LMFDB - Embedded newform 29.6.a.a.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [29,6,Mod(1,29)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(29, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0]))

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("29.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 29 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	29.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:	$4.65113077458$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.3257317.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 34x^{2} - 27x + 10 $ x^4 - 34x^2 - 27x + 10
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$6.17343$ of defining polynomial
Character	$\chi$	$=$	29.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-0.863638 q^{2} +7.07413 q^{3} -31.2541 q^{4} -44.4758 q^{5} -6.10949 q^{6} -36.7447 q^{7} +54.6287 q^{8} -192.957 q^{9} +38.4110 q^{10} -302.283 q^{11} -221.096 q^{12} -373.472 q^{13} +31.7341 q^{14} -314.627 q^{15} +952.953 q^{16} +280.365 q^{17} +166.645 q^{18} +1371.41 q^{19} +1390.05 q^{20} -259.937 q^{21} +261.063 q^{22} +1861.10 q^{23} +386.450 q^{24} -1146.91 q^{25} +322.545 q^{26} -3084.01 q^{27} +1148.42 q^{28} -841.000 q^{29} +271.724 q^{30} +1472.03 q^{31} -2571.12 q^{32} -2138.39 q^{33} -242.134 q^{34} +1634.25 q^{35} +6030.69 q^{36} -11730.4 q^{37} -1184.40 q^{38} -2641.99 q^{39} -2429.65 q^{40} -2177.39 q^{41} +224.491 q^{42} -9679.03 q^{43} +9447.58 q^{44} +8581.90 q^{45} -1607.32 q^{46} -15909.1 q^{47} +6741.31 q^{48} -15456.8 q^{49} +990.512 q^{50} +1983.34 q^{51} +11672.6 q^{52} +24359.3 q^{53} +2663.47 q^{54} +13444.3 q^{55} -2007.32 q^{56} +9701.53 q^{57} +726.320 q^{58} +36304.7 q^{59} +9833.40 q^{60} -22316.1 q^{61} -1271.30 q^{62} +7090.14 q^{63} -28274.0 q^{64} +16610.5 q^{65} +1846.79 q^{66} -54808.6 q^{67} -8762.58 q^{68} +13165.7 q^{69} -1411.40 q^{70} +27790.4 q^{71} -10541.0 q^{72} +31685.5 q^{73} +10130.8 q^{74} -8113.36 q^{75} -42862.2 q^{76} +11107.3 q^{77} +2281.73 q^{78} -55328.4 q^{79} -42383.3 q^{80} +25071.8 q^{81} +1880.47 q^{82} -46888.8 q^{83} +8124.10 q^{84} -12469.5 q^{85} +8359.18 q^{86} -5949.34 q^{87} -16513.3 q^{88} +2564.30 q^{89} -7411.65 q^{90} +13723.1 q^{91} -58167.2 q^{92} +10413.3 q^{93} +13739.7 q^{94} -60994.5 q^{95} -18188.5 q^{96} +34940.3 q^{97} +13349.1 q^{98} +58327.5 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 28 q^{3} + 10 q^{4} - 68 q^{5} - 194 q^{6} - 208 q^{7} - 504 q^{8} - 280 q^{9} - 788 q^{10} - 124 q^{11} + 20 q^{12} - 460 q^{13} + 768 q^{14} + 932 q^{15} - 414 q^{16} + 184 q^{17} + 3208 q^{18}+ \cdots + 166720 q^{99}+O(q^{100}) $ 4 * q - 28 * q^3 + 10 * q^4 - 68 * q^5 - 194 * q^6 - 208 * q^7 - 504 * q^8 - 280 * q^9 - 788 * q^10 - 124 * q^11 + 20 * q^12 - 460 * q^13 + 768 * q^14 + 932 * q^15 - 414 * q^16 + 184 * q^17 + 3208 * q^18 - 2392 * q^19 + 2822 * q^20 + 992 * q^21 + 5538 * q^22 - 1192 * q^23 + 6786 * q^24 + 1824 * q^25 + 4724 * q^26 + 2468 * q^27 + 44 * q^28 - 3364 * q^29 + 8186 * q^30 - 19212 * q^31 + 6552 * q^32 - 10580 * q^33 - 7612 * q^34 - 22944 * q^35 - 7468 * q^36 - 10928 * q^37 - 456 * q^38 - 8732 * q^39 - 20 * q^40 - 1120 * q^41 + 1844 * q^42 - 21420 * q^43 - 1932 * q^44 - 8344 * q^45 - 7588 * q^46 + 23772 * q^47 + 33060 * q^48 + 10452 * q^49 + 43240 * q^50 + 12744 * q^51 - 29062 * q^52 + 8860 * q^53 + 35410 * q^54 - 52652 * q^55 + 34304 * q^56 + 48944 * q^57 - 10840 * q^59 + 25200 * q^60 + 49448 * q^61 + 18518 * q^62 + 27488 * q^63 - 20734 * q^64 + 97836 * q^65 - 47744 * q^66 - 7840 * q^67 + 20724 * q^68 + 58792 * q^69 - 77496 * q^70 - 48744 * q^71 + 8088 * q^72 - 74992 * q^73 - 35920 * q^74 - 90448 * q^75 - 140792 * q^76 + 128656 * q^77 + 2982 * q^78 - 106076 * q^79 + 58638 * q^80 - 59692 * q^81 - 234132 * q^82 + 62888 * q^83 - 59832 * q^84 + 23848 * q^85 - 216014 * q^86 + 23548 * q^87 - 39426 * q^88 + 107568 * q^89 + 41552 * q^90 - 268896 * q^91 - 26268 * q^92 + 221460 * q^93 + 30542 * q^94 + 147352 * q^95 - 78606 * q^96 - 49520 * q^97 + 242304 * q^98 + 166720 * 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 $

$2$	−0.863638	−0.152671	−0.0763356	−	0.997082i	$-0.524322\pi$
$2$	−0.863638	−0.152671	−0.0763356	+	0.997082i	$0.524322\pi$
$3$	7.07413	0.453806	0.226903	−	0.973917i	$-0.427140\pi$
$3$	7.07413	0.453806	0.226903	+	0.973917i	$0.427140\pi$
$4$	−31.2541	−0.976692
$5$	−44.4758	−0.795607	−0.397803	−	0.917471i	$-0.630227\pi$
$5$	−44.4758	−0.795607	−0.397803	+	0.917471i	$0.630227\pi$
$6$	−6.10949	−0.0692830
$7$	−36.7447	−0.283433	−0.141716	−	0.989907i	$-0.545262\pi$
$7$	−36.7447	−0.283433	−0.141716	+	0.989907i	$0.545262\pi$
$8$	54.6287	0.301784
$9$	−192.957	−0.794060
$10$	38.4110	0.121466
$11$	−302.283	−0.753237	−0.376619	−	0.926368i	$-0.622913\pi$
$11$	−302.283	−0.753237	−0.376619	+	0.926368i	$0.622913\pi$
$12$	−221.096	−0.443228
$13$	−373.472	−0.612915	−0.306457	−	0.951884i	$-0.599144\pi$
$13$	−373.472	−0.612915	−0.306457	+	0.951884i	$0.599144\pi$
$14$	31.7341	0.0432720
$15$	−314.627	−0.361051
$16$	952.953	0.930618
$17$	280.365	0.235289	0.117645	−	0.993056i	$-0.462466\pi$
$17$	280.365	0.235289	0.117645	+	0.993056i	$0.462466\pi$
$18$	166.645	0.121230
$19$	1371.41	0.871532	0.435766	−	0.900060i	$-0.356478\pi$
$19$	1371.41	0.871532	0.435766	+	0.900060i	$0.356478\pi$
$20$	1390.05	0.777062
$21$	−259.937	−0.128623
$22$	261.063	0.114998
$23$	1861.10	0.733586	0.366793	−	0.930303i	$-0.380456\pi$
$23$	1861.10	0.733586	0.366793	+	0.930303i	$0.380456\pi$
$24$	386.450	0.136951
$25$	−1146.91	−0.367010
$26$	322.545	0.0935744
$27$	−3084.01	−0.814155
$28$	1148.42	0.276826
$29$	−841.000	−0.185695
$29$	−841.000	−0.185695
$30$	271.724	0.0551220
$31$	1472.03	0.275114	0.137557	−	0.990494i	$-0.456075\pi$
$31$	1472.03	0.275114	0.137557	+	0.990494i	$0.456075\pi$
$32$	−2571.12	−0.443862
$33$	−2138.39	−0.341823
$34$	−242.134	−0.0359219
$35$	1634.25	0.225501
$36$	6030.69	0.775552
$37$	−11730.4	−1.40867	−0.704335	−	0.709868i	$-0.748755\pi$
$37$	−11730.4	−1.40867	−0.704335	+	0.709868i	$0.748755\pi$
$38$	−1184.40	−0.133058
$39$	−2641.99	−0.278144
$40$	−2429.65	−0.240101
$41$	−2177.39	−0.202291	−0.101145	−	0.994872i	$-0.532251\pi$
$41$	−2177.39	−0.202291	−0.101145	+	0.994872i	$0.532251\pi$
$42$	224.491	0.0196371
$43$	−9679.03	−0.798290	−0.399145	−	0.916888i	$-0.630693\pi$
$43$	−9679.03	−0.798290	−0.399145	+	0.916888i	$0.630693\pi$
$44$	9447.58	0.735680
$45$	8581.90	0.631760
$46$	−1607.32	−0.111997
$47$	−15909.1	−1.05051	−0.525255	−	0.850945i	$-0.676030\pi$
$47$	−15909.1	−1.05051	−0.525255	+	0.850945i	$0.676030\pi$
$48$	6741.31	0.422320
$49$	−15456.8	−0.919666
$50$	990.512	0.0560318
$51$	1983.34	0.106776
$52$	11672.6	0.598629
$53$	24359.3	1.19117	0.595586	−	0.803291i	$-0.296920\pi$
$53$	24359.3	1.19117	0.595586	+	0.803291i	$0.296920\pi$
$54$	2663.47	0.124298
$55$	13444.3	0.599280
$56$	−2007.32	−0.0855353
$57$	9701.53	0.395506
$58$	726.320	0.0283503
$59$	36304.7	1.35779	0.678894	−	0.734236i	$-0.262459\pi$
$59$	36304.7	1.35779	0.678894	+	0.734236i	$0.262459\pi$
$60$	9833.40	0.352635
$61$	−22316.1	−0.767880	−0.383940	−	0.923358i	$-0.625433\pi$
$61$	−22316.1	−0.767880	−0.383940	+	0.923358i	$0.625433\pi$
$62$	−1271.30	−0.0420019
$63$	7090.14	0.225063
$64$	−28274.0	−0.862853
$65$	16610.5	0.487639
$66$	1846.79	0.0521865
$67$	−54808.6	−1.49163	−0.745817	−	0.666151i	$-0.767940\pi$
$67$	−54808.6	−1.49163	−0.745817	+	0.666151i	$0.767940\pi$
$68$	−8762.58	−0.229805
$69$	13165.7	0.332905
$70$	−1411.40	−0.0344275
$71$	27790.4	0.654258	0.327129	−	0.944980i	$-0.393919\pi$
$71$	27790.4	0.654258	0.327129	+	0.944980i	$0.393919\pi$
$72$	−10541.0	−0.239635
$73$	31685.5	0.695910	0.347955	−	0.937511i	$-0.386876\pi$
$73$	31685.5	0.695910	0.347955	+	0.937511i	$0.386876\pi$
$74$	10130.8	0.215063
$75$	−8113.36	−0.166551
$76$	−42862.2	−0.851218
$77$	11107.3	0.213492
$78$	2281.73	0.0424646
$79$	−55328.4	−0.997426	−0.498713	−	0.866767i	$-0.666194\pi$
$79$	−55328.4	−0.997426	−0.498713	+	0.866767i	$0.666194\pi$
$80$	−42383.3	−0.740406
$81$	25071.8	0.424592
$82$	1880.47	0.0308839
$83$	−46888.8	−0.747092	−0.373546	−	0.927612i	$-0.621858\pi$
$83$	−46888.8	−0.747092	−0.373546	+	0.927612i	$0.621858\pi$
$84$	8124.10	0.125625
$85$	−12469.5	−0.187198
$86$	8359.18	0.121876
$87$	−5949.34	−0.0842696
$88$	−16513.3	−0.227315
$89$	2564.30	0.0343157	0.0171579	−	0.999853i	$-0.494538\pi$
$89$	2564.30	0.0343157	0.0171579	+	0.999853i	$0.494538\pi$
$90$	−7411.65	−0.0964515
$91$	13723.1	0.173720
$92$	−58167.2	−0.716487
$93$	10413.3	0.124848
$94$	13739.7	0.160382
$95$	−60994.5	−0.693396
$96$	−18188.5	−0.201427
$97$	34940.3	0.377048	0.188524	−	0.982069i	$-0.439630\pi$
$97$	34940.3	0.377048	0.188524	+	0.982069i	$0.439630\pi$
$98$	13349.1	0.140406
$99$	58327.5	0.598116
$100$	35845.6	0.358456
$101$	170224.	1.66042	0.830210	−	0.557451i	$-0.188221\pi$
$101$	170224.	1.66042	0.830210	+	0.557451i	$0.188221\pi$
$102$	−1712.89	−0.0163015
$103$	83962.9	0.779820	0.389910	−	0.920853i	$-0.372506\pi$
$103$	83962.9	0.779820	0.389910	+	0.920853i	$0.372506\pi$
$104$	−20402.3	−0.184968
$105$	11560.9	0.102334
$106$	−21037.6	−0.181858
$107$	16842.4	0.142215	0.0711074	−	0.997469i	$-0.477347\pi$
$107$	16842.4	0.142215	0.0711074	+	0.997469i	$0.477347\pi$
$108$	96388.2	0.795178
$109$	−3855.63	−0.0310834	−0.0155417	−	0.999879i	$-0.504947\pi$
$109$	−3855.63	−0.0310834	−0.0155417	+	0.999879i	$0.504947\pi$
$110$	−11611.0	−0.0914928
$111$	−82982.5	−0.639262
$112$	−35016.0	−0.263767
$113$	107319.	0.790644	0.395322	−	0.918543i	$-0.370633\pi$
$113$	107319.	0.790644	0.395322	+	0.918543i	$0.370633\pi$
$114$	−8378.61	−0.0603823
$115$	−82774.0	−0.583646
$116$	26284.7	0.181367
$117$	72064.0	0.486691
$118$	−31354.1	−0.207295
$119$	−10301.9	−0.0666886
$120$	−17187.7	−0.108959
$121$	−69676.1	−0.432634
$122$	19273.0	0.117233
$123$	−15403.1	−0.0918006
$124$	−46007.0	−0.268701
$125$	189996.	1.08760
$126$	−6123.31	−0.0343606
$127$	−14823.9	−0.0815554	−0.0407777	−	0.999168i	$-0.512984\pi$
$127$	−14823.9	−0.0815554	−0.0407777	+	0.999168i	$0.512984\pi$
$128$	106694.	0.575595
$129$	−68470.7	−0.362269
$130$	−14345.4	−0.0744484
$131$	−313171.	−1.59442	−0.797211	−	0.603701i	$-0.793692\pi$
$131$	−313171.	−1.59442	−0.797211	+	0.603701i	$0.793692\pi$
$132$	66833.4	0.333856
$133$	−50392.1	−0.247020
$134$	47334.8	0.227729
$135$	137164.	0.647747
$136$	15316.0	0.0710065
$137$	−254312.	−1.15762	−0.578810	−	0.815463i	$-0.696483\pi$
$137$	−254312.	−1.15762	−0.578810	+	0.815463i	$0.696483\pi$
$138$	−11370.4	−0.0508250
$139$	−390931.	−1.71618	−0.858091	−	0.513498i	$-0.828349\pi$
$139$	−390931.	−1.71618	−0.858091	+	0.513498i	$0.828349\pi$
$140$	−51077.0	−0.220245
$141$	−112543.	−0.476727
$142$	−24000.9	−0.0998863
$143$	112894.	0.461670
$144$	−183879.	−0.738967
$145$	37404.1	0.147740
$146$	−27364.8	−0.106245
$147$	−109344.	−0.417350
$148$	366624.	1.37584
$149$	72345.7	0.266961	0.133480	−	0.991051i	$-0.457385\pi$
$149$	72345.7	0.266961	0.133480	+	0.991051i	$0.457385\pi$
$150$	7007.01	0.0254276
$151$	−227419.	−0.811679	−0.405840	−	0.913944i	$-0.633021\pi$
$151$	−227419.	−0.811679	−0.405840	+	0.913944i	$0.633021\pi$
$152$	74918.3	0.263014
$153$	−54098.4	−0.186834
$154$	−9592.68	−0.0325940
$155$	−65469.6	−0.218882
$156$	82573.2	0.271661
$157$	539281.	1.74609	0.873044	−	0.487641i	$-0.162143\pi$
$157$	539281.	1.74609	0.873044	+	0.487641i	$0.162143\pi$
$158$	47783.7	0.152278
$159$	172321.	0.540561
$160$	114353.	0.353140
$161$	−68385.7	−0.207922
$162$	−21652.9	−0.0648230
$163$	87996.5	0.259416	0.129708	−	0.991552i	$-0.458596\pi$
$163$	87996.5	0.259416	0.129708	+	0.991552i	$0.458596\pi$
$164$	68052.3	0.197576
$165$	95106.4	0.271957
$166$	40495.0	0.114059
$167$	199897.	0.554644	0.277322	−	0.960777i	$-0.410553\pi$
$167$	199897.	0.554644	0.277322	+	0.960777i	$0.410553\pi$
$168$	−14200.0	−0.0388164
$169$	−231811.	−0.624335
$170$	10769.1	0.0285797
$171$	−264623.	−0.692049
$172$	302510.	0.779683
$173$	−313797.	−0.797139	−0.398569	−	0.917138i	$-0.630493\pi$
$173$	−313797.	−0.797139	−0.398569	+	0.917138i	$0.630493\pi$
$174$	5138.08	0.0128655
$175$	42142.7	0.104023
$176$	−288061.	−0.700976
$177$	256824.	0.616172
$178$	−2214.63	−0.00523902
$179$	−59814.7	−0.139533	−0.0697663	−	0.997563i	$-0.522225\pi$
$179$	−59814.7	−0.139533	−0.0697663	+	0.997563i	$0.522225\pi$
$180$	−268220.	−0.617034
$181$	402231.	0.912597	0.456298	−	0.889827i	$-0.349175\pi$
$181$	402231.	0.912597	0.456298	+	0.889827i	$0.349175\pi$
$182$	−11851.8	−0.0265220
$183$	−157867.	−0.348468
$184$	101670.	0.221384
$185$	521719.	1.12075
$186$	−8993.35	−0.0190607
$187$	−84749.6	−0.177229
$188$	497224.	1.02602
$189$	113321.	0.230758
$190$	52677.2	0.105862
$191$	−903529.	−1.79209	−0.896043	−	0.443967i	$-0.853571\pi$
$191$	−903529.	−1.79209	−0.896043	+	0.443967i	$0.853571\pi$
$192$	−200014.	−0.391568
$193$	1.02870e6	1.98791	0.993956	−	0.109780i	$-0.0350148\pi$
$193$	1.02870e6	1.98791	0.993956	+	0.109780i	$0.0350148\pi$
$194$	−30175.7	−0.0575644
$195$	117505.	0.221293
$196$	483090.	0.898230
$197$	−866099.	−1.59002	−0.795009	−	0.606598i	$-0.792534\pi$
$197$	−866099.	−1.59002	−0.795009	+	0.606598i	$0.792534\pi$
$198$	−50373.8	−0.0913150
$199$	285836.	0.511663	0.255832	−	0.966721i	$-0.417651\pi$
$199$	285836.	0.511663	0.255832	+	0.966721i	$0.417651\pi$
$200$	−62654.0	−0.110758
$201$	−387723.	−0.676911
$202$	−147012.	−0.253498
$203$	30902.3	0.0526321
$204$	−61987.6	−0.104287
$205$	96840.9	0.160944
$206$	−72513.6	−0.119056
$207$	−359112.	−0.582512
$208$	−355901.	−0.570389
$209$	−414554.	−0.656470
$210$	−9984.43	−0.0156234
$211$	−918930.	−1.42094	−0.710471	−	0.703727i	$-0.751518\pi$
$211$	−918930.	−1.42094	−0.710471	+	0.703727i	$0.751518\pi$
$212$	−761328.	−1.16341
$213$	196593.	0.296906
$214$	−14545.8	−0.0217121
$215$	430482.	0.635125
$216$	−168476.	−0.245699
$217$	−54089.3	−0.0779762
$218$	3329.87	0.00474554
$219$	224147.	0.315808
$220$	−420189.	−0.585312
$221$	−104709.	−0.144212
$222$	71666.8	0.0975969
$223$	486134.	0.654626	0.327313	−	0.944916i	$-0.393857\pi$
$223$	486134.	0.654626	0.327313	+	0.944916i	$0.393857\pi$
$224$	94475.2	0.125805
$225$	221303.	0.291428
$226$	−92684.9	−0.120708
$227$	−557639.	−0.718271	−0.359136	−	0.933285i	$-0.616928\pi$
$227$	−557639.	−0.718271	−0.359136	+	0.933285i	$0.616928\pi$
$228$	−303213.	−0.386287
$229$	440289.	0.554816	0.277408	−	0.960752i	$-0.410525\pi$
$229$	440289.	0.554816	0.277408	+	0.960752i	$0.410525\pi$
$230$	71486.8	0.0891059
$231$	78574.4	0.0968838
$232$	−45942.7	−0.0560398
$233$	1.46183e6	1.76403	0.882015	−	0.471221i	$-0.156187\pi$
$233$	1.46183e6	1.76403	0.882015	+	0.471221i	$0.156187\pi$
$234$	−62237.2	−0.0743037
$235$	707568.	0.835792
$236$	−1.13467e6	−1.32614
$237$	−391400.	−0.452637
$238$	8897.16	0.0101814
$239$	1.51292e6	1.71325	0.856627	−	0.515936i	$-0.172556\pi$
$239$	1.51292e6	1.71325	0.856627	+	0.515936i	$0.172556\pi$
$240$	−299825.	−0.336000
$241$	−800679.	−0.888006	−0.444003	−	0.896025i	$-0.646442\pi$
$241$	−800679.	−0.888006	−0.444003	+	0.896025i	$0.646442\pi$
$242$	60175.0	0.0660507
$243$	926776.	1.00684
$244$	697470.	0.749982
$245$	687454.	0.731692
$246$	13302.7	0.0140153
$247$	−512184.	−0.534175
$248$	80415.0	0.0830249
$249$	−331698.	−0.339035
$250$	−164088.	−0.166045
$251$	1.66023e6	1.66335	0.831676	−	0.555261i	$-0.187382\pi$
$251$	1.66023e6	1.66335	0.831676	+	0.555261i	$0.187382\pi$
$252$	−221596.	−0.219817
$253$	−562580.	−0.552564
$254$	12802.5	0.0124512
$255$	−88210.6	−0.0849514
$256$	812621.	0.774976
$257$	−1.00723e6	−0.951250	−0.475625	−	0.879648i	$-0.657778\pi$
$257$	−1.00723e6	−0.951250	−0.475625	+	0.879648i	$0.657778\pi$
$258$	59133.9	0.0553079
$259$	431031.	0.399263
$260$	−519146.	−0.476273
$261$	162277.	0.147453
$262$	270466.	0.243422
$263$	1.01326e6	0.903298	0.451649	−	0.892196i	$-0.350836\pi$
$263$	1.01326e6	0.903298	0.451649	+	0.892196i	$0.350836\pi$
$264$	−116817.	−0.103157
$265$	−1.08340e6	−0.947705
$266$	43520.5	0.0377129
$267$	18140.2	0.0155727
$268$	1.71300e6	1.45687
$269$	−1.45518e6	−1.22613	−0.613064	−	0.790034i	$-0.710063\pi$
$269$	−1.45518e6	−1.22613	−0.613064	+	0.790034i	$0.710063\pi$
$270$	−118460.	−0.0988922
$271$	−1.61146e6	−1.33290	−0.666450	−	0.745550i	$-0.732187\pi$
$271$	−1.61146e6	−1.33290	−0.666450	+	0.745550i	$0.732187\pi$
$272$	267175.	0.218964
$273$	97079.2	0.0788351
$274$	219634.	0.176735
$275$	346690.	0.276446
$276$	−411482.	−0.325146
$277$	2.06358e6	1.61593	0.807966	−	0.589230i	$-0.200569\pi$
$277$	2.06358e6	1.61593	0.807966	+	0.589230i	$0.200569\pi$
$278$	337623.	0.262011
$279$	−284038.	−0.218457
$280$	89276.9	0.0680525
$281$	21664.7	0.0163676	0.00818382	−	0.999967i	$-0.497395\pi$
$281$	21664.7	0.0163676	0.00818382	+	0.999967i	$0.497395\pi$
$282$	97196.2	0.0727824
$283$	−552726.	−0.410245	−0.205123	−	0.978736i	$-0.565759\pi$
$283$	−552726.	−0.410245	−0.205123	+	0.978736i	$0.565759\pi$
$284$	−868565.	−0.639008
$285$	−431483.	−0.314667
$286$	−97499.8	−0.0704837
$287$	80007.4	0.0573358
$288$	496116.	0.352453
$289$	−1.34125e6	−0.944639
$290$	−32303.6	−0.0225557
$291$	247172.	0.171107
$292$	−990301.	−0.679689
$293$	−1.53503e6	−1.04459	−0.522297	−	0.852764i	$-0.674925\pi$
$293$	−1.53503e6	−1.04459	−0.522297	+	0.852764i	$0.674925\pi$
$294$	94433.3	0.0637172
$295$	−1.61468e6	−1.08027
$296$	−640817.	−0.425114
$297$	932244.	0.613251
$298$	−62480.5	−0.0407572
$299$	−695071.	−0.449626
$300$	253576.	0.162669
$301$	355653.	0.226261
$302$	196408.	0.123920
$303$	1.20419e6	0.753508
$304$	1.30689e6	0.811063
$305$	992525.	0.610930
$306$	46721.4	0.0285241
$307$	−2.69140e6	−1.62979	−0.814895	−	0.579609i	$-0.803205\pi$
$307$	−2.69140e6	−1.62979	−0.814895	+	0.579609i	$0.803205\pi$
$308$	−347149.	−0.208516
$309$	593964.	0.353887
$310$	56542.1	0.0334170
$311$	−1.88553e6	−1.10543	−0.552717	−	0.833369i	$-0.686409\pi$
$311$	−1.88553e6	−1.10543	−0.552717	+	0.833369i	$0.686409\pi$
$312$	−144329.	−0.0839394
$313$	−2.48142e6	−1.43166	−0.715828	−	0.698276i	$-0.753951\pi$
$313$	−2.48142e6	−1.43166	−0.715828	+	0.698276i	$0.753951\pi$
$314$	−465744.	−0.266577
$315$	−315339.	−0.179061
$316$	1.72924e6	0.974177
$317$	204476.	0.114286	0.0571432	−	0.998366i	$-0.481801\pi$
$317$	204476.	0.114286	0.0571432	+	0.998366i	$0.481801\pi$
$318$	−148823.	−0.0825280
$319$	254220.	0.139873
$320$	1.25751e6	0.686492
$321$	119145.	0.0645379
$322$	59060.5	0.0317437
$323$	384496.	0.205062
$324$	−783596.	−0.414696
$325$	428338.	0.224946
$326$	−75997.1	−0.0396053
$327$	−27275.2	−0.0141058
$328$	−118948.	−0.0610480
$329$	584574.	0.297749
$330$	−82137.5	−0.0415199
$331$	1.58800e6	0.796672	0.398336	−	0.917240i	$-0.369588\pi$
$331$	1.58800e6	0.796672	0.398336	+	0.917240i	$0.369588\pi$
$332$	1.46547e6	0.729678
$333$	2.26346e6	1.11857
$334$	−172638.	−0.0846782
$335$	2.43766e6	1.18675
$336$	−247708.	−0.119699
$337$	−813616.	−0.390252	−0.195126	−	0.980778i	$-0.562512\pi$
$337$	−813616.	−0.390252	−0.195126	+	0.980778i	$0.562512\pi$
$338$	200201.	0.0953180
$339$	759189.	0.358799
$340$	389722.	0.182834
$341$	−444969.	−0.207226
$342$	228538.	0.105656
$343$	1.18553e6	0.544096
$344$	−528753.	−0.240911
$345$	−585554.	−0.264862
$346$	271007.	0.121700
$347$	516038.	0.230069	0.115034	−	0.993362i	$-0.463302\pi$
$347$	516038.	0.230069	0.115034	+	0.993362i	$0.463302\pi$
$348$	185942.	0.0823054
$349$	−2.65164e6	−1.16533	−0.582667	−	0.812711i	$-0.697991\pi$
$349$	−2.65164e6	−1.16533	−0.582667	+	0.812711i	$0.697991\pi$
$350$	−36396.1	−0.0158812
$351$	1.15179e6	0.499007
$352$	777207.	0.334333
$353$	391323.	0.167147	0.0835734	−	0.996502i	$-0.473367\pi$
$353$	391323.	0.167147	0.0835734	+	0.996502i	$0.473367\pi$
$354$	−221803.	−0.0940717
$355$	−1.23600e6	−0.520532
$356$	−80144.9	−0.0335159
$357$	−72877.3	−0.0302637
$358$	51658.3	0.0213026
$359$	3.03988e6	1.24486	0.622430	−	0.782675i	$-0.286146\pi$
$359$	3.03988e6	1.24486	0.622430	+	0.782675i	$0.286146\pi$
$360$	468818.	0.190655
$361$	−595334.	−0.240432
$362$	−347382.	−0.139327
$363$	−492898.	−0.196332
$364$	−428905.	−0.169671
$365$	−1.40924e6	−0.553670
$366$	136340.	0.0532010
$367$	1.02897e6	0.398785	0.199392	−	0.979920i	$-0.436103\pi$
$367$	1.02897e6	0.398785	0.199392	+	0.979920i	$0.436103\pi$
$368$	1.77354e6	0.682688
$369$	420141.	0.160631
$370$	−450577.	−0.171106
$371$	−895074.	−0.337617
$372$	−325460.	−0.121938
$373$	−4.42358e6	−1.64627	−0.823136	−	0.567844i	$-0.807778\pi$
$373$	−4.42358e6	−1.64627	−0.823136	+	0.567844i	$0.807778\pi$
$374$	73193.0	0.0270577
$375$	1.34406e6	0.493560
$376$	−869091.	−0.317027
$377$	314090.	0.113815
$378$	−97868.5	−0.0352301
$379$	−4.23499e6	−1.51445	−0.757223	−	0.653156i	$-0.773445\pi$
$379$	−4.23499e6	−1.51445	−0.757223	+	0.653156i	$0.773445\pi$
$380$	1.90633e6	0.677234
$381$	−104866.	−0.0370103
$382$	780323.	0.273600
$383$	−894484.	−0.311584	−0.155792	−	0.987790i	$-0.549793\pi$
$383$	−894484.	−0.311584	−0.155792	+	0.987790i	$0.549793\pi$
$384$	754770.	0.261208
$385$	−494005.	−0.169856
$386$	−888428.	−0.303497
$387$	1.86763e6	0.633891
$388$	−1.09203e6	−0.368260
$389$	3.60641e6	1.20837	0.604186	−	0.796843i	$-0.293498\pi$
$389$	3.60641e6	1.20837	0.604186	+	0.796843i	$0.293498\pi$
$390$	−101481.	−0.0337851
$391$	521789.	0.172605
$392$	−844386.	−0.277540
$393$	−2.21541e6	−0.723557
$394$	747996.	0.242750
$395$	2.46077e6	0.793559
$396$	−1.82297e6	−0.584175
$397$	−1.96195e6	−0.624757	−0.312379	−	0.949958i	$-0.601126\pi$
$397$	−1.96195e6	−0.624757	−0.312379	+	0.949958i	$0.601126\pi$
$398$	−246859.	−0.0781162
$399$	−356480.	−0.112099
$400$	−1.09295e6	−0.341546
$401$	−721116.	−0.223947	−0.111973	−	0.993711i	$-0.535717\pi$
$401$	−721116.	−0.223947	−0.111973	+	0.993711i	$0.535717\pi$
$402$	334853.	0.103345
$403$	−549762.	−0.168621
$404$	−5.32021e6	−1.62172
$405$	−1.11509e6	−0.337809
$406$	−26688.4	−0.00803540
$407$	3.54590e6	1.06106
$408$	108347.	0.0322231
$409$	1.33383e6	0.394269	0.197135	−	0.980376i	$-0.436836\pi$
$409$	1.33383e6	0.394269	0.197135	+	0.980376i	$0.436836\pi$
$410$	−83635.5	−0.0245715
$411$	−1.79904e6	−0.525334
$412$	−2.62419e6	−0.761644
$413$	−1.33400e6	−0.384842
$414$	310143.	0.0889327
$415$	2.08542e6	0.594391
$416$	960244.	0.272050
$417$	−2.76550e6	−0.778813
$418$	358024.	0.100224
$419$	−4.51294e6	−1.25581	−0.627905	−	0.778290i	$-0.716087\pi$
$419$	−4.51294e6	−1.25581	−0.627905	+	0.778290i	$0.716087\pi$
$420$	−361326.	−0.0999483
$421$	982181.	0.270076	0.135038	−	0.990840i	$-0.456884\pi$
$421$	982181.	0.270076	0.135038	+	0.990840i	$0.456884\pi$
$422$	793623.	0.216937
$423$	3.06976e6	0.834168
$424$	1.33071e6	0.359476
$425$	−321553.	−0.0863535
$426$	−169785.	−0.0453290
$427$	819998.	0.217642
$428$	−526395.	−0.138900
$429$	798629.	0.209508
$430$	−371781.	−0.0969652
$431$	5.31784e6	1.37893	0.689465	−	0.724319i	$-0.257846\pi$
$431$	5.31784e6	1.37893	0.689465	+	0.724319i	$0.257846\pi$
$432$	−2.93892e6	−0.757667
$433$	679278.	0.174112	0.0870558	−	0.996203i	$-0.472254\pi$
$433$	679278.	0.174112	0.0870558	+	0.996203i	$0.472254\pi$
$434$	46713.6	0.0119047
$435$	264602.	0.0670454
$436$	120504.	0.0303589
$437$	2.55234e6	0.639344
$438$	−193582.	−0.0482147
$439$	3.53619e6	0.875739	0.437870	−	0.899039i	$-0.355733\pi$
$439$	3.53619e6	0.875739	0.437870	+	0.899039i	$0.355733\pi$
$440$	734442.	0.180853
$441$	2.98250e6	0.730270
$442$	90430.5	0.0220170
$443$	−1.35298e6	−0.327553	−0.163776	−	0.986497i	$-0.552368\pi$
$443$	−1.35298e6	−0.327553	−0.163776	+	0.986497i	$0.552368\pi$
$444$	2.59355e6	0.624362
$445$	−114049.	−0.0273018
$446$	−419844.	−0.0999425
$447$	511783.	0.121148
$448$	1.03892e6	0.244561
$449$	−2.04554e6	−0.478843	−0.239421	−	0.970916i	$-0.576958\pi$
$449$	−2.04554e6	−0.478843	−0.239421	+	0.970916i	$0.576958\pi$
$450$	−191126.	−0.0444927
$451$	658186.	0.152373
$452$	−3.35416e6	−0.772215
$453$	−1.60879e6	−0.368345
$454$	481598.	0.109659
$455$	−610347.	−0.138213
$456$	529982.	0.119357
$457$	8.48379e6	1.90020	0.950100	−	0.311944i	$-0.100980\pi$
$457$	8.48379e6	1.90020	0.950100	+	0.311944i	$0.100980\pi$
$458$	−380250.	−0.0847044
$459$	−864651.	−0.191562
$460$	2.58703e6	0.570042
$461$	−2.06795e6	−0.453198	−0.226599	−	0.973988i	$-0.572761\pi$
$461$	−2.06795e6	−0.453198	−0.226599	+	0.973988i	$0.572761\pi$
$462$	−67859.9	−0.0147914
$463$	−368290.	−0.0798430	−0.0399215	−	0.999203i	$-0.512711\pi$
$463$	−368290.	−0.0798430	−0.0399215	+	0.999203i	$0.512711\pi$
$464$	−801433.	−0.172811
$465$	−463141.	−0.0993300
$466$	−1.26249e6	−0.269317
$467$	−2.78365e6	−0.590640	−0.295320	−	0.955398i	$-0.595426\pi$
$467$	−2.78365e6	−0.590640	−0.295320	+	0.955398i	$0.595426\pi$
$468$	−2.25230e6	−0.475347
$469$	2.01393e6	0.422777
$470$	−611082.	−0.127601
$471$	3.81494e6	0.792385
$472$	1.98328e6	0.409759
$473$	2.92580e6	0.601302
$474$	338028.	0.0691047
$475$	−1.57288e6	−0.319861
$476$	321978.	0.0651342
$477$	−4.70028e6	−0.945863
$478$	−1.30662e6	−0.261564
$479$	1.49820e6	0.298352	0.149176	−	0.988811i	$-0.452338\pi$
$479$	1.49820e6	0.298352	0.149176	+	0.988811i	$0.452338\pi$
$480$	808946.	0.160257
$481$	4.38099e6	0.863394
$482$	691497.	0.135573
$483$	−483770.	−0.0943563
$484$	2.17767e6	0.422550
$485$	−1.55399e6	−0.299982
$486$	−800399.	−0.153715
$487$	7.95675e6	1.52024	0.760122	−	0.649781i	$-0.225139\pi$
$487$	7.95675e6	1.52024	0.760122	+	0.649781i	$0.225139\pi$
$488$	−1.21910e6	−0.231734
$489$	622498.	0.117724
$490$	−593712.	−0.111708
$491$	−1.87342e6	−0.350697	−0.175348	−	0.984506i	$-0.556105\pi$
$491$	−1.87342e6	−0.350697	−0.175348	+	0.984506i	$0.556105\pi$
$492$	481411.	0.0896609
$493$	−235787.	−0.0436921
$494$	442341.	0.0815530
$495$	−2.59416e6	−0.475865
$496$	1.40277e6	0.256026
$497$	−1.02115e6	−0.185438
$498$	286467.	0.0517608
$499$	−352286.	−0.0633350	−0.0316675	−	0.999498i	$-0.510082\pi$
$499$	−352286.	−0.0633350	−0.0316675	+	0.999498i	$0.510082\pi$
$500$	−5.93817e6	−1.06225
$501$	1.41410e6	0.251701
$502$	−1.43384e6	−0.253946
$503$	−3.57266e6	−0.629611	−0.314805	−	0.949156i	$-0.601939\pi$
$503$	−3.57266e6	−0.629611	−0.314805	+	0.949156i	$0.601939\pi$
$504$	387325.	0.0679202
$505$	−7.57085e6	−1.32104
$506$	485865.	0.0843606
$507$	−1.63986e6	−0.283327
$508$	463308.	0.0796545
$509$	−1.05719e7	−1.80867	−0.904337	−	0.426819i	$-0.859634\pi$
$509$	−1.05719e7	−1.80867	−0.904337	+	0.426819i	$0.859634\pi$
$510$	76182.1	0.0129696
$511$	−1.16427e6	−0.197243
$512$	−4.11603e6	−0.693911
$513$	−4.22945e6	−0.709562
$514$	869880.	0.145228
$515$	−3.73431e6	−0.620430
$516$	2.13999e6	0.353825
$517$	4.80903e6	0.791282
$518$	−372255.	−0.0609559
$519$	−2.21984e6	−0.361746
$520$	907408.	0.147162
$521$	7.42701e6	1.19872	0.599362	−	0.800478i	$-0.295421\pi$
$521$	7.42701e6	1.19872	0.599362	+	0.800478i	$0.295421\pi$
$522$	−140148.	−0.0225119
$523$	−1.03292e7	−1.65125	−0.825623	−	0.564222i	$-0.809176\pi$
$523$	−1.03292e7	−1.65125	−0.825623	+	0.564222i	$0.809176\pi$
$524$	9.78788e6	1.55726
$525$	298123.	0.0472060
$526$	−875089.	−0.137908
$527$	412706.	0.0647313
$528$	−2.03778e6	−0.318107
$529$	−2.97263e6	−0.461851
$530$	935663.	0.144687
$531$	−7.00523e6	−1.07817
$532$	1.57496e6	0.241263
$533$	813193.	0.123987
$534$	−15666.5	−0.00237750
$535$	−749079.	−0.113147
$536$	−2.99412e6	−0.450151
$537$	−423137.	−0.0633207
$538$	1.25675e6	0.187194
$539$	4.67233e6	0.692726
$540$	−4.28694e6	−0.632649
$541$	4.60833e6	0.676941	0.338470	−	0.940977i	$-0.390091\pi$
$541$	4.60833e6	0.676941	0.338470	+	0.940977i	$0.390091\pi$
$542$	1.39172e6	0.203495
$543$	2.84543e6	0.414142
$544$	−720854.	−0.104436
$545$	171482.	0.0247302
$546$	−83841.3	−0.0120358
$547$	−1.79146e6	−0.255999	−0.127999	−	0.991774i	$-0.540856\pi$
$547$	−1.79146e6	−0.255999	−0.127999	+	0.991774i	$0.540856\pi$
$548$	7.94831e6	1.13064
$549$	4.30604e6	0.609743
$550$	−299415.	−0.0422052
$551$	−1.15336e6	−0.161839
$552$	719224.	0.100465
$553$	2.03303e6	0.282703
$554$	−1.78219e6	−0.246706
$555$	3.69071e6	0.508601
$556$	1.22182e7	1.67618
$557$	1.07840e7	1.47280	0.736399	−	0.676548i	$-0.236525\pi$
$557$	1.07840e7	1.47280	0.736399	+	0.676548i	$0.236525\pi$
$558$	245306.	0.0333521
$559$	3.61485e6	0.489284
$560$	1.55736e6	0.209855
$561$	−599530.	−0.0804273
$562$	−18710.4	−0.00249887
$563$	−1.40598e7	−1.86942	−0.934710	−	0.355411i	$-0.884341\pi$
$563$	−1.40598e7	−1.86942	−0.934710	+	0.355411i	$0.884341\pi$
$564$	3.51743e6	0.465615
$565$	−4.77310e6	−0.629041
$566$	477355.	0.0626326
$567$	−921255.	−0.120343
$568$	1.51815e6	0.197444
$569$	4.09890e6	0.530746	0.265373	−	0.964146i	$-0.414505\pi$
$569$	4.09890e6	0.530746	0.265373	+	0.964146i	$0.414505\pi$
$570$	372645.	0.0480406
$571$	−2.46148e6	−0.315941	−0.157971	−	0.987444i	$-0.550495\pi$
$571$	−2.46148e6	−0.315941	−0.157971	+	0.987444i	$0.550495\pi$
$572$	−3.52841e6	−0.450909
$573$	−6.39168e6	−0.813259
$574$	−69097.5	−0.00875351
$575$	−2.13451e6	−0.269233
$576$	5.45565e6	0.685157
$577$	7.09785e6	0.887539	0.443770	−	0.896141i	$-0.353641\pi$
$577$	7.09785e6	0.887539	0.443770	+	0.896141i	$0.353641\pi$
$578$	1.15836e6	0.144219
$579$	7.27718e6	0.902125
$580$	−1.16903e6	−0.144297
$581$	1.72292e6	0.211750
$582$	−213467.	−0.0261230
$583$	−7.36339e6	−0.897235
$584$	1.73094e6	0.210014
$585$	−3.20510e6	−0.387215
$586$	1.32571e6	0.159479
$587$	−1.43549e7	−1.71951	−0.859753	−	0.510710i	$-0.829383\pi$
$587$	−1.43549e7	−1.71951	−0.859753	+	0.510710i	$0.829383\pi$
$588$	3.41744e6	0.407622
$589$	2.01876e6	0.239770
$590$	1.39450e6	0.164925
$591$	−6.12690e6	−0.721559
$592$	−1.11785e7	−1.31093
$593$	1.01492e7	1.18521	0.592606	−	0.805493i	$-0.298099\pi$
$593$	1.01492e7	1.18521	0.592606	+	0.805493i	$0.298099\pi$
$594$	−805122.	−0.0936258
$595$	458187.	0.0530579
$596$	−2.26110e6	−0.260738
$597$	2.02204e6	0.232196
$598$	600290.	0.0686449
$599$	−8.76777e6	−0.998440	−0.499220	−	0.866475i	$-0.666380\pi$
$599$	−8.76777e6	−0.998440	−0.499220	+	0.866475i	$0.666380\pi$
$600$	−443222.	−0.0502624
$601$	1.49619e7	1.68966	0.844831	−	0.535034i	$-0.179701\pi$
$601$	1.49619e7	1.68966	0.844831	+	0.535034i	$0.179701\pi$
$602$	−307156.	−0.0345436
$603$	1.05757e7	1.18445
$604$	7.10779e6	0.792760
$605$	3.09890e6	0.344206
$606$	−1.03998e6	−0.115039
$607$	7.03290e6	0.774752	0.387376	−	0.921922i	$-0.373382\pi$
$607$	7.03290e6	0.774752	0.387376	+	0.921922i	$0.373382\pi$
$608$	−3.52606e6	−0.386840
$609$	218607.	0.0238847
$610$	−857182.	−0.0932714
$611$	5.94159e6	0.643873
$612$	1.69080e6	0.182479
$613$	−806845.	−0.0867239	−0.0433620	−	0.999059i	$-0.513807\pi$
$613$	−806845.	−0.0867239	−0.0433620	+	0.999059i	$0.513807\pi$
$614$	2.32439e6	0.248822
$615$	685065.	0.0730372
$616$	606777.	0.0644284
$617$	7.36483e6	0.778843	0.389421	−	0.921060i	$-0.372675\pi$
$617$	7.36483e6	0.778843	0.389421	+	0.921060i	$0.372675\pi$
$618$	−512970.	−0.0540283
$619$	−1.30528e7	−1.36923	−0.684617	−	0.728903i	$-0.740031\pi$
$619$	−1.30528e7	−1.36923	−0.684617	+	0.728903i	$0.740031\pi$
$620$	2.04620e6	0.213781
$621$	−5.73967e6	−0.597253
$622$	1.62842e6	0.168768
$623$	−94224.4	−0.00972620
$624$	−2.51769e6	−0.258846
$625$	−4.86615e6	−0.498294
$626$	2.14305e6	0.218573
$627$	−2.93261e6	−0.297910
$628$	−1.68548e7	−1.70539
$629$	−3.28880e6	−0.331445
$630$	272339.	0.0273375
$631$	2.89105e6	0.289056	0.144528	−	0.989501i	$-0.453834\pi$
$631$	2.89105e6	0.289056	0.144528	+	0.989501i	$0.453834\pi$
$632$	−3.02252e6	−0.301007
$633$	−6.50063e6	−0.644831
$634$	−176594.	−0.0174482
$635$	659304.	0.0648860
$636$	−5.38573e6	−0.527961
$637$	5.77270e6	0.563677
$638$	−219554.	−0.0213545
$639$	−5.36234e6	−0.519520
$640$	−4.74532e6	−0.457947
$641$	−1.31397e7	−1.26311	−0.631555	−	0.775331i	$-0.717583\pi$
$641$	−1.31397e7	−1.26311	−0.631555	+	0.775331i	$0.717583\pi$
$642$	−102899.	−0.00985307
$643$	−1.17266e7	−1.11852	−0.559259	−	0.828993i	$-0.688914\pi$
$643$	−1.17266e7	−1.11852	−0.559259	+	0.828993i	$0.688914\pi$
$644$	2.13734e6	0.203076
$645$	3.04529e6	0.288223
$646$	−332065.	−0.0313071
$647$	1.10662e7	1.03929	0.519646	−	0.854382i	$-0.326064\pi$
$647$	1.10662e7	1.03929	0.519646	+	0.854382i	$0.326064\pi$
$648$	1.36964e6	0.128135
$649$	−1.09743e7	−1.02274
$650$	−369929.	−0.0343427
$651$	−382635.	−0.0353860
$652$	−2.75025e6	−0.253369
$653$	−3.44597e6	−0.316249	−0.158124	−	0.987419i	$-0.550545\pi$
$653$	−3.44597e6	−0.316249	−0.158124	+	0.987419i	$0.550545\pi$
$654$	23555.9	0.00215355
$655$	1.39285e7	1.26853
$656$	−2.07495e6	−0.188255
$657$	−6.11392e6	−0.552594
$658$	−504860.	−0.0454576
$659$	−9.12181e6	−0.818215	−0.409108	−	0.912486i	$-0.634160\pi$
$659$	−9.12181e6	−0.818215	−0.409108	+	0.912486i	$0.634160\pi$
$660$	−2.97247e6	−0.265618
$661$	1.68113e7	1.49657	0.748287	−	0.663375i	$-0.230877\pi$
$661$	1.68113e7	1.49657	0.748287	+	0.663375i	$0.230877\pi$
$662$	−1.37145e6	−0.121629
$663$	−740723.	−0.0654443
$664$	−2.56147e6	−0.225460
$665$	2.24123e6	0.196531
$666$	−1.95481e6	−0.170773
$667$	−1.56519e6	−0.136224
$668$	−6.24760e6	−0.541716
$669$	3.43897e6	0.297073
$670$	−2.10525e6	−0.181183
$671$	6.74577e6	0.578396
$672$	668330.	0.0570910
$673$	8.53511e6	0.726393	0.363196	−	0.931713i	$-0.381685\pi$
$673$	8.53511e6	0.726393	0.363196	+	0.931713i	$0.381685\pi$
$674$	702670.	0.0595802
$675$	3.53708e6	0.298803
$676$	7.24506e6	0.609783
$677$	−912690.	−0.0765335	−0.0382667	−	0.999268i	$-0.512184\pi$
$677$	−912690.	−0.0765335	−0.0382667	+	0.999268i	$0.512184\pi$
$678$	−655665.	−0.0547782
$679$	−1.28387e6	−0.106868
$680$	−681191.	−0.0564932
$681$	−3.94481e6	−0.325955
$682$	384292.	0.0316374
$683$	−4.20567e6	−0.344972	−0.172486	−	0.985012i	$-0.555180\pi$
$683$	−4.20567e6	−0.344972	−0.172486	+	0.985012i	$0.555180\pi$
$684$	8.27055e6	0.675918
$685$	1.13107e7	0.921010
$686$	−1.02386e6	−0.0830677
$687$	3.11466e6	0.251779
$688$	−9.22366e6	−0.742903
$689$	−9.09751e6	−0.730087
$690$	505707.	0.0404367
$691$	6.23030e6	0.496379	0.248190	−	0.968711i	$-0.420164\pi$
$691$	6.23030e6	0.496379	0.248190	+	0.968711i	$0.420164\pi$
$692$	9.80746e6	0.778559
$693$	−2.14323e6	−0.169525
$694$	−445670.	−0.0351249
$695$	1.73870e7	1.36541
$696$	−325005.	−0.0254312
$697$	−610464.	−0.0475968
$698$	2.29005e6	0.177913
$699$	1.03412e7	0.800527
$700$	−1.31713e6	−0.101598
$701$	1.69453e7	1.30243	0.651215	−	0.758893i	$-0.274259\pi$
$701$	1.69453e7	1.30243	0.651215	+	0.758893i	$0.274259\pi$
$702$	−994733.	−0.0761840
$703$	−1.60872e7	−1.22770
$704$	8.54673e6	0.649933
$705$	5.00542e6	0.379287
$706$	−337961.	−0.0255185
$707$	−6.25484e6	−0.470617
$708$	−8.02680e6	−0.601810
$709$	−1.87326e7	−1.39953	−0.699766	−	0.714372i	$-0.746712\pi$
$709$	−1.87326e7	−1.39953	−0.699766	+	0.714372i	$0.746712\pi$
$710$	1.06746e6	0.0794702
$711$	1.06760e7	0.792016
$712$	140084.	0.0103559
$713$	2.73960e6	0.201820
$714$	62939.6	0.00462039
$715$	−5.02106e6	−0.367308
$716$	1.86946e6	0.136280
$717$	1.07026e7	0.777485
$718$	−2.62536e6	−0.190054
$719$	2.15611e6	0.155542	0.0777711	−	0.996971i	$-0.475220\pi$
$719$	2.15611e6	0.155542	0.0777711	+	0.996971i	$0.475220\pi$
$720$	8.17814e6	0.587927
$721$	−3.08519e6	−0.221026
$722$	514154.	0.0367071
$723$	−5.66410e6	−0.402982
$724$	−1.25714e7	−0.891326
$725$	964548.	0.0681520
$726$	425686.	0.0299742
$727$	4.87466e6	0.342065	0.171032	−	0.985265i	$-0.445290\pi$
$727$	4.87466e6	0.342065	0.171032	+	0.985265i	$0.445290\pi$
$728$	749677.	0.0524259
$729$	463697.	0.0323159
$730$	1.21707e6	0.0845295
$731$	−2.71367e6	−0.187829
$732$	4.93399e6	0.340346
$733$	5.15920e6	0.354669	0.177334	−	0.984151i	$-0.443253\pi$
$733$	5.15920e6	0.354669	0.177334	+	0.984151i	$0.443253\pi$
$734$	−888660.	−0.0608829
$735$	4.86314e6	0.332046
$736$	−4.78513e6	−0.325611
$737$	1.65677e7	1.12355
$738$	−362850.	−0.0245237
$739$	−4.60821e6	−0.310400	−0.155200	−	0.987883i	$-0.549602\pi$
$739$	−4.60821e6	−0.310400	−0.155200	+	0.987883i	$0.549602\pi$
$740$	−1.63059e7	−1.09462
$741$	−3.62325e6	−0.242411
$742$	773020.	0.0515444
$743$	2.50473e7	1.66452	0.832261	−	0.554384i	$-0.187046\pi$
$743$	2.50473e7	1.66452	0.832261	+	0.554384i	$0.187046\pi$
$744$	568866.	0.0376771
$745$	−3.21763e6	−0.212396
$746$	3.82037e6	0.251338
$747$	9.04751e6	0.593236
$748$	2.64878e6	0.173098
$749$	−618870.	−0.0403083
$750$	−1.16078e6	−0.0753524
$751$	−1.87279e7	−1.21168	−0.605841	−	0.795586i	$-0.707163\pi$
$751$	−1.87279e7	−1.21168	−0.605841	+	0.795586i	$0.707163\pi$
$752$	−1.51606e7	−0.977623
$753$	1.17447e7	0.754839
$754$	−271260.	−0.0173763
$755$	1.01146e7	0.645778
$756$	−3.54176e6	−0.225379
$757$	−5.50814e6	−0.349354	−0.174677	−	0.984626i	$-0.555888\pi$
$757$	−5.50814e6	−0.349354	−0.174677	+	0.984626i	$0.555888\pi$
$758$	3.65750e6	0.231212
$759$	−3.97976e6	−0.250757
$760$	−3.33205e6	−0.209256
$761$	−2.67826e7	−1.67645	−0.838226	−	0.545323i	$-0.816407\pi$
$761$	−2.67826e7	−1.67645	−0.838226	+	0.545323i	$0.816407\pi$
$762$	90566.4	0.00565040
$763$	141674.	0.00881005
$764$	2.82390e7	1.75032
$765$	2.40607e6	0.148646
$766$	772511.	0.0475699
$767$	−1.35588e7	−0.832209
$768$	5.74859e6	0.351689
$769$	1.76374e7	1.07552	0.537761	−	0.843097i	$-0.319270\pi$
$769$	1.76374e7	1.07552	0.537761	+	0.843097i	$0.319270\pi$
$770$	426642.	0.0259320
$771$	−7.12526e6	−0.431683
$772$	−3.21512e7	−1.94158
$773$	−6.59079e6	−0.396724	−0.198362	−	0.980129i	$-0.563562\pi$
$773$	−6.59079e6	−0.396724	−0.198362	+	0.980129i	$0.563562\pi$
$774$	−1.61296e6	−0.0967768
$775$	−1.68828e6	−0.100970
$776$	1.90874e6	0.113787
$777$	3.04917e6	0.181188
$778$	−3.11463e6	−0.184484
$779$	−2.98609e6	−0.176303
$780$	−3.67250e6	−0.216135
$781$	−8.40056e6	−0.492811
$782$	−450637.	−0.0263518
$783$	2.59366e6	0.151185
$784$	−1.47296e7	−0.855858
$785$	−2.39849e7	−1.38920
$786$	1.91331e6	0.110466
$787$	1.83730e7	1.05741	0.528704	−	0.848806i	$-0.322678\pi$
$787$	1.83730e7	1.05741	0.528704	+	0.848806i	$0.322678\pi$
$788$	2.70692e7	1.55296
$789$	7.16792e6	0.409922
$790$	−2.12522e6	−0.121153
$791$	−3.94341e6	−0.224094
$792$	3.18635e6	0.180502
$793$	8.33444e6	0.470645
$794$	1.69441e6	0.0953824
$795$	−7.66409e6	−0.430074
$796$	−8.93356e6	−0.499737
$797$	−3.06769e7	−1.71067	−0.855334	−	0.518078i	$-0.826648\pi$
$797$	−3.06769e7	−1.71067	−0.855334	+	0.518078i	$0.826648\pi$
$798$	307870.	0.0171143
$799$	−4.46035e6	−0.247174
$800$	2.94884e6	0.162902
$801$	−494798.	−0.0272488
$802$	622784.	0.0341902
$803$	−9.57797e6	−0.524185
$804$	1.21180e7	0.661134
$805$	3.04151e6	0.165424
$806$	474796.	0.0257436
$807$	−1.02941e7	−0.556423
$808$	9.29912e6	0.501088
$809$	−2.81720e6	−0.151338	−0.0756688	−	0.997133i	$-0.524109\pi$
$809$	−2.81720e6	−0.151338	−0.0756688	+	0.997133i	$0.524109\pi$
$810$	963031.	0.0515736
$811$	−1.57819e7	−0.842574	−0.421287	−	0.906927i	$-0.638421\pi$
$811$	−1.57819e7	−0.842574	−0.421287	+	0.906927i	$0.638421\pi$
$812$	−965825.	−0.0514053
$813$	−1.13997e7	−0.604877
$814$	−3.06238e6	−0.161994
$815$	−3.91371e6	−0.206393
$816$	1.89003e6	0.0993673
$817$	−1.32739e7	−0.695735
$818$	−1.15195e6	−0.0601935
$819$	−2.64797e6	−0.137944
$820$	−3.02668e6	−0.157192
$821$	−2.40563e6	−0.124558	−0.0622789	−	0.998059i	$-0.519837\pi$
$821$	−2.40563e6	−0.124558	−0.0622789	+	0.998059i	$0.519837\pi$
$822$	1.55372e6	0.0802034
$823$	4.47397e6	0.230247	0.115123	−	0.993351i	$-0.463274\pi$
$823$	4.47397e6	0.230247	0.115123	+	0.993351i	$0.463274\pi$
$824$	4.58678e6	0.235337
$825$	2.45253e6	0.125453
$826$	1.15210e6	0.0587542
$827$	2.36054e7	1.20018	0.600091	−	0.799932i	$-0.295131\pi$
$827$	2.36054e7	1.20018	0.600091	+	0.799932i	$0.295131\pi$
$828$	1.12237e7	0.568934
$829$	−7.52668e6	−0.380380	−0.190190	−	0.981747i	$-0.560910\pi$
$829$	−7.52668e6	−0.380380	−0.190190	+	0.981747i	$0.560910\pi$
$830$	−1.80105e6	−0.0907464
$831$	1.45981e7	0.733319
$832$	1.05595e7	0.528855
$833$	−4.33356e6	−0.216388
$834$	2.38839e6	0.118902
$835$	−8.89056e6	−0.441279
$836$	1.29565e7	0.641169
$837$	−4.53976e6	−0.223985
$838$	3.89754e6	0.191726
$839$	−7.28878e6	−0.357478	−0.178739	−	0.983896i	$-0.557202\pi$
$839$	−7.28878e6	−0.357478	−0.178739	+	0.983896i	$0.557202\pi$
$840$	631556.	0.0308826
$841$	707281.	0.0344828
$842$	−848249.	−0.0412328
$843$	153259.	0.00742773
$844$	2.87204e7	1.38782
$845$	1.03100e7	0.496725
$846$	−2.65116e6	−0.127353
$847$	2.56023e6	0.122623
$848$	2.32132e7	1.10853
$849$	−3.91005e6	−0.186172
$850$	277705.	0.0131837
$851$	−2.18315e7	−1.03338
$852$	−6.14434e6	−0.289985
$853$	2.18767e7	1.02946	0.514729	−	0.857353i	$-0.327893\pi$
$853$	2.18767e7	1.02946	0.514729	+	0.857353i	$0.327893\pi$
$854$	−708182.	−0.0332277
$855$	1.17693e7	0.550599
$856$	920079.	0.0429181
$857$	−1.62738e7	−0.756897	−0.378448	−	0.925622i	$-0.623542\pi$
$857$	−1.62738e7	−0.756897	−0.378448	+	0.925622i	$0.623542\pi$
$858$	−689726.	−0.0319859
$859$	1.50220e7	0.694615	0.347308	−	0.937751i	$-0.387096\pi$
$859$	1.50220e7	0.694615	0.347308	+	0.937751i	$0.387096\pi$
$860$	−1.34544e7	−0.620321
$861$	565983.	0.0260193
$862$	−4.59269e6	−0.210523
$863$	2.24278e7	1.02508	0.512542	−	0.858662i	$-0.328704\pi$
$863$	2.24278e7	1.02508	0.512542	+	0.858662i	$0.328704\pi$
$864$	7.92938e6	0.361372
$865$	1.39564e7	0.634209
$866$	−586650.	−0.0265818
$867$	−9.48819e6	−0.428682
$868$	1.69051e6	0.0761587
$869$	1.67248e7	0.751298
$870$	−228520.	−0.0102359
$871$	2.04695e7	0.914244
$872$	−210628.	−0.00938047
$873$	−6.74196e6	−0.299399
$874$	−2.20430e6	−0.0976093
$875$	−6.98136e6	−0.308262
$876$	−7.00552e6	−0.308447
$877$	1.25450e7	0.550771	0.275385	−	0.961334i	$-0.411195\pi$
$877$	1.25450e7	0.550771	0.275385	+	0.961334i	$0.411195\pi$
$878$	−3.05399e6	−0.133700
$879$	−1.08590e7	−0.474043
$880$	1.28117e7	0.557701
$881$	7.60341e6	0.330042	0.165021	−	0.986290i	$-0.447231\pi$
$881$	7.60341e6	0.330042	0.165021	+	0.986290i	$0.447231\pi$
$882$	−2.57580e6	−0.111491
$883$	2.15220e7	0.928923	0.464462	−	0.885593i	$-0.346248\pi$
$883$	2.15220e7	0.928923	0.464462	+	0.885593i	$0.346248\pi$
$884$	3.27258e6	0.140851
$885$	−1.14224e7	−0.490231
$886$	1.16848e6	0.0500079
$887$	−333279.	−0.0142232	−0.00711162	−	0.999975i	$-0.502264\pi$
$887$	−333279.	−0.0142232	−0.00711162	+	0.999975i	$0.502264\pi$
$888$	−4.53322e6	−0.192919
$889$	544700.	0.0231155
$890$	98497.2	0.00416820
$891$	−7.57876e6	−0.319819
$892$	−1.51937e7	−0.639368
$893$	−2.18178e7	−0.915552
$894$	−441995.	−0.0184958
$895$	2.66031e6	0.111013
$896$	−3.92046e6	−0.163142
$897$	−4.91702e6	−0.204043
$898$	1.76661e6	0.0731055
$899$	−1.23798e6	−0.0510873
$900$	−6.91664e6	−0.284635
$901$	6.82950e6	0.280270
$902$	−568435.	−0.0232629
$903$	2.51594e6	0.102679
$904$	5.86270e6	0.238603
$905$	−1.78895e7	−0.726068
$906$	1.38941e6	0.0562356
$907$	1.65687e7	0.668759	0.334380	−	0.942439i	$-0.391473\pi$
$907$	1.65687e7	0.668759	0.334380	+	0.942439i	$0.391473\pi$
$908$	1.74285e7	0.701529
$909$	−3.28459e7	−1.31847
$910$	527119.	0.0211011
$911$	1.30648e7	0.521561	0.260781	−	0.965398i	$-0.416020\pi$
$911$	1.30648e7	0.521561	0.260781	+	0.965398i	$0.416020\pi$
$912$	9.24510e6	0.368065
$913$	1.41737e7	0.562737
$914$	−7.32693e6	−0.290106
$915$	7.02125e6	0.277244
$916$	−1.37608e7	−0.541884
$917$	1.15074e7	0.451911
$918$	746746.	0.0292460
$919$	−7.30805e6	−0.285439	−0.142719	−	0.989763i	$-0.545585\pi$
$919$	−7.30805e6	−0.285439	−0.142719	+	0.989763i	$0.545585\pi$
$920$	−4.52184e6	−0.176135
$921$	−1.90393e7	−0.739608
$922$	1.78596e6	0.0691902
$923$	−1.03789e7	−0.401004
$924$	−2.45578e6	−0.0946256
$925$	1.34537e7	0.516996
$926$	318069.	0.0121897
$927$	−1.62012e7	−0.619224
$928$	2.16232e6	0.0824231
$929$	2.69866e7	1.02591	0.512955	−	0.858416i	$-0.328551\pi$
$929$	2.69866e7	1.02591	0.512955	+	0.858416i	$0.328551\pi$
$930$	399986.	0.0151648
$931$	−2.11976e7	−0.801518
$932$	−4.56881e7	−1.72291
$933$	−1.33385e7	−0.501652
$934$	2.40407e6	0.0901737
$935$	3.76930e6	0.141004
$936$	3.93676e6	0.146876
$937$	−2.15536e7	−0.801993	−0.400997	−	0.916080i	$-0.631336\pi$
$937$	−2.15536e7	−0.801993	−0.400997	+	0.916080i	$0.631336\pi$
$938$	−1.73930e6	−0.0645459
$939$	−1.75539e7	−0.649694
$940$	−2.21144e7	−0.816311
$941$	−1.22668e7	−0.451602	−0.225801	−	0.974173i	$-0.572500\pi$
$941$	−1.22668e7	−0.451602	−0.225801	+	0.974173i	$0.572500\pi$
$942$	−3.29473e6	−0.120974
$943$	−4.05234e6	−0.148398
$944$	3.45966e7	1.26358
$945$	−5.04005e6	−0.183593
$946$	−2.52684e6	−0.0918014
$947$	5.13280e7	1.85986	0.929928	−	0.367741i	$-0.119869\pi$
$947$	5.13280e7	1.85986	0.929928	+	0.367741i	$0.119869\pi$
$948$	1.22329e7	0.442087
$949$	−1.18336e7	−0.426533
$950$	1.35840e6	0.0488335
$951$	1.44649e6	0.0518638
$952$	−562782.	−0.0201255
$953$	−3.30301e7	−1.17809	−0.589044	−	0.808101i	$-0.700496\pi$
$953$	−3.30301e7	−1.17809	−0.589044	+	0.808101i	$0.700496\pi$
$954$	4.05934e6	0.144406
$955$	4.01852e7	1.42580
$956$	−4.72851e7	−1.67332
$957$	1.79838e6	0.0634750
$958$	−1.29390e6	−0.0455498
$959$	9.34463e6	0.328107
$960$	8.89576e6	0.311534
$961$	−2.64623e7	−0.924312
$962$	−3.78359e6	−0.131815
$963$	−3.24986e6	−0.112927
$964$	2.50245e7	0.867307
$965$	−4.57524e7	−1.58160
$966$	417802.	0.0144055
$967$	−7.77281e6	−0.267308	−0.133654	−	0.991028i	$-0.542671\pi$
$967$	−7.77281e6	−0.267308	−0.133654	+	0.991028i	$0.542671\pi$
$968$	−3.80632e6	−0.130562
$969$	2.71997e6	0.0930583
$970$	1.34209e6	0.0457986
$971$	2.31111e7	0.786635	0.393317	−	0.919403i	$-0.371327\pi$
$971$	2.31111e7	0.786635	0.393317	+	0.919403i	$0.371327\pi$
$972$	−2.89656e7	−0.983369
$973$	1.43647e7	0.486422
$974$	−6.87175e6	−0.232097
$975$	3.03012e6	0.102082
$976$	−2.12662e7	−0.714603
$977$	1.04792e7	0.351231	0.175616	−	0.984459i	$-0.443808\pi$
$977$	1.04792e7	0.351231	0.175616	+	0.984459i	$0.443808\pi$
$978$	−537613.	−0.0179731
$979$	−775143.	−0.0258479
$980$	−2.14858e7	−0.714638
$981$	743969.	0.0246821
$982$	1.61796e6	0.0535413
$983$	3.94189e7	1.30113	0.650565	−	0.759450i	$-0.274532\pi$
$983$	3.94189e7	1.30113	0.650565	+	0.759450i	$0.274532\pi$
$984$	−841452.	−0.0277039
$985$	3.85204e7	1.26503
$986$	203635.	0.00667052
$987$	4.13535e6	0.135120
$988$	1.60079e7	0.521724
$989$	−1.80137e7	−0.585615
$990$	2.24042e6	0.0726508
$991$	−4.53294e7	−1.46621	−0.733104	−	0.680117i	$-0.761929\pi$
$991$	−4.53294e7	−1.46621	−0.733104	+	0.680117i	$0.761929\pi$
$992$	−3.78477e6	−0.122113
$993$	1.12337e7	0.361534
$994$	881905.	0.0283110
$995$	−1.27128e7	−0.407083
$996$	1.03669e7	0.331132
$997$	4.74723e7	1.51253	0.756263	−	0.654268i	$-0.227023\pi$
$997$	4.74723e7	1.51253	0.756263	+	0.654268i	$0.227023\pi$
$998$	304247.	0.00966942
$999$	3.61768e7	1.14687

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	29.6.a.a.1.2	✓	4
3.2	odd	2		261.6.a.a.1.3		4
4.3	odd	2		464.6.a.i.1.1		4
5.4	even	2		725.6.a.a.1.3		4
29.28	even	2		841.6.a.a.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
29.6.a.a.1.2	✓	4	1.1	even	1	trivial
261.6.a.a.1.3		4	3.2	odd	2
464.6.a.i.1.1		4	4.3	odd	2
725.6.a.a.1.3		4	5.4	even	2
841.6.a.a.1.3		4	29.28	even	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:	\( N \)	\(=\)	\( 29 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	29.a (trivial)

\(f(q)\)	\(=\)	\(q-0.863638 q^{2} +7.07413 q^{3} -31.2541 q^{4} -44.4758 q^{5} -6.10949 q^{6} -36.7447 q^{7} +54.6287 q^{8} -192.957 q^{9} +38.4110 q^{10} -302.283 q^{11} -221.096 q^{12} -373.472 q^{13} +31.7341 q^{14} -314.627 q^{15} +952.953 q^{16} +280.365 q^{17} +166.645 q^{18} +1371.41 q^{19} +1390.05 q^{20} -259.937 q^{21} +261.063 q^{22} +1861.10 q^{23} +386.450 q^{24} -1146.91 q^{25} +322.545 q^{26} -3084.01 q^{27} +1148.42 q^{28} -841.000 q^{29} +271.724 q^{30} +1472.03 q^{31} -2571.12 q^{32} -2138.39 q^{33} -242.134 q^{34} +1634.25 q^{35} +6030.69 q^{36} -11730.4 q^{37} -1184.40 q^{38} -2641.99 q^{39} -2429.65 q^{40} -2177.39 q^{41} +224.491 q^{42} -9679.03 q^{43} +9447.58 q^{44} +8581.90 q^{45} -1607.32 q^{46} -15909.1 q^{47} +6741.31 q^{48} -15456.8 q^{49} +990.512 q^{50} +1983.34 q^{51} +11672.6 q^{52} +24359.3 q^{53} +2663.47 q^{54} +13444.3 q^{55} -2007.32 q^{56} +9701.53 q^{57} +726.320 q^{58} +36304.7 q^{59} +9833.40 q^{60} -22316.1 q^{61} -1271.30 q^{62} +7090.14 q^{63} -28274.0 q^{64} +16610.5 q^{65} +1846.79 q^{66} -54808.6 q^{67} -8762.58 q^{68} +13165.7 q^{69} -1411.40 q^{70} +27790.4 q^{71} -10541.0 q^{72} +31685.5 q^{73} +10130.8 q^{74} -8113.36 q^{75} -42862.2 q^{76} +11107.3 q^{77} +2281.73 q^{78} -55328.4 q^{79} -42383.3 q^{80} +25071.8 q^{81} +1880.47 q^{82} -46888.8 q^{83} +8124.10 q^{84} -12469.5 q^{85} +8359.18 q^{86} -5949.34 q^{87} -16513.3 q^{88} +2564.30 q^{89} -7411.65 q^{90} +13723.1 q^{91} -58167.2 q^{92} +10413.3 q^{93} +13739.7 q^{94} -60994.5 q^{95} -18188.5 q^{96} +34940.3 q^{97} +13349.1 q^{98} +58327.5 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 28 q^{3} + 10 q^{4} - 68 q^{5} - 194 q^{6} - 208 q^{7} - 504 q^{8} - 280 q^{9} - 788 q^{10} - 124 q^{11} + 20 q^{12} - 460 q^{13} + 768 q^{14} + 932 q^{15} - 414 q^{16} + 184 q^{17} + 3208 q^{18}+ \cdots + 166720 q^{99}+O(q^{100}) \) 4 * q - 28 * q^3 + 10 * q^4 - 68 * q^5 - 194 * q^6 - 208 * q^7 - 504 * q^8 - 280 * q^9 - 788 * q^10 - 124 * q^11 + 20 * q^12 - 460 * q^13 + 768 * q^14 + 932 * q^15 - 414 * q^16 + 184 * q^17 + 3208 * q^18 - 2392 * q^19 + 2822 * q^20 + 992 * q^21 + 5538 * q^22 - 1192 * q^23 + 6786 * q^24 + 1824 * q^25 + 4724 * q^26 + 2468 * q^27 + 44 * q^28 - 3364 * q^29 + 8186 * q^30 - 19212 * q^31 + 6552 * q^32 - 10580 * q^33 - 7612 * q^34 - 22944 * q^35 - 7468 * q^36 - 10928 * q^37 - 456 * q^38 - 8732 * q^39 - 20 * q^40 - 1120 * q^41 + 1844 * q^42 - 21420 * q^43 - 1932 * q^44 - 8344 * q^45 - 7588 * q^46 + 23772 * q^47 + 33060 * q^48 + 10452 * q^49 + 43240 * q^50 + 12744 * q^51 - 29062 * q^52 + 8860 * q^53 + 35410 * q^54 - 52652 * q^55 + 34304 * q^56 + 48944 * q^57 - 10840 * q^59 + 25200 * q^60 + 49448 * q^61 + 18518 * q^62 + 27488 * q^63 - 20734 * q^64 + 97836 * q^65 - 47744 * q^66 - 7840 * q^67 + 20724 * q^68 + 58792 * q^69 - 77496 * q^70 - 48744 * q^71 + 8088 * q^72 - 74992 * q^73 - 35920 * q^74 - 90448 * q^75 - 140792 * q^76 + 128656 * q^77 + 2982 * q^78 - 106076 * q^79 + 58638 * q^80 - 59692 * q^81 - 234132 * q^82 + 62888 * q^83 - 59832 * q^84 + 23848 * q^85 - 216014 * q^86 + 23548 * q^87 - 39426 * q^88 + 107568 * q^89 + 41552 * q^90 - 268896 * q^91 - 26268 * q^92 + 221460 * q^93 + 30542 * q^94 + 147352 * q^95 - 78606 * q^96 - 49520 * q^97 + 242304 * q^98 + 166720 * q^99

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