LMFDB - Embedded newform 115.2.a.c.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [115,2,Mod(1,115)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(115, 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("115.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 115 = 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	115.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:	$0.918279623245$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.15317.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 4x^{2} + 5x + 2 $ x^4 - 2x^3 - 4x^2 + 5*x + 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$2.69353$ of defining polynomial
Character	$\chi$	$=$	115.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.69353 q^{2} -2.56155 q^{3} +5.25508 q^{4} +1.00000 q^{5} -6.89961 q^{6} -2.74252 q^{7} +8.76763 q^{8} +3.56155 q^{9} +O(q^{10})$	\(q+2.69353 q^{2} -2.56155 q^{3} +5.25508 q^{4} +1.00000 q^{5} -6.89961 q^{6} -2.74252 q^{7} +8.76763 q^{8} +3.56155 q^{9} +2.69353 q^{10} -3.38705 q^{11} -13.4612 q^{12} -2.46356 q^{13} -7.38705 q^{14} -2.56155 q^{15} +13.1057 q^{16} -2.64453 q^{17} +9.59313 q^{18} +3.38705 q^{19} +5.25508 q^{20} +7.02511 q^{21} -9.12311 q^{22} -1.00000 q^{23} -22.4588 q^{24} +1.00000 q^{25} -6.63566 q^{26} -1.43845 q^{27} -14.4122 q^{28} +9.20608 q^{29} -6.89961 q^{30} -5.10809 q^{31} +17.7652 q^{32} +8.67611 q^{33} -7.12311 q^{34} -2.74252 q^{35} +18.7162 q^{36} +5.50369 q^{37} +9.12311 q^{38} +6.31054 q^{39} +8.76763 q^{40} -1.20608 q^{41} +18.9223 q^{42} +3.02511 q^{43} -17.7992 q^{44} +3.56155 q^{45} -2.69353 q^{46} +8.21255 q^{47} -33.5709 q^{48} +0.521423 q^{49} +2.69353 q^{50} +6.77410 q^{51} -12.9462 q^{52} -10.5417 q^{53} -3.87449 q^{54} -3.38705 q^{55} -24.0454 q^{56} -8.67611 q^{57} +24.7968 q^{58} +8.28259 q^{59} -13.4612 q^{60} +0.263945 q^{61} -13.7588 q^{62} -9.76763 q^{63} +21.6397 q^{64} -2.46356 q^{65} +23.3693 q^{66} -7.66964 q^{67} -13.8972 q^{68} +2.56155 q^{69} -7.38705 q^{70} -0.0150161 q^{71} +31.2264 q^{72} -5.53644 q^{73} +14.8243 q^{74} -2.56155 q^{75} +17.7992 q^{76} +9.28906 q^{77} +16.9976 q^{78} -8.67611 q^{79} +13.1057 q^{80} -7.00000 q^{81} -3.24861 q^{82} -3.52142 q^{83} +36.9175 q^{84} -2.64453 q^{85} +8.14822 q^{86} -23.5819 q^{87} -29.6964 q^{88} -4.66318 q^{89} +9.59313 q^{90} +6.75637 q^{91} -5.25508 q^{92} +13.0846 q^{93} +22.1207 q^{94} +3.38705 q^{95} -45.5066 q^{96} -4.09799 q^{97} +1.40447 q^{98} -12.0632 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{2} - 2 q^{3} + 4 q^{4} + 4 q^{5} - q^{6} - 3 q^{7} + 9 q^{8} + 6 q^{9}+O(q^{10}) $ 4 * q + 2 * q^2 - 2 * q^3 + 4 * q^4 + 4 * q^5 - q^6 - 3 * q^7 + 9 * q^8 + 6 * q^9	\( 4 q + 2 q^{2} - 2 q^{3} + 4 q^{4} + 4 q^{5} - q^{6} - 3 q^{7} + 9 q^{8} + 6 q^{9} + 2 q^{10} + 4 q^{11} - 19 q^{12} - 12 q^{14} - 2 q^{15} + 8 q^{16} - q^{17} + 3 q^{18} - 4 q^{19} + 4 q^{20} + 10 q^{21} - 20 q^{22} - 4 q^{23} - 30 q^{24} + 4 q^{25} - q^{26} - 14 q^{27} - 22 q^{28} + 19 q^{29} - q^{30} - q^{31} + 20 q^{32} - 2 q^{33} - 12 q^{34} - 3 q^{35} + 23 q^{36} - 3 q^{37} + 20 q^{38} + 9 q^{40} + 13 q^{41} + 6 q^{42} - 6 q^{43} - 18 q^{44} + 6 q^{45} - 2 q^{46} + 6 q^{47} - 21 q^{48} + 9 q^{49} + 2 q^{50} - 8 q^{51} - q^{52} + 19 q^{53} - 7 q^{54} + 4 q^{55} - 10 q^{56} + 2 q^{57} + 21 q^{58} + 23 q^{59} - 19 q^{60} - 13 q^{62} - 13 q^{63} + 27 q^{64} + 44 q^{66} - 3 q^{67} - 4 q^{68} + 2 q^{69} - 12 q^{70} - 3 q^{71} + 39 q^{72} - 32 q^{73} - 12 q^{74} - 2 q^{75} + 18 q^{76} + 18 q^{77} + 43 q^{78} + 2 q^{79} + 8 q^{80} - 28 q^{81} - 5 q^{82} - 21 q^{83} + 28 q^{84} - q^{85} - 2 q^{86} - 18 q^{87} - 14 q^{88} + 3 q^{90} - 40 q^{91} - 4 q^{92} - 8 q^{93} + 47 q^{94} - 4 q^{95} - 61 q^{96} - 18 q^{97} + 16 q^{98} + 6 q^{99}+O(q^{100}) \) 4 * q + 2 * q^2 - 2 * q^3 + 4 * q^4 + 4 * q^5 - q^6 - 3 * q^7 + 9 * q^8 + 6 * q^9 + 2 * q^10 + 4 * q^11 - 19 * q^12 - 12 * q^14 - 2 * q^15 + 8 * q^16 - q^17 + 3 * q^18 - 4 * q^19 + 4 * q^20 + 10 * q^21 - 20 * q^22 - 4 * q^23 - 30 * q^24 + 4 * q^25 - q^26 - 14 * q^27 - 22 * q^28 + 19 * q^29 - q^30 - q^31 + 20 * q^32 - 2 * q^33 - 12 * q^34 - 3 * q^35 + 23 * q^36 - 3 * q^37 + 20 * q^38 + 9 * q^40 + 13 * q^41 + 6 * q^42 - 6 * q^43 - 18 * q^44 + 6 * q^45 - 2 * q^46 + 6 * q^47 - 21 * q^48 + 9 * q^49 + 2 * q^50 - 8 * q^51 - q^52 + 19 * q^53 - 7 * q^54 + 4 * q^55 - 10 * q^56 + 2 * q^57 + 21 * q^58 + 23 * q^59 - 19 * q^60 - 13 * q^62 - 13 * q^63 + 27 * q^64 + 44 * q^66 - 3 * q^67 - 4 * q^68 + 2 * q^69 - 12 * q^70 - 3 * q^71 + 39 * q^72 - 32 * q^73 - 12 * q^74 - 2 * q^75 + 18 * q^76 + 18 * q^77 + 43 * q^78 + 2 * q^79 + 8 * q^80 - 28 * q^81 - 5 * q^82 - 21 * q^83 + 28 * q^84 - q^85 - 2 * q^86 - 18 * q^87 - 14 * q^88 + 3 * q^90 - 40 * q^91 - 4 * q^92 - 8 * q^93 + 47 * q^94 - 4 * q^95 - 61 * q^96 - 18 * q^97 + 16 * 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 $

$2$	2.69353	1.90461	0.952305	−	0.305148i	$-0.0987059\pi$
$2$	2.69353	1.90461	0.952305	+	0.305148i	$0.0987059\pi$
$3$	−2.56155	−1.47891	−0.739457	−	0.673204i	$-0.764917\pi$
$3$	−2.56155	−1.47891	−0.739457	+	0.673204i	$0.764917\pi$
$4$	5.25508	2.62754
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	−6.89961	−2.81675
$7$	−2.74252	−1.03658	−0.518288	−	0.855206i	$-0.673430\pi$
$7$	−2.74252	−1.03658	−0.518288	+	0.855206i	$0.673430\pi$
$8$	8.76763	3.09983
$9$	3.56155	1.18718
$10$	2.69353	0.851767
$11$	−3.38705	−1.02123	−0.510617	−	0.859808i	$-0.670583\pi$
$11$	−3.38705	−1.02123	−0.510617	+	0.859808i	$0.670583\pi$
$12$	−13.4612	−3.88590
$13$	−2.46356	−0.683269	−0.341634	−	0.939833i	$-0.610980\pi$
$13$	−2.46356	−0.683269	−0.341634	+	0.939833i	$0.610980\pi$
$14$	−7.38705	−1.97427
$15$	−2.56155	−0.661390
$16$	13.1057	3.27642
$17$	−2.64453	−0.641393	−0.320696	−	0.947182i	$-0.603917\pi$
$17$	−2.64453	−0.641393	−0.320696	+	0.947182i	$0.603917\pi$
$18$	9.59313	2.26112
$19$	3.38705	0.777043	0.388521	−	0.921440i	$-0.372986\pi$
$19$	3.38705	0.777043	0.388521	+	0.921440i	$0.372986\pi$
$20$	5.25508	1.17507
$21$	7.02511	1.53301
$22$	−9.12311	−1.94505
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	−22.4588	−4.58438
$25$	1.00000	0.200000
$26$	−6.63566	−1.30136
$27$	−1.43845	−0.276829
$28$	−14.4122	−2.72364
$29$	9.20608	1.70953	0.854763	−	0.519018i	$-0.173702\pi$
$29$	9.20608	1.70953	0.854763	+	0.519018i	$0.173702\pi$
$30$	−6.89961	−1.25969
$31$	−5.10809	−0.917440	−0.458720	−	0.888581i	$-0.651692\pi$
$31$	−5.10809	−0.917440	−0.458720	+	0.888581i	$0.651692\pi$
$32$	17.7652	3.14048
$33$	8.67611	1.51032
$34$	−7.12311	−1.22160
$35$	−2.74252	−0.463571
$36$	18.7162	3.11937
$37$	5.50369	0.904801	0.452401	−	0.891815i	$-0.350568\pi$
$37$	5.50369	0.904801	0.452401	+	0.891815i	$0.350568\pi$
$38$	9.12311	1.47996
$39$	6.31054	1.01050
$40$	8.76763	1.38628
$41$	−1.20608	−0.188358	−0.0941792	−	0.995555i	$-0.530023\pi$
$41$	−1.20608	−0.188358	−0.0941792	+	0.995555i	$0.530023\pi$
$42$	18.9223	2.91978
$43$	3.02511	0.461325	0.230663	−	0.973034i	$-0.425911\pi$
$43$	3.02511	0.461325	0.230663	+	0.973034i	$0.425911\pi$
$44$	−17.7992	−2.68333
$45$	3.56155	0.530925
$46$	−2.69353	−0.397139
$47$	8.21255	1.19792	0.598962	−	0.800778i	$-0.295580\pi$
$47$	8.21255	1.19792	0.598962	+	0.800778i	$0.295580\pi$
$48$	−33.5709	−4.84554
$49$	0.521423	0.0744891
$50$	2.69353	0.380922
$51$	6.77410	0.948564
$52$	−12.9462	−1.79532
$53$	−10.5417	−1.44802	−0.724009	−	0.689790i	$-0.757703\pi$
$53$	−10.5417	−1.44802	−0.724009	+	0.689790i	$0.757703\pi$
$54$	−3.87449	−0.527252
$55$	−3.38705	−0.456710
$56$	−24.0454	−3.21321
$57$	−8.67611	−1.14918
$58$	24.7968	3.25598
$59$	8.28259	1.07830	0.539151	−	0.842209i	$-0.318745\pi$
$59$	8.28259	1.07830	0.539151	+	0.842209i	$0.318745\pi$
$60$	−13.4612	−1.73783
$61$	0.263945	0.0337947	0.0168973	−	0.999857i	$-0.494621\pi$
$61$	0.263945	0.0337947	0.0168973	+	0.999857i	$0.494621\pi$
$62$	−13.7588	−1.74737
$63$	−9.76763	−1.23061
$64$	21.6397	2.70497
$65$	−2.46356	−0.305567
$66$	23.3693	2.87656
$67$	−7.66964	−0.936996	−0.468498	−	0.883465i	$-0.655205\pi$
$67$	−7.66964	−0.936996	−0.468498	+	0.883465i	$0.655205\pi$
$68$	−13.8972	−1.68528
$69$	2.56155	0.308375
$70$	−7.38705	−0.882921
$71$	−0.0150161	−0.00178208	−0.000891041	−	1.00000i	$-0.500284\pi$
$71$	−0.0150161	−0.00178208	−0.000891041		1.00000i	$0.500284\pi$
$72$	31.2264	3.68007
$73$	−5.53644	−0.647991	−0.323996	−	0.946059i	$-0.605026\pi$
$73$	−5.53644	−0.647991	−0.323996	+	0.946059i	$0.605026\pi$
$74$	14.8243	1.72329
$75$	−2.56155	−0.295783
$76$	17.7992	2.04171
$77$	9.28906	1.05859
$78$	16.9976	1.92460
$79$	−8.67611	−0.976138	−0.488069	−	0.872805i	$-0.662299\pi$
$79$	−8.67611	−0.976138	−0.488069	+	0.872805i	$0.662299\pi$
$80$	13.1057	1.46526
$81$	−7.00000	−0.777778
$82$	−3.24861	−0.358749
$83$	−3.52142	−0.386526	−0.193263	−	0.981147i	$-0.561907\pi$
$83$	−3.52142	−0.386526	−0.193263	+	0.981147i	$0.561907\pi$
$84$	36.9175	4.02803
$85$	−2.64453	−0.286839
$86$	8.14822	0.878645
$87$	−23.5819	−2.52824
$88$	−29.6964	−3.16565
$89$	−4.66318	−0.494296	−0.247148	−	0.968978i	$-0.579493\pi$
$89$	−4.66318	−0.494296	−0.247148	+	0.968978i	$0.579493\pi$
$90$	9.59313	1.01120
$91$	6.75637	0.708260
$92$	−5.25508	−0.547880
$93$	13.0846	1.35681
$94$	22.1207	2.28158
$95$	3.38705	0.347504
$96$	−45.5066	−4.64450
$97$	−4.09799	−0.416088	−0.208044	−	0.978119i	$-0.566710\pi$
$97$	−4.09799	−0.416088	−0.208044	+	0.978119i	$0.566710\pi$
$98$	1.40447	0.141873
$99$	−12.0632	−1.21239
$100$	5.25508	0.525508
$101$	12.2955	1.22345	0.611725	−	0.791070i	$-0.290476\pi$
$101$	12.2955	1.22345	0.611725	+	0.791070i	$0.290476\pi$
$102$	18.2462	1.80664
$103$	6.05023	0.596147	0.298073	−	0.954543i	$-0.403656\pi$
$103$	6.05023	0.596147	0.298073	+	0.954543i	$0.403656\pi$
$104$	−21.5996	−2.11801
$105$	7.02511	0.685581
$106$	−28.3944	−2.75791
$107$	13.1547	1.27171	0.635856	−	0.771808i	$-0.280647\pi$
$107$	13.1547	1.27171	0.635856	+	0.771808i	$0.280647\pi$
$108$	−7.55915	−0.727380
$109$	−17.1183	−1.63964	−0.819818	−	0.572624i	$-0.805925\pi$
$109$	−17.1183	−1.63964	−0.819818	+	0.572624i	$0.805925\pi$
$110$	−9.12311	−0.869854
$111$	−14.0980	−1.33812
$112$	−35.9426	−3.39626
$113$	4.38058	0.412091	0.206045	−	0.978542i	$-0.433941\pi$
$113$	4.38058	0.412091	0.206045	+	0.978542i	$0.433941\pi$
$114$	−23.3693	−2.18874
$115$	−1.00000	−0.0932505
$116$	48.3787	4.49185
$117$	−8.77410	−0.811166
$118$	22.3094	2.05374
$119$	7.25268	0.664852
$120$	−22.4588	−2.05019
$121$	0.472110	0.0429191
$122$	0.710942	0.0643657
$123$	3.08944	0.278566
$124$	−26.8434	−2.41061
$125$	1.00000	0.0894427
$126$	−26.3094	−2.34382
$127$	−18.4588	−1.63795	−0.818975	−	0.573829i	$-0.805457\pi$
$127$	−18.4588	−1.63795	−0.818975	+	0.573829i	$0.805457\pi$
$128$	22.7567	2.01143
$129$	−7.74899	−0.682260
$130$	−6.63566	−0.581986
$131$	−3.86834	−0.337979	−0.168989	−	0.985618i	$-0.554050\pi$
$131$	−3.86834	−0.337979	−0.168989	+	0.985618i	$0.554050\pi$
$132$	45.5936	3.96842
$133$	−9.28906	−0.805463
$134$	−20.6584	−1.78461
$135$	−1.43845	−0.123802
$136$	−23.1863	−1.98821
$137$	20.6713	1.76607	0.883034	−	0.469308i	$-0.155497\pi$
$137$	20.6713	1.76607	0.883034	+	0.469308i	$0.155497\pi$
$138$	6.89961	0.587334
$139$	12.5802	1.06704	0.533519	−	0.845788i	$-0.320869\pi$
$139$	12.5802	1.06704	0.533519	+	0.845788i	$0.320869\pi$
$140$	−14.4122	−1.21805
$141$	−21.0369	−1.77162
$142$	−0.0404462	−0.00339417
$143$	8.34420	0.697777
$144$	46.6766	3.88972
$145$	9.20608	0.764523
$146$	−14.9125	−1.23417
$147$	−1.33565	−0.110163
$148$	28.9223	2.37740
$149$	11.4850	0.940891	0.470446	−	0.882429i	$-0.344093\pi$
$149$	11.4850	0.940891	0.470446	+	0.882429i	$0.344093\pi$
$150$	−6.89961	−0.563351
$151$	−5.58667	−0.454636	−0.227318	−	0.973821i	$-0.572996\pi$
$151$	−5.58667	−0.454636	−0.227318	+	0.973821i	$0.572996\pi$
$152$	29.6964	2.40870
$153$	−9.41863	−0.761451
$154$	25.0203	2.01619
$155$	−5.10809	−0.410292
$156$	33.1624	2.65512
$157$	5.86563	0.468128	0.234064	−	0.972221i	$-0.424797\pi$
$157$	5.86563	0.468128	0.234064	+	0.972221i	$0.424797\pi$
$158$	−23.3693	−1.85916
$159$	27.0032	2.14149
$160$	17.7652	1.40447
$161$	2.74252	0.216141
$162$	−18.8547	−1.48136
$163$	0.0465955	0.00364964	0.00182482	−	0.999998i	$-0.499419\pi$
$163$	0.0465955	0.00364964	0.00182482	+	0.999998i	$0.499419\pi$
$164$	−6.33805	−0.494919
$165$	8.67611	0.675434
$166$	−9.48504	−0.736182
$167$	−2.24621	−0.173817	−0.0869085	−	0.996216i	$-0.527699\pi$
$167$	−2.24621	−0.173817	−0.0869085	+	0.996216i	$0.527699\pi$
$168$	61.5936	4.75205
$169$	−6.93087	−0.533144
$170$	−7.12311	−0.546317
$171$	12.0632	0.922493
$172$	15.8972	1.21215
$173$	8.01293	0.609212	0.304606	−	0.952478i	$-0.401475\pi$
$173$	8.01293	0.609212	0.304606	+	0.952478i	$0.401475\pi$
$174$	−63.5183	−4.81531
$175$	−2.74252	−0.207315
$176$	−44.3896	−3.34599
$177$	−21.2163	−1.59471
$178$	−12.5604	−0.941440
$179$	−11.9615	−0.894047	−0.447024	−	0.894522i	$-0.647516\pi$
$179$	−11.9615	−0.894047	−0.447024	+	0.894522i	$0.647516\pi$
$180$	18.7162	1.39503
$181$	−17.4802	−1.29930	−0.649648	−	0.760235i	$-0.725084\pi$
$181$	−17.4802	−1.29930	−0.649648	+	0.760235i	$0.725084\pi$
$182$	18.1984	1.34896
$183$	−0.676108	−0.0499794
$184$	−8.76763	−0.646359
$185$	5.50369	0.404639
$186$	35.2438	2.58420
$187$	8.95715	0.655012
$188$	43.1576	3.14759
$189$	3.94497	0.286954
$190$	9.12311	0.661860
$191$	−13.1231	−0.949555	−0.474777	−	0.880106i	$-0.657471\pi$
$191$	−13.1231	−0.949555	−0.474777	+	0.880106i	$0.657471\pi$
$192$	−55.4313	−4.00041
$193$	−12.5438	−0.902924	−0.451462	−	0.892290i	$-0.649097\pi$
$193$	−12.5438	−0.902924	−0.451462	+	0.892290i	$0.649097\pi$
$194$	−11.0380	−0.792485
$195$	6.31054	0.451907
$196$	2.74012	0.195723
$197$	3.88544	0.276826	0.138413	−	0.990375i	$-0.455800\pi$
$197$	3.88544	0.276826	0.138413	+	0.990375i	$0.455800\pi$
$198$	−32.4924	−2.30914
$199$	22.1612	1.57096	0.785481	−	0.618886i	$-0.212416\pi$
$199$	22.1612	1.57096	0.785481	+	0.618886i	$0.212416\pi$
$200$	8.76763	0.619965
$201$	19.6462	1.38574
$202$	33.1183	2.33020
$203$	−25.2479	−1.77205
$204$	35.5984	2.49239
$205$	−1.20608	−0.0842364
$206$	16.2964	1.13543
$207$	−3.56155	−0.247545
$208$	−32.2867	−2.23868
$209$	−11.4721	−0.793542
$210$	18.9223	1.30576
$211$	4.81048	0.331167	0.165584	−	0.986196i	$-0.447049\pi$
$211$	4.81048	0.331167	0.165584	+	0.986196i	$0.447049\pi$
$212$	−55.3976	−3.80473
$213$	0.0384645	0.00263554
$214$	35.4325	2.42211
$215$	3.02511	0.206311
$216$	−12.6118	−0.858123
$217$	14.0090	0.950996
$218$	−46.1086	−3.12287
$219$	14.1819	0.958323
$220$	−17.7992	−1.20002
$221$	6.51496	0.438243
$222$	−37.9733	−2.54860
$223$	13.7815	0.922876	0.461438	−	0.887172i	$-0.347334\pi$
$223$	13.7815	0.922876	0.461438	+	0.887172i	$0.347334\pi$
$224$	−48.7215	−3.25534
$225$	3.56155	0.237437
$226$	11.7992	0.784872
$227$	29.1012	1.93151	0.965757	−	0.259447i	$-0.0835402\pi$
$227$	29.1012	1.93151	0.965757	+	0.259447i	$0.0835402\pi$
$228$	−45.5936	−3.01951
$229$	−9.38705	−0.620314	−0.310157	−	0.950685i	$-0.600382\pi$
$229$	−9.38705	−0.620314	−0.310157	+	0.950685i	$0.600382\pi$
$230$	−2.69353	−0.177606
$231$	−23.7944	−1.56556
$232$	80.7156	5.29924
$233$	−12.6725	−0.830202	−0.415101	−	0.909775i	$-0.636254\pi$
$233$	−12.6725	−0.830202	−0.415101	+	0.909775i	$0.636254\pi$
$234$	−23.6333	−1.54495
$235$	8.21255	0.535728
$236$	43.5257	2.83328
$237$	22.2243	1.44362
$238$	19.5353	1.26628
$239$	19.1883	1.24119	0.620596	−	0.784131i	$-0.286891\pi$
$239$	19.1883	1.24119	0.620596	+	0.784131i	$0.286891\pi$
$240$	−33.5709	−2.16699
$241$	16.7741	1.08051	0.540257	−	0.841500i	$-0.318327\pi$
$241$	16.7741	1.08051	0.540257	+	0.841500i	$0.318327\pi$
$242$	1.27164	0.0817442
$243$	22.2462	1.42710
$244$	1.38705	0.0887968
$245$	0.521423	0.0333125
$246$	8.32149	0.530559
$247$	−8.34420	−0.530929
$248$	−44.7859	−2.84391
$249$	9.02031	0.571639
$250$	2.69353	0.170353
$251$	−22.8114	−1.43984	−0.719921	−	0.694056i	$-0.755822\pi$
$251$	−22.8114	−1.43984	−0.719921	+	0.694056i	$0.755822\pi$
$252$	−51.3297	−3.23347
$253$	3.38705	0.212942
$254$	−49.7191	−3.11966
$255$	6.77410	0.424211
$256$	18.0162	1.12602
$257$	−23.7526	−1.48165	−0.740824	−	0.671699i	$-0.765565\pi$
$257$	−23.7526	−1.48165	−0.740824	+	0.671699i	$0.765565\pi$
$258$	−20.8721	−1.29944
$259$	−15.0940	−0.937895
$260$	−12.9462	−0.802889
$261$	32.7879	2.02952
$262$	−10.4195	−0.643718
$263$	16.5895	1.02295	0.511476	−	0.859297i	$-0.329099\pi$
$263$	16.5895	1.02295	0.511476	+	0.859297i	$0.329099\pi$
$264$	76.0689	4.68172
$265$	−10.5417	−0.647574
$266$	−25.0203	−1.53409
$267$	11.9450	0.731020
$268$	−40.3046	−2.46199
$269$	−8.47495	−0.516727	−0.258363	−	0.966048i	$-0.583183\pi$
$269$	−8.47495	−0.516727	−0.258363	+	0.966048i	$0.583183\pi$
$270$	−3.87449	−0.235794
$271$	−21.3029	−1.29406	−0.647030	−	0.762465i	$-0.723989\pi$
$271$	−21.3029	−1.29406	−0.647030	+	0.762465i	$0.723989\pi$
$272$	−34.6584	−2.10147
$273$	−17.3068	−1.04745
$274$	55.6787	3.36367
$275$	−3.38705	−0.204247
$276$	13.4612	0.810267
$277$	21.9615	1.31954	0.659770	−	0.751467i	$-0.270654\pi$
$277$	21.9615	1.31954	0.659770	+	0.751467i	$0.270654\pi$
$278$	33.8851	2.03229
$279$	−18.1927	−1.08917
$280$	−24.0454	−1.43699
$281$	19.9020	1.18725	0.593627	−	0.804740i	$-0.297695\pi$
$281$	19.9020	1.18725	0.593627	+	0.804740i	$0.297695\pi$
$282$	−56.6634	−3.37425
$283$	−8.87781	−0.527731	−0.263865	−	0.964559i	$-0.584997\pi$
$283$	−8.87781	−0.527731	−0.263865	+	0.964559i	$0.584997\pi$
$284$	−0.0789107	−0.00468249
$285$	−8.67611	−0.513928
$286$	22.4753	1.32899
$287$	3.30771	0.195248
$288$	63.2718	3.72833
$289$	−10.0065	−0.588616
$290$	24.7968	1.45612
$291$	10.4972	0.615358
$292$	−29.0944	−1.70262
$293$	23.9709	1.40039	0.700197	−	0.713950i	$-0.253096\pi$
$293$	23.9709	1.40039	0.700197	+	0.713950i	$0.253096\pi$
$294$	−3.59762	−0.209817
$295$	8.28259	0.482231
$296$	48.2543	2.80473
$297$	4.87209	0.282708
$298$	30.9353	1.79203
$299$	2.46356	0.142471
$300$	−13.4612	−0.777180
$301$	−8.29644	−0.478199
$302$	−15.0478	−0.865905
$303$	−31.4956	−1.80938
$304$	44.3896	2.54592
$305$	0.263945	0.0151134
$306$	−25.3693	−1.45027
$307$	12.0632	0.688481	0.344240	−	0.938882i	$-0.388136\pi$
$307$	12.0632	0.688481	0.344240	+	0.938882i	$0.388136\pi$
$308$	48.8147	2.78148
$309$	−15.4980	−0.881649
$310$	−13.7588	−0.781446
$311$	−6.81257	−0.386305	−0.193153	−	0.981169i	$-0.561871\pi$
$311$	−6.81257	−0.386305	−0.193153	+	0.981169i	$0.561871\pi$
$312$	55.3285	3.13236
$313$	−6.38539	−0.360923	−0.180462	−	0.983582i	$-0.557759\pi$
$313$	−6.38539	−0.360923	−0.180462	+	0.983582i	$0.557759\pi$
$314$	15.7992	0.891601
$315$	−9.76763	−0.550344
$316$	−45.5936	−2.56484
$317$	16.8114	0.944222	0.472111	−	0.881539i	$-0.343492\pi$
$317$	16.8114	0.944222	0.472111	+	0.881539i	$0.343492\pi$
$318$	72.7338	4.07871
$319$	−31.1815	−1.74583
$320$	21.6397	1.20970
$321$	−33.6964	−1.88075
$322$	7.38705	0.411664
$323$	−8.95715	−0.498389
$324$	−36.7855	−2.04364
$325$	−2.46356	−0.136654
$326$	0.125506	0.00695115
$327$	43.8494	2.42488
$328$	−10.5745	−0.583878
$329$	−22.5231	−1.24174
$330$	23.3693	1.28644
$331$	3.29114	0.180898	0.0904488	−	0.995901i	$-0.471170\pi$
$331$	3.29114	0.180898	0.0904488	+	0.995901i	$0.471170\pi$
$332$	−18.5054	−1.01561
$333$	19.6017	1.07417
$334$	−6.05023	−0.331054
$335$	−7.66964	−0.419037
$336$	92.0689	5.02277
$337$	−7.90201	−0.430450	−0.215225	−	0.976565i	$-0.569048\pi$
$337$	−7.90201	−0.430450	−0.215225	+	0.976565i	$0.569048\pi$
$338$	−18.6685	−1.01543
$339$	−11.2211	−0.609446
$340$	−13.8972	−0.753682
$341$	17.3014	0.936921
$342$	32.4924	1.75699
$343$	17.7676	0.959362
$344$	26.5231	1.43003
$345$	2.56155	0.137909
$346$	21.5830	1.16031
$347$	−19.4024	−1.04158	−0.520789	−	0.853686i	$-0.674362\pi$
$347$	−19.4024	−1.04158	−0.520789	+	0.853686i	$0.674362\pi$
$348$	−123.925	−6.64305
$349$	−33.1664	−1.77536	−0.887680	−	0.460462i	$-0.847684\pi$
$349$	−33.1664	−1.77536	−0.887680	+	0.460462i	$0.847684\pi$
$350$	−7.38705	−0.394854
$351$	3.54370	0.189149
$352$	−60.1717	−3.20716
$353$	−29.8960	−1.59121	−0.795603	−	0.605819i	$-0.792846\pi$
$353$	−29.8960	−1.59121	−0.795603	+	0.605819i	$0.792846\pi$
$354$	−57.1466	−3.03731
$355$	−0.0150161	−0.000796971 0
$356$	−24.5054	−1.29878
$357$	−18.5781	−0.983258
$358$	−32.2187	−1.70281
$359$	−24.9725	−1.31800	−0.659000	−	0.752143i	$-0.729020\pi$
$359$	−24.9725	−1.31800	−0.659000	+	0.752143i	$0.729020\pi$
$360$	31.2264	1.64578
$361$	−7.52789	−0.396205
$362$	−47.0835	−2.47465
$363$	−1.20934	−0.0634737
$364$	35.5052	1.86098
$365$	−5.53644	−0.289790
$366$	−1.82112	−0.0951912
$367$	15.8179	0.825686	0.412843	−	0.910802i	$-0.364536\pi$
$367$	15.8179	0.825686	0.412843	+	0.910802i	$0.364536\pi$
$368$	−13.1057	−0.683181
$369$	−4.29552	−0.223616
$370$	14.8243	0.770680
$371$	28.9109	1.50098
$372$	68.7608	3.56508
$373$	22.6283	1.17165	0.585826	−	0.810437i	$-0.300770\pi$
$373$	22.6283	1.17165	0.585826	+	0.810437i	$0.300770\pi$
$374$	24.1263	1.24754
$375$	−2.56155	−0.132278
$376$	72.0046	3.71335
$377$	−22.6797	−1.16807
$378$	10.6259	0.546536
$379$	10.2721	0.527641	0.263821	−	0.964572i	$-0.415017\pi$
$379$	10.2721	0.527641	0.263821	+	0.964572i	$0.415017\pi$
$380$	17.7992	0.913080
$381$	47.2831	2.42239
$382$	−35.3474	−1.80853
$383$	6.21463	0.317553	0.158776	−	0.987315i	$-0.449245\pi$
$383$	6.21463	0.317553	0.158776	+	0.987315i	$0.449245\pi$
$384$	−58.2924	−2.97472
$385$	9.28906	0.473414
$386$	−33.7871	−1.71972
$387$	10.7741	0.547678
$388$	−21.5353	−1.09329
$389$	27.6414	1.40147	0.700737	−	0.713420i	$-0.252855\pi$
$389$	27.6414	1.40147	0.700737	+	0.713420i	$0.252855\pi$
$390$	16.9976	0.860707
$391$	2.64453	0.133740
$392$	4.57165	0.230903
$393$	9.90897	0.499841
$394$	10.4655	0.527246
$395$	−8.67611	−0.436542
$396$	−63.3928	−3.18561
$397$	1.07171	0.0537875	0.0268938	−	0.999638i	$-0.491438\pi$
$397$	1.07171	0.0537875	0.0268938	+	0.999638i	$0.491438\pi$
$398$	59.6916	2.99207
$399$	23.7944	1.19121
$400$	13.1057	0.655284
$401$	1.49798	0.0748053	0.0374027	−	0.999300i	$-0.488092\pi$
$401$	1.49798	0.0748053	0.0374027	+	0.999300i	$0.488092\pi$
$402$	52.9175	2.63929
$403$	12.5841	0.626858
$404$	64.6139	3.21466
$405$	−7.00000	−0.347833
$406$	−68.0058	−3.37507
$407$	−18.6413	−0.924014
$408$	59.3928	2.94038
$409$	20.9373	1.03528	0.517642	−	0.855597i	$-0.326810\pi$
$409$	20.9373	1.03528	0.517642	+	0.855597i	$0.326810\pi$
$410$	−3.24861	−0.160438
$411$	−52.9506	−2.61186
$412$	31.7944	1.56640
$413$	−22.7152	−1.11774
$414$	−9.59313	−0.471477
$415$	−3.52142	−0.172860
$416$	−43.7657	−2.14579
$417$	−32.2248	−1.57806
$418$	−30.9004	−1.51139
$419$	−18.7564	−0.916309	−0.458154	−	0.888873i	$-0.651489\pi$
$419$	−18.7564	−0.916309	−0.458154	+	0.888873i	$0.651489\pi$
$420$	36.9175	1.80139
$421$	20.6163	1.00478	0.502388	−	0.864642i	$-0.332455\pi$
$421$	20.6163	1.00478	0.502388	+	0.864642i	$0.332455\pi$
$422$	12.9572	0.630744
$423$	29.2494	1.42216
$424$	−92.4261	−4.48861
$425$	−2.64453	−0.128279
$426$	0.103605	0.00501968
$427$	−0.723874	−0.0350307
$428$	69.1289	3.34147
$429$	−21.3741	−1.03195
$430$	8.14822	0.392942
$431$	27.0155	1.30129	0.650646	−	0.759381i	$-0.274498\pi$
$431$	27.0155	1.30129	0.650646	+	0.759381i	$0.274498\pi$
$432$	−18.8518	−0.907010
$433$	−25.2900	−1.21536	−0.607679	−	0.794183i	$-0.707899\pi$
$433$	−25.2900	−1.21536	−0.607679	+	0.794183i	$0.707899\pi$
$434$	37.7337	1.81128
$435$	−23.5819	−1.13066
$436$	−89.9580	−4.30821
$437$	−3.38705	−0.162025
$438$	38.1993	1.82523
$439$	34.4110	1.64235	0.821174	−	0.570679i	$-0.193320\pi$
$439$	34.4110	1.64235	0.821174	+	0.570679i	$0.193320\pi$
$440$	−29.6964	−1.41572
$441$	1.85708	0.0884322
$442$	17.5482	0.834683
$443$	−29.8281	−1.41717	−0.708587	−	0.705623i	$-0.750667\pi$
$443$	−29.8281	−1.41717	−0.708587	+	0.705623i	$0.750667\pi$
$444$	−74.0860	−3.51597
$445$	−4.66318	−0.221056
$446$	37.1208	1.75772
$447$	−29.4195	−1.39150
$448$	−59.3474	−2.80390
$449$	−2.67456	−0.126220	−0.0631102	−	0.998007i	$-0.520102\pi$
$449$	−2.67456	−0.126220	−0.0631102	+	0.998007i	$0.520102\pi$
$450$	9.59313	0.452225
$451$	4.08506	0.192358
$452$	23.0203	1.08278
$453$	14.3105	0.672368
$454$	78.3848	3.67878
$455$	6.75637	0.316743
$456$	−76.0689	−3.56225
$457$	−30.9037	−1.44561	−0.722806	−	0.691051i	$-0.757148\pi$
$457$	−30.9037	−1.44561	−0.722806	+	0.691051i	$0.757148\pi$
$458$	−25.2843	−1.18146
$459$	3.80402	0.177556
$460$	−5.25508	−0.245019
$461$	−26.6022	−1.23899	−0.619493	−	0.785002i	$-0.712662\pi$
$461$	−26.6022	−1.23899	−0.619493	+	0.785002i	$0.712662\pi$
$462$	−64.0909	−2.98178
$463$	26.7013	1.24092	0.620458	−	0.784239i	$-0.286947\pi$
$463$	26.7013	1.24092	0.620458	+	0.784239i	$0.286947\pi$
$464$	120.652	5.60113
$465$	13.0846	0.606786
$466$	−34.1336	−1.58121
$467$	35.2503	1.63119	0.815596	−	0.578622i	$-0.196410\pi$
$467$	35.2503	1.63119	0.815596	+	0.578622i	$0.196410\pi$
$468$	−46.1086	−2.13137
$469$	21.0342	0.971267
$470$	22.1207	1.02035
$471$	−15.0251	−0.692321
$472$	72.6187	3.34255
$473$	−10.2462	−0.471121
$474$	59.8617	2.74954
$475$	3.38705	0.155409
$476$	38.1134	1.74692
$477$	−37.5449	−1.71907
$478$	51.6843	2.36398
$479$	−39.0705	−1.78518	−0.892589	−	0.450871i	$-0.851114\pi$
$479$	−39.0705	−1.78518	−0.892589	+	0.450871i	$0.851114\pi$
$480$	−45.5066	−2.07708
$481$	−13.5587	−0.618222
$482$	45.1815	2.05796
$483$	−7.02511	−0.319654
$484$	2.48098	0.112772
$485$	−4.09799	−0.186080
$486$	59.9207	2.71806
$487$	24.0839	1.09135	0.545673	−	0.837998i	$-0.316274\pi$
$487$	24.0839	1.09135	0.545673	+	0.837998i	$0.316274\pi$
$488$	2.31417	0.104758
$489$	−0.119357	−0.00539751
$490$	1.40447	0.0634474
$491$	5.60051	0.252748	0.126374	−	0.991983i	$-0.459666\pi$
$491$	5.60051	0.252748	0.126374	+	0.991983i	$0.459666\pi$
$492$	16.2353	0.731942
$493$	−24.3458	−1.09648
$494$	−22.4753	−1.01121
$495$	−12.0632	−0.542199
$496$	−66.9450	−3.00592
$497$	0.0411819	0.00184726
$498$	24.2964	1.08875
$499$	9.53319	0.426764	0.213382	−	0.976969i	$-0.431552\pi$
$499$	9.53319	0.426764	0.213382	+	0.976969i	$0.431552\pi$
$500$	5.25508	0.235014
$501$	5.75379	0.257060
$502$	−61.4431	−2.74234
$503$	14.0568	0.626762	0.313381	−	0.949627i	$-0.398538\pi$
$503$	14.0568	0.626762	0.313381	+	0.949627i	$0.398538\pi$
$504$	−85.6391	−3.81467
$505$	12.2955	0.547144
$506$	9.12311	0.405572
$507$	17.7538	0.788473
$508$	−97.0022	−4.30378
$509$	21.7105	0.962302	0.481151	−	0.876638i	$-0.340219\pi$
$509$	21.7105	0.962302	0.481151	+	0.876638i	$0.340219\pi$
$510$	18.2462	0.807956
$511$	15.1838	0.671692
$512$	3.01385	0.133194
$513$	−4.87209	−0.215108
$514$	−63.9783	−2.82196
$515$	6.05023	0.266605
$516$	−40.7215	−1.79267
$517$	−27.8163	−1.22336
$518$	−40.6560	−1.78632
$519$	−20.5255	−0.900972
$520$	−21.5996	−0.947205
$521$	−8.87689	−0.388904	−0.194452	−	0.980912i	$-0.562293\pi$
$521$	−8.87689	−0.388904	−0.194452	+	0.980912i	$0.562293\pi$
$522$	88.3152	3.86545
$523$	−0.0550279	−0.00240620	−0.00120310	−	0.999999i	$-0.500383\pi$
$523$	−0.0550279	−0.00240620	−0.00120310	+	0.999999i	$0.500383\pi$
$524$	−20.3285	−0.888053
$525$	7.02511	0.306601
$526$	44.6842	1.94833
$527$	13.5085	0.588439
$528$	113.706	4.94843
$529$	1.00000	0.0434783
$530$	−28.3944	−1.23338
$531$	29.4989	1.28014
$532$	−48.8147	−2.11639
$533$	2.97126	0.128699
$534$	32.1741	1.39231
$535$	13.1547	0.568727
$536$	−67.2446	−2.90453
$537$	30.6401	1.32222
$538$	−22.8275	−0.984163
$539$	−1.76609	−0.0760708
$540$	−7.55915	−0.325294
$541$	5.99070	0.257560	0.128780	−	0.991673i	$-0.458894\pi$
$541$	5.99070	0.257560	0.128780	+	0.991673i	$0.458894\pi$
$542$	−57.3799	−2.46468
$543$	44.7766	1.92155
$544$	−46.9807	−2.01428
$545$	−17.1183	−0.733268
$546$	−46.6163	−1.99499
$547$	−0.408533	−0.0174676	−0.00873380	−	0.999962i	$-0.502780\pi$
$547$	−0.408533	−0.0174676	−0.00873380	+	0.999962i	$0.502780\pi$
$548$	108.629	4.64042
$549$	0.940053	0.0401205
$550$	−9.12311	−0.389011
$551$	31.1815	1.32837
$552$	22.4588	0.955908
$553$	23.7944	1.01184
$554$	59.1539	2.51321
$555$	−14.0980	−0.598426
$556$	66.1099	2.80369
$557$	−5.46406	−0.231519	−0.115760	−	0.993277i	$-0.536930\pi$
$557$	−5.46406	−0.231519	−0.115760	+	0.993277i	$0.536930\pi$
$558$	−49.0026	−2.07444
$559$	−7.45255	−0.315209
$560$	−35.9426	−1.51885
$561$	−22.9442	−0.968706
$562$	53.6066	2.26126
$563$	−41.8212	−1.76255	−0.881277	−	0.472601i	$-0.843315\pi$
$563$	−41.8212	−1.76255	−0.881277	+	0.472601i	$0.843315\pi$
$564$	−110.550	−4.65501
$565$	4.38058	0.184293
$566$	−23.9126	−1.00512
$567$	19.1976	0.806225
$568$	−0.131656	−0.00552415
$569$	39.5127	1.65646	0.828230	−	0.560388i	$-0.189348\pi$
$569$	39.5127	1.65646	0.828230	+	0.560388i	$0.189348\pi$
$570$	−23.3693	−0.978833
$571$	−24.4924	−1.02498	−0.512488	−	0.858694i	$-0.671276\pi$
$571$	−24.4924	−1.02498	−0.512488	+	0.858694i	$0.671276\pi$
$572$	43.8494	1.83344
$573$	33.6155	1.40431
$574$	8.90939	0.371871
$575$	−1.00000	−0.0417029
$576$	77.0710	3.21129
$577$	−36.3382	−1.51278	−0.756390	−	0.654121i	$-0.773039\pi$
$577$	−36.3382	−1.51278	−0.756390	+	0.654121i	$0.773039\pi$
$578$	−26.9527	−1.12108
$579$	32.1317	1.33535
$580$	48.3787	2.00882
$581$	9.65758	0.400664
$582$	28.2745	1.17202
$583$	35.7054	1.47877
$584$	−48.5415	−2.00866
$585$	−8.77410	−0.362764
$586$	64.5662	2.66720
$587$	5.89358	0.243254	0.121627	−	0.992576i	$-0.461189\pi$
$587$	5.89358	0.243254	0.121627	+	0.992576i	$0.461189\pi$
$588$	−7.01896	−0.289457
$589$	−17.3014	−0.712890
$590$	22.3094	0.918462
$591$	−9.95277	−0.409402
$592$	72.1296	2.96451
$593$	−11.0931	−0.455538	−0.227769	−	0.973715i	$-0.573143\pi$
$593$	−11.0931	−0.455538	−0.227769	+	0.973715i	$0.573143\pi$
$594$	13.1231	0.538448
$595$	7.25268	0.297331
$596$	60.3548	2.47223
$597$	−56.7670	−2.32332
$598$	6.63566	0.271352
$599$	40.1864	1.64197	0.820986	−	0.570949i	$-0.193425\pi$
$599$	40.1864	1.64197	0.820986	+	0.570949i	$0.193425\pi$
$600$	−22.4588	−0.916875
$601$	41.1965	1.68044	0.840220	−	0.542246i	$-0.182426\pi$
$601$	41.1965	1.68044	0.840220	+	0.542246i	$0.182426\pi$
$602$	−22.3467	−0.910782
$603$	−27.3158	−1.11239
$604$	−29.3584	−1.19457
$605$	0.472110	0.0191940
$606$	−84.8343	−3.44616
$607$	−16.6283	−0.674924	−0.337462	−	0.941339i	$-0.609568\pi$
$607$	−16.6283	−0.674924	−0.337462	+	0.941339i	$0.609568\pi$
$608$	60.1717	2.44029
$609$	64.6738	2.62071
$610$	0.710942	0.0287852
$611$	−20.2321	−0.818504
$612$	−49.4956	−2.00074
$613$	25.5052	1.03015	0.515073	−	0.857146i	$-0.327765\pi$
$613$	25.5052	1.03015	0.515073	+	0.857146i	$0.327765\pi$
$614$	32.4924	1.31129
$615$	3.08944	0.124578
$616$	81.4431	3.28143
$617$	−3.14742	−0.126710	−0.0633552	−	0.997991i	$-0.520180\pi$
$617$	−3.14742	−0.126710	−0.0633552	+	0.997991i	$0.520180\pi$
$618$	−41.7442	−1.67920
$619$	−39.2665	−1.57825	−0.789127	−	0.614230i	$-0.789467\pi$
$619$	−39.2665	−1.57825	−0.789127	+	0.614230i	$0.789467\pi$
$620$	−26.8434	−1.07806
$621$	1.43845	0.0577229
$622$	−18.3498	−0.735761
$623$	12.7889	0.512375
$624$	82.7040	3.31081
$625$	1.00000	0.0400000
$626$	−17.1992	−0.687418
$627$	29.3864	1.17358
$628$	30.8243	1.23002
$629$	−14.5547	−0.580333
$630$	−26.3094	−1.04819
$631$	18.2916	0.728179	0.364089	−	0.931364i	$-0.381380\pi$
$631$	18.2916	0.728179	0.364089	+	0.931364i	$0.381380\pi$
$632$	−76.0689	−3.02586
$633$	−12.3223	−0.489768
$634$	45.2819	1.79837
$635$	−18.4588	−0.732514
$636$	141.904	5.62686
$637$	−1.28456	−0.0508960
$638$	−83.9881	−3.32512
$639$	−0.0534806	−0.00211566
$640$	22.7567	0.899537
$641$	28.5304	1.12688	0.563441	−	0.826157i	$-0.309477\pi$
$641$	28.5304	1.12688	0.563441	+	0.826157i	$0.309477\pi$
$642$	−90.7622	−3.58210
$643$	−12.2430	−0.482815	−0.241408	−	0.970424i	$-0.577609\pi$
$643$	−12.2430	−0.482815	−0.241408	+	0.970424i	$0.577609\pi$
$644$	14.4122	0.567919
$645$	−7.74899	−0.305116
$646$	−24.1263	−0.949237
$647$	3.11947	0.122639	0.0613196	−	0.998118i	$-0.480469\pi$
$647$	3.11947	0.122639	0.0613196	+	0.998118i	$0.480469\pi$
$648$	−61.3734	−2.41098
$649$	−28.0536	−1.10120
$650$	−6.63566	−0.260272
$651$	−35.8849	−1.40644
$652$	0.244863	0.00958958
$653$	−21.7301	−0.850364	−0.425182	−	0.905108i	$-0.639790\pi$
$653$	−21.7301	−0.850364	−0.425182	+	0.905108i	$0.639790\pi$
$654$	118.110	4.61845
$655$	−3.86834	−0.151149
$656$	−15.8065	−0.617141
$657$	−19.7183	−0.769285
$658$	−60.6665	−2.36503
$659$	−12.2333	−0.476541	−0.238270	−	0.971199i	$-0.576580\pi$
$659$	−12.2333	−0.476541	−0.238270	+	0.971199i	$0.576580\pi$
$660$	45.5936	1.77473
$661$	22.0000	0.855701	0.427850	−	0.903850i	$-0.359271\pi$
$661$	22.0000	0.855701	0.427850	+	0.903850i	$0.359271\pi$
$662$	8.86477	0.344539
$663$	−16.6884	−0.648124
$664$	−30.8746	−1.19817
$665$	−9.28906	−0.360214
$666$	52.7976	2.04587
$667$	−9.20608	−0.356461
$668$	−11.8040	−0.456711
$669$	−35.3020	−1.36485
$670$	−20.6584	−0.798103
$671$	−0.893994	−0.0345123
$672$	124.803	4.81437
$673$	10.7900	0.415925	0.207963	−	0.978137i	$-0.433317\pi$
$673$	10.7900	0.415925	0.207963	+	0.978137i	$0.433317\pi$
$674$	−21.2843	−0.819839
$675$	−1.43845	−0.0553659
$676$	−36.4223	−1.40086
$677$	23.6721	0.909793	0.454896	−	0.890544i	$-0.349676\pi$
$677$	23.6721	0.909793	0.454896	+	0.890544i	$0.349676\pi$
$678$	−30.2243	−1.16076
$679$	11.2388	0.431307
$680$	−23.1863	−0.889153
$681$	−74.5443	−2.85654
$682$	46.6016	1.78447
$683$	−10.3583	−0.396350	−0.198175	−	0.980167i	$-0.563501\pi$
$683$	−10.3583	−0.396350	−0.198175	+	0.980167i	$0.563501\pi$
$684$	63.3928	2.42389
$685$	20.6713	0.789810
$686$	47.8576	1.82721
$687$	24.0454	0.917390
$688$	39.6462	1.51150
$689$	25.9702	0.989386
$690$	6.89961	0.262664
$691$	13.5578	0.515763	0.257882	−	0.966177i	$-0.416976\pi$
$691$	13.5578	0.515763	0.257882	+	0.966177i	$0.416976\pi$
$692$	42.1086	1.60073
$693$	33.0835	1.25674
$694$	−52.2610	−1.98380
$695$	12.5802	0.477194
$696$	−206.757	−7.83711
$697$	3.18952	0.120812
$698$	−89.3347	−3.38137
$699$	32.4612	1.22780
$700$	−14.4122	−0.544729
$701$	−21.3774	−0.807415	−0.403708	−	0.914888i	$-0.632279\pi$
$701$	−21.3774	−0.807415	−0.403708	+	0.914888i	$0.632279\pi$
$702$	9.54505	0.360255
$703$	18.6413	0.703069
$704$	−73.2948	−2.76240
$705$	−21.0369	−0.792295
$706$	−80.5257	−3.03063
$707$	−33.7207	−1.26820
$708$	−111.493	−4.19017
$709$	30.3719	1.14064	0.570320	−	0.821422i	$-0.306819\pi$
$709$	30.3719	1.14064	0.570320	+	0.821422i	$0.306819\pi$
$710$	−0.0404462	−0.00151792
$711$	−30.9004	−1.15886
$712$	−40.8850	−1.53223
$713$	5.10809	0.191299
$714$	−50.0406	−1.87272
$715$	8.34420	0.312056
$716$	−62.8588	−2.34914
$717$	−49.1520	−1.83561
$718$	−67.2642	−2.51028
$719$	8.41851	0.313958	0.156979	−	0.987602i	$-0.449825\pi$
$719$	8.41851	0.313958	0.156979	+	0.987602i	$0.449825\pi$
$720$	46.6766	1.73953
$721$	−16.5929	−0.617951
$722$	−20.2766	−0.754615
$723$	−42.9677	−1.59799
$724$	−91.8600	−3.41395
$725$	9.20608	0.341905
$726$	−3.25738	−0.120893
$727$	19.7505	0.732507	0.366253	−	0.930515i	$-0.380640\pi$
$727$	19.7505	0.732507	0.366253	+	0.930515i	$0.380640\pi$
$728$	59.2374	2.19548
$729$	−35.9848	−1.33277
$730$	−14.9125	−0.551938
$731$	−8.00000	−0.295891
$732$	−3.55300	−0.131323
$733$	19.7280	0.728670	0.364335	−	0.931268i	$-0.381296\pi$
$733$	19.7280	0.728670	0.364335	+	0.931268i	$0.381296\pi$
$734$	42.6058	1.57261
$735$	−1.33565	−0.0492663
$736$	−17.7652	−0.654835
$737$	25.9775	0.956892
$738$	−11.5701	−0.425901
$739$	−0.180969	−0.00665704	−0.00332852	−	0.999994i	$-0.501060\pi$
$739$	−0.180969	−0.00665704	−0.00332852	+	0.999994i	$0.501060\pi$
$740$	28.9223	1.06321
$741$	21.3741	0.785198
$742$	77.8723	2.85878
$743$	6.05023	0.221961	0.110981	−	0.993823i	$-0.464601\pi$
$743$	6.05023	0.221961	0.110981	+	0.993823i	$0.464601\pi$
$744$	114.721	4.20589
$745$	11.4850	0.420779
$746$	60.9500	2.23154
$747$	−12.5417	−0.458878
$748$	47.0705	1.72107
$749$	−36.0770	−1.31823
$750$	−6.89961	−0.251938
$751$	−21.5659	−0.786952	−0.393476	−	0.919335i	$-0.628728\pi$
$751$	−21.5659	−0.786952	−0.393476	+	0.919335i	$0.628728\pi$
$752$	107.631	3.92490
$753$	58.4326	2.12940
$754$	−61.0885	−2.22471
$755$	−5.58667	−0.203320
$756$	20.7311	0.753984
$757$	−24.7798	−0.900638	−0.450319	−	0.892868i	$-0.648690\pi$
$757$	−24.7798	−0.900638	−0.450319	+	0.892868i	$0.648690\pi$
$758$	27.6681	1.00495
$759$	−8.67611	−0.314923
$760$	29.6964	1.07720
$761$	2.29916	0.0833443	0.0416722	−	0.999131i	$-0.486731\pi$
$761$	2.29916	0.0833443	0.0416722	+	0.999131i	$0.486731\pi$
$762$	127.358	4.61370
$763$	46.9473	1.69961
$764$	−68.9629	−2.49499
$765$	−9.41863	−0.340531
$766$	16.7393	0.604814
$767$	−20.4047	−0.736770
$768$	−46.1496	−1.66528
$769$	−10.3692	−0.373923	−0.186961	−	0.982367i	$-0.559864\pi$
$769$	−10.3692	−0.373923	−0.186961	+	0.982367i	$0.559864\pi$
$770$	25.0203	0.901669
$771$	60.8436	2.19123
$772$	−65.9187	−2.37247
$773$	12.7864	0.459895	0.229947	−	0.973203i	$-0.426145\pi$
$773$	12.7864	0.459895	0.229947	+	0.973203i	$0.426145\pi$
$774$	29.0203	1.04311
$775$	−5.10809	−0.183488
$776$	−35.9297	−1.28980
$777$	38.6640	1.38706
$778$	74.4528	2.66926
$779$	−4.08506	−0.146362
$780$	33.1624	1.18740
$781$	0.0508602	0.00181992
$782$	7.12311	0.254722
$783$	−13.2425	−0.473247
$784$	6.83361	0.244058
$785$	5.86563	0.209353
$786$	26.6901	0.952003
$787$	−34.9239	−1.24490	−0.622451	−	0.782659i	$-0.713863\pi$
$787$	−34.9239	−1.24490	−0.622451	+	0.782659i	$0.713863\pi$
$788$	20.4183	0.727372
$789$	−42.4949	−1.51286
$790$	−23.3693	−0.831443
$791$	−12.0138	−0.427163
$792$	−105.765	−3.75821
$793$	−0.650244	−0.0230908
$794$	2.88667	0.102444
$795$	27.0032	0.957705
$796$	116.459	4.12776
$797$	4.87844	0.172803	0.0864016	−	0.996260i	$-0.472463\pi$
$797$	4.87844	0.172803	0.0864016	+	0.996260i	$0.472463\pi$
$798$	64.0909	2.26879
$799$	−21.7183	−0.768339
$800$	17.7652	0.628096
$801$	−16.6081	−0.586820
$802$	4.03483	0.142475
$803$	18.7522	0.661751
$804$	103.242	3.64107
$805$	2.74252	0.0966612
$806$	33.8956	1.19392
$807$	21.7090	0.764194
$808$	107.803	3.79248
$809$	18.1498	0.638112	0.319056	−	0.947736i	$-0.396634\pi$
$809$	18.1498	0.638112	0.319056	+	0.947736i	$0.396634\pi$
$810$	−18.8547	−0.662486
$811$	17.8019	0.625110	0.312555	−	0.949900i	$-0.398815\pi$
$811$	17.8019	0.625110	0.312555	+	0.949900i	$0.398815\pi$
$812$	−132.680	−4.65614
$813$	54.5685	1.91380
$814$	−50.2107	−1.75989
$815$	0.0465955	0.00163217
$816$	88.7793	3.10790
$817$	10.2462	0.358470
$818$	56.3952	1.97181
$819$	24.0632	0.840835
$820$	−6.33805	−0.221334
$821$	32.9701	1.15066	0.575332	−	0.817920i	$-0.304873\pi$
$821$	32.9701	1.15066	0.575332	+	0.817920i	$0.304873\pi$
$822$	−142.624	−4.97458
$823$	−39.5819	−1.37974	−0.689869	−	0.723935i	$-0.742332\pi$
$823$	−39.5819	−1.37974	−0.689869	+	0.723935i	$0.742332\pi$
$824$	53.0462	1.84795
$825$	8.67611	0.302063
$826$	−61.1839	−2.12886
$827$	−1.50849	−0.0524554	−0.0262277	−	0.999656i	$-0.508349\pi$
$827$	−1.50849	−0.0524554	−0.0262277	+	0.999656i	$0.508349\pi$
$828$	−18.7162	−0.650434
$829$	−7.66718	−0.266292	−0.133146	−	0.991096i	$-0.542508\pi$
$829$	−7.66718	−0.266292	−0.133146	+	0.991096i	$0.542508\pi$
$830$	−9.48504	−0.329231
$831$	−56.2556	−1.95149
$832$	−53.3108	−1.84822
$833$	−1.37892	−0.0477767
$834$	−86.7984	−3.00558
$835$	−2.24621	−0.0777333
$836$	−60.2868	−2.08506
$837$	7.34772	0.253974
$838$	−50.5207	−1.74521
$839$	−8.53602	−0.294696	−0.147348	−	0.989085i	$-0.547074\pi$
$839$	−8.53602	−0.294696	−0.147348	+	0.989085i	$0.547074\pi$
$840$	61.5936	2.12518
$841$	55.7519	1.92248
$842$	55.5305	1.91371
$843$	−50.9800	−1.75585
$844$	25.2795	0.870155
$845$	−6.93087	−0.238429
$846$	78.7841	2.70865
$847$	−1.29477	−0.0444889
$848$	−138.157	−4.74432
$849$	22.7410	0.780468
$850$	−7.12311	−0.244321
$851$	−5.50369	−0.188664
$852$	0.202134	0.00692500
$853$	−48.0665	−1.64577	−0.822883	−	0.568211i	$-0.807636\pi$
$853$	−48.0665	−1.64577	−0.822883	+	0.568211i	$0.807636\pi$
$854$	−1.94977	−0.0667199
$855$	12.0632	0.412551
$856$	115.335	3.94209
$857$	29.4313	1.00535	0.502677	−	0.864474i	$-0.332348\pi$
$857$	29.4313	1.00535	0.502677	+	0.864474i	$0.332348\pi$
$858$	−57.5717	−1.96547
$859$	−33.5308	−1.14406	−0.572029	−	0.820234i	$-0.693843\pi$
$859$	−33.5308	−1.14406	−0.572029	+	0.820234i	$0.693843\pi$
$860$	15.8972	0.542090
$861$	−8.47286	−0.288754
$862$	72.7670	2.47845
$863$	6.79483	0.231299	0.115649	−	0.993290i	$-0.463105\pi$
$863$	6.79483	0.231299	0.115649	+	0.993290i	$0.463105\pi$
$864$	−25.5544	−0.869377
$865$	8.01293	0.272448
$866$	−68.1192	−2.31478
$867$	25.6321	0.870511
$868$	73.6186	2.49878
$869$	29.3864	0.996866
$870$	−63.5183	−2.15347
$871$	18.8946	0.640220
$872$	−150.087	−5.08259
$873$	−14.5952	−0.493973
$874$	−9.12311	−0.308594
$875$	−2.74252	−0.0927141
$876$	74.5269	2.51803
$877$	−21.9904	−0.742563	−0.371281	−	0.928520i	$-0.621082\pi$
$877$	−21.9904	−0.742563	−0.371281	+	0.928520i	$0.621082\pi$
$878$	92.6869	3.12803
$879$	−61.4027	−2.07106
$880$	−44.3896	−1.49637
$881$	3.66242	0.123390	0.0616951	−	0.998095i	$-0.480349\pi$
$881$	3.66242	0.123390	0.0616951	+	0.998095i	$0.480349\pi$
$882$	5.00208	0.168429
$883$	10.6640	0.358874	0.179437	−	0.983769i	$-0.442572\pi$
$883$	10.6640	0.358874	0.179437	+	0.983769i	$0.442572\pi$
$884$	34.2366	1.15150
$885$	−21.2163	−0.713178
$886$	−80.3427	−2.69916
$887$	0.481294	0.0161603	0.00808014	−	0.999967i	$-0.497428\pi$
$887$	0.481294	0.0161603	0.00808014	+	0.999967i	$0.497428\pi$
$888$	−123.606	−4.14795
$889$	50.6235	1.69786
$890$	−12.5604	−0.421025
$891$	23.7094	0.794293
$892$	72.4228	2.42489
$893$	27.8163	0.930837
$894$	−79.2423	−2.65026
$895$	−11.9615	−0.399830
$896$	−62.4107	−2.08499
$897$	−6.31054	−0.210703
$898$	−7.20400	−0.240401
$899$	−47.0255	−1.56839
$900$	18.7162	0.623875
$901$	27.8779	0.928748
$902$	11.0032	0.366367
$903$	21.2518	0.707214
$904$	38.4074	1.27741
$905$	−17.4802	−0.581063
$906$	38.5458	1.28060
$907$	−50.3078	−1.67044	−0.835222	−	0.549913i	$-0.814661\pi$
$907$	−50.3078	−1.67044	−0.835222	+	0.549913i	$0.814661\pi$
$908$	152.929	5.07513
$909$	43.7912	1.45246
$910$	18.1984	0.603273
$911$	−39.5103	−1.30903	−0.654517	−	0.756047i	$-0.727128\pi$
$911$	−39.5103	−1.30903	−0.654517	+	0.756047i	$0.727128\pi$
$912$	−113.706	−3.76519
$913$	11.9272	0.394734
$914$	−83.2398	−2.75333
$915$	−0.676108	−0.0223515
$916$	−49.3297	−1.62990
$917$	10.6090	0.350341
$918$	10.2462	0.338175
$919$	−10.0307	−0.330881	−0.165441	−	0.986220i	$-0.552905\pi$
$919$	−10.0307	−0.330881	−0.165441	+	0.986220i	$0.552905\pi$
$920$	−8.76763	−0.289060
$921$	−30.9004	−1.01820
$922$	−71.6536	−2.35979
$923$	0.0369930	0.00121764
$924$	−125.041	−4.11356
$925$	5.50369	0.180960
$926$	71.9207	2.36346
$927$	21.5482	0.707736
$928$	163.548	5.36873
$929$	5.83676	0.191498	0.0957490	−	0.995406i	$-0.469475\pi$
$929$	5.83676	0.191498	0.0957490	+	0.995406i	$0.469475\pi$
$930$	35.2438	1.15569
$931$	1.76609	0.0578812
$932$	−66.5949	−2.18139
$933$	17.4507	0.571312
$934$	94.9477	3.10678
$935$	8.95715	0.292930
$936$	−76.9281	−2.51447
$937$	37.5175	1.22564	0.612822	−	0.790221i	$-0.290034\pi$
$937$	37.5175	1.22564	0.612822	+	0.790221i	$0.290034\pi$
$938$	56.6560	1.84989
$939$	16.3565	0.533774
$940$	43.1576	1.40765
$941$	45.6189	1.48713	0.743566	−	0.668662i	$-0.233133\pi$
$941$	45.6189	1.48713	0.743566	+	0.668662i	$0.233133\pi$
$942$	−40.4705	−1.31860
$943$	1.20608	0.0392754
$944$	108.549	3.53297
$945$	3.94497	0.128330
$946$	−27.5984	−0.897302
$947$	−17.3357	−0.563333	−0.281667	−	0.959512i	$-0.590887\pi$
$947$	−17.3357	−0.563333	−0.281667	+	0.959512i	$0.590887\pi$
$948$	116.790	3.79318
$949$	13.6394	0.442752
$950$	9.12311	0.295993
$951$	−43.0633	−1.39642
$952$	63.5888	2.06093
$953$	14.2706	0.462269	0.231135	−	0.972922i	$-0.425756\pi$
$953$	14.2706	0.462269	0.231135	+	0.972922i	$0.425756\pi$
$954$	−101.128	−3.27415
$955$	−13.1231	−0.424654
$956$	100.836	3.26128
$957$	79.8730	2.58193
$958$	−105.237	−3.40007
$959$	−56.6915	−1.83066
$960$	−55.4313	−1.78904
$961$	−4.90742	−0.158304
$962$	−36.5206	−1.17747
$963$	46.8511	1.50976
$964$	88.1492	2.83909
$965$	−12.5438	−0.403800
$966$	−18.9223	−0.608816
$967$	−4.07663	−0.131096	−0.0655478	−	0.997849i	$-0.520879\pi$
$967$	−4.07663	−0.131096	−0.0655478	+	0.997849i	$0.520879\pi$
$968$	4.13929	0.133042
$969$	22.9442	0.737075
$970$	−11.0380	−0.354410
$971$	−20.2988	−0.651419	−0.325709	−	0.945470i	$-0.605603\pi$
$971$	−20.2988	−0.651419	−0.325709	+	0.945470i	$0.605603\pi$
$972$	116.906	3.74975
$973$	−34.5015	−1.10607
$974$	64.8706	2.07859
$975$	6.31054	0.202099
$976$	3.45918	0.110726
$977$	−40.3782	−1.29181	−0.645907	−	0.763416i	$-0.723521\pi$
$977$	−40.3782	−1.29181	−0.645907	+	0.763416i	$0.723521\pi$
$978$	−0.321491	−0.0102801
$979$	15.7944	0.504792
$980$	2.74012	0.0875299
$981$	−60.9677	−1.94655
$982$	15.0851	0.481386
$983$	−53.1628	−1.69563	−0.847815	−	0.530292i	$-0.822082\pi$
$983$	−53.1628	−1.69563	−0.847815	+	0.530292i	$0.822082\pi$
$984$	27.0871	0.863505
$985$	3.88544	0.123801
$986$	−65.5759	−2.08836
$987$	57.6941	1.83642
$988$	−43.8494	−1.39504
$989$	−3.02511	−0.0961930
$990$	−32.4924	−1.03268
$991$	36.6746	1.16501	0.582503	−	0.812829i	$-0.302073\pi$
$991$	36.6746	1.16501	0.582503	+	0.812829i	$0.302073\pi$
$992$	−90.7464	−2.88120
$993$	−8.43043	−0.267532
$994$	0.110925	0.00351831
$995$	22.1612	0.702556
$996$	47.4024	1.50200
$997$	−11.1189	−0.352140	−0.176070	−	0.984378i	$-0.556339\pi$
$997$	−11.1189	−0.352140	−0.176070	+	0.984378i	$0.556339\pi$
$998$	25.6779	0.812819
$999$	−7.91677	−0.250475

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	115.2.a.c.1.4	✓	4
3.2	odd	2		1035.2.a.o.1.1		4
4.3	odd	2		1840.2.a.u.1.4		4
5.2	odd	4		575.2.b.e.24.8		8
5.3	odd	4		575.2.b.e.24.1		8
5.4	even	2		575.2.a.h.1.1		4
7.6	odd	2		5635.2.a.v.1.4		4
8.3	odd	2		7360.2.a.cg.1.2		4
8.5	even	2		7360.2.a.cj.1.3		4
15.14	odd	2		5175.2.a.bx.1.4		4
20.19	odd	2		9200.2.a.cl.1.1		4
23.22	odd	2		2645.2.a.m.1.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
115.2.a.c.1.4	✓	4	1.1	even	1	trivial
575.2.a.h.1.1		4	5.4	even	2
575.2.b.e.24.1		8	5.3	odd	4
575.2.b.e.24.8		8	5.2	odd	4
1035.2.a.o.1.1		4	3.2	odd	2
1840.2.a.u.1.4		4	4.3	odd	2
2645.2.a.m.1.4		4	23.22	odd	2
5175.2.a.bx.1.4		4	15.14	odd	2
5635.2.a.v.1.4		4	7.6	odd	2
7360.2.a.cg.1.2		4	8.3	odd	2
7360.2.a.cj.1.3		4	8.5	even	2
9200.2.a.cl.1.1		4	20.19	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 115 = 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	115.a (trivial)

\(f(q)\)	\(=\)	\(q+2.69353 q^{2} -2.56155 q^{3} +5.25508 q^{4} +1.00000 q^{5} -6.89961 q^{6} -2.74252 q^{7} +8.76763 q^{8} +3.56155 q^{9} +O(q^{10})\)	\(q+2.69353 q^{2} -2.56155 q^{3} +5.25508 q^{4} +1.00000 q^{5} -6.89961 q^{6} -2.74252 q^{7} +8.76763 q^{8} +3.56155 q^{9} +2.69353 q^{10} -3.38705 q^{11} -13.4612 q^{12} -2.46356 q^{13} -7.38705 q^{14} -2.56155 q^{15} +13.1057 q^{16} -2.64453 q^{17} +9.59313 q^{18} +3.38705 q^{19} +5.25508 q^{20} +7.02511 q^{21} -9.12311 q^{22} -1.00000 q^{23} -22.4588 q^{24} +1.00000 q^{25} -6.63566 q^{26} -1.43845 q^{27} -14.4122 q^{28} +9.20608 q^{29} -6.89961 q^{30} -5.10809 q^{31} +17.7652 q^{32} +8.67611 q^{33} -7.12311 q^{34} -2.74252 q^{35} +18.7162 q^{36} +5.50369 q^{37} +9.12311 q^{38} +6.31054 q^{39} +8.76763 q^{40} -1.20608 q^{41} +18.9223 q^{42} +3.02511 q^{43} -17.7992 q^{44} +3.56155 q^{45} -2.69353 q^{46} +8.21255 q^{47} -33.5709 q^{48} +0.521423 q^{49} +2.69353 q^{50} +6.77410 q^{51} -12.9462 q^{52} -10.5417 q^{53} -3.87449 q^{54} -3.38705 q^{55} -24.0454 q^{56} -8.67611 q^{57} +24.7968 q^{58} +8.28259 q^{59} -13.4612 q^{60} +0.263945 q^{61} -13.7588 q^{62} -9.76763 q^{63} +21.6397 q^{64} -2.46356 q^{65} +23.3693 q^{66} -7.66964 q^{67} -13.8972 q^{68} +2.56155 q^{69} -7.38705 q^{70} -0.0150161 q^{71} +31.2264 q^{72} -5.53644 q^{73} +14.8243 q^{74} -2.56155 q^{75} +17.7992 q^{76} +9.28906 q^{77} +16.9976 q^{78} -8.67611 q^{79} +13.1057 q^{80} -7.00000 q^{81} -3.24861 q^{82} -3.52142 q^{83} +36.9175 q^{84} -2.64453 q^{85} +8.14822 q^{86} -23.5819 q^{87} -29.6964 q^{88} -4.66318 q^{89} +9.59313 q^{90} +6.75637 q^{91} -5.25508 q^{92} +13.0846 q^{93} +22.1207 q^{94} +3.38705 q^{95} -45.5066 q^{96} -4.09799 q^{97} +1.40447 q^{98} -12.0632 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{2} - 2 q^{3} + 4 q^{4} + 4 q^{5} - q^{6} - 3 q^{7} + 9 q^{8} + 6 q^{9}+O(q^{10}) \) 4 * q + 2 * q^2 - 2 * q^3 + 4 * q^4 + 4 * q^5 - q^6 - 3 * q^7 + 9 * q^8 + 6 * q^9	\( 4 q + 2 q^{2} - 2 q^{3} + 4 q^{4} + 4 q^{5} - q^{6} - 3 q^{7} + 9 q^{8} + 6 q^{9} + 2 q^{10} + 4 q^{11} - 19 q^{12} - 12 q^{14} - 2 q^{15} + 8 q^{16} - q^{17} + 3 q^{18} - 4 q^{19} + 4 q^{20} + 10 q^{21} - 20 q^{22} - 4 q^{23} - 30 q^{24} + 4 q^{25} - q^{26} - 14 q^{27} - 22 q^{28} + 19 q^{29} - q^{30} - q^{31} + 20 q^{32} - 2 q^{33} - 12 q^{34} - 3 q^{35} + 23 q^{36} - 3 q^{37} + 20 q^{38} + 9 q^{40} + 13 q^{41} + 6 q^{42} - 6 q^{43} - 18 q^{44} + 6 q^{45} - 2 q^{46} + 6 q^{47} - 21 q^{48} + 9 q^{49} + 2 q^{50} - 8 q^{51} - q^{52} + 19 q^{53} - 7 q^{54} + 4 q^{55} - 10 q^{56} + 2 q^{57} + 21 q^{58} + 23 q^{59} - 19 q^{60} - 13 q^{62} - 13 q^{63} + 27 q^{64} + 44 q^{66} - 3 q^{67} - 4 q^{68} + 2 q^{69} - 12 q^{70} - 3 q^{71} + 39 q^{72} - 32 q^{73} - 12 q^{74} - 2 q^{75} + 18 q^{76} + 18 q^{77} + 43 q^{78} + 2 q^{79} + 8 q^{80} - 28 q^{81} - 5 q^{82} - 21 q^{83} + 28 q^{84} - q^{85} - 2 q^{86} - 18 q^{87} - 14 q^{88} + 3 q^{90} - 40 q^{91} - 4 q^{92} - 8 q^{93} + 47 q^{94} - 4 q^{95} - 61 q^{96} - 18 q^{97} + 16 q^{98} + 6 q^{99}+O(q^{100}) \) 4 * q + 2 * q^2 - 2 * q^3 + 4 * q^4 + 4 * q^5 - q^6 - 3 * q^7 + 9 * q^8 + 6 * q^9 + 2 * q^10 + 4 * q^11 - 19 * q^12 - 12 * q^14 - 2 * q^15 + 8 * q^16 - q^17 + 3 * q^18 - 4 * q^19 + 4 * q^20 + 10 * q^21 - 20 * q^22 - 4 * q^23 - 30 * q^24 + 4 * q^25 - q^26 - 14 * q^27 - 22 * q^28 + 19 * q^29 - q^30 - q^31 + 20 * q^32 - 2 * q^33 - 12 * q^34 - 3 * q^35 + 23 * q^36 - 3 * q^37 + 20 * q^38 + 9 * q^40 + 13 * q^41 + 6 * q^42 - 6 * q^43 - 18 * q^44 + 6 * q^45 - 2 * q^46 + 6 * q^47 - 21 * q^48 + 9 * q^49 + 2 * q^50 - 8 * q^51 - q^52 + 19 * q^53 - 7 * q^54 + 4 * q^55 - 10 * q^56 + 2 * q^57 + 21 * q^58 + 23 * q^59 - 19 * q^60 - 13 * q^62 - 13 * q^63 + 27 * q^64 + 44 * q^66 - 3 * q^67 - 4 * q^68 + 2 * q^69 - 12 * q^70 - 3 * q^71 + 39 * q^72 - 32 * q^73 - 12 * q^74 - 2 * q^75 + 18 * q^76 + 18 * q^77 + 43 * q^78 + 2 * q^79 + 8 * q^80 - 28 * q^81 - 5 * q^82 - 21 * q^83 + 28 * q^84 - q^85 - 2 * q^86 - 18 * q^87 - 14 * q^88 + 3 * q^90 - 40 * q^91 - 4 * q^92 - 8 * q^93 + 47 * q^94 - 4 * q^95 - 61 * q^96 - 18 * q^97 + 16 * q^98 + 6 * q^99

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