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Properties

Label 1450.2.c.g
Level $1450$
Weight $2$
Character orbit 1450.c
Analytic conductor $11.578$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1450,2,Mod(1101,1450)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1450, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("1450.1101");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1450 = 2 \cdot 5^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1450.c (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: \(11.5783082931\)
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: \( x^{16} + 28x^{14} + 288x^{12} + 1370x^{10} + 3196x^{8} + 3636x^{6} + 1833x^{4} + 320x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{10} \)
Twist minimal: no (minimal twist has level 290)
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.

\(f(q)\) \(=\) \( q - \beta_{8} q^{2} + ( - \beta_{8} + \beta_{7}) q^{3} - q^{4} + ( - \beta_1 - 1) q^{6} + (\beta_{14} + \beta_{12}) q^{7} + \beta_{8} q^{8} + (\beta_{13} - \beta_1 - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{8} q^{2} + ( - \beta_{8} + \beta_{7}) q^{3} - q^{4} + ( - \beta_1 - 1) q^{6} + (\beta_{14} + \beta_{12}) q^{7} + \beta_{8} q^{8} + (\beta_{13} - \beta_1 - 1) q^{9} + \beta_{15} q^{11} + (\beta_{8} - \beta_{7}) q^{12} + ( - 2 \beta_{14} - \beta_{10}) q^{13} + (\beta_{15} + \beta_{4}) q^{14} + q^{16} + (\beta_{9} + \beta_{7} - \beta_{3}) q^{17} + ( - \beta_{9} + \beta_{8} - \beta_{7}) q^{18} + (\beta_{15} + \beta_{11} + \cdots + \beta_{4}) q^{19}+ \cdots + ( - 3 \beta_{15} + \beta_{11} + 3 \beta_{4}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 16 q^{4} - 8 q^{6} - 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 16 q^{4} - 8 q^{6} - 16 q^{9} + 16 q^{16} + 8 q^{24} + 8 q^{29} + 4 q^{34} + 16 q^{36} + 12 q^{49} - 48 q^{51} + 20 q^{54} + 36 q^{59} - 16 q^{64} - 24 q^{71} + 32 q^{74} - 48 q^{81} - 16 q^{86} - 40 q^{91} + 12 q^{94} - 8 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 28x^{14} + 288x^{12} + 1370x^{10} + 3196x^{8} + 3636x^{6} + 1833x^{4} + 320x^{2} + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 644 \nu^{14} - 32714 \nu^{12} - 563020 \nu^{10} - 4233299 \nu^{8} - 14139332 \nu^{6} + \cdots + 216153 ) / 392659 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 14682 \nu^{14} - 377548 \nu^{12} - 3351019 \nu^{10} - 12081108 \nu^{8} - 16017870 \nu^{6} + \cdots - 1568060 ) / 785318 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 29236 \nu^{15} + 863221 \nu^{13} + 9641167 \nu^{11} + 52121499 \nu^{9} + 146669501 \nu^{7} + \cdots + 31962694 \nu ) / 785318 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 76867 \nu^{15} + 2154807 \nu^{13} + 22199197 \nu^{11} + 105823071 \nu^{9} + 247652167 \nu^{7} + \cdots + 34426468 \nu ) / 1570636 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 80257 \nu^{15} - 2144097 \nu^{13} - 20319313 \nu^{11} - 82720793 \nu^{9} - 138755845 \nu^{7} + \cdots + 19063044 \nu ) / 1570636 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 10620 \nu^{14} + 268761 \nu^{12} + 2304522 \nu^{10} + 7530861 \nu^{8} + 5917678 \nu^{6} + \cdots - 1908514 ) / 392659 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 86965 \nu^{15} + 2404368 \nu^{13} + 24225396 \nu^{11} + 111313972 \nu^{9} + 245311326 \nu^{7} + \cdots + 11651758 \nu ) / 1570636 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 88253 \nu^{15} - 2469796 \nu^{13} - 25351436 \nu^{11} - 119780570 \nu^{9} - 273589990 \nu^{7} + \cdots - 15146042 \nu ) / 1570636 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 116329 \nu^{15} - 3159464 \nu^{13} - 30927434 \nu^{11} - 135476188 \nu^{9} + \cdots - 24211694 \nu ) / 1570636 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 67141 \nu^{14} - 1824149 \nu^{12} - 17803263 \nu^{10} - 76759565 \nu^{8} - 147120873 \nu^{6} + \cdots + 2010324 ) / 1570636 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 86321 \nu^{15} - 2371654 \nu^{13} - 23662376 \nu^{11} - 107080673 \nu^{9} - 231171994 \nu^{7} + \cdots - 14616524 \nu ) / 785318 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 95775 \nu^{14} - 2588501 \nu^{12} - 25125555 \nu^{10} - 108338275 \nu^{8} - 214691821 \nu^{6} + \cdots + 2052148 ) / 1570636 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 31296 \nu^{14} + 853238 \nu^{12} + 8391098 \nu^{10} + 36862113 \nu^{8} + 74453736 \nu^{6} + \cdots + 917025 ) / 392659 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 127715 \nu^{14} + 3474453 \nu^{12} + 34079673 \nu^{10} + 149433687 \nu^{8} + 303284889 \nu^{6} + \cdots + 4931268 ) / 1570636 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 272281 \nu^{15} + 7528093 \nu^{13} + 75828427 \nu^{11} + 347899415 \nu^{9} + 761871801 \nu^{7} + \cdots + 28239936 \nu ) / 1570636 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{15} + \beta_{11} + 2\beta_{8} + \beta_{4} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{14} + \beta_{12} + \beta_{2} + \beta _1 - 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -3\beta_{15} - 4\beta_{11} - 7\beta_{8} - 3\beta_{7} - 3\beta_{4} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -8\beta_{14} - \beta_{13} - 8\beta_{12} - 12\beta_{2} - 11\beta _1 + 20 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 24\beta_{15} + 40\beta_{11} - 5\beta_{9} + 56\beta_{8} + 35\beta_{7} + \beta_{5} + 23\beta_{4} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 66\beta_{14} + 23\beta_{13} + 72\beta_{12} + 6\beta_{10} + \beta_{6} + 128\beta_{2} + 114\beta _1 - 165 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -224\beta_{15} - 417\beta_{11} + 91\beta_{9} - 491\beta_{8} - 364\beta_{7} - 31\beta_{5} - 195\beta_{4} + 7\beta_{3} \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -584\beta_{14} - 344\beta_{13} - 704\beta_{12} - 136\beta_{10} - 38\beta_{6} - 1352\beta_{2} - 1167\beta _1 + 1489 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 2242 \beta_{15} + 4401 \beta_{11} - 1248 \beta_{9} + 4561 \beta_{8} + 3735 \beta_{7} + \cdots - 174 \beta_{3} \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 5476 \beta_{14} + 4427 \beta_{13} + 7188 \beta_{12} + 2152 \beta_{10} + 730 \beta_{6} + 14330 \beta_{2} + \cdots - 14112 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 23178 \beta_{15} - 46728 \beta_{11} + 15433 \beta_{9} - 43990 \beta_{8} - 38489 \beta_{7} + \cdots + 2882 \beta_{3} \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 53444 \beta_{14} - 53161 \beta_{13} - 75016 \beta_{12} - 29236 \beta_{10} - 10921 \beta_{6} + \cdots + 138035 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 243488 \beta_{15} + 498027 \beta_{11} - 181441 \beta_{9} + 435527 \beta_{8} + 399646 \beta_{7} + \cdots - 40157 \beta_{3} \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 535778 \beta_{14} + 614952 \beta_{13} + 791952 \beta_{12} + 365994 \beta_{10} + 144396 \beta_{6} + \cdots - 1380807 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 2579958 \beta_{15} - 5321997 \beta_{11} + 2072746 \beta_{9} - 4396079 \beta_{8} + \cdots + 510390 \beta_{3} \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1450\mathbb{Z}\right)^\times\).

\(n\) \(901\) \(1277\)
\(\chi(n)\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1101.1
1.42803i
0.571974i
2.11627i
− 0.116272i
2.83507i
− 0.835068i
− 1.28106i
3.28106i
1.28106i
− 3.28106i
− 2.83507i
0.835068i
− 2.11627i
0.116272i
− 1.42803i
− 0.571974i
− 1.00000i − 2.81679i −1.00000 0 −2.81679 −2.06171 1.00000i −4.93433 0
1101.2 − 1.00000i − 2.81679i −1.00000 0 −2.81679 2.06171 1.00000i −4.93433 0
1101.3 − 1.00000i − 1.75394i −1.00000 0 −1.75394 −4.19041 1.00000i −0.0762945 0
1101.4 − 1.00000i − 1.75394i −1.00000 0 −1.75394 4.19041 1.00000i −0.0762945 0
1101.5 − 1.00000i 0.367473i −1.00000 0 0.367473 −2.99580 1.00000i 2.86496 0
1101.6 − 1.00000i 0.367473i −1.00000 0 0.367473 2.99580 1.00000i 2.86496 0
1101.7 − 1.00000i 2.20326i −1.00000 0 2.20326 −0.463643 1.00000i −1.85434 0
1101.8 − 1.00000i 2.20326i −1.00000 0 2.20326 0.463643 1.00000i −1.85434 0
1101.9 1.00000i − 2.20326i −1.00000 0 2.20326 −0.463643 − 1.00000i −1.85434 0
1101.10 1.00000i − 2.20326i −1.00000 0 2.20326 0.463643 − 1.00000i −1.85434 0
1101.11 1.00000i − 0.367473i −1.00000 0 0.367473 −2.99580 − 1.00000i 2.86496 0
1101.12 1.00000i − 0.367473i −1.00000 0 0.367473 2.99580 − 1.00000i 2.86496 0
1101.13 1.00000i 1.75394i −1.00000 0 −1.75394 −4.19041 − 1.00000i −0.0762945 0
1101.14 1.00000i 1.75394i −1.00000 0 −1.75394 4.19041 − 1.00000i −0.0762945 0
1101.15 1.00000i 2.81679i −1.00000 0 −2.81679 −2.06171 − 1.00000i −4.93433 0
1101.16 1.00000i 2.81679i −1.00000 0 −2.81679 2.06171 − 1.00000i −4.93433 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1101.16
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1450.2.c.g 16
5.b even 2 1 inner 1450.2.c.g 16
5.c odd 4 1 290.2.d.a ✓ 8
5.c odd 4 1 290.2.d.b yes 8
15.e even 4 1 2610.2.b.d 8
15.e even 4 1 2610.2.b.f 8
20.e even 4 1 2320.2.j.d 8
20.e even 4 1 2320.2.j.e 8
29.b even 2 1 inner 1450.2.c.g 16
145.d even 2 1 inner 1450.2.c.g 16
145.h odd 4 1 290.2.d.a ✓ 8
145.h odd 4 1 290.2.d.b yes 8
435.p even 4 1 2610.2.b.d 8
435.p even 4 1 2610.2.b.f 8
580.o even 4 1 2320.2.j.d 8
580.o even 4 1 2320.2.j.e 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
290.2.d.a ✓ 8 5.c odd 4 1
290.2.d.a ✓ 8 145.h odd 4 1
290.2.d.b yes 8 5.c odd 4 1
290.2.d.b yes 8 145.h odd 4 1
1450.2.c.g 16 1.a even 1 1 trivial
1450.2.c.g 16 5.b even 2 1 inner
1450.2.c.g 16 29.b even 2 1 inner
1450.2.c.g 16 145.d even 2 1 inner
2320.2.j.d 8 20.e even 4 1
2320.2.j.d 8 580.o even 4 1
2320.2.j.e 8 20.e even 4 1
2320.2.j.e 8 580.o even 4 1
2610.2.b.d 8 15.e even 4 1
2610.2.b.d 8 435.p even 4 1
2610.2.b.f 8 15.e even 4 1
2610.2.b.f 8 435.p even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1450, [\chi])\):

\( T_{3}^{8} + 16T_{3}^{6} + 80T_{3}^{4} + 129T_{3}^{2} + 16 \) Copy content Toggle raw display
\( T_{7}^{8} - 31T_{7}^{6} + 277T_{7}^{4} - 728T_{7}^{2} + 144 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 1)^{8} \) Copy content Toggle raw display
$3$ \( (T^{8} + 16 T^{6} + \cdots + 16)^{2} \) Copy content Toggle raw display
$5$ \( T^{16} \) Copy content Toggle raw display
$7$ \( (T^{8} - 31 T^{6} + \cdots + 144)^{2} \) Copy content Toggle raw display
$11$ \( (T^{8} + 27 T^{6} + \cdots + 64)^{2} \) Copy content Toggle raw display
$13$ \( (T^{8} - 82 T^{6} + \cdots + 11664)^{2} \) Copy content Toggle raw display
$17$ \( (T^{8} + 71 T^{6} + \cdots + 144)^{2} \) Copy content Toggle raw display
$19$ \( (T^{8} + 104 T^{6} + \cdots + 36864)^{2} \) Copy content Toggle raw display
$23$ \( (T^{8} - 111 T^{6} + \cdots + 21904)^{2} \) Copy content Toggle raw display
$29$ \( (T^{8} - 4 T^{7} + \cdots + 707281)^{2} \) Copy content Toggle raw display
$31$ \( (T^{8} + 138 T^{6} + \cdots + 26244)^{2} \) Copy content Toggle raw display
$37$ \( (T^{8} + 144 T^{6} + \cdots + 1024)^{2} \) Copy content Toggle raw display
$41$ \( (T^{8} + 120 T^{6} + \cdots + 4096)^{2} \) Copy content Toggle raw display
$43$ \( (T^{8} + 244 T^{6} + \cdots + 8596624)^{2} \) Copy content Toggle raw display
$47$ \( (T^{8} + 93 T^{6} + \cdots + 36864)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} - 238 T^{6} + \cdots + 2972176)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} - 9 T^{3} - 3 T^{2} + \cdots + 72)^{4} \) Copy content Toggle raw display
$61$ \( (T^{8} + 127 T^{6} + \cdots + 5184)^{2} \) Copy content Toggle raw display
$67$ \( (T^{8} - 232 T^{6} + \cdots + 6718464)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + 6 T^{3} + \cdots - 3456)^{4} \) Copy content Toggle raw display
$73$ \( (T^{8} + 155 T^{6} + \cdots + 1201216)^{2} \) Copy content Toggle raw display
$79$ \( (T^{8} + 98 T^{6} + \cdots + 2916)^{2} \) Copy content Toggle raw display
$83$ \( (T^{8} - 516 T^{6} + \cdots + 8202496)^{2} \) Copy content Toggle raw display
$89$ \( (T^{8} + 356 T^{6} + \cdots + 2985984)^{2} \) Copy content Toggle raw display
$97$ \( (T^{8} + 487 T^{6} + \cdots + 3225616)^{2} \) Copy content Toggle raw display
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