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Properties

Label 950.2.e.n
Level $950$
Weight $2$
Character orbit 950.e
Analytic conductor $7.586$
Analytic rank $0$
Dimension $10$
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] = [950,2,Mod(201,950)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(950, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("950.201");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 950 = 2 \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 950.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: \(7.58578819202\)
Analytic rank: \(0\)
Dimension: \(10\)
Relative dimension: \(5\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} + 10x^{8} - 12x^{7} + 85x^{6} - 70x^{5} + 186x^{4} - 110x^{3} + 285x^{2} - 150x + 100 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 190)
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} + 10x^{8} - 12x^{7} + 85x^{6} - 70x^{5} + 186x^{4} - 110x^{3} + 285x^{2} - 150x + 100 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 47 \nu^{9} + 582 \nu^{8} - 1940 \nu^{7} + 5464 \nu^{6} - 22892 \nu^{5} + 61110 \nu^{4} + \cdots + 154050 ) / 224695 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 535 \nu^{9} - 888 \nu^{8} + 2960 \nu^{7} - 13433 \nu^{6} + 34928 \nu^{5} - 93240 \nu^{4} + \cdots - 740030 ) / 224695 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 864 \nu^{9} - 4962 \nu^{8} + 16540 \nu^{7} - 51681 \nu^{6} + 195172 \nu^{5} - 521010 \nu^{4} + \cdots - 919600 ) / 224695 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 3081 \nu^{9} - 94 \nu^{8} - 29646 \nu^{7} + 33092 \nu^{6} - 250957 \nu^{5} + 169886 \nu^{4} + \cdots - 103640 ) / 449390 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 1752 \nu^{9} - 2572 \nu^{8} + 23553 \nu^{7} - 41134 \nu^{6} + 250962 \nu^{5} - 359938 \nu^{4} + \cdots - 866100 ) / 224695 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 3642 \nu^{9} + 7809 \nu^{8} - 26030 \nu^{7} + 128908 \nu^{6} - 307154 \nu^{5} + 819945 \nu^{4} + \cdots + 1304860 ) / 224695 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 5580 \nu^{9} + 1658 \nu^{8} + 54392 \nu^{7} - 47287 \nu^{6} + 444094 \nu^{5} - 185422 \nu^{4} + \cdots + 202580 ) / 224695 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 6198 \nu^{9} + 11216 \nu^{8} + 67471 \nu^{7} + 27158 \nu^{6} + 427658 \nu^{5} + 323839 \nu^{4} + \cdots + 318760 ) / 224695 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{8} - 4\beta_{5} - \beta_{3} + \beta_{2} - \beta _1 - 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{4} + \beta_{3} - 7\beta_{2} + 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{9} + 9\beta_{8} - \beta_{6} + 28\beta_{5} + \beta_{4} + 13\beta_1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -16\beta_{8} + 10\beta_{6} - 54\beta_{5} - 16\beta_{3} + 61\beta_{2} - 61\beta _1 - 54 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 10\beta_{7} - 16\beta_{4} + 81\beta_{3} - 147\beta_{2} + 244 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -6\beta_{9} + 189\beta_{8} - 91\beta_{6} + 608\beta_{5} + 91\beta_{4} + 573\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 85 \beta_{9} - 761 \beta_{8} - 85 \beta_{7} + 195 \beta_{6} - 2304 \beta_{5} - 761 \beta_{3} + \cdots - 2304 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 110\beta_{7} - 846\beta_{4} + 2046\beta_{3} - 5567\beta_{2} + 6454 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/950\mathbb{Z}\right)^\times\).

\(n\) \(77\) \(401\)
\(\chi(n)\) \(1\) \(-1 - \beta_{5}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
201.1
−1.58826 + 2.75095i
−0.664633 + 1.15118i
0.341187 − 0.590953i
0.741409 − 1.28416i
1.17030 − 2.02701i
−1.58826 − 2.75095i
−0.664633 − 1.15118i
0.341187 + 0.590953i
0.741409 + 1.28416i
1.17030 + 2.02701i
−0.500000 + 0.866025i −1.58826 + 2.75095i −0.500000 − 0.866025i 0 −1.58826 − 2.75095i −4.17652 1.00000 −3.54514 − 6.14037i 0
201.2 −0.500000 + 0.866025i −0.664633 + 1.15118i −0.500000 − 0.866025i 0 −0.664633 − 1.15118i −2.32927 1.00000 0.616527 + 1.06786i 0
201.3 −0.500000 + 0.866025i 0.341187 − 0.590953i −0.500000 − 0.866025i 0 0.341187 + 0.590953i −0.317626 1.00000 1.26718 + 2.19482i 0
201.4 −0.500000 + 0.866025i 0.741409 − 1.28416i −0.500000 − 0.866025i 0 0.741409 + 1.28416i 0.482818 1.00000 0.400626 + 0.693904i 0
201.5 −0.500000 + 0.866025i 1.17030 − 2.02701i −0.500000 − 0.866025i 0 1.17030 + 2.02701i 1.34059 1.00000 −1.23919 − 2.14634i 0
501.1 −0.500000 − 0.866025i −1.58826 − 2.75095i −0.500000 + 0.866025i 0 −1.58826 + 2.75095i −4.17652 1.00000 −3.54514 + 6.14037i 0
501.2 −0.500000 − 0.866025i −0.664633 − 1.15118i −0.500000 + 0.866025i 0 −0.664633 + 1.15118i −2.32927 1.00000 0.616527 − 1.06786i 0
501.3 −0.500000 − 0.866025i 0.341187 + 0.590953i −0.500000 + 0.866025i 0 0.341187 − 0.590953i −0.317626 1.00000 1.26718 − 2.19482i 0
501.4 −0.500000 − 0.866025i 0.741409 + 1.28416i −0.500000 + 0.866025i 0 0.741409 − 1.28416i 0.482818 1.00000 0.400626 − 0.693904i 0
501.5 −0.500000 − 0.866025i 1.17030 + 2.02701i −0.500000 + 0.866025i 0 1.17030 − 2.02701i 1.34059 1.00000 −1.23919 + 2.14634i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 201.5
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 950.2.e.n 10
5.b even 2 1 950.2.e.o 10
5.c odd 4 2 190.2.i.a ✓ 20
15.e even 4 2 1710.2.t.d 20
19.c even 3 1 inner 950.2.e.n 10
95.i even 6 1 950.2.e.o 10
95.m odd 12 2 190.2.i.a ✓ 20
285.v even 12 2 1710.2.t.d 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
190.2.i.a ✓ 20 5.c odd 4 2
190.2.i.a ✓ 20 95.m odd 12 2
950.2.e.n 10 1.a even 1 1 trivial
950.2.e.n 10 19.c even 3 1 inner
950.2.e.o 10 5.b even 2 1
950.2.e.o 10 95.i even 6 1
1710.2.t.d 20 15.e even 4 2
1710.2.t.d 20 285.v even 12 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(950, [\chi])\):

\( T_{3}^{10} + 10T_{3}^{8} - 12T_{3}^{7} + 85T_{3}^{6} - 70T_{3}^{5} + 186T_{3}^{4} - 110T_{3}^{3} + 285T_{3}^{2} - 150T_{3} + 100 \) Copy content Toggle raw display
\( T_{7}^{5} + 5T_{7}^{4} - 14T_{7}^{2} + 2T_{7} + 2 \) Copy content Toggle raw display
\( T_{11}^{5} + 3T_{11}^{4} - 48T_{11}^{3} - 176T_{11}^{2} + 389T_{11} + 1555 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + T + 1)^{5} \) Copy content Toggle raw display
$3$ \( T^{10} + 10 T^{8} + \cdots + 100 \) Copy content Toggle raw display
$5$ \( T^{10} \) Copy content Toggle raw display
$7$ \( (T^{5} + 5 T^{4} - 14 T^{2} + \cdots + 2)^{2} \) Copy content Toggle raw display
$11$ \( (T^{5} + 3 T^{4} + \cdots + 1555)^{2} \) Copy content Toggle raw display
$13$ \( T^{10} + 2 T^{9} + \cdots + 256 \) Copy content Toggle raw display
$17$ \( T^{10} - 4 T^{9} + \cdots + 256 \) Copy content Toggle raw display
$19$ \( T^{10} - 11 T^{9} + \cdots + 2476099 \) Copy content Toggle raw display
$23$ \( T^{10} - 13 T^{9} + \cdots + 917764 \) Copy content Toggle raw display
$29$ \( T^{10} - 2 T^{9} + \cdots + 4000000 \) Copy content Toggle raw display
$31$ \( (T^{5} + 4 T^{4} + \cdots + 1252)^{2} \) Copy content Toggle raw display
$37$ \( (T^{5} - 5 T^{4} + \cdots - 7758)^{2} \) Copy content Toggle raw display
$41$ \( T^{10} - T^{9} + \cdots + 6561 \) Copy content Toggle raw display
$43$ \( T^{10} + 88 T^{8} + \cdots + 1327104 \) Copy content Toggle raw display
$47$ \( T^{10} + 10 T^{9} + \cdots + 129600 \) Copy content Toggle raw display
$53$ \( T^{10} - 5 T^{9} + \cdots + 62726400 \) Copy content Toggle raw display
$59$ \( T^{10} - 22 T^{9} + \cdots + 15376 \) Copy content Toggle raw display
$61$ \( T^{10} + 2 T^{9} + \cdots + 62853184 \) Copy content Toggle raw display
$67$ \( T^{10} - 4 T^{9} + \cdots + 656100 \) Copy content Toggle raw display
$71$ \( T^{10} + 22 T^{9} + \cdots + 952576 \) Copy content Toggle raw display
$73$ \( T^{10} + \cdots + 360620100 \) Copy content Toggle raw display
$79$ \( T^{10} - 2 T^{9} + \cdots + 19360000 \) Copy content Toggle raw display
$83$ \( (T^{5} - 6 T^{4} + \cdots + 792)^{2} \) Copy content Toggle raw display
$89$ \( T^{10} + \cdots + 397922704 \) Copy content Toggle raw display
$97$ \( T^{10} - 8 T^{9} + \cdots + 83905600 \) Copy content Toggle raw display
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