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Properties

Label 100.7.b.f
Level $100$
Weight $7$
Character orbit 100.b
Analytic conductor $23.005$
Analytic rank $0$
Dimension $12$
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(51,100)]
 
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, 0]))
 
N = Newforms(chi, 7, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("100.51");
 
S:= CuspForms(chi, 7);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 100 = 2^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 100.b (of order \(2\), degree \(1\), 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: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 16x^{10} - 10x^{8} + 1775x^{6} - 1000x^{4} - 160000x^{2} + 1000000 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{32}\cdot 3^{2}\cdot 5^{2}\cdot 7^{2} \)
Twist minimal: no (minimal twist has level 20)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + ( - \beta_{2} + \beta_1) q^{3} + ( - \beta_{7} + \beta_{5} - 6) q^{4} + (6 \beta_{5} + \beta_{3} - 58) q^{6} + ( - \beta_{8} - \beta_{4} + 5 \beta_{2}) q^{7} + (2 \beta_{10} + \beta_{8} + \cdots - 9 \beta_1) q^{8}+ \cdots + ( - 3 \beta_{11} + 6 \beta_{7} + \cdots - 162) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + ( - \beta_{2} + \beta_1) q^{3} + ( - \beta_{7} + \beta_{5} - 6) q^{4} + (6 \beta_{5} + \beta_{3} - 58) q^{6} + ( - \beta_{8} - \beta_{4} + 5 \beta_{2}) q^{7} + (2 \beta_{10} + \beta_{8} + \cdots - 9 \beta_1) q^{8}+ \cdots + ( - 1611 \beta_{11} - 357 \beta_{9} + \cdots - 2667) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 64 q^{4} - 672 q^{6} - 1956 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 64 q^{4} - 672 q^{6} - 1956 q^{9} - 512 q^{14} - 20928 q^{16} + 51216 q^{21} + 20928 q^{24} + 14496 q^{26} + 16072 q^{29} - 257216 q^{34} - 144960 q^{36} - 192136 q^{41} + 165120 q^{44} - 49472 q^{46} - 145796 q^{49} + 118656 q^{54} + 1078208 q^{56} + 215384 q^{61} + 6656 q^{64} - 1403520 q^{66} - 1015824 q^{69} + 1020384 q^{74} + 2515200 q^{76} - 2327652 q^{81} - 424704 q^{84} - 5268832 q^{86} + 4346152 q^{89} - 4292992 q^{94} + 7673088 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 16x^{10} - 10x^{8} + 1775x^{6} - 1000x^{4} - 160000x^{2} + 1000000 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -9\nu^{11} + 64\nu^{9} + 970\nu^{7} - 8775\nu^{5} - 89000\nu^{3} + 770000\nu ) / 50000 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -2\nu^{11} - 33\nu^{9} + 260\nu^{7} + 2900\nu^{5} - 28375\nu^{3} - 230000\nu ) / 12500 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -9\nu^{10} - 16\nu^{8} + 1050\nu^{6} - 775\nu^{4} - 87000\nu^{2} - 20000 ) / 1250 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -3\nu^{11} + 28\nu^{9} + 350\nu^{7} - 3125\nu^{5} - 24500\nu^{3} + 290000\nu ) / 6250 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -3\nu^{10} + 28\nu^{8} + 350\nu^{6} - 3125\nu^{4} - 34500\nu^{2} + 327500 ) / 2500 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -49\nu^{11} + 464\nu^{9} + 4410\nu^{7} - 56575\nu^{5} - 355000\nu^{3} + 6070000\nu ) / 25000 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -\nu^{10} + 56\nu^{8} + 170\nu^{6} - 5975\nu^{4} - 5000\nu^{2} + 586250 ) / 1250 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 103\nu^{11} + 2512\nu^{9} - 5590\nu^{7} - 226775\nu^{5} + 957000\nu^{3} + 26490000\nu ) / 50000 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 22\nu^{10} + 28\nu^{8} - 1900\nu^{6} + 7850\nu^{4} + 225500\nu^{2} - 238125 ) / 625 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 209\nu^{11} - 1744\nu^{9} - 20890\nu^{7} + 182175\nu^{5} + 2027000\nu^{3} - 17970000\nu ) / 50000 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 61\nu^{10} - 436\nu^{8} - 6050\nu^{6} + 49675\nu^{4} + 547500\nu^{2} - 4712500 ) / 2500 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( 5\beta_{8} + 21\beta_{6} - 41\beta_{4} + 100\beta_{2} - 151\beta_1 ) / 1680 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -33\beta_{11} + 8\beta_{9} + 130\beta_{7} - 979\beta_{5} + 76\beta_{3} + 9338 ) / 3360 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 35\beta_{10} - 10\beta_{8} + 42\beta_{6} + 383\beta_{4} - 60\beta_{2} - 727\beta_1 ) / 560 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -333\beta_{11} + 208\beta_{9} - 1870\beta_{7} + 1321\beta_{5} - 124\beta_{3} + 153538 ) / 3360 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -1470\beta_{10} + 295\beta_{8} + 651\beta_{6} + 1949\beta_{4} + 8420\beta_{2} - 50531\beta_1 ) / 1680 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 507\beta_{11} + 243\beta_{9} + 1980\beta_{7} + 1741\beta_{5} + 2396\beta_{3} - 70077 ) / 560 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 9030\beta_{10} + 2795\beta_{8} - 9849\beta_{6} + 123199\beta_{4} + 58420\beta_{2} - 31531\beta_1 ) / 1680 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( -4869\beta_{11} + 2744\beta_{9} - 19910\beta_{7} + 29153\beta_{5} - 5732\beta_{3} - 2705566 ) / 480 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( -53865\beta_{10} - 1360\beta_{8} - 47908\beta_{6} + 151733\beta_{4} - 113860\beta_{2} - 1048177\beta_1 ) / 560 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 763167\beta_{11} + 40708\beta_{9} + 538130\beta_{7} + 10205821\beta_{5} + 557876\beta_{3} - 126339962 ) / 3360 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 219030 \beta_{10} + 709045 \beta_{8} - 2167599 \beta_{6} - 255301 \beta_{4} + 5993420 \beta_{2} + 22823719 \beta_1 ) / 1680 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
51.1
0.771446 − 3.06674i
0.771446 + 3.06674i
3.01463 + 0.954985i
3.01463 − 0.954985i
−3.05235 + 0.826519i
−3.05235 − 0.826519i
3.05235 + 0.826519i
3.05235 − 0.826519i
−3.01463 + 0.954985i
−3.01463 − 0.954985i
−0.771446 − 3.06674i
−0.771446 + 3.06674i
−7.09927 − 3.68788i 31.7642i 36.7991 + 52.3624i 0 117.142 − 225.502i 88.8045i −68.1408 − 507.445i −279.961 0
51.2 −7.09927 + 3.68788i − 31.7642i 36.7991 − 52.3624i 0 117.142 + 225.502i − 88.8045i −68.1408 + 507.445i −279.961 0
51.3 −5.35482 − 5.94356i − 40.3729i −6.65182 + 63.6534i 0 −239.959 + 216.189i 450.705i 413.947 − 301.317i −900.969 0
51.4 −5.35482 + 5.94356i 40.3729i −6.65182 − 63.6534i 0 −239.959 − 216.189i − 450.705i 413.947 + 301.317i −900.969 0
51.5 −2.98770 − 7.42116i − 6.08848i −46.1473 + 44.3444i 0 −45.1836 + 18.1905i − 422.345i 466.961 + 209.979i 691.930 0
51.6 −2.98770 + 7.42116i 6.08848i −46.1473 − 44.3444i 0 −45.1836 − 18.1905i 422.345i 466.961 − 209.979i 691.930 0
51.7 2.98770 − 7.42116i − 6.08848i −46.1473 − 44.3444i 0 −45.1836 − 18.1905i − 422.345i −466.961 + 209.979i 691.930 0
51.8 2.98770 + 7.42116i 6.08848i −46.1473 + 44.3444i 0 −45.1836 + 18.1905i 422.345i −466.961 − 209.979i 691.930 0
51.9 5.35482 − 5.94356i − 40.3729i −6.65182 − 63.6534i 0 −239.959 − 216.189i 450.705i −413.947 − 301.317i −900.969 0
51.10 5.35482 + 5.94356i 40.3729i −6.65182 + 63.6534i 0 −239.959 + 216.189i − 450.705i −413.947 + 301.317i −900.969 0
51.11 7.09927 − 3.68788i 31.7642i 36.7991 − 52.3624i 0 117.142 + 225.502i 88.8045i 68.1408 − 507.445i −279.961 0
51.12 7.09927 + 3.68788i − 31.7642i 36.7991 + 52.3624i 0 117.142 − 225.502i − 88.8045i 68.1408 + 507.445i −279.961 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 51.12
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.b.f 12
4.b odd 2 1 inner 100.7.b.f 12
5.b even 2 1 inner 100.7.b.f 12
5.c odd 4 2 20.7.d.d ✓ 12
15.e even 4 2 180.7.f.e 12
20.d odd 2 1 inner 100.7.b.f 12
20.e even 4 2 20.7.d.d ✓ 12
40.i odd 4 2 320.7.h.f 12
40.k even 4 2 320.7.h.f 12
60.l odd 4 2 180.7.f.e 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
20.7.d.d ✓ 12 5.c odd 4 2
20.7.d.d ✓ 12 20.e even 4 2
100.7.b.f 12 1.a even 1 1 trivial
100.7.b.f 12 4.b odd 2 1 inner
100.7.b.f 12 5.b even 2 1 inner
100.7.b.f 12 20.d odd 2 1 inner
180.7.f.e 12 15.e even 4 2
180.7.f.e 12 60.l odd 4 2
320.7.h.f 12 40.i odd 4 2
320.7.h.f 12 40.k even 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{7}^{\mathrm{new}}(100, [\chi])\):

\( T_{3}^{6} + 2676T_{3}^{4} + 1742400T_{3}^{2} + 60963840 \) Copy content Toggle raw display
\( T_{13}^{6} - 22348032T_{13}^{4} + 133299351330816T_{13}^{2} - 93610387146807705600 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} + \cdots + 68719476736 \) Copy content Toggle raw display
$3$ \( (T^{6} + 2676 T^{4} + \cdots + 60963840)^{2} \) Copy content Toggle raw display
$5$ \( T^{12} \) Copy content Toggle raw display
$7$ \( (T^{6} + \cdots + 285751234944000)^{2} \) Copy content Toggle raw display
$11$ \( (T^{6} + \cdots + 14\!\cdots\!00)^{2} \) Copy content Toggle raw display
$13$ \( (T^{6} + \cdots - 93\!\cdots\!00)^{2} \) Copy content Toggle raw display
$17$ \( (T^{6} + \cdots - 67\!\cdots\!00)^{2} \) Copy content Toggle raw display
$19$ \( (T^{6} + \cdots + 14\!\cdots\!00)^{2} \) Copy content Toggle raw display
$23$ \( (T^{6} + \cdots + 56\!\cdots\!40)^{2} \) Copy content Toggle raw display
$29$ \( (T^{3} + \cdots + 9262562663592)^{4} \) Copy content Toggle raw display
$31$ \( (T^{6} + \cdots + 14\!\cdots\!00)^{2} \) Copy content Toggle raw display
$37$ \( (T^{6} + \cdots - 16\!\cdots\!00)^{2} \) Copy content Toggle raw display
$41$ \( (T^{3} + \cdots - 3327456655368)^{4} \) Copy content Toggle raw display
$43$ \( (T^{6} + \cdots + 22\!\cdots\!00)^{2} \) Copy content Toggle raw display
$47$ \( (T^{6} + \cdots + 29\!\cdots\!40)^{2} \) Copy content Toggle raw display
$53$ \( (T^{6} + \cdots - 51\!\cdots\!00)^{2} \) Copy content Toggle raw display
$59$ \( (T^{6} + \cdots + 13\!\cdots\!00)^{2} \) Copy content Toggle raw display
$61$ \( (T^{3} + \cdots - 21\!\cdots\!88)^{4} \) Copy content Toggle raw display
$67$ \( (T^{6} + \cdots + 18\!\cdots\!40)^{2} \) Copy content Toggle raw display
$71$ \( (T^{6} + \cdots + 13\!\cdots\!00)^{2} \) Copy content Toggle raw display
$73$ \( (T^{6} + \cdots - 40\!\cdots\!00)^{2} \) Copy content Toggle raw display
$79$ \( (T^{6} + \cdots + 17\!\cdots\!00)^{2} \) Copy content Toggle raw display
$83$ \( (T^{6} + \cdots + 29\!\cdots\!40)^{2} \) Copy content Toggle raw display
$89$ \( (T^{3} + \cdots + 46\!\cdots\!72)^{4} \) Copy content Toggle raw display
$97$ \( (T^{6} + \cdots - 59\!\cdots\!00)^{2} \) Copy content Toggle raw display
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