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Properties

Label 900.2.n.c
Level $900$
Weight $2$
Character orbit 900.n
Analytic conductor $7.187$
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] = [900,2,Mod(181,900)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(900, base_ring=CyclotomicField(10))
 
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("900.181");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 900.n (of order \(5\), degree \(4\), 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: \(7.18653618192\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(3\) over \(\Q(\zeta_{5})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 3 x^{11} + 13 x^{10} - 24 x^{9} + 93 x^{8} - 6 x^{7} + 342 x^{6} + 786 x^{5} + 1473 x^{4} + \cdots + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 100)
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( - \beta_{11} - \beta_{10} - \beta_{9} + \cdots + 1) q^{5}+ \cdots + ( - \beta_{11} + \beta_{10} - \beta_{6} + \cdots - 1) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{11} - \beta_{10} - \beta_{9} + \cdots + 1) q^{5}+ \cdots + ( - 3 \beta_{11} - 4 \beta_{10} + \cdots + 2) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 4 q^{5} - 2 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 4 q^{5} - 2 q^{7} + 5 q^{11} - 2 q^{13} - q^{17} - 8 q^{19} + 6 q^{23} - 26 q^{25} + 18 q^{29} + 12 q^{31} + 3 q^{35} + 13 q^{37} + 23 q^{41} + 50 q^{43} - q^{47} + 34 q^{49} - 21 q^{53} + 5 q^{55} - 9 q^{59} - 26 q^{61} + 18 q^{65} - 37 q^{67} - 21 q^{71} + 18 q^{73} + 60 q^{77} - 24 q^{79} + 46 q^{83} - 16 q^{85} + 2 q^{89} - 32 q^{91} - 6 q^{95} - 7 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 3 x^{11} + 13 x^{10} - 24 x^{9} + 93 x^{8} - 6 x^{7} + 342 x^{6} + 786 x^{5} + 1473 x^{4} + \cdots + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 260325772 \nu^{11} - 576792708 \nu^{10} + 3020952123 \nu^{9} - 4900209927 \nu^{8} + \cdots + 112273499910 ) / 438461548785 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 3457672033 \nu^{11} - 11783661403 \nu^{10} + 48283397793 \nu^{9} - 97783045044 \nu^{8} + \cdots + 178985069048 ) / 1753846195140 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 5369632561 \nu^{11} - 15433734165 \nu^{10} + 67292419131 \nu^{9} - 119576878026 \nu^{8} + \cdots + 391154012184 ) / 1753846195140 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 22373133631 \nu^{11} - 74034744959 \nu^{10} + 314418060009 \nu^{9} - 633522002730 \nu^{8} + \cdots - 273874062464 ) / 3507692390280 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 29615632063 \nu^{11} + 87105581309 \nu^{10} - 382317412959 \nu^{9} + 696924743820 \nu^{8} + \cdots - 5593310307856 ) / 3507692390280 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 99321102659 \nu^{11} - 361617020679 \nu^{10} + 1484218111509 \nu^{9} - 3218484118998 \nu^{8} + \cdots - 17878470653712 ) / 8769230975700 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 48894251523 \nu^{11} - 157422019691 \nu^{10} + 666492738129 \nu^{9} - 1308046874814 \nu^{8} + \cdots + 2898411243400 ) / 3507692390280 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 167923324379 \nu^{11} + 478101391109 \nu^{10} - 2090376253539 \nu^{9} + \cdots - 37003359759088 ) / 8769230975700 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 542891618887 \nu^{11} - 1717504490237 \nu^{10} + 7294073305527 \nu^{9} + \cdots + 77595685775224 ) / 17538461951400 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 572643183207 \nu^{11} - 1777970044487 \nu^{10} + 7631251682697 \nu^{9} + \cdots + 26266096925344 ) / 17538461951400 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{10} - 4\beta_{8} + \beta_{7} - \beta_{6} + \beta_{4} + \beta_{3} + \beta _1 - 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{11} + 2\beta_{10} + \beta_{9} - 6\beta_{8} - \beta_{7} + 4\beta_{6} + 5\beta_{5} + 7\beta_{4} + \beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 8 \beta_{11} + 8 \beta_{10} + 8 \beta_{9} - 2 \beta_{8} - 16 \beta_{7} + 24 \beta_{6} + 10 \beta_{5} + \cdots - 8 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 3 \beta_{11} + 3 \beta_{10} - 12 \beta_{9} - 12 \beta_{7} + 29 \beta_{6} + 29 \beta_{5} - 69 \beta_{4} + \cdots - 47 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 75 \beta_{11} - 66 \beta_{10} - 132 \beta_{9} + 54 \beta_{8} + 57 \beta_{7} + 66 \beta_{6} + \cdots + 12 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 256 \beta_{11} - 178 \beta_{10} - 128 \beta_{9} + 844 \beta_{8} + 78 \beta_{7} - 220 \beta_{6} + \cdots - 220 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 753 \beta_{11} - 368 \beta_{10} + 385 \beta_{9} + 3112 \beta_{8} + 184 \beta_{7} - 2744 \beta_{6} + \cdots - 1852 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 652 \beta_{11} - 2009 \beta_{10} + 705 \beta_{9} + 4658 \beta_{8} + 1304 \beta_{7} - 7165 \beta_{6} + \cdots + 2009 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 5100 \beta_{11} - 5100 \beta_{10} + 2700 \beta_{9} + 2700 \beta_{7} - 7110 \beta_{6} - 7110 \beta_{5} + \cdots + 25394 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 22335 \beta_{11} + 7815 \beta_{10} + 15630 \beta_{9} - 53280 \beta_{8} + 6705 \beta_{7} - 7815 \beta_{6} + \cdots + 45465 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/900\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(451\) \(577\)
\(\chi(n)\) \(1\) \(1\) \(-1 - \beta_{5} - \beta_{6} + \beta_{8}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
181.1
0.838695 − 2.58124i
0.213831 − 0.658105i
−0.861543 + 2.65156i
−1.25005 − 0.908212i
−0.120753 − 0.0877319i
2.67982 + 1.94700i
−1.25005 + 0.908212i
−0.120753 + 0.0877319i
2.67982 − 1.94700i
0.838695 + 2.58124i
0.213831 + 0.658105i
−0.861543 − 2.65156i
0 0 0 −0.548020 − 2.16787i 0 −4.40288 0 0 0
181.2 0 0 0 0.463031 + 2.18760i 0 3.25686 0 0 0
181.3 0 0 0 2.20302 − 0.383000i 0 −2.70809 0 0 0
361.1 0 0 0 −1.08159 − 1.95708i 0 −2.43270 0 0 0
361.2 0 0 0 −0.383646 + 2.20291i 0 3.58696 0 0 0
361.3 0 0 0 1.34720 − 1.78467i 0 1.69984 0 0 0
541.1 0 0 0 −1.08159 + 1.95708i 0 −2.43270 0 0 0
541.2 0 0 0 −0.383646 − 2.20291i 0 3.58696 0 0 0
541.3 0 0 0 1.34720 + 1.78467i 0 1.69984 0 0 0
721.1 0 0 0 −0.548020 + 2.16787i 0 −4.40288 0 0 0
721.2 0 0 0 0.463031 − 2.18760i 0 3.25686 0 0 0
721.3 0 0 0 2.20302 + 0.383000i 0 −2.70809 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 181.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
25.d even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 900.2.n.c 12
3.b odd 2 1 100.2.g.a ✓ 12
12.b even 2 1 400.2.u.f 12
15.d odd 2 1 500.2.g.a 12
15.e even 4 2 500.2.i.b 24
25.d even 5 1 inner 900.2.n.c 12
75.h odd 10 1 500.2.g.a 12
75.h odd 10 1 2500.2.a.c 6
75.j odd 10 1 100.2.g.a ✓ 12
75.j odd 10 1 2500.2.a.d 6
75.l even 20 2 500.2.i.b 24
75.l even 20 2 2500.2.c.c 12
300.n even 10 1 400.2.u.f 12
300.n even 10 1 10000.2.a.bc 6
300.r even 10 1 10000.2.a.bd 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
100.2.g.a ✓ 12 3.b odd 2 1
100.2.g.a ✓ 12 75.j odd 10 1
400.2.u.f 12 12.b even 2 1
400.2.u.f 12 300.n even 10 1
500.2.g.a 12 15.d odd 2 1
500.2.g.a 12 75.h odd 10 1
500.2.i.b 24 15.e even 4 2
500.2.i.b 24 75.l even 20 2
900.2.n.c 12 1.a even 1 1 trivial
900.2.n.c 12 25.d even 5 1 inner
2500.2.a.c 6 75.h odd 10 1
2500.2.a.d 6 75.j odd 10 1
2500.2.c.c 12 75.l even 20 2
10000.2.a.bc 6 300.n even 10 1
10000.2.a.bd 6 300.r even 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{6} + T_{7}^{5} - 29T_{7}^{4} - 18T_{7}^{3} + 244T_{7}^{2} + 96T_{7} - 576 \) acting on \(S_{2}^{\mathrm{new}}(900, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( T^{12} - 4 T^{11} + \cdots + 15625 \) Copy content Toggle raw display
$7$ \( (T^{6} + T^{5} - 29 T^{4} + \cdots - 576)^{2} \) Copy content Toggle raw display
$11$ \( T^{12} - 5 T^{11} + \cdots + 160000 \) Copy content Toggle raw display
$13$ \( T^{12} + 2 T^{11} + \cdots + 32761 \) Copy content Toggle raw display
$17$ \( T^{12} + T^{11} + \cdots + 1296 \) Copy content Toggle raw display
$19$ \( T^{12} + 8 T^{11} + \cdots + 10471696 \) Copy content Toggle raw display
$23$ \( T^{12} - 6 T^{11} + \cdots + 4096 \) Copy content Toggle raw display
$29$ \( T^{12} - 18 T^{11} + \cdots + 1042441 \) Copy content Toggle raw display
$31$ \( T^{12} + \cdots + 102495376 \) Copy content Toggle raw display
$37$ \( T^{12} - 13 T^{11} + \cdots + 2128681 \) Copy content Toggle raw display
$41$ \( T^{12} - 23 T^{11} + \cdots + 41422096 \) Copy content Toggle raw display
$43$ \( (T^{6} - 25 T^{5} + \cdots + 6400)^{2} \) Copy content Toggle raw display
$47$ \( T^{12} + T^{11} + \cdots + 256 \) Copy content Toggle raw display
$53$ \( T^{12} + 21 T^{11} + \cdots + 1745041 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 130051216 \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 1661296081 \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 58285547776 \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 5547866256 \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 4336354201 \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 56261942416 \) Copy content Toggle raw display
$83$ \( T^{12} - 46 T^{11} + \cdots + 36048016 \) Copy content Toggle raw display
$89$ \( T^{12} - 2 T^{11} + \cdots + 63744256 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 121807282081 \) Copy content Toggle raw display
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