Properties

Label 1815.4.a.m
Level $1815$
Weight $4$
Character orbit 1815.a
Self dual yes
Analytic conductor $107.088$
Analytic rank $1$
Dimension $2$
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] = [1815,4,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: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{15}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 15 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
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 \(\beta = \sqrt{15}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{2} + 3 q^{3} + 7 q^{4} - 5 q^{5} + 3 \beta q^{6} - 4 \beta q^{7} - \beta q^{8} + 9 q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + \beta q^{2} + 3 q^{3} + 7 q^{4} - 5 q^{5} + 3 \beta q^{6} - 4 \beta q^{7} - \beta q^{8} + 9 q^{9} - 5 \beta q^{10} + 21 q^{12} + 18 \beta q^{13} - 60 q^{14} - 15 q^{15} - 71 q^{16} - 4 \beta q^{17} + 9 \beta q^{18} + 2 \beta q^{19} - 35 q^{20} - 12 \beta q^{21} + 48 q^{23} - 3 \beta q^{24} + 25 q^{25} + 270 q^{26} + 27 q^{27} - 28 \beta q^{28} - 54 \beta q^{29} - 15 \beta q^{30} + 160 q^{31} - 63 \beta q^{32} - 60 q^{34} + 20 \beta q^{35} + 63 q^{36} - 266 q^{37} + 30 q^{38} + 54 \beta q^{39} + 5 \beta q^{40} + 46 \beta q^{41} - 180 q^{42} - 80 \beta q^{43} - 45 q^{45} + 48 \beta q^{46} - 504 q^{47} - 213 q^{48} - 103 q^{49} + 25 \beta q^{50} - 12 \beta q^{51} + 126 \beta q^{52} - 342 q^{53} + 27 \beta q^{54} + 60 q^{56} + 6 \beta q^{57} - 810 q^{58} - 660 q^{59} - 105 q^{60} - 56 \beta q^{61} + 160 \beta q^{62} - 36 \beta q^{63} - 377 q^{64} - 90 \beta q^{65} + 496 q^{67} - 28 \beta q^{68} + 144 q^{69} + 300 q^{70} - 708 q^{71} - 9 \beta q^{72} - 166 \beta q^{73} - 266 \beta q^{74} + 75 q^{75} + 14 \beta q^{76} + 810 q^{78} - 46 \beta q^{79} + 355 q^{80} + 81 q^{81} + 690 q^{82} - 22 \beta q^{83} - 84 \beta q^{84} + 20 \beta q^{85} - 1200 q^{86} - 162 \beta q^{87} - 606 q^{89} - 45 \beta q^{90} - 1080 q^{91} + 336 q^{92} + 480 q^{93} - 504 \beta q^{94} - 10 \beta q^{95} - 189 \beta q^{96} + 254 q^{97} - 103 \beta q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 6 q^{3} + 14 q^{4} - 10 q^{5} + 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 6 q^{3} + 14 q^{4} - 10 q^{5} + 18 q^{9} + 42 q^{12} - 120 q^{14} - 30 q^{15} - 142 q^{16} - 70 q^{20} + 96 q^{23} + 50 q^{25} + 540 q^{26} + 54 q^{27} + 320 q^{31} - 120 q^{34} + 126 q^{36} - 532 q^{37} + 60 q^{38} - 360 q^{42} - 90 q^{45} - 1008 q^{47} - 426 q^{48} - 206 q^{49} - 684 q^{53} + 120 q^{56} - 1620 q^{58} - 1320 q^{59} - 210 q^{60} - 754 q^{64} + 992 q^{67} + 288 q^{69} + 600 q^{70} - 1416 q^{71} + 150 q^{75} + 1620 q^{78} + 710 q^{80} + 162 q^{81} + 1380 q^{82} - 2400 q^{86} - 1212 q^{89} - 2160 q^{91} + 672 q^{92} + 960 q^{93} + 508 q^{97}+O(q^{100}) \) 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
−3.87298
3.87298
−3.87298 3.00000 7.00000 −5.00000 −11.6190 15.4919 3.87298 9.00000 19.3649
1.2 3.87298 3.00000 7.00000 −5.00000 11.6190 −15.4919 −3.87298 9.00000 −19.3649
\(n\): e.g. 2-40 or 990-1000
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.m 2
11.b odd 2 1 inner 1815.4.a.m 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1815.4.a.m 2 1.a even 1 1 trivial
1815.4.a.m 2 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(1815))\):

\( T_{2}^{2} - 15 \) Copy content Toggle raw display
\( T_{7}^{2} - 240 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - 15 \) Copy content Toggle raw display
$3$ \( (T - 3)^{2} \) Copy content Toggle raw display
$5$ \( (T + 5)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} - 240 \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( T^{2} - 4860 \) Copy content Toggle raw display
$17$ \( T^{2} - 240 \) Copy content Toggle raw display
$19$ \( T^{2} - 60 \) Copy content Toggle raw display
$23$ \( (T - 48)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} - 43740 \) Copy content Toggle raw display
$31$ \( (T - 160)^{2} \) Copy content Toggle raw display
$37$ \( (T + 266)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} - 31740 \) Copy content Toggle raw display
$43$ \( T^{2} - 96000 \) Copy content Toggle raw display
$47$ \( (T + 504)^{2} \) Copy content Toggle raw display
$53$ \( (T + 342)^{2} \) Copy content Toggle raw display
$59$ \( (T + 660)^{2} \) Copy content Toggle raw display
$61$ \( T^{2} - 47040 \) Copy content Toggle raw display
$67$ \( (T - 496)^{2} \) Copy content Toggle raw display
$71$ \( (T + 708)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} - 413340 \) Copy content Toggle raw display
$79$ \( T^{2} - 31740 \) Copy content Toggle raw display
$83$ \( T^{2} - 7260 \) Copy content Toggle raw display
$89$ \( (T + 606)^{2} \) Copy content Toggle raw display
$97$ \( (T - 254)^{2} \) Copy content Toggle raw display
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