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Properties

Label 100.4.e.d
Level $100$
Weight $4$
Character orbit 100.e
Analytic conductor $5.900$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [100,4,Mod(7,100)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(100, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("100.7");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 100 = 2^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 100.e (of order \(4\), degree \(2\), 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: \(5.90019100057\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: 8.0.2342560000.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 4x^{7} + 24x^{6} - 58x^{5} + 141x^{4} - 190x^{3} + 186x^{2} - 100x + 20 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 2^{8} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{5} q^{2} + ( - \beta_{6} - \beta_{3}) q^{3} + ( - 3 \beta_{4} - \beta_1) q^{4} + ( - \beta_{7} + 5) q^{6} + (5 \beta_{5} - 5 \beta_{2}) q^{7} + ( - 8 \beta_{6} - 6 \beta_{3}) q^{8} + 17 \beta_{4} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{5} q^{2} + ( - \beta_{6} - \beta_{3}) q^{3} + ( - 3 \beta_{4} - \beta_1) q^{4} + ( - \beta_{7} + 5) q^{6} + (5 \beta_{5} - 5 \beta_{2}) q^{7} + ( - 8 \beta_{6} - 6 \beta_{3}) q^{8} + 17 \beta_{4} q^{9} - 8 \beta_{7} q^{11} + (2 \beta_{5} + 8 \beta_{2}) q^{12} + (8 \beta_{6} - 8 \beta_{3}) q^{13} + (25 \beta_{4} - 5 \beta_1) q^{14} + ( - 6 \beta_{7} + 46) q^{16} + ( - 24 \beta_{5} - 24 \beta_{2}) q^{17} + 17 \beta_{3} q^{18} - 8 \beta_1 q^{19} + 50 q^{21} + ( - 24 \beta_{5} + 64 \beta_{2}) q^{22} + (47 \beta_{6} + 47 \beta_{3}) q^{23} + ( - 70 \beta_{4} - 2 \beta_1) q^{24} + ( - 8 \beta_{7} - 88) q^{26} + (44 \beta_{5} - 44 \beta_{2}) q^{27} + ( - 40 \beta_{6} + 10 \beta_{3}) q^{28} - 46 \beta_{4} q^{29} + 16 \beta_{7} q^{31} + (28 \beta_{5} + 48 \beta_{2}) q^{32} + (40 \beta_{6} - 40 \beta_{3}) q^{33} + (264 \beta_{4} + 24 \beta_1) q^{34} + (17 \beta_{7} + 51) q^{36} + ( - 64 \beta_{5} - 64 \beta_{2}) q^{37} + ( - 64 \beta_{6} - 24 \beta_{3}) q^{38} + 16 \beta_1 q^{39} - 188 q^{41} + 50 \beta_{5} q^{42} + ( - 55 \beta_{6} - 55 \beta_{3}) q^{43} + ( - 440 \beta_{4} + 24 \beta_1) q^{44} + (47 \beta_{7} - 235) q^{46} + (87 \beta_{5} - 87 \beta_{2}) q^{47} + ( - 16 \beta_{6} - 76 \beta_{3}) q^{48} - 93 \beta_{4} q^{49} + 48 \beta_{7} q^{51} + ( - 112 \beta_{5} + 64 \beta_{2}) q^{52} + (8 \beta_{6} - 8 \beta_{3}) q^{53} + (220 \beta_{4} - 44 \beta_1) q^{54} + (10 \beta_{7} + 350) q^{56} + (40 \beta_{5} + 40 \beta_{2}) q^{57} - 46 \beta_{3} q^{58} + 56 \beta_1 q^{59} + 72 q^{61} + (48 \beta_{5} - 128 \beta_{2}) q^{62} + (85 \beta_{6} + 85 \beta_{3}) q^{63} + ( - 468 \beta_{4} - 28 \beta_1) q^{64} + ( - 40 \beta_{7} - 440) q^{66} + ( - 267 \beta_{5} + 267 \beta_{2}) q^{67} + (192 \beta_{6} + 336 \beta_{3}) q^{68} + 470 \beta_{4} q^{69} - 96 \beta_{7} q^{71} + (102 \beta_{5} - 136 \beta_{2}) q^{72} + ( - 152 \beta_{6} + 152 \beta_{3}) q^{73} + (704 \beta_{4} + 64 \beta_1) q^{74} + ( - 24 \beta_{7} + 440) q^{76} + (200 \beta_{5} + 200 \beta_{2}) q^{77} + (128 \beta_{6} + 48 \beta_{3}) q^{78} - 112 \beta_1 q^{79} - 19 q^{81} - 188 \beta_{5} q^{82} + ( - 99 \beta_{6} - 99 \beta_{3}) q^{83} + ( - 150 \beta_{4} - 50 \beta_1) q^{84} + ( - 55 \beta_{7} + 275) q^{86} + ( - 46 \beta_{5} + 46 \beta_{2}) q^{87} + (192 \beta_{6} - 368 \beta_{3}) q^{88} - 726 \beta_{4} q^{89} - 80 \beta_{7} q^{91} + ( - 94 \beta_{5} - 376 \beta_{2}) q^{92} + ( - 80 \beta_{6} + 80 \beta_{3}) q^{93} + (435 \beta_{4} - 87 \beta_1) q^{94} + ( - 76 \beta_{7} - 100) q^{96} + (136 \beta_{5} + 136 \beta_{2}) q^{97} - 93 \beta_{3} q^{98} - 136 \beta_1 q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 40 q^{6}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 40 q^{6} + 368 q^{16} + 400 q^{21} - 704 q^{26} + 408 q^{36} - 1504 q^{41} - 1880 q^{46} + 2800 q^{56} + 576 q^{61} - 3520 q^{66} + 3520 q^{76} - 152 q^{81} + 2200 q^{86} - 800 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 4x^{7} + 24x^{6} - 58x^{5} + 141x^{4} - 190x^{3} + 186x^{2} - 100x + 20 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{6} + 3\nu^{5} - 17\nu^{4} + 29\nu^{3} - 34\nu^{2} + 20\nu + 70 ) / 10 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 33\nu^{7} - 98\nu^{6} + 678\nu^{5} - 1190\nu^{4} + 3213\nu^{3} - 2756\nu^{2} + 2760\nu - 620 ) / 150 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -33\nu^{7} + 133\nu^{6} - 783\nu^{5} + 1885\nu^{4} - 4428\nu^{3} + 5746\nu^{2} - 5160\nu + 2020 ) / 150 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -22\nu^{7} + 77\nu^{6} - 487\nu^{5} + 1025\nu^{4} - 2547\nu^{3} + 2834\nu^{2} - 2540\nu + 830 ) / 50 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 163\nu^{7} - 563\nu^{6} + 3613\nu^{5} - 7535\nu^{4} + 19108\nu^{3} - 21206\nu^{2} + 19960\nu - 6620 ) / 150 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 163\nu^{7} - 578\nu^{6} + 3658\nu^{5} - 7790\nu^{4} + 19543\nu^{3} - 22016\nu^{2} + 20560\nu - 6920 ) / 150 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 16\nu^{7} - 56\nu^{6} + 356\nu^{5} - 750\nu^{4} + 1876\nu^{3} - 2092\nu^{2} + 1880\nu - 615 ) / 5 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{4} - \beta_{3} + \beta_{2} + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{6} + \beta_{5} + \beta_{4} - \beta_{3} + \beta_{2} + \beta _1 - 8 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -3\beta_{7} + 6\beta_{5} - 23\beta_{4} + 16\beta_{3} - 16\beta_{2} + 3\beta _1 - 25 ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -3\beta_{7} + 18\beta_{6} - 12\beta_{5} - 24\beta_{4} + 20\beta_{3} - 14\beta_{2} - 8\beta _1 + 59 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 35\beta_{7} + 56\beta_{6} - 104\beta_{5} + 257\beta_{4} - 106\beta_{3} + 136\beta_{2} - 45\beta _1 + 337 ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 60\beta_{7} - 188\beta_{6} + 101\beta_{5} + 446\beta_{4} - 253\beta_{3} + 196\beta_{2} + 58\beta _1 - 428 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 287 \beta_{7} - 1136 \beta_{6} + 1454 \beta_{5} - 2123 \beta_{4} + 560 \beta_{3} - 1064 \beta_{2} + \cdots - 4205 ) / 4 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(51\) \(77\)
\(\chi(n)\) \(-1\) \(-\beta_{4}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
7.1
0.500000 − 3.27635i
0.500000 − 1.04028i
0.500000 + 0.0402784i
0.500000 + 2.27635i
0.500000 + 3.27635i
0.500000 + 1.04028i
0.500000 − 0.0402784i
0.500000 − 2.27635i
−2.77635 − 0.540278i −2.23607 − 2.23607i 7.41620 + 3.00000i 0 5.00000 + 7.41620i −11.1803 + 11.1803i −18.9691 − 12.3359i − 17.0000i 0
7.2 −0.540278 − 2.77635i 2.23607 + 2.23607i −7.41620 + 3.00000i 0 5.00000 − 7.41620i 11.1803 − 11.1803i 12.3359 + 18.9691i − 17.0000i 0
7.3 0.540278 + 2.77635i −2.23607 − 2.23607i −7.41620 + 3.00000i 0 5.00000 − 7.41620i −11.1803 + 11.1803i −12.3359 − 18.9691i − 17.0000i 0
7.4 2.77635 + 0.540278i 2.23607 + 2.23607i 7.41620 + 3.00000i 0 5.00000 + 7.41620i 11.1803 − 11.1803i 18.9691 + 12.3359i − 17.0000i 0
43.1 −2.77635 + 0.540278i −2.23607 + 2.23607i 7.41620 − 3.00000i 0 5.00000 − 7.41620i −11.1803 − 11.1803i −18.9691 + 12.3359i 17.0000i 0
43.2 −0.540278 + 2.77635i 2.23607 − 2.23607i −7.41620 − 3.00000i 0 5.00000 + 7.41620i 11.1803 + 11.1803i 12.3359 − 18.9691i 17.0000i 0
43.3 0.540278 − 2.77635i −2.23607 + 2.23607i −7.41620 − 3.00000i 0 5.00000 + 7.41620i −11.1803 − 11.1803i −12.3359 + 18.9691i 17.0000i 0
43.4 2.77635 − 0.540278i 2.23607 − 2.23607i 7.41620 − 3.00000i 0 5.00000 − 7.41620i 11.1803 + 11.1803i 18.9691 − 12.3359i 17.0000i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 7.4
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
5.c odd 4 2 inner
20.d odd 2 1 inner
20.e even 4 2 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 100.4.e.d ✓ 8
4.b odd 2 1 inner 100.4.e.d ✓ 8
5.b even 2 1 inner 100.4.e.d ✓ 8
5.c odd 4 2 inner 100.4.e.d ✓ 8
20.d odd 2 1 inner 100.4.e.d ✓ 8
20.e even 4 2 inner 100.4.e.d ✓ 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
100.4.e.d ✓ 8 1.a even 1 1 trivial
100.4.e.d ✓ 8 4.b odd 2 1 inner
100.4.e.d ✓ 8 5.b even 2 1 inner
100.4.e.d ✓ 8 5.c odd 4 2 inner
100.4.e.d ✓ 8 20.d odd 2 1 inner
100.4.e.d ✓ 8 20.e even 4 2 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} + 100 \) acting on \(S_{4}^{\mathrm{new}}(100, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} - 92T^{4} + 4096 \) Copy content Toggle raw display
$3$ \( (T^{4} + 100)^{2} \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( (T^{4} + 62500)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} + 3520)^{4} \) Copy content Toggle raw display
$13$ \( (T^{4} + 1982464)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} + 160579584)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} - 3520)^{4} \) Copy content Toggle raw display
$23$ \( (T^{4} + 487968100)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 2116)^{4} \) Copy content Toggle raw display
$31$ \( (T^{2} + 14080)^{4} \) Copy content Toggle raw display
$37$ \( (T^{4} + 8120172544)^{2} \) Copy content Toggle raw display
$41$ \( (T + 188)^{8} \) Copy content Toggle raw display
$43$ \( (T^{4} + 915062500)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} + 5728976100)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} + 1982464)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} - 172480)^{4} \) Copy content Toggle raw display
$61$ \( (T - 72)^{8} \) Copy content Toggle raw display
$67$ \( (T^{4} + 508212152100)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} + 506880)^{4} \) Copy content Toggle raw display
$73$ \( (T^{4} + 258356690944)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} - 689920)^{4} \) Copy content Toggle raw display
$83$ \( (T^{4} + 9605960100)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 527076)^{4} \) Copy content Toggle raw display
$97$ \( (T^{4} + 165577375744)^{2} \) Copy content Toggle raw display
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