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Properties

Label 350.4.e.m
Level $350$
Weight $4$
Character orbit 350.e
Analytic conductor $20.651$
Analytic rank $0$
Dimension $8$
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: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 33x^{6} - 74x^{5} + 1019x^{4} - 1221x^{3} + 3679x^{2} + 2590x + 4900 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3^{3} \)
Twist minimal: yes
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (2 \beta_{2} + 2) q^{2} - \beta_{3} q^{3} + 4 \beta_{2} q^{4} + 2 \beta_{5} q^{6} + ( - 3 \beta_{5} - 3 \beta_{4} + \cdots + 1) q^{7}+ \cdots + ( - \beta_{7} + \beta_{5} + 2 \beta_{3} + \cdots - 18) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (2 \beta_{2} + 2) q^{2} - \beta_{3} q^{3} + 4 \beta_{2} q^{4} + 2 \beta_{5} q^{6} + ( - 3 \beta_{5} - 3 \beta_{4} + \cdots + 1) q^{7}+ \cdots + ( - 22 \beta_{6} + 251 \beta_{5} + \cdots - 1252) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{2} - q^{3} - 16 q^{4} - 4 q^{6} - 7 q^{7} - 64 q^{8} - 73 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 8 q^{2} - q^{3} - 16 q^{4} - 4 q^{6} - 7 q^{7} - 64 q^{8} - 73 q^{9} + 10 q^{11} - 4 q^{12} - 188 q^{13} - 70 q^{14} - 64 q^{16} + 99 q^{17} + 146 q^{18} - 246 q^{19} - 535 q^{21} + 40 q^{22} - 2 q^{23} + 8 q^{24} - 188 q^{26} + 392 q^{27} - 112 q^{28} - 196 q^{29} - 304 q^{31} + 128 q^{32} + 526 q^{33} + 396 q^{34} + 584 q^{36} - 82 q^{37} + 492 q^{38} + 214 q^{39} + 704 q^{41} - 634 q^{42} - 262 q^{43} + 40 q^{44} + 4 q^{46} - 491 q^{47} + 32 q^{48} + 1283 q^{49} + 1437 q^{51} + 376 q^{52} - 140 q^{53} + 392 q^{54} + 56 q^{56} - 438 q^{57} - 196 q^{58} - 673 q^{59} - 1425 q^{61} - 1216 q^{62} + 1226 q^{63} + 512 q^{64} - 1052 q^{66} - 666 q^{67} + 396 q^{68} + 1876 q^{69} - 12 q^{71} + 584 q^{72} - 78 q^{73} + 164 q^{74} + 1968 q^{76} + 2575 q^{77} + 856 q^{78} - 1744 q^{79} - 1708 q^{81} + 704 q^{82} - 1852 q^{83} + 872 q^{84} - 262 q^{86} + 1931 q^{87} - 80 q^{88} + 871 q^{89} - 797 q^{91} + 16 q^{92} - 3106 q^{93} + 982 q^{94} + 32 q^{96} + 2924 q^{97} + 1244 q^{98} - 10562 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 33x^{6} - 74x^{5} + 1019x^{4} - 1221x^{3} + 3679x^{2} + 2590x + 4900 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 64535 \nu^{7} + 4861745 \nu^{6} - 23861212 \nu^{5} + 119495228 \nu^{4} - 967304561 \nu^{3} + \cdots + 11230630950 ) / 4265905805 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 1329669 \nu^{7} - 2353890 \nu^{6} + 41058567 \nu^{5} - 171080776 \nu^{4} + 1442026641 \nu^{3} + \cdots - 5437485900 ) / 8531811610 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 1112410 \nu^{7} - 3451490 \nu^{6} + 60203847 \nu^{5} - 163042193 \nu^{4} + 2114432081 \nu^{3} + \cdots - 7972941900 ) / 4265905805 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 2289355 \nu^{7} + 2041235 \nu^{6} - 96546482 \nu^{5} + 206589158 \nu^{4} - 3261559601 \nu^{3} + \cdots + 4715252850 ) / 4265905805 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 1035771 \nu^{7} + 1360706 \nu^{6} + 31983353 \nu^{5} - 38323527 \nu^{4} + 883221504 \nu^{3} + \cdots + 4362061090 ) / 609415115 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 9217276 \nu^{7} - 21330946 \nu^{6} - 310472285 \nu^{5} + 462922235 \nu^{4} + \cdots + 134159464355 ) / 4265905805 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 48179783 \nu^{7} - 119890848 \nu^{6} + 1384317401 \nu^{5} - 7267388006 \nu^{4} + \cdots + 106983663570 ) / 8531811610 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{4} + 2\beta_{3} + \beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{7} + 2\beta_{5} + 2\beta_{4} - \beta_{3} - 49\beta_{2} - 5\beta _1 - 49 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -\beta_{6} - 18\beta_{5} - 69\beta_{4} - 35\beta_{3} + 35\beta _1 + 79 ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -33\beta_{7} + 33\beta_{6} + 33\beta_{5} + 136\beta_{4} + 206\beta_{3} + 1407\beta_{2} + 103\beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 70\beta_{7} + 635\beta_{5} + 1126\beta_{4} - 561\beta_{3} - 4420\beta_{2} - 2322\beta _1 - 4420 ) / 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -1056\beta_{6} - 3723\beta_{5} - 10090\beta_{4} - 5573\beta_{3} + 5573\beta _1 + 45924 ) / 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 3461 \beta_{7} + 3461 \beta_{6} + 3461 \beta_{5} + 42050 \beta_{4} + 61893 \beta_{3} + \cdots + 38589 \beta_1 ) / 3 \) Copy content Toggle raw display

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

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
51.1
1.21436 + 2.10334i
2.33965 + 4.05240i
−3.04898 − 5.28099i
−0.505037 − 0.874749i
1.21436 − 2.10334i
2.33965 − 4.05240i
−3.04898 + 5.28099i
−0.505037 + 0.874749i
1.00000 − 1.73205i −4.74942 − 8.22624i −2.00000 − 3.46410i 0 −18.9977 11.3178 − 14.6597i −8.00000 −31.6140 + 54.7570i 0
51.2 1.00000 − 1.73205i −1.53843 − 2.66464i −2.00000 − 3.46410i 0 −6.15371 −14.8647 − 11.0472i −8.00000 8.76648 − 15.1840i 0
51.3 1.00000 − 1.73205i 1.55664 + 2.69618i −2.00000 − 3.46410i 0 6.22657 18.1576 + 3.64704i −8.00000 8.65373 − 14.9887i 0
51.4 1.00000 − 1.73205i 4.23121 + 7.32867i −2.00000 − 3.46410i 0 16.9248 −18.1107 + 3.87329i −8.00000 −22.3062 + 38.6355i 0
151.1 1.00000 + 1.73205i −4.74942 + 8.22624i −2.00000 + 3.46410i 0 −18.9977 11.3178 + 14.6597i −8.00000 −31.6140 − 54.7570i 0
151.2 1.00000 + 1.73205i −1.53843 + 2.66464i −2.00000 + 3.46410i 0 −6.15371 −14.8647 + 11.0472i −8.00000 8.76648 + 15.1840i 0
151.3 1.00000 + 1.73205i 1.55664 − 2.69618i −2.00000 + 3.46410i 0 6.22657 18.1576 − 3.64704i −8.00000 8.65373 + 14.9887i 0
151.4 1.00000 + 1.73205i 4.23121 − 7.32867i −2.00000 + 3.46410i 0 16.9248 −18.1107 − 3.87329i −8.00000 −22.3062 − 38.6355i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 51.4
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.m yes 8
5.b even 2 1 350.4.e.l ✓ 8
5.c odd 4 2 350.4.j.j 16
7.c even 3 1 inner 350.4.e.m yes 8
7.c even 3 1 2450.4.a.co 4
7.d odd 6 1 2450.4.a.ck 4
35.i odd 6 1 2450.4.a.cu 4
35.j even 6 1 350.4.e.l ✓ 8
35.j even 6 1 2450.4.a.cq 4
35.l odd 12 2 350.4.j.j 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
350.4.e.l ✓ 8 5.b even 2 1
350.4.e.l ✓ 8 35.j even 6 1
350.4.e.m yes 8 1.a even 1 1 trivial
350.4.e.m yes 8 7.c even 3 1 inner
350.4.j.j 16 5.c odd 4 2
350.4.j.j 16 35.l odd 12 2
2450.4.a.ck 4 7.d odd 6 1
2450.4.a.co 4 7.c even 3 1
2450.4.a.cq 4 35.j even 6 1
2450.4.a.cu 4 35.i odd 6 1

Hecke kernels

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

\( T_{3}^{8} + T_{3}^{7} + 91T_{3}^{6} - 76T_{3}^{5} + 7337T_{3}^{4} - 910T_{3}^{3} + 69349T_{3}^{2} - 5390T_{3} + 592900 \) 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

$p$ $F_p(T)$
$2$ \( (T^{2} - 2 T + 4)^{4} \) Copy content Toggle raw display
$3$ \( T^{8} + T^{7} + \cdots + 592900 \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( T^{8} + \cdots + 13841287201 \) Copy content Toggle raw display
$11$ \( T^{8} + \cdots + 15291795600 \) Copy content Toggle raw display
$13$ \( (T^{4} + 94 T^{3} + \cdots - 210070)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 316672524137025 \) Copy content Toggle raw display
$19$ \( T^{8} + \cdots + 203151429734400 \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 155878995492801 \) Copy content Toggle raw display
$29$ \( (T^{4} + 98 T^{3} + \cdots - 38101482)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} + \cdots + 86\!\cdots\!25 \) Copy content Toggle raw display
$37$ \( T^{8} + \cdots + 11\!\cdots\!16 \) Copy content Toggle raw display
$41$ \( (T^{4} - 352 T^{3} + \cdots + 84396627)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + 131 T^{3} + \cdots + 79091830)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 48\!\cdots\!25 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 35\!\cdots\!56 \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 66\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{8} + \cdots + 23\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 17\!\cdots\!00 \) Copy content Toggle raw display
$71$ \( (T^{4} + 6 T^{3} + \cdots - 5901875865)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 13\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( T^{8} + \cdots + 19\!\cdots\!25 \) Copy content Toggle raw display
$83$ \( (T^{4} + 926 T^{3} + \cdots + 82998924750)^{2} \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 90\!\cdots\!25 \) Copy content Toggle raw display
$97$ \( (T^{4} - 1462 T^{3} + \cdots - 7337974175)^{2} \) Copy content Toggle raw display
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