LMFDB - Embedded newform 825.6.a.v.1.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [825,6,Mod(1,825)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(825, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("825.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 825 = 3 \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	825.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:	$132.316651346$
Analytic rank:	$1$
Dimension:	$13$
Coefficient field:	$\mathbb{Q}[x]/(x^{13} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{13} - 306 x^{11} - 206 x^{10} + 34574 x^{9} + 39928 x^{8} - 1788312 x^{7} - 2591628 x^{6} + \cdots + 522579400 $ x^13 - 306x^11 - 206x^10 + 34574x^9 + 39928x^8 - 1788312x^7 - 2591628x^6 + 42852537x^5 + 63733360x^4 - 448113518x^3 - 549984598x^2 + 1518551280*x + 522579400
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{9}\cdot 3^{2}\cdot 5^{7} $
Twist minimal:	no (minimal twist has level 165)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.12
Root			$8.81495$ of defining polynomial
Character	$\chi$	$=$	825.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+7.81495 q^{2} -9.00000 q^{3} +29.0734 q^{4} -70.3345 q^{6} -11.0326 q^{7} -22.8710 q^{8} +81.0000 q^{9} +O(q^{10})$	\(q+7.81495 q^{2} -9.00000 q^{3} +29.0734 q^{4} -70.3345 q^{6} -11.0326 q^{7} -22.8710 q^{8} +81.0000 q^{9} +121.000 q^{11} -261.661 q^{12} -248.951 q^{13} -86.2194 q^{14} -1109.09 q^{16} +669.264 q^{17} +633.011 q^{18} +2327.66 q^{19} +99.2936 q^{21} +945.609 q^{22} -867.792 q^{23} +205.839 q^{24} -1945.54 q^{26} -729.000 q^{27} -320.756 q^{28} +2640.61 q^{29} -3381.17 q^{31} -7935.57 q^{32} -1089.00 q^{33} +5230.26 q^{34} +2354.95 q^{36} +434.439 q^{37} +18190.5 q^{38} +2240.56 q^{39} +6114.13 q^{41} +775.975 q^{42} -6682.47 q^{43} +3517.89 q^{44} -6781.75 q^{46} -20195.8 q^{47} +9981.77 q^{48} -16685.3 q^{49} -6023.37 q^{51} -7237.87 q^{52} -3335.16 q^{53} -5697.10 q^{54} +252.327 q^{56} -20948.9 q^{57} +20636.2 q^{58} -35671.0 q^{59} +16705.5 q^{61} -26423.7 q^{62} -893.642 q^{63} -26525.4 q^{64} -8510.48 q^{66} -15071.1 q^{67} +19457.8 q^{68} +7810.13 q^{69} -21017.5 q^{71} -1852.55 q^{72} -5360.83 q^{73} +3395.12 q^{74} +67673.1 q^{76} -1334.95 q^{77} +17509.9 q^{78} +11717.1 q^{79} +6561.00 q^{81} +47781.6 q^{82} -51987.1 q^{83} +2886.81 q^{84} -52223.2 q^{86} -23765.5 q^{87} -2767.39 q^{88} +89693.5 q^{89} +2746.58 q^{91} -25229.7 q^{92} +30430.5 q^{93} -157829. q^{94} +71420.2 q^{96} -19443.4 q^{97} -130395. q^{98} +9801.00 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 13 q - 13 q^{2} - 117 q^{3} + 209 q^{4} + 117 q^{6} - 304 q^{7} - 399 q^{8} + 1053 q^{9}+O(q^{10}) $ 13 * q - 13 * q^2 - 117 * q^3 + 209 * q^4 + 117 * q^6 - 304 * q^7 - 399 * q^8 + 1053 * q^9	\( 13 q - 13 q^{2} - 117 q^{3} + 209 q^{4} + 117 q^{6} - 304 q^{7} - 399 q^{8} + 1053 q^{9} + 1573 q^{11} - 1881 q^{12} - 986 q^{13} - 610 q^{14} + 3501 q^{16} - 1476 q^{17} - 1053 q^{18} + 270 q^{19} + 2736 q^{21} - 1573 q^{22} - 9084 q^{23} + 3591 q^{24} + 2652 q^{26} - 9477 q^{27} - 10920 q^{28} + 11952 q^{29} + 19096 q^{31} - 11661 q^{32} - 14157 q^{33} - 1302 q^{34} + 16929 q^{36} - 39964 q^{37} - 1574 q^{38} + 8874 q^{39} + 35184 q^{41} + 5490 q^{42} + 96 q^{43} + 25289 q^{44} - 4120 q^{46} - 34984 q^{47} - 31509 q^{48} + 14557 q^{49} + 13284 q^{51} - 39002 q^{52} - 22984 q^{53} + 9477 q^{54} + 59802 q^{56} - 2430 q^{57} - 18896 q^{58} - 9192 q^{59} + 5438 q^{61} - 272 q^{62} - 24624 q^{63} + 106557 q^{64} + 14157 q^{66} - 71508 q^{67} - 127948 q^{68} + 81756 q^{69} + 101700 q^{71} - 32319 q^{72} - 77390 q^{73} + 13676 q^{74} + 139966 q^{76} - 36784 q^{77} - 23868 q^{78} + 93954 q^{79} + 85293 q^{81} - 53284 q^{82} - 185918 q^{83} + 98280 q^{84} + 370930 q^{86} - 107568 q^{87} - 48279 q^{88} - 18418 q^{89} + 174536 q^{91} - 274264 q^{92} - 171864 q^{93} + 64520 q^{94} + 104949 q^{96} - 94312 q^{97} - 145677 q^{98} + 127413 q^{99}+O(q^{100}) \) 13 * q - 13 * q^2 - 117 * q^3 + 209 * q^4 + 117 * q^6 - 304 * q^7 - 399 * q^8 + 1053 * q^9 + 1573 * q^11 - 1881 * q^12 - 986 * q^13 - 610 * q^14 + 3501 * q^16 - 1476 * q^17 - 1053 * q^18 + 270 * q^19 + 2736 * q^21 - 1573 * q^22 - 9084 * q^23 + 3591 * q^24 + 2652 * q^26 - 9477 * q^27 - 10920 * q^28 + 11952 * q^29 + 19096 * q^31 - 11661 * q^32 - 14157 * q^33 - 1302 * q^34 + 16929 * q^36 - 39964 * q^37 - 1574 * q^38 + 8874 * q^39 + 35184 * q^41 + 5490 * q^42 + 96 * q^43 + 25289 * q^44 - 4120 * q^46 - 34984 * q^47 - 31509 * q^48 + 14557 * q^49 + 13284 * q^51 - 39002 * q^52 - 22984 * q^53 + 9477 * q^54 + 59802 * q^56 - 2430 * q^57 - 18896 * q^58 - 9192 * q^59 + 5438 * q^61 - 272 * q^62 - 24624 * q^63 + 106557 * q^64 + 14157 * q^66 - 71508 * q^67 - 127948 * q^68 + 81756 * q^69 + 101700 * q^71 - 32319 * q^72 - 77390 * q^73 + 13676 * q^74 + 139966 * q^76 - 36784 * q^77 - 23868 * q^78 + 93954 * q^79 + 85293 * q^81 - 53284 * q^82 - 185918 * q^83 + 98280 * q^84 + 370930 * q^86 - 107568 * q^87 - 48279 * q^88 - 18418 * q^89 + 174536 * q^91 - 274264 * q^92 - 171864 * q^93 + 64520 * q^94 + 104949 * q^96 - 94312 * q^97 - 145677 * q^98 + 127413 * 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 $

$2$	7.81495	1.38150	0.690750	−	0.723093i	$-0.257280\pi$
$2$	7.81495	1.38150	0.690750	+	0.723093i	$0.257280\pi$
$3$	−9.00000	−0.577350
$3$	−9.00000	−0.577350
$4$	29.0734	0.908545
$5$	0	0
$5$	0	0
$6$	−70.3345	−0.797610
$7$	−11.0326	−0.0851008	−0.0425504	−	0.999094i	$-0.513548\pi$
$7$	−11.0326	−0.0851008	−0.0425504	+	0.999094i	$0.513548\pi$
$8$	−22.8710	−0.126346
$9$	81.0000	0.333333
$10$	0	0
$11$	121.000	0.301511
$11$	121.000	0.301511
$12$	−261.661	−0.524549
$13$	−248.951	−0.408560	−0.204280	−	0.978913i	$-0.565485\pi$
$13$	−248.951	−0.408560	−0.204280	+	0.978913i	$0.565485\pi$
$14$	−86.2194	−0.117567
$15$	0	0
$16$	−1109.09	−1.08309
$17$	669.264	0.561662	0.280831	−	0.959757i	$-0.409390\pi$
$17$	669.264	0.561662	0.280831	+	0.959757i	$0.409390\pi$
$18$	633.011	0.460500
$19$	2327.66	1.47923	0.739615	−	0.673030i	$-0.235008\pi$
$19$	2327.66	1.47923	0.739615	+	0.673030i	$0.235008\pi$
$20$	0	0
$21$	99.2936	0.0491330
$22$	945.609	0.416538
$23$	−867.792	−0.342055	−0.171028	−	0.985266i	$-0.554709\pi$
$23$	−867.792	−0.342055	−0.171028	+	0.985266i	$0.554709\pi$
$24$	205.839	0.0729456
$25$	0	0
$26$	−1945.54	−0.564426
$27$	−729.000	−0.192450
$28$	−320.756	−0.0773179
$29$	2640.61	0.583055	0.291527	−	0.956562i	$-0.405837\pi$
$29$	2640.61	0.583055	0.291527	+	0.956562i	$0.405837\pi$
$30$	0	0
$31$	−3381.17	−0.631921	−0.315961	−	0.948772i	$-0.602327\pi$
$31$	−3381.17	−0.631921	−0.315961	+	0.948772i	$0.602327\pi$
$32$	−7935.57	−1.36995
$33$	−1089.00	−0.174078
$34$	5230.26	0.775937
$35$	0	0
$36$	2354.95	0.302848
$37$	434.439	0.0521705	0.0260852	−	0.999660i	$-0.491696\pi$
$37$	434.439	0.0521705	0.0260852	+	0.999660i	$0.491696\pi$
$38$	18190.5	2.04356
$39$	2240.56	0.235882
$40$	0	0
$41$	6114.13	0.568035	0.284018	−	0.958819i	$-0.408333\pi$
$41$	6114.13	0.568035	0.284018	+	0.958819i	$0.408333\pi$
$42$	775.975	0.0678772
$43$	−6682.47	−0.551145	−0.275573	−	0.961280i	$-0.588867\pi$
$43$	−6682.47	−0.551145	−0.275573	+	0.961280i	$0.588867\pi$
$44$	3517.89	0.273937
$45$	0	0
$46$	−6781.75	−0.472550
$47$	−20195.8	−1.33357	−0.666787	−	0.745249i	$-0.732331\pi$
$47$	−20195.8	−1.33357	−0.666787	+	0.745249i	$0.732331\pi$
$48$	9981.77	0.625323
$49$	−16685.3	−0.992758
$50$	0	0
$51$	−6023.37	−0.324276
$52$	−7237.87	−0.371195
$53$	−3335.16	−0.163090	−0.0815450	−	0.996670i	$-0.525985\pi$
$53$	−3335.16	−0.163090	−0.0815450	+	0.996670i	$0.525985\pi$
$54$	−5697.10	−0.265870
$55$	0	0
$56$	252.327	0.0107521
$57$	−20948.9	−0.854034
$58$	20636.2	0.805491
$59$	−35671.0	−1.33409	−0.667045	−	0.745017i	$-0.732441\pi$
$59$	−35671.0	−1.33409	−0.667045	+	0.745017i	$0.732441\pi$
$60$	0	0
$61$	16705.5	0.574825	0.287412	−	0.957807i	$-0.407205\pi$
$61$	16705.5	0.574825	0.287412	+	0.957807i	$0.407205\pi$
$62$	−26423.7	−0.873000
$63$	−893.642	−0.0283669
$64$	−26525.4	−0.809490
$65$	0	0
$66$	−8510.48	−0.240488
$67$	−15071.1	−0.410164	−0.205082	−	0.978745i	$-0.565746\pi$
$67$	−15071.1	−0.410164	−0.205082	+	0.978745i	$0.565746\pi$
$68$	19457.8	0.510295
$69$	7810.13	0.197486
$70$	0	0
$71$	−21017.5	−0.494806	−0.247403	−	0.968913i	$-0.579577\pi$
$71$	−21017.5	−0.494806	−0.247403	+	0.968913i	$0.579577\pi$
$72$	−1852.55	−0.0421152
$73$	−5360.83	−0.117740	−0.0588701	−	0.998266i	$-0.518750\pi$
$73$	−5360.83	−0.117740	−0.0588701	+	0.998266i	$0.518750\pi$
$74$	3395.12	0.0720735
$75$	0	0
$76$	67673.1	1.34395
$77$	−1334.95	−0.0256589
$78$	17509.9	0.325871
$79$	11717.1	0.211228	0.105614	−	0.994407i	$-0.466319\pi$
$79$	11717.1	0.211228	0.105614	+	0.994407i	$0.466319\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	47781.6	0.784741
$83$	−51987.1	−0.828325	−0.414162	−	0.910203i	$-0.635925\pi$
$83$	−51987.1	−0.828325	−0.414162	+	0.910203i	$0.635925\pi$
$84$	2886.81	0.0446395
$85$	0	0
$86$	−52223.2	−0.761408
$87$	−23765.5	−0.336627
$88$	−2767.39	−0.0380946
$89$	89693.5	1.20029	0.600145	−	0.799892i	$-0.295110\pi$
$89$	89693.5	1.20029	0.600145	+	0.799892i	$0.295110\pi$
$90$	0	0
$91$	2746.58	0.0347688
$92$	−25229.7	−0.310772
$93$	30430.5	0.364840
$94$	−157829.	−1.84233
$95$	0	0
$96$	71420.2	0.790939
$97$	−19443.4	−0.209819	−0.104909	−	0.994482i	$-0.533455\pi$
$97$	−19443.4	−0.209819	−0.104909	+	0.994482i	$0.533455\pi$
$98$	−130395.	−1.37150
$99$	9801.00	0.100504
$100$	0	0
$101$	−144534.	−1.40983	−0.704917	−	0.709290i	$-0.749016\pi$
$101$	−144534.	−1.40983	−0.704917	+	0.709290i	$0.749016\pi$
$102$	−47072.4	−0.447987
$103$	−156184.	−1.45059	−0.725293	−	0.688441i	$-0.758296\pi$
$103$	−156184.	−1.45059	−0.725293	+	0.688441i	$0.758296\pi$
$104$	5693.76	0.0516197
$105$	0	0
$106$	−26064.1	−0.225309
$107$	53993.7	0.455915	0.227957	−	0.973671i	$-0.426795\pi$
$107$	53993.7	0.455915	0.227957	+	0.973671i	$0.426795\pi$
$108$	−21194.5	−0.174850
$109$	−119467.	−0.963120	−0.481560	−	0.876413i	$-0.659930\pi$
$109$	−119467.	−0.963120	−0.481560	+	0.876413i	$0.659930\pi$
$110$	0	0
$111$	−3909.95	−0.0301206
$112$	12236.1	0.0921719
$113$	−143349.	−1.05609	−0.528043	−	0.849217i	$-0.677074\pi$
$113$	−143349.	−1.05609	−0.528043	+	0.849217i	$0.677074\pi$
$114$	−163715.	−1.17985
$115$	0	0
$116$	76771.6	0.529731
$117$	−20165.0	−0.136187
$118$	−278767.	−1.84305
$119$	−7383.74	−0.0477979
$120$	0	0
$121$	14641.0	0.0909091
$122$	130553.	0.794121
$123$	−55027.2	−0.327955
$124$	−98302.3	−0.574129
$125$	0	0
$126$	−6983.77	−0.0391889
$127$	160600.	0.883562	0.441781	−	0.897123i	$-0.354347\pi$
$127$	160600.	0.883562	0.441781	+	0.897123i	$0.354347\pi$
$128$	46643.9	0.251634
$129$	60142.2	0.318204
$130$	0	0
$131$	−6734.56	−0.0342871	−0.0171436	−	0.999853i	$-0.505457\pi$
$131$	−6734.56	−0.0342871	−0.0171436	+	0.999853i	$0.505457\pi$
$132$	−31661.0	−0.158157
$133$	−25680.2	−0.125884
$134$	−117780.	−0.566642
$135$	0	0
$136$	−15306.7	−0.0709635
$137$	117299.	0.533942	0.266971	−	0.963705i	$-0.413977\pi$
$137$	117299.	0.533942	0.266971	+	0.963705i	$0.413977\pi$
$138$	61035.8	0.272827
$139$	−364129.	−1.59852	−0.799261	−	0.600985i	$-0.794775\pi$
$139$	−364129.	−1.59852	−0.799261	+	0.600985i	$0.794775\pi$
$140$	0	0
$141$	181762.	0.769939
$142$	−164251.	−0.683575
$143$	−30123.1	−0.123185
$144$	−89835.9	−0.361030
$145$	0	0
$146$	−41894.6	−0.162658
$147$	150168.	0.573169
$148$	12630.6	0.0473992
$149$	−296786.	−1.09516	−0.547580	−	0.836753i	$-0.684451\pi$
$149$	−296786.	−1.09516	−0.547580	+	0.836753i	$0.684451\pi$
$150$	0	0
$151$	−464852.	−1.65910	−0.829549	−	0.558433i	$-0.811403\pi$
$151$	−464852.	−1.65910	−0.829549	+	0.558433i	$0.811403\pi$
$152$	−53235.9	−0.186894
$153$	54210.4	0.187221
$154$	−10432.5	−0.0354477
$155$	0	0
$156$	65140.8	0.214310
$157$	188905.	0.611639	0.305820	−	0.952089i	$-0.401070\pi$
$157$	188905.	0.611639	0.305820	+	0.952089i	$0.401070\pi$
$158$	91568.4	0.291812
$159$	30016.5	0.0941601
$160$	0	0
$161$	9574.02	0.0291092
$162$	51273.9	0.153500
$163$	547363.	1.61364	0.806820	−	0.590797i	$-0.201187\pi$
$163$	547363.	1.61364	0.806820	+	0.590797i	$0.201187\pi$
$164$	177759.	0.516085
$165$	0	0
$166$	−406277.	−1.14433
$167$	−146681.	−0.406990	−0.203495	−	0.979076i	$-0.565230\pi$
$167$	−146681.	−0.406990	−0.203495	+	0.979076i	$0.565230\pi$
$168$	−2270.94	−0.00620773
$169$	−309316.	−0.833079
$170$	0	0
$171$	188540.	0.493076
$172$	−194282.	−0.500740
$173$	−625665.	−1.58937	−0.794687	−	0.607019i	$-0.792365\pi$
$173$	−625665.	−1.58937	−0.794687	+	0.607019i	$0.792365\pi$
$174$	−185726.	−0.465050
$175$	0	0
$176$	−134199.	−0.326564
$177$	321039.	0.770238
$178$	700950.	1.65820
$179$	233383.	0.544422	0.272211	−	0.962238i	$-0.412245\pi$
$179$	233383.	0.544422	0.272211	+	0.962238i	$0.412245\pi$
$180$	0	0
$181$	503730.	1.14288	0.571441	−	0.820643i	$-0.306385\pi$
$181$	503730.	1.14288	0.571441	+	0.820643i	$0.306385\pi$
$182$	21464.4	0.0480331
$183$	−150350.	−0.331875
$184$	19847.3	0.0432171
$185$	0	0
$186$	237813.	0.504027
$187$	80980.9	0.169347
$188$	−587162.	−1.21161
$189$	8042.78	0.0163777
$190$	0	0
$191$	953992.	1.89217	0.946087	−	0.323912i	$-0.104998\pi$
$191$	953992.	1.89217	0.946087	+	0.323912i	$0.104998\pi$
$192$	238728.	0.467359
$193$	−862260.	−1.66627	−0.833135	−	0.553070i	$-0.813456\pi$
$193$	−862260.	−1.66627	−0.833135	+	0.553070i	$0.813456\pi$
$194$	−151950.	−0.289865
$195$	0	0
$196$	−485098.	−0.901965
$197$	−497679.	−0.913658	−0.456829	−	0.889554i	$-0.651015\pi$
$197$	−497679.	−0.913658	−0.456829	+	0.889554i	$0.651015\pi$
$198$	76594.3	0.138846
$199$	−387099.	−0.692930	−0.346465	−	0.938063i	$-0.612618\pi$
$199$	−387099.	−0.692930	−0.346465	+	0.938063i	$0.612618\pi$
$200$	0	0
$201$	135640.	0.236808
$202$	−1.12953e6	−1.94769
$203$	−29132.9	−0.0496184
$204$	−175120.	−0.294619
$205$	0	0
$206$	−1.22057e6	−2.00399
$207$	−70291.2	−0.114018
$208$	276108.	0.442508
$209$	281647.	0.446004
$210$	0	0
$211$	−717688.	−1.10976	−0.554880	−	0.831930i	$-0.687236\pi$
$211$	−717688.	−1.10976	−0.554880	+	0.831930i	$0.687236\pi$
$212$	−96964.6	−0.148175
$213$	189158.	0.285677
$214$	421958.	0.629846
$215$	0	0
$216$	16672.9	0.0243152
$217$	37303.2	0.0537770
$218$	−933626.	−1.33055
$219$	48247.5	0.0679773
$220$	0	0
$221$	−166614.	−0.229473
$222$	−30556.1	−0.0416117
$223$	113180.	0.152409	0.0762043	−	0.997092i	$-0.475720\pi$
$223$	113180.	0.152409	0.0762043	+	0.997092i	$0.475720\pi$
$224$	87550.2	0.116584
$225$	0	0
$226$	−1.12027e6	−1.45898
$227$	−616694.	−0.794337	−0.397169	−	0.917746i	$-0.630007\pi$
$227$	−616694.	−0.794337	−0.397169	+	0.917746i	$0.630007\pi$
$228$	−609058.	−0.775928
$229$	502825.	0.633620	0.316810	−	0.948489i	$-0.397388\pi$
$229$	502825.	0.633620	0.316810	+	0.948489i	$0.397388\pi$
$230$	0	0
$231$	12014.5	0.0148141
$232$	−60393.4	−0.0736664
$233$	−141935.	−0.171277	−0.0856383	−	0.996326i	$-0.527293\pi$
$233$	−141935.	−0.171277	−0.0856383	+	0.996326i	$0.527293\pi$
$234$	−157589.	−0.188142
$235$	0	0
$236$	−1.03708e6	−1.21208
$237$	−105454.	−0.121953
$238$	−57703.5	−0.0660328
$239$	733743.	0.830901	0.415450	−	0.909616i	$-0.363624\pi$
$239$	733743.	0.830901	0.415450	+	0.909616i	$0.363624\pi$
$240$	0	0
$241$	−327132.	−0.362811	−0.181406	−	0.983408i	$-0.558065\pi$
$241$	−327132.	−0.362811	−0.181406	+	0.983408i	$0.558065\pi$
$242$	114419.	0.125591
$243$	−59049.0	−0.0641500
$244$	485687.	0.522254
$245$	0	0
$246$	−430035.	−0.453071
$247$	−579474.	−0.604354
$248$	77330.7	0.0798404
$249$	467884.	0.478233
$250$	0	0
$251$	36093.3	0.0361612	0.0180806	−	0.999837i	$-0.494244\pi$
$251$	36093.3	0.0361612	0.0180806	+	0.999837i	$0.494244\pi$
$252$	−25981.3	−0.0257726
$253$	−105003.	−0.103134
$254$	1.25508e6	1.22064
$255$	0	0
$256$	1.21333e6	1.15712
$257$	−1.70289e6	−1.60825	−0.804126	−	0.594459i	$-0.797366\pi$
$257$	−1.70289e6	−1.60825	−0.804126	+	0.594459i	$0.797366\pi$
$258$	470009.	0.439599
$259$	−4793.00	−0.00443975
$260$	0	0
$261$	213889.	0.194352
$262$	−52630.3	−0.0473677
$263$	−411013.	−0.366409	−0.183205	−	0.983075i	$-0.558647\pi$
$263$	−411013.	−0.366409	−0.183205	+	0.983075i	$0.558647\pi$
$264$	24906.5	0.0219939
$265$	0	0
$266$	−200689.	−0.173908
$267$	−807241.	−0.692987
$268$	−438169.	−0.372653
$269$	−100472.	−0.0846571	−0.0423286	−	0.999104i	$-0.513478\pi$
$269$	−100472.	−0.0846571	−0.0423286	+	0.999104i	$0.513478\pi$
$270$	0	0
$271$	1.74029e6	1.43945	0.719727	−	0.694257i	$-0.244267\pi$
$271$	1.74029e6	1.43945	0.719727	+	0.694257i	$0.244267\pi$
$272$	−742271.	−0.608331
$273$	−24719.3	−0.0200738
$274$	916689.	0.737642
$275$	0	0
$276$	227067.	0.179425
$277$	1.99970e6	1.56590	0.782951	−	0.622083i	$-0.213714\pi$
$277$	1.99970e6	1.56590	0.782951	+	0.622083i	$0.213714\pi$
$278$	−2.84565e6	−2.20836
$279$	−273875.	−0.210640
$280$	0	0
$281$	1.76621e6	1.33437	0.667184	−	0.744893i	$-0.267499\pi$
$281$	1.76621e6	1.33437	0.667184	+	0.744893i	$0.267499\pi$
$282$	1.42046e6	1.06367
$283$	−2.31041e6	−1.71484	−0.857419	−	0.514619i	$-0.827933\pi$
$283$	−2.31041e6	−1.71484	−0.857419	+	0.514619i	$0.827933\pi$
$284$	−611051.	−0.449554
$285$	0	0
$286$	−235410.	−0.170181
$287$	−67454.9	−0.0483403
$288$	−642782.	−0.456649
$289$	−971943.	−0.684536
$290$	0	0
$291$	174991.	0.121139
$292$	−155858.	−0.106972
$293$	−480796.	−0.327183	−0.163592	−	0.986528i	$-0.552308\pi$
$293$	−480796.	−0.327183	−0.163592	+	0.986528i	$0.552308\pi$
$294$	1.17355e6	0.791834
$295$	0	0
$296$	−9936.05	−0.00659150
$297$	−88209.0	−0.0580259
$298$	−2.31937e6	−1.51296
$299$	216038.	0.139750
$300$	0	0
$301$	73725.2	0.0469029
$302$	−3.63279e6	−2.29205
$303$	1.30081e6	0.813968
$304$	−2.58157e6	−1.60214
$305$	0	0
$306$	423651.	0.258646
$307$	1.27658e6	0.773043	0.386521	−	0.922280i	$-0.373677\pi$
$307$	1.27658e6	0.773043	0.386521	+	0.922280i	$0.373677\pi$
$308$	−38811.5	−0.0233122
$309$	1.40566e6	0.837496
$310$	0	0
$311$	2.50885e6	1.47087	0.735434	−	0.677596i	$-0.236978\pi$
$311$	2.50885e6	1.47087	0.735434	+	0.677596i	$0.236978\pi$
$312$	−51243.8	−0.0298027
$313$	−1.85827e6	−1.07213	−0.536065	−	0.844177i	$-0.680090\pi$
$313$	−1.85827e6	−1.07213	−0.536065	+	0.844177i	$0.680090\pi$
$314$	1.47629e6	0.844980
$315$	0	0
$316$	340656.	0.191910
$317$	−2.48818e6	−1.39070	−0.695352	−	0.718670i	$-0.744751\pi$
$317$	−2.48818e6	−1.39070	−0.695352	+	0.718670i	$0.744751\pi$
$318$	234577.	0.130082
$319$	319514.	0.175798
$320$	0	0
$321$	−485943.	−0.263222
$322$	74820.5	0.0402143
$323$	1.55782e6	0.830827
$324$	190751.	0.100949
$325$	0	0
$326$	4.27762e6	2.22924
$327$	1.07520e6	0.556058
$328$	−139836.	−0.0717687
$329$	222813.	0.113488
$330$	0	0
$331$	2.19276e6	1.10007	0.550037	−	0.835141i	$-0.314614\pi$
$331$	2.19276e6	1.10007	0.550037	+	0.835141i	$0.314614\pi$
$332$	−1.51144e6	−0.752570
$333$	35189.6	0.0173902
$334$	−1.14631e6	−0.562257
$335$	0	0
$336$	−110125.	−0.0532155
$337$	1.08814e6	0.521926	0.260963	−	0.965349i	$-0.415960\pi$
$337$	1.08814e6	0.521926	0.260963	+	0.965349i	$0.415960\pi$
$338$	−2.41729e6	−1.15090
$339$	1.29014e6	0.609732
$340$	0	0
$341$	−409122.	−0.190531
$342$	1.47343e6	0.681186
$343$	369508.	0.169585
$344$	152835.	0.0696347
$345$	0	0
$346$	−4.88954e6	−2.19572
$347$	−1.48961e6	−0.664123	−0.332061	−	0.943258i	$-0.607744\pi$
$347$	−1.48961e6	−0.664123	−0.332061	+	0.943258i	$0.607744\pi$
$348$	−690944.	−0.305841
$349$	3.61825e6	1.59014	0.795070	−	0.606518i	$-0.207434\pi$
$349$	3.61825e6	1.59014	0.795070	+	0.606518i	$0.207434\pi$
$350$	0	0
$351$	181485.	0.0786274
$352$	−960205.	−0.413054
$353$	346438.	0.147975	0.0739875	−	0.997259i	$-0.476428\pi$
$353$	346438.	0.147975	0.0739875	+	0.997259i	$0.476428\pi$
$354$	2.50890e6	1.06408
$355$	0	0
$356$	2.60770e6	1.09052
$357$	66453.6	0.0275961
$358$	1.82387e6	0.752120
$359$	−1.75327e6	−0.717981	−0.358991	−	0.933341i	$-0.616879\pi$
$359$	−1.75327e6	−0.717981	−0.358991	+	0.933341i	$0.616879\pi$
$360$	0	0
$361$	2.94190e6	1.18812
$362$	3.93663e6	1.57889
$363$	−131769.	−0.0524864
$364$	79852.6	0.0315890
$365$	0	0
$366$	−1.17498e6	−0.458486
$367$	2.68865e6	1.04200	0.521001	−	0.853556i	$-0.325559\pi$
$367$	2.68865e6	1.04200	0.521001	+	0.853556i	$0.325559\pi$
$368$	962456.	0.370477
$369$	495245.	0.189345
$370$	0	0
$371$	36795.6	0.0138791
$372$	884720.	0.331473
$373$	−2.73768e6	−1.01885	−0.509426	−	0.860515i	$-0.670142\pi$
$373$	−2.73768e6	−1.01885	−0.509426	+	0.860515i	$0.670142\pi$
$374$	632862.	0.233954
$375$	0	0
$376$	461898.	0.168491
$377$	−657383.	−0.238213
$378$	62853.9	0.0226257
$379$	1.99354e6	0.712898	0.356449	−	0.934315i	$-0.383987\pi$
$379$	1.99354e6	0.712898	0.356449	+	0.934315i	$0.383987\pi$
$380$	0	0
$381$	−1.44540e6	−0.510125
$382$	7.45540e6	2.61404
$383$	3.32117e6	1.15690	0.578448	−	0.815720i	$-0.303659\pi$
$383$	3.32117e6	1.15690	0.578448	+	0.815720i	$0.303659\pi$
$384$	−419795.	−0.145281
$385$	0	0
$386$	−6.73852e6	−2.30195
$387$	−541280.	−0.183715
$388$	−565288.	−0.190630
$389$	640065.	0.214462	0.107231	−	0.994234i	$-0.465802\pi$
$389$	640065.	0.214462	0.107231	+	0.994234i	$0.465802\pi$
$390$	0	0
$391$	−580782.	−0.192119
$392$	381609.	0.125431
$393$	60611.1	0.0197957
$394$	−3.88934e6	−1.26222
$395$	0	0
$396$	284949.	0.0913122
$397$	806457.	0.256806	0.128403	−	0.991722i	$-0.459015\pi$
$397$	806457.	0.256806	0.128403	+	0.991722i	$0.459015\pi$
$398$	−3.02516e6	−0.957284
$399$	231122.	0.0726789
$400$	0	0
$401$	2.17447e6	0.675292	0.337646	−	0.941273i	$-0.390369\pi$
$401$	2.17447e6	0.675292	0.337646	+	0.941273i	$0.390369\pi$
$402$	1.06002e6	0.327151
$403$	841747.	0.258178
$404$	−4.20211e6	−1.28090
$405$	0	0
$406$	−227672.	−0.0685479
$407$	52567.1	0.0157300
$408$	137760.	0.0409708
$409$	4.98556e6	1.47369	0.736844	−	0.676063i	$-0.236315\pi$
$409$	4.98556e6	1.47369	0.736844	+	0.676063i	$0.236315\pi$
$410$	0	0
$411$	−1.05569e6	−0.308272
$412$	−4.54080e6	−1.31792
$413$	393545.	0.113532
$414$	−549322.	−0.157517
$415$	0	0
$416$	1.97557e6	0.559705
$417$	3.27716e6	0.922907
$418$	2.20106e6	0.616156
$419$	5.22690e6	1.45448	0.727242	−	0.686382i	$-0.240802\pi$
$419$	5.22690e6	1.45448	0.727242	+	0.686382i	$0.240802\pi$
$420$	0	0
$421$	1.48521e6	0.408398	0.204199	−	0.978929i	$-0.434541\pi$
$421$	1.48521e6	0.408398	0.204199	+	0.978929i	$0.434541\pi$
$422$	−5.60869e6	−1.53314
$423$	−1.63586e6	−0.444524
$424$	76278.5	0.0206057
$425$	0	0
$426$	1.47826e6	0.394662
$427$	−184306.	−0.0489181
$428$	1.56978e6	0.414219
$429$	271108.	0.0711212
$430$	0	0
$431$	−301670.	−0.0782238	−0.0391119	−	0.999235i	$-0.512453\pi$
$431$	−301670.	−0.0782238	−0.0391119	+	0.999235i	$0.512453\pi$
$432$	808523.	0.208441
$433$	1.78448e6	0.457396	0.228698	−	0.973497i	$-0.426553\pi$
$433$	1.78448e6	0.457396	0.228698	+	0.973497i	$0.426553\pi$
$434$	291523.	0.0742930
$435$	0	0
$436$	−3.47331e6	−0.875038
$437$	−2.01993e6	−0.505978
$438$	377051.	0.0939108
$439$	−499339.	−0.123661	−0.0618307	−	0.998087i	$-0.519694\pi$
$439$	−499339.	−0.123661	−0.0618307	+	0.998087i	$0.519694\pi$
$440$	0	0
$441$	−1.35151e6	−0.330919
$442$	−1.30208e6	−0.317017
$443$	−4.62775e6	−1.12037	−0.560184	−	0.828369i	$-0.689269\pi$
$443$	−4.62775e6	−1.12037	−0.560184	+	0.828369i	$0.689269\pi$
$444$	−113676.	−0.0273659
$445$	0	0
$446$	884500.	0.210553
$447$	2.67107e6	0.632291
$448$	292645.	0.0688883
$449$	3.31281e6	0.775499	0.387749	−	0.921765i	$-0.373253\pi$
$449$	3.31281e6	0.775499	0.387749	+	0.921765i	$0.373253\pi$
$450$	0	0
$451$	739810.	0.171269
$452$	−4.16766e6	−0.959502
$453$	4.18367e6	0.957881
$454$	−4.81943e6	−1.09738
$455$	0	0
$456$	479123.	0.107903
$457$	7.09134e6	1.58832	0.794160	−	0.607709i	$-0.207911\pi$
$457$	7.09134e6	1.58832	0.794160	+	0.607709i	$0.207911\pi$
$458$	3.92956e6	0.875346
$459$	−487893.	−0.108092
$460$	0	0
$461$	−3.68457e6	−0.807485	−0.403742	−	0.914873i	$-0.632291\pi$
$461$	−3.68457e6	−0.807485	−0.403742	+	0.914873i	$0.632291\pi$
$462$	93892.9	0.0204658
$463$	5.32110e6	1.15358	0.576791	−	0.816892i	$-0.304305\pi$
$463$	5.32110e6	1.15358	0.576791	+	0.816892i	$0.304305\pi$
$464$	−2.92866e6	−0.631501
$465$	0	0
$466$	−1.10921e6	−0.236619
$467$	−9.27946e6	−1.96893	−0.984465	−	0.175581i	$-0.943820\pi$
$467$	−9.27946e6	−1.96893	−0.984465	+	0.175581i	$0.943820\pi$
$468$	−586267.	−0.123732
$469$	166274.	0.0349053
$470$	0	0
$471$	−1.70015e6	−0.353130
$472$	815831.	0.168556
$473$	−808579.	−0.166177
$474$	−824116.	−0.168478
$475$	0	0
$476$	−214671.	−0.0434265
$477$	−270148.	−0.0543633
$478$	5.73416e6	1.14789
$479$	−4.22915e6	−0.842199	−0.421100	−	0.907014i	$-0.638356\pi$
$479$	−4.22915e6	−0.842199	−0.421100	+	0.907014i	$0.638356\pi$
$480$	0	0
$481$	−108154.	−0.0213148
$482$	−2.55652e6	−0.501224
$483$	−86166.2	−0.0168062
$484$	425664.	0.0825950
$485$	0	0
$486$	−461465.	−0.0886233
$487$	−8.16043e6	−1.55916	−0.779580	−	0.626302i	$-0.784568\pi$
$487$	−8.16043e6	−1.55916	−0.779580	+	0.626302i	$0.784568\pi$
$488$	−382072.	−0.0726265
$489$	−4.92627e6	−0.931635
$490$	0	0
$491$	−1.72620e6	−0.323138	−0.161569	−	0.986861i	$-0.551655\pi$
$491$	−1.72620e6	−0.323138	−0.161569	+	0.986861i	$0.551655\pi$
$492$	−1.59983e6	−0.297962
$493$	1.76727e6	0.327480
$494$	−4.52856e6	−0.834916
$495$	0	0
$496$	3.75001e6	0.684428
$497$	231878.	0.0421084
$498$	3.65649e6	0.660680
$499$	568851.	0.102270	0.0511349	−	0.998692i	$-0.483716\pi$
$499$	568851.	0.102270	0.0511349	+	0.998692i	$0.483716\pi$
$500$	0	0
$501$	1.32013e6	0.234976
$502$	282068.	0.0499567
$503$	−2.06495e6	−0.363906	−0.181953	−	0.983307i	$-0.558242\pi$
$503$	−2.06495e6	−0.363906	−0.181953	+	0.983307i	$0.558242\pi$
$504$	20438.5	0.00358404
$505$	0	0
$506$	−820592.	−0.142479
$507$	2.78385e6	0.480978
$508$	4.66920e6	0.802755
$509$	−4.03961e6	−0.691107	−0.345554	−	0.938399i	$-0.612309\pi$
$509$	−4.03961e6	−0.691107	−0.345554	+	0.938399i	$0.612309\pi$
$510$	0	0
$511$	59144.0	0.0100198
$512$	7.98952e6	1.34693
$513$	−1.69686e6	−0.284678
$514$	−1.33080e7	−2.22180
$515$	0	0
$516$	1.74854e6	0.289102
$517$	−2.44369e6	−0.402087
$518$	−37457.1	−0.00613352
$519$	5.63098e6	0.917626
$520$	0	0
$521$	2.55223e6	0.411932	0.205966	−	0.978559i	$-0.433966\pi$
$521$	2.55223e6	0.411932	0.205966	+	0.978559i	$0.433966\pi$
$522$	1.67154e6	0.268497
$523$	1.54874e6	0.247586	0.123793	−	0.992308i	$-0.460494\pi$
$523$	1.54874e6	0.247586	0.123793	+	0.992308i	$0.460494\pi$
$524$	−195797.	−0.0311514
$525$	0	0
$526$	−3.21205e6	−0.506195
$527$	−2.26290e6	−0.354926
$528$	1.20779e6	0.188542
$529$	−5.68328e6	−0.882998
$530$	0	0
$531$	−2.88935e6	−0.444697
$532$	−746611.	−0.114371
$533$	−1.52212e6	−0.232076
$534$	−6.30855e6	−0.957362
$535$	0	0
$536$	344691.	0.0518224
$537$	−2.10044e6	−0.314322
$538$	−785182.	−0.116954
$539$	−2.01892e6	−0.299328
$540$	0	0
$541$	171030.	0.0251235	0.0125617	−	0.999921i	$-0.496001\pi$
$541$	171030.	0.0251235	0.0125617	+	0.999921i	$0.496001\pi$
$542$	1.36003e7	1.98861
$543$	−4.53357e6	−0.659843
$544$	−5.31099e6	−0.769447
$545$	0	0
$546$	−193180.	−0.0277319
$547$	−1.12874e7	−1.61297	−0.806487	−	0.591252i	$-0.798634\pi$
$547$	−1.12874e7	−1.61297	−0.806487	+	0.591252i	$0.798634\pi$
$548$	3.41029e6	0.485110
$549$	1.35315e6	0.191608
$550$	0	0
$551$	6.14644e6	0.862472
$552$	−178625.	−0.0249514
$553$	−129270.	−0.0179757
$554$	1.56275e7	2.16329
$555$	0	0
$556$	−1.05865e7	−1.45233
$557$	164004.	0.0223984	0.0111992	−	0.999937i	$-0.496435\pi$
$557$	164004.	0.0223984	0.0111992	+	0.999937i	$0.496435\pi$
$558$	−2.14032e6	−0.291000
$559$	1.66361e6	0.225176
$560$	0	0
$561$	−728828.	−0.0977728
$562$	1.38028e7	1.84343
$563$	9.26696e6	1.23216	0.616079	−	0.787685i	$-0.288720\pi$
$563$	9.26696e6	1.23216	0.616079	+	0.787685i	$0.288720\pi$
$564$	5.28446e6	0.699524
$565$	0	0
$566$	−1.80557e7	−2.36905
$567$	−72385.0	−0.00945565
$568$	480691.	0.0625166
$569$	−8.70558e6	−1.12724	−0.563621	−	0.826034i	$-0.690592\pi$
$569$	−8.70558e6	−1.12724	−0.563621	+	0.826034i	$0.690592\pi$
$570$	0	0
$571$	−3.36751e6	−0.432233	−0.216117	−	0.976368i	$-0.569339\pi$
$571$	−3.36751e6	−0.432233	−0.216117	+	0.976368i	$0.569339\pi$
$572$	−875782.	−0.111920
$573$	−8.58593e6	−1.09245
$574$	−527157.	−0.0667821
$575$	0	0
$576$	−2.14856e6	−0.269830
$577$	702817.	0.0878826	0.0439413	−	0.999034i	$-0.486009\pi$
$577$	702817.	0.0878826	0.0439413	+	0.999034i	$0.486009\pi$
$578$	−7.59568e6	−0.945687
$579$	7.76034e6	0.962021
$580$	0	0
$581$	573554.	0.0704911
$582$	1.36755e6	0.167353
$583$	−403555.	−0.0491735
$584$	122607.	0.0148759
$585$	0	0
$586$	−3.75739e6	−0.452004
$587$	3.09026e6	0.370168	0.185084	−	0.982723i	$-0.440744\pi$
$587$	3.09026e6	0.370168	0.185084	+	0.982723i	$0.440744\pi$
$588$	4.36589e6	0.520750
$589$	−7.87022e6	−0.934757
$590$	0	0
$591$	4.47911e6	0.527501
$592$	−481830.	−0.0565054
$593$	1.16268e7	1.35777	0.678883	−	0.734246i	$-0.262464\pi$
$593$	1.16268e7	1.35777	0.678883	+	0.734246i	$0.262464\pi$
$594$	−689349.	−0.0801628
$595$	0	0
$596$	−8.62859e6	−0.995002
$597$	3.48389e6	0.400063
$598$	1.68833e6	0.193065
$599$	−2.09163e6	−0.238187	−0.119093	−	0.992883i	$-0.537999\pi$
$599$	−2.09163e6	−0.238187	−0.119093	+	0.992883i	$0.537999\pi$
$600$	0	0
$601$	2.26126e6	0.255367	0.127683	−	0.991815i	$-0.459246\pi$
$601$	2.26126e6	0.255367	0.127683	+	0.991815i	$0.459246\pi$
$602$	576159.	0.0647964
$603$	−1.22076e6	−0.136721
$604$	−1.35148e7	−1.50737
$605$	0	0
$606$	1.01658e7	1.12450
$607$	−1.26470e7	−1.39321	−0.696603	−	0.717457i	$-0.745306\pi$
$607$	−1.26470e7	−1.39321	−0.696603	+	0.717457i	$0.745306\pi$
$608$	−1.84713e7	−2.02646
$609$	262196.	0.0286472
$610$	0	0
$611$	5.02777e6	0.544845
$612$	1.57608e6	0.170098
$613$	−9.20580e6	−0.989488	−0.494744	−	0.869039i	$-0.664738\pi$
$613$	−9.20580e6	−0.989488	−0.494744	+	0.869039i	$0.664738\pi$
$614$	9.97644e6	1.06796
$615$	0	0
$616$	30531.6	0.00324188
$617$	−1.90063e6	−0.200995	−0.100497	−	0.994937i	$-0.532043\pi$
$617$	−1.90063e6	−0.200995	−0.100497	+	0.994937i	$0.532043\pi$
$618$	1.09851e7	1.15700
$619$	−3.94913e6	−0.414261	−0.207131	−	0.978313i	$-0.566413\pi$
$619$	−3.94913e6	−0.414261	−0.207131	+	0.978313i	$0.566413\pi$
$620$	0	0
$621$	632620.	0.0658285
$622$	1.96065e7	2.03201
$623$	−989554.	−0.102146
$624$	−2.48497e6	−0.255482
$625$	0	0
$626$	−1.45223e7	−1.48115
$627$	−2.53482e6	−0.257501
$628$	5.49213e6	0.555702
$629$	290754.	0.0293022
$630$	0	0
$631$	−1.50561e7	−1.50536	−0.752680	−	0.658387i	$-0.771239\pi$
$631$	−1.50561e7	−1.50536	−0.752680	+	0.658387i	$0.771239\pi$
$632$	−267981.	−0.0266877
$633$	6.45919e6	0.640721
$634$	−1.94450e7	−1.92126
$635$	0	0
$636$	872682.	0.0855486
$637$	4.15382e6	0.405601
$638$	2.49698e6	0.242865
$639$	−1.70242e6	−0.164935
$640$	0	0
$641$	1.51936e7	1.46055	0.730276	−	0.683153i	$-0.239392\pi$
$641$	1.51936e7	1.46055	0.730276	+	0.683153i	$0.239392\pi$
$642$	−3.79762e6	−0.363642
$643$	1.61149e6	0.153709	0.0768547	−	0.997042i	$-0.475512\pi$
$643$	1.61149e6	0.153709	0.0768547	+	0.997042i	$0.475512\pi$
$644$	278350.	0.0264470
$645$	0	0
$646$	1.21743e7	1.14779
$647$	−2.05398e7	−1.92902	−0.964508	−	0.264055i	$-0.914940\pi$
$647$	−2.05398e7	−1.92902	−0.964508	+	0.264055i	$0.914940\pi$
$648$	−150057.	−0.0140384
$649$	−4.31619e6	−0.402244
$650$	0	0
$651$	−335729.	−0.0310482
$652$	1.59137e7	1.46606
$653$	−7.96056e6	−0.730568	−0.365284	−	0.930896i	$-0.619028\pi$
$653$	−7.96056e6	−0.730568	−0.365284	+	0.930896i	$0.619028\pi$
$654$	8.40263e6	0.768194
$655$	0	0
$656$	−6.78110e6	−0.615234
$657$	−434227.	−0.0392467
$658$	1.74127e6	0.156784
$659$	2.91266e6	0.261262	0.130631	−	0.991431i	$-0.458300\pi$
$659$	2.91266e6	0.261262	0.130631	+	0.991431i	$0.458300\pi$
$660$	0	0
$661$	1.89585e7	1.68772	0.843860	−	0.536564i	$-0.180278\pi$
$661$	1.89585e7	1.68772	0.843860	+	0.536564i	$0.180278\pi$
$662$	1.71363e7	1.51975
$663$	1.49953e6	0.132486
$664$	1.18900e6	0.104655
$665$	0	0
$666$	275005.	0.0240245
$667$	−2.29150e6	−0.199437
$668$	−4.26453e6	−0.369768
$669$	−1.01862e6	−0.0879931
$670$	0	0
$671$	2.02137e6	0.173316
$672$	−787952.	−0.0673095
$673$	2.14478e7	1.82535	0.912674	−	0.408688i	$-0.134014\pi$
$673$	2.14478e7	1.82535	0.912674	+	0.408688i	$0.134014\pi$
$674$	8.50374e6	0.721041
$675$	0	0
$676$	−8.99289e6	−0.756889
$677$	1.84040e7	1.54326	0.771631	−	0.636071i	$-0.219441\pi$
$677$	1.84040e7	1.54326	0.771631	+	0.636071i	$0.219441\pi$
$678$	1.00824e7	0.842345
$679$	214512.	0.0178557
$680$	0	0
$681$	5.55024e6	0.458611
$682$	−3.19727e6	−0.263219
$683$	2.04252e7	1.67538	0.837692	−	0.546143i	$-0.183905\pi$
$683$	2.04252e7	1.67538	0.837692	+	0.546143i	$0.183905\pi$
$684$	5.48152e6	0.447982
$685$	0	0
$686$	2.88768e6	0.234282
$687$	−4.52543e6	−0.365820
$688$	7.41143e6	0.596940
$689$	830293.	0.0666321
$690$	0	0
$691$	1.16275e6	0.0926388	0.0463194	−	0.998927i	$-0.485251\pi$
$691$	1.16275e6	0.0926388	0.0463194	+	0.998927i	$0.485251\pi$
$692$	−1.81902e7	−1.44402
$693$	−108131.	−0.00855295
$694$	−1.16412e7	−0.917486
$695$	0	0
$696$	543540.	0.0425313
$697$	4.09197e6	0.319044
$698$	2.82765e7	2.19678
$699$	1.27741e6	0.0988866
$700$	0	0
$701$	1.80087e7	1.38416	0.692082	−	0.721818i	$-0.256693\pi$
$701$	1.80087e7	1.38416	0.692082	+	0.721818i	$0.256693\pi$
$702$	1.41830e6	0.108624
$703$	1.01123e6	0.0771721
$704$	−3.20957e6	−0.244071
$705$	0	0
$706$	2.70739e6	0.204428
$707$	1.59459e6	0.119978
$708$	9.33371e6	0.699795
$709$	−1.64523e7	−1.22917	−0.614583	−	0.788852i	$-0.710676\pi$
$709$	−1.64523e7	−1.22917	−0.614583	+	0.788852i	$0.710676\pi$
$710$	0	0
$711$	949084.	0.0704094
$712$	−2.05138e6	−0.151651
$713$	2.93415e6	0.216152
$714$	519332.	0.0381241
$715$	0	0
$716$	6.78523e6	0.494632
$717$	−6.60369e6	−0.479721
$718$	−1.37017e7	−0.991892
$719$	2.09659e7	1.51249	0.756243	−	0.654290i	$-0.227033\pi$
$719$	2.09659e7	1.51249	0.756243	+	0.654290i	$0.227033\pi$
$720$	0	0
$721$	1.72312e6	0.123446
$722$	2.29908e7	1.64139
$723$	2.94419e6	0.209469
$724$	1.46452e7	1.03836
$725$	0	0
$726$	−1.02977e6	−0.0725100
$727$	1.23702e7	0.868040	0.434020	−	0.900903i	$-0.357095\pi$
$727$	1.23702e7	0.868040	0.434020	+	0.900903i	$0.357095\pi$
$728$	−62817.1	−0.00439288
$729$	531441.	0.0370370
$730$	0	0
$731$	−4.47234e6	−0.309557
$732$	−4.37118e6	−0.301524
$733$	−1.78981e7	−1.23040	−0.615200	−	0.788371i	$-0.710925\pi$
$733$	−1.78981e7	−1.23040	−0.615200	+	0.788371i	$0.710925\pi$
$734$	2.10116e7	1.43953
$735$	0	0
$736$	6.88643e6	0.468597
$737$	−1.82360e6	−0.123669
$738$	3.87031e6	0.261580
$739$	−2.08324e6	−0.140323	−0.0701615	−	0.997536i	$-0.522351\pi$
$739$	−2.08324e6	−0.140323	−0.0701615	+	0.997536i	$0.522351\pi$
$740$	0	0
$741$	5.21526e6	0.348924
$742$	287556.	0.0191740
$743$	7.09623e6	0.471580	0.235790	−	0.971804i	$-0.424232\pi$
$743$	7.09623e6	0.471580	0.235790	+	0.971804i	$0.424232\pi$
$744$	−695977.	−0.0460959
$745$	0	0
$746$	−2.13948e7	−1.40754
$747$	−4.21096e6	−0.276108
$748$	2.35439e6	0.153860
$749$	−595692.	−0.0387987
$750$	0	0
$751$	−1.82817e7	−1.18282	−0.591408	−	0.806373i	$-0.701428\pi$
$751$	−1.82817e7	−1.18282	−0.591408	+	0.806373i	$0.701428\pi$
$752$	2.23989e7	1.44438
$753$	−324840.	−0.0208777
$754$	−5.13742e6	−0.329091
$755$	0	0
$756$	233831.	0.0148798
$757$	1.02358e6	0.0649206	0.0324603	−	0.999473i	$-0.489666\pi$
$757$	1.02358e6	0.0649206	0.0324603	+	0.999473i	$0.489666\pi$
$758$	1.55794e7	0.984869
$759$	945026.	0.0595442
$760$	0	0
$761$	9.52820e6	0.596416	0.298208	−	0.954501i	$-0.403611\pi$
$761$	9.52820e6	0.596416	0.298208	+	0.954501i	$0.403611\pi$
$762$	−1.12957e7	−0.704738
$763$	1.31803e6	0.0819623
$764$	2.77358e7	1.71913
$765$	0	0
$766$	2.59548e7	1.59825
$767$	8.88034e6	0.545056
$768$	−1.09200e7	−0.668065
$769$	−1.07522e6	−0.0655666	−0.0327833	−	0.999462i	$-0.510437\pi$
$769$	−1.07522e6	−0.0655666	−0.0327833	+	0.999462i	$0.510437\pi$
$770$	0	0
$771$	1.53260e7	0.928524
$772$	−2.50689e7	−1.51388
$773$	−6.01805e6	−0.362249	−0.181124	−	0.983460i	$-0.557974\pi$
$773$	−6.01805e6	−0.362249	−0.181124	+	0.983460i	$0.557974\pi$
$774$	−4.23008e6	−0.253803
$775$	0	0
$776$	444691.	0.0265096
$777$	43137.0	0.00256329
$778$	5.00207e6	0.296279
$779$	1.42316e7	0.840254
$780$	0	0
$781$	−2.54312e6	−0.149190
$782$	−4.53878e6	−0.265413
$783$	−1.92500e6	−0.112209
$784$	1.85054e7	1.07525
$785$	0	0
$786$	473673.	0.0273478
$787$	−2.82761e7	−1.62736	−0.813679	−	0.581314i	$-0.802539\pi$
$787$	−2.82761e7	−1.62736	−0.813679	+	0.581314i	$0.802539\pi$
$788$	−1.44692e7	−0.830100
$789$	3.69912e6	0.211547
$790$	0	0
$791$	1.58152e6	0.0898738
$792$	−224158.	−0.0126982
$793$	−4.15886e6	−0.234850
$794$	6.30242e6	0.354777
$795$	0	0
$796$	−1.12543e7	−0.629558
$797$	−2.70959e7	−1.51098	−0.755490	−	0.655161i	$-0.772601\pi$
$797$	−2.70959e7	−1.51098	−0.755490	+	0.655161i	$0.772601\pi$
$798$	1.80620e6	0.100406
$799$	−1.35163e7	−0.749017
$800$	0	0
$801$	7.26517e6	0.400096
$802$	1.69933e7	0.932917
$803$	−648660.	−0.0355000
$804$	3.94352e6	0.215151
$805$	0	0
$806$	6.57821e6	0.356673
$807$	904246.	0.0488768
$808$	3.30564e6	0.178126
$809$	1.78310e7	0.957868	0.478934	−	0.877851i	$-0.341023\pi$
$809$	1.78310e7	0.957868	0.478934	+	0.877851i	$0.341023\pi$
$810$	0	0
$811$	−2.28534e7	−1.22011	−0.610055	−	0.792359i	$-0.708853\pi$
$811$	−2.28534e7	−1.22011	−0.610055	+	0.792359i	$0.708853\pi$
$812$	−846992.	−0.0450806
$813$	−1.56626e7	−0.831070
$814$	410810.	0.0217310
$815$	0	0
$816$	6.68044e6	0.351220
$817$	−1.55545e7	−0.815270
$818$	3.89619e7	2.03590
$819$	222473.	0.0115896
$820$	0	0
$821$	−3.84556e6	−0.199114	−0.0995570	−	0.995032i	$-0.531743\pi$
$821$	−3.84556e6	−0.199114	−0.0995570	+	0.995032i	$0.531743\pi$
$822$	−8.25020e6	−0.425878
$823$	−2.40452e7	−1.23746	−0.618728	−	0.785605i	$-0.712352\pi$
$823$	−2.40452e7	−1.23746	−0.618728	+	0.785605i	$0.712352\pi$
$824$	3.57208e6	0.183275
$825$	0	0
$826$	3.07553e6	0.156845
$827$	3.67928e7	1.87068	0.935339	−	0.353754i	$-0.115095\pi$
$827$	3.67928e7	1.87068	0.935339	+	0.353754i	$0.115095\pi$
$828$	−2.04361e6	−0.103591
$829$	3.38058e7	1.70846	0.854231	−	0.519894i	$-0.174029\pi$
$829$	3.38058e7	1.70846	0.854231	+	0.519894i	$0.174029\pi$
$830$	0	0
$831$	−1.79973e7	−0.904074
$832$	6.60352e6	0.330725
$833$	−1.11669e7	−0.557594
$834$	2.56109e7	1.27500
$835$	0	0
$836$	8.18844e6	0.405215
$837$	2.46487e6	0.121613
$838$	4.08479e7	2.00937
$839$	−2.59170e7	−1.27110	−0.635550	−	0.772060i	$-0.719227\pi$
$839$	−2.59170e7	−1.27110	−0.635550	+	0.772060i	$0.719227\pi$
$840$	0	0
$841$	−1.35383e7	−0.660047
$842$	1.16069e7	0.564202
$843$	−1.58959e7	−0.770398
$844$	−2.08656e7	−1.00827
$845$	0	0
$846$	−1.27842e7	−0.614111
$847$	−161529.	−0.00773644
$848$	3.69898e6	0.176641
$849$	2.07937e7	0.990062
$850$	0	0
$851$	−377003.	−0.0178452
$852$	5.49946e6	0.259550
$853$	−1.25062e7	−0.588510	−0.294255	−	0.955727i	$-0.595071\pi$
$853$	−1.25062e7	−0.588510	−0.294255	+	0.955727i	$0.595071\pi$
$854$	−1.44034e6	−0.0675803
$855$	0	0
$856$	−1.23489e6	−0.0576028
$857$	3.10464e7	1.44398	0.721988	−	0.691906i	$-0.243229\pi$
$857$	3.10464e7	1.44398	0.721988	+	0.691906i	$0.243229\pi$
$858$	2.11869e6	0.0982539
$859$	3.06277e7	1.41622	0.708111	−	0.706102i	$-0.249548\pi$
$859$	3.06277e7	1.41622	0.708111	+	0.706102i	$0.249548\pi$
$860$	0	0
$861$	607094.	0.0279093
$862$	−2.35754e6	−0.108066
$863$	5.27975e6	0.241316	0.120658	−	0.992694i	$-0.461500\pi$
$863$	5.27975e6	0.241316	0.120658	+	0.992694i	$0.461500\pi$
$864$	5.78503e6	0.263646
$865$	0	0
$866$	1.39456e7	0.631893
$867$	8.74749e6	0.395217
$868$	1.08453e6	0.0488588
$869$	1.41777e6	0.0636877
$870$	0	0
$871$	3.75197e6	0.167577
$872$	2.73232e6	0.121686
$873$	−1.57492e6	−0.0699395
$874$	−1.57856e7	−0.699009
$875$	0	0
$876$	1.40272e6	0.0617605
$877$	−2.47729e7	−1.08762	−0.543811	−	0.839207i	$-0.683019\pi$
$877$	−2.47729e7	−1.08762	−0.543811	+	0.839207i	$0.683019\pi$
$878$	−3.90231e6	−0.170838
$879$	4.32716e6	0.188899
$880$	0	0
$881$	3.14613e7	1.36564	0.682822	−	0.730585i	$-0.260753\pi$
$881$	3.14613e7	1.36564	0.682822	+	0.730585i	$0.260753\pi$
$882$	−1.05620e7	−0.457165
$883$	5.34805e6	0.230831	0.115415	−	0.993317i	$-0.463180\pi$
$883$	5.34805e6	0.230831	0.115415	+	0.993317i	$0.463180\pi$
$884$	−4.84404e6	−0.208486
$885$	0	0
$886$	−3.61656e7	−1.54779
$887$	5.37540e6	0.229404	0.114702	−	0.993400i	$-0.463409\pi$
$887$	5.37540e6	0.229404	0.114702	+	0.993400i	$0.463409\pi$
$888$	89424.5	0.00380561
$889$	−1.77184e6	−0.0751918
$890$	0	0
$891$	793881.	0.0335013
$892$	3.29054e6	0.138470
$893$	−4.70090e7	−1.97266
$894$	2.08743e7	0.873511
$895$	0	0
$896$	−514604.	−0.0214143
$897$	−1.94434e6	−0.0806847
$898$	2.58895e7	1.07135
$899$	−8.92836e6	−0.368445
$900$	0	0
$901$	−2.23210e6	−0.0916015
$902$	5.78158e6	0.236608
$903$	−663527.	−0.0270794
$904$	3.27854e6	0.133432
$905$	0	0
$906$	3.26951e7	1.32331
$907$	−1.34668e7	−0.543558	−0.271779	−	0.962360i	$-0.587612\pi$
$907$	−1.34668e7	−0.543558	−0.271779	+	0.962360i	$0.587612\pi$
$908$	−1.79294e7	−0.721691
$909$	−1.17073e7	−0.469944
$910$	0	0
$911$	5.76380e6	0.230098	0.115049	−	0.993360i	$-0.463297\pi$
$911$	5.76380e6	0.230098	0.115049	+	0.993360i	$0.463297\pi$
$912$	2.32342e7	0.924996
$913$	−6.29044e6	−0.249749
$914$	5.54185e7	2.19427
$915$	0	0
$916$	1.46189e7	0.575672
$917$	74299.9	0.00291786
$918$	−3.81286e6	−0.149329
$919$	−9.91711e6	−0.387344	−0.193672	−	0.981066i	$-0.562040\pi$
$919$	−9.91711e6	−0.387344	−0.193672	+	0.981066i	$0.562040\pi$
$920$	0	0
$921$	−1.14893e7	−0.446316
$922$	−2.87947e7	−1.11554
$923$	5.23233e6	0.202158
$924$	349304.	0.0134593
$925$	0	0
$926$	4.15841e7	1.59368
$927$	−1.26509e7	−0.483529
$928$	−2.09548e7	−0.798753
$929$	−2.50662e7	−0.952905	−0.476453	−	0.879200i	$-0.658078\pi$
$929$	−2.50662e7	−0.952905	−0.476453	+	0.879200i	$0.658078\pi$
$930$	0	0
$931$	−3.88377e7	−1.46852
$932$	−4.12652e6	−0.155613
$933$	−2.25796e7	−0.849206
$934$	−7.25185e7	−2.72008
$935$	0	0
$936$	461194.	0.0172066
$937$	−782497.	−0.0291161	−0.0145581	−	0.999894i	$-0.504634\pi$
$937$	−782497.	−0.0291161	−0.0145581	+	0.999894i	$0.504634\pi$
$938$	1.29942e6	0.0482217
$939$	1.67244e7	0.618995
$940$	0	0
$941$	1.08811e7	0.400589	0.200295	−	0.979736i	$-0.435810\pi$
$941$	1.08811e7	0.400589	0.200295	+	0.979736i	$0.435810\pi$
$942$	−1.32866e7	−0.487850
$943$	−5.30580e6	−0.194299
$944$	3.95622e7	1.44494
$945$	0	0
$946$	−6.31900e6	−0.229573
$947$	−3.52427e7	−1.27701	−0.638505	−	0.769617i	$-0.720447\pi$
$947$	−3.52427e7	−1.27701	−0.638505	+	0.769617i	$0.720447\pi$
$948$	−3.06590e6	−0.110799
$949$	1.33458e6	0.0481039
$950$	0	0
$951$	2.23937e7	0.802923
$952$	168873.	0.00603905
$953$	6.73891e6	0.240358	0.120179	−	0.992752i	$-0.461653\pi$
$953$	6.73891e6	0.240358	0.120179	+	0.992752i	$0.461653\pi$
$954$	−2.11119e6	−0.0751030
$955$	0	0
$956$	2.13324e7	0.754911
$957$	−2.87562e6	−0.101497
$958$	−3.30506e7	−1.16350
$959$	−1.29412e6	−0.0454389
$960$	0	0
$961$	−1.71968e7	−0.600675
$962$	−845219.	−0.0294464
$963$	4.37349e6	0.151972
$964$	−9.51086e6	−0.329630
$965$	0	0
$966$	−673385.	−0.0232178
$967$	8.67300e6	0.298266	0.149133	−	0.988817i	$-0.452352\pi$
$967$	8.67300e6	0.298266	0.149133	+	0.988817i	$0.452352\pi$
$968$	−334854.	−0.0114860
$969$	−1.40204e7	−0.479678
$970$	0	0
$971$	1.57795e7	0.537089	0.268545	−	0.963267i	$-0.413457\pi$
$971$	1.57795e7	0.537089	0.268545	+	0.963267i	$0.413457\pi$
$972$	−1.71676e6	−0.0582832
$973$	4.01730e6	0.136035
$974$	−6.37734e7	−2.15398
$975$	0	0
$976$	−1.85279e7	−0.622588
$977$	−1.20786e6	−0.0404836	−0.0202418	−	0.999795i	$-0.506444\pi$
$977$	−1.20786e6	−0.0404836	−0.0202418	+	0.999795i	$0.506444\pi$
$978$	−3.84985e7	−1.28706
$979$	1.08529e7	0.361901
$980$	0	0
$981$	−9.67680e6	−0.321040
$982$	−1.34902e7	−0.446416
$983$	7.42794e6	0.245180	0.122590	−	0.992457i	$-0.460880\pi$
$983$	7.42794e6	0.245180	0.122590	+	0.992457i	$0.460880\pi$
$984$	1.25853e6	0.0414357
$985$	0	0
$986$	1.38111e7	0.452413
$987$	−2.00532e6	−0.0655224
$988$	−1.68473e7	−0.549083
$989$	5.79900e6	0.188522
$990$	0	0
$991$	−4.26703e7	−1.38020	−0.690099	−	0.723715i	$-0.742433\pi$
$991$	−4.26703e7	−1.38020	−0.690099	+	0.723715i	$0.742433\pi$
$992$	2.68315e7	0.865698
$993$	−1.97349e7	−0.635128
$994$	1.81212e6	0.0581728
$995$	0	0
$996$	1.36030e7	0.434496
$997$	−2.10754e7	−0.671486	−0.335743	−	0.941954i	$-0.608987\pi$
$997$	−2.10754e7	−0.671486	−0.335743	+	0.941954i	$0.608987\pi$
$998$	4.44554e6	0.141286
$999$	−316706.	−0.0100402

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	825.6.a.v.1.12		13
5.2	odd	4		165.6.c.b.34.22	yes	26
5.3	odd	4		165.6.c.b.34.5	✓	26
5.4	even	2		825.6.a.y.1.2		13

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
165.6.c.b.34.5	✓	26	5.3	odd	4
165.6.c.b.34.22	yes	26	5.2	odd	4
825.6.a.v.1.12		13	1.1	even	1	trivial
825.6.a.y.1.2		13	5.4	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:	\( N \)	\(=\)	\( 825 = 3 \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	825.a (trivial)

\(f(q)\)	\(=\)	\(q+7.81495 q^{2} -9.00000 q^{3} +29.0734 q^{4} -70.3345 q^{6} -11.0326 q^{7} -22.8710 q^{8} +81.0000 q^{9} +O(q^{10})\)	\(q+7.81495 q^{2} -9.00000 q^{3} +29.0734 q^{4} -70.3345 q^{6} -11.0326 q^{7} -22.8710 q^{8} +81.0000 q^{9} +121.000 q^{11} -261.661 q^{12} -248.951 q^{13} -86.2194 q^{14} -1109.09 q^{16} +669.264 q^{17} +633.011 q^{18} +2327.66 q^{19} +99.2936 q^{21} +945.609 q^{22} -867.792 q^{23} +205.839 q^{24} -1945.54 q^{26} -729.000 q^{27} -320.756 q^{28} +2640.61 q^{29} -3381.17 q^{31} -7935.57 q^{32} -1089.00 q^{33} +5230.26 q^{34} +2354.95 q^{36} +434.439 q^{37} +18190.5 q^{38} +2240.56 q^{39} +6114.13 q^{41} +775.975 q^{42} -6682.47 q^{43} +3517.89 q^{44} -6781.75 q^{46} -20195.8 q^{47} +9981.77 q^{48} -16685.3 q^{49} -6023.37 q^{51} -7237.87 q^{52} -3335.16 q^{53} -5697.10 q^{54} +252.327 q^{56} -20948.9 q^{57} +20636.2 q^{58} -35671.0 q^{59} +16705.5 q^{61} -26423.7 q^{62} -893.642 q^{63} -26525.4 q^{64} -8510.48 q^{66} -15071.1 q^{67} +19457.8 q^{68} +7810.13 q^{69} -21017.5 q^{71} -1852.55 q^{72} -5360.83 q^{73} +3395.12 q^{74} +67673.1 q^{76} -1334.95 q^{77} +17509.9 q^{78} +11717.1 q^{79} +6561.00 q^{81} +47781.6 q^{82} -51987.1 q^{83} +2886.81 q^{84} -52223.2 q^{86} -23765.5 q^{87} -2767.39 q^{88} +89693.5 q^{89} +2746.58 q^{91} -25229.7 q^{92} +30430.5 q^{93} -157829. q^{94} +71420.2 q^{96} -19443.4 q^{97} -130395. q^{98} +9801.00 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 13 q - 13 q^{2} - 117 q^{3} + 209 q^{4} + 117 q^{6} - 304 q^{7} - 399 q^{8} + 1053 q^{9}+O(q^{10}) \) 13 * q - 13 * q^2 - 117 * q^3 + 209 * q^4 + 117 * q^6 - 304 * q^7 - 399 * q^8 + 1053 * q^9	\( 13 q - 13 q^{2} - 117 q^{3} + 209 q^{4} + 117 q^{6} - 304 q^{7} - 399 q^{8} + 1053 q^{9} + 1573 q^{11} - 1881 q^{12} - 986 q^{13} - 610 q^{14} + 3501 q^{16} - 1476 q^{17} - 1053 q^{18} + 270 q^{19} + 2736 q^{21} - 1573 q^{22} - 9084 q^{23} + 3591 q^{24} + 2652 q^{26} - 9477 q^{27} - 10920 q^{28} + 11952 q^{29} + 19096 q^{31} - 11661 q^{32} - 14157 q^{33} - 1302 q^{34} + 16929 q^{36} - 39964 q^{37} - 1574 q^{38} + 8874 q^{39} + 35184 q^{41} + 5490 q^{42} + 96 q^{43} + 25289 q^{44} - 4120 q^{46} - 34984 q^{47} - 31509 q^{48} + 14557 q^{49} + 13284 q^{51} - 39002 q^{52} - 22984 q^{53} + 9477 q^{54} + 59802 q^{56} - 2430 q^{57} - 18896 q^{58} - 9192 q^{59} + 5438 q^{61} - 272 q^{62} - 24624 q^{63} + 106557 q^{64} + 14157 q^{66} - 71508 q^{67} - 127948 q^{68} + 81756 q^{69} + 101700 q^{71} - 32319 q^{72} - 77390 q^{73} + 13676 q^{74} + 139966 q^{76} - 36784 q^{77} - 23868 q^{78} + 93954 q^{79} + 85293 q^{81} - 53284 q^{82} - 185918 q^{83} + 98280 q^{84} + 370930 q^{86} - 107568 q^{87} - 48279 q^{88} - 18418 q^{89} + 174536 q^{91} - 274264 q^{92} - 171864 q^{93} + 64520 q^{94} + 104949 q^{96} - 94312 q^{97} - 145677 q^{98} + 127413 q^{99}+O(q^{100}) \) 13 * q - 13 * q^2 - 117 * q^3 + 209 * q^4 + 117 * q^6 - 304 * q^7 - 399 * q^8 + 1053 * q^9 + 1573 * q^11 - 1881 * q^12 - 986 * q^13 - 610 * q^14 + 3501 * q^16 - 1476 * q^17 - 1053 * q^18 + 270 * q^19 + 2736 * q^21 - 1573 * q^22 - 9084 * q^23 + 3591 * q^24 + 2652 * q^26 - 9477 * q^27 - 10920 * q^28 + 11952 * q^29 + 19096 * q^31 - 11661 * q^32 - 14157 * q^33 - 1302 * q^34 + 16929 * q^36 - 39964 * q^37 - 1574 * q^38 + 8874 * q^39 + 35184 * q^41 + 5490 * q^42 + 96 * q^43 + 25289 * q^44 - 4120 * q^46 - 34984 * q^47 - 31509 * q^48 + 14557 * q^49 + 13284 * q^51 - 39002 * q^52 - 22984 * q^53 + 9477 * q^54 + 59802 * q^56 - 2430 * q^57 - 18896 * q^58 - 9192 * q^59 + 5438 * q^61 - 272 * q^62 - 24624 * q^63 + 106557 * q^64 + 14157 * q^66 - 71508 * q^67 - 127948 * q^68 + 81756 * q^69 + 101700 * q^71 - 32319 * q^72 - 77390 * q^73 + 13676 * q^74 + 139966 * q^76 - 36784 * q^77 - 23868 * q^78 + 93954 * q^79 + 85293 * q^81 - 53284 * q^82 - 185918 * q^83 + 98280 * q^84 + 370930 * q^86 - 107568 * q^87 - 48279 * q^88 - 18418 * q^89 + 174536 * q^91 - 274264 * q^92 - 171864 * q^93 + 64520 * q^94 + 104949 * q^96 - 94312 * q^97 - 145677 * q^98 + 127413 * q^99

Introduction

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