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Properties

Label 650.6.b.o
Level $650$
Weight $6$
Character orbit 650.b
Analytic conductor $104.249$
Analytic rank $0$
Dimension $12$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [650,6,Mod(599,650)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(650, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("650.599");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 650 = 2 \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 650.b (of order \(2\), degree \(1\), not 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: \(104.249482878\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 2075 x^{10} + 1596139 x^{8} + 560320345 x^{6} + 89526738200 x^{4} + 5863900282000 x^{2} + 131166627840000 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{8}\cdot 3^{4}\cdot 5^{4} \)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 4 \beta_{7} q^{2} + (2 \beta_{7} + \beta_1) q^{3} - 16 q^{4} + ( - 4 \beta_{3} + 4) q^{6} + (\beta_{8} + 9 \beta_{7} - 2 \beta_1) q^{7} + 64 \beta_{7} q^{8} + (4 \beta_{3} + \beta_{2} - 103) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 4 \beta_{7} q^{2} + (2 \beta_{7} + \beta_1) q^{3} - 16 q^{4} + ( - 4 \beta_{3} + 4) q^{6} + (\beta_{8} + 9 \beta_{7} - 2 \beta_1) q^{7} + 64 \beta_{7} q^{8} + (4 \beta_{3} + \beta_{2} - 103) q^{9} + ( - \beta_{5} + \beta_{4} - 4 \beta_{3} + \cdots + 59) q^{11}+ \cdots + ( - 41 \beta_{6} + 61 \beta_{5} + \cdots - 63266) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 192 q^{4} + 72 q^{6} - 1258 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 192 q^{4} + 72 q^{6} - 1258 q^{9} + 740 q^{11} + 504 q^{14} + 3072 q^{16} + 3318 q^{19} + 8644 q^{21} - 1152 q^{24} + 8112 q^{26} + 11822 q^{29} + 2806 q^{31} - 6704 q^{34} + 20128 q^{36} - 3042 q^{39} + 2974 q^{41} - 11840 q^{44} + 12528 q^{46} - 57954 q^{49} + 14106 q^{51} - 60552 q^{54} - 8064 q^{56} - 44734 q^{59} + 13422 q^{61} - 49152 q^{64} + 71944 q^{66} - 268676 q^{69} + 30440 q^{71} - 84784 q^{74} - 53088 q^{76} - 190856 q^{79} - 16268 q^{81} - 138304 q^{84} + 128624 q^{86} - 584622 q^{89} - 21294 q^{91} - 57080 q^{94} + 18432 q^{96} - 772238 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 2075 x^{10} + 1596139 x^{8} + 560320345 x^{6} + 89526738200 x^{4} + 5863900282000 x^{2} + 131166627840000 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 346 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 19733 \nu^{10} + 39822560 \nu^{8} + 29735096937 \nu^{6} + 9918450660950 \nu^{4} + \cdots + 50\!\cdots\!00 ) / 266061505788000 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 6881171 \nu^{10} - 14959851470 \nu^{8} - 11735455544619 \nu^{6} + \cdots - 17\!\cdots\!00 ) / 74\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 3169303 \nu^{10} - 6184839685 \nu^{8} - 4354797096417 \nu^{6} + \cdots - 57\!\cdots\!00 ) / 18\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 9823253 \nu^{10} + 19943566010 \nu^{8} + 14743997352717 \nu^{6} + \cdots + 23\!\cdots\!00 ) / 10\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 37126727 \nu^{11} - 75154641005 \nu^{9} - 55458751780653 \nu^{7} + \cdots - 87\!\cdots\!00 \nu ) / 25\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 1415489533 \nu^{11} - 2860475606545 \nu^{9} + \cdots - 24\!\cdots\!00 \nu ) / 22\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 137083729 \nu^{11} - 271894870255 \nu^{9} - 195359122153731 \nu^{7} + \cdots - 27\!\cdots\!00 \nu ) / 43\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 1827088337 \nu^{11} - 3700473443555 \nu^{9} + \cdots - 42\!\cdots\!00 \nu ) / 42\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 4667921521 \nu^{11} - 9391985272615 \nu^{9} + \cdots - 11\!\cdots\!00 \nu ) / 84\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 346 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -5\beta_{11} + 6\beta_{10} + 3\beta_{9} - 8\beta_{8} - 178\beta_{7} - 533\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 38\beta_{6} + 92\beta_{5} + 102\beta_{4} - 1348\beta_{3} - 757\beta_{2} + 185022 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 3877\beta_{11} - 6570\beta_{10} - 2451\beta_{9} + 9448\beta_{8} + 589514\beta_{7} + 325113\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -42298\beta_{6} - 90172\beta_{5} - 103914\beta_{4} + 1901000\beta_{3} + 524161\beta_{2} - 112341914 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 2710977 \beta_{11} + 5821302 \beta_{10} + 1777095 \beta_{9} - 7713528 \beta_{8} + \cdots - 209249149 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 36820038 \beta_{6} + 73566732 \beta_{5} + 79638822 \beta_{4} - 1902575148 \beta_{3} - 360667117 \beta_{2} + 71914733782 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 1876902317 \beta_{11} - 4797514050 \beta_{10} - 1185467691 \beta_{9} + 5680202888 \beta_{8} + \cdots + 138884012513 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 29667773138 \beta_{6} - 58827339932 \beta_{5} - 55400135394 \beta_{4} + 1665997400560 \beta_{3} + \cdots - 47493288156594 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 1305650063737 \beta_{11} + 3810722955822 \beta_{10} + 739045744815 \beta_{9} + \cdots - 94114482067749 \beta_1 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(27\) \(301\)
\(\chi(n)\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
599.1
− 24.3019i
− 20.7411i
− 7.34457i
7.89583i
14.5346i
26.9571i
− 26.9571i
− 14.5346i
− 7.89583i
7.34457i
20.7411i
24.3019i
− 4.00000i − 22.3019i −16.0000 0 −89.2074 − 57.6251i 64.0000i −254.373 0
599.2 − 4.00000i − 18.7411i −16.0000 0 −74.9644 230.695i 64.0000i −108.229 0
599.3 − 4.00000i − 5.34457i −16.0000 0 −21.3783 − 50.8857i 64.0000i 214.436 0
599.4 − 4.00000i 9.89583i −16.0000 0 39.5833 138.510i 64.0000i 145.073 0
599.5 − 4.00000i 16.5346i −16.0000 0 66.1385 − 225.264i 64.0000i −30.3936 0
599.6 − 4.00000i 28.9571i −16.0000 0 115.828 27.5699i 64.0000i −595.513 0
599.7 4.00000i − 28.9571i −16.0000 0 115.828 − 27.5699i − 64.0000i −595.513 0
599.8 4.00000i − 16.5346i −16.0000 0 66.1385 225.264i − 64.0000i −30.3936 0
599.9 4.00000i − 9.89583i −16.0000 0 39.5833 − 138.510i − 64.0000i 145.073 0
599.10 4.00000i 5.34457i −16.0000 0 −21.3783 50.8857i − 64.0000i 214.436 0
599.11 4.00000i 18.7411i −16.0000 0 −74.9644 − 230.695i − 64.0000i −108.229 0
599.12 4.00000i 22.3019i −16.0000 0 −89.2074 57.6251i − 64.0000i −254.373 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 599.12
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 650.6.b.o 12
5.b even 2 1 inner 650.6.b.o 12
5.c odd 4 1 650.6.a.s ✓ 6
5.c odd 4 1 650.6.a.t yes 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
650.6.a.s ✓ 6 5.c odd 4 1
650.6.a.t yes 6 5.c odd 4 1
650.6.b.o 12 1.a even 1 1 trivial
650.6.b.o 12 5.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{12} + 2087 T_{3}^{10} + 1598287 T_{3}^{8} + 564709677 T_{3}^{6} + 92993520132 T_{3}^{4} + \cdots + 112021056000000 \) acting on \(S_{6}^{\mathrm{new}}(650, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 16)^{6} \) Copy content Toggle raw display
$3$ \( T^{12} + \cdots + 112021056000000 \) Copy content Toggle raw display
$5$ \( T^{12} \) Copy content Toggle raw display
$7$ \( T^{12} + \cdots + 33\!\cdots\!00 \) Copy content Toggle raw display
$11$ \( (T^{6} + \cdots - 13218208681500)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 28561)^{6} \) Copy content Toggle raw display
$17$ \( T^{12} + \cdots + 24\!\cdots\!16 \) Copy content Toggle raw display
$19$ \( (T^{6} + \cdots - 20\!\cdots\!40)^{2} \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 10\!\cdots\!96 \) Copy content Toggle raw display
$29$ \( (T^{6} + \cdots - 42\!\cdots\!40)^{2} \) Copy content Toggle raw display
$31$ \( (T^{6} + \cdots + 19\!\cdots\!00)^{2} \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 14\!\cdots\!24 \) Copy content Toggle raw display
$41$ \( (T^{6} + \cdots - 52\!\cdots\!00)^{2} \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 32\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 52\!\cdots\!76 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 66\!\cdots\!96 \) Copy content Toggle raw display
$59$ \( (T^{6} + \cdots - 36\!\cdots\!00)^{2} \) Copy content Toggle raw display
$61$ \( (T^{6} + \cdots - 11\!\cdots\!64)^{2} \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 48\!\cdots\!00 \) Copy content Toggle raw display
$71$ \( (T^{6} + \cdots + 15\!\cdots\!00)^{2} \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( (T^{6} + \cdots + 87\!\cdots\!00)^{2} \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 61\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( (T^{6} + \cdots + 20\!\cdots\!00)^{2} \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 46\!\cdots\!04 \) Copy content Toggle raw display
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