Properties

Label 100.5.d.c
Level $100$
Weight $5$
Character orbit 100.d
Analytic conductor $10.337$
Analytic rank $0$
Dimension $16$
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] = [100,5,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, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("100.99");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 100 = 2^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 5 \)
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: \(10.3369963084\)
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: \( x^{16} - 5x^{14} + 21x^{12} + 35x^{10} - 199x^{8} + 560x^{6} + 5376x^{4} - 20480x^{2} + 65536 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{37}\cdot 5^{8} \)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{2} q^{2} + \beta_{8} q^{3} + ( - \beta_{4} + 2) q^{4} + (\beta_{7} + \beta_{5} + 6) q^{6} + (\beta_{14} + \beta_{2}) q^{7} + (\beta_{15} + 2 \beta_{11} - \beta_{10} - 2 \beta_{8} + 4 \beta_{2} - \beta_1) q^{8} + (2 \beta_{9} + 2 \beta_{5} + 2 \beta_{4} + \beta_{3} + 41) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{2} + \beta_{8} q^{3} + ( - \beta_{4} + 2) q^{4} + (\beta_{7} + \beta_{5} + 6) q^{6} + (\beta_{14} + \beta_{2}) q^{7} + (\beta_{15} + 2 \beta_{11} - \beta_{10} - 2 \beta_{8} + 4 \beta_{2} - \beta_1) q^{8} + (2 \beta_{9} + 2 \beta_{5} + 2 \beta_{4} + \beta_{3} + 41) q^{9} + (\beta_{13} - \beta_{9} + \beta_{5} - 4 \beta_{4} + \beta_{3} - 1) q^{11} + (4 \beta_{15} - 2 \beta_{14} + 2 \beta_{12} + \beta_{11} + \beta_{10} + 6 \beta_{2} + 3 \beta_1) q^{12} + (3 \beta_{11} + 4 \beta_{10} + 17 \beta_{2} + 5 \beta_1) q^{13} + (2 \beta_{13} - \beta_{9} + 3 \beta_{7} - 2 \beta_{6} - 3 \beta_{4} + 7 \beta_{3} + 20) q^{14} + ( - 2 \beta_{13} + \beta_{9} - 4 \beta_{7} + \beta_{5} - 5 \beta_{4} + 7 \beta_{3} + \cdots - 38) q^{16}+ \cdots + ( - 33 \beta_{13} - 55 \beta_{9} + 216 \beta_{7} - 40 \beta_{6} + 33 \beta_{5} + \cdots + 11) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 40 q^{4} + 96 q^{6} + 656 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 40 q^{4} + 96 q^{6} + 656 q^{9} + 336 q^{14} - 544 q^{16} + 32 q^{21} - 3104 q^{24} - 4344 q^{26} - 2400 q^{29} + 4264 q^{34} - 2088 q^{36} + 9792 q^{41} - 15840 q^{44} + 1456 q^{46} + 11536 q^{49} + 35552 q^{54} + 96 q^{56} + 15872 q^{61} - 37760 q^{64} - 16160 q^{66} + 4512 q^{69} + 36984 q^{74} + 24000 q^{76} - 1872 q^{81} - 100928 q^{84} - 14784 q^{86} - 47520 q^{89} + 86736 q^{94} + 5376 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 5x^{14} + 21x^{12} + 35x^{10} - 199x^{8} + 560x^{6} + 5376x^{4} - 20480x^{2} + 65536 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{15} - 5\nu^{13} + 21\nu^{11} + 35\nu^{9} - 199\nu^{7} + 560\nu^{5} + 5376\nu^{3} + 45056\nu ) / 8192 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{15} - 5\nu^{13} + 21\nu^{11} + 35\nu^{9} - 199\nu^{7} + 560\nu^{5} + 5376\nu^{3} - 20480\nu ) / 8192 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -23\nu^{14} - 13\nu^{12} - 99\nu^{10} - 2213\nu^{8} - 1183\nu^{6} + 15920\nu^{4} - 128000\nu^{2} - 26624 ) / 15360 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{14} - 5\nu^{12} + 21\nu^{10} + 35\nu^{8} - 199\nu^{6} + 560\nu^{4} + 5376\nu^{2} - 18432 ) / 1024 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{14} + 59\nu^{12} - 43\nu^{10} + 99\nu^{8} + 3321\nu^{6} + 880\nu^{4} - 42496\nu^{2} + 207872 ) / 3072 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 7\nu^{14} + 117\nu^{12} - 229\nu^{10} + 1517\nu^{8} + 16087\nu^{6} - 600\nu^{4} + 21760\nu^{2} + 810496 ) / 7680 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -13\nu^{14} + 45\nu^{12} - 45\nu^{10} - 491\nu^{8} + 479\nu^{6} + 9372\nu^{4} - 33856\nu^{2} + 81664 ) / 3840 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 47 \nu^{15} - 475 \nu^{13} + 395 \nu^{11} + 1469 \nu^{9} - 18521 \nu^{7} - 41888 \nu^{5} + 347904 \nu^{3} - 1458176 \nu ) / 122880 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 33 \nu^{14} + 357 \nu^{12} - 629 \nu^{10} - 2243 \nu^{8} + 18407 \nu^{6} + 11920 \nu^{4} - 175360 \nu^{2} + 1135616 ) / 15360 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 13 \nu^{15} + 113 \nu^{13} - 257 \nu^{11} - 727 \nu^{9} + 9643 \nu^{7} + 8512 \nu^{5} - 126720 \nu^{3} + 503808 \nu ) / 24576 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( -9\nu^{15} + 39\nu^{13} - 63\nu^{11} + 103\nu^{9} + 2573\nu^{7} + 6682\nu^{5} + 14144\nu^{3} + 219136\nu ) / 15360 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 45 \nu^{15} + 1041 \nu^{13} - 1697 \nu^{11} - 9271 \nu^{9} + 50059 \nu^{7} + 2624 \nu^{5} - 733952 \nu^{3} + 2543616 \nu ) / 122880 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 61\nu^{14} - 81\nu^{12} - 223\nu^{10} + 4663\nu^{8} + 1973\nu^{6} - 21808\nu^{4} + 251264\nu^{2} + 124416 ) / 7680 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 49 \nu^{15} + 275 \nu^{13} + 605 \nu^{11} + 3803 \nu^{9} + 35713 \nu^{7} - 55176 \nu^{5} + 185088 \nu^{3} + 1128448 \nu ) / 61440 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 107 \nu^{15} - 331 \nu^{13} + 267 \nu^{11} + 6685 \nu^{9} - 11785 \nu^{7} - 49092 \nu^{5} + 436096 \nu^{3} - 745472 \nu ) / 30720 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -\beta_{2} + \beta_1 ) / 8 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{9} - \beta_{5} + \beta_{4} - \beta_{3} + 10 ) / 16 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{12} + 2\beta_{11} - \beta_{10} + 2\beta_{8} + 2\beta_{2} ) / 8 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 2\beta_{13} + 4\beta_{7} + \beta_{5} + 8\beta_{4} + 7\beta_{3} - 29 ) / 16 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( \beta_{15} - 3\beta_{14} + \beta_{12} + 4\beta_{11} + 4\beta_{10} - 3\beta_{8} + 42\beta_{2} - 2\beta_1 ) / 8 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -4\beta_{13} - \beta_{9} - 8\beta_{7} + 12\beta_{6} - 4\beta_{5} + 5\beta_{4} + 8\beta_{3} - 583 ) / 16 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -2\beta_{15} + 6\beta_{14} - 3\beta_{12} + 5\beta_{11} + 26\beta_{10} + 34\beta_{8} + 64\beta_{2} - 25\beta_1 ) / 8 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 3\beta_{13} - 51\beta_{9} + 36\beta_{7} + 12\beta_{6} + 18\beta_{5} - 52\beta_{4} - 15\beta_{3} - 490 ) / 8 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 21 \beta_{15} + 9 \beta_{14} - 99 \beta_{12} - 21 \beta_{11} + 21 \beta_{10} - 201 \beta_{8} - 215 \beta_{2} - 120 \beta_1 ) / 8 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( -90\beta_{13} - 57\beta_{9} - 30\beta_{7} + 30\beta_{6} + 39\beta_{5} + 135\beta_{4} - 213\beta_{3} + 2566 ) / 8 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 144 \beta_{15} + 216 \beta_{14} - 204 \beta_{12} - 336 \beta_{11} - 204 \beta_{10} - 384 \beta_{8} + 397 \beta_{2} + 395 \beta_1 ) / 8 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 24 \beta_{13} + 791 \beta_{9} + 288 \beta_{7} - 624 \beta_{6} + 277 \beta_{5} + 551 \beta_{4} - 1619 \beta_{3} - 10774 ) / 16 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 132 \beta_{15} + 84 \beta_{14} + 803 \beta_{12} + 310 \beta_{11} - 2039 \beta_{10} - 2654 \beta_{8} + 490 \beta_{2} - 1428 \beta_1 ) / 8 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 1774 \beta_{13} + 4344 \beta_{9} - 3652 \beta_{7} - 2832 \beta_{6} + 2507 \beta_{5} + 8248 \beta_{4} + 4949 \beta_{3} + 14033 ) / 16 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 1991 \beta_{15} - 1557 \beta_{14} + 5231 \beta_{12} - 2656 \beta_{11} + 1664 \beta_{10} - 477 \beta_{8} + 25158 \beta_{2} + 5390 \beta_1 ) / 8 \) 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
1.89449 0.641015i
1.89449 + 0.641015i
1.71641 1.02661i
1.71641 + 1.02661i
1.58665 1.21760i
1.58665 + 1.21760i
0.444269 1.95003i
0.444269 + 1.95003i
−0.444269 1.95003i
−0.444269 + 1.95003i
−1.58665 1.21760i
−1.58665 + 1.21760i
−1.71641 1.02661i
−1.71641 + 1.02661i
−1.89449 0.641015i
−1.89449 + 0.641015i
−3.78898 1.28203i 8.14153 12.7128 + 9.71518i 0 −30.8481 10.4377i −63.6032 −35.7134 53.1089i −14.7155 0
99.2 −3.78898 + 1.28203i 8.14153 12.7128 9.71518i 0 −30.8481 + 10.4377i −63.6032 −35.7134 + 53.1089i −14.7155 0
99.3 −3.43282 2.05323i −15.5779 7.56853 + 14.0967i 0 53.4761 + 31.9849i 37.6230 2.96235 63.9314i 161.671 0
99.4 −3.43282 + 2.05323i −15.5779 7.56853 14.0967i 0 53.4761 31.9849i 37.6230 2.96235 + 63.9314i 161.671 0
99.5 −3.17330 2.43519i 3.20523 4.13968 + 15.4552i 0 −10.1712 7.80536i 30.6227 24.4999 59.1249i −70.7265 0
99.6 −3.17330 + 2.43519i 3.20523 4.13968 15.4552i 0 −10.1712 + 7.80536i 30.6227 24.4999 + 59.1249i −70.7265 0
99.7 −0.888538 3.90006i −12.9912 −14.4210 + 6.93071i 0 11.5432 + 50.6665i −78.0345 39.8438 + 50.0846i 87.7712 0
99.8 −0.888538 + 3.90006i −12.9912 −14.4210 6.93071i 0 11.5432 50.6665i −78.0345 39.8438 50.0846i 87.7712 0
99.9 0.888538 3.90006i 12.9912 −14.4210 6.93071i 0 11.5432 50.6665i 78.0345 −39.8438 + 50.0846i 87.7712 0
99.10 0.888538 + 3.90006i 12.9912 −14.4210 + 6.93071i 0 11.5432 + 50.6665i 78.0345 −39.8438 50.0846i 87.7712 0
99.11 3.17330 2.43519i −3.20523 4.13968 15.4552i 0 −10.1712 + 7.80536i −30.6227 −24.4999 59.1249i −70.7265 0
99.12 3.17330 + 2.43519i −3.20523 4.13968 + 15.4552i 0 −10.1712 7.80536i −30.6227 −24.4999 + 59.1249i −70.7265 0
99.13 3.43282 2.05323i 15.5779 7.56853 14.0967i 0 53.4761 31.9849i −37.6230 −2.96235 63.9314i 161.671 0
99.14 3.43282 + 2.05323i 15.5779 7.56853 + 14.0967i 0 53.4761 + 31.9849i −37.6230 −2.96235 + 63.9314i 161.671 0
99.15 3.78898 1.28203i −8.14153 12.7128 9.71518i 0 −30.8481 + 10.4377i 63.6032 35.7134 53.1089i −14.7155 0
99.16 3.78898 + 1.28203i −8.14153 12.7128 + 9.71518i 0 −30.8481 10.4377i 63.6032 35.7134 + 53.1089i −14.7155 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 99.16
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 100.5.d.c 16
4.b odd 2 1 inner 100.5.d.c 16
5.b even 2 1 inner 100.5.d.c 16
5.c odd 4 1 20.5.b.a 8
5.c odd 4 1 100.5.b.c 8
15.e even 4 1 180.5.c.a 8
20.d odd 2 1 inner 100.5.d.c 16
20.e even 4 1 20.5.b.a 8
20.e even 4 1 100.5.b.c 8
40.i odd 4 1 320.5.b.d 8
40.k even 4 1 320.5.b.d 8
60.l odd 4 1 180.5.c.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
20.5.b.a 8 5.c odd 4 1
20.5.b.a 8 20.e even 4 1
100.5.b.c 8 5.c odd 4 1
100.5.b.c 8 20.e even 4 1
100.5.d.c 16 1.a even 1 1 trivial
100.5.d.c 16 4.b odd 2 1 inner
100.5.d.c 16 5.b even 2 1 inner
100.5.d.c 16 20.d odd 2 1 inner
180.5.c.a 8 15.e even 4 1
180.5.c.a 8 60.l odd 4 1
320.5.b.d 8 40.i odd 4 1
320.5.b.d 8 40.k even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{8} - 488T_{3}^{6} + 73136T_{3}^{4} - 3415680T_{3}^{2} + 27889920 \) acting on \(S_{5}^{\mathrm{new}}(100, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} - 20 T^{14} + \cdots + 4294967296 \) Copy content Toggle raw display
$3$ \( (T^{8} - 488 T^{6} + 73136 T^{4} + \cdots + 27889920)^{2} \) Copy content Toggle raw display
$5$ \( T^{16} \) Copy content Toggle raw display
$7$ \( (T^{8} - 12488 T^{6} + \cdots + 32698357408000)^{2} \) Copy content Toggle raw display
$11$ \( (T^{8} + 39200 T^{6} + \cdots + 204994355200000)^{2} \) Copy content Toggle raw display
$13$ \( (T^{8} + 104208 T^{6} + \cdots + 64249843360000)^{2} \) Copy content Toggle raw display
$17$ \( (T^{8} + 353168 T^{6} + \cdots + 47\!\cdots\!00)^{2} \) Copy content Toggle raw display
$19$ \( (T^{8} + 311680 T^{6} + \cdots + 11\!\cdots\!00)^{2} \) Copy content Toggle raw display
$23$ \( (T^{8} - 1352968 T^{6} + \cdots + 15\!\cdots\!20)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} + 600 T^{3} + \cdots - 98595968624)^{4} \) Copy content Toggle raw display
$31$ \( (T^{8} + 5480480 T^{6} + \cdots + 18\!\cdots\!00)^{2} \) Copy content Toggle raw display
$37$ \( (T^{8} + 12321168 T^{6} + \cdots + 39\!\cdots\!00)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} - 2448 T^{3} + \cdots + 1586334915856)^{4} \) Copy content Toggle raw display
$43$ \( (T^{8} - 8441128 T^{6} + \cdots + 13\!\cdots\!00)^{2} \) Copy content Toggle raw display
$47$ \( (T^{8} - 19967368 T^{6} + \cdots + 43\!\cdots\!20)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} + 31106448 T^{6} + \cdots + 23\!\cdots\!00)^{2} \) Copy content Toggle raw display
$59$ \( (T^{8} + 39075840 T^{6} + \cdots + 35\!\cdots\!00)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} - 3968 T^{3} + \cdots + 17262940540816)^{4} \) Copy content Toggle raw display
$67$ \( (T^{8} - 87547688 T^{6} + \cdots + 10\!\cdots\!20)^{2} \) Copy content Toggle raw display
$71$ \( (T^{8} + 109932320 T^{6} + \cdots + 16\!\cdots\!00)^{2} \) Copy content Toggle raw display
$73$ \( (T^{8} + 47333648 T^{6} + \cdots + 15\!\cdots\!00)^{2} \) Copy content Toggle raw display
$79$ \( (T^{8} + 57226880 T^{6} + \cdots + 26\!\cdots\!00)^{2} \) Copy content Toggle raw display
$83$ \( (T^{8} - 63696808 T^{6} + \cdots + 13\!\cdots\!20)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} + 11880 T^{3} + \cdots + 26555598339856)^{4} \) Copy content Toggle raw display
$97$ \( (T^{8} + 432840528 T^{6} + \cdots + 15\!\cdots\!00)^{2} \) Copy content Toggle raw display
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