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Properties

Label 1815.2.c.i
Level $1815$
Weight $2$
Character orbit 1815.c
Analytic conductor $14.493$
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] = [1815,2,Mod(364,1815)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1815, 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("1815.364");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1815 = 3 \cdot 5 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1815.c (of order \(2\), degree \(1\), 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: \(14.4928479669\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 21x^{10} + 164x^{8} + 589x^{6} + 965x^{4} + 576x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} - \beta_{4} q^{3} + (\beta_{2} - 2) q^{4} - \beta_{5} q^{5} + \beta_{3} q^{6} + ( - \beta_{8} - \beta_{6} + \cdots - \beta_1) q^{7}+ \cdots - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} - \beta_{4} q^{3} + (\beta_{2} - 2) q^{4} - \beta_{5} q^{5} + \beta_{3} q^{6} + ( - \beta_{8} - \beta_{6} + \cdots - \beta_1) q^{7}+ \cdots + ( - \beta_{11} - \beta_{10} + \cdots - 3 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 18 q^{4} - 2 q^{5} + 2 q^{6} - 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 18 q^{4} - 2 q^{5} + 2 q^{6} - 12 q^{9} + 12 q^{10} + 20 q^{14} + 22 q^{16} + 4 q^{19} - 2 q^{20} - 8 q^{21} + 2 q^{25} - 24 q^{29} - 8 q^{30} + 36 q^{31} + 2 q^{34} + 24 q^{35} + 18 q^{36} - 4 q^{39} - 22 q^{40} - 12 q^{41} + 2 q^{45} - 22 q^{46} - 24 q^{49} - 58 q^{50} + 4 q^{51} - 2 q^{54} - 84 q^{56} + 36 q^{59} + 22 q^{60} + 8 q^{61} - 44 q^{64} - 14 q^{65} - 24 q^{69} - 16 q^{70} + 8 q^{74} - 12 q^{75} - 40 q^{76} - 4 q^{79} - 58 q^{80} + 12 q^{81} + 48 q^{84} - 2 q^{85} + 56 q^{86} + 32 q^{89} - 12 q^{90} + 48 q^{91} - 6 q^{94} + 62 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 21x^{10} + 164x^{8} + 589x^{6} + 965x^{4} + 576x^{2} + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 2\nu^{10} + 37\nu^{8} + 224\nu^{6} + 480\nu^{4} + 247\nu^{2} - 6 ) / 92 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -3\nu^{11} - 67\nu^{9} - 566\nu^{7} - 2215\nu^{5} - 3855\nu^{3} - 2222\nu ) / 184 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 7 \nu^{11} - 5 \nu^{10} + 141 \nu^{9} - 81 \nu^{8} + 1014 \nu^{7} - 422 \nu^{6} + 3083 \nu^{5} + \cdots + 406 ) / 184 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 7 \nu^{11} - 5 \nu^{10} - 141 \nu^{9} - 81 \nu^{8} - 1014 \nu^{7} - 422 \nu^{6} - 3083 \nu^{5} + \cdots + 406 ) / 184 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 8 \nu^{11} - 14 \nu^{10} - 171 \nu^{9} - 259 \nu^{8} - 1356 \nu^{7} - 1660 \nu^{6} - 4910 \nu^{5} + \cdots + 42 ) / 184 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 13\nu^{11} + 275\nu^{9} + 2146\nu^{7} + 7605\nu^{5} + 12059\nu^{3} + 6654\nu ) / 184 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 8 \nu^{11} + 14 \nu^{10} - 171 \nu^{9} + 259 \nu^{8} - 1356 \nu^{7} + 1660 \nu^{6} - 4910 \nu^{5} + \cdots - 42 ) / 184 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 25 \nu^{11} - 11 \nu^{10} - 497 \nu^{9} - 215 \nu^{8} - 3582 \nu^{7} - 1462 \nu^{6} - 11405 \nu^{5} + \cdots + 562 ) / 184 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 25 \nu^{11} + 11 \nu^{10} - 497 \nu^{9} + 215 \nu^{8} - 3582 \nu^{7} + 1462 \nu^{6} - 11405 \nu^{5} + \cdots - 562 ) / 184 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{9} - \beta_{8} - \beta_{7} + \beta_{4} - 5\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{11} + \beta_{10} + \beta_{9} - \beta_{7} + \beta_{6} + \beta_{5} + \beta_{3} - 8\beta_{2} + 22 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 9\beta_{9} + 11\beta_{8} + 9\beta_{7} + \beta_{6} - \beta_{5} - 5\beta_{4} + 30\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 10\beta_{11} - 10\beta_{10} - 9\beta_{9} + 9\beta_{7} - 10\beta_{6} - 10\beta_{5} - 17\beta_{3} + 59\beta_{2} - 136 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - \beta_{11} - \beta_{10} - 68 \beta_{9} - 96 \beta_{8} - 68 \beta_{7} - 10 \beta_{6} + \cdots - 195 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 78 \beta_{11} + 78 \beta_{10} + 66 \beta_{9} - 66 \beta_{7} + 86 \beta_{6} + 86 \beta_{5} + \cdots + 894 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 20 \beta_{11} + 20 \beta_{10} + 493 \beta_{9} + 781 \beta_{8} + 493 \beta_{7} + 70 \beta_{6} + \cdots + 1321 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 563 \beta_{11} - 563 \beta_{10} - 453 \beta_{9} + 453 \beta_{7} - 711 \beta_{6} - 711 \beta_{5} + \cdots - 6090 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 258 \beta_{11} - 258 \beta_{10} - 3541 \beta_{9} - 6167 \beta_{8} - 3541 \beta_{7} + \cdots - 9178 \beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1815\mathbb{Z}\right)^\times\).

\(n\) \(727\) \(1211\) \(1696\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
364.1
− 2.73519i
− 2.42911i
− 1.95455i
− 1.53672i
− 1.19557i
− 0.0838261i
0.0838261i
1.19557i
1.53672i
1.95455i
2.42911i
2.73519i
− 2.73519i 1.00000i −5.48125 −1.65271 + 1.50617i 2.73519 4.20410i 9.52186i −1.00000 4.11965 + 4.52048i
364.2 − 2.42911i − 1.00000i −3.90056 2.04405 − 0.906565i −2.42911 1.34168i 4.61665i −1.00000 −2.20214 − 4.96521i
364.3 − 1.95455i − 1.00000i −1.82026 −1.24280 + 1.85889i −1.95455 − 2.58348i − 0.351308i −1.00000 3.63328 + 2.42911i
364.4 − 1.53672i 1.00000i −0.361523 1.20807 + 1.88164i 1.53672 − 2.86503i − 2.51789i −1.00000 2.89156 − 1.85647i
364.5 − 1.19557i 1.00000i 0.570614 0.849151 − 2.06856i 1.19557 3.76208i − 3.07335i −1.00000 −2.47311 − 1.01522i
364.6 − 0.0838261i − 1.00000i 1.99297 −2.20576 + 0.366926i −0.0838261 2.34295i − 0.334715i −1.00000 0.0307580 + 0.184900i
364.7 0.0838261i 1.00000i 1.99297 −2.20576 − 0.366926i −0.0838261 − 2.34295i 0.334715i −1.00000 0.0307580 − 0.184900i
364.8 1.19557i − 1.00000i 0.570614 0.849151 + 2.06856i 1.19557 − 3.76208i 3.07335i −1.00000 −2.47311 + 1.01522i
364.9 1.53672i − 1.00000i −0.361523 1.20807 − 1.88164i 1.53672 2.86503i 2.51789i −1.00000 2.89156 + 1.85647i
364.10 1.95455i 1.00000i −1.82026 −1.24280 − 1.85889i −1.95455 2.58348i 0.351308i −1.00000 3.63328 − 2.42911i
364.11 2.42911i 1.00000i −3.90056 2.04405 + 0.906565i −2.42911 − 1.34168i − 4.61665i −1.00000 −2.20214 + 4.96521i
364.12 2.73519i − 1.00000i −5.48125 −1.65271 − 1.50617i 2.73519 − 4.20410i − 9.52186i −1.00000 4.11965 − 4.52048i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 364.12
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 1815.2.c.i yes 12
5.b even 2 1 inner 1815.2.c.i yes 12
5.c odd 4 1 9075.2.a.do 6
5.c odd 4 1 9075.2.a.ds 6
11.b odd 2 1 1815.2.c.h ✓ 12
55.d odd 2 1 1815.2.c.h ✓ 12
55.e even 4 1 9075.2.a.dp 6
55.e even 4 1 9075.2.a.dr 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1815.2.c.h ✓ 12 11.b odd 2 1
1815.2.c.h ✓ 12 55.d odd 2 1
1815.2.c.i yes 12 1.a even 1 1 trivial
1815.2.c.i yes 12 5.b even 2 1 inner
9075.2.a.do 6 5.c odd 4 1
9075.2.a.dp 6 55.e even 4 1
9075.2.a.dr 6 55.e even 4 1
9075.2.a.ds 6 5.c odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1815, [\chi])\):

\( T_{2}^{12} + 21T_{2}^{10} + 164T_{2}^{8} + 589T_{2}^{6} + 965T_{2}^{4} + 576T_{2}^{2} + 4 \) Copy content Toggle raw display
\( T_{19}^{6} - 2T_{19}^{5} - 47T_{19}^{4} + 64T_{19}^{3} + 368T_{19}^{2} - 128T_{19} - 512 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} + 21 T^{10} + \cdots + 4 \) Copy content Toggle raw display
$3$ \( (T^{2} + 1)^{6} \) Copy content Toggle raw display
$5$ \( T^{12} + 2 T^{11} + \cdots + 15625 \) Copy content Toggle raw display
$7$ \( T^{12} + 54 T^{10} + \cdots + 135424 \) Copy content Toggle raw display
$11$ \( T^{12} \) Copy content Toggle raw display
$13$ \( T^{12} + 98 T^{10} + \cdots + 1024 \) Copy content Toggle raw display
$17$ \( T^{12} + 140 T^{10} + \cdots + 2116 \) Copy content Toggle raw display
$19$ \( (T^{6} - 2 T^{5} + \cdots - 512)^{2} \) Copy content Toggle raw display
$23$ \( T^{12} + 186 T^{10} + \cdots + 1149184 \) Copy content Toggle raw display
$29$ \( (T^{6} + 12 T^{5} + \cdots - 7688)^{2} \) Copy content Toggle raw display
$31$ \( (T^{6} - 18 T^{5} + \cdots - 2656)^{2} \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 9710131600 \) Copy content Toggle raw display
$41$ \( (T^{6} + 6 T^{5} + \cdots - 52544)^{2} \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 287641600 \) Copy content Toggle raw display
$47$ \( T^{12} + 94 T^{10} + \cdots + 25600 \) Copy content Toggle raw display
$53$ \( T^{12} + 200 T^{10} + \cdots + 84419344 \) Copy content Toggle raw display
$59$ \( (T^{6} - 18 T^{5} + \cdots - 8224)^{2} \) Copy content Toggle raw display
$61$ \( (T^{6} - 4 T^{5} + \cdots - 104284)^{2} \) Copy content Toggle raw display
$67$ \( T^{12} + 262 T^{10} + \cdots + 215296 \) Copy content Toggle raw display
$71$ \( (T^{6} - 130 T^{4} + \cdots + 11744)^{2} \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 1961426944 \) Copy content Toggle raw display
$79$ \( (T^{6} + 2 T^{5} + \cdots - 196624)^{2} \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 70883737600 \) Copy content Toggle raw display
$89$ \( (T^{6} - 16 T^{5} + \cdots - 80776)^{2} \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 555639184 \) Copy content Toggle raw display
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