Properties

Label 207.10.a.b
Level $207$
Weight $10$
Character orbit 207.a
Self dual yes
Analytic conductor $106.612$
Analytic rank $1$
Dimension $7$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [207,10,Mod(1,207)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(207, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 10, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("207.1");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 207 = 3^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 207.a (trivial)

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: yes
Analytic conductor: \(106.612418086\)
Analytic rank: \(1\)
Dimension: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{7} - 640x^{5} - 1455x^{4} + 114552x^{3} + 321544x^{2} - 5741296x - 13379024 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{7}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 23)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 \(1,\beta_1,\ldots,\beta_{6}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + (\beta_{5} + 2 \beta_{4} + 6 \beta_1 + 220) q^{4} + (2 \beta_{5} - 2 \beta_{4} + \beta_{3} + 13 \beta_1 + 341) q^{5} + ( - 6 \beta_{6} - 8 \beta_{5} + 9 \beta_{4} + 4 \beta_{3} + 7 \beta_{2} + \cdots - 1410) q^{7}+ \cdots + ( - 4 \beta_{6} - 30 \beta_{5} - 28 \beta_{4} + 16 \beta_{3} + 12 \beta_{2} + \cdots - 4988) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + (\beta_{5} + 2 \beta_{4} + 6 \beta_1 + 220) q^{4} + (2 \beta_{5} - 2 \beta_{4} + \beta_{3} + 13 \beta_1 + 341) q^{5} + ( - 6 \beta_{6} - 8 \beta_{5} + 9 \beta_{4} + 4 \beta_{3} + 7 \beta_{2} + \cdots - 1410) q^{7}+ \cdots + ( - 465512 \beta_{6} - 2467342 \beta_{5} + \cdots - 354802284) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7 q + 1536 q^{4} + 2388 q^{5} - 9896 q^{7} - 34920 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 7 q + 1536 q^{4} + 2388 q^{5} - 9896 q^{7} - 34920 q^{8} - 60820 q^{10} + 78484 q^{11} - 296769 q^{13} + 711120 q^{14} - 253440 q^{16} + 1128820 q^{17} - 1301252 q^{19} + 3482704 q^{20} - 1562088 q^{22} + 1958887 q^{23} - 1320899 q^{25} - 692230 q^{26} - 8371144 q^{28} - 2813849 q^{29} + 7334751 q^{31} - 26028800 q^{32} + 14981564 q^{34} - 23410104 q^{35} - 13324320 q^{37} - 37578632 q^{38} - 45307920 q^{40} + 15691573 q^{41} - 46474818 q^{43} - 43428040 q^{44} - 8232227 q^{47} + 29219031 q^{49} - 50366304 q^{50} - 100922292 q^{52} + 53545400 q^{53} - 181608484 q^{55} + 420111696 q^{56} - 39304854 q^{58} + 341275144 q^{59} - 277157656 q^{61} - 464777594 q^{62} + 340566208 q^{64} - 106659278 q^{65} + 89654580 q^{67} - 62700400 q^{68} + 1187910040 q^{70} + 286098961 q^{71} - 637495039 q^{73} - 189880036 q^{74} + 228563936 q^{76} - 511682536 q^{77} + 274469546 q^{79} + 345318560 q^{80} - 570256066 q^{82} - 1164579762 q^{83} - 18639492 q^{85} - 415245796 q^{86} + 103329440 q^{88} + 504153000 q^{89} - 1692320156 q^{91} + 429835776 q^{92} - 2214048622 q^{94} - 162962164 q^{95} - 3519929016 q^{97} - 2474592568 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{7} - 640x^{5} - 1455x^{4} + 114552x^{3} + 321544x^{2} - 5741296x - 13379024 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 2\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 3199 \nu^{6} - 79358 \nu^{5} - 1488724 \nu^{4} + 30366663 \nu^{3} + 273707722 \nu^{2} - 2328138428 \nu - 14876298328 ) / 11419280 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 1391 \nu^{6} - 6842 \nu^{5} - 682136 \nu^{4} - 269473 \nu^{3} + 87308078 \nu^{2} + 269497648 \nu - 2521122692 ) / 2854820 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 7713 \nu^{6} + 105666 \nu^{5} + 3328828 \nu^{4} - 32964761 \nu^{3} - 360613014 \nu^{2} + 2265197396 \nu + 6770868456 ) / 11419280 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 7713 \nu^{6} - 105666 \nu^{5} - 3328828 \nu^{4} + 32964761 \nu^{3} + 383451574 \nu^{2} - 2333713076 \nu - 10950324936 ) / 5709640 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 29851 \nu^{6} + 497782 \nu^{5} + 11250276 \nu^{4} - 154091107 \nu^{3} - 1009430098 \nu^{2} + 10082064652 \nu + 17652094632 ) / 11419280 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{5} + 2\beta_{4} + 6\beta _1 + 732 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 2\beta_{6} + 15\beta_{5} + 14\beta_{4} - 8\beta_{3} - 6\beta_{2} + 560\beta _1 + 2494 ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -4\beta_{6} + 243\beta_{5} + 456\beta_{4} - 32\beta_{3} - 54\beta_{2} + 2205\beta _1 + 103240 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 964\beta_{6} + 9063\beta_{5} + 10258\beta_{4} - 3564\beta_{3} - 3776\beta_{2} + 192494\beta _1 + 1742612 ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 1206 \beta_{6} + 223049 \beta_{5} + 374874 \beta_{4} - 42256 \beta_{3} - 72698 \beta_{2} + 2453864 \beta _1 + 71615698 ) / 4 \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
21.2915
13.1537
8.70295
−2.27545
−10.3773
−15.0730
−15.4224
−42.5829 0 1301.31 1780.57 0 −11661.0 −33611.0 0 −75821.7
1.2 −26.3074 0 180.082 −92.1465 0 2337.45 8731.92 0 2424.14
1.3 −17.4059 0 −209.035 1221.93 0 −4757.43 12550.3 0 −21268.8
1.4 4.55091 0 −491.289 −2146.10 0 615.663 −4565.88 0 −9766.68
1.5 20.7546 0 −81.2472 680.714 0 −8213.17 −12312.6 0 14127.9
1.6 30.1460 0 396.783 −568.966 0 3614.33 −3473.33 0 −17152.1
1.7 30.8448 0 439.399 1512.00 0 8168.19 −2239.35 0 46637.2
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.7
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(23\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 207.10.a.b 7
3.b odd 2 1 23.10.a.a 7
12.b even 2 1 368.10.a.f 7
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
23.10.a.a 7 3.b odd 2 1
207.10.a.b 7 1.a even 1 1 trivial
368.10.a.f 7 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{7} - 2560T_{2}^{5} + 11640T_{2}^{4} + 1832832T_{2}^{3} - 10289408T_{2}^{2} - 367442944T_{2} + 1712515072 \) acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(207))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{7} - 2560 T^{5} + \cdots + 1712515072 \) Copy content Toggle raw display
$3$ \( T^{7} \) Copy content Toggle raw display
$5$ \( T^{7} - 2388 T^{6} + \cdots + 25\!\cdots\!80 \) Copy content Toggle raw display
$7$ \( T^{7} + 9896 T^{6} + \cdots + 19\!\cdots\!28 \) Copy content Toggle raw display
$11$ \( T^{7} - 78484 T^{6} + \cdots + 10\!\cdots\!48 \) Copy content Toggle raw display
$13$ \( T^{7} + 296769 T^{6} + \cdots - 72\!\cdots\!88 \) Copy content Toggle raw display
$17$ \( T^{7} - 1128820 T^{6} + \cdots - 48\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( T^{7} + 1301252 T^{6} + \cdots - 58\!\cdots\!40 \) Copy content Toggle raw display
$23$ \( (T - 279841)^{7} \) Copy content Toggle raw display
$29$ \( T^{7} + 2813849 T^{6} + \cdots + 15\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{7} - 7334751 T^{6} + \cdots - 89\!\cdots\!28 \) Copy content Toggle raw display
$37$ \( T^{7} + 13324320 T^{6} + \cdots + 12\!\cdots\!92 \) Copy content Toggle raw display
$41$ \( T^{7} - 15691573 T^{6} + \cdots - 21\!\cdots\!20 \) Copy content Toggle raw display
$43$ \( T^{7} + 46474818 T^{6} + \cdots - 77\!\cdots\!68 \) Copy content Toggle raw display
$47$ \( T^{7} + 8232227 T^{6} + \cdots + 30\!\cdots\!40 \) Copy content Toggle raw display
$53$ \( T^{7} - 53545400 T^{6} + \cdots + 18\!\cdots\!72 \) Copy content Toggle raw display
$59$ \( T^{7} - 341275144 T^{6} + \cdots + 23\!\cdots\!60 \) Copy content Toggle raw display
$61$ \( T^{7} + 277157656 T^{6} + \cdots + 16\!\cdots\!04 \) Copy content Toggle raw display
$67$ \( T^{7} - 89654580 T^{6} + \cdots - 12\!\cdots\!56 \) Copy content Toggle raw display
$71$ \( T^{7} - 286098961 T^{6} + \cdots - 17\!\cdots\!20 \) Copy content Toggle raw display
$73$ \( T^{7} + 637495039 T^{6} + \cdots + 95\!\cdots\!60 \) Copy content Toggle raw display
$79$ \( T^{7} - 274469546 T^{6} + \cdots + 26\!\cdots\!56 \) Copy content Toggle raw display
$83$ \( T^{7} + 1164579762 T^{6} + \cdots - 71\!\cdots\!08 \) Copy content Toggle raw display
$89$ \( T^{7} - 504153000 T^{6} + \cdots - 63\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{7} + 3519929016 T^{6} + \cdots - 62\!\cdots\!00 \) Copy content Toggle raw display
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