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Properties

Label 100.11.b.e
Level $100$
Weight $11$
Character orbit 100.b
Analytic conductor $63.536$
Analytic rank $0$
Dimension $20$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [100,11,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, 11, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("100.51");
 
S:= CuspForms(chi, 11);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 100 = 2^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 11 \)
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: \(63.5357252674\)
Analytic rank: \(0\)
Dimension: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} + 199481 x^{18} + 16413464051 x^{16} + 725560177607766 x^{14} + \cdots + 21\!\cdots\!25 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{97}\cdot 3^{4}\cdot 5^{29} \)
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 \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_1 - 1) q^{2} + (\beta_{2} - \beta_1) q^{3} + ( - \beta_{5} + \beta_{2} + \beta_1 - 32) q^{4} + (\beta_{6} - \beta_{5} + \beta_{3} + \cdots - 739) q^{6}+ \cdots + (\beta_{9} + \beta_{6} - 17 \beta_{5} + \cdots - 20721) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 - 1) q^{2} + (\beta_{2} - \beta_1) q^{3} + ( - \beta_{5} + \beta_{2} + \beta_1 - 32) q^{4} + (\beta_{6} - \beta_{5} + \beta_{3} + \cdots - 739) q^{6}+ \cdots + (11903 \beta_{19} + 17874 \beta_{18} + \cdots + 9653837) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 22 q^{2} - 644 q^{4} - 14784 q^{6} - 3448 q^{8} - 414868 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q - 22 q^{2} - 644 q^{4} - 14784 q^{6} - 3448 q^{8} - 414868 q^{9} - 1329640 q^{12} + 278864 q^{13} - 2240504 q^{14} + 4261360 q^{16} + 1921656 q^{17} + 3556082 q^{18} + 4157512 q^{21} + 5811280 q^{22} - 19112144 q^{24} + 25066884 q^{26} + 87415400 q^{28} - 66014888 q^{29} + 33171328 q^{32} - 85980560 q^{33} - 27236084 q^{34} + 355456476 q^{36} + 153620656 q^{37} - 250352720 q^{38} + 477406160 q^{41} + 570662040 q^{42} + 339141040 q^{44} - 897549304 q^{46} + 479727360 q^{48} + 333772012 q^{49} + 110465096 q^{52} + 1669491824 q^{53} + 706139792 q^{54} - 1362290224 q^{56} - 3973032960 q^{57} - 2075027916 q^{58} - 4283166080 q^{61} - 1664032240 q^{62} + 340459456 q^{64} + 1884031760 q^{66} - 3042411896 q^{68} - 5321669928 q^{69} - 1632326712 q^{72} - 2474287656 q^{73} + 188682276 q^{74} + 2323171200 q^{76} - 410885040 q^{77} + 19914223760 q^{78} + 9939722652 q^{81} + 3197757116 q^{82} + 2383099552 q^{84} + 19648321456 q^{86} - 2774318240 q^{88} + 3011851592 q^{89} + 27349072440 q^{92} + 11861394640 q^{93} + 15684681576 q^{94} - 1990377984 q^{96} + 39984502056 q^{97} - 38416891998 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} + 199481 x^{18} + 16413464051 x^{16} + 725560177607766 x^{14} + \cdots + 21\!\cdots\!25 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 18\!\cdots\!50 \nu^{19} + \cdots - 96\!\cdots\!75 ) / 40\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 18\!\cdots\!50 \nu^{19} + \cdots - 96\!\cdots\!75 ) / 40\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 18\!\cdots\!50 \nu^{19} + \cdots + 88\!\cdots\!25 ) / 13\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 15\!\cdots\!50 \nu^{19} + \cdots - 84\!\cdots\!75 ) / 13\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 30\!\cdots\!85 \nu^{19} + \cdots + 11\!\cdots\!50 ) / 20\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 37\!\cdots\!86 \nu^{19} + \cdots + 16\!\cdots\!75 ) / 20\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 56\!\cdots\!66 \nu^{19} + \cdots + 10\!\cdots\!75 ) / 40\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 10\!\cdots\!70 \nu^{19} + \cdots + 20\!\cdots\!75 ) / 40\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 57\!\cdots\!81 \nu^{19} + \cdots + 16\!\cdots\!50 ) / 20\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 23\!\cdots\!40 \nu^{19} + \cdots + 81\!\cdots\!75 ) / 40\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 26\!\cdots\!01 \nu^{19} + \cdots - 36\!\cdots\!50 ) / 33\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 14\!\cdots\!81 \nu^{19} + \cdots + 73\!\cdots\!75 ) / 10\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 42\!\cdots\!25 \nu^{19} + \cdots - 26\!\cdots\!25 ) / 22\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 47\!\cdots\!54 \nu^{19} + \cdots + 13\!\cdots\!25 ) / 20\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 42\!\cdots\!41 \nu^{19} + \cdots + 79\!\cdots\!50 ) / 10\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 17\!\cdots\!18 \nu^{19} + \cdots + 48\!\cdots\!25 ) / 40\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 19\!\cdots\!88 \nu^{19} + \cdots + 16\!\cdots\!75 ) / 44\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( - 22\!\cdots\!70 \nu^{19} + \cdots - 48\!\cdots\!75 ) / 40\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( 12\!\cdots\!06 \nu^{19} + \cdots + 11\!\cdots\!25 ) / 20\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{2} - \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{9} + \beta_{6} - 17\beta_{5} - \beta_{4} - \beta_{3} + \beta_{2} - 149\beta _1 - 79770 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 18 \beta_{19} - 17 \beta_{18} - 140 \beta_{17} - 25 \beta_{16} + 42 \beta_{15} - 36 \beta_{14} + \cdots - 1549 ) / 8 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 1001 \beta_{19} - 2307 \beta_{18} - 3273 \beta_{17} - 2341 \beta_{16} + 2745 \beta_{15} + \cdots + 2785837821 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 1582900 \beta_{19} + 765580 \beta_{18} + 8489000 \beta_{17} + 1492375 \beta_{16} - 2653280 \beta_{15} + \cdots - 35539745 ) / 8 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 108060431 \beta_{19} + 373226541 \beta_{18} + 544085658 \beta_{17} + 376735486 \beta_{16} + \cdots - 233621488853961 ) / 8 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 92413589696 \beta_{19} - 36734890024 \beta_{18} - 442661606313 \beta_{17} - 81670400505 \beta_{16} + \cdots + 5307894750012 ) / 8 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 2573064261856 \beta_{19} - 10856118679092 \beta_{18} - 16319506238160 \beta_{17} + \cdots + 53\!\cdots\!90 ) / 4 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 49\!\cdots\!28 \beta_{19} + \cdots - 33\!\cdots\!81 ) / 8 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 25\!\cdots\!83 \beta_{19} + \cdots - 50\!\cdots\!85 ) / 8 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 25\!\cdots\!02 \beta_{19} + \cdots + 16\!\cdots\!25 ) / 8 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 65\!\cdots\!61 \beta_{19} + \cdots + 12\!\cdots\!45 ) / 4 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 32\!\cdots\!02 \beta_{19} + \cdots - 17\!\cdots\!77 ) / 2 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 34\!\cdots\!44 \beta_{19} + \cdots - 60\!\cdots\!40 ) / 4 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 67\!\cdots\!42 \beta_{19} + \cdots + 27\!\cdots\!25 ) / 8 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( ( - 18\!\cdots\!16 \beta_{19} + \cdots + 29\!\cdots\!10 ) / 4 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( ( - 34\!\cdots\!08 \beta_{19} + \cdots - 90\!\cdots\!31 ) / 8 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( ( 19\!\cdots\!93 \beta_{19} + \cdots - 29\!\cdots\!35 ) / 8 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( ( 17\!\cdots\!92 \beta_{19} + \cdots + 20\!\cdots\!64 ) / 8 \) 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
103.213i
− 103.213i
− 160.986i
160.986i
104.388i
− 104.388i
− 165.390i
165.390i
221.211i
− 221.211i
40.2810i
− 40.2810i
− 224.354i
224.354i
− 8.56570i
8.56570i
− 68.5823i
68.5823i
− 137.813i
137.813i
−30.9316 − 8.19968i 206.425i 889.530 + 507.259i 0 1692.62 − 6385.07i 1527.38i −23355.3 − 22984.2i 16437.6 0
51.2 −30.9316 + 8.19968i − 206.425i 889.530 − 507.259i 0 1692.62 + 6385.07i − 1527.38i −23355.3 + 22984.2i 16437.6 0
51.3 −30.5298 − 9.58811i − 321.971i 840.136 + 585.446i 0 −3087.10 + 9829.71i − 9880.78i −20035.9 − 25928.9i −44616.5 0
51.4 −30.5298 + 9.58811i 321.971i 840.136 − 585.446i 0 −3087.10 − 9829.71i 9880.78i −20035.9 + 25928.9i −44616.5 0
51.5 −24.2624 − 20.8647i 208.777i 153.329 + 1012.46i 0 4356.06 − 5065.43i − 17557.3i 17404.4 − 27763.8i 15461.2 0
51.6 −24.2624 + 20.8647i − 208.777i 153.329 − 1012.46i 0 4356.06 + 5065.43i 17557.3i 17404.4 + 27763.8i 15461.2 0
51.7 −9.98863 − 30.4011i − 330.781i −824.454 + 607.331i 0 −10056.1 + 3304.05i − 29449.0i 26698.7 + 18997.9i −50366.8 0
51.8 −9.98863 + 30.4011i 330.781i −824.454 − 607.331i 0 −10056.1 − 3304.05i 29449.0i 26698.7 − 18997.9i −50366.8 0
51.9 −7.05603 − 31.2124i 442.423i −924.425 + 440.471i 0 13809.1 − 3121.75i − 2455.96i 20270.9 + 25745.5i −136689. 0
51.10 −7.05603 + 31.2124i − 442.423i −924.425 − 440.471i 0 13809.1 + 3121.75i 2455.96i 20270.9 − 25745.5i −136689. 0
51.11 0.770283 − 31.9907i 80.5620i −1022.81 − 49.2838i 0 2577.24 + 62.0555i − 345.112i −2364.48 + 32682.6i 52558.8 0
51.12 0.770283 + 31.9907i − 80.5620i −1022.81 + 49.2838i 0 2577.24 − 62.0555i 345.112i −2364.48 − 32682.6i 52558.8 0
51.13 13.0154 − 29.2335i − 448.707i −685.197 − 760.974i 0 −13117.3 − 5840.12i 19334.7i −31164.1 + 10126.3i −142289. 0
51.14 13.0154 + 29.2335i 448.707i −685.197 + 760.974i 0 −13117.3 + 5840.12i − 19334.7i −31164.1 − 10126.3i −142289. 0
51.15 16.5576 − 27.3833i − 17.1314i −475.690 − 906.805i 0 −469.114 − 283.655i − 2883.30i −32707.6 − 1988.60i 58755.5 0
51.16 16.5576 + 27.3833i 17.1314i −475.690 + 906.805i 0 −469.114 + 283.655i 2883.30i −32707.6 + 1988.60i 58755.5 0
51.17 29.9163 − 11.3585i − 137.165i 765.971 − 679.606i 0 −1557.98 − 4103.46i − 24910.8i 15195.7 − 29031.6i 40234.9 0
51.18 29.9163 + 11.3585i 137.165i 765.971 + 679.606i 0 −1557.98 + 4103.46i 24910.8i 15195.7 + 29031.6i 40234.9 0
51.19 31.5088 − 5.58516i − 275.626i 961.612 − 351.964i 0 −1539.42 − 8684.66i 19327.7i 28333.5 − 16460.7i −16920.9 0
51.20 31.5088 + 5.58516i 275.626i 961.612 + 351.964i 0 −1539.42 + 8684.66i − 19327.7i 28333.5 + 16460.7i −16920.9 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 51.20
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 100.11.b.e 20
4.b odd 2 1 inner 100.11.b.e 20
5.b even 2 1 20.11.b.a ✓ 20
5.c odd 4 2 100.11.d.c 40
15.d odd 2 1 180.11.c.a 20
20.d odd 2 1 20.11.b.a ✓ 20
20.e even 4 2 100.11.d.c 40
40.e odd 2 1 320.11.b.d 20
40.f even 2 1 320.11.b.d 20
60.h even 2 1 180.11.c.a 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
20.11.b.a ✓ 20 5.b even 2 1
20.11.b.a ✓ 20 20.d odd 2 1
100.11.b.e 20 1.a even 1 1 trivial
100.11.b.e 20 4.b odd 2 1 inner
100.11.d.c 40 5.c odd 4 2
100.11.d.c 40 20.e even 4 2
180.11.c.a 20 15.d odd 2 1
180.11.c.a 20 60.h even 2 1
320.11.b.d 20 40.e odd 2 1
320.11.b.d 20 40.f even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{11}^{\mathrm{new}}(100, [\chi])\):

\( T_{3}^{20} + 797924 T_{3}^{18} + 262615424816 T_{3}^{16} + \cdots + 22\!\cdots\!00 \) Copy content Toggle raw display
\( T_{13}^{10} - 139432 T_{13}^{9} - 838980325396 T_{13}^{8} + \cdots + 16\!\cdots\!00 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} + \cdots + 12\!\cdots\!76 \) Copy content Toggle raw display
$3$ \( T^{20} + \cdots + 22\!\cdots\!00 \) Copy content Toggle raw display
$5$ \( T^{20} \) Copy content Toggle raw display
$7$ \( T^{20} + \cdots + 31\!\cdots\!00 \) Copy content Toggle raw display
$11$ \( T^{20} + \cdots + 28\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( (T^{10} + \cdots + 16\!\cdots\!00)^{2} \) Copy content Toggle raw display
$17$ \( (T^{10} + \cdots + 82\!\cdots\!00)^{2} \) Copy content Toggle raw display
$19$ \( T^{20} + \cdots + 21\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{20} + \cdots + 79\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( (T^{10} + \cdots - 19\!\cdots\!56)^{2} \) Copy content Toggle raw display
$31$ \( T^{20} + \cdots + 44\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( (T^{10} + \cdots - 54\!\cdots\!00)^{2} \) Copy content Toggle raw display
$41$ \( (T^{10} + \cdots - 10\!\cdots\!16)^{2} \) Copy content Toggle raw display
$43$ \( T^{20} + \cdots + 64\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( T^{20} + \cdots + 37\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( (T^{10} + \cdots - 26\!\cdots\!00)^{2} \) Copy content Toggle raw display
$59$ \( T^{20} + \cdots + 46\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( (T^{10} + \cdots + 11\!\cdots\!44)^{2} \) Copy content Toggle raw display
$67$ \( T^{20} + \cdots + 47\!\cdots\!00 \) Copy content Toggle raw display
$71$ \( T^{20} + \cdots + 78\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( (T^{10} + \cdots - 36\!\cdots\!00)^{2} \) Copy content Toggle raw display
$79$ \( T^{20} + \cdots + 12\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{20} + \cdots + 18\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( (T^{10} + \cdots + 10\!\cdots\!24)^{2} \) Copy content Toggle raw display
$97$ \( (T^{10} + \cdots + 47\!\cdots\!00)^{2} \) Copy content Toggle raw display
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