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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 2.16
Character \(\chi\) \(=\) 75.2
Dual form 75.8.l.a.38.16

$q$-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+(-2.19123 + 13.8349i) q^{2} +(26.4684 + 38.5542i) q^{3} +(-64.8671 - 21.0766i) q^{4} +(-278.769 - 20.3187i) q^{5} +(-591.390 + 281.706i) q^{6} +(-491.802 - 491.802i) q^{7} +(-380.246 + 746.275i) q^{8} +(-785.845 + 2040.94i) q^{9} +(891.953 - 3812.21i) q^{10} +(-1366.95 - 1881.45i) q^{11} +(-904.340 - 3058.76i) q^{12} +(6221.00 - 985.310i) q^{13} +(7881.67 - 5726.37i) q^{14} +(-6595.21 - 11285.5i) q^{15} +(-16554.4 - 12027.4i) q^{16} +(-6093.21 - 3104.64i) q^{17} +(-26514.1 - 15344.2i) q^{18} +(34235.0 - 11123.6i) q^{19} +(17654.7 + 7193.52i) q^{20} +(5943.78 - 31978.2i) q^{21} +(29024.9 - 14788.9i) q^{22} +(-95910.7 - 15190.8i) q^{23} +(-38836.5 + 5092.66i) q^{24} +(77299.3 + 11328.4i) q^{25} +88225.9i q^{26} +(-99486.6 + 23722.8i) q^{27} +(21536.3 + 42267.3i) q^{28} +(11761.3 - 36197.5i) q^{29} +(170585. - 66514.8i) q^{30} +(-10401.3 - 32011.9i) q^{31} +(126865. - 126865. i) q^{32} +(36356.5 - 102501. i) q^{33} +(56304.0 - 77495.8i) q^{34} +(127106. + 147092. i) q^{35} +(93991.5 - 115827. i) q^{36} +(-5795.60 - 36592.0i) q^{37} +(78877.3 + 498012. i) q^{38} +(202648. + 213766. i) q^{39} +(121164. - 200312. i) q^{40} +(143886. - 198042. i) q^{41} +(429390. + 152303. i) q^{42} +(58329.3 - 58329.3i) q^{43} +(49015.7 + 150855. i) q^{44} +(260538. - 552982. i) q^{45} +(420325. - 1.29363e6i) q^{46} +(294037. + 577081. i) q^{47} +(25540.2 - 956587. i) q^{48} -339805. i q^{49} +(-326108. + 1.04460e6i) q^{50} +(-41580.7 - 317093. i) q^{51} +(-424306. - 67203.4i) q^{52} +(-1.16522e6 + 593712. i) q^{53} +(-110203. - 1.42837e6i) q^{54} +(342835. + 552263. i) q^{55} +(554025. - 180014. i) q^{56} +(1.33501e6 + 1.02548e6i) q^{57} +(475016. + 242033. i) q^{58} +(-770744. - 559978. i) q^{59} +(189952. + 871063. i) q^{60} +(-904940. + 657477. i) q^{61} +(465673. - 73755.4i) q^{62} +(1.39022e6 - 617255. i) q^{63} +(-62340.9 - 85804.9i) q^{64} +(-1.75424e6 + 148271. i) q^{65} +(1.33842e6 + 727590. i) q^{66} +(-1.85156e6 + 3.63390e6i) q^{67} +(329813. + 329813. i) q^{68} +(-1.95294e6 - 4.09983e6i) q^{69} +(-2.31352e6 + 1.43619e6i) q^{70} +(-3.70056e6 - 1.20239e6i) q^{71} +(-1.22428e6 - 1.36251e6i) q^{72} +(655500. - 4.13867e6i) q^{73} +518945. q^{74} +(1.60923e6 + 3.28005e6i) q^{75} -2.45518e6 q^{76} +(-253030. + 1.59757e6i) q^{77} +(-3.40147e6 + 2.33520e6i) q^{78} +(-6.91779e6 - 2.24773e6i) q^{79} +(4.37046e6 + 3.68924e6i) q^{80} +(-3.54786e6 - 3.20772e6i) q^{81} +(2.42460e6 + 2.42460e6i) q^{82} +(1.45993e6 - 2.86527e6i) q^{83} +(-1.05955e6 + 1.94906e6i) q^{84} +(1.63551e6 + 989284. i) q^{85} +(679166. + 934792. i) q^{86} +(1.70687e6 - 504645. i) q^{87} +(1.92385e6 - 304708. i) q^{88} +(2.78394e6 - 2.02265e6i) q^{89} +(7.07954e6 + 4.81623e6i) q^{90} +(-3.54408e6 - 2.57492e6i) q^{91} +(5.90128e6 + 3.00685e6i) q^{92} +(958887. - 1.24832e6i) q^{93} +(-8.62814e6 + 2.80345e6i) q^{94} +(-9.76967e6 + 2.40531e6i) q^{95} +(8.24909e6 + 1.53326e6i) q^{96} +(-1.42528e7 + 7.26217e6i) q^{97} +(4.70116e6 + 744591. i) q^{98} +(4.91412e6 - 1.31133e6i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.19123 + 13.8349i −0.193679 + 1.22284i 0.678849 + 0.734278i \(0.262479\pi\)
−0.872528 + 0.488564i \(0.837521\pi\)
\(3\) 26.4684 + 38.5542i 0.565983 + 0.824417i
\(4\) −64.8671 − 21.0766i −0.506775 − 0.164661i
\(5\) −278.769 − 20.3187i −0.997354 − 0.0726943i
\(6\) −591.390 + 281.706i −1.11775 + 0.532436i
\(7\) −491.802 − 491.802i −0.541935 − 0.541935i 0.382161 − 0.924096i \(-0.375180\pi\)
−0.924096 + 0.382161i \(0.875180\pi\)
\(8\) −380.246 + 746.275i −0.262573 + 0.515328i
\(9\) −785.845 + 2040.94i −0.359326 + 0.933212i
\(10\) 891.953 − 3812.21i 0.282060 − 1.20553i
\(11\) −1366.95 − 1881.45i −0.309655 − 0.426204i 0.625619 − 0.780129i \(-0.284847\pi\)
−0.935274 + 0.353925i \(0.884847\pi\)
\(12\) −904.340 − 3058.76i −0.151077 − 0.510989i
\(13\) 6221.00 − 985.310i 0.785342 − 0.124386i 0.249131 − 0.968470i \(-0.419855\pi\)
0.536211 + 0.844084i \(0.319855\pi\)
\(14\) 7881.67 − 5726.37i 0.767662 − 0.557739i
\(15\) −6595.21 − 11285.5i −0.504556 − 0.863379i
\(16\) −16554.4 − 12027.4i −1.01040 − 0.734097i
\(17\) −6093.21 − 3104.64i −0.300798 − 0.153264i 0.297077 − 0.954853i \(-0.403988\pi\)
−0.597875 + 0.801589i \(0.703988\pi\)
\(18\) −26514.1 − 15344.2i −1.07158 − 0.620142i
\(19\) 34235.0 − 11123.6i 1.14507 − 0.372056i 0.325787 − 0.945443i \(-0.394371\pi\)
0.819285 + 0.573387i \(0.194371\pi\)
\(20\) 17654.7 + 7193.52i 0.493464 + 0.201065i
\(21\) 5943.78 − 31978.2i 0.140054 − 0.753506i
\(22\) 29024.9 − 14788.9i 0.581154 − 0.296113i
\(23\) −95910.7 − 15190.8i −1.64369 − 0.260335i −0.735079 − 0.677981i \(-0.762855\pi\)
−0.908610 + 0.417647i \(0.862855\pi\)
\(24\) −38836.5 + 5092.66i −0.573457 + 0.0751978i
\(25\) 77299.3 + 11328.4i 0.989431 + 0.145004i
\(26\) 88225.9i 0.984440i
\(27\) −99486.6 + 23722.8i −0.972728 + 0.231949i
\(28\) 21536.3 + 42267.3i 0.185403 + 0.363874i
\(29\) 11761.3 − 36197.5i 0.0895492 − 0.275604i −0.896246 − 0.443558i \(-0.853716\pi\)
0.985795 + 0.167954i \(0.0537160\pi\)
\(30\) 170585. − 66514.8i 1.15350 − 0.449773i
\(31\) −10401.3 − 32011.9i −0.0627079 − 0.192995i 0.914794 − 0.403920i \(-0.132353\pi\)
−0.977502 + 0.210925i \(0.932353\pi\)
\(32\) 126865. − 126865.i 0.684411 − 0.684411i
\(33\) 36356.5 − 102501.i 0.176110 − 0.496509i
\(34\) 56304.0 − 77495.8i 0.245676 − 0.338144i
\(35\) 127106. + 147092.i 0.501105 + 0.579896i
\(36\) 93991.5 − 115827.i 0.335761 − 0.413761i
\(37\) −5795.60 − 36592.0i −0.0188101 − 0.118763i 0.976497 − 0.215533i \(-0.0691488\pi\)
−0.995307 + 0.0967702i \(0.969149\pi\)
\(38\) 78877.3 + 498012.i 0.233190 + 1.47230i
\(39\) 202648. + 213766.i 0.547036 + 0.577049i
\(40\) 121164. − 200312.i 0.299339 − 0.494877i
\(41\) 143886. − 198042.i 0.326044 − 0.448760i −0.614257 − 0.789106i \(-0.710544\pi\)
0.940300 + 0.340346i \(0.110544\pi\)
\(42\) 429390. + 152303.i 0.894293 + 0.317202i
\(43\) 58329.3 − 58329.3i 0.111879 − 0.111879i −0.648951 − 0.760830i \(-0.724792\pi\)
0.760830 + 0.648951i \(0.224792\pi\)
\(44\) 49015.7 + 150855.i 0.0867462 + 0.266977i
\(45\) 260538. − 552982.i 0.426214 − 0.904622i
\(46\) 420325. − 1.29363e6i 0.636696 − 1.95955i
\(47\) 294037. + 577081.i 0.413104 + 0.810763i 0.999999 + 0.00110003i \(0.000350150\pi\)
−0.586895 + 0.809663i \(0.699650\pi\)
\(48\) 25540.2 − 956587.i 0.0333334 − 1.24848i
\(49\) − 339805.i − 0.412614i
\(50\) −326108. + 1.04460e6i −0.368949 + 1.18183i
\(51\) −41580.7 − 317093.i −0.0438931 − 0.334728i
\(52\) −424306. − 67203.4i −0.418473 − 0.0662796i
\(53\) −1.16522e6 + 593712.i −1.07509 + 0.547785i −0.899608 − 0.436698i \(-0.856148\pi\)
−0.175480 + 0.984483i \(0.556148\pi\)
\(54\) −110203. − 1.42837e6i −0.0952395 − 1.23442i
\(55\) 342835. + 552263.i 0.277853 + 0.447586i
\(56\) 554025. − 180014.i 0.421571 − 0.136977i
\(57\) 1.33501e6 + 1.02548e6i 0.954821 + 0.733439i
\(58\) 475016. + 242033.i 0.319676 + 0.162883i
\(59\) −770744. − 559978.i −0.488572 − 0.354968i 0.316063 − 0.948738i \(-0.397639\pi\)
−0.804635 + 0.593770i \(0.797639\pi\)
\(60\) 189952. + 871063.i 0.113531 + 0.520619i
\(61\) −904940. + 657477.i −0.510464 + 0.370874i −0.813000 − 0.582264i \(-0.802167\pi\)
0.302536 + 0.953138i \(0.402167\pi\)
\(62\) 465673. − 73755.4i 0.248148 − 0.0393027i
\(63\) 1.39022e6 − 617255.i 0.700471 − 0.311009i
\(64\) −62340.9 − 85804.9i −0.0297265 − 0.0409150i
\(65\) −1.75424e6 + 148271.i −0.792306 + 0.0669670i
\(66\) 1.33842e6 + 727590.i 0.573043 + 0.311518i
\(67\) −1.85156e6 + 3.63390e6i −0.752102 + 1.47608i 0.123135 + 0.992390i \(0.460705\pi\)
−0.875237 + 0.483694i \(0.839295\pi\)
\(68\) 329813. + 329813.i 0.127200 + 0.127200i
\(69\) −1.95294e6 − 4.09983e6i −0.715676 − 1.50243i
\(70\) −2.31352e6 + 1.43619e6i −0.806175 + 0.500459i
\(71\) −3.70056e6 − 1.20239e6i −1.22705 − 0.398694i −0.377407 − 0.926047i \(-0.623184\pi\)
−0.849646 + 0.527354i \(0.823184\pi\)
\(72\) −1.22428e6 − 1.36251e6i −0.386561 − 0.430207i
\(73\) 655500. − 4.13867e6i 0.197216 − 1.24518i −0.668147 − 0.744030i \(-0.732912\pi\)
0.865363 − 0.501146i \(-0.167088\pi\)
\(74\) 518945. 0.148871
\(75\) 1.60923e6 + 3.28005e6i 0.440458 + 0.897773i
\(76\) −2.45518e6 −0.641556
\(77\) −253030. + 1.59757e6i −0.0631617 + 0.398787i
\(78\) −3.40147e6 + 2.33520e6i −0.811589 + 0.557177i
\(79\) −6.91779e6 − 2.24773e6i −1.57860 − 0.512919i −0.616906 − 0.787037i \(-0.711614\pi\)
−0.961696 + 0.274118i \(0.911614\pi\)
\(80\) 4.37046e6 + 3.68924e6i 0.954360 + 0.805605i
\(81\) −3.54786e6 − 3.20772e6i −0.741770 − 0.670654i
\(82\) 2.42460e6 + 2.42460e6i 0.485615 + 0.485615i
\(83\) 1.45993e6 − 2.86527e6i 0.280258 − 0.550037i −0.707372 − 0.706841i \(-0.750119\pi\)
0.987630 + 0.156805i \(0.0501194\pi\)
\(84\) −1.05955e6 + 1.94906e6i −0.195049 + 0.358796i
\(85\) 1.63551e6 + 989284.i 0.288861 + 0.174725i
\(86\) 679166. + 934792.i 0.115141 + 0.158478i
\(87\) 1.70687e6 − 504645.i 0.277896 − 0.0821615i
\(88\) 1.92385e6 − 304708.i 0.300942 − 0.0476645i
\(89\) 2.78394e6 − 2.02265e6i 0.418596 − 0.304128i −0.358477 − 0.933539i \(-0.616704\pi\)
0.777073 + 0.629411i \(0.216704\pi\)
\(90\) 7.07954e6 + 4.81623e6i 1.02366 + 0.696399i
\(91\) −3.54408e6 − 2.57492e6i −0.493013 − 0.358195i
\(92\) 5.90128e6 + 3.00685e6i 0.790112 + 0.402582i
\(93\) 958887. − 1.24832e6i 0.123617 − 0.160929i
\(94\) −8.62814e6 + 2.80345e6i −1.07144 + 0.348134i
\(95\) −9.76967e6 + 2.40531e6i −1.16909 + 0.287832i
\(96\) 8.24909e6 + 1.53326e6i 0.951605 + 0.176875i
\(97\) −1.42528e7 + 7.26217e6i −1.58562 + 0.807914i −0.999996 − 0.00266796i \(-0.999151\pi\)
−0.585625 + 0.810582i \(0.699151\pi\)
\(98\) 4.70116e6 + 744591.i 0.504562 + 0.0799147i
\(99\) 4.91412e6 − 1.31133e6i 0.509006 − 0.135828i
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.2.16 ✓ 544
3.2 odd 2 inner 75.8.l.a.2.53 yes 544
25.13 odd 20 inner 75.8.l.a.38.53 yes 544
75.38 even 20 inner 75.8.l.a.38.16 yes 544
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
75.8.l.a.2.16 ✓ 544 1.1 even 1 trivial
75.8.l.a.2.53 yes 544 3.2 odd 2 inner
75.8.l.a.38.16 yes 544 75.38 even 20 inner
75.8.l.a.38.53 yes 544 25.13 odd 20 inner