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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [75,6,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, 6); 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, 6, names="a")
 
Level: \( N \) \(=\) \( 75 = 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 6 \)
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: \(12.0287864860\)
Analytic rank: \(0\)
Dimension: \(384\)
Relative dimension: \(48\) over \(\Q(\zeta_{20})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{20}]$

Embedding invariants

Embedding label 2.9
Character \(\chi\) \(=\) 75.2
Dual form 75.6.l.a.38.9

$q$-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+(-1.22902 + 7.75970i) q^{2} +(5.37355 - 14.6330i) q^{3} +(-28.2686 - 9.18502i) q^{4} +(55.1156 - 9.34170i) q^{5} +(106.944 + 59.6813i) q^{6} +(-124.351 - 124.351i) q^{7} +(-8.12019 + 15.9368i) q^{8} +(-185.250 - 157.262i) q^{9} +(4.75079 + 439.162i) q^{10} +(-256.008 - 352.365i) q^{11} +(-286.307 + 364.298i) q^{12} +(-165.585 + 26.2260i) q^{13} +(1117.76 - 812.097i) q^{14} +(159.469 - 856.706i) q^{15} +(-883.179 - 641.667i) q^{16} +(-1719.24 - 875.996i) q^{17} +(1447.98 - 1244.21i) q^{18} +(173.199 - 56.2757i) q^{19} +(-1643.85 - 242.162i) q^{20} +(-2487.84 + 1151.42i) q^{21} +(3048.88 - 1553.48i) q^{22} +(4456.21 + 705.795i) q^{23} +(189.569 + 204.460i) q^{24} +(2950.47 - 1029.75i) q^{25} -1317.12i q^{26} +(-3296.67 + 1865.71i) q^{27} +(2373.06 + 4657.40i) q^{28} +(508.174 - 1564.00i) q^{29} +(6451.78 + 2290.34i) q^{30} +(1667.36 + 5131.61i) q^{31} +(5659.86 - 5659.86i) q^{32} +(-6531.82 + 1852.72i) q^{33} +(8910.44 - 12264.2i) q^{34} +(-8015.33 - 5692.03i) q^{35} +(3792.30 + 6147.11i) q^{36} +(-1029.69 - 6501.19i) q^{37} +(223.818 + 1413.13i) q^{38} +(-506.011 + 2563.93i) q^{39} +(-298.673 + 954.221i) q^{40} +(10906.8 - 15012.0i) q^{41} +(-5877.11 - 20720.0i) q^{42} +(-11870.1 + 11870.1i) q^{43} +(4000.50 + 12312.3i) q^{44} +(-11679.3 - 6937.07i) q^{45} +(-10953.5 + 33711.4i) q^{46} +(-6225.68 - 12218.6i) q^{47} +(-14135.3 + 9475.54i) q^{48} +14119.3i q^{49} +(4364.36 + 24160.3i) q^{50} +(-22056.9 + 20450.4i) q^{51} +(4921.73 + 779.525i) q^{52} +(-3786.77 + 1929.45i) q^{53} +(-10425.7 - 27874.2i) q^{54} +(-17401.7 - 17029.2i) q^{55} +(2991.51 - 971.999i) q^{56} +(107.209 - 2836.82i) q^{57} +(11511.6 + 5865.46i) q^{58} +(-2325.68 - 1689.70i) q^{59} +(-12376.8 + 22753.1i) q^{60} +(18073.0 - 13130.8i) q^{61} +(-41869.0 + 6631.39i) q^{62} +(3480.28 + 42591.8i) q^{63} +(16429.4 + 22613.1i) q^{64} +(-8881.30 + 2992.30i) q^{65} +(-6348.81 - 52962.0i) q^{66} +(-10880.4 + 21354.1i) q^{67} +(40554.4 + 40554.4i) q^{68} +(34273.6 - 61415.2i) q^{69} +(54019.4 - 55201.0i) q^{70} +(72088.3 + 23422.9i) q^{71} +(4010.52 - 1675.28i) q^{72} +(-2104.59 + 13287.9i) q^{73} +51712.8 q^{74} +(786.172 - 48707.6i) q^{75} -5412.98 q^{76} +(-11982.1 + 75651.7i) q^{77} +(-19273.4 - 7077.60i) q^{78} +(-21949.1 - 7131.71i) q^{79} +(-54671.2 - 27115.5i) q^{80} +(9586.08 + 58265.7i) q^{81} +(103084. + 103084. i) q^{82} +(-32427.0 + 63641.5i) q^{83} +(80903.5 - 9698.29i) q^{84} +(-102940. - 32220.5i) q^{85} +(-77520.0 - 106697. i) q^{86} +(-20155.3 - 15840.3i) q^{87} +(7694.38 - 1218.67i) q^{88} +(-2658.01 + 1931.16i) q^{89} +(68183.5 - 82101.8i) q^{90} +(23851.8 + 17329.4i) q^{91} +(-119488. - 60882.3i) q^{92} +(84050.6 + 3176.45i) q^{93} +(102464. - 33292.6i) q^{94} +(9020.25 - 4719.64i) q^{95} +(-52407.3 - 113234. i) q^{96} +(-26729.8 + 13619.5i) q^{97} +(-109562. - 17352.9i) q^{98} +(-7988.25 + 105536. i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 384 q - 10 q^{3} - 20 q^{4} - 6 q^{6} + 60 q^{7} - 10 q^{9} - 1080 q^{10} - 430 q^{12} + 2100 q^{13} + 2990 q^{15} + 21492 q^{16} - 6010 q^{18} - 8800 q^{19} - 6 q^{21} + 6040 q^{22} + 23880 q^{25} + 12950 q^{27}+ \cdots + 715720 q^{97}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.22902 + 7.75970i −0.217261 + 1.37173i 0.602084 + 0.798433i \(0.294337\pi\)
−0.819345 + 0.573301i \(0.805663\pi\)
\(3\) 5.37355 − 14.6330i 0.344713 − 0.938708i
\(4\) −28.2686 − 9.18502i −0.883394 − 0.287032i
\(5\) 55.1156 − 9.34170i 0.985938 − 0.167109i
\(6\) 106.944 + 59.6813i 1.21276 + 0.676800i
\(7\) −124.351 − 124.351i −0.959189 − 0.959189i 0.0400102 − 0.999199i \(-0.487261\pi\)
−0.999199 + 0.0400102i \(0.987261\pi\)
\(8\) −8.12019 + 15.9368i −0.0448581 + 0.0880390i
\(9\) −185.250 − 157.262i −0.762345 − 0.647170i
\(10\) 4.75079 + 439.162i 0.0150233 + 1.38875i
\(11\) −256.008 − 352.365i −0.637928 − 0.878032i 0.360575 − 0.932730i \(-0.382580\pi\)
−0.998503 + 0.0546979i \(0.982580\pi\)
\(12\) −286.307 + 364.298i −0.573957 + 0.730305i
\(13\) −165.585 + 26.2260i −0.271745 + 0.0430402i −0.290820 − 0.956778i \(-0.593928\pi\)
0.0190747 + 0.999818i \(0.493928\pi\)
\(14\) 1117.76 − 812.097i 1.52415 − 1.10736i
\(15\) 159.469 − 856.706i 0.182999 − 0.983113i
\(16\) −883.179 − 641.667i −0.862480 − 0.626628i
\(17\) −1719.24 − 875.996i −1.44283 − 0.735157i −0.454967 − 0.890508i \(-0.650349\pi\)
−0.987859 + 0.155351i \(0.950349\pi\)
\(18\) 1447.98 − 1244.21i 1.05337 − 0.905130i
\(19\) 173.199 − 56.2757i 0.110068 − 0.0357632i −0.253465 − 0.967345i \(-0.581570\pi\)
0.363533 + 0.931581i \(0.381570\pi\)
\(20\) −1643.85 − 242.162i −0.918937 − 0.135373i
\(21\) −2487.84 + 1151.42i −1.23104 + 0.569753i
\(22\) 3048.88 − 1553.48i 1.34302 − 0.684304i
\(23\) 4456.21 + 705.795i 1.75649 + 0.278201i 0.949820 − 0.312797i \(-0.101266\pi\)
0.806673 + 0.590998i \(0.201266\pi\)
\(24\) 189.569 + 204.460i 0.0671797 + 0.0724569i
\(25\) 2950.47 − 1029.75i 0.944149 − 0.329519i
\(26\) − 1317.12i − 0.382113i
\(27\) −3296.67 + 1865.71i −0.870295 + 0.492532i
\(28\) 2373.06 + 4657.40i 0.572024 + 1.12266i
\(29\) 508.174 − 1564.00i 0.112206 − 0.345336i −0.879148 − 0.476549i \(-0.841887\pi\)
0.991354 + 0.131214i \(0.0418874\pi\)
\(30\) 6451.78 + 2290.34i 1.30881 + 0.464618i
\(31\) 1667.36 + 5131.61i 0.311620 + 0.959068i 0.977123 + 0.212673i \(0.0682170\pi\)
−0.665503 + 0.746395i \(0.731783\pi\)
\(32\) 5659.86 − 5659.86i 0.977082 − 0.977082i
\(33\) −6531.82 + 1852.72i −1.04412 + 0.296158i
\(34\) 8910.44 − 12264.2i 1.32191 − 1.81945i
\(35\) −8015.33 − 5692.03i −1.10599 − 0.785412i
\(36\) 3792.30 + 6147.11i 0.487693 + 0.790524i
\(37\) −1029.69 − 6501.19i −0.123652 − 0.780708i −0.969103 − 0.246656i \(-0.920668\pi\)
0.845451 − 0.534053i \(-0.179332\pi\)
\(38\) 223.818 + 1413.13i 0.0251441 + 0.158754i
\(39\) −506.011 + 2563.93i −0.0532719 + 0.269926i
\(40\) −298.673 + 954.221i −0.0295152 + 0.0942972i
\(41\) 10906.8 − 15012.0i 1.01330 − 1.39469i 0.0965099 − 0.995332i \(-0.469232\pi\)
0.916794 − 0.399361i \(-0.130768\pi\)
\(42\) −5877.11 − 20720.0i −0.514091 − 1.81245i
\(43\) −11870.1 + 11870.1i −0.979004 + 0.979004i −0.999784 − 0.0207804i \(-0.993385\pi\)
0.0207804 + 0.999784i \(0.493385\pi\)
\(44\) 4000.50 + 12312.3i 0.311518 + 0.958754i
\(45\) −11679.3 − 6937.07i −0.859774 − 0.510675i
\(46\) −10953.5 + 33711.4i −0.763236 + 2.34900i
\(47\) −6225.68 − 12218.6i −0.411095 − 0.806820i 0.588904 − 0.808203i \(-0.299560\pi\)
−0.999999 + 0.00138350i \(0.999560\pi\)
\(48\) −14135.3 + 9475.54i −0.885529 + 0.593609i
\(49\) 14119.3i 0.840087i
\(50\) 4364.36 + 24160.3i 0.246885 + 1.36671i
\(51\) −22056.9 + 20450.4i −1.18746 + 1.10097i
\(52\) 4921.73 + 779.525i 0.252412 + 0.0399781i
\(53\) −3786.77 + 1929.45i −0.185173 + 0.0943506i −0.544120 − 0.839008i \(-0.683136\pi\)
0.358946 + 0.933358i \(0.383136\pi\)
\(54\) −10425.7 − 27874.2i −0.486541 − 1.30082i
\(55\) −17401.7 − 17029.2i −0.775685 − 0.759082i
\(56\) 2991.51 − 971.999i 0.127473 − 0.0414186i
\(57\) 107.209 − 2836.82i 0.00437065 − 0.115650i
\(58\) 11511.6 + 5865.46i 0.449331 + 0.228945i
\(59\) −2325.68 − 1689.70i −0.0869799 − 0.0631946i 0.543445 − 0.839445i \(-0.317119\pi\)
−0.630425 + 0.776250i \(0.717119\pi\)
\(60\) −12376.8 + 22753.1i −0.443845 + 0.815949i
\(61\) 18073.0 − 13130.8i 0.621880 − 0.451822i −0.231698 − 0.972788i \(-0.574428\pi\)
0.853578 + 0.520966i \(0.174428\pi\)
\(62\) −41869.0 + 6631.39i −1.38329 + 0.219091i
\(63\) 3480.28 + 42591.8i 0.110475 + 1.35199i
\(64\) 16429.4 + 22613.1i 0.501386 + 0.690098i
\(65\) −8881.30 + 2992.30i −0.260731 + 0.0878461i
\(66\) −6348.81 − 52962.0i −0.179404 − 1.49660i
\(67\) −10880.4 + 21354.1i −0.296114 + 0.581157i −0.990350 − 0.138591i \(-0.955743\pi\)
0.694235 + 0.719748i \(0.255743\pi\)
\(68\) 40554.4 + 40554.4i 1.06357 + 1.06357i
\(69\) 34273.6 − 61415.2i 0.866636 − 1.55293i
\(70\) 54019.4 − 55201.0i 1.31766 − 1.34648i
\(71\) 72088.3 + 23422.9i 1.69715 + 0.551436i 0.988112 − 0.153739i \(-0.0491314\pi\)
0.709034 + 0.705175i \(0.249131\pi\)
\(72\) 4010.52 − 1675.28i 0.0911736 − 0.0380853i
\(73\) −2104.59 + 13287.9i −0.0462233 + 0.291843i −0.999961 − 0.00879646i \(-0.997200\pi\)
0.953738 + 0.300639i \(0.0972000\pi\)
\(74\) 51712.8 1.09779
\(75\) 786.172 − 48707.6i 0.0161385 − 0.999870i
\(76\) −5412.98 −0.107499
\(77\) −11982.1 + 75651.7i −0.230306 + 1.45409i
\(78\) −19273.4 − 7077.60i −0.358692 − 0.131719i
\(79\) −21949.1 − 7131.71i −0.395685 − 0.128566i 0.104414 − 0.994534i \(-0.466703\pi\)
−0.500100 + 0.865968i \(0.666703\pi\)
\(80\) −54671.2 − 27115.5i −0.955067 − 0.473688i
\(81\) 9586.08 + 58265.7i 0.162341 + 0.986735i
\(82\) 103084. + 103084.i 1.69300 + 1.69300i
\(83\) −32427.0 + 63641.5i −0.516668 + 1.01402i 0.474356 + 0.880333i \(0.342681\pi\)
−0.991024 + 0.133684i \(0.957319\pi\)
\(84\) 80903.5 − 9698.29i 1.25103 − 0.149967i
\(85\) −102940. − 32220.5i −1.54539 − 0.483710i
\(86\) −77520.0 − 106697.i −1.13023 − 1.55563i
\(87\) −20155.3 − 15840.3i −0.285490 − 0.224371i
\(88\) 7694.38 − 1218.67i 0.105917 − 0.0167757i
\(89\) −2658.01 + 1931.16i −0.0355698 + 0.0258430i −0.605428 − 0.795900i \(-0.706998\pi\)
0.569858 + 0.821743i \(0.306998\pi\)
\(90\) 68183.5 − 82101.8i 0.887305 − 1.06843i
\(91\) 23851.8 + 17329.4i 0.301938 + 0.219371i
\(92\) −119488. − 60882.3i −1.47182 − 0.749931i
\(93\) 84050.6 + 3176.45i 1.00770 + 0.0380833i
\(94\) 102464. − 33292.6i 1.19606 − 0.388622i
\(95\) 9020.25 − 4719.64i 0.102544 − 0.0536537i
\(96\) −52407.3 − 113234.i −0.580381 − 1.25401i
\(97\) −26729.8 + 13619.5i −0.288448 + 0.146971i −0.592227 − 0.805771i \(-0.701751\pi\)
0.303780 + 0.952742i \(0.401751\pi\)
\(98\) −109562. − 17352.9i −1.15238 − 0.182518i
\(99\) −7988.25 + 105536.i −0.0819151 + 1.08221i
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 75.6.l.a.2.9 ✓ 384
3.2 odd 2 inner 75.6.l.a.2.40 yes 384
25.13 odd 20 inner 75.6.l.a.38.40 yes 384
75.38 even 20 inner 75.6.l.a.38.9 yes 384
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
75.6.l.a.2.9 ✓ 384 1.1 even 1 trivial
75.6.l.a.2.40 yes 384 3.2 odd 2 inner
75.6.l.a.38.9 yes 384 75.38 even 20 inner
75.6.l.a.38.40 yes 384 25.13 odd 20 inner