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Properties

Label 190.3.g.a
Level $190$
Weight $3$
Character orbit 190.g
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(77,190)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(190, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([1, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("190.77");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 190 = 2 \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 190.g (of order \(4\), 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: \(5.17712502285\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 6 x^{15} - 12 x^{14} + 258 x^{13} - 209 x^{12} - 2408 x^{11} + 11590 x^{10} - 19384 x^{9} + \cdots + 192420 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{8}\cdot 5 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( - \beta_{5} - 1) q^{2} + ( - \beta_{5} - \beta_{2}) q^{3} + 2 \beta_{5} q^{4} + (\beta_{6} + \beta_{5}) q^{5} + (\beta_{2} + \beta_1 - 1) q^{6} + (\beta_{11} + \beta_{9} - \beta_{5} + \cdots - 1) q^{7}+ \cdots + ( - \beta_{15} + \beta_{13} + \beta_{12} + \cdots - 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{5} - 1) q^{2} + ( - \beta_{5} - \beta_{2}) q^{3} + 2 \beta_{5} q^{4} + (\beta_{6} + \beta_{5}) q^{5} + (\beta_{2} + \beta_1 - 1) q^{6} + (\beta_{11} + \beta_{9} - \beta_{5} + \cdots - 1) q^{7}+ \cdots + ( - 9 \beta_{15} - 5 \beta_{14} + \cdots - 28) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 16 q^{2} + 8 q^{3} + 2 q^{5} - 16 q^{6} - 10 q^{7} + 32 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 16 q^{2} + 8 q^{3} + 2 q^{5} - 16 q^{6} - 10 q^{7} + 32 q^{8} + 18 q^{10} + 56 q^{11} + 16 q^{12} - 4 q^{13} - 32 q^{15} - 64 q^{16} - 6 q^{17} - 32 q^{18} - 40 q^{20} + 16 q^{21} - 56 q^{22} - 64 q^{23} + 2 q^{25} + 8 q^{26} - 52 q^{27} + 20 q^{28} + 28 q^{30} - 160 q^{31} + 64 q^{32} + 136 q^{33} - 134 q^{35} + 64 q^{36} + 140 q^{37} + 44 q^{40} + 88 q^{41} - 16 q^{42} - 18 q^{43} + 196 q^{45} + 128 q^{46} + 102 q^{47} - 32 q^{48} + 46 q^{50} - 32 q^{51} - 8 q^{52} - 296 q^{53} + 128 q^{55} - 40 q^{56} - 160 q^{58} + 8 q^{60} - 64 q^{61} + 160 q^{62} + 234 q^{63} - 96 q^{65} - 272 q^{66} - 184 q^{67} + 12 q^{68} + 110 q^{70} + 144 q^{71} - 64 q^{72} + 234 q^{73} - 272 q^{75} - 134 q^{77} + 24 q^{78} - 8 q^{80} - 416 q^{81} - 88 q^{82} - 152 q^{83} + 352 q^{85} + 36 q^{86} + 208 q^{87} + 112 q^{88} - 498 q^{90} + 280 q^{91} - 128 q^{92} + 272 q^{93} + 38 q^{95} + 64 q^{96} + 184 q^{97} - 104 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 6 x^{15} - 12 x^{14} + 258 x^{13} - 209 x^{12} - 2408 x^{11} + 11590 x^{10} - 19384 x^{9} + \cdots + 192420 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 81\!\cdots\!81 \nu^{15} + \cdots + 85\!\cdots\!80 ) / 75\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 16\!\cdots\!83 \nu^{15} + \cdots - 28\!\cdots\!90 ) / 90\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 26\!\cdots\!81 \nu^{15} + \cdots + 40\!\cdots\!70 ) / 41\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 36\!\cdots\!44 \nu^{15} + \cdots + 10\!\cdots\!30 ) / 52\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 10\!\cdots\!47 \nu^{15} + \cdots - 20\!\cdots\!30 ) / 84\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 31\!\cdots\!22 \nu^{15} + \cdots + 65\!\cdots\!60 ) / 18\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 19\!\cdots\!36 \nu^{15} + \cdots + 28\!\cdots\!30 ) / 90\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 16\!\cdots\!09 \nu^{15} + \cdots + 26\!\cdots\!20 ) / 45\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 37\!\cdots\!67 \nu^{15} + \cdots - 63\!\cdots\!60 ) / 90\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 22\!\cdots\!68 \nu^{15} + \cdots - 40\!\cdots\!40 ) / 52\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 10\!\cdots\!70 \nu^{15} + \cdots - 14\!\cdots\!90 ) / 23\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 41\!\cdots\!44 \nu^{15} + \cdots + 73\!\cdots\!20 ) / 90\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 15\!\cdots\!25 \nu^{15} + \cdots - 29\!\cdots\!40 ) / 25\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 11\!\cdots\!13 \nu^{15} + \cdots + 18\!\cdots\!40 ) / 12\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 93\!\cdots\!13 \nu^{15} + \cdots + 16\!\cdots\!40 ) / 75\!\cdots\!50 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{14} - \beta_{13} - 3 \beta_{12} + \beta_{9} + 2 \beta_{8} + 2 \beta_{7} - 3 \beta_{6} + 2 \beta_{5} + \cdots + 6 ) / 10 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 5 \beta_{15} - 12 \beta_{14} + 2 \beta_{13} - 9 \beta_{12} - 5 \beta_{11} + 8 \beta_{9} + 11 \beta_{8} + \cdots + 53 ) / 10 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 25 \beta_{15} - 58 \beta_{14} - 62 \beta_{13} - 51 \beta_{12} + 25 \beta_{11} + 62 \beta_{9} + \cdots - 203 ) / 10 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2 \beta_{15} - 60 \beta_{14} - 33 \beta_{13} + 35 \beta_{12} - 22 \beta_{11} - 13 \beta_{10} + \cdots - 125 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 100 \beta_{15} - 912 \beta_{14} - 1778 \beta_{13} + 1046 \beta_{12} + 570 \beta_{11} - 365 \beta_{10} + \cdots - 6587 ) / 5 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 2095 \beta_{15} - 516 \beta_{14} - 4304 \beta_{13} + 10288 \beta_{12} - 1895 \beta_{11} + \cdots - 31021 ) / 5 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 11360 \beta_{15} + 35626 \beta_{14} - 9526 \beta_{13} + 37422 \beta_{12} + 18905 \beta_{11} + \cdots - 62574 ) / 5 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 18614 \beta_{15} + 77176 \beta_{14} + 50096 \beta_{13} + 12113 \beta_{12} - 1250 \beta_{11} + \cdots + 77955 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 234180 \beta_{15} + 2190523 \beta_{14} + 2061012 \beta_{13} - 978229 \beta_{12} + 297010 \beta_{11} + \cdots + 8400078 ) / 5 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 400200 \beta_{15} + 9143479 \beta_{14} + 13975446 \beta_{13} - 12784497 \beta_{12} - 623910 \beta_{11} + \cdots + 53859659 ) / 5 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 15240950 \beta_{15} - 7882819 \beta_{14} + 48666009 \beta_{13} - 80052473 \beta_{12} - 7900400 \beta_{11} + \cdots + 298891781 ) / 5 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 27194218 \beta_{15} - 70051559 \beta_{14} - 5861936 \beta_{13} - 75038407 \beta_{12} - 10976134 \beta_{11} + \cdots + 84263165 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 712309295 \beta_{15} - 3597835762 \beta_{14} - 2150717368 \beta_{13} - 188607074 \beta_{12} + \cdots - 5009074127 ) / 5 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 2327914900 \beta_{15} - 20302007021 \beta_{14} - 21044498064 \beta_{13} + 10093819613 \beta_{12} + \cdots - 76640312201 ) / 5 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 7663668145 \beta_{15} - 71623911204 \beta_{14} - 119494424526 \beta_{13} + 120382568792 \beta_{12} + \cdots - 525798785899 ) / 5 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(21\) \(77\)
\(\chi(n)\) \(1\) \(\beta_{5}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
77.1
−4.67816 + 1.39035i
−0.619376 − 1.53379i
−1.72551 + 1.11725i
0.459764 + 0.238948i
1.13325 + 0.0246848i
4.97394 − 3.72185i
2.25043 + 0.627370i
1.20568 + 2.85704i
−4.67816 − 1.39035i
−0.619376 + 1.53379i
−1.72551 − 1.11725i
0.459764 − 0.238948i
1.13325 − 0.0246848i
4.97394 + 3.72185i
2.25043 − 0.627370i
1.20568 − 2.85704i
−1.00000 − 1.00000i −3.28782 + 3.28782i 2.00000i 4.26936 + 2.60242i 6.57563 3.47266 + 3.47266i 2.00000 − 2.00000i − 12.6195i −1.66694 − 6.87178i
77.2 −1.00000 − 1.00000i −2.15317 + 2.15317i 2.00000i −2.31834 + 4.43004i 4.30633 −2.67809 − 2.67809i 2.00000 − 2.00000i − 0.272240i 6.74839 − 2.11170i
77.3 −1.00000 − 1.00000i −0.608266 + 0.608266i 2.00000i 4.89107 + 1.03799i 1.21653 −7.97749 − 7.97749i 2.00000 − 2.00000i 8.26002i −3.85308 − 5.92906i
77.4 −1.00000 − 1.00000i 0.698711 − 0.698711i 2.00000i −4.10268 − 2.85797i −1.39742 −4.63629 − 4.63629i 2.00000 − 2.00000i 8.02360i 1.24471 + 6.96065i
77.5 −1.00000 − 1.00000i 1.15793 − 1.15793i 2.00000i −4.99360 − 0.252934i −2.31586 7.14893 + 7.14893i 2.00000 − 2.00000i 6.31840i 4.74066 + 5.24653i
77.6 −1.00000 − 1.00000i 1.25208 − 1.25208i 2.00000i −1.16793 + 4.86168i −2.50417 0.944808 + 0.944808i 2.00000 − 2.00000i 5.86457i 6.02961 − 3.69375i
77.7 −1.00000 − 1.00000i 2.87780 − 2.87780i 2.00000i 2.39217 − 4.39062i −5.75560 −3.16915 − 3.16915i 2.00000 − 2.00000i − 7.56349i −6.78279 + 1.99844i
77.8 −1.00000 − 1.00000i 4.06272 − 4.06272i 2.00000i 2.02995 + 4.56939i −8.12544 1.89462 + 1.89462i 2.00000 − 2.00000i − 24.0114i 2.53943 − 6.59934i
153.1 −1.00000 + 1.00000i −3.28782 − 3.28782i − 2.00000i 4.26936 − 2.60242i 6.57563 3.47266 − 3.47266i 2.00000 + 2.00000i 12.6195i −1.66694 + 6.87178i
153.2 −1.00000 + 1.00000i −2.15317 − 2.15317i − 2.00000i −2.31834 − 4.43004i 4.30633 −2.67809 + 2.67809i 2.00000 + 2.00000i 0.272240i 6.74839 + 2.11170i
153.3 −1.00000 + 1.00000i −0.608266 − 0.608266i − 2.00000i 4.89107 − 1.03799i 1.21653 −7.97749 + 7.97749i 2.00000 + 2.00000i − 8.26002i −3.85308 + 5.92906i
153.4 −1.00000 + 1.00000i 0.698711 + 0.698711i − 2.00000i −4.10268 + 2.85797i −1.39742 −4.63629 + 4.63629i 2.00000 + 2.00000i − 8.02360i 1.24471 − 6.96065i
153.5 −1.00000 + 1.00000i 1.15793 + 1.15793i − 2.00000i −4.99360 + 0.252934i −2.31586 7.14893 − 7.14893i 2.00000 + 2.00000i − 6.31840i 4.74066 − 5.24653i
153.6 −1.00000 + 1.00000i 1.25208 + 1.25208i − 2.00000i −1.16793 − 4.86168i −2.50417 0.944808 − 0.944808i 2.00000 + 2.00000i − 5.86457i 6.02961 + 3.69375i
153.7 −1.00000 + 1.00000i 2.87780 + 2.87780i − 2.00000i 2.39217 + 4.39062i −5.75560 −3.16915 + 3.16915i 2.00000 + 2.00000i 7.56349i −6.78279 − 1.99844i
153.8 −1.00000 + 1.00000i 4.06272 + 4.06272i − 2.00000i 2.02995 − 4.56939i −8.12544 1.89462 − 1.89462i 2.00000 + 2.00000i 24.0114i 2.53943 + 6.59934i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 77.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 190.3.g.a ✓ 16
5.c odd 4 1 inner 190.3.g.a ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
190.3.g.a ✓ 16 1.a even 1 1 trivial
190.3.g.a ✓ 16 5.c odd 4 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{16} - 8 T_{3}^{15} + 32 T_{3}^{14} - 20 T_{3}^{13} + 616 T_{3}^{12} - 4520 T_{3}^{11} + \cdots + 665856 \) acting on \(S_{3}^{\mathrm{new}}(190, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 2 T + 2)^{8} \) Copy content Toggle raw display
$3$ \( T^{16} - 8 T^{15} + \cdots + 665856 \) Copy content Toggle raw display
$5$ \( T^{16} + \cdots + 152587890625 \) Copy content Toggle raw display
$7$ \( T^{16} + \cdots + 49818240000 \) Copy content Toggle raw display
$11$ \( (T^{8} - 28 T^{7} + \cdots + 12820240)^{2} \) Copy content Toggle raw display
$13$ \( T^{16} + \cdots + 47\!\cdots\!44 \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 29\!\cdots\!64 \) Copy content Toggle raw display
$19$ \( (T^{2} + 19)^{8} \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 48\!\cdots\!76 \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots + 43\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( (T^{8} + 80 T^{7} + \cdots - 458059802624)^{2} \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 73\!\cdots\!00 \) Copy content Toggle raw display
$41$ \( (T^{8} - 44 T^{7} + \cdots + 60021705600)^{2} \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 51\!\cdots\!44 \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 26\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 20\!\cdots\!24 \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 54\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( (T^{8} + \cdots - 1136227165872)^{2} \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 13\!\cdots\!56 \) Copy content Toggle raw display
$71$ \( (T^{8} + \cdots - 32319984079360)^{2} \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 48\!\cdots\!16 \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 28\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 27\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
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