LMFDB - Embedded newform 2415.4.a.q.1.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2415,4,Mod(1,2415)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2415, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("2415.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2415 = 3 \cdot 5 \cdot 7 \cdot 23 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2415.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:	$142.489612664$
Analytic rank:	$0$
Dimension:	$19$
Coefficient field:	$\mathbb{Q}[x]/(x^{19} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{19} - 120 x^{17} + 5938 x^{15} - 5 x^{14} - 157040 x^{13} + 1378 x^{12} + 2407387 x^{11} + \cdots + 1059840 $ x^19 - 120x^17 + 5938x^15 - 5x^14 - 157040x^13 + 1378x^12 + 2407387x^11 - 79876x^10 - 21686250x^9 + 1877594x^8 + 110550562x^7 - 20574807x^6 - 286581166x^5 + 99696228x^4 + 291858384x^3 - 150111168x^2 - 26481024x + 1059840
Coefficient ring:	$\Z[a_1, \ldots, a_{29}]$
Coefficient ring index:	$ 2^{12}\cdot 5 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
Root			$0.911725$ of defining polynomial
Character	$\chi$	$=$	2415.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.911725 q^{2} +3.00000 q^{3} -7.16876 q^{4} -5.00000 q^{5} -2.73518 q^{6} +7.00000 q^{7} +13.8297 q^{8} +9.00000 q^{9} +O(q^{10})$	\(q-0.911725 q^{2} +3.00000 q^{3} -7.16876 q^{4} -5.00000 q^{5} -2.73518 q^{6} +7.00000 q^{7} +13.8297 q^{8} +9.00000 q^{9} +4.55863 q^{10} -12.5400 q^{11} -21.5063 q^{12} -0.115906 q^{13} -6.38208 q^{14} -15.0000 q^{15} +44.7411 q^{16} -106.056 q^{17} -8.20553 q^{18} +68.4702 q^{19} +35.8438 q^{20} +21.0000 q^{21} +11.4331 q^{22} -23.0000 q^{23} +41.4892 q^{24} +25.0000 q^{25} +0.105674 q^{26} +27.0000 q^{27} -50.1813 q^{28} -13.4728 q^{29} +13.6759 q^{30} +133.708 q^{31} -151.430 q^{32} -37.6201 q^{33} +96.6935 q^{34} -35.0000 q^{35} -64.5188 q^{36} -87.0880 q^{37} -62.4260 q^{38} -0.347717 q^{39} -69.1487 q^{40} -259.254 q^{41} -19.1462 q^{42} +443.421 q^{43} +89.8966 q^{44} -45.0000 q^{45} +20.9697 q^{46} -219.310 q^{47} +134.223 q^{48} +49.0000 q^{49} -22.7931 q^{50} -318.167 q^{51} +0.830899 q^{52} +220.884 q^{53} -24.6166 q^{54} +62.7002 q^{55} +96.8082 q^{56} +205.411 q^{57} +12.2835 q^{58} -749.707 q^{59} +107.531 q^{60} -228.551 q^{61} -121.905 q^{62} +63.0000 q^{63} -219.867 q^{64} +0.579528 q^{65} +34.2992 q^{66} +207.486 q^{67} +760.286 q^{68} -69.0000 q^{69} +31.9104 q^{70} -39.5579 q^{71} +124.468 q^{72} +423.050 q^{73} +79.4003 q^{74} +75.0000 q^{75} -490.846 q^{76} -87.7803 q^{77} +0.317022 q^{78} -515.886 q^{79} -223.706 q^{80} +81.0000 q^{81} +236.369 q^{82} +723.362 q^{83} -150.544 q^{84} +530.278 q^{85} -404.278 q^{86} -40.4184 q^{87} -173.426 q^{88} -553.418 q^{89} +41.0276 q^{90} -0.811339 q^{91} +164.881 q^{92} +401.124 q^{93} +199.950 q^{94} -342.351 q^{95} -454.289 q^{96} +653.236 q^{97} -44.6745 q^{98} -112.860 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 19 q + 57 q^{3} + 88 q^{4} - 95 q^{5} + 133 q^{7} + 171 q^{9}+O(q^{10}) $ 19 * q + 57 * q^3 + 88 * q^4 - 95 * q^5 + 133 * q^7 + 171 * q^9	\( 19 q + 57 q^{3} + 88 q^{4} - 95 q^{5} + 133 q^{7} + 171 q^{9} + 31 q^{11} + 264 q^{12} + 139 q^{13} - 285 q^{15} + 504 q^{16} + 55 q^{17} + 257 q^{19} - 440 q^{20} + 399 q^{21} + 218 q^{22} - 437 q^{23} + 475 q^{25} + 275 q^{26} + 513 q^{27} + 616 q^{28} - 84 q^{29} + 164 q^{31} - 25 q^{32} + 93 q^{33} + 1005 q^{34} - 665 q^{35} + 792 q^{36} + 775 q^{37} + 760 q^{38} + 417 q^{39} + 321 q^{41} + 1265 q^{43} + 351 q^{44} - 855 q^{45} - 542 q^{47} + 1512 q^{48} + 931 q^{49} + 165 q^{51} + 2696 q^{52} + 644 q^{53} - 155 q^{55} + 771 q^{57} - 6 q^{58} + 789 q^{59} - 1320 q^{60} + 977 q^{61} + 674 q^{62} + 1197 q^{63} + 3392 q^{64} - 695 q^{65} + 654 q^{66} + 4025 q^{67} + 62 q^{68} - 1311 q^{69} + 402 q^{71} + 789 q^{73} - 1495 q^{74} + 1425 q^{75} + 2941 q^{76} + 217 q^{77} + 825 q^{78} + 2850 q^{79} - 2520 q^{80} + 1539 q^{81} + 5062 q^{82} - 615 q^{83} + 1848 q^{84} - 275 q^{85} + 1246 q^{86} - 252 q^{87} + 563 q^{88} - 6 q^{89} + 973 q^{91} - 2024 q^{92} + 492 q^{93} + 2754 q^{94} - 1285 q^{95} - 75 q^{96} + 4050 q^{97} + 279 q^{99}+O(q^{100}) \) 19 * q + 57 * q^3 + 88 * q^4 - 95 * q^5 + 133 * q^7 + 171 * q^9 + 31 * q^11 + 264 * q^12 + 139 * q^13 - 285 * q^15 + 504 * q^16 + 55 * q^17 + 257 * q^19 - 440 * q^20 + 399 * q^21 + 218 * q^22 - 437 * q^23 + 475 * q^25 + 275 * q^26 + 513 * q^27 + 616 * q^28 - 84 * q^29 + 164 * q^31 - 25 * q^32 + 93 * q^33 + 1005 * q^34 - 665 * q^35 + 792 * q^36 + 775 * q^37 + 760 * q^38 + 417 * q^39 + 321 * q^41 + 1265 * q^43 + 351 * q^44 - 855 * q^45 - 542 * q^47 + 1512 * q^48 + 931 * q^49 + 165 * q^51 + 2696 * q^52 + 644 * q^53 - 155 * q^55 + 771 * q^57 - 6 * q^58 + 789 * q^59 - 1320 * q^60 + 977 * q^61 + 674 * q^62 + 1197 * q^63 + 3392 * q^64 - 695 * q^65 + 654 * q^66 + 4025 * q^67 + 62 * q^68 - 1311 * q^69 + 402 * q^71 + 789 * q^73 - 1495 * q^74 + 1425 * q^75 + 2941 * q^76 + 217 * q^77 + 825 * q^78 + 2850 * q^79 - 2520 * q^80 + 1539 * q^81 + 5062 * q^82 - 615 * q^83 + 1848 * q^84 - 275 * q^85 + 1246 * q^86 - 252 * q^87 + 563 * q^88 - 6 * q^89 + 973 * q^91 - 2024 * q^92 + 492 * q^93 + 2754 * q^94 - 1285 * q^95 - 75 * q^96 + 4050 * q^97 + 279 * 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 $

$2$	−0.911725	−0.322344	−0.161172	−	0.986926i	$-0.551527\pi$
$2$	−0.911725	−0.322344	−0.161172	+	0.986926i	$0.551527\pi$
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	−7.16876	−0.896095
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	−2.73518	−0.186105
$7$	7.00000	0.377964
$7$	7.00000	0.377964
$8$	13.8297	0.611194
$9$	9.00000	0.333333
$10$	4.55863	0.144156
$11$	−12.5400	−0.343724	−0.171862	−	0.985121i	$-0.554978\pi$
$11$	−12.5400	−0.343724	−0.171862	+	0.985121i	$0.554978\pi$
$12$	−21.5063	−0.517360
$13$	−0.115906	−0.00247280	−0.00123640	−	0.999999i	$-0.500394\pi$
$13$	−0.115906	−0.00247280	−0.00123640	+	0.999999i	$0.500394\pi$
$14$	−6.38208	−0.121834
$15$	−15.0000	−0.258199
$16$	44.7411	0.699080
$17$	−106.056	−1.51307	−0.756537	−	0.653951i	$-0.773110\pi$
$17$	−106.056	−1.51307	−0.756537	+	0.653951i	$0.773110\pi$
$18$	−8.20553	−0.107448
$19$	68.4702	0.826744	0.413372	−	0.910562i	$-0.364351\pi$
$19$	68.4702	0.826744	0.413372	+	0.910562i	$0.364351\pi$
$20$	35.8438	0.400746
$21$	21.0000	0.218218
$22$	11.4331	0.110797
$23$	−23.0000	−0.208514
$23$	−23.0000	−0.208514
$24$	41.4892	0.352873
$25$	25.0000	0.200000
$26$	0.105674	0.000797092 0
$27$	27.0000	0.192450
$28$	−50.1813	−0.338692
$29$	−13.4728	−0.0862702	−0.0431351	−	0.999069i	$-0.513735\pi$
$29$	−13.4728	−0.0862702	−0.0431351	+	0.999069i	$0.513735\pi$
$30$	13.6759	0.0832288
$31$	133.708	0.774666	0.387333	−	0.921940i	$-0.373396\pi$
$31$	133.708	0.774666	0.387333	+	0.921940i	$0.373396\pi$
$32$	−151.430	−0.836538
$33$	−37.6201	−0.198449
$34$	96.6935	0.487730
$35$	−35.0000	−0.169031
$36$	−64.5188	−0.298698
$37$	−87.0880	−0.386951	−0.193475	−	0.981105i	$-0.561976\pi$
$37$	−87.0880	−0.386951	−0.193475	+	0.981105i	$0.561976\pi$
$38$	−62.4260	−0.266496
$39$	−0.347717	−0.00142767
$40$	−69.1487	−0.273334
$41$	−259.254	−0.987530	−0.493765	−	0.869595i	$-0.664380\pi$
$41$	−259.254	−0.987530	−0.493765	+	0.869595i	$0.664380\pi$
$42$	−19.1462	−0.0703411
$43$	443.421	1.57258	0.786292	−	0.617856i	$-0.211998\pi$
$43$	443.421	1.57258	0.786292	+	0.617856i	$0.211998\pi$
$44$	89.8966	0.308009
$45$	−45.0000	−0.149071
$46$	20.9697	0.0672133
$47$	−219.310	−0.680629	−0.340315	−	0.940312i	$-0.610534\pi$
$47$	−219.310	−0.680629	−0.340315	+	0.940312i	$0.610534\pi$
$48$	134.223	0.403614
$49$	49.0000	0.142857
$50$	−22.7931	−0.0644687
$51$	−318.167	−0.873573
$52$	0.830899	0.00221587
$53$	220.884	0.572467	0.286233	−	0.958160i	$-0.407597\pi$
$53$	220.884	0.572467	0.286233	+	0.958160i	$0.407597\pi$
$54$	−24.6166	−0.0620351
$55$	62.7002	0.153718
$56$	96.8082	0.231010
$57$	205.411	0.477321
$58$	12.2835	0.0278087
$59$	−749.707	−1.65430	−0.827149	−	0.561983i	$-0.810039\pi$
$59$	−749.707	−1.65430	−0.827149	+	0.561983i	$0.810039\pi$
$60$	107.531	0.231371
$61$	−228.551	−0.479721	−0.239860	−	0.970807i	$-0.577102\pi$
$61$	−228.551	−0.479721	−0.239860	+	0.970807i	$0.577102\pi$
$62$	−121.905	−0.249709
$63$	63.0000	0.125988
$64$	−219.867	−0.429427
$65$	0.579528	0.00110587
$66$	34.2992	0.0639689
$67$	207.486	0.378335	0.189168	−	0.981945i	$-0.439421\pi$
$67$	207.486	0.378335	0.189168	+	0.981945i	$0.439421\pi$
$68$	760.286	1.35586
$69$	−69.0000	−0.120386
$70$	31.9104	0.0544860
$71$	−39.5579	−0.0661219	−0.0330609	−	0.999453i	$-0.510526\pi$
$71$	−39.5579	−0.0661219	−0.0330609	+	0.999453i	$0.510526\pi$
$72$	124.468	0.203731
$73$	423.050	0.678277	0.339139	−	0.940736i	$-0.389864\pi$
$73$	423.050	0.678277	0.339139	+	0.940736i	$0.389864\pi$
$74$	79.4003	0.124731
$75$	75.0000	0.115470
$76$	−490.846	−0.740841
$77$	−87.7803	−0.129916
$78$	0.317022	0.000460201 0
$79$	−515.886	−0.734705	−0.367352	−	0.930082i	$-0.619736\pi$
$79$	−515.886	−0.734705	−0.367352	+	0.930082i	$0.619736\pi$
$80$	−223.706	−0.312638
$81$	81.0000	0.111111
$82$	236.369	0.318324
$83$	723.362	0.956618	0.478309	−	0.878192i	$-0.341250\pi$
$83$	723.362	0.956618	0.478309	+	0.878192i	$0.341250\pi$
$84$	−150.544	−0.195544
$85$	530.278	0.676667
$86$	−404.278	−0.506912
$87$	−40.4184	−0.0498081
$88$	−173.426	−0.210082
$89$	−553.418	−0.659125	−0.329563	−	0.944134i	$-0.606901\pi$
$89$	−553.418	−0.659125	−0.329563	+	0.944134i	$0.606901\pi$
$90$	41.0276	0.0480521
$91$	−0.811339	−0.000934632 0
$92$	164.881	0.186849
$93$	401.124	0.447254
$94$	199.950	0.219397
$95$	−342.351	−0.369731
$96$	−454.289	−0.482975
$97$	653.236	0.683774	0.341887	−	0.939741i	$-0.388934\pi$
$97$	653.236	0.683774	0.341887	+	0.939741i	$0.388934\pi$
$98$	−44.6745	−0.0460491
$99$	−112.860	−0.114575
$100$	−179.219	−0.179219
$101$	1657.55	1.63300	0.816499	−	0.577347i	$-0.195912\pi$
$101$	1657.55	1.63300	0.816499	+	0.577347i	$0.195912\pi$
$102$	290.081	0.281591
$103$	−1348.47	−1.28998	−0.644992	−	0.764189i	$-0.723139\pi$
$103$	−1348.47	−1.28998	−0.644992	+	0.764189i	$0.723139\pi$
$104$	−1.60294	−0.00151136
$105$	−105.000	−0.0975900
$106$	−201.385	−0.184531
$107$	935.201	0.844947	0.422473	−	0.906375i	$-0.361162\pi$
$107$	935.201	0.844947	0.422473	+	0.906375i	$0.361162\pi$
$108$	−193.556	−0.172453
$109$	1007.33	0.885182	0.442591	−	0.896724i	$-0.354059\pi$
$109$	1007.33	0.885182	0.442591	+	0.896724i	$0.354059\pi$
$110$	−57.1654	−0.0495501
$111$	−261.264	−0.223406
$112$	313.188	0.264227
$113$	−1319.96	−1.09886	−0.549432	−	0.835538i	$-0.685156\pi$
$113$	−1319.96	−1.09886	−0.549432	+	0.835538i	$0.685156\pi$
$114$	−187.278	−0.153861
$115$	115.000	0.0932505
$116$	96.5833	0.0773063
$117$	−1.04315	−0.000824268 0
$118$	683.527	0.533252
$119$	−742.389	−0.571888
$120$	−207.446	−0.157810
$121$	−1173.75	−0.881854
$122$	208.376	0.154635
$123$	−777.763	−0.570151
$124$	−958.520	−0.694174
$125$	−125.000	−0.0894427
$126$	−57.4387	−0.0406115
$127$	265.657	0.185616	0.0928082	−	0.995684i	$-0.470416\pi$
$127$	265.657	0.185616	0.0928082	+	0.995684i	$0.470416\pi$
$128$	1411.89	0.974961
$129$	1330.26	0.907931
$130$	−0.528370	−0.000356470 0
$131$	−700.334	−0.467088	−0.233544	−	0.972346i	$-0.575032\pi$
$131$	−700.334	−0.467088	−0.233544	+	0.972346i	$0.575032\pi$
$132$	269.690	0.177829
$133$	479.291	0.312480
$134$	−189.170	−0.121954
$135$	−135.000	−0.0860663
$136$	−1466.72	−0.924781
$137$	−3109.64	−1.93923	−0.969614	−	0.244640i	$-0.921330\pi$
$137$	−3109.64	−1.93923	−0.969614	+	0.244640i	$0.921330\pi$
$138$	62.9091	0.0388056
$139$	1296.23	0.790967	0.395483	−	0.918473i	$-0.370577\pi$
$139$	1296.23	0.790967	0.395483	+	0.918473i	$0.370577\pi$
$140$	250.906	0.151468
$141$	−657.929	−0.392962
$142$	36.0659	0.0213140
$143$	1.45346	0.000849962 0
$144$	402.670	0.233027
$145$	67.3640	0.0385812
$146$	−385.705	−0.218638
$147$	147.000	0.0824786
$148$	624.312	0.346744
$149$	321.523	0.176780	0.0883898	−	0.996086i	$-0.471828\pi$
$149$	321.523	0.176780	0.0883898	+	0.996086i	$0.471828\pi$
$150$	−68.3794	−0.0372210
$151$	2300.32	1.23972	0.619858	−	0.784714i	$-0.287190\pi$
$151$	2300.32	1.23972	0.619858	+	0.784714i	$0.287190\pi$
$152$	946.925	0.505301
$153$	−954.500	−0.504358
$154$	80.0316	0.0418775
$155$	−668.540	−0.346441
$156$	2.49270	0.00127933
$157$	−3819.47	−1.94157	−0.970786	−	0.239946i	$-0.922870\pi$
$157$	−3819.47	−1.94157	−0.970786	+	0.239946i	$0.922870\pi$
$158$	470.346	0.236827
$159$	662.652	0.330514
$160$	757.148	0.374111
$161$	−161.000	−0.0788110
$162$	−73.8498	−0.0358160
$163$	−214.367	−0.103009	−0.0515046	−	0.998673i	$-0.516402\pi$
$163$	−214.367	−0.103009	−0.0515046	+	0.998673i	$0.516402\pi$
$164$	1858.53	0.884920
$165$	188.101	0.0887492
$166$	−659.507	−0.308360
$167$	187.958	0.0870935	0.0435467	−	0.999051i	$-0.486134\pi$
$167$	187.958	0.0870935	0.0435467	+	0.999051i	$0.486134\pi$
$168$	290.425	0.133373
$169$	−2196.99	−0.999994
$170$	−483.468	−0.218119
$171$	616.232	0.275581
$172$	−3178.78	−1.40918
$173$	−1948.51	−0.856314	−0.428157	−	0.903704i	$-0.640837\pi$
$173$	−1948.51	−0.856314	−0.428157	+	0.903704i	$0.640837\pi$
$174$	36.8505	0.0160553
$175$	175.000	0.0755929
$176$	−561.056	−0.240291
$177$	−2249.12	−0.955109
$178$	504.565	0.212465
$179$	3826.20	1.59767	0.798837	−	0.601548i	$-0.205449\pi$
$179$	3826.20	1.59767	0.798837	+	0.601548i	$0.205449\pi$
$180$	322.594	0.133582
$181$	431.175	0.177066	0.0885330	−	0.996073i	$-0.471782\pi$
$181$	431.175	0.177066	0.0885330	+	0.996073i	$0.471782\pi$
$182$	0.739719	0.000301272 0
$183$	−685.653	−0.276967
$184$	−318.084	−0.127443
$185$	435.440	0.173050
$186$	−365.715	−0.144169
$187$	1329.94	0.520080
$188$	1572.18	0.609908
$189$	189.000	0.0727393
$190$	312.130	0.119180
$191$	2498.15	0.946386	0.473193	−	0.880959i	$-0.343101\pi$
$191$	2498.15	0.946386	0.473193	+	0.880959i	$0.343101\pi$
$192$	−659.601	−0.247930
$193$	4651.03	1.73465	0.867327	−	0.497738i	$-0.165836\pi$
$193$	4651.03	1.73465	0.867327	+	0.497738i	$0.165836\pi$
$194$	−595.572	−0.220410
$195$	1.73858	0.000638475 0
$196$	−351.269	−0.128014
$197$	3352.57	1.21249	0.606245	−	0.795278i	$-0.292675\pi$
$197$	3352.57	1.21249	0.606245	+	0.795278i	$0.292675\pi$
$198$	102.898	0.0369324
$199$	4620.48	1.64592	0.822958	−	0.568102i	$-0.192322\pi$
$199$	4620.48	1.64592	0.822958	+	0.568102i	$0.192322\pi$
$200$	345.744	0.122239
$201$	622.459	0.218432
$202$	−1511.23	−0.526386
$203$	−94.3096	−0.0326071
$204$	2280.86	0.782804
$205$	1296.27	0.441637
$206$	1229.43	0.415818
$207$	−207.000	−0.0695048
$208$	−5.18575	−0.00172869
$209$	−858.619	−0.284172
$210$	95.7312	0.0314575
$211$	4109.89	1.34093	0.670466	−	0.741940i	$-0.266094\pi$
$211$	4109.89	1.34093	0.670466	+	0.741940i	$0.266094\pi$
$212$	−1583.46	−0.512984
$213$	−118.674	−0.0381755
$214$	−852.647	−0.272363
$215$	−2217.11	−0.703281
$216$	373.403	0.117624
$217$	935.956	0.292796
$218$	−918.409	−0.285333
$219$	1269.15	0.391603
$220$	−449.483	−0.137746
$221$	12.2924	0.00374153
$222$	238.201	0.0720135
$223$	−1574.25	−0.472734	−0.236367	−	0.971664i	$-0.575957\pi$
$223$	−1574.25	−0.472734	−0.236367	+	0.971664i	$0.575957\pi$
$224$	−1060.01	−0.316182
$225$	225.000	0.0666667
$226$	1203.44	0.354212
$227$	5428.47	1.58723	0.793613	−	0.608423i	$-0.208198\pi$
$227$	5428.47	1.58723	0.793613	+	0.608423i	$0.208198\pi$
$228$	−1472.54	−0.427725
$229$	498.244	0.143777	0.0718883	−	0.997413i	$-0.477097\pi$
$229$	498.244	0.143777	0.0718883	+	0.997413i	$0.477097\pi$
$230$	−104.848	−0.0300587
$231$	−263.341	−0.0750068
$232$	−186.325	−0.0527279
$233$	10.3467	0.00290915	0.00145458	−	0.999999i	$-0.499537\pi$
$233$	10.3467	0.00290915	0.00145458	+	0.999999i	$0.499537\pi$
$234$	0.951067	0.000265697 0
$235$	1096.55	0.304387
$236$	5374.47	1.48241
$237$	−1547.66	−0.424182
$238$	676.855	0.184344
$239$	3915.07	1.05960	0.529802	−	0.848122i	$-0.322266\pi$
$239$	3915.07	1.05960	0.529802	+	0.848122i	$0.322266\pi$
$240$	−671.117	−0.180502
$241$	−5998.02	−1.60318	−0.801590	−	0.597874i	$-0.796012\pi$
$241$	−5998.02	−1.60318	−0.801590	+	0.597874i	$0.796012\pi$
$242$	1070.14	0.284260
$243$	243.000	0.0641500
$244$	1638.43	0.429875
$245$	−245.000	−0.0638877
$246$	709.106	0.183784
$247$	−7.93608	−0.00204437
$248$	1849.15	0.473471
$249$	2170.09	0.552303
$250$	113.966	0.0288313
$251$	6323.76	1.59025	0.795124	−	0.606447i	$-0.207406\pi$
$251$	6323.76	1.59025	0.795124	+	0.606447i	$0.207406\pi$
$252$	−451.632	−0.112897
$253$	288.421	0.0716715
$254$	−242.207	−0.0598323
$255$	1590.83	0.390674
$256$	471.675	0.115155
$257$	−169.113	−0.0410465	−0.0205233	−	0.999789i	$-0.506533\pi$
$257$	−169.113	−0.0410465	−0.0205233	+	0.999789i	$0.506533\pi$
$258$	−1212.83	−0.292666
$259$	−609.616	−0.146254
$260$	−4.15450	−0.000990965 0
$261$	−121.255	−0.0287567
$262$	638.513	0.150563
$263$	5498.88	1.28926	0.644630	−	0.764495i	$-0.277012\pi$
$263$	5498.88	1.28926	0.644630	+	0.764495i	$0.277012\pi$
$264$	−520.277	−0.121291
$265$	−1104.42	−0.256015
$266$	−436.982	−0.100726
$267$	−1660.25	−0.380546
$268$	−1487.42	−0.339024
$269$	−1593.51	−0.361183	−0.180591	−	0.983558i	$-0.557801\pi$
$269$	−1593.51	−0.361183	−0.180591	+	0.983558i	$0.557801\pi$
$270$	123.083	0.0277429
$271$	−3696.09	−0.828493	−0.414246	−	0.910165i	$-0.635955\pi$
$271$	−3696.09	−0.828493	−0.414246	+	0.910165i	$0.635955\pi$
$272$	−4745.04	−1.05776
$273$	−2.43402	−0.000539610 0
$274$	2835.13	0.625098
$275$	−313.501	−0.0687449
$276$	494.644	0.107877
$277$	6264.99	1.35894	0.679471	−	0.733702i	$-0.262209\pi$
$277$	6264.99	1.35894	0.679471	+	0.733702i	$0.262209\pi$
$278$	−1181.80	−0.254963
$279$	1203.37	0.258222
$280$	−484.041	−0.103311
$281$	4297.56	0.912352	0.456176	−	0.889889i	$-0.349219\pi$
$281$	4297.56	0.912352	0.456176	+	0.889889i	$0.349219\pi$
$282$	599.850	0.126669
$283$	−201.441	−0.0423124	−0.0211562	−	0.999776i	$-0.506735\pi$
$283$	−201.441	−0.0423124	−0.0211562	+	0.999776i	$0.506735\pi$
$284$	283.581	0.0592515
$285$	−1027.05	−0.213464
$286$	−1.32516	−0.000273980 0
$287$	−1814.78	−0.373251
$288$	−1362.87	−0.278846
$289$	6334.78	1.28939
$290$	−61.4175	−0.0124364
$291$	1959.71	0.394777
$292$	−3032.74	−0.607800
$293$	8428.23	1.68049	0.840243	−	0.542210i	$-0.182412\pi$
$293$	8428.23	1.68049	0.840243	+	0.542210i	$0.182412\pi$
$294$	−134.024	−0.0265865
$295$	3748.54	0.739825
$296$	−1204.40	−0.236502
$297$	−338.581	−0.0661498
$298$	−293.140	−0.0569838
$299$	2.66583	0.000515615 0
$300$	−537.657	−0.103472
$301$	3103.95	0.594381
$302$	−2097.26	−0.399615
$303$	4972.66	0.942811
$304$	3063.43	0.577960
$305$	1142.76	0.214538
$306$	870.242	0.162577
$307$	8681.07	1.61386	0.806930	−	0.590647i	$-0.201128\pi$
$307$	8681.07	1.61386	0.806930	+	0.590647i	$0.201128\pi$
$308$	629.276	0.116417
$309$	−4045.40	−0.744773
$310$	609.525	0.111673
$311$	−8242.38	−1.50284	−0.751419	−	0.659825i	$-0.770630\pi$
$311$	−8242.38	−1.50284	−0.751419	+	0.659825i	$0.770630\pi$
$312$	−4.80883	−0.000872585 0
$313$	96.9882	0.0175147	0.00875735	−	0.999962i	$-0.497212\pi$
$313$	96.9882	0.0175147	0.00875735	+	0.999962i	$0.497212\pi$
$314$	3482.31	0.625853
$315$	−315.000	−0.0563436
$316$	3698.26	0.658365
$317$	2448.83	0.433880	0.216940	−	0.976185i	$-0.430392\pi$
$317$	2448.83	0.433880	0.216940	+	0.976185i	$0.430392\pi$
$318$	−604.156	−0.106539
$319$	168.950	0.0296532
$320$	1099.33	0.192046
$321$	2805.60	0.487830
$322$	146.788	0.0254042
$323$	−7261.64	−1.25092
$324$	−580.669	−0.0995661
$325$	−2.89764	−0.000494561 0
$326$	195.444	0.0332044
$327$	3021.99	0.511060
$328$	−3585.42	−0.603572
$329$	−1535.17	−0.257254
$330$	−171.496	−0.0286077
$331$	1631.96	0.270999	0.135500	−	0.990777i	$-0.456736\pi$
$331$	1631.96	0.270999	0.135500	+	0.990777i	$0.456736\pi$
$332$	−5185.60	−0.857220
$333$	−783.792	−0.128984
$334$	−171.366	−0.0280740
$335$	−1037.43	−0.169197
$336$	939.564	0.152552
$337$	−1516.50	−0.245130	−0.122565	−	0.992460i	$-0.539112\pi$
$337$	−1516.50	−0.245130	−0.122565	+	0.992460i	$0.539112\pi$
$338$	2003.05	0.322342
$339$	−3959.89	−0.634430
$340$	−3801.43	−0.606358
$341$	−1676.70	−0.266272
$342$	−561.834	−0.0888319
$343$	343.000	0.0539949
$344$	6132.40	0.961153
$345$	345.000	0.0538382
$346$	1776.50	0.276027
$347$	−2673.76	−0.413646	−0.206823	−	0.978378i	$-0.566312\pi$
$347$	−2673.76	−0.413646	−0.206823	+	0.978378i	$0.566312\pi$
$348$	289.750	0.0446328
$349$	10680.0	1.63807	0.819035	−	0.573743i	$-0.194509\pi$
$349$	10680.0	1.63807	0.819035	+	0.573743i	$0.194509\pi$
$350$	−159.552	−0.0243669
$351$	−3.12945	−0.000475891 0
$352$	1898.93	0.287538
$353$	−7993.63	−1.20526	−0.602632	−	0.798019i	$-0.705881\pi$
$353$	−7993.63	−1.20526	−0.602632	+	0.798019i	$0.705881\pi$
$354$	2050.58	0.307873
$355$	197.789	0.0295706
$356$	3967.32	0.590639
$357$	−2227.17	−0.330180
$358$	−3488.44	−0.515000
$359$	3533.68	0.519500	0.259750	−	0.965676i	$-0.416360\pi$
$359$	3533.68	0.519500	0.259750	+	0.965676i	$0.416360\pi$
$360$	−622.338	−0.0911114
$361$	−2170.83	−0.316494
$362$	−393.113	−0.0570761
$363$	−3521.24	−0.509138
$364$	5.81629	0.000837518 0
$365$	−2115.25	−0.303335
$366$	625.127	0.0892785
$367$	−10332.1	−1.46956	−0.734782	−	0.678303i	$-0.762716\pi$
$367$	−10332.1	−1.46956	−0.734782	+	0.678303i	$0.762716\pi$
$368$	−1029.05	−0.145768
$369$	−2333.29	−0.329177
$370$	−397.002	−0.0557814
$371$	1546.19	0.216372
$372$	−2875.56	−0.400782
$373$	3447.66	0.478587	0.239293	−	0.970947i	$-0.423084\pi$
$373$	3447.66	0.478587	0.239293	+	0.970947i	$0.423084\pi$
$374$	−1212.54	−0.167644
$375$	−375.000	−0.0516398
$376$	−3032.99	−0.415997
$377$	1.56157	0.000213329 0
$378$	−172.316	−0.0234470
$379$	7763.74	1.05223	0.526117	−	0.850412i	$-0.323648\pi$
$379$	7763.74	1.05223	0.526117	+	0.850412i	$0.323648\pi$
$380$	2454.23	0.331314
$381$	796.972	0.107166
$382$	−2277.62	−0.305061
$383$	7918.34	1.05642	0.528209	−	0.849114i	$-0.322864\pi$
$383$	7918.34	1.05642	0.528209	+	0.849114i	$0.322864\pi$
$384$	4235.68	0.562894
$385$	438.902	0.0581000
$386$	−4240.46	−0.559155
$387$	3990.79	0.524194
$388$	−4682.89	−0.612726
$389$	−3227.35	−0.420651	−0.210326	−	0.977631i	$-0.567452\pi$
$389$	−3227.35	−0.420651	−0.210326	+	0.977631i	$0.567452\pi$
$390$	−1.58511	−0.000205808 0
$391$	2439.28	0.315498
$392$	677.657	0.0873134
$393$	−2101.00	−0.269673
$394$	−3056.62	−0.390839
$395$	2579.43	0.328570
$396$	809.069	0.102670
$397$	12355.2	1.56194	0.780972	−	0.624566i	$-0.214724\pi$
$397$	12355.2	1.56194	0.780972	+	0.624566i	$0.214724\pi$
$398$	−4212.61	−0.530551
$399$	1437.87	0.180410
$400$	1118.53	0.139816
$401$	1812.57	0.225724	0.112862	−	0.993611i	$-0.463998\pi$
$401$	1812.57	0.225724	0.112862	+	0.993611i	$0.463998\pi$
$402$	−567.511	−0.0704102
$403$	−15.4975	−0.00191560
$404$	−11882.6	−1.46332
$405$	−405.000	−0.0496904
$406$	85.9845	0.0105107
$407$	1092.09	0.133004
$408$	−4400.16	−0.533923
$409$	−676.285	−0.0817607	−0.0408804	−	0.999164i	$-0.513016\pi$
$409$	−676.285	−0.0817607	−0.0408804	+	0.999164i	$0.513016\pi$
$410$	−1181.84	−0.142359
$411$	−9328.91	−1.11961
$412$	9666.83	1.15595
$413$	−5247.95	−0.625266
$414$	188.727	0.0224044
$415$	−3616.81	−0.427812
$416$	17.5515	0.00206859
$417$	3888.68	0.456665
$418$	782.825	0.0916010
$419$	5453.34	0.635831	0.317915	−	0.948119i	$-0.397017\pi$
$419$	5453.34	0.635831	0.317915	+	0.948119i	$0.397017\pi$
$420$	752.719	0.0874499
$421$	−11503.8	−1.33174	−0.665870	−	0.746068i	$-0.731939\pi$
$421$	−11503.8	−1.33174	−0.665870	+	0.746068i	$0.731939\pi$
$422$	−3747.09	−0.432241
$423$	−1973.79	−0.226876
$424$	3054.77	0.349888
$425$	−2651.39	−0.302615
$426$	108.198	0.0123056
$427$	−1599.86	−0.181317
$428$	−6704.23	−0.757152
$429$	4.36039	0.000490726 0
$430$	2021.39	0.226698
$431$	10387.1	1.16086	0.580430	−	0.814310i	$-0.302885\pi$
$431$	10387.1	1.16086	0.580430	+	0.814310i	$0.302885\pi$
$432$	1208.01	0.134538
$433$	16805.6	1.86519	0.932595	−	0.360924i	$-0.117539\pi$
$433$	16805.6	1.86519	0.932595	+	0.360924i	$0.117539\pi$
$434$	−853.335	−0.0943810
$435$	202.092	0.0222749
$436$	−7221.31	−0.793207
$437$	−1574.81	−0.172388
$438$	−1157.12	−0.126231
$439$	10816.8	1.17599	0.587993	−	0.808866i	$-0.299918\pi$
$439$	10816.8	1.17599	0.587993	+	0.808866i	$0.299918\pi$
$440$	867.128	0.0939516
$441$	441.000	0.0476190
$442$	−11.2073	−0.00120606
$443$	−17700.3	−1.89834	−0.949172	−	0.314757i	$-0.898077\pi$
$443$	−17700.3	−1.89834	−0.949172	+	0.314757i	$0.898077\pi$
$444$	1872.94	0.200193
$445$	2767.09	0.294770
$446$	1435.29	0.152383
$447$	964.568	0.102064
$448$	−1539.07	−0.162308
$449$	207.213	0.0217795	0.0108897	−	0.999941i	$-0.496534\pi$
$449$	207.213	0.0217795	0.0108897	+	0.999941i	$0.496534\pi$
$450$	−205.138	−0.0214896
$451$	3251.06	0.339438
$452$	9462.50	0.984687
$453$	6900.95	0.715750
$454$	−4949.27	−0.511632
$455$	4.05670	0.000417980 0
$456$	2840.77	0.291736
$457$	12529.8	1.28254	0.641268	−	0.767317i	$-0.278409\pi$
$457$	12529.8	1.28254	0.641268	+	0.767317i	$0.278409\pi$
$458$	−454.261	−0.0463455
$459$	−2863.50	−0.291191
$460$	−824.407	−0.0835613
$461$	−4271.94	−0.431592	−0.215796	−	0.976438i	$-0.569235\pi$
$461$	−4271.94	−0.431592	−0.215796	+	0.976438i	$0.569235\pi$
$462$	240.095	0.0241780
$463$	−8176.03	−0.820675	−0.410338	−	0.911934i	$-0.634589\pi$
$463$	−8176.03	−0.820675	−0.410338	+	0.911934i	$0.634589\pi$
$464$	−602.788	−0.0603098
$465$	−2005.62	−0.200018
$466$	−9.43332	−0.000937747 0
$467$	9567.30	0.948013	0.474006	−	0.880521i	$-0.342807\pi$
$467$	9567.30	0.948013	0.474006	+	0.880521i	$0.342807\pi$
$468$	7.47809	0.000738622 0
$469$	1452.40	0.142997
$470$	−999.750	−0.0981171
$471$	−11458.4	−1.12097
$472$	−10368.3	−1.01110
$473$	−5560.52	−0.540535
$474$	1411.04	0.136732
$475$	1711.75	0.165349
$476$	5322.00	0.512466
$477$	1987.96	0.190822
$478$	−3569.47	−0.341556
$479$	−1179.30	−0.112492	−0.0562459	−	0.998417i	$-0.517913\pi$
$479$	−1179.30	−0.112492	−0.0562459	+	0.998417i	$0.517913\pi$
$480$	2271.44	0.215993
$481$	10.0940	0.000956853 0
$482$	5468.54	0.516775
$483$	−483.000	−0.0455016
$484$	8414.31	0.790224
$485$	−3266.18	−0.305793
$486$	−221.549	−0.0206784
$487$	−5769.89	−0.536876	−0.268438	−	0.963297i	$-0.586507\pi$
$487$	−5769.89	−0.536876	−0.268438	+	0.963297i	$0.586507\pi$
$488$	−3160.80	−0.293202
$489$	−643.101	−0.0594724
$490$	223.373	0.0205938
$491$	−14300.5	−1.31440	−0.657201	−	0.753715i	$-0.728260\pi$
$491$	−14300.5	−1.31440	−0.657201	+	0.753715i	$0.728260\pi$
$492$	5575.60	0.510909
$493$	1428.87	0.130533
$494$	7.23552	0.000658991 0
$495$	564.302	0.0512394
$496$	5982.24	0.541554
$497$	−276.905	−0.0249917
$498$	−1978.52	−0.178031
$499$	11810.6	1.05955	0.529773	−	0.848139i	$-0.322277\pi$
$499$	11810.6	1.05955	0.529773	+	0.848139i	$0.322277\pi$
$500$	896.095	0.0801491
$501$	563.873	0.0502834
$502$	−5765.53	−0.512606
$503$	19578.4	1.73550	0.867752	−	0.496998i	$-0.165564\pi$
$503$	19578.4	1.73550	0.867752	+	0.496998i	$0.165564\pi$
$504$	871.274	0.0770032
$505$	−8287.77	−0.730299
$506$	−262.961	−0.0231028
$507$	−6590.96	−0.577347
$508$	−1904.43	−0.166330
$509$	−7603.62	−0.662131	−0.331065	−	0.943608i	$-0.607408\pi$
$509$	−7603.62	−0.662131	−0.331065	+	0.943608i	$0.607408\pi$
$510$	−1450.40	−0.125931
$511$	2961.35	0.256365
$512$	−11725.2	−1.01208
$513$	1848.69	0.159107
$514$	154.184	0.0132311
$515$	6742.33	0.576898
$516$	−9536.33	−0.813592
$517$	2750.15	0.233949
$518$	555.802	0.0471439
$519$	−5845.53	−0.494393
$520$	8.01472	0.000675902 0
$521$	7430.29	0.624812	0.312406	−	0.949949i	$-0.398865\pi$
$521$	7430.29	0.624812	0.312406	+	0.949949i	$0.398865\pi$
$522$	110.551	0.00926955
$523$	11638.3	0.973055	0.486528	−	0.873665i	$-0.338263\pi$
$523$	11638.3	0.973055	0.486528	+	0.873665i	$0.338263\pi$
$524$	5020.53	0.418555
$525$	525.000	0.0436436
$526$	−5013.46	−0.415585
$527$	−14180.5	−1.17213
$528$	−1683.17	−0.138732
$529$	529.000	0.0434783
$530$	1006.93	0.0825248
$531$	−6747.37	−0.551433
$532$	−3435.92	−0.280012
$533$	30.0490	0.00244197
$534$	1513.69	0.122667
$535$	−4676.01	−0.377872
$536$	2869.48	0.231236
$537$	11478.6	0.922417
$538$	1452.85	0.116425
$539$	−614.462	−0.0491035
$540$	967.782	0.0771235
$541$	4922.74	0.391211	0.195605	−	0.980683i	$-0.437333\pi$
$541$	4922.74	0.391211	0.195605	+	0.980683i	$0.437333\pi$
$542$	3369.82	0.267059
$543$	1293.52	0.102229
$544$	16059.9	1.26574
$545$	−5036.65	−0.395865
$546$	2.21916	0.000173940 0
$547$	12724.2	0.994604	0.497302	−	0.867577i	$-0.334324\pi$
$547$	12724.2	0.994604	0.497302	+	0.867577i	$0.334324\pi$
$548$	22292.2	1.73773
$549$	−2056.96	−0.159907
$550$	285.827	0.0221595
$551$	−922.485	−0.0713234
$552$	−954.252	−0.0735791
$553$	−3611.20	−0.277692
$554$	−5711.95	−0.438046
$555$	1306.32	0.0999102
$556$	−9292.32	−0.708781
$557$	−22803.9	−1.73471	−0.867353	−	0.497693i	$-0.834181\pi$
$557$	−22803.9	−1.73471	−0.867353	+	0.497693i	$0.834181\pi$
$558$	−1097.14	−0.0832363
$559$	−51.3950	−0.00388869
$560$	−1565.94	−0.118166
$561$	3989.83	0.300268
$562$	−3918.20	−0.294091
$563$	18424.0	1.37918	0.689592	−	0.724198i	$-0.257790\pi$
$563$	18424.0	1.37918	0.689592	+	0.724198i	$0.257790\pi$
$564$	4716.53	0.352131
$565$	6599.82	0.491427
$566$	183.659	0.0136391
$567$	567.000	0.0419961
$568$	−547.075	−0.0404133
$569$	−1160.90	−0.0855317	−0.0427659	−	0.999085i	$-0.513617\pi$
$569$	−1160.90	−0.0855317	−0.0427659	+	0.999085i	$0.513617\pi$
$570$	936.390	0.0688089
$571$	8517.55	0.624253	0.312126	−	0.950041i	$-0.398959\pi$
$571$	8517.55	0.624253	0.312126	+	0.950041i	$0.398959\pi$
$572$	−10.4195	−0.000761647 0
$573$	7494.44	0.546396
$574$	1654.58	0.120315
$575$	−575.000	−0.0417029
$576$	−1978.80	−0.143142
$577$	−22342.1	−1.61198	−0.805990	−	0.591929i	$-0.798366\pi$
$577$	−22342.1	−1.61198	−0.805990	+	0.591929i	$0.798366\pi$
$578$	−5775.58	−0.415627
$579$	13953.1	1.00150
$580$	−482.916	−0.0345724
$581$	5063.53	0.361568
$582$	−1786.72	−0.127254
$583$	−2769.90	−0.196771
$584$	5850.67	0.414559
$585$	5.21575	0.000368624 0
$586$	−7684.23	−0.541694
$587$	−16052.3	−1.12870	−0.564351	−	0.825535i	$-0.690874\pi$
$587$	−16052.3	−1.12870	−0.564351	+	0.825535i	$0.690874\pi$
$588$	−1053.81	−0.0739086
$589$	9155.01	0.640451
$590$	−3417.64	−0.238478
$591$	10057.7	0.700032
$592$	−3896.41	−0.270510
$593$	4318.72	0.299070	0.149535	−	0.988756i	$-0.452222\pi$
$593$	4318.72	0.299070	0.149535	+	0.988756i	$0.452222\pi$
$594$	308.693	0.0213230
$595$	3711.94	0.255756
$596$	−2304.92	−0.158411
$597$	13861.5	0.950271
$598$	−2.43050	−0.000166205 0
$599$	8361.59	0.570360	0.285180	−	0.958474i	$-0.407947\pi$
$599$	8361.59	0.570360	0.285180	+	0.958474i	$0.407947\pi$
$600$	1037.23	0.0705746
$601$	3827.25	0.259762	0.129881	−	0.991530i	$-0.458541\pi$
$601$	3827.25	0.259762	0.129881	+	0.991530i	$0.458541\pi$
$602$	−2829.95	−0.191595
$603$	1867.38	0.126112
$604$	−16490.4	−1.11090
$605$	5868.74	0.394377
$606$	−4533.70	−0.303909
$607$	−2915.80	−0.194973	−0.0974866	−	0.995237i	$-0.531080\pi$
$607$	−2915.80	−0.194973	−0.0974866	+	0.995237i	$0.531080\pi$
$608$	−10368.4	−0.691603
$609$	−282.929	−0.0188257
$610$	−1041.88	−0.0691548
$611$	25.4192	0.00168306
$612$	6842.58	0.451952
$613$	−14696.6	−0.968339	−0.484169	−	0.874974i	$-0.660878\pi$
$613$	−14696.6	−0.968339	−0.484169	+	0.874974i	$0.660878\pi$
$614$	−7914.75	−0.520217
$615$	3888.82	0.254979
$616$	−1213.98	−0.0794036
$617$	7708.79	0.502989	0.251495	−	0.967859i	$-0.419078\pi$
$617$	7708.79	0.502989	0.251495	+	0.967859i	$0.419078\pi$
$618$	3688.29	0.240073
$619$	−2646.96	−0.171875	−0.0859373	−	0.996301i	$-0.527388\pi$
$619$	−2646.96	−0.171875	−0.0859373	+	0.996301i	$0.527388\pi$
$620$	4792.60	0.310444
$621$	−621.000	−0.0401286
$622$	7514.79	0.484430
$623$	−3873.92	−0.249126
$624$	−15.5572	−0.000998058 0
$625$	625.000	0.0400000
$626$	−88.4266	−0.00564575
$627$	−2575.86	−0.164067
$628$	27380.8	1.73983
$629$	9236.16	0.585485
$630$	287.193	0.0181620
$631$	−14200.3	−0.895887	−0.447943	−	0.894062i	$-0.647843\pi$
$631$	−14200.3	−0.895887	−0.447943	+	0.894062i	$0.647843\pi$
$632$	−7134.56	−0.449047
$633$	12329.7	0.774188
$634$	−2232.66	−0.139858
$635$	−1328.29	−0.0830102
$636$	−4750.39	−0.296172
$637$	−5.67938	−0.000353258 0
$638$	−154.036	−0.00955851
$639$	−356.021	−0.0220406
$640$	−7059.47	−0.436016
$641$	−16514.2	−1.01758	−0.508792	−	0.860890i	$-0.669908\pi$
$641$	−16514.2	−1.01758	−0.508792	+	0.860890i	$0.669908\pi$
$642$	−2557.94	−0.157249
$643$	3523.45	0.216099	0.108049	−	0.994146i	$-0.465540\pi$
$643$	3523.45	0.216099	0.108049	+	0.994146i	$0.465540\pi$
$644$	1154.17	0.0706221
$645$	−6651.32	−0.406039
$646$	6620.62	0.403227
$647$	29444.6	1.78916	0.894579	−	0.446911i	$-0.147476\pi$
$647$	29444.6	1.78916	0.894579	+	0.446911i	$0.147476\pi$
$648$	1120.21	0.0679104
$649$	9401.37	0.568622
$650$	2.64185	0.000159418 0
$651$	2807.87	0.169046
$652$	1536.74	0.0923060
$653$	−1169.70	−0.0700979	−0.0350490	−	0.999386i	$-0.511159\pi$
$653$	−1169.70	−0.0700979	−0.0350490	+	0.999386i	$0.511159\pi$
$654$	−2755.23	−0.164737
$655$	3501.67	0.208888
$656$	−11599.3	−0.690363
$657$	3807.45	0.226092
$658$	1399.65	0.0829241
$659$	16310.4	0.964134	0.482067	−	0.876134i	$-0.339886\pi$
$659$	16310.4	0.964134	0.482067	+	0.876134i	$0.339886\pi$
$660$	−1348.45	−0.0795277
$661$	−15802.4	−0.929870	−0.464935	−	0.885345i	$-0.653922\pi$
$661$	−15802.4	−0.929870	−0.464935	+	0.885345i	$0.653922\pi$
$662$	−1487.90	−0.0873549
$663$	36.8773	0.00216017
$664$	10003.9	0.584679
$665$	−2396.46	−0.139745
$666$	714.603	0.0415770
$667$	309.875	0.0179886
$668$	−1347.42	−0.0780440
$669$	−4722.76	−0.272933
$670$	945.852	0.0545395
$671$	2866.04	0.164892
$672$	−3180.02	−0.182548
$673$	22014.2	1.26090	0.630451	−	0.776229i	$-0.282870\pi$
$673$	22014.2	1.26090	0.630451	+	0.776229i	$0.282870\pi$
$674$	1382.63	0.0790162
$675$	675.000	0.0384900
$676$	15749.7	0.896089
$677$	12575.1	0.713884	0.356942	−	0.934127i	$-0.383819\pi$
$677$	12575.1	0.713884	0.356942	+	0.934127i	$0.383819\pi$
$678$	3610.33	0.204504
$679$	4572.65	0.258442
$680$	7333.60	0.413575
$681$	16285.4	0.916385
$682$	1528.69	0.0858310
$683$	19478.5	1.09125	0.545624	−	0.838030i	$-0.316293\pi$
$683$	19478.5	1.09125	0.545624	+	0.838030i	$0.316293\pi$
$684$	−4417.61	−0.246947
$685$	15548.2	0.867249
$686$	−312.722	−0.0174049
$687$	1494.73	0.0830095
$688$	19839.2	1.09936
$689$	−25.6017	−0.00141560
$690$	−314.545	−0.0173544
$691$	−18430.2	−1.01464	−0.507321	−	0.861757i	$-0.669364\pi$
$691$	−18430.2	−1.01464	−0.507321	+	0.861757i	$0.669364\pi$
$692$	13968.4	0.767339
$693$	−790.023	−0.0433052
$694$	2437.74	0.133336
$695$	−6481.13	−0.353731
$696$	−558.976	−0.0304424
$697$	27495.4	1.49421
$698$	−9737.22	−0.528022
$699$	31.0400	0.00167960
$700$	−1254.53	−0.0677384
$701$	−121.944	−0.00657028	−0.00328514	−	0.999995i	$-0.501046\pi$
$701$	−121.944	−0.00657028	−0.00328514	+	0.999995i	$0.501046\pi$
$702$	2.85320	0.000153400 0
$703$	−5962.93	−0.319909
$704$	2757.14	0.147605
$705$	3289.64	0.175738
$706$	7287.99	0.388509
$707$	11602.9	0.617215
$708$	16123.4	0.855868
$709$	276.724	0.0146581	0.00732905	−	0.999973i	$-0.497667\pi$
$709$	276.724	0.0146581	0.00732905	+	0.999973i	$0.497667\pi$
$710$	−180.330	−0.00953190
$711$	−4642.97	−0.244902
$712$	−7653.62	−0.402853
$713$	−3075.28	−0.161529
$714$	2030.56	0.106431
$715$	−7.26731	−0.000380115 0
$716$	−27429.1	−1.43167
$717$	11745.2	0.611762
$718$	−3221.74	−0.167457
$719$	1033.48	0.0536053	0.0268027	−	0.999641i	$-0.491467\pi$
$719$	1033.48	0.0536053	0.0268027	+	0.999641i	$0.491467\pi$
$720$	−2013.35	−0.104213
$721$	−9439.27	−0.487568
$722$	1979.20	0.102020
$723$	−17994.1	−0.925596
$724$	−3090.99	−0.158668
$725$	−336.820	−0.0172540
$726$	3210.41	0.164118
$727$	3331.14	0.169938	0.0849692	−	0.996384i	$-0.472921\pi$
$727$	3331.14	0.169938	0.0849692	+	0.996384i	$0.472921\pi$
$728$	−11.2206	−0.000571241 0
$729$	729.000	0.0370370
$730$	1928.53	0.0977780
$731$	−47027.3	−2.37943
$732$	4915.28	0.248188
$733$	−2177.32	−0.109715	−0.0548575	−	0.998494i	$-0.517470\pi$
$733$	−2177.32	−0.109715	−0.0548575	+	0.998494i	$0.517470\pi$
$734$	9420.02	0.473705
$735$	−735.000	−0.0368856
$736$	3482.88	0.174430
$737$	−2601.89	−0.130043
$738$	2127.32	0.106108
$739$	8495.09	0.422865	0.211432	−	0.977393i	$-0.432187\pi$
$739$	8495.09	0.422865	0.211432	+	0.977393i	$0.432187\pi$
$740$	−3121.56	−0.155069
$741$	−23.8082	−0.00118032
$742$	−1409.70	−0.0697462
$743$	3992.04	0.197111	0.0985556	−	0.995132i	$-0.468578\pi$
$743$	3992.04	0.197111	0.0985556	+	0.995132i	$0.468578\pi$
$744$	5547.44	0.273359
$745$	−1607.61	−0.0790583
$746$	−3143.31	−0.154269
$747$	6510.26	0.318873
$748$	−9534.03	−0.466041
$749$	6546.41	0.319360
$750$	341.897	0.0166458
$751$	10342.5	0.502536	0.251268	−	0.967918i	$-0.419152\pi$
$751$	10342.5	0.502536	0.251268	+	0.967918i	$0.419152\pi$
$752$	−9812.16	−0.475815
$753$	18971.3	0.918130
$754$	−1.42373	−6.87653e−5 0
$755$	−11501.6	−0.554418
$756$	−1354.90	−0.0651813
$757$	7047.84	0.338386	0.169193	−	0.985583i	$-0.445884\pi$
$757$	7047.84	0.338386	0.169193	+	0.985583i	$0.445884\pi$
$758$	−7078.40	−0.339181
$759$	865.263	0.0413795
$760$	−4734.62	−0.225977
$761$	3234.73	0.154085	0.0770426	−	0.997028i	$-0.475452\pi$
$761$	3234.73	0.154085	0.0770426	+	0.997028i	$0.475452\pi$
$762$	−726.620	−0.0345442
$763$	7051.32	0.334567
$764$	−17908.6	−0.848051
$765$	4772.50	0.225556
$766$	−7219.35	−0.340530
$767$	86.8953	0.00409075
$768$	1415.02	0.0664848
$769$	16097.8	0.754876	0.377438	−	0.926035i	$-0.376805\pi$
$769$	16097.8	0.754876	0.377438	+	0.926035i	$0.376805\pi$
$770$	−400.158	−0.0187282
$771$	−507.338	−0.0236982
$772$	−33342.1	−1.55441
$773$	−15638.7	−0.727663	−0.363831	−	0.931465i	$-0.618532\pi$
$773$	−15638.7	−0.727663	−0.363831	+	0.931465i	$0.618532\pi$
$774$	−3638.50	−0.168971
$775$	3342.70	0.154933
$776$	9034.09	0.417919
$777$	−1828.85	−0.0844396
$778$	2942.46	0.135594
$779$	−17751.2	−0.816435
$780$	−12.4635	−0.000572134 0
$781$	496.057	0.0227277
$782$	−2223.95	−0.101699
$783$	−363.766	−0.0166027
$784$	2192.32	0.0998686
$785$	19097.3	0.868298
$786$	1915.54	0.0869274
$787$	−5466.30	−0.247589	−0.123795	−	0.992308i	$-0.539506\pi$
$787$	−5466.30	−0.247589	−0.123795	+	0.992308i	$0.539506\pi$
$788$	−24033.7	−1.08651
$789$	16496.6	0.744354
$790$	−2351.73	−0.105912
$791$	−9239.74	−0.415332
$792$	−1560.83	−0.0700274
$793$	26.4903	0.00118625
$794$	−11264.6	−0.503483
$795$	−3313.26	−0.147810
$796$	−33123.1	−1.47490
$797$	−11847.4	−0.526544	−0.263272	−	0.964722i	$-0.584802\pi$
$797$	−11847.4	−0.526544	−0.263272	+	0.964722i	$0.584802\pi$
$798$	−1310.95	−0.0581541
$799$	23259.0	1.02984
$800$	−3785.74	−0.167308
$801$	−4980.76	−0.219708
$802$	−1652.56	−0.0727607
$803$	−5305.06	−0.233140
$804$	−4462.25	−0.195736
$805$	805.000	0.0352454
$806$	14.1295	0.000617480 0
$807$	−4780.54	−0.208529
$808$	22923.5	0.998078
$809$	−7274.05	−0.316121	−0.158061	−	0.987429i	$-0.550524\pi$
$809$	−7274.05	−0.316121	−0.158061	+	0.987429i	$0.550524\pi$
$810$	369.249	0.0160174
$811$	−6860.52	−0.297047	−0.148524	−	0.988909i	$-0.547452\pi$
$811$	−6860.52	−0.297047	−0.148524	+	0.988909i	$0.547452\pi$
$812$	676.083	0.0292190
$813$	−11088.3	−0.478330
$814$	−995.684	−0.0428731
$815$	1071.83	0.0460671
$816$	−14235.1	−0.610698
$817$	30361.1	1.30012
$818$	616.586	0.0263550
$819$	−7.30205	−0.000311544 0
$820$	−9292.66	−0.395748
$821$	−23548.6	−1.00104	−0.500518	−	0.865726i	$-0.666857\pi$
$821$	−23548.6	−1.00104	−0.500518	+	0.865726i	$0.666857\pi$
$822$	8505.40	0.360900
$823$	1553.41	0.0657941	0.0328971	−	0.999459i	$-0.489527\pi$
$823$	1553.41	0.0657941	0.0328971	+	0.999459i	$0.489527\pi$
$824$	−18648.9	−0.788430
$825$	−940.504	−0.0396899
$826$	4784.69	0.201550
$827$	−13960.6	−0.587012	−0.293506	−	0.955957i	$-0.594822\pi$
$827$	−13960.6	−0.587012	−0.293506	+	0.955957i	$0.594822\pi$
$828$	1483.93	0.0622829
$829$	33428.4	1.40050	0.700251	−	0.713897i	$-0.253072\pi$
$829$	33428.4	1.40050	0.700251	+	0.713897i	$0.253072\pi$
$830$	3297.54	0.137903
$831$	18795.0	0.784586
$832$	25.4838	0.00106189
$833$	−5196.72	−0.216153
$834$	−3545.40	−0.147203
$835$	−939.789	−0.0389494
$836$	6155.23	0.254645
$837$	3610.11	0.149085
$838$	−4971.95	−0.204956
$839$	6280.20	0.258423	0.129211	−	0.991617i	$-0.458755\pi$
$839$	6280.20	0.258423	0.129211	+	0.991617i	$0.458755\pi$
$840$	−1452.12	−0.0596464
$841$	−24207.5	−0.992557
$842$	10488.3	0.429278
$843$	12892.7	0.526747
$844$	−29462.8	−1.20160
$845$	10984.9	0.447211
$846$	1799.55	0.0731322
$847$	−8216.23	−0.333309
$848$	9882.60	0.400200
$849$	−604.322	−0.0244291
$850$	2417.34	0.0975459
$851$	2003.02	0.0806848
$852$	850.742	0.0342088
$853$	10721.0	0.430338	0.215169	−	0.976577i	$-0.430970\pi$
$853$	10721.0	0.430338	0.215169	+	0.976577i	$0.430970\pi$
$854$	1458.63	0.0584465
$855$	−3081.16	−0.123244
$856$	12933.6	0.516426
$857$	−26883.2	−1.07154	−0.535772	−	0.844363i	$-0.679979\pi$
$857$	−26883.2	−1.07154	−0.535772	+	0.844363i	$0.679979\pi$
$858$	−3.97548	−0.000158182 0
$859$	−8161.36	−0.324170	−0.162085	−	0.986777i	$-0.551822\pi$
$859$	−8161.36	−0.324170	−0.162085	+	0.986777i	$0.551822\pi$
$860$	15893.9	0.630206
$861$	−5444.34	−0.215497
$862$	−9470.21	−0.374196
$863$	−17205.7	−0.678665	−0.339332	−	0.940667i	$-0.610201\pi$
$863$	−17205.7	−0.678665	−0.339332	+	0.940667i	$0.610201\pi$
$864$	−4088.60	−0.160992
$865$	9742.54	0.382955
$866$	−15322.1	−0.601232
$867$	19004.3	0.744430
$868$	−6709.64	−0.262373
$869$	6469.23	0.252536
$870$	−184.252	−0.00718017
$871$	−24.0488	−0.000935549 0
$872$	13931.1	0.541018
$873$	5879.13	0.227925
$874$	1435.80	0.0555682
$875$	−875.000	−0.0338062
$876$	−9098.22	−0.350914
$877$	−40488.2	−1.55894	−0.779470	−	0.626440i	$-0.784511\pi$
$877$	−40488.2	−1.55894	−0.779470	+	0.626440i	$0.784511\pi$
$878$	−9861.95	−0.379072
$879$	25284.7	0.970229
$880$	2805.28	0.107461
$881$	−41690.8	−1.59432	−0.797161	−	0.603767i	$-0.793666\pi$
$881$	−41690.8	−1.59432	−0.797161	+	0.603767i	$0.793666\pi$
$882$	−402.071	−0.0153497
$883$	30811.3	1.17427	0.587136	−	0.809489i	$-0.300256\pi$
$883$	30811.3	1.17427	0.587136	+	0.809489i	$0.300256\pi$
$884$	−88.1215	−0.00335277
$885$	11245.6	0.427138
$886$	16137.8	0.611919
$887$	37707.6	1.42739	0.713697	−	0.700454i	$-0.247019\pi$
$887$	37707.6	1.42739	0.713697	+	0.700454i	$0.247019\pi$
$888$	−3613.21	−0.136544
$889$	1859.60	0.0701564
$890$	−2522.82	−0.0950172
$891$	−1015.74	−0.0381916
$892$	11285.4	0.423615
$893$	−15016.2	−0.562706
$894$	−879.421	−0.0328996
$895$	−19131.0	−0.714501
$896$	9883.26	0.368501
$897$	7.99749	0.000297690 0
$898$	−188.921	−0.00702047
$899$	−1801.42	−0.0668307
$900$	−1612.97	−0.0597396
$901$	−23426.0	−0.866184
$902$	−2964.08	−0.109416
$903$	9311.84	0.343166
$904$	−18254.7	−0.671619
$905$	−2155.87	−0.0791864
$906$	−6291.77	−0.230718
$907$	37813.8	1.38433	0.692165	−	0.721740i	$-0.256657\pi$
$907$	37813.8	1.38433	0.692165	+	0.721740i	$0.256657\pi$
$908$	−38915.4	−1.42230
$909$	14918.0	0.544332
$910$	−3.69859	−0.000134733 0
$911$	−23978.0	−0.872040	−0.436020	−	0.899937i	$-0.643612\pi$
$911$	−23978.0	−0.872040	−0.436020	+	0.899937i	$0.643612\pi$
$912$	9190.30	0.333686
$913$	−9070.99	−0.328813
$914$	−11423.7	−0.413417
$915$	3428.27	0.123863
$916$	−3571.79	−0.128838
$917$	−4902.34	−0.176543
$918$	2610.73	0.0938636
$919$	20359.5	0.730792	0.365396	−	0.930852i	$-0.380934\pi$
$919$	20359.5	0.730792	0.365396	+	0.930852i	$0.380934\pi$
$920$	1590.42	0.0569941
$921$	26043.2	0.931762
$922$	3894.83	0.139121
$923$	4.58498	0.000163506 0
$924$	1887.83	0.0672132
$925$	−2177.20	−0.0773901
$926$	7454.30	0.264539
$927$	−12136.2	−0.429995
$928$	2040.18	0.0721683
$929$	10442.5	0.368792	0.184396	−	0.982852i	$-0.440967\pi$
$929$	10442.5	0.368792	0.184396	+	0.982852i	$0.440967\pi$
$930$	1828.57	0.0644745
$931$	3355.04	0.118106
$932$	−74.1728	−0.00260688
$933$	−24727.2	−0.867664
$934$	−8722.75	−0.305586
$935$	−6649.71	−0.232587
$936$	−14.4265	−0.000503787 0
$937$	−51524.7	−1.79641	−0.898207	−	0.439573i	$-0.855130\pi$
$937$	−51524.7	−1.79641	−0.898207	+	0.439573i	$0.855130\pi$
$938$	−1324.19	−0.0460943
$939$	290.965	0.0101121
$940$	−7860.88	−0.272759
$941$	36540.3	1.26586	0.632932	−	0.774207i	$-0.281851\pi$
$941$	36540.3	1.26586	0.632932	+	0.774207i	$0.281851\pi$
$942$	10446.9	0.361337
$943$	5962.85	0.205914
$944$	−33542.8	−1.15649
$945$	−945.000	−0.0325300
$946$	5069.67	0.174238
$947$	−2735.28	−0.0938592	−0.0469296	−	0.998898i	$-0.514944\pi$
$947$	−2735.28	−0.0938592	−0.0469296	+	0.998898i	$0.514944\pi$
$948$	11094.8	0.380107
$949$	−49.0338	−0.00167725
$950$	−1560.65	−0.0532991
$951$	7346.48	0.250501
$952$	−10267.0	−0.349534
$953$	−50950.3	−1.73184	−0.865919	−	0.500184i	$-0.833266\pi$
$953$	−50950.3	−1.73184	−0.865919	+	0.500184i	$0.833266\pi$
$954$	−1812.47	−0.0615103
$955$	−12490.7	−0.423236
$956$	−28066.2	−0.949505
$957$	506.849	0.0171203
$958$	1075.20	0.0362610
$959$	−21767.5	−0.732959
$960$	3298.00	0.110878
$961$	−11913.2	−0.399892
$962$	−9.20294	−0.000308435 0
$963$	8416.81	0.281649
$964$	42998.3	1.43660
$965$	−23255.1	−0.775761
$966$	440.363	0.0146671
$967$	8344.39	0.277495	0.138747	−	0.990328i	$-0.455692\pi$
$967$	8344.39	0.277495	0.138747	+	0.990328i	$0.455692\pi$
$968$	−16232.6	−0.538984
$969$	−21784.9	−0.722222
$970$	2977.86	0.0985704
$971$	−55962.3	−1.84955	−0.924776	−	0.380512i	$-0.875748\pi$
$971$	−55962.3	−1.84955	−0.924776	+	0.380512i	$0.875748\pi$
$972$	−1742.01	−0.0574845
$973$	9073.58	0.298957
$974$	5260.55	0.173059
$975$	−8.69292	−0.000285535 0
$976$	−10225.6	−0.335363
$977$	−37772.4	−1.23689	−0.618447	−	0.785827i	$-0.712238\pi$
$977$	−37772.4	−1.23689	−0.618447	+	0.785827i	$0.712238\pi$
$978$	586.331	0.0191706
$979$	6939.89	0.226557
$980$	1756.35	0.0572494
$981$	9065.98	0.295061
$982$	13038.1	0.423689
$983$	−11325.4	−0.367472	−0.183736	−	0.982976i	$-0.558819\pi$
$983$	−11325.4	−0.367472	−0.183736	+	0.982976i	$0.558819\pi$
$984$	−10756.3	−0.348473
$985$	−16762.8	−0.542242
$986$	−1302.73	−0.0420765
$987$	−4605.50	−0.148526
$988$	56.8918	0.00183195
$989$	−10198.7	−0.327906
$990$	−514.489	−0.0165167
$991$	10538.9	0.337819	0.168909	−	0.985632i	$-0.445976\pi$
$991$	10538.9	0.337819	0.168909	+	0.985632i	$0.445976\pi$
$992$	−20247.3	−0.648038
$993$	4895.89	0.156461
$994$	252.461	0.00805592
$995$	−23102.4	−0.736076
$996$	−15556.8	−0.494916
$997$	−17439.2	−0.553966	−0.276983	−	0.960875i	$-0.589335\pi$
$997$	−17439.2	−0.553966	−0.276983	+	0.960875i	$0.589335\pi$
$998$	−10768.0	−0.341538
$999$	−2351.38	−0.0744687

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2415.4.a.q.1.9	✓	19

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2415.4.a.q.1.9	✓	19	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:	\( N \)	\(=\)	\( 2415 = 3 \cdot 5 \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2415.a (trivial)

\(f(q)\)	\(=\)	\(q-0.911725 q^{2} +3.00000 q^{3} -7.16876 q^{4} -5.00000 q^{5} -2.73518 q^{6} +7.00000 q^{7} +13.8297 q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q-0.911725 q^{2} +3.00000 q^{3} -7.16876 q^{4} -5.00000 q^{5} -2.73518 q^{6} +7.00000 q^{7} +13.8297 q^{8} +9.00000 q^{9} +4.55863 q^{10} -12.5400 q^{11} -21.5063 q^{12} -0.115906 q^{13} -6.38208 q^{14} -15.0000 q^{15} +44.7411 q^{16} -106.056 q^{17} -8.20553 q^{18} +68.4702 q^{19} +35.8438 q^{20} +21.0000 q^{21} +11.4331 q^{22} -23.0000 q^{23} +41.4892 q^{24} +25.0000 q^{25} +0.105674 q^{26} +27.0000 q^{27} -50.1813 q^{28} -13.4728 q^{29} +13.6759 q^{30} +133.708 q^{31} -151.430 q^{32} -37.6201 q^{33} +96.6935 q^{34} -35.0000 q^{35} -64.5188 q^{36} -87.0880 q^{37} -62.4260 q^{38} -0.347717 q^{39} -69.1487 q^{40} -259.254 q^{41} -19.1462 q^{42} +443.421 q^{43} +89.8966 q^{44} -45.0000 q^{45} +20.9697 q^{46} -219.310 q^{47} +134.223 q^{48} +49.0000 q^{49} -22.7931 q^{50} -318.167 q^{51} +0.830899 q^{52} +220.884 q^{53} -24.6166 q^{54} +62.7002 q^{55} +96.8082 q^{56} +205.411 q^{57} +12.2835 q^{58} -749.707 q^{59} +107.531 q^{60} -228.551 q^{61} -121.905 q^{62} +63.0000 q^{63} -219.867 q^{64} +0.579528 q^{65} +34.2992 q^{66} +207.486 q^{67} +760.286 q^{68} -69.0000 q^{69} +31.9104 q^{70} -39.5579 q^{71} +124.468 q^{72} +423.050 q^{73} +79.4003 q^{74} +75.0000 q^{75} -490.846 q^{76} -87.7803 q^{77} +0.317022 q^{78} -515.886 q^{79} -223.706 q^{80} +81.0000 q^{81} +236.369 q^{82} +723.362 q^{83} -150.544 q^{84} +530.278 q^{85} -404.278 q^{86} -40.4184 q^{87} -173.426 q^{88} -553.418 q^{89} +41.0276 q^{90} -0.811339 q^{91} +164.881 q^{92} +401.124 q^{93} +199.950 q^{94} -342.351 q^{95} -454.289 q^{96} +653.236 q^{97} -44.6745 q^{98} -112.860 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 19 q + 57 q^{3} + 88 q^{4} - 95 q^{5} + 133 q^{7} + 171 q^{9}+O(q^{10}) \) 19 * q + 57 * q^3 + 88 * q^4 - 95 * q^5 + 133 * q^7 + 171 * q^9	\( 19 q + 57 q^{3} + 88 q^{4} - 95 q^{5} + 133 q^{7} + 171 q^{9} + 31 q^{11} + 264 q^{12} + 139 q^{13} - 285 q^{15} + 504 q^{16} + 55 q^{17} + 257 q^{19} - 440 q^{20} + 399 q^{21} + 218 q^{22} - 437 q^{23} + 475 q^{25} + 275 q^{26} + 513 q^{27} + 616 q^{28} - 84 q^{29} + 164 q^{31} - 25 q^{32} + 93 q^{33} + 1005 q^{34} - 665 q^{35} + 792 q^{36} + 775 q^{37} + 760 q^{38} + 417 q^{39} + 321 q^{41} + 1265 q^{43} + 351 q^{44} - 855 q^{45} - 542 q^{47} + 1512 q^{48} + 931 q^{49} + 165 q^{51} + 2696 q^{52} + 644 q^{53} - 155 q^{55} + 771 q^{57} - 6 q^{58} + 789 q^{59} - 1320 q^{60} + 977 q^{61} + 674 q^{62} + 1197 q^{63} + 3392 q^{64} - 695 q^{65} + 654 q^{66} + 4025 q^{67} + 62 q^{68} - 1311 q^{69} + 402 q^{71} + 789 q^{73} - 1495 q^{74} + 1425 q^{75} + 2941 q^{76} + 217 q^{77} + 825 q^{78} + 2850 q^{79} - 2520 q^{80} + 1539 q^{81} + 5062 q^{82} - 615 q^{83} + 1848 q^{84} - 275 q^{85} + 1246 q^{86} - 252 q^{87} + 563 q^{88} - 6 q^{89} + 973 q^{91} - 2024 q^{92} + 492 q^{93} + 2754 q^{94} - 1285 q^{95} - 75 q^{96} + 4050 q^{97} + 279 q^{99}+O(q^{100}) \) 19 * q + 57 * q^3 + 88 * q^4 - 95 * q^5 + 133 * q^7 + 171 * q^9 + 31 * q^11 + 264 * q^12 + 139 * q^13 - 285 * q^15 + 504 * q^16 + 55 * q^17 + 257 * q^19 - 440 * q^20 + 399 * q^21 + 218 * q^22 - 437 * q^23 + 475 * q^25 + 275 * q^26 + 513 * q^27 + 616 * q^28 - 84 * q^29 + 164 * q^31 - 25 * q^32 + 93 * q^33 + 1005 * q^34 - 665 * q^35 + 792 * q^36 + 775 * q^37 + 760 * q^38 + 417 * q^39 + 321 * q^41 + 1265 * q^43 + 351 * q^44 - 855 * q^45 - 542 * q^47 + 1512 * q^48 + 931 * q^49 + 165 * q^51 + 2696 * q^52 + 644 * q^53 - 155 * q^55 + 771 * q^57 - 6 * q^58 + 789 * q^59 - 1320 * q^60 + 977 * q^61 + 674 * q^62 + 1197 * q^63 + 3392 * q^64 - 695 * q^65 + 654 * q^66 + 4025 * q^67 + 62 * q^68 - 1311 * q^69 + 402 * q^71 + 789 * q^73 - 1495 * q^74 + 1425 * q^75 + 2941 * q^76 + 217 * q^77 + 825 * q^78 + 2850 * q^79 - 2520 * q^80 + 1539 * q^81 + 5062 * q^82 - 615 * q^83 + 1848 * q^84 - 275 * q^85 + 1246 * q^86 - 252 * q^87 + 563 * q^88 - 6 * q^89 + 973 * q^91 - 2024 * q^92 + 492 * q^93 + 2754 * q^94 - 1285 * q^95 - 75 * q^96 + 4050 * q^97 + 279 * q^99

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