LMFDB - Embedded newform 825.6.a.m.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("825.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

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

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$132.316651346$
Analytic rank:	$1$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - 2x^{6} - 152x^{5} + 358x^{4} + 5771x^{3} - 13444x^{2} - 51316x + 92576 $ x^7 - 2x^6 - 152x^5 + 358x^4 + 5771x^3 - 13444x^2 - 51316x + 92576
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{2}\cdot 5 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$-6.33680$ 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)

$f(q)$	$=$	$q+5.33680 q^{2} +9.00000 q^{3} -3.51852 q^{4} +48.0312 q^{6} +73.9998 q^{7} -189.555 q^{8} +81.0000 q^{9} +O(q^{10})$	\(q+5.33680 q^{2} +9.00000 q^{3} -3.51852 q^{4} +48.0312 q^{6} +73.9998 q^{7} -189.555 q^{8} +81.0000 q^{9} +121.000 q^{11} -31.6667 q^{12} -66.6080 q^{13} +394.923 q^{14} -899.027 q^{16} -808.876 q^{17} +432.281 q^{18} -900.233 q^{19} +665.998 q^{21} +645.753 q^{22} +3372.25 q^{23} -1706.00 q^{24} -355.474 q^{26} +729.000 q^{27} -260.370 q^{28} -5030.88 q^{29} +2599.04 q^{31} +1267.84 q^{32} +1089.00 q^{33} -4316.81 q^{34} -285.000 q^{36} -6950.65 q^{37} -4804.37 q^{38} -599.472 q^{39} +197.788 q^{41} +3554.30 q^{42} -13340.3 q^{43} -425.741 q^{44} +17997.0 q^{46} -3822.58 q^{47} -8091.25 q^{48} -11331.0 q^{49} -7279.88 q^{51} +234.362 q^{52} +4858.62 q^{53} +3890.53 q^{54} -14027.1 q^{56} -8102.10 q^{57} -26848.8 q^{58} -6936.31 q^{59} -10538.4 q^{61} +13870.5 q^{62} +5993.99 q^{63} +35535.1 q^{64} +5811.78 q^{66} -30768.1 q^{67} +2846.05 q^{68} +30350.3 q^{69} +13974.9 q^{71} -15354.0 q^{72} -13733.4 q^{73} -37094.3 q^{74} +3167.49 q^{76} +8953.98 q^{77} -3199.26 q^{78} +28954.9 q^{79} +6561.00 q^{81} +1055.56 q^{82} -103828. q^{83} -2343.33 q^{84} -71194.4 q^{86} -45277.9 q^{87} -22936.2 q^{88} -63069.9 q^{89} -4928.98 q^{91} -11865.3 q^{92} +23391.3 q^{93} -20400.4 q^{94} +11410.6 q^{96} -62964.9 q^{97} -60471.5 q^{98} +9801.00 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q - 9 q^{2} + 63 q^{3} + 95 q^{4} - 81 q^{6} - 65 q^{7} - 207 q^{8} + 567 q^{9}+O(q^{10}) $ 7 * q - 9 * q^2 + 63 * q^3 + 95 * q^4 - 81 * q^6 - 65 * q^7 - 207 * q^8 + 567 * q^9	\( 7 q - 9 q^{2} + 63 q^{3} + 95 q^{4} - 81 q^{6} - 65 q^{7} - 207 q^{8} + 567 q^{9} + 847 q^{11} + 855 q^{12} + 113 q^{13} - 1212 q^{14} + 499 q^{16} - 1030 q^{17} - 729 q^{18} - 3803 q^{19} - 585 q^{21} - 1089 q^{22} - 514 q^{23} - 1863 q^{24} - 12111 q^{26} + 5103 q^{27} + 342 q^{28} - 2698 q^{29} - 17233 q^{31} - 9943 q^{32} + 7623 q^{33} + 4090 q^{34} + 7695 q^{36} - 23182 q^{37} + 11943 q^{38} + 1017 q^{39} - 16158 q^{41} - 10908 q^{42} + 4249 q^{43} + 11495 q^{44} - 28769 q^{46} - 7580 q^{47} + 4491 q^{48} + 37140 q^{49} - 9270 q^{51} + 23887 q^{52} - 20574 q^{53} - 6561 q^{54} - 73276 q^{56} - 34227 q^{57} - 8733 q^{58} - 364 q^{59} - 28127 q^{61} + 71917 q^{62} - 5265 q^{63} + 43379 q^{64} - 9801 q^{66} + 21493 q^{67} - 160660 q^{68} - 4626 q^{69} - 177084 q^{71} - 16767 q^{72} - 78670 q^{73} + 196750 q^{74} - 32701 q^{76} - 7865 q^{77} - 108999 q^{78} - 187432 q^{79} + 45927 q^{81} - 179552 q^{82} + 44592 q^{83} + 3078 q^{84} - 110433 q^{86} - 24282 q^{87} - 25047 q^{88} - 151168 q^{89} - 230153 q^{91} + 44767 q^{92} - 155097 q^{93} + 54040 q^{94} - 89487 q^{96} + 55589 q^{97} - 478341 q^{98} + 68607 q^{99}+O(q^{100}) \) 7 * q - 9 * q^2 + 63 * q^3 + 95 * q^4 - 81 * q^6 - 65 * q^7 - 207 * q^8 + 567 * q^9 + 847 * q^11 + 855 * q^12 + 113 * q^13 - 1212 * q^14 + 499 * q^16 - 1030 * q^17 - 729 * q^18 - 3803 * q^19 - 585 * q^21 - 1089 * q^22 - 514 * q^23 - 1863 * q^24 - 12111 * q^26 + 5103 * q^27 + 342 * q^28 - 2698 * q^29 - 17233 * q^31 - 9943 * q^32 + 7623 * q^33 + 4090 * q^34 + 7695 * q^36 - 23182 * q^37 + 11943 * q^38 + 1017 * q^39 - 16158 * q^41 - 10908 * q^42 + 4249 * q^43 + 11495 * q^44 - 28769 * q^46 - 7580 * q^47 + 4491 * q^48 + 37140 * q^49 - 9270 * q^51 + 23887 * q^52 - 20574 * q^53 - 6561 * q^54 - 73276 * q^56 - 34227 * q^57 - 8733 * q^58 - 364 * q^59 - 28127 * q^61 + 71917 * q^62 - 5265 * q^63 + 43379 * q^64 - 9801 * q^66 + 21493 * q^67 - 160660 * q^68 - 4626 * q^69 - 177084 * q^71 - 16767 * q^72 - 78670 * q^73 + 196750 * q^74 - 32701 * q^76 - 7865 * q^77 - 108999 * q^78 - 187432 * q^79 + 45927 * q^81 - 179552 * q^82 + 44592 * q^83 + 3078 * q^84 - 110433 * q^86 - 24282 * q^87 - 25047 * q^88 - 151168 * q^89 - 230153 * q^91 + 44767 * q^92 - 155097 * q^93 + 54040 * q^94 - 89487 * q^96 + 55589 * q^97 - 478341 * q^98 + 68607 * 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 $

$2$	5.33680	0.943423	0.471711	−	0.881753i	$-0.343636\pi$
$2$	5.33680	0.943423	0.471711	+	0.881753i	$0.343636\pi$
$3$	9.00000	0.577350
$3$	9.00000	0.577350
$4$	−3.51852	−0.109954
$5$	0	0
$5$	0	0
$6$	48.0312	0.544685
$7$	73.9998	0.570802	0.285401	−	0.958408i	$-0.407873\pi$
$7$	73.9998	0.570802	0.285401	+	0.958408i	$0.407873\pi$
$8$	−189.555	−1.04716
$9$	81.0000	0.333333
$10$	0	0
$11$	121.000	0.301511
$11$	121.000	0.301511
$12$	−31.6667	−0.0634819
$13$	−66.6080	−0.109312	−0.0546560	−	0.998505i	$-0.517406\pi$
$13$	−66.6080	−0.109312	−0.0546560	+	0.998505i	$0.517406\pi$
$14$	394.923	0.538508
$15$	0	0
$16$	−899.027	−0.877956
$17$	−808.876	−0.678828	−0.339414	−	0.940637i	$-0.610229\pi$
$17$	−808.876	−0.678828	−0.339414	+	0.940637i	$0.610229\pi$
$18$	432.281	0.314474
$19$	−900.233	−0.572099	−0.286049	−	0.958215i	$-0.592342\pi$
$19$	−900.233	−0.572099	−0.286049	+	0.958215i	$0.592342\pi$
$20$	0	0
$21$	665.998	0.329553
$22$	645.753	0.284453
$23$	3372.25	1.32923	0.664615	−	0.747186i	$-0.268595\pi$
$23$	3372.25	1.32923	0.664615	+	0.747186i	$0.268595\pi$
$24$	−1706.00	−0.604576
$25$	0	0
$26$	−355.474	−0.103127
$27$	729.000	0.192450
$28$	−260.370	−0.0627619
$29$	−5030.88	−1.11083	−0.555417	−	0.831572i	$-0.687441\pi$
$29$	−5030.88	−1.11083	−0.555417	+	0.831572i	$0.687441\pi$
$30$	0	0
$31$	2599.04	0.485745	0.242872	−	0.970058i	$-0.421910\pi$
$31$	2599.04	0.485745	0.242872	+	0.970058i	$0.421910\pi$
$32$	1267.84	0.218872
$33$	1089.00	0.174078
$34$	−4316.81	−0.640422
$35$	0	0
$36$	−285.000	−0.0366513
$37$	−6950.65	−0.834682	−0.417341	−	0.908750i	$-0.637038\pi$
$37$	−6950.65	−0.834682	−0.417341	+	0.908750i	$0.637038\pi$
$38$	−4804.37	−0.539731
$39$	−599.472	−0.0631113
$40$	0	0
$41$	197.788	0.0183756	0.00918778	−	0.999958i	$-0.497075\pi$
$41$	197.788	0.0183756	0.00918778	+	0.999958i	$0.497075\pi$
$42$	3554.30	0.310908
$43$	−13340.3	−1.10025	−0.550127	−	0.835081i	$-0.685421\pi$
$43$	−13340.3	−1.10025	−0.550127	+	0.835081i	$0.685421\pi$
$44$	−425.741	−0.0331523
$45$	0	0
$46$	17997.0	1.25403
$47$	−3822.58	−0.252413	−0.126207	−	0.992004i	$-0.540280\pi$
$47$	−3822.58	−0.252413	−0.126207	+	0.992004i	$0.540280\pi$
$48$	−8091.25	−0.506888
$49$	−11331.0	−0.674185
$50$	0	0
$51$	−7279.88	−0.391921
$52$	234.362	0.0120193
$53$	4858.62	0.237588	0.118794	−	0.992919i	$-0.462097\pi$
$53$	4858.62	0.237588	0.118794	+	0.992919i	$0.462097\pi$
$54$	3890.53	0.181562
$55$	0	0
$56$	−14027.1	−0.597719
$57$	−8102.10	−0.330301
$58$	−26848.8	−1.04799
$59$	−6936.31	−0.259417	−0.129708	−	0.991552i	$-0.541404\pi$
$59$	−6936.31	−0.259417	−0.129708	+	0.991552i	$0.541404\pi$
$60$	0	0
$61$	−10538.4	−0.362619	−0.181309	−	0.983426i	$-0.558033\pi$
$61$	−10538.4	−0.362619	−0.181309	+	0.983426i	$0.558033\pi$
$62$	13870.5	0.458263
$63$	5993.99	0.190267
$64$	35535.1	1.08444
$65$	0	0
$66$	5811.78	0.164229
$67$	−30768.1	−0.837362	−0.418681	−	0.908133i	$-0.637507\pi$
$67$	−30768.1	−0.837362	−0.418681	+	0.908133i	$0.637507\pi$
$68$	2846.05	0.0746397
$69$	30350.3	0.767432
$70$	0	0
$71$	13974.9	0.329006	0.164503	−	0.986377i	$-0.447398\pi$
$71$	13974.9	0.329006	0.164503	+	0.986377i	$0.447398\pi$
$72$	−15354.0	−0.349052
$73$	−13733.4	−0.301627	−0.150813	−	0.988562i	$-0.548189\pi$
$73$	−13733.4	−0.301627	−0.150813	+	0.988562i	$0.548189\pi$
$74$	−37094.3	−0.787458
$75$	0	0
$76$	3167.49	0.0629045
$77$	8953.98	0.172103
$78$	−3199.26	−0.0595407
$79$	28954.9	0.521980	0.260990	−	0.965342i	$-0.415951\pi$
$79$	28954.9	0.521980	0.260990	+	0.965342i	$0.415951\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	1055.56	0.0173359
$83$	−103828.	−1.65432	−0.827161	−	0.561965i	$-0.810046\pi$
$83$	−103828.	−1.65432	−0.827161	+	0.561965i	$0.810046\pi$
$84$	−2343.33	−0.0362356
$85$	0	0
$86$	−71194.4	−1.03801
$87$	−45277.9	−0.641340
$88$	−22936.2	−0.315729
$89$	−63069.9	−0.844010	−0.422005	−	0.906594i	$-0.638673\pi$
$89$	−63069.9	−0.844010	−0.422005	+	0.906594i	$0.638673\pi$
$90$	0	0
$91$	−4928.98	−0.0623955
$92$	−11865.3	−0.146154
$93$	23391.3	0.280445
$94$	−20400.4	−0.238132
$95$	0	0
$96$	11410.6	0.126366
$97$	−62964.9	−0.679468	−0.339734	−	0.940522i	$-0.610337\pi$
$97$	−62964.9	−0.679468	−0.339734	+	0.940522i	$0.610337\pi$
$98$	−60471.5	−0.636041
$99$	9801.00	0.100504
$100$	0	0
$101$	32233.2	0.314413	0.157206	−	0.987566i	$-0.449751\pi$
$101$	32233.2	0.314413	0.157206	+	0.987566i	$0.449751\pi$
$102$	−38851.3	−0.369748
$103$	45990.8	0.427148	0.213574	−	0.976927i	$-0.431490\pi$
$103$	45990.8	0.427148	0.213574	+	0.976927i	$0.431490\pi$
$104$	12625.9	0.114467
$105$	0	0
$106$	25929.5	0.224145
$107$	40102.2	0.338617	0.169309	−	0.985563i	$-0.445847\pi$
$107$	40102.2	0.338617	0.169309	+	0.985563i	$0.445847\pi$
$108$	−2565.00	−0.0211606
$109$	−44635.8	−0.359847	−0.179923	−	0.983681i	$-0.557585\pi$
$109$	−44635.8	−0.359847	−0.179923	+	0.983681i	$0.557585\pi$
$110$	0	0
$111$	−62555.8	−0.481904
$112$	−66527.9	−0.501139
$113$	−251257.	−1.85106	−0.925532	−	0.378669i	$-0.876382\pi$
$113$	−251257.	−1.85106	−0.925532	+	0.378669i	$0.876382\pi$
$114$	−43239.3	−0.311614
$115$	0	0
$116$	17701.3	0.122140
$117$	−5395.25	−0.0364373
$118$	−37017.7	−0.244740
$119$	−59856.7	−0.387476
$120$	0	0
$121$	14641.0	0.0909091
$122$	−56241.4	−0.342103
$123$	1780.09	0.0106091
$124$	−9144.77	−0.0534095
$125$	0	0
$126$	31988.7	0.179503
$127$	214460.	1.17988	0.589939	−	0.807448i	$-0.299152\pi$
$127$	214460.	1.17988	0.589939	+	0.807448i	$0.299152\pi$
$128$	149073.	0.804218
$129$	−120062.	−0.635232
$130$	0	0
$131$	157394.	0.801325	0.400662	−	0.916226i	$-0.368780\pi$
$131$	157394.	0.801325	0.400662	+	0.916226i	$0.368780\pi$
$132$	−3831.67	−0.0191405
$133$	−66617.1	−0.326555
$134$	−164203.	−0.789986
$135$	0	0
$136$	153327.	0.710838
$137$	−385559.	−1.75505	−0.877525	−	0.479532i	$-0.840807\pi$
$137$	−385559.	−1.75505	−0.877525	+	0.479532i	$0.840807\pi$
$138$	161973.	0.724012
$139$	−181540.	−0.796959	−0.398480	−	0.917177i	$-0.630462\pi$
$139$	−181540.	−0.796959	−0.398480	+	0.917177i	$0.630462\pi$
$140$	0	0
$141$	−34403.2	−0.145731
$142$	74581.5	0.310392
$143$	−8059.57	−0.0329588
$144$	−72821.2	−0.292652
$145$	0	0
$146$	−73292.3	−0.284561
$147$	−101979.	−0.389241
$148$	24456.0	0.0917765
$149$	−189099.	−0.697790	−0.348895	−	0.937162i	$-0.613443\pi$
$149$	−189099.	−0.697790	−0.348895	+	0.937162i	$0.613443\pi$
$150$	0	0
$151$	182551.	0.651541	0.325770	−	0.945449i	$-0.394376\pi$
$151$	182551.	0.651541	0.325770	+	0.945449i	$0.394376\pi$
$152$	170644.	0.599076
$153$	−65519.0	−0.226276
$154$	47785.6	0.162366
$155$	0	0
$156$	2109.26	0.00693933
$157$	−299449.	−0.969559	−0.484779	−	0.874636i	$-0.661100\pi$
$157$	−299449.	−0.969559	−0.484779	+	0.874636i	$0.661100\pi$
$158$	154526.	0.492447
$159$	43727.6	0.137171
$160$	0	0
$161$	249546.	0.758728
$162$	35014.8	0.104825
$163$	−105810.	−0.311930	−0.155965	−	0.987763i	$-0.549849\pi$
$163$	−105810.	−0.311930	−0.155965	+	0.987763i	$0.549849\pi$
$164$	−695.922	−0.00202046
$165$	0	0
$166$	−554111.	−1.56072
$167$	44979.5	0.124802	0.0624012	−	0.998051i	$-0.480124\pi$
$167$	44979.5	0.124802	0.0624012	+	0.998051i	$0.480124\pi$
$168$	−126244.	−0.345093
$169$	−366856.	−0.988051
$170$	0	0
$171$	−72918.9	−0.190700
$172$	46938.0	0.120977
$173$	−445425.	−1.13151	−0.565756	−	0.824573i	$-0.691415\pi$
$173$	−445425.	−1.13151	−0.565756	+	0.824573i	$0.691415\pi$
$174$	−241639.	−0.605055
$175$	0	0
$176$	−108782.	−0.264714
$177$	−62426.8	−0.149774
$178$	−336592.	−0.796258
$179$	−552254.	−1.28827	−0.644134	−	0.764913i	$-0.722782\pi$
$179$	−552254.	−1.28827	−0.644134	+	0.764913i	$0.722782\pi$
$180$	0	0
$181$	52408.5	0.118906	0.0594532	−	0.998231i	$-0.481064\pi$
$181$	52408.5	0.118906	0.0594532	+	0.998231i	$0.481064\pi$
$182$	−26305.0	−0.0588654
$183$	−94845.6	−0.209358
$184$	−639228.	−1.39191
$185$	0	0
$186$	124835.	0.264578
$187$	−97874.0	−0.204674
$188$	13449.8	0.0277538
$189$	53945.9	0.109851
$190$	0	0
$191$	135687.	0.269125	0.134562	−	0.990905i	$-0.457037\pi$
$191$	135687.	0.269125	0.134562	+	0.990905i	$0.457037\pi$
$192$	319816.	0.626105
$193$	−561304.	−1.08469	−0.542344	−	0.840157i	$-0.682463\pi$
$193$	−561304.	−1.08469	−0.542344	+	0.840157i	$0.682463\pi$
$194$	−336031.	−0.641026
$195$	0	0
$196$	39868.5	0.0741292
$197$	774322.	1.42153	0.710765	−	0.703429i	$-0.248349\pi$
$197$	774322.	1.42153	0.710765	+	0.703429i	$0.248349\pi$
$198$	52306.0	0.0948175
$199$	−494275.	−0.884780	−0.442390	−	0.896823i	$-0.645869\pi$
$199$	−494275.	−0.884780	−0.442390	+	0.896823i	$0.645869\pi$
$200$	0	0
$201$	−276913.	−0.483451
$202$	172022.	0.296624
$203$	−372284.	−0.634066
$204$	25614.4	0.0430933
$205$	0	0
$206$	245444.	0.402981
$207$	273152.	0.443077
$208$	59882.4	0.0959712
$209$	−108928.	−0.172494
$210$	0	0
$211$	−30102.4	−0.0465474	−0.0232737	−	0.999729i	$-0.507409\pi$
$211$	−30102.4	−0.0465474	−0.0232737	+	0.999729i	$0.507409\pi$
$212$	−17095.2	−0.0261237
$213$	125774.	0.189952
$214$	214018.	0.319459
$215$	0	0
$216$	−138186.	−0.201525
$217$	192328.	0.277264
$218$	−238213.	−0.339487
$219$	−123600.	−0.174144
$220$	0	0
$221$	53877.6	0.0742041
$222$	−333848.	−0.454639
$223$	539464.	0.726441	0.363221	−	0.931703i	$-0.381677\pi$
$223$	539464.	0.726441	0.363221	+	0.931703i	$0.381677\pi$
$224$	93820.0	0.124932
$225$	0	0
$226$	−1.34091e6	−1.74634
$227$	972036.	1.25204	0.626019	−	0.779808i	$-0.284683\pi$
$227$	972036.	1.25204	0.626019	+	0.779808i	$0.284683\pi$
$228$	28507.4	0.0363179
$229$	172701.	0.217624	0.108812	−	0.994062i	$-0.465295\pi$
$229$	172701.	0.217624	0.108812	+	0.994062i	$0.465295\pi$
$230$	0	0
$231$	80585.8	0.0993639
$232$	953631.	1.16322
$233$	18356.9	0.0221519	0.0110759	−	0.999939i	$-0.496474\pi$
$233$	18356.9	0.0221519	0.0110759	+	0.999939i	$0.496474\pi$
$234$	−28793.4	−0.0343758
$235$	0	0
$236$	24405.6	0.0285239
$237$	260594.	0.301365
$238$	−319443.	−0.365554
$239$	−50802.6	−0.0575296	−0.0287648	−	0.999586i	$-0.509157\pi$
$239$	−50802.6	−0.0575296	−0.0287648	+	0.999586i	$0.509157\pi$
$240$	0	0
$241$	57284.6	0.0635324	0.0317662	−	0.999495i	$-0.489887\pi$
$241$	57284.6	0.0635324	0.0317662	+	0.999495i	$0.489887\pi$
$242$	78136.1	0.0857657
$243$	59049.0	0.0641500
$244$	37079.6	0.0398713
$245$	0	0
$246$	9500.01	0.0100089
$247$	59962.7	0.0625373
$248$	−492661.	−0.508650
$249$	−934454.	−0.955123
$250$	0	0
$251$	275746.	0.276265	0.138132	−	0.990414i	$-0.455890\pi$
$251$	275746.	0.276265	0.138132	+	0.990414i	$0.455890\pi$
$252$	−21090.0	−0.0209206
$253$	408042.	0.400778
$254$	1.14453e6	1.11312
$255$	0	0
$256$	−341550.	−0.325727
$257$	−55323.5	−0.0522489	−0.0261244	−	0.999659i	$-0.508317\pi$
$257$	−55323.5	−0.0522489	−0.0261244	+	0.999659i	$0.508317\pi$
$258$	−640749.	−0.599293
$259$	−514347.	−0.476438
$260$	0	0
$261$	−407501.	−0.370278
$262$	839978.	0.755988
$263$	1.50652e6	1.34303	0.671515	−	0.740991i	$-0.265644\pi$
$263$	1.50652e6	1.34303	0.671515	+	0.740991i	$0.265644\pi$
$264$	−206426.	−0.182286
$265$	0	0
$266$	−355522.	−0.308080
$267$	−567630.	−0.487289
$268$	108258.	0.0920711
$269$	300252.	0.252991	0.126496	−	0.991967i	$-0.459627\pi$
$269$	300252.	0.252991	0.126496	+	0.991967i	$0.459627\pi$
$270$	0	0
$271$	−1.88992e6	−1.56322	−0.781608	−	0.623770i	$-0.785600\pi$
$271$	−1.88992e6	−1.56322	−0.781608	+	0.623770i	$0.785600\pi$
$272$	727202.	0.595981
$273$	−44360.8	−0.0360241
$274$	−2.05765e6	−1.65575
$275$	0	0
$276$	−106788.	−0.0843821
$277$	493383.	0.386354	0.193177	−	0.981164i	$-0.438121\pi$
$277$	493383.	0.386354	0.193177	+	0.981164i	$0.438121\pi$
$278$	−968845.	−0.751869
$279$	210522.	0.161915
$280$	0	0
$281$	1.39118e6	1.05104	0.525518	−	0.850783i	$-0.323872\pi$
$281$	1.39118e6	1.05104	0.525518	+	0.850783i	$0.323872\pi$
$282$	−183603.	−0.137486
$283$	1.92592e6	1.42946	0.714730	−	0.699400i	$-0.246549\pi$
$283$	1.92592e6	1.42946	0.714730	+	0.699400i	$0.246549\pi$
$284$	−49171.1	−0.0361755
$285$	0	0
$286$	−43012.3	−0.0310941
$287$	14636.3	0.0104888
$288$	102695.	0.0729572
$289$	−765577.	−0.539193
$290$	0	0
$291$	−566684.	−0.392291
$292$	48321.1	0.0331650
$293$	−1.53666e6	−1.04571	−0.522853	−	0.852423i	$-0.675132\pi$
$293$	−1.53666e6	−1.04571	−0.522853	+	0.852423i	$0.675132\pi$
$294$	−544243.	−0.367219
$295$	0	0
$296$	1.31753e6	0.874042
$297$	88209.0	0.0580259
$298$	−1.00919e6	−0.658311
$299$	−224619.	−0.145301
$300$	0	0
$301$	−987177.	−0.628028
$302$	974238.	0.614678
$303$	290099.	0.181526
$304$	809334.	0.502278
$305$	0	0
$306$	−349662.	−0.213474
$307$	2.17758e6	1.31865	0.659324	−	0.751859i	$-0.270842\pi$
$307$	2.17758e6	1.31865	0.659324	+	0.751859i	$0.270842\pi$
$308$	−31504.8	−0.0189234
$309$	413917.	0.246614
$310$	0	0
$311$	624920.	0.366373	0.183186	−	0.983078i	$-0.441359\pi$
$311$	624920.	0.366373	0.183186	+	0.983078i	$0.441359\pi$
$312$	113633.	0.0660874
$313$	757413.	0.436990	0.218495	−	0.975838i	$-0.429885\pi$
$313$	757413.	0.436990	0.218495	+	0.975838i	$0.429885\pi$
$314$	−1.59810e6	−0.914704
$315$	0	0
$316$	−101878.	−0.0573937
$317$	3.10220e6	1.73389	0.866944	−	0.498406i	$-0.166081\pi$
$317$	3.10220e6	1.73389	0.866944	+	0.498406i	$0.166081\pi$
$318$	233366.	0.129410
$319$	−608737.	−0.334929
$320$	0	0
$321$	360920.	0.195501
$322$	1.33178e6	0.715801
$323$	728177.	0.388357
$324$	−23085.0	−0.0122171
$325$	0	0
$326$	−564687.	−0.294282
$327$	−401723.	−0.207758
$328$	−37491.8	−0.0192421
$329$	−282870.	−0.144078
$330$	0	0
$331$	−1.59132e6	−0.798339	−0.399169	−	0.916877i	$-0.630701\pi$
$331$	−1.59132e6	−0.798339	−0.399169	+	0.916877i	$0.630701\pi$
$332$	365322.	0.181899
$333$	−563003.	−0.278227
$334$	240047.	0.117741
$335$	0	0
$336$	−598751.	−0.289333
$337$	−2.25270e6	−1.08051	−0.540254	−	0.841502i	$-0.681672\pi$
$337$	−2.25270e6	−1.08051	−0.540254	+	0.841502i	$0.681672\pi$
$338$	−1.95784e6	−0.932150
$339$	−2.26131e6	−1.06871
$340$	0	0
$341$	314483.	0.146458
$342$	−389154.	−0.179910
$343$	−2.08221e6	−0.955628
$344$	2.52872e6	1.15214
$345$	0	0
$346$	−2.37714e6	−1.06749
$347$	171210.	0.0763318	0.0381659	−	0.999271i	$-0.487848\pi$
$347$	171210.	0.0763318	0.0381659	+	0.999271i	$0.487848\pi$
$348$	159311.	0.0705178
$349$	3.09886e6	1.36188	0.680940	−	0.732339i	$-0.261571\pi$
$349$	3.09886e6	1.36188	0.680940	+	0.732339i	$0.261571\pi$
$350$	0	0
$351$	−48557.2	−0.0210371
$352$	153409.	0.0659923
$353$	−2.33605e6	−0.997805	−0.498903	−	0.866658i	$-0.666263\pi$
$353$	−2.33605e6	−0.997805	−0.498903	+	0.866658i	$0.666263\pi$
$354$	−333159.	−0.141301
$355$	0	0
$356$	221913.	0.0928021
$357$	−538710.	−0.223710
$358$	−2.94727e6	−1.21538
$359$	161176.	0.0660029	0.0330015	−	0.999455i	$-0.489493\pi$
$359$	161176.	0.0660029	0.0330015	+	0.999455i	$0.489493\pi$
$360$	0	0
$361$	−1.66568e6	−0.672703
$362$	279694.	0.112179
$363$	131769.	0.0524864
$364$	17342.7	0.00686063
$365$	0	0
$366$	−506172.	−0.197513
$367$	2.18840e6	0.848129	0.424064	−	0.905632i	$-0.360603\pi$
$367$	2.18840e6	0.848129	0.424064	+	0.905632i	$0.360603\pi$
$368$	−3.03175e6	−1.16701
$369$	16020.8	0.00612519
$370$	0	0
$371$	359537.	0.135615
$372$	−82302.9	−0.0308360
$373$	3.63629e6	1.35328	0.676639	−	0.736315i	$-0.263436\pi$
$373$	3.63629e6	1.35328	0.676639	+	0.736315i	$0.263436\pi$
$374$	−522334.	−0.193094
$375$	0	0
$376$	724591.	0.264316
$377$	335097.	0.121428
$378$	287899.	0.103636
$379$	1.67725e6	0.599791	0.299895	−	0.953972i	$-0.403048\pi$
$379$	1.67725e6	0.599791	0.299895	+	0.953972i	$0.403048\pi$
$380$	0	0
$381$	1.93014e6	0.681203
$382$	724134.	0.253899
$383$	−2.08391e6	−0.725908	−0.362954	−	0.931807i	$-0.618232\pi$
$383$	−2.08391e6	−0.725908	−0.362954	+	0.931807i	$0.618232\pi$
$384$	1.34166e6	0.464315
$385$	0	0
$386$	−2.99557e6	−1.02332
$387$	−1.08056e6	−0.366752
$388$	221543.	0.0747101
$389$	−5.80280e6	−1.94430	−0.972151	−	0.234356i	$-0.924702\pi$
$389$	−5.80280e6	−1.94430	−0.972151	+	0.234356i	$0.924702\pi$
$390$	0	0
$391$	−2.72773e6	−0.902319
$392$	2.14786e6	0.705976
$393$	1.41654e6	0.462645
$394$	4.13241e6	1.34110
$395$	0	0
$396$	−34485.0	−0.0110508
$397$	1.25839e6	0.400718	0.200359	−	0.979723i	$-0.435789\pi$
$397$	1.25839e6	0.400718	0.200359	+	0.979723i	$0.435789\pi$
$398$	−2.63785e6	−0.834722
$399$	−599554.	−0.188537
$400$	0	0
$401$	−1.29837e6	−0.403216	−0.201608	−	0.979466i	$-0.564617\pi$
$401$	−1.29837e6	−0.403216	−0.201608	+	0.979466i	$0.564617\pi$
$402$	−1.47783e6	−0.456099
$403$	−173117.	−0.0530977
$404$	−113413.	−0.0345709
$405$	0	0
$406$	−1.98681e6	−0.598193
$407$	−841029.	−0.251666
$408$	1.37994e6	0.410403
$409$	−4.45805e6	−1.31776	−0.658880	−	0.752248i	$-0.728970\pi$
$409$	−4.45805e6	−1.31776	−0.658880	+	0.752248i	$0.728970\pi$
$410$	0	0
$411$	−3.47003e6	−1.01328
$412$	−161820.	−0.0469665
$413$	−513286.	−0.148076
$414$	1.45776e6	0.418009
$415$	0	0
$416$	−84448.3	−0.0239253
$417$	−1.63386e6	−0.460125
$418$	−581329.	−0.162735
$419$	2.33339e6	0.649312	0.324656	−	0.945832i	$-0.394752\pi$
$419$	2.33339e6	0.649312	0.324656	+	0.945832i	$0.394752\pi$
$420$	0	0
$421$	4.04741e6	1.11294	0.556470	−	0.830868i	$-0.312156\pi$
$421$	4.04741e6	1.11294	0.556470	+	0.830868i	$0.312156\pi$
$422$	−160651.	−0.0439139
$423$	−309629.	−0.0841377
$424$	−920978.	−0.248791
$425$	0	0
$426$	671234.	0.179205
$427$	−779840.	−0.206983
$428$	−141101.	−0.0372322
$429$	−72536.1	−0.0190288
$430$	0	0
$431$	2.38060e6	0.617295	0.308647	−	0.951177i	$-0.400124\pi$
$431$	2.38060e6	0.617295	0.308647	+	0.951177i	$0.400124\pi$
$432$	−655391.	−0.168963
$433$	5.02364e6	1.28765	0.643826	−	0.765172i	$-0.277346\pi$
$433$	5.02364e6	1.28765	0.643826	+	0.765172i	$0.277346\pi$
$434$	1.02642e6	0.261577
$435$	0	0
$436$	157052.	0.0395665
$437$	−3.03581e6	−0.760451
$438$	−659630.	−0.164292
$439$	−2.27414e6	−0.563192	−0.281596	−	0.959533i	$-0.590864\pi$
$439$	−2.27414e6	−0.563192	−0.281596	+	0.959533i	$0.590864\pi$
$440$	0	0
$441$	−917813.	−0.224728
$442$	287534.	0.0700058
$443$	−93691.6	−0.0226825	−0.0113413	−	0.999936i	$-0.503610\pi$
$443$	−93691.6	−0.0226825	−0.0113413	+	0.999936i	$0.503610\pi$
$444$	220104.	0.0529872
$445$	0	0
$446$	2.87901e6	0.685341
$447$	−1.70190e6	−0.402869
$448$	2.62959e6	0.619003
$449$	−5.96118e6	−1.39546	−0.697728	−	0.716363i	$-0.745806\pi$
$449$	−5.96118e6	−1.39546	−0.697728	+	0.716363i	$0.745806\pi$
$450$	0	0
$451$	23932.4	0.00554044
$452$	884052.	0.203532
$453$	1.64296e6	0.376167
$454$	5.18756e6	1.18120
$455$	0	0
$456$	1.53580e6	0.345877
$457$	878215.	0.196703	0.0983514	−	0.995152i	$-0.468643\pi$
$457$	878215.	0.196703	0.0983514	+	0.995152i	$0.468643\pi$
$458$	921673.	0.205311
$459$	−589671.	−0.130640
$460$	0	0
$461$	−2960.95	−0.000648901 0	−0.000324450	−	1.00000i	$-0.500103\pi$
$461$	−2960.95	−0.000648901 0	−0.000324450		1.00000i	$0.500103\pi$
$462$	430071.	0.0937421
$463$	−1.73957e6	−0.377129	−0.188564	−	0.982061i	$-0.560383\pi$
$463$	−1.73957e6	−0.377129	−0.188564	+	0.982061i	$0.560383\pi$
$464$	4.52290e6	0.975264
$465$	0	0
$466$	97967.3	0.0208986
$467$	236495.	0.0501800	0.0250900	−	0.999685i	$-0.492013\pi$
$467$	236495.	0.0501800	0.0250900	+	0.999685i	$0.492013\pi$
$468$	18983.3	0.00400643
$469$	−2.27683e6	−0.477968
$470$	0	0
$471$	−2.69504e6	−0.559775
$472$	1.31481e6	0.271650
$473$	−1.61417e6	−0.331739
$474$	1.39074e6	0.284315
$475$	0	0
$476$	210607.	0.0426045
$477$	393549.	0.0791958
$478$	−271124.	−0.0542747
$479$	−1.23533e6	−0.246006	−0.123003	−	0.992406i	$-0.539252\pi$
$479$	−1.23533e6	−0.246006	−0.123003	+	0.992406i	$0.539252\pi$
$480$	0	0
$481$	462969.	0.0912408
$482$	305717.	0.0599379
$483$	2.24591e6	0.438052
$484$	−51514.7	−0.00999580
$485$	0	0
$486$	315133.	0.0605206
$487$	−4.77244e6	−0.911839	−0.455920	−	0.890021i	$-0.650690\pi$
$487$	−4.77244e6	−0.911839	−0.455920	+	0.890021i	$0.650690\pi$
$488$	1.99761e6	0.379718
$489$	−952289.	−0.180093
$490$	0	0
$491$	9.12682e6	1.70850	0.854251	−	0.519860i	$-0.174016\pi$
$491$	9.12682e6	1.70850	0.854251	+	0.519860i	$0.174016\pi$
$492$	−6263.30	−0.00116652
$493$	4.06936e6	0.754065
$494$	320009.	0.0589991
$495$	0	0
$496$	−2.33660e6	−0.426463
$497$	1.03414e6	0.187797
$498$	−4.98700e6	−0.901085
$499$	8.42908e6	1.51541	0.757703	−	0.652600i	$-0.226322\pi$
$499$	8.42908e6	1.51541	0.757703	+	0.652600i	$0.226322\pi$
$500$	0	0
$501$	404815.	0.0720547
$502$	1.47160e6	0.260634
$503$	9.13714e6	1.61024	0.805120	−	0.593113i	$-0.202101\pi$
$503$	9.13714e6	1.61024	0.805120	+	0.593113i	$0.202101\pi$
$504$	−1.13619e6	−0.199240
$505$	0	0
$506$	2.17764e6	0.378103
$507$	−3.30171e6	−0.570451
$508$	−754583.	−0.129732
$509$	3.02787e6	0.518015	0.259008	−	0.965875i	$-0.416605\pi$
$509$	3.02787e6	0.518015	0.259008	+	0.965875i	$0.416605\pi$
$510$	0	0
$511$	−1.01627e6	−0.172169
$512$	−6.59312e6	−1.11152
$513$	−656270.	−0.110100
$514$	−295251.	−0.0492928
$515$	0	0
$516$	422442.	0.0698462
$517$	−462532.	−0.0761054
$518$	−2.74497e6	−0.449483
$519$	−4.00882e6	−0.653278
$520$	0	0
$521$	1.19443e7	1.92782	0.963912	−	0.266221i	$-0.0857751\pi$
$521$	1.19443e7	1.92782	0.963912	+	0.266221i	$0.0857751\pi$
$522$	−2.17476e6	−0.349329
$523$	8.53138e6	1.36384	0.681922	−	0.731425i	$-0.261144\pi$
$523$	8.53138e6	1.36384	0.681922	+	0.731425i	$0.261144\pi$
$524$	−553793.	−0.0881087
$525$	0	0
$526$	8.04000e6	1.26704
$527$	−2.10230e6	−0.329737
$528$	−979041.	−0.152833
$529$	4.93574e6	0.766854
$530$	0	0
$531$	−561841.	−0.0864723
$532$	234394.	0.0359060
$533$	−13174.3	−0.00200867
$534$	−3.02933e6	−0.459720
$535$	0	0
$536$	5.83225e6	0.876848
$537$	−4.97029e6	−0.743782
$538$	1.60239e6	0.238678
$539$	−1.37105e6	−0.203274
$540$	0	0
$541$	−2.84250e6	−0.417548	−0.208774	−	0.977964i	$-0.566947\pi$
$541$	−2.84250e6	−0.417548	−0.208774	+	0.977964i	$0.566947\pi$
$542$	−1.00861e7	−1.47477
$543$	471677.	0.0686507
$544$	−1.02553e6	−0.148576
$545$	0	0
$546$	−236745.	−0.0339859
$547$	1.40460e6	0.200717	0.100358	−	0.994951i	$-0.468001\pi$
$547$	1.40460e6	0.200717	0.100358	+	0.994951i	$0.468001\pi$
$548$	1.35660e6	0.192974
$549$	−853610.	−0.120873
$550$	0	0
$551$	4.52897e6	0.635507
$552$	−5.75306e6	−0.803620
$553$	2.14265e6	0.297947
$554$	2.63309e6	0.364495
$555$	0	0
$556$	638754.	0.0876287
$557$	−4.17495e6	−0.570182	−0.285091	−	0.958500i	$-0.592024\pi$
$557$	−4.17495e6	−0.570182	−0.285091	+	0.958500i	$0.592024\pi$
$558$	1.12351e6	0.152754
$559$	888568.	0.120271
$560$	0	0
$561$	−880866.	−0.118169
$562$	7.42445e6	0.991570
$563$	7.54584e6	1.00331	0.501657	−	0.865067i	$-0.332724\pi$
$563$	7.54584e6	1.00331	0.501657	+	0.865067i	$0.332724\pi$
$564$	121049.	0.0160237
$565$	0	0
$566$	1.02783e7	1.34859
$567$	485513.	0.0634225
$568$	−2.64903e6	−0.344521
$569$	1.62724e6	0.210703	0.105351	−	0.994435i	$-0.466403\pi$
$569$	1.62724e6	0.210703	0.105351	+	0.994435i	$0.466403\pi$
$570$	0	0
$571$	2.56277e6	0.328942	0.164471	−	0.986382i	$-0.447408\pi$
$571$	2.56277e6	0.328942	0.164471	+	0.986382i	$0.447408\pi$
$572$	28357.8	0.00362395
$573$	1.22118e6	0.155379
$574$	78111.0	0.00989538
$575$	0	0
$576$	2.87834e6	0.361482
$577$	−6.16261e6	−0.770594	−0.385297	−	0.922793i	$-0.625901\pi$
$577$	−6.16261e6	−0.770594	−0.385297	+	0.922793i	$0.625901\pi$
$578$	−4.08573e6	−0.508687
$579$	−5.05173e6	−0.626245
$580$	0	0
$581$	−7.68327e6	−0.944290
$582$	−3.02428e6	−0.370096
$583$	587893.	0.0716353
$584$	2.60323e6	0.315850
$585$	0	0
$586$	−8.20086e6	−0.986542
$587$	−9.16105e6	−1.09736	−0.548681	−	0.836032i	$-0.684870\pi$
$587$	−9.16105e6	−1.09736	−0.548681	+	0.836032i	$0.684870\pi$
$588$	358816.	0.0427985
$589$	−2.33974e6	−0.277894
$590$	0	0
$591$	6.96890e6	0.820721
$592$	6.24882e6	0.732814
$593$	−6.23563e6	−0.728188	−0.364094	−	0.931362i	$-0.618621\pi$
$593$	−6.23563e6	−0.728188	−0.364094	+	0.931362i	$0.618621\pi$
$594$	470754.	0.0547429
$595$	0	0
$596$	665351.	0.0767247
$597$	−4.44847e6	−0.510828
$598$	−1.19875e6	−0.137080
$599$	−1.48855e6	−0.169510	−0.0847552	−	0.996402i	$-0.527011\pi$
$599$	−1.48855e6	−0.169510	−0.0847552	+	0.996402i	$0.527011\pi$
$600$	0	0
$601$	−3.40965e6	−0.385055	−0.192528	−	0.981292i	$-0.561668\pi$
$601$	−3.40965e6	−0.385055	−0.192528	+	0.981292i	$0.561668\pi$
$602$	−5.26837e6	−0.592496
$603$	−2.49221e6	−0.279121
$604$	−642309.	−0.0716394
$605$	0	0
$606$	1.54820e6	0.171256
$607$	4.05477e6	0.446678	0.223339	−	0.974741i	$-0.428304\pi$
$607$	4.05477e6	0.446678	0.223339	+	0.974741i	$0.428304\pi$
$608$	−1.14135e6	−0.125216
$609$	−3.35056e6	−0.366078
$610$	0	0
$611$	254614.	0.0275918
$612$	230530.	0.0248799
$613$	−1.60665e6	−0.172691	−0.0863456	−	0.996265i	$-0.527519\pi$
$613$	−1.60665e6	−0.172691	−0.0863456	+	0.996265i	$0.527519\pi$
$614$	1.16213e7	1.24404
$615$	0	0
$616$	−1.69728e6	−0.180219
$617$	−1.01598e7	−1.07441	−0.537207	−	0.843451i	$-0.680520\pi$
$617$	−1.01598e7	−1.07441	−0.537207	+	0.843451i	$0.680520\pi$
$618$	2.20900e6	0.232661
$619$	3.04114e6	0.319014	0.159507	−	0.987197i	$-0.449009\pi$
$619$	3.04114e6	0.319014	0.159507	+	0.987197i	$0.449009\pi$
$620$	0	0
$621$	2.45837e6	0.255811
$622$	3.33508e6	0.345645
$623$	−4.66717e6	−0.481763
$624$	538942.	0.0554090
$625$	0	0
$626$	4.04216e6	0.412267
$627$	−980354.	−0.0995896
$628$	1.05362e6	0.106607
$629$	5.62221e6	0.566605
$630$	0	0
$631$	−1.07109e6	−0.107091	−0.0535455	−	0.998565i	$-0.517052\pi$
$631$	−1.07109e6	−0.107091	−0.0535455	+	0.998565i	$0.517052\pi$
$632$	−5.48855e6	−0.546594
$633$	−270922.	−0.0268742
$634$	1.65558e7	1.63579
$635$	0	0
$636$	−153857.	−0.0150825
$637$	754737.	0.0736965
$638$	−3.24871e6	−0.315980
$639$	1.13197e6	0.109669
$640$	0	0
$641$	−902752.	−0.0867807	−0.0433904	−	0.999058i	$-0.513816\pi$
$641$	−902752.	−0.0867807	−0.0433904	+	0.999058i	$0.513816\pi$
$642$	1.92616e6	0.184440
$643$	1.24048e7	1.18321	0.591607	−	0.806227i	$-0.298494\pi$
$643$	1.24048e7	1.18321	0.591607	+	0.806227i	$0.298494\pi$
$644$	−878033.	−0.0834250
$645$	0	0
$646$	3.88614e6	0.366384
$647$	1.32160e7	1.24119	0.620594	−	0.784132i	$-0.286891\pi$
$647$	1.32160e7	1.24119	0.620594	+	0.784132i	$0.286891\pi$
$648$	−1.24367e6	−0.116351
$649$	−839293.	−0.0782172
$650$	0	0
$651$	1.73095e6	0.160079
$652$	372294.	0.0342979
$653$	1.61119e6	0.147864	0.0739321	−	0.997263i	$-0.476445\pi$
$653$	1.61119e6	0.147864	0.0739321	+	0.997263i	$0.476445\pi$
$654$	−2.14391e6	−0.196003
$655$	0	0
$656$	−177817.	−0.0161329
$657$	−1.11240e6	−0.100542
$658$	−1.50962e6	−0.135926
$659$	2.72261e6	0.244215	0.122108	−	0.992517i	$-0.461035\pi$
$659$	2.72261e6	0.244215	0.122108	+	0.992517i	$0.461035\pi$
$660$	0	0
$661$	−6.62010e6	−0.589333	−0.294666	−	0.955600i	$-0.595209\pi$
$661$	−6.62010e6	−0.589333	−0.294666	+	0.955600i	$0.595209\pi$
$662$	−8.49256e6	−0.753171
$663$	484898.	0.0428417
$664$	1.96812e7	1.73233
$665$	0	0
$666$	−3.00463e6	−0.262486
$667$	−1.69654e7	−1.47655
$668$	−158261.	−0.0137225
$669$	4.85518e6	0.419411
$670$	0	0
$671$	−1.27515e6	−0.109334
$672$	844380.	0.0721298
$673$	4.49731e6	0.382750	0.191375	−	0.981517i	$-0.438705\pi$
$673$	4.49731e6	0.382750	0.191375	+	0.981517i	$0.438705\pi$
$674$	−1.20222e7	−1.01938
$675$	0	0
$676$	1.29079e6	0.108640
$677$	1.05097e6	0.0881288	0.0440644	−	0.999029i	$-0.485969\pi$
$677$	1.05097e6	0.0881288	0.0440644	+	0.999029i	$0.485969\pi$
$678$	−1.20682e7	−1.00825
$679$	−4.65939e6	−0.387842
$680$	0	0
$681$	8.74832e6	0.722865
$682$	1.67834e6	0.138171
$683$	158619.	0.0130108	0.00650540	−	0.999979i	$-0.497929\pi$
$683$	158619.	0.0130108	0.00650540	+	0.999979i	$0.497929\pi$
$684$	256567.	0.0209682
$685$	0	0
$686$	−1.11123e7	−0.901561
$687$	1.55431e6	0.125645
$688$	1.19933e7	0.965976
$689$	−323623.	−0.0259712
$690$	0	0
$691$	−1.14939e7	−0.915741	−0.457870	−	0.889019i	$-0.651388\pi$
$691$	−1.14939e7	−0.915741	−0.457870	+	0.889019i	$0.651388\pi$
$692$	1.56724e6	0.124414
$693$	725272.	0.0573678
$694$	913714.	0.0720131
$695$	0	0
$696$	8.58268e6	0.671583
$697$	−159986.	−0.0124738
$698$	1.65380e7	1.28483
$699$	165212.	0.0127894
$700$	0	0
$701$	−1.15908e7	−0.890879	−0.445440	−	0.895312i	$-0.646953\pi$
$701$	−1.15908e7	−0.890879	−0.445440	+	0.895312i	$0.646953\pi$
$702$	−259140.	−0.0198469
$703$	6.25721e6	0.477521
$704$	4.29975e6	0.326972
$705$	0	0
$706$	−1.24671e7	−0.941352
$707$	2.38525e6	0.179467
$708$	219650.	0.0164683
$709$	8.43220e6	0.629978	0.314989	−	0.949095i	$-0.397999\pi$
$709$	8.43220e6	0.629978	0.314989	+	0.949095i	$0.397999\pi$
$710$	0	0
$711$	2.34534e6	0.173993
$712$	1.19552e7	0.883809
$713$	8.76461e6	0.645667
$714$	−2.87499e6	−0.211053
$715$	0	0
$716$	1.94312e6	0.141650
$717$	−457223.	−0.0332147
$718$	860163.	0.0622687
$719$	2.06959e6	0.149301	0.0746505	−	0.997210i	$-0.476216\pi$
$719$	2.06959e6	0.149301	0.0746505	+	0.997210i	$0.476216\pi$
$720$	0	0
$721$	3.40331e6	0.243817
$722$	−8.88940e6	−0.634643
$723$	515562.	0.0366805
$724$	−184401.	−0.0130742
$725$	0	0
$726$	703225.	0.0495168
$727$	1.69013e7	1.18600	0.593001	−	0.805202i	$-0.297943\pi$
$727$	1.69013e7	1.18600	0.593001	+	0.805202i	$0.297943\pi$
$728$	934315.	0.0653378
$729$	531441.	0.0370370
$730$	0	0
$731$	1.07906e7	0.746884
$732$	333716.	0.0230197
$733$	9.45646e6	0.650083	0.325041	−	0.945700i	$-0.394622\pi$
$733$	9.45646e6	0.650083	0.325041	+	0.945700i	$0.394622\pi$
$734$	1.16791e7	0.800144
$735$	0	0
$736$	4.27548e6	0.290931
$737$	−3.72294e6	−0.252474
$738$	85500.1	0.00577864
$739$	1.41110e7	0.950486	0.475243	−	0.879855i	$-0.342360\pi$
$739$	1.41110e7	0.950486	0.475243	+	0.879855i	$0.342360\pi$
$740$	0	0
$741$	539665.	0.0361059
$742$	1.91878e6	0.127943
$743$	3.86361e6	0.256757	0.128378	−	0.991725i	$-0.459023\pi$
$743$	3.86361e6	0.256757	0.128378	+	0.991725i	$0.459023\pi$
$744$	−4.43395e6	−0.293669
$745$	0	0
$746$	1.94062e7	1.27671
$747$	−8.41008e6	−0.551441
$748$	344372.	0.0225047
$749$	2.96756e6	0.193283
$750$	0	0
$751$	5.68914e6	0.368084	0.184042	−	0.982918i	$-0.441082\pi$
$751$	5.68914e6	0.368084	0.184042	+	0.982918i	$0.441082\pi$
$752$	3.43660e6	0.221608
$753$	2.48172e6	0.159502
$754$	1.78835e6	0.114557
$755$	0	0
$756$	−189810.	−0.0120785
$757$	2.31729e7	1.46974	0.734870	−	0.678208i	$-0.237243\pi$
$757$	2.31729e7	1.46974	0.734870	+	0.678208i	$0.237243\pi$
$758$	8.95116e6	0.565856
$759$	3.67238e6	0.231389
$760$	0	0
$761$	−1.16856e7	−0.731455	−0.365728	−	0.930722i	$-0.619180\pi$
$761$	−1.16856e7	−0.731455	−0.365728	+	0.930722i	$0.619180\pi$
$762$	1.03008e7	0.642662
$763$	−3.30304e6	−0.205401
$764$	−477417.	−0.0295913
$765$	0	0
$766$	−1.11214e7	−0.684838
$767$	462014.	0.0283574
$768$	−3.07395e6	−0.188059
$769$	7.02112e6	0.428145	0.214072	−	0.976818i	$-0.431327\pi$
$769$	7.02112e6	0.428145	0.214072	+	0.976818i	$0.431327\pi$
$770$	0	0
$771$	−497912.	−0.0301659
$772$	1.97496e6	0.119266
$773$	4.45003e6	0.267864	0.133932	−	0.990991i	$-0.457240\pi$
$773$	4.45003e6	0.267864	0.133932	+	0.990991i	$0.457240\pi$
$774$	−5.76674e6	−0.346002
$775$	0	0
$776$	1.19353e7	0.711509
$777$	−4.62912e6	−0.275072
$778$	−3.09684e7	−1.83430
$779$	−178056.	−0.0105126
$780$	0	0
$781$	1.69097e6	0.0991991
$782$	−1.45574e7	−0.851268
$783$	−3.66751e6	−0.213780
$784$	1.01869e7	0.591905
$785$	0	0
$786$	7.55980e6	0.436470
$787$	1.96811e7	1.13269	0.566347	−	0.824167i	$-0.308356\pi$
$787$	1.96811e7	1.13269	0.566347	+	0.824167i	$0.308356\pi$
$788$	−2.72447e6	−0.156303
$789$	1.35587e7	0.775398
$790$	0	0
$791$	−1.85929e7	−1.05659
$792$	−1.85783e6	−0.105243
$793$	701941.	0.0396386
$794$	6.71578e6	0.378047
$795$	0	0
$796$	1.73912e6	0.0972850
$797$	−3.00192e7	−1.67399	−0.836996	−	0.547208i	$-0.815690\pi$
$797$	−3.00192e7	−1.67399	−0.836996	+	0.547208i	$0.815690\pi$
$798$	−3.19970e6	−0.177870
$799$	3.09199e6	0.171345
$800$	0	0
$801$	−5.10867e6	−0.281337
$802$	−6.92915e6	−0.380403
$803$	−1.66174e6	−0.0909439
$804$	974323.	0.0531573
$805$	0	0
$806$	−923889.	−0.0500936
$807$	2.70227e6	0.146065
$808$	−6.10998e6	−0.329239
$809$	−8.99554e6	−0.483233	−0.241616	−	0.970372i	$-0.577678\pi$
$809$	−8.99554e6	−0.483233	−0.241616	+	0.970372i	$0.577678\pi$
$810$	0	0
$811$	−2.96119e7	−1.58093	−0.790467	−	0.612505i	$-0.790162\pi$
$811$	−2.96119e7	−1.58093	−0.790467	+	0.612505i	$0.790162\pi$
$812$	1.30989e6	0.0697180
$813$	−1.70092e7	−0.902523
$814$	−4.48840e6	−0.237427
$815$	0	0
$816$	6.54481e6	0.344090
$817$	1.20093e7	0.629455
$818$	−2.37917e7	−1.24320
$819$	−399247.	−0.0207985
$820$	0	0
$821$	−2.05145e7	−1.06219	−0.531097	−	0.847311i	$-0.678220\pi$
$821$	−2.05145e7	−1.06219	−0.531097	+	0.847311i	$0.678220\pi$
$822$	−1.85189e7	−0.955949
$823$	2.38536e7	1.22759	0.613796	−	0.789465i	$-0.289642\pi$
$823$	2.38536e7	1.22759	0.613796	+	0.789465i	$0.289642\pi$
$824$	−8.71781e6	−0.447290
$825$	0	0
$826$	−2.73930e6	−0.139698
$827$	1.89103e7	0.961468	0.480734	−	0.876866i	$-0.340370\pi$
$827$	1.89103e7	0.961468	0.480734	+	0.876866i	$0.340370\pi$
$828$	−961093.	−0.0487180
$829$	3.48075e7	1.75908	0.879541	−	0.475823i	$-0.157850\pi$
$829$	3.48075e7	1.75908	0.879541	+	0.475823i	$0.157850\pi$
$830$	0	0
$831$	4.44045e6	0.223061
$832$	−2.36692e6	−0.118543
$833$	9.16540e6	0.457656
$834$	−8.71961e6	−0.434092
$835$	0	0
$836$	383266.	0.0189664
$837$	1.89470e6	0.0934816
$838$	1.24529e7	0.612575
$839$	−2.32033e7	−1.13800	−0.569002	−	0.822336i	$-0.692670\pi$
$839$	−2.32033e7	−1.13800	−0.569002	+	0.822336i	$0.692670\pi$
$840$	0	0
$841$	4.79863e6	0.233952
$842$	2.16002e7	1.04997
$843$	1.25206e7	0.606815
$844$	105916.	0.00511807
$845$	0	0
$846$	−1.65243e6	−0.0793774
$847$	1.08343e6	0.0518911
$848$	−4.36804e6	−0.208591
$849$	1.73333e7	0.825299
$850$	0	0
$851$	−2.34393e7	−1.10949
$852$	−442540.	−0.0208859
$853$	2.30500e7	1.08467	0.542336	−	0.840162i	$-0.317540\pi$
$853$	2.30500e7	1.08467	0.542336	+	0.840162i	$0.317540\pi$
$854$	−4.16185e6	−0.195273
$855$	0	0
$856$	−7.60159e6	−0.354585
$857$	−1.45809e7	−0.678158	−0.339079	−	0.940758i	$-0.610115\pi$
$857$	−1.45809e7	−0.678158	−0.339079	+	0.940758i	$0.610115\pi$
$858$	−387111.	−0.0179522
$859$	4.26393e6	0.197164	0.0985818	−	0.995129i	$-0.468569\pi$
$859$	4.26393e6	0.197164	0.0985818	+	0.995129i	$0.468569\pi$
$860$	0	0
$861$	131727.	0.00605572
$862$	1.27048e7	0.582370
$863$	9.80948e6	0.448352	0.224176	−	0.974549i	$-0.428031\pi$
$863$	9.80948e6	0.448352	0.224176	+	0.974549i	$0.428031\pi$
$864$	924256.	0.0421219
$865$	0	0
$866$	2.68102e7	1.21480
$867$	−6.89019e6	−0.311303
$868$	−676711.	−0.0304863
$869$	3.50354e6	0.157383
$870$	0	0
$871$	2.04940e6	0.0915337
$872$	8.46096e6	0.376815
$873$	−5.10016e6	−0.226489
$874$	−1.62015e7	−0.717427
$875$	0	0
$876$	434890.	0.0191478
$877$	1.86437e7	0.818525	0.409263	−	0.912417i	$-0.365786\pi$
$877$	1.86437e7	0.818525	0.409263	+	0.912417i	$0.365786\pi$
$878$	−1.21367e7	−0.531328
$879$	−1.38300e7	−0.603738
$880$	0	0
$881$	−3.42571e7	−1.48700	−0.743500	−	0.668736i	$-0.766836\pi$
$881$	−3.42571e7	−1.48700	−0.743500	+	0.668736i	$0.766836\pi$
$882$	−4.89819e6	−0.212014
$883$	1.57969e7	0.681818	0.340909	−	0.940096i	$-0.389265\pi$
$883$	1.57969e7	0.681818	0.340909	+	0.940096i	$0.389265\pi$
$884$	−189570.	−0.00815902
$885$	0	0
$886$	−500014.	−0.0213992
$887$	−2.96846e7	−1.26684	−0.633420	−	0.773808i	$-0.718349\pi$
$887$	−2.96846e7	−1.26684	−0.633420	+	0.773808i	$0.718349\pi$
$888$	1.18578e7	0.504628
$889$	1.58700e7	0.673477
$890$	0	0
$891$	793881.	0.0335013
$892$	−1.89812e6	−0.0798750
$893$	3.44121e6	0.144405
$894$	−9.08268e6	−0.380076
$895$	0	0
$896$	1.10314e7	0.459049
$897$	−2.02157e6	−0.0838895
$898$	−3.18136e7	−1.31650
$899$	−1.30754e7	−0.539582
$900$	0	0
$901$	−3.93002e6	−0.161281
$902$	127722.	0.00522698
$903$	−8.88460e6	−0.362592
$904$	4.76271e7	1.93835
$905$	0	0
$906$	8.76814e6	0.354885
$907$	2.52650e7	1.01977	0.509884	−	0.860243i	$-0.329688\pi$
$907$	2.52650e7	1.01977	0.509884	+	0.860243i	$0.329688\pi$
$908$	−3.42013e6	−0.137666
$909$	2.61089e6	0.104804
$910$	0	0
$911$	−2.64889e7	−1.05747	−0.528734	−	0.848787i	$-0.677333\pi$
$911$	−2.64889e7	−1.05747	−0.528734	+	0.848787i	$0.677333\pi$
$912$	7.28401e6	0.289990
$913$	−1.25632e7	−0.498797
$914$	4.68686e6	0.185574
$915$	0	0
$916$	−607653.	−0.0239286
$917$	1.16471e7	0.457398
$918$	−3.14696e6	−0.123249
$919$	4.82111e6	0.188303	0.0941517	−	0.995558i	$-0.469986\pi$
$919$	4.82111e6	0.188303	0.0941517	+	0.995558i	$0.469986\pi$
$920$	0	0
$921$	1.95983e7	0.761322
$922$	−15802.0	−0.000612188 0
$923$	−930843.	−0.0359643
$924$	−283543.	−0.0109254
$925$	0	0
$926$	−9.28375e6	−0.355792
$927$	3.72526e6	0.142383
$928$	−6.37836e6	−0.243130
$929$	−5.00384e7	−1.90224	−0.951118	−	0.308828i	$-0.900063\pi$
$929$	−5.00384e7	−1.90224	−0.951118	+	0.308828i	$0.900063\pi$
$930$	0	0
$931$	1.02006e7	0.385700
$932$	−64589.3	−0.00243568
$933$	5.62428e6	0.211526
$934$	1.26213e6	0.0473409
$935$	0	0
$936$	1.02270e6	0.0381556
$937$	3.70330e7	1.37797	0.688985	−	0.724775i	$-0.258056\pi$
$937$	3.70330e7	1.37797	0.688985	+	0.724775i	$0.258056\pi$
$938$	−1.21510e7	−0.450926
$939$	6.81672e6	0.252296
$940$	0	0
$941$	3.30398e7	1.21636	0.608182	−	0.793798i	$-0.291899\pi$
$941$	3.30398e7	1.21636	0.608182	+	0.793798i	$0.291899\pi$
$942$	−1.43829e7	−0.528104
$943$	666992.	0.0244254
$944$	6.23593e6	0.227757
$945$	0	0
$946$	−8.61452e6	−0.312970
$947$	410586.	0.0148775	0.00743873	−	0.999972i	$-0.497632\pi$
$947$	410586.	0.0148775	0.00743873	+	0.999972i	$0.497632\pi$
$948$	−916905.	−0.0331363
$949$	914752.	0.0329714
$950$	0	0
$951$	2.79198e7	1.00106
$952$	1.13462e7	0.405748
$953$	−4.08733e7	−1.45783	−0.728915	−	0.684604i	$-0.759975\pi$
$953$	−4.08733e7	−1.45783	−0.728915	+	0.684604i	$0.759975\pi$
$954$	2.10029e6	0.0747151
$955$	0	0
$956$	178750.	0.00632560
$957$	−5.47863e6	−0.193371
$958$	−6.59272e6	−0.232087
$959$	−2.85313e7	−1.00179
$960$	0	0
$961$	−2.18742e7	−0.764052
$962$	2.47077e6	0.0860786
$963$	3.24828e6	0.112872
$964$	−201557.	−0.00698564
$965$	0	0
$966$	1.19860e7	0.413268
$967$	1.34815e7	0.463629	0.231815	−	0.972760i	$-0.425534\pi$
$967$	1.34815e7	0.463629	0.231815	+	0.972760i	$0.425534\pi$
$968$	−2.77528e6	−0.0951960
$969$	6.55359e6	0.224218
$970$	0	0
$971$	−4.78750e7	−1.62952	−0.814761	−	0.579797i	$-0.803132\pi$
$971$	−4.78750e7	−1.62952	−0.814761	+	0.579797i	$0.803132\pi$
$972$	−207765.	−0.00705354
$973$	−1.34340e7	−0.454906
$974$	−2.54696e7	−0.860250
$975$	0	0
$976$	9.47431e6	0.318363
$977$	−1.03468e7	−0.346793	−0.173396	−	0.984852i	$-0.555474\pi$
$977$	−1.03468e7	−0.346793	−0.173396	+	0.984852i	$0.555474\pi$
$978$	−5.08218e6	−0.169904
$979$	−7.63146e6	−0.254479
$980$	0	0
$981$	−3.61550e6	−0.119949
$982$	4.87080e7	1.61184
$983$	1.81075e7	0.597689	0.298844	−	0.954302i	$-0.403399\pi$
$983$	1.81075e7	0.597689	0.298844	+	0.954302i	$0.403399\pi$
$984$	−337426.	−0.0111094
$985$	0	0
$986$	2.17174e7	0.711402
$987$	−2.54583e6	−0.0831835
$988$	−210980.	−0.00687621
$989$	−4.49867e7	−1.46249
$990$	0	0
$991$	−4.58378e7	−1.48265	−0.741327	−	0.671144i	$-0.765803\pi$
$991$	−4.58378e7	−1.48265	−0.741327	+	0.671144i	$0.765803\pi$
$992$	3.29516e6	0.106316
$993$	−1.43219e7	−0.460921
$994$	5.51902e6	0.177172
$995$	0	0
$996$	3.28790e6	0.105019
$997$	−3.02752e7	−0.964605	−0.482302	−	0.876005i	$-0.660199\pi$
$997$	−3.02752e7	−0.964605	−0.482302	+	0.876005i	$0.660199\pi$
$998$	4.49844e7	1.42967
$999$	−5.06702e6	−0.160635

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	825.6.a.m.1.6	✓	7
5.4	even	2		825.6.a.o.1.2	yes	7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
825.6.a.m.1.6	✓	7	1.1	even	1	trivial
825.6.a.o.1.2	yes	7	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)

\(f(q)\)	\(=\)	\(q+5.33680 q^{2} +9.00000 q^{3} -3.51852 q^{4} +48.0312 q^{6} +73.9998 q^{7} -189.555 q^{8} +81.0000 q^{9} +O(q^{10})\)	\(q+5.33680 q^{2} +9.00000 q^{3} -3.51852 q^{4} +48.0312 q^{6} +73.9998 q^{7} -189.555 q^{8} +81.0000 q^{9} +121.000 q^{11} -31.6667 q^{12} -66.6080 q^{13} +394.923 q^{14} -899.027 q^{16} -808.876 q^{17} +432.281 q^{18} -900.233 q^{19} +665.998 q^{21} +645.753 q^{22} +3372.25 q^{23} -1706.00 q^{24} -355.474 q^{26} +729.000 q^{27} -260.370 q^{28} -5030.88 q^{29} +2599.04 q^{31} +1267.84 q^{32} +1089.00 q^{33} -4316.81 q^{34} -285.000 q^{36} -6950.65 q^{37} -4804.37 q^{38} -599.472 q^{39} +197.788 q^{41} +3554.30 q^{42} -13340.3 q^{43} -425.741 q^{44} +17997.0 q^{46} -3822.58 q^{47} -8091.25 q^{48} -11331.0 q^{49} -7279.88 q^{51} +234.362 q^{52} +4858.62 q^{53} +3890.53 q^{54} -14027.1 q^{56} -8102.10 q^{57} -26848.8 q^{58} -6936.31 q^{59} -10538.4 q^{61} +13870.5 q^{62} +5993.99 q^{63} +35535.1 q^{64} +5811.78 q^{66} -30768.1 q^{67} +2846.05 q^{68} +30350.3 q^{69} +13974.9 q^{71} -15354.0 q^{72} -13733.4 q^{73} -37094.3 q^{74} +3167.49 q^{76} +8953.98 q^{77} -3199.26 q^{78} +28954.9 q^{79} +6561.00 q^{81} +1055.56 q^{82} -103828. q^{83} -2343.33 q^{84} -71194.4 q^{86} -45277.9 q^{87} -22936.2 q^{88} -63069.9 q^{89} -4928.98 q^{91} -11865.3 q^{92} +23391.3 q^{93} -20400.4 q^{94} +11410.6 q^{96} -62964.9 q^{97} -60471.5 q^{98} +9801.00 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q - 9 q^{2} + 63 q^{3} + 95 q^{4} - 81 q^{6} - 65 q^{7} - 207 q^{8} + 567 q^{9}+O(q^{10}) \) 7 * q - 9 * q^2 + 63 * q^3 + 95 * q^4 - 81 * q^6 - 65 * q^7 - 207 * q^8 + 567 * q^9	\( 7 q - 9 q^{2} + 63 q^{3} + 95 q^{4} - 81 q^{6} - 65 q^{7} - 207 q^{8} + 567 q^{9} + 847 q^{11} + 855 q^{12} + 113 q^{13} - 1212 q^{14} + 499 q^{16} - 1030 q^{17} - 729 q^{18} - 3803 q^{19} - 585 q^{21} - 1089 q^{22} - 514 q^{23} - 1863 q^{24} - 12111 q^{26} + 5103 q^{27} + 342 q^{28} - 2698 q^{29} - 17233 q^{31} - 9943 q^{32} + 7623 q^{33} + 4090 q^{34} + 7695 q^{36} - 23182 q^{37} + 11943 q^{38} + 1017 q^{39} - 16158 q^{41} - 10908 q^{42} + 4249 q^{43} + 11495 q^{44} - 28769 q^{46} - 7580 q^{47} + 4491 q^{48} + 37140 q^{49} - 9270 q^{51} + 23887 q^{52} - 20574 q^{53} - 6561 q^{54} - 73276 q^{56} - 34227 q^{57} - 8733 q^{58} - 364 q^{59} - 28127 q^{61} + 71917 q^{62} - 5265 q^{63} + 43379 q^{64} - 9801 q^{66} + 21493 q^{67} - 160660 q^{68} - 4626 q^{69} - 177084 q^{71} - 16767 q^{72} - 78670 q^{73} + 196750 q^{74} - 32701 q^{76} - 7865 q^{77} - 108999 q^{78} - 187432 q^{79} + 45927 q^{81} - 179552 q^{82} + 44592 q^{83} + 3078 q^{84} - 110433 q^{86} - 24282 q^{87} - 25047 q^{88} - 151168 q^{89} - 230153 q^{91} + 44767 q^{92} - 155097 q^{93} + 54040 q^{94} - 89487 q^{96} + 55589 q^{97} - 478341 q^{98} + 68607 q^{99}+O(q^{100}) \) 7 * q - 9 * q^2 + 63 * q^3 + 95 * q^4 - 81 * q^6 - 65 * q^7 - 207 * q^8 + 567 * q^9 + 847 * q^11 + 855 * q^12 + 113 * q^13 - 1212 * q^14 + 499 * q^16 - 1030 * q^17 - 729 * q^18 - 3803 * q^19 - 585 * q^21 - 1089 * q^22 - 514 * q^23 - 1863 * q^24 - 12111 * q^26 + 5103 * q^27 + 342 * q^28 - 2698 * q^29 - 17233 * q^31 - 9943 * q^32 + 7623 * q^33 + 4090 * q^34 + 7695 * q^36 - 23182 * q^37 + 11943 * q^38 + 1017 * q^39 - 16158 * q^41 - 10908 * q^42 + 4249 * q^43 + 11495 * q^44 - 28769 * q^46 - 7580 * q^47 + 4491 * q^48 + 37140 * q^49 - 9270 * q^51 + 23887 * q^52 - 20574 * q^53 - 6561 * q^54 - 73276 * q^56 - 34227 * q^57 - 8733 * q^58 - 364 * q^59 - 28127 * q^61 + 71917 * q^62 - 5265 * q^63 + 43379 * q^64 - 9801 * q^66 + 21493 * q^67 - 160660 * q^68 - 4626 * q^69 - 177084 * q^71 - 16767 * q^72 - 78670 * q^73 + 196750 * q^74 - 32701 * q^76 - 7865 * q^77 - 108999 * q^78 - 187432 * q^79 + 45927 * q^81 - 179552 * q^82 + 44592 * q^83 + 3078 * q^84 - 110433 * q^86 - 24282 * q^87 - 25047 * q^88 - 151168 * q^89 - 230153 * q^91 + 44767 * q^92 - 155097 * q^93 + 54040 * q^94 - 89487 * q^96 + 55589 * q^97 - 478341 * q^98 + 68607 * q^99

Introduction

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