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Properties

Label 1900.2.i.f
Level $1900$
Weight $2$
Character orbit 1900.i
Analytic conductor $15.172$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1900,2,Mod(201,1900)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1900, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1900.201");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1900 = 2^{2} \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1900.i (of order \(3\), degree \(2\), 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: \(15.1715763840\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 3 x^{11} + 15 x^{10} - 16 x^{9} + 79 x^{8} - 65 x^{7} + 298 x^{6} + 13 x^{5} + 233 x^{4} + \cdots + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{6} - \beta_1 + 1) q^{3} - \beta_{4} q^{7} + (\beta_{10} + \beta_{6} - \beta_{3} + \cdots + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{6} - \beta_1 + 1) q^{3} - \beta_{4} q^{7} + (\beta_{10} + \beta_{6} - \beta_{3} + \cdots + 1) q^{9}+ \cdots + (2 \beta_{10} + \beta_{7} + 4 \beta_{6} + \cdots + 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 3 q^{3} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 3 q^{3} - 3 q^{9} + 2 q^{11} + 7 q^{13} - q^{17} + q^{21} - 2 q^{23} - 24 q^{27} + q^{29} + 2 q^{31} + 10 q^{33} + 20 q^{37} + 36 q^{39} - 7 q^{41} + 19 q^{43} - 14 q^{47} + 8 q^{49} + 11 q^{51} + 6 q^{53} - 28 q^{57} - 5 q^{61} - 11 q^{63} + 14 q^{67} - 14 q^{69} + 8 q^{71} - 9 q^{73} + 2 q^{77} + q^{79} + 2 q^{81} - 26 q^{83} + 30 q^{87} - 8 q^{89} + 3 q^{91} + 9 q^{93} - 11 q^{97} - 18 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 3 x^{11} + 15 x^{10} - 16 x^{9} + 79 x^{8} - 65 x^{7} + 298 x^{6} + 13 x^{5} + 233 x^{4} + \cdots + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 638495 \nu^{11} - 5059670 \nu^{10} + 2185864 \nu^{9} - 61632615 \nu^{8} - 86473310 \nu^{7} + \cdots - 2681192682 ) / 2135068398 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 2112220 \nu^{11} + 5607653 \nu^{10} - 24678133 \nu^{9} + 8533470 \nu^{8} - 89367331 \nu^{7} + \cdots - 1951111485 ) / 711689466 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 3389210 \nu^{11} + 4511687 \nu^{10} + 20306405 \nu^{9} + 114731760 \nu^{8} + 262313951 \nu^{7} + \cdots + 1619981121 ) / 711689466 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 35281678 \nu^{11} + 70497029 \nu^{10} - 385595812 \nu^{9} - 77727516 \nu^{8} + \cdots - 5196018312 ) / 2135068398 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 60680476 \nu^{11} + 182679923 \nu^{10} - 905147470 \nu^{9} + 968701752 \nu^{8} + \cdots - 1794617748 ) / 2135068398 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 62867497 \nu^{11} - 203695424 \nu^{10} + 980933620 \nu^{9} - 1201195899 \nu^{8} + \cdots - 397480590 ) / 2135068398 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 102677425 \nu^{11} - 177003953 \nu^{10} + 1095480130 \nu^{9} + 493877715 \nu^{8} + \cdots + 17785843362 ) / 2135068398 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 131028322 \nu^{11} + 444681245 \nu^{10} - 2136716515 \nu^{9} + 2925836544 \nu^{8} + \cdots + 924096825 ) / 2135068398 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 175066273 \nu^{11} - 536276480 \nu^{10} + 2643593875 \nu^{9} - 2942137461 \nu^{8} + \cdots - 1015605495 ) / 2135068398 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 137992489 \nu^{11} - 469201302 \nu^{10} + 2243299983 \nu^{9} - 3049379221 \nu^{8} + \cdots - 965971179 ) / 711689466 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{10} + 3\beta_{6} - \beta_{3} - \beta_{2} + \beta _1 - 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{4} - \beta_{3} - 6\beta_{2} - 8 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -7\beta_{10} - \beta_{9} + \beta_{7} - 17\beta_{6} - 10\beta _1 - 17 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - \beta_{11} - 10 \beta_{10} - 3 \beta_{9} - \beta_{8} + 8 \beta_{7} - 21 \beta_{6} - 3 \beta_{5} + \cdots + 42 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -3\beta_{8} - 15\beta_{5} + 13\beta_{4} + 47\beta_{3} + 87\beta_{2} + 197 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 15\beta_{11} + 85\beta_{10} + 46\beta_{9} - 62\beta_{7} + 184\beta_{6} + 309\beta _1 + 184 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 46 \beta_{11} + 325 \beta_{10} + 169 \beta_{9} + 46 \beta_{8} - 131 \beta_{7} + 750 \beta_{6} + \cdots - 724 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 169\beta_{8} + 515\beta_{5} - 494\beta_{4} - 686\beta_{3} - 2339\beta_{2} - 3848 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( -515\beta_{11} - 2318\beta_{10} - 1693\beta_{9} + 1201\beta_{7} - 5301\beta_{6} - 5904\beta _1 - 5301 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 1693 \beta_{11} - 5412 \beta_{10} - 5102 \beta_{9} - 1693 \beta_{8} + 4011 \beta_{7} - 12008 \beta_{6} + \cdots + 18049 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1900\mathbb{Z}\right)^\times\).

\(n\) \(77\) \(401\) \(951\)
\(\chi(n)\) \(1\) \(\beta_{6}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
201.1
1.42323 − 2.46510i
1.28913 − 2.23284i
0.430479 − 0.745611i
−0.126563 + 0.219213i
−0.432807 + 0.749643i
−1.08347 + 1.87662i
1.42323 + 2.46510i
1.28913 + 2.23284i
0.430479 + 0.745611i
−0.126563 − 0.219213i
−0.432807 − 0.749643i
−1.08347 − 1.87662i
0 −0.923228 + 1.59908i 0 0 0 −1.72830 0 −0.204702 − 0.354554i 0
201.2 0 −0.789132 + 1.36682i 0 0 0 1.39989 0 0.254541 + 0.440878i 0
201.3 0 0.0695210 − 0.120414i 0 0 0 3.40785 0 1.49033 + 2.58133i 0
201.4 0 0.626563 − 1.08524i 0 0 0 −2.18534 0 0.714838 + 1.23814i 0
201.5 0 0.932807 − 1.61567i 0 0 0 −3.93019 0 −0.240257 − 0.416137i 0
201.6 0 1.58347 − 2.74265i 0 0 0 3.03607 0 −3.51475 − 6.08773i 0
501.1 0 −0.923228 − 1.59908i 0 0 0 −1.72830 0 −0.204702 + 0.354554i 0
501.2 0 −0.789132 − 1.36682i 0 0 0 1.39989 0 0.254541 − 0.440878i 0
501.3 0 0.0695210 + 0.120414i 0 0 0 3.40785 0 1.49033 − 2.58133i 0
501.4 0 0.626563 + 1.08524i 0 0 0 −2.18534 0 0.714838 − 1.23814i 0
501.5 0 0.932807 + 1.61567i 0 0 0 −3.93019 0 −0.240257 + 0.416137i 0
501.6 0 1.58347 + 2.74265i 0 0 0 3.03607 0 −3.51475 + 6.08773i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 201.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1900.2.i.f yes 12
5.b even 2 1 1900.2.i.e ✓ 12
5.c odd 4 2 1900.2.s.e 24
19.c even 3 1 inner 1900.2.i.f yes 12
95.i even 6 1 1900.2.i.e ✓ 12
95.m odd 12 2 1900.2.s.e 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1900.2.i.e ✓ 12 5.b even 2 1
1900.2.i.e ✓ 12 95.i even 6 1
1900.2.i.f yes 12 1.a even 1 1 trivial
1900.2.i.f yes 12 19.c even 3 1 inner
1900.2.s.e 24 5.c odd 4 2
1900.2.s.e 24 95.m odd 12 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{12} - 3 T_{3}^{11} + 15 T_{3}^{10} - 16 T_{3}^{9} + 79 T_{3}^{8} - 77 T_{3}^{7} + 274 T_{3}^{6} + \cdots + 9 \) acting on \(S_{2}^{\mathrm{new}}(1900, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( T^{12} - 3 T^{11} + \cdots + 9 \) Copy content Toggle raw display
$5$ \( T^{12} \) Copy content Toggle raw display
$7$ \( (T^{6} - 23 T^{4} + \cdots - 215)^{2} \) Copy content Toggle raw display
$11$ \( (T^{6} - T^{5} - 22 T^{4} + \cdots + 27)^{2} \) Copy content Toggle raw display
$13$ \( T^{12} - 7 T^{11} + \cdots + 199809 \) Copy content Toggle raw display
$17$ \( T^{12} + T^{11} + \cdots + 762129 \) Copy content Toggle raw display
$19$ \( T^{12} + 33 T^{10} + \cdots + 47045881 \) Copy content Toggle raw display
$23$ \( T^{12} + 2 T^{11} + \cdots + 455625 \) Copy content Toggle raw display
$29$ \( T^{12} - T^{11} + \cdots + 729 \) Copy content Toggle raw display
$31$ \( (T^{6} - T^{5} - 120 T^{4} + \cdots + 2213)^{2} \) Copy content Toggle raw display
$37$ \( (T^{6} - 10 T^{5} + \cdots - 2880)^{2} \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots + 303282225 \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 1154300625 \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 2546514369 \) Copy content Toggle raw display
$53$ \( T^{12} - 6 T^{11} + \cdots + 19210689 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 624650049 \) Copy content Toggle raw display
$61$ \( T^{12} + 5 T^{11} + \cdots + 3932289 \) Copy content Toggle raw display
$67$ \( T^{12} - 14 T^{11} + \cdots + 5089536 \) Copy content Toggle raw display
$71$ \( T^{12} - 8 T^{11} + \cdots + 531441 \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 352125225 \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 526105969 \) Copy content Toggle raw display
$83$ \( (T^{6} + 13 T^{5} + \cdots + 27345)^{2} \) Copy content Toggle raw display
$89$ \( T^{12} + 8 T^{11} + \cdots + 3908529 \) Copy content Toggle raw display
$97$ \( T^{12} + 11 T^{11} + \cdots + 22325625 \) Copy content Toggle raw display
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