Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [5175,2,Mod(1,5175)] 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("5175.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(5175, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 5175 = 3^{2} \cdot 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5175.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,6,0,0,-2,0,0,0,-8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(41.3225830460\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{10})^+\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 69)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-0.618034\) of defining polynomial
Character \(\chi\) \(=\) 5175.1

$q$-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)
 
\(f(q)\) \(=\) \(q+2.23607 q^{2} +3.00000 q^{4} +1.23607 q^{7} +2.23607 q^{8} -4.00000 q^{11} -4.47214 q^{13} +2.76393 q^{14} -1.00000 q^{16} -7.23607 q^{17} +2.76393 q^{19} -8.94427 q^{22} +1.00000 q^{23} -10.0000 q^{26} +3.70820 q^{28} +4.47214 q^{29} +2.47214 q^{31} -6.70820 q^{32} -16.1803 q^{34} +4.47214 q^{37} +6.18034 q^{38} -6.94427 q^{41} -7.70820 q^{43} -12.0000 q^{44} +2.23607 q^{46} -4.00000 q^{47} -5.47214 q^{49} -13.4164 q^{52} -0.763932 q^{53} +2.76393 q^{56} +10.0000 q^{58} -12.9443 q^{59} -4.47214 q^{61} +5.52786 q^{62} -13.0000 q^{64} -5.23607 q^{67} -21.7082 q^{68} +8.00000 q^{71} +10.9443 q^{73} +10.0000 q^{74} +8.29180 q^{76} -4.94427 q^{77} -3.70820 q^{79} -15.5279 q^{82} +4.00000 q^{83} -17.2361 q^{86} -8.94427 q^{88} -3.23607 q^{89} -5.52786 q^{91} +3.00000 q^{92} -8.94427 q^{94} +0.472136 q^{97} -12.2361 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 6 q^{4} - 2 q^{7} - 8 q^{11} + 10 q^{14} - 2 q^{16} - 10 q^{17} + 10 q^{19} + 2 q^{23} - 20 q^{26} - 6 q^{28} - 4 q^{31} - 10 q^{34} - 10 q^{38} + 4 q^{41} - 2 q^{43} - 24 q^{44} - 8 q^{47} - 2 q^{49}+ \cdots - 20 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.23607 1.58114 0.790569 − 0.612372i \(-0.209785\pi\)
0.790569 + 0.612372i \(0.209785\pi\)
\(3\) 0 0
\(4\) 3.00000 1.50000
\(5\) 0 0
\(6\) 0 0
\(7\) 1.23607 0.467190 0.233595 − 0.972334i \(-0.424951\pi\)
0.233595 + 0.972334i \(0.424951\pi\)
\(8\) 2.23607 0.790569
\(9\) 0 0
\(10\) 0 0
\(11\) −4.00000 −1.20605 −0.603023 − 0.797724i \(-0.706037\pi\)
−0.603023 + 0.797724i \(0.706037\pi\)
\(12\) 0 0
\(13\) −4.47214 −1.24035 −0.620174 − 0.784465i \(-0.712938\pi\)
−0.620174 + 0.784465i \(0.712938\pi\)
\(14\) 2.76393 0.738692
\(15\) 0 0
\(16\) −1.00000 −0.250000
\(17\) −7.23607 −1.75500 −0.877502 − 0.479573i \(-0.840792\pi\)
−0.877502 + 0.479573i \(0.840792\pi\)
\(18\) 0 0
\(19\) 2.76393 0.634089 0.317045 − 0.948411i \(-0.397309\pi\)
0.317045 + 0.948411i \(0.397309\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −8.94427 −1.90693
\(23\) 1.00000 0.208514
\(24\) 0 0
\(25\) 0 0
\(26\) −10.0000 −1.96116
\(27\) 0 0
\(28\) 3.70820 0.700785
\(29\) 4.47214 0.830455 0.415227 − 0.909718i \(-0.363702\pi\)
0.415227 + 0.909718i \(0.363702\pi\)
\(30\) 0 0
\(31\) 2.47214 0.444009 0.222004 − 0.975046i \(-0.428740\pi\)
0.222004 + 0.975046i \(0.428740\pi\)
\(32\) −6.70820 −1.18585
\(33\) 0 0
\(34\) −16.1803 −2.77491
\(35\) 0 0
\(36\) 0 0
\(37\) 4.47214 0.735215 0.367607 − 0.929981i \(-0.380177\pi\)
0.367607 + 0.929981i \(0.380177\pi\)
\(38\) 6.18034 1.00258
\(39\) 0 0
\(40\) 0 0
\(41\) −6.94427 −1.08451 −0.542257 − 0.840213i \(-0.682430\pi\)
−0.542257 + 0.840213i \(0.682430\pi\)
\(42\) 0 0
\(43\) −7.70820 −1.17549 −0.587745 − 0.809046i \(-0.699984\pi\)
−0.587745 + 0.809046i \(0.699984\pi\)
\(44\) −12.0000 −1.80907
\(45\) 0 0
\(46\) 2.23607 0.329690
\(47\) −4.00000 −0.583460 −0.291730 − 0.956501i \(-0.594231\pi\)
−0.291730 + 0.956501i \(0.594231\pi\)
\(48\) 0 0
\(49\) −5.47214 −0.781734
\(50\) 0 0
\(51\) 0 0
\(52\) −13.4164 −1.86052
\(53\) −0.763932 −0.104934 −0.0524671 − 0.998623i \(-0.516708\pi\)
−0.0524671 + 0.998623i \(0.516708\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 2.76393 0.369346
\(57\) 0 0
\(58\) 10.0000 1.31306
\(59\) −12.9443 −1.68520 −0.842600 − 0.538539i \(-0.818976\pi\)
−0.842600 + 0.538539i \(0.818976\pi\)
\(60\) 0 0
\(61\) −4.47214 −0.572598 −0.286299 − 0.958140i \(-0.592425\pi\)
−0.286299 + 0.958140i \(0.592425\pi\)
\(62\) 5.52786 0.702039
\(63\) 0 0
\(64\) −13.0000 −1.62500
\(65\) 0 0
\(66\) 0 0
\(67\) −5.23607 −0.639688 −0.319844 − 0.947470i \(-0.603630\pi\)
−0.319844 + 0.947470i \(0.603630\pi\)
\(68\) −21.7082 −2.63251
\(69\) 0 0
\(70\) 0 0
\(71\) 8.00000 0.949425 0.474713 − 0.880141i \(-0.342552\pi\)
0.474713 + 0.880141i \(0.342552\pi\)
\(72\) 0 0
\(73\) 10.9443 1.28093 0.640465 − 0.767987i \(-0.278742\pi\)
0.640465 + 0.767987i \(0.278742\pi\)
\(74\) 10.0000 1.16248
\(75\) 0 0
\(76\) 8.29180 0.951134
\(77\) −4.94427 −0.563452
\(78\) 0 0
\(79\) −3.70820 −0.417206 −0.208603 − 0.978000i \(-0.566892\pi\)
−0.208603 + 0.978000i \(0.566892\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) −15.5279 −1.71477
\(83\) 4.00000 0.439057 0.219529 − 0.975606i \(-0.429548\pi\)
0.219529 + 0.975606i \(0.429548\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −17.2361 −1.85861
\(87\) 0 0
\(88\) −8.94427 −0.953463
\(89\) −3.23607 −0.343023 −0.171511 − 0.985182i \(-0.554865\pi\)
−0.171511 + 0.985182i \(0.554865\pi\)
\(90\) 0 0
\(91\) −5.52786 −0.579478
\(92\) 3.00000 0.312772
\(93\) 0 0
\(94\) −8.94427 −0.922531
\(95\) 0 0
\(96\) 0 0
\(97\) 0.472136 0.0479381 0.0239691 − 0.999713i \(-0.492370\pi\)
0.0239691 + 0.999713i \(0.492370\pi\)
\(98\) −12.2361 −1.23603
\(99\) 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 5175.2.a.bk.1.2 2
3.2 odd 2 1725.2.a.ba.1.1 2
5.4 even 2 207.2.a.c.1.1 2
15.2 even 4 1725.2.b.o.1174.1 4
15.8 even 4 1725.2.b.o.1174.4 4
15.14 odd 2 69.2.a.b.1.2 ✓ 2
20.19 odd 2 3312.2.a.bb.1.2 2
60.59 even 2 1104.2.a.m.1.1 2
105.104 even 2 3381.2.a.t.1.2 2
115.114 odd 2 4761.2.a.v.1.1 2
120.29 odd 2 4416.2.a.bm.1.2 2
120.59 even 2 4416.2.a.bg.1.2 2
165.164 even 2 8349.2.a.i.1.1 2
345.344 even 2 1587.2.a.i.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
69.2.a.b.1.2 ✓ 2 15.14 odd 2
207.2.a.c.1.1 2 5.4 even 2
1104.2.a.m.1.1 2 60.59 even 2
1587.2.a.i.1.2 2 345.344 even 2
1725.2.a.ba.1.1 2 3.2 odd 2
1725.2.b.o.1174.1 4 15.2 even 4
1725.2.b.o.1174.4 4 15.8 even 4
3312.2.a.bb.1.2 2 20.19 odd 2
3381.2.a.t.1.2 2 105.104 even 2
4416.2.a.bg.1.2 2 120.59 even 2
4416.2.a.bm.1.2 2 120.29 odd 2
4761.2.a.v.1.1 2 115.114 odd 2
5175.2.a.bk.1.2 2 1.1 even 1 trivial
8349.2.a.i.1.1 2 165.164 even 2