LMFDB - Embedded newform 1680.2.f.l.881.15

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1680,2,Mod(881,1680)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1680, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1, 0, 1]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("1680.881");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1680 = 2^{4} \cdot 3 \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1680.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:	$13.4148675396$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 2 x^{15} + x^{14} - 4 x^{13} + 10 x^{12} - 32 x^{11} + 71 x^{10} - 70 x^{9} + 74 x^{8} + \cdots + 6561 $ x^16 - 2x^15 + x^14 - 4x^13 + 10x^12 - 32x^11 + 71x^10 - 70x^9 + 74x^8 - 210x^7 + 639x^6 - 864x^5 + 810x^4 - 972x^3 + 729x^2 - 4374x + 6561
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{16} $
Twist minimal:	no (minimal twist has level 840)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			881.15
Root			$0.348228 - 1.69668i$ of defining polynomial
Character	$\chi$	$=$	1680.881
Dual form			1680.2.f.l.881.16

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.69668 - 0.348228i) q^{3} +1.00000 q^{5} +(2.63166 - 0.272689i) q^{7} +(2.75748 - 1.18166i) q^{9} +O(q^{10})$	\(q+(1.69668 - 0.348228i) q^{3} +1.00000 q^{5} +(2.63166 - 0.272689i) q^{7} +(2.75748 - 1.18166i) q^{9} +3.13162i q^{11} +4.34145i q^{13} +(1.69668 - 0.348228i) q^{15} -0.791039 q^{17} +0.768290i q^{19} +(4.37014 - 1.37908i) q^{21} +6.32787i q^{23} +1.00000 q^{25} +(4.26708 - 2.96514i) q^{27} -5.96574i q^{29} +5.49220i q^{31} +(1.09052 + 5.31337i) q^{33} +(2.63166 - 0.272689i) q^{35} +1.83534 q^{37} +(1.51181 + 7.36607i) q^{39} -1.60516 q^{41} -6.18441 q^{43} +(2.75748 - 1.18166i) q^{45} -3.24695 q^{47} +(6.85128 - 1.43525i) q^{49} +(-1.34214 + 0.275462i) q^{51} -10.0560i q^{53} +3.13162i q^{55} +(0.267540 + 1.30354i) q^{57} -9.30114 q^{59} -13.1365i q^{61} +(6.93451 - 3.86167i) q^{63} +4.34145i q^{65} +2.64216 q^{67} +(2.20354 + 10.7364i) q^{69} -0.227035i q^{71} -15.7756i q^{73} +(1.69668 - 0.348228i) q^{75} +(0.853957 + 8.24136i) q^{77} +13.6345 q^{79} +(6.20734 - 6.51682i) q^{81} -2.74892 q^{83} -0.791039 q^{85} +(-2.07744 - 10.1220i) q^{87} -16.6351 q^{89} +(1.18386 + 11.4252i) q^{91} +(1.91254 + 9.31853i) q^{93} +0.768290i q^{95} +6.11396i q^{97} +(3.70052 + 8.63536i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 16 q^{5} - 2 q^{7} - 2 q^{9}+O(q^{10}) $ 16 * q + 16 * q^5 - 2 * q^7 - 2 * q^9	$ 16 q + 16 q^{5} - 2 q^{7} - 2 q^{9} + 10 q^{21} + 16 q^{25} - 6 q^{27} + 6 q^{33} - 2 q^{35} + 12 q^{37} - 6 q^{39} + 32 q^{41} - 32 q^{43} - 2 q^{45} - 4 q^{47} - 4 q^{49} - 6 q^{51} + 24 q^{59} + 24 q^{63} + 8 q^{69} - 32 q^{77} + 4 q^{79} - 6 q^{81} - 20 q^{83} - 6 q^{87} - 24 q^{89} - 20 q^{91} - 32 q^{93} + 58 q^{99}+O(q^{100}) $ 16 * q + 16 * q^5 - 2 * q^7 - 2 * q^9 + 10 * q^21 + 16 * q^25 - 6 * q^27 + 6 * q^33 - 2 * q^35 + 12 * q^37 - 6 * q^39 + 32 * q^41 - 32 * q^43 - 2 * q^45 - 4 * q^47 - 4 * q^49 - 6 * q^51 + 24 * q^59 + 24 * q^63 + 8 * q^69 - 32 * q^77 + 4 * q^79 - 6 * q^81 - 20 * q^83 - 6 * q^87 - 24 * q^89 - 20 * q^91 - 32 * q^93 + 58 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1680\mathbb{Z}\right)^\times$.

$n$	$241$	$337$	$421$	$1121$	$1471$
$\chi(n)$	$-1$	$1$	$1$	$-1$	$1$

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
$3$	1.69668	−	0.348228i	0.979581	−	0.201049i
$3$	1.69668	−	0.348228i	0.979581	−	0.201049i
$4$		0			0
$5$	1.00000			0.447214
$5$	1.00000			0.447214
$6$		0			0
$7$	2.63166	−	0.272689i	0.994674	−	0.103067i
$7$	2.63166	−	0.272689i	0.994674	−	0.103067i
$8$		0			0
$9$	2.75748	−	1.18166i	0.919158	−	0.393888i
$10$		0			0
$11$			3.13162i			0.944219i	0.881540	+	0.472109i	$0.156507\pi$
$11$			3.13162i			0.944219i	−0.881540	+	0.472109i	$0.843493\pi$
$12$		0			0
$13$			4.34145i			1.20410i	0.798458	+	0.602051i	$0.205650\pi$
$13$			4.34145i			1.20410i	−0.798458	+	0.602051i	$0.794350\pi$
$14$		0			0
$15$	1.69668	−	0.348228i	0.438082	−	0.0899120i
$16$		0			0
$17$	−0.791039			−0.191855			−0.0959276	−	0.995388i	$-0.530582\pi$
$17$	−0.791039			−0.191855			−0.0959276	+	0.995388i	$0.530582\pi$
$18$		0			0
$19$			0.768290i			0.176258i	0.996109	+	0.0881289i	$0.0280887\pi$
$19$			0.768290i			0.176258i	−0.996109	+	0.0881289i	$0.971911\pi$
$20$		0			0
$21$	4.37014	−	1.37908i	0.953643	−	0.300941i
$22$		0			0
$23$			6.32787i			1.31945i	0.751506	+	0.659726i	$0.229328\pi$
$23$			6.32787i			1.31945i	−0.751506	+	0.659726i	$0.770672\pi$
$24$		0			0
$25$	1.00000			0.200000
$26$		0			0
$27$	4.26708	−	2.96514i	0.821199	−	0.570642i
$28$		0			0
$29$		−	5.96574i		−	1.10781i	−0.832580	−	0.553905i	$-0.813137\pi$
$29$		−	5.96574i		−	1.10781i	0.832580	−	0.553905i	$-0.186863\pi$
$30$		0			0
$31$			5.49220i			0.986428i	0.869908	+	0.493214i	$0.164178\pi$
$31$			5.49220i			0.986428i	−0.869908	+	0.493214i	$0.835822\pi$
$32$		0			0
$33$	1.09052	+	5.31337i	0.189834	+	0.924939i
$34$		0			0
$35$	2.63166	−	0.272689i	0.444832	−	0.0460928i
$36$		0			0
$37$	1.83534			0.301728			0.150864	−	0.988555i	$-0.451794\pi$
$37$	1.83534			0.301728			0.150864	+	0.988555i	$0.451794\pi$
$38$		0			0
$39$	1.51181	+	7.36607i	0.242084	+	1.17952i
$40$		0			0
$41$	−1.60516			−0.250684			−0.125342	−	0.992114i	$-0.540003\pi$
$41$	−1.60516			−0.250684			−0.125342	+	0.992114i	$0.540003\pi$
$42$		0			0
$43$	−6.18441			−0.943113			−0.471557	−	0.881836i	$-0.656308\pi$
$43$	−6.18441			−0.943113			−0.471557	+	0.881836i	$0.656308\pi$
$44$		0			0
$45$	2.75748	−	1.18166i	0.411060	−	0.176152i
$46$		0			0
$47$	−3.24695			−0.473617			−0.236808	−	0.971556i	$-0.576101\pi$
$47$	−3.24695			−0.473617			−0.236808	+	0.971556i	$0.576101\pi$
$48$		0			0
$49$	6.85128	−	1.43525i	0.978755	−	0.205035i
$50$		0			0
$51$	−1.34214	+	0.275462i	−0.187938	+	0.0385724i
$52$		0			0
$53$		−	10.0560i		−	1.38130i	−0.723189	−	0.690651i	$-0.757324\pi$
$53$		−	10.0560i		−	1.38130i	0.723189	−	0.690651i	$-0.242676\pi$
$54$		0			0
$55$			3.13162i			0.422267i
$56$		0			0
$57$	0.267540	+	1.30354i	0.0354365	+	0.172659i
$58$		0			0
$59$	−9.30114			−1.21091			−0.605453	−	0.795881i	$-0.707008\pi$
$59$	−9.30114			−1.21091			−0.605453	+	0.795881i	$0.707008\pi$
$60$		0			0
$61$		−	13.1365i		−	1.68195i	−0.541071	−	0.840977i	$-0.681981\pi$
$61$		−	13.1365i		−	1.68195i	0.541071	−	0.840977i	$-0.318019\pi$
$62$		0			0
$63$	6.93451	−	3.86167i	0.873667	−	0.486525i
$64$		0			0
$65$			4.34145i			0.538491i
$66$		0			0
$67$	2.64216			0.322792			0.161396	−	0.986890i	$-0.448400\pi$
$67$	2.64216			0.322792			0.161396	+	0.986890i	$0.448400\pi$
$68$		0			0
$69$	2.20354	+	10.7364i	0.265275	+	1.29251i
$70$		0			0
$71$		−	0.227035i		−	0.0269441i	−0.999909	−	0.0134721i	$-0.995712\pi$
$71$		−	0.227035i		−	0.0269441i	0.999909	−	0.0134721i	$-0.00428842\pi$
$72$		0			0
$73$		−	15.7756i		−	1.84639i	−0.384327	−	0.923197i	$-0.625566\pi$
$73$		−	15.7756i		−	1.84639i	0.384327	−	0.923197i	$-0.374434\pi$
$74$		0			0
$75$	1.69668	−	0.348228i	0.195916	−	0.0402099i
$76$		0			0
$77$	0.853957	+	8.24136i	0.0973174	+	0.939190i
$78$		0			0
$79$	13.6345			1.53400			0.767002	−	0.641645i	$-0.221748\pi$
$79$	13.6345			1.53400			0.767002	+	0.641645i	$0.221748\pi$
$80$		0			0
$81$	6.20734	−	6.51682i	0.689704	−	0.724091i
$82$		0			0
$83$	−2.74892			−0.301733			−0.150866	−	0.988554i	$-0.548206\pi$
$83$	−2.74892			−0.301733			−0.150866	+	0.988554i	$0.548206\pi$
$84$		0			0
$85$	−0.791039			−0.0858002
$86$		0			0
$87$	−2.07744	−	10.1220i	−0.222725	−	1.08519i
$88$		0			0
$89$	−16.6351			−1.76331			−0.881657	−	0.471892i	$-0.843571\pi$
$89$	−16.6351			−1.76331			−0.881657	+	0.471892i	$0.843571\pi$
$90$		0			0
$91$	1.18386	+	11.4252i	0.124103	+	1.19769i
$92$		0			0
$93$	1.91254	+	9.31853i	0.198321	+	0.966287i
$94$		0			0
$95$			0.768290i			0.0788248i
$96$		0			0
$97$			6.11396i			0.620779i	0.950609	+	0.310389i	$0.100459\pi$
$97$			6.11396i			0.620779i	−0.950609	+	0.310389i	$0.899541\pi$
$98$		0			0
$99$	3.70052	+	8.63536i	0.371917	+	0.867886i
$100$		0			0
$101$	−1.20483			−0.119885			−0.0599425	−	0.998202i	$-0.519092\pi$
$101$	−1.20483			−0.119885			−0.0599425	+	0.998202i	$0.519092\pi$
$102$		0			0
$103$			1.42563i			0.140471i	0.997530	+	0.0702355i	$0.0223751\pi$
$103$			1.42563i			0.140471i	−0.997530	+	0.0702355i	$0.977625\pi$
$104$		0			0
$105$	4.37014	−	1.37908i	0.426482	−	0.134585i
$106$		0			0
$107$		−	11.2184i		−	1.08453i	−0.840208	−	0.542264i	$-0.817567\pi$
$107$		−	11.2184i		−	1.08453i	0.840208	−	0.542264i	$-0.182433\pi$
$108$		0			0
$109$	16.1783			1.54960			0.774798	−	0.632209i	$-0.217851\pi$
$109$	16.1783			1.54960			0.774798	+	0.632209i	$0.217851\pi$
$110$		0			0
$111$	3.11399	−	0.639115i	0.295567	−	0.0606621i
$112$		0			0
$113$		−	4.91036i		−	0.461928i	−0.972962	−	0.230964i	$-0.925812\pi$
$113$		−	4.91036i		−	0.461928i	0.972962	−	0.230964i	$-0.0741879\pi$
$114$		0			0
$115$			6.32787i			0.590077i
$116$		0			0
$117$	5.13014	+	11.9714i	0.474282	+	1.10676i
$118$		0			0
$119$	−2.08175	+	0.215707i	−0.190833	+	0.0197739i
$120$		0			0
$121$	1.19296			0.108451
$122$		0			0
$123$	−2.72345	+	0.558962i	−0.245565	+	0.0503999i
$124$		0			0
$125$	1.00000			0.0894427
$126$		0			0
$127$	6.25616			0.555144			0.277572	−	0.960705i	$-0.410470\pi$
$127$	6.25616			0.555144			0.277572	+	0.960705i	$0.410470\pi$
$128$		0			0
$129$	−10.4930	+	2.15358i	−0.923856	+	0.189612i
$130$		0			0
$131$	−9.05057			−0.790752			−0.395376	−	0.918519i	$-0.629386\pi$
$131$	−9.05057			−0.790752			−0.395376	+	0.918519i	$0.629386\pi$
$132$		0			0
$133$	0.209504	+	2.02188i	0.0181663	+	0.175319i
$134$		0			0
$135$	4.26708	−	2.96514i	0.367251	−	0.255199i
$136$		0			0
$137$		−	5.29111i		−	0.452050i	−0.974122	−	0.226025i	$-0.927427\pi$
$137$		−	5.29111i		−	0.452050i	0.974122	−	0.226025i	$-0.0725731\pi$
$138$		0			0
$139$		−	18.3215i		−	1.55401i	−0.629494	−	0.777006i	$-0.716738\pi$
$139$		−	18.3215i		−	1.55401i	0.629494	−	0.777006i	$-0.283262\pi$
$140$		0			0
$141$	−5.50905	+	1.13068i	−0.463946	+	0.0952203i
$142$		0			0
$143$	−13.5958			−1.13694
$144$		0			0
$145$		−	5.96574i		−	0.495428i
$146$		0			0
$147$	11.1247	−	4.82097i	0.917547	−	0.397627i
$148$		0			0
$149$			23.3480i			1.91274i	0.292160	+	0.956370i	$0.405626\pi$
$149$			23.3480i			1.91274i	−0.292160	+	0.956370i	$0.594374\pi$
$150$		0			0
$151$	9.14170			0.743941			0.371971	−	0.928245i	$-0.378682\pi$
$151$	9.14170			0.743941			0.371971	+	0.928245i	$0.378682\pi$
$152$		0			0
$153$	−2.18127	+	0.934743i	−0.176345	+	0.0755695i
$154$		0			0
$155$			5.49220i			0.441144i
$156$		0			0
$157$			15.2610i			1.21796i	0.793185	+	0.608981i	$0.208421\pi$
$157$			15.2610i			1.21796i	−0.793185	+	0.608981i	$0.791579\pi$
$158$		0			0
$159$	−3.50179	−	17.0619i	−0.277710	−	1.35310i
$160$		0			0
$161$	1.72554	+	16.6528i	0.135991	+	1.31242i
$162$		0			0
$163$	−5.02683			−0.393732			−0.196866	−	0.980430i	$-0.563076\pi$
$163$	−5.02683			−0.393732			−0.196866	+	0.980430i	$0.563076\pi$
$164$		0			0
$165$	1.09052	+	5.31337i	0.0848966	+	0.413645i
$166$		0			0
$167$	21.9872			1.70142			0.850710	−	0.525635i	$-0.176172\pi$
$167$	21.9872			1.70142			0.850710	+	0.525635i	$0.176172\pi$
$168$		0			0
$169$	−5.84820			−0.449862
$170$		0			0
$171$	0.907861	+	2.11854i	0.0694258	+	0.162009i
$172$		0			0
$173$	−18.9508			−1.44081			−0.720403	−	0.693556i	$-0.756043\pi$
$173$	−18.9508			−1.44081			−0.720403	+	0.693556i	$0.756043\pi$
$174$		0			0
$175$	2.63166	−	0.272689i	0.198935	−	0.0206133i
$176$		0			0
$177$	−15.7811	+	3.23892i	−1.18618	+	0.243452i
$178$		0			0
$179$		−	1.39906i		−	0.104571i	−0.998632	−	0.0522853i	$-0.983349\pi$
$179$		−	1.39906i		−	0.104571i	0.998632	−	0.0522853i	$-0.0166505\pi$
$180$		0			0
$181$		−	13.3999i		−	0.996006i	−0.867175	−	0.498003i	$-0.834067\pi$
$181$		−	13.3999i		−	0.996006i	0.867175	−	0.498003i	$-0.165933\pi$
$182$		0			0
$183$	−4.57449	−	22.2885i	−0.338156	−	1.64761i
$184$		0			0
$185$	1.83534			0.134937
$186$		0			0
$187$		−	2.47723i		−	0.181153i
$188$		0			0
$189$	10.4209	−	8.96683i	0.758012	−	0.652241i
$190$		0			0
$191$		−	22.3132i		−	1.61452i	−0.590193	−	0.807262i	$-0.700948\pi$
$191$		−	22.3132i		−	1.61452i	0.590193	−	0.807262i	$-0.299052\pi$
$192$		0			0
$193$	−10.2441			−0.737385			−0.368692	−	0.929551i	$-0.620194\pi$
$193$	−10.2441			−0.737385			−0.368692	+	0.929551i	$0.620194\pi$
$194$		0			0
$195$	1.51181	+	7.36607i	0.108263	+	0.527495i
$196$		0			0
$197$			1.79870i			0.128152i	0.997945	+	0.0640761i	$0.0204100\pi$
$197$			1.79870i			0.128152i	−0.997945	+	0.0640761i	$0.979590\pi$
$198$		0			0
$199$			14.1953i			1.00628i	0.864204	+	0.503141i	$0.167822\pi$
$199$			14.1953i			1.00628i	−0.864204	+	0.503141i	$0.832178\pi$
$200$		0			0
$201$	4.48292	−	0.920074i	0.316201	−	0.0648970i
$202$		0			0
$203$	−1.62679	−	15.6998i	−0.114178	−	1.10191i
$204$		0			0
$205$	−1.60516			−0.112109
$206$		0			0
$207$	7.47742	+	17.4489i	0.519716	+	1.21278i
$208$		0			0
$209$	−2.40599			−0.166426
$210$		0			0
$211$	−21.9123			−1.50851			−0.754254	−	0.656583i	$-0.772001\pi$
$211$	−21.9123			−1.50851			−0.754254	+	0.656583i	$0.772001\pi$
$212$		0			0
$213$	−0.0790599	−	0.385207i	−0.00541710	−	0.0263940i
$214$		0			0
$215$	−6.18441			−0.421773
$216$		0			0
$217$	1.49766	+	14.4536i	0.101668	+	0.981175i
$218$		0			0
$219$	−5.49350	−	26.7662i	−0.371216	−	1.80869i
$220$		0			0
$221$		−	3.43426i		−	0.231013i
$222$		0			0
$223$		−	4.77721i		−	0.319905i	−0.987125	−	0.159953i	$-0.948866\pi$
$223$		−	4.77721i		−	0.319905i	0.987125	−	0.159953i	$-0.0511342\pi$
$224$		0			0
$225$	2.75748	−	1.18166i	0.183832	−	0.0787776i
$226$		0			0
$227$	19.9326			1.32297			0.661487	−	0.749957i	$-0.269926\pi$
$227$	19.9326			1.32297			0.661487	+	0.749957i	$0.269926\pi$
$228$		0			0
$229$		−	6.04167i		−	0.399245i	−0.979873	−	0.199622i	$-0.936028\pi$
$229$		−	6.04167i		−	0.399245i	0.979873	−	0.199622i	$-0.0639715\pi$
$230$		0			0
$231$	4.31876	+	13.6856i	0.284154	+	0.900447i
$232$		0			0
$233$		−	6.52570i		−	0.427513i	−0.976887	−	0.213756i	$-0.931430\pi$
$233$		−	6.52570i		−	0.427513i	0.976887	−	0.213756i	$-0.0685699\pi$
$234$		0			0
$235$	−3.24695			−0.211808
$236$		0			0
$237$	23.1335	−	4.74792i	1.50268	−	0.308410i
$238$		0			0
$239$			3.42646i			0.221639i	0.993841	+	0.110820i	$0.0353476\pi$
$239$			3.42646i			0.221639i	−0.993841	+	0.110820i	$0.964652\pi$
$240$		0			0
$241$		−	11.6681i		−	0.751607i	−0.926699	−	0.375804i	$-0.877367\pi$
$241$		−	11.6681i		−	0.751607i	0.926699	−	0.375804i	$-0.122633\pi$
$242$		0			0
$243$	8.26255	−	13.2186i	0.530043	−	0.847971i
$244$		0			0
$245$	6.85128	−	1.43525i	0.437712	−	0.0916947i
$246$		0			0
$247$	−3.33549			−0.212232
$248$		0			0
$249$	−4.66404	+	0.957248i	−0.295572	+	0.0606631i
$250$		0			0
$251$	−21.5770			−1.36193			−0.680966	−	0.732315i	$-0.738440\pi$
$251$	−21.5770			−1.36193			−0.680966	+	0.732315i	$0.738440\pi$
$252$		0			0
$253$	−19.8165			−1.24585
$254$		0			0
$255$	−1.34214	+	0.275462i	−0.0840483	+	0.0172501i
$256$		0			0
$257$	17.2123			1.07367			0.536837	−	0.843686i	$-0.319619\pi$
$257$	17.2123			1.07367			0.536837	+	0.843686i	$0.319619\pi$
$258$		0			0
$259$	4.82999	−	0.500476i	0.300121	−	0.0310981i
$260$		0			0
$261$	−7.04951	−	16.4504i	−0.436354	−	1.01825i
$262$		0			0
$263$			24.1528i			1.48933i	0.667440	+	0.744663i	$0.267390\pi$
$263$			24.1528i			1.48933i	−0.667440	+	0.744663i	$0.732610\pi$
$264$		0			0
$265$		−	10.0560i		−	0.617737i
$266$		0			0
$267$	−28.2244	+	5.79279i	−1.72731	+	0.354513i
$268$		0			0
$269$	−15.8931			−0.969021			−0.484511	−	0.874785i	$-0.661002\pi$
$269$	−15.8931			−0.969021			−0.484511	+	0.874785i	$0.661002\pi$
$270$		0			0
$271$			27.5408i			1.67299i	0.547977	+	0.836494i	$0.315398\pi$
$271$			27.5408i			1.67299i	−0.547977	+	0.836494i	$0.684602\pi$
$272$		0			0
$273$	5.98722	+	18.9728i	0.362363	+	1.14828i
$274$		0			0
$275$			3.13162i			0.188844i
$276$		0			0
$277$	29.5866			1.77769			0.888843	−	0.458212i	$-0.151510\pi$
$277$	29.5866			1.77769			0.888843	+	0.458212i	$0.151510\pi$
$278$		0			0
$279$	6.48994	+	15.1446i	0.388542	+	0.906684i
$280$		0			0
$281$			9.58240i			0.571638i	0.958284	+	0.285819i	$0.0922656\pi$
$281$			9.58240i			0.571638i	−0.958284	+	0.285819i	$0.907734\pi$
$282$		0			0
$283$		−	5.41762i		−	0.322044i	−0.986951	−	0.161022i	$-0.948521\pi$
$283$		−	5.41762i		−	0.322044i	0.986951	−	0.161022i	$-0.0514790\pi$
$284$		0			0
$285$	0.267540	+	1.30354i	0.0158477	+	0.0772153i
$286$		0			0
$287$	−4.22424	+	0.437709i	−0.249349	+	0.0258372i
$288$		0			0
$289$	−16.3743			−0.963192
$290$		0			0
$291$	2.12905	+	10.3735i	0.124807	+	0.608103i
$292$		0			0
$293$	−3.92094			−0.229064			−0.114532	−	0.993420i	$-0.536537\pi$
$293$	−3.92094			−0.229064			−0.114532	+	0.993420i	$0.536537\pi$
$294$		0			0
$295$	−9.30114			−0.541534
$296$		0			0
$297$	9.28569	+	13.3629i	0.538810	+	0.775392i
$298$		0			0
$299$	−27.4721			−1.58875
$300$		0			0
$301$	−16.2753	+	1.68642i	−0.938091	+	0.0972035i
$302$		0			0
$303$	−2.04421	+	0.419555i	−0.117437	+	0.0241028i
$304$		0			0
$305$		−	13.1365i		−	0.752193i
$306$		0			0
$307$			20.1163i			1.14810i	0.818820	+	0.574050i	$0.194628\pi$
$307$			20.1163i			1.14810i	−0.818820	+	0.574050i	$0.805372\pi$
$308$		0			0
$309$	0.496442	+	2.41884i	0.0282416	+	0.137603i
$310$		0			0
$311$	7.92125			0.449173			0.224586	−	0.974454i	$-0.427897\pi$
$311$	7.92125			0.449173			0.224586	+	0.974454i	$0.427897\pi$
$312$		0			0
$313$			7.13951i			0.403549i	0.979432	+	0.201774i	$0.0646708\pi$
$313$			7.13951i			0.403549i	−0.979432	+	0.201774i	$0.935329\pi$
$314$		0			0
$315$	6.93451	−	3.86167i	0.390716	−	0.217581i
$316$		0			0
$317$		−	10.9008i		−	0.612249i	−0.951991	−	0.306125i	$-0.900968\pi$
$317$		−	10.9008i		−	0.612249i	0.951991	−	0.306125i	$-0.0990324\pi$
$318$		0			0
$319$	18.6824			1.04602
$320$		0			0
$321$	−3.90657	−	19.0341i	−0.218043	−	1.06238i
$322$		0			0
$323$		−	0.607747i		−	0.0338160i
$324$		0			0
$325$			4.34145i			0.240820i
$326$		0			0
$327$	27.4494	−	5.63372i	1.51796	−	0.311545i
$328$		0			0
$329$	−8.54488	+	0.885407i	−0.471094	+	0.0488141i
$330$		0			0
$331$	−24.5105			−1.34722			−0.673608	−	0.739088i	$-0.735257\pi$
$331$	−24.5105			−1.34722			−0.673608	+	0.739088i	$0.735257\pi$
$332$		0			0
$333$	5.06090	−	2.16875i	0.277336	−	0.118847i
$334$		0			0
$335$	2.64216			0.144357
$336$		0			0
$337$	−24.4917			−1.33415			−0.667074	−	0.744991i	$-0.732454\pi$
$337$	−24.4917			−1.33415			−0.667074	+	0.744991i	$0.732454\pi$
$338$		0			0
$339$	−1.70992	−	8.33133i	−0.0928703	−	0.452496i
$340$		0			0
$341$	−17.1995			−0.931404
$342$		0			0
$343$	17.6389	−	5.64535i	0.952410	−	0.304821i
$344$		0			0
$345$	2.20354	+	10.7364i	0.118634	+	0.578028i
$346$		0			0
$347$			13.9427i			0.748482i	0.927332	+	0.374241i	$0.122097\pi$
$347$			13.9427i			0.748482i	−0.927332	+	0.374241i	$0.877903\pi$
$348$		0			0
$349$			22.2380i			1.19037i	0.803588	+	0.595185i	$0.202921\pi$
$349$			22.2380i			1.19037i	−0.803588	+	0.595185i	$0.797079\pi$
$350$		0			0
$351$	12.8730	+	18.5253i	0.687111	+	0.988808i
$352$		0			0
$353$	7.30187			0.388639			0.194320	−	0.980938i	$-0.437750\pi$
$353$	7.30187			0.388639			0.194320	+	0.980938i	$0.437750\pi$
$354$		0			0
$355$		−	0.227035i		−	0.0120498i
$356$		0			0
$357$	−3.45695	+	1.09091i	−0.182961	+	0.0577370i
$358$		0			0
$359$		−	22.4577i		−	1.18527i	−0.805471	−	0.592635i	$-0.798088\pi$
$359$		−	22.4577i		−	1.18527i	0.805471	−	0.592635i	$-0.201912\pi$
$360$		0			0
$361$	18.4097			0.968933
$362$		0			0
$363$	2.02408	−	0.415423i	0.106237	−	0.0218041i
$364$		0			0
$365$		−	15.7756i		−	0.825733i
$366$		0			0
$367$		−	34.5325i		−	1.80258i	−0.433216	−	0.901290i	$-0.642621\pi$
$367$		−	34.5325i		−	1.80258i	0.433216	−	0.901290i	$-0.357379\pi$
$368$		0			0
$369$	−4.42619	+	1.89676i	−0.230418	+	0.0987415i
$370$		0			0
$371$	−2.74216	−	26.4641i	−0.142366	−	1.37395i
$372$		0			0
$373$	7.87066			0.407527			0.203764	−	0.979020i	$-0.434683\pi$
$373$	7.87066			0.407527			0.203764	+	0.979020i	$0.434683\pi$
$374$		0			0
$375$	1.69668	−	0.348228i	0.0876164	−	0.0179824i
$376$		0			0
$377$	25.9000			1.33392
$378$		0			0
$379$	−33.3198			−1.71152			−0.855761	−	0.517371i	$-0.826911\pi$
$379$	−33.3198			−1.71152			−0.855761	+	0.517371i	$0.826911\pi$
$380$		0			0
$381$	10.6147	−	2.17857i	0.543809	−	0.111611i
$382$		0			0
$383$	−13.9800			−0.714345			−0.357172	−	0.934038i	$-0.616259\pi$
$383$	−13.9800			−0.714345			−0.357172	+	0.934038i	$0.616259\pi$
$384$		0			0
$385$	0.853957	+	8.24136i	0.0435217	+	0.420019i
$386$		0			0
$387$	−17.0534	+	7.30790i	−0.866871	+	0.371481i
$388$		0			0
$389$		−	17.5098i		−	0.887783i	−0.896081	−	0.443892i	$-0.853598\pi$
$389$		−	17.5098i		−	0.887783i	0.896081	−	0.443892i	$-0.146402\pi$
$390$		0			0
$391$		−	5.00559i		−	0.253144i
$392$		0			0
$393$	−15.3560	+	3.15166i	−0.774606	+	0.158980i
$394$		0			0
$395$	13.6345			0.686027
$396$		0			0
$397$		−	18.3263i		−	0.919769i	−0.887979	−	0.459884i	$-0.847891\pi$
$397$		−	18.3263i		−	0.919769i	0.887979	−	0.459884i	$-0.152109\pi$
$398$		0			0
$399$	1.05954	+	3.35753i	0.0530431	+	0.168087i
$400$		0			0
$401$		−	0.472174i		−	0.0235793i	−0.999930	−	0.0117896i	$-0.996247\pi$
$401$		−	0.472174i		−	0.0235793i	0.999930	−	0.0117896i	$-0.00375284\pi$
$402$		0			0
$403$	−23.8441			−1.18776
$404$		0			0
$405$	6.20734	−	6.51682i	0.308445	−	0.323823i
$406$		0			0
$407$			5.74758i			0.284897i
$408$		0			0
$409$			7.30432i			0.361176i	0.983559	+	0.180588i	$0.0578000\pi$
$409$			7.30432i			0.361176i	−0.983559	+	0.180588i	$0.942200\pi$
$410$		0			0
$411$	−1.84251	−	8.97734i	−0.0908843	−	0.442819i
$412$		0			0
$413$	−24.4775	+	2.53632i	−1.20446	+	0.124804i
$414$		0			0
$415$	−2.74892			−0.134939
$416$		0			0
$417$	−6.38006	−	31.0858i	−0.312433	−	1.52228i
$418$		0			0
$419$	0.396150			0.0193532			0.00967659	−	0.999953i	$-0.496920\pi$
$419$	0.396150			0.0193532			0.00967659	+	0.999953i	$0.496920\pi$
$420$		0			0
$421$	−25.0098			−1.21891			−0.609453	−	0.792823i	$-0.708611\pi$
$421$	−25.0098			−1.21891			−0.609453	+	0.792823i	$0.708611\pi$
$422$		0			0
$423$	−8.95339	+	3.83681i	−0.435329	+	0.186552i
$424$		0			0
$425$	−0.791039			−0.0383710
$426$		0			0
$427$	−3.58217	−	34.5708i	−0.173353	−	1.67300i
$428$		0			0
$429$	−23.0677	+	4.73442i	−1.11372	+	0.228580i
$430$		0			0
$431$			25.2731i			1.21736i	0.793415	+	0.608681i	$0.208301\pi$
$431$			25.2731i			1.21736i	−0.793415	+	0.608681i	$0.791699\pi$
$432$		0			0
$433$			13.3865i			0.643316i	0.946856	+	0.321658i	$0.104240\pi$
$433$			13.3865i			0.643316i	−0.946856	+	0.321658i	$0.895760\pi$
$434$		0			0
$435$	−2.07744	−	10.1220i	−0.0996055	−	0.485312i
$436$		0			0
$437$	−4.86163			−0.232563
$438$		0			0
$439$		−	17.3616i		−	0.828622i	−0.910135	−	0.414311i	$-0.864023\pi$
$439$		−	17.3616i		−	0.828622i	0.910135	−	0.414311i	$-0.135977\pi$
$440$		0			0
$441$	17.1963	−	12.0536i	0.818869	−	0.573980i
$442$		0			0
$443$			30.2288i			1.43621i	0.695933	+	0.718107i	$0.254991\pi$
$443$			30.2288i			1.43621i	−0.695933	+	0.718107i	$0.745009\pi$
$444$		0			0
$445$	−16.6351			−0.788578
$446$		0			0
$447$	8.13040	+	39.6141i	0.384555	+	1.87368i
$448$		0			0
$449$		−	37.1531i		−	1.75336i	−0.481071	−	0.876681i	$-0.659752\pi$
$449$		−	37.1531i		−	1.75336i	0.481071	−	0.876681i	$-0.340248\pi$
$450$		0			0
$451$		−	5.02675i		−	0.236701i
$452$		0			0
$453$	15.5106	−	3.18339i	0.728751	−	0.149569i
$454$		0			0
$455$	1.18386	+	11.4252i	0.0555004	+	0.535623i
$456$		0			0
$457$	7.31615			0.342235			0.171118	−	0.985251i	$-0.445262\pi$
$457$	7.31615			0.342235			0.171118	+	0.985251i	$0.445262\pi$
$458$		0			0
$459$	−3.37542	+	2.34554i	−0.157551	+	0.109481i
$460$		0			0
$461$	−20.8550			−0.971313			−0.485656	−	0.874150i	$-0.661419\pi$
$461$	−20.8550			−0.971313			−0.485656	+	0.874150i	$0.661419\pi$
$462$		0			0
$463$	−4.79424			−0.222807			−0.111404	−	0.993775i	$-0.535535\pi$
$463$	−4.79424			−0.222807			−0.111404	+	0.993775i	$0.535535\pi$
$464$		0			0
$465$	1.91254	+	9.31853i	0.0886917	+	0.432136i
$466$		0			0
$467$	10.3144			0.477292			0.238646	−	0.971107i	$-0.423296\pi$
$467$	10.3144			0.477292			0.238646	+	0.971107i	$0.423296\pi$
$468$		0			0
$469$	6.95328	−	0.720488i	0.321073	−	0.0332691i
$470$		0			0
$471$	5.31431	+	25.8931i	0.244870	+	1.19309i
$472$		0			0
$473$		−	19.3672i		−	0.890505i
$474$		0			0
$475$			0.768290i			0.0352515i
$476$		0			0
$477$	−11.8828	−	27.7292i	−0.544078	−	1.26963i
$478$		0			0
$479$	−27.1291			−1.23956			−0.619780	−	0.784775i	$-0.712778\pi$
$479$	−27.1291			−1.23956			−0.619780	+	0.784775i	$0.712778\pi$
$480$		0			0
$481$			7.96803i			0.363311i
$482$		0			0
$483$	8.72666	+	27.6537i	0.397077	+	1.25829i
$484$		0			0
$485$			6.11396i			0.277621i
$486$		0			0
$487$	10.5407			0.477646			0.238823	−	0.971063i	$-0.423238\pi$
$487$	10.5407			0.477646			0.238823	+	0.971063i	$0.423238\pi$
$488$		0			0
$489$	−8.52895	+	1.75048i	−0.385692	+	0.0791595i
$490$		0			0
$491$		−	26.2238i		−	1.18346i	−0.806134	−	0.591732i	$-0.798444\pi$
$491$		−	26.2238i		−	1.18346i	0.806134	−	0.591732i	$-0.201556\pi$
$492$		0			0
$493$			4.71914i			0.212539i
$494$		0			0
$495$	3.70052	+	8.63536i	0.166326	+	0.388131i
$496$		0			0
$497$	−0.0619099	−	0.597480i	−0.00277704	−	0.0268006i
$498$		0			0
$499$	−16.7873			−0.751505			−0.375752	−	0.926720i	$-0.622616\pi$
$499$	−16.7873			−0.751505			−0.375752	+	0.926720i	$0.622616\pi$
$500$		0			0
$501$	37.3053	−	7.65655i	1.66668	−	0.342069i
$502$		0			0
$503$	−18.4191			−0.821267			−0.410634	−	0.911800i	$-0.634692\pi$
$503$	−18.4191			−0.821267			−0.410634	+	0.911800i	$0.634692\pi$
$504$		0			0
$505$	−1.20483			−0.0536142
$506$		0			0
$507$	−9.92255	+	2.03650i	−0.440676	+	0.0904444i
$508$		0			0
$509$	−31.5649			−1.39909			−0.699544	−	0.714589i	$-0.746614\pi$
$509$	−31.5649			−1.39909			−0.699544	+	0.714589i	$0.746614\pi$
$510$		0			0
$511$	−4.30183	−	41.5160i	−0.190302	−	1.83656i
$512$		0			0
$513$	2.27809	+	3.27835i	0.100580	+	0.144743i
$514$		0			0
$515$			1.42563i			0.0628205i
$516$		0			0
$517$		−	10.1682i		−	0.447198i
$518$		0			0
$519$	−32.1536	+	6.59921i	−1.41139	+	0.289673i
$520$		0			0
$521$	19.0356			0.833963			0.416981	−	0.908915i	$-0.363088\pi$
$521$	19.0356			0.833963			0.416981	+	0.908915i	$0.363088\pi$
$522$		0			0
$523$			32.7943i			1.43400i	0.697075	+	0.716998i	$0.254484\pi$
$523$			32.7943i			1.43400i	−0.697075	+	0.716998i	$0.745516\pi$
$524$		0			0
$525$	4.37014	−	1.37908i	0.190729	−	0.0601881i
$526$		0			0
$527$		−	4.34455i		−	0.189251i
$528$		0			0
$529$	−17.0419			−0.740952
$530$		0			0
$531$	−25.6477	+	10.9908i	−1.11301	+	0.476962i
$532$		0			0
$533$		−	6.96873i		−	0.301849i
$534$		0			0
$535$		−	11.2184i		−	0.485015i
$536$		0			0
$537$	−0.487191	−	2.37376i	−0.0210239	−	0.102435i
$538$		0			0
$539$	4.49465	+	21.4556i	0.193598	+	0.924158i
$540$		0			0
$541$	−43.0191			−1.84954			−0.924769	−	0.380530i	$-0.875742\pi$
$541$	−43.0191			−1.84954			−0.924769	+	0.380530i	$0.875742\pi$
$542$		0			0
$543$	−4.66621	−	22.7354i	−0.200246	−	0.975669i
$544$		0			0
$545$	16.1783			0.693001
$546$		0			0
$547$	−1.93124			−0.0825739			−0.0412869	−	0.999147i	$-0.513146\pi$
$547$	−1.93124			−0.0825739			−0.0412869	+	0.999147i	$0.513146\pi$
$548$		0			0
$549$	−15.5229	−	36.2235i	−0.662502	−	1.54598i
$550$		0			0
$551$	4.58342			0.195260
$552$		0			0
$553$	35.8814	−	3.71798i	1.52583	−	0.158105i
$554$		0			0
$555$	3.11399	−	0.639115i	0.132181	−	0.0271289i
$556$		0			0
$557$			41.2956i			1.74975i	0.484347	+	0.874876i	$0.339057\pi$
$557$			41.2956i			1.74975i	−0.484347	+	0.874876i	$0.660943\pi$
$558$		0			0
$559$		−	26.8493i		−	1.13560i
$560$		0			0
$561$	−0.862641	−	4.20308i	−0.0364207	−	0.177454i
$562$		0			0
$563$	39.7986			1.67731			0.838656	−	0.544662i	$-0.183342\pi$
$563$	39.7986			1.67731			0.838656	+	0.544662i	$0.183342\pi$
$564$		0			0
$565$		−	4.91036i		−	0.206580i
$566$		0			0
$567$	14.5585	−	18.8427i	0.611401	−	0.791321i
$568$		0			0
$569$			21.6458i			0.907439i	0.891145	+	0.453720i	$0.149903\pi$
$569$			21.6458i			0.907439i	−0.891145	+	0.453720i	$0.850097\pi$
$570$		0			0
$571$	19.2486			0.805531			0.402765	−	0.915303i	$-0.368049\pi$
$571$	19.2486			0.805531			0.402765	+	0.915303i	$0.368049\pi$
$572$		0			0
$573$	−7.77006	−	37.8584i	−0.324599	−	1.58156i
$574$		0			0
$575$			6.32787i			0.263890i
$576$		0			0
$577$			0.0237908i			0.000990422i	1.00000		0.000495211i	$0.000157631\pi$
$577$			0.0237908i			0.000990422i	−1.00000		0.000495211i	$0.999842\pi$
$578$		0			0
$579$	−17.3810	+	3.56727i	−0.722328	+	0.148251i
$580$		0			0
$581$	−7.23421	+	0.749598i	−0.300126	+	0.0310986i
$582$		0			0
$583$	31.4916			1.30425
$584$		0			0
$585$	5.13014	+	11.9714i	0.212105	+	0.494958i
$586$		0			0
$587$	−28.4193			−1.17299			−0.586495	−	0.809953i	$-0.699492\pi$
$587$	−28.4193			−1.17299			−0.586495	+	0.809953i	$0.699492\pi$
$588$		0			0
$589$	−4.21960			−0.173866
$590$		0			0
$591$	0.626358	+	3.05183i	0.0257649	+	0.125536i
$592$		0			0
$593$	40.6292			1.66844			0.834221	−	0.551430i	$-0.185918\pi$
$593$	40.6292			1.66844			0.834221	+	0.551430i	$0.185918\pi$
$594$		0			0
$595$	−2.08175	+	0.215707i	−0.0853433	+	0.00884314i
$596$		0			0
$597$	4.94321	+	24.0850i	0.202312	+	0.985734i
$598$		0			0
$599$		−	5.26519i		−	0.215130i	−0.994198	−	0.107565i	$-0.965695\pi$
$599$		−	5.26519i		−	0.215130i	0.994198	−	0.107565i	$-0.0343054\pi$
$600$		0			0
$601$			40.1818i			1.63905i	0.573042	+	0.819526i	$0.305763\pi$
$601$			40.1818i			1.63905i	−0.573042	+	0.819526i	$0.694237\pi$
$602$		0			0
$603$	7.28570	−	3.12215i	0.296697	−	0.127144i
$604$		0			0
$605$	1.19296			0.0485009
$606$		0			0
$607$			39.9601i			1.62193i	0.585095	+	0.810965i	$0.301057\pi$
$607$			39.9601i			1.62193i	−0.585095	+	0.810965i	$0.698943\pi$
$608$		0			0
$609$	−8.22726	−	26.0711i	−0.333385	−	1.05646i
$610$		0			0
$611$		−	14.0965i		−	0.570283i
$612$		0			0
$613$	26.7062			1.07865			0.539327	−	0.842096i	$-0.318679\pi$
$613$	26.7062			1.07865			0.539327	+	0.842096i	$0.318679\pi$
$614$		0			0
$615$	−2.72345	+	0.558962i	−0.109820	+	0.0225395i
$616$		0			0
$617$			25.8390i			1.04024i	0.854093	+	0.520120i	$0.174113\pi$
$617$			25.8390i			1.04024i	−0.854093	+	0.520120i	$0.825887\pi$
$618$		0			0
$619$			23.7203i			0.953398i	0.879066	+	0.476699i	$0.158167\pi$
$619$			23.7203i			0.953398i	−0.879066	+	0.476699i	$0.841833\pi$
$620$		0			0
$621$	18.7630	+	27.0015i	0.752934	+	1.08353i
$622$		0			0
$623$	−43.7778	+	4.53619i	−1.75392	+	0.181739i
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$	−4.08221	+	0.837832i	−0.163028	+	0.0334598i
$628$		0			0
$629$	−1.45182			−0.0578880
$630$		0			0
$631$	−15.2580			−0.607411			−0.303706	−	0.952766i	$-0.598224\pi$
$631$	−15.2580			−0.607411			−0.303706	+	0.952766i	$0.598224\pi$
$632$		0			0
$633$	−37.1783	+	7.63048i	−1.47771	+	0.303284i
$634$		0			0
$635$	6.25616			0.248268
$636$		0			0
$637$	6.23106	+	29.7445i	0.246884	+	1.17852i
$638$		0			0
$639$	−0.268279	−	0.626044i	−0.0106130	−	0.0247659i
$640$		0			0
$641$			20.3042i			0.801967i	0.916085	+	0.400984i	$0.131332\pi$
$641$			20.3042i			0.801967i	−0.916085	+	0.400984i	$0.868668\pi$
$642$		0			0
$643$		−	35.2942i		−	1.39187i	−0.718105	−	0.695934i	$-0.754990\pi$
$643$		−	35.2942i		−	1.39187i	0.718105	−	0.695934i	$-0.245010\pi$
$644$		0			0
$645$	−10.4930	+	2.15358i	−0.413161	+	0.0847972i
$646$		0			0
$647$	36.1983			1.42310			0.711551	−	0.702635i	$-0.247993\pi$
$647$	36.1983			1.42310			0.711551	+	0.702635i	$0.247993\pi$
$648$		0			0
$649$		−	29.1276i		−	1.14336i
$650$		0			0
$651$	7.57420	+	24.0017i	0.296856	+	0.940700i
$652$		0			0
$653$		−	29.0293i		−	1.13601i	−0.823027	−	0.568003i	$-0.807716\pi$
$653$		−	29.0293i		−	1.13601i	0.823027	−	0.568003i	$-0.192284\pi$
$654$		0			0
$655$	−9.05057			−0.353635
$656$		0			0
$657$	−18.6415	−	43.5008i	−0.727273	−	1.69713i
$658$		0			0
$659$		−	1.26171i		−	0.0491491i	−0.999698	−	0.0245745i	$-0.992177\pi$
$659$		−	1.26171i		−	0.0491491i	0.999698	−	0.0245745i	$-0.00782310\pi$
$660$		0			0
$661$			10.5711i			0.411170i	0.978639	+	0.205585i	$0.0659097\pi$
$661$			10.5711i			0.411170i	−0.978639	+	0.205585i	$0.934090\pi$
$662$		0			0
$663$	−1.19590	−	5.82685i	−0.0464450	−	0.226296i
$664$		0			0
$665$	0.209504	+	2.02188i	0.00812421	+	0.0784051i
$666$		0			0
$667$	37.7504			1.46170
$668$		0			0
$669$	−1.66356	−	8.10541i	−0.0643168	−	0.313373i
$670$		0			0
$671$	41.1385			1.58813
$672$		0			0
$673$	−18.2901			−0.705030			−0.352515	−	0.935806i	$-0.614673\pi$
$673$	−18.2901			−0.705030			−0.352515	+	0.935806i	$0.614673\pi$
$674$		0			0
$675$	4.26708	−	2.96514i	0.164240	−	0.114128i
$676$		0			0
$677$	−8.52037			−0.327464			−0.163732	−	0.986505i	$-0.552353\pi$
$677$	−8.52037			−0.327464			−0.163732	+	0.986505i	$0.552353\pi$
$678$		0			0
$679$	1.66721	+	16.0899i	0.0639816	+	0.617473i
$680$		0			0
$681$	33.8193	−	6.94108i	1.29596	−	0.265983i
$682$		0			0
$683$			14.6010i			0.558692i	0.960190	+	0.279346i	$0.0901177\pi$
$683$			14.6010i			0.558692i	−0.960190	+	0.279346i	$0.909882\pi$
$684$		0			0
$685$		−	5.29111i		−	0.202163i
$686$		0			0
$687$	−2.10387	−	10.2508i	−0.0802678	−	0.391092i
$688$		0			0
$689$	43.6577			1.66323
$690$		0			0
$691$		−	33.8458i		−	1.28756i	−0.765212	−	0.643778i	$-0.777366\pi$
$691$		−	33.8458i		−	1.28756i	0.765212	−	0.643778i	$-0.222634\pi$
$692$		0			0
$693$	12.0933	+	21.7163i	0.459386	+	0.824932i
$694$		0			0
$695$		−	18.3215i		−	0.694975i
$696$		0			0
$697$	1.26975			0.0480951
$698$		0			0
$699$	−2.27243	−	11.0721i	−0.0859512	−	0.418784i
$700$		0			0
$701$			1.51324i			0.0571545i	0.999592	+	0.0285772i	$0.00909765\pi$
$701$			1.51324i			0.0571545i	−0.999592	+	0.0285772i	$0.990902\pi$
$702$		0			0
$703$			1.41007i			0.0531818i
$704$		0			0
$705$	−5.50905	+	1.13068i	−0.207483	+	0.0425838i
$706$		0			0
$707$	−3.17070	+	0.328543i	−0.119246	+	0.0123561i
$708$		0			0
$709$	2.87428			0.107946			0.0539729	−	0.998542i	$-0.482812\pi$
$709$	2.87428			0.107946			0.0539729	+	0.998542i	$0.482812\pi$
$710$		0			0
$711$	37.5968	−	16.1114i	1.40999	−	0.604226i
$712$		0			0
$713$	−34.7539			−1.30154
$714$		0			0
$715$	−13.5958			−0.508453
$716$		0			0
$717$	1.19319	+	5.81361i	0.0445604	+	0.217113i
$718$		0			0
$719$	−24.5502			−0.915566			−0.457783	−	0.889064i	$-0.651356\pi$
$719$	−24.5502			−0.915566			−0.457783	+	0.889064i	$0.651356\pi$
$720$		0			0
$721$	0.388752	+	3.75176i	0.0144779	+	0.139723i
$722$		0			0
$723$	−4.06315	−	19.7970i	−0.151110	−	0.736260i
$724$		0			0
$725$		−	5.96574i		−	0.221562i
$726$		0			0
$727$			43.1067i			1.59874i	0.600839	+	0.799370i	$0.294833\pi$
$727$			43.1067i			1.59874i	−0.600839	+	0.799370i	$0.705167\pi$
$728$		0			0
$729$	9.41588	−	25.3050i	0.348736	−	0.937221i
$730$		0			0
$731$	4.89211			0.180941
$732$		0			0
$733$		−	14.8194i		−	0.547367i	−0.961820	−	0.273684i	$-0.911758\pi$
$733$		−	14.8194i		−	0.547367i	0.961820	−	0.273684i	$-0.0882421\pi$
$734$		0			0
$735$	11.1247	−	4.82097i	0.410340	−	0.177824i
$736$		0			0
$737$			8.27425i			0.304786i
$738$		0			0
$739$	14.3445			0.527672			0.263836	−	0.964568i	$-0.415012\pi$
$739$	14.3445			0.527672			0.263836	+	0.964568i	$0.415012\pi$
$740$		0			0
$741$	−5.65928	+	1.16151i	−0.207899	+	0.0426691i
$742$		0			0
$743$			3.81720i			0.140040i	0.997546	+	0.0700198i	$0.0223062\pi$
$743$			3.81720i			0.140040i	−0.997546	+	0.0700198i	$0.977694\pi$
$744$		0			0
$745$			23.3480i			0.855403i
$746$		0			0
$747$	−7.58007	+	3.24830i	−0.277340	+	0.118849i
$748$		0			0
$749$	−3.05914	−	29.5231i	−0.111779	−	1.07875i
$750$		0			0
$751$	−23.2840			−0.849644			−0.424822	−	0.905277i	$-0.639663\pi$
$751$	−23.2840			−0.849644			−0.424822	+	0.905277i	$0.639663\pi$
$752$		0			0
$753$	−36.6094	+	7.51372i	−1.33412	+	0.273815i
$754$		0			0
$755$	9.14170			0.332701
$756$		0			0
$757$	43.1461			1.56817			0.784086	−	0.620652i	$-0.213132\pi$
$757$	43.1461			1.56817			0.784086	+	0.620652i	$0.213132\pi$
$758$		0			0
$759$	−33.6223	+	6.90064i	−1.22041	+	0.250477i
$760$		0			0
$761$	19.4395			0.704682			0.352341	−	0.935872i	$-0.385386\pi$
$761$	19.4395			0.704682			0.352341	+	0.935872i	$0.385386\pi$
$762$		0			0
$763$	42.5757	−	4.41163i	1.54134	−	0.159712i
$764$		0			0
$765$	−2.18127	+	0.934743i	−0.0788640	+	0.0337957i
$766$		0			0
$767$		−	40.3805i		−	1.45805i
$768$		0			0
$769$			29.2444i			1.05458i	0.849686	+	0.527290i	$0.176792\pi$
$769$			29.2444i			1.05458i	−0.849686	+	0.527290i	$0.823208\pi$
$770$		0			0
$771$	29.2038	−	5.99380i	1.05175	−	0.215861i
$772$		0			0
$773$	17.6142			0.633539			0.316770	−	0.948503i	$-0.397402\pi$
$773$	17.6142			0.633539			0.316770	+	0.948503i	$0.397402\pi$
$774$		0			0
$775$			5.49220i			0.197286i
$776$		0			0
$777$	8.02068	−	2.53108i	0.287740	−	0.0908022i
$778$		0			0
$779$		−	1.23323i		−	0.0441850i
$780$		0			0
$781$	0.710988			0.0254411
$782$		0			0
$783$	−17.6893	−	25.4563i	−0.632163	−	0.909733i
$784$		0			0
$785$			15.2610i			0.544689i
$786$		0			0
$787$			14.8565i			0.529578i	0.964306	+	0.264789i	$0.0853023\pi$
$787$			14.8565i			0.529578i	−0.964306	+	0.264789i	$0.914698\pi$
$788$		0			0
$789$	8.41068	+	40.9797i	0.299428	+	1.45892i
$790$		0			0
$791$	−1.33900	−	12.9224i	−0.0476093	−	0.459468i
$792$		0			0
$793$	57.0314			2.02524
$794$		0			0
$795$	−3.50179	−	17.0619i	−0.124196	−	0.605123i
$796$		0			0
$797$	23.6741			0.838579			0.419290	−	0.907853i	$-0.362279\pi$
$797$	23.6741			0.838579			0.419290	+	0.907853i	$0.362279\pi$
$798$		0			0
$799$	2.56847			0.0908658
$800$		0			0
$801$	−45.8708	+	19.6571i	−1.62076	+	0.694548i
$802$		0			0
$803$	49.4032			1.74340
$804$		0			0
$805$	1.72554	+	16.6528i	0.0608172	+	0.586934i
$806$		0			0
$807$	−26.9656	+	5.53443i	−0.949235	+	0.194821i
$808$		0			0
$809$			38.1376i			1.34085i	0.741979	+	0.670423i	$0.233887\pi$
$809$			38.1376i			1.34085i	−0.741979	+	0.670423i	$0.766113\pi$
$810$		0			0
$811$		−	32.4174i		−	1.13833i	−0.822224	−	0.569164i	$-0.807267\pi$
$811$		−	32.4174i		−	1.13833i	0.822224	−	0.569164i	$-0.192733\pi$
$812$		0			0
$813$	9.59048	+	46.7281i	0.336353	+	1.63883i
$814$		0			0
$815$	−5.02683			−0.176082
$816$		0			0
$817$		−	4.75142i		−	0.166231i
$818$		0			0
$819$	16.7653	+	30.1059i	0.585826	+	1.05198i
$820$		0			0
$821$			30.2789i			1.05674i	0.849014	+	0.528370i	$0.177197\pi$
$821$			30.2789i			1.05674i	−0.849014	+	0.528370i	$0.822803\pi$
$822$		0			0
$823$	20.2258			0.705027			0.352513	−	0.935807i	$-0.385327\pi$
$823$	20.2258			0.705027			0.352513	+	0.935807i	$0.385327\pi$
$824$		0			0
$825$	1.09052	+	5.31337i	0.0379669	+	0.184988i
$826$		0			0
$827$		−	35.8039i		−	1.24502i	−0.782611	−	0.622512i	$-0.786112\pi$
$827$		−	35.8039i		−	1.24502i	0.782611	−	0.622512i	$-0.213888\pi$
$828$		0			0
$829$		−	51.3995i		−	1.78518i	−0.450871	−	0.892589i	$-0.648887\pi$
$829$		−	51.3995i		−	1.78518i	0.450871	−	0.892589i	$-0.351113\pi$
$830$		0			0
$831$	50.1991	−	10.3029i	1.74139	−	0.357403i
$832$		0			0
$833$	−5.41963	+	1.13534i	−0.187779	+	0.0393371i
$834$		0			0
$835$	21.9872			0.760898
$836$		0			0
$837$	16.2851	+	23.4356i	0.562897	+	0.810054i
$838$		0			0
$839$	39.5046			1.36385			0.681925	−	0.731422i	$-0.261143\pi$
$839$	39.5046			1.36385			0.681925	+	0.731422i	$0.261143\pi$
$840$		0			0
$841$	−6.59010			−0.227245
$842$		0			0
$843$	3.33686	+	16.2583i	0.114927	+	0.559966i
$844$		0			0
$845$	−5.84820			−0.201184
$846$		0			0
$847$	3.13948	−	0.325308i	0.107874	−	0.0111777i
$848$		0			0
$849$	−1.88656	−	9.19199i	−0.0647467	−	0.315468i
$850$		0			0
$851$			11.6138i			0.398115i
$852$		0			0
$853$			50.6339i			1.73367i	0.498594	+	0.866836i	$0.333850\pi$
$853$			50.6339i			1.73367i	−0.498594	+	0.866836i	$0.666150\pi$
$854$		0			0
$855$	0.907861	+	2.11854i	0.0310482	+	0.0724525i
$856$		0			0
$857$	39.4930			1.34905			0.674527	−	0.738250i	$-0.264348\pi$
$857$	39.4930			1.34905			0.674527	+	0.738250i	$0.264348\pi$
$858$		0			0
$859$			16.7650i			0.572014i	0.958228	+	0.286007i	$0.0923280\pi$
$859$			16.7650i			0.572014i	−0.958228	+	0.286007i	$0.907672\pi$
$860$		0			0
$861$	−7.01478	+	2.21365i	−0.239063	+	0.0754411i
$862$		0			0
$863$			28.8252i			0.981220i	0.871379	+	0.490610i	$0.163226\pi$
$863$			28.8252i			0.981220i	−0.871379	+	0.490610i	$0.836774\pi$
$864$		0			0
$865$	−18.9508			−0.644348
$866$		0			0
$867$	−27.7819	+	5.70197i	−0.943524	+	0.193649i
$868$		0			0
$869$			42.6981i			1.44843i
$870$		0			0
$871$			11.4708i			0.388674i
$872$		0			0
$873$	7.22465	+	16.8591i	0.244517	+	0.570594i
$874$		0			0
$875$	2.63166	−	0.272689i	0.0889664	−	0.00921856i
$876$		0			0
$877$	−26.8225			−0.905733			−0.452866	−	0.891578i	$-0.649599\pi$
$877$	−26.8225			−0.905733			−0.452866	+	0.891578i	$0.649599\pi$
$878$		0			0
$879$	−6.65260	+	1.36538i	−0.224387	+	0.0460531i
$880$		0			0
$881$	12.0356			0.405491			0.202745	−	0.979232i	$-0.435014\pi$
$881$	12.0356			0.405491			0.202745	+	0.979232i	$0.435014\pi$
$882$		0			0
$883$	−5.41892			−0.182361			−0.0911807	−	0.995834i	$-0.529064\pi$
$883$	−5.41892			−0.182361			−0.0911807	+	0.995834i	$0.529064\pi$
$884$		0			0
$885$	−15.7811	+	3.23892i	−0.530476	+	0.108875i
$886$		0			0
$887$	26.8010			0.899889			0.449945	−	0.893056i	$-0.351444\pi$
$887$	26.8010			0.899889			0.449945	+	0.893056i	$0.351444\pi$
$888$		0			0
$889$	16.4641	−	1.70598i	0.552188	−	0.0572168i
$890$		0			0
$891$	20.4082	+	19.4390i	0.683700	+	0.651231i
$892$		0			0
$893$		−	2.49460i		−	0.0834786i
$894$		0			0
$895$		−	1.39906i		−	0.0467654i
$896$		0			0
$897$	−46.6115	+	9.56655i	−1.55631	+	0.319418i
$898$		0			0
$899$	32.7651			1.09278
$900$		0			0
$901$			7.95471i			0.265010i
$902$		0			0
$903$	−27.0267	+	8.52882i	−0.899393	+	0.283821i
$904$		0			0
$905$		−	13.3999i		−	0.445427i
$906$		0			0
$907$	50.7667			1.68568			0.842840	−	0.538165i	$-0.180882\pi$
$907$	50.7667			1.68568			0.842840	+	0.538165i	$0.180882\pi$
$908$		0			0
$909$	−3.32228	+	1.42370i	−0.110193	+	0.0472213i
$910$		0			0
$911$		−	9.33951i		−	0.309432i	−0.987959	−	0.154716i	$-0.950554\pi$
$911$		−	9.33951i		−	0.309432i	0.987959	−	0.154716i	$-0.0494462\pi$
$912$		0			0
$913$		−	8.60856i		−	0.284902i
$914$		0			0
$915$	−4.57449	−	22.2885i	−0.151228	−	0.736834i
$916$		0			0
$917$	−23.8180	+	2.46799i	−0.786541	+	0.0815002i
$918$		0			0
$919$	6.07625			0.200437			0.100219	−	0.994965i	$-0.468046\pi$
$919$	6.07625			0.200437			0.100219	+	0.994965i	$0.468046\pi$
$920$		0			0
$921$	7.00506	+	34.1311i	0.230825	+	1.12466i
$922$		0			0
$923$	0.985662			0.0324435
$924$		0			0
$925$	1.83534			0.0603455
$926$		0			0
$927$	1.68461	+	3.93113i	0.0553299	+	0.129115i
$928$		0			0
$929$	−18.6804			−0.612883			−0.306442	−	0.951889i	$-0.599138\pi$
$929$	−18.6804			−0.612883			−0.306442	+	0.951889i	$0.599138\pi$
$930$		0			0
$931$	1.10269	+	5.26377i	0.0361391	+	0.172513i
$932$		0			0
$933$	13.4399	−	2.75840i	0.440001	−	0.0903059i
$934$		0			0
$935$		−	2.47723i		−	0.0810142i
$936$		0			0
$937$		−	60.6614i		−	1.98172i	−0.134889	−	0.990861i	$-0.543068\pi$
$937$		−	60.6614i		−	1.98172i	0.134889	−	0.990861i	$-0.456932\pi$
$938$		0			0
$939$	2.48617	+	12.1135i	0.0811332	+	0.395309i
$940$		0			0
$941$	37.6348			1.22686			0.613429	−	0.789750i	$-0.289790\pi$
$941$	37.6348			1.22686			0.613429	+	0.789750i	$0.289790\pi$
$942$		0			0
$943$		−	10.1573i		−	0.330766i
$944$		0			0
$945$	10.4209	−	8.96683i	0.338993	−	0.291691i
$946$		0			0
$947$			9.53338i			0.309793i	0.987931	+	0.154897i	$0.0495044\pi$
$947$			9.53338i			0.309793i	−0.987931	+	0.154897i	$0.950496\pi$
$948$		0			0
$949$	68.4890			2.22325
$950$		0			0
$951$	−3.79596	−	18.4952i	−0.123092	−	0.599748i
$952$		0			0
$953$			0.837324i			0.0271236i	0.999908	+	0.0135618i	$0.00431699\pi$
$953$			0.837324i			0.0271236i	−0.999908	+	0.0135618i	$0.995683\pi$
$954$		0			0
$955$		−	22.3132i		−	0.722037i
$956$		0			0
$957$	31.6982	−	6.50574i	1.02466	−	0.210301i
$958$		0			0
$959$	−1.44282	−	13.9244i	−0.0465912	−	0.449642i
$960$		0			0
$961$	0.835734			0.0269592
$962$		0			0
$963$	−13.2564	−	30.9346i	−0.427183	−	0.996852i
$964$		0			0
$965$	−10.2441			−0.329769
$966$		0			0
$967$	17.8394			0.573677			0.286838	−	0.957979i	$-0.407396\pi$
$967$	17.8394			0.573677			0.286838	+	0.957979i	$0.407396\pi$
$968$		0			0
$969$	−0.211634	−	1.03116i	−0.00679867	−	0.0331255i
$970$		0			0
$971$	17.5383			0.562830			0.281415	−	0.959586i	$-0.409196\pi$
$971$	17.5383			0.562830			0.281415	+	0.959586i	$0.409196\pi$
$972$		0			0
$973$	−4.99607	−	48.2160i	−0.160167	−	1.54574i
$974$		0			0
$975$	1.51181	+	7.36607i	0.0484168	+	0.235903i
$976$		0			0
$977$		−	0.391252i		−	0.0125173i	−0.999980	−	0.00625863i	$-0.998008\pi$
$977$		−	0.391252i		−	0.0125173i	0.999980	−	0.00625863i	$-0.00199220\pi$
$978$		0			0
$979$		−	52.0947i		−	1.66495i
$980$		0			0
$981$	44.6112	−	19.1173i	1.42432	−	0.610368i
$982$		0			0
$983$	−29.5956			−0.943952			−0.471976	−	0.881611i	$-0.656459\pi$
$983$	−29.5956			−0.943952			−0.471976	+	0.881611i	$0.656459\pi$
$984$		0			0
$985$			1.79870i			0.0573114i
$986$		0			0
$987$	−14.1896	+	4.47782i	−0.451661	+	0.142531i
$988$		0			0
$989$		−	39.1341i		−	1.24439i
$990$		0			0
$991$	−9.06251			−0.287880			−0.143940	−	0.989586i	$-0.545977\pi$
$991$	−9.06251			−0.287880			−0.143940	+	0.989586i	$0.545977\pi$
$992$		0			0
$993$	−41.5865	+	8.53522i	−1.31971	+	0.270857i
$994$		0			0
$995$			14.1953i			0.450023i
$996$		0			0
$997$		−	0.0383191i		−	0.00121358i	−1.00000		0.000606789i	$-0.999807\pi$
$997$		−	0.0383191i		−	0.00121358i	1.00000		0.000606789i	$-0.000193147\pi$
$998$		0			0
$999$	7.83153	−	5.44204i	0.247779	−	0.172178i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1680.2.f.l.881.15		16
3.2	odd	2		1680.2.f.k.881.1		16
4.3	odd	2		840.2.f.b.41.2	yes	16
7.6	odd	2		1680.2.f.k.881.2		16
12.11	even	2		840.2.f.a.41.16	yes	16
21.20	even	2	inner	1680.2.f.l.881.16		16
28.27	even	2		840.2.f.a.41.15	✓	16
84.83	odd	2		840.2.f.b.41.1	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
840.2.f.a.41.15	✓	16	28.27	even	2
840.2.f.a.41.16	yes	16	12.11	even	2
840.2.f.b.41.1	yes	16	84.83	odd	2
840.2.f.b.41.2	yes	16	4.3	odd	2
1680.2.f.k.881.1		16	3.2	odd	2
1680.2.f.k.881.2		16	7.6	odd	2
1680.2.f.l.881.15		16	1.1	even	1	trivial
1680.2.f.l.881.16		16	21.20	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:	\( N \)	\(=\)	\( 1680 = 2^{4} \cdot 3 \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1680.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+(1.69668 - 0.348228i) q^{3} +1.00000 q^{5} +(2.63166 - 0.272689i) q^{7} +(2.75748 - 1.18166i) q^{9} +O(q^{10})\)	\(q+(1.69668 - 0.348228i) q^{3} +1.00000 q^{5} +(2.63166 - 0.272689i) q^{7} +(2.75748 - 1.18166i) q^{9} +3.13162i q^{11} +4.34145i q^{13} +(1.69668 - 0.348228i) q^{15} -0.791039 q^{17} +0.768290i q^{19} +(4.37014 - 1.37908i) q^{21} +6.32787i q^{23} +1.00000 q^{25} +(4.26708 - 2.96514i) q^{27} -5.96574i q^{29} +5.49220i q^{31} +(1.09052 + 5.31337i) q^{33} +(2.63166 - 0.272689i) q^{35} +1.83534 q^{37} +(1.51181 + 7.36607i) q^{39} -1.60516 q^{41} -6.18441 q^{43} +(2.75748 - 1.18166i) q^{45} -3.24695 q^{47} +(6.85128 - 1.43525i) q^{49} +(-1.34214 + 0.275462i) q^{51} -10.0560i q^{53} +3.13162i q^{55} +(0.267540 + 1.30354i) q^{57} -9.30114 q^{59} -13.1365i q^{61} +(6.93451 - 3.86167i) q^{63} +4.34145i q^{65} +2.64216 q^{67} +(2.20354 + 10.7364i) q^{69} -0.227035i q^{71} -15.7756i q^{73} +(1.69668 - 0.348228i) q^{75} +(0.853957 + 8.24136i) q^{77} +13.6345 q^{79} +(6.20734 - 6.51682i) q^{81} -2.74892 q^{83} -0.791039 q^{85} +(-2.07744 - 10.1220i) q^{87} -16.6351 q^{89} +(1.18386 + 11.4252i) q^{91} +(1.91254 + 9.31853i) q^{93} +0.768290i q^{95} +6.11396i q^{97} +(3.70052 + 8.63536i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 16 q^{5} - 2 q^{7} - 2 q^{9}+O(q^{10}) \) 16 * q + 16 * q^5 - 2 * q^7 - 2 * q^9	\( 16 q + 16 q^{5} - 2 q^{7} - 2 q^{9} + 10 q^{21} + 16 q^{25} - 6 q^{27} + 6 q^{33} - 2 q^{35} + 12 q^{37} - 6 q^{39} + 32 q^{41} - 32 q^{43} - 2 q^{45} - 4 q^{47} - 4 q^{49} - 6 q^{51} + 24 q^{59} + 24 q^{63} + 8 q^{69} - 32 q^{77} + 4 q^{79} - 6 q^{81} - 20 q^{83} - 6 q^{87} - 24 q^{89} - 20 q^{91} - 32 q^{93} + 58 q^{99}+O(q^{100}) \) 16 * q + 16 * q^5 - 2 * q^7 - 2 * q^9 + 10 * q^21 + 16 * q^25 - 6 * q^27 + 6 * q^33 - 2 * q^35 + 12 * q^37 - 6 * q^39 + 32 * q^41 - 32 * q^43 - 2 * q^45 - 4 * q^47 - 4 * q^49 - 6 * q^51 + 24 * q^59 + 24 * q^63 + 8 * q^69 - 32 * q^77 + 4 * q^79 - 6 * q^81 - 20 * q^83 - 6 * q^87 - 24 * q^89 - 20 * q^91 - 32 * q^93 + 58 * q^99

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