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Properties

Label 100.9.b.f
Level $100$
Weight $9$
Character orbit 100.b
Analytic conductor $40.738$
Analytic rank $0$
Dimension $16$
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [100,9,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, 9, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("100.51");
 
S:= CuspForms(chi, 9);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 100 = 2^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 9 \)
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: \(40.7378610061\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 8 x^{15} + 17288 x^{14} - 120876 x^{13} + 118671360 x^{12} - 710457136 x^{11} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{56}\cdot 5^{8} \)
Twist minimal: yes
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{2} q^{2} - \beta_1 q^{3} + (\beta_{4} - 21) q^{4} + (\beta_{3} + \beta_1 - 30) q^{6} + ( - \beta_{8} + \beta_{4} - 5 \beta_{2} + 1) q^{7} + (\beta_{5} - 21 \beta_{2} - 5 \beta_1 - 43) q^{8} + (\beta_{6} + \beta_{4} + 32 \beta_{2} - 2074) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{2} - \beta_1 q^{3} + (\beta_{4} - 21) q^{4} + (\beta_{3} + \beta_1 - 30) q^{6} + ( - \beta_{8} + \beta_{4} - 5 \beta_{2} + 1) q^{7} + (\beta_{5} - 21 \beta_{2} - 5 \beta_1 - 43) q^{8} + (\beta_{6} + \beta_{4} + 32 \beta_{2} - 2074) q^{9} + (\beta_{15} - \beta_{7} + \beta_{4} + \cdots + 31) q^{11}+ \cdots + ( - 2777 \beta_{15} - 1070 \beta_{14} + \cdots - 1003605) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 3 q^{2} - 331 q^{4} - 483 q^{6} - 747 q^{8} - 33088 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 3 q^{2} - 331 q^{4} - 483 q^{6} - 747 q^{8} - 33088 q^{9} - 19105 q^{12} + 25696 q^{13} + 20358 q^{14} + 127361 q^{16} + 13776 q^{17} + 126278 q^{18} + 49024 q^{21} + 670415 q^{22} - 734177 q^{24} - 1118700 q^{26} - 1049270 q^{28} - 433632 q^{29} - 3731427 q^{32} - 675920 q^{33} + 2226965 q^{34} + 1871778 q^{36} - 1928704 q^{37} - 445905 q^{38} + 1740432 q^{41} - 1617750 q^{42} - 473955 q^{44} + 5513602 q^{46} + 3724275 q^{48} - 5501488 q^{49} + 11236516 q^{52} + 9264096 q^{53} - 16426009 q^{54} - 5411718 q^{56} - 20489520 q^{57} - 18275784 q^{58} + 32528192 q^{61} - 55584390 q^{62} + 5696249 q^{64} + 12325505 q^{66} + 12173691 q^{68} - 46720224 q^{69} - 2487342 q^{72} - 110501904 q^{73} + 15172110 q^{74} + 52834005 q^{76} - 20554560 q^{77} - 51381220 q^{78} - 67764240 q^{81} + 53011531 q^{82} - 134015234 q^{84} - 96677688 q^{86} + 76847405 q^{88} + 80609808 q^{89} - 128870010 q^{92} + 257159200 q^{93} + 205910268 q^{94} + 239873127 q^{96} - 82969824 q^{97} - 110647377 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 8 x^{15} + 17288 x^{14} - 120876 x^{13} + 118671360 x^{12} - 710457136 x^{11} + \cdots + 11\!\cdots\!00 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 2\nu - 1 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 16\!\cdots\!75 \nu^{15} + \cdots - 13\!\cdots\!00 ) / 11\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 88\!\cdots\!73 \nu^{15} + \cdots + 25\!\cdots\!00 ) / 76\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 21\!\cdots\!61 \nu^{15} + \cdots + 41\!\cdots\!00 ) / 17\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 34\!\cdots\!75 \nu^{15} + \cdots + 14\!\cdots\!00 ) / 11\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 60\!\cdots\!39 \nu^{15} + \cdots + 17\!\cdots\!00 ) / 17\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 99\!\cdots\!25 \nu^{15} + \cdots + 76\!\cdots\!00 ) / 58\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 88\!\cdots\!47 \nu^{15} + \cdots + 15\!\cdots\!00 ) / 35\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 93\!\cdots\!91 \nu^{15} + \cdots - 20\!\cdots\!00 ) / 35\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 16\!\cdots\!23 \nu^{15} + \cdots + 76\!\cdots\!00 ) / 35\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 17\!\cdots\!69 \nu^{15} + \cdots + 11\!\cdots\!00 ) / 35\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 20\!\cdots\!82 \nu^{15} + \cdots + 13\!\cdots\!00 ) / 29\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 69\!\cdots\!37 \nu^{15} + \cdots + 49\!\cdots\!00 ) / 87\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 44\!\cdots\!27 \nu^{15} + \cdots + 48\!\cdots\!00 ) / 35\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 31\!\cdots\!70 \nu^{15} + \cdots - 28\!\cdots\!00 ) / 17\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{6} + \beta_{4} + 32\beta_{2} + 2\beta _1 - 8634 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 5 \beta_{15} - 3 \beta_{14} + 6 \beta_{13} - 2 \beta_{12} + 2 \beta_{11} - 7 \beta_{10} + 2 \beta_{9} + \cdots - 25305 ) / 8 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 20 \beta_{15} - 360 \beta_{14} + 378 \beta_{13} - 568 \beta_{12} - 346 \beta_{11} - 19 \beta_{10} + \cdots + 122702328 ) / 16 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 181560 \beta_{15} + 81125 \beta_{14} - 180000 \beta_{13} + 39870 \beta_{12} - 62360 \beta_{11} + \cdots + 583734514 ) / 32 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 544780 \beta_{15} + 4300947 \beta_{14} - 5020536 \beta_{13} + 10686834 \beta_{12} + \cdots - 968545182070 ) / 32 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 1255371760 \beta_{15} - 402895891 \beta_{14} + 1042691628 \beta_{13} - 131710818 \beta_{12} + \cdots - 3175757506414 ) / 32 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 2512014720 \beta_{15} - 10973259685 \beta_{14} + 13647409704 \beta_{13} - 35755427662 \beta_{12} + \cdots + 20\!\cdots\!06 ) / 16 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 3787990680550 \beta_{15} + 917699720513 \beta_{14} - 2811832423020 \beta_{13} + \cdots + 84\!\cdots\!16 ) / 16 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 18958795419980 \beta_{15} + 55378887663955 \beta_{14} - 69881711647614 \beta_{13} + \cdots - 87\!\cdots\!98 ) / 16 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 42\!\cdots\!40 \beta_{15} + \cdots - 88\!\cdots\!82 ) / 32 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 25\!\cdots\!80 \beta_{15} + \cdots + 77\!\cdots\!06 ) / 32 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 22\!\cdots\!00 \beta_{15} + \cdots + 46\!\cdots\!06 ) / 32 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 19\!\cdots\!50 \beta_{15} + \cdots - 43\!\cdots\!12 ) / 4 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 29\!\cdots\!45 \beta_{15} + \cdots - 61\!\cdots\!61 ) / 8 \) 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
0.500000 + 59.1340i
0.500000 − 59.1340i
0.500000 − 8.86333i
0.500000 + 8.86333i
0.500000 + 53.5585i
0.500000 − 53.5585i
0.500000 − 70.1071i
0.500000 + 70.1071i
0.500000 + 9.75682i
0.500000 − 9.75682i
0.500000 − 24.1578i
0.500000 + 24.1578i
0.500000 + 61.3929i
0.500000 − 61.3929i
0.500000 − 38.0948i
0.500000 + 38.0948i
−15.9575 − 1.16525i − 118.268i 253.284 + 37.1889i 0 −137.812 + 1887.26i − 1474.52i −3998.45 − 888.582i −7426.34 0
51.2 −15.9575 + 1.16525i 118.268i 253.284 − 37.1889i 0 −137.812 − 1887.26i 1474.52i −3998.45 + 888.582i −7426.34 0
51.3 −12.8126 − 9.58313i 17.7267i 72.3273 + 245.570i 0 169.877 − 227.125i − 536.344i 1426.63 − 3839.52i 6246.77 0
51.4 −12.8126 + 9.58313i − 17.7267i 72.3273 − 245.570i 0 169.877 + 227.125i 536.344i 1426.63 + 3839.52i 6246.77 0
51.5 −5.23963 − 15.1177i − 107.117i −201.093 + 158.423i 0 −1619.37 + 561.253i 4212.76i 3448.65 + 2209.99i −4913.04 0
51.6 −5.23963 + 15.1177i 107.117i −201.093 − 158.423i 0 −1619.37 − 561.253i − 4212.76i 3448.65 − 2209.99i −4913.04 0
51.7 −2.90464 − 15.7341i 140.214i −239.126 + 91.4039i 0 2206.15 − 407.271i 132.843i 2132.74 + 3496.95i −13099.0 0
51.8 −2.90464 + 15.7341i − 140.214i −239.126 − 91.4039i 0 2206.15 + 407.271i − 132.843i 2132.74 − 3496.95i −13099.0 0
51.9 0.948150 − 15.9719i − 19.5136i −254.202 − 30.2875i 0 −311.670 − 18.5019i − 3691.10i −724.770 + 4031.37i 6180.22 0
51.10 0.948150 + 15.9719i 19.5136i −254.202 + 30.2875i 0 −311.670 + 18.5019i 3691.10i −724.770 − 4031.37i 6180.22 0
51.11 10.0150 − 12.4780i 48.3156i −55.3985 − 249.934i 0 602.880 + 483.882i 2732.84i −3673.48 − 1811.84i 4226.60 0
51.12 10.0150 + 12.4780i − 48.3156i −55.3985 + 249.934i 0 602.880 − 483.882i − 2732.84i −3673.48 + 1811.84i 4226.60 0
51.13 11.6558 − 10.9610i − 122.786i 15.7140 − 255.517i 0 −1345.85 − 1431.16i − 1671.99i −2617.56 − 3150.49i −8515.34 0
51.14 11.6558 + 10.9610i 122.786i 15.7140 + 255.517i 0 −1345.85 + 1431.16i 1671.99i −2617.56 + 3150.49i −8515.34 0
51.15 15.7955 − 2.55014i 76.1896i 242.994 − 80.5612i 0 194.294 + 1203.45i − 2180.40i 3632.76 − 1892.17i 756.150 0
51.16 15.7955 + 2.55014i − 76.1896i 242.994 + 80.5612i 0 194.294 − 1203.45i 2180.40i 3632.76 + 1892.17i 756.150 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 51.16
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 100.9.b.f yes 16
4.b odd 2 1 inner 100.9.b.f yes 16
5.b even 2 1 100.9.b.e ✓ 16
5.c odd 4 2 100.9.d.d 32
20.d odd 2 1 100.9.b.e ✓ 16
20.e even 4 2 100.9.d.d 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
100.9.b.e ✓ 16 5.b even 2 1
100.9.b.e ✓ 16 20.d odd 2 1
100.9.b.f yes 16 1.a even 1 1 trivial
100.9.b.f yes 16 4.b odd 2 1 inner
100.9.d.d 32 5.c odd 4 2
100.9.d.d 32 20.e even 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{9}^{\mathrm{new}}(100, [\chi])\):

\( T_{3}^{16} + 69032 T_{3}^{14} + 1892458028 T_{3}^{12} + 26145166260984 T_{3}^{10} + \cdots + 77\!\cdots\!25 \) Copy content Toggle raw display
\( T_{13}^{8} - 12848 T_{13}^{7} - 3306571168 T_{13}^{6} + 13816363581184 T_{13}^{5} + \cdots + 17\!\cdots\!00 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} + \cdots + 18\!\cdots\!16 \) Copy content Toggle raw display
$3$ \( T^{16} + \cdots + 77\!\cdots\!25 \) Copy content Toggle raw display
$5$ \( T^{16} \) Copy content Toggle raw display
$7$ \( T^{16} + \cdots + 26\!\cdots\!00 \) Copy content Toggle raw display
$11$ \( T^{16} + \cdots + 18\!\cdots\!25 \) Copy content Toggle raw display
$13$ \( (T^{8} + \cdots + 17\!\cdots\!00)^{2} \) Copy content Toggle raw display
$17$ \( (T^{8} + \cdots - 17\!\cdots\!75)^{2} \) Copy content Toggle raw display
$19$ \( T^{16} + \cdots + 88\!\cdots\!25 \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 21\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( (T^{8} + \cdots + 11\!\cdots\!24)^{2} \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots + 27\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( (T^{8} + \cdots + 12\!\cdots\!00)^{2} \) Copy content Toggle raw display
$41$ \( (T^{8} + \cdots + 45\!\cdots\!41)^{2} \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 63\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 67\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( (T^{8} + \cdots - 26\!\cdots\!00)^{2} \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 34\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( (T^{8} + \cdots - 28\!\cdots\!44)^{2} \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 61\!\cdots\!25 \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 13\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( (T^{8} + \cdots + 97\!\cdots\!25)^{2} \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 83\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 29\!\cdots\!25 \) Copy content Toggle raw display
$89$ \( (T^{8} + \cdots - 15\!\cdots\!31)^{2} \) Copy content Toggle raw display
$97$ \( (T^{8} + \cdots + 37\!\cdots\!00)^{2} \) Copy content Toggle raw display
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