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Properties

Label 430.2.b.b
Level $430$
Weight $2$
Character orbit 430.b
Analytic conductor $3.434$
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] = [430,2,Mod(259,430)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(430, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("430.259");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 430 = 2 \cdot 5 \cdot 43 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 430.b (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: \(3.43356728692\)
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: \( x^{16} + 38x^{14} + 525x^{12} + 3518x^{10} + 12216x^{8} + 20990x^{6} + 15229x^{4} + 4754x^{2} + 529 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q - \beta_{7} q^{2} + (\beta_{7} + \beta_1) q^{3} - q^{4} - \beta_{5} q^{5} + \beta_{4} q^{6} + \beta_{8} q^{7} + \beta_{7} q^{8} + ( - \beta_{6} + \beta_{5} - \beta_{4} + \cdots - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{7} q^{2} + (\beta_{7} + \beta_1) q^{3} - q^{4} - \beta_{5} q^{5} + \beta_{4} q^{6} + \beta_{8} q^{7} + \beta_{7} q^{8} + ( - \beta_{6} + \beta_{5} - \beta_{4} + \cdots - 1) q^{9}+ \cdots + (2 \beta_{15} - 2 \beta_{14} + \cdots + 4) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 16 q^{4} + 2 q^{5} + 8 q^{6} - 28 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 16 q^{4} + 2 q^{5} + 8 q^{6} - 28 q^{9} + 4 q^{11} - 6 q^{14} - 4 q^{15} + 16 q^{16} - 30 q^{19} - 2 q^{20} + 32 q^{21} - 8 q^{24} - 10 q^{25} - 6 q^{26} + 6 q^{29} - 12 q^{30} + 50 q^{31} - 36 q^{35} + 28 q^{36} - 4 q^{39} + 38 q^{41} - 4 q^{44} - 50 q^{45} + 24 q^{46} - 38 q^{49} - 8 q^{50} + 8 q^{51} - 20 q^{54} - 28 q^{55} + 6 q^{56} + 24 q^{59} + 4 q^{60} + 58 q^{61} - 16 q^{64} - 32 q^{65} + 36 q^{66} - 4 q^{69} - 22 q^{70} + 24 q^{71} + 4 q^{74} - 36 q^{75} + 30 q^{76} - 10 q^{79} + 2 q^{80} + 80 q^{81} - 32 q^{84} - 56 q^{85} + 16 q^{86} + 40 q^{89} - 22 q^{90} + 46 q^{91} - 12 q^{94} - 52 q^{95} + 8 q^{96} + 32 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 38x^{14} + 525x^{12} + 3518x^{10} + 12216x^{8} + 20990x^{6} + 15229x^{4} + 4754x^{2} + 529 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{14} + 25\nu^{12} + 95\nu^{10} - 1182\nu^{8} - 10113\nu^{6} - 24981\nu^{4} - 17383\nu^{2} - 3262 ) / 100 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{14} + 45\nu^{12} + 750\nu^{10} + 5838\nu^{8} + 22052\nu^{6} + 36294\nu^{4} + 16597\nu^{2} + 2023 ) / 100 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -2\nu^{14} - 70\nu^{12} - 850\nu^{10} - 4811\nu^{8} - 13439\nu^{6} - 17188\nu^{4} - 7809\nu^{2} - 1031 ) / 50 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 102 \nu^{15} + 69 \nu^{14} + 3830 \nu^{13} + 2645 \nu^{12} + 51940 \nu^{11} + 36915 \nu^{10} + \cdots + 133952 ) / 4600 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 102 \nu^{15} - 69 \nu^{14} + 3830 \nu^{13} - 2645 \nu^{12} + 51940 \nu^{11} - 36915 \nu^{10} + \cdots - 133952 ) / 4600 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 47 \nu^{15} - 1740 \nu^{13} - 23065 \nu^{11} - 145796 \nu^{9} - 463499 \nu^{7} - 677433 \nu^{5} + \cdots - 43831 \nu ) / 1150 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 109 \nu^{15} - 4165 \nu^{13} - 57800 \nu^{11} - 385532 \nu^{9} - 1301368 \nu^{7} + \cdots - 174267 \nu ) / 2300 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 18 \nu^{15} + 276 \nu^{14} + 845 \nu^{13} + 10810 \nu^{12} + 14625 \nu^{11} + 154790 \nu^{10} + \cdots + 484518 ) / 4600 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 84 \nu^{15} + 391 \nu^{14} + 2985 \nu^{13} + 14605 \nu^{12} + 37315 \nu^{11} + 196075 \nu^{10} + \cdots + 468418 ) / 4600 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 131 \nu^{15} - 368 \nu^{14} + 4840 \nu^{13} - 13570 \nu^{12} + 64060 \nu^{11} - 178940 \nu^{10} + \cdots - 388424 ) / 4600 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 131 \nu^{15} + 368 \nu^{14} + 4840 \nu^{13} + 13570 \nu^{12} + 64060 \nu^{11} + 178940 \nu^{10} + \cdots + 388424 ) / 4600 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 135 \nu^{15} - 437 \nu^{14} - 4900 \nu^{13} - 16560 \nu^{12} - 62940 \nu^{11} - 227010 \nu^{10} + \cdots - 585281 ) / 4600 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 564 \nu^{15} - 69 \nu^{14} + 21340 \nu^{13} - 2990 \nu^{12} + 291960 \nu^{11} - 48070 \nu^{10} + \cdots - 196857 ) / 4600 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 564 \nu^{15} + 69 \nu^{14} + 21340 \nu^{13} + 2990 \nu^{12} + 291960 \nu^{11} + 48070 \nu^{10} + \cdots + 196857 ) / 4600 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{6} + \beta_{5} + \beta_{4} + \beta_{2} - 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( - \beta_{15} + 2 \beta_{14} - 3 \beta_{13} - 3 \beta_{12} + 2 \beta_{10} - 2 \beta_{9} - 2 \beta_{8} + \cdots - 8 \beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 8 \beta_{15} - 8 \beta_{14} + \beta_{12} - \beta_{11} - 3 \beta_{10} - 3 \beta_{9} + 18 \beta_{6} + \cdots + 55 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 19 \beta_{15} - 43 \beta_{14} + 62 \beta_{13} + 70 \beta_{12} + 8 \beta_{11} - 44 \beta_{10} + \cdots + 91 \beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 182 \beta_{15} + 182 \beta_{14} - 27 \beta_{12} + 27 \beta_{11} + 70 \beta_{10} + 70 \beta_{9} + \cdots - 799 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 328 \beta_{15} + 773 \beta_{14} - 1101 \beta_{13} - 1297 \beta_{12} - 196 \beta_{11} + \cdots - 1336 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 3362 \beta_{15} - 3362 \beta_{14} + 531 \beta_{12} - 531 \beta_{11} - 1304 \beta_{10} - 1304 \beta_{9} + \cdots + 12979 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 5615 \beta_{15} - 13413 \beta_{14} + 19028 \beta_{13} + 22776 \beta_{12} + 3748 \beta_{11} + \cdots + 21833 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 59022 \beta_{15} + 59022 \beta_{14} - 9536 \beta_{12} + 9536 \beta_{11} + 22949 \beta_{10} + \cdots - 219133 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 96271 \beta_{15} + 230980 \beta_{14} - 327251 \beta_{13} - 394019 \beta_{12} - 66768 \beta_{11} + \cdots - 369610 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 1021100 \beta_{15} - 1021100 \beta_{14} + 166282 \beta_{12} - 166282 \beta_{11} - 397262 \beta_{10} + \cdots + 3745693 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 1653014 \beta_{15} - 3971476 \beta_{14} + 5624490 \beta_{13} + 6786286 \beta_{12} + 1161796 \beta_{11} + \cdots + 6324493 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 17587244 \beta_{15} + 17587244 \beta_{14} - 2871558 \beta_{12} + 2871558 \beta_{11} + 6843034 \beta_{10} + \cdots - 64273497 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 28402583 \beta_{15} + 68267666 \beta_{14} - 96670249 \beta_{13} - 116724253 \beta_{12} + \cdots - 108566368 \beta_1 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(87\) \(261\)
\(\chi(n)\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
259.1
− 4.14603i
− 2.39034i
− 2.20024i
− 0.689100i
0.530860i
0.639453i
1.99373i
2.26167i
− 2.26167i
− 1.99373i
− 0.639453i
− 0.530860i
0.689100i
2.20024i
2.39034i
4.14603i
− 1.00000i − 3.14603i −1.00000 2.21633 + 0.296419i −3.14603 0.417000i 1.00000i −6.89749 0.296419 − 2.21633i
259.2 − 1.00000i − 1.39034i −1.00000 −1.86897 + 1.22758i −1.39034 4.65953i 1.00000i 1.06694 1.22758 + 1.86897i
259.3 − 1.00000i − 1.20024i −1.00000 1.77243 − 1.36327i −1.20024 − 4.33187i 1.00000i 1.55943 −1.36327 − 1.77243i
259.4 − 1.00000i 0.310900i −1.00000 −0.205955 − 2.22656i 0.310900 2.43727i 1.00000i 2.90334 −2.22656 + 0.205955i
259.5 − 1.00000i 1.53086i −1.00000 −2.23604 + 0.0111723i 1.53086 − 2.29871i 1.00000i 0.656468 0.0111723 + 2.23604i
259.6 − 1.00000i 1.63945i −1.00000 0.156651 + 2.23057i 1.63945 − 0.479890i 1.00000i 0.312193 2.23057 − 0.156651i
259.7 − 1.00000i 2.99373i −1.00000 0.260869 − 2.22080i 2.99373 − 4.62603i 1.00000i −5.96240 −2.22080 − 0.260869i
259.8 − 1.00000i 3.26167i −1.00000 0.904683 + 2.04488i 3.26167 1.22270i 1.00000i −7.63849 2.04488 − 0.904683i
259.9 1.00000i − 3.26167i −1.00000 0.904683 − 2.04488i 3.26167 − 1.22270i − 1.00000i −7.63849 2.04488 + 0.904683i
259.10 1.00000i − 2.99373i −1.00000 0.260869 + 2.22080i 2.99373 4.62603i − 1.00000i −5.96240 −2.22080 + 0.260869i
259.11 1.00000i − 1.63945i −1.00000 0.156651 − 2.23057i 1.63945 0.479890i − 1.00000i 0.312193 2.23057 + 0.156651i
259.12 1.00000i − 1.53086i −1.00000 −2.23604 − 0.0111723i 1.53086 2.29871i − 1.00000i 0.656468 0.0111723 − 2.23604i
259.13 1.00000i − 0.310900i −1.00000 −0.205955 + 2.22656i 0.310900 − 2.43727i − 1.00000i 2.90334 −2.22656 − 0.205955i
259.14 1.00000i 1.20024i −1.00000 1.77243 + 1.36327i −1.20024 4.33187i − 1.00000i 1.55943 −1.36327 + 1.77243i
259.15 1.00000i 1.39034i −1.00000 −1.86897 − 1.22758i −1.39034 − 4.65953i − 1.00000i 1.06694 1.22758 − 1.86897i
259.16 1.00000i 3.14603i −1.00000 2.21633 − 0.296419i −3.14603 − 0.417000i − 1.00000i −6.89749 0.296419 + 2.21633i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 259.16
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 430.2.b.b ✓ 16
5.b even 2 1 inner 430.2.b.b ✓ 16
5.c odd 4 1 2150.2.a.bg 8
5.c odd 4 1 2150.2.a.bh 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
430.2.b.b ✓ 16 1.a even 1 1 trivial
430.2.b.b ✓ 16 5.b even 2 1 inner
2150.2.a.bg 8 5.c odd 4 1
2150.2.a.bh 8 5.c odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{16} + 38T_{3}^{14} + 567T_{3}^{12} + 4234T_{3}^{10} + 16933T_{3}^{8} + 36908T_{3}^{6} + 41764T_{3}^{4} + 20260T_{3}^{2} + 1600 \) acting on \(S_{2}^{\mathrm{new}}(430, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 1)^{8} \) Copy content Toggle raw display
$3$ \( T^{16} + 38 T^{14} + \cdots + 1600 \) Copy content Toggle raw display
$5$ \( T^{16} - 2 T^{15} + \cdots + 390625 \) Copy content Toggle raw display
$7$ \( T^{16} + 75 T^{14} + \cdots + 16384 \) Copy content Toggle raw display
$11$ \( (T^{8} - 2 T^{7} + \cdots + 2560)^{2} \) Copy content Toggle raw display
$13$ \( T^{16} + 151 T^{14} + \cdots + 19219456 \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 676000000 \) Copy content Toggle raw display
$19$ \( (T^{8} + 15 T^{7} + \cdots + 9050)^{2} \) Copy content Toggle raw display
$23$ \( T^{16} + 154 T^{14} + \cdots + 1373584 \) Copy content Toggle raw display
$29$ \( (T^{8} - 3 T^{7} + \cdots - 52120)^{2} \) Copy content Toggle raw display
$31$ \( (T^{8} - 25 T^{7} + \cdots - 920)^{2} \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 137827600 \) Copy content Toggle raw display
$41$ \( (T^{8} - 19 T^{7} + \cdots + 6350)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 1)^{8} \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 2203363600 \) Copy content Toggle raw display
$53$ \( T^{16} + 272 T^{14} + \cdots + 409600 \) Copy content Toggle raw display
$59$ \( (T^{8} - 12 T^{7} + \cdots - 40960)^{2} \) Copy content Toggle raw display
$61$ \( (T^{8} - 29 T^{7} + \cdots + 3896320)^{2} \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 608312896000000 \) Copy content Toggle raw display
$71$ \( (T^{8} - 12 T^{7} + \cdots - 7445248)^{2} \) Copy content Toggle raw display
$73$ \( T^{16} + 263 T^{14} + \cdots + 50922496 \) Copy content Toggle raw display
$79$ \( (T^{8} + 5 T^{7} + \cdots + 2000000)^{2} \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 69542256640000 \) Copy content Toggle raw display
$89$ \( (T^{8} - 20 T^{7} + \cdots - 36334400)^{2} \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 203357706304 \) Copy content Toggle raw display
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