LMFDB - Embedded newform 242.3.b.d.241.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [242,3,Mod(241,242)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("242.241"); S:= CuspForms(chi, 3); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(242, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1])) N = Newforms(chi, 3, names="a")

Level:	$ N $	$=$	$ 242 = 2 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	242.b (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [8,0,-12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$6.59402239752$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.64000000.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{6} + 4x^{4} - 8x^{2} + 16 $ x^8 - 2x^6 + 4x^4 - 8*x^2 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{4}\cdot 11^{2} $
Twist minimal:	no (minimal twist has level 22)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			241.3
Root			$1.34500 + 0.437016i$ of defining polynomial
Character	$\chi$	$=$	242.241
Dual form			242.3.b.d.241.7

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.41421i q^{2} -0.955526 q^{3} -2.00000 q^{4} +2.99802 q^{5} +1.35132i q^{6} -3.89538i q^{7} +2.82843i q^{8} -8.08697 q^{9} -4.23984i q^{10} +1.91105 q^{12} -24.5565i q^{13} -5.50890 q^{14} -2.86469 q^{15} +4.00000 q^{16} +6.45970i q^{17} +11.4367i q^{18} -15.6814i q^{19} -5.99604 q^{20} +3.72214i q^{21} -30.3518 q^{23} -2.70264i q^{24} -16.0119 q^{25} -34.7282 q^{26} +16.3270 q^{27} +7.79076i q^{28} -14.9898i q^{29} +4.05128i q^{30} -26.5316 q^{31} -5.65685i q^{32} +9.13540 q^{34} -11.6784i q^{35} +16.1739 q^{36} -4.17592 q^{37} -22.1768 q^{38} +23.4644i q^{39} +8.47969i q^{40} -43.3537i q^{41} +5.26390 q^{42} +56.0236i q^{43} -24.2449 q^{45} +42.9240i q^{46} +30.2362 q^{47} -3.82210 q^{48} +33.8260 q^{49} +22.6442i q^{50} -6.17242i q^{51} +49.1131i q^{52} +52.0630 q^{53} -23.0899i q^{54} +11.0178 q^{56} +14.9840i q^{57} -21.1987 q^{58} +94.4081 q^{59} +5.72938 q^{60} -93.3079i q^{61} +37.5214i q^{62} +31.5018i q^{63} -8.00000 q^{64} -73.6210i q^{65} +54.0771 q^{67} -12.9194i q^{68} +29.0020 q^{69} -16.5158 q^{70} -93.8933 q^{71} -22.8734i q^{72} -24.5355i q^{73} +5.90564i q^{74} +15.2998 q^{75} +31.3627i q^{76} +33.1837 q^{78} +77.3099i q^{79} +11.9921 q^{80} +57.1818 q^{81} -61.3114 q^{82} -28.6246i q^{83} -7.44427i q^{84} +19.3663i q^{85} +79.2293 q^{86} +14.3231i q^{87} +68.2705 q^{89} +34.2875i q^{90} -95.6570 q^{91} +60.7037 q^{92} +25.3517 q^{93} -42.7605i q^{94} -47.0131i q^{95} +5.40527i q^{96} +36.0119 q^{97} -47.8372i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 12 q^{3} - 16 q^{4} - 28 q^{5} - 4 q^{9} + 24 q^{12} - 24 q^{14} + 32 q^{15} + 32 q^{16} + 56 q^{20} - 104 q^{23} + 68 q^{25} - 16 q^{26} + 24 q^{27} - 124 q^{31} + 112 q^{34} + 8 q^{36} + 36 q^{37}+ \cdots + 92 q^{97}+O(q^{100}) $ 8 * q - 12 * q^3 - 16 * q^4 - 28 * q^5 - 4 * q^9 + 24 * q^12 - 24 * q^14 + 32 * q^15 + 32 * q^16 + 56 * q^20 - 104 * q^23 + 68 * q^25 - 16 * q^26 + 24 * q^27 - 124 * q^31 + 112 * q^34 + 8 * q^36 + 36 * q^37 + 8 * q^38 + 96 * q^42 - 136 * q^45 + 76 * q^47 - 48 * q^48 + 76 * q^49 - 44 * q^53 + 48 * q^56 - 16 * q^58 + 100 * q^59 - 64 * q^60 - 64 * q^64 + 112 * q^67 - 224 * q^69 - 136 * q^70 - 276 * q^71 + 48 * q^75 + 104 * q^78 - 112 * q^80 - 280 * q^81 + 64 * q^82 + 128 * q^86 + 24 * q^89 + 176 * q^91 + 208 * q^92 + 96 * q^93 + 92 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/242\mathbb{Z}\right)^\times$.

$n$	$123$
$\chi(n)$	$-1$

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 $

$2$	− 1.41421i	− 0.707107i
$2$	− 1.41421i	− 0.707107i
$3$	−0.955526	−0.318509	−0.159254	−	0.987238i	$-0.550909\pi$
$3$	−0.955526	−0.318509	−0.159254	+	0.987238i	$0.550909\pi$
$4$	−2.00000	−0.500000
$5$	2.99802	0.599604	0.299802	−	0.954001i	$-0.403079\pi$
$5$	2.99802	0.599604	0.299802	+	0.954001i	$0.403079\pi$
$6$	1.35132i	0.225220i
$7$	− 3.89538i	− 0.556483i	−0.960511	−	0.278241i	$-0.910248\pi$
$7$	− 3.89538i	− 0.556483i	0.960511	−	0.278241i	$-0.0897515\pi$
$8$	2.82843i	0.353553i
$9$	−8.08697	−0.898552
$10$	− 4.23984i	− 0.423984i
$11$	0	0
$11$	0	0
$12$	1.91105	0.159254
$13$	− 24.5565i	− 1.88896i	−0.328562	−	0.944482i	$-0.606564\pi$
$13$	− 24.5565i	− 1.88896i	0.328562	−	0.944482i	$-0.393436\pi$
$14$	−5.50890	−0.393493
$15$	−2.86469	−0.190979
$16$	4.00000	0.250000
$17$	6.45970i	0.379983i	0.981786	+	0.189991i	$0.0608460\pi$
$17$	6.45970i	0.379983i	−0.981786	+	0.189991i	$0.939154\pi$
$18$	11.4367i	0.635372i
$19$	− 15.6814i	− 0.825335i	−0.910882	−	0.412667i	$-0.864597\pi$
$19$	− 15.6814i	− 0.825335i	0.910882	−	0.412667i	$-0.135403\pi$
$20$	−5.99604	−0.299802
$21$	3.72214i	0.177245i
$22$	0	0
$23$	−30.3518	−1.31964	−0.659822	−	0.751422i	$-0.729369\pi$
$23$	−30.3518	−1.31964	−0.659822	+	0.751422i	$0.729369\pi$
$24$	− 2.70264i	− 0.112610i
$25$	−16.0119	−0.640475
$26$	−34.7282	−1.33570
$27$	16.3270	0.604705
$28$	7.79076i	0.278241i
$29$	− 14.9898i	− 0.516888i	−0.966026	−	0.258444i	$-0.916790\pi$
$29$	− 14.9898i	− 0.516888i	0.966026	−	0.258444i	$-0.0832098\pi$
$30$	4.05128i	0.135043i
$31$	−26.5316	−0.855859	−0.427929	−	0.903812i	$-0.640757\pi$
$31$	−26.5316	−0.855859	−0.427929	+	0.903812i	$0.640757\pi$
$32$	− 5.65685i	− 0.176777i
$33$	0	0
$34$	9.13540	0.268688
$35$	− 11.6784i	− 0.333670i
$36$	16.1739	0.449276
$37$	−4.17592	−0.112863	−0.0564313	−	0.998406i	$-0.517972\pi$
$37$	−4.17592	−0.112863	−0.0564313	+	0.998406i	$0.517972\pi$
$38$	−22.1768	−0.583600
$39$	23.4644i	0.601652i
$40$	8.47969i	0.211992i
$41$	− 43.3537i	− 1.05741i	−0.848806	−	0.528704i	$-0.822678\pi$
$41$	− 43.3537i	− 1.05741i	0.848806	−	0.528704i	$-0.177322\pi$
$42$	5.26390	0.125331
$43$	56.0236i	1.30287i	0.758703	+	0.651437i	$0.225833\pi$
$43$	56.0236i	1.30287i	−0.758703	+	0.651437i	$0.774167\pi$
$44$	0	0
$45$	−24.2449	−0.538776
$46$	42.9240i	0.933130i
$47$	30.2362	0.643324	0.321662	−	0.946855i	$-0.395759\pi$
$47$	30.2362	0.643324	0.321662	+	0.946855i	$0.395759\pi$
$48$	−3.82210	−0.0796272
$49$	33.8260	0.690327
$50$	22.6442i	0.452884i
$51$	− 6.17242i	− 0.121028i
$52$	49.1131i	0.944482i
$53$	52.0630	0.982321	0.491161	−	0.871069i	$-0.336573\pi$
$53$	52.0630	0.982321	0.491161	+	0.871069i	$0.336573\pi$
$54$	− 23.0899i	− 0.427591i
$55$	0	0
$56$	11.0178	0.196746
$57$	14.9840i	0.262876i
$58$	−21.1987	−0.365495
$59$	94.4081	1.60014	0.800069	−	0.599908i	$-0.204796\pi$
$59$	94.4081	1.60014	0.800069	+	0.599908i	$0.204796\pi$
$60$	5.72938	0.0954896
$61$	− 93.3079i	− 1.52964i	−0.644246	−	0.764819i	$-0.722828\pi$
$61$	− 93.3079i	− 1.52964i	0.644246	−	0.764819i	$-0.277172\pi$
$62$	37.5214i	0.605183i
$63$	31.5018i	0.500029i
$64$	−8.00000	−0.125000
$65$	− 73.6210i	− 1.13263i
$66$	0	0
$67$	54.0771	0.807121	0.403560	−	0.914953i	$-0.367773\pi$
$67$	54.0771	0.807121	0.403560	+	0.914953i	$0.367773\pi$
$68$	− 12.9194i	− 0.189991i
$69$	29.0020	0.420318
$70$	−16.5158	−0.235940
$71$	−93.8933	−1.32244	−0.661221	−	0.750191i	$-0.729961\pi$
$71$	−93.8933	−1.32244	−0.661221	+	0.750191i	$0.729961\pi$
$72$	− 22.8734i	− 0.317686i
$73$	− 24.5355i	− 0.336103i	−0.985778	−	0.168051i	$-0.946253\pi$
$73$	− 24.5355i	− 0.336103i	0.985778	−	0.168051i	$-0.0537475\pi$
$74$	5.90564i	0.0798059i
$75$	15.2998	0.203997
$76$	31.3627i	0.412667i
$77$	0	0
$78$	33.1837	0.425432
$79$	77.3099i	0.978606i	0.872114	+	0.489303i	$0.162749\pi$
$79$	77.3099i	0.978606i	−0.872114	+	0.489303i	$0.837251\pi$
$80$	11.9921	0.149901
$81$	57.1818	0.705948
$82$	−61.3114	−0.747700
$83$	− 28.6246i	− 0.344875i	−0.985020	−	0.172437i	$-0.944836\pi$
$83$	− 28.6246i	− 0.344875i	0.985020	−	0.172437i	$-0.0551642\pi$
$84$	− 7.44427i	− 0.0886223i
$85$	19.3663i	0.227839i
$86$	79.2293	0.921271
$87$	14.3231i	0.164633i
$88$	0	0
$89$	68.2705	0.767085	0.383542	−	0.923523i	$-0.374704\pi$
$89$	68.2705	0.767085	0.383542	+	0.923523i	$0.374704\pi$
$90$	34.2875i	0.380972i
$91$	−95.6570	−1.05118
$92$	60.7037	0.659822
$93$	25.3517	0.272598
$94$	− 42.7605i	− 0.454898i
$95$	− 47.0131i	− 0.494874i
$96$	5.40527i	0.0563049i
$97$	36.0119	0.371256	0.185628	−	0.982620i	$-0.440568\pi$
$97$	36.0119	0.371256	0.185628	+	0.982620i	$0.440568\pi$
$98$	− 47.8372i	− 0.488135i
$99$	0	0
$100$	32.0237	0.320237
$101$	− 2.47858i	− 0.0245404i	−0.999925	−	0.0122702i	$-0.996094\pi$
$101$	− 2.47858i	− 0.0245404i	0.999925	−	0.0122702i	$-0.00390583\pi$
$102$	−8.72911	−0.0855795
$103$	−30.3548	−0.294707	−0.147353	−	0.989084i	$-0.547075\pi$
$103$	−30.3548	−0.294707	−0.147353	+	0.989084i	$0.547075\pi$
$104$	69.4564	0.667850
$105$	11.1591i	0.106277i
$106$	− 73.6282i	− 0.694606i
$107$	76.5766i	0.715669i	0.933785	+	0.357835i	$0.116485\pi$
$107$	76.5766i	0.715669i	−0.933785	+	0.357835i	$0.883515\pi$
$108$	−32.6541	−0.302353
$109$	− 12.1116i	− 0.111116i	−0.998455	−	0.0555580i	$-0.982306\pi$
$109$	− 12.1116i	− 0.111116i	0.998455	−	0.0555580i	$-0.0176938\pi$
$110$	0	0
$111$	3.99020	0.0359477
$112$	− 15.5815i	− 0.139121i
$113$	94.8750	0.839602	0.419801	−	0.907616i	$-0.362100\pi$
$113$	94.8750	0.839602	0.419801	+	0.907616i	$0.362100\pi$
$114$	21.1905	0.185882
$115$	−90.9955	−0.791265
$116$	29.9795i	0.258444i
$117$	198.588i	1.69733i
$118$	− 133.513i	− 1.13147i
$119$	25.1630	0.211454
$120$	− 8.10256i	− 0.0675214i
$121$	0	0
$122$	−131.957	−1.08162
$123$	41.4256i	0.336794i
$124$	53.0632	0.427929
$125$	−122.954	−0.983636
$126$	44.5503	0.353574
$127$	133.421i	1.05056i	0.850930	+	0.525279i	$0.176039\pi$
$127$	133.421i	1.05056i	−0.850930	+	0.525279i	$0.823961\pi$
$128$	11.3137i	0.0883883i
$129$	− 53.5320i	− 0.414977i
$130$	−104.116	−0.800891
$131$	42.3750i	0.323473i	0.986834	+	0.161737i	$0.0517095\pi$
$131$	42.3750i	0.323473i	−0.986834	+	0.161737i	$0.948290\pi$
$132$	0	0
$133$	−61.0849	−0.459285
$134$	− 76.4765i	− 0.570720i
$135$	48.9488	0.362584
$136$	−18.2708	−0.134344
$137$	136.020	0.992844	0.496422	−	0.868081i	$-0.334647\pi$
$137$	136.020	0.992844	0.496422	+	0.868081i	$0.334647\pi$
$138$	− 41.0150i	− 0.297210i
$139$	− 101.228i	− 0.728262i	−0.931348	−	0.364131i	$-0.881366\pi$
$139$	− 101.228i	− 0.728262i	0.931348	−	0.364131i	$-0.118634\pi$
$140$	23.3569i	0.166835i
$141$	−28.8915	−0.204904
$142$	132.785i	0.935107i
$143$	0	0
$144$	−32.3479	−0.224638
$145$	− 44.9396i	− 0.309929i
$146$	−34.6984	−0.237661
$147$	−32.3216	−0.219875
$148$	8.35183	0.0564313
$149$	51.6735i	0.346802i	0.984851	+	0.173401i	$0.0554757\pi$
$149$	51.6735i	0.346802i	−0.984851	+	0.173401i	$0.944524\pi$
$150$	− 21.6371i	− 0.144247i
$151$	83.7944i	0.554930i	0.960736	+	0.277465i	$0.0894943\pi$
$151$	83.7944i	0.554930i	−0.960736	+	0.277465i	$0.910506\pi$
$152$	44.3536	0.291800
$153$	− 52.2394i	− 0.341434i
$154$	0	0
$155$	−79.5424	−0.513177
$156$	− 46.9288i	− 0.300826i
$157$	−232.506	−1.48093	−0.740464	−	0.672096i	$-0.765394\pi$
$157$	−232.506	−1.48093	−0.740464	+	0.672096i	$0.765394\pi$
$158$	109.333	0.691979
$159$	−49.7476	−0.312878
$160$	− 16.9594i	− 0.105996i
$161$	118.232i	0.734360i
$162$	− 80.8673i	− 0.499181i
$163$	−321.632	−1.97320	−0.986602	−	0.163145i	$-0.947836\pi$
$163$	−321.632	−1.97320	−0.986602	+	0.163145i	$0.947836\pi$
$164$	86.7074i	0.528704i
$165$	0	0
$166$	−40.4813	−0.243863
$167$	− 125.085i	− 0.749010i	−0.927225	−	0.374505i	$-0.877813\pi$
$167$	− 125.085i	− 0.749010i	0.927225	−	0.374505i	$-0.122187\pi$
$168$	−10.5278	−0.0626654
$169$	−434.024	−2.56819
$170$	27.3881	0.161107
$171$	126.815i	0.741606i
$172$	− 112.047i	− 0.651437i
$173$	− 126.724i	− 0.732511i	−0.930514	−	0.366256i	$-0.880640\pi$
$173$	− 126.724i	− 0.732511i	0.930514	−	0.366256i	$-0.119360\pi$
$174$	20.2559	0.116413
$175$	62.3723i	0.356413i
$176$	0	0
$177$	−90.2094	−0.509658
$178$	− 96.5491i	− 0.542411i
$179$	198.341	1.10805	0.554025	−	0.832500i	$-0.313091\pi$
$179$	198.341	1.10805	0.554025	+	0.832500i	$0.313091\pi$
$180$	48.4898	0.269388
$181$	106.851	0.590338	0.295169	−	0.955445i	$-0.404624\pi$
$181$	106.851	0.590338	0.295169	+	0.955445i	$0.404624\pi$
$182$	135.279i	0.743294i
$183$	89.1581i	0.487203i
$184$	− 85.8480i	− 0.466565i
$185$	−12.5195	−0.0676729
$186$	− 35.8527i	− 0.192756i
$187$	0	0
$188$	−60.4724	−0.321662
$189$	− 63.6000i	− 0.336508i
$190$	−66.4865	−0.349929
$191$	34.4315	0.180270	0.0901348	−	0.995930i	$-0.471270\pi$
$191$	34.4315	0.180270	0.0901348	+	0.995930i	$0.471270\pi$
$192$	7.64421	0.0398136
$193$	− 213.990i	− 1.10876i	−0.832265	−	0.554378i	$-0.812956\pi$
$193$	− 213.990i	− 1.10876i	0.832265	−	0.554378i	$-0.187044\pi$
$194$	− 50.9285i	− 0.262518i
$195$	70.3468i	0.360753i
$196$	−67.6520	−0.345163
$197$	− 271.384i	− 1.37758i	−0.724960	−	0.688791i	$-0.758142\pi$
$197$	− 271.384i	− 1.37758i	0.724960	−	0.688791i	$-0.241858\pi$
$198$	0	0
$199$	249.930	1.25593	0.627966	−	0.778241i	$-0.283888\pi$
$199$	249.930	1.25593	0.627966	+	0.778241i	$0.283888\pi$
$200$	− 45.2884i	− 0.226442i
$201$	−51.6721	−0.257075
$202$	−3.50525	−0.0173527
$203$	−58.3908	−0.287639
$204$	12.3448i	0.0605139i
$205$	− 129.975i	− 0.634026i
$206$	42.9281i	0.208389i
$207$	245.454	1.18577
$208$	− 98.2262i	− 0.472241i
$209$	0	0
$210$	15.7813	0.0751490
$211$	8.91058i	0.0422302i	0.999777	+	0.0211151i	$0.00672165\pi$
$211$	8.91058i	0.0422302i	−0.999777	+	0.0211151i	$0.993278\pi$
$212$	−104.126	−0.491161
$213$	89.7175	0.421209
$214$	108.296	0.506055
$215$	167.960i	0.781209i
$216$	46.1799i	0.213796i
$217$	103.351i	0.476271i
$218$	−17.1285	−0.0785709
$219$	23.4443i	0.107052i
$220$	0	0
$221$	158.628	0.717774
$222$	− 5.64299i	− 0.0254189i
$223$	−207.051	−0.928480	−0.464240	−	0.885709i	$-0.653673\pi$
$223$	−207.051	−0.928480	−0.464240	+	0.885709i	$0.653673\pi$
$224$	−22.0356	−0.0983732
$225$	129.487	0.575500
$226$	− 134.174i	− 0.593688i
$227$	97.0796i	0.427663i	0.976871	+	0.213832i	$0.0685944\pi$
$227$	97.0796i	0.427663i	−0.976871	+	0.213832i	$0.931406\pi$
$228$	− 29.9679i	− 0.131438i
$229$	57.8258	0.252514	0.126257	−	0.991998i	$-0.459704\pi$
$229$	57.8258	0.252514	0.126257	+	0.991998i	$0.459704\pi$
$230$	128.687i	0.559509i
$231$	0	0
$232$	42.3975	0.182748
$233$	70.2380i	0.301450i	0.988576	+	0.150725i	$0.0481609\pi$
$233$	70.2380i	0.301450i	−0.988576	+	0.150725i	$0.951839\pi$
$234$	280.846	1.20020
$235$	90.6488	0.385740
$236$	−188.816	−0.800069
$237$	− 73.8716i	− 0.311695i
$238$	− 35.5858i	− 0.149520i
$239$	− 66.0238i	− 0.276250i	−0.990415	−	0.138125i	$-0.955892\pi$
$239$	− 66.0238i	− 0.276250i	0.990415	−	0.138125i	$-0.0441076\pi$
$240$	−11.4588	−0.0477448
$241$	49.6438i	0.205991i	0.994682	+	0.102995i	$0.0328427\pi$
$241$	49.6438i	0.205991i	−0.994682	+	0.102995i	$0.967157\pi$
$242$	0	0
$243$	−201.582	−0.829556
$244$	186.616i	0.764819i
$245$	101.411	0.413923
$246$	58.5847	0.238149
$247$	−385.080	−1.55903
$248$	− 75.0428i	− 0.302592i
$249$	27.3516i	0.109846i
$250$	173.884i	0.695536i
$251$	−66.0050	−0.262968	−0.131484	−	0.991318i	$-0.541974\pi$
$251$	−66.0050	−0.262968	−0.131484	+	0.991318i	$0.541974\pi$
$252$	− 63.0036i	− 0.250014i
$253$	0	0
$254$	188.685	0.742856
$255$	− 18.5050i	− 0.0725688i
$256$	16.0000	0.0625000
$257$	272.459	1.06015	0.530075	−	0.847951i	$-0.322164\pi$
$257$	272.459	1.06015	0.530075	+	0.847951i	$0.322164\pi$
$258$	−75.7056	−0.293433
$259$	16.2668i	0.0628061i
$260$	147.242i	0.566316i
$261$	121.222i	0.464451i
$262$	59.9273	0.228730
$263$	66.3839i	0.252410i	0.992004	+	0.126205i	$0.0402797\pi$
$263$	66.3839i	0.252410i	−0.992004	+	0.126205i	$0.959720\pi$
$264$	0	0
$265$	156.086	0.589004
$266$	86.3870i	0.324763i
$267$	−65.2343	−0.244323
$268$	−108.154	−0.403560
$269$	182.728	0.679285	0.339643	−	0.940555i	$-0.389694\pi$
$269$	182.728	0.679285	0.339643	+	0.940555i	$0.389694\pi$
$270$	− 69.2241i	− 0.256386i
$271$	− 400.000i	− 1.47601i	−0.674794	−	0.738006i	$-0.735767\pi$
$271$	− 400.000i	− 1.47601i	0.674794	−	0.738006i	$-0.264233\pi$
$272$	25.8388i	0.0949956i
$273$	91.4028	0.334809
$274$	− 192.361i	− 0.702047i
$275$	0	0
$276$	−58.0039	−0.210159
$277$	46.6210i	0.168307i	0.996453	+	0.0841534i	$0.0268186\pi$
$277$	46.6210i	0.168307i	−0.996453	+	0.0841534i	$0.973181\pi$
$278$	−143.159	−0.514959
$279$	214.560	0.769034
$280$	33.0316	0.117970
$281$	− 503.658i	− 1.79238i	−0.443674	−	0.896188i	$-0.646325\pi$
$281$	− 503.658i	− 1.79238i	0.443674	−	0.896188i	$-0.353675\pi$
$282$	40.8587i	0.144889i
$283$	56.2125i	0.198631i	0.995056	+	0.0993153i	$0.0316652\pi$
$283$	56.2125i	0.198631i	−0.995056	+	0.0993153i	$0.968335\pi$
$284$	187.787	0.661221
$285$	44.9222i	0.157622i
$286$	0	0
$287$	−168.879	−0.588429
$288$	45.7468i	0.158843i
$289$	247.272	0.855613
$290$	−63.5542	−0.219153
$291$	−34.4103	−0.118248
$292$	49.0710i	0.168051i
$293$	186.647i	0.637021i	0.947919	+	0.318510i	$0.103183\pi$
$293$	186.647i	0.637021i	−0.947919	+	0.318510i	$0.896817\pi$
$294$	45.7097i	0.155475i
$295$	283.038	0.959450
$296$	− 11.8113i	− 0.0399030i
$297$	0	0
$298$	73.0774	0.245226
$299$	745.336i	2.49276i
$300$	−30.5995	−0.101998
$301$	218.233	0.725027
$302$	118.503	0.392395
$303$	2.36835i	0.00781634i
$304$	− 62.7255i	− 0.206334i
$305$	− 279.739i	− 0.917177i
$306$	−73.8777	−0.241430
$307$	495.856i	1.61517i	0.589755	+	0.807583i	$0.299224\pi$
$307$	495.856i	1.61517i	−0.589755	+	0.807583i	$0.700776\pi$
$308$	0	0
$309$	29.0048	0.0938666
$310$	112.490i	0.362871i
$311$	372.119	1.19653	0.598263	−	0.801300i	$-0.295858\pi$
$311$	372.119	1.19653	0.598263	+	0.801300i	$0.295858\pi$
$312$	−66.3674	−0.212716
$313$	−245.314	−0.783751	−0.391875	−	0.920018i	$-0.628174\pi$
$313$	−245.314	−0.783751	−0.391875	+	0.920018i	$0.628174\pi$
$314$	328.813i	1.04717i
$315$	94.4431i	0.299819i
$316$	− 154.620i	− 0.489303i
$317$	−60.1541	−0.189760	−0.0948802	−	0.995489i	$-0.530247\pi$
$317$	−60.1541	−0.189760	−0.0948802	+	0.995489i	$0.530247\pi$
$318$	70.3537i	0.221238i
$319$	0	0
$320$	−23.9842	−0.0749506
$321$	− 73.1710i	− 0.227947i
$322$	167.205	0.519271
$323$	101.297	0.313613
$324$	−114.364	−0.352974
$325$	393.196i	1.20983i
$326$	454.857i	1.39527i
$327$	11.5730i	0.0353914i
$328$	122.623	0.373850
$329$	− 117.782i	− 0.357999i
$330$	0	0
$331$	−89.7487	−0.271144	−0.135572	−	0.990767i	$-0.543287\pi$
$331$	−89.7487	−0.271144	−0.135572	+	0.990767i	$0.543287\pi$
$332$	57.2492i	0.172437i
$333$	33.7705	0.101413
$334$	−176.896	−0.529630
$335$	162.124	0.483953
$336$	14.8885i	0.0443112i
$337$	− 123.139i	− 0.365396i	−0.983169	−	0.182698i	$-0.941517\pi$
$337$	− 123.139i	− 0.365396i	0.983169	−	0.182698i	$-0.0584831\pi$
$338$	613.802i	1.81598i
$339$	−90.6556	−0.267421
$340$	− 38.7327i	− 0.113920i
$341$	0	0
$342$	179.343	0.524395
$343$	− 322.639i	− 0.940638i
$344$	−158.459	−0.460635
$345$	86.9486	0.252025
$346$	−179.215	−0.517964
$347$	− 318.041i	− 0.916544i	−0.888812	−	0.458272i	$-0.848469\pi$
$347$	− 318.041i	− 0.916544i	0.888812	−	0.458272i	$-0.151531\pi$
$348$	− 28.6462i	− 0.0823167i
$349$	− 265.834i	− 0.761702i	−0.924637	−	0.380851i	$-0.875631\pi$
$349$	− 265.834i	− 0.761702i	0.924637	−	0.380851i	$-0.124369\pi$
$350$	88.2077	0.252022
$351$	− 400.936i	− 1.14227i
$352$	0	0
$353$	−249.501	−0.706802	−0.353401	−	0.935472i	$-0.614975\pi$
$353$	−249.501	−0.706802	−0.353401	+	0.935472i	$0.614975\pi$
$354$	127.575i	0.360383i
$355$	−281.494	−0.792942
$356$	−136.541	−0.383542
$357$	−24.0439	−0.0673499
$358$	− 280.497i	− 0.783510i
$359$	− 562.573i	− 1.56705i	−0.621358	−	0.783527i	$-0.713419\pi$
$359$	− 562.573i	− 1.56705i	0.621358	−	0.783527i	$-0.286581\pi$
$360$	− 68.5750i	− 0.190486i
$361$	115.095	0.318822
$362$	− 151.110i	− 0.417432i
$363$	0	0
$364$	191.314	0.525588
$365$	− 73.5580i	− 0.201529i
$366$	126.089	0.344504
$367$	28.0962	0.0765564	0.0382782	−	0.999267i	$-0.487813\pi$
$367$	28.0962	0.0765564	0.0382782	+	0.999267i	$0.487813\pi$
$368$	−121.407	−0.329911
$369$	350.600i	0.950136i
$370$	17.7052i	0.0478520i
$371$	− 202.805i	− 0.546645i
$372$	−50.7033	−0.136299
$373$	157.016i	0.420953i	0.977599	+	0.210477i	$0.0675016\pi$
$373$	157.016i	0.420953i	−0.977599	+	0.210477i	$0.932498\pi$
$374$	0	0
$375$	117.486	0.313297
$376$	85.5209i	0.227449i
$377$	−368.097	−0.976384
$378$	−89.9441	−0.237947
$379$	622.335	1.64204	0.821022	−	0.570897i	$-0.193404\pi$
$379$	622.335	1.64204	0.821022	+	0.570897i	$0.193404\pi$
$380$	94.0261i	0.247437i
$381$	− 127.487i	− 0.334612i
$382$	− 48.6935i	− 0.127470i
$383$	−344.480	−0.899427	−0.449713	−	0.893173i	$-0.648474\pi$
$383$	−344.480	−0.899427	−0.449713	+	0.893173i	$0.648474\pi$
$384$	− 10.8105i	− 0.0281525i
$385$	0	0
$386$	−302.627	−0.784009
$387$	− 453.061i	− 1.17070i
$388$	−72.0237	−0.185628
$389$	−290.021	−0.745555	−0.372777	−	0.927921i	$-0.621595\pi$
$389$	−290.021	−0.745555	−0.372777	+	0.927921i	$0.621595\pi$
$390$	99.4855	0.255091
$391$	− 196.064i	− 0.501442i
$392$	95.6744i	0.244067i
$393$	− 40.4904i	− 0.103029i
$394$	−383.794	−0.974097
$395$	231.777i	0.586777i
$396$	0	0
$397$	−238.817	−0.601554	−0.300777	−	0.953694i	$-0.597246\pi$
$397$	−238.817	−0.601554	−0.300777	+	0.953694i	$0.597246\pi$
$398$	− 353.455i	− 0.888077i
$399$	58.3682	0.146286
$400$	−64.0475	−0.160119
$401$	−135.417	−0.337699	−0.168850	−	0.985642i	$-0.554005\pi$
$401$	−135.417	−0.337699	−0.168850	+	0.985642i	$0.554005\pi$
$402$	73.0753i	0.181779i
$403$	651.525i	1.61669i
$404$	4.95717i	0.0122702i
$405$	171.432	0.423290
$406$	82.5771i	0.203392i
$407$	0	0
$408$	17.4582	0.0427898
$409$	− 304.319i	− 0.744055i	−0.928222	−	0.372028i	$-0.878663\pi$
$409$	− 304.319i	− 0.744055i	0.928222	−	0.372028i	$-0.121337\pi$
$410$	−183.813	−0.448324
$411$	−129.970	−0.316229
$412$	60.7096	0.147353
$413$	− 367.755i	− 0.890449i
$414$	− 347.125i	− 0.838466i
$415$	− 85.8172i	− 0.206788i
$416$	−138.913	−0.333925
$417$	96.7264i	0.231958i
$418$	0	0
$419$	594.131	1.41797	0.708987	−	0.705221i	$-0.249152\pi$
$419$	594.131	1.41797	0.708987	+	0.705221i	$0.249152\pi$
$420$	− 22.3181i	− 0.0531383i
$421$	−255.156	−0.606070	−0.303035	−	0.952979i	$-0.598000\pi$
$421$	−255.156	−0.606070	−0.303035	+	0.952979i	$0.598000\pi$
$422$	12.6015	0.0298613
$423$	−244.519	−0.578060
$424$	147.256i	0.347303i
$425$	− 103.432i	− 0.243369i
$426$	− 126.880i	− 0.297840i
$427$	−363.470	−0.851217
$428$	− 153.153i	− 0.357835i
$429$	0	0
$430$	237.531	0.552398
$431$	− 661.532i	− 1.53488i	−0.641122	−	0.767439i	$-0.721531\pi$
$431$	− 661.532i	− 1.53488i	0.641122	−	0.767439i	$-0.278469\pi$
$432$	65.3082	0.151176
$433$	110.026	0.254101	0.127051	−	0.991896i	$-0.459449\pi$
$433$	110.026	0.254101	0.127051	+	0.991896i	$0.459449\pi$
$434$	146.160	0.336774
$435$	42.9410i	0.0987150i
$436$	24.2233i	0.0555580i
$437$	475.958i	1.08915i
$438$	33.1553	0.0756970
$439$	− 549.236i	− 1.25111i	−0.780181	−	0.625553i	$-0.784873\pi$
$439$	− 549.236i	− 1.25111i	0.780181	−	0.625553i	$-0.215127\pi$
$440$	0	0
$441$	−273.550	−0.620295
$442$	− 224.334i	− 0.507543i
$443$	−35.1773	−0.0794069	−0.0397035	−	0.999212i	$-0.512641\pi$
$443$	−35.1773	−0.0794069	−0.0397035	+	0.999212i	$0.512641\pi$
$444$	−7.98040	−0.0179739
$445$	204.677	0.459947
$446$	292.814i	0.656535i
$447$	− 49.3754i	− 0.110460i
$448$	31.1630i	0.0695604i
$449$	−564.036	−1.25621	−0.628103	−	0.778130i	$-0.716168\pi$
$449$	−564.036	−1.25621	−0.628103	+	0.778130i	$0.716168\pi$
$450$	− 183.123i	− 0.406940i
$451$	0	0
$452$	−189.750	−0.419801
$453$	− 80.0678i	− 0.176750i
$454$	137.291	0.302404
$455$	−286.782	−0.630290
$456$	−42.3810	−0.0929408
$457$	680.352i	1.48873i	0.667771	+	0.744367i	$0.267249\pi$
$457$	680.352i	1.48873i	−0.667771	+	0.744367i	$0.732751\pi$
$458$	− 81.7780i	− 0.178555i
$459$	105.468i	0.229778i
$460$	181.991	0.395632
$461$	202.839i	0.439998i	0.975500	+	0.219999i	$0.0706053\pi$
$461$	202.839i	0.439998i	−0.975500	+	0.219999i	$0.929395\pi$
$462$	0	0
$463$	−746.270	−1.61182	−0.805908	−	0.592041i	$-0.798322\pi$
$463$	−746.270	−1.61182	−0.805908	+	0.592041i	$0.798322\pi$
$464$	− 59.9591i	− 0.129222i
$465$	76.0048	0.163451
$466$	99.3315	0.213158
$467$	372.512	0.797669	0.398835	−	0.917023i	$-0.369415\pi$
$467$	372.512	0.797669	0.398835	+	0.917023i	$0.369415\pi$
$468$	− 397.176i	− 0.848667i
$469$	− 210.651i	− 0.449149i
$470$	− 128.197i	− 0.272759i
$471$	222.165	0.471688
$472$	267.027i	0.565734i
$473$	0	0
$474$	−104.470	−0.220401
$475$	251.088i	0.528606i
$476$	−50.3260	−0.105727
$477$	−421.032	−0.882667
$478$	−93.3718	−0.195338
$479$	573.125i	1.19650i	0.801309	+	0.598251i	$0.204138\pi$
$479$	573.125i	1.19650i	−0.801309	+	0.598251i	$0.795862\pi$
$480$	16.2051i	0.0337607i
$481$	102.546i	0.213194i
$482$	70.2069	0.145657
$483$	− 112.974i	− 0.233900i
$484$	0	0
$485$	107.964	0.222607
$486$	285.080i	0.586585i
$487$	42.3202	0.0868999	0.0434499	−	0.999056i	$-0.486165\pi$
$487$	42.3202	0.0868999	0.0434499	+	0.999056i	$0.486165\pi$
$488$	263.915	0.540809
$489$	307.328	0.628483
$490$	− 143.417i	− 0.292688i
$491$	612.678i	1.24782i	0.781497	+	0.623909i	$0.214456\pi$
$491$	612.678i	1.24782i	−0.781497	+	0.623909i	$0.785544\pi$
$492$	− 82.8512i	− 0.168397i
$493$	96.8294	0.196409
$494$	544.585i	1.10240i
$495$	0	0
$496$	−106.126	−0.213965
$497$	365.750i	0.735916i
$498$	38.6809	0.0776726
$499$	−463.167	−0.928189	−0.464095	−	0.885786i	$-0.653620\pi$
$499$	−463.167	−0.928189	−0.464095	+	0.885786i	$0.653620\pi$
$500$	245.909	0.491818
$501$	119.522i	0.238566i
$502$	93.3451i	0.185946i
$503$	− 754.755i	− 1.50051i	−0.661151	−	0.750253i	$-0.729932\pi$
$503$	− 754.755i	− 1.50051i	0.661151	−	0.750253i	$-0.270068\pi$
$504$	−89.1006	−0.176787
$505$	− 7.43085i	− 0.0147145i
$506$	0	0
$507$	414.721	0.817990
$508$	− 266.842i	− 0.525279i
$509$	700.304	1.37584	0.687921	−	0.725785i	$-0.258523\pi$
$509$	700.304	1.37584	0.687921	+	0.725785i	$0.258523\pi$
$510$	−26.1701	−0.0513139
$511$	−95.5751	−0.187035
$512$	− 22.6274i	− 0.0441942i
$513$	− 256.030i	− 0.499085i
$514$	− 385.315i	− 0.749640i
$515$	−91.0043	−0.176707
$516$	107.064i	0.207488i
$517$	0	0
$518$	23.0047	0.0444106
$519$	121.089i	0.233311i
$520$	208.232	0.400446
$521$	235.041	0.451135	0.225567	−	0.974228i	$-0.427576\pi$
$521$	235.041	0.451135	0.225567	+	0.974228i	$0.427576\pi$
$522$	171.433	0.328417
$523$	− 648.324i	− 1.23963i	−0.784750	−	0.619813i	$-0.787209\pi$
$523$	− 648.324i	− 1.23963i	0.784750	−	0.619813i	$-0.212791\pi$
$524$	− 84.7500i	− 0.161737i
$525$	− 59.5984i	− 0.113521i
$526$	93.8810	0.178481
$527$	− 171.386i	− 0.325211i
$528$	0	0
$529$	392.234	0.741463
$530$	− 220.739i	− 0.416489i
$531$	−763.476	−1.43781
$532$	122.170	0.229642
$533$	−1064.62	−1.99741
$534$	92.2552i	0.172763i
$535$	229.578i	0.429119i
$536$	152.953i	0.285360i
$537$	−189.520	−0.352924
$538$	− 258.416i	− 0.480327i
$539$	0	0
$540$	−97.8977	−0.181292
$541$	330.782i	0.611427i	0.952124	+	0.305714i	$0.0988950\pi$
$541$	330.782i	0.611427i	−0.952124	+	0.305714i	$0.901105\pi$
$542$	−565.685	−1.04370
$543$	−102.099	−0.188028
$544$	36.5416	0.0671721
$545$	− 36.3110i	− 0.0666256i
$546$	− 129.263i	− 0.236746i
$547$	605.112i	1.10624i	0.833102	+	0.553119i	$0.186562\pi$
$547$	605.112i	1.10624i	−0.833102	+	0.553119i	$0.813438\pi$
$548$	−272.039	−0.496422
$549$	754.578i	1.37446i
$550$	0	0
$551$	−235.060	−0.426606
$552$	82.0300i	0.148605i
$553$	301.151	0.544578
$554$	65.9320	0.119011
$555$	11.9627	0.0215544
$556$	202.457i	0.364131i
$557$	− 254.770i	− 0.457397i	−0.973497	−	0.228698i	$-0.926553\pi$
$557$	− 254.770i	− 0.457397i	0.973497	−	0.228698i	$-0.0734470\pi$
$558$	− 303.434i	− 0.543789i
$559$	1375.74	2.46108
$560$	− 46.7137i	− 0.0834174i
$561$	0	0
$562$	−712.280	−1.26740
$563$	1011.87i	1.79728i	0.438682	+	0.898642i	$0.355446\pi$
$563$	1011.87i	1.79728i	−0.438682	+	0.898642i	$0.644554\pi$
$564$	57.7830	0.102452
$565$	284.437	0.503429
$566$	79.4964	0.140453
$567$	− 222.745i	− 0.392848i
$568$	− 265.570i	− 0.467554i
$569$	440.476i	0.774124i	0.922054	+	0.387062i	$0.126510\pi$
$569$	440.476i	0.774124i	−0.922054	+	0.387062i	$0.873490\pi$
$570$	63.5296	0.111455
$571$	− 577.275i	− 1.01099i	−0.862830	−	0.505495i	$-0.831310\pi$
$571$	− 577.275i	− 1.01099i	0.862830	−	0.505495i	$-0.168690\pi$
$572$	0	0
$573$	−32.9002	−0.0574175
$574$	238.831i	0.416082i
$575$	485.989	0.845199
$576$	64.6958	0.112319
$577$	−823.852	−1.42782	−0.713910	−	0.700238i	$-0.753077\pi$
$577$	−823.852	−1.42782	−0.713910	+	0.700238i	$0.753077\pi$
$578$	− 349.696i	− 0.605010i
$579$	204.473i	0.353148i
$580$	89.8793i	0.154964i
$581$	−111.504	−0.191917
$582$	48.6635i	0.0836142i
$583$	0	0
$584$	69.3969	0.118830
$585$	595.371i	1.01773i
$586$	263.959	0.450442
$587$	−227.466	−0.387506	−0.193753	−	0.981050i	$-0.562066\pi$
$587$	−227.466	−0.387506	−0.193753	+	0.981050i	$0.562066\pi$
$588$	64.6433	0.109938
$589$	416.052i	0.706370i
$590$	− 400.276i	− 0.678433i
$591$	259.314i	0.438772i
$592$	−16.7037	−0.0282157
$593$	154.112i	0.259885i	0.991522	+	0.129942i	$0.0414792\pi$
$593$	154.112i	0.259885i	−0.991522	+	0.129942i	$0.958521\pi$
$594$	0	0
$595$	75.4392	0.126789
$596$	− 103.347i	− 0.173401i
$597$	−238.815	−0.400025
$598$	1054.06	1.76265
$599$	441.354	0.736819	0.368409	−	0.929664i	$-0.379903\pi$
$599$	441.354	0.736819	0.368409	+	0.929664i	$0.379903\pi$
$600$	43.2742i	0.0721237i
$601$	1129.59i	1.87952i	0.341841	+	0.939758i	$0.388950\pi$
$601$	1129.59i	1.87952i	−0.341841	+	0.939758i	$0.611050\pi$
$602$	− 308.628i	− 0.512671i
$603$	−437.320	−0.725240
$604$	− 167.589i	− 0.277465i
$605$	0	0
$606$	3.34935	0.00552699
$607$	− 266.019i	− 0.438253i	−0.975696	−	0.219126i	$-0.929679\pi$
$607$	− 266.019i	− 0.438253i	0.975696	−	0.219126i	$-0.0703207\pi$
$608$	−88.7072	−0.145900
$609$	55.7940	0.0916157
$610$	−395.611	−0.648542
$611$	− 742.497i	− 1.21522i
$612$	104.479i	0.170717i
$613$	217.536i	0.354871i	0.984132	+	0.177436i	$0.0567801\pi$
$613$	217.536i	0.354871i	−0.984132	+	0.177436i	$0.943220\pi$
$614$	701.246	1.14209
$615$	124.195i	0.201943i
$616$	0	0
$617$	828.147	1.34222	0.671108	−	0.741360i	$-0.265819\pi$
$617$	828.147	1.34222	0.671108	+	0.741360i	$0.265819\pi$
$618$	− 41.0190i	− 0.0663737i
$619$	357.924	0.578229	0.289114	−	0.957295i	$-0.406639\pi$
$619$	357.924	0.578229	0.289114	+	0.957295i	$0.406639\pi$
$620$	159.085	0.256588
$621$	−495.556	−0.797997
$622$	− 526.256i	− 0.846071i
$623$	− 265.940i	− 0.426869i
$624$	93.8577i	0.150413i
$625$	31.6764	0.0506822
$626$	346.926i	0.554195i
$627$	0	0
$628$	465.011	0.740464
$629$	− 26.9752i	− 0.0428858i
$630$	133.563	0.212004
$631$	625.439	0.991186	0.495593	−	0.868555i	$-0.334951\pi$
$631$	625.439	0.991186	0.495593	+	0.868555i	$0.334951\pi$
$632$	−218.665	−0.345990
$633$	− 8.51429i	− 0.0134507i
$634$	85.0707i	0.134181i
$635$	399.998i	0.629919i
$636$	99.4952	0.156439
$637$	− 830.650i	− 1.30400i
$638$	0	0
$639$	759.313	1.18828
$640$	33.9187i	0.0529980i
$641$	−595.524	−0.929054	−0.464527	−	0.885559i	$-0.653776\pi$
$641$	−595.524	−0.929054	−0.464527	+	0.885559i	$0.653776\pi$
$642$	−103.479	−0.161183
$643$	734.360	1.14208	0.571042	−	0.820921i	$-0.306539\pi$
$643$	734.360	1.14208	0.571042	+	0.820921i	$0.306539\pi$
$644$	− 236.464i	− 0.367180i
$645$	− 160.490i	− 0.248822i
$646$	− 143.256i	− 0.221758i
$647$	915.268	1.41463	0.707317	−	0.706896i	$-0.249905\pi$
$647$	915.268	1.41463	0.707317	+	0.706896i	$0.249905\pi$
$648$	161.735i	0.249590i
$649$	0	0
$650$	556.063	0.855482
$651$	− 98.7543i	− 0.151696i
$652$	643.265	0.986602
$653$	474.092	0.726021	0.363011	−	0.931785i	$-0.381749\pi$
$653$	474.092	0.726021	0.363011	+	0.931785i	$0.381749\pi$
$654$	16.3667	0.0250255
$655$	127.041i	0.193956i
$656$	− 173.415i	− 0.264352i
$657$	198.418i	0.302006i
$658$	−166.568	−0.253143
$659$	− 482.603i	− 0.732326i	−0.930551	−	0.366163i	$-0.880671\pi$
$659$	− 482.603i	− 0.732326i	0.930551	−	0.366163i	$-0.119329\pi$
$660$	0	0
$661$	45.0509	0.0681557	0.0340779	−	0.999419i	$-0.489151\pi$
$661$	45.0509	0.0681557	0.0340779	+	0.999419i	$0.489151\pi$
$662$	126.924i	0.191728i
$663$	−151.573	−0.228617
$664$	80.9626	0.121932
$665$	−183.134	−0.275389
$666$	− 47.7587i	− 0.0717098i
$667$	454.967i	0.682109i
$668$	250.169i	0.374505i
$669$	197.843	0.295729
$670$	− 229.278i	− 0.342207i
$671$	0	0
$672$	21.0556	0.0313327
$673$	306.030i	0.454725i	0.973810	+	0.227362i	$0.0730102\pi$
$673$	306.030i	0.454725i	−0.973810	+	0.227362i	$0.926990\pi$
$674$	−174.144	−0.258374
$675$	−261.426	−0.387298
$676$	868.047	1.28409
$677$	− 109.975i	− 0.162444i	−0.996696	−	0.0812222i	$-0.974118\pi$
$677$	− 109.975i	− 0.162444i	0.996696	−	0.0812222i	$-0.0258823\pi$
$678$	128.206i	0.189095i
$679$	− 140.280i	− 0.206598i
$680$	−54.7763	−0.0805533
$681$	− 92.7621i	− 0.136215i
$682$	0	0
$683$	−490.810	−0.718610	−0.359305	−	0.933220i	$-0.616986\pi$
$683$	−490.810	−0.718610	−0.359305	+	0.933220i	$0.616986\pi$
$684$	− 253.629i	− 0.370803i
$685$	407.790	0.595313
$686$	−456.280	−0.665131
$687$	−55.2541	−0.0804280
$688$	224.094i	0.325718i
$689$	− 1278.49i	− 1.85557i
$690$	− 122.964i	− 0.178208i
$691$	−782.332	−1.13217	−0.566087	−	0.824346i	$-0.691543\pi$
$691$	−782.332	−1.13217	−0.566087	+	0.824346i	$0.691543\pi$
$692$	253.449i	0.366256i
$693$	0	0
$694$	−449.778	−0.648095
$695$	− 303.485i	− 0.436669i
$696$	−40.5119	−0.0582067
$697$	280.052	0.401796
$698$	−375.946	−0.538604
$699$	− 67.1142i	− 0.0960146i
$700$	− 124.745i	− 0.178207i
$701$	314.270i	0.448316i	0.974553	+	0.224158i	$0.0719632\pi$
$701$	314.270i	0.448316i	−0.974553	+	0.224158i	$0.928037\pi$
$702$	−567.009	−0.807705
$703$	65.4841i	0.0931495i
$704$	0	0
$705$	−86.6173	−0.122861
$706$	352.848i	0.499784i
$707$	−9.65502	−0.0136563
$708$	180.419	0.254829
$709$	66.2454	0.0934349	0.0467175	−	0.998908i	$-0.485124\pi$
$709$	66.2454	0.0934349	0.0467175	+	0.998908i	$0.485124\pi$
$710$	398.093i	0.560694i
$711$	− 625.203i	− 0.879329i
$712$	193.098i	0.271205i
$713$	805.283	1.12943
$714$	34.0032i	0.0476235i
$715$	0	0
$716$	−396.682	−0.554025
$717$	63.0875i	0.0879881i
$718$	−795.598	−1.10807
$719$	−1003.44	−1.39561	−0.697804	−	0.716289i	$-0.745839\pi$
$719$	−1003.44	−1.39561	−0.697804	+	0.716289i	$0.745839\pi$
$720$	−96.9797	−0.134694
$721$	118.243i	0.163999i
$722$	− 162.769i	− 0.225441i
$723$	− 47.4359i	− 0.0656099i
$724$	−213.702	−0.295169
$725$	240.014i	0.331054i
$726$	0	0
$727$	364.375	0.501204	0.250602	−	0.968090i	$-0.419371\pi$
$727$	364.375	0.501204	0.250602	+	0.968090i	$0.419371\pi$
$728$	− 270.559i	− 0.371647i
$729$	−322.019	−0.441727
$730$	−104.027	−0.142502
$731$	−361.895	−0.495069
$732$	− 178.316i	− 0.243601i
$733$	946.396i	1.29113i	0.763707	+	0.645563i	$0.223377\pi$
$733$	946.396i	1.29113i	−0.763707	+	0.645563i	$0.776623\pi$
$734$	− 39.7340i	− 0.0541335i
$735$	−96.9010	−0.131838
$736$	171.696i	0.233282i
$737$	0	0
$738$	495.823	0.671847
$739$	− 214.579i	− 0.290364i	−0.989405	−	0.145182i	$-0.953623\pi$
$739$	− 214.579i	− 0.290364i	0.989405	−	0.145182i	$-0.0463767\pi$
$740$	25.0390	0.0338365
$741$	367.954	0.496564
$742$	−286.810	−0.386536
$743$	43.6715i	0.0587773i	0.999568	+	0.0293886i	$0.00935604\pi$
$743$	43.6715i	0.0587773i	−0.999568	+	0.0293886i	$0.990644\pi$
$744$	71.7053i	0.0963781i
$745$	154.918i	0.207944i
$746$	222.054	0.297659
$747$	231.486i	0.309888i
$748$	0	0
$749$	298.295	0.398258
$750$	− 166.151i	− 0.221534i
$751$	511.229	0.680731	0.340366	−	0.940293i	$-0.389449\pi$
$751$	511.229	0.680731	0.340366	+	0.940293i	$0.389449\pi$
$752$	120.945	0.160831
$753$	63.0695	0.0837576
$754$	520.567i	0.690408i
$755$	251.218i	0.332739i
$756$	127.200i	0.168254i
$757$	−866.438	−1.14457	−0.572284	−	0.820055i	$-0.693943\pi$
$757$	−866.438	−1.14457	−0.572284	+	0.820055i	$0.693943\pi$
$758$	− 880.114i	− 1.16110i
$759$	0	0
$760$	132.973	0.174965
$761$	− 1310.39i	− 1.72193i	−0.508663	−	0.860966i	$-0.669860\pi$
$761$	− 1310.39i	− 1.72193i	0.508663	−	0.860966i	$-0.330140\pi$
$762$	−180.294	−0.236606
$763$	−47.1795	−0.0618341
$764$	−68.8630	−0.0901348
$765$	− 156.615i	− 0.204725i
$766$	487.169i	0.635991i
$767$	− 2318.34i	− 3.02260i
$768$	−15.2884	−0.0199068
$769$	1207.32i	1.56999i	0.619503	+	0.784994i	$0.287334\pi$
$769$	1207.32i	1.56999i	−0.619503	+	0.784994i	$0.712666\pi$
$770$	0	0
$771$	−260.341	−0.337667
$772$	427.980i	0.554378i
$773$	−802.013	−1.03753	−0.518767	−	0.854916i	$-0.673609\pi$
$773$	−802.013	−1.03753	−0.518767	+	0.854916i	$0.673609\pi$
$774$	−640.725	−0.827810
$775$	424.821	0.548156
$776$	101.857i	0.131259i
$777$	− 15.5433i	− 0.0200043i
$778$	410.151i	0.527187i
$779$	−679.845	−0.872715
$780$	− 140.694i	− 0.180377i
$781$	0	0
$782$	−277.276	−0.354573
$783$	− 244.739i	− 0.312565i
$784$	135.304	0.172582
$785$	−697.057	−0.887971
$786$	−57.2621	−0.0728526
$787$	− 333.915i	− 0.424288i	−0.977238	−	0.212144i	$-0.931955\pi$
$787$	− 333.915i	− 0.424288i	0.977238	−	0.212144i	$-0.0680446\pi$
$788$	542.767i	0.688791i
$789$	− 63.4315i	− 0.0803949i
$790$	327.782	0.414914
$791$	− 369.574i	− 0.467224i
$792$	0	0
$793$	−2291.32	−2.88943
$794$	337.738i	0.425363i
$795$	−149.144	−0.187603
$796$	−499.861	−0.627966
$797$	805.075	1.01013	0.505066	−	0.863081i	$-0.331468\pi$
$797$	805.075	1.01013	0.505066	+	0.863081i	$0.331468\pi$
$798$	− 82.5451i	− 0.103440i
$799$	195.317i	0.244452i
$800$	90.5768i	0.113221i
$801$	−552.102	−0.689266
$802$	191.509i	0.238790i
$803$	0	0
$804$	103.344	0.128537
$805$	354.462i	0.440325i
$806$	921.395	1.14317
$807$	−174.601	−0.216358
$808$	7.01049	0.00867635
$809$	1188.62i	1.46924i	0.678477	+	0.734622i	$0.262640\pi$
$809$	1188.62i	1.46924i	−0.678477	+	0.734622i	$0.737360\pi$
$810$	− 242.442i	− 0.299311i
$811$	− 881.303i	− 1.08669i	−0.839510	−	0.543344i	$-0.817158\pi$
$811$	− 881.303i	− 1.08669i	0.839510	−	0.543344i	$-0.182842\pi$
$812$	116.782	0.143820
$813$	382.210i	0.470123i
$814$	0	0
$815$	−964.261	−1.18314
$816$	− 24.6897i	− 0.0302569i
$817$	878.526	1.07531
$818$	−430.371	−0.526126
$819$	773.576	0.944537
$820$	259.951i	0.317013i
$821$	− 1032.25i	− 1.25730i	−0.777687	−	0.628652i	$-0.783607\pi$
$821$	− 1032.25i	− 1.25730i	0.777687	−	0.628652i	$-0.216393\pi$
$822$	183.806i	0.223608i
$823$	804.556	0.977589	0.488795	−	0.872399i	$-0.337437\pi$
$823$	804.556	0.977589	0.488795	+	0.872399i	$0.337437\pi$
$824$	− 85.8563i	− 0.104195i
$825$	0	0
$826$	−520.085	−0.629643
$827$	64.5511i	0.0780546i	0.999238	+	0.0390273i	$0.0124259\pi$
$827$	64.5511i	0.0780546i	−0.999238	+	0.0390273i	$0.987574\pi$
$828$	−490.909	−0.592885
$829$	610.188	0.736053	0.368026	−	0.929815i	$-0.380034\pi$
$829$	610.188	0.736053	0.368026	+	0.929815i	$0.380034\pi$
$830$	−121.364	−0.146221
$831$	− 44.5475i	− 0.0536072i
$832$	196.452i	0.236121i
$833$	218.506i	0.262312i
$834$	136.792	0.164019
$835$	− 375.006i	− 0.449109i
$836$	0	0
$837$	−433.183	−0.517542
$838$	− 840.228i	− 1.00266i
$839$	−631.399	−0.752562	−0.376281	−	0.926506i	$-0.622797\pi$
$839$	−631.399	−0.752562	−0.376281	+	0.926506i	$0.622797\pi$
$840$	−31.5626	−0.0375745
$841$	616.307	0.732826
$842$	360.845i	0.428557i
$843$	481.258i	0.570887i
$844$	− 17.8212i	− 0.0211151i
$845$	−1301.21	−1.53990
$846$	345.803i	0.408750i
$847$	0	0
$848$	208.252	0.245580
$849$	− 53.7125i	− 0.0632656i
$850$	−146.275	−0.172088
$851$	126.747	0.148939
$852$	−179.435	−0.210605
$853$	431.983i	0.506428i	0.967410	+	0.253214i	$0.0814876\pi$
$853$	431.983i	0.506428i	−0.967410	+	0.253214i	$0.918512\pi$
$854$	514.024i	0.601901i
$855$	380.193i	0.444671i
$856$	−216.591	−0.253027
$857$	137.536i	0.160485i	0.996775	+	0.0802425i	$0.0255695\pi$
$857$	137.536i	0.160485i	−0.996775	+	0.0802425i	$0.974431\pi$
$858$	0	0
$859$	−64.8174	−0.0754568	−0.0377284	−	0.999288i	$-0.512012\pi$
$859$	−64.8174	−0.0754568	−0.0377284	+	0.999288i	$0.512012\pi$
$860$	− 335.920i	− 0.390604i
$861$	161.368	0.187420
$862$	−935.548	−1.08532
$863$	−1010.74	−1.17119	−0.585595	−	0.810604i	$-0.699139\pi$
$863$	−1010.74	−1.17119	−0.585595	+	0.810604i	$0.699139\pi$
$864$	− 92.3597i	− 0.106898i
$865$	− 379.923i	− 0.439217i
$866$	− 155.600i	− 0.179677i
$867$	−236.275	−0.272520
$868$	− 206.701i	− 0.238135i
$869$	0	0
$870$	60.7278	0.0698020
$871$	− 1327.95i	− 1.52462i
$872$	34.2569	0.0392854
$873$	−291.227	−0.333593
$874$	673.106	0.770145
$875$	478.954i	0.547376i
$876$	− 46.8886i	− 0.0535258i
$877$	− 660.462i	− 0.753092i	−0.926398	−	0.376546i	$-0.877112\pi$
$877$	− 660.462i	− 0.753092i	0.926398	−	0.376546i	$-0.122888\pi$
$878$	−776.737	−0.884666
$879$	− 178.346i	− 0.202897i
$880$	0	0
$881$	1527.49	1.73382	0.866909	−	0.498466i	$-0.166103\pi$
$881$	1527.49	1.73382	0.866909	+	0.498466i	$0.166103\pi$
$882$	386.858i	0.438615i
$883$	−0.432414	−0.000489710 0	−0.000244855	−	1.00000i	$-0.500078\pi$
$883$	−0.432414	−0.000489710 0	−0.000244855		1.00000i	$0.500078\pi$
$884$	−317.256	−0.358887
$885$	−270.450	−0.305593
$886$	49.7482i	0.0561492i
$887$	− 987.674i	− 1.11350i	−0.830680	−	0.556750i	$-0.812048\pi$
$887$	− 987.674i	− 1.11350i	0.830680	−	0.556750i	$-0.187952\pi$
$888$	11.2860i	0.0127094i
$889$	519.725	0.584617
$890$	− 289.456i	− 0.325232i
$891$	0	0
$892$	414.102	0.464240
$893$	− 474.145i	− 0.530957i
$894$	−69.8274	−0.0781067
$895$	594.631	0.664392
$896$	44.0712	0.0491866
$897$	− 712.188i	− 0.793967i
$898$	797.668i	0.888271i
$899$	397.703i	0.442383i
$900$	−258.975	−0.287750
$901$	336.312i	0.373265i
$902$	0	0
$903$	−208.527	−0.230927
$904$	268.347i	0.296844i
$905$	320.342	0.353970
$906$	−113.233	−0.124981
$907$	718.281	0.791931	0.395965	−	0.918265i	$-0.370410\pi$
$907$	718.281	0.791931	0.395965	+	0.918265i	$0.370410\pi$
$908$	− 194.159i	− 0.213832i
$909$	20.0442i	0.0220509i
$910$	405.571i	0.445682i
$911$	340.456	0.373717	0.186858	−	0.982387i	$-0.440169\pi$
$911$	340.456	0.373717	0.186858	+	0.982387i	$0.440169\pi$
$912$	59.9358i	0.0657191i
$913$	0	0
$914$	962.162	1.05269
$915$	267.298i	0.292129i
$916$	−115.652	−0.126257
$917$	165.067	0.180007
$918$	149.154	0.162477
$919$	1004.57i	1.09312i	0.837421	+	0.546558i	$0.184062\pi$
$919$	1004.57i	1.09312i	−0.837421	+	0.546558i	$0.815938\pi$
$920$	− 257.374i	− 0.279754i
$921$	− 473.803i	− 0.514444i
$922$	286.858	0.311125
$923$	2305.70i	2.49804i
$924$	0	0
$925$	66.8642	0.0722856
$926$	1055.39i	1.13973i
$927$	245.478	0.264809
$928$	−84.7949	−0.0913738
$929$	−338.481	−0.364349	−0.182175	−	0.983266i	$-0.558314\pi$
$929$	−338.481	−0.364349	−0.182175	+	0.983266i	$0.558314\pi$
$930$	− 107.487i	− 0.115577i
$931$	− 530.438i	− 0.569751i
$932$	− 140.476i	− 0.150725i
$933$	−355.570	−0.381104
$934$	− 526.811i	− 0.564037i
$935$	0	0
$936$	−561.692	−0.600098
$937$	− 27.5060i	− 0.0293554i	−0.999892	−	0.0146777i	$-0.995328\pi$
$937$	− 27.5060i	− 0.0293554i	0.999892	−	0.0146777i	$-0.00467223\pi$
$938$	−297.905	−0.317596
$939$	234.404	0.249631
$940$	−181.298	−0.192870
$941$	− 663.333i	− 0.704924i	−0.935826	−	0.352462i	$-0.885345\pi$
$941$	− 663.333i	− 0.704924i	0.935826	−	0.352462i	$-0.114655\pi$
$942$	− 314.189i	− 0.333534i
$943$	1315.86i	1.39540i
$944$	377.633	0.400034
$945$	− 190.674i	− 0.201772i
$946$	0	0
$947$	−730.987	−0.771898	−0.385949	−	0.922520i	$-0.626126\pi$
$947$	−730.987	−0.771898	−0.385949	+	0.922520i	$0.626126\pi$
$948$	147.743i	0.155847i
$949$	−602.507	−0.634886
$950$	355.092	0.373781
$951$	57.4788	0.0604404
$952$	71.1717i	0.0747602i
$953$	381.328i	0.400134i	0.979782	+	0.200067i	$0.0641160\pi$
$953$	381.328i	0.400134i	−0.979782	+	0.200067i	$0.935884\pi$
$954$	595.429i	0.624140i
$955$	103.226	0.108091
$956$	132.048i	0.138125i
$957$	0	0
$958$	810.521	0.846055
$959$	− 529.848i	− 0.552500i
$960$	22.9175	0.0238724
$961$	−257.073	−0.267506
$962$	145.022	0.150751
$963$	− 619.273i	− 0.643066i
$964$	− 99.2875i	− 0.102995i
$965$	− 641.546i	− 0.664815i
$966$	−159.769	−0.165392
$967$	− 1791.77i	− 1.85292i	−0.376396	−	0.926459i	$-0.622837\pi$
$967$	− 1791.77i	− 1.85292i	0.376396	−	0.926459i	$-0.377163\pi$
$968$	0	0
$969$	−96.7919	−0.0998884
$970$	− 152.685i	− 0.157407i
$971$	−627.826	−0.646577	−0.323288	−	0.946300i	$-0.604788\pi$
$971$	−627.826	−0.646577	−0.323288	+	0.946300i	$0.604788\pi$
$972$	403.164	0.414778
$973$	−394.323	−0.405265
$974$	− 59.8499i	− 0.0614475i
$975$	− 375.709i	− 0.385343i
$976$	− 373.232i	− 0.382409i
$977$	472.303	0.483422	0.241711	−	0.970348i	$-0.422291\pi$
$977$	472.303	0.483422	0.241711	+	0.970348i	$0.422291\pi$
$978$	− 434.628i	− 0.444404i
$979$	0	0
$980$	−202.822	−0.206962
$981$	97.9465i	0.0998435i
$982$	866.458	0.882340
$983$	470.545	0.478682	0.239341	−	0.970936i	$-0.423069\pi$
$983$	470.545	0.478682	0.239341	+	0.970936i	$0.423069\pi$
$984$	−117.169	−0.119074
$985$	− 813.614i	− 0.826004i
$986$	− 136.937i	− 0.138882i
$987$	112.543i	0.114026i
$988$	770.160	0.779514
$989$	− 1700.42i	− 1.71933i
$990$	0	0
$991$	1639.02	1.65391	0.826955	−	0.562269i	$-0.190071\pi$
$991$	1639.02	1.65391	0.826955	+	0.562269i	$0.190071\pi$
$992$	150.086i	0.151296i
$993$	85.7573	0.0863618
$994$	517.249	0.520371
$995$	749.297	0.753062
$996$	− 54.7031i	− 0.0549228i
$997$	130.326i	0.130718i	0.997862	+	0.0653590i	$0.0208192\pi$
$997$	130.326i	0.130718i	−0.997862	+	0.0653590i	$0.979181\pi$
$998$	655.016i	0.656329i
$999$	−68.1804	−0.0682486

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	242.3.b.d.241.3		8
3.2	odd	2		2178.3.d.l.1693.5		8
11.2	odd	10		22.3.d.a.7.2	✓	8
11.3	even	5		242.3.d.e.233.1		8
11.4	even	5		242.3.d.d.215.2		8
11.5	even	5		22.3.d.a.19.2	yes	8
11.6	odd	10		242.3.d.c.239.1		8
11.7	odd	10		242.3.d.e.215.1		8
11.8	odd	10		242.3.d.d.233.2		8
11.9	even	5		242.3.d.c.161.1		8
11.10	odd	2	inner	242.3.b.d.241.7		8
33.2	even	10		198.3.j.a.73.1		8
33.5	odd	10		198.3.j.a.19.1		8
33.32	even	2		2178.3.d.l.1693.1		8
44.27	odd	10		176.3.n.b.129.1		8
44.35	even	10		176.3.n.b.161.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
22.3.d.a.7.2	✓	8	11.2	odd	10
22.3.d.a.19.2	yes	8	11.5	even	5
176.3.n.b.129.1		8	44.27	odd	10
176.3.n.b.161.1		8	44.35	even	10
198.3.j.a.19.1		8	33.5	odd	10
198.3.j.a.73.1		8	33.2	even	10
242.3.b.d.241.3		8	1.1	even	1	trivial
242.3.b.d.241.7		8	11.10	odd	2	inner
242.3.d.c.161.1		8	11.9	even	5
242.3.d.c.239.1		8	11.6	odd	10
242.3.d.d.215.2		8	11.4	even	5
242.3.d.d.233.2		8	11.8	odd	10
242.3.d.e.215.1		8	11.7	odd	10
242.3.d.e.233.1		8	11.3	even	5
2178.3.d.l.1693.1		8	33.32	even	2
2178.3.d.l.1693.5		8	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 242 = 2 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	242.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.41421i q^{2} -0.955526 q^{3} -2.00000 q^{4} +2.99802 q^{5} +1.35132i q^{6} -3.89538i q^{7} +2.82843i q^{8} -8.08697 q^{9} -4.23984i q^{10} +1.91105 q^{12} -24.5565i q^{13} -5.50890 q^{14} -2.86469 q^{15} +4.00000 q^{16} +6.45970i q^{17} +11.4367i q^{18} -15.6814i q^{19} -5.99604 q^{20} +3.72214i q^{21} -30.3518 q^{23} -2.70264i q^{24} -16.0119 q^{25} -34.7282 q^{26} +16.3270 q^{27} +7.79076i q^{28} -14.9898i q^{29} +4.05128i q^{30} -26.5316 q^{31} -5.65685i q^{32} +9.13540 q^{34} -11.6784i q^{35} +16.1739 q^{36} -4.17592 q^{37} -22.1768 q^{38} +23.4644i q^{39} +8.47969i q^{40} -43.3537i q^{41} +5.26390 q^{42} +56.0236i q^{43} -24.2449 q^{45} +42.9240i q^{46} +30.2362 q^{47} -3.82210 q^{48} +33.8260 q^{49} +22.6442i q^{50} -6.17242i q^{51} +49.1131i q^{52} +52.0630 q^{53} -23.0899i q^{54} +11.0178 q^{56} +14.9840i q^{57} -21.1987 q^{58} +94.4081 q^{59} +5.72938 q^{60} -93.3079i q^{61} +37.5214i q^{62} +31.5018i q^{63} -8.00000 q^{64} -73.6210i q^{65} +54.0771 q^{67} -12.9194i q^{68} +29.0020 q^{69} -16.5158 q^{70} -93.8933 q^{71} -22.8734i q^{72} -24.5355i q^{73} +5.90564i q^{74} +15.2998 q^{75} +31.3627i q^{76} +33.1837 q^{78} +77.3099i q^{79} +11.9921 q^{80} +57.1818 q^{81} -61.3114 q^{82} -28.6246i q^{83} -7.44427i q^{84} +19.3663i q^{85} +79.2293 q^{86} +14.3231i q^{87} +68.2705 q^{89} +34.2875i q^{90} -95.6570 q^{91} +60.7037 q^{92} +25.3517 q^{93} -42.7605i q^{94} -47.0131i q^{95} +5.40527i q^{96} +36.0119 q^{97} -47.8372i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 12 q^{3} - 16 q^{4} - 28 q^{5} - 4 q^{9} + 24 q^{12} - 24 q^{14} + 32 q^{15} + 32 q^{16} + 56 q^{20} - 104 q^{23} + 68 q^{25} - 16 q^{26} + 24 q^{27} - 124 q^{31} + 112 q^{34} + 8 q^{36} + 36 q^{37}+ \cdots + 92 q^{97}+O(q^{100}) \) 8 * q - 12 * q^3 - 16 * q^4 - 28 * q^5 - 4 * q^9 + 24 * q^12 - 24 * q^14 + 32 * q^15 + 32 * q^16 + 56 * q^20 - 104 * q^23 + 68 * q^25 - 16 * q^26 + 24 * q^27 - 124 * q^31 + 112 * q^34 + 8 * q^36 + 36 * q^37 + 8 * q^38 + 96 * q^42 - 136 * q^45 + 76 * q^47 - 48 * q^48 + 76 * q^49 - 44 * q^53 + 48 * q^56 - 16 * q^58 + 100 * q^59 - 64 * q^60 - 64 * q^64 + 112 * q^67 - 224 * q^69 - 136 * q^70 - 276 * q^71 + 48 * q^75 + 104 * q^78 - 112 * q^80 - 280 * q^81 + 64 * q^82 + 128 * q^86 + 24 * q^89 + 176 * q^91 + 208 * q^92 + 96 * q^93 + 92 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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