LMFDB - Embedded newform 768.4.f.h.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.h.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	$-0.616636\pi$
$5$	−8.01133			−0.716555			−0.358278	+	0.933615i	$0.616636\pi$
$6$		0			0
$7$		−	12.6015i		−	0.680418i	−0.940350	−	0.340209i	$-0.889502\pi$
$7$		−	12.6015i		−	0.680418i	0.940350	−	0.340209i	$-0.110498\pi$
$8$		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	$0.221374\pi$
$13$			60.0661i			1.28149i	−0.767755	+	0.640744i	$0.778626\pi$
$14$		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	$-0.583075\pi$
$23$	−56.9249			−0.516072			−0.258036	+	0.966135i	$0.583075\pi$
$24$		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	$-0.250259\pi$
$29$	220.677			1.41306			0.706530	+	0.707683i	$0.250259\pi$
$30$		0			0
$31$		−	2.26106i		−	0.0130999i	−0.999979	−	0.00654997i	$-0.997915\pi$
$31$		−	2.26106i		−	0.0130999i	0.999979	−	0.00654997i	$-0.00208493\pi$
$32$		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	$0.120645\pi$
$37$			166.549i			0.740015i	−0.929029	+	0.370007i	$0.879355\pi$
$38$		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	$-0.870529\pi$
$47$	−591.855			−1.83683			−0.918413	+	0.395622i	$0.870529\pi$
$48$		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	$-0.745914\pi$
$53$	−538.619			−1.39594			−0.697971	+	0.716126i	$0.745914\pi$
$54$		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	$0.149589\pi$
$61$			431.489i			0.905681i	−0.891592	+	0.452840i	$0.850411\pi$
$62$		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	$-0.492147\pi$
$71$	29.5180			0.0493400			0.0246700	+	0.999696i	$0.492147\pi$
$72$		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	$-0.905747\pi$
$79$		−	409.780i		−	0.583593i	0.956480	−	0.291796i	$-0.0942530\pi$
$80$		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	$-0.818385\pi$
$101$	−1708.51			−1.68320			−0.841598	+	0.540105i	$0.818385\pi$
$102$		0			0
$103$		−	354.419i		−	0.339048i	−0.985526	−	0.169524i	$-0.945777\pi$
$103$		−	354.419i		−	0.339048i	0.985526	−	0.169524i	$-0.0542230\pi$
$104$		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	$0.373797\pi$
$109$			2099.43i			1.84485i	−0.386171	+	0.922427i	$0.626203\pi$
$110$		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	$-0.887679\pi$
$127$		−	989.228i		−	0.691180i	0.938386	−	0.345590i	$-0.112321\pi$
$128$		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	$-0.417222\pi$
$149$	935.337			0.514267			0.257134	+	0.966376i	$0.417222\pi$
$150$		0			0
$151$		−	3637.89i		−	1.96058i	−0.197572	−	0.980288i	$-0.563306\pi$
$151$		−	3637.89i		−	1.96058i	0.197572	−	0.980288i	$-0.436694\pi$
$152$		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	$-0.811614\pi$
$157$		−	2194.94i		−	1.11577i	0.829919	−	0.557883i	$-0.188386\pi$
$158$		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	$-0.755775\pi$
$167$	−3106.90			−1.43964			−0.719818	+	0.694163i	$0.755775\pi$
$168$		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	$-0.718303\pi$
$173$	−2882.13			−1.26662			−0.633308	+	0.773900i	$0.718303\pi$
$174$		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	$0.0368858\pi$
$181$			563.098i			0.231242i	−0.993293	+	0.115621i	$0.963114\pi$
$182$		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	$-0.333499\pi$
$191$	2637.29			0.999098			0.499549	+	0.866286i	$0.333499\pi$
$192$		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	$-0.522708\pi$
$197$	−394.173			−0.142557			−0.0712783	+	0.997456i	$0.522708\pi$
$198$		0			0
$199$			3133.05i			1.11606i	0.829821	+	0.558030i	$0.188443\pi$
$199$			3133.05i			1.11606i	−0.829821	+	0.558030i	$0.811557\pi$
$200$		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	$-0.868404\pi$
$223$		−	2675.69i		−	0.803488i	0.915752	−	0.401744i	$-0.131596\pi$
$224$		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	$0.0643813\pi$
$229$			1392.28i			0.401767i	−0.979615	+	0.200884i	$0.935619\pi$
$230$		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	$-0.626446\pi$
$239$	−2858.90			−0.773752			−0.386876	+	0.922132i	$0.626446\pi$
$240$		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	$-0.605870\pi$
$263$	−2785.15			−0.653002			−0.326501	+	0.945197i	$0.605870\pi$
$264$		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	$-0.348860\pi$
$269$	4034.08			0.914358			0.457179	+	0.889375i	$0.348860\pi$
$270$		0			0
$271$			2226.42i			0.499060i	0.968367	+	0.249530i	$0.0802761\pi$
$271$			2226.42i			0.499060i	−0.968367	+	0.249530i	$0.919724\pi$
$272$		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	$-0.754901\pi$
$277$		−	6418.65i		−	1.39227i	0.717910	−	0.696136i	$-0.245099\pi$
$278$		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	$-0.524362\pi$
$293$	−766.953			−0.152921			−0.0764605	+	0.997073i	$0.524362\pi$
$294$		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	$-0.211223\pi$
$311$	8641.39			1.57559			0.787795	+	0.615938i	$0.211223\pi$
$312$		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	$-0.452388\pi$
$317$	1682.16			0.298043			0.149021	+	0.988834i	$0.452388\pi$
$318$		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	$-0.935205\pi$
$349$		−	2636.08i		−	0.404316i	0.979353	−	0.202158i	$-0.0647955\pi$
$350$		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	$-0.493481\pi$
$359$	278.615			0.0409603			0.0204802	+	0.999790i	$0.493481\pi$
$360$		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	$-0.670901\pi$
$367$		−	12082.9i		−	1.71860i	0.511476	−	0.859298i	$-0.329099\pi$
$368$		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	$-0.953398\pi$
$373$		−	2101.82i		−	0.291764i	0.989302	−	0.145882i	$-0.0466020\pi$
$374$		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	$-0.701834\pi$
$383$	−8881.17			−1.18487			−0.592437	+	0.805617i	$0.701834\pi$
$384$		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	$-0.435350\pi$
$389$	3095.15			0.403420			0.201710	+	0.979445i	$0.435350\pi$
$390$		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	$-0.762046\pi$
$397$		−	10755.4i		−	1.35969i	0.733354	−	0.679847i	$-0.237954\pi$
$398$		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	$-0.839707\pi$
$421$		−	8336.91i		−	0.965121i	0.875863	−	0.482561i	$-0.160293\pi$
$422$		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	$-0.513436\pi$
$431$	−755.175			−0.0843979			−0.0421990	+	0.999109i	$0.513436\pi$
$432$		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	$-0.673443\pi$
$439$		−	15732.1i		−	1.71037i	0.518322	−	0.855186i	$-0.326557\pi$
$440$		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	$-0.433388\pi$
$461$	4112.54			0.415488			0.207744	+	0.978183i	$0.433388\pi$
$462$		0			0
$463$			16061.5i			1.61219i	0.591788	+	0.806094i	$0.298422\pi$
$463$			16061.5i			1.61219i	−0.591788	+	0.806094i	$0.701578\pi$
$464$		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	$-0.547568\pi$
$479$	−3121.62			−0.297767			−0.148883	+	0.988855i	$0.547568\pi$
$480$		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	$0.0234566\pi$
$487$			1582.51i			0.147249i	−0.997286	+	0.0736244i	$0.976543\pi$
$488$		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	$-0.338136\pi$
$503$	10985.0			0.973753			0.486876	+	0.873471i	$0.338136\pi$
$504$		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	$-0.874026\pi$
$509$	−21191.9			−1.84541			−0.922704	+	0.385509i	$0.874026\pi$
$510$		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	$0.0334548\pi$
$541$			2640.19i			0.209816i	−0.994482	+	0.104908i	$0.966545\pi$
$542$		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	$-0.699949\pi$
$557$	−15450.3			−1.17531			−0.587657	+	0.809110i	$0.699949\pi$
$558$		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	$-0.452384\pi$
$599$	4369.74			0.298068			0.149034	+	0.988832i	$0.452384\pi$
$600$		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	$-0.838584\pi$
$607$		−	14525.6i		−	0.971293i	0.874155	−	0.485647i	$-0.161416\pi$
$608$		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	$0.163904\pi$
$613$			14948.5i			0.984932i	−0.870332	+	0.492466i	$0.836096\pi$
$614$		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	$-0.703557\pi$
$631$		−	25436.9i		−	1.60480i	0.596790	−	0.802398i	$-0.296443\pi$
$632$		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	$-0.165557\pi$
$647$	28562.0			1.73553			0.867764	+	0.496977i	$0.165557\pi$
$648$		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	$-0.592847\pi$
$653$	−9597.11			−0.575136			−0.287568	+	0.957760i	$0.592847\pi$
$654$		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	$-0.994527\pi$
$661$		−	584.379i		−	0.0343869i	0.999852	−	0.0171934i	$-0.00547311\pi$
$662$		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	$-0.729657\pi$
$677$	−23269.5			−1.32101			−0.660503	+	0.750824i	$0.729657\pi$
$678$		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	$-0.505886\pi$
$701$	−686.418			−0.0369838			−0.0184919	+	0.999829i	$0.505886\pi$
$702$		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	$0.223236\pi$
$709$			24361.8i			1.29045i	−0.763993	+	0.645224i	$0.776764\pi$
$710$		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	$-0.590052\pi$
$719$	−10763.6			−0.558293			−0.279147	+	0.960248i	$0.590052\pi$
$720$		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	$-0.885635\pi$
$727$		−	13784.5i		−	0.703217i	0.936147	−	0.351609i	$-0.114365\pi$
$728$		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	$0.230725\pi$
$733$			26315.5i			1.32604i	−0.748602	+	0.663020i	$0.769275\pi$
$734$		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	$-0.304854\pi$
$743$	23306.0			1.15076			0.575381	+	0.817886i	$0.304854\pi$
$744$		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	$0.151358\pi$
$751$			18843.1i			0.915572i	−0.889062	+	0.457786i	$0.848642\pi$
$752$		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	$-0.934554\pi$
$757$		−	8504.43i		−	0.408320i	0.978937	−	0.204160i	$-0.0654464\pi$
$758$		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	$-0.705020\pi$
$773$	−25810.2			−1.20094			−0.600470	+	0.799647i	$0.705020\pi$
$774$		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	$-0.472390\pi$
$797$	3898.40			0.173260			0.0866302	+	0.996241i	$0.472390\pi$
$798$		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	$-0.369639\pi$
$821$	18734.2			0.796379			0.398190	+	0.917303i	$0.369639\pi$
$822$		0			0
$823$			31710.5i			1.34309i	0.740965	+	0.671543i	$0.234368\pi$
$823$			31710.5i			1.34309i	−0.740965	+	0.671543i	$0.765632\pi$
$824$		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	$-0.930917\pi$
$829$		−	10279.5i		−	0.430665i	0.976541	−	0.215332i	$-0.0690835\pi$
$830$		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	$-0.713909\pi$
$839$	−30259.2			−1.24513			−0.622564	+	0.782569i	$0.713909\pi$
$840$		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	$-0.877442\pi$
$853$		−	18713.7i		−	0.751167i	0.926789	−	0.375584i	$-0.122558\pi$
$854$		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	$-0.703505\pi$
$863$	−30253.2			−1.19331			−0.596657	+	0.802497i	$0.703505\pi$
$864$		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	$-0.926423\pi$
$877$		−	11900.1i		−	0.458194i	0.973404	−	0.229097i	$-0.0735773\pi$
$878$		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	$-0.518833\pi$
$887$	−3124.21			−0.118264			−0.0591322	+	0.998250i	$0.518833\pi$
$888$		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	$-0.401547\pi$
$911$	16739.3			0.608781			0.304390	+	0.952547i	$0.401547\pi$
$912$		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	$0.229430\pi$
$919$			36772.7i			1.31994i	−0.751294	+	0.659968i	$0.770570\pi$
$920$		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	$-0.727005\pi$
$941$	−37769.5			−1.30845			−0.654224	+	0.756300i	$0.727005\pi$
$942$		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	$-0.936362\pi$
$967$		−	11943.8i		−	0.397193i	0.980081	−	0.198597i	$-0.0636384\pi$
$968$		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	$-0.628583\pi$
$983$	−24228.1			−0.786120			−0.393060	+	0.919513i	$0.628583\pi$
$984$		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	$0.0889618\pi$
$991$			17211.8i			0.551715i	−0.961198	+	0.275858i	$0.911038\pi$
$992$		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	$0.338393\pi$
$997$			55019.5i			1.74773i	−0.486171	+	0.873864i	$0.661607\pi$
$998$		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.h.383.11		12
3.2	odd	2		768.4.f.g.383.12		12
4.3	odd	2		768.4.f.f.383.2		12
8.3	odd	2		768.4.f.g.383.11		12
8.5	even	2		768.4.f.e.383.2		12
12.11	even	2		768.4.f.e.383.1		12
16.3	odd	4		384.4.c.d.383.3	yes	12
16.5	even	4		384.4.c.c.383.3	yes	12
16.11	odd	4		384.4.c.b.383.10	yes	12
16.13	even	4		384.4.c.a.383.10	yes	12
24.5	odd	2		768.4.f.f.383.1		12
24.11	even	2	inner	768.4.f.h.383.12		12
48.5	odd	4		384.4.c.b.383.9	yes	12
48.11	even	4		384.4.c.c.383.4	yes	12
48.29	odd	4		384.4.c.d.383.4	yes	12
48.35	even	4		384.4.c.a.383.9	✓	12

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

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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