LMFDB - Embedded newform 245.6.a.i.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [245,6,Mod(1,245)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(245, 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("245.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 245 = 5 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	245.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:	$39.2940358542$
Analytic rank:	$1$
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} - x^{5} - 109x^{4} + 41x^{3} + 2208x^{2} - 3204x + 560 $ x^6 - x^5 - 109x^4 + 41x^3 + 2208x^2 - 3204x + 560
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 35)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$4.12050$ of defining polynomial
Character	$\chi$	$=$	245.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.12050 q^{2} +27.8717 q^{3} -22.2625 q^{4} -25.0000 q^{5} +86.9736 q^{6} -169.326 q^{8} +533.832 q^{9} +O(q^{10})$	\(q+3.12050 q^{2} +27.8717 q^{3} -22.2625 q^{4} -25.0000 q^{5} +86.9736 q^{6} -169.326 q^{8} +533.832 q^{9} -78.0125 q^{10} -405.258 q^{11} -620.494 q^{12} -931.313 q^{13} -696.793 q^{15} +184.018 q^{16} -736.293 q^{17} +1665.82 q^{18} -994.229 q^{19} +556.562 q^{20} -1264.61 q^{22} +1224.83 q^{23} -4719.41 q^{24} +625.000 q^{25} -2906.16 q^{26} +8106.00 q^{27} -5585.88 q^{29} -2174.34 q^{30} +3149.58 q^{31} +5992.66 q^{32} -11295.2 q^{33} -2297.60 q^{34} -11884.4 q^{36} -9613.78 q^{37} -3102.49 q^{38} -25957.3 q^{39} +4233.15 q^{40} -14680.2 q^{41} +3526.05 q^{43} +9022.04 q^{44} -13345.8 q^{45} +3822.09 q^{46} +18567.6 q^{47} +5128.90 q^{48} +1950.31 q^{50} -20521.7 q^{51} +20733.3 q^{52} -5457.99 q^{53} +25294.8 q^{54} +10131.4 q^{55} -27710.9 q^{57} -17430.7 q^{58} -7767.95 q^{59} +15512.3 q^{60} +18874.7 q^{61} +9828.25 q^{62} +12811.5 q^{64} +23282.8 q^{65} -35246.7 q^{66} -66704.4 q^{67} +16391.7 q^{68} +34138.2 q^{69} +30994.2 q^{71} -90391.7 q^{72} +37314.5 q^{73} -29999.8 q^{74} +17419.8 q^{75} +22134.0 q^{76} -80999.7 q^{78} -15661.5 q^{79} -4600.45 q^{80} +96206.8 q^{81} -45809.5 q^{82} +70001.9 q^{83} +18407.3 q^{85} +11003.0 q^{86} -155688. q^{87} +68620.7 q^{88} -7904.17 q^{89} -41645.6 q^{90} -27267.8 q^{92} +87784.1 q^{93} +57940.3 q^{94} +24855.7 q^{95} +167026. q^{96} -149459. q^{97} -216340. q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 5 q^{2} + 20 q^{3} + 31 q^{4} - 150 q^{5} + 96 q^{6} - 135 q^{8} + 378 q^{9}+O(q^{10}) $ 6 * q - 5 * q^2 + 20 * q^3 + 31 * q^4 - 150 * q^5 + 96 * q^6 - 135 * q^8 + 378 * q^9	\( 6 q - 5 q^{2} + 20 q^{3} + 31 q^{4} - 150 q^{5} + 96 q^{6} - 135 q^{8} + 378 q^{9} + 125 q^{10} - 924 q^{11} + 370 q^{12} - 150 q^{13} - 500 q^{15} + 435 q^{16} + 1540 q^{17} + 195 q^{18} + 92 q^{19} - 775 q^{20} - 6855 q^{22} - 3920 q^{23} - 7200 q^{24} + 3750 q^{25} + 2635 q^{26} + 2060 q^{27} + 1264 q^{29} - 2400 q^{30} - 7160 q^{31} + 9105 q^{32} + 4460 q^{33} + 2166 q^{34} - 26375 q^{36} - 14170 q^{37} - 46215 q^{38} - 15376 q^{39} + 3375 q^{40} + 4098 q^{41} - 24460 q^{43} - 27873 q^{44} - 9450 q^{45} + 6815 q^{46} + 42940 q^{47} + 11610 q^{48} - 3125 q^{50} - 42008 q^{51} - 36115 q^{52} - 2450 q^{53} + 19566 q^{54} + 23100 q^{55} - 97100 q^{57} - 36110 q^{58} - 64600 q^{59} - 9250 q^{60} + 73620 q^{61} + 111440 q^{62} - 157997 q^{64} + 3750 q^{65} - 139138 q^{66} - 142620 q^{67} + 124330 q^{68} + 17344 q^{69} - 154256 q^{71} - 117495 q^{72} - 5120 q^{73} + 2785 q^{74} + 12500 q^{75} + 7775 q^{76} - 214090 q^{78} - 222504 q^{79} - 10875 q^{80} - 43986 q^{81} + 31665 q^{82} + 179580 q^{83} - 38500 q^{85} - 207160 q^{86} - 209300 q^{87} - 45145 q^{88} + 41648 q^{89} - 4875 q^{90} - 292185 q^{92} - 198520 q^{93} - 333699 q^{94} - 2300 q^{95} + 61824 q^{96} - 73980 q^{97} - 190772 q^{99}+O(q^{100}) \) 6 * q - 5 * q^2 + 20 * q^3 + 31 * q^4 - 150 * q^5 + 96 * q^6 - 135 * q^8 + 378 * q^9 + 125 * q^10 - 924 * q^11 + 370 * q^12 - 150 * q^13 - 500 * q^15 + 435 * q^16 + 1540 * q^17 + 195 * q^18 + 92 * q^19 - 775 * q^20 - 6855 * q^22 - 3920 * q^23 - 7200 * q^24 + 3750 * q^25 + 2635 * q^26 + 2060 * q^27 + 1264 * q^29 - 2400 * q^30 - 7160 * q^31 + 9105 * q^32 + 4460 * q^33 + 2166 * q^34 - 26375 * q^36 - 14170 * q^37 - 46215 * q^38 - 15376 * q^39 + 3375 * q^40 + 4098 * q^41 - 24460 * q^43 - 27873 * q^44 - 9450 * q^45 + 6815 * q^46 + 42940 * q^47 + 11610 * q^48 - 3125 * q^50 - 42008 * q^51 - 36115 * q^52 - 2450 * q^53 + 19566 * q^54 + 23100 * q^55 - 97100 * q^57 - 36110 * q^58 - 64600 * q^59 - 9250 * q^60 + 73620 * q^61 + 111440 * q^62 - 157997 * q^64 + 3750 * q^65 - 139138 * q^66 - 142620 * q^67 + 124330 * q^68 + 17344 * q^69 - 154256 * q^71 - 117495 * q^72 - 5120 * q^73 + 2785 * q^74 + 12500 * q^75 + 7775 * q^76 - 214090 * q^78 - 222504 * q^79 - 10875 * q^80 - 43986 * q^81 + 31665 * q^82 + 179580 * q^83 - 38500 * q^85 - 207160 * q^86 - 209300 * q^87 - 45145 * q^88 + 41648 * q^89 - 4875 * q^90 - 292185 * q^92 - 198520 * q^93 - 333699 * q^94 - 2300 * q^95 + 61824 * q^96 - 73980 * q^97 - 190772 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	3.12050	0.551631	0.275816	−	0.961211i	$-0.411052\pi$
$2$	3.12050	0.551631	0.275816	+	0.961211i	$0.411052\pi$
$3$	27.8717	1.78797	0.893986	−	0.448096i	$-0.147898\pi$
$3$	27.8717	1.78797	0.893986	+	0.448096i	$0.147898\pi$
$4$	−22.2625	−0.695703
$5$	−25.0000	−0.447214
$5$	−25.0000	−0.447214
$6$	86.9736	0.986301
$7$	0	0
$7$	0	0
$8$	−169.326	−0.935403
$9$	533.832	2.19684
$10$	−78.0125	−0.246697
$11$	−405.258	−1.00983	−0.504916	−	0.863168i	$-0.668477\pi$
$11$	−405.258	−1.00983	−0.504916	+	0.863168i	$0.668477\pi$
$12$	−620.494	−1.24390
$13$	−931.313	−1.52840	−0.764200	−	0.644979i	$-0.776866\pi$
$13$	−931.313	−1.52840	−0.764200	+	0.644979i	$0.776866\pi$
$14$	0	0
$15$	−696.793	−0.799605
$16$	184.018	0.179705
$17$	−736.293	−0.617914	−0.308957	−	0.951076i	$-0.599980\pi$
$17$	−736.293	−0.617914	−0.308957	+	0.951076i	$0.599980\pi$
$18$	1665.82	1.21185
$19$	−994.229	−0.631833	−0.315917	−	0.948787i	$-0.602312\pi$
$19$	−994.229	−0.631833	−0.315917	+	0.948787i	$0.602312\pi$
$20$	556.562	0.311128
$21$	0	0
$22$	−1264.61	−0.557055
$23$	1224.83	0.482789	0.241394	−	0.970427i	$-0.422395\pi$
$23$	1224.83	0.482789	0.241394	+	0.970427i	$0.422395\pi$
$24$	−4719.41	−1.67247
$25$	625.000	0.200000
$26$	−2906.16	−0.843114
$27$	8106.00	2.13992
$28$	0	0
$29$	−5585.88	−1.23338	−0.616689	−	0.787207i	$-0.711527\pi$
$29$	−5585.88	−1.23338	−0.616689	+	0.787207i	$0.711527\pi$
$30$	−2174.34	−0.441087
$31$	3149.58	0.588638	0.294319	−	0.955707i	$-0.404907\pi$
$31$	3149.58	0.588638	0.294319	+	0.955707i	$0.404907\pi$
$32$	5992.66	1.03453
$33$	−11295.2	−1.80555
$34$	−2297.60	−0.340861
$35$	0	0
$36$	−11884.4	−1.52835
$37$	−9613.78	−1.15449	−0.577245	−	0.816571i	$-0.695872\pi$
$37$	−9613.78	−1.15449	−0.577245	+	0.816571i	$0.695872\pi$
$38$	−3102.49	−0.348539
$39$	−25957.3	−2.73274
$40$	4233.15	0.418325
$41$	−14680.2	−1.36387	−0.681933	−	0.731414i	$-0.738861\pi$
$41$	−14680.2	−1.36387	−0.681933	+	0.731414i	$0.738861\pi$
$42$	0	0
$43$	3526.05	0.290815	0.145408	−	0.989372i	$-0.453551\pi$
$43$	3526.05	0.290815	0.145408	+	0.989372i	$0.453551\pi$
$44$	9022.04	0.702544
$45$	−13345.8	−0.982457
$46$	3822.09	0.266321
$47$	18567.6	1.22606	0.613031	−	0.790059i	$-0.289950\pi$
$47$	18567.6	1.22606	0.613031	+	0.790059i	$0.289950\pi$
$48$	5128.90	0.321308
$49$	0	0
$50$	1950.31	0.110326
$51$	−20521.7	−1.10481
$52$	20733.3	1.06331
$53$	−5457.99	−0.266896	−0.133448	−	0.991056i	$-0.542605\pi$
$53$	−5457.99	−0.266896	−0.133448	+	0.991056i	$0.542605\pi$
$54$	25294.8	1.18045
$55$	10131.4	0.451611
$56$	0	0
$57$	−27710.9	−1.12970
$58$	−17430.7	−0.680371
$59$	−7767.95	−0.290520	−0.145260	−	0.989393i	$-0.546402\pi$
$59$	−7767.95	−0.290520	−0.145260	+	0.989393i	$0.546402\pi$
$60$	15512.3	0.556287
$61$	18874.7	0.649465	0.324733	−	0.945806i	$-0.394726\pi$
$61$	18874.7	0.649465	0.324733	+	0.945806i	$0.394726\pi$
$62$	9828.25	0.324711
$63$	0	0
$64$	12811.5	0.390976
$65$	23282.8	0.683522
$66$	−35246.7	−0.995999
$67$	−66704.4	−1.81538	−0.907691	−	0.419640i	$-0.862156\pi$
$67$	−66704.4	−1.81538	−0.907691	+	0.419640i	$0.862156\pi$
$68$	16391.7	0.429885
$69$	34138.2	0.863212
$70$	0	0
$71$	30994.2	0.729683	0.364842	−	0.931070i	$-0.381123\pi$
$71$	30994.2	0.729683	0.364842	+	0.931070i	$0.381123\pi$
$72$	−90391.7	−2.05493
$73$	37314.5	0.819541	0.409771	−	0.912189i	$-0.365609\pi$
$73$	37314.5	0.819541	0.409771	+	0.912189i	$0.365609\pi$
$74$	−29999.8	−0.636852
$75$	17419.8	0.357594
$76$	22134.0	0.439568
$77$	0	0
$78$	−80999.7	−1.50746
$79$	−15661.5	−0.282335	−0.141167	−	0.989986i	$-0.545086\pi$
$79$	−15661.5	−0.282335	−0.141167	+	0.989986i	$0.545086\pi$
$80$	−4600.45	−0.0803666
$81$	96206.8	1.62927
$82$	−45809.5	−0.752352
$83$	70001.9	1.11536	0.557679	−	0.830057i	$-0.311692\pi$
$83$	70001.9	1.11536	0.557679	+	0.830057i	$0.311692\pi$
$84$	0	0
$85$	18407.3	0.276340
$86$	11003.0	0.160423
$87$	−155688.	−2.20525
$88$	68620.7	0.944601
$89$	−7904.17	−0.105775	−0.0528873	−	0.998600i	$-0.516842\pi$
$89$	−7904.17	−0.105775	−0.0528873	+	0.998600i	$0.516842\pi$
$90$	−41645.6	−0.541954
$91$	0	0
$92$	−27267.8	−0.335877
$93$	87784.1	1.05247
$94$	57940.3	0.676334
$95$	24855.7	0.282564
$96$	167026.	1.84972
$97$	−149459.	−1.61284	−0.806420	−	0.591343i	$-0.798598\pi$
$97$	−149459.	−1.61284	−0.806420	+	0.591343i	$0.798598\pi$
$98$	0	0
$99$	−216340.	−2.21844
$100$	−13914.1	−0.139141
$101$	−126933.	−1.23814	−0.619070	−	0.785336i	$-0.712490\pi$
$101$	−126933.	−1.23814	−0.619070	+	0.785336i	$0.712490\pi$
$102$	−64038.1	−0.609450
$103$	13500.5	0.125388	0.0626941	−	0.998033i	$-0.480031\pi$
$103$	13500.5	0.125388	0.0626941	+	0.998033i	$0.480031\pi$
$104$	157696.	1.42967
$105$	0	0
$106$	−17031.6	−0.147228
$107$	74431.6	0.628489	0.314245	−	0.949342i	$-0.398249\pi$
$107$	74431.6	0.628489	0.314245	+	0.949342i	$0.398249\pi$
$108$	−180460.	−1.48875
$109$	137681.	1.10996	0.554982	−	0.831862i	$-0.312725\pi$
$109$	137681.	1.10996	0.554982	+	0.831862i	$0.312725\pi$
$110$	31615.1	0.249123
$111$	−267953.	−2.06419
$112$	0	0
$113$	62258.8	0.458674	0.229337	−	0.973347i	$-0.426344\pi$
$113$	62258.8	0.458674	0.229337	+	0.973347i	$0.426344\pi$
$114$	−86471.7	−0.623178
$115$	−30620.8	−0.215910
$116$	124356.	0.858065
$117$	−497165.	−3.35765
$118$	−24239.9	−0.160260
$119$	0	0
$120$	117985.	0.747953
$121$	3182.74	0.0197623
$122$	58898.5	0.358265
$123$	−409162.	−2.43855
$124$	−70117.4	−0.409517
$125$	−15625.0	−0.0894427
$126$	0	0
$127$	−264361.	−1.45442	−0.727208	−	0.686417i	$-0.759183\pi$
$127$	−264361.	−1.45442	−0.727208	+	0.686417i	$0.759183\pi$
$128$	−151787.	−0.818859
$129$	98277.0	0.519969
$130$	72654.0	0.377052
$131$	109994.	0.560006	0.280003	−	0.959999i	$-0.409665\pi$
$131$	109994.	0.560006	0.280003	+	0.959999i	$0.409665\pi$
$132$	251460.	1.25613
$133$	0	0
$134$	−208151.	−1.00142
$135$	−202650.	−0.957000
$136$	124674.	0.577999
$137$	−281879.	−1.28310	−0.641552	−	0.767080i	$-0.721709\pi$
$137$	−281879.	−1.28310	−0.641552	+	0.767080i	$0.721709\pi$
$138$	106528.	0.476175
$139$	236209.	1.03696	0.518478	−	0.855091i	$-0.326499\pi$
$139$	236209.	1.03696	0.518478	+	0.855091i	$0.326499\pi$
$140$	0	0
$141$	517512.	2.19216
$142$	96717.3	0.402516
$143$	377422.	1.54343
$144$	98234.8	0.394784
$145$	139647.	0.551584
$146$	116440.	0.452085
$147$	0	0
$148$	214027.	0.803181
$149$	−25639.9	−0.0946128	−0.0473064	−	0.998880i	$-0.515064\pi$
$149$	−25639.9	−0.0946128	−0.0473064	+	0.998880i	$0.515064\pi$
$150$	54358.5	0.197260
$151$	262577.	0.937162	0.468581	−	0.883420i	$-0.344765\pi$
$151$	262577.	0.937162	0.468581	+	0.883420i	$0.344765\pi$
$152$	168349.	0.591019
$153$	−393057.	−1.35746
$154$	0	0
$155$	−78739.4	−0.263247
$156$	577874.	1.90117
$157$	262263.	0.849157	0.424578	−	0.905391i	$-0.360422\pi$
$157$	262263.	0.849157	0.424578	+	0.905391i	$0.360422\pi$
$158$	−48871.6	−0.155745
$159$	−152123.	−0.477203
$160$	−149817.	−0.462658
$161$	0	0
$162$	300213.	0.898757
$163$	285382.	0.841312	0.420656	−	0.907220i	$-0.361800\pi$
$163$	285382.	0.841312	0.420656	+	0.907220i	$0.361800\pi$
$164$	326817.	0.948846
$165$	282381.	0.807467
$166$	218441.	0.615267
$167$	467133.	1.29613	0.648067	−	0.761583i	$-0.275578\pi$
$167$	467133.	1.29613	0.648067	+	0.761583i	$0.275578\pi$
$168$	0	0
$169$	496051.	1.33601
$170$	57440.0	0.152438
$171$	−530752.	−1.38804
$172$	−78498.6	−0.202321
$173$	−30164.3	−0.0766263	−0.0383132	−	0.999266i	$-0.512198\pi$
$173$	−30164.3	−0.0766263	−0.0383132	+	0.999266i	$0.512198\pi$
$174$	−485824.	−1.21648
$175$	0	0
$176$	−74574.8	−0.181472
$177$	−216506.	−0.519442
$178$	−24664.9	−0.0583486
$179$	−594377.	−1.38653	−0.693265	−	0.720683i	$-0.743828\pi$
$179$	−594377.	−1.38653	−0.693265	+	0.720683i	$0.743828\pi$
$180$	297111.	0.683498
$181$	155232.	0.352197	0.176099	−	0.984373i	$-0.443652\pi$
$181$	155232.	0.352197	0.176099	+	0.984373i	$0.443652\pi$
$182$	0	0
$183$	526071.	1.16123
$184$	−207396.	−0.451602
$185$	240344.	0.516303
$186$	273930.	0.580574
$187$	298388.	0.623990
$188$	−413362.	−0.852974
$189$	0	0
$190$	77562.2	0.155871
$191$	320961.	0.636603	0.318302	−	0.947990i	$-0.396888\pi$
$191$	320961.	0.636603	0.318302	+	0.947990i	$0.396888\pi$
$192$	357079.	0.699054
$193$	−727235.	−1.40534	−0.702670	−	0.711516i	$-0.748009\pi$
$193$	−727235.	−1.40534	−0.702670	+	0.711516i	$0.748009\pi$
$194$	−466385.	−0.889693
$195$	648932.	1.22212
$196$	0	0
$197$	−752919.	−1.38224	−0.691119	−	0.722741i	$-0.742882\pi$
$197$	−752919.	−1.38224	−0.691119	+	0.722741i	$0.742882\pi$
$198$	−675088.	−1.22376
$199$	96635.3	0.172983	0.0864915	−	0.996253i	$-0.472434\pi$
$199$	96635.3	0.172983	0.0864915	+	0.996253i	$0.472434\pi$
$200$	−105829.	−0.187081
$201$	−1.85917e6	−3.24585
$202$	−396093.	−0.682996
$203$	0	0
$204$	456865.	0.768622
$205$	367005.	0.609940
$206$	42128.3	0.0691681
$207$	653855.	1.06061
$208$	−171378.	−0.274662
$209$	402919.	0.638046
$210$	0	0
$211$	−373868.	−0.578112	−0.289056	−	0.957312i	$-0.593341\pi$
$211$	−373868.	−0.578112	−0.289056	+	0.957312i	$0.593341\pi$
$212$	121508.	0.185681
$213$	863861.	1.30465
$214$	232264.	0.346694
$215$	−88151.2	−0.130057
$216$	−1.37256e6	−2.00169
$217$	0	0
$218$	429634.	0.612291
$219$	1.04002e6	1.46532
$220$	−225551.	−0.314187
$221$	685719.	0.944421
$222$	−836145.	−1.13867
$223$	545786.	0.734954	0.367477	−	0.930033i	$-0.380222\pi$
$223$	545786.	0.734954	0.367477	+	0.930033i	$0.380222\pi$
$224$	0	0
$225$	333645.	0.439368
$226$	194278.	0.253019
$227$	−123277.	−0.158787	−0.0793937	−	0.996843i	$-0.525298\pi$
$227$	−123277.	−0.158787	−0.0793937	+	0.996843i	$0.525298\pi$
$228$	616913.	0.785935
$229$	1.07833e6	1.35882	0.679412	−	0.733757i	$-0.262235\pi$
$229$	1.07833e6	1.35882	0.679412	+	0.733757i	$0.262235\pi$
$230$	−95552.2	−0.119103
$231$	0	0
$232$	945835.	1.15371
$233$	141208.	0.170400	0.0852002	−	0.996364i	$-0.472847\pi$
$233$	141208.	0.170400	0.0852002	+	0.996364i	$0.472847\pi$
$234$	−1.55140e6	−1.85219
$235$	−464191.	−0.548311
$236$	172934.	0.202116
$237$	−436512.	−0.504807
$238$	0	0
$239$	−508655.	−0.576008	−0.288004	−	0.957629i	$-0.592992\pi$
$239$	−508655.	−0.576008	−0.288004	+	0.957629i	$0.592992\pi$
$240$	−128223.	−0.143693
$241$	−25696.1	−0.0284987	−0.0142493	−	0.999898i	$-0.504536\pi$
$241$	−25696.1	−0.0284987	−0.0142493	+	0.999898i	$0.504536\pi$
$242$	9931.75	0.0109015
$243$	711691.	0.773171
$244$	−420198.	−0.451835
$245$	0	0
$246$	−1.27679e6	−1.34518
$247$	925938.	0.965694
$248$	−533305.	−0.550613
$249$	1.95107e6	1.99423
$250$	−48757.8	−0.0493394
$251$	−30939.4	−0.0309975	−0.0154988	−	0.999880i	$-0.504934\pi$
$251$	−30939.4	−0.0309975	−0.0154988	+	0.999880i	$0.504934\pi$
$252$	0	0
$253$	−496373.	−0.487536
$254$	−824939.	−0.802302
$255$	513044.	0.494087
$256$	−883619.	−0.842685
$257$	256109.	0.241876	0.120938	−	0.992660i	$-0.461410\pi$
$257$	256109.	0.241876	0.120938	+	0.992660i	$0.461410\pi$
$258$	306673.	0.286831
$259$	0	0
$260$	−518334.	−0.475528
$261$	−2.98192e6	−2.70954
$262$	343238.	0.308917
$263$	1.02074e6	0.909969	0.454985	−	0.890499i	$-0.349645\pi$
$263$	1.02074e6	0.909969	0.454985	+	0.890499i	$0.349645\pi$
$264$	1.91258e6	1.68892
$265$	136450.	0.119360
$266$	0	0
$267$	−220303.	−0.189122
$268$	1.48501e6	1.26297
$269$	−493765.	−0.416045	−0.208022	−	0.978124i	$-0.566703\pi$
$269$	−493765.	−0.416045	−0.208022	+	0.978124i	$0.566703\pi$
$270$	−632369.	−0.527911
$271$	−1.47579e6	−1.22068	−0.610340	−	0.792140i	$-0.708967\pi$
$271$	−1.47579e6	−1.22068	−0.610340	+	0.792140i	$0.708967\pi$
$272$	−135491.	−0.111042
$273$	0	0
$274$	−879604.	−0.707800
$275$	−253286.	−0.201967
$276$	−760001.	−0.600539
$277$	−877911.	−0.687466	−0.343733	−	0.939067i	$-0.611692\pi$
$277$	−877911.	−0.687466	−0.343733	+	0.939067i	$0.611692\pi$
$278$	737091.	0.572017
$279$	1.68135e6	1.29314
$280$	0	0
$281$	−1.46598e6	−1.10755	−0.553774	−	0.832667i	$-0.686813\pi$
$281$	−1.46598e6	−1.10755	−0.553774	+	0.832667i	$0.686813\pi$
$282$	1.61490e6	1.20927
$283$	1.73108e6	1.28484	0.642422	−	0.766351i	$-0.277929\pi$
$283$	1.73108e6	1.28484	0.642422	+	0.766351i	$0.277929\pi$
$284$	−690008.	−0.507643
$285$	692772.	0.505217
$286$	1.17774e6	0.851404
$287$	0	0
$288$	3.19908e6	2.27271
$289$	−877730.	−0.618182
$290$	435768.	0.304271
$291$	−4.16566e6	−2.88371
$292$	−830714.	−0.570157
$293$	−623726.	−0.424449	−0.212224	−	0.977221i	$-0.568071\pi$
$293$	−623726.	−0.424449	−0.212224	+	0.977221i	$0.568071\pi$
$294$	0	0
$295$	194199.	0.129925
$296$	1.62786e6	1.07991
$297$	−3.28502e6	−2.16096
$298$	−80009.1	−0.0521914
$299$	−1.14070e6	−0.737895
$300$	−387809.	−0.248779
$301$	0	0
$302$	819372.	0.516968
$303$	−3.53783e6	−2.21376
$304$	−182956.	−0.113544
$305$	−471868.	−0.290450
$306$	−1.22653e6	−0.748817
$307$	−1.92297e6	−1.16446	−0.582232	−	0.813022i	$-0.697821\pi$
$307$	−1.92297e6	−1.16446	−0.582232	+	0.813022i	$0.697821\pi$
$308$	0	0
$309$	376282.	0.224191
$310$	−245706.	−0.145215
$311$	2.10305e6	1.23296	0.616478	−	0.787372i	$-0.288559\pi$
$311$	2.10305e6	1.23296	0.616478	+	0.787372i	$0.288559\pi$
$312$	4.39524e6	2.55621
$313$	2.74351e6	1.58287	0.791437	−	0.611250i	$-0.209333\pi$
$313$	2.74351e6	1.58287	0.791437	+	0.611250i	$0.209333\pi$
$314$	818391.	0.468422
$315$	0	0
$316$	348663.	0.196421
$317$	−1.51211e6	−0.845150	−0.422575	−	0.906328i	$-0.638874\pi$
$317$	−1.51211e6	−0.845150	−0.422575	+	0.906328i	$0.638874\pi$
$318$	−474701.	−0.263240
$319$	2.26372e6	1.24551
$320$	−320288.	−0.174850
$321$	2.07454e6	1.12372
$322$	0	0
$323$	732044.	0.390419
$324$	−2.14180e6	−1.13349
$325$	−582071.	−0.305680
$326$	890533.	0.464094
$327$	3.83742e6	1.98458
$328$	2.48574e6	1.27576
$329$	0	0
$330$	881168.	0.445424
$331$	782778.	0.392707	0.196353	−	0.980533i	$-0.437090\pi$
$331$	782778.	0.392707	0.196353	+	0.980533i	$0.437090\pi$
$332$	−1.55842e6	−0.775958
$333$	−5.13215e6	−2.53623
$334$	1.45769e6	0.714988
$335$	1.66761e6	0.811863
$336$	0	0
$337$	−2.98847e6	−1.43342	−0.716711	−	0.697370i	$-0.754354\pi$
$337$	−2.98847e6	−1.43342	−0.716711	+	0.697370i	$0.754354\pi$
$338$	1.54793e6	0.736985
$339$	1.73526e6	0.820097
$340$	−409793.	−0.192250
$341$	−1.27639e6	−0.594425
$342$	−1.65621e6	−0.765685
$343$	0	0
$344$	−597052.	−0.272029
$345$	−853455.	−0.386040
$346$	−94127.6	−0.0422695
$347$	−614063.	−0.273772	−0.136886	−	0.990587i	$-0.543709\pi$
$347$	−614063.	−0.273772	−0.136886	+	0.990587i	$0.543709\pi$
$348$	3.46600e6	1.53420
$349$	−3.34125e6	−1.46841	−0.734203	−	0.678930i	$-0.762444\pi$
$349$	−3.34125e6	−1.46841	−0.734203	+	0.678930i	$0.762444\pi$
$350$	0	0
$351$	−7.54922e6	−3.27065
$352$	−2.42857e6	−1.04471
$353$	−3.62123e6	−1.54675	−0.773374	−	0.633950i	$-0.781433\pi$
$353$	−3.62123e6	−1.54675	−0.773374	+	0.633950i	$0.781433\pi$
$354$	−675607.	−0.286541
$355$	−774855.	−0.326324
$356$	175966.	0.0735876
$357$	0	0
$358$	−1.85475e6	−0.764854
$359$	3.58244e6	1.46704	0.733522	−	0.679666i	$-0.237875\pi$
$359$	3.58244e6	1.46704	0.733522	+	0.679666i	$0.237875\pi$
$360$	2.25979e6	0.918993
$361$	−1.48761e6	−0.600787
$362$	484402.	0.194283
$363$	88708.6	0.0353345
$364$	0	0
$365$	−932863.	−0.366510
$366$	1.64160e6	0.640568
$367$	2.88029e6	1.11628	0.558138	−	0.829748i	$-0.311516\pi$
$367$	2.88029e6	1.11628	0.558138	+	0.829748i	$0.311516\pi$
$368$	225391.	0.0867597
$369$	−7.83676e6	−2.99620
$370$	749995.	0.284809
$371$	0	0
$372$	−1.95429e6	−0.732204
$373$	−3.91807e6	−1.45814	−0.729071	−	0.684438i	$-0.760048\pi$
$373$	−3.91807e6	−1.45814	−0.729071	+	0.684438i	$0.760048\pi$
$374$	931120.	0.344213
$375$	−435496.	−0.159921
$376$	−3.14398e6	−1.14686
$377$	5.20220e6	1.88510
$378$	0	0
$379$	−795965.	−0.284640	−0.142320	−	0.989821i	$-0.545456\pi$
$379$	−795965.	−0.284640	−0.142320	+	0.989821i	$0.545456\pi$
$380$	−553350.	−0.196581
$381$	−7.36821e6	−2.60046
$382$	1.00156e6	0.351170
$383$	2.02575e6	0.705651	0.352825	−	0.935689i	$-0.385221\pi$
$383$	2.02575e6	0.705651	0.352825	+	0.935689i	$0.385221\pi$
$384$	−4.23056e6	−1.46410
$385$	0	0
$386$	−2.26934e6	−0.775230
$387$	1.88232e6	0.638875
$388$	3.32732e6	1.12206
$389$	−2.60319e6	−0.872233	−0.436117	−	0.899890i	$-0.643646\pi$
$389$	−2.60319e6	−0.872233	−0.436117	+	0.899890i	$0.643646\pi$
$390$	2.02499e6	0.674158
$391$	−901836.	−0.298322
$392$	0	0
$393$	3.06573e6	1.00127
$394$	−2.34948e6	−0.762486
$395$	391537.	0.126264
$396$	4.81626e6	1.54338
$397$	−5.16774e6	−1.64560	−0.822800	−	0.568331i	$-0.807589\pi$
$397$	−5.16774e6	−1.64560	−0.822800	+	0.568331i	$0.807589\pi$
$398$	301550.	0.0954228
$399$	0	0
$400$	115011.	0.0359410
$401$	1.35062e6	0.419441	0.209720	−	0.977761i	$-0.432745\pi$
$401$	1.35062e6	0.419441	0.209720	+	0.977761i	$0.432745\pi$
$402$	−5.80153e6	−1.79051
$403$	−2.93324e6	−0.899674
$404$	2.82583e6	0.861377
$405$	−2.40517e6	−0.728632
$406$	0	0
$407$	3.89606e6	1.16584
$408$	3.47487e6	1.03345
$409$	−2.77803e6	−0.821163	−0.410581	−	0.911824i	$-0.634674\pi$
$409$	−2.77803e6	−0.821163	−0.410581	+	0.911824i	$0.634674\pi$
$410$	1.14524e6	0.336462
$411$	−7.85646e6	−2.29415
$412$	−300555.	−0.0872330
$413$	0	0
$414$	2.04035e6	0.585066
$415$	−1.75005e6	−0.498803
$416$	−5.58104e6	−1.58118
$417$	6.58356e6	1.85405
$418$	1.25731e6	0.351966
$419$	1.67370e6	0.465739	0.232869	−	0.972508i	$-0.425189\pi$
$419$	1.67370e6	0.465739	0.232869	+	0.972508i	$0.425189\pi$
$420$	0	0
$421$	3.23947e6	0.890776	0.445388	−	0.895338i	$-0.353066\pi$
$421$	3.23947e6	0.890776	0.445388	+	0.895338i	$0.353066\pi$
$422$	−1.16665e6	−0.318905
$423$	9.91201e6	2.69346
$424$	924179.	0.249656
$425$	−460183.	−0.123583
$426$	2.69568e6	0.719687
$427$	0	0
$428$	−1.65703e6	−0.437242
$429$	1.05194e7	2.75961
$430$	−275076.	−0.0717433
$431$	3.16320e6	0.820225	0.410112	−	0.912035i	$-0.365489\pi$
$431$	3.16320e6	0.820225	0.410112	+	0.912035i	$0.365489\pi$
$432$	1.49165e6	0.384554
$433$	−5.42491e6	−1.39050	−0.695252	−	0.718766i	$-0.744707\pi$
$433$	−5.42491e6	−1.39050	−0.695252	+	0.718766i	$0.744707\pi$
$434$	0	0
$435$	3.89220e6	0.986216
$436$	−3.06513e6	−0.772205
$437$	−1.21776e6	−0.305042
$438$	3.24538e6	0.808314
$439$	507432.	0.125666	0.0628328	−	0.998024i	$-0.479987\pi$
$439$	507432.	0.125666	0.0628328	+	0.998024i	$0.479987\pi$
$440$	−1.71552e6	−0.422438
$441$	0	0
$442$	2.13979e6	0.520972
$443$	−3.38396e6	−0.819248	−0.409624	−	0.912254i	$-0.634340\pi$
$443$	−3.38396e6	−0.819248	−0.409624	+	0.912254i	$0.634340\pi$
$444$	5.96529e6	1.43607
$445$	197604.	0.0473038
$446$	1.70312e6	0.405424
$447$	−714627.	−0.169165
$448$	0	0
$449$	2.00014e6	0.468215	0.234108	−	0.972211i	$-0.424783\pi$
$449$	2.00014e6	0.468215	0.234108	+	0.972211i	$0.424783\pi$
$450$	1.04114e6	0.242369
$451$	5.94926e6	1.37728
$452$	−1.38604e6	−0.319101
$453$	7.31848e6	1.67562
$454$	−384685.	−0.0875922
$455$	0	0
$456$	4.69217e6	1.05672
$457$	−1.18467e6	−0.265342	−0.132671	−	0.991160i	$-0.542355\pi$
$457$	−1.18467e6	−0.265342	−0.132671	+	0.991160i	$0.542355\pi$
$458$	3.36493e6	0.749570
$459$	−5.96839e6	−1.32229
$460$	681696.	0.150209
$461$	−8.48869e6	−1.86032	−0.930161	−	0.367152i	$-0.880333\pi$
$461$	−8.48869e6	−1.86032	−0.930161	+	0.367152i	$0.880333\pi$
$462$	0	0
$463$	4.81635e6	1.04416	0.522079	−	0.852897i	$-0.325157\pi$
$463$	4.81635e6	1.04416	0.522079	+	0.852897i	$0.325157\pi$
$464$	−1.02790e6	−0.221645
$465$	−2.19460e6	−0.470678
$466$	440640.	0.0939982
$467$	−4.54734e6	−0.964863	−0.482431	−	0.875934i	$-0.660246\pi$
$467$	−4.54734e6	−0.964863	−0.482431	+	0.875934i	$0.660246\pi$
$468$	1.10681e7	2.33593
$469$	0	0
$470$	−1.44851e6	−0.302466
$471$	7.30972e6	1.51827
$472$	1.31532e6	0.271754
$473$	−1.42896e6	−0.293675
$474$	−1.36213e6	−0.278467
$475$	−621393.	−0.126367
$476$	0	0
$477$	−2.91365e6	−0.586329
$478$	−1.58726e6	−0.317744
$479$	−8.74449e6	−1.74139	−0.870694	−	0.491825i	$-0.836330\pi$
$479$	−8.74449e6	−1.74139	−0.870694	+	0.491825i	$0.836330\pi$
$480$	−4.17564e6	−0.827219
$481$	8.95344e6	1.76452
$482$	−80184.6	−0.0157208
$483$	0	0
$484$	−70855.8	−0.0137487
$485$	3.73646e6	0.721284
$486$	2.22083e6	0.426505
$487$	−9.05551e6	−1.73018	−0.865088	−	0.501620i	$-0.832738\pi$
$487$	−9.05551e6	−1.73018	−0.865088	+	0.501620i	$0.832738\pi$
$488$	−3.19598e6	−0.607512
$489$	7.95408e6	1.50424
$490$	0	0
$491$	−4.29340e6	−0.803706	−0.401853	−	0.915704i	$-0.631634\pi$
$491$	−4.29340e6	−0.803706	−0.401853	+	0.915704i	$0.631634\pi$
$492$	9.10896e6	1.69651
$493$	4.11284e6	0.762123
$494$	2.88939e6	0.532707
$495$	5.40849e6	0.992118
$496$	579579.	0.105781
$497$	0	0
$498$	6.08832e6	1.10008
$499$	−3.65835e6	−0.657709	−0.328855	−	0.944381i	$-0.606663\pi$
$499$	−3.65835e6	−0.657709	−0.328855	+	0.944381i	$0.606663\pi$
$500$	347851.	0.0622256
$501$	1.30198e7	2.31745
$502$	−96546.2	−0.0170992
$503$	−248395.	−0.0437746	−0.0218873	−	0.999760i	$-0.506968\pi$
$503$	−248395.	−0.0437746	−0.0218873	+	0.999760i	$0.506968\pi$
$504$	0	0
$505$	3.17331e6	0.553713
$506$	−1.54893e6	−0.268940
$507$	1.38258e7	2.38875
$508$	5.88534e6	1.01184
$509$	8.12053e6	1.38928	0.694640	−	0.719357i	$-0.255564\pi$
$509$	8.12053e6	1.38928	0.694640	+	0.719357i	$0.255564\pi$
$510$	1.60095e6	0.272554
$511$	0	0
$512$	2.09985e6	0.354008
$513$	−8.05922e6	−1.35207
$514$	799188.	0.133426
$515$	−337513.	−0.0560753
$516$	−2.18789e6	−0.361744
$517$	−7.52468e6	−1.23812
$518$	0	0
$519$	−840731.	−0.137006
$520$	−3.94239e6	−0.639368
$521$	7.02461e6	1.13378	0.566889	−	0.823794i	$-0.308147\pi$
$521$	7.02461e6	1.13378	0.566889	+	0.823794i	$0.308147\pi$
$522$	−9.30509e6	−1.49467
$523$	−403325.	−0.0644764	−0.0322382	−	0.999480i	$-0.510264\pi$
$523$	−403325.	−0.0644764	−0.0322382	+	0.999480i	$0.510264\pi$
$524$	−2.44875e6	−0.389598
$525$	0	0
$526$	3.18522e6	0.501968
$527$	−2.31901e6	−0.363728
$528$	−2.07853e6	−0.324467
$529$	−4.93613e6	−0.766915
$530$	425791.	0.0658426
$531$	−4.14679e6	−0.638227
$532$	0	0
$533$	1.36718e7	2.08453
$534$	−687454.	−0.104326
$535$	−1.86079e6	−0.281069
$536$	1.12948e7	1.69811
$537$	−1.65663e7	−2.47908
$538$	−1.54079e6	−0.229503
$539$	0	0
$540$	4.51149e6	0.665788
$541$	1.28608e6	0.188919	0.0944593	−	0.995529i	$-0.469888\pi$
$541$	1.28608e6	0.188919	0.0944593	+	0.995529i	$0.469888\pi$
$542$	−4.60520e6	−0.673365
$543$	4.32659e6	0.629718
$544$	−4.41235e6	−0.639253
$545$	−3.44203e6	−0.496391
$546$	0	0
$547$	−7.76839e6	−1.11010	−0.555051	−	0.831816i	$-0.687301\pi$
$547$	−7.76839e6	−1.11010	−0.555051	+	0.831816i	$0.687301\pi$
$548$	6.27534e6	0.892659
$549$	1.00759e7	1.42677
$550$	−790379.	−0.111411
$551$	5.55364e6	0.779290
$552$	−5.78048e6	−0.807451
$553$	0	0
$554$	−2.73952e6	−0.379228
$555$	6.69881e6	0.923135
$556$	−5.25861e6	−0.721413
$557$	−6.66949e6	−0.910867	−0.455433	−	0.890270i	$-0.650516\pi$
$557$	−6.66949e6	−0.910867	−0.455433	+	0.890270i	$0.650516\pi$
$558$	5.24664e6	0.713338
$559$	−3.28386e6	−0.444482
$560$	0	0
$561$	8.31659e6	1.11568
$562$	−4.57459e6	−0.610958
$563$	−1.70653e6	−0.226905	−0.113452	−	0.993543i	$-0.536191\pi$
$563$	−1.70653e6	−0.226905	−0.113452	+	0.993543i	$0.536191\pi$
$564$	−1.15211e7	−1.52509
$565$	−1.55647e6	−0.205125
$566$	5.40183e6	0.708761
$567$	0	0
$568$	−5.24812e6	−0.682548
$569$	−2.21728e6	−0.287104	−0.143552	−	0.989643i	$-0.545852\pi$
$569$	−2.21728e6	−0.287104	−0.143552	+	0.989643i	$0.545852\pi$
$570$	2.16179e6	0.278694
$571$	−6.37563e6	−0.818338	−0.409169	−	0.912459i	$-0.634181\pi$
$571$	−6.37563e6	−0.818338	−0.409169	+	0.912459i	$0.634181\pi$
$572$	−8.40235e6	−1.07377
$573$	8.94573e6	1.13823
$574$	0	0
$575$	765520.	0.0965578
$576$	6.83920e6	0.858913
$577$	−2.91197e6	−0.364122	−0.182061	−	0.983287i	$-0.558277\pi$
$577$	−2.91197e6	−0.364122	−0.182061	+	0.983287i	$0.558277\pi$
$578$	−2.73895e6	−0.341008
$579$	−2.02693e7	−2.51271
$580$	−3.10889e6	−0.383738
$581$	0	0
$582$	−1.29989e7	−1.59075
$583$	2.21189e6	0.269521
$584$	−6.31832e6	−0.766601
$585$	1.24291e7	1.50159
$586$	−1.94634e6	−0.234139
$587$	1.61871e7	1.93899	0.969494	−	0.245116i	$-0.0788259\pi$
$587$	1.61871e7	1.93899	0.969494	+	0.245116i	$0.0788259\pi$
$588$	0	0
$589$	−3.13140e6	−0.371921
$590$	605997.	0.0716705
$591$	−2.09852e7	−2.47140
$592$	−1.76911e6	−0.207468
$593$	−8.34065e6	−0.974010	−0.487005	−	0.873399i	$-0.661911\pi$
$593$	−8.34065e6	−0.974010	−0.487005	+	0.873399i	$0.661911\pi$
$594$	−1.02509e7	−1.19205
$595$	0	0
$596$	570807.	0.0658224
$597$	2.69339e6	0.309289
$598$	−3.55956e6	−0.407046
$599$	1.40996e7	1.60561	0.802805	−	0.596242i	$-0.203340\pi$
$599$	1.40996e7	1.60561	0.802805	+	0.596242i	$0.203340\pi$
$600$	−2.94963e6	−0.334495
$601$	−6.75584e6	−0.762945	−0.381472	−	0.924380i	$-0.624583\pi$
$601$	−6.75584e6	−0.762945	−0.381472	+	0.924380i	$0.624583\pi$
$602$	0	0
$603$	−3.56090e7	−3.98810
$604$	−5.84562e6	−0.651986
$605$	−79568.6	−0.00883799
$606$	−1.10398e7	−1.22118
$607$	1.20208e7	1.32422	0.662110	−	0.749406i	$-0.269661\pi$
$607$	1.20208e7	1.32422	0.662110	+	0.749406i	$0.269661\pi$
$608$	−5.95808e6	−0.653653
$609$	0	0
$610$	−1.47246e6	−0.160221
$611$	−1.72923e7	−1.87391
$612$	8.75043e6	0.944389
$613$	−1.45941e6	−0.156865	−0.0784323	−	0.996919i	$-0.524991\pi$
$613$	−1.45941e6	−0.156865	−0.0784323	+	0.996919i	$0.524991\pi$
$614$	−6.00062e6	−0.642355
$615$	1.02290e7	1.09055
$616$	0	0
$617$	−6.88916e6	−0.728540	−0.364270	−	0.931293i	$-0.618681\pi$
$617$	−6.88916e6	−0.728540	−0.364270	+	0.931293i	$0.618681\pi$
$618$	1.17419e6	0.123671
$619$	−1.17088e7	−1.22824	−0.614121	−	0.789212i	$-0.710489\pi$
$619$	−1.17088e7	−1.22824	−0.614121	+	0.789212i	$0.710489\pi$
$620$	1.75294e6	0.183141
$621$	9.92849e6	1.03313
$622$	6.56255e6	0.680137
$623$	0	0
$624$	−4.77661e6	−0.491087
$625$	390625.	0.0400000
$626$	8.56113e6	0.873163
$627$	1.12300e7	1.14081
$628$	−5.83863e6	−0.590761
$629$	7.07856e6	0.713375
$630$	0	0
$631$	2.08769e6	0.208734	0.104367	−	0.994539i	$-0.466718\pi$
$631$	2.08769e6	0.208734	0.104367	+	0.994539i	$0.466718\pi$
$632$	2.65189e6	0.264097
$633$	−1.04203e7	−1.03365
$634$	−4.71852e6	−0.466211
$635$	6.60904e6	0.650435
$636$	3.38665e6	0.331992
$637$	0	0
$638$	7.06394e6	0.687061
$639$	1.65457e7	1.60300
$640$	3.79467e6	0.366205
$641$	1.25720e7	1.20853	0.604266	−	0.796783i	$-0.293466\pi$
$641$	1.25720e7	1.20853	0.604266	+	0.796783i	$0.293466\pi$
$642$	6.47358e6	0.619879
$643$	−5.47387e6	−0.522116	−0.261058	−	0.965323i	$-0.584071\pi$
$643$	−5.47387e6	−0.522116	−0.261058	+	0.965323i	$0.584071\pi$
$644$	0	0
$645$	−2.45693e6	−0.232537
$646$	2.28434e6	0.215367
$647$	33033.0	0.00310233	0.00155116	−	0.999999i	$-0.499506\pi$
$647$	33033.0	0.00310233	0.00155116	+	0.999999i	$0.499506\pi$
$648$	−1.62903e7	−1.52402
$649$	3.14802e6	0.293377
$650$	−1.81635e6	−0.168623
$651$	0	0
$652$	−6.35331e6	−0.585303
$653$	6.78770e6	0.622930	0.311465	−	0.950258i	$-0.399180\pi$
$653$	6.78770e6	0.622930	0.311465	+	0.950258i	$0.399180\pi$
$654$	1.19746e7	1.09476
$655$	−2.74986e6	−0.250442
$656$	−2.70142e6	−0.245094
$657$	1.99197e7	1.80040
$658$	0	0
$659$	1.30322e7	1.16898	0.584488	−	0.811403i	$-0.301296\pi$
$659$	1.30322e7	1.16898	0.584488	+	0.811403i	$0.301296\pi$
$660$	−6.28650e6	−0.561757
$661$	−1.58081e7	−1.40726	−0.703632	−	0.710564i	$-0.748440\pi$
$661$	−1.58081e7	−1.40726	−0.703632	+	0.710564i	$0.748440\pi$
$662$	2.44266e6	0.216629
$663$	1.91122e7	1.68860
$664$	−1.18531e7	−1.04331
$665$	0	0
$666$	−1.60149e7	−1.39906
$667$	−6.84177e6	−0.595462
$668$	−1.03996e7	−0.901724
$669$	1.52120e7	1.31408
$670$	5.20378e6	0.447849
$671$	−7.64912e6	−0.655851
$672$	0	0
$673$	1.94847e7	1.65827	0.829137	−	0.559045i	$-0.188832\pi$
$673$	1.94847e7	1.65827	0.829137	+	0.559045i	$0.188832\pi$
$674$	−9.32552e6	−0.790721
$675$	5.06625e6	0.427984
$676$	−1.10433e7	−0.929465
$677$	7.97082e6	0.668392	0.334196	−	0.942504i	$-0.391535\pi$
$677$	7.97082e6	0.668392	0.334196	+	0.942504i	$0.391535\pi$
$678$	5.41487e6	0.452391
$679$	0	0
$680$	−3.11684e6	−0.258489
$681$	−3.43593e6	−0.283907
$682$	−3.98297e6	−0.327904
$683$	−9.93020e6	−0.814528	−0.407264	−	0.913310i	$-0.633517\pi$
$683$	−9.93020e6	−0.814528	−0.407264	+	0.913310i	$0.633517\pi$
$684$	1.18159e7	0.965661
$685$	7.04698e6	0.573821
$686$	0	0
$687$	3.00549e7	2.42954
$688$	648857.	0.0522610
$689$	5.08309e6	0.407925
$690$	−2.66320e6	−0.212952
$691$	6.39227e6	0.509284	0.254642	−	0.967035i	$-0.418042\pi$
$691$	6.39227e6	0.509284	0.254642	+	0.967035i	$0.418042\pi$
$692$	671532.	0.0533091
$693$	0	0
$694$	−1.91618e6	−0.151021
$695$	−5.90524e6	−0.463741
$696$	2.63620e7	2.06279
$697$	1.08089e7	0.842753
$698$	−1.04264e7	−0.810018
$699$	3.93572e6	0.304671
$700$	0	0
$701$	−5.78581e6	−0.444702	−0.222351	−	0.974967i	$-0.571373\pi$
$701$	−5.78581e6	−0.444702	−0.222351	+	0.974967i	$0.571373\pi$
$702$	−2.35573e7	−1.80419
$703$	9.55830e6	0.729444
$704$	−5.19196e6	−0.394821
$705$	−1.29378e7	−0.980365
$706$	−1.13001e7	−0.853235
$707$	0	0
$708$	4.81997e6	0.361377
$709$	−2.42588e7	−1.81240	−0.906198	−	0.422854i	$-0.861028\pi$
$709$	−2.42588e7	−1.81240	−0.906198	+	0.422854i	$0.861028\pi$
$710$	−2.41793e6	−0.180011
$711$	−8.36060e6	−0.620245
$712$	1.33838e6	0.0989418
$713$	3.85770e6	0.284188
$714$	0	0
$715$	−9.43554e6	−0.690243
$716$	1.32323e7	0.964613
$717$	−1.41771e7	−1.02989
$718$	1.11790e7	0.809267
$719$	6.70633e6	0.483797	0.241898	−	0.970302i	$-0.422230\pi$
$719$	6.70633e6	0.483797	0.241898	+	0.970302i	$0.422230\pi$
$720$	−2.45587e6	−0.176553
$721$	0	0
$722$	−4.64208e6	−0.331413
$723$	−716194.	−0.0509548
$724$	−3.45586e6	−0.245025
$725$	−3.49117e6	−0.246676
$726$	276815.	0.0194916
$727$	−1.44972e7	−1.01730	−0.508649	−	0.860974i	$-0.669855\pi$
$727$	−1.44972e7	−1.01730	−0.508649	+	0.860974i	$0.669855\pi$
$728$	0	0
$729$	−3.54221e6	−0.246863
$730$	−2.91100e6	−0.202178
$731$	−2.59621e6	−0.179699
$732$	−1.17116e7	−0.807868
$733$	1.07253e7	0.737308	0.368654	−	0.929567i	$-0.379819\pi$
$733$	1.07253e7	0.737308	0.368654	+	0.929567i	$0.379819\pi$
$734$	8.98795e6	0.615773
$735$	0	0
$736$	7.34001e6	0.499461
$737$	2.70325e7	1.83323
$738$	−2.44546e7	−1.65280
$739$	−3.54704e6	−0.238921	−0.119461	−	0.992839i	$-0.538116\pi$
$739$	−3.54704e6	−0.238921	−0.119461	+	0.992839i	$0.538116\pi$
$740$	−5.35067e6	−0.359194
$741$	2.58075e7	1.72663
$742$	0	0
$743$	−8.10764e6	−0.538794	−0.269397	−	0.963029i	$-0.586824\pi$
$743$	−8.10764e6	−0.538794	−0.269397	+	0.963029i	$0.586824\pi$
$744$	−1.48641e7	−0.984481
$745$	640996.	0.0423121
$746$	−1.22263e7	−0.804357
$747$	3.73693e7	2.45027
$748$	−6.64287e6	−0.434112
$749$	0	0
$750$	−1.35896e6	−0.0882175
$751$	−2.43974e7	−1.57850	−0.789250	−	0.614072i	$-0.789530\pi$
$751$	−2.43974e7	−1.57850	−0.789250	+	0.614072i	$0.789530\pi$
$752$	3.41678e6	0.220330
$753$	−862333.	−0.0554227
$754$	1.62335e7	1.03988
$755$	−6.56443e6	−0.419112
$756$	0	0
$757$	−9.35579e6	−0.593391	−0.296695	−	0.954972i	$-0.595885\pi$
$757$	−9.35579e6	−0.593391	−0.296695	+	0.954972i	$0.595885\pi$
$758$	−2.48381e6	−0.157016
$759$	−1.38348e7	−0.871700
$760$	−4.20872e6	−0.264312
$761$	−1.73774e7	−1.08773	−0.543867	−	0.839171i	$-0.683040\pi$
$761$	−1.73774e7	−1.08773	−0.543867	+	0.839171i	$0.683040\pi$
$762$	−2.29925e7	−1.43449
$763$	0	0
$764$	−7.14539e6	−0.442886
$765$	9.82643e6	0.607075
$766$	6.32136e6	0.389259
$767$	7.23440e6	0.444032
$768$	−2.46280e7	−1.50670
$769$	4.58080e6	0.279335	0.139667	−	0.990198i	$-0.455397\pi$
$769$	4.58080e6	0.279335	0.139667	+	0.990198i	$0.455397\pi$
$770$	0	0
$771$	7.13820e6	0.432467
$772$	1.61901e7	0.977699
$773$	2.37429e7	1.42917	0.714585	−	0.699548i	$-0.246615\pi$
$773$	2.37429e7	1.42917	0.714585	+	0.699548i	$0.246615\pi$
$774$	5.87378e6	0.352424
$775$	1.96849e6	0.117728
$776$	2.53072e7	1.50866
$777$	0	0
$778$	−8.12326e6	−0.481151
$779$	1.45955e7	0.861736
$780$	−1.44468e7	−0.850230
$781$	−1.25606e7	−0.736858
$782$	−2.81418e6	−0.164564
$783$	−4.52791e7	−2.63933
$784$	0	0
$785$	−6.55657e6	−0.379754
$786$	9.56662e6	0.552335
$787$	1.48241e7	0.853162	0.426581	−	0.904449i	$-0.359718\pi$
$787$	1.48241e7	0.853162	0.426581	+	0.904449i	$0.359718\pi$
$788$	1.67619e7	0.961627
$789$	2.84498e7	1.62700
$790$	1.22179e6	0.0696512
$791$	0	0
$792$	3.66319e7	2.07514
$793$	−1.75783e7	−0.992643
$794$	−1.61259e7	−0.907764
$795$	3.80309e6	0.213412
$796$	−2.15134e6	−0.120345
$797$	−1.29629e7	−0.722866	−0.361433	−	0.932398i	$-0.617712\pi$
$797$	−1.29629e7	−0.722866	−0.361433	+	0.932398i	$0.617712\pi$
$798$	0	0
$799$	−1.36712e7	−0.757601
$800$	3.74541e6	0.206907
$801$	−4.21950e6	−0.232370
$802$	4.21459e6	0.231377
$803$	−1.51220e7	−0.827600
$804$	4.13897e7	2.25815
$805$	0	0
$806$	−9.15318e6	−0.496288
$807$	−1.37621e7	−0.743876
$808$	2.14930e7	1.15816
$809$	3.00021e7	1.61169	0.805843	−	0.592130i	$-0.201713\pi$
$809$	3.00021e7	1.61169	0.805843	+	0.592130i	$0.201713\pi$
$810$	−7.50533e6	−0.401936
$811$	−9.00717e6	−0.480880	−0.240440	−	0.970664i	$-0.577292\pi$
$811$	−9.00717e6	−0.480880	−0.240440	+	0.970664i	$0.577292\pi$
$812$	0	0
$813$	−4.11328e7	−2.18254
$814$	1.21576e7	0.643115
$815$	−7.13454e6	−0.376246
$816$	−3.77637e6	−0.198541
$817$	−3.50570e6	−0.183747
$818$	−8.66885e6	−0.452979
$819$	0	0
$820$	−8.17044e6	−0.424337
$821$	3.10235e7	1.60633	0.803163	−	0.595760i	$-0.203149\pi$
$821$	3.10235e7	1.60633	0.803163	+	0.595760i	$0.203149\pi$
$822$	−2.45161e7	−1.26553
$823$	1.18388e7	0.609270	0.304635	−	0.952469i	$-0.401466\pi$
$823$	1.18388e7	0.609270	0.304635	+	0.952469i	$0.401466\pi$
$824$	−2.28599e6	−0.117289
$825$	−7.05952e6	−0.361110
$826$	0	0
$827$	3.17753e7	1.61557	0.807785	−	0.589478i	$-0.200666\pi$
$827$	3.17753e7	1.61557	0.807785	+	0.589478i	$0.200666\pi$
$828$	−1.45564e7	−0.737870
$829$	1.31310e7	0.663608	0.331804	−	0.943348i	$-0.392343\pi$
$829$	1.31310e7	0.663608	0.331804	+	0.943348i	$0.392343\pi$
$830$	−5.46102e6	−0.275156
$831$	−2.44689e7	−1.22917
$832$	−1.19315e7	−0.597568
$833$	0	0
$834$	2.05440e7	1.02275
$835$	−1.16783e7	−0.579649
$836$	−8.96998e6	−0.443890
$837$	2.55305e7	1.25964
$838$	5.22278e6	0.256916
$839$	1.51638e7	0.743707	0.371853	−	0.928291i	$-0.378722\pi$
$839$	1.51638e7	0.743707	0.371853	+	0.928291i	$0.378722\pi$
$840$	0	0
$841$	1.06909e7	0.521224
$842$	1.01088e7	0.491380
$843$	−4.08594e7	−1.98026
$844$	8.32323e6	0.402194
$845$	−1.24013e7	−0.597481
$846$	3.09304e7	1.48580
$847$	0	0
$848$	−1.00437e6	−0.0479627
$849$	4.82481e7	2.29727
$850$	−1.43600e6	−0.0681722
$851$	−1.17753e7	−0.557374
$852$	−1.92317e7	−0.907650
$853$	1.27022e7	0.597733	0.298866	−	0.954295i	$-0.403392\pi$
$853$	1.27022e7	0.597733	0.298866	+	0.954295i	$0.403392\pi$
$854$	0	0
$855$	1.32688e7	0.620749
$856$	−1.26032e7	−0.587890
$857$	2.53820e6	0.118052	0.0590260	−	0.998256i	$-0.481201\pi$
$857$	2.53820e6	0.118052	0.0590260	+	0.998256i	$0.481201\pi$
$858$	3.28257e7	1.52229
$859$	−1.82606e6	−0.0844367	−0.0422184	−	0.999108i	$-0.513443\pi$
$859$	−1.82606e6	−0.0844367	−0.0422184	+	0.999108i	$0.513443\pi$
$860$	1.96247e6	0.0904807
$861$	0	0
$862$	9.87075e6	0.452462
$863$	3.79256e7	1.73343	0.866714	−	0.498805i	$-0.166227\pi$
$863$	3.79256e7	1.73343	0.866714	+	0.498805i	$0.166227\pi$
$864$	4.85765e7	2.21382
$865$	754107.	0.0342683
$866$	−1.69284e7	−0.767046
$867$	−2.44638e7	−1.10529
$868$	0	0
$869$	6.34693e6	0.285111
$870$	1.21456e7	0.544028
$871$	6.21227e7	2.77463
$872$	−2.33130e7	−1.03826
$873$	−7.97858e7	−3.54315
$874$	−3.80003e6	−0.168271
$875$	0	0
$876$	−2.31534e7	−1.01942
$877$	−1.60537e7	−0.704816	−0.352408	−	0.935847i	$-0.614637\pi$
$877$	−1.60537e7	−0.704816	−0.352408	+	0.935847i	$0.614637\pi$
$878$	1.58344e6	0.0693211
$879$	−1.73843e7	−0.758902
$880$	1.86437e6	0.0811568
$881$	−2.93698e6	−0.127486	−0.0637428	−	0.997966i	$-0.520304\pi$
$881$	−2.93698e6	−0.127486	−0.0637428	+	0.997966i	$0.520304\pi$
$882$	0	0
$883$	5.42066e6	0.233965	0.116982	−	0.993134i	$-0.462678\pi$
$883$	5.42066e6	0.233965	0.116982	+	0.993134i	$0.462678\pi$
$884$	−1.52658e7	−0.657036
$885$	5.41265e6	0.232302
$886$	−1.05596e7	−0.451923
$887$	3.93701e6	0.168018	0.0840092	−	0.996465i	$-0.473227\pi$
$887$	3.93701e6	0.168018	0.0840092	+	0.996465i	$0.473227\pi$
$888$	4.53713e7	1.93085
$889$	0	0
$890$	616624.	0.0260943
$891$	−3.89885e7	−1.64529
$892$	−1.21506e7	−0.511310
$893$	−1.84605e7	−0.774666
$894$	−2.22999e6	−0.0933167
$895$	1.48594e7	0.620075
$896$	0	0
$897$	−3.17933e7	−1.31933
$898$	6.24145e6	0.258282
$899$	−1.75932e7	−0.726013
$900$	−7.42777e6	−0.305670
$901$	4.01868e6	0.164919
$902$	1.85646e7	0.759749
$903$	0	0
$904$	−1.05420e7	−0.429045
$905$	−3.88081e6	−0.157507
$906$	2.28373e7	0.924324
$907$	1.95342e6	0.0788455	0.0394228	−	0.999223i	$-0.487448\pi$
$907$	1.95342e6	0.0788455	0.0394228	+	0.999223i	$0.487448\pi$
$908$	2.74445e6	0.110469
$909$	−6.77607e7	−2.72000
$910$	0	0
$911$	3.17289e7	1.26666	0.633329	−	0.773882i	$-0.281688\pi$
$911$	3.17289e7	1.26666	0.633329	+	0.773882i	$0.281688\pi$
$912$	−5.09930e6	−0.203013
$913$	−2.83688e7	−1.12633
$914$	−3.69675e6	−0.146371
$915$	−1.31518e7	−0.519316
$916$	−2.40063e7	−0.945338
$917$	0	0
$918$	−1.86243e7	−0.729415
$919$	−491150.	−0.0191834	−0.00959169	−	0.999954i	$-0.503053\pi$
$919$	−491150.	−0.0191834	−0.00959169	+	0.999954i	$0.503053\pi$
$920$	5.18490e6	0.201963
$921$	−5.35964e7	−2.08203
$922$	−2.64889e7	−1.02621
$923$	−2.88653e7	−1.11525
$924$	0	0
$925$	−6.00861e6	−0.230898
$926$	1.50294e7	0.575990
$927$	7.20701e6	0.275458
$928$	−3.34743e7	−1.27597
$929$	2.77950e7	1.05664	0.528320	−	0.849045i	$-0.322822\pi$
$929$	2.77950e7	1.05664	0.528320	+	0.849045i	$0.322822\pi$
$930$	−6.84825e6	−0.259641
$931$	0	0
$932$	−3.14365e6	−0.118548
$933$	5.86155e7	2.20449
$934$	−1.41900e7	−0.532249
$935$	−7.45971e6	−0.279057
$936$	8.41830e7	3.14076
$937$	2.96127e7	1.10187	0.550934	−	0.834549i	$-0.314272\pi$
$937$	2.96127e7	1.10187	0.550934	+	0.834549i	$0.314272\pi$
$938$	0	0
$939$	7.64665e7	2.83013
$940$	1.03340e7	0.381462
$941$	4.93546e7	1.81699	0.908497	−	0.417890i	$-0.137230\pi$
$941$	4.93546e7	1.81699	0.908497	+	0.417890i	$0.137230\pi$
$942$	2.28100e7	0.837524
$943$	−1.79808e7	−0.658460
$944$	−1.42944e6	−0.0522080
$945$	0	0
$946$	−4.45906e6	−0.162000
$947$	−1.16917e7	−0.423647	−0.211823	−	0.977308i	$-0.567940\pi$
$947$	−1.16917e7	−0.423647	−0.211823	+	0.977308i	$0.567940\pi$
$948$	9.71784e6	0.351195
$949$	−3.47515e7	−1.25259
$950$	−1.93906e6	−0.0697078
$951$	−4.21450e7	−1.51110
$952$	0	0
$953$	5.11241e7	1.82345	0.911725	−	0.410802i	$-0.134751\pi$
$953$	5.11241e7	1.82345	0.911725	+	0.410802i	$0.134751\pi$
$954$	−9.09204e6	−0.323438
$955$	−8.02402e6	−0.284698
$956$	1.13239e7	0.400730
$957$	6.30938e7	2.22693
$958$	−2.72872e7	−0.960604
$959$	0	0
$960$	−8.92697e6	−0.312627
$961$	−1.87093e7	−0.653506
$962$	2.79392e7	0.973366
$963$	3.97340e7	1.38069
$964$	572059.	0.0198266
$965$	1.81809e7	0.628487
$966$	0	0
$967$	3.09492e7	1.06435	0.532173	−	0.846636i	$-0.321376\pi$
$967$	3.09492e7	1.06435	0.532173	+	0.846636i	$0.321376\pi$
$968$	−538921.	−0.0184858
$969$	2.04033e7	0.698058
$970$	1.16596e7	0.397883
$971$	4.69439e7	1.59783	0.798917	−	0.601442i	$-0.205407\pi$
$971$	4.69439e7	1.59783	0.798917	+	0.601442i	$0.205407\pi$
$972$	−1.58440e7	−0.537897
$973$	0	0
$974$	−2.82577e7	−0.954420
$975$	−1.62233e7	−0.546547
$976$	3.47329e6	0.116712
$977$	3.39213e7	1.13693	0.568467	−	0.822706i	$-0.307537\pi$
$977$	3.39213e7	1.13693	0.568467	+	0.822706i	$0.307537\pi$
$978$	2.48207e7	0.829787
$979$	3.20322e6	0.106815
$980$	0	0
$981$	7.34988e7	2.43841
$982$	−1.33975e7	−0.443349
$983$	1.47261e7	0.486077	0.243038	−	0.970017i	$-0.421856\pi$
$983$	1.47261e7	0.486077	0.243038	+	0.970017i	$0.421856\pi$
$984$	6.92818e7	2.28103
$985$	1.88230e7	0.618156
$986$	1.28341e7	0.420411
$987$	0	0
$988$	−2.06137e7	−0.671836
$989$	4.31882e6	0.140402
$990$	1.68772e7	0.547283
$991$	822773.	0.0266131	0.0133066	−	0.999911i	$-0.495764\pi$
$991$	822773.	0.0266131	0.0133066	+	0.999911i	$0.495764\pi$
$992$	1.88743e7	0.608965
$993$	2.18174e7	0.702148
$994$	0	0
$995$	−2.41588e6	−0.0773603
$996$	−4.34357e7	−1.38739
$997$	1.10066e7	0.350685	0.175342	−	0.984508i	$-0.443897\pi$
$997$	1.10066e7	0.350685	0.175342	+	0.984508i	$0.443897\pi$
$998$	−1.14159e7	−0.362813
$999$	−7.79293e7	−2.47051

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	245.6.a.i.1.5		6
7.2	even	3		35.6.e.a.11.2	✓	12
7.4	even	3		35.6.e.a.16.2	yes	12
7.6	odd	2		245.6.a.h.1.5		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
35.6.e.a.11.2	✓	12	7.2	even	3
35.6.e.a.16.2	yes	12	7.4	even	3
245.6.a.h.1.5		6	7.6	odd	2
245.6.a.i.1.5		6	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 245 = 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	245.a (trivial)

\(f(q)\)	\(=\)	\(q+3.12050 q^{2} +27.8717 q^{3} -22.2625 q^{4} -25.0000 q^{5} +86.9736 q^{6} -169.326 q^{8} +533.832 q^{9} +O(q^{10})\)	\(q+3.12050 q^{2} +27.8717 q^{3} -22.2625 q^{4} -25.0000 q^{5} +86.9736 q^{6} -169.326 q^{8} +533.832 q^{9} -78.0125 q^{10} -405.258 q^{11} -620.494 q^{12} -931.313 q^{13} -696.793 q^{15} +184.018 q^{16} -736.293 q^{17} +1665.82 q^{18} -994.229 q^{19} +556.562 q^{20} -1264.61 q^{22} +1224.83 q^{23} -4719.41 q^{24} +625.000 q^{25} -2906.16 q^{26} +8106.00 q^{27} -5585.88 q^{29} -2174.34 q^{30} +3149.58 q^{31} +5992.66 q^{32} -11295.2 q^{33} -2297.60 q^{34} -11884.4 q^{36} -9613.78 q^{37} -3102.49 q^{38} -25957.3 q^{39} +4233.15 q^{40} -14680.2 q^{41} +3526.05 q^{43} +9022.04 q^{44} -13345.8 q^{45} +3822.09 q^{46} +18567.6 q^{47} +5128.90 q^{48} +1950.31 q^{50} -20521.7 q^{51} +20733.3 q^{52} -5457.99 q^{53} +25294.8 q^{54} +10131.4 q^{55} -27710.9 q^{57} -17430.7 q^{58} -7767.95 q^{59} +15512.3 q^{60} +18874.7 q^{61} +9828.25 q^{62} +12811.5 q^{64} +23282.8 q^{65} -35246.7 q^{66} -66704.4 q^{67} +16391.7 q^{68} +34138.2 q^{69} +30994.2 q^{71} -90391.7 q^{72} +37314.5 q^{73} -29999.8 q^{74} +17419.8 q^{75} +22134.0 q^{76} -80999.7 q^{78} -15661.5 q^{79} -4600.45 q^{80} +96206.8 q^{81} -45809.5 q^{82} +70001.9 q^{83} +18407.3 q^{85} +11003.0 q^{86} -155688. q^{87} +68620.7 q^{88} -7904.17 q^{89} -41645.6 q^{90} -27267.8 q^{92} +87784.1 q^{93} +57940.3 q^{94} +24855.7 q^{95} +167026. q^{96} -149459. q^{97} -216340. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 5 q^{2} + 20 q^{3} + 31 q^{4} - 150 q^{5} + 96 q^{6} - 135 q^{8} + 378 q^{9}+O(q^{10}) \) 6 * q - 5 * q^2 + 20 * q^3 + 31 * q^4 - 150 * q^5 + 96 * q^6 - 135 * q^8 + 378 * q^9	\( 6 q - 5 q^{2} + 20 q^{3} + 31 q^{4} - 150 q^{5} + 96 q^{6} - 135 q^{8} + 378 q^{9} + 125 q^{10} - 924 q^{11} + 370 q^{12} - 150 q^{13} - 500 q^{15} + 435 q^{16} + 1540 q^{17} + 195 q^{18} + 92 q^{19} - 775 q^{20} - 6855 q^{22} - 3920 q^{23} - 7200 q^{24} + 3750 q^{25} + 2635 q^{26} + 2060 q^{27} + 1264 q^{29} - 2400 q^{30} - 7160 q^{31} + 9105 q^{32} + 4460 q^{33} + 2166 q^{34} - 26375 q^{36} - 14170 q^{37} - 46215 q^{38} - 15376 q^{39} + 3375 q^{40} + 4098 q^{41} - 24460 q^{43} - 27873 q^{44} - 9450 q^{45} + 6815 q^{46} + 42940 q^{47} + 11610 q^{48} - 3125 q^{50} - 42008 q^{51} - 36115 q^{52} - 2450 q^{53} + 19566 q^{54} + 23100 q^{55} - 97100 q^{57} - 36110 q^{58} - 64600 q^{59} - 9250 q^{60} + 73620 q^{61} + 111440 q^{62} - 157997 q^{64} + 3750 q^{65} - 139138 q^{66} - 142620 q^{67} + 124330 q^{68} + 17344 q^{69} - 154256 q^{71} - 117495 q^{72} - 5120 q^{73} + 2785 q^{74} + 12500 q^{75} + 7775 q^{76} - 214090 q^{78} - 222504 q^{79} - 10875 q^{80} - 43986 q^{81} + 31665 q^{82} + 179580 q^{83} - 38500 q^{85} - 207160 q^{86} - 209300 q^{87} - 45145 q^{88} + 41648 q^{89} - 4875 q^{90} - 292185 q^{92} - 198520 q^{93} - 333699 q^{94} - 2300 q^{95} + 61824 q^{96} - 73980 q^{97} - 190772 q^{99}+O(q^{100}) \) 6 * q - 5 * q^2 + 20 * q^3 + 31 * q^4 - 150 * q^5 + 96 * q^6 - 135 * q^8 + 378 * q^9 + 125 * q^10 - 924 * q^11 + 370 * q^12 - 150 * q^13 - 500 * q^15 + 435 * q^16 + 1540 * q^17 + 195 * q^18 + 92 * q^19 - 775 * q^20 - 6855 * q^22 - 3920 * q^23 - 7200 * q^24 + 3750 * q^25 + 2635 * q^26 + 2060 * q^27 + 1264 * q^29 - 2400 * q^30 - 7160 * q^31 + 9105 * q^32 + 4460 * q^33 + 2166 * q^34 - 26375 * q^36 - 14170 * q^37 - 46215 * q^38 - 15376 * q^39 + 3375 * q^40 + 4098 * q^41 - 24460 * q^43 - 27873 * q^44 - 9450 * q^45 + 6815 * q^46 + 42940 * q^47 + 11610 * q^48 - 3125 * q^50 - 42008 * q^51 - 36115 * q^52 - 2450 * q^53 + 19566 * q^54 + 23100 * q^55 - 97100 * q^57 - 36110 * q^58 - 64600 * q^59 - 9250 * q^60 + 73620 * q^61 + 111440 * q^62 - 157997 * q^64 + 3750 * q^65 - 139138 * q^66 - 142620 * q^67 + 124330 * q^68 + 17344 * q^69 - 154256 * q^71 - 117495 * q^72 - 5120 * q^73 + 2785 * q^74 + 12500 * q^75 + 7775 * q^76 - 214090 * q^78 - 222504 * q^79 - 10875 * q^80 - 43986 * q^81 + 31665 * q^82 + 179580 * q^83 - 38500 * q^85 - 207160 * q^86 - 209300 * q^87 - 45145 * q^88 + 41648 * q^89 - 4875 * q^90 - 292185 * q^92 - 198520 * q^93 - 333699 * q^94 - 2300 * q^95 + 61824 * q^96 - 73980 * q^97 - 190772 * q^99

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