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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [75,8,Mod(2,75)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("75.2"); S:= CuspForms(chi, 8); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(75, base_ring=CyclotomicField(20)) chi = DirichletCharacter(H, H._module([10, 1])) N = Newforms(chi, 8, names="a")
 
Level: \( N \) \(=\) \( 75 = 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 75.l (of order \(20\), degree \(8\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(23.4288769113\)
Analytic rank: \(0\)
Dimension: \(544\)
Relative dimension: \(68\) over \(\Q(\zeta_{20})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{20}]$

Embedding invariants

Embedding label 2.18
Character \(\chi\) \(=\) 75.2
Dual form 75.8.l.a.38.18

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-1.88257 + 11.8861i) q^{2} +(33.2863 - 32.8484i) q^{3} +(-15.9995 - 5.19856i) q^{4} +(279.504 - 1.66925i) q^{5} +(327.775 + 457.484i) q^{6} +(759.321 + 759.321i) q^{7} +(-607.408 + 1192.11i) q^{8} +(28.9621 - 2186.81i) q^{9} +(-506.344 + 3325.34i) q^{10} +(-761.465 - 1048.07i) q^{11} +(-703.330 + 352.518i) q^{12} +(2204.77 - 349.201i) q^{13} +(-10454.8 + 7595.87i) q^{14} +(9248.82 - 9236.81i) q^{15} +(-14768.1 - 10729.6i) q^{16} +(14393.7 + 7333.97i) q^{17} +(25938.0 + 4461.06i) q^{18} +(5428.52 - 1763.83i) q^{19} +(-4480.60 - 1426.31i) q^{20} +(50217.5 + 332.526i) q^{21} +(13890.9 - 7077.78i) q^{22} +(31110.2 + 4927.38i) q^{23} +(18940.4 + 59633.3i) q^{24} +(78119.4 - 933.123i) q^{25} +26863.4i q^{26} +(-70869.2 - 73742.2i) q^{27} +(-8201.39 - 16096.1i) q^{28} +(-43411.9 + 133608. i) q^{29} +(92377.9 + 127321. i) q^{30} +(-79941.5 - 246035. i) q^{31} +(34239.2 - 34239.2i) q^{32} +(-59773.7 - 9873.39i) q^{33} +(-114269. + 157278. i) q^{34} +(213500. + 210965. i) q^{35} +(-11831.6 + 34837.3i) q^{36} +(23326.9 + 147280. i) q^{37} +(10745.5 + 67844.4i) q^{38} +(61917.9 - 84046.7i) q^{39} +(-167783. + 334212. i) q^{40} +(5909.80 - 8134.15i) q^{41} +(-98490.4 + 596263. i) q^{42} +(-255520. + 255520. i) q^{43} +(6734.64 + 20727.1i) q^{44} +(4444.69 - 611269. i) q^{45} +(-117134. + 360503. i) q^{46} +(578176. + 1.13473e6i) q^{47} +(-844026. + 127957. i) q^{48} +329593. i q^{49} +(-135974. + 930290. i) q^{50} +(720024. - 228690. i) q^{51} +(-37090.5 - 5874.56i) q^{52} +(-821185. + 418415. i) q^{53} +(1.00992e6 - 703531. i) q^{54} +(-214582. - 291667. i) q^{55} +(-1.36641e6 + 443973. i) q^{56} +(122757. - 237030. i) q^{57} +(-1.50635e6 - 767524. i) q^{58} +(1.67531e6 + 1.21718e6i) q^{59} +(-195995. + 99704.0i) q^{60} +(1.84948e6 - 1.34373e6i) q^{61} +(3.07488e6 - 487014. i) q^{62} +(1.68248e6 - 1.63850e6i) q^{63} +(-1.03088e6 - 1.41888e6i) q^{64} +(615657. - 101283. i) q^{65} +(229884. - 691888. i) q^{66} +(548804. - 1.07709e6i) q^{67} +(-192167. - 192167. i) q^{68} +(1.19740e6 - 857908. i) q^{69} +(-2.90948e6 + 2.14052e6i) q^{70} +(-4.91708e6 - 1.59766e6i) q^{71} +(2.58932e6 + 1.36281e6i) q^{72} +(790329. - 4.98994e6i) q^{73} -1.79450e6 q^{74} +(2.56966e6 - 2.59716e6i) q^{75} -96023.1 q^{76} +(217623. - 1.37402e6i) q^{77} +(882421. + 894185. i) q^{78} +(2.60425e6 + 846172. i) q^{79} +(-4.14563e6 - 2.97431e6i) q^{80} +(-4.78129e6 - 126669. i) q^{81} +(85557.5 + 85557.5i) q^{82} +(440823. - 865164. i) q^{83} +(-801727. - 266379. i) q^{84} +(4.03534e6 + 2.02584e6i) q^{85} +(-2.55609e6 - 3.51816e6i) q^{86} +(2.94379e6 + 5.87334e6i) q^{87} +(1.71193e6 - 271143. i) q^{88} +(-4.66162e6 + 3.38686e6i) q^{89} +(7.25722e6 + 1.20359e6i) q^{90} +(1.93928e6 + 1.40897e6i) q^{91} +(-472133. - 240564. i) q^{92} +(-1.07428e7 - 5.56364e6i) q^{93} +(-1.45760e7 + 4.73603e6i) q^{94} +(1.51435e6 - 502059. i) q^{95} +(14994.2 - 2.26440e6i) q^{96} +(-9.49913e6 + 4.84005e6i) q^{97} +(-3.91757e6 - 620482. i) q^{98} +(-2.31398e6 + 1.63482e6i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 544 q - 34 q^{3} - 20 q^{4} - 6 q^{6} - 1364 q^{7} - 10 q^{9} + 3160 q^{10} + 2018 q^{12} - 16868 q^{13} - 37570 q^{15} + 507892 q^{16} + 106550 q^{18} + 70800 q^{19} - 6 q^{21} - 195160 q^{22} - 444200 q^{25}+ \cdots - 22089384 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/75\mathbb{Z}\right)^\times\).

\(n\) \(26\) \(52\)
\(\chi(n)\) \(-1\) \(e\left(\frac{1}{20}\right)\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\). You can download additional coefficients here.



Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.88257 + 11.8861i −0.166397 + 1.05059i 0.753218 + 0.657771i \(0.228500\pi\)
−0.919615 + 0.392820i \(0.871500\pi\)
\(3\) 33.2863 − 32.8484i 0.711773 − 0.702409i
\(4\) −15.9995 − 5.19856i −0.124996 − 0.0406137i
\(5\) 279.504 − 1.66925i 0.999982 − 0.00597209i
\(6\) 327.775 + 457.484i 0.619507 + 0.864661i
\(7\) 759.321 + 759.321i 0.836724 + 0.836724i 0.988426 − 0.151702i \(-0.0484755\pi\)
−0.151702 + 0.988426i \(0.548475\pi\)
\(8\) −607.408 + 1192.11i −0.419436 + 0.823190i
\(9\) 28.9621 − 2186.81i 0.0132429 − 0.999912i
\(10\) −506.344 + 3325.34i −0.160120 + 1.05157i
\(11\) −761.465 − 1048.07i −0.172495 − 0.237419i 0.714013 − 0.700132i \(-0.246876\pi\)
−0.886508 + 0.462714i \(0.846876\pi\)
\(12\) −703.330 + 352.518i −0.117496 + 0.0588907i
\(13\) 2204.77 − 349.201i 0.278331 − 0.0440832i −0.0157086 − 0.999877i \(-0.505000\pi\)
0.294039 + 0.955793i \(0.405000\pi\)
\(14\) −10454.8 + 7595.87i −1.01828 + 0.739826i
\(15\) 9248.82 − 9236.81i 0.707566 − 0.706647i
\(16\) −14768.1 − 10729.6i −0.901371 − 0.654884i
\(17\) 14393.7 + 7333.97i 0.710562 + 0.362050i 0.771619 − 0.636085i \(-0.219447\pi\)
−0.0610570 + 0.998134i \(0.519447\pi\)
\(18\) 25938.0 + 4461.06i 1.04829 + 0.180295i
\(19\) 5428.52 − 1763.83i 0.181570 − 0.0589957i −0.216821 − 0.976211i \(-0.569569\pi\)
0.398391 + 0.917216i \(0.369569\pi\)
\(20\) −4480.60 − 1426.31i −0.125237 − 0.0398665i
\(21\) 50217.5 + 332.526i 1.18328 + 0.00783536i
\(22\) 13890.9 − 7077.78i 0.278132 − 0.141716i
\(23\) 31110.2 + 4927.38i 0.533158 + 0.0844439i 0.417208 − 0.908811i \(-0.363009\pi\)
0.115950 + 0.993255i \(0.463009\pi\)
\(24\) 18940.4 + 59633.3i 0.279672 + 0.880540i
\(25\) 78119.4 − 933.123i 0.999929 − 0.0119440i
\(26\) 26863.4i 0.299747i
\(27\) −70869.2 − 73742.2i −0.692922 − 0.721013i
\(28\) −8201.39 − 16096.1i −0.0706048 − 0.138570i
\(29\) −43411.9 + 133608.i −0.330534 + 1.01728i 0.638347 + 0.769749i \(0.279619\pi\)
−0.968880 + 0.247529i \(0.920381\pi\)
\(30\) 92377.9 + 127321.i 0.624660 + 0.860946i
\(31\) −79941.5 − 246035.i −0.481955 − 1.48331i −0.836343 − 0.548207i \(-0.815311\pi\)
0.354388 − 0.935099i \(-0.384689\pi\)
\(32\) 34239.2 − 34239.2i 0.184714 − 0.184714i
\(33\) −59773.7 − 9873.39i −0.289542 − 0.0478264i
\(34\) −114269. + 157278.i −0.498601 + 0.686266i
\(35\) 213500. + 210965.i 0.841706 + 0.831712i
\(36\) −11831.6 + 34837.3i −0.0422655 + 0.124447i
\(37\) 23326.9 + 147280.i 0.0757097 + 0.478012i 0.996189 + 0.0872210i \(0.0277986\pi\)
−0.920479 + 0.390791i \(0.872201\pi\)
\(38\) 10745.5 + 67844.4i 0.0317676 + 0.200572i
\(39\) 61917.9 − 84046.7i 0.167144 − 0.226879i
\(40\) −167783. + 334212.i −0.414512 + 0.825680i
\(41\) 5909.80 − 8134.15i 0.0133915 − 0.0184318i −0.802269 − 0.596963i \(-0.796374\pi\)
0.815660 + 0.578531i \(0.196374\pi\)
\(42\) −98490.4 + 596263.i −0.205126 + 1.24184i
\(43\) −255520. + 255520.i −0.490100 + 0.490100i −0.908338 − 0.418237i \(-0.862648\pi\)
0.418237 + 0.908338i \(0.362648\pi\)
\(44\) 6734.64 + 20727.1i 0.0119187 + 0.0366821i
\(45\) 4444.69 − 611269.i 0.00727106 − 0.999974i
\(46\) −117134. + 360503.i −0.177432 + 0.546080i
\(47\) 578176. + 1.13473e6i 0.812302 + 1.59423i 0.804269 + 0.594265i \(0.202557\pi\)
0.00803308 + 0.999968i \(0.497443\pi\)
\(48\) −844026. + 127957.i −1.10157 + 0.167002i
\(49\) 329593.i 0.400214i
\(50\) −135974. + 930290.i −0.153837 + 1.05250i
\(51\) 720024. − 228690.i 0.760066 − 0.241408i
\(52\) −37090.5 − 5874.56i −0.0365806 − 0.00579381i
\(53\) −821185. + 418415.i −0.757662 + 0.386048i −0.789744 − 0.613437i \(-0.789786\pi\)
0.0320814 + 0.999485i \(0.489786\pi\)
\(54\) 1.00992e6 − 703531.i 0.872790 − 0.608002i
\(55\) −214582. − 291667.i −0.173909 − 0.236384i
\(56\) −1.36641e6 + 443973.i −1.03973 + 0.337830i
\(57\) 122757. − 237030.i 0.0877976 − 0.169528i
\(58\) −1.50635e6 − 767524.i −1.01374 − 0.516528i
\(59\) 1.67531e6 + 1.21718e6i 1.06197 + 0.771566i 0.974452 − 0.224597i \(-0.0721066\pi\)
0.0875176 + 0.996163i \(0.472107\pi\)
\(60\) −195995. + 99704.0i −0.117143 + 0.0595913i
\(61\) 1.84948e6 − 1.34373e6i 1.04327 − 0.757979i 0.0723476 − 0.997379i \(-0.476951\pi\)
0.970921 + 0.239401i \(0.0769509\pi\)
\(62\) 3.07488e6 − 487014.i 1.63854 − 0.259520i
\(63\) 1.68248e6 − 1.63850e6i 0.847731 − 0.825570i
\(64\) −1.03088e6 − 1.41888e6i −0.491562 − 0.676576i
\(65\) 615657. − 101283.i 0.278062 − 0.0457447i
\(66\) 229884. − 691888.i 0.0984250 − 0.296232i
\(67\) 548804. − 1.07709e6i 0.222923 − 0.437512i −0.752273 − 0.658851i \(-0.771043\pi\)
0.975197 + 0.221339i \(0.0710428\pi\)
\(68\) −192167. − 192167.i −0.0741134 − 0.0741134i
\(69\) 1.19740e6 − 857908.i 0.438802 − 0.314390i
\(70\) −2.90948e6 + 2.14052e6i −1.01385 + 0.745894i
\(71\) −4.91708e6 − 1.59766e6i −1.63043 − 0.529760i −0.656062 − 0.754707i \(-0.727779\pi\)
−0.974371 + 0.224947i \(0.927779\pi\)
\(72\) 2.58932e6 + 1.36281e6i 0.817563 + 0.430301i
\(73\) 790329. − 4.98994e6i 0.237781 − 1.50129i −0.523027 − 0.852316i \(-0.675197\pi\)
0.760809 − 0.648976i \(-0.224803\pi\)
\(74\) −1.79450e6 −0.514793
\(75\) 2.56966e6 − 2.59716e6i 0.703333 − 0.710860i
\(76\) −96023.1 −0.0250916
\(77\) 217623. − 1.37402e6i 0.0543234 − 0.342984i
\(78\) 882421. + 894185.i 0.210545 + 0.213352i
\(79\) 2.60425e6 + 846172.i 0.594275 + 0.193092i 0.590686 − 0.806902i \(-0.298857\pi\)
0.00358972 + 0.999994i \(0.498857\pi\)
\(80\) −4.14563e6 − 2.97431e6i −0.905265 − 0.649489i
\(81\) −4.78129e6 − 126669.i −0.999649 − 0.0264834i
\(82\) 85557.5 + 85557.5i 0.0171360 + 0.0171360i
\(83\) 440823. − 865164.i 0.0846235 − 0.166083i −0.844817 − 0.535056i \(-0.820291\pi\)
0.929440 + 0.368973i \(0.120291\pi\)
\(84\) −801727. − 266379.i −0.147587 − 0.0490368i
\(85\) 4.03534e6 + 2.02584e6i 0.712712 + 0.357800i
\(86\) −2.55609e6 − 3.51816e6i −0.433344 − 0.596446i
\(87\) 2.94379e6 + 5.87334e6i 0.479281 + 0.956242i
\(88\) 1.71193e6 − 271143.i 0.267791 − 0.0424139i
\(89\) −4.66162e6 + 3.38686e6i −0.700925 + 0.509252i −0.880233 − 0.474541i \(-0.842614\pi\)
0.179308 + 0.983793i \(0.442614\pi\)
\(90\) 7.25722e6 + 1.20359e6i 1.04935 + 0.174032i
\(91\) 1.93928e6 + 1.40897e6i 0.269771 + 0.196000i
\(92\) −472133. − 240564.i −0.0632131 − 0.0322087i
\(93\) −1.07428e7 − 5.56364e6i −1.38493 − 0.717248i
\(94\) −1.45760e7 + 4.73603e6i −1.81005 + 0.588121i
\(95\) 1.51435e6 − 502059.i 0.181214 − 0.0600790i
\(96\) 14994.2 − 2.26440e6i 0.00172972 − 0.261219i
\(97\) −9.49913e6 + 4.84005e6i −1.05678 + 0.538454i −0.893934 − 0.448198i \(-0.852066\pi\)
−0.162841 + 0.986652i \(0.552066\pi\)
\(98\) −3.91757e6 − 620482.i −0.420461 − 0.0665945i
\(99\) −2.31398e6 + 1.63482e6i −0.239682 + 0.169335i
Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 75.8.l.a.2.18 ✓ 544
3.2 odd 2 inner 75.8.l.a.2.51 yes 544
25.13 odd 20 inner 75.8.l.a.38.51 yes 544
75.38 even 20 inner 75.8.l.a.38.18 yes 544
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
75.8.l.a.2.18 ✓ 544 1.1 even 1 trivial
75.8.l.a.2.51 yes 544 3.2 odd 2 inner
75.8.l.a.38.18 yes 544 75.38 even 20 inner
75.8.l.a.38.51 yes 544 25.13 odd 20 inner