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Properties

Label 2175.2.c.k
Level $2175$
Weight $2$
Character orbit 2175.c
Analytic conductor $17.367$
Analytic rank $0$
Dimension $4$
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] = [2175,2,Mod(349,2175)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2175, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2175.349");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2175 = 3 \cdot 5^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2175.c (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: \(17.3674624396\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{5})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 3x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 87)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + \beta_{3} q^{3} + (\beta_{2} + 1) q^{4} - \beta_{2} q^{6} + (\beta_{3} - 2 \beta_1) q^{7} + (\beta_{3} + 2 \beta_1) q^{8} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + \beta_{3} q^{3} + (\beta_{2} + 1) q^{4} - \beta_{2} q^{6} + (\beta_{3} - 2 \beta_1) q^{7} + (\beta_{3} + 2 \beta_1) q^{8} - q^{9} + ( - 2 \beta_{2} + 1) q^{11} + (\beta_{3} + \beta_1) q^{12} + ( - 3 \beta_{3} - 4 \beta_1) q^{13} + ( - 3 \beta_{2} + 2) q^{14} + 3 \beta_{2} q^{16} - 3 \beta_{3} q^{17} - \beta_1 q^{18} + (2 \beta_{2} + 6) q^{19} + (2 \beta_{2} - 1) q^{21} + ( - 2 \beta_{3} + 3 \beta_1) q^{22} + ( - 4 \beta_{3} - 6 \beta_1) q^{23} + ( - 2 \beta_{2} - 1) q^{24} + ( - \beta_{2} + 4) q^{26} - \beta_{3} q^{27} + ( - \beta_{3} + \beta_1) q^{28} + q^{29} + 6 \beta_{2} q^{31} + (5 \beta_{3} + \beta_1) q^{32} + (\beta_{3} - 2 \beta_1) q^{33} + 3 \beta_{2} q^{34} + ( - \beta_{2} - 1) q^{36} + ( - 4 \beta_{3} - 2 \beta_1) q^{37} + (2 \beta_{3} + 4 \beta_1) q^{38} + (4 \beta_{2} + 3) q^{39} + 2 q^{41} + (2 \beta_{3} - 3 \beta_1) q^{42} + 4 \beta_{3} q^{43} + (\beta_{2} - 1) q^{44} + ( - 2 \beta_{2} + 6) q^{46} + ( - \beta_{3} - 6 \beta_1) q^{47} + 3 \beta_1 q^{48} + (8 \beta_{2} + 2) q^{49} + 3 q^{51} + ( - 7 \beta_{3} - 3 \beta_1) q^{52} + (10 \beta_{3} + 2 \beta_1) q^{53} + \beta_{2} q^{54} + ( - 4 \beta_{2} + 3) q^{56} + (6 \beta_{3} + 2 \beta_1) q^{57} + \beta_1 q^{58} + ( - 4 \beta_{2} - 2) q^{59} + ( - 2 \beta_{2} - 4) q^{61} + (6 \beta_{3} - 6 \beta_1) q^{62} + ( - \beta_{3} + 2 \beta_1) q^{63} + (2 \beta_{2} - 1) q^{64} + ( - 3 \beta_{2} + 2) q^{66} + (7 \beta_{3} + 10 \beta_1) q^{67} + ( - 3 \beta_{3} - 3 \beta_1) q^{68} + (6 \beta_{2} + 4) q^{69} + ( - 2 \beta_{2} - 4) q^{71} + ( - \beta_{3} - 2 \beta_1) q^{72} + (8 \beta_{3} - 2 \beta_1) q^{73} + (2 \beta_{2} + 2) q^{74} + (6 \beta_{2} + 8) q^{76} + (5 \beta_{3} - 8 \beta_1) q^{77} + (4 \beta_{3} - \beta_1) q^{78} + ( - 2 \beta_{2} + 14) q^{79} + q^{81} + 2 \beta_1 q^{82} + ( - 2 \beta_{3} + 8 \beta_1) q^{83} + ( - \beta_{2} + 1) q^{84} - 4 \beta_{2} q^{86} + \beta_{3} q^{87} + ( - 3 \beta_{3} + 4 \beta_1) q^{88} - 5 q^{89} + (6 \beta_{2} - 5) q^{91} + ( - 10 \beta_{3} - 4 \beta_1) q^{92} + 6 \beta_1 q^{93} + ( - 5 \beta_{2} + 6) q^{94} + ( - \beta_{2} - 5) q^{96} + ( - 10 \beta_{3} - 14 \beta_1) q^{97} + (8 \beta_{3} - 6 \beta_1) q^{98} + (2 \beta_{2} - 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{4} + 2 q^{6} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{4} + 2 q^{6} - 4 q^{9} + 8 q^{11} + 14 q^{14} - 6 q^{16} + 20 q^{19} - 8 q^{21} + 18 q^{26} + 4 q^{29} - 12 q^{31} - 6 q^{34} - 2 q^{36} + 4 q^{39} + 8 q^{41} - 6 q^{44} + 28 q^{46} - 8 q^{49} + 12 q^{51} - 2 q^{54} + 20 q^{56} - 12 q^{61} - 8 q^{64} + 14 q^{66} + 4 q^{69} - 12 q^{71} + 4 q^{74} + 20 q^{76} + 60 q^{79} + 4 q^{81} + 6 q^{84} + 8 q^{86} - 20 q^{89} - 32 q^{91} + 34 q^{94} - 18 q^{96} - 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 3x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 1 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} + 2\nu \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} - 2\beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2175\mathbb{Z}\right)^\times\).

\(n\) \(901\) \(1451\) \(2002\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
349.1
− 1.61803i
− 0.618034i
0.618034i
1.61803i
− 1.61803i 1.00000i −0.618034 0 1.61803 4.23607i − 2.23607i −1.00000 0
349.2 − 0.618034i − 1.00000i 1.61803 0 −0.618034 0.236068i − 2.23607i −1.00000 0
349.3 0.618034i 1.00000i 1.61803 0 −0.618034 − 0.236068i 2.23607i −1.00000 0
349.4 1.61803i − 1.00000i −0.618034 0 1.61803 − 4.23607i 2.23607i −1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2175.2.c.k 4
5.b even 2 1 inner 2175.2.c.k 4
5.c odd 4 1 87.2.a.a ✓ 2
5.c odd 4 1 2175.2.a.l 2
15.e even 4 1 261.2.a.b 2
15.e even 4 1 6525.2.a.ba 2
20.e even 4 1 1392.2.a.q 2
35.f even 4 1 4263.2.a.j 2
40.i odd 4 1 5568.2.a.bl 2
40.k even 4 1 5568.2.a.bs 2
60.l odd 4 1 4176.2.a.bn 2
145.h odd 4 1 2523.2.a.c 2
435.p even 4 1 7569.2.a.k 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
87.2.a.a ✓ 2 5.c odd 4 1
261.2.a.b 2 15.e even 4 1
1392.2.a.q 2 20.e even 4 1
2175.2.a.l 2 5.c odd 4 1
2175.2.c.k 4 1.a even 1 1 trivial
2175.2.c.k 4 5.b even 2 1 inner
2523.2.a.c 2 145.h odd 4 1
4176.2.a.bn 2 60.l odd 4 1
4263.2.a.j 2 35.f even 4 1
5568.2.a.bl 2 40.i odd 4 1
5568.2.a.bs 2 40.k even 4 1
6525.2.a.ba 2 15.e even 4 1
7569.2.a.k 2 435.p even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2175, [\chi])\):

\( T_{2}^{4} + 3T_{2}^{2} + 1 \) Copy content Toggle raw display
\( T_{7}^{4} + 18T_{7}^{2} + 1 \) Copy content Toggle raw display
\( T_{11}^{2} - 4T_{11} - 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 3T^{2} + 1 \) Copy content Toggle raw display
$3$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} + 18T^{2} + 1 \) Copy content Toggle raw display
$11$ \( (T^{2} - 4 T - 1)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} + 42T^{2} + 361 \) Copy content Toggle raw display
$17$ \( (T^{2} + 9)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} - 10 T + 20)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + 92T^{2} + 1936 \) Copy content Toggle raw display
$29$ \( (T - 1)^{4} \) Copy content Toggle raw display
$31$ \( (T^{2} + 6 T - 36)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} + 28T^{2} + 16 \) Copy content Toggle raw display
$41$ \( (T - 2)^{4} \) Copy content Toggle raw display
$43$ \( (T^{2} + 16)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + 98T^{2} + 1681 \) Copy content Toggle raw display
$53$ \( T^{4} + 172T^{2} + 5776 \) Copy content Toggle raw display
$59$ \( (T^{2} - 20)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 6 T + 4)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + 258 T^{2} + 14641 \) Copy content Toggle raw display
$71$ \( (T^{2} + 6 T + 4)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + 172T^{2} + 5776 \) Copy content Toggle raw display
$79$ \( (T^{2} - 30 T + 220)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 232T^{2} + 1936 \) Copy content Toggle raw display
$89$ \( (T + 5)^{4} \) Copy content Toggle raw display
$97$ \( T^{4} + 508 T^{2} + 55696 \) Copy content Toggle raw display
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