LMFDB - Embedded newform 5054.2.a.d.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:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$1.61803$ 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)

$f(q)$	$=$	$q-1.00000 q^{2} -1.61803 q^{3} +1.00000 q^{4} -3.85410 q^{5} +1.61803 q^{6} -1.00000 q^{7} -1.00000 q^{8} -0.381966 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -1.61803 q^{3} +1.00000 q^{4} -3.85410 q^{5} +1.61803 q^{6} -1.00000 q^{7} -1.00000 q^{8} -0.381966 q^{9} +3.85410 q^{10} +1.61803 q^{11} -1.61803 q^{12} -2.00000 q^{13} +1.00000 q^{14} +6.23607 q^{15} +1.00000 q^{16} -3.23607 q^{17} +0.381966 q^{18} -3.85410 q^{20} +1.61803 q^{21} -1.61803 q^{22} +1.61803 q^{24} +9.85410 q^{25} +2.00000 q^{26} +5.47214 q^{27} -1.00000 q^{28} +1.09017 q^{29} -6.23607 q^{30} -8.47214 q^{31} -1.00000 q^{32} -2.61803 q^{33} +3.23607 q^{34} +3.85410 q^{35} -0.381966 q^{36} -9.09017 q^{37} +3.23607 q^{39} +3.85410 q^{40} +2.14590 q^{41} -1.61803 q^{42} +1.38197 q^{43} +1.61803 q^{44} +1.47214 q^{45} +8.38197 q^{47} -1.61803 q^{48} +1.00000 q^{49} -9.85410 q^{50} +5.23607 q^{51} -2.00000 q^{52} +13.3262 q^{53} -5.47214 q^{54} -6.23607 q^{55} +1.00000 q^{56} -1.09017 q^{58} -9.38197 q^{59} +6.23607 q^{60} +12.5623 q^{61} +8.47214 q^{62} +0.381966 q^{63} +1.00000 q^{64} +7.70820 q^{65} +2.61803 q^{66} -2.76393 q^{67} -3.23607 q^{68} -3.85410 q^{70} +12.5623 q^{71} +0.381966 q^{72} +12.9443 q^{73} +9.09017 q^{74} -15.9443 q^{75} -1.61803 q^{77} -3.23607 q^{78} -11.0902 q^{79} -3.85410 q^{80} -7.70820 q^{81} -2.14590 q^{82} +13.7082 q^{83} +1.61803 q^{84} +12.4721 q^{85} -1.38197 q^{86} -1.76393 q^{87} -1.61803 q^{88} -11.3262 q^{89} -1.47214 q^{90} +2.00000 q^{91} +13.7082 q^{93} -8.38197 q^{94} +1.61803 q^{96} -5.61803 q^{97} -1.00000 q^{98} -0.618034 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} - q^{3} + 2 q^{4} - q^{5} + q^{6} - 2 q^{7} - 2 q^{8} - 3 q^{9}+O(q^{10}) $ 2 * q - 2 * q^2 - q^3 + 2 * q^4 - q^5 + q^6 - 2 * q^7 - 2 * q^8 - 3 * q^9	$ 2 q - 2 q^{2} - q^{3} + 2 q^{4} - q^{5} + q^{6} - 2 q^{7} - 2 q^{8} - 3 q^{9} + q^{10} + q^{11} - q^{12} - 4 q^{13} + 2 q^{14} + 8 q^{15} + 2 q^{16} - 2 q^{17} + 3 q^{18} - q^{20} + q^{21} - q^{22} + q^{24} + 13 q^{25} + 4 q^{26} + 2 q^{27} - 2 q^{28} - 9 q^{29} - 8 q^{30} - 8 q^{31} - 2 q^{32} - 3 q^{33} + 2 q^{34} + q^{35} - 3 q^{36} - 7 q^{37} + 2 q^{39} + q^{40} + 11 q^{41} - q^{42} + 5 q^{43} + q^{44} - 6 q^{45} + 19 q^{47} - q^{48} + 2 q^{49} - 13 q^{50} + 6 q^{51} - 4 q^{52} + 11 q^{53} - 2 q^{54} - 8 q^{55} + 2 q^{56} + 9 q^{58} - 21 q^{59} + 8 q^{60} + 5 q^{61} + 8 q^{62} + 3 q^{63} + 2 q^{64} + 2 q^{65} + 3 q^{66} - 10 q^{67} - 2 q^{68} - q^{70} + 5 q^{71} + 3 q^{72} + 8 q^{73} + 7 q^{74} - 14 q^{75} - q^{77} - 2 q^{78} - 11 q^{79} - q^{80} - 2 q^{81} - 11 q^{82} + 14 q^{83} + q^{84} + 16 q^{85} - 5 q^{86} - 8 q^{87} - q^{88} - 7 q^{89} + 6 q^{90} + 4 q^{91} + 14 q^{93} - 19 q^{94} + q^{96} - 9 q^{97} - 2 q^{98} + q^{99}+O(q^{100}) $ 2 * q - 2 * q^2 - q^3 + 2 * q^4 - q^5 + q^6 - 2 * q^7 - 2 * q^8 - 3 * q^9 + q^10 + q^11 - q^12 - 4 * q^13 + 2 * q^14 + 8 * q^15 + 2 * q^16 - 2 * q^17 + 3 * q^18 - q^20 + q^21 - q^22 + q^24 + 13 * q^25 + 4 * q^26 + 2 * q^27 - 2 * q^28 - 9 * q^29 - 8 * q^30 - 8 * q^31 - 2 * q^32 - 3 * q^33 + 2 * q^34 + q^35 - 3 * q^36 - 7 * q^37 + 2 * q^39 + q^40 + 11 * q^41 - q^42 + 5 * q^43 + q^44 - 6 * q^45 + 19 * q^47 - q^48 + 2 * q^49 - 13 * q^50 + 6 * q^51 - 4 * q^52 + 11 * q^53 - 2 * q^54 - 8 * q^55 + 2 * q^56 + 9 * q^58 - 21 * q^59 + 8 * q^60 + 5 * q^61 + 8 * q^62 + 3 * q^63 + 2 * q^64 + 2 * q^65 + 3 * q^66 - 10 * q^67 - 2 * q^68 - q^70 + 5 * q^71 + 3 * q^72 + 8 * q^73 + 7 * q^74 - 14 * q^75 - q^77 - 2 * q^78 - 11 * q^79 - q^80 - 2 * q^81 - 11 * q^82 + 14 * q^83 + q^84 + 16 * q^85 - 5 * q^86 - 8 * q^87 - q^88 - 7 * q^89 + 6 * q^90 + 4 * q^91 + 14 * q^93 - 19 * q^94 + q^96 - 9 * q^97 - 2 * q^98 + 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
$3$	−1.61803	−0.934172	−0.467086	−	0.884212i	$-0.654696\pi$
$3$	−1.61803	−0.934172	−0.467086	+	0.884212i	$0.654696\pi$
$4$	1.00000	0.500000
$5$	−3.85410	−1.72361	−0.861803	−	0.507242i	$-0.830665\pi$
$5$	−3.85410	−1.72361	−0.861803	+	0.507242i	$0.830665\pi$
$6$	1.61803	0.660560
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	−1.00000	−0.353553
$9$	−0.381966	−0.127322
$10$	3.85410	1.21877
$11$	1.61803	0.487856	0.243928	−	0.969793i	$-0.421564\pi$
$11$	1.61803	0.487856	0.243928	+	0.969793i	$0.421564\pi$
$12$	−1.61803	−0.467086
$13$	−2.00000	−0.554700	−0.277350	−	0.960769i	$-0.589456\pi$
$13$	−2.00000	−0.554700	−0.277350	+	0.960769i	$0.589456\pi$
$14$	1.00000	0.267261
$15$	6.23607	1.61015
$16$	1.00000	0.250000
$17$	−3.23607	−0.784862	−0.392431	−	0.919781i	$-0.628366\pi$
$17$	−3.23607	−0.784862	−0.392431	+	0.919781i	$0.628366\pi$
$18$	0.381966	0.0900303
$19$	0	0
$19$	0	0
$20$	−3.85410	−0.861803
$21$	1.61803	0.353084
$22$	−1.61803	−0.344966
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	1.61803	0.330280
$25$	9.85410	1.97082
$26$	2.00000	0.392232
$27$	5.47214	1.05311
$28$	−1.00000	−0.188982
$29$	1.09017	0.202439	0.101220	−	0.994864i	$-0.467725\pi$
$29$	1.09017	0.202439	0.101220	+	0.994864i	$0.467725\pi$
$30$	−6.23607	−1.13855
$31$	−8.47214	−1.52164	−0.760820	−	0.648963i	$-0.775203\pi$
$31$	−8.47214	−1.52164	−0.760820	+	0.648963i	$0.775203\pi$
$32$	−1.00000	−0.176777
$33$	−2.61803	−0.455741
$34$	3.23607	0.554981
$35$	3.85410	0.651462
$36$	−0.381966	−0.0636610
$37$	−9.09017	−1.49441	−0.747207	−	0.664591i	$-0.768606\pi$
$37$	−9.09017	−1.49441	−0.747207	+	0.664591i	$0.768606\pi$
$38$	0	0
$39$	3.23607	0.518186
$40$	3.85410	0.609387
$41$	2.14590	0.335133	0.167566	−	0.985861i	$-0.446409\pi$
$41$	2.14590	0.335133	0.167566	+	0.985861i	$0.446409\pi$
$42$	−1.61803	−0.249668
$43$	1.38197	0.210748	0.105374	−	0.994433i	$-0.466396\pi$
$43$	1.38197	0.210748	0.105374	+	0.994433i	$0.466396\pi$
$44$	1.61803	0.243928
$45$	1.47214	0.219453
$46$	0	0
$47$	8.38197	1.22264	0.611318	−	0.791385i	$-0.290640\pi$
$47$	8.38197	1.22264	0.611318	+	0.791385i	$0.290640\pi$
$48$	−1.61803	−0.233543
$49$	1.00000	0.142857
$50$	−9.85410	−1.39358
$51$	5.23607	0.733196
$52$	−2.00000	−0.277350
$53$	13.3262	1.83050	0.915250	−	0.402887i	$-0.131993\pi$
$53$	13.3262	1.83050	0.915250	+	0.402887i	$0.131993\pi$
$54$	−5.47214	−0.744663
$55$	−6.23607	−0.840871
$56$	1.00000	0.133631
$57$	0	0
$58$	−1.09017	−0.143146
$59$	−9.38197	−1.22143	−0.610714	−	0.791851i	$-0.709117\pi$
$59$	−9.38197	−1.22143	−0.610714	+	0.791851i	$0.709117\pi$
$60$	6.23607	0.805073
$61$	12.5623	1.60844	0.804219	−	0.594333i	$-0.202584\pi$
$61$	12.5623	1.60844	0.804219	+	0.594333i	$0.202584\pi$
$62$	8.47214	1.07596
$63$	0.381966	0.0481232
$64$	1.00000	0.125000
$65$	7.70820	0.956085
$66$	2.61803	0.322258
$67$	−2.76393	−0.337668	−0.168834	−	0.985644i	$-0.554000\pi$
$67$	−2.76393	−0.337668	−0.168834	+	0.985644i	$0.554000\pi$
$68$	−3.23607	−0.392431
$69$	0	0
$70$	−3.85410	−0.460653
$71$	12.5623	1.49087	0.745436	−	0.666578i	$-0.232241\pi$
$71$	12.5623	1.49087	0.745436	+	0.666578i	$0.232241\pi$
$72$	0.381966	0.0450151
$73$	12.9443	1.51501	0.757506	−	0.652828i	$-0.226418\pi$
$73$	12.9443	1.51501	0.757506	+	0.652828i	$0.226418\pi$
$74$	9.09017	1.05671
$75$	−15.9443	−1.84109
$76$	0	0
$77$	−1.61803	−0.184392
$78$	−3.23607	−0.366413
$79$	−11.0902	−1.24774	−0.623871	−	0.781527i	$-0.714441\pi$
$79$	−11.0902	−1.24774	−0.623871	+	0.781527i	$0.714441\pi$
$80$	−3.85410	−0.430902
$81$	−7.70820	−0.856467
$82$	−2.14590	−0.236975
$83$	13.7082	1.50467	0.752335	−	0.658780i	$-0.228927\pi$
$83$	13.7082	1.50467	0.752335	+	0.658780i	$0.228927\pi$
$84$	1.61803	0.176542
$85$	12.4721	1.35279
$86$	−1.38197	−0.149021
$87$	−1.76393	−0.189113
$88$	−1.61803	−0.172483
$89$	−11.3262	−1.20058	−0.600289	−	0.799783i	$-0.704948\pi$
$89$	−11.3262	−1.20058	−0.600289	+	0.799783i	$0.704948\pi$
$90$	−1.47214	−0.155177
$91$	2.00000	0.209657
$92$	0	0
$93$	13.7082	1.42147
$94$	−8.38197	−0.864534
$95$	0	0
$96$	1.61803	0.165140
$97$	−5.61803	−0.570425	−0.285212	−	0.958464i	$-0.592064\pi$
$97$	−5.61803	−0.570425	−0.285212	+	0.958464i	$0.592064\pi$
$98$	−1.00000	−0.101015
$99$	−0.618034	−0.0621148
$100$	9.85410	0.985410
$101$	10.0000	0.995037	0.497519	−	0.867453i	$-0.334245\pi$
$101$	10.0000	0.995037	0.497519	+	0.867453i	$0.334245\pi$
$102$	−5.23607	−0.518448
$103$	−1.52786	−0.150545	−0.0752725	−	0.997163i	$-0.523983\pi$
$103$	−1.52786	−0.150545	−0.0752725	+	0.997163i	$0.523983\pi$
$104$	2.00000	0.196116
$105$	−6.23607	−0.608578
$106$	−13.3262	−1.29436
$107$	−0.291796	−0.0282090	−0.0141045	−	0.999901i	$-0.504490\pi$
$107$	−0.291796	−0.0282090	−0.0141045	+	0.999901i	$0.504490\pi$
$108$	5.47214	0.526557
$109$	8.38197	0.802847	0.401423	−	0.915893i	$-0.368516\pi$
$109$	8.38197	0.802847	0.401423	+	0.915893i	$0.368516\pi$
$110$	6.23607	0.594586
$111$	14.7082	1.39604
$112$	−1.00000	−0.0944911
$113$	15.8885	1.49467	0.747334	−	0.664448i	$-0.231333\pi$
$113$	15.8885	1.49467	0.747334	+	0.664448i	$0.231333\pi$
$114$	0	0
$115$	0	0
$116$	1.09017	0.101220
$117$	0.763932	0.0706255
$118$	9.38197	0.863680
$119$	3.23607	0.296650
$120$	−6.23607	−0.569273
$121$	−8.38197	−0.761997
$122$	−12.5623	−1.13734
$123$	−3.47214	−0.313072
$124$	−8.47214	−0.760820
$125$	−18.7082	−1.67331
$126$	−0.381966	−0.0340282
$127$	15.0344	1.33409	0.667045	−	0.745017i	$-0.267559\pi$
$127$	15.0344	1.33409	0.667045	+	0.745017i	$0.267559\pi$
$128$	−1.00000	−0.0883883
$129$	−2.23607	−0.196875
$130$	−7.70820	−0.676054
$131$	−20.4721	−1.78866	−0.894329	−	0.447409i	$-0.852347\pi$
$131$	−20.4721	−1.78866	−0.894329	+	0.447409i	$0.852347\pi$
$132$	−2.61803	−0.227871
$133$	0	0
$134$	2.76393	0.238767
$135$	−21.0902	−1.81515
$136$	3.23607	0.277491
$137$	13.2705	1.13378	0.566888	−	0.823795i	$-0.308147\pi$
$137$	13.2705	1.13378	0.566888	+	0.823795i	$0.308147\pi$
$138$	0	0
$139$	−2.94427	−0.249730	−0.124865	−	0.992174i	$-0.539850\pi$
$139$	−2.94427	−0.249730	−0.124865	+	0.992174i	$0.539850\pi$
$140$	3.85410	0.325731
$141$	−13.5623	−1.14215
$142$	−12.5623	−1.05421
$143$	−3.23607	−0.270614
$144$	−0.381966	−0.0318305
$145$	−4.20163	−0.348926
$146$	−12.9443	−1.07128
$147$	−1.61803	−0.133453
$148$	−9.09017	−0.747207
$149$	−5.23607	−0.428955	−0.214478	−	0.976729i	$-0.568805\pi$
$149$	−5.23607	−0.428955	−0.214478	+	0.976729i	$0.568805\pi$
$150$	15.9443	1.30184
$151$	−3.05573	−0.248672	−0.124336	−	0.992240i	$-0.539680\pi$
$151$	−3.05573	−0.248672	−0.124336	+	0.992240i	$0.539680\pi$
$152$	0	0
$153$	1.23607	0.0999302
$154$	1.61803	0.130385
$155$	32.6525	2.62271
$156$	3.23607	0.259093
$157$	−1.90983	−0.152421	−0.0762105	−	0.997092i	$-0.524282\pi$
$157$	−1.90983	−0.152421	−0.0762105	+	0.997092i	$0.524282\pi$
$158$	11.0902	0.882287
$159$	−21.5623	−1.71000
$160$	3.85410	0.304694
$161$	0	0
$162$	7.70820	0.605614
$163$	12.0902	0.946975	0.473488	−	0.880800i	$-0.342995\pi$
$163$	12.0902	0.946975	0.473488	+	0.880800i	$0.342995\pi$
$164$	2.14590	0.167566
$165$	10.0902	0.785519
$166$	−13.7082	−1.06396
$167$	−12.4721	−0.965123	−0.482561	−	0.875862i	$-0.660293\pi$
$167$	−12.4721	−0.965123	−0.482561	+	0.875862i	$0.660293\pi$
$168$	−1.61803	−0.124834
$169$	−9.00000	−0.692308
$170$	−12.4721	−0.956569
$171$	0	0
$172$	1.38197	0.105374
$173$	6.94427	0.527963	0.263982	−	0.964528i	$-0.414964\pi$
$173$	6.94427	0.527963	0.263982	+	0.964528i	$0.414964\pi$
$174$	1.76393	0.133723
$175$	−9.85410	−0.744900
$176$	1.61803	0.121964
$177$	15.1803	1.14102
$178$	11.3262	0.848937
$179$	−4.18034	−0.312453	−0.156227	−	0.987721i	$-0.549933\pi$
$179$	−4.18034	−0.312453	−0.156227	+	0.987721i	$0.549933\pi$
$180$	1.47214	0.109727
$181$	−25.8885	−1.92428	−0.962140	−	0.272555i	$-0.912131\pi$
$181$	−25.8885	−1.92428	−0.962140	+	0.272555i	$0.912131\pi$
$182$	−2.00000	−0.148250
$183$	−20.3262	−1.50256
$184$	0	0
$185$	35.0344	2.57578
$186$	−13.7082	−1.00513
$187$	−5.23607	−0.382899
$188$	8.38197	0.611318
$189$	−5.47214	−0.398039
$190$	0	0
$191$	17.4164	1.26021	0.630104	−	0.776511i	$-0.283012\pi$
$191$	17.4164	1.26021	0.630104	+	0.776511i	$0.283012\pi$
$192$	−1.61803	−0.116772
$193$	18.4721	1.32965	0.664827	−	0.746998i	$-0.268505\pi$
$193$	18.4721	1.32965	0.664827	+	0.746998i	$0.268505\pi$
$194$	5.61803	0.403351
$195$	−12.4721	−0.893148
$196$	1.00000	0.0714286
$197$	19.1246	1.36257	0.681286	−	0.732017i	$-0.261421\pi$
$197$	19.1246	1.36257	0.681286	+	0.732017i	$0.261421\pi$
$198$	0.618034	0.0439218
$199$	−0.618034	−0.0438113	−0.0219056	−	0.999760i	$-0.506973\pi$
$199$	−0.618034	−0.0438113	−0.0219056	+	0.999760i	$0.506973\pi$
$200$	−9.85410	−0.696790
$201$	4.47214	0.315440
$202$	−10.0000	−0.703598
$203$	−1.09017	−0.0765149
$204$	5.23607	0.366598
$205$	−8.27051	−0.577637
$206$	1.52786	0.106451
$207$	0	0
$208$	−2.00000	−0.138675
$209$	0	0
$210$	6.23607	0.430330
$211$	−23.1246	−1.59196	−0.795982	−	0.605320i	$-0.793045\pi$
$211$	−23.1246	−1.59196	−0.795982	+	0.605320i	$0.793045\pi$
$212$	13.3262	0.915250
$213$	−20.3262	−1.39273
$214$	0.291796	0.0199468
$215$	−5.32624	−0.363246
$216$	−5.47214	−0.372332
$217$	8.47214	0.575126
$218$	−8.38197	−0.567698
$219$	−20.9443	−1.41528
$220$	−6.23607	−0.420436
$221$	6.47214	0.435363
$222$	−14.7082	−0.987150
$223$	−8.65248	−0.579413	−0.289706	−	0.957116i	$-0.593558\pi$
$223$	−8.65248	−0.579413	−0.289706	+	0.957116i	$0.593558\pi$
$224$	1.00000	0.0668153
$225$	−3.76393	−0.250929
$226$	−15.8885	−1.05689
$227$	24.3607	1.61688	0.808438	−	0.588582i	$-0.200314\pi$
$227$	24.3607	1.61688	0.808438	+	0.588582i	$0.200314\pi$
$228$	0	0
$229$	−11.2705	−0.744776	−0.372388	−	0.928077i	$-0.621461\pi$
$229$	−11.2705	−0.744776	−0.372388	+	0.928077i	$0.621461\pi$
$230$	0	0
$231$	2.61803	0.172254
$232$	−1.09017	−0.0715732
$233$	−5.79837	−0.379864	−0.189932	−	0.981797i	$-0.560827\pi$
$233$	−5.79837	−0.379864	−0.189932	+	0.981797i	$0.560827\pi$
$234$	−0.763932	−0.0499398
$235$	−32.3050	−2.10734
$236$	−9.38197	−0.610714
$237$	17.9443	1.16561
$238$	−3.23607	−0.209763
$239$	22.1803	1.43473	0.717363	−	0.696699i	$-0.245349\pi$
$239$	22.1803	1.43473	0.717363	+	0.696699i	$0.245349\pi$
$240$	6.23607	0.402536
$241$	−18.1459	−1.16888	−0.584440	−	0.811437i	$-0.698686\pi$
$241$	−18.1459	−1.16888	−0.584440	+	0.811437i	$0.698686\pi$
$242$	8.38197	0.538813
$243$	−3.94427	−0.253025
$244$	12.5623	0.804219
$245$	−3.85410	−0.246230
$246$	3.47214	0.221375
$247$	0	0
$248$	8.47214	0.537981
$249$	−22.1803	−1.40562
$250$	18.7082	1.18321
$251$	−22.3607	−1.41139	−0.705697	−	0.708514i	$-0.749366\pi$
$251$	−22.3607	−1.41139	−0.705697	+	0.708514i	$0.749366\pi$
$252$	0.381966	0.0240616
$253$	0	0
$254$	−15.0344	−0.943345
$255$	−20.1803	−1.26374
$256$	1.00000	0.0625000
$257$	−30.2148	−1.88475	−0.942373	−	0.334564i	$-0.891411\pi$
$257$	−30.2148	−1.88475	−0.942373	+	0.334564i	$0.891411\pi$
$258$	2.23607	0.139212
$259$	9.09017	0.564836
$260$	7.70820	0.478043
$261$	−0.416408	−0.0257750
$262$	20.4721	1.26477
$263$	−8.65248	−0.533535	−0.266767	−	0.963761i	$-0.585956\pi$
$263$	−8.65248	−0.533535	−0.266767	+	0.963761i	$0.585956\pi$
$264$	2.61803	0.161129
$265$	−51.3607	−3.15506
$266$	0	0
$267$	18.3262	1.12155
$268$	−2.76393	−0.168834
$269$	−4.00000	−0.243884	−0.121942	−	0.992537i	$-0.538912\pi$
$269$	−4.00000	−0.243884	−0.121942	+	0.992537i	$0.538912\pi$
$270$	21.0902	1.28351
$271$	1.56231	0.0949033	0.0474517	−	0.998874i	$-0.484890\pi$
$271$	1.56231	0.0949033	0.0474517	+	0.998874i	$0.484890\pi$
$272$	−3.23607	−0.196215
$273$	−3.23607	−0.195856
$274$	−13.2705	−0.801701
$275$	15.9443	0.961476
$276$	0	0
$277$	4.47214	0.268705	0.134352	−	0.990934i	$-0.457105\pi$
$277$	4.47214	0.268705	0.134352	+	0.990934i	$0.457105\pi$
$278$	2.94427	0.176586
$279$	3.23607	0.193738
$280$	−3.85410	−0.230327
$281$	2.47214	0.147475	0.0737376	−	0.997278i	$-0.476507\pi$
$281$	2.47214	0.147475	0.0737376	+	0.997278i	$0.476507\pi$
$282$	13.5623	0.807624
$283$	−27.4164	−1.62974	−0.814868	−	0.579646i	$-0.803191\pi$
$283$	−27.4164	−1.62974	−0.814868	+	0.579646i	$0.803191\pi$
$284$	12.5623	0.745436
$285$	0	0
$286$	3.23607	0.191353
$287$	−2.14590	−0.126668
$288$	0.381966	0.0225076
$289$	−6.52786	−0.383992
$290$	4.20163	0.246728
$291$	9.09017	0.532875
$292$	12.9443	0.757506
$293$	−1.52786	−0.0892588	−0.0446294	−	0.999004i	$-0.514211\pi$
$293$	−1.52786	−0.0892588	−0.0446294	+	0.999004i	$0.514211\pi$
$294$	1.61803	0.0943657
$295$	36.1591	2.10526
$296$	9.09017	0.528355
$297$	8.85410	0.513767
$298$	5.23607	0.303317
$299$	0	0
$300$	−15.9443	−0.920543
$301$	−1.38197	−0.0796552
$302$	3.05573	0.175837
$303$	−16.1803	−0.929536
$304$	0	0
$305$	−48.4164	−2.77232
$306$	−1.23607	−0.0706613
$307$	14.9098	0.850949	0.425474	−	0.904970i	$-0.360107\pi$
$307$	14.9098	0.850949	0.425474	+	0.904970i	$0.360107\pi$
$308$	−1.61803	−0.0921960
$309$	2.47214	0.140635
$310$	−32.6525	−1.85454
$311$	5.90983	0.335116	0.167558	−	0.985862i	$-0.446412\pi$
$311$	5.90983	0.335116	0.167558	+	0.985862i	$0.446412\pi$
$312$	−3.23607	−0.183206
$313$	−8.47214	−0.478873	−0.239437	−	0.970912i	$-0.576963\pi$
$313$	−8.47214	−0.478873	−0.239437	+	0.970912i	$0.576963\pi$
$314$	1.90983	0.107778
$315$	−1.47214	−0.0829455
$316$	−11.0902	−0.623871
$317$	−26.2705	−1.47550	−0.737749	−	0.675075i	$-0.764111\pi$
$317$	−26.2705	−1.47550	−0.737749	+	0.675075i	$0.764111\pi$
$318$	21.5623	1.20915
$319$	1.76393	0.0987612
$320$	−3.85410	−0.215451
$321$	0.472136	0.0263521
$322$	0	0
$323$	0	0
$324$	−7.70820	−0.428234
$325$	−19.7082	−1.09321
$326$	−12.0902	−0.669613
$327$	−13.5623	−0.749997
$328$	−2.14590	−0.118487
$329$	−8.38197	−0.462113
$330$	−10.0902	−0.555446
$331$	13.2361	0.727520	0.363760	−	0.931493i	$-0.381493\pi$
$331$	13.2361	0.727520	0.363760	+	0.931493i	$0.381493\pi$
$332$	13.7082	0.752335
$333$	3.47214	0.190272
$334$	12.4721	0.682445
$335$	10.6525	0.582007
$336$	1.61803	0.0882710
$337$	−20.7639	−1.13108	−0.565542	−	0.824720i	$-0.691333\pi$
$337$	−20.7639	−1.13108	−0.565542	+	0.824720i	$0.691333\pi$
$338$	9.00000	0.489535
$339$	−25.7082	−1.39628
$340$	12.4721	0.676397
$341$	−13.7082	−0.742341
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	−1.38197	−0.0745106
$345$	0	0
$346$	−6.94427	−0.373326
$347$	13.8885	0.745576	0.372788	−	0.927917i	$-0.378402\pi$
$347$	13.8885	0.745576	0.372788	+	0.927917i	$0.378402\pi$
$348$	−1.76393	−0.0945567
$349$	35.8885	1.92107	0.960535	−	0.278160i	$-0.0897244\pi$
$349$	35.8885	1.92107	0.960535	+	0.278160i	$0.0897244\pi$
$350$	9.85410	0.526724
$351$	−10.9443	−0.584162
$352$	−1.61803	−0.0862415
$353$	−18.3607	−0.977240	−0.488620	−	0.872497i	$-0.662500\pi$
$353$	−18.3607	−0.977240	−0.488620	+	0.872497i	$0.662500\pi$
$354$	−15.1803	−0.806826
$355$	−48.4164	−2.56968
$356$	−11.3262	−0.600289
$357$	−5.23607	−0.277122
$358$	4.18034	0.220938
$359$	−5.34752	−0.282232	−0.141116	−	0.989993i	$-0.545069\pi$
$359$	−5.34752	−0.282232	−0.141116	+	0.989993i	$0.545069\pi$
$360$	−1.47214	−0.0775884
$361$	0	0
$362$	25.8885	1.36067
$363$	13.5623	0.711836
$364$	2.00000	0.104828
$365$	−49.8885	−2.61129
$366$	20.3262	1.06247
$367$	−19.0344	−0.993590	−0.496795	−	0.867868i	$-0.665490\pi$
$367$	−19.0344	−0.993590	−0.496795	+	0.867868i	$0.665490\pi$
$368$	0	0
$369$	−0.819660	−0.0426698
$370$	−35.0344	−1.82135
$371$	−13.3262	−0.691864
$372$	13.7082	0.710737
$373$	5.67376	0.293776	0.146888	−	0.989153i	$-0.453074\pi$
$373$	5.67376	0.293776	0.146888	+	0.989153i	$0.453074\pi$
$374$	5.23607	0.270751
$375$	30.2705	1.56316
$376$	−8.38197	−0.432267
$377$	−2.18034	−0.112293
$378$	5.47214	0.281456
$379$	17.1246	0.879632	0.439816	−	0.898088i	$-0.355044\pi$
$379$	17.1246	0.879632	0.439816	+	0.898088i	$0.355044\pi$
$380$	0	0
$381$	−24.3262	−1.24627
$382$	−17.4164	−0.891101
$383$	1.34752	0.0688553	0.0344276	−	0.999407i	$-0.489039\pi$
$383$	1.34752	0.0688553	0.0344276	+	0.999407i	$0.489039\pi$
$384$	1.61803	0.0825700
$385$	6.23607	0.317819
$386$	−18.4721	−0.940207
$387$	−0.527864	−0.0268328
$388$	−5.61803	−0.285212
$389$	24.9443	1.26472	0.632362	−	0.774673i	$-0.282085\pi$
$389$	24.9443	1.26472	0.632362	+	0.774673i	$0.282085\pi$
$390$	12.4721	0.631551
$391$	0	0
$392$	−1.00000	−0.0505076
$393$	33.1246	1.67092
$394$	−19.1246	−0.963484
$395$	42.7426	2.15062
$396$	−0.618034	−0.0310574
$397$	24.5623	1.23275	0.616373	−	0.787454i	$-0.288601\pi$
$397$	24.5623	1.23275	0.616373	+	0.787454i	$0.288601\pi$
$398$	0.618034	0.0309792
$399$	0	0
$400$	9.85410	0.492705
$401$	−2.58359	−0.129018	−0.0645092	−	0.997917i	$-0.520548\pi$
$401$	−2.58359	−0.129018	−0.0645092	+	0.997917i	$0.520548\pi$
$402$	−4.47214	−0.223050
$403$	16.9443	0.844054
$404$	10.0000	0.497519
$405$	29.7082	1.47621
$406$	1.09017	0.0541042
$407$	−14.7082	−0.729059
$408$	−5.23607	−0.259224
$409$	8.27051	0.408950	0.204475	−	0.978872i	$-0.434451\pi$
$409$	8.27051	0.408950	0.204475	+	0.978872i	$0.434451\pi$
$410$	8.27051	0.408451
$411$	−21.4721	−1.05914
$412$	−1.52786	−0.0752725
$413$	9.38197	0.461656
$414$	0	0
$415$	−52.8328	−2.59346
$416$	2.00000	0.0980581
$417$	4.76393	0.233291
$418$	0	0
$419$	−3.34752	−0.163537	−0.0817686	−	0.996651i	$-0.526057\pi$
$419$	−3.34752	−0.163537	−0.0817686	+	0.996651i	$0.526057\pi$
$420$	−6.23607	−0.304289
$421$	16.4721	0.802803	0.401401	−	0.915902i	$-0.368523\pi$
$421$	16.4721	0.802803	0.401401	+	0.915902i	$0.368523\pi$
$422$	23.1246	1.12569
$423$	−3.20163	−0.155668
$424$	−13.3262	−0.647179
$425$	−31.8885	−1.54682
$426$	20.3262	0.984809
$427$	−12.5623	−0.607933
$428$	−0.291796	−0.0141045
$429$	5.23607	0.252800
$430$	5.32624	0.256854
$431$	26.8541	1.29352	0.646758	−	0.762695i	$-0.276124\pi$
$431$	26.8541	1.29352	0.646758	+	0.762695i	$0.276124\pi$
$432$	5.47214	0.263278
$433$	5.85410	0.281330	0.140665	−	0.990057i	$-0.455076\pi$
$433$	5.85410	0.281330	0.140665	+	0.990057i	$0.455076\pi$
$434$	−8.47214	−0.406676
$435$	6.79837	0.325957
$436$	8.38197	0.401423
$437$	0	0
$438$	20.9443	1.00076
$439$	31.7082	1.51335	0.756675	−	0.653791i	$-0.226823\pi$
$439$	31.7082	1.51335	0.756675	+	0.653791i	$0.226823\pi$
$440$	6.23607	0.297293
$441$	−0.381966	−0.0181889
$442$	−6.47214	−0.307848
$443$	−22.3262	−1.06075	−0.530376	−	0.847763i	$-0.677949\pi$
$443$	−22.3262	−1.06075	−0.530376	+	0.847763i	$0.677949\pi$
$444$	14.7082	0.698020
$445$	43.6525	2.06933
$446$	8.65248	0.409707
$447$	8.47214	0.400718
$448$	−1.00000	−0.0472456
$449$	−36.6525	−1.72974	−0.864869	−	0.501998i	$-0.832598\pi$
$449$	−36.6525	−1.72974	−0.864869	+	0.501998i	$0.832598\pi$
$450$	3.76393	0.177433
$451$	3.47214	0.163496
$452$	15.8885	0.747334
$453$	4.94427	0.232302
$454$	−24.3607	−1.14330
$455$	−7.70820	−0.361366
$456$	0	0
$457$	4.09017	0.191330	0.0956650	−	0.995414i	$-0.469502\pi$
$457$	4.09017	0.191330	0.0956650	+	0.995414i	$0.469502\pi$
$458$	11.2705	0.526636
$459$	−17.7082	−0.826548
$460$	0	0
$461$	−12.3820	−0.576686	−0.288343	−	0.957527i	$-0.593104\pi$
$461$	−12.3820	−0.576686	−0.288343	+	0.957527i	$0.593104\pi$
$462$	−2.61803	−0.121802
$463$	−40.3607	−1.87572	−0.937860	−	0.347014i	$-0.887196\pi$
$463$	−40.3607	−1.87572	−0.937860	+	0.347014i	$0.887196\pi$
$464$	1.09017	0.0506099
$465$	−52.8328	−2.45006
$466$	5.79837	0.268604
$467$	28.0000	1.29569	0.647843	−	0.761774i	$-0.275671\pi$
$467$	28.0000	1.29569	0.647843	+	0.761774i	$0.275671\pi$
$468$	0.763932	0.0353128
$469$	2.76393	0.127627
$470$	32.3050	1.49012
$471$	3.09017	0.142388
$472$	9.38197	0.431840
$473$	2.23607	0.102815
$474$	−17.9443	−0.824208
$475$	0	0
$476$	3.23607	0.148325
$477$	−5.09017	−0.233063
$478$	−22.1803	−1.01451
$479$	28.6180	1.30759	0.653796	−	0.756671i	$-0.273176\pi$
$479$	28.6180	1.30759	0.653796	+	0.756671i	$0.273176\pi$
$480$	−6.23607	−0.284636
$481$	18.1803	0.828952
$482$	18.1459	0.826523
$483$	0	0
$484$	−8.38197	−0.380998
$485$	21.6525	0.983188
$486$	3.94427	0.178916
$487$	−18.5066	−0.838613	−0.419307	−	0.907845i	$-0.637727\pi$
$487$	−18.5066	−0.838613	−0.419307	+	0.907845i	$0.637727\pi$
$488$	−12.5623	−0.568669
$489$	−19.5623	−0.884638
$490$	3.85410	0.174111
$491$	4.58359	0.206855	0.103427	−	0.994637i	$-0.467019\pi$
$491$	4.58359	0.206855	0.103427	+	0.994637i	$0.467019\pi$
$492$	−3.47214	−0.156536
$493$	−3.52786	−0.158887
$494$	0	0
$495$	2.38197	0.107061
$496$	−8.47214	−0.380410
$497$	−12.5623	−0.563496
$498$	22.1803	0.993925
$499$	−33.3820	−1.49438	−0.747191	−	0.664609i	$-0.768598\pi$
$499$	−33.3820	−1.49438	−0.747191	+	0.664609i	$0.768598\pi$
$500$	−18.7082	−0.836656
$501$	20.1803	0.901591
$502$	22.3607	0.998006
$503$	−13.2705	−0.591703	−0.295851	−	0.955234i	$-0.595603\pi$
$503$	−13.2705	−0.591703	−0.295851	+	0.955234i	$0.595603\pi$
$504$	−0.381966	−0.0170141
$505$	−38.5410	−1.71505
$506$	0	0
$507$	14.5623	0.646735
$508$	15.0344	0.667045
$509$	21.5967	0.957259	0.478630	−	0.878017i	$-0.341134\pi$
$509$	21.5967	0.957259	0.478630	+	0.878017i	$0.341134\pi$
$510$	20.1803	0.893600
$511$	−12.9443	−0.572621
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	30.2148	1.33272
$515$	5.88854	0.259480
$516$	−2.23607	−0.0984374
$517$	13.5623	0.596470
$518$	−9.09017	−0.399399
$519$	−11.2361	−0.493209
$520$	−7.70820	−0.338027
$521$	16.4721	0.721657	0.360829	−	0.932632i	$-0.382494\pi$
$521$	16.4721	0.721657	0.360829	+	0.932632i	$0.382494\pi$
$522$	0.416408	0.0182257
$523$	16.0000	0.699631	0.349816	−	0.936819i	$-0.386244\pi$
$523$	16.0000	0.699631	0.349816	+	0.936819i	$0.386244\pi$
$524$	−20.4721	−0.894329
$525$	15.9443	0.695865
$526$	8.65248	0.377266
$527$	27.4164	1.19428
$528$	−2.61803	−0.113935
$529$	−23.0000	−1.00000
$530$	51.3607	2.23097
$531$	3.58359	0.155515
$532$	0	0
$533$	−4.29180	−0.185898
$534$	−18.3262	−0.793054
$535$	1.12461	0.0486212
$536$	2.76393	0.119384
$537$	6.76393	0.291885
$538$	4.00000	0.172452
$539$	1.61803	0.0696937
$540$	−21.0902	−0.907576
$541$	16.0689	0.690855	0.345428	−	0.938445i	$-0.387734\pi$
$541$	16.0689	0.690855	0.345428	+	0.938445i	$0.387734\pi$
$542$	−1.56231	−0.0671068
$543$	41.8885	1.79761
$544$	3.23607	0.138745
$545$	−32.3050	−1.38379
$546$	3.23607	0.138491
$547$	2.18034	0.0932246	0.0466123	−	0.998913i	$-0.485157\pi$
$547$	2.18034	0.0932246	0.0466123	+	0.998913i	$0.485157\pi$
$548$	13.2705	0.566888
$549$	−4.79837	−0.204790
$550$	−15.9443	−0.679866
$551$	0	0
$552$	0	0
$553$	11.0902	0.471602
$554$	−4.47214	−0.190003
$555$	−56.6869	−2.40623
$556$	−2.94427	−0.124865
$557$	−28.6525	−1.21404	−0.607022	−	0.794685i	$-0.707636\pi$
$557$	−28.6525	−1.21404	−0.607022	+	0.794685i	$0.707636\pi$
$558$	−3.23607	−0.136994
$559$	−2.76393	−0.116902
$560$	3.85410	0.162866
$561$	8.47214	0.357694
$562$	−2.47214	−0.104281
$563$	−0.270510	−0.0114006	−0.00570032	−	0.999984i	$-0.501814\pi$
$563$	−0.270510	−0.0114006	−0.00570032	+	0.999984i	$0.501814\pi$
$564$	−13.5623	−0.571076
$565$	−61.2361	−2.57622
$566$	27.4164	1.15240
$567$	7.70820	0.323714
$568$	−12.5623	−0.527103
$569$	−26.6525	−1.11733	−0.558665	−	0.829393i	$-0.688686\pi$
$569$	−26.6525	−1.11733	−0.558665	+	0.829393i	$0.688686\pi$
$570$	0	0
$571$	−23.5066	−0.983720	−0.491860	−	0.870674i	$-0.663683\pi$
$571$	−23.5066	−0.983720	−0.491860	+	0.870674i	$0.663683\pi$
$572$	−3.23607	−0.135307
$573$	−28.1803	−1.17725
$574$	2.14590	0.0895681
$575$	0	0
$576$	−0.381966	−0.0159153
$577$	−30.6525	−1.27608	−0.638040	−	0.770004i	$-0.720254\pi$
$577$	−30.6525	−1.27608	−0.638040	+	0.770004i	$0.720254\pi$
$578$	6.52786	0.271523
$579$	−29.8885	−1.24213
$580$	−4.20163	−0.174463
$581$	−13.7082	−0.568712
$582$	−9.09017	−0.376800
$583$	21.5623	0.893019
$584$	−12.9443	−0.535638
$585$	−2.94427	−0.121731
$586$	1.52786	0.0631155
$587$	7.52786	0.310708	0.155354	−	0.987859i	$-0.450348\pi$
$587$	7.52786	0.310708	0.155354	+	0.987859i	$0.450348\pi$
$588$	−1.61803	−0.0667266
$589$	0	0
$590$	−36.1591	−1.48864
$591$	−30.9443	−1.27288
$592$	−9.09017	−0.373604
$593$	−18.5836	−0.763137	−0.381568	−	0.924341i	$-0.624616\pi$
$593$	−18.5836	−0.763137	−0.381568	+	0.924341i	$0.624616\pi$
$594$	−8.85410	−0.363288
$595$	−12.4721	−0.511308
$596$	−5.23607	−0.214478
$597$	1.00000	0.0409273
$598$	0	0
$599$	−14.8541	−0.606922	−0.303461	−	0.952844i	$-0.598142\pi$
$599$	−14.8541	−0.606922	−0.303461	+	0.952844i	$0.598142\pi$
$600$	15.9443	0.650922
$601$	−37.4164	−1.52625	−0.763124	−	0.646253i	$-0.776335\pi$
$601$	−37.4164	−1.52625	−0.763124	+	0.646253i	$0.776335\pi$
$602$	1.38197	0.0563247
$603$	1.05573	0.0429926
$604$	−3.05573	−0.124336
$605$	32.3050	1.31338
$606$	16.1803	0.657281
$607$	−16.9443	−0.687747	−0.343873	−	0.939016i	$-0.611739\pi$
$607$	−16.9443	−0.687747	−0.343873	+	0.939016i	$0.611739\pi$
$608$	0	0
$609$	1.76393	0.0714781
$610$	48.4164	1.96032
$611$	−16.7639	−0.678196
$612$	1.23607	0.0499651
$613$	17.5967	0.710726	0.355363	−	0.934728i	$-0.384357\pi$
$613$	17.5967	0.710726	0.355363	+	0.934728i	$0.384357\pi$
$614$	−14.9098	−0.601712
$615$	13.3820	0.539613
$616$	1.61803	0.0651924
$617$	−32.6869	−1.31593	−0.657963	−	0.753050i	$-0.728582\pi$
$617$	−32.6869	−1.31593	−0.657963	+	0.753050i	$0.728582\pi$
$618$	−2.47214	−0.0994439
$619$	−4.00000	−0.160774	−0.0803868	−	0.996764i	$-0.525616\pi$
$619$	−4.00000	−0.160774	−0.0803868	+	0.996764i	$0.525616\pi$
$620$	32.6525	1.31135
$621$	0	0
$622$	−5.90983	−0.236963
$623$	11.3262	0.453776
$624$	3.23607	0.129546
$625$	22.8328	0.913313
$626$	8.47214	0.338615
$627$	0	0
$628$	−1.90983	−0.0762105
$629$	29.4164	1.17291
$630$	1.47214	0.0586513
$631$	−25.5279	−1.01625	−0.508124	−	0.861284i	$-0.669661\pi$
$631$	−25.5279	−1.01625	−0.508124	+	0.861284i	$0.669661\pi$
$632$	11.0902	0.441143
$633$	37.4164	1.48717
$634$	26.2705	1.04334
$635$	−57.9443	−2.29945
$636$	−21.5623	−0.855001
$637$	−2.00000	−0.0792429
$638$	−1.76393	−0.0698347
$639$	−4.79837	−0.189821
$640$	3.85410	0.152347
$641$	8.58359	0.339032	0.169516	−	0.985527i	$-0.445780\pi$
$641$	8.58359	0.339032	0.169516	+	0.985527i	$0.445780\pi$
$642$	−0.472136	−0.0186337
$643$	−4.83282	−0.190588	−0.0952938	−	0.995449i	$-0.530379\pi$
$643$	−4.83282	−0.190588	−0.0952938	+	0.995449i	$0.530379\pi$
$644$	0	0
$645$	8.61803	0.339335
$646$	0	0
$647$	−5.56231	−0.218677	−0.109338	−	0.994005i	$-0.534873\pi$
$647$	−5.56231	−0.218677	−0.109338	+	0.994005i	$0.534873\pi$
$648$	7.70820	0.302807
$649$	−15.1803	−0.595880
$650$	19.7082	0.773019
$651$	−13.7082	−0.537267
$652$	12.0902	0.473488
$653$	12.3607	0.483711	0.241855	−	0.970312i	$-0.422244\pi$
$653$	12.3607	0.483711	0.241855	+	0.970312i	$0.422244\pi$
$654$	13.5623	0.530328
$655$	78.9017	3.08294
$656$	2.14590	0.0837832
$657$	−4.94427	−0.192894
$658$	8.38197	0.326763
$659$	−17.2361	−0.671422	−0.335711	−	0.941965i	$-0.608977\pi$
$659$	−17.2361	−0.671422	−0.335711	+	0.941965i	$0.608977\pi$
$660$	10.0902	0.392759
$661$	−14.4721	−0.562901	−0.281450	−	0.959576i	$-0.590815\pi$
$661$	−14.4721	−0.562901	−0.281450	+	0.959576i	$0.590815\pi$
$662$	−13.2361	−0.514434
$663$	−10.4721	−0.406704
$664$	−13.7082	−0.531981
$665$	0	0
$666$	−3.47214	−0.134543
$667$	0	0
$668$	−12.4721	−0.482561
$669$	14.0000	0.541271
$670$	−10.6525	−0.411541
$671$	20.3262	0.784686
$672$	−1.61803	−0.0624170
$673$	−12.0000	−0.462566	−0.231283	−	0.972887i	$-0.574292\pi$
$673$	−12.0000	−0.462566	−0.231283	+	0.972887i	$0.574292\pi$
$674$	20.7639	0.799797
$675$	53.9230	2.07550
$676$	−9.00000	−0.346154
$677$	−35.2361	−1.35423	−0.677116	−	0.735876i	$-0.736771\pi$
$677$	−35.2361	−1.35423	−0.677116	+	0.735876i	$0.736771\pi$
$678$	25.7082	0.987318
$679$	5.61803	0.215600
$680$	−12.4721	−0.478285
$681$	−39.4164	−1.51044
$682$	13.7082	0.524914
$683$	46.6525	1.78511	0.892554	−	0.450941i	$-0.148912\pi$
$683$	46.6525	1.78511	0.892554	+	0.450941i	$0.148912\pi$
$684$	0	0
$685$	−51.1459	−1.95418
$686$	1.00000	0.0381802
$687$	18.2361	0.695749
$688$	1.38197	0.0526870
$689$	−26.6525	−1.01538
$690$	0	0
$691$	8.18034	0.311195	0.155597	−	0.987821i	$-0.450270\pi$
$691$	8.18034	0.311195	0.155597	+	0.987821i	$0.450270\pi$
$692$	6.94427	0.263982
$693$	0.618034	0.0234772
$694$	−13.8885	−0.527202
$695$	11.3475	0.430436
$696$	1.76393	0.0668617
$697$	−6.94427	−0.263033
$698$	−35.8885	−1.35840
$699$	9.38197	0.354859
$700$	−9.85410	−0.372450
$701$	−38.5410	−1.45567	−0.727837	−	0.685750i	$-0.759474\pi$
$701$	−38.5410	−1.45567	−0.727837	+	0.685750i	$0.759474\pi$
$702$	10.9443	0.413065
$703$	0	0
$704$	1.61803	0.0609820
$705$	52.2705	1.96862
$706$	18.3607	0.691013
$707$	−10.0000	−0.376089
$708$	15.1803	0.570512
$709$	8.18034	0.307219	0.153610	−	0.988132i	$-0.450910\pi$
$709$	8.18034	0.307219	0.153610	+	0.988132i	$0.450910\pi$
$710$	48.4164	1.81704
$711$	4.23607	0.158865
$712$	11.3262	0.424469
$713$	0	0
$714$	5.23607	0.195955
$715$	12.4721	0.466431
$716$	−4.18034	−0.156227
$717$	−35.8885	−1.34028
$718$	5.34752	0.199568
$719$	35.4164	1.32081	0.660405	−	0.750910i	$-0.270385\pi$
$719$	35.4164	1.32081	0.660405	+	0.750910i	$0.270385\pi$
$720$	1.47214	0.0548633
$721$	1.52786	0.0569006
$722$	0	0
$723$	29.3607	1.09194
$724$	−25.8885	−0.962140
$725$	10.7426	0.398972
$726$	−13.5623	−0.503344
$727$	−35.9787	−1.33438	−0.667188	−	0.744889i	$-0.732502\pi$
$727$	−35.9787	−1.33438	−0.667188	+	0.744889i	$0.732502\pi$
$728$	−2.00000	−0.0741249
$729$	29.5066	1.09284
$730$	49.8885	1.84646
$731$	−4.47214	−0.165408
$732$	−20.3262	−0.751279
$733$	27.9230	1.03136	0.515680	−	0.856782i	$-0.327540\pi$
$733$	27.9230	1.03136	0.515680	+	0.856782i	$0.327540\pi$
$734$	19.0344	0.702574
$735$	6.23607	0.230021
$736$	0	0
$737$	−4.47214	−0.164733
$738$	0.819660	0.0301721
$739$	32.7984	1.20651	0.603254	−	0.797549i	$-0.293871\pi$
$739$	32.7984	1.20651	0.603254	+	0.797549i	$0.293871\pi$
$740$	35.0344	1.28789
$741$	0	0
$742$	13.3262	0.489222
$743$	32.6180	1.19664	0.598320	−	0.801257i	$-0.295835\pi$
$743$	32.6180	1.19664	0.598320	+	0.801257i	$0.295835\pi$
$744$	−13.7082	−0.502567
$745$	20.1803	0.739350
$746$	−5.67376	−0.207731
$747$	−5.23607	−0.191578
$748$	−5.23607	−0.191450
$749$	0.291796	0.0106620
$750$	−30.2705	−1.10532
$751$	−35.7426	−1.30427	−0.652134	−	0.758104i	$-0.726126\pi$
$751$	−35.7426	−1.30427	−0.652134	+	0.758104i	$0.726126\pi$
$752$	8.38197	0.305659
$753$	36.1803	1.31848
$754$	2.18034	0.0794033
$755$	11.7771	0.428612
$756$	−5.47214	−0.199020
$757$	−4.58359	−0.166593	−0.0832967	−	0.996525i	$-0.526545\pi$
$757$	−4.58359	−0.166593	−0.0832967	+	0.996525i	$0.526545\pi$
$758$	−17.1246	−0.621994
$759$	0	0
$760$	0	0
$761$	−30.3607	−1.10057	−0.550287	−	0.834976i	$-0.685482\pi$
$761$	−30.3607	−1.10057	−0.550287	+	0.834976i	$0.685482\pi$
$762$	24.3262	0.881247
$763$	−8.38197	−0.303448
$764$	17.4164	0.630104
$765$	−4.76393	−0.172240
$766$	−1.34752	−0.0486880
$767$	18.7639	0.677526
$768$	−1.61803	−0.0583858
$769$	8.29180	0.299010	0.149505	−	0.988761i	$-0.452232\pi$
$769$	8.29180	0.299010	0.149505	+	0.988761i	$0.452232\pi$
$770$	−6.23607	−0.224732
$771$	48.8885	1.76068
$772$	18.4721	0.664827
$773$	32.7639	1.17844	0.589218	−	0.807974i	$-0.299436\pi$
$773$	32.7639	1.17844	0.589218	+	0.807974i	$0.299436\pi$
$774$	0.527864	0.0189737
$775$	−83.4853	−2.99888
$776$	5.61803	0.201676
$777$	−14.7082	−0.527654
$778$	−24.9443	−0.894295
$779$	0	0
$780$	−12.4721	−0.446574
$781$	20.3262	0.727330
$782$	0	0
$783$	5.96556	0.213192
$784$	1.00000	0.0357143
$785$	7.36068	0.262714
$786$	−33.1246	−1.18152
$787$	25.3820	0.904769	0.452385	−	0.891823i	$-0.350573\pi$
$787$	25.3820	0.904769	0.452385	+	0.891823i	$0.350573\pi$
$788$	19.1246	0.681286
$789$	14.0000	0.498413
$790$	−42.7426	−1.52072
$791$	−15.8885	−0.564932
$792$	0.618034	0.0219609
$793$	−25.1246	−0.892201
$794$	−24.5623	−0.871684
$795$	83.1033	2.94737
$796$	−0.618034	−0.0219056
$797$	50.9443	1.80454	0.902269	−	0.431173i	$-0.141900\pi$
$797$	50.9443	1.80454	0.902269	+	0.431173i	$0.141900\pi$
$798$	0	0
$799$	−27.1246	−0.959600
$800$	−9.85410	−0.348395
$801$	4.32624	0.152860
$802$	2.58359	0.0912298
$803$	20.9443	0.739107
$804$	4.47214	0.157720
$805$	0	0
$806$	−16.9443	−0.596837
$807$	6.47214	0.227830
$808$	−10.0000	−0.351799
$809$	−1.20163	−0.0422469	−0.0211235	−	0.999777i	$-0.506724\pi$
$809$	−1.20163	−0.0422469	−0.0211235	+	0.999777i	$0.506724\pi$
$810$	−29.7082	−1.04384
$811$	9.03444	0.317242	0.158621	−	0.987340i	$-0.449295\pi$
$811$	9.03444	0.317242	0.158621	+	0.987340i	$0.449295\pi$
$812$	−1.09017	−0.0382575
$813$	−2.52786	−0.0886561
$814$	14.7082	0.515522
$815$	−46.5967	−1.63221
$816$	5.23607	0.183299
$817$	0	0
$818$	−8.27051	−0.289172
$819$	−0.763932	−0.0266939
$820$	−8.27051	−0.288819
$821$	−9.70820	−0.338819	−0.169409	−	0.985546i	$-0.554186\pi$
$821$	−9.70820	−0.338819	−0.169409	+	0.985546i	$0.554186\pi$
$822$	21.4721	0.748927
$823$	20.7639	0.723785	0.361893	−	0.932220i	$-0.382131\pi$
$823$	20.7639	0.723785	0.361893	+	0.932220i	$0.382131\pi$
$824$	1.52786	0.0532257
$825$	−25.7984	−0.898184
$826$	−9.38197	−0.326440
$827$	9.34752	0.325045	0.162523	−	0.986705i	$-0.448037\pi$
$827$	9.34752	0.325045	0.162523	+	0.986705i	$0.448037\pi$
$828$	0	0
$829$	21.2361	0.737559	0.368780	−	0.929517i	$-0.379776\pi$
$829$	21.2361	0.737559	0.368780	+	0.929517i	$0.379776\pi$
$830$	52.8328	1.83385
$831$	−7.23607	−0.251016
$832$	−2.00000	−0.0693375
$833$	−3.23607	−0.112123
$834$	−4.76393	−0.164961
$835$	48.0689	1.66349
$836$	0	0
$837$	−46.3607	−1.60246
$838$	3.34752	0.115638
$839$	1.41641	0.0488998	0.0244499	−	0.999701i	$-0.492217\pi$
$839$	1.41641	0.0488998	0.0244499	+	0.999701i	$0.492217\pi$
$840$	6.23607	0.215165
$841$	−27.8115	−0.959018
$842$	−16.4721	−0.567667
$843$	−4.00000	−0.137767
$844$	−23.1246	−0.795982
$845$	34.6869	1.19327
$846$	3.20163	0.110074
$847$	8.38197	0.288008
$848$	13.3262	0.457625
$849$	44.3607	1.52245
$850$	31.8885	1.09377
$851$	0	0
$852$	−20.3262	−0.696365
$853$	29.7426	1.01837	0.509184	−	0.860657i	$-0.329947\pi$
$853$	29.7426	1.01837	0.509184	+	0.860657i	$0.329947\pi$
$854$	12.5623	0.429873
$855$	0	0
$856$	0.291796	0.00997338
$857$	−17.4164	−0.594933	−0.297467	−	0.954732i	$-0.596142\pi$
$857$	−17.4164	−0.594933	−0.297467	+	0.954732i	$0.596142\pi$
$858$	−5.23607	−0.178756
$859$	31.8885	1.08802	0.544012	−	0.839078i	$-0.316905\pi$
$859$	31.8885	1.08802	0.544012	+	0.839078i	$0.316905\pi$
$860$	−5.32624	−0.181623
$861$	3.47214	0.118330
$862$	−26.8541	−0.914654
$863$	11.6738	0.397379	0.198690	−	0.980062i	$-0.436331\pi$
$863$	11.6738	0.397379	0.198690	+	0.980062i	$0.436331\pi$
$864$	−5.47214	−0.186166
$865$	−26.7639	−0.910001
$866$	−5.85410	−0.198930
$867$	10.5623	0.358715
$868$	8.47214	0.287563
$869$	−17.9443	−0.608718
$870$	−6.79837	−0.230486
$871$	5.52786	0.187305
$872$	−8.38197	−0.283849
$873$	2.14590	0.0726276
$874$	0	0
$875$	18.7082	0.632453
$876$	−20.9443	−0.707641
$877$	−30.0902	−1.01607	−0.508036	−	0.861336i	$-0.669628\pi$
$877$	−30.0902	−1.01607	−0.508036	+	0.861336i	$0.669628\pi$
$878$	−31.7082	−1.07010
$879$	2.47214	0.0833831
$880$	−6.23607	−0.210218
$881$	12.2918	0.414121	0.207061	−	0.978328i	$-0.433610\pi$
$881$	12.2918	0.414121	0.207061	+	0.978328i	$0.433610\pi$
$882$	0.381966	0.0128615
$883$	45.3951	1.52767	0.763834	−	0.645413i	$-0.223315\pi$
$883$	45.3951	1.52767	0.763834	+	0.645413i	$0.223315\pi$
$884$	6.47214	0.217681
$885$	−58.5066	−1.96668
$886$	22.3262	0.750065
$887$	−3.63932	−0.122196	−0.0610982	−	0.998132i	$-0.519460\pi$
$887$	−3.63932	−0.122196	−0.0610982	+	0.998132i	$0.519460\pi$
$888$	−14.7082	−0.493575
$889$	−15.0344	−0.504239
$890$	−43.6525	−1.46323
$891$	−12.4721	−0.417832
$892$	−8.65248	−0.289706
$893$	0	0
$894$	−8.47214	−0.283351
$895$	16.1115	0.538547
$896$	1.00000	0.0334077
$897$	0	0
$898$	36.6525	1.22311
$899$	−9.23607	−0.308040
$900$	−3.76393	−0.125464
$901$	−43.1246	−1.43669
$902$	−3.47214	−0.115609
$903$	2.23607	0.0744117
$904$	−15.8885	−0.528445
$905$	99.7771	3.31670
$906$	−4.94427	−0.164262
$907$	−8.94427	−0.296990	−0.148495	−	0.988913i	$-0.547443\pi$
$907$	−8.94427	−0.296990	−0.148495	+	0.988913i	$0.547443\pi$
$908$	24.3607	0.808438
$909$	−3.81966	−0.126690
$910$	7.70820	0.255524
$911$	28.6180	0.948158	0.474079	−	0.880482i	$-0.342781\pi$
$911$	28.6180	0.948158	0.474079	+	0.880482i	$0.342781\pi$
$912$	0	0
$913$	22.1803	0.734062
$914$	−4.09017	−0.135291
$915$	78.3394	2.58982
$916$	−11.2705	−0.372388
$917$	20.4721	0.676049
$918$	17.7082	0.584458
$919$	41.9574	1.38405	0.692024	−	0.721875i	$-0.256719\pi$
$919$	41.9574	1.38405	0.692024	+	0.721875i	$0.256719\pi$
$920$	0	0
$921$	−24.1246	−0.794933
$922$	12.3820	0.407778
$923$	−25.1246	−0.826987
$924$	2.61803	0.0861270
$925$	−89.5755	−2.94522
$926$	40.3607	1.32633
$927$	0.583592	0.0191677
$928$	−1.09017	−0.0357866
$929$	−1.23607	−0.0405541	−0.0202770	−	0.999794i	$-0.506455\pi$
$929$	−1.23607	−0.0405541	−0.0202770	+	0.999794i	$0.506455\pi$
$930$	52.8328	1.73246
$931$	0	0
$932$	−5.79837	−0.189932
$933$	−9.56231	−0.313056
$934$	−28.0000	−0.916188
$935$	20.1803	0.659968
$936$	−0.763932	−0.0249699
$937$	44.1803	1.44331	0.721654	−	0.692254i	$-0.243382\pi$
$937$	44.1803	1.44331	0.721654	+	0.692254i	$0.243382\pi$
$938$	−2.76393	−0.0902456
$939$	13.7082	0.447350
$940$	−32.3050	−1.05367
$941$	−31.0132	−1.01100	−0.505500	−	0.862827i	$-0.668692\pi$
$941$	−31.0132	−1.01100	−0.505500	+	0.862827i	$0.668692\pi$
$942$	−3.09017	−0.100683
$943$	0	0
$944$	−9.38197	−0.305357
$945$	21.0902	0.686063
$946$	−2.23607	−0.0727008
$947$	19.3262	0.628018	0.314009	−	0.949420i	$-0.398328\pi$
$947$	19.3262	0.628018	0.314009	+	0.949420i	$0.398328\pi$
$948$	17.9443	0.582803
$949$	−25.8885	−0.840378
$950$	0	0
$951$	42.5066	1.37837
$952$	−3.23607	−0.104882
$953$	1.70820	0.0553342	0.0276671	−	0.999617i	$-0.491192\pi$
$953$	1.70820	0.0553342	0.0276671	+	0.999617i	$0.491192\pi$
$954$	5.09017	0.164800
$955$	−67.1246	−2.17210
$956$	22.1803	0.717363
$957$	−2.85410	−0.0922600
$958$	−28.6180	−0.924607
$959$	−13.2705	−0.428527
$960$	6.23607	0.201268
$961$	40.7771	1.31539
$962$	−18.1803	−0.586158
$963$	0.111456	0.00359163
$964$	−18.1459	−0.584440
$965$	−71.1935	−2.29180
$966$	0	0
$967$	40.7214	1.30951	0.654755	−	0.755841i	$-0.272772\pi$
$967$	40.7214	1.30951	0.654755	+	0.755841i	$0.272772\pi$
$968$	8.38197	0.269407
$969$	0	0
$970$	−21.6525	−0.695219
$971$	16.4934	0.529299	0.264650	−	0.964345i	$-0.414744\pi$
$971$	16.4934	0.529299	0.264650	+	0.964345i	$0.414744\pi$
$972$	−3.94427	−0.126513
$973$	2.94427	0.0943890
$974$	18.5066	0.592989
$975$	31.8885	1.02125
$976$	12.5623	0.402110
$977$	−0.652476	−0.0208746	−0.0104373	−	0.999946i	$-0.503322\pi$
$977$	−0.652476	−0.0208746	−0.0104373	+	0.999946i	$0.503322\pi$
$978$	19.5623	0.625534
$979$	−18.3262	−0.585709
$980$	−3.85410	−0.123115
$981$	−3.20163	−0.102220
$982$	−4.58359	−0.146268
$983$	−49.0132	−1.56328	−0.781638	−	0.623732i	$-0.785616\pi$
$983$	−49.0132	−1.56328	−0.781638	+	0.623732i	$0.785616\pi$
$984$	3.47214	0.110688
$985$	−73.7082	−2.34854
$986$	3.52786	0.112350
$987$	13.5623	0.431693
$988$	0	0
$989$	0	0
$990$	−2.38197	−0.0757038
$991$	−61.3262	−1.94809	−0.974046	−	0.226350i	$-0.927321\pi$
$991$	−61.3262	−1.94809	−0.974046	+	0.226350i	$0.927321\pi$
$992$	8.47214	0.268991
$993$	−21.4164	−0.679629
$994$	12.5623	0.398452
$995$	2.38197	0.0755134
$996$	−22.1803	−0.702811
$997$	−31.5623	−0.999588	−0.499794	−	0.866144i	$-0.666591\pi$
$997$	−31.5623	−0.999588	−0.499794	+	0.866144i	$0.666591\pi$
$998$	33.3820	1.05669
$999$	−49.7426	−1.57379

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5054.2.a.d.1.1	✓	2
19.18	odd	2		5054.2.a.o.1.2	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5054.2.a.d.1.1	✓	2	1.1	even	1	trivial
5054.2.a.o.1.2	yes	2	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -1.61803 q^{3} +1.00000 q^{4} -3.85410 q^{5} +1.61803 q^{6} -1.00000 q^{7} -1.00000 q^{8} -0.381966 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -1.61803 q^{3} +1.00000 q^{4} -3.85410 q^{5} +1.61803 q^{6} -1.00000 q^{7} -1.00000 q^{8} -0.381966 q^{9} +3.85410 q^{10} +1.61803 q^{11} -1.61803 q^{12} -2.00000 q^{13} +1.00000 q^{14} +6.23607 q^{15} +1.00000 q^{16} -3.23607 q^{17} +0.381966 q^{18} -3.85410 q^{20} +1.61803 q^{21} -1.61803 q^{22} +1.61803 q^{24} +9.85410 q^{25} +2.00000 q^{26} +5.47214 q^{27} -1.00000 q^{28} +1.09017 q^{29} -6.23607 q^{30} -8.47214 q^{31} -1.00000 q^{32} -2.61803 q^{33} +3.23607 q^{34} +3.85410 q^{35} -0.381966 q^{36} -9.09017 q^{37} +3.23607 q^{39} +3.85410 q^{40} +2.14590 q^{41} -1.61803 q^{42} +1.38197 q^{43} +1.61803 q^{44} +1.47214 q^{45} +8.38197 q^{47} -1.61803 q^{48} +1.00000 q^{49} -9.85410 q^{50} +5.23607 q^{51} -2.00000 q^{52} +13.3262 q^{53} -5.47214 q^{54} -6.23607 q^{55} +1.00000 q^{56} -1.09017 q^{58} -9.38197 q^{59} +6.23607 q^{60} +12.5623 q^{61} +8.47214 q^{62} +0.381966 q^{63} +1.00000 q^{64} +7.70820 q^{65} +2.61803 q^{66} -2.76393 q^{67} -3.23607 q^{68} -3.85410 q^{70} +12.5623 q^{71} +0.381966 q^{72} +12.9443 q^{73} +9.09017 q^{74} -15.9443 q^{75} -1.61803 q^{77} -3.23607 q^{78} -11.0902 q^{79} -3.85410 q^{80} -7.70820 q^{81} -2.14590 q^{82} +13.7082 q^{83} +1.61803 q^{84} +12.4721 q^{85} -1.38197 q^{86} -1.76393 q^{87} -1.61803 q^{88} -11.3262 q^{89} -1.47214 q^{90} +2.00000 q^{91} +13.7082 q^{93} -8.38197 q^{94} +1.61803 q^{96} -5.61803 q^{97} -1.00000 q^{98} -0.618034 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} - q^{3} + 2 q^{4} - q^{5} + q^{6} - 2 q^{7} - 2 q^{8} - 3 q^{9}+O(q^{10}) \) 2 * q - 2 * q^2 - q^3 + 2 * q^4 - q^5 + q^6 - 2 * q^7 - 2 * q^8 - 3 * q^9	\( 2 q - 2 q^{2} - q^{3} + 2 q^{4} - q^{5} + q^{6} - 2 q^{7} - 2 q^{8} - 3 q^{9} + q^{10} + q^{11} - q^{12} - 4 q^{13} + 2 q^{14} + 8 q^{15} + 2 q^{16} - 2 q^{17} + 3 q^{18} - q^{20} + q^{21} - q^{22} + q^{24} + 13 q^{25} + 4 q^{26} + 2 q^{27} - 2 q^{28} - 9 q^{29} - 8 q^{30} - 8 q^{31} - 2 q^{32} - 3 q^{33} + 2 q^{34} + q^{35} - 3 q^{36} - 7 q^{37} + 2 q^{39} + q^{40} + 11 q^{41} - q^{42} + 5 q^{43} + q^{44} - 6 q^{45} + 19 q^{47} - q^{48} + 2 q^{49} - 13 q^{50} + 6 q^{51} - 4 q^{52} + 11 q^{53} - 2 q^{54} - 8 q^{55} + 2 q^{56} + 9 q^{58} - 21 q^{59} + 8 q^{60} + 5 q^{61} + 8 q^{62} + 3 q^{63} + 2 q^{64} + 2 q^{65} + 3 q^{66} - 10 q^{67} - 2 q^{68} - q^{70} + 5 q^{71} + 3 q^{72} + 8 q^{73} + 7 q^{74} - 14 q^{75} - q^{77} - 2 q^{78} - 11 q^{79} - q^{80} - 2 q^{81} - 11 q^{82} + 14 q^{83} + q^{84} + 16 q^{85} - 5 q^{86} - 8 q^{87} - q^{88} - 7 q^{89} + 6 q^{90} + 4 q^{91} + 14 q^{93} - 19 q^{94} + q^{96} - 9 q^{97} - 2 q^{98} + q^{99}+O(q^{100}) \) 2 * q - 2 * q^2 - q^3 + 2 * q^4 - q^5 + q^6 - 2 * q^7 - 2 * q^8 - 3 * q^9 + q^10 + q^11 - q^12 - 4 * q^13 + 2 * q^14 + 8 * q^15 + 2 * q^16 - 2 * q^17 + 3 * q^18 - q^20 + q^21 - q^22 + q^24 + 13 * q^25 + 4 * q^26 + 2 * q^27 - 2 * q^28 - 9 * q^29 - 8 * q^30 - 8 * q^31 - 2 * q^32 - 3 * q^33 + 2 * q^34 + q^35 - 3 * q^36 - 7 * q^37 + 2 * q^39 + q^40 + 11 * q^41 - q^42 + 5 * q^43 + q^44 - 6 * q^45 + 19 * q^47 - q^48 + 2 * q^49 - 13 * q^50 + 6 * q^51 - 4 * q^52 + 11 * q^53 - 2 * q^54 - 8 * q^55 + 2 * q^56 + 9 * q^58 - 21 * q^59 + 8 * q^60 + 5 * q^61 + 8 * q^62 + 3 * q^63 + 2 * q^64 + 2 * q^65 + 3 * q^66 - 10 * q^67 - 2 * q^68 - q^70 + 5 * q^71 + 3 * q^72 + 8 * q^73 + 7 * q^74 - 14 * q^75 - q^77 - 2 * q^78 - 11 * q^79 - q^80 - 2 * q^81 - 11 * q^82 + 14 * q^83 + q^84 + 16 * q^85 - 5 * q^86 - 8 * q^87 - q^88 - 7 * q^89 + 6 * q^90 + 4 * q^91 + 14 * q^93 - 19 * q^94 + q^96 - 9 * q^97 - 2 * q^98 + q^99

Introduction

L-functions

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