LMFDB - Embedded newform 768.2.f.c.383.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [768,2,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, 2, names="a")

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

chi := DirichletCharacter("768.383");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 768 = 2^{8} \cdot 3 $
Weight:	$ k $	$=$	$ 2 $
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:	$6.13251087523$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{5})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 3x^{2} + 1 $ x^4 + 3*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 384)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			383.3
Root			$1.61803i$ of defining polynomial
Character	$\chi$	$=$	768.383
Dual form			768.2.f.c.383.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.618034 - 1.61803i) q^{3} -1.23607 q^{5} -3.23607i q^{7} +(-2.23607 - 2.00000i) q^{9} +O(q^{10})$	\(q+(0.618034 - 1.61803i) q^{3} -1.23607 q^{5} -3.23607i q^{7} +(-2.23607 - 2.00000i) q^{9} -0.763932i q^{11} +4.47214i q^{13} +(-0.763932 + 2.00000i) q^{15} -6.47214i q^{17} +5.23607 q^{19} +(-5.23607 - 2.00000i) q^{21} -6.47214 q^{23} -3.47214 q^{25} +(-4.61803 + 2.38197i) q^{27} -9.23607 q^{29} -0.763932i q^{31} +(-1.23607 - 0.472136i) q^{33} +4.00000i q^{35} +0.472136i q^{37} +(7.23607 + 2.76393i) q^{39} -2.47214i q^{41} +2.76393 q^{43} +(2.76393 + 2.47214i) q^{45} +8.00000 q^{47} -3.47214 q^{49} +(-10.4721 - 4.00000i) q^{51} -1.23607 q^{53} +0.944272i q^{55} +(3.23607 - 8.47214i) q^{57} -3.23607i q^{59} -8.47214i q^{61} +(-6.47214 + 7.23607i) q^{63} -5.52786i q^{65} -3.70820 q^{67} +(-4.00000 + 10.4721i) q^{69} +11.4164 q^{71} +2.00000 q^{73} +(-2.14590 + 5.61803i) q^{75} -2.47214 q^{77} -13.7082i q^{79} +(1.00000 + 8.94427i) q^{81} +7.23607i q^{83} +8.00000i q^{85} +(-5.70820 + 14.9443i) q^{87} +4.00000i q^{89} +14.4721 q^{91} +(-1.23607 - 0.472136i) q^{93} -6.47214 q^{95} -8.47214 q^{97} +(-1.52786 + 1.70820i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 2 q^{3} + 4 q^{5}+O(q^{10}) $ 4 * q - 2 * q^3 + 4 * q^5	$ 4 q - 2 q^{3} + 4 q^{5} - 12 q^{15} + 12 q^{19} - 12 q^{21} - 8 q^{23} + 4 q^{25} - 14 q^{27} - 28 q^{29} + 4 q^{33} + 20 q^{39} + 20 q^{43} + 20 q^{45} + 32 q^{47} + 4 q^{49} - 24 q^{51} + 4 q^{53} + 4 q^{57} - 8 q^{63} + 12 q^{67} - 16 q^{69} - 8 q^{71} + 8 q^{73} - 22 q^{75} + 8 q^{77} + 4 q^{81} + 4 q^{87} + 40 q^{91} + 4 q^{93} - 8 q^{95} - 16 q^{97} - 24 q^{99}+O(q^{100}) $ 4 * q - 2 * q^3 + 4 * q^5 - 12 * q^15 + 12 * q^19 - 12 * q^21 - 8 * q^23 + 4 * q^25 - 14 * q^27 - 28 * q^29 + 4 * q^33 + 20 * q^39 + 20 * q^43 + 20 * q^45 + 32 * q^47 + 4 * q^49 - 24 * q^51 + 4 * q^53 + 4 * q^57 - 8 * q^63 + 12 * q^67 - 16 * q^69 - 8 * q^71 + 8 * q^73 - 22 * q^75 + 8 * q^77 + 4 * q^81 + 4 * q^87 + 40 * q^91 + 4 * q^93 - 8 * q^95 - 16 * q^97 - 24 * 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
$3$	0.618034	−	1.61803i	0.356822	−	0.934172i
$3$	0.618034	−	1.61803i	0.356822	−	0.934172i
$4$		0			0
$5$	−1.23607			−0.552786			−0.276393	−	0.961045i	$-0.589139\pi$
$5$	−1.23607			−0.552786			−0.276393	+	0.961045i	$0.589139\pi$
$6$		0			0
$7$		−	3.23607i		−	1.22312i	−0.791199	−	0.611559i	$-0.790543\pi$
$7$		−	3.23607i		−	1.22312i	0.791199	−	0.611559i	$-0.209457\pi$
$8$		0			0
$9$	−2.23607	−	2.00000i	−0.745356	−	0.666667i
$10$		0			0
$11$		−	0.763932i		−	0.230334i	−0.993346	−	0.115167i	$-0.963260\pi$
$11$		−	0.763932i		−	0.230334i	0.993346	−	0.115167i	$-0.0367403\pi$
$12$		0			0
$13$			4.47214i			1.24035i	0.784465	+	0.620174i	$0.212938\pi$
$13$			4.47214i			1.24035i	−0.784465	+	0.620174i	$0.787062\pi$
$14$		0			0
$15$	−0.763932	+	2.00000i	−0.197246	+	0.516398i
$16$		0			0
$17$		−	6.47214i		−	1.56972i	−0.619671	−	0.784862i	$-0.712734\pi$
$17$		−	6.47214i		−	1.56972i	0.619671	−	0.784862i	$-0.287266\pi$
$18$		0			0
$19$	5.23607			1.20124			0.600618	−	0.799536i	$-0.294921\pi$
$19$	5.23607			1.20124			0.600618	+	0.799536i	$0.294921\pi$
$20$		0			0
$21$	−5.23607	−	2.00000i	−1.14260	−	0.436436i
$22$		0			0
$23$	−6.47214			−1.34953			−0.674767	−	0.738031i	$-0.735756\pi$
$23$	−6.47214			−1.34953			−0.674767	+	0.738031i	$0.735756\pi$
$24$		0			0
$25$	−3.47214			−0.694427
$26$		0			0
$27$	−4.61803	+	2.38197i	−0.888741	+	0.458410i
$28$		0			0
$29$	−9.23607			−1.71509			−0.857547	−	0.514405i	$-0.828013\pi$
$29$	−9.23607			−1.71509			−0.857547	+	0.514405i	$0.828013\pi$
$30$		0			0
$31$		−	0.763932i		−	0.137206i	−0.997644	−	0.0686031i	$-0.978146\pi$
$31$		−	0.763932i		−	0.137206i	0.997644	−	0.0686031i	$-0.0218542\pi$
$32$		0			0
$33$	−1.23607	−	0.472136i	−0.215172	−	0.0821883i
$34$		0			0
$35$			4.00000i			0.676123i
$36$		0			0
$37$			0.472136i			0.0776187i	0.999247	+	0.0388093i	$0.0123565\pi$
$37$			0.472136i			0.0776187i	−0.999247	+	0.0388093i	$0.987644\pi$
$38$		0			0
$39$	7.23607	+	2.76393i	1.15870	+	0.442583i
$40$		0			0
$41$		−	2.47214i		−	0.386083i	−0.981191	−	0.193041i	$-0.938165\pi$
$41$		−	2.47214i		−	0.386083i	0.981191	−	0.193041i	$-0.0618352\pi$
$42$		0			0
$43$	2.76393			0.421496			0.210748	−	0.977540i	$-0.432410\pi$
$43$	2.76393			0.421496			0.210748	+	0.977540i	$0.432410\pi$
$44$		0			0
$45$	2.76393	+	2.47214i	0.412023	+	0.368524i
$46$		0			0
$47$	8.00000			1.16692			0.583460	−	0.812142i	$-0.301699\pi$
$47$	8.00000			1.16692			0.583460	+	0.812142i	$0.301699\pi$
$48$		0			0
$49$	−3.47214			−0.496019
$50$		0			0
$51$	−10.4721	−	4.00000i	−1.46639	−	0.560112i
$52$		0			0
$53$	−1.23607			−0.169787			−0.0848935	−	0.996390i	$-0.527055\pi$
$53$	−1.23607			−0.169787			−0.0848935	+	0.996390i	$0.527055\pi$
$54$		0			0
$55$			0.944272i			0.127326i
$56$		0			0
$57$	3.23607	−	8.47214i	0.428628	−	1.12216i
$58$		0			0
$59$		−	3.23607i		−	0.421300i	−0.977562	−	0.210650i	$-0.932442\pi$
$59$		−	3.23607i		−	0.421300i	0.977562	−	0.210650i	$-0.0675581\pi$
$60$		0			0
$61$		−	8.47214i		−	1.08475i	−0.840138	−	0.542373i	$-0.817526\pi$
$61$		−	8.47214i		−	1.08475i	0.840138	−	0.542373i	$-0.182474\pi$
$62$		0			0
$63$	−6.47214	+	7.23607i	−0.815412	+	0.911659i
$64$		0			0
$65$		−	5.52786i		−	0.685647i
$66$		0			0
$67$	−3.70820			−0.453029			−0.226515	−	0.974008i	$-0.572733\pi$
$67$	−3.70820			−0.453029			−0.226515	+	0.974008i	$0.572733\pi$
$68$		0			0
$69$	−4.00000	+	10.4721i	−0.481543	+	1.26070i
$70$		0			0
$71$	11.4164			1.35488			0.677439	−	0.735579i	$-0.263090\pi$
$71$	11.4164			1.35488			0.677439	+	0.735579i	$0.263090\pi$
$72$		0			0
$73$	2.00000			0.234082			0.117041	−	0.993127i	$-0.462659\pi$
$73$	2.00000			0.234082			0.117041	+	0.993127i	$0.462659\pi$
$74$		0			0
$75$	−2.14590	+	5.61803i	−0.247787	+	0.648715i
$76$		0			0
$77$	−2.47214			−0.281726
$78$		0			0
$79$		−	13.7082i		−	1.54229i	−0.636657	−	0.771147i	$-0.719683\pi$
$79$		−	13.7082i		−	1.54229i	0.636657	−	0.771147i	$-0.280317\pi$
$80$		0			0
$81$	1.00000	+	8.94427i	0.111111	+	0.993808i
$82$		0			0
$83$			7.23607i			0.794262i	0.917762	+	0.397131i	$0.129994\pi$
$83$			7.23607i			0.794262i	−0.917762	+	0.397131i	$0.870006\pi$
$84$		0			0
$85$			8.00000i			0.867722i
$86$		0			0
$87$	−5.70820	+	14.9443i	−0.611984	+	1.60219i
$88$		0			0
$89$			4.00000i			0.423999i	0.977270	+	0.212000i	$0.0679975\pi$
$89$			4.00000i			0.423999i	−0.977270	+	0.212000i	$0.932002\pi$
$90$		0			0
$91$	14.4721			1.51709
$92$		0			0
$93$	−1.23607	−	0.472136i	−0.128174	−	0.0489582i
$94$		0			0
$95$	−6.47214			−0.664027
$96$		0			0
$97$	−8.47214			−0.860215			−0.430108	−	0.902778i	$-0.641524\pi$
$97$	−8.47214			−0.860215			−0.430108	+	0.902778i	$0.641524\pi$
$98$		0			0
$99$	−1.52786	+	1.70820i	−0.153556	+	0.171681i
$100$		0			0
$101$	11.7082			1.16501			0.582505	−	0.812827i	$-0.302073\pi$
$101$	11.7082			1.16501			0.582505	+	0.812827i	$0.302073\pi$
$102$		0			0
$103$		−	8.18034i		−	0.806033i	−0.915193	−	0.403016i	$-0.867962\pi$
$103$		−	8.18034i		−	0.806033i	0.915193	−	0.403016i	$-0.132038\pi$
$104$		0			0
$105$	6.47214	+	2.47214i	0.631616	+	0.241256i
$106$		0			0
$107$		−	8.18034i		−	0.790823i	−0.918504	−	0.395412i	$-0.870602\pi$
$107$		−	8.18034i		−	0.790823i	0.918504	−	0.395412i	$-0.129398\pi$
$108$		0			0
$109$		−	0.472136i		−	0.0452224i	−0.999744	−	0.0226112i	$-0.992802\pi$
$109$		−	0.472136i		−	0.0452224i	0.999744	−	0.0226112i	$-0.00719799\pi$
$110$		0			0
$111$	0.763932	+	0.291796i	0.0725092	+	0.0276961i
$112$		0			0
$113$			8.00000i			0.752577i	0.926503	+	0.376288i	$0.122800\pi$
$113$			8.00000i			0.752577i	−0.926503	+	0.376288i	$0.877200\pi$
$114$		0			0
$115$	8.00000			0.746004
$116$		0			0
$117$	8.94427	−	10.0000i	0.826898	−	0.924500i
$118$		0			0
$119$	−20.9443			−1.91996
$120$		0			0
$121$	10.4164			0.946946
$122$		0			0
$123$	−4.00000	−	1.52786i	−0.360668	−	0.137763i
$124$		0			0
$125$	10.4721			0.936656
$126$		0			0
$127$		−	8.76393i		−	0.777673i	−0.921307	−	0.388837i	$-0.872877\pi$
$127$		−	8.76393i		−	0.777673i	0.921307	−	0.388837i	$-0.127123\pi$
$128$		0			0
$129$	1.70820	−	4.47214i	0.150399	−	0.393750i
$130$		0			0
$131$		−	11.2361i		−	0.981700i	−0.871244	−	0.490850i	$-0.836686\pi$
$131$		−	11.2361i		−	0.981700i	0.871244	−	0.490850i	$-0.163314\pi$
$132$		0			0
$133$		−	16.9443i		−	1.46925i
$134$		0			0
$135$	5.70820	−	2.94427i	0.491284	−	0.253403i
$136$		0			0
$137$			10.4721i			0.894695i	0.894360	+	0.447347i	$0.147631\pi$
$137$			10.4721i			0.894695i	−0.894360	+	0.447347i	$0.852369\pi$
$138$		0			0
$139$	3.70820			0.314526			0.157263	−	0.987557i	$-0.449733\pi$
$139$	3.70820			0.314526			0.157263	+	0.987557i	$0.449733\pi$
$140$		0			0
$141$	4.94427	−	12.9443i	0.416383	−	1.09010i
$142$		0			0
$143$	3.41641			0.285694
$144$		0			0
$145$	11.4164			0.948081
$146$		0			0
$147$	−2.14590	+	5.61803i	−0.176991	+	0.463368i
$148$		0			0
$149$	3.70820			0.303788			0.151894	−	0.988397i	$-0.451463\pi$
$149$	3.70820			0.303788			0.151894	+	0.988397i	$0.451463\pi$
$150$		0			0
$151$		−	6.29180i		−	0.512019i	−0.966674	−	0.256010i	$-0.917592\pi$
$151$		−	6.29180i		−	0.512019i	0.966674	−	0.256010i	$-0.0824079\pi$
$152$		0			0
$153$	−12.9443	+	14.4721i	−1.04648	+	1.17000i
$154$		0			0
$155$			0.944272i			0.0758457i
$156$		0			0
$157$			4.47214i			0.356915i	0.983948	+	0.178458i	$0.0571108\pi$
$157$			4.47214i			0.356915i	−0.983948	+	0.178458i	$0.942889\pi$
$158$		0			0
$159$	−0.763932	+	2.00000i	−0.0605838	+	0.158610i
$160$		0			0
$161$			20.9443i			1.65064i
$162$		0			0
$163$	18.1803			1.42399			0.711997	−	0.702182i	$-0.247791\pi$
$163$	18.1803			1.42399			0.711997	+	0.702182i	$0.247791\pi$
$164$		0			0
$165$	1.52786	+	0.583592i	0.118944	+	0.0454326i
$166$		0			0
$167$	22.4721			1.73895			0.869473	−	0.493980i	$-0.164459\pi$
$167$	22.4721			1.73895			0.869473	+	0.493980i	$0.164459\pi$
$168$		0			0
$169$	−7.00000			−0.538462
$170$		0			0
$171$	−11.7082	−	10.4721i	−0.895349	−	0.800824i
$172$		0			0
$173$	8.65248			0.657836			0.328918	−	0.944359i	$-0.393316\pi$
$173$	8.65248			0.657836			0.328918	+	0.944359i	$0.393316\pi$
$174$		0			0
$175$			11.2361i			0.849367i
$176$		0			0
$177$	−5.23607	−	2.00000i	−0.393567	−	0.150329i
$178$		0			0
$179$		−	16.1803i		−	1.20938i	−0.796463	−	0.604688i	$-0.793298\pi$
$179$		−	16.1803i		−	1.20938i	0.796463	−	0.604688i	$-0.206702\pi$
$180$		0			0
$181$			11.5279i			0.856859i	0.903575	+	0.428430i	$0.140933\pi$
$181$			11.5279i			0.856859i	−0.903575	+	0.428430i	$0.859067\pi$
$182$		0			0
$183$	−13.7082	−	5.23607i	−1.01334	−	0.387061i
$184$		0			0
$185$		−	0.583592i		−	0.0429065i
$186$		0			0
$187$	−4.94427			−0.361561
$188$		0			0
$189$	7.70820	+	14.9443i	0.560689	+	1.08704i
$190$		0			0
$191$	−4.94427			−0.357755			−0.178877	−	0.983871i	$-0.557247\pi$
$191$	−4.94427			−0.357755			−0.178877	+	0.983871i	$0.557247\pi$
$192$		0			0
$193$	23.8885			1.71954			0.859768	−	0.510686i	$-0.170608\pi$
$193$	23.8885			1.71954			0.859768	+	0.510686i	$0.170608\pi$
$194$		0			0
$195$	−8.94427	−	3.41641i	−0.640513	−	0.244654i
$196$		0			0
$197$	−9.23607			−0.658043			−0.329021	−	0.944323i	$-0.606719\pi$
$197$	−9.23607			−0.658043			−0.329021	+	0.944323i	$0.606719\pi$
$198$		0			0
$199$		−	16.1803i		−	1.14699i	−0.819208	−	0.573497i	$-0.805586\pi$
$199$		−	16.1803i		−	1.14699i	0.819208	−	0.573497i	$-0.194414\pi$
$200$		0			0
$201$	−2.29180	+	6.00000i	−0.161651	+	0.423207i
$202$		0			0
$203$			29.8885i			2.09776i
$204$		0			0
$205$			3.05573i			0.213421i
$206$		0			0
$207$	14.4721	+	12.9443i	1.00588	+	0.899689i
$208$		0			0
$209$		−	4.00000i		−	0.276686i
$210$		0			0
$211$	−6.76393			−0.465648			−0.232824	−	0.972519i	$-0.574797\pi$
$211$	−6.76393			−0.465648			−0.232824	+	0.972519i	$0.574797\pi$
$212$		0			0
$213$	7.05573	−	18.4721i	0.483451	−	1.26569i
$214$		0			0
$215$	−3.41641			−0.232997
$216$		0			0
$217$	−2.47214			−0.167820
$218$		0			0
$219$	1.23607	−	3.23607i	0.0835257	−	0.218673i
$220$		0			0
$221$	28.9443			1.94700
$222$		0			0
$223$			12.1803i			0.815656i	0.913059	+	0.407828i	$0.133714\pi$
$223$			12.1803i			0.815656i	−0.913059	+	0.407828i	$0.866286\pi$
$224$		0			0
$225$	7.76393	+	6.94427i	0.517595	+	0.462951i
$226$		0			0
$227$			20.1803i			1.33942i	0.742624	+	0.669708i	$0.233581\pi$
$227$			20.1803i			1.33942i	−0.742624	+	0.669708i	$0.766419\pi$
$228$		0			0
$229$		−	20.4721i		−	1.35284i	−0.736518	−	0.676418i	$-0.763531\pi$
$229$		−	20.4721i		−	1.35284i	0.736518	−	0.676418i	$-0.236469\pi$
$230$		0			0
$231$	−1.52786	+	4.00000i	−0.100526	+	0.263181i
$232$		0			0
$233$		−	7.05573i		−	0.462236i	−0.972926	−	0.231118i	$-0.925762\pi$
$233$		−	7.05573i		−	0.462236i	0.972926	−	0.231118i	$-0.0742384\pi$
$234$		0			0
$235$	−9.88854			−0.645057
$236$		0			0
$237$	−22.1803	−	8.47214i	−1.44077	−	0.550324i
$238$		0			0
$239$	−8.00000			−0.517477			−0.258738	−	0.965947i	$-0.583307\pi$
$239$	−8.00000			−0.517477			−0.258738	+	0.965947i	$0.583307\pi$
$240$		0			0
$241$	−2.00000			−0.128831			−0.0644157	−	0.997923i	$-0.520518\pi$
$241$	−2.00000			−0.128831			−0.0644157	+	0.997923i	$0.520518\pi$
$242$		0			0
$243$	15.0902	+	3.90983i	0.968035	+	0.250816i
$244$		0			0
$245$	4.29180			0.274193
$246$		0			0
$247$			23.4164i			1.48995i
$248$		0			0
$249$	11.7082	+	4.47214i	0.741977	+	0.283410i
$250$		0			0
$251$			10.2918i			0.649612i	0.945781	+	0.324806i	$0.105299\pi$
$251$			10.2918i			0.649612i	−0.945781	+	0.324806i	$0.894701\pi$
$252$		0			0
$253$			4.94427i			0.310844i
$254$		0			0
$255$	12.9443	+	4.94427i	0.810602	+	0.309622i
$256$		0			0
$257$			3.05573i			0.190611i	0.995448	+	0.0953055i	$0.0303828\pi$
$257$			3.05573i			0.190611i	−0.995448	+	0.0953055i	$0.969617\pi$
$258$		0			0
$259$	1.52786			0.0949369
$260$		0			0
$261$	20.6525	+	18.4721i	1.27836	+	1.14340i
$262$		0			0
$263$	1.52786			0.0942121			0.0471061	−	0.998890i	$-0.485000\pi$
$263$	1.52786			0.0942121			0.0471061	+	0.998890i	$0.485000\pi$
$264$		0			0
$265$	1.52786			0.0938559
$266$		0			0
$267$	6.47214	+	2.47214i	0.396088	+	0.151292i
$268$		0			0
$269$	−6.18034			−0.376822			−0.188411	−	0.982090i	$-0.560334\pi$
$269$	−6.18034			−0.376822			−0.188411	+	0.982090i	$0.560334\pi$
$270$		0			0
$271$		−	13.7082i		−	0.832714i	−0.909201	−	0.416357i	$-0.863307\pi$
$271$		−	13.7082i		−	0.832714i	0.909201	−	0.416357i	$-0.136693\pi$
$272$		0			0
$273$	8.94427	−	23.4164i	0.541332	−	1.41723i
$274$		0			0
$275$			2.65248i			0.159950i
$276$		0			0
$277$			16.4721i			0.989715i	0.868974	+	0.494857i	$0.164780\pi$
$277$			16.4721i			0.989715i	−0.868974	+	0.494857i	$0.835220\pi$
$278$		0			0
$279$	−1.52786	+	1.70820i	−0.0914708	+	0.102267i
$280$		0			0
$281$			7.05573i			0.420909i	0.977604	+	0.210455i	$0.0674945\pi$
$281$			7.05573i			0.420909i	−0.977604	+	0.210455i	$0.932506\pi$
$282$		0			0
$283$	−25.2361			−1.50013			−0.750064	−	0.661365i	$-0.769977\pi$
$283$	−25.2361			−1.50013			−0.750064	+	0.661365i	$0.769977\pi$
$284$		0			0
$285$	−4.00000	+	10.4721i	−0.236940	+	0.620316i
$286$		0			0
$287$	−8.00000			−0.472225
$288$		0			0
$289$	−24.8885			−1.46403
$290$		0			0
$291$	−5.23607	+	13.7082i	−0.306944	+	0.803589i
$292$		0			0
$293$	−25.2361			−1.47431			−0.737153	−	0.675725i	$-0.763831\pi$
$293$	−25.2361			−1.47431			−0.737153	+	0.675725i	$0.763831\pi$
$294$		0			0
$295$			4.00000i			0.232889i
$296$		0			0
$297$	1.81966	+	3.52786i	0.105587	+	0.204707i
$298$		0			0
$299$		−	28.9443i		−	1.67389i
$300$		0			0
$301$		−	8.94427i		−	0.515539i
$302$		0			0
$303$	7.23607	−	18.9443i	0.415701	−	1.08832i
$304$		0			0
$305$			10.4721i			0.599633i
$306$		0			0
$307$	−21.5967			−1.23259			−0.616296	−	0.787515i	$-0.711367\pi$
$307$	−21.5967			−1.23259			−0.616296	+	0.787515i	$0.711367\pi$
$308$		0			0
$309$	−13.2361	−	5.05573i	−0.752974	−	0.287610i
$310$		0			0
$311$	14.4721			0.820640			0.410320	−	0.911942i	$-0.365417\pi$
$311$	14.4721			0.820640			0.410320	+	0.911942i	$0.365417\pi$
$312$		0			0
$313$	−8.47214			−0.478873			−0.239437	−	0.970912i	$-0.576963\pi$
$313$	−8.47214			−0.478873			−0.239437	+	0.970912i	$0.576963\pi$
$314$		0			0
$315$	8.00000	−	8.94427i	0.450749	−	0.503953i
$316$		0			0
$317$	−27.1246			−1.52347			−0.761735	−	0.647889i	$-0.775652\pi$
$317$	−27.1246			−1.52347			−0.761735	+	0.647889i	$0.775652\pi$
$318$		0			0
$319$			7.05573i			0.395045i
$320$		0			0
$321$	−13.2361	−	5.05573i	−0.738765	−	0.282183i
$322$		0			0
$323$		−	33.8885i		−	1.88561i
$324$		0			0
$325$		−	15.5279i		−	0.861331i
$326$		0			0
$327$	−0.763932	−	0.291796i	−0.0422455	−	0.0161364i
$328$		0			0
$329$		−	25.8885i		−	1.42728i
$330$		0			0
$331$	34.5410			1.89855			0.949273	−	0.314453i	$-0.101821\pi$
$331$	34.5410			1.89855			0.949273	+	0.314453i	$0.101821\pi$
$332$		0			0
$333$	0.944272	−	1.05573i	0.0517458	−	0.0578535i
$334$		0			0
$335$	4.58359			0.250428
$336$		0			0
$337$	−22.3607			−1.21806			−0.609032	−	0.793146i	$-0.708442\pi$
$337$	−22.3607			−1.21806			−0.609032	+	0.793146i	$0.708442\pi$
$338$		0			0
$339$	12.9443	+	4.94427i	0.703036	+	0.268536i
$340$		0			0
$341$	−0.583592			−0.0316033
$342$		0			0
$343$		−	11.4164i		−	0.616428i
$344$		0			0
$345$	4.94427	−	12.9443i	0.266191	−	0.696896i
$346$		0			0
$347$			10.2918i			0.552493i	0.961087	+	0.276246i	$0.0890905\pi$
$347$			10.2918i			0.552493i	−0.961087	+	0.276246i	$0.910909\pi$
$348$		0			0
$349$		−	19.5279i		−	1.04530i	−0.852547	−	0.522651i	$-0.824943\pi$
$349$		−	19.5279i		−	1.04530i	0.852547	−	0.522651i	$-0.175057\pi$
$350$		0			0
$351$	−10.6525	−	20.6525i	−0.568587	−	1.10235i
$352$		0			0
$353$		−	25.8885i		−	1.37791i	−0.724805	−	0.688954i	$-0.758070\pi$
$353$		−	25.8885i		−	1.37791i	0.724805	−	0.688954i	$-0.241930\pi$
$354$		0			0
$355$	−14.1115			−0.748958
$356$		0			0
$357$	−12.9443	+	33.8885i	−0.685084	+	1.79357i
$358$		0			0
$359$	−4.58359			−0.241913			−0.120956	−	0.992658i	$-0.538596\pi$
$359$	−4.58359			−0.241913			−0.120956	+	0.992658i	$0.538596\pi$
$360$		0			0
$361$	8.41641			0.442969
$362$		0			0
$363$	6.43769	−	16.8541i	0.337891	−	0.884611i
$364$		0			0
$365$	−2.47214			−0.129398
$366$		0			0
$367$		−	0.763932i		−	0.0398769i	−0.999801	−	0.0199385i	$-0.993653\pi$
$367$		−	0.763932i		−	0.0398769i	0.999801	−	0.0199385i	$-0.00634703\pi$
$368$		0			0
$369$	−4.94427	+	5.52786i	−0.257389	+	0.287769i
$370$		0			0
$371$			4.00000i			0.207670i
$372$		0			0
$373$			29.4164i			1.52312i	0.648092	+	0.761562i	$0.275567\pi$
$373$			29.4164i			1.52312i	−0.648092	+	0.761562i	$0.724433\pi$
$374$		0			0
$375$	6.47214	−	16.9443i	0.334220	−	0.874998i
$376$		0			0
$377$		−	41.3050i		−	2.12731i
$378$		0			0
$379$	20.6525			1.06085			0.530423	−	0.847733i	$-0.322033\pi$
$379$	20.6525			1.06085			0.530423	+	0.847733i	$0.322033\pi$
$380$		0			0
$381$	−14.1803	−	5.41641i	−0.726481	−	0.277491i
$382$		0			0
$383$	−11.0557			−0.564921			−0.282461	−	0.959279i	$-0.591151\pi$
$383$	−11.0557			−0.564921			−0.282461	+	0.959279i	$0.591151\pi$
$384$		0			0
$385$	3.05573			0.155734
$386$		0			0
$387$	−6.18034	−	5.52786i	−0.314164	−	0.280997i
$388$		0			0
$389$	9.81966			0.497877			0.248938	−	0.968519i	$-0.419918\pi$
$389$	9.81966			0.497877			0.248938	+	0.968519i	$0.419918\pi$
$390$		0			0
$391$			41.8885i			2.11839i
$392$		0			0
$393$	−18.1803	−	6.94427i	−0.917077	−	0.350292i
$394$		0			0
$395$			16.9443i			0.852559i
$396$		0			0
$397$			9.41641i			0.472596i	0.971681	+	0.236298i	$0.0759341\pi$
$397$			9.41641i			0.472596i	−0.971681	+	0.236298i	$0.924066\pi$
$398$		0			0
$399$	−27.4164	−	10.4721i	−1.37254	−	0.524263i
$400$		0			0
$401$			16.3607i			0.817013i	0.912755	+	0.408507i	$0.133950\pi$
$401$			16.3607i			0.817013i	−0.912755	+	0.408507i	$0.866050\pi$
$402$		0			0
$403$	3.41641			0.170183
$404$		0			0
$405$	−1.23607	−	11.0557i	−0.0614207	−	0.549364i
$406$		0			0
$407$	0.360680			0.0178782
$408$		0			0
$409$	−3.88854			−0.192276			−0.0961381	−	0.995368i	$-0.530649\pi$
$409$	−3.88854			−0.192276			−0.0961381	+	0.995368i	$0.530649\pi$
$410$		0			0
$411$	16.9443	+	6.47214i	0.835799	+	0.319247i
$412$		0			0
$413$	−10.4721			−0.515300
$414$		0			0
$415$		−	8.94427i		−	0.439057i
$416$		0			0
$417$	2.29180	−	6.00000i	0.112230	−	0.293821i
$418$		0			0
$419$			12.1803i			0.595049i	0.954714	+	0.297524i	$0.0961609\pi$
$419$			12.1803i			0.595049i	−0.954714	+	0.297524i	$0.903839\pi$
$420$		0			0
$421$			5.41641i			0.263980i	0.991251	+	0.131990i	$0.0421366\pi$
$421$			5.41641i			0.263980i	−0.991251	+	0.131990i	$0.957863\pi$
$422$		0			0
$423$	−17.8885	−	16.0000i	−0.869771	−	0.777947i
$424$		0			0
$425$			22.4721i			1.09006i
$426$		0			0
$427$	−27.4164			−1.32677
$428$		0			0
$429$	2.11146	−	5.52786i	0.101942	−	0.266888i
$430$		0			0
$431$	−8.00000			−0.385346			−0.192673	−	0.981263i	$-0.561716\pi$
$431$	−8.00000			−0.385346			−0.192673	+	0.981263i	$0.561716\pi$
$432$		0			0
$433$	7.52786			0.361766			0.180883	−	0.983505i	$-0.442104\pi$
$433$	7.52786			0.361766			0.180883	+	0.983505i	$0.442104\pi$
$434$		0			0
$435$	7.05573	−	18.4721i	0.338296	−	0.885671i
$436$		0			0
$437$	−33.8885			−1.62111
$438$		0			0
$439$		−	0.180340i		−	0.00860715i	−0.999991	−	0.00430358i	$-0.998630\pi$
$439$		−	0.180340i		−	0.00860715i	0.999991	−	0.00430358i	$-0.00136988\pi$
$440$		0			0
$441$	7.76393	+	6.94427i	0.369711	+	0.330680i
$442$		0			0
$443$		−	29.7082i		−	1.41148i	−0.708471	−	0.705740i	$-0.750615\pi$
$443$		−	29.7082i		−	1.41148i	0.708471	−	0.705740i	$-0.249385\pi$
$444$		0			0
$445$		−	4.94427i		−	0.234381i
$446$		0			0
$447$	2.29180	−	6.00000i	0.108398	−	0.283790i
$448$		0			0
$449$		−	27.4164i		−	1.29386i	−0.762549	−	0.646930i	$-0.776053\pi$
$449$		−	27.4164i		−	1.29386i	0.762549	−	0.646930i	$-0.223947\pi$
$450$		0			0
$451$	−1.88854			−0.0889281
$452$		0			0
$453$	−10.1803	−	3.88854i	−0.478314	−	0.182700i
$454$		0			0
$455$	−17.8885			−0.838628
$456$		0			0
$457$	22.0000			1.02912			0.514558	−	0.857455i	$-0.327956\pi$
$457$	22.0000			1.02912			0.514558	+	0.857455i	$0.327956\pi$
$458$		0			0
$459$	15.4164	+	29.8885i	0.719576	+	1.39508i
$460$		0			0
$461$	−14.1803			−0.660444			−0.330222	−	0.943903i	$-0.607124\pi$
$461$	−14.1803			−0.660444			−0.330222	+	0.943903i	$0.607124\pi$
$462$		0			0
$463$			39.2361i			1.82345i	0.410796	+	0.911727i	$0.365251\pi$
$463$			39.2361i			1.82345i	−0.410796	+	0.911727i	$0.634749\pi$
$464$		0			0
$465$	1.52786	+	0.583592i	0.0708530	+	0.0270634i
$466$		0			0
$467$			15.2361i			0.705041i	0.935804	+	0.352521i	$0.114675\pi$
$467$			15.2361i			0.705041i	−0.935804	+	0.352521i	$0.885325\pi$
$468$		0			0
$469$			12.0000i			0.554109i
$470$		0			0
$471$	7.23607	+	2.76393i	0.333420	+	0.127355i
$472$		0			0
$473$		−	2.11146i		−	0.0970849i
$474$		0			0
$475$	−18.1803			−0.834171
$476$		0			0
$477$	2.76393	+	2.47214i	0.126552	+	0.113191i
$478$		0			0
$479$	41.8885			1.91394			0.956968	−	0.290193i	$-0.0937194\pi$
$479$	41.8885			1.91394			0.956968	+	0.290193i	$0.0937194\pi$
$480$		0			0
$481$	−2.11146			−0.0962741
$482$		0			0
$483$	33.8885	+	12.9443i	1.54198	+	0.588985i
$484$		0			0
$485$	10.4721			0.475515
$486$		0			0
$487$		−	1.34752i		−	0.0610621i	−0.999534	−	0.0305311i	$-0.990280\pi$
$487$		−	1.34752i		−	0.0610621i	0.999534	−	0.0305311i	$-0.00971985\pi$
$488$		0			0
$489$	11.2361	−	29.4164i	0.508113	−	1.33026i
$490$		0			0
$491$		−	26.0689i		−	1.17647i	−0.808689	−	0.588236i	$-0.799823\pi$
$491$		−	26.0689i		−	1.17647i	0.808689	−	0.588236i	$-0.200177\pi$
$492$		0			0
$493$			59.7771i			2.69222i
$494$		0			0
$495$	1.88854	−	2.11146i	0.0848837	−	0.0949029i
$496$		0			0
$497$		−	36.9443i		−	1.65718i
$498$		0			0
$499$	−22.7639			−1.01905			−0.509527	−	0.860455i	$-0.670180\pi$
$499$	−22.7639			−1.01905			−0.509527	+	0.860455i	$0.670180\pi$
$500$		0			0
$501$	13.8885	−	36.3607i	0.620494	−	1.62448i
$502$		0			0
$503$	30.4721			1.35869			0.679343	−	0.733821i	$-0.262265\pi$
$503$	30.4721			1.35869			0.679343	+	0.733821i	$0.262265\pi$
$504$		0			0
$505$	−14.4721			−0.644002
$506$		0			0
$507$	−4.32624	+	11.3262i	−0.192135	+	0.503016i
$508$		0			0
$509$	−12.2918			−0.544824			−0.272412	−	0.962181i	$-0.587821\pi$
$509$	−12.2918			−0.544824			−0.272412	+	0.962181i	$0.587821\pi$
$510$		0			0
$511$		−	6.47214i		−	0.286310i
$512$		0			0
$513$	−24.1803	+	12.4721i	−1.06759	+	0.550658i
$514$		0			0
$515$			10.1115i			0.445564i
$516$		0			0
$517$		−	6.11146i		−	0.268782i
$518$		0			0
$519$	5.34752	−	14.0000i	0.234730	−	0.614532i
$520$		0			0
$521$			20.3607i			0.892018i	0.895029	+	0.446009i	$0.147155\pi$
$521$			20.3607i			0.892018i	−0.895029	+	0.446009i	$0.852845\pi$
$522$		0			0
$523$	−34.1803			−1.49460			−0.747301	−	0.664486i	$-0.768651\pi$
$523$	−34.1803			−1.49460			−0.747301	+	0.664486i	$0.768651\pi$
$524$		0			0
$525$	18.1803	+	6.94427i	0.793455	+	0.303073i
$526$		0			0
$527$	−4.94427			−0.215376
$528$		0			0
$529$	18.8885			0.821241
$530$		0			0
$531$	−6.47214	+	7.23607i	−0.280867	+	0.314019i
$532$		0			0
$533$	11.0557			0.478877
$534$		0			0
$535$			10.1115i			0.437156i
$536$		0			0
$537$	−26.1803	−	10.0000i	−1.12977	−	0.431532i
$538$		0			0
$539$			2.65248i			0.114250i
$540$		0			0
$541$		−	18.3607i		−	0.789387i	−0.918813	−	0.394694i	$-0.870851\pi$
$541$		−	18.3607i		−	0.789387i	0.918813	−	0.394694i	$-0.129149\pi$
$542$		0			0
$543$	18.6525	+	7.12461i	0.800454	+	0.305746i
$544$		0			0
$545$			0.583592i			0.0249983i
$546$		0			0
$547$	−2.76393			−0.118177			−0.0590886	−	0.998253i	$-0.518819\pi$
$547$	−2.76393			−0.118177			−0.0590886	+	0.998253i	$0.518819\pi$
$548$		0			0
$549$	−16.9443	+	18.9443i	−0.723164	+	0.808522i
$550$		0			0
$551$	−48.3607			−2.06023
$552$		0			0
$553$	−44.3607			−1.88641
$554$		0			0
$555$	−0.944272	−	0.360680i	−0.0400821	−	0.0153100i
$556$		0			0
$557$	16.6525			0.705588			0.352794	−	0.935701i	$-0.385232\pi$
$557$	16.6525			0.705588			0.352794	+	0.935701i	$0.385232\pi$
$558$		0			0
$559$			12.3607i			0.522801i
$560$		0			0
$561$	−3.05573	+	8.00000i	−0.129013	+	0.337760i
$562$		0			0
$563$		−	36.5410i		−	1.54002i	−0.638032	−	0.770010i	$-0.720251\pi$
$563$		−	36.5410i		−	1.54002i	0.638032	−	0.770010i	$-0.279749\pi$
$564$		0			0
$565$		−	9.88854i		−	0.416014i
$566$		0			0
$567$	28.9443	−	3.23607i	1.21555	−	0.135902i
$568$		0			0
$569$			37.5279i			1.57325i	0.617431	+	0.786625i	$0.288173\pi$
$569$			37.5279i			1.57325i	−0.617431	+	0.786625i	$0.711827\pi$
$570$		0			0
$571$	−35.1246			−1.46992			−0.734960	−	0.678111i	$-0.762799\pi$
$571$	−35.1246			−1.46992			−0.734960	+	0.678111i	$0.762799\pi$
$572$		0			0
$573$	−3.05573	+	8.00000i	−0.127655	+	0.334205i
$574$		0			0
$575$	22.4721			0.937153
$576$		0			0
$577$	19.5279			0.812956			0.406478	−	0.913661i	$-0.366757\pi$
$577$	19.5279			0.812956			0.406478	+	0.913661i	$0.366757\pi$
$578$		0			0
$579$	14.7639	−	38.6525i	0.613568	−	1.60634i
$580$		0			0
$581$	23.4164			0.971476
$582$		0			0
$583$			0.944272i			0.0391077i
$584$		0			0
$585$	−11.0557	+	12.3607i	−0.457098	+	0.511051i
$586$		0			0
$587$			27.5967i			1.13904i	0.821978	+	0.569520i	$0.192871\pi$
$587$			27.5967i			1.13904i	−0.821978	+	0.569520i	$0.807129\pi$
$588$		0			0
$589$		−	4.00000i		−	0.164817i
$590$		0			0
$591$	−5.70820	+	14.9443i	−0.234804	+	0.614725i
$592$		0			0
$593$			11.0557i			0.454004i	0.973894	+	0.227002i	$0.0728924\pi$
$593$			11.0557i			0.454004i	−0.973894	+	0.227002i	$0.927108\pi$
$594$		0			0
$595$	25.8885			1.06133
$596$		0			0
$597$	−26.1803	−	10.0000i	−1.07149	−	0.409273i
$598$		0			0
$599$	19.4164			0.793333			0.396666	−	0.917963i	$-0.370167\pi$
$599$	19.4164			0.793333			0.396666	+	0.917963i	$0.370167\pi$
$600$		0			0
$601$	37.7771			1.54096			0.770480	−	0.637464i	$-0.220017\pi$
$601$	37.7771			1.54096			0.770480	+	0.637464i	$0.220017\pi$
$602$		0			0
$603$	8.29180	+	7.41641i	0.337668	+	0.302019i
$604$		0			0
$605$	−12.8754			−0.523459
$606$		0			0
$607$			39.2361i			1.59254i	0.604940	+	0.796271i	$0.293197\pi$
$607$			39.2361i			1.59254i	−0.604940	+	0.796271i	$0.706803\pi$
$608$		0			0
$609$	48.3607	+	18.4721i	1.95967	+	0.748529i
$610$		0			0
$611$			35.7771i			1.44739i
$612$		0			0
$613$		−	43.3050i		−	1.74907i	−0.484962	−	0.874535i	$-0.661167\pi$
$613$		−	43.3050i		−	1.74907i	0.484962	−	0.874535i	$-0.338833\pi$
$614$		0			0
$615$	4.94427	+	1.88854i	0.199372	+	0.0761534i
$616$		0			0
$617$		−	13.8885i		−	0.559132i	−0.960127	−	0.279566i	$-0.909809\pi$
$617$		−	13.8885i		−	0.559132i	0.960127	−	0.279566i	$-0.0901905\pi$
$618$		0			0
$619$	−11.1246			−0.447136			−0.223568	−	0.974688i	$-0.571770\pi$
$619$	−11.1246			−0.447136			−0.223568	+	0.974688i	$0.571770\pi$
$620$		0			0
$621$	29.8885	−	15.4164i	1.19939	−	0.618639i
$622$		0			0
$623$	12.9443			0.518601
$624$		0			0
$625$	4.41641			0.176656
$626$		0			0
$627$	−6.47214	−	2.47214i	−0.258472	−	0.0987276i
$628$		0			0
$629$	3.05573			0.121840
$630$		0			0
$631$		−	21.1246i		−	0.840958i	−0.907302	−	0.420479i	$-0.861862\pi$
$631$		−	21.1246i		−	0.840958i	0.907302	−	0.420479i	$-0.138138\pi$
$632$		0			0
$633$	−4.18034	+	10.9443i	−0.166154	+	0.434996i
$634$		0			0
$635$			10.8328i			0.429887i
$636$		0			0
$637$		−	15.5279i		−	0.615236i
$638$		0			0
$639$	−25.5279	−	22.8328i	−1.00987	−	0.903252i
$640$		0			0
$641$			37.3050i			1.47346i	0.676189	+	0.736729i	$0.263630\pi$
$641$			37.3050i			1.47346i	−0.676189	+	0.736729i	$0.736370\pi$
$642$		0			0
$643$	−44.6525			−1.76092			−0.880461	−	0.474119i	$-0.842767\pi$
$643$	−44.6525			−1.76092			−0.880461	+	0.474119i	$0.842767\pi$
$644$		0			0
$645$	−2.11146	+	5.52786i	−0.0831385	+	0.217659i
$646$		0			0
$647$	−29.3050			−1.15210			−0.576048	−	0.817416i	$-0.695406\pi$
$647$	−29.3050			−1.15210			−0.576048	+	0.817416i	$0.695406\pi$
$648$		0			0
$649$	−2.47214			−0.0970398
$650$		0			0
$651$	−1.52786	+	4.00000i	−0.0598817	+	0.156772i
$652$		0			0
$653$	27.7082			1.08431			0.542153	−	0.840280i	$-0.317609\pi$
$653$	27.7082			1.08431			0.542153	+	0.840280i	$0.317609\pi$
$654$		0			0
$655$			13.8885i			0.542670i
$656$		0			0
$657$	−4.47214	−	4.00000i	−0.174475	−	0.156055i
$658$		0			0
$659$			41.7082i			1.62472i	0.583156	+	0.812360i	$0.301818\pi$
$659$			41.7082i			1.62472i	−0.583156	+	0.812360i	$0.698182\pi$
$660$		0			0
$661$		−	11.3050i		−	0.439712i	−0.975532	−	0.219856i	$-0.929441\pi$
$661$		−	11.3050i		−	0.439712i	0.975532	−	0.219856i	$-0.0705587\pi$
$662$		0			0
$663$	17.8885	−	46.8328i	0.694733	−	1.81884i
$664$		0			0
$665$			20.9443i			0.812184i
$666$		0			0
$667$	59.7771			2.31458
$668$		0			0
$669$	19.7082	+	7.52786i	0.761963	+	0.291044i
$670$		0			0
$671$	−6.47214			−0.249854
$672$		0			0
$673$	1.41641			0.0545985			0.0272993	−	0.999627i	$-0.491309\pi$
$673$	1.41641			0.0545985			0.0272993	+	0.999627i	$0.491309\pi$
$674$		0			0
$675$	16.0344	−	8.27051i	0.617166	−	0.318332i
$676$		0			0
$677$	27.7082			1.06491			0.532456	−	0.846457i	$-0.321269\pi$
$677$	27.7082			1.06491			0.532456	+	0.846457i	$0.321269\pi$
$678$		0			0
$679$			27.4164i			1.05215i
$680$		0			0
$681$	32.6525	+	12.4721i	1.25125	+	0.477933i
$682$		0			0
$683$		−	8.76393i		−	0.335343i	−0.985843	−	0.167671i	$-0.946375\pi$
$683$		−	8.76393i		−	0.335343i	0.985843	−	0.167671i	$-0.0536247\pi$
$684$		0			0
$685$		−	12.9443i		−	0.494575i
$686$		0			0
$687$	−33.1246	−	12.6525i	−1.26378	−	0.482722i
$688$		0			0
$689$		−	5.52786i		−	0.210595i
$690$		0			0
$691$	16.2918			0.619769			0.309885	−	0.950774i	$-0.399710\pi$
$691$	16.2918			0.619769			0.309885	+	0.950774i	$0.399710\pi$
$692$		0			0
$693$	5.52786	+	4.94427i	0.209986	+	0.187817i
$694$		0			0
$695$	−4.58359			−0.173866
$696$		0			0
$697$	−16.0000			−0.606043
$698$		0			0
$699$	−11.4164	−	4.36068i	−0.431808	−	0.164936i
$700$		0			0
$701$	27.7082			1.04652			0.523262	−	0.852172i	$-0.324715\pi$
$701$	27.7082			1.04652			0.523262	+	0.852172i	$0.324715\pi$
$702$		0			0
$703$			2.47214i			0.0932384i
$704$		0			0
$705$	−6.11146	+	16.0000i	−0.230171	+	0.602595i
$706$		0			0
$707$		−	37.8885i		−	1.42495i
$708$		0			0
$709$			16.4721i			0.618624i	0.950961	+	0.309312i	$0.100099\pi$
$709$			16.4721i			0.618624i	−0.950961	+	0.309312i	$0.899901\pi$
$710$		0			0
$711$	−27.4164	+	30.6525i	−1.02820	+	1.14956i
$712$		0			0
$713$			4.94427i			0.185164i
$714$		0			0
$715$	−4.22291			−0.157928
$716$		0			0
$717$	−4.94427	+	12.9443i	−0.184647	+	0.483413i
$718$		0			0
$719$	−22.8328			−0.851520			−0.425760	−	0.904836i	$-0.639993\pi$
$719$	−22.8328			−0.851520			−0.425760	+	0.904836i	$0.639993\pi$
$720$		0			0
$721$	−26.4721			−0.985874
$722$		0			0
$723$	−1.23607	+	3.23607i	−0.0459699	+	0.120351i
$724$		0			0
$725$	32.0689			1.19101
$726$		0			0
$727$		−	8.18034i		−	0.303392i	−0.988427	−	0.151696i	$-0.951527\pi$
$727$		−	8.18034i		−	0.303392i	0.988427	−	0.151696i	$-0.0484735\pi$
$728$		0			0
$729$	15.6525	−	22.0000i	0.579721	−	0.814815i
$730$		0			0
$731$		−	17.8885i		−	0.661632i
$732$		0			0
$733$			2.58359i			0.0954272i	0.998861	+	0.0477136i	$0.0151935\pi$
$733$			2.58359i			0.0954272i	−0.998861	+	0.0477136i	$0.984807\pi$
$734$		0			0
$735$	2.65248	−	6.94427i	0.0978380	−	0.256143i
$736$		0			0
$737$			2.83282i			0.104348i
$738$		0			0
$739$	27.1246			0.997795			0.498897	−	0.866661i	$-0.333738\pi$
$739$	27.1246			0.997795			0.498897	+	0.866661i	$0.333738\pi$
$740$		0			0
$741$	37.8885	+	14.4721i	1.39187	+	0.531647i
$742$		0			0
$743$	−19.4164			−0.712319			−0.356159	−	0.934425i	$-0.615914\pi$
$743$	−19.4164			−0.712319			−0.356159	+	0.934425i	$0.615914\pi$
$744$		0			0
$745$	−4.58359			−0.167930
$746$		0			0
$747$	14.4721	−	16.1803i	0.529508	−	0.592008i
$748$		0			0
$749$	−26.4721			−0.967271
$750$		0			0
$751$		−	31.5967i		−	1.15298i	−0.817104	−	0.576491i	$-0.804422\pi$
$751$		−	31.5967i		−	1.15298i	0.817104	−	0.576491i	$-0.195578\pi$
$752$		0			0
$753$	16.6525	+	6.36068i	0.606850	+	0.231796i
$754$		0			0
$755$			7.77709i			0.283037i
$756$		0			0
$757$			21.4164i			0.778393i	0.921155	+	0.389196i	$0.127247\pi$
$757$			21.4164i			0.778393i	−0.921155	+	0.389196i	$0.872753\pi$
$758$		0			0
$759$	8.00000	+	3.05573i	0.290382	+	0.110916i
$760$		0			0
$761$			38.2492i			1.38653i	0.720681	+	0.693267i	$0.243829\pi$
$761$			38.2492i			1.38653i	−0.720681	+	0.693267i	$0.756171\pi$
$762$		0			0
$763$	−1.52786			−0.0553124
$764$		0			0
$765$	16.0000	−	17.8885i	0.578481	−	0.646762i
$766$		0			0
$767$	14.4721			0.522559
$768$		0			0
$769$	−21.7771			−0.785302			−0.392651	−	0.919688i	$-0.628442\pi$
$769$	−21.7771			−0.785302			−0.392651	+	0.919688i	$0.628442\pi$
$770$		0			0
$771$	4.94427	+	1.88854i	0.178064	+	0.0680142i
$772$		0			0
$773$	−30.1803			−1.08551			−0.542756	−	0.839891i	$-0.682619\pi$
$773$	−30.1803			−1.08551			−0.542756	+	0.839891i	$0.682619\pi$
$774$		0			0
$775$			2.65248i			0.0952797i
$776$		0			0
$777$	0.944272	−	2.47214i	0.0338756	−	0.0886874i
$778$		0			0
$779$		−	12.9443i		−	0.463777i
$780$		0			0
$781$		−	8.72136i		−	0.312075i
$782$		0			0
$783$	42.6525	−	22.0000i	1.52428	−	0.786216i
$784$		0			0
$785$		−	5.52786i		−	0.197298i
$786$		0			0
$787$	−5.81966			−0.207448			−0.103724	−	0.994606i	$-0.533076\pi$
$787$	−5.81966			−0.207448			−0.103724	+	0.994606i	$0.533076\pi$
$788$		0			0
$789$	0.944272	−	2.47214i	0.0336170	−	0.0880104i
$790$		0			0
$791$	25.8885			0.920491
$792$		0			0
$793$	37.8885			1.34546
$794$		0			0
$795$	0.944272	−	2.47214i	0.0334899	−	0.0876776i
$796$		0			0
$797$	−8.06888			−0.285815			−0.142907	−	0.989736i	$-0.545645\pi$
$797$	−8.06888			−0.285815			−0.142907	+	0.989736i	$0.545645\pi$
$798$		0			0
$799$		−	51.7771i		−	1.83174i
$800$		0			0
$801$	8.00000	−	8.94427i	0.282666	−	0.316030i
$802$		0			0
$803$		−	1.52786i		−	0.0539172i
$804$		0			0
$805$		−	25.8885i		−	0.912451i
$806$		0			0
$807$	−3.81966	+	10.0000i	−0.134458	+	0.352017i
$808$		0			0
$809$			23.4164i			0.823277i	0.911347	+	0.411639i	$0.135043\pi$
$809$			23.4164i			0.823277i	−0.911347	+	0.411639i	$0.864957\pi$
$810$		0			0
$811$	29.8197			1.04711			0.523555	−	0.851992i	$-0.324605\pi$
$811$	29.8197			1.04711			0.523555	+	0.851992i	$0.324605\pi$
$812$		0			0
$813$	−22.1803	−	8.47214i	−0.777898	−	0.297131i
$814$		0			0
$815$	−22.4721			−0.787165
$816$		0			0
$817$	14.4721			0.506316
$818$		0			0
$819$	−32.3607	−	28.9443i	−1.13077	−	1.01139i
$820$		0			0
$821$	52.4296			1.82980			0.914902	−	0.403676i	$-0.132268\pi$
$821$	52.4296			1.82980			0.914902	+	0.403676i	$0.132268\pi$
$822$		0			0
$823$			10.8754i			0.379092i	0.981872	+	0.189546i	$0.0607016\pi$
$823$			10.8754i			0.379092i	−0.981872	+	0.189546i	$0.939298\pi$
$824$		0			0
$825$	4.29180	+	1.63932i	0.149421	+	0.0570738i
$826$		0			0
$827$		−	21.1246i		−	0.734575i	−0.930108	−	0.367287i	$-0.880287\pi$
$827$		−	21.1246i		−	0.734575i	0.930108	−	0.367287i	$-0.119713\pi$
$828$		0			0
$829$			4.47214i			0.155324i	0.996980	+	0.0776619i	$0.0247455\pi$
$829$			4.47214i			0.155324i	−0.996980	+	0.0776619i	$0.975255\pi$
$830$		0			0
$831$	26.6525	+	10.1803i	0.924564	+	0.353152i
$832$		0			0
$833$			22.4721i			0.778613i
$834$		0			0
$835$	−27.7771			−0.961266
$836$		0			0
$837$	1.81966	+	3.52786i	0.0628967	+	0.121941i
$838$		0			0
$839$	−4.58359			−0.158243			−0.0791216	−	0.996865i	$-0.525212\pi$
$839$	−4.58359			−0.158243			−0.0791216	+	0.996865i	$0.525212\pi$
$840$		0			0
$841$	56.3050			1.94155
$842$		0			0
$843$	11.4164	+	4.36068i	0.393202	+	0.150190i
$844$		0			0
$845$	8.65248			0.297654
$846$		0			0
$847$		−	33.7082i		−	1.15823i
$848$		0			0
$849$	−15.5967	+	40.8328i	−0.535279	+	1.40138i
$850$		0			0
$851$		−	3.05573i		−	0.104749i
$852$		0			0
$853$			23.3050i			0.797946i	0.916963	+	0.398973i	$0.130633\pi$
$853$			23.3050i			0.797946i	−0.916963	+	0.398973i	$0.869367\pi$
$854$		0			0
$855$	14.4721	+	12.9443i	0.494937	+	0.442685i
$856$		0			0
$857$		−	52.3607i		−	1.78861i	−0.447461	−	0.894303i	$-0.647672\pi$
$857$		−	52.3607i		−	1.78861i	0.447461	−	0.894303i	$-0.352328\pi$
$858$		0			0
$859$	−21.2361			−0.724565			−0.362283	−	0.932068i	$-0.618002\pi$
$859$	−21.2361			−0.724565			−0.362283	+	0.932068i	$0.618002\pi$
$860$		0			0
$861$	−4.94427	+	12.9443i	−0.168500	+	0.441140i
$862$		0			0
$863$	−30.8328			−1.04956			−0.524781	−	0.851238i	$-0.675853\pi$
$863$	−30.8328			−1.04956			−0.524781	+	0.851238i	$0.675853\pi$
$864$		0			0
$865$	−10.6950			−0.363643
$866$		0			0
$867$	−15.3820	+	40.2705i	−0.522399	+	1.36766i
$868$		0			0
$869$	−10.4721			−0.355243
$870$		0			0
$871$		−	16.5836i		−	0.561914i
$872$		0			0
$873$	18.9443	+	16.9443i	0.641166	+	0.573477i
$874$		0			0
$875$		−	33.8885i		−	1.14564i
$876$		0			0
$877$		−	45.4164i		−	1.53360i	−0.641884	−	0.766802i	$-0.721847\pi$
$877$		−	45.4164i		−	1.53360i	0.641884	−	0.766802i	$-0.278153\pi$
$878$		0			0
$879$	−15.5967	+	40.8328i	−0.526065	+	1.37726i
$880$		0			0
$881$			24.0000i			0.808581i	0.914631	+	0.404290i	$0.132481\pi$
$881$			24.0000i			0.808581i	−0.914631	+	0.404290i	$0.867519\pi$
$882$		0			0
$883$	5.23607			0.176208			0.0881039	−	0.996111i	$-0.471919\pi$
$883$	5.23607			0.176208			0.0881039	+	0.996111i	$0.471919\pi$
$884$		0			0
$885$	6.47214	+	2.47214i	0.217558	+	0.0830999i
$886$		0			0
$887$	24.3607			0.817952			0.408976	−	0.912545i	$-0.365886\pi$
$887$	24.3607			0.817952			0.408976	+	0.912545i	$0.365886\pi$
$888$		0			0
$889$	−28.3607			−0.951187
$890$		0			0
$891$	6.83282	−	0.763932i	0.228908	−	0.0255927i
$892$		0			0
$893$	41.8885			1.40175
$894$		0			0
$895$			20.0000i			0.668526i
$896$		0			0
$897$	−46.8328	−	17.8885i	−1.56370	−	0.597281i
$898$		0			0
$899$			7.05573i			0.235322i
$900$		0			0
$901$			8.00000i			0.266519i
$902$		0			0
$903$	−14.4721	−	5.52786i	−0.481603	−	0.183956i
$904$		0			0
$905$		−	14.2492i		−	0.473660i
$906$		0			0
$907$	34.7639			1.15432			0.577159	−	0.816632i	$-0.304161\pi$
$907$	34.7639			1.15432			0.577159	+	0.816632i	$0.304161\pi$
$908$		0			0
$909$	−26.1803	−	23.4164i	−0.868347	−	0.776673i
$910$		0			0
$911$	1.88854			0.0625702			0.0312851	−	0.999511i	$-0.490040\pi$
$911$	1.88854			0.0625702			0.0312851	+	0.999511i	$0.490040\pi$
$912$		0			0
$913$	5.52786			0.182946
$914$		0			0
$915$	16.9443	+	6.47214i	0.560160	+	0.213962i
$916$		0			0
$917$	−36.3607			−1.20074
$918$		0			0
$919$			8.54102i			0.281742i	0.990028	+	0.140871i	$0.0449903\pi$
$919$			8.54102i			0.281742i	−0.990028	+	0.140871i	$0.955010\pi$
$920$		0			0
$921$	−13.3475	+	34.9443i	−0.439816	+	1.15145i
$922$		0			0
$923$			51.0557i			1.68052i
$924$		0			0
$925$		−	1.63932i		−	0.0539005i
$926$		0			0
$927$	−16.3607	+	18.2918i	−0.537355	+	0.600781i
$928$		0			0
$929$		−	20.5836i		−	0.675326i	−0.941267	−	0.337663i	$-0.890364\pi$
$929$		−	20.5836i		−	0.675326i	0.941267	−	0.337663i	$-0.109636\pi$
$930$		0			0
$931$	−18.1803			−0.595837
$932$		0			0
$933$	8.94427	−	23.4164i	0.292822	−	0.766619i
$934$		0			0
$935$	6.11146			0.199866
$936$		0			0
$937$	−30.3607			−0.991840			−0.495920	−	0.868368i	$-0.665169\pi$
$937$	−30.3607			−0.991840			−0.495920	+	0.868368i	$0.665169\pi$
$938$		0			0
$939$	−5.23607	+	13.7082i	−0.170873	+	0.447350i
$940$		0			0
$941$	11.7082			0.381677			0.190838	−	0.981621i	$-0.438879\pi$
$941$	11.7082			0.381677			0.190838	+	0.981621i	$0.438879\pi$
$942$		0			0
$943$			16.0000i			0.521032i
$944$		0			0
$945$	−9.52786	−	18.4721i	−0.309941	−	0.600899i
$946$		0			0
$947$		−	34.0689i		−	1.10709i	−0.832819	−	0.553545i	$-0.813275\pi$
$947$		−	34.0689i		−	1.10709i	0.832819	−	0.553545i	$-0.186725\pi$
$948$		0			0
$949$			8.94427i			0.290343i
$950$		0			0
$951$	−16.7639	+	43.8885i	−0.543608	+	1.42318i
$952$		0			0
$953$		−	20.3607i		−	0.659547i	−0.944060	−	0.329774i	$-0.893028\pi$
$953$		−	20.3607i		−	0.659547i	0.944060	−	0.329774i	$-0.106972\pi$
$954$		0			0
$955$	6.11146			0.197762
$956$		0			0
$957$	11.4164	+	4.36068i	0.369040	+	0.140961i
$958$		0			0
$959$	33.8885			1.09432
$960$		0			0
$961$	30.4164			0.981174
$962$		0			0
$963$	−16.3607	+	18.2918i	−0.527216	+	0.589445i
$964$		0			0
$965$	−29.5279			−0.950536
$966$		0			0
$967$			4.76393i			0.153198i	0.997062	+	0.0765989i	$0.0244061\pi$
$967$			4.76393i			0.153198i	−0.997062	+	0.0765989i	$0.975594\pi$
$968$		0			0
$969$	−54.8328	−	20.9443i	−1.76148	−	0.672827i
$970$		0			0
$971$			59.0132i			1.89382i	0.321495	+	0.946911i	$0.395815\pi$
$971$			59.0132i			1.89382i	−0.321495	+	0.946911i	$0.604185\pi$
$972$		0			0
$973$		−	12.0000i		−	0.384702i
$974$		0			0
$975$	−25.1246	−	9.59675i	−0.804632	−	0.307342i
$976$		0			0
$977$			15.6393i			0.500346i	0.968201	+	0.250173i	$0.0804875\pi$
$977$			15.6393i			0.500346i	−0.968201	+	0.250173i	$0.919512\pi$
$978$		0			0
$979$	3.05573			0.0976615
$980$		0			0
$981$	−0.944272	+	1.05573i	−0.0301483	+	0.0337068i
$982$		0			0
$983$	30.4721			0.971910			0.485955	−	0.873984i	$-0.338472\pi$
$983$	30.4721			0.971910			0.485955	+	0.873984i	$0.338472\pi$
$984$		0			0
$985$	11.4164			0.363757
$986$		0			0
$987$	−41.8885	−	16.0000i	−1.33333	−	0.509286i
$988$		0			0
$989$	−17.8885			−0.568823
$990$		0			0
$991$		−	27.8197i		−	0.883721i	−0.897084	−	0.441860i	$-0.854319\pi$
$991$		−	27.8197i		−	0.883721i	0.897084	−	0.441860i	$-0.145681\pi$
$992$		0			0
$993$	21.3475	−	55.8885i	0.677443	−	1.77357i
$994$		0			0
$995$			20.0000i			0.634043i
$996$		0			0
$997$			26.3607i			0.834851i	0.908711	+	0.417426i	$0.137068\pi$
$997$			26.3607i			0.834851i	−0.908711	+	0.417426i	$0.862932\pi$
$998$		0			0
$999$	−1.12461	−	2.18034i	−0.0355811	−	0.0689829i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	768.2.f.c.383.3		4
3.2	odd	2		768.2.f.b.383.4		4
4.3	odd	2		768.2.f.f.383.2		4
8.3	odd	2		768.2.f.b.383.3		4
8.5	even	2		768.2.f.e.383.2		4
12.11	even	2		768.2.f.e.383.1		4
16.3	odd	4		384.2.c.b.383.1	yes	4
16.5	even	4		384.2.c.a.383.1	✓	4
16.11	odd	4		384.2.c.d.383.4	yes	4
16.13	even	4		384.2.c.c.383.4	yes	4
24.5	odd	2		768.2.f.f.383.1		4
24.11	even	2	inner	768.2.f.c.383.4		4
48.5	odd	4		384.2.c.d.383.3	yes	4
48.11	even	4		384.2.c.a.383.2	yes	4
48.29	odd	4		384.2.c.b.383.2	yes	4
48.35	even	4		384.2.c.c.383.3	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.2.c.a.383.1	✓	4	16.5	even	4
384.2.c.a.383.2	yes	4	48.11	even	4
384.2.c.b.383.1	yes	4	16.3	odd	4
384.2.c.b.383.2	yes	4	48.29	odd	4
384.2.c.c.383.3	yes	4	48.35	even	4
384.2.c.c.383.4	yes	4	16.13	even	4
384.2.c.d.383.3	yes	4	48.5	odd	4
384.2.c.d.383.4	yes	4	16.11	odd	4
768.2.f.b.383.3		4	8.3	odd	2
768.2.f.b.383.4		4	3.2	odd	2
768.2.f.c.383.3		4	1.1	even	1	trivial
768.2.f.c.383.4		4	24.11	even	2	inner
768.2.f.e.383.1		4	12.11	even	2
768.2.f.e.383.2		4	8.5	even	2
768.2.f.f.383.1		4	24.5	odd	2
768.2.f.f.383.2		4	4.3	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:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	768.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+(0.618034 - 1.61803i) q^{3} -1.23607 q^{5} -3.23607i q^{7} +(-2.23607 - 2.00000i) q^{9} +O(q^{10})\)	\(q+(0.618034 - 1.61803i) q^{3} -1.23607 q^{5} -3.23607i q^{7} +(-2.23607 - 2.00000i) q^{9} -0.763932i q^{11} +4.47214i q^{13} +(-0.763932 + 2.00000i) q^{15} -6.47214i q^{17} +5.23607 q^{19} +(-5.23607 - 2.00000i) q^{21} -6.47214 q^{23} -3.47214 q^{25} +(-4.61803 + 2.38197i) q^{27} -9.23607 q^{29} -0.763932i q^{31} +(-1.23607 - 0.472136i) q^{33} +4.00000i q^{35} +0.472136i q^{37} +(7.23607 + 2.76393i) q^{39} -2.47214i q^{41} +2.76393 q^{43} +(2.76393 + 2.47214i) q^{45} +8.00000 q^{47} -3.47214 q^{49} +(-10.4721 - 4.00000i) q^{51} -1.23607 q^{53} +0.944272i q^{55} +(3.23607 - 8.47214i) q^{57} -3.23607i q^{59} -8.47214i q^{61} +(-6.47214 + 7.23607i) q^{63} -5.52786i q^{65} -3.70820 q^{67} +(-4.00000 + 10.4721i) q^{69} +11.4164 q^{71} +2.00000 q^{73} +(-2.14590 + 5.61803i) q^{75} -2.47214 q^{77} -13.7082i q^{79} +(1.00000 + 8.94427i) q^{81} +7.23607i q^{83} +8.00000i q^{85} +(-5.70820 + 14.9443i) q^{87} +4.00000i q^{89} +14.4721 q^{91} +(-1.23607 - 0.472136i) q^{93} -6.47214 q^{95} -8.47214 q^{97} +(-1.52786 + 1.70820i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 2 q^{3} + 4 q^{5}+O(q^{10}) \) 4 * q - 2 * q^3 + 4 * q^5	\( 4 q - 2 q^{3} + 4 q^{5} - 12 q^{15} + 12 q^{19} - 12 q^{21} - 8 q^{23} + 4 q^{25} - 14 q^{27} - 28 q^{29} + 4 q^{33} + 20 q^{39} + 20 q^{43} + 20 q^{45} + 32 q^{47} + 4 q^{49} - 24 q^{51} + 4 q^{53} + 4 q^{57} - 8 q^{63} + 12 q^{67} - 16 q^{69} - 8 q^{71} + 8 q^{73} - 22 q^{75} + 8 q^{77} + 4 q^{81} + 4 q^{87} + 40 q^{91} + 4 q^{93} - 8 q^{95} - 16 q^{97} - 24 q^{99}+O(q^{100}) \) 4 * q - 2 * q^3 + 4 * q^5 - 12 * q^15 + 12 * q^19 - 12 * q^21 - 8 * q^23 + 4 * q^25 - 14 * q^27 - 28 * q^29 + 4 * q^33 + 20 * q^39 + 20 * q^43 + 20 * q^45 + 32 * q^47 + 4 * q^49 - 24 * q^51 + 4 * q^53 + 4 * q^57 - 8 * q^63 + 12 * q^67 - 16 * q^69 - 8 * q^71 + 8 * q^73 - 22 * q^75 + 8 * q^77 + 4 * q^81 + 4 * q^87 + 40 * q^91 + 4 * q^93 - 8 * q^95 - 16 * q^97 - 24 * q^99

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