Properties

Label 405.4.b.e
Level $405$
Weight $4$
Character orbit 405.b
Analytic conductor $23.896$
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] = [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: \( x^{16} + 91 x^{14} + 3268 x^{12} + 59128 x^{10} + 571975 x^{8} + 2881141 x^{6} + 6555196 x^{4} + 4069504 x^{2} + 614656 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{12}\cdot 7^{2} \)
Twist minimal: no (minimal twist has level 45)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{2} - 3) q^{4} + \beta_{9} q^{5} + ( - \beta_{13} - \beta_1) q^{7} + (\beta_{3} - 3 \beta_1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{2} - 3) q^{4} + \beta_{9} q^{5} + ( - \beta_{13} - \beta_1) q^{7} + (\beta_{3} - 3 \beta_1) q^{8} + ( - \beta_{10} - 1) q^{10} + (\beta_{5} + \beta_{2} - 5) q^{11} + ( - \beta_{15} + \beta_{14} - \beta_{13} - \beta_{3} + 2 \beta_1) q^{13} + ( - \beta_{12} - \beta_{10} + \beta_{7} - 2 \beta_{2} + 5) q^{14} + (\beta_{9} - \beta_{8} - \beta_{5} + \beta_{4} - 3 \beta_{2} + 8) q^{16} + ( - \beta_{15} - \beta_{12} + \beta_{11} + \beta_{10} - \beta_{3} + \beta_1) q^{17} + (\beta_{12} + \beta_{10} + \beta_{9} - \beta_{8} - 2 \beta_{7} - \beta_{6} + \beta_{4} - \beta_{2}) q^{19} + (\beta_{15} - 2 \beta_{14} + 3 \beta_{13} - \beta_{12} - 2 \beta_{9} - \beta_{7} - \beta_{6} - \beta_{5} + \cdots - 1) q^{20}+ \cdots + ( - 6 \beta_{15} + 29 \beta_{14} + 15 \beta_{13} + 9 \beta_{12} - 7 \beta_{11} + \cdots + 39 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 54 q^{4} - 3 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 54 q^{4} - 3 q^{5} - 10 q^{10} - 90 q^{11} + 102 q^{14} + 146 q^{16} - 4 q^{19} + 6 q^{20} - 71 q^{25} - 468 q^{26} + 516 q^{29} + 38 q^{31} - 212 q^{34} - 267 q^{35} - 44 q^{40} - 576 q^{41} + 1644 q^{44} - 290 q^{46} + 4 q^{49} - 558 q^{50} + 15 q^{55} - 2430 q^{56} + 2202 q^{59} + 20 q^{61} + 322 q^{64} - 339 q^{65} - 636 q^{70} - 2952 q^{71} + 4080 q^{74} - 396 q^{76} + 218 q^{79} + 1266 q^{80} + 704 q^{85} - 6108 q^{86} + 4074 q^{89} - 942 q^{91} + 1078 q^{94} + 1692 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 91 x^{14} + 3268 x^{12} + 59128 x^{10} + 571975 x^{8} + 2881141 x^{6} + 6555196 x^{4} + 4069504 x^{2} + 614656 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 11 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} + 19\nu \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 13 \nu^{14} - 1022 \nu^{12} - 29466 \nu^{10} - 370126 \nu^{8} - 1713581 \nu^{6} + 1226376 \nu^{4} + 22574480 \nu^{2} + 15782656 ) / 176256 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - \nu^{14} + 94 \nu^{12} + 11574 \nu^{10} + 387218 \nu^{8} + 5621959 \nu^{6} + 36266220 \nu^{4} + 85224032 \nu^{2} + 17199616 ) / 352512 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 27 \nu^{14} + 2222 \nu^{12} + 68574 \nu^{10} + 968298 \nu^{8} + 5902643 \nu^{6} + 8366676 \nu^{4} - 30262848 \nu^{2} - 13692160 ) / 117504 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 73 \nu^{14} - 6194 \nu^{12} - 200394 \nu^{10} - 3084622 \nu^{8} - 22981937 \nu^{6} - 73691748 \nu^{4} - 64228384 \nu^{2} - 8294528 ) / 176256 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 849 \nu^{15} - 1225 \nu^{14} - 75642 \nu^{13} - 104762 \nu^{12} - 2626650 \nu^{11} - 3454794 \nu^{10} - 45044382 \nu^{9} - 55246030 \nu^{8} + \cdots - 833385728 ) / 34546176 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 849 \nu^{15} + 1225 \nu^{14} - 75642 \nu^{13} + 104762 \nu^{12} - 2626650 \nu^{11} + 3454794 \nu^{10} - 45044382 \nu^{9} + 55246030 \nu^{8} + \cdots + 833385728 ) / 34546176 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 25 \nu^{15} - 33 \nu^{14} - 2138 \nu^{13} - 3018 \nu^{12} - 70506 \nu^{11} - 105210 \nu^{10} - 1127470 \nu^{9} - 1750686 \nu^{8} - 9049121 \nu^{7} - 14215017 \nu^{6} + \cdots - 11354880 ) / 705024 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 101 \nu^{15} - 8407 \nu^{13} - 263820 \nu^{11} - 3850424 \nu^{9} - 25802659 \nu^{7} - 63425913 \nu^{5} - 13608404 \nu^{3} - 100002400 \nu ) / 2878848 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 25 \nu^{15} - 33 \nu^{14} + 2138 \nu^{13} - 3018 \nu^{12} + 70506 \nu^{11} - 105210 \nu^{10} + 1127470 \nu^{9} - 1750686 \nu^{8} + 9049121 \nu^{7} - 14215017 \nu^{6} + \cdots - 11354880 ) / 705024 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 895 \nu^{15} - 75908 \nu^{13} - 2465730 \nu^{11} - 38436238 \nu^{9} - 295408283 \nu^{7} - 1022927202 \nu^{5} - 1116316120 \nu^{3} + \cdots + 79369792 \nu ) / 17273088 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 365 \nu^{15} - 31990 \nu^{13} - 1088058 \nu^{11} - 18206894 \nu^{9} - 157923085 \nu^{7} - 689377056 \nu^{5} - 1314355088 \nu^{3} + \cdots - 640762624 \nu ) / 5757696 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 137 \nu^{15} - 12712 \nu^{13} - 468198 \nu^{11} - 8743514 \nu^{9} - 87674005 \nu^{7} - 455445594 \nu^{5} - 1030303640 \nu^{3} - 475010752 \nu ) / 1016064 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 11 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} - 19\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{9} - \beta_{8} - \beta_{5} + \beta_{4} - 27\beta_{2} + 208 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -\beta_{15} + 3\beta_{14} - \beta_{12} - 5\beta_{11} + \beta_{10} + \beta_{9} + \beta_{8} - 33\beta_{3} + 418\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 4 \beta_{12} - 4 \beta_{10} - 20 \beta_{9} + 20 \beta_{8} + 4 \beta_{7} - \beta_{6} + 33 \beta_{5} - 41 \beta_{4} + 687 \beta_{2} - 4588 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 45 \beta_{15} - 143 \beta_{14} + 30 \beta_{13} + 57 \beta_{12} + 211 \beta_{11} - 57 \beta_{10} - 39 \beta_{9} - 39 \beta_{8} + 918 \beta_{3} - 9830 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 200 \beta_{12} + 200 \beta_{10} + 235 \beta_{9} - 235 \beta_{8} - 212 \beta_{7} + 29 \beta_{6} - 912 \beta_{5} + 1288 \beta_{4} - 17333 \beta_{2} + 108335 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 1476 \beta_{15} + 4732 \beta_{14} - 1518 \beta_{13} - 2112 \beta_{12} - 6662 \beta_{11} + 2112 \beta_{10} + 1254 \beta_{9} + 1254 \beta_{8} - 24314 \beta_{3} + 238799 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 6868 \beta_{12} - 6868 \beta_{10} + 212 \beta_{9} - 212 \beta_{8} + 7672 \beta_{7} - 508 \beta_{6} + 24260 \beta_{5} - 36676 \beta_{4} + 437433 \beta_{2} - 2640943 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 42740 \beta_{15} - 136604 \beta_{14} + 52824 \beta_{13} + 66188 \beta_{12} + 189356 \beta_{11} - 66188 \beta_{10} - 36404 \beta_{9} - 36404 \beta_{8} + 631557 \beta_{3} - 5902367 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 202384 \beta_{12} + 202384 \beta_{10} - 125815 \beta_{9} + 125815 \beta_{8} - 237952 \beta_{7} + 4228 \beta_{6} - 637341 \beta_{5} + 996029 \beta_{4} - 11058627 \beta_{2} + \cdots + 65437500 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 1162845 \beta_{15} + 3697319 \beta_{14} - 1577928 \beta_{13} - 1911261 \beta_{12} - 5122729 \beta_{11} + 1911261 \beta_{10} + 983541 \beta_{9} + 983541 \beta_{8} + \cdots + 147362278 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 5510372 \beta_{12} - 5510372 \beta_{10} + 5450348 \beta_{9} - 5450348 \beta_{8} + 6825860 \beta_{7} + 125203 \beta_{6} + 16638285 \beta_{5} - 26358613 \beta_{4} + \cdots - 1636326120 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 30553497 \beta_{15} - 96685123 \beta_{14} + 43569078 \beta_{13} + 52782405 \beta_{12} + 134973239 \beta_{11} - 52782405 \beta_{10} - 25248315 \beta_{9} - 25248315 \beta_{8} + \cdots - 3702009370 \beta_1 \) Copy content Toggle raw display

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
5.05435i
5.02371i
4.07626i
3.08740i
2.67506i
2.46385i
0.785333i
0.473990i
0.473990i
0.785333i
2.46385i
2.67506i
3.08740i
4.07626i
5.02371i
5.05435i
5.05435i 0 −17.5465 −7.77185 + 8.03731i 0 21.0117i 48.2513i 0 40.6234 + 39.2817i
244.2 5.02371i 0 −17.2377 5.75603 9.58479i 0 5.38197i 46.4074i 0 −48.1512 28.9166i
244.3 4.07626i 0 −8.61587 10.3994 + 4.10522i 0 13.3430i 2.51043i 0 16.7339 42.3906i
244.4 3.08740i 0 −1.53204 −5.83763 9.53531i 0 31.3204i 19.9692i 0 −29.4393 + 18.0231i
244.5 2.67506i 0 0.844033 −10.9788 2.11350i 0 15.4153i 23.6584i 0 −5.65374 + 29.3689i
244.6 2.46385i 0 1.92944 −0.0773702 + 11.1801i 0 19.2401i 24.4647i 0 27.5460 + 0.190629i
244.7 0.785333i 0 7.38325 −3.61462 10.5799i 0 20.9136i 12.0810i 0 −8.30875 + 2.83868i
244.8 0.473990i 0 7.77533 10.6248 + 3.48041i 0 8.20657i 7.47735i 0 1.64968 5.03606i
244.9 0.473990i 0 7.77533 10.6248 3.48041i 0 8.20657i 7.47735i 0 1.64968 + 5.03606i
244.10 0.785333i 0 7.38325 −3.61462 + 10.5799i 0 20.9136i 12.0810i 0 −8.30875 2.83868i
244.11 2.46385i 0 1.92944 −0.0773702 11.1801i 0 19.2401i 24.4647i 0 27.5460 0.190629i
244.12 2.67506i 0 0.844033 −10.9788 + 2.11350i 0 15.4153i 23.6584i 0 −5.65374 29.3689i
244.13 3.08740i 0 −1.53204 −5.83763 + 9.53531i 0 31.3204i 19.9692i 0 −29.4393 18.0231i
244.14 4.07626i 0 −8.61587 10.3994 4.10522i 0 13.3430i 2.51043i 0 16.7339 + 42.3906i
244.15 5.02371i 0 −17.2377 5.75603 + 9.58479i 0 5.38197i 46.4074i 0 −48.1512 + 28.9166i
244.16 5.05435i 0 −17.5465 −7.77185 8.03731i 0 21.0117i 48.2513i 0 40.6234 39.2817i
\(n\): 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
5.b even 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.e 16
3.b odd 2 1 405.4.b.f 16
5.b even 2 1 inner 405.4.b.e 16
5.c odd 4 2 2025.4.a.bk 16
9.c even 3 2 45.4.j.a 32
9.d odd 6 2 135.4.j.a 32
15.d odd 2 1 405.4.b.f 16
15.e even 4 2 2025.4.a.bl 16
45.h odd 6 2 135.4.j.a 32
45.j even 6 2 45.4.j.a 32
45.k odd 12 4 225.4.e.g 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
45.4.j.a 32 9.c even 3 2
45.4.j.a 32 45.j even 6 2
135.4.j.a 32 9.d odd 6 2
135.4.j.a 32 45.h odd 6 2
225.4.e.g 32 45.k odd 12 4
405.4.b.e 16 1.a even 1 1 trivial
405.4.b.e 16 5.b even 2 1 inner
405.4.b.f 16 3.b odd 2 1
405.4.b.f 16 15.d odd 2 1
2025.4.a.bk 16 5.c odd 4 2
2025.4.a.bl 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 \(S_{4}^{\mathrm{new}}(405, [\chi])\):

\( T_{2}^{16} + 91 T_{2}^{14} + 3268 T_{2}^{12} + 59128 T_{2}^{10} + 571975 T_{2}^{8} + 2881141 T_{2}^{6} + 6555196 T_{2}^{4} + 4069504 T_{2}^{2} + 614656 \) Copy content Toggle raw display
\( T_{11}^{8} + 45 T_{11}^{7} - 3975 T_{11}^{6} - 160551 T_{11}^{5} + 4235328 T_{11}^{4} + 158752116 T_{11}^{3} - 809472312 T_{11}^{2} - 26757336852 T_{11} + 137637673416 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} + 91 T^{14} + 3268 T^{12} + \cdots + 614656 \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( T^{16} + 3 T^{15} + \cdots + 59\!\cdots\!25 \) Copy content Toggle raw display
$7$ \( T^{16} + 2742 T^{14} + \cdots + 57\!\cdots\!16 \) Copy content Toggle raw display
$11$ \( (T^{8} + 45 T^{7} + \cdots + 137637673416)^{2} \) Copy content Toggle raw display
$13$ \( T^{16} + 18483 T^{14} + \cdots + 53\!\cdots\!96 \) Copy content Toggle raw display
$17$ \( T^{16} + 36958 T^{14} + \cdots + 18\!\cdots\!56 \) Copy content Toggle raw display
$19$ \( (T^{8} + 2 T^{7} + \cdots + 11815095359200)^{2} \) Copy content Toggle raw display
$23$ \( T^{16} + 123130 T^{14} + \cdots + 30\!\cdots\!64 \) Copy content Toggle raw display
$29$ \( (T^{8} - 258 T^{7} + \cdots - 16\!\cdots\!16)^{2} \) Copy content Toggle raw display
$31$ \( (T^{8} - 19 T^{7} + \cdots - 12\!\cdots\!60)^{2} \) Copy content Toggle raw display
$37$ \( T^{16} + 369900 T^{14} + \cdots + 12\!\cdots\!24 \) Copy content Toggle raw display
$41$ \( (T^{8} + 288 T^{7} + \cdots - 50\!\cdots\!35)^{2} \) Copy content Toggle raw display
$43$ \( T^{16} + 322017 T^{14} + \cdots + 22\!\cdots\!56 \) Copy content Toggle raw display
$47$ \( T^{16} + 594514 T^{14} + \cdots + 89\!\cdots\!44 \) Copy content Toggle raw display
$53$ \( T^{16} + 564124 T^{14} + \cdots + 45\!\cdots\!44 \) Copy content Toggle raw display
$59$ \( (T^{8} - 1101 T^{7} + \cdots - 79\!\cdots\!32)^{2} \) Copy content Toggle raw display
$61$ \( (T^{8} - 10 T^{7} + \cdots - 49\!\cdots\!22)^{2} \) Copy content Toggle raw display
$67$ \( T^{16} + 2156100 T^{14} + \cdots + 26\!\cdots\!89 \) Copy content Toggle raw display
$71$ \( (T^{8} + 1476 T^{7} + \cdots - 15\!\cdots\!24)^{2} \) Copy content Toggle raw display
$73$ \( T^{16} + 2819826 T^{14} + \cdots + 43\!\cdots\!96 \) Copy content Toggle raw display
$79$ \( (T^{8} - 109 T^{7} + \cdots + 78\!\cdots\!32)^{2} \) Copy content Toggle raw display
$83$ \( T^{16} + 2411074 T^{14} + \cdots + 15\!\cdots\!76 \) Copy content Toggle raw display
$89$ \( (T^{8} - 2037 T^{7} + \cdots + 15\!\cdots\!00)^{2} \) Copy content Toggle raw display
$97$ \( T^{16} + 8277621 T^{14} + \cdots + 17\!\cdots\!96 \) Copy content Toggle raw display
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