Show commands: Magma / Pari/GP / SageMath

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 = [4,0,0,-1082] 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: \(4\)
Coefficient field: \(\Q(i, \sqrt{241})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 121x^{2} + 3600 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 3^{2} \)
Twist minimal: no (minimal twist has level 15)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 49.2
Root \(8.26209i\) of defining polynomial
Character \(\chi\) \(=\) 75.49
Dual form 75.10.b.f.49.3

$q$-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)
 
\(f(q)\) \(=\) \(q-7.78626i q^{2} +81.0000i q^{3} +451.374 q^{4} +630.687 q^{6} +1839.88i q^{7} -7501.08i q^{8} -6561.00 q^{9} +44385.9 q^{11} +36561.3i q^{12} -136584. i q^{13} +14325.8 q^{14} +172698. q^{16} -253591. i q^{17} +51085.7i q^{18} -85435.7 q^{19} -149030. q^{21} -345600. i q^{22} +979409. i q^{23} +607588. q^{24} -1.06348e6 q^{26} -531441. i q^{27} +830473. i q^{28} -2.58640e6 q^{29} +8.94787e6 q^{31} -5.18523e6i q^{32} +3.59526e6i q^{33} -1.97452e6 q^{34} -2.96147e6 q^{36} +1.56064e7i q^{37} +665225. i q^{38} +1.10633e7 q^{39} +2.44893e7 q^{41} +1.16039e6i q^{42} -1.27592e7i q^{43} +2.00346e7 q^{44} +7.62593e6 q^{46} -6.16764e7i q^{47} +1.39886e7i q^{48} +3.69685e7 q^{49} +2.05408e7 q^{51} -6.16504e7i q^{52} -5.70418e6i q^{53} -4.13794e6 q^{54} +1.38011e7 q^{56} -6.92029e6i q^{57} +2.01384e7i q^{58} -8.35095e7 q^{59} +1.48622e8 q^{61} -6.96704e7i q^{62} -1.20714e7i q^{63} +4.80479e7 q^{64} +2.79936e7 q^{66} -1.68003e8i q^{67} -1.14464e8i q^{68} -7.93321e7 q^{69} +2.10986e8 q^{71} +4.92146e7i q^{72} +1.43534e8i q^{73} +1.21515e8 q^{74} -3.85635e7 q^{76} +8.16646e7i q^{77} -8.61416e7i q^{78} +4.55960e8 q^{79} +4.30467e7 q^{81} -1.90680e8i q^{82} +3.55106e8i q^{83} -6.72683e7 q^{84} -9.93465e7 q^{86} -2.09498e8i q^{87} -3.32942e8i q^{88} +4.24540e8 q^{89} +2.51297e8 q^{91} +4.42080e8i q^{92} +7.24777e8i q^{93} -4.80228e8 q^{94} +4.20003e8 q^{96} +1.19905e9i q^{97} -2.87846e8i q^{98} -2.91216e8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 1082 q^{4} - 5022 q^{6} - 26244 q^{9} - 43024 q^{11} - 923328 q^{14} + 774530 q^{16} + 191792 q^{19} - 2286144 q^{21} + 4233546 q^{24} - 10838332 q^{26} + 5356424 q^{29} + 21564864 q^{31} - 11445244 q^{34}+ \cdots + 282280464 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 7.78626i − 0.344107i −0.985088 − 0.172054i \(-0.944960\pi\)
0.985088 − 0.172054i \(-0.0550403\pi\)
\(3\) 81.0000i 0.577350i
\(4\) 451.374 0.881590
\(5\) 0 0
\(6\) 630.687 0.198671
\(7\) 1839.88i 0.289633i 0.989459 + 0.144816i \(0.0462591\pi\)
−0.989459 + 0.144816i \(0.953741\pi\)
\(8\) − 7501.08i − 0.647469i
\(9\) −6561.00 −0.333333
\(10\) 0 0
\(11\) 44385.9 0.914066 0.457033 − 0.889450i \(-0.348912\pi\)
0.457033 + 0.889450i \(0.348912\pi\)
\(12\) 36561.3i 0.508986i
\(13\) − 136584.i − 1.32634i −0.748470 − 0.663169i \(-0.769211\pi\)
0.748470 − 0.663169i \(-0.230789\pi\)
\(14\) 14325.8 0.0996648
\(15\) 0 0
\(16\) 172698. 0.658791
\(17\) − 253591.i − 0.736399i −0.929747 − 0.368199i \(-0.879974\pi\)
0.929747 − 0.368199i \(-0.120026\pi\)
\(18\) 51085.7i 0.114702i
\(19\) −85435.7 −0.150400 −0.0752001 − 0.997168i \(-0.523960\pi\)
−0.0752001 + 0.997168i \(0.523960\pi\)
\(20\) 0 0
\(21\) −149030. −0.167220
\(22\) − 345600.i − 0.314537i
\(23\) 979409.i 0.729775i 0.931052 + 0.364887i \(0.118892\pi\)
−0.931052 + 0.364887i \(0.881108\pi\)
\(24\) 607588. 0.373816
\(25\) 0 0
\(26\) −1.06348e6 −0.456403
\(27\) − 531441.i − 0.192450i
\(28\) 830473.i 0.255337i
\(29\) −2.58640e6 −0.679054 −0.339527 − 0.940596i \(-0.610267\pi\)
−0.339527 + 0.940596i \(0.610267\pi\)
\(30\) 0 0
\(31\) 8.94787e6 1.74017 0.870085 − 0.492901i \(-0.164064\pi\)
0.870085 + 0.492901i \(0.164064\pi\)
\(32\) − 5.18523e6i − 0.874164i
\(33\) 3.59526e6i 0.527736i
\(34\) −1.97452e6 −0.253400
\(35\) 0 0
\(36\) −2.96147e6 −0.293863
\(37\) 1.56064e7i 1.36897i 0.729026 + 0.684486i \(0.239974\pi\)
−0.729026 + 0.684486i \(0.760026\pi\)
\(38\) 665225.i 0.0517538i
\(39\) 1.10633e7 0.765761
\(40\) 0 0
\(41\) 2.44893e7 1.35347 0.676735 − 0.736227i \(-0.263394\pi\)
0.676735 + 0.736227i \(0.263394\pi\)
\(42\) 1.16039e6i 0.0575415i
\(43\) − 1.27592e7i − 0.569135i −0.958656 − 0.284568i \(-0.908150\pi\)
0.958656 − 0.284568i \(-0.0918500\pi\)
\(44\) 2.00346e7 0.805832
\(45\) 0 0
\(46\) 7.62593e6 0.251121
\(47\) − 6.16764e7i − 1.84365i −0.387607 − 0.921825i \(-0.626698\pi\)
0.387607 − 0.921825i \(-0.373302\pi\)
\(48\) 1.39886e7i 0.380353i
\(49\) 3.69685e7 0.916113
\(50\) 0 0
\(51\) 2.05408e7 0.425160
\(52\) − 6.16504e7i − 1.16929i
\(53\) − 5.70418e6i − 0.0993006i −0.998767 − 0.0496503i \(-0.984189\pi\)
0.998767 − 0.0496503i \(-0.0158107\pi\)
\(54\) −4.13794e6 −0.0662235
\(55\) 0 0
\(56\) 1.38011e7 0.187528
\(57\) − 6.92029e6i − 0.0868336i
\(58\) 2.01384e7i 0.233668i
\(59\) −8.35095e7 −0.897226 −0.448613 − 0.893726i \(-0.648082\pi\)
−0.448613 + 0.893726i \(0.648082\pi\)
\(60\) 0 0
\(61\) 1.48622e8 1.37435 0.687177 − 0.726490i \(-0.258850\pi\)
0.687177 + 0.726490i \(0.258850\pi\)
\(62\) − 6.96704e7i − 0.598806i
\(63\) − 1.20714e7i − 0.0965443i
\(64\) 4.80479e7 0.357985
\(65\) 0 0
\(66\) 2.79936e7 0.181598
\(67\) − 1.68003e8i − 1.01854i −0.860606 − 0.509272i \(-0.829915\pi\)
0.860606 − 0.509272i \(-0.170085\pi\)
\(68\) − 1.14464e8i − 0.649202i
\(69\) −7.93321e7 −0.421336
\(70\) 0 0
\(71\) 2.10986e8 0.985349 0.492675 − 0.870214i \(-0.336019\pi\)
0.492675 + 0.870214i \(0.336019\pi\)
\(72\) 4.92146e7i 0.215823i
\(73\) 1.43534e8i 0.591566i 0.955255 + 0.295783i \(0.0955805\pi\)
−0.955255 + 0.295783i \(0.904420\pi\)
\(74\) 1.21515e8 0.471074
\(75\) 0 0
\(76\) −3.85635e7 −0.132591
\(77\) 8.16646e7i 0.264744i
\(78\) − 8.61416e7i − 0.263504i
\(79\) 4.55960e8 1.31706 0.658529 − 0.752555i \(-0.271179\pi\)
0.658529 + 0.752555i \(0.271179\pi\)
\(80\) 0 0
\(81\) 4.30467e7 0.111111
\(82\) − 1.90680e8i − 0.465739i
\(83\) 3.55106e8i 0.821309i 0.911791 + 0.410655i \(0.134700\pi\)
−0.911791 + 0.410655i \(0.865300\pi\)
\(84\) −6.72683e7 −0.147419
\(85\) 0 0
\(86\) −9.93465e7 −0.195844
\(87\) − 2.09498e8i − 0.392052i
\(88\) − 3.32942e8i − 0.591830i
\(89\) 4.24540e8 0.717239 0.358620 − 0.933484i \(-0.383248\pi\)
0.358620 + 0.933484i \(0.383248\pi\)
\(90\) 0 0
\(91\) 2.51297e8 0.384151
\(92\) 4.42080e8i 0.643362i
\(93\) 7.24777e8i 1.00469i
\(94\) −4.80228e8 −0.634413
\(95\) 0 0
\(96\) 4.20003e8 0.504699
\(97\) 1.19905e9i 1.37520i 0.726092 + 0.687598i \(0.241335\pi\)
−0.726092 + 0.687598i \(0.758665\pi\)
\(98\) − 2.87846e8i − 0.315241i
\(99\) −2.91216e8 −0.304689
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.f.49.2 4
3.2 odd 2 225.10.b.i.199.3 4
5.2 odd 4 75.10.a.f.1.2 2
5.3 odd 4 15.10.a.d.1.1 ✓ 2
5.4 even 2 inner 75.10.b.f.49.3 4
15.2 even 4 225.10.a.k.1.1 2
15.8 even 4 45.10.a.d.1.2 2
15.14 odd 2 225.10.b.i.199.2 4
20.3 even 4 240.10.a.r.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
15.10.a.d.1.1 ✓ 2 5.3 odd 4
45.10.a.d.1.2 2 15.8 even 4
75.10.a.f.1.2 2 5.2 odd 4
75.10.b.f.49.2 4 1.1 even 1 trivial
75.10.b.f.49.3 4 5.4 even 2 inner
225.10.a.k.1.1 2 15.2 even 4
225.10.b.i.199.2 4 15.14 odd 2
225.10.b.i.199.3 4 3.2 odd 2
240.10.a.r.1.2 2 20.3 even 4