LMFDB - Embedded newform 768.4.f.e.383.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [768,4,Mod(383,768)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(768, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("768.383");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 768 = 2^{8} \cdot 3 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	768.f (of order $2$, degree $1$, 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:	$45.3134668844$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} + 39x^{10} + 549x^{8} + 3500x^{6} + 10236x^{4} + 11952x^{2} + 4624 $ x^12 + 39x^10 + 549x^8 + 3500x^6 + 10236x^4 + 11952*x^2 + 4624
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{26}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 384)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			383.4
Root			$1.08600i$ of defining polynomial
Character	$\chi$	$=$	768.383
Dual form			768.4.f.e.383.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-4.21320 + 3.04120i) q^{3} -9.33303 q^{5} -36.3792i q^{7} +(8.50216 - 25.6264i) q^{9} +O(q^{10})$	\(q+(-4.21320 + 3.04120i) q^{3} -9.33303 q^{5} -36.3792i q^{7} +(8.50216 - 25.6264i) q^{9} -48.4408i q^{11} -25.8988i q^{13} +(39.3219 - 28.3836i) q^{15} -74.2219i q^{17} -82.9826 q^{19} +(110.637 + 153.273i) q^{21} +179.084 q^{23} -37.8946 q^{25} +(42.1138 + 133.826i) q^{27} -122.309 q^{29} -64.1893i q^{31} +(147.318 + 204.091i) q^{33} +339.528i q^{35} +5.01487i q^{37} +(78.7635 + 109.117i) q^{39} -325.933i q^{41} +321.853 q^{43} +(-79.3509 + 239.172i) q^{45} +95.9780 q^{47} -980.446 q^{49} +(225.724 + 312.712i) q^{51} +185.069 q^{53} +452.099i q^{55} +(349.623 - 252.367i) q^{57} -226.316i q^{59} -198.851i q^{61} +(-932.268 - 309.302i) q^{63} +241.714i q^{65} +23.9524 q^{67} +(-754.518 + 544.632i) q^{69} +399.741 q^{71} -669.631 q^{73} +(159.658 - 115.245i) q^{75} -1762.24 q^{77} -229.453i q^{79} +(-584.427 - 435.760i) q^{81} -321.150i q^{83} +692.715i q^{85} +(515.313 - 371.967i) q^{87} -131.446i q^{89} -942.177 q^{91} +(195.213 + 270.443i) q^{93} +774.479 q^{95} +136.371 q^{97} +(-1241.36 - 411.851i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 2 q^{3} - 12 q^{5}+O(q^{10}) $ 12 * q - 2 * q^3 - 12 * q^5	$ 12 q - 2 q^{3} - 12 q^{5} - 84 q^{15} - 180 q^{19} - 156 q^{21} - 120 q^{23} + 300 q^{25} + 130 q^{27} - 588 q^{29} - 116 q^{33} + 620 q^{39} + 372 q^{43} + 740 q^{45} + 1248 q^{47} - 948 q^{49} + 360 q^{51} + 948 q^{53} + 172 q^{57} - 2744 q^{63} - 2292 q^{67} - 3280 q^{69} - 2040 q^{71} + 216 q^{73} + 2522 q^{75} - 4824 q^{77} - 1076 q^{81} + 4156 q^{87} - 3480 q^{91} + 4180 q^{93} + 5448 q^{95} - 48 q^{97} + 3048 q^{99}+O(q^{100}) $ 12 * q - 2 * q^3 - 12 * q^5 - 84 * q^15 - 180 * q^19 - 156 * q^21 - 120 * q^23 + 300 * q^25 + 130 * q^27 - 588 * q^29 - 116 * q^33 + 620 * q^39 + 372 * q^43 + 740 * q^45 + 1248 * q^47 - 948 * q^49 + 360 * q^51 + 948 * q^53 + 172 * q^57 - 2744 * q^63 - 2292 * q^67 - 3280 * q^69 - 2040 * q^71 + 216 * q^73 + 2522 * q^75 - 4824 * q^77 - 1076 * q^81 + 4156 * q^87 - 3480 * q^91 + 4180 * q^93 + 5448 * q^95 - 48 * q^97 + 3048 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/768\mathbb{Z}\right)^\times$.

$n$	$257$	$511$	$517$
$\chi(n)$	$-1$	$-1$	$-1$

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
$3$	−4.21320	+	3.04120i	−0.810831	+	0.585280i
$3$	−4.21320	+	3.04120i	−0.810831	+	0.585280i
$4$		0			0
$5$	−9.33303			−0.834771			−0.417386	−	0.908729i	$-0.637054\pi$
$5$	−9.33303			−0.834771			−0.417386	+	0.908729i	$0.637054\pi$
$6$		0			0
$7$		−	36.3792i		−	1.96429i	−0.188120	−	0.982146i	$-0.560240\pi$
$7$		−	36.3792i		−	1.96429i	0.188120	−	0.982146i	$-0.439760\pi$
$8$		0			0
$9$	8.50216	−	25.6264i	0.314895	−	0.949127i
$10$		0			0
$11$		−	48.4408i		−	1.32777i	−0.747836	−	0.663884i	$-0.768907\pi$
$11$		−	48.4408i		−	1.32777i	0.747836	−	0.663884i	$-0.231093\pi$
$12$		0			0
$13$		−	25.8988i		−	0.552541i	−0.961080	−	0.276270i	$-0.910901\pi$
$13$		−	25.8988i		−	0.552541i	0.961080	−	0.276270i	$-0.0890985\pi$
$14$		0			0
$15$	39.3219	−	28.3836i	0.676859	−	0.488575i
$16$		0			0
$17$		−	74.2219i		−	1.05891i	−0.848338	−	0.529455i	$-0.822397\pi$
$17$		−	74.2219i		−	1.05891i	0.848338	−	0.529455i	$-0.177603\pi$
$18$		0			0
$19$	−82.9826			−1.00197			−0.500987	−	0.865455i	$-0.667030\pi$
$19$	−82.9826			−1.00197			−0.500987	+	0.865455i	$0.667030\pi$
$20$		0			0
$21$	110.637	+	153.273i	1.14966	+	1.59271i
$22$		0			0
$23$	179.084			1.62355			0.811775	−	0.583971i	$-0.198502\pi$
$23$	179.084			1.62355			0.811775	+	0.583971i	$0.198502\pi$
$24$		0			0
$25$	−37.8946			−0.303157
$26$		0			0
$27$	42.1138	+	133.826i	0.300178	+	0.953883i
$28$		0			0
$29$	−122.309			−0.783180			−0.391590	−	0.920140i	$-0.628075\pi$
$29$	−122.309			−0.783180			−0.391590	+	0.920140i	$0.628075\pi$
$30$		0			0
$31$		−	64.1893i		−	0.371895i	−0.982560	−	0.185947i	$-0.940465\pi$
$31$		−	64.1893i		−	0.371895i	0.982560	−	0.185947i	$-0.0595354\pi$
$32$		0			0
$33$	147.318	+	204.091i	0.777116	+	1.07660i
$34$		0			0
$35$			339.528i			1.63973i
$36$		0			0
$37$			5.01487i			0.0222822i	0.999938	+	0.0111411i	$0.00354639\pi$
$37$			5.01487i			0.0222822i	−0.999938	+	0.0111411i	$0.996454\pi$
$38$		0			0
$39$	78.7635	+	109.117i	0.323391	+	0.448017i
$40$		0			0
$41$		−	325.933i		−	1.24152i	−0.784003	−	0.620758i	$-0.786825\pi$
$41$		−	325.933i		−	1.24152i	0.784003	−	0.620758i	$-0.213175\pi$
$42$		0			0
$43$	321.853			1.14144			0.570722	−	0.821143i	$-0.306663\pi$
$43$	321.853			1.14144			0.570722	+	0.821143i	$0.306663\pi$
$44$		0			0
$45$	−79.3509	+	239.172i	−0.262865	+	0.792304i
$46$		0			0
$47$	95.9780			0.297869			0.148934	−	0.988847i	$-0.452416\pi$
$47$	95.9780			0.297869			0.148934	+	0.988847i	$0.452416\pi$
$48$		0			0
$49$	−980.446			−2.85844
$50$		0			0
$51$	225.724	+	312.712i	0.619758	+	0.858597i
$52$		0			0
$53$	185.069			0.479645			0.239822	−	0.970817i	$-0.422911\pi$
$53$	185.069			0.479645			0.239822	+	0.970817i	$0.422911\pi$
$54$		0			0
$55$			452.099i			1.10838i
$56$		0			0
$57$	349.623	−	252.367i	0.812432	−	0.586436i
$58$		0			0
$59$		−	226.316i		−	0.499386i	−0.968325	−	0.249693i	$-0.919670\pi$
$59$		−	226.316i		−	0.499386i	0.968325	−	0.249693i	$-0.0803297\pi$
$60$		0			0
$61$		−	198.851i		−	0.417382i	−0.977982	−	0.208691i	$-0.933080\pi$
$61$		−	198.851i		−	0.417382i	0.977982	−	0.208691i	$-0.0669203\pi$
$62$		0			0
$63$	−932.268	−	309.302i	−1.86436	−	0.618545i
$64$		0			0
$65$			241.714i			0.461245i
$66$		0			0
$67$	23.9524			0.0436753			0.0218377	−	0.999762i	$-0.493048\pi$
$67$	23.9524			0.0436753			0.0218377	+	0.999762i	$0.493048\pi$
$68$		0			0
$69$	−754.518	+	544.632i	−1.31642	+	0.950231i
$70$		0			0
$71$	399.741			0.668176			0.334088	−	0.942542i	$-0.391572\pi$
$71$	399.741			0.668176			0.334088	+	0.942542i	$0.391572\pi$
$72$		0			0
$73$	−669.631			−1.07362			−0.536811	−	0.843703i	$-0.680371\pi$
$73$	−669.631			−1.07362			−0.536811	+	0.843703i	$0.680371\pi$
$74$		0			0
$75$	159.658	−	115.245i	0.245809	−	0.177432i
$76$		0			0
$77$	−1762.24			−2.60812
$78$		0			0
$79$		−	229.453i		−	0.326778i	−0.986562	−	0.163389i	$-0.947757\pi$
$79$		−	229.453i		−	0.326778i	0.986562	−	0.163389i	$-0.0522426\pi$
$80$		0			0
$81$	−584.427	−	435.760i	−0.801683	−	0.597750i
$82$		0			0
$83$		−	321.150i		−	0.424709i	−0.977193	−	0.212354i	$-0.931887\pi$
$83$		−	321.150i		−	0.424709i	0.977193	−	0.212354i	$-0.0681131\pi$
$84$		0			0
$85$			692.715i			0.883947i
$86$		0			0
$87$	515.313	−	371.967i	0.635027	−	0.458380i
$88$		0			0
$89$		−	131.446i		−	0.156554i	−0.996932	−	0.0782768i	$-0.975058\pi$
$89$		−	131.446i		−	0.156554i	0.996932	−	0.0782768i	$-0.0249418\pi$
$90$		0			0
$91$	−942.177			−1.08535
$92$		0			0
$93$	195.213	+	270.443i	0.217663	+	0.301544i
$94$		0			0
$95$	774.479			0.836420
$96$		0			0
$97$	136.371			0.142746			0.0713732	−	0.997450i	$-0.477262\pi$
$97$	136.371			0.142746			0.0713732	+	0.997450i	$0.477262\pi$
$98$		0			0
$99$	−1241.36	−	411.851i	−1.26022	−	0.418107i
$100$		0			0
$101$	411.095			0.405005			0.202502	−	0.979282i	$-0.435093\pi$
$101$	411.095			0.405005			0.202502	+	0.979282i	$0.435093\pi$
$102$		0			0
$103$		−	137.507i		−	0.131543i	−0.997835	−	0.0657716i	$-0.979049\pi$
$103$		−	137.507i		−	0.131543i	0.997835	−	0.0657716i	$-0.0209509\pi$
$104$		0			0
$105$	−1032.57	−	1430.50i	−0.959704	−	1.32955i
$106$		0			0
$107$		−	2113.08i		−	1.90915i	−0.297964	−	0.954577i	$-0.596308\pi$
$107$		−	2113.08i		−	1.90915i	0.297964	−	0.954577i	$-0.403692\pi$
$108$		0			0
$109$			1310.07i			1.15121i	0.817727	+	0.575606i	$0.195234\pi$
$109$			1310.07i			1.15121i	−0.817727	+	0.575606i	$0.804766\pi$
$110$		0			0
$111$	−15.2512	−	21.1287i	−0.0130413	−	0.0180671i
$112$		0			0
$113$		−	1403.62i		−	1.16851i	−0.811571	−	0.584254i	$-0.801387\pi$
$113$		−	1403.62i		−	1.16851i	0.811571	−	0.584254i	$-0.198613\pi$
$114$		0			0
$115$	−1671.40			−1.35529
$116$		0			0
$117$	−663.693	−	220.196i	−0.524431	−	0.173992i
$118$		0			0
$119$	−2700.13			−2.08001
$120$		0			0
$121$	−1015.51			−0.762967
$122$		0			0
$123$	991.227	+	1373.22i	0.726634	+	1.00666i
$124$		0			0
$125$	1520.30			1.08784
$126$		0			0
$127$			1583.69i			1.10653i	0.833004	+	0.553267i	$0.186619\pi$
$127$			1583.69i			1.10653i	−0.833004	+	0.553267i	$0.813381\pi$
$128$		0			0
$129$	−1356.03	+	978.820i	−0.925519	+	0.668065i
$130$		0			0
$131$			160.326i			0.106929i	0.998570	+	0.0534646i	$0.0170264\pi$
$131$			160.326i			0.106929i	−0.998570	+	0.0534646i	$0.982974\pi$
$132$		0			0
$133$			3018.84i			1.96817i
$134$		0			0
$135$	−393.050	−	1249.00i	−0.250580	−	0.796274i
$136$		0			0
$137$			1487.47i			0.927614i	0.885936	+	0.463807i	$0.153517\pi$
$137$			1487.47i			0.927614i	−0.885936	+	0.463807i	$0.846483\pi$
$138$		0			0
$139$	−58.7660			−0.0358595			−0.0179298	−	0.999839i	$-0.505708\pi$
$139$	−58.7660			−0.0358595			−0.0179298	+	0.999839i	$0.505708\pi$
$140$		0			0
$141$	−404.375	+	291.889i	−0.241521	+	0.174337i
$142$		0			0
$143$	−1254.56			−0.733646
$144$		0			0
$145$	1141.51			0.653776
$146$		0			0
$147$	4130.82	−	2981.74i	2.31771	−	1.67299i
$148$		0			0
$149$	−940.146			−0.516911			−0.258456	−	0.966023i	$-0.583214\pi$
$149$	−940.146			−0.516911			−0.258456	+	0.966023i	$0.583214\pi$
$150$		0			0
$151$			1800.43i			0.970308i	0.874429	+	0.485154i	$0.161237\pi$
$151$			1800.43i			0.970308i	−0.874429	+	0.485154i	$0.838763\pi$
$152$		0			0
$153$	−1902.04	−	631.046i	−1.00504	−	0.333445i
$154$		0			0
$155$			599.081i			0.310447i
$156$		0			0
$157$			3024.10i			1.53726i	0.639695	+	0.768629i	$0.279061\pi$
$157$			3024.10i			1.53726i	−0.639695	+	0.768629i	$0.720939\pi$
$158$		0			0
$159$	−779.733	+	562.832i	−0.388911	+	0.280726i
$160$		0			0
$161$		−	6514.94i		−	3.18913i
$162$		0			0
$163$	3124.46			1.50139			0.750694	−	0.660650i	$-0.229719\pi$
$163$	3124.46			1.50139			0.750694	+	0.660650i	$0.229719\pi$
$164$		0			0
$165$	−1374.93	−	1904.79i	−0.648714	−	0.898711i
$166$		0			0
$167$	−371.384			−0.172087			−0.0860435	−	0.996291i	$-0.527422\pi$
$167$	−371.384			−0.172087			−0.0860435	+	0.996291i	$0.527422\pi$
$168$		0			0
$169$	1526.25			0.694699
$170$		0			0
$171$	−705.531	+	2126.55i	−0.315517	+	0.951001i
$172$		0			0
$173$	−2931.24			−1.28820			−0.644099	−	0.764943i	$-0.722767\pi$
$173$	−2931.24			−1.28820			−0.644099	+	0.764943i	$0.722767\pi$
$174$		0			0
$175$			1378.58i			0.595489i
$176$		0			0
$177$	688.272	+	953.513i	0.292281	+	0.404918i
$178$		0			0
$179$			2834.46i			1.18356i	0.806098	+	0.591781i	$0.201575\pi$
$179$			2834.46i			1.18356i	−0.806098	+	0.591781i	$0.798425\pi$
$180$		0			0
$181$		−	1819.98i		−	0.747392i	−0.927551	−	0.373696i	$-0.878090\pi$
$181$		−	1819.98i		−	0.747392i	0.927551	−	0.373696i	$-0.121910\pi$
$182$		0			0
$183$	604.747	+	837.800i	0.244285	+	0.338426i
$184$		0			0
$185$		−	46.8039i		−	0.0186005i
$186$		0			0
$187$	−3595.37			−1.40599
$188$		0			0
$189$	4868.49	−	1532.07i	1.87370	−	0.589638i
$190$		0			0
$191$	4627.06			1.75289			0.876446	−	0.481500i	$-0.159908\pi$
$191$	4627.06			1.75289			0.876446	+	0.481500i	$0.159908\pi$
$192$		0			0
$193$	−2770.29			−1.03321			−0.516606	−	0.856223i	$-0.672805\pi$
$193$	−2770.29			−1.03321			−0.516606	+	0.856223i	$0.672805\pi$
$194$		0			0
$195$	−735.102	−	1018.39i	−0.269958	−	0.373992i
$196$		0			0
$197$	−3880.01			−1.40324			−0.701622	−	0.712550i	$-0.747540\pi$
$197$	−3880.01			−1.40324			−0.701622	+	0.712550i	$0.747540\pi$
$198$		0			0
$199$		−	2015.00i		−	0.717785i	−0.933379	−	0.358893i	$-0.883154\pi$
$199$		−	2015.00i		−	0.717785i	0.933379	−	0.358893i	$-0.116846\pi$
$200$		0			0
$201$	−100.916	+	72.8440i	−0.0354133	+	0.0255623i
$202$		0			0
$203$			4449.51i			1.53839i
$204$		0			0
$205$			3041.94i			1.03638i
$206$		0			0
$207$	1522.60	−	4589.29i	0.511247	−	1.54095i
$208$		0			0
$209$			4019.74i			1.33039i
$210$		0			0
$211$	−5611.42			−1.83084			−0.915418	−	0.402505i	$-0.868140\pi$
$211$	−5611.42			−1.83084			−0.915418	+	0.402505i	$0.868140\pi$
$212$		0			0
$213$	−1684.19	+	1215.69i	−0.541778	+	0.391070i
$214$		0			0
$215$	−3003.86			−0.952845
$216$		0			0
$217$	−2335.16			−0.730510
$218$		0			0
$219$	2821.29	−	2036.48i	0.870526	−	0.628369i
$220$		0			0
$221$	−1922.26			−0.585091
$222$		0			0
$223$			2025.65i			0.608285i	0.952627	+	0.304142i	$0.0983699\pi$
$223$			2025.65i			0.608285i	−0.952627	+	0.304142i	$0.901630\pi$
$224$		0			0
$225$	−322.186	+	971.103i	−0.0954625	+	0.287734i
$226$		0			0
$227$		−	404.049i		−	0.118140i	−0.998254	−	0.0590698i	$-0.981187\pi$
$227$		−	404.049i		−	0.118140i	0.998254	−	0.0590698i	$-0.0188135\pi$
$228$		0			0
$229$		−	4735.42i		−	1.36649i	−0.730191	−	0.683243i	$-0.760569\pi$
$229$		−	4735.42i		−	1.36649i	0.730191	−	0.683243i	$-0.239431\pi$
$230$		0			0
$231$	7424.66	−	5359.32i	2.11475	−	1.52648i
$232$		0			0
$233$			3703.79i			1.04139i	0.853744	+	0.520694i	$0.174327\pi$
$233$			3703.79i			1.04139i	−0.853744	+	0.520694i	$0.825673\pi$
$234$		0			0
$235$	−895.765			−0.248652
$236$		0			0
$237$	697.814	+	966.733i	0.191257	+	0.264962i
$238$		0			0
$239$	6398.83			1.73182			0.865912	−	0.500197i	$-0.166739\pi$
$239$	6398.83			1.73182			0.865912	+	0.500197i	$0.166739\pi$
$240$		0			0
$241$	3766.48			1.00672			0.503362	−	0.864076i	$-0.332096\pi$
$241$	3766.48			1.00672			0.503362	+	0.864076i	$0.332096\pi$
$242$		0			0
$243$	3787.54	+	58.5842i	0.999880	+	0.0154658i
$244$		0			0
$245$	9150.53			2.38615
$246$		0			0
$247$			2149.15i			0.553632i
$248$		0			0
$249$	976.683	+	1353.07i	0.248573	+	0.344367i
$250$		0			0
$251$		−	1979.71i		−	0.497841i	−0.968524	−	0.248921i	$-0.919924\pi$
$251$		−	1979.71i		−	0.497841i	0.968524	−	0.248921i	$-0.0800758\pi$
$252$		0			0
$253$		−	8674.98i		−	2.15570i
$254$		0			0
$255$	−2106.69	−	2918.55i	−0.517356	−	0.716732i
$256$		0			0
$257$			1081.54i			0.262508i	0.991349	+	0.131254i	$0.0419004\pi$
$257$			1081.54i			0.262508i	−0.991349	+	0.131254i	$0.958100\pi$
$258$		0			0
$259$	182.437			0.0437687
$260$		0			0
$261$	−1039.89	+	3134.34i	−0.246619	+	0.743337i
$262$		0			0
$263$	1121.20			0.262875			0.131438	−	0.991324i	$-0.458041\pi$
$263$	1121.20			0.262875			0.131438	+	0.991324i	$0.458041\pi$
$264$		0			0
$265$	−1727.25			−0.400394
$266$		0			0
$267$	399.755	+	553.810i	0.0916277	+	0.126939i
$268$		0			0
$269$	1844.86			0.418153			0.209076	−	0.977899i	$-0.432954\pi$
$269$	1844.86			0.418153			0.209076	+	0.977899i	$0.432954\pi$
$270$		0			0
$271$			938.970i			0.210474i	0.994447	+	0.105237i	$0.0335601\pi$
$271$			938.970i			0.210474i	−0.994447	+	0.105237i	$0.966440\pi$
$272$		0			0
$273$	3969.58	−	2865.35i	0.880037	−	0.635234i
$274$		0			0
$275$			1835.64i			0.402522i
$276$		0			0
$277$			1100.93i			0.238803i	0.992846	+	0.119401i	$0.0380975\pi$
$277$			1100.93i			0.238803i	−0.992846	+	0.119401i	$0.961902\pi$
$278$		0			0
$279$	−1644.94	−	545.748i	−0.352975	−	0.117108i
$280$		0			0
$281$			748.208i			0.158841i	0.996841	+	0.0794205i	$0.0253070\pi$
$281$			748.208i			0.158841i	−0.996841	+	0.0794205i	$0.974693\pi$
$282$		0			0
$283$	2150.15			0.451636			0.225818	−	0.974170i	$-0.427495\pi$
$283$	2150.15			0.451636			0.225818	+	0.974170i	$0.427495\pi$
$284$		0			0
$285$	−3263.04	+	2355.35i	−0.678195	+	0.489540i
$286$		0			0
$287$	−11857.2			−2.43870
$288$		0			0
$289$	−595.891			−0.121289
$290$		0			0
$291$	−574.559	+	414.733i	−0.115743	+	0.0835466i
$292$		0			0
$293$	5252.83			1.04735			0.523675	−	0.851918i	$-0.324561\pi$
$293$	5252.83			1.04735			0.523675	+	0.851918i	$0.324561\pi$
$294$		0			0
$295$			2112.21i			0.416873i
$296$		0			0
$297$	6482.64	−	2040.03i	1.26654	−	0.398567i
$298$		0			0
$299$		−	4638.06i		−	0.897077i
$300$		0			0
$301$		−	11708.8i		−	2.24213i
$302$		0			0
$303$	−1732.03	+	1250.22i	−0.328390	+	0.237041i
$304$		0			0
$305$			1855.88i			0.348418i
$306$		0			0
$307$	−5337.06			−0.992190			−0.496095	−	0.868268i	$-0.665233\pi$
$307$	−5337.06			−0.992190			−0.496095	+	0.868268i	$0.665233\pi$
$308$		0			0
$309$	418.186	+	579.344i	0.0769896	+	0.106659i
$310$		0			0
$311$	−1034.85			−0.188685			−0.0943424	−	0.995540i	$-0.530075\pi$
$311$	−1034.85			−0.188685			−0.0943424	+	0.995540i	$0.530075\pi$
$312$		0			0
$313$	9570.33			1.72826			0.864132	−	0.503265i	$-0.167868\pi$
$313$	9570.33			1.72826			0.864132	+	0.503265i	$0.167868\pi$
$314$		0			0
$315$	8700.89	+	2886.72i	1.55632	+	0.516344i
$316$		0			0
$317$	3593.05			0.636612			0.318306	−	0.947988i	$-0.396886\pi$
$317$	3593.05			0.636612			0.318306	+	0.947988i	$0.396886\pi$
$318$		0			0
$319$			5924.75i			1.03988i
$320$		0			0
$321$	6426.32	+	8902.85i	1.11739	+	1.54800i
$322$		0			0
$323$			6159.13i			1.06100i
$324$		0			0
$325$			981.425i			0.167507i
$326$		0			0
$327$	−3984.19	−	5519.60i	−0.673781	−	0.933438i
$328$		0			0
$329$		−	3491.60i		−	0.585101i
$330$		0			0
$331$	−4119.28			−0.684037			−0.342018	−	0.939693i	$-0.611110\pi$
$331$	−4119.28			−0.684037			−0.342018	+	0.939693i	$0.611110\pi$
$332$		0			0
$333$	128.513	+	42.6372i	0.0211486	+	0.00701654i
$334$		0			0
$335$	−223.548			−0.0364589
$336$		0			0
$337$	−6488.93			−1.04889			−0.524443	−	0.851446i	$-0.675726\pi$
$337$	−6488.93			−1.04889			−0.524443	+	0.851446i	$0.675726\pi$
$338$		0			0
$339$	4268.69	+	5913.73i	0.683904	+	0.947462i
$340$		0			0
$341$	−3109.38			−0.493790
$342$		0			0
$343$			23189.8i			3.65052i
$344$		0			0
$345$	7041.94	−	5083.06i	1.09891	−	0.793225i
$346$		0			0
$347$		−	7085.97i		−	1.09624i	−0.836400	−	0.548119i	$-0.815344\pi$
$347$		−	7085.97i		−	1.09624i	0.836400	−	0.548119i	$-0.184656\pi$
$348$		0			0
$349$			6075.79i			0.931891i	0.884814	+	0.465945i	$0.154286\pi$
$349$			6075.79i			0.931891i	−0.884814	+	0.465945i	$0.845714\pi$
$350$		0			0
$351$	3465.93	−	1090.70i	0.527059	−	0.165861i
$352$		0			0
$353$			4451.86i			0.671242i	0.941997	+	0.335621i	$0.108946\pi$
$353$			4451.86i			0.671242i	−0.941997	+	0.335621i	$0.891054\pi$
$354$		0			0
$355$	−3730.79			−0.557774
$356$		0			0
$357$	11376.2	−	8211.66i	1.68653	−	1.21739i
$358$		0			0
$359$	10771.6			1.58357			0.791785	−	0.610800i	$-0.209152\pi$
$359$	10771.6			1.58357			0.791785	+	0.610800i	$0.209152\pi$
$360$		0			0
$361$	27.1131			0.00395293
$362$		0			0
$363$	4278.54	−	3088.37i	0.618637	−	0.446549i
$364$		0			0
$365$	6249.68			0.896228
$366$		0			0
$367$		−	6671.95i		−	0.948973i	−0.880263	−	0.474486i	$-0.842634\pi$
$367$		−	6671.95i		−	0.948973i	0.880263	−	0.474486i	$-0.157366\pi$
$368$		0			0
$369$	−8352.48	−	2771.13i	−1.17835	−	0.390947i
$370$		0			0
$371$		−	6732.66i		−	0.942162i
$372$		0			0
$373$		−	12618.5i		−	1.75164i	−0.482641	−	0.875819i	$-0.660322\pi$
$373$		−	12618.5i		−	1.75164i	0.482641	−	0.875819i	$-0.339678\pi$
$374$		0			0
$375$	−6405.33	+	4623.54i	−0.882053	+	0.636690i
$376$		0			0
$377$			3167.66i			0.432739i
$378$		0			0
$379$	−7915.38			−1.07279			−0.536393	−	0.843968i	$-0.680213\pi$
$379$	−7915.38			−1.07279			−0.536393	+	0.843968i	$0.680213\pi$
$380$		0			0
$381$	−4816.32	−	6672.40i	−0.647632	−	0.897212i
$382$		0			0
$383$	−10820.5			−1.44360			−0.721802	−	0.692099i	$-0.756686\pi$
$383$	−10820.5			−1.44360			−0.721802	+	0.692099i	$0.756686\pi$
$384$		0			0
$385$	16447.0			2.17719
$386$		0			0
$387$	2736.45	−	8247.94i	0.359435	−	1.08338i
$388$		0			0
$389$	7990.57			1.04149			0.520743	−	0.853714i	$-0.325655\pi$
$389$	7990.57			1.04149			0.520743	+	0.853714i	$0.325655\pi$
$390$		0			0
$391$		−	13292.0i		−	1.71919i
$392$		0			0
$393$	−487.583	−	675.485i	−0.0625835	−	0.0867016i
$394$		0			0
$395$			2141.49i			0.272785i
$396$		0			0
$397$		−	5526.21i		−	0.698620i	−0.937007	−	0.349310i	$-0.886416\pi$
$397$		−	5526.21i		−	0.698620i	0.937007	−	0.349310i	$-0.113584\pi$
$398$		0			0
$399$	−9180.91	−	12719.0i	−1.15193	−	1.59585i
$400$		0			0
$401$			7844.55i			0.976903i	0.872591	+	0.488451i	$0.162438\pi$
$401$			7844.55i			0.976903i	−0.872591	+	0.488451i	$0.837562\pi$
$402$		0			0
$403$	−1662.43			−0.205487
$404$		0			0
$405$	5454.47	+	4066.96i	0.669222	+	0.498985i
$406$		0			0
$407$	242.924			0.0295855
$408$		0			0
$409$	4407.65			0.532871			0.266435	−	0.963853i	$-0.414154\pi$
$409$	4407.65			0.532871			0.266435	+	0.963853i	$0.414154\pi$
$410$		0			0
$411$	−4523.70	−	6267.01i	−0.542914	−	0.752138i
$412$		0			0
$413$	−8233.18			−0.980940
$414$		0			0
$415$			2997.30i			0.354534i
$416$		0			0
$417$	247.593	−	178.720i	0.0290760	−	0.0209878i
$418$		0			0
$419$		−	12032.8i		−	1.40296i	−0.712688	−	0.701481i	$-0.752523\pi$
$419$		−	12032.8i		−	1.40296i	0.712688	−	0.701481i	$-0.247477\pi$
$420$		0			0
$421$			10320.3i			1.19473i	0.801969	+	0.597365i	$0.203786\pi$
$421$			10320.3i			1.19473i	−0.801969	+	0.597365i	$0.796214\pi$
$422$		0			0
$423$	816.021	−	2459.57i	0.0937973	−	0.282715i
$424$		0			0
$425$			2812.61i			0.321016i
$426$		0			0
$427$	−7234.05			−0.819860
$428$		0			0
$429$	5285.70	−	3815.36i	0.594863	−	0.429388i
$430$		0			0
$431$	8068.52			0.901733			0.450866	−	0.892591i	$-0.351115\pi$
$431$	8068.52			0.901733			0.450866	+	0.892591i	$0.351115\pi$
$432$		0			0
$433$	−3236.75			−0.359234			−0.179617	−	0.983737i	$-0.557486\pi$
$433$	−3236.75			−0.359234			−0.179617	+	0.983737i	$0.557486\pi$
$434$		0			0
$435$	−4809.43	+	3471.58i	−0.530102	+	0.382642i
$436$		0			0
$437$	−14860.9			−1.62676
$438$		0			0
$439$			2207.71i			0.240019i	0.992773	+	0.120009i	$0.0382924\pi$
$439$			2207.71i			0.240019i	−0.992773	+	0.120009i	$0.961708\pi$
$440$		0			0
$441$	−8335.91	+	25125.3i	−0.900109	+	2.71302i
$442$		0			0
$443$		−	1058.18i		−	0.113489i	−0.998389	−	0.0567447i	$-0.981928\pi$
$443$		−	1058.18i		−	0.113489i	0.998389	−	0.0567447i	$-0.0180721\pi$
$444$		0			0
$445$			1226.79i			0.130686i
$446$		0			0
$447$	3961.03	−	2859.18i	0.419128	−	0.302538i
$448$		0			0
$449$			10836.4i			1.13898i	0.822000	+	0.569488i	$0.192858\pi$
$449$			10836.4i			1.13898i	−0.822000	+	0.569488i	$0.807142\pi$
$450$		0			0
$451$	−15788.4			−1.64844
$452$		0			0
$453$	−5475.46	−	7585.56i	−0.567902	−	0.786756i
$454$		0			0
$455$	8793.36			0.906020
$456$		0			0
$457$	9827.12			1.00589			0.502947	−	0.864317i	$-0.332249\pi$
$457$	9827.12			1.00589			0.502947	+	0.864317i	$0.332249\pi$
$458$		0			0
$459$	9932.83	−	3125.77i	1.01008	−	0.317861i
$460$		0			0
$461$	9311.74			0.940761			0.470381	−	0.882464i	$-0.344117\pi$
$461$	9311.74			0.940761			0.470381	+	0.882464i	$0.344117\pi$
$462$		0			0
$463$			7601.37i			0.762993i	0.924370	+	0.381497i	$0.124591\pi$
$463$			7601.37i			0.762993i	−0.924370	+	0.381497i	$0.875409\pi$
$464$		0			0
$465$	−1821.93	−	2524.05i	−0.181699	−	0.251720i
$466$		0			0
$467$		−	4446.20i		−	0.440569i	−0.975436	−	0.220284i	$-0.929301\pi$
$467$		−	4446.20i		−	0.440569i	0.975436	−	0.220284i	$-0.0706985\pi$
$468$		0			0
$469$		−	871.368i		−	0.0857911i
$470$		0			0
$471$	−9196.90	−	12741.1i	−0.899726	−	1.24646i
$472$		0			0
$473$		−	15590.8i		−	1.51557i
$474$		0			0
$475$	3144.59			0.303756
$476$		0			0
$477$	1573.49	−	4742.65i	0.151038	−	0.455244i
$478$		0			0
$479$	12285.9			1.17194			0.585970	−	0.810333i	$-0.300714\pi$
$479$	12285.9			1.17194			0.585970	+	0.810333i	$0.300714\pi$
$480$		0			0
$481$	129.879			0.0123118
$482$		0			0
$483$	19813.3	+	27448.8i	1.86653	+	2.58584i
$484$		0			0
$485$	−1272.76			−0.119161
$486$		0			0
$487$		−	8464.78i		−	0.787630i	−0.919190	−	0.393815i	$-0.871155\pi$
$487$		−	8464.78i		−	0.787630i	0.919190	−	0.393815i	$-0.128845\pi$
$488$		0			0
$489$	−13164.0	+	9502.11i	−1.21737	+	0.878732i
$490$		0			0
$491$			6356.37i			0.584234i	0.956383	+	0.292117i	$0.0943596\pi$
$491$			6356.37i			0.584234i	−0.956383	+	0.292117i	$0.905640\pi$
$492$		0			0
$493$			9078.02i			0.829317i
$494$		0			0
$495$	11585.7	+	3843.82i	1.05200	+	0.349024i
$496$		0			0
$497$		−	14542.2i		−	1.31249i
$498$		0			0
$499$	11856.6			1.06368			0.531840	−	0.846845i	$-0.321501\pi$
$499$	11856.6			1.06368			0.531840	+	0.846845i	$0.321501\pi$
$500$		0			0
$501$	1564.72	−	1129.45i	0.139534	−	0.100719i
$502$		0			0
$503$	−17032.5			−1.50982			−0.754912	−	0.655826i	$-0.772321\pi$
$503$	−17032.5			−1.50982			−0.754912	+	0.655826i	$0.772321\pi$
$504$		0			0
$505$	−3836.76			−0.338086
$506$		0			0
$507$	−6430.41	+	4641.65i	−0.563283	+	0.406593i
$508$		0			0
$509$	−1563.18			−0.136123			−0.0680617	−	0.997681i	$-0.521681\pi$
$509$	−1563.18			−0.136123			−0.0680617	+	0.997681i	$0.521681\pi$
$510$		0			0
$511$			24360.6i			2.10891i
$512$		0			0
$513$	−3494.72	−	11105.2i	−0.300771	−	0.955767i
$514$		0			0
$515$			1283.36i			0.109809i
$516$		0			0
$517$		−	4649.25i		−	0.395501i
$518$		0			0
$519$	12349.9	−	8914.50i	1.04451	−	0.753956i
$520$		0			0
$521$		−	9096.23i		−	0.764900i	−0.923976	−	0.382450i	$-0.875080\pi$
$521$		−	9096.23i		−	0.764900i	0.923976	−	0.382450i	$-0.124920\pi$
$522$		0			0
$523$	10108.4			0.845143			0.422572	−	0.906330i	$-0.361127\pi$
$523$	10108.4			0.845143			0.422572	+	0.906330i	$0.361127\pi$
$524$		0			0
$525$	−4192.53	−	5808.22i	−0.348528	−	0.482841i
$526$		0			0
$527$	−4764.25			−0.393803
$528$		0			0
$529$	19904.2			1.63591
$530$		0			0
$531$	−5799.66	−	1924.17i	−0.473980	−	0.157254i
$532$		0			0
$533$	−8441.26			−0.685988
$534$		0			0
$535$			19721.5i			1.59371i
$536$		0			0
$537$	−8620.18	−	11942.2i	−0.692716	−	0.959670i
$538$		0			0
$539$			47493.6i			3.79535i
$540$		0			0
$541$			10097.6i			0.802462i	0.915977	+	0.401231i	$0.131417\pi$
$541$			10097.6i			0.802462i	−0.915977	+	0.401231i	$0.868583\pi$
$542$		0			0
$543$	5534.93	+	7667.94i	0.437434	+	0.606009i
$544$		0			0
$545$		−	12226.9i		−	0.960998i
$546$		0			0
$547$	−2064.83			−0.161400			−0.0806998	−	0.996738i	$-0.525716\pi$
$547$	−2064.83			−0.161400			−0.0806998	+	0.996738i	$0.525716\pi$
$548$		0			0
$549$	−5095.84	−	1690.66i	−0.396148	−	0.131431i
$550$		0			0
$551$	10149.5			0.784727
$552$		0			0
$553$	−8347.32			−0.641888
$554$		0			0
$555$	142.340	+	197.194i	0.0108865	+	0.0150819i
$556$		0			0
$557$	12125.7			0.922409			0.461204	−	0.887294i	$-0.347417\pi$
$557$	12125.7			0.922409			0.461204	+	0.887294i	$0.347417\pi$
$558$		0			0
$559$		−	8335.60i		−	0.630695i
$560$		0			0
$561$	15148.0	−	10934.2i	1.14002	−	0.822895i
$562$		0			0
$563$		−	7278.58i		−	0.544859i	−0.962176	−	0.272429i	$-0.912173\pi$
$563$		−	7278.58i		−	0.544859i	0.962176	−	0.272429i	$-0.0878271\pi$
$564$		0			0
$565$			13100.0i			0.975436i
$566$		0			0
$567$	−15852.6	+	21261.0i	−1.17416	+	1.57474i
$568$		0			0
$569$			17428.1i			1.28405i	0.766685	+	0.642024i	$0.221905\pi$
$569$			17428.1i			1.28405i	−0.766685	+	0.642024i	$0.778095\pi$
$570$		0			0
$571$	2423.73			0.177635			0.0888177	−	0.996048i	$-0.471691\pi$
$571$	2423.73			0.177635			0.0888177	+	0.996048i	$0.471691\pi$
$572$		0			0
$573$	−19494.7	+	14071.8i	−1.42130	+	1.02593i
$574$		0			0
$575$	−6786.33			−0.492190
$576$		0			0
$577$	−17471.1			−1.26054			−0.630268	−	0.776377i	$-0.717055\pi$
$577$	−17471.1			−1.26054			−0.630268	+	0.776377i	$0.717055\pi$
$578$		0			0
$579$	11671.8	−	8425.03i	0.837761	−	0.604719i
$580$		0			0
$581$	−11683.2			−0.834252
$582$		0			0
$583$		−	8964.88i		−	0.636857i
$584$		0			0
$585$	6194.27	+	2055.09i	0.437780	+	0.145244i
$586$		0			0
$587$		−	7273.06i		−	0.511399i	−0.966756	−	0.255700i	$-0.917694\pi$
$587$		−	7273.06i		−	0.511399i	0.966756	−	0.255700i	$-0.0823058\pi$
$588$		0			0
$589$			5326.60i			0.372629i
$590$		0			0
$591$	16347.3	−	11799.9i	1.13779	−	0.821290i
$592$		0			0
$593$			19957.9i			1.38208i	0.722818	+	0.691038i	$0.242846\pi$
$593$			19957.9i			1.38208i	−0.722818	+	0.691038i	$0.757154\pi$
$594$		0			0
$595$	25200.4			1.73633
$596$		0			0
$597$	6128.01	+	8489.59i	0.420105	+	0.582003i
$598$		0			0
$599$	6101.48			0.416193			0.208097	−	0.978108i	$-0.433273\pi$
$599$	6101.48			0.416193			0.208097	+	0.978108i	$0.433273\pi$
$600$		0			0
$601$	−16659.6			−1.13071			−0.565357	−	0.824846i	$-0.691262\pi$
$601$	−16659.6			−1.13071			−0.565357	+	0.824846i	$0.691262\pi$
$602$		0			0
$603$	203.647	−	613.813i	0.0137531	−	0.0414534i
$604$		0			0
$605$	9477.77			0.636903
$606$		0			0
$607$		−	13728.6i		−	0.917999i	−0.888437	−	0.458999i	$-0.848208\pi$
$607$		−	13728.6i		−	0.917999i	0.888437	−	0.458999i	$-0.151792\pi$
$608$		0			0
$609$	−13531.9	−	18746.7i	−0.900392	−	1.24738i
$610$		0			0
$611$		−	2485.71i		−	0.164585i
$612$		0			0
$613$			7514.10i			0.495093i	0.968876	+	0.247546i	$0.0796242\pi$
$613$			7514.10i			0.495093i	−0.968876	+	0.247546i	$0.920376\pi$
$614$		0			0
$615$	−9251.15	−	12816.3i	−0.606573	−	0.840330i
$616$		0			0
$617$			8052.78i			0.525434i	0.964873	+	0.262717i	$0.0846186\pi$
$617$			8052.78i			0.525434i	−0.964873	+	0.262717i	$0.915381\pi$
$618$		0			0
$619$	−25194.4			−1.63594			−0.817970	−	0.575261i	$-0.804900\pi$
$619$	−25194.4			−1.63594			−0.817970	+	0.575261i	$0.804900\pi$
$620$		0			0
$621$	7541.92	+	23966.1i	0.487354	+	1.54868i
$622$		0			0
$623$	−4781.91			−0.307517
$624$		0			0
$625$	−9452.17			−0.604939
$626$		0			0
$627$	−12224.9	−	16936.0i	−0.778650	−	1.07872i
$628$		0			0
$629$	372.213			0.0235948
$630$		0			0
$631$		−	14945.1i		−	0.942876i	−0.881899	−	0.471438i	$-0.843735\pi$
$631$		−	14945.1i		−	0.942876i	0.881899	−	0.471438i	$-0.156265\pi$
$632$		0			0
$633$	23642.1	−	17065.5i	1.48450	−	1.07155i
$634$		0			0
$635$		−	14780.6i		−	0.923702i
$636$		0			0
$637$			25392.4i			1.57941i
$638$		0			0
$639$	3398.66	−	10243.9i	0.210405	−	0.634184i
$640$		0			0
$641$		−	13909.9i		−	0.857109i	−0.903516	−	0.428555i	$-0.859023\pi$
$641$		−	13909.9i		−	0.857109i	0.903516	−	0.428555i	$-0.140977\pi$
$642$		0			0
$643$	−13338.7			−0.818084			−0.409042	−	0.912516i	$-0.634137\pi$
$643$	−13338.7			−0.818084			−0.409042	+	0.912516i	$0.634137\pi$
$644$		0			0
$645$	12655.9	−	9135.36i	0.772597	−	0.557681i
$646$		0			0
$647$	719.190			0.0437006			0.0218503	−	0.999761i	$-0.493044\pi$
$647$	719.190			0.0437006			0.0218503	+	0.999761i	$0.493044\pi$
$648$		0			0
$649$	−10962.9			−0.663069
$650$		0			0
$651$	9838.49	−	7101.69i	0.592321	−	0.427553i
$652$		0			0
$653$	19019.4			1.13980			0.569899	−	0.821714i	$-0.306982\pi$
$653$	19019.4			1.13980			0.569899	+	0.821714i	$0.306982\pi$
$654$		0			0
$655$		−	1496.32i		−	0.0892615i
$656$		0			0
$657$	−5693.31	+	17160.2i	−0.338078	+	1.01900i
$658$		0			0
$659$			22748.7i			1.34471i	0.740230	+	0.672354i	$0.234717\pi$
$659$			22748.7i			1.34471i	−0.740230	+	0.672354i	$0.765283\pi$
$660$		0			0
$661$		−	23275.3i		−	1.36960i	−0.728733	−	0.684798i	$-0.759890\pi$
$661$		−	23275.3i		−	1.36960i	0.728733	−	0.684798i	$-0.240110\pi$
$662$		0			0
$663$	8098.86	−	5845.98i	0.474410	−	0.342442i
$664$		0			0
$665$		−	28174.9i		−	1.64297i
$666$		0			0
$667$	−21903.6			−1.27153
$668$		0			0
$669$	−6160.41	−	8534.47i	−0.356017	−	0.493216i
$670$		0			0
$671$	−9632.51			−0.554186
$672$		0			0
$673$	−28195.5			−1.61495			−0.807473	−	0.589905i	$-0.799165\pi$
$673$	−28195.5			−1.61495			−0.807473	+	0.589905i	$0.799165\pi$
$674$		0			0
$675$	−1595.89	−	5071.29i	−0.0910011	−	0.289176i
$676$		0			0
$677$	−32369.7			−1.83762			−0.918809	−	0.394701i	$-0.870848\pi$
$677$	−32369.7			−1.83762			−0.918809	+	0.394701i	$0.870848\pi$
$678$		0			0
$679$		−	4961.07i		−	0.280396i
$680$		0			0
$681$	1228.80	+	1702.34i	0.0691447	+	0.0957913i
$682$		0			0
$683$			1227.33i			0.0687592i	0.999409	+	0.0343796i	$0.0109455\pi$
$683$			1227.33i			0.0687592i	−0.999409	+	0.0343796i	$0.989054\pi$
$684$		0			0
$685$		−	13882.6i		−	0.774345i
$686$		0			0
$687$	14401.4	+	19951.3i	0.799776	+	1.10799i
$688$		0			0
$689$		−	4793.06i		−	0.265023i
$690$		0			0
$691$	−3563.29			−0.196170			−0.0980852	−	0.995178i	$-0.531272\pi$
$691$	−3563.29			−0.196170			−0.0980852	+	0.995178i	$0.531272\pi$
$692$		0			0
$693$	−14982.8	+	45159.8i	−0.821284	+	2.47544i
$694$		0			0
$695$	548.465			0.0299345
$696$		0			0
$697$	−24191.3			−1.31465
$698$		0			0
$699$	−11264.0	−	15604.8i	−0.609503	−	0.844389i
$700$		0			0
$701$	−11942.4			−0.643448			−0.321724	−	0.946834i	$-0.604262\pi$
$701$	−11942.4			−0.643448			−0.321724	+	0.946834i	$0.604262\pi$
$702$		0			0
$703$		−	416.147i		−	0.0223262i
$704$		0			0
$705$	3774.04	−	2724.21i	0.201615	−	0.145531i
$706$		0			0
$707$		−	14955.3i		−	0.795547i
$708$		0			0
$709$			22547.9i			1.19436i	0.802105	+	0.597182i	$0.203713\pi$
$709$			22547.9i			1.19436i	−0.802105	+	0.597182i	$0.796287\pi$
$710$		0			0
$711$	−5880.06	−	1950.85i	−0.310154	−	0.102901i
$712$		0			0
$713$		−	11495.3i		−	0.603790i
$714$		0			0
$715$	11708.8			0.612426
$716$		0			0
$717$	−26959.6	+	19460.1i	−1.40422	+	1.01360i
$718$		0			0
$719$	−2034.54			−0.105529			−0.0527647	−	0.998607i	$-0.516803\pi$
$719$	−2034.54			−0.105529			−0.0527647	+	0.998607i	$0.516803\pi$
$720$		0			0
$721$	−5002.39			−0.258389
$722$		0			0
$723$	−15869.0	+	11454.6i	−0.816283	+	0.589215i
$724$		0			0
$725$	4634.86			0.237427
$726$		0			0
$727$			26487.5i			1.35126i	0.737240	+	0.675631i	$0.236129\pi$
$727$			26487.5i			1.35126i	−0.737240	+	0.675631i	$0.763871\pi$
$728$		0			0
$729$	−16135.9	+	11271.9i	−0.819786	+	0.572670i
$730$		0			0
$731$		−	23888.5i		−	1.20869i
$732$		0			0
$733$		−	38967.7i		−	1.96358i	−0.189964	−	0.981791i	$-0.560837\pi$
$733$		−	38967.7i		−	1.96358i	0.189964	−	0.981791i	$-0.439163\pi$
$734$		0			0
$735$	−38553.0	+	27828.6i	−1.93476	+	1.39656i
$736$		0			0
$737$		−	1160.27i		−	0.0579907i
$738$		0			0
$739$	−14312.2			−0.712425			−0.356212	−	0.934405i	$-0.615932\pi$
$739$	−14312.2			−0.712425			−0.356212	+	0.934405i	$0.615932\pi$
$740$		0			0
$741$	−6536.00	−	9054.80i	−0.324030	−	0.448902i
$742$		0			0
$743$	12790.0			0.631520			0.315760	−	0.948839i	$-0.397741\pi$
$743$	12790.0			0.631520			0.315760	+	0.948839i	$0.397741\pi$
$744$		0			0
$745$	8774.41			0.431503
$746$		0			0
$747$	−8229.93	−	2730.47i	−0.403102	−	0.133739i
$748$		0			0
$749$	−76872.3			−3.75014
$750$		0			0
$751$		−	19441.6i		−	0.944653i	−0.881424	−	0.472327i	$-0.843414\pi$
$751$		−	19441.6i		−	0.944653i	0.881424	−	0.472327i	$-0.156586\pi$
$752$		0			0
$753$	6020.70	+	8340.92i	0.291376	+	0.403665i
$754$		0			0
$755$		−	16803.4i		−	0.809985i
$756$		0			0
$757$		−	13901.0i		−	0.667424i	−0.942675	−	0.333712i	$-0.891699\pi$
$757$		−	13901.0i		−	0.667424i	0.942675	−	0.333712i	$-0.108301\pi$
$758$		0			0
$759$	26382.4	+	36549.4i	1.26169	+	1.74791i
$760$		0			0
$761$			8818.58i			0.420070i	0.977694	+	0.210035i	$0.0673578\pi$
$761$			8818.58i			0.420070i	−0.977694	+	0.210035i	$0.932642\pi$
$762$		0			0
$763$	47659.3			2.26132
$764$		0			0
$765$	17751.8	+	5889.57i	0.838978	+	0.278350i
$766$		0			0
$767$	−5861.30			−0.275931
$768$		0			0
$769$	1323.39			0.0620582			0.0310291	−	0.999518i	$-0.490122\pi$
$769$	1323.39			0.0620582			0.0310291	+	0.999518i	$0.490122\pi$
$770$		0			0
$771$	−3289.18	−	4556.75i	−0.153641	−	0.212850i
$772$		0			0
$773$	34327.1			1.59723			0.798617	−	0.601840i	$-0.205565\pi$
$773$	34327.1			1.59723			0.798617	+	0.601840i	$0.205565\pi$
$774$		0			0
$775$			2432.43i			0.112743i
$776$		0			0
$777$	−768.644	+	554.828i	−0.0354890	+	0.0256169i
$778$		0			0
$779$			27046.7i			1.24397i
$780$		0			0
$781$		−	19363.8i		−	0.887183i
$782$		0			0
$783$	−5150.91	−	16368.2i	−0.235094	−	0.747063i
$784$		0			0
$785$		−	28224.0i		−	1.28326i
$786$		0			0
$787$	15375.3			0.696405			0.348203	−	0.937419i	$-0.386792\pi$
$787$	15375.3			0.696405			0.348203	+	0.937419i	$0.386792\pi$
$788$		0			0
$789$	−4723.84	+	3409.80i	−0.213147	+	0.153856i
$790$		0			0
$791$	−51062.5			−2.29529
$792$		0			0
$793$	−5150.00			−0.230620
$794$		0			0
$795$	7277.27	−	5252.93i	0.324652	−	0.234342i
$796$		0			0
$797$	3347.25			0.148765			0.0743826	−	0.997230i	$-0.476301\pi$
$797$	3347.25			0.148765			0.0743826	+	0.997230i	$0.476301\pi$
$798$		0			0
$799$		−	7123.67i		−	0.315416i
$800$		0			0
$801$	−3368.50	−	1117.58i	−0.148589	−	0.0492979i
$802$		0			0
$803$			32437.4i			1.42552i
$804$		0			0
$805$			60804.1i			2.66219i
$806$		0			0
$807$	−7772.76	+	5610.59i	−0.339051	+	0.244736i
$808$		0			0
$809$		−	39577.3i		−	1.71998i	−0.510309	−	0.859991i	$-0.670469\pi$
$809$		−	39577.3i		−	1.71998i	0.510309	−	0.859991i	$-0.329531\pi$
$810$		0			0
$811$	12614.4			0.546180			0.273090	−	0.961989i	$-0.411954\pi$
$811$	12614.4			0.546180			0.273090	+	0.961989i	$0.411954\pi$
$812$		0			0
$813$	−2855.60	−	3956.07i	−0.123186	−	0.170659i
$814$		0			0
$815$	−29160.6			−1.25332
$816$		0			0
$817$	−26708.2			−1.14370
$818$		0			0
$819$	−8010.54	+	24144.6i	−0.341772	+	1.03014i
$820$		0			0
$821$	18243.6			0.775526			0.387763	−	0.921759i	$-0.373248\pi$
$821$	18243.6			0.775526			0.387763	+	0.921759i	$0.373248\pi$
$822$		0			0
$823$		−	18350.6i		−	0.777230i	−0.921400	−	0.388615i	$-0.872954\pi$
$823$		−	18350.6i		−	0.777230i	0.921400	−	0.388615i	$-0.127046\pi$
$824$		0			0
$825$	−5582.57	−	7733.94i	−0.235588	−	0.326377i
$826$		0			0
$827$		−	15785.4i		−	0.663739i	−0.943325	−	0.331869i	$-0.892321\pi$
$827$		−	15785.4i		−	0.663739i	0.943325	−	0.331869i	$-0.107679\pi$
$828$		0			0
$829$			36102.8i			1.51255i	0.654255	+	0.756274i	$0.272982\pi$
$829$			36102.8i			1.51255i	−0.654255	+	0.756274i	$0.727018\pi$
$830$		0			0
$831$	−3348.15	−	4638.43i	−0.139766	−	0.193629i
$832$		0			0
$833$			72770.6i			3.02683i
$834$		0			0
$835$	3466.13			0.143653
$836$		0			0
$837$	8590.21	−	2703.26i	0.354744	−	0.111635i
$838$		0			0
$839$	−18961.8			−0.780255			−0.390127	−	0.920761i	$-0.627569\pi$
$839$	−18961.8			−0.780255			−0.390127	+	0.920761i	$0.627569\pi$
$840$		0			0
$841$	−9429.48			−0.386628
$842$		0			0
$843$	−2275.45	−	3152.35i	−0.0929665	−	0.128793i
$844$		0			0
$845$	−14244.6			−0.579914
$846$		0			0
$847$			36943.4i			1.49869i
$848$		0			0
$849$	−9059.00	+	6539.03i	−0.366201	+	0.264333i
$850$		0			0
$851$			898.084i			0.0361762i
$852$		0			0
$853$		−	27761.8i		−	1.11435i	−0.830394	−	0.557177i	$-0.811884\pi$
$853$		−	27761.8i		−	1.11435i	0.830394	−	0.557177i	$-0.188116\pi$
$854$		0			0
$855$	6584.74	−	19847.1i	0.263384	−	0.793868i
$856$		0			0
$857$			2082.01i			0.0829874i	0.999139	+	0.0414937i	$0.0132117\pi$
$857$			2082.01i			0.0829874i	−0.999139	+	0.0414937i	$0.986788\pi$
$858$		0			0
$859$	34995.2			1.39001			0.695005	−	0.719004i	$-0.255402\pi$
$859$	34995.2			1.39001			0.695005	+	0.719004i	$0.255402\pi$
$860$		0			0
$861$	49956.6	−	36060.1i	1.97737	−	1.42732i
$862$		0			0
$863$	6045.46			0.238459			0.119229	−	0.992867i	$-0.461958\pi$
$863$	6045.46			0.238459			0.119229	+	0.992867i	$0.461958\pi$
$864$		0			0
$865$	27357.3			1.07535
$866$		0			0
$867$	2510.61	−	1812.23i	0.0983446	−	0.0709878i
$868$		0			0
$869$	−11114.9			−0.433886
$870$		0			0
$871$		−	620.337i		−	0.0241324i
$872$		0			0
$873$	1159.45	−	3494.70i	0.0449501	−	0.135484i
$874$		0			0
$875$		−	55307.3i		−	2.13683i
$876$		0			0
$877$			30101.8i			1.15903i	0.814963	+	0.579514i	$0.196757\pi$
$877$			30101.8i			1.15903i	−0.814963	+	0.579514i	$0.803243\pi$
$878$		0			0
$879$	−22131.2	+	15974.9i	−0.849224	+	0.612993i
$880$		0			0
$881$		−	16655.3i		−	0.636926i	−0.947935	−	0.318463i	$-0.896833\pi$
$881$		−	16655.3i		−	0.636926i	0.947935	−	0.318463i	$-0.103167\pi$
$882$		0			0
$883$	−24940.5			−0.950528			−0.475264	−	0.879843i	$-0.657647\pi$
$883$	−24940.5			−0.950528			−0.475264	+	0.879843i	$0.657647\pi$
$884$		0			0
$885$	−6423.66	−	8899.16i	−0.243987	−	0.338014i
$886$		0			0
$887$	32143.9			1.21678			0.608391	−	0.793637i	$-0.291815\pi$
$887$	32143.9			1.21678			0.608391	+	0.793637i	$0.291815\pi$
$888$		0			0
$889$	57613.3			2.17355
$890$		0			0
$891$	−21108.5	+	28310.1i	−0.793673	+	1.06445i
$892$		0			0
$893$	−7964.51			−0.298457
$894$		0			0
$895$		−	26454.1i		−	0.988004i
$896$		0			0
$897$	14105.3	+	19541.1i	0.525041	+	0.727378i
$898$		0			0
$899$			7850.94i			0.291261i
$900$		0			0
$901$		−	13736.2i		−	0.507900i
$902$		0			0
$903$	35608.7	+	49331.3i	1.31227	+	1.81799i
$904$		0			0
$905$			16985.9i			0.623902i
$906$		0			0
$907$	3376.58			0.123613			0.0618067	−	0.998088i	$-0.480314\pi$
$907$	3376.58			0.123613			0.0618067	+	0.998088i	$0.480314\pi$
$908$		0			0
$909$	3495.19	−	10534.9i	0.127534	−	0.384401i
$910$		0			0
$911$	45248.4			1.64561			0.822803	−	0.568326i	$-0.192409\pi$
$911$	45248.4			1.64561			0.822803	+	0.568326i	$0.192409\pi$
$912$		0			0
$913$	−15556.8			−0.563914
$914$		0			0
$915$	−5644.12	−	7819.21i	−0.203922	−	0.282508i
$916$		0			0
$917$	5832.52			0.210040
$918$		0			0
$919$		−	18545.2i		−	0.665668i	−0.942985	−	0.332834i	$-0.891995\pi$
$919$		−	18545.2i		−	0.665668i	0.942985	−	0.332834i	$-0.108005\pi$
$920$		0			0
$921$	22486.1	−	16231.1i	0.804498	−	0.580709i
$922$		0			0
$923$		−	10352.8i		−	0.369195i
$924$		0			0
$925$		−	190.037i		−	0.00675499i
$926$		0			0
$927$	−3523.81	−	1169.11i	−0.124851	−	0.0414223i
$928$		0			0
$929$		−	33976.7i		−	1.19993i	−0.800024	−	0.599967i	$-0.795180\pi$
$929$		−	33976.7i		−	1.19993i	0.800024	−	0.599967i	$-0.204820\pi$
$930$		0			0
$931$	81360.0			2.86409
$932$		0			0
$933$	4360.03	−	3147.19i	0.152992	−	0.110433i
$934$		0			0
$935$	33555.7			1.17368
$936$		0			0
$937$	22803.1			0.795031			0.397516	−	0.917595i	$-0.369872\pi$
$937$	22803.1			0.795031			0.397516	+	0.917595i	$0.369872\pi$
$938$		0			0
$939$	−40321.7	+	29105.3i	−1.40133	+	1.01152i
$940$		0			0
$941$	24174.7			0.837484			0.418742	−	0.908105i	$-0.362471\pi$
$941$	24174.7			0.837484			0.418742	+	0.908105i	$0.362471\pi$
$942$		0			0
$943$		−	58369.4i		−	2.01566i
$944$		0			0
$945$	−45437.7	+	14298.8i	−1.56412	+	0.492213i
$946$		0			0
$947$			8044.99i			0.276058i	0.990428	+	0.138029i	$0.0440767\pi$
$947$			8044.99i			0.276058i	−0.990428	+	0.138029i	$0.955923\pi$
$948$		0			0
$949$			17342.6i			0.593220i
$950$		0			0
$951$	−15138.3	+	10927.2i	−0.516185	+	0.372596i
$952$		0			0
$953$			39730.8i			1.35048i	0.737598	+	0.675240i	$0.235960\pi$
$953$			39730.8i			1.35048i	−0.737598	+	0.675240i	$0.764040\pi$
$954$		0			0
$955$	−43184.5			−1.46326
$956$		0			0
$957$	−18018.4	−	24962.2i	−0.608622	−	0.843169i
$958$		0			0
$959$	54112.9			1.82210
$960$		0			0
$961$	25670.7			0.861694
$962$		0			0
$963$	−54150.8	−	17965.8i	−1.81203	−	0.601183i
$964$		0			0
$965$	25855.2			0.862496
$966$		0			0
$967$		−	17685.2i		−	0.588126i	−0.955786	−	0.294063i	$-0.904992\pi$
$967$		−	17685.2i		−	0.588126i	0.955786	−	0.294063i	$-0.0950076\pi$
$968$		0			0
$969$	−18731.2	−	25949.7i	−0.620982	−	0.860292i
$970$		0			0
$971$			44010.1i			1.45453i	0.686355	+	0.727267i	$0.259210\pi$
$971$			44010.1i			1.45453i	−0.686355	+	0.727267i	$0.740790\pi$
$972$		0			0
$973$			2137.86i			0.0704385i
$974$		0			0
$975$	−2984.71	−	4134.94i	−0.0980382	−	0.135820i
$976$		0			0
$977$		−	33518.4i		−	1.09759i	−0.835956	−	0.548796i	$-0.815086\pi$
$977$		−	33518.4i		−	1.09759i	0.835956	−	0.548796i	$-0.184914\pi$
$978$		0			0
$979$	−6367.36			−0.207867
$980$		0			0
$981$	33572.4	+	11138.4i	1.09265	+	0.362511i
$982$		0			0
$983$	43740.3			1.41923			0.709613	−	0.704592i	$-0.248870\pi$
$983$	43740.3			1.41923			0.709613	+	0.704592i	$0.248870\pi$
$984$		0			0
$985$	36212.2			1.17139
$986$		0			0
$987$	10618.7	+	14710.8i	0.342448	+	0.474418i
$988$		0			0
$989$	57638.8			1.85319
$990$		0			0
$991$		−	10468.7i		−	0.335568i	−0.985824	−	0.167784i	$-0.946339\pi$
$991$		−	10468.7i		−	0.335568i	0.985824	−	0.167784i	$-0.0536611\pi$
$992$		0			0
$993$	17355.4	−	12527.6i	0.554638	−	0.400353i
$994$		0			0
$995$			18806.0i			0.599186i
$996$		0			0
$997$		−	9561.56i		−	0.303729i	−0.988401	−	0.151864i	$-0.951472\pi$
$997$		−	9561.56i		−	0.303729i	0.988401	−	0.151864i	$-0.0485277\pi$
$998$		0			0
$999$	−671.121	+	211.195i	−0.0212546	+	0.00668862i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	768.4.f.e.383.4		12
3.2	odd	2		768.4.f.f.383.3		12
4.3	odd	2		768.4.f.g.383.9		12
8.3	odd	2		768.4.f.f.383.4		12
8.5	even	2		768.4.f.h.383.9		12
12.11	even	2		768.4.f.h.383.10		12
16.3	odd	4		384.4.c.d.383.8	yes	12
16.5	even	4		384.4.c.c.383.8	yes	12
16.11	odd	4		384.4.c.b.383.5	yes	12
16.13	even	4		384.4.c.a.383.5	✓	12
24.5	odd	2		768.4.f.g.383.10		12
24.11	even	2	inner	768.4.f.e.383.3		12
48.5	odd	4		384.4.c.b.383.6	yes	12
48.11	even	4		384.4.c.c.383.7	yes	12
48.29	odd	4		384.4.c.d.383.7	yes	12
48.35	even	4		384.4.c.a.383.6	yes	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.4.c.a.383.5	✓	12	16.13	even	4
384.4.c.a.383.6	yes	12	48.35	even	4
384.4.c.b.383.5	yes	12	16.11	odd	4
384.4.c.b.383.6	yes	12	48.5	odd	4
384.4.c.c.383.7	yes	12	48.11	even	4
384.4.c.c.383.8	yes	12	16.5	even	4
384.4.c.d.383.7	yes	12	48.29	odd	4
384.4.c.d.383.8	yes	12	16.3	odd	4
768.4.f.e.383.3		12	24.11	even	2	inner
768.4.f.e.383.4		12	1.1	even	1	trivial
768.4.f.f.383.3		12	3.2	odd	2
768.4.f.f.383.4		12	8.3	odd	2
768.4.f.g.383.9		12	4.3	odd	2
768.4.f.g.383.10		12	24.5	odd	2
768.4.f.h.383.9		12	8.5	even	2
768.4.f.h.383.10		12	12.11	even	2

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:	\( N \)	\(=\)	\( 768 = 2^{8} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	768.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+(-4.21320 + 3.04120i) q^{3} -9.33303 q^{5} -36.3792i q^{7} +(8.50216 - 25.6264i) q^{9} +O(q^{10})\)	\(q+(-4.21320 + 3.04120i) q^{3} -9.33303 q^{5} -36.3792i q^{7} +(8.50216 - 25.6264i) q^{9} -48.4408i q^{11} -25.8988i q^{13} +(39.3219 - 28.3836i) q^{15} -74.2219i q^{17} -82.9826 q^{19} +(110.637 + 153.273i) q^{21} +179.084 q^{23} -37.8946 q^{25} +(42.1138 + 133.826i) q^{27} -122.309 q^{29} -64.1893i q^{31} +(147.318 + 204.091i) q^{33} +339.528i q^{35} +5.01487i q^{37} +(78.7635 + 109.117i) q^{39} -325.933i q^{41} +321.853 q^{43} +(-79.3509 + 239.172i) q^{45} +95.9780 q^{47} -980.446 q^{49} +(225.724 + 312.712i) q^{51} +185.069 q^{53} +452.099i q^{55} +(349.623 - 252.367i) q^{57} -226.316i q^{59} -198.851i q^{61} +(-932.268 - 309.302i) q^{63} +241.714i q^{65} +23.9524 q^{67} +(-754.518 + 544.632i) q^{69} +399.741 q^{71} -669.631 q^{73} +(159.658 - 115.245i) q^{75} -1762.24 q^{77} -229.453i q^{79} +(-584.427 - 435.760i) q^{81} -321.150i q^{83} +692.715i q^{85} +(515.313 - 371.967i) q^{87} -131.446i q^{89} -942.177 q^{91} +(195.213 + 270.443i) q^{93} +774.479 q^{95} +136.371 q^{97} +(-1241.36 - 411.851i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 2 q^{3} - 12 q^{5}+O(q^{10}) \) 12 * q - 2 * q^3 - 12 * q^5	\( 12 q - 2 q^{3} - 12 q^{5} - 84 q^{15} - 180 q^{19} - 156 q^{21} - 120 q^{23} + 300 q^{25} + 130 q^{27} - 588 q^{29} - 116 q^{33} + 620 q^{39} + 372 q^{43} + 740 q^{45} + 1248 q^{47} - 948 q^{49} + 360 q^{51} + 948 q^{53} + 172 q^{57} - 2744 q^{63} - 2292 q^{67} - 3280 q^{69} - 2040 q^{71} + 216 q^{73} + 2522 q^{75} - 4824 q^{77} - 1076 q^{81} + 4156 q^{87} - 3480 q^{91} + 4180 q^{93} + 5448 q^{95} - 48 q^{97} + 3048 q^{99}+O(q^{100}) \) 12 * q - 2 * q^3 - 12 * q^5 - 84 * q^15 - 180 * q^19 - 156 * q^21 - 120 * q^23 + 300 * q^25 + 130 * q^27 - 588 * q^29 - 116 * q^33 + 620 * q^39 + 372 * q^43 + 740 * q^45 + 1248 * q^47 - 948 * q^49 + 360 * q^51 + 948 * q^53 + 172 * q^57 - 2744 * q^63 - 2292 * q^67 - 3280 * q^69 - 2040 * q^71 + 216 * q^73 + 2522 * q^75 - 4824 * q^77 - 1076 * q^81 + 4156 * q^87 - 3480 * q^91 + 4180 * q^93 + 5448 * q^95 - 48 * q^97 + 3048 * q^99

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