↑

Properties

Label 364.2.u.a
Level $364$
Weight $2$
Character orbit 364.u
Analytic conductor $2.907$
Analytic rank $0$
Dimension $16$
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [364,2,Mod(225,364)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(364, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("364.225");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 364 = 2^{2} \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 364.u (of order \(6\), degree \(2\), 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: \(2.90655463357\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 38x^{14} + 587x^{12} + 4762x^{10} + 21849x^{8} + 56552x^{6} + 76456x^{4} + 42624x^{2} + 2704 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q + (\beta_{14} - \beta_{2} + \beta_1) q^{3} + ( - \beta_{13} + \beta_{12} + \beta_{2}) q^{5} - \beta_{3} q^{7} + (\beta_{15} + \beta_{14} - \beta_{10} + \cdots - 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{14} - \beta_{2} + \beta_1) q^{3} + ( - \beta_{13} + \beta_{12} + \beta_{2}) q^{5} - \beta_{3} q^{7} + (\beta_{15} + \beta_{14} - \beta_{10} + \cdots - 2) q^{9}+ \cdots + (\beta_{13} - 2 \beta_{12} + \beta_{11} + \cdots + 5) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 14 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 14 q^{9} + 6 q^{11} + 10 q^{13} + 6 q^{15} + 2 q^{17} - 44 q^{25} - 12 q^{27} - 22 q^{29} + 42 q^{33} - 6 q^{35} + 12 q^{37} + 24 q^{39} + 36 q^{41} + 6 q^{43} - 30 q^{45} + 8 q^{49} - 4 q^{51} + 8 q^{53} + 2 q^{55} - 18 q^{59} + 4 q^{61} - 12 q^{63} - 30 q^{65} + 24 q^{67} - 52 q^{69} + 36 q^{71} - 10 q^{75} - 24 q^{77} + 8 q^{79} + 42 q^{85} + 26 q^{87} - 36 q^{89} - 2 q^{91} - 42 q^{93} - 30 q^{95} - 42 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 38x^{14} + 587x^{12} + 4762x^{10} + 21849x^{8} + 56552x^{6} + 76456x^{4} + 42624x^{2} + 2704 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 125 \nu^{14} + 4204 \nu^{12} + 55047 \nu^{10} + 355436 \nu^{8} + 1184157 \nu^{6} + 1944230 \nu^{4} + \cdots + 104520 ) / 70144 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 1005 \nu^{15} - 34940 \nu^{13} - 480631 \nu^{11} - 3354588 \nu^{9} - 12716909 \nu^{7} + \cdots - 8550920 \nu ) / 1823744 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 1005 \nu^{15} - 56472 \nu^{14} + 34940 \nu^{13} - 1972672 \nu^{12} + 480631 \nu^{11} + \cdots - 53684800 ) / 3647488 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 29 \nu^{14} - 1012 \nu^{12} - 13879 \nu^{10} - 94948 \nu^{8} - 338349 \nu^{6} - 588174 \nu^{4} + \cdots - 25064 ) / 1024 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 29 \nu^{14} + 1012 \nu^{12} + 13879 \nu^{10} + 94948 \nu^{8} + 338349 \nu^{6} + 588174 \nu^{4} + \cdots + 25064 ) / 1024 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 6205 \nu^{15} - 37973 \nu^{14} - 205552 \nu^{13} - 1333644 \nu^{12} - 2636655 \nu^{11} + \cdots - 33629960 ) / 1823744 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 14471 \nu^{15} - 26962 \nu^{14} - 505724 \nu^{13} - 935168 \nu^{12} - 6952053 \nu^{11} + \cdots - 16802032 ) / 1823744 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 15621 \nu^{15} + 56823 \nu^{14} + 542976 \nu^{13} + 1976156 \nu^{12} + 7402151 \nu^{11} + \cdots + 49562552 ) / 1823744 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 241 \nu^{15} - 8404 \nu^{13} - 115155 \nu^{11} - 786788 \nu^{9} - 2796961 \nu^{7} - 4831958 \nu^{5} + \cdots - 13312 ) / 26624 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 15621 \nu^{15} - 56823 \nu^{14} + 542976 \nu^{13} - 1976156 \nu^{12} + 7402151 \nu^{11} + \cdots - 49562552 ) / 1823744 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 19521 \nu^{15} - 39260 \nu^{14} + 686964 \nu^{13} - 1365416 \nu^{12} + 9521411 \nu^{11} + \cdots - 33180992 ) / 1823744 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 25726 \nu^{15} - 77233 \nu^{14} - 892516 \nu^{13} - 2699060 \nu^{12} - 12158066 \nu^{11} + \cdots - 66810952 ) / 1823744 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 25726 \nu^{15} - 77233 \nu^{14} + 892516 \nu^{13} - 2699060 \nu^{12} + 12158066 \nu^{11} + \cdots - 66810952 ) / 1823744 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 29241 \nu^{15} - 1625 \nu^{14} - 1021276 \nu^{13} - 54652 \nu^{12} - 14031675 \nu^{11} + \cdots - 1358760 ) / 1823744 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 30092 \nu^{15} + 29861 \nu^{14} + 1048700 \nu^{13} + 1040988 \nu^{12} + 14354204 \nu^{11} + \cdots + 32760520 ) / 1823744 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_{5} + \beta_{4} \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{15} - \beta_{10} + \beta_{7} + 2\beta_{3} + \beta_{2} - \beta _1 - 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2 \beta_{14} + \beta_{13} - 2 \beta_{12} + \beta_{11} + 2 \beta_{9} + \beta_{6} - 6 \beta_{5} - 6 \beta_{4} + \cdots + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 10 \beta_{15} + \beta_{13} + 2 \beta_{12} + \beta_{11} + 9 \beta_{10} + \beta_{8} - 10 \beta_{7} + \cdots + 38 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 2 \beta_{15} - 26 \beta_{14} - 12 \beta_{13} + 25 \beta_{12} - 13 \beta_{11} - \beta_{10} - 22 \beta_{9} + \cdots - 11 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 99 \beta_{15} - 15 \beta_{13} - 28 \beta_{12} - 13 \beta_{11} - 82 \beta_{10} - 17 \beta_{8} + \cdots - 331 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 28 \beta_{15} + 280 \beta_{14} + 123 \beta_{13} - 265 \beta_{12} + 142 \beta_{11} + 27 \beta_{10} + \cdots + 124 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 977 \beta_{15} + 188 \beta_{13} + 320 \beta_{12} + 132 \beta_{11} + 771 \beta_{10} + 206 \beta_{8} + \cdots + 3065 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 312 \beta_{15} - 2826 \beta_{14} - 1223 \beta_{13} + 2708 \beta_{12} - 1485 \beta_{11} - 460 \beta_{10} + \cdots - 1463 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 9654 \beta_{15} - 2207 \beta_{13} - 3480 \beta_{12} - 1273 \beta_{11} - 7405 \beta_{10} - 2249 \beta_{8} + \cdots - 29296 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 3360 \beta_{15} + 27686 \beta_{14} + 12158 \beta_{13} - 27499 \beta_{12} + 15341 \beta_{11} + \cdots + 17351 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 95721 \beta_{15} + 24959 \beta_{13} + 37294 \beta_{12} + 12335 \beta_{11} + 72152 \beta_{10} + \cdots + 285301 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 36406 \beta_{15} - 267484 \beta_{14} - 121781 \beta_{13} + 279755 \beta_{12} - 157974 \beta_{11} + \cdots - 202962 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 953123 \beta_{15} - 275542 \beta_{13} - 397536 \beta_{12} - 121994 \beta_{11} - 710589 \beta_{10} + \cdots - 2812765 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 397768 \beta_{15} + 2570258 \beta_{14} + 1230119 \beta_{13} - 2856320 \beta_{12} + 1626201 \beta_{11} + \cdots + 2331497 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/364\mathbb{Z}\right)^\times\).

\(n\) \(157\) \(183\) \(197\)
\(\chi(n)\) \(1\) \(1\) \(1 + \beta_{9}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
225.1
− 3.23100i
1.77673i
− 1.75101i
1.17707i
0.268953i
− 2.25607i
− 2.42977i
2.98100i
3.23100i
− 1.77673i
1.75101i
− 1.17707i
− 0.268953i
2.25607i
2.42977i
− 2.98100i
0 −1.61550 + 2.79813i 0 − 3.14769i 0 0.866025 − 0.500000i 0 −3.71968 − 6.44268i 0
225.2 0 −0.888364 + 1.53869i 0 3.82804i 0 −0.866025 + 0.500000i 0 −0.0783804 − 0.135759i 0
225.3 0 −0.875503 + 1.51642i 0 1.38536i 0 0.866025 − 0.500000i 0 −0.0330100 − 0.0571750i 0
225.4 0 −0.588533 + 1.01937i 0 − 1.46614i 0 −0.866025 + 0.500000i 0 0.807258 + 1.39821i 0
225.5 0 0.134476 − 0.232920i 0 − 1.35585i 0 0.866025 − 0.500000i 0 1.46383 + 2.53543i 0
225.6 0 1.12804 − 1.95381i 0 4.27591i 0 −0.866025 + 0.500000i 0 −1.04493 − 1.80987i 0
225.7 0 1.21489 − 2.10425i 0 − 3.63781i 0 −0.866025 + 0.500000i 0 −1.45190 − 2.51476i 0
225.8 0 1.49050 − 2.58162i 0 0.118179i 0 0.866025 − 0.500000i 0 −2.94319 − 5.09775i 0
309.1 0 −1.61550 − 2.79813i 0 3.14769i 0 0.866025 + 0.500000i 0 −3.71968 + 6.44268i 0
309.2 0 −0.888364 − 1.53869i 0 − 3.82804i 0 −0.866025 − 0.500000i 0 −0.0783804 + 0.135759i 0
309.3 0 −0.875503 − 1.51642i 0 − 1.38536i 0 0.866025 + 0.500000i 0 −0.0330100 + 0.0571750i 0
309.4 0 −0.588533 − 1.01937i 0 1.46614i 0 −0.866025 − 0.500000i 0 0.807258 − 1.39821i 0
309.5 0 0.134476 + 0.232920i 0 1.35585i 0 0.866025 + 0.500000i 0 1.46383 − 2.53543i 0
309.6 0 1.12804 + 1.95381i 0 − 4.27591i 0 −0.866025 − 0.500000i 0 −1.04493 + 1.80987i 0
309.7 0 1.21489 + 2.10425i 0 3.63781i 0 −0.866025 − 0.500000i 0 −1.45190 + 2.51476i 0
309.8 0 1.49050 + 2.58162i 0 − 0.118179i 0 0.866025 + 0.500000i 0 −2.94319 + 5.09775i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 225.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.e even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 364.2.u.a ✓ 16
3.b odd 2 1 3276.2.cf.c 16
4.b odd 2 1 1456.2.cc.f 16
7.b odd 2 1 2548.2.u.c 16
7.c even 3 1 2548.2.bb.d 16
7.c even 3 1 2548.2.bq.e 16
7.d odd 6 1 2548.2.bb.c 16
7.d odd 6 1 2548.2.bq.c 16
13.c even 3 1 4732.2.g.k 16
13.e even 6 1 inner 364.2.u.a ✓ 16
13.e even 6 1 4732.2.g.k 16
13.f odd 12 1 4732.2.a.s 8
13.f odd 12 1 4732.2.a.t 8
39.h odd 6 1 3276.2.cf.c 16
52.i odd 6 1 1456.2.cc.f 16
91.k even 6 1 2548.2.bq.e 16
91.l odd 6 1 2548.2.bq.c 16
91.p odd 6 1 2548.2.bb.c 16
91.t odd 6 1 2548.2.u.c 16
91.u even 6 1 2548.2.bb.d 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
364.2.u.a ✓ 16 1.a even 1 1 trivial
364.2.u.a ✓ 16 13.e even 6 1 inner
1456.2.cc.f 16 4.b odd 2 1
1456.2.cc.f 16 52.i odd 6 1
2548.2.u.c 16 7.b odd 2 1
2548.2.u.c 16 91.t odd 6 1
2548.2.bb.c 16 7.d odd 6 1
2548.2.bb.c 16 91.p odd 6 1
2548.2.bb.d 16 7.c even 3 1
2548.2.bb.d 16 91.u even 6 1
2548.2.bq.c 16 7.d odd 6 1
2548.2.bq.c 16 91.l odd 6 1
2548.2.bq.e 16 7.c even 3 1
2548.2.bq.e 16 91.k even 6 1
3276.2.cf.c 16 3.b odd 2 1
3276.2.cf.c 16 39.h odd 6 1
4732.2.a.s 8 13.f odd 12 1
4732.2.a.t 8 13.f odd 12 1
4732.2.g.k 16 13.c even 3 1
4732.2.g.k 16 13.e even 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(364, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( T^{16} + 19 T^{14} + \cdots + 2704 \) Copy content Toggle raw display
$5$ \( T^{16} + 62 T^{14} + \cdots + 3721 \) Copy content Toggle raw display
$7$ \( (T^{4} - T^{2} + 1)^{4} \) Copy content Toggle raw display
$11$ \( T^{16} + \cdots + 185722384 \) Copy content Toggle raw display
$13$ \( T^{16} + \cdots + 815730721 \) Copy content Toggle raw display
$17$ \( T^{16} - 2 T^{15} + \cdots + 2982529 \) Copy content Toggle raw display
$19$ \( T^{16} + \cdots + 1235663104 \) Copy content Toggle raw display
$23$ \( T^{16} + 92 T^{14} + \cdots + 88510464 \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots + 530426961 \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots + 1643965037584 \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 592240896 \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 1637621852416 \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 5967799496464 \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 5725040896 \) Copy content Toggle raw display
$53$ \( (T^{8} - 4 T^{7} + \cdots + 117909)^{2} \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 119508624 \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 1654989623296 \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 62066753424 \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 7345574313984 \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 89262317824 \) Copy content Toggle raw display
$79$ \( (T^{8} - 4 T^{7} + \cdots - 1070784)^{2} \) Copy content Toggle raw display
$83$ \( T^{16} + 656 T^{14} + \cdots + 1971216 \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 33199755264 \) Copy content Toggle raw display
$97$ \( T^{16} + 42 T^{15} + \cdots + 44302336 \) Copy content Toggle raw display
show more
show less