LMFDB - Embedded newform 1344.3.d.f.449.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1344,3,Mod(449,1344)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1344, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("1344.449");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1344 = 2^{6} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1344.d (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:	$36.6213475300$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.0.65856.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 14x^{2} + 21 $ x^4 + 14*x^2 + 21
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 21)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			449.2
Root			$3.50592i$ of defining polynomial
Character	$\chi$	$=$	1344.449
Dual form			1344.3.d.f.449.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.822876 + 2.88494i) q^{3} +1.24197i q^{5} +2.64575 q^{7} +(-7.64575 - 4.74789i) q^{9} +O(q^{10})$	\(q+(-0.822876 + 2.88494i) q^{3} +1.24197i q^{5} +2.64575 q^{7} +(-7.64575 - 4.74789i) q^{9} -7.01185i q^{11} +11.6458 q^{13} +(-3.58301 - 1.02199i) q^{15} -4.52791i q^{17} -16.2288 q^{19} +(-2.17712 + 7.63283i) q^{21} +25.5635i q^{23} +23.4575 q^{25} +(19.9889 - 18.1506i) q^{27} +9.49579i q^{29} +28.7085 q^{31} +(20.2288 + 5.76988i) q^{33} +3.28594i q^{35} +33.0405 q^{37} +(-9.58301 + 33.5973i) q^{39} -67.1946i q^{41} +24.1255 q^{43} +(5.89674 - 9.49579i) q^{45} +33.0153i q^{47} +7.00000 q^{49} +(13.0627 + 3.72591i) q^{51} -15.1877i q^{53} +8.70850 q^{55} +(13.3542 - 46.8190i) q^{57} +92.3960i q^{59} +57.5203 q^{61} +(-20.2288 - 12.5617i) q^{63} +14.4637i q^{65} -15.1660 q^{67} +(-73.7490 - 21.0355i) q^{69} +70.5584i q^{71} -76.7895 q^{73} +(-19.3026 + 67.6735i) q^{75} -18.5516i q^{77} +127.247 q^{79} +(35.9150 + 72.6024i) q^{81} +74.2844i q^{83} +5.62352 q^{85} +(-27.3948 - 7.81385i) q^{87} +127.377i q^{89} +30.8118 q^{91} +(-23.6235 + 82.8223i) q^{93} -20.1556i q^{95} -23.1660 q^{97} +(-33.2915 + 53.6108i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{3} - 20 q^{9}+O(q^{10}) $ 4 * q + 2 * q^3 - 20 * q^9	$ 4 q + 2 q^{3} - 20 q^{9} + 36 q^{13} + 28 q^{15} - 12 q^{19} - 14 q^{21} - 12 q^{25} - 10 q^{27} + 136 q^{31} + 28 q^{33} - 16 q^{37} + 4 q^{39} + 160 q^{43} + 140 q^{45} + 28 q^{49} + 84 q^{51} + 56 q^{55} + 64 q^{57} + 156 q^{61} - 28 q^{63} + 24 q^{67} - 168 q^{69} - 32 q^{73} - 146 q^{75} + 128 q^{79} - 68 q^{81} - 168 q^{85} + 28 q^{87} + 28 q^{91} + 96 q^{93} - 8 q^{97} - 112 q^{99}+O(q^{100}) $ 4 * q + 2 * q^3 - 20 * q^9 + 36 * q^13 + 28 * q^15 - 12 * q^19 - 14 * q^21 - 12 * q^25 - 10 * q^27 + 136 * q^31 + 28 * q^33 - 16 * q^37 + 4 * q^39 + 160 * q^43 + 140 * q^45 + 28 * q^49 + 84 * q^51 + 56 * q^55 + 64 * q^57 + 156 * q^61 - 28 * q^63 + 24 * q^67 - 168 * q^69 - 32 * q^73 - 146 * q^75 + 128 * q^79 - 68 * q^81 - 168 * q^85 + 28 * q^87 + 28 * q^91 + 96 * q^93 - 8 * q^97 - 112 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1344\mathbb{Z}\right)^\times$.

$n$	$127$	$449$	$577$	$1093$
$\chi(n)$	$1$	$-1$	$1$	$1$

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
$3$	−0.822876	+	2.88494i	−0.274292	+	0.961646i
$3$	−0.822876	+	2.88494i	−0.274292	+	0.961646i
$4$		0			0
$5$			1.24197i			0.248394i	0.992258	+	0.124197i	$0.0396354\pi$
$5$			1.24197i			0.248394i	−0.992258	+	0.124197i	$0.960365\pi$
$6$		0			0
$7$	2.64575			0.377964
$7$	2.64575			0.377964
$8$		0			0
$9$	−7.64575	−	4.74789i	−0.849528	−	0.527544i
$10$		0			0
$11$		−	7.01185i		−	0.637441i	−0.947849	−	0.318720i	$-0.896747\pi$
$11$		−	7.01185i		−	0.637441i	0.947849	−	0.318720i	$-0.103253\pi$
$12$		0			0
$13$	11.6458			0.895827			0.447914	−	0.894077i	$-0.352167\pi$
$13$	11.6458			0.895827			0.447914	+	0.894077i	$0.352167\pi$
$14$		0			0
$15$	−3.58301	−	1.02199i	−0.238867	−	0.0681324i
$16$		0			0
$17$		−	4.52791i		−	0.266348i	−0.991093	−	0.133174i	$-0.957483\pi$
$17$		−	4.52791i		−	0.266348i	0.991093	−	0.133174i	$-0.0425169\pi$
$18$		0			0
$19$	−16.2288			−0.854145			−0.427073	−	0.904217i	$-0.640455\pi$
$19$	−16.2288			−0.854145			−0.427073	+	0.904217i	$0.640455\pi$
$20$		0			0
$21$	−2.17712	+	7.63283i	−0.103673	+	0.363468i
$22$		0			0
$23$			25.5635i			1.11145i	0.831365	+	0.555727i	$0.187560\pi$
$23$			25.5635i			1.11145i	−0.831365	+	0.555727i	$0.812440\pi$
$24$		0			0
$25$	23.4575			0.938301
$26$		0			0
$27$	19.9889	−	18.1506i	0.740329	−	0.672245i
$28$		0			0
$29$			9.49579i			0.327441i	0.986507	+	0.163720i	$0.0523495\pi$
$29$			9.49579i			0.327441i	−0.986507	+	0.163720i	$0.947650\pi$
$30$		0			0
$31$	28.7085			0.926081			0.463040	−	0.886337i	$-0.346759\pi$
$31$	28.7085			0.926081			0.463040	+	0.886337i	$0.346759\pi$
$32$		0			0
$33$	20.2288	+	5.76988i	0.612993	+	0.174845i
$34$		0			0
$35$			3.28594i			0.0938840i
$36$		0			0
$37$	33.0405			0.892987			0.446493	−	0.894787i	$-0.352673\pi$
$37$	33.0405			0.892987			0.446493	+	0.894787i	$0.352673\pi$
$38$		0			0
$39$	−9.58301	+	33.5973i	−0.245718	+	0.861469i
$40$		0			0
$41$		−	67.1946i		−	1.63889i	−0.573156	−	0.819446i	$-0.694281\pi$
$41$		−	67.1946i		−	1.63889i	0.573156	−	0.819446i	$-0.305719\pi$
$42$		0			0
$43$	24.1255			0.561058			0.280529	−	0.959846i	$-0.409490\pi$
$43$	24.1255			0.561058			0.280529	+	0.959846i	$0.409490\pi$
$44$		0			0
$45$	5.89674	−	9.49579i	0.131039	−	0.211017i
$46$		0			0
$47$			33.0153i			0.702452i	0.936291	+	0.351226i	$0.114235\pi$
$47$			33.0153i			0.702452i	−0.936291	+	0.351226i	$0.885765\pi$
$48$		0			0
$49$	7.00000			0.142857
$50$		0			0
$51$	13.0627	+	3.72591i	0.256132	+	0.0730570i
$52$		0			0
$53$		−	15.1877i		−	0.286561i	−0.989682	−	0.143281i	$-0.954235\pi$
$53$		−	15.1877i		−	0.286561i	0.989682	−	0.143281i	$-0.0457651\pi$
$54$		0			0
$55$	8.70850			0.158336
$56$		0			0
$57$	13.3542	−	46.8190i	0.234285	−	0.821386i
$58$		0			0
$59$			92.3960i			1.56603i	0.622000	+	0.783017i	$0.286320\pi$
$59$			92.3960i			1.56603i	−0.622000	+	0.783017i	$0.713680\pi$
$60$		0			0
$61$	57.5203			0.942955			0.471478	−	0.881878i	$-0.343721\pi$
$61$	57.5203			0.942955			0.471478	+	0.881878i	$0.343721\pi$
$62$		0			0
$63$	−20.2288	−	12.5617i	−0.321091	−	0.199393i
$64$		0			0
$65$			14.4637i			0.222518i
$66$		0			0
$67$	−15.1660			−0.226358			−0.113179	−	0.993575i	$-0.536103\pi$
$67$	−15.1660			−0.226358			−0.113179	+	0.993575i	$0.536103\pi$
$68$		0			0
$69$	−73.7490	−	21.0355i	−1.06883	−	0.304863i
$70$		0			0
$71$			70.5584i			0.993781i	0.867813	+	0.496890i	$0.165525\pi$
$71$			70.5584i			0.993781i	−0.867813	+	0.496890i	$0.834475\pi$
$72$		0			0
$73$	−76.7895			−1.05191			−0.525956	−	0.850512i	$-0.676292\pi$
$73$	−76.7895			−1.05191			−0.525956	+	0.850512i	$0.676292\pi$
$74$		0			0
$75$	−19.3026	+	67.6735i	−0.257368	+	0.902313i
$76$		0			0
$77$		−	18.5516i		−	0.240930i
$78$		0			0
$79$	127.247			1.61072			0.805361	−	0.592785i	$-0.201971\pi$
$79$	127.247			1.61072			0.805361	+	0.592785i	$0.201971\pi$
$80$		0			0
$81$	35.9150	+	72.6024i	0.443395	+	0.896326i
$82$		0			0
$83$			74.2844i			0.894992i	0.894286	+	0.447496i	$0.147684\pi$
$83$			74.2844i			0.894992i	−0.894286	+	0.447496i	$0.852316\pi$
$84$		0			0
$85$	5.62352			0.0661591
$86$		0			0
$87$	−27.3948	−	7.81385i	−0.314882	−	0.0898144i
$88$		0			0
$89$			127.377i			1.43121i	0.698507	+	0.715603i	$0.253848\pi$
$89$			127.377i			1.43121i	−0.698507	+	0.715603i	$0.746152\pi$
$90$		0			0
$91$	30.8118			0.338591
$92$		0			0
$93$	−23.6235	+	82.8223i	−0.254016	+	0.890562i
$94$		0			0
$95$		−	20.1556i		−	0.212164i
$96$		0			0
$97$	−23.1660			−0.238825			−0.119412	−	0.992845i	$-0.538101\pi$
$97$	−23.1660			−0.238825			−0.119412	+	0.992845i	$0.538101\pi$
$98$		0			0
$99$	−33.2915	+	53.6108i	−0.336278	+	0.541524i
$100$		0			0
$101$		−	134.907i		−	1.33571i	−0.744290	−	0.667857i	$-0.767212\pi$
$101$		−	134.907i		−	1.33571i	0.744290	−	0.667857i	$-0.232788\pi$
$102$		0			0
$103$	−119.749			−1.16261			−0.581306	−	0.813685i	$-0.697458\pi$
$103$	−119.749			−1.16261			−0.581306	+	0.813685i	$0.697458\pi$
$104$		0			0
$105$	−9.47974	−	2.70392i	−0.0902832	−	0.0257516i
$106$		0			0
$107$		−	77.8544i		−	0.727611i	−0.931475	−	0.363806i	$-0.881477\pi$
$107$		−	77.8544i		−	0.727611i	0.931475	−	0.363806i	$-0.118523\pi$
$108$		0			0
$109$	36.5385			0.335216			0.167608	−	0.985854i	$-0.446396\pi$
$109$	36.5385			0.335216			0.167608	+	0.985854i	$0.446396\pi$
$110$		0			0
$111$	−27.1882	+	95.3199i	−0.244939	+	0.858738i
$112$		0			0
$113$		−	21.7596i		−	0.192563i	−0.995354	−	0.0962815i	$-0.969305\pi$
$113$		−	21.7596i		−	0.192563i	0.995354	−	0.0962815i	$-0.0306949\pi$
$114$		0			0
$115$	−31.7490			−0.276078
$116$		0			0
$117$	−89.0405	−	55.2928i	−0.761030	−	0.472588i
$118$		0			0
$119$		−	11.9797i		−	0.100670i
$120$		0			0
$121$	71.8340			0.593669
$122$		0			0
$123$	193.852	+	55.2928i	1.57603	+	0.449535i
$124$		0			0
$125$			60.1827i			0.481462i
$126$		0			0
$127$	−15.4170			−0.121394			−0.0606968	−	0.998156i	$-0.519332\pi$
$127$	−15.4170			−0.121394			−0.0606968	+	0.998156i	$0.519332\pi$
$128$		0			0
$129$	−19.8523	+	69.6006i	−0.153894	+	0.539539i
$130$		0			0
$131$			183.110i			1.39779i	0.715226	+	0.698893i	$0.246324\pi$
$131$			183.110i			1.39779i	−0.715226	+	0.698893i	$0.753676\pi$
$132$		0			0
$133$	−42.9373			−0.322836
$134$		0			0
$135$	22.5425	+	24.8256i	0.166981	+	0.183893i
$136$		0			0
$137$		−	33.0153i		−	0.240987i	−0.992714	−	0.120494i	$-0.961552\pi$
$137$		−	33.0153i		−	0.240987i	0.992714	−	0.120494i	$-0.0384477\pi$
$138$		0			0
$139$	−64.6418			−0.465049			−0.232525	−	0.972591i	$-0.574699\pi$
$139$	−64.6418			−0.465049			−0.232525	+	0.972591i	$0.574699\pi$
$140$		0			0
$141$	−95.2470	−	27.1675i	−0.675511	−	0.192677i
$142$		0			0
$143$		−	81.6582i		−	0.571037i
$144$		0			0
$145$	−11.7935			−0.0813343
$146$		0			0
$147$	−5.76013	+	20.1946i	−0.0391846	+	0.137378i
$148$		0			0
$149$			195.736i			1.31366i	0.754037	+	0.656832i	$0.228104\pi$
$149$			195.736i			1.31366i	−0.754037	+	0.656832i	$0.771896\pi$
$150$		0			0
$151$	102.251			0.677159			0.338579	−	0.940938i	$-0.390054\pi$
$151$	102.251			0.677159			0.338579	+	0.940938i	$0.390054\pi$
$152$		0			0
$153$	−21.4980	+	34.6193i	−0.140510	+	0.226270i
$154$		0			0
$155$			35.6551i			0.230033i
$156$		0			0
$157$	−104.723			−0.667025			−0.333512	−	0.942746i	$-0.608234\pi$
$157$	−104.723			−0.667025			−0.333512	+	0.942746i	$0.608234\pi$
$158$		0			0
$159$	43.8157	+	12.4976i	0.275570	+	0.0786014i
$160$		0			0
$161$			67.6345i			0.420090i
$162$		0			0
$163$	70.9595			0.435334			0.217667	−	0.976023i	$-0.430155\pi$
$163$	70.9595			0.435334			0.217667	+	0.976023i	$0.430155\pi$
$164$		0			0
$165$	−7.16601	+	25.1235i	−0.0434304	+	0.152264i
$166$		0			0
$167$			206.992i			1.23947i	0.784811	+	0.619735i	$0.212760\pi$
$167$			206.992i			1.23947i	−0.784811	+	0.619735i	$0.787240\pi$
$168$		0			0
$169$	−33.3765			−0.197494
$170$		0			0
$171$	124.081	+	77.0524i	0.725620	+	0.450599i
$172$		0			0
$173$			108.464i			0.626958i	0.949595	+	0.313479i	$0.101494\pi$
$173$			108.464i			0.626958i	−0.949595	+	0.313479i	$0.898506\pi$
$174$		0			0
$175$	62.0627			0.354644
$176$		0			0
$177$	−266.557	−	76.0304i	−1.50597	−	0.429550i
$178$		0			0
$179$			159.357i			0.890261i	0.895466	+	0.445131i	$0.146843\pi$
$179$			159.357i			0.890261i	−0.895466	+	0.445131i	$0.853157\pi$
$180$		0			0
$181$	233.889			1.29220			0.646102	−	0.763251i	$-0.276398\pi$
$181$	233.889			1.29220			0.646102	+	0.763251i	$0.276398\pi$
$182$		0			0
$183$	−47.3320	+	165.942i	−0.258645	+	0.906789i
$184$		0			0
$185$			41.0353i			0.221812i
$186$		0			0
$187$	−31.7490			−0.169781
$188$		0			0
$189$	52.8856	−	48.0220i	0.279818	−	0.254085i
$190$		0			0
$191$			288.210i			1.50895i	0.656328	+	0.754476i	$0.272109\pi$
$191$			288.210i			1.50895i	−0.656328	+	0.754476i	$0.727891\pi$
$192$		0			0
$193$	77.1216			0.399594			0.199797	−	0.979837i	$-0.435972\pi$
$193$	77.1216			0.399594			0.199797	+	0.979837i	$0.435972\pi$
$194$		0			0
$195$	−41.7268	−	11.9018i	−0.213984	−	0.0610348i
$196$		0			0
$197$		−	136.433i		−	0.692554i	−0.938132	−	0.346277i	$-0.887446\pi$
$197$		−	136.433i		−	0.692554i	0.938132	−	0.346277i	$-0.112554\pi$
$198$		0			0
$199$	86.5830			0.435090			0.217545	−	0.976050i	$-0.430195\pi$
$199$	86.5830			0.435090			0.217545	+	0.976050i	$0.430195\pi$
$200$		0			0
$201$	12.4797	−	43.7530i	0.0620883	−	0.217677i
$202$		0			0
$203$			25.1235i			0.123761i
$204$		0			0
$205$	83.4536			0.407091
$206$		0			0
$207$	121.373	−	195.452i	0.586341	−	0.944212i
$208$		0			0
$209$			113.794i			0.544467i
$210$		0			0
$211$	−19.4170			−0.0920237			−0.0460118	−	0.998941i	$-0.514651\pi$
$211$	−19.4170			−0.0920237			−0.0460118	+	0.998941i	$0.514651\pi$
$212$		0			0
$213$	−203.557	−	58.0608i	−0.955666	−	0.272586i
$214$		0			0
$215$			29.9631i			0.139363i
$216$		0			0
$217$	75.9555			0.350026
$218$		0			0
$219$	63.1882	−	221.533i	0.288531	−	1.01157i
$220$		0			0
$221$		−	52.7309i		−	0.238601i
$222$		0			0
$223$	175.041			0.784935			0.392468	−	0.919766i	$-0.371622\pi$
$223$	175.041			0.784935			0.392468	+	0.919766i	$0.371622\pi$
$224$		0			0
$225$	−179.350	−	111.374i	−0.797112	−	0.494994i
$226$		0			0
$227$		−	177.574i		−	0.782264i	−0.920335	−	0.391132i	$-0.872084\pi$
$227$		−	177.574i		−	0.782264i	0.920335	−	0.391132i	$-0.127916\pi$
$228$		0			0
$229$	−40.8118			−0.178217			−0.0891087	−	0.996022i	$-0.528402\pi$
$229$	−40.8118			−0.178217			−0.0891087	+	0.996022i	$0.528402\pi$
$230$		0			0
$231$	53.5203	+	15.2657i	0.231689	+	0.0660851i
$232$		0			0
$233$		−	387.696i		−	1.66393i	−0.554828	−	0.831965i	$-0.687216\pi$
$233$		−	387.696i		−	1.66393i	0.554828	−	0.831965i	$-0.312784\pi$
$234$		0			0
$235$	−41.0039			−0.174485
$236$		0			0
$237$	−104.708	+	367.100i	−0.441808	+	1.54895i
$238$		0			0
$239$			49.5229i			0.207209i	0.994619	+	0.103604i	$0.0330376\pi$
$239$			49.5229i			0.207209i	−0.994619	+	0.103604i	$0.966962\pi$
$240$		0			0
$241$	325.247			1.34957			0.674786	−	0.738013i	$-0.264236\pi$
$241$	325.247			1.34957			0.674786	+	0.738013i	$0.264236\pi$
$242$		0			0
$243$	−239.007	+	43.8699i	−0.983569	+	0.180535i
$244$		0			0
$245$			8.69378i			0.0354848i
$246$		0			0
$247$	−188.996			−0.765166
$248$		0			0
$249$	−214.306	−	61.1268i	−0.860666	−	0.245489i
$250$		0			0
$251$		−	263.732i		−	1.05073i	−0.850878	−	0.525364i	$-0.823929\pi$
$251$		−	263.732i		−	1.05073i	0.850878	−	0.525364i	$-0.176071\pi$
$252$		0			0
$253$	179.247			0.708486
$254$		0			0
$255$	−4.62746	+	16.2235i	−0.0181469	+	0.0636217i
$256$		0			0
$257$			151.181i			0.588252i	0.955767	+	0.294126i	$0.0950286\pi$
$257$			151.181i			0.588252i	−0.955767	+	0.294126i	$0.904971\pi$
$258$		0			0
$259$	87.4170			0.337517
$260$		0			0
$261$	45.0850	−	72.6024i	0.172739	−	0.278170i
$262$		0			0
$263$		−	114.389i		−	0.434941i	−0.976067	−	0.217470i	$-0.930219\pi$
$263$		−	114.389i		−	0.434941i	0.976067	−	0.217470i	$-0.0697805\pi$
$264$		0			0
$265$	18.8627			0.0711800
$266$		0			0
$267$	−367.476	−	104.816i	−1.37631	−	0.392568i
$268$		0			0
$269$			4.76170i			0.0177015i	0.999961	+	0.00885074i	$0.00281731\pi$
$269$			4.76170i			0.0177015i	−0.999961	+	0.00885074i	$0.997183\pi$
$270$		0			0
$271$	−518.701			−1.91402			−0.957012	−	0.290048i	$-0.906329\pi$
$271$	−518.701			−1.91402			−0.957012	+	0.290048i	$0.906329\pi$
$272$		0			0
$273$	−25.3542	+	88.8901i	−0.0928727	+	0.325605i
$274$		0			0
$275$		−	164.481i		−	0.598111i
$276$		0			0
$277$	121.085			0.437130			0.218565	−	0.975822i	$-0.429862\pi$
$277$	121.085			0.437130			0.218565	+	0.975822i	$0.429862\pi$
$278$		0			0
$279$	−219.498	−	136.305i	−0.786731	−	0.488548i
$280$		0			0
$281$		−	407.255i		−	1.44931i	−0.689113	−	0.724654i	$-0.742000\pi$
$281$		−	407.255i		−	1.44931i	0.689113	−	0.724654i	$-0.258000\pi$
$282$		0			0
$283$	398.634			1.40860			0.704300	−	0.709902i	$-0.251261\pi$
$283$	398.634			1.40860			0.704300	+	0.709902i	$0.251261\pi$
$284$		0			0
$285$	58.1477	+	16.5856i	0.204027	+	0.0581950i
$286$		0			0
$287$		−	177.780i		−	0.619443i
$288$		0			0
$289$	268.498			0.929059
$290$		0			0
$291$	19.0627	−	66.8325i	0.0655077	−	0.229665i
$292$		0			0
$293$			2.53426i			0.00864935i	0.999991	+	0.00432468i	$0.00137659\pi$
$293$			2.53426i			0.00864935i	−0.999991	+	0.00432468i	$0.998623\pi$
$294$		0			0
$295$	−114.753			−0.388993
$296$		0			0
$297$	−127.269	−	140.159i	−0.428516	−	0.471916i
$298$		0			0
$299$			297.706i			0.995671i
$300$		0			0
$301$	63.8301			0.212060
$302$		0			0
$303$	389.199	+	111.012i	1.28448	+	0.366375i
$304$		0			0
$305$			71.4384i			0.234224i
$306$		0			0
$307$	−86.2366			−0.280901			−0.140451	−	0.990088i	$-0.544855\pi$
$307$	−86.2366			−0.280901			−0.140451	+	0.990088i	$0.544855\pi$
$308$		0			0
$309$	98.5385	−	345.469i	0.318895	−	1.11802i
$310$		0			0
$311$		−	151.777i		−	0.488028i	−0.969772	−	0.244014i	$-0.921536\pi$
$311$		−	151.777i		−	0.488028i	0.969772	−	0.244014i	$-0.0784643\pi$
$312$		0			0
$313$	318.118			1.01635			0.508175	−	0.861254i	$-0.330320\pi$
$313$	318.118			1.01635			0.508175	+	0.861254i	$0.330320\pi$
$314$		0			0
$315$	15.6013	−	25.1235i	0.0495279	−	0.0797571i
$316$		0			0
$317$			364.020i			1.14833i	0.818740	+	0.574164i	$0.194673\pi$
$317$			364.020i			1.14833i	−0.818740	+	0.574164i	$0.805327\pi$
$318$		0			0
$319$	66.5830			0.208724
$320$		0			0
$321$	224.605	+	64.0645i	0.699705	+	0.199578i
$322$		0			0
$323$			73.4823i			0.227500i
$324$		0			0
$325$	273.180			0.840555
$326$		0			0
$327$	−30.0667	+	105.412i	−0.0919470	+	0.322359i
$328$		0			0
$329$			87.3502i			0.265502i
$330$		0			0
$331$	−154.369			−0.466370			−0.233185	−	0.972432i	$-0.574915\pi$
$331$	−154.369			−0.466370			−0.233185	+	0.972432i	$0.574915\pi$
$332$		0			0
$333$	−252.620	−	156.873i	−0.758617	−	0.471090i
$334$		0			0
$335$		−	18.8357i		−	0.0562260i
$336$		0			0
$337$	403.041			1.19597			0.597983	−	0.801509i	$-0.295969\pi$
$337$	403.041			1.19597			0.597983	+	0.801509i	$0.295969\pi$
$338$		0			0
$339$	62.7752	+	17.9055i	0.185178	+	0.0528185i
$340$		0			0
$341$		−	201.300i		−	0.590321i
$342$		0			0
$343$	18.5203			0.0539949
$344$		0			0
$345$	26.1255	−	91.5940i	0.0757261	−	0.265490i
$346$		0			0
$347$		−	471.242i		−	1.35805i	−0.734117	−	0.679023i	$-0.762404\pi$
$347$		−	471.242i		−	1.35805i	0.734117	−	0.679023i	$-0.237596\pi$
$348$		0			0
$349$	364.516			1.04446			0.522230	−	0.852805i	$-0.325100\pi$
$349$	364.516			1.04446			0.522230	+	0.852805i	$0.325100\pi$
$350$		0			0
$351$	232.786	−	211.377i	0.663207	−	0.602215i
$352$		0			0
$353$			86.3420i			0.244595i	0.992493	+	0.122297i	$0.0390262\pi$
$353$			86.3420i			0.244595i	−0.992493	+	0.122297i	$0.960974\pi$
$354$		0			0
$355$	−87.6314			−0.246849
$356$		0			0
$357$	34.5608	+	9.85782i	0.0968089	+	0.0276129i
$358$		0			0
$359$		−	372.068i		−	1.03640i	−0.855259	−	0.518200i	$-0.826602\pi$
$359$		−	372.068i		−	1.03640i	0.855259	−	0.518200i	$-0.173398\pi$
$360$		0			0
$361$	−97.6275			−0.270436
$362$		0			0
$363$	−59.1104	+	207.237i	−0.162839	+	0.570900i
$364$		0			0
$365$		−	95.3702i		−	0.261288i
$366$		0			0
$367$	−161.786			−0.440833			−0.220416	−	0.975406i	$-0.570742\pi$
$367$	−161.786			−0.440833			−0.220416	+	0.975406i	$0.570742\pi$
$368$		0			0
$369$	−319.033	+	513.753i	−0.864587	+	1.39228i
$370$		0			0
$371$		−	40.1830i		−	0.108310i
$372$		0			0
$373$	−378.251			−1.01408			−0.507039	−	0.861923i	$-0.669260\pi$
$373$	−378.251			−1.01408			−0.507039	+	0.861923i	$0.669260\pi$
$374$		0			0
$375$	−173.624	−	49.5229i	−0.462996	−	0.132061i
$376$		0			0
$377$			110.586i			0.293330i
$378$		0			0
$379$	50.7974			0.134030			0.0670151	−	0.997752i	$-0.478652\pi$
$379$	50.7974			0.134030			0.0670151	+	0.997752i	$0.478652\pi$
$380$		0			0
$381$	12.6863	−	44.4771i	0.0332973	−	0.116738i
$382$		0			0
$383$		−	113.381i		−	0.296034i	−0.988985	−	0.148017i	$-0.952711\pi$
$383$		−	113.381i		−	0.296034i	0.988985	−	0.148017i	$-0.0472891\pi$
$384$		0			0
$385$	23.0405			0.0598455
$386$		0			0
$387$	−184.458	−	114.545i	−0.476634	−	0.295983i
$388$		0			0
$389$		−	725.584i		−	1.86526i	−0.360841	−	0.932628i	$-0.617510\pi$
$389$		−	725.584i		−	1.86526i	0.360841	−	0.932628i	$-0.382490\pi$
$390$		0			0
$391$	115.749			0.296033
$392$		0			0
$393$	−528.261	−	150.677i	−1.34418	−	0.383402i
$394$		0			0
$395$			158.037i			0.400093i
$396$		0			0
$397$	94.3464			0.237648			0.118824	−	0.992915i	$-0.462088\pi$
$397$	94.3464			0.237648			0.118824	+	0.992915i	$0.462088\pi$
$398$		0			0
$399$	35.3320	−	123.871i	0.0885514	−	0.310455i
$400$		0			0
$401$			677.665i			1.68994i	0.534815	+	0.844969i	$0.320381\pi$
$401$			677.665i			1.68994i	−0.534815	+	0.844969i	$0.679619\pi$
$402$		0			0
$403$	334.332			0.829608
$404$		0			0
$405$	−90.1699	+	44.6053i	−0.222642	+	0.110137i
$406$		0			0
$407$		−	231.675i		−	0.569226i
$408$		0			0
$409$	17.3647			0.0424564			0.0212282	−	0.999775i	$-0.493242\pi$
$409$	17.3647			0.0424564			0.0212282	+	0.999775i	$0.493242\pi$
$410$		0			0
$411$	95.2470	+	27.1675i	0.231745	+	0.0661009i
$412$		0			0
$413$			244.457i			0.591905i
$414$		0			0
$415$	−92.2589			−0.222311
$416$		0			0
$417$	53.1922	−	186.488i	0.127559	−	0.447213i
$418$		0			0
$419$			136.071i			0.324752i	0.986729	+	0.162376i	$0.0519157\pi$
$419$			136.071i			0.324752i	−0.986729	+	0.162376i	$0.948084\pi$
$420$		0			0
$421$	−423.992			−1.00711			−0.503554	−	0.863964i	$-0.667974\pi$
$421$	−423.992			−1.00711			−0.503554	+	0.863964i	$0.667974\pi$
$422$		0			0
$423$	156.753	−	252.427i	0.370574	−	0.596753i
$424$		0			0
$425$		−	106.214i		−	0.249914i
$426$		0			0
$427$	152.184			0.356404
$428$		0			0
$429$	235.579	+	67.1946i	0.549135	+	0.156631i
$430$		0			0
$431$		−	340.244i		−	0.789430i	−0.918804	−	0.394715i	$-0.870843\pi$
$431$		−	340.244i		−	0.789430i	0.918804	−	0.394715i	$-0.129157\pi$
$432$		0			0
$433$	159.166			0.367589			0.183794	−	0.982965i	$-0.441162\pi$
$433$	159.166			0.367589			0.183794	+	0.982965i	$0.441162\pi$
$434$		0			0
$435$	9.70456	−	34.0235i	0.0223093	−	0.0782148i
$436$		0			0
$437$		−	414.863i		−	0.949343i
$438$		0			0
$439$	128.073			0.291738			0.145869	−	0.989304i	$-0.453402\pi$
$439$	128.073			0.291738			0.145869	+	0.989304i	$0.453402\pi$
$440$		0			0
$441$	−53.5203	−	33.2353i	−0.121361	−	0.0753634i
$442$		0			0
$443$		−	197.340i		−	0.445463i	−0.974880	−	0.222731i	$-0.928503\pi$
$443$		−	197.340i		−	0.445463i	0.974880	−	0.222731i	$-0.0714973\pi$
$444$		0			0
$445$	−158.199			−0.355503
$446$		0			0
$447$	−564.686	−	161.066i	−1.26328	−	0.360327i
$448$		0			0
$449$		−	148.101i		−	0.329847i	−0.986306	−	0.164923i	$-0.947262\pi$
$449$		−	148.101i		−	0.329847i	0.986306	−	0.164923i	$-0.0527377\pi$
$450$		0			0
$451$	−471.158			−1.04470
$452$		0			0
$453$	−84.1398	+	294.988i	−0.185739	+	0.651187i
$454$		0			0
$455$			38.2673i			0.0841038i
$456$		0			0
$457$	−122.214			−0.267428			−0.133714	−	0.991020i	$-0.542690\pi$
$457$	−122.214			−0.267428			−0.133714	+	0.991020i	$0.542690\pi$
$458$		0			0
$459$	−82.1843	−	90.5079i	−0.179051	−	0.197185i
$460$		0			0
$461$			602.089i			1.30605i	0.757337	+	0.653025i	$0.226500\pi$
$461$			602.089i			1.30605i	−0.757337	+	0.653025i	$0.773500\pi$
$462$		0			0
$463$	−637.061			−1.37594			−0.687971	−	0.725738i	$-0.741498\pi$
$463$	−637.061			−1.37594			−0.687971	+	0.725738i	$0.741498\pi$
$464$		0			0
$465$	−102.863	−	29.3397i	−0.221210	−	0.0630961i
$466$		0			0
$467$		−	767.706i		−	1.64391i	−0.569553	−	0.821955i	$-0.692884\pi$
$467$		−	767.706i		−	1.64391i	0.569553	−	0.821955i	$-0.307116\pi$
$468$		0			0
$469$	−40.1255			−0.0855554
$470$		0			0
$471$	86.1739	−	302.119i	0.182959	−	0.641442i
$472$		0			0
$473$		−	169.164i		−	0.357641i
$474$		0			0
$475$	−380.686			−0.801445
$476$		0			0
$477$	−72.1097	+	116.122i	−0.151173	+	0.243442i
$478$		0			0
$479$		−	393.855i		−	0.822245i	−0.911580	−	0.411122i	$-0.865137\pi$
$479$		−	393.855i		−	0.822245i	0.911580	−	0.411122i	$-0.134863\pi$
$480$		0			0
$481$	384.782			0.799962
$482$		0			0
$483$	−195.122	−	55.6548i	−0.403978	−	0.115227i
$484$		0			0
$485$		−	28.7715i		−	0.0593226i
$486$		0			0
$487$	573.409			1.17743			0.588716	−	0.808340i	$-0.299634\pi$
$487$	573.409			1.17743			0.588716	+	0.808340i	$0.299634\pi$
$488$		0			0
$489$	−58.3908	+	204.714i	−0.119409	+	0.418638i
$490$		0			0
$491$		−	170.796i		−	0.347853i	−0.984759	−	0.173927i	$-0.944354\pi$
$491$		−	170.796i		−	0.347853i	0.984759	−	0.173927i	$-0.0556455\pi$
$492$		0			0
$493$	42.9961			0.0872131
$494$		0			0
$495$	−66.5830	−	41.3470i	−0.134511	−	0.0835293i
$496$		0			0
$497$			186.680i			0.375614i
$498$		0			0
$499$	847.814			1.69903			0.849513	−	0.527567i	$-0.176896\pi$
$499$	847.814			1.69903			0.849513	+	0.527567i	$0.176896\pi$
$500$		0			0
$501$	−597.158	−	170.328i	−1.19193	−	0.339977i
$502$		0			0
$503$		−	197.624i		−	0.392891i	−0.980515	−	0.196445i	$-0.937060\pi$
$503$		−	197.624i		−	0.392891i	0.980515	−	0.196445i	$-0.0629398\pi$
$504$		0			0
$505$	167.550			0.331783
$506$		0			0
$507$	27.4647	−	96.2891i	0.0541710	−	0.189919i
$508$		0			0
$509$			491.448i			0.965516i	0.875754	+	0.482758i	$0.160365\pi$
$509$			491.448i			0.965516i	−0.875754	+	0.482758i	$0.839635\pi$
$510$		0			0
$511$	−203.166			−0.397585
$512$		0			0
$513$	−324.395	+	294.562i	−0.632348	+	0.574194i
$514$		0			0
$515$		−	148.725i		−	0.288786i
$516$		0			0
$517$	231.498			0.447772
$518$		0			0
$519$	−312.911	−	89.2521i	−0.602912	−	0.171969i
$520$		0			0
$521$		−	870.010i		−	1.66988i	−0.550338	−	0.834942i	$-0.685501\pi$
$521$		−	870.010i		−	1.66988i	0.550338	−	0.834942i	$-0.314499\pi$
$522$		0			0
$523$	−798.707			−1.52716			−0.763582	−	0.645710i	$-0.776561\pi$
$523$	−798.707			−1.52716			−0.763582	+	0.645710i	$0.776561\pi$
$524$		0			0
$525$	−51.0699	+	179.047i	−0.0972760	+	0.341042i
$526$		0			0
$527$		−	129.989i		−	0.246659i
$528$		0			0
$529$	−124.490			−0.235331
$530$		0			0
$531$	438.686	−	706.437i	0.826151	−	1.33039i
$532$		0			0
$533$		−	782.531i		−	1.46816i
$534$		0			0
$535$	96.6927			0.180734
$536$		0			0
$537$	−459.735	−	131.131i	−0.856117	−	0.244191i
$538$		0			0
$539$		−	49.0829i		−	0.0910630i
$540$		0			0
$541$	736.243			1.36089			0.680446	−	0.732798i	$-0.261786\pi$
$541$	736.243			1.36089			0.680446	+	0.732798i	$0.261786\pi$
$542$		0			0
$543$	−192.461	+	674.755i	−0.354441	+	1.24264i
$544$		0			0
$545$			45.3797i			0.0832656i
$546$		0			0
$547$	228.952			0.418559			0.209279	−	0.977856i	$-0.432888\pi$
$547$	228.952			0.418559			0.209279	+	0.977856i	$0.432888\pi$
$548$		0			0
$549$	−439.786	−	273.100i	−0.801067	−	0.497450i
$550$		0			0
$551$		−	154.105i		−	0.279682i
$552$		0			0
$553$	336.664			0.608796
$554$		0			0
$555$	−118.384	−	33.7669i	−0.213305	−	0.0608413i
$556$		0			0
$557$			906.288i			1.62709i	0.581503	+	0.813544i	$0.302465\pi$
$557$			906.288i			1.62709i	−0.581503	+	0.813544i	$0.697535\pi$
$558$		0			0
$559$	280.959			0.502611
$560$		0			0
$561$	26.1255	−	91.5940i	0.0465695	−	0.163269i
$562$		0			0
$563$		−	458.616i		−	0.814593i	−0.913296	−	0.407297i	$-0.866471\pi$
$563$		−	458.616i		−	0.814593i	0.913296	−	0.407297i	$-0.133529\pi$
$564$		0			0
$565$	27.0248			0.0478315
$566$		0			0
$567$	95.0222	+	192.088i	0.167588	+	0.338779i
$568$		0			0
$569$			577.428i			1.01481i	0.861707	+	0.507406i	$0.169395\pi$
$569$			577.428i			1.01481i	−0.861707	+	0.507406i	$0.830605\pi$
$570$		0			0
$571$	103.122			0.180598			0.0902991	−	0.995915i	$-0.471218\pi$
$571$	103.122			0.180598			0.0902991	+	0.995915i	$0.471218\pi$
$572$		0			0
$573$	−831.468	−	237.161i	−1.45108	−	0.413893i
$574$		0			0
$575$			599.655i			1.04288i
$576$		0			0
$577$	−676.583			−1.17259			−0.586294	−	0.810099i	$-0.699414\pi$
$577$	−676.583			−1.17259			−0.586294	+	0.810099i	$0.699414\pi$
$578$		0			0
$579$	−63.4615	+	222.491i	−0.109605	+	0.384268i
$580$		0			0
$581$			196.538i			0.338275i
$582$		0			0
$583$	−106.494			−0.182666
$584$		0			0
$585$	68.6719	−	110.586i	0.117388	−	0.189035i
$586$		0			0
$587$			158.683i			0.270329i	0.990823	+	0.135164i	$0.0431563\pi$
$587$			158.683i			0.270329i	−0.990823	+	0.135164i	$0.956844\pi$
$588$		0			0
$589$	−465.903			−0.791007
$590$		0			0
$591$	393.601	+	112.267i	0.665992	+	0.189962i
$592$		0			0
$593$			935.371i			1.57735i	0.614807	+	0.788677i	$0.289234\pi$
$593$			935.371i			1.57735i	−0.614807	+	0.788677i	$0.710766\pi$
$594$		0			0
$595$	14.8784			0.0250058
$596$		0			0
$597$	−71.2470	+	249.787i	−0.119342	+	0.418403i
$598$		0			0
$599$		−	73.7665i		−	0.123149i	−0.998102	−	0.0615747i	$-0.980388\pi$
$599$		−	73.7665i		−	0.123149i	0.998102	−	0.0615747i	$-0.0196122\pi$
$600$		0			0
$601$	−934.280			−1.55454			−0.777271	−	0.629166i	$-0.783397\pi$
$601$	−934.280			−1.55454			−0.777271	+	0.629166i	$0.783397\pi$
$602$		0			0
$603$	115.956	+	72.0066i	0.192298	+	0.119414i
$604$		0			0
$605$			89.2156i			0.147464i
$606$		0			0
$607$	−181.608			−0.299189			−0.149595	−	0.988747i	$-0.547797\pi$
$607$	−181.608			−0.299189			−0.149595	+	0.988747i	$0.547797\pi$
$608$		0			0
$609$	−72.4797	−	20.6735i	−0.119014	−	0.0339466i
$610$		0			0
$611$			384.488i			0.629276i
$612$		0			0
$613$	897.940			1.46483			0.732414	−	0.680859i	$-0.238394\pi$
$613$	897.940			1.46483			0.732414	+	0.680859i	$0.238394\pi$
$614$		0			0
$615$	−68.6719	+	240.759i	−0.111662	+	0.391477i
$616$		0			0
$617$		−	1169.69i		−	1.89576i	−0.318622	−	0.947882i	$-0.603220\pi$
$617$		−	1169.69i		−	1.89576i	0.318622	−	0.947882i	$-0.396780\pi$
$618$		0			0
$619$	−1208.97			−1.95310			−0.976548	−	0.215301i	$-0.930927\pi$
$619$	−1208.97			−1.95310			−0.976548	+	0.215301i	$0.930927\pi$
$620$		0			0
$621$	463.992	+	510.985i	0.747169	+	0.822842i
$622$		0			0
$623$			337.009i			0.540945i
$624$		0			0
$625$	511.693			0.818708
$626$		0			0
$627$	−328.288	−	93.6380i	−0.523585	−	0.149343i
$628$		0			0
$629$		−	149.604i		−	0.237845i
$630$		0			0
$631$	901.223			1.42825			0.714123	−	0.700020i	$-0.246826\pi$
$631$	901.223			1.42825			0.714123	+	0.700020i	$0.246826\pi$
$632$		0			0
$633$	15.9778	−	56.0169i	0.0252413	−	0.0884942i
$634$		0			0
$635$		−	19.1474i		−	0.0301534i
$636$		0			0
$637$	81.5203			0.127975
$638$		0			0
$639$	335.004	−	539.472i	0.524263	−	0.844245i
$640$		0			0
$641$		−	528.629i		−	0.824694i	−0.911027	−	0.412347i	$-0.864709\pi$
$641$		−	528.629i		−	0.824694i	0.911027	−	0.412347i	$-0.135291\pi$
$642$		0			0
$643$	−33.4392			−0.0520050			−0.0260025	−	0.999662i	$-0.508278\pi$
$643$	−33.4392			−0.0520050			−0.0260025	+	0.999662i	$0.508278\pi$
$644$		0			0
$645$	−86.4418	−	24.6559i	−0.134018	−	0.0382262i
$646$		0			0
$647$			786.308i			1.21531i	0.794200	+	0.607657i	$0.207890\pi$
$647$			786.308i			1.21531i	−0.794200	+	0.607657i	$0.792110\pi$
$648$		0			0
$649$	647.867			0.998254
$650$		0			0
$651$	−62.5020	+	219.127i	−0.0960092	+	0.336601i
$652$		0			0
$653$		−	385.807i		−	0.590823i	−0.955370	−	0.295412i	$-0.904543\pi$
$653$		−	385.807i		−	0.590823i	0.955370	−	0.295412i	$-0.0954568\pi$
$654$		0			0
$655$	−227.417			−0.347202
$656$		0			0
$657$	587.114	+	364.588i	0.893628	+	0.554929i
$658$		0			0
$659$		−	97.2583i		−	0.147585i	−0.997274	−	0.0737924i	$-0.976490\pi$
$659$		−	97.2583i		−	0.147585i	0.997274	−	0.0737924i	$-0.0235102\pi$
$660$		0			0
$661$	961.505			1.45462			0.727311	−	0.686309i	$-0.240770\pi$
$661$	961.505			1.45462			0.727311	+	0.686309i	$0.240770\pi$
$662$		0			0
$663$	152.125	+	43.3910i	0.229450	+	0.0654464i
$664$		0			0
$665$		−	53.3267i		−	0.0801906i
$666$		0			0
$667$	−242.745			−0.363936
$668$		0			0
$669$	−144.037	+	504.981i	−0.215301	+	0.754830i
$670$		0			0
$671$		−	403.323i		−	0.601078i
$672$		0			0
$673$	−1089.81			−1.61933			−0.809663	−	0.586895i	$-0.800350\pi$
$673$	−1089.81			−1.61933			−0.809663	+	0.586895i	$0.800350\pi$
$674$		0			0
$675$	468.890	−	425.768i	0.694651	−	0.630767i
$676$		0			0
$677$			1252.56i			1.85016i	0.379771	+	0.925080i	$0.376003\pi$
$677$			1252.56i			1.85016i	−0.379771	+	0.925080i	$0.623997\pi$
$678$		0			0
$679$	−61.2915			−0.0902673
$680$		0			0
$681$	512.290	+	146.121i	0.752262	+	0.214569i
$682$		0			0
$683$			341.097i			0.499409i	0.968322	+	0.249705i	$0.0803335\pi$
$683$			341.097i			0.499409i	−0.968322	+	0.249705i	$0.919666\pi$
$684$		0			0
$685$	41.0039			0.0598598
$686$		0			0
$687$	33.5830	−	117.739i	0.0488836	−	0.171382i
$688$		0			0
$689$		−	176.873i		−	0.256709i
$690$		0			0
$691$	783.667			1.13411			0.567053	−	0.823682i	$-0.308084\pi$
$691$	783.667			1.13411			0.567053	+	0.823682i	$0.308084\pi$
$692$		0			0
$693$	−88.0810	+	141.841i	−0.127101	+	0.204677i
$694$		0			0
$695$		−	80.2831i		−	0.115515i
$696$		0			0
$697$	−304.251			−0.436515
$698$		0			0
$699$	1118.48	+	319.025i	1.60011	+	0.456402i
$700$		0			0
$701$		−	1331.76i		−	1.89979i	−0.312562	−	0.949897i	$-0.601187\pi$
$701$		−	1331.76i		−	1.89979i	0.312562	−	0.949897i	$-0.398813\pi$
$702$		0			0
$703$	−536.207			−0.762740
$704$		0			0
$705$	33.7411	−	118.294i	0.0478598	−	0.167793i
$706$		0			0
$707$		−	356.930i		−	0.504852i
$708$		0			0
$709$	−763.963			−1.07752			−0.538761	−	0.842459i	$-0.681107\pi$
$709$	−763.963			−1.07752			−0.538761	+	0.842459i	$0.681107\pi$
$710$		0			0
$711$	−972.899	−	604.155i	−1.36835	−	0.849726i
$712$		0			0
$713$			733.888i			1.02930i
$714$		0			0
$715$	101.417			0.141842
$716$		0			0
$717$	−142.871	−	40.7512i	−0.199262	−	0.0568357i
$718$		0			0
$719$			623.715i			0.867476i	0.901039	+	0.433738i	$0.142806\pi$
$719$			623.715i			0.867476i	−0.901039	+	0.433738i	$0.857194\pi$
$720$		0			0
$721$	−316.826			−0.439426
$722$		0			0
$723$	−267.638	+	938.318i	−0.370177	+	1.29781i
$724$		0			0
$725$			222.748i			0.307238i
$726$		0			0
$727$	678.494			0.933279			0.466640	−	0.884448i	$-0.345464\pi$
$727$	678.494			0.933279			0.466640	+	0.884448i	$0.345464\pi$
$728$		0			0
$729$	70.1111	−	725.621i	0.0961744	−	0.995364i
$730$		0			0
$731$		−	109.238i		−	0.149436i
$732$		0			0
$733$	394.966			0.538835			0.269417	−	0.963023i	$-0.413169\pi$
$733$	394.966			0.538835			0.269417	+	0.963023i	$0.413169\pi$
$734$		0			0
$735$	−25.0810	−	7.15390i	−0.0341239	−	0.00973320i
$736$		0			0
$737$			106.342i			0.144290i
$738$		0			0
$739$	−292.199			−0.395397			−0.197699	−	0.980263i	$-0.563347\pi$
$739$	−292.199			−0.395397			−0.197699	+	0.980263i	$0.563347\pi$
$740$		0			0
$741$	155.520	−	545.242i	0.209879	−	0.735819i
$742$		0			0
$743$		−	383.452i		−	0.516086i	−0.966133	−	0.258043i	$-0.916922\pi$
$743$		−	383.452i		−	0.516086i	0.966133	−	0.258043i	$-0.0830776\pi$
$744$		0			0
$745$	−243.098			−0.326306
$746$		0			0
$747$	352.694	−	567.960i	0.472147	−	0.760321i
$748$		0			0
$749$		−	205.983i		−	0.275011i
$750$		0			0
$751$	−696.332			−0.927206			−0.463603	−	0.886043i	$-0.653444\pi$
$751$	−696.332			−0.927206			−0.463603	+	0.886043i	$0.653444\pi$
$752$		0			0
$753$	760.852	+	217.019i	1.01043	+	0.288206i
$754$		0			0
$755$			126.993i			0.168202i
$756$		0			0
$757$	−967.357			−1.27788			−0.638941	−	0.769256i	$-0.720627\pi$
$757$	−967.357			−1.27788			−0.638941	+	0.769256i	$0.720627\pi$
$758$		0			0
$759$	−147.498	+	517.117i	−0.194332	+	0.681313i
$760$		0			0
$761$		−	89.7059i		−	0.117879i	−0.998262	−	0.0589395i	$-0.981228\pi$
$761$		−	89.7059i		−	0.117879i	0.998262	−	0.0589395i	$-0.0187719\pi$
$762$		0			0
$763$	96.6719			0.126700
$764$		0			0
$765$	−42.9961	−	26.6999i	−0.0562040	−	0.0349018i
$766$		0			0
$767$			1076.02i			1.40290i
$768$		0			0
$769$	926.219			1.20445			0.602223	−	0.798328i	$-0.294282\pi$
$769$	926.219			1.20445			0.602223	+	0.798328i	$0.294282\pi$
$770$		0			0
$771$	−436.148	−	124.403i	−0.565691	−	0.161353i
$772$		0			0
$773$		−	424.125i		−	0.548674i	−0.961634	−	0.274337i	$-0.911542\pi$
$773$		−	424.125i		−	0.548674i	0.961634	−	0.274337i	$-0.0884584\pi$
$774$		0			0
$775$	673.430			0.868942
$776$		0			0
$777$	−71.9333	+	252.193i	−0.0925783	+	0.324572i
$778$		0			0
$779$			1090.48i			1.39985i
$780$		0			0
$781$	494.745			0.633476
$782$		0			0
$783$	172.354	+	189.810i	0.220120	+	0.242414i
$784$		0			0
$785$		−	130.063i		−	0.165685i
$786$		0			0
$787$	155.889			0.198080			0.0990399	−	0.995083i	$-0.468423\pi$
$787$	155.889			0.198080			0.0990399	+	0.995083i	$0.468423\pi$
$788$		0			0
$789$	330.006	+	94.1282i	0.418259	+	0.119301i
$790$		0			0
$791$		−	57.5705i		−	0.0727820i
$792$		0			0
$793$	669.867			0.844725
$794$		0			0
$795$	−15.5217	+	54.4177i	−0.0195241	+	0.0684500i
$796$		0			0
$797$			719.191i			0.902373i	0.892430	+	0.451186i	$0.148999\pi$
$797$			719.191i			0.902373i	−0.892430	+	0.451186i	$0.851001\pi$
$798$		0			0
$799$	149.490			0.187097
$800$		0			0
$801$	604.774	−	973.895i	0.755023	−	1.21585i
$802$		0			0
$803$			538.437i			0.670531i
$804$		0			0
$805$	−84.0000			−0.104348
$806$		0			0
$807$	−13.7372	−	3.91828i	−0.0170226	−	0.00485537i
$808$		0			0
$809$		−	212.244i		−	0.262353i	−0.991359	−	0.131176i	$-0.958125\pi$
$809$		−	212.244i		−	0.262353i	0.991359	−	0.131176i	$-0.0418755\pi$
$810$		0			0
$811$	−1058.66			−1.30538			−0.652690	−	0.757625i	$-0.726359\pi$
$811$	−1058.66			−1.30538			−0.652690	+	0.757625i	$0.726359\pi$
$812$		0			0
$813$	426.826	−	1496.42i	0.525001	−	1.84061i
$814$		0			0
$815$			88.1295i			0.108134i
$816$		0			0
$817$	−391.527			−0.479225
$818$		0			0
$819$	−235.579	−	146.291i	−0.287642	−	0.178621i
$820$		0			0
$821$		−	818.571i		−	0.997042i	−0.866878	−	0.498521i	$-0.833877\pi$
$821$		−	818.571i		−	0.997042i	0.866878	−	0.498521i	$-0.166123\pi$
$822$		0			0
$823$	206.850			0.251336			0.125668	−	0.992072i	$-0.459893\pi$
$823$	206.850			0.251336			0.125668	+	0.992072i	$0.459893\pi$
$824$		0			0
$825$	474.516	+	135.347i	0.575171	+	0.164057i
$826$		0			0
$827$			438.639i			0.530398i	0.964194	+	0.265199i	$0.0854376\pi$
$827$			438.639i			0.530398i	−0.964194	+	0.265199i	$0.914562\pi$
$828$		0			0
$829$	−654.804			−0.789872			−0.394936	−	0.918709i	$-0.629233\pi$
$829$	−654.804			−0.789872			−0.394936	+	0.918709i	$0.629233\pi$
$830$		0			0
$831$	−99.6379	+	349.323i	−0.119901	+	0.420364i
$832$		0			0
$833$		−	31.6954i		−	0.0380497i
$834$		0			0
$835$	−257.077			−0.307877
$836$		0			0
$837$	573.851	−	521.077i	0.685604	−	0.622553i
$838$		0			0
$839$			50.9710i			0.0607521i	0.999539	+	0.0303761i	$0.00967049\pi$
$839$			50.9710i			0.0607521i	−0.999539	+	0.0303761i	$0.990330\pi$
$840$		0			0
$841$	750.830			0.892782
$842$		0			0
$843$	1174.91	+	335.121i	1.39372	+	0.397533i
$844$		0			0
$845$		−	41.4525i		−	0.0490563i
$846$		0			0
$847$	190.055			0.224386
$848$		0			0
$849$	−328.026	+	1150.03i	−0.386368	+	1.35458i
$850$		0			0
$851$			844.630i			0.992514i
$852$		0			0
$853$	883.941			1.03627			0.518137	−	0.855298i	$-0.326626\pi$
$853$	883.941			1.03627			0.518137	+	0.855298i	$0.326626\pi$
$854$		0			0
$855$	−95.6967	+	154.105i	−0.111926	+	0.180240i
$856$		0			0
$857$		−	556.521i		−	0.649382i	−0.945820	−	0.324691i	$-0.894740\pi$
$857$		−	556.521i		−	0.649382i	0.945820	−	0.324691i	$-0.105260\pi$
$858$		0			0
$859$	643.078			0.748636			0.374318	−	0.927300i	$-0.377877\pi$
$859$	643.078			0.748636			0.374318	+	0.927300i	$0.377877\pi$
$860$		0			0
$861$	512.885	+	146.291i	0.595685	+	0.169908i
$862$		0			0
$863$		−	204.892i		−	0.237419i	−0.992929	−	0.118709i	$-0.962124\pi$
$863$		−	204.892i		−	0.237419i	0.992929	−	0.118709i	$-0.0378757\pi$
$864$		0			0
$865$	−134.708			−0.155732
$866$		0			0
$867$	−220.940	+	774.601i	−0.254833	+	0.893426i
$868$		0			0
$869$		−	892.237i		−	1.02674i
$870$		0			0
$871$	−176.620			−0.202778
$872$		0			0
$873$	177.122	+	109.990i	0.202888	+	0.125991i
$874$		0			0
$875$			159.229i			0.181975i
$876$		0			0
$877$	207.210			0.236272			0.118136	−	0.992997i	$-0.462308\pi$
$877$	207.210			0.236272			0.118136	+	0.992997i	$0.462308\pi$
$878$		0			0
$879$	−7.31119	−	2.08538i	−0.00831762	−	0.00237245i
$880$		0			0
$881$			1391.37i			1.57931i	0.613552	+	0.789654i	$0.289740\pi$
$881$			1391.37i			1.57931i	−0.613552	+	0.789654i	$0.710260\pi$
$882$		0			0
$883$	1091.99			1.23668			0.618342	−	0.785909i	$-0.287805\pi$
$883$	1091.99			1.23668			0.618342	+	0.785909i	$0.287805\pi$
$884$		0			0
$885$	94.4274	−	331.055i	0.106698	−	0.374074i
$886$		0			0
$887$		−	149.449i		−	0.168488i	−0.996445	−	0.0842439i	$-0.973153\pi$
$887$		−	149.449i		−	0.168488i	0.996445	−	0.0842439i	$-0.0268475\pi$
$888$		0			0
$889$	−40.7895			−0.0458825
$890$		0			0
$891$	509.077	−	251.831i	0.571355	−	0.282638i
$892$		0			0
$893$		−	535.797i		−	0.599996i
$894$		0			0
$895$	−197.916			−0.221135
$896$		0			0
$897$	−858.863	−	244.975i	−0.957483	−	0.273104i
$898$		0			0
$899$			272.610i			0.303237i
$900$		0			0
$901$	−68.7687			−0.0763249
$902$		0			0
$903$	−52.5242	+	184.146i	−0.0581663	+	0.203927i
$904$		0			0
$905$			290.483i			0.320975i
$906$		0			0
$907$	−593.718			−0.654595			−0.327297	−	0.944921i	$-0.606138\pi$
$907$	−593.718			−0.654595			−0.327297	+	0.944921i	$0.606138\pi$
$908$		0			0
$909$	−640.524	+	1031.47i	−0.704647	+	1.13473i
$910$		0			0
$911$		−	1133.75i		−	1.24451i	−0.782815	−	0.622254i	$-0.786217\pi$
$911$		−	1133.75i		−	1.24451i	0.782815	−	0.622254i	$-0.213783\pi$
$912$		0			0
$913$	520.871			0.570504
$914$		0			0
$915$	−206.095	−	58.7849i	−0.225241	−	0.0642458i
$916$		0			0
$917$			484.464i			0.528314i
$918$		0			0
$919$	−684.988			−0.745363			−0.372681	−	0.927959i	$-0.621562\pi$
$919$	−684.988			−0.745363			−0.372681	+	0.927959i	$0.621562\pi$
$920$		0			0
$921$	70.9620	−	248.787i	0.0770489	−	0.270128i
$922$		0			0
$923$			821.706i			0.890256i
$924$		0			0
$925$	775.048			0.837890
$926$		0			0
$927$	915.571	+	568.556i	0.987671	+	0.613328i
$928$		0			0
$929$			192.317i			0.207015i	0.994629	+	0.103507i	$0.0330066\pi$
$929$			192.317i			0.207015i	−0.994629	+	0.103507i	$0.966993\pi$
$930$		0			0
$931$	−113.601			−0.122021
$932$		0			0
$933$	437.867	+	124.893i	0.469310	+	0.133862i
$934$		0			0
$935$		−	39.4313i		−	0.0421725i
$936$		0			0
$937$	−1270.28			−1.35569			−0.677844	−	0.735206i	$-0.737086\pi$
$937$	−1270.28			−1.35569			−0.677844	+	0.735206i	$0.737086\pi$
$938$		0			0
$939$	−261.771	+	917.750i	−0.278777	+	0.977370i
$940$		0			0
$941$			156.951i			0.166791i	0.996517	+	0.0833957i	$0.0265766\pi$
$941$			156.951i			0.166791i	−0.996517	+	0.0833957i	$0.973423\pi$
$942$		0			0
$943$	1717.73			1.82155
$944$		0			0
$945$	59.6418	+	65.6823i	0.0631130	+	0.0695051i
$946$		0			0
$947$			879.945i			0.929193i	0.885523	+	0.464596i	$0.153801\pi$
$947$			879.945i			0.929193i	−0.885523	+	0.464596i	$0.846199\pi$
$948$		0			0
$949$	−894.272			−0.942331
$950$		0			0
$951$	−1050.18	−	299.543i	−1.10429	−	0.314977i
$952$		0			0
$953$		−	563.276i		−	0.591056i	−0.955334	−	0.295528i	$-0.904505\pi$
$953$		−	563.276i		−	0.591056i	0.955334	−	0.295528i	$-0.0954955\pi$
$954$		0			0
$955$	−357.948			−0.374814
$956$		0			0
$957$	−54.7895	+	192.088i	−0.0572513	+	0.200719i
$958$		0			0
$959$		−	87.3502i		−	0.0910847i
$960$		0			0
$961$	−136.822			−0.142375
$962$		0			0
$963$	−369.644	+	595.255i	−0.383847	+	0.618126i
$964$		0			0
$965$			95.7826i			0.0992566i
$966$		0			0
$967$	237.676			0.245787			0.122893	−	0.992420i	$-0.460783\pi$
$967$	237.676			0.245787			0.122893	+	0.992420i	$0.460783\pi$
$968$		0			0
$969$	−211.992	−	60.4668i	−0.218774	−	0.0624013i
$970$		0			0
$971$		−	1355.00i		−	1.39546i	−0.716359	−	0.697732i	$-0.754192\pi$
$971$		−	1355.00i		−	1.39546i	0.716359	−	0.697732i	$-0.245808\pi$
$972$		0			0
$973$	−171.026			−0.175772
$974$		0			0
$975$	−224.793	+	788.109i	−0.230557	+	0.808317i
$976$		0			0
$977$			493.726i			0.505349i	0.967551	+	0.252674i	$0.0813101\pi$
$977$			493.726i			0.505349i	−0.967551	+	0.252674i	$0.918690\pi$
$978$		0			0
$979$	893.150			0.912309
$980$		0			0
$981$	−279.365	−	173.481i	−0.284775	−	0.176841i
$982$		0			0
$983$			1538.05i			1.56465i	0.622870	+	0.782325i	$0.285966\pi$
$983$			1538.05i			1.56465i	−0.622870	+	0.782325i	$0.714034\pi$
$984$		0			0
$985$	169.446			0.172026
$986$		0			0
$987$	−252.000	−	71.8783i	−0.255319	−	0.0728251i
$988$		0			0
$989$			616.731i			0.623590i
$990$		0			0
$991$	1514.73			1.52849			0.764243	−	0.644929i	$-0.223113\pi$
$991$	1514.73			1.52849			0.764243	+	0.644929i	$0.223113\pi$
$992$		0			0
$993$	127.026	−	445.344i	0.127922	−	0.448483i
$994$		0			0
$995$			107.533i			0.108074i
$996$		0			0
$997$	−1826.43			−1.83193			−0.915964	−	0.401260i	$-0.868572\pi$
$997$	−1826.43			−1.83193			−0.915964	+	0.401260i	$0.868572\pi$
$998$		0			0
$999$	660.443	−	599.705i	0.661104	−	0.600306i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1344.3.d.f.449.2		4
3.2	odd	2	inner	1344.3.d.f.449.1		4
4.3	odd	2		1344.3.d.b.449.3		4
8.3	odd	2		336.3.d.c.113.2		4
8.5	even	2		21.3.b.a.8.4	yes	4
12.11	even	2		1344.3.d.b.449.4		4
24.5	odd	2		21.3.b.a.8.1	✓	4
24.11	even	2		336.3.d.c.113.1		4
40.13	odd	4		525.3.f.a.449.8		8
40.29	even	2		525.3.c.a.176.1		4
40.37	odd	4		525.3.f.a.449.1		8
56.5	odd	6		147.3.h.c.116.4		8
56.13	odd	2		147.3.b.f.50.4		4
56.37	even	6		147.3.h.e.116.4		8
56.45	odd	6		147.3.h.c.128.1		8
56.53	even	6		147.3.h.e.128.1		8
72.5	odd	6		567.3.r.c.512.4		8
72.13	even	6		567.3.r.c.512.1		8
72.29	odd	6		567.3.r.c.134.1		8
72.61	even	6		567.3.r.c.134.4		8
120.29	odd	2		525.3.c.a.176.4		4
120.53	even	4		525.3.f.a.449.2		8
120.77	even	4		525.3.f.a.449.7		8
168.5	even	6		147.3.h.c.116.1		8
168.53	odd	6		147.3.h.e.128.4		8
168.101	even	6		147.3.h.c.128.4		8
168.125	even	2		147.3.b.f.50.1		4
168.149	odd	6		147.3.h.e.116.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
21.3.b.a.8.1	✓	4	24.5	odd	2
21.3.b.a.8.4	yes	4	8.5	even	2
147.3.b.f.50.1		4	168.125	even	2
147.3.b.f.50.4		4	56.13	odd	2
147.3.h.c.116.1		8	168.5	even	6
147.3.h.c.116.4		8	56.5	odd	6
147.3.h.c.128.1		8	56.45	odd	6
147.3.h.c.128.4		8	168.101	even	6
147.3.h.e.116.1		8	168.149	odd	6
147.3.h.e.116.4		8	56.37	even	6
147.3.h.e.128.1		8	56.53	even	6
147.3.h.e.128.4		8	168.53	odd	6
336.3.d.c.113.1		4	24.11	even	2
336.3.d.c.113.2		4	8.3	odd	2
525.3.c.a.176.1		4	40.29	even	2
525.3.c.a.176.4		4	120.29	odd	2
525.3.f.a.449.1		8	40.37	odd	4
525.3.f.a.449.2		8	120.53	even	4
525.3.f.a.449.7		8	120.77	even	4
525.3.f.a.449.8		8	40.13	odd	4
567.3.r.c.134.1		8	72.29	odd	6
567.3.r.c.134.4		8	72.61	even	6
567.3.r.c.512.1		8	72.13	even	6
567.3.r.c.512.4		8	72.5	odd	6
1344.3.d.b.449.3		4	4.3	odd	2
1344.3.d.b.449.4		4	12.11	even	2
1344.3.d.f.449.1		4	3.2	odd	2	inner
1344.3.d.f.449.2		4	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 1344 = 2^{6} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1344.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+(-0.822876 + 2.88494i) q^{3} +1.24197i q^{5} +2.64575 q^{7} +(-7.64575 - 4.74789i) q^{9} +O(q^{10})\)	\(q+(-0.822876 + 2.88494i) q^{3} +1.24197i q^{5} +2.64575 q^{7} +(-7.64575 - 4.74789i) q^{9} -7.01185i q^{11} +11.6458 q^{13} +(-3.58301 - 1.02199i) q^{15} -4.52791i q^{17} -16.2288 q^{19} +(-2.17712 + 7.63283i) q^{21} +25.5635i q^{23} +23.4575 q^{25} +(19.9889 - 18.1506i) q^{27} +9.49579i q^{29} +28.7085 q^{31} +(20.2288 + 5.76988i) q^{33} +3.28594i q^{35} +33.0405 q^{37} +(-9.58301 + 33.5973i) q^{39} -67.1946i q^{41} +24.1255 q^{43} +(5.89674 - 9.49579i) q^{45} +33.0153i q^{47} +7.00000 q^{49} +(13.0627 + 3.72591i) q^{51} -15.1877i q^{53} +8.70850 q^{55} +(13.3542 - 46.8190i) q^{57} +92.3960i q^{59} +57.5203 q^{61} +(-20.2288 - 12.5617i) q^{63} +14.4637i q^{65} -15.1660 q^{67} +(-73.7490 - 21.0355i) q^{69} +70.5584i q^{71} -76.7895 q^{73} +(-19.3026 + 67.6735i) q^{75} -18.5516i q^{77} +127.247 q^{79} +(35.9150 + 72.6024i) q^{81} +74.2844i q^{83} +5.62352 q^{85} +(-27.3948 - 7.81385i) q^{87} +127.377i q^{89} +30.8118 q^{91} +(-23.6235 + 82.8223i) q^{93} -20.1556i q^{95} -23.1660 q^{97} +(-33.2915 + 53.6108i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{3} - 20 q^{9}+O(q^{10}) \) 4 * q + 2 * q^3 - 20 * q^9	\( 4 q + 2 q^{3} - 20 q^{9} + 36 q^{13} + 28 q^{15} - 12 q^{19} - 14 q^{21} - 12 q^{25} - 10 q^{27} + 136 q^{31} + 28 q^{33} - 16 q^{37} + 4 q^{39} + 160 q^{43} + 140 q^{45} + 28 q^{49} + 84 q^{51} + 56 q^{55} + 64 q^{57} + 156 q^{61} - 28 q^{63} + 24 q^{67} - 168 q^{69} - 32 q^{73} - 146 q^{75} + 128 q^{79} - 68 q^{81} - 168 q^{85} + 28 q^{87} + 28 q^{91} + 96 q^{93} - 8 q^{97} - 112 q^{99}+O(q^{100}) \) 4 * q + 2 * q^3 - 20 * q^9 + 36 * q^13 + 28 * q^15 - 12 * q^19 - 14 * q^21 - 12 * q^25 - 10 * q^27 + 136 * q^31 + 28 * q^33 - 16 * q^37 + 4 * q^39 + 160 * q^43 + 140 * q^45 + 28 * q^49 + 84 * q^51 + 56 * q^55 + 64 * q^57 + 156 * q^61 - 28 * q^63 + 24 * q^67 - 168 * q^69 - 32 * q^73 - 146 * q^75 + 128 * q^79 - 68 * q^81 - 168 * q^85 + 28 * q^87 + 28 * q^91 + 96 * q^93 - 8 * q^97 - 112 * q^99

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