LMFDB - Embedded newform 845.6.a.h.1.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [845,6,Mod(1,845)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("845.1"); S:= CuspForms(chi, 6); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(845, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 6, names="a")

Level:	$ N $	$=$	$ 845 = 5 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	845.a (trivial)

Newform invariants

comment:select newform

sage:traces = [6,0,38] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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

Self dual:	yes
Analytic conductor:	$135.524327742$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 163x^{4} - 8x^{3} + 6120x^{2} + 6624x - 19440 $ x^6 - 163x^4 - 8x^3 + 6120x^2 + 6624x - 19440
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{4}\cdot 3 $
Twist minimal:	no (minimal twist has level 65)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-2.75663$ of defining polynomial
Character	$\chi$	$=$	845.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+2.75663 q^{2} +23.9621 q^{3} -24.4010 q^{4} +25.0000 q^{5} +66.0547 q^{6} -85.7300 q^{7} -155.477 q^{8} +331.182 q^{9} +68.9158 q^{10} -431.046 q^{11} -584.699 q^{12} -236.326 q^{14} +599.052 q^{15} +352.239 q^{16} -438.894 q^{17} +912.946 q^{18} -1617.28 q^{19} -610.025 q^{20} -2054.27 q^{21} -1188.23 q^{22} +2173.44 q^{23} -3725.55 q^{24} +625.000 q^{25} +2113.02 q^{27} +2091.90 q^{28} +8289.72 q^{29} +1651.37 q^{30} +2745.88 q^{31} +5946.25 q^{32} -10328.8 q^{33} -1209.87 q^{34} -2143.25 q^{35} -8081.16 q^{36} +2137.03 q^{37} -4458.24 q^{38} -3886.92 q^{40} +19520.9 q^{41} -5662.87 q^{42} +8153.29 q^{43} +10517.9 q^{44} +8279.55 q^{45} +5991.39 q^{46} +13235.5 q^{47} +8440.39 q^{48} -9457.36 q^{49} +1722.89 q^{50} -10516.8 q^{51} +1753.17 q^{53} +5824.83 q^{54} -10776.1 q^{55} +13329.0 q^{56} -38753.4 q^{57} +22851.7 q^{58} +1976.46 q^{59} -14617.5 q^{60} +45578.3 q^{61} +7569.39 q^{62} -28392.2 q^{63} +5119.96 q^{64} -28472.6 q^{66} +19457.7 q^{67} +10709.5 q^{68} +52080.3 q^{69} -5908.15 q^{70} +64224.9 q^{71} -51491.1 q^{72} -1029.22 q^{73} +5891.00 q^{74} +14976.3 q^{75} +39463.2 q^{76} +36953.6 q^{77} -107661. q^{79} +8805.99 q^{80} -29844.7 q^{81} +53812.1 q^{82} +46473.4 q^{83} +50126.2 q^{84} -10972.4 q^{85} +22475.6 q^{86} +198639. q^{87} +67017.6 q^{88} +3410.51 q^{89} +22823.7 q^{90} -53034.2 q^{92} +65797.1 q^{93} +36485.3 q^{94} -40431.9 q^{95} +142485. q^{96} -133264. q^{97} -26070.5 q^{98} -142755. q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 38 q^{3} + 134 q^{4} + 150 q^{5} - 318 q^{6} - 220 q^{7} - 24 q^{8} + 518 q^{9} + 170 q^{11} + 2238 q^{12} - 1440 q^{14} + 950 q^{15} + 3506 q^{16} + 728 q^{17} - 7788 q^{18} - 1218 q^{19} + 3350 q^{20}+ \cdots + 32270 q^{99}+O(q^{100}) $ 6 * q + 38 * q^3 + 134 * q^4 + 150 * q^5 - 318 * q^6 - 220 * q^7 - 24 * q^8 + 518 * q^9 + 170 * q^11 + 2238 * q^12 - 1440 * q^14 + 950 * q^15 + 3506 * q^16 + 728 * q^17 - 7788 * q^18 - 1218 * q^19 + 3350 * q^20 + 396 * q^21 + 5154 * q^22 + 8954 * q^23 - 12618 * q^24 + 3750 * q^25 + 13112 * q^27 - 13212 * q^28 + 8364 * q^29 - 7950 * q^30 - 2862 * q^31 + 29672 * q^32 + 3772 * q^33 + 21484 * q^34 - 5500 * q^35 - 2138 * q^36 - 13840 * q^37 - 27706 * q^38 - 600 * q^40 - 2248 * q^41 - 47028 * q^42 + 3822 * q^43 + 73482 * q^44 + 12950 * q^45 + 39886 * q^46 + 10860 * q^47 + 13962 * q^48 + 85902 * q^49 - 55028 * q^51 + 6584 * q^53 - 45504 * q^54 + 4250 * q^55 - 188800 * q^56 - 11692 * q^57 + 88500 * q^58 + 28874 * q^59 + 55950 * q^60 + 972 * q^61 + 123466 * q^62 + 125236 * q^63 + 14022 * q^64 - 158040 * q^66 - 60040 * q^67 + 3504 * q^68 + 153088 * q^69 - 36000 * q^70 + 74174 * q^71 - 298668 * q^72 - 48960 * q^73 + 226112 * q^74 + 23750 * q^75 + 93422 * q^76 + 231588 * q^77 + 214508 * q^79 + 87650 * q^80 + 116006 * q^81 + 58732 * q^82 - 123260 * q^83 + 235268 * q^84 + 18200 * q^85 + 304666 * q^86 + 228144 * q^87 + 127622 * q^88 - 94028 * q^89 - 194700 * q^90 + 430298 * q^92 - 192892 * q^93 - 125920 * q^94 - 30450 * q^95 + 385858 * q^96 - 246284 * q^97 + 4736 * q^98 + 32270 * 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 $

$2$	2.75663	0.487308	0.243654	−	0.969862i	$-0.421654\pi$
$2$	2.75663	0.487308	0.243654	+	0.969862i	$0.421654\pi$
$3$	23.9621	1.53717	0.768584	−	0.639748i	$-0.220961\pi$
$3$	23.9621	1.53717	0.768584	+	0.639748i	$0.220961\pi$
$4$	−24.4010	−0.762531
$5$	25.0000	0.447214
$5$	25.0000	0.447214
$6$	66.0547	0.749075
$7$	−85.7300	−0.661284	−0.330642	−	0.943756i	$-0.607265\pi$
$7$	−85.7300	−0.661284	−0.330642	+	0.943756i	$0.607265\pi$
$8$	−155.477	−0.858896
$9$	331.182	1.36289
$10$	68.9158	0.217931
$11$	−431.046	−1.07409	−0.537046	−	0.843553i	$-0.680460\pi$
$11$	−431.046	−1.07409	−0.537046	+	0.843553i	$0.680460\pi$
$12$	−584.699	−1.17214
$13$	0	0
$13$	0	0
$14$	−236.326	−0.322249
$15$	599.052	0.687443
$16$	352.239	0.343984
$17$	−438.894	−0.368331	−0.184165	−	0.982895i	$-0.558958\pi$
$17$	−438.894	−0.368331	−0.184165	+	0.982895i	$0.558958\pi$
$18$	912.946	0.664147
$19$	−1617.28	−1.02778	−0.513891	−	0.857856i	$-0.671796\pi$
$19$	−1617.28	−1.02778	−0.513891	+	0.857856i	$0.671796\pi$
$20$	−610.025	−0.341014
$21$	−2054.27	−1.01650
$22$	−1188.23	−0.523414
$23$	2173.44	0.856700	0.428350	−	0.903613i	$-0.359095\pi$
$23$	2173.44	0.856700	0.428350	+	0.903613i	$0.359095\pi$
$24$	−3725.55	−1.32027
$25$	625.000	0.200000
$26$	0	0
$27$	2113.02	0.557821
$28$	2091.90	0.504249
$29$	8289.72	1.83040	0.915198	−	0.403005i	$-0.132034\pi$
$29$	8289.72	1.83040	0.915198	+	0.403005i	$0.132034\pi$
$30$	1651.37	0.334997
$31$	2745.88	0.513190	0.256595	−	0.966519i	$-0.417399\pi$
$31$	2745.88	0.513190	0.256595	+	0.966519i	$0.417399\pi$
$32$	5946.25	1.02652
$33$	−10328.8	−1.65106
$34$	−1209.87	−0.179491
$35$	−2143.25	−0.295735
$36$	−8081.16	−1.03924
$37$	2137.03	0.256629	0.128315	−	0.991734i	$-0.459043\pi$
$37$	2137.03	0.256629	0.128315	+	0.991734i	$0.459043\pi$
$38$	−4458.24	−0.500846
$39$	0	0
$40$	−3886.92	−0.384110
$41$	19520.9	1.81360	0.906799	−	0.421562i	$-0.138518\pi$
$41$	19520.9	1.81360	0.906799	+	0.421562i	$0.138518\pi$
$42$	−5662.87	−0.495351
$43$	8153.29	0.672452	0.336226	−	0.941781i	$-0.390849\pi$
$43$	8153.29	0.672452	0.336226	+	0.941781i	$0.390849\pi$
$44$	10517.9	0.819029
$45$	8279.55	0.609502
$46$	5991.39	0.417477
$47$	13235.5	0.873966	0.436983	−	0.899470i	$-0.356047\pi$
$47$	13235.5	0.873966	0.436983	+	0.899470i	$0.356047\pi$
$48$	8440.39	0.528761
$49$	−9457.36	−0.562704
$50$	1722.89	0.0974616
$51$	−10516.8	−0.566186
$52$	0	0
$53$	1753.17	0.0857303	0.0428651	−	0.999081i	$-0.486351\pi$
$53$	1753.17	0.0857303	0.0428651	+	0.999081i	$0.486351\pi$
$54$	5824.83	0.271831
$55$	−10776.1	−0.480349
$56$	13329.0	0.567974
$57$	−38753.4	−1.57987
$58$	22851.7	0.891967
$59$	1976.46	0.0739192	0.0369596	−	0.999317i	$-0.488233\pi$
$59$	1976.46	0.0739192	0.0369596	+	0.999317i	$0.488233\pi$
$60$	−14617.5	−0.524196
$61$	45578.3	1.56832	0.784158	−	0.620561i	$-0.213095\pi$
$61$	45578.3	1.56832	0.784158	+	0.620561i	$0.213095\pi$
$62$	7569.39	0.250082
$63$	−28392.2	−0.901256
$64$	5119.96	0.156249
$65$	0	0
$66$	−28472.6	−0.804576
$67$	19457.7	0.529546	0.264773	−	0.964311i	$-0.414703\pi$
$67$	19457.7	0.529546	0.264773	+	0.964311i	$0.414703\pi$
$68$	10709.5	0.280863
$69$	52080.3	1.31689
$70$	−5908.15	−0.144114
$71$	64224.9	1.51202	0.756010	−	0.654561i	$-0.227146\pi$
$71$	64224.9	1.51202	0.756010	+	0.654561i	$0.227146\pi$
$72$	−51491.1	−1.17058
$73$	−1029.22	−0.0226048	−0.0113024	−	0.999936i	$-0.503598\pi$
$73$	−1029.22	−0.0226048	−0.0113024	+	0.999936i	$0.503598\pi$
$74$	5891.00	0.125058
$75$	14976.3	0.307434
$76$	39463.2	0.783715
$77$	36953.6	0.710280
$78$	0	0
$79$	−107661.	−1.94085	−0.970424	−	0.241407i	$-0.922391\pi$
$79$	−107661.	−1.94085	−0.970424	+	0.241407i	$0.922391\pi$
$80$	8805.99	0.153834
$81$	−29844.7	−0.505423
$82$	53812.1	0.883782
$83$	46473.4	0.740473	0.370237	−	0.928938i	$-0.379277\pi$
$83$	46473.4	0.740473	0.370237	+	0.928938i	$0.379277\pi$
$84$	50126.2	0.775116
$85$	−10972.4	−0.164722
$86$	22475.6	0.327692
$87$	198639.	2.81363
$88$	67017.6	0.922534
$89$	3410.51	0.0456398	0.0228199	−	0.999740i	$-0.492736\pi$
$89$	3410.51	0.0456398	0.0228199	+	0.999740i	$0.492736\pi$
$90$	22823.7	0.297015
$91$	0	0
$92$	−53034.2	−0.653260
$93$	65797.1	0.788859
$94$	36485.3	0.425891
$95$	−40431.9	−0.459638
$96$	142485.	1.57794
$97$	−133264.	−1.43808	−0.719041	−	0.694967i	$-0.755419\pi$
$97$	−133264.	−1.43808	−0.719041	+	0.694967i	$0.755419\pi$
$98$	−26070.5	−0.274210
$99$	−142755.	−1.46387
$100$	−15250.6	−0.152506
$101$	112628.	1.09860	0.549302	−	0.835624i	$-0.314894\pi$
$101$	112628.	1.09860	0.549302	+	0.835624i	$0.314894\pi$
$102$	−28991.0	−0.275907
$103$	102456.	0.951581	0.475791	−	0.879559i	$-0.342162\pi$
$103$	102456.	0.951581	0.475791	+	0.879559i	$0.342162\pi$
$104$	0	0
$105$	−51356.8	−0.454595
$106$	4832.84	0.0417771
$107$	−90417.9	−0.763475	−0.381738	−	0.924271i	$-0.624674\pi$
$107$	−90417.9	−0.763475	−0.381738	+	0.924271i	$0.624674\pi$
$108$	−51559.8	−0.425356
$109$	−164916.	−1.32953	−0.664764	−	0.747053i	$-0.731468\pi$
$109$	−164916.	−1.32953	−0.664764	+	0.747053i	$0.731468\pi$
$110$	−29705.9	−0.234078
$111$	51207.7	0.394483
$112$	−30197.5	−0.227471
$113$	−36453.6	−0.268562	−0.134281	−	0.990943i	$-0.542872\pi$
$113$	−36453.6	−0.268562	−0.134281	+	0.990943i	$0.542872\pi$
$114$	−106829.	−0.769885
$115$	54336.1	0.383128
$116$	−202277.	−1.39573
$117$	0	0
$118$	5448.36	0.0360214
$119$	37626.4	0.243571
$120$	−93138.7	−0.590442
$121$	24749.6	0.153676
$122$	125643.	0.764253
$123$	467763.	2.78781
$124$	−67002.3	−0.391323
$125$	15625.0	0.0894427
$126$	−78266.9	−0.439189
$127$	178579.	0.982476	0.491238	−	0.871025i	$-0.336544\pi$
$127$	178579.	0.982476	0.491238	+	0.871025i	$0.336544\pi$
$128$	−176166.	−0.950381
$129$	195370.	1.03367
$130$	0	0
$131$	43555.7	0.221752	0.110876	−	0.993834i	$-0.464634\pi$
$131$	43555.7	0.221752	0.110876	+	0.993834i	$0.464634\pi$
$132$	252032.	1.25899
$133$	138649.	0.679655
$134$	53637.6	0.258052
$135$	52825.6	0.249465
$136$	68237.9	0.316358
$137$	347518.	1.58189	0.790944	−	0.611888i	$-0.209590\pi$
$137$	347518.	1.58189	0.790944	+	0.611888i	$0.209590\pi$
$138$	143566.	0.641733
$139$	54594.1	0.239667	0.119834	−	0.992794i	$-0.461764\pi$
$139$	54594.1	0.239667	0.119834	+	0.992794i	$0.461764\pi$
$140$	52297.4	0.225507
$141$	317150.	1.34343
$142$	177044.	0.736819
$143$	0	0
$144$	116655.	0.468812
$145$	207243.	0.818578
$146$	−2837.18	−0.0110155
$147$	−226618.	−0.864971
$148$	−52145.6	−0.195688
$149$	249528.	0.920777	0.460388	−	0.887718i	$-0.347710\pi$
$149$	249528.	0.920777	0.460388	+	0.887718i	$0.347710\pi$
$150$	41284.2	0.149815
$151$	−398788.	−1.42331	−0.711655	−	0.702529i	$-0.752054\pi$
$151$	−398788.	−1.42331	−0.711655	+	0.702529i	$0.752054\pi$
$152$	251449.	0.882757
$153$	−145354.	−0.501994
$154$	101867.	0.346125
$155$	68647.1	0.229505
$156$	0	0
$157$	432099.	1.39905	0.699527	−	0.714606i	$-0.253394\pi$
$157$	432099.	1.39905	0.699527	+	0.714606i	$0.253394\pi$
$158$	−296782.	−0.945791
$159$	42009.6	0.131782
$160$	148656.	0.459075
$161$	−186329.	−0.566522
$162$	−82271.0	−0.246297
$163$	−56309.9	−0.166003	−0.0830014	−	0.996549i	$-0.526451\pi$
$163$	−56309.9	−0.166003	−0.0830014	+	0.996549i	$0.526451\pi$
$164$	−476330.	−1.38292
$165$	−258219.	−0.738378
$166$	128110.	0.360839
$167$	512611.	1.42232	0.711160	−	0.703031i	$-0.248170\pi$
$167$	512611.	1.42232	0.711160	+	0.703031i	$0.248170\pi$
$168$	319391.	0.873072
$169$	0	0
$170$	−30246.8	−0.0802706
$171$	−535613.	−1.40075
$172$	−198948.	−0.512766
$173$	−363030.	−0.922204	−0.461102	−	0.887347i	$-0.652546\pi$
$173$	−363030.	−0.922204	−0.461102	+	0.887347i	$0.652546\pi$
$174$	547575.	1.37110
$175$	−53581.3	−0.132257
$176$	−151831.	−0.369471
$177$	47360.0	0.113626
$178$	9401.51	0.0222406
$179$	−812489.	−1.89533	−0.947665	−	0.319265i	$-0.896564\pi$
$179$	−812489.	−1.89533	−0.947665	+	0.319265i	$0.896564\pi$
$180$	−202029.	−0.464764
$181$	−464974.	−1.05495	−0.527475	−	0.849570i	$-0.676861\pi$
$181$	−464974.	−1.05495	−0.527475	+	0.849570i	$0.676861\pi$
$182$	0	0
$183$	1.09215e6	2.41077
$184$	−337920.	−0.735816
$185$	53425.7	0.114768
$186$	181378.	0.384418
$187$	189184.	0.395621
$188$	−322959.	−0.666426
$189$	−181150.	−0.368878
$190$	−111456.	−0.223985
$191$	−15537.6	−0.0308177	−0.0154088	−	0.999881i	$-0.504905\pi$
$191$	−15537.6	−0.0308177	−0.0154088	+	0.999881i	$0.504905\pi$
$192$	122685.	0.240181
$193$	319408.	0.617238	0.308619	−	0.951186i	$-0.400133\pi$
$193$	319408.	0.617238	0.308619	+	0.951186i	$0.400133\pi$
$194$	−367360.	−0.700789
$195$	0	0
$196$	230769.	0.429079
$197$	−635878.	−1.16737	−0.583685	−	0.811980i	$-0.698390\pi$
$197$	−635878.	−1.16737	−0.583685	+	0.811980i	$0.698390\pi$
$198$	−393522.	−0.713355
$199$	57426.3	0.102797	0.0513983	−	0.998678i	$-0.483632\pi$
$199$	57426.3	0.102797	0.0513983	+	0.998678i	$0.483632\pi$
$200$	−97173.0	−0.171779
$201$	466246.	0.814002
$202$	310473.	0.535359
$203$	−710678.	−1.21041
$204$	256621.	0.431734
$205$	488024.	0.811066
$206$	282434.	0.463713
$207$	719806.	1.16759
$208$	0	0
$209$	697121.	1.10393
$210$	−141572.	−0.221528
$211$	−407094.	−0.629489	−0.314745	−	0.949176i	$-0.601919\pi$
$211$	−407094.	−0.629489	−0.314745	+	0.949176i	$0.601919\pi$
$212$	−42779.1	−0.0653720
$213$	1.53896e6	2.32423
$214$	−249249.	−0.372048
$215$	203832.	0.300730
$216$	−328526.	−0.479110
$217$	−235405.	−0.339364
$218$	−454614.	−0.647890
$219$	−24662.3	−0.0347474
$220$	262949.	0.366281
$221$	0	0
$222$	141161.	0.192235
$223$	−94338.9	−0.127037	−0.0635183	−	0.997981i	$-0.520232\pi$
$223$	−94338.9	−0.127037	−0.0635183	+	0.997981i	$0.520232\pi$
$224$	−509772.	−0.678822
$225$	206989.	0.272578
$226$	−100489.	−0.130872
$227$	−251896.	−0.324456	−0.162228	−	0.986753i	$-0.551868\pi$
$227$	−251896.	−0.324456	−0.162228	+	0.986753i	$0.551868\pi$
$228$	945620.	1.20470
$229$	428033.	0.539372	0.269686	−	0.962948i	$-0.413080\pi$
$229$	428033.	0.539372	0.269686	+	0.962948i	$0.413080\pi$
$230$	149785.	0.186701
$231$	885485.	1.09182
$232$	−1.28886e6	−1.57212
$233$	47973.8	0.0578914	0.0289457	−	0.999581i	$-0.490785\pi$
$233$	47973.8	0.0578914	0.0289457	+	0.999581i	$0.490785\pi$
$234$	0	0
$235$	330887.	0.390850
$236$	−48227.5	−0.0563657
$237$	−2.57979e6	−2.98341
$238$	103722.	0.118694
$239$	1.39917e6	1.58444	0.792220	−	0.610236i	$-0.208925\pi$
$239$	1.39917e6	1.58444	0.792220	+	0.610236i	$0.208925\pi$
$240$	211010.	0.236469
$241$	576792.	0.639700	0.319850	−	0.947468i	$-0.396367\pi$
$241$	576792.	0.639700	0.319850	+	0.947468i	$0.396367\pi$
$242$	68225.6	0.0748875
$243$	−1.22861e6	−1.33474
$244$	−1.11216e6	−1.19589
$245$	−236434.	−0.251649
$246$	1.28945e6	1.35852
$247$	0	0
$248$	−426921.	−0.440776
$249$	1.11360e6	1.13823
$250$	43072.4	0.0435862
$251$	676974.	0.678247	0.339124	−	0.940742i	$-0.389870\pi$
$251$	676974.	0.678247	0.339124	+	0.940742i	$0.389870\pi$
$252$	692798.	0.687235
$253$	−936855.	−0.920176
$254$	492278.	0.478769
$255$	−262921.	−0.253206
$256$	−649464.	−0.619377
$257$	−2.05124e6	−1.93724	−0.968620	−	0.248545i	$-0.920047\pi$
$257$	−2.05124e6	−1.93724	−0.968620	+	0.248545i	$0.920047\pi$
$258$	538562.	0.503717
$259$	−183208.	−0.169705
$260$	0	0
$261$	2.74541e6	2.49462
$262$	120067.	0.108061
$263$	1.51723e6	1.35258	0.676289	−	0.736636i	$-0.263587\pi$
$263$	1.51723e6	1.35258	0.676289	+	0.736636i	$0.263587\pi$
$264$	1.60588e6	1.41809
$265$	43829.2	0.0383397
$266$	382205.	0.331201
$267$	81722.8	0.0701561
$268$	−474786.	−0.403795
$269$	180817.	0.152355	0.0761777	−	0.997094i	$-0.475728\pi$
$269$	180817.	0.152355	0.0761777	+	0.997094i	$0.475728\pi$
$270$	145621.	0.121566
$271$	1.58203e6	1.30856	0.654278	−	0.756254i	$-0.272972\pi$
$271$	1.58203e6	1.30856	0.654278	+	0.756254i	$0.272972\pi$
$272$	−154596.	−0.126700
$273$	0	0
$274$	957979.	0.770867
$275$	−269404.	−0.214819
$276$	−1.27081e6	−1.00417
$277$	−1.53514e6	−1.20212	−0.601062	−	0.799202i	$-0.705256\pi$
$277$	−1.53514e6	−1.20212	−0.601062	+	0.799202i	$0.705256\pi$
$278$	150496.	0.116792
$279$	909387.	0.699420
$280$	333226.	0.254006
$281$	−959201.	−0.724676	−0.362338	−	0.932047i	$-0.618021\pi$
$281$	−959201.	−0.724676	−0.362338	+	0.932047i	$0.618021\pi$
$282$	874265.	0.654666
$283$	2.63997e6	1.95945	0.979724	−	0.200353i	$-0.0642089\pi$
$283$	2.63997e6	1.95945	0.979724	+	0.200353i	$0.0642089\pi$
$284$	−1.56715e6	−1.15296
$285$	−968834.	−0.706541
$286$	0	0
$287$	−1.67353e6	−1.19930
$288$	1.96929e6	1.39903
$289$	−1.22723e6	−0.864333
$290$	571293.	0.398900
$291$	−3.19329e6	−2.21058
$292$	25114.0	0.0172369
$293$	−286709.	−0.195106	−0.0975532	−	0.995230i	$-0.531102\pi$
$293$	−286709.	−0.195106	−0.0975532	+	0.995230i	$0.531102\pi$
$294$	−624703.	−0.421507
$295$	49411.4	0.0330577
$296$	−332258.	−0.220418
$297$	−910810.	−0.599152
$298$	687858.	0.448702
$299$	0	0
$300$	−365437.	−0.234428
$301$	−698981.	−0.444682
$302$	−1.09931e6	−0.693591
$303$	2.69879e6	1.68874
$304$	−569669.	−0.353540
$305$	1.13946e6	0.701372
$306$	−400687.	−0.244626
$307$	−1.89896e6	−1.14993	−0.574963	−	0.818179i	$-0.694984\pi$
$307$	−1.89896e6	−1.14993	−0.574963	+	0.818179i	$0.694984\pi$
$308$	−901704.	−0.541610
$309$	2.45507e6	1.46274
$310$	189235.	0.111840
$311$	1.74416e6	1.02255	0.511277	−	0.859416i	$-0.329173\pi$
$311$	1.74416e6	1.02255	0.511277	+	0.859416i	$0.329173\pi$
$312$	0	0
$313$	605507.	0.349348	0.174674	−	0.984626i	$-0.444113\pi$
$313$	605507.	0.349348	0.174674	+	0.984626i	$0.444113\pi$
$314$	1.19114e6	0.681770
$315$	−709806.	−0.403054
$316$	2.62704e6	1.47996
$317$	2.19697e6	1.22793	0.613967	−	0.789332i	$-0.289573\pi$
$317$	2.19697e6	1.22793	0.613967	+	0.789332i	$0.289573\pi$
$318$	115805.	0.0642184
$319$	−3.57325e6	−1.96601
$320$	127999.	0.0698765
$321$	−2.16660e6	−1.17359
$322$	−513642.	−0.276071
$323$	709814.	0.378563
$324$	728241.	0.385401
$325$	0	0
$326$	−155226.	−0.0808945
$327$	−3.95174e6	−2.04371
$328$	−3.03505e6	−1.55769
$329$	−1.13468e6	−0.577940
$330$	−711815.	−0.359817
$331$	3.68205e6	1.84722	0.923612	−	0.383329i	$-0.125223\pi$
$331$	3.68205e6	1.84722	0.923612	+	0.383329i	$0.125223\pi$
$332$	−1.13400e6	−0.564633
$333$	707745.	0.349757
$334$	1.41308e6	0.693108
$335$	486442.	0.236820
$336$	−723595.	−0.349661
$337$	−3.46729e6	−1.66309	−0.831543	−	0.555460i	$-0.812542\pi$
$337$	−3.46729e6	−1.66309	−0.831543	+	0.555460i	$0.812542\pi$
$338$	0	0
$339$	−873504.	−0.412825
$340$	267736.	0.125606
$341$	−1.18360e6	−0.551214
$342$	−1.47649e6	−0.682597
$343$	2.25164e6	1.03339
$344$	−1.26765e6	−0.577566
$345$	1.30201e6	0.588933
$346$	−1.00074e6	−0.449398
$347$	3.91532e6	1.74559	0.872797	−	0.488083i	$-0.162304\pi$
$347$	3.91532e6	1.74559	0.872797	+	0.488083i	$0.162304\pi$
$348$	−4.84699e6	−2.14548
$349$	3.32051e6	1.45929	0.729644	−	0.683827i	$-0.239686\pi$
$349$	3.32051e6	1.45929	0.729644	+	0.683827i	$0.239686\pi$
$350$	−147704.	−0.0644498
$351$	0	0
$352$	−2.56311e6	−1.10258
$353$	−3.09575e6	−1.32230	−0.661148	−	0.750255i	$-0.729931\pi$
$353$	−3.09575e6	−1.32230	−0.661148	+	0.750255i	$0.729931\pi$
$354$	130554.	0.0553710
$355$	1.60562e6	0.676196
$356$	−83219.7	−0.0348017
$357$	901608.	0.374410
$358$	−2.23973e6	−0.923610
$359$	2.71429e6	1.11153	0.555764	−	0.831340i	$-0.312426\pi$
$359$	2.71429e6	1.11153	0.555764	+	0.831340i	$0.312426\pi$
$360$	−1.28728e6	−0.523499
$361$	139488.	0.0563340
$362$	−1.28176e6	−0.514086
$363$	593053.	0.236226
$364$	0	0
$365$	−25730.5	−0.0101092
$366$	3.01066e6	1.17479
$367$	2.54964e6	0.988130	0.494065	−	0.869425i	$-0.335511\pi$
$367$	2.54964e6	0.988130	0.494065	+	0.869425i	$0.335511\pi$
$368$	765573.	0.294691
$369$	6.46498e6	2.47173
$370$	147275.	0.0559274
$371$	−150299.	−0.0566920
$372$	−1.60551e6	−0.601530
$373$	3.67946e6	1.36934	0.684671	−	0.728852i	$-0.259946\pi$
$373$	3.67946e6	1.36934	0.684671	+	0.728852i	$0.259946\pi$
$374$	521510.	0.192790
$375$	374408.	0.137489
$376$	−2.05781e6	−0.750646
$377$	0	0
$378$	−499362.	−0.179757
$379$	1.06822e6	0.382000	0.191000	−	0.981590i	$-0.438827\pi$
$379$	1.06822e6	0.382000	0.191000	+	0.981590i	$0.438827\pi$
$380$	986579.	0.350488
$381$	4.27914e6	1.51023
$382$	−42831.4	−0.0150177
$383$	601784.	0.209625	0.104813	−	0.994492i	$-0.466576\pi$
$383$	601784.	0.209625	0.104813	+	0.994492i	$0.466576\pi$
$384$	−4.22131e6	−1.46090
$385$	923839.	0.317647
$386$	880490.	0.300785
$387$	2.70022e6	0.916478
$388$	3.25178e6	1.09658
$389$	5.00396e6	1.67664	0.838319	−	0.545180i	$-0.183539\pi$
$389$	5.00396e6	1.67664	0.838319	+	0.545180i	$0.183539\pi$
$390$	0	0
$391$	−953913.	−0.315549
$392$	1.47040e6	0.483304
$393$	1.04369e6	0.340870
$394$	−1.75288e6	−0.568869
$395$	−2.69153e6	−0.867973
$396$	3.48335e6	1.11625
$397$	−3.47583e6	−1.10683	−0.553417	−	0.832904i	$-0.686677\pi$
$397$	−3.47583e6	−1.10683	−0.553417	+	0.832904i	$0.686677\pi$
$398$	158303.	0.0500936
$399$	3.32233e6	1.04474
$400$	220150.	0.0687968
$401$	5.15091e6	1.59964	0.799822	−	0.600238i	$-0.204927\pi$
$401$	5.15091e6	1.59964	0.799822	+	0.600238i	$0.204927\pi$
$402$	1.28527e6	0.396670
$403$	0	0
$404$	−2.74822e6	−0.837719
$405$	−746119.	−0.226032
$406$	−1.95908e6	−0.589843
$407$	−921158.	−0.275644
$408$	1.63512e6	0.486295
$409$	1.11646e6	0.330015	0.165008	−	0.986292i	$-0.447235\pi$
$409$	1.11646e6	0.330015	0.165008	+	0.986292i	$0.447235\pi$
$410$	1.34530e6	0.395239
$411$	8.32726e6	2.43163
$412$	−2.50004e6	−0.725610
$413$	−169442.	−0.0488816
$414$	1.98424e6	0.568975
$415$	1.16184e6	0.331150
$416$	0	0
$417$	1.30819e6	0.368409
$418$	1.92171e6	0.537955
$419$	1.79740e6	0.500160	0.250080	−	0.968225i	$-0.419543\pi$
$419$	1.79740e6	0.500160	0.250080	+	0.968225i	$0.419543\pi$
$420$	1.25316e6	0.346642
$421$	−5.31302e6	−1.46095	−0.730477	−	0.682937i	$-0.760702\pi$
$421$	−5.31302e6	−1.46095	−0.730477	+	0.682937i	$0.760702\pi$
$422$	−1.12221e6	−0.306755
$423$	4.38335e6	1.19112
$424$	−272577.	−0.0736334
$425$	−274309.	−0.0736661
$426$	4.24235e6	1.13262
$427$	−3.90743e6	−1.03710
$428$	2.20629e6	0.582173
$429$	0	0
$430$	561890.	0.146548
$431$	4.63630e6	1.20221	0.601103	−	0.799172i	$-0.294728\pi$
$431$	4.63630e6	1.20221	0.601103	+	0.799172i	$0.294728\pi$
$432$	744290.	0.191881
$433$	−3.68664e6	−0.944955	−0.472478	−	0.881343i	$-0.656640\pi$
$433$	−3.68664e6	−0.944955	−0.472478	+	0.881343i	$0.656640\pi$
$434$	−648924.	−0.165375
$435$	4.96598e6	1.25829
$436$	4.02412e6	1.01381
$437$	−3.51506e6	−0.880501
$438$	−67984.8	−0.0169327
$439$	−1.67032e6	−0.413656	−0.206828	−	0.978377i	$-0.566314\pi$
$439$	−1.67032e6	−0.413656	−0.206828	+	0.978377i	$0.566314\pi$
$440$	1.67544e6	0.412570
$441$	−3.13211e6	−0.766903
$442$	0	0
$443$	3.59209e6	0.869637	0.434819	−	0.900518i	$-0.356813\pi$
$443$	3.59209e6	0.869637	0.434819	+	0.900518i	$0.356813\pi$
$444$	−1.24952e6	−0.300805
$445$	85262.6	0.0204107
$446$	−260058.	−0.0619060
$447$	5.97922e6	1.41539
$448$	−438934.	−0.103325
$449$	−80455.2	−0.0188338	−0.00941690	−	0.999956i	$-0.502998\pi$
$449$	−80455.2	−0.0188338	−0.00941690	+	0.999956i	$0.502998\pi$
$450$	570592.	0.132829
$451$	−8.41443e6	−1.94797
$452$	889503.	0.204786
$453$	−9.55579e6	−2.18787
$454$	−694384.	−0.158110
$455$	0	0
$456$	6.02525e6	1.35695
$457$	−2.87687e6	−0.644362	−0.322181	−	0.946678i	$-0.604416\pi$
$457$	−2.87687e6	−0.644362	−0.322181	+	0.946678i	$0.604416\pi$
$458$	1.17993e6	0.262840
$459$	−927394.	−0.205463
$460$	−1.32585e6	−0.292147
$461$	−81644.7	−0.0178927	−0.00894635	−	0.999960i	$-0.502848\pi$
$461$	−81644.7	−0.0178927	−0.00894635	+	0.999960i	$0.502848\pi$
$462$	2.44096e6	0.532053
$463$	2.50214e6	0.542450	0.271225	−	0.962516i	$-0.412571\pi$
$463$	2.50214e6	0.542450	0.271225	+	0.962516i	$0.412571\pi$
$464$	2.91997e6	0.629626
$465$	1.64493e6	0.352789
$466$	132246.	0.0282109
$467$	1.38773e6	0.294452	0.147226	−	0.989103i	$-0.452966\pi$
$467$	1.38773e6	0.294452	0.147226	+	0.989103i	$0.452966\pi$
$468$	0	0
$469$	−1.66811e6	−0.350180
$470$	912133.	0.190464
$471$	1.03540e7	2.15058
$472$	−307293.	−0.0634889
$473$	−3.51444e6	−0.722276
$474$	−7.11152e6	−1.45384
$475$	−1.01080e6	−0.205556
$476$	−918122.	−0.185730
$477$	580618.	0.116841
$478$	3.85700e6	0.772111
$479$	4.96374e6	0.988485	0.494243	−	0.869324i	$-0.335445\pi$
$479$	4.96374e6	0.988485	0.494243	+	0.869324i	$0.335445\pi$
$480$	3.56211e6	0.705675
$481$	0	0
$482$	1.59000e6	0.311731
$483$	−4.46484e6	−0.870840
$484$	−603915.	−0.117182
$485$	−3.33160e6	−0.643130
$486$	−3.38682e6	−0.650431
$487$	−8.57371e6	−1.63812	−0.819061	−	0.573706i	$-0.805505\pi$
$487$	−8.57371e6	−1.63812	−0.819061	+	0.573706i	$0.805505\pi$
$488$	−7.08637e6	−1.34702
$489$	−1.34930e6	−0.255174
$490$	−651762.	−0.122631
$491$	2.55302e6	0.477915	0.238958	−	0.971030i	$-0.423194\pi$
$491$	2.55302e6	0.477915	0.238958	+	0.971030i	$0.423194\pi$
$492$	−1.14139e7	−2.12579
$493$	−3.63831e6	−0.674191
$494$	0	0
$495$	−3.56887e6	−0.654662
$496$	967209.	0.176529
$497$	−5.50600e6	−0.999874
$498$	3.06979e6	0.554670
$499$	−161861.	−0.0290999	−0.0145500	−	0.999894i	$-0.504632\pi$
$499$	−161861.	−0.0290999	−0.0145500	+	0.999894i	$0.504632\pi$
$500$	−381265.	−0.0682028
$501$	1.22832e7	2.18634
$502$	1.86617e6	0.330515
$503$	5.40474e6	0.952477	0.476239	−	0.879316i	$-0.342000\pi$
$503$	5.40474e6	0.952477	0.476239	+	0.879316i	$0.342000\pi$
$504$	4.41433e6	0.774085
$505$	2.81569e6	0.491311
$506$	−2.58256e6	−0.448409
$507$	0	0
$508$	−4.35751e6	−0.749169
$509$	−1.72999e6	−0.295971	−0.147985	−	0.988990i	$-0.547279\pi$
$509$	−1.72999e6	−0.295971	−0.147985	+	0.988990i	$0.547279\pi$
$510$	−724776.	−0.123389
$511$	88235.1	0.0149482
$512$	3.84698e6	0.648553
$513$	−3.41735e6	−0.573318
$514$	−5.65451e6	−0.944033
$515$	2.56141e6	0.425560
$516$	−4.76721e6	−0.788207
$517$	−5.70510e6	−0.938721
$518$	−505036.	−0.0826985
$519$	−8.69896e6	−1.41758
$520$	0	0
$521$	8.15642e6	1.31645	0.658226	−	0.752820i	$-0.271307\pi$
$521$	8.15642e6	1.31645	0.658226	+	0.752820i	$0.271307\pi$
$522$	7.56807e6	1.21565
$523$	−7.57602e6	−1.21112	−0.605559	−	0.795800i	$-0.707051\pi$
$523$	−7.57602e6	−1.21112	−0.605559	+	0.795800i	$0.707051\pi$
$524$	−1.06280e6	−0.169093
$525$	−1.28392e6	−0.203301
$526$	4.18245e6	0.659123
$527$	−1.20515e6	−0.189023
$528$	−3.63820e6	−0.567939
$529$	−1.71248e6	−0.266064
$530$	120821.	0.0186833
$531$	654567.	0.100744
$532$	−3.38318e6	−0.518258
$533$	0	0
$534$	225280.	0.0341876
$535$	−2.26045e6	−0.341436
$536$	−3.02521e6	−0.454825
$537$	−1.94689e7	−2.91344
$538$	498445.	0.0742440
$539$	4.07656e6	0.604396
$540$	−1.28900e6	−0.190225
$541$	−1.29055e7	−1.89575	−0.947876	−	0.318640i	$-0.896774\pi$
$541$	−1.29055e7	−1.89575	−0.947876	+	0.318640i	$0.896774\pi$
$542$	4.36108e6	0.637670
$543$	−1.11417e7	−1.62164
$544$	−2.60978e6	−0.378099
$545$	−4.12291e6	−0.594583
$546$	0	0
$547$	−7.61965e6	−1.08885	−0.544423	−	0.838811i	$-0.683252\pi$
$547$	−7.61965e6	−1.08885	−0.544423	+	0.838811i	$0.683252\pi$
$548$	−8.47978e6	−1.20624
$549$	1.50947e7	2.13744
$550$	−742647.	−0.104683
$551$	−1.34068e7	−1.88125
$552$	−8.09727e6	−1.13107
$553$	9.22980e6	1.28345
$554$	−4.23182e6	−0.585805
$555$	1.28019e6	0.176418
$556$	−1.33215e6	−0.182754
$557$	7.35984e6	1.00515	0.502574	−	0.864534i	$-0.332386\pi$
$557$	7.35984e6	1.00515	0.502574	+	0.864534i	$0.332386\pi$
$558$	2.50685e6	0.340833
$559$	0	0
$560$	−754937.	−0.101728
$561$	4.53324e6	0.608137
$562$	−2.64416e6	−0.353140
$563$	−5.68526e6	−0.755927	−0.377963	−	0.925821i	$-0.623375\pi$
$563$	−5.68526e6	−0.755927	−0.377963	+	0.925821i	$0.623375\pi$
$564$	−7.73876e6	−1.02441
$565$	−911339.	−0.120104
$566$	7.27744e6	0.954855
$567$	2.55859e6	0.334228
$568$	−9.98547e6	−1.29867
$569$	−1.43108e7	−1.85304	−0.926519	−	0.376249i	$-0.877214\pi$
$569$	−1.43108e7	−1.85304	−0.926519	+	0.376249i	$0.877214\pi$
$570$	−2.67072e6	−0.344303
$571$	−1.95049e6	−0.250354	−0.125177	−	0.992134i	$-0.539950\pi$
$571$	−1.95049e6	−0.250354	−0.125177	+	0.992134i	$0.539950\pi$
$572$	0	0
$573$	−372313.	−0.0473720
$574$	−4.61331e6	−0.584430
$575$	1.35840e6	0.171340
$576$	1.69564e6	0.212950
$577$	987263.	0.123451	0.0617253	−	0.998093i	$-0.480340\pi$
$577$	987263.	0.123451	0.0617253	+	0.998093i	$0.480340\pi$
$578$	−3.38302e6	−0.421196
$579$	7.65368e6	0.948798
$580$	−5.05693e6	−0.624191
$581$	−3.98417e6	−0.489663
$582$	−8.80272e6	−1.07723
$583$	−755697.	−0.0920823
$584$	160020.	0.0194152
$585$	0	0
$586$	−790350.	−0.0950770
$587$	−1.11813e6	−0.133936	−0.0669679	−	0.997755i	$-0.521332\pi$
$587$	−1.11813e6	−0.133936	−0.0669679	+	0.997755i	$0.521332\pi$
$588$	5.52971e6	0.659567
$589$	−4.44086e6	−0.527447
$590$	136209.	0.0161093
$591$	−1.52370e7	−1.79445
$592$	752746.	0.0882763
$593$	905001.	0.105685	0.0528424	−	0.998603i	$-0.483172\pi$
$593$	905001.	0.105685	0.0528424	+	0.998603i	$0.483172\pi$
$594$	−2.51077e6	−0.291971
$595$	940661.	0.108928
$596$	−6.08874e6	−0.702121
$597$	1.37606e6	0.158016
$598$	0	0
$599$	−5.11859e6	−0.582885	−0.291443	−	0.956588i	$-0.594135\pi$
$599$	−5.11859e6	−0.582885	−0.291443	+	0.956588i	$0.594135\pi$
$600$	−2.32847e6	−0.264054
$601$	8.84991e6	0.999431	0.499716	−	0.866190i	$-0.333438\pi$
$601$	8.84991e6	0.999431	0.499716	+	0.866190i	$0.333438\pi$
$602$	−1.92683e6	−0.216697
$603$	6.44403e6	0.721712
$604$	9.73082e6	1.08532
$605$	618741.	0.0687259
$606$	7.43957e6	0.822937
$607$	1.37995e7	1.52017	0.760085	−	0.649824i	$-0.225157\pi$
$607$	1.37995e7	1.52017	0.760085	+	0.649824i	$0.225157\pi$
$608$	−9.61674e6	−1.05504
$609$	−1.70293e7	−1.86061
$610$	3.14107e6	0.341785
$611$	0	0
$612$	3.54678e6	0.382785
$613$	−1.55631e7	−1.67280	−0.836401	−	0.548118i	$-0.815345\pi$
$613$	−1.55631e7	−1.67280	−0.836401	+	0.548118i	$0.815345\pi$
$614$	−5.23473e6	−0.560368
$615$	1.16941e7	1.24675
$616$	−5.74542e6	−0.610057
$617$	−1.57277e7	−1.66323	−0.831615	−	0.555353i	$-0.812583\pi$
$617$	−1.57277e7	−1.66323	−0.831615	+	0.555353i	$0.812583\pi$
$618$	6.76772e6	0.712806
$619$	5.68261e6	0.596103	0.298051	−	0.954550i	$-0.403663\pi$
$619$	5.68261e6	0.596103	0.298051	+	0.954550i	$0.403663\pi$
$620$	−1.67506e6	−0.175005
$621$	4.59254e6	0.477885
$622$	4.80802e6	0.498299
$623$	−292383.	−0.0301809
$624$	0	0
$625$	390625.	0.0400000
$626$	1.66916e6	0.170240
$627$	1.67045e7	1.69693
$628$	−1.05436e7	−1.06682
$629$	−937930.	−0.0945244
$630$	−1.95667e6	−0.196411
$631$	−7.92146e6	−0.792012	−0.396006	−	0.918248i	$-0.629604\pi$
$631$	−7.92146e6	−0.792012	−0.396006	+	0.918248i	$0.629604\pi$
$632$	1.67388e7	1.66699
$633$	−9.75482e6	−0.967631
$634$	6.05622e6	0.598382
$635$	4.46449e6	0.439377
$636$	−1.02508e6	−0.100488
$637$	0	0
$638$	−9.85014e6	−0.958055
$639$	2.12701e7	2.06071
$640$	−4.40415e6	−0.425023
$641$	8.63330e6	0.829911	0.414956	−	0.909842i	$-0.363797\pi$
$641$	8.63330e6	0.829911	0.414956	+	0.909842i	$0.363797\pi$
$642$	−5.97252e6	−0.571900
$643$	−5.41901e6	−0.516884	−0.258442	−	0.966027i	$-0.583209\pi$
$643$	−5.41901e6	−0.516884	−0.258442	+	0.966027i	$0.583209\pi$
$644$	4.54662e6	0.431990
$645$	4.88424e6	0.462273
$646$	1.95670e6	0.184477
$647$	2.85189e6	0.267838	0.133919	−	0.990992i	$-0.457244\pi$
$647$	2.85189e6	0.267838	0.133919	+	0.990992i	$0.457244\pi$
$648$	4.64016e6	0.434106
$649$	−851944.	−0.0793961
$650$	0	0
$651$	−5.64079e6	−0.521660
$652$	1.37402e6	0.126582
$653$	1.31018e7	1.20239	0.601197	−	0.799101i	$-0.294691\pi$
$653$	1.31018e7	1.20239	0.601197	+	0.799101i	$0.294691\pi$
$654$	−1.08935e7	−0.995916
$655$	1.08889e6	0.0991704
$656$	6.87605e6	0.623849
$657$	−340859.	−0.0308079
$658$	−3.12789e6	−0.281635
$659$	−987681.	−0.0885937	−0.0442969	−	0.999018i	$-0.514105\pi$
$659$	−987681.	−0.0885937	−0.0442969	+	0.999018i	$0.514105\pi$
$660$	6.30080e6	0.563036
$661$	−2.30367e6	−0.205077	−0.102538	−	0.994729i	$-0.532696\pi$
$661$	−2.30367e6	−0.205077	−0.102538	+	0.994729i	$0.532696\pi$
$662$	1.01500e7	0.900167
$663$	0	0
$664$	−7.22553e6	−0.635989
$665$	3.46623e6	0.303951
$666$	1.95099e6	0.170439
$667$	1.80172e7	1.56810
$668$	−1.25082e7	−1.08456
$669$	−2.26056e6	−0.195277
$670$	1.34094e6	0.115404
$671$	−1.96464e7	−1.68452
$672$	−1.22152e7	−1.04346
$673$	−3.89370e6	−0.331379	−0.165689	−	0.986178i	$-0.552985\pi$
$673$	−3.89370e6	−0.331379	−0.165689	+	0.986178i	$0.552985\pi$
$674$	−9.55803e6	−0.810436
$675$	1.32064e6	0.111564
$676$	0	0
$677$	2.83801e6	0.237981	0.118991	−	0.992895i	$-0.462034\pi$
$677$	2.83801e6	0.237981	0.118991	+	0.992895i	$0.462034\pi$
$678$	−2.40793e6	−0.201173
$679$	1.14247e7	0.950981
$680$	1.70595e6	0.141479
$681$	−6.03595e6	−0.498744
$682$	−3.26276e6	−0.268611
$683$	6.98711e6	0.573121	0.286560	−	0.958062i	$-0.407488\pi$
$683$	6.98711e6	0.573121	0.286560	+	0.958062i	$0.407488\pi$
$684$	1.30695e7	1.06812
$685$	8.68795e6	0.707442
$686$	6.20695e6	0.503580
$687$	1.02566e7	0.829105
$688$	2.87191e6	0.231313
$689$	0	0
$690$	3.58915e6	0.286992
$691$	−3.58854e6	−0.285905	−0.142953	−	0.989730i	$-0.545660\pi$
$691$	−3.58854e6	−0.285905	−0.142953	+	0.989730i	$0.545660\pi$
$692$	8.85829e6	0.703209
$693$	1.22384e7	0.968033
$694$	1.07931e7	0.850643
$695$	1.36485e6	0.107182
$696$	−3.08838e7	−2.41661
$697$	−8.56764e6	−0.668004
$698$	9.15342e6	0.711123
$699$	1.14955e6	0.0889888
$700$	1.30744e6	0.100850
$701$	−1.07723e7	−0.827971	−0.413986	−	0.910283i	$-0.635864\pi$
$701$	−1.07723e7	−0.827971	−0.413986	+	0.910283i	$0.635864\pi$
$702$	0	0
$703$	−3.45617e6	−0.263759
$704$	−2.20694e6	−0.167826
$705$	7.92874e6	0.600802
$706$	−8.53384e6	−0.644366
$707$	−9.65556e6	−0.726489
$708$	−1.15563e6	−0.0866436
$709$	−1.68465e7	−1.25862	−0.629309	−	0.777155i	$-0.716662\pi$
$709$	−1.68465e7	−1.25862	−0.629309	+	0.777155i	$0.716662\pi$
$710$	4.42611e6	0.329516
$711$	−3.56554e7	−2.64516
$712$	−530254.	−0.0391998
$713$	5.96803e6	0.439650
$714$	2.48540e6	0.182453
$715$	0	0
$716$	1.98255e7	1.44525
$717$	3.35270e7	2.43555
$718$	7.48230e6	0.541656
$719$	7.10379e6	0.512470	0.256235	−	0.966615i	$-0.417518\pi$
$719$	7.10379e6	0.512470	0.256235	+	0.966615i	$0.417518\pi$
$720$	2.91638e6	0.209659
$721$	−8.78359e6	−0.629265
$722$	384518.	0.0274520
$723$	1.38211e7	0.983327
$724$	1.13458e7	0.804432
$725$	5.18108e6	0.366079
$726$	1.63483e6	0.115115
$727$	−2.50048e7	−1.75464	−0.877319	−	0.479908i	$-0.840670\pi$
$727$	−2.50048e7	−1.75464	−0.877319	+	0.479908i	$0.840670\pi$
$728$	0	0
$729$	−2.21877e7	−1.54630
$730$	−70929.5	−0.00492629
$731$	−3.57843e6	−0.247685
$732$	−2.66496e7	−1.83828
$733$	7.77802e6	0.534699	0.267349	−	0.963600i	$-0.413852\pi$
$733$	7.77802e6	0.534699	0.267349	+	0.963600i	$0.413852\pi$
$734$	7.02842e6	0.481524
$735$	−5.66546e6	−0.386827
$736$	1.29238e7	0.879422
$737$	−8.38715e6	−0.568782
$738$	1.78216e7	1.20450
$739$	−1.00573e7	−0.677439	−0.338720	−	0.940887i	$-0.609994\pi$
$739$	−1.00573e7	−0.677439	−0.338720	+	0.940887i	$0.609994\pi$
$740$	−1.30364e6	−0.0875142
$741$	0	0
$742$	−414320.	−0.0276265
$743$	3.40838e6	0.226504	0.113252	−	0.993566i	$-0.463873\pi$
$743$	3.40838e6	0.226504	0.113252	+	0.993566i	$0.463873\pi$
$744$	−1.02299e7	−0.677548
$745$	6.23821e6	0.411784
$746$	1.01429e7	0.667292
$747$	1.53912e7	1.00918
$748$	−4.61627e6	−0.301673
$749$	7.75153e6	0.504874
$750$	1.03210e6	0.0669993
$751$	1.57084e7	1.01633	0.508163	−	0.861261i	$-0.330325\pi$
$751$	1.57084e7	1.01633	0.508163	+	0.861261i	$0.330325\pi$
$752$	4.66205e6	0.300630
$753$	1.62217e7	1.04258
$754$	0	0
$755$	−9.96970e6	−0.636524
$756$	4.42023e6	0.281281
$757$	−2.52846e7	−1.60367	−0.801836	−	0.597544i	$-0.796143\pi$
$757$	−2.52846e7	−1.60367	−0.801836	+	0.597544i	$0.796143\pi$
$758$	2.94469e6	0.186152
$759$	−2.24490e7	−1.41447
$760$	6.28623e6	0.394781
$761$	−1.34808e7	−0.843825	−0.421913	−	0.906636i	$-0.638641\pi$
$761$	−1.34808e7	−0.843825	−0.421913	+	0.906636i	$0.638641\pi$
$762$	1.17960e7	0.735949
$763$	1.41383e7	0.879195
$764$	379132.	0.0234994
$765$	−3.63385e6	−0.224498
$766$	1.65890e6	0.102152
$767$	0	0
$768$	−1.55625e7	−0.952087
$769$	−8.09020e6	−0.493337	−0.246668	−	0.969100i	$-0.579336\pi$
$769$	−8.09020e6	−0.493337	−0.246668	+	0.969100i	$0.579336\pi$
$770$	2.54668e6	0.154792
$771$	−4.91520e7	−2.97787
$772$	−7.79387e6	−0.470663
$773$	−2.75295e6	−0.165710	−0.0828551	−	0.996562i	$-0.526404\pi$
$773$	−2.75295e6	−0.165710	−0.0828551	+	0.996562i	$0.526404\pi$
$774$	7.44351e6	0.446607
$775$	1.71618e6	0.102638
$776$	2.07195e7	1.23516
$777$	−4.39004e6	−0.260865
$778$	1.37941e7	0.817040
$779$	−3.15708e7	−1.86398
$780$	0	0
$781$	−2.76839e7	−1.62405
$782$	−2.62959e6	−0.153770
$783$	1.75164e7	1.02103
$784$	−3.33126e6	−0.193561
$785$	1.08025e7	0.625676
$786$	2.87706e6	0.166109
$787$	−2.06536e7	−1.18867	−0.594333	−	0.804219i	$-0.702584\pi$
$787$	−2.06536e7	−1.18867	−0.594333	+	0.804219i	$0.702584\pi$
$788$	1.55161e7	0.890156
$789$	3.63560e7	2.07914
$790$	−7.41956e6	−0.422971
$791$	3.12516e6	0.177595
$792$	2.21950e7	1.25731
$793$	0	0
$794$	−9.58160e6	−0.539370
$795$	1.05024e6	0.0589347
$796$	−1.40126e6	−0.0783855
$797$	5.27093e6	0.293928	0.146964	−	0.989142i	$-0.453050\pi$
$797$	5.27093e6	0.293928	0.146964	+	0.989142i	$0.453050\pi$
$798$	9.15843e6	0.509113
$799$	−5.80897e6	−0.321909
$800$	3.71641e6	0.205304
$801$	1.12950e6	0.0622020
$802$	1.41992e7	0.779519
$803$	443641.	0.0242797
$804$	−1.13769e7	−0.620701
$805$	−4.65824e6	−0.253356
$806$	0	0
$807$	4.33275e6	0.234196
$808$	−1.75110e7	−0.943586
$809$	−2.62432e7	−1.40976	−0.704882	−	0.709325i	$-0.749000\pi$
$809$	−2.62432e7	−1.40976	−0.704882	+	0.709325i	$0.749000\pi$
$810$	−2.05677e6	−0.110147
$811$	−1.50454e7	−0.803251	−0.401626	−	0.915804i	$-0.631555\pi$
$811$	−1.50454e7	−0.803251	−0.401626	+	0.915804i	$0.631555\pi$
$812$	1.73412e7	0.922975
$813$	3.79088e7	2.01147
$814$	−2.53929e6	−0.134323
$815$	−1.40775e6	−0.0742387
$816$	−3.70444e6	−0.194759
$817$	−1.31861e7	−0.691134
$818$	3.07766e6	0.160819
$819$	0	0
$820$	−1.19083e7	−0.618463
$821$	−1.82770e6	−0.0946338	−0.0473169	−	0.998880i	$-0.515067\pi$
$821$	−1.82770e6	−0.0946338	−0.0473169	+	0.998880i	$0.515067\pi$
$822$	2.29552e7	1.18495
$823$	2.45074e7	1.26124	0.630619	−	0.776093i	$-0.282801\pi$
$823$	2.45074e7	1.26124	0.630619	+	0.776093i	$0.282801\pi$
$824$	−1.59296e7	−0.817309
$825$	−6.45548e6	−0.330212
$826$	−467088.	−0.0238204
$827$	7.03772e6	0.357823	0.178911	−	0.983865i	$-0.442742\pi$
$827$	7.03772e6	0.357823	0.178911	+	0.983865i	$0.442742\pi$
$828$	−1.75640e7	−0.890321
$829$	3.44967e7	1.74338	0.871689	−	0.490059i	$-0.163025\pi$
$829$	3.44967e7	1.74338	0.871689	+	0.490059i	$0.163025\pi$
$830$	3.20275e6	0.161372
$831$	−3.67852e7	−1.84787
$832$	0	0
$833$	4.15078e6	0.207261
$834$	3.60619e6	0.179529
$835$	1.28153e7	0.636080
$836$	−1.70104e7	−0.841782
$837$	5.80212e6	0.286268
$838$	4.95476e6	0.243732
$839$	−2.79128e6	−0.136899	−0.0684493	−	0.997655i	$-0.521805\pi$
$839$	−2.79128e6	−0.136899	−0.0684493	+	0.997655i	$0.521805\pi$
$840$	7.98478e6	0.390449
$841$	4.82083e7	2.35035
$842$	−1.46460e7	−0.711935
$843$	−2.29845e7	−1.11395
$844$	9.93349e6	0.480005
$845$	0	0
$846$	1.20833e7	0.580442
$847$	−2.12179e6	−0.101623
$848$	617535.	0.0294898
$849$	6.32593e7	3.01200
$850$	−756169.	−0.0358981
$851$	4.64471e6	0.219854
$852$	−3.75522e7	−1.77230
$853$	2.48047e7	1.16724	0.583621	−	0.812026i	$-0.301635\pi$
$853$	2.48047e7	1.16724	0.583621	+	0.812026i	$0.301635\pi$
$854$	−1.07713e7	−0.505388
$855$	−1.33903e7	−0.626435
$856$	1.40579e7	0.655746
$857$	−1.83304e7	−0.852550	−0.426275	−	0.904594i	$-0.640174\pi$
$857$	−1.83304e7	−0.852550	−0.426275	+	0.904594i	$0.640174\pi$
$858$	0	0
$859$	1.67092e7	0.772632	0.386316	−	0.922366i	$-0.373747\pi$
$859$	1.67092e7	0.772632	0.386316	+	0.922366i	$0.373747\pi$
$860$	−4.97370e6	−0.229316
$861$	−4.01013e7	−1.84353
$862$	1.27806e7	0.585844
$863$	−8.29285e6	−0.379033	−0.189516	−	0.981878i	$-0.560692\pi$
$863$	−8.29285e6	−0.379033	−0.189516	+	0.981878i	$0.560692\pi$
$864$	1.25646e7	0.572615
$865$	−9.07575e6	−0.412422
$866$	−1.01627e7	−0.460484
$867$	−2.94070e7	−1.32863
$868$	5.74411e6	0.258775
$869$	4.64069e7	2.08465
$870$	1.36894e7	0.613176
$871$	0	0
$872$	2.56407e7	1.14193
$873$	−4.41347e7	−1.95995
$874$	−9.68974e6	−0.429075
$875$	−1.33953e6	−0.0591470
$876$	601784.	0.0264960
$877$	1.44569e7	0.634711	0.317356	−	0.948307i	$-0.397205\pi$
$877$	1.44569e7	0.634711	0.317356	+	0.948307i	$0.397205\pi$
$878$	−4.60447e6	−0.201578
$879$	−6.87014e6	−0.299912
$880$	−3.79578e6	−0.165232
$881$	−3.42081e7	−1.48487	−0.742437	−	0.669916i	$-0.766330\pi$
$881$	−3.42081e7	−1.48487	−0.742437	+	0.669916i	$0.766330\pi$
$882$	−8.63407e6	−0.373718
$883$	−1.41341e7	−0.610051	−0.305026	−	0.952344i	$-0.598665\pi$
$883$	−1.41341e7	−0.610051	−0.305026	+	0.952344i	$0.598665\pi$
$884$	0	0
$885$	1.18400e6	0.0508152
$886$	9.90207e6	0.423781
$887$	2.15650e7	0.920325	0.460163	−	0.887835i	$-0.347791\pi$
$887$	2.15650e7	0.920325	0.460163	+	0.887835i	$0.347791\pi$
$888$	−7.96160e6	−0.338819
$889$	−1.53096e7	−0.649696
$890$	235038.	0.00994632
$891$	1.28645e7	0.542872
$892$	2.30196e6	0.0968693
$893$	−2.14054e7	−0.898246
$894$	1.64825e7	0.689731
$895$	−2.03122e7	−0.847618
$896$	1.51027e7	0.628471
$897$	0	0
$898$	−221785.	−0.00917787
$899$	2.27626e7	0.939340
$900$	−5.05073e6	−0.207849
$901$	−769456.	−0.0315771
$902$	−2.31955e7	−0.949264
$903$	−1.67491e7	−0.683551
$904$	5.66768e6	0.230666
$905$	−1.16243e7	−0.471788
$906$	−2.63418e7	−1.06617
$907$	1.53494e7	0.619547	0.309773	−	0.950810i	$-0.399747\pi$
$907$	1.53494e7	0.619547	0.309773	+	0.950810i	$0.399747\pi$
$908$	6.14650e6	0.247408
$909$	3.73002e7	1.49727
$910$	0	0
$911$	−8.00925e6	−0.319739	−0.159870	−	0.987138i	$-0.551107\pi$
$911$	−8.00925e6	−0.319739	−0.159870	+	0.987138i	$0.551107\pi$
$912$	−1.36505e7	−0.543451
$913$	−2.00322e7	−0.795337
$914$	−7.93047e6	−0.314003
$915$	2.73038e7	1.07813
$916$	−1.04444e7	−0.411287
$917$	−3.73403e6	−0.146641
$918$	−2.55648e6	−0.100124
$919$	−1.00787e7	−0.393655	−0.196828	−	0.980438i	$-0.563064\pi$
$919$	−1.00787e7	−0.393655	−0.196828	+	0.980438i	$0.563064\pi$
$920$	−8.44800e6	−0.329067
$921$	−4.55031e7	−1.76763
$922$	−225064.	−0.00871926
$923$	0	0
$924$	−2.16067e7	−0.832547
$925$	1.33564e6	0.0513258
$926$	6.89748e6	0.264340
$927$	3.39317e7	1.29690
$928$	4.92927e7	1.87894
$929$	1.20808e7	0.459257	0.229629	−	0.973278i	$-0.426249\pi$
$929$	1.20808e7	0.459257	0.229629	+	0.973278i	$0.426249\pi$
$930$	4.53446e6	0.171917
$931$	1.52952e7	0.578336
$932$	−1.17061e6	−0.0441440
$933$	4.17938e7	1.57184
$934$	3.82547e6	0.143489
$935$	4.72959e6	0.176927
$936$	0	0
$937$	1.85069e7	0.688628	0.344314	−	0.938854i	$-0.388111\pi$
$937$	1.85069e7	0.688628	0.344314	+	0.938854i	$0.388111\pi$
$938$	−4.59835e6	−0.170646
$939$	1.45092e7	0.537007
$940$	−8.07396e6	−0.298035
$941$	1.31161e7	0.482871	0.241436	−	0.970417i	$-0.422382\pi$
$941$	1.31161e7	0.482871	0.241436	+	0.970417i	$0.422382\pi$
$942$	2.85422e7	1.04800
$943$	4.24277e7	1.55371
$944$	696186.	0.0254270
$945$	−4.52874e6	−0.164967
$946$	−9.68802e6	−0.351971
$947$	2.46835e7	0.894399	0.447199	−	0.894434i	$-0.352421\pi$
$947$	2.46835e7	0.894399	0.447199	+	0.894434i	$0.352421\pi$
$948$	6.29494e7	2.27494
$949$	0	0
$950$	−2.78640e6	−0.100169
$951$	5.26439e7	1.88754
$952$	−5.85003e6	−0.209202
$953$	3.21978e7	1.14840	0.574201	−	0.818715i	$-0.305313\pi$
$953$	3.21978e7	1.14840	0.574201	+	0.818715i	$0.305313\pi$
$954$	1.60055e6	0.0569375
$955$	−388439.	−0.0137821
$956$	−3.41411e7	−1.20818
$957$	−8.56226e7	−3.02210
$958$	1.36832e7	0.481697
$959$	−2.97927e7	−1.04608
$960$	3.06712e6	0.107412
$961$	−2.10893e7	−0.736636
$962$	0	0
$963$	−2.99448e7	−1.04053
$964$	−1.40743e7	−0.487791
$965$	7.98520e6	0.276037
$966$	−1.23079e7	−0.424368
$967$	−2.28108e7	−0.784465	−0.392232	−	0.919866i	$-0.628297\pi$
$967$	−2.28108e7	−0.784465	−0.392232	+	0.919866i	$0.628297\pi$
$968$	−3.84799e6	−0.131991
$969$	1.70086e7	0.581916
$970$	−9.18400e6	−0.313403
$971$	4.02182e7	1.36891	0.684455	−	0.729055i	$-0.260040\pi$
$971$	4.02182e7	1.36891	0.684455	+	0.729055i	$0.260040\pi$
$972$	2.99792e7	1.01778
$973$	−4.68035e6	−0.158488
$974$	−2.36346e7	−0.798271
$975$	0	0
$976$	1.60545e7	0.539475
$977$	−3.17409e7	−1.06386	−0.531928	−	0.846790i	$-0.678532\pi$
$977$	−3.17409e7	−1.06386	−0.531928	+	0.846790i	$0.678532\pi$
$978$	−3.71953e6	−0.124349
$979$	−1.47008e6	−0.0490214
$980$	5.76922e6	0.191890
$981$	−5.46173e7	−1.81200
$982$	7.03774e6	0.232892
$983$	−4.52276e6	−0.149286	−0.0746431	−	0.997210i	$-0.523782\pi$
$983$	−4.52276e6	−0.149286	−0.0746431	+	0.997210i	$0.523782\pi$
$984$	−7.27262e7	−2.39444
$985$	−1.58970e7	−0.522064
$986$	−1.00295e7	−0.328539
$987$	−2.71892e7	−0.888391
$988$	0	0
$989$	1.77207e7	0.576090
$990$	−9.83805e6	−0.319022
$991$	−2.04730e7	−0.662213	−0.331107	−	0.943593i	$-0.607422\pi$
$991$	−2.04730e7	−0.662213	−0.331107	+	0.943593i	$0.607422\pi$
$992$	1.63277e7	0.526801
$993$	8.82296e7	2.83949
$994$	−1.51780e7	−0.487247
$995$	1.43566e6	0.0459720
$996$	−2.71729e7	−0.867937
$997$	−3.49736e7	−1.11430	−0.557150	−	0.830412i	$-0.688105\pi$
$997$	−3.49736e7	−1.11430	−0.557150	+	0.830412i	$0.688105\pi$
$998$	−446192.	−0.0141806
$999$	4.51559e6	0.143153

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	845.6.a.h.1.4		6
13.12	even	2		65.6.a.d.1.3	✓	6
39.38	odd	2		585.6.a.m.1.4		6
52.51	odd	2		1040.6.a.q.1.2		6
65.12	odd	4		325.6.b.g.274.5		12
65.38	odd	4		325.6.b.g.274.8		12
65.64	even	2		325.6.a.g.1.4		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
65.6.a.d.1.3	✓	6	13.12	even	2
325.6.a.g.1.4		6	65.64	even	2
325.6.b.g.274.5		12	65.12	odd	4
325.6.b.g.274.8		12	65.38	odd	4
585.6.a.m.1.4		6	39.38	odd	2
845.6.a.h.1.4		6	1.1	even	1	trivial
1040.6.a.q.1.2		6	52.51	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 845 = 5 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	845.a (trivial)

\(f(q)\)	\(=\)	\(q+2.75663 q^{2} +23.9621 q^{3} -24.4010 q^{4} +25.0000 q^{5} +66.0547 q^{6} -85.7300 q^{7} -155.477 q^{8} +331.182 q^{9} +68.9158 q^{10} -431.046 q^{11} -584.699 q^{12} -236.326 q^{14} +599.052 q^{15} +352.239 q^{16} -438.894 q^{17} +912.946 q^{18} -1617.28 q^{19} -610.025 q^{20} -2054.27 q^{21} -1188.23 q^{22} +2173.44 q^{23} -3725.55 q^{24} +625.000 q^{25} +2113.02 q^{27} +2091.90 q^{28} +8289.72 q^{29} +1651.37 q^{30} +2745.88 q^{31} +5946.25 q^{32} -10328.8 q^{33} -1209.87 q^{34} -2143.25 q^{35} -8081.16 q^{36} +2137.03 q^{37} -4458.24 q^{38} -3886.92 q^{40} +19520.9 q^{41} -5662.87 q^{42} +8153.29 q^{43} +10517.9 q^{44} +8279.55 q^{45} +5991.39 q^{46} +13235.5 q^{47} +8440.39 q^{48} -9457.36 q^{49} +1722.89 q^{50} -10516.8 q^{51} +1753.17 q^{53} +5824.83 q^{54} -10776.1 q^{55} +13329.0 q^{56} -38753.4 q^{57} +22851.7 q^{58} +1976.46 q^{59} -14617.5 q^{60} +45578.3 q^{61} +7569.39 q^{62} -28392.2 q^{63} +5119.96 q^{64} -28472.6 q^{66} +19457.7 q^{67} +10709.5 q^{68} +52080.3 q^{69} -5908.15 q^{70} +64224.9 q^{71} -51491.1 q^{72} -1029.22 q^{73} +5891.00 q^{74} +14976.3 q^{75} +39463.2 q^{76} +36953.6 q^{77} -107661. q^{79} +8805.99 q^{80} -29844.7 q^{81} +53812.1 q^{82} +46473.4 q^{83} +50126.2 q^{84} -10972.4 q^{85} +22475.6 q^{86} +198639. q^{87} +67017.6 q^{88} +3410.51 q^{89} +22823.7 q^{90} -53034.2 q^{92} +65797.1 q^{93} +36485.3 q^{94} -40431.9 q^{95} +142485. q^{96} -133264. q^{97} -26070.5 q^{98} -142755. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 38 q^{3} + 134 q^{4} + 150 q^{5} - 318 q^{6} - 220 q^{7} - 24 q^{8} + 518 q^{9} + 170 q^{11} + 2238 q^{12} - 1440 q^{14} + 950 q^{15} + 3506 q^{16} + 728 q^{17} - 7788 q^{18} - 1218 q^{19} + 3350 q^{20}+ \cdots + 32270 q^{99}+O(q^{100}) \) 6 * q + 38 * q^3 + 134 * q^4 + 150 * q^5 - 318 * q^6 - 220 * q^7 - 24 * q^8 + 518 * q^9 + 170 * q^11 + 2238 * q^12 - 1440 * q^14 + 950 * q^15 + 3506 * q^16 + 728 * q^17 - 7788 * q^18 - 1218 * q^19 + 3350 * q^20 + 396 * q^21 + 5154 * q^22 + 8954 * q^23 - 12618 * q^24 + 3750 * q^25 + 13112 * q^27 - 13212 * q^28 + 8364 * q^29 - 7950 * q^30 - 2862 * q^31 + 29672 * q^32 + 3772 * q^33 + 21484 * q^34 - 5500 * q^35 - 2138 * q^36 - 13840 * q^37 - 27706 * q^38 - 600 * q^40 - 2248 * q^41 - 47028 * q^42 + 3822 * q^43 + 73482 * q^44 + 12950 * q^45 + 39886 * q^46 + 10860 * q^47 + 13962 * q^48 + 85902 * q^49 - 55028 * q^51 + 6584 * q^53 - 45504 * q^54 + 4250 * q^55 - 188800 * q^56 - 11692 * q^57 + 88500 * q^58 + 28874 * q^59 + 55950 * q^60 + 972 * q^61 + 123466 * q^62 + 125236 * q^63 + 14022 * q^64 - 158040 * q^66 - 60040 * q^67 + 3504 * q^68 + 153088 * q^69 - 36000 * q^70 + 74174 * q^71 - 298668 * q^72 - 48960 * q^73 + 226112 * q^74 + 23750 * q^75 + 93422 * q^76 + 231588 * q^77 + 214508 * q^79 + 87650 * q^80 + 116006 * q^81 + 58732 * q^82 - 123260 * q^83 + 235268 * q^84 + 18200 * q^85 + 304666 * q^86 + 228144 * q^87 + 127622 * q^88 - 94028 * q^89 - 194700 * q^90 + 430298 * q^92 - 192892 * q^93 - 125920 * q^94 - 30450 * q^95 + 385858 * q^96 - 246284 * q^97 + 4736 * q^98 + 32270 * q^99

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