LMFDB - Embedded newform 825.6.a.h.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("825.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

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

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$132.316651346$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.788.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 7x - 3 $ x^3 - x^2 - 7*x - 3
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 165)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.87740$ 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)

$f(q)$	$=$	$q-6.55890 q^{2} +9.00000 q^{3} +11.0192 q^{4} -59.0301 q^{6} -146.487 q^{7} +137.611 q^{8} +81.0000 q^{9} +O(q^{10})$	\(q-6.55890 q^{2} +9.00000 q^{3} +11.0192 q^{4} -59.0301 q^{6} -146.487 q^{7} +137.611 q^{8} +81.0000 q^{9} -121.000 q^{11} +99.1731 q^{12} -170.238 q^{13} +960.792 q^{14} -1255.19 q^{16} -1564.81 q^{17} -531.271 q^{18} +569.786 q^{19} -1318.38 q^{21} +793.627 q^{22} -3153.25 q^{23} +1238.50 q^{24} +1116.57 q^{26} +729.000 q^{27} -1614.17 q^{28} +3982.58 q^{29} +2990.78 q^{31} +3829.14 q^{32} -1089.00 q^{33} +10263.5 q^{34} +892.558 q^{36} -7858.92 q^{37} -3737.17 q^{38} -1532.14 q^{39} -5206.60 q^{41} +8647.13 q^{42} -13874.3 q^{43} -1333.33 q^{44} +20681.9 q^{46} -6852.01 q^{47} -11296.7 q^{48} +4651.35 q^{49} -14083.3 q^{51} -1875.89 q^{52} +3834.15 q^{53} -4781.44 q^{54} -20158.2 q^{56} +5128.08 q^{57} -26121.4 q^{58} +9649.38 q^{59} -21131.8 q^{61} -19616.2 q^{62} -11865.4 q^{63} +15051.2 q^{64} +7142.65 q^{66} +43499.9 q^{67} -17243.0 q^{68} -28379.2 q^{69} -52607.2 q^{71} +11146.5 q^{72} +64367.9 q^{73} +51545.9 q^{74} +6278.61 q^{76} +17724.9 q^{77} +10049.2 q^{78} +28935.7 q^{79} +6561.00 q^{81} +34149.6 q^{82} +4648.64 q^{83} -14527.5 q^{84} +91000.1 q^{86} +35843.2 q^{87} -16650.9 q^{88} -103458. q^{89} +24937.6 q^{91} -34746.4 q^{92} +26917.0 q^{93} +44941.6 q^{94} +34462.2 q^{96} -75809.5 q^{97} -30507.8 q^{98} -9801.00 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 2 q^{2} + 27 q^{3} - 20 q^{4} - 18 q^{6} - 152 q^{7} + 24 q^{8} + 243 q^{9}+O(q^{10}) $ 3 * q - 2 * q^2 + 27 * q^3 - 20 * q^4 - 18 * q^6 - 152 * q^7 + 24 * q^8 + 243 * q^9	\( 3 q - 2 q^{2} + 27 q^{3} - 20 q^{4} - 18 q^{6} - 152 q^{7} + 24 q^{8} + 243 q^{9} - 363 q^{11} - 180 q^{12} + 546 q^{13} - 8 q^{14} - 1360 q^{16} + 314 q^{17} - 162 q^{18} + 1808 q^{19} - 1368 q^{21} + 242 q^{22} - 4288 q^{23} + 216 q^{24} + 812 q^{26} + 2187 q^{27} - 5888 q^{28} + 5582 q^{29} + 6328 q^{31} + 736 q^{32} - 3267 q^{33} + 11596 q^{34} - 1620 q^{36} - 16866 q^{37} - 9584 q^{38} + 4914 q^{39} + 23282 q^{41} - 72 q^{42} - 20572 q^{43} + 2420 q^{44} + 16592 q^{46} - 3432 q^{47} - 12240 q^{48} + 11531 q^{49} + 2826 q^{51} - 21816 q^{52} - 16138 q^{53} - 1458 q^{54} + 15648 q^{56} + 16272 q^{57} - 17460 q^{58} + 21972 q^{59} + 8322 q^{61} - 5056 q^{62} - 12312 q^{63} + 22208 q^{64} + 2178 q^{66} + 84332 q^{67} - 59832 q^{68} - 38592 q^{69} + 50528 q^{71} + 1944 q^{72} + 53838 q^{73} + 79212 q^{74} - 52448 q^{76} + 18392 q^{77} + 7308 q^{78} + 6364 q^{79} + 19683 q^{81} + 68020 q^{82} - 96272 q^{83} - 52992 q^{84} + 143152 q^{86} + 50238 q^{87} - 2904 q^{88} - 38938 q^{89} + 104968 q^{91} - 24000 q^{92} + 56952 q^{93} + 49088 q^{94} + 6624 q^{96} + 103242 q^{97} - 9554 q^{98} - 29403 q^{99}+O(q^{100}) \) 3 * q - 2 * q^2 + 27 * q^3 - 20 * q^4 - 18 * q^6 - 152 * q^7 + 24 * q^8 + 243 * q^9 - 363 * q^11 - 180 * q^12 + 546 * q^13 - 8 * q^14 - 1360 * q^16 + 314 * q^17 - 162 * q^18 + 1808 * q^19 - 1368 * q^21 + 242 * q^22 - 4288 * q^23 + 216 * q^24 + 812 * q^26 + 2187 * q^27 - 5888 * q^28 + 5582 * q^29 + 6328 * q^31 + 736 * q^32 - 3267 * q^33 + 11596 * q^34 - 1620 * q^36 - 16866 * q^37 - 9584 * q^38 + 4914 * q^39 + 23282 * q^41 - 72 * q^42 - 20572 * q^43 + 2420 * q^44 + 16592 * q^46 - 3432 * q^47 - 12240 * q^48 + 11531 * q^49 + 2826 * q^51 - 21816 * q^52 - 16138 * q^53 - 1458 * q^54 + 15648 * q^56 + 16272 * q^57 - 17460 * q^58 + 21972 * q^59 + 8322 * q^61 - 5056 * q^62 - 12312 * q^63 + 22208 * q^64 + 2178 * q^66 + 84332 * q^67 - 59832 * q^68 - 38592 * q^69 + 50528 * q^71 + 1944 * q^72 + 53838 * q^73 + 79212 * q^74 - 52448 * q^76 + 18392 * q^77 + 7308 * q^78 + 6364 * q^79 + 19683 * q^81 + 68020 * q^82 - 96272 * q^83 - 52992 * q^84 + 143152 * q^86 + 50238 * q^87 - 2904 * q^88 - 38938 * q^89 + 104968 * q^91 - 24000 * q^92 + 56952 * q^93 + 49088 * q^94 + 6624 * q^96 + 103242 * q^97 - 9554 * 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 $

$2$	−6.55890	−1.15946	−0.579731	−	0.814808i	$-0.696842\pi$
$2$	−6.55890	−1.15946	−0.579731	+	0.814808i	$0.696842\pi$
$3$	9.00000	0.577350
$3$	9.00000	0.577350
$4$	11.0192	0.344351
$5$	0	0
$5$	0	0
$6$	−59.0301	−0.669415
$7$	−146.487	−1.12993	−0.564967	−	0.825113i	$-0.691111\pi$
$7$	−146.487	−1.12993	−0.564967	+	0.825113i	$0.691111\pi$
$8$	137.611	0.760200
$9$	81.0000	0.333333
$10$	0	0
$11$	−121.000	−0.301511
$11$	−121.000	−0.301511
$12$	99.1731	0.198811
$13$	−170.238	−0.279382	−0.139691	−	0.990195i	$-0.544611\pi$
$13$	−170.238	−0.279382	−0.139691	+	0.990195i	$0.544611\pi$
$14$	960.792	1.31012
$15$	0	0
$16$	−1255.19	−1.22577
$17$	−1564.81	−1.31323	−0.656614	−	0.754227i	$-0.728012\pi$
$17$	−1564.81	−1.31323	−0.656614	+	0.754227i	$0.728012\pi$
$18$	−531.271	−0.386487
$19$	569.786	0.362100	0.181050	−	0.983474i	$-0.442050\pi$
$19$	569.786	0.362100	0.181050	+	0.983474i	$0.442050\pi$
$20$	0	0
$21$	−1318.38	−0.652368
$22$	793.627	0.349591
$23$	−3153.25	−1.24291	−0.621454	−	0.783451i	$-0.713458\pi$
$23$	−3153.25	−1.24291	−0.621454	+	0.783451i	$0.713458\pi$
$24$	1238.50	0.438902
$25$	0	0
$26$	1116.57	0.323932
$27$	729.000	0.192450
$28$	−1614.17	−0.389094
$29$	3982.58	0.879366	0.439683	−	0.898153i	$-0.355091\pi$
$29$	3982.58	0.879366	0.439683	+	0.898153i	$0.355091\pi$
$30$	0	0
$31$	2990.78	0.558959	0.279480	−	0.960152i	$-0.409838\pi$
$31$	2990.78	0.558959	0.279480	+	0.960152i	$0.409838\pi$
$32$	3829.14	0.661037
$33$	−1089.00	−0.174078
$34$	10263.5	1.52264
$35$	0	0
$36$	892.558	0.114784
$37$	−7858.92	−0.943753	−0.471877	−	0.881665i	$-0.656423\pi$
$37$	−7858.92	−0.943753	−0.471877	+	0.881665i	$0.656423\pi$
$38$	−3737.17	−0.419841
$39$	−1532.14	−0.161301
$40$	0	0
$41$	−5206.60	−0.483721	−0.241860	−	0.970311i	$-0.577758\pi$
$41$	−5206.60	−0.483721	−0.241860	+	0.970311i	$0.577758\pi$
$42$	8647.13	0.756395
$43$	−13874.3	−1.14430	−0.572149	−	0.820149i	$-0.693890\pi$
$43$	−13874.3	−1.14430	−0.572149	+	0.820149i	$0.693890\pi$
$44$	−1333.33	−0.103826
$45$	0	0
$46$	20681.9	1.44110
$47$	−6852.01	−0.452453	−0.226226	−	0.974075i	$-0.572639\pi$
$47$	−6852.01	−0.452453	−0.226226	+	0.974075i	$0.572639\pi$
$48$	−11296.7	−0.707701
$49$	4651.35	0.276751
$50$	0	0
$51$	−14083.3	−0.758192
$52$	−1875.89	−0.0962054
$53$	3834.15	0.187490	0.0937452	−	0.995596i	$-0.470116\pi$
$53$	3834.15	0.187490	0.0937452	+	0.995596i	$0.470116\pi$
$54$	−4781.44	−0.223138
$55$	0	0
$56$	−20158.2	−0.858976
$57$	5128.08	0.209058
$58$	−26121.4	−1.01959
$59$	9649.38	0.360885	0.180443	−	0.983586i	$-0.442247\pi$
$59$	9649.38	0.360885	0.180443	+	0.983586i	$0.442247\pi$
$60$	0	0
$61$	−21131.8	−0.727128	−0.363564	−	0.931569i	$-0.618440\pi$
$61$	−21131.8	−0.727128	−0.363564	+	0.931569i	$0.618440\pi$
$62$	−19616.2	−0.648092
$63$	−11865.4	−0.376645
$64$	15051.2	0.459326
$65$	0	0
$66$	7142.65	0.201836
$67$	43499.9	1.18386	0.591931	−	0.805989i	$-0.298366\pi$
$67$	43499.9	1.18386	0.591931	+	0.805989i	$0.298366\pi$
$68$	−17243.0	−0.452211
$69$	−28379.2	−0.717593
$70$	0	0
$71$	−52607.2	−1.23851	−0.619255	−	0.785190i	$-0.712565\pi$
$71$	−52607.2	−1.23851	−0.619255	+	0.785190i	$0.712565\pi$
$72$	11146.5	0.253400
$73$	64367.9	1.41372	0.706858	−	0.707355i	$-0.250112\pi$
$73$	64367.9	1.41372	0.706858	+	0.707355i	$0.250112\pi$
$74$	51545.9	1.09425
$75$	0	0
$76$	6278.61	0.124689
$77$	17724.9	0.340688
$78$	10049.2	0.187022
$79$	28935.7	0.521635	0.260817	−	0.965388i	$-0.416008\pi$
$79$	28935.7	0.521635	0.260817	+	0.965388i	$0.416008\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	34149.6	0.560855
$83$	4648.64	0.0740681	0.0370340	−	0.999314i	$-0.488209\pi$
$83$	4648.64	0.0740681	0.0370340	+	0.999314i	$0.488209\pi$
$84$	−14527.5	−0.224643
$85$	0	0
$86$	91000.1	1.32677
$87$	35843.2	0.507702
$88$	−16650.9	−0.229209
$89$	−103458.	−1.38448	−0.692242	−	0.721666i	$-0.743377\pi$
$89$	−103458.	−1.38448	−0.692242	+	0.721666i	$0.743377\pi$
$90$	0	0
$91$	24937.6	0.315683
$92$	−34746.4	−0.427996
$93$	26917.0	0.322715
$94$	44941.6	0.524601
$95$	0	0
$96$	34462.2	0.381650
$97$	−75809.5	−0.818077	−0.409039	−	0.912517i	$-0.634136\pi$
$97$	−75809.5	−0.818077	−0.409039	+	0.912517i	$0.634136\pi$
$98$	−30507.8	−0.320882
$99$	−9801.00	−0.100504
$100$	0	0
$101$	35023.6	0.341631	0.170816	−	0.985303i	$-0.445360\pi$
$101$	35023.6	0.341631	0.170816	+	0.985303i	$0.445360\pi$
$102$	92371.1	0.879095
$103$	−16965.7	−0.157572	−0.0787859	−	0.996892i	$-0.525104\pi$
$103$	−16965.7	−0.157572	−0.0787859	+	0.996892i	$0.525104\pi$
$104$	−23426.6	−0.212386
$105$	0	0
$106$	−25147.8	−0.217388
$107$	−12190.4	−0.102934	−0.0514671	−	0.998675i	$-0.516390\pi$
$107$	−12190.4	−0.102934	−0.0514671	+	0.998675i	$0.516390\pi$
$108$	8033.02	0.0662704
$109$	12138.7	0.0978606	0.0489303	−	0.998802i	$-0.484419\pi$
$109$	12138.7	0.0978606	0.0489303	+	0.998802i	$0.484419\pi$
$110$	0	0
$111$	−70730.3	−0.544876
$112$	183869.	1.38504
$113$	−204649.	−1.50770	−0.753848	−	0.657048i	$-0.771805\pi$
$113$	−204649.	−1.50770	−0.753848	+	0.657048i	$0.771805\pi$
$114$	−33634.6	−0.242395
$115$	0	0
$116$	43885.0	0.302810
$117$	−13789.3	−0.0931273
$118$	−63289.3	−0.418433
$119$	229224.	1.48386
$120$	0	0
$121$	14641.0	0.0909091
$122$	138601.	0.843077
$123$	−46859.4	−0.279276
$124$	32956.1	0.192478
$125$	0	0
$126$	77824.2	0.436705
$127$	112241.	0.617506	0.308753	−	0.951142i	$-0.400088\pi$
$127$	112241.	0.617506	0.308753	+	0.951142i	$0.400088\pi$
$128$	−221252.	−1.19361
$129$	−124869.	−0.660661
$130$	0	0
$131$	−11679.0	−0.0594602	−0.0297301	−	0.999558i	$-0.509465\pi$
$131$	−11679.0	−0.0594602	−0.0297301	+	0.999558i	$0.509465\pi$
$132$	−11999.9	−0.0599438
$133$	−83466.1	−0.409149
$134$	−285311.	−1.37264
$135$	0	0
$136$	−215335.	−0.998315
$137$	384935.	1.75221	0.876105	−	0.482121i	$-0.160133\pi$
$137$	384935.	1.75221	0.876105	+	0.482121i	$0.160133\pi$
$138$	186137.	0.832021
$139$	−310605.	−1.36355	−0.681775	−	0.731562i	$-0.738792\pi$
$139$	−310605.	−1.36355	−0.681775	+	0.731562i	$0.738792\pi$
$140$	0	0
$141$	−61668.0	−0.261224
$142$	345046.	1.43600
$143$	20598.8	0.0842368
$144$	−101671.	−0.408591
$145$	0	0
$146$	−422183.	−1.63915
$147$	41862.2	0.159782
$148$	−86599.2	−0.324982
$149$	285203.	1.05242	0.526209	−	0.850355i	$-0.323613\pi$
$149$	285203.	1.05242	0.526209	+	0.850355i	$0.323613\pi$
$150$	0	0
$151$	−522357.	−1.86434	−0.932170	−	0.362020i	$-0.882087\pi$
$151$	−522357.	−1.86434	−0.932170	+	0.362020i	$0.882087\pi$
$152$	78408.8	0.275268
$153$	−126750.	−0.437743
$154$	−116256.	−0.395015
$155$	0	0
$156$	−16883.0	−0.0555442
$157$	2221.66	0.00719330	0.00359665	−	0.999994i	$-0.498855\pi$
$157$	2221.66	0.00719330	0.00359665	+	0.999994i	$0.498855\pi$
$158$	−189787.	−0.604816
$159$	34507.3	0.108248
$160$	0	0
$161$	461909.	1.40440
$162$	−43033.0	−0.128829
$163$	−250146.	−0.737436	−0.368718	−	0.929541i	$-0.620203\pi$
$163$	−250146.	−0.737436	−0.368718	+	0.929541i	$0.620203\pi$
$164$	−57372.7	−0.166570
$165$	0	0
$166$	−30490.0	−0.0858791
$167$	42083.4	0.116767	0.0583834	−	0.998294i	$-0.481405\pi$
$167$	42083.4	0.116767	0.0583834	+	0.998294i	$0.481405\pi$
$168$	−181423.	−0.495930
$169$	−342312.	−0.921946
$170$	0	0
$171$	46152.7	0.120700
$172$	−152884.	−0.394040
$173$	117206.	0.297737	0.148869	−	0.988857i	$-0.452437\pi$
$173$	117206.	0.297737	0.148869	+	0.988857i	$0.452437\pi$
$174$	−235092.	−0.588661
$175$	0	0
$176$	151878.	0.369585
$177$	86844.4	0.208357
$178$	678569.	1.60526
$179$	582985.	1.35996	0.679978	−	0.733233i	$-0.261989\pi$
$179$	582985.	1.35996	0.679978	+	0.733233i	$0.261989\pi$
$180$	0	0
$181$	235291.	0.533837	0.266919	−	0.963719i	$-0.413994\pi$
$181$	235291.	0.533837	0.266919	+	0.963719i	$0.413994\pi$
$182$	−163563.	−0.366022
$183$	−190186.	−0.419808
$184$	−433921.	−0.944858
$185$	0	0
$186$	−176546.	−0.374176
$187$	189342.	0.395953
$188$	−75503.8	−0.155802
$189$	−106789.	−0.217456
$190$	0	0
$191$	591788.	1.17377	0.586885	−	0.809670i	$-0.300354\pi$
$191$	591788.	1.17377	0.586885	+	0.809670i	$0.300354\pi$
$192$	135461.	0.265192
$193$	540825.	1.04511	0.522557	−	0.852605i	$-0.324978\pi$
$193$	540825.	1.04511	0.522557	+	0.852605i	$0.324978\pi$
$194$	497227.	0.948529
$195$	0	0
$196$	51254.3	0.0952994
$197$	−481711.	−0.884344	−0.442172	−	0.896930i	$-0.645792\pi$
$197$	−481711.	−0.884344	−0.442172	+	0.896930i	$0.645792\pi$
$198$	64283.8	0.116530
$199$	−1.01427e6	−1.81560	−0.907801	−	0.419401i	$-0.862240\pi$
$199$	−1.01427e6	−1.81560	−0.907801	+	0.419401i	$0.862240\pi$
$200$	0	0
$201$	391499.	0.683503
$202$	−229717.	−0.396108
$203$	−583395.	−0.993625
$204$	−155187.	−0.261084
$205$	0	0
$206$	111276.	0.182699
$207$	−255413.	−0.414302
$208$	213681.	0.342459
$209$	−68944.2	−0.109177
$210$	0	0
$211$	619718.	0.958270	0.479135	−	0.877741i	$-0.340950\pi$
$211$	619718.	0.958270	0.479135	+	0.877741i	$0.340950\pi$
$212$	42249.3	0.0645625
$213$	−473465.	−0.715054
$214$	79955.8	0.119348
$215$	0	0
$216$	100318.	0.146301
$217$	−438109.	−0.631587
$218$	−79616.9	−0.113466
$219$	579311.	0.816210
$220$	0	0
$221$	266391.	0.366892
$222$	463913.	0.631763
$223$	−894252.	−1.20420	−0.602099	−	0.798422i	$-0.705669\pi$
$223$	−894252.	−1.20420	−0.602099	+	0.798422i	$0.705669\pi$
$224$	−560918.	−0.746928
$225$	0	0
$226$	1.34227e6	1.74812
$227$	−851468.	−1.09674	−0.548370	−	0.836236i	$-0.684751\pi$
$227$	−851468.	−1.09674	−0.548370	+	0.836236i	$0.684751\pi$
$228$	56507.5	0.0719894
$229$	137840.	0.173695	0.0868475	−	0.996222i	$-0.472321\pi$
$229$	137840.	0.173695	0.0868475	+	0.996222i	$0.472321\pi$
$230$	0	0
$231$	159524.	0.196696
$232$	548046.	0.668494
$233$	1.31590e6	1.58793	0.793965	−	0.607963i	$-0.208013\pi$
$233$	1.31590e6	1.58793	0.793965	+	0.607963i	$0.208013\pi$
$234$	90442.6	0.107977
$235$	0	0
$236$	106329.	0.124271
$237$	260422.	0.301166
$238$	−1.50346e6	−1.72048
$239$	1.11778e6	1.26579	0.632896	−	0.774237i	$-0.281866\pi$
$239$	1.11778e6	1.26579	0.632896	+	0.774237i	$0.281866\pi$
$240$	0	0
$241$	−471581.	−0.523015	−0.261507	−	0.965201i	$-0.584220\pi$
$241$	−471581.	−0.523015	−0.261507	+	0.965201i	$0.584220\pi$
$242$	−96028.9	−0.105406
$243$	59049.0	0.0641500
$244$	−232856.	−0.250387
$245$	0	0
$246$	307346.	0.323810
$247$	−96999.3	−0.101164
$248$	411564.	0.424921
$249$	41837.8	0.0427632
$250$	0	0
$251$	1.76541e6	1.76873	0.884363	−	0.466799i	$-0.154593\pi$
$251$	1.76541e6	1.76873	0.884363	+	0.466799i	$0.154593\pi$
$252$	−130748.	−0.129698
$253$	381543.	0.374751
$254$	−736176.	−0.715974
$255$	0	0
$256$	969531.	0.924617
$257$	19920.1	0.0188130	0.00940652	−	0.999956i	$-0.497006\pi$
$257$	19920.1	0.0188130	0.00940652	+	0.999956i	$0.497006\pi$
$258$	819001.	0.766011
$259$	1.15123e6	1.06638
$260$	0	0
$261$	322589.	0.293122
$262$	76601.2	0.0689418
$263$	2.18348e6	1.94652	0.973262	−	0.229697i	$-0.0737734\pi$
$263$	2.18348e6	1.94652	0.973262	+	0.229697i	$0.0737734\pi$
$264$	−149858.	−0.132334
$265$	0	0
$266$	547446.	0.474392
$267$	−931119.	−0.799332
$268$	479335.	0.407664
$269$	839300.	0.707190	0.353595	−	0.935399i	$-0.384959\pi$
$269$	839300.	0.707190	0.353595	+	0.935399i	$0.384959\pi$
$270$	0	0
$271$	1.42641e6	1.17984	0.589918	−	0.807463i	$-0.299160\pi$
$271$	1.42641e6	1.17984	0.589918	+	0.807463i	$0.299160\pi$
$272$	1.96414e6	1.60972
$273$	224438.	0.182260
$274$	−2.52475e6	−2.03162
$275$	0	0
$276$	−312717.	−0.247104
$277$	1.30448e6	1.02150	0.510749	−	0.859730i	$-0.329368\pi$
$277$	1.30448e6	1.02150	0.510749	+	0.859730i	$0.329368\pi$
$278$	2.03723e6	1.58098
$279$	242253.	0.186320
$280$	0	0
$281$	−580646.	−0.438678	−0.219339	−	0.975649i	$-0.570390\pi$
$281$	−580646.	−0.438678	−0.219339	+	0.975649i	$0.570390\pi$
$282$	404475.	0.302879
$283$	601974.	0.446799	0.223399	−	0.974727i	$-0.428285\pi$
$283$	601974.	0.446799	0.223399	+	0.974727i	$0.428285\pi$
$284$	−579691.	−0.426482
$285$	0	0
$286$	−135106.	−0.0976693
$287$	762698.	0.546572
$288$	310160.	0.220346
$289$	1.02878e6	0.724567
$290$	0	0
$291$	−682286.	−0.472317
$292$	709285.	0.486815
$293$	2.40993e6	1.63996	0.819982	−	0.572389i	$-0.193983\pi$
$293$	2.40993e6	1.63996	0.819982	+	0.572389i	$0.193983\pi$
$294$	−274570.	−0.185261
$295$	0	0
$296$	−1.08147e6	−0.717441
$297$	−88209.0	−0.0580259
$298$	−1.87062e6	−1.22024
$299$	536803.	0.347246
$300$	0	0
$301$	2.03240e6	1.29298
$302$	3.42609e6	2.16163
$303$	315213.	0.197241
$304$	−715191.	−0.443852
$305$	0	0
$306$	831340.	0.507546
$307$	1.83380e6	1.11047	0.555235	−	0.831694i	$-0.312629\pi$
$307$	1.83380e6	1.11047	0.555235	+	0.831694i	$0.312629\pi$
$308$	195315.	0.117316
$309$	−152691.	−0.0909742
$310$	0	0
$311$	2.27507e6	1.33381	0.666906	−	0.745142i	$-0.267618\pi$
$311$	2.27507e6	1.33381	0.666906	+	0.745142i	$0.267618\pi$
$312$	−210839.	−0.122621
$313$	−159585.	−0.0920728	−0.0460364	−	0.998940i	$-0.514659\pi$
$313$	−159585.	−0.0920728	−0.0460364	+	0.998940i	$0.514659\pi$
$314$	−14571.6	−0.00834035
$315$	0	0
$316$	318849.	0.179625
$317$	1.24338e6	0.694955	0.347477	−	0.937688i	$-0.387038\pi$
$317$	1.24338e6	0.694955	0.347477	+	0.937688i	$0.387038\pi$
$318$	−226330.	−0.125509
$319$	−481892.	−0.265139
$320$	0	0
$321$	−109714.	−0.0594290
$322$	−3.02962e6	−1.62835
$323$	−891609.	−0.475519
$324$	72297.2	0.0382612
$325$	0	0
$326$	1.64068e6	0.855029
$327$	109249.	0.0564998
$328$	−716485.	−0.367724
$329$	1.00373e6	0.511242
$330$	0	0
$331$	−958448.	−0.480838	−0.240419	−	0.970669i	$-0.577285\pi$
$331$	−958448.	−0.480838	−0.240419	+	0.970669i	$0.577285\pi$
$332$	51224.5	0.0255054
$333$	−636573.	−0.314584
$334$	−276021.	−0.135387
$335$	0	0
$336$	1.65482e6	0.799655
$337$	1.51074e6	0.724626	0.362313	−	0.932056i	$-0.381987\pi$
$337$	1.51074e6	0.724626	0.362313	+	0.932056i	$0.381987\pi$
$338$	2.24519e6	1.06896
$339$	−1.84184e6	−0.870469
$340$	0	0
$341$	−361884.	−0.168533
$342$	−302711.	−0.139947
$343$	1.78064e6	0.817224
$344$	−1.90925e6	−0.869896
$345$	0	0
$346$	−768741.	−0.345215
$347$	1.93847e6	0.864243	0.432122	−	0.901815i	$-0.357765\pi$
$347$	1.93847e6	0.864243	0.432122	+	0.901815i	$0.357765\pi$
$348$	394965.	0.174828
$349$	−1.62462e6	−0.713985	−0.356992	−	0.934107i	$-0.616198\pi$
$349$	−1.62462e6	−0.713985	−0.356992	+	0.934107i	$0.616198\pi$
$350$	0	0
$351$	−124103.	−0.0537671
$352$	−463325.	−0.199310
$353$	23347.5	0.00997249	0.00498624	−	0.999988i	$-0.498413\pi$
$353$	23347.5	0.00997249	0.00498624	+	0.999988i	$0.498413\pi$
$354$	−569604.	−0.241582
$355$	0	0
$356$	−1.14002e6	−0.476748
$357$	2.06302e6	0.856707
$358$	−3.82374e6	−1.57682
$359$	3.63918e6	1.49028	0.745139	−	0.666909i	$-0.232383\pi$
$359$	3.63918e6	1.49028	0.745139	+	0.666909i	$0.232383\pi$
$360$	0	0
$361$	−2.15144e6	−0.868884
$362$	−1.54325e6	−0.618964
$363$	131769.	0.0524864
$364$	274793.	0.108706
$365$	0	0
$366$	1.24741e6	0.486751
$367$	−2.24713e6	−0.870890	−0.435445	−	0.900215i	$-0.643409\pi$
$367$	−2.24713e6	−0.870890	−0.435445	+	0.900215i	$0.643409\pi$
$368$	3.95793e6	1.52352
$369$	−421735.	−0.161240
$370$	0	0
$371$	−561651.	−0.211852
$372$	296605.	0.111127
$373$	−3.82745e6	−1.42442	−0.712208	−	0.701968i	$-0.752305\pi$
$373$	−3.82745e6	−1.42442	−0.712208	+	0.701968i	$0.752305\pi$
$374$	−1.24188e6	−0.459092
$375$	0	0
$376$	−942910.	−0.343954
$377$	−677986.	−0.245679
$378$	700418.	0.252132
$379$	−4.26443e6	−1.52498	−0.762489	−	0.647002i	$-0.776023\pi$
$379$	−4.26443e6	−1.52498	−0.762489	+	0.647002i	$0.776023\pi$
$380$	0	0
$381$	1.01017e6	0.356517
$382$	−3.88148e6	−1.36094
$383$	−2.40920e6	−0.839221	−0.419611	−	0.907704i	$-0.637833\pi$
$383$	−2.40920e6	−0.839221	−0.419611	+	0.907704i	$0.637833\pi$
$384$	−1.99127e6	−0.689130
$385$	0	0
$386$	−3.54722e6	−1.21177
$387$	−1.12382e6	−0.381433
$388$	−835362.	−0.281706
$389$	3.73204e6	1.25047	0.625233	−	0.780438i	$-0.285004\pi$
$389$	3.73204e6	1.25047	0.625233	+	0.780438i	$0.285004\pi$
$390$	0	0
$391$	4.93424e6	1.63222
$392$	640077.	0.210386
$393$	−105111.	−0.0343293
$394$	3.15950e6	1.02536
$395$	0	0
$396$	−107999.	−0.0346086
$397$	4.89288e6	1.55808	0.779038	−	0.626977i	$-0.215708\pi$
$397$	4.89288e6	1.55808	0.779038	+	0.626977i	$0.215708\pi$
$398$	6.65250e6	2.10512
$399$	−751195.	−0.236222
$400$	0	0
$401$	−3.93524e6	−1.22211	−0.611055	−	0.791588i	$-0.709255\pi$
$401$	−3.93524e6	−1.22211	−0.611055	+	0.791588i	$0.709255\pi$
$402$	−2.56780e6	−0.792495
$403$	−509144.	−0.156163
$404$	385933.	0.117641
$405$	0	0
$406$	3.82643e6	1.15207
$407$	950929.	0.284552
$408$	−1.93802e6	−0.576378
$409$	657037.	0.194215	0.0971073	−	0.995274i	$-0.469041\pi$
$409$	657037.	0.194215	0.0971073	+	0.995274i	$0.469041\pi$
$410$	0	0
$411$	3.46442e6	1.01164
$412$	−186949.	−0.0542600
$413$	−1.41351e6	−0.407777
$414$	1.67523e6	0.480368
$415$	0	0
$416$	−651864.	−0.184682
$417$	−2.79544e6	−0.787246
$418$	452198.	0.126587
$419$	4.38504e6	1.22022	0.610110	−	0.792317i	$-0.291125\pi$
$419$	4.38504e6	1.22022	0.610110	+	0.792317i	$0.291125\pi$
$420$	0	0
$421$	2.34058e6	0.643604	0.321802	−	0.946807i	$-0.395711\pi$
$421$	2.34058e6	0.643604	0.321802	+	0.946807i	$0.395711\pi$
$422$	−4.06467e6	−1.11108
$423$	−555012.	−0.150818
$424$	527620.	0.142530
$425$	0	0
$426$	3.10541e6	0.829078
$427$	3.09552e6	0.821607
$428$	−134329.	−0.0354455
$429$	185389.	0.0486341
$430$	0	0
$431$	−2.18215e6	−0.565838	−0.282919	−	0.959144i	$-0.591303\pi$
$431$	−2.18215e6	−0.565838	−0.282919	+	0.959144i	$0.591303\pi$
$432$	−915035.	−0.235900
$433$	3.61080e6	0.925515	0.462757	−	0.886485i	$-0.346860\pi$
$433$	3.61080e6	0.925515	0.462757	+	0.886485i	$0.346860\pi$
$434$	2.87352e6	0.732301
$435$	0	0
$436$	133760.	0.0336984
$437$	−1.79668e6	−0.450056
$438$	−3.79965e6	−0.946364
$439$	6.48191e6	1.60525	0.802624	−	0.596486i	$-0.203437\pi$
$439$	6.48191e6	1.60525	0.802624	+	0.596486i	$0.203437\pi$
$440$	0	0
$441$	376760.	0.0922503
$442$	−1.74723e6	−0.425397
$443$	6.81523e6	1.64995	0.824976	−	0.565167i	$-0.191188\pi$
$443$	6.81523e6	1.64995	0.824976	+	0.565167i	$0.191188\pi$
$444$	−779393.	−0.187629
$445$	0	0
$446$	5.86531e6	1.39622
$447$	2.56683e6	0.607614
$448$	−2.20480e6	−0.519008
$449$	−2.98356e6	−0.698423	−0.349211	−	0.937044i	$-0.613551\pi$
$449$	−2.98356e6	−0.698423	−0.349211	+	0.937044i	$0.613551\pi$
$450$	0	0
$451$	629999.	0.145847
$452$	−2.25508e6	−0.519177
$453$	−4.70122e6	−1.07638
$454$	5.58469e6	1.27163
$455$	0	0
$456$	705679.	0.158926
$457$	6.11184e6	1.36893	0.684466	−	0.729045i	$-0.260036\pi$
$457$	6.11184e6	1.36893	0.684466	+	0.729045i	$0.260036\pi$
$458$	−904081.	−0.201393
$459$	−1.14075e6	−0.252731
$460$	0	0
$461$	−3.41772e6	−0.749003	−0.374502	−	0.927226i	$-0.622186\pi$
$461$	−3.41772e6	−0.749003	−0.374502	+	0.927226i	$0.622186\pi$
$462$	−1.04630e6	−0.228062
$463$	4.60011e6	0.997277	0.498639	−	0.866810i	$-0.333834\pi$
$463$	4.60011e6	0.997277	0.498639	+	0.866810i	$0.333834\pi$
$464$	−4.99890e6	−1.07790
$465$	0	0
$466$	−8.63083e6	−1.84114
$467$	−5.03617e6	−1.06858	−0.534291	−	0.845301i	$-0.679421\pi$
$467$	−5.03617e6	−1.06858	−0.534291	+	0.845301i	$0.679421\pi$
$468$	−151947.	−0.0320685
$469$	−6.37215e6	−1.33769
$470$	0	0
$471$	19994.9	0.00415305
$472$	1.32786e6	0.274345
$473$	1.67879e6	0.345019
$474$	−1.70808e6	−0.349190
$475$	0	0
$476$	2.52587e6	0.510969
$477$	310566.	0.0624968
$478$	−7.33142e6	−1.46764
$479$	−2.82760e6	−0.563091	−0.281545	−	0.959548i	$-0.590847\pi$
$479$	−2.82760e6	−0.563091	−0.281545	+	0.959548i	$0.590847\pi$
$480$	0	0
$481$	1.33789e6	0.263668
$482$	3.09306e6	0.606416
$483$	4.15718e6	0.810833
$484$	161333.	0.0313046
$485$	0	0
$486$	−387297.	−0.0743795
$487$	5.68931e6	1.08702	0.543510	−	0.839403i	$-0.317095\pi$
$487$	5.68931e6	1.08702	0.543510	+	0.839403i	$0.317095\pi$
$488$	−2.90796e6	−0.552763
$489$	−2.25131e6	−0.425759
$490$	0	0
$491$	3.23715e6	0.605981	0.302990	−	0.952994i	$-0.402015\pi$
$491$	3.23715e6	0.605981	0.302990	+	0.952994i	$0.402015\pi$
$492$	−516355.	−0.0961690
$493$	−6.23199e6	−1.15481
$494$	636209.	0.117296
$495$	0	0
$496$	−3.75400e6	−0.685157
$497$	7.70626e6	1.39943
$498$	−274410.	−0.0495823
$499$	−5.08564e6	−0.914312	−0.457156	−	0.889386i	$-0.651132\pi$
$499$	−5.08564e6	−0.914312	−0.457156	+	0.889386i	$0.651132\pi$
$500$	0	0
$501$	378751.	0.0674154
$502$	−1.15791e7	−2.05077
$503$	−9.02252e6	−1.59004	−0.795019	−	0.606584i	$-0.792539\pi$
$503$	−9.02252e6	−1.59004	−0.795019	+	0.606584i	$0.792539\pi$
$504$	−1.63281e6	−0.286325
$505$	0	0
$506$	−2.50251e6	−0.434509
$507$	−3.08081e6	−0.532286
$508$	1.23681e6	0.212639
$509$	8.98909e6	1.53788	0.768938	−	0.639323i	$-0.220785\pi$
$509$	8.98909e6	1.53788	0.768938	+	0.639323i	$0.220785\pi$
$510$	0	0
$511$	−9.42904e6	−1.59741
$512$	720997.	0.121551
$513$	415374.	0.0696861
$514$	−130654.	−0.0218130
$515$	0	0
$516$	−1.37596e6	−0.227499
$517$	829093.	0.136420
$518$	−7.55079e6	−1.23643
$519$	1.05485e6	0.171899
$520$	0	0
$521$	−1.11219e7	−1.79508	−0.897539	−	0.440936i	$-0.854647\pi$
$521$	−1.11219e7	−1.79508	−0.897539	+	0.440936i	$0.854647\pi$
$522$	−2.11583e6	−0.339863
$523$	−3.67510e6	−0.587510	−0.293755	−	0.955881i	$-0.594905\pi$
$523$	−3.67510e6	−0.587510	−0.293755	+	0.955881i	$0.594905\pi$
$524$	−128693.	−0.0204752
$525$	0	0
$526$	−1.43212e7	−2.25692
$527$	−4.68001e6	−0.734040
$528$	1.36690e6	0.213380
$529$	3.50664e6	0.544818
$530$	0	0
$531$	781600.	0.120295
$532$	−919733.	−0.140891
$533$	886361.	0.135143
$534$	6.10712e6	0.926795
$535$	0	0
$536$	5.98605e6	0.899971
$537$	5.24687e6	0.785171
$538$	−5.50489e6	−0.819960
$539$	−562814.	−0.0834436
$540$	0	0
$541$	−4.31097e6	−0.633260	−0.316630	−	0.948549i	$-0.602551\pi$
$541$	−4.31097e6	−0.633260	−0.316630	+	0.948549i	$0.602551\pi$
$542$	−9.35569e6	−1.36797
$543$	2.11762e6	0.308211
$544$	−5.99188e6	−0.868092
$545$	0	0
$546$	−1.47207e6	−0.211323
$547$	1.80626e6	0.258114	0.129057	−	0.991637i	$-0.458805\pi$
$547$	1.80626e6	0.258114	0.129057	+	0.991637i	$0.458805\pi$
$548$	4.24169e6	0.603375
$549$	−1.71167e6	−0.242376
$550$	0	0
$551$	2.26922e6	0.318418
$552$	−3.90529e6	−0.545514
$553$	−4.23870e6	−0.589413
$554$	−8.55595e6	−1.18439
$555$	0	0
$556$	−3.42262e6	−0.469540
$557$	−8.42670e6	−1.15085	−0.575426	−	0.817854i	$-0.695164\pi$
$557$	−8.42670e6	−1.15085	−0.575426	+	0.817854i	$0.695164\pi$
$558$	−1.58891e6	−0.216031
$559$	2.36193e6	0.319696
$560$	0	0
$561$	1.70408e6	0.228604
$562$	3.80840e6	0.508630
$563$	9.19966e6	1.22321	0.611605	−	0.791163i	$-0.290524\pi$
$563$	9.19966e6	1.22321	0.611605	+	0.791163i	$0.290524\pi$
$564$	−679534.	−0.0899526
$565$	0	0
$566$	−3.94829e6	−0.518046
$567$	−961099.	−0.125548
$568$	−7.23932e6	−0.941515
$569$	5.18644e6	0.671566	0.335783	−	0.941939i	$-0.390999\pi$
$569$	5.18644e6	0.671566	0.335783	+	0.941939i	$0.390999\pi$
$570$	0	0
$571$	−3.29390e6	−0.422786	−0.211393	−	0.977401i	$-0.567800\pi$
$571$	−3.29390e6	−0.422786	−0.211393	+	0.977401i	$0.567800\pi$
$572$	226983.	0.0290070
$573$	5.32609e6	0.677676
$574$	−5.00246e6	−0.633730
$575$	0	0
$576$	1.21915e6	0.153109
$577$	1.02043e7	1.27598	0.637989	−	0.770046i	$-0.279767\pi$
$577$	1.02043e7	1.27598	0.637989	+	0.770046i	$0.279767\pi$
$578$	−6.74768e6	−0.840107
$579$	4.86742e6	0.603396
$580$	0	0
$581$	−680965.	−0.0836921
$582$	4.47505e6	0.547633
$583$	−463932.	−0.0565305
$584$	8.85773e6	1.07471
$585$	0	0
$586$	−1.58065e7	−1.90148
$587$	1.33951e7	1.60454	0.802272	−	0.596958i	$-0.203624\pi$
$587$	1.33951e7	1.60454	0.802272	+	0.596958i	$0.203624\pi$
$588$	461289.	0.0550212
$589$	1.70411e6	0.202399
$590$	0	0
$591$	−4.33540e6	−0.510576
$592$	9.86445e6	1.15683
$593$	−3.76550e6	−0.439730	−0.219865	−	0.975530i	$-0.570562\pi$
$593$	−3.76550e6	−0.439730	−0.219865	+	0.975530i	$0.570562\pi$
$594$	578554.	0.0672788
$595$	0	0
$596$	3.14272e6	0.362401
$597$	−9.12843e6	−1.04824
$598$	−3.52084e6	−0.402618
$599$	1.21252e6	0.138077	0.0690384	−	0.997614i	$-0.478007\pi$
$599$	1.21252e6	0.138077	0.0690384	+	0.997614i	$0.478007\pi$
$600$	0	0
$601$	−7.81714e6	−0.882799	−0.441399	−	0.897311i	$-0.645518\pi$
$601$	−7.81714e6	−0.882799	−0.441399	+	0.897311i	$0.645518\pi$
$602$	−1.33303e7	−1.49916
$603$	3.52349e6	0.394620
$604$	−5.75597e6	−0.641987
$605$	0	0
$606$	−2.06745e6	−0.228693
$607$	−5.43853e6	−0.599114	−0.299557	−	0.954078i	$-0.596839\pi$
$607$	−5.43853e6	−0.599114	−0.299557	+	0.954078i	$0.596839\pi$
$608$	2.18179e6	0.239361
$609$	−5.25055e6	−0.573670
$610$	0	0
$611$	1.16647e6	0.126407
$612$	−1.39669e6	−0.150737
$613$	−1.70637e7	−1.83409	−0.917045	−	0.398783i	$-0.869433\pi$
$613$	−1.70637e7	−1.83409	−0.917045	+	0.398783i	$0.869433\pi$
$614$	−1.20277e7	−1.28755
$615$	0	0
$616$	2.43914e6	0.258991
$617$	−7.80667e6	−0.825568	−0.412784	−	0.910829i	$-0.635444\pi$
$617$	−7.80667e6	−0.825568	−0.412784	+	0.910829i	$0.635444\pi$
$618$	1.00149e6	0.105481
$619$	7.31658e6	0.767506	0.383753	−	0.923436i	$-0.374631\pi$
$619$	7.31658e6	0.767506	0.383753	+	0.923436i	$0.374631\pi$
$620$	0	0
$621$	−2.29872e6	−0.239198
$622$	−1.49220e7	−1.54650
$623$	1.51552e7	1.56438
$624$	1.92313e6	0.197719
$625$	0	0
$626$	1.04670e6	0.106755
$627$	−620497.	−0.0630335
$628$	24481.0	0.00247702
$629$	1.22977e7	1.23936
$630$	0	0
$631$	−1.18729e7	−1.18709	−0.593543	−	0.804802i	$-0.702271\pi$
$631$	−1.18729e7	−1.18709	−0.593543	+	0.804802i	$0.702271\pi$
$632$	3.98187e6	0.396547
$633$	5.57746e6	0.553257
$634$	−8.15523e6	−0.805773
$635$	0	0
$636$	380244.	0.0372752
$637$	−791837.	−0.0773192
$638$	3.16068e6	0.307418
$639$	−4.26118e6	−0.412837
$640$	0	0
$641$	−1.85963e7	−1.78764	−0.893821	−	0.448424i	$-0.851985\pi$
$641$	−1.85963e7	−1.78764	−0.893821	+	0.448424i	$0.851985\pi$
$642$	719603.	0.0689057
$643$	1.18028e7	1.12579	0.562894	−	0.826529i	$-0.309688\pi$
$643$	1.18028e7	1.12579	0.562894	+	0.826529i	$0.309688\pi$
$644$	5.08988e6	0.483608
$645$	0	0
$646$	5.84798e6	0.551346
$647$	−2.05252e7	−1.92764	−0.963821	−	0.266551i	$-0.914116\pi$
$647$	−2.05252e7	−1.92764	−0.963821	+	0.266551i	$0.914116\pi$
$648$	902865.	0.0844666
$649$	−1.16757e6	−0.108811
$650$	0	0
$651$	−3.94298e6	−0.364647
$652$	−2.75642e6	−0.253937
$653$	5.78511e6	0.530919	0.265460	−	0.964122i	$-0.414476\pi$
$653$	5.78511e6	0.530919	0.265460	+	0.964122i	$0.414476\pi$
$654$	−716552.	−0.0655094
$655$	0	0
$656$	6.53528e6	0.592932
$657$	5.21380e6	0.471239
$658$	−6.58335e6	−0.592765
$659$	1.88598e7	1.69170	0.845852	−	0.533418i	$-0.179093\pi$
$659$	1.88598e7	1.69170	0.845852	+	0.533418i	$0.179093\pi$
$660$	0	0
$661$	−1.12652e7	−1.00285	−0.501426	−	0.865201i	$-0.667191\pi$
$661$	−1.12652e7	−1.00285	−0.501426	+	0.865201i	$0.667191\pi$
$662$	6.28637e6	0.557513
$663$	2.39751e6	0.211825
$664$	639704.	0.0563065
$665$	0	0
$666$	4.17522e6	0.364749
$667$	−1.25581e7	−1.09297
$668$	463727.	0.0402088
$669$	−8.04827e6	−0.695244
$670$	0	0
$671$	2.55694e6	0.219237
$672$	−5.04826e6	−0.431239
$673$	−1.51301e7	−1.28767	−0.643833	−	0.765166i	$-0.722657\pi$
$673$	−1.51301e7	−1.28767	−0.643833	+	0.765166i	$0.722657\pi$
$674$	−9.90878e6	−0.840176
$675$	0	0
$676$	−3.77201e6	−0.317473
$677$	1.12001e7	0.939187	0.469593	−	0.882883i	$-0.344401\pi$
$677$	1.12001e7	0.939187	0.469593	+	0.882883i	$0.344401\pi$
$678$	1.20805e7	1.00928
$679$	1.11051e7	0.924373
$680$	0	0
$681$	−7.66321e6	−0.633203
$682$	2.37356e6	0.195407
$683$	−2.12907e7	−1.74638	−0.873190	−	0.487379i	$-0.837953\pi$
$683$	−2.12907e7	−1.74638	−0.873190	+	0.487379i	$0.837953\pi$
$684$	508567.	0.0415631
$685$	0	0
$686$	−1.16791e7	−0.947539
$687$	1.24056e6	0.100283
$688$	1.74149e7	1.40265
$689$	−652717.	−0.0523814
$690$	0	0
$691$	1.77648e6	0.141536	0.0707678	−	0.997493i	$-0.477455\pi$
$691$	1.77648e6	0.141536	0.0707678	+	0.997493i	$0.477455\pi$
$692$	1.29152e6	0.102526
$693$	1.43572e6	0.113563
$694$	−1.27143e7	−1.00206
$695$	0	0
$696$	4.93242e6	0.385955
$697$	8.14735e6	0.635235
$698$	1.06557e7	0.827838
$699$	1.18431e7	0.916792
$700$	0	0
$701$	94420.1	0.00725720	0.00362860	−	0.999993i	$-0.498845\pi$
$701$	94420.1	0.00725720	0.00362860	+	0.999993i	$0.498845\pi$
$702$	813983.	0.0623408
$703$	−4.47791e6	−0.341733
$704$	−1.82120e6	−0.138492
$705$	0	0
$706$	−153134.	−0.0115627
$707$	−5.13050e6	−0.386021
$708$	956958.	0.0717480
$709$	−1.46333e7	−1.09327	−0.546633	−	0.837372i	$-0.684091\pi$
$709$	−1.46333e7	−1.09327	−0.546633	+	0.837372i	$0.684091\pi$
$710$	0	0
$711$	2.34379e6	0.173878
$712$	−1.42369e7	−1.05248
$713$	−9.43067e6	−0.694734
$714$	−1.35311e7	−0.993319
$715$	0	0
$716$	6.42405e6	0.468302
$717$	1.00600e7	0.730805
$718$	−2.38690e7	−1.72792
$719$	1.15215e7	0.831161	0.415580	−	0.909556i	$-0.363578\pi$
$719$	1.15215e7	0.831161	0.415580	+	0.909556i	$0.363578\pi$
$720$	0	0
$721$	2.48525e6	0.178046
$722$	1.41111e7	1.00744
$723$	−4.24423e6	−0.301963
$724$	2.59273e6	0.183827
$725$	0	0
$726$	−864260.	−0.0608559
$727$	−1.73993e7	−1.22095	−0.610473	−	0.792037i	$-0.709021\pi$
$727$	−1.73993e7	−1.22095	−0.610473	+	0.792037i	$0.709021\pi$
$728$	3.43168e6	0.239982
$729$	531441.	0.0370370
$730$	0	0
$731$	2.17107e7	1.50272
$732$	−2.09570e6	−0.144561
$733$	1.21356e7	0.834262	0.417131	−	0.908846i	$-0.363036\pi$
$733$	1.21356e7	0.834262	0.417131	+	0.908846i	$0.363036\pi$
$734$	1.47387e7	1.00976
$735$	0	0
$736$	−1.20742e7	−0.821608
$737$	−5.26348e6	−0.356948
$738$	2.76612e6	0.186952
$739$	1.30600e7	0.879692	0.439846	−	0.898073i	$-0.355033\pi$
$739$	1.30600e7	0.879692	0.439846	+	0.898073i	$0.355033\pi$
$740$	0	0
$741$	−872994.	−0.0584071
$742$	3.68382e6	0.245634
$743$	2.36143e6	0.156929	0.0784644	−	0.996917i	$-0.474998\pi$
$743$	2.36143e6	0.156929	0.0784644	+	0.996917i	$0.474998\pi$
$744$	3.70407e6	0.245328
$745$	0	0
$746$	2.51038e7	1.65156
$747$	376540.	0.0246894
$748$	2.08641e6	0.136347
$749$	1.78574e6	0.116309
$750$	0	0
$751$	4.12172e6	0.266673	0.133336	−	0.991071i	$-0.457431\pi$
$751$	4.12172e6	0.266673	0.133336	+	0.991071i	$0.457431\pi$
$752$	8.60058e6	0.554604
$753$	1.58887e7	1.02117
$754$	4.44685e6	0.284855
$755$	0	0
$756$	−1.17673e6	−0.0748811
$757$	−4.12364e6	−0.261542	−0.130771	−	0.991413i	$-0.541745\pi$
$757$	−4.12364e6	−0.261542	−0.130771	+	0.991413i	$0.541745\pi$
$758$	2.79700e7	1.76815
$759$	3.43389e6	0.216362
$760$	0	0
$761$	−2.08230e7	−1.30341	−0.651706	−	0.758472i	$-0.725946\pi$
$761$	−2.08230e7	−1.30341	−0.651706	+	0.758472i	$0.725946\pi$
$762$	−6.62558e6	−0.413368
$763$	−1.77817e6	−0.110576
$764$	6.52105e6	0.404189
$765$	0	0
$766$	1.58017e7	0.973045
$767$	−1.64269e6	−0.100825
$768$	8.72578e6	0.533828
$769$	1.58156e7	0.964427	0.482213	−	0.876054i	$-0.339833\pi$
$769$	1.58156e7	0.964427	0.482213	+	0.876054i	$0.339833\pi$
$770$	0	0
$771$	179281.	0.0108617
$772$	5.95947e6	0.359886
$773$	2.58350e6	0.155510	0.0777552	−	0.996972i	$-0.475225\pi$
$773$	2.58350e6	0.155510	0.0777552	+	0.996972i	$0.475225\pi$
$774$	7.37101e6	0.442257
$775$	0	0
$776$	−1.04322e7	−0.621902
$777$	1.03610e7	0.615674
$778$	−2.44781e7	−1.44987
$779$	−2.96665e6	−0.175155
$780$	0	0
$781$	6.36547e6	0.373425
$782$	−3.23632e7	−1.89250
$783$	2.90330e6	0.169234
$784$	−5.83834e6	−0.339234
$785$	0	0
$786$	689411.	0.0398035
$787$	5.64885e6	0.325105	0.162552	−	0.986700i	$-0.448027\pi$
$787$	5.64885e6	0.325105	0.162552	+	0.986700i	$0.448027\pi$
$788$	−5.30809e6	−0.304525
$789$	1.96513e7	1.12383
$790$	0	0
$791$	2.99784e7	1.70360
$792$	−1.34872e6	−0.0764030
$793$	3.59743e6	0.203146
$794$	−3.20920e7	−1.80653
$795$	0	0
$796$	−1.11765e7	−0.625204
$797$	3.72876e6	0.207931	0.103965	−	0.994581i	$-0.466847\pi$
$797$	3.72876e6	0.207931	0.103965	+	0.994581i	$0.466847\pi$
$798$	4.92702e6	0.273890
$799$	1.07221e7	0.594173
$800$	0	0
$801$	−8.38007e6	−0.461495
$802$	2.58109e7	1.41699
$803$	−7.78852e6	−0.426252
$804$	4.31401e6	0.235365
$805$	0	0
$806$	3.33943e6	0.181065
$807$	7.55370e6	0.408296
$808$	4.81963e6	0.259708
$809$	6.35004e6	0.341119	0.170559	−	0.985347i	$-0.445443\pi$
$809$	6.35004e6	0.341119	0.170559	+	0.985347i	$0.445443\pi$
$810$	0	0
$811$	1.34156e7	0.716241	0.358120	−	0.933675i	$-0.383418\pi$
$811$	1.34156e7	0.716241	0.358120	+	0.933675i	$0.383418\pi$
$812$	−6.42856e6	−0.342156
$813$	1.28377e7	0.681178
$814$	−6.23705e6	−0.329928
$815$	0	0
$816$	1.76773e7	0.929372
$817$	−7.90538e6	−0.414350
$818$	−4.30945e6	−0.225184
$819$	2.01995e6	0.105228
$820$	0	0
$821$	2.40201e7	1.24370	0.621851	−	0.783135i	$-0.286381\pi$
$821$	2.40201e7	1.24370	0.621851	+	0.783135i	$0.286381\pi$
$822$	−2.27228e7	−1.17296
$823$	1.46523e6	0.0754058	0.0377029	−	0.999289i	$-0.487996\pi$
$823$	1.46523e6	0.0754058	0.0377029	+	0.999289i	$0.487996\pi$
$824$	−2.33466e6	−0.119786
$825$	0	0
$826$	9.27105e6	0.472801
$827$	2.32918e7	1.18424	0.592120	−	0.805850i	$-0.298291\pi$
$827$	2.32918e7	1.18424	0.592120	+	0.805850i	$0.298291\pi$
$828$	−2.81446e6	−0.142665
$829$	−2.02821e6	−0.102501	−0.0512503	−	0.998686i	$-0.516321\pi$
$829$	−2.02821e6	−0.102501	−0.0512503	+	0.998686i	$0.516321\pi$
$830$	0	0
$831$	1.17403e7	0.589762
$832$	−2.56229e6	−0.128327
$833$	−7.27850e6	−0.363437
$834$	1.83350e7	0.912781
$835$	0	0
$836$	−759712.	−0.0375953
$837$	2.18028e6	0.107572
$838$	−2.87610e7	−1.41480
$839$	−3.07518e7	−1.50822	−0.754111	−	0.656747i	$-0.771932\pi$
$839$	−3.07518e7	−1.50822	−0.754111	+	0.656747i	$0.771932\pi$
$840$	0	0
$841$	−4.65021e6	−0.226716
$842$	−1.53517e7	−0.746234
$843$	−5.22582e6	−0.253271
$844$	6.82881e6	0.329981
$845$	0	0
$846$	3.64027e6	0.174867
$847$	−2.14471e6	−0.102721
$848$	−4.81259e6	−0.229821
$849$	5.41777e6	0.257959
$850$	0	0
$851$	2.47811e7	1.17300
$852$	−5.21722e6	−0.246229
$853$	2.05598e7	0.967491	0.483745	−	0.875209i	$-0.339276\pi$
$853$	2.05598e7	0.967491	0.483745	+	0.875209i	$0.339276\pi$
$854$	−2.03032e7	−0.952621
$855$	0	0
$856$	−1.67753e6	−0.0782505
$857$	−3.20463e7	−1.49048	−0.745240	−	0.666796i	$-0.767665\pi$
$857$	−3.20463e7	−1.49048	−0.745240	+	0.666796i	$0.767665\pi$
$858$	−1.21595e6	−0.0563894
$859$	2.34894e6	0.108615	0.0543075	−	0.998524i	$-0.482705\pi$
$859$	2.34894e6	0.108615	0.0543075	+	0.998524i	$0.482705\pi$
$860$	0	0
$861$	6.86428e6	0.315564
$862$	1.43125e7	0.656068
$863$	1.44887e7	0.662219	0.331110	−	0.943592i	$-0.392577\pi$
$863$	1.44887e7	0.662219	0.331110	+	0.943592i	$0.392577\pi$
$864$	2.79144e6	0.127217
$865$	0	0
$866$	−2.36829e7	−1.07310
$867$	9.25903e6	0.418329
$868$	−4.82763e6	−0.217488
$869$	−3.50122e6	−0.157279
$870$	0	0
$871$	−7.40533e6	−0.330749
$872$	1.67042e6	0.0743936
$873$	−6.14057e6	−0.272692
$874$	1.17842e7	0.521823
$875$	0	0
$876$	6.38356e6	0.281063
$877$	3.17684e7	1.39475	0.697376	−	0.716705i	$-0.254351\pi$
$877$	3.17684e7	1.39475	0.697376	+	0.716705i	$0.254351\pi$
$878$	−4.25143e7	−1.86122
$879$	2.16893e7	0.946834
$880$	0	0
$881$	1.06777e7	0.463486	0.231743	−	0.972777i	$-0.425557\pi$
$881$	1.06777e7	0.463486	0.231743	+	0.972777i	$0.425557\pi$
$882$	−2.47113e6	−0.106961
$883$	−1.77890e7	−0.767804	−0.383902	−	0.923374i	$-0.625420\pi$
$883$	−1.77890e7	−0.767804	−0.383902	+	0.923374i	$0.625420\pi$
$884$	2.93542e6	0.126340
$885$	0	0
$886$	−4.47005e7	−1.91306
$887$	−3.13731e7	−1.33890	−0.669450	−	0.742857i	$-0.733470\pi$
$887$	−3.13731e7	−1.33890	−0.669450	+	0.742857i	$0.733470\pi$
$888$	−9.73326e6	−0.414215
$889$	−1.64418e7	−0.697741
$890$	0	0
$891$	−793881.	−0.0335013
$892$	−9.85397e6	−0.414667
$893$	−3.90418e6	−0.163833
$894$	−1.68356e7	−0.704505
$895$	0	0
$896$	3.24104e7	1.34870
$897$	4.83123e6	0.200482
$898$	1.95689e7	0.809795
$899$	1.19110e7	0.491529
$900$	0	0
$901$	−5.99972e6	−0.246218
$902$	−4.13210e6	−0.169104
$903$	1.82916e7	0.746504
$904$	−2.81620e7	−1.14615
$905$	0	0
$906$	3.08348e7	1.24802
$907$	8.46355e6	0.341613	0.170807	−	0.985305i	$-0.445363\pi$
$907$	8.46355e6	0.341613	0.170807	+	0.985305i	$0.445363\pi$
$908$	−9.38252e6	−0.377663
$909$	2.83691e6	0.113877
$910$	0	0
$911$	−2.18411e7	−0.871923	−0.435961	−	0.899965i	$-0.643592\pi$
$911$	−2.18411e7	−0.871923	−0.435961	+	0.899965i	$0.643592\pi$
$912$	−6.43672e6	−0.256258
$913$	−562486.	−0.0223324
$914$	−4.00870e7	−1.58722
$915$	0	0
$916$	1.51889e6	0.0598120
$917$	1.71081e6	0.0671861
$918$	7.48206e6	0.293032
$919$	4.18752e7	1.63557	0.817784	−	0.575525i	$-0.195202\pi$
$919$	4.18752e7	1.63557	0.817784	+	0.575525i	$0.195202\pi$
$920$	0	0
$921$	1.65042e7	0.641130
$922$	2.24165e7	0.868440
$923$	8.95575e6	0.346017
$924$	1.75783e6	0.0677325
$925$	0	0
$926$	−3.01717e7	−1.15630
$927$	−1.37422e6	−0.0525240
$928$	1.52498e7	0.581293
$929$	8.03740e6	0.305546	0.152773	−	0.988261i	$-0.451180\pi$
$929$	8.03740e6	0.305546	0.152773	+	0.988261i	$0.451180\pi$
$930$	0	0
$931$	2.65028e6	0.100211
$932$	1.45002e7	0.546805
$933$	2.04757e7	0.770077
$934$	3.30317e7	1.23898
$935$	0	0
$936$	−1.89755e6	−0.0707953
$937$	1.78192e6	0.0663038	0.0331519	−	0.999450i	$-0.489445\pi$
$937$	1.78192e6	0.0663038	0.0331519	+	0.999450i	$0.489445\pi$
$938$	4.17943e7	1.55099
$939$	−1.43627e6	−0.0531583
$940$	0	0
$941$	2.97245e7	1.09431	0.547154	−	0.837032i	$-0.315711\pi$
$941$	2.97245e7	1.09431	0.547154	+	0.837032i	$0.315711\pi$
$942$	−131145.	−0.00481531
$943$	1.64177e7	0.601220
$944$	−1.21118e7	−0.442364
$945$	0	0
$946$	−1.10110e7	−0.400036
$947$	−931083.	−0.0337375	−0.0168688	−	0.999858i	$-0.505370\pi$
$947$	−931083.	−0.0337375	−0.0168688	+	0.999858i	$0.505370\pi$
$948$	2.86965e6	0.103707
$949$	−1.09579e7	−0.394967
$950$	0	0
$951$	1.11904e7	0.401232
$952$	3.15437e7	1.12803
$953$	2.74983e7	0.980784	0.490392	−	0.871502i	$-0.336854\pi$
$953$	2.74983e7	0.980784	0.490392	+	0.871502i	$0.336854\pi$
$954$	−2.03697e6	−0.0724626
$955$	0	0
$956$	1.23171e7	0.435876
$957$	−4.33703e6	−0.153078
$958$	1.85459e7	0.652882
$959$	−5.63879e7	−1.97988
$960$	0	0
$961$	−1.96844e7	−0.687565
$962$	−8.77507e6	−0.305712
$963$	−987424.	−0.0343114
$964$	−5.19646e6	−0.180101
$965$	0	0
$966$	−2.72666e7	−0.940129
$967$	−4.33623e6	−0.149123	−0.0745617	−	0.997216i	$-0.523756\pi$
$967$	−4.33623e6	−0.149123	−0.0745617	+	0.997216i	$0.523756\pi$
$968$	2.01476e6	0.0691091
$969$	−8.02448e6	−0.274541
$970$	0	0
$971$	−5.44123e7	−1.85204	−0.926018	−	0.377480i	$-0.876791\pi$
$971$	−5.44123e7	−1.85204	−0.926018	+	0.377480i	$0.876791\pi$
$972$	650675.	0.0220901
$973$	4.54995e7	1.54072
$974$	−3.73157e7	−1.26036
$975$	0	0
$976$	2.65244e7	0.891294
$977$	3.16213e7	1.05985	0.529924	−	0.848045i	$-0.322220\pi$
$977$	3.16213e7	1.05985	0.529924	+	0.848045i	$0.322220\pi$
$978$	1.47662e7	0.493651
$979$	1.25184e7	0.417438
$980$	0	0
$981$	983239.	0.0326202
$982$	−2.12321e7	−0.702611
$983$	−1.31368e7	−0.433617	−0.216808	−	0.976214i	$-0.569565\pi$
$983$	−1.31368e7	−0.433617	−0.216808	+	0.976214i	$0.569565\pi$
$984$	−6.44836e6	−0.212306
$985$	0	0
$986$	4.08750e7	1.33895
$987$	9.03355e6	0.295165
$988$	−1.06886e6	−0.0348359
$989$	4.37491e7	1.42226
$990$	0	0
$991$	−3.53335e7	−1.14288	−0.571442	−	0.820643i	$-0.693616\pi$
$991$	−3.53335e7	−1.14288	−0.571442	+	0.820643i	$0.693616\pi$
$992$	1.14521e7	0.369493
$993$	−8.62603e6	−0.277612
$994$	−5.05446e7	−1.62259
$995$	0	0
$996$	461020.	0.0147256
$997$	−3.72145e7	−1.18570	−0.592850	−	0.805313i	$-0.701997\pi$
$997$	−3.72145e7	−1.18570	−0.592850	+	0.805313i	$0.701997\pi$
$998$	3.33563e7	1.06011
$999$	−5.72915e6	−0.181625

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	825.6.a.h.1.1		3
5.4	even	2		165.6.a.d.1.3	✓	3
15.14	odd	2		495.6.a.c.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
165.6.a.d.1.3	✓	3	5.4	even	2
495.6.a.c.1.1		3	15.14	odd	2
825.6.a.h.1.1		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)

\(f(q)\)	\(=\)	\(q-6.55890 q^{2} +9.00000 q^{3} +11.0192 q^{4} -59.0301 q^{6} -146.487 q^{7} +137.611 q^{8} +81.0000 q^{9} +O(q^{10})\)	\(q-6.55890 q^{2} +9.00000 q^{3} +11.0192 q^{4} -59.0301 q^{6} -146.487 q^{7} +137.611 q^{8} +81.0000 q^{9} -121.000 q^{11} +99.1731 q^{12} -170.238 q^{13} +960.792 q^{14} -1255.19 q^{16} -1564.81 q^{17} -531.271 q^{18} +569.786 q^{19} -1318.38 q^{21} +793.627 q^{22} -3153.25 q^{23} +1238.50 q^{24} +1116.57 q^{26} +729.000 q^{27} -1614.17 q^{28} +3982.58 q^{29} +2990.78 q^{31} +3829.14 q^{32} -1089.00 q^{33} +10263.5 q^{34} +892.558 q^{36} -7858.92 q^{37} -3737.17 q^{38} -1532.14 q^{39} -5206.60 q^{41} +8647.13 q^{42} -13874.3 q^{43} -1333.33 q^{44} +20681.9 q^{46} -6852.01 q^{47} -11296.7 q^{48} +4651.35 q^{49} -14083.3 q^{51} -1875.89 q^{52} +3834.15 q^{53} -4781.44 q^{54} -20158.2 q^{56} +5128.08 q^{57} -26121.4 q^{58} +9649.38 q^{59} -21131.8 q^{61} -19616.2 q^{62} -11865.4 q^{63} +15051.2 q^{64} +7142.65 q^{66} +43499.9 q^{67} -17243.0 q^{68} -28379.2 q^{69} -52607.2 q^{71} +11146.5 q^{72} +64367.9 q^{73} +51545.9 q^{74} +6278.61 q^{76} +17724.9 q^{77} +10049.2 q^{78} +28935.7 q^{79} +6561.00 q^{81} +34149.6 q^{82} +4648.64 q^{83} -14527.5 q^{84} +91000.1 q^{86} +35843.2 q^{87} -16650.9 q^{88} -103458. q^{89} +24937.6 q^{91} -34746.4 q^{92} +26917.0 q^{93} +44941.6 q^{94} +34462.2 q^{96} -75809.5 q^{97} -30507.8 q^{98} -9801.00 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 2 q^{2} + 27 q^{3} - 20 q^{4} - 18 q^{6} - 152 q^{7} + 24 q^{8} + 243 q^{9}+O(q^{10}) \) 3 * q - 2 * q^2 + 27 * q^3 - 20 * q^4 - 18 * q^6 - 152 * q^7 + 24 * q^8 + 243 * q^9	\( 3 q - 2 q^{2} + 27 q^{3} - 20 q^{4} - 18 q^{6} - 152 q^{7} + 24 q^{8} + 243 q^{9} - 363 q^{11} - 180 q^{12} + 546 q^{13} - 8 q^{14} - 1360 q^{16} + 314 q^{17} - 162 q^{18} + 1808 q^{19} - 1368 q^{21} + 242 q^{22} - 4288 q^{23} + 216 q^{24} + 812 q^{26} + 2187 q^{27} - 5888 q^{28} + 5582 q^{29} + 6328 q^{31} + 736 q^{32} - 3267 q^{33} + 11596 q^{34} - 1620 q^{36} - 16866 q^{37} - 9584 q^{38} + 4914 q^{39} + 23282 q^{41} - 72 q^{42} - 20572 q^{43} + 2420 q^{44} + 16592 q^{46} - 3432 q^{47} - 12240 q^{48} + 11531 q^{49} + 2826 q^{51} - 21816 q^{52} - 16138 q^{53} - 1458 q^{54} + 15648 q^{56} + 16272 q^{57} - 17460 q^{58} + 21972 q^{59} + 8322 q^{61} - 5056 q^{62} - 12312 q^{63} + 22208 q^{64} + 2178 q^{66} + 84332 q^{67} - 59832 q^{68} - 38592 q^{69} + 50528 q^{71} + 1944 q^{72} + 53838 q^{73} + 79212 q^{74} - 52448 q^{76} + 18392 q^{77} + 7308 q^{78} + 6364 q^{79} + 19683 q^{81} + 68020 q^{82} - 96272 q^{83} - 52992 q^{84} + 143152 q^{86} + 50238 q^{87} - 2904 q^{88} - 38938 q^{89} + 104968 q^{91} - 24000 q^{92} + 56952 q^{93} + 49088 q^{94} + 6624 q^{96} + 103242 q^{97} - 9554 q^{98} - 29403 q^{99}+O(q^{100}) \) 3 * q - 2 * q^2 + 27 * q^3 - 20 * q^4 - 18 * q^6 - 152 * q^7 + 24 * q^8 + 243 * q^9 - 363 * q^11 - 180 * q^12 + 546 * q^13 - 8 * q^14 - 1360 * q^16 + 314 * q^17 - 162 * q^18 + 1808 * q^19 - 1368 * q^21 + 242 * q^22 - 4288 * q^23 + 216 * q^24 + 812 * q^26 + 2187 * q^27 - 5888 * q^28 + 5582 * q^29 + 6328 * q^31 + 736 * q^32 - 3267 * q^33 + 11596 * q^34 - 1620 * q^36 - 16866 * q^37 - 9584 * q^38 + 4914 * q^39 + 23282 * q^41 - 72 * q^42 - 20572 * q^43 + 2420 * q^44 + 16592 * q^46 - 3432 * q^47 - 12240 * q^48 + 11531 * q^49 + 2826 * q^51 - 21816 * q^52 - 16138 * q^53 - 1458 * q^54 + 15648 * q^56 + 16272 * q^57 - 17460 * q^58 + 21972 * q^59 + 8322 * q^61 - 5056 * q^62 - 12312 * q^63 + 22208 * q^64 + 2178 * q^66 + 84332 * q^67 - 59832 * q^68 - 38592 * q^69 + 50528 * q^71 + 1944 * q^72 + 53838 * q^73 + 79212 * q^74 - 52448 * q^76 + 18392 * q^77 + 7308 * q^78 + 6364 * q^79 + 19683 * q^81 + 68020 * q^82 - 96272 * q^83 - 52992 * q^84 + 143152 * q^86 + 50238 * q^87 - 2904 * q^88 - 38938 * q^89 + 104968 * q^91 - 24000 * q^92 + 56952 * q^93 + 49088 * q^94 + 6624 * q^96 + 103242 * q^97 - 9554 * q^98 - 29403 * q^99

Introduction

L-functions

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Database

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