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Properties

Label 4450.2.a.bm
Level $4450$
Weight $2$
Character orbit 4450.a
Self dual yes
Analytic conductor $35.533$
Analytic rank $1$
Dimension $14$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [4450,2,Mod(1,4450)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4450, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4450.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4450 = 2 \cdot 5^{2} \cdot 89 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4450.a (trivial)

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: yes
Analytic conductor: \(35.5334288995\)
Analytic rank: \(1\)
Dimension: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} - 6 x^{13} - 12 x^{12} + 122 x^{11} - 8 x^{10} - 938 x^{9} + 630 x^{8} + 3470 x^{7} - 3133 x^{6} + \cdots + 1000 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 890)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} - \beta_1 q^{3} + q^{4} + \beta_1 q^{6} + \beta_{7} q^{7} - q^{8} + (\beta_{2} + \beta_1 + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} - \beta_1 q^{3} + q^{4} + \beta_1 q^{6} + \beta_{7} q^{7} - q^{8} + (\beta_{2} + \beta_1 + 1) q^{9} + \beta_{3} q^{11} - \beta_1 q^{12} + ( - \beta_{10} + \beta_{8} - \beta_{5} + \cdots - 1) q^{13}+ \cdots + (2 \beta_{13} + \beta_{12} - \beta_{10} + \cdots + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q - 14 q^{2} - 6 q^{3} + 14 q^{4} + 6 q^{6} - 6 q^{7} - 14 q^{8} + 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 14 q - 14 q^{2} - 6 q^{3} + 14 q^{4} + 6 q^{6} - 6 q^{7} - 14 q^{8} + 18 q^{9} + 4 q^{11} - 6 q^{12} - 10 q^{13} + 6 q^{14} + 14 q^{16} - 10 q^{17} - 18 q^{18} + 10 q^{19} + 10 q^{21} - 4 q^{22} - 24 q^{23} + 6 q^{24} + 10 q^{26} - 30 q^{27} - 6 q^{28} + 8 q^{29} - 12 q^{31} - 14 q^{32} - 22 q^{33} + 10 q^{34} + 18 q^{36} - 12 q^{37} - 10 q^{38} - 10 q^{39} - 4 q^{41} - 10 q^{42} - 22 q^{43} + 4 q^{44} + 24 q^{46} - 24 q^{47} - 6 q^{48} + 38 q^{49} - 4 q^{51} - 10 q^{52} - 42 q^{53} + 30 q^{54} + 6 q^{56} + 8 q^{57} - 8 q^{58} + 10 q^{59} + 20 q^{61} + 12 q^{62} - 46 q^{63} + 14 q^{64} + 22 q^{66} - 10 q^{67} - 10 q^{68} + 10 q^{69} - 12 q^{71} - 18 q^{72} - 6 q^{73} + 12 q^{74} + 10 q^{76} - 30 q^{77} + 10 q^{78} + 2 q^{79} + 2 q^{81} + 4 q^{82} - 30 q^{83} + 10 q^{84} + 22 q^{86} - 6 q^{87} - 4 q^{88} + 14 q^{89} - 32 q^{91} - 24 q^{92} - 22 q^{93} + 24 q^{94} + 6 q^{96} + 16 q^{97} - 38 q^{98} + 40 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} - 6 x^{13} - 12 x^{12} + 122 x^{11} - 8 x^{10} - 938 x^{9} + 630 x^{8} + 3470 x^{7} - 3133 x^{6} + \cdots + 1000 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 14 \nu^{13} - 69 \nu^{12} - 183 \nu^{11} + 1233 \nu^{10} + 388 \nu^{9} - 7782 \nu^{8} + 3405 \nu^{7} + \cdots + 500 ) / 1100 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 2 \nu^{13} - 57 \nu^{12} + 241 \nu^{11} + 624 \nu^{10} - 4486 \nu^{9} + 169 \nu^{8} + 27075 \nu^{7} + \cdots + 7300 ) / 1100 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 48 \nu^{13} + 268 \nu^{12} + 541 \nu^{11} - 4801 \nu^{10} - 136 \nu^{9} + 30594 \nu^{8} + \cdots + 5200 ) / 1100 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 51 \nu^{13} + 271 \nu^{12} + 702 \nu^{11} - 5132 \nu^{10} - 2317 \nu^{9} + 35373 \nu^{8} + \cdots + 12400 ) / 1100 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 23 \nu^{13} + 243 \nu^{12} - 324 \nu^{11} - 3766 \nu^{10} + 10284 \nu^{9} + 17884 \nu^{8} + \cdots - 17400 ) / 1100 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 37 \nu^{13} + 92 \nu^{12} + 1179 \nu^{11} - 2964 \nu^{10} - 13314 \nu^{9} + 32981 \nu^{8} + \cdots + 50300 ) / 1100 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 18 \nu^{13} + 73 \nu^{12} + 372 \nu^{11} - 1579 \nu^{10} - 2955 \nu^{9} + 12922 \nu^{8} + \cdots + 12620 ) / 220 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 93 \nu^{13} + 423 \nu^{12} + 1801 \nu^{11} - 9106 \nu^{10} - 13271 \nu^{9} + 74394 \nu^{8} + \cdots + 70300 ) / 1100 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 153 \nu^{13} + 758 \nu^{12} + 2381 \nu^{11} - 14791 \nu^{10} - 11571 \nu^{9} + 106889 \nu^{8} + \cdots + 69100 ) / 1100 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 194 \nu^{13} + 964 \nu^{12} + 3023 \nu^{11} - 18838 \nu^{10} - 14868 \nu^{9} + 136617 \nu^{8} + \cdots + 87200 ) / 1100 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 116 \nu^{13} - 446 \nu^{12} - 2687 \nu^{11} + 10397 \nu^{10} + 24492 \nu^{9} - 92278 \nu^{8} + \cdots - 124400 ) / 1100 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{12} - \beta_{10} - \beta_{9} + \beta_{8} - \beta_{5} + 7\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{13} - \beta_{12} + \beta_{11} - 2\beta_{10} + \beta_{8} + \beta_{7} + \beta_{6} + 9\beta_{2} + 12\beta _1 + 26 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 2 \beta_{13} + 9 \beta_{12} - \beta_{11} - 15 \beta_{10} - 8 \beta_{9} + 13 \beta_{8} + 3 \beta_{7} + \cdots + 19 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 16 \beta_{13} - 14 \beta_{12} + 12 \beta_{11} - 36 \beta_{10} + 22 \beta_{8} + 18 \beta_{7} + \cdots + 208 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 38 \beta_{13} + 65 \beta_{12} - 12 \beta_{11} - 187 \beta_{10} - 53 \beta_{9} + 151 \beta_{8} + \cdots + 255 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 195 \beta_{13} - 147 \beta_{12} + 115 \beta_{11} - 504 \beta_{10} + 8 \beta_{9} + 347 \beta_{8} + \cdots + 1836 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 502 \beta_{13} + 435 \beta_{12} - 91 \beta_{11} - 2211 \beta_{10} - 316 \beta_{9} + 1767 \beta_{8} + \cdots + 3043 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 2136 \beta_{13} - 1382 \beta_{12} + 1056 \beta_{11} - 6422 \beta_{10} + 212 \beta_{9} + 4818 \beta_{8} + \cdots + 17118 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 5762 \beta_{13} + 2803 \beta_{12} - 416 \beta_{11} - 25595 \beta_{10} - 1543 \beta_{9} + 20917 \beta_{8} + \cdots + 34555 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 22177 \beta_{13} - 12221 \beta_{12} + 9825 \beta_{11} - 78090 \beta_{10} + 3724 \beta_{9} + \cdots + 165582 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 61750 \beta_{13} + 17789 \beta_{12} + 1459 \beta_{11} - 293467 \beta_{10} - 2692 \beta_{9} + \cdots + 383523 \) Copy content Toggle raw display

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
3.36926
3.08367
2.58265
2.51983
1.54429
1.44445
1.19695
0.432441
−0.587278
−1.29284
−1.64320
−1.69371
−2.19556
−2.76095
−1.00000 −3.36926 1.00000 0 3.36926 −1.39634 −1.00000 8.35193 0
1.2 −1.00000 −3.08367 1.00000 0 3.08367 −1.45974 −1.00000 6.50901 0
1.3 −1.00000 −2.58265 1.00000 0 2.58265 2.75127 −1.00000 3.67008 0
1.4 −1.00000 −2.51983 1.00000 0 2.51983 −4.62924 −1.00000 3.34955 0
1.5 −1.00000 −1.54429 1.00000 0 1.54429 1.43916 −1.00000 −0.615168 0
1.6 −1.00000 −1.44445 1.00000 0 1.44445 −0.395130 −1.00000 −0.913551 0
1.7 −1.00000 −1.19695 1.00000 0 1.19695 −4.95996 −1.00000 −1.56731 0
1.8 −1.00000 −0.432441 1.00000 0 0.432441 4.16861 −1.00000 −2.81300 0
1.9 −1.00000 0.587278 1.00000 0 −0.587278 −0.319407 −1.00000 −2.65510 0
1.10 −1.00000 1.29284 1.00000 0 −1.29284 2.46610 −1.00000 −1.32856 0
1.11 −1.00000 1.64320 1.00000 0 −1.64320 −4.15025 −1.00000 −0.299884 0
1.12 −1.00000 1.69371 1.00000 0 −1.69371 4.97557 −1.00000 −0.131335 0
1.13 −1.00000 2.19556 1.00000 0 −2.19556 −2.72829 −1.00000 1.82049 0
1.14 −1.00000 2.76095 1.00000 0 −2.76095 −1.76236 −1.00000 4.62284 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.14
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( +1 \)
\(5\) \( -1 \)
\(89\) \( -1 \)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4450.2.a.bm 14
5.b even 2 1 4450.2.a.bn 14
5.c odd 4 2 890.2.b.b ✓ 28
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
890.2.b.b ✓ 28 5.c odd 4 2
4450.2.a.bm 14 1.a even 1 1 trivial
4450.2.a.bn 14 5.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4450))\):

\( T_{3}^{14} + 6 T_{3}^{13} - 12 T_{3}^{12} - 122 T_{3}^{11} - 8 T_{3}^{10} + 938 T_{3}^{9} + 630 T_{3}^{8} + \cdots + 1000 \) Copy content Toggle raw display
\( T_{7}^{14} + 6 T_{7}^{13} - 50 T_{7}^{12} - 342 T_{7}^{11} + 710 T_{7}^{10} + 6712 T_{7}^{9} + \cdots - 23872 \) Copy content Toggle raw display
\( T_{11}^{14} - 4 T_{11}^{13} - 86 T_{11}^{12} + 276 T_{11}^{11} + 2877 T_{11}^{10} - 6996 T_{11}^{9} + \cdots + 180736 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{14} \) Copy content Toggle raw display
$3$ \( T^{14} + 6 T^{13} + \cdots + 1000 \) Copy content Toggle raw display
$5$ \( T^{14} \) Copy content Toggle raw display
$7$ \( T^{14} + 6 T^{13} + \cdots - 23872 \) Copy content Toggle raw display
$11$ \( T^{14} - 4 T^{13} + \cdots + 180736 \) Copy content Toggle raw display
$13$ \( T^{14} + 10 T^{13} + \cdots + 17408 \) Copy content Toggle raw display
$17$ \( T^{14} + 10 T^{13} + \cdots + 5354560 \) Copy content Toggle raw display
$19$ \( T^{14} - 10 T^{13} + \cdots - 1360000 \) Copy content Toggle raw display
$23$ \( T^{14} + 24 T^{13} + \cdots - 414208 \) Copy content Toggle raw display
$29$ \( T^{14} + \cdots - 780779520 \) Copy content Toggle raw display
$31$ \( T^{14} + \cdots + 6362111216 \) Copy content Toggle raw display
$37$ \( T^{14} + 12 T^{13} + \cdots + 5760000 \) Copy content Toggle raw display
$41$ \( T^{14} + \cdots + 1200619520 \) Copy content Toggle raw display
$43$ \( T^{14} + 22 T^{13} + \cdots + 6643384 \) Copy content Toggle raw display
$47$ \( T^{14} + \cdots - 130580480 \) Copy content Toggle raw display
$53$ \( T^{14} + \cdots + 58846457056 \) Copy content Toggle raw display
$59$ \( T^{14} + \cdots - 292037611520 \) Copy content Toggle raw display
$61$ \( T^{14} + \cdots + 867184000 \) Copy content Toggle raw display
$67$ \( T^{14} + \cdots - 251781120 \) Copy content Toggle raw display
$71$ \( T^{14} + 12 T^{13} + \cdots - 37519360 \) Copy content Toggle raw display
$73$ \( T^{14} + \cdots - 135722289152 \) Copy content Toggle raw display
$79$ \( T^{14} + \cdots + 19404369756160 \) Copy content Toggle raw display
$83$ \( T^{14} + \cdots + 165760300000 \) Copy content Toggle raw display
$89$ \( (T - 1)^{14} \) Copy content Toggle raw display
$97$ \( T^{14} + \cdots - 7440772352 \) Copy content Toggle raw display
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