LMFDB - Embedded newform 825.6.a.u.1.1

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			1.1
Root			$-10.9094$ 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)

$f(q)$	$=$	$q-10.9094 q^{2} -9.00000 q^{3} +87.0148 q^{4} +98.1845 q^{6} -180.635 q^{7} -600.178 q^{8} +81.0000 q^{9} +O(q^{10})$	\(q-10.9094 q^{2} -9.00000 q^{3} +87.0148 q^{4} +98.1845 q^{6} -180.635 q^{7} -600.178 q^{8} +81.0000 q^{9} +121.000 q^{11} -783.133 q^{12} -61.0567 q^{13} +1970.61 q^{14} +3763.10 q^{16} +133.450 q^{17} -883.661 q^{18} +1994.80 q^{19} +1625.71 q^{21} -1320.04 q^{22} +445.111 q^{23} +5401.60 q^{24} +666.092 q^{26} -729.000 q^{27} -15717.9 q^{28} -6712.28 q^{29} -8738.84 q^{31} -21847.4 q^{32} -1089.00 q^{33} -1455.86 q^{34} +7048.20 q^{36} +14900.0 q^{37} -21762.1 q^{38} +549.511 q^{39} +2981.70 q^{41} -17735.5 q^{42} +6573.14 q^{43} +10528.8 q^{44} -4855.89 q^{46} +19188.0 q^{47} -33867.9 q^{48} +15821.9 q^{49} -1201.05 q^{51} -5312.84 q^{52} -20004.1 q^{53} +7952.95 q^{54} +108413. q^{56} -17953.2 q^{57} +73226.9 q^{58} -10643.6 q^{59} +8344.36 q^{61} +95335.4 q^{62} -14631.4 q^{63} +117923. q^{64} +11880.3 q^{66} -44102.8 q^{67} +11612.2 q^{68} -4006.00 q^{69} +47876.2 q^{71} -48614.4 q^{72} -27978.1 q^{73} -162550. q^{74} +173578. q^{76} -21856.8 q^{77} -5994.83 q^{78} +21302.9 q^{79} +6561.00 q^{81} -32528.6 q^{82} -93039.7 q^{83} +141461. q^{84} -71709.0 q^{86} +60410.5 q^{87} -72621.5 q^{88} -130595. q^{89} +11029.0 q^{91} +38731.2 q^{92} +78649.6 q^{93} -209330. q^{94} +196627. q^{96} +168032. q^{97} -172607. q^{98} +9801.00 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−10.9094	−1.92853	−0.964263	−	0.264947i	$-0.914646\pi$
$2$	−10.9094	−1.92853	−0.964263	+	0.264947i	$0.914646\pi$
$3$	−9.00000	−0.577350
$3$	−9.00000	−0.577350
$4$	87.0148	2.71921
$5$	0	0
$5$	0	0
$6$	98.1845	1.11343
$7$	−180.635	−1.39334	−0.696668	−	0.717393i	$-0.745335\pi$
$7$	−180.635	−1.39334	−0.696668	+	0.717393i	$0.745335\pi$
$8$	−600.178	−3.31555
$9$	81.0000	0.333333
$10$	0	0
$11$	121.000	0.301511
$11$	121.000	0.301511
$12$	−783.133	−1.56994
$13$	−61.0567	−0.100202	−0.0501009	−	0.998744i	$-0.515954\pi$
$13$	−61.0567	−0.100202	−0.0501009	+	0.998744i	$0.515954\pi$
$14$	1970.61	2.68709
$15$	0	0
$16$	3763.10	3.67490
$17$	133.450	0.111995	0.0559974	−	0.998431i	$-0.482166\pi$
$17$	133.450	0.111995	0.0559974	+	0.998431i	$0.482166\pi$
$18$	−883.661	−0.642842
$19$	1994.80	1.26770	0.633850	−	0.773456i	$-0.281474\pi$
$19$	1994.80	1.26770	0.633850	+	0.773456i	$0.281474\pi$
$20$	0	0
$21$	1625.71	0.804443
$22$	−1320.04	−0.581472
$23$	445.111	0.175448	0.0877241	−	0.996145i	$-0.472041\pi$
$23$	445.111	0.175448	0.0877241	+	0.996145i	$0.472041\pi$
$24$	5401.60	1.91423
$25$	0	0
$26$	666.092	0.193242
$27$	−729.000	−0.192450
$28$	−15717.9	−3.78878
$29$	−6712.28	−1.48209	−0.741046	−	0.671454i	$-0.765670\pi$
$29$	−6712.28	−1.48209	−0.741046	+	0.671454i	$0.765670\pi$
$30$	0	0
$31$	−8738.84	−1.63324	−0.816619	−	0.577177i	$-0.804154\pi$
$31$	−8738.84	−1.63324	−0.816619	+	0.577177i	$0.804154\pi$
$32$	−21847.4	−3.77160
$33$	−1089.00	−0.174078
$34$	−1455.86	−0.215985
$35$	0	0
$36$	7048.20	0.906404
$37$	14900.0	1.78930	0.894650	−	0.446769i	$-0.147425\pi$
$37$	14900.0	1.78930	0.894650	+	0.446769i	$0.147425\pi$
$38$	−21762.1	−2.44479
$39$	549.511	0.0578515
$40$	0	0
$41$	2981.70	0.277016	0.138508	−	0.990361i	$-0.455769\pi$
$41$	2981.70	0.277016	0.138508	+	0.990361i	$0.455769\pi$
$42$	−17735.5	−1.55139
$43$	6573.14	0.542128	0.271064	−	0.962561i	$-0.412624\pi$
$43$	6573.14	0.542128	0.271064	+	0.962561i	$0.412624\pi$
$44$	10528.8	0.819873
$45$	0	0
$46$	−4855.89	−0.338356
$47$	19188.0	1.26703	0.633514	−	0.773731i	$-0.281612\pi$
$47$	19188.0	1.26703	0.633514	+	0.773731i	$0.281612\pi$
$48$	−33867.9	−2.12171
$49$	15821.9	0.941388
$50$	0	0
$51$	−1201.05	−0.0646602
$52$	−5312.84	−0.272470
$53$	−20004.1	−0.978205	−0.489102	−	0.872226i	$-0.662676\pi$
$53$	−20004.1	−0.978205	−0.489102	+	0.872226i	$0.662676\pi$
$54$	7952.95	0.371145
$55$	0	0
$56$	108413.	4.61967
$57$	−17953.2	−0.731907
$58$	73226.9	2.85825
$59$	−10643.6	−0.398070	−0.199035	−	0.979992i	$-0.563781\pi$
$59$	−10643.6	−0.398070	−0.199035	+	0.979992i	$0.563781\pi$
$60$	0	0
$61$	8344.36	0.287123	0.143562	−	0.989641i	$-0.454144\pi$
$61$	8344.36	0.287123	0.143562	+	0.989641i	$0.454144\pi$
$62$	95335.4	3.14974
$63$	−14631.4	−0.464446
$64$	117923.	3.59873
$65$	0	0
$66$	11880.3	0.335713
$67$	−44102.8	−1.20027	−0.600136	−	0.799898i	$-0.704887\pi$
$67$	−44102.8	−1.20027	−0.600136	+	0.799898i	$0.704887\pi$
$68$	11612.2	0.304537
$69$	−4006.00	−0.101295
$70$	0	0
$71$	47876.2	1.12713	0.563565	−	0.826072i	$-0.309430\pi$
$71$	47876.2	1.12713	0.563565	+	0.826072i	$0.309430\pi$
$72$	−48614.4	−1.10518
$73$	−27978.1	−0.614486	−0.307243	−	0.951631i	$-0.599406\pi$
$73$	−27978.1	−0.614486	−0.307243	+	0.951631i	$0.599406\pi$
$74$	−162550.	−3.45071
$75$	0	0
$76$	173578.	3.44714
$77$	−21856.8	−0.420107
$78$	−5994.83	−0.111568
$79$	21302.9	0.384035	0.192018	−	0.981391i	$-0.438497\pi$
$79$	21302.9	0.384035	0.192018	+	0.981391i	$0.438497\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	−32528.6	−0.534232
$83$	−93039.7	−1.48243	−0.741213	−	0.671270i	$-0.765749\pi$
$83$	−93039.7	−1.48243	−0.741213	+	0.671270i	$0.765749\pi$
$84$	141461.	2.18745
$85$	0	0
$86$	−71709.0	−1.04551
$87$	60410.5	0.855686
$88$	−72621.5	−0.999675
$89$	−130595.	−1.74764	−0.873820	−	0.486249i	$-0.838365\pi$
$89$	−130595.	−1.74764	−0.873820	+	0.486249i	$0.838365\pi$
$90$	0	0
$91$	11029.0	0.139615
$92$	38731.2	0.477081
$93$	78649.6	0.942950
$94$	−209330.	−2.44350
$95$	0	0
$96$	196627.	2.17753
$97$	168032.	1.81328	0.906638	−	0.421910i	$-0.138640\pi$
$97$	168032.	1.81328	0.906638	+	0.421910i	$0.138640\pi$
$98$	−172607.	−1.81549
$99$	9801.00	0.100504
$100$	0	0
$101$	29236.1	0.285178	0.142589	−	0.989782i	$-0.454457\pi$
$101$	29236.1	0.285178	0.142589	+	0.989782i	$0.454457\pi$
$102$	13102.8	0.124699
$103$	152844.	1.41956	0.709781	−	0.704423i	$-0.248794\pi$
$103$	152844.	1.41956	0.709781	+	0.704423i	$0.248794\pi$
$104$	36644.9	0.332223
$105$	0	0
$106$	218233.	1.88649
$107$	−78730.7	−0.664790	−0.332395	−	0.943140i	$-0.607857\pi$
$107$	−78730.7	−0.664790	−0.332395	+	0.943140i	$0.607857\pi$
$108$	−63433.8	−0.523313
$109$	−76836.5	−0.619443	−0.309722	−	0.950827i	$-0.600236\pi$
$109$	−76836.5	−0.619443	−0.309722	+	0.950827i	$0.600236\pi$
$110$	0	0
$111$	−134100.	−1.03305
$112$	−679747.	−5.12038
$113$	−17298.0	−0.127438	−0.0637190	−	0.997968i	$-0.520296\pi$
$113$	−17298.0	−0.127438	−0.0637190	+	0.997968i	$0.520296\pi$
$114$	195859.	1.41150
$115$	0	0
$116$	−584068.	−4.03012
$117$	−4945.60	−0.0334006
$118$	116115.	0.767688
$119$	−24105.8	−0.156046
$120$	0	0
$121$	14641.0	0.0909091
$122$	−91031.9	−0.553725
$123$	−26835.3	−0.159935
$124$	−760408.	−4.44112
$125$	0	0
$126$	159620.	0.895695
$127$	−213760.	−1.17603	−0.588014	−	0.808851i	$-0.700090\pi$
$127$	−213760.	−1.17603	−0.588014	+	0.808851i	$0.700090\pi$
$128$	−587350.	−3.16864
$129$	−59158.3	−0.312998
$130$	0	0
$131$	−230756.	−1.17483	−0.587415	−	0.809286i	$-0.699855\pi$
$131$	−230756.	−1.17483	−0.587415	+	0.809286i	$0.699855\pi$
$132$	−94759.1	−0.473354
$133$	−360331.	−1.76633
$134$	481135.	2.31476
$135$	0	0
$136$	−80094.0	−0.371324
$137$	−256086.	−1.16570	−0.582848	−	0.812581i	$-0.698062\pi$
$137$	−256086.	−1.16570	−0.582848	+	0.812581i	$0.698062\pi$
$138$	43703.0	0.195350
$139$	361970.	1.58904	0.794521	−	0.607237i	$-0.207722\pi$
$139$	361970.	1.58904	0.794521	+	0.607237i	$0.207722\pi$
$140$	0	0
$141$	−172692.	−0.731519
$142$	−522300.	−2.17370
$143$	−7387.87	−0.0302120
$144$	304811.	1.22497
$145$	0	0
$146$	305224.	1.18505
$147$	−142397.	−0.543510
$148$	1.29652e6	4.86548
$149$	−85280.8	−0.314692	−0.157346	−	0.987544i	$-0.550294\pi$
$149$	−85280.8	−0.314692	−0.157346	+	0.987544i	$0.550294\pi$
$150$	0	0
$151$	−540222.	−1.92810	−0.964050	−	0.265721i	$-0.914390\pi$
$151$	−540222.	−1.92810	−0.964050	+	0.265721i	$0.914390\pi$
$152$	−1.19724e6	−4.20312
$153$	10809.5	0.0373316
$154$	238444.	0.810187
$155$	0	0
$156$	47815.6	0.157311
$157$	−28945.4	−0.0937197	−0.0468598	−	0.998901i	$-0.514921\pi$
$157$	−28945.4	−0.0937197	−0.0468598	+	0.998901i	$0.514921\pi$
$158$	−232402.	−0.740622
$159$	180037.	0.564767
$160$	0	0
$161$	−80402.5	−0.244458
$162$	−71576.5	−0.214281
$163$	350161.	1.03228	0.516141	−	0.856504i	$-0.327368\pi$
$163$	350161.	1.03228	0.516141	+	0.856504i	$0.327368\pi$
$164$	259452.	0.753265
$165$	0	0
$166$	1.01501e6	2.85890
$167$	−320597.	−0.889545	−0.444772	−	0.895644i	$-0.646715\pi$
$167$	−320597.	−0.889545	−0.444772	+	0.895644i	$0.646715\pi$
$168$	−975717.	−2.66717
$169$	−367565.	−0.989960
$170$	0	0
$171$	161579.	0.422567
$172$	571961.	1.47416
$173$	−572203.	−1.45357	−0.726784	−	0.686866i	$-0.758986\pi$
$173$	−572203.	−1.45357	−0.726784	+	0.686866i	$0.758986\pi$
$174$	−659042.	−1.65021
$175$	0	0
$176$	455335.	1.10802
$177$	95792.5	0.229826
$178$	1.42471e6	3.37037
$179$	−327870.	−0.764837	−0.382419	−	0.923989i	$-0.624909\pi$
$179$	−327870.	−0.764837	−0.382419	+	0.923989i	$0.624909\pi$
$180$	0	0
$181$	709192.	1.60904	0.804521	−	0.593924i	$-0.202422\pi$
$181$	709192.	1.60904	0.804521	+	0.593924i	$0.202422\pi$
$182$	−120319.	−0.269251
$183$	−75099.2	−0.165771
$184$	−267146.	−0.581706
$185$	0	0
$186$	−858019.	−1.81850
$187$	16147.5	0.0337677
$188$	1.66964e6	3.44532
$189$	131683.	0.268148
$190$	0	0
$191$	283572.	0.562445	0.281223	−	0.959643i	$-0.409260\pi$
$191$	283572.	0.562445	0.281223	+	0.959643i	$0.409260\pi$
$192$	−1.06131e6	−2.07773
$193$	452910.	0.875222	0.437611	−	0.899164i	$-0.355825\pi$
$193$	452910.	0.875222	0.437611	+	0.899164i	$0.355825\pi$
$194$	−1.83313e6	−3.49695
$195$	0	0
$196$	1.37674e6	2.55983
$197$	−950326.	−1.74465	−0.872323	−	0.488931i	$-0.837387\pi$
$197$	−950326.	−1.74465	−0.872323	+	0.488931i	$0.837387\pi$
$198$	−106923.	−0.193824
$199$	−490008.	−0.877143	−0.438571	−	0.898696i	$-0.644515\pi$
$199$	−490008.	−0.877143	−0.438571	+	0.898696i	$0.644515\pi$
$200$	0	0
$201$	396926.	0.692977
$202$	−318948.	−0.549973
$203$	1.21247e6	2.06505
$204$	−104509.	−0.175825
$205$	0	0
$206$	−1.66743e6	−2.73766
$207$	36054.0	0.0584827
$208$	−229763.	−0.368232
$209$	241371.	0.382226
$210$	0	0
$211$	−875775.	−1.35421	−0.677106	−	0.735886i	$-0.736766\pi$
$211$	−875775.	−1.35421	−0.677106	+	0.735886i	$0.736766\pi$
$212$	−1.74065e6	−2.65995
$213$	−430886.	−0.650748
$214$	858904.	1.28207
$215$	0	0
$216$	437530.	0.638077
$217$	1.57854e6	2.27565
$218$	838240.	1.19461
$219$	251803.	0.354773
$220$	0	0
$221$	−8148.05	−0.0112221
$222$	1.46295e6	1.99227
$223$	769593.	1.03633	0.518166	−	0.855280i	$-0.326615\pi$
$223$	769593.	1.03633	0.518166	+	0.855280i	$0.326615\pi$
$224$	3.94641e6	5.25511
$225$	0	0
$226$	188710.	0.245767
$227$	438703.	0.565075	0.282537	−	0.959256i	$-0.408824\pi$
$227$	438703.	0.565075	0.282537	+	0.959256i	$0.408824\pi$
$228$	−1.56220e6	−1.99021
$229$	561279.	0.707277	0.353639	−	0.935382i	$-0.384944\pi$
$229$	561279.	0.707277	0.353639	+	0.935382i	$0.384944\pi$
$230$	0	0
$231$	196711.	0.242549
$232$	4.02856e6	4.91394
$233$	419090.	0.505728	0.252864	−	0.967502i	$-0.418627\pi$
$233$	419090.	0.505728	0.252864	+	0.967502i	$0.418627\pi$
$234$	53953.4	0.0644139
$235$	0	0
$236$	−926152.	−1.08244
$237$	−191726.	−0.221723
$238$	262979.	0.300939
$239$	903602.	1.02325	0.511626	−	0.859208i	$-0.329043\pi$
$239$	903602.	1.02325	0.511626	+	0.859208i	$0.329043\pi$
$240$	0	0
$241$	571945.	0.634324	0.317162	−	0.948371i	$-0.397270\pi$
$241$	571945.	0.634324	0.317162	+	0.948371i	$0.397270\pi$
$242$	−159724.	−0.175321
$243$	−59049.0	−0.0641500
$244$	726083.	0.780749
$245$	0	0
$246$	292757.	0.308439
$247$	−121796.	−0.127026
$248$	5.24486e6	5.41507
$249$	837357.	0.855879
$250$	0	0
$251$	842162.	0.843746	0.421873	−	0.906655i	$-0.361373\pi$
$251$	842162.	0.843746	0.421873	+	0.906655i	$0.361373\pi$
$252$	−1.27315e6	−1.26293
$253$	53858.4	0.0528996
$254$	2.33199e6	2.26800
$255$	0	0
$256$	2.63410e6	2.51207
$257$	−886698.	−0.837419	−0.418709	−	0.908120i	$-0.637517\pi$
$257$	−886698.	−0.837419	−0.418709	+	0.908120i	$0.637517\pi$
$258$	645381.	0.603624
$259$	−2.69146e6	−2.49310
$260$	0	0
$261$	−543695.	−0.494031
$262$	2.51741e6	2.26569
$263$	−980651.	−0.874229	−0.437114	−	0.899406i	$-0.643999\pi$
$263$	−980651.	−0.874229	−0.437114	+	0.899406i	$0.643999\pi$
$264$	653594.	0.577162
$265$	0	0
$266$	3.93099e6	3.40642
$267$	1.17536e6	1.00900
$268$	−3.83760e6	−3.26379
$269$	−1.22085e6	−1.02868	−0.514342	−	0.857585i	$-0.671964\pi$
$269$	−1.22085e6	−1.02868	−0.514342	+	0.857585i	$0.671964\pi$
$270$	0	0
$271$	270251.	0.223534	0.111767	−	0.993734i	$-0.464349\pi$
$271$	270251.	0.223534	0.111767	+	0.993734i	$0.464349\pi$
$272$	502187.	0.411570
$273$	−99260.7	−0.0806066
$274$	2.79375e6	2.24807
$275$	0	0
$276$	−348581.	−0.275443
$277$	1.09672e6	0.858812	0.429406	−	0.903112i	$-0.358723\pi$
$277$	1.09672e6	0.858812	0.429406	+	0.903112i	$0.358723\pi$
$278$	−3.94887e6	−3.06451
$279$	−707846.	−0.544413
$280$	0	0
$281$	781595.	0.590494	0.295247	−	0.955421i	$-0.404598\pi$
$281$	781595.	0.590494	0.295247	+	0.955421i	$0.404598\pi$
$282$	1.88397e6	1.41075
$283$	−2.19428e6	−1.62864	−0.814320	−	0.580416i	$-0.802890\pi$
$283$	−2.19428e6	−1.62864	−0.814320	+	0.580416i	$0.802890\pi$
$284$	4.16594e6	3.06490
$285$	0	0
$286$	80597.1	0.0582646
$287$	−538599.	−0.385976
$288$	−1.76964e6	−1.25720
$289$	−1.40205e6	−0.987457
$290$	0	0
$291$	−1.51229e6	−1.04690
$292$	−2.43451e6	−1.67092
$293$	−221759.	−0.150908	−0.0754541	−	0.997149i	$-0.524041\pi$
$293$	−221759.	−0.150908	−0.0754541	+	0.997149i	$0.524041\pi$
$294$	1.55347e6	1.04817
$295$	0	0
$296$	−8.94267e6	−5.93250
$297$	−88209.0	−0.0580259
$298$	930361.	0.606891
$299$	−27177.0	−0.0175802
$300$	0	0
$301$	−1.18734e6	−0.755367
$302$	5.89349e6	3.71839
$303$	−263125.	−0.164648
$304$	7.50665e6	4.65867
$305$	0	0
$306$	−117925.	−0.0719949
$307$	519503.	0.314588	0.157294	−	0.987552i	$-0.449723\pi$
$307$	519503.	0.314588	0.157294	+	0.987552i	$0.449723\pi$
$308$	−1.90186e6	−1.14236
$309$	−1.37559e6	−0.819584
$310$	0	0
$311$	558483.	0.327423	0.163711	−	0.986508i	$-0.447653\pi$
$311$	558483.	0.327423	0.163711	+	0.986508i	$0.447653\pi$
$312$	−329804.	−0.191809
$313$	1.24474e6	0.718156	0.359078	−	0.933308i	$-0.383091\pi$
$313$	1.24474e6	0.718156	0.359078	+	0.933308i	$0.383091\pi$
$314$	315777.	0.180741
$315$	0	0
$316$	1.85367e6	1.04427
$317$	828735.	0.463199	0.231600	−	0.972811i	$-0.425604\pi$
$317$	828735.	0.463199	0.231600	+	0.972811i	$0.425604\pi$
$318$	−1.96409e6	−1.08917
$319$	−812186.	−0.446867
$320$	0	0
$321$	708576.	0.383817
$322$	877142.	0.471444
$323$	266207.	0.141976
$324$	570904.	0.302135
$325$	0	0
$326$	−3.82004e6	−1.99078
$327$	691529.	0.357636
$328$	−1.78955e6	−0.918459
$329$	−3.46603e6	−1.76540
$330$	0	0
$331$	3.26287e6	1.63693	0.818464	−	0.574558i	$-0.194826\pi$
$331$	3.26287e6	1.63693	0.818464	+	0.574558i	$0.194826\pi$
$332$	−8.09583e6	−4.03103
$333$	1.20690e6	0.596433
$334$	3.49751e6	1.71551
$335$	0	0
$336$	6.11772e6	2.95625
$337$	−1.48364e6	−0.711627	−0.355814	−	0.934557i	$-0.615796\pi$
$337$	−1.48364e6	−0.711627	−0.355814	+	0.934557i	$0.615796\pi$
$338$	4.00991e6	1.90916
$339$	155682.	0.0735763
$340$	0	0
$341$	−1.05740e6	−0.492440
$342$	−1.76273e6	−0.814931
$343$	177943.	0.0816668
$344$	−3.94505e6	−1.79745
$345$	0	0
$346$	6.24239e6	2.80324
$347$	3.91376e6	1.74490	0.872451	−	0.488702i	$-0.162529\pi$
$347$	3.91376e6	1.74490	0.872451	+	0.488702i	$0.162529\pi$
$348$	5.25661e6	2.32679
$349$	426649.	0.187503	0.0937513	−	0.995596i	$-0.470114\pi$
$349$	426649.	0.187503	0.0937513	+	0.995596i	$0.470114\pi$
$350$	0	0
$351$	44510.4	0.0192838
$352$	−2.64354e6	−1.13718
$353$	3.34984e6	1.43083	0.715414	−	0.698701i	$-0.246238\pi$
$353$	3.34984e6	1.43083	0.715414	+	0.698701i	$0.246238\pi$
$354$	−1.04504e6	−0.443225
$355$	0	0
$356$	−1.13637e7	−4.75221
$357$	216952.	0.0900934
$358$	3.57686e6	1.47501
$359$	−390605.	−0.159957	−0.0799783	−	0.996797i	$-0.525485\pi$
$359$	−390605.	−0.159957	−0.0799783	+	0.996797i	$0.525485\pi$
$360$	0	0
$361$	1.50315e6	0.607063
$362$	−7.73686e6	−3.10308
$363$	−131769.	−0.0524864
$364$	959683.	0.379642
$365$	0	0
$366$	819287.	0.319693
$367$	2.56846e6	0.995423	0.497712	−	0.867343i	$-0.334174\pi$
$367$	2.56846e6	0.995423	0.497712	+	0.867343i	$0.334174\pi$
$368$	1.67500e6	0.644755
$369$	241518.	0.0923386
$370$	0	0
$371$	3.61344e6	1.36297
$372$	6.84367e6	2.56408
$373$	−562503.	−0.209340	−0.104670	−	0.994507i	$-0.533379\pi$
$373$	−562503.	−0.209340	−0.104670	+	0.994507i	$0.533379\pi$
$374$	−176159.	−0.0651218
$375$	0	0
$376$	−1.15162e7	−4.20089
$377$	409830.	0.148508
$378$	−1.43658e6	−0.517130
$379$	−4.40309e6	−1.57456	−0.787281	−	0.616595i	$-0.788512\pi$
$379$	−4.40309e6	−1.57456	−0.787281	+	0.616595i	$0.788512\pi$
$380$	0	0
$381$	1.92384e6	0.678980
$382$	−3.09360e6	−1.08469
$383$	1.62003e6	0.564322	0.282161	−	0.959367i	$-0.408949\pi$
$383$	1.62003e6	0.564322	0.282161	+	0.959367i	$0.408949\pi$
$384$	5.28615e6	1.82941
$385$	0	0
$386$	−4.94097e6	−1.68789
$387$	532425.	0.180709
$388$	1.46213e7	4.93068
$389$	2.38756e6	0.799981	0.399991	−	0.916519i	$-0.369013\pi$
$389$	2.38756e6	0.799981	0.399991	+	0.916519i	$0.369013\pi$
$390$	0	0
$391$	59400.2	0.0196493
$392$	−9.49595e6	−3.12121
$393$	2.07681e6	0.678289
$394$	1.03675e7	3.36459
$395$	0	0
$396$	852832.	0.273291
$397$	799422.	0.254566	0.127283	−	0.991866i	$-0.459374\pi$
$397$	799422.	0.254566	0.127283	+	0.991866i	$0.459374\pi$
$398$	5.34569e6	1.69159
$399$	3.24298e6	1.01979
$400$	0	0
$401$	1.55040e6	0.481485	0.240743	−	0.970589i	$-0.422609\pi$
$401$	1.55040e6	0.481485	0.240743	+	0.970589i	$0.422609\pi$
$402$	−4.33022e6	−1.33642
$403$	533565.	0.163653
$404$	2.54397e6	0.775459
$405$	0	0
$406$	−1.32273e7	−3.98251
$407$	1.80290e6	0.539494
$408$	720846.	0.214384
$409$	−5.50372e6	−1.62685	−0.813426	−	0.581669i	$-0.802400\pi$
$409$	−5.50372e6	−1.62685	−0.813426	+	0.581669i	$0.802400\pi$
$410$	0	0
$411$	2.30478e6	0.673015
$412$	1.32997e7	3.86009
$413$	1.92261e6	0.554645
$414$	−393327.	−0.112785
$415$	0	0
$416$	1.33393e6	0.377921
$417$	−3.25773e6	−0.917433
$418$	−2.63321e6	−0.737132
$419$	−5.79996e6	−1.61395	−0.806975	−	0.590586i	$-0.798897\pi$
$419$	−5.79996e6	−1.61395	−0.806975	+	0.590586i	$0.798897\pi$
$420$	0	0
$421$	5.04301e6	1.38671	0.693353	−	0.720598i	$-0.256132\pi$
$421$	5.04301e6	1.38671	0.693353	+	0.720598i	$0.256132\pi$
$422$	9.55417e6	2.61163
$423$	1.55423e6	0.422343
$424$	1.20060e7	3.24328
$425$	0	0
$426$	4.70070e6	1.25498
$427$	−1.50728e6	−0.400059
$428$	−6.85074e6	−1.80771
$429$	66490.8	0.0174429
$430$	0	0
$431$	−784866.	−0.203518	−0.101759	−	0.994809i	$-0.532447\pi$
$431$	−784866.	−0.203518	−0.101759	+	0.994809i	$0.532447\pi$
$432$	−2.74330e6	−0.707235
$433$	−3.31581e6	−0.849903	−0.424952	−	0.905216i	$-0.639709\pi$
$433$	−3.31581e6	−0.849903	−0.424952	+	0.905216i	$0.639709\pi$
$434$	−1.72209e7	−4.38865
$435$	0	0
$436$	−6.68591e6	−1.68440
$437$	887909.	0.222416
$438$	−2.74702e6	−0.684190
$439$	6.22581e6	1.54182	0.770912	−	0.636942i	$-0.219801\pi$
$439$	6.22581e6	1.54182	0.770912	+	0.636942i	$0.219801\pi$
$440$	0	0
$441$	1.28157e6	0.313796
$442$	88890.2	0.0216420
$443$	6.60216e6	1.59837	0.799183	−	0.601087i	$-0.205266\pi$
$443$	6.60216e6	1.59837	0.799183	+	0.601087i	$0.205266\pi$
$444$	−1.16687e7	−2.80909
$445$	0	0
$446$	−8.39579e6	−1.99859
$447$	767527.	0.181687
$448$	−2.13010e7	−5.01424
$449$	943642.	0.220898	0.110449	−	0.993882i	$-0.464771\pi$
$449$	943642.	0.220898	0.110449	+	0.993882i	$0.464771\pi$
$450$	0	0
$451$	360786.	0.0835234
$452$	−1.50518e6	−0.346531
$453$	4.86199e6	1.11319
$454$	−4.78598e6	−1.08976
$455$	0	0
$456$	1.07751e7	2.42667
$457$	−4.58271e6	−1.02644	−0.513218	−	0.858258i	$-0.671547\pi$
$457$	−4.58271e6	−1.02644	−0.513218	+	0.858258i	$0.671547\pi$
$458$	−6.12321e6	−1.36400
$459$	−97285.3	−0.0215534
$460$	0	0
$461$	5.96368e6	1.30696	0.653480	−	0.756944i	$-0.273308\pi$
$461$	5.96368e6	1.30696	0.653480	+	0.756944i	$0.273308\pi$
$462$	−2.14600e6	−0.467762
$463$	3.33367e6	0.722721	0.361361	−	0.932426i	$-0.382312\pi$
$463$	3.33367e6	0.722721	0.361361	+	0.932426i	$0.382312\pi$
$464$	−2.52590e7	−5.44654
$465$	0	0
$466$	−4.57201e6	−0.975310
$467$	5.51891e6	1.17101	0.585505	−	0.810669i	$-0.300896\pi$
$467$	5.51891e6	1.17101	0.585505	+	0.810669i	$0.300896\pi$
$468$	−430340.	−0.0908233
$469$	7.96651e6	1.67238
$470$	0	0
$471$	260509.	0.0541091
$472$	6.38806e6	1.31982
$473$	795350.	0.163458
$474$	2.09162e6	0.427598
$475$	0	0
$476$	−2.09756e6	−0.424323
$477$	−1.62033e6	−0.326068
$478$	−9.85775e6	−1.97337
$479$	6.32499e6	1.25957	0.629783	−	0.776771i	$-0.283144\pi$
$479$	6.32499e6	1.25957	0.629783	+	0.776771i	$0.283144\pi$
$480$	0	0
$481$	−909748.	−0.179291
$482$	−6.23957e6	−1.22331
$483$	723622.	0.141138
$484$	1.27398e6	0.247201
$485$	0	0
$486$	644189.	0.123715
$487$	−2.28585e6	−0.436743	−0.218371	−	0.975866i	$-0.570074\pi$
$487$	−2.28585e6	−0.436743	−0.218371	+	0.975866i	$0.570074\pi$
$488$	−5.00810e6	−0.951970
$489$	−3.15145e6	−0.595988
$490$	0	0
$491$	9.30686e6	1.74221	0.871103	−	0.491101i	$-0.163405\pi$
$491$	9.30686e6	1.74221	0.871103	+	0.491101i	$0.163405\pi$
$492$	−2.33507e6	−0.434898
$493$	−895756.	−0.165986
$494$	1.32872e6	0.244972
$495$	0	0
$496$	−3.28851e7	−6.00199
$497$	−8.64810e6	−1.57047
$498$	−9.13506e6	−1.65059
$499$	−3.44825e6	−0.619937	−0.309969	−	0.950747i	$-0.600319\pi$
$499$	−3.44825e6	−0.619937	−0.309969	+	0.950747i	$0.600319\pi$
$500$	0	0
$501$	2.88537e6	0.513579
$502$	−9.18748e6	−1.62719
$503$	−3.22740e6	−0.568764	−0.284382	−	0.958711i	$-0.591788\pi$
$503$	−3.22740e6	−0.568764	−0.284382	+	0.958711i	$0.591788\pi$
$504$	8.78145e6	1.53989
$505$	0	0
$506$	−587563.	−0.102018
$507$	3.30809e6	0.571553
$508$	−1.86003e7	−3.19787
$509$	−516724.	−0.0884025	−0.0442012	−	0.999023i	$-0.514074\pi$
$509$	−516724.	−0.0884025	−0.0442012	+	0.999023i	$0.514074\pi$
$510$	0	0
$511$	5.05382e6	0.856185
$512$	−9.94118e6	−1.67596
$513$	−1.45421e6	−0.243969
$514$	9.67333e6	1.61498
$515$	0	0
$516$	−5.14765e6	−0.851108
$517$	2.32175e6	0.382023
$518$	2.93622e7	4.80800
$519$	5.14983e6	0.839218
$520$	0	0
$521$	−9.12427e6	−1.47266	−0.736332	−	0.676620i	$-0.763444\pi$
$521$	−9.12427e6	−1.47266	−0.736332	+	0.676620i	$0.763444\pi$
$522$	5.93138e6	0.952751
$523$	731400.	0.116923	0.0584616	−	0.998290i	$-0.481380\pi$
$523$	731400.	0.116923	0.0584616	+	0.998290i	$0.481380\pi$
$524$	−2.00792e7	−3.19461
$525$	0	0
$526$	1.06983e7	1.68597
$527$	−1.16620e6	−0.182914
$528$	−4.09802e6	−0.639718
$529$	−6.23822e6	−0.969218
$530$	0	0
$531$	−862133.	−0.132690
$532$	−3.13541e7	−4.80303
$533$	−182053.	−0.0277575
$534$	−1.28224e7	−1.94588
$535$	0	0
$536$	2.64696e7	3.97956
$537$	2.95083e6	0.441579
$538$	1.33188e7	1.98385
$539$	1.91445e6	0.283839
$540$	0	0
$541$	6.38698e6	0.938215	0.469108	−	0.883141i	$-0.344576\pi$
$541$	6.38698e6	0.938215	0.469108	+	0.883141i	$0.344576\pi$
$542$	−2.94827e6	−0.431091
$543$	−6.38273e6	−0.928981
$544$	−2.91555e6	−0.422399
$545$	0	0
$546$	1.08287e6	0.155452
$547$	1.06541e7	1.52247	0.761233	−	0.648479i	$-0.224594\pi$
$547$	1.06541e7	1.52247	0.761233	+	0.648479i	$0.224594\pi$
$548$	−2.22833e7	−3.16977
$549$	675893.	0.0957078
$550$	0	0
$551$	−1.33897e7	−1.87885
$552$	2.40431e6	0.335848
$553$	−3.84804e6	−0.535091
$554$	−1.19646e7	−1.65624
$555$	0	0
$556$	3.14967e7	4.32094
$557$	−1.41194e7	−1.92831	−0.964157	−	0.265331i	$-0.914519\pi$
$557$	−1.41194e7	−1.92831	−0.964157	+	0.265331i	$0.914519\pi$
$558$	7.72217e6	1.04991
$559$	−401335.	−0.0543222
$560$	0	0
$561$	−145327.	−0.0194958
$562$	−8.52672e6	−1.13878
$563$	1.31948e7	1.75441	0.877204	−	0.480118i	$-0.159406\pi$
$563$	1.31948e7	1.75441	0.877204	+	0.480118i	$0.159406\pi$
$564$	−1.50268e7	−1.98916
$565$	0	0
$566$	2.39382e7	3.14088
$567$	−1.18514e6	−0.154815
$568$	−2.87342e7	−3.73705
$569$	6.06676e6	0.785555	0.392777	−	0.919634i	$-0.371514\pi$
$569$	6.06676e6	0.785555	0.392777	+	0.919634i	$0.371514\pi$
$570$	0	0
$571$	−3.92244e6	−0.503461	−0.251731	−	0.967797i	$-0.581000\pi$
$571$	−3.92244e6	−0.503461	−0.251731	+	0.967797i	$0.581000\pi$
$572$	−642854.	−0.0821527
$573$	−2.55215e6	−0.324728
$574$	5.87579e6	0.744366
$575$	0	0
$576$	9.55177e6	1.19958
$577$	−4.38499e6	−0.548314	−0.274157	−	0.961685i	$-0.588399\pi$
$577$	−4.38499e6	−0.548314	−0.274157	+	0.961685i	$0.588399\pi$
$578$	1.52955e7	1.90434
$579$	−4.07619e6	−0.505310
$580$	0	0
$581$	1.68062e7	2.06552
$582$	1.64982e7	2.01896
$583$	−2.42050e6	−0.294940
$584$	1.67919e7	2.03735
$585$	0	0
$586$	2.41926e6	0.291031
$587$	1.27493e7	1.52718	0.763591	−	0.645700i	$-0.223434\pi$
$587$	1.27493e7	1.52718	0.763591	+	0.645700i	$0.223434\pi$
$588$	−1.23907e7	−1.47792
$589$	−1.74323e7	−2.07046
$590$	0	0
$591$	8.55294e6	1.00727
$592$	5.60703e7	6.57550
$593$	2.95784e6	0.345412	0.172706	−	0.984973i	$-0.444749\pi$
$593$	2.95784e6	0.345412	0.172706	+	0.984973i	$0.444749\pi$
$594$	962306.	0.111904
$595$	0	0
$596$	−7.42069e6	−0.855714
$597$	4.41007e6	0.506419
$598$	296485.	0.0339039
$599$	1.54469e7	1.75904	0.879519	−	0.475864i	$-0.157864\pi$
$599$	1.54469e7	1.75904	0.879519	+	0.475864i	$0.157864\pi$
$600$	0	0
$601$	7.65566e6	0.864563	0.432281	−	0.901739i	$-0.357709\pi$
$601$	7.65566e6	0.864563	0.432281	+	0.901739i	$0.357709\pi$
$602$	1.29531e7	1.45675
$603$	−3.57233e6	−0.400091
$604$	−4.70073e7	−5.24291
$605$	0	0
$606$	2.87053e6	0.317527
$607$	9.38604e6	1.03398	0.516988	−	0.855993i	$-0.327053\pi$
$607$	9.38604e6	1.03398	0.516988	+	0.855993i	$0.327053\pi$
$608$	−4.35814e7	−4.78126
$609$	−1.09122e7	−1.19226
$610$	0	0
$611$	−1.17156e6	−0.126958
$612$	940585.	0.101512
$613$	−6.41005e6	−0.688985	−0.344493	−	0.938789i	$-0.611949\pi$
$613$	−6.41005e6	−0.688985	−0.344493	+	0.938789i	$0.611949\pi$
$614$	−5.66746e6	−0.606691
$615$	0	0
$616$	1.31180e7	1.39288
$617$	1.69261e7	1.78996	0.894980	−	0.446107i	$-0.147190\pi$
$617$	1.69261e7	1.78996	0.894980	+	0.446107i	$0.147190\pi$
$618$	1.50069e7	1.58059
$619$	−5.36423e6	−0.562705	−0.281352	−	0.959605i	$-0.590783\pi$
$619$	−5.36423e6	−0.562705	−0.281352	+	0.959605i	$0.590783\pi$
$620$	0	0
$621$	−324486.	−0.0337650
$622$	−6.09270e6	−0.631443
$623$	2.35900e7	2.43505
$624$	2.06786e6	0.212599
$625$	0	0
$626$	−1.35794e7	−1.38498
$627$	−2.17234e6	−0.220678
$628$	−2.51868e6	−0.254844
$629$	1.98841e6	0.200392
$630$	0	0
$631$	1.37293e7	1.37270	0.686350	−	0.727271i	$-0.259212\pi$
$631$	1.37293e7	1.37270	0.686350	+	0.727271i	$0.259212\pi$
$632$	−1.27855e7	−1.27329
$633$	7.88197e6	0.781854
$634$	−9.04100e6	−0.893291
$635$	0	0
$636$	1.56659e7	1.53572
$637$	−966034.	−0.0943287
$638$	8.86045e6	0.861795
$639$	3.87797e6	0.375710
$640$	0	0
$641$	−5.14127e6	−0.494226	−0.247113	−	0.968987i	$-0.579482\pi$
$641$	−5.14127e6	−0.494226	−0.247113	+	0.968987i	$0.579482\pi$
$642$	−7.73014e6	−0.740201
$643$	−5.96167e6	−0.568644	−0.284322	−	0.958729i	$-0.591769\pi$
$643$	−5.96167e6	−0.568644	−0.284322	+	0.958729i	$0.591769\pi$
$644$	−6.99621e6	−0.664734
$645$	0	0
$646$	−2.90416e6	−0.273804
$647$	1.63201e6	0.153272	0.0766361	−	0.997059i	$-0.475582\pi$
$647$	1.63201e6	0.153272	0.0766361	+	0.997059i	$0.475582\pi$
$648$	−3.93777e6	−0.368394
$649$	−1.28788e6	−0.120022
$650$	0	0
$651$	−1.42068e7	−1.31385
$652$	3.04692e7	2.80699
$653$	−1.98987e6	−0.182617	−0.0913086	−	0.995823i	$-0.529105\pi$
$653$	−1.98987e6	−0.182617	−0.0913086	+	0.995823i	$0.529105\pi$
$654$	−7.54416e6	−0.689710
$655$	0	0
$656$	1.12204e7	1.01801
$657$	−2.26623e6	−0.204829
$658$	3.78123e7	3.40461
$659$	1.40827e7	1.26320	0.631600	−	0.775294i	$-0.282399\pi$
$659$	1.40827e7	1.26320	0.631600	+	0.775294i	$0.282399\pi$
$660$	0	0
$661$	−7.86389e6	−0.700058	−0.350029	−	0.936739i	$-0.613828\pi$
$661$	−7.86389e6	−0.700058	−0.350029	+	0.936739i	$0.613828\pi$
$662$	−3.55959e7	−3.15686
$663$	73332.4	0.00647906
$664$	5.58404e7	4.91505
$665$	0	0
$666$	−1.31666e7	−1.15024
$667$	−2.98771e6	−0.260030
$668$	−2.78967e7	−2.41886
$669$	−6.92634e6	−0.598327
$670$	0	0
$671$	1.00967e6	0.0865709
$672$	−3.55177e7	−3.03404
$673$	−1.03441e7	−0.880354	−0.440177	−	0.897911i	$-0.645084\pi$
$673$	−1.03441e7	−0.880354	−0.440177	+	0.897911i	$0.645084\pi$
$674$	1.61856e7	1.37239
$675$	0	0
$676$	−3.19836e7	−2.69191
$677$	1.12364e7	0.942223	0.471111	−	0.882074i	$-0.343853\pi$
$677$	1.12364e7	0.942223	0.471111	+	0.882074i	$0.343853\pi$
$678$	−1.69839e6	−0.141894
$679$	−3.03525e7	−2.52650
$680$	0	0
$681$	−3.94833e6	−0.326246
$682$	1.15356e7	0.949683
$683$	−1.83216e7	−1.50284	−0.751419	−	0.659825i	$-0.770630\pi$
$683$	−1.83216e7	−1.50284	−0.751419	+	0.659825i	$0.770630\pi$
$684$	1.40598e7	1.14905
$685$	0	0
$686$	−1.94125e6	−0.157497
$687$	−5.05151e6	−0.408347
$688$	2.47354e7	1.99227
$689$	1.22139e6	0.0980178
$690$	0	0
$691$	1.48541e7	1.18345	0.591725	−	0.806140i	$-0.298447\pi$
$691$	1.48541e7	1.18345	0.591725	+	0.806140i	$0.298447\pi$
$692$	−4.97902e7	−3.95256
$693$	−1.77040e6	−0.140036
$694$	−4.26968e7	−3.36509
$695$	0	0
$696$	−3.62571e7	−2.83707
$697$	397909.	0.0310243
$698$	−4.65448e6	−0.361604
$699$	−3.77181e6	−0.291982
$700$	0	0
$701$	1.72338e7	1.32461	0.662303	−	0.749236i	$-0.269579\pi$
$701$	1.72338e7	1.32461	0.662303	+	0.749236i	$0.269579\pi$
$702$	−485581.	−0.0371894
$703$	2.97227e7	2.26829
$704$	1.42687e7	1.08506
$705$	0	0
$706$	−3.65447e7	−2.75939
$707$	−5.28105e6	−0.397349
$708$	8.33537e6	0.624945
$709$	1.89467e7	1.41553	0.707765	−	0.706448i	$-0.249704\pi$
$709$	1.89467e7	1.41553	0.707765	+	0.706448i	$0.249704\pi$
$710$	0	0
$711$	1.72554e6	0.128012
$712$	7.83803e7	5.79438
$713$	−3.88975e6	−0.286549
$714$	−2.36681e6	−0.173747
$715$	0	0
$716$	−2.85295e7	−2.07975
$717$	−8.13242e6	−0.590775
$718$	4.26127e6	0.308480
$719$	−2.15583e7	−1.55522	−0.777610	−	0.628747i	$-0.783568\pi$
$719$	−2.15583e7	−1.55522	−0.777610	+	0.628747i	$0.783568\pi$
$720$	0	0
$721$	−2.76089e7	−1.97793
$722$	−1.63984e7	−1.17074
$723$	−5.14750e6	−0.366227
$724$	6.17102e7	4.37533
$725$	0	0
$726$	1.43752e6	0.101221
$727$	−9.26019e6	−0.649806	−0.324903	−	0.945747i	$-0.605332\pi$
$727$	−9.26019e6	−0.649806	−0.324903	+	0.945747i	$0.605332\pi$
$728$	−6.61934e6	−0.462899
$729$	531441.	0.0370370
$730$	0	0
$731$	877188.	0.0607155
$732$	−6.53474e6	−0.450766
$733$	2.30127e7	1.58200	0.791001	−	0.611815i	$-0.209560\pi$
$733$	2.30127e7	1.58200	0.791001	+	0.611815i	$0.209560\pi$
$734$	−2.80203e7	−1.91970
$735$	0	0
$736$	−9.72454e6	−0.661720
$737$	−5.33644e6	−0.361896
$738$	−2.63481e6	−0.178077
$739$	2.36244e7	1.59129	0.795644	−	0.605764i	$-0.207133\pi$
$739$	2.36244e7	1.59129	0.795644	+	0.605764i	$0.207133\pi$
$740$	0	0
$741$	1.09617e6	0.0733383
$742$	−3.94204e7	−2.62852
$743$	−1.56870e7	−1.04248	−0.521239	−	0.853411i	$-0.674530\pi$
$743$	−1.56870e7	−1.04248	−0.521239	+	0.853411i	$0.674530\pi$
$744$	−4.72037e7	−3.12640
$745$	0	0
$746$	6.13656e6	0.403718
$747$	−7.53622e6	−0.494142
$748$	1.40507e6	0.0918215
$749$	1.42215e7	0.926277
$750$	0	0
$751$	1.78084e6	0.115219	0.0576095	−	0.998339i	$-0.481652\pi$
$751$	1.78084e6	0.115219	0.0576095	+	0.998339i	$0.481652\pi$
$752$	7.22066e7	4.65620
$753$	−7.57946e6	−0.487137
$754$	−4.47100e6	−0.286402
$755$	0	0
$756$	1.14583e7	0.729151
$757$	8.44958e6	0.535914	0.267957	−	0.963431i	$-0.413651\pi$
$757$	8.44958e6	0.535914	0.267957	+	0.963431i	$0.413651\pi$
$758$	4.80350e7	3.03658
$759$	−484726.	−0.0305416
$760$	0	0
$761$	1.72176e6	0.107773	0.0538867	−	0.998547i	$-0.482839\pi$
$761$	1.72176e6	0.107773	0.0538867	+	0.998547i	$0.482839\pi$
$762$	−2.09879e7	−1.30943
$763$	1.38793e7	0.863093
$764$	2.46750e7	1.52941
$765$	0	0
$766$	−1.76736e7	−1.08831
$767$	649864.	0.0398873
$768$	−2.37069e7	−1.45034
$769$	−1.72877e6	−0.105420	−0.0527098	−	0.998610i	$-0.516786\pi$
$769$	−1.72877e6	−0.105420	−0.0527098	+	0.998610i	$0.516786\pi$
$770$	0	0
$771$	7.98028e6	0.483484
$772$	3.94098e7	2.37991
$773$	−7.19192e6	−0.432909	−0.216454	−	0.976293i	$-0.569449\pi$
$773$	−7.19192e6	−0.432909	−0.216454	+	0.976293i	$0.569449\pi$
$774$	−5.80843e6	−0.348503
$775$	0	0
$776$	−1.00849e8	−6.01200
$777$	2.42232e7	1.43939
$778$	−2.60468e7	−1.54278
$779$	5.94792e6	0.351173
$780$	0	0
$781$	5.79302e6	0.339842
$782$	−648020.	−0.0378941
$783$	4.89325e6	0.285229
$784$	5.95394e7	3.45951
$785$	0	0
$786$	−2.26567e7	−1.30810
$787$	1.17240e7	0.674745	0.337373	−	0.941371i	$-0.390462\pi$
$787$	1.17240e7	0.674745	0.337373	+	0.941371i	$0.390462\pi$
$788$	−8.26924e7	−4.74406
$789$	8.82586e6	0.504736
$790$	0	0
$791$	3.12461e6	0.177564
$792$	−5.88234e6	−0.333225
$793$	−509479.	−0.0287703
$794$	−8.72121e6	−0.490937
$795$	0	0
$796$	−4.26379e7	−2.38514
$797$	−2.89699e7	−1.61548	−0.807739	−	0.589540i	$-0.799309\pi$
$797$	−2.89699e7	−1.61548	−0.807739	+	0.589540i	$0.799309\pi$
$798$	−3.53789e7	−1.96670
$799$	2.56065e6	0.141900
$800$	0	0
$801$	−1.05782e7	−0.582547
$802$	−1.69139e7	−0.928557
$803$	−3.38536e6	−0.185274
$804$	3.45384e7	1.88435
$805$	0	0
$806$	−5.82087e6	−0.315610
$807$	1.09877e7	0.593912
$808$	−1.75469e7	−0.945520
$809$	−2.21775e7	−1.19135	−0.595676	−	0.803225i	$-0.703116\pi$
$809$	−2.21775e7	−1.19135	−0.595676	+	0.803225i	$0.703116\pi$
$810$	0	0
$811$	−1.64052e7	−0.875848	−0.437924	−	0.899012i	$-0.644286\pi$
$811$	−1.64052e7	−0.875848	−0.437924	+	0.899012i	$0.644286\pi$
$812$	1.05503e8	5.61532
$813$	−2.43226e6	−0.129057
$814$	−1.96686e7	−1.04043
$815$	0	0
$816$	−4.51968e6	−0.237620
$817$	1.31121e7	0.687256
$818$	6.00422e7	3.13743
$819$	893346.	0.0465383
$820$	0	0
$821$	6.29819e6	0.326105	0.163053	−	0.986617i	$-0.447866\pi$
$821$	6.29819e6	0.326105	0.163053	+	0.986617i	$0.447866\pi$
$822$	−2.51437e7	−1.29793
$823$	2.19234e6	0.112826	0.0564128	−	0.998408i	$-0.482034\pi$
$823$	2.19234e6	0.112826	0.0564128	+	0.998408i	$0.482034\pi$
$824$	−9.17334e7	−4.70662
$825$	0	0
$826$	−2.09745e7	−1.06965
$827$	1.06952e7	0.543784	0.271892	−	0.962328i	$-0.412351\pi$
$827$	1.06952e7	0.543784	0.271892	+	0.962328i	$0.412351\pi$
$828$	3.13723e6	0.159027
$829$	−2.28630e7	−1.15544	−0.577720	−	0.816235i	$-0.696057\pi$
$829$	−2.28630e7	−1.15544	−0.577720	+	0.816235i	$0.696057\pi$
$830$	0	0
$831$	−9.87052e6	−0.495835
$832$	−7.20000e6	−0.360599
$833$	2.11144e6	0.105430
$834$	3.55398e7	1.76929
$835$	0	0
$836$	2.10029e7	1.03935
$837$	6.37061e6	0.314317
$838$	6.32740e7	3.11254
$839$	−9.91234e6	−0.486151	−0.243075	−	0.970007i	$-0.578156\pi$
$839$	−9.91234e6	−0.486151	−0.243075	+	0.970007i	$0.578156\pi$
$840$	0	0
$841$	2.45436e7	1.19660
$842$	−5.50162e7	−2.67430
$843$	−7.03435e6	−0.340922
$844$	−7.62054e7	−3.68239
$845$	0	0
$846$	−1.69557e7	−0.814499
$847$	−2.64467e6	−0.126667
$848$	−7.52775e7	−3.59481
$849$	1.97485e7	0.940296
$850$	0	0
$851$	6.63217e6	0.313929
$852$	−3.74934e7	−1.76952
$853$	−6.21622e6	−0.292519	−0.146259	−	0.989246i	$-0.546723\pi$
$853$	−6.21622e6	−0.292519	−0.146259	+	0.989246i	$0.546723\pi$
$854$	1.64435e7	0.771525
$855$	0	0
$856$	4.72524e7	2.20414
$857$	1.10815e7	0.515404	0.257702	−	0.966224i	$-0.417035\pi$
$857$	1.10815e7	0.515404	0.257702	+	0.966224i	$0.417035\pi$
$858$	−725374.	−0.0336391
$859$	6.50304e6	0.300700	0.150350	−	0.988633i	$-0.451960\pi$
$859$	6.50304e6	0.300700	0.150350	+	0.988633i	$0.451960\pi$
$860$	0	0
$861$	4.84739e6	0.222844
$862$	8.56241e6	0.392489
$863$	1.72118e7	0.786684	0.393342	−	0.919392i	$-0.371319\pi$
$863$	1.72118e7	0.786684	0.393342	+	0.919392i	$0.371319\pi$
$864$	1.59268e7	0.725845
$865$	0	0
$866$	3.61734e7	1.63906
$867$	1.26184e7	0.570109
$868$	1.37356e8	6.18798
$869$	2.57765e6	0.115791
$870$	0	0
$871$	2.69278e6	0.120269
$872$	4.61156e7	2.05379
$873$	1.36106e7	0.604425
$874$	−9.68655e6	−0.428934
$875$	0	0
$876$	2.19106e7	0.964704
$877$	−1.22474e7	−0.537706	−0.268853	−	0.963181i	$-0.586645\pi$
$877$	−1.22474e7	−0.537706	−0.268853	+	0.963181i	$0.586645\pi$
$878$	−6.79198e7	−2.97345
$879$	1.99584e6	0.0871269
$880$	0	0
$881$	−664437.	−0.0288412	−0.0144206	−	0.999896i	$-0.504590\pi$
$881$	−664437.	−0.0288412	−0.0144206	+	0.999896i	$0.504590\pi$
$882$	−1.39812e7	−0.605163
$883$	1.97793e7	0.853707	0.426853	−	0.904321i	$-0.359622\pi$
$883$	1.97793e7	0.853707	0.426853	+	0.904321i	$0.359622\pi$
$884$	−709001.	−0.0305152
$885$	0	0
$886$	−7.20255e7	−3.08249
$887$	6.69427e6	0.285689	0.142845	−	0.989745i	$-0.454375\pi$
$887$	6.69427e6	0.285689	0.142845	+	0.989745i	$0.454375\pi$
$888$	8.04840e7	3.42513
$889$	3.86125e7	1.63860
$890$	0	0
$891$	793881.	0.0335013
$892$	6.69660e7	2.81801
$893$	3.82764e7	1.60621
$894$	−8.37325e6	−0.350389
$895$	0	0
$896$	1.06096e8	4.41498
$897$	244593.	0.0101499
$898$	−1.02946e7	−0.426007
$899$	5.86575e7	2.42061
$900$	0	0
$901$	−2.66956e6	−0.109554
$902$	−3.93596e6	−0.161077
$903$	1.06860e7	0.436111
$904$	1.03818e7	0.422526
$905$	0	0
$906$	−5.30414e7	−2.14681
$907$	4.08070e7	1.64709	0.823543	−	0.567255i	$-0.191994\pi$
$907$	4.08070e7	1.64709	0.823543	+	0.567255i	$0.191994\pi$
$908$	3.81736e7	1.53656
$909$	2.36812e6	0.0950593
$910$	0	0
$911$	2.08656e7	0.832979	0.416490	−	0.909140i	$-0.363260\pi$
$911$	2.08656e7	0.832979	0.416490	+	0.909140i	$0.363260\pi$
$912$	−6.75599e7	−2.68969
$913$	−1.12578e7	−0.446968
$914$	4.99946e7	1.97951
$915$	0	0
$916$	4.88395e7	1.92324
$917$	4.16826e7	1.63694
$918$	1.06132e6	0.0415663
$919$	−4.95432e6	−0.193506	−0.0967531	−	0.995308i	$-0.530846\pi$
$919$	−4.95432e6	−0.193506	−0.0967531	+	0.995308i	$0.530846\pi$
$920$	0	0
$921$	−4.67553e6	−0.181627
$922$	−6.50602e7	−2.52051
$923$	−2.92316e6	−0.112940
$924$	1.71168e7	0.659542
$925$	0	0
$926$	−3.63684e7	−1.39379
$927$	1.23803e7	0.473187
$928$	1.46646e8	5.58986
$929$	−1.06264e6	−0.0403968	−0.0201984	−	0.999796i	$-0.506430\pi$
$929$	−1.06264e6	−0.0403968	−0.0201984	+	0.999796i	$0.506430\pi$
$930$	0	0
$931$	3.15616e7	1.19340
$932$	3.64670e7	1.37518
$933$	−5.02634e6	−0.189038
$934$	−6.02079e7	−2.25832
$935$	0	0
$936$	2.96824e6	0.110741
$937$	−3.46900e7	−1.29079	−0.645395	−	0.763849i	$-0.723307\pi$
$937$	−3.46900e7	−1.29079	−0.645395	+	0.763849i	$0.723307\pi$
$938$	−8.69097e7	−3.22523
$939$	−1.12027e7	−0.414628
$940$	0	0
$941$	1.98978e7	0.732538	0.366269	−	0.930509i	$-0.380635\pi$
$941$	1.98978e7	0.732538	0.366269	+	0.930509i	$0.380635\pi$
$942$	−2.84199e6	−0.104351
$943$	1.32719e6	0.0486019
$944$	−4.00530e7	−1.46287
$945$	0	0
$946$	−8.67679e6	−0.315233
$947$	−3.55257e7	−1.28727	−0.643633	−	0.765335i	$-0.722574\pi$
$947$	−3.55257e7	−1.28727	−0.643633	+	0.765335i	$0.722574\pi$
$948$	−1.66830e7	−0.602912
$949$	1.70825e6	0.0615725
$950$	0	0
$951$	−7.45862e6	−0.267428
$952$	1.44677e7	0.517379
$953$	−6.77400e6	−0.241609	−0.120804	−	0.992676i	$-0.538547\pi$
$953$	−6.77400e6	−0.241609	−0.120804	+	0.992676i	$0.538547\pi$
$954$	1.76769e7	0.628831
$955$	0	0
$956$	7.86268e7	2.78244
$957$	7.30967e6	0.257999
$958$	−6.90017e7	−2.42911
$959$	4.62581e7	1.62421
$960$	0	0
$961$	4.77382e7	1.66747
$962$	9.92479e6	0.345767
$963$	−6.37719e6	−0.221597
$964$	4.97676e7	1.72486
$965$	0	0
$966$	−7.89428e6	−0.272188
$967$	2.51293e7	0.864201	0.432100	−	0.901826i	$-0.357773\pi$
$967$	2.51293e7	0.864201	0.432100	+	0.901826i	$0.357773\pi$
$968$	−8.78720e6	−0.301413
$969$	−2.39587e6	−0.0819697
$970$	0	0
$971$	−1.15749e7	−0.393975	−0.196987	−	0.980406i	$-0.563116\pi$
$971$	−1.15749e7	−0.393975	−0.196987	+	0.980406i	$0.563116\pi$
$972$	−5.13814e6	−0.174438
$973$	−6.53843e7	−2.21407
$974$	2.49373e7	0.842270
$975$	0	0
$976$	3.14007e7	1.05515
$977$	−2.63836e7	−0.884297	−0.442149	−	0.896942i	$-0.645784\pi$
$977$	−2.63836e7	−0.884297	−0.442149	+	0.896942i	$0.645784\pi$
$978$	3.43803e7	1.14938
$979$	−1.58020e7	−0.526933
$980$	0	0
$981$	−6.22376e6	−0.206481
$982$	−1.01532e8	−3.35989
$983$	−2.08150e7	−0.687058	−0.343529	−	0.939142i	$-0.611622\pi$
$983$	−2.08150e7	−0.687058	−0.343529	+	0.939142i	$0.611622\pi$
$984$	1.61060e7	0.530272
$985$	0	0
$986$	9.77215e6	0.320109
$987$	3.11943e7	1.01925
$988$	−1.05981e7	−0.345410
$989$	2.92578e6	0.0951154
$990$	0	0
$991$	−2.37994e7	−0.769807	−0.384904	−	0.922957i	$-0.625765\pi$
$991$	−2.37994e7	−0.769807	−0.384904	+	0.922957i	$0.625765\pi$
$992$	1.90921e8	6.15992
$993$	−2.93658e7	−0.945081
$994$	9.43455e7	3.02869
$995$	0	0
$996$	7.28625e7	2.32732
$997$	−3.26637e7	−1.04070	−0.520352	−	0.853952i	$-0.674199\pi$
$997$	−3.26637e7	−1.04070	−0.520352	+	0.853952i	$0.674199\pi$
$998$	3.76183e7	1.19557
$999$	−1.08621e7	−0.344351

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	825.6.a.u.1.1	yes	10
5.4	even	2		825.6.a.t.1.10	✓	10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
825.6.a.t.1.10	✓	10	5.4	even	2
825.6.a.u.1.1	yes	10	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 825 = 3 \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	825.a (trivial)

\(f(q)\)	\(=\)	\(q-10.9094 q^{2} -9.00000 q^{3} +87.0148 q^{4} +98.1845 q^{6} -180.635 q^{7} -600.178 q^{8} +81.0000 q^{9} +O(q^{10})\)	\(q-10.9094 q^{2} -9.00000 q^{3} +87.0148 q^{4} +98.1845 q^{6} -180.635 q^{7} -600.178 q^{8} +81.0000 q^{9} +121.000 q^{11} -783.133 q^{12} -61.0567 q^{13} +1970.61 q^{14} +3763.10 q^{16} +133.450 q^{17} -883.661 q^{18} +1994.80 q^{19} +1625.71 q^{21} -1320.04 q^{22} +445.111 q^{23} +5401.60 q^{24} +666.092 q^{26} -729.000 q^{27} -15717.9 q^{28} -6712.28 q^{29} -8738.84 q^{31} -21847.4 q^{32} -1089.00 q^{33} -1455.86 q^{34} +7048.20 q^{36} +14900.0 q^{37} -21762.1 q^{38} +549.511 q^{39} +2981.70 q^{41} -17735.5 q^{42} +6573.14 q^{43} +10528.8 q^{44} -4855.89 q^{46} +19188.0 q^{47} -33867.9 q^{48} +15821.9 q^{49} -1201.05 q^{51} -5312.84 q^{52} -20004.1 q^{53} +7952.95 q^{54} +108413. q^{56} -17953.2 q^{57} +73226.9 q^{58} -10643.6 q^{59} +8344.36 q^{61} +95335.4 q^{62} -14631.4 q^{63} +117923. q^{64} +11880.3 q^{66} -44102.8 q^{67} +11612.2 q^{68} -4006.00 q^{69} +47876.2 q^{71} -48614.4 q^{72} -27978.1 q^{73} -162550. q^{74} +173578. q^{76} -21856.8 q^{77} -5994.83 q^{78} +21302.9 q^{79} +6561.00 q^{81} -32528.6 q^{82} -93039.7 q^{83} +141461. q^{84} -71709.0 q^{86} +60410.5 q^{87} -72621.5 q^{88} -130595. q^{89} +11029.0 q^{91} +38731.2 q^{92} +78649.6 q^{93} -209330. q^{94} +196627. q^{96} +168032. q^{97} -172607. q^{98} +9801.00 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + q^{2} - 90 q^{3} + 223 q^{4} - 9 q^{6} - 188 q^{7} - 177 q^{8} + 810 q^{9}+O(q^{10}) \) 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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