LMFDB - Embedded newform 768.4.f.g.383.11

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			383.11
Root			$-4.03251i$ of defining polynomial
Character	$\chi$	$=$	768.383
Dual form			768.4.f.g.383.12

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(4.54315 - 2.52186i) q^{3} +8.01133 q^{5} +12.6015i q^{7} +(14.2805 - 22.9144i) q^{9} +O(q^{10})$	\(q+(4.54315 - 2.52186i) q^{3} +8.01133 q^{5} +12.6015i q^{7} +(14.2805 - 22.9144i) q^{9} -37.7546i q^{11} -60.0661i q^{13} +(36.3967 - 20.2034i) q^{15} -37.0747i q^{17} -127.761 q^{19} +(31.7792 + 57.2506i) q^{21} +56.9249 q^{23} -60.8186 q^{25} +(7.09157 - 140.117i) q^{27} -220.677 q^{29} +2.26106i q^{31} +(-95.2117 - 171.525i) q^{33} +100.955i q^{35} -166.549i q^{37} +(-151.478 - 272.890i) q^{39} +154.816i q^{41} +53.4029 q^{43} +(114.406 - 183.575i) q^{45} +591.855 q^{47} +184.202 q^{49} +(-93.4972 - 168.436i) q^{51} +538.619 q^{53} -302.465i q^{55} +(-580.437 + 322.195i) q^{57} -586.798i q^{59} -431.489i q^{61} +(288.756 + 179.956i) q^{63} -481.209i q^{65} +175.719 q^{67} +(258.619 - 143.557i) q^{69} -29.5180 q^{71} -937.448 q^{73} +(-276.308 + 153.376i) q^{75} +475.765 q^{77} +409.780i q^{79} +(-321.136 - 654.456i) q^{81} +1299.81i q^{83} -297.018i q^{85} +(-1002.57 + 556.517i) q^{87} -921.475i q^{89} +756.924 q^{91} +(5.70207 + 10.2723i) q^{93} -1023.53 q^{95} -1644.22 q^{97} +(-865.123 - 539.153i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	4.54315	−	2.52186i	0.874330	−	0.485332i
$3$	4.54315	−	2.52186i	0.874330	−	0.485332i
$4$		0			0
$5$	8.01133			0.716555			0.358278	−	0.933615i	$-0.383364\pi$
$5$	8.01133			0.716555			0.358278	+	0.933615i	$0.383364\pi$
$6$		0			0
$7$			12.6015i			0.680418i	0.940350	+	0.340209i	$0.110498\pi$
$7$			12.6015i			0.680418i	−0.940350	+	0.340209i	$0.889502\pi$
$8$		0			0
$9$	14.2805	−	22.9144i	0.528906	−	0.848680i
$10$		0			0
$11$		−	37.7546i		−	1.03486i	−0.855726	−	0.517429i	$-0.826889\pi$
$11$		−	37.7546i		−	1.03486i	0.855726	−	0.517429i	$-0.173111\pi$
$12$		0			0
$13$		−	60.0661i		−	1.28149i	−0.767755	−	0.640744i	$-0.778626\pi$
$13$		−	60.0661i		−	1.28149i	0.767755	−	0.640744i	$-0.221374\pi$
$14$		0			0
$15$	36.3967	−	20.2034i	0.626506	−	0.347767i
$16$		0			0
$17$		−	37.0747i		−	0.528938i	−0.964394	−	0.264469i	$-0.914803\pi$
$17$		−	37.0747i		−	0.528938i	0.964394	−	0.264469i	$-0.0851967\pi$
$18$		0			0
$19$	−127.761			−1.54265			−0.771325	−	0.636441i	$-0.780406\pi$
$19$	−127.761			−1.54265			−0.771325	+	0.636441i	$0.780406\pi$
$20$		0			0
$21$	31.7792	+	57.2506i	0.330228	+	0.594910i
$22$		0			0
$23$	56.9249			0.516072			0.258036	−	0.966135i	$-0.416925\pi$
$23$	56.9249			0.516072			0.258036	+	0.966135i	$0.416925\pi$
$24$		0			0
$25$	−60.8186			−0.486549
$26$		0			0
$27$	7.09157	−	140.117i	0.0505471	−	0.998722i
$28$		0			0
$29$	−220.677			−1.41306			−0.706530	−	0.707683i	$-0.749741\pi$
$29$	−220.677			−1.41306			−0.706530	+	0.707683i	$0.749741\pi$
$30$		0			0
$31$			2.26106i			0.0130999i	0.999979	+	0.00654997i	$0.00208493\pi$
$31$			2.26106i			0.0130999i	−0.999979	+	0.00654997i	$0.997915\pi$
$32$		0			0
$33$	−95.2117	−	171.525i	−0.502250	−	0.904808i
$34$		0			0
$35$			100.955i			0.487557i
$36$		0			0
$37$		−	166.549i		−	0.740015i	−0.929029	−	0.370007i	$-0.879355\pi$
$37$		−	166.549i		−	0.740015i	0.929029	−	0.370007i	$-0.120645\pi$
$38$		0			0
$39$	−151.478	−	272.890i	−0.621947	−	1.12044i
$40$		0			0
$41$			154.816i			0.589713i	0.955542	+	0.294857i	$0.0952719\pi$
$41$			154.816i			0.589713i	−0.955542	+	0.294857i	$0.904728\pi$
$42$		0			0
$43$	53.4029			0.189392			0.0946961	−	0.995506i	$-0.469812\pi$
$43$	53.4029			0.189392			0.0946961	+	0.995506i	$0.469812\pi$
$44$		0			0
$45$	114.406	−	183.575i	0.378990	−	0.608126i
$46$		0			0
$47$	591.855			1.83683			0.918413	−	0.395622i	$-0.129471\pi$
$47$	591.855			1.83683			0.918413	+	0.395622i	$0.129471\pi$
$48$		0			0
$49$	184.202			0.537032
$50$		0			0
$51$	−93.4972	−	168.436i	−0.256710	−	0.462466i
$52$		0			0
$53$	538.619			1.39594			0.697971	−	0.716126i	$-0.254086\pi$
$53$	538.619			1.39594			0.697971	+	0.716126i	$0.254086\pi$
$54$		0			0
$55$		−	302.465i		−	0.741533i
$56$		0			0
$57$	−580.437	+	322.195i	−1.34879	+	0.748697i
$58$		0			0
$59$		−	586.798i		−	1.29482i	−0.762141	−	0.647412i	$-0.775852\pi$
$59$		−	586.798i		−	1.29482i	0.762141	−	0.647412i	$-0.224148\pi$
$60$		0			0
$61$		−	431.489i		−	0.905681i	−0.891592	−	0.452840i	$-0.850411\pi$
$61$		−	431.489i		−	0.905681i	0.891592	−	0.452840i	$-0.149589\pi$
$62$		0			0
$63$	288.756	+	179.956i	0.577457	+	0.359877i
$64$		0			0
$65$		−	481.209i		−	0.918257i
$66$		0			0
$67$	175.719			0.320411			0.160205	−	0.987084i	$-0.448784\pi$
$67$	175.719			0.320411			0.160205	+	0.987084i	$0.448784\pi$
$68$		0			0
$69$	258.619	−	143.557i	0.451218	−	0.250466i
$70$		0			0
$71$	−29.5180			−0.0493400			−0.0246700	−	0.999696i	$-0.507853\pi$
$71$	−29.5180			−0.0493400			−0.0246700	+	0.999696i	$0.507853\pi$
$72$		0			0
$73$	−937.448			−1.50301			−0.751506	−	0.659726i	$-0.770672\pi$
$73$	−937.448			−1.50301			−0.751506	+	0.659726i	$0.770672\pi$
$74$		0			0
$75$	−276.308	+	153.376i	−0.425404	+	0.236138i
$76$		0			0
$77$	475.765			0.704136
$78$		0			0
$79$			409.780i			0.583593i	0.956480	+	0.291796i	$0.0942530\pi$
$79$			409.780i			0.583593i	−0.956480	+	0.291796i	$0.905747\pi$
$80$		0			0
$81$	−321.136	−	654.456i	−0.440516	−	0.897745i
$82$		0			0
$83$			1299.81i			1.71895i	0.511174	+	0.859477i	$0.329211\pi$
$83$			1299.81i			1.71895i	−0.511174	+	0.859477i	$0.670789\pi$
$84$		0			0
$85$		−	297.018i		−	0.379013i
$86$		0			0
$87$	−1002.57	+	556.517i	−1.23548	+	0.685803i
$88$		0			0
$89$		−	921.475i		−	1.09748i	−0.835992	−	0.548742i	$-0.815107\pi$
$89$		−	921.475i		−	1.09748i	0.835992	−	0.548742i	$-0.184893\pi$
$90$		0			0
$91$	756.924			0.871947
$92$		0			0
$93$	5.70207	+	10.2723i	0.00635781	+	0.0114537i
$94$		0			0
$95$	−1023.53			−1.10539
$96$		0			0
$97$	−1644.22			−1.72108			−0.860542	−	0.509379i	$-0.829875\pi$
$97$	−1644.22			−1.72108			−0.860542	+	0.509379i	$0.829875\pi$
$98$		0			0
$99$	−865.123	−	539.153i	−0.878264	−	0.547343i
$100$		0			0
$101$	1708.51			1.68320			0.841598	−	0.540105i	$-0.181615\pi$
$101$	1708.51			1.68320			0.841598	+	0.540105i	$0.181615\pi$
$102$		0			0
$103$			354.419i			0.339048i	0.985526	+	0.169524i	$0.0542230\pi$
$103$			354.419i			0.339048i	−0.985526	+	0.169524i	$0.945777\pi$
$104$		0			0
$105$	254.594	+	458.653i	0.236627	+	0.426286i
$106$		0			0
$107$			532.454i			0.481068i	0.970641	+	0.240534i	$0.0773226\pi$
$107$			532.454i			0.481068i	−0.970641	+	0.240534i	$0.922677\pi$
$108$		0			0
$109$		−	2099.43i		−	1.84485i	−0.386171	−	0.922427i	$-0.626203\pi$
$109$		−	2099.43i		−	1.84485i	0.386171	−	0.922427i	$-0.373797\pi$
$110$		0			0
$111$	−420.014	−	756.659i	−0.359153	−	0.647017i
$112$		0			0
$113$			510.544i			0.425026i	0.977158	+	0.212513i	$0.0681647\pi$
$113$			510.544i			0.425026i	−0.977158	+	0.212513i	$0.931835\pi$
$114$		0			0
$115$	456.044			0.369794
$116$		0			0
$117$	−1376.38	−	857.772i	−1.08757	−	0.677787i
$118$		0			0
$119$	467.198			0.359899
$120$		0			0
$121$	−94.4098			−0.0709315
$122$		0			0
$123$	390.425	+	703.354i	0.286207	+	0.515604i
$124$		0			0
$125$	−1488.65			−1.06519
$126$		0			0
$127$			989.228i			0.691180i	0.938386	+	0.345590i	$0.112321\pi$
$127$			989.228i			0.691180i	−0.938386	+	0.345590i	$0.887679\pi$
$128$		0			0
$129$	242.617	−	134.674i	0.165591	−	0.0919180i
$130$		0			0
$131$			519.522i			0.346495i	0.984878	+	0.173248i	$0.0554261\pi$
$131$			519.522i			0.346495i	−0.984878	+	0.173248i	$0.944574\pi$
$132$		0			0
$133$		−	1609.98i		−	1.04965i
$134$		0			0
$135$	56.8129	−	1122.52i	0.0362198	−	0.715639i
$136$		0			0
$137$		−	2731.83i		−	1.70362i	−0.523853	−	0.851809i	$-0.675506\pi$
$137$		−	2731.83i		−	1.70362i	0.523853	−	0.851809i	$-0.324494\pi$
$138$		0			0
$139$	182.811			0.111553			0.0557765	−	0.998443i	$-0.482237\pi$
$139$	182.811			0.111553			0.0557765	+	0.998443i	$0.482237\pi$
$140$		0			0
$141$	2688.89	−	1492.57i	1.60599	−	0.891470i
$142$		0			0
$143$	−2267.77			−1.32616
$144$		0			0
$145$	−1767.92			−1.01254
$146$		0			0
$147$	836.857	−	464.531i	0.469543	−	0.260638i
$148$		0			0
$149$	−935.337			−0.514267			−0.257134	−	0.966376i	$-0.582778\pi$
$149$	−935.337			−0.514267			−0.257134	+	0.966376i	$0.582778\pi$
$150$		0			0
$151$			3637.89i			1.96058i	0.197572	+	0.980288i	$0.436694\pi$
$151$			3637.89i			1.96058i	−0.197572	+	0.980288i	$0.563306\pi$
$152$		0			0
$153$	−849.544	−	529.445i	−0.448899	−	0.279759i
$154$		0			0
$155$			18.1141i			0.00938682i
$156$		0			0
$157$			2194.94i			1.11577i	0.829919	+	0.557883i	$0.188386\pi$
$157$			2194.94i			1.11577i	−0.829919	+	0.557883i	$0.811614\pi$
$158$		0			0
$159$	2447.03	−	1358.32i	1.22051	−	0.677495i
$160$		0			0
$161$			717.340i			0.351145i
$162$		0			0
$163$	2938.89			1.41222			0.706109	−	0.708103i	$-0.250449\pi$
$163$	2938.89			1.41222			0.706109	+	0.708103i	$0.250449\pi$
$164$		0			0
$165$	−762.773	−	1374.14i	−0.359889	−	0.648345i
$166$		0			0
$167$	3106.90			1.43964			0.719818	−	0.694163i	$-0.244225\pi$
$167$	3106.90			1.43964			0.719818	+	0.694163i	$0.244225\pi$
$168$		0			0
$169$	−1410.94			−0.642211
$170$		0			0
$171$	−1824.49	+	2927.56i	−0.815918	+	1.30922i
$172$		0			0
$173$	2882.13			1.26662			0.633308	−	0.773900i	$-0.281697\pi$
$173$	2882.13			1.26662			0.633308	+	0.773900i	$0.281697\pi$
$174$		0			0
$175$		−	766.406i		−	0.331056i
$176$		0			0
$177$	−1479.82	−	2665.91i	−0.628419	−	1.13210i
$178$		0			0
$179$		−	2571.39i		−	1.07371i	−0.843673	−	0.536857i	$-0.819611\pi$
$179$		−	2571.39i		−	1.07371i	0.843673	−	0.536857i	$-0.180389\pi$
$180$		0			0
$181$		−	563.098i		−	0.231242i	−0.993293	−	0.115621i	$-0.963114\pi$
$181$		−	563.098i		−	0.231242i	0.993293	−	0.115621i	$-0.0368858\pi$
$182$		0			0
$183$	−1088.15	−	1960.32i	−0.439556	−	0.791864i
$184$		0			0
$185$		−	1334.28i		−	0.530261i
$186$		0			0
$187$	−1399.74			−0.547376
$188$		0			0
$189$	1765.68	+	89.3645i	0.679548	+	0.0343932i
$190$		0			0
$191$	−2637.29			−0.999098			−0.499549	−	0.866286i	$-0.666501\pi$
$191$	−2637.29			−0.999098			−0.499549	+	0.866286i	$0.666501\pi$
$192$		0			0
$193$	1498.22			0.558777			0.279388	−	0.960178i	$-0.409868\pi$
$193$	1498.22			0.558777			0.279388	+	0.960178i	$0.409868\pi$
$194$		0			0
$195$	−1213.54	−	2186.21i	−0.445659	−	0.802860i
$196$		0			0
$197$	394.173			0.142557			0.0712783	−	0.997456i	$-0.477292\pi$
$197$	394.173			0.142557			0.0712783	+	0.997456i	$0.477292\pi$
$198$		0			0
$199$		−	3133.05i		−	1.11606i	−0.829821	−	0.558030i	$-0.811557\pi$
$199$		−	3133.05i		−	1.11606i	0.829821	−	0.558030i	$-0.188443\pi$
$200$		0			0
$201$	798.320	−	443.139i	0.280145	−	0.155506i
$202$		0			0
$203$		−	2780.87i		−	0.961471i
$204$		0			0
$205$			1240.28i			0.422562i
$206$		0			0
$207$	812.915	−	1304.40i	0.272954	−	0.437981i
$208$		0			0
$209$			4823.56i			1.59642i
$210$		0			0
$211$	−1665.67			−0.543458			−0.271729	−	0.962374i	$-0.587595\pi$
$211$	−1665.67			−0.543458			−0.271729	+	0.962374i	$0.587595\pi$
$212$		0			0
$213$	−134.105	+	74.4401i	−0.0431394	+	0.0239462i
$214$		0			0
$215$	427.828			0.135710
$216$		0			0
$217$	−28.4927			−0.00891343
$218$		0			0
$219$	−4258.97	+	2364.11i	−1.31413	+	0.729460i
$220$		0			0
$221$	−2226.94			−0.677828
$222$		0			0
$223$			2675.69i			0.803488i	0.915752	+	0.401744i	$0.131596\pi$
$223$			2675.69i			0.803488i	−0.915752	+	0.401744i	$0.868404\pi$
$224$		0			0
$225$	−868.518	+	1393.62i	−0.257339	+	0.412924i
$226$		0			0
$227$		−	47.7540i		−	0.0139627i	−0.999976	−	0.00698137i	$-0.997778\pi$
$227$		−	47.7540i		−	0.0139627i	0.999976	−	0.00698137i	$-0.00222226\pi$
$228$		0			0
$229$		−	1392.28i		−	0.401767i	−0.979615	−	0.200884i	$-0.935619\pi$
$229$		−	1392.28i		−	0.401767i	0.979615	−	0.200884i	$-0.0643813\pi$
$230$		0			0
$231$	2161.47	−	1199.81i	0.615647	−	0.341740i
$232$		0			0
$233$			1627.49i			0.457598i	0.973474	+	0.228799i	$0.0734799\pi$
$233$			1627.49i			0.457598i	−0.973474	+	0.228799i	$0.926520\pi$
$234$		0			0
$235$	4741.54			1.31619
$236$		0			0
$237$	1033.41	+	1861.69i	0.283236	+	0.510253i
$238$		0			0
$239$	2858.90			0.773752			0.386876	−	0.922132i	$-0.373554\pi$
$239$	2858.90			0.773752			0.386876	+	0.922132i	$0.373554\pi$
$240$		0			0
$241$	−1021.31			−0.272982			−0.136491	−	0.990641i	$-0.543582\pi$
$241$	−1021.31			−0.272982			−0.136491	+	0.990641i	$0.543582\pi$
$242$		0			0
$243$	−3109.42	−	2163.43i	−0.820861	−	0.571128i
$244$		0			0
$245$	1475.70			0.384813
$246$		0			0
$247$			7674.10i			1.97689i
$248$		0			0
$249$	3277.95	+	5905.26i	0.834263	+	1.50293i
$250$		0			0
$251$		−	3017.49i		−	0.758814i	−0.925230	−	0.379407i	$-0.876128\pi$
$251$		−	3017.49i		−	0.758814i	0.925230	−	0.379407i	$-0.123872\pi$
$252$		0			0
$253$		−	2149.18i		−	0.534062i
$254$		0			0
$255$	−749.037	−	1349.40i	−0.183947	−	0.331383i
$256$		0			0
$257$			1097.93i			0.266486i	0.991083	+	0.133243i	$0.0425390\pi$
$257$			1097.93i			0.266486i	−0.991083	+	0.133243i	$0.957461\pi$
$258$		0			0
$259$	2098.77			0.503519
$260$		0			0
$261$	−3151.37	+	5056.68i	−0.747376	+	1.19924i
$262$		0			0
$263$	2785.15			0.653002			0.326501	−	0.945197i	$-0.394130\pi$
$263$	2785.15			0.653002			0.326501	+	0.945197i	$0.394130\pi$
$264$		0			0
$265$	4315.05			1.00027
$266$		0			0
$267$	−2323.83	−	4186.40i	−0.532644	−	0.959564i
$268$		0			0
$269$	−4034.08			−0.914358			−0.457179	−	0.889375i	$-0.651140\pi$
$269$	−4034.08			−0.914358			−0.457179	+	0.889375i	$0.651140\pi$
$270$		0			0
$271$		−	2226.42i		−	0.499060i	−0.968367	−	0.249530i	$-0.919724\pi$
$271$		−	2226.42i		−	0.499060i	0.968367	−	0.249530i	$-0.0802761\pi$
$272$		0			0
$273$	3438.82	−	1908.85i	0.762370	−	0.423184i
$274$		0			0
$275$			2296.18i			0.503509i
$276$		0			0
$277$			6418.65i			1.39227i	0.717910	+	0.696136i	$0.245099\pi$
$277$			6418.65i			1.39227i	−0.717910	+	0.696136i	$0.754901\pi$
$278$		0			0
$279$	51.8107	+	32.2890i	0.0111177	+	0.00692864i
$280$		0			0
$281$		−	3917.54i		−	0.831675i	−0.909439	−	0.415838i	$-0.863488\pi$
$281$		−	3917.54i		−	0.831675i	0.909439	−	0.415838i	$-0.136512\pi$
$282$		0			0
$283$	4219.70			0.886343			0.443171	−	0.896437i	$-0.353853\pi$
$283$	4219.70			0.886343			0.443171	+	0.896437i	$0.353853\pi$
$284$		0			0
$285$	−4650.08	+	2581.21i	−0.966480	+	0.536483i
$286$		0			0
$287$	−1950.92			−0.401251
$288$		0			0
$289$	3538.46			0.720225
$290$		0			0
$291$	−7469.94	+	4146.49i	−1.50480	+	0.835297i
$292$		0			0
$293$	766.953			0.152921			0.0764605	−	0.997073i	$-0.475638\pi$
$293$	766.953			0.152921			0.0764605	+	0.997073i	$0.475638\pi$
$294$		0			0
$295$		−	4701.03i		−	0.927812i
$296$		0			0
$297$	−5290.05	−	267.739i	−1.03354	−	0.0523091i
$298$		0			0
$299$		−	3419.26i		−	0.661341i
$300$		0			0
$301$			672.957i			0.128866i
$302$		0			0
$303$	7762.01	−	4308.61i	1.47167	−	0.816908i
$304$		0			0
$305$		−	3456.80i		−	0.648970i
$306$		0			0
$307$	7296.62			1.35648			0.678242	−	0.734839i	$-0.262742\pi$
$307$	7296.62			1.35648			0.678242	+	0.734839i	$0.262742\pi$
$308$		0			0
$309$	893.794	+	1610.18i	0.164551	+	0.296440i
$310$		0			0
$311$	−8641.39			−1.57559			−0.787795	−	0.615938i	$-0.788777\pi$
$311$	−8641.39			−1.57559			−0.787795	+	0.615938i	$0.788777\pi$
$312$		0			0
$313$	2171.54			0.392149			0.196075	−	0.980589i	$-0.437181\pi$
$313$	2171.54			0.392149			0.196075	+	0.980589i	$0.437181\pi$
$314$		0			0
$315$	2313.32	+	1441.68i	0.413780	+	0.257872i
$316$		0			0
$317$	−1682.16			−0.298043			−0.149021	−	0.988834i	$-0.547612\pi$
$317$	−1682.16			−0.298043			−0.149021	+	0.988834i	$0.547612\pi$
$318$		0			0
$319$			8331.58i			1.46232i
$320$		0			0
$321$	1342.77	+	2419.02i	0.233478	+	0.420613i
$322$		0			0
$323$			4736.70i			0.815967i
$324$		0			0
$325$			3653.14i			0.623506i
$326$		0			0
$327$	−5294.47	−	9538.04i	−0.895366	−	1.61301i
$328$		0			0
$329$			7458.26i			1.24981i
$330$		0			0
$331$	2389.41			0.396779			0.198389	−	0.980123i	$-0.436429\pi$
$331$	2389.41			0.396779			0.198389	+	0.980123i	$0.436429\pi$
$332$		0			0
$333$	−3816.37	−	2378.40i	−0.628036	−	0.391398i
$334$		0			0
$335$	1407.75			0.229592
$336$		0			0
$337$	11043.0			1.78501			0.892505	−	0.451037i	$-0.148946\pi$
$337$	11043.0			1.78501			0.892505	+	0.451037i	$0.148946\pi$
$338$		0			0
$339$	1287.52	+	2319.48i	0.206278	+	0.371613i
$340$		0			0
$341$	85.3653			0.0135566
$342$		0			0
$343$			6643.54i			1.04582i
$344$		0			0
$345$	2071.88	−	1150.08i	0.323322	−	0.179473i
$346$		0			0
$347$			8790.06i			1.35987i	0.733272	+	0.679935i	$0.237992\pi$
$347$			8790.06i			1.35987i	−0.733272	+	0.679935i	$0.762008\pi$
$348$		0			0
$349$			2636.08i			0.404316i	0.979353	+	0.202158i	$0.0647955\pi$
$349$			2636.08i			0.404316i	−0.979353	+	0.202158i	$0.935205\pi$
$350$		0			0
$351$	−8416.27	−	425.963i	−1.27985	−	0.0647756i
$352$		0			0
$353$		−	10828.6i		−	1.63272i	−0.577544	−	0.816359i	$-0.695989\pi$
$353$		−	10828.6i		−	1.63272i	0.577544	−	0.816359i	$-0.304011\pi$
$354$		0			0
$355$	−236.478			−0.0353548
$356$		0			0
$357$	2122.55	−	1178.21i	0.314670	−	0.174670i
$358$		0			0
$359$	−278.615			−0.0409603			−0.0204802	−	0.999790i	$-0.506519\pi$
$359$	−278.615			−0.0409603			−0.0204802	+	0.999790i	$0.506519\pi$
$360$		0			0
$361$	9463.85			1.37977
$362$		0			0
$363$	−428.918	+	238.088i	−0.0620176	+	0.0344253i
$364$		0			0
$365$	−7510.20			−1.07699
$366$		0			0
$367$			12082.9i			1.71860i	0.511476	+	0.859298i	$0.329099\pi$
$367$			12082.9i			1.71860i	−0.511476	+	0.859298i	$0.670901\pi$
$368$		0			0
$369$	3547.52	+	2210.85i	0.500478	+	0.311903i
$370$		0			0
$371$			6787.41i			0.949824i
$372$		0			0
$373$			2101.82i			0.291764i	0.989302	+	0.145882i	$0.0466020\pi$
$373$			2101.82i			0.291764i	−0.989302	+	0.145882i	$0.953398\pi$
$374$		0			0
$375$	−6763.18	+	3754.17i	−0.931331	+	0.516973i
$376$		0			0
$377$			13255.2i			1.81082i
$378$		0			0
$379$	2146.64			0.290938			0.145469	−	0.989363i	$-0.453531\pi$
$379$	2146.64			0.290938			0.145469	+	0.989363i	$0.453531\pi$
$380$		0			0
$381$	2494.69	+	4494.21i	0.335451	+	0.604319i
$382$		0			0
$383$	8881.17			1.18487			0.592437	−	0.805617i	$-0.298166\pi$
$383$	8881.17			1.18487			0.592437	+	0.805617i	$0.298166\pi$
$384$		0			0
$385$	3811.51			0.504552
$386$		0			0
$387$	762.618	−	1223.69i	0.100171	−	0.160733i
$388$		0			0
$389$	−3095.15			−0.403420			−0.201710	−	0.979445i	$-0.564650\pi$
$389$	−3095.15			−0.403420			−0.201710	+	0.979445i	$0.564650\pi$
$390$		0			0
$391$		−	2110.48i		−	0.272970i
$392$		0			0
$393$	1310.16	+	2360.27i	0.168165	+	0.302951i
$394$		0			0
$395$			3282.88i			0.418176i
$396$		0			0
$397$			10755.4i			1.35969i	0.733354	+	0.679847i	$0.237954\pi$
$397$			10755.4i			1.35969i	−0.733354	+	0.679847i	$0.762046\pi$
$398$		0			0
$399$	−4060.14	−	7314.39i	−0.509427	−	0.917738i
$400$		0			0
$401$			8333.04i			1.03774i	0.854854	+	0.518868i	$0.173646\pi$
$401$			8333.04i			1.03774i	−0.854854	+	0.518868i	$0.826354\pi$
$402$		0			0
$403$	135.813			0.0167874
$404$		0			0
$405$	−2572.73	−	5243.06i	−0.315654	−	0.643283i
$406$		0			0
$407$	−6288.00			−0.765810
$408$		0			0
$409$	−8921.31			−1.07856			−0.539279	−	0.842127i	$-0.681303\pi$
$409$	−8921.31			−1.07856			−0.539279	+	0.842127i	$0.681303\pi$
$410$		0			0
$411$	−6889.27	−	12411.1i	−0.826820	−	1.48952i
$412$		0			0
$413$	7394.54			0.881021
$414$		0			0
$415$			10413.2i			1.23173i
$416$		0			0
$417$	830.540	−	461.024i	0.0975341	−	0.0541402i
$418$		0			0
$419$		−	6810.32i		−	0.794048i	−0.917808	−	0.397024i	$-0.870043\pi$
$419$		−	6810.32i		−	0.794048i	0.917808	−	0.397024i	$-0.129957\pi$
$420$		0			0
$421$			8336.91i			0.965121i	0.875863	+	0.482561i	$0.160293\pi$
$421$			8336.91i			0.965121i	−0.875863	+	0.482561i	$0.839707\pi$
$422$		0			0
$423$	8451.96	−	13562.0i	0.971509	−	1.55888i
$424$		0			0
$425$			2254.83i			0.257354i
$426$		0			0
$427$	5437.42			0.616241
$428$		0			0
$429$	−10302.8	+	5719.00i	−1.15950	+	0.643627i
$430$		0			0
$431$	755.175			0.0843979			0.0421990	−	0.999109i	$-0.486564\pi$
$431$	755.175			0.0843979			0.0421990	+	0.999109i	$0.486564\pi$
$432$		0			0
$433$	−2736.36			−0.303697			−0.151849	−	0.988404i	$-0.548523\pi$
$433$	−2736.36			−0.303697			−0.151849	+	0.988404i	$0.548523\pi$
$434$		0			0
$435$	−8031.92	+	4458.44i	−0.885290	+	0.491416i
$436$		0			0
$437$	−7272.78			−0.796120
$438$		0			0
$439$			15732.1i			1.71037i	0.518322	+	0.855186i	$0.326557\pi$
$439$			15732.1i			1.71037i	−0.518322	+	0.855186i	$0.673443\pi$
$440$		0			0
$441$	2630.49	−	4220.87i	0.284039	−	0.455768i
$442$		0			0
$443$		−	1568.89i		−	0.168263i	−0.996455	−	0.0841313i	$-0.973188\pi$
$443$		−	1568.89i		−	0.168263i	0.996455	−	0.0841313i	$-0.0268115\pi$
$444$		0			0
$445$		−	7382.24i		−	0.786408i
$446$		0			0
$447$	−4249.38	+	2358.79i	−0.449639	+	0.249590i
$448$		0			0
$449$			8882.13i			0.933571i	0.884371	+	0.466786i	$0.154588\pi$
$449$			8882.13i			0.933571i	−0.884371	+	0.466786i	$0.845412\pi$
$450$		0			0
$451$	5845.03			0.610270
$452$		0			0
$453$	9174.23	+	16527.5i	0.951530	+	1.71419i
$454$		0			0
$455$	6063.97			0.624798
$456$		0			0
$457$	−6441.62			−0.659357			−0.329679	−	0.944093i	$-0.606940\pi$
$457$	−6441.62			−0.659357			−0.329679	+	0.944093i	$0.606940\pi$
$458$		0			0
$459$	−5194.79	−	262.918i	−0.528262	−	0.0267363i
$460$		0			0
$461$	−4112.54			−0.415488			−0.207744	−	0.978183i	$-0.566612\pi$
$461$	−4112.54			−0.415488			−0.207744	+	0.978183i	$0.566612\pi$
$462$		0			0
$463$		−	16061.5i		−	1.61219i	−0.591788	−	0.806094i	$-0.701578\pi$
$463$		−	16061.5i		−	1.61219i	0.591788	−	0.806094i	$-0.298422\pi$
$464$		0			0
$465$	45.6811	+	82.2950i	0.00455572	+	0.00820718i
$466$		0			0
$467$		−	12155.4i		−	1.20447i	−0.798321	−	0.602233i	$-0.794278\pi$
$467$		−	12155.4i		−	1.20447i	0.798321	−	0.602233i	$-0.205722\pi$
$468$		0			0
$469$			2214.33i			0.218013i
$470$		0			0
$471$	5535.33	+	9971.95i	0.541517	+	0.975548i
$472$		0			0
$473$		−	2016.20i		−	0.195994i
$474$		0			0
$475$	7770.24			0.750575
$476$		0			0
$477$	7691.73	−	12342.1i	0.738323	−	1.18471i
$478$		0			0
$479$	3121.62			0.297767			0.148883	−	0.988855i	$-0.452432\pi$
$479$	3121.62			0.297767			0.148883	+	0.988855i	$0.452432\pi$
$480$		0			0
$481$	−10004.0			−0.948320
$482$		0			0
$483$	1809.03	+	3258.99i	0.170422	+	0.307017i
$484$		0			0
$485$	−13172.4			−1.23325
$486$		0			0
$487$		−	1582.51i		−	0.147249i	−0.997286	−	0.0736244i	$-0.976543\pi$
$487$		−	1582.51i		−	0.147249i	0.997286	−	0.0736244i	$-0.0234566\pi$
$488$		0			0
$489$	13351.8	−	7411.46i	1.23474	−	0.685394i
$490$		0			0
$491$			12211.6i			1.12241i	0.827677	+	0.561205i	$0.189662\pi$
$491$			12211.6i			1.12241i	−0.827677	+	0.561205i	$0.810338\pi$
$492$		0			0
$493$			8181.55i			0.747421i
$494$		0			0
$495$	−6930.78	−	4319.34i	−0.629324	−	0.392201i
$496$		0			0
$497$		−	371.971i		−	0.0335718i
$498$		0			0
$499$	3871.87			0.347352			0.173676	−	0.984803i	$-0.444435\pi$
$499$	3871.87			0.347352			0.173676	+	0.984803i	$0.444435\pi$
$500$		0			0
$501$	14115.1	−	7835.16i	1.25872	−	0.698701i
$502$		0			0
$503$	−10985.0			−0.973753			−0.486876	−	0.873471i	$-0.661864\pi$
$503$	−10985.0			−0.973753			−0.486876	+	0.873471i	$0.661864\pi$
$504$		0			0
$505$	13687.4			1.20610
$506$		0			0
$507$	−6410.11	+	3558.19i	−0.561505	+	0.311686i
$508$		0			0
$509$	21191.9			1.84541			0.922704	−	0.385509i	$-0.125974\pi$
$509$	21191.9			1.84541			0.922704	+	0.385509i	$0.125974\pi$
$510$		0			0
$511$		−	11813.3i		−	1.02268i
$512$		0			0
$513$	−906.025	+	17901.4i	−0.0779766	+	1.54068i
$514$		0			0
$515$			2839.37i			0.242947i
$516$		0			0
$517$		−	22345.2i		−	1.90086i
$518$		0			0
$519$	13094.0	−	7268.33i	1.10744	−	0.614729i
$520$		0			0
$521$		−	8953.89i		−	0.752931i	−0.926431	−	0.376465i	$-0.877139\pi$
$521$		−	8953.89i		−	0.752931i	0.926431	−	0.376465i	$-0.122861\pi$
$522$		0			0
$523$	−8410.95			−0.703223			−0.351611	−	0.936146i	$-0.614366\pi$
$523$	−8410.95			−0.703223			−0.351611	+	0.936146i	$0.614366\pi$
$524$		0			0
$525$	−1932.77	−	3481.90i	−0.160672	−	0.289453i
$526$		0			0
$527$	83.8281			0.00692905
$528$		0			0
$529$	−8926.55			−0.733669
$530$		0			0
$531$	−13446.1	−	8379.75i	−1.09889	−	0.684840i
$532$		0			0
$533$	9299.21			0.755710
$534$		0			0
$535$			4265.67i			0.344712i
$536$		0			0
$537$	−6484.69	−	11682.2i	−0.521108	−	0.938781i
$538$		0			0
$539$		−	6954.47i		−	0.555752i
$540$		0			0
$541$		−	2640.19i		−	0.209816i	−0.994482	−	0.104908i	$-0.966545\pi$
$541$		−	2640.19i		−	0.209816i	0.994482	−	0.104908i	$-0.0334548\pi$
$542$		0			0
$543$	−1420.05	−	2558.24i	−0.112229	−	0.202182i
$544$		0			0
$545$		−	16819.2i		−	1.32194i
$546$		0			0
$547$	−133.631			−0.0104455			−0.00522273	−	0.999986i	$-0.501662\pi$
$547$	−133.631			−0.0104455			−0.00522273	+	0.999986i	$0.501662\pi$
$548$		0			0
$549$	−9887.30	−	6161.87i	−0.768633	−	0.479020i
$550$		0			0
$551$	28193.9			2.17986
$552$		0			0
$553$	−5163.85			−0.397087
$554$		0			0
$555$	−3364.87	−	6061.84i	−0.257353	−	0.463623i
$556$		0			0
$557$	15450.3			1.17531			0.587657	−	0.809110i	$-0.300051\pi$
$557$	15450.3			1.17531			0.587657	+	0.809110i	$0.300051\pi$
$558$		0			0
$559$		−	3207.70i		−	0.242704i
$560$		0			0
$561$	−6359.24	+	3529.95i	−0.478587	+	0.265659i
$562$		0			0
$563$		−	11670.9i		−	0.873662i	−0.899544	−	0.436831i	$-0.856101\pi$
$563$		−	11670.9i		−	0.873662i	0.899544	−	0.436831i	$-0.143899\pi$
$564$		0			0
$565$			4090.13i			0.304554i
$566$		0			0
$567$	8247.13	−	4046.81i	0.610841	−	0.299735i
$568$		0			0
$569$		−	3730.57i		−	0.274857i	−0.990512	−	0.137428i	$-0.956116\pi$
$569$		−	3730.57i		−	0.274857i	0.990512	−	0.137428i	$-0.0438837\pi$
$570$		0			0
$571$	1257.89			0.0921907			0.0460954	−	0.998937i	$-0.485322\pi$
$571$	1257.89			0.0921907			0.0460954	+	0.998937i	$0.485322\pi$
$572$		0			0
$573$	−11981.6	+	6650.87i	−0.873541	+	0.484894i
$574$		0			0
$575$	−3462.09			−0.251094
$576$		0			0
$577$	3631.50			0.262013			0.131006	−	0.991382i	$-0.458179\pi$
$577$	3631.50			0.262013			0.131006	+	0.991382i	$0.458179\pi$
$578$		0			0
$579$	6806.62	−	3778.29i	0.488555	−	0.271192i
$580$		0			0
$581$	−16379.6			−1.16961
$582$		0			0
$583$		−	20335.3i		−	1.44460i
$584$		0			0
$585$	−11026.6	−	6871.90i	−0.779306	−	0.485672i
$586$		0			0
$587$			15455.9i			1.08677i	0.839485	+	0.543383i	$0.182857\pi$
$587$			15455.9i			1.08677i	−0.839485	+	0.543383i	$0.817143\pi$
$588$		0			0
$589$		−	288.875i		−	0.0202086i
$590$		0			0
$591$	1790.79	−	994.048i	0.124642	−	0.0691873i
$592$		0			0
$593$		−	15871.0i		−	1.09906i	−0.835474	−	0.549530i	$-0.814807\pi$
$593$		−	15871.0i		−	1.09906i	0.835474	−	0.549530i	$-0.185193\pi$
$594$		0			0
$595$	3742.88			0.257887
$596$		0			0
$597$	−7901.11	−	14233.9i	−0.541660	−	0.975805i
$598$		0			0
$599$	−4369.74			−0.298068			−0.149034	−	0.988832i	$-0.547616\pi$
$599$	−4369.74			−0.298068			−0.149034	+	0.988832i	$0.547616\pi$
$600$		0			0
$601$	−507.273			−0.0344295			−0.0172147	−	0.999852i	$-0.505480\pi$
$601$	−507.273			−0.0344295			−0.0172147	+	0.999852i	$0.505480\pi$
$602$		0			0
$603$	2509.35	−	4026.50i	0.169467	−	0.271926i
$604$		0			0
$605$	−756.348			−0.0508263
$606$		0			0
$607$			14525.6i			0.971293i	0.874155	+	0.485647i	$0.161416\pi$
$607$			14525.6i			0.971293i	−0.874155	+	0.485647i	$0.838584\pi$
$608$		0			0
$609$	−7012.95	−	12633.9i	−0.466633	−	0.840643i
$610$		0			0
$611$		−	35550.4i		−	2.35387i
$612$		0			0
$613$		−	14948.5i		−	0.984932i	−0.870332	−	0.492466i	$-0.836096\pi$
$613$		−	14948.5i		−	0.984932i	0.870332	−	0.492466i	$-0.163904\pi$
$614$		0			0
$615$	3127.82	+	5634.80i	0.205083	+	0.369459i
$616$		0			0
$617$			3250.68i			0.212103i	0.994361	+	0.106052i	$0.0338209\pi$
$617$			3250.68i			0.212103i	−0.994361	+	0.106052i	$0.966179\pi$
$618$		0			0
$619$	−4111.34			−0.266961			−0.133480	−	0.991051i	$-0.542615\pi$
$619$	−4111.34			−0.266961			−0.133480	+	0.991051i	$0.542615\pi$
$620$		0			0
$621$	403.687	−	7976.14i	0.0260860	−	0.515413i
$622$		0			0
$623$	11612.0			0.746748
$624$		0			0
$625$	−4323.78			−0.276722
$626$		0			0
$627$	12164.3	+	21914.2i	0.774796	+	1.39580i
$628$		0			0
$629$	−6174.77			−0.391422
$630$		0			0
$631$			25436.9i			1.60480i	0.596790	+	0.802398i	$0.296443\pi$
$631$			25436.9i			1.60480i	−0.596790	+	0.802398i	$0.703557\pi$
$632$		0			0
$633$	−7567.41	+	4200.59i	−0.475162	+	0.263757i
$634$		0			0
$635$			7925.03i			0.495268i
$636$		0			0
$637$		−	11064.3i		−	0.688199i
$638$		0			0
$639$	−421.530	+	676.385i	−0.0260962	+	0.0418738i
$640$		0			0
$641$			22085.5i			1.36088i	0.732803	+	0.680440i	$0.238211\pi$
$641$			22085.5i			1.36088i	−0.732803	+	0.680440i	$0.761789\pi$
$642$		0			0
$643$	−930.199			−0.0570505			−0.0285252	−	0.999593i	$-0.509081\pi$
$643$	−930.199			−0.0570505			−0.0285252	+	0.999593i	$0.509081\pi$
$644$		0			0
$645$	1943.69	−	1078.92i	0.118655	−	0.0658644i
$646$		0			0
$647$	−28562.0			−1.73553			−0.867764	−	0.496977i	$-0.834443\pi$
$647$	−28562.0			−1.73553			−0.867764	+	0.496977i	$0.834443\pi$
$648$		0			0
$649$	−22154.3			−1.33996
$650$		0			0
$651$	−129.447	+	71.8547i	−0.00779328	+	0.00432597i
$652$		0			0
$653$	9597.11			0.575136			0.287568	−	0.957760i	$-0.407153\pi$
$653$	9597.11			0.575136			0.287568	+	0.957760i	$0.407153\pi$
$654$		0			0
$655$			4162.07i			0.248283i
$656$		0			0
$657$	−13387.2	+	21481.0i	−0.794953	+	1.27558i
$658$		0			0
$659$		−	477.222i		−	0.0282093i	−0.999901	−	0.0141046i	$-0.995510\pi$
$659$		−	477.222i		−	0.0282093i	0.999901	−	0.0141046i	$-0.00448980\pi$
$660$		0			0
$661$			584.379i			0.0343869i	0.999852	+	0.0171934i	$0.00547311\pi$
$661$			584.379i			0.0343869i	−0.999852	+	0.0171934i	$0.994527\pi$
$662$		0			0
$663$	−10117.3	+	5616.02i	−0.592645	+	0.328971i
$664$		0			0
$665$		−	12898.1i		−	0.752130i
$666$		0			0
$667$	−12562.0			−0.729241
$668$		0			0
$669$	6747.72	+	12156.1i	0.389958	+	0.702513i
$670$		0			0
$671$	−16290.7			−0.937251
$672$		0			0
$673$	−8219.98			−0.470813			−0.235406	−	0.971897i	$-0.575642\pi$
$673$	−8219.98			−0.470813			−0.235406	+	0.971897i	$0.575642\pi$
$674$		0			0
$675$	−431.299	+	8521.70i	−0.0245936	+	0.485927i
$676$		0			0
$677$	23269.5			1.32101			0.660503	−	0.750824i	$-0.270343\pi$
$677$	23269.5			1.32101			0.660503	+	0.750824i	$0.270343\pi$
$678$		0			0
$679$		−	20719.7i		−	1.17106i
$680$		0			0
$681$	−120.429	−	216.954i	−0.00677656	−	0.0122080i
$682$		0			0
$683$		−	12757.4i		−	0.714713i	−0.933968	−	0.357357i	$-0.883678\pi$
$683$		−	12757.4i		−	0.714713i	0.933968	−	0.357357i	$-0.116322\pi$
$684$		0			0
$685$		−	21885.6i		−	1.22074i
$686$		0			0
$687$	−3511.14	−	6325.36i	−0.194990	−	0.351277i
$688$		0			0
$689$		−	32352.7i		−	1.78888i
$690$		0			0
$691$	13188.1			0.726049			0.363025	−	0.931780i	$-0.381744\pi$
$691$	13188.1			0.726049			0.363025	+	0.931780i	$0.381744\pi$
$692$		0			0
$693$	6794.15	−	10901.9i	0.372422	−	0.597586i
$694$		0			0
$695$	1464.56			0.0799338
$696$		0			0
$697$	5739.77			0.311922
$698$		0			0
$699$	4104.30	+	7393.93i	0.222087	+	0.400092i
$700$		0			0
$701$	686.418			0.0369838			0.0184919	−	0.999829i	$-0.494114\pi$
$701$	686.418			0.0369838			0.0184919	+	0.999829i	$0.494114\pi$
$702$		0			0
$703$			21278.5i			1.14158i
$704$		0			0
$705$	21541.6	−	11957.5i	1.15078	−	0.638788i
$706$		0			0
$707$			21529.8i			1.14528i
$708$		0			0
$709$		−	24361.8i		−	1.29045i	−0.763993	−	0.645224i	$-0.776764\pi$
$709$		−	24361.8i		−	1.29045i	0.763993	−	0.645224i	$-0.223236\pi$
$710$		0			0
$711$	9389.85	+	5851.85i	0.495284	+	0.308666i
$712$		0			0
$713$			128.711i			0.00676051i
$714$		0			0
$715$	−18167.9			−0.950266
$716$		0			0
$717$	12988.4	−	7209.74i	0.676515	−	0.375527i
$718$		0			0
$719$	10763.6			0.558293			0.279147	−	0.960248i	$-0.409948\pi$
$719$	10763.6			0.558293			0.279147	+	0.960248i	$0.409948\pi$
$720$		0			0
$721$	−4466.22			−0.230694
$722$		0			0
$723$	−4639.98	+	2575.61i	−0.238676	+	0.132487i
$724$		0			0
$725$	13421.3			0.687523
$726$		0			0
$727$			13784.5i			0.703217i	0.936147	+	0.351609i	$0.114365\pi$
$727$			13784.5i			0.703217i	−0.936147	+	0.351609i	$0.885635\pi$
$728$		0			0
$729$	−19582.4	−	1987.30i	−0.994890	−	0.100965i
$730$		0			0
$731$		−	1979.90i		−	0.100177i
$732$		0			0
$733$		−	26315.5i		−	1.32604i	−0.748602	−	0.663020i	$-0.769275\pi$
$733$		−	26315.5i		−	1.32604i	0.748602	−	0.663020i	$-0.230725\pi$
$734$		0			0
$735$	6704.34	−	3721.51i	0.336453	−	0.186762i
$736$		0			0
$737$		−	6634.21i		−	0.331580i
$738$		0			0
$739$	−37873.7			−1.88526			−0.942629	−	0.333843i	$-0.891654\pi$
$739$	−37873.7			−1.88526			−0.942629	+	0.333843i	$0.891654\pi$
$740$		0			0
$741$	19353.0	+	34864.6i	0.959447	+	1.72845i
$742$		0			0
$743$	−23306.0			−1.15076			−0.575381	−	0.817886i	$-0.695146\pi$
$743$	−23306.0			−1.15076			−0.575381	+	0.817886i	$0.695146\pi$
$744$		0			0
$745$	−7493.29			−0.368501
$746$		0			0
$747$	29784.4	+	18562.0i	1.45884	+	0.909166i
$748$		0			0
$749$	−6709.73			−0.327327
$750$		0			0
$751$		−	18843.1i		−	0.915572i	−0.889062	−	0.457786i	$-0.848642\pi$
$751$		−	18843.1i		−	0.915572i	0.889062	−	0.457786i	$-0.151358\pi$
$752$		0			0
$753$	−7609.68	−	13708.9i	−0.368277	−	0.663454i
$754$		0			0
$755$			29144.3i			1.40486i
$756$		0			0
$757$			8504.43i			0.408320i	0.978937	+	0.204160i	$0.0654464\pi$
$757$			8504.43i			0.408320i	−0.978937	+	0.204160i	$0.934554\pi$
$758$		0			0
$759$	−5419.92	−	9764.04i	−0.259197	−	0.466946i
$760$		0			0
$761$			26943.9i			1.28346i	0.766929	+	0.641732i	$0.221784\pi$
$761$			26943.9i			1.28346i	−0.766929	+	0.641732i	$0.778216\pi$
$762$		0			0
$763$	26456.0			1.25527
$764$		0			0
$765$	−6805.98	−	4241.56i	−0.321661	−	0.200462i
$766$		0			0
$767$	−35246.7			−1.65930
$768$		0			0
$769$	−20451.5			−0.959040			−0.479520	−	0.877531i	$-0.659189\pi$
$769$	−20451.5			−0.959040			−0.479520	+	0.877531i	$0.659189\pi$
$770$		0			0
$771$	2768.81	+	4988.05i	0.129334	+	0.232996i
$772$		0			0
$773$	25810.2			1.20094			0.600470	−	0.799647i	$-0.294980\pi$
$773$	25810.2			1.20094			0.600470	+	0.799647i	$0.294980\pi$
$774$		0			0
$775$		−	137.514i		−	0.00637376i
$776$		0			0
$777$	9535.05	−	5292.81i	0.440242	−	0.244374i
$778$		0			0
$779$		−	19779.5i		−	0.909722i
$780$		0			0
$781$			1114.44i			0.0510599i
$782$		0			0
$783$	−1564.95	+	30920.6i	−0.0714262	+	1.41125i
$784$		0			0
$785$			17584.4i			0.799508i
$786$		0			0
$787$	−9016.03			−0.408370			−0.204185	−	0.978932i	$-0.565454\pi$
$787$	−9016.03			−0.408370			−0.204185	+	0.978932i	$0.565454\pi$
$788$		0			0
$789$	12653.3	−	7023.74i	0.570939	−	0.316922i
$790$		0			0
$791$	−6433.62			−0.289195
$792$		0			0
$793$	−25917.9			−1.16062
$794$		0			0
$795$	19603.9	−	10881.9i	0.874566	−	0.485463i
$796$		0			0
$797$	−3898.40			−0.173260			−0.0866302	−	0.996241i	$-0.527610\pi$
$797$	−3898.40			−0.173260			−0.0866302	+	0.996241i	$0.527610\pi$
$798$		0			0
$799$		−	21942.9i		−	0.971568i
$800$		0			0
$801$	−21115.0	−	13159.1i	−0.931413	−	0.580466i
$802$		0			0
$803$			35393.0i			1.55541i
$804$		0			0
$805$			5746.85i			0.251615i
$806$		0			0
$807$	−18327.5	+	10173.4i	−0.799451	+	0.443767i
$808$		0			0
$809$		−	5567.95i		−	0.241976i	−0.992654	−	0.120988i	$-0.961394\pi$
$809$		−	5567.95i		−	0.241976i	0.992654	−	0.120988i	$-0.0386063\pi$
$810$		0			0
$811$	42428.3			1.83706			0.918532	−	0.395346i	$-0.129375\pi$
$811$	42428.3			1.83706			0.918532	+	0.395346i	$0.129375\pi$
$812$		0			0
$813$	−5614.71	−	10115.0i	−0.242210	−	0.436343i
$814$		0			0
$815$	23544.4			1.01193
$816$		0			0
$817$	−6822.80			−0.292166
$818$		0			0
$819$	10809.2	−	17344.4i	0.461178	−	0.740004i
$820$		0			0
$821$	−18734.2			−0.796379			−0.398190	−	0.917303i	$-0.630361\pi$
$821$	−18734.2			−0.796379			−0.398190	+	0.917303i	$0.630361\pi$
$822$		0			0
$823$		−	31710.5i		−	1.34309i	−0.740965	−	0.671543i	$-0.765632\pi$
$823$		−	31710.5i		−	1.34309i	0.740965	−	0.671543i	$-0.234368\pi$
$824$		0			0
$825$	5790.64	+	10431.9i	0.244369	+	0.440233i
$826$		0			0
$827$			5275.05i			0.221803i	0.993831	+	0.110902i	$0.0353739\pi$
$827$			5275.05i			0.221803i	−0.993831	+	0.110902i	$0.964626\pi$
$828$		0			0
$829$			10279.5i			0.430665i	0.976541	+	0.215332i	$0.0690835\pi$
$829$			10279.5i			0.430665i	−0.976541	+	0.215332i	$0.930917\pi$
$830$		0			0
$831$	16186.9	+	29160.9i	0.675713	+	1.21730i
$832$		0			0
$833$		−	6829.24i		−	0.284056i
$834$		0			0
$835$	24890.4			1.03158
$836$		0			0
$837$	316.812	+	16.0344i	0.0130832	+	0.000662164i
$838$		0			0
$839$	30259.2			1.24513			0.622564	−	0.782569i	$-0.286091\pi$
$839$	30259.2			1.24513			0.622564	+	0.782569i	$0.286091\pi$
$840$		0			0
$841$	24309.5			0.996739
$842$		0			0
$843$	−9879.48	−	17798.0i	−0.403638	−	0.727159i
$844$		0			0
$845$	−11303.5			−0.460180
$846$		0			0
$847$		−	1189.71i		−	0.0482631i
$848$		0			0
$849$	19170.7	−	10641.5i	0.774956	−	0.430170i
$850$		0			0
$851$		−	9480.81i		−	0.381901i
$852$		0			0
$853$			18713.7i			0.751167i	0.926789	+	0.375584i	$0.122558\pi$
$853$			18713.7i			0.751167i	−0.926789	+	0.375584i	$0.877442\pi$
$854$		0			0
$855$	−14616.6	+	23453.7i	−0.584650	+	0.938126i
$856$		0			0
$857$		−	10924.0i		−	0.435423i	−0.976013	−	0.217711i	$-0.930141\pi$
$857$		−	10924.0i		−	0.435423i	0.976013	−	0.217711i	$-0.0698591\pi$
$858$		0			0
$859$	−33510.2			−1.33103			−0.665514	−	0.746385i	$-0.731788\pi$
$859$	−33510.2			−1.33103			−0.665514	+	0.746385i	$0.731788\pi$
$860$		0			0
$861$	−8863.33	+	4919.94i	−0.350826	+	0.194740i
$862$		0			0
$863$	30253.2			1.19331			0.596657	−	0.802497i	$-0.296495\pi$
$863$	30253.2			1.19331			0.596657	+	0.802497i	$0.296495\pi$
$864$		0			0
$865$	23089.7			0.907600
$866$		0			0
$867$	16075.8	−	8923.50i	0.629714	−	0.349548i
$868$		0			0
$869$	15471.1			0.603936
$870$		0			0
$871$		−	10554.8i		−	0.410603i
$872$		0			0
$873$	−23480.2	+	37676.2i	−0.910292	+	1.46065i
$874$		0			0
$875$		−	18759.3i		−	0.724777i
$876$		0			0
$877$			11900.1i			0.458194i	0.973404	+	0.229097i	$0.0735773\pi$
$877$			11900.1i			0.458194i	−0.973404	+	0.229097i	$0.926423\pi$
$878$		0			0
$879$	3484.38	−	1934.15i	0.133704	−	0.0742175i
$880$		0			0
$881$			8633.31i			0.330152i	0.986281	+	0.165076i	$0.0527869\pi$
$881$			8633.31i			0.330152i	−0.986281	+	0.165076i	$0.947213\pi$
$882$		0			0
$883$	−27438.6			−1.04573			−0.522867	−	0.852414i	$-0.675138\pi$
$883$	−27438.6			−1.04573			−0.522867	+	0.852414i	$0.675138\pi$
$884$		0			0
$885$	−11855.3	−	21357.5i	−0.450297	−	0.811214i
$886$		0			0
$887$	3124.21			0.118264			0.0591322	−	0.998250i	$-0.481167\pi$
$887$	3124.21			0.118264			0.0591322	+	0.998250i	$0.481167\pi$
$888$		0			0
$889$	−12465.8			−0.470291
$890$		0			0
$891$	−24708.7	+	12124.4i	−0.929038	+	0.455872i
$892$		0			0
$893$	−75615.9			−2.83358
$894$		0			0
$895$		−	20600.3i		−	0.769376i
$896$		0			0
$897$	−8622.88	−	15534.2i	−0.320970	−	0.578230i
$898$		0			0
$899$		−	498.964i		−	0.0185110i
$900$		0			0
$901$		−	19969.1i		−	0.738367i
$902$		0			0
$903$	1697.10	+	3057.35i	0.0625427	+	0.112671i
$904$		0			0
$905$		−	4511.17i		−	0.165698i
$906$		0			0
$907$	−21850.0			−0.799909			−0.399955	−	0.916535i	$-0.630974\pi$
$907$	−21850.0			−0.799909			−0.399955	+	0.916535i	$0.630974\pi$
$908$		0			0
$909$	24398.3	−	39149.3i	0.890253	−	1.42849i
$910$		0			0
$911$	−16739.3			−0.608781			−0.304390	−	0.952547i	$-0.598453\pi$
$911$	−16739.3			−0.608781			−0.304390	+	0.952547i	$0.598453\pi$
$912$		0			0
$913$	49074.0			1.77887
$914$		0			0
$915$	−8717.56	−	15704.8i	−0.314966	−	0.567414i
$916$		0			0
$917$	−6546.77			−0.235762
$918$		0			0
$919$		−	36772.7i		−	1.31994i	−0.751294	−	0.659968i	$-0.770570\pi$
$919$		−	36772.7i		−	1.31994i	0.751294	−	0.659968i	$-0.229430\pi$
$920$		0			0
$921$	33149.7	−	18401.0i	1.18601	−	0.658344i
$922$		0			0
$923$			1773.03i			0.0632286i
$924$		0			0
$925$			10129.3i			0.360053i
$926$		0			0
$927$	8121.29	+	5061.27i	0.287743	+	0.179325i
$928$		0			0
$929$			28060.4i			0.990991i	0.868611	+	0.495495i	$0.165013\pi$
$929$			28060.4i			0.990991i	−0.868611	+	0.495495i	$0.834987\pi$
$930$		0			0
$931$	−23533.8			−0.828452
$932$		0			0
$933$	−39259.1	+	21792.4i	−1.37758	+	0.764683i
$934$		0			0
$935$	−11213.8			−0.392225
$936$		0			0
$937$	−5258.29			−0.183331			−0.0916653	−	0.995790i	$-0.529219\pi$
$937$	−5258.29			−0.183331			−0.0916653	+	0.995790i	$0.529219\pi$
$938$		0			0
$939$	9865.64	−	5476.32i	0.342868	−	0.190323i
$940$		0			0
$941$	37769.5			1.30845			0.654224	−	0.756300i	$-0.272995\pi$
$941$	37769.5			1.30845			0.654224	+	0.756300i	$0.272995\pi$
$942$		0			0
$943$			8812.91i			0.304335i
$944$		0			0
$945$	14145.5	+	715.928i	0.486934	+	0.0246446i
$946$		0			0
$947$			9839.83i			0.337647i	0.985646	+	0.168823i	$0.0539968\pi$
$947$			9839.83i			0.337647i	−0.985646	+	0.168823i	$0.946003\pi$
$948$		0			0
$949$			56308.8i			1.92609i
$950$		0			0
$951$	−7642.31	+	4242.17i	−0.260588	+	0.144650i
$952$		0			0
$953$			21903.4i			0.744514i	0.928130	+	0.372257i	$0.121416\pi$
$953$			21903.4i			0.744514i	−0.928130	+	0.372257i	$0.878584\pi$
$954$		0			0
$955$	−21128.2			−0.715909
$956$		0			0
$957$	21011.1	+	37851.7i	0.709709	+	1.27855i
$958$		0			0
$959$	34425.1			1.15917
$960$		0			0
$961$	29785.9			0.999828
$962$		0			0
$963$	12200.9	+	7603.70i	0.408273	+	0.254440i
$964$		0			0
$965$	12002.7			0.400394
$966$		0			0
$967$			11943.8i			0.397193i	0.980081	+	0.198597i	$0.0636384\pi$
$967$			11943.8i			0.397193i	−0.980081	+	0.198597i	$0.936362\pi$
$968$		0			0
$969$	11945.3	+	21519.6i	0.396015	+	0.713424i
$970$		0			0
$971$		−	15658.6i		−	0.517517i	−0.965942	−	0.258758i	$-0.916687\pi$
$971$		−	15658.6i		−	0.517517i	0.965942	−	0.258758i	$-0.0833133\pi$
$972$		0			0
$973$			2303.70i			0.0759026i
$974$		0			0
$975$	9212.69	+	16596.8i	0.302607	+	0.545150i
$976$		0			0
$977$		−	14022.1i		−	0.459168i	−0.973289	−	0.229584i	$-0.926263\pi$
$977$		−	14022.1i		−	0.459168i	0.973289	−	0.229584i	$-0.0737366\pi$
$978$		0			0
$979$	−34789.9			−1.13574
$980$		0			0
$981$	−48107.2	−	29980.9i	−1.56569	−	0.975755i
$982$		0			0
$983$	24228.1			0.786120			0.393060	−	0.919513i	$-0.371417\pi$
$983$	24228.1			0.786120			0.393060	+	0.919513i	$0.371417\pi$
$984$		0			0
$985$	3157.85			0.102150
$986$		0			0
$987$	18808.7	+	33884.0i	0.606572	+	1.09275i
$988$		0			0
$989$	3039.96			0.0977401
$990$		0			0
$991$		−	17211.8i		−	0.551715i	−0.961198	−	0.275858i	$-0.911038\pi$
$991$		−	17211.8i		−	0.551715i	0.961198	−	0.275858i	$-0.0889618\pi$
$992$		0			0
$993$	10855.4	−	6025.75i	0.346916	−	0.192569i
$994$		0			0
$995$		−	25099.9i		−	0.799719i
$996$		0			0
$997$		−	55019.5i		−	1.74773i	−0.486171	−	0.873864i	$-0.661607\pi$
$997$		−	55019.5i		−	1.74773i	0.486171	−	0.873864i	$-0.338393\pi$
$998$		0			0
$999$	−23336.4	−	1181.10i	−0.739069	−	0.0374056i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	768.4.f.g.383.11		12
3.2	odd	2		768.4.f.h.383.12		12
4.3	odd	2		768.4.f.e.383.2		12
8.3	odd	2		768.4.f.h.383.11		12
8.5	even	2		768.4.f.f.383.2		12
12.11	even	2		768.4.f.f.383.1		12
16.3	odd	4		384.4.c.c.383.3	yes	12
16.5	even	4		384.4.c.d.383.3	yes	12
16.11	odd	4		384.4.c.a.383.10	yes	12
16.13	even	4		384.4.c.b.383.10	yes	12
24.5	odd	2		768.4.f.e.383.1		12
24.11	even	2	inner	768.4.f.g.383.12		12
48.5	odd	4		384.4.c.a.383.9	✓	12
48.11	even	4		384.4.c.d.383.4	yes	12
48.29	odd	4		384.4.c.c.383.4	yes	12
48.35	even	4		384.4.c.b.383.9	yes	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.4.c.a.383.9	✓	12	48.5	odd	4
384.4.c.a.383.10	yes	12	16.11	odd	4
384.4.c.b.383.9	yes	12	48.35	even	4
384.4.c.b.383.10	yes	12	16.13	even	4
384.4.c.c.383.3	yes	12	16.3	odd	4
384.4.c.c.383.4	yes	12	48.29	odd	4
384.4.c.d.383.3	yes	12	16.5	even	4
384.4.c.d.383.4	yes	12	48.11	even	4
768.4.f.e.383.1		12	24.5	odd	2
768.4.f.e.383.2		12	4.3	odd	2
768.4.f.f.383.1		12	12.11	even	2
768.4.f.f.383.2		12	8.5	even	2
768.4.f.g.383.11		12	1.1	even	1	trivial
768.4.f.g.383.12		12	24.11	even	2	inner
768.4.f.h.383.11		12	8.3	odd	2
768.4.f.h.383.12		12	3.2	odd	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)

\(f(q)\)	\(=\)	\(q+(4.54315 - 2.52186i) q^{3} +8.01133 q^{5} +12.6015i q^{7} +(14.2805 - 22.9144i) q^{9} +O(q^{10})\)	\(q+(4.54315 - 2.52186i) q^{3} +8.01133 q^{5} +12.6015i q^{7} +(14.2805 - 22.9144i) q^{9} -37.7546i q^{11} -60.0661i q^{13} +(36.3967 - 20.2034i) q^{15} -37.0747i q^{17} -127.761 q^{19} +(31.7792 + 57.2506i) q^{21} +56.9249 q^{23} -60.8186 q^{25} +(7.09157 - 140.117i) q^{27} -220.677 q^{29} +2.26106i q^{31} +(-95.2117 - 171.525i) q^{33} +100.955i q^{35} -166.549i q^{37} +(-151.478 - 272.890i) q^{39} +154.816i q^{41} +53.4029 q^{43} +(114.406 - 183.575i) q^{45} +591.855 q^{47} +184.202 q^{49} +(-93.4972 - 168.436i) q^{51} +538.619 q^{53} -302.465i q^{55} +(-580.437 + 322.195i) q^{57} -586.798i q^{59} -431.489i q^{61} +(288.756 + 179.956i) q^{63} -481.209i q^{65} +175.719 q^{67} +(258.619 - 143.557i) q^{69} -29.5180 q^{71} -937.448 q^{73} +(-276.308 + 153.376i) q^{75} +475.765 q^{77} +409.780i q^{79} +(-321.136 - 654.456i) q^{81} +1299.81i q^{83} -297.018i q^{85} +(-1002.57 + 556.517i) q^{87} -921.475i q^{89} +756.924 q^{91} +(5.70207 + 10.2723i) q^{93} -1023.53 q^{95} -1644.22 q^{97} +(-865.123 - 539.153i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 2 q^{3} - 12 q^{5}+O(q^{10}) \) 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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