Properties

Label 35.4.f.b
Level $35$
Weight $4$
Character orbit 35.f
Analytic conductor $2.065$
Analytic rank $0$
Dimension $16$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [35,4,Mod(13,35)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(35, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([3, 2]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("35.13");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 35 = 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 35.f (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: \(2.06506685020\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 10654x^{12} + 22102125x^{8} + 5700572500x^{4} + 44626562500 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2}\cdot 5^{2} \)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{7} q^{2} - \beta_{8} q^{3} + (\beta_{9} - \beta_{7} + \cdots - 4 \beta_{3}) q^{4}+ \cdots + ( - \beta_{9} - 2 \beta_{7} + 2 \beta_{6} + \cdots + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{7} q^{2} - \beta_{8} q^{3} + (\beta_{9} - \beta_{7} + \cdots - 4 \beta_{3}) q^{4}+ \cdots + ( - 30 \beta_{9} + 112 \beta_{7} + \cdots + 32) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 4 q^{2} + 32 q^{7} + 176 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 4 q^{2} + 32 q^{7} + 176 q^{8} - 152 q^{11} - 480 q^{15} - 504 q^{16} + 288 q^{18} + 328 q^{21} + 348 q^{22} - 72 q^{23} - 160 q^{25} - 528 q^{28} + 1780 q^{30} + 432 q^{32} + 160 q^{35} + 344 q^{36} - 256 q^{37} - 1300 q^{42} - 312 q^{43} - 1856 q^{46} - 20 q^{50} + 696 q^{51} + 1768 q^{53} + 1304 q^{56} - 3920 q^{57} - 4764 q^{58} - 2000 q^{60} + 2544 q^{63} - 1000 q^{65} + 4504 q^{67} - 180 q^{70} + 6368 q^{71} + 7848 q^{72} - 3016 q^{77} + 5340 q^{78} - 3088 q^{81} - 2280 q^{85} - 9336 q^{86} - 2048 q^{88} + 1608 q^{91} - 6328 q^{92} - 3960 q^{93} + 1240 q^{95} + 3308 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 10654x^{12} + 22102125x^{8} + 5700572500x^{4} + 44626562500 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 13637\nu^{12} + 141541523\nu^{8} + 268260815950\nu^{4} + 33643813420000 ) / 2595065895000 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 103709\nu^{14} + 1108253436\nu^{10} + 2329722745125\nu^{6} + 699405434108750\nu^{2} ) / 1934850601125000 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 922187 \nu^{14} + 16951545 \nu^{12} + 10180112673 \nu^{10} + 177015939555 \nu^{8} + \cdots - 45\!\cdots\!00 ) / 13\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 2712347 \nu^{14} + 28252575 \nu^{12} - 29527060063 \nu^{10} + 295026565925 \nu^{8} + \cdots - 54\!\cdots\!00 ) / 21\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 26023741 \nu^{14} + 53408225 \nu^{12} + 277997050989 \nu^{10} + 614189471025 \nu^{8} + \cdots + 41\!\cdots\!00 ) / 13\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 26023741 \nu^{14} + 53408225 \nu^{12} - 277997050989 \nu^{10} + 614189471025 \nu^{8} + \cdots + 41\!\cdots\!00 ) / 13\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 103709\nu^{15} + 1108253436\nu^{11} + 2329722745125\nu^{7} + 699405434108750\nu^{3} ) / 1934850601125000 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 11171152 \nu^{14} - 118324616783 \nu^{10} - 240090289254625 \nu^{6} - 52\!\cdots\!50 \nu^{2} ) / 10\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 274103521 \nu^{15} - 616072600 \nu^{13} - 2911318125984 \nu^{11} + \cdots - 52\!\cdots\!00 \nu ) / 85\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 274103521 \nu^{15} + 616072600 \nu^{13} - 2911318125984 \nu^{11} + \cdots + 52\!\cdots\!00 \nu ) / 85\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 12698211 \nu^{15} - 46505420 \nu^{13} - 135427260114 \nu^{11} + \cdots - 15\!\cdots\!00 \nu ) / 34\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 12698211 \nu^{15} - 46505420 \nu^{13} + 135427260114 \nu^{11} - 492481264080 \nu^{9} + \cdots - 15\!\cdots\!00 \nu ) / 34\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 753130203 \nu^{15} + 2863109275 \nu^{13} + 8000341557387 \nu^{11} + \cdots + 11\!\cdots\!00 \nu ) / 17\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 753130203 \nu^{15} - 2863109275 \nu^{13} + 8000341557387 \nu^{11} + \cdots - 11\!\cdots\!00 \nu ) / 17\!\cdots\!00 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{9} + 2\beta_{7} - 2\beta_{6} + 2\beta_{5} - 2\beta_{4} + 41\beta_{3} - 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -4\beta_{15} - 4\beta_{14} - 7\beta_{13} + 7\beta_{12} - 8\beta_{11} - 8\beta_{10} + 67\beta_{8} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 36\beta_{7} + 36\beta_{6} - 201\beta_{5} - 201\beta_{4} - 201\beta_{3} + 93\beta_{2} - 2635 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -552\beta_{15} + 552\beta_{14} + 969\beta_{13} + 969\beta_{12} + 546\beta_{11} - 546\beta_{10} - 5011\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -9439\beta_{9} + 4672\beta_{7} - 4672\beta_{6} - 18173\beta_{5} + 18173\beta_{4} - 210644\beta_{3} + 18173 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 59896 \beta_{15} + 59896 \beta_{14} + 40408 \beta_{13} - 40408 \beta_{12} + 95537 \beta_{11} + \cdots - 401203 \beta_{8} \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 808956 \beta_{7} + 808956 \beta_{6} + 1619979 \beta_{5} + 1619979 \beta_{4} + 1619979 \beta_{3} + \cdots + 16461715 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 5887608 \beta_{15} - 5887608 \beta_{14} - 8908851 \beta_{13} - 8908851 \beta_{12} + \cdots + 33582919 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 86195731 \beta_{9} - 91446988 \beta_{7} + 91446988 \beta_{6} + 144279767 \beta_{5} - 144279767 \beta_{4} + \cdots - 144279767 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 552397984 \beta_{15} - 552397984 \beta_{14} - 255196582 \beta_{13} + 255196582 \beta_{12} + \cdots + 2889367387 \beta_{8} \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 9104513724 \beta_{7} - 9104513724 \beta_{6} - 12860150391 \beta_{5} - 12860150391 \beta_{4} + \cdots - 121492467235 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 50630722632 \beta_{15} + 50630722632 \beta_{14} + 73405265679 \beta_{13} + 73405265679 \beta_{12} + \cdots - 252648048151 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 715809307699 \beta_{9} + 858779499652 \beta_{7} - 858779499652 \beta_{6} - 1147049021243 \beta_{5} + \cdots + 1147049021243 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 4584496157536 \beta_{15} + 4584496157536 \beta_{14} + 1866558256378 \beta_{13} + \cdots - 22296858454723 \beta_{8} \) Copy content Toggle raw display

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
13.1
−4.94129 4.94129i
4.94129 + 4.94129i
−2.91939 2.91939i
2.91939 + 2.91939i
−6.68128 6.68128i
6.68128 + 6.68128i
−1.19219 1.19219i
1.19219 + 1.19219i
−4.94129 + 4.94129i
4.94129 4.94129i
−2.91939 + 2.91939i
2.91939 2.91939i
−6.68128 + 6.68128i
6.68128 6.68128i
−1.19219 + 1.19219i
1.19219 1.19219i
−3.57033 + 3.57033i −4.94129 + 4.94129i 17.4944i 3.01167 + 10.7671i 35.2840i 0.826888 18.5018i 33.8983 + 33.8983i 21.8328i −49.1946 27.6893i
13.2 −3.57033 + 3.57033i 4.94129 4.94129i 17.4944i −3.01167 10.7671i 35.2840i 18.5018 0.826888i 33.8983 + 33.8983i 21.8328i 49.1946 + 27.6893i
13.3 0.172516 0.172516i −2.91939 + 2.91939i 7.94048i −9.78787 + 5.40347i 1.00728i 18.1737 3.56610i 2.74998 + 2.74998i 9.95432i −0.756378 + 2.62074i
13.4 0.172516 0.172516i 2.91939 2.91939i 7.94048i 9.78787 5.40347i 1.00728i 3.56610 18.1737i 2.74998 + 2.74998i 9.95432i 0.756378 2.62074i
13.5 1.31781 1.31781i −6.68128 + 6.68128i 4.52674i 11.1616 + 0.646864i 17.6093i −17.1776 + 6.92319i 16.5079 + 16.5079i 62.2789i 15.5614 13.8565i
13.6 1.31781 1.31781i 6.68128 6.68128i 4.52674i −11.1616 0.646864i 17.6093i −6.92319 + 17.1776i 16.5079 + 16.5079i 62.2789i −15.5614 + 13.8565i
13.7 3.08000 3.08000i −1.19219 + 1.19219i 10.9728i −0.738762 11.1559i 7.34390i 12.6030 + 13.5707i −9.15614 9.15614i 24.1574i −36.6356 32.0848i
13.8 3.08000 3.08000i 1.19219 1.19219i 10.9728i 0.738762 + 11.1559i 7.34390i −13.5707 12.6030i −9.15614 9.15614i 24.1574i 36.6356 + 32.0848i
27.1 −3.57033 3.57033i −4.94129 4.94129i 17.4944i 3.01167 10.7671i 35.2840i 0.826888 + 18.5018i 33.8983 33.8983i 21.8328i −49.1946 + 27.6893i
27.2 −3.57033 3.57033i 4.94129 + 4.94129i 17.4944i −3.01167 + 10.7671i 35.2840i 18.5018 + 0.826888i 33.8983 33.8983i 21.8328i 49.1946 27.6893i
27.3 0.172516 + 0.172516i −2.91939 2.91939i 7.94048i −9.78787 5.40347i 1.00728i 18.1737 + 3.56610i 2.74998 2.74998i 9.95432i −0.756378 2.62074i
27.4 0.172516 + 0.172516i 2.91939 + 2.91939i 7.94048i 9.78787 + 5.40347i 1.00728i 3.56610 + 18.1737i 2.74998 2.74998i 9.95432i 0.756378 + 2.62074i
27.5 1.31781 + 1.31781i −6.68128 6.68128i 4.52674i 11.1616 0.646864i 17.6093i −17.1776 6.92319i 16.5079 16.5079i 62.2789i 15.5614 + 13.8565i
27.6 1.31781 + 1.31781i 6.68128 + 6.68128i 4.52674i −11.1616 + 0.646864i 17.6093i −6.92319 17.1776i 16.5079 16.5079i 62.2789i −15.5614 13.8565i
27.7 3.08000 + 3.08000i −1.19219 1.19219i 10.9728i −0.738762 + 11.1559i 7.34390i 12.6030 13.5707i −9.15614 + 9.15614i 24.1574i −36.6356 + 32.0848i
27.8 3.08000 + 3.08000i 1.19219 + 1.19219i 10.9728i 0.738762 11.1559i 7.34390i −13.5707 + 12.6030i −9.15614 + 9.15614i 24.1574i 36.6356 32.0848i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 13.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
7.b odd 2 1 inner
35.f even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 35.4.f.b 16
5.b even 2 1 175.4.f.g 16
5.c odd 4 1 inner 35.4.f.b 16
5.c odd 4 1 175.4.f.g 16
7.b odd 2 1 inner 35.4.f.b 16
7.c even 3 2 245.4.l.b 32
7.d odd 6 2 245.4.l.b 32
35.c odd 2 1 175.4.f.g 16
35.f even 4 1 inner 35.4.f.b 16
35.f even 4 1 175.4.f.g 16
35.k even 12 2 245.4.l.b 32
35.l odd 12 2 245.4.l.b 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.4.f.b 16 1.a even 1 1 trivial
35.4.f.b 16 5.c odd 4 1 inner
35.4.f.b 16 7.b odd 2 1 inner
35.4.f.b 16 35.f even 4 1 inner
175.4.f.g 16 5.b even 2 1
175.4.f.g 16 5.c odd 4 1
175.4.f.g 16 35.c odd 2 1
175.4.f.g 16 35.f even 4 1
245.4.l.b 32 7.c even 3 2
245.4.l.b 32 7.d odd 6 2
245.4.l.b 32 35.k even 12 2
245.4.l.b 32 35.l odd 12 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} - 2T_{2}^{7} + 2T_{2}^{6} - 20T_{2}^{5} + 549T_{2}^{4} - 1538T_{2}^{3} + 2178T_{2}^{2} - 660T_{2} + 100 \) acting on \(S_{4}^{\mathrm{new}}(35, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{8} - 2 T^{7} + \cdots + 100)^{2} \) Copy content Toggle raw display
$3$ \( T^{16} + \cdots + 44626562500 \) Copy content Toggle raw display
$5$ \( T^{16} + \cdots + 59\!\cdots\!25 \) Copy content Toggle raw display
$7$ \( T^{16} + \cdots + 19\!\cdots\!01 \) Copy content Toggle raw display
$11$ \( (T^{4} + 38 T^{3} + \cdots - 199336)^{4} \) Copy content Toggle raw display
$13$ \( T^{16} + \cdots + 68\!\cdots\!00 \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 30\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( (T^{8} + \cdots + 132690195125000)^{2} \) Copy content Toggle raw display
$23$ \( (T^{8} + \cdots + 1537004857600)^{2} \) Copy content Toggle raw display
$29$ \( (T^{8} + \cdots + 12\!\cdots\!76)^{2} \) Copy content Toggle raw display
$31$ \( (T^{8} + \cdots + 353301362000000)^{2} \) Copy content Toggle raw display
$37$ \( (T^{8} + \cdots + 13\!\cdots\!00)^{2} \) Copy content Toggle raw display
$41$ \( (T^{8} + \cdots + 37\!\cdots\!00)^{2} \) Copy content Toggle raw display
$43$ \( (T^{8} + \cdots + 12\!\cdots\!00)^{2} \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 12\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( (T^{8} + \cdots + 31\!\cdots\!00)^{2} \) Copy content Toggle raw display
$59$ \( (T^{8} + \cdots + 18\!\cdots\!00)^{2} \) Copy content Toggle raw display
$61$ \( (T^{8} + \cdots + 35\!\cdots\!00)^{2} \) Copy content Toggle raw display
$67$ \( (T^{8} + \cdots + 56\!\cdots\!00)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} - 1592 T^{3} + \cdots - 101976596376)^{4} \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 16\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( (T^{8} + \cdots + 89\!\cdots\!16)^{2} \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 33\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( (T^{8} + \cdots + 19\!\cdots\!00)^{2} \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 80\!\cdots\!00 \) Copy content Toggle raw display
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