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Properties

Label 405.3.c.a
Level $405$
Weight $3$
Character orbit 405.c
Analytic conductor $11.035$
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] = [405,3,Mod(161,405)]
 
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([1, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("405.161");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 405 = 3^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 405.c (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: \(11.0354507066\)
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: \( x^{16} + 48x^{14} + 912x^{12} + 8704x^{10} + 43602x^{8} + 109032x^{6} + 117844x^{4} + 36000x^{2} + 81 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{8}\cdot 3^{14} \)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{2} - 2) q^{4} + \beta_{10} q^{5} - \beta_{6} q^{7} + ( - \beta_{10} + \beta_{9} - 2 \beta_1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{2} - 2) q^{4} + \beta_{10} q^{5} - \beta_{6} q^{7} + ( - \beta_{10} + \beta_{9} - 2 \beta_1) q^{8} + \beta_{3} q^{10} - \beta_{14} q^{11} + ( - \beta_{5} + 1) q^{13} + ( - \beta_{10} - \beta_{7}) q^{14} + (\beta_{4} - \beta_{3} - \beta_{2} + 4) q^{16} + ( - \beta_{15} + \beta_{13} + \cdots + 2 \beta_1) q^{17}+ \cdots + ( - 2 \beta_{15} + 5 \beta_{13} + \cdots - 3 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 32 q^{4} - 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 32 q^{4} - 4 q^{7} + 20 q^{13} + 64 q^{16} - 52 q^{19} + 48 q^{22} - 80 q^{25} + 32 q^{28} - 64 q^{31} - 108 q^{34} + 44 q^{37} + 60 q^{40} + 248 q^{43} - 108 q^{46} + 108 q^{49} - 124 q^{52} - 180 q^{58} - 124 q^{61} + 256 q^{64} - 28 q^{67} + 120 q^{70} - 268 q^{73} + 212 q^{76} + 80 q^{79} - 204 q^{82} - 60 q^{85} - 288 q^{88} + 136 q^{91} + 300 q^{94} + 284 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 48x^{14} + 912x^{12} + 8704x^{10} + 43602x^{8} + 109032x^{6} + 117844x^{4} + 36000x^{2} + 81 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 6 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{14} - 21\nu^{12} + 99\nu^{10} + 4607\nu^{8} + 27813\nu^{6} + 21801\nu^{4} - 63115\nu^{2} - 2223 ) / 19536 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{14} - 21\nu^{12} + 99\nu^{10} + 4607\nu^{8} + 27813\nu^{6} + 41337\nu^{4} + 190853\nu^{2} + 349425 ) / 19536 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 3\nu^{14} + 137\nu^{12} + 2367\nu^{10} + 18813\nu^{8} + 63525\nu^{6} + 38567\nu^{4} - 128115\nu^{2} - 55417 ) / 3256 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 19 \nu^{14} + 991 \nu^{12} + 20245 \nu^{10} + 203657 \nu^{8} + 1034729 \nu^{6} + 2408585 \nu^{4} + \cdots + 116163 ) / 29304 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 133 \nu^{15} - 5901 \nu^{13} - 103605 \nu^{11} - 928837 \nu^{9} - 4602783 \nu^{7} + \cdots - 7626987 \nu ) / 175824 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 139 \nu^{14} + 6619 \nu^{12} + 122695 \nu^{10} + 1111799 \nu^{8} + 5061353 \nu^{6} + 10747721 \nu^{4} + \cdots + 1743561 ) / 58608 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 247 \nu^{15} + 11847 \nu^{13} + 225075 \nu^{11} + 2150779 \nu^{9} + 10811157 \nu^{7} + \cdots + 10082205 \nu ) / 175824 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 247 \nu^{15} + 11847 \nu^{13} + 225075 \nu^{11} + 2150779 \nu^{9} + 10811157 \nu^{7} + \cdots + 8323965 \nu ) / 175824 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 85 \nu^{15} + 4227 \nu^{13} + 84381 \nu^{11} + 863593 \nu^{9} + 4783629 \nu^{7} + 13858743 \nu^{5} + \cdots + 7873929 \nu ) / 43956 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 313 \nu^{14} - 13825 \nu^{12} - 238225 \nu^{10} - 2028017 \nu^{8} - 8868347 \nu^{6} + \cdots - 1753407 ) / 58608 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 491 \nu^{15} - 23631 \nu^{13} - 450447 \nu^{11} - 4315379 \nu^{9} - 21705753 \nu^{7} + \cdots - 16699869 \nu ) / 58608 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 72 \nu^{15} + 3473 \nu^{13} + 66317 \nu^{11} + 636068 \nu^{9} + 3202957 \nu^{7} + 8067496 \nu^{5} + \cdots + 2862351 \nu ) / 7326 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 2245 \nu^{15} - 107085 \nu^{13} - 2018613 \nu^{11} - 19065181 \nu^{9} - 94080471 \nu^{7} + \cdots - 68122107 \nu ) / 175824 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 6 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{10} + \beta_{9} - 10\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{4} - \beta_{3} - 13\beta_{2} + 60 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -\beta_{15} + 2\beta_{13} + \beta_{11} + 18\beta_{10} - 16\beta_{9} + \beta_{7} + 112\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 2\beta_{8} - 4\beta_{6} - 2\beta_{5} - 21\beta_{4} + 27\beta_{3} + 162\beta_{2} - 671 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 23\beta_{15} + 2\beta_{14} - 50\beta_{13} - 23\beta_{11} - 309\beta_{10} + 223\beta_{9} - 27\beta_{7} - 1327\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 2\beta_{12} - 48\beta_{8} + 106\beta_{6} + 50\beta_{5} + 346\beta_{4} - 536\beta_{3} - 2044\beta_{2} + 7933 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 390 \beta_{15} - 42 \beta_{14} + 932 \beta_{13} + 392 \beta_{11} + 5072 \beta_{10} + \cdots + 16343 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 74 \beta_{12} + 824 \beta_{8} - 1982 \beta_{6} - 930 \beta_{5} - 5252 \beta_{4} + 9306 \beta_{3} + \cdots - 97472 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 5928 \beta_{15} + 570 \beta_{14} - 15488 \beta_{13} - 5970 \beta_{11} - 79833 \beta_{10} + \cdots - 207350 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 1782 \beta_{12} - 12468 \beta_{8} + 32550 \beta_{6} + 15446 \beta_{5} + 76787 \beta_{4} - 150065 \beta_{3} + \cdots + 1233944 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 85691 \beta_{15} - 5926 \beta_{14} + 242298 \beta_{13} + 86277 \beta_{11} + 1216578 \beta_{10} + \cdots + 2692502 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 35534 \beta_{12} + 177894 \beta_{8} - 502678 \beta_{6} - 241712 \beta_{5} - 1100725 \beta_{4} + \cdots - 15993217 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 1207551 \beta_{15} + 43008 \beta_{14} - 3655358 \beta_{13} - 1214801 \beta_{11} - 18106369 \beta_{10} + \cdots - 35595667 \beta_1 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
161.1
− 3.73655i
− 3.27064i
− 3.09125i
− 2.82877i
− 1.83391i
− 1.39204i
− 0.692902i
− 0.0476108i
0.0476108i
0.692902i
1.39204i
1.83391i
2.82877i
3.09125i
3.27064i
3.73655i
− 3.73655i 0 −9.96177 − 2.23607i 0 −7.26122 22.2764i 0 −8.35517
161.2 − 3.27064i 0 −6.69706 2.23607i 0 6.33861 8.82112i 0 7.31337
161.3 − 3.09125i 0 −5.55585 2.23607i 0 −2.20591 4.80953i 0 6.91225
161.4 − 2.82877i 0 −4.00192 − 2.23607i 0 7.94944 0.00541780i 0 −6.32531
161.5 − 1.83391i 0 0.636762 − 2.23607i 0 −6.46796 − 8.50342i 0 −4.10076
161.6 − 1.39204i 0 2.06222 2.23607i 0 −8.82009 − 8.43886i 0 3.11270
161.7 − 0.692902i 0 3.51989 2.23607i 0 12.2822 − 5.21055i 0 1.54938
161.8 − 0.0476108i 0 3.99773 − 2.23607i 0 −3.81512 − 0.380778i 0 −0.106461
161.9 0.0476108i 0 3.99773 2.23607i 0 −3.81512 0.380778i 0 −0.106461
161.10 0.692902i 0 3.51989 − 2.23607i 0 12.2822 5.21055i 0 1.54938
161.11 1.39204i 0 2.06222 − 2.23607i 0 −8.82009 8.43886i 0 3.11270
161.12 1.83391i 0 0.636762 2.23607i 0 −6.46796 8.50342i 0 −4.10076
161.13 2.82877i 0 −4.00192 2.23607i 0 7.94944 − 0.00541780i 0 −6.32531
161.14 3.09125i 0 −5.55585 − 2.23607i 0 −2.20591 − 4.80953i 0 6.91225
161.15 3.27064i 0 −6.69706 − 2.23607i 0 6.33861 − 8.82112i 0 7.31337
161.16 3.73655i 0 −9.96177 2.23607i 0 −7.26122 − 22.2764i 0 −8.35517
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 161.16
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 405.3.c.a 16
3.b odd 2 1 inner 405.3.c.a 16
9.c even 3 1 45.3.i.a ✓ 16
9.c even 3 1 135.3.i.a 16
9.d odd 6 1 45.3.i.a ✓ 16
9.d odd 6 1 135.3.i.a 16
36.f odd 6 1 720.3.bs.c 16
36.f odd 6 1 2160.3.bs.c 16
36.h even 6 1 720.3.bs.c 16
36.h even 6 1 2160.3.bs.c 16
45.h odd 6 1 225.3.j.b 16
45.h odd 6 1 675.3.j.b 16
45.j even 6 1 225.3.j.b 16
45.j even 6 1 675.3.j.b 16
45.k odd 12 2 225.3.i.b 32
45.k odd 12 2 675.3.i.c 32
45.l even 12 2 225.3.i.b 32
45.l even 12 2 675.3.i.c 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
45.3.i.a ✓ 16 9.c even 3 1
45.3.i.a ✓ 16 9.d odd 6 1
135.3.i.a 16 9.c even 3 1
135.3.i.a 16 9.d odd 6 1
225.3.i.b 32 45.k odd 12 2
225.3.i.b 32 45.l even 12 2
225.3.j.b 16 45.h odd 6 1
225.3.j.b 16 45.j even 6 1
405.3.c.a 16 1.a even 1 1 trivial
405.3.c.a 16 3.b odd 2 1 inner
675.3.i.c 32 45.k odd 12 2
675.3.i.c 32 45.l even 12 2
675.3.j.b 16 45.h odd 6 1
675.3.j.b 16 45.j even 6 1
720.3.bs.c 16 36.f odd 6 1
720.3.bs.c 16 36.h even 6 1
2160.3.bs.c 16 36.f odd 6 1
2160.3.bs.c 16 36.h even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{16} + 48T_{2}^{14} + 912T_{2}^{12} + 8704T_{2}^{10} + 43602T_{2}^{8} + 109032T_{2}^{6} + 117844T_{2}^{4} + 36000T_{2}^{2} + 81 \) acting on \(S_{3}^{\mathrm{new}}(405, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} + 48 T^{14} + \cdots + 81 \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( (T^{2} + 5)^{8} \) Copy content Toggle raw display
$7$ \( (T^{8} + 2 T^{7} + \cdots - 2157516)^{2} \) Copy content Toggle raw display
$11$ \( T^{16} + \cdots + 112356358416 \) Copy content Toggle raw display
$13$ \( (T^{8} - 10 T^{7} + \cdots + 8329504)^{2} \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 17\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( (T^{8} + 26 T^{7} + \cdots - 3133657244)^{2} \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 64\!\cdots\!16 \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots + 21\!\cdots\!56 \) Copy content Toggle raw display
$31$ \( (T^{8} + 32 T^{7} + \cdots - 629856)^{2} \) Copy content Toggle raw display
$37$ \( (T^{8} - 22 T^{7} + \cdots - 143779124336)^{2} \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 44\!\cdots\!61 \) Copy content Toggle raw display
$43$ \( (T^{8} - 124 T^{7} + \cdots + 607888940116)^{2} \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 28\!\cdots\!76 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 32\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 25\!\cdots\!76 \) Copy content Toggle raw display
$61$ \( (T^{8} + \cdots + 4314637266184)^{2} \) Copy content Toggle raw display
$67$ \( (T^{8} + \cdots - 19953331060419)^{2} \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 34\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( (T^{8} + \cdots - 3873480104384)^{2} \) Copy content Toggle raw display
$79$ \( (T^{8} + \cdots - 389393894172096)^{2} \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 23\!\cdots\!96 \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 23\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( (T^{8} + \cdots - 415070817801776)^{2} \) Copy content Toggle raw display
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