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Properties

Label 350.4.j.i
Level $350$
Weight $4$
Character orbit 350.j
Analytic conductor $20.651$
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] = [350,4,Mod(149,350)]
 
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([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("350.149");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 350 = 2 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 350.j (of order \(6\), degree \(2\), not 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: \(12\)
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: \( x^{12} - 133x^{10} + 13153x^{8} - 587088x^{6} + 19497996x^{4} - 36741600x^{2} + 65610000 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 2^{6}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 70)
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.

\(f(q)\) \(=\) \( q + \beta_{3} q^{2} + ( - \beta_{9} - \beta_{7}) q^{3} - 4 \beta_{2} q^{4} + ( - \beta_{5} - \beta_{4} + 3) q^{6} + (\beta_{10} - 3 \beta_{9} + \cdots - \beta_1) q^{7}+ \cdots + (\beta_{8} + 2 \beta_{6} + 19 \beta_{2} + 19) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{3} q^{2} + ( - \beta_{9} - \beta_{7}) q^{3} - 4 \beta_{2} q^{4} + ( - \beta_{5} - \beta_{4} + 3) q^{6} + (\beta_{10} - 3 \beta_{9} + \cdots - \beta_1) q^{7}+ \cdots + ( - \beta_{8} + \beta_{6} - 113 \beta_{5} + \cdots + 908) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 24 q^{4} + 32 q^{6} + 114 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 24 q^{4} + 32 q^{6} + 114 q^{9} - 82 q^{11} + 148 q^{14} - 96 q^{16} + 98 q^{19} + 96 q^{21} + 64 q^{24} + 4 q^{26} - 1072 q^{29} + 56 q^{31} - 240 q^{34} + 912 q^{36} + 228 q^{39} + 1252 q^{41} + 328 q^{44} - 580 q^{46} + 3720 q^{49} - 3264 q^{51} + 472 q^{54} + 224 q^{56} - 544 q^{59} + 1804 q^{61} - 768 q^{64} + 2504 q^{66} - 9056 q^{69} - 4320 q^{71} + 3252 q^{74} + 784 q^{76} - 620 q^{79} - 294 q^{81} - 528 q^{84} - 1440 q^{86} - 800 q^{89} - 240 q^{91} - 3908 q^{94} - 256 q^{96} + 10444 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 133x^{10} + 13153x^{8} - 587088x^{6} + 19497996x^{4} - 36741600x^{2} + 65610000 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 3211 \nu^{11} + 317551 \nu^{9} + 7198073 \nu^{7} + 470737332 \nu^{5} + \cdots - 3018903196392 \nu ) / 11089638745920 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 3758 \nu^{10} + 490789 \nu^{8} - 48536449 \nu^{6} + 2118010679 \nu^{4} - 71950390668 \nu^{2} + 7229452500 ) / 128352300300 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 3211 \nu^{11} + 317551 \nu^{9} + 7198073 \nu^{7} + 470737332 \nu^{5} + \cdots + 8070735549528 \nu ) / 5544819372960 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 1798 \nu^{10} - 467159 \nu^{8} + 23222069 \nu^{6} - 1013353699 \nu^{4} - 32040353142 \nu^{2} - 3458902500 ) / 36672085800 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 221359 \nu^{10} - 9738697 \nu^{8} + 963105877 \nu^{6} - 961457142 \nu^{4} + \cdots - 2690340674400 ) / 1540227603600 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 124939 \nu^{10} - 19861792 \nu^{8} + 2060732182 \nu^{6} - 104207198967 \nu^{4} + \cdots - 12791372766300 ) / 462068281080 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 2114753 \nu^{11} - 288486899 \nu^{9} + 28529835959 \nu^{7} - 1225350445014 \nu^{5} + \cdots - 79695265024800 \nu ) / 277240968648000 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 492983 \nu^{10} - 63765389 \nu^{8} + 5920039409 \nu^{6} - 244053949194 \nu^{4} + \cdots + 11602645318800 ) / 924136562160 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 6002527 \nu^{11} - 771617341 \nu^{9} + 76308893881 \nu^{7} - 3349552621626 \nu^{5} + \cdots - 213161321023200 \nu ) / 138620484324000 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 74933137 \nu^{11} - 10408596721 \nu^{9} + 1043831256961 \nu^{7} + \cdots - 60\!\cdots\!00 \nu ) / 277240968648000 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 32653079 \nu^{11} - 4228585607 \nu^{9} + 408855996887 \nu^{7} - 17332698775752 \nu^{5} + \cdots + 938010700863000 \nu ) / 92413656216000 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{3} - 2\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -2\beta_{8} - \beta_{6} + \beta_{4} - 44\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -2\beta_{11} - \beta_{10} + 11\beta_{9} + 64\beta_{7} + 11\beta_{3} - 65\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -67\beta_{8} - 134\beta_{6} - 79\beta_{5} - 2846\beta_{2} - 2846 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -43\beta_{11} - 86\beta_{10} + 140\beta_{9} + 4246\beta_{7} + 43\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 4375\beta_{8} - 4375\beta_{6} - 5971\beta_{5} - 5971\beta_{4} - 187034 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 1183\beta_{11} - 1183\beta_{10} + 27226\beta_{3} + 284880\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 572126\beta_{8} + 286063\beta_{6} - 443899\beta_{4} + 12322466\beta_{2} \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( -59218\beta_{11} - 29609\beta_{10} + 4166998\beta_{9} - 18832906\beta_{7} + 4166998\beta_{3} + 18803297\beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 18744079\beta_{8} + 37488158\beta_{6} + 32594011\beta_{5} + 813554354\beta_{2} + 813554354 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( -8955785\beta_{11} - 17911570\beta_{10} + 429579598\beta_{9} - 1257685594\beta_{7} + 8955785\beta_1 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(101\) \(127\)
\(\chi(n)\) \(-1 - \beta_{2}\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
149.1
−6.84792 − 3.95365i
−1.19022 − 0.687175i
7.17212 + 4.14083i
−7.17212 − 4.14083i
1.19022 + 0.687175i
6.84792 + 3.95365i
−6.84792 + 3.95365i
−1.19022 + 0.687175i
7.17212 − 4.14083i
−7.17212 + 4.14083i
1.19022 − 0.687175i
6.84792 − 3.95365i
−1.73205 − 1.00000i −7.71395 + 4.45365i 2.00000 + 3.46410i 0 17.8146 −18.4631 − 1.45365i − 8.00000i 26.1700 − 45.3278i 0
149.2 −1.73205 − 1.00000i −2.05625 + 1.18717i 2.00000 + 3.46410i 0 4.74870 18.4313 + 1.81283i − 8.00000i −10.6812 + 18.5004i 0
149.3 −1.73205 − 1.00000i 6.30609 − 3.64083i 2.00000 + 3.46410i 0 −14.5633 −17.2887 + 6.64083i − 8.00000i 13.0112 − 22.5361i 0
149.4 1.73205 + 1.00000i −6.30609 + 3.64083i 2.00000 + 3.46410i 0 −14.5633 17.2887 − 6.64083i 8.00000i 13.0112 − 22.5361i 0
149.5 1.73205 + 1.00000i 2.05625 − 1.18717i 2.00000 + 3.46410i 0 4.74870 −18.4313 − 1.81283i 8.00000i −10.6812 + 18.5004i 0
149.6 1.73205 + 1.00000i 7.71395 − 4.45365i 2.00000 + 3.46410i 0 17.8146 18.4631 + 1.45365i 8.00000i 26.1700 − 45.3278i 0
249.1 −1.73205 + 1.00000i −7.71395 − 4.45365i 2.00000 − 3.46410i 0 17.8146 −18.4631 + 1.45365i 8.00000i 26.1700 + 45.3278i 0
249.2 −1.73205 + 1.00000i −2.05625 − 1.18717i 2.00000 − 3.46410i 0 4.74870 18.4313 − 1.81283i 8.00000i −10.6812 − 18.5004i 0
249.3 −1.73205 + 1.00000i 6.30609 + 3.64083i 2.00000 − 3.46410i 0 −14.5633 −17.2887 − 6.64083i 8.00000i 13.0112 + 22.5361i 0
249.4 1.73205 − 1.00000i −6.30609 − 3.64083i 2.00000 − 3.46410i 0 −14.5633 17.2887 + 6.64083i − 8.00000i 13.0112 + 22.5361i 0
249.5 1.73205 − 1.00000i 2.05625 + 1.18717i 2.00000 − 3.46410i 0 4.74870 −18.4313 + 1.81283i − 8.00000i −10.6812 − 18.5004i 0
249.6 1.73205 − 1.00000i 7.71395 + 4.45365i 2.00000 − 3.46410i 0 17.8146 18.4631 − 1.45365i − 8.00000i 26.1700 + 45.3278i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 149.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
7.c even 3 1 inner
35.j even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 350.4.j.i 12
5.b even 2 1 inner 350.4.j.i 12
5.c odd 4 1 70.4.e.e ✓ 6
5.c odd 4 1 350.4.e.k 6
7.c even 3 1 inner 350.4.j.i 12
15.e even 4 1 630.4.k.r 6
20.e even 4 1 560.4.q.m 6
35.f even 4 1 490.4.e.y 6
35.j even 6 1 inner 350.4.j.i 12
35.k even 12 1 490.4.a.v 3
35.k even 12 1 490.4.e.y 6
35.k even 12 1 2450.4.a.ce 3
35.l odd 12 1 70.4.e.e ✓ 6
35.l odd 12 1 350.4.e.k 6
35.l odd 12 1 490.4.a.w 3
35.l odd 12 1 2450.4.a.cb 3
105.x even 12 1 630.4.k.r 6
140.w even 12 1 560.4.q.m 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
70.4.e.e ✓ 6 5.c odd 4 1
70.4.e.e ✓ 6 35.l odd 12 1
350.4.e.k 6 5.c odd 4 1
350.4.e.k 6 35.l odd 12 1
350.4.j.i 12 1.a even 1 1 trivial
350.4.j.i 12 5.b even 2 1 inner
350.4.j.i 12 7.c even 3 1 inner
350.4.j.i 12 35.j even 6 1 inner
490.4.a.v 3 35.k even 12 1
490.4.a.w 3 35.l odd 12 1
490.4.e.y 6 35.f even 4 1
490.4.e.y 6 35.k even 12 1
560.4.q.m 6 20.e even 4 1
560.4.q.m 6 140.w even 12 1
630.4.k.r 6 15.e even 4 1
630.4.k.r 6 105.x even 12 1
2450.4.a.cb 3 35.l odd 12 1
2450.4.a.ce 3 35.k even 12 1

Hecke kernels

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

\( T_{3}^{12} - 138T_{3}^{10} + 14091T_{3}^{8} - 636082T_{3}^{6} + 21259401T_{3}^{4} - 117465348T_{3}^{2} + 562448656 \) 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

$p$ $F_p(T)$
$2$ \( (T^{4} - 4 T^{2} + 16)^{3} \) Copy content Toggle raw display
$3$ \( T^{12} + \cdots + 562448656 \) Copy content Toggle raw display
$5$ \( T^{12} \) Copy content Toggle raw display
$7$ \( T^{12} + \cdots + 16\!\cdots\!49 \) Copy content Toggle raw display
$11$ \( (T^{6} + 41 T^{5} + \cdots + 10489856400)^{2} \) Copy content Toggle raw display
$13$ \( (T^{6} + 2721 T^{4} + \cdots + 367872400)^{2} \) Copy content Toggle raw display
$17$ \( T^{12} + \cdots + 20\!\cdots\!56 \) Copy content Toggle raw display
$19$ \( (T^{6} - 49 T^{5} + \cdots + 348912313344)^{2} \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 32\!\cdots\!81 \) Copy content Toggle raw display
$29$ \( (T^{3} + 268 T^{2} + \cdots - 562914)^{4} \) Copy content Toggle raw display
$31$ \( (T^{6} + \cdots + 1155367014400)^{2} \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 13\!\cdots\!56 \) Copy content Toggle raw display
$41$ \( (T^{3} - 313 T^{2} + \cdots + 15201837)^{4} \) Copy content Toggle raw display
$43$ \( (T^{6} + \cdots + 461104760222500)^{2} \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 20\!\cdots\!36 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 79\!\cdots\!96 \) Copy content Toggle raw display
$59$ \( (T^{6} + \cdots + 24\!\cdots\!00)^{2} \) Copy content Toggle raw display
$61$ \( (T^{6} + \cdots + 252484981330176)^{2} \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 45\!\cdots\!96 \) Copy content Toggle raw display
$71$ \( (T^{3} + 1080 T^{2} + \cdots - 45563904)^{4} \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 65\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( (T^{6} + \cdots + 696242104960000)^{2} \) Copy content Toggle raw display
$83$ \( (T^{6} + \cdots + 21\!\cdots\!24)^{2} \) Copy content Toggle raw display
$89$ \( (T^{6} + \cdots + 22\!\cdots\!04)^{2} \) Copy content Toggle raw display
$97$ \( (T^{6} + \cdots + 12\!\cdots\!00)^{2} \) Copy content Toggle raw display
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