LMFDB - Embedded newform 115.6.a.e.1.10

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [115,6,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, 6, names="a")

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

chi := DirichletCharacter("115.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 115 = 5 \cdot 23 $
Weight:	$ k $	$=$	$ 6 $
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:	$18.4441392785$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 4 x^{11} - 329 x^{10} + 1059 x^{9} + 41059 x^{8} - 99023 x^{7} - 2392947 x^{6} + 3889937 x^{5} + \cdots + 4039776 $ x^12 - 4x^11 - 329x^10 + 1059x^9 + 41059x^8 - 99023x^7 - 2392947x^6 + 3889937x^5 + 63665212x^4 - 59034365x^3 - 601032676x^2 + 161487612*x + 4039776
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.10
Root			$-7.26134$ 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)

$f(q)$	$=$	\(q+8.26134 q^{2} +12.5950 q^{3} +36.2497 q^{4} +25.0000 q^{5} +104.052 q^{6} +63.2337 q^{7} +35.1081 q^{8} -84.3653 q^{9} +206.533 q^{10} +468.606 q^{11} +456.566 q^{12} +1123.23 q^{13} +522.395 q^{14} +314.876 q^{15} -869.950 q^{16} -601.631 q^{17} -696.970 q^{18} -608.068 q^{19} +906.242 q^{20} +796.430 q^{21} +3871.31 q^{22} -529.000 q^{23} +442.187 q^{24} +625.000 q^{25} +9279.36 q^{26} -4123.17 q^{27} +2292.20 q^{28} +1892.66 q^{29} +2601.29 q^{30} -4457.82 q^{31} -8310.41 q^{32} +5902.10 q^{33} -4970.28 q^{34} +1580.84 q^{35} -3058.22 q^{36} +134.186 q^{37} -5023.45 q^{38} +14147.1 q^{39} +877.702 q^{40} -8546.11 q^{41} +6579.58 q^{42} +1466.08 q^{43} +16986.8 q^{44} -2109.13 q^{45} -4370.25 q^{46} +4434.31 q^{47} -10957.0 q^{48} -12808.5 q^{49} +5163.34 q^{50} -7577.56 q^{51} +40716.6 q^{52} -13348.2 q^{53} -34062.9 q^{54} +11715.1 q^{55} +2220.02 q^{56} -7658.63 q^{57} +15635.9 q^{58} +48451.0 q^{59} +11414.1 q^{60} -37607.9 q^{61} -36827.6 q^{62} -5334.73 q^{63} -40816.7 q^{64} +28080.7 q^{65} +48759.2 q^{66} -12159.5 q^{67} -21808.9 q^{68} -6662.77 q^{69} +13059.9 q^{70} -67887.7 q^{71} -2961.90 q^{72} +69090.1 q^{73} +1108.56 q^{74} +7871.89 q^{75} -22042.3 q^{76} +29631.7 q^{77} +116874. q^{78} -75657.9 q^{79} -21748.8 q^{80} -31430.7 q^{81} -70602.3 q^{82} +57852.2 q^{83} +28870.4 q^{84} -15040.8 q^{85} +12111.8 q^{86} +23838.1 q^{87} +16451.9 q^{88} -70027.0 q^{89} -17424.3 q^{90} +71025.8 q^{91} -19176.1 q^{92} -56146.4 q^{93} +36633.4 q^{94} -15201.7 q^{95} -104670. q^{96} +10083.5 q^{97} -105815. q^{98} -39534.1 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 8 q^{2} + 22 q^{3} + 294 q^{4} + 300 q^{5} + 454 q^{6} + 16 q^{7} + 675 q^{8} + 1598 q^{9} + 200 q^{10} + 132 q^{11} + 728 q^{12} - 236 q^{13} + 359 q^{14} + 550 q^{15} + 4514 q^{16} + 1666 q^{17}+ \cdots - 740784 q^{99}+O(q^{100}) $ 12 * q + 8 * q^2 + 22 * q^3 + 294 * q^4 + 300 * q^5 + 454 * q^6 + 16 * q^7 + 675 * q^8 + 1598 * q^9 + 200 * q^10 + 132 * q^11 + 728 * q^12 - 236 * q^13 + 359 * q^14 + 550 * q^15 + 4514 * q^16 + 1666 * q^17 - 3096 * q^18 + 616 * q^19 + 7350 * q^20 + 2732 * q^21 + 305 * q^22 - 6348 * q^23 + 18873 * q^24 + 7500 * q^25 - 4502 * q^26 + 11584 * q^27 + 5407 * q^28 + 23722 * q^29 + 11350 * q^30 + 18446 * q^31 + 35808 * q^32 + 4416 * q^33 + 53123 * q^34 + 400 * q^35 + 68916 * q^36 + 10394 * q^37 + 18681 * q^38 + 27032 * q^39 + 16875 * q^40 + 48232 * q^41 - 18980 * q^42 + 10732 * q^43 - 4765 * q^44 + 39950 * q^45 - 4232 * q^46 - 30448 * q^47 - 2052 * q^48 + 26948 * q^49 + 5000 * q^50 + 1524 * q^51 - 55346 * q^52 + 36494 * q^53 + 55567 * q^54 + 3300 * q^55 - 50981 * q^56 + 37572 * q^57 - 83373 * q^58 - 23870 * q^59 + 18200 * q^60 + 30862 * q^61 + 63582 * q^62 - 49698 * q^63 + 29965 * q^64 - 5900 * q^65 - 235225 * q^66 - 71910 * q^67 - 39371 * q^68 - 11638 * q^69 + 8975 * q^70 + 167158 * q^71 - 296052 * q^72 + 52152 * q^73 - 59356 * q^74 + 13750 * q^75 - 230417 * q^76 + 4808 * q^77 - 469771 * q^78 - 123092 * q^79 + 112850 * q^80 + 159868 * q^81 - 140098 * q^82 + 89322 * q^83 - 488082 * q^84 + 41650 * q^85 - 55318 * q^86 - 334376 * q^87 - 104551 * q^88 - 46184 * q^89 - 77400 * q^90 - 153444 * q^91 - 155526 * q^92 - 16576 * q^93 - 456595 * q^94 + 15400 * q^95 + 330540 * q^96 - 94220 * q^97 + 413841 * q^98 - 740784 * q^99

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 $

$2$	8.26134	1.46041	0.730206	−	0.683227i	$-0.239424\pi$
$2$	8.26134	1.46041	0.730206	+	0.683227i	$0.239424\pi$
$3$	12.5950	0.807971	0.403986	−	0.914765i	$-0.367625\pi$
$3$	12.5950	0.807971	0.403986	+	0.914765i	$0.367625\pi$
$4$	36.2497	1.13280
$5$	25.0000	0.447214
$5$	25.0000	0.447214
$6$	104.052	1.17997
$7$	63.2337	0.487757	0.243879	−	0.969806i	$-0.421580\pi$
$7$	63.2337	0.487757	0.243879	+	0.969806i	$0.421580\pi$
$8$	35.1081	0.193947
$9$	−84.3653	−0.347182
$10$	206.533	0.653116
$11$	468.606	1.16769	0.583843	−	0.811867i	$-0.301548\pi$
$11$	468.606	1.16769	0.583843	+	0.811867i	$0.301548\pi$
$12$	456.566	0.915272
$13$	1123.23	1.84336	0.921678	−	0.387956i	$-0.126819\pi$
$13$	1123.23	1.84336	0.921678	+	0.387956i	$0.126819\pi$
$14$	522.395	0.712326
$15$	314.876	0.361336
$16$	−869.950	−0.849561
$17$	−601.631	−0.504903	−0.252452	−	0.967610i	$-0.581237\pi$
$17$	−601.631	−0.504903	−0.252452	+	0.967610i	$0.581237\pi$
$18$	−696.970	−0.507029
$19$	−608.068	−0.386427	−0.193214	−	0.981157i	$-0.561891\pi$
$19$	−608.068	−0.386427	−0.193214	+	0.981157i	$0.561891\pi$
$20$	906.242	0.506605
$21$	796.430	0.394094
$22$	3871.31	1.70530
$23$	−529.000	−0.208514
$23$	−529.000	−0.208514
$24$	442.187	0.156703
$25$	625.000	0.200000
$26$	9279.36	2.69206
$27$	−4123.17	−1.08848
$28$	2292.20	0.552533
$29$	1892.66	0.417905	0.208952	−	0.977926i	$-0.432995\pi$
$29$	1892.66	0.417905	0.208952	+	0.977926i	$0.432995\pi$
$30$	2601.29	0.527699
$31$	−4457.82	−0.833141	−0.416570	−	0.909103i	$-0.636768\pi$
$31$	−4457.82	−0.833141	−0.416570	+	0.909103i	$0.636768\pi$
$32$	−8310.41	−1.43466
$33$	5902.10	0.943456
$34$	−4970.28	−0.737366
$35$	1580.84	0.218132
$36$	−3058.22	−0.393289
$37$	134.186	0.0161140	0.00805701	−	0.999968i	$-0.497435\pi$
$37$	134.186	0.0161140	0.00805701	+	0.999968i	$0.497435\pi$
$38$	−5023.45	−0.564343
$39$	14147.1	1.48938
$40$	877.702	0.0867356
$41$	−8546.11	−0.793979	−0.396989	−	0.917823i	$-0.629945\pi$
$41$	−8546.11	−0.793979	−0.396989	+	0.917823i	$0.629945\pi$
$42$	6579.58	0.575539
$43$	1466.08	0.120917	0.0604583	−	0.998171i	$-0.480744\pi$
$43$	1466.08	0.120917	0.0604583	+	0.998171i	$0.480744\pi$
$44$	16986.8	1.32276
$45$	−2109.13	−0.155265
$46$	−4370.25	−0.304517
$47$	4434.31	0.292807	0.146404	−	0.989225i	$-0.453230\pi$
$47$	4434.31	0.292807	0.146404	+	0.989225i	$0.453230\pi$
$48$	−10957.0	−0.686421
$49$	−12808.5	−0.762093
$50$	5163.34	0.292082
$51$	−7577.56	−0.407947
$52$	40716.6	2.08816
$53$	−13348.2	−0.652728	−0.326364	−	0.945244i	$-0.605824\pi$
$53$	−13348.2	−0.652728	−0.326364	+	0.945244i	$0.605824\pi$
$54$	−34062.9	−1.58964
$55$	11715.1	0.522205
$56$	2220.02	0.0945989
$57$	−7658.63	−0.312222
$58$	15635.9	0.610313
$59$	48451.0	1.81206	0.906031	−	0.423212i	$-0.139098\pi$
$59$	48451.0	1.81206	0.906031	+	0.423212i	$0.139098\pi$
$60$	11414.1	0.409322
$61$	−37607.9	−1.29406	−0.647029	−	0.762465i	$-0.723989\pi$
$61$	−37607.9	−1.29406	−0.647029	+	0.762465i	$0.723989\pi$
$62$	−36827.6	−1.21673
$63$	−5334.73	−0.169341
$64$	−40816.7	−1.24563
$65$	28080.7	0.824374
$66$	48759.2	1.37783
$67$	−12159.5	−0.330925	−0.165463	−	0.986216i	$-0.552912\pi$
$67$	−12159.5	−0.330925	−0.165463	+	0.986216i	$0.552912\pi$
$68$	−21808.9	−0.571956
$69$	−6662.77	−0.168474
$70$	13059.9	0.318562
$71$	−67887.7	−1.59825	−0.799126	−	0.601163i	$-0.794704\pi$
$71$	−67887.7	−1.59825	−0.799126	+	0.601163i	$0.794704\pi$
$72$	−2961.90	−0.0673348
$73$	69090.1	1.51743	0.758715	−	0.651423i	$-0.225828\pi$
$73$	69090.1	1.51743	0.758715	+	0.651423i	$0.225828\pi$
$74$	1108.56	0.0235331
$75$	7871.89	0.161594
$76$	−22042.3	−0.437746
$77$	29631.7	0.569547
$78$	116874.	2.17511
$79$	−75657.9	−1.36391	−0.681956	−	0.731393i	$-0.738870\pi$
$79$	−75657.9	−1.36391	−0.681956	+	0.731393i	$0.738870\pi$
$80$	−21748.8	−0.379935
$81$	−31430.7	−0.532282
$82$	−70602.3	−1.15954
$83$	57852.2	0.921774	0.460887	−	0.887459i	$-0.347531\pi$
$83$	57852.2	0.921774	0.460887	+	0.887459i	$0.347531\pi$
$84$	28870.4	0.446431
$85$	−15040.8	−0.225799
$86$	12111.8	0.176588
$87$	23838.1	0.337655
$88$	16451.9	0.226469
$89$	−70027.0	−0.937110	−0.468555	−	0.883434i	$-0.655225\pi$
$89$	−70027.0	−0.937110	−0.468555	+	0.883434i	$0.655225\pi$
$90$	−17424.3	−0.226750
$91$	71025.8	0.899110
$92$	−19176.1	−0.236206
$93$	−56146.4	−0.673154
$94$	36633.4	0.427619
$95$	−15201.7	−0.172816
$96$	−104670.	−1.15916
$97$	10083.5	0.108813	0.0544064	−	0.998519i	$-0.482673\pi$
$97$	10083.5	0.108813	0.0544064	+	0.998519i	$0.482673\pi$
$98$	−105815.	−1.11297
$99$	−39534.1	−0.405400
$100$	22656.1	0.226561
$101$	42192.8	0.411562	0.205781	−	0.978598i	$-0.434027\pi$
$101$	42192.8	0.411562	0.205781	+	0.978598i	$0.434027\pi$
$102$	−62600.8	−0.595771
$103$	139491.	1.29554	0.647772	−	0.761834i	$-0.275701\pi$
$103$	139491.	1.29554	0.647772	+	0.761834i	$0.275701\pi$
$104$	39434.4	0.357513
$105$	19910.8	0.176244
$106$	−110274.	−0.953252
$107$	−183948.	−1.55323	−0.776616	−	0.629974i	$-0.783065\pi$
$107$	−183948.	−1.55323	−0.776616	+	0.629974i	$0.783065\pi$
$108$	−149464.	−1.23304
$109$	−225031.	−1.81416	−0.907081	−	0.420955i	$-0.861695\pi$
$109$	−225031.	−1.81416	−0.907081	+	0.420955i	$0.861695\pi$
$110$	96782.7	0.762634
$111$	1690.08	0.0130197
$112$	−55010.2	−0.414379
$113$	186586.	1.37462	0.687310	−	0.726364i	$-0.258791\pi$
$113$	186586.	1.37462	0.687310	+	0.726364i	$0.258791\pi$
$114$	−63270.5	−0.455973
$115$	−13225.0	−0.0932505
$116$	68608.3	0.473403
$117$	−94761.4	−0.639981
$118$	400270.	2.64636
$119$	−38043.4	−0.246270
$120$	11054.7	0.0700799
$121$	58540.3	0.363489
$122$	−310691.	−1.88986
$123$	−107638.	−0.641512
$124$	−161595.	−0.943784
$125$	15625.0	0.0894427
$126$	−44072.0	−0.247307
$127$	111792.	0.615040	0.307520	−	0.951542i	$-0.400501\pi$
$127$	111792.	0.615040	0.307520	+	0.951542i	$0.400501\pi$
$128$	−71267.3	−0.384473
$129$	18465.3	0.0976972
$130$	231984.	1.20393
$131$	117161.	0.596490	0.298245	−	0.954489i	$-0.403599\pi$
$131$	117161.	0.596490	0.298245	+	0.954489i	$0.403599\pi$
$132$	213949.	1.06875
$133$	−38450.4	−0.188483
$134$	−100454.	−0.483287
$135$	−103079.	−0.486785
$136$	−21122.1	−0.0979242
$137$	26002.1	0.118360	0.0591802	−	0.998247i	$-0.481151\pi$
$137$	26002.1	0.118360	0.0591802	+	0.998247i	$0.481151\pi$
$138$	−55043.4	−0.246041
$139$	210639.	0.924702	0.462351	−	0.886697i	$-0.347006\pi$
$139$	210639.	0.924702	0.462351	+	0.886697i	$0.347006\pi$
$140$	57305.1	0.247100
$141$	55850.3	0.236580
$142$	−560843.	−2.33411
$143$	526351.	2.15246
$144$	73393.6	0.294952
$145$	47316.4	0.186893
$146$	570777.	2.21607
$147$	−161323.	−0.615749
$148$	4864.21	0.0182540
$149$	233149.	0.860337	0.430169	−	0.902749i	$-0.358454\pi$
$149$	233149.	0.860337	0.430169	+	0.902749i	$0.358454\pi$
$150$	65032.4	0.235994
$151$	515259.	1.83901	0.919503	−	0.393084i	$-0.128592\pi$
$151$	515259.	1.83901	0.919503	+	0.393084i	$0.128592\pi$
$152$	−21348.1	−0.0749463
$153$	50756.8	0.175293
$154$	244797.	0.831773
$155$	−111446.	−0.372592
$156$	512827.	1.68717
$157$	−43150.9	−0.139714	−0.0698571	−	0.997557i	$-0.522254\pi$
$157$	−43150.9	−0.139714	−0.0698571	+	0.997557i	$0.522254\pi$
$158$	−625035.	−1.99187
$159$	−168121.	−0.527386
$160$	−207760.	−0.641597
$161$	−33450.6	−0.101704
$162$	−259660.	−0.777351
$163$	314107.	0.925995	0.462997	−	0.886360i	$-0.346774\pi$
$163$	314107.	0.925995	0.462997	+	0.886360i	$0.346774\pi$
$164$	−309794.	−0.899421
$165$	147553.	0.421926
$166$	477936.	1.34617
$167$	−220148.	−0.610835	−0.305418	−	0.952219i	$-0.598796\pi$
$167$	−220148.	−0.610835	−0.305418	+	0.952219i	$0.598796\pi$
$168$	27961.2	0.0764332
$169$	890346.	2.39796
$170$	−124257.	−0.329760
$171$	51299.8	0.134161
$172$	53144.9	0.136975
$173$	−290644.	−0.738324	−0.369162	−	0.929365i	$-0.620355\pi$
$173$	−290644.	−0.738324	−0.369162	+	0.929365i	$0.620355\pi$
$174$	196934.	0.493115
$175$	39521.1	0.0975514
$176$	−407664.	−0.992020
$177$	610242.	1.46409
$178$	−578517.	−1.36857
$179$	178931.	0.417401	0.208700	−	0.977980i	$-0.433077\pi$
$179$	178931.	0.417401	0.208700	+	0.977980i	$0.433077\pi$
$180$	−76455.4	−0.175884
$181$	589666.	1.33786	0.668928	−	0.743327i	$-0.266753\pi$
$181$	589666.	1.33786	0.668928	+	0.743327i	$0.266753\pi$
$182$	586768.	1.31307
$183$	−473672.	−1.04556
$184$	−18572.2	−0.0404407
$185$	3354.66	0.00720641
$186$	−463844.	−0.983082
$187$	−281928.	−0.589568
$188$	160743.	0.331693
$189$	−260724.	−0.530916
$190$	−125586.	−0.252382
$191$	−260213.	−0.516115	−0.258057	−	0.966130i	$-0.583082\pi$
$191$	−260213.	−0.516115	−0.258057	+	0.966130i	$0.583082\pi$
$192$	−514087.	−1.00643
$193$	377194.	0.728906	0.364453	−	0.931222i	$-0.381256\pi$
$193$	377194.	0.728906	0.364453	+	0.931222i	$0.381256\pi$
$194$	83302.9	0.158912
$195$	353677.	0.666070
$196$	−464304.	−0.863301
$197$	653546.	1.19980	0.599902	−	0.800073i	$-0.295206\pi$
$197$	653546.	1.19980	0.599902	+	0.800073i	$0.295206\pi$
$198$	−326604.	−0.592050
$199$	662519.	1.18595	0.592974	−	0.805221i	$-0.297954\pi$
$199$	662519.	1.18595	0.592974	+	0.805221i	$0.297954\pi$
$200$	21942.6	0.0387893
$201$	−153150.	−0.267378
$202$	348569.	0.601050
$203$	119680.	0.203836
$204$	−274684.	−0.462124
$205$	−213653.	−0.355078
$206$	1.15238e6	1.89203
$207$	44629.2	0.0723925
$208$	−977152.	−1.56604
$209$	−284944.	−0.451226
$210$	164490.	0.257389
$211$	−221802.	−0.342973	−0.171487	−	0.985186i	$-0.554857\pi$
$211$	−221802.	−0.342973	−0.171487	+	0.985186i	$0.554857\pi$
$212$	−483867.	−0.739412
$213$	−855048.	−1.29134
$214$	−1.51966e6	−2.26836
$215$	36652.0	0.0540756
$216$	−144757.	−0.211108
$217$	−281885.	−0.406370
$218$	−1.85906e6	−2.64943
$219$	870192.	1.22604
$220$	424670.	0.591555
$221$	−675768.	−0.930716
$222$	13962.3	0.0190141
$223$	−47494.5	−0.0639560	−0.0319780	−	0.999489i	$-0.510181\pi$
$223$	−47494.5	−0.0639560	−0.0319780	+	0.999489i	$0.510181\pi$
$224$	−525498.	−0.699763
$225$	−52728.3	−0.0694365
$226$	1.54145e6	2.00751
$227$	−1.08504e6	−1.39760	−0.698799	−	0.715318i	$-0.746282\pi$
$227$	−1.08504e6	−1.39760	−0.698799	+	0.715318i	$0.746282\pi$
$228$	−277623.	−0.353686
$229$	−1.48081e6	−1.86599	−0.932996	−	0.359886i	$-0.882815\pi$
$229$	−1.48081e6	−1.86599	−0.932996	+	0.359886i	$0.882815\pi$
$230$	−109256.	−0.136184
$231$	373212.	0.460178
$232$	66447.6	0.0810512
$233$	165975.	0.200287	0.100144	−	0.994973i	$-0.468070\pi$
$233$	165975.	0.200287	0.100144	+	0.994973i	$0.468070\pi$
$234$	−782856.	−0.934635
$235$	110858.	0.130947
$236$	1.75633e6	2.05271
$237$	−952913.	−1.10200
$238$	−314289.	−0.359656
$239$	−175263.	−0.198470	−0.0992352	−	0.995064i	$-0.531640\pi$
$239$	−175263.	−0.198470	−0.0992352	+	0.995064i	$0.531640\pi$
$240$	−273926.	−0.306977
$241$	170515.	0.189113	0.0945563	−	0.995520i	$-0.469857\pi$
$241$	170515.	0.189113	0.0945563	+	0.995520i	$0.469857\pi$
$242$	483621.	0.530844
$243$	606061.	0.658416
$244$	−1.36327e6	−1.46591
$245$	−320212.	−0.340818
$246$	−889238.	−0.936872
$247$	−682998.	−0.712323
$248$	−156506.	−0.161585
$249$	728650.	0.744767
$250$	129083.	0.130623
$251$	−571498.	−0.572572	−0.286286	−	0.958144i	$-0.592421\pi$
$251$	−571498.	−0.572572	−0.286286	+	0.958144i	$0.592421\pi$
$252$	−193382.	−0.191830
$253$	−247892.	−0.243479
$254$	923555.	0.898211
$255$	−189439.	−0.182440
$256$	717371.	0.684138
$257$	832132.	0.785886	0.392943	−	0.919563i	$-0.371457\pi$
$257$	832132.	0.785886	0.392943	+	0.919563i	$0.371457\pi$
$258$	152548.	0.142678
$259$	8485.10	0.00785973
$260$	1.01792e6	0.933853
$261$	−159675.	−0.145089
$262$	967903.	0.871121
$263$	235655.	0.210081	0.105041	−	0.994468i	$-0.466503\pi$
$263$	235655.	0.210081	0.105041	+	0.994468i	$0.466503\pi$
$264$	207212.	0.182980
$265$	−333704.	−0.291909
$266$	−317652.	−0.275262
$267$	−881992.	−0.757158
$268$	−440779.	−0.374873
$269$	2.07196e6	1.74582	0.872911	−	0.487879i	$-0.162229\pi$
$269$	2.07196e6	1.74582	0.872911	+	0.487879i	$0.162229\pi$
$270$	−851573.	−0.710907
$271$	1.96875e6	1.62842	0.814212	−	0.580568i	$-0.197169\pi$
$271$	1.96875e6	1.62842	0.814212	+	0.580568i	$0.197169\pi$
$272$	523389.	0.428946
$273$	894572.	0.726455
$274$	214812.	0.172855
$275$	292879.	0.233537
$276$	−241523.	−0.190847
$277$	−754291.	−0.590662	−0.295331	−	0.955395i	$-0.595430\pi$
$277$	−754291.	−0.590662	−0.295331	+	0.955395i	$0.595430\pi$
$278$	1.74016e6	1.35045
$279$	376085.	0.289252
$280$	55500.4	0.0423059
$281$	252483.	0.190751	0.0953754	−	0.995441i	$-0.469595\pi$
$281$	252483.	0.190751	0.0953754	+	0.995441i	$0.469595\pi$
$282$	461398.	0.345504
$283$	−1.23325e6	−0.915342	−0.457671	−	0.889122i	$-0.651316\pi$
$283$	−1.23325e6	−0.915342	−0.457671	+	0.889122i	$0.651316\pi$
$284$	−2.46091e6	−1.81050
$285$	−191466.	−0.139630
$286$	4.34836e6	3.14348
$287$	−540402.	−0.387269
$288$	701110.	0.498087
$289$	−1.05790e6	−0.745073
$290$	390897.	0.272940
$291$	127001.	0.0879177
$292$	2.50449e6	1.71895
$293$	350407.	0.238453	0.119227	−	0.992867i	$-0.461958\pi$
$293$	350407.	0.238453	0.119227	+	0.992867i	$0.461958\pi$
$294$	−1.33275e6	−0.899248
$295$	1.21128e6	0.810378
$296$	4711.03	0.00312526
$297$	−1.93214e6	−1.27101
$298$	1.92613e6	1.25645
$299$	−594187.	−0.384366
$300$	285354.	0.183054
$301$	92705.6	0.0589780
$302$	4.25673e6	2.68571
$303$	531420.	0.332530
$304$	528989.	0.328294
$305$	−940197.	−0.578721
$306$	419319.	0.256001
$307$	−708307.	−0.428920	−0.214460	−	0.976733i	$-0.568799\pi$
$307$	−708307.	−0.428920	−0.214460	+	0.976733i	$0.568799\pi$
$308$	1.07414e6	0.645184
$309$	1.75689e6	1.04676
$310$	−920689.	−0.544138
$311$	−46185.3	−0.0270771	−0.0135386	−	0.999908i	$-0.504310\pi$
$311$	−46185.3	−0.0270771	−0.0135386	+	0.999908i	$0.504310\pi$
$312$	496677.	0.288860
$313$	−3.32846e6	−1.92036	−0.960179	−	0.279386i	$-0.909869\pi$
$313$	−3.32846e6	−1.92036	−0.960179	+	0.279386i	$0.909869\pi$
$314$	−356484.	−0.204040
$315$	−133368.	−0.0757314
$316$	−2.74257e6	−1.54504
$317$	1.61877e6	0.904768	0.452384	−	0.891823i	$-0.350574\pi$
$317$	1.61877e6	0.904768	0.452384	+	0.891823i	$0.350574\pi$
$318$	−1.38890e6	−0.770200
$319$	886910.	0.487981
$320$	−1.02042e6	−0.557061
$321$	−2.31683e6	−1.25497
$322$	−276347.	−0.148530
$323$	365832.	0.195108
$324$	−1.13935e6	−0.602971
$325$	702017.	0.368671
$326$	2.59494e6	1.35233
$327$	−2.83427e6	−1.46579
$328$	−300038.	−0.153989
$329$	280398.	0.142819
$330$	1.21898e6	0.616186
$331$	−3.25860e6	−1.63479	−0.817393	−	0.576081i	$-0.804581\pi$
$331$	−3.25860e6	−1.63479	−0.817393	+	0.576081i	$0.804581\pi$
$332$	2.09712e6	1.04419
$333$	−11320.7	−0.00559450
$334$	−1.81872e6	−0.892071
$335$	−303988.	−0.147994
$336$	−692855.	−0.334807
$337$	370879.	0.177893	0.0889463	−	0.996036i	$-0.471650\pi$
$337$	370879.	0.177893	0.0889463	+	0.996036i	$0.471650\pi$
$338$	7.35545e6	3.50201
$339$	2.35005e6	1.11065
$340$	−545223.	−0.255786
$341$	−2.08896e6	−0.972846
$342$	423805.	0.195930
$343$	−1.87270e6	−0.859473
$344$	51471.2	0.0234514
$345$	−166569.	−0.0753437
$346$	−2.40111e6	−1.07826
$347$	−958308.	−0.427249	−0.213625	−	0.976916i	$-0.568527\pi$
$347$	−958308.	−0.427249	−0.213625	+	0.976916i	$0.568527\pi$
$348$	864123.	0.382496
$349$	−3.02928e6	−1.33130	−0.665650	−	0.746265i	$-0.731845\pi$
$349$	−3.02928e6	−1.33130	−0.665650	+	0.746265i	$0.731845\pi$
$350$	326497.	0.142465
$351$	−4.63126e6	−2.00646
$352$	−3.89431e6	−1.67523
$353$	1.62357e6	0.693479	0.346739	−	0.937961i	$-0.387289\pi$
$353$	1.62357e6	0.693479	0.346739	+	0.937961i	$0.387289\pi$
$354$	5.04141e6	2.13818
$355$	−1.69719e6	−0.714760
$356$	−2.53846e6	−1.06156
$357$	−479157.	−0.198979
$358$	1.47821e6	0.609577
$359$	−97815.9	−0.0400565	−0.0200283	−	0.999799i	$-0.506376\pi$
$359$	−97815.9	−0.0400565	−0.0200283	+	0.999799i	$0.506376\pi$
$360$	−74047.6	−0.0301131
$361$	−2.10635e6	−0.850674
$362$	4.87143e6	1.95382
$363$	737316.	0.293689
$364$	2.57466e6	1.01851
$365$	1.72725e6	0.678615
$366$	−3.91316e6	−1.52695
$367$	−2.42352e6	−0.939249	−0.469624	−	0.882866i	$-0.655611\pi$
$367$	−2.42352e6	−0.939249	−0.469624	+	0.882866i	$0.655611\pi$
$368$	460204.	0.177146
$369$	720995.	0.275655
$370$	27714.0	0.0105243
$371$	−844055.	−0.318373
$372$	−2.03529e6	−0.762551
$373$	1.31572e6	0.489658	0.244829	−	0.969566i	$-0.421268\pi$
$373$	1.31572e6	0.489658	0.244829	+	0.969566i	$0.421268\pi$
$374$	−2.32910e6	−0.861012
$375$	196797.	0.0722672
$376$	155680.	0.0567890
$377$	2.12588e6	0.770347
$378$	−2.15393e6	−0.775356
$379$	−206664.	−0.0739039	−0.0369519	−	0.999317i	$-0.511765\pi$
$379$	−206664.	−0.0739039	−0.0369519	+	0.999317i	$0.511765\pi$
$380$	−551057.	−0.195766
$381$	1.40803e6	0.496934
$382$	−2.14971e6	−0.753740
$383$	822751.	0.286597	0.143298	−	0.989680i	$-0.454229\pi$
$383$	822751.	0.286597	0.143298	+	0.989680i	$0.454229\pi$
$384$	−897614.	−0.310643
$385$	740792.	0.254709
$386$	3.11613e6	1.06450
$387$	−123686.	−0.0419801
$388$	365522.	0.123264
$389$	3.91075e6	1.31034	0.655172	−	0.755479i	$-0.272596\pi$
$389$	3.91075e6	1.31034	0.655172	+	0.755479i	$0.272596\pi$
$390$	2.92184e6	0.972737
$391$	318263.	0.105280
$392$	−449682.	−0.147805
$393$	1.47564e6	0.481947
$394$	5.39916e6	1.75221
$395$	−1.89145e6	−0.609960
$396$	−1.43310e6	−0.459238
$397$	−3.62697e6	−1.15496	−0.577480	−	0.816404i	$-0.695964\pi$
$397$	−3.62697e6	−1.15496	−0.577480	+	0.816404i	$0.695964\pi$
$398$	5.47330e6	1.73197
$399$	−484284.	−0.152289
$400$	−543719.	−0.169912
$401$	4.76269e6	1.47908	0.739540	−	0.673113i	$-0.235043\pi$
$401$	4.76269e6	1.47908	0.739540	+	0.673113i	$0.235043\pi$
$402$	−1.26522e6	−0.390482
$403$	−5.00715e6	−1.53578
$404$	1.52948e6	0.466218
$405$	−785768.	−0.238044
$406$	988715.	0.297684
$407$	62880.5	0.0188161
$408$	−266034.	−0.0791200
$409$	3.11683e6	0.921307	0.460654	−	0.887580i	$-0.347615\pi$
$409$	3.11683e6	0.921307	0.460654	+	0.887580i	$0.347615\pi$
$410$	−1.76506e6	−0.518560
$411$	327497.	0.0956318
$412$	5.05649e6	1.46760
$413$	3.06374e6	0.883846
$414$	368697.	0.105723
$415$	1.44630e6	0.412230
$416$	−9.33448e6	−2.64458
$417$	2.65300e6	0.747133
$418$	−2.35402e6	−0.658975
$419$	3.22635e6	0.897793	0.448897	−	0.893584i	$-0.351817\pi$
$419$	3.22635e6	0.897793	0.448897	+	0.893584i	$0.351817\pi$
$420$	721759.	0.199650
$421$	638097.	0.175461	0.0877307	−	0.996144i	$-0.472039\pi$
$421$	638097.	0.175461	0.0877307	+	0.996144i	$0.472039\pi$
$422$	−1.83238e6	−0.500882
$423$	−374102.	−0.101658
$424$	−468629.	−0.126594
$425$	−376019.	−0.100981
$426$	−7.06384e6	−1.88589
$427$	−2.37809e6	−0.631187
$428$	−6.66807e6	−1.75951
$429$	6.62940e6	1.73913
$430$	302794.	0.0789726
$431$	1.08234e6	0.280654	0.140327	−	0.990105i	$-0.455185\pi$
$431$	1.08234e6	0.280654	0.140327	+	0.990105i	$0.455185\pi$
$432$	3.58696e6	0.924734
$433$	3.47077e6	0.889623	0.444812	−	0.895624i	$-0.353271\pi$
$433$	3.47077e6	0.889623	0.444812	+	0.895624i	$0.353271\pi$
$434$	−2.32874e6	−0.593468
$435$	595952.	0.151004
$436$	−8.15731e6	−2.05509
$437$	321668.	0.0805757
$438$	7.18895e6	1.79052
$439$	−5.62564e6	−1.39319	−0.696595	−	0.717464i	$-0.745303\pi$
$439$	−5.62564e6	−1.39319	−0.696595	+	0.717464i	$0.745303\pi$
$440$	411296.	0.101280
$441$	1.08059e6	0.264585
$442$	−5.58275e6	−1.35923
$443$	269762.	0.0653086	0.0326543	−	0.999467i	$-0.489604\pi$
$443$	269762.	0.0653086	0.0326543	+	0.999467i	$0.489604\pi$
$444$	61264.9	0.0147487
$445$	−1.75068e6	−0.419088
$446$	−392368.	−0.0934020
$447$	2.93652e6	0.695128
$448$	−2.58099e6	−0.607563
$449$	271606.	0.0635805	0.0317902	−	0.999495i	$-0.489879\pi$
$449$	271606.	0.0635805	0.0317902	+	0.999495i	$0.489879\pi$
$450$	−435606.	−0.101406
$451$	−4.00476e6	−0.927117
$452$	6.76368e6	1.55717
$453$	6.48970e6	1.48586
$454$	−8.96390e6	−2.04107
$455$	1.77565e6	0.402094
$456$	−268880.	−0.0605545
$457$	−5.71667e6	−1.28042	−0.640211	−	0.768199i	$-0.721153\pi$
$457$	−5.71667e6	−1.28042	−0.640211	+	0.768199i	$0.721153\pi$
$458$	−1.22334e7	−2.72512
$459$	2.48063e6	0.549579
$460$	−479402.	−0.105634
$461$	1.34860e6	0.295549	0.147774	−	0.989021i	$-0.452789\pi$
$461$	1.34860e6	0.295549	0.147774	+	0.989021i	$0.452789\pi$
$462$	3.08323e6	0.672049
$463$	−796195.	−0.172611	−0.0863053	−	0.996269i	$-0.527506\pi$
$463$	−796195.	−0.172611	−0.0863053	+	0.996269i	$0.527506\pi$
$464$	−1.64652e6	−0.355035
$465$	−1.40366e6	−0.301044
$466$	1.37118e6	0.292502
$467$	922263.	0.195687	0.0978436	−	0.995202i	$-0.468806\pi$
$467$	922263.	0.195687	0.0978436	+	0.995202i	$0.468806\pi$
$468$	−3.43507e6	−0.724972
$469$	−768893.	−0.161411
$470$	915834.	0.191237
$471$	−543487.	−0.112885
$472$	1.70102e6	0.351443
$473$	687013.	0.141193
$474$	−7.87233e6	−1.60938
$475$	−380042.	−0.0772855
$476$	−1.37906e6	−0.278975
$477$	1.12612e6	0.226616
$478$	−1.44791e6	−0.289848
$479$	−9.14492e6	−1.82113	−0.910565	−	0.413365i	$-0.864353\pi$
$479$	−9.14492e6	−1.82113	−0.910565	+	0.413365i	$0.864353\pi$
$480$	−2.61675e6	−0.518392
$481$	150722.	0.0297039
$482$	1.40868e6	0.276182
$483$	−421312.	−0.0821742
$484$	2.12207e6	0.411761
$485$	252086.	0.0486626
$486$	5.00687e6	0.961558
$487$	1.61494e6	0.308556	0.154278	−	0.988028i	$-0.450695\pi$
$487$	1.61494e6	0.308556	0.154278	+	0.988028i	$0.450695\pi$
$488$	−1.32034e6	−0.250978
$489$	3.95619e6	0.748177
$490$	−2.64538e6	−0.497735
$491$	−3.36320e6	−0.629577	−0.314788	−	0.949162i	$-0.601934\pi$
$491$	−3.36320e6	−0.629577	−0.314788	+	0.949162i	$0.601934\pi$
$492$	−3.90186e6	−0.726707
$493$	−1.13868e6	−0.211001
$494$	−5.64248e6	−1.04029
$495$	−988351.	−0.181300
$496$	3.87808e6	0.707804
$497$	−4.29279e6	−0.779559
$498$	6.01962e6	1.08767
$499$	4.12967e6	0.742445	0.371222	−	0.928544i	$-0.378939\pi$
$499$	4.12967e6	0.742445	0.371222	+	0.928544i	$0.378939\pi$
$500$	566401.	0.101321
$501$	−2.77277e6	−0.493537
$502$	−4.72134e6	−0.836191
$503$	−7.88098e6	−1.38887	−0.694433	−	0.719557i	$-0.744345\pi$
$503$	−7.88098e6	−1.38887	−0.694433	+	0.719557i	$0.744345\pi$
$504$	−187292.	−0.0328430
$505$	1.05482e6	0.184056
$506$	−2.04792e6	−0.355580
$507$	1.12139e7	1.93748
$508$	4.05244e6	0.696718
$509$	7.94501e6	1.35925	0.679626	−	0.733559i	$-0.262142\pi$
$509$	7.94501e6	1.35925	0.679626	+	0.733559i	$0.262142\pi$
$510$	−1.56502e6	−0.266437
$511$	4.36882e6	0.740137
$512$	8.20700e6	1.38360
$513$	2.50717e6	0.420620
$514$	6.87452e6	1.14772
$515$	3.48727e6	0.579385
$516$	669361.	0.110672
$517$	2.07795e6	0.341907
$518$	70098.3	0.0114784
$519$	−3.66068e6	−0.596544
$520$	985859.	0.159885
$521$	1.21250e7	1.95698	0.978491	−	0.206288i	$-0.0661385\pi$
$521$	1.21250e7	1.95698	0.978491	+	0.206288i	$0.0661385\pi$
$522$	−1.31913e6	−0.211890
$523$	4.37509e6	0.699412	0.349706	−	0.936860i	$-0.386282\pi$
$523$	4.37509e6	0.699412	0.349706	+	0.936860i	$0.386282\pi$
$524$	4.24703e6	0.675706
$525$	497769.	0.0788188
$526$	1.94682e6	0.306805
$527$	2.68196e6	0.420655
$528$	−5.13453e6	−0.801523
$529$	279841.	0.0434783
$530$	−2.75684e6	−0.426307
$531$	−4.08758e6	−0.629116
$532$	−1.39381e6	−0.213514
$533$	−9.59922e6	−1.46359
$534$	−7.28644e6	−1.10576
$535$	−4.59871e6	−0.694627
$536$	−426898.	−0.0641819
$537$	2.25364e6	0.337248
$538$	1.71171e7	2.54962
$539$	−6.00213e6	−0.889885
$540$	−3.73659e6	−0.551432
$541$	4.37003e6	0.641936	0.320968	−	0.947090i	$-0.395992\pi$
$541$	4.37003e6	0.641936	0.320968	+	0.947090i	$0.395992\pi$
$542$	1.62645e7	2.37817
$543$	7.42685e6	1.08095
$544$	4.99980e6	0.724362
$545$	−5.62578e6	−0.811318
$546$	7.39036e6	1.06092
$547$	2.33085e6	0.333078	0.166539	−	0.986035i	$-0.446741\pi$
$547$	2.33085e6	0.333078	0.166539	+	0.986035i	$0.446741\pi$
$548$	942566.	0.134079
$549$	3.17280e6	0.449274
$550$	2.41957e6	0.341060
$551$	−1.15086e6	−0.161490
$552$	−233917.	−0.0326749
$553$	−4.78413e6	−0.665258
$554$	−6.23145e6	−0.862610
$555$	42252.0	0.00582257
$556$	7.63560e6	1.04751
$557$	−4.33859e6	−0.592530	−0.296265	−	0.955106i	$-0.595741\pi$
$557$	−4.33859e6	−0.592530	−0.296265	+	0.955106i	$0.595741\pi$
$558$	3.10697e6	0.422427
$559$	1.64674e6	0.222892
$560$	−1.37525e6	−0.185316
$561$	−3.55089e6	−0.476354
$562$	2.08585e6	0.278575
$563$	−1.38036e7	−1.83536	−0.917681	−	0.397317i	$-0.869941\pi$
$563$	−1.38036e7	−1.83536	−0.917681	+	0.397317i	$0.869941\pi$
$564$	2.02456e6	0.267998
$565$	4.66465e6	0.614749
$566$	−1.01883e7	−1.33678
$567$	−1.98748e6	−0.259624
$568$	−2.38341e6	−0.309976
$569$	6.68069e6	0.865049	0.432524	−	0.901622i	$-0.357623\pi$
$569$	6.68069e6	0.865049	0.432524	+	0.901622i	$0.357623\pi$
$570$	−1.58176e6	−0.203917
$571$	−1.00163e7	−1.28563	−0.642817	−	0.766019i	$-0.722235\pi$
$571$	−1.00163e7	−1.28563	−0.642817	+	0.766019i	$0.722235\pi$
$572$	1.90800e7	2.43831
$573$	−3.27739e6	−0.417006
$574$	−4.46445e6	−0.565572
$575$	−330625.	−0.0417029
$576$	3.44351e6	0.432459
$577$	−1.05421e7	−1.31822	−0.659108	−	0.752048i	$-0.729066\pi$
$577$	−1.05421e7	−1.31822	−0.659108	+	0.752048i	$0.729066\pi$
$578$	−8.73964e6	−1.08811
$579$	4.75077e6	0.588935
$580$	1.71521e6	0.211712
$581$	3.65821e6	0.449602
$582$	1.04920e6	0.128396
$583$	−6.25503e6	−0.762181
$584$	2.42562e6	0.294300
$585$	−2.36903e6	−0.286208
$586$	2.89483e6	0.348240
$587$	−1.22129e7	−1.46293	−0.731465	−	0.681880i	$-0.761163\pi$
$587$	−1.22129e7	−1.46293	−0.731465	+	0.681880i	$0.761163\pi$
$588$	−5.84792e6	−0.697522
$589$	2.71066e6	0.321949
$590$	1.00068e7	1.18349
$591$	8.23143e6	0.969407
$592$	−116735.	−0.0136898
$593$	−6.09666e6	−0.711959	−0.355980	−	0.934494i	$-0.615853\pi$
$593$	−6.09666e6	−0.711959	−0.355980	+	0.934494i	$0.615853\pi$
$594$	−1.59621e7	−1.85619
$595$	−951084.	−0.110135
$596$	8.45159e6	0.974592
$597$	8.34445e6	0.958213
$598$	−4.90878e6	−0.561333
$599$	−5.61018e6	−0.638865	−0.319433	−	0.947609i	$-0.603492\pi$
$599$	−5.61018e6	−0.638865	−0.319433	+	0.947609i	$0.603492\pi$
$600$	276367.	0.0313407
$601$	1.21279e7	1.36962	0.684809	−	0.728723i	$-0.259886\pi$
$601$	1.21279e7	1.36962	0.684809	+	0.728723i	$0.259886\pi$
$602$	765872.	0.0861321
$603$	1.02584e6	0.114891
$604$	1.86780e7	2.08323
$605$	1.46351e6	0.162557
$606$	4.39024e6	0.485631
$607$	1.46018e7	1.60855	0.804276	−	0.594255i	$-0.202553\pi$
$607$	1.46018e7	1.60855	0.804276	+	0.594255i	$0.202553\pi$
$608$	5.05329e6	0.554390
$609$	1.50737e6	0.164694
$610$	−7.76728e6	−0.845171
$611$	4.98074e6	0.539748
$612$	1.83992e6	0.198573
$613$	1.34736e7	1.44821	0.724104	−	0.689691i	$-0.242254\pi$
$613$	1.34736e7	1.44821	0.724104	+	0.689691i	$0.242254\pi$
$614$	−5.85157e6	−0.626399
$615$	−2.69096e6	−0.286893
$616$	1.04031e6	0.110462
$617$	1.86160e7	1.96867	0.984334	−	0.176314i	$-0.0564174\pi$
$617$	1.86160e7	1.96867	0.984334	+	0.176314i	$0.0564174\pi$
$618$	1.45142e7	1.52870
$619$	−5.17151e6	−0.542489	−0.271244	−	0.962510i	$-0.587435\pi$
$619$	−5.17151e6	−0.542489	−0.271244	+	0.962510i	$0.587435\pi$
$620$	−4.03987e6	−0.422073
$621$	2.18116e6	0.226965
$622$	−381552.	−0.0395437
$623$	−4.42807e6	−0.457082
$624$	−1.23073e7	−1.26532
$625$	390625.	0.0400000
$626$	−2.74975e7	−2.80451
$627$	−3.58888e6	−0.364577
$628$	−1.56421e6	−0.158269
$629$	−80730.7	−0.00813602
$630$	−1.10180e6	−0.110599
$631$	−5.55004e6	−0.554910	−0.277455	−	0.960739i	$-0.589491\pi$
$631$	−5.55004e6	−0.554910	−0.277455	+	0.960739i	$0.589491\pi$
$632$	−2.65620e6	−0.264526
$633$	−2.79361e6	−0.277113
$634$	1.33732e7	1.32133
$635$	2.79481e6	0.275054
$636$	−6.09432e6	−0.597424
$637$	−1.43869e7	−1.40481
$638$	7.32706e6	0.712653
$639$	5.72737e6	0.554885
$640$	−1.78168e6	−0.171941
$641$	1.49507e7	1.43720	0.718599	−	0.695425i	$-0.244784\pi$
$641$	1.49507e7	1.43720	0.718599	+	0.695425i	$0.244784\pi$
$642$	−1.91401e7	−1.83277
$643$	−1.09430e7	−1.04378	−0.521891	−	0.853012i	$-0.674773\pi$
$643$	−1.09430e7	−1.04378	−0.521891	+	0.853012i	$0.674773\pi$
$644$	−1.21258e6	−0.115211
$645$	461632.	0.0436915
$646$	3.02227e6	0.284939
$647$	1.88954e7	1.77458	0.887291	−	0.461210i	$-0.152585\pi$
$647$	1.88954e7	1.77458	0.887291	+	0.461210i	$0.152585\pi$
$648$	−1.10347e6	−0.103234
$649$	2.27044e7	2.11592
$650$	5.79960e6	0.538412
$651$	−3.55035e6	−0.328336
$652$	1.13863e7	1.04897
$653$	593534.	0.0544706	0.0272353	−	0.999629i	$-0.491330\pi$
$653$	593534.	0.0544706	0.0272353	+	0.999629i	$0.491330\pi$
$654$	−2.34149e7	−2.14066
$655$	2.92901e6	0.266758
$656$	7.43469e6	0.674533
$657$	−5.82881e6	−0.526825
$658$	2.31646e6	0.208574
$659$	890782.	0.0799020	0.0399510	−	0.999202i	$-0.487280\pi$
$659$	890782.	0.0799020	0.0399510	+	0.999202i	$0.487280\pi$
$660$	5.34873e6	0.477959
$661$	−5.10517e6	−0.454472	−0.227236	−	0.973840i	$-0.572969\pi$
$661$	−5.10517e6	−0.454472	−0.227236	+	0.973840i	$0.572969\pi$
$662$	−2.69204e7	−2.38746
$663$	−8.51132e6	−0.751992
$664$	2.03108e6	0.178775
$665$	−961260.	−0.0842921
$666$	−93523.9	−0.00817028
$667$	−1.00122e6	−0.0871391
$668$	−7.98030e6	−0.691956
$669$	−598194.	−0.0516746
$670$	−2.51135e6	−0.216133
$671$	−1.76233e7	−1.51105
$672$	−6.61866e6	−0.565389
$673$	−1.41252e7	−1.20215	−0.601073	−	0.799194i	$-0.705260\pi$
$673$	−1.41252e7	−1.20215	−0.601073	+	0.799194i	$0.705260\pi$
$674$	3.06396e6	0.259796
$675$	−2.57698e6	−0.217697
$676$	3.22748e7	2.71642
$677$	−1.21847e7	−1.02174	−0.510872	−	0.859657i	$-0.670677\pi$
$677$	−1.21847e7	−1.02174	−0.510872	+	0.859657i	$0.670677\pi$
$678$	1.94146e7	1.62201
$679$	637615.	0.0530743
$680$	−528053.	−0.0437931
$681$	−1.36661e7	−1.12922
$682$	−1.72576e7	−1.42076
$683$	−2.12485e7	−1.74292	−0.871458	−	0.490470i	$-0.836825\pi$
$683$	−2.12485e7	−1.74292	−0.871458	+	0.490470i	$0.836825\pi$
$684$	1.85960e6	0.151978
$685$	650051.	0.0529324
$686$	−1.54710e7	−1.25519
$687$	−1.86508e7	−1.50767
$688$	−1.27542e6	−0.102726
$689$	−1.49930e7	−1.20321
$690$	−1.37608e6	−0.110033
$691$	−1.59280e6	−0.126902	−0.0634508	−	0.997985i	$-0.520211\pi$
$691$	−1.59280e6	−0.126902	−0.0634508	+	0.997985i	$0.520211\pi$
$692$	−1.05358e7	−0.836375
$693$	−2.49989e6	−0.197737
$694$	−7.91691e6	−0.623960
$695$	5.26598e6	0.413539
$696$	836909.	0.0654870
$697$	5.14161e6	0.400882
$698$	−2.50259e7	−1.94424
$699$	2.09046e6	0.161826
$700$	1.43263e6	0.110507
$701$	2.27090e7	1.74543	0.872716	−	0.488228i	$-0.162356\pi$
$701$	2.27090e7	1.74543	0.872716	+	0.488228i	$0.162356\pi$
$702$	−3.82604e7	−2.93026
$703$	−81594.4	−0.00622690
$704$	−1.91269e7	−1.45450
$705$	1.39626e6	0.105802
$706$	1.34128e7	1.01276
$707$	2.66801e6	0.200742
$708$	2.21211e7	1.65853
$709$	7.00561e6	0.523396	0.261698	−	0.965150i	$-0.415718\pi$
$709$	7.00561e6	0.523396	0.261698	+	0.965150i	$0.415718\pi$
$710$	−1.40211e7	−1.04384
$711$	6.38290e6	0.473526
$712$	−2.45852e6	−0.181749
$713$	2.35819e6	0.173722
$714$	−3.95848e6	−0.290592
$715$	1.31588e7	0.962609
$716$	6.48620e6	0.472833
$717$	−2.20744e6	−0.160358
$718$	−808090.	−0.0584990
$719$	2.00896e7	1.44927	0.724633	−	0.689135i	$-0.242009\pi$
$719$	2.00896e7	1.44927	0.724633	+	0.689135i	$0.242009\pi$
$720$	1.83484e6	0.131907
$721$	8.82051e6	0.631911
$722$	−1.74013e7	−1.24233
$723$	2.14764e6	0.152798
$724$	2.13752e7	1.51553
$725$	1.18291e6	0.0835809
$726$	6.09122e6	0.428907
$727$	−2.05660e7	−1.44316	−0.721578	−	0.692333i	$-0.756583\pi$
$727$	−2.05660e7	−1.44316	−0.721578	+	0.692333i	$0.756583\pi$
$728$	2.49358e6	0.174379
$729$	1.52710e7	1.06426
$730$	1.42694e7	0.991058
$731$	−882038.	−0.0610512
$732$	−1.71705e7	−1.18442
$733$	−5.53202e6	−0.380298	−0.190149	−	0.981755i	$-0.560897\pi$
$733$	−5.53202e6	−0.380298	−0.190149	+	0.981755i	$0.560897\pi$
$734$	−2.00215e7	−1.37169
$735$	−4.03308e6	−0.275371
$736$	4.39621e6	0.299146
$737$	−5.69803e6	−0.386417
$738$	5.95638e6	0.402570
$739$	−1.41125e7	−0.950591	−0.475296	−	0.879826i	$-0.657659\pi$
$739$	−1.41125e7	−0.950591	−0.475296	+	0.879826i	$0.657659\pi$
$740$	121605.	0.00816344
$741$	−8.60238e6	−0.575537
$742$	−6.97302e6	−0.464955
$743$	2.36569e7	1.57212	0.786062	−	0.618148i	$-0.212117\pi$
$743$	2.36569e7	1.57212	0.786062	+	0.618148i	$0.212117\pi$
$744$	−1.97119e6	−0.130556
$745$	5.82874e6	0.384754
$746$	1.08696e7	0.715102
$747$	−4.88071e6	−0.320024
$748$	−1.02198e7	−0.667864
$749$	−1.16317e7	−0.757600
$750$	1.62581e6	0.105540
$751$	−1.64044e7	−1.06135	−0.530677	−	0.847574i	$-0.678062\pi$
$751$	−1.64044e7	−1.06135	−0.530677	+	0.847574i	$0.678062\pi$
$752$	−3.85763e6	−0.248758
$753$	−7.19803e6	−0.462622
$754$	1.75626e7	1.12502
$755$	1.28815e7	0.822428
$756$	−9.45115e6	−0.601423
$757$	2.16303e7	1.37190	0.685949	−	0.727649i	$-0.259387\pi$
$757$	2.16303e7	1.37190	0.685949	+	0.727649i	$0.259387\pi$
$758$	−1.70732e6	−0.107930
$759$	−3.12221e6	−0.196724
$760$	−533702.	−0.0335170
$761$	7.82013e6	0.489500	0.244750	−	0.969586i	$-0.421294\pi$
$761$	7.82013e6	0.489500	0.244750	+	0.969586i	$0.421294\pi$
$762$	1.16322e7	0.725729
$763$	−1.42296e7	−0.884871
$764$	−9.43265e6	−0.584656
$765$	1.26892e6	0.0783936
$766$	6.79702e6	0.418549
$767$	5.44215e7	3.34027
$768$	9.03531e6	0.552764
$769$	2.00595e7	1.22322	0.611608	−	0.791161i	$-0.290523\pi$
$769$	2.00595e7	1.22322	0.611608	+	0.791161i	$0.290523\pi$
$770$	6.11993e6	0.371980
$771$	1.04807e7	0.634973
$772$	1.36732e7	0.825706
$773$	6.86430e6	0.413188	0.206594	−	0.978427i	$-0.433762\pi$
$773$	6.86430e6	0.413188	0.206594	+	0.978427i	$0.433762\pi$
$774$	−1.02181e6	−0.0613082
$775$	−2.78614e6	−0.166628
$776$	354011.	0.0211039
$777$	106870.	0.00635044
$778$	3.23080e7	1.91364
$779$	5.19661e6	0.306815
$780$	1.28207e7	0.754526
$781$	−3.18126e7	−1.86626
$782$	2.62928e6	0.153752
$783$	−7.80376e6	−0.454883
$784$	1.11428e7	0.647444
$785$	−1.07877e6	−0.0624821
$786$	1.21908e7	0.703841
$787$	−2.32633e7	−1.33886	−0.669430	−	0.742875i	$-0.733462\pi$
$787$	−2.32633e7	−1.33886	−0.669430	+	0.742875i	$0.733462\pi$
$788$	2.36908e7	1.35914
$789$	2.96808e6	0.169740
$790$	−1.56259e7	−0.890793
$791$	1.17985e7	0.670481
$792$	−1.38797e6	−0.0786259
$793$	−4.22422e7	−2.38541
$794$	−2.99636e7	−1.68672
$795$	−4.20302e6	−0.235854
$796$	2.40161e7	1.34345
$797$	−1.67412e7	−0.933556	−0.466778	−	0.884375i	$-0.654585\pi$
$797$	−1.67412e7	−0.933556	−0.466778	+	0.884375i	$0.654585\pi$
$798$	−4.00083e6	−0.222404
$799$	−2.66782e6	−0.147839
$800$	−5.19401e6	−0.286931
$801$	5.90785e6	0.325348
$802$	3.93462e7	2.16006
$803$	3.23760e7	1.77188
$804$	−5.55163e6	−0.302887
$805$	−836266.	−0.0454836
$806$	−4.13657e7	−2.24286
$807$	2.60964e7	1.41057
$808$	1.48131e6	0.0798210
$809$	4.90961e6	0.263740	0.131870	−	0.991267i	$-0.457902\pi$
$809$	4.90961e6	0.263740	0.131870	+	0.991267i	$0.457902\pi$
$810$	−6.49150e6	−0.347642
$811$	−1.43179e7	−0.764411	−0.382206	−	0.924077i	$-0.624835\pi$
$811$	−1.43179e7	−0.764411	−0.382206	+	0.924077i	$0.624835\pi$
$812$	4.33836e6	0.230906
$813$	2.47965e7	1.31572
$814$	519477.	0.0274793
$815$	7.85267e6	0.414117
$816$	6.59210e6	0.346576
$817$	−891475.	−0.0467255
$818$	2.57492e7	1.34549
$819$	−5.99212e6	−0.312155
$820$	−7.74485e6	−0.402233
$821$	−2.36515e7	−1.22462	−0.612310	−	0.790618i	$-0.709759\pi$
$821$	−2.36515e7	−1.22462	−0.612310	+	0.790618i	$0.709759\pi$
$822$	2.70556e6	0.139662
$823$	2.95081e7	1.51859	0.759297	−	0.650744i	$-0.225543\pi$
$823$	2.95081e7	1.51859	0.759297	+	0.650744i	$0.225543\pi$
$824$	4.89725e6	0.251266
$825$	3.68881e6	0.188691
$826$	2.53106e7	1.29078
$827$	1.08770e7	0.553025	0.276513	−	0.961010i	$-0.410821\pi$
$827$	1.08770e7	0.553025	0.276513	+	0.961010i	$0.410821\pi$
$828$	1.61780e6	0.0820064
$829$	−3.03437e7	−1.53349	−0.766746	−	0.641951i	$-0.778125\pi$
$829$	−3.03437e7	−1.53349	−0.766746	+	0.641951i	$0.778125\pi$
$830$	1.19484e7	0.602025
$831$	−9.50031e6	−0.477238
$832$	−4.58464e7	−2.29613
$833$	7.70599e6	0.384783
$834$	2.19174e7	1.09112
$835$	−5.50371e6	−0.273174
$836$	−1.03291e7	−0.511150
$837$	1.83804e7	0.906861
$838$	2.66540e7	1.31115
$839$	−747249.	−0.0366488	−0.0183244	−	0.999832i	$-0.505833\pi$
$839$	−747249.	−0.0366488	−0.0183244	+	0.999832i	$0.505833\pi$
$840$	699029.	0.0341820
$841$	−1.69290e7	−0.825356
$842$	5.27154e6	0.256246
$843$	3.18003e6	0.154121
$844$	−8.04027e6	−0.388521
$845$	2.22587e7	1.07240
$846$	−3.09058e6	−0.148462
$847$	3.70172e6	0.177294
$848$	1.16122e7	0.554532
$849$	−1.55328e7	−0.739570
$850$	−3.10642e6	−0.147473
$851$	−70984.6	−0.00336001
$852$	−3.09952e7	−1.46284
$853$	1.54381e7	0.726474	0.363237	−	0.931697i	$-0.381672\pi$
$853$	1.54381e7	0.726474	0.363237	+	0.931697i	$0.381672\pi$
$854$	−1.96462e7	−0.921792
$855$	1.28250e6	0.0599985
$856$	−6.45807e6	−0.301244
$857$	−1.91262e7	−0.889563	−0.444782	−	0.895639i	$-0.646719\pi$
$857$	−1.91262e7	−0.889563	−0.444782	+	0.895639i	$0.646719\pi$
$858$	5.47677e7	2.53984
$859$	2.27475e7	1.05184	0.525922	−	0.850533i	$-0.323720\pi$
$859$	2.27475e7	1.05184	0.525922	+	0.850533i	$0.323720\pi$
$860$	1.32862e6	0.0612569
$861$	−6.80638e6	−0.312902
$862$	8.94160e6	0.409871
$863$	1.36087e7	0.621998	0.310999	−	0.950410i	$-0.399336\pi$
$863$	1.36087e7	0.621998	0.310999	+	0.950410i	$0.399336\pi$
$864$	3.42653e7	1.56160
$865$	−7.26611e6	−0.330188
$866$	2.86732e7	1.29922
$867$	−1.33242e7	−0.601998
$868$	−1.02182e7	−0.460338
$869$	−3.54537e7	−1.59262
$870$	4.92336e6	0.220528
$871$	−1.36579e7	−0.610013
$872$	−7.90041e6	−0.351851
$873$	−850694.	−0.0377779
$874$	2.65741e6	0.117674
$875$	988027.	0.0436263
$876$	3.15442e7	1.38886
$877$	−2.75272e7	−1.20855	−0.604273	−	0.796778i	$-0.706536\pi$
$877$	−2.75272e7	−1.20855	−0.604273	+	0.796778i	$0.706536\pi$
$878$	−4.64753e7	−2.03463
$879$	4.41338e6	0.192664
$880$	−1.01916e7	−0.443645
$881$	1.73692e6	0.0753947	0.0376974	−	0.999289i	$-0.487998\pi$
$881$	1.73692e6	0.0753947	0.0376974	+	0.999289i	$0.487998\pi$
$882$	8.92714e6	0.386403
$883$	−3.26232e6	−0.140807	−0.0704035	−	0.997519i	$-0.522429\pi$
$883$	−3.26232e6	−0.140807	−0.0704035	+	0.997519i	$0.522429\pi$
$884$	−2.44964e7	−1.05432
$885$	1.52560e7	0.654763
$886$	2.22859e6	0.0953775
$887$	−7.87377e6	−0.336027	−0.168013	−	0.985785i	$-0.553735\pi$
$887$	−7.87377e6	−0.336027	−0.168013	+	0.985785i	$0.553735\pi$
$888$	59335.5	0.00252512
$889$	7.06905e6	0.299990
$890$	−1.44629e7	−0.612042
$891$	−1.47286e7	−0.621538
$892$	−1.72166e6	−0.0724495
$893$	−2.69636e6	−0.113149
$894$	2.42596e7	1.01517
$895$	4.47328e6	0.186667
$896$	−4.50650e6	−0.187529
$897$	−7.48380e6	−0.310557
$898$	2.24383e6	0.0928537
$899$	−8.43713e6	−0.348173
$900$	−1.91138e6	−0.0786578
$901$	8.03068e6	0.329564
$902$	−3.30846e7	−1.35397
$903$	1.16763e6	0.0476525
$904$	6.55067e6	0.266603
$905$	1.47416e7	0.598307
$906$	5.36136e7	2.16997
$907$	3.31823e7	1.33933	0.669666	−	0.742662i	$-0.266437\pi$
$907$	3.31823e7	1.33933	0.669666	+	0.742662i	$0.266437\pi$
$908$	−3.93325e7	−1.58320
$909$	−3.55961e6	−0.142887
$910$	1.46692e7	0.587223
$911$	1.25433e7	0.500744	0.250372	−	0.968150i	$-0.419447\pi$
$911$	1.25433e7	0.500744	0.250372	+	0.968150i	$0.419447\pi$
$912$	6.66263e6	0.265252
$913$	2.71099e7	1.07634
$914$	−4.72274e7	−1.86994
$915$	−1.18418e7	−0.467590
$916$	−5.36788e7	−2.11380
$917$	7.40850e6	0.290942
$918$	2.04933e7	0.802612
$919$	−2.37627e7	−0.928125	−0.464062	−	0.885803i	$-0.653609\pi$
$919$	−2.37627e7	−0.928125	−0.464062	+	0.885803i	$0.653609\pi$
$920$	−464305.	−0.0180856
$921$	−8.92115e6	−0.346555
$922$	1.11412e7	0.431623
$923$	−7.62533e7	−2.94615
$924$	1.35288e7	0.521290
$925$	83866.5	0.00322280
$926$	−6.57764e6	−0.252082
$927$	−1.17682e7	−0.449790
$928$	−1.57288e7	−0.599549
$929$	−2.78476e7	−1.05864	−0.529321	−	0.848422i	$-0.677553\pi$
$929$	−2.78476e7	−1.05864	−0.529321	+	0.848422i	$0.677553\pi$
$930$	−1.15961e7	−0.439648
$931$	7.78843e6	0.294494
$932$	6.01655e6	0.226886
$933$	−581705.	−0.0218775
$934$	7.61912e6	0.285784
$935$	−7.04819e6	−0.263663
$936$	−3.32689e6	−0.124122
$937$	1.36877e7	0.509310	0.254655	−	0.967032i	$-0.418038\pi$
$937$	1.36877e7	0.509310	0.254655	+	0.967032i	$0.418038\pi$
$938$	−6.35208e6	−0.235727
$939$	−4.19220e7	−1.55159
$940$	4.01856e6	0.148338
$941$	6.63715e6	0.244347	0.122174	−	0.992509i	$-0.461013\pi$
$941$	6.63715e6	0.244347	0.122174	+	0.992509i	$0.461013\pi$
$942$	−4.48993e6	−0.164859
$943$	4.52089e6	0.165556
$944$	−4.21500e7	−1.53946
$945$	−6.51809e6	−0.237433
$946$	5.67564e6	0.206199
$947$	2.64105e7	0.956979	0.478489	−	0.878093i	$-0.341185\pi$
$947$	2.64105e7	0.956979	0.478489	+	0.878093i	$0.341185\pi$
$948$	−3.45428e7	−1.24835
$949$	7.76039e7	2.79716
$950$	−3.13966e6	−0.112869
$951$	2.03885e7	0.731027
$952$	−1.33563e6	−0.0477633
$953$	−3.65284e7	−1.30286	−0.651430	−	0.758708i	$-0.725831\pi$
$953$	−3.65284e7	−1.30286	−0.651430	+	0.758708i	$0.725831\pi$
$954$	9.30328e6	0.330952
$955$	−6.50533e6	−0.230813
$956$	−6.35323e6	−0.224828
$957$	1.11707e7	0.394275
$958$	−7.55493e7	−2.65960
$959$	1.64421e6	0.0577311
$960$	−1.28522e7	−0.450089
$961$	−8.75698e6	−0.305876
$962$	1.24516e6	0.0433799
$963$	1.55189e7	0.539255
$964$	6.18112e6	0.214227
$965$	9.42985e6	0.325977
$966$	−3.48060e6	−0.120008
$967$	1.98036e7	0.681050	0.340525	−	0.940235i	$-0.389395\pi$
$967$	1.98036e7	0.681050	0.340525	+	0.940235i	$0.389395\pi$
$968$	2.05524e6	0.0704975
$969$	4.60767e6	0.157642
$970$	2.08257e6	0.0710674
$971$	−3.57591e7	−1.21714	−0.608568	−	0.793502i	$-0.708256\pi$
$971$	−3.57591e7	−1.21714	−0.608568	+	0.793502i	$0.708256\pi$
$972$	2.19695e7	0.745855
$973$	1.33195e7	0.451030
$974$	1.33415e7	0.450618
$975$	8.84192e6	0.297876
$976$	3.27170e7	1.09938
$977$	3.41411e7	1.14430	0.572152	−	0.820148i	$-0.306109\pi$
$977$	3.41411e7	1.14430	0.572152	+	0.820148i	$0.306109\pi$
$978$	3.26834e7	1.09265
$979$	−3.28151e7	−1.09425
$980$	−1.16076e7	−0.386080
$981$	1.89848e7	0.629845
$982$	−2.77845e7	−0.919441
$983$	−3.35916e7	−1.10878	−0.554392	−	0.832256i	$-0.687049\pi$
$983$	−3.35916e7	−1.10878	−0.554392	+	0.832256i	$0.687049\pi$
$984$	−3.77898e6	−0.124419
$985$	1.63386e7	0.536569
$986$	−9.40703e6	−0.308149
$987$	3.53162e6	0.115394
$988$	−2.47585e7	−0.806922
$989$	−775555.	−0.0252129
$990$	−8.16510e6	−0.264773
$991$	5.16931e7	1.67205	0.836024	−	0.548693i	$-0.184874\pi$
$991$	5.16931e7	1.67205	0.836024	+	0.548693i	$0.184874\pi$
$992$	3.70463e7	1.19527
$993$	−4.10421e7	−1.32086
$994$	−3.54642e7	−1.13848
$995$	1.65630e7	0.530372
$996$	2.64133e7	0.843674
$997$	−2.00025e7	−0.637302	−0.318651	−	0.947872i	$-0.603230\pi$
$997$	−2.00025e7	−0.637302	−0.318651	+	0.947872i	$0.603230\pi$
$998$	3.41166e7	1.08428
$999$	−553274.	−0.0175399

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	115.6.a.e.1.10	✓	12
3.2	odd	2		1035.6.a.m.1.3		12
5.4	even	2		575.6.a.g.1.3		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
115.6.a.e.1.10	✓	12	1.1	even	1	trivial
575.6.a.g.1.3		12	5.4	even	2
1035.6.a.m.1.3		12	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 115 = 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	115.a (trivial)

\(f(q)\)	\(=\)	\(q+8.26134 q^{2} +12.5950 q^{3} +36.2497 q^{4} +25.0000 q^{5} +104.052 q^{6} +63.2337 q^{7} +35.1081 q^{8} -84.3653 q^{9} +206.533 q^{10} +468.606 q^{11} +456.566 q^{12} +1123.23 q^{13} +522.395 q^{14} +314.876 q^{15} -869.950 q^{16} -601.631 q^{17} -696.970 q^{18} -608.068 q^{19} +906.242 q^{20} +796.430 q^{21} +3871.31 q^{22} -529.000 q^{23} +442.187 q^{24} +625.000 q^{25} +9279.36 q^{26} -4123.17 q^{27} +2292.20 q^{28} +1892.66 q^{29} +2601.29 q^{30} -4457.82 q^{31} -8310.41 q^{32} +5902.10 q^{33} -4970.28 q^{34} +1580.84 q^{35} -3058.22 q^{36} +134.186 q^{37} -5023.45 q^{38} +14147.1 q^{39} +877.702 q^{40} -8546.11 q^{41} +6579.58 q^{42} +1466.08 q^{43} +16986.8 q^{44} -2109.13 q^{45} -4370.25 q^{46} +4434.31 q^{47} -10957.0 q^{48} -12808.5 q^{49} +5163.34 q^{50} -7577.56 q^{51} +40716.6 q^{52} -13348.2 q^{53} -34062.9 q^{54} +11715.1 q^{55} +2220.02 q^{56} -7658.63 q^{57} +15635.9 q^{58} +48451.0 q^{59} +11414.1 q^{60} -37607.9 q^{61} -36827.6 q^{62} -5334.73 q^{63} -40816.7 q^{64} +28080.7 q^{65} +48759.2 q^{66} -12159.5 q^{67} -21808.9 q^{68} -6662.77 q^{69} +13059.9 q^{70} -67887.7 q^{71} -2961.90 q^{72} +69090.1 q^{73} +1108.56 q^{74} +7871.89 q^{75} -22042.3 q^{76} +29631.7 q^{77} +116874. q^{78} -75657.9 q^{79} -21748.8 q^{80} -31430.7 q^{81} -70602.3 q^{82} +57852.2 q^{83} +28870.4 q^{84} -15040.8 q^{85} +12111.8 q^{86} +23838.1 q^{87} +16451.9 q^{88} -70027.0 q^{89} -17424.3 q^{90} +71025.8 q^{91} -19176.1 q^{92} -56146.4 q^{93} +36633.4 q^{94} -15201.7 q^{95} -104670. q^{96} +10083.5 q^{97} -105815. q^{98} -39534.1 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 8 q^{2} + 22 q^{3} + 294 q^{4} + 300 q^{5} + 454 q^{6} + 16 q^{7} + 675 q^{8} + 1598 q^{9} + 200 q^{10} + 132 q^{11} + 728 q^{12} - 236 q^{13} + 359 q^{14} + 550 q^{15} + 4514 q^{16} + 1666 q^{17}+ \cdots - 740784 q^{99}+O(q^{100}) \) 12 * q + 8 * q^2 + 22 * q^3 + 294 * q^4 + 300 * q^5 + 454 * q^6 + 16 * q^7 + 675 * q^8 + 1598 * q^9 + 200 * q^10 + 132 * q^11 + 728 * q^12 - 236 * q^13 + 359 * q^14 + 550 * q^15 + 4514 * q^16 + 1666 * q^17 - 3096 * q^18 + 616 * q^19 + 7350 * q^20 + 2732 * q^21 + 305 * q^22 - 6348 * q^23 + 18873 * q^24 + 7500 * q^25 - 4502 * q^26 + 11584 * q^27 + 5407 * q^28 + 23722 * q^29 + 11350 * q^30 + 18446 * q^31 + 35808 * q^32 + 4416 * q^33 + 53123 * q^34 + 400 * q^35 + 68916 * q^36 + 10394 * q^37 + 18681 * q^38 + 27032 * q^39 + 16875 * q^40 + 48232 * q^41 - 18980 * q^42 + 10732 * q^43 - 4765 * q^44 + 39950 * q^45 - 4232 * q^46 - 30448 * q^47 - 2052 * q^48 + 26948 * q^49 + 5000 * q^50 + 1524 * q^51 - 55346 * q^52 + 36494 * q^53 + 55567 * q^54 + 3300 * q^55 - 50981 * q^56 + 37572 * q^57 - 83373 * q^58 - 23870 * q^59 + 18200 * q^60 + 30862 * q^61 + 63582 * q^62 - 49698 * q^63 + 29965 * q^64 - 5900 * q^65 - 235225 * q^66 - 71910 * q^67 - 39371 * q^68 - 11638 * q^69 + 8975 * q^70 + 167158 * q^71 - 296052 * q^72 + 52152 * q^73 - 59356 * q^74 + 13750 * q^75 - 230417 * q^76 + 4808 * q^77 - 469771 * q^78 - 123092 * q^79 + 112850 * q^80 + 159868 * q^81 - 140098 * q^82 + 89322 * q^83 - 488082 * q^84 + 41650 * q^85 - 55318 * q^86 - 334376 * q^87 - 104551 * q^88 - 46184 * q^89 - 77400 * q^90 - 153444 * q^91 - 155526 * q^92 - 16576 * q^93 - 456595 * q^94 + 15400 * q^95 + 330540 * q^96 - 94220 * q^97 + 413841 * q^98 - 740784 * q^99

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