LMFDB - Embedded newform 115.2.a.c.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(115, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("115.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

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

Embedding invariants

Embedding label			1.2
Root			$-0.329727$ of defining polynomial
Character	$\chi$	$=$	115.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.329727 q^{2} +1.56155 q^{3} -1.89128 q^{4} +1.00000 q^{5} -0.514886 q^{6} +4.06562 q^{7} +1.28306 q^{8} -0.561553 q^{9} +O(q^{10})$	\(q-0.329727 q^{2} +1.56155 q^{3} -1.89128 q^{4} +1.00000 q^{5} -0.514886 q^{6} +4.06562 q^{7} +1.28306 q^{8} -0.561553 q^{9} -0.329727 q^{10} +2.65945 q^{11} -2.95333 q^{12} -5.91023 q^{13} -1.34055 q^{14} +1.56155 q^{15} +3.35950 q^{16} -3.40617 q^{17} +0.185159 q^{18} -2.65945 q^{19} -1.89128 q^{20} +6.34868 q^{21} -0.876894 q^{22} -1.00000 q^{23} +2.00357 q^{24} +1.00000 q^{25} +1.94876 q^{26} -5.56155 q^{27} -7.68923 q^{28} +5.84461 q^{29} -0.514886 q^{30} -9.31640 q^{31} -3.67384 q^{32} +4.15288 q^{33} +1.12311 q^{34} +4.06562 q^{35} +1.06205 q^{36} -4.18059 q^{37} +0.876894 q^{38} -9.22914 q^{39} +1.28306 q^{40} +2.15539 q^{41} -2.09333 q^{42} +2.34868 q^{43} -5.02977 q^{44} -0.561553 q^{45} +0.329727 q^{46} +0.242644 q^{47} +5.24604 q^{48} +9.52927 q^{49} -0.329727 q^{50} -5.31891 q^{51} +11.1779 q^{52} +9.03585 q^{53} +1.83380 q^{54} +2.65945 q^{55} +5.21644 q^{56} -4.15288 q^{57} -1.92713 q^{58} +14.4143 q^{59} -2.95333 q^{60} +2.46365 q^{61} +3.07187 q^{62} -2.28306 q^{63} -5.50764 q^{64} -5.91023 q^{65} -1.36932 q^{66} -7.75485 q^{67} +6.44201 q^{68} -1.56155 q^{69} -1.34055 q^{70} +12.4395 q^{71} -0.720506 q^{72} -2.08977 q^{73} +1.37845 q^{74} +1.56155 q^{75} +5.02977 q^{76} +10.8123 q^{77} +3.04310 q^{78} -4.15288 q^{79} +3.35950 q^{80} -7.00000 q^{81} -0.710689 q^{82} -12.5293 q^{83} -12.0071 q^{84} -3.40617 q^{85} -0.774424 q^{86} +9.12667 q^{87} +3.41224 q^{88} -9.35682 q^{89} +0.185159 q^{90} -24.0288 q^{91} +1.89128 q^{92} -14.5481 q^{93} -0.0800064 q^{94} -2.65945 q^{95} -5.73690 q^{96} +3.47179 q^{97} -3.14206 q^{98} -1.49342 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{2} - 2 q^{3} + 4 q^{4} + 4 q^{5} - q^{6} - 3 q^{7} + 9 q^{8} + 6 q^{9}+O(q^{10}) $ 4 * q + 2 * q^2 - 2 * q^3 + 4 * q^4 + 4 * q^5 - q^6 - 3 * q^7 + 9 * q^8 + 6 * q^9	\( 4 q + 2 q^{2} - 2 q^{3} + 4 q^{4} + 4 q^{5} - q^{6} - 3 q^{7} + 9 q^{8} + 6 q^{9} + 2 q^{10} + 4 q^{11} - 19 q^{12} - 12 q^{14} - 2 q^{15} + 8 q^{16} - q^{17} + 3 q^{18} - 4 q^{19} + 4 q^{20} + 10 q^{21} - 20 q^{22} - 4 q^{23} - 30 q^{24} + 4 q^{25} - q^{26} - 14 q^{27} - 22 q^{28} + 19 q^{29} - q^{30} - q^{31} + 20 q^{32} - 2 q^{33} - 12 q^{34} - 3 q^{35} + 23 q^{36} - 3 q^{37} + 20 q^{38} + 9 q^{40} + 13 q^{41} + 6 q^{42} - 6 q^{43} - 18 q^{44} + 6 q^{45} - 2 q^{46} + 6 q^{47} - 21 q^{48} + 9 q^{49} + 2 q^{50} - 8 q^{51} - q^{52} + 19 q^{53} - 7 q^{54} + 4 q^{55} - 10 q^{56} + 2 q^{57} + 21 q^{58} + 23 q^{59} - 19 q^{60} - 13 q^{62} - 13 q^{63} + 27 q^{64} + 44 q^{66} - 3 q^{67} - 4 q^{68} + 2 q^{69} - 12 q^{70} - 3 q^{71} + 39 q^{72} - 32 q^{73} - 12 q^{74} - 2 q^{75} + 18 q^{76} + 18 q^{77} + 43 q^{78} + 2 q^{79} + 8 q^{80} - 28 q^{81} - 5 q^{82} - 21 q^{83} + 28 q^{84} - q^{85} - 2 q^{86} - 18 q^{87} - 14 q^{88} + 3 q^{90} - 40 q^{91} - 4 q^{92} - 8 q^{93} + 47 q^{94} - 4 q^{95} - 61 q^{96} - 18 q^{97} + 16 q^{98} + 6 q^{99}+O(q^{100}) \) 4 * q + 2 * q^2 - 2 * q^3 + 4 * q^4 + 4 * q^5 - q^6 - 3 * q^7 + 9 * q^8 + 6 * q^9 + 2 * q^10 + 4 * q^11 - 19 * q^12 - 12 * q^14 - 2 * q^15 + 8 * q^16 - q^17 + 3 * q^18 - 4 * q^19 + 4 * q^20 + 10 * q^21 - 20 * q^22 - 4 * q^23 - 30 * q^24 + 4 * q^25 - q^26 - 14 * q^27 - 22 * q^28 + 19 * q^29 - q^30 - q^31 + 20 * q^32 - 2 * q^33 - 12 * q^34 - 3 * q^35 + 23 * q^36 - 3 * q^37 + 20 * q^38 + 9 * q^40 + 13 * q^41 + 6 * q^42 - 6 * q^43 - 18 * q^44 + 6 * q^45 - 2 * q^46 + 6 * q^47 - 21 * q^48 + 9 * q^49 + 2 * q^50 - 8 * q^51 - q^52 + 19 * q^53 - 7 * q^54 + 4 * q^55 - 10 * q^56 + 2 * q^57 + 21 * q^58 + 23 * q^59 - 19 * q^60 - 13 * q^62 - 13 * q^63 + 27 * q^64 + 44 * q^66 - 3 * q^67 - 4 * q^68 + 2 * q^69 - 12 * q^70 - 3 * q^71 + 39 * q^72 - 32 * q^73 - 12 * q^74 - 2 * q^75 + 18 * q^76 + 18 * q^77 + 43 * q^78 + 2 * q^79 + 8 * q^80 - 28 * q^81 - 5 * q^82 - 21 * q^83 + 28 * q^84 - q^85 - 2 * q^86 - 18 * q^87 - 14 * q^88 + 3 * q^90 - 40 * q^91 - 4 * q^92 - 8 * q^93 + 47 * q^94 - 4 * q^95 - 61 * q^96 - 18 * q^97 + 16 * q^98 + 6 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−0.329727	−0.233152	−0.116576	−	0.993182i	$-0.537192\pi$
$2$	−0.329727	−0.233152	−0.116576	+	0.993182i	$0.537192\pi$
$3$	1.56155	0.901563	0.450781	−	0.892634i	$-0.351145\pi$
$3$	1.56155	0.901563	0.450781	+	0.892634i	$0.351145\pi$
$4$	−1.89128	−0.945640
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	−0.514886	−0.210201
$7$	4.06562	1.53666	0.768330	−	0.640054i	$-0.221088\pi$
$7$	4.06562	1.53666	0.768330	+	0.640054i	$0.221088\pi$
$8$	1.28306	0.453630
$9$	−0.561553	−0.187184
$10$	−0.329727	−0.104269
$11$	2.65945	0.801856	0.400928	−	0.916110i	$-0.368688\pi$
$11$	2.65945	0.801856	0.400928	+	0.916110i	$0.368688\pi$
$12$	−2.95333	−0.852554
$13$	−5.91023	−1.63920	−0.819602	−	0.572933i	$-0.805805\pi$
$13$	−5.91023	−1.63920	−0.819602	+	0.572933i	$0.805805\pi$
$14$	−1.34055	−0.358276
$15$	1.56155	0.403191
$16$	3.35950	0.839875
$17$	−3.40617	−0.826117	−0.413058	−	0.910705i	$-0.635539\pi$
$17$	−3.40617	−0.826117	−0.413058	+	0.910705i	$0.635539\pi$
$18$	0.185159	0.0436424
$19$	−2.65945	−0.610121	−0.305060	−	0.952333i	$-0.598677\pi$
$19$	−2.65945	−0.610121	−0.305060	+	0.952333i	$0.598677\pi$
$20$	−1.89128	−0.422903
$21$	6.34868	1.38540
$22$	−0.876894	−0.186955
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	2.00357	0.408976
$25$	1.00000	0.200000
$26$	1.94876	0.382184
$27$	−5.56155	−1.07032
$28$	−7.68923	−1.45313
$29$	5.84461	1.08532	0.542659	−	0.839953i	$-0.317418\pi$
$29$	5.84461	1.08532	0.542659	+	0.839953i	$0.317418\pi$
$30$	−0.514886	−0.0940050
$31$	−9.31640	−1.67327	−0.836637	−	0.547757i	$-0.815482\pi$
$31$	−9.31640	−1.67327	−0.836637	+	0.547757i	$0.815482\pi$
$32$	−3.67384	−0.649449
$33$	4.15288	0.722923
$34$	1.12311	0.192611
$35$	4.06562	0.687215
$36$	1.06205	0.177009
$37$	−4.18059	−0.687285	−0.343642	−	0.939101i	$-0.611661\pi$
$37$	−4.18059	−0.687285	−0.343642	+	0.939101i	$0.611661\pi$
$38$	0.876894	0.142251
$39$	−9.22914	−1.47785
$40$	1.28306	0.202870
$41$	2.15539	0.336615	0.168307	−	0.985735i	$-0.446170\pi$
$41$	2.15539	0.336615	0.168307	+	0.985735i	$0.446170\pi$
$42$	−2.09333	−0.323008
$43$	2.34868	0.358171	0.179085	−	0.983834i	$-0.442686\pi$
$43$	2.34868	0.358171	0.179085	+	0.983834i	$0.442686\pi$
$44$	−5.02977	−0.758267
$45$	−0.561553	−0.0837114
$46$	0.329727	0.0486156
$47$	0.242644	0.0353933	0.0176966	−	0.999843i	$-0.494367\pi$
$47$	0.242644	0.0353933	0.0176966	+	0.999843i	$0.494367\pi$
$48$	5.24604	0.757200
$49$	9.52927	1.36132
$50$	−0.329727	−0.0466305
$51$	−5.31891	−0.744796
$52$	11.1779	1.55010
$53$	9.03585	1.24117	0.620585	−	0.784140i	$-0.286895\pi$
$53$	9.03585	1.24117	0.620585	+	0.784140i	$0.286895\pi$
$54$	1.83380	0.249548
$55$	2.65945	0.358601
$56$	5.21644	0.697076
$57$	−4.15288	−0.550062
$58$	−1.92713	−0.253044
$59$	14.4143	1.87658	0.938291	−	0.345846i	$-0.112408\pi$
$59$	14.4143	1.87658	0.938291	+	0.345846i	$0.112408\pi$
$60$	−2.95333	−0.381274
$61$	2.46365	0.315438	0.157719	−	0.987484i	$-0.449586\pi$
$61$	2.46365	0.315438	0.157719	+	0.987484i	$0.449586\pi$
$62$	3.07187	0.390128
$63$	−2.28306	−0.287639
$64$	−5.50764	−0.688454
$65$	−5.91023	−0.733074
$66$	−1.36932	−0.168551
$67$	−7.75485	−0.947405	−0.473703	−	0.880685i	$-0.657083\pi$
$67$	−7.75485	−0.947405	−0.473703	+	0.880685i	$0.657083\pi$
$68$	6.44201	0.781209
$69$	−1.56155	−0.187989
$70$	−1.34055	−0.160226
$71$	12.4395	1.47630	0.738149	−	0.674638i	$-0.235700\pi$
$71$	12.4395	1.47630	0.738149	+	0.674638i	$0.235700\pi$
$72$	−0.720506	−0.0849125
$73$	−2.08977	−0.244589	−0.122294	−	0.992494i	$-0.539025\pi$
$73$	−2.08977	−0.244589	−0.122294	+	0.992494i	$0.539025\pi$
$74$	1.37845	0.160242
$75$	1.56155	0.180313
$76$	5.02977	0.576955
$77$	10.8123	1.23218
$78$	3.04310	0.344563
$79$	−4.15288	−0.467235	−0.233618	−	0.972329i	$-0.575056\pi$
$79$	−4.15288	−0.467235	−0.233618	+	0.972329i	$0.575056\pi$
$80$	3.35950	0.375604
$81$	−7.00000	−0.777778
$82$	−0.710689	−0.0784825
$83$	−12.5293	−1.37527	−0.687633	−	0.726058i	$-0.741350\pi$
$83$	−12.5293	−1.37527	−0.687633	+	0.726058i	$0.741350\pi$
$84$	−12.0071	−1.31009
$85$	−3.40617	−0.369451
$86$	−0.774424	−0.0835083
$87$	9.12667	0.978482
$88$	3.41224	0.363746
$89$	−9.35682	−0.991821	−0.495910	−	0.868374i	$-0.665166\pi$
$89$	−9.35682	−0.991821	−0.495910	+	0.868374i	$0.665166\pi$
$90$	0.185159	0.0195175
$91$	−24.0288	−2.51890
$92$	1.89128	0.197180
$93$	−14.5481	−1.50856
$94$	−0.0800064	−0.00825203
$95$	−2.65945	−0.272854
$96$	−5.73690	−0.585519
$97$	3.47179	0.352507	0.176253	−	0.984345i	$-0.443602\pi$
$97$	3.47179	0.352507	0.176253	+	0.984345i	$0.443602\pi$
$98$	−3.14206	−0.317396
$99$	−1.49342	−0.150095
$100$	−1.89128	−0.189128
$101$	9.21036	0.916465	0.458233	−	0.888832i	$-0.348483\pi$
$101$	9.21036	0.916465	0.458233	+	0.888832i	$0.348483\pi$
$102$	1.75379	0.173651
$103$	4.69736	0.462845	0.231422	−	0.972853i	$-0.425662\pi$
$103$	4.69736	0.462845	0.231422	+	0.972853i	$0.425662\pi$
$104$	−7.58319	−0.743593
$105$	6.34868	0.619568
$106$	−2.97936	−0.289381
$107$	−0.376394	−0.0363873	−0.0181937	−	0.999834i	$-0.505792\pi$
$107$	−0.376394	−0.0363873	−0.0181937	+	0.999834i	$0.505792\pi$
$108$	10.5185	1.01214
$109$	19.0369	1.82340	0.911702	−	0.410851i	$-0.134768\pi$
$109$	19.0369	1.82340	0.911702	+	0.410851i	$0.134768\pi$
$110$	−0.876894	−0.0836086
$111$	−6.52821	−0.619631
$112$	13.6585	1.29060
$113$	2.94252	0.276809	0.138404	−	0.990376i	$-0.455803\pi$
$113$	2.94252	0.276809	0.138404	+	0.990376i	$0.455803\pi$
$114$	1.36932	0.128248
$115$	−1.00000	−0.0932505
$116$	−11.0538	−1.02632
$117$	3.31891	0.306833
$118$	−4.75279	−0.437530
$119$	−13.8482	−1.26946
$120$	2.00357	0.182900
$121$	−3.92730	−0.357028
$122$	−0.812333	−0.0735452
$123$	3.36575	0.303479
$124$	17.6199	1.58232
$125$	1.00000	0.0894427
$126$	0.752787	0.0670636
$127$	6.00357	0.532730	0.266365	−	0.963872i	$-0.414177\pi$
$127$	6.00357	0.532730	0.266365	+	0.963872i	$0.414177\pi$
$128$	9.16370	0.809964
$129$	3.66759	0.322913
$130$	1.94876	0.170918
$131$	−19.9606	−1.74397	−0.871985	−	0.489533i	$-0.837167\pi$
$131$	−19.9606	−1.74397	−0.871985	+	0.489533i	$0.837167\pi$
$132$	−7.85426	−0.683625
$133$	−10.8123	−0.937548
$134$	2.55698	0.220890
$135$	−5.56155	−0.478662
$136$	−4.37032	−0.374752
$137$	−11.7609	−1.00480	−0.502402	−	0.864634i	$-0.667550\pi$
$137$	−11.7609	−1.00480	−0.502402	+	0.864634i	$0.667550\pi$
$138$	0.514886	0.0438300
$139$	12.3891	1.05083	0.525415	−	0.850846i	$-0.323910\pi$
$139$	12.3891	1.05083	0.525415	+	0.850846i	$0.323910\pi$
$140$	−7.68923	−0.649858
$141$	0.378902	0.0319093
$142$	−4.10164	−0.344202
$143$	−15.7180	−1.31441
$144$	−1.88654	−0.157211
$145$	5.84461	0.485369
$146$	0.689053	0.0570264
$147$	14.8805	1.22732
$148$	7.90667	0.649924
$149$	−2.13124	−0.174598	−0.0872991	−	0.996182i	$-0.527824\pi$
$149$	−2.13124	−0.174598	−0.0872991	+	0.996182i	$0.527824\pi$
$150$	−0.514886	−0.0420403
$151$	−0.787129	−0.0640556	−0.0320278	−	0.999487i	$-0.510197\pi$
$151$	−0.787129	−0.0640556	−0.0320278	+	0.999487i	$0.510197\pi$
$152$	−3.41224	−0.276769
$153$	1.91274	0.154636
$154$	−3.56512	−0.287286
$155$	−9.31640	−0.748311
$156$	17.4549	1.39751
$157$	−9.18873	−0.733340	−0.366670	−	0.930351i	$-0.619502\pi$
$157$	−9.18873	−0.733340	−0.366670	+	0.930351i	$0.619502\pi$
$158$	1.36932	0.108937
$159$	14.1100	1.11899
$160$	−3.67384	−0.290443
$161$	−4.06562	−0.320416
$162$	2.30809	0.181341
$163$	−17.6928	−1.38581	−0.692903	−	0.721031i	$-0.743669\pi$
$163$	−17.6928	−1.38581	−0.692903	+	0.721031i	$0.743669\pi$
$164$	−4.07644	−0.318316
$165$	4.15288	0.323301
$166$	4.13124	0.320647
$167$	14.2462	1.10240	0.551202	−	0.834372i	$-0.314169\pi$
$167$	14.2462	1.10240	0.551202	+	0.834372i	$0.314169\pi$
$168$	8.14574	0.628458
$169$	21.9309	1.68699
$170$	1.12311	0.0861383
$171$	1.49342	0.114205
$172$	−4.44201	−0.338700
$173$	−1.20394	−0.0915338	−0.0457669	−	0.998952i	$-0.514573\pi$
$173$	−1.20394	−0.0915338	−0.0457669	+	0.998952i	$0.514573\pi$
$174$	−3.00931	−0.228135
$175$	4.06562	0.307332
$176$	8.93444	0.673459
$177$	22.5087	1.69186
$178$	3.08520	0.231245
$179$	7.42495	0.554967	0.277483	−	0.960730i	$-0.410500\pi$
$179$	7.42495	0.554967	0.277483	+	0.960730i	$0.410500\pi$
$180$	1.06205	0.0791608
$181$	24.0450	1.78725	0.893627	−	0.448810i	$-0.148152\pi$
$181$	24.0450	1.78725	0.893627	+	0.448810i	$0.148152\pi$
$182$	7.92294	0.587287
$183$	3.84712	0.284387
$184$	−1.28306	−0.0945885
$185$	−4.18059	−0.307363
$186$	4.79689	0.351725
$187$	−9.05854	−0.662426
$188$	−0.458908	−0.0334693
$189$	−22.6112	−1.64472
$190$	0.876894	0.0636166
$191$	−4.87689	−0.352880	−0.176440	−	0.984311i	$-0.556458\pi$
$191$	−4.87689	−0.352880	−0.176440	+	0.984311i	$0.556458\pi$
$192$	−8.60046	−0.620685
$193$	10.2714	0.739353	0.369676	−	0.929161i	$-0.379469\pi$
$193$	10.2714	0.739353	0.369676	+	0.929161i	$0.379469\pi$
$194$	−1.14474	−0.0821877
$195$	−9.22914	−0.660913
$196$	−18.0225	−1.28732
$197$	4.28557	0.305334	0.152667	−	0.988278i	$-0.451214\pi$
$197$	4.28557	0.305334	0.152667	+	0.988278i	$0.451214\pi$
$198$	0.492423	0.0349949
$199$	4.02164	0.285086	0.142543	−	0.989789i	$-0.454472\pi$
$199$	4.02164	0.285086	0.142543	+	0.989789i	$0.454472\pi$
$200$	1.28306	0.0907261
$201$	−12.1096	−0.854146
$202$	−3.03691	−0.213676
$203$	23.7620	1.66776
$204$	10.0595	0.704309
$205$	2.15539	0.150539
$206$	−1.54885	−0.107913
$207$	0.561553	0.0390306
$208$	−19.8554	−1.37673
$209$	−7.07270	−0.489229
$210$	−2.09333	−0.144454
$211$	15.3416	1.05616	0.528080	−	0.849195i	$-0.322912\pi$
$211$	15.3416	1.05616	0.528080	+	0.849195i	$0.322912\pi$
$212$	−17.0893	−1.17370
$213$	19.4249	1.33098
$214$	0.124107	0.00848379
$215$	2.34868	0.160179
$216$	−7.13581	−0.485530
$217$	−37.8770	−2.57126
$218$	−6.27699	−0.425131
$219$	−3.26328	−0.220512
$220$	−5.02977	−0.339107
$221$	20.1312	1.35417
$222$	2.15253	0.144468
$223$	−17.6801	−1.18395	−0.591973	−	0.805958i	$-0.701651\pi$
$223$	−17.6801	−1.18395	−0.591973	+	0.805958i	$0.701651\pi$
$224$	−14.9364	−0.997983
$225$	−0.561553	−0.0374369
$226$	−0.970227	−0.0645386
$227$	8.63817	0.573335	0.286668	−	0.958030i	$-0.407452\pi$
$227$	8.63817	0.573335	0.286668	+	0.958030i	$0.407452\pi$
$228$	7.85426	0.520161
$229$	−3.34055	−0.220749	−0.110375	−	0.993890i	$-0.535205\pi$
$229$	−3.34055	−0.220749	−0.110375	+	0.993890i	$0.535205\pi$
$230$	0.329727	0.0217416
$231$	16.8840	1.11089
$232$	7.49899	0.492333
$233$	8.23728	0.539642	0.269821	−	0.962910i	$-0.413035\pi$
$233$	8.23728	0.539642	0.269821	+	0.962910i	$0.413035\pi$
$234$	−1.09433	−0.0715389
$235$	0.242644	0.0158284
$236$	−27.2615	−1.77457
$237$	−6.48494	−0.421242
$238$	4.56612	0.295978
$239$	−2.86525	−0.185338	−0.0926688	−	0.995697i	$-0.529540\pi$
$239$	−2.86525	−0.185338	−0.0926688	+	0.995697i	$0.529540\pi$
$240$	5.24604	0.338630
$241$	4.68109	0.301536	0.150768	−	0.988569i	$-0.451825\pi$
$241$	4.68109	0.301536	0.150768	+	0.988569i	$0.451825\pi$
$242$	1.29494	0.0832418
$243$	5.75379	0.369106
$244$	−4.65945	−0.298291
$245$	9.52927	0.608803
$246$	−1.10978	−0.0707569
$247$	15.7180	1.00011
$248$	−11.9535	−0.759049
$249$	−19.5651	−1.23989
$250$	−0.329727	−0.0208538
$251$	−18.5824	−1.17291	−0.586455	−	0.809982i	$-0.699477\pi$
$251$	−18.5824	−1.17291	−0.586455	+	0.809982i	$0.699477\pi$
$252$	4.31791	0.272003
$253$	−2.65945	−0.167198
$254$	−1.97954	−0.124207
$255$	−5.31891	−0.333083
$256$	7.99375	0.499609
$257$	−28.7226	−1.79166	−0.895832	−	0.444392i	$-0.853420\pi$
$257$	−28.7226	−1.79166	−0.895832	+	0.444392i	$0.853420\pi$
$258$	−1.20930	−0.0752880
$259$	−16.9967	−1.05612
$260$	11.1779	0.693224
$261$	−3.28206	−0.203154
$262$	6.58157	0.406611
$263$	−9.20500	−0.567604	−0.283802	−	0.958883i	$-0.591596\pi$
$263$	−9.20500	−0.567604	−0.283802	+	0.958883i	$0.591596\pi$
$264$	5.32840	0.327940
$265$	9.03585	0.555068
$266$	3.56512	0.218592
$267$	−14.6112	−0.894189
$268$	14.6666	0.895904
$269$	16.9194	1.03160	0.515798	−	0.856710i	$-0.327496\pi$
$269$	16.9194	1.03160	0.515798	+	0.856710i	$0.327496\pi$
$270$	1.83380	0.111601
$271$	1.15082	0.0699072	0.0349536	−	0.999389i	$-0.488872\pi$
$271$	1.15082	0.0699072	0.0349536	+	0.999389i	$0.488872\pi$
$272$	−11.4430	−0.693835
$273$	−37.5222	−2.27095
$274$	3.87790	0.234272
$275$	2.65945	0.160371
$276$	2.95333	0.177770
$277$	2.57505	0.154720	0.0773600	−	0.997003i	$-0.475351\pi$
$277$	2.57505	0.154720	0.0773600	+	0.997003i	$0.475351\pi$
$278$	−4.08502	−0.245003
$279$	5.23165	0.313211
$280$	5.21644	0.311742
$281$	27.4718	1.63883	0.819415	−	0.573201i	$-0.194299\pi$
$281$	27.4718	1.63883	0.819415	+	0.573201i	$0.194299\pi$
$282$	−0.124934	−0.00743972
$283$	−2.36389	−0.140519	−0.0702595	−	0.997529i	$-0.522383\pi$
$283$	−2.36389	−0.140519	−0.0702595	+	0.997529i	$0.522383\pi$
$284$	−23.5266	−1.39605
$285$	−4.15288	−0.245995
$286$	5.18265	0.306457
$287$	8.76298	0.517263
$288$	2.06306	0.121567
$289$	−5.39803	−0.317531
$290$	−1.92713	−0.113165
$291$	5.42138	0.317807
$292$	3.95233	0.231293
$293$	34.1198	1.99330	0.996650	−	0.0817846i	$-0.0260619\pi$
$293$	34.1198	1.99330	0.996650	+	0.0817846i	$0.0260619\pi$
$294$	−4.90649	−0.286152
$295$	14.4143	0.839233
$296$	−5.36395	−0.311773
$297$	−14.7907	−0.858243
$298$	0.702728	0.0407080
$299$	5.91023	0.341798
$300$	−2.95333	−0.170511
$301$	9.54885	0.550386
$302$	0.259538	0.0149347
$303$	14.3825	0.826251
$304$	−8.93444	−0.512425
$305$	2.46365	0.141068
$306$	−0.630683	−0.0360538
$307$	1.49342	0.0852342	0.0426171	−	0.999091i	$-0.486430\pi$
$307$	1.49342	0.0852342	0.0426171	+	0.999091i	$0.486430\pi$
$308$	−20.4491	−1.16520
$309$	7.33518	0.417284
$310$	3.07187	0.174471
$311$	−14.1060	−0.799880	−0.399940	−	0.916541i	$-0.630969\pi$
$311$	−14.1060	−0.799880	−0.399940	+	0.916541i	$0.630969\pi$
$312$	−11.8416	−0.670396
$313$	−32.8563	−1.85715	−0.928574	−	0.371146i	$-0.878965\pi$
$313$	−32.8563	−1.85715	−0.928574	+	0.371146i	$0.878965\pi$
$314$	3.02977	0.170980
$315$	−2.28306	−0.128636
$316$	7.85426	0.441836
$317$	12.5824	0.706698	0.353349	−	0.935492i	$-0.385043\pi$
$317$	12.5824	0.706698	0.353349	+	0.935492i	$0.385043\pi$
$318$	−4.65244	−0.260896
$319$	15.5435	0.870268
$320$	−5.50764	−0.307886
$321$	−0.587758	−0.0328055
$322$	1.34055	0.0747057
$323$	9.05854	0.504031
$324$	13.2390	0.735498
$325$	−5.91023	−0.327841
$326$	5.83380	0.323104
$327$	29.7271	1.64391
$328$	2.76549	0.152699
$329$	0.986499	0.0543874
$330$	−1.36932	−0.0753784
$331$	1.57677	0.0866669	0.0433334	−	0.999061i	$-0.486202\pi$
$331$	1.57677	0.0866669	0.0433334	+	0.999061i	$0.486202\pi$
$332$	23.6964	1.30051
$333$	2.34762	0.128649
$334$	−4.69736	−0.257028
$335$	−7.75485	−0.423693
$336$	21.3284	1.16356
$337$	−15.4718	−0.842802	−0.421401	−	0.906874i	$-0.638461\pi$
$337$	−15.4718	−0.842802	−0.421401	+	0.906874i	$0.638461\pi$
$338$	−7.23120	−0.393326
$339$	4.59489	0.249560
$340$	6.44201	0.349367
$341$	−24.7765	−1.34172
$342$	−0.492423	−0.0266272
$343$	10.2831	0.555233
$344$	3.01350	0.162477
$345$	−1.56155	−0.0840712
$346$	0.396971	0.0213413
$347$	−9.00312	−0.483313	−0.241656	−	0.970362i	$-0.577691\pi$
$347$	−9.00312	−0.483313	−0.241656	+	0.970362i	$0.577691\pi$
$348$	−17.2611	−0.925292
$349$	1.10398	0.0590945	0.0295473	−	0.999563i	$-0.490593\pi$
$349$	1.10398	0.0590945	0.0295473	+	0.999563i	$0.490593\pi$
$350$	−1.34055	−0.0716552
$351$	32.8701	1.75448
$352$	−9.77041	−0.520765
$353$	1.96566	0.104621	0.0523107	−	0.998631i	$-0.483341\pi$
$353$	1.96566	0.104621	0.0523107	+	0.998631i	$0.483341\pi$
$354$	−7.42173	−0.394460
$355$	12.4395	0.660220
$356$	17.6964	0.937905
$357$	−21.6247	−1.14450
$358$	−2.44821	−0.129392
$359$	−2.60403	−0.137435	−0.0687177	−	0.997636i	$-0.521891\pi$
$359$	−2.60403	−0.137435	−0.0687177	+	0.997636i	$0.521891\pi$
$360$	−0.720506	−0.0379740
$361$	−11.9273	−0.627753
$362$	−7.92830	−0.416702
$363$	−6.13269	−0.321883
$364$	45.4451	2.38197
$365$	−2.08977	−0.109383
$366$	−1.26850	−0.0663056
$367$	6.98042	0.364375	0.182188	−	0.983264i	$-0.441682\pi$
$367$	6.98042	0.364375	0.182188	+	0.983264i	$0.441682\pi$
$368$	−3.35950	−0.175126
$369$	−1.21036	−0.0630090
$370$	1.37845	0.0716624
$371$	36.7363	1.90726
$372$	27.5144	1.42656
$373$	24.3220	1.25935	0.629673	−	0.776860i	$-0.283189\pi$
$373$	24.3220	1.25935	0.629673	+	0.776860i	$0.283189\pi$
$374$	2.98685	0.154446
$375$	1.56155	0.0806382
$376$	0.311327	0.0160555
$377$	−34.5430	−1.77906
$378$	7.45552	0.383470
$379$	−24.6541	−1.26640	−0.633198	−	0.773990i	$-0.718258\pi$
$379$	−24.6541	−1.26640	−0.633198	+	0.773990i	$0.718258\pi$
$380$	5.02977	0.258022
$381$	9.37489	0.480290
$382$	1.60804	0.0822747
$383$	−4.99292	−0.255126	−0.127563	−	0.991830i	$-0.540716\pi$
$383$	−4.99292	−0.255126	−0.127563	+	0.991830i	$0.540716\pi$
$384$	14.3096	0.730234
$385$	10.8123	0.551048
$386$	−3.38676	−0.172382
$387$	−1.31891	−0.0670439
$388$	−6.56612	−0.333344
$389$	−32.0234	−1.62365	−0.811826	−	0.583900i	$-0.801526\pi$
$389$	−32.0234	−1.62365	−0.811826	+	0.583900i	$0.801526\pi$
$390$	3.04310	0.154093
$391$	3.40617	0.172257
$392$	12.2266	0.617538
$393$	−31.1696	−1.57230
$394$	−1.41307	−0.0711894
$395$	−4.15288	−0.208954
$396$	2.82448	0.141936
$397$	−17.3441	−0.870476	−0.435238	−	0.900315i	$-0.643336\pi$
$397$	−17.3441	−0.870476	−0.435238	+	0.900315i	$0.643336\pi$
$398$	−1.32604	−0.0664685
$399$	−16.8840	−0.845259
$400$	3.35950	0.167975
$401$	−21.3352	−1.06543	−0.532714	−	0.846295i	$-0.678828\pi$
$401$	−21.3352	−1.06543	−0.532714	+	0.846295i	$0.678828\pi$
$402$	3.99287	0.199146
$403$	55.0621	2.74284
$404$	−17.4194	−0.866646
$405$	−7.00000	−0.347833
$406$	−7.83497	−0.388843
$407$	−11.1181	−0.551103
$408$	−6.82448	−0.337862
$409$	−12.5328	−0.619709	−0.309855	−	0.950784i	$-0.600280\pi$
$409$	−12.5328	−0.619709	−0.309855	+	0.950784i	$0.600280\pi$
$410$	−0.710689	−0.0350985
$411$	−18.3653	−0.905894
$412$	−8.88403	−0.437685
$413$	58.6031	2.88367
$414$	−0.185159	−0.00910008
$415$	−12.5293	−0.615038
$416$	21.7133	1.06458
$417$	19.3462	0.947389
$418$	2.33206	0.114065
$419$	12.0288	0.587644	0.293822	−	0.955860i	$-0.405073\pi$
$419$	12.0288	0.587644	0.293822	+	0.955860i	$0.405073\pi$
$420$	−12.0071	−0.585888
$421$	−38.3721	−1.87014	−0.935071	−	0.354462i	$-0.884664\pi$
$421$	−38.3721	−1.87014	−0.935071	+	0.354462i	$0.884664\pi$
$422$	−5.05854	−0.246246
$423$	−0.136257	−0.00662507
$424$	11.5935	0.563032
$425$	−3.40617	−0.165223
$426$	−6.40493	−0.310320
$427$	10.0163	0.484721
$428$	0.711866	0.0344093
$429$	−24.5445	−1.18502
$430$	−0.774424	−0.0373460
$431$	−29.4789	−1.41995	−0.709975	−	0.704227i	$-0.751294\pi$
$431$	−29.4789	−1.41995	−0.709975	+	0.704227i	$0.751294\pi$
$432$	−18.6840	−0.898936
$433$	−12.0531	−0.579236	−0.289618	−	0.957142i	$-0.593528\pi$
$433$	−12.0531	−0.579236	−0.289618	+	0.957142i	$0.593528\pi$
$434$	12.4891	0.599494
$435$	9.12667	0.437590
$436$	−36.0041	−1.72428
$437$	2.65945	0.127219
$438$	1.07599	0.0514129
$439$	16.1656	0.771541	0.385771	−	0.922595i	$-0.373936\pi$
$439$	16.1656	0.771541	0.385771	+	0.922595i	$0.373936\pi$
$440$	3.41224	0.162672
$441$	−5.35119	−0.254819
$442$	−6.63782	−0.315729
$443$	19.3729	0.920433	0.460217	−	0.887807i	$-0.347772\pi$
$443$	19.3729	0.920433	0.460217	+	0.887807i	$0.347772\pi$
$444$	12.3467	0.585947
$445$	−9.35682	−0.443556
$446$	5.82961	0.276040
$447$	−3.32805	−0.157411
$448$	−22.3920	−1.05792
$449$	21.4728	1.01337	0.506683	−	0.862132i	$-0.330871\pi$
$449$	21.4728	1.01337	0.506683	+	0.862132i	$0.330871\pi$
$450$	0.185159	0.00872849
$451$	5.73215	0.269916
$452$	−5.56512	−0.261761
$453$	−1.22914	−0.0577502
$454$	−2.84824	−0.133674
$455$	−24.0288	−1.12649
$456$	−5.32840	−0.249525
$457$	−5.95602	−0.278611	−0.139305	−	0.990249i	$-0.544487\pi$
$457$	−5.95602	−0.278611	−0.139305	+	0.990249i	$0.544487\pi$
$458$	1.10147	0.0514683
$459$	18.9436	0.884210
$460$	1.89128	0.0881814
$461$	34.6918	1.61576	0.807879	−	0.589348i	$-0.200615\pi$
$461$	34.6918	1.61576	0.807879	+	0.589348i	$0.200615\pi$
$462$	−5.56712	−0.259006
$463$	−30.6399	−1.42396	−0.711979	−	0.702200i	$-0.752201\pi$
$463$	−30.6399	−1.42396	−0.711979	+	0.702200i	$0.752201\pi$
$464$	19.6350	0.911531
$465$	−14.5481	−0.674650
$466$	−2.71605	−0.125819
$467$	−8.89547	−0.411633	−0.205817	−	0.978591i	$-0.565985\pi$
$467$	−8.89547	−0.411633	−0.205817	+	0.978591i	$0.565985\pi$
$468$	−6.27699	−0.290154
$469$	−31.5283	−1.45584
$470$	−0.0800064	−0.00369042
$471$	−14.3487	−0.661152
$472$	18.4944	0.851275
$473$	6.24621	0.287201
$474$	2.13826	0.0982136
$475$	−2.65945	−0.122024
$476$	26.1908	1.20045
$477$	−5.07411	−0.232327
$478$	0.944750	0.0432119
$479$	−9.13224	−0.417263	−0.208631	−	0.977994i	$-0.566901\pi$
$479$	−9.13224	−0.417263	−0.208631	+	0.977994i	$0.566901\pi$
$480$	−5.73690	−0.261852
$481$	24.7083	1.12660
$482$	−1.54348	−0.0703037
$483$	−6.34868	−0.288875
$484$	7.42763	0.337619
$485$	3.47179	0.157646
$486$	−1.89718	−0.0860579
$487$	14.2085	0.643849	0.321924	−	0.946765i	$-0.395670\pi$
$487$	14.2085	0.643849	0.321924	+	0.946765i	$0.395670\pi$
$488$	3.16101	0.143092
$489$	−27.6282	−1.24939
$490$	−3.14206	−0.141944
$491$	−23.1760	−1.04592	−0.522960	−	0.852357i	$-0.675172\pi$
$491$	−23.1760	−1.04592	−0.522960	+	0.852357i	$0.675172\pi$
$492$	−6.36558	−0.286982
$493$	−19.9077	−0.896599
$494$	−5.18265	−0.233179
$495$	−1.49342	−0.0671244
$496$	−31.2984	−1.40534
$497$	50.5743	2.26857
$498$	6.45115	0.289083
$499$	−2.19831	−0.0984099	−0.0492050	−	0.998789i	$-0.515669\pi$
$499$	−2.19831	−0.0984099	−0.0492050	+	0.998789i	$0.515669\pi$
$500$	−1.89128	−0.0845806
$501$	22.2462	0.993887
$502$	6.12712	0.273467
$503$	−44.0461	−1.96392	−0.981959	−	0.189092i	$-0.939445\pi$
$503$	−44.0461	−1.96392	−0.981959	+	0.189092i	$0.939445\pi$
$504$	−2.92931	−0.130482
$505$	9.21036	0.409856
$506$	0.876894	0.0389827
$507$	34.2462	1.52093
$508$	−11.3544	−0.503771
$509$	−9.09254	−0.403020	−0.201510	−	0.979486i	$-0.564585\pi$
$509$	−9.09254	−0.403020	−0.201510	+	0.979486i	$0.564585\pi$
$510$	1.75379	0.0776591
$511$	−8.49619	−0.375850
$512$	−20.9632	−0.926449
$513$	14.7907	0.653025
$514$	9.47061	0.417731
$515$	4.69736	0.206991
$516$	−6.93644	−0.305360
$517$	0.645301	0.0283803
$518$	5.60427	0.246238
$519$	−1.88001	−0.0825235
$520$	−7.58319	−0.332545
$521$	−17.1231	−0.750177	−0.375088	−	0.926989i	$-0.622388\pi$
$521$	−17.1231	−0.750177	−0.375088	+	0.926989i	$0.622388\pi$
$522$	1.08218	0.0473659
$523$	−26.6112	−1.16362	−0.581812	−	0.813323i	$-0.697656\pi$
$523$	−26.6112	−1.16362	−0.581812	+	0.813323i	$0.697656\pi$
$524$	37.7512	1.64917
$525$	6.34868	0.277079
$526$	3.03514	0.132338
$527$	31.7332	1.38232
$528$	13.9516	0.607165
$529$	1.00000	0.0434783
$530$	−2.97936	−0.129415
$531$	−8.09439	−0.351267
$532$	20.4491	0.886583
$533$	−12.7388	−0.551780
$534$	4.81770	0.208482
$535$	−0.376394	−0.0162729
$536$	−9.94994	−0.429772
$537$	11.5944	0.500337
$538$	−5.57880	−0.240519
$539$	25.3427	1.09159
$540$	10.5185	0.452642
$541$	31.5941	1.35834	0.679168	−	0.733983i	$-0.262341\pi$
$541$	31.5941	1.35834	0.679168	+	0.733983i	$0.262341\pi$
$542$	−0.379456	−0.0162990
$543$	37.5476	1.61132
$544$	12.5137	0.536521
$545$	19.0369	0.815452
$546$	12.3721	0.529476
$547$	22.7009	0.970622	0.485311	−	0.874342i	$-0.338706\pi$
$547$	22.7009	0.970622	0.485311	+	0.874342i	$0.338706\pi$
$548$	22.2432	0.950182
$549$	−1.38347	−0.0590451
$550$	−0.876894	−0.0373909
$551$	−15.5435	−0.662175
$552$	−2.00357	−0.0852775
$553$	−16.8840	−0.717982
$554$	−0.849065	−0.0360733
$555$	−6.52821	−0.277107
$556$	−23.4313	−0.993706
$557$	35.1292	1.48847	0.744236	−	0.667917i	$-0.232814\pi$
$557$	35.1292	1.48847	0.744236	+	0.667917i	$0.232814\pi$
$558$	−1.72502	−0.0730258
$559$	−13.8813	−0.587115
$560$	13.6585	0.577175
$561$	−14.1454	−0.597219
$562$	−9.05819	−0.382097
$563$	32.0511	1.35079	0.675397	−	0.737455i	$-0.263972\pi$
$563$	32.0511	1.35079	0.675397	+	0.737455i	$0.263972\pi$
$564$	−0.716609	−0.0301747
$565$	2.94252	0.123793
$566$	0.779440	0.0327623
$567$	−28.4593	−1.19518
$568$	15.9606	0.669694
$569$	−22.0575	−0.924700	−0.462350	−	0.886697i	$-0.652994\pi$
$569$	−22.0575	−0.924700	−0.462350	+	0.886697i	$0.652994\pi$
$570$	1.36932	0.0573544
$571$	8.49242	0.355397	0.177698	−	0.984085i	$-0.443135\pi$
$571$	8.49242	0.355397	0.177698	+	0.984085i	$0.443135\pi$
$572$	29.7271	1.24295
$573$	−7.61553	−0.318143
$574$	−2.88939	−0.120601
$575$	−1.00000	−0.0417029
$576$	3.09283	0.128868
$577$	27.1554	1.13050	0.565248	−	0.824921i	$-0.308780\pi$
$577$	27.1554	1.13050	0.565248	+	0.824921i	$0.308780\pi$
$578$	1.77988	0.0740331
$579$	16.0394	0.666573
$580$	−11.0538	−0.458984
$581$	−50.9393	−2.11332
$582$	−1.78758	−0.0740974
$583$	24.0304	0.995239
$584$	−2.68130	−0.110953
$585$	3.31891	0.137220
$586$	−11.2502	−0.464743
$587$	−30.8322	−1.27258	−0.636290	−	0.771450i	$-0.719532\pi$
$587$	−30.8322	−1.27258	−0.636290	+	0.771450i	$0.719532\pi$
$588$	−28.1431	−1.16060
$589$	24.7765	1.02090
$590$	−4.75279	−0.195669
$591$	6.69214	0.275278
$592$	−14.0447	−0.577233
$593$	−27.7559	−1.13980	−0.569899	−	0.821715i	$-0.693018\pi$
$593$	−27.7559	−1.13980	−0.569899	+	0.821715i	$0.693018\pi$
$594$	4.87689	0.200101
$595$	−13.8482	−0.567720
$596$	4.03077	0.165107
$597$	6.28000	0.257023
$598$	−1.94876	−0.0796909
$599$	−30.7712	−1.25728	−0.628638	−	0.777698i	$-0.716387\pi$
$599$	−30.7712	−1.25728	−0.628638	+	0.777698i	$0.716387\pi$
$600$	2.00357	0.0817953
$601$	−17.9830	−0.733541	−0.366771	−	0.930311i	$-0.619537\pi$
$601$	−17.9830	−0.733541	−0.366771	+	0.930311i	$0.619537\pi$
$602$	−3.14851	−0.128324
$603$	4.35476	0.177339
$604$	1.48868	0.0605736
$605$	−3.92730	−0.159668
$606$	−4.74229	−0.192642
$607$	−18.3220	−0.743668	−0.371834	−	0.928299i	$-0.621271\pi$
$607$	−18.3220	−0.743668	−0.371834	+	0.928299i	$0.621271\pi$
$608$	9.77041	0.396242
$609$	37.1056	1.50359
$610$	−0.812333	−0.0328904
$611$	−1.43408	−0.0580168
$612$	−3.61753	−0.146230
$613$	35.4451	1.43162	0.715808	−	0.698297i	$-0.246059\pi$
$613$	35.4451	1.43162	0.715808	+	0.698297i	$0.246059\pi$
$614$	−0.492423	−0.0198726
$615$	3.36575	0.135720
$616$	13.8729	0.558954
$617$	43.1567	1.73742	0.868712	−	0.495318i	$-0.164948\pi$
$617$	43.1567	1.73742	0.868712	+	0.495318i	$0.164948\pi$
$618$	−2.41861	−0.0972907
$619$	5.81133	0.233577	0.116789	−	0.993157i	$-0.462740\pi$
$619$	5.81133	0.233577	0.116789	+	0.993157i	$0.462740\pi$
$620$	17.6199	0.707633
$621$	5.56155	0.223177
$622$	4.65114	0.186494
$623$	−38.0413	−1.52409
$624$	−31.0053	−1.24121
$625$	1.00000	0.0400000
$626$	10.8336	0.432999
$627$	−11.0444	−0.441070
$628$	17.3785	0.693476
$629$	14.2398	0.567777
$630$	0.752787	0.0299918
$631$	−27.4626	−1.09327	−0.546635	−	0.837371i	$-0.684092\pi$
$631$	−27.4626	−1.09327	−0.546635	+	0.837371i	$0.684092\pi$
$632$	−5.32840	−0.211952
$633$	23.9567	0.952194
$634$	−4.14876	−0.164768
$635$	6.00357	0.238244
$636$	−26.6859	−1.05816
$637$	−56.3202	−2.23149
$638$	−5.12511	−0.202905
$639$	−6.98544	−0.276340
$640$	9.16370	0.362227
$641$	37.7938	1.49277	0.746383	−	0.665517i	$-0.231789\pi$
$641$	37.7938	1.49277	0.746383	+	0.665517i	$0.231789\pi$
$642$	0.193800	0.00764867
$643$	12.5343	0.494304	0.247152	−	0.968977i	$-0.420505\pi$
$643$	12.5343	0.494304	0.247152	+	0.968977i	$0.420505\pi$
$644$	7.68923	0.302998
$645$	3.66759	0.144411
$646$	−2.98685	−0.117516
$647$	−21.5133	−0.845774	−0.422887	−	0.906183i	$-0.638983\pi$
$647$	−21.5133	−0.845774	−0.422887	+	0.906183i	$0.638983\pi$
$648$	−8.98143	−0.352824
$649$	38.3342	1.50475
$650$	1.94876	0.0764368
$651$	−59.1469	−2.31815
$652$	33.4620	1.31047
$653$	19.9011	0.778790	0.389395	−	0.921071i	$-0.372684\pi$
$653$	19.9011	0.778790	0.389395	+	0.921071i	$0.372684\pi$
$654$	−9.80184	−0.383282
$655$	−19.9606	−0.779927
$656$	7.24102	0.282714
$657$	1.17351	0.0457831
$658$	−0.325276	−0.0126806
$659$	−4.95773	−0.193126	−0.0965628	−	0.995327i	$-0.530785\pi$
$659$	−4.95773	−0.193126	−0.0965628	+	0.995327i	$0.530785\pi$
$660$	−7.85426	−0.305726
$661$	22.0000	0.855701	0.427850	−	0.903850i	$-0.359271\pi$
$661$	22.0000	0.855701	0.427850	+	0.903850i	$0.359271\pi$
$662$	−0.519902	−0.0202066
$663$	31.4360	1.22087
$664$	−16.0758	−0.623863
$665$	−10.8123	−0.419284
$666$	−0.774075	−0.0299948
$667$	−5.84461	−0.226304
$668$	−26.9436	−1.04248
$669$	−27.6084	−1.06740
$670$	2.55698	0.0987849
$671$	6.55197	0.252936
$672$	−23.3240	−0.899744
$673$	−28.5176	−1.09927	−0.549637	−	0.835404i	$-0.685234\pi$
$673$	−28.5176	−1.09927	−0.549637	+	0.835404i	$0.685234\pi$
$674$	5.10147	0.196501
$675$	−5.56155	−0.214064
$676$	−41.4774	−1.59529
$677$	28.6214	1.10001	0.550004	−	0.835162i	$-0.314626\pi$
$677$	28.6214	1.10001	0.550004	+	0.835162i	$0.314626\pi$
$678$	−1.51506	−0.0581856
$679$	14.1150	0.541683
$680$	−4.37032	−0.167594
$681$	13.4890	0.516898
$682$	8.16950	0.312826
$683$	11.3983	0.436144	0.218072	−	0.975933i	$-0.430023\pi$
$683$	11.3983	0.436144	0.218072	+	0.975933i	$0.430023\pi$
$684$	−2.82448	−0.107997
$685$	−11.7609	−0.449362
$686$	−3.39060	−0.129454
$687$	−5.21644	−0.199020
$688$	7.89040	0.300819
$689$	−53.4040	−2.03453
$690$	0.514886	0.0196014
$691$	45.1898	1.71910	0.859550	−	0.511051i	$-0.170744\pi$
$691$	45.1898	1.71910	0.859550	+	0.511051i	$0.170744\pi$
$692$	2.27699	0.0865580
$693$	−6.07170	−0.230645
$694$	2.96857	0.112686
$695$	12.3891	0.469945
$696$	11.7101	0.443869
$697$	−7.34160	−0.278083
$698$	−0.364011	−0.0137780
$699$	12.8629	0.486521
$700$	−7.68923	−0.290625
$701$	40.4871	1.52918	0.764588	−	0.644520i	$-0.222943\pi$
$701$	40.4871	1.52918	0.764588	+	0.644520i	$0.222943\pi$
$702$	−10.8382	−0.409060
$703$	11.1181	0.419327
$704$	−14.6473	−0.552041
$705$	0.378902	0.0142703
$706$	−0.648131	−0.0243927
$707$	37.4458	1.40830
$708$	−42.5702	−1.59989
$709$	−41.6443	−1.56398	−0.781992	−	0.623288i	$-0.785796\pi$
$709$	−41.6443	−1.56398	−0.781992	+	0.623288i	$0.785796\pi$
$710$	−4.10164	−0.153932
$711$	2.33206	0.0874591
$712$	−12.0054	−0.449920
$713$	9.31640	0.348902
$714$	7.13024	0.266843
$715$	−15.7180	−0.587820
$716$	−14.0427	−0.524799
$717$	−4.47424	−0.167093
$718$	0.858620	0.0320434
$719$	49.2288	1.83592	0.917961	−	0.396670i	$-0.129834\pi$
$719$	49.2288	1.83592	0.917961	+	0.396670i	$0.129834\pi$
$720$	−1.88654	−0.0703071
$721$	19.0977	0.711235
$722$	3.93276	0.146362
$723$	7.30977	0.271853
$724$	−45.4759	−1.69010
$725$	5.84461	0.217063
$726$	2.02211	0.0750477
$727$	27.9581	1.03691	0.518455	−	0.855105i	$-0.326507\pi$
$727$	27.9581	1.03691	0.518455	+	0.855105i	$0.326507\pi$
$728$	−30.8304	−1.14265
$729$	29.9848	1.11055
$730$	0.689053	0.0255030
$731$	−8.00000	−0.295891
$732$	−7.27598	−0.268928
$733$	−18.6655	−0.689427	−0.344714	−	0.938708i	$-0.612024\pi$
$733$	−18.6655	−0.689427	−0.344714	+	0.938708i	$0.612024\pi$
$734$	−2.30164	−0.0849549
$735$	14.8805	0.548874
$736$	3.67384	0.135420
$737$	−20.6237	−0.759682
$738$	0.399090	0.0146907
$739$	2.50407	0.0921136	0.0460568	−	0.998939i	$-0.485334\pi$
$739$	2.50407	0.0921136	0.0460568	+	0.998939i	$0.485334\pi$
$740$	7.90667	0.290655
$741$	24.5445	0.901664
$742$	−12.1130	−0.444681
$743$	4.69736	0.172330	0.0861648	−	0.996281i	$-0.472539\pi$
$743$	4.69736	0.172330	0.0861648	+	0.996281i	$0.472539\pi$
$744$	−18.6660	−0.684330
$745$	−2.13124	−0.0780826
$746$	−8.01963	−0.293620
$747$	7.03585	0.257428
$748$	17.1322	0.626417
$749$	−1.53027	−0.0559150
$750$	−0.514886	−0.0188010
$751$	−16.0720	−0.586477	−0.293239	−	0.956039i	$-0.594733\pi$
$751$	−16.0720	−0.586477	−0.293239	+	0.956039i	$0.594733\pi$
$752$	0.815163	0.0297259
$753$	−29.0174	−1.05745
$754$	11.3898	0.414791
$755$	−0.787129	−0.0286465
$756$	42.7640	1.55531
$757$	−25.8357	−0.939014	−0.469507	−	0.882929i	$-0.655568\pi$
$757$	−25.8357	−0.939014	−0.469507	+	0.882929i	$0.655568\pi$
$758$	8.12912	0.295263
$759$	−4.15288	−0.150740
$760$	−3.41224	−0.123775
$761$	15.6005	0.565518	0.282759	−	0.959191i	$-0.408750\pi$
$761$	15.6005	0.565518	0.282759	+	0.959191i	$0.408750\pi$
$762$	−3.09116	−0.111981
$763$	77.3968	2.80195
$764$	9.22357	0.333697
$765$	1.91274	0.0691553
$766$	1.64630	0.0594833
$767$	−85.1919	−3.07610
$768$	12.4827	0.450429
$769$	−37.7722	−1.36210	−0.681050	−	0.732237i	$-0.738476\pi$
$769$	−37.7722	−1.36210	−0.681050	+	0.732237i	$0.738476\pi$
$770$	−3.56512	−0.128478
$771$	−44.8518	−1.61530
$772$	−19.4261	−0.699161
$773$	−42.9078	−1.54329	−0.771643	−	0.636056i	$-0.780565\pi$
$773$	−42.9078	−1.54329	−0.771643	+	0.636056i	$0.780565\pi$
$774$	0.434880	0.0156314
$775$	−9.31640	−0.334655
$776$	4.45451	0.159908
$777$	−26.5412	−0.952162
$778$	10.5590	0.378558
$779$	−5.73215	−0.205376
$780$	17.4549	0.624985
$781$	33.0823	1.18378
$782$	−1.12311	−0.0401622
$783$	−32.5051	−1.16164
$784$	32.0136	1.14334
$785$	−9.18873	−0.327960
$786$	10.2775	0.366585
$787$	−33.5324	−1.19530	−0.597650	−	0.801757i	$-0.703899\pi$
$787$	−33.5324	−1.19530	−0.597650	+	0.801757i	$0.703899\pi$
$788$	−8.10521	−0.288736
$789$	−14.3741	−0.511731
$790$	1.36932	0.0487181
$791$	11.9632	0.425361
$792$	−1.91615	−0.0680876
$793$	−14.5608	−0.517068
$794$	5.71883	0.202954
$795$	14.1100	0.500428
$796$	−7.60604	−0.269589
$797$	32.7488	1.16002	0.580012	−	0.814608i	$-0.303048\pi$
$797$	32.7488	1.16002	0.580012	+	0.814608i	$0.303048\pi$
$798$	5.56712	0.197074
$799$	−0.826486	−0.0292390
$800$	−3.67384	−0.129890
$801$	5.25435	0.185653
$802$	7.03479	0.248407
$803$	−5.55764	−0.196125
$804$	22.9027	0.807714
$805$	−4.06562	−0.143294
$806$	−18.1555	−0.639499
$807$	26.4206	0.930049
$808$	11.8175	0.415737
$809$	28.8513	1.01436	0.507179	−	0.861841i	$-0.330688\pi$
$809$	28.8513	1.01436	0.507179	+	0.861841i	$0.330688\pi$
$810$	2.30809	0.0810980
$811$	36.1791	1.27042	0.635211	−	0.772339i	$-0.280913\pi$
$811$	36.1791	1.27042	0.635211	+	0.772339i	$0.280913\pi$
$812$	−44.9406	−1.57710
$813$	1.79706	0.0630257
$814$	3.66594	0.128491
$815$	−17.6928	−0.619752
$816$	−17.8689	−0.625536
$817$	−6.24621	−0.218527
$818$	4.13242	0.144487
$819$	13.4934	0.471498
$820$	−4.07644	−0.142355
$821$	5.73752	0.200241	0.100120	−	0.994975i	$-0.468077\pi$
$821$	5.73752	0.200241	0.100120	+	0.994975i	$0.468077\pi$
$822$	6.05554	0.211211
$823$	−6.87333	−0.239589	−0.119795	−	0.992799i	$-0.538224\pi$
$823$	−6.87333	−0.239589	−0.119795	+	0.992799i	$0.538224\pi$
$824$	6.02700	0.209961
$825$	4.15288	0.144585
$826$	−19.3230	−0.672334
$827$	−19.7332	−0.686191	−0.343095	−	0.939301i	$-0.611475\pi$
$827$	−19.7332	−0.686191	−0.343095	+	0.939301i	$0.611475\pi$
$828$	−1.06205	−0.0369089
$829$	−2.88833	−0.100316	−0.0501580	−	0.998741i	$-0.515972\pi$
$829$	−2.88833	−0.100316	−0.0501580	+	0.998741i	$0.515972\pi$
$830$	4.13124	0.143397
$831$	4.02108	0.139490
$832$	32.5514	1.12852
$833$	−32.4583	−1.12461
$834$	−6.37898	−0.220886
$835$	14.2462	0.493010
$836$	13.3765	0.462634
$837$	51.8137	1.79094
$838$	−3.96621	−0.137011
$839$	24.1904	0.835147	0.417573	−	0.908643i	$-0.362881\pi$
$839$	24.1904	0.835147	0.417573	+	0.908643i	$0.362881\pi$
$840$	8.14574	0.281055
$841$	5.15951	0.177914
$842$	12.6523	0.436028
$843$	42.8986	1.47751
$844$	−29.0153	−0.998747
$845$	21.9309	0.754445
$846$	0.0449278	0.00154465
$847$	−15.9669	−0.548630
$848$	30.3559	1.04243
$849$	−3.69135	−0.126687
$850$	1.12311	0.0385222
$851$	4.18059	0.143309
$852$	−36.7380	−1.25862
$853$	27.5381	0.942887	0.471444	−	0.881896i	$-0.343733\pi$
$853$	27.5381	0.942887	0.471444	+	0.881896i	$0.343733\pi$
$854$	−3.30264	−0.113014
$855$	1.49342	0.0510740
$856$	−0.482936	−0.0165064
$857$	−17.3995	−0.594357	−0.297178	−	0.954822i	$-0.596046\pi$
$857$	−17.3995	−0.594357	−0.297178	+	0.954822i	$0.596046\pi$
$858$	8.09298	0.276290
$859$	1.24560	0.0424993	0.0212497	−	0.999774i	$-0.493236\pi$
$859$	1.24560	0.0424993	0.0212497	+	0.999774i	$0.493236\pi$
$860$	−4.44201	−0.151471
$861$	13.6839	0.466345
$862$	9.72000	0.331065
$863$	−4.60383	−0.156716	−0.0783580	−	0.996925i	$-0.524968\pi$
$863$	−4.60383	−0.156716	−0.0783580	+	0.996925i	$0.524968\pi$
$864$	20.4323	0.695119
$865$	−1.20394	−0.0409352
$866$	3.97424	0.135050
$867$	−8.42931	−0.286274
$868$	71.6359	2.43148
$869$	−11.0444	−0.374655
$870$	−3.00931	−0.102025
$871$	45.8330	1.55299
$872$	24.4255	0.827152
$873$	−1.94959	−0.0659837
$874$	−0.876894	−0.0296614
$875$	4.06562	0.137443
$876$	6.17178	0.208525
$877$	33.8276	1.14228	0.571138	−	0.820854i	$-0.306502\pi$
$877$	33.8276	1.14228	0.571138	+	0.820854i	$0.306502\pi$
$878$	−5.33023	−0.179887
$879$	53.2799	1.79709
$880$	8.93444	0.301180
$881$	26.1134	0.879782	0.439891	−	0.898051i	$-0.355017\pi$
$881$	26.1134	0.879782	0.439891	+	0.898051i	$0.355017\pi$
$882$	1.76443	0.0594115
$883$	−54.5412	−1.83546	−0.917729	−	0.397206i	$-0.869980\pi$
$883$	−54.5412	−1.83546	−0.917729	+	0.397206i	$0.869980\pi$
$884$	−38.0738	−1.28056
$885$	22.5087	0.756621
$886$	−6.38777	−0.214601
$887$	22.6201	0.759509	0.379754	−	0.925087i	$-0.376009\pi$
$887$	22.6201	0.759509	0.379754	+	0.925087i	$0.376009\pi$
$888$	−8.37609	−0.281083
$889$	24.4082	0.818626
$890$	3.08520	0.103416
$891$	−18.6162	−0.623666
$892$	33.4380	1.11959
$893$	−0.645301	−0.0215942
$894$	1.09735	0.0367008
$895$	7.42495	0.248189
$896$	37.2561	1.24464
$897$	9.22914	0.308152
$898$	−7.08018	−0.236269
$899$	−54.4508	−1.81603
$900$	1.06205	0.0354018
$901$	−30.7776	−1.02535
$902$	−1.89005	−0.0629317
$903$	14.9110	0.496208
$904$	3.77543	0.125569
$905$	24.0450	0.799284
$906$	0.405282	0.0134646
$907$	−3.62149	−0.120250	−0.0601248	−	0.998191i	$-0.519150\pi$
$907$	−3.62149	−0.120250	−0.0601248	+	0.998191i	$0.519150\pi$
$908$	−16.3372	−0.542169
$909$	−5.17211	−0.171548
$910$	7.92294	0.262643
$911$	26.9241	0.892034	0.446017	−	0.895025i	$-0.352842\pi$
$911$	26.9241	0.892034	0.446017	+	0.895025i	$0.352842\pi$
$912$	−13.9516	−0.461983
$913$	−33.3210	−1.10277
$914$	1.96386	0.0649587
$915$	3.84712	0.127182
$916$	6.31791	0.208750
$917$	−81.1524	−2.67989
$918$	−6.24621	−0.206156
$919$	−19.5059	−0.643441	−0.321721	−	0.946835i	$-0.604261\pi$
$919$	−19.5059	−0.643441	−0.321721	+	0.946835i	$0.604261\pi$
$920$	−1.28306	−0.0423013
$921$	2.33206	0.0768440
$922$	−11.4388	−0.376718
$923$	−73.5204	−2.41995
$924$	−31.9324	−1.05050
$925$	−4.18059	−0.137457
$926$	10.1028	0.331999
$927$	−2.63782	−0.0866373
$928$	−21.4722	−0.704859
$929$	27.2139	0.892860	0.446430	−	0.894819i	$-0.352695\pi$
$929$	27.2139	0.892860	0.446430	+	0.894819i	$0.352695\pi$
$930$	4.79689	0.157296
$931$	−25.3427	−0.830572
$932$	−15.5790	−0.510307
$933$	−22.0273	−0.721142
$934$	2.93308	0.0959732
$935$	−9.05854	−0.296246
$936$	4.25836	0.139189
$937$	3.85626	0.125978	0.0629892	−	0.998014i	$-0.479937\pi$
$937$	3.85626	0.125978	0.0629892	+	0.998014i	$0.479937\pi$
$938$	10.3957	0.339433
$939$	−51.3069	−1.67434
$940$	−0.458908	−0.0149679
$941$	−60.6471	−1.97704	−0.988519	−	0.151097i	$-0.951720\pi$
$941$	−60.6471	−1.97704	−0.988519	+	0.151097i	$0.951720\pi$
$942$	4.73115	0.154149
$943$	−2.15539	−0.0701890
$944$	48.4248	1.57609
$945$	−22.6112	−0.735541
$946$	−2.05955	−0.0669616
$947$	−1.11954	−0.0363801	−0.0181901	−	0.999835i	$-0.505790\pi$
$947$	−1.11954	−0.0363801	−0.0181901	+	0.999835i	$0.505790\pi$
$948$	12.2648	0.398343
$949$	12.3510	0.400931
$950$	0.876894	0.0284502
$951$	19.6481	0.637132
$952$	−17.7681	−0.575866
$953$	14.8590	0.481331	0.240666	−	0.970608i	$-0.422634\pi$
$953$	14.8590	0.481331	0.240666	+	0.970608i	$0.422634\pi$
$954$	1.67307	0.0541677
$955$	−4.87689	−0.157813
$956$	5.41899	0.175263
$957$	24.2720	0.784601
$958$	3.01115	0.0972858
$959$	−47.8155	−1.54404
$960$	−8.60046	−0.277579
$961$	55.7953	1.79985
$962$	−8.14699	−0.262669
$963$	0.211365	0.00681114
$964$	−8.85325	−0.285144
$965$	10.2714	0.330649
$966$	2.09333	0.0673519
$967$	38.5718	1.24039	0.620193	−	0.784449i	$-0.287054\pi$
$967$	38.5718	1.24039	0.620193	+	0.784449i	$0.287054\pi$
$968$	−5.03897	−0.161959
$969$	14.1454	0.454416
$970$	−1.14474	−0.0367555
$971$	−25.4984	−0.818284	−0.409142	−	0.912471i	$-0.634172\pi$
$971$	−25.4984	−0.818284	−0.409142	+	0.912471i	$0.634172\pi$
$972$	−10.8820	−0.349041
$973$	50.3694	1.61477
$974$	−4.68493	−0.150115
$975$	−9.22914	−0.295569
$976$	8.27664	0.264929
$977$	−15.8952	−0.508533	−0.254267	−	0.967134i	$-0.581834\pi$
$977$	−15.8952	−0.508533	−0.254267	+	0.967134i	$0.581834\pi$
$978$	9.10978	0.291299
$979$	−24.8840	−0.795297
$980$	−18.0225	−0.575708
$981$	−10.6902	−0.341313
$982$	7.64176	0.243858
$983$	−2.50587	−0.0799247	−0.0399624	−	0.999201i	$-0.512724\pi$
$983$	−2.50587	−0.0799247	−0.0399624	+	0.999201i	$0.512724\pi$
$984$	4.31846	0.137668
$985$	4.28557	0.136550
$986$	6.56412	0.209044
$987$	1.54047	0.0490337
$988$	−29.7271	−0.945746
$989$	−2.34868	−0.0746837
$990$	0.492423	0.0156502
$991$	12.5272	0.397938	0.198969	−	0.980006i	$-0.436241\pi$
$991$	12.5272	0.397938	0.198969	+	0.980006i	$0.436241\pi$
$992$	34.2270	1.08671
$993$	2.46220	0.0781356
$994$	−16.6757	−0.528922
$995$	4.02164	0.127494
$996$	37.0031	1.17249
$997$	−9.34803	−0.296055	−0.148028	−	0.988983i	$-0.547292\pi$
$997$	−9.34803	−0.296055	−0.148028	+	0.988983i	$0.547292\pi$
$998$	0.724843	0.0229445
$999$	23.2506	0.735616

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	115.2.a.c.1.2	✓	4
3.2	odd	2		1035.2.a.o.1.3		4
4.3	odd	2		1840.2.a.u.1.1		4
5.2	odd	4		575.2.b.e.24.4		8
5.3	odd	4		575.2.b.e.24.5		8
5.4	even	2		575.2.a.h.1.3		4
7.6	odd	2		5635.2.a.v.1.2		4
8.3	odd	2		7360.2.a.cg.1.3		4
8.5	even	2		7360.2.a.cj.1.2		4
15.14	odd	2		5175.2.a.bx.1.2		4
20.19	odd	2		9200.2.a.cl.1.4		4
23.22	odd	2		2645.2.a.m.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
115.2.a.c.1.2	✓	4	1.1	even	1	trivial
575.2.a.h.1.3		4	5.4	even	2
575.2.b.e.24.4		8	5.2	odd	4
575.2.b.e.24.5		8	5.3	odd	4
1035.2.a.o.1.3		4	3.2	odd	2
1840.2.a.u.1.1		4	4.3	odd	2
2645.2.a.m.1.2		4	23.22	odd	2
5175.2.a.bx.1.2		4	15.14	odd	2
5635.2.a.v.1.2		4	7.6	odd	2
7360.2.a.cg.1.3		4	8.3	odd	2
7360.2.a.cj.1.2		4	8.5	even	2
9200.2.a.cl.1.4		4	20.19	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-0.329727 q^{2} +1.56155 q^{3} -1.89128 q^{4} +1.00000 q^{5} -0.514886 q^{6} +4.06562 q^{7} +1.28306 q^{8} -0.561553 q^{9} +O(q^{10})\)	\(q-0.329727 q^{2} +1.56155 q^{3} -1.89128 q^{4} +1.00000 q^{5} -0.514886 q^{6} +4.06562 q^{7} +1.28306 q^{8} -0.561553 q^{9} -0.329727 q^{10} +2.65945 q^{11} -2.95333 q^{12} -5.91023 q^{13} -1.34055 q^{14} +1.56155 q^{15} +3.35950 q^{16} -3.40617 q^{17} +0.185159 q^{18} -2.65945 q^{19} -1.89128 q^{20} +6.34868 q^{21} -0.876894 q^{22} -1.00000 q^{23} +2.00357 q^{24} +1.00000 q^{25} +1.94876 q^{26} -5.56155 q^{27} -7.68923 q^{28} +5.84461 q^{29} -0.514886 q^{30} -9.31640 q^{31} -3.67384 q^{32} +4.15288 q^{33} +1.12311 q^{34} +4.06562 q^{35} +1.06205 q^{36} -4.18059 q^{37} +0.876894 q^{38} -9.22914 q^{39} +1.28306 q^{40} +2.15539 q^{41} -2.09333 q^{42} +2.34868 q^{43} -5.02977 q^{44} -0.561553 q^{45} +0.329727 q^{46} +0.242644 q^{47} +5.24604 q^{48} +9.52927 q^{49} -0.329727 q^{50} -5.31891 q^{51} +11.1779 q^{52} +9.03585 q^{53} +1.83380 q^{54} +2.65945 q^{55} +5.21644 q^{56} -4.15288 q^{57} -1.92713 q^{58} +14.4143 q^{59} -2.95333 q^{60} +2.46365 q^{61} +3.07187 q^{62} -2.28306 q^{63} -5.50764 q^{64} -5.91023 q^{65} -1.36932 q^{66} -7.75485 q^{67} +6.44201 q^{68} -1.56155 q^{69} -1.34055 q^{70} +12.4395 q^{71} -0.720506 q^{72} -2.08977 q^{73} +1.37845 q^{74} +1.56155 q^{75} +5.02977 q^{76} +10.8123 q^{77} +3.04310 q^{78} -4.15288 q^{79} +3.35950 q^{80} -7.00000 q^{81} -0.710689 q^{82} -12.5293 q^{83} -12.0071 q^{84} -3.40617 q^{85} -0.774424 q^{86} +9.12667 q^{87} +3.41224 q^{88} -9.35682 q^{89} +0.185159 q^{90} -24.0288 q^{91} +1.89128 q^{92} -14.5481 q^{93} -0.0800064 q^{94} -2.65945 q^{95} -5.73690 q^{96} +3.47179 q^{97} -3.14206 q^{98} -1.49342 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{2} - 2 q^{3} + 4 q^{4} + 4 q^{5} - q^{6} - 3 q^{7} + 9 q^{8} + 6 q^{9}+O(q^{10}) \) 4 * q + 2 * q^2 - 2 * q^3 + 4 * q^4 + 4 * q^5 - q^6 - 3 * q^7 + 9 * q^8 + 6 * q^9	\( 4 q + 2 q^{2} - 2 q^{3} + 4 q^{4} + 4 q^{5} - q^{6} - 3 q^{7} + 9 q^{8} + 6 q^{9} + 2 q^{10} + 4 q^{11} - 19 q^{12} - 12 q^{14} - 2 q^{15} + 8 q^{16} - q^{17} + 3 q^{18} - 4 q^{19} + 4 q^{20} + 10 q^{21} - 20 q^{22} - 4 q^{23} - 30 q^{24} + 4 q^{25} - q^{26} - 14 q^{27} - 22 q^{28} + 19 q^{29} - q^{30} - q^{31} + 20 q^{32} - 2 q^{33} - 12 q^{34} - 3 q^{35} + 23 q^{36} - 3 q^{37} + 20 q^{38} + 9 q^{40} + 13 q^{41} + 6 q^{42} - 6 q^{43} - 18 q^{44} + 6 q^{45} - 2 q^{46} + 6 q^{47} - 21 q^{48} + 9 q^{49} + 2 q^{50} - 8 q^{51} - q^{52} + 19 q^{53} - 7 q^{54} + 4 q^{55} - 10 q^{56} + 2 q^{57} + 21 q^{58} + 23 q^{59} - 19 q^{60} - 13 q^{62} - 13 q^{63} + 27 q^{64} + 44 q^{66} - 3 q^{67} - 4 q^{68} + 2 q^{69} - 12 q^{70} - 3 q^{71} + 39 q^{72} - 32 q^{73} - 12 q^{74} - 2 q^{75} + 18 q^{76} + 18 q^{77} + 43 q^{78} + 2 q^{79} + 8 q^{80} - 28 q^{81} - 5 q^{82} - 21 q^{83} + 28 q^{84} - q^{85} - 2 q^{86} - 18 q^{87} - 14 q^{88} + 3 q^{90} - 40 q^{91} - 4 q^{92} - 8 q^{93} + 47 q^{94} - 4 q^{95} - 61 q^{96} - 18 q^{97} + 16 q^{98} + 6 q^{99}+O(q^{100}) \) 4 * q + 2 * q^2 - 2 * q^3 + 4 * q^4 + 4 * q^5 - q^6 - 3 * q^7 + 9 * q^8 + 6 * q^9 + 2 * q^10 + 4 * q^11 - 19 * q^12 - 12 * q^14 - 2 * q^15 + 8 * q^16 - q^17 + 3 * q^18 - 4 * q^19 + 4 * q^20 + 10 * q^21 - 20 * q^22 - 4 * q^23 - 30 * q^24 + 4 * q^25 - q^26 - 14 * q^27 - 22 * q^28 + 19 * q^29 - q^30 - q^31 + 20 * q^32 - 2 * q^33 - 12 * q^34 - 3 * q^35 + 23 * q^36 - 3 * q^37 + 20 * q^38 + 9 * q^40 + 13 * q^41 + 6 * q^42 - 6 * q^43 - 18 * q^44 + 6 * q^45 - 2 * q^46 + 6 * q^47 - 21 * q^48 + 9 * q^49 + 2 * q^50 - 8 * q^51 - q^52 + 19 * q^53 - 7 * q^54 + 4 * q^55 - 10 * q^56 + 2 * q^57 + 21 * q^58 + 23 * q^59 - 19 * q^60 - 13 * q^62 - 13 * q^63 + 27 * q^64 + 44 * q^66 - 3 * q^67 - 4 * q^68 + 2 * q^69 - 12 * q^70 - 3 * q^71 + 39 * q^72 - 32 * q^73 - 12 * q^74 - 2 * q^75 + 18 * q^76 + 18 * q^77 + 43 * q^78 + 2 * q^79 + 8 * q^80 - 28 * q^81 - 5 * q^82 - 21 * q^83 + 28 * q^84 - q^85 - 2 * q^86 - 18 * q^87 - 14 * q^88 + 3 * q^90 - 40 * q^91 - 4 * q^92 - 8 * q^93 + 47 * q^94 - 4 * q^95 - 61 * q^96 - 18 * q^97 + 16 * q^98 + 6 * q^99

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