Properties

Label 1305.4.a.f
Level $1305$
Weight $4$
Character orbit 1305.a
Self dual yes
Analytic conductor $76.997$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1305,4,Mod(1,1305)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1305, 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("1305.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1305 = 3^{2} \cdot 5 \cdot 29 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1305.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: \(76.9974925575\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{41}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 10 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 435)
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 \(\beta = \frac{1}{2}(1 + \sqrt{41})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{2} + (\beta + 2) q^{4} - 5 q^{5} + (\beta - 7) q^{7} + ( - 5 \beta + 10) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta q^{2} + (\beta + 2) q^{4} - 5 q^{5} + (\beta - 7) q^{7} + ( - 5 \beta + 10) q^{8} - 5 \beta q^{10} + ( - \beta + 27) q^{11} + (9 \beta - 41) q^{13} + ( - 6 \beta + 10) q^{14} + ( - 3 \beta - 66) q^{16} + ( - \beta + 53) q^{17} + (10 \beta + 14) q^{19} + ( - 5 \beta - 10) q^{20} + (26 \beta - 10) q^{22} + (20 \beta + 20) q^{23} + 25 q^{25} + ( - 32 \beta + 90) q^{26} + ( - 4 \beta - 4) q^{28} - 29 q^{29} + ( - 12 \beta - 116) q^{31} + ( - 29 \beta - 110) q^{32} + (52 \beta - 10) q^{34} + ( - 5 \beta + 35) q^{35} + ( - 54 \beta - 72) q^{37} + (24 \beta + 100) q^{38} + (25 \beta - 50) q^{40} + ( - 48 \beta - 42) q^{41} + ( - 60 \beta + 168) q^{43} + (24 \beta + 44) q^{44} + (40 \beta + 200) q^{46} + ( - 157 \beta + 107) q^{47} + ( - 13 \beta - 284) q^{49} + 25 \beta q^{50} + ( - 14 \beta + 8) q^{52} + ( - 52 \beta - 386) q^{53} + (5 \beta - 135) q^{55} + (40 \beta - 120) q^{56} - 29 \beta q^{58} + ( - 102 \beta + 150) q^{59} + ( - 106 \beta - 308) q^{61} + ( - 128 \beta - 120) q^{62} + ( - 115 \beta + 238) q^{64} + ( - 45 \beta + 205) q^{65} + ( - 69 \beta + 55) q^{67} + (50 \beta + 96) q^{68} + (30 \beta - 50) q^{70} + (184 \beta + 336) q^{71} + ( - 162 \beta - 312) q^{73} + ( - 126 \beta - 540) q^{74} + (44 \beta + 128) q^{76} + (33 \beta - 199) q^{77} + ( - 102 \beta - 318) q^{79} + (15 \beta + 330) q^{80} + ( - 90 \beta - 480) q^{82} + ( - 62 \beta + 878) q^{83} + (5 \beta - 265) q^{85} + (108 \beta - 600) q^{86} + ( - 140 \beta + 320) q^{88} + (29 \beta + 1131) q^{89} + ( - 95 \beta + 377) q^{91} + (80 \beta + 240) q^{92} + ( - 50 \beta - 1570) q^{94} + ( - 50 \beta - 70) q^{95} + (228 \beta + 422) q^{97} + ( - 297 \beta - 130) q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{2} + 5 q^{4} - 10 q^{5} - 13 q^{7} + 15 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{2} + 5 q^{4} - 10 q^{5} - 13 q^{7} + 15 q^{8} - 5 q^{10} + 53 q^{11} - 73 q^{13} + 14 q^{14} - 135 q^{16} + 105 q^{17} + 38 q^{19} - 25 q^{20} + 6 q^{22} + 60 q^{23} + 50 q^{25} + 148 q^{26} - 12 q^{28} - 58 q^{29} - 244 q^{31} - 249 q^{32} + 32 q^{34} + 65 q^{35} - 198 q^{37} + 224 q^{38} - 75 q^{40} - 132 q^{41} + 276 q^{43} + 112 q^{44} + 440 q^{46} + 57 q^{47} - 581 q^{49} + 25 q^{50} + 2 q^{52} - 824 q^{53} - 265 q^{55} - 200 q^{56} - 29 q^{58} + 198 q^{59} - 722 q^{61} - 368 q^{62} + 361 q^{64} + 365 q^{65} + 41 q^{67} + 242 q^{68} - 70 q^{70} + 856 q^{71} - 786 q^{73} - 1206 q^{74} + 300 q^{76} - 365 q^{77} - 738 q^{79} + 675 q^{80} - 1050 q^{82} + 1694 q^{83} - 525 q^{85} - 1092 q^{86} + 500 q^{88} + 2291 q^{89} + 659 q^{91} + 560 q^{92} - 3190 q^{94} - 190 q^{95} + 1072 q^{97} - 557 q^{98}+O(q^{100}) \) 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
−2.70156
3.70156
−2.70156 0 −0.701562 −5.00000 0 −9.70156 23.5078 0 13.5078
1.2 3.70156 0 5.70156 −5.00000 0 −3.29844 −8.50781 0 −18.5078
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \( -1 \)
\(5\) \( +1 \)
\(29\) \( +1 \)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1305.4.a.f 2
3.b odd 2 1 435.4.a.d 2
15.d odd 2 1 2175.4.a.e 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
435.4.a.d 2 3.b odd 2 1
1305.4.a.f 2 1.a even 1 1 trivial
2175.4.a.e 2 15.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} - T_{2} - 10 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1305))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - T - 10 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( (T + 5)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 13T + 32 \) Copy content Toggle raw display
$11$ \( T^{2} - 53T + 692 \) Copy content Toggle raw display
$13$ \( T^{2} + 73T + 502 \) Copy content Toggle raw display
$17$ \( T^{2} - 105T + 2746 \) Copy content Toggle raw display
$19$ \( T^{2} - 38T - 664 \) Copy content Toggle raw display
$23$ \( T^{2} - 60T - 3200 \) Copy content Toggle raw display
$29$ \( (T + 29)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 244T + 13408 \) Copy content Toggle raw display
$37$ \( T^{2} + 198T - 20088 \) Copy content Toggle raw display
$41$ \( T^{2} + 132T - 19260 \) Copy content Toggle raw display
$43$ \( T^{2} - 276T - 17856 \) Copy content Toggle raw display
$47$ \( T^{2} - 57T - 251840 \) Copy content Toggle raw display
$53$ \( T^{2} + 824T + 142028 \) Copy content Toggle raw display
$59$ \( T^{2} - 198T - 96840 \) Copy content Toggle raw display
$61$ \( T^{2} + 722T + 15152 \) Copy content Toggle raw display
$67$ \( T^{2} - 41T - 48380 \) Copy content Toggle raw display
$71$ \( T^{2} - 856T - 163840 \) Copy content Toggle raw display
$73$ \( T^{2} + 786T - 114552 \) Copy content Toggle raw display
$79$ \( T^{2} + 738T + 29520 \) Copy content Toggle raw display
$83$ \( T^{2} - 1694 T + 678008 \) Copy content Toggle raw display
$89$ \( T^{2} - 2291 T + 1303550 \) Copy content Toggle raw display
$97$ \( T^{2} - 1072 T - 245540 \) Copy content Toggle raw display
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