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Properties

Label 3234.2.e.a
Level $3234$
Weight $2$
Character orbit 3234.e
Analytic conductor $25.824$
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] = [3234,2,Mod(2155,3234)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3234, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3234.2155");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3234 = 2 \cdot 3 \cdot 7^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3234.e (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: \(25.8236200137\)
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: \( x^{16} - 8 x^{15} + 74 x^{14} - 378 x^{13} + 1878 x^{12} - 6718 x^{11} + 22086 x^{10} - 56904 x^{9} + \cdots + 13417 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: no (minimal twist has level 462)
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.

\(f(q)\) \(=\) \( q - \beta_{12} q^{2} - \beta_{12} q^{3} - q^{4} + (\beta_{13} - \beta_{11}) q^{5} - q^{6} + \beta_{12} q^{8} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{12} q^{2} - \beta_{12} q^{3} - q^{4} + (\beta_{13} - \beta_{11}) q^{5} - q^{6} + \beta_{12} q^{8} - q^{9} - \beta_{3} q^{10} + ( - \beta_{14} + \beta_{2}) q^{11} + \beta_{12} q^{12} + (\beta_{4} - \beta_{3} - \beta_{2} - \beta_1) q^{13} - \beta_{3} q^{15} + q^{16} + ( - \beta_{5} + \beta_{3} + \beta_{2}) q^{17} + \beta_{12} q^{18} + (\beta_{10} - \beta_{9} - \beta_{4} + 1) q^{19} + ( - \beta_{13} + \beta_{11}) q^{20} + (\beta_{8} + \beta_{6}) q^{22} + (\beta_{10} - \beta_{9} + \beta_{5} + \cdots + \beta_1) q^{23}+ \cdots + (\beta_{14} - \beta_{2}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 16 q^{4} - 16 q^{6} - 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 16 q^{4} - 16 q^{6} - 16 q^{9} - 4 q^{10} + 8 q^{11} - 4 q^{15} + 16 q^{16} + 20 q^{19} + 2 q^{22} + 8 q^{23} + 16 q^{24} - 20 q^{25} + 2 q^{33} + 16 q^{36} - 28 q^{37} + 4 q^{40} + 32 q^{41} - 8 q^{44} + 16 q^{54} - 14 q^{55} + 4 q^{60} - 56 q^{61} - 8 q^{62} - 16 q^{64} - 8 q^{66} + 32 q^{67} - 56 q^{71} + 88 q^{73} - 20 q^{76} + 16 q^{81} + 8 q^{83} + 24 q^{86} - 2 q^{88} + 4 q^{90} - 8 q^{92} - 8 q^{93} - 28 q^{94} - 16 q^{96} - 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 8 x^{15} + 74 x^{14} - 378 x^{13} + 1878 x^{12} - 6718 x^{11} + 22086 x^{10} - 56904 x^{9} + \cdots + 13417 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 6983 \nu^{14} - 48881 \nu^{13} + 474100 \nu^{12} - 2209147 \nu^{11} + 11140707 \nu^{10} + \cdots + 32362633 ) / 22807968 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 3325577 \nu^{14} + 23279039 \nu^{13} - 218220364 \nu^{12} + 1006694677 \nu^{11} + \cdots - 13304183911 ) / 3261539424 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 189803 \nu^{14} + 1328621 \nu^{13} - 12040024 \nu^{12} + 54968071 \nu^{11} + \cdots - 1021023169 ) / 148251792 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 1252325 \nu^{14} + 8766275 \nu^{13} - 81315274 \nu^{12} + 373930069 \nu^{11} + \cdots - 14998863325 ) / 815384856 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 250035 \nu^{14} + 1750245 \nu^{13} - 16051742 \nu^{12} + 73557267 \nu^{11} - 351447497 \nu^{10} + \cdots + 128107011 ) / 135897476 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 31288770 \nu^{15} - 16748485 \nu^{14} + 797619205 \nu^{13} + 2631204646 \nu^{12} + \cdots - 181576943117 ) / 158184662064 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 31288770 \nu^{15} + 452583065 \nu^{14} - 3848461265 \nu^{13} + 25712532526 \nu^{12} + \cdots + 4642794315301 ) / 158184662064 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 61841878 \nu^{15} + 463814085 \nu^{14} - 4332228469 \nu^{13} + 21124971426 \nu^{12} + \cdots - 306749906363 ) / 158184662064 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 124647224 \nu^{15} - 792790405 \nu^{14} + 7684579947 \nu^{13} - 32966828504 \nu^{12} + \cdots - 1247802075679 ) / 158184662064 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 124647224 \nu^{15} - 1076917955 \nu^{14} + 9673472797 \nu^{13} - 51503270872 \nu^{12} + \cdots - 4344274484769 ) / 158184662064 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 15958 \nu^{15} - 119685 \nu^{14} + 1130119 \nu^{13} - 5530551 \nu^{12} + 27795985 \nu^{11} + \cdots - 518381770 ) / 18627492 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 13342 \nu^{15} + 100065 \nu^{14} - 928497 \nu^{13} + 4517578 \nu^{12} - 22245945 \nu^{11} + \cdots + 261797265 ) / 7787936 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 15958 \nu^{15} - 119685 \nu^{14} + 1130119 \nu^{13} - 5530551 \nu^{12} + 27795985 \nu^{11} + \cdots - 509068024 ) / 9313746 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 575875964 \nu^{15} - 4436099745 \nu^{14} + 41319402047 \nu^{13} - 205410283332 \nu^{12} + \cdots - 16963072781711 ) / 158184662064 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 158954387 \nu^{15} - 1172652900 \nu^{14} + 11041151095 \nu^{13} - 53296410441 \nu^{12} + \cdots - 3970555117103 ) / 26364110344 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{13} + 2\beta_{11} + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - \beta_{13} + 2 \beta_{11} + 2 \beta_{10} - 2 \beta_{9} - 2 \beta_{7} - 2 \beta_{6} + 2 \beta_{5} + \cdots - 9 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 2 \beta_{15} + \beta_{14} + 9 \beta_{13} - 2 \beta_{12} - 17 \beta_{11} + 3 \beta_{10} - 2 \beta_{8} + \cdots - 14 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 4 \beta_{15} + 2 \beta_{14} + 19 \beta_{13} - 4 \beta_{12} - 36 \beta_{11} - 22 \beta_{10} + \cdots + 71 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 19 \beta_{15} + 3 \beta_{14} - 83 \beta_{13} + 34 \beta_{12} + 145 \beta_{11} - 69 \beta_{10} + \cdots + 201 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 67 \beta_{15} + 4 \beta_{14} - 297 \beta_{13} + 112 \beta_{12} + 526 \beta_{11} + 176 \beta_{10} + \cdots - 586 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 150 \beta_{15} - 121 \beta_{14} + 707 \beta_{13} - 360 \beta_{12} - 1095 \beta_{11} + 1072 \beta_{10} + \cdots - 2771 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 922 \beta_{15} - 498 \beta_{14} + 4259 \beta_{13} - 1972 \beta_{12} - 6920 \beta_{11} - 826 \beta_{10} + \cdots + 3975 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 927 \beta_{15} + 1353 \beta_{14} - 4574 \beta_{13} + 2602 \beta_{12} + 5587 \beta_{11} - 14119 \beta_{10} + \cdots + 35386 ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 12039 \beta_{15} + 10518 \beta_{14} - 56988 \beta_{13} + 28604 \beta_{12} + 83700 \beta_{11} + \cdots - 11089 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 50 \beta_{15} - 6198 \beta_{14} + 796 \beta_{13} - 3010 \beta_{12} + 18558 \beta_{11} + 166047 \beta_{10} + \cdots - 418131 ) / 2 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 74741 \beta_{15} - 80579 \beta_{14} + 354385 \beta_{13} - 183886 \beta_{12} - 467854 \beta_{11} + \cdots - 141469 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 144088 \beta_{15} - 108010 \beta_{14} + 697161 \beta_{13} - 343180 \beta_{12} - 1133136 \beta_{11} + \cdots + 4562117 ) / 2 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 1733162 \beta_{15} + 2074966 \beta_{14} - 8106763 \beta_{13} + 4282548 \beta_{12} + 9602618 \beta_{11} + \cdots + 7785995 ) / 2 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 3522094 \beta_{15} + 3795721 \beta_{14} - 16818405 \beta_{13} + 8683710 \beta_{12} + 22482349 \beta_{11} + \cdots - 45325614 ) / 2 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3234\mathbb{Z}\right)^\times\).

\(n\) \(199\) \(1079\) \(2059\)
\(\chi(n)\) \(-1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2155.1
0.500000 + 3.43554i
0.500000 + 3.19339i
0.500000 + 0.921602i
0.500000 − 0.0286340i
0.500000 + 1.56688i
0.500000 − 1.35798i
0.500000 − 3.32851i
0.500000 − 2.40229i
0.500000 + 2.40229i
0.500000 + 3.32851i
0.500000 + 1.35798i
0.500000 − 1.56688i
0.500000 + 0.0286340i
0.500000 − 0.921602i
0.500000 − 3.19339i
0.500000 − 3.43554i
− 1.00000i − 1.00000i −1.00000 − 4.30156i −1.00000 0 1.00000i −1.00000 −4.30156
2155.2 − 1.00000i − 1.00000i −1.00000 − 2.32736i −1.00000 0 1.00000i −1.00000 −2.32736
2155.3 − 1.00000i − 1.00000i −1.00000 − 1.78763i −1.00000 0 1.00000i −1.00000 −1.78763
2155.4 − 1.00000i − 1.00000i −1.00000 − 0.837391i −1.00000 0 1.00000i −1.00000 −0.837391
2155.5 − 1.00000i − 1.00000i −1.00000 − 0.700858i −1.00000 0 1.00000i −1.00000 −0.700858
2155.6 − 1.00000i − 1.00000i −1.00000 2.22400i −1.00000 0 1.00000i −1.00000 2.22400
2155.7 − 1.00000i − 1.00000i −1.00000 2.46248i −1.00000 0 1.00000i −1.00000 2.46248
2155.8 − 1.00000i − 1.00000i −1.00000 3.26832i −1.00000 0 1.00000i −1.00000 3.26832
2155.9 1.00000i 1.00000i −1.00000 − 3.26832i −1.00000 0 − 1.00000i −1.00000 3.26832
2155.10 1.00000i 1.00000i −1.00000 − 2.46248i −1.00000 0 − 1.00000i −1.00000 2.46248
2155.11 1.00000i 1.00000i −1.00000 − 2.22400i −1.00000 0 − 1.00000i −1.00000 2.22400
2155.12 1.00000i 1.00000i −1.00000 0.700858i −1.00000 0 − 1.00000i −1.00000 −0.700858
2155.13 1.00000i 1.00000i −1.00000 0.837391i −1.00000 0 − 1.00000i −1.00000 −0.837391
2155.14 1.00000i 1.00000i −1.00000 1.78763i −1.00000 0 − 1.00000i −1.00000 −1.78763
2155.15 1.00000i 1.00000i −1.00000 2.32736i −1.00000 0 − 1.00000i −1.00000 −2.32736
2155.16 1.00000i 1.00000i −1.00000 4.30156i −1.00000 0 − 1.00000i −1.00000 −4.30156
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2155.16
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3234.2.e.a 16
7.b odd 2 1 3234.2.e.b 16
7.c even 3 1 462.2.p.a ✓ 16
7.d odd 6 1 462.2.p.b yes 16
11.b odd 2 1 3234.2.e.b 16
21.g even 6 1 1386.2.bk.b 16
21.h odd 6 1 1386.2.bk.a 16
77.b even 2 1 inner 3234.2.e.a 16
77.h odd 6 1 462.2.p.b yes 16
77.i even 6 1 462.2.p.a ✓ 16
231.k odd 6 1 1386.2.bk.a 16
231.l even 6 1 1386.2.bk.b 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
462.2.p.a ✓ 16 7.c even 3 1
462.2.p.a ✓ 16 77.i even 6 1
462.2.p.b yes 16 7.d odd 6 1
462.2.p.b yes 16 77.h odd 6 1
1386.2.bk.a 16 21.h odd 6 1
1386.2.bk.a 16 231.k odd 6 1
1386.2.bk.b 16 21.g even 6 1
1386.2.bk.b 16 231.l even 6 1
3234.2.e.a 16 1.a even 1 1 trivial
3234.2.e.a 16 77.b even 2 1 inner
3234.2.e.b 16 7.b odd 2 1
3234.2.e.b 16 11.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(3234, [\chi])\):

\( T_{5}^{16} + 50 T_{5}^{14} + 971 T_{5}^{12} + 9580 T_{5}^{10} + 52131 T_{5}^{8} + 156530 T_{5}^{6} + \cdots + 35344 \) Copy content Toggle raw display
\( T_{13}^{8} - 64T_{13}^{6} - 8T_{13}^{5} + 836T_{13}^{4} + 1168T_{13}^{3} - 592T_{13}^{2} - 1216T_{13} - 128 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 1)^{8} \) Copy content Toggle raw display
$3$ \( (T^{2} + 1)^{8} \) Copy content Toggle raw display
$5$ \( T^{16} + 50 T^{14} + \cdots + 35344 \) Copy content Toggle raw display
$7$ \( T^{16} \) Copy content Toggle raw display
$11$ \( T^{16} + \cdots + 214358881 \) Copy content Toggle raw display
$13$ \( (T^{8} - 64 T^{6} + \cdots - 128)^{2} \) Copy content Toggle raw display
$17$ \( (T^{8} - 53 T^{6} + \cdots + 24)^{2} \) Copy content Toggle raw display
$19$ \( (T^{8} - 10 T^{7} + \cdots - 1664)^{2} \) Copy content Toggle raw display
$23$ \( (T^{8} - 4 T^{7} + \cdots - 155048)^{2} \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots + 12810617856 \) Copy content Toggle raw display
$31$ \( T^{16} + 110 T^{14} + \cdots + 262144 \) Copy content Toggle raw display
$37$ \( (T^{8} + 14 T^{7} + \cdots - 354432)^{2} \) Copy content Toggle raw display
$41$ \( (T^{8} - 16 T^{7} + \cdots + 256)^{2} \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 23084548096 \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 1344737098384 \) Copy content Toggle raw display
$53$ \( (T^{8} - 235 T^{6} + \cdots - 162752)^{2} \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 109280491776 \) Copy content Toggle raw display
$61$ \( (T^{8} + 28 T^{7} + \cdots - 59072)^{2} \) Copy content Toggle raw display
$67$ \( (T^{8} - 16 T^{7} + \cdots + 49024)^{2} \) Copy content Toggle raw display
$71$ \( (T^{8} + 28 T^{7} + \cdots + 4193232)^{2} \) Copy content Toggle raw display
$73$ \( (T^{8} - 44 T^{7} + \cdots - 346112)^{2} \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 1130708969104 \) Copy content Toggle raw display
$83$ \( (T^{8} - 4 T^{7} + \cdots + 5604)^{2} \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 96546188427264 \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 68597371921 \) Copy content Toggle raw display
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