LMFDB - Embedded newform 1045.4.a.c.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1045,4,Mod(1,1045)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1045, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("1045.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1045 = 5 \cdot 11 \cdot 19 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1045.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:	$61.6569959560$
Analytic rank:	$1$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - x^{19} - 105 x^{18} + 103 x^{17} + 4500 x^{16} - 4345 x^{15} - 101844 x^{14} + 95592 x^{13} + \cdots + 150528 $ x^20 - x^19 - 105x^18 + 103x^17 + 4500x^16 - 4345x^15 - 101844x^14 + 95592x^13 + 1317797x^12 - 1160501x^11 - 9914845x^10 + 7570653x^9 + 42786958x^8 - 23777633x^7 - 102801526x^6 + 28436356x^5 + 122325928x^4 + 411232x^3 - 47350496x^2 - 4782848x + 150528
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$5.49185$ of defining polynomial
Character	$\chi$	$=$	1045.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-5.49185 q^{2} +3.98529 q^{3} +22.1605 q^{4} -5.00000 q^{5} -21.8866 q^{6} +23.4740 q^{7} -77.7672 q^{8} -11.1175 q^{9} +O(q^{10})$	\(q-5.49185 q^{2} +3.98529 q^{3} +22.1605 q^{4} -5.00000 q^{5} -21.8866 q^{6} +23.4740 q^{7} -77.7672 q^{8} -11.1175 q^{9} +27.4593 q^{10} -11.0000 q^{11} +88.3159 q^{12} -8.78884 q^{13} -128.916 q^{14} -19.9265 q^{15} +249.802 q^{16} -67.5364 q^{17} +61.0555 q^{18} +19.0000 q^{19} -110.802 q^{20} +93.5508 q^{21} +60.4104 q^{22} +110.224 q^{23} -309.925 q^{24} +25.0000 q^{25} +48.2670 q^{26} -151.909 q^{27} +520.195 q^{28} +203.695 q^{29} +109.433 q^{30} -239.344 q^{31} -749.740 q^{32} -43.8382 q^{33} +370.900 q^{34} -117.370 q^{35} -246.368 q^{36} +315.815 q^{37} -104.345 q^{38} -35.0261 q^{39} +388.836 q^{40} +320.047 q^{41} -513.767 q^{42} -181.836 q^{43} -243.765 q^{44} +55.5873 q^{45} -605.334 q^{46} -473.487 q^{47} +995.535 q^{48} +208.029 q^{49} -137.296 q^{50} -269.152 q^{51} -194.765 q^{52} -21.6858 q^{53} +834.263 q^{54} +55.0000 q^{55} -1825.51 q^{56} +75.7205 q^{57} -1118.66 q^{58} -839.431 q^{59} -441.579 q^{60} -527.019 q^{61} +1314.44 q^{62} -260.971 q^{63} +2119.05 q^{64} +43.9442 q^{65} +240.753 q^{66} +63.2738 q^{67} -1496.64 q^{68} +439.274 q^{69} +644.579 q^{70} -717.930 q^{71} +864.573 q^{72} +787.870 q^{73} -1734.41 q^{74} +99.6323 q^{75} +421.049 q^{76} -258.214 q^{77} +192.358 q^{78} +337.557 q^{79} -1249.01 q^{80} -305.231 q^{81} -1757.65 q^{82} -676.463 q^{83} +2073.13 q^{84} +337.682 q^{85} +998.615 q^{86} +811.782 q^{87} +855.439 q^{88} -669.037 q^{89} -305.277 q^{90} -206.309 q^{91} +2442.61 q^{92} -953.853 q^{93} +2600.32 q^{94} -95.0000 q^{95} -2987.93 q^{96} +1410.86 q^{97} -1142.47 q^{98} +122.292 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q - q^{2} - 8 q^{3} + 51 q^{4} - 100 q^{5} - 54 q^{6} + 49 q^{7} + 9 q^{8} + 146 q^{9}+O(q^{10}) $ 20 * q - q^2 - 8 * q^3 + 51 * q^4 - 100 * q^5 - 54 * q^6 + 49 * q^7 + 9 * q^8 + 146 * q^9	\( 20 q - q^{2} - 8 q^{3} + 51 q^{4} - 100 q^{5} - 54 q^{6} + 49 q^{7} + 9 q^{8} + 146 q^{9} + 5 q^{10} - 220 q^{11} - 59 q^{12} + 60 q^{13} - 89 q^{14} + 40 q^{15} + 275 q^{16} - 155 q^{17} + 45 q^{18} + 380 q^{19} - 255 q^{20} + 105 q^{21} + 11 q^{22} - 154 q^{23} - 397 q^{24} + 500 q^{25} + 176 q^{26} - 206 q^{27} + 155 q^{28} - 305 q^{29} + 270 q^{30} - 759 q^{31} - 254 q^{32} + 88 q^{33} - 565 q^{34} - 245 q^{35} + 705 q^{36} + 698 q^{37} - 19 q^{38} - 758 q^{39} - 45 q^{40} + 547 q^{41} + 106 q^{42} - 925 q^{43} - 561 q^{44} - 730 q^{45} - 254 q^{46} - 681 q^{47} - 540 q^{48} + 213 q^{49} - 25 q^{50} - 899 q^{51} + 889 q^{52} - 419 q^{53} - 2241 q^{54} + 1100 q^{55} - 2473 q^{56} - 152 q^{57} - 1440 q^{58} - 2829 q^{59} + 295 q^{60} - 959 q^{61} + 1575 q^{62} - 426 q^{63} + 93 q^{64} - 300 q^{65} + 594 q^{66} - 1020 q^{67} - 4218 q^{68} - 572 q^{69} + 445 q^{70} + 106 q^{71} + 210 q^{72} + 558 q^{73} - 3439 q^{74} - 200 q^{75} + 969 q^{76} - 539 q^{77} - 3599 q^{78} + 536 q^{79} - 1375 q^{80} - 2128 q^{81} - 1255 q^{82} - 4179 q^{83} - 2024 q^{84} + 775 q^{85} - 1119 q^{86} - 557 q^{87} - 99 q^{88} - 4120 q^{89} - 225 q^{90} - 111 q^{91} - 2831 q^{92} + 801 q^{93} + 1213 q^{94} - 1900 q^{95} - 6147 q^{96} + 1414 q^{97} - 7869 q^{98} - 1606 q^{99}+O(q^{100}) \) 20 * q - q^2 - 8 * q^3 + 51 * q^4 - 100 * q^5 - 54 * q^6 + 49 * q^7 + 9 * q^8 + 146 * q^9 + 5 * q^10 - 220 * q^11 - 59 * q^12 + 60 * q^13 - 89 * q^14 + 40 * q^15 + 275 * q^16 - 155 * q^17 + 45 * q^18 + 380 * q^19 - 255 * q^20 + 105 * q^21 + 11 * q^22 - 154 * q^23 - 397 * q^24 + 500 * q^25 + 176 * q^26 - 206 * q^27 + 155 * q^28 - 305 * q^29 + 270 * q^30 - 759 * q^31 - 254 * q^32 + 88 * q^33 - 565 * q^34 - 245 * q^35 + 705 * q^36 + 698 * q^37 - 19 * q^38 - 758 * q^39 - 45 * q^40 + 547 * q^41 + 106 * q^42 - 925 * q^43 - 561 * q^44 - 730 * q^45 - 254 * q^46 - 681 * q^47 - 540 * q^48 + 213 * q^49 - 25 * q^50 - 899 * q^51 + 889 * q^52 - 419 * q^53 - 2241 * q^54 + 1100 * q^55 - 2473 * q^56 - 152 * q^57 - 1440 * q^58 - 2829 * q^59 + 295 * q^60 - 959 * q^61 + 1575 * q^62 - 426 * q^63 + 93 * q^64 - 300 * q^65 + 594 * q^66 - 1020 * q^67 - 4218 * q^68 - 572 * q^69 + 445 * q^70 + 106 * q^71 + 210 * q^72 + 558 * q^73 - 3439 * q^74 - 200 * q^75 + 969 * q^76 - 539 * q^77 - 3599 * q^78 + 536 * q^79 - 1375 * q^80 - 2128 * q^81 - 1255 * q^82 - 4179 * q^83 - 2024 * q^84 + 775 * q^85 - 1119 * q^86 - 557 * q^87 - 99 * q^88 - 4120 * q^89 - 225 * q^90 - 111 * q^91 - 2831 * q^92 + 801 * q^93 + 1213 * q^94 - 1900 * q^95 - 6147 * q^96 + 1414 * q^97 - 7869 * q^98 - 1606 * 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 $

$2$	−5.49185	−1.94166	−0.970832	−	0.239762i	$-0.922931\pi$
$2$	−5.49185	−1.94166	−0.970832	+	0.239762i	$0.922931\pi$
$3$	3.98529	0.766970	0.383485	−	0.923547i	$-0.374724\pi$
$3$	3.98529	0.766970	0.383485	+	0.923547i	$0.374724\pi$
$4$	22.1605	2.77006
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	−21.8866	−1.48920
$7$	23.4740	1.26748	0.633739	−	0.773547i	$-0.281519\pi$
$7$	23.4740	1.26748	0.633739	+	0.773547i	$0.281519\pi$
$8$	−77.7672	−3.43686
$9$	−11.1175	−0.411758
$10$	27.4593	0.868338
$11$	−11.0000	−0.301511
$11$	−11.0000	−0.301511
$12$	88.3159	2.12455
$13$	−8.78884	−0.187507	−0.0937533	−	0.995595i	$-0.529886\pi$
$13$	−8.78884	−0.187507	−0.0937533	+	0.995595i	$0.529886\pi$
$14$	−128.916	−2.46102
$15$	−19.9265	−0.342999
$16$	249.802	3.90316
$17$	−67.5364	−0.963528	−0.481764	−	0.876301i	$-0.660004\pi$
$17$	−67.5364	−0.963528	−0.481764	+	0.876301i	$0.660004\pi$
$18$	61.0555	0.799495
$19$	19.0000	0.229416
$19$	19.0000	0.229416
$20$	−110.802	−1.23881
$21$	93.5508	0.972117
$22$	60.4104	0.585434
$23$	110.224	0.999273	0.499636	−	0.866235i	$-0.333467\pi$
$23$	110.224	0.999273	0.499636	+	0.866235i	$0.333467\pi$
$24$	−309.925	−2.63596
$25$	25.0000	0.200000
$26$	48.2670	0.364075
$27$	−151.909	−1.08278
$28$	520.195	3.51099
$29$	203.695	1.30431	0.652157	−	0.758084i	$-0.273864\pi$
$29$	203.695	1.30431	0.652157	+	0.758084i	$0.273864\pi$
$30$	109.433	0.665989
$31$	−239.344	−1.38669	−0.693345	−	0.720606i	$-0.743864\pi$
$31$	−239.344	−1.38669	−0.693345	+	0.720606i	$0.743864\pi$
$32$	−749.740	−4.14177
$33$	−43.8382	−0.231250
$34$	370.900	1.87085
$35$	−117.370	−0.566833
$36$	−246.368	−1.14059
$37$	315.815	1.40323	0.701617	−	0.712554i	$-0.252462\pi$
$37$	315.815	1.40323	0.701617	+	0.712554i	$0.252462\pi$
$38$	−104.345	−0.445448
$39$	−35.0261	−0.143812
$40$	388.836	1.53701
$41$	320.047	1.21910	0.609548	−	0.792749i	$-0.291351\pi$
$41$	320.047	1.21910	0.609548	+	0.792749i	$0.291351\pi$
$42$	−513.767	−1.88752
$43$	−181.836	−0.644876	−0.322438	−	0.946591i	$-0.604502\pi$
$43$	−181.836	−0.644876	−0.322438	+	0.946591i	$0.604502\pi$
$44$	−243.765	−0.835204
$45$	55.5873	0.184144
$46$	−605.334	−1.94025
$47$	−473.487	−1.46947	−0.734735	−	0.678354i	$-0.762694\pi$
$47$	−473.487	−1.46947	−0.734735	+	0.678354i	$0.762694\pi$
$48$	995.535	2.99360
$49$	208.029	0.606500
$50$	−137.296	−0.388333
$51$	−269.152	−0.738997
$52$	−194.765	−0.519404
$53$	−21.6858	−0.0562033	−0.0281016	−	0.999605i	$-0.508946\pi$
$53$	−21.6858	−0.0562033	−0.0281016	+	0.999605i	$0.508946\pi$
$54$	834.263	2.10239
$55$	55.0000	0.134840
$56$	−1825.51	−4.35614
$57$	75.7205	0.175955
$58$	−1118.66	−2.53254
$59$	−839.431	−1.85228	−0.926141	−	0.377177i	$-0.876895\pi$
$59$	−839.431	−1.85228	−0.926141	+	0.377177i	$0.876895\pi$
$60$	−441.579	−0.950127
$61$	−527.019	−1.10619	−0.553097	−	0.833117i	$-0.686554\pi$
$61$	−527.019	−1.10619	−0.553097	+	0.833117i	$0.686554\pi$
$62$	1314.44	2.69248
$63$	−260.971	−0.521894
$64$	2119.05	4.13876
$65$	43.9442	0.0838555
$66$	240.753	0.449010
$67$	63.2738	0.115375	0.0576875	−	0.998335i	$-0.481627\pi$
$67$	63.2738	0.115375	0.0576875	+	0.998335i	$0.481627\pi$
$68$	−1496.64	−2.66903
$69$	439.274	0.766412
$70$	644.579	1.10060
$71$	−717.930	−1.20004	−0.600019	−	0.799986i	$-0.704840\pi$
$71$	−717.930	−1.20004	−0.600019	+	0.799986i	$0.704840\pi$
$72$	864.573	1.41515
$73$	787.870	1.26320	0.631598	−	0.775296i	$-0.282400\pi$
$73$	787.870	1.26320	0.631598	+	0.775296i	$0.282400\pi$
$74$	−1734.41	−2.72461
$75$	99.6323	0.153394
$76$	421.049	0.635495
$77$	−258.214	−0.382159
$78$	192.358	0.279234
$79$	337.557	0.480736	0.240368	−	0.970682i	$-0.422732\pi$
$79$	337.557	0.480736	0.240368	+	0.970682i	$0.422732\pi$
$80$	−1249.01	−1.74555
$81$	−305.231	−0.418698
$82$	−1757.65	−2.36707
$83$	−676.463	−0.894597	−0.447298	−	0.894385i	$-0.647614\pi$
$83$	−676.463	−0.894597	−0.447298	+	0.894385i	$0.647614\pi$
$84$	2073.13	2.69282
$85$	337.682	0.430903
$86$	998.615	1.25213
$87$	811.782	1.00037
$88$	855.439	1.03625
$89$	−669.037	−0.796829	−0.398414	−	0.917206i	$-0.630439\pi$
$89$	−669.037	−0.796829	−0.398414	+	0.917206i	$0.630439\pi$
$90$	−305.277	−0.357545
$91$	−206.309	−0.237660
$92$	2442.61	2.76804
$93$	−953.853	−1.06355
$94$	2600.32	2.85322
$95$	−95.0000	−0.102598
$96$	−2987.93	−3.17661
$97$	1410.86	1.47681	0.738406	−	0.674356i	$-0.235579\pi$
$97$	1410.86	1.47681	0.738406	+	0.674356i	$0.235579\pi$
$98$	−1142.47	−1.17762
$99$	122.292	0.124150
$100$	554.011	0.554011
$101$	−1383.84	−1.36334	−0.681672	−	0.731658i	$-0.738747\pi$
$101$	−1383.84	−1.36334	−0.681672	+	0.731658i	$0.738747\pi$
$102$	1478.14	1.43488
$103$	−51.8532	−0.0496043	−0.0248022	−	0.999692i	$-0.507896\pi$
$103$	−51.8532	−0.0496043	−0.0248022	+	0.999692i	$0.507896\pi$
$104$	683.483	0.644433
$105$	−467.754	−0.434744
$106$	119.095	0.109128
$107$	−216.278	−0.195405	−0.0977026	−	0.995216i	$-0.531149\pi$
$107$	−216.278	−0.195405	−0.0977026	+	0.995216i	$0.531149\pi$
$108$	−3366.38	−2.99935
$109$	1098.50	0.965296	0.482648	−	0.875814i	$-0.339675\pi$
$109$	1098.50	0.965296	0.482648	+	0.875814i	$0.339675\pi$
$110$	−302.052	−0.261814
$111$	1258.61	1.07624
$112$	5863.86	4.94717
$113$	−237.561	−0.197768	−0.0988842	−	0.995099i	$-0.531527\pi$
$113$	−237.561	−0.197768	−0.0988842	+	0.995099i	$0.531527\pi$
$114$	−415.846	−0.341645
$115$	−551.120	−0.446888
$116$	4513.96	3.61303
$117$	97.7095	0.0772073
$118$	4610.03	3.59651
$119$	−1585.35	−1.22125
$120$	1549.62	1.17884
$121$	121.000	0.0909091
$122$	2894.31	2.14786
$123$	1275.48	0.935009
$124$	−5303.96	−3.84121
$125$	−125.000	−0.0894427
$126$	1433.22	1.01334
$127$	1617.06	1.12985	0.564925	−	0.825143i	$-0.308905\pi$
$127$	1617.06	1.12985	0.564925	+	0.825143i	$0.308905\pi$
$128$	−5639.56	−3.89431
$129$	−724.668	−0.494600
$130$	−241.335	−0.162819
$131$	1540.60	1.02750	0.513750	−	0.857940i	$-0.328256\pi$
$131$	1540.60	1.02750	0.513750	+	0.857940i	$0.328256\pi$
$132$	−971.475	−0.640576
$133$	446.006	0.290779
$134$	−347.491	−0.224020
$135$	759.546	0.484232
$136$	5252.11	3.31151
$137$	246.055	0.153445	0.0767224	−	0.997052i	$-0.475554\pi$
$137$	246.055	0.153445	0.0767224	+	0.997052i	$0.475554\pi$
$138$	−2412.43	−1.48811
$139$	−1628.12	−0.993494	−0.496747	−	0.867895i	$-0.665472\pi$
$139$	−1628.12	−0.993494	−0.496747	+	0.867895i	$0.665472\pi$
$140$	−2600.97	−1.57016
$141$	−1886.98	−1.12704
$142$	3942.77	2.33007
$143$	96.6772	0.0565353
$144$	−2777.17	−1.60716
$145$	−1018.47	−0.583307
$146$	−4326.87	−2.45270
$147$	829.057	0.465167
$148$	6998.60	3.88704
$149$	−1966.37	−1.08115	−0.540575	−	0.841296i	$-0.681793\pi$
$149$	−1966.37	−1.08115	−0.540575	+	0.841296i	$0.681793\pi$
$150$	−547.166	−0.297839
$151$	−1723.70	−0.928959	−0.464479	−	0.885584i	$-0.653758\pi$
$151$	−1723.70	−0.928959	−0.464479	+	0.885584i	$0.653758\pi$
$152$	−1477.58	−0.788469
$153$	750.833	0.396740
$154$	1418.07	0.742024
$155$	1196.72	0.620146
$156$	−776.194	−0.398367
$157$	−1262.72	−0.641885	−0.320943	−	0.947099i	$-0.604000\pi$
$157$	−1262.72	−0.641885	−0.320943	+	0.947099i	$0.604000\pi$
$158$	−1853.82	−0.933428
$159$	−86.4242	−0.0431062
$160$	3748.70	1.85225
$161$	2587.40	1.26656
$162$	1676.28	0.812970
$163$	−863.324	−0.414851	−0.207426	−	0.978251i	$-0.566509\pi$
$163$	−863.324	−0.414851	−0.207426	+	0.978251i	$0.566509\pi$
$164$	7092.39	3.37697
$165$	219.191	0.103418
$166$	3715.04	1.73701
$167$	3683.95	1.70702	0.853510	−	0.521076i	$-0.174469\pi$
$167$	3683.95	1.70702	0.853510	+	0.521076i	$0.174469\pi$
$168$	−7275.18	−3.34102
$169$	−2119.76	−0.964841
$170$	−1854.50	−0.836668
$171$	−211.232	−0.0944637
$172$	−4029.56	−1.78634
$173$	1622.83	0.713190	0.356595	−	0.934259i	$-0.383938\pi$
$173$	1622.83	0.713190	0.356595	+	0.934259i	$0.383938\pi$
$174$	−4458.19	−1.94238
$175$	586.850	0.253496
$176$	−2747.82	−1.17685
$177$	−3345.38	−1.42064
$178$	3674.25	1.54717
$179$	324.415	0.135463	0.0677317	−	0.997704i	$-0.478424\pi$
$179$	324.415	0.135463	0.0677317	+	0.997704i	$0.478424\pi$
$180$	1231.84	0.510089
$181$	−4009.21	−1.64642	−0.823210	−	0.567736i	$-0.807819\pi$
$181$	−4009.21	−1.64642	−0.823210	+	0.567736i	$0.807819\pi$
$182$	1133.02	0.461456
$183$	−2100.32	−0.848418
$184$	−8571.80	−3.43436
$185$	−1579.07	−0.627545
$186$	5238.42	2.06505
$187$	742.900	0.290515
$188$	−10492.7	−4.07052
$189$	−3565.92	−1.37239
$190$	521.726	0.199210
$191$	−1317.95	−0.499285	−0.249642	−	0.968338i	$-0.580313\pi$
$191$	−1317.95	−0.499285	−0.249642	+	0.968338i	$0.580313\pi$
$192$	8445.01	3.17430
$193$	−3068.38	−1.14439	−0.572195	−	0.820118i	$-0.693908\pi$
$193$	−3068.38	−1.14439	−0.572195	+	0.820118i	$0.693908\pi$
$194$	−7748.22	−2.86747
$195$	175.130	0.0643146
$196$	4610.03	1.68004
$197$	−3281.14	−1.18666	−0.593329	−	0.804960i	$-0.702187\pi$
$197$	−3281.14	−1.18666	−0.593329	+	0.804960i	$0.702187\pi$
$198$	−671.610	−0.241057
$199$	−198.822	−0.0708247	−0.0354123	−	0.999373i	$-0.511274\pi$
$199$	−198.822	−0.0708247	−0.0354123	+	0.999373i	$0.511274\pi$
$200$	−1944.18	−0.687371
$201$	252.165	0.0884892
$202$	7599.87	2.64715
$203$	4781.53	1.65319
$204$	−5964.53	−2.04706
$205$	−1600.23	−0.545196
$206$	284.770	0.0963150
$207$	−1225.41	−0.411458
$208$	−2195.47	−0.731868
$209$	−209.000	−0.0691714
$210$	2568.84	0.844126
$211$	−3515.65	−1.14705	−0.573525	−	0.819188i	$-0.694424\pi$
$211$	−3515.65	−1.14705	−0.573525	+	0.819188i	$0.694424\pi$
$212$	−480.567	−0.155686
$213$	−2861.16	−0.920392
$214$	1187.77	0.379411
$215$	909.178	0.288397
$216$	11813.5	3.72134
$217$	−5618.35	−1.75760
$218$	−6032.80	−1.87428
$219$	3139.89	0.968832
$220$	1218.83	0.373514
$221$	593.566	0.180668
$222$	−6912.12	−2.08969
$223$	−5407.62	−1.62386	−0.811930	−	0.583755i	$-0.801583\pi$
$223$	−5407.62	−1.62386	−0.811930	+	0.583755i	$0.801583\pi$
$224$	−17599.4	−5.24960
$225$	−277.936	−0.0823516
$226$	1304.65	0.384000
$227$	−3899.47	−1.14016	−0.570081	−	0.821589i	$-0.693088\pi$
$227$	−3899.47	−1.14016	−0.570081	+	0.821589i	$0.693088\pi$
$228$	1678.00	0.487405
$229$	−2026.15	−0.584680	−0.292340	−	0.956314i	$-0.594434\pi$
$229$	−2026.15	−0.584680	−0.292340	+	0.956314i	$0.594434\pi$
$230$	3026.67	0.867707
$231$	−1029.06	−0.293104
$232$	−15840.7	−4.48274
$233$	4071.30	1.14472	0.572360	−	0.820002i	$-0.306028\pi$
$233$	4071.30	1.14472	0.572360	+	0.820002i	$0.306028\pi$
$234$	−536.606	−0.149911
$235$	2367.43	0.657167
$236$	−18602.2	−5.13093
$237$	1345.26	0.368710
$238$	8706.51	2.37126
$239$	−4905.19	−1.32757	−0.663787	−	0.747921i	$-0.731052\pi$
$239$	−4905.19	−1.32757	−0.663787	+	0.747921i	$0.731052\pi$
$240$	−4977.67	−1.33878
$241$	2265.09	0.605423	0.302712	−	0.953082i	$-0.402108\pi$
$241$	2265.09	0.605423	0.302712	+	0.953082i	$0.402108\pi$
$242$	−664.514	−0.176515
$243$	2885.11	0.761647
$244$	−11679.0	−3.06422
$245$	−1040.15	−0.271235
$246$	−7004.75	−1.81547
$247$	−166.988	−0.0430169
$248$	18613.1	4.76585
$249$	−2695.90	−0.686128
$250$	686.482	0.173668
$251$	−4935.28	−1.24108	−0.620542	−	0.784173i	$-0.713087\pi$
$251$	−4935.28	−1.24108	−0.620542	+	0.784173i	$0.713087\pi$
$252$	−5783.25	−1.44568
$253$	−1212.46	−0.301292
$254$	−8880.65	−2.19379
$255$	1345.76	0.330489
$256$	14019.3	3.42268
$257$	−3476.11	−0.843711	−0.421855	−	0.906663i	$-0.638621\pi$
$257$	−3476.11	−0.843711	−0.421855	+	0.906663i	$0.638621\pi$
$258$	3979.77	0.960347
$259$	7413.44	1.77857
$260$	973.823	0.232284
$261$	−2264.57	−0.537062
$262$	−8460.73	−1.99506
$263$	3603.01	0.844757	0.422379	−	0.906420i	$-0.361195\pi$
$263$	3603.01	0.844757	0.422379	+	0.906420i	$0.361195\pi$
$264$	3409.17	0.794773
$265$	108.429	0.0251349
$266$	−2449.40	−0.564596
$267$	−2666.31	−0.611143
$268$	1402.18	0.319596
$269$	2755.99	0.624669	0.312335	−	0.949972i	$-0.398889\pi$
$269$	2755.99	0.624669	0.312335	+	0.949972i	$0.398889\pi$
$270$	−4171.31	−0.940215
$271$	1038.47	0.232777	0.116388	−	0.993204i	$-0.462868\pi$
$271$	1038.47	0.232777	0.116388	+	0.993204i	$0.462868\pi$
$272$	−16870.7	−3.76080
$273$	−822.202	−0.182278
$274$	−1351.30	−0.297938
$275$	−275.000	−0.0603023
$276$	9734.52	2.12300
$277$	6546.14	1.41993	0.709963	−	0.704239i	$-0.248712\pi$
$277$	6546.14	1.41993	0.709963	+	0.704239i	$0.248712\pi$
$278$	8941.42	1.92903
$279$	2660.89	0.570980
$280$	9127.54	1.94812
$281$	3697.77	0.785020	0.392510	−	0.919748i	$-0.371607\pi$
$281$	3697.77	0.785020	0.392510	+	0.919748i	$0.371607\pi$
$282$	10363.0	2.18833
$283$	6192.80	1.30079	0.650395	−	0.759596i	$-0.274603\pi$
$283$	6192.80	1.30079	0.650395	+	0.759596i	$0.274603\pi$
$284$	−15909.7	−3.32417
$285$	−378.603	−0.0786894
$286$	−530.937	−0.109773
$287$	7512.79	1.54518
$288$	8335.21	1.70541
$289$	−351.839	−0.0716138
$290$	5593.30	1.13259
$291$	5622.68	1.13267
$292$	17459.6	3.49912
$293$	2436.53	0.485815	0.242907	−	0.970050i	$-0.421899\pi$
$293$	2436.53	0.485815	0.242907	+	0.970050i	$0.421899\pi$
$294$	−4553.06	−0.903197
$295$	4197.16	0.828366
$296$	−24560.0	−4.82271
$297$	1671.00	0.326469
$298$	10799.0	2.09923
$299$	−968.740	−0.187370
$300$	2207.90	0.424910
$301$	−4268.41	−0.817366
$302$	9466.31	1.80373
$303$	−5515.02	−1.04564
$304$	4746.24	0.895446
$305$	2635.10	0.494705
$306$	−4123.46	−0.770336
$307$	−9393.52	−1.74631	−0.873154	−	0.487445i	$-0.837929\pi$
$307$	−9393.52	−1.74631	−0.873154	+	0.487445i	$0.837929\pi$
$308$	−5722.14	−1.05860
$309$	−206.650	−0.0380450
$310$	−6572.20	−1.20412
$311$	−8860.09	−1.61547	−0.807733	−	0.589549i	$-0.799306\pi$
$311$	−8860.09	−1.61547	−0.807733	+	0.589549i	$0.799306\pi$
$312$	2723.88	0.494260
$313$	−6626.51	−1.19665	−0.598326	−	0.801253i	$-0.704167\pi$
$313$	−6626.51	−1.19665	−0.598326	+	0.801253i	$0.704167\pi$
$314$	6934.67	1.24633
$315$	1304.86	0.233398
$316$	7480.43	1.33167
$317$	5218.59	0.924622	0.462311	−	0.886718i	$-0.347020\pi$
$317$	5218.59	0.924622	0.462311	+	0.886718i	$0.347020\pi$
$318$	474.629	0.0836977
$319$	−2240.64	−0.393266
$320$	−10595.2	−1.85091
$321$	−861.930	−0.149870
$322$	−14209.6	−2.45923
$323$	−1283.19	−0.221048
$324$	−6764.05	−1.15982
$325$	−219.721	−0.0375013
$326$	4741.25	0.805502
$327$	4377.84	0.740353
$328$	−24889.1	−4.18986
$329$	−11114.6	−1.86252
$330$	−1203.76	−0.200803
$331$	1207.65	0.200539	0.100269	−	0.994960i	$-0.468030\pi$
$331$	1207.65	0.200539	0.100269	+	0.994960i	$0.468030\pi$
$332$	−14990.7	−2.47808
$333$	−3511.06	−0.577792
$334$	−20231.7	−3.31446
$335$	−316.369	−0.0515973
$336$	23369.2	3.79433
$337$	3389.35	0.547862	0.273931	−	0.961749i	$-0.411676\pi$
$337$	3389.35	0.547862	0.273931	+	0.961749i	$0.411676\pi$
$338$	11641.4	1.87340
$339$	−946.748	−0.151682
$340$	7483.18	1.19363
$341$	2632.78	0.418103
$342$	1160.05	0.183417
$343$	−3168.30	−0.498753
$344$	14140.8	2.21635
$345$	−2196.37	−0.342750
$346$	−8912.37	−1.38477
$347$	7849.30	1.21433	0.607165	−	0.794576i	$-0.292307\pi$
$347$	7849.30	1.21433	0.607165	+	0.794576i	$0.292307\pi$
$348$	17989.5	2.77108
$349$	−2085.00	−0.319793	−0.159896	−	0.987134i	$-0.551116\pi$
$349$	−2085.00	−0.319793	−0.159896	+	0.987134i	$0.551116\pi$
$350$	−3222.90	−0.492203
$351$	1335.10	0.203027
$352$	8247.14	1.24879
$353$	−6855.73	−1.03369	−0.516847	−	0.856078i	$-0.672894\pi$
$353$	−6855.73	−1.03369	−0.516847	+	0.856078i	$0.672894\pi$
$354$	18372.3	2.75841
$355$	3589.65	0.536673
$356$	−14826.2	−2.20726
$357$	−6318.08	−0.936662
$358$	−1781.64	−0.263024
$359$	−10187.2	−1.49766	−0.748830	−	0.662762i	$-0.769384\pi$
$359$	−10187.2	−1.49766	−0.748830	+	0.662762i	$0.769384\pi$
$360$	−4322.87	−0.632875
$361$	361.000	0.0526316
$362$	22018.0	3.19680
$363$	482.220	0.0697245
$364$	−4571.91	−0.658333
$365$	−3939.35	−0.564918
$366$	11534.7	1.64734
$367$	−4320.26	−0.614485	−0.307242	−	0.951631i	$-0.599406\pi$
$367$	−4320.26	−0.614485	−0.307242	+	0.951631i	$0.599406\pi$
$368$	27534.2	3.90032
$369$	−3558.11	−0.501972
$370$	8672.05	1.21848
$371$	−509.053	−0.0712364
$372$	−21137.8	−2.94609
$373$	−7662.67	−1.06369	−0.531847	−	0.846840i	$-0.678502\pi$
$373$	−7662.67	−1.06369	−0.531847	+	0.846840i	$0.678502\pi$
$374$	−4079.90	−0.564082
$375$	−498.161	−0.0685998
$376$	36821.7	5.05036
$377$	−1790.24	−0.244567
$378$	19583.5	2.66473
$379$	−3750.17	−0.508268	−0.254134	−	0.967169i	$-0.581790\pi$
$379$	−3750.17	−0.508268	−0.254134	+	0.967169i	$0.581790\pi$
$380$	−2105.24	−0.284202
$381$	6444.45	0.866560
$382$	7237.98	0.969443
$383$	2162.83	0.288552	0.144276	−	0.989538i	$-0.453915\pi$
$383$	2162.83	0.288552	0.144276	+	0.989538i	$0.453915\pi$
$384$	−22475.3	−2.98682
$385$	1291.07	0.170907
$386$	16851.1	2.22202
$387$	2021.55	0.265533
$388$	31265.2	4.09086
$389$	−999.503	−0.130275	−0.0651373	−	0.997876i	$-0.520749\pi$
$389$	−999.503	−0.130275	−0.0651373	+	0.997876i	$0.520749\pi$
$390$	−961.790	−0.124877
$391$	−7444.12	−0.962827
$392$	−16177.9	−2.08445
$393$	6139.73	0.788062
$394$	18019.5	2.30409
$395$	−1687.79	−0.214992
$396$	2710.05	0.343902
$397$	−10068.8	−1.27289	−0.636444	−	0.771323i	$-0.719595\pi$
$397$	−10068.8	−1.27289	−0.636444	+	0.771323i	$0.719595\pi$
$398$	1091.90	0.137518
$399$	1777.46	0.223019
$400$	6245.06	0.780632
$401$	13583.9	1.69165	0.845823	−	0.533464i	$-0.179110\pi$
$401$	13583.9	1.69165	0.845823	+	0.533464i	$0.179110\pi$
$402$	−1384.85	−0.171816
$403$	2103.55	0.260013
$404$	−30666.6	−3.77654
$405$	1526.15	0.187247
$406$	−26259.5	−3.20994
$407$	−3473.96	−0.423091
$408$	20931.2	2.53982
$409$	8845.82	1.06943	0.534716	−	0.845032i	$-0.320419\pi$
$409$	8845.82	1.06943	0.534716	+	0.845032i	$0.320419\pi$
$410$	8788.25	1.05859
$411$	980.602	0.117687
$412$	−1149.09	−0.137407
$413$	−19704.8	−2.34773
$414$	6729.77	0.798914
$415$	3382.32	0.400076
$416$	6589.34	0.776608
$417$	−6488.55	−0.761980
$418$	1147.80	0.134308
$419$	2614.86	0.304879	0.152439	−	0.988313i	$-0.451287\pi$
$419$	2614.86	0.304879	0.152439	+	0.988313i	$0.451287\pi$
$420$	−10365.6	−1.20427
$421$	12134.0	1.40469	0.702344	−	0.711838i	$-0.252137\pi$
$421$	12134.0	1.40469	0.702344	+	0.711838i	$0.252137\pi$
$422$	19307.4	2.22718
$423$	5263.97	0.605066
$424$	1686.44	0.193163
$425$	−1688.41	−0.192706
$426$	15713.1	1.78709
$427$	−12371.3	−1.40208
$428$	−4792.81	−0.541284
$429$	385.287	0.0433609
$430$	−4993.07	−0.559971
$431$	4377.45	0.489221	0.244611	−	0.969621i	$-0.421340\pi$
$431$	4377.45	0.489221	0.244611	+	0.969621i	$0.421340\pi$
$432$	−37947.2	−4.22624
$433$	−9922.42	−1.10125	−0.550625	−	0.834753i	$-0.685611\pi$
$433$	−9922.42	−1.10125	−0.550625	+	0.834753i	$0.685611\pi$
$434$	30855.2	3.41266
$435$	−4058.91	−0.447379
$436$	24343.3	2.67393
$437$	2094.25	0.229249
$438$	−17243.8	−1.88115
$439$	−14898.4	−1.61973	−0.809863	−	0.586619i	$-0.800459\pi$
$439$	−14898.4	−1.61973	−0.809863	+	0.586619i	$0.800459\pi$
$440$	−4277.19	−0.463426
$441$	−2312.76	−0.249731
$442$	−3259.78	−0.350796
$443$	4179.04	0.448199	0.224100	−	0.974566i	$-0.428056\pi$
$443$	4179.04	0.448199	0.224100	+	0.974566i	$0.428056\pi$
$444$	27891.5	2.98124
$445$	3345.18	0.356353
$446$	29697.8	3.15299
$447$	−7836.56	−0.829210
$448$	49742.5	5.24579
$449$	−373.753	−0.0392839	−0.0196420	−	0.999807i	$-0.506253\pi$
$449$	−373.753	−0.0392839	−0.0196420	+	0.999807i	$0.506253\pi$
$450$	1526.39	0.159899
$451$	−3520.52	−0.367571
$452$	−5264.45	−0.547830
$453$	−6869.45	−0.712483
$454$	21415.3	2.21381
$455$	1031.55	0.106285
$456$	−5888.57	−0.604732
$457$	10028.6	1.02652	0.513260	−	0.858233i	$-0.328438\pi$
$457$	10028.6	1.02652	0.513260	+	0.858233i	$0.328438\pi$
$458$	11127.3	1.13525
$459$	10259.4	1.04328
$460$	−12213.1	−1.23791
$461$	10757.3	1.08680	0.543401	−	0.839473i	$-0.317136\pi$
$461$	10757.3	1.08680	0.543401	+	0.839473i	$0.317136\pi$
$462$	5651.44	0.569110
$463$	−2248.94	−0.225739	−0.112869	−	0.993610i	$-0.536004\pi$
$463$	−2248.94	−0.225739	−0.112869	+	0.993610i	$0.536004\pi$
$464$	50883.4	5.09095
$465$	4769.27	0.475633
$466$	−22359.0	−2.22266
$467$	18757.8	1.85869	0.929343	−	0.369217i	$-0.120374\pi$
$467$	18757.8	1.85869	0.929343	+	0.369217i	$0.120374\pi$
$468$	2165.29	0.213869
$469$	1485.29	0.146235
$470$	−13001.6	−1.27600
$471$	−5032.30	−0.492306
$472$	65280.2	6.36603
$473$	2000.19	0.194437
$474$	−7387.99	−0.715911
$475$	475.000	0.0458831
$476$	−35132.1	−3.38293
$477$	241.091	0.0231421
$478$	26938.6	2.57770
$479$	−15736.1	−1.50104	−0.750521	−	0.660847i	$-0.770197\pi$
$479$	−15736.1	−1.50104	−0.750521	+	0.660847i	$0.770197\pi$
$480$	14939.7	1.42062
$481$	−2775.65	−0.263115
$482$	−12439.5	−1.17553
$483$	10311.5	0.971410
$484$	2681.42	0.251823
$485$	−7054.29	−0.660451
$486$	−15844.6	−1.47886
$487$	−12215.2	−1.13660	−0.568299	−	0.822822i	$-0.692398\pi$
$487$	−12215.2	−1.13660	−0.568299	+	0.822822i	$0.692398\pi$
$488$	40984.8	3.80183
$489$	−3440.60	−0.318178
$490$	5712.33	0.526647
$491$	1905.67	0.175157	0.0875783	−	0.996158i	$-0.472087\pi$
$491$	1905.67	0.175157	0.0875783	+	0.996158i	$0.472087\pi$
$492$	28265.2	2.59003
$493$	−13756.8	−1.25674
$494$	917.073	0.0835244
$495$	−611.460	−0.0555214
$496$	−59788.5	−5.41247
$497$	−16852.7	−1.52102
$498$	14805.5	1.33223
$499$	2816.83	0.252702	0.126351	−	0.991986i	$-0.459673\pi$
$499$	2816.83	0.252702	0.126351	+	0.991986i	$0.459673\pi$
$500$	−2770.06	−0.247761
$501$	14681.6	1.30923
$502$	27103.8	2.40977
$503$	−16316.7	−1.44638	−0.723188	−	0.690651i	$-0.757324\pi$
$503$	−16316.7	−1.44638	−0.723188	+	0.690651i	$0.757324\pi$
$504$	20295.0	1.79367
$505$	6919.22	0.609706
$506$	6658.67	0.585008
$507$	−8447.84	−0.740004
$508$	35834.8	3.12975
$509$	−22911.0	−1.99512	−0.997558	−	0.0698477i	$-0.977749\pi$
$509$	−22911.0	−1.99512	−0.997558	+	0.0698477i	$0.977749\pi$
$510$	−7390.72	−0.641699
$511$	18494.5	1.60107
$512$	−31875.5	−2.75139
$513$	−2886.27	−0.248406
$514$	19090.3	1.63820
$515$	259.266	0.0221837
$516$	−16059.0	−1.37007
$517$	5208.35	0.443062
$518$	−40713.5	−3.45338
$519$	6467.47	0.546995
$520$	−3417.41	−0.288199
$521$	5625.85	0.473077	0.236538	−	0.971622i	$-0.423987\pi$
$521$	5625.85	0.473077	0.236538	+	0.971622i	$0.423987\pi$
$522$	12436.7	1.04279
$523$	−15929.0	−1.33179	−0.665894	−	0.746046i	$-0.731950\pi$
$523$	−15929.0	−1.33179	−0.665894	+	0.746046i	$0.731950\pi$
$524$	34140.3	2.84624
$525$	2338.77	0.194423
$526$	−19787.2	−1.64023
$527$	16164.4	1.33611
$528$	−10950.9	−0.902606
$529$	−17.6871	−0.00145369
$530$	−595.476	−0.0488034
$531$	9332.35	0.762692
$532$	9883.70	0.805475
$533$	−2812.84	−0.228588
$534$	14643.0	1.18663
$535$	1081.39	0.0873879
$536$	−4920.63	−0.396528
$537$	1292.89	0.103896
$538$	−15135.5	−1.21290
$539$	−2288.32	−0.182867
$540$	16831.9	1.34135
$541$	5156.51	0.409789	0.204894	−	0.978784i	$-0.434315\pi$
$541$	5156.51	0.409789	0.204894	+	0.978784i	$0.434315\pi$
$542$	−5703.12	−0.451974
$543$	−15977.9	−1.26275
$544$	50634.7	3.99071
$545$	−5492.50	−0.431694
$546$	4515.42	0.353923
$547$	20142.0	1.57443	0.787213	−	0.616681i	$-0.211523\pi$
$547$	20142.0	1.57443	0.787213	+	0.616681i	$0.211523\pi$
$548$	5452.70	0.425051
$549$	5859.11	0.455484
$550$	1510.26	0.117087
$551$	3870.20	0.299230
$552$	−34161.1	−2.63405
$553$	7923.83	0.609323
$554$	−35950.5	−2.75702
$555$	−6293.07	−0.481308
$556$	−36080.0	−2.75203
$557$	−22951.9	−1.74597	−0.872984	−	0.487749i	$-0.837818\pi$
$557$	−22951.9	−1.74597	−0.872984	+	0.487749i	$0.837818\pi$
$558$	−14613.2	−1.10865
$559$	1598.12	0.120918
$560$	−29319.3	−2.21244
$561$	2960.67	0.222816
$562$	−20307.6	−1.52425
$563$	10996.3	0.823160	0.411580	−	0.911374i	$-0.364977\pi$
$563$	10996.3	0.823160	0.411580	+	0.911374i	$0.364977\pi$
$564$	−41816.4	−3.12196
$565$	1187.80	0.0884447
$566$	−34009.9	−2.52570
$567$	−7164.99	−0.530690
$568$	55831.4	4.12435
$569$	−8014.62	−0.590492	−0.295246	−	0.955421i	$-0.595402\pi$
$569$	−8014.62	−0.590492	−0.295246	+	0.955421i	$0.595402\pi$
$570$	2079.23	0.152788
$571$	3059.41	0.224224	0.112112	−	0.993696i	$-0.464238\pi$
$571$	3059.41	0.224224	0.112112	+	0.993696i	$0.464238\pi$
$572$	2142.41	0.156606
$573$	−5252.41	−0.382936
$574$	−41259.1	−3.00021
$575$	2755.60	0.199855
$576$	−23558.4	−1.70417
$577$	−12621.3	−0.910626	−0.455313	−	0.890331i	$-0.650473\pi$
$577$	−12621.3	−0.910626	−0.455313	+	0.890331i	$0.650473\pi$
$578$	1932.25	0.139050
$579$	−12228.4	−0.877712
$580$	−22569.8	−1.61579
$581$	−15879.3	−1.13388
$582$	−30878.9	−2.19926
$583$	238.544	0.0169459
$584$	−61270.4	−4.34142
$585$	−488.548	−0.0345281
$586$	−13381.1	−0.943289
$587$	−20179.0	−1.41887	−0.709436	−	0.704770i	$-0.751050\pi$
$587$	−20179.0	−1.41887	−0.709436	+	0.704770i	$0.751050\pi$
$588$	18372.3	1.28854
$589$	−4547.53	−0.318128
$590$	−23050.2	−1.60841
$591$	−13076.3	−0.910130
$592$	78891.3	5.47705
$593$	−7204.55	−0.498913	−0.249456	−	0.968386i	$-0.580252\pi$
$593$	−7204.55	−0.498913	−0.249456	+	0.968386i	$0.580252\pi$
$594$	−9176.89	−0.633893
$595$	7926.75	0.546160
$596$	−43575.7	−2.99485
$597$	−792.363	−0.0543204
$598$	5320.18	0.363810
$599$	−1822.69	−0.124329	−0.0621645	−	0.998066i	$-0.519800\pi$
$599$	−1822.69	−0.124329	−0.0621645	+	0.998066i	$0.519800\pi$
$600$	−7748.12	−0.527193
$601$	−14839.0	−1.00715	−0.503573	−	0.863953i	$-0.667982\pi$
$601$	−14839.0	−1.00715	−0.503573	+	0.863953i	$0.667982\pi$
$602$	23441.5	1.58705
$603$	−703.444	−0.0475066
$604$	−38198.0	−2.57327
$605$	−605.000	−0.0406558
$606$	30287.7	2.03029
$607$	−9115.94	−0.609562	−0.304781	−	0.952422i	$-0.598583\pi$
$607$	−9115.94	−0.609562	−0.304781	+	0.952422i	$0.598583\pi$
$608$	−14245.1	−0.950187
$609$	19055.8	1.26795
$610$	−14471.6	−0.960552
$611$	4161.40	0.275535
$612$	16638.8	1.09899
$613$	6548.39	0.431463	0.215732	−	0.976453i	$-0.430786\pi$
$613$	6548.39	0.431463	0.215732	+	0.976453i	$0.430786\pi$
$614$	51587.8	3.39074
$615$	−6377.40	−0.418149
$616$	20080.6	1.31342
$617$	2256.63	0.147242	0.0736212	−	0.997286i	$-0.476544\pi$
$617$	2256.63	0.147242	0.0736212	+	0.997286i	$0.476544\pi$
$618$	1134.89	0.0738706
$619$	−5141.30	−0.333839	−0.166919	−	0.985971i	$-0.553382\pi$
$619$	−5141.30	−0.333839	−0.166919	+	0.985971i	$0.553382\pi$
$620$	26519.8	1.71784
$621$	−16744.0	−1.08199
$622$	48658.3	3.13669
$623$	−15705.0	−1.00996
$624$	−8749.59	−0.561320
$625$	625.000	0.0400000
$626$	36391.8	2.32350
$627$	−832.926	−0.0530524
$628$	−27982.4	−1.77806
$629$	−21329.0	−1.35205
$630$	−7166.08	−0.453180
$631$	−13482.9	−0.850628	−0.425314	−	0.905046i	$-0.639836\pi$
$631$	−13482.9	−0.850628	−0.425314	+	0.905046i	$0.639836\pi$
$632$	−26250.9	−1.65222
$633$	−14010.9	−0.879752
$634$	−28659.7	−1.79530
$635$	−8085.30	−0.505284
$636$	−1915.20	−0.119407
$637$	−1828.34	−0.113723
$638$	12305.3	0.763590
$639$	7981.56	0.494125
$640$	28197.8	1.74159
$641$	3637.53	0.224140	0.112070	−	0.993700i	$-0.464252\pi$
$641$	3637.53	0.224140	0.112070	+	0.993700i	$0.464252\pi$
$642$	4733.59	0.290997
$643$	−3865.03	−0.237048	−0.118524	−	0.992951i	$-0.537816\pi$
$643$	−3865.03	−0.237048	−0.118524	+	0.992951i	$0.537816\pi$
$644$	57337.9	3.50843
$645$	3623.34	0.221192
$646$	7047.10	0.429202
$647$	−795.734	−0.0483517	−0.0241758	−	0.999708i	$-0.507696\pi$
$647$	−795.734	−0.0483517	−0.0241758	+	0.999708i	$0.507696\pi$
$648$	23736.9	1.43900
$649$	9233.75	0.558484
$650$	1206.68	0.0728149
$651$	−22390.8	−1.34802
$652$	−19131.7	−1.14916
$653$	15688.6	0.940186	0.470093	−	0.882617i	$-0.344220\pi$
$653$	15688.6	0.940186	0.470093	+	0.882617i	$0.344220\pi$
$654$	−24042.5	−1.43752
$655$	−7702.98	−0.459512
$656$	79948.4	4.75833
$657$	−8759.12	−0.520130
$658$	61039.9	3.61639
$659$	−12671.8	−0.749050	−0.374525	−	0.927217i	$-0.622194\pi$
$659$	−12671.8	−0.749050	−0.374525	+	0.927217i	$0.622194\pi$
$660$	4857.37	0.286474
$661$	13458.9	0.791967	0.395983	−	0.918258i	$-0.370404\pi$
$661$	13458.9	0.791967	0.395983	+	0.918258i	$0.370404\pi$
$662$	−6632.22	−0.389378
$663$	2365.53	0.138567
$664$	52606.6	3.07460
$665$	−2230.03	−0.130040
$666$	19282.2	1.12188
$667$	22452.0	1.30337
$668$	81638.0	4.72855
$669$	−21550.9	−1.24545
$670$	1737.45	0.100185
$671$	5797.21	0.333530
$672$	−70138.8	−4.02628
$673$	16193.8	0.927524	0.463762	−	0.885960i	$-0.346499\pi$
$673$	16193.8	0.927524	0.463762	+	0.885960i	$0.346499\pi$
$674$	−18613.8	−1.06376
$675$	−3797.73	−0.216555
$676$	−46974.8	−2.67267
$677$	17029.3	0.966749	0.483374	−	0.875414i	$-0.339411\pi$
$677$	17029.3	0.966749	0.483374	+	0.875414i	$0.339411\pi$
$678$	5199.40	0.294516
$679$	33118.5	1.87183
$680$	−26260.6	−1.48095
$681$	−15540.5	−0.874469
$682$	−14458.8	−0.811814
$683$	34294.4	1.92129	0.960643	−	0.277786i	$-0.0896008\pi$
$683$	34294.4	1.92129	0.960643	+	0.277786i	$0.0896008\pi$
$684$	−4680.99	−0.261670
$685$	−1230.28	−0.0686226
$686$	17399.9	0.968410
$687$	−8074.80	−0.448432
$688$	−45422.9	−2.51705
$689$	190.593	0.0105385
$690$	12062.2	0.665505
$691$	−16318.6	−0.898394	−0.449197	−	0.893433i	$-0.648290\pi$
$691$	−16318.6	−0.898394	−0.449197	+	0.893433i	$0.648290\pi$
$692$	35962.8	1.97558
$693$	2870.69	0.157357
$694$	−43107.2	−2.35782
$695$	8140.62	0.444304
$696$	−63130.0	−3.43813
$697$	−21614.8	−1.17463
$698$	11450.5	0.620930
$699$	16225.3	0.877966
$700$	13004.9	0.702197
$701$	30823.8	1.66077	0.830384	−	0.557192i	$-0.188121\pi$
$701$	30823.8	1.66077	0.830384	+	0.557192i	$0.188121\pi$
$702$	−7332.20	−0.394211
$703$	6000.48	0.321924
$704$	−23309.5	−1.24788
$705$	9434.91	0.504027
$706$	37650.7	2.00708
$707$	−32484.4	−1.72801
$708$	−74135.1	−3.93527
$709$	−9897.96	−0.524296	−0.262148	−	0.965028i	$-0.584431\pi$
$709$	−9897.96	−0.524296	−0.262148	+	0.965028i	$0.584431\pi$
$710$	−19713.8	−1.04204
$711$	−3752.78	−0.197947
$712$	52029.1	2.73858
$713$	−26381.4	−1.38568
$714$	34698.0	1.81868
$715$	−483.386	−0.0252834
$716$	7189.20	0.375241
$717$	−19548.6	−1.01821
$718$	55946.7	2.90795
$719$	15902.1	0.824824	0.412412	−	0.910997i	$-0.364686\pi$
$719$	15902.1	0.824824	0.412412	+	0.910997i	$0.364686\pi$
$720$	13885.8	0.718742
$721$	−1217.20	−0.0628724
$722$	−1982.56	−0.102193
$723$	9027.03	0.464341
$724$	−88845.9	−4.56068
$725$	5092.36	0.260863
$726$	−2648.28	−0.135382
$727$	29088.2	1.48394	0.741968	−	0.670435i	$-0.233893\pi$
$727$	29088.2	1.48394	0.741968	+	0.670435i	$0.233893\pi$
$728$	16044.1	0.816804
$729$	19739.2	1.00286
$730$	21634.3	1.09688
$731$	12280.5	0.621356
$732$	−46544.2	−2.35017
$733$	−22443.8	−1.13094	−0.565470	−	0.824769i	$-0.691305\pi$
$733$	−22443.8	−1.13094	−0.565470	+	0.824769i	$0.691305\pi$
$734$	23726.3	1.19312
$735$	−4145.29	−0.208029
$736$	−82639.3	−4.13876
$737$	−696.012	−0.0347869
$738$	19540.6	0.974661
$739$	212.662	0.0105858	0.00529289	−	0.999986i	$-0.498315\pi$
$739$	212.662	0.0105858	0.00529289	+	0.999986i	$0.498315\pi$
$740$	−34993.0	−1.73834
$741$	−665.495	−0.0329927
$742$	2795.64	0.138317
$743$	−2585.45	−0.127659	−0.0638297	−	0.997961i	$-0.520331\pi$
$743$	−2585.45	−0.127659	−0.0638297	+	0.997961i	$0.520331\pi$
$744$	74178.5	3.65526
$745$	9831.86	0.483505
$746$	42082.3	2.06534
$747$	7520.56	0.368357
$748$	16463.0	0.804742
$749$	−5076.91	−0.247672
$750$	2735.83	0.133198
$751$	36821.0	1.78910	0.894552	−	0.446964i	$-0.147495\pi$
$751$	36821.0	1.78910	0.894552	+	0.446964i	$0.147495\pi$
$752$	−118278.	−5.73558
$753$	−19668.5	−0.951873
$754$	9831.72	0.474868
$755$	8618.50	0.415443
$756$	−79022.4	−3.80161
$757$	−5075.69	−0.243698	−0.121849	−	0.992549i	$-0.538882\pi$
$757$	−5075.69	−0.243698	−0.121849	+	0.992549i	$0.538882\pi$
$758$	20595.4	0.986885
$759$	−4832.02	−0.231082
$760$	7387.88	0.352614
$761$	15128.7	0.720652	0.360326	−	0.932827i	$-0.382666\pi$
$761$	15128.7	0.720652	0.360326	+	0.932827i	$0.382666\pi$
$762$	−35392.0	−1.68257
$763$	25786.2	1.22349
$764$	−29206.3	−1.38305
$765$	−3754.16	−0.177428
$766$	−11877.9	−0.560270
$767$	7377.63	0.347315
$768$	55871.0	2.62509
$769$	−22435.1	−1.05206	−0.526028	−	0.850468i	$-0.676319\pi$
$769$	−22435.1	−1.05206	−0.526028	+	0.850468i	$0.676319\pi$
$770$	−7090.37	−0.331843
$771$	−13853.3	−0.647100
$772$	−67996.8	−3.17002
$773$	−31386.2	−1.46039	−0.730196	−	0.683237i	$-0.760571\pi$
$773$	−31386.2	−1.46039	−0.730196	+	0.683237i	$0.760571\pi$
$774$	−11102.1	−0.515575
$775$	−5983.59	−0.277338
$776$	−109718.	−5.07559
$777$	29544.7	1.36411
$778$	5489.13	0.252949
$779$	6080.89	0.279680
$780$	3880.97	0.178155
$781$	7897.23	0.361825
$782$	40882.0	1.86949
$783$	−30943.1	−1.41228
$784$	51966.2	2.36727
$785$	6313.60	0.287060
$786$	−33718.5	−1.53015
$787$	17511.8	0.793175	0.396587	−	0.917997i	$-0.370194\pi$
$787$	17511.8	0.793175	0.396587	+	0.917997i	$0.370194\pi$
$788$	−72711.6	−3.28711
$789$	14359.0	0.647903
$790$	9269.08	0.417442
$791$	−5576.50	−0.250667
$792$	−9510.31	−0.426684
$793$	4631.89	0.207419
$794$	55296.2	2.47152
$795$	432.121	0.0192777
$796$	−4405.99	−0.196188
$797$	−8596.69	−0.382071	−0.191035	−	0.981583i	$-0.561185\pi$
$797$	−8596.69	−0.382071	−0.191035	+	0.981583i	$0.561185\pi$
$798$	−9761.58	−0.433028
$799$	31977.6	1.41588
$800$	−18743.5	−0.828354
$801$	7437.99	0.328100
$802$	−74601.1	−3.28461
$803$	−8666.57	−0.380868
$804$	5588.08	0.245120
$805$	−12937.0	−0.566421
$806$	−11552.4	−0.504858
$807$	10983.4	0.479102
$808$	107618.	4.68562
$809$	34548.0	1.50141	0.750705	−	0.660637i	$-0.229714\pi$
$809$	34548.0	1.50141	0.750705	+	0.660637i	$0.229714\pi$
$810$	−8381.41	−0.363571
$811$	−7381.58	−0.319608	−0.159804	−	0.987149i	$-0.551086\pi$
$811$	−7381.58	−0.319608	−0.159804	+	0.987149i	$0.551086\pi$
$812$	105961.	4.57943
$813$	4138.60	0.178533
$814$	19078.5	0.821500
$815$	4316.62	0.185527
$816$	−67234.8	−2.88442
$817$	−3454.88	−0.147945
$818$	−48579.9	−2.07648
$819$	2293.64	0.0978585
$820$	−35461.9	−1.51022
$821$	12793.8	0.543856	0.271928	−	0.962318i	$-0.412339\pi$
$821$	12793.8	0.543856	0.271928	+	0.962318i	$0.412339\pi$
$822$	−5385.33	−0.228509
$823$	−30776.1	−1.30351	−0.651754	−	0.758430i	$-0.725967\pi$
$823$	−30776.1	−1.30351	−0.651754	+	0.758430i	$0.725967\pi$
$824$	4032.48	0.170483
$825$	−1095.95	−0.0462500
$826$	108216.	4.55850
$827$	44495.6	1.87094	0.935468	−	0.353413i	$-0.114979\pi$
$827$	44495.6	1.87094	0.935468	+	0.353413i	$0.114979\pi$
$828$	−27155.6	−1.13976
$829$	21307.1	0.892673	0.446337	−	0.894865i	$-0.352728\pi$
$829$	21307.1	0.892673	0.446337	+	0.894865i	$0.352728\pi$
$830$	−18575.2	−0.776812
$831$	26088.3	1.08904
$832$	−18623.9	−0.776044
$833$	−14049.5	−0.584379
$834$	35634.1	1.47951
$835$	−18419.7	−0.763403
$836$	−4631.54	−0.191609
$837$	36358.5	1.50147
$838$	−14360.4	−0.591971
$839$	−22987.8	−0.945920	−0.472960	−	0.881084i	$-0.656814\pi$
$839$	−22987.8	−0.945920	−0.472960	+	0.881084i	$0.656814\pi$
$840$	36375.9	1.49415
$841$	17102.5	0.701237
$842$	−66638.0	−2.72743
$843$	14736.7	0.602087
$844$	−77908.4	−3.17739
$845$	10598.8	0.431490
$846$	−28908.9	−1.17483
$847$	2840.36	0.115225
$848$	−5417.16	−0.219370
$849$	24680.1	0.997666
$850$	9272.50	0.374169
$851$	34810.4	1.40221
$852$	−63404.6	−2.54954
$853$	2854.66	0.114586	0.0572930	−	0.998357i	$-0.481753\pi$
$853$	2854.66	0.114586	0.0572930	+	0.998357i	$0.481753\pi$
$854$	67941.1	2.72236
$855$	1056.16	0.0422455
$856$	16819.3	0.671580
$857$	38412.9	1.53111	0.765554	−	0.643371i	$-0.222465\pi$
$857$	38412.9	1.53111	0.765554	+	0.643371i	$0.222465\pi$
$858$	−2115.94	−0.0841922
$859$	18619.0	0.739547	0.369774	−	0.929122i	$-0.379435\pi$
$859$	18619.0	0.739547	0.369774	+	0.929122i	$0.379435\pi$
$860$	20147.8	0.798877
$861$	29940.6	1.18510
$862$	−24040.3	−0.949903
$863$	38079.3	1.50201	0.751005	−	0.660296i	$-0.229569\pi$
$863$	38079.3	1.50201	0.751005	+	0.660296i	$0.229569\pi$
$864$	113892.	4.48460
$865$	−8114.17	−0.318948
$866$	54492.5	2.13826
$867$	−1402.18	−0.0549256
$868$	−124505.	−4.86865
$869$	−3713.13	−0.144947
$870$	22290.9	0.868659
$871$	−556.103	−0.0216336
$872$	−85427.3	−3.31758
$873$	−15685.1	−0.608089
$874$	−11501.3	−0.445124
$875$	−2934.25	−0.113367
$876$	69581.4	2.68372
$877$	3488.29	0.134312	0.0671558	−	0.997742i	$-0.478608\pi$
$877$	3488.29	0.134312	0.0671558	+	0.997742i	$0.478608\pi$
$878$	81819.6	3.14496
$879$	9710.29	0.372605
$880$	13739.1	0.526302
$881$	5509.45	0.210690	0.105345	−	0.994436i	$-0.466405\pi$
$881$	5509.45	0.210690	0.105345	+	0.994436i	$0.466405\pi$
$882$	12701.3	0.484893
$883$	−42291.1	−1.61179	−0.805894	−	0.592060i	$-0.798315\pi$
$883$	−42291.1	−1.61179	−0.805894	+	0.592060i	$0.798315\pi$
$884$	13153.7	0.500460
$885$	16726.9	0.635331
$886$	−22950.7	−0.870253
$887$	−22017.5	−0.833458	−0.416729	−	0.909031i	$-0.636824\pi$
$887$	−22017.5	−0.833458	−0.416729	+	0.909031i	$0.636824\pi$
$888$	−97878.8	−3.69887
$889$	37958.9	1.43206
$890$	−18371.3	−0.691917
$891$	3357.54	0.126242
$892$	−119835.	−4.49819
$893$	−8996.25	−0.337120
$894$	43037.3	1.61005
$895$	−1622.08	−0.0605811
$896$	−132383.	−4.93595
$897$	−3860.71	−0.143707
$898$	2052.60	0.0762762
$899$	−48753.0	−1.80868
$900$	−6159.20	−0.228119
$901$	1464.58	0.0541534
$902$	19334.2	0.713700
$903$	−17010.9	−0.626895
$904$	18474.4	0.679701
$905$	20046.0	0.736302
$906$	37726.0	1.38340
$907$	−28739.7	−1.05213	−0.526067	−	0.850443i	$-0.676334\pi$
$907$	−28739.7	−1.05213	−0.526067	+	0.850443i	$0.676334\pi$
$908$	−86414.0	−3.15831
$909$	15384.8	0.561367
$910$	−5665.10	−0.206370
$911$	42713.9	1.55343	0.776715	−	0.629852i	$-0.216884\pi$
$911$	42713.9	1.55343	0.776715	+	0.629852i	$0.216884\pi$
$912$	18915.2	0.686780
$913$	7441.10	0.269731
$914$	−55075.8	−1.99316
$915$	10501.6	0.379424
$916$	−44900.4	−1.61960
$917$	36164.0	1.30233
$918$	−56343.1	−2.02571
$919$	−55660.1	−1.99789	−0.998943	−	0.0459701i	$-0.985362\pi$
$919$	−55660.1	−1.99789	−0.998943	+	0.0459701i	$0.985362\pi$
$920$	42859.0	1.53589
$921$	−37435.9	−1.33936
$922$	−59077.3	−2.11020
$923$	6309.77	0.225015
$924$	−22804.4	−0.811915
$925$	7895.37	0.280647
$926$	12350.8	0.438309
$927$	576.476	0.0204250
$928$	−152718.	−5.40217
$929$	34915.0	1.23307	0.616535	−	0.787327i	$-0.288536\pi$
$929$	34915.0	1.23307	0.616535	+	0.787327i	$0.288536\pi$
$930$	−26192.1	−0.923520
$931$	3952.56	0.139141
$932$	90221.9	3.17094
$933$	−35310.0	−1.23901
$934$	−103015.	−3.60894
$935$	−3714.50	−0.129922
$936$	−7598.59	−0.265350
$937$	−4378.29	−0.152649	−0.0763247	−	0.997083i	$-0.524319\pi$
$937$	−4378.29	−0.152649	−0.0763247	+	0.997083i	$0.524319\pi$
$938$	−8157.00	−0.283940
$939$	−26408.5	−0.917796
$940$	52463.4	1.82039
$941$	−37364.1	−1.29440	−0.647202	−	0.762318i	$-0.724061\pi$
$941$	−37364.1	−1.29440	−0.647202	+	0.762318i	$0.724061\pi$
$942$	27636.7	0.955893
$943$	35276.8	1.21821
$944$	−209692.	−7.22976
$945$	17829.6	0.613753
$946$	−10984.8	−0.377532
$947$	−15488.3	−0.531469	−0.265735	−	0.964046i	$-0.585614\pi$
$947$	−15488.3	−0.531469	−0.265735	+	0.964046i	$0.585614\pi$
$948$	29811.7	1.02135
$949$	−6924.46	−0.236857
$950$	−2608.63	−0.0890896
$951$	20797.6	0.709157
$952$	123288.	4.19726
$953$	−4993.46	−0.169731	−0.0848657	−	0.996392i	$-0.527046\pi$
$953$	−4993.46	−0.169731	−0.0848657	+	0.996392i	$0.527046\pi$
$954$	−1324.04	−0.0449342
$955$	6589.74	0.223287
$956$	−108701.	−3.67746
$957$	−8929.60	−0.301623
$958$	86420.1	2.91452
$959$	5775.91	0.194488
$960$	−42225.1	−1.41959
$961$	27494.3	0.922907
$962$	15243.4	0.510882
$963$	2404.46	0.0804596
$964$	50195.4	1.67706
$965$	15341.9	0.511787
$966$	−56629.4	−1.88615
$967$	16144.0	0.536873	0.268437	−	0.963297i	$-0.413493\pi$
$967$	16144.0	0.536873	0.268437	+	0.963297i	$0.413493\pi$
$968$	−9409.83	−0.312441
$969$	−5113.89	−0.169537
$970$	38741.1	1.28237
$971$	36223.7	1.19719	0.598596	−	0.801051i	$-0.295725\pi$
$971$	36223.7	1.19719	0.598596	+	0.801051i	$0.295725\pi$
$972$	63935.5	2.10981
$973$	−38218.6	−1.25923
$974$	67084.1	2.20689
$975$	−875.652	−0.0287624
$976$	−131651.	−4.31766
$977$	38326.3	1.25503	0.627516	−	0.778604i	$-0.284072\pi$
$977$	38326.3	1.25503	0.627516	+	0.778604i	$0.284072\pi$
$978$	18895.3	0.617795
$979$	7359.40	0.240253
$980$	−23050.1	−0.751336
$981$	−12212.5	−0.397468
$982$	−10465.7	−0.340095
$983$	55460.1	1.79949	0.899747	−	0.436412i	$-0.143751\pi$
$983$	55460.1	1.79949	0.899747	+	0.436412i	$0.143751\pi$
$984$	−99190.5	−3.21349
$985$	16405.7	0.530689
$986$	75550.3	2.44017
$987$	−44295.0	−1.42850
$988$	−3700.53	−0.119159
$989$	−20042.6	−0.644407
$990$	3358.05	0.107804
$991$	10116.0	0.324263	0.162132	−	0.986769i	$-0.448163\pi$
$991$	10116.0	0.324263	0.162132	+	0.986769i	$0.448163\pi$
$992$	179445.	5.74334
$993$	4812.82	0.153807
$994$	92552.6	2.95331
$995$	994.110	0.0316738
$996$	−59742.5	−1.90061
$997$	26645.6	0.846414	0.423207	−	0.906033i	$-0.360904\pi$
$997$	26645.6	0.846414	0.423207	+	0.906033i	$0.360904\pi$
$998$	−15469.6	−0.490663
$999$	−47975.2	−1.51939

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1045.4.a.c.1.1	✓	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1045.4.a.c.1.1	✓	20	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 1045 = 5 \cdot 11 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1045.a (trivial)

\(f(q)\)	\(=\)	\(q-5.49185 q^{2} +3.98529 q^{3} +22.1605 q^{4} -5.00000 q^{5} -21.8866 q^{6} +23.4740 q^{7} -77.7672 q^{8} -11.1175 q^{9} +O(q^{10})\)	\(q-5.49185 q^{2} +3.98529 q^{3} +22.1605 q^{4} -5.00000 q^{5} -21.8866 q^{6} +23.4740 q^{7} -77.7672 q^{8} -11.1175 q^{9} +27.4593 q^{10} -11.0000 q^{11} +88.3159 q^{12} -8.78884 q^{13} -128.916 q^{14} -19.9265 q^{15} +249.802 q^{16} -67.5364 q^{17} +61.0555 q^{18} +19.0000 q^{19} -110.802 q^{20} +93.5508 q^{21} +60.4104 q^{22} +110.224 q^{23} -309.925 q^{24} +25.0000 q^{25} +48.2670 q^{26} -151.909 q^{27} +520.195 q^{28} +203.695 q^{29} +109.433 q^{30} -239.344 q^{31} -749.740 q^{32} -43.8382 q^{33} +370.900 q^{34} -117.370 q^{35} -246.368 q^{36} +315.815 q^{37} -104.345 q^{38} -35.0261 q^{39} +388.836 q^{40} +320.047 q^{41} -513.767 q^{42} -181.836 q^{43} -243.765 q^{44} +55.5873 q^{45} -605.334 q^{46} -473.487 q^{47} +995.535 q^{48} +208.029 q^{49} -137.296 q^{50} -269.152 q^{51} -194.765 q^{52} -21.6858 q^{53} +834.263 q^{54} +55.0000 q^{55} -1825.51 q^{56} +75.7205 q^{57} -1118.66 q^{58} -839.431 q^{59} -441.579 q^{60} -527.019 q^{61} +1314.44 q^{62} -260.971 q^{63} +2119.05 q^{64} +43.9442 q^{65} +240.753 q^{66} +63.2738 q^{67} -1496.64 q^{68} +439.274 q^{69} +644.579 q^{70} -717.930 q^{71} +864.573 q^{72} +787.870 q^{73} -1734.41 q^{74} +99.6323 q^{75} +421.049 q^{76} -258.214 q^{77} +192.358 q^{78} +337.557 q^{79} -1249.01 q^{80} -305.231 q^{81} -1757.65 q^{82} -676.463 q^{83} +2073.13 q^{84} +337.682 q^{85} +998.615 q^{86} +811.782 q^{87} +855.439 q^{88} -669.037 q^{89} -305.277 q^{90} -206.309 q^{91} +2442.61 q^{92} -953.853 q^{93} +2600.32 q^{94} -95.0000 q^{95} -2987.93 q^{96} +1410.86 q^{97} -1142.47 q^{98} +122.292 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - q^{2} - 8 q^{3} + 51 q^{4} - 100 q^{5} - 54 q^{6} + 49 q^{7} + 9 q^{8} + 146 q^{9}+O(q^{10}) \) 20 * q - q^2 - 8 * q^3 + 51 * q^4 - 100 * q^5 - 54 * q^6 + 49 * q^7 + 9 * q^8 + 146 * q^9	\( 20 q - q^{2} - 8 q^{3} + 51 q^{4} - 100 q^{5} - 54 q^{6} + 49 q^{7} + 9 q^{8} + 146 q^{9} + 5 q^{10} - 220 q^{11} - 59 q^{12} + 60 q^{13} - 89 q^{14} + 40 q^{15} + 275 q^{16} - 155 q^{17} + 45 q^{18} + 380 q^{19} - 255 q^{20} + 105 q^{21} + 11 q^{22} - 154 q^{23} - 397 q^{24} + 500 q^{25} + 176 q^{26} - 206 q^{27} + 155 q^{28} - 305 q^{29} + 270 q^{30} - 759 q^{31} - 254 q^{32} + 88 q^{33} - 565 q^{34} - 245 q^{35} + 705 q^{36} + 698 q^{37} - 19 q^{38} - 758 q^{39} - 45 q^{40} + 547 q^{41} + 106 q^{42} - 925 q^{43} - 561 q^{44} - 730 q^{45} - 254 q^{46} - 681 q^{47} - 540 q^{48} + 213 q^{49} - 25 q^{50} - 899 q^{51} + 889 q^{52} - 419 q^{53} - 2241 q^{54} + 1100 q^{55} - 2473 q^{56} - 152 q^{57} - 1440 q^{58} - 2829 q^{59} + 295 q^{60} - 959 q^{61} + 1575 q^{62} - 426 q^{63} + 93 q^{64} - 300 q^{65} + 594 q^{66} - 1020 q^{67} - 4218 q^{68} - 572 q^{69} + 445 q^{70} + 106 q^{71} + 210 q^{72} + 558 q^{73} - 3439 q^{74} - 200 q^{75} + 969 q^{76} - 539 q^{77} - 3599 q^{78} + 536 q^{79} - 1375 q^{80} - 2128 q^{81} - 1255 q^{82} - 4179 q^{83} - 2024 q^{84} + 775 q^{85} - 1119 q^{86} - 557 q^{87} - 99 q^{88} - 4120 q^{89} - 225 q^{90} - 111 q^{91} - 2831 q^{92} + 801 q^{93} + 1213 q^{94} - 1900 q^{95} - 6147 q^{96} + 1414 q^{97} - 7869 q^{98} - 1606 q^{99}+O(q^{100}) \) 20 * q - q^2 - 8 * q^3 + 51 * q^4 - 100 * q^5 - 54 * q^6 + 49 * q^7 + 9 * q^8 + 146 * q^9 + 5 * q^10 - 220 * q^11 - 59 * q^12 + 60 * q^13 - 89 * q^14 + 40 * q^15 + 275 * q^16 - 155 * q^17 + 45 * q^18 + 380 * q^19 - 255 * q^20 + 105 * q^21 + 11 * q^22 - 154 * q^23 - 397 * q^24 + 500 * q^25 + 176 * q^26 - 206 * q^27 + 155 * q^28 - 305 * q^29 + 270 * q^30 - 759 * q^31 - 254 * q^32 + 88 * q^33 - 565 * q^34 - 245 * q^35 + 705 * q^36 + 698 * q^37 - 19 * q^38 - 758 * q^39 - 45 * q^40 + 547 * q^41 + 106 * q^42 - 925 * q^43 - 561 * q^44 - 730 * q^45 - 254 * q^46 - 681 * q^47 - 540 * q^48 + 213 * q^49 - 25 * q^50 - 899 * q^51 + 889 * q^52 - 419 * q^53 - 2241 * q^54 + 1100 * q^55 - 2473 * q^56 - 152 * q^57 - 1440 * q^58 - 2829 * q^59 + 295 * q^60 - 959 * q^61 + 1575 * q^62 - 426 * q^63 + 93 * q^64 - 300 * q^65 + 594 * q^66 - 1020 * q^67 - 4218 * q^68 - 572 * q^69 + 445 * q^70 + 106 * q^71 + 210 * q^72 + 558 * q^73 - 3439 * q^74 - 200 * q^75 + 969 * q^76 - 539 * q^77 - 3599 * q^78 + 536 * q^79 - 1375 * q^80 - 2128 * q^81 - 1255 * q^82 - 4179 * q^83 - 2024 * q^84 + 775 * q^85 - 1119 * q^86 - 557 * q^87 - 99 * q^88 - 4120 * q^89 - 225 * q^90 - 111 * q^91 - 2831 * q^92 + 801 * q^93 + 1213 * q^94 - 1900 * q^95 - 6147 * q^96 + 1414 * q^97 - 7869 * q^98 - 1606 * q^99

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