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Properties

Label 350.4.e.k
Level $350$
Weight $4$
Character orbit 350.e
Analytic conductor $20.651$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [350,4,Mod(51,350)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(350, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 2]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("350.51");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 350 = 2 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 350.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: \(20.6506685020\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: 6.0.362560708800.2
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{5} + 67x^{4} - 114x^{3} + 4446x^{2} - 5940x + 8100 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 70)
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - 2 \beta_{3} + 2) q^{2} + (\beta_{3} + \beta_1) q^{3} - 4 \beta_{3} q^{4} + (2 \beta_{2} + 2) q^{6} + ( - \beta_{5} - \beta_{4} - 7 \beta_{3} + \cdots + 1) q^{7}+ \cdots + (2 \beta_{5} + \beta_{4} + 19 \beta_{3} + \cdots - 19) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - 2 \beta_{3} + 2) q^{2} + (\beta_{3} + \beta_1) q^{3} - 4 \beta_{3} q^{4} + (2 \beta_{2} + 2) q^{6} + ( - \beta_{5} - \beta_{4} - 7 \beta_{3} + \cdots + 1) q^{7}+ \cdots + (\beta_{5} - \beta_{4} - 228 \beta_{2} - 795) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 6 q^{2} + 4 q^{3} - 12 q^{4} + 16 q^{6} - 14 q^{7} - 48 q^{8} - 57 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 6 q^{2} + 4 q^{3} - 12 q^{4} + 16 q^{6} - 14 q^{7} - 48 q^{8} - 57 q^{9} - 41 q^{11} + 16 q^{12} + 2 q^{13} - 74 q^{14} - 48 q^{16} + 30 q^{17} + 114 q^{18} - 49 q^{19} + 48 q^{21} - 164 q^{22} + 145 q^{23} - 32 q^{24} + 2 q^{26} - 236 q^{27} - 92 q^{28} + 536 q^{29} + 28 q^{31} + 96 q^{32} - 626 q^{33} + 120 q^{34} + 456 q^{36} + 813 q^{37} + 98 q^{38} - 114 q^{39} + 626 q^{41} + 228 q^{42} - 720 q^{43} - 164 q^{44} - 290 q^{46} - 977 q^{47} - 128 q^{48} - 1860 q^{49} - 1632 q^{51} - 4 q^{52} + 325 q^{53} - 236 q^{54} + 112 q^{56} - 500 q^{57} + 536 q^{58} + 272 q^{59} + 902 q^{61} + 112 q^{62} + 3193 q^{63} + 384 q^{64} + 1252 q^{66} - 170 q^{67} + 120 q^{68} + 4528 q^{69} - 2160 q^{71} + 456 q^{72} - 584 q^{73} - 1626 q^{74} + 392 q^{76} + 2395 q^{77} - 456 q^{78} + 310 q^{79} - 147 q^{81} + 626 q^{82} - 1252 q^{83} + 264 q^{84} - 720 q^{86} - 1846 q^{87} + 328 q^{88} + 400 q^{89} - 120 q^{91} - 1160 q^{92} - 372 q^{93} + 1954 q^{94} - 128 q^{96} - 3252 q^{97} - 1482 q^{98} - 5222 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - x^{5} + 67x^{4} - 114x^{3} + 4446x^{2} - 5940x + 8100 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{5} + 67\nu^{4} - 4489\nu^{3} + 4446\nu^{2} - 5940\nu + 397980 ) / 291942 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 737\nu^{5} - 722\nu^{4} + 48374\nu^{3} - 16683\nu^{2} + 3210012\nu + 90450 ) / 4379130 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -2078\nu^{5} - 6745\nu^{4} - 131969\nu^{3} - 249327\nu^{2} - 8256132\nu - 13790520 ) / 875826 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -2276\nu^{5} + 6521\nu^{4} - 144965\nu^{3} + 630981\nu^{2} - 9432252\nu + 25597350 ) / 875826 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 2\beta_{5} + \beta_{4} + 45\beta_{3} + \beta_{2} - 2\beta _1 - 45 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{5} - \beta_{4} - 66\beta_{2} + 45 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -67\beta_{5} - 134\beta_{4} - 2925\beta_{3} - 67\beta_{2} + 158\beta_1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -86\beta_{5} - 43\beta_{4} + 4095\beta_{3} + 4289\beta_{2} - 4246\beta _1 - 4095 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/350\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(127\)
\(\chi(n)\) \(-1 + \beta_{3}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
51.1
−4.14083 − 7.17212i
0.687175 + 1.19022i
3.95365 + 6.84792i
−4.14083 + 7.17212i
0.687175 − 1.19022i
3.95365 − 6.84792i
1.00000 − 1.73205i −3.64083 − 6.30609i −2.00000 − 3.46410i 0 −14.5633 −6.64083 − 17.2887i −8.00000 −13.0112 + 22.5361i 0
51.2 1.00000 − 1.73205i 1.18717 + 2.05625i −2.00000 − 3.46410i 0 4.74870 −1.81283 + 18.4313i −8.00000 10.6812 − 18.5004i 0
51.3 1.00000 − 1.73205i 4.45365 + 7.71395i −2.00000 − 3.46410i 0 17.8146 1.45365 − 18.4631i −8.00000 −26.1700 + 45.3278i 0
151.1 1.00000 + 1.73205i −3.64083 + 6.30609i −2.00000 + 3.46410i 0 −14.5633 −6.64083 + 17.2887i −8.00000 −13.0112 − 22.5361i 0
151.2 1.00000 + 1.73205i 1.18717 − 2.05625i −2.00000 + 3.46410i 0 4.74870 −1.81283 − 18.4313i −8.00000 10.6812 + 18.5004i 0
151.3 1.00000 + 1.73205i 4.45365 − 7.71395i −2.00000 + 3.46410i 0 17.8146 1.45365 + 18.4631i −8.00000 −26.1700 − 45.3278i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 51.3
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 350.4.e.k 6
5.b even 2 1 70.4.e.e ✓ 6
5.c odd 4 2 350.4.j.i 12
7.c even 3 1 inner 350.4.e.k 6
7.c even 3 1 2450.4.a.cb 3
7.d odd 6 1 2450.4.a.ce 3
15.d odd 2 1 630.4.k.r 6
20.d odd 2 1 560.4.q.m 6
35.c odd 2 1 490.4.e.y 6
35.i odd 6 1 490.4.a.v 3
35.i odd 6 1 490.4.e.y 6
35.j even 6 1 70.4.e.e ✓ 6
35.j even 6 1 490.4.a.w 3
35.l odd 12 2 350.4.j.i 12
105.o odd 6 1 630.4.k.r 6
140.p odd 6 1 560.4.q.m 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
70.4.e.e ✓ 6 5.b even 2 1
70.4.e.e ✓ 6 35.j even 6 1
350.4.e.k 6 1.a even 1 1 trivial
350.4.e.k 6 7.c even 3 1 inner
350.4.j.i 12 5.c odd 4 2
350.4.j.i 12 35.l odd 12 2
490.4.a.v 3 35.i odd 6 1
490.4.a.w 3 35.j even 6 1
490.4.e.y 6 35.c odd 2 1
490.4.e.y 6 35.i odd 6 1
560.4.q.m 6 20.d odd 2 1
560.4.q.m 6 140.p odd 6 1
630.4.k.r 6 15.d odd 2 1
630.4.k.r 6 105.o odd 6 1
2450.4.a.cb 3 7.c even 3 1
2450.4.a.ce 3 7.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(350, [\chi])\):

\( T_{3}^{6} - 4T_{3}^{5} + 77T_{3}^{4} - 64T_{3}^{3} + 4337T_{3}^{2} - 9394T_{3} + 23716 \) Copy content Toggle raw display
\( T_{11}^{6} + 41T_{11}^{5} + 4177T_{11}^{4} + 102504T_{11}^{3} + 10429236T_{11}^{2} + 255640320T_{11} + 10489856400 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - 2 T + 4)^{3} \) Copy content Toggle raw display
$3$ \( T^{6} - 4 T^{5} + \cdots + 23716 \) Copy content Toggle raw display
$5$ \( T^{6} \) Copy content Toggle raw display
$7$ \( T^{6} + 14 T^{5} + \cdots + 40353607 \) Copy content Toggle raw display
$11$ \( T^{6} + \cdots + 10489856400 \) Copy content Toggle raw display
$13$ \( (T^{3} - T^{2} + \cdots + 19180)^{2} \) Copy content Toggle raw display
$17$ \( T^{6} + \cdots + 44899914816 \) Copy content Toggle raw display
$19$ \( T^{6} + \cdots + 348912313344 \) Copy content Toggle raw display
$23$ \( T^{6} + \cdots + 5665366321209 \) Copy content Toggle raw display
$29$ \( (T^{3} - 268 T^{2} + \cdots + 562914)^{2} \) Copy content Toggle raw display
$31$ \( T^{6} + \cdots + 1155367014400 \) Copy content Toggle raw display
$37$ \( T^{6} + \cdots + 364355408905216 \) Copy content Toggle raw display
$41$ \( (T^{3} - 313 T^{2} + \cdots + 15201837)^{2} \) Copy content Toggle raw display
$43$ \( (T^{3} + 360 T^{2} + \cdots - 21473350)^{2} \) Copy content Toggle raw display
$47$ \( T^{6} + \cdots + 14\!\cdots\!56 \) Copy content Toggle raw display
$53$ \( T^{6} + \cdots + 89\!\cdots\!64 \) Copy content Toggle raw display
$59$ \( T^{6} + \cdots + 24\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{6} + \cdots + 252484981330176 \) Copy content Toggle raw display
$67$ \( T^{6} + \cdots + 212379790441536 \) Copy content Toggle raw display
$71$ \( (T^{3} + 1080 T^{2} + \cdots - 45563904)^{2} \) Copy content Toggle raw display
$73$ \( T^{6} + \cdots + 25\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( T^{6} + \cdots + 696242104960000 \) Copy content Toggle raw display
$83$ \( (T^{3} + 626 T^{2} + \cdots - 46561032)^{2} \) Copy content Toggle raw display
$89$ \( T^{6} + \cdots + 22\!\cdots\!04 \) Copy content Toggle raw display
$97$ \( (T^{3} + 1626 T^{2} + \cdots - 35869960)^{2} \) Copy content Toggle raw display
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