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Properties

Label 190.3.c.a
Level $190$
Weight $3$
Character orbit 190.c
Analytic conductor $5.177$
Analytic rank $0$
Dimension $16$
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] = [190,3,Mod(151,190)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(190, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("190.151");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 190 = 2 \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 190.c (of order \(2\), degree \(1\), 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: \(5.17712502285\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 104 x^{14} + 4112 x^{12} + 76896 x^{10} + 699096 x^{8} + 3068640 x^{6} + 6595840 x^{4} + \cdots + 2624400 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{12} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{9} q^{2} - \beta_1 q^{3} - 2 q^{4} - \beta_{5} q^{5} + (\beta_{4} + 1) q^{6} + ( - \beta_{7} - 1) q^{7} - 2 \beta_{9} q^{8} + (\beta_{6} - \beta_{5} - \beta_{4} - 4) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{9} q^{2} - \beta_1 q^{3} - 2 q^{4} - \beta_{5} q^{5} + (\beta_{4} + 1) q^{6} + ( - \beta_{7} - 1) q^{7} - 2 \beta_{9} q^{8} + (\beta_{6} - \beta_{5} - \beta_{4} - 4) q^{9} + \beta_{2} q^{10} + ( - \beta_{13} + \beta_{12} + \beta_{6} - 2) q^{11} + 2 \beta_1 q^{12} + (\beta_{15} - \beta_{14} + \cdots - 2 \beta_1) q^{13}+ \cdots + (2 \beta_{13} - 2 \beta_{12} + \cdots + 10) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 32 q^{4} + 16 q^{6} - 8 q^{7} - 64 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 32 q^{4} + 16 q^{6} - 8 q^{7} - 64 q^{9} - 32 q^{11} + 64 q^{16} - 16 q^{19} - 56 q^{23} - 32 q^{24} + 80 q^{25} - 64 q^{26} + 16 q^{28} + 40 q^{35} + 128 q^{36} + 104 q^{38} - 224 q^{39} + 144 q^{42} - 264 q^{43} + 64 q^{44} + 80 q^{45} + 120 q^{47} + 336 q^{49} - 176 q^{54} - 304 q^{57} + 240 q^{58} + 48 q^{61} - 112 q^{62} - 72 q^{63} - 128 q^{64} + 176 q^{66} - 32 q^{73} - 144 q^{74} + 32 q^{76} - 128 q^{77} + 864 q^{81} - 432 q^{82} + 440 q^{83} - 624 q^{87} + 112 q^{92} - 64 q^{93} + 120 q^{95} + 64 q^{96} + 176 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 104 x^{14} + 4112 x^{12} + 76896 x^{10} + 699096 x^{8} + 3068640 x^{6} + 6595840 x^{4} + \cdots + 2624400 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 60125 \nu^{15} - 6061192 \nu^{13} - 228000388 \nu^{11} - 3907703304 \nu^{9} + \cdots - 145829795040 \nu ) / 19991979360 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 3543463 \nu^{14} - 363492644 \nu^{12} - 14055171704 \nu^{10} - 252549868842 \nu^{8} + \cdots - 6572525603400 ) / 1110665520 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 1537 \nu^{14} + 157499 \nu^{12} + 6079538 \nu^{10} + 108905184 \nu^{8} + 908251788 \nu^{6} + \cdots + 2642729040 ) / 424080 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 8753 \nu^{14} + 896947 \nu^{12} + 34622626 \nu^{10} + 620193888 \nu^{8} + 5171945748 \nu^{6} + \cdots + 15049411200 ) / 1885680 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 1147615 \nu^{14} + 117598958 \nu^{12} + 4539379592 \nu^{10} + 81314609616 \nu^{8} + \cdots + 1974981150540 ) / 138833190 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 3685855 \nu^{14} + 377680718 \nu^{12} + 14577413702 \nu^{10} + 261084258036 \nu^{8} + \cdots + 6306223427280 ) / 246814560 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 508669 \nu^{14} + 52123076 \nu^{12} + 2011874378 \nu^{10} + 36035448984 \nu^{8} + \cdots + 871371519840 ) / 27423840 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 407894 \nu^{15} + 41798491 \nu^{13} + 1613473033 \nu^{11} + 28903204134 \nu^{9} + \cdots + 701761887000 \nu ) / 171752400 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 202120787 \nu^{15} - 20717995663 \nu^{13} - 800100771964 \nu^{11} + \cdots - 354674216154600 \nu ) / 49979948400 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 8753 \nu^{15} + 896947 \nu^{13} + 34622626 \nu^{11} + 620193888 \nu^{9} + 5171945748 \nu^{7} + \cdots + 15049411200 \nu ) / 1885680 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 474232246 \nu^{15} - 464405535 \nu^{14} + 48594622004 \nu^{13} - 47583522360 \nu^{12} + \cdots - 793236617946000 ) / 49979948400 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 474232246 \nu^{15} + 696403800 \nu^{14} + 48594622004 \nu^{13} + 71357102595 \nu^{12} + \cdots + 11\!\cdots\!00 ) / 49979948400 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 341803973 \nu^{15} - 35029722682 \nu^{13} - 1352423496976 \nu^{11} + \cdots - 592907918914800 \nu ) / 33319965600 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 1041402551 \nu^{15} - 106710596224 \nu^{13} - 4118783110222 \nu^{11} + \cdots - 17\!\cdots\!00 \nu ) / 49979948400 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{6} - \beta_{5} - \beta_{4} - 13 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{15} + \beta_{14} - 3\beta_{11} - 2\beta_{10} - 2\beta_{9} - 3\beta_{2} - 23\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -2\beta_{13} + 2\beta_{12} - 2\beta_{8} - 2\beta_{7} - 28\beta_{6} + 46\beta_{5} + 38\beta_{4} + 2\beta_{3} + 322 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 42 \beta_{15} - 28 \beta_{14} - 2 \beta_{13} - 2 \beta_{12} + 114 \beta_{11} + 70 \beta_{10} + \cdots + 622 \beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 106 \beta_{13} - 106 \beta_{12} + 122 \beta_{8} + 26 \beta_{7} + 766 \beta_{6} - 1514 \beta_{5} + \cdots - 9002 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 1542 \beta_{15} + 702 \beta_{14} + 26 \beta_{13} + 26 \beta_{12} - 3826 \beta_{11} + \cdots - 17466 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 4184 \beta_{13} + 4184 \beta_{12} - 5680 \beta_{8} + 1216 \beta_{7} - 21480 \beta_{6} + 46536 \beta_{5} + \cdots + 259484 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 53928 \beta_{15} - 17088 \beta_{14} + 1216 \beta_{13} + 1216 \beta_{12} + 124880 \beta_{11} + \cdots + 497812 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 151176 \beta_{13} - 151176 \beta_{12} + 232240 \beta_{8} - 104112 \beta_{7} + 611460 \beta_{6} + \cdots - 7577844 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 1837252 \beta_{15} + 401988 \beta_{14} - 104112 \beta_{13} - 104112 \beta_{12} + \cdots - 14312988 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 5268360 \beta_{13} + 5268360 \beta_{12} - 8804200 \beta_{8} + 5179192 \beta_{7} - 17554208 \beta_{6} + \cdots + 223116168 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 61544552 \beta_{15} - 8943840 \beta_{14} + 5179192 \beta_{13} + 5179192 \beta_{12} + \cdots + 414327672 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 180042936 \beta_{13} - 180042936 \beta_{12} + 318266568 \beta_{8} - 216101336 \beta_{7} + \cdots - 6612282056 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 2037010840 \beta_{15} + 179773400 \beta_{14} - 216101336 \beta_{13} - 216101336 \beta_{12} + \cdots - 12067878568 \beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/190\mathbb{Z}\right)^\times\).

\(n\) \(21\) \(77\)
\(\chi(n)\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
151.1
5.15315i
3.15928i
1.23540i
− 1.12134i
− 1.23917i
− 1.95901i
− 5.26410i
− 5.62107i
5.62107i
5.26410i
1.95901i
1.23917i
1.12134i
− 1.23540i
− 3.15928i
− 5.15315i
− 1.41421i − 5.15315i −2.00000 −2.23607 −7.28765 0.0751428 2.82843i −17.5549 3.16228i
151.2 − 1.41421i − 3.15928i −2.00000 2.23607 −4.46790 −13.6136 2.82843i −0.981053 − 3.16228i
151.3 − 1.41421i − 1.23540i −2.00000 2.23607 −1.74712 8.54633 2.82843i 7.47378 − 3.16228i
151.4 − 1.41421i 1.12134i −2.00000 −2.23607 1.58581 6.17053 2.82843i 7.74260 3.16228i
151.5 − 1.41421i 1.23917i −2.00000 −2.23607 1.75244 −6.19957 2.82843i 7.46447 3.16228i
151.6 − 1.41421i 1.95901i −2.00000 2.23607 2.77046 −5.01111 2.82843i 5.16227 − 3.16228i
151.7 − 1.41421i 5.26410i −2.00000 2.23607 7.44456 12.5505 2.82843i −18.7107 − 3.16228i
151.8 − 1.41421i 5.62107i −2.00000 −2.23607 7.94939 −6.51823 2.82843i −22.5964 3.16228i
151.9 1.41421i − 5.62107i −2.00000 −2.23607 7.94939 −6.51823 − 2.82843i −22.5964 − 3.16228i
151.10 1.41421i − 5.26410i −2.00000 2.23607 7.44456 12.5505 − 2.82843i −18.7107 3.16228i
151.11 1.41421i − 1.95901i −2.00000 2.23607 2.77046 −5.01111 − 2.82843i 5.16227 3.16228i
151.12 1.41421i − 1.23917i −2.00000 −2.23607 1.75244 −6.19957 − 2.82843i 7.46447 − 3.16228i
151.13 1.41421i − 1.12134i −2.00000 −2.23607 1.58581 6.17053 − 2.82843i 7.74260 − 3.16228i
151.14 1.41421i 1.23540i −2.00000 2.23607 −1.74712 8.54633 − 2.82843i 7.47378 3.16228i
151.15 1.41421i 3.15928i −2.00000 2.23607 −4.46790 −13.6136 − 2.82843i −0.981053 3.16228i
151.16 1.41421i 5.15315i −2.00000 −2.23607 −7.28765 0.0751428 − 2.82843i −17.5549 − 3.16228i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 151.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 190.3.c.a ✓ 16
3.b odd 2 1 1710.3.h.a 16
4.b odd 2 1 1520.3.h.c 16
5.b even 2 1 950.3.c.d 16
5.c odd 4 2 950.3.d.c 32
19.b odd 2 1 inner 190.3.c.a ✓ 16
57.d even 2 1 1710.3.h.a 16
76.d even 2 1 1520.3.h.c 16
95.d odd 2 1 950.3.c.d 16
95.g even 4 2 950.3.d.c 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
190.3.c.a ✓ 16 1.a even 1 1 trivial
190.3.c.a ✓ 16 19.b odd 2 1 inner
950.3.c.d 16 5.b even 2 1
950.3.c.d 16 95.d odd 2 1
950.3.d.c 32 5.c odd 4 2
950.3.d.c 32 95.g even 4 2
1520.3.h.c 16 4.b odd 2 1
1520.3.h.c 16 76.d even 2 1
1710.3.h.a 16 3.b odd 2 1
1710.3.h.a 16 57.d even 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(190, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 2)^{8} \) Copy content Toggle raw display
$3$ \( T^{16} + 104 T^{14} + \cdots + 2624400 \) Copy content Toggle raw display
$5$ \( (T^{2} - 5)^{8} \) Copy content Toggle raw display
$7$ \( (T^{8} + 4 T^{7} + \cdots + 137104)^{2} \) Copy content Toggle raw display
$11$ \( (T^{8} + 16 T^{7} + \cdots - 9695664)^{2} \) Copy content Toggle raw display
$13$ \( T^{16} + \cdots + 89\!\cdots\!96 \) Copy content Toggle raw display
$17$ \( (T^{8} - 1120 T^{6} + \cdots + 22155264)^{2} \) Copy content Toggle raw display
$19$ \( T^{16} + \cdots + 28\!\cdots\!81 \) Copy content Toggle raw display
$23$ \( (T^{8} + 28 T^{7} + \cdots + 101918736)^{2} \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots + 19\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots + 43\!\cdots\!56 \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 10\!\cdots\!76 \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 22\!\cdots\!76 \) Copy content Toggle raw display
$43$ \( (T^{8} + 132 T^{7} + \cdots + 6504465680)^{2} \) Copy content Toggle raw display
$47$ \( (T^{8} + \cdots - 3549720077424)^{2} \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 16\!\cdots\!56 \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( (T^{8} + \cdots - 20869925488624)^{2} \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 19\!\cdots\!96 \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 12\!\cdots\!76 \) Copy content Toggle raw display
$73$ \( (T^{8} + \cdots + 30481328390400)^{2} \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 61\!\cdots\!56 \) Copy content Toggle raw display
$83$ \( (T^{8} + \cdots + 95722283134224)^{2} \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 19\!\cdots\!56 \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 31\!\cdots\!96 \) Copy content Toggle raw display
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