Properties

Label 49.7.d.d
Level $49$
Weight $7$
Character orbit 49.d
Analytic conductor $11.273$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [49,7,Mod(19,49)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(49, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([5]))
 
N = Newforms(chi, 7, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("49.19");
 
S:= CuspForms(chi, 7);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 49 = 7^{2} \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 49.d (of order \(6\), degree \(2\), not 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: \(11.2726500974\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{170})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 170x^{2} + 28900 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 2^{2}\cdot 3 \)
Twist minimal: no (minimal twist has level 7)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (8 \beta_1 + 8) q^{2} + \beta_{3} q^{3} + (\beta_{3} - \beta_{2}) q^{5} + 8 \beta_{2} q^{6} + 512 q^{8} + (1311 \beta_1 + 1311) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (8 \beta_1 + 8) q^{2} + \beta_{3} q^{3} + (\beta_{3} - \beta_{2}) q^{5} + 8 \beta_{2} q^{6} + 512 q^{8} + (1311 \beta_1 + 1311) q^{9} + 8 \beta_{3} q^{10} + 874 \beta_1 q^{11} - 49 \beta_{2} q^{13} + 2040 q^{15} + (4096 \beta_1 + 4096) q^{16} - 132 \beta_{3} q^{17} + 10488 \beta_1 q^{18} + ( - 69 \beta_{3} + 69 \beta_{2}) q^{19} - 6992 q^{22} + ( - 4738 \beta_1 - 4738) q^{23} + 512 \beta_{3} q^{24} + 13585 \beta_1 q^{25} + (392 \beta_{3} - 392 \beta_{2}) q^{26} + 582 \beta_{2} q^{27} + 11146 q^{29} + (16320 \beta_1 + 16320) q^{30} - 608 \beta_{3} q^{31} + ( - 874 \beta_{3} + 874 \beta_{2}) q^{33} - 1056 \beta_{2} q^{34} + ( - 3002 \beta_1 - 3002) q^{37} - 552 \beta_{3} q^{38} - 99960 \beta_1 q^{39} + (512 \beta_{3} - 512 \beta_{2}) q^{40} - 1274 \beta_{2} q^{41} + 31418 q^{43} + 1311 \beta_{3} q^{45} - 37904 \beta_1 q^{46} + (1604 \beta_{3} - 1604 \beta_{2}) q^{47} + 4096 \beta_{2} q^{48} - 108680 q^{50} + ( - 269280 \beta_1 - 269280) q^{51} - 76406 \beta_1 q^{53} + ( - 4656 \beta_{3} + 4656 \beta_{2}) q^{54} + 874 \beta_{2} q^{55} - 140760 q^{57} + (89168 \beta_1 + 89168) q^{58} + 2507 \beta_{3} q^{59} + (6091 \beta_{3} - 6091 \beta_{2}) q^{61} - 4864 \beta_{2} q^{62} + 262144 q^{64} + ( - 99960 \beta_1 - 99960) q^{65} - 6992 \beta_{3} q^{66} + 495242 \beta_1 q^{67} - 4738 \beta_{2} q^{69} - 184406 q^{71} + (671232 \beta_1 + 671232) q^{72} - 1350 \beta_{3} q^{73} - 24016 \beta_1 q^{74} + ( - 13585 \beta_{3} + 13585 \beta_{2}) q^{75} + 799680 q^{78} + (534934 \beta_1 + 534934) q^{79} + 4096 \beta_{3} q^{80} + 231561 \beta_1 q^{81} + (10192 \beta_{3} - 10192 \beta_{2}) q^{82} + 15827 \beta_{2} q^{83} - 269280 q^{85} + (251344 \beta_1 + 251344) q^{86} + 11146 \beta_{3} q^{87} + 447488 \beta_1 q^{88} + (13938 \beta_{3} - 13938 \beta_{2}) q^{89} + 10488 \beta_{2} q^{90} + ( - 1240320 \beta_1 - 1240320) q^{93} + 12832 \beta_{3} q^{94} + 140760 \beta_1 q^{95} - 18032 \beta_{2} q^{97} - 1145814 q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 16 q^{2} + 2048 q^{8} + 2622 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 16 q^{2} + 2048 q^{8} + 2622 q^{9} - 1748 q^{11} + 8160 q^{15} + 8192 q^{16} - 20976 q^{18} - 27968 q^{22} - 9476 q^{23} - 27170 q^{25} + 44584 q^{29} + 32640 q^{30} - 6004 q^{37} + 199920 q^{39} + 125672 q^{43} + 75808 q^{46} - 434720 q^{50} - 538560 q^{51} + 152812 q^{53} - 563040 q^{57} + 178336 q^{58} + 1048576 q^{64} - 199920 q^{65} - 990484 q^{67} - 737624 q^{71} + 1342464 q^{72} + 48032 q^{74} + 3198720 q^{78} + 1069868 q^{79} - 463122 q^{81} - 1077120 q^{85} + 502688 q^{86} - 894976 q^{88} - 2480640 q^{93} - 281520 q^{95} - 4583256 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 170x^{2} + 28900 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{2} ) / 170 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} + 340\nu ) / 85 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{3} + 170\nu ) / 85 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} + \beta_{2} ) / 6 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 170\beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -170\beta_{3} + 85\beta_{2} ) / 3 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/49\mathbb{Z}\right)^\times\).

\(n\) \(3\)
\(\chi(n)\) \(1 + \beta_{1}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1
−6.51920 + 11.2916i
6.51920 11.2916i
−6.51920 11.2916i
6.51920 + 11.2916i
4.00000 6.92820i −39.1152 + 22.5832i 0 −39.1152 22.5832i 361.331i 0 512.000 655.500 1135.36i −312.922 + 180.665i
19.2 4.00000 6.92820i 39.1152 22.5832i 0 39.1152 + 22.5832i 361.331i 0 512.000 655.500 1135.36i 312.922 180.665i
31.1 4.00000 + 6.92820i −39.1152 22.5832i 0 −39.1152 + 22.5832i 361.331i 0 512.000 655.500 + 1135.36i −312.922 180.665i
31.2 4.00000 + 6.92820i 39.1152 + 22.5832i 0 39.1152 22.5832i 361.331i 0 512.000 655.500 + 1135.36i 312.922 + 180.665i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
7.c even 3 1 inner
7.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 49.7.d.d 4
7.b odd 2 1 inner 49.7.d.d 4
7.c even 3 1 7.7.b.b 2
7.c even 3 1 inner 49.7.d.d 4
7.d odd 6 1 7.7.b.b 2
7.d odd 6 1 inner 49.7.d.d 4
21.g even 6 1 63.7.d.d 2
21.h odd 6 1 63.7.d.d 2
28.f even 6 1 112.7.c.b 2
28.g odd 6 1 112.7.c.b 2
35.i odd 6 1 175.7.d.e 2
35.j even 6 1 175.7.d.e 2
35.k even 12 2 175.7.c.c 4
35.l odd 12 2 175.7.c.c 4
56.j odd 6 1 448.7.c.d 2
56.k odd 6 1 448.7.c.c 2
56.m even 6 1 448.7.c.c 2
56.p even 6 1 448.7.c.d 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7.7.b.b 2 7.c even 3 1
7.7.b.b 2 7.d odd 6 1
49.7.d.d 4 1.a even 1 1 trivial
49.7.d.d 4 7.b odd 2 1 inner
49.7.d.d 4 7.c even 3 1 inner
49.7.d.d 4 7.d odd 6 1 inner
63.7.d.d 2 21.g even 6 1
63.7.d.d 2 21.h odd 6 1
112.7.c.b 2 28.f even 6 1
112.7.c.b 2 28.g odd 6 1
175.7.c.c 4 35.k even 12 2
175.7.c.c 4 35.l odd 12 2
175.7.d.e 2 35.i odd 6 1
175.7.d.e 2 35.j even 6 1
448.7.c.c 2 56.k odd 6 1
448.7.c.c 2 56.m even 6 1
448.7.c.d 2 56.j odd 6 1
448.7.c.d 2 56.p even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} - 8T_{2} + 64 \) acting on \(S_{7}^{\mathrm{new}}(49, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - 8 T + 64)^{2} \) Copy content Toggle raw display
$3$ \( T^{4} - 2040 T^{2} + \cdots + 4161600 \) Copy content Toggle raw display
$5$ \( T^{4} - 2040 T^{2} + \cdots + 4161600 \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( (T^{2} + 874 T + 763876)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 4898040)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} - 35544960 T^{2} + \cdots + 12\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( T^{4} - 9712440 T^{2} + \cdots + 94331490753600 \) Copy content Toggle raw display
$23$ \( (T^{2} + 4738 T + 22448644)^{2} \) Copy content Toggle raw display
$29$ \( (T - 11146)^{4} \) Copy content Toggle raw display
$31$ \( T^{4} - 754114560 T^{2} + \cdots + 56\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( (T^{2} + 3002 T + 9012004)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} + 3311075040)^{2} \) Copy content Toggle raw display
$43$ \( (T - 31418)^{4} \) Copy content Toggle raw display
$47$ \( T^{4} - 5248544640 T^{2} + \cdots + 27\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( (T^{2} - 76406 T + 5837876836)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} - 12821499960 T^{2} + \cdots + 16\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{4} - 75684573240 T^{2} + \cdots + 57\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( (T^{2} + 495242 T + 245264638564)^{2} \) Copy content Toggle raw display
$71$ \( (T + 184406)^{4} \) Copy content Toggle raw display
$73$ \( T^{4} - 3717900000 T^{2} + \cdots + 13\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( (T^{2} - 534934 T + 286154384356)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} + 511007615160)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} - 396306401760 T^{2} + \cdots + 15\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( (T^{2} + 663312168960)^{2} \) Copy content Toggle raw display
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