Properties

Label 224.6.a.c
Level $224$
Weight $6$
Character orbit 224.a
Self dual yes
Analytic conductor $35.926$
Analytic rank $0$
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] = [224,6,Mod(1,224)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(224, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("224.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 224 = 2^{5} \cdot 7 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 224.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: \(35.9259756381\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{61}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 15 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
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 \(\beta = \sqrt{61}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta - 7) q^{3} + ( - 7 \beta + 17) q^{5} - 49 q^{7} + (14 \beta - 133) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta - 7) q^{3} + ( - 7 \beta + 17) q^{5} - 49 q^{7} + (14 \beta - 133) q^{9} + ( - 46 \beta + 210) q^{11} + ( - 49 \beta - 245) q^{13} + (32 \beta + 308) q^{15} + ( - 14 \beta - 528) q^{17} + ( - 247 \beta + 623) q^{19} + (49 \beta + 343) q^{21} + ( - 16 \beta - 252) q^{23} + ( - 238 \beta + 153) q^{25} + (278 \beta + 1778) q^{27} + ( - 546 \beta - 1952) q^{29} + (642 \beta + 1022) q^{31} + (112 \beta + 1336) q^{33} + (343 \beta - 833) q^{35} + (14 \beta - 3744) q^{37} + (588 \beta + 4704) q^{39} + (1134 \beta + 3916) q^{41} + ( - 1090 \beta + 5166) q^{43} + (1169 \beta - 8239) q^{45} + ( - 850 \beta + 20986) q^{47} + 2401 q^{49} + (626 \beta + 4550) q^{51} + ( - 2688 \beta + 16406) q^{53} + ( - 2252 \beta + 23212) q^{55} + (1106 \beta + 10706) q^{57} + ( - 1639 \beta + 24199) q^{59} + (441 \beta - 359) q^{61} + ( - 686 \beta + 6517) q^{63} + (882 \beta + 16758) q^{65} + (3696 \beta + 6412) q^{67} + (364 \beta + 2740) q^{69} + ( - 396 \beta + 51996) q^{71} + (5404 \beta - 27050) q^{73} + (1513 \beta + 13447) q^{75} + (2254 \beta - 10290) q^{77} + ( - 1108 \beta + 32284) q^{79} + ( - 7126 \beta + 2915) q^{81} + ( - 527 \beta - 23905) q^{83} + (3458 \beta - 2998) q^{85} + (5774 \beta + 46970) q^{87} + ( - 14448 \beta - 8694) q^{89} + (2401 \beta + 12005) q^{91} + ( - 5516 \beta - 46316) q^{93} + ( - 8560 \beta + 116060) q^{95} + (6146 \beta - 48648) q^{97} + (9058 \beta - 67214) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 14 q^{3} + 34 q^{5} - 98 q^{7} - 266 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 14 q^{3} + 34 q^{5} - 98 q^{7} - 266 q^{9} + 420 q^{11} - 490 q^{13} + 616 q^{15} - 1056 q^{17} + 1246 q^{19} + 686 q^{21} - 504 q^{23} + 306 q^{25} + 3556 q^{27} - 3904 q^{29} + 2044 q^{31} + 2672 q^{33} - 1666 q^{35} - 7488 q^{37} + 9408 q^{39} + 7832 q^{41} + 10332 q^{43} - 16478 q^{45} + 41972 q^{47} + 4802 q^{49} + 9100 q^{51} + 32812 q^{53} + 46424 q^{55} + 21412 q^{57} + 48398 q^{59} - 718 q^{61} + 13034 q^{63} + 33516 q^{65} + 12824 q^{67} + 5480 q^{69} + 103992 q^{71} - 54100 q^{73} + 26894 q^{75} - 20580 q^{77} + 64568 q^{79} + 5830 q^{81} - 47810 q^{83} - 5996 q^{85} + 93940 q^{87} - 17388 q^{89} + 24010 q^{91} - 92632 q^{93} + 232120 q^{95} - 97296 q^{97} - 134428 q^{99}+O(q^{100}) \) 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
4.40512
−3.40512
0 −14.8102 0 −37.6717 0 −49.0000 0 −23.6565 0
1.2 0 0.810250 0 71.6717 0 −49.0000 0 −242.343 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( -1 \)
\(7\) \( +1 \)

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 224.6.a.c 2
4.b odd 2 1 224.6.a.d yes 2
8.b even 2 1 448.6.a.y 2
8.d odd 2 1 448.6.a.s 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
224.6.a.c 2 1.a even 1 1 trivial
224.6.a.d yes 2 4.b odd 2 1
448.6.a.s 2 8.d odd 2 1
448.6.a.y 2 8.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} + 14T_{3} - 12 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(224))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 14T - 12 \) Copy content Toggle raw display
$5$ \( T^{2} - 34T - 2700 \) Copy content Toggle raw display
$7$ \( (T + 49)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} - 420T - 84976 \) Copy content Toggle raw display
$13$ \( T^{2} + 490T - 86436 \) Copy content Toggle raw display
$17$ \( T^{2} + 1056 T + 266828 \) Copy content Toggle raw display
$19$ \( T^{2} - 1246 T - 3333420 \) Copy content Toggle raw display
$23$ \( T^{2} + 504T + 47888 \) Copy content Toggle raw display
$29$ \( T^{2} + 3904 T - 14374772 \) Copy content Toggle raw display
$31$ \( T^{2} - 2044 T - 24097520 \) Copy content Toggle raw display
$37$ \( T^{2} + 7488 T + 14005580 \) Copy content Toggle raw display
$41$ \( T^{2} - 7832 T - 63108260 \) Copy content Toggle raw display
$43$ \( T^{2} - 10332 T - 45786544 \) Copy content Toggle raw display
$47$ \( T^{2} - 41972 T + 396339696 \) Copy content Toggle raw display
$53$ \( T^{2} - 32812 T - 171589148 \) Copy content Toggle raw display
$59$ \( T^{2} - 48398 T + 421726020 \) Copy content Toggle raw display
$61$ \( T^{2} + 718 T - 11734460 \) Copy content Toggle raw display
$67$ \( T^{2} - 12824 T - 792171632 \) Copy content Toggle raw display
$71$ \( T^{2} + \cdots + 2694018240 \) Copy content Toggle raw display
$73$ \( T^{2} + \cdots - 1049693676 \) Copy content Toggle raw display
$79$ \( T^{2} - 64568 T + 967369152 \) Copy content Toggle raw display
$83$ \( T^{2} + 47810 T + 554507556 \) Copy content Toggle raw display
$89$ \( T^{2} + \cdots - 12657841308 \) Copy content Toggle raw display
$97$ \( T^{2} + 97296 T + 62455628 \) Copy content Toggle raw display
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