Properties

Label 100.7.d.a
Level $100$
Weight $7$
Character orbit 100.d
Analytic conductor $23.005$
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] = [100,7,Mod(99,100)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(100, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 7, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("100.99");
 
S:= CuspForms(chi, 7);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 100 = 2^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 100.d (of order \(2\), degree \(1\), 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: \(23.0054083620\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{15})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 7x^{2} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: no (minimal twist has level 4)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

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

\(f(q)\) \(=\) \( q + ( - \beta_{2} + \beta_1) q^{2} - 4 \beta_{2} q^{3} + ( - 2 \beta_{3} + 56) q^{4} + ( - 4 \beta_{3} + 240) q^{6} - 40 \beta_{2} q^{7} + ( - 48 \beta_{2} + 176 \beta_1) q^{8} + 231 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{2} + \beta_1) q^{2} - 4 \beta_{2} q^{3} + ( - 2 \beta_{3} + 56) q^{4} + ( - 4 \beta_{3} + 240) q^{6} - 40 \beta_{2} q^{7} + ( - 48 \beta_{2} + 176 \beta_1) q^{8} + 231 q^{9} - 62 \beta_{3} q^{11} + ( - 224 \beta_{2} + 480 \beta_1) q^{12} - 733 \beta_1 q^{13} + ( - 40 \beta_{3} + 2400) q^{14} + ( - 224 \beta_{3} + 2176) q^{16} - 2383 \beta_1 q^{17} + ( - 231 \beta_{2} + 231 \beta_1) q^{18} - 486 \beta_{3} q^{19} + 9600 q^{21} + (248 \beta_{2} + 3720 \beta_1) q^{22} - 1352 \beta_{2} q^{23} + ( - 704 \beta_{3} + 11520) q^{24} + (733 \beta_{3} + 2932) q^{26} + 1992 \beta_{2} q^{27} + ( - 2240 \beta_{2} + 4800 \beta_1) q^{28} - 25498 q^{29} + 2704 \beta_{3} q^{31} + ( - 1280 \beta_{2} + 15616 \beta_1) q^{32} + 14880 \beta_1 q^{33} + (2383 \beta_{3} + 9532) q^{34} + ( - 462 \beta_{3} + 12936) q^{36} + 997 \beta_1 q^{37} + (1944 \beta_{2} + 29160 \beta_1) q^{38} + 2932 \beta_{3} q^{39} + 29362 q^{41} + ( - 9600 \beta_{2} + 9600 \beta_1) q^{42} - 2780 \beta_{2} q^{43} + ( - 3472 \beta_{3} - 29760) q^{44} + ( - 1352 \beta_{3} + 81120) q^{46} + 976 \beta_{2} q^{47} + ( - 8704 \beta_{2} + 53760 \beta_1) q^{48} - 21649 q^{49} + 9532 \beta_{3} q^{51} + ( - 5864 \beta_{2} - 41048 \beta_1) q^{52} + 96427 \beta_1 q^{53} + (1992 \beta_{3} - 119520) q^{54} + ( - 7040 \beta_{3} + 115200) q^{56} + 116640 \beta_1 q^{57} + (25498 \beta_{2} - 25498 \beta_1) q^{58} + 5062 \beta_{3} q^{59} - 10918 q^{61} + ( - 10816 \beta_{2} - 162240 \beta_1) q^{62} - 9240 \beta_{2} q^{63} + ( - 16896 \beta_{3} + 14336) q^{64} + ( - 14880 \beta_{3} - 59520) q^{66} + 50884 \beta_{2} q^{67} + ( - 19064 \beta_{2} - 133448 \beta_1) q^{68} + 324480 q^{69} + 34356 \beta_{3} q^{71} + ( - 11088 \beta_{2} + 40656 \beta_1) q^{72} - 144313 \beta_1 q^{73} + ( - 997 \beta_{3} - 3988) q^{74} + ( - 27216 \beta_{3} - 233280) q^{76} + 148800 \beta_1 q^{77} + ( - 11728 \beta_{2} - 175920 \beta_1) q^{78} + 20056 \beta_{3} q^{79} - 646479 q^{81} + ( - 29362 \beta_{2} + 29362 \beta_1) q^{82} - 26356 \beta_{2} q^{83} + ( - 19200 \beta_{3} + 537600) q^{84} + ( - 2780 \beta_{3} + 166800) q^{86} + 101992 \beta_{2} q^{87} + (43648 \beta_{2} + 178560 \beta_1) q^{88} - 310738 q^{89} + 29320 \beta_{3} q^{91} + ( - 75712 \beta_{2} + 162240 \beta_1) q^{92} - 648960 \beta_1 q^{93} + (976 \beta_{3} - 58560) q^{94} + ( - 62464 \beta_{3} + 307200) q^{96} - 728543 \beta_1 q^{97} + (21649 \beta_{2} - 21649 \beta_1) q^{98} - 14322 \beta_{3} q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 224 q^{4} + 960 q^{6} + 924 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 224 q^{4} + 960 q^{6} + 924 q^{9} + 9600 q^{14} + 8704 q^{16} + 38400 q^{21} + 46080 q^{24} + 11728 q^{26} - 101992 q^{29} + 38128 q^{34} + 51744 q^{36} + 117448 q^{41} - 119040 q^{44} + 324480 q^{46} - 86596 q^{49} - 478080 q^{54} + 460800 q^{56} - 43672 q^{61} + 57344 q^{64} - 238080 q^{66} + 1297920 q^{69} - 15952 q^{74} - 933120 q^{76} - 2585916 q^{81} + 2150400 q^{84} + 667200 q^{86} - 1242952 q^{89} - 234240 q^{94} + 1228800 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 7x^{2} + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{3} - 3\nu ) / 2 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{3} + 11\nu ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 8\nu^{2} - 28 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} + \beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} + 28 ) / 8 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 3\beta_{2} + 11\beta_1 ) / 4 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/100\mathbb{Z}\right)^\times\).

\(n\) \(51\) \(77\)
\(\chi(n)\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
99.1
1.93649 0.500000i
1.93649 + 0.500000i
−1.93649 0.500000i
−1.93649 + 0.500000i
−7.74597 2.00000i −30.9839 56.0000 + 30.9839i 0 240.000 + 61.9677i −309.839 −371.806 352.000i 231.000 0
99.2 −7.74597 + 2.00000i −30.9839 56.0000 30.9839i 0 240.000 61.9677i −309.839 −371.806 + 352.000i 231.000 0
99.3 7.74597 2.00000i 30.9839 56.0000 30.9839i 0 240.000 61.9677i 309.839 371.806 352.000i 231.000 0
99.4 7.74597 + 2.00000i 30.9839 56.0000 + 30.9839i 0 240.000 + 61.9677i 309.839 371.806 + 352.000i 231.000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
5.b even 2 1 inner
20.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 100.7.d.a 4
4.b odd 2 1 inner 100.7.d.a 4
5.b even 2 1 inner 100.7.d.a 4
5.c odd 4 1 4.7.b.a 2
5.c odd 4 1 100.7.b.c 2
15.e even 4 1 36.7.d.c 2
20.d odd 2 1 inner 100.7.d.a 4
20.e even 4 1 4.7.b.a 2
20.e even 4 1 100.7.b.c 2
40.i odd 4 1 64.7.c.c 2
40.k even 4 1 64.7.c.c 2
60.l odd 4 1 36.7.d.c 2
80.i odd 4 1 256.7.d.f 4
80.j even 4 1 256.7.d.f 4
80.s even 4 1 256.7.d.f 4
80.t odd 4 1 256.7.d.f 4
120.q odd 4 1 576.7.g.h 2
120.w even 4 1 576.7.g.h 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4.7.b.a 2 5.c odd 4 1
4.7.b.a 2 20.e even 4 1
36.7.d.c 2 15.e even 4 1
36.7.d.c 2 60.l odd 4 1
64.7.c.c 2 40.i odd 4 1
64.7.c.c 2 40.k even 4 1
100.7.b.c 2 5.c odd 4 1
100.7.b.c 2 20.e even 4 1
100.7.d.a 4 1.a even 1 1 trivial
100.7.d.a 4 4.b odd 2 1 inner
100.7.d.a 4 5.b even 2 1 inner
100.7.d.a 4 20.d odd 2 1 inner
256.7.d.f 4 80.i odd 4 1
256.7.d.f 4 80.j even 4 1
256.7.d.f 4 80.s even 4 1
256.7.d.f 4 80.t odd 4 1
576.7.g.h 2 120.q odd 4 1
576.7.g.h 2 120.w even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} - 960 \) acting on \(S_{7}^{\mathrm{new}}(100, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - 112T^{2} + 4096 \) Copy content Toggle raw display
$3$ \( (T^{2} - 960)^{2} \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( (T^{2} - 96000)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} + 922560)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 2149156)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} + 22714756)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} + 56687040)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} - 109674240)^{2} \) Copy content Toggle raw display
$29$ \( (T + 25498)^{4} \) Copy content Toggle raw display
$31$ \( (T^{2} + 1754787840)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 3976036)^{2} \) Copy content Toggle raw display
$41$ \( (T - 29362)^{4} \) Copy content Toggle raw display
$43$ \( (T^{2} - 463704000)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} - 57154560)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} + 37192665316)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} + 6149722560)^{2} \) Copy content Toggle raw display
$61$ \( (T + 10918)^{4} \) Copy content Toggle raw display
$67$ \( (T^{2} - 155350887360)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} + 283280336640)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} + 83304967876)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} + 96538352640)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} - 41678324160)^{2} \) Copy content Toggle raw display
$89$ \( (T + 310738)^{4} \) Copy content Toggle raw display
$97$ \( (T^{2} + 2123099611396)^{2} \) Copy content Toggle raw display
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