↑

Properties

Label 430.2.k.b
Level $430$
Weight $2$
Character orbit 430.k
Analytic conductor $3.434$
Analytic rank $0$
Dimension $18$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [430,2,Mod(11,430)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(430, base_ring=CyclotomicField(14))
 
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("430.11");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 430 = 2 \cdot 5 \cdot 43 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 430.k (of order \(7\), degree \(6\), 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: \(3.43356728692\)
Analytic rank: \(0\)
Dimension: \(18\)
Relative dimension: \(3\) over \(\Q(\zeta_{7})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{18} - 5 x^{17} + 17 x^{16} - 43 x^{15} + 90 x^{14} - 114 x^{13} + 135 x^{12} + 2 x^{11} + 313 x^{10} + \cdots + 169 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{7}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{17}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{3} q^{2} + \beta_{5} q^{3} + \beta_{4} q^{4} - \beta_{10} q^{5} - \beta_{11} q^{6} + (\beta_{13} - \beta_{12} + \cdots - \beta_{2}) q^{7}+ \cdots + ( - \beta_{16} - \beta_{13} + \beta_{6}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{3} q^{2} + \beta_{5} q^{3} + \beta_{4} q^{4} - \beta_{10} q^{5} - \beta_{11} q^{6} + (\beta_{13} - \beta_{12} + \cdots - \beta_{2}) q^{7}+ \cdots + ( - 2 \beta_{17} - 2 \beta_{16} + \cdots + 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q - 3 q^{2} + 2 q^{3} - 3 q^{4} + 3 q^{5} + 2 q^{6} + 6 q^{7} - 3 q^{8} - 7 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 18 q - 3 q^{2} + 2 q^{3} - 3 q^{4} + 3 q^{5} + 2 q^{6} + 6 q^{7} - 3 q^{8} - 7 q^{9} + 3 q^{10} + 9 q^{11} + 2 q^{12} + 4 q^{13} + 13 q^{14} + 5 q^{15} - 3 q^{16} - q^{17} + 7 q^{18} - 2 q^{19} + 3 q^{20} - 2 q^{21} - 12 q^{22} - 19 q^{23} - 5 q^{24} - 3 q^{25} + 4 q^{26} + 17 q^{27} - q^{28} - 4 q^{29} - 2 q^{30} - 13 q^{31} - 3 q^{32} - 42 q^{33} - 8 q^{34} + 15 q^{35} + 2 q^{37} + 12 q^{38} - 39 q^{39} + 3 q^{40} - 4 q^{41} - 2 q^{42} - 13 q^{43} - 12 q^{44} - 7 q^{45} + 16 q^{46} + 16 q^{47} + 2 q^{48} + 4 q^{49} + 18 q^{50} + 7 q^{51} - 10 q^{52} + 14 q^{53} - 4 q^{54} + 12 q^{55} - 15 q^{56} + 45 q^{57} + 24 q^{58} - 16 q^{59} - 2 q^{60} + 24 q^{61} + 50 q^{62} - 33 q^{63} - 3 q^{64} - 4 q^{65} + 5 q^{67} + 13 q^{68} + 4 q^{69} + 15 q^{70} - 3 q^{71} + 7 q^{72} - 22 q^{73} - 12 q^{74} - 5 q^{75} + 12 q^{76} - 24 q^{77} + 31 q^{78} - 38 q^{79} - 18 q^{80} + 22 q^{81} - 11 q^{82} - 26 q^{83} - 2 q^{84} + 8 q^{85} - 41 q^{86} - 22 q^{87} - 12 q^{88} + 52 q^{89} + 7 q^{90} - 14 q^{91} + 2 q^{92} - 6 q^{93} + 2 q^{94} + 2 q^{95} + 2 q^{96} + 27 q^{97} + 32 q^{98} + 36 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{18} - 5 x^{17} + 17 x^{16} - 43 x^{15} + 90 x^{14} - 114 x^{13} + 135 x^{12} + 2 x^{11} + 313 x^{10} + \cdots + 169 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 10\!\cdots\!89 \nu^{17} + \cdots - 19\!\cdots\!71 ) / 38\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 13\!\cdots\!11 \nu^{17} + \cdots + 37\!\cdots\!93 ) / 19\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 52\!\cdots\!21 \nu^{17} + \cdots + 89\!\cdots\!47 ) / 38\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 38\!\cdots\!43 \nu^{17} + \cdots - 23\!\cdots\!59 ) / 19\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 42\!\cdots\!03 \nu^{17} + \cdots + 37\!\cdots\!03 ) / 19\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 95\!\cdots\!49 \nu^{17} + \cdots - 29\!\cdots\!63 ) / 38\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 49\!\cdots\!51 \nu^{17} + \cdots - 92\!\cdots\!73 ) / 19\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 25\!\cdots\!31 \nu^{17} + \cdots + 13\!\cdots\!13 ) / 97\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 11\!\cdots\!59 \nu^{17} + \cdots - 61\!\cdots\!33 ) / 38\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 12\!\cdots\!41 \nu^{17} + \cdots + 88\!\cdots\!49 ) / 38\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 17\!\cdots\!27 \nu^{17} + \cdots - 27\!\cdots\!09 ) / 38\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 19\!\cdots\!97 \nu^{17} + \cdots - 74\!\cdots\!59 ) / 38\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 82\!\cdots\!81 \nu^{17} + \cdots - 36\!\cdots\!01 ) / 97\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 11\!\cdots\!33 \nu^{17} + \cdots + 23\!\cdots\!68 ) / 97\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 30\!\cdots\!61 \nu^{17} + \cdots - 15\!\cdots\!73 ) / 19\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 76\!\cdots\!05 \nu^{17} + \cdots - 16\!\cdots\!63 ) / 38\!\cdots\!84 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{16} + \beta_{15} - \beta_{13} - \beta_{12} + 3 \beta_{10} - \beta_{9} - \beta_{7} - \beta_{5} + \cdots + \beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{17} + \beta_{16} + \beta_{15} - \beta_{14} - \beta_{13} - 3 \beta_{12} + \beta_{11} + 2 \beta_{10} + \cdots + 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 6 \beta_{17} + \beta_{15} - 6 \beta_{14} - 14 \beta_{12} + 7 \beta_{11} - 5 \beta_{9} + \beta_{8} + \cdots + 6 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 8 \beta_{17} + 2 \beta_{15} - 8 \beta_{14} - 6 \beta_{13} - 9 \beta_{12} + 9 \beta_{11} + 8 \beta_{10} + \cdots - 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 2 \beta_{17} - 2 \beta_{16} - 11 \beta_{14} + 2 \beta_{13} + 22 \beta_{11} + 33 \beta_{10} + 3 \beta_{9} + \cdots - 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 57 \beta_{16} - 23 \beta_{15} - 23 \beta_{14} + 86 \beta_{13} + 65 \beta_{12} + 135 \beta_{11} + \cdots - 34 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 28 \beta_{17} - 227 \beta_{16} - 96 \beta_{15} + 265 \beta_{13} + 492 \beta_{12} + 143 \beta_{11} + \cdots - 473 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 398 \beta_{17} - 415 \beta_{16} - 415 \beta_{15} + 197 \beta_{14} + 459 \beta_{13} + 1427 \beta_{12} + \cdots - 1012 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 1485 \beta_{17} - 767 \beta_{16} - 1485 \beta_{15} + 767 \beta_{14} + 1485 \beta_{13} + 2856 \beta_{12} + \cdots - 823 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 2966 \beta_{17} - 1517 \beta_{16} - 2774 \beta_{15} + 2966 \beta_{14} + 2463 \beta_{13} + 6416 \beta_{12} + \cdots - 2966 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 5854 \beta_{17} - 2389 \beta_{15} + 9957 \beta_{14} - 3598 \beta_{13} + 13729 \beta_{12} + \cdots - 8406 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 11127 \beta_{17} + 11127 \beta_{16} + 19379 \beta_{14} - 18169 \beta_{13} - 46663 \beta_{11} + \cdots + 18169 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 43517 \beta_{16} + 19216 \beta_{15} + 19216 \beta_{14} - 43404 \beta_{13} - 93431 \beta_{12} + \cdots + 112521 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 79662 \beta_{17} + 151782 \beta_{16} + 135767 \beta_{15} - 191427 \beta_{13} - 343209 \beta_{12} + \cdots + 227417 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 318365 \beta_{17} + 467946 \beta_{16} + 467946 \beta_{15} - 148473 \beta_{14} - 639933 \beta_{13} + \cdots + 656695 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( 1085848 \beta_{17} + 953557 \beta_{16} + 1085848 \beta_{15} - 953557 \beta_{14} - 1085848 \beta_{13} + \cdots + 2178051 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/430\mathbb{Z}\right)^\times\).

\(n\) \(87\) \(261\)
\(\chi(n)\) \(1\) \(-1 + \beta_{3} - \beta_{4} - \beta_{10} + \beta_{12} + \beta_{13}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
11.1
−0.532432 − 2.33273i
0.163144 + 0.714782i
0.468318 + 2.05184i
−0.767481 + 0.962391i
0.257241 − 0.322570i
1.28772 − 1.61475i
−0.767481 − 0.962391i
0.257241 + 0.322570i
1.28772 + 1.61475i
2.39969 − 1.15563i
0.695279 − 0.334829i
−1.47148 + 0.708626i
2.39969 + 1.15563i
0.695279 + 0.334829i
−1.47148 − 0.708626i
−0.532432 + 2.33273i
0.163144 − 0.714782i
0.468318 − 2.05184i
0.623490 + 0.781831i −1.49184 + 1.87071i −0.222521 + 0.974928i 0.900969 − 0.433884i −2.39273 4.52707 −0.900969 + 0.433884i −0.606400 − 2.65681i 0.900969 + 0.433884i
11.2 0.623490 + 0.781831i 0.457120 − 0.573211i −0.222521 + 0.974928i 0.900969 − 0.433884i 0.733164 −0.660533 −0.900969 + 0.433884i 0.547951 + 2.40073i 0.900969 + 0.433884i
11.3 0.623490 + 0.781831i 1.31220 − 1.64545i −0.222521 + 0.974928i 0.900969 − 0.433884i 2.10460 3.23129 −0.900969 + 0.433884i −0.318062 − 1.39352i 0.900969 + 0.433884i
21.1 −0.900969 − 0.433884i −1.10904 + 0.534087i 0.623490 + 0.781831i 0.222521 + 0.974928i 1.23094 −1.03973 −0.222521 − 0.974928i −0.925743 + 1.16084i 0.222521 − 0.974928i
21.2 −0.900969 − 0.433884i 0.371724 − 0.179013i 0.623490 + 0.781831i 0.222521 + 0.974928i −0.412583 −2.38473 −0.222521 − 0.974928i −1.76434 + 2.21241i 0.222521 − 0.974928i
21.3 −0.900969 − 0.433884i 1.86081 − 0.896118i 0.623490 + 0.781831i 0.222521 + 0.974928i −2.06534 1.71068 −0.222521 − 0.974928i 0.789110 − 0.989513i 0.222521 − 0.974928i
41.1 −0.900969 + 0.433884i −1.10904 − 0.534087i 0.623490 − 0.781831i 0.222521 − 0.974928i 1.23094 −1.03973 −0.222521 + 0.974928i −0.925743 − 1.16084i 0.222521 + 0.974928i
41.2 −0.900969 + 0.433884i 0.371724 + 0.179013i 0.623490 − 0.781831i 0.222521 − 0.974928i −0.412583 −2.38473 −0.222521 + 0.974928i −1.76434 − 2.21241i 0.222521 + 0.974928i
41.3 −0.900969 + 0.433884i 1.86081 + 0.896118i 0.623490 − 0.781831i 0.222521 − 0.974928i −2.06534 1.71068 −0.222521 + 0.974928i 0.789110 + 0.989513i 0.222521 + 0.974928i
121.1 −0.222521 + 0.974928i −0.592674 − 2.59667i −0.900969 − 0.433884i −0.623490 + 0.781831i 2.66345 2.84700 0.623490 − 0.781831i −3.68855 + 1.77631i −0.623490 − 0.781831i
121.2 −0.222521 + 0.974928i −0.171720 − 0.752353i −0.900969 − 0.433884i −0.623490 + 0.781831i 0.771701 −3.65146 0.623490 − 0.781831i 2.16636 − 1.04326i −0.623490 − 0.781831i
121.3 −0.222521 + 0.974928i 0.363425 + 1.59227i −0.900969 − 0.433884i −0.623490 + 0.781831i −1.63322 −1.57959 0.623490 − 0.781831i 0.299667 − 0.144312i −0.623490 − 0.781831i
231.1 −0.222521 − 0.974928i −0.592674 + 2.59667i −0.900969 + 0.433884i −0.623490 − 0.781831i 2.66345 2.84700 0.623490 + 0.781831i −3.68855 − 1.77631i −0.623490 + 0.781831i
231.2 −0.222521 − 0.974928i −0.171720 + 0.752353i −0.900969 + 0.433884i −0.623490 − 0.781831i 0.771701 −3.65146 0.623490 + 0.781831i 2.16636 + 1.04326i −0.623490 + 0.781831i
231.3 −0.222521 − 0.974928i 0.363425 − 1.59227i −0.900969 + 0.433884i −0.623490 − 0.781831i −1.63322 −1.57959 0.623490 + 0.781831i 0.299667 + 0.144312i −0.623490 + 0.781831i
391.1 0.623490 − 0.781831i −1.49184 − 1.87071i −0.222521 − 0.974928i 0.900969 + 0.433884i −2.39273 4.52707 −0.900969 − 0.433884i −0.606400 + 2.65681i 0.900969 − 0.433884i
391.2 0.623490 − 0.781831i 0.457120 + 0.573211i −0.222521 − 0.974928i 0.900969 + 0.433884i 0.733164 −0.660533 −0.900969 − 0.433884i 0.547951 − 2.40073i 0.900969 − 0.433884i
391.3 0.623490 − 0.781831i 1.31220 + 1.64545i −0.222521 − 0.974928i 0.900969 + 0.433884i 2.10460 3.23129 −0.900969 − 0.433884i −0.318062 + 1.39352i 0.900969 − 0.433884i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 11.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
43.e even 7 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 430.2.k.b ✓ 18
43.e even 7 1 inner 430.2.k.b ✓ 18
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
430.2.k.b ✓ 18 1.a even 1 1 trivial
430.2.k.b ✓ 18 43.e even 7 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{18} - 2 T_{3}^{17} + 10 T_{3}^{16} - 27 T_{3}^{15} + 76 T_{3}^{14} - 138 T_{3}^{13} + 345 T_{3}^{12} + \cdots + 169 \) acting on \(S_{2}^{\mathrm{new}}(430, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{6} + T^{5} + T^{4} + \cdots + 1)^{3} \) Copy content Toggle raw display
$3$ \( T^{18} - 2 T^{17} + \cdots + 169 \) Copy content Toggle raw display
$5$ \( (T^{6} - T^{5} + T^{4} + \cdots + 1)^{3} \) Copy content Toggle raw display
$7$ \( (T^{9} - 3 T^{8} + \cdots + 673)^{2} \) Copy content Toggle raw display
$11$ \( T^{18} - 9 T^{17} + \cdots + 28217344 \) Copy content Toggle raw display
$13$ \( T^{18} - 4 T^{17} + \cdots + 31449664 \) Copy content Toggle raw display
$17$ \( T^{18} + \cdots + 114233344 \) Copy content Toggle raw display
$19$ \( T^{18} + 2 T^{17} + \cdots + 29506624 \) Copy content Toggle raw display
$23$ \( T^{18} + \cdots + 1589696641 \) Copy content Toggle raw display
$29$ \( T^{18} + \cdots + 392951329 \) Copy content Toggle raw display
$31$ \( T^{18} + \cdots + 14176068372544 \) Copy content Toggle raw display
$37$ \( (T^{9} - T^{8} + \cdots - 19256)^{2} \) Copy content Toggle raw display
$41$ \( T^{18} + \cdots + 91538146948249 \) Copy content Toggle raw display
$43$ \( T^{18} + \cdots + 502592611936843 \) Copy content Toggle raw display
$47$ \( T^{18} + \cdots + 2735557218304 \) Copy content Toggle raw display
$53$ \( T^{18} + \cdots + 358493629509184 \) Copy content Toggle raw display
$59$ \( T^{18} + \cdots + 99431747584 \) Copy content Toggle raw display
$61$ \( T^{18} + \cdots + 17614863841 \) Copy content Toggle raw display
$67$ \( T^{18} + \cdots + 3333585807721 \) Copy content Toggle raw display
$71$ \( T^{18} + \cdots + 10922390449216 \) Copy content Toggle raw display
$73$ \( T^{18} + 22 T^{17} + \cdots + 19998784 \) Copy content Toggle raw display
$79$ \( (T^{9} + 19 T^{8} + \cdots + 31576)^{2} \) Copy content Toggle raw display
$83$ \( T^{18} + \cdots + 1340452212841 \) Copy content Toggle raw display
$89$ \( T^{18} + \cdots + 26\!\cdots\!01 \) Copy content Toggle raw display
$97$ \( T^{18} + \cdots + 12\!\cdots\!64 \) Copy content Toggle raw display
show more
show less