LMFDB - Embedded newform 825.6.a.t.1.5

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:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - x^{9} - 271 x^{8} + 309 x^{7} + 24456 x^{6} - 33410 x^{5} - 822204 x^{4} + 1367872 x^{3} + \cdots - 7036608 $ x^10 - x^9 - 271x^8 + 309x^7 + 24456x^6 - 33410x^5 - 822204x^4 + 1367872x^3 + 7443872x^2 - 12856224x - 7036608
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{4}\cdot 5^{4} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$2.70074$ 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)

$f(q)$	$=$	$q-2.70074 q^{2} +9.00000 q^{3} -24.7060 q^{4} -24.3067 q^{6} -73.4900 q^{7} +153.148 q^{8} +81.0000 q^{9} +O(q^{10})$	\(q-2.70074 q^{2} +9.00000 q^{3} -24.7060 q^{4} -24.3067 q^{6} -73.4900 q^{7} +153.148 q^{8} +81.0000 q^{9} +121.000 q^{11} -222.354 q^{12} +1058.30 q^{13} +198.478 q^{14} +376.977 q^{16} +1854.09 q^{17} -218.760 q^{18} +1753.76 q^{19} -661.410 q^{21} -326.790 q^{22} +572.513 q^{23} +1378.33 q^{24} -2858.21 q^{26} +729.000 q^{27} +1815.64 q^{28} +2446.73 q^{29} -6035.15 q^{31} -5918.86 q^{32} +1089.00 q^{33} -5007.41 q^{34} -2001.19 q^{36} -4859.75 q^{37} -4736.45 q^{38} +9524.74 q^{39} -12742.8 q^{41} +1786.30 q^{42} -1876.74 q^{43} -2989.42 q^{44} -1546.21 q^{46} +15242.3 q^{47} +3392.80 q^{48} -11406.2 q^{49} +16686.8 q^{51} -26146.5 q^{52} -397.221 q^{53} -1968.84 q^{54} -11254.9 q^{56} +15783.8 q^{57} -6607.98 q^{58} -14886.7 q^{59} +16880.6 q^{61} +16299.4 q^{62} -5952.69 q^{63} +3922.05 q^{64} -2941.11 q^{66} +19400.8 q^{67} -45807.0 q^{68} +5152.61 q^{69} -57338.6 q^{71} +12405.0 q^{72} +52587.5 q^{73} +13124.9 q^{74} -43328.3 q^{76} -8892.30 q^{77} -25723.9 q^{78} +57323.5 q^{79} +6561.00 q^{81} +34415.1 q^{82} +39354.7 q^{83} +16340.8 q^{84} +5068.58 q^{86} +22020.5 q^{87} +18530.9 q^{88} +7491.74 q^{89} -77774.8 q^{91} -14144.5 q^{92} -54316.4 q^{93} -41165.6 q^{94} -53269.8 q^{96} +46219.5 q^{97} +30805.2 q^{98} +9801.00 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - q^{2} + 90 q^{3} + 223 q^{4} - 9 q^{6} + 188 q^{7} + 177 q^{8} + 810 q^{9}+O(q^{10}) $ 10 * q - q^2 + 90 * q^3 + 223 * q^4 - 9 * q^6 + 188 * q^7 + 177 * q^8 + 810 * q^9	\( 10 q - q^{2} + 90 q^{3} + 223 q^{4} - 9 q^{6} + 188 q^{7} + 177 q^{8} + 810 q^{9} + 1210 q^{11} + 2007 q^{12} - 1102 q^{13} + 684 q^{14} + 7019 q^{16} + 1106 q^{17} - 81 q^{18} + 2586 q^{19} + 1692 q^{21} - 121 q^{22} + 2206 q^{23} + 1593 q^{24} + 12001 q^{26} + 7290 q^{27} + 14452 q^{28} + 5824 q^{29} - 4586 q^{31} - 10627 q^{32} + 10890 q^{33} + 7426 q^{34} + 18063 q^{36} - 18362 q^{37} + 44001 q^{38} - 9918 q^{39} - 5474 q^{41} + 6156 q^{42} + 20496 q^{43} + 26983 q^{44} + 19981 q^{46} - 14970 q^{47} + 63171 q^{48} + 68582 q^{49} + 9954 q^{51} - 58603 q^{52} + 61980 q^{53} - 729 q^{54} + 7132 q^{56} + 23274 q^{57} + 7161 q^{58} + 61190 q^{59} + 8230 q^{61} - 4509 q^{62} + 15228 q^{63} + 152223 q^{64} - 1089 q^{66} - 11930 q^{67} + 105598 q^{68} + 19854 q^{69} + 59822 q^{71} + 14337 q^{72} - 20680 q^{73} + 132564 q^{74} + 68165 q^{76} + 22748 q^{77} + 108009 q^{78} + 234494 q^{79} + 65610 q^{81} + 151948 q^{82} + 185478 q^{83} + 130068 q^{84} - 17825 q^{86} + 52416 q^{87} + 21417 q^{88} + 181834 q^{89} + 206274 q^{91} + 98373 q^{92} - 41274 q^{93} - 64998 q^{94} - 95643 q^{96} + 304358 q^{97} + 153453 q^{98} + 98010 q^{99}+O(q^{100}) \) 10 * q - q^2 + 90 * q^3 + 223 * q^4 - 9 * q^6 + 188 * q^7 + 177 * q^8 + 810 * q^9 + 1210 * q^11 + 2007 * q^12 - 1102 * q^13 + 684 * q^14 + 7019 * q^16 + 1106 * q^17 - 81 * q^18 + 2586 * q^19 + 1692 * q^21 - 121 * q^22 + 2206 * q^23 + 1593 * q^24 + 12001 * q^26 + 7290 * q^27 + 14452 * q^28 + 5824 * q^29 - 4586 * q^31 - 10627 * q^32 + 10890 * q^33 + 7426 * q^34 + 18063 * q^36 - 18362 * q^37 + 44001 * q^38 - 9918 * q^39 - 5474 * q^41 + 6156 * q^42 + 20496 * q^43 + 26983 * q^44 + 19981 * q^46 - 14970 * q^47 + 63171 * q^48 + 68582 * q^49 + 9954 * q^51 - 58603 * q^52 + 61980 * q^53 - 729 * q^54 + 7132 * q^56 + 23274 * q^57 + 7161 * q^58 + 61190 * q^59 + 8230 * q^61 - 4509 * q^62 + 15228 * q^63 + 152223 * q^64 - 1089 * q^66 - 11930 * q^67 + 105598 * q^68 + 19854 * q^69 + 59822 * q^71 + 14337 * q^72 - 20680 * q^73 + 132564 * q^74 + 68165 * q^76 + 22748 * q^77 + 108009 * q^78 + 234494 * q^79 + 65610 * q^81 + 151948 * q^82 + 185478 * q^83 + 130068 * q^84 - 17825 * q^86 + 52416 * q^87 + 21417 * q^88 + 181834 * q^89 + 206274 * q^91 + 98373 * q^92 - 41274 * q^93 - 64998 * q^94 - 95643 * q^96 + 304358 * q^97 + 153453 * q^98 + 98010 * 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 $

$2$	−2.70074	−0.477428	−0.238714	−	0.971090i	$-0.576726\pi$
$2$	−2.70074	−0.477428	−0.238714	+	0.971090i	$0.576726\pi$
$3$	9.00000	0.577350
$3$	9.00000	0.577350
$4$	−24.7060	−0.772062
$5$	0	0
$5$	0	0
$6$	−24.3067	−0.275643
$7$	−73.4900	−0.566870	−0.283435	−	0.958991i	$-0.591474\pi$
$7$	−73.4900	−0.566870	−0.283435	+	0.958991i	$0.591474\pi$
$8$	153.148	0.846033
$9$	81.0000	0.333333
$10$	0	0
$11$	121.000	0.301511
$11$	121.000	0.301511
$12$	−222.354	−0.445750
$13$	1058.30	1.73681	0.868405	−	0.495856i	$-0.165146\pi$
$13$	1058.30	1.73681	0.868405	+	0.495856i	$0.165146\pi$
$14$	198.478	0.270640
$15$	0	0
$16$	376.977	0.368142
$17$	1854.09	1.55599	0.777996	−	0.628269i	$-0.216236\pi$
$17$	1854.09	1.55599	0.777996	+	0.628269i	$0.216236\pi$
$18$	−218.760	−0.159143
$19$	1753.76	1.11451	0.557257	−	0.830340i	$-0.311854\pi$
$19$	1753.76	1.11451	0.557257	+	0.830340i	$0.311854\pi$
$20$	0	0
$21$	−661.410	−0.327283
$22$	−326.790	−0.143950
$23$	572.513	0.225666	0.112833	−	0.993614i	$-0.464008\pi$
$23$	572.513	0.225666	0.112833	+	0.993614i	$0.464008\pi$
$24$	1378.33	0.488457
$25$	0	0
$26$	−2858.21	−0.829202
$27$	729.000	0.192450
$28$	1815.64	0.437659
$29$	2446.73	0.540245	0.270122	−	0.962826i	$-0.412936\pi$
$29$	2446.73	0.540245	0.270122	+	0.962826i	$0.412936\pi$
$30$	0	0
$31$	−6035.15	−1.12793	−0.563967	−	0.825797i	$-0.690726\pi$
$31$	−6035.15	−1.12793	−0.563967	+	0.825797i	$0.690726\pi$
$32$	−5918.86	−1.02179
$33$	1089.00	0.174078
$34$	−5007.41	−0.742875
$35$	0	0
$36$	−2001.19	−0.257354
$37$	−4859.75	−0.583593	−0.291796	−	0.956480i	$-0.594253\pi$
$37$	−4859.75	−0.583593	−0.291796	+	0.956480i	$0.594253\pi$
$38$	−4736.45	−0.532101
$39$	9524.74	1.00275
$40$	0	0
$41$	−12742.8	−1.18388	−0.591939	−	0.805983i	$-0.701637\pi$
$41$	−12742.8	−1.18388	−0.591939	+	0.805983i	$0.701637\pi$
$42$	1786.30	0.156254
$43$	−1876.74	−0.154786	−0.0773931	−	0.997001i	$-0.524660\pi$
$43$	−1876.74	−0.154786	−0.0773931	+	0.997001i	$0.524660\pi$
$44$	−2989.42	−0.232785
$45$	0	0
$46$	−1546.21	−0.107739
$47$	15242.3	1.00648	0.503241	−	0.864146i	$-0.332141\pi$
$47$	15242.3	1.00648	0.503241	+	0.864146i	$0.332141\pi$
$48$	3392.80	0.212547
$49$	−11406.2	−0.678658
$50$	0	0
$51$	16686.8	0.898353
$52$	−26146.5	−1.34092
$53$	−397.221	−0.0194242	−0.00971209	−	0.999953i	$-0.503092\pi$
$53$	−397.221	−0.0194242	−0.00971209	+	0.999953i	$0.503092\pi$
$54$	−1968.84	−0.0918811
$55$	0	0
$56$	−11254.9	−0.479591
$57$	15783.8	0.643465
$58$	−6607.98	−0.257928
$59$	−14886.7	−0.556759	−0.278380	−	0.960471i	$-0.589797\pi$
$59$	−14886.7	−0.556759	−0.278380	+	0.960471i	$0.589797\pi$
$60$	0	0
$61$	16880.6	0.580849	0.290425	−	0.956898i	$-0.406203\pi$
$61$	16880.6	0.580849	0.290425	+	0.956898i	$0.406203\pi$
$62$	16299.4	0.538508
$63$	−5952.69	−0.188957
$64$	3922.05	0.119691
$65$	0	0
$66$	−2941.11	−0.0831096
$67$	19400.8	0.527999	0.264000	−	0.964523i	$-0.414958\pi$
$67$	19400.8	0.527999	0.264000	+	0.964523i	$0.414958\pi$
$68$	−45807.0	−1.20132
$69$	5152.61	0.130288
$70$	0	0
$71$	−57338.6	−1.34990	−0.674949	−	0.737865i	$-0.735834\pi$
$71$	−57338.6	−1.34990	−0.674949	+	0.737865i	$0.735834\pi$
$72$	12405.0	0.282011
$73$	52587.5	1.15498	0.577491	−	0.816397i	$-0.304032\pi$
$73$	52587.5	1.15498	0.577491	+	0.816397i	$0.304032\pi$
$74$	13124.9	0.278624
$75$	0	0
$76$	−43328.3	−0.860475
$77$	−8892.30	−0.170918
$78$	−25723.9	−0.478740
$79$	57323.5	1.03339	0.516695	−	0.856169i	$-0.327162\pi$
$79$	57323.5	1.03339	0.516695	+	0.856169i	$0.327162\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	34415.1	0.565217
$83$	39354.7	0.627048	0.313524	−	0.949580i	$-0.398490\pi$
$83$	39354.7	0.627048	0.313524	+	0.949580i	$0.398490\pi$
$84$	16340.8	0.252682
$85$	0	0
$86$	5068.58	0.0738993
$87$	22020.5	0.311910
$88$	18530.9	0.255088
$89$	7491.74	0.100255	0.0501277	−	0.998743i	$-0.484037\pi$
$89$	7491.74	0.100255	0.0501277	+	0.998743i	$0.484037\pi$
$90$	0	0
$91$	−77774.8	−0.984545
$92$	−14144.5	−0.174228
$93$	−54316.4	−0.651213
$94$	−41165.6	−0.480523
$95$	0	0
$96$	−53269.8	−0.589933
$97$	46219.5	0.498765	0.249382	−	0.968405i	$-0.419772\pi$
$97$	46219.5	0.498765	0.249382	+	0.968405i	$0.419772\pi$
$98$	30805.2	0.324011
$99$	9801.00	0.100504
$100$	0	0
$101$	137105.	1.33736	0.668682	−	0.743549i	$-0.266859\pi$
$101$	137105.	1.33736	0.668682	+	0.743549i	$0.266859\pi$
$102$	−45066.7	−0.428899
$103$	1693.09	0.0157248	0.00786241	−	0.999969i	$-0.497497\pi$
$103$	1693.09	0.0157248	0.00786241	+	0.999969i	$0.497497\pi$
$104$	162077.	1.46940
$105$	0	0
$106$	1072.79	0.00927365
$107$	−146935.	−1.24070	−0.620348	−	0.784327i	$-0.713009\pi$
$107$	−146935.	−1.24070	−0.620348	+	0.784327i	$0.713009\pi$
$108$	−18010.7	−0.148583
$109$	221030.	1.78191	0.890955	−	0.454092i	$-0.150036\pi$
$109$	221030.	1.78191	0.890955	+	0.454092i	$0.150036\pi$
$110$	0	0
$111$	−43737.8	−0.336937
$112$	−27704.1	−0.208689
$113$	−134869.	−0.993607	−0.496803	−	0.867863i	$-0.665493\pi$
$113$	−134869.	−0.993607	−0.496803	+	0.867863i	$0.665493\pi$
$114$	−42628.1	−0.307209
$115$	0	0
$116$	−60448.8	−0.417102
$117$	85722.6	0.578936
$118$	40205.0	0.265813
$119$	−136257.	−0.882045
$120$	0	0
$121$	14641.0	0.0909091
$122$	−45590.2	−0.277314
$123$	−114686.	−0.683512
$124$	149104.	0.870836
$125$	0	0
$126$	16076.7	0.0902133
$127$	171356.	0.942738	0.471369	−	0.881936i	$-0.343760\pi$
$127$	171356.	0.942738	0.471369	+	0.881936i	$0.343760\pi$
$128$	178811.	0.964650
$129$	−16890.6	−0.0893658
$130$	0	0
$131$	166556.	0.847973	0.423987	−	0.905668i	$-0.360630\pi$
$131$	166556.	0.847973	0.423987	+	0.905668i	$0.360630\pi$
$132$	−26904.8	−0.134399
$133$	−128884.	−0.631785
$134$	−52396.7	−0.252082
$135$	0	0
$136$	283950.	1.31642
$137$	−94610.5	−0.430663	−0.215332	−	0.976541i	$-0.569083\pi$
$137$	−94610.5	−0.430663	−0.215332	+	0.976541i	$0.569083\pi$
$138$	−13915.9	−0.0622032
$139$	301241.	1.32244	0.661222	−	0.750190i	$-0.270038\pi$
$139$	301241.	1.32244	0.661222	+	0.750190i	$0.270038\pi$
$140$	0	0
$141$	137181.	0.581093
$142$	154857.	0.644479
$143$	128055.	0.523668
$144$	30535.2	0.122714
$145$	0	0
$146$	−142025.	−0.551421
$147$	−102656.	−0.391824
$148$	120065.	0.450570
$149$	−39922.6	−0.147317	−0.0736586	−	0.997284i	$-0.523468\pi$
$149$	−39922.6	−0.147317	−0.0736586	+	0.997284i	$0.523468\pi$
$150$	0	0
$151$	114191.	0.407557	0.203778	−	0.979017i	$-0.434678\pi$
$151$	114191.	0.407557	0.203778	+	0.979017i	$0.434678\pi$
$152$	268585.	0.942916
$153$	150181.	0.518664
$154$	24015.8	0.0816010
$155$	0	0
$156$	−235318.	−0.774183
$157$	−597139.	−1.93342	−0.966711	−	0.255872i	$-0.917638\pi$
$157$	−597139.	−1.93342	−0.966711	+	0.255872i	$0.917638\pi$
$158$	−154816.	−0.493370
$159$	−3574.99	−0.0112146
$160$	0	0
$161$	−42074.0	−0.127923
$162$	−17719.6	−0.0530476
$163$	−342738.	−1.01040	−0.505200	−	0.863002i	$-0.668581\pi$
$163$	−342738.	−1.01040	−0.505200	+	0.863002i	$0.668581\pi$
$164$	314825.	0.914027
$165$	0	0
$166$	−106287.	−0.299371
$167$	37584.6	0.104284	0.0521422	−	0.998640i	$-0.483395\pi$
$167$	37584.6	0.104284	0.0521422	+	0.998640i	$0.483395\pi$
$168$	−101294.	−0.276892
$169$	748715.	2.01651
$170$	0	0
$171$	142054.	0.371505
$172$	46366.6	0.119505
$173$	−172329.	−0.437767	−0.218884	−	0.975751i	$-0.570242\pi$
$173$	−172329.	−0.437767	−0.218884	+	0.975751i	$0.570242\pi$
$174$	−59471.8	−0.148915
$175$	0	0
$176$	45614.3	0.110999
$177$	−133980.	−0.321445
$178$	−20233.3	−0.0478648
$179$	422737.	0.986138	0.493069	−	0.869990i	$-0.335875\pi$
$179$	422737.	0.986138	0.493069	+	0.869990i	$0.335875\pi$
$180$	0	0
$181$	−584562.	−1.32628	−0.663139	−	0.748497i	$-0.730776\pi$
$181$	−584562.	−1.32628	−0.663139	+	0.748497i	$0.730776\pi$
$182$	210050.	0.470050
$183$	151925.	0.335354
$184$	87679.3	0.190921
$185$	0	0
$186$	146695.	0.310908
$187$	224344.	0.469149
$188$	−376576.	−0.777067
$189$	−53574.2	−0.109094
$190$	0	0
$191$	52330.3	0.103793	0.0518967	−	0.998652i	$-0.483473\pi$
$191$	52330.3	0.103793	0.0518967	+	0.998652i	$0.483473\pi$
$192$	35298.5	0.0691039
$193$	−225739.	−0.436227	−0.218114	−	0.975923i	$-0.569990\pi$
$193$	−225739.	−0.436227	−0.218114	+	0.975923i	$0.569990\pi$
$194$	−124827.	−0.238124
$195$	0	0
$196$	281802.	0.523966
$197$	−592389.	−1.08753	−0.543765	−	0.839238i	$-0.683002\pi$
$197$	−592389.	−1.08753	−0.543765	+	0.839238i	$0.683002\pi$
$198$	−26470.0	−0.0479834
$199$	−108410.	−0.194060	−0.0970299	−	0.995281i	$-0.530934\pi$
$199$	−108410.	−0.194060	−0.0970299	+	0.995281i	$0.530934\pi$
$200$	0	0
$201$	174608.	0.304841
$202$	−370285.	−0.638495
$203$	−179810.	−0.306248
$204$	−412263.	−0.693584
$205$	0	0
$206$	−4572.59	−0.00750748
$207$	46373.5	0.0752219
$208$	398957.	0.639393
$209$	212205.	0.336039
$210$	0	0
$211$	514952.	0.796270	0.398135	−	0.917327i	$-0.369658\pi$
$211$	514952.	0.796270	0.398135	+	0.917327i	$0.369658\pi$
$212$	9813.74	0.0149967
$213$	−516047.	−0.779364
$214$	396833.	0.592344
$215$	0	0
$216$	111645.	0.162819
$217$	443524.	0.639392
$218$	−596946.	−0.850734
$219$	473287.	0.666829
$220$	0	0
$221$	1.96219e6	2.70246
$222$	118124.	0.160863
$223$	−572648.	−0.771127	−0.385563	−	0.922681i	$-0.625993\pi$
$223$	−572648.	−0.771127	−0.385563	+	0.922681i	$0.625993\pi$
$224$	434978.	0.579224
$225$	0	0
$226$	364245.	0.474376
$227$	1.45788e6	1.87784	0.938918	−	0.344142i	$-0.111830\pi$
$227$	1.45788e6	1.87784	0.938918	+	0.344142i	$0.111830\pi$
$228$	−389955.	−0.496795
$229$	−979092.	−1.23377	−0.616886	−	0.787053i	$-0.711606\pi$
$229$	−979092.	−1.23377	−0.616886	+	0.787053i	$0.711606\pi$
$230$	0	0
$231$	−80030.7	−0.0986794
$232$	374712.	0.457065
$233$	−1.58285e6	−1.91007	−0.955036	−	0.296490i	$-0.904184\pi$
$233$	−1.58285e6	−1.91007	−0.955036	+	0.296490i	$0.904184\pi$
$234$	−231515.	−0.276401
$235$	0	0
$236$	367790.	0.429853
$237$	515911.	0.596628
$238$	367995.	0.421114
$239$	342941.	0.388351	0.194176	−	0.980967i	$-0.437797\pi$
$239$	342941.	0.388351	0.194176	+	0.980967i	$0.437797\pi$
$240$	0	0
$241$	−1.12369e6	−1.24625	−0.623124	−	0.782123i	$-0.714137\pi$
$241$	−1.12369e6	−1.24625	−0.623124	+	0.782123i	$0.714137\pi$
$242$	−39541.6	−0.0434026
$243$	59049.0	0.0641500
$244$	−417052.	−0.448452
$245$	0	0
$246$	309736.	0.326328
$247$	1.85601e6	1.93570
$248$	−924273.	−0.954270
$249$	354192.	0.362026
$250$	0	0
$251$	767281.	0.768724	0.384362	−	0.923183i	$-0.374422\pi$
$251$	767281.	0.768724	0.384362	+	0.923183i	$0.374422\pi$
$252$	147067.	0.145886
$253$	69274.0	0.0680407
$254$	−462790.	−0.450090
$255$	0	0
$256$	−608429.	−0.580243
$257$	448786.	0.423844	0.211922	−	0.977287i	$-0.432028\pi$
$257$	448786.	0.423844	0.211922	+	0.977287i	$0.432028\pi$
$258$	45617.2	0.0426658
$259$	357143.	0.330821
$260$	0	0
$261$	198185.	0.180082
$262$	−449825.	−0.404847
$263$	921492.	0.821490	0.410745	−	0.911750i	$-0.365269\pi$
$263$	921492.	0.821490	0.410745	+	0.911750i	$0.365269\pi$
$264$	166778.	0.147275
$265$	0	0
$266$	348082.	0.301632
$267$	67425.7	0.0578825
$268$	−479317.	−0.407648
$269$	1.40136e6	1.18078	0.590388	−	0.807119i	$-0.298975\pi$
$269$	1.40136e6	1.18078	0.590388	+	0.807119i	$0.298975\pi$
$270$	0	0
$271$	−2.02100e6	−1.67164	−0.835821	−	0.549003i	$-0.815008\pi$
$271$	−2.02100e6	−1.67164	−0.835821	+	0.549003i	$0.815008\pi$
$272$	698949.	0.572826
$273$	−699974.	−0.568427
$274$	255519.	0.205611
$275$	0	0
$276$	−127300.	−0.100591
$277$	−74470.0	−0.0583152	−0.0291576	−	0.999575i	$-0.509282\pi$
$277$	−74470.0	−0.0583152	−0.0291576	+	0.999575i	$0.509282\pi$
$278$	−813575.	−0.631372
$279$	−488847.	−0.375978
$280$	0	0
$281$	951506.	0.718862	0.359431	−	0.933172i	$-0.382971\pi$
$281$	951506.	0.718862	0.359431	+	0.933172i	$0.382971\pi$
$282$	−370490.	−0.277430
$283$	535770.	0.397660	0.198830	−	0.980034i	$-0.436286\pi$
$283$	535770.	0.397660	0.198830	+	0.980034i	$0.436286\pi$
$284$	1.41661e6	1.04220
$285$	0	0
$286$	−345843.	−0.250014
$287$	936472.	0.671105
$288$	−479428.	−0.340598
$289$	2.01778e6	1.42111
$290$	0	0
$291$	415975.	0.287962
$292$	−1.29923e6	−0.891718
$293$	2.61667e6	1.78065	0.890326	−	0.455323i	$-0.150476\pi$
$293$	2.61667e6	1.78065	0.890326	+	0.455323i	$0.150476\pi$
$294$	277247.	0.187068
$295$	0	0
$296$	−744263.	−0.493738
$297$	88209.0	0.0580259
$298$	107821.	0.0703334
$299$	605892.	0.391938
$300$	0	0
$301$	137921.	0.0877436
$302$	−308399.	−0.194579
$303$	1.23394e6	0.772127
$304$	661128.	0.410300
$305$	0	0
$306$	−405600.	−0.247625
$307$	2.42951e6	1.47120	0.735601	−	0.677415i	$-0.236900\pi$
$307$	2.42951e6	1.47120	0.735601	+	0.677415i	$0.236900\pi$
$308$	219693.	0.131959
$309$	15237.8	0.00907873
$310$	0	0
$311$	571145.	0.334846	0.167423	−	0.985885i	$-0.446455\pi$
$311$	571145.	0.334846	0.167423	+	0.985885i	$0.446455\pi$
$312$	1.45870e6	0.848357
$313$	−2.21026e6	−1.27521	−0.637605	−	0.770363i	$-0.720075\pi$
$313$	−2.21026e6	−1.27521	−0.637605	+	0.770363i	$0.720075\pi$
$314$	1.61272e6	0.923070
$315$	0	0
$316$	−1.41623e6	−0.797842
$317$	1.19719e6	0.669137	0.334568	−	0.942371i	$-0.391409\pi$
$317$	1.19719e6	0.669137	0.334568	+	0.942371i	$0.391409\pi$
$318$	9655.13	0.00535415
$319$	296054.	0.162890
$320$	0	0
$321$	−1.32241e6	−0.716316
$322$	113631.	0.0610741
$323$	3.25162e6	1.73418
$324$	−162096.	−0.0857847
$325$	0	0
$326$	925648.	0.482394
$327$	1.98927e6	1.02879
$328$	−1.95154e6	−1.00160
$329$	−1.12016e6	−0.570545
$330$	0	0
$331$	1.82817e6	0.917162	0.458581	−	0.888653i	$-0.348358\pi$
$331$	1.82817e6	0.917162	0.458581	+	0.888653i	$0.348358\pi$
$332$	−972296.	−0.484120
$333$	−393640.	−0.194531
$334$	−101506.	−0.0497883
$335$	0	0
$336$	−249337.	−0.120486
$337$	1.38711e6	0.665329	0.332665	−	0.943045i	$-0.392052\pi$
$337$	1.38711e6	0.665329	0.332665	+	0.943045i	$0.392052\pi$
$338$	−2.02209e6	−0.962738
$339$	−1.21382e6	−0.573659
$340$	0	0
$341$	−730253.	−0.340085
$342$	−383653.	−0.177367
$343$	2.07339e6	0.951581
$344$	−287419.	−0.130954
$345$	0	0
$346$	465416.	0.209002
$347$	−3.00936e6	−1.34169	−0.670843	−	0.741599i	$-0.734068\pi$
$347$	−3.00936e6	−1.34169	−0.670843	+	0.741599i	$0.734068\pi$
$348$	−544039.	−0.240814
$349$	2.44051e6	1.07255	0.536275	−	0.844043i	$-0.319831\pi$
$349$	2.44051e6	1.07255	0.536275	+	0.844043i	$0.319831\pi$
$350$	0	0
$351$	771504.	0.334249
$352$	−716183.	−0.308083
$353$	−1.65608e6	−0.707368	−0.353684	−	0.935365i	$-0.615071\pi$
$353$	−1.65608e6	−0.707368	−0.353684	+	0.935365i	$0.615071\pi$
$354$	361845.	0.153467
$355$	0	0
$356$	−185091.	−0.0774034
$357$	−1.22631e6	−0.509249
$358$	−1.14170e6	−0.470810
$359$	1.24299e6	0.509018	0.254509	−	0.967070i	$-0.418086\pi$
$359$	1.24299e6	0.509018	0.254509	+	0.967070i	$0.418086\pi$
$360$	0	0
$361$	599571.	0.242143
$362$	1.57875e6	0.633202
$363$	131769.	0.0524864
$364$	1.92150e6	0.760130
$365$	0	0
$366$	−410312.	−0.160107
$367$	2.26767e6	0.878852	0.439426	−	0.898279i	$-0.355182\pi$
$367$	2.26767e6	0.878852	0.439426	+	0.898279i	$0.355182\pi$
$368$	215824.	0.0830770
$369$	−1.03217e6	−0.394626
$370$	0	0
$371$	29191.8	0.0110110
$372$	1.34194e6	0.502777
$373$	−1.92708e6	−0.717180	−0.358590	−	0.933495i	$-0.616742\pi$
$373$	−1.92708e6	−0.717180	−0.358590	+	0.933495i	$0.616742\pi$
$374$	−605896.	−0.223985
$375$	0	0
$376$	2.33433e6	0.851517
$377$	2.58938e6	0.938302
$378$	144690.	0.0520847
$379$	−4.49386e6	−1.60702	−0.803511	−	0.595291i	$-0.797037\pi$
$379$	−4.49386e6	−1.60702	−0.803511	+	0.595291i	$0.797037\pi$
$380$	0	0
$381$	1.54221e6	0.544290
$382$	−141331.	−0.0495539
$383$	2.69103e6	0.937391	0.468696	−	0.883360i	$-0.344724\pi$
$383$	2.69103e6	0.937391	0.468696	+	0.883360i	$0.344724\pi$
$384$	1.60930e6	0.556941
$385$	0	0
$386$	609662.	0.208267
$387$	−152016.	−0.0515954
$388$	−1.14190e6	−0.385077
$389$	−1.35749e6	−0.454843	−0.227422	−	0.973796i	$-0.573030\pi$
$389$	−1.35749e6	−0.454843	−0.227422	+	0.973796i	$0.573030\pi$
$390$	0	0
$391$	1.06149e6	0.351134
$392$	−1.74684e6	−0.574167
$393$	1.49900e6	0.489578
$394$	1.59989e6	0.519218
$395$	0	0
$396$	−242143.	−0.0775952
$397$	−3.44695e6	−1.09764	−0.548819	−	0.835941i	$-0.684922\pi$
$397$	−3.44695e6	−1.09764	−0.548819	+	0.835941i	$0.684922\pi$
$398$	292787.	0.0926496
$399$	−1.15995e6	−0.364761
$400$	0	0
$401$	5.61209e6	1.74287	0.871433	−	0.490515i	$-0.163191\pi$
$401$	5.61209e6	1.74287	0.871433	+	0.490515i	$0.163191\pi$
$402$	−471570.	−0.145540
$403$	−6.38703e6	−1.95901
$404$	−3.38731e6	−1.03253
$405$	0	0
$406$	485621.	0.146212
$407$	−588030.	−0.175960
$408$	2.55555e6	0.760036
$409$	4.41294e6	1.30443	0.652214	−	0.758035i	$-0.273840\pi$
$409$	4.41294e6	1.30443	0.652214	+	0.758035i	$0.273840\pi$
$410$	0	0
$411$	−851494.	−0.248644
$412$	−41829.4	−0.0121405
$413$	1.09402e6	0.315610
$414$	−125243.	−0.0359131
$415$	0	0
$416$	−6.26396e6	−1.77466
$417$	2.71117e6	0.763513
$418$	−573111.	−0.160435
$419$	659874.	0.183623	0.0918113	−	0.995776i	$-0.470734\pi$
$419$	659874.	0.183623	0.0918113	+	0.995776i	$0.470734\pi$
$420$	0	0
$421$	3.30334e6	0.908339	0.454170	−	0.890915i	$-0.349936\pi$
$421$	3.30334e6	0.908339	0.454170	+	0.890915i	$0.349936\pi$
$422$	−1.39075e6	−0.380162
$423$	1.23463e6	0.335494
$424$	−60833.7	−0.0164335
$425$	0	0
$426$	1.39371e6	0.372090
$427$	−1.24056e6	−0.329266
$428$	3.63017e6	0.957895
$429$	1.15249e6	0.302340
$430$	0	0
$431$	−1.04442e6	−0.270819	−0.135410	−	0.990790i	$-0.543235\pi$
$431$	−1.04442e6	−0.270819	−0.135410	+	0.990790i	$0.543235\pi$
$432$	274817.	0.0708490
$433$	3.34631e6	0.857722	0.428861	−	0.903371i	$-0.358915\pi$
$433$	3.34631e6	0.857722	0.428861	+	0.903371i	$0.358915\pi$
$434$	−1.19784e6	−0.305264
$435$	0	0
$436$	−5.46077e6	−1.37575
$437$	1.00405e6	0.251508
$438$	−1.27823e6	−0.318363
$439$	5.24252e6	1.29831	0.649155	−	0.760656i	$-0.275123\pi$
$439$	5.24252e6	1.29831	0.649155	+	0.760656i	$0.275123\pi$
$440$	0	0
$441$	−923903.	−0.226219
$442$	−5.29936e6	−1.29023
$443$	3.96633e6	0.960239	0.480119	−	0.877203i	$-0.340593\pi$
$443$	3.96633e6	0.960239	0.480119	+	0.877203i	$0.340593\pi$
$444$	1.08058e6	0.260137
$445$	0	0
$446$	1.54658e6	0.368158
$447$	−359304.	−0.0850536
$448$	−288232.	−0.0678495
$449$	−3.69739e6	−0.865524	−0.432762	−	0.901508i	$-0.642461\pi$
$449$	−3.69739e6	−0.865524	−0.432762	+	0.901508i	$0.642461\pi$
$450$	0	0
$451$	−1.54188e6	−0.356952
$452$	3.33206e6	0.767126
$453$	1.02772e6	0.235303
$454$	−3.93736e6	−0.896532
$455$	0	0
$456$	2.41727e6	0.544393
$457$	−6.44119e6	−1.44270	−0.721349	−	0.692571i	$-0.756478\pi$
$457$	−6.44119e6	−1.44270	−0.721349	+	0.692571i	$0.756478\pi$
$458$	2.64428e6	0.589038
$459$	1.35163e6	0.299451
$460$	0	0
$461$	3.40133e6	0.745413	0.372706	−	0.927949i	$-0.378430\pi$
$461$	3.40133e6	0.745413	0.372706	+	0.927949i	$0.378430\pi$
$462$	216142.	0.0471123
$463$	6.02105e6	1.30533	0.652664	−	0.757647i	$-0.273651\pi$
$463$	6.02105e6	1.30533	0.652664	+	0.757647i	$0.273651\pi$
$464$	922361.	0.198887
$465$	0	0
$466$	4.27487e6	0.911923
$467$	1.98722e6	0.421652	0.210826	−	0.977524i	$-0.432385\pi$
$467$	1.98722e6	0.421652	0.210826	+	0.977524i	$0.432385\pi$
$468$	−2.11786e6	−0.446975
$469$	−1.42577e6	−0.299307
$470$	0	0
$471$	−5.37425e6	−1.11626
$472$	−2.27987e6	−0.471036
$473$	−227085.	−0.0466698
$474$	−1.39334e6	−0.284847
$475$	0	0
$476$	3.36636e6	0.680994
$477$	−32174.9	−0.00647472
$478$	−926196.	−0.185410
$479$	1.78305e6	0.355078	0.177539	−	0.984114i	$-0.443186\pi$
$479$	1.78305e6	0.355078	0.177539	+	0.984114i	$0.443186\pi$
$480$	0	0
$481$	−5.14310e6	−1.01359
$482$	3.03480e6	0.594994
$483$	−378666.	−0.0738564
$484$	−361720.	−0.0701875
$485$	0	0
$486$	−159476.	−0.0306270
$487$	−6.00734e6	−1.14778	−0.573891	−	0.818931i	$-0.694567\pi$
$487$	−6.00734e6	−1.14778	−0.573891	+	0.818931i	$0.694567\pi$
$488$	2.58524e6	0.491418
$489$	−3.08465e6	−0.583355
$490$	0	0
$491$	3.33196e6	0.623729	0.311865	−	0.950127i	$-0.399046\pi$
$491$	3.33196e6	0.623729	0.311865	+	0.950127i	$0.399046\pi$
$492$	2.83342e6	0.527714
$493$	4.53644e6	0.840617
$494$	−5.01261e6	−0.924158
$495$	0	0
$496$	−2.27512e6	−0.415240
$497$	4.21381e6	0.765216
$498$	−956581.	−0.172842
$499$	7.11288e6	1.27878	0.639388	−	0.768885i	$-0.279188\pi$
$499$	7.11288e6	1.27878	0.639388	+	0.768885i	$0.279188\pi$
$500$	0	0
$501$	338262.	0.0602086
$502$	−2.07223e6	−0.367010
$503$	−2.89775e6	−0.510671	−0.255335	−	0.966853i	$-0.582186\pi$
$503$	−2.89775e6	−0.510671	−0.255335	+	0.966853i	$0.582186\pi$
$504$	−911645.	−0.159864
$505$	0	0
$506$	−187091.	−0.0324846
$507$	6.73843e6	1.16423
$508$	−4.23353e6	−0.727853
$509$	9.15158e6	1.56568	0.782838	−	0.622226i	$-0.213771\pi$
$509$	9.15158e6	1.56568	0.782838	+	0.622226i	$0.213771\pi$
$510$	0	0
$511$	−3.86466e6	−0.654724
$512$	−4.07875e6	−0.687626
$513$	1.27849e6	0.214488
$514$	−1.21205e6	−0.202355
$515$	0	0
$516$	417300.	0.0689960
$517$	1.84432e6	0.303466
$518$	−964552.	−0.157943
$519$	−1.55096e6	−0.252745
$520$	0	0
$521$	−1.17851e7	−1.90213	−0.951066	−	0.308989i	$-0.900009\pi$
$521$	−1.17851e7	−1.90213	−0.951066	+	0.308989i	$0.900009\pi$
$522$	−535246.	−0.0859760
$523$	−2.89889e6	−0.463423	−0.231711	−	0.972785i	$-0.574433\pi$
$523$	−2.89889e6	−0.463423	−0.231711	+	0.972785i	$0.574433\pi$
$524$	−4.11493e6	−0.654688
$525$	0	0
$526$	−2.48871e6	−0.392203
$527$	−1.11897e7	−1.75506
$528$	410528.	0.0640853
$529$	−6.10857e6	−0.949075
$530$	0	0
$531$	−1.20582e6	−0.185586
$532$	3.18420e6	0.487777
$533$	−1.34858e7	−2.05617
$534$	−182099.	−0.0276347
$535$	0	0
$536$	2.97121e6	0.446705
$537$	3.80463e6	0.569347
$538$	−3.78470e6	−0.563736
$539$	−1.38015e6	−0.204623
$540$	0	0
$541$	−9.80373e6	−1.44012	−0.720059	−	0.693913i	$-0.755885\pi$
$541$	−9.80373e6	−1.44012	−0.720059	+	0.693913i	$0.755885\pi$
$542$	5.45820e6	0.798089
$543$	−5.26106e6	−0.765727
$544$	−1.09741e7	−1.58990
$545$	0	0
$546$	1.89045e6	0.271383
$547$	−7.81347e6	−1.11654	−0.558272	−	0.829658i	$-0.688535\pi$
$547$	−7.81347e6	−1.11654	−0.558272	+	0.829658i	$0.688535\pi$
$548$	2.33745e6	0.332499
$549$	1.36733e6	0.193616
$550$	0	0
$551$	4.29097e6	0.602111
$552$	789114.	0.110228
$553$	−4.21270e6	−0.585798
$554$	201124.	0.0278414
$555$	0	0
$556$	−7.44246e6	−1.02101
$557$	−1.08730e7	−1.48494	−0.742472	−	0.669877i	$-0.766347\pi$
$557$	−1.08730e7	−1.48494	−0.742472	+	0.669877i	$0.766347\pi$
$558$	1.32025e6	0.179503
$559$	−1.98616e6	−0.268834
$560$	0	0
$561$	2.01910e6	0.270864
$562$	−2.56977e6	−0.343205
$563$	1.10093e7	1.46382	0.731910	−	0.681401i	$-0.238629\pi$
$563$	1.10093e7	1.46382	0.731910	+	0.681401i	$0.238629\pi$
$564$	−3.38919e6	−0.448640
$565$	0	0
$566$	−1.44698e6	−0.189854
$567$	−482168.	−0.0629856
$568$	−8.78130e6	−1.14206
$569$	−1.03276e7	−1.33727	−0.668635	−	0.743590i	$-0.733121\pi$
$569$	−1.03276e7	−1.33727	−0.668635	+	0.743590i	$0.733121\pi$
$570$	0	0
$571$	1.13717e7	1.45961	0.729803	−	0.683657i	$-0.239612\pi$
$571$	1.13717e7	1.45961	0.729803	+	0.683657i	$0.239612\pi$
$572$	−3.16372e6	−0.404304
$573$	470973.	0.0599252
$574$	−2.52917e6	−0.320404
$575$	0	0
$576$	317686.	0.0398972
$577$	−1.35247e7	−1.69117	−0.845585	−	0.533841i	$-0.820748\pi$
$577$	−1.35247e7	−1.69117	−0.845585	+	0.533841i	$0.820748\pi$
$578$	−5.44950e6	−0.678480
$579$	−2.03165e6	−0.251856
$580$	0	0
$581$	−2.89218e6	−0.355455
$582$	−1.12344e6	−0.137481
$583$	−48063.7	−0.00585661
$584$	8.05368e6	0.977152
$585$	0	0
$586$	−7.06694e6	−0.850134
$587$	−5.40752e6	−0.647743	−0.323871	−	0.946101i	$-0.604985\pi$
$587$	−5.40752e6	−0.647743	−0.323871	+	0.946101i	$0.604985\pi$
$588$	2.53622e6	0.302512
$589$	−1.05842e7	−1.25710
$590$	0	0
$591$	−5.33150e6	−0.627886
$592$	−1.83202e6	−0.214845
$593$	1.18513e7	1.38398	0.691991	−	0.721906i	$-0.256734\pi$
$593$	1.18513e7	1.38398	0.691991	+	0.721906i	$0.256734\pi$
$594$	−238230.	−0.0277032
$595$	0	0
$596$	986328.	0.113738
$597$	−975687.	−0.112040
$598$	−1.63636e6	−0.187122
$599$	8.77521e6	0.999287	0.499644	−	0.866231i	$-0.333464\pi$
$599$	8.77521e6	0.999287	0.499644	+	0.866231i	$0.333464\pi$
$600$	0	0
$601$	−5.36530e6	−0.605910	−0.302955	−	0.953005i	$-0.597973\pi$
$601$	−5.36530e6	−0.605910	−0.302955	+	0.953005i	$0.597973\pi$
$602$	−372490.	−0.0418913
$603$	1.57147e6	0.176000
$604$	−2.82119e6	−0.314659
$605$	0	0
$606$	−3.33257e6	−0.368635
$607$	−1.28875e7	−1.41971	−0.709853	−	0.704350i	$-0.751239\pi$
$607$	−1.28875e7	−1.41971	−0.709853	+	0.704350i	$0.751239\pi$
$608$	−1.03803e7	−1.13880
$609$	−1.61829e6	−0.176813
$610$	0	0
$611$	1.61310e7	1.74807
$612$	−3.71037e6	−0.400441
$613$	−2.15633e6	−0.231773	−0.115887	−	0.993262i	$-0.536971\pi$
$613$	−2.15633e6	−0.231773	−0.115887	+	0.993262i	$0.536971\pi$
$614$	−6.56148e6	−0.702394
$615$	0	0
$616$	−1.36184e6	−0.144602
$617$	5.17630e6	0.547402	0.273701	−	0.961815i	$-0.411752\pi$
$617$	5.17630e6	0.547402	0.273701	+	0.961815i	$0.411752\pi$
$618$	−41153.3	−0.00433445
$619$	5.56308e6	0.583564	0.291782	−	0.956485i	$-0.405752\pi$
$619$	5.56308e6	0.583564	0.291782	+	0.956485i	$0.405752\pi$
$620$	0	0
$621$	417362.	0.0434294
$622$	−1.54252e6	−0.159865
$623$	−550568.	−0.0568318
$624$	3.59061e6	0.369154
$625$	0	0
$626$	5.96934e6	0.608822
$627$	1.90984e6	0.194012
$628$	1.47529e7	1.49272
$629$	−9.01040e6	−0.908066
$630$	0	0
$631$	4.18668e6	0.418597	0.209298	−	0.977852i	$-0.432882\pi$
$631$	4.18668e6	0.418597	0.209298	+	0.977852i	$0.432882\pi$
$632$	8.77899e6	0.874283
$633$	4.63457e6	0.459727
$634$	−3.23330e6	−0.319465
$635$	0	0
$636$	88323.6	0.00865833
$637$	−1.20712e7	−1.17870
$638$	−799565.	−0.0777682
$639$	−4.64442e6	−0.449966
$640$	0	0
$641$	−1.85710e7	−1.78521	−0.892606	−	0.450837i	$-0.851125\pi$
$641$	−1.85710e7	−1.78521	−0.892606	+	0.450837i	$0.851125\pi$
$642$	3.57150e6	0.341990
$643$	−1.57847e7	−1.50560	−0.752799	−	0.658250i	$-0.771297\pi$
$643$	−1.57847e7	−1.50560	−0.752799	+	0.658250i	$0.771297\pi$
$644$	1.03948e6	0.0987646
$645$	0	0
$646$	−8.78179e6	−0.827945
$647$	1.26849e7	1.19132	0.595658	−	0.803238i	$-0.296891\pi$
$647$	1.26849e7	1.19132	0.595658	+	0.803238i	$0.296891\pi$
$648$	1.00481e6	0.0940036
$649$	−1.80129e6	−0.167869
$650$	0	0
$651$	3.99171e6	0.369153
$652$	8.46769e6	0.780092
$653$	1.85337e7	1.70090	0.850451	−	0.526055i	$-0.176329\pi$
$653$	1.85337e7	1.70090	0.850451	+	0.526055i	$0.176329\pi$
$654$	−5.37252e6	−0.491172
$655$	0	0
$656$	−4.80377e6	−0.435835
$657$	4.25958e6	0.384994
$658$	3.02526e6	0.272394
$659$	−1.80309e7	−1.61735	−0.808673	−	0.588258i	$-0.799814\pi$
$659$	−1.80309e7	−1.61735	−0.808673	+	0.588258i	$0.799814\pi$
$660$	0	0
$661$	−1.68970e6	−0.150420	−0.0752099	−	0.997168i	$-0.523963\pi$
$661$	−1.68970e6	−0.150420	−0.0752099	+	0.997168i	$0.523963\pi$
$662$	−4.93741e6	−0.437879
$663$	1.76597e7	1.56027
$664$	6.02710e6	0.530503
$665$	0	0
$666$	1.06312e6	0.0928746
$667$	1.40078e6	0.121915
$668$	−928566.	−0.0805140
$669$	−5.15384e6	−0.445210
$670$	0	0
$671$	2.04255e6	0.175133
$672$	3.91480e6	0.334415
$673$	2.02346e7	1.72209	0.861046	−	0.508527i	$-0.169810\pi$
$673$	2.02346e7	1.72209	0.861046	+	0.508527i	$0.169810\pi$
$674$	−3.74623e6	−0.317647
$675$	0	0
$676$	−1.84977e7	−1.55687
$677$	1.68122e7	1.40978	0.704892	−	0.709314i	$-0.250995\pi$
$677$	1.68122e7	1.40978	0.704892	+	0.709314i	$0.250995\pi$
$678$	3.27821e6	0.273881
$679$	−3.39667e6	−0.282735
$680$	0	0
$681$	1.31209e7	1.08417
$682$	1.97223e6	0.162366
$683$	1.07248e7	0.879706	0.439853	−	0.898070i	$-0.355031\pi$
$683$	1.07248e7	0.879706	0.439853	+	0.898070i	$0.355031\pi$
$684$	−3.50960e6	−0.286825
$685$	0	0
$686$	−5.59969e6	−0.454312
$687$	−8.81183e6	−0.712319
$688$	−707487.	−0.0569833
$689$	−420381.	−0.0337361
$690$	0	0
$691$	6.49615e6	0.517561	0.258780	−	0.965936i	$-0.416679\pi$
$691$	6.49615e6	0.517561	0.258780	+	0.965936i	$0.416679\pi$
$692$	4.25756e6	0.337983
$693$	−720276.	−0.0569726
$694$	8.12752e6	0.640559
$695$	0	0
$696$	3.37241e6	0.263886
$697$	−2.36263e7	−1.84210
$698$	−6.59120e6	−0.512066
$699$	−1.42456e7	−1.10278
$700$	0	0
$701$	−5.20523e6	−0.400078	−0.200039	−	0.979788i	$-0.564107\pi$
$701$	−5.20523e6	−0.400078	−0.200039	+	0.979788i	$0.564107\pi$
$702$	−2.08363e6	−0.159580
$703$	−8.52283e6	−0.650423
$704$	474568.	0.0360883
$705$	0	0
$706$	4.47266e6	0.337718
$707$	−1.00758e7	−0.758111
$708$	3.31011e6	0.248176
$709$	1.89444e7	1.41536	0.707679	−	0.706534i	$-0.249742\pi$
$709$	1.89444e7	1.41536	0.707679	+	0.706534i	$0.249742\pi$
$710$	0	0
$711$	4.64320e6	0.344464
$712$	1.14735e6	0.0848193
$713$	−3.45520e6	−0.254536
$714$	3.31195e6	0.243130
$715$	0	0
$716$	−1.04441e7	−0.761360
$717$	3.08647e6	0.224215
$718$	−3.35701e6	−0.243019
$719$	−2.52342e7	−1.82040	−0.910199	−	0.414170i	$-0.864072\pi$
$719$	−2.52342e7	−1.82040	−0.910199	+	0.414170i	$0.864072\pi$
$720$	0	0
$721$	−124425.	−0.00891393
$722$	−1.61929e6	−0.115606
$723$	−1.01132e7	−0.719521
$724$	1.44422e7	1.02397
$725$	0	0
$726$	−355874.	−0.0250585
$727$	−1.83133e7	−1.28508	−0.642541	−	0.766251i	$-0.722120\pi$
$727$	−1.83133e7	−1.28508	−0.642541	+	0.766251i	$0.722120\pi$
$728$	−1.19111e7	−0.832957
$729$	531441.	0.0370370
$730$	0	0
$731$	−3.47963e6	−0.240846
$732$	−3.75347e6	−0.258914
$733$	2.77295e7	1.90626	0.953130	−	0.302561i	$-0.0978415\pi$
$733$	2.77295e7	1.90626	0.953130	+	0.302561i	$0.0978415\pi$
$734$	−6.12441e6	−0.419589
$735$	0	0
$736$	−3.38862e6	−0.230584
$737$	2.34750e6	0.159198
$738$	2.78763e6	0.188406
$739$	6.45701e6	0.434931	0.217466	−	0.976068i	$-0.430221\pi$
$739$	6.45701e6	0.434931	0.217466	+	0.976068i	$0.430221\pi$
$740$	0	0
$741$	1.67041e7	1.11758
$742$	−78839.5	−0.00525695
$743$	1.64452e7	1.09287	0.546435	−	0.837502i	$-0.315985\pi$
$743$	1.64452e7	1.09287	0.546435	+	0.837502i	$0.315985\pi$
$744$	−8.31846e6	−0.550948
$745$	0	0
$746$	5.20455e6	0.342402
$747$	3.18773e6	0.209016
$748$	−5.54265e6	−0.362212
$749$	1.07983e7	0.703314
$750$	0	0
$751$	−1.41589e7	−0.916072	−0.458036	−	0.888934i	$-0.651447\pi$
$751$	−1.41589e7	−0.916072	−0.458036	+	0.888934i	$0.651447\pi$
$752$	5.74601e6	0.370529
$753$	6.90553e6	0.443823
$754$	−6.99325e6	−0.447972
$755$	0	0
$756$	1.32360e6	0.0842275
$757$	1.64472e7	1.04316	0.521581	−	0.853202i	$-0.325343\pi$
$757$	1.64472e7	1.04316	0.521581	+	0.853202i	$0.325343\pi$
$758$	1.21368e7	0.767237
$759$	623466.	0.0392833
$760$	0	0
$761$	2.78426e7	1.74280	0.871400	−	0.490573i	$-0.163213\pi$
$761$	2.78426e7	1.74280	0.871400	+	0.490573i	$0.163213\pi$
$762$	−4.16511e6	−0.259860
$763$	−1.62435e7	−1.01011
$764$	−1.29287e6	−0.0801350
$765$	0	0
$766$	−7.26777e6	−0.447537
$767$	−1.57546e7	−0.966984
$768$	−5.47586e6	−0.335003
$769$	1.96418e7	1.19775	0.598873	−	0.800844i	$-0.295615\pi$
$769$	1.96418e7	1.19775	0.598873	+	0.800844i	$0.295615\pi$
$770$	0	0
$771$	4.03907e6	0.244707
$772$	5.57710e6	0.336794
$773$	7.95055e6	0.478573	0.239287	−	0.970949i	$-0.423086\pi$
$773$	7.95055e6	0.478573	0.239287	+	0.970949i	$0.423086\pi$
$774$	410555.	0.0246331
$775$	0	0
$776$	7.07843e6	0.421971
$777$	3.21429e6	0.191000
$778$	3.66622e6	0.217155
$779$	−2.23479e7	−1.31945
$780$	0	0
$781$	−6.93796e6	−0.407009
$782$	−2.86680e6	−0.167641
$783$	1.78366e6	0.103970
$784$	−4.29989e6	−0.249843
$785$	0	0
$786$	−4.04843e6	−0.233738
$787$	4.35212e6	0.250475	0.125237	−	0.992127i	$-0.460031\pi$
$787$	4.35212e6	0.250475	0.125237	+	0.992127i	$0.460031\pi$
$788$	1.46355e7	0.839641
$789$	8.29343e6	0.474287
$790$	0	0
$791$	9.91150e6	0.563246
$792$	1.50101e6	0.0850295
$793$	1.78648e7	1.00882
$794$	9.30933e6	0.524043
$795$	0	0
$796$	2.67837e6	0.149826
$797$	1.48951e7	0.830609	0.415305	−	0.909682i	$-0.363675\pi$
$797$	1.48951e7	0.830609	0.415305	+	0.909682i	$0.363675\pi$
$798$	3.13274e6	0.174147
$799$	2.82606e7	1.56608
$800$	0	0
$801$	606831.	0.0334185
$802$	−1.51568e7	−0.832094
$803$	6.36308e6	0.348240
$804$	−4.31385e6	−0.235356
$805$	0	0
$806$	1.72497e7	0.935286
$807$	1.26122e7	0.681721
$808$	2.09974e7	1.13145
$809$	−1.06835e7	−0.573906	−0.286953	−	0.957945i	$-0.592642\pi$
$809$	−1.06835e7	−0.573906	−0.286953	+	0.957945i	$0.592642\pi$
$810$	0	0
$811$	1.64454e6	0.0877998	0.0438999	−	0.999036i	$-0.486022\pi$
$811$	1.64454e6	0.0877998	0.0438999	+	0.999036i	$0.486022\pi$
$812$	4.44238e6	0.236443
$813$	−1.81890e7	−0.965123
$814$	1.58812e6	0.0840082
$815$	0	0
$816$	6.29054e6	0.330721
$817$	−3.29134e6	−0.172511
$818$	−1.19182e7	−0.622771
$819$	−6.29976e6	−0.328182
$820$	0	0
$821$	−5.82113e6	−0.301404	−0.150702	−	0.988579i	$-0.548153\pi$
$821$	−5.82113e6	−0.301404	−0.150702	+	0.988579i	$0.548153\pi$
$822$	2.29967e6	0.118709
$823$	−2.30028e7	−1.18381	−0.591903	−	0.806009i	$-0.701623\pi$
$823$	−2.30028e7	−1.18381	−0.591903	+	0.806009i	$0.701623\pi$
$824$	259293.	0.0133037
$825$	0	0
$826$	−2.95467e6	−0.150681
$827$	1.19553e7	0.607853	0.303926	−	0.952696i	$-0.401702\pi$
$827$	1.19553e7	0.607853	0.303926	+	0.952696i	$0.401702\pi$
$828$	−1.14570e6	−0.0580760
$829$	2.22758e7	1.12576	0.562880	−	0.826538i	$-0.309693\pi$
$829$	2.22758e7	1.12576	0.562880	+	0.826538i	$0.309693\pi$
$830$	0	0
$831$	−670230.	−0.0336683
$832$	4.15072e6	0.207881
$833$	−2.11481e7	−1.05599
$834$	−7.32217e6	−0.364523
$835$	0	0
$836$	−5.24273e6	−0.259443
$837$	−4.39962e6	−0.217071
$838$	−1.78215e6	−0.0876666
$839$	1.07814e7	0.528773	0.264387	−	0.964417i	$-0.414830\pi$
$839$	1.07814e7	0.528773	0.264387	+	0.964417i	$0.414830\pi$
$840$	0	0
$841$	−1.45247e7	−0.708136
$842$	−8.92147e6	−0.433667
$843$	8.56355e6	0.415035
$844$	−1.27224e7	−0.614770
$845$	0	0
$846$	−3.33441e6	−0.160174
$847$	−1.07597e6	−0.0515336
$848$	−149743.	−0.00715086
$849$	4.82193e6	0.229589
$850$	0	0
$851$	−2.78227e6	−0.131697
$852$	1.27495e7	0.601717
$853$	1.90081e7	0.894470	0.447235	−	0.894416i	$-0.352409\pi$
$853$	1.90081e7	0.894470	0.447235	+	0.894416i	$0.352409\pi$
$854$	3.35042e6	0.157201
$855$	0	0
$856$	−2.25028e7	−1.04967
$857$	−1.95218e7	−0.907963	−0.453981	−	0.891011i	$-0.649997\pi$
$857$	−1.95218e7	−0.907963	−0.453981	+	0.891011i	$0.649997\pi$
$858$	−3.11259e6	−0.144346
$859$	−1.63370e7	−0.755420	−0.377710	−	0.925924i	$-0.623288\pi$
$859$	−1.63370e7	−0.755420	−0.377710	+	0.925924i	$0.623288\pi$
$860$	0	0
$861$	8.42825e6	0.387462
$862$	2.82070e6	0.129297
$863$	2.85401e7	1.30445	0.652226	−	0.758024i	$-0.273835\pi$
$863$	2.85401e7	1.30445	0.652226	+	0.758024i	$0.273835\pi$
$864$	−4.31485e6	−0.196644
$865$	0	0
$866$	−9.03752e6	−0.409501
$867$	1.81600e7	0.820480
$868$	−1.09577e7	−0.493650
$869$	6.93614e6	0.311579
$870$	0	0
$871$	2.05320e7	0.917034
$872$	3.38504e7	1.50755
$873$	3.74378e6	0.166255
$874$	−2.71168e6	−0.120077
$875$	0	0
$876$	−1.16930e7	−0.514833
$877$	−1.20298e7	−0.528153	−0.264076	−	0.964502i	$-0.585067\pi$
$877$	−1.20298e7	−0.528153	−0.264076	+	0.964502i	$0.585067\pi$
$878$	−1.41587e7	−0.619850
$879$	2.35500e7	1.02806
$880$	0	0
$881$	1.12631e7	0.488896	0.244448	−	0.969662i	$-0.421393\pi$
$881$	1.12631e7	0.488896	0.244448	+	0.969662i	$0.421393\pi$
$882$	2.49522e6	0.108004
$883$	8.83933e6	0.381520	0.190760	−	0.981637i	$-0.438905\pi$
$883$	8.83933e6	0.381520	0.190760	+	0.981637i	$0.438905\pi$
$884$	−4.84778e7	−2.08647
$885$	0	0
$886$	−1.07120e7	−0.458445
$887$	1.56404e7	0.667481	0.333741	−	0.942665i	$-0.391689\pi$
$887$	1.56404e7	0.667481	0.333741	+	0.942665i	$0.391689\pi$
$888$	−6.69836e6	−0.285060
$889$	−1.25930e7	−0.534410
$890$	0	0
$891$	793881.	0.0335013
$892$	1.41478e7	0.595358
$893$	2.67313e7	1.12174
$894$	970387.	0.0406070
$895$	0	0
$896$	−1.31408e7	−0.546831
$897$	5.45303e6	0.226286
$898$	9.98569e6	0.413226
$899$	−1.47664e7	−0.609360
$900$	0	0
$901$	−736482.	−0.0302239
$902$	4.16423e6	0.170419
$903$	1.24129e6	0.0506588
$904$	−2.06549e7	−0.840624
$905$	0	0
$906$	−2.77559e6	−0.112340
$907$	−4.34658e7	−1.75440	−0.877202	−	0.480121i	$-0.840593\pi$
$907$	−4.34658e7	−1.75440	−0.877202	+	0.480121i	$0.840593\pi$
$908$	−3.60184e7	−1.44981
$909$	1.11055e7	0.445788
$910$	0	0
$911$	2.58339e7	1.03132	0.515661	−	0.856792i	$-0.327546\pi$
$911$	2.58339e7	1.03132	0.515661	+	0.856792i	$0.327546\pi$
$912$	5.95015e6	0.236887
$913$	4.76191e6	0.189062
$914$	1.73960e7	0.688785
$915$	0	0
$916$	2.41894e7	0.952549
$917$	−1.22402e7	−0.480691
$918$	−3.65040e6	−0.142966
$919$	3.14288e7	1.22755	0.613775	−	0.789481i	$-0.289650\pi$
$919$	3.14288e7	1.22755	0.613775	+	0.789481i	$0.289650\pi$
$920$	0	0
$921$	2.18656e7	0.849399
$922$	−9.18612e6	−0.355881
$923$	−6.06816e7	−2.34451
$924$	1.97724e6	0.0761866
$925$	0	0
$926$	−1.62613e7	−0.623201
$927$	137140.	0.00524161
$928$	−1.44818e7	−0.552019
$929$	3.42592e7	1.30238	0.651190	−	0.758915i	$-0.274270\pi$
$929$	3.42592e7	1.30238	0.651190	+	0.758915i	$0.274270\pi$
$930$	0	0
$931$	−2.00037e7	−0.756375
$932$	3.91059e7	1.47469
$933$	5.14030e6	0.193323
$934$	−5.36698e6	−0.201309
$935$	0	0
$936$	1.31283e7	0.489799
$937$	−1.58561e7	−0.589995	−0.294997	−	0.955498i	$-0.595319\pi$
$937$	−1.58561e7	−0.589995	−0.294997	+	0.955498i	$0.595319\pi$
$938$	3.85063e6	0.142898
$939$	−1.98923e7	−0.736243
$940$	0	0
$941$	1.34320e7	0.494500	0.247250	−	0.968952i	$-0.420473\pi$
$941$	1.34320e7	0.494500	0.247250	+	0.968952i	$0.420473\pi$
$942$	1.45145e7	0.532935
$943$	−7.29544e6	−0.267160
$944$	−5.61194e6	−0.204966
$945$	0	0
$946$	613298.	0.0222815
$947$	1.43177e7	0.518796	0.259398	−	0.965770i	$-0.416476\pi$
$947$	1.43177e7	0.518796	0.259398	+	0.965770i	$0.416476\pi$
$948$	−1.27461e7	−0.460634
$949$	5.56535e7	2.00598
$950$	0	0
$951$	1.07747e7	0.386326
$952$	−2.08675e7	−0.746239
$953$	1.89929e6	0.0677420	0.0338710	−	0.999426i	$-0.489216\pi$
$953$	1.89929e6	0.0677420	0.0338710	+	0.999426i	$0.489216\pi$
$954$	86896.1	0.00309122
$955$	0	0
$956$	−8.47270e6	−0.299831
$957$	2.66449e6	0.0940445
$958$	−4.81555e6	−0.169524
$959$	6.95293e6	0.244130
$960$	0	0
$961$	7.79389e6	0.272236
$962$	1.38902e7	0.483916
$963$	−1.19017e7	−0.413565
$964$	2.77619e7	0.962180
$965$	0	0
$966$	1.02268e6	0.0352611
$967$	−2.40545e6	−0.0827237	−0.0413618	−	0.999144i	$-0.513170\pi$
$967$	−2.40545e6	−0.0827237	−0.0413618	+	0.999144i	$0.513170\pi$
$968$	2.24224e6	0.0769121
$969$	2.92646e7	1.00123
$970$	0	0
$971$	−1.46923e6	−0.0500083	−0.0250042	−	0.999687i	$-0.507960\pi$
$971$	−1.46923e6	−0.0500083	−0.0250042	+	0.999687i	$0.507960\pi$
$972$	−1.45886e6	−0.0495278
$973$	−2.21382e7	−0.749654
$974$	1.62243e7	0.547984
$975$	0	0
$976$	6.36361e6	0.213835
$977$	−2.02265e7	−0.677929	−0.338964	−	0.940799i	$-0.610077\pi$
$977$	−2.02265e7	−0.677929	−0.338964	+	0.940799i	$0.610077\pi$
$978$	8.33083e6	0.278510
$979$	906501.	0.0302281
$980$	0	0
$981$	1.79035e7	0.593970
$982$	−8.99877e6	−0.297786
$983$	−1.06114e7	−0.350257	−0.175129	−	0.984546i	$-0.556034\pi$
$983$	−1.06114e7	−0.350257	−0.175129	+	0.984546i	$0.556034\pi$
$984$	−1.75639e7	−0.578274
$985$	0	0
$986$	−1.22518e7	−0.401334
$987$	−1.00814e7	−0.329404
$988$	−4.58546e7	−1.49448
$989$	−1.07446e6	−0.0349299
$990$	0	0
$991$	−4.32715e6	−0.139965	−0.0699823	−	0.997548i	$-0.522294\pi$
$991$	−4.32715e6	−0.139965	−0.0699823	+	0.997548i	$0.522294\pi$
$992$	3.57212e7	1.15252
$993$	1.64535e7	0.529524
$994$	−1.13804e7	−0.365336
$995$	0	0
$996$	−8.75066e6	−0.279507
$997$	−4.80656e7	−1.53143	−0.765714	−	0.643181i	$-0.777614\pi$
$997$	−4.80656e7	−1.53143	−0.765714	+	0.643181i	$0.777614\pi$
$998$	−1.92101e7	−0.610524
$999$	−3.54276e6	−0.112312

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	825.6.a.t.1.5	✓	10
5.4	even	2		825.6.a.u.1.6	yes	10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
825.6.a.t.1.5	✓	10	1.1	even	1	trivial
825.6.a.u.1.6	yes	10	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)

\(f(q)\)	\(=\)	\(q-2.70074 q^{2} +9.00000 q^{3} -24.7060 q^{4} -24.3067 q^{6} -73.4900 q^{7} +153.148 q^{8} +81.0000 q^{9} +O(q^{10})\)	\(q-2.70074 q^{2} +9.00000 q^{3} -24.7060 q^{4} -24.3067 q^{6} -73.4900 q^{7} +153.148 q^{8} +81.0000 q^{9} +121.000 q^{11} -222.354 q^{12} +1058.30 q^{13} +198.478 q^{14} +376.977 q^{16} +1854.09 q^{17} -218.760 q^{18} +1753.76 q^{19} -661.410 q^{21} -326.790 q^{22} +572.513 q^{23} +1378.33 q^{24} -2858.21 q^{26} +729.000 q^{27} +1815.64 q^{28} +2446.73 q^{29} -6035.15 q^{31} -5918.86 q^{32} +1089.00 q^{33} -5007.41 q^{34} -2001.19 q^{36} -4859.75 q^{37} -4736.45 q^{38} +9524.74 q^{39} -12742.8 q^{41} +1786.30 q^{42} -1876.74 q^{43} -2989.42 q^{44} -1546.21 q^{46} +15242.3 q^{47} +3392.80 q^{48} -11406.2 q^{49} +16686.8 q^{51} -26146.5 q^{52} -397.221 q^{53} -1968.84 q^{54} -11254.9 q^{56} +15783.8 q^{57} -6607.98 q^{58} -14886.7 q^{59} +16880.6 q^{61} +16299.4 q^{62} -5952.69 q^{63} +3922.05 q^{64} -2941.11 q^{66} +19400.8 q^{67} -45807.0 q^{68} +5152.61 q^{69} -57338.6 q^{71} +12405.0 q^{72} +52587.5 q^{73} +13124.9 q^{74} -43328.3 q^{76} -8892.30 q^{77} -25723.9 q^{78} +57323.5 q^{79} +6561.00 q^{81} +34415.1 q^{82} +39354.7 q^{83} +16340.8 q^{84} +5068.58 q^{86} +22020.5 q^{87} +18530.9 q^{88} +7491.74 q^{89} -77774.8 q^{91} -14144.5 q^{92} -54316.4 q^{93} -41165.6 q^{94} -53269.8 q^{96} +46219.5 q^{97} +30805.2 q^{98} +9801.00 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - q^{2} + 90 q^{3} + 223 q^{4} - 9 q^{6} + 188 q^{7} + 177 q^{8} + 810 q^{9}+O(q^{10}) \) 10 * q - q^2 + 90 * q^3 + 223 * q^4 - 9 * q^6 + 188 * q^7 + 177 * q^8 + 810 * q^9	\( 10 q - q^{2} + 90 q^{3} + 223 q^{4} - 9 q^{6} + 188 q^{7} + 177 q^{8} + 810 q^{9} + 1210 q^{11} + 2007 q^{12} - 1102 q^{13} + 684 q^{14} + 7019 q^{16} + 1106 q^{17} - 81 q^{18} + 2586 q^{19} + 1692 q^{21} - 121 q^{22} + 2206 q^{23} + 1593 q^{24} + 12001 q^{26} + 7290 q^{27} + 14452 q^{28} + 5824 q^{29} - 4586 q^{31} - 10627 q^{32} + 10890 q^{33} + 7426 q^{34} + 18063 q^{36} - 18362 q^{37} + 44001 q^{38} - 9918 q^{39} - 5474 q^{41} + 6156 q^{42} + 20496 q^{43} + 26983 q^{44} + 19981 q^{46} - 14970 q^{47} + 63171 q^{48} + 68582 q^{49} + 9954 q^{51} - 58603 q^{52} + 61980 q^{53} - 729 q^{54} + 7132 q^{56} + 23274 q^{57} + 7161 q^{58} + 61190 q^{59} + 8230 q^{61} - 4509 q^{62} + 15228 q^{63} + 152223 q^{64} - 1089 q^{66} - 11930 q^{67} + 105598 q^{68} + 19854 q^{69} + 59822 q^{71} + 14337 q^{72} - 20680 q^{73} + 132564 q^{74} + 68165 q^{76} + 22748 q^{77} + 108009 q^{78} + 234494 q^{79} + 65610 q^{81} + 151948 q^{82} + 185478 q^{83} + 130068 q^{84} - 17825 q^{86} + 52416 q^{87} + 21417 q^{88} + 181834 q^{89} + 206274 q^{91} + 98373 q^{92} - 41274 q^{93} - 64998 q^{94} - 95643 q^{96} + 304358 q^{97} + 153453 q^{98} + 98010 q^{99}+O(q^{100}) \) 10 * q - q^2 + 90 * q^3 + 223 * q^4 - 9 * q^6 + 188 * q^7 + 177 * q^8 + 810 * q^9 + 1210 * q^11 + 2007 * q^12 - 1102 * q^13 + 684 * q^14 + 7019 * q^16 + 1106 * q^17 - 81 * q^18 + 2586 * q^19 + 1692 * q^21 - 121 * q^22 + 2206 * q^23 + 1593 * q^24 + 12001 * q^26 + 7290 * q^27 + 14452 * q^28 + 5824 * q^29 - 4586 * q^31 - 10627 * q^32 + 10890 * q^33 + 7426 * q^34 + 18063 * q^36 - 18362 * q^37 + 44001 * q^38 - 9918 * q^39 - 5474 * q^41 + 6156 * q^42 + 20496 * q^43 + 26983 * q^44 + 19981 * q^46 - 14970 * q^47 + 63171 * q^48 + 68582 * q^49 + 9954 * q^51 - 58603 * q^52 + 61980 * q^53 - 729 * q^54 + 7132 * q^56 + 23274 * q^57 + 7161 * q^58 + 61190 * q^59 + 8230 * q^61 - 4509 * q^62 + 15228 * q^63 + 152223 * q^64 - 1089 * q^66 - 11930 * q^67 + 105598 * q^68 + 19854 * q^69 + 59822 * q^71 + 14337 * q^72 - 20680 * q^73 + 132564 * q^74 + 68165 * q^76 + 22748 * q^77 + 108009 * q^78 + 234494 * q^79 + 65610 * q^81 + 151948 * q^82 + 185478 * q^83 + 130068 * q^84 - 17825 * q^86 + 52416 * q^87 + 21417 * q^88 + 181834 * q^89 + 206274 * q^91 + 98373 * q^92 - 41274 * q^93 - 64998 * q^94 - 95643 * q^96 + 304358 * q^97 + 153453 * q^98 + 98010 * q^99

Introduction

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