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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [75,10,Mod(49,75)] 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("75.49"); S:= CuspForms(chi, 10); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(75, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 10, names="a")
 
Level: \( N \) \(=\) \( 75 = 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 75.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,0,-3584] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(38.6276877123\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2x^{7} + 2x^{6} + 1470x^{5} + 317749x^{4} + 221032x^{3} + 2888x^{2} + 10631640x + 19569212100 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{10}\cdot 3^{2}\cdot 5^{4}\cdot 23^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 49.8
Root \(-13.2852 - 13.2852i\) of defining polynomial
Character \(\chi\) \(=\) 75.49
Dual form 75.10.b.h.49.1

$q$-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)
 
\(f(q)\) \(=\) \(q+44.2736i q^{2} +81.0000i q^{3} -1448.15 q^{4} -3586.16 q^{6} -3879.20i q^{7} -41447.0i q^{8} -6561.00 q^{9} +84809.6 q^{11} -117301. i q^{12} +58662.8i q^{13} +171746. q^{14} +1.09355e6 q^{16} -176199. i q^{17} -290479. i q^{18} -800041. q^{19} +314215. q^{21} +3.75483e6i q^{22} -1.65723e6i q^{23} +3.35720e6 q^{24} -2.59721e6 q^{26} -531441. i q^{27} +5.61768e6i q^{28} +4.70073e6 q^{29} +5.05198e6 q^{31} +2.71947e7i q^{32} +6.86958e6i q^{33} +7.80099e6 q^{34} +9.50134e6 q^{36} -4.68753e6i q^{37} -3.54207e7i q^{38} -4.75168e6 q^{39} -4.71307e6 q^{41} +1.39114e7i q^{42} -1.65548e7i q^{43} -1.22817e8 q^{44} +7.33718e7 q^{46} -1.79860e7i q^{47} +8.85778e7i q^{48} +2.53054e7 q^{49} +1.42722e7 q^{51} -8.49527e7i q^{52} +1.50727e7i q^{53} +2.35288e7 q^{54} -1.60781e8 q^{56} -6.48033e7i q^{57} +2.08118e8i q^{58} -1.15379e7 q^{59} -3.48684e7 q^{61} +2.23669e8i q^{62} +2.54514e7i q^{63} -6.44110e8 q^{64} -3.04141e8 q^{66} -1.51756e7i q^{67} +2.55164e8i q^{68} +1.34236e8 q^{69} -3.98346e7 q^{71} +2.71934e8i q^{72} -7.02902e7i q^{73} +2.07534e8 q^{74} +1.15858e9 q^{76} -3.28993e8i q^{77} -2.10374e8i q^{78} -1.16781e8 q^{79} +4.30467e7 q^{81} -2.08665e8i q^{82} +7.69619e8i q^{83} -4.55032e8 q^{84} +7.32941e8 q^{86} +3.80759e8i q^{87} -3.51510e9i q^{88} +9.57076e8 q^{89} +2.27564e8 q^{91} +2.39993e9i q^{92} +4.09210e8i q^{93} +7.96305e8 q^{94} -2.20277e9 q^{96} -4.93418e8i q^{97} +1.12036e9i q^{98} -5.56436e8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 3584 q^{4} - 324 q^{6} - 52488 q^{9} + 209392 q^{11} + 362124 q^{14} + 2639600 q^{16} - 786616 q^{19} + 2111832 q^{21} + 3991032 q^{24} - 12955004 q^{26} - 9853232 q^{29} - 195032 q^{31} - 33657288 q^{34}+ \cdots - 1373820912 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(26\) \(52\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 44.2736i 1.95664i 0.207107 + 0.978318i \(0.433595\pi\)
−0.207107 + 0.978318i \(0.566405\pi\)
\(3\) 81.0000i 0.577350i
\(4\) −1448.15 −2.82843
\(5\) 0 0
\(6\) −3586.16 −1.12966
\(7\) − 3879.20i − 0.610662i −0.952246 − 0.305331i \(-0.901233\pi\)
0.952246 − 0.305331i \(-0.0987671\pi\)
\(8\) − 41447.0i − 3.57757i
\(9\) −6561.00 −0.333333
\(10\) 0 0
\(11\) 84809.6 1.74654 0.873269 − 0.487239i \(-0.161996\pi\)
0.873269 + 0.487239i \(0.161996\pi\)
\(12\) − 117301.i − 1.63299i
\(13\) 58662.8i 0.569662i 0.958578 + 0.284831i \(0.0919375\pi\)
−0.958578 + 0.284831i \(0.908062\pi\)
\(14\) 171746. 1.19484
\(15\) 0 0
\(16\) 1.09355e6 4.17157
\(17\) − 176199.i − 0.511663i −0.966721 − 0.255832i \(-0.917651\pi\)
0.966721 − 0.255832i \(-0.0823493\pi\)
\(18\) − 290479.i − 0.652212i
\(19\) −800041. −1.40838 −0.704192 − 0.710009i \(-0.748691\pi\)
−0.704192 + 0.710009i \(0.748691\pi\)
\(20\) 0 0
\(21\) 314215. 0.352566
\(22\) 3.75483e6i 3.41734i
\(23\) − 1.65723e6i − 1.23483i −0.786636 − 0.617417i \(-0.788179\pi\)
0.786636 − 0.617417i \(-0.211821\pi\)
\(24\) 3.35720e6 2.06551
\(25\) 0 0
\(26\) −2.59721e6 −1.11462
\(27\) − 531441.i − 0.192450i
\(28\) 5.61768e6i 1.72721i
\(29\) 4.70073e6 1.23417 0.617084 − 0.786897i \(-0.288314\pi\)
0.617084 + 0.786897i \(0.288314\pi\)
\(30\) 0 0
\(31\) 5.05198e6 0.982503 0.491251 − 0.871018i \(-0.336540\pi\)
0.491251 + 0.871018i \(0.336540\pi\)
\(32\) 2.71947e7i 4.58469i
\(33\) 6.86958e6i 1.00836i
\(34\) 7.80099e6 1.00114
\(35\) 0 0
\(36\) 9.50134e6 0.942809
\(37\) − 4.68753e6i − 0.411184i −0.978638 − 0.205592i \(-0.934088\pi\)
0.978638 − 0.205592i \(-0.0659120\pi\)
\(38\) − 3.54207e7i − 2.75570i
\(39\) −4.75168e6 −0.328895
\(40\) 0 0
\(41\) −4.71307e6 −0.260481 −0.130241 − 0.991482i \(-0.541575\pi\)
−0.130241 + 0.991482i \(0.541575\pi\)
\(42\) 1.39114e7i 0.689843i
\(43\) − 1.65548e7i − 0.738441i −0.929342 − 0.369221i \(-0.879625\pi\)
0.929342 − 0.369221i \(-0.120375\pi\)
\(44\) −1.22817e8 −4.93995
\(45\) 0 0
\(46\) 7.33718e7 2.41612
\(47\) − 1.79860e7i − 0.537643i −0.963190 − 0.268821i \(-0.913366\pi\)
0.963190 − 0.268821i \(-0.0866341\pi\)
\(48\) 8.85778e7i 2.40846i
\(49\) 2.53054e7 0.627092
\(50\) 0 0
\(51\) 1.42722e7 0.295409
\(52\) − 8.49527e7i − 1.61125i
\(53\) 1.50727e7i 0.262391i 0.991357 + 0.131195i \(0.0418815\pi\)
−0.991357 + 0.131195i \(0.958118\pi\)
\(54\) 2.35288e7 0.376555
\(55\) 0 0
\(56\) −1.60781e8 −2.18468
\(57\) − 6.48033e7i − 0.813131i
\(58\) 2.08118e8i 2.41482i
\(59\) −1.15379e7 −0.123963 −0.0619814 − 0.998077i \(-0.519742\pi\)
−0.0619814 + 0.998077i \(0.519742\pi\)
\(60\) 0 0
\(61\) −3.48684e7 −0.322439 −0.161219 − 0.986919i \(-0.551543\pi\)
−0.161219 + 0.986919i \(0.551543\pi\)
\(62\) 2.23669e8i 1.92240i
\(63\) 2.54514e7i 0.203554i
\(64\) −6.44110e8 −4.79899
\(65\) 0 0
\(66\) −3.04141e8 −1.97300
\(67\) − 1.51756e7i − 0.0920044i −0.998941 − 0.0460022i \(-0.985352\pi\)
0.998941 − 0.0460022i \(-0.0146481\pi\)
\(68\) 2.55164e8i 1.44720i
\(69\) 1.34236e8 0.712932
\(70\) 0 0
\(71\) −3.98346e7 −0.186036 −0.0930182 − 0.995664i \(-0.529651\pi\)
−0.0930182 + 0.995664i \(0.529651\pi\)
\(72\) 2.71934e8i 1.19252i
\(73\) − 7.02902e7i − 0.289696i −0.989454 − 0.144848i \(-0.953731\pi\)
0.989454 − 0.144848i \(-0.0462693\pi\)
\(74\) 2.07534e8 0.804538
\(75\) 0 0
\(76\) 1.15858e9 3.98351
\(77\) − 3.28993e8i − 1.06654i
\(78\) − 2.10374e8i − 0.643527i
\(79\) −1.16781e8 −0.337327 −0.168663 − 0.985674i \(-0.553945\pi\)
−0.168663 + 0.985674i \(0.553945\pi\)
\(80\) 0 0
\(81\) 4.30467e7 0.111111
\(82\) − 2.08665e8i − 0.509667i
\(83\) 7.69619e8i 1.78002i 0.455943 + 0.890009i \(0.349302\pi\)
−0.455943 + 0.890009i \(0.650698\pi\)
\(84\) −4.55032e8 −0.997207
\(85\) 0 0
\(86\) 7.32941e8 1.44486
\(87\) 3.80759e8i 0.712547i
\(88\) − 3.51510e9i − 6.24836i
\(89\) 9.57076e8 1.61693 0.808466 − 0.588543i \(-0.200298\pi\)
0.808466 + 0.588543i \(0.200298\pi\)
\(90\) 0 0
\(91\) 2.27564e8 0.347871
\(92\) 2.39993e9i 3.49264i
\(93\) 4.09210e8i 0.567248i
\(94\) 7.96305e8 1.05197
\(95\) 0 0
\(96\) −2.20277e9 −2.64697
\(97\) − 4.93418e8i − 0.565903i −0.959134 − 0.282952i \(-0.908686\pi\)
0.959134 − 0.282952i \(-0.0913136\pi\)
\(98\) 1.12036e9i 1.22699i
\(99\) −5.56436e8 −0.582179
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 75.10.b.h.49.8 8
3.2 odd 2 225.10.b.n.199.1 8
5.2 odd 4 75.10.a.j.1.1 ✓ 4
5.3 odd 4 75.10.a.k.1.4 yes 4
5.4 even 2 inner 75.10.b.h.49.1 8
15.2 even 4 225.10.a.t.1.4 4
15.8 even 4 225.10.a.r.1.1 4
15.14 odd 2 225.10.b.n.199.8 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
75.10.a.j.1.1 ✓ 4 5.2 odd 4
75.10.a.k.1.4 yes 4 5.3 odd 4
75.10.b.h.49.1 8 5.4 even 2 inner
75.10.b.h.49.8 8 1.1 even 1 trivial
225.10.a.r.1.1 4 15.8 even 4
225.10.a.t.1.4 4 15.2 even 4
225.10.b.n.199.1 8 3.2 odd 2
225.10.b.n.199.8 8 15.14 odd 2