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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [75,14,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, 14); 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, 14, names="a")
 
Level: \( N \) \(=\) \( 75 = 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 14 \)
Character orbit: \([\chi]\) \(=\) 75.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,16096] 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: \(80.4231967139\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 3)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 49.2
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 75.49
Dual form 75.14.b.b.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)
 
\(f(q)\) \(=\) \(q+12.0000i q^{2} -729.000i q^{3} +8048.00 q^{4} +8748.00 q^{6} -235088. i q^{7} +194880. i q^{8} -531441. q^{9} -1.11829e7 q^{11} -5.86699e6i q^{12} +8.04961e6i q^{13} +2.82106e6 q^{14} +6.35907e7 q^{16} +1.17495e8i q^{17} -6.37729e6i q^{18} +2.14061e8 q^{19} -1.71379e8 q^{21} -1.34195e8i q^{22} +8.30556e8i q^{23} +1.42068e8 q^{24} -9.65954e7 q^{26} +3.87420e8i q^{27} -1.89199e9i q^{28} +1.25240e9 q^{29} +6.15935e9 q^{31} +2.35954e9i q^{32} +8.15234e9i q^{33} -1.40994e9 q^{34} -4.27704e9 q^{36} +5.49819e9i q^{37} +2.56874e9i q^{38} +5.86817e9 q^{39} -4.67869e9 q^{41} -2.05655e9i q^{42} +7.11501e9i q^{43} -9.00000e10 q^{44} -9.96667e9 q^{46} +2.95288e10i q^{47} -4.63576e10i q^{48} +4.16226e10 q^{49} +8.56536e10 q^{51} +6.47833e10i q^{52} -2.04125e11i q^{53} -4.64905e9 q^{54} +4.58139e10 q^{56} -1.56051e11i q^{57} +1.50288e10i q^{58} +2.99098e10 q^{59} -1.34392e11 q^{61} +7.39122e10i q^{62} +1.24935e11i q^{63} +4.92620e11 q^{64} -9.78281e10 q^{66} -3.48519e11i q^{67} +9.45597e11i q^{68} +6.05475e11 q^{69} +1.31434e12 q^{71} -1.03567e11i q^{72} -1.17888e12i q^{73} -6.59783e10 q^{74} +1.72277e12 q^{76} +2.62897e12i q^{77} +7.04180e10i q^{78} +1.07242e12 q^{79} +2.82430e11 q^{81} -5.61443e10i q^{82} +1.12403e12i q^{83} -1.37926e12 q^{84} -8.53802e10 q^{86} -9.13000e11i q^{87} -2.17933e12i q^{88} -2.23561e12 q^{89} +1.89237e12 q^{91} +6.68431e12i q^{92} -4.49017e12i q^{93} -3.54345e11 q^{94} +1.72011e12 q^{96} +1.42153e13i q^{97} +4.99472e11i q^{98} +5.94306e12 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 16096 q^{4} + 17496 q^{6} - 1062882 q^{9} - 22365816 q^{11} + 5642112 q^{14} + 127181312 q^{16} + 428122760 q^{19} - 342758304 q^{21} + 284135040 q^{24} - 193190736 q^{26} + 2504800500 q^{29} + 12318701104 q^{31}+ \cdots + 11886111620856 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 12.0000i 0.132583i 0.997800 + 0.0662913i \(0.0211166\pi\)
−0.997800 + 0.0662913i \(0.978883\pi\)
\(3\) − 729.000i − 0.577350i
\(4\) 8048.00 0.982422
\(5\) 0 0
\(6\) 8748.00 0.0765466
\(7\) − 235088.i − 0.755254i −0.925958 − 0.377627i \(-0.876740\pi\)
0.925958 − 0.377627i \(-0.123260\pi\)
\(8\) 194880.i 0.262834i
\(9\) −531441. −0.333333
\(10\) 0 0
\(11\) −1.11829e7 −1.90328 −0.951639 − 0.307218i \(-0.900602\pi\)
−0.951639 + 0.307218i \(0.900602\pi\)
\(12\) − 5.86699e6i − 0.567202i
\(13\) 8.04961e6i 0.462534i 0.972890 + 0.231267i \(0.0742870\pi\)
−0.972890 + 0.231267i \(0.925713\pi\)
\(14\) 2.82106e6 0.100134
\(15\) 0 0
\(16\) 6.35907e7 0.947575
\(17\) 1.17495e8i 1.18059i 0.807187 + 0.590296i \(0.200989\pi\)
−0.807187 + 0.590296i \(0.799011\pi\)
\(18\) − 6.37729e6i − 0.0441942i
\(19\) 2.14061e8 1.04385 0.521927 − 0.852990i \(-0.325213\pi\)
0.521927 + 0.852990i \(0.325213\pi\)
\(20\) 0 0
\(21\) −1.71379e8 −0.436046
\(22\) − 1.34195e8i − 0.252341i
\(23\) 8.30556e8i 1.16987i 0.811080 + 0.584935i \(0.198880\pi\)
−0.811080 + 0.584935i \(0.801120\pi\)
\(24\) 1.42068e8 0.151748
\(25\) 0 0
\(26\) −9.65954e7 −0.0613239
\(27\) 3.87420e8i 0.192450i
\(28\) − 1.89199e9i − 0.741978i
\(29\) 1.25240e9 0.390981 0.195491 − 0.980706i \(-0.437370\pi\)
0.195491 + 0.980706i \(0.437370\pi\)
\(30\) 0 0
\(31\) 6.15935e9 1.24648 0.623238 − 0.782032i \(-0.285817\pi\)
0.623238 + 0.782032i \(0.285817\pi\)
\(32\) 2.35954e9i 0.388466i
\(33\) 8.15234e9i 1.09886i
\(34\) −1.40994e9 −0.156526
\(35\) 0 0
\(36\) −4.27704e9 −0.327474
\(37\) 5.49819e9i 0.352297i 0.984364 + 0.176148i \(0.0563639\pi\)
−0.984364 + 0.176148i \(0.943636\pi\)
\(38\) 2.56874e9i 0.138397i
\(39\) 5.86817e9 0.267044
\(40\) 0 0
\(41\) −4.67869e9 −0.153826 −0.0769129 − 0.997038i \(-0.524506\pi\)
−0.0769129 + 0.997038i \(0.524506\pi\)
\(42\) − 2.05655e9i − 0.0578121i
\(43\) 7.11501e9i 0.171645i 0.996310 + 0.0858224i \(0.0273518\pi\)
−0.996310 + 0.0858224i \(0.972648\pi\)
\(44\) −9.00000e10 −1.86982
\(45\) 0 0
\(46\) −9.96667e9 −0.155104
\(47\) 2.95288e10i 0.399585i 0.979838 + 0.199793i \(0.0640268\pi\)
−0.979838 + 0.199793i \(0.935973\pi\)
\(48\) − 4.63576e10i − 0.547082i
\(49\) 4.16226e10 0.429591
\(50\) 0 0
\(51\) 8.56536e10 0.681615
\(52\) 6.47833e10i 0.454403i
\(53\) − 2.04125e11i − 1.26504i −0.774545 − 0.632518i \(-0.782021\pi\)
0.774545 − 0.632518i \(-0.217979\pi\)
\(54\) −4.64905e9 −0.0255155
\(55\) 0 0
\(56\) 4.58139e10 0.198507
\(57\) − 1.56051e11i − 0.602670i
\(58\) 1.50288e10i 0.0518373i
\(59\) 2.99098e10 0.0923157 0.0461579 − 0.998934i \(-0.485302\pi\)
0.0461579 + 0.998934i \(0.485302\pi\)
\(60\) 0 0
\(61\) −1.34392e11 −0.333987 −0.166993 − 0.985958i \(-0.553406\pi\)
−0.166993 + 0.985958i \(0.553406\pi\)
\(62\) 7.39122e10i 0.165261i
\(63\) 1.24935e11i 0.251751i
\(64\) 4.92620e11 0.896071
\(65\) 0 0
\(66\) −9.78281e10 −0.145689
\(67\) − 3.48519e11i − 0.470695i −0.971911 − 0.235348i \(-0.924377\pi\)
0.971911 − 0.235348i \(-0.0756229\pi\)
\(68\) 9.45597e11i 1.15984i
\(69\) 6.05475e11 0.675425
\(70\) 0 0
\(71\) 1.31434e12 1.21766 0.608831 − 0.793300i \(-0.291639\pi\)
0.608831 + 0.793300i \(0.291639\pi\)
\(72\) − 1.03567e11i − 0.0876115i
\(73\) − 1.17888e12i − 0.911737i −0.890047 − 0.455868i \(-0.849329\pi\)
0.890047 − 0.455868i \(-0.150671\pi\)
\(74\) −6.59783e10 −0.0467084
\(75\) 0 0
\(76\) 1.72277e12 1.02551
\(77\) 2.62897e12i 1.43746i
\(78\) 7.04180e10i 0.0354054i
\(79\) 1.07242e12 0.496351 0.248176 − 0.968715i \(-0.420169\pi\)
0.248176 + 0.968715i \(0.420169\pi\)
\(80\) 0 0
\(81\) 2.82430e11 0.111111
\(82\) − 5.61443e10i − 0.0203946i
\(83\) 1.12403e12i 0.377371i 0.982038 + 0.188685i \(0.0604227\pi\)
−0.982038 + 0.188685i \(0.939577\pi\)
\(84\) −1.37926e12 −0.428381
\(85\) 0 0
\(86\) −8.53802e10 −0.0227571
\(87\) − 9.13000e11i − 0.225733i
\(88\) − 2.17933e12i − 0.500247i
\(89\) −2.23561e12 −0.476827 −0.238414 − 0.971164i \(-0.576627\pi\)
−0.238414 + 0.971164i \(0.576627\pi\)
\(90\) 0 0
\(91\) 1.89237e12 0.349330
\(92\) 6.68431e12i 1.14931i
\(93\) − 4.49017e12i − 0.719653i
\(94\) −3.54345e11 −0.0529780
\(95\) 0 0
\(96\) 1.72011e12 0.224281
\(97\) 1.42153e13i 1.73276i 0.499385 + 0.866380i \(0.333559\pi\)
−0.499385 + 0.866380i \(0.666441\pi\)
\(98\) 4.99472e11i 0.0569563i
\(99\) 5.94306e12 0.634426
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 75.14.b.b.49.2 2
5.2 odd 4 3.14.a.a.1.1 ✓ 1
5.3 odd 4 75.14.a.a.1.1 1
5.4 even 2 inner 75.14.b.b.49.1 2
15.2 even 4 9.14.a.a.1.1 1
20.7 even 4 48.14.a.c.1.1 1
35.27 even 4 147.14.a.a.1.1 1
40.27 even 4 192.14.a.e.1.1 1
40.37 odd 4 192.14.a.j.1.1 1
60.47 odd 4 144.14.a.k.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3.14.a.a.1.1 ✓ 1 5.2 odd 4
9.14.a.a.1.1 1 15.2 even 4
48.14.a.c.1.1 1 20.7 even 4
75.14.a.a.1.1 1 5.3 odd 4
75.14.b.b.49.1 2 5.4 even 2 inner
75.14.b.b.49.2 2 1.1 even 1 trivial
144.14.a.k.1.1 1 60.47 odd 4
147.14.a.a.1.1 1 35.27 even 4
192.14.a.e.1.1 1 40.27 even 4
192.14.a.j.1.1 1 40.37 odd 4