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Properties

Label 414.2.i.h
Level $414$
Weight $2$
Character orbit 414.i
Analytic conductor $3.306$
Analytic rank $0$
Dimension $20$
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] = [414,2,Mod(55,414)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(414, base_ring=CyclotomicField(22))
 
chi = DirichletCharacter(H, H._module([0, 10]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("414.55");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 414 = 2 \cdot 3^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 414.i (of order \(11\), degree \(10\), 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: \(3.30580664368\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(2\) over \(\Q(\zeta_{11})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} + 8 x^{18} + 53 x^{16} + 358 x^{14} + 1753 x^{12} + 7149 x^{10} + 23268 x^{8} + 37292 x^{6} + \cdots + 58081 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{11}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{3} q^{2} + \beta_{7} q^{4} + ( - \beta_{13} + \beta_{5} - \beta_{2}) q^{5} + ( - \beta_{17} + \beta_{16} + \cdots + \beta_{8}) q^{7}+ \cdots + ( - \beta_{15} + \beta_{14} - \beta_{12} + \cdots + 1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{3} q^{2} + \beta_{7} q^{4} + ( - \beta_{13} + \beta_{5} - \beta_{2}) q^{5} + ( - \beta_{17} + \beta_{16} + \cdots + \beta_{8}) q^{7}+ \cdots + ( - 2 \beta_{19} + 2 \beta_{17} + \cdots - 2) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 2 q^{2} - 2 q^{4} + 2 q^{5} - 4 q^{7} + 2 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 20 q + 2 q^{2} - 2 q^{4} + 2 q^{5} - 4 q^{7} + 2 q^{8} - 2 q^{10} - 2 q^{11} - 18 q^{14} - 2 q^{16} + 18 q^{17} + 16 q^{19} + 2 q^{20} + 24 q^{22} - 2 q^{23} + 38 q^{25} + 18 q^{28} - 30 q^{29} + 14 q^{31} + 2 q^{32} + 4 q^{34} + 48 q^{35} - 20 q^{37} - 16 q^{38} - 2 q^{40} - 12 q^{41} - 28 q^{43} - 2 q^{44} + 2 q^{46} + 32 q^{47} + 6 q^{49} + 6 q^{50} - 46 q^{53} - 28 q^{55} + 4 q^{56} - 14 q^{58} - 50 q^{61} + 8 q^{62} - 2 q^{64} + 16 q^{65} - 8 q^{67} - 48 q^{68} - 48 q^{70} + 12 q^{71} - 18 q^{73} + 20 q^{74} - 6 q^{76} - 4 q^{77} - 18 q^{79} + 2 q^{80} - 10 q^{82} - 44 q^{83} + 32 q^{85} + 28 q^{86} + 2 q^{88} + 44 q^{91} - 2 q^{92} + 12 q^{94} + 64 q^{95} + 14 q^{97} - 28 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} + 8 x^{18} + 53 x^{16} + 358 x^{14} + 1753 x^{12} + 7149 x^{10} + 23268 x^{8} + 37292 x^{6} + \cdots + 58081 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 62\!\cdots\!54 \nu^{19} + \cdots + 22\!\cdots\!34 \nu ) / 99\!\cdots\!17 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 26\!\cdots\!65 \nu^{18} + \cdots + 27\!\cdots\!88 ) / 99\!\cdots\!17 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 26\!\cdots\!65 \nu^{19} + \cdots + 27\!\cdots\!88 \nu ) / 99\!\cdots\!17 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 51\!\cdots\!23 \nu^{18} + \cdots + 62\!\cdots\!97 ) / 99\!\cdots\!17 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 77\!\cdots\!69 \nu^{18} + \cdots + 96\!\cdots\!44 ) / 99\!\cdots\!17 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 77\!\cdots\!86 \nu^{18} + \cdots - 23\!\cdots\!60 ) / 99\!\cdots\!17 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 77\!\cdots\!86 \nu^{19} + \cdots + 23\!\cdots\!60 \nu ) / 99\!\cdots\!17 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 10\!\cdots\!45 \nu^{18} + \cdots + 93\!\cdots\!83 ) / 99\!\cdots\!17 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 10\!\cdots\!45 \nu^{19} + \cdots - 93\!\cdots\!83 \nu ) / 99\!\cdots\!17 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 11\!\cdots\!14 \nu^{18} + \cdots - 11\!\cdots\!82 ) / 99\!\cdots\!17 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 11\!\cdots\!58 \nu^{18} + \cdots - 63\!\cdots\!04 ) / 99\!\cdots\!17 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 11\!\cdots\!58 \nu^{19} + \cdots + 63\!\cdots\!04 \nu ) / 99\!\cdots\!17 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 12\!\cdots\!94 \nu^{18} + \cdots - 15\!\cdots\!15 ) / 99\!\cdots\!17 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 18\!\cdots\!30 \nu^{18} + \cdots - 10\!\cdots\!54 ) / 99\!\cdots\!17 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 37\!\cdots\!68 \nu^{19} + \cdots - 38\!\cdots\!46 \nu ) / 99\!\cdots\!17 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 39\!\cdots\!35 \nu^{19} + \cdots + 83\!\cdots\!85 \nu ) / 99\!\cdots\!17 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( - 47\!\cdots\!04 \nu^{19} + \cdots + 18\!\cdots\!29 \nu ) / 99\!\cdots\!17 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( 52\!\cdots\!27 \nu^{19} + \cdots - 34\!\cdots\!43 \nu ) / 99\!\cdots\!17 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{14} - \beta_{12} - \beta_{6} - \beta_{5} + 4\beta_{3} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{17} + \beta_{16} + \beta_{13} + 5\beta_{4} + \beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2\beta_{15} + 5\beta_{14} - 5\beta_{12} - \beta_{11} + 5\beta_{9} + 17\beta_{7} + 2\beta_{6} - 5\beta_{5} - 1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 7\beta_{18} - \beta_{17} + 8\beta_{16} + 8\beta_{13} - 7\beta_{10} - 17\beta_{8} + 8\beta_{4} + \beta_{2} + 7\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 54 \beta_{15} - 54 \beta_{14} + 52 \beta_{12} - 75 \beta_{11} - 52 \beta_{9} - 75 \beta_{7} + 85 \beta_{6} + \cdots - 94 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 40 \beta_{19} - 9 \beta_{18} - 10 \beta_{17} + 75 \beta_{16} + 77 \beta_{13} - 2 \beta_{10} + \cdots - 59 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 109 \beta_{15} - 132 \beta_{14} - 82 \beta_{12} + 340 \beta_{11} + 109 \beta_{9} - 73 \beta_{7} + \cdots + 82 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 59 \beta_{19} - 73 \beta_{18} - 340 \beta_{17} - 581 \beta_{16} - 367 \beta_{13} + 73 \beta_{8} + \cdots - 440 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 1009 \beta_{15} + 345 \beta_{12} + 474 \beta_{11} + 769 \beta_{9} - 345 \beta_{7} - 2048 \beta_{6} + \cdots + 1009 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 474 \beta_{19} - 1574 \beta_{18} + 1574 \beta_{17} - 1483 \beta_{16} - 1828 \beta_{13} + \cdots - 474 \beta_{2} \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 5735 \beta_{15} + 4820 \beta_{14} - 1915 \beta_{12} + 1915 \beta_{11} + 5735 \beta_{7} - 4820 \beta_{6} + \cdots + 7514 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 7410 \beta_{19} + 7410 \beta_{18} + 2905 \beta_{17} - 2830 \beta_{16} - 5735 \beta_{13} + \cdots + 12094 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 29435 \beta_{15} + 33513 \beta_{14} + 1862 \beta_{12} - 29435 \beta_{11} - 19081 \beta_{9} + \cdots + 11936 \beta_{3} \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 35375 \beta_{19} + 17219 \beta_{18} + 10354 \beta_{17} + 92383 \beta_{16} + 57008 \beta_{13} + \cdots + 57008 \beta_1 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 171356 \beta_{15} - 71463 \beta_{14} - 99472 \beta_{12} - 75317 \beta_{11} - 71463 \beta_{9} + \cdots - 226570 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( - 99893 \beta_{19} + 55214 \beta_{18} - 151253 \beta_{17} + 175210 \beta_{16} + 346145 \beta_{13} + \cdots - 151253 \beta_1 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( 372929 \beta_{15} - 943857 \beta_{14} + 372929 \beta_{12} + 462064 \beta_{9} - 456375 \beta_{7} + \cdots - 943857 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( - 292319 \beta_{19} - 779801 \beta_{18} - 456375 \beta_{17} - 570928 \beta_{16} - 834993 \beta_{10} + \cdots - 1807104 \beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/414\mathbb{Z}\right)^\times\).

\(n\) \(47\) \(235\)
\(\chi(n)\) \(1\) \(\beta_{3}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
55.1
−0.303441 − 2.11048i
0.303441 + 2.11048i
−1.95451 − 1.25609i
1.95451 + 1.25609i
0.842658 − 0.247427i
−0.842658 + 0.247427i
0.842658 + 0.247427i
−0.842658 − 0.247427i
−0.303441 + 2.11048i
0.303441 − 2.11048i
0.708869 − 1.55221i
−0.708869 + 1.55221i
−1.36939 − 1.58036i
1.36939 + 1.58036i
−1.36939 + 1.58036i
1.36939 − 1.58036i
0.708869 + 1.55221i
−0.708869 − 1.55221i
−1.95451 + 1.25609i
1.95451 − 1.25609i
0.959493 − 0.281733i 0 0.841254 − 0.540641i −0.0234248 − 0.162923i 0 1.32938 − 2.91093i 0.654861 − 0.755750i 0 −0.0683767 − 0.149724i
55.2 0.959493 − 0.281733i 0 0.841254 − 0.540641i 0.308054 + 2.14257i 0 −1.56585 + 3.42874i 0.654861 − 0.755750i 0 0.899207 + 1.96899i
73.1 −0.415415 − 0.909632i 0 −0.654861 + 0.755750i −3.57342 − 2.29650i 0 0.0421569 + 0.293207i 0.959493 + 0.281733i 0 −0.604516 + 4.20450i
73.2 −0.415415 − 0.909632i 0 −0.654861 + 0.755750i 1.89092 + 1.21522i 0 −0.521048 − 3.62397i 0.959493 + 0.281733i 0 0.319886 − 2.22486i
127.1 −0.841254 + 0.540641i 0 0.415415 − 0.909632i −1.76120 + 0.517136i 0 −0.814622 − 0.940124i 0.142315 + 0.989821i 0 1.20203 − 1.38722i
127.2 −0.841254 + 0.540641i 0 0.415415 − 0.909632i 3.68019 − 1.08060i 0 3.32796 + 3.84067i 0.142315 + 0.989821i 0 −2.51176 + 2.89872i
163.1 −0.841254 − 0.540641i 0 0.415415 + 0.909632i −1.76120 − 0.517136i 0 −0.814622 + 0.940124i 0.142315 − 0.989821i 0 1.20203 + 1.38722i
163.2 −0.841254 − 0.540641i 0 0.415415 + 0.909632i 3.68019 + 1.08060i 0 3.32796 − 3.84067i 0.142315 − 0.989821i 0 −2.51176 − 2.89872i
271.1 0.959493 + 0.281733i 0 0.841254 + 0.540641i −0.0234248 + 0.162923i 0 1.32938 + 2.91093i 0.654861 + 0.755750i 0 −0.0683767 + 0.149724i
271.2 0.959493 + 0.281733i 0 0.841254 + 0.540641i 0.308054 − 2.14257i 0 −1.56585 − 3.42874i 0.654861 + 0.755750i 0 0.899207 − 1.96899i
289.1 0.654861 + 0.755750i 0 −0.142315 + 0.989821i −1.18678 + 2.59868i 0 −4.30188 − 1.26315i −0.841254 + 0.540641i 0 −2.74112 + 0.804866i
289.2 0.654861 + 0.755750i 0 −0.142315 + 0.989821i 0.355945 − 0.779412i 0 2.70753 + 0.795003i −0.841254 + 0.540641i 0 0.822135 − 0.241401i
307.1 0.142315 − 0.989821i 0 −0.959493 − 0.281733i 0.144371 + 0.166613i 0 −2.90511 − 1.86700i −0.415415 + 0.909632i 0 0.185463 − 0.119190i
307.2 0.142315 − 0.989821i 0 −0.959493 − 0.281733i 1.16535 + 1.34489i 0 0.701495 + 0.450824i −0.415415 + 0.909632i 0 1.49704 − 0.962092i
325.1 0.142315 + 0.989821i 0 −0.959493 + 0.281733i 0.144371 − 0.166613i 0 −2.90511 + 1.86700i −0.415415 − 0.909632i 0 0.185463 + 0.119190i
325.2 0.142315 + 0.989821i 0 −0.959493 + 0.281733i 1.16535 − 1.34489i 0 0.701495 − 0.450824i −0.415415 − 0.909632i 0 1.49704 + 0.962092i
361.1 0.654861 − 0.755750i 0 −0.142315 − 0.989821i −1.18678 − 2.59868i 0 −4.30188 + 1.26315i −0.841254 − 0.540641i 0 −2.74112 − 0.804866i
361.2 0.654861 − 0.755750i 0 −0.142315 − 0.989821i 0.355945 + 0.779412i 0 2.70753 − 0.795003i −0.841254 − 0.540641i 0 0.822135 + 0.241401i
397.1 −0.415415 + 0.909632i 0 −0.654861 − 0.755750i −3.57342 + 2.29650i 0 0.0421569 − 0.293207i 0.959493 − 0.281733i 0 −0.604516 − 4.20450i
397.2 −0.415415 + 0.909632i 0 −0.654861 − 0.755750i 1.89092 − 1.21522i 0 −0.521048 + 3.62397i 0.959493 − 0.281733i 0 0.319886 + 2.22486i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 55.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
23.c even 11 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 414.2.i.h yes 20
3.b odd 2 1 414.2.i.g ✓ 20
23.c even 11 1 inner 414.2.i.h yes 20
23.c even 11 1 9522.2.a.cg 10
23.d odd 22 1 9522.2.a.ch 10
69.g even 22 1 9522.2.a.ci 10
69.h odd 22 1 414.2.i.g ✓ 20
69.h odd 22 1 9522.2.a.cj 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
414.2.i.g ✓ 20 3.b odd 2 1
414.2.i.g ✓ 20 69.h odd 22 1
414.2.i.h yes 20 1.a even 1 1 trivial
414.2.i.h yes 20 23.c even 11 1 inner
9522.2.a.cg 10 23.c even 11 1
9522.2.a.ch 10 23.d odd 22 1
9522.2.a.ci 10 69.g even 22 1
9522.2.a.cj 10 69.h odd 22 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{20} - 2 T_{5}^{19} - 12 T_{5}^{18} - 10 T_{5}^{17} + 212 T_{5}^{16} - 374 T_{5}^{15} + 1547 T_{5}^{14} + \cdots + 529 \) acting on \(S_{2}^{\mathrm{new}}(414, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{10} - T^{9} + T^{8} + \cdots + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{20} \) Copy content Toggle raw display
$5$ \( T^{20} - 2 T^{19} + \cdots + 529 \) Copy content Toggle raw display
$7$ \( T^{20} + 4 T^{19} + \cdots + 9078169 \) Copy content Toggle raw display
$11$ \( T^{20} + 2 T^{19} + \cdots + 1 \) Copy content Toggle raw display
$13$ \( T^{20} + \cdots + 454414489 \) Copy content Toggle raw display
$17$ \( T^{20} - 18 T^{19} + \cdots + 6285049 \) Copy content Toggle raw display
$19$ \( T^{20} + \cdots + 2091873169 \) Copy content Toggle raw display
$23$ \( T^{20} + \cdots + 41426511213649 \) Copy content Toggle raw display
$29$ \( T^{20} + \cdots + 17610086209 \) Copy content Toggle raw display
$31$ \( T^{20} + \cdots + 8128284649 \) Copy content Toggle raw display
$37$ \( T^{20} + \cdots + 745344601 \) Copy content Toggle raw display
$41$ \( T^{20} + \cdots + 7825374027769 \) Copy content Toggle raw display
$43$ \( T^{20} + \cdots + 489903404761 \) Copy content Toggle raw display
$47$ \( (T^{10} - 16 T^{9} + \cdots + 1948123)^{2} \) Copy content Toggle raw display
$53$ \( T^{20} + \cdots + 9306653841 \) Copy content Toggle raw display
$59$ \( T^{20} + \cdots + 22\!\cdots\!49 \) Copy content Toggle raw display
$61$ \( T^{20} + \cdots + 79\!\cdots\!41 \) Copy content Toggle raw display
$67$ \( T^{20} + \cdots + 36487869789361 \) Copy content Toggle raw display
$71$ \( T^{20} + \cdots + 71\!\cdots\!89 \) Copy content Toggle raw display
$73$ \( T^{20} + \cdots + 452068349086201 \) Copy content Toggle raw display
$79$ \( T^{20} + \cdots + 14\!\cdots\!29 \) Copy content Toggle raw display
$83$ \( T^{20} + \cdots + 30\!\cdots\!41 \) Copy content Toggle raw display
$89$ \( T^{20} + \cdots + 77\!\cdots\!49 \) Copy content Toggle raw display
$97$ \( T^{20} + \cdots + 82\!\cdots\!09 \) Copy content Toggle raw display
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