LMFDB - Embedded newform 309.2.a.d.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [309,2,Mod(1,309)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(309, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("309.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 309 = 3 \cdot 103 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	309.a (trivial)

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:	yes
Analytic conductor:	$2.46737742246$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{7} - 13x^{6} + 11x^{5} + 52x^{4} - 35x^{3} - 59x^{2} + 27x + 1 $ x^8 - x^7 - 13x^6 + 11x^5 + 52x^4 - 35x^3 - 59x^2 + 27x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.49857$ of defining polynomial
Character	$\chi$	$=$	309.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.49857 q^{2} +1.00000 q^{3} +4.24286 q^{4} +0.808843 q^{5} -2.49857 q^{6} +4.57320 q^{7} -5.60396 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-2.49857 q^{2} +1.00000 q^{3} +4.24286 q^{4} +0.808843 q^{5} -2.49857 q^{6} +4.57320 q^{7} -5.60396 q^{8} +1.00000 q^{9} -2.02095 q^{10} +0.362508 q^{11} +4.24286 q^{12} -1.82643 q^{13} -11.4265 q^{14} +0.808843 q^{15} +5.51616 q^{16} +2.93571 q^{17} -2.49857 q^{18} -2.76779 q^{19} +3.43181 q^{20} +4.57320 q^{21} -0.905752 q^{22} -5.40105 q^{23} -5.60396 q^{24} -4.34577 q^{25} +4.56347 q^{26} +1.00000 q^{27} +19.4035 q^{28} +5.73411 q^{29} -2.02095 q^{30} +7.71995 q^{31} -2.57462 q^{32} +0.362508 q^{33} -7.33508 q^{34} +3.69900 q^{35} +4.24286 q^{36} +4.62297 q^{37} +6.91553 q^{38} -1.82643 q^{39} -4.53272 q^{40} +2.08044 q^{41} -11.4265 q^{42} -1.28921 q^{43} +1.53807 q^{44} +0.808843 q^{45} +13.4949 q^{46} -8.73420 q^{47} +5.51616 q^{48} +13.9142 q^{49} +10.8582 q^{50} +2.93571 q^{51} -7.74930 q^{52} +2.23057 q^{53} -2.49857 q^{54} +0.293212 q^{55} -25.6280 q^{56} -2.76779 q^{57} -14.3271 q^{58} +13.3912 q^{59} +3.43181 q^{60} -11.8465 q^{61} -19.2889 q^{62} +4.57320 q^{63} -4.59946 q^{64} -1.47730 q^{65} -0.905752 q^{66} -7.83115 q^{67} +12.4558 q^{68} -5.40105 q^{69} -9.24222 q^{70} -12.8686 q^{71} -5.60396 q^{72} -12.1237 q^{73} -11.5508 q^{74} -4.34577 q^{75} -11.7434 q^{76} +1.65782 q^{77} +4.56347 q^{78} +14.8944 q^{79} +4.46171 q^{80} +1.00000 q^{81} -5.19813 q^{82} +0.977981 q^{83} +19.4035 q^{84} +2.37453 q^{85} +3.22119 q^{86} +5.73411 q^{87} -2.03148 q^{88} -9.74090 q^{89} -2.02095 q^{90} -8.35264 q^{91} -22.9159 q^{92} +7.71995 q^{93} +21.8230 q^{94} -2.23871 q^{95} -2.57462 q^{96} -17.4247 q^{97} -34.7656 q^{98} +0.362508 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - q^{2} + 8 q^{3} + 11 q^{4} - q^{5} - q^{6} + 6 q^{7} - 3 q^{8} + 8 q^{9}+O(q^{10}) $ 8 * q - q^2 + 8 * q^3 + 11 * q^4 - q^5 - q^6 + 6 * q^7 - 3 * q^8 + 8 * q^9	\( 8 q - q^{2} + 8 q^{3} + 11 q^{4} - q^{5} - q^{6} + 6 q^{7} - 3 q^{8} + 8 q^{9} + 3 q^{10} + 6 q^{11} + 11 q^{12} + 9 q^{13} - 6 q^{14} - q^{15} + 9 q^{16} - 4 q^{17} - q^{18} + 16 q^{19} - 23 q^{20} + 6 q^{21} - 4 q^{22} - 11 q^{23} - 3 q^{24} + 15 q^{25} - 14 q^{26} + 8 q^{27} - 5 q^{28} + 3 q^{30} + 17 q^{31} - 12 q^{32} + 6 q^{33} - 8 q^{34} + 4 q^{35} + 11 q^{36} - 6 q^{37} - 3 q^{38} + 9 q^{39} + 3 q^{40} + 12 q^{41} - 6 q^{42} + 9 q^{43} + 8 q^{44} - q^{45} - 30 q^{46} - 6 q^{47} + 9 q^{48} + 18 q^{49} - 36 q^{50} - 4 q^{51} + 23 q^{52} - 16 q^{53} - q^{54} - 10 q^{55} - 13 q^{56} + 16 q^{57} - 22 q^{58} + 11 q^{59} - 23 q^{60} + 5 q^{61} - 25 q^{62} + 6 q^{63} - 35 q^{64} - 41 q^{65} - 4 q^{66} + 5 q^{67} - 19 q^{68} - 11 q^{69} - 48 q^{70} - 10 q^{71} - 3 q^{72} + 14 q^{73} + 4 q^{74} + 15 q^{75} - 12 q^{76} - 40 q^{77} - 14 q^{78} + 14 q^{79} - 19 q^{80} + 8 q^{81} - 13 q^{82} - 23 q^{83} - 5 q^{84} - 4 q^{85} - 3 q^{86} - 30 q^{88} - 14 q^{89} + 3 q^{90} - 6 q^{91} - 21 q^{92} + 17 q^{93} + 22 q^{94} + 6 q^{95} - 12 q^{96} + 3 q^{97} - 18 q^{98} + 6 q^{99}+O(q^{100}) \) 8 * q - q^2 + 8 * q^3 + 11 * q^4 - q^5 - q^6 + 6 * q^7 - 3 * q^8 + 8 * q^9 + 3 * q^10 + 6 * q^11 + 11 * q^12 + 9 * q^13 - 6 * q^14 - q^15 + 9 * q^16 - 4 * q^17 - q^18 + 16 * q^19 - 23 * q^20 + 6 * q^21 - 4 * q^22 - 11 * q^23 - 3 * q^24 + 15 * q^25 - 14 * q^26 + 8 * q^27 - 5 * q^28 + 3 * q^30 + 17 * q^31 - 12 * q^32 + 6 * q^33 - 8 * q^34 + 4 * q^35 + 11 * q^36 - 6 * q^37 - 3 * q^38 + 9 * q^39 + 3 * q^40 + 12 * q^41 - 6 * q^42 + 9 * q^43 + 8 * q^44 - q^45 - 30 * q^46 - 6 * q^47 + 9 * q^48 + 18 * q^49 - 36 * q^50 - 4 * q^51 + 23 * q^52 - 16 * q^53 - q^54 - 10 * q^55 - 13 * q^56 + 16 * q^57 - 22 * q^58 + 11 * q^59 - 23 * q^60 + 5 * q^61 - 25 * q^62 + 6 * q^63 - 35 * q^64 - 41 * q^65 - 4 * q^66 + 5 * q^67 - 19 * q^68 - 11 * q^69 - 48 * q^70 - 10 * q^71 - 3 * q^72 + 14 * q^73 + 4 * q^74 + 15 * q^75 - 12 * q^76 - 40 * q^77 - 14 * q^78 + 14 * q^79 - 19 * q^80 + 8 * q^81 - 13 * q^82 - 23 * q^83 - 5 * q^84 - 4 * q^85 - 3 * q^86 - 30 * q^88 - 14 * q^89 + 3 * q^90 - 6 * q^91 - 21 * q^92 + 17 * q^93 + 22 * q^94 + 6 * q^95 - 12 * q^96 + 3 * q^97 - 18 * q^98 + 6 * q^99

Display 100 coefficients 10 coefficients

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 $

$2$	−2.49857	−1.76676	−0.883379	−	0.468660i	$-0.844737\pi$
$2$	−2.49857	−1.76676	−0.883379	+	0.468660i	$0.844737\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	4.24286	2.12143
$5$	0.808843	0.361725	0.180863	−	0.983508i	$-0.442111\pi$
$5$	0.808843	0.361725	0.180863	+	0.983508i	$0.442111\pi$
$6$	−2.49857	−1.02004
$7$	4.57320	1.72851	0.864254	−	0.503056i	$-0.167791\pi$
$7$	4.57320	1.72851	0.864254	+	0.503056i	$0.167791\pi$
$8$	−5.60396	−1.98130
$9$	1.00000	0.333333
$10$	−2.02095	−0.639081
$11$	0.362508	0.109300	0.0546501	−	0.998506i	$-0.482596\pi$
$11$	0.362508	0.109300	0.0546501	+	0.998506i	$0.482596\pi$
$12$	4.24286	1.22481
$13$	−1.82643	−0.506561	−0.253281	−	0.967393i	$-0.581510\pi$
$13$	−1.82643	−0.506561	−0.253281	+	0.967393i	$0.581510\pi$
$14$	−11.4265	−3.05385
$15$	0.808843	0.208842
$16$	5.51616	1.37904
$17$	2.93571	0.712014	0.356007	−	0.934483i	$-0.384138\pi$
$17$	2.93571	0.712014	0.356007	+	0.934483i	$0.384138\pi$
$18$	−2.49857	−0.588919
$19$	−2.76779	−0.634976	−0.317488	−	0.948262i	$-0.602839\pi$
$19$	−2.76779	−0.634976	−0.317488	+	0.948262i	$0.602839\pi$
$20$	3.43181	0.767376
$21$	4.57320	0.997954
$22$	−0.905752	−0.193107
$23$	−5.40105	−1.12620	−0.563098	−	0.826390i	$-0.690391\pi$
$23$	−5.40105	−1.12620	−0.563098	+	0.826390i	$0.690391\pi$
$24$	−5.60396	−1.14390
$25$	−4.34577	−0.869155
$26$	4.56347	0.894971
$27$	1.00000	0.192450
$28$	19.4035	3.66691
$29$	5.73411	1.06480	0.532398	−	0.846494i	$-0.321291\pi$
$29$	5.73411	1.06480	0.532398	+	0.846494i	$0.321291\pi$
$30$	−2.02095	−0.368974
$31$	7.71995	1.38654	0.693272	−	0.720676i	$-0.256168\pi$
$31$	7.71995	1.38654	0.693272	+	0.720676i	$0.256168\pi$
$32$	−2.57462	−0.455132
$33$	0.362508	0.0631045
$34$	−7.33508	−1.25796
$35$	3.69900	0.625245
$36$	4.24286	0.707144
$37$	4.62297	0.760011	0.380005	−	0.924984i	$-0.375922\pi$
$37$	4.62297	0.760011	0.380005	+	0.924984i	$0.375922\pi$
$38$	6.91553	1.12185
$39$	−1.82643	−0.292463
$40$	−4.53272	−0.716686
$41$	2.08044	0.324910	0.162455	−	0.986716i	$-0.448059\pi$
$41$	2.08044	0.324910	0.162455	+	0.986716i	$0.448059\pi$
$42$	−11.4265	−1.76314
$43$	−1.28921	−0.196603	−0.0983015	−	0.995157i	$-0.531341\pi$
$43$	−1.28921	−0.196603	−0.0983015	+	0.995157i	$0.531341\pi$
$44$	1.53807	0.231873
$45$	0.808843	0.120575
$46$	13.4949	1.98972
$47$	−8.73420	−1.27401	−0.637007	−	0.770858i	$-0.719828\pi$
$47$	−8.73420	−1.27401	−0.637007	+	0.770858i	$0.719828\pi$
$48$	5.51616	0.796189
$49$	13.9142	1.98774
$50$	10.8582	1.53559
$51$	2.93571	0.411082
$52$	−7.74930	−1.07463
$53$	2.23057	0.306393	0.153196	−	0.988196i	$-0.451043\pi$
$53$	2.23057	0.306393	0.153196	+	0.988196i	$0.451043\pi$
$54$	−2.49857	−0.340013
$55$	0.293212	0.0395367
$56$	−25.6280	−3.42469
$57$	−2.76779	−0.366603
$58$	−14.3271	−1.88124
$59$	13.3912	1.74338	0.871692	−	0.490055i	$-0.163023\pi$
$59$	13.3912	1.74338	0.871692	+	0.490055i	$0.163023\pi$
$60$	3.43181	0.443045
$61$	−11.8465	−1.51678	−0.758391	−	0.651800i	$-0.774014\pi$
$61$	−11.8465	−1.51678	−0.758391	+	0.651800i	$0.774014\pi$
$62$	−19.2889	−2.44969
$63$	4.57320	0.576169
$64$	−4.59946	−0.574932
$65$	−1.47730	−0.183236
$66$	−0.905752	−0.111490
$67$	−7.83115	−0.956727	−0.478364	−	0.878162i	$-0.658770\pi$
$67$	−7.83115	−0.956727	−0.478364	+	0.878162i	$0.658770\pi$
$68$	12.4558	1.51049
$69$	−5.40105	−0.650210
$70$	−9.24222	−1.10466
$71$	−12.8686	−1.52722	−0.763609	−	0.645679i	$-0.776574\pi$
$71$	−12.8686	−1.52722	−0.763609	+	0.645679i	$0.776574\pi$
$72$	−5.60396	−0.660433
$73$	−12.1237	−1.41898	−0.709488	−	0.704717i	$-0.751074\pi$
$73$	−12.1237	−1.41898	−0.709488	+	0.704717i	$0.751074\pi$
$74$	−11.5508	−1.34275
$75$	−4.34577	−0.501807
$76$	−11.7434	−1.34706
$77$	1.65782	0.188926
$78$	4.56347	0.516711
$79$	14.8944	1.67575	0.837874	−	0.545864i	$-0.183799\pi$
$79$	14.8944	1.67575	0.837874	+	0.545864i	$0.183799\pi$
$80$	4.46171	0.498834
$81$	1.00000	0.111111
$82$	−5.19813	−0.574037
$83$	0.977981	0.107347	0.0536737	−	0.998559i	$-0.482907\pi$
$83$	0.977981	0.107347	0.0536737	+	0.998559i	$0.482907\pi$
$84$	19.4035	2.11709
$85$	2.37453	0.257554
$86$	3.22119	0.347350
$87$	5.73411	0.614761
$88$	−2.03148	−0.216556
$89$	−9.74090	−1.03253	−0.516267	−	0.856428i	$-0.672679\pi$
$89$	−9.74090	−1.03253	−0.516267	+	0.856428i	$0.672679\pi$
$90$	−2.02095	−0.213027
$91$	−8.35264	−0.875595
$92$	−22.9159	−2.38915
$93$	7.71995	0.800522
$94$	21.8230	2.25087
$95$	−2.23871	−0.229687
$96$	−2.57462	−0.262771
$97$	−17.4247	−1.76921	−0.884603	−	0.466345i	$-0.845571\pi$
$97$	−17.4247	−1.76921	−0.884603	+	0.466345i	$0.845571\pi$
$98$	−34.7656	−3.51185
$99$	0.362508	0.0364334
$100$	−18.4385	−1.84385
$101$	14.8717	1.47979	0.739895	−	0.672722i	$-0.234875\pi$
$101$	14.8717	1.47979	0.739895	+	0.672722i	$0.234875\pi$
$102$	−7.33508	−0.726281
$103$	−1.00000	−0.0985329
$103$	−1.00000	−0.0985329
$104$	10.2352	1.00365
$105$	3.69900	0.360985
$106$	−5.57325	−0.541322
$107$	6.22636	0.601925	0.300963	−	0.953636i	$-0.402692\pi$
$107$	6.22636	0.601925	0.300963	+	0.953636i	$0.402692\pi$
$108$	4.24286	0.408270
$109$	14.6190	1.40025	0.700124	−	0.714021i	$-0.253128\pi$
$109$	14.6190	1.40025	0.700124	+	0.714021i	$0.253128\pi$
$110$	−0.732611	−0.0698517
$111$	4.62297	0.438793
$112$	25.2265	2.38368
$113$	7.69721	0.724093	0.362046	−	0.932160i	$-0.382078\pi$
$113$	7.69721	0.724093	0.362046	+	0.932160i	$0.382078\pi$
$114$	6.91553	0.647699
$115$	−4.36860	−0.407374
$116$	24.3290	2.25889
$117$	−1.82643	−0.168854
$118$	−33.4588	−3.08014
$119$	13.4256	1.23072
$120$	−4.53272	−0.413779
$121$	−10.8686	−0.988053
$122$	29.5992	2.67979
$123$	2.08044	0.187587
$124$	32.7547	2.94146
$125$	−7.55926	−0.676121
$126$	−11.4265	−1.01795
$127$	−7.48528	−0.664211	−0.332106	−	0.943242i	$-0.607759\pi$
$127$	−7.48528	−0.664211	−0.332106	+	0.943242i	$0.607759\pi$
$128$	16.6413	1.47090
$129$	−1.28921	−0.113509
$130$	3.69113	0.323734
$131$	10.2219	0.893091	0.446545	−	0.894761i	$-0.352654\pi$
$131$	10.2219	0.893091	0.446545	+	0.894761i	$0.352654\pi$
$132$	1.53807	0.133872
$133$	−12.6577	−1.09756
$134$	19.5667	1.69031
$135$	0.808843	0.0696141
$136$	−16.4516	−1.41071
$137$	−21.3999	−1.82832	−0.914158	−	0.405357i	$-0.867147\pi$
$137$	−21.3999	−1.82832	−0.914158	+	0.405357i	$0.867147\pi$
$138$	13.4949	1.14876
$139$	13.3485	1.13221	0.566103	−	0.824335i	$-0.308450\pi$
$139$	13.3485	1.13221	0.566103	+	0.824335i	$0.308450\pi$
$140$	15.6944	1.32641
$141$	−8.73420	−0.735553
$142$	32.1530	2.69822
$143$	−0.662096	−0.0553672
$144$	5.51616	0.459680
$145$	4.63799	0.385164
$146$	30.2920	2.50699
$147$	13.9142	1.14762
$148$	19.6146	1.61231
$149$	−10.6806	−0.874988	−0.437494	−	0.899221i	$-0.644134\pi$
$149$	−10.6806	−0.874988	−0.437494	+	0.899221i	$0.644134\pi$
$150$	10.8582	0.886571
$151$	−11.7813	−0.958752	−0.479376	−	0.877610i	$-0.659137\pi$
$151$	−11.7813	−0.958752	−0.479376	+	0.877610i	$0.659137\pi$
$152$	15.5106	1.25808
$153$	2.93571	0.237338
$154$	−4.14219	−0.333787
$155$	6.24423	0.501548
$156$	−7.74930	−0.620441
$157$	−4.07956	−0.325585	−0.162792	−	0.986660i	$-0.552050\pi$
$157$	−4.07956	−0.325585	−0.162792	+	0.986660i	$0.552050\pi$
$158$	−37.2146	−2.96064
$159$	2.23057	0.176896
$160$	−2.08246	−0.164633
$161$	−24.7001	−1.94664
$162$	−2.49857	−0.196306
$163$	10.6467	0.833916	0.416958	−	0.908926i	$-0.363096\pi$
$163$	10.6467	0.833916	0.416958	+	0.908926i	$0.363096\pi$
$164$	8.82702	0.689275
$165$	0.293212	0.0228265
$166$	−2.44356	−0.189657
$167$	−4.34354	−0.336113	−0.168057	−	0.985777i	$-0.553749\pi$
$167$	−4.34354	−0.336113	−0.168057	+	0.985777i	$0.553749\pi$
$168$	−25.6280	−1.97724
$169$	−9.66415	−0.743396
$170$	−5.93293	−0.455035
$171$	−2.76779	−0.211659
$172$	−5.46995	−0.417080
$173$	−14.0523	−1.06837	−0.534187	−	0.845366i	$-0.679382\pi$
$173$	−14.0523	−1.06837	−0.534187	+	0.845366i	$0.679382\pi$
$174$	−14.3271	−1.08613
$175$	−19.8741	−1.50234
$176$	1.99965	0.150729
$177$	13.3912	1.00654
$178$	24.3384	1.82424
$179$	−7.75600	−0.579711	−0.289855	−	0.957070i	$-0.593607\pi$
$179$	−7.75600	−0.579711	−0.289855	+	0.957070i	$0.593607\pi$
$180$	3.43181	0.255792
$181$	−16.5860	−1.23282	−0.616412	−	0.787424i	$-0.711415\pi$
$181$	−16.5860	−1.23282	−0.616412	+	0.787424i	$0.711415\pi$
$182$	20.8697	1.54696
$183$	−11.8465	−0.875715
$184$	30.2672	2.23133
$185$	3.73925	0.274915
$186$	−19.2889	−1.41433
$187$	1.06422	0.0778233
$188$	−37.0580	−2.70273
$189$	4.57320	0.332651
$190$	5.59358	0.405801
$191$	−8.68943	−0.628745	−0.314373	−	0.949300i	$-0.601794\pi$
$191$	−8.68943	−0.628745	−0.314373	+	0.949300i	$0.601794\pi$
$192$	−4.59946	−0.331937
$193$	12.2131	0.879115	0.439558	−	0.898214i	$-0.355135\pi$
$193$	12.2131	0.879115	0.439558	+	0.898214i	$0.355135\pi$
$194$	43.5368	3.12576
$195$	−1.47730	−0.105791
$196$	59.0359	4.21685
$197$	−9.59274	−0.683454	−0.341727	−	0.939799i	$-0.611012\pi$
$197$	−9.59274	−0.683454	−0.341727	+	0.939799i	$0.611012\pi$
$198$	−0.905752	−0.0643690
$199$	10.5816	0.750113	0.375056	−	0.927002i	$-0.377623\pi$
$199$	10.5816	0.750113	0.375056	+	0.927002i	$0.377623\pi$
$200$	24.3535	1.72205
$201$	−7.83115	−0.552367
$202$	−37.1580	−2.61443
$203$	26.2232	1.84051
$204$	12.4558	0.872081
$205$	1.68275	0.117528
$206$	2.49857	0.174084
$207$	−5.40105	−0.375399
$208$	−10.0749	−0.698568
$209$	−1.00335	−0.0694030
$210$	−9.24222	−0.637774
$211$	23.0149	1.58441	0.792207	−	0.610253i	$-0.208932\pi$
$211$	23.0149	1.58441	0.792207	+	0.610253i	$0.208932\pi$
$212$	9.46402	0.649992
$213$	−12.8686	−0.881739
$214$	−15.5570	−1.06346
$215$	−1.04277	−0.0711163
$216$	−5.60396	−0.381301
$217$	35.3049	2.39665
$218$	−36.5267	−2.47390
$219$	−12.1237	−0.819246
$220$	1.24406	0.0838743
$221$	−5.36187	−0.360679
$222$	−11.5508	−0.775240
$223$	−6.50148	−0.435371	−0.217686	−	0.976019i	$-0.569851\pi$
$223$	−6.50148	−0.435371	−0.217686	+	0.976019i	$0.569851\pi$
$224$	−11.7742	−0.786700
$225$	−4.34577	−0.289718
$226$	−19.2320	−1.27930
$227$	−10.4555	−0.693953	−0.346976	−	0.937874i	$-0.612792\pi$
$227$	−10.4555	−0.693953	−0.346976	+	0.937874i	$0.612792\pi$
$228$	−11.7434	−0.777724
$229$	12.4378	0.821911	0.410955	−	0.911655i	$-0.365195\pi$
$229$	12.4378	0.821911	0.410955	+	0.911655i	$0.365195\pi$
$230$	10.9153	0.719731
$231$	1.65782	0.109077
$232$	−32.1337	−2.10968
$233$	−10.5268	−0.689633	−0.344816	−	0.938670i	$-0.612059\pi$
$233$	−10.5268	−0.689633	−0.344816	+	0.938670i	$0.612059\pi$
$234$	4.56347	0.298324
$235$	−7.06460	−0.460843
$236$	56.8170	3.69847
$237$	14.8944	0.967493
$238$	−33.5448	−2.17439
$239$	−18.9448	−1.22544	−0.612720	−	0.790300i	$-0.709925\pi$
$239$	−18.9448	−1.22544	−0.612720	+	0.790300i	$0.709925\pi$
$240$	4.46171	0.288002
$241$	−13.2580	−0.854020	−0.427010	−	0.904247i	$-0.640433\pi$
$241$	−13.2580	−0.854020	−0.427010	+	0.904247i	$0.640433\pi$
$242$	27.1560	1.74565
$243$	1.00000	0.0641500
$244$	−50.2629	−3.21775
$245$	11.2544	0.719016
$246$	−5.19813	−0.331421
$247$	5.05519	0.321654
$248$	−43.2623	−2.74716
$249$	0.977981	0.0619770
$250$	18.8874	1.19454
$251$	−14.6689	−0.925891	−0.462946	−	0.886387i	$-0.653207\pi$
$251$	−14.6689	−0.925891	−0.462946	+	0.886387i	$0.653207\pi$
$252$	19.4035	1.22230
$253$	−1.95792	−0.123094
$254$	18.7025	1.17350
$255$	2.37453	0.148699
$256$	−32.3806	−2.02379
$257$	21.2168	1.32347	0.661735	−	0.749738i	$-0.269820\pi$
$257$	21.2168	1.32347	0.661735	+	0.749738i	$0.269820\pi$
$258$	3.22119	0.200542
$259$	21.1418	1.31368
$260$	−6.26797	−0.388723
$261$	5.73411	0.354932
$262$	−25.5401	−1.57787
$263$	5.89986	0.363801	0.181901	−	0.983317i	$-0.441775\pi$
$263$	5.89986	0.363801	0.181901	+	0.983317i	$0.441775\pi$
$264$	−2.03148	−0.125029
$265$	1.80418	0.110830
$266$	31.6261	1.93912
$267$	−9.74090	−0.596134
$268$	−33.2265	−2.02963
$269$	−7.89725	−0.481504	−0.240752	−	0.970587i	$-0.577394\pi$
$269$	−7.89725	−0.481504	−0.240752	+	0.970587i	$0.577394\pi$
$270$	−2.02095	−0.122991
$271$	12.5597	0.762948	0.381474	−	0.924380i	$-0.375417\pi$
$271$	12.5597	0.762948	0.381474	+	0.924380i	$0.375417\pi$
$272$	16.1938	0.981896
$273$	−8.35264	−0.505525
$274$	53.4692	3.23019
$275$	−1.57538	−0.0949988
$276$	−22.9159	−1.37938
$277$	15.9453	0.958059	0.479029	−	0.877799i	$-0.340989\pi$
$277$	15.9453	0.958059	0.479029	+	0.877799i	$0.340989\pi$
$278$	−33.3522	−2.00033
$279$	7.71995	0.462181
$280$	−20.7290	−1.23880
$281$	21.8428	1.30303	0.651517	−	0.758634i	$-0.274133\pi$
$281$	21.8428	1.30303	0.651517	+	0.758634i	$0.274133\pi$
$282$	21.8230	1.29954
$283$	−12.2766	−0.729769	−0.364884	−	0.931053i	$-0.618891\pi$
$283$	−12.2766	−0.729769	−0.364884	+	0.931053i	$0.618891\pi$
$284$	−54.5996	−3.23989
$285$	−2.23871	−0.132610
$286$	1.65429	0.0978205
$287$	9.51427	0.561610
$288$	−2.57462	−0.151711
$289$	−8.38161	−0.493036
$290$	−11.5884	−0.680491
$291$	−17.4247	−1.02145
$292$	−51.4394	−3.01026
$293$	−25.6845	−1.50050	−0.750251	−	0.661153i	$-0.770068\pi$
$293$	−25.6845	−1.50050	−0.750251	+	0.661153i	$0.770068\pi$
$294$	−34.7656	−2.02757
$295$	10.8314	0.630626
$296$	−25.9069	−1.50581
$297$	0.362508	0.0210348
$298$	26.6862	1.54589
$299$	9.86465	0.570487
$300$	−18.4385	−1.06455
$301$	−5.89582	−0.339830
$302$	29.4365	1.69388
$303$	14.8717	0.854358
$304$	−15.2676	−0.875657
$305$	−9.58191	−0.548659
$306$	−7.33508	−0.419319
$307$	12.9178	0.737258	0.368629	−	0.929577i	$-0.379827\pi$
$307$	12.9178	0.737258	0.368629	+	0.929577i	$0.379827\pi$
$308$	7.03391	0.400794
$309$	−1.00000	−0.0568880
$310$	−15.6017	−0.886114
$311$	3.47771	0.197203	0.0986015	−	0.995127i	$-0.468563\pi$
$311$	3.47771	0.197203	0.0986015	+	0.995127i	$0.468563\pi$
$312$	10.2352	0.579457
$313$	−6.25339	−0.353462	−0.176731	−	0.984259i	$-0.556552\pi$
$313$	−6.25339	−0.353462	−0.176731	+	0.984259i	$0.556552\pi$
$314$	10.1931	0.575229
$315$	3.69900	0.208415
$316$	63.1948	3.55498
$317$	−15.8817	−0.892007	−0.446004	−	0.895031i	$-0.647153\pi$
$317$	−15.8817	−0.892007	−0.446004	+	0.895031i	$0.647153\pi$
$318$	−5.57325	−0.312532
$319$	2.07866	0.116383
$320$	−3.72024	−0.207968
$321$	6.22636	0.347522
$322$	61.7149	3.43924
$323$	−8.12544	−0.452112
$324$	4.24286	0.235715
$325$	7.93726	0.440280
$326$	−26.6016	−1.47333
$327$	14.6190	0.808434
$328$	−11.6587	−0.643744
$329$	−39.9433	−2.20214
$330$	−0.732611	−0.0403289
$331$	12.4977	0.686935	0.343467	−	0.939165i	$-0.388398\pi$
$331$	12.4977	0.686935	0.343467	+	0.939165i	$0.388398\pi$
$332$	4.14944	0.227730
$333$	4.62297	0.253337
$334$	10.8527	0.593831
$335$	−6.33417	−0.346073
$336$	25.2265	1.37622
$337$	10.1626	0.553592	0.276796	−	0.960929i	$-0.410727\pi$
$337$	10.1626	0.553592	0.276796	+	0.960929i	$0.410727\pi$
$338$	24.1466	1.31340
$339$	7.69721	0.418055
$340$	10.0748	0.546382
$341$	2.79854	0.151550
$342$	6.91553	0.373949
$343$	31.6199	1.70731
$344$	7.22468	0.389529
$345$	−4.36860	−0.235197
$346$	35.1106	1.88756
$347$	10.4443	0.560679	0.280339	−	0.959901i	$-0.409553\pi$
$347$	10.4443	0.560679	0.280339	+	0.959901i	$0.409553\pi$
$348$	24.3290	1.30417
$349$	12.6768	0.678572	0.339286	−	0.940683i	$-0.389814\pi$
$349$	12.6768	0.678572	0.339286	+	0.940683i	$0.389814\pi$
$350$	49.6569	2.65427
$351$	−1.82643	−0.0974877
$352$	−0.933319	−0.0497461
$353$	−2.31987	−0.123474	−0.0617371	−	0.998092i	$-0.519664\pi$
$353$	−2.31987	−0.123474	−0.0617371	+	0.998092i	$0.519664\pi$
$354$	−33.4588	−1.77832
$355$	−10.4086	−0.552433
$356$	−41.3293	−2.19045
$357$	13.4256	0.710558
$358$	19.3789	1.02421
$359$	8.95381	0.472564	0.236282	−	0.971685i	$-0.424071\pi$
$359$	8.95381	0.472564	0.236282	+	0.971685i	$0.424071\pi$
$360$	−4.53272	−0.238895
$361$	−11.3393	−0.596806
$362$	41.4412	2.17810
$363$	−10.8686	−0.570453
$364$	−35.4391	−1.85751
$365$	−9.80620	−0.513280
$366$	29.5992	1.54718
$367$	−4.03891	−0.210830	−0.105415	−	0.994428i	$-0.533617\pi$
$367$	−4.03891	−0.210830	−0.105415	+	0.994428i	$0.533617\pi$
$368$	−29.7931	−1.55307
$369$	2.08044	0.108303
$370$	−9.34279	−0.485709
$371$	10.2009	0.529602
$372$	32.7547	1.69825
$373$	−5.71904	−0.296121	−0.148060	−	0.988978i	$-0.547303\pi$
$373$	−5.71904	−0.296121	−0.148060	+	0.988978i	$0.547303\pi$
$374$	−2.65902	−0.137495
$375$	−7.55926	−0.390359
$376$	48.9461	2.52420
$377$	−10.4730	−0.539385
$378$	−11.4265	−0.587714
$379$	−13.8349	−0.710650	−0.355325	−	0.934743i	$-0.615630\pi$
$379$	−13.8349	−0.710650	−0.355325	+	0.934743i	$0.615630\pi$
$380$	−9.49854	−0.487265
$381$	−7.48528	−0.383483
$382$	21.7112	1.11084
$383$	32.9537	1.68385	0.841927	−	0.539591i	$-0.181421\pi$
$383$	32.9537	1.68385	0.841927	+	0.539591i	$0.181421\pi$
$384$	16.6413	0.849223
$385$	1.34092	0.0683394
$386$	−30.5152	−1.55318
$387$	−1.28921	−0.0655343
$388$	−73.9304	−3.75325
$389$	−31.2901	−1.58647	−0.793235	−	0.608916i	$-0.791605\pi$
$389$	−31.2901	−1.58647	−0.793235	+	0.608916i	$0.791605\pi$
$390$	3.69113	0.186908
$391$	−15.8559	−0.801868
$392$	−77.9744	−3.93830
$393$	10.2219	0.515626
$394$	23.9681	1.20750
$395$	12.0472	0.606160
$396$	1.53807	0.0772910
$397$	−23.3843	−1.17363	−0.586813	−	0.809723i	$-0.699618\pi$
$397$	−23.3843	−1.17363	−0.586813	+	0.809723i	$0.699618\pi$
$398$	−26.4390	−1.32527
$399$	−12.6577	−0.633677
$400$	−23.9720	−1.19860
$401$	9.72523	0.485655	0.242827	−	0.970070i	$-0.421925\pi$
$401$	9.72523	0.485655	0.242827	+	0.970070i	$0.421925\pi$
$402$	19.5667	0.975898
$403$	−14.1000	−0.702369
$404$	63.0986	3.13927
$405$	0.808843	0.0401917
$406$	−65.5206	−3.25173
$407$	1.67586	0.0830694
$408$	−16.4516	−0.814475
$409$	24.4695	1.20994	0.604969	−	0.796249i	$-0.293185\pi$
$409$	24.4695	1.20994	0.604969	+	0.796249i	$0.293185\pi$
$410$	−4.20447	−0.207644
$411$	−21.3999	−1.05558
$412$	−4.24286	−0.209031
$413$	61.2406	3.01345
$414$	13.4949	0.663239
$415$	0.791033	0.0388303
$416$	4.70236	0.230552
$417$	13.3485	0.653679
$418$	2.50694	0.122618
$419$	21.0962	1.03062	0.515308	−	0.857005i	$-0.327677\pi$
$419$	21.0962	1.03062	0.515308	+	0.857005i	$0.327677\pi$
$420$	15.6944	0.765806
$421$	−9.46115	−0.461108	−0.230554	−	0.973059i	$-0.574054\pi$
$421$	−9.46115	−0.461108	−0.230554	+	0.973059i	$0.574054\pi$
$422$	−57.5045	−2.79927
$423$	−8.73420	−0.424672
$424$	−12.5000	−0.607055
$425$	−12.7579	−0.618850
$426$	32.1530	1.55782
$427$	−54.1762	−2.62177
$428$	26.4176	1.27694
$429$	−0.662096	−0.0319663
$430$	2.60543	0.125645
$431$	−26.0304	−1.25384	−0.626919	−	0.779084i	$-0.715684\pi$
$431$	−26.0304	−1.25384	−0.626919	+	0.779084i	$0.715684\pi$
$432$	5.51616	0.265396
$433$	27.1752	1.30596	0.652979	−	0.757376i	$-0.273519\pi$
$433$	27.1752	1.30596	0.652979	+	0.757376i	$0.273519\pi$
$434$	−88.2118	−4.23430
$435$	4.63799	0.222375
$436$	62.0265	2.97053
$437$	14.9490	0.715107
$438$	30.2920	1.44741
$439$	30.8548	1.47262	0.736309	−	0.676646i	$-0.236567\pi$
$439$	30.8548	1.47262	0.736309	+	0.676646i	$0.236567\pi$
$440$	−1.64315	−0.0783339
$441$	13.9142	0.662580
$442$	13.3970	0.637232
$443$	27.0385	1.28464	0.642318	−	0.766438i	$-0.277973\pi$
$443$	27.0385	1.28464	0.642318	+	0.766438i	$0.277973\pi$
$444$	19.6146	0.930868
$445$	−7.87886	−0.373494
$446$	16.2444	0.769195
$447$	−10.6806	−0.505175
$448$	−21.0342	−0.993775
$449$	−12.5667	−0.593059	−0.296529	−	0.955024i	$-0.595829\pi$
$449$	−12.5667	−0.593059	−0.296529	+	0.955024i	$0.595829\pi$
$450$	10.8582	0.511862
$451$	0.754176	0.0355127
$452$	32.6582	1.53611
$453$	−11.7813	−0.553536
$454$	26.1237	1.22605
$455$	−6.75597	−0.316725
$456$	15.5106	0.726350
$457$	7.19896	0.336753	0.168377	−	0.985723i	$-0.446148\pi$
$457$	7.19896	0.336753	0.168377	+	0.985723i	$0.446148\pi$
$458$	−31.0767	−1.45212
$459$	2.93571	0.137027
$460$	−18.5354	−0.864216
$461$	−37.4615	−1.74476	−0.872378	−	0.488832i	$-0.837423\pi$
$461$	−37.4615	−1.74476	−0.872378	+	0.488832i	$0.837423\pi$
$462$	−4.14219	−0.192712
$463$	−23.7033	−1.10159	−0.550793	−	0.834642i	$-0.685675\pi$
$463$	−23.7033	−1.10159	−0.550793	+	0.834642i	$0.685675\pi$
$464$	31.6303	1.46840
$465$	6.24423	0.289569
$466$	26.3019	1.21841
$467$	27.1565	1.25665	0.628326	−	0.777950i	$-0.283741\pi$
$467$	27.1565	1.25665	0.628326	+	0.777950i	$0.283741\pi$
$468$	−7.74930	−0.358212
$469$	−35.8134	−1.65371
$470$	17.6514	0.814199
$471$	−4.07956	−0.187976
$472$	−75.0436	−3.45416
$473$	−0.467349	−0.0214887
$474$	−37.2146	−1.70933
$475$	12.0282	0.551892
$476$	56.9629	2.61089
$477$	2.23057	0.102131
$478$	47.3351	2.16506
$479$	−5.35682	−0.244759	−0.122380	−	0.992483i	$-0.539053\pi$
$479$	−5.35682	−0.244759	−0.122380	+	0.992483i	$0.539053\pi$
$480$	−2.08246	−0.0950509
$481$	−8.44353	−0.384992
$482$	33.1259	1.50885
$483$	−24.7001	−1.12389
$484$	−46.1139	−2.09609
$485$	−14.0938	−0.639967
$486$	−2.49857	−0.113338
$487$	−22.4820	−1.01876	−0.509378	−	0.860543i	$-0.670124\pi$
$487$	−22.4820	−1.01876	−0.509378	+	0.860543i	$0.670124\pi$
$488$	66.3870	3.00520
$489$	10.6467	0.481462
$490$	−28.1199	−1.27033
$491$	38.8567	1.75358	0.876790	−	0.480874i	$-0.159680\pi$
$491$	38.8567	1.75358	0.876790	+	0.480874i	$0.159680\pi$
$492$	8.82702	0.397953
$493$	16.8337	0.758150
$494$	−12.6308	−0.568284
$495$	0.293212	0.0131789
$496$	42.5845	1.91210
$497$	−58.8505	−2.63981
$498$	−2.44356	−0.109498
$499$	16.3365	0.731321	0.365660	−	0.930748i	$-0.380843\pi$
$499$	16.3365	0.731321	0.365660	+	0.930748i	$0.380843\pi$
$500$	−32.0729	−1.43434
$501$	−4.34354	−0.194055
$502$	36.6512	1.63583
$503$	6.16697	0.274972	0.137486	−	0.990504i	$-0.456098\pi$
$503$	6.16697	0.274972	0.137486	+	0.990504i	$0.456098\pi$
$504$	−25.6280	−1.14156
$505$	12.0289	0.535278
$506$	4.89201	0.217476
$507$	−9.66415	−0.429200
$508$	−31.7590	−1.40908
$509$	5.74752	0.254754	0.127377	−	0.991854i	$-0.459344\pi$
$509$	5.74752	0.254754	0.127377	+	0.991854i	$0.459344\pi$
$510$	−5.93293	−0.262714
$511$	−55.4443	−2.45271
$512$	47.6226	2.10464
$513$	−2.76779	−0.122201
$514$	−53.0118	−2.33825
$515$	−0.808843	−0.0356419
$516$	−5.46995	−0.240801
$517$	−3.16622	−0.139250
$518$	−52.8242	−2.32096
$519$	−14.0523	−0.616826
$520$	8.27870	0.363045
$521$	11.3310	0.496419	0.248209	−	0.968706i	$-0.420158\pi$
$521$	11.3310	0.496419	0.248209	+	0.968706i	$0.420158\pi$
$522$	−14.3271	−0.627079
$523$	7.09113	0.310074	0.155037	−	0.987909i	$-0.450450\pi$
$523$	7.09113	0.310074	0.155037	+	0.987909i	$0.450450\pi$
$524$	43.3701	1.89463
$525$	−19.8741	−0.867377
$526$	−14.7412	−0.642748
$527$	22.6635	0.987239
$528$	1.99965	0.0870237
$529$	6.17133	0.268319
$530$	−4.50788	−0.195810
$531$	13.3912	0.581128
$532$	−53.7048	−2.32840
$533$	−3.79978	−0.164587
$534$	24.3384	1.05322
$535$	5.03615	0.217732
$536$	43.8854	1.89556
$537$	−7.75600	−0.334696
$538$	19.7318	0.850700
$539$	5.04400	0.217260
$540$	3.43181	0.147682
$541$	6.05538	0.260341	0.130171	−	0.991492i	$-0.458447\pi$
$541$	6.05538	0.260341	0.130171	+	0.991492i	$0.458447\pi$
$542$	−31.3813	−1.34794
$543$	−16.5860	−0.711772
$544$	−7.55833	−0.324061
$545$	11.8245	0.506505
$546$	20.8697	0.893140
$547$	27.2723	1.16608	0.583040	−	0.812444i	$-0.301863\pi$
$547$	27.2723	1.16608	0.583040	+	0.812444i	$0.301863\pi$
$548$	−90.7968	−3.87865
$549$	−11.8465	−0.505594
$550$	3.93619	0.167840
$551$	−15.8708	−0.676120
$552$	30.2672	1.28826
$553$	68.1149	2.89654
$554$	−39.8404	−1.69266
$555$	3.73925	0.158722
$556$	56.6359	2.40190
$557$	10.3942	0.440418	0.220209	−	0.975453i	$-0.429326\pi$
$557$	10.3942	0.440418	0.220209	+	0.975453i	$0.429326\pi$
$558$	−19.2889	−0.816562
$559$	2.35466	0.0995914
$560$	20.4043	0.862238
$561$	1.06422	0.0449313
$562$	−54.5759	−2.30214
$563$	−26.0023	−1.09587	−0.547933	−	0.836522i	$-0.684585\pi$
$563$	−26.0023	−1.09587	−0.547933	+	0.836522i	$0.684585\pi$
$564$	−37.0580	−1.56042
$565$	6.22583	0.261923
$566$	30.6740	1.28932
$567$	4.57320	0.192056
$568$	72.1149	3.02587
$569$	−26.6440	−1.11698	−0.558488	−	0.829512i	$-0.688618\pi$
$569$	−26.6440	−1.11698	−0.558488	+	0.829512i	$0.688618\pi$
$570$	5.59358	0.234289
$571$	−2.83010	−0.118436	−0.0592181	−	0.998245i	$-0.518861\pi$
$571$	−2.83010	−0.118436	−0.0592181	+	0.998245i	$0.518861\pi$
$572$	−2.80918	−0.117458
$573$	−8.68943	−0.363006
$574$	−23.7721	−0.992228
$575$	23.4717	0.978839
$576$	−4.59946	−0.191644
$577$	35.3410	1.47127	0.735633	−	0.677381i	$-0.236885\pi$
$577$	35.3410	1.47127	0.735633	+	0.677381i	$0.236885\pi$
$578$	20.9421	0.871075
$579$	12.2131	0.507557
$580$	19.6784	0.817099
$581$	4.47251	0.185551
$582$	43.5368	1.80466
$583$	0.808600	0.0334888
$584$	67.9409	2.81141
$585$	−1.47730	−0.0610787
$586$	64.1745	2.65102
$587$	−17.1604	−0.708287	−0.354143	−	0.935191i	$-0.615228\pi$
$587$	−17.1604	−0.708287	−0.354143	+	0.935191i	$0.615228\pi$
$588$	59.0359	2.43460
$589$	−21.3672	−0.880422
$590$	−27.0629	−1.11416
$591$	−9.59274	−0.394593
$592$	25.5010	1.04809
$593$	21.1251	0.867506	0.433753	−	0.901032i	$-0.357189\pi$
$593$	21.1251	0.867506	0.433753	+	0.901032i	$0.357189\pi$
$594$	−0.905752	−0.0371635
$595$	10.8592	0.445183
$596$	−45.3163	−1.85623
$597$	10.5816	0.433078
$598$	−24.6475	−1.00791
$599$	46.2134	1.88823	0.944115	−	0.329617i	$-0.106920\pi$
$599$	46.2134	1.88823	0.944115	+	0.329617i	$0.106920\pi$
$600$	24.3535	0.994228
$601$	−32.0370	−1.30681	−0.653407	−	0.757007i	$-0.726661\pi$
$601$	−32.0370	−1.30681	−0.653407	+	0.757007i	$0.726661\pi$
$602$	14.7311	0.600397
$603$	−7.83115	−0.318909
$604$	−49.9866	−2.03393
$605$	−8.79098	−0.357404
$606$	−37.1580	−1.50944
$607$	8.55945	0.347417	0.173709	−	0.984797i	$-0.444425\pi$
$607$	8.55945	0.347417	0.173709	+	0.984797i	$0.444425\pi$
$608$	7.12601	0.288998
$609$	26.2232	1.06262
$610$	23.9411	0.969347
$611$	15.9524	0.645366
$612$	12.4558	0.503496
$613$	31.3516	1.26628	0.633140	−	0.774037i	$-0.281766\pi$
$613$	31.3516	1.26628	0.633140	+	0.774037i	$0.281766\pi$
$614$	−32.2761	−1.30256
$615$	1.68275	0.0678550
$616$	−9.29036	−0.374319
$617$	−17.4735	−0.703457	−0.351729	−	0.936102i	$-0.614406\pi$
$617$	−17.4735	−0.703457	−0.351729	+	0.936102i	$0.614406\pi$
$618$	2.49857	0.100507
$619$	23.7575	0.954896	0.477448	−	0.878660i	$-0.341562\pi$
$619$	23.7575	0.954896	0.477448	+	0.878660i	$0.341562\pi$
$620$	26.4934	1.06400
$621$	−5.40105	−0.216737
$622$	−8.68932	−0.348410
$623$	−44.5471	−1.78474
$624$	−10.0749	−0.403319
$625$	15.6146	0.624585
$626$	15.6245	0.624482
$627$	−1.00335	−0.0400698
$628$	−17.3090	−0.690706
$629$	13.5717	0.541138
$630$	−9.24222	−0.368219
$631$	23.2019	0.923654	0.461827	−	0.886970i	$-0.347194\pi$
$631$	23.2019	0.923654	0.461827	+	0.886970i	$0.347194\pi$
$632$	−83.4674	−3.32015
$633$	23.0149	0.914762
$634$	39.6817	1.57596
$635$	−6.05441	−0.240262
$636$	9.46402	0.375273
$637$	−25.4133	−1.00691
$638$	−5.19368	−0.205620
$639$	−12.8686	−0.509072
$640$	13.4602	0.532061
$641$	44.4111	1.75413	0.877066	−	0.480369i	$-0.159497\pi$
$641$	44.4111	1.75413	0.877066	+	0.480369i	$0.159497\pi$
$642$	−15.5570	−0.613986
$643$	−19.4790	−0.768176	−0.384088	−	0.923297i	$-0.625484\pi$
$643$	−19.4790	−0.768176	−0.384088	+	0.923297i	$0.625484\pi$
$644$	−104.799	−4.12966
$645$	−1.04277	−0.0410590
$646$	20.3020	0.798771
$647$	−15.5267	−0.610416	−0.305208	−	0.952286i	$-0.598726\pi$
$647$	−15.5267	−0.610416	−0.305208	+	0.952286i	$0.598726\pi$
$648$	−5.60396	−0.220144
$649$	4.85441	0.190552
$650$	−19.8318	−0.777868
$651$	35.3049	1.38371
$652$	45.1726	1.76910
$653$	48.7327	1.90706	0.953530	−	0.301299i	$-0.0974202\pi$
$653$	48.7327	1.90706	0.953530	+	0.301299i	$0.0974202\pi$
$654$	−36.5267	−1.42831
$655$	8.26790	0.323054
$656$	11.4760	0.448064
$657$	−12.1237	−0.472992
$658$	99.8012	3.89065
$659$	38.7020	1.50762	0.753808	−	0.657095i	$-0.228215\pi$
$659$	38.7020	1.50762	0.753808	+	0.657095i	$0.228215\pi$
$660$	1.24406	0.0484249
$661$	2.15610	0.0838624	0.0419312	−	0.999121i	$-0.486649\pi$
$661$	2.15610	0.0838624	0.0419312	+	0.999121i	$0.486649\pi$
$662$	−31.2264	−1.21365
$663$	−5.36187	−0.208238
$664$	−5.48056	−0.212687
$665$	−10.2381	−0.397015
$666$	−11.5508	−0.447585
$667$	−30.9702	−1.19917
$668$	−18.4290	−0.713041
$669$	−6.50148	−0.251362
$670$	15.8264	0.611426
$671$	−4.29443	−0.165785
$672$	−11.7742	−0.454201
$673$	4.19588	0.161739	0.0808696	−	0.996725i	$-0.474230\pi$
$673$	4.19588	0.161739	0.0808696	+	0.996725i	$0.474230\pi$
$674$	−25.3920	−0.978063
$675$	−4.34577	−0.167269
$676$	−41.0036	−1.57706
$677$	−29.2228	−1.12312	−0.561561	−	0.827435i	$-0.689799\pi$
$677$	−29.2228	−1.12312	−0.561561	+	0.827435i	$0.689799\pi$
$678$	−19.2320	−0.738602
$679$	−79.6865	−3.05809
$680$	−13.3067	−0.510290
$681$	−10.4555	−0.400654
$682$	−6.99236	−0.267751
$683$	−1.78658	−0.0683614	−0.0341807	−	0.999416i	$-0.510882\pi$
$683$	−1.78658	−0.0683614	−0.0341807	+	0.999416i	$0.510882\pi$
$684$	−11.7434	−0.449019
$685$	−17.3092	−0.661349
$686$	−79.0046	−3.01641
$687$	12.4378	0.474530
$688$	−7.11150	−0.271123
$689$	−4.07399	−0.155207
$690$	10.9153	0.415537
$691$	−1.69010	−0.0642944	−0.0321472	−	0.999483i	$-0.510235\pi$
$691$	−1.69010	−0.0642944	−0.0321472	+	0.999483i	$0.510235\pi$
$692$	−59.6218	−2.26648
$693$	1.65782	0.0629754
$694$	−26.0958	−0.990584
$695$	10.7968	0.409548
$696$	−32.1337	−1.21802
$697$	6.10757	0.231341
$698$	−31.6738	−1.19887
$699$	−10.5268	−0.398160
$700$	−84.3231	−3.18711
$701$	−38.0381	−1.43668	−0.718339	−	0.695693i	$-0.755097\pi$
$701$	−38.0381	−1.43668	−0.718339	+	0.695693i	$0.755097\pi$
$702$	4.56347	0.172237
$703$	−12.7954	−0.482588
$704$	−1.66734	−0.0628402
$705$	−7.06460	−0.266068
$706$	5.79637	0.218149
$707$	68.0113	2.55783
$708$	56.8170	2.13531
$709$	−2.57051	−0.0965375	−0.0482687	−	0.998834i	$-0.515370\pi$
$709$	−2.57051	−0.0965375	−0.0482687	+	0.998834i	$0.515370\pi$
$710$	26.0067	0.976016
$711$	14.8944	0.558582
$712$	54.5876	2.04576
$713$	−41.6958	−1.56152
$714$	−33.5448	−1.25538
$715$	−0.535531	−0.0200277
$716$	−32.9076	−1.22982
$717$	−18.9448	−0.707508
$718$	−22.3717	−0.834906
$719$	−31.7544	−1.18424	−0.592120	−	0.805850i	$-0.701709\pi$
$719$	−31.7544	−1.18424	−0.592120	+	0.805850i	$0.701709\pi$
$720$	4.46171	0.166278
$721$	−4.57320	−0.170315
$722$	28.3321	1.05441
$723$	−13.2580	−0.493069
$724$	−70.3720	−2.61535
$725$	−24.9191	−0.925473
$726$	27.1560	1.00785
$727$	3.75292	0.139188	0.0695940	−	0.997575i	$-0.477830\pi$
$727$	3.75292	0.139188	0.0695940	+	0.997575i	$0.477830\pi$
$728$	46.8078	1.73481
$729$	1.00000	0.0370370
$730$	24.5015	0.906841
$731$	−3.78475	−0.139984
$732$	−50.2629	−1.85777
$733$	−2.09440	−0.0773585	−0.0386793	−	0.999252i	$-0.512315\pi$
$733$	−2.09440	−0.0773585	−0.0386793	+	0.999252i	$0.512315\pi$
$734$	10.0915	0.372485
$735$	11.2544	0.415124
$736$	13.9056	0.512568
$737$	−2.83885	−0.104571
$738$	−5.19813	−0.191346
$739$	51.0764	1.87888	0.939439	−	0.342717i	$-0.111347\pi$
$739$	51.0764	1.87888	0.939439	+	0.342717i	$0.111347\pi$
$740$	15.8651	0.583214
$741$	5.05519	0.185707
$742$	−25.4876	−0.935679
$743$	−22.7882	−0.836019	−0.418010	−	0.908443i	$-0.637272\pi$
$743$	−22.7882	−0.836019	−0.418010	+	0.908443i	$0.637272\pi$
$744$	−43.2623	−1.58607
$745$	−8.63892	−0.316506
$746$	14.2894	0.523173
$747$	0.977981	0.0357825
$748$	4.51533	0.165097
$749$	28.4744	1.04043
$750$	18.8874	0.689669
$751$	−10.1643	−0.370900	−0.185450	−	0.982654i	$-0.559374\pi$
$751$	−10.1643	−0.370900	−0.185450	+	0.982654i	$0.559374\pi$
$752$	−48.1793	−1.75692
$753$	−14.6689	−0.534564
$754$	26.1674	0.952962
$755$	−9.52925	−0.346805
$756$	19.4035	0.705697
$757$	7.53120	0.273726	0.136863	−	0.990590i	$-0.456298\pi$
$757$	7.53120	0.273726	0.136863	+	0.990590i	$0.456298\pi$
$758$	34.5674	1.25555
$759$	−1.95792	−0.0710681
$760$	12.5456	0.455078
$761$	−30.9134	−1.12061	−0.560305	−	0.828287i	$-0.689316\pi$
$761$	−30.9134	−1.12061	−0.560305	+	0.828287i	$0.689316\pi$
$762$	18.7025	0.677521
$763$	66.8557	2.42034
$764$	−36.8681	−1.33384
$765$	2.37453	0.0858512
$766$	−82.3371	−2.97496
$767$	−24.4581	−0.883130
$768$	−32.3806	−1.16843
$769$	−11.3123	−0.407930	−0.203965	−	0.978978i	$-0.565383\pi$
$769$	−11.3123	−0.407930	−0.203965	+	0.978978i	$0.565383\pi$
$770$	−3.35038	−0.120739
$771$	21.2168	0.764106
$772$	51.8183	1.86498
$773$	31.0522	1.11687	0.558435	−	0.829549i	$-0.311402\pi$
$773$	31.0522	1.11687	0.558435	+	0.829549i	$0.311402\pi$
$774$	3.22119	0.115783
$775$	−33.5492	−1.20512
$776$	97.6470	3.50532
$777$	21.1418	0.758456
$778$	78.1805	2.80291
$779$	−5.75823	−0.206310
$780$	−6.26797	−0.224429
$781$	−4.66495	−0.166925
$782$	39.6171	1.41671
$783$	5.73411	0.204920
$784$	76.7528	2.74117
$785$	−3.29973	−0.117772
$786$	−25.5401	−0.910986
$787$	8.46575	0.301771	0.150886	−	0.988551i	$-0.451787\pi$
$787$	8.46575	0.301771	0.150886	+	0.988551i	$0.451787\pi$
$788$	−40.7007	−1.44990
$789$	5.89986	0.210041
$790$	−30.1008	−1.07094
$791$	35.2009	1.25160
$792$	−2.03148	−0.0721854
$793$	21.6367	0.768343
$794$	58.4274	2.07351
$795$	1.80418	0.0639878
$796$	44.8965	1.59131
$797$	44.5897	1.57945	0.789724	−	0.613463i	$-0.210224\pi$
$797$	44.5897	1.57945	0.789724	+	0.613463i	$0.210224\pi$
$798$	31.6261	1.11955
$799$	−25.6411	−0.907116
$800$	11.1887	0.395580
$801$	−9.74090	−0.344178
$802$	−24.2992	−0.858034
$803$	−4.39495	−0.155094
$804$	−33.2265	−1.17181
$805$	−19.9785	−0.704149
$806$	35.2298	1.24092
$807$	−7.89725	−0.277996
$808$	−83.3404	−2.93191
$809$	25.5947	0.899861	0.449931	−	0.893063i	$-0.351449\pi$
$809$	25.5947	0.899861	0.449931	+	0.893063i	$0.351449\pi$
$810$	−2.02095	−0.0710090
$811$	−24.8540	−0.872743	−0.436371	−	0.899767i	$-0.643737\pi$
$811$	−24.8540	−0.872743	−0.436371	+	0.899767i	$0.643737\pi$
$812$	111.262	3.90451
$813$	12.5597	0.440488
$814$	−4.18726	−0.146763
$815$	8.61153	0.301649
$816$	16.1938	0.566898
$817$	3.56827	0.124838
$818$	−61.1388	−2.13767
$819$	−8.35264	−0.291865
$820$	7.13967	0.249328
$821$	−5.17889	−0.180744	−0.0903722	−	0.995908i	$-0.528806\pi$
$821$	−5.17889	−0.180744	−0.0903722	+	0.995908i	$0.528806\pi$
$822$	53.4692	1.86495
$823$	8.87107	0.309226	0.154613	−	0.987975i	$-0.450587\pi$
$823$	8.87107	0.309226	0.154613	+	0.987975i	$0.450587\pi$
$824$	5.60396	0.195223
$825$	−1.57538	−0.0548476
$826$	−153.014	−5.32404
$827$	14.2579	0.495795	0.247898	−	0.968786i	$-0.420260\pi$
$827$	14.2579	0.495795	0.247898	+	0.968786i	$0.420260\pi$
$828$	−22.9159	−0.796383
$829$	29.8001	1.03500	0.517499	−	0.855684i	$-0.326863\pi$
$829$	29.8001	1.03500	0.517499	+	0.855684i	$0.326863\pi$
$830$	−1.97645	−0.0686037
$831$	15.9453	0.553135
$832$	8.40060	0.291238
$833$	40.8480	1.41530
$834$	−33.3522	−1.15489
$835$	−3.51324	−0.121581
$836$	−4.25706	−0.147234
$837$	7.71995	0.266841
$838$	−52.7104	−1.82085
$839$	20.0702	0.692902	0.346451	−	0.938068i	$-0.387387\pi$
$839$	20.0702	0.692902	0.346451	+	0.938068i	$0.387387\pi$
$840$	−20.7290	−0.715220
$841$	3.87997	0.133792
$842$	23.6394	0.814667
$843$	21.8428	0.752307
$844$	97.6492	3.36123
$845$	−7.81677	−0.268905
$846$	21.8230	0.750292
$847$	−49.7042	−1.70786
$848$	12.3042	0.422528
$849$	−12.2766	−0.421332
$850$	31.8766	1.09336
$851$	−24.9689	−0.855922
$852$	−54.5996	−1.87055
$853$	26.8605	0.919686	0.459843	−	0.888000i	$-0.347906\pi$
$853$	26.8605	0.919686	0.459843	+	0.888000i	$0.347906\pi$
$854$	135.363	4.63203
$855$	−2.23871	−0.0765623
$856$	−34.8923	−1.19259
$857$	18.2101	0.622046	0.311023	−	0.950402i	$-0.399328\pi$
$857$	18.2101	0.622046	0.311023	+	0.950402i	$0.399328\pi$
$858$	1.65429	0.0564767
$859$	−34.8253	−1.18822	−0.594112	−	0.804382i	$-0.702496\pi$
$859$	−34.8253	−1.18822	−0.594112	+	0.804382i	$0.702496\pi$
$860$	−4.42433	−0.150868
$861$	9.51427	0.324245
$862$	65.0387	2.21523
$863$	−41.1726	−1.40153	−0.700767	−	0.713391i	$-0.747159\pi$
$863$	−41.1726	−1.40153	−0.700767	+	0.713391i	$0.747159\pi$
$864$	−2.57462	−0.0875902
$865$	−11.3661	−0.386458
$866$	−67.8993	−2.30731
$867$	−8.38161	−0.284654
$868$	149.794	5.08433
$869$	5.39932	0.183160
$870$	−11.5884	−0.392882
$871$	14.3031	0.484641
$872$	−81.9244	−2.77431
$873$	−17.4247	−0.589735
$874$	−37.3511	−1.26342
$875$	−34.5700	−1.16868
$876$	−51.4394	−1.73798
$877$	−20.3506	−0.687190	−0.343595	−	0.939118i	$-0.611645\pi$
$877$	−20.3506	−0.687190	−0.343595	+	0.939118i	$0.611645\pi$
$878$	−77.0928	−2.60176
$879$	−25.6845	−0.866316
$880$	1.61740	0.0545227
$881$	34.8381	1.17372	0.586862	−	0.809687i	$-0.300363\pi$
$881$	34.8381	1.17372	0.586862	+	0.809687i	$0.300363\pi$
$882$	−34.7656	−1.17062
$883$	−23.8788	−0.803584	−0.401792	−	0.915731i	$-0.631613\pi$
$883$	−23.8788	−0.803584	−0.401792	+	0.915731i	$0.631613\pi$
$884$	−22.7497	−0.765155
$885$	10.8314	0.364092
$886$	−67.5575	−2.26964
$887$	−3.81355	−0.128047	−0.0640233	−	0.997948i	$-0.520393\pi$
$887$	−3.81355	−0.128047	−0.0640233	+	0.997948i	$0.520393\pi$
$888$	−25.9069	−0.869379
$889$	−34.2317	−1.14809
$890$	19.6859	0.659873
$891$	0.362508	0.0121445
$892$	−27.5849	−0.923610
$893$	24.1745	0.808968
$894$	26.6862	0.892521
$895$	−6.27338	−0.209696
$896$	76.1041	2.54246
$897$	9.86465	0.329371
$898$	31.3988	1.04779
$899$	44.2670	1.47639
$900$	−18.4385	−0.614617
$901$	6.54832	0.218156
$902$	−1.88436	−0.0627424
$903$	−5.89582	−0.196201
$904$	−43.1348	−1.43464
$905$	−13.4154	−0.445944
$906$	29.4365	0.977963
$907$	17.9084	0.594637	0.297319	−	0.954778i	$-0.403908\pi$
$907$	17.9084	0.594637	0.297319	+	0.954778i	$0.403908\pi$
$908$	−44.3611	−1.47217
$909$	14.8717	0.493264
$910$	16.8803	0.559576
$911$	23.7124	0.785625	0.392813	−	0.919618i	$-0.371502\pi$
$911$	23.7124	0.785625	0.392813	+	0.919618i	$0.371502\pi$
$912$	−15.2676	−0.505561
$913$	0.354526	0.0117331
$914$	−17.9871	−0.594961
$915$	−9.58191	−0.316768
$916$	52.7717	1.74363
$917$	46.7468	1.54371
$918$	−7.33508	−0.242094
$919$	20.3626	0.671701	0.335851	−	0.941915i	$-0.390976\pi$
$919$	20.3626	0.671701	0.335851	+	0.941915i	$0.390976\pi$
$920$	24.4814	0.807129
$921$	12.9178	0.425656
$922$	93.6003	3.08256
$923$	23.5036	0.773629
$924$	7.03391	0.231399
$925$	−20.0904	−0.660567
$926$	59.2244	1.94624
$927$	−1.00000	−0.0328443
$928$	−14.7631	−0.484623
$929$	32.9413	1.08077	0.540385	−	0.841418i	$-0.318279\pi$
$929$	32.9413	1.08077	0.540385	+	0.841418i	$0.318279\pi$
$930$	−15.6017	−0.511598
$931$	−38.5116	−1.26217
$932$	−44.6637	−1.46301
$933$	3.47771	0.113855
$934$	−67.8524	−2.22020
$935$	0.860785	0.0281507
$936$	10.2352	0.334549
$937$	46.0318	1.50379	0.751897	−	0.659281i	$-0.229139\pi$
$937$	46.0318	1.50379	0.751897	+	0.659281i	$0.229139\pi$
$938$	89.4825	2.92171
$939$	−6.25339	−0.204072
$940$	−29.9741	−0.977648
$941$	−15.7306	−0.512803	−0.256402	−	0.966570i	$-0.582537\pi$
$941$	−15.7306	−0.512803	−0.256402	+	0.966570i	$0.582537\pi$
$942$	10.1931	0.332109
$943$	−11.2366	−0.365913
$944$	73.8679	2.40420
$945$	3.69900	0.120328
$946$	1.16771	0.0379654
$947$	40.3557	1.31139	0.655693	−	0.755028i	$-0.272377\pi$
$947$	40.3557	1.31139	0.655693	+	0.755028i	$0.272377\pi$
$948$	63.1948	2.05247
$949$	22.1432	0.718798
$950$	−30.0533	−0.975059
$951$	−15.8817	−0.515001
$952$	−75.2364	−2.43843
$953$	−8.41563	−0.272609	−0.136305	−	0.990667i	$-0.543523\pi$
$953$	−8.41563	−0.272609	−0.136305	+	0.990667i	$0.543523\pi$
$954$	−5.57325	−0.180441
$955$	−7.02838	−0.227433
$956$	−80.3804	−2.59969
$957$	2.07866	0.0671935
$958$	13.3844	0.432431
$959$	−97.8661	−3.16026
$960$	−3.72024	−0.120070
$961$	28.5977	0.922505
$962$	21.0968	0.680187
$963$	6.22636	0.200642
$964$	−56.2517	−1.81174
$965$	9.87844	0.317998
$966$	61.7149	1.98565
$967$	11.7316	0.377264	0.188632	−	0.982048i	$-0.439595\pi$
$967$	11.7316	0.377264	0.188632	+	0.982048i	$0.439595\pi$
$968$	60.9071	1.95763
$969$	−8.12544	−0.261027
$970$	35.2144	1.13067
$971$	31.2437	1.00266	0.501328	−	0.865257i	$-0.332845\pi$
$971$	31.2437	1.00266	0.501328	+	0.865257i	$0.332845\pi$
$972$	4.24286	0.136090
$973$	61.0454	1.95703
$974$	56.1728	1.79989
$975$	7.93726	0.254196
$976$	−65.3469	−2.09170
$977$	43.4508	1.39012	0.695058	−	0.718954i	$-0.255379\pi$
$977$	43.4508	1.39012	0.695058	+	0.718954i	$0.255379\pi$
$978$	−26.6016	−0.850626
$979$	−3.53115	−0.112856
$980$	47.7508	1.52534
$981$	14.6190	0.466749
$982$	−97.0864	−3.09815
$983$	52.7600	1.68278	0.841391	−	0.540427i	$-0.181737\pi$
$983$	52.7600	1.68278	0.841391	+	0.540427i	$0.181737\pi$
$984$	−11.6587	−0.371666
$985$	−7.75902	−0.247223
$986$	−42.0601	−1.33947
$987$	−39.9433	−1.27141
$988$	21.4485	0.682367
$989$	6.96309	0.221414
$990$	−0.732611	−0.0232839
$991$	−46.1284	−1.46532	−0.732658	−	0.680597i	$-0.761721\pi$
$991$	−46.1284	−1.46532	−0.732658	+	0.680597i	$0.761721\pi$
$992$	−19.8759	−0.631061
$993$	12.4977	0.396602
$994$	147.042	4.66390
$995$	8.55889	0.271335
$996$	4.14944	0.131480
$997$	−49.7220	−1.57471	−0.787356	−	0.616499i	$-0.788550\pi$
$997$	−49.7220	−1.57471	−0.787356	+	0.616499i	$0.788550\pi$
$998$	−40.8179	−1.29207
$999$	4.62297	0.146264

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	309.2.a.d.1.1	✓	8
3.2	odd	2		927.2.a.g.1.8		8
4.3	odd	2		4944.2.a.bf.1.5		8
5.4	even	2		7725.2.a.z.1.8		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
309.2.a.d.1.1	✓	8	1.1	even	1	trivial
927.2.a.g.1.8		8	3.2	odd	2
4944.2.a.bf.1.5		8	4.3	odd	2
7725.2.a.z.1.8		8	5.4	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 309 = 3 \cdot 103 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	309.a (trivial)

\(f(q)\)	\(=\)	\(q-2.49857 q^{2} +1.00000 q^{3} +4.24286 q^{4} +0.808843 q^{5} -2.49857 q^{6} +4.57320 q^{7} -5.60396 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-2.49857 q^{2} +1.00000 q^{3} +4.24286 q^{4} +0.808843 q^{5} -2.49857 q^{6} +4.57320 q^{7} -5.60396 q^{8} +1.00000 q^{9} -2.02095 q^{10} +0.362508 q^{11} +4.24286 q^{12} -1.82643 q^{13} -11.4265 q^{14} +0.808843 q^{15} +5.51616 q^{16} +2.93571 q^{17} -2.49857 q^{18} -2.76779 q^{19} +3.43181 q^{20} +4.57320 q^{21} -0.905752 q^{22} -5.40105 q^{23} -5.60396 q^{24} -4.34577 q^{25} +4.56347 q^{26} +1.00000 q^{27} +19.4035 q^{28} +5.73411 q^{29} -2.02095 q^{30} +7.71995 q^{31} -2.57462 q^{32} +0.362508 q^{33} -7.33508 q^{34} +3.69900 q^{35} +4.24286 q^{36} +4.62297 q^{37} +6.91553 q^{38} -1.82643 q^{39} -4.53272 q^{40} +2.08044 q^{41} -11.4265 q^{42} -1.28921 q^{43} +1.53807 q^{44} +0.808843 q^{45} +13.4949 q^{46} -8.73420 q^{47} +5.51616 q^{48} +13.9142 q^{49} +10.8582 q^{50} +2.93571 q^{51} -7.74930 q^{52} +2.23057 q^{53} -2.49857 q^{54} +0.293212 q^{55} -25.6280 q^{56} -2.76779 q^{57} -14.3271 q^{58} +13.3912 q^{59} +3.43181 q^{60} -11.8465 q^{61} -19.2889 q^{62} +4.57320 q^{63} -4.59946 q^{64} -1.47730 q^{65} -0.905752 q^{66} -7.83115 q^{67} +12.4558 q^{68} -5.40105 q^{69} -9.24222 q^{70} -12.8686 q^{71} -5.60396 q^{72} -12.1237 q^{73} -11.5508 q^{74} -4.34577 q^{75} -11.7434 q^{76} +1.65782 q^{77} +4.56347 q^{78} +14.8944 q^{79} +4.46171 q^{80} +1.00000 q^{81} -5.19813 q^{82} +0.977981 q^{83} +19.4035 q^{84} +2.37453 q^{85} +3.22119 q^{86} +5.73411 q^{87} -2.03148 q^{88} -9.74090 q^{89} -2.02095 q^{90} -8.35264 q^{91} -22.9159 q^{92} +7.71995 q^{93} +21.8230 q^{94} -2.23871 q^{95} -2.57462 q^{96} -17.4247 q^{97} -34.7656 q^{98} +0.362508 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - q^{2} + 8 q^{3} + 11 q^{4} - q^{5} - q^{6} + 6 q^{7} - 3 q^{8} + 8 q^{9}+O(q^{10}) \) 8 * q - q^2 + 8 * q^3 + 11 * q^4 - q^5 - q^6 + 6 * q^7 - 3 * q^8 + 8 * q^9	\( 8 q - q^{2} + 8 q^{3} + 11 q^{4} - q^{5} - q^{6} + 6 q^{7} - 3 q^{8} + 8 q^{9} + 3 q^{10} + 6 q^{11} + 11 q^{12} + 9 q^{13} - 6 q^{14} - q^{15} + 9 q^{16} - 4 q^{17} - q^{18} + 16 q^{19} - 23 q^{20} + 6 q^{21} - 4 q^{22} - 11 q^{23} - 3 q^{24} + 15 q^{25} - 14 q^{26} + 8 q^{27} - 5 q^{28} + 3 q^{30} + 17 q^{31} - 12 q^{32} + 6 q^{33} - 8 q^{34} + 4 q^{35} + 11 q^{36} - 6 q^{37} - 3 q^{38} + 9 q^{39} + 3 q^{40} + 12 q^{41} - 6 q^{42} + 9 q^{43} + 8 q^{44} - q^{45} - 30 q^{46} - 6 q^{47} + 9 q^{48} + 18 q^{49} - 36 q^{50} - 4 q^{51} + 23 q^{52} - 16 q^{53} - q^{54} - 10 q^{55} - 13 q^{56} + 16 q^{57} - 22 q^{58} + 11 q^{59} - 23 q^{60} + 5 q^{61} - 25 q^{62} + 6 q^{63} - 35 q^{64} - 41 q^{65} - 4 q^{66} + 5 q^{67} - 19 q^{68} - 11 q^{69} - 48 q^{70} - 10 q^{71} - 3 q^{72} + 14 q^{73} + 4 q^{74} + 15 q^{75} - 12 q^{76} - 40 q^{77} - 14 q^{78} + 14 q^{79} - 19 q^{80} + 8 q^{81} - 13 q^{82} - 23 q^{83} - 5 q^{84} - 4 q^{85} - 3 q^{86} - 30 q^{88} - 14 q^{89} + 3 q^{90} - 6 q^{91} - 21 q^{92} + 17 q^{93} + 22 q^{94} + 6 q^{95} - 12 q^{96} + 3 q^{97} - 18 q^{98} + 6 q^{99}+O(q^{100}) \) 8 * q - q^2 + 8 * q^3 + 11 * q^4 - q^5 - q^6 + 6 * q^7 - 3 * q^8 + 8 * q^9 + 3 * q^10 + 6 * q^11 + 11 * q^12 + 9 * q^13 - 6 * q^14 - q^15 + 9 * q^16 - 4 * q^17 - q^18 + 16 * q^19 - 23 * q^20 + 6 * q^21 - 4 * q^22 - 11 * q^23 - 3 * q^24 + 15 * q^25 - 14 * q^26 + 8 * q^27 - 5 * q^28 + 3 * q^30 + 17 * q^31 - 12 * q^32 + 6 * q^33 - 8 * q^34 + 4 * q^35 + 11 * q^36 - 6 * q^37 - 3 * q^38 + 9 * q^39 + 3 * q^40 + 12 * q^41 - 6 * q^42 + 9 * q^43 + 8 * q^44 - q^45 - 30 * q^46 - 6 * q^47 + 9 * q^48 + 18 * q^49 - 36 * q^50 - 4 * q^51 + 23 * q^52 - 16 * q^53 - q^54 - 10 * q^55 - 13 * q^56 + 16 * q^57 - 22 * q^58 + 11 * q^59 - 23 * q^60 + 5 * q^61 - 25 * q^62 + 6 * q^63 - 35 * q^64 - 41 * q^65 - 4 * q^66 + 5 * q^67 - 19 * q^68 - 11 * q^69 - 48 * q^70 - 10 * q^71 - 3 * q^72 + 14 * q^73 + 4 * q^74 + 15 * q^75 - 12 * q^76 - 40 * q^77 - 14 * q^78 + 14 * q^79 - 19 * q^80 + 8 * q^81 - 13 * q^82 - 23 * q^83 - 5 * q^84 - 4 * q^85 - 3 * q^86 - 30 * q^88 - 14 * q^89 + 3 * q^90 - 6 * q^91 - 21 * q^92 + 17 * q^93 + 22 * q^94 + 6 * q^95 - 12 * q^96 + 3 * q^97 - 18 * q^98 + 6 * q^99

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