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Properties

Label 4950.2.f.h
Level $4950$
Weight $2$
Character orbit 4950.f
Analytic conductor $39.526$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [4950,2,Mod(4949,4950)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4950, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4950.4949");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4950 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4950.f (of order \(2\), degree \(1\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(39.5259490005\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 20 x^{14} - 20 x^{13} + 224 x^{12} + 96 x^{11} - 1194 x^{10} - 948 x^{9} + 5477 x^{8} + \cdots + 139876 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{10} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{10} q^{2} - q^{4} + ( - \beta_{6} - \beta_{4}) q^{7} + \beta_{10} q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{10} q^{2} - q^{4} + ( - \beta_{6} - \beta_{4}) q^{7} + \beta_{10} q^{8} + (\beta_{9} - \beta_{5}) q^{11} - \beta_{3} q^{13} + ( - \beta_{14} + \beta_{11}) q^{14} + q^{16} + ( - \beta_{15} - \beta_{10}) q^{17} + (\beta_{14} - \beta_{11} + \cdots + \beta_{8}) q^{19}+ \cdots + ( - \beta_{15} + 2 \beta_{13} + \cdots - 4 \beta_{10}) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 16 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 16 q^{4} + 16 q^{16} + 32 q^{29} - 32 q^{31} - 8 q^{34} + 40 q^{41} + 56 q^{49} - 16 q^{64} - 24 q^{74} - 48 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 20 x^{14} - 20 x^{13} + 224 x^{12} + 96 x^{11} - 1194 x^{10} - 948 x^{9} + 5477 x^{8} + \cdots + 139876 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 42\!\cdots\!24 \nu^{15} + \cdots + 67\!\cdots\!40 ) / 50\!\cdots\!86 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 481348746782012 \nu^{15} + \cdots + 38\!\cdots\!58 ) / 12\!\cdots\!10 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 12\!\cdots\!17 \nu^{15} + \cdots + 11\!\cdots\!04 ) / 22\!\cdots\!30 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 8106451142 \nu^{15} + 1539955076 \nu^{14} + 167604923732 \nu^{13} + \cdots - 14\!\cdots\!60 ) / 930455977173030 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 34\!\cdots\!18 \nu^{15} + \cdots + 86\!\cdots\!44 ) / 25\!\cdots\!30 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 20\!\cdots\!88 \nu^{15} - 472499417881202 \nu^{14} + \cdots - 50\!\cdots\!98 ) / 12\!\cdots\!10 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 527977472944 \nu^{15} + 141599127784 \nu^{14} + 11063309775556 \nu^{13} + \cdots - 10\!\cdots\!77 ) / 18\!\cdots\!99 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 14\!\cdots\!57 \nu^{15} + \cdots - 20\!\cdots\!88 ) / 25\!\cdots\!30 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 63899078032620 \nu^{15} + 71567999155928 \nu^{14} + \cdots - 94\!\cdots\!71 ) / 10\!\cdots\!45 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 287232506915542 \nu^{15} + 248774449861366 \nu^{14} + \cdots - 43\!\cdots\!66 ) / 37\!\cdots\!30 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 3485764132538 \nu^{15} - 2857995289308 \nu^{14} - 66583878332332 \nu^{13} + \cdots + 58\!\cdots\!28 ) / 34\!\cdots\!30 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 98\!\cdots\!78 \nu^{15} + \cdots + 14\!\cdots\!00 ) / 68\!\cdots\!54 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 24\!\cdots\!98 \nu^{15} + \cdots + 39\!\cdots\!12 ) / 10\!\cdots\!81 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 16\!\cdots\!00 \nu^{15} + \cdots - 26\!\cdots\!86 ) / 60\!\cdots\!70 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 12\!\cdots\!54 \nu^{15} + \cdots - 20\!\cdots\!72 ) / 40\!\cdots\!70 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( \beta_{14} + \beta_{13} - \beta_{12} + \beta_{11} - 2 \beta_{10} + \beta_{9} - \beta_{6} - 3 \beta_{5} + \cdots + \beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - \beta_{14} - 2 \beta_{13} + \beta_{12} + 3 \beta_{11} + 5 \beta_{10} - 2 \beta_{8} - \beta_{6} + \cdots + 5 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 10 \beta_{14} + 8 \beta_{13} - \beta_{12} + 2 \beta_{11} - 14 \beta_{10} + 2 \beta_{9} + 3 \beta_{8} + \cdots + 9 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 3 \beta_{15} - 10 \beta_{14} - 7 \beta_{13} + 2 \beta_{12} + 10 \beta_{11} + 27 \beta_{10} - 2 \beta_{9} + \cdots - 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 5 \beta_{15} + 46 \beta_{14} + 53 \beta_{13} + 9 \beta_{12} - 38 \beta_{11} - 56 \beta_{10} + \cdots + 200 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 39 \beta_{15} - 30 \beta_{14} - \beta_{13} - 26 \beta_{12} + 80 \beta_{11} + 34 \beta_{10} - 50 \beta_{9} + \cdots - 259 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 14 \beta_{15} - 413 \beta_{14} - 383 \beta_{13} + 196 \beta_{12} - 81 \beta_{11} + 694 \beta_{10} + \cdots + 2247 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 152 \beta_{15} + 1084 \beta_{14} + 864 \beta_{13} - 184 \beta_{12} - 672 \beta_{11} - 2736 \beta_{10} + \cdots + 283 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 1665 \beta_{15} - 9605 \beta_{14} - 9482 \beta_{13} + 548 \beta_{12} + 7289 \beta_{11} + 15386 \beta_{10} + \cdots + 10284 ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 8145 \beta_{15} + 28594 \beta_{14} + 22473 \beta_{13} + 3990 \beta_{12} - 29948 \beta_{11} + \cdots + 57975 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 27126 \beta_{15} - 69663 \beta_{14} - 65523 \beta_{13} - 14322 \beta_{12} + 98621 \beta_{11} + \cdots - 30943 ) / 2 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 24486 \beta_{15} + 41792 \beta_{14} + 28417 \beta_{13} + 44882 \beta_{12} - 87420 \beta_{11} + \cdots + 439012 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 76349 \beta_{15} + 269383 \beta_{14} + 226358 \beta_{13} - 91712 \beta_{12} + 181291 \beta_{11} + \cdots - 492700 ) / 2 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 600467 \beta_{15} - 2390674 \beta_{14} - 2218789 \beta_{13} + 369854 \beta_{12} + 2274788 \beta_{11} + \cdots + 7047353 ) / 2 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 2739500 \beta_{15} + 11553299 \beta_{14} + 9715743 \beta_{13} + 2117462 \beta_{12} - 10712457 \beta_{11} + \cdots + 7043139 ) / 2 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4950\mathbb{Z}\right)^\times\).

\(n\) \(551\) \(2377\) \(4501\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4949.1
−2.95227 − 1.58940i
−0.0663649 + 1.29650i
1.57695 − 1.67080i
−2.50433 − 1.67080i
−0.640742 − 1.58940i
2.24516 + 1.29650i
3.21144 − 0.0363060i
−0.869840 − 0.0363060i
−2.95227 + 1.58940i
−0.0663649 − 1.29650i
1.57695 + 1.67080i
−2.50433 + 1.67080i
−0.640742 + 1.58940i
2.24516 − 1.29650i
3.21144 + 0.0363060i
−0.869840 + 0.0363060i
− 1.00000i 0 −1.00000 0 0 −5.04973 1.00000i 0 0
4949.2 − 1.00000i 0 −1.00000 0 0 −3.35921 1.00000i 0 0
4949.3 − 1.00000i 0 −1.00000 0 0 −2.22130 1.00000i 0 0
4949.4 − 1.00000i 0 −1.00000 0 0 −0.530782 1.00000i 0 0
4949.5 − 1.00000i 0 −1.00000 0 0 0.530782 1.00000i 0 0
4949.6 − 1.00000i 0 −1.00000 0 0 2.22130 1.00000i 0 0
4949.7 − 1.00000i 0 −1.00000 0 0 3.35921 1.00000i 0 0
4949.8 − 1.00000i 0 −1.00000 0 0 5.04973 1.00000i 0 0
4949.9 1.00000i 0 −1.00000 0 0 −5.04973 − 1.00000i 0 0
4949.10 1.00000i 0 −1.00000 0 0 −3.35921 − 1.00000i 0 0
4949.11 1.00000i 0 −1.00000 0 0 −2.22130 − 1.00000i 0 0
4949.12 1.00000i 0 −1.00000 0 0 −0.530782 − 1.00000i 0 0
4949.13 1.00000i 0 −1.00000 0 0 0.530782 − 1.00000i 0 0
4949.14 1.00000i 0 −1.00000 0 0 2.22130 − 1.00000i 0 0
4949.15 1.00000i 0 −1.00000 0 0 3.35921 − 1.00000i 0 0
4949.16 1.00000i 0 −1.00000 0 0 5.04973 − 1.00000i 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 4949.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
33.d even 2 1 inner
165.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4950.2.f.h 16
3.b odd 2 1 4950.2.f.g 16
5.b even 2 1 inner 4950.2.f.h 16
5.c odd 4 1 4950.2.d.i ✓ 8
5.c odd 4 1 4950.2.d.k yes 8
11.b odd 2 1 4950.2.f.g 16
15.d odd 2 1 4950.2.f.g 16
15.e even 4 1 4950.2.d.j yes 8
15.e even 4 1 4950.2.d.l yes 8
33.d even 2 1 inner 4950.2.f.h 16
55.d odd 2 1 4950.2.f.g 16
55.e even 4 1 4950.2.d.j yes 8
55.e even 4 1 4950.2.d.l yes 8
165.d even 2 1 inner 4950.2.f.h 16
165.l odd 4 1 4950.2.d.i ✓ 8
165.l odd 4 1 4950.2.d.k yes 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4950.2.d.i ✓ 8 5.c odd 4 1
4950.2.d.i ✓ 8 165.l odd 4 1
4950.2.d.j yes 8 15.e even 4 1
4950.2.d.j yes 8 55.e even 4 1
4950.2.d.k yes 8 5.c odd 4 1
4950.2.d.k yes 8 165.l odd 4 1
4950.2.d.l yes 8 15.e even 4 1
4950.2.d.l yes 8 55.e even 4 1
4950.2.f.g 16 3.b odd 2 1
4950.2.f.g 16 11.b odd 2 1
4950.2.f.g 16 15.d odd 2 1
4950.2.f.g 16 55.d odd 2 1
4950.2.f.h 16 1.a even 1 1 trivial
4950.2.f.h 16 5.b even 2 1 inner
4950.2.f.h 16 33.d even 2 1 inner
4950.2.f.h 16 165.d even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(4950, [\chi])\):

\( T_{7}^{8} - 42T_{7}^{6} + 481T_{7}^{4} - 1552T_{7}^{2} + 400 \) Copy content Toggle raw display
\( T_{23}^{8} - 62T_{23}^{6} + 970T_{23}^{4} - 3038T_{23}^{2} + 2401 \) Copy content Toggle raw display
\( T_{29}^{4} - 8T_{29}^{3} - 43T_{29}^{2} + 236T_{29} - 220 \) Copy content Toggle raw display
\( T_{41}^{4} - 10T_{41}^{3} - 75T_{41}^{2} + 856T_{41} - 1772 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 1)^{8} \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( T^{16} \) Copy content Toggle raw display
$7$ \( (T^{8} - 42 T^{6} + \cdots + 400)^{2} \) Copy content Toggle raw display
$11$ \( (T^{8} - 114 T^{4} + 14641)^{2} \) Copy content Toggle raw display
$13$ \( (T^{8} - 56 T^{6} + \cdots + 2500)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} + 45 T^{2} + 484)^{4} \) Copy content Toggle raw display
$19$ \( (T^{8} + 98 T^{6} + \cdots + 7921)^{2} \) Copy content Toggle raw display
$23$ \( (T^{8} - 62 T^{6} + \cdots + 2401)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} - 8 T^{3} + \cdots - 220)^{4} \) Copy content Toggle raw display
$31$ \( (T^{4} + 8 T^{3} + \cdots + 398)^{4} \) Copy content Toggle raw display
$37$ \( (T^{8} + 142 T^{6} + \cdots + 25600)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} - 10 T^{3} + \cdots - 1772)^{4} \) Copy content Toggle raw display
$43$ \( (T^{8} - 124 T^{6} + \cdots + 256)^{2} \) Copy content Toggle raw display
$47$ \( (T^{8} - 214 T^{6} + \cdots + 1716100)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} - 76 T^{6} + \cdots + 18496)^{2} \) Copy content Toggle raw display
$59$ \( (T^{8} + 278 T^{6} + \cdots + 48400)^{2} \) Copy content Toggle raw display
$61$ \( (T^{8} + 208 T^{6} + \cdots + 10000)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} + 146 T^{2} + 968)^{4} \) Copy content Toggle raw display
$71$ \( (T^{8} + 494 T^{6} + \cdots + 4950625)^{2} \) Copy content Toggle raw display
$73$ \( (T^{8} - 116 T^{6} + \cdots + 1600)^{2} \) Copy content Toggle raw display
$79$ \( (T^{8} + 346 T^{6} + \cdots + 32672656)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} + 49 T^{2} + 578)^{4} \) Copy content Toggle raw display
$89$ \( (T^{8} + 472 T^{6} + \cdots + 1562500)^{2} \) Copy content Toggle raw display
$97$ \( (T^{8} + 616 T^{6} + \cdots + 226291849)^{2} \) Copy content Toggle raw display
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