LMFDB - Embedded newform 825.6.a.f.1.2

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:	$3$
Coefficient field:	3.3.307532.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 76x + 168 $ x^3 - x^2 - 76*x + 168
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 165)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.30119$ 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)

$f(q)$	$=$	$q-4.30119 q^{2} +9.00000 q^{3} -13.4998 q^{4} -38.7107 q^{6} +148.216 q^{7} +195.703 q^{8} +81.0000 q^{9} +O(q^{10})$	\(q-4.30119 q^{2} +9.00000 q^{3} -13.4998 q^{4} -38.7107 q^{6} +148.216 q^{7} +195.703 q^{8} +81.0000 q^{9} +121.000 q^{11} -121.498 q^{12} -234.137 q^{13} -637.504 q^{14} -409.764 q^{16} -1031.76 q^{17} -348.396 q^{18} +1266.29 q^{19} +1333.94 q^{21} -520.444 q^{22} -384.710 q^{23} +1761.33 q^{24} +1007.07 q^{26} +729.000 q^{27} -2000.88 q^{28} -4484.85 q^{29} -997.235 q^{31} -4500.03 q^{32} +1089.00 q^{33} +4437.78 q^{34} -1093.48 q^{36} -5168.11 q^{37} -5446.56 q^{38} -2107.23 q^{39} +2259.17 q^{41} -5737.54 q^{42} -15818.9 q^{43} -1633.47 q^{44} +1654.71 q^{46} -12033.2 q^{47} -3687.87 q^{48} +5160.90 q^{49} -9285.81 q^{51} +3160.79 q^{52} +3851.67 q^{53} -3135.57 q^{54} +29006.3 q^{56} +11396.6 q^{57} +19290.2 q^{58} -20261.0 q^{59} +2006.88 q^{61} +4289.30 q^{62} +12005.5 q^{63} +32467.9 q^{64} -4684.00 q^{66} -40945.6 q^{67} +13928.5 q^{68} -3462.39 q^{69} +16970.8 q^{71} +15852.0 q^{72} +56640.5 q^{73} +22229.0 q^{74} -17094.6 q^{76} +17934.1 q^{77} +9063.59 q^{78} +58507.9 q^{79} +6561.00 q^{81} -9717.12 q^{82} +52243.8 q^{83} -18007.9 q^{84} +68040.2 q^{86} -40363.6 q^{87} +23680.1 q^{88} +55114.4 q^{89} -34702.7 q^{91} +5193.50 q^{92} -8975.11 q^{93} +51757.0 q^{94} -40500.3 q^{96} -99383.8 q^{97} -22198.0 q^{98} +9801.00 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 7 q^{2} + 27 q^{3} + 73 q^{4} - 63 q^{6} - 92 q^{7} - 231 q^{8} + 243 q^{9}+O(q^{10}) $ 3 * q - 7 * q^2 + 27 * q^3 + 73 * q^4 - 63 * q^6 - 92 * q^7 - 231 * q^8 + 243 * q^9	\( 3 q - 7 q^{2} + 27 q^{3} + 73 q^{4} - 63 q^{6} - 92 q^{7} - 231 q^{8} + 243 q^{9} + 363 q^{11} + 657 q^{12} + 90 q^{13} - 784 q^{14} - 415 q^{16} - 1934 q^{17} - 567 q^{18} + 2084 q^{19} - 828 q^{21} - 847 q^{22} - 1220 q^{23} - 2079 q^{24} + 17062 q^{26} + 2187 q^{27} - 11120 q^{28} + 4402 q^{29} - 10688 q^{31} - 12439 q^{32} + 3267 q^{33} - 4094 q^{34} + 5913 q^{36} + 8190 q^{37} - 13792 q^{38} + 810 q^{39} + 5974 q^{41} - 7056 q^{42} - 18868 q^{43} + 8833 q^{44} + 46220 q^{46} - 55500 q^{47} - 3735 q^{48} + 1907 q^{49} - 17406 q^{51} - 27330 q^{52} - 9206 q^{53} - 5103 q^{54} + 73248 q^{56} + 18756 q^{57} - 15366 q^{58} - 59196 q^{59} + 79902 q^{61} - 64616 q^{62} - 7452 q^{63} + 2129 q^{64} - 7623 q^{66} - 4468 q^{67} + 1218 q^{68} - 10980 q^{69} - 75164 q^{71} - 18711 q^{72} + 61290 q^{73} - 56766 q^{74} + 37816 q^{76} - 11132 q^{77} + 153558 q^{78} - 83564 q^{79} + 19683 q^{81} - 147410 q^{82} - 74764 q^{83} - 100080 q^{84} - 253432 q^{86} + 39618 q^{87} - 27951 q^{88} + 37342 q^{89} - 126488 q^{91} - 148164 q^{92} - 96192 q^{93} + 59252 q^{94} - 111951 q^{96} - 33486 q^{97} + 95249 q^{98} + 29403 q^{99}+O(q^{100}) \) 3 * q - 7 * q^2 + 27 * q^3 + 73 * q^4 - 63 * q^6 - 92 * q^7 - 231 * q^8 + 243 * q^9 + 363 * q^11 + 657 * q^12 + 90 * q^13 - 784 * q^14 - 415 * q^16 - 1934 * q^17 - 567 * q^18 + 2084 * q^19 - 828 * q^21 - 847 * q^22 - 1220 * q^23 - 2079 * q^24 + 17062 * q^26 + 2187 * q^27 - 11120 * q^28 + 4402 * q^29 - 10688 * q^31 - 12439 * q^32 + 3267 * q^33 - 4094 * q^34 + 5913 * q^36 + 8190 * q^37 - 13792 * q^38 + 810 * q^39 + 5974 * q^41 - 7056 * q^42 - 18868 * q^43 + 8833 * q^44 + 46220 * q^46 - 55500 * q^47 - 3735 * q^48 + 1907 * q^49 - 17406 * q^51 - 27330 * q^52 - 9206 * q^53 - 5103 * q^54 + 73248 * q^56 + 18756 * q^57 - 15366 * q^58 - 59196 * q^59 + 79902 * q^61 - 64616 * q^62 - 7452 * q^63 + 2129 * q^64 - 7623 * q^66 - 4468 * q^67 + 1218 * q^68 - 10980 * q^69 - 75164 * q^71 - 18711 * q^72 + 61290 * q^73 - 56766 * q^74 + 37816 * q^76 - 11132 * q^77 + 153558 * q^78 - 83564 * q^79 + 19683 * q^81 - 147410 * q^82 - 74764 * q^83 - 100080 * q^84 - 253432 * q^86 + 39618 * q^87 - 27951 * q^88 + 37342 * q^89 - 126488 * q^91 - 148164 * q^92 - 96192 * q^93 + 59252 * q^94 - 111951 * q^96 - 33486 * q^97 + 95249 * q^98 + 29403 * 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 $

$2$	−4.30119	−0.760350	−0.380175	−	0.924915i	$-0.624136\pi$
$2$	−4.30119	−0.760350	−0.380175	+	0.924915i	$0.624136\pi$
$3$	9.00000	0.577350
$3$	9.00000	0.577350
$4$	−13.4998	−0.421868
$5$	0	0
$5$	0	0
$6$	−38.7107	−0.438988
$7$	148.216	1.14327	0.571635	−	0.820508i	$-0.306309\pi$
$7$	148.216	1.14327	0.571635	+	0.820508i	$0.306309\pi$
$8$	195.703	1.08112
$9$	81.0000	0.333333
$10$	0	0
$11$	121.000	0.301511
$11$	121.000	0.301511
$12$	−121.498	−0.243565
$13$	−234.137	−0.384247	−0.192124	−	0.981371i	$-0.561537\pi$
$13$	−234.137	−0.384247	−0.192124	+	0.981371i	$0.561537\pi$
$14$	−637.504	−0.869286
$15$	0	0
$16$	−409.764	−0.400160
$17$	−1031.76	−0.865875	−0.432937	−	0.901424i	$-0.642523\pi$
$17$	−1031.76	−0.865875	−0.432937	+	0.901424i	$0.642523\pi$
$18$	−348.396	−0.253450
$19$	1266.29	0.804729	0.402365	−	0.915480i	$-0.368188\pi$
$19$	1266.29	0.804729	0.402365	+	0.915480i	$0.368188\pi$
$20$	0	0
$21$	1333.94	0.660068
$22$	−520.444	−0.229254
$23$	−384.710	−0.151640	−0.0758201	−	0.997122i	$-0.524157\pi$
$23$	−384.710	−0.151640	−0.0758201	+	0.997122i	$0.524157\pi$
$24$	1761.33	0.624183
$25$	0	0
$26$	1007.07	0.292163
$27$	729.000	0.192450
$28$	−2000.88	−0.482309
$29$	−4484.85	−0.990268	−0.495134	−	0.868817i	$-0.664881\pi$
$29$	−4484.85	−0.990268	−0.495134	+	0.868817i	$0.664881\pi$
$30$	0	0
$31$	−997.235	−0.186377	−0.0931887	−	0.995648i	$-0.529706\pi$
$31$	−997.235	−0.186377	−0.0931887	+	0.995648i	$0.529706\pi$
$32$	−4500.03	−0.776856
$33$	1089.00	0.174078
$34$	4437.78	0.658368
$35$	0	0
$36$	−1093.48	−0.140623
$37$	−5168.11	−0.620622	−0.310311	−	0.950635i	$-0.600433\pi$
$37$	−5168.11	−0.620622	−0.310311	+	0.950635i	$0.600433\pi$
$38$	−5446.56	−0.611876
$39$	−2107.23	−0.221845
$40$	0	0
$41$	2259.17	0.209889	0.104944	−	0.994478i	$-0.466534\pi$
$41$	2259.17	0.209889	0.104944	+	0.994478i	$0.466534\pi$
$42$	−5737.54	−0.501883
$43$	−15818.9	−1.30469	−0.652343	−	0.757924i	$-0.726214\pi$
$43$	−15818.9	−1.30469	−0.652343	+	0.757924i	$0.726214\pi$
$44$	−1633.47	−0.127198
$45$	0	0
$46$	1654.71	0.115300
$47$	−12033.2	−0.794577	−0.397289	−	0.917694i	$-0.630049\pi$
$47$	−12033.2	−0.794577	−0.397289	+	0.917694i	$0.630049\pi$
$48$	−3687.87	−0.231032
$49$	5160.90	0.307069
$50$	0	0
$51$	−9285.81	−0.499913
$52$	3160.79	0.162102
$53$	3851.67	0.188347	0.0941737	−	0.995556i	$-0.469979\pi$
$53$	3851.67	0.188347	0.0941737	+	0.995556i	$0.469979\pi$
$54$	−3135.57	−0.146329
$55$	0	0
$56$	29006.3	1.23601
$57$	11396.6	0.464611
$58$	19290.2	0.752950
$59$	−20261.0	−0.757760	−0.378880	−	0.925446i	$-0.623691\pi$
$59$	−20261.0	−0.757760	−0.378880	+	0.925446i	$0.623691\pi$
$60$	0	0
$61$	2006.88	0.0690553	0.0345276	−	0.999404i	$-0.489007\pi$
$61$	2006.88	0.0690553	0.0345276	+	0.999404i	$0.489007\pi$
$62$	4289.30	0.141712
$63$	12005.5	0.381090
$64$	32467.9	0.990842
$65$	0	0
$66$	−4684.00	−0.132360
$67$	−40945.6	−1.11435	−0.557173	−	0.830397i	$-0.688114\pi$
$67$	−40945.6	−1.11435	−0.557173	+	0.830397i	$0.688114\pi$
$68$	13928.5	0.365285
$69$	−3462.39	−0.0875495
$70$	0	0
$71$	16970.8	0.399537	0.199769	−	0.979843i	$-0.435981\pi$
$71$	16970.8	0.399537	0.199769	+	0.979843i	$0.435981\pi$
$72$	15852.0	0.360372
$73$	56640.5	1.24400	0.621999	−	0.783018i	$-0.286321\pi$
$73$	56640.5	1.24400	0.621999	+	0.783018i	$0.286321\pi$
$74$	22229.0	0.471890
$75$	0	0
$76$	−17094.6	−0.339489
$77$	17934.1	0.344709
$78$	9063.59	0.168680
$79$	58507.9	1.05474	0.527372	−	0.849635i	$-0.323178\pi$
$79$	58507.9	1.05474	0.527372	+	0.849635i	$0.323178\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	−9717.12	−0.159589
$83$	52243.8	0.832415	0.416207	−	0.909270i	$-0.363359\pi$
$83$	52243.8	0.832415	0.416207	+	0.909270i	$0.363359\pi$
$84$	−18007.9	−0.278461
$85$	0	0
$86$	68040.2	0.992018
$87$	−40363.6	−0.571732
$88$	23680.1	0.325969
$89$	55114.4	0.737548	0.368774	−	0.929519i	$-0.379778\pi$
$89$	55114.4	0.737548	0.368774	+	0.929519i	$0.379778\pi$
$90$	0	0
$91$	−34702.7	−0.439299
$92$	5193.50	0.0639721
$93$	−8975.11	−0.107605
$94$	51757.0	0.604157
$95$	0	0
$96$	−40500.3	−0.448518
$97$	−99383.8	−1.07247	−0.536237	−	0.844068i	$-0.680154\pi$
$97$	−99383.8	−1.07247	−0.536237	+	0.844068i	$0.680154\pi$
$98$	−22198.0	−0.233480
$99$	9801.00	0.100504
$100$	0	0
$101$	−25334.3	−0.247118	−0.123559	−	0.992337i	$-0.539431\pi$
$101$	−25334.3	−0.247118	−0.123559	+	0.992337i	$0.539431\pi$
$102$	39940.0	0.380109
$103$	93216.6	0.865766	0.432883	−	0.901450i	$-0.357496\pi$
$103$	93216.6	0.865766	0.432883	+	0.901450i	$0.357496\pi$
$104$	−45821.3	−0.415416
$105$	0	0
$106$	−16566.8	−0.143210
$107$	−205830.	−1.73799	−0.868997	−	0.494817i	$-0.835235\pi$
$107$	−205830.	−1.73799	−0.868997	+	0.494817i	$0.835235\pi$
$108$	−9841.33	−0.0811885
$109$	−155901.	−1.25684	−0.628422	−	0.777872i	$-0.716299\pi$
$109$	−155901.	−1.25684	−0.628422	+	0.777872i	$0.716299\pi$
$110$	0	0
$111$	−46513.0	−0.358316
$112$	−60733.4	−0.457491
$113$	−40304.5	−0.296932	−0.148466	−	0.988917i	$-0.547434\pi$
$113$	−40304.5	−0.296932	−0.148466	+	0.988917i	$0.547434\pi$
$114$	−49019.0	−0.353267
$115$	0	0
$116$	60544.4	0.417762
$117$	−18965.1	−0.128082
$118$	87146.6	0.576163
$119$	−152923.	−0.989929
$120$	0	0
$121$	14641.0	0.0909091
$122$	−8631.97	−0.0525062
$123$	20332.5	0.121179
$124$	13462.4	0.0786266
$125$	0	0
$126$	−51637.8	−0.289762
$127$	−120825.	−0.664733	−0.332367	−	0.943150i	$-0.607847\pi$
$127$	−120825.	−0.664733	−0.332367	+	0.943150i	$0.607847\pi$
$128$	4350.25	0.0234687
$129$	−142370.	−0.753260
$130$	0	0
$131$	266474.	1.35668	0.678338	−	0.734750i	$-0.262701\pi$
$131$	266474.	1.35668	0.678338	+	0.734750i	$0.262701\pi$
$132$	−14701.2	−0.0734377
$133$	187684.	0.920023
$134$	176115.	0.847292
$135$	0	0
$136$	−201918.	−0.936112
$137$	−17176.5	−0.0781866	−0.0390933	−	0.999236i	$-0.512447\pi$
$137$	−17176.5	−0.0781866	−0.0390933	+	0.999236i	$0.512447\pi$
$138$	14892.4	0.0665682
$139$	−5031.99	−0.0220903	−0.0110452	−	0.999939i	$-0.503516\pi$
$139$	−5031.99	−0.0220903	−0.0110452	+	0.999939i	$0.503516\pi$
$140$	0	0
$141$	−108299.	−0.458749
$142$	−72994.7	−0.303788
$143$	−28330.5	−0.115855
$144$	−33190.9	−0.133387
$145$	0	0
$146$	−243621.	−0.945874
$147$	46448.1	0.177286
$148$	69768.3	0.261820
$149$	−37407.9	−0.138038	−0.0690188	−	0.997615i	$-0.521987\pi$
$149$	−37407.9	−0.138038	−0.0690188	+	0.997615i	$0.521987\pi$
$150$	0	0
$151$	−225853.	−0.806091	−0.403046	−	0.915180i	$-0.632048\pi$
$151$	−225853.	−0.806091	−0.403046	+	0.915180i	$0.632048\pi$
$152$	247817.	0.870006
$153$	−83572.3	−0.288625
$154$	−77138.0	−0.262100
$155$	0	0
$156$	28447.1	0.0935894
$157$	205537.	0.665489	0.332744	−	0.943017i	$-0.392025\pi$
$157$	205537.	0.665489	0.332744	+	0.943017i	$0.392025\pi$
$158$	−251654.	−0.801974
$159$	34665.1	0.108742
$160$	0	0
$161$	−57020.1	−0.173366
$162$	−28220.1	−0.0844833
$163$	639398.	1.88496	0.942481	−	0.334261i	$-0.108487\pi$
$163$	639398.	1.88496	0.942481	+	0.334261i	$0.108487\pi$
$164$	−30498.3	−0.0885453
$165$	0	0
$166$	−224711.	−0.632927
$167$	−416990.	−1.15700	−0.578502	−	0.815681i	$-0.696363\pi$
$167$	−416990.	−1.15700	−0.578502	+	0.815681i	$0.696363\pi$
$168$	261057.	0.713611
$169$	−316473.	−0.852354
$170$	0	0
$171$	102570.	0.268243
$172$	213552.	0.550405
$173$	−608651.	−1.54615	−0.773077	−	0.634312i	$-0.781284\pi$
$173$	−608651.	−1.54615	−0.773077	+	0.634312i	$0.781284\pi$
$174$	173612.	0.434716
$175$	0	0
$176$	−49581.4	−0.120653
$177$	−182349.	−0.437493
$178$	−237058.	−0.560795
$179$	511566.	1.19335	0.596676	−	0.802482i	$-0.296488\pi$
$179$	511566.	1.19335	0.596676	+	0.802482i	$0.296488\pi$
$180$	0	0
$181$	56369.0	0.127892	0.0639460	−	0.997953i	$-0.479631\pi$
$181$	56369.0	0.127892	0.0639460	+	0.997953i	$0.479631\pi$
$182$	149263.	0.334021
$183$	18061.9	0.0398691
$184$	−75289.0	−0.163941
$185$	0	0
$186$	38603.7	0.0818175
$187$	−124843.	−0.261071
$188$	162445.	0.335206
$189$	108049.	0.220023
$190$	0	0
$191$	45986.9	0.0912117	0.0456059	−	0.998960i	$-0.485478\pi$
$191$	45986.9	0.0912117	0.0456059	+	0.998960i	$0.485478\pi$
$192$	292211.	0.572063
$193$	−360889.	−0.697398	−0.348699	−	0.937235i	$-0.613376\pi$
$193$	−360889.	−0.697398	−0.348699	+	0.937235i	$0.613376\pi$
$194$	427469.	0.815455
$195$	0	0
$196$	−69671.0	−0.129542
$197$	−978922.	−1.79714	−0.898571	−	0.438828i	$-0.855394\pi$
$197$	−978922.	−1.79714	−0.898571	+	0.438828i	$0.855394\pi$
$198$	−42156.0	−0.0764181
$199$	902328.	1.61522	0.807610	−	0.589717i	$-0.200761\pi$
$199$	902328.	1.61522	0.807610	+	0.589717i	$0.200761\pi$
$200$	0	0
$201$	−368510.	−0.643367
$202$	108968.	0.187897
$203$	−664725.	−1.13214
$204$	125356.	0.210897
$205$	0	0
$206$	−400942.	−0.658285
$207$	−31161.5	−0.0505467
$208$	95940.7	0.153760
$209$	153221.	0.242635
$210$	0	0
$211$	318392.	0.492330	0.246165	−	0.969228i	$-0.420830\pi$
$211$	318392.	0.492330	0.246165	+	0.969228i	$0.420830\pi$
$212$	−51996.7	−0.0794577
$213$	152737.	0.230673
$214$	885312.	1.32148
$215$	0	0
$216$	142668.	0.208061
$217$	−147806.	−0.213080
$218$	670558.	0.955642
$219$	509764.	0.718223
$220$	0	0
$221$	241572.	0.332710
$222$	200061.	0.272446
$223$	−661352.	−0.890575	−0.445287	−	0.895388i	$-0.646899\pi$
$223$	−661352.	−0.890575	−0.445287	+	0.895388i	$0.646899\pi$
$224$	−666975.	−0.888157
$225$	0	0
$226$	173357.	0.225773
$227$	−606704.	−0.781470	−0.390735	−	0.920503i	$-0.627779\pi$
$227$	−606704.	−0.781470	−0.390735	+	0.920503i	$0.627779\pi$
$228$	−153852.	−0.196004
$229$	−352377.	−0.444036	−0.222018	−	0.975043i	$-0.571264\pi$
$229$	−352377.	−0.444036	−0.222018	+	0.975043i	$0.571264\pi$
$230$	0	0
$231$	161407.	0.199018
$232$	−877699.	−1.07060
$233$	−1.49288e6	−1.80151	−0.900755	−	0.434328i	$-0.856986\pi$
$233$	−1.49288e6	−1.80151	−0.900755	+	0.434328i	$0.856986\pi$
$234$	81572.3	0.0973875
$235$	0	0
$236$	273519.	0.319675
$237$	526571.	0.608956
$238$	657749.	0.752693
$239$	388141.	0.439536	0.219768	−	0.975552i	$-0.429470\pi$
$239$	388141.	0.439536	0.219768	+	0.975552i	$0.429470\pi$
$240$	0	0
$241$	−75337.2	−0.0835540	−0.0417770	−	0.999127i	$-0.513302\pi$
$241$	−75337.2	−0.0835540	−0.0417770	+	0.999127i	$0.513302\pi$
$242$	−62973.7	−0.0691227
$243$	59049.0	0.0641500
$244$	−27092.4	−0.0291322
$245$	0	0
$246$	−87454.1	−0.0921388
$247$	−296485.	−0.309215
$248$	−195162.	−0.201496
$249$	470194.	0.480595
$250$	0	0
$251$	−346975.	−0.347627	−0.173814	−	0.984779i	$-0.555609\pi$
$251$	−346975.	−0.347627	−0.173814	+	0.984779i	$0.555609\pi$
$252$	−162071.	−0.160770
$253$	−46549.9	−0.0457212
$254$	519691.	0.505430
$255$	0	0
$256$	−1.05768e6	−1.00869
$257$	−1.97267e6	−1.86304	−0.931518	−	0.363696i	$-0.881515\pi$
$257$	−1.97267e6	−1.86304	−0.931518	+	0.363696i	$0.881515\pi$
$258$	612362.	0.572742
$259$	−765995.	−0.709539
$260$	0	0
$261$	−363273.	−0.330089
$262$	−1.14615e6	−1.03155
$263$	190986.	0.170260	0.0851300	−	0.996370i	$-0.472869\pi$
$263$	190986.	0.170260	0.0851300	+	0.996370i	$0.472869\pi$
$264$	213121.	0.188198
$265$	0	0
$266$	−807266.	−0.699540
$267$	496030.	0.425823
$268$	552755.	0.470106
$269$	933912.	0.786911	0.393455	−	0.919344i	$-0.371280\pi$
$269$	933912.	0.786911	0.393455	+	0.919344i	$0.371280\pi$
$270$	0	0
$271$	−1.17171e6	−0.969163	−0.484582	−	0.874746i	$-0.661028\pi$
$271$	−1.17171e6	−0.969163	−0.484582	+	0.874746i	$0.661028\pi$
$272$	422776.	0.346488
$273$	−312325.	−0.253629
$274$	73879.3	0.0594492
$275$	0	0
$276$	46741.5	0.0369343
$277$	472512.	0.370010	0.185005	−	0.982738i	$-0.440770\pi$
$277$	472512.	0.370010	0.185005	+	0.982738i	$0.440770\pi$
$278$	21643.5	0.0167964
$279$	−80776.0	−0.0621258
$280$	0	0
$281$	−2.05374e6	−1.55160	−0.775798	−	0.630981i	$-0.782652\pi$
$281$	−2.05374e6	−1.55160	−0.775798	+	0.630981i	$0.782652\pi$
$282$	465813.	0.348810
$283$	−465578.	−0.345562	−0.172781	−	0.984960i	$-0.555275\pi$
$283$	−465578.	−0.345562	−0.172781	+	0.984960i	$0.555275\pi$
$284$	−229102.	−0.168552
$285$	0	0
$286$	121855.	0.0880903
$287$	334845.	0.239960
$288$	−364502.	−0.258952
$289$	−355335.	−0.250261
$290$	0	0
$291$	−894454.	−0.619193
$292$	−764633.	−0.524803
$293$	1.19434e6	0.812751	0.406376	−	0.913706i	$-0.366792\pi$
$293$	1.19434e6	0.812751	0.406376	+	0.913706i	$0.366792\pi$
$294$	−199782.	−0.134800
$295$	0	0
$296$	−1.01141e6	−0.670965
$297$	88209.0	0.0580259
$298$	160898.	0.104957
$299$	90074.7	0.0582673
$300$	0	0
$301$	−2.34461e6	−1.49161
$302$	971438.	0.612912
$303$	−228009.	−0.142674
$304$	−518880.	−0.322020
$305$	0	0
$306$	359460.	0.219456
$307$	−1.72251e6	−1.04308	−0.521538	−	0.853228i	$-0.674641\pi$
$307$	−1.72251e6	−1.04308	−0.521538	+	0.853228i	$0.674641\pi$
$308$	−242106.	−0.145422
$309$	838950.	0.499850
$310$	0	0
$311$	−2.60616e6	−1.52792	−0.763960	−	0.645263i	$-0.776748\pi$
$311$	−2.60616e6	−1.52792	−0.763960	+	0.645263i	$0.776748\pi$
$312$	−412391.	−0.239841
$313$	−1.22311e6	−0.705677	−0.352839	−	0.935684i	$-0.614784\pi$
$313$	−1.22311e6	−0.705677	−0.352839	+	0.935684i	$0.614784\pi$
$314$	−884053.	−0.506005
$315$	0	0
$316$	−789843.	−0.444962
$317$	700043.	0.391270	0.195635	−	0.980677i	$-0.437323\pi$
$317$	700043.	0.391270	0.195635	+	0.980677i	$0.437323\pi$
$318$	−149101.	−0.0826824
$319$	−542667.	−0.298577
$320$	0	0
$321$	−1.85247e6	−1.00343
$322$	245254.	0.131819
$323$	−1.30650e6	−0.696794
$324$	−88572.0	−0.0468742
$325$	0	0
$326$	−2.75017e6	−1.43323
$327$	−1.40311e6	−0.725640
$328$	442127.	0.226914
$329$	−1.78351e6	−0.908417
$330$	0	0
$331$	2.78738e6	1.39839	0.699193	−	0.714933i	$-0.253543\pi$
$331$	2.78738e6	1.39839	0.699193	+	0.714933i	$0.253543\pi$
$332$	−705279.	−0.351169
$333$	−418617.	−0.206874
$334$	1.79355e6	0.879728
$335$	0	0
$336$	−546601.	−0.264133
$337$	795333.	0.381482	0.190741	−	0.981640i	$-0.438911\pi$
$337$	795333.	0.381482	0.190741	+	0.981640i	$0.438911\pi$
$338$	1.36121e6	0.648087
$339$	−362741.	−0.171434
$340$	0	0
$341$	−120665.	−0.0561949
$342$	−441171.	−0.203959
$343$	−1.72613e6	−0.792208
$344$	−3.09581e6	−1.41052
$345$	0	0
$346$	2.61792e6	1.17562
$347$	3.80533e6	1.69656	0.848278	−	0.529551i	$-0.177639\pi$
$347$	3.80533e6	1.69656	0.848278	+	0.529551i	$0.177639\pi$
$348$	544900.	0.241195
$349$	−798008.	−0.350706	−0.175353	−	0.984506i	$-0.556107\pi$
$349$	−798008.	−0.350706	−0.175353	+	0.984506i	$0.556107\pi$
$350$	0	0
$351$	−170686.	−0.0739484
$352$	−544503.	−0.234231
$353$	−2.48978e6	−1.06347	−0.531733	−	0.846912i	$-0.678459\pi$
$353$	−2.48978e6	−1.06347	−0.531733	+	0.846912i	$0.678459\pi$
$354$	784319.	0.332648
$355$	0	0
$356$	−744032.	−0.311148
$357$	−1.37630e6	−0.571536
$358$	−2.20034e6	−0.907366
$359$	−2.15148e6	−0.881050	−0.440525	−	0.897740i	$-0.645208\pi$
$359$	−2.15148e6	−0.881050	−0.440525	+	0.897740i	$0.645208\pi$
$360$	0	0
$361$	−872605.	−0.352411
$362$	−242454.	−0.0972427
$363$	131769.	0.0524864
$364$	468479.	0.185326
$365$	0	0
$366$	−77687.8	−0.0303145
$367$	−1.47515e6	−0.571705	−0.285853	−	0.958274i	$-0.592277\pi$
$367$	−1.47515e6	−0.571705	−0.285853	+	0.958274i	$0.592277\pi$
$368$	157640.	0.0606803
$369$	182993.	0.0699630
$370$	0	0
$371$	570879.	0.215332
$372$	121162.	0.0453951
$373$	4.71290e6	1.75394	0.876972	−	0.480541i	$-0.159560\pi$
$373$	4.71290e6	1.75394	0.876972	+	0.480541i	$0.159560\pi$
$374$	536972.	0.198505
$375$	0	0
$376$	−2.35493e6	−0.859031
$377$	1.05007e6	0.380508
$378$	−464740.	−0.167294
$379$	4.47900e6	1.60171	0.800853	−	0.598861i	$-0.204380\pi$
$379$	4.47900e6	1.60171	0.800853	+	0.598861i	$0.204380\pi$
$380$	0	0
$381$	−1.08742e6	−0.383784
$382$	−197798.	−0.0693528
$383$	57968.7	0.0201928	0.0100964	−	0.999949i	$-0.496786\pi$
$383$	57968.7	0.0201928	0.0100964	+	0.999949i	$0.496786\pi$
$384$	39152.3	0.0135497
$385$	0	0
$386$	1.55225e6	0.530267
$387$	−1.28133e6	−0.434895
$388$	1.34166e6	0.452442
$389$	−4.53325e6	−1.51892	−0.759461	−	0.650553i	$-0.774537\pi$
$389$	−4.53325e6	−1.51892	−0.759461	+	0.650553i	$0.774537\pi$
$390$	0	0
$391$	396927.	0.131301
$392$	1.01000e6	0.331977
$393$	2.39826e6	0.783277
$394$	4.21053e6	1.36646
$395$	0	0
$396$	−132311.	−0.0423993
$397$	−621573.	−0.197932	−0.0989660	−	0.995091i	$-0.531554\pi$
$397$	−621573.	−0.197932	−0.0989660	+	0.995091i	$0.531554\pi$
$398$	−3.88108e6	−1.22813
$399$	1.68916e6	0.531176
$400$	0	0
$401$	954326.	0.296371	0.148186	−	0.988960i	$-0.452657\pi$
$401$	954326.	0.296371	0.148186	+	0.988960i	$0.452657\pi$
$402$	1.58503e6	0.489185
$403$	233489.	0.0716150
$404$	342007.	0.104251
$405$	0	0
$406$	2.85911e6	0.860826
$407$	−625341.	−0.187125
$408$	−1.81726e6	−0.540465
$409$	3.19864e6	0.945491	0.472746	−	0.881199i	$-0.343263\pi$
$409$	3.19864e6	0.945491	0.472746	+	0.881199i	$0.343263\pi$
$410$	0	0
$411$	−154588.	−0.0451411
$412$	−1.25840e6	−0.365239
$413$	−3.00301e6	−0.866325
$414$	134032.	0.0384332
$415$	0	0
$416$	1.05362e6	0.298505
$417$	−45287.9	−0.0127539
$418$	−659034.	−0.184487
$419$	−6.60122e6	−1.83691	−0.918457	−	0.395520i	$-0.870564\pi$
$419$	−6.60122e6	−1.83691	−0.918457	+	0.395520i	$0.870564\pi$
$420$	0	0
$421$	−4.22510e6	−1.16180	−0.580901	−	0.813974i	$-0.697300\pi$
$421$	−4.22510e6	−1.16180	−0.580901	+	0.813974i	$0.697300\pi$
$422$	−1.36946e6	−0.374343
$423$	−974688.	−0.264859
$424$	753785.	0.203626
$425$	0	0
$426$	−656953.	−0.175392
$427$	297451.	0.0789489
$428$	2.77865e6	0.733204
$429$	−254975.	−0.0668889
$430$	0	0
$431$	4.72300e6	1.22469	0.612344	−	0.790592i	$-0.290227\pi$
$431$	4.72300e6	1.22469	0.612344	+	0.790592i	$0.290227\pi$
$432$	−298718.	−0.0770108
$433$	882129.	0.226106	0.113053	−	0.993589i	$-0.463937\pi$
$433$	882129.	0.226106	0.113053	+	0.993589i	$0.463937\pi$
$434$	635741.	0.162015
$435$	0	0
$436$	2.10462e6	0.530222
$437$	−487155.	−0.122029
$438$	−2.19259e6	−0.546101
$439$	−3.07514e6	−0.761560	−0.380780	−	0.924666i	$-0.624344\pi$
$439$	−3.07514e6	−0.761560	−0.380780	+	0.924666i	$0.624344\pi$
$440$	0	0
$441$	418033.	0.102356
$442$	−1.03905e6	−0.252976
$443$	−3.40381e6	−0.824055	−0.412028	−	0.911171i	$-0.635179\pi$
$443$	−3.40381e6	−0.824055	−0.412028	+	0.911171i	$0.635179\pi$
$444$	627914.	0.151162
$445$	0	0
$446$	2.84460e6	0.677149
$447$	−336671.	−0.0796961
$448$	4.81226e6	1.13280
$449$	−605757.	−0.141802	−0.0709010	−	0.997483i	$-0.522587\pi$
$449$	−605757.	−0.141802	−0.0709010	+	0.997483i	$0.522587\pi$
$450$	0	0
$451$	273360.	0.0632839
$452$	544102.	0.125266
$453$	−2.03268e6	−0.465397
$454$	2.60955e6	0.594191
$455$	0	0
$456$	2.23035e6	0.502298
$457$	5.46329e6	1.22367	0.611834	−	0.790986i	$-0.290432\pi$
$457$	5.46329e6	1.22367	0.611834	+	0.790986i	$0.290432\pi$
$458$	1.51564e6	0.337623
$459$	−752151.	−0.166638
$460$	0	0
$461$	−2.84223e6	−0.622883	−0.311441	−	0.950265i	$-0.600812\pi$
$461$	−2.84223e6	−0.622883	−0.311441	+	0.950265i	$0.600812\pi$
$462$	−694242.	−0.151323
$463$	8.51652e6	1.84633	0.923166	−	0.384401i	$-0.125592\pi$
$463$	8.51652e6	1.84633	0.923166	+	0.384401i	$0.125592\pi$
$464$	1.83773e6	0.396266
$465$	0	0
$466$	6.42118e6	1.36978
$467$	−7.00140e6	−1.48557	−0.742784	−	0.669531i	$-0.766495\pi$
$467$	−7.00140e6	−1.48557	−0.742784	+	0.669531i	$0.766495\pi$
$468$	256024.	0.0540339
$469$	−6.06877e6	−1.27400
$470$	0	0
$471$	1.84983e6	0.384220
$472$	−3.96515e6	−0.819228
$473$	−1.91409e6	−0.393377
$474$	−2.26488e6	−0.463020
$475$	0	0
$476$	2.06442e6	0.417619
$477$	311986.	0.0627825
$478$	−1.66947e6	−0.334201
$479$	1.06826e6	0.212734	0.106367	−	0.994327i	$-0.466078\pi$
$479$	1.06826e6	0.212734	0.106367	+	0.994327i	$0.466078\pi$
$480$	0	0
$481$	1.21004e6	0.238472
$482$	324040.	0.0635303
$483$	−513181.	−0.100093
$484$	−197650.	−0.0383516
$485$	0	0
$486$	−253981.	−0.0487765
$487$	−1.00308e6	−0.191652	−0.0958260	−	0.995398i	$-0.530549\pi$
$487$	−1.00308e6	−0.191652	−0.0958260	+	0.995398i	$0.530549\pi$
$488$	392753.	0.0746569
$489$	5.75458e6	1.08828
$490$	0	0
$491$	9.23719e6	1.72916	0.864582	−	0.502491i	$-0.167583\pi$
$491$	9.23719e6	1.72916	0.864582	+	0.502491i	$0.167583\pi$
$492$	−274484.	−0.0511217
$493$	4.62727e6	0.857448
$494$	1.27524e6	0.235112
$495$	0	0
$496$	408631.	0.0745807
$497$	2.51534e6	0.456779
$498$	−2.02240e6	−0.365420
$499$	4.31532e6	0.775822	0.387911	−	0.921697i	$-0.373197\pi$
$499$	4.31532e6	0.775822	0.387911	+	0.921697i	$0.373197\pi$
$500$	0	0
$501$	−3.75291e6	−0.667997
$502$	1.49240e6	0.264318
$503$	−3.28052e6	−0.578127	−0.289064	−	0.957310i	$-0.593344\pi$
$503$	−3.28052e6	−0.578127	−0.289064	+	0.957310i	$0.593344\pi$
$504$	2.34951e6	0.412003
$505$	0	0
$506$	200220.	0.0347641
$507$	−2.84826e6	−0.492107
$508$	1.63111e6	0.280430
$509$	−2.50882e6	−0.429215	−0.214608	−	0.976700i	$-0.568847\pi$
$509$	−2.50882e6	−0.429215	−0.214608	+	0.976700i	$0.568847\pi$
$510$	0	0
$511$	8.39501e6	1.42223
$512$	4.41009e6	0.743486
$513$	923126.	0.154870
$514$	8.48482e6	1.41656
$515$	0	0
$516$	1.92197e6	0.317776
$517$	−1.45602e6	−0.239574
$518$	3.29469e6	0.539498
$519$	−5.47786e6	−0.892673
$520$	0	0
$521$	−3.63347e6	−0.586444	−0.293222	−	0.956044i	$-0.594728\pi$
$521$	−3.63347e6	−0.586444	−0.293222	+	0.956044i	$0.594728\pi$
$522$	1.56250e6	0.250983
$523$	7.95990e6	1.27249	0.636243	−	0.771488i	$-0.280487\pi$
$523$	7.95990e6	1.27249	0.636243	+	0.771488i	$0.280487\pi$
$524$	−3.59733e6	−0.572338
$525$	0	0
$526$	−821468.	−0.129457
$527$	1.02890e6	0.161379
$528$	−446233.	−0.0696589
$529$	−6.28834e6	−0.977005
$530$	0	0
$531$	−1.64114e6	−0.252587
$532$	−2.53369e6	−0.388128
$533$	−528954.	−0.0806492
$534$	−2.13352e6	−0.323775
$535$	0	0
$536$	−8.01317e6	−1.20474
$537$	4.60409e6	0.688983
$538$	−4.01693e6	−0.598328
$539$	624469.	0.0925847
$540$	0	0
$541$	1.20412e6	0.176879	0.0884395	−	0.996082i	$-0.471812\pi$
$541$	1.20412e6	0.176879	0.0884395	+	0.996082i	$0.471812\pi$
$542$	5.03975e6	0.736903
$543$	507321.	0.0738385
$544$	4.64293e6	0.672660
$545$	0	0
$546$	1.34337e6	0.192847
$547$	1.04446e7	1.49254	0.746268	−	0.665646i	$-0.231844\pi$
$547$	1.04446e7	1.49254	0.746268	+	0.665646i	$0.231844\pi$
$548$	231878.	0.0329844
$549$	162557.	0.0230184
$550$	0	0
$551$	−5.67912e6	−0.796897
$552$	−677601.	−0.0946512
$553$	8.67179e6	1.20586
$554$	−2.03236e6	−0.281337
$555$	0	0
$556$	67930.6	0.00931920
$557$	−8.45756e6	−1.15507	−0.577534	−	0.816367i	$-0.695985\pi$
$557$	−8.45756e6	−1.15507	−0.577534	+	0.816367i	$0.695985\pi$
$558$	347433.	0.0472373
$559$	3.70379e6	0.501322
$560$	0	0
$561$	−1.12358e6	−0.150729
$562$	8.83350e6	1.17976
$563$	−9.96372e6	−1.32480	−0.662401	−	0.749150i	$-0.730462\pi$
$563$	−9.96372e6	−1.32480	−0.662401	+	0.749150i	$0.730462\pi$
$564$	1.46201e6	0.193532
$565$	0	0
$566$	2.00254e6	0.262748
$567$	972443.	0.127030
$568$	3.32124e6	0.431946
$569$	−1.05157e7	−1.36163	−0.680813	−	0.732458i	$-0.738373\pi$
$569$	−1.05157e7	−1.36163	−0.680813	+	0.732458i	$0.738373\pi$
$570$	0	0
$571$	9.02900e6	1.15891	0.579454	−	0.815005i	$-0.303266\pi$
$571$	9.02900e6	1.15891	0.579454	+	0.815005i	$0.303266\pi$
$572$	382456.	0.0488755
$573$	413882.	0.0526611
$574$	−1.44023e6	−0.182453
$575$	0	0
$576$	2.62990e6	0.330281
$577$	−6.65688e6	−0.832398	−0.416199	−	0.909273i	$-0.636638\pi$
$577$	−6.65688e6	−0.832398	−0.416199	+	0.909273i	$0.636638\pi$
$578$	1.52836e6	0.190286
$579$	−3.24800e6	−0.402643
$580$	0	0
$581$	7.74336e6	0.951676
$582$	3.84722e6	0.470803
$583$	466053.	0.0567889
$584$	1.10847e7	1.34491
$585$	0	0
$586$	−5.13707e6	−0.617975
$587$	−2.20946e6	−0.264661	−0.132331	−	0.991206i	$-0.542246\pi$
$587$	−2.20946e6	−0.264661	−0.132331	+	0.991206i	$0.542246\pi$
$588$	−627039.	−0.0747913
$589$	−1.26279e6	−0.149983
$590$	0	0
$591$	−8.81030e6	−1.03758
$592$	2.11770e6	0.248348
$593$	261524.	0.0305404	0.0152702	−	0.999883i	$-0.495139\pi$
$593$	261524.	0.0305404	0.0152702	+	0.999883i	$0.495139\pi$
$594$	−379404.	−0.0441200
$595$	0	0
$596$	504998.	0.0582336
$597$	8.12095e6	0.932548
$598$	−387429.	−0.0443036
$599$	7.09631e6	0.808101	0.404050	−	0.914737i	$-0.367602\pi$
$599$	7.09631e6	0.808101	0.404050	+	0.914737i	$0.367602\pi$
$600$	0	0
$601$	6.04793e6	0.683000	0.341500	−	0.939882i	$-0.389065\pi$
$601$	6.04793e6	0.683000	0.341500	+	0.939882i	$0.389065\pi$
$602$	1.00846e7	1.13414
$603$	−3.31659e6	−0.371448
$604$	3.04897e6	0.340064
$605$	0	0
$606$	980708.	0.108482
$607$	1.05203e7	1.15892	0.579461	−	0.815000i	$-0.303263\pi$
$607$	1.05203e7	1.15892	0.579461	+	0.815000i	$0.303263\pi$
$608$	−5.69835e6	−0.625158
$609$	−5.98253e6	−0.653644
$610$	0	0
$611$	2.81741e6	0.305314
$612$	1.12821e6	0.121762
$613$	8.55961e6	0.920032	0.460016	−	0.887911i	$-0.347844\pi$
$613$	8.55961e6	0.920032	0.460016	+	0.887911i	$0.347844\pi$
$614$	7.40884e6	0.793103
$615$	0	0
$616$	3.50976e6	0.372671
$617$	6.18055e6	0.653604	0.326802	−	0.945093i	$-0.394029\pi$
$617$	6.18055e6	0.653604	0.326802	+	0.945093i	$0.394029\pi$
$618$	−3.60848e6	−0.380061
$619$	−1.12540e7	−1.18053	−0.590267	−	0.807208i	$-0.700978\pi$
$619$	−1.12540e7	−1.18053	−0.590267	+	0.807208i	$0.700978\pi$
$620$	0	0
$621$	−280454.	−0.0291832
$622$	1.12096e7	1.16175
$623$	8.16882e6	0.843217
$624$	863466.	0.0887736
$625$	0	0
$626$	5.26085e6	0.536562
$627$	1.37899e6	0.140085
$628$	−2.77470e6	−0.280748
$629$	5.33223e6	0.537381
$630$	0	0
$631$	1.17369e7	1.17349	0.586746	−	0.809771i	$-0.300409\pi$
$631$	1.17369e7	1.17349	0.586746	+	0.809771i	$0.300409\pi$
$632$	1.14502e7	1.14030
$633$	2.86553e6	0.284247
$634$	−3.01102e6	−0.297502
$635$	0	0
$636$	−467970.	−0.0458749
$637$	−1.20836e6	−0.117990
$638$	2.33411e6	0.227023
$639$	1.37464e6	0.133179
$640$	0	0
$641$	6.80878e6	0.654522	0.327261	−	0.944934i	$-0.393874\pi$
$641$	6.80878e6	0.654522	0.327261	+	0.944934i	$0.393874\pi$
$642$	7.96781e6	0.762959
$643$	1.16456e7	1.11080	0.555400	−	0.831583i	$-0.312565\pi$
$643$	1.16456e7	1.11080	0.555400	+	0.831583i	$0.312565\pi$
$644$	769758.	0.0731374
$645$	0	0
$646$	5.61952e6	0.529808
$647$	−2.61990e6	−0.246050	−0.123025	−	0.992404i	$-0.539260\pi$
$647$	−2.61990e6	−0.246050	−0.123025	+	0.992404i	$0.539260\pi$
$648$	1.28401e6	0.120124
$649$	−2.45159e6	−0.228473
$650$	0	0
$651$	−1.33025e6	−0.123022
$652$	−8.63173e6	−0.795204
$653$	−1.48492e7	−1.36276	−0.681379	−	0.731931i	$-0.738619\pi$
$653$	−1.48492e7	−1.36276	−0.681379	+	0.731931i	$0.738619\pi$
$654$	6.03502e6	0.551740
$655$	0	0
$656$	−925726.	−0.0839891
$657$	4.58788e6	0.414666
$658$	7.67120e6	0.690715
$659$	−2.47470e6	−0.221977	−0.110989	−	0.993822i	$-0.535402\pi$
$659$	−2.47470e6	−0.221977	−0.110989	+	0.993822i	$0.535402\pi$
$660$	0	0
$661$	7.61212e6	0.677645	0.338822	−	0.940850i	$-0.389971\pi$
$661$	7.61212e6	0.677645	0.338822	+	0.940850i	$0.389971\pi$
$662$	−1.19891e7	−1.06326
$663$	2.17415e6	0.192090
$664$	1.02243e7	0.899938
$665$	0	0
$666$	1.80055e6	0.157297
$667$	1.72537e6	0.150164
$668$	5.62927e6	0.488103
$669$	−5.95217e6	−0.514174
$670$	0	0
$671$	242833.	0.0208210
$672$	−6.00278e6	−0.512777
$673$	−6.62472e6	−0.563807	−0.281903	−	0.959443i	$-0.590966\pi$
$673$	−6.62472e6	−0.563807	−0.281903	+	0.959443i	$0.590966\pi$
$674$	−3.42088e6	−0.290060
$675$	0	0
$676$	4.27231e6	0.359581
$677$	−228067.	−0.0191245	−0.00956227	−	0.999954i	$-0.503044\pi$
$677$	−228067.	−0.0191245	−0.00956227	+	0.999954i	$0.503044\pi$
$678$	1.56022e6	0.130350
$679$	−1.47302e7	−1.22613
$680$	0	0
$681$	−5.46034e6	−0.451182
$682$	519005.	0.0427278
$683$	1.58370e6	0.129903	0.0649517	−	0.997888i	$-0.479311\pi$
$683$	1.58370e6	0.129903	0.0649517	+	0.997888i	$0.479311\pi$
$684$	−1.38467e6	−0.113163
$685$	0	0
$686$	7.42443e6	0.602356
$687$	−3.17139e6	−0.256364
$688$	6.48202e6	0.522083
$689$	−901818.	−0.0723720
$690$	0	0
$691$	2.07411e7	1.65248	0.826241	−	0.563317i	$-0.190475\pi$
$691$	2.07411e7	1.65248	0.826241	+	0.563317i	$0.190475\pi$
$692$	8.21665e6	0.652273
$693$	1.45266e6	0.114903
$694$	−1.63674e7	−1.28998
$695$	0	0
$696$	−7.89929e6	−0.618109
$697$	−2.33091e6	−0.181737
$698$	3.43238e6	0.266660
$699$	−1.34360e7	−1.04010
$700$	0	0
$701$	−1.54594e7	−1.18822	−0.594109	−	0.804384i	$-0.702495\pi$
$701$	−1.54594e7	−1.18822	−0.594109	+	0.804384i	$0.702495\pi$
$702$	734151.	0.0562267
$703$	−6.54433e6	−0.499433
$704$	3.92862e6	0.298750
$705$	0	0
$706$	1.07090e7	0.808606
$707$	−3.75494e6	−0.282523
$708$	2.46167e6	0.184564
$709$	−4.71237e6	−0.352066	−0.176033	−	0.984384i	$-0.556327\pi$
$709$	−4.71237e6	−0.352066	−0.176033	+	0.984384i	$0.556327\pi$
$710$	0	0
$711$	4.73914e6	0.351581
$712$	1.07861e7	0.797376
$713$	383646.	0.0282623
$714$	5.91974e6	0.434567
$715$	0	0
$716$	−6.90602e6	−0.503437
$717$	3.49327e6	0.253766
$718$	9.25391e6	0.669906
$719$	2.29858e6	0.165820	0.0829099	−	0.996557i	$-0.473579\pi$
$719$	2.29858e6	0.165820	0.0829099	+	0.996557i	$0.473579\pi$
$720$	0	0
$721$	1.38162e7	0.989805
$722$	3.75324e6	0.267956
$723$	−678035.	−0.0482399
$724$	−760968.	−0.0539535
$725$	0	0
$726$	−566763.	−0.0399080
$727$	−2.81705e7	−1.97678	−0.988391	−	0.151933i	$-0.951450\pi$
$727$	−2.81705e7	−1.97678	−0.988391	+	0.151933i	$0.951450\pi$
$728$	−6.79143e6	−0.474934
$729$	531441.	0.0370370
$730$	0	0
$731$	1.63213e7	1.12969
$732$	−243832.	−0.0168195
$733$	1.32094e7	0.908078	0.454039	−	0.890982i	$-0.349983\pi$
$733$	1.32094e7	0.908078	0.454039	+	0.890982i	$0.349983\pi$
$734$	6.34492e6	0.434696
$735$	0	0
$736$	1.73121e6	0.117802
$737$	−4.95441e6	−0.335988
$738$	−787087.	−0.0531963
$739$	−6.63649e6	−0.447021	−0.223510	−	0.974702i	$-0.571752\pi$
$739$	−6.63649e6	−0.447021	−0.223510	+	0.974702i	$0.571752\pi$
$740$	0	0
$741$	−2.66837e6	−0.178525
$742$	−2.45546e6	−0.163728
$743$	7.93013e6	0.526997	0.263499	−	0.964660i	$-0.415124\pi$
$743$	7.93013e6	0.526997	0.263499	+	0.964660i	$0.415124\pi$
$744$	−1.75646e6	−0.116334
$745$	0	0
$746$	−2.02711e7	−1.33361
$747$	4.23175e6	0.277472
$748$	1.68535e6	0.110137
$749$	−3.05072e7	−1.98700
$750$	0	0
$751$	−1.43446e7	−0.928086	−0.464043	−	0.885813i	$-0.653602\pi$
$751$	−1.43446e7	−0.928086	−0.464043	+	0.885813i	$0.653602\pi$
$752$	4.93076e6	0.317958
$753$	−3.12277e6	−0.200703
$754$	−4.51654e6	−0.289319
$755$	0	0
$756$	−1.45864e6	−0.0928204
$757$	−8.38496e6	−0.531816	−0.265908	−	0.963998i	$-0.585672\pi$
$757$	−8.38496e6	−0.531816	−0.265908	+	0.963998i	$0.585672\pi$
$758$	−1.92650e7	−1.21786
$759$	−418949.	−0.0263972
$760$	0	0
$761$	2.68159e7	1.67853	0.839267	−	0.543719i	$-0.182984\pi$
$761$	2.68159e7	1.67853	0.839267	+	0.543719i	$0.182984\pi$
$762$	4.67722e6	0.291810
$763$	−2.31069e7	−1.43691
$764$	−620812.	−0.0384793
$765$	0	0
$766$	−249334.	−0.0153536
$767$	4.74385e6	0.291167
$768$	−9.51916e6	−0.582365
$769$	−2.82141e7	−1.72049	−0.860243	−	0.509884i	$-0.829688\pi$
$769$	−2.82141e7	−1.72049	−0.860243	+	0.509884i	$0.829688\pi$
$770$	0	0
$771$	−1.77540e7	−1.07562
$772$	4.87192e6	0.294210
$773$	−2.41553e7	−1.45400	−0.727000	−	0.686637i	$-0.759086\pi$
$773$	−2.41553e7	−1.45400	−0.727000	+	0.686637i	$0.759086\pi$
$774$	5.51125e6	0.330673
$775$	0	0
$776$	−1.94497e7	−1.15947
$777$	−6.89395e6	−0.409653
$778$	1.94983e7	1.15491
$779$	2.86077e6	0.168904
$780$	0	0
$781$	2.05347e6	0.120465
$782$	−1.70726e6	−0.0998350
$783$	−3.26945e6	−0.190577
$784$	−2.11475e6	−0.122877
$785$	0	0
$786$	−1.03154e7	−0.595565
$787$	−1.96829e7	−1.13280	−0.566400	−	0.824131i	$-0.691664\pi$
$787$	−1.96829e7	−1.13280	−0.566400	+	0.824131i	$0.691664\pi$
$788$	1.32152e7	0.758156
$789$	1.71888e6	0.0982997
$790$	0	0
$791$	−5.97376e6	−0.339474
$792$	1.91809e6	0.108656
$793$	−469884.	−0.0265343
$794$	2.67351e6	0.150498
$795$	0	0
$796$	−1.21812e7	−0.681409
$797$	−2.41989e6	−0.134943	−0.0674714	−	0.997721i	$-0.521493\pi$
$797$	−2.41989e6	−0.134943	−0.0674714	+	0.997721i	$0.521493\pi$
$798$	−7.26539e6	−0.403880
$799$	1.24153e7	0.688004
$800$	0	0
$801$	4.46427e6	0.245849
$802$	−4.10474e6	−0.225346
$803$	6.85350e6	0.375080
$804$	4.97480e6	0.271416
$805$	0	0
$806$	−1.00428e6	−0.0544525
$807$	8.40521e6	0.454323
$808$	−4.95800e6	−0.267164
$809$	2.06594e7	1.10981	0.554903	−	0.831915i	$-0.312755\pi$
$809$	2.06594e7	1.10981	0.554903	+	0.831915i	$0.312755\pi$
$810$	0	0
$811$	1.38880e7	0.741461	0.370730	−	0.928741i	$-0.379107\pi$
$811$	1.38880e7	0.741461	0.370730	+	0.928741i	$0.379107\pi$
$812$	8.97363e6	0.477615
$813$	−1.05454e7	−0.559547
$814$	2.68971e6	0.142280
$815$	0	0
$816$	3.80499e6	0.200045
$817$	−2.00314e7	−1.04992
$818$	−1.37580e7	−0.718904
$819$	−2.81092e6	−0.146433
$820$	0	0
$821$	5.26859e6	0.272795	0.136398	−	0.990654i	$-0.456448\pi$
$821$	5.26859e6	0.272795	0.136398	+	0.990654i	$0.456448\pi$
$822$	664913.	0.0343230
$823$	−3.61766e7	−1.86178	−0.930890	−	0.365299i	$-0.880967\pi$
$823$	−3.61766e7	−1.86178	−0.930890	+	0.365299i	$0.880967\pi$
$824$	1.82428e7	0.935994
$825$	0	0
$826$	1.29165e7	0.658710
$827$	−3.01917e7	−1.53505	−0.767527	−	0.641017i	$-0.778513\pi$
$827$	−3.01917e7	−1.53505	−0.767527	+	0.641017i	$0.778513\pi$
$828$	420673.	0.0213240
$829$	2.93884e7	1.48522	0.742608	−	0.669727i	$-0.233589\pi$
$829$	2.93884e7	1.48522	0.742608	+	0.669727i	$0.233589\pi$
$830$	0	0
$831$	4.25261e6	0.213625
$832$	−7.60193e6	−0.380728
$833$	−5.32479e6	−0.265883
$834$	194792.	0.00969740
$835$	0	0
$836$	−2.06845e6	−0.102360
$837$	−726984.	−0.0358683
$838$	2.83931e7	1.39670
$839$	1.57845e7	0.774150	0.387075	−	0.922048i	$-0.373485\pi$
$839$	1.57845e7	0.774150	0.387075	+	0.922048i	$0.373485\pi$
$840$	0	0
$841$	−397286.	−0.0193692
$842$	1.81730e7	0.883376
$843$	−1.84836e7	−0.895814
$844$	−4.29822e6	−0.207698
$845$	0	0
$846$	4.19232e6	0.201386
$847$	2.17003e6	0.103934
$848$	−1.57828e6	−0.0753691
$849$	−4.19020e6	−0.199510
$850$	0	0
$851$	1.98822e6	0.0941112
$852$	−2.06192e6	−0.0973134
$853$	−2.68137e7	−1.26178	−0.630891	−	0.775871i	$-0.717311\pi$
$853$	−2.68137e7	−1.26178	−0.630891	+	0.775871i	$0.717311\pi$
$854$	−1.27939e6	−0.0600288
$855$	0	0
$856$	−4.02815e7	−1.87898
$857$	2.59115e7	1.20515	0.602574	−	0.798063i	$-0.294142\pi$
$857$	2.59115e7	1.20515	0.602574	+	0.798063i	$0.294142\pi$
$858$	1.09669e6	0.0508590
$859$	1.20430e7	0.556870	0.278435	−	0.960455i	$-0.410184\pi$
$859$	1.20430e7	0.556870	0.278435	+	0.960455i	$0.410184\pi$
$860$	0	0
$861$	3.01360e6	0.138541
$862$	−2.03145e7	−0.931191
$863$	−2.28033e7	−1.04225	−0.521125	−	0.853481i	$-0.674487\pi$
$863$	−2.28033e7	−1.04225	−0.521125	+	0.853481i	$0.674487\pi$
$864$	−3.28052e6	−0.149506
$865$	0	0
$866$	−3.79420e6	−0.171920
$867$	−3.19801e6	−0.144488
$868$	1.99534e6	0.0898915
$869$	7.07945e6	0.318017
$870$	0	0
$871$	9.58685e6	0.428184
$872$	−3.05102e7	−1.35880
$873$	−8.05009e6	−0.357491
$874$	2.09535e6	0.0927849
$875$	0	0
$876$	−6.88170e6	−0.302995
$877$	3.10668e7	1.36395	0.681974	−	0.731376i	$-0.261122\pi$
$877$	3.10668e7	1.36395	0.681974	+	0.731376i	$0.261122\pi$
$878$	1.32268e7	0.579052
$879$	1.07490e7	0.469242
$880$	0	0
$881$	−3.45208e7	−1.49844	−0.749222	−	0.662319i	$-0.769573\pi$
$881$	−3.45208e7	−1.49844	−0.749222	+	0.662319i	$0.769573\pi$
$882$	−1.79804e6	−0.0778265
$883$	3.29428e7	1.42186	0.710932	−	0.703260i	$-0.248273\pi$
$883$	3.29428e7	1.42186	0.710932	+	0.703260i	$0.248273\pi$
$884$	−3.26117e6	−0.140360
$885$	0	0
$886$	1.46404e7	0.626571
$887$	2.11009e7	0.900519	0.450260	−	0.892898i	$-0.351331\pi$
$887$	2.11009e7	0.900519	0.450260	+	0.892898i	$0.351331\pi$
$888$	−9.10273e6	−0.387382
$889$	−1.79082e7	−0.759970
$890$	0	0
$891$	793881.	0.0335013
$892$	8.92810e6	0.375705
$893$	−1.52375e7	−0.639419
$894$	1.44809e6	0.0605969
$895$	0	0
$896$	644775.	0.0268311
$897$	810673.	0.0336406
$898$	2.60547e6	0.107819
$899$	4.47245e6	0.184564
$900$	0	0
$901$	−3.97399e6	−0.163085
$902$	−1.17577e6	−0.0481179
$903$	−2.11015e7	−0.861181
$904$	−7.88772e6	−0.321019
$905$	0	0
$906$	8.74294e6	0.353865
$907$	−1.30426e7	−0.526438	−0.263219	−	0.964736i	$-0.584784\pi$
$907$	−1.30426e7	−0.526438	−0.263219	+	0.964736i	$0.584784\pi$
$908$	8.19036e6	0.329677
$909$	−2.05208e6	−0.0823728
$910$	0	0
$911$	−4.09652e7	−1.63538	−0.817690	−	0.575658i	$-0.804746\pi$
$911$	−4.09652e7	−1.63538	−0.817690	+	0.575658i	$0.804746\pi$
$912$	−4.66992e6	−0.185918
$913$	6.32150e6	0.250982
$914$	−2.34987e7	−0.930417
$915$	0	0
$916$	4.75700e6	0.187325
$917$	3.94956e7	1.55105
$918$	3.23514e6	0.126703
$919$	5.59367e6	0.218478	0.109239	−	0.994016i	$-0.465159\pi$
$919$	5.59367e6	0.218478	0.109239	+	0.994016i	$0.465159\pi$
$920$	0	0
$921$	−1.55026e7	−0.602220
$922$	1.22250e7	0.473609
$923$	−3.97349e6	−0.153521
$924$	−2.17896e6	−0.0839592
$925$	0	0
$926$	−3.66312e7	−1.40386
$927$	7.55055e6	0.288589
$928$	2.01819e7	0.769295
$929$	−4.15645e7	−1.58010	−0.790048	−	0.613045i	$-0.789944\pi$
$929$	−4.15645e7	−1.58010	−0.790048	+	0.613045i	$0.789944\pi$
$930$	0	0
$931$	6.53520e6	0.247107
$932$	2.01536e7	0.759999
$933$	−2.34555e7	−0.882145
$934$	3.01144e7	1.12955
$935$	0	0
$936$	−3.71152e6	−0.138472
$937$	1.62578e7	0.604941	0.302470	−	0.953159i	$-0.402189\pi$
$937$	1.62578e7	0.604941	0.302470	+	0.953159i	$0.402189\pi$
$938$	2.61030e7	0.968685
$939$	−1.10080e7	−0.407423
$940$	0	0
$941$	1.33538e7	0.491622	0.245811	−	0.969318i	$-0.420946\pi$
$941$	1.33538e7	0.491622	0.245811	+	0.969318i	$0.420946\pi$
$942$	−7.95648e6	−0.292142
$943$	−869126.	−0.0318276
$944$	8.30224e6	0.303225
$945$	0	0
$946$	8.23286e6	0.299105
$947$	−1.36642e7	−0.495117	−0.247559	−	0.968873i	$-0.579628\pi$
$947$	−1.36642e7	−0.495117	−0.247559	+	0.968873i	$0.579628\pi$
$948$	−7.10859e6	−0.256899
$949$	−1.32616e7	−0.478003
$950$	0	0
$951$	6.30038e6	0.225900
$952$	−2.99274e7	−1.07023
$953$	2.92858e7	1.04454	0.522270	−	0.852780i	$-0.325085\pi$
$953$	2.92858e7	1.04454	0.522270	+	0.852780i	$0.325085\pi$
$954$	−1.34191e6	−0.0477367
$955$	0	0
$956$	−5.23981e6	−0.185426
$957$	−4.88400e6	−0.172384
$958$	−4.59478e6	−0.161752
$959$	−2.54582e6	−0.0893885
$960$	0	0
$961$	−2.76347e7	−0.965263
$962$	−5.20462e6	−0.181323
$963$	−1.66722e7	−0.579332
$964$	1.01704e6	0.0352487
$965$	0	0
$966$	2.20729e6	0.0761055
$967$	4.41255e7	1.51748	0.758741	−	0.651393i	$-0.225815\pi$
$967$	4.41255e7	1.51748	0.758741	+	0.651393i	$0.225815\pi$
$968$	2.86529e6	0.0982834
$969$	−1.17585e7	−0.402294
$970$	0	0
$971$	2.28334e7	0.777183	0.388592	−	0.921410i	$-0.372962\pi$
$971$	2.28334e7	0.777183	0.388592	+	0.921410i	$0.372962\pi$
$972$	−797148.	−0.0270628
$973$	−745819.	−0.0252552
$974$	4.31444e6	0.145723
$975$	0	0
$976$	−822347.	−0.0276332
$977$	5.35594e7	1.79515	0.897573	−	0.440866i	$-0.145329\pi$
$977$	5.35594e7	1.79515	0.897573	+	0.440866i	$0.145329\pi$
$978$	−2.47516e7	−0.827476
$979$	6.66884e6	0.222379
$980$	0	0
$981$	−1.26279e7	−0.418948
$982$	−3.97309e7	−1.31477
$983$	2.25516e7	0.744378	0.372189	−	0.928157i	$-0.378607\pi$
$983$	2.25516e7	0.744378	0.372189	+	0.928157i	$0.378607\pi$
$984$	3.97914e6	0.131009
$985$	0	0
$986$	−1.99028e7	−0.651961
$987$	−1.60516e7	−0.524475
$988$	4.00248e6	0.130448
$989$	6.08570e6	0.197843
$990$	0	0
$991$	7.10889e6	0.229942	0.114971	−	0.993369i	$-0.463323\pi$
$991$	7.10889e6	0.229942	0.114971	+	0.993369i	$0.463323\pi$
$992$	4.48758e6	0.144788
$993$	2.50865e7	0.807358
$994$	−1.08190e7	−0.347312
$995$	0	0
$996$	−6.34752e6	−0.202747
$997$	−3.59868e7	−1.14658	−0.573291	−	0.819352i	$-0.694334\pi$
$997$	−3.59868e7	−1.14658	−0.573291	+	0.819352i	$0.694334\pi$
$998$	−1.85610e7	−0.589896
$999$	−3.76755e6	−0.119439

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	825.6.a.f.1.2		3
5.4	even	2		165.6.a.e.1.2	✓	3
15.14	odd	2		495.6.a.a.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
165.6.a.e.1.2	✓	3	5.4	even	2
495.6.a.a.1.2		3	15.14	odd	2
825.6.a.f.1.2		3	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)

\(f(q)\)	\(=\)	\(q-4.30119 q^{2} +9.00000 q^{3} -13.4998 q^{4} -38.7107 q^{6} +148.216 q^{7} +195.703 q^{8} +81.0000 q^{9} +O(q^{10})\)	\(q-4.30119 q^{2} +9.00000 q^{3} -13.4998 q^{4} -38.7107 q^{6} +148.216 q^{7} +195.703 q^{8} +81.0000 q^{9} +121.000 q^{11} -121.498 q^{12} -234.137 q^{13} -637.504 q^{14} -409.764 q^{16} -1031.76 q^{17} -348.396 q^{18} +1266.29 q^{19} +1333.94 q^{21} -520.444 q^{22} -384.710 q^{23} +1761.33 q^{24} +1007.07 q^{26} +729.000 q^{27} -2000.88 q^{28} -4484.85 q^{29} -997.235 q^{31} -4500.03 q^{32} +1089.00 q^{33} +4437.78 q^{34} -1093.48 q^{36} -5168.11 q^{37} -5446.56 q^{38} -2107.23 q^{39} +2259.17 q^{41} -5737.54 q^{42} -15818.9 q^{43} -1633.47 q^{44} +1654.71 q^{46} -12033.2 q^{47} -3687.87 q^{48} +5160.90 q^{49} -9285.81 q^{51} +3160.79 q^{52} +3851.67 q^{53} -3135.57 q^{54} +29006.3 q^{56} +11396.6 q^{57} +19290.2 q^{58} -20261.0 q^{59} +2006.88 q^{61} +4289.30 q^{62} +12005.5 q^{63} +32467.9 q^{64} -4684.00 q^{66} -40945.6 q^{67} +13928.5 q^{68} -3462.39 q^{69} +16970.8 q^{71} +15852.0 q^{72} +56640.5 q^{73} +22229.0 q^{74} -17094.6 q^{76} +17934.1 q^{77} +9063.59 q^{78} +58507.9 q^{79} +6561.00 q^{81} -9717.12 q^{82} +52243.8 q^{83} -18007.9 q^{84} +68040.2 q^{86} -40363.6 q^{87} +23680.1 q^{88} +55114.4 q^{89} -34702.7 q^{91} +5193.50 q^{92} -8975.11 q^{93} +51757.0 q^{94} -40500.3 q^{96} -99383.8 q^{97} -22198.0 q^{98} +9801.00 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 7 q^{2} + 27 q^{3} + 73 q^{4} - 63 q^{6} - 92 q^{7} - 231 q^{8} + 243 q^{9}+O(q^{10}) \) 3 * q - 7 * q^2 + 27 * q^3 + 73 * q^4 - 63 * q^6 - 92 * q^7 - 231 * q^8 + 243 * q^9	\( 3 q - 7 q^{2} + 27 q^{3} + 73 q^{4} - 63 q^{6} - 92 q^{7} - 231 q^{8} + 243 q^{9} + 363 q^{11} + 657 q^{12} + 90 q^{13} - 784 q^{14} - 415 q^{16} - 1934 q^{17} - 567 q^{18} + 2084 q^{19} - 828 q^{21} - 847 q^{22} - 1220 q^{23} - 2079 q^{24} + 17062 q^{26} + 2187 q^{27} - 11120 q^{28} + 4402 q^{29} - 10688 q^{31} - 12439 q^{32} + 3267 q^{33} - 4094 q^{34} + 5913 q^{36} + 8190 q^{37} - 13792 q^{38} + 810 q^{39} + 5974 q^{41} - 7056 q^{42} - 18868 q^{43} + 8833 q^{44} + 46220 q^{46} - 55500 q^{47} - 3735 q^{48} + 1907 q^{49} - 17406 q^{51} - 27330 q^{52} - 9206 q^{53} - 5103 q^{54} + 73248 q^{56} + 18756 q^{57} - 15366 q^{58} - 59196 q^{59} + 79902 q^{61} - 64616 q^{62} - 7452 q^{63} + 2129 q^{64} - 7623 q^{66} - 4468 q^{67} + 1218 q^{68} - 10980 q^{69} - 75164 q^{71} - 18711 q^{72} + 61290 q^{73} - 56766 q^{74} + 37816 q^{76} - 11132 q^{77} + 153558 q^{78} - 83564 q^{79} + 19683 q^{81} - 147410 q^{82} - 74764 q^{83} - 100080 q^{84} - 253432 q^{86} + 39618 q^{87} - 27951 q^{88} + 37342 q^{89} - 126488 q^{91} - 148164 q^{92} - 96192 q^{93} + 59252 q^{94} - 111951 q^{96} - 33486 q^{97} + 95249 q^{98} + 29403 q^{99}+O(q^{100}) \) 3 * q - 7 * q^2 + 27 * q^3 + 73 * q^4 - 63 * q^6 - 92 * q^7 - 231 * q^8 + 243 * q^9 + 363 * q^11 + 657 * q^12 + 90 * q^13 - 784 * q^14 - 415 * q^16 - 1934 * q^17 - 567 * q^18 + 2084 * q^19 - 828 * q^21 - 847 * q^22 - 1220 * q^23 - 2079 * q^24 + 17062 * q^26 + 2187 * q^27 - 11120 * q^28 + 4402 * q^29 - 10688 * q^31 - 12439 * q^32 + 3267 * q^33 - 4094 * q^34 + 5913 * q^36 + 8190 * q^37 - 13792 * q^38 + 810 * q^39 + 5974 * q^41 - 7056 * q^42 - 18868 * q^43 + 8833 * q^44 + 46220 * q^46 - 55500 * q^47 - 3735 * q^48 + 1907 * q^49 - 17406 * q^51 - 27330 * q^52 - 9206 * q^53 - 5103 * q^54 + 73248 * q^56 + 18756 * q^57 - 15366 * q^58 - 59196 * q^59 + 79902 * q^61 - 64616 * q^62 - 7452 * q^63 + 2129 * q^64 - 7623 * q^66 - 4468 * q^67 + 1218 * q^68 - 10980 * q^69 - 75164 * q^71 - 18711 * q^72 + 61290 * q^73 - 56766 * q^74 + 37816 * q^76 - 11132 * q^77 + 153558 * q^78 - 83564 * q^79 + 19683 * q^81 - 147410 * q^82 - 74764 * q^83 - 100080 * q^84 - 253432 * q^86 + 39618 * q^87 - 27951 * q^88 + 37342 * q^89 - 126488 * q^91 - 148164 * q^92 - 96192 * q^93 + 59252 * q^94 - 111951 * q^96 - 33486 * q^97 + 95249 * q^98 + 29403 * q^99

Introduction

L-functions

Modular forms

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Database

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