Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [8325,2,Mod(1,8325)] 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("8325.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(8325, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 8325 = 3^{2} \cdot 5^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8325.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,2,0,-2,0,0,8,0,0,0,4,0,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(66.4754596827\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{14 +2 \sqrt{5}})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 7x^{2} + 11 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 555)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.54336\) of defining polynomial
Character \(\chi\) \(=\) 8325.1

$q$-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)
 
\(f(q)\) \(=\) \(q-0.618034 q^{2} -1.61803 q^{4} +4.49721 q^{7} +2.23607 q^{8} +2.54336 q^{11} -1.57188 q^{13} -2.77943 q^{14} +1.85410 q^{16} +2.46869 q^{17} +3.04057 q^{19} -1.57188 q^{22} +7.23049 q^{23} +0.971478 q^{26} -7.27664 q^{28} +4.21499 q^{29} +4.39746 q^{31} -5.61803 q^{32} -1.52573 q^{34} +1.00000 q^{37} -1.87918 q^{38} +4.42812 q^{41} -0.207546 q^{43} -4.11525 q^{44} -4.46869 q^{46} +0.746304 q^{47} +13.2249 q^{49} +2.54336 q^{52} -8.08115 q^{53} +10.0561 q^{56} -2.60501 q^{58} +9.79893 q^{59} -3.31074 q^{61} -2.71778 q^{62} -0.236068 q^{64} +6.63009 q^{67} -3.99442 q^{68} +4.03410 q^{71} +11.7989 q^{73} -0.618034 q^{74} -4.91975 q^{76} +11.4380 q^{77} -16.9046 q^{79} -2.73673 q^{82} +1.72983 q^{83} +0.128270 q^{86} +5.68713 q^{88} +17.9638 q^{89} -7.06910 q^{91} -11.6992 q^{92} -0.461241 q^{94} -10.0440 q^{97} -8.17345 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{2} - 2 q^{4} + 8 q^{7} + 4 q^{11} + 2 q^{13} + 4 q^{14} - 6 q^{16} + 2 q^{17} - 4 q^{19} + 2 q^{22} + 6 q^{26} - 4 q^{28} + 12 q^{29} - 2 q^{31} - 18 q^{32} + 6 q^{34} + 4 q^{37} - 2 q^{38} + 26 q^{41}+ \cdots - 8 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.618034 −0.437016 −0.218508 − 0.975835i \(-0.570119\pi\)
−0.218508 + 0.975835i \(0.570119\pi\)
\(3\) 0 0
\(4\) −1.61803 −0.809017
\(5\) 0 0
\(6\) 0 0
\(7\) 4.49721 1.69979 0.849893 − 0.526955i \(-0.176666\pi\)
0.849893 + 0.526955i \(0.176666\pi\)
\(8\) 2.23607 0.790569
\(9\) 0 0
\(10\) 0 0
\(11\) 2.54336 0.766852 0.383426 − 0.923572i \(-0.374744\pi\)
0.383426 + 0.923572i \(0.374744\pi\)
\(12\) 0 0
\(13\) −1.57188 −0.435962 −0.217981 − 0.975953i \(-0.569947\pi\)
−0.217981 + 0.975953i \(0.569947\pi\)
\(14\) −2.77943 −0.742834
\(15\) 0 0
\(16\) 1.85410 0.463525
\(17\) 2.46869 0.598745 0.299373 − 0.954136i \(-0.403223\pi\)
0.299373 + 0.954136i \(0.403223\pi\)
\(18\) 0 0
\(19\) 3.04057 0.697556 0.348778 − 0.937205i \(-0.386597\pi\)
0.348778 + 0.937205i \(0.386597\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −1.57188 −0.335127
\(23\) 7.23049 1.50766 0.753831 − 0.657068i \(-0.228204\pi\)
0.753831 + 0.657068i \(0.228204\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0.971478 0.190522
\(27\) 0 0
\(28\) −7.27664 −1.37516
\(29\) 4.21499 0.782705 0.391352 − 0.920241i \(-0.372007\pi\)
0.391352 + 0.920241i \(0.372007\pi\)
\(30\) 0 0
\(31\) 4.39746 0.789808 0.394904 − 0.918722i \(-0.370778\pi\)
0.394904 + 0.918722i \(0.370778\pi\)
\(32\) −5.61803 −0.993137
\(33\) 0 0
\(34\) −1.52573 −0.261661
\(35\) 0 0
\(36\) 0 0
\(37\) 1.00000 0.164399
\(38\) −1.87918 −0.304843
\(39\) 0 0
\(40\) 0 0
\(41\) 4.42812 0.691556 0.345778 − 0.938316i \(-0.387615\pi\)
0.345778 + 0.938316i \(0.387615\pi\)
\(42\) 0 0
\(43\) −0.207546 −0.0316504 −0.0158252 − 0.999875i \(-0.505038\pi\)
−0.0158252 + 0.999875i \(0.505038\pi\)
\(44\) −4.11525 −0.620397
\(45\) 0 0
\(46\) −4.46869 −0.658872
\(47\) 0.746304 0.108860 0.0544298 − 0.998518i \(-0.482666\pi\)
0.0544298 + 0.998518i \(0.482666\pi\)
\(48\) 0 0
\(49\) 13.2249 1.88927
\(50\) 0 0
\(51\) 0 0
\(52\) 2.54336 0.352701
\(53\) −8.08115 −1.11003 −0.555016 − 0.831840i \(-0.687288\pi\)
−0.555016 + 0.831840i \(0.687288\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 10.0561 1.34380
\(57\) 0 0
\(58\) −2.60501 −0.342055
\(59\) 9.79893 1.27571 0.637856 − 0.770156i \(-0.279821\pi\)
0.637856 + 0.770156i \(0.279821\pi\)
\(60\) 0 0
\(61\) −3.31074 −0.423897 −0.211948 − 0.977281i \(-0.567981\pi\)
−0.211948 + 0.977281i \(0.567981\pi\)
\(62\) −2.71778 −0.345159
\(63\) 0 0
\(64\) −0.236068 −0.0295085
\(65\) 0 0
\(66\) 0 0
\(67\) 6.63009 0.809994 0.404997 − 0.914318i \(-0.367273\pi\)
0.404997 + 0.914318i \(0.367273\pi\)
\(68\) −3.99442 −0.484395
\(69\) 0 0
\(70\) 0 0
\(71\) 4.03410 0.478759 0.239380 − 0.970926i \(-0.423056\pi\)
0.239380 + 0.970926i \(0.423056\pi\)
\(72\) 0 0
\(73\) 11.7989 1.38096 0.690480 − 0.723351i \(-0.257399\pi\)
0.690480 + 0.723351i \(0.257399\pi\)
\(74\) −0.618034 −0.0718450
\(75\) 0 0
\(76\) −4.91975 −0.564334
\(77\) 11.4380 1.30349
\(78\) 0 0
\(79\) −16.9046 −1.90192 −0.950958 − 0.309320i \(-0.899898\pi\)
−0.950958 + 0.309320i \(0.899898\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) −2.73673 −0.302221
\(83\) 1.72983 0.189874 0.0949370 − 0.995483i \(-0.469735\pi\)
0.0949370 + 0.995483i \(0.469735\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0.128270 0.0138317
\(87\) 0 0
\(88\) 5.68713 0.606250
\(89\) 17.9638 1.90416 0.952078 − 0.305855i \(-0.0989424\pi\)
0.952078 + 0.305855i \(0.0989424\pi\)
\(90\) 0 0
\(91\) −7.06910 −0.741043
\(92\) −11.6992 −1.21972
\(93\) 0 0
\(94\) −0.461241 −0.0475734
\(95\) 0 0
\(96\) 0 0
\(97\) −10.0440 −1.01982 −0.509908 − 0.860229i \(-0.670321\pi\)
−0.509908 + 0.860229i \(0.670321\pi\)
\(98\) −8.17345 −0.825643
\(99\) 0 0
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 8325.2.a.bw.1.2 4
3.2 odd 2 2775.2.a.w.1.4 4
5.2 odd 4 1665.2.c.d.334.4 8
5.3 odd 4 1665.2.c.d.334.6 8
5.4 even 2 8325.2.a.bt.1.3 4
15.2 even 4 555.2.c.b.334.5 yes 8
15.8 even 4 555.2.c.b.334.3 ✓ 8
15.14 odd 2 2775.2.a.y.1.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
555.2.c.b.334.3 ✓ 8 15.8 even 4
555.2.c.b.334.5 yes 8 15.2 even 4
1665.2.c.d.334.4 8 5.2 odd 4
1665.2.c.d.334.6 8 5.3 odd 4
2775.2.a.w.1.4 4 3.2 odd 2
2775.2.a.y.1.1 4 15.14 odd 2
8325.2.a.bt.1.3 4 5.4 even 2
8325.2.a.bw.1.2 4 1.1 even 1 trivial