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Properties

Label 400.3.bg.a
Level $400$
Weight $3$
Character orbit 400.bg
Analytic conductor $10.899$
Analytic rank $0$
Dimension $16$
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] = [400,3,Mod(17,400)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(400, base_ring=CyclotomicField(20))
 
chi = DirichletCharacter(H, H._module([0, 0, 13]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("400.17");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 400 = 2^{4} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 400.bg (of order \(20\), degree \(8\), 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: \(10.8992105744\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(2\) over \(\Q(\zeta_{20})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 30x^{14} + 351x^{12} + 2130x^{10} + 7341x^{8} + 14480x^{6} + 15196x^{4} + 6560x^{2} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 5^{2} \)
Twist minimal: no (minimal twist has level 50)
Sato-Tate group: $\mathrm{SU}(2)[C_{20}]$

$q$-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.

\(f(q)\) \(=\) \( q + (\beta_{15} - \beta_{13} - \beta_{10} + \cdots - 1) q^{3}+ \cdots + ( - \beta_{15} - \beta_{14} - 2 \beta_{9} + \cdots - 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{15} - \beta_{13} - \beta_{10} + \cdots - 1) q^{3}+ \cdots + ( - 9 \beta_{15} + 9 \beta_{14} + \cdots + 8) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 2 q^{3} + 2 q^{7} - 40 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 2 q^{3} + 2 q^{7} - 40 q^{9} - 32 q^{11} - 8 q^{13} - 62 q^{17} - 30 q^{19} - 68 q^{21} + 18 q^{23} + 70 q^{25} + 40 q^{27} + 100 q^{29} - 132 q^{31} - 36 q^{33} - 150 q^{35} + 138 q^{37} + 320 q^{39} - 88 q^{41} + 78 q^{43} - 20 q^{45} + 22 q^{47} + 168 q^{51} + 182 q^{53} - 280 q^{55} + 280 q^{57} + 350 q^{59} + 372 q^{61} - 22 q^{63} - 910 q^{65} + 112 q^{67} - 30 q^{69} - 122 q^{71} - 248 q^{73} + 80 q^{75} + 16 q^{77} - 760 q^{79} - 144 q^{81} - 132 q^{83} - 30 q^{85} - 770 q^{87} + 550 q^{89} + 798 q^{91} + 54 q^{93} - 40 q^{95} - 292 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 30x^{14} + 351x^{12} + 2130x^{10} + 7341x^{8} + 14480x^{6} + 15196x^{4} + 6560x^{2} + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 61 \nu^{15} + 364 \nu^{14} - 2016 \nu^{13} + 10504 \nu^{12} - 25551 \nu^{11} + 114532 \nu^{10} + \cdots + 6528 ) / 5728 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 61 \nu^{15} + 364 \nu^{14} + 2016 \nu^{13} + 10504 \nu^{12} + 25551 \nu^{11} + 114532 \nu^{10} + \cdots + 6528 ) / 5728 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 297 \nu^{15} + 550 \nu^{14} + 8366 \nu^{13} + 14644 \nu^{12} + 88651 \nu^{11} + 143378 \nu^{10} + \cdots - 3040 ) / 5728 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 297 \nu^{15} + 550 \nu^{14} - 8366 \nu^{13} + 14644 \nu^{12} - 88651 \nu^{11} + 143378 \nu^{10} + \cdots - 3040 ) / 5728 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 219 \nu^{15} - 1228 \nu^{14} - 5348 \nu^{13} - 33084 \nu^{12} - 45697 \nu^{11} - 329596 \nu^{10} + \cdots - 21488 ) / 5728 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 219 \nu^{15} - 1228 \nu^{14} + 5348 \nu^{13} - 33084 \nu^{12} + 45697 \nu^{11} - 329596 \nu^{10} + \cdots - 21488 ) / 5728 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 347 \nu^{15} - 1584 \nu^{14} + 9502 \nu^{13} - 42948 \nu^{12} + 97129 \nu^{11} - 432232 \nu^{10} + \cdots - 10720 ) / 5728 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 347 \nu^{15} + 1584 \nu^{14} + 9502 \nu^{13} + 42948 \nu^{12} + 97129 \nu^{11} + 432232 \nu^{10} + \cdots + 10720 ) / 5728 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 1259 \nu^{15} - 1584 \nu^{14} - 34132 \nu^{13} - 42948 \nu^{12} - 343473 \nu^{11} - 432232 \nu^{10} + \cdots - 13584 ) / 5728 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 583 \nu^{15} - 1480 \nu^{14} - 15494 \nu^{13} - 39998 \nu^{12} - 151637 \nu^{11} - 400736 \nu^{10} + \cdots + 760 ) / 2864 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 165 \nu^{15} - 1998 \nu^{14} - 4429 \nu^{13} - 54946 \nu^{12} - 44159 \nu^{11} - 563834 \nu^{10} + \cdots - 11504 ) / 2864 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 622 \nu^{15} - 1819 \nu^{14} + 17003 \nu^{13} - 49218 \nu^{12} + 173114 \nu^{11} - 493845 \nu^{10} + \cdots - 8640 ) / 2864 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 33 \nu^{15} - 4546 \nu^{14} + 492 \nu^{13} - 124536 \nu^{12} - 333 \nu^{11} - 1271046 \nu^{10} + \cdots - 19968 ) / 5728 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 1820 \nu^{15} + 1365 \nu^{14} - 49298 \nu^{13} + 36884 \nu^{12} - 495332 \nu^{11} + 369351 \nu^{10} + \cdots + 7296 ) / 2864 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 1551 \nu^{15} - 5454 \nu^{14} + 42456 \nu^{13} - 149204 \nu^{12} + 432565 \nu^{11} - 1520034 \nu^{10} + \cdots - 25520 ) / 5728 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( - \beta_{15} + \beta_{14} - 3 \beta_{13} + 2 \beta_{12} + 4 \beta_{11} - \beta_{10} - 3 \beta_{9} + \cdots - 3 ) / 5 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - 2 \beta_{15} - 2 \beta_{14} + 4 \beta_{13} - 4 \beta_{12} + 2 \beta_{11} - 2 \beta_{10} + 2 \beta_{9} + \cdots - 17 ) / 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 2 \beta_{15} - 2 \beta_{14} + 21 \beta_{13} - 14 \beta_{12} - 23 \beta_{11} + 12 \beta_{10} + 6 \beta_{9} + \cdots + 16 ) / 5 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 8 \beta_{15} + 8 \beta_{14} - 11 \beta_{13} + 13 \beta_{12} - 3 \beta_{11} + 5 \beta_{10} - 8 \beta_{9} + \cdots + 21 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 9 \beta_{15} - 9 \beta_{14} - 208 \beta_{13} + 137 \beta_{12} + 199 \beta_{11} - 146 \beta_{10} + \cdots - 133 ) / 5 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 542 \beta_{15} - 542 \beta_{14} + 669 \beta_{13} - 819 \beta_{12} + 127 \beta_{11} - 277 \beta_{10} + \cdots - 912 ) / 5 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 233 \beta_{15} + 233 \beta_{14} + 2321 \beta_{13} - 1509 \beta_{12} - 2088 \beta_{11} + 1742 \beta_{10} + \cdots + 1351 ) / 5 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 1320 \beta_{15} + 1320 \beta_{14} - 1568 \beta_{13} + 1948 \beta_{12} - 248 \beta_{11} + 628 \beta_{10} + \cdots + 1901 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 3264 \beta_{15} - 3264 \beta_{14} - 26693 \beta_{13} + 17232 \beta_{12} + 23429 \beta_{11} + \cdots - 14848 ) / 5 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 77742 \beta_{15} - 77742 \beta_{14} + 90989 \beta_{13} - 113869 \beta_{12} + 13247 \beta_{11} + \cdots - 106197 ) / 5 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 40423 \beta_{15} + 40423 \beta_{14} + 309006 \beta_{13} - 198849 \beta_{12} - 268583 \beta_{11} + \cdots + 168331 ) / 5 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 181168 \beta_{15} + 181168 \beta_{14} - 210773 \beta_{13} + 264731 \beta_{12} - 29605 \beta_{11} + \cdots + 242974 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 480759 \beta_{15} - 480759 \beta_{14} - 3581023 \beta_{13} + 2301347 \beta_{12} + 3100264 \beta_{11} + \cdots - 1932503 ) / 5 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 10515952 \beta_{15} - 10515952 \beta_{14} + 12204034 \beta_{13} - 15354984 \beta_{12} + \cdots - 14009687 ) / 5 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 5630388 \beta_{15} + 5630388 \beta_{14} + 41499681 \beta_{13} - 26655034 \beta_{12} - 35869293 \beta_{11} + \cdots + 22301706 ) / 5 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/400\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(177\) \(351\)
\(\chi(n)\) \(1\) \(\beta_{3}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
17.1
− 2.26402i
1.64599i
1.84816i
− 1.23012i
− 1.84816i
1.23012i
1.78563i
− 3.40366i
− 1.78563i
3.40366i
− 0.0495271i
− 1.56851i
0.0495271i
1.56851i
2.26402i
− 1.64599i
0 −0.569657 + 3.59668i 0 2.10649 − 4.53461i 0 −0.635779 − 0.635779i 0 −4.05206 − 1.31659i 0
17.2 0 0.711697 − 4.49348i 0 −4.53886 − 2.09733i 0 −3.58690 − 3.58690i 0 −11.1253 − 3.61484i 0
33.1 0 −2.50268 − 0.396386i 0 −3.83232 − 3.21144i 0 1.84619 − 1.84619i 0 −2.45322 − 0.797099i 0
33.2 0 0.742607 + 0.117617i 0 4.02861 + 2.96147i 0 −2.71368 + 2.71368i 0 −8.02188 − 2.60647i 0
97.1 0 −2.50268 + 0.396386i 0 −3.83232 + 3.21144i 0 1.84619 + 1.84619i 0 −2.45322 + 0.797099i 0
97.2 0 0.742607 − 0.117617i 0 4.02861 − 2.96147i 0 −2.71368 − 2.71368i 0 −8.02188 + 2.60647i 0
113.1 0 −0.252608 + 0.495771i 0 3.68015 + 3.38474i 0 −7.20385 + 7.20385i 0 5.10809 + 7.03068i 0
113.2 0 1.14941 − 2.25584i 0 −4.68874 + 1.73657i 0 6.58346 − 6.58346i 0 1.52238 + 2.09537i 0
177.1 0 −0.252608 − 0.495771i 0 3.68015 − 3.38474i 0 −7.20385 − 7.20385i 0 5.10809 − 7.03068i 0
177.2 0 1.14941 + 2.25584i 0 −4.68874 − 1.73657i 0 6.58346 + 6.58346i 0 1.52238 − 2.09537i 0
273.1 0 −2.68205 + 1.36657i 0 4.84361 − 1.24072i 0 3.67488 − 3.67488i 0 0.0358006 − 0.0492753i 0
273.2 0 2.40328 − 1.22453i 0 −1.59895 + 4.73744i 0 3.03568 − 3.03568i 0 −1.01379 + 1.39536i 0
337.1 0 −2.68205 − 1.36657i 0 4.84361 + 1.24072i 0 3.67488 + 3.67488i 0 0.0358006 + 0.0492753i 0
337.2 0 2.40328 + 1.22453i 0 −1.59895 − 4.73744i 0 3.03568 + 3.03568i 0 −1.01379 − 1.39536i 0
353.1 0 −0.569657 − 3.59668i 0 2.10649 + 4.53461i 0 −0.635779 + 0.635779i 0 −4.05206 + 1.31659i 0
353.2 0 0.711697 + 4.49348i 0 −4.53886 + 2.09733i 0 −3.58690 + 3.58690i 0 −11.1253 + 3.61484i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 17.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
25.f odd 20 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 400.3.bg.a 16
4.b odd 2 1 50.3.f.a ✓ 16
20.d odd 2 1 250.3.f.b 16
20.e even 4 1 250.3.f.a 16
20.e even 4 1 250.3.f.c 16
25.f odd 20 1 inner 400.3.bg.a 16
100.h odd 10 1 250.3.f.c 16
100.j odd 10 1 250.3.f.a 16
100.l even 20 1 50.3.f.a ✓ 16
100.l even 20 1 250.3.f.b 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
50.3.f.a ✓ 16 4.b odd 2 1
50.3.f.a ✓ 16 100.l even 20 1
250.3.f.a 16 20.e even 4 1
250.3.f.a 16 100.j odd 10 1
250.3.f.b 16 20.d odd 2 1
250.3.f.b 16 100.l even 20 1
250.3.f.c 16 20.e even 4 1
250.3.f.c 16 100.h odd 10 1
400.3.bg.a 16 1.a even 1 1 trivial
400.3.bg.a 16 25.f odd 20 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{16} + 2 T_{3}^{15} + 22 T_{3}^{14} + 76 T_{3}^{13} + 76 T_{3}^{12} + 322 T_{3}^{11} + \cdots + 130321 \) acting on \(S_{3}^{\mathrm{new}}(400, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( T^{16} + 2 T^{15} + \cdots + 130321 \) Copy content Toggle raw display
$5$ \( T^{16} + \cdots + 152587890625 \) Copy content Toggle raw display
$7$ \( T^{16} + \cdots + 9353984656 \) Copy content Toggle raw display
$11$ \( T^{16} + \cdots + 21080623380496 \) Copy content Toggle raw display
$13$ \( T^{16} + \cdots + 79\!\cdots\!21 \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 12\!\cdots\!16 \) Copy content Toggle raw display
$19$ \( T^{16} + \cdots + 15\!\cdots\!25 \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 58\!\cdots\!81 \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots + 52\!\cdots\!25 \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots + 26\!\cdots\!21 \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 11\!\cdots\!81 \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 10\!\cdots\!36 \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 47\!\cdots\!56 \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 93\!\cdots\!21 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 17\!\cdots\!61 \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 38\!\cdots\!25 \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 17\!\cdots\!01 \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 73\!\cdots\!36 \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 15\!\cdots\!01 \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 21\!\cdots\!61 \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 27\!\cdots\!25 \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 84\!\cdots\!01 \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 71\!\cdots\!81 \) Copy content Toggle raw display
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