LMFDB - Embedded newform 65.6.a.d.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [65,6,Mod(1,65)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(65, 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("65.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 65 = 5 \cdot 13 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	65.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:	$10.4249482878$
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:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-2.75663$ of defining polynomial
Character	$\chi$	$=$	65.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-2.75663 q^{2} +23.9621 q^{3} -24.4010 q^{4} -25.0000 q^{5} -66.0547 q^{6} +85.7300 q^{7} +155.477 q^{8} +331.182 q^{9} +68.9158 q^{10} +431.046 q^{11} -584.699 q^{12} +169.000 q^{13} -236.326 q^{14} -599.052 q^{15} +352.239 q^{16} -438.894 q^{17} -912.946 q^{18} +1617.28 q^{19} +610.025 q^{20} +2054.27 q^{21} -1188.23 q^{22} +2173.44 q^{23} +3725.55 q^{24} +625.000 q^{25} -465.871 q^{26} +2113.02 q^{27} -2091.90 q^{28} +8289.72 q^{29} +1651.37 q^{30} -2745.88 q^{31} -5946.25 q^{32} +10328.8 q^{33} +1209.87 q^{34} -2143.25 q^{35} -8081.16 q^{36} -2137.03 q^{37} -4458.24 q^{38} +4049.59 q^{39} -3886.92 q^{40} -19520.9 q^{41} -5662.87 q^{42} +8153.29 q^{43} -10517.9 q^{44} -8279.55 q^{45} -5991.39 q^{46} -13235.5 q^{47} +8440.39 q^{48} -9457.36 q^{49} -1722.89 q^{50} -10516.8 q^{51} -4123.77 q^{52} +1753.17 q^{53} -5824.83 q^{54} -10776.1 q^{55} +13329.0 q^{56} +38753.4 q^{57} -22851.7 q^{58} -1976.46 q^{59} +14617.5 q^{60} +45578.3 q^{61} +7569.39 q^{62} +28392.2 q^{63} +5119.96 q^{64} -4225.00 q^{65} -28472.6 q^{66} -19457.7 q^{67} +10709.5 q^{68} +52080.3 q^{69} +5908.15 q^{70} -64224.9 q^{71} +51491.1 q^{72} +1029.22 q^{73} +5891.00 q^{74} +14976.3 q^{75} -39463.2 q^{76} +36953.6 q^{77} -11163.2 q^{78} -107661. q^{79} -8805.99 q^{80} -29844.7 q^{81} +53812.1 q^{82} -46473.4 q^{83} -50126.2 q^{84} +10972.4 q^{85} -22475.6 q^{86} +198639. q^{87} +67017.6 q^{88} -3410.51 q^{89} +22823.7 q^{90} +14488.4 q^{91} -53034.2 q^{92} -65797.1 q^{93} +36485.3 q^{94} -40431.9 q^{95} -142485. q^{96} +133264. q^{97} +26070.5 q^{98} +142755. q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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} + 1014 q^{13} - 1440 q^{14} - 950 q^{15} + 3506 q^{16} + 728 q^{17} + 7788 q^{18} + 1218 q^{19}+ \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 + 1014 * q^13 - 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 + 6422 * q^39 - 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 + 22646 * q^52 + 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 - 25350 * q^65 - 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 + 53742 * q^78 + 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 + 37180 * q^91 + 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	$-0.578346\pi$
$2$	−2.75663	−0.487308	−0.243654	+	0.969862i	$0.578346\pi$
$3$	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	$-0.392735\pi$
$7$	85.7300	0.661284	0.330642	+	0.943756i	$0.392735\pi$
$8$	155.477	0.858896
$9$	331.182	1.36289
$10$	68.9158	0.217931
$11$	431.046	1.07409	0.537046	−	0.843553i	$-0.319540\pi$
$11$	431.046	1.07409	0.537046	+	0.843553i	$0.319540\pi$
$12$	−584.699	−1.17214
$13$	169.000	0.277350
$13$	169.000	0.277350
$14$	−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	$-0.328204\pi$
$19$	1617.28	1.02778	0.513891	+	0.857856i	$0.328204\pi$
$20$	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
$26$	−465.871	−0.135155
$27$	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	$-0.582601\pi$
$31$	−2745.88	−0.513190	−0.256595	+	0.966519i	$0.582601\pi$
$32$	−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	$-0.540957\pi$
$37$	−2137.03	−0.256629	−0.128315	+	0.991734i	$0.540957\pi$
$38$	−4458.24	−0.500846
$39$	4049.59	0.426334
$40$	−3886.92	−0.384110
$41$	−19520.9	−1.81360	−0.906799	−	0.421562i	$-0.861482\pi$
$41$	−19520.9	−1.81360	−0.906799	+	0.421562i	$0.861482\pi$
$42$	−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	$-0.643953\pi$
$47$	−13235.5	−0.873966	−0.436983	+	0.899470i	$0.643953\pi$
$48$	8440.39	0.528761
$49$	−9457.36	−0.562704
$50$	−1722.89	−0.0974616
$51$	−10516.8	−0.566186
$52$	−4123.77	−0.211488
$53$	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	$-0.511767\pi$
$59$	−1976.46	−0.0739192	−0.0369596	+	0.999317i	$0.511767\pi$
$60$	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
$65$	−4225.00	−0.124035
$66$	−28472.6	−0.804576
$67$	−19457.7	−0.529546	−0.264773	−	0.964311i	$-0.585297\pi$
$67$	−19457.7	−0.529546	−0.264773	+	0.964311i	$0.585297\pi$
$68$	10709.5	0.280863
$69$	52080.3	1.31689
$70$	5908.15	0.144114
$71$	−64224.9	−1.51202	−0.756010	−	0.654561i	$-0.772854\pi$
$71$	−64224.9	−1.51202	−0.756010	+	0.654561i	$0.772854\pi$
$72$	51491.1	1.17058
$73$	1029.22	0.0226048	0.0113024	−	0.999936i	$-0.496402\pi$
$73$	1029.22	0.0226048	0.0113024	+	0.999936i	$0.496402\pi$
$74$	5891.00	0.125058
$75$	14976.3	0.307434
$76$	−39463.2	−0.783715
$77$	36953.6	0.710280
$78$	−11163.2	−0.207756
$79$	−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	$-0.620723\pi$
$83$	−46473.4	−0.740473	−0.370237	+	0.928938i	$0.620723\pi$
$84$	−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	$-0.507264\pi$
$89$	−3410.51	−0.0456398	−0.0228199	+	0.999740i	$0.507264\pi$
$90$	22823.7	0.297015
$91$	14488.4	0.183407
$92$	−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	$-0.244581\pi$
$97$	133264.	1.43808	0.719041	+	0.694967i	$0.244581\pi$
$98$	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	$0.342162\pi$
$104$	26275.6	0.238215
$105$	−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	$-0.268532\pi$
$109$	164916.	1.32953	0.664764	+	0.747053i	$0.268532\pi$
$110$	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
$117$	55969.7	0.377997
$118$	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
$130$	11646.8	0.0604431
$131$	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	$-0.790410\pi$
$137$	−347518.	−1.58189	−0.790944	+	0.611888i	$0.790410\pi$
$138$	−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
$143$	72846.8	0.297900
$144$	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	$-0.652290\pi$
$149$	−249528.	−0.920777	−0.460388	+	0.887718i	$0.652290\pi$
$150$	−41284.2	−0.149815
$151$	398788.	1.42331	0.711655	−	0.702529i	$-0.247946\pi$
$151$	398788.	1.42331	0.711655	+	0.702529i	$0.247946\pi$
$152$	251449.	0.882757
$153$	−145354.	−0.501994
$154$	−101867.	−0.346125
$155$	68647.1	0.229505
$156$	−98814.1	−0.325093
$157$	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	$-0.473549\pi$
$163$	56309.9	0.166003	0.0830014	+	0.996549i	$0.473549\pi$
$164$	476330.	1.38292
$165$	−258219.	−0.738378
$166$	128110.	0.360839
$167$	−512611.	−1.42232	−0.711160	−	0.703031i	$-0.751830\pi$
$167$	−512611.	−1.42232	−0.711160	+	0.703031i	$0.751830\pi$
$168$	319391.	0.873072
$169$	28561.0	0.0769231
$170$	−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	$0.676861\pi$
$182$	−39939.1	−0.0893758
$183$	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	$-0.599867\pi$
$193$	−319408.	−0.617238	−0.308619	+	0.951186i	$0.599867\pi$
$194$	−367360.	−0.700789
$195$	−101240.	−0.190662
$196$	230769.	0.429079
$197$	635878.	1.16737	0.583685	−	0.811980i	$-0.301610\pi$
$197$	635878.	1.16737	0.583685	+	0.811980i	$0.301610\pi$
$198$	−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
$208$	59528.5	0.0954039
$209$	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
$221$	−74173.2	−0.102157
$222$	141161.	0.192235
$223$	94338.9	0.127037	0.0635183	−	0.997981i	$-0.479768\pi$
$223$	94338.9	0.127037	0.0635183	+	0.997981i	$0.479768\pi$
$224$	−509772.	−0.678822
$225$	206989.	0.272578
$226$	100489.	0.130872
$227$	251896.	0.324456	0.162228	−	0.986753i	$-0.448132\pi$
$227$	251896.	0.324456	0.162228	+	0.986753i	$0.448132\pi$
$228$	−945620.	−1.20470
$229$	−428033.	−0.539372	−0.269686	−	0.962948i	$-0.586920\pi$
$229$	−428033.	−0.539372	−0.269686	+	0.962948i	$0.586920\pi$
$230$	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	$0.490785\pi$
$234$	−154288.	−0.184201
$235$	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	$-0.791075\pi$
$239$	−1.39917e6	−1.58444	−0.792220	+	0.610236i	$0.791075\pi$
$240$	−211010.	−0.236469
$241$	−576792.	−0.639700	−0.319850	−	0.947468i	$-0.603633\pi$
$241$	−576792.	−0.639700	−0.319850	+	0.947468i	$0.603633\pi$
$242$	−68225.6	−0.0748875
$243$	−1.22861e6	−1.33474
$244$	−1.11216e6	−1.19589
$245$	236434.	0.251649
$246$	1.28945e6	1.35852
$247$	273320.	0.285055
$248$	−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
$260$	103094.	0.0945803
$261$	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	$-0.727028\pi$
$271$	−1.58203e6	−1.30856	−0.654278	+	0.756254i	$0.727028\pi$
$272$	−154596.	−0.126700
$273$	347172.	0.281928
$274$	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	$-0.381979\pi$
$281$	959201.	0.724676	0.362338	+	0.932047i	$0.381979\pi$
$282$	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
$286$	−200812.	−0.145169
$287$	−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	$-0.468898\pi$
$293$	286709.	0.195106	0.0975532	+	0.995230i	$0.468898\pi$
$294$	624703.	0.421507
$295$	49411.4	0.0330577
$296$	−332258.	−0.220418
$297$	910810.	0.599152
$298$	687858.	0.448702
$299$	367312.	0.237606
$300$	−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	$-0.305016\pi$
$307$	1.89896e6	1.14993	0.574963	+	0.818179i	$0.305016\pi$
$308$	−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	$0.329173\pi$
$312$	629618.	0.366176
$313$	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	$-0.710427\pi$
$317$	−2.19697e6	−1.22793	−0.613967	+	0.789332i	$0.710427\pi$
$318$	−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
$325$	105625.	0.0554700
$326$	−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	$-0.874777\pi$
$331$	−3.68205e6	−1.84722	−0.923612	+	0.383329i	$0.874777\pi$
$332$	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	$0.812542\pi$
$338$	−78732.2	−0.0374852
$339$	−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	$-0.760314\pi$
$349$	−3.32051e6	−1.45929	−0.729644	+	0.683827i	$0.760314\pi$
$350$	−147704.	−0.0644498
$351$	357101.	0.154712
$352$	−2.56311e6	−1.10258
$353$	3.09575e6	1.32230	0.661148	−	0.750255i	$-0.270069\pi$
$353$	3.09575e6	1.32230	0.661148	+	0.750255i	$0.270069\pi$
$354$	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	$-0.687574\pi$
$359$	−2.71429e6	−1.11153	−0.555764	+	0.831340i	$0.687574\pi$
$360$	−1.28728e6	−0.523499
$361$	139488.	0.0563340
$362$	1.28176e6	0.514086
$363$	593053.	0.236226
$364$	−353531.	−0.139854
$365$	−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
$377$	1.40096e6	0.507660
$378$	−499362.	−0.179757
$379$	−1.06822e6	−0.382000	−0.191000	−	0.981590i	$-0.561173\pi$
$379$	−1.06822e6	−0.382000	−0.191000	+	0.981590i	$0.561173\pi$
$380$	986579.	0.350488
$381$	4.27914e6	1.51023
$382$	42831.4	0.0150177
$383$	−601784.	−0.209625	−0.104813	−	0.994492i	$-0.533424\pi$
$383$	−601784.	−0.209625	−0.104813	+	0.994492i	$0.533424\pi$
$384$	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	$0.183539\pi$
$390$	279081.	0.0929113
$391$	−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	$-0.313323\pi$
$397$	3.47583e6	1.10683	0.553417	+	0.832904i	$0.313323\pi$
$398$	−158303.	−0.0500936
$399$	3.32233e6	1.04474
$400$	220150.	0.0687968
$401$	−5.15091e6	−1.59964	−0.799822	−	0.600238i	$-0.795073\pi$
$401$	−5.15091e6	−1.59964	−0.799822	+	0.600238i	$0.795073\pi$
$402$	1.28527e6	0.396670
$403$	−464054.	−0.142333
$404$	−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	$-0.552765\pi$
$409$	−1.11646e6	−0.330015	−0.165008	+	0.986292i	$0.552765\pi$
$410$	−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
$416$	−1.00492e6	−0.284706
$417$	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	$-0.239298\pi$
$421$	5.31302e6	1.46095	0.730477	+	0.682937i	$0.239298\pi$
$422$	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
$429$	1.74556e6	0.457922
$430$	561890.	0.146548
$431$	−4.63630e6	−1.20221	−0.601103	−	0.799172i	$-0.705272\pi$
$431$	−4.63630e6	−1.20221	−0.601103	+	0.799172i	$0.705272\pi$
$432$	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
$442$	204468.	0.0497817
$443$	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	$-0.497002\pi$
$449$	80455.2	0.0188338	0.00941690	+	0.999956i	$0.497002\pi$
$450$	−570592.	−0.132829
$451$	−8.41443e6	−1.94797
$452$	889503.	0.204786
$453$	9.55579e6	2.18787
$454$	−694384.	−0.158110
$455$	−362209.	−0.0820221
$456$	6.02525e6	1.35695
$457$	2.87687e6	0.644362	0.322181	−	0.946678i	$-0.395584\pi$
$457$	2.87687e6	0.644362	0.322181	+	0.946678i	$0.395584\pi$
$458$	1.17993e6	0.262840
$459$	−927394.	−0.205463
$460$	1.32585e6	0.292147
$461$	81644.7	0.0178927	0.00894635	−	0.999960i	$-0.497152\pi$
$461$	81644.7	0.0178927	0.00894635	+	0.999960i	$0.497152\pi$
$462$	−2.44096e6	−0.532053
$463$	−2.50214e6	−0.542450	−0.271225	−	0.962516i	$-0.587429\pi$
$463$	−2.50214e6	−0.542450	−0.271225	+	0.962516i	$0.587429\pi$
$464$	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	$0.452966\pi$
$468$	−1.36572e6	−0.288235
$469$	−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	$-0.664555\pi$
$479$	−4.96374e6	−0.988485	−0.494243	+	0.869324i	$0.664555\pi$
$480$	3.56211e6	0.705675
$481$	−361158.	−0.0711761
$482$	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	$-0.194495\pi$
$487$	8.57371e6	1.63812	0.819061	+	0.573706i	$0.194495\pi$
$488$	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
$494$	−753442.	−0.138910
$495$	−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	$-0.495368\pi$
$499$	161861.	0.0290999	0.0145500	+	0.999894i	$0.495368\pi$
$500$	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
$507$	684381.	0.118244
$508$	−4.35751e6	−0.749169
$509$	1.72999e6	0.295971	0.147985	−	0.988990i	$-0.452721\pi$
$509$	1.72999e6	0.295971	0.147985	+	0.988990i	$0.452721\pi$
$510$	−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
$520$	−656889.	−0.106533
$521$	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
$533$	−3.29904e6	−0.503002
$534$	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	$-0.103226\pi$
$541$	1.29055e7	1.89575	0.947876	+	0.318640i	$0.103226\pi$
$542$	4.36108e6	0.637670
$543$	−1.11417e7	−1.62164
$544$	2.60978e6	0.378099
$545$	−4.12291e6	−0.594583
$546$	−957025.	−0.137386
$547$	−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	$-0.667614\pi$
$557$	−7.35984e6	−1.00515	−0.502574	+	0.864534i	$0.667614\pi$
$558$	2.50685e6	0.340833
$559$	1.37791e6	0.186505
$560$	−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	$0.539950\pi$
$572$	−1.77753e6	−0.227158
$573$	−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	$-0.519660\pi$
$577$	−987263.	−0.123451	−0.0617253	+	0.998093i	$0.519660\pi$
$578$	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
$585$	−1.39924e6	−0.169046
$586$	−790350.	−0.0950770
$587$	1.11813e6	0.133936	0.0669679	−	0.997755i	$-0.478668\pi$
$587$	1.11813e6	0.133936	0.0669679	+	0.997755i	$0.478668\pi$
$588$	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	$-0.516828\pi$
$593$	−905001.	−0.105685	−0.0528424	+	0.998603i	$0.516828\pi$
$594$	−2.51077e6	−0.291971
$595$	940661.	0.108928
$596$	6.08874e6	0.702121
$597$	1.37606e6	0.158016
$598$	−1.01254e6	−0.115787
$599$	−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
$611$	−2.23679e6	−0.242395
$612$	3.54678e6	0.382785
$613$	1.55631e7	1.67280	0.836401	−	0.548118i	$-0.184655\pi$
$613$	1.55631e7	1.67280	0.836401	+	0.548118i	$0.184655\pi$
$614$	−5.23473e6	−0.560368
$615$	1.16941e7	1.24675
$616$	5.74542e6	0.610057
$617$	1.57277e7	1.66323	0.831615	−	0.555353i	$-0.187417\pi$
$617$	1.57277e7	1.66323	0.831615	+	0.555353i	$0.187417\pi$
$618$	−6.76772e6	−0.712806
$619$	−5.68261e6	−0.596103	−0.298051	−	0.954550i	$-0.596337\pi$
$619$	−5.68261e6	−0.596103	−0.298051	+	0.954550i	$0.596337\pi$
$620$	−1.67506e6	−0.175005
$621$	4.59254e6	0.477885
$622$	−4.80802e6	−0.498299
$623$	−292383.	−0.0301809
$624$	1.42643e6	0.146652
$625$	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	$-0.370396\pi$
$631$	7.92146e6	0.792012	0.396006	+	0.918248i	$0.370396\pi$
$632$	−1.67388e7	−1.66699
$633$	−9.75482e6	−0.967631
$634$	6.05622e6	0.598382
$635$	−4.46449e6	−0.439377
$636$	−1.02508e6	−0.100488
$637$	−1.59829e6	−0.156066
$638$	−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	$-0.416791\pi$
$643$	5.41901e6	0.516884	0.258442	+	0.966027i	$0.416791\pi$
$644$	−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
$650$	−291169.	−0.0270310
$651$	−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	$-0.467304\pi$
$661$	2.30367e6	0.205077	0.102538	+	0.994729i	$0.467304\pi$
$662$	1.01500e7	0.900167
$663$	−1.77734e6	−0.157032
$664$	−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
$676$	−696916.	−0.0586562
$677$	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	$-0.592512\pi$
$683$	−6.98711e6	−0.573121	−0.286560	+	0.958062i	$0.592512\pi$
$684$	−1.30695e7	−1.06812
$685$	8.68795e6	0.707442
$686$	6.20695e6	0.503580
$687$	−1.02566e7	−0.829105
$688$	2.87191e6	0.231313
$689$	296286.	0.0237773
$690$	3.58915e6	0.286992
$691$	3.58854e6	0.285905	0.142953	−	0.989730i	$-0.454340\pi$
$691$	3.58854e6	0.285905	0.142953	+	0.989730i	$0.454340\pi$
$692$	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	$0.635864\pi$
$702$	−984396.	−0.0753923
$703$	−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	$-0.283338\pi$
$709$	1.68465e7	1.25862	0.629309	+	0.777155i	$0.283338\pi$
$710$	−4.42611e6	−0.329516
$711$	−3.56554e7	−2.64516
$712$	−530254.	−0.0391998
$713$	−5.96803e6	−0.439650
$714$	2.48540e6	0.182453
$715$	−1.82117e6	−0.133225
$716$	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	$0.840670\pi$
$728$	2.25260e6	0.157528
$729$	−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	$-0.586148\pi$
$733$	−7.77802e6	−0.534699	−0.267349	+	0.963600i	$0.586148\pi$
$734$	−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	$-0.390006\pi$
$739$	1.00573e7	0.677439	0.338720	+	0.940887i	$0.390006\pi$
$740$	−1.30364e6	−0.0875142
$741$	6.54932e6	0.438178
$742$	−414320.	−0.0276265
$743$	−3.40838e6	−0.226504	−0.113252	−	0.993566i	$-0.536127\pi$
$743$	−3.40838e6	−0.226504	−0.113252	+	0.993566i	$0.536127\pi$
$744$	−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
$754$	−3.86194e6	−0.247387
$755$	−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	$-0.361359\pi$
$761$	1.34808e7	0.843825	0.421913	+	0.906636i	$0.361359\pi$
$762$	−1.17960e7	−0.735949
$763$	1.41383e7	0.879195
$764$	379132.	0.0234994
$765$	3.63385e6	0.224498
$766$	1.65890e6	0.102152
$767$	−334021.	−0.0205015
$768$	−1.55625e7	−0.952087
$769$	8.09020e6	0.493337	0.246668	−	0.969100i	$-0.420664\pi$
$769$	8.09020e6	0.493337	0.246668	+	0.969100i	$0.420664\pi$
$770$	2.54668e6	0.154792
$771$	−4.91520e7	−2.97787
$772$	7.79387e6	0.470663
$773$	2.75295e6	0.165710	0.0828551	−	0.996562i	$-0.473596\pi$
$773$	2.75295e6	0.165710	0.0828551	+	0.996562i	$0.473596\pi$
$774$	−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
$780$	2.47035e6	0.145386
$781$	−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	$-0.297416\pi$
$787$	2.06536e7	1.18867	0.594333	+	0.804219i	$0.297416\pi$
$788$	−1.55161e7	−0.890156
$789$	3.63560e7	2.07914
$790$	−7.41956e6	−0.422971
$791$	−3.12516e6	−0.177595
$792$	2.21950e7	1.25731
$793$	7.70274e6	0.434973
$794$	−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
$806$	1.27923e6	0.0693602
$807$	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	$-0.368445\pi$
$811$	1.50454e7	0.803251	0.401626	+	0.915804i	$0.368445\pi$
$812$	−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
$819$	4.79829e6	0.249963
$820$	−1.19083e7	−0.618463
$821$	1.82770e6	0.0946338	0.0473169	−	0.998880i	$-0.484933\pi$
$821$	1.82770e6	0.0946338	0.0473169	+	0.998880i	$0.484933\pi$
$822$	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	$-0.557258\pi$
$827$	−7.03772e6	−0.357823	−0.178911	+	0.983865i	$0.557258\pi$
$828$	−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
$832$	865273.	0.0433356
$833$	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	$-0.478195\pi$
$839$	2.79128e6	0.136899	0.0684493	+	0.997655i	$0.478195\pi$
$840$	−7.98478e6	−0.390449
$841$	4.82083e7	2.35035
$842$	−1.46460e7	−0.711935
$843$	2.29845e7	1.11395
$844$	9.93349e6	0.480005
$845$	−714025.	−0.0344010
$846$	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	$-0.698365\pi$
$853$	−2.48047e7	−1.16724	−0.583621	+	0.812026i	$0.698365\pi$
$854$	−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	$0.640174\pi$
$858$	−4.81187e6	−0.223149
$859$	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	$-0.439308\pi$
$863$	8.29285e6	0.379033	0.189516	+	0.981878i	$0.439308\pi$
$864$	−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
$871$	−3.28834e6	−0.146870
$872$	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	$-0.602795\pi$
$877$	−1.44569e7	−0.634711	−0.317356	+	0.948307i	$0.602795\pi$
$878$	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	$0.598665\pi$
$884$	1.80990e6	0.0778975
$885$	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
$897$	8.80157e6	0.365241
$898$	−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
$910$	998478.	0.0399701
$911$	−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
$923$	−1.08540e7	−0.419359
$924$	−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	$-0.573751\pi$
$929$	−1.20808e7	−0.459257	−0.229629	+	0.973278i	$0.573751\pi$
$930$	−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
$936$	8.70199e6	0.324660
$937$	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	$-0.577618\pi$
$941$	−1.31161e7	−0.482871	−0.241436	+	0.970417i	$0.577618\pi$
$942$	−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	$-0.647579\pi$
$947$	−2.46835e7	−0.894399	−0.447199	+	0.894434i	$0.647579\pi$
$948$	6.29494e7	2.27494
$949$	173938.	0.00626945
$950$	−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
$962$	995579.	0.0346847
$963$	−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	$-0.371703\pi$
$967$	2.28108e7	0.784465	0.392232	+	0.919866i	$0.371703\pi$
$968$	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
$975$	2.53100e6	0.0852668
$976$	1.60545e7	0.539475
$977$	3.17409e7	1.06386	0.531928	−	0.846790i	$-0.321468\pi$
$977$	3.17409e7	1.06386	0.531928	+	0.846790i	$0.321468\pi$
$978$	−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	$-0.476218\pi$
$983$	4.52276e6	0.149286	0.0746431	+	0.997210i	$0.476218\pi$
$984$	−7.27262e7	−2.39444
$985$	−1.58970e7	−0.522064
$986$	1.00295e7	0.328539
$987$	−2.71892e7	−0.888391
$988$	−6.66928e6	−0.217363
$989$	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	65.6.a.d.1.3	✓	6
3.2	odd	2		585.6.a.m.1.4		6
4.3	odd	2		1040.6.a.q.1.2		6
5.2	odd	4		325.6.b.g.274.5		12
5.3	odd	4		325.6.b.g.274.8		12
5.4	even	2		325.6.a.g.1.4		6
13.12	even	2		845.6.a.h.1.4		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
65.6.a.d.1.3	✓	6	1.1	even	1	trivial
325.6.a.g.1.4		6	5.4	even	2
325.6.b.g.274.5		12	5.2	odd	4
325.6.b.g.274.8		12	5.3	odd	4
585.6.a.m.1.4		6	3.2	odd	2
845.6.a.h.1.4		6	13.12	even	2
1040.6.a.q.1.2		6	4.3	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:	\( N \)	\(=\)	\( 65 = 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	65.a (trivial)

\(f(q)\)	\(=\)	\(q-2.75663 q^{2} +23.9621 q^{3} -24.4010 q^{4} -25.0000 q^{5} -66.0547 q^{6} +85.7300 q^{7} +155.477 q^{8} +331.182 q^{9} +68.9158 q^{10} +431.046 q^{11} -584.699 q^{12} +169.000 q^{13} -236.326 q^{14} -599.052 q^{15} +352.239 q^{16} -438.894 q^{17} -912.946 q^{18} +1617.28 q^{19} +610.025 q^{20} +2054.27 q^{21} -1188.23 q^{22} +2173.44 q^{23} +3725.55 q^{24} +625.000 q^{25} -465.871 q^{26} +2113.02 q^{27} -2091.90 q^{28} +8289.72 q^{29} +1651.37 q^{30} -2745.88 q^{31} -5946.25 q^{32} +10328.8 q^{33} +1209.87 q^{34} -2143.25 q^{35} -8081.16 q^{36} -2137.03 q^{37} -4458.24 q^{38} +4049.59 q^{39} -3886.92 q^{40} -19520.9 q^{41} -5662.87 q^{42} +8153.29 q^{43} -10517.9 q^{44} -8279.55 q^{45} -5991.39 q^{46} -13235.5 q^{47} +8440.39 q^{48} -9457.36 q^{49} -1722.89 q^{50} -10516.8 q^{51} -4123.77 q^{52} +1753.17 q^{53} -5824.83 q^{54} -10776.1 q^{55} +13329.0 q^{56} +38753.4 q^{57} -22851.7 q^{58} -1976.46 q^{59} +14617.5 q^{60} +45578.3 q^{61} +7569.39 q^{62} +28392.2 q^{63} +5119.96 q^{64} -4225.00 q^{65} -28472.6 q^{66} -19457.7 q^{67} +10709.5 q^{68} +52080.3 q^{69} +5908.15 q^{70} -64224.9 q^{71} +51491.1 q^{72} +1029.22 q^{73} +5891.00 q^{74} +14976.3 q^{75} -39463.2 q^{76} +36953.6 q^{77} -11163.2 q^{78} -107661. q^{79} -8805.99 q^{80} -29844.7 q^{81} +53812.1 q^{82} -46473.4 q^{83} -50126.2 q^{84} +10972.4 q^{85} -22475.6 q^{86} +198639. q^{87} +67017.6 q^{88} -3410.51 q^{89} +22823.7 q^{90} +14488.4 q^{91} -53034.2 q^{92} -65797.1 q^{93} +36485.3 q^{94} -40431.9 q^{95} -142485. q^{96} +133264. q^{97} +26070.5 q^{98} +142755. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 38 q^{3} + 134 q^{4} - 150 q^{5} + 318 q^{6} + 220 q^{7} + 24 q^{8} + 518 q^{9} - 170 q^{11} + 2238 q^{12} + 1014 q^{13} - 1440 q^{14} - 950 q^{15} + 3506 q^{16} + 728 q^{17} + 7788 q^{18} + 1218 q^{19}+ \cdots - 32270 q^{99}+O(q^{100}) \) 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 + 1014 * q^13 - 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 + 6422 * q^39 - 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 + 22646 * q^52 + 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 - 25350 * q^65 - 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 + 53742 * q^78 + 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 + 37180 * q^91 + 430298 * q^92 + 192892 * q^93 - 125920 * q^94 - 30450 * q^95 - 385858 * q^96 + 246284 * q^97 - 4736 * q^98 - 32270 * q^99

Introduction

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