Properties

Label 1815.4.a.bb
Level $1815$
Weight $4$
Character orbit 1815.a
Self dual yes
Analytic conductor $107.088$
Analytic rank $0$
Dimension $6$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1815,4,Mod(1,1815)]
 
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, 0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1815.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1815 = 3 \cdot 5 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1815.a (trivial)

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: yes
Analytic conductor: \(107.088466660\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 44x^{4} + 495x^{2} - 1200 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4}\cdot 5 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + 3 q^{3} + (\beta_{2} + 7) q^{4} + 5 q^{5} + 3 \beta_1 q^{6} + ( - \beta_{3} + \beta_1) q^{7} + (\beta_{4} + \beta_{3} + 6 \beta_1) q^{8} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + 3 q^{3} + (\beta_{2} + 7) q^{4} + 5 q^{5} + 3 \beta_1 q^{6} + ( - \beta_{3} + \beta_1) q^{7} + (\beta_{4} + \beta_{3} + 6 \beta_1) q^{8} + 9 q^{9} + 5 \beta_1 q^{10} + (3 \beta_{2} + 21) q^{12} + ( - \beta_{3} + 3 \beta_1) q^{13} + ( - \beta_{5} - \beta_{2} + 19) q^{14} + 15 q^{15} + (\beta_{5} + 8 \beta_{2} + 30) q^{16} + ( - 2 \beta_{4} - 5 \beta_{3} - 3 \beta_1) q^{17} + 9 \beta_1 q^{18} + (2 \beta_{4} + 6 \beta_1) q^{19} + (5 \beta_{2} + 35) q^{20} + ( - 3 \beta_{3} + 3 \beta_1) q^{21} + (2 \beta_{5} + 2 \beta_{2} + 10) q^{23} + (3 \beta_{4} + 3 \beta_{3} + 18 \beta_1) q^{24} + 25 q^{25} + ( - \beta_{5} + \beta_{2} + 49) q^{26} + 27 q^{27} + ( - 3 \beta_{4} - 5 \beta_{3} + \beta_1) q^{28} + (2 \beta_{4} - 2 \beta_{3} + 20 \beta_1) q^{29} + 15 \beta_1 q^{30} + (12 \beta_{2} + 100) q^{31} + (2 \beta_{4} + 12 \beta_{3} + 41 \beta_1) q^{32} + ( - 5 \beta_{5} - 29 \beta_{2} - 25) q^{34} + ( - 5 \beta_{3} + 5 \beta_1) q^{35} + (9 \beta_{2} + 63) q^{36} + (2 \beta_{5} + 2 \beta_{2} + 56) q^{37} + (22 \beta_{2} + 90) q^{38} + ( - 3 \beta_{3} + 9 \beta_1) q^{39} + (5 \beta_{4} + 5 \beta_{3} + 30 \beta_1) q^{40} + (2 \beta_{4} - 2 \beta_{3}) q^{41} + ( - 3 \beta_{5} - 3 \beta_{2} + 57) q^{42} + (11 \beta_{3} + 65 \beta_1) q^{43} + 45 q^{45} + (6 \beta_{4} + 26 \beta_{3} + 30 \beta_1) q^{46} + (4 \beta_{5} + 12 \beta_{2} + 20) q^{47} + (3 \beta_{5} + 24 \beta_{2} + 90) q^{48} + ( - 6 \beta_{5} - 14 \beta_{2} + 71) q^{49} + 25 \beta_1 q^{50} + ( - 6 \beta_{4} - 15 \beta_{3} - 9 \beta_1) q^{51} + ( - \beta_{4} - 3 \beta_{3} + 29 \beta_1) q^{52} + (2 \beta_{5} - 34 \beta_{2} + 40) q^{53} + 27 \beta_1 q^{54} + (3 \beta_{5} - 25 \beta_{2} - 117) q^{56} + (6 \beta_{4} + 18 \beta_1) q^{57} + ( - 2 \beta_{5} + 32 \beta_{2} + 308) q^{58} + ( - 6 \beta_{5} + 22 \beta_{2} + 354) q^{59} + (15 \beta_{2} + 105) q^{60} + ( - 14 \beta_{3} + 14 \beta_1) q^{61} + (12 \beta_{4} + 12 \beta_{3} + 184 \beta_1) q^{62} + ( - 9 \beta_{3} + 9 \beta_1) q^{63} + (4 \beta_{5} + 17 \beta_{2} + 327) q^{64} + ( - 5 \beta_{3} + 15 \beta_1) q^{65} + (6 \beta_{5} - 70 \beta_{2} - 218) q^{67} + ( - 23 \beta_{4} - 49 \beta_{3} - 219 \beta_1) q^{68} + (6 \beta_{5} + 6 \beta_{2} + 30) q^{69} + ( - 5 \beta_{5} - 5 \beta_{2} + 95) q^{70} + (6 \beta_{5} + 62 \beta_{2} + 198) q^{71} + (9 \beta_{4} + 9 \beta_{3} + 54 \beta_1) q^{72} + ( - 12 \beta_{4} + 13 \beta_{3} - 35 \beta_1) q^{73} + (6 \beta_{4} + 26 \beta_{3} + 76 \beta_1) q^{74} + 75 q^{75} + (6 \beta_{4} + 22 \beta_{3} + 196 \beta_1) q^{76} + ( - 3 \beta_{5} + 3 \beta_{2} + 147) q^{78} + (2 \beta_{4} - 12 \beta_{3} - 222 \beta_1) q^{79} + (5 \beta_{5} + 40 \beta_{2} + 150) q^{80} + 81 q^{81} + ( - 2 \beta_{5} + 12 \beta_{2} + 8) q^{82} + ( - 14 \beta_{4} + 3 \beta_{3} + 75 \beta_1) q^{83} + ( - 9 \beta_{4} - 15 \beta_{3} + 3 \beta_1) q^{84} + ( - 10 \beta_{4} - 25 \beta_{3} - 15 \beta_1) q^{85} + (11 \beta_{5} + 87 \beta_{2} + 931) q^{86} + (6 \beta_{4} - 6 \beta_{3} + 60 \beta_1) q^{87} + ( - 4 \beta_{5} + 12 \beta_{2} - 110) q^{89} + 45 \beta_1 q^{90} + ( - 8 \beta_{5} - 16 \beta_{2} + 452) q^{91} + (10 \beta_{5} + 114 \beta_{2} + 266) q^{92} + (36 \beta_{2} + 300) q^{93} + (20 \beta_{4} + 60 \beta_{3} + 116 \beta_1) q^{94} + (10 \beta_{4} + 30 \beta_1) q^{95} + (6 \beta_{4} + 36 \beta_{3} + 123 \beta_1) q^{96} + ( - 10 \beta_{5} - 74 \beta_{2} + 444) q^{97} + ( - 26 \beta_{4} - 86 \beta_{3} - 45 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 18 q^{3} + 40 q^{4} + 30 q^{5} + 54 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 18 q^{3} + 40 q^{4} + 30 q^{5} + 54 q^{9} + 120 q^{12} + 116 q^{14} + 90 q^{15} + 164 q^{16} + 200 q^{20} + 56 q^{23} + 150 q^{25} + 292 q^{26} + 162 q^{27} + 576 q^{31} - 92 q^{34} + 360 q^{36} + 332 q^{37} + 496 q^{38} + 348 q^{42} + 270 q^{45} + 96 q^{47} + 492 q^{48} + 454 q^{49} + 308 q^{53} - 652 q^{56} + 1784 q^{58} + 2080 q^{59} + 600 q^{60} + 1928 q^{64} - 1168 q^{67} + 168 q^{69} + 580 q^{70} + 1064 q^{71} + 450 q^{75} + 876 q^{78} + 820 q^{80} + 486 q^{81} + 24 q^{82} + 5412 q^{86} - 684 q^{89} + 2744 q^{91} + 1368 q^{92} + 1728 q^{93} + 2812 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 44x^{4} + 495x^{2} - 1200 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 15 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{5} - 34\nu^{3} + 195\nu ) / 10 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{5} + 44\nu^{3} - 415\nu ) / 10 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( \nu^{4} - 32\nu^{2} + 154 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 15 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{4} + \beta_{3} + 22\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{5} + 32\beta_{2} + 326 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 34\beta_{4} + 44\beta_{3} + 553\beta_1 \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−5.26184
−3.60164
−1.82790
1.82790
3.60164
5.26184
−5.26184 3.00000 19.6870 5.00000 −15.7855 5.37309 −61.4950 9.00000 −26.3092
1.2 −3.60164 3.00000 4.97180 5.00000 −10.8049 −31.6129 10.9065 9.00000 −18.0082
1.3 −1.82790 3.00000 −4.65878 5.00000 −5.48370 15.0916 23.1390 9.00000 −9.13951
1.4 1.82790 3.00000 −4.65878 5.00000 5.48370 −15.0916 −23.1390 9.00000 9.13951
1.5 3.60164 3.00000 4.97180 5.00000 10.8049 31.6129 −10.9065 9.00000 18.0082
1.6 5.26184 3.00000 19.6870 5.00000 15.7855 −5.37309 61.4950 9.00000 26.3092
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.6
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \( -1 \)
\(5\) \( -1 \)
\(11\) \( +1 \)

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1815.4.a.bb 6
11.b odd 2 1 inner 1815.4.a.bb 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1815.4.a.bb 6 1.a even 1 1 trivial
1815.4.a.bb 6 11.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1815))\):

\( T_{2}^{6} - 44T_{2}^{4} + 495T_{2}^{2} - 1200 \) Copy content Toggle raw display
\( T_{7}^{6} - 1256T_{7}^{4} + 263040T_{7}^{2} - 6571200 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} - 44 T^{4} + \cdots - 1200 \) Copy content Toggle raw display
$3$ \( (T - 3)^{6} \) Copy content Toggle raw display
$5$ \( (T - 5)^{6} \) Copy content Toggle raw display
$7$ \( T^{6} - 1256 T^{4} + \cdots - 6571200 \) Copy content Toggle raw display
$11$ \( T^{6} \) Copy content Toggle raw display
$13$ \( T^{6} - 1664 T^{4} + \cdots - 5227200 \) Copy content Toggle raw display
$17$ \( T^{6} + \cdots - 31678852800 \) Copy content Toggle raw display
$19$ \( T^{6} + \cdots - 6481171200 \) Copy content Toggle raw display
$23$ \( (T^{3} - 28 T^{2} + \cdots + 1938432)^{2} \) Copy content Toggle raw display
$29$ \( T^{6} + \cdots - 2099899468800 \) Copy content Toggle raw display
$31$ \( (T^{3} - 288 T^{2} + \cdots + 761600)^{2} \) Copy content Toggle raw display
$37$ \( (T^{3} - 166 T^{2} + \cdots + 3002872)^{2} \) Copy content Toggle raw display
$41$ \( T^{6} + \cdots - 12226636800 \) Copy content Toggle raw display
$43$ \( T^{6} + \cdots - 107322824836800 \) Copy content Toggle raw display
$47$ \( (T^{3} - 48 T^{2} + \cdots + 13192704)^{2} \) Copy content Toggle raw display
$53$ \( (T^{3} - 154 T^{2} + \cdots - 13663032)^{2} \) Copy content Toggle raw display
$59$ \( (T^{3} - 1040 T^{2} + \cdots + 92904000)^{2} \) Copy content Toggle raw display
$61$ \( T^{6} + \cdots - 49478086963200 \) Copy content Toggle raw display
$67$ \( (T^{3} + 584 T^{2} + \cdots - 538988864)^{2} \) Copy content Toggle raw display
$71$ \( (T^{3} - 532 T^{2} + \cdots - 101340672)^{2} \) Copy content Toggle raw display
$73$ \( T^{6} + \cdots - 86\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( T^{6} + \cdots - 17\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{6} + \cdots - 12\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( (T^{3} + 342 T^{2} + \cdots - 10996920)^{2} \) Copy content Toggle raw display
$97$ \( (T^{3} - 1406 T^{2} + \cdots + 927003032)^{2} \) Copy content Toggle raw display
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