LMFDB - Embedded newform 5054.2.a.bf.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5054,2,Mod(1,5054)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5054, 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("5054.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5054 = 2 \cdot 7 \cdot 19^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5054.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:	$40.3563931816$
Analytic rank:	$1$
Dimension:	$8$
Coefficient field:	8.8.19520000000.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{7} - 12x^{6} + 16x^{5} + 50x^{4} - 24x^{3} - 72x^{2} - 32x - 4 $ x^8 - 2x^7 - 12x^6 + 16x^5 + 50x^4 - 24x^3 - 72x^2 - 32*x - 4
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			$-1.91631$ of defining polynomial
Character	$\chi$	$=$	5054.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} -3.35992 q^{3} +1.00000 q^{4} -1.35877 q^{5} +3.35992 q^{6} -1.00000 q^{7} -1.00000 q^{8} +8.28904 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -3.35992 q^{3} +1.00000 q^{4} -1.35877 q^{5} +3.35992 q^{6} -1.00000 q^{7} -1.00000 q^{8} +8.28904 q^{9} +1.35877 q^{10} +0.672249 q^{11} -3.35992 q^{12} +1.04304 q^{13} +1.00000 q^{14} +4.56537 q^{15} +1.00000 q^{16} -2.46184 q^{17} -8.28904 q^{18} -1.35877 q^{20} +3.35992 q^{21} -0.672249 q^{22} +9.22420 q^{23} +3.35992 q^{24} -3.15373 q^{25} -1.04304 q^{26} -17.7707 q^{27} -1.00000 q^{28} -9.26173 q^{29} -4.56537 q^{30} -7.38217 q^{31} -1.00000 q^{32} -2.25870 q^{33} +2.46184 q^{34} +1.35877 q^{35} +8.28904 q^{36} +6.98489 q^{37} -3.50451 q^{39} +1.35877 q^{40} -5.46835 q^{41} -3.35992 q^{42} +0.0812913 q^{43} +0.672249 q^{44} -11.2629 q^{45} -9.22420 q^{46} -0.226152 q^{47} -3.35992 q^{48} +1.00000 q^{49} +3.15373 q^{50} +8.27156 q^{51} +1.04304 q^{52} +6.14086 q^{53} +17.7707 q^{54} -0.913435 q^{55} +1.00000 q^{56} +9.26173 q^{58} +10.2103 q^{59} +4.56537 q^{60} -9.07391 q^{61} +7.38217 q^{62} -8.28904 q^{63} +1.00000 q^{64} -1.41725 q^{65} +2.25870 q^{66} +12.4558 q^{67} -2.46184 q^{68} -30.9925 q^{69} -1.35877 q^{70} -0.358308 q^{71} -8.28904 q^{72} +5.86115 q^{73} -6.98489 q^{74} +10.5963 q^{75} -0.672249 q^{77} +3.50451 q^{78} -10.0492 q^{79} -1.35877 q^{80} +34.8410 q^{81} +5.46835 q^{82} +2.73042 q^{83} +3.35992 q^{84} +3.34508 q^{85} -0.0812913 q^{86} +31.1186 q^{87} -0.672249 q^{88} -9.58916 q^{89} +11.2629 q^{90} -1.04304 q^{91} +9.22420 q^{92} +24.8035 q^{93} +0.226152 q^{94} +3.35992 q^{96} -13.2610 q^{97} -1.00000 q^{98} +5.57230 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{2} - 4 q^{3} + 8 q^{4} + 6 q^{5} + 4 q^{6} - 8 q^{7} - 8 q^{8} + 8 q^{9}+O(q^{10}) $ 8 * q - 8 * q^2 - 4 * q^3 + 8 * q^4 + 6 * q^5 + 4 * q^6 - 8 * q^7 - 8 * q^8 + 8 * q^9	\( 8 q - 8 q^{2} - 4 q^{3} + 8 q^{4} + 6 q^{5} + 4 q^{6} - 8 q^{7} - 8 q^{8} + 8 q^{9} - 6 q^{10} + 4 q^{11} - 4 q^{12} - 6 q^{13} + 8 q^{14} - 8 q^{15} + 8 q^{16} + 2 q^{17} - 8 q^{18} + 6 q^{20} + 4 q^{21} - 4 q^{22} + 20 q^{23} + 4 q^{24} - 8 q^{25} + 6 q^{26} - 22 q^{27} - 8 q^{28} - 16 q^{29} + 8 q^{30} - 22 q^{31} - 8 q^{32} + 8 q^{33} - 2 q^{34} - 6 q^{35} + 8 q^{36} + 12 q^{37} + 8 q^{39} - 6 q^{40} - 36 q^{41} - 4 q^{42} + 4 q^{44} - 4 q^{45} - 20 q^{46} + 6 q^{47} - 4 q^{48} + 8 q^{49} + 8 q^{50} + 4 q^{51} - 6 q^{52} - 16 q^{53} + 22 q^{54} + 18 q^{55} + 8 q^{56} + 16 q^{58} - 14 q^{59} - 8 q^{60} + 10 q^{61} + 22 q^{62} - 8 q^{63} + 8 q^{64} - 32 q^{65} - 8 q^{66} + 20 q^{67} + 2 q^{68} - 40 q^{69} + 6 q^{70} - 8 q^{72} - 18 q^{73} - 12 q^{74} - 16 q^{75} - 4 q^{77} - 8 q^{78} - 4 q^{79} + 6 q^{80} + 12 q^{81} + 36 q^{82} + 36 q^{83} + 4 q^{84} - 16 q^{85} + 28 q^{87} - 4 q^{88} - 18 q^{89} + 4 q^{90} + 6 q^{91} + 20 q^{92} + 16 q^{93} - 6 q^{94} + 4 q^{96} - 26 q^{97} - 8 q^{98} + 4 q^{99}+O(q^{100}) \) 8 * q - 8 * q^2 - 4 * q^3 + 8 * q^4 + 6 * q^5 + 4 * q^6 - 8 * q^7 - 8 * q^8 + 8 * q^9 - 6 * q^10 + 4 * q^11 - 4 * q^12 - 6 * q^13 + 8 * q^14 - 8 * q^15 + 8 * q^16 + 2 * q^17 - 8 * q^18 + 6 * q^20 + 4 * q^21 - 4 * q^22 + 20 * q^23 + 4 * q^24 - 8 * q^25 + 6 * q^26 - 22 * q^27 - 8 * q^28 - 16 * q^29 + 8 * q^30 - 22 * q^31 - 8 * q^32 + 8 * q^33 - 2 * q^34 - 6 * q^35 + 8 * q^36 + 12 * q^37 + 8 * q^39 - 6 * q^40 - 36 * q^41 - 4 * q^42 + 4 * q^44 - 4 * q^45 - 20 * q^46 + 6 * q^47 - 4 * q^48 + 8 * q^49 + 8 * q^50 + 4 * q^51 - 6 * q^52 - 16 * q^53 + 22 * q^54 + 18 * q^55 + 8 * q^56 + 16 * q^58 - 14 * q^59 - 8 * q^60 + 10 * q^61 + 22 * q^62 - 8 * q^63 + 8 * q^64 - 32 * q^65 - 8 * q^66 + 20 * q^67 + 2 * q^68 - 40 * q^69 + 6 * q^70 - 8 * q^72 - 18 * q^73 - 12 * q^74 - 16 * q^75 - 4 * q^77 - 8 * q^78 - 4 * q^79 + 6 * q^80 + 12 * q^81 + 36 * q^82 + 36 * q^83 + 4 * q^84 - 16 * q^85 + 28 * q^87 - 4 * q^88 - 18 * q^89 + 4 * q^90 + 6 * q^91 + 20 * q^92 + 16 * q^93 - 6 * q^94 + 4 * q^96 - 26 * q^97 - 8 * q^98 + 4 * 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 $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	−3.35992	−1.93985	−0.969924	−	0.243407i	$-0.921735\pi$
$3$	−3.35992	−1.93985	−0.969924	+	0.243407i	$0.921735\pi$
$4$	1.00000	0.500000
$5$	−1.35877	−0.607663	−0.303831	−	0.952726i	$-0.598266\pi$
$5$	−1.35877	−0.607663	−0.303831	+	0.952726i	$0.598266\pi$
$6$	3.35992	1.37168
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	−1.00000	−0.353553
$9$	8.28904	2.76301
$10$	1.35877	0.429682
$11$	0.672249	0.202691	0.101345	−	0.994851i	$-0.467685\pi$
$11$	0.672249	0.202691	0.101345	+	0.994851i	$0.467685\pi$
$12$	−3.35992	−0.969924
$13$	1.04304	0.289286	0.144643	−	0.989484i	$-0.453797\pi$
$13$	1.04304	0.289286	0.144643	+	0.989484i	$0.453797\pi$
$14$	1.00000	0.267261
$15$	4.56537	1.17877
$16$	1.00000	0.250000
$17$	−2.46184	−0.597083	−0.298542	−	0.954397i	$-0.596500\pi$
$17$	−2.46184	−0.597083	−0.298542	+	0.954397i	$0.596500\pi$
$18$	−8.28904	−1.95374
$19$	0	0
$19$	0	0
$20$	−1.35877	−0.303831
$21$	3.35992	0.733194
$22$	−0.672249	−0.143324
$23$	9.22420	1.92338	0.961689	−	0.274142i	$-0.0883938\pi$
$23$	9.22420	1.92338	0.961689	+	0.274142i	$0.0883938\pi$
$24$	3.35992	0.685840
$25$	−3.15373	−0.630746
$26$	−1.04304	−0.204556
$27$	−17.7707	−3.41998
$28$	−1.00000	−0.188982
$29$	−9.26173	−1.71986	−0.859930	−	0.510411i	$-0.829493\pi$
$29$	−9.26173	−1.71986	−0.859930	+	0.510411i	$0.829493\pi$
$30$	−4.56537	−0.833519
$31$	−7.38217	−1.32588	−0.662938	−	0.748674i	$-0.730691\pi$
$31$	−7.38217	−1.32588	−0.662938	+	0.748674i	$0.730691\pi$
$32$	−1.00000	−0.176777
$33$	−2.25870	−0.393189
$34$	2.46184	0.422201
$35$	1.35877	0.229675
$36$	8.28904	1.38151
$37$	6.98489	1.14831	0.574155	−	0.818747i	$-0.305331\pi$
$37$	6.98489	1.14831	0.574155	+	0.818747i	$0.305331\pi$
$38$	0	0
$39$	−3.50451	−0.561171
$40$	1.35877	0.214841
$41$	−5.46835	−0.854012	−0.427006	−	0.904249i	$-0.640432\pi$
$41$	−5.46835	−0.854012	−0.427006	+	0.904249i	$0.640432\pi$
$42$	−3.35992	−0.518446
$43$	0.0812913	0.0123968	0.00619841	−	0.999981i	$-0.498027\pi$
$43$	0.0812913	0.0123968	0.00619841	+	0.999981i	$0.498027\pi$
$44$	0.672249	0.101345
$45$	−11.2629	−1.67898
$46$	−9.22420	−1.36003
$47$	−0.226152	−0.0329876	−0.0164938	−	0.999864i	$-0.505250\pi$
$47$	−0.226152	−0.0329876	−0.0164938	+	0.999864i	$0.505250\pi$
$48$	−3.35992	−0.484962
$49$	1.00000	0.142857
$50$	3.15373	0.446005
$51$	8.27156	1.15825
$52$	1.04304	0.144643
$53$	6.14086	0.843512	0.421756	−	0.906709i	$-0.361414\pi$
$53$	6.14086	0.843512	0.421756	+	0.906709i	$0.361414\pi$
$54$	17.7707	2.41829
$55$	−0.913435	−0.123168
$56$	1.00000	0.133631
$57$	0	0
$58$	9.26173	1.21613
$59$	10.2103	1.32926	0.664632	−	0.747171i	$-0.268588\pi$
$59$	10.2103	1.32926	0.664632	+	0.747171i	$0.268588\pi$
$60$	4.56537	0.589387
$61$	−9.07391	−1.16179	−0.580897	−	0.813977i	$-0.697298\pi$
$61$	−9.07391	−1.16179	−0.580897	+	0.813977i	$0.697298\pi$
$62$	7.38217	0.937536
$63$	−8.28904	−1.04432
$64$	1.00000	0.125000
$65$	−1.41725	−0.175788
$66$	2.25870	0.278027
$67$	12.4558	1.52172	0.760862	−	0.648914i	$-0.224776\pi$
$67$	12.4558	1.52172	0.760862	+	0.648914i	$0.224776\pi$
$68$	−2.46184	−0.298542
$69$	−30.9925	−3.73106
$70$	−1.35877	−0.162405
$71$	−0.358308	−0.0425234	−0.0212617	−	0.999774i	$-0.506768\pi$
$71$	−0.358308	−0.0425234	−0.0212617	+	0.999774i	$0.506768\pi$
$72$	−8.28904	−0.976872
$73$	5.86115	0.685996	0.342998	−	0.939336i	$-0.388558\pi$
$73$	5.86115	0.685996	0.342998	+	0.939336i	$0.388558\pi$
$74$	−6.98489	−0.811978
$75$	10.5963	1.22355
$76$	0	0
$77$	−0.672249	−0.0766099
$78$	3.50451	0.396808
$79$	−10.0492	−1.13062	−0.565311	−	0.824878i	$-0.691244\pi$
$79$	−10.0492	−1.13062	−0.565311	+	0.824878i	$0.691244\pi$
$80$	−1.35877	−0.151916
$81$	34.8410	3.87122
$82$	5.46835	0.603878
$83$	2.73042	0.299702	0.149851	−	0.988709i	$-0.452121\pi$
$83$	2.73042	0.299702	0.149851	+	0.988709i	$0.452121\pi$
$84$	3.35992	0.366597
$85$	3.34508	0.362825
$86$	−0.0812913	−0.00876587
$87$	31.1186	3.33627
$88$	−0.672249	−0.0716620
$89$	−9.58916	−1.01645	−0.508225	−	0.861225i	$-0.669698\pi$
$89$	−9.58916	−1.01645	−0.508225	+	0.861225i	$0.669698\pi$
$90$	11.2629	1.18722
$91$	−1.04304	−0.109340
$92$	9.22420	0.961689
$93$	24.8035	2.57200
$94$	0.226152	0.0233258
$95$	0	0
$96$	3.35992	0.342920
$97$	−13.2610	−1.34646	−0.673228	−	0.739435i	$-0.735093\pi$
$97$	−13.2610	−1.34646	−0.673228	+	0.739435i	$0.735093\pi$
$98$	−1.00000	−0.101015
$99$	5.57230	0.560037
$100$	−3.15373	−0.315373
$101$	18.5225	1.84305	0.921527	−	0.388314i	$-0.126942\pi$
$101$	18.5225	1.84305	0.921527	+	0.388314i	$0.126942\pi$
$102$	−8.27156	−0.819007
$103$	7.88935	0.777361	0.388681	−	0.921373i	$-0.372931\pi$
$103$	7.88935	0.777361	0.388681	+	0.921373i	$0.372931\pi$
$104$	−1.04304	−0.102278
$105$	−4.56537	−0.445534
$106$	−6.14086	−0.596453
$107$	5.44998	0.526869	0.263435	−	0.964677i	$-0.415145\pi$
$107$	5.44998	0.526869	0.263435	+	0.964677i	$0.415145\pi$
$108$	−17.7707	−1.70999
$109$	−0.525792	−0.0503617	−0.0251809	−	0.999683i	$-0.508016\pi$
$109$	−0.525792	−0.0503617	−0.0251809	+	0.999683i	$0.508016\pi$
$110$	0.913435	0.0870926
$111$	−23.4687	−2.22755
$112$	−1.00000	−0.0944911
$113$	17.3578	1.63289	0.816443	−	0.577426i	$-0.195943\pi$
$113$	17.3578	1.63289	0.816443	+	0.577426i	$0.195943\pi$
$114$	0	0
$115$	−12.5336	−1.16877
$116$	−9.26173	−0.859930
$117$	8.64576	0.799301
$118$	−10.2103	−0.939932
$119$	2.46184	0.225676
$120$	−4.56537	−0.416759
$121$	−10.5481	−0.958916
$122$	9.07391	0.821513
$123$	18.3732	1.65665
$124$	−7.38217	−0.662938
$125$	11.0791	0.990943
$126$	8.28904	0.738446
$127$	7.06798	0.627182	0.313591	−	0.949558i	$-0.398468\pi$
$127$	7.06798	0.627182	0.313591	+	0.949558i	$0.398468\pi$
$128$	−1.00000	−0.0883883
$129$	−0.273132	−0.0240479
$130$	1.41725	0.124301
$131$	9.79698	0.855966	0.427983	−	0.903787i	$-0.359224\pi$
$131$	9.79698	0.855966	0.427983	+	0.903787i	$0.359224\pi$
$132$	−2.25870	−0.196595
$133$	0	0
$134$	−12.4558	−1.07602
$135$	24.1464	2.07819
$136$	2.46184	0.211101
$137$	−20.2972	−1.73411	−0.867054	−	0.498214i	$-0.833989\pi$
$137$	−20.2972	−1.73411	−0.867054	+	0.498214i	$0.833989\pi$
$138$	30.9925	2.63826
$139$	5.15852	0.437540	0.218770	−	0.975776i	$-0.429796\pi$
$139$	5.15852	0.437540	0.218770	+	0.975776i	$0.429796\pi$
$140$	1.35877	0.114837
$141$	0.759851	0.0639910
$142$	0.358308	0.0300685
$143$	0.701180	0.0586356
$144$	8.28904	0.690753
$145$	12.5846	1.04509
$146$	−5.86115	−0.485073
$147$	−3.35992	−0.277121
$148$	6.98489	0.574155
$149$	9.78505	0.801623	0.400811	−	0.916161i	$-0.368728\pi$
$149$	9.78505	0.801623	0.400811	+	0.916161i	$0.368728\pi$
$150$	−10.5963	−0.865182
$151$	−6.87697	−0.559640	−0.279820	−	0.960052i	$-0.590275\pi$
$151$	−6.87697	−0.559640	−0.279820	+	0.960052i	$0.590275\pi$
$152$	0	0
$153$	−20.4063	−1.64975
$154$	0.672249	0.0541714
$155$	10.0307	0.805685
$156$	−3.50451	−0.280585
$157$	12.1056	0.966133	0.483066	−	0.875584i	$-0.339523\pi$
$157$	12.1056	0.966133	0.483066	+	0.875584i	$0.339523\pi$
$158$	10.0492	0.799470
$159$	−20.6328	−1.63629
$160$	1.35877	0.107421
$161$	−9.22420	−0.726969
$162$	−34.8410	−2.73737
$163$	−11.3384	−0.888089	−0.444045	−	0.896005i	$-0.646457\pi$
$163$	−11.3384	−0.888089	−0.444045	+	0.896005i	$0.646457\pi$
$164$	−5.46835	−0.427006
$165$	3.06907	0.238926
$166$	−2.73042	−0.211922
$167$	11.8009	0.913184	0.456592	−	0.889676i	$-0.349070\pi$
$167$	11.8009	0.913184	0.456592	+	0.889676i	$0.349070\pi$
$168$	−3.35992	−0.259223
$169$	−11.9121	−0.916314
$170$	−3.34508	−0.256556
$171$	0	0
$172$	0.0812913	0.00619841
$173$	−18.0756	−1.37426	−0.687132	−	0.726533i	$-0.741131\pi$
$173$	−18.0756	−1.37426	−0.687132	+	0.726533i	$0.741131\pi$
$174$	−31.1186	−2.35910
$175$	3.15373	0.238400
$176$	0.672249	0.0506727
$177$	−34.3057	−2.57857
$178$	9.58916	0.718738
$179$	−4.30866	−0.322044	−0.161022	−	0.986951i	$-0.551479\pi$
$179$	−4.30866	−0.322044	−0.161022	+	0.986951i	$0.551479\pi$
$180$	−11.2629	−0.839489
$181$	−3.51816	−0.261503	−0.130751	−	0.991415i	$-0.541739\pi$
$181$	−3.51816	−0.261503	−0.130751	+	0.991415i	$0.541739\pi$
$182$	1.04304	0.0773149
$183$	30.4876	2.25371
$184$	−9.22420	−0.680017
$185$	−9.49090	−0.697785
$186$	−24.8035	−1.81868
$187$	−1.65497	−0.121023
$188$	−0.226152	−0.0164938
$189$	17.7707	1.29263
$190$	0	0
$191$	15.7576	1.14018	0.570090	−	0.821582i	$-0.306908\pi$
$191$	15.7576	1.14018	0.570090	+	0.821582i	$0.306908\pi$
$192$	−3.35992	−0.242481
$193$	11.2836	0.812214	0.406107	−	0.913826i	$-0.366886\pi$
$193$	11.2836	0.812214	0.406107	+	0.913826i	$0.366886\pi$
$194$	13.2610	0.952088
$195$	4.76184	0.341003
$196$	1.00000	0.0714286
$197$	−14.9533	−1.06538	−0.532691	−	0.846310i	$-0.678819\pi$
$197$	−14.9533	−1.06538	−0.532691	+	0.846310i	$0.678819\pi$
$198$	−5.57230	−0.396006
$199$	−1.85467	−0.131474	−0.0657370	−	0.997837i	$-0.520940\pi$
$199$	−1.85467	−0.131474	−0.0657370	+	0.997837i	$0.520940\pi$
$200$	3.15373	0.223002
$201$	−41.8506	−2.95191
$202$	−18.5225	−1.30324
$203$	9.26173	0.650046
$204$	8.27156	0.579125
$205$	7.43025	0.518951
$206$	−7.88935	−0.549677
$207$	76.4597	5.31432
$208$	1.04304	0.0723215
$209$	0	0
$210$	4.56537	0.315040
$211$	27.3986	1.88620	0.943100	−	0.332510i	$-0.107895\pi$
$211$	27.3986	1.88620	0.943100	+	0.332510i	$0.107895\pi$
$212$	6.14086	0.421756
$213$	1.20389	0.0824889
$214$	−5.44998	−0.372553
$215$	−0.110457	−0.00753308
$216$	17.7707	1.20914
$217$	7.38217	0.501134
$218$	0.525792	0.0356111
$219$	−19.6930	−1.33073
$220$	−0.913435	−0.0615838
$221$	−2.56778	−0.172728
$222$	23.4687	1.57511
$223$	1.56448	0.104766	0.0523828	−	0.998627i	$-0.483318\pi$
$223$	1.56448	0.104766	0.0523828	+	0.998627i	$0.483318\pi$
$224$	1.00000	0.0668153
$225$	−26.1414	−1.74276
$226$	−17.3578	−1.15463
$227$	24.2868	1.61197	0.805986	−	0.591935i	$-0.201636\pi$
$227$	24.2868	1.61197	0.805986	+	0.591935i	$0.201636\pi$
$228$	0	0
$229$	12.8094	0.846471	0.423236	−	0.906020i	$-0.360894\pi$
$229$	12.8094	0.846471	0.423236	+	0.906020i	$0.360894\pi$
$230$	12.5336	0.826442
$231$	2.25870	0.148612
$232$	9.26173	0.608063
$233$	−16.1943	−1.06092	−0.530460	−	0.847710i	$-0.677981\pi$
$233$	−16.1943	−1.06092	−0.530460	+	0.847710i	$0.677981\pi$
$234$	−8.64576	−0.565191
$235$	0.307289	0.0200453
$236$	10.2103	0.664632
$237$	33.7644	2.19323
$238$	−2.46184	−0.159577
$239$	−11.1682	−0.722410	−0.361205	−	0.932487i	$-0.617634\pi$
$239$	−11.1682	−0.722410	−0.361205	+	0.932487i	$0.617634\pi$
$240$	4.56537	0.294693
$241$	18.5754	1.19655	0.598274	−	0.801292i	$-0.295853\pi$
$241$	18.5754	1.19655	0.598274	+	0.801292i	$0.295853\pi$
$242$	10.5481	0.678056
$243$	−63.7507	−4.08961
$244$	−9.07391	−0.580897
$245$	−1.35877	−0.0868089
$246$	−18.3732	−1.17143
$247$	0	0
$248$	7.38217	0.468768
$249$	−9.17397	−0.581377
$250$	−11.0791	−0.700703
$251$	−6.42172	−0.405335	−0.202668	−	0.979248i	$-0.564961\pi$
$251$	−6.42172	−0.405335	−0.202668	+	0.979248i	$0.564961\pi$
$252$	−8.28904	−0.522160
$253$	6.20096	0.389851
$254$	−7.06798	−0.443484
$255$	−11.2392	−0.703826
$256$	1.00000	0.0625000
$257$	5.64329	0.352019	0.176009	−	0.984389i	$-0.443681\pi$
$257$	5.64329	0.352019	0.176009	+	0.984389i	$0.443681\pi$
$258$	0.273132	0.0170045
$259$	−6.98489	−0.434020
$260$	−1.41725	−0.0878941
$261$	−76.7708	−4.75200
$262$	−9.79698	−0.605259
$263$	−6.80169	−0.419410	−0.209705	−	0.977765i	$-0.567250\pi$
$263$	−6.80169	−0.419410	−0.209705	+	0.977765i	$0.567250\pi$
$264$	2.25870	0.139013
$265$	−8.34405	−0.512571
$266$	0	0
$267$	32.2188	1.97176
$268$	12.4558	0.760862
$269$	−16.3966	−0.999715	−0.499858	−	0.866108i	$-0.666614\pi$
$269$	−16.3966	−0.999715	−0.499858	+	0.866108i	$0.666614\pi$
$270$	−24.1464	−1.46950
$271$	−16.7984	−1.02043	−0.510216	−	0.860046i	$-0.670435\pi$
$271$	−16.7984	−1.02043	−0.510216	+	0.860046i	$0.670435\pi$
$272$	−2.46184	−0.149271
$273$	3.50451	0.212103
$274$	20.2972	1.22620
$275$	−2.12009	−0.127846
$276$	−30.9925	−1.86553
$277$	−18.1789	−1.09227	−0.546133	−	0.837699i	$-0.683901\pi$
$277$	−18.1789	−1.09227	−0.546133	+	0.837699i	$0.683901\pi$
$278$	−5.15852	−0.309387
$279$	−61.1910	−3.66341
$280$	−1.35877	−0.0812023
$281$	16.7577	0.999682	0.499841	−	0.866117i	$-0.333392\pi$
$281$	16.7577	0.999682	0.499841	+	0.866117i	$0.333392\pi$
$282$	−0.759851	−0.0452485
$283$	−3.36785	−0.200198	−0.100099	−	0.994977i	$-0.531916\pi$
$283$	−3.36785	−0.200198	−0.100099	+	0.994977i	$0.531916\pi$
$284$	−0.358308	−0.0212617
$285$	0	0
$286$	−0.701180	−0.0414616
$287$	5.46835	0.322786
$288$	−8.28904	−0.488436
$289$	−10.9394	−0.643492
$290$	−12.5846	−0.738994
$291$	44.5560	2.61192
$292$	5.86115	0.342998
$293$	8.95240	0.523005	0.261502	−	0.965203i	$-0.415782\pi$
$293$	8.95240	0.523005	0.261502	+	0.965203i	$0.415782\pi$
$294$	3.35992	0.195954
$295$	−13.8735	−0.807744
$296$	−6.98489	−0.405989
$297$	−11.9464	−0.693198
$298$	−9.78505	−0.566833
$299$	9.62117	0.556406
$300$	10.5963	0.611776
$301$	−0.0812913	−0.00468556
$302$	6.87697	0.395725
$303$	−62.2339	−3.57525
$304$	0	0
$305$	12.3294	0.705979
$306$	20.4063	1.16655
$307$	−10.2203	−0.583303	−0.291651	−	0.956525i	$-0.594205\pi$
$307$	−10.2203	−0.583303	−0.291651	+	0.956525i	$0.594205\pi$
$308$	−0.672249	−0.0383050
$309$	−26.5076	−1.50796
$310$	−10.0307	−0.569705
$311$	−11.7748	−0.667688	−0.333844	−	0.942628i	$-0.608346\pi$
$311$	−11.7748	−0.667688	−0.333844	+	0.942628i	$0.608346\pi$
$312$	3.50451	0.198404
$313$	−8.34493	−0.471683	−0.235842	−	0.971792i	$-0.575785\pi$
$313$	−8.34493	−0.471683	−0.235842	+	0.971792i	$0.575785\pi$
$314$	−12.1056	−0.683159
$315$	11.2629	0.634594
$316$	−10.0492	−0.565311
$317$	21.3660	1.20003	0.600016	−	0.799988i	$-0.295161\pi$
$317$	21.3660	1.20003	0.600016	+	0.799988i	$0.295161\pi$
$318$	20.6328	1.15703
$319$	−6.22619	−0.348600
$320$	−1.35877	−0.0759578
$321$	−18.3115	−1.02205
$322$	9.22420	0.514045
$323$	0	0
$324$	34.8410	1.93561
$325$	−3.28945	−0.182466
$326$	11.3384	0.627974
$327$	1.76662	0.0976941
$328$	5.46835	0.301939
$329$	0.226152	0.0124681
$330$	−3.06907	−0.168947
$331$	−12.2945	−0.675769	−0.337885	−	0.941188i	$-0.609711\pi$
$331$	−12.2945	−0.675769	−0.337885	+	0.941188i	$0.609711\pi$
$332$	2.73042	0.149851
$333$	57.8980	3.17279
$334$	−11.8009	−0.645718
$335$	−16.9247	−0.924695
$336$	3.35992	0.183298
$337$	−29.3162	−1.59695	−0.798477	−	0.602025i	$-0.794361\pi$
$337$	−29.3162	−1.59695	−0.798477	+	0.602025i	$0.794361\pi$
$338$	11.9121	0.647932
$339$	−58.3208	−3.16755
$340$	3.34508	0.181413
$341$	−4.96265	−0.268743
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	−0.0812913	−0.00438294
$345$	42.1119	2.26723
$346$	18.0756	0.971752
$347$	4.66549	0.250457	0.125228	−	0.992128i	$-0.460034\pi$
$347$	4.66549	0.250457	0.125228	+	0.992128i	$0.460034\pi$
$348$	31.1186	1.66813
$349$	10.9979	0.588705	0.294353	−	0.955697i	$-0.404896\pi$
$349$	10.9979	0.588705	0.294353	+	0.955697i	$0.404896\pi$
$350$	−3.15373	−0.168574
$351$	−18.5355	−0.989351
$352$	−0.672249	−0.0358310
$353$	−8.33356	−0.443551	−0.221775	−	0.975098i	$-0.571185\pi$
$353$	−8.33356	−0.443551	−0.221775	+	0.975098i	$0.571185\pi$
$354$	34.3057	1.82333
$355$	0.486860	0.0258398
$356$	−9.58916	−0.508225
$357$	−8.27156	−0.437778
$358$	4.30866	0.227720
$359$	−4.47179	−0.236012	−0.118006	−	0.993013i	$-0.537650\pi$
$359$	−4.47179	−0.236012	−0.118006	+	0.993013i	$0.537650\pi$
$360$	11.2629	0.593609
$361$	0	0
$362$	3.51816	0.184910
$363$	35.4407	1.86015
$364$	−1.04304	−0.0546699
$365$	−7.96399	−0.416854
$366$	−30.4876	−1.59361
$367$	−6.11166	−0.319026	−0.159513	−	0.987196i	$-0.550992\pi$
$367$	−6.11166	−0.319026	−0.159513	+	0.987196i	$0.550992\pi$
$368$	9.22420	0.480845
$369$	−45.3273	−2.35965
$370$	9.49090	0.493408
$371$	−6.14086	−0.318818
$372$	24.8035	1.28600
$373$	−12.6405	−0.654502	−0.327251	−	0.944937i	$-0.606122\pi$
$373$	−12.6405	−0.654502	−0.327251	+	0.944937i	$0.606122\pi$
$374$	1.65497	0.0855763
$375$	−37.2248	−1.92228
$376$	0.226152	0.0116629
$377$	−9.66032	−0.497532
$378$	−17.7707	−0.914027
$379$	−1.93824	−0.0995607	−0.0497803	−	0.998760i	$-0.515852\pi$
$379$	−1.93824	−0.0995607	−0.0497803	+	0.998760i	$0.515852\pi$
$380$	0	0
$381$	−23.7478	−1.21664
$382$	−15.7576	−0.806229
$383$	24.3608	1.24478	0.622388	−	0.782709i	$-0.286163\pi$
$383$	24.3608	1.24478	0.622388	+	0.782709i	$0.286163\pi$
$384$	3.35992	0.171460
$385$	0.913435	0.0465530
$386$	−11.2836	−0.574322
$387$	0.673827	0.0342525
$388$	−13.2610	−0.673228
$389$	−10.0046	−0.507252	−0.253626	−	0.967302i	$-0.581623\pi$
$389$	−10.0046	−0.507252	−0.253626	+	0.967302i	$0.581623\pi$
$390$	−4.76184	−0.241125
$391$	−22.7085	−1.14842
$392$	−1.00000	−0.0505076
$393$	−32.9170	−1.66044
$394$	14.9533	0.753339
$395$	13.6546	0.687036
$396$	5.57230	0.280019
$397$	−31.1380	−1.56277	−0.781385	−	0.624050i	$-0.785486\pi$
$397$	−31.1380	−1.56277	−0.781385	+	0.624050i	$0.785486\pi$
$398$	1.85467	0.0929661
$399$	0	0
$400$	−3.15373	−0.157687
$401$	−4.25458	−0.212464	−0.106232	−	0.994341i	$-0.533879\pi$
$401$	−4.25458	−0.212464	−0.106232	+	0.994341i	$0.533879\pi$
$402$	41.8506	2.08732
$403$	−7.69986	−0.383557
$404$	18.5225	0.921527
$405$	−47.3411	−2.35240
$406$	−9.26173	−0.459652
$407$	4.69559	0.232752
$408$	−8.27156	−0.409503
$409$	−35.8582	−1.77307	−0.886537	−	0.462658i	$-0.846896\pi$
$409$	−35.8582	−1.77307	−0.886537	+	0.462658i	$0.846896\pi$
$410$	−7.43025	−0.366954
$411$	68.1969	3.36391
$412$	7.88935	0.388681
$413$	−10.2103	−0.502415
$414$	−76.4597	−3.75779
$415$	−3.71002	−0.182118
$416$	−1.04304	−0.0511390
$417$	−17.3322	−0.848761
$418$	0	0
$419$	−21.8254	−1.06624	−0.533121	−	0.846039i	$-0.678981\pi$
$419$	−21.8254	−1.06624	−0.533121	+	0.846039i	$0.678981\pi$
$420$	−4.56537	−0.222767
$421$	−27.7218	−1.35108	−0.675540	−	0.737324i	$-0.736089\pi$
$421$	−27.7218	−1.35108	−0.675540	+	0.737324i	$0.736089\pi$
$422$	−27.3986	−1.33374
$423$	−1.87458	−0.0911452
$424$	−6.14086	−0.298227
$425$	7.76397	0.376608
$426$	−1.20389	−0.0583284
$427$	9.07391	0.439117
$428$	5.44998	0.263435
$429$	−2.35591	−0.113744
$430$	0.110457	0.00532669
$431$	28.6460	1.37983	0.689915	−	0.723890i	$-0.257648\pi$
$431$	28.6460	1.37983	0.689915	+	0.723890i	$0.257648\pi$
$432$	−17.7707	−0.854994
$433$	−3.35163	−0.161069	−0.0805345	−	0.996752i	$-0.525663\pi$
$433$	−3.35163	−0.161069	−0.0805345	+	0.996752i	$0.525663\pi$
$434$	−7.38217	−0.354355
$435$	−42.2832	−2.02733
$436$	−0.525792	−0.0251809
$437$	0	0
$438$	19.6930	0.940967
$439$	−34.8282	−1.66226	−0.831130	−	0.556078i	$-0.812306\pi$
$439$	−34.8282	−1.66226	−0.831130	+	0.556078i	$0.812306\pi$
$440$	0.913435	0.0435463
$441$	8.28904	0.394716
$442$	2.56778	0.122137
$443$	27.7162	1.31683	0.658417	−	0.752653i	$-0.271226\pi$
$443$	27.7162	1.31683	0.658417	+	0.752653i	$0.271226\pi$
$444$	−23.4687	−1.11377
$445$	13.0295	0.617658
$446$	−1.56448	−0.0740804
$447$	−32.8770	−1.55503
$448$	−1.00000	−0.0472456
$449$	2.10506	0.0993438	0.0496719	−	0.998766i	$-0.484182\pi$
$449$	2.10506	0.0993438	0.0496719	+	0.998766i	$0.484182\pi$
$450$	26.1414	1.23232
$451$	−3.67609	−0.173100
$452$	17.3578	0.816443
$453$	23.1061	1.08562
$454$	−24.2868	−1.13984
$455$	1.41725	0.0664417
$456$	0	0
$457$	−13.1492	−0.615093	−0.307547	−	0.951533i	$-0.599508\pi$
$457$	−13.1492	−0.615093	−0.307547	+	0.951533i	$0.599508\pi$
$458$	−12.8094	−0.598546
$459$	43.7486	2.04201
$460$	−12.5336	−0.584383
$461$	−13.9604	−0.650200	−0.325100	−	0.945680i	$-0.605398\pi$
$461$	−13.9604	−0.650200	−0.325100	+	0.945680i	$0.605398\pi$
$462$	−2.25870	−0.105084
$463$	−5.45782	−0.253647	−0.126823	−	0.991925i	$-0.540478\pi$
$463$	−5.45782	−0.253647	−0.126823	+	0.991925i	$0.540478\pi$
$464$	−9.26173	−0.429965
$465$	−33.7023	−1.56291
$466$	16.1943	0.750184
$467$	40.0931	1.85529	0.927643	−	0.373469i	$-0.121832\pi$
$467$	40.0931	1.85529	0.927643	+	0.373469i	$0.121832\pi$
$468$	8.64576	0.399650
$469$	−12.4558	−0.575158
$470$	−0.307289	−0.0141742
$471$	−40.6738	−1.87415
$472$	−10.2103	−0.469966
$473$	0.0546480	0.00251272
$474$	−33.7644	−1.55085
$475$	0	0
$476$	2.46184	0.112838
$477$	50.9018	2.33063
$478$	11.1682	0.510821
$479$	−14.5050	−0.662751	−0.331375	−	0.943499i	$-0.607513\pi$
$479$	−14.5050	−0.662751	−0.331375	+	0.943499i	$0.607513\pi$
$480$	−4.56537	−0.208380
$481$	7.28549	0.332190
$482$	−18.5754	−0.846087
$483$	30.9925	1.41021
$484$	−10.5481	−0.479458
$485$	18.0188	0.818190
$486$	63.7507	2.89179
$487$	6.25818	0.283585	0.141793	−	0.989896i	$-0.454713\pi$
$487$	6.25818	0.283585	0.141793	+	0.989896i	$0.454713\pi$
$488$	9.07391	0.410756
$489$	38.0959	1.72276
$490$	1.35877	0.0613832
$491$	−24.5017	−1.10575	−0.552873	−	0.833266i	$-0.686468\pi$
$491$	−24.5017	−1.10575	−0.552873	+	0.833266i	$0.686468\pi$
$492$	18.3732	0.828327
$493$	22.8009	1.02690
$494$	0	0
$495$	−7.57150	−0.340314
$496$	−7.38217	−0.331469
$497$	0.358308	0.0160723
$498$	9.17397	0.411096
$499$	3.36082	0.150451	0.0752255	−	0.997167i	$-0.476032\pi$
$499$	3.36082	0.150451	0.0752255	+	0.997167i	$0.476032\pi$
$500$	11.0791	0.495472
$501$	−39.6501	−1.77144
$502$	6.42172	0.286615
$503$	−17.1646	−0.765332	−0.382666	−	0.923887i	$-0.624994\pi$
$503$	−17.1646	−0.765332	−0.382666	+	0.923887i	$0.624994\pi$
$504$	8.28904	0.369223
$505$	−25.1679	−1.11996
$506$	−6.20096	−0.275666
$507$	40.0236	1.77751
$508$	7.06798	0.313591
$509$	−34.1490	−1.51363	−0.756813	−	0.653631i	$-0.773245\pi$
$509$	−34.1490	−1.51363	−0.756813	+	0.653631i	$0.773245\pi$
$510$	11.2392	0.497680
$511$	−5.86115	−0.259282
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	−5.64329	−0.248915
$515$	−10.7199	−0.472373
$516$	−0.273132	−0.0120240
$517$	−0.152030	−0.00668629
$518$	6.98489	0.306899
$519$	60.7326	2.66586
$520$	1.41725	0.0621505
$521$	5.10628	0.223710	0.111855	−	0.993725i	$-0.464321\pi$
$521$	5.10628	0.223710	0.111855	+	0.993725i	$0.464321\pi$
$522$	76.7708	3.36017
$523$	9.09139	0.397539	0.198769	−	0.980046i	$-0.436306\pi$
$523$	9.09139	0.397539	0.198769	+	0.980046i	$0.436306\pi$
$524$	9.79698	0.427983
$525$	−10.5963	−0.462459
$526$	6.80169	0.296568
$527$	18.1737	0.791658
$528$	−2.25870	−0.0982973
$529$	62.0858	2.69938
$530$	8.34405	0.362442
$531$	84.6333	3.67277
$532$	0	0
$533$	−5.70368	−0.247054
$534$	−32.2188	−1.39424
$535$	−7.40529	−0.320159
$536$	−12.4558	−0.538011
$537$	14.4767	0.624717
$538$	16.3966	0.706906
$539$	0.672249	0.0289558
$540$	24.1464	1.03910
$541$	26.4006	1.13505	0.567525	−	0.823356i	$-0.307901\pi$
$541$	26.4006	1.13505	0.567525	+	0.823356i	$0.307901\pi$
$542$	16.7984	0.721555
$543$	11.8207	0.507276
$544$	2.46184	0.105550
$545$	0.714433	0.0306029
$546$	−3.50451	−0.149979
$547$	7.01748	0.300046	0.150023	−	0.988683i	$-0.452065\pi$
$547$	7.01748	0.300046	0.150023	+	0.988683i	$0.452065\pi$
$548$	−20.2972	−0.867054
$549$	−75.2139	−3.21005
$550$	2.12009	0.0904011
$551$	0	0
$552$	30.9925	1.31913
$553$	10.0492	0.427335
$554$	18.1789	0.772348
$555$	31.8886	1.35360
$556$	5.15852	0.218770
$557$	15.3486	0.650342	0.325171	−	0.945655i	$-0.394578\pi$
$557$	15.3486	0.650342	0.325171	+	0.945655i	$0.394578\pi$
$558$	61.1910	2.59042
$559$	0.0847898	0.00358622
$560$	1.35877	0.0574187
$561$	5.56055	0.234767
$562$	−16.7577	−0.706882
$563$	−5.92162	−0.249567	−0.124783	−	0.992184i	$-0.539824\pi$
$563$	−5.92162	−0.249567	−0.124783	+	0.992184i	$0.539824\pi$
$564$	0.759851	0.0319955
$565$	−23.5854	−0.992244
$566$	3.36785	0.141561
$567$	−34.8410	−1.46318
$568$	0.358308	0.0150343
$569$	23.2519	0.974770	0.487385	−	0.873187i	$-0.337951\pi$
$569$	23.2519	0.974770	0.487385	+	0.873187i	$0.337951\pi$
$570$	0	0
$571$	−19.4012	−0.811916	−0.405958	−	0.913892i	$-0.633062\pi$
$571$	−19.4012	−0.811916	−0.405958	+	0.913892i	$0.633062\pi$
$572$	0.701180	0.0293178
$573$	−52.9442	−2.21178
$574$	−5.46835	−0.228244
$575$	−29.0906	−1.21316
$576$	8.28904	0.345376
$577$	−2.39813	−0.0998353	−0.0499177	−	0.998753i	$-0.515896\pi$
$577$	−2.39813	−0.0998353	−0.0499177	+	0.998753i	$0.515896\pi$
$578$	10.9394	0.455017
$579$	−37.9121	−1.57557
$580$	12.5846	0.522547
$581$	−2.73042	−0.113277
$582$	−44.5560	−1.84691
$583$	4.12819	0.170972
$584$	−5.86115	−0.242536
$585$	−11.7476	−0.485705
$586$	−8.95240	−0.369820
$587$	−2.32808	−0.0960903	−0.0480452	−	0.998845i	$-0.515299\pi$
$587$	−2.32808	−0.0960903	−0.0480452	+	0.998845i	$0.515299\pi$
$588$	−3.35992	−0.138561
$589$	0	0
$590$	13.8735	0.571161
$591$	50.2420	2.06668
$592$	6.98489	0.287077
$593$	9.50805	0.390449	0.195224	−	0.980759i	$-0.437457\pi$
$593$	9.50805	0.390449	0.195224	+	0.980759i	$0.437457\pi$
$594$	11.9464	0.490165
$595$	−3.34508	−0.137135
$596$	9.78505	0.400811
$597$	6.23153	0.255040
$598$	−9.62117	−0.393439
$599$	11.9202	0.487045	0.243523	−	0.969895i	$-0.421697\pi$
$599$	11.9202	0.487045	0.243523	+	0.969895i	$0.421697\pi$
$600$	−10.5963	−0.432591
$601$	35.8525	1.46245	0.731227	−	0.682134i	$-0.238948\pi$
$601$	35.8525	1.46245	0.731227	+	0.682134i	$0.238948\pi$
$602$	0.0812913	0.00331319
$603$	103.247	4.20454
$604$	−6.87697	−0.279820
$605$	14.3325	0.582698
$606$	62.2339	2.52808
$607$	−16.0032	−0.649549	−0.324774	−	0.945792i	$-0.605288\pi$
$607$	−16.0032	−0.649549	−0.324774	+	0.945792i	$0.605288\pi$
$608$	0	0
$609$	−31.1186	−1.26099
$610$	−12.3294	−0.499203
$611$	−0.235884	−0.00954286
$612$	−20.4063	−0.824874
$613$	14.1692	0.572290	0.286145	−	0.958186i	$-0.407626\pi$
$613$	14.1692	0.572290	0.286145	+	0.958186i	$0.407626\pi$
$614$	10.2203	0.412457
$615$	−24.9650	−1.00669
$616$	0.672249	0.0270857
$617$	−27.0031	−1.08710	−0.543552	−	0.839375i	$-0.682921\pi$
$617$	−27.0031	−1.08710	−0.543552	+	0.839375i	$0.682921\pi$
$618$	26.5076	1.06629
$619$	−31.9044	−1.28235	−0.641173	−	0.767397i	$-0.721552\pi$
$619$	−31.9044	−1.28235	−0.641173	+	0.767397i	$0.721552\pi$
$620$	10.0307	0.402843
$621$	−163.921	−6.57791
$622$	11.7748	0.472126
$623$	9.58916	0.384182
$624$	−3.50451	−0.140293
$625$	0.714673	0.0285869
$626$	8.34493	0.333530
$627$	0	0
$628$	12.1056	0.483066
$629$	−17.1957	−0.685636
$630$	−11.2629	−0.448726
$631$	−19.9626	−0.794699	−0.397350	−	0.917667i	$-0.630070\pi$
$631$	−19.9626	−0.794699	−0.397350	+	0.917667i	$0.630070\pi$
$632$	10.0492	0.399735
$633$	−92.0571	−3.65894
$634$	−21.3660	−0.848551
$635$	−9.60379	−0.381115
$636$	−20.6328	−0.818143
$637$	1.04304	0.0413266
$638$	6.22619	0.246497
$639$	−2.97003	−0.117493
$640$	1.35877	0.0537103
$641$	−14.7339	−0.581953	−0.290976	−	0.956730i	$-0.593980\pi$
$641$	−14.7339	−0.581953	−0.290976	+	0.956730i	$0.593980\pi$
$642$	18.3115	0.722696
$643$	31.6423	1.24785	0.623926	−	0.781484i	$-0.285537\pi$
$643$	31.6423	1.24785	0.623926	+	0.781484i	$0.285537\pi$
$644$	−9.22420	−0.363484
$645$	0.371125	0.0146130
$646$	0	0
$647$	4.39065	0.172614	0.0863072	−	0.996269i	$-0.472493\pi$
$647$	4.39065	0.172614	0.0863072	+	0.996269i	$0.472493\pi$
$648$	−34.8410	−1.36868
$649$	6.86385	0.269430
$650$	3.28945	0.129023
$651$	−24.8035	−0.972124
$652$	−11.3384	−0.444045
$653$	9.91627	0.388054	0.194027	−	0.980996i	$-0.437845\pi$
$653$	9.91627	0.388054	0.194027	+	0.980996i	$0.437845\pi$
$654$	−1.76662	−0.0690802
$655$	−13.3119	−0.520138
$656$	−5.46835	−0.213503
$657$	48.5833	1.89542
$658$	−0.226152	−0.00881631
$659$	−19.9626	−0.777633	−0.388816	−	0.921315i	$-0.627116\pi$
$659$	−19.9626	−0.777633	−0.388816	+	0.921315i	$0.627116\pi$
$660$	3.06907	0.119463
$661$	9.49464	0.369299	0.184649	−	0.982804i	$-0.440885\pi$
$661$	9.49464	0.369299	0.184649	+	0.982804i	$0.440885\pi$
$662$	12.2945	0.477841
$663$	8.62753	0.335066
$664$	−2.73042	−0.105961
$665$	0	0
$666$	−57.8980	−2.24350
$667$	−85.4321	−3.30794
$668$	11.8009	0.456592
$669$	−5.25653	−0.203229
$670$	16.9247	0.653858
$671$	−6.09993	−0.235485
$672$	−3.35992	−0.129612
$673$	−48.6884	−1.87680	−0.938400	−	0.345552i	$-0.887692\pi$
$673$	−48.6884	−1.87680	−0.938400	+	0.345552i	$0.887692\pi$
$674$	29.3162	1.12922
$675$	56.0441	2.15714
$676$	−11.9121	−0.458157
$677$	−35.6077	−1.36851	−0.684257	−	0.729241i	$-0.739873\pi$
$677$	−35.6077	−1.36851	−0.684257	+	0.729241i	$0.739873\pi$
$678$	58.3208	2.23980
$679$	13.2610	0.508912
$680$	−3.34508	−0.128278
$681$	−81.6016	−3.12698
$682$	4.96265	0.190030
$683$	1.59061	0.0608629	0.0304314	−	0.999537i	$-0.490312\pi$
$683$	1.59061	0.0608629	0.0304314	+	0.999537i	$0.490312\pi$
$684$	0	0
$685$	27.5793	1.05375
$686$	1.00000	0.0381802
$687$	−43.0386	−1.64203
$688$	0.0812913	0.00309920
$689$	6.40514	0.244016
$690$	−42.1119	−1.60317
$691$	7.59300	0.288851	0.144426	−	0.989516i	$-0.453867\pi$
$691$	7.59300	0.288851	0.144426	+	0.989516i	$0.453867\pi$
$692$	−18.0756	−0.687132
$693$	−5.57230	−0.211674
$694$	−4.66549	−0.177099
$695$	−7.00926	−0.265877
$696$	−31.1186	−1.17955
$697$	13.4622	0.509916
$698$	−10.9979	−0.416278
$699$	54.4113	2.05803
$700$	3.15373	0.119200
$701$	20.5770	0.777183	0.388592	−	0.921410i	$-0.372962\pi$
$701$	20.5770	0.777183	0.388592	+	0.921410i	$0.372962\pi$
$702$	18.5355	0.699577
$703$	0	0
$704$	0.672249	0.0253363
$705$	−1.03247	−0.0388849
$706$	8.33356	0.313638
$707$	−18.5225	−0.696609
$708$	−34.3057	−1.28929
$709$	16.2673	0.610931	0.305465	−	0.952203i	$-0.401188\pi$
$709$	16.2673	0.610931	0.305465	+	0.952203i	$0.401188\pi$
$710$	−0.486860	−0.0182715
$711$	−83.2980	−3.12392
$712$	9.58916	0.359369
$713$	−68.0946	−2.55016
$714$	8.27156	0.309555
$715$	−0.952745	−0.0356307
$716$	−4.30866	−0.161022
$717$	37.5242	1.40137
$718$	4.47179	0.166886
$719$	49.1755	1.83394	0.916969	−	0.398959i	$-0.130628\pi$
$719$	49.1755	1.83394	0.916969	+	0.398959i	$0.130628\pi$
$720$	−11.2629	−0.419745
$721$	−7.88935	−0.293815
$722$	0	0
$723$	−62.4119	−2.32112
$724$	−3.51816	−0.130751
$725$	29.2090	1.08480
$726$	−35.4407	−1.31533
$727$	−37.8702	−1.40453	−0.702264	−	0.711917i	$-0.747827\pi$
$727$	−37.8702	−1.40453	−0.702264	+	0.711917i	$0.747827\pi$
$728$	1.04304	0.0386575
$729$	109.674	4.06200
$730$	7.96399	0.294760
$731$	−0.200126	−0.00740193
$732$	30.4876	1.12685
$733$	−46.4071	−1.71409	−0.857044	−	0.515244i	$-0.827701\pi$
$733$	−46.4071	−1.71409	−0.857044	+	0.515244i	$0.827701\pi$
$734$	6.11166	0.225585
$735$	4.56537	0.168396
$736$	−9.22420	−0.340008
$737$	8.37343	0.308439
$738$	45.3273	1.66852
$739$	36.1093	1.32830	0.664151	−	0.747598i	$-0.268793\pi$
$739$	36.1093	1.32830	0.664151	+	0.747598i	$0.268793\pi$
$740$	−9.49090	−0.348892
$741$	0	0
$742$	6.14086	0.225438
$743$	−21.2678	−0.780241	−0.390120	−	0.920764i	$-0.627567\pi$
$743$	−21.2678	−0.780241	−0.390120	+	0.920764i	$0.627567\pi$
$744$	−24.8035	−0.909339
$745$	−13.2957	−0.487116
$746$	12.6405	0.462803
$747$	22.6325	0.828081
$748$	−1.65497	−0.0605116
$749$	−5.44998	−0.199138
$750$	37.2248	1.35926
$751$	−7.42960	−0.271110	−0.135555	−	0.990770i	$-0.543282\pi$
$751$	−7.42960	−0.271110	−0.135555	+	0.990770i	$0.543282\pi$
$752$	−0.226152	−0.00824690
$753$	21.5764	0.786289
$754$	9.66032	0.351808
$755$	9.34426	0.340072
$756$	17.7707	0.646315
$757$	−30.1366	−1.09533	−0.547666	−	0.836697i	$-0.684484\pi$
$757$	−30.1366	−1.09533	−0.547666	+	0.836697i	$0.684484\pi$
$758$	1.93824	0.0704000
$759$	−20.8347	−0.756252
$760$	0	0
$761$	−35.5985	−1.29044	−0.645222	−	0.763995i	$-0.723235\pi$
$761$	−35.5985	−1.29044	−0.645222	+	0.763995i	$0.723235\pi$
$762$	23.7478	0.860292
$763$	0.525792	0.0190349
$764$	15.7576	0.570090
$765$	27.7275	1.00249
$766$	−24.3608	−0.880190
$767$	10.6497	0.384537
$768$	−3.35992	−0.121241
$769$	29.7011	1.07105	0.535525	−	0.844519i	$-0.320114\pi$
$769$	29.7011	1.07105	0.535525	+	0.844519i	$0.320114\pi$
$770$	−0.913435	−0.0329179
$771$	−18.9610	−0.682863
$772$	11.2836	0.406107
$773$	−22.7091	−0.816790	−0.408395	−	0.912805i	$-0.633911\pi$
$773$	−22.7091	−0.816790	−0.408395	+	0.912805i	$0.633911\pi$
$774$	−0.673827	−0.0242202
$775$	23.2814	0.836291
$776$	13.2610	0.476044
$777$	23.4687	0.841933
$778$	10.0046	0.358682
$779$	0	0
$780$	4.76184	0.170501
$781$	−0.240872	−0.00861909
$782$	22.7085	0.812053
$783$	164.588	5.88188
$784$	1.00000	0.0357143
$785$	−16.4488	−0.587083
$786$	32.9170	1.17411
$787$	9.88614	0.352403	0.176201	−	0.984354i	$-0.443619\pi$
$787$	9.88614	0.352403	0.176201	+	0.984354i	$0.443619\pi$
$788$	−14.9533	−0.532691
$789$	22.8531	0.813592
$790$	−13.6546	−0.485808
$791$	−17.3578	−0.617173
$792$	−5.57230	−0.198003
$793$	−9.46440	−0.336091
$794$	31.1380	1.10504
$795$	28.0353	0.994310
$796$	−1.85467	−0.0657370
$797$	13.4346	0.475879	0.237939	−	0.971280i	$-0.423528\pi$
$797$	13.4346	0.475879	0.237939	+	0.971280i	$0.423528\pi$
$798$	0	0
$799$	0.556749	0.0196963
$800$	3.15373	0.111501
$801$	−79.4849	−2.80846
$802$	4.25458	0.150234
$803$	3.94016	0.139045
$804$	−41.8506	−1.47596
$805$	12.5336	0.441752
$806$	7.69986	0.271216
$807$	55.0910	1.93930
$808$	−18.5225	−0.651618
$809$	−29.6717	−1.04320	−0.521601	−	0.853190i	$-0.674665\pi$
$809$	−29.6717	−1.04320	−0.521601	+	0.853190i	$0.674665\pi$
$810$	47.3411	1.66340
$811$	−46.9132	−1.64734	−0.823672	−	0.567067i	$-0.808078\pi$
$811$	−46.9132	−1.64734	−0.823672	+	0.567067i	$0.808078\pi$
$812$	9.26173	0.325023
$813$	56.4413	1.97948
$814$	−4.69559	−0.164580
$815$	15.4063	0.539658
$816$	8.27156	0.289563
$817$	0	0
$818$	35.8582	1.25375
$819$	−8.64576	−0.302107
$820$	7.43025	0.259476
$821$	8.20066	0.286205	0.143103	−	0.989708i	$-0.454292\pi$
$821$	8.20066	0.286205	0.143103	+	0.989708i	$0.454292\pi$
$822$	−68.1969	−2.37864
$823$	−13.2650	−0.462388	−0.231194	−	0.972908i	$-0.574263\pi$
$823$	−13.2650	−0.462388	−0.231194	+	0.972908i	$0.574263\pi$
$824$	−7.88935	−0.274839
$825$	7.12334	0.248003
$826$	10.2103	0.355261
$827$	−7.03178	−0.244519	−0.122260	−	0.992498i	$-0.539014\pi$
$827$	−7.03178	−0.244519	−0.122260	+	0.992498i	$0.539014\pi$
$828$	76.4597	2.65716
$829$	9.85280	0.342202	0.171101	−	0.985254i	$-0.445268\pi$
$829$	9.85280	0.342202	0.171101	+	0.985254i	$0.445268\pi$
$830$	3.71002	0.128777
$831$	61.0796	2.11883
$832$	1.04304	0.0361607
$833$	−2.46184	−0.0852976
$834$	17.3322	0.600164
$835$	−16.0348	−0.554907
$836$	0	0
$837$	131.186	4.53446
$838$	21.8254	0.753946
$839$	23.8269	0.822595	0.411298	−	0.911501i	$-0.365076\pi$
$839$	23.8269	0.822595	0.411298	+	0.911501i	$0.365076\pi$
$840$	4.56537	0.157520
$841$	56.7797	1.95792
$842$	27.7218	0.955358
$843$	−56.3046	−1.93923
$844$	27.3986	0.943100
$845$	16.1858	0.556809
$846$	1.87458	0.0644494
$847$	10.5481	0.362436
$848$	6.14086	0.210878
$849$	11.3157	0.388353
$850$	−7.76397	−0.266302
$851$	64.4301	2.20863
$852$	1.20389	0.0412444
$853$	−16.3690	−0.560465	−0.280233	−	0.959932i	$-0.590412\pi$
$853$	−16.3690	−0.560465	−0.280233	+	0.959932i	$0.590412\pi$
$854$	−9.07391	−0.310503
$855$	0	0
$856$	−5.44998	−0.186276
$857$	14.3213	0.489207	0.244604	−	0.969623i	$-0.421342\pi$
$857$	14.3213	0.489207	0.244604	+	0.969623i	$0.421342\pi$
$858$	2.35591	0.0804293
$859$	−6.06335	−0.206879	−0.103439	−	0.994636i	$-0.532985\pi$
$859$	−6.06335	−0.206879	−0.103439	+	0.994636i	$0.532985\pi$
$860$	−0.110457	−0.00376654
$861$	−18.3732	−0.626156
$862$	−28.6460	−0.975688
$863$	34.3964	1.17087	0.585434	−	0.810720i	$-0.300924\pi$
$863$	34.3964	1.17087	0.585434	+	0.810720i	$0.300924\pi$
$864$	17.7707	0.604572
$865$	24.5607	0.835089
$866$	3.35163	0.113893
$867$	36.7553	1.24828
$868$	7.38217	0.250567
$869$	−6.75555	−0.229166
$870$	42.2832	1.43354
$871$	12.9919	0.440213
$872$	0.525792	0.0178056
$873$	−109.921	−3.72027
$874$	0	0
$875$	−11.0791	−0.374541
$876$	−19.6930	−0.665364
$877$	−9.81095	−0.331292	−0.165646	−	0.986185i	$-0.552971\pi$
$877$	−9.81095	−0.331292	−0.165646	+	0.986185i	$0.552971\pi$
$878$	34.8282	1.17540
$879$	−30.0793	−1.01455
$880$	−0.913435	−0.0307919
$881$	−29.8319	−1.00506	−0.502531	−	0.864559i	$-0.667598\pi$
$881$	−29.8319	−1.00506	−0.502531	+	0.864559i	$0.667598\pi$
$882$	−8.28904	−0.279106
$883$	−25.5339	−0.859283	−0.429642	−	0.902999i	$-0.641360\pi$
$883$	−25.5339	−0.859283	−0.429642	+	0.902999i	$0.641360\pi$
$884$	−2.56778	−0.0863639
$885$	46.6137	1.56690
$886$	−27.7162	−0.931143
$887$	−1.66452	−0.0558893	−0.0279446	−	0.999609i	$-0.508896\pi$
$887$	−1.66452	−0.0558893	−0.0279446	+	0.999609i	$0.508896\pi$
$888$	23.4687	0.787557
$889$	−7.06798	−0.237052
$890$	−13.0295	−0.436750
$891$	23.4218	0.784661
$892$	1.56448	0.0523828
$893$	0	0
$894$	32.8770	1.09957
$895$	5.85450	0.195694
$896$	1.00000	0.0334077
$897$	−32.3263	−1.07934
$898$	−2.10506	−0.0702467
$899$	68.3716	2.28032
$900$	−26.1414	−0.871380
$901$	−15.1178	−0.503647
$902$	3.67609	0.122400
$903$	0.273132	0.00908927
$904$	−17.3578	−0.577313
$905$	4.78039	0.158906
$906$	−23.1061	−0.767647
$907$	−42.3441	−1.40601	−0.703007	−	0.711183i	$-0.748160\pi$
$907$	−42.3441	−1.40601	−0.703007	+	0.711183i	$0.748160\pi$
$908$	24.2868	0.805986
$909$	153.533	5.09238
$910$	−1.41725	−0.0469814
$911$	−2.81275	−0.0931905	−0.0465953	−	0.998914i	$-0.514837\pi$
$911$	−2.81275	−0.0931905	−0.0465953	+	0.998914i	$0.514837\pi$
$912$	0	0
$913$	1.83552	0.0607469
$914$	13.1492	0.434937
$915$	−41.4257	−1.36949
$916$	12.8094	0.423236
$917$	−9.79698	−0.323525
$918$	−43.7486	−1.44392
$919$	42.3008	1.39537	0.697687	−	0.716403i	$-0.254213\pi$
$919$	42.3008	1.39537	0.697687	+	0.716403i	$0.254213\pi$
$920$	12.5336	0.413221
$921$	34.3393	1.13152
$922$	13.9604	0.459761
$923$	−0.373728	−0.0123014
$924$	2.25870	0.0743058
$925$	−22.0285	−0.724292
$926$	5.45782	0.179355
$927$	65.3951	2.14786
$928$	9.26173	0.304031
$929$	23.2939	0.764247	0.382123	−	0.924111i	$-0.375193\pi$
$929$	23.2939	0.764247	0.382123	+	0.924111i	$0.375193\pi$
$930$	33.7023	1.10514
$931$	0	0
$932$	−16.1943	−0.530460
$933$	39.5623	1.29521
$934$	−40.0931	−1.31188
$935$	2.24873	0.0735413
$936$	−8.64576	−0.282595
$937$	−59.3078	−1.93750	−0.968750	−	0.248038i	$-0.920214\pi$
$937$	−59.3078	−1.93750	−0.968750	+	0.248038i	$0.920214\pi$
$938$	12.4558	0.406698
$939$	28.0382	0.914994
$940$	0.307289	0.0100227
$941$	0.727116	0.0237033	0.0118517	−	0.999930i	$-0.496227\pi$
$941$	0.727116	0.0237033	0.0118517	+	0.999930i	$0.496227\pi$
$942$	40.6738	1.32522
$943$	−50.4411	−1.64259
$944$	10.2103	0.332316
$945$	−24.1464	−0.785483
$946$	−0.0546480	−0.00177676
$947$	22.0588	0.716814	0.358407	−	0.933565i	$-0.383320\pi$
$947$	22.0588	0.716814	0.358407	+	0.933565i	$0.383320\pi$
$948$	33.7644	1.09662
$949$	6.11339	0.198449
$950$	0	0
$951$	−71.7879	−2.32788
$952$	−2.46184	−0.0797886
$953$	−32.6219	−1.05673	−0.528364	−	0.849018i	$-0.677194\pi$
$953$	−32.6219	−1.05673	−0.528364	+	0.849018i	$0.677194\pi$
$954$	−50.9018	−1.64801
$955$	−21.4110	−0.692844
$956$	−11.1682	−0.361205
$957$	20.9195	0.676231
$958$	14.5050	0.468636
$959$	20.2972	0.655431
$960$	4.56537	0.147347
$961$	23.4964	0.757947
$962$	−7.28549	−0.234894
$963$	45.1751	1.45575
$964$	18.5754	0.598274
$965$	−15.3319	−0.493552
$966$	−30.9925	−0.997168
$967$	−34.4438	−1.10764	−0.553820	−	0.832637i	$-0.686830\pi$
$967$	−34.4438	−1.10764	−0.553820	+	0.832637i	$0.686830\pi$
$968$	10.5481	0.339028
$969$	0	0
$970$	−18.0188	−0.578548
$971$	−10.0122	−0.321307	−0.160653	−	0.987011i	$-0.551360\pi$
$971$	−10.0122	−0.321307	−0.160653	+	0.987011i	$0.551360\pi$
$972$	−63.7507	−2.04481
$973$	−5.15852	−0.165374
$974$	−6.25818	−0.200525
$975$	11.0523	0.353956
$976$	−9.07391	−0.290449
$977$	0.947408	0.0303103	0.0151551	−	0.999885i	$-0.495176\pi$
$977$	0.947408	0.0303103	0.0151551	+	0.999885i	$0.495176\pi$
$978$	−38.0959	−1.21817
$979$	−6.44631	−0.206025
$980$	−1.35877	−0.0434045
$981$	−4.35831	−0.139150
$982$	24.5017	0.781880
$983$	19.4148	0.619237	0.309619	−	0.950861i	$-0.399799\pi$
$983$	19.4148	0.619237	0.309619	+	0.950861i	$0.399799\pi$
$984$	−18.3732	−0.585716
$985$	20.3182	0.647393
$986$	−22.8009	−0.726128
$987$	−0.759851	−0.0241863
$988$	0	0
$989$	0.749848	0.0238438
$990$	7.57150	0.240638
$991$	−31.2746	−0.993471	−0.496736	−	0.867902i	$-0.665468\pi$
$991$	−31.2746	−0.993471	−0.496736	+	0.867902i	$0.665468\pi$
$992$	7.38217	0.234384
$993$	41.3086	1.31089
$994$	−0.358308	−0.0113648
$995$	2.52008	0.0798918
$996$	−9.17397	−0.290689
$997$	20.5619	0.651202	0.325601	−	0.945507i	$-0.394433\pi$
$997$	20.5619	0.651202	0.325601	+	0.945507i	$0.394433\pi$
$998$	−3.36082	−0.106385
$999$	−124.127	−3.92719

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5054.2.a.bf.1.1	✓	8
19.18	odd	2		5054.2.a.bi.1.8	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5054.2.a.bf.1.1	✓	8	1.1	even	1	trivial
5054.2.a.bi.1.8	yes	8	19.18	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:	\( N \)	\(=\)	\( 5054 = 2 \cdot 7 \cdot 19^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5054.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -3.35992 q^{3} +1.00000 q^{4} -1.35877 q^{5} +3.35992 q^{6} -1.00000 q^{7} -1.00000 q^{8} +8.28904 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -3.35992 q^{3} +1.00000 q^{4} -1.35877 q^{5} +3.35992 q^{6} -1.00000 q^{7} -1.00000 q^{8} +8.28904 q^{9} +1.35877 q^{10} +0.672249 q^{11} -3.35992 q^{12} +1.04304 q^{13} +1.00000 q^{14} +4.56537 q^{15} +1.00000 q^{16} -2.46184 q^{17} -8.28904 q^{18} -1.35877 q^{20} +3.35992 q^{21} -0.672249 q^{22} +9.22420 q^{23} +3.35992 q^{24} -3.15373 q^{25} -1.04304 q^{26} -17.7707 q^{27} -1.00000 q^{28} -9.26173 q^{29} -4.56537 q^{30} -7.38217 q^{31} -1.00000 q^{32} -2.25870 q^{33} +2.46184 q^{34} +1.35877 q^{35} +8.28904 q^{36} +6.98489 q^{37} -3.50451 q^{39} +1.35877 q^{40} -5.46835 q^{41} -3.35992 q^{42} +0.0812913 q^{43} +0.672249 q^{44} -11.2629 q^{45} -9.22420 q^{46} -0.226152 q^{47} -3.35992 q^{48} +1.00000 q^{49} +3.15373 q^{50} +8.27156 q^{51} +1.04304 q^{52} +6.14086 q^{53} +17.7707 q^{54} -0.913435 q^{55} +1.00000 q^{56} +9.26173 q^{58} +10.2103 q^{59} +4.56537 q^{60} -9.07391 q^{61} +7.38217 q^{62} -8.28904 q^{63} +1.00000 q^{64} -1.41725 q^{65} +2.25870 q^{66} +12.4558 q^{67} -2.46184 q^{68} -30.9925 q^{69} -1.35877 q^{70} -0.358308 q^{71} -8.28904 q^{72} +5.86115 q^{73} -6.98489 q^{74} +10.5963 q^{75} -0.672249 q^{77} +3.50451 q^{78} -10.0492 q^{79} -1.35877 q^{80} +34.8410 q^{81} +5.46835 q^{82} +2.73042 q^{83} +3.35992 q^{84} +3.34508 q^{85} -0.0812913 q^{86} +31.1186 q^{87} -0.672249 q^{88} -9.58916 q^{89} +11.2629 q^{90} -1.04304 q^{91} +9.22420 q^{92} +24.8035 q^{93} +0.226152 q^{94} +3.35992 q^{96} -13.2610 q^{97} -1.00000 q^{98} +5.57230 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{2} - 4 q^{3} + 8 q^{4} + 6 q^{5} + 4 q^{6} - 8 q^{7} - 8 q^{8} + 8 q^{9}+O(q^{10}) \) 8 * q - 8 * q^2 - 4 * q^3 + 8 * q^4 + 6 * q^5 + 4 * q^6 - 8 * q^7 - 8 * q^8 + 8 * q^9	\( 8 q - 8 q^{2} - 4 q^{3} + 8 q^{4} + 6 q^{5} + 4 q^{6} - 8 q^{7} - 8 q^{8} + 8 q^{9} - 6 q^{10} + 4 q^{11} - 4 q^{12} - 6 q^{13} + 8 q^{14} - 8 q^{15} + 8 q^{16} + 2 q^{17} - 8 q^{18} + 6 q^{20} + 4 q^{21} - 4 q^{22} + 20 q^{23} + 4 q^{24} - 8 q^{25} + 6 q^{26} - 22 q^{27} - 8 q^{28} - 16 q^{29} + 8 q^{30} - 22 q^{31} - 8 q^{32} + 8 q^{33} - 2 q^{34} - 6 q^{35} + 8 q^{36} + 12 q^{37} + 8 q^{39} - 6 q^{40} - 36 q^{41} - 4 q^{42} + 4 q^{44} - 4 q^{45} - 20 q^{46} + 6 q^{47} - 4 q^{48} + 8 q^{49} + 8 q^{50} + 4 q^{51} - 6 q^{52} - 16 q^{53} + 22 q^{54} + 18 q^{55} + 8 q^{56} + 16 q^{58} - 14 q^{59} - 8 q^{60} + 10 q^{61} + 22 q^{62} - 8 q^{63} + 8 q^{64} - 32 q^{65} - 8 q^{66} + 20 q^{67} + 2 q^{68} - 40 q^{69} + 6 q^{70} - 8 q^{72} - 18 q^{73} - 12 q^{74} - 16 q^{75} - 4 q^{77} - 8 q^{78} - 4 q^{79} + 6 q^{80} + 12 q^{81} + 36 q^{82} + 36 q^{83} + 4 q^{84} - 16 q^{85} + 28 q^{87} - 4 q^{88} - 18 q^{89} + 4 q^{90} + 6 q^{91} + 20 q^{92} + 16 q^{93} - 6 q^{94} + 4 q^{96} - 26 q^{97} - 8 q^{98} + 4 q^{99}+O(q^{100}) \) 8 * q - 8 * q^2 - 4 * q^3 + 8 * q^4 + 6 * q^5 + 4 * q^6 - 8 * q^7 - 8 * q^8 + 8 * q^9 - 6 * q^10 + 4 * q^11 - 4 * q^12 - 6 * q^13 + 8 * q^14 - 8 * q^15 + 8 * q^16 + 2 * q^17 - 8 * q^18 + 6 * q^20 + 4 * q^21 - 4 * q^22 + 20 * q^23 + 4 * q^24 - 8 * q^25 + 6 * q^26 - 22 * q^27 - 8 * q^28 - 16 * q^29 + 8 * q^30 - 22 * q^31 - 8 * q^32 + 8 * q^33 - 2 * q^34 - 6 * q^35 + 8 * q^36 + 12 * q^37 + 8 * q^39 - 6 * q^40 - 36 * q^41 - 4 * q^42 + 4 * q^44 - 4 * q^45 - 20 * q^46 + 6 * q^47 - 4 * q^48 + 8 * q^49 + 8 * q^50 + 4 * q^51 - 6 * q^52 - 16 * q^53 + 22 * q^54 + 18 * q^55 + 8 * q^56 + 16 * q^58 - 14 * q^59 - 8 * q^60 + 10 * q^61 + 22 * q^62 - 8 * q^63 + 8 * q^64 - 32 * q^65 - 8 * q^66 + 20 * q^67 + 2 * q^68 - 40 * q^69 + 6 * q^70 - 8 * q^72 - 18 * q^73 - 12 * q^74 - 16 * q^75 - 4 * q^77 - 8 * q^78 - 4 * q^79 + 6 * q^80 + 12 * q^81 + 36 * q^82 + 36 * q^83 + 4 * q^84 - 16 * q^85 + 28 * q^87 - 4 * q^88 - 18 * q^89 + 4 * q^90 + 6 * q^91 + 20 * q^92 + 16 * q^93 - 6 * q^94 + 4 * q^96 - 26 * q^97 - 8 * q^98 + 4 * q^99

Introduction

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