Show commands: Magma / Pari/GP / SageMath

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 2.20
Character \(\chi\) \(=\) 75.2
Dual form 75.8.l.a.38.20

$q$-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+(-1.79599 + 11.3395i) q^{2} +(-28.9982 - 36.6893i) q^{3} +(-3.62242 - 1.17700i) q^{4} +(245.145 + 134.273i) q^{5} +(468.117 - 262.930i) q^{6} +(-1082.77 - 1082.77i) q^{7} +(-647.306 + 1270.41i) q^{8} +(-505.206 + 2127.85i) q^{9} +(-1962.86 + 2538.65i) q^{10} +(-1048.01 - 1442.46i) q^{11} +(61.8606 + 167.035i) q^{12} +(12341.2 - 1954.65i) q^{13} +(14222.7 - 10333.4i) q^{14} +(-2182.39 - 12887.8i) q^{15} +(-13637.6 - 9908.31i) q^{16} +(24795.9 + 12634.1i) q^{17} +(-23221.3 - 9550.37i) q^{18} +(-9446.90 + 3069.48i) q^{19} +(-729.978 - 774.926i) q^{20} +(-8327.68 + 71124.7i) q^{21} +(18239.0 - 9293.22i) q^{22} +(-27113.4 - 4294.35i) q^{23} +(65381.1 - 13090.4i) q^{24} +(42066.7 + 65832.4i) q^{25} +143453. i q^{26} +(92719.3 - 43168.1i) q^{27} +(2647.84 + 5196.68i) q^{28} +(-30422.4 + 93630.6i) q^{29} +(150061. - 1600.62i) q^{30} +(79487.8 + 244638. i) q^{31} +(7797.93 - 7797.93i) q^{32} +(-22532.5 + 80279.7i) q^{33} +(-187797. + 258481. i) q^{34} +(-120049. - 410823. i) q^{35} +(4334.54 - 7113.33i) q^{36} +(30730.2 + 194023. i) q^{37} +(-17839.7 - 112635. i) q^{38} +(-429586. - 396107. i) q^{39} +(-329265. + 224519. i) q^{40} +(-269017. + 370270. i) q^{41} +(-791559. - 222171. i) q^{42} +(354271. - 354271. i) q^{43} +(2098.56 + 6458.71i) q^{44} +(-409560. + 453795. i) q^{45} +(97391.1 - 299739. i) q^{46} +(572588. + 1.12377e6i) q^{47} +(31937.9 + 787678. i) q^{48} +1.52126e6i q^{49} +(-822055. + 358779. i) q^{50} +(-255499. - 1.27611e6i) q^{51} +(-47005.5 - 7444.94i) q^{52} +(1.24027e6 - 631950. i) q^{53} +(322980. + 1.12892e6i) q^{54} +(-63230.8 - 494331. i) q^{55} +(2.07645e6 - 674681. i) q^{56} +(386560. + 257590. i) q^{57} +(-1.00708e6 - 513134. i) q^{58} +(-654819. - 475754. i) q^{59} +(-7263.40 + 49253.8i) q^{60} +(-293166. + 212998. i) q^{61} +(-2.91682e6 + 461979. i) q^{62} +(2.85100e6 - 1.75695e6i) q^{63} +(-1.19384e6 - 1.64319e6i) q^{64} +(3.28782e6 + 1.17791e6i) q^{65} +(-869859. - 399688. i) q^{66} +(-1.05929e6 + 2.07898e6i) q^{67} +(-74950.8 - 74950.8i) q^{68} +(628685. + 1.11930e6i) q^{69} +(4.87412e6 - 623457. i) q^{70} +(5.36074e6 + 1.74181e6i) q^{71} +(-2.37622e6 - 2.01919e6i) q^{72} +(-82679.1 + 522015. i) q^{73} -2.25530e6 q^{74} +(1.19548e6 - 3.45242e6i) q^{75} +37833.4 q^{76} +(-427103. + 2.69662e6i) q^{77} +(5.26317e6 - 4.15987e6i) q^{78} +(-3.47536e6 - 1.12921e6i) q^{79} +(-2.01277e6 - 4.26013e6i) q^{80} +(-4.27250e6 - 2.15000e6i) q^{81} +(-3.71551e6 - 3.71551e6i) q^{82} +(-786549. + 1.54369e6i) q^{83} +(113880. - 247842. i) q^{84} +(4.38216e6 + 6.42660e6i) q^{85} +(3.38098e6 + 4.65351e6i) q^{86} +(4.31743e6 - 1.59894e6i) q^{87} +(2.51090e6 - 397688. i) q^{88} +(3.56902e6 - 2.59305e6i) q^{89} +(-4.41022e6 - 5.45920e6i) q^{90} +(-1.54791e7 - 1.12462e7i) q^{91} +(93161.8 + 47468.3i) q^{92} +(6.67059e6 - 1.00104e7i) q^{93} +(-1.37713e7 + 4.47456e6i) q^{94} +(-2.72800e6 - 515993. i) q^{95} +(-512227. - 59974.4i) q^{96} +(5.31864e6 - 2.70998e6i) q^{97} +(-1.72502e7 - 2.73217e6i) q^{98} +(3.59881e6 - 1.50127e6i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.79599 + 11.3395i −0.158745 + 1.00228i 0.771738 + 0.635941i \(0.219388\pi\)
−0.930483 + 0.366335i \(0.880612\pi\)
\(3\) −28.9982 − 36.6893i −0.620079 − 0.784539i
\(4\) −3.62242 − 1.17700i −0.0283002 − 0.00919528i
\(5\) 245.145 + 134.273i 0.877056 + 0.480388i
\(6\) 468.117 − 262.930i 0.884759 − 0.496948i
\(7\) −1082.77 − 1082.77i −1.19315 − 1.19315i −0.976178 − 0.216971i \(-0.930382\pi\)
−0.216971 − 0.976178i \(-0.569618\pi\)
\(8\) −647.306 + 1270.41i −0.446987 + 0.877261i
\(9\) −505.206 + 2127.85i −0.231004 + 0.972953i
\(10\) −1962.86 + 2538.65i −0.620710 + 0.802793i
\(11\) −1048.01 − 1442.46i −0.237406 − 0.326761i 0.673645 − 0.739055i \(-0.264728\pi\)
−0.911051 + 0.412294i \(0.864728\pi\)
\(12\) 61.8606 + 167.035i 0.0103343 + 0.0279044i
\(13\) 12341.2 − 1954.65i 1.55795 − 0.246755i 0.682801 − 0.730604i \(-0.260762\pi\)
0.875152 + 0.483849i \(0.160762\pi\)
\(14\) 14222.7 − 10333.4i 1.38527 − 1.00646i
\(15\) −2182.39 − 12887.8i −0.166960 − 0.985964i
\(16\) −13637.6 − 9908.31i −0.832374 − 0.604755i
\(17\) 24795.9 + 12634.1i 1.22408 + 0.623698i 0.941974 − 0.335687i \(-0.108968\pi\)
0.282102 + 0.959384i \(0.408968\pi\)
\(18\) −23221.3 − 9550.37i −0.938496 − 0.385981i
\(19\) −9446.90 + 3069.48i −0.315974 + 0.102666i −0.462711 − 0.886509i \(-0.653123\pi\)
0.146736 + 0.989176i \(0.453123\pi\)
\(20\) −729.978 − 774.926i −0.0204035 − 0.0216598i
\(21\) −8327.68 + 71124.7i −0.196226 + 1.67592i
\(22\) 18239.0 − 9293.22i 0.365192 − 0.186075i
\(23\) −27113.4 − 4294.35i −0.464662 − 0.0735952i −0.0802860 − 0.996772i \(-0.525583\pi\)
−0.384376 + 0.923177i \(0.625583\pi\)
\(24\) 65381.1 − 13090.4i 0.965413 − 0.193292i
\(25\) 42066.7 + 65832.4i 0.538454 + 0.842655i
\(26\) 143453.i 1.60067i
\(27\) 92719.3 − 43168.1i 0.906561 − 0.422075i
\(28\) 2647.84 + 5196.68i 0.0227950 + 0.0447376i
\(29\) −30422.4 + 93630.6i −0.231633 + 0.712894i 0.765917 + 0.642939i \(0.222285\pi\)
−0.997550 + 0.0699542i \(0.977715\pi\)
\(30\) 150061. − 1600.62i 1.01471 − 0.0108234i
\(31\) 79487.8 + 244638.i 0.479219 + 1.47489i 0.840182 + 0.542305i \(0.182448\pi\)
−0.360962 + 0.932580i \(0.617552\pi\)
\(32\) 7797.93 − 7797.93i 0.0420683 − 0.0420683i
\(33\) −22532.5 + 80279.7i −0.109147 + 0.388872i
\(34\) −187797. + 258481.i −0.819433 + 1.12785i
\(35\) −120049. − 410823.i −0.473283 − 1.61963i
\(36\) 4334.54 − 7113.33i 0.0154840 − 0.0254106i
\(37\) 30730.2 + 194023.i 0.0997375 + 0.629718i 0.986027 + 0.166586i \(0.0532744\pi\)
−0.886289 + 0.463132i \(0.846726\pi\)
\(38\) −17839.7 − 112635.i −0.0527406 − 0.332991i
\(39\) −429586. − 396107.i −1.15964 − 1.06927i
\(40\) −329265. + 224519.i −0.813459 + 0.554680i
\(41\) −269017. + 370270.i −0.609588 + 0.839026i −0.996544 − 0.0830722i \(-0.973527\pi\)
0.386955 + 0.922098i \(0.373527\pi\)
\(42\) −791559. − 222171.i −1.64858 − 0.462716i
\(43\) 354271. − 354271.i 0.679511 − 0.679511i −0.280379 − 0.959890i \(-0.590460\pi\)
0.959890 + 0.280379i \(0.0904600\pi\)
\(44\) 2098.56 + 6458.71i 0.00371396 + 0.0114304i
\(45\) −409560. + 453795.i −0.669999 + 0.742362i
\(46\) 97391.1 − 299739.i 0.147525 − 0.454037i
\(47\) 572588. + 1.12377e6i 0.804451 + 1.57882i 0.815409 + 0.578885i \(0.196512\pi\)
−0.0109578 + 0.999940i \(0.503488\pi\)
\(48\) 31937.9 + 787678.i 0.0416833 + 1.02803i
\(49\) 1.52126e6i 1.84721i
\(50\) −822055. + 358779.i −0.930049 + 0.405912i
\(51\) −255499. − 1.27611e6i −0.269708 − 1.34708i
\(52\) −47005.5 − 7444.94i −0.0463593 − 0.00734259i
\(53\) 1.24027e6 − 631950.i 1.14433 − 0.583065i 0.224148 − 0.974555i \(-0.428040\pi\)
0.920183 + 0.391490i \(0.128040\pi\)
\(54\) 322980. + 1.12892e6i 0.279124 + 0.975626i
\(55\) −63230.8 − 494331.i −0.0512459 − 0.400635i
\(56\) 2.07645e6 − 674681.i 1.58003 − 0.513381i
\(57\) 386560. + 257590.i 0.276475 + 0.184233i
\(58\) −1.00708e6 − 513134.i −0.677745 − 0.345329i
\(59\) −654819. − 475754.i −0.415087 − 0.301579i 0.360571 − 0.932732i \(-0.382582\pi\)
−0.775658 + 0.631153i \(0.782582\pi\)
\(60\) −7263.40 + 49253.8i −0.00434121 + 0.0294382i
\(61\) −293166. + 212998.i −0.165371 + 0.120149i −0.667393 − 0.744706i \(-0.732590\pi\)
0.502022 + 0.864855i \(0.332590\pi\)
\(62\) −2.91682e6 + 461979.i −1.55432 + 0.246179i
\(63\) 2.85100e6 − 1.75695e6i 1.43650 − 0.885255i
\(64\) −1.19384e6 − 1.64319e6i −0.569269 − 0.783532i
\(65\) 3.28782e6 + 1.17791e6i 1.48495 + 0.532004i
\(66\) −869859. − 399688.i −0.372431 − 0.171127i
\(67\) −1.05929e6 + 2.07898e6i −0.430283 + 0.844478i 0.569464 + 0.822016i \(0.307151\pi\)
−0.999748 + 0.0224624i \(0.992849\pi\)
\(68\) −74950.8 − 74950.8i −0.0289065 − 0.0289065i
\(69\) 628685. + 1.11930e6i 0.230389 + 0.410181i
\(70\) 4.87412e6 − 623457.i 1.69845 − 0.217252i
\(71\) 5.36074e6 + 1.74181e6i 1.77754 + 0.577559i 0.998762 − 0.0497352i \(-0.0158377\pi\)
0.778782 + 0.627294i \(0.215838\pi\)
\(72\) −2.37622e6 − 2.01919e6i −0.750278 − 0.637548i
\(73\) −82679.1 + 522015.i −0.0248751 + 0.157055i −0.996999 − 0.0774115i \(-0.975334\pi\)
0.972124 + 0.234467i \(0.0753345\pi\)
\(74\) −2.25530e6 −0.646984
\(75\) 1.19548e6 − 3.45242e6i 0.327212 − 0.944951i
\(76\) 37833.4 0.00988616
\(77\) −427103. + 2.69662e6i −0.106614 + 0.673136i
\(78\) 5.26317e6 − 4.15987e6i 1.25579 − 0.992541i
\(79\) −3.47536e6 − 1.12921e6i −0.793059 − 0.257680i −0.115653 − 0.993290i \(-0.536896\pi\)
−0.677406 + 0.735609i \(0.736896\pi\)
\(80\) −2.01277e6 − 4.26013e6i −0.439521 − 0.930267i
\(81\) −4.27250e6 − 2.15000e6i −0.893274 − 0.449513i
\(82\) −3.71551e6 − 3.71551e6i −0.744167 − 0.744167i
\(83\) −786549. + 1.54369e6i −0.150991 + 0.296337i −0.954096 − 0.299501i \(-0.903180\pi\)
0.803105 + 0.595838i \(0.203180\pi\)
\(84\) 113880. − 247842.i 0.0209638 − 0.0456244i
\(85\) 4.38216e6 + 6.42660e6i 0.773966 + 1.13505i
\(86\) 3.38098e6 + 4.65351e6i 0.573188 + 0.788926i
\(87\) 4.31743e6 − 1.59894e6i 0.702924 − 0.260325i
\(88\) 2.51090e6 − 397688.i 0.392772 − 0.0622090i
\(89\) 3.56902e6 − 2.59305e6i 0.536641 − 0.389893i −0.286195 − 0.958171i \(-0.592390\pi\)
0.822836 + 0.568279i \(0.192390\pi\)
\(90\) −4.41022e6 − 5.45920e6i −0.637693 − 0.789370i
\(91\) −1.54791e7 − 1.12462e7i −2.15329 − 1.56445i
\(92\) 93161.8 + 47468.3i 0.0124733 + 0.00635545i
\(93\) 6.67059e6 − 1.00104e7i 0.859952 − 1.29051i
\(94\) −1.37713e7 + 4.47456e6i −1.71012 + 0.555652i
\(95\) −2.72800e6 − 515993.i −0.326447 − 0.0617463i
\(96\) −512227. − 59974.4i −0.0590899 − 0.00691857i
\(97\) 5.31864e6 − 2.70998e6i 0.591698 − 0.301485i −0.132385 − 0.991198i \(-0.542263\pi\)
0.724082 + 0.689713i \(0.242263\pi\)
\(98\) −1.72502e7 − 2.73217e6i −1.85141 − 0.293235i
\(99\) 3.59881e6 − 1.50127e6i 0.372765 − 0.155502i
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.2.20 ✓ 544
3.2 odd 2 inner 75.8.l.a.2.49 yes 544
25.13 odd 20 inner 75.8.l.a.38.49 yes 544
75.38 even 20 inner 75.8.l.a.38.20 yes 544
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
75.8.l.a.2.20 ✓ 544 1.1 even 1 trivial
75.8.l.a.2.49 yes 544 3.2 odd 2 inner
75.8.l.a.38.20 yes 544 75.38 even 20 inner
75.8.l.a.38.49 yes 544 25.13 odd 20 inner