Properties

Label 1450.2.e.f
Level $1450$
Weight $2$
Character orbit 1450.e
Analytic conductor $11.578$
Analytic rank $0$
Dimension $10$
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] = [1450,2,Mod(307,1450)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1450, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([1, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1450.307");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1450 = 2 \cdot 5^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1450.e (of order \(4\), degree \(2\), 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: \(10\)
Relative dimension: \(5\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} + 18x^{8} + 99x^{6} + 186x^{4} + 105x^{2} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} + 18x^{8} + 99x^{6} + 186x^{4} + 105x^{2} + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -2\nu^{8} - 33\nu^{6} - 155\nu^{4} - 211\nu^{2} - 56 ) / 26 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 2\nu^{8} + 33\nu^{6} + 155\nu^{4} + 237\nu^{2} + 134 ) / 26 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 3\nu^{9} + \nu^{8} + 56\nu^{7} + 23\nu^{6} + 330\nu^{5} + 175\nu^{4} + 687\nu^{3} + 450\nu^{2} + 318\nu + 184 ) / 52 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -7\nu^{9} - 122\nu^{7} - 627\nu^{5} - 992\nu^{3} - 313\nu ) / 52 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 5 \nu^{9} - 11 \nu^{8} - 89 \nu^{7} - 188 \nu^{6} - 472 \nu^{5} - 924 \nu^{4} - 755 \nu^{3} - 1297 \nu^{2} - 75 \nu - 308 ) / 52 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 9 \nu^{9} + 12 \nu^{8} + 155 \nu^{7} + 211 \nu^{6} + 769 \nu^{5} + 1099 \nu^{4} + 1060 \nu^{3} + 1747 \nu^{2} + 70 \nu + 440 ) / 52 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 17\nu^{9} + 300\nu^{7} + 1571\nu^{5} + 2554\nu^{3} + 879\nu ) / 52 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 9 \nu^{9} - 12 \nu^{8} + 155 \nu^{7} - 211 \nu^{6} + 769 \nu^{5} - 1099 \nu^{4} + 1060 \nu^{3} - 1747 \nu^{2} + 70 \nu - 440 ) / 52 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + \beta_{2} - 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{9} + \beta_{8} + \beta_{6} - \beta_{4} - 8\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{9} + \beta_{7} + 2\beta_{6} + 2\beta_{5} + 2\beta_{4} - 8\beta_{3} - 13\beta_{2} + 18 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 9\beta_{9} - 13\beta_{8} - 2\beta_{7} - 11\beta_{6} - 10\beta_{5} + 11\beta_{4} + 67\beta _1 - 11 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -15\beta_{9} - 11\beta_{7} - 26\beta_{6} - 26\beta_{5} - 26\beta_{4} + 67\beta_{3} + 144\beta_{2} - 131 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -78\beta_{9} + 144\beta_{8} + 26\beta_{7} + 104\beta_{6} + 164\beta_{5} - 104\beta_{4} - 591\beta _1 + 104 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 170\beta_{9} + 104\beta_{7} + 274\beta_{6} + 274\beta_{5} + 274\beta_{4} - 591\beta_{3} - 1487\beta_{2} + 1055 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 695\beta_{9} - 1487\beta_{8} - 274\beta_{7} - 969\beta_{6} - 1970\beta_{5} + 969\beta_{4} + 5388\beta _1 - 969 \) Copy content Toggle raw display

Character values

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

\(n\) \(901\) \(1277\)
\(\chi(n)\) \(-\beta_{5}\) \(\beta_{5}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
307.1
3.09258i
1.48865i
0.496555i
0.752486i
2.32530i
2.32530i
0.752486i
0.496555i
1.48865i
3.09258i
−1.00000 3.09258i 1.00000 0 3.09258i 3.27535 + 3.27535i −1.00000 −6.56407 0
307.2 −1.00000 1.48865i 1.00000 0 1.48865i −2.18018 2.18018i −1.00000 0.783913 0
307.3 −1.00000 0.496555i 1.00000 0 0.496555i −0.303453 0.303453i −1.00000 2.75343 0
307.4 −1.00000 0.752486i 1.00000 0 0.752486i 2.38506 + 2.38506i −1.00000 2.43376 0
307.5 −1.00000 2.32530i 1.00000 0 2.32530i −2.17677 2.17677i −1.00000 −2.40704 0
1143.1 −1.00000 2.32530i 1.00000 0 2.32530i −2.17677 + 2.17677i −1.00000 −2.40704 0
1143.2 −1.00000 0.752486i 1.00000 0 0.752486i 2.38506 2.38506i −1.00000 2.43376 0
1143.3 −1.00000 0.496555i 1.00000 0 0.496555i −0.303453 + 0.303453i −1.00000 2.75343 0
1143.4 −1.00000 1.48865i 1.00000 0 1.48865i −2.18018 + 2.18018i −1.00000 0.783913 0
1143.5 −1.00000 3.09258i 1.00000 0 3.09258i 3.27535 3.27535i −1.00000 −6.56407 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 307.5
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
145.j even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1450.2.e.f 10
5.b even 2 1 1450.2.e.g yes 10
5.c odd 4 1 1450.2.j.f yes 10
5.c odd 4 1 1450.2.j.g yes 10
29.c odd 4 1 1450.2.j.g yes 10
145.e even 4 1 1450.2.e.g yes 10
145.f odd 4 1 1450.2.j.f yes 10
145.j even 4 1 inner 1450.2.e.f 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1450.2.e.f 10 1.a even 1 1 trivial
1450.2.e.f 10 145.j even 4 1 inner
1450.2.e.g yes 10 5.b even 2 1
1450.2.e.g yes 10 145.e even 4 1
1450.2.j.f yes 10 5.c odd 4 1
1450.2.j.f yes 10 145.f odd 4 1
1450.2.j.g yes 10 5.c odd 4 1
1450.2.j.g yes 10 29.c odd 4 1

Hecke kernels

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

\( T_{3}^{10} + 18T_{3}^{8} + 99T_{3}^{6} + 186T_{3}^{4} + 105T_{3}^{2} + 16 \) Copy content Toggle raw display
\( T_{7}^{10} - 2 T_{7}^{9} + 2 T_{7}^{8} + 36 T_{7}^{7} + 311 T_{7}^{6} - 126 T_{7}^{5} + 278 T_{7}^{4} + 4448 T_{7}^{3} + 24649 T_{7}^{2} + 14130 T_{7} + 4050 \) Copy content Toggle raw display
\( T_{11}^{10} + 4 T_{11}^{9} + 8 T_{11}^{8} - 16 T_{11}^{7} + 156 T_{11}^{6} + 448 T_{11}^{5} + 672 T_{11}^{4} - 352 T_{11}^{3} + 400 T_{11}^{2} + 640 T_{11} + 512 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{10} \) Copy content Toggle raw display
$3$ \( T^{10} + 18 T^{8} + 99 T^{6} + 186 T^{4} + \cdots + 16 \) Copy content Toggle raw display
$5$ \( T^{10} \) Copy content Toggle raw display
$7$ \( T^{10} - 2 T^{9} + 2 T^{8} + 36 T^{7} + \cdots + 4050 \) Copy content Toggle raw display
$11$ \( T^{10} + 4 T^{9} + 8 T^{8} - 16 T^{7} + \cdots + 512 \) Copy content Toggle raw display
$13$ \( T^{10} - 4 T^{9} + 8 T^{8} + 104 T^{7} + \cdots + 392 \) Copy content Toggle raw display
$17$ \( (T^{5} - 4 T^{4} - 33 T^{3} + 68 T^{2} + \cdots - 6)^{2} \) Copy content Toggle raw display
$19$ \( T^{10} - 4 T^{9} + 8 T^{8} + 32 T^{7} + \cdots + 128 \) Copy content Toggle raw display
$23$ \( T^{10} + 6 T^{9} + 18 T^{8} + \cdots + 1250 \) Copy content Toggle raw display
$29$ \( T^{10} + 8 T^{9} - 19 T^{8} + \cdots + 20511149 \) Copy content Toggle raw display
$31$ \( T^{10} + 4 T^{9} + 8 T^{8} + \cdots + 127008 \) Copy content Toggle raw display
$37$ \( T^{10} + 208 T^{8} + 13228 T^{6} + \cdots + 576 \) Copy content Toggle raw display
$41$ \( T^{10} + 6 T^{9} + 18 T^{8} + \cdots + 35212832 \) Copy content Toggle raw display
$43$ \( T^{10} + 470 T^{8} + \cdots + 1660562500 \) Copy content Toggle raw display
$47$ \( T^{10} + 240 T^{8} + 15260 T^{6} + \cdots + 2383936 \) Copy content Toggle raw display
$53$ \( T^{10} - 144 T^{7} + \cdots + 12500000 \) Copy content Toggle raw display
$59$ \( T^{10} + 182 T^{8} + 8983 T^{6} + \cdots + 430336 \) Copy content Toggle raw display
$61$ \( T^{10} + 6 T^{9} + 18 T^{8} + \cdots + 16971138 \) Copy content Toggle raw display
$67$ \( T^{10} + 6 T^{9} + \cdots + 2259533088 \) Copy content Toggle raw display
$71$ \( T^{10} + 392 T^{8} + \cdots + 36000000 \) Copy content Toggle raw display
$73$ \( (T^{5} + 4 T^{4} - 113 T^{3} - 316 T^{2} + \cdots - 1766)^{2} \) Copy content Toggle raw display
$79$ \( T^{10} + 8 T^{9} + 32 T^{8} + \cdots + 922608968 \) Copy content Toggle raw display
$83$ \( T^{10} - 14 T^{9} + 98 T^{8} + \cdots + 17287200 \) Copy content Toggle raw display
$89$ \( T^{10} + 34 T^{9} + 578 T^{8} + \cdots + 25088 \) Copy content Toggle raw display
$97$ \( T^{10} + 702 T^{8} + \cdots + 2006502436 \) Copy content Toggle raw display
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