LMFDB - Embedded newform 507.4.a.k.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [507,4,Mod(1,507)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(507, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("507.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 507 = 3 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	507.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:	$29.9139683729$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{3}, \sqrt{17})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 13x^{2} + 14x - 2 $ x^4 - 2x^3 - 13x^2 + 14*x - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 39)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-3.29360$ of defining polynomial
Character	$\chi$	$=$	507.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-4.12311 q^{2} -3.00000 q^{3} +9.00000 q^{4} +3.05006 q^{5} +12.3693 q^{6} +6.68324 q^{7} -4.12311 q^{8} +9.00000 q^{9} +O(q^{10})$	\(q-4.12311 q^{2} -3.00000 q^{3} +9.00000 q^{4} +3.05006 q^{5} +12.3693 q^{6} +6.68324 q^{7} -4.12311 q^{8} +9.00000 q^{9} -12.5757 q^{10} -32.2500 q^{11} -27.0000 q^{12} -27.5557 q^{14} -9.15018 q^{15} -55.0000 q^{16} -28.8586 q^{17} -37.1080 q^{18} +101.194 q^{19} +27.4505 q^{20} -20.0497 q^{21} +132.970 q^{22} -118.990 q^{23} +12.3693 q^{24} -115.697 q^{25} -27.0000 q^{27} +60.1492 q^{28} +160.111 q^{29} +37.7271 q^{30} +38.0705 q^{31} +259.756 q^{32} +96.7499 q^{33} +118.987 q^{34} +20.3843 q^{35} +81.0000 q^{36} +327.568 q^{37} -417.233 q^{38} -12.5757 q^{40} -56.0846 q^{41} +82.6671 q^{42} +127.879 q^{43} -290.250 q^{44} +27.4505 q^{45} +490.608 q^{46} +517.983 q^{47} +165.000 q^{48} -298.334 q^{49} +477.032 q^{50} +86.5757 q^{51} -695.546 q^{53} +111.324 q^{54} -98.3643 q^{55} -27.5557 q^{56} -303.581 q^{57} -660.156 q^{58} -656.523 q^{59} -82.3516 q^{60} +701.304 q^{61} -156.969 q^{62} +60.1492 q^{63} -631.000 q^{64} -398.910 q^{66} +57.1750 q^{67} -259.727 q^{68} +356.970 q^{69} -84.0465 q^{70} -309.226 q^{71} -37.1080 q^{72} +389.711 q^{73} -1350.60 q^{74} +347.091 q^{75} +910.744 q^{76} -215.534 q^{77} +901.820 q^{79} -167.753 q^{80} +81.0000 q^{81} +231.243 q^{82} -687.095 q^{83} -180.448 q^{84} -88.0203 q^{85} -527.257 q^{86} -480.334 q^{87} +132.970 q^{88} -1070.54 q^{89} -113.181 q^{90} -1070.91 q^{92} -114.211 q^{93} -2135.70 q^{94} +308.647 q^{95} -779.267 q^{96} -1754.48 q^{97} +1230.06 q^{98} -290.250 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 12 q^{3} + 36 q^{4} + 36 q^{9}+O(q^{10}) $ 4 * q - 12 * q^3 + 36 * q^4 + 36 * q^9	$ 4 q - 12 q^{3} + 36 q^{4} + 36 q^{9} - 136 q^{10} - 108 q^{12} + 204 q^{14} - 220 q^{16} - 144 q^{17} - 68 q^{22} - 276 q^{23} - 120 q^{25} - 108 q^{27} + 12 q^{29} + 408 q^{30} - 804 q^{35} + 324 q^{36} - 612 q^{38} - 136 q^{40} - 612 q^{42} + 940 q^{43} + 660 q^{48} + 692 q^{49} + 432 q^{51} - 2268 q^{53} + 892 q^{55} + 204 q^{56} + 320 q^{61} - 2856 q^{62} - 2524 q^{64} + 204 q^{66} - 1296 q^{68} + 828 q^{69} - 3060 q^{74} + 360 q^{75} - 2976 q^{77} + 8 q^{79} + 324 q^{81} + 68 q^{82} - 36 q^{87} - 68 q^{88} - 1224 q^{90} - 2484 q^{92} - 5372 q^{94} - 108 q^{95}+O(q^{100}) $ 4 * q - 12 * q^3 + 36 * q^4 + 36 * q^9 - 136 * q^10 - 108 * q^12 + 204 * q^14 - 220 * q^16 - 144 * q^17 - 68 * q^22 - 276 * q^23 - 120 * q^25 - 108 * q^27 + 12 * q^29 + 408 * q^30 - 804 * q^35 + 324 * q^36 - 612 * q^38 - 136 * q^40 - 612 * q^42 + 940 * q^43 + 660 * q^48 + 692 * q^49 + 432 * q^51 - 2268 * q^53 + 892 * q^55 + 204 * q^56 + 320 * q^61 - 2856 * q^62 - 2524 * q^64 + 204 * q^66 - 1296 * q^68 + 828 * q^69 - 3060 * q^74 + 360 * q^75 - 2976 * q^77 + 8 * q^79 + 324 * q^81 + 68 * q^82 - 36 * q^87 - 68 * q^88 - 1224 * q^90 - 2484 * q^92 - 5372 * q^94 - 108 * q^95

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 $

$2$	−4.12311	−1.45774	−0.728869	−	0.684653i	$-0.759954\pi$
$2$	−4.12311	−1.45774	−0.728869	+	0.684653i	$0.759954\pi$
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	9.00000	1.12500
$5$	3.05006	0.272806	0.136403	−	0.990653i	$-0.456446\pi$
$5$	3.05006	0.272806	0.136403	+	0.990653i	$0.456446\pi$
$6$	12.3693	0.841625
$7$	6.68324	0.360861	0.180431	−	0.983588i	$-0.442251\pi$
$7$	6.68324	0.360861	0.180431	+	0.983588i	$0.442251\pi$
$8$	−4.12311	−0.182217
$9$	9.00000	0.333333
$10$	−12.5757	−0.397679
$11$	−32.2500	−0.883975	−0.441988	−	0.897021i	$-0.645727\pi$
$11$	−32.2500	−0.883975	−0.441988	+	0.897021i	$0.645727\pi$
$12$	−27.0000	−0.649519
$13$	0	0
$13$	0	0
$14$	−27.5557	−0.526041
$15$	−9.15018	−0.157504
$16$	−55.0000	−0.859375
$17$	−28.8586	−0.411720	−0.205860	−	0.978582i	$-0.565999\pi$
$17$	−28.8586	−0.411720	−0.205860	+	0.978582i	$0.565999\pi$
$18$	−37.1080	−0.485913
$19$	101.194	1.22187	0.610933	−	0.791682i	$-0.290794\pi$
$19$	101.194	1.22187	0.610933	+	0.791682i	$0.290794\pi$
$20$	27.4505	0.306906
$21$	−20.0497	−0.208343
$22$	132.970	1.28860
$23$	−118.990	−1.07874	−0.539372	−	0.842067i	$-0.681338\pi$
$23$	−118.990	−1.07874	−0.539372	+	0.842067i	$0.681338\pi$
$24$	12.3693	0.105203
$25$	−115.697	−0.925577
$26$	0	0
$27$	−27.0000	−0.192450
$28$	60.1492	0.405969
$29$	160.111	1.02524	0.512620	−	0.858616i	$-0.328675\pi$
$29$	160.111	1.02524	0.512620	+	0.858616i	$0.328675\pi$
$30$	37.7271	0.229600
$31$	38.0705	0.220570	0.110285	−	0.993900i	$-0.464824\pi$
$31$	38.0705	0.220570	0.110285	+	0.993900i	$0.464824\pi$
$32$	259.756	1.43496
$33$	96.7499	0.510363
$34$	118.987	0.600179
$35$	20.3843	0.0984449
$36$	81.0000	0.375000
$37$	327.568	1.45545	0.727727	−	0.685866i	$-0.240577\pi$
$37$	327.568	1.45545	0.727727	+	0.685866i	$0.240577\pi$
$38$	−417.233	−1.78116
$39$	0	0
$40$	−12.5757	−0.0497099
$41$	−56.0846	−0.213633	−0.106816	−	0.994279i	$-0.534066\pi$
$41$	−56.0846	−0.213633	−0.106816	+	0.994279i	$0.534066\pi$
$42$	82.6671	0.303710
$43$	127.879	0.453519	0.226759	−	0.973951i	$-0.427187\pi$
$43$	127.879	0.453519	0.226759	+	0.973951i	$0.427187\pi$
$44$	−290.250	−0.994472
$45$	27.4505	0.0909352
$46$	490.608	1.57253
$47$	517.983	1.60757	0.803783	−	0.594923i	$-0.202817\pi$
$47$	517.983	1.60757	0.803783	+	0.594923i	$0.202817\pi$
$48$	165.000	0.496160
$49$	−298.334	−0.869779
$50$	477.032	1.34925
$51$	86.5757	0.237706
$52$	0	0
$53$	−695.546	−1.80265	−0.901326	−	0.433141i	$-0.857405\pi$
$53$	−695.546	−1.80265	−0.901326	+	0.433141i	$0.857405\pi$
$54$	111.324	0.280542
$55$	−98.3643	−0.241153
$56$	−27.5557	−0.0657551
$57$	−303.581	−0.705445
$58$	−660.156	−1.49453
$59$	−656.523	−1.44868	−0.724339	−	0.689444i	$-0.757855\pi$
$59$	−656.523	−1.44868	−0.724339	+	0.689444i	$0.757855\pi$
$60$	−82.3516	−0.177192
$61$	701.304	1.47201	0.736007	−	0.676974i	$-0.236709\pi$
$61$	701.304	1.47201	0.736007	+	0.676974i	$0.236709\pi$
$62$	−156.969	−0.321533
$63$	60.1492	0.120287
$64$	−631.000	−1.23242
$65$	0	0
$66$	−398.910	−0.743976
$67$	57.1750	0.104254	0.0521271	−	0.998640i	$-0.483400\pi$
$67$	57.1750	0.104254	0.0521271	+	0.998640i	$0.483400\pi$
$68$	−259.727	−0.463184
$69$	356.970	0.622814
$70$	−84.0465	−0.143507
$71$	−309.226	−0.516879	−0.258440	−	0.966027i	$-0.583208\pi$
$71$	−309.226	−0.516879	−0.258440	+	0.966027i	$0.583208\pi$
$72$	−37.1080	−0.0607391
$73$	389.711	0.624826	0.312413	−	0.949946i	$-0.398863\pi$
$73$	389.711	0.624826	0.312413	+	0.949946i	$0.398863\pi$
$74$	−1350.60	−2.12167
$75$	347.091	0.534382
$76$	910.744	1.37460
$77$	−215.534	−0.318992
$78$	0	0
$79$	901.820	1.28434	0.642169	−	0.766563i	$-0.278035\pi$
$79$	901.820	1.28434	0.642169	+	0.766563i	$0.278035\pi$
$80$	−167.753	−0.234442
$81$	81.0000	0.111111
$82$	231.243	0.311421
$83$	−687.095	−0.908657	−0.454328	−	0.890834i	$-0.650121\pi$
$83$	−687.095	−0.908657	−0.454328	+	0.890834i	$0.650121\pi$
$84$	−180.448	−0.234386
$85$	−88.0203	−0.112319
$86$	−527.257	−0.661111
$87$	−480.334	−0.591922
$88$	132.970	0.161076
$89$	−1070.54	−1.27502	−0.637510	−	0.770442i	$-0.720036\pi$
$89$	−1070.54	−1.27502	−0.637510	+	0.770442i	$0.720036\pi$
$90$	−113.181	−0.132560
$91$	0	0
$92$	−1070.91	−1.21359
$93$	−114.211	−0.127346
$94$	−2135.70	−2.34341
$95$	308.647	0.333332
$96$	−779.267	−0.828475
$97$	−1754.48	−1.83650	−0.918251	−	0.395998i	$-0.870399\pi$
$97$	−1754.48	−1.83650	−0.918251	+	0.395998i	$0.870399\pi$
$98$	1230.06	1.26791
$99$	−290.250	−0.294658
$100$	−1041.27	−1.04127
$101$	−640.840	−0.631346	−0.315673	−	0.948868i	$-0.602230\pi$
$101$	−640.840	−0.631346	−0.315673	+	0.948868i	$0.602230\pi$
$102$	−356.961	−0.346514
$103$	−693.153	−0.663091	−0.331546	−	0.943439i	$-0.607570\pi$
$103$	−693.153	−0.663091	−0.331546	+	0.943439i	$0.607570\pi$
$104$	0	0
$105$	−61.1528	−0.0568372
$106$	2867.81	2.62779
$107$	−405.676	−0.366525	−0.183262	−	0.983064i	$-0.558666\pi$
$107$	−405.676	−0.366525	−0.183262	+	0.983064i	$0.558666\pi$
$108$	−243.000	−0.216506
$109$	479.516	0.421370	0.210685	−	0.977554i	$-0.432431\pi$
$109$	479.516	0.421370	0.210685	+	0.977554i	$0.432431\pi$
$110$	405.566	0.351538
$111$	−982.704	−0.840307
$112$	−367.578	−0.310115
$113$	−1547.20	−1.28804	−0.644021	−	0.765008i	$-0.722735\pi$
$113$	−1547.20	−1.28804	−0.644021	+	0.765008i	$0.722735\pi$
$114$	1251.70	1.02835
$115$	−362.926	−0.294288
$116$	1441.00	1.15339
$117$	0	0
$118$	2706.91	2.11179
$119$	−192.869	−0.148574
$120$	37.7271	0.0287000
$121$	−290.940	−0.218588
$122$	−2891.55	−2.14581
$123$	168.254	0.123341
$124$	342.634	0.248141
$125$	−734.140	−0.525308
$126$	−248.001	−0.175347
$127$	−2495.15	−1.74338	−0.871690	−	0.490059i	$-0.836975\pi$
$127$	−2495.15	−1.74338	−0.871690	+	0.490059i	$0.836975\pi$
$128$	523.634	0.361587
$129$	−383.636	−0.261839
$130$	0	0
$131$	43.7571	0.0291838	0.0145919	−	0.999894i	$-0.495355\pi$
$131$	43.7571	0.0291838	0.0145919	+	0.999894i	$0.495355\pi$
$132$	870.749	0.574159
$133$	676.303	0.440924
$134$	−235.739	−0.151975
$135$	−82.3516	−0.0525015
$136$	118.987	0.0750224
$137$	206.194	0.128586	0.0642932	−	0.997931i	$-0.479521\pi$
$137$	206.194	0.128586	0.0642932	+	0.997931i	$0.479521\pi$
$138$	−1471.83	−0.907899
$139$	−100.000	−0.0610208	−0.0305104	−	0.999534i	$-0.509713\pi$
$139$	−100.000	−0.0610208	−0.0305104	+	0.999534i	$0.509713\pi$
$140$	183.459	0.110751
$141$	−1553.95	−0.928128
$142$	1274.97	0.753474
$143$	0	0
$144$	−495.000	−0.286458
$145$	488.349	0.279691
$146$	−1606.82	−0.910832
$147$	895.003	0.502167
$148$	2948.11	1.63739
$149$	−380.451	−0.209179	−0.104590	−	0.994515i	$-0.533353\pi$
$149$	−380.451	−0.209179	−0.104590	+	0.994515i	$0.533353\pi$
$150$	−1431.09	−0.778989
$151$	1517.45	0.817805	0.408902	−	0.912578i	$-0.365912\pi$
$151$	1517.45	0.817805	0.408902	+	0.912578i	$0.365912\pi$
$152$	−417.233	−0.222645
$153$	−259.727	−0.137240
$154$	888.671	0.465007
$155$	116.117	0.0601726
$156$	0	0
$157$	1450.16	0.737166	0.368583	−	0.929595i	$-0.379843\pi$
$157$	1450.16	0.737166	0.368583	+	0.929595i	$0.379843\pi$
$158$	−3718.30	−1.87223
$159$	2086.64	1.04076
$160$	792.270	0.391465
$161$	−795.239	−0.389277
$162$	−333.972	−0.161971
$163$	−2342.36	−1.12557	−0.562785	−	0.826603i	$-0.690270\pi$
$163$	−2342.36	−1.12557	−0.562785	+	0.826603i	$0.690270\pi$
$164$	−504.762	−0.240337
$165$	295.093	0.139230
$166$	2832.97	1.32458
$167$	40.0731	0.0185686	0.00928428	−	0.999957i	$-0.497045\pi$
$167$	40.0731	0.0185686	0.00928428	+	0.999957i	$0.497045\pi$
$168$	82.6671	0.0379637
$169$	0	0
$170$	362.917	0.163732
$171$	910.744	0.407289
$172$	1150.91	0.510208
$173$	1909.54	0.839189	0.419594	−	0.907712i	$-0.362172\pi$
$173$	1909.54	0.839189	0.419594	+	0.907712i	$0.362172\pi$
$174$	1980.47	0.862868
$175$	−773.232	−0.334005
$176$	1773.75	0.759666
$177$	1969.57	0.836395
$178$	4413.94	1.85864
$179$	−509.959	−0.212939	−0.106470	−	0.994316i	$-0.533955\pi$
$179$	−509.959	−0.212939	−0.106470	+	0.994316i	$0.533955\pi$
$180$	247.055	0.102302
$181$	−2136.88	−0.877531	−0.438766	−	0.898602i	$-0.644584\pi$
$181$	−2136.88	−0.877531	−0.438766	+	0.898602i	$0.644584\pi$
$182$	0	0
$183$	−2103.91	−0.849867
$184$	490.608	0.196566
$185$	999.101	0.397056
$186$	470.906	0.185637
$187$	930.688	0.363950
$188$	4661.85	1.80851
$189$	−180.448	−0.0694478
$190$	−1272.58	−0.485911
$191$	4057.74	1.53721	0.768607	−	0.639721i	$-0.220950\pi$
$191$	4057.74	1.53721	0.768607	+	0.639721i	$0.220950\pi$
$192$	1893.00	0.711539
$193$	−873.394	−0.325742	−0.162871	−	0.986647i	$-0.552075\pi$
$193$	−873.394	−0.325742	−0.162871	+	0.986647i	$0.552075\pi$
$194$	7233.92	2.67714
$195$	0	0
$196$	−2685.01	−0.978502
$197$	−4147.25	−1.49989	−0.749947	−	0.661498i	$-0.769921\pi$
$197$	−4147.25	−1.49989	−0.749947	+	0.661498i	$0.769921\pi$
$198$	1196.73	0.429535
$199$	−2404.06	−0.856379	−0.428189	−	0.903689i	$-0.640848\pi$
$199$	−2404.06	−0.856379	−0.428189	+	0.903689i	$0.640848\pi$
$200$	477.032	0.168656
$201$	−171.525	−0.0601912
$202$	2642.25	0.920337
$203$	1070.06	0.369969
$204$	779.181	0.267420
$205$	−171.061	−0.0582802
$206$	2857.94	0.966613
$207$	−1070.91	−0.359582
$208$	0	0
$209$	−3263.50	−1.08010
$210$	252.140	0.0828538
$211$	3868.85	1.26229	0.631144	−	0.775665i	$-0.282586\pi$
$211$	3868.85	1.26229	0.631144	+	0.775665i	$0.282586\pi$
$212$	−6259.91	−2.02798
$213$	927.679	0.298420
$214$	1672.64	0.534297
$215$	390.037	0.123722
$216$	111.324	0.0350677
$217$	254.434	0.0795950
$218$	−1977.09	−0.614246
$219$	−1169.13	−0.360743
$220$	−885.279	−0.271298
$221$	0	0
$222$	4051.79	1.22495
$223$	−2813.55	−0.844885	−0.422443	−	0.906390i	$-0.638827\pi$
$223$	−2813.55	−0.844885	−0.422443	+	0.906390i	$0.638827\pi$
$224$	1736.01	0.517822
$225$	−1041.27	−0.308526
$226$	6379.29	1.87763
$227$	4518.37	1.32112	0.660561	−	0.750772i	$-0.270318\pi$
$227$	4518.37	1.32112	0.660561	+	0.750772i	$0.270318\pi$
$228$	−2732.23	−0.793625
$229$	1305.27	0.376658	0.188329	−	0.982106i	$-0.439693\pi$
$229$	1305.27	0.376658	0.188329	+	0.982106i	$0.439693\pi$
$230$	1496.38	0.428994
$231$	646.603	0.184170
$232$	−660.156	−0.186816
$233$	−3360.55	−0.944879	−0.472440	−	0.881363i	$-0.656627\pi$
$233$	−3360.55	−0.944879	−0.472440	+	0.881363i	$0.656627\pi$
$234$	0	0
$235$	1579.88	0.438553
$236$	−5908.71	−1.62976
$237$	−2705.46	−0.741513
$238$	795.219	0.216581
$239$	4737.17	1.28210	0.641050	−	0.767499i	$-0.278499\pi$
$239$	4737.17	1.28210	0.641050	+	0.767499i	$0.278499\pi$
$240$	503.260	0.135355
$241$	−4785.28	−1.27903	−0.639516	−	0.768778i	$-0.720865\pi$
$241$	−4785.28	−1.27903	−0.639516	+	0.768778i	$0.720865\pi$
$242$	1199.58	0.318643
$243$	−243.000	−0.0641500
$244$	6311.74	1.65601
$245$	−909.937	−0.237281
$246$	−693.729	−0.179799
$247$	0	0
$248$	−156.969	−0.0401916
$249$	2061.29	0.524613
$250$	3026.94	0.765762
$251$	−3273.86	−0.823284	−0.411642	−	0.911346i	$-0.635045\pi$
$251$	−3273.86	−0.823284	−0.411642	+	0.911346i	$0.635045\pi$
$252$	541.343	0.135323
$253$	3837.42	0.953584
$254$	10287.8	2.54139
$255$	264.061	0.0648476
$256$	2889.00	0.705322
$257$	−6545.81	−1.58878	−0.794390	−	0.607408i	$-0.792209\pi$
$257$	−6545.81	−1.58878	−0.794390	+	0.607408i	$0.792209\pi$
$258$	1581.77	0.381693
$259$	2189.22	0.525217
$260$	0	0
$261$	1441.00	0.341746
$262$	−180.415	−0.0425424
$263$	88.2014	0.0206796	0.0103398	−	0.999947i	$-0.496709\pi$
$263$	88.2014	0.0206796	0.0103398	+	0.999947i	$0.496709\pi$
$264$	−398.910	−0.0929970
$265$	−2121.46	−0.491773
$266$	−2788.47	−0.642752
$267$	3211.61	0.736133
$268$	514.575	0.117286
$269$	−4527.60	−1.02622	−0.513109	−	0.858324i	$-0.671506\pi$
$269$	−4527.60	−1.02622	−0.513109	+	0.858324i	$0.671506\pi$
$270$	339.544	0.0765334
$271$	−8321.82	−1.86537	−0.932684	−	0.360695i	$-0.882540\pi$
$271$	−8321.82	−1.86537	−0.932684	+	0.360695i	$0.882540\pi$
$272$	1587.22	0.353821
$273$	0	0
$274$	−850.160	−0.187445
$275$	3731.23	0.818187
$276$	3212.73	0.700665
$277$	−2881.31	−0.624986	−0.312493	−	0.949920i	$-0.601164\pi$
$277$	−2881.31	−0.624986	−0.312493	+	0.949920i	$0.601164\pi$
$278$	412.311	0.0889523
$279$	342.634	0.0735232
$280$	−84.0465	−0.0179384
$281$	2817.99	0.598247	0.299123	−	0.954214i	$-0.403306\pi$
$281$	2817.99	0.598247	0.299123	+	0.954214i	$0.403306\pi$
$282$	6407.10	1.35297
$283$	264.601	0.0555792	0.0277896	−	0.999614i	$-0.491153\pi$
$283$	264.601	0.0555792	0.0277896	+	0.999614i	$0.491153\pi$
$284$	−2783.04	−0.581489
$285$	−925.941	−0.192449
$286$	0	0
$287$	−374.827	−0.0770918
$288$	2337.80	0.478320
$289$	−4080.18	−0.830487
$290$	−2013.52	−0.407716
$291$	5263.45	1.06031
$292$	3507.40	0.702929
$293$	−4292.52	−0.855877	−0.427938	−	0.903808i	$-0.640760\pi$
$293$	−4292.52	−0.855877	−0.427938	+	0.903808i	$0.640760\pi$
$294$	−3690.19	−0.732028
$295$	−2002.43	−0.395208
$296$	−1350.60	−0.265209
$297$	870.749	0.170121
$298$	1568.64	0.304929
$299$	0	0
$300$	3123.82	0.601180
$301$	854.643	0.163657
$302$	−6256.62	−1.19214
$303$	1922.52	0.364508
$304$	−5565.66	−1.05004
$305$	2139.02	0.401573
$306$	1070.88	0.200060
$307$	−7026.26	−1.30622	−0.653110	−	0.757263i	$-0.726536\pi$
$307$	−7026.26	−1.30622	−0.653110	+	0.757263i	$0.726536\pi$
$308$	−1939.81	−0.358866
$309$	2079.46	0.382836
$310$	−478.763	−0.0877159
$311$	−1133.21	−0.206618	−0.103309	−	0.994649i	$-0.532943\pi$
$311$	−1133.21	−0.206618	−0.103309	+	0.994649i	$0.532943\pi$
$312$	0	0
$313$	−5285.95	−0.954566	−0.477283	−	0.878750i	$-0.658378\pi$
$313$	−5285.95	−0.954566	−0.477283	+	0.878750i	$0.658378\pi$
$314$	−5979.15	−1.07459
$315$	183.459	0.0328150
$316$	8116.38	1.44488
$317$	4782.16	0.847296	0.423648	−	0.905827i	$-0.360749\pi$
$317$	4782.16	0.847296	0.423648	+	0.905827i	$0.360749\pi$
$318$	−8603.43	−1.51716
$319$	−5163.59	−0.906286
$320$	−1924.59	−0.336212
$321$	1217.03	0.211613
$322$	3278.85	0.567464
$323$	−2920.31	−0.503066
$324$	729.000	0.125000
$325$	0	0
$326$	9657.80	1.64079
$327$	−1438.55	−0.243278
$328$	231.243	0.0389276
$329$	3461.81	0.580108
$330$	−1216.70	−0.202961
$331$	−8669.98	−1.43971	−0.719857	−	0.694122i	$-0.755793\pi$
$331$	−8669.98	−1.43971	−0.719857	+	0.694122i	$0.755793\pi$
$332$	−6183.86	−1.02224
$333$	2948.11	0.485152
$334$	−165.226	−0.0270681
$335$	174.387	0.0284412
$336$	1102.73	0.179045
$337$	8526.59	1.37826	0.689129	−	0.724639i	$-0.257993\pi$
$337$	8526.59	1.37826	0.689129	+	0.724639i	$0.257993\pi$
$338$	0	0
$339$	4641.61	0.743651
$340$	−792.183	−0.126359
$341$	−1227.77	−0.194978
$342$	−3755.10	−0.593720
$343$	−4286.19	−0.674731
$344$	−527.257	−0.0826389
$345$	1088.78	0.169907
$346$	−7873.23	−1.22332
$347$	−12581.5	−1.94643	−0.973213	−	0.229907i	$-0.926158\pi$
$347$	−12581.5	−1.94643	−0.973213	+	0.229907i	$0.926158\pi$
$348$	−4323.01	−0.665913
$349$	8961.18	1.37444	0.687222	−	0.726447i	$-0.258830\pi$
$349$	8961.18	1.37444	0.687222	+	0.726447i	$0.258830\pi$
$350$	3188.12	0.486892
$351$	0	0
$352$	−8377.11	−1.26847
$353$	5357.99	0.807868	0.403934	−	0.914788i	$-0.367643\pi$
$353$	5357.99	0.807868	0.403934	+	0.914788i	$0.367643\pi$
$354$	−8120.74	−1.21924
$355$	−943.159	−0.141007
$356$	−9634.84	−1.43440
$357$	578.606	0.0857790
$358$	2102.61	0.310409
$359$	2705.40	0.397731	0.198866	−	0.980027i	$-0.436274\pi$
$359$	2705.40	0.397731	0.198866	+	0.980027i	$0.436274\pi$
$360$	−113.181	−0.0165700
$361$	3381.19	0.492957
$362$	8810.59	1.27921
$363$	872.820	0.126202
$364$	0	0
$365$	1188.64	0.170456
$366$	8674.65	1.23888
$367$	10473.8	1.48972	0.744858	−	0.667223i	$-0.232517\pi$
$367$	10473.8	1.48972	0.744858	+	0.667223i	$0.232517\pi$
$368$	6544.45	0.927046
$369$	−504.762	−0.0712110
$370$	−4119.40	−0.578804
$371$	−4648.50	−0.650507
$372$	−1027.90	−0.143264
$373$	−12763.0	−1.77170	−0.885850	−	0.463973i	$-0.846424\pi$
$373$	−12763.0	−1.77170	−0.885850	+	0.463973i	$0.846424\pi$
$374$	−3837.32	−0.530544
$375$	2202.42	0.303287
$376$	−2135.70	−0.292926
$377$	0	0
$378$	744.004	0.101237
$379$	2318.02	0.314166	0.157083	−	0.987585i	$-0.449791\pi$
$379$	2318.02	0.314166	0.157083	+	0.987585i	$0.449791\pi$
$380$	2777.82	0.374998
$381$	7485.46	1.00654
$382$	−16730.5	−2.24086
$383$	−1983.34	−0.264606	−0.132303	−	0.991209i	$-0.542237\pi$
$383$	−1983.34	−0.264606	−0.132303	+	0.991209i	$0.542237\pi$
$384$	−1570.90	−0.208763
$385$	−657.392	−0.0870229
$386$	3601.10	0.474847
$387$	1150.91	0.151173
$388$	−15790.3	−2.06607
$389$	−3244.51	−0.422887	−0.211444	−	0.977390i	$-0.567816\pi$
$389$	−3244.51	−0.422887	−0.211444	+	0.977390i	$0.567816\pi$
$390$	0	0
$391$	3433.88	0.444140
$392$	1230.06	0.158489
$393$	−131.271	−0.0168493
$394$	17099.5	2.18645
$395$	2750.60	0.350374
$396$	−2612.25	−0.331491
$397$	−3759.72	−0.475302	−0.237651	−	0.971351i	$-0.576378\pi$
$397$	−3759.72	−0.475302	−0.237651	+	0.971351i	$0.576378\pi$
$398$	9912.20	1.24838
$399$	−2028.91	−0.254568
$400$	6363.34	0.795418
$401$	1997.55	0.248760	0.124380	−	0.992235i	$-0.460306\pi$
$401$	1997.55	0.248760	0.124380	+	0.992235i	$0.460306\pi$
$402$	707.216	0.0877431
$403$	0	0
$404$	−5767.56	−0.710264
$405$	247.055	0.0303117
$406$	−4411.98	−0.539318
$407$	−10564.1	−1.28659
$408$	−356.961	−0.0433142
$409$	5195.23	0.628087	0.314044	−	0.949409i	$-0.398316\pi$
$409$	5195.23	0.628087	0.314044	+	0.949409i	$0.398316\pi$
$410$	705.304	0.0849573
$411$	−618.582	−0.0742394
$412$	−6238.38	−0.745977
$413$	−4387.70	−0.522772
$414$	4415.48	0.524176
$415$	−2095.68	−0.247887
$416$	0	0
$417$	300.000	0.0352304
$418$	13455.7	1.57450
$419$	−6822.11	−0.795422	−0.397711	−	0.917511i	$-0.630195\pi$
$419$	−6822.11	−0.795422	−0.397711	+	0.917511i	$0.630195\pi$
$420$	−550.376	−0.0639419
$421$	−7537.70	−0.872601	−0.436300	−	0.899801i	$-0.643712\pi$
$421$	−7537.70	−0.872601	−0.436300	+	0.899801i	$0.643712\pi$
$422$	−15951.7	−1.84009
$423$	4661.85	0.535855
$424$	2867.81	0.328474
$425$	3338.85	0.381078
$426$	−3824.92	−0.435019
$427$	4686.99	0.531192
$428$	−3651.08	−0.412340
$429$	0	0
$430$	−1608.16	−0.180355
$431$	13404.2	1.49805	0.749023	−	0.662544i	$-0.230523\pi$
$431$	13404.2	1.49805	0.749023	+	0.662544i	$0.230523\pi$
$432$	1485.00	0.165387
$433$	−17715.9	−1.96622	−0.983110	−	0.183014i	$-0.941415\pi$
$433$	−17715.9	−1.96622	−0.983110	+	0.183014i	$0.941415\pi$
$434$	−1049.06	−0.116029
$435$	−1465.05	−0.161480
$436$	4315.64	0.474041
$437$	−12041.1	−1.31808
$438$	4820.46	0.525869
$439$	7163.47	0.778801	0.389401	−	0.921068i	$-0.372682\pi$
$439$	7163.47	0.778801	0.389401	+	0.921068i	$0.372682\pi$
$440$	405.566	0.0439423
$441$	−2685.01	−0.289926
$442$	0	0
$443$	−10169.2	−1.09064	−0.545321	−	0.838227i	$-0.683592\pi$
$443$	−10169.2	−1.09064	−0.545321	+	0.838227i	$0.683592\pi$
$444$	−8844.33	−0.945346
$445$	−3265.20	−0.347833
$446$	11600.6	1.23162
$447$	1141.35	0.120770
$448$	−4217.13	−0.444733
$449$	17142.5	1.80179	0.900895	−	0.434037i	$-0.142911\pi$
$449$	17142.5	1.80179	0.900895	+	0.434037i	$0.142911\pi$
$450$	4293.28	0.449750
$451$	1808.73	0.188846
$452$	−13924.8	−1.44905
$453$	−4552.36	−0.472160
$454$	−18629.7	−1.92585
$455$	0	0
$456$	1251.70	0.128544
$457$	−14091.1	−1.44235	−0.721177	−	0.692750i	$-0.756399\pi$
$457$	−14091.1	−1.44235	−0.721177	+	0.692750i	$0.756399\pi$
$458$	−5381.76	−0.549068
$459$	779.181	0.0792355
$460$	−3266.34	−0.331074
$461$	2922.22	0.295231	0.147616	−	0.989045i	$-0.452840\pi$
$461$	2922.22	0.295231	0.147616	+	0.989045i	$0.452840\pi$
$462$	−2666.01	−0.268472
$463$	2072.61	0.208040	0.104020	−	0.994575i	$-0.466829\pi$
$463$	2072.61	0.208040	0.104020	+	0.994575i	$0.466829\pi$
$464$	−8806.13	−0.881065
$465$	−348.352	−0.0347407
$466$	13855.9	1.37739
$467$	−2664.19	−0.263992	−0.131996	−	0.991250i	$-0.542139\pi$
$467$	−2664.19	−0.263992	−0.131996	+	0.991250i	$0.542139\pi$
$468$	0	0
$469$	382.114	0.0376213
$470$	−6514.01	−0.639295
$471$	−4350.47	−0.425603
$472$	2706.91	0.263974
$473$	−4124.08	−0.400899
$474$	11154.9	1.08093
$475$	−11707.8	−1.13093
$476$	−1735.82	−0.167145
$477$	−6259.91	−0.600884
$478$	−19531.8	−1.86897
$479$	5220.70	0.497995	0.248998	−	0.968504i	$-0.419899\pi$
$479$	5220.70	0.497995	0.248998	+	0.968504i	$0.419899\pi$
$480$	−2376.81	−0.226013
$481$	0	0
$482$	19730.2	1.86449
$483$	2385.72	0.224749
$484$	−2618.46	−0.245911
$485$	−5351.28	−0.501008
$486$	1001.91	0.0935139
$487$	12224.6	1.13747	0.568737	−	0.822520i	$-0.307432\pi$
$487$	12224.6	1.13747	0.568737	+	0.822520i	$0.307432\pi$
$488$	−2891.55	−0.268226
$489$	7027.08	0.649848
$490$	3751.77	0.345893
$491$	19653.2	1.80639	0.903195	−	0.429231i	$-0.141216\pi$
$491$	19653.2	1.80639	0.903195	+	0.429231i	$0.141216\pi$
$492$	1514.29	0.138759
$493$	−4620.59	−0.422111
$494$	0	0
$495$	−885.279	−0.0803845
$496$	−2093.88	−0.189552
$497$	−2066.63	−0.186522
$498$	−8498.90	−0.764749
$499$	11713.6	1.05084	0.525422	−	0.850842i	$-0.323907\pi$
$499$	11713.6	1.05084	0.525422	+	0.850842i	$0.323907\pi$
$500$	−6607.26	−0.590972
$501$	−120.219	−0.0107206
$502$	13498.5	1.20013
$503$	13003.3	1.15266	0.576332	−	0.817216i	$-0.304483\pi$
$503$	13003.3	1.15266	0.576332	+	0.817216i	$0.304483\pi$
$504$	−248.001	−0.0219184
$505$	−1954.60	−0.172235
$506$	−15822.1	−1.39008
$507$	0	0
$508$	−22456.4	−1.96130
$509$	5328.93	0.464049	0.232024	−	0.972710i	$-0.425465\pi$
$509$	5328.93	0.464049	0.232024	+	0.972710i	$0.425465\pi$
$510$	−1088.75	−0.0945308
$511$	2604.54	0.225475
$512$	−16100.7	−1.38976
$513$	−2732.23	−0.235148
$514$	26989.1	2.31603
$515$	−2114.16	−0.180895
$516$	−3452.72	−0.294569
$517$	−16704.9	−1.42105
$518$	−9026.37	−0.765629
$519$	−5728.62	−0.484506
$520$	0	0
$521$	−11700.3	−0.983876	−0.491938	−	0.870630i	$-0.663711\pi$
$521$	−11700.3	−0.983876	−0.491938	+	0.870630i	$0.663711\pi$
$522$	−5941.41	−0.498177
$523$	−4535.04	−0.379165	−0.189583	−	0.981865i	$-0.560714\pi$
$523$	−4535.04	−0.379165	−0.189583	+	0.981865i	$0.560714\pi$
$524$	393.814	0.0328318
$525$	2319.70	0.192838
$526$	−363.664	−0.0301454
$527$	−1098.66	−0.0908128
$528$	−5321.24	−0.438594
$529$	1991.62	0.163690
$530$	8746.98	0.716877
$531$	−5908.71	−0.482893
$532$	6086.73	0.496040
$533$	0	0
$534$	−13241.8	−1.07309
$535$	−1237.33	−0.0999900
$536$	−235.739	−0.0189969
$537$	1529.88	0.122940
$538$	18667.8	1.49596
$539$	9621.27	0.768863
$540$	−741.164	−0.0590641
$541$	5184.89	0.412044	0.206022	−	0.978547i	$-0.433948\pi$
$541$	5184.89	0.412044	0.206022	+	0.978547i	$0.433948\pi$
$542$	34311.7	2.71922
$543$	6410.64	0.506643
$544$	−7496.18	−0.590801
$545$	1462.55	0.114952
$546$	0	0
$547$	5609.12	0.438443	0.219222	−	0.975675i	$-0.429648\pi$
$547$	5609.12	0.438443	0.219222	+	0.975675i	$0.429648\pi$
$548$	1855.75	0.144660
$549$	6311.74	0.490671
$550$	−15384.2	−1.19270
$551$	16202.3	1.25271
$552$	−1471.83	−0.113487
$553$	6027.08	0.463468
$554$	11879.9	0.911066
$555$	−2997.30	−0.229241
$556$	−900.000	−0.0686484
$557$	20150.5	1.53286	0.766432	−	0.642326i	$-0.222030\pi$
$557$	20150.5	1.53286	0.766432	+	0.642326i	$0.222030\pi$
$558$	−1412.72	−0.107178
$559$	0	0
$560$	−1121.14	−0.0846011
$561$	−2792.06	−0.210127
$562$	−11618.9	−0.872087
$563$	16292.2	1.21960	0.609800	−	0.792556i	$-0.291250\pi$
$563$	16292.2	1.21960	0.609800	+	0.792556i	$0.291250\pi$
$564$	−13985.5	−1.04414
$565$	−4719.06	−0.351385
$566$	−1090.98	−0.0810200
$567$	541.343	0.0400957
$568$	1274.97	0.0941843
$569$	10460.5	0.770700	0.385350	−	0.922770i	$-0.374081\pi$
$569$	10460.5	0.770700	0.385350	+	0.922770i	$0.374081\pi$
$570$	3817.75	0.280541
$571$	2225.96	0.163141	0.0815705	−	0.996668i	$-0.474006\pi$
$571$	2225.96	0.163141	0.0815705	+	0.996668i	$0.474006\pi$
$572$	0	0
$573$	−12173.2	−0.887511
$574$	1545.45	0.112380
$575$	13766.8	0.998461
$576$	−5679.00	−0.410807
$577$	−4686.23	−0.338112	−0.169056	−	0.985606i	$-0.554072\pi$
$577$	−4686.23	−0.338112	−0.169056	+	0.985606i	$0.554072\pi$
$578$	16823.0	1.21063
$579$	2620.18	0.188067
$580$	4395.14	0.314652
$581$	−4592.03	−0.327899
$582$	−21701.8	−1.54565
$583$	22431.3	1.59350
$584$	−1606.82	−0.113854
$585$	0	0
$586$	17698.5	1.24764
$587$	−12090.6	−0.850138	−0.425069	−	0.905161i	$-0.639750\pi$
$587$	−12090.6	−0.850138	−0.425069	+	0.905161i	$0.639750\pi$
$588$	8055.03	0.564938
$589$	3852.50	0.269507
$590$	8256.25	0.576109
$591$	12441.7	0.865964
$592$	−18016.2	−1.25078
$593$	−6135.97	−0.424914	−0.212457	−	0.977170i	$-0.568147\pi$
$593$	−6135.97	−0.424914	−0.212457	+	0.977170i	$0.568147\pi$
$594$	−3590.19	−0.247992
$595$	−588.261	−0.0405317
$596$	−3424.06	−0.235327
$597$	7212.18	0.494431
$598$	0	0
$599$	−6198.80	−0.422831	−0.211416	−	0.977396i	$-0.567807\pi$
$599$	−6198.80	−0.422831	−0.211416	+	0.977396i	$0.567807\pi$
$600$	−1431.09	−0.0973737
$601$	18345.4	1.24513	0.622565	−	0.782568i	$-0.286091\pi$
$601$	18345.4	1.24513	0.622565	+	0.782568i	$0.286091\pi$
$602$	−3523.79	−0.238569
$603$	514.575	0.0347514
$604$	13657.1	0.920030
$605$	−887.384	−0.0596319
$606$	−7926.75	−0.531357
$607$	10388.1	0.694631	0.347315	−	0.937748i	$-0.387093\pi$
$607$	10388.1	0.694631	0.347315	+	0.937748i	$0.387093\pi$
$608$	26285.7	1.75333
$609$	−3210.19	−0.213602
$610$	−8819.40	−0.585389
$611$	0	0
$612$	−2337.54	−0.154395
$613$	804.480	0.0530060	0.0265030	−	0.999649i	$-0.491563\pi$
$613$	804.480	0.0530060	0.0265030	+	0.999649i	$0.491563\pi$
$614$	28970.0	1.90413
$615$	513.184	0.0336481
$616$	888.671	0.0581259
$617$	−15218.3	−0.992973	−0.496486	−	0.868044i	$-0.665377\pi$
$617$	−15218.3	−0.992973	−0.496486	+	0.868044i	$0.665377\pi$
$618$	−8573.83	−0.558074
$619$	11462.5	0.744291	0.372145	−	0.928174i	$-0.378622\pi$
$619$	11462.5	0.744291	0.372145	+	0.928174i	$0.378622\pi$
$620$	1045.05	0.0676942
$621$	3212.73	0.207605
$622$	4672.33	0.301195
$623$	−7154.66	−0.460105
$624$	0	0
$625$	12223.0	0.782270
$626$	21794.5	1.39151
$627$	9790.49	0.623596
$628$	13051.4	0.829311
$629$	−9453.14	−0.599239
$630$	−756.419	−0.0478356
$631$	4468.68	0.281926	0.140963	−	0.990015i	$-0.454980\pi$
$631$	4468.68	0.281926	0.140963	+	0.990015i	$0.454980\pi$
$632$	−3718.30	−0.234028
$633$	−11606.6	−0.728783
$634$	−19717.3	−1.23514
$635$	−7610.37	−0.475603
$636$	18779.7	1.17086
$637$	0	0
$638$	21290.0	1.32113
$639$	−2783.04	−0.172293
$640$	1597.12	0.0986430
$641$	6142.36	0.378484	0.189242	−	0.981930i	$-0.439397\pi$
$641$	6142.36	0.378484	0.189242	+	0.981930i	$0.439397\pi$
$642$	−5017.93	−0.308477
$643$	−20738.2	−1.27190	−0.635951	−	0.771729i	$-0.719392\pi$
$643$	−20738.2	−1.27190	−0.635951	+	0.771729i	$0.719392\pi$
$644$	−7157.15	−0.437937
$645$	−1170.11	−0.0714312
$646$	12040.7	0.733339
$647$	852.757	0.0518166	0.0259083	−	0.999664i	$-0.491752\pi$
$647$	852.757	0.0518166	0.0259083	+	0.999664i	$0.491752\pi$
$648$	−333.972	−0.0202464
$649$	21172.8	1.28060
$650$	0	0
$651$	−763.303	−0.0459542
$652$	−21081.3	−1.26627
$653$	−7345.75	−0.440217	−0.220108	−	0.975475i	$-0.570641\pi$
$653$	−7345.75	−0.440217	−0.220108	+	0.975475i	$0.570641\pi$
$654$	5931.28	0.354635
$655$	133.462	0.00796151
$656$	3084.65	0.183591
$657$	3507.40	0.208275
$658$	−14273.4	−0.845645
$659$	12540.7	0.741297	0.370648	−	0.928773i	$-0.379136\pi$
$659$	12540.7	0.741297	0.370648	+	0.928773i	$0.379136\pi$
$660$	2655.84	0.156634
$661$	2242.95	0.131983	0.0659915	−	0.997820i	$-0.478979\pi$
$661$	2242.95	0.131983	0.0659915	+	0.997820i	$0.478979\pi$
$662$	35747.3	2.09873
$663$	0	0
$664$	2832.97	0.165573
$665$	2062.76	0.120287
$666$	−12155.4	−0.707224
$667$	−19051.7	−1.10597
$668$	360.658	0.0208896
$669$	8440.66	0.487795
$670$	−719.016	−0.0414597
$671$	−22617.0	−1.30122
$672$	−5208.03	−0.298964
$673$	−4776.46	−0.273579	−0.136790	−	0.990600i	$-0.543678\pi$
$673$	−4776.46	−0.273579	−0.136790	+	0.990600i	$0.543678\pi$
$674$	−35156.0	−2.00914
$675$	3123.82	0.178127
$676$	0	0
$677$	−7933.57	−0.450387	−0.225193	−	0.974314i	$-0.572301\pi$
$677$	−7933.57	−0.450387	−0.225193	+	0.974314i	$0.572301\pi$
$678$	−19137.9	−1.08405
$679$	−11725.6	−0.662723
$680$	362.917	0.0204665
$681$	−13555.1	−0.762750
$682$	5062.23	0.284227
$683$	−23573.8	−1.32068	−0.660340	−	0.750967i	$-0.729588\pi$
$683$	−23573.8	−1.32068	−0.660340	+	0.750967i	$0.729588\pi$
$684$	8196.70	0.458200
$685$	628.904	0.0350791
$686$	17672.4	0.983581
$687$	−3915.81	−0.217463
$688$	−7033.32	−0.389743
$689$	0	0
$690$	−4489.15	−0.247680
$691$	12543.8	0.690575	0.345288	−	0.938497i	$-0.387781\pi$
$691$	12543.8	0.690575	0.345288	+	0.938497i	$0.387781\pi$
$692$	17185.9	0.944087
$693$	−1939.81	−0.106331
$694$	51874.8	2.83738
$695$	−305.006	−0.0166468
$696$	1980.47	0.107858
$697$	1618.52	0.0879568
$698$	−36947.9	−2.00358
$699$	10081.6	0.545526
$700$	−6959.09	−0.375755
$701$	−581.786	−0.0313463	−0.0156731	−	0.999877i	$-0.504989\pi$
$701$	−581.786	−0.0313463	−0.0156731	+	0.999877i	$0.504989\pi$
$702$	0	0
$703$	33147.9	1.77837
$704$	20349.7	1.08943
$705$	−4739.64	−0.253199
$706$	−22091.6	−1.17766
$707$	−4282.89	−0.227828
$708$	17726.1	0.940944
$709$	20742.0	1.09871	0.549353	−	0.835590i	$-0.314874\pi$
$709$	20742.0	1.09871	0.549353	+	0.835590i	$0.314874\pi$
$710$	3888.74	0.205552
$711$	8116.38	0.428113
$712$	4413.94	0.232331
$713$	−4530.01	−0.237938
$714$	−2385.66	−0.125043
$715$	0	0
$716$	−4589.63	−0.239556
$717$	−14211.5	−0.740221
$718$	−11154.6	−0.579788
$719$	25350.2	1.31489	0.657443	−	0.753504i	$-0.271638\pi$
$719$	25350.2	1.31489	0.657443	+	0.753504i	$0.271638\pi$
$720$	−1509.78	−0.0781474
$721$	−4632.51	−0.239284
$722$	−13941.0	−0.718602
$723$	14355.8	0.738449
$724$	−19231.9	−0.987223
$725$	−18524.4	−0.948938
$726$	−3598.73	−0.183969
$727$	33428.2	1.70534	0.852672	−	0.522447i	$-0.174981\pi$
$727$	33428.2	1.70534	0.852672	+	0.522447i	$0.174981\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	−4900.90	−0.248480
$731$	−3690.39	−0.186722
$732$	−18935.2	−0.956101
$733$	3842.67	0.193632	0.0968160	−	0.995302i	$-0.469134\pi$
$733$	3842.67	0.193632	0.0968160	+	0.995302i	$0.469134\pi$
$734$	−43184.4	−2.17162
$735$	2729.81	0.136994
$736$	−30908.3	−1.54796
$737$	−1843.89	−0.0921582
$738$	2081.19	0.103807
$739$	−29029.6	−1.44502	−0.722511	−	0.691359i	$-0.757012\pi$
$739$	−29029.6	−1.44502	−0.722511	+	0.691359i	$0.757012\pi$
$740$	8991.91	0.446688
$741$	0	0
$742$	19166.3	0.948269
$743$	34996.7	1.72800	0.864000	−	0.503492i	$-0.167952\pi$
$743$	34996.7	1.72800	0.864000	+	0.503492i	$0.167952\pi$
$744$	470.906	0.0232046
$745$	−1160.40	−0.0570653
$746$	52623.2	2.58267
$747$	−6183.86	−0.302886
$748$	8376.19	0.409444
$749$	−2711.23	−0.132265
$750$	−9080.82	−0.442113
$751$	10454.1	0.507957	0.253979	−	0.967210i	$-0.418261\pi$
$751$	10454.1	0.507957	0.253979	+	0.967210i	$0.418261\pi$
$752$	−28489.1	−1.38150
$753$	9821.58	0.475323
$754$	0	0
$755$	4628.32	0.223102
$756$	−1624.03	−0.0781287
$757$	−28130.4	−1.35062	−0.675308	−	0.737536i	$-0.735989\pi$
$757$	−28130.4	−1.35062	−0.675308	+	0.737536i	$0.735989\pi$
$758$	−9557.45	−0.457971
$759$	−11512.3	−0.550552
$760$	−1272.58	−0.0607388
$761$	−21087.0	−1.00447	−0.502236	−	0.864731i	$-0.667489\pi$
$761$	−21087.0	−1.00447	−0.502236	+	0.864731i	$0.667489\pi$
$762$	−30863.4	−1.46727
$763$	3204.72	0.152056
$764$	36519.7	1.72937
$765$	−792.183	−0.0374398
$766$	8177.54	0.385727
$767$	0	0
$768$	−8667.00	−0.407218
$769$	−19527.9	−0.915728	−0.457864	−	0.889022i	$-0.651385\pi$
$769$	−19527.9	−0.915728	−0.457864	+	0.889022i	$0.651385\pi$
$770$	2710.50	0.126857
$771$	19637.4	0.917283
$772$	−7860.55	−0.366460
$773$	29352.0	1.36574	0.682870	−	0.730540i	$-0.260731\pi$
$773$	29352.0	1.36574	0.682870	+	0.730540i	$0.260731\pi$
$774$	−4745.31	−0.220370
$775$	−4404.64	−0.204154
$776$	7233.92	0.334643
$777$	−6567.65	−0.303234
$778$	13377.5	0.616459
$779$	−5675.42	−0.261031
$780$	0	0
$781$	9972.54	0.456908
$782$	−14158.3	−0.647440
$783$	−4323.01	−0.197307
$784$	16408.4	0.747467
$785$	4423.06	0.201103
$786$	541.246	0.0245618
$787$	−4463.12	−0.202151	−0.101076	−	0.994879i	$-0.532228\pi$
$787$	−4463.12	−0.202151	−0.101076	+	0.994879i	$0.532228\pi$
$788$	−37325.2	−1.68738
$789$	−264.604	−0.0119394
$790$	−11341.0	−0.510754
$791$	−10340.3	−0.464804
$792$	1196.73	0.0536919
$793$	0	0
$794$	15501.7	0.692866
$795$	6364.37	0.283926
$796$	−21636.6	−0.963426
$797$	−34785.5	−1.54600	−0.773002	−	0.634404i	$-0.781246\pi$
$797$	−34785.5	−1.54600	−0.773002	+	0.634404i	$0.781246\pi$
$798$	8365.40	0.371093
$799$	−14948.2	−0.661866
$800$	−30053.0	−1.32817
$801$	−9634.84	−0.425007
$802$	−8236.09	−0.362627
$803$	−12568.2	−0.552330
$804$	−1543.72	−0.0677152
$805$	−2425.53	−0.106197
$806$	0	0
$807$	13582.8	0.592487
$808$	2642.25	0.115042
$809$	−10620.0	−0.461530	−0.230765	−	0.973010i	$-0.574123\pi$
$809$	−10620.0	−0.461530	−0.230765	+	0.973010i	$0.574123\pi$
$810$	−1018.63	−0.0441866
$811$	−5497.87	−0.238047	−0.119024	−	0.992891i	$-0.537976\pi$
$811$	−5497.87	−0.238047	−0.119024	+	0.992891i	$0.537976\pi$
$812$	9630.57	0.416215
$813$	24965.5	1.07697
$814$	43556.7	1.87551
$815$	−7144.34	−0.307062
$816$	−4761.66	−0.204279
$817$	12940.5	0.554139
$818$	−21420.5	−0.915587
$819$	0	0
$820$	−1539.55	−0.0655653
$821$	−21305.3	−0.905678	−0.452839	−	0.891592i	$-0.649589\pi$
$821$	−21305.3	−0.905678	−0.452839	+	0.891592i	$0.649589\pi$
$822$	2550.48	0.108222
$823$	17342.6	0.734537	0.367268	−	0.930115i	$-0.380293\pi$
$823$	17342.6	0.734537	0.367268	+	0.930115i	$0.380293\pi$
$824$	2857.94	0.120827
$825$	−11193.7	−0.472381
$826$	18091.0	0.762064
$827$	−5129.96	−0.215703	−0.107851	−	0.994167i	$-0.534397\pi$
$827$	−5129.96	−0.215703	−0.107851	+	0.994167i	$0.534397\pi$
$828$	−9638.19	−0.404529
$829$	−8471.81	−0.354931	−0.177466	−	0.984127i	$-0.556790\pi$
$829$	−8471.81	−0.354931	−0.177466	+	0.984127i	$0.556790\pi$
$830$	8640.72	0.361354
$831$	8643.93	0.360836
$832$	0	0
$833$	8609.50	0.358105
$834$	−1236.93	−0.0513566
$835$	122.225	0.00506561
$836$	−29371.5	−1.21511
$837$	−1027.90	−0.0424486
$838$	28128.3	1.15952
$839$	19155.0	0.788207	0.394103	−	0.919066i	$-0.371055\pi$
$839$	19155.0	0.788207	0.394103	+	0.919066i	$0.371055\pi$
$840$	252.140	0.0103567
$841$	1246.67	0.0511160
$842$	31078.7	1.27202
$843$	−8453.98	−0.345398
$844$	34819.7	1.42007
$845$	0	0
$846$	−19221.3	−0.781136
$847$	−1944.42	−0.0788797
$848$	38255.0	1.54915
$849$	−793.804	−0.0320887
$850$	−13766.4	−0.555512
$851$	−38977.3	−1.57006
$852$	8349.11	0.335723
$853$	18075.1	0.725532	0.362766	−	0.931880i	$-0.381832\pi$
$853$	18075.1	0.725532	0.362766	+	0.931880i	$0.381832\pi$
$854$	−19324.9	−0.774339
$855$	2777.82	0.111111
$856$	1672.64	0.0667871
$857$	−21054.6	−0.839219	−0.419609	−	0.907705i	$-0.637833\pi$
$857$	−21054.6	−0.839219	−0.419609	+	0.907705i	$0.637833\pi$
$858$	0	0
$859$	920.322	0.0365553	0.0182776	−	0.999833i	$-0.494182\pi$
$859$	920.322	0.0365553	0.0182776	+	0.999833i	$0.494182\pi$
$860$	3510.33	0.139188
$861$	1124.48	0.0445090
$862$	−55267.0	−2.18376
$863$	19427.5	0.766304	0.383152	−	0.923685i	$-0.374839\pi$
$863$	19427.5	0.766304	0.383152	+	0.923685i	$0.374839\pi$
$864$	−7013.40	−0.276158
$865$	5824.21	0.228935
$866$	73044.7	2.86623
$867$	12240.5	0.479482
$868$	2289.91	0.0895444
$869$	−29083.7	−1.13532
$870$	6040.55	0.235395
$871$	0	0
$872$	−1977.09	−0.0767808
$873$	−15790.3	−0.612168
$874$	49646.5	1.92142
$875$	−4906.44	−0.189563
$876$	−10522.2	−0.405836
$877$	−14872.2	−0.572632	−0.286316	−	0.958135i	$-0.592431\pi$
$877$	−14872.2	−0.572632	−0.286316	+	0.958135i	$0.592431\pi$
$878$	−29535.7	−1.13529
$879$	12877.6	0.494141
$880$	5410.04	0.207241
$881$	−12940.6	−0.494870	−0.247435	−	0.968905i	$-0.579588\pi$
$881$	−12940.6	−0.494870	−0.247435	+	0.968905i	$0.579588\pi$
$882$	11070.6	0.422637
$883$	−25585.5	−0.975108	−0.487554	−	0.873093i	$-0.662111\pi$
$883$	−25585.5	−0.975108	−0.487554	+	0.873093i	$0.662111\pi$
$884$	0	0
$885$	6007.30	0.228173
$886$	41928.8	1.58987
$887$	3716.46	0.140684	0.0703418	−	0.997523i	$-0.477591\pi$
$887$	3716.46	0.140684	0.0703418	+	0.997523i	$0.477591\pi$
$888$	4051.79	0.153118
$889$	−16675.7	−0.629118
$890$	13462.8	0.507049
$891$	−2612.25	−0.0982195
$892$	−25322.0	−0.950496
$893$	52416.7	1.96423
$894$	−4705.92	−0.176051
$895$	−1555.40	−0.0580910
$896$	3499.58	0.130483
$897$	0	0
$898$	−70680.3	−2.62654
$899$	6095.52	0.226137
$900$	−9371.47	−0.347091
$901$	20072.5	0.742187
$902$	−7457.57	−0.275288
$903$	−2563.93	−0.0944876
$904$	6379.29	0.234703
$905$	−6517.61	−0.239395
$906$	18769.8	0.688285
$907$	−12960.4	−0.474469	−0.237235	−	0.971452i	$-0.576241\pi$
$907$	−12960.4	−0.474469	−0.237235	+	0.971452i	$0.576241\pi$
$908$	40665.3	1.48626
$909$	−5767.56	−0.210449
$910$	0	0
$911$	−36607.1	−1.33134	−0.665668	−	0.746248i	$-0.731853\pi$
$911$	−36607.1	−1.33134	−0.665668	+	0.746248i	$0.731853\pi$
$912$	16697.0	0.606242
$913$	22158.8	0.803230
$914$	58099.3	2.10258
$915$	−6417.06	−0.231849
$916$	11747.4	0.423740
$917$	292.440	0.0105313
$918$	−3212.65	−0.115505
$919$	20356.3	0.730676	0.365338	−	0.930875i	$-0.380953\pi$
$919$	20356.3	0.730676	0.365338	+	0.930875i	$0.380953\pi$
$920$	1496.38	0.0536243
$921$	21078.8	0.754147
$922$	−12048.6	−0.430370
$923$	0	0
$924$	5819.43	0.207192
$925$	−37898.7	−1.34714
$926$	−8545.61	−0.303268
$927$	−6238.38	−0.221030
$928$	41589.8	1.47118
$929$	45069.7	1.59170	0.795849	−	0.605495i	$-0.207025\pi$
$929$	45069.7	1.59170	0.795849	+	0.605495i	$0.207025\pi$
$930$	1436.29	0.0506428
$931$	−30189.6	−1.06275
$932$	−30244.9	−1.06299
$933$	3399.62	0.119291
$934$	10984.7	0.384831
$935$	2838.65	0.0992876
$936$	0	0
$937$	−6771.10	−0.236075	−0.118037	−	0.993009i	$-0.537660\pi$
$937$	−6771.10	−0.236075	−0.118037	+	0.993009i	$0.537660\pi$
$938$	−1575.50	−0.0548420
$939$	15857.8	0.551119
$940$	14218.9	0.493372
$941$	−36690.7	−1.27108	−0.635538	−	0.772070i	$-0.719222\pi$
$941$	−36690.7	−1.27108	−0.635538	+	0.772070i	$0.719222\pi$
$942$	17937.4	0.620417
$943$	6673.51	0.230455
$944$	36108.8	1.24496
$945$	−550.376	−0.0189457
$946$	17004.0	0.584406
$947$	50861.2	1.74527	0.872634	−	0.488375i	$-0.162410\pi$
$947$	50861.2	1.74527	0.872634	+	0.488375i	$0.162410\pi$
$948$	−24349.1	−0.834202
$949$	0	0
$950$	48272.6	1.64860
$951$	−14346.5	−0.489187
$952$	795.219	0.0270727
$953$	11855.6	0.402980	0.201490	−	0.979491i	$-0.435422\pi$
$953$	11855.6	0.402980	0.201490	+	0.979491i	$0.435422\pi$
$954$	25810.3	0.875931
$955$	12376.4	0.419361
$956$	42634.5	1.44236
$957$	15490.8	0.523245
$958$	−21525.5	−0.725947
$959$	1378.04	0.0464019
$960$	5773.76	0.194112
$961$	−28341.6	−0.951349
$962$	0	0
$963$	−3651.08	−0.122175
$964$	−43067.5	−1.43891
$965$	−2663.90	−0.0888643
$966$	−9836.56	−0.327625
$967$	−40661.7	−1.35221	−0.676107	−	0.736803i	$-0.736334\pi$
$967$	−40661.7	−1.35221	−0.676107	+	0.736803i	$0.736334\pi$
$968$	1199.58	0.0398304
$969$	8760.93	0.290445
$970$	22063.9	0.730339
$971$	57318.3	1.89437	0.947184	−	0.320690i	$-0.103915\pi$
$971$	57318.3	1.89437	0.947184	+	0.320690i	$0.103915\pi$
$972$	−2187.00	−0.0721688
$973$	−668.324	−0.0220200
$974$	−50403.3	−1.65814
$975$	0	0
$976$	−38571.7	−1.26501
$977$	3026.73	0.0991134	0.0495567	−	0.998771i	$-0.484219\pi$
$977$	3026.73	0.0991134	0.0495567	+	0.998771i	$0.484219\pi$
$978$	−28973.4	−0.947308
$979$	34524.8	1.12709
$980$	−8189.43	−0.266941
$981$	4315.64	0.140457
$982$	−81032.3	−2.63324
$983$	−33942.5	−1.10132	−0.550659	−	0.834730i	$-0.685624\pi$
$983$	−33942.5	−1.10132	−0.550659	+	0.834730i	$0.685624\pi$
$984$	−693.729	−0.0224749
$985$	−12649.3	−0.409179
$986$	19051.2	0.615327
$987$	−10385.4	−0.334925
$988$	0	0
$989$	−15216.3	−0.489231
$990$	3650.10	0.117179
$991$	21637.5	0.693580	0.346790	−	0.937943i	$-0.387272\pi$
$991$	21637.5	0.693580	0.346790	+	0.937943i	$0.387272\pi$
$992$	9889.02	0.316509
$993$	26009.9	0.831219
$994$	8520.95	0.271900
$995$	−7332.53	−0.233625
$996$	18551.6	0.590190
$997$	19624.6	0.623388	0.311694	−	0.950182i	$-0.399104\pi$
$997$	19624.6	0.623388	0.311694	+	0.950182i	$0.399104\pi$
$998$	−48296.3	−1.53186
$999$	−8844.33	−0.280102

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	507.4.a.k.1.1		4
3.2	odd	2		1521.4.a.z.1.4		4
13.5	odd	4		507.4.b.e.337.4		4
13.6	odd	12		39.4.j.b.10.2	yes	4
13.8	odd	4		507.4.b.e.337.1		4
13.11	odd	12		39.4.j.b.4.2	✓	4
13.12	even	2	inner	507.4.a.k.1.4		4
39.11	even	12		117.4.q.d.82.1		4
39.32	even	12		117.4.q.d.10.1		4
39.38	odd	2		1521.4.a.z.1.1		4
52.11	even	12		624.4.bv.c.433.2		4
52.19	even	12		624.4.bv.c.49.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
39.4.j.b.4.2	✓	4	13.11	odd	12
39.4.j.b.10.2	yes	4	13.6	odd	12
117.4.q.d.10.1		4	39.32	even	12
117.4.q.d.82.1		4	39.11	even	12
507.4.a.k.1.1		4	1.1	even	1	trivial
507.4.a.k.1.4		4	13.12	even	2	inner
507.4.b.e.337.1		4	13.8	odd	4
507.4.b.e.337.4		4	13.5	odd	4
624.4.bv.c.49.1		4	52.19	even	12
624.4.bv.c.433.2		4	52.11	even	12
1521.4.a.z.1.1		4	39.38	odd	2
1521.4.a.z.1.4		4	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 507 = 3 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	507.a (trivial)

\(f(q)\)	\(=\)	\(q-4.12311 q^{2} -3.00000 q^{3} +9.00000 q^{4} +3.05006 q^{5} +12.3693 q^{6} +6.68324 q^{7} -4.12311 q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q-4.12311 q^{2} -3.00000 q^{3} +9.00000 q^{4} +3.05006 q^{5} +12.3693 q^{6} +6.68324 q^{7} -4.12311 q^{8} +9.00000 q^{9} -12.5757 q^{10} -32.2500 q^{11} -27.0000 q^{12} -27.5557 q^{14} -9.15018 q^{15} -55.0000 q^{16} -28.8586 q^{17} -37.1080 q^{18} +101.194 q^{19} +27.4505 q^{20} -20.0497 q^{21} +132.970 q^{22} -118.990 q^{23} +12.3693 q^{24} -115.697 q^{25} -27.0000 q^{27} +60.1492 q^{28} +160.111 q^{29} +37.7271 q^{30} +38.0705 q^{31} +259.756 q^{32} +96.7499 q^{33} +118.987 q^{34} +20.3843 q^{35} +81.0000 q^{36} +327.568 q^{37} -417.233 q^{38} -12.5757 q^{40} -56.0846 q^{41} +82.6671 q^{42} +127.879 q^{43} -290.250 q^{44} +27.4505 q^{45} +490.608 q^{46} +517.983 q^{47} +165.000 q^{48} -298.334 q^{49} +477.032 q^{50} +86.5757 q^{51} -695.546 q^{53} +111.324 q^{54} -98.3643 q^{55} -27.5557 q^{56} -303.581 q^{57} -660.156 q^{58} -656.523 q^{59} -82.3516 q^{60} +701.304 q^{61} -156.969 q^{62} +60.1492 q^{63} -631.000 q^{64} -398.910 q^{66} +57.1750 q^{67} -259.727 q^{68} +356.970 q^{69} -84.0465 q^{70} -309.226 q^{71} -37.1080 q^{72} +389.711 q^{73} -1350.60 q^{74} +347.091 q^{75} +910.744 q^{76} -215.534 q^{77} +901.820 q^{79} -167.753 q^{80} +81.0000 q^{81} +231.243 q^{82} -687.095 q^{83} -180.448 q^{84} -88.0203 q^{85} -527.257 q^{86} -480.334 q^{87} +132.970 q^{88} -1070.54 q^{89} -113.181 q^{90} -1070.91 q^{92} -114.211 q^{93} -2135.70 q^{94} +308.647 q^{95} -779.267 q^{96} -1754.48 q^{97} +1230.06 q^{98} -290.250 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 12 q^{3} + 36 q^{4} + 36 q^{9}+O(q^{10}) \) 4 * q - 12 * q^3 + 36 * q^4 + 36 * q^9	\( 4 q - 12 q^{3} + 36 q^{4} + 36 q^{9} - 136 q^{10} - 108 q^{12} + 204 q^{14} - 220 q^{16} - 144 q^{17} - 68 q^{22} - 276 q^{23} - 120 q^{25} - 108 q^{27} + 12 q^{29} + 408 q^{30} - 804 q^{35} + 324 q^{36} - 612 q^{38} - 136 q^{40} - 612 q^{42} + 940 q^{43} + 660 q^{48} + 692 q^{49} + 432 q^{51} - 2268 q^{53} + 892 q^{55} + 204 q^{56} + 320 q^{61} - 2856 q^{62} - 2524 q^{64} + 204 q^{66} - 1296 q^{68} + 828 q^{69} - 3060 q^{74} + 360 q^{75} - 2976 q^{77} + 8 q^{79} + 324 q^{81} + 68 q^{82} - 36 q^{87} - 68 q^{88} - 1224 q^{90} - 2484 q^{92} - 5372 q^{94} - 108 q^{95}+O(q^{100}) \) 4 * q - 12 * q^3 + 36 * q^4 + 36 * q^9 - 136 * q^10 - 108 * q^12 + 204 * q^14 - 220 * q^16 - 144 * q^17 - 68 * q^22 - 276 * q^23 - 120 * q^25 - 108 * q^27 + 12 * q^29 + 408 * q^30 - 804 * q^35 + 324 * q^36 - 612 * q^38 - 136 * q^40 - 612 * q^42 + 940 * q^43 + 660 * q^48 + 692 * q^49 + 432 * q^51 - 2268 * q^53 + 892 * q^55 + 204 * q^56 + 320 * q^61 - 2856 * q^62 - 2524 * q^64 + 204 * q^66 - 1296 * q^68 + 828 * q^69 - 3060 * q^74 + 360 * q^75 - 2976 * q^77 + 8 * q^79 + 324 * q^81 + 68 * q^82 - 36 * q^87 - 68 * q^88 - 1224 * q^90 - 2484 * q^92 - 5372 * q^94 - 108 * q^95

Introduction

L-functions

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