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Properties

Label 405.4.b.d
Level $405$
Weight $4$
Character orbit 405.b
Analytic conductor $23.896$
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] = [405,4,Mod(244,405)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(405, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("405.244");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 405 = 3^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 405.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: \(23.8957735523\)
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: \( x^{16} + 70 x^{14} + 2191 x^{12} + 30544 x^{10} + 285007 x^{8} + 513982 x^{6} + 1874737 x^{4} + \cdots + 149328400 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{6}\cdot 3^{16} \)
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.

\(f(q)\) \(=\) \( q - \beta_{4} q^{2} + ( - \beta_1 - 4) q^{4} + (\beta_{11} - \beta_{4}) q^{5} - \beta_{5} q^{7} + ( - \beta_{10} + 7 \beta_{4}) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{4} q^{2} + ( - \beta_1 - 4) q^{4} + (\beta_{11} - \beta_{4}) q^{5} - \beta_{5} q^{7} + ( - \beta_{10} + 7 \beta_{4}) q^{8} + ( - \beta_{2} - \beta_1 - 14) q^{10} + ( - \beta_{13} - \beta_{12} + \beta_{11}) q^{11} + \beta_{8} q^{13} + ( - \beta_{15} + \beta_{12} + \cdots + 3 \beta_{9}) q^{14}+ \cdots + (10 \beta_{14} + 2 \beta_{12} + \cdots + 92 \beta_{4}) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 60 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 60 q^{4} - 220 q^{10} + 788 q^{16} + 356 q^{19} + 550 q^{25} + 1520 q^{31} - 392 q^{34} + 1132 q^{40} + 724 q^{46} + 4 q^{49} + 1716 q^{55} + 1268 q^{61} - 9476 q^{64} - 672 q^{70} + 228 q^{76} - 472 q^{79} - 3562 q^{85} + 1524 q^{91} - 10700 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 70 x^{14} + 2191 x^{12} + 30544 x^{10} + 285007 x^{8} + 513982 x^{6} + 1874737 x^{4} + \cdots + 149328400 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 6111164 \nu^{14} + 389525593 \nu^{12} + 11114984720 \nu^{10} + 124965476326 \nu^{8} + \cdots - 14\!\cdots\!00 ) / 127514435185800 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 769031519 \nu^{14} + 47220420803 \nu^{12} + 1159339499670 \nu^{10} + 6172043855546 \nu^{8} + \cdots - 491391059146000 ) / 30\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 4364290711 \nu^{14} + 405874116327 \nu^{12} + 17438734845990 \nu^{10} + \cdots - 64\!\cdots\!00 ) / 12\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 144493201 \nu^{15} + 9827299362 \nu^{13} + 298276900520 \nu^{11} + \cdots - 23\!\cdots\!40 \nu ) / 11\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 2293638707 \nu^{14} + 162782827659 \nu^{12} + 5162877439110 \nu^{10} + \cdots - 85\!\cdots\!00 ) / 30\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 4478251607 \nu^{14} + 320413998559 \nu^{12} + 10366833728310 \nu^{10} + \cdots - 20\!\cdots\!00 ) / 30\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 1485591283 \nu^{14} + 106934227971 \nu^{12} + 3549305158590 \nu^{10} + \cdots - 91\!\cdots\!00 ) / 941645059833600 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 14530340289 \nu^{14} + 1098711625753 \nu^{12} + 37731589903050 \nu^{10} + \cdots - 76\!\cdots\!00 ) / 61\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 144493201 \nu^{15} - 9827299362 \nu^{13} - 298276900520 \nu^{11} + \cdots + 14\!\cdots\!40 \nu ) / 13\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 2895149102 \nu^{15} + 200586674799 \nu^{13} + 6187986568000 \nu^{11} + \cdots - 54\!\cdots\!40 \nu ) / 11\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 169986205537 \nu^{15} - 12205352174289 \nu^{13} - 395835255429930 \nu^{11} + \cdots + 72\!\cdots\!60 \nu ) / 57\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 171087580601 \nu^{15} - 12720776908377 \nu^{13} - 428122638723930 \nu^{11} + \cdots + 80\!\cdots\!80 \nu ) / 57\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 2541846674 \nu^{15} + 192102101435 \nu^{13} + 6483781105452 \nu^{11} + \cdots - 19\!\cdots\!16 \nu ) / 71\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 37948059229 \nu^{15} - 2706640727013 \nu^{13} - 83576643681890 \nu^{11} + \cdots + 76\!\cdots\!00 \nu ) / 95\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 6220914005 \nu^{15} - 416502539157 \nu^{13} - 12345315789234 \nu^{11} + \cdots + 52\!\cdots\!68 \nu ) / 14\!\cdots\!40 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{9} + 9\beta_{4} ) / 9 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2\beta_{6} - 4\beta_{5} + 2\beta_{2} - 9\beta _1 - 81 ) / 9 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 3\beta_{15} + 3\beta_{12} - 3\beta_{11} + 3\beta_{10} - 10\beta_{9} - 42\beta_{4} ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -24\beta_{8} - 9\beta_{7} - 32\beta_{6} + 184\beta_{5} + 9\beta_{3} - 116\beta_{2} + 126\beta _1 + 612 ) / 9 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 330 \beta_{15} - 18 \beta_{14} - 135 \beta_{13} - 618 \beta_{12} + 582 \beta_{11} + \cdots - 144 \beta_{4} ) / 9 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 312\beta_{8} + 6\beta_{7} + 148\beta_{6} - 1832\beta_{5} + 18\beta_{3} + 1402\beta_{2} + 993\beta _1 + 9807 ) / 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 8757 \beta_{15} - 288 \beta_{14} + 4536 \beta_{13} + 19665 \beta_{12} - 20529 \beta_{11} + \cdots + 128034 \beta_{4} ) / 9 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 22788 \beta_{8} + 15111 \beta_{7} - 8644 \beta_{6} + 114128 \beta_{5} - 12447 \beta_{3} + \cdots - 2230488 ) / 9 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 14226 \beta_{15} + 4606 \beta_{14} - 8019 \beta_{13} - 32790 \beta_{12} + 40050 \beta_{11} + \cdots - 772272 \beta_{4} \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 199812 \beta_{8} - 773730 \beta_{7} + 98224 \beta_{6} - 439544 \beta_{5} + 570978 \beta_{3} + \cdots + 93160341 ) / 9 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 2537007 \beta_{15} - 1641312 \beta_{14} - 1152360 \beta_{13} - 6379287 \beta_{12} + \cdots + 241982946 \beta_{4} ) / 9 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 4934928 \beta_{8} + 7645413 \beta_{7} + 1866896 \beta_{6} - 33839704 \beta_{5} - 5534613 \beta_{3} + \cdots - 892918692 ) / 3 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 280605078 \beta_{15} + 38178126 \beta_{14} + 139287681 \beta_{13} + 661180446 \beta_{12} + \cdots - 5562109440 \beta_{4} ) / 9 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 1031654448 \beta_{8} - 372255174 \beta_{7} - 452575468 \beta_{6} + 6063116168 \beta_{5} + \cdots + 43503619581 ) / 9 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 4173644433 \beta_{15} - 83950464 \beta_{14} - 2080233576 \beta_{13} - 9533184645 \beta_{12} + \cdots + 11980045302 \beta_{4} ) / 3 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/405\mathbb{Z}\right)^\times\).

\(n\) \(82\) \(326\)
\(\chi(n)\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
244.1
−1.73205 + 5.49509i
1.73205 + 5.49509i
1.73205 + 3.35627i
−1.73205 + 3.35627i
−1.73205 + 2.34632i
1.73205 + 2.34632i
1.73205 + 0.184872i
−1.73205 + 0.184872i
1.73205 − 0.184872i
−1.73205 − 0.184872i
−1.73205 − 2.34632i
1.73205 − 2.34632i
1.73205 − 3.35627i
−1.73205 − 3.35627i
−1.73205 − 5.49509i
1.73205 − 5.49509i
− 5.49509i 0 −22.1960 −9.52216 − 5.85905i 0 − 19.7308i 78.0086i 0 −32.1960 + 52.3251i
244.2 − 5.49509i 0 −22.1960 9.52216 − 5.85905i 0 19.7308i 78.0086i 0 −32.1960 − 52.3251i
244.3 − 3.35627i 0 −3.26456 −10.4585 − 3.95217i 0 26.8077i − 15.8934i 0 −13.2646 + 35.1016i
244.4 − 3.35627i 0 −3.26456 10.4585 − 3.95217i 0 − 26.8077i − 15.8934i 0 −13.2646 − 35.1016i
244.5 − 2.34632i 0 2.49478 −10.7130 − 3.19872i 0 10.5008i − 24.6241i 0 −7.50522 + 25.1361i
244.6 − 2.34632i 0 2.49478 10.7130 − 3.19872i 0 − 10.5008i − 24.6241i 0 −7.50522 − 25.1361i
244.7 − 0.184872i 0 7.96582 −1.98241 − 11.0032i 0 − 12.3602i − 2.95163i 0 −2.03418 + 0.366493i
244.8 − 0.184872i 0 7.96582 1.98241 − 11.0032i 0 12.3602i − 2.95163i 0 −2.03418 − 0.366493i
244.9 0.184872i 0 7.96582 −1.98241 + 11.0032i 0 12.3602i 2.95163i 0 −2.03418 − 0.366493i
244.10 0.184872i 0 7.96582 1.98241 + 11.0032i 0 − 12.3602i 2.95163i 0 −2.03418 + 0.366493i
244.11 2.34632i 0 2.49478 −10.7130 + 3.19872i 0 − 10.5008i 24.6241i 0 −7.50522 − 25.1361i
244.12 2.34632i 0 2.49478 10.7130 + 3.19872i 0 10.5008i 24.6241i 0 −7.50522 + 25.1361i
244.13 3.35627i 0 −3.26456 −10.4585 + 3.95217i 0 − 26.8077i 15.8934i 0 −13.2646 − 35.1016i
244.14 3.35627i 0 −3.26456 10.4585 + 3.95217i 0 26.8077i 15.8934i 0 −13.2646 + 35.1016i
244.15 5.49509i 0 −22.1960 −9.52216 + 5.85905i 0 19.7308i − 78.0086i 0 −32.1960 − 52.3251i
244.16 5.49509i 0 −22.1960 9.52216 + 5.85905i 0 − 19.7308i − 78.0086i 0 −32.1960 + 52.3251i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 244.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.b even 2 1 inner
15.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 405.4.b.d ✓ 16
3.b odd 2 1 inner 405.4.b.d ✓ 16
5.b even 2 1 inner 405.4.b.d ✓ 16
5.c odd 4 2 2025.4.a.bm 16
15.d odd 2 1 inner 405.4.b.d ✓ 16
15.e even 4 2 2025.4.a.bm 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
405.4.b.d ✓ 16 1.a even 1 1 trivial
405.4.b.d ✓ 16 3.b odd 2 1 inner
405.4.b.d ✓ 16 5.b even 2 1 inner
405.4.b.d ✓ 16 15.d odd 2 1 inner
2025.4.a.bm 16 5.c odd 4 2
2025.4.a.bm 16 15.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 \(S_{4}^{\mathrm{new}}(405, [\chi])\):

\( T_{2}^{8} + 47T_{2}^{6} + 570T_{2}^{4} + 1892T_{2}^{2} + 64 \) Copy content Toggle raw display
\( T_{11}^{8} - 10518T_{11}^{6} + 40539825T_{11}^{4} - 68135375448T_{11}^{2} + 42262844976144 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{8} + 47 T^{6} + \cdots + 64)^{2} \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( T^{16} + \cdots + 59\!\cdots\!25 \) Copy content Toggle raw display
$7$ \( (T^{8} + 1371 T^{6} + \cdots + 4713097104)^{2} \) Copy content Toggle raw display
$11$ \( (T^{8} + \cdots + 42262844976144)^{2} \) Copy content Toggle raw display
$13$ \( (T^{8} + 8238 T^{6} + \cdots + 74975201856)^{2} \) Copy content Toggle raw display
$17$ \( (T^{8} + \cdots + 375826921697536)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} - 89 T^{3} + \cdots + 9481450)^{4} \) Copy content Toggle raw display
$23$ \( (T^{8} + \cdots + 43\!\cdots\!76)^{2} \) Copy content Toggle raw display
$29$ \( (T^{8} + \cdots + 14\!\cdots\!96)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} - 380 T^{3} + \cdots - 341909504)^{4} \) Copy content Toggle raw display
$37$ \( (T^{8} + \cdots + 13\!\cdots\!84)^{2} \) Copy content Toggle raw display
$41$ \( (T^{8} + \cdots + 79\!\cdots\!24)^{2} \) Copy content Toggle raw display
$43$ \( (T^{8} + \cdots + 20\!\cdots\!00)^{2} \) Copy content Toggle raw display
$47$ \( (T^{8} + \cdots + 319341188814400)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} + \cdots + 29\!\cdots\!00)^{2} \) Copy content Toggle raw display
$59$ \( (T^{8} + \cdots + 23\!\cdots\!44)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} - 317 T^{3} + \cdots - 1659106592)^{4} \) Copy content Toggle raw display
$67$ \( (T^{8} + \cdots + 49\!\cdots\!00)^{2} \) Copy content Toggle raw display
$71$ \( (T^{8} + \cdots + 22\!\cdots\!76)^{2} \) Copy content Toggle raw display
$73$ \( (T^{8} + \cdots + 29\!\cdots\!96)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} + 118 T^{3} + \cdots + 26021380864)^{4} \) Copy content Toggle raw display
$83$ \( (T^{8} + \cdots + 43\!\cdots\!16)^{2} \) Copy content Toggle raw display
$89$ \( (T^{8} + \cdots + 13\!\cdots\!24)^{2} \) Copy content Toggle raw display
$97$ \( (T^{8} + \cdots + 68\!\cdots\!84)^{2} \) Copy content Toggle raw display
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