Properties

Label 740.2.d.a
Level $740$
Weight $2$
Character orbit 740.d
Analytic conductor $5.909$
Analytic rank $0$
Dimension $18$
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [740,2,Mod(149,740)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(740, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("740.149");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 740 = 2^{2} \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 740.d (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: \(5.90892974957\)
Analytic rank: \(0\)
Dimension: \(18\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{18} + 26 x^{16} + 283 x^{14} + 1674 x^{12} + 5841 x^{10} + 12196 x^{8} + 14736 x^{6} + 9408 x^{4} + \cdots + 144 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{4}\cdot 3^{2} \)
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 \(1,\beta_1,\ldots,\beta_{17}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{13} q^{3} + \beta_{10} q^{5} - \beta_{8} q^{7} + ( - \beta_{11} + \beta_{10} + \beta_{7} + \cdots - 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{13} q^{3} + \beta_{10} q^{5} - \beta_{8} q^{7} + ( - \beta_{11} + \beta_{10} + \beta_{7} + \cdots - 2) q^{9}+ \cdots + (\beta_{17} + \beta_{16} + \beta_{11} + \cdots + 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q + 2 q^{5} - 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 18 q + 2 q^{5} - 18 q^{9} + 6 q^{15} + 4 q^{19} - 16 q^{21} - 2 q^{25} - 4 q^{29} + 8 q^{31} - 2 q^{35} + 8 q^{39} - 4 q^{41} + 8 q^{45} + 6 q^{49} - 40 q^{51} - 6 q^{55} + 8 q^{59} - 12 q^{65} + 28 q^{69} - 24 q^{71} - 16 q^{75} - 24 q^{79} + 34 q^{81} + 36 q^{91} + 24 q^{95} + 16 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{18} + 26 x^{16} + 283 x^{14} + 1674 x^{12} + 5841 x^{10} + 12196 x^{8} + 14736 x^{6} + 9408 x^{4} + \cdots + 144 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{17} - 20\nu^{15} - 148\nu^{13} - 459\nu^{11} - 273\nu^{9} + 1475\nu^{7} + 2595\nu^{5} + 714\nu^{3} - 252\nu ) / 36 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{16} + 20\nu^{14} + 151\nu^{12} + 510\nu^{10} + 591\nu^{8} - 620\nu^{6} - 1800\nu^{4} - 912\nu^{2} - 48 ) / 24 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{16} + 20\nu^{14} + 151\nu^{12} + 510\nu^{10} + 579\nu^{8} - 752\nu^{6} - 2244\nu^{4} - 1392\nu^{2} - 144 ) / 24 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{17} + 26 \nu^{15} + 277 \nu^{13} + 1554 \nu^{11} + 4917 \nu^{9} + 8794 \nu^{7} + 8724 \nu^{5} + \cdots + 1368 \nu ) / 72 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{17} + 26 \nu^{15} + 277 \nu^{13} + 1554 \nu^{11} + 4899 \nu^{9} + 8524 \nu^{7} + 7338 \nu^{5} + \cdots - 144 \nu ) / 72 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{16} + 23\nu^{14} + 214\nu^{12} + 1032\nu^{10} + 2742\nu^{8} + 3931\nu^{6} + 2778\nu^{4} + 852\nu^{2} + 96 ) / 12 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{16} + 23\nu^{14} + 214\nu^{12} + 1035\nu^{10} + 2787\nu^{8} + 4162\nu^{6} + 3234\nu^{4} + 1092\nu^{2} + 60 ) / 12 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 2 \nu^{17} + 43 \nu^{15} + 365 \nu^{13} + 1542 \nu^{11} + 3327 \nu^{9} + 3251 \nu^{7} + 798 \nu^{5} + \cdots - 72 \nu ) / 36 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( \nu^{16} + 22\nu^{14} + 193\nu^{12} + 858\nu^{10} + 2025\nu^{8} + 2414\nu^{6} + 1244\nu^{4} + 208\nu^{2} - 8 ) / 8 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( \nu^{17} + 9 \nu^{16} + 20 \nu^{15} + 207 \nu^{14} + 148 \nu^{13} + 1935 \nu^{12} + 459 \nu^{11} + \cdots + 720 ) / 72 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( \nu^{17} - 9 \nu^{16} + 20 \nu^{15} - 207 \nu^{14} + 148 \nu^{13} - 1935 \nu^{12} + 459 \nu^{11} + \cdots - 720 ) / 72 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( -\nu^{16} - 23\nu^{14} - 214\nu^{12} - 1032\nu^{10} - 2745\nu^{8} - 3958\nu^{6} - 2829\nu^{4} - 804\nu^{2} - 24 ) / 6 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( \nu^{17} + 24 \nu^{15} + 237 \nu^{13} + 1246 \nu^{11} + 3771 \nu^{9} + 6622 \nu^{7} + 6412 \nu^{5} + \cdots + 384 \nu ) / 24 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 7 \nu^{17} + 158 \nu^{15} + 1441 \nu^{13} + 6804 \nu^{11} + 17661 \nu^{9} + 24442 \nu^{7} + \cdots - 1152 \nu ) / 72 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 7 \nu^{17} + 158 \nu^{15} + 1441 \nu^{13} + 6822 \nu^{11} + 17967 \nu^{9} + 26368 \nu^{7} + \cdots + 1440 \nu ) / 72 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 4 \nu^{17} + 9 \nu^{16} - 92 \nu^{15} + 207 \nu^{14} - 859 \nu^{13} + 1935 \nu^{12} - 4179 \nu^{11} + \cdots + 504 ) / 72 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 4 \nu^{17} + 9 \nu^{16} + 92 \nu^{15} + 207 \nu^{14} + 859 \nu^{13} + 1935 \nu^{12} + 4179 \nu^{11} + \cdots + 504 ) / 72 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -\beta_{17} + \beta_{16} + 2\beta_{15} + 2\beta_{14} - 2\beta_{13} - \beta_{11} - \beta_{10} - 4\beta_{8} - 2\beta_1 ) / 6 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{17} - \beta_{16} - \beta_{11} + \beta_{10} - 6 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{17} - \beta_{16} - \beta_{15} - 2\beta_{14} + \beta_{13} + \beta_{11} + \beta_{10} + 3\beta_{8} + 2\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 7\beta_{17} + 7\beta_{16} + 7\beta_{11} - 7\beta_{10} - 2\beta_{9} + 2\beta_{6} + 2\beta_{2} + 28 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 11 \beta_{17} + 11 \beta_{16} + 6 \beta_{15} + 24 \beta_{14} - 6 \beta_{13} - 9 \beta_{11} + \cdots - 22 \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 21 \beta_{17} - 21 \beta_{16} + \beta_{12} - 21 \beta_{11} + 21 \beta_{10} + 10 \beta_{9} - 8 \beta_{6} + \cdots - 72 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 59 \beta_{17} - 59 \beta_{16} - 16 \beta_{15} - 140 \beta_{14} + 16 \beta_{13} + 41 \beta_{11} + \cdots + 122 \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 243 \beta_{17} + 243 \beta_{16} - 22 \beta_{12} + 243 \beta_{11} - 243 \beta_{10} - 146 \beta_{9} + \cdots + 772 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 157 \beta_{17} + 157 \beta_{16} + 13 \beta_{15} + 402 \beta_{14} - 13 \beta_{13} - 99 \beta_{11} + \cdots - 344 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 1395 \beta_{17} - 1395 \beta_{16} + 176 \beta_{12} - 1395 \beta_{11} + 1395 \beta_{10} + 954 \beta_{9} + \cdots - 4244 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 1667 \beta_{17} - 1667 \beta_{16} + 98 \beta_{15} - 4584 \beta_{14} - 90 \beta_{13} + 1017 \beta_{11} + \cdots + 3926 \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 4009 \beta_{17} + 4009 \beta_{16} - 621 \beta_{12} + 4005 \beta_{11} - 4005 \beta_{10} - 2966 \beta_{9} + \cdots + 11860 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 8843 \beta_{17} + 8843 \beta_{16} - 1528 \beta_{15} + 26052 \beta_{14} + 1392 \beta_{13} + \cdots - 22554 \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 46259 \beta_{17} - 46259 \beta_{16} + 8198 \beta_{12} - 46091 \beta_{11} + 46091 \beta_{10} + \cdots - 134036 ) / 2 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 23445 \beta_{17} - 23445 \beta_{16} + 6331 \beta_{15} - 73934 \beta_{14} - 5599 \beta_{13} + \cdots + 65004 \beta_1 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( ( 267947 \beta_{17} + 267947 \beta_{16} - 51936 \beta_{12} + 265795 \beta_{11} - 265795 \beta_{10} + \cdots + 762932 ) / 2 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( ( - 248475 \beta_{17} + 248475 \beta_{16} - 88802 \beta_{15} + 838984 \beta_{14} + 75978 \beta_{13} + \cdots - 750566 \beta_1 ) / 2 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/740\mathbb{Z}\right)^\times\).

\(n\) \(261\) \(297\) \(371\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
149.1
1.87047i
0.807220i
2.04568i
0.269568i
2.35298i
0.981412i
1.78123i
2.40934i
1.45428i
1.45428i
2.40934i
1.78123i
0.981412i
2.35298i
0.269568i
2.04568i
0.807220i
1.87047i
0 3.16661i 0 −1.81068 + 1.31203i 0 3.31356i 0 −7.02744 0
149.2 0 3.05716i 0 0.115303 2.23309i 0 0.448446i 0 −6.34624 0
149.3 0 2.45165i 0 0.873577 + 2.05836i 0 2.99321i 0 −3.01060 0
149.4 0 2.26639i 0 2.06173 + 0.865603i 0 0.375867i 0 −2.13650 0
149.5 0 1.51208i 0 −2.21246 + 0.324035i 0 1.74419i 0 0.713617 0
149.6 0 1.31739i 0 2.22789 0.191060i 0 3.90250i 0 1.26449 0
149.7 0 1.06683i 0 1.49239 1.66516i 0 0.712098i 0 1.86188 0
149.8 0 0.430822i 0 −0.672341 + 2.13259i 0 3.62709i 0 2.81439 0
149.9 0 0.365498i 0 −1.07541 1.96049i 0 2.78997i 0 2.86641 0
149.10 0 0.365498i 0 −1.07541 + 1.96049i 0 2.78997i 0 2.86641 0
149.11 0 0.430822i 0 −0.672341 2.13259i 0 3.62709i 0 2.81439 0
149.12 0 1.06683i 0 1.49239 + 1.66516i 0 0.712098i 0 1.86188 0
149.13 0 1.31739i 0 2.22789 + 0.191060i 0 3.90250i 0 1.26449 0
149.14 0 1.51208i 0 −2.21246 0.324035i 0 1.74419i 0 0.713617 0
149.15 0 2.26639i 0 2.06173 0.865603i 0 0.375867i 0 −2.13650 0
149.16 0 2.45165i 0 0.873577 2.05836i 0 2.99321i 0 −3.01060 0
149.17 0 3.05716i 0 0.115303 + 2.23309i 0 0.448446i 0 −6.34624 0
149.18 0 3.16661i 0 −1.81068 1.31203i 0 3.31356i 0 −7.02744 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 149.18
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 740.2.d.a 18
3.b odd 2 1 6660.2.f.c 18
5.b even 2 1 inner 740.2.d.a 18
5.c odd 4 1 3700.2.a.o 9
5.c odd 4 1 3700.2.a.p 9
15.d odd 2 1 6660.2.f.c 18
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
740.2.d.a 18 1.a even 1 1 trivial
740.2.d.a 18 5.b even 2 1 inner
3700.2.a.o 9 5.c odd 4 1
3700.2.a.p 9 5.c odd 4 1
6660.2.f.c 18 3.b odd 2 1
6660.2.f.c 18 15.d odd 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(740, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{18} \) Copy content Toggle raw display
$3$ \( T^{18} + 36 T^{16} + \cdots + 324 \) Copy content Toggle raw display
$5$ \( T^{18} - 2 T^{17} + \cdots + 1953125 \) Copy content Toggle raw display
$7$ \( T^{18} + 60 T^{16} + \cdots + 6724 \) Copy content Toggle raw display
$11$ \( (T^{9} - 50 T^{7} + \cdots + 1152)^{2} \) Copy content Toggle raw display
$13$ \( T^{18} + 116 T^{16} + \cdots + 9935104 \) Copy content Toggle raw display
$17$ \( T^{18} + 176 T^{16} + \cdots + 4665600 \) Copy content Toggle raw display
$19$ \( (T^{9} - 2 T^{8} + \cdots - 1368)^{2} \) Copy content Toggle raw display
$23$ \( T^{18} + \cdots + 122242535424 \) Copy content Toggle raw display
$29$ \( (T^{9} + 2 T^{8} + \cdots + 236160)^{2} \) Copy content Toggle raw display
$31$ \( (T^{9} - 4 T^{8} - 60 T^{7} + \cdots - 8)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 1)^{9} \) Copy content Toggle raw display
$41$ \( (T^{9} + 2 T^{8} + \cdots - 6128)^{2} \) Copy content Toggle raw display
$43$ \( T^{18} + \cdots + 46269730816 \) Copy content Toggle raw display
$47$ \( T^{18} + \cdots + 561621343396 \) Copy content Toggle raw display
$53$ \( T^{18} + 420 T^{16} + \cdots + 70829056 \) Copy content Toggle raw display
$59$ \( (T^{9} - 4 T^{8} + \cdots - 1829376)^{2} \) Copy content Toggle raw display
$61$ \( (T^{9} - 126 T^{7} + \cdots + 350208)^{2} \) Copy content Toggle raw display
$67$ \( T^{18} + \cdots + 20250428416 \) Copy content Toggle raw display
$71$ \( (T^{9} + 12 T^{8} + \cdots + 50596992)^{2} \) Copy content Toggle raw display
$73$ \( T^{18} + \cdots + 894480807568384 \) Copy content Toggle raw display
$79$ \( (T^{9} + 12 T^{8} + \cdots + 23212328)^{2} \) Copy content Toggle raw display
$83$ \( T^{18} + \cdots + 32497272900 \) Copy content Toggle raw display
$89$ \( (T^{9} - 432 T^{7} + \cdots - 20536704)^{2} \) Copy content Toggle raw display
$97$ \( T^{18} + \cdots + 84\!\cdots\!36 \) Copy content Toggle raw display
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