Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [6975,2,Mod(1,6975)] 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("6975.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(6975, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 6975 = 3^{2} \cdot 5^{2} \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6975.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [3,0,0,2,0,0,-4,-3,0,0,2,0,-4,5,0,-4,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(55.6956554098\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.229.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - 4x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 93)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-0.254102\) of defining polynomial
Character \(\chi\) \(=\) 6975.1

$q$-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)
 
\(f(q)\) \(=\) \(q+0.254102 q^{2} -1.93543 q^{4} -3.68133 q^{7} -1.00000 q^{8} +5.87086 q^{11} +3.87086 q^{13} -0.935432 q^{14} +3.61676 q^{16} -5.36266 q^{17} +4.69774 q^{19} +1.49180 q^{22} -2.50820 q^{23} +0.983593 q^{26} +7.12497 q^{28} -7.23353 q^{29} -1.00000 q^{31} +2.91903 q^{32} -1.36266 q^{34} -0.508203 q^{37} +1.19370 q^{38} +6.69774 q^{41} -8.85446 q^{43} -11.3627 q^{44} -0.637339 q^{46} +5.01641 q^{47} +6.55220 q^{49} -7.49180 q^{52} +2.37907 q^{53} +3.68133 q^{56} -1.83805 q^{58} -5.81047 q^{59} -4.34625 q^{61} -0.254102 q^{62} -6.49180 q^{64} -4.00000 q^{67} +10.3791 q^{68} +4.15672 q^{71} +5.52461 q^{73} -0.129135 q^{74} -9.09215 q^{76} -21.6126 q^{77} +7.36266 q^{79} +1.70191 q^{82} +11.3627 q^{83} -2.24993 q^{86} -5.87086 q^{88} +6.00000 q^{89} -14.2499 q^{91} +4.85446 q^{92} +1.27468 q^{94} -3.17313 q^{97} +1.66492 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 2 q^{4} - 4 q^{7} - 3 q^{8} + 2 q^{11} - 4 q^{13} + 5 q^{14} - 4 q^{16} - 2 q^{17} + 4 q^{19} + 6 q^{22} - 6 q^{23} + 6 q^{26} + 5 q^{28} + 8 q^{29} - 3 q^{31} + 4 q^{32} + 10 q^{34} + 27 q^{38} + 10 q^{41}+ \cdots + q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.254102 0.179677 0.0898385 − 0.995956i \(-0.471365\pi\)
0.0898385 + 0.995956i \(0.471365\pi\)
\(3\) 0 0
\(4\) −1.93543 −0.967716
\(5\) 0 0
\(6\) 0 0
\(7\) −3.68133 −1.39141 −0.695706 − 0.718327i \(-0.744908\pi\)
−0.695706 + 0.718327i \(0.744908\pi\)
\(8\) −1.00000 −0.353553
\(9\) 0 0
\(10\) 0 0
\(11\) 5.87086 1.77013 0.885066 − 0.465465i \(-0.154113\pi\)
0.885066 + 0.465465i \(0.154113\pi\)
\(12\) 0 0
\(13\) 3.87086 1.07358 0.536792 − 0.843714i \(-0.319636\pi\)
0.536792 + 0.843714i \(0.319636\pi\)
\(14\) −0.935432 −0.250005
\(15\) 0 0
\(16\) 3.61676 0.904191
\(17\) −5.36266 −1.30064 −0.650318 − 0.759662i \(-0.725364\pi\)
−0.650318 + 0.759662i \(0.725364\pi\)
\(18\) 0 0
\(19\) 4.69774 1.07773 0.538867 − 0.842391i \(-0.318852\pi\)
0.538867 + 0.842391i \(0.318852\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 1.49180 0.318052
\(23\) −2.50820 −0.522997 −0.261498 − 0.965204i \(-0.584217\pi\)
−0.261498 + 0.965204i \(0.584217\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0.983593 0.192899
\(27\) 0 0
\(28\) 7.12497 1.34649
\(29\) −7.23353 −1.34323 −0.671616 − 0.740899i \(-0.734400\pi\)
−0.671616 + 0.740899i \(0.734400\pi\)
\(30\) 0 0
\(31\) −1.00000 −0.179605
\(32\) 2.91903 0.516016
\(33\) 0 0
\(34\) −1.36266 −0.233694
\(35\) 0 0
\(36\) 0 0
\(37\) −0.508203 −0.0835481 −0.0417741 − 0.999127i \(-0.513301\pi\)
−0.0417741 + 0.999127i \(0.513301\pi\)
\(38\) 1.19370 0.193644
\(39\) 0 0
\(40\) 0 0
\(41\) 6.69774 1.04601 0.523005 − 0.852329i \(-0.324811\pi\)
0.523005 + 0.852329i \(0.324811\pi\)
\(42\) 0 0
\(43\) −8.85446 −1.35029 −0.675146 − 0.737684i \(-0.735919\pi\)
−0.675146 + 0.737684i \(0.735919\pi\)
\(44\) −11.3627 −1.71299
\(45\) 0 0
\(46\) −0.637339 −0.0939705
\(47\) 5.01641 0.731718 0.365859 − 0.930670i \(-0.380775\pi\)
0.365859 + 0.930670i \(0.380775\pi\)
\(48\) 0 0
\(49\) 6.55220 0.936028
\(50\) 0 0
\(51\) 0 0
\(52\) −7.49180 −1.03893
\(53\) 2.37907 0.326790 0.163395 − 0.986561i \(-0.447755\pi\)
0.163395 + 0.986561i \(0.447755\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 3.68133 0.491938
\(57\) 0 0
\(58\) −1.83805 −0.241348
\(59\) −5.81047 −0.756458 −0.378229 − 0.925712i \(-0.623467\pi\)
−0.378229 + 0.925712i \(0.623467\pi\)
\(60\) 0 0
\(61\) −4.34625 −0.556481 −0.278240 − 0.960511i \(-0.589751\pi\)
−0.278240 + 0.960511i \(0.589751\pi\)
\(62\) −0.254102 −0.0322709
\(63\) 0 0
\(64\) −6.49180 −0.811475
\(65\) 0 0
\(66\) 0 0
\(67\) −4.00000 −0.488678 −0.244339 − 0.969690i \(-0.578571\pi\)
−0.244339 + 0.969690i \(0.578571\pi\)
\(68\) 10.3791 1.25865
\(69\) 0 0
\(70\) 0 0
\(71\) 4.15672 0.493312 0.246656 − 0.969103i \(-0.420668\pi\)
0.246656 + 0.969103i \(0.420668\pi\)
\(72\) 0 0
\(73\) 5.52461 0.646607 0.323303 − 0.946295i \(-0.395207\pi\)
0.323303 + 0.946295i \(0.395207\pi\)
\(74\) −0.129135 −0.0150117
\(75\) 0 0
\(76\) −9.09215 −1.04294
\(77\) −21.6126 −2.46298
\(78\) 0 0
\(79\) 7.36266 0.828364 0.414182 − 0.910194i \(-0.364068\pi\)
0.414182 + 0.910194i \(0.364068\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 1.70191 0.187944
\(83\) 11.3627 1.24721 0.623607 − 0.781738i \(-0.285667\pi\)
0.623607 + 0.781738i \(0.285667\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −2.24993 −0.242616
\(87\) 0 0
\(88\) −5.87086 −0.625836
\(89\) 6.00000 0.635999 0.317999 − 0.948091i \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) 0 0
\(91\) −14.2499 −1.49380
\(92\) 4.85446 0.506112
\(93\) 0 0
\(94\) 1.27468 0.131473
\(95\) 0 0
\(96\) 0 0
\(97\) −3.17313 −0.322182 −0.161091 − 0.986940i \(-0.551501\pi\)
−0.161091 + 0.986940i \(0.551501\pi\)
\(98\) 1.66492 0.168183
\(99\) 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 6975.2.a.bb.1.2 3
3.2 odd 2 2325.2.a.s.1.2 3
5.4 even 2 279.2.a.c.1.2 3
15.2 even 4 2325.2.c.n.1024.3 6
15.8 even 4 2325.2.c.n.1024.4 6
15.14 odd 2 93.2.a.b.1.2 ✓ 3
20.19 odd 2 4464.2.a.bq.1.1 3
60.59 even 2 1488.2.a.t.1.3 3
105.104 even 2 4557.2.a.v.1.2 3
120.29 odd 2 5952.2.a.bz.1.1 3
120.59 even 2 5952.2.a.cf.1.1 3
155.154 odd 2 8649.2.a.p.1.2 3
465.464 even 2 2883.2.a.f.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
93.2.a.b.1.2 ✓ 3 15.14 odd 2
279.2.a.c.1.2 3 5.4 even 2
1488.2.a.t.1.3 3 60.59 even 2
2325.2.a.s.1.2 3 3.2 odd 2
2325.2.c.n.1024.3 6 15.2 even 4
2325.2.c.n.1024.4 6 15.8 even 4
2883.2.a.f.1.2 3 465.464 even 2
4464.2.a.bq.1.1 3 20.19 odd 2
4557.2.a.v.1.2 3 105.104 even 2
5952.2.a.bz.1.1 3 120.29 odd 2
5952.2.a.cf.1.1 3 120.59 even 2
6975.2.a.bb.1.2 3 1.1 even 1 trivial
8649.2.a.p.1.2 3 155.154 odd 2