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Properties

Label 350.4.j.j
Level $350$
Weight $4$
Character orbit 350.j
Analytic conductor $20.651$
Analytic rank $0$
Dimension $16$
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: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 66 x^{14} + 3127 x^{12} - 69136 x^{10} + 1110267 x^{8} - 6713681 x^{6} + 29846021 x^{4} + \cdots + 24010000 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3^{6} \)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (2 \beta_{11} - 2 \beta_{8}) q^{2} + \beta_{10} q^{3} + 4 \beta_1 q^{4} - 2 \beta_{4} q^{6} + ( - 3 \beta_{13} + \beta_{12} + \cdots + 3 \beta_{7}) q^{7}+ \cdots + ( - \beta_{9} + \beta_{6} + 2 \beta_{4} + \cdots + 18) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (2 \beta_{11} - 2 \beta_{8}) q^{2} + \beta_{10} q^{3} + 4 \beta_1 q^{4} - 2 \beta_{4} q^{6} + ( - 3 \beta_{13} + \beta_{12} + \cdots + 3 \beta_{7}) q^{7}+ \cdots + (12 \beta_{5} + 271 \beta_{4} + \cdots + 1252) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 32 q^{4} - 8 q^{6} + 146 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 32 q^{4} - 8 q^{6} + 146 q^{9} + 20 q^{11} + 140 q^{14} - 128 q^{16} + 492 q^{19} - 1070 q^{21} - 16 q^{24} - 376 q^{26} + 392 q^{29} - 608 q^{31} - 792 q^{34} + 1168 q^{36} - 428 q^{39} + 1408 q^{41} - 80 q^{44} + 8 q^{46} - 2566 q^{49} + 2874 q^{51} - 784 q^{54} + 112 q^{56} + 1346 q^{59} - 2850 q^{61} - 1024 q^{64} - 2104 q^{66} - 3752 q^{69} - 24 q^{71} - 328 q^{74} + 3936 q^{76} + 3488 q^{79} - 3416 q^{81} - 1744 q^{84} - 524 q^{86} - 1742 q^{89} - 1594 q^{91} - 1964 q^{94} + 64 q^{96} + 21124 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 66 x^{14} + 3127 x^{12} - 69136 x^{10} + 1110267 x^{8} - 6713681 x^{6} + 29846021 x^{4} + \cdots + 24010000 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 2604153807301 \nu^{14} + 168903785729166 \nu^{12} + \cdots + 71\!\cdots\!00 ) / 59\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 2703728513065 \nu^{14} - 170548554989742 \nu^{12} + \cdots + 11\!\cdots\!70 ) / 31\!\cdots\!10 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 2749547048257 \nu^{14} - 158589969005046 \nu^{12} + \cdots + 24\!\cdots\!60 ) / 31\!\cdots\!10 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 3150820367474 \nu^{14} + 192746704071381 \nu^{12} + \cdots - 12\!\cdots\!30 ) / 15\!\cdots\!55 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 6331351818859 \nu^{14} + 397962452359848 \nu^{12} + \cdots - 18\!\cdots\!60 ) / 31\!\cdots\!10 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 366360683944851 \nu^{14} + \cdots + 11\!\cdots\!00 ) / 10\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 10569562844623 \nu^{15} + 273214207938718 \nu^{13} + \cdots - 16\!\cdots\!20 \nu ) / 21\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 489319736751 \nu^{15} - 29917136473135 \nu^{13} + \cdots + 30\!\cdots\!31 \nu ) / 31\!\cdots\!10 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 36\!\cdots\!79 \nu^{14} + \cdots + 15\!\cdots\!00 ) / 21\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 763737526534 \nu^{15} + 46590387713699 \nu^{13} + \cdots - 13\!\cdots\!73 \nu ) / 15\!\cdots\!55 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 70864573 \nu^{15} - 4628664518 \nu^{13} + 218218634971 \nu^{11} + \cdots - 349879794530000 \nu ) / 12\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 203276944584897 \nu^{15} + \cdots + 13\!\cdots\!20 \nu ) / 21\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 31\!\cdots\!57 \nu^{15} + \cdots - 15\!\cdots\!00 \nu ) / 76\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 13\!\cdots\!47 \nu^{15} + \cdots + 76\!\cdots\!90 \nu ) / 21\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 22\!\cdots\!73 \nu^{15} + \cdots + 70\!\cdots\!00 \nu ) / 15\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{12} - 2\beta_{10} + \beta_{7} ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{9} + 2\beta_{6} - 2\beta_{5} - \beta_{4} - 5\beta_{3} - 49\beta _1 + 49 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{15} - \beta_{14} - 17\beta_{13} + 35\beta_{12} + 79\beta_{11} - 34\beta_{10} + 69\beta_{7} ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 33\beta_{9} + 66\beta_{6} - 206\beta_{5} + 103\beta_{4} - 103\beta_{3} + 33\beta_{2} - 1407\beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 70\beta_{15} - 635\beta_{13} + 2322\beta_{12} + 4420\beta_{11} + 561\beta_{10} - 4420\beta_{8} + 1126\beta_{7} ) / 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -4517\beta_{5} + 7423\beta_{4} + 5573\beta_{3} + 1056\beta_{2} - 45924 ) / 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 3461\beta_{14} + 38589\beta_{12} + 58432\beta_{10} - 192389\beta_{8} - 42050\beta_{7} ) / 3 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 35128 \beta_{9} - 106886 \beta_{6} + 183513 \beta_{5} + 111755 \beta_{4} + 402154 \beta_{3} + \cdots - 1580542 ) / 3 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 148385 \beta_{15} + 148385 \beta_{14} + 787060 \beta_{13} - 1510131 \beta_{12} + \cdots - 2871877 \beta_{7} ) / 3 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 1213361 \beta_{9} - 4162207 \beta_{6} + 14335064 \beta_{5} - 7167532 \beta_{4} + \cdots + 56061599 \beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 5954171 \beta_{15} + 28373272 \beta_{13} - 104006909 \beta_{12} - 300387449 \beta_{11} + \cdots - 49026369 \beta_{7} ) / 3 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 274067248\beta_{5} - 432214663\beta_{4} - 317139446\beta_{3} - 43072198\beta_{2} + 2025723952 ) / 3 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( -76998350\beta_{14} - 595915547\beta_{12} - 923754299\beta_{10} + 3815120410\beta_{8} + 672913897\beta_{7} \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 1556751591 \beta_{9} + 6005193597 \beta_{6} - 10356467597 \beta_{5} - 5908025591 \beta_{4} + \cdots + 74038914099 ) / 3 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 8799716006 \beta_{15} - 8799716006 \beta_{14} - 38059015462 \beta_{13} + 74503997505 \beta_{12} + \cdots + 140208279004 \beta_{7} ) / 3 \) Copy content Toggle raw display

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

Character values

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

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

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
0.874749 − 0.505037i
5.28099 − 3.04898i
−4.05240 + 2.33965i
−2.10334 + 1.21436i
2.10334 − 1.21436i
4.05240 − 2.33965i
−5.28099 + 3.04898i
−0.874749 + 0.505037i
0.874749 + 0.505037i
5.28099 + 3.04898i
−4.05240 − 2.33965i
−2.10334 − 1.21436i
2.10334 + 1.21436i
4.05240 + 2.33965i
−5.28099 − 3.04898i
−0.874749 − 0.505037i
−1.73205 − 1.00000i −7.32867 + 4.23121i 2.00000 + 3.46410i 0 16.9248 3.87329 + 18.1107i − 8.00000i 22.3062 − 38.6355i 0
149.2 −1.73205 − 1.00000i −2.69618 + 1.55664i 2.00000 + 3.46410i 0 6.22657 3.64704 − 18.1576i − 8.00000i −8.65373 + 14.9887i 0
149.3 −1.73205 − 1.00000i 2.66464 − 1.53843i 2.00000 + 3.46410i 0 −6.15371 −11.0472 + 14.8647i − 8.00000i −8.76648 + 15.1840i 0
149.4 −1.73205 − 1.00000i 8.22624 − 4.74942i 2.00000 + 3.46410i 0 −18.9977 −14.6597 − 11.3178i − 8.00000i 31.6140 − 54.7570i 0
149.5 1.73205 + 1.00000i −8.22624 + 4.74942i 2.00000 + 3.46410i 0 −18.9977 14.6597 + 11.3178i 8.00000i 31.6140 − 54.7570i 0
149.6 1.73205 + 1.00000i −2.66464 + 1.53843i 2.00000 + 3.46410i 0 −6.15371 11.0472 − 14.8647i 8.00000i −8.76648 + 15.1840i 0
149.7 1.73205 + 1.00000i 2.69618 − 1.55664i 2.00000 + 3.46410i 0 6.22657 −3.64704 + 18.1576i 8.00000i −8.65373 + 14.9887i 0
149.8 1.73205 + 1.00000i 7.32867 − 4.23121i 2.00000 + 3.46410i 0 16.9248 −3.87329 − 18.1107i 8.00000i 22.3062 − 38.6355i 0
249.1 −1.73205 + 1.00000i −7.32867 − 4.23121i 2.00000 − 3.46410i 0 16.9248 3.87329 − 18.1107i 8.00000i 22.3062 + 38.6355i 0
249.2 −1.73205 + 1.00000i −2.69618 − 1.55664i 2.00000 − 3.46410i 0 6.22657 3.64704 + 18.1576i 8.00000i −8.65373 − 14.9887i 0
249.3 −1.73205 + 1.00000i 2.66464 + 1.53843i 2.00000 − 3.46410i 0 −6.15371 −11.0472 − 14.8647i 8.00000i −8.76648 − 15.1840i 0
249.4 −1.73205 + 1.00000i 8.22624 + 4.74942i 2.00000 − 3.46410i 0 −18.9977 −14.6597 + 11.3178i 8.00000i 31.6140 + 54.7570i 0
249.5 1.73205 − 1.00000i −8.22624 − 4.74942i 2.00000 − 3.46410i 0 −18.9977 14.6597 − 11.3178i − 8.00000i 31.6140 + 54.7570i 0
249.6 1.73205 − 1.00000i −2.66464 − 1.53843i 2.00000 − 3.46410i 0 −6.15371 11.0472 + 14.8647i − 8.00000i −8.76648 − 15.1840i 0
249.7 1.73205 − 1.00000i 2.69618 + 1.55664i 2.00000 − 3.46410i 0 6.22657 −3.64704 − 18.1576i − 8.00000i −8.65373 − 14.9887i 0
249.8 1.73205 − 1.00000i 7.32867 + 4.23121i 2.00000 − 3.46410i 0 16.9248 −3.87329 + 18.1107i − 8.00000i 22.3062 + 38.6355i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 149.8
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.j 16
5.b even 2 1 inner 350.4.j.j 16
5.c odd 4 1 350.4.e.l ✓ 8
5.c odd 4 1 350.4.e.m yes 8
7.c even 3 1 inner 350.4.j.j 16
35.j even 6 1 inner 350.4.j.j 16
35.k even 12 1 2450.4.a.ck 4
35.k even 12 1 2450.4.a.cu 4
35.l odd 12 1 350.4.e.l ✓ 8
35.l odd 12 1 350.4.e.m yes 8
35.l odd 12 1 2450.4.a.co 4
35.l odd 12 1 2450.4.a.cq 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
350.4.e.l ✓ 8 5.c odd 4 1
350.4.e.l ✓ 8 35.l odd 12 1
350.4.e.m yes 8 5.c odd 4 1
350.4.e.m yes 8 35.l odd 12 1
350.4.j.j 16 1.a even 1 1 trivial
350.4.j.j 16 5.b even 2 1 inner
350.4.j.j 16 7.c even 3 1 inner
350.4.j.j 16 35.j even 6 1 inner
2450.4.a.ck 4 35.k even 12 1
2450.4.a.co 4 35.l odd 12 1
2450.4.a.cq 4 35.l odd 12 1
2450.4.a.cu 4 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 \(S_{4}^{\mathrm{new}}(350, [\chi])\):

\( T_{3}^{16} - 181 T_{3}^{14} + 23107 T_{3}^{12} - 1470076 T_{3}^{10} + 67511347 T_{3}^{8} + \cdots + 351530410000 \) Copy content Toggle raw display
\( T_{11}^{8} - 10 T_{11}^{7} + 4537 T_{11}^{6} + 49068 T_{11}^{5} + 19539819 T_{11}^{4} + \cdots + 15291795600 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - 4 T^{2} + 16)^{4} \) Copy content Toggle raw display
$3$ \( T^{16} + \cdots + 351530410000 \) Copy content Toggle raw display
$5$ \( T^{16} \) Copy content Toggle raw display
$7$ \( T^{16} + \cdots + 19\!\cdots\!01 \) Copy content Toggle raw display
$11$ \( (T^{8} - 10 T^{7} + \cdots + 15291795600)^{2} \) Copy content Toggle raw display
$13$ \( (T^{8} + 7474 T^{6} + \cdots + 44129404900)^{2} \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 10\!\cdots\!25 \) Copy content Toggle raw display
$19$ \( (T^{8} + \cdots + 203151429734400)^{2} \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 24\!\cdots\!01 \) Copy content Toggle raw display
$29$ \( (T^{4} - 98 T^{3} + \cdots - 38101482)^{4} \) Copy content Toggle raw display
$31$ \( (T^{8} + \cdots + 86\!\cdots\!25)^{2} \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 13\!\cdots\!56 \) Copy content Toggle raw display
$41$ \( (T^{4} - 352 T^{3} + \cdots + 84396627)^{4} \) Copy content Toggle raw display
$43$ \( (T^{8} + \cdots + 62\!\cdots\!00)^{2} \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 23\!\cdots\!25 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 12\!\cdots\!36 \) Copy content Toggle raw display
$59$ \( (T^{8} + \cdots + 66\!\cdots\!00)^{2} \) Copy content Toggle raw display
$61$ \( (T^{8} + \cdots + 23\!\cdots\!00)^{2} \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 31\!\cdots\!00 \) Copy content Toggle raw display
$71$ \( (T^{4} + 6 T^{3} + \cdots - 5901875865)^{4} \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 19\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( (T^{8} + \cdots + 19\!\cdots\!25)^{2} \) Copy content Toggle raw display
$83$ \( (T^{8} + \cdots + 68\!\cdots\!00)^{2} \) Copy content Toggle raw display
$89$ \( (T^{8} + \cdots + 90\!\cdots\!25)^{2} \) Copy content Toggle raw display
$97$ \( (T^{8} + \cdots + 53\!\cdots\!25)^{2} \) Copy content Toggle raw display
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