$	−9.13772e6	+	2.88585e6i	−1.54805	+	0.488901i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	225262.	+	390165.i	0.0374257	+	0.0648232i
$516$		0			0
$517$	3.01702e6			0.496422
$518$		0			0
$519$		0			0
$520$		0			0
$521$	5.83316e6	+	1.01033e7i	0.941477	+	1.63069i	0.762656	+	0.646805i	$0.223895\pi$
$521$	5.83316e6	+	1.01033e7i	0.941477	+	1.63069i	0.178821	+	0.983882i	$0.442772\pi$
$522$		0			0
$523$	−5.41109e6	+	9.37228e6i	−0.865029	+	1.49827i	0.00199012	+	0.999998i	$0.499367\pi$
$523$	−5.41109e6	+	9.37228e6i	−0.865029	+	1.49827i	−0.867019	+	0.498276i	$0.833967\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−1.06744e6	+	1.84887e6i	−0.167424	+	0.289988i
$528$		0			0
$529$	2.02199e6	+	3.50219e6i	0.314152	+	0.544128i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−1.89476e6			−0.288892
$534$		0			0
$535$	2.29863e6	+	3.98135e6i	0.347204	+	0.601376i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−163982.	−	1.85420e6i	−0.0243122	−	0.274906i
$540$		0			0
$541$	−543570.	+	941491.i	−0.0798477	+	0.138300i	−0.903184	−	0.429253i	$-0.858777\pi$
$541$	−543570.	+	941491.i	−0.0798477	+	0.138300i	0.823336	+	0.567554i	$0.192110\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	510578.			0.0736327
$546$		0			0
$547$	413333.			0.0590652			0.0295326	−	0.999564i	$-0.490598\pi$
$547$	413333.			0.0590652			0.0295326	+	0.999564i	$0.490598\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	6.13580e6	−	1.06275e7i	0.860978	−	1.49126i
$552$		0			0
$553$	−4.20103e6	−	3.84589e6i	−0.584174	−	0.534791i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−6.00908e6	−	1.04080e7i	−0.820673	−	1.42145i	−0.905182	−	0.425024i	$-0.860266\pi$
$557$	−6.00908e6	−	1.04080e7i	−0.820673	−	1.42145i	0.0845091	−	0.996423i	$-0.473068\pi$
$558$		0			0
$559$	1.20307e6			0.162840
$560$		0			0
$561$		0			0
$562$		0			0
$563$	7.12659e6	+	1.23436e7i	0.947569	+	1.64124i	0.750524	+	0.660843i	$0.229801\pi$
$563$	7.12659e6	+	1.23436e7i	0.947569	+	1.64124i	0.197045	+	0.980394i	$0.436865\pi$
$564$		0			0
$565$	−2.54670e6	+	4.41101e6i	−0.335626	+	0.581322i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	1.36695e6	−	2.36762e6i	0.176999	−	0.306571i	−0.763852	−	0.645391i	$-0.776694\pi$
$569$	1.36695e6	−	2.36762e6i	0.176999	−	0.306571i	0.940851	+	0.338820i	$0.110028\pi$
$570$		0			0
$571$	1.80448e6	+	3.12545e6i	0.231612	+	0.401164i	0.958283	−	0.285822i	$-0.0922666\pi$
$571$	1.80448e6	+	3.12545e6i	0.231612	+	0.401164i	−0.726670	+	0.686986i	$0.758933\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	2.15143e6			0.271367
$576$		0			0
$577$	812927.	+	1.40803e6i	0.101651	+	0.176065i	0.912365	−	0.409378i	$-0.134254\pi$
$577$	812927.	+	1.40803e6i	0.101651	+	0.176065i	−0.810714	+	0.585443i	$0.800921\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−1.15344e6	+	5.21445e6i	−0.141760	+	0.640867i
$582$		0			0
$583$	1.80968e6	−	3.13446e6i	0.220511	−	0.381936i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−9.69220e6			−1.16099			−0.580493	−	0.814265i	$-0.697140\pi$
$587$	−9.69220e6			−1.16099			−0.580493	+	0.814265i	$0.697140\pi$
$588$		0			0
$589$	1.17439e7			1.39484
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−1.82133e6	+	3.15463e6i	−0.212692	+	0.368394i	−0.952556	−	0.304363i	$-0.901556\pi$
$593$	−1.82133e6	+	3.15463e6i	−0.212692	+	0.368394i	0.739864	+	0.672756i	$0.234890\pi$
$594$		0			0
$595$	414581.	−	1.87423e6i	0.0480084	−	0.217036i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	328643.	+	569226.i	0.0374246	+	0.0648213i	0.884131	−	0.467239i	$-0.154751\pi$
$599$	328643.	+	569226.i	0.0374246	+	0.0648213i	−0.846706	+	0.532060i	$0.821418\pi$
$600$		0			0
$601$	−1.45544e7			−1.64365			−0.821824	−	0.569742i	$-0.807043\pi$
$601$	−1.45544e7			−1.64365			−0.821824	+	0.569742i	$0.807043\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	3.09783e6	+	5.36560e6i	0.344088	+	0.595978i
$606$		0			0
$607$	7.04214e6	−	1.21974e7i	0.775770	−	1.34367i	−0.158590	−	0.987344i	$-0.550695\pi$
$607$	7.04214e6	−	1.21974e7i	0.775770	−	1.34367i	0.934361	−	0.356329i	$-0.115972\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	2.43989e6	−	4.22601e6i	0.264403	−	0.457960i
$612$		0			0
$613$	−1.72048e6	−	2.97996e6i	−0.184926	−	0.320302i	0.758625	−	0.651527i	$-0.225871\pi$
$613$	−1.72048e6	−	2.97996e6i	−0.184926	−	0.320302i	−0.943552	+	0.331225i	$0.892538\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−6.03183e6			−0.637876			−0.318938	−	0.947776i	$-0.603326\pi$
$617$	−6.03183e6			−0.637876			−0.318938	+	0.947776i	$0.603326\pi$
$618$		0			0
$619$	−5.73176e6	−	9.92770e6i	−0.601259	−	1.04141i	−0.992631	−	0.121178i	$-0.961333\pi$
$619$	−5.73176e6	−	9.92770e6i	−0.601259	−	1.04141i	0.391372	−	0.920233i	$-0.372001\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−6.28479e6	−	5.75351e6i	−0.648740	−	0.593899i
$624$		0			0
$625$	1.74206e6	−	3.01734e6i	0.178387	−	0.308975i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−3.44490e6			−0.347176
$630$		0			0
$631$	536107.			0.0536016			0.0268008	−	0.999641i	$-0.491468\pi$
$631$	536107.			0.0536016			0.0268008	+	0.999641i	$0.491468\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	3.52025e6	−	6.09725e6i	0.346449	−	0.600067i
$636$		0			0
$637$	−2.72983e6	−	1.26981e6i	−0.266555	−	0.123991i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	1.45587e6	+	2.52164e6i	0.139952	+	0.242403i	0.927478	−	0.373877i	$-0.121972\pi$
$641$	1.45587e6	+	2.52164e6i	0.139952	+	0.242403i	−0.787526	+	0.616281i	$0.788639\pi$
$642$		0			0
$643$	1.93974e7			1.85019			0.925096	−	0.379732i	$-0.123984\pi$
$643$	1.93974e7			1.85019			0.925096	+	0.379732i	$0.123984\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−7.00425e6	−	1.21317e7i	−0.657811	−	1.13936i	−0.981181	−	0.193090i	$-0.938149\pi$
$647$	−7.00425e6	−	1.21317e7i	−0.657811	−	1.13936i	0.323370	−	0.946273i	$-0.395184\pi$
$648$		0			0
$649$	27282.1	−	47254.0i	0.00254253	−	0.00440379i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	5.55312e6	−	9.61829e6i	0.509629	−	0.882703i	−0.490309	−	0.871549i	$-0.663116\pi$
$653$	5.55312e6	−	9.61829e6i	0.509629	−	0.882703i	0.999938	−	0.0111546i	$-0.00355069\pi$
$654$		0			0
$655$	−7.85887e6	−	1.36120e7i	−0.715743	−	1.23970i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	1.00615e7			0.902502			0.451251	−	0.892397i	$-0.350978\pi$
$659$	1.00615e7			0.902502			0.451251	+	0.892397i	$0.350978\pi$
$660$		0			0
$661$	7.14337e6	+	1.23727e7i	0.635916	+	1.10144i	0.986320	+	0.164840i	$0.0527108\pi$
$661$	7.14337e6	+	1.23727e7i	0.635916	+	1.10144i	−0.350405	+	0.936598i	$0.613956\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−1.00691e7	+	3.17999e6i	−0.882951	+	0.278851i
$666$		0			0
$667$	−4.85203e6	+	8.40397e6i	−0.422288	+	0.731425i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−4.49130e6			−0.385094
$672$		0			0
$673$	−2.05348e7			−1.74764			−0.873822	−	0.486245i	$-0.838366\pi$
$673$	−2.05348e7			−1.74764			−0.873822	+	0.486245i	$0.838366\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−1.79648e6	+	3.11159e6i	−0.150643	+	0.260922i	−0.931464	−	0.363833i	$-0.881468\pi$
$677$	−1.79648e6	+	3.11159e6i	−0.150643	+	0.260922i	0.780821	+	0.624755i	$0.214801\pi$
$678$		0			0
$679$	−1.91897e6	+	8.67524e6i	−0.159732	+	0.722116i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−5.96018e6	−	1.03233e7i	−0.488886	−	0.846776i	0.511032	−	0.859562i	$-0.329263\pi$
$683$	−5.96018e6	−	1.03233e7i	−0.488886	−	0.846776i	−0.999918	+	0.0127857i	$0.995930\pi$
$684$		0			0
$685$	−1.31850e7			−1.07363
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−2.92701e6	−	5.06973e6i	−0.234896	−	0.406852i
$690$		0			0
$691$	2.14050e6	−	3.70746e6i	0.170538	−	0.295380i	−0.768070	−	0.640366i	$-0.778783\pi$
$691$	2.14050e6	−	3.70746e6i	0.170538	−	0.295380i	0.938608	+	0.344985i	$0.112116\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	4.77730e6	−	8.27452e6i	0.375163	−	0.649802i
$696$		0			0
$697$	1.88047e6	+	3.25707e6i	0.146617	+	0.253948i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	8.79081e6			0.675669			0.337834	−	0.941206i	$-0.390306\pi$
$701$	8.79081e6			0.675669			0.337834	+	0.941206i	$0.390306\pi$
$702$		0			0
$703$	9.47512e6	+	1.64114e7i	0.723097	+	1.25244i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−1.45082e7	+	4.58194e6i	−1.09161	+	0.344748i
$708$		0			0
$709$	−1.98988e6	+	3.44658e6i	−0.148666	+	0.257497i	−0.930735	−	0.365695i	$-0.880831\pi$
$709$	−1.98988e6	+	3.44658e6i	−0.148666	+	0.257497i	0.782069	+	0.623192i	$0.214165\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−9.28681e6			−0.684136
$714$		0			0
$715$	−826166.			−0.0604369
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−1.44409e6	+	2.50124e6i	−0.104177	+	0.180440i	−0.913402	−	0.407059i	$-0.866554\pi$
$719$	−1.44409e6	+	2.50124e6i	−0.104177	+	0.180440i	0.809225	+	0.587499i	$0.199888\pi$
$720$		0			0
$721$	−1.03455e6	−	947094.i	−0.0741162	−	0.0678508i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−4.36339e6	−	7.55762e6i	−0.308304	−	0.533998i
$726$		0			0
$727$	−2.12040e7			−1.48793			−0.743963	−	0.668221i	$-0.767056\pi$
$727$	−2.12040e7			−1.48793			−0.743963	+	0.668221i	$0.767056\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−1.19400e6	−	2.06806e6i	−0.0826436	−	0.143143i
$732$		0			0
$733$	9.79859e6	−	1.69716e7i	0.673602	−	1.16671i	−0.303273	−	0.952904i	$-0.598079\pi$
$733$	9.79859e6	−	1.69716e7i	0.673602	−	1.16671i	0.976875	−	0.213810i	$-0.0685872\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−425745.	+	737412.i	−0.0288723	+	0.0500082i
$738$		0			0
$739$	−1.20566e7	−	2.08827e7i	−0.812108	−	1.40661i	−0.911386	−	0.411553i	$-0.864986\pi$
$739$	−1.20566e7	−	2.08827e7i	−0.812108	−	1.40661i	0.0992773	−	0.995060i	$-0.468347\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	7.79774e6			0.518199			0.259100	−	0.965851i	$-0.416574\pi$
$743$	7.79774e6			0.518199			0.259100	+	0.965851i	$0.416574\pi$
$744$		0			0
$745$	3.27933e6	+	5.67996e6i	0.216468	+	0.374934i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−1.05568e7	−	9.66440e6i	−0.687588	−	0.629463i
$750$		0			0
$751$	1.00843e7	−	1.74666e7i	0.652450	−	1.13008i	−0.330077	−	0.943954i	$-0.607075\pi$
$751$	1.00843e7	−	1.74666e7i	0.652450	−	1.13008i	0.982527	−	0.186122i	$-0.0595920\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−6.11238e6			−0.390250
$756$		0			0
$757$	−2.80945e6			−0.178190			−0.0890948	−	0.996023i	$-0.528397\pi$
$757$	−2.80945e6			−0.178190			−0.0890948	+	0.996023i	$0.528397\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−9.35273e6	+	1.61994e7i	−0.585433	+	1.01400i	0.409389	+	0.912360i	$0.365742\pi$
$761$	−9.35273e6	+	1.61994e7i	−0.585433	+	1.01400i	−0.994821	+	0.101639i	$0.967591\pi$
$762$		0			0
$763$	−1.51577e6	+	478704.i	−0.0942586	+	0.0297684i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−44126.6	−	76429.4i	−0.00270839	−	0.00469107i
$768$		0			0
$769$	2.25184e7			1.37316			0.686580	−	0.727054i	$-0.259111\pi$
$769$	2.25184e7			1.37316			0.686580	+	0.727054i	$0.259111\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	1.12470e7	+	1.94804e7i	0.677001	+	1.17260i	0.975880	+	0.218309i	$0.0700541\pi$
$773$	1.12470e7	+	1.94804e7i	0.677001	+	1.17260i	−0.298879	+	0.954291i	$0.596613\pi$
$774$		0			0
$775$	4.17578e6	−	7.23266e6i	0.249737	−	0.432557i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	1.03444e7	−	1.79170e7i	0.610747	−	1.05784i
$780$		0			0
$781$	−4.31271e6	−	7.46983e6i	−0.253001	−	0.438211i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−1.36371e7			−0.789854
$786$		0			0
$787$	−4.90014e6	−	8.48729e6i	−0.282014	−	0.488463i	0.689866	−	0.723937i	$-0.257669\pi$
$787$	−4.90014e6	−	8.48729e6i	−0.282014	−	0.488463i	−0.971881	+	0.235473i	$0.924336\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	3.42480e6	−	1.54828e7i	0.194623	−	0.879848i
$792$		0			0
$793$	−3.63216e6	+	6.29108e6i	−0.205108	+	0.355257i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	1.21486e7			0.677456			0.338728	−	0.940884i	$-0.390003\pi$
$797$	1.21486e7			0.677456			0.338728	+	0.940884i	$0.390003\pi$
$798$		0			0
$799$	−9.68594e6			−0.536753
$800$		0			0
$801$		0			0
$802$		0			0
$803$	4.09323e6	−	7.08967e6i	0.224015	−	0.388005i
$804$		0			0
$805$	7.96238e6	−	2.51465e6i	0.433065	−	0.136769i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−2.86932e6	−	4.96981e6i	−0.154137	−	0.266974i	0.778607	−	0.627512i	$-0.215926\pi$
$809$	−2.86932e6	−	4.96981e6i	−0.154137	−	0.266974i	−0.932745	+	0.360538i	$0.882593\pi$
$810$		0			0
$811$	1.14111e7			0.609220			0.304610	−	0.952477i	$-0.401474\pi$
$811$	1.14111e7			0.609220			0.304610	+	0.952477i	$0.401474\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−7.24956e6	−	1.25566e7i	−0.382312	−	0.662184i
$816$		0			0
$817$	−6.56812e6	+	1.13763e7i	−0.344260	+	0.596275i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−7.69674e6	+	1.33311e7i	−0.398519	+	0.690255i	−0.993543	−	0.113453i	$-0.963809\pi$
$821$	−7.69674e6	+	1.33311e7i	−0.398519	+	0.690255i	0.595025	+	0.803708i	$0.297142\pi$
$822$		0			0
$823$	8.45959e6	+	1.46524e7i	0.435361	+	0.754068i	0.997325	−	0.0730941i	$-0.0232873\pi$
$823$	8.45959e6	+	1.46524e7i	0.435361	+	0.754068i	−0.561964	+	0.827162i	$0.689954\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	2.64225e7			1.34341			0.671707	−	0.740817i	$-0.265561\pi$
$827$	2.64225e7			1.34341			0.671707	+	0.740817i	$0.265561\pi$
$828$		0			0
$829$	989105.	+	1.71318e6i	0.0499869	+	0.0865798i	0.889936	−	0.456085i	$-0.150749\pi$
$829$	989105.	+	1.71318e6i	0.0499869	+	0.0865798i	−0.839949	+	0.542665i	$0.817415\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	526453.	+	5.95279e6i	0.0262874	+	0.297240i
$834$		0			0
$835$	−1.14051e7	+	1.97542e7i	−0.566086	+	0.980490i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	2.20079e7			1.07938			0.539689	−	0.841865i	$-0.318542\pi$
$839$	2.20079e7			1.07938			0.539689	+	0.841865i	$0.318542\pi$
$840$		0			0
$841$	1.88512e7			0.919071
$842$		0			0
$843$		0			0
$844$		0			0
$845$	7.06253e6	−	1.22327e7i	0.340266	−	0.589358i
$846$		0			0
$847$	−1.42273e7	−	1.30246e7i	−0.681417	−	0.623813i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−7.49268e6	−	1.29777e7i	−0.354661	−	0.614291i
$852$		0			0
$853$	−2.53601e6			−0.119338			−0.0596689	−	0.998218i	$-0.519004\pi$
$853$	−2.53601e6			−0.119338			−0.0596689	+	0.998218i	$0.519004\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−573156.	−	992735.i	−0.0266576	−	0.0461723i	0.852389	−	0.522908i	$-0.175153\pi$
$857$	−573156.	−	992735.i	−0.0266576	−	0.0461723i	−0.879046	+	0.476736i	$0.841820\pi$
$858$		0			0
$859$	−8.38795e6	+	1.45284e7i	−0.387858	+	0.671790i	−0.992161	−	0.124964i	$-0.960118\pi$
$859$	−8.38795e6	+	1.45284e7i	−0.387858	+	0.671790i	0.604303	+	0.796755i	$0.293452\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−9.24028e6	+	1.60046e7i	−0.422336	+	0.731507i	−0.996168	−	0.0874660i	$-0.972123\pi$
$863$	−9.24028e6	+	1.60046e7i	−0.422336	+	0.731507i	0.573832	+	0.818973i	$0.305456\pi$
$864$		0			0
$865$	3.80067e6	+	6.58295e6i	0.172711	+	0.299144i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	4.86575e6			0.218575
$870$		0			0
$871$	688607.	+	1.19270e6i	0.0307557	+	0.0532705i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−5.26548e6	+	2.38041e7i	−0.232497	+	1.05107i
$876$		0			0
$877$	3.30705e6	−	5.72798e6i	0.145192	−	0.251479i	−0.784253	−	0.620441i	$-0.786953\pi$
$877$	3.30705e6	−	5.72798e6i	0.145192	−	0.251479i	0.929444	+	0.368962i	$0.120287\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−3.43282e7			−1.49009			−0.745043	−	0.667016i	$-0.767571\pi$
$881$	−3.43282e7			−1.49009			−0.745043	+	0.667016i	$0.767571\pi$
$882$		0			0
$883$	5.51963e6			0.238236			0.119118	−	0.992880i	$-0.461993\pi$
$883$	5.51963e6			0.238236			0.119118	+	0.992880i	$0.461993\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−3.77735e6	+	6.54256e6i	−0.161205	+	0.279215i	−0.935301	−	0.353853i	$-0.884871\pi$
$887$	−3.77735e6	+	6.54256e6i	−0.161205	+	0.279215i	0.774096	+	0.633068i	$0.218205\pi$
$888$		0			0
$889$	−4.73404e6	+	2.14016e7i	−0.200899	+	0.908220i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	2.66410e7	+	4.61436e7i	1.11795	+	1.93634i
$894$		0			0
$895$	−5.66842e6			−0.236540
$896$		0			0
$897$		0			0
$898$		0			0
$899$	1.88349e7	+	3.26230e7i	0.777257	+	1.34625i
$900$		0			0
$901$	−5.80987e6	+	1.00630e7i	−0.238426	+	0.412966i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−1.23131e7	+	2.13269e7i	−0.499742	+	0.865578i
$906$		0			0
$907$	1.69135e7	+	2.92951e7i	0.682678	+	1.18243i	0.974161	+	0.225857i	$0.0725181\pi$
$907$	1.69135e7	+	2.92951e7i	0.682678	+	1.18243i	−0.291483	+	0.956576i	$0.594149\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−3.27859e7			−1.30885			−0.654426	−	0.756126i	$-0.727090\pi$
$911$	−3.27859e7			−1.30885			−0.654426	+	0.756126i	$0.727090\pi$
$912$		0			0
$913$	−2.28120e6	−	3.95116e6i	−0.0905706	−	0.156873i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	3.60931e7	+	3.30419e7i	1.41743	+	1.29760i
$918$		0			0
$919$	2.34936e7	−	4.06921e7i	0.917615	−	1.58936i	0.114588	−	0.993413i	$-0.463445\pi$
$919$	2.34936e7	−	4.06921e7i	0.917615	−	1.58936i	0.803027	−	0.595943i	$-0.203222\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−1.39509e7			−0.539011
$924$		0			0
$925$	1.34762e7			0.517862
$926$		0			0
$927$		0			0
$928$		0			0
$929$	1.97801e7	−	3.42602e7i	0.751952	−	1.30242i	−0.194923	−	0.980819i	$-0.562446\pi$
$929$	1.97801e7	−	3.42602e7i	0.751952	−	1.30242i	0.946875	−	0.321601i	$-0.104221\pi$
$930$		0			0
$931$	2.69109e7	−	1.88810e7i	1.01755	−	0.713923i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	819935.	+	1.42017e6i	0.0306726	+	0.0531265i
$936$		0			0
$937$	−2.90483e7			−1.08086			−0.540432	−	0.841387i	$-0.681739\pi$
$937$	−2.90483e7			−1.08086			−0.540432	+	0.841387i	$0.681739\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	1.65600e7	+	2.86827e7i	0.609656	+	1.05596i	0.991297	+	0.131645i	$0.0420258\pi$
$941$	1.65600e7	+	2.86827e7i	0.609656	+	1.05596i	−0.381641	+	0.924311i	$0.624641\pi$
$942$		0			0
$943$	−8.18008e6	+	1.41683e7i	−0.299556	+	0.518847i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	1.99566e7	−	3.45658e7i	0.723121	−	1.25248i	−0.236622	−	0.971602i	$-0.576040\pi$
$947$	1.99566e7	−	3.45658e7i	0.723121	−	1.25248i	0.959743	−	0.280880i	$-0.0906264\pi$
$948$		0			0
$949$	−6.62045e6	−	1.14670e7i	−0.238628	−	0.413317i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−1.23250e7			−0.439596			−0.219798	−	0.975545i	$-0.570540\pi$
$953$	−1.23250e7			−0.439596			−0.219798	+	0.975545i	$0.570540\pi$
$954$		0			0
$955$	−9.18600e6	−	1.59106e7i	−0.325925	−	0.564519i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	3.91428e7	−	1.23619e7i	1.37437	−	0.434051i
$960$		0			0
$961$	−3.71048e6	+	6.42675e6i	−0.129605	+	0.224483i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	1.86712e7			0.645436
$966$		0			0
$967$	4.13540e7			1.42217			0.711084	−	0.703107i	$-0.248204\pi$
$967$	4.13540e7			1.42217			0.711084	+	0.703107i	$0.248204\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−1.44158e7	+	2.49690e7i	−0.490672	+	0.849870i	−0.999942	−	0.0107372i	$-0.996582\pi$
$971$	−1.44158e7	+	2.49690e7i	−0.490672	+	0.849870i	0.509270	+	0.860607i	$0.329916\pi$
$972$		0			0
$973$	−6.42451e6	+	2.90439e7i	−0.217550	+	0.983495i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−4.42571e6	−	7.66556e6i	−0.148336	−	0.256926i	0.782277	−	0.622931i	$-0.214058\pi$
$977$	−4.42571e6	−	7.66556e6i	−0.148336	−	0.256926i	−0.930613	+	0.366006i	$0.880725\pi$
$978$		0			0
$979$	7.27923e6			0.242733
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−1.38853e7	−	2.40500e7i	−0.458322	−	0.793836i	0.540551	−	0.841311i	$-0.318216\pi$
$983$	−1.38853e7	−	2.40500e7i	−0.458322	−	0.793836i	−0.998872	+	0.0474751i	$0.984883\pi$
$984$		0			0
$985$	−1.12853e7	+	1.95468e7i	−0.370616	+	0.641925i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	5.19390e6	−	8.99610e6i	0.168851	−	0.292458i
$990$		0			0
$991$	2.72862e7	+	4.72612e7i	0.882591	+	1.52869i	0.848450	+	0.529276i	$0.177536\pi$
$991$	2.72862e7	+	4.72612e7i	0.882591	+	1.52869i	0.0341418	+	0.999417i	$0.489130\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−8.24336e6			−0.263965
$996$		0			0
$997$	−8.19953e6	−	1.42020e7i	−0.261247	−	0.452493i	0.705327	−	0.708882i	$-0.250800\pi$
$997$	−8.19953e6	−	1.42020e7i	−0.261247	−	0.452493i	−0.966574	+	0.256390i	$0.917467\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	252.6.k.d.37.1		4
3.2	odd	2		28.6.e.b.9.2	✓	4
7.4	even	3	inner	252.6.k.d.109.1		4
12.11	even	2		112.6.i.e.65.1		4
21.2	odd	6		196.6.a.j.1.1		2
21.5	even	6		196.6.a.h.1.2		2
21.11	odd	6		28.6.e.b.25.2	yes	4
21.17	even	6		196.6.e.k.165.1		4
21.20	even	2		196.6.e.k.177.1		4
84.11	even	6		112.6.i.e.81.1		4
84.23	even	6		784.6.a.o.1.2		2
84.47	odd	6		784.6.a.bd.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
28.6.e.b.9.2	✓	4	3.2	odd	2
28.6.e.b.25.2	yes	4	21.11	odd	6
112.6.i.e.65.1		4	12.11	even	2
112.6.i.e.81.1		4	84.11	even	6
196.6.a.h.1.2		2	21.5	even	6
196.6.a.j.1.1		2	21.2	odd	6
196.6.e.k.165.1		4	21.17	even	6
196.6.e.k.177.1		4	21.20	even	2
252.6.k.d.37.1		4	1.1	even	1	trivial
252.6.k.d.109.1		4	7.4	even	3	inner
784.6.a.o.1.2		2	84.23	even	6
784.6.a.bd.1.1		2	84.47	odd	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 252 = 2^{2} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	252.k (of order \(3\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-20.8209 + 36.0629i) q^{5} +(28.0000 - 126.582i) q^{7} +(55.3768 + 95.9154i) q^{11} +179.135 q^{13} +(-177.784 - 307.930i) q^{17} +(-977.981 + 1693.91i) q^{19} +(773.363 - 1339.50i) q^{23} +(695.479 + 1204.60i) q^{25} -6273.94 q^{29} +(-3002.09 - 5199.77i) q^{31} +(3981.93 + 3645.31i) q^{35} +(4844.23 - 8390.44i) q^{37} -10577.3 q^{41} +6716.00 q^{43} +(13620.4 - 23591.2i) q^{47} +(-15239.0 - 7088.59i) q^{49} +(-16339.7 - 28301.2i) q^{53} -4611.98 q^{55} +(-246.332 - 426.659i) q^{59} +(-20276.1 + 35119.3i) q^{61} +(-3729.75 + 6460.12i) q^{65} +(3844.07 + 6658.13i) q^{67} -77879.3 q^{71} +(-36958.0 - 64013.1i) q^{73} +(13691.7 - 4324.07i) q^{77} +(21966.6 - 38047.2i) q^{79} -41194.2 q^{83} +14806.5 q^{85} +(32862.3 - 56919.1i) q^{89} +(5015.77 - 22675.2i) q^{91} +(-40724.9 - 70537.7i) q^{95} -68534.5 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 42 q^{5} + 112 q^{7} + 660 q^{11} - 1288 q^{13} - 210 q^{17} - 3724 q^{19} + 24 q^{23} - 2480 q^{25} - 11064 q^{29} - 2800 q^{31} + 28644 q^{35} + 13238 q^{37} - 8232 q^{41} + 26864 q^{43} + 8064 q^{47}+ \cdots + 16520 q^{97}+O(q^{100}) \) 4 * q + 42 * q^5 + 112 * q^7 + 660 * q^11 - 1288 * q^13 - 210 * q^17 - 3724 * q^19 + 24 * q^23 - 2480 * q^25 - 11064 * q^29 - 2800 * q^31 + 28644 * q^35 + 13238 * q^37 - 8232 * q^41 + 26864 * q^43 + 8064 * q^47 - 60956 * q^49 - 53958 * q^53 + 82656 * q^55 + 36036 * q^59 - 83986 * q^61 - 76308 * q^65 + 2660 * q^67 - 143136 * q^71 - 31318 * q^73 - 77658 * q^77 - 51136 * q^79 - 12432 * q^83 + 53964 * q^85 + 166278 * q^89 - 36064 * q^91 + 66432 * q^95 + 16520 * q^97

Introduction

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