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Properties

Label 35.3.i.a
Level $35$
Weight $3$
Character orbit 35.i
Analytic conductor $0.954$
Analytic rank $0$
Dimension $12$
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] = [35,3,Mod(19,35)]
 
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([3, 5]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("35.19");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 35 = 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 35.i (of order \(6\), 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: \(0.953680925261\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 15x^{10} + 180x^{8} - 669x^{6} + 1980x^{4} - 135x^{2} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + ( - \beta_{11} - \beta_1) q^{3} + (\beta_{7} + \beta_{3} + 1) q^{4} + ( - \beta_{9} + \beta_{8} + \cdots + \beta_1) q^{5}+ \cdots + (\beta_{7} - \beta_{6} + \cdots - \beta_{2}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + ( - \beta_{11} - \beta_1) q^{3} + (\beta_{7} + \beta_{3} + 1) q^{4} + ( - \beta_{9} + \beta_{8} + \cdots + \beta_1) q^{5}+ \cdots + (3 \beta_{6} + 3 \beta_{5} - 4 \beta_{2} + 33) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 6 q^{4} - 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 6 q^{4} - 18 q^{9} - 54 q^{10} + 6 q^{11} - 66 q^{14} + 48 q^{15} - 6 q^{16} + 18 q^{19} + 12 q^{21} + 216 q^{24} + 18 q^{25} + 18 q^{26} + 48 q^{30} - 108 q^{31} + 222 q^{35} - 204 q^{36} - 240 q^{39} - 162 q^{40} - 42 q^{44} - 216 q^{45} + 114 q^{46} - 324 q^{49} - 192 q^{50} + 180 q^{51} + 252 q^{54} + 336 q^{56} + 396 q^{59} + 384 q^{60} - 108 q^{61} + 372 q^{64} - 54 q^{65} - 108 q^{66} + 300 q^{70} + 192 q^{71} - 594 q^{74} - 216 q^{75} - 192 q^{79} - 504 q^{80} + 294 q^{81} - 1200 q^{84} - 192 q^{85} - 384 q^{86} + 684 q^{89} - 72 q^{91} + 990 q^{94} + 288 q^{95} - 540 q^{96} + 396 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 15x^{10} + 180x^{8} - 669x^{6} + 1980x^{4} - 135x^{2} + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{10} + 12\nu^{8} - 144\nu^{6} + 132\nu^{4} - 9\nu^{2} - 7767 ) / 1575 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 12\nu^{10} - 179\nu^{8} + 2148\nu^{6} - 7884\nu^{4} + 23628\nu^{2} - 1611 ) / 1575 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -12\nu^{11} + 179\nu^{9} - 2148\nu^{7} + 7884\nu^{5} - 23628\nu^{3} + 1611\nu ) / 1575 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 53\nu^{10} - 811\nu^{8} + 9767\nu^{6} - 37971\nu^{4} + 111777\nu^{2} - 15069 ) / 3150 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -62\nu^{10} + 919\nu^{8} - 10958\nu^{6} + 39159\nu^{4} - 111858\nu^{2} - 7269 ) / 3150 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -12\nu^{10} + 179\nu^{8} - 2148\nu^{6} + 7884\nu^{4} - 23313\nu^{2} + 36 ) / 315 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 53\nu^{11} - 801\nu^{9} + 9627\nu^{7} - 36471\nu^{5} + 108207\nu^{3} - 15939\nu ) / 1350 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -404\nu^{11} + 6003\nu^{9} - 71826\nu^{7} + 259653\nu^{5} - 757746\nu^{3} - 58743\nu ) / 9450 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( -583\nu^{11} + 8781\nu^{9} - 105477\nu^{7} + 396681\nu^{5} - 1179567\nu^{3} + 168489\nu ) / 9450 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( -622\nu^{11} + 9249\nu^{9} - 110778\nu^{7} + 401829\nu^{5} - 1179918\nu^{3} - 86229\nu ) / 9450 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{7} + 5\beta_{3} + 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{11} + \beta_{10} - 2\beta_{9} + \beta_{8} - 9\beta_{4} + 9\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 12\beta_{7} - 2\beta_{6} + 4\beta_{5} + 45\beta_{3} - 12\beta_{2} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 14\beta_{11} + 28\beta_{10} - 16\beta_{9} + 32\beta_{8} - 93\beta_{4} + 2\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 30\beta_{6} + 30\beta_{5} - 135\beta_{2} - 453 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -165\beta_{11} + 165\beta_{10} + 195\beta_{9} + 195\beta_{8} - 933\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -1488\beta_{7} + 720\beta_{6} - 360\beta_{5} - 4785\beta_{3} - 4785 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( -3696\beta_{11} - 1848\beta_{10} + 4416\beta_{9} - 2208\beta_{8} + 10737\beta_{4} - 10377\beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( -16281\beta_{7} + 4056\beta_{6} - 8112\beta_{5} - 51525\beta_{3} + 16281\beta_{2} \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( -20337\beta_{11} - 40674\beta_{10} + 24393\beta_{9} - 48786\beta_{8} + 116649\beta_{4} - 4056\beta_1 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(22\) \(31\)
\(\chi(n)\) \(-1\) \(-\beta_{3}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1
−2.85853 − 1.65037i
−1.74002 − 1.00460i
−0.226181 − 0.130586i
0.226181 + 0.130586i
1.74002 + 1.00460i
2.85853 + 1.65037i
−2.85853 + 1.65037i
−1.74002 + 1.00460i
−0.226181 + 0.130586i
0.226181 − 0.130586i
1.74002 − 1.00460i
2.85853 − 1.65037i
−2.85853 − 1.65037i 2.20085 + 3.81198i 3.44745 + 5.97116i 3.20324 − 3.83918i − 14.5289i 5.32175 + 4.54742i − 9.55533i −5.18747 + 8.98496i −15.4926 + 5.68786i
19.2 −1.74002 − 1.00460i −0.784824 − 1.35935i 0.0184439 + 0.0319457i −4.64034 − 1.86205i 3.15374i 1.38623 − 6.86137i 7.96269i 3.26810 − 5.66052i 6.20367 + 7.90169i
19.3 −0.226181 − 0.130586i −1.88157 − 3.25898i −1.96589 − 3.40503i 4.98089 − 0.436783i 0.982826i 1.66053 + 6.80019i 2.07156i −2.58064 + 4.46979i −1.18362 − 0.551640i
19.4 0.226181 + 0.130586i 1.88157 + 3.25898i −1.96589 − 3.40503i 2.11218 + 4.53196i 0.982826i −1.66053 − 6.80019i − 2.07156i −2.58064 + 4.46979i −0.114075 + 1.30086i
19.5 1.74002 + 1.00460i 0.784824 + 1.35935i 0.0184439 + 0.0319457i −3.93275 − 3.08763i 3.15374i −1.38623 + 6.86137i − 7.96269i 3.26810 − 5.66052i −3.74123 − 9.32338i
19.6 2.85853 + 1.65037i −2.20085 − 3.81198i 3.44745 + 5.97116i −1.72321 + 4.69367i − 14.5289i −5.32175 − 4.54742i 9.55533i −5.18747 + 8.98496i −12.6721 + 10.5731i
24.1 −2.85853 + 1.65037i 2.20085 − 3.81198i 3.44745 − 5.97116i 3.20324 + 3.83918i 14.5289i 5.32175 − 4.54742i 9.55533i −5.18747 − 8.98496i −15.4926 − 5.68786i
24.2 −1.74002 + 1.00460i −0.784824 + 1.35935i 0.0184439 − 0.0319457i −4.64034 + 1.86205i − 3.15374i 1.38623 + 6.86137i − 7.96269i 3.26810 + 5.66052i 6.20367 − 7.90169i
24.3 −0.226181 + 0.130586i −1.88157 + 3.25898i −1.96589 + 3.40503i 4.98089 + 0.436783i − 0.982826i 1.66053 − 6.80019i − 2.07156i −2.58064 − 4.46979i −1.18362 + 0.551640i
24.4 0.226181 − 0.130586i 1.88157 − 3.25898i −1.96589 + 3.40503i 2.11218 − 4.53196i − 0.982826i −1.66053 + 6.80019i 2.07156i −2.58064 − 4.46979i −0.114075 − 1.30086i
24.5 1.74002 − 1.00460i 0.784824 − 1.35935i 0.0184439 − 0.0319457i −3.93275 + 3.08763i − 3.15374i −1.38623 − 6.86137i 7.96269i 3.26810 + 5.66052i −3.74123 + 9.32338i
24.6 2.85853 − 1.65037i −2.20085 + 3.81198i 3.44745 − 5.97116i −1.72321 − 4.69367i 14.5289i −5.32175 + 4.54742i − 9.55533i −5.18747 − 8.98496i −12.6721 − 10.5731i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 19.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
7.d odd 6 1 inner
35.i odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 35.3.i.a ✓ 12
3.b odd 2 1 315.3.bi.c 12
4.b odd 2 1 560.3.br.a 12
5.b even 2 1 inner 35.3.i.a ✓ 12
5.c odd 4 2 175.3.i.c 12
7.b odd 2 1 245.3.i.d 12
7.c even 3 1 245.3.c.a 12
7.c even 3 1 245.3.i.d 12
7.d odd 6 1 inner 35.3.i.a ✓ 12
7.d odd 6 1 245.3.c.a 12
15.d odd 2 1 315.3.bi.c 12
20.d odd 2 1 560.3.br.a 12
21.g even 6 1 315.3.bi.c 12
28.f even 6 1 560.3.br.a 12
35.c odd 2 1 245.3.i.d 12
35.i odd 6 1 inner 35.3.i.a ✓ 12
35.i odd 6 1 245.3.c.a 12
35.j even 6 1 245.3.c.a 12
35.j even 6 1 245.3.i.d 12
35.k even 12 2 175.3.i.c 12
105.p even 6 1 315.3.bi.c 12
140.s even 6 1 560.3.br.a 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.3.i.a ✓ 12 1.a even 1 1 trivial
35.3.i.a ✓ 12 5.b even 2 1 inner
35.3.i.a ✓ 12 7.d odd 6 1 inner
35.3.i.a ✓ 12 35.i odd 6 1 inner
175.3.i.c 12 5.c odd 4 2
175.3.i.c 12 35.k even 12 2
245.3.c.a 12 7.c even 3 1
245.3.c.a 12 7.d odd 6 1
245.3.c.a 12 35.i odd 6 1
245.3.c.a 12 35.j even 6 1
245.3.i.d 12 7.b odd 2 1
245.3.i.d 12 7.c even 3 1
245.3.i.d 12 35.c odd 2 1
245.3.i.d 12 35.j even 6 1
315.3.bi.c 12 3.b odd 2 1
315.3.bi.c 12 15.d odd 2 1
315.3.bi.c 12 21.g even 6 1
315.3.bi.c 12 105.p even 6 1
560.3.br.a 12 4.b odd 2 1
560.3.br.a 12 20.d odd 2 1
560.3.br.a 12 28.f even 6 1
560.3.br.a 12 140.s even 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(35, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} - 15 T^{10} + \cdots + 9 \) Copy content Toggle raw display
$3$ \( T^{12} + 36 T^{10} + \cdots + 456976 \) Copy content Toggle raw display
$5$ \( T^{12} + \cdots + 244140625 \) Copy content Toggle raw display
$7$ \( T^{12} + \cdots + 13841287201 \) Copy content Toggle raw display
$11$ \( (T^{6} - 3 T^{5} + \cdots + 784)^{2} \) Copy content Toggle raw display
$13$ \( (T^{6} - 489 T^{4} + \cdots - 4104676)^{2} \) Copy content Toggle raw display
$17$ \( T^{12} + \cdots + 119168138285056 \) Copy content Toggle raw display
$19$ \( (T^{6} - 9 T^{5} + \cdots + 2333772)^{2} \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 59129994297609 \) Copy content Toggle raw display
$29$ \( (T^{3} - 543 T - 4508)^{4} \) Copy content Toggle raw display
$31$ \( (T^{6} + 54 T^{5} + \cdots + 12288)^{2} \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 79\!\cdots\!04 \) Copy content Toggle raw display
$41$ \( (T^{6} + 1077 T^{4} + \cdots + 30432675)^{2} \) Copy content Toggle raw display
$43$ \( (T^{6} + 6792 T^{4} + \cdots + 2881200)^{2} \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 20\!\cdots\!96 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( (T^{6} - 198 T^{5} + \cdots + 139491478272)^{2} \) Copy content Toggle raw display
$61$ \( (T^{6} + 54 T^{5} + \cdots + 4280171952)^{2} \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 15\!\cdots\!44 \) Copy content Toggle raw display
$71$ \( (T^{3} - 48 T^{2} + \cdots + 15508)^{4} \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 21\!\cdots\!16 \) Copy content Toggle raw display
$79$ \( (T^{6} + 96 T^{5} + \cdots + 330803344)^{2} \) Copy content Toggle raw display
$83$ \( (T^{6} - 23520 T^{4} + \cdots - 118402057216)^{2} \) Copy content Toggle raw display
$89$ \( (T^{6} - 342 T^{5} + \cdots + 285265069488)^{2} \) Copy content Toggle raw display
$97$ \( (T^{6} - 18228 T^{4} + \cdots - 207129664)^{2} \) Copy content Toggle raw display
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