Properties

Label 35.6.a.d
Level $35$
Weight $6$
Character orbit 35.a
Self dual yes
Analytic conductor $5.613$
Analytic rank $0$
Dimension $4$
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] = [35,6,Mod(1,35)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(35, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("35.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 35 = 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 35.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: \(5.61343369345\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 82x^{2} + 58x + 1168 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_1 + 2) q^{2} + (\beta_{3} - \beta_1 + 4) q^{3} + (3 \beta_{2} - 5 \beta_1 + 15) q^{4} + 25 q^{5} + ( - 11 \beta_{2} - 2 \beta_1 + 29) q^{6} - 49 q^{7} + ( - 6 \beta_{3} + 21 \beta_{2} + \cdots + 133) q^{8}+ \cdots + ( - \beta_{3} - 40 \beta_{2} + \cdots + 165) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 + 2) q^{2} + (\beta_{3} - \beta_1 + 4) q^{3} + (3 \beta_{2} - 5 \beta_1 + 15) q^{4} + 25 q^{5} + ( - 11 \beta_{2} - 2 \beta_1 + 29) q^{6} - 49 q^{7} + ( - 6 \beta_{3} + 21 \beta_{2} + \cdots + 133) q^{8}+ \cdots + ( - 3650 \beta_{3} + 6224 \beta_{2} + \cdots - 106900) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 7 q^{2} + 14 q^{3} + 49 q^{4} + 100 q^{5} + 136 q^{6} - 196 q^{7} + 489 q^{8} + 774 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 7 q^{2} + 14 q^{3} + 49 q^{4} + 100 q^{5} + 136 q^{6} - 196 q^{7} + 489 q^{8} + 774 q^{9} + 175 q^{10} + 770 q^{11} + 840 q^{12} + 58 q^{13} - 343 q^{14} + 350 q^{15} - 615 q^{16} + 2006 q^{17} - 1409 q^{18} + 564 q^{19} + 1225 q^{20} - 686 q^{21} - 1736 q^{22} - 6340 q^{23} - 6244 q^{24} + 2500 q^{25} - 8730 q^{26} - 7438 q^{27} - 2401 q^{28} + 8066 q^{29} + 3400 q^{30} - 5856 q^{31} - 3495 q^{32} - 8130 q^{33} + 3402 q^{34} - 4900 q^{35} - 28759 q^{36} + 29544 q^{37} - 36860 q^{38} + 57466 q^{39} + 12225 q^{40} + 13156 q^{41} - 6664 q^{42} - 5692 q^{43} + 44952 q^{44} + 19350 q^{45} + 30928 q^{46} + 39926 q^{47} - 57156 q^{48} + 9604 q^{49} + 4375 q^{50} + 9830 q^{51} - 23398 q^{52} + 20300 q^{53} + 77292 q^{54} + 19250 q^{55} - 23961 q^{56} - 50876 q^{57} - 22234 q^{58} + 8432 q^{59} + 21000 q^{60} + 30540 q^{61} - 137568 q^{62} - 37926 q^{63} + 37121 q^{64} + 1450 q^{65} - 166756 q^{66} + 32792 q^{67} - 72554 q^{68} - 36540 q^{69} - 8575 q^{70} - 83920 q^{71} - 71795 q^{72} - 75424 q^{73} + 168762 q^{74} + 8750 q^{75} - 125092 q^{76} - 37730 q^{77} - 89204 q^{78} + 129486 q^{79} - 15375 q^{80} + 146308 q^{81} + 247230 q^{82} - 187520 q^{83} - 41160 q^{84} + 50150 q^{85} + 36156 q^{86} - 238670 q^{87} + 475556 q^{88} - 30324 q^{89} - 35225 q^{90} - 2842 q^{91} + 124464 q^{92} + 8256 q^{93} + 245756 q^{94} + 14100 q^{95} + 172340 q^{96} + 180270 q^{97} + 16807 q^{98} - 420668 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} - 82x^{2} + 58x + 1168 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} + \nu - 43 ) / 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} + \nu^{2} - 52\nu - 48 ) / 6 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 3\beta_{2} - \beta _1 + 43 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 6\beta_{3} - 3\beta_{2} + 53\beta _1 + 5 \) Copy content Toggle raw display

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
8.05190
4.73688
−3.86448
−7.92431
−6.05190 15.9755 4.62555 25.0000 −96.6824 −49.0000 165.668 12.2177 −151.298
1.2 −2.73688 −28.3358 −24.5095 25.0000 77.5516 −49.0000 154.660 559.917 −68.4220
1.3 5.86448 26.2268 2.39209 25.0000 153.807 −49.0000 −173.635 444.847 146.612
1.4 9.92431 0.133419 66.4919 25.0000 1.32409 −49.0000 342.308 −242.982 248.108
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \( -1 \)
\(7\) \( +1 \)

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 35.6.a.d 4
3.b odd 2 1 315.6.a.l 4
4.b odd 2 1 560.6.a.v 4
5.b even 2 1 175.6.a.f 4
5.c odd 4 2 175.6.b.f 8
7.b odd 2 1 245.6.a.e 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.6.a.d 4 1.a even 1 1 trivial
175.6.a.f 4 5.b even 2 1
175.6.b.f 8 5.c odd 4 2
245.6.a.e 4 7.b odd 2 1
315.6.a.l 4 3.b odd 2 1
560.6.a.v 4 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} - 7T_{2}^{3} - 64T_{2}^{2} + 250T_{2} + 964 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(35))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - 7 T^{3} + \cdots + 964 \) Copy content Toggle raw display
$3$ \( T^{4} - 14 T^{3} + \cdots - 1584 \) Copy content Toggle raw display
$5$ \( (T - 25)^{4} \) Copy content Toggle raw display
$7$ \( (T + 49)^{4} \) Copy content Toggle raw display
$11$ \( T^{4} + \cdots - 20510223536 \) Copy content Toggle raw display
$13$ \( T^{4} + \cdots + 430417376452 \) Copy content Toggle raw display
$17$ \( T^{4} + \cdots + 617989800868 \) Copy content Toggle raw display
$19$ \( T^{4} + \cdots + 5221683964480 \) Copy content Toggle raw display
$23$ \( T^{4} + \cdots - 19307649395456 \) Copy content Toggle raw display
$29$ \( T^{4} - 8066 T^{3} + \cdots + 831691300 \) Copy content Toggle raw display
$31$ \( T^{4} + \cdots - 738407035699200 \) Copy content Toggle raw display
$37$ \( T^{4} + \cdots - 57\!\cdots\!96 \) Copy content Toggle raw display
$41$ \( T^{4} + \cdots + 225655412958976 \) Copy content Toggle raw display
$43$ \( T^{4} + \cdots + 16\!\cdots\!68 \) Copy content Toggle raw display
$47$ \( T^{4} + \cdots - 16\!\cdots\!88 \) Copy content Toggle raw display
$53$ \( T^{4} + \cdots - 15\!\cdots\!96 \) Copy content Toggle raw display
$59$ \( T^{4} + \cdots + 13\!\cdots\!40 \) Copy content Toggle raw display
$61$ \( T^{4} + \cdots - 25\!\cdots\!36 \) Copy content Toggle raw display
$67$ \( T^{4} + \cdots - 74\!\cdots\!96 \) Copy content Toggle raw display
$71$ \( T^{4} + \cdots - 17\!\cdots\!24 \) Copy content Toggle raw display
$73$ \( T^{4} + \cdots - 32\!\cdots\!52 \) Copy content Toggle raw display
$79$ \( T^{4} + \cdots + 39\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{4} + \cdots - 29\!\cdots\!64 \) Copy content Toggle raw display
$89$ \( T^{4} + \cdots + 21\!\cdots\!20 \) Copy content Toggle raw display
$97$ \( T^{4} + \cdots - 75\!\cdots\!24 \) Copy content Toggle raw display
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