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Properties

Label 350.4.e.n
Level $350$
Weight $4$
Character orbit 350.e
Analytic conductor $20.651$
Analytic rank $0$
Dimension $12$
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: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - x^{11} + 93 x^{10} + 40 x^{9} + 6374 x^{8} + 1920 x^{7} + 179828 x^{6} + 77536 x^{5} + \cdots + 60840000 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2}\cdot 3 \)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 2 \beta_{2} q^{2} + ( - \beta_{2} - \beta_1 - 1) q^{3} + ( - 4 \beta_{2} - 4) q^{4} + ( - 2 \beta_{3} + 2) q^{6} + (\beta_{10} - 2 \beta_{2} - 2) q^{7} + 8 q^{8} + ( - \beta_{9} + \beta_{8} + \cdots + 3 \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 2 \beta_{2} q^{2} + ( - \beta_{2} - \beta_1 - 1) q^{3} + ( - 4 \beta_{2} - 4) q^{4} + ( - 2 \beta_{3} + 2) q^{6} + (\beta_{10} - 2 \beta_{2} - 2) q^{7} + 8 q^{8} + ( - \beta_{9} + \beta_{8} + \cdots + 3 \beta_1) q^{9}+ \cdots + ( - 22 \beta_{11} + 6 \beta_{10} + \cdots + 683) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 12 q^{2} - 7 q^{3} - 24 q^{4} + 28 q^{6} - 9 q^{7} + 96 q^{8} - 31 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 12 q^{2} - 7 q^{3} - 24 q^{4} + 28 q^{6} - 9 q^{7} + 96 q^{8} - 31 q^{9} - 31 q^{11} - 28 q^{12} + 118 q^{13} + 42 q^{14} - 96 q^{16} - 68 q^{17} - 62 q^{18} - 93 q^{19} + 175 q^{21} + 124 q^{22} + 94 q^{23} - 56 q^{24} - 118 q^{26} + 770 q^{27} - 48 q^{28} + 338 q^{29} - 326 q^{31} - 192 q^{32} - 400 q^{33} + 272 q^{34} + 248 q^{36} + 253 q^{37} - 186 q^{38} + 434 q^{39} + 396 q^{41} - 154 q^{42} - 198 q^{43} - 124 q^{44} + 188 q^{46} - 901 q^{47} + 224 q^{48} - 801 q^{49} - 724 q^{51} - 236 q^{52} - 233 q^{53} - 770 q^{54} - 72 q^{56} - 1036 q^{57} - 338 q^{58} - 668 q^{59} - 157 q^{61} + 1304 q^{62} + 2413 q^{63} + 768 q^{64} - 800 q^{66} - 1193 q^{67} - 272 q^{68} - 90 q^{69} + 2216 q^{71} - 248 q^{72} - 1458 q^{73} + 506 q^{74} + 744 q^{76} + 655 q^{77} - 1736 q^{78} + 886 q^{79} + 614 q^{81} - 396 q^{82} + 3598 q^{83} - 392 q^{84} + 198 q^{86} - 789 q^{87} - 248 q^{88} - 3047 q^{89} - 2182 q^{91} - 752 q^{92} - 1902 q^{93} - 1802 q^{94} - 224 q^{96} + 6808 q^{97} + 1452 q^{98} + 8546 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - x^{11} + 93 x^{10} + 40 x^{9} + 6374 x^{8} + 1920 x^{7} + 179828 x^{6} + 77536 x^{5} + \cdots + 60840000 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 4324910085583 \nu^{11} + 357021186377847 \nu^{10} + 41262874601889 \nu^{9} + \cdots + 95\!\cdots\!00 ) / 46\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 2779585357411 \nu^{11} - 2776567410441 \nu^{10} + 244908397759503 \nu^{9} + \cdots - 20\!\cdots\!00 ) / 36\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 425570503381453 \nu^{11} + \cdots - 14\!\cdots\!00 ) / 28\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 818478818346983 \nu^{11} + 776574627175153 \nu^{10} + \cdots - 47\!\cdots\!00 ) / 46\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 33\!\cdots\!87 \nu^{11} - 295597050435097 \nu^{10} + \cdots + 68\!\cdots\!00 ) / 14\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 92\!\cdots\!73 \nu^{11} + \cdots - 25\!\cdots\!00 ) / 21\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 18\!\cdots\!91 \nu^{11} + \cdots - 43\!\cdots\!00 ) / 42\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 48\!\cdots\!45 \nu^{11} + \cdots + 39\!\cdots\!00 ) / 84\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 53\!\cdots\!03 \nu^{11} + \cdots + 12\!\cdots\!00 ) / 84\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 33\!\cdots\!09 \nu^{11} + \cdots + 63\!\cdots\!00 ) / 46\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{9} + \beta_{8} - \beta_{6} + \beta_{3} + 31\beta_{2} + \beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 4 \beta_{11} + 2 \beta_{10} - \beta_{9} + \beta_{8} - \beta_{7} + 2 \beta_{6} + 4 \beta_{5} + 2 \beta_{4} + \cdots - 25 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 63 \beta_{10} - 13 \beta_{9} - 63 \beta_{8} - 50 \beta_{7} + 26 \beta_{5} - 48 \beta_{4} - 63 \beta_{3} + \cdots - 1443 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 314 \beta_{11} - 29 \beta_{10} + 178 \beta_{9} - 178 \beta_{8} - 29 \beta_{7} - 172 \beta_{6} + \cdots - 2446 \beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 2602 \beta_{11} - 2610 \beta_{10} + 3803 \beta_{9} + 1193 \beta_{8} + 3803 \beta_{7} + 2180 \beta_{6} + \cdots + 77147 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 13677 \beta_{10} + 303 \beta_{9} + 13677 \beta_{8} + 13374 \beta_{7} - 22350 \beta_{5} + \cdots + 245253 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 198234 \beta_{11} - 83385 \beta_{10} - 150706 \beta_{9} + 150706 \beta_{8} - 83385 \beta_{7} + \cdots + 667846 \beta_1 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 1551478 \beta_{11} + 948206 \beta_{10} - 1080973 \beta_{9} - 132767 \beta_{8} - 1080973 \beta_{7} + \cdots - 19089349 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 14694075 \beta_{10} - 5375641 \beta_{9} - 14694075 \beta_{8} - 9318434 \beta_{7} + 13879754 \beta_{5} + \cdots - 270397443 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 106410086 \beta_{11} + 14459407 \beta_{10} + 65515246 \beta_{9} - 65515246 \beta_{8} + \cdots - 543865786 \beta_1 \) Copy content Toggle raw display

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

Character values

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

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

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
4.09383 + 7.09072i
2.72116 + 4.71319i
1.03379 + 1.79059i
−1.10015 − 1.90552i
−2.74742 − 4.75867i
−3.50121 − 6.06428i
4.09383 − 7.09072i
2.72116 − 4.71319i
1.03379 − 1.79059i
−1.10015 + 1.90552i
−2.74742 + 4.75867i
−3.50121 + 6.06428i
−1.00000 + 1.73205i −4.59383 − 7.95674i −2.00000 − 3.46410i 0 18.3753 −8.37265 − 16.5196i 8.00000 −28.7065 + 49.7211i 0
51.2 −1.00000 + 1.73205i −3.22116 − 5.57922i −2.00000 − 3.46410i 0 12.8846 −3.64221 + 18.1586i 8.00000 −7.25176 + 12.5604i 0
51.3 −1.00000 + 1.73205i −1.53379 − 2.65661i −2.00000 − 3.46410i 0 6.13518 18.4345 − 1.78031i 8.00000 8.79495 − 15.2333i 0
51.4 −1.00000 + 1.73205i 0.600154 + 1.03950i −2.00000 − 3.46410i 0 −2.40061 −8.00236 + 16.7022i 8.00000 12.7796 − 22.1350i 0
51.5 −1.00000 + 1.73205i 2.24742 + 3.89264i −2.00000 − 3.46410i 0 −8.98967 −14.4444 − 11.5914i 8.00000 3.39822 − 5.88590i 0
51.6 −1.00000 + 1.73205i 3.00121 + 5.19825i −2.00000 − 3.46410i 0 −12.0048 11.5271 − 14.4957i 8.00000 −4.51454 + 7.81942i 0
151.1 −1.00000 − 1.73205i −4.59383 + 7.95674i −2.00000 + 3.46410i 0 18.3753 −8.37265 + 16.5196i 8.00000 −28.7065 − 49.7211i 0
151.2 −1.00000 − 1.73205i −3.22116 + 5.57922i −2.00000 + 3.46410i 0 12.8846 −3.64221 − 18.1586i 8.00000 −7.25176 − 12.5604i 0
151.3 −1.00000 − 1.73205i −1.53379 + 2.65661i −2.00000 + 3.46410i 0 6.13518 18.4345 + 1.78031i 8.00000 8.79495 + 15.2333i 0
151.4 −1.00000 − 1.73205i 0.600154 − 1.03950i −2.00000 + 3.46410i 0 −2.40061 −8.00236 − 16.7022i 8.00000 12.7796 + 22.1350i 0
151.5 −1.00000 − 1.73205i 2.24742 − 3.89264i −2.00000 + 3.46410i 0 −8.98967 −14.4444 + 11.5914i 8.00000 3.39822 + 5.88590i 0
151.6 −1.00000 − 1.73205i 3.00121 − 5.19825i −2.00000 + 3.46410i 0 −12.0048 11.5271 + 14.4957i 8.00000 −4.51454 − 7.81942i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 51.6
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.n 12
5.b even 2 1 350.4.e.o 12
5.c odd 4 2 70.4.i.a ✓ 24
7.c even 3 1 inner 350.4.e.n 12
7.c even 3 1 2450.4.a.cy 6
7.d odd 6 1 2450.4.a.cx 6
35.i odd 6 1 2450.4.a.cw 6
35.j even 6 1 350.4.e.o 12
35.j even 6 1 2450.4.a.cv 6
35.k even 12 2 490.4.c.e 12
35.l odd 12 2 70.4.i.a ✓ 24
35.l odd 12 2 490.4.c.f 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
70.4.i.a ✓ 24 5.c odd 4 2
70.4.i.a ✓ 24 35.l odd 12 2
350.4.e.n 12 1.a even 1 1 trivial
350.4.e.n 12 7.c even 3 1 inner
350.4.e.o 12 5.b even 2 1
350.4.e.o 12 35.j even 6 1
490.4.c.e 12 35.k even 12 2
490.4.c.f 12 35.l odd 12 2
2450.4.a.cv 6 35.j even 6 1
2450.4.a.cw 6 35.i odd 6 1
2450.4.a.cx 6 7.d odd 6 1
2450.4.a.cy 6 7.c even 3 1

Hecke kernels

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

\( T_{3}^{12} + 7 T_{3}^{11} + 121 T_{3}^{10} + 224 T_{3}^{9} + 6221 T_{3}^{8} + 8771 T_{3}^{7} + \cdots + 34574400 \) Copy content Toggle raw display
\( T_{11}^{12} + 31 T_{11}^{11} + 3994 T_{11}^{10} + 99479 T_{11}^{9} + 10853162 T_{11}^{8} + \cdots + 82\!\cdots\!00 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 2 T + 4)^{6} \) Copy content Toggle raw display
$3$ \( T^{12} + 7 T^{11} + \cdots + 34574400 \) 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
$11$ \( T^{12} + \cdots + 82\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( (T^{6} - 59 T^{5} + \cdots - 26675500800)^{2} \) Copy content Toggle raw display
$17$ \( T^{12} + \cdots + 18\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( T^{12} + \cdots + 21\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 27\!\cdots\!41 \) Copy content Toggle raw display
$29$ \( (T^{6} + \cdots - 1796409740592)^{2} \) Copy content Toggle raw display
$31$ \( T^{12} + \cdots + 24\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 19\!\cdots\!16 \) Copy content Toggle raw display
$41$ \( (T^{6} + \cdots - 1146896772372)^{2} \) Copy content Toggle raw display
$43$ \( (T^{6} + \cdots - 75964400923000)^{2} \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 81\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 84\!\cdots\!56 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 27\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 15\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 26\!\cdots\!00 \) Copy content Toggle raw display
$71$ \( (T^{6} + \cdots + 16\!\cdots\!40)^{2} \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 99\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 21\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( (T^{6} + \cdots - 76\!\cdots\!00)^{2} \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( (T^{6} + \cdots - 81\!\cdots\!00)^{2} \) Copy content Toggle raw display
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