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Properties

Label 350.2.h.b
Level $350$
Weight $2$
Character orbit 350.h
Analytic conductor $2.795$
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] = [350,2,Mod(71,350)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(350, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([6, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("350.71");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 350 = 2 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 350.h (of order \(5\), degree \(4\), 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: \(2.79476407074\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(3\) over \(\Q(\zeta_{5})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 4 x^{11} + 6 x^{10} + x^{9} - 14 x^{8} + 10 x^{7} + 35 x^{6} - 110 x^{5} + 230 x^{4} + \cdots + 125 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 5 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{5} q^{2} + (\beta_{7} - \beta_{3} + \beta_1) q^{3} - \beta_{3} q^{4} + ( - \beta_{11} + \beta_{9} + \beta_{3} + \cdots - 1) q^{5}+ \cdots + (\beta_{10} + 2 \beta_{9} + \beta_{6} + \cdots - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{5} q^{2} + (\beta_{7} - \beta_{3} + \beta_1) q^{3} - \beta_{3} q^{4} + ( - \beta_{11} + \beta_{9} + \beta_{3} + \cdots - 1) q^{5}+ \cdots + ( - \beta_{4} - \beta_{3} - 3 \beta_1 - 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 3 q^{2} + q^{3} - 3 q^{4} - 5 q^{5} - q^{6} - 12 q^{7} + 3 q^{8} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 3 q^{2} + q^{3} - 3 q^{4} - 5 q^{5} - q^{6} - 12 q^{7} + 3 q^{8} + 6 q^{9} + 7 q^{11} - 4 q^{12} + 3 q^{13} - 3 q^{14} - 10 q^{15} - 3 q^{16} + 4 q^{17} - 6 q^{18} + 4 q^{19} + 5 q^{20} - q^{21} - 2 q^{22} - q^{23} - 6 q^{24} - 5 q^{25} + 12 q^{26} + 10 q^{27} + 3 q^{28} + 22 q^{29} + 15 q^{30} + 31 q^{31} - 12 q^{32} - 21 q^{33} + 6 q^{34} + 5 q^{35} + 6 q^{36} + 9 q^{37} - 4 q^{38} - 20 q^{39} - 19 q^{41} + q^{42} + 50 q^{43} + 2 q^{44} - 25 q^{45} + 16 q^{46} - 24 q^{47} - 4 q^{48} + 12 q^{49} - 58 q^{51} + 3 q^{52} + 35 q^{53} + 25 q^{54} - 10 q^{55} - 3 q^{56} - 44 q^{57} - 22 q^{58} + q^{59} - 5 q^{60} + 8 q^{61} + 19 q^{62} - 6 q^{63} - 3 q^{64} - 25 q^{65} - 14 q^{66} - 36 q^{67} + 4 q^{68} - 31 q^{69} + q^{71} + 9 q^{72} - 31 q^{73} - 14 q^{74} + 55 q^{75} - 16 q^{76} - 7 q^{77} - 30 q^{78} + 2 q^{79} - 8 q^{81} - 6 q^{82} - 19 q^{83} + 4 q^{84} + 20 q^{85} + 10 q^{86} + 28 q^{87} - 7 q^{88} + 40 q^{89} + 20 q^{90} - 3 q^{91} - 16 q^{92} + 50 q^{93} + 24 q^{94} - q^{96} + 28 q^{97} + 3 q^{98} - 24 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 4 x^{11} + 6 x^{10} + x^{9} - 14 x^{8} + 10 x^{7} + 35 x^{6} - 110 x^{5} + 230 x^{4} + \cdots + 125 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 8500 \nu^{11} + 27691 \nu^{10} - 20739 \nu^{9} - 55289 \nu^{8} + 111206 \nu^{7} + 70641 \nu^{6} + \cdots + 675625 ) / 374525 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 8797 \nu^{11} - 32897 \nu^{10} + 41413 \nu^{9} + 21743 \nu^{8} - 123267 \nu^{7} + 62971 \nu^{6} + \cdots - 1238725 ) / 374525 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 40926 \nu^{11} - 112175 \nu^{10} + 101300 \nu^{9} + 175115 \nu^{8} - 345870 \nu^{7} + \cdots - 4105300 ) / 374525 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 42604 \nu^{11} + 116457 \nu^{10} - 103013 \nu^{9} - 180213 \nu^{8} + 353992 \nu^{7} + \cdots + 4795375 ) / 374525 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 67540 \nu^{11} - 184923 \nu^{10} + 163897 \nu^{9} + 284782 \nu^{8} - 564428 \nu^{7} + \cdots - 7323600 ) / 374525 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 102504 \nu^{11} - 303172 \nu^{10} + 308798 \nu^{9} + 405938 \nu^{8} - 1025517 \nu^{7} + \cdots - 11045025 ) / 374525 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 132868 \nu^{11} - 375458 \nu^{10} + 364552 \nu^{9} + 534927 \nu^{8} - 1217313 \nu^{7} + \cdots - 14403300 ) / 374525 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 136311 \nu^{11} - 386732 \nu^{10} + 370398 \nu^{9} + 564408 \nu^{8} - 1256242 \nu^{7} + \cdots - 14601675 ) / 374525 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 154033 \nu^{11} + 447428 \nu^{10} - 442987 \nu^{9} - 633032 \nu^{8} + 1498533 \nu^{7} + \cdots + 16535300 ) / 374525 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 155098 \nu^{11} - 444623 \nu^{10} + 433467 \nu^{9} + 643337 \nu^{8} - 1489778 \nu^{7} + \cdots - 16066775 ) / 374525 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 158566 \nu^{11} + 460173 \nu^{10} - 458557 \nu^{9} - 645092 \nu^{8} + 1543428 \nu^{7} + \cdots + 16705350 ) / 374525 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( 2 \beta_{11} + \beta_{10} - 4 \beta_{9} - \beta_{8} - \beta_{7} - 2 \beta_{6} + \beta_{5} - \beta_{4} + \cdots + 4 ) / 5 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2 \beta_{11} + \beta_{10} - 4 \beta_{9} - 6 \beta_{8} - \beta_{7} + 3 \beta_{6} + \beta_{5} - 6 \beta_{4} + \cdots + 4 ) / 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 3 \beta_{11} + \beta_{10} - 4 \beta_{9} - 11 \beta_{8} - \beta_{7} - 2 \beta_{6} + 16 \beta_{5} + \cdots - 16 ) / 5 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 17 \beta_{11} + \beta_{10} + \beta_{9} - \beta_{8} + 4 \beta_{7} + 23 \beta_{6} + 11 \beta_{5} + \cdots - 21 ) / 5 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 8 \beta_{11} - 9 \beta_{10} + \beta_{9} + 14 \beta_{8} - 6 \beta_{7} - 2 \beta_{6} + 46 \beta_{5} + \cdots - 21 ) / 5 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 57 \beta_{11} + 31 \beta_{10} + 6 \beta_{9} + 44 \beta_{8} - 36 \beta_{7} + 48 \beta_{6} + \beta_{5} + \cdots - 26 ) / 5 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 103 \beta_{11} - 69 \beta_{10} - 44 \beta_{9} + 19 \beta_{8} - 96 \beta_{7} - 52 \beta_{6} + \cdots + 99 ) / 5 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 63 \beta_{11} + 21 \beta_{10} - 89 \beta_{9} - \beta_{8} - 141 \beta_{7} - 142 \beta_{6} - 119 \beta_{5} + \cdots - 256 ) / 5 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 333 \beta_{11} - 299 \beta_{10} + 76 \beta_{9} - 26 \beta_{8} + 154 \beta_{7} - 52 \beta_{6} + \cdots - 121 ) / 5 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 508 \beta_{11} - 84 \beta_{10} + 576 \beta_{9} + 264 \beta_{8} + 669 \beta_{7} - 677 \beta_{6} + \cdots - 1321 ) / 5 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 782 \beta_{11} + 131 \beta_{10} + 1906 \beta_{9} + 1319 \beta_{8} + 1764 \beta_{7} + 1048 \beta_{6} + \cdots + 799 ) / 5 \) Copy content Toggle raw display

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

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
71.1
0.0830190 − 1.17264i
−0.258306 + 1.14684i
1.17529 + 0.0257946i
1.70682 + 0.839517i
−1.74662 + 0.753236i
1.03979 − 1.59275i
1.70682 − 0.839517i
−1.74662 − 0.753236i
1.03979 + 1.59275i
0.0830190 + 1.17264i
−0.258306 − 1.14684i
1.17529 − 0.0257946i
−0.309017 + 0.951057i −2.43768 − 1.77108i −0.809017 − 0.587785i 0.539075 + 2.17011i 2.43768 − 1.77108i −1.00000 0.809017 − 0.587785i 1.87852 + 5.78148i −2.23049 − 0.157911i
71.2 −0.309017 + 0.951057i 0.408688 + 0.296929i −0.809017 − 0.587785i −0.206815 − 2.22648i −0.408688 + 0.296929i −1.00000 0.809017 − 0.587785i −0.848192 − 2.61047i 2.18142 + 0.491328i
71.3 −0.309017 + 0.951057i 1.71998 + 1.24964i −0.809017 − 0.587785i −2.14128 + 0.644154i −1.71998 + 1.24964i −1.00000 0.809017 − 0.587785i 0.469677 + 1.44552i 0.0490643 − 2.23553i
141.1 0.809017 + 0.587785i −0.197419 − 0.607592i 0.309017 + 0.951057i −0.380957 − 2.20338i 0.197419 − 0.607592i −1.00000 −0.309017 + 0.951057i 2.09686 − 1.52346i 0.986912 − 2.00649i
141.2 0.809017 + 0.587785i 0.241548 + 0.743409i 0.309017 + 0.951057i 1.92325 + 1.14066i −0.241548 + 0.743409i −1.00000 −0.309017 + 0.951057i 1.93274 − 1.40422i 0.885482 + 2.05327i
141.3 0.809017 + 0.587785i 0.764888 + 2.35408i 0.309017 + 0.951057i −2.23328 + 0.111663i −0.764888 + 2.35408i −1.00000 −0.309017 + 0.951057i −2.52960 + 1.83786i −1.87239 − 1.22235i
211.1 0.809017 − 0.587785i −0.197419 + 0.607592i 0.309017 − 0.951057i −0.380957 + 2.20338i 0.197419 + 0.607592i −1.00000 −0.309017 − 0.951057i 2.09686 + 1.52346i 0.986912 + 2.00649i
211.2 0.809017 − 0.587785i 0.241548 − 0.743409i 0.309017 − 0.951057i 1.92325 − 1.14066i −0.241548 − 0.743409i −1.00000 −0.309017 − 0.951057i 1.93274 + 1.40422i 0.885482 − 2.05327i
211.3 0.809017 − 0.587785i 0.764888 − 2.35408i 0.309017 − 0.951057i −2.23328 − 0.111663i −0.764888 − 2.35408i −1.00000 −0.309017 − 0.951057i −2.52960 − 1.83786i −1.87239 + 1.22235i
281.1 −0.309017 − 0.951057i −2.43768 + 1.77108i −0.809017 + 0.587785i 0.539075 − 2.17011i 2.43768 + 1.77108i −1.00000 0.809017 + 0.587785i 1.87852 − 5.78148i −2.23049 + 0.157911i
281.2 −0.309017 − 0.951057i 0.408688 − 0.296929i −0.809017 + 0.587785i −0.206815 + 2.22648i −0.408688 − 0.296929i −1.00000 0.809017 + 0.587785i −0.848192 + 2.61047i 2.18142 − 0.491328i
281.3 −0.309017 − 0.951057i 1.71998 − 1.24964i −0.809017 + 0.587785i −2.14128 − 0.644154i −1.71998 − 1.24964i −1.00000 0.809017 + 0.587785i 0.469677 − 1.44552i 0.0490643 + 2.23553i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 71.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
25.d even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 350.2.h.b ✓ 12
25.d even 5 1 inner 350.2.h.b ✓ 12
25.d even 5 1 8750.2.a.p 6
25.e even 10 1 8750.2.a.q 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
350.2.h.b ✓ 12 1.a even 1 1 trivial
350.2.h.b ✓ 12 25.d even 5 1 inner
8750.2.a.p 6 25.d even 5 1
8750.2.a.q 6 25.e even 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{12} - T_{3}^{11} + 2 T_{3}^{10} + 3 T_{3}^{9} + 33 T_{3}^{8} - 147 T_{3}^{7} + 398 T_{3}^{6} + \cdots + 16 \) acting on \(S_{2}^{\mathrm{new}}(350, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - T^{3} + T^{2} + \cdots + 1)^{3} \) Copy content Toggle raw display
$3$ \( T^{12} - T^{11} + \cdots + 16 \) Copy content Toggle raw display
$5$ \( T^{12} + 5 T^{11} + \cdots + 15625 \) Copy content Toggle raw display
$7$ \( (T + 1)^{12} \) Copy content Toggle raw display
$11$ \( T^{12} - 7 T^{11} + \cdots + 16 \) Copy content Toggle raw display
$13$ \( T^{12} - 3 T^{11} + \cdots + 22201 \) Copy content Toggle raw display
$17$ \( T^{12} - 4 T^{11} + \cdots + 167281 \) Copy content Toggle raw display
$19$ \( T^{12} - 4 T^{11} + \cdots + 400 \) Copy content Toggle raw display
$23$ \( T^{12} + T^{11} + \cdots + 17808400 \) Copy content Toggle raw display
$29$ \( T^{12} - 22 T^{11} + \cdots + 126025 \) Copy content Toggle raw display
$31$ \( T^{12} + \cdots + 1740224656 \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 16065816001 \) Copy content Toggle raw display
$41$ \( T^{12} + 19 T^{11} + \cdots + 212521 \) Copy content Toggle raw display
$43$ \( (T^{6} - 25 T^{5} + \cdots - 14284)^{2} \) Copy content Toggle raw display
$47$ \( T^{12} + 24 T^{11} + \cdots + 1478656 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 1592089801 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 174768400 \) Copy content Toggle raw display
$61$ \( T^{12} - 8 T^{11} + \cdots + 25 \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 202663696 \) Copy content Toggle raw display
$71$ \( T^{12} - T^{11} + \cdots + 18524416 \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 438693025 \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 968295360400 \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 251412736 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 596022600625 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 2280540025 \) Copy content Toggle raw display
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