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Properties

Label 650.2.m.e
Level $650$
Weight $2$
Character orbit 650.m
Analytic conductor $5.190$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [650,2,Mod(101,650)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(650, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("650.101");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 650 = 2 \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 650.m (of order \(6\), 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: \(5.19027613138\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 4 x^{15} + 8 x^{14} - 60 x^{13} + 92 x^{12} + 292 x^{11} - 104 x^{10} + 936 x^{9} + 664 x^{8} + \cdots + 256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 130)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q + (\beta_{6} - \beta_1) q^{2} - \beta_{5} q^{3} - \beta_{8} q^{4} - \beta_{10} q^{6} + ( - \beta_{15} + \beta_{12} - 2 \beta_1) q^{7} + \beta_{6} q^{8} + (\beta_{13} + 2 \beta_{8} + \beta_{3}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{6} - \beta_1) q^{2} - \beta_{5} q^{3} - \beta_{8} q^{4} - \beta_{10} q^{6} + ( - \beta_{15} + \beta_{12} - 2 \beta_1) q^{7} + \beta_{6} q^{8} + (\beta_{13} + 2 \beta_{8} + \beta_{3}) q^{9} + (\beta_{13} + \beta_{9} + \cdots + \beta_{3}) q^{11}+ \cdots + ( - 2 \beta_{14} - 2 \beta_{13} + \cdots - 3) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 8 q^{4} - 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 8 q^{4} - 16 q^{9} - 6 q^{11} + 20 q^{14} - 8 q^{16} - 18 q^{19} + 2 q^{26} + 6 q^{29} + 16 q^{36} + 60 q^{39} + 42 q^{41} + 12 q^{46} + 30 q^{49} - 40 q^{51} - 36 q^{54} + 10 q^{56} - 60 q^{59} - 10 q^{61} - 16 q^{64} + 40 q^{66} + 4 q^{69} - 40 q^{74} - 18 q^{76} - 8 q^{79} + 16 q^{81} + 12 q^{84} + 78 q^{89} + 38 q^{91} + 6 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 4 x^{15} + 8 x^{14} - 60 x^{13} + 92 x^{12} + 292 x^{11} - 104 x^{10} + 936 x^{9} + 664 x^{8} + \cdots + 256 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 2065578893166 \nu^{15} - 6111766661753 \nu^{14} + 6120662015756 \nu^{13} + \cdots - 80997325765120 ) / 53\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 13397656256353 \nu^{15} - 9016531902152 \nu^{14} - 103008017245240 \nu^{13} + \cdots - 90\!\cdots\!72 ) / 31\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 22580243702731 \nu^{15} + 98005936147114 \nu^{14} - 204452125829736 \nu^{13} + \cdots + 55\!\cdots\!72 ) / 31\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 27874286643835 \nu^{15} + 115563014713402 \nu^{14} - 256230263859000 \nu^{13} + \cdots - 82\!\cdots\!72 ) / 31\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 4724485253054 \nu^{15} - 25966685637649 \nu^{14} + 68728556980573 \nu^{13} + \cdots + 26\!\cdots\!72 ) / 39\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 48028030533 \nu^{15} + 232497597480 \nu^{14} - 556156244592 \nu^{13} + \cdots - 20248659992896 ) / 32556190177088 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 56141077577939 \nu^{15} + 200713994460202 \nu^{14} - 328005231218680 \nu^{13} + \cdots - 63\!\cdots\!60 ) / 31\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 550055329 \nu^{15} + 2332659166 \nu^{14} - 5017965216 \nu^{13} + 34498573860 \nu^{12} + \cdots - 454245598336 ) / 286495174464 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 8015009402188 \nu^{15} + 34029622170307 \nu^{14} - 73132623048273 \nu^{13} + \cdots - 10\!\cdots\!96 ) / 39\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 4085162631071 \nu^{15} + 17298688988309 \nu^{14} - 37280037848952 \nu^{13} + \cdots + 887599405731968 ) / 19\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 9427336476743 \nu^{15} + 39626746566937 \nu^{14} - 81445703602405 \nu^{13} + \cdots - 25\!\cdots\!60 ) / 39\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 42864733185952 \nu^{15} - 212199751759007 \nu^{14} + 523041919014800 \nu^{13} + \cdots + 20\!\cdots\!08 ) / 15\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 31344348417259 \nu^{15} - 133281206159050 \nu^{14} + 285592859171934 \nu^{13} + \cdots + 66\!\cdots\!28 ) / 79\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 32202738967789 \nu^{15} - 136066538220838 \nu^{14} + 294095136247026 \nu^{13} + \cdots + 28\!\cdots\!76 ) / 79\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 65461821600364 \nu^{15} - 280028758731041 \nu^{14} + 594599182252508 \nu^{13} + \cdots + 19\!\cdots\!04 ) / 15\!\cdots\!64 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{14} - \beta_{12} + \beta_{8} - \beta_{6} + \beta_{4} - \beta_{2} + \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{15} - \beta_{12} + \beta_{11} + \beta_{7} + \beta_{5} - \beta_{2} + 5\beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 4 \beta_{15} + 5 \beta_{14} - 5 \beta_{13} - \beta_{12} - 2 \beta_{11} - 2 \beta_{10} + 2 \beta_{9} + \cdots + 10 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 17\beta_{14} - 4\beta_{13} - 28\beta_{10} - 14\beta_{9} + 61\beta_{8} + 21\beta_{4} + 4\beta_{3} + 61 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 65 \beta_{15} - 14 \beta_{14} + 51 \beta_{13} - 65 \beta_{12} + 35 \beta_{11} - 35 \beta_{10} + \cdots + 155 \beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 262\beta_{15} - 70\beta_{12} - 200\beta_{11} + 332\beta_{7} + 866\beta_{6} + 400\beta_{5} + 70\beta_{2} \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 200 \beta_{15} + 729 \beta_{14} - 200 \beta_{13} + 729 \beta_{12} - 1064 \beta_{11} - 1064 \beta_{10} + \cdots + 2317 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -1064\beta_{14} + 3910\beta_{13} - 2922\beta_{10} - 5844\beta_{9} + 12670\beta_{8} + 1064\beta_{4} + 4974\beta_{3} \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 10680 \beta_{15} - 13602 \beta_{14} + 13602 \beta_{13} - 2922 \beta_{12} - 7896 \beta_{11} + \cdots - 34292 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 15792 \beta_{15} + 57842 \beta_{12} - 85992 \beta_{11} + 15792 \beta_{7} + 186554 \beta_{6} + \cdots - 186554 \beta_1 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 200294 \beta_{15} - 42996 \beta_{14} + 157298 \beta_{13} + 200294 \beta_{12} - 116630 \beta_{11} + \cdots - 506242 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 1087064 \beta_{14} + 1087064 \beta_{13} + 633848 \beta_{10} - 633848 \beta_{9} - 853804 \beta_{4} + \cdots - 2750692 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 633848 \beta_{15} - 2319466 \beta_{14} + 633848 \beta_{13} + 2319466 \beta_{12} - 3441824 \beta_{11} + \cdots - 7468898 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 16038164 \beta_{15} + 16038164 \beta_{12} - 9348452 \beta_{11} - 12596340 \beta_{7} + \cdots - 40571108 \beta_1 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 34211264 \beta_{15} - 43559716 \beta_{14} + 43559716 \beta_{13} + 9348452 \beta_{12} + \cdots - 110177176 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/650\mathbb{Z}\right)^\times\).

\(n\) \(27\) \(301\)
\(\chi(n)\) \(1\) \(-\beta_{8}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
101.1
3.70981 + 0.994041i
−1.78675 − 0.478758i
−0.425524 − 0.114019i
1.23451 + 0.330787i
−0.330787 + 1.23451i
0.114019 − 0.425524i
0.478758 − 1.78675i
−0.994041 + 3.70981i
3.70981 − 0.994041i
−1.78675 + 0.478758i
−0.425524 + 0.114019i
1.23451 − 0.330787i
−0.330787 − 1.23451i
0.114019 + 0.425524i
0.478758 + 1.78675i
−0.994041 − 3.70981i
−0.866025 + 0.500000i −1.53554 − 2.65963i 0.500000 − 0.866025i 0 2.65963 + 1.53554i −3.20005 − 1.84755i 1.00000i −3.21577 + 5.56988i 0
101.2 −0.866025 + 0.500000i −0.588222 − 1.01883i 0.500000 − 0.866025i 0 1.01883 + 0.588222i 1.80732 + 1.04346i 1.00000i 0.807991 − 1.39948i 0
101.3 −0.866025 + 0.500000i 0.918829 + 1.59146i 0.500000 − 0.866025i 0 −1.59146 − 0.918829i −4.15469 − 2.39871i 1.00000i −0.188495 + 0.326482i 0
101.4 −0.866025 + 0.500000i 1.20493 + 2.08700i 0.500000 − 0.866025i 0 −2.08700 − 1.20493i 1.21729 + 0.702803i 1.00000i −1.40373 + 2.43133i 0
101.5 0.866025 − 0.500000i −1.20493 − 2.08700i 0.500000 − 0.866025i 0 −2.08700 − 1.20493i −1.21729 − 0.702803i − 1.00000i −1.40373 + 2.43133i 0
101.6 0.866025 − 0.500000i −0.918829 − 1.59146i 0.500000 − 0.866025i 0 −1.59146 − 0.918829i 4.15469 + 2.39871i − 1.00000i −0.188495 + 0.326482i 0
101.7 0.866025 − 0.500000i 0.588222 + 1.01883i 0.500000 − 0.866025i 0 1.01883 + 0.588222i −1.80732 − 1.04346i − 1.00000i 0.807991 − 1.39948i 0
101.8 0.866025 − 0.500000i 1.53554 + 2.65963i 0.500000 − 0.866025i 0 2.65963 + 1.53554i 3.20005 + 1.84755i − 1.00000i −3.21577 + 5.56988i 0
251.1 −0.866025 − 0.500000i −1.53554 + 2.65963i 0.500000 + 0.866025i 0 2.65963 − 1.53554i −3.20005 + 1.84755i − 1.00000i −3.21577 − 5.56988i 0
251.2 −0.866025 − 0.500000i −0.588222 + 1.01883i 0.500000 + 0.866025i 0 1.01883 − 0.588222i 1.80732 − 1.04346i − 1.00000i 0.807991 + 1.39948i 0
251.3 −0.866025 − 0.500000i 0.918829 − 1.59146i 0.500000 + 0.866025i 0 −1.59146 + 0.918829i −4.15469 + 2.39871i − 1.00000i −0.188495 − 0.326482i 0
251.4 −0.866025 − 0.500000i 1.20493 − 2.08700i 0.500000 + 0.866025i 0 −2.08700 + 1.20493i 1.21729 − 0.702803i − 1.00000i −1.40373 − 2.43133i 0
251.5 0.866025 + 0.500000i −1.20493 + 2.08700i 0.500000 + 0.866025i 0 −2.08700 + 1.20493i −1.21729 + 0.702803i 1.00000i −1.40373 − 2.43133i 0
251.6 0.866025 + 0.500000i −0.918829 + 1.59146i 0.500000 + 0.866025i 0 −1.59146 + 0.918829i 4.15469 − 2.39871i 1.00000i −0.188495 − 0.326482i 0
251.7 0.866025 + 0.500000i 0.588222 − 1.01883i 0.500000 + 0.866025i 0 1.01883 − 0.588222i −1.80732 + 1.04346i 1.00000i 0.807991 + 1.39948i 0
251.8 0.866025 + 0.500000i 1.53554 − 2.65963i 0.500000 + 0.866025i 0 2.65963 − 1.53554i 3.20005 − 1.84755i 1.00000i −3.21577 − 5.56988i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 101.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
13.e even 6 1 inner
65.l even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 650.2.m.e 16
5.b even 2 1 inner 650.2.m.e 16
5.c odd 4 1 130.2.m.a ✓ 8
5.c odd 4 1 130.2.m.b yes 8
13.e even 6 1 inner 650.2.m.e 16
13.f odd 12 1 8450.2.a.cr 8
13.f odd 12 1 8450.2.a.cs 8
15.e even 4 1 1170.2.bj.a 8
15.e even 4 1 1170.2.bj.b 8
20.e even 4 1 1040.2.df.a 8
20.e even 4 1 1040.2.df.c 8
65.l even 6 1 inner 650.2.m.e 16
65.o even 12 2 1690.2.b.e 16
65.q odd 12 1 1690.2.c.e 8
65.q odd 12 1 1690.2.c.f 8
65.r odd 12 1 130.2.m.a ✓ 8
65.r odd 12 1 130.2.m.b yes 8
65.r odd 12 1 1690.2.c.e 8
65.r odd 12 1 1690.2.c.f 8
65.s odd 12 1 8450.2.a.cr 8
65.s odd 12 1 8450.2.a.cs 8
65.t even 12 2 1690.2.b.e 16
195.bf even 12 1 1170.2.bj.a 8
195.bf even 12 1 1170.2.bj.b 8
260.bg even 12 1 1040.2.df.a 8
260.bg even 12 1 1040.2.df.c 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
130.2.m.a ✓ 8 5.c odd 4 1
130.2.m.a ✓ 8 65.r odd 12 1
130.2.m.b yes 8 5.c odd 4 1
130.2.m.b yes 8 65.r odd 12 1
650.2.m.e 16 1.a even 1 1 trivial
650.2.m.e 16 5.b even 2 1 inner
650.2.m.e 16 13.e even 6 1 inner
650.2.m.e 16 65.l even 6 1 inner
1040.2.df.a 8 20.e even 4 1
1040.2.df.a 8 260.bg even 12 1
1040.2.df.c 8 20.e even 4 1
1040.2.df.c 8 260.bg even 12 1
1170.2.bj.a 8 15.e even 4 1
1170.2.bj.a 8 195.bf even 12 1
1170.2.bj.b 8 15.e even 4 1
1170.2.bj.b 8 195.bf even 12 1
1690.2.b.e 16 65.o even 12 2
1690.2.b.e 16 65.t even 12 2
1690.2.c.e 8 65.q odd 12 1
1690.2.c.e 8 65.r odd 12 1
1690.2.c.f 8 65.q odd 12 1
1690.2.c.f 8 65.r odd 12 1
8450.2.a.cr 8 13.f odd 12 1
8450.2.a.cr 8 65.s odd 12 1
8450.2.a.cs 8 13.f odd 12 1
8450.2.a.cs 8 65.s odd 12 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{16} + 20 T_{3}^{14} + 268 T_{3}^{12} + 1976 T_{3}^{10} + 10528 T_{3}^{8} + 33584 T_{3}^{6} + \cdots + 65536 \) acting on \(S_{2}^{\mathrm{new}}(650, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - T^{2} + 1)^{4} \) Copy content Toggle raw display
$3$ \( T^{16} + 20 T^{14} + \cdots + 65536 \) Copy content Toggle raw display
$5$ \( T^{16} \) Copy content Toggle raw display
$7$ \( T^{16} - 43 T^{14} + \cdots + 7311616 \) Copy content Toggle raw display
$11$ \( (T^{8} + 3 T^{7} + \cdots + 784)^{2} \) Copy content Toggle raw display
$13$ \( T^{16} + \cdots + 815730721 \) Copy content Toggle raw display
$17$ \( T^{16} + 59 T^{14} + \cdots + 16 \) Copy content Toggle raw display
$19$ \( (T^{8} + 9 T^{7} + \cdots + 9216)^{2} \) Copy content Toggle raw display
$23$ \( T^{16} + 44 T^{14} + \cdots + 65536 \) Copy content Toggle raw display
$29$ \( (T^{8} - 3 T^{7} + \cdots + 576)^{2} \) Copy content Toggle raw display
$31$ \( (T^{8} + 60 T^{6} + \cdots + 576)^{2} \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 141158161 \) Copy content Toggle raw display
$41$ \( (T^{8} - 21 T^{7} + \cdots + 1106704)^{2} \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 5566277615616 \) Copy content Toggle raw display
$47$ \( (T^{8} + 243 T^{6} + \cdots + 9216)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} - 356 T^{6} + \cdots + 17783089)^{2} \) Copy content Toggle raw display
$59$ \( (T^{8} + 30 T^{7} + \cdots + 369664)^{2} \) Copy content Toggle raw display
$61$ \( (T^{8} + 5 T^{7} + \cdots + 28751044)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} - 16 T^{2} + 256)^{4} \) Copy content Toggle raw display
$71$ \( (T^{8} - 160 T^{6} + \cdots + 16777216)^{2} \) Copy content Toggle raw display
$73$ \( (T^{8} + 397 T^{6} + \cdots + 1976836)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} + 2 T^{3} + \cdots + 1384)^{4} \) Copy content Toggle raw display
$83$ \( (T^{8} + 420 T^{6} + \cdots + 94945536)^{2} \) Copy content Toggle raw display
$89$ \( (T^{8} - 39 T^{7} + \cdots + 59474944)^{2} \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 107049369856 \) Copy content Toggle raw display
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