Properties

Label 35.4.e.b
Level $35$
Weight $4$
Character orbit 35.e
Analytic conductor $2.065$
Analytic rank $0$
Dimension $4$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [35,4,Mod(11,35)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(35, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 4]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("35.11");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 35 = 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 35.e (of order \(3\), 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: \(2.06506685020\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{-3})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 2x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{3} - 3 \beta_{2} + \beta_1) q^{2} + (\beta_{2} + 3 \beta_1 + 1) q^{3} + ( - 3 \beta_{2} + 6 \beta_1 - 3) q^{4} + 5 \beta_{2} q^{5} + ( - 8 \beta_{3} - 3) q^{6} + (2 \beta_{3} - 5 \beta_{2} - 11 \beta_1 + 3) q^{7} + ( - 13 \beta_{3} + 3) q^{8} + (6 \beta_{3} - 8 \beta_{2} + 6 \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{3} - 3 \beta_{2} + \beta_1) q^{2} + (\beta_{2} + 3 \beta_1 + 1) q^{3} + ( - 3 \beta_{2} + 6 \beta_1 - 3) q^{4} + 5 \beta_{2} q^{5} + ( - 8 \beta_{3} - 3) q^{6} + (2 \beta_{3} - 5 \beta_{2} - 11 \beta_1 + 3) q^{7} + ( - 13 \beta_{3} + 3) q^{8} + (6 \beta_{3} - 8 \beta_{2} + 6 \beta_1) q^{9} + (15 \beta_{2} - 5 \beta_1 + 15) q^{10} + ( - 14 \beta_{2} - 10 \beta_1 - 14) q^{11} + ( - 3 \beta_{3} + 33 \beta_{2} - 3 \beta_1) q^{12} + ( - 10 \beta_{3} - 18) q^{13} + (42 \beta_{3} - 28 \beta_{2} + \cdots + 7) q^{14}+ \cdots + ( - 4 \beta_{3} + 8) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 6 q^{2} + 2 q^{3} - 6 q^{4} - 10 q^{5} - 12 q^{6} + 22 q^{7} + 12 q^{8} + 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 6 q^{2} + 2 q^{3} - 6 q^{4} - 10 q^{5} - 12 q^{6} + 22 q^{7} + 12 q^{8} + 16 q^{9} + 30 q^{10} - 28 q^{11} - 66 q^{12} - 72 q^{13} + 84 q^{14} - 20 q^{15} + 14 q^{16} + 76 q^{17} - 72 q^{18} - 160 q^{19} + 60 q^{20} + 134 q^{21} - 88 q^{22} + 22 q^{23} + 162 q^{24} - 50 q^{25} - 148 q^{26} - 4 q^{27} + 138 q^{28} - 500 q^{29} + 30 q^{30} + 132 q^{31} - 42 q^{32} + 148 q^{33} + 888 q^{34} + 20 q^{35} - 384 q^{36} + 416 q^{37} + 376 q^{38} + 84 q^{39} - 30 q^{40} - 212 q^{41} - 546 q^{42} - 1332 q^{43} + 156 q^{44} + 80 q^{45} + 402 q^{46} + 196 q^{47} - 260 q^{48} - 230 q^{49} - 300 q^{50} + 572 q^{51} + 348 q^{52} + 952 q^{53} - 402 q^{54} + 280 q^{55} - 714 q^{56} - 944 q^{57} - 510 q^{58} - 840 q^{59} - 330 q^{60} - 98 q^{61} - 8 q^{62} + 544 q^{63} - 1204 q^{64} + 180 q^{65} - 236 q^{66} + 1286 q^{67} + 1524 q^{68} + 2852 q^{69} + 210 q^{70} + 2128 q^{71} - 264 q^{72} + 172 q^{73} - 1792 q^{74} + 50 q^{75} - 288 q^{76} - 724 q^{77} + 856 q^{78} + 1240 q^{79} + 70 q^{80} + 754 q^{81} - 438 q^{82} - 3812 q^{83} - 156 q^{84} - 760 q^{85} - 2018 q^{86} - 970 q^{87} - 604 q^{88} - 650 q^{89} + 720 q^{90} - 996 q^{91} + 5484 q^{92} - 1332 q^{93} - 332 q^{94} - 800 q^{95} - 1722 q^{96} - 1256 q^{97} + 3066 q^{98} + 32 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 2x^{2} + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} ) / 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 2\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{3} \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/35\mathbb{Z}\right)^\times\).

\(n\) \(22\) \(31\)
\(\chi(n)\) \(1\) \(\beta_{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
11.1
0.707107 1.22474i
−0.707107 + 1.22474i
0.707107 + 1.22474i
−0.707107 1.22474i
0.792893 + 1.37333i 2.62132 4.54026i 2.74264 4.75039i −2.50000 4.33013i 8.31371 −5.10660 + 17.8023i 21.3848 −0.242641 0.420266i 3.96447 6.86666i
11.2 2.20711 + 3.82282i −1.62132 + 2.80821i −5.74264 + 9.94655i −2.50000 4.33013i −14.3137 16.1066 9.14207i −15.3848 8.24264 + 14.2767i 11.0355 19.1141i
16.1 0.792893 1.37333i 2.62132 + 4.54026i 2.74264 + 4.75039i −2.50000 + 4.33013i 8.31371 −5.10660 17.8023i 21.3848 −0.242641 + 0.420266i 3.96447 + 6.86666i
16.2 2.20711 3.82282i −1.62132 2.80821i −5.74264 9.94655i −2.50000 + 4.33013i −14.3137 16.1066 + 9.14207i −15.3848 8.24264 14.2767i 11.0355 + 19.1141i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 35.4.e.b 4
3.b odd 2 1 315.4.j.c 4
4.b odd 2 1 560.4.q.i 4
5.b even 2 1 175.4.e.c 4
5.c odd 4 2 175.4.k.c 8
7.b odd 2 1 245.4.e.l 4
7.c even 3 1 inner 35.4.e.b 4
7.c even 3 1 245.4.a.g 2
7.d odd 6 1 245.4.a.h 2
7.d odd 6 1 245.4.e.l 4
21.g even 6 1 2205.4.a.bg 2
21.h odd 6 1 315.4.j.c 4
21.h odd 6 1 2205.4.a.bf 2
28.g odd 6 1 560.4.q.i 4
35.i odd 6 1 1225.4.a.v 2
35.j even 6 1 175.4.e.c 4
35.j even 6 1 1225.4.a.x 2
35.l odd 12 2 175.4.k.c 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.4.e.b 4 1.a even 1 1 trivial
35.4.e.b 4 7.c even 3 1 inner
175.4.e.c 4 5.b even 2 1
175.4.e.c 4 35.j even 6 1
175.4.k.c 8 5.c odd 4 2
175.4.k.c 8 35.l odd 12 2
245.4.a.g 2 7.c even 3 1
245.4.a.h 2 7.d odd 6 1
245.4.e.l 4 7.b odd 2 1
245.4.e.l 4 7.d odd 6 1
315.4.j.c 4 3.b odd 2 1
315.4.j.c 4 21.h odd 6 1
560.4.q.i 4 4.b odd 2 1
560.4.q.i 4 28.g odd 6 1
1225.4.a.v 2 35.i odd 6 1
1225.4.a.x 2 35.j even 6 1
2205.4.a.bf 2 21.h odd 6 1
2205.4.a.bg 2 21.g even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} - 6T_{2}^{3} + 29T_{2}^{2} - 42T_{2} + 49 \) acting on \(S_{4}^{\mathrm{new}}(35, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - 6 T^{3} + \cdots + 49 \) Copy content Toggle raw display
$3$ \( T^{4} - 2 T^{3} + \cdots + 289 \) Copy content Toggle raw display
$5$ \( (T^{2} + 5 T + 25)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} - 22 T^{3} + \cdots + 117649 \) Copy content Toggle raw display
$11$ \( T^{4} + 28 T^{3} + \cdots + 16 \) Copy content Toggle raw display
$13$ \( (T^{2} + 36 T + 124)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} - 76 T^{3} + \cdots + 19254544 \) Copy content Toggle raw display
$19$ \( T^{4} + 160 T^{3} + \cdots + 25482304 \) Copy content Toggle raw display
$23$ \( T^{4} - 22 T^{3} + \cdots + 742944049 \) Copy content Toggle raw display
$29$ \( (T^{2} + 250 T + 8425)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} - 132 T^{3} + \cdots + 244734736 \) Copy content Toggle raw display
$37$ \( T^{4} - 416 T^{3} + \cdots + 39337984 \) Copy content Toggle raw display
$41$ \( (T^{2} + 106 T + 1009)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 666 T + 110839)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} - 196 T^{3} + \cdots + 1993744 \) Copy content Toggle raw display
$53$ \( T^{4} + \cdots + 34688317504 \) Copy content Toggle raw display
$59$ \( T^{4} + \cdots + 1217172544 \) Copy content Toggle raw display
$61$ \( T^{4} + \cdots + 419125465201 \) Copy content Toggle raw display
$67$ \( T^{4} + \cdots + 163157021329 \) Copy content Toggle raw display
$71$ \( (T^{2} - 1064 T + 38024)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} - 172 T^{3} + \cdots + 418284304 \) Copy content Toggle raw display
$79$ \( T^{4} + \cdots + 85736524864 \) Copy content Toggle raw display
$83$ \( (T^{2} + 1906 T + 908159)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} + \cdots + 10884122929 \) Copy content Toggle raw display
$97$ \( (T^{2} + 628 T - 401404)^{2} \) Copy content Toggle raw display
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