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(1,75)] 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("75.1"); 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, 0])) N = Newforms(chi, 10, names="a")
 
Level: \( N \) \(=\) \( 75 = 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 75.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(38.6276877123\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{3} - 1546x^{2} + 152x + 559560 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{3}\cdot 3\cdot 5^{2}\cdot 23 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(23.0287\) of defining polynomial
Character \(\chi\) \(=\) 75.1

$q$-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)
 
\(f(q)\) \(=\) \(q+11.8931 q^{2} -81.0000 q^{3} -370.553 q^{4} -963.344 q^{6} -10946.0 q^{7} -10496.3 q^{8} +6561.00 q^{9} -39453.3 q^{11} +30014.8 q^{12} -50257.6 q^{13} -130183. q^{14} +64888.9 q^{16} +461329. q^{17} +78030.9 q^{18} -370082. q^{19} +886628. q^{21} -469223. q^{22} -2.29014e6 q^{23} +850203. q^{24} -597721. q^{26} -531441. q^{27} +4.05608e6 q^{28} -1.28309e6 q^{29} +6.51912e6 q^{31} +6.14585e6 q^{32} +3.19571e6 q^{33} +5.48665e6 q^{34} -2.43120e6 q^{36} +1.45530e7 q^{37} -4.40144e6 q^{38} +4.07086e6 q^{39} +1.34566e7 q^{41} +1.05448e7 q^{42} -2.24211e7 q^{43} +1.46195e7 q^{44} -2.72369e7 q^{46} -1.45871e7 q^{47} -5.25600e6 q^{48} +7.94618e7 q^{49} -3.73677e7 q^{51} +1.86231e7 q^{52} -6.49845e7 q^{53} -6.32050e6 q^{54} +1.14893e8 q^{56} +2.99766e7 q^{57} -1.52599e7 q^{58} +1.04242e8 q^{59} +1.46085e8 q^{61} +7.75328e7 q^{62} -7.18168e7 q^{63} +3.98704e7 q^{64} +3.80071e7 q^{66} -9.72944e7 q^{67} -1.70947e8 q^{68} +1.85501e8 q^{69} -2.89431e8 q^{71} -6.88664e7 q^{72} -6.21359e7 q^{73} +1.73081e8 q^{74} +1.37135e8 q^{76} +4.31856e8 q^{77} +4.84154e7 q^{78} +3.55247e8 q^{79} +4.30467e7 q^{81} +1.60042e8 q^{82} +2.13970e8 q^{83} -3.28543e8 q^{84} -2.66658e8 q^{86} +1.03930e8 q^{87} +4.14114e8 q^{88} -8.61695e8 q^{89} +5.50121e8 q^{91} +8.48617e8 q^{92} -5.28049e8 q^{93} -1.73486e8 q^{94} -4.97814e8 q^{96} -1.00809e9 q^{97} +9.45050e8 q^{98} -2.58853e8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{2} - 324 q^{3} + 1792 q^{4} - 162 q^{6} - 13036 q^{7} + 24636 q^{8} + 26244 q^{9} + 104696 q^{11} - 145152 q^{12} - 140812 q^{13} - 181062 q^{14} + 1319800 q^{16} - 489352 q^{17} + 13122 q^{18}+ \cdots + 686910456 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 11.8931 0.525608 0.262804 − 0.964849i \(-0.415353\pi\)
0.262804 + 0.964849i \(0.415353\pi\)
\(3\) −81.0000 −0.577350
\(4\) −370.553 −0.723737
\(5\) 0 0
\(6\) −963.344 −0.303460
\(7\) −10946.0 −1.72312 −0.861559 − 0.507657i \(-0.830512\pi\)
−0.861559 + 0.507657i \(0.830512\pi\)
\(8\) −10496.3 −0.906009
\(9\) 6561.00 0.333333
\(10\) 0 0
\(11\) −39453.3 −0.812486 −0.406243 − 0.913765i \(-0.633161\pi\)
−0.406243 + 0.913765i \(0.633161\pi\)
\(12\) 30014.8 0.417850
\(13\) −50257.6 −0.488041 −0.244021 − 0.969770i \(-0.578466\pi\)
−0.244021 + 0.969770i \(0.578466\pi\)
\(14\) −130183. −0.905684
\(15\) 0 0
\(16\) 64888.9 0.247532
\(17\) 461329. 1.33965 0.669824 − 0.742520i \(-0.266369\pi\)
0.669824 + 0.742520i \(0.266369\pi\)
\(18\) 78030.9 0.175203
\(19\) −370082. −0.651489 −0.325744 − 0.945458i \(-0.605615\pi\)
−0.325744 + 0.945458i \(0.605615\pi\)
\(20\) 0 0
\(21\) 886628. 0.994843
\(22\) −469223. −0.427049
\(23\) −2.29014e6 −1.70642 −0.853210 − 0.521567i \(-0.825348\pi\)
−0.853210 + 0.521567i \(0.825348\pi\)
\(24\) 850203. 0.523085
\(25\) 0 0
\(26\) −597721. −0.256518
\(27\) −531441. −0.192450
\(28\) 4.05608e6 1.24708
\(29\) −1.28309e6 −0.336872 −0.168436 − 0.985713i \(-0.553872\pi\)
−0.168436 + 0.985713i \(0.553872\pi\)
\(30\) 0 0
\(31\) 6.51912e6 1.26783 0.633916 − 0.773402i \(-0.281447\pi\)
0.633916 + 0.773402i \(0.281447\pi\)
\(32\) 6.14585e6 1.03611
\(33\) 3.19571e6 0.469089
\(34\) 5.48665e6 0.704129
\(35\) 0 0
\(36\) −2.43120e6 −0.241246
\(37\) 1.45530e7 1.27657 0.638286 − 0.769800i \(-0.279644\pi\)
0.638286 + 0.769800i \(0.279644\pi\)
\(38\) −4.40144e6 −0.342427
\(39\) 4.07086e6 0.281771
\(40\) 0 0
\(41\) 1.34566e7 0.743720 0.371860 − 0.928289i \(-0.378720\pi\)
0.371860 + 0.928289i \(0.378720\pi\)
\(42\) 1.05448e7 0.522897
\(43\) −2.24211e7 −1.00011 −0.500057 − 0.865992i \(-0.666688\pi\)
−0.500057 + 0.865992i \(0.666688\pi\)
\(44\) 1.46195e7 0.588026
\(45\) 0 0
\(46\) −2.72369e7 −0.896907
\(47\) −1.45871e7 −0.436042 −0.218021 − 0.975944i \(-0.569960\pi\)
−0.218021 + 0.975944i \(0.569960\pi\)
\(48\) −5.25600e6 −0.142912
\(49\) 7.94618e7 1.96914
\(50\) 0 0
\(51\) −3.73677e7 −0.773447
\(52\) 1.86231e7 0.353213
\(53\) −6.49845e7 −1.13128 −0.565638 − 0.824654i \(-0.691370\pi\)
−0.565638 + 0.824654i \(0.691370\pi\)
\(54\) −6.32050e6 −0.101153
\(55\) 0 0
\(56\) 1.14893e8 1.56116
\(57\) 2.99766e7 0.376137
\(58\) −1.52599e7 −0.177063
\(59\) 1.04242e8 1.11997 0.559986 − 0.828502i \(-0.310807\pi\)
0.559986 + 0.828502i \(0.310807\pi\)
\(60\) 0 0
\(61\) 1.46085e8 1.35089 0.675445 − 0.737410i \(-0.263952\pi\)
0.675445 + 0.737410i \(0.263952\pi\)
\(62\) 7.75328e7 0.666382
\(63\) −7.18168e7 −0.574373
\(64\) 3.98704e7 0.297058
\(65\) 0 0
\(66\) 3.80071e7 0.246557
\(67\) −9.72944e7 −0.589863 −0.294932 − 0.955518i \(-0.595297\pi\)
−0.294932 + 0.955518i \(0.595297\pi\)
\(68\) −1.70947e8 −0.969553
\(69\) 1.85501e8 0.985202
\(70\) 0 0
\(71\) −2.89431e8 −1.35171 −0.675854 − 0.737036i \(-0.736225\pi\)
−0.675854 + 0.737036i \(0.736225\pi\)
\(72\) −6.88664e7 −0.302003
\(73\) −6.21359e7 −0.256088 −0.128044 − 0.991768i \(-0.540870\pi\)
−0.128044 + 0.991768i \(0.540870\pi\)
\(74\) 1.73081e8 0.670975
\(75\) 0 0
\(76\) 1.37135e8 0.471506
\(77\) 4.31856e8 1.40001
\(78\) 4.84154e7 0.148101
\(79\) 3.55247e8 1.02614 0.513072 − 0.858346i \(-0.328507\pi\)
0.513072 + 0.858346i \(0.328507\pi\)
\(80\) 0 0
\(81\) 4.30467e7 0.111111
\(82\) 1.60042e8 0.390905
\(83\) 2.13970e8 0.494881 0.247440 − 0.968903i \(-0.420411\pi\)
0.247440 + 0.968903i \(0.420411\pi\)
\(84\) −3.28543e8 −0.720004
\(85\) 0 0
\(86\) −2.66658e8 −0.525668
\(87\) 1.03930e8 0.194493
\(88\) 4.14114e8 0.736120
\(89\) −8.61695e8 −1.45579 −0.727895 − 0.685689i \(-0.759501\pi\)
−0.727895 + 0.685689i \(0.759501\pi\)
\(90\) 0 0
\(91\) 5.50121e8 0.840953
\(92\) 8.48617e8 1.23500
\(93\) −5.28049e8 −0.731983
\(94\) −1.73486e8 −0.229187
\(95\) 0 0
\(96\) −4.97814e8 −0.598200
\(97\) −1.00809e9 −1.15618 −0.578089 − 0.815974i \(-0.696201\pi\)
−0.578089 + 0.815974i \(0.696201\pi\)
\(98\) 9.45050e8 1.03499
\(99\) −2.58853e8 −0.270829
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.a.k.1.3 yes 4
3.2 odd 2 225.10.a.r.1.2 4
5.2 odd 4 75.10.b.h.49.5 8
5.3 odd 4 75.10.b.h.49.4 8
5.4 even 2 75.10.a.j.1.2 ✓ 4
15.2 even 4 225.10.b.n.199.4 8
15.8 even 4 225.10.b.n.199.5 8
15.14 odd 2 225.10.a.t.1.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
75.10.a.j.1.2 ✓ 4 5.4 even 2
75.10.a.k.1.3 yes 4 1.1 even 1 trivial
75.10.b.h.49.4 8 5.3 odd 4
75.10.b.h.49.5 8 5.2 odd 4
225.10.a.r.1.2 4 3.2 odd 2
225.10.a.t.1.3 4 15.14 odd 2
225.10.b.n.199.4 8 15.2 even 4
225.10.b.n.199.5 8 15.8 even 4