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Properties

Label 1008.4.h.a
Level $1008$
Weight $4$
Character orbit 1008.h
Analytic conductor $59.474$
Analytic rank $0$
Dimension $12$
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] = [1008,4,Mod(575,1008)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1008, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 1, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1008.575");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1008 = 2^{4} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1008.h (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: \(59.4739252858\)
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: \( x^{12} - 144x^{10} + 12024x^{8} - 766296x^{6} + 11751192x^{4} + 565147728x^{2} + 9666232489 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{13}\cdot 3^{4}\cdot 7^{4} \)
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.

\(f(q)\) \(=\) \( q + \beta_{6} q^{5} + \beta_1 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{6} q^{5} + \beta_1 q^{7} + \beta_{10} q^{11} + ( - \beta_{3} + 16) q^{13} + (2 \beta_{9} - 4 \beta_{7} - 3 \beta_{6}) q^{17} + ( - \beta_{5} + \beta_1) q^{19} + (2 \beta_{11} - \beta_{10} + 2 \beta_{8}) q^{23} + ( - \beta_{3} + 2 \beta_{2} + 7) q^{25} + ( - 5 \beta_{9} - 12 \beta_{7}) q^{29} + ( - \beta_{5} - 2 \beta_{4} + 5 \beta_1) q^{31} + ( - \beta_{11} - \beta_{10}) q^{35} + (7 \beta_{3} - 6 \beta_{2} + 6) q^{37} + (\beta_{9} + 17 \beta_{7} + 13 \beta_{6}) q^{41} + ( - 2 \beta_{5} + 2 \beta_{4} + 14 \beta_1) q^{43} + (\beta_{11} + 8 \beta_{10} + 5 \beta_{8}) q^{47} - 49 q^{49} + (4 \beta_{9} - 11 \beta_{7} - 24 \beta_{6}) q^{53} + ( - 3 \beta_{5} + 2 \beta_{4} + 23 \beta_1) q^{55} + (\beta_{11} - 8 \beta_{10} + 13 \beta_{8}) q^{59} + (4 \beta_{3} + 11 \beta_{2} - 150) q^{61} + ( - 7 \beta_{9} + 45 \beta_{7} + 12 \beta_{6}) q^{65} + (10 \beta_{5} - \beta_{4} + 52 \beta_1) q^{67} + ( - 4 \beta_{11} + 5 \beta_{10} + 12 \beta_{8}) q^{71} + ( - 19 \beta_{3} + 3 \beta_{2}) q^{73} + (7 \beta_{9} + 14 \beta_{7} + 7 \beta_{6}) q^{77} + (8 \beta_{5} - 5 \beta_{4} + 62 \beta_1) q^{79} + ( - 5 \beta_{11} - 2 \beta_{10} + 23 \beta_{8}) q^{83} + ( - 19 \beta_{3} - 12 \beta_{2} + 262) q^{85} + (15 \beta_{9} + 51 \beta_{7} - 19 \beta_{6}) q^{89} + (3 \beta_{5} - \beta_{4} + 15 \beta_1) q^{91} + (4 \beta_{11} - 12 \beta_{10} + 28 \beta_{8}) q^{95} + (13 \beta_{3} - 27 \beta_{2} - 96) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 192 q^{13} + 84 q^{25} + 72 q^{37} - 588 q^{49} - 1800 q^{61} + 3144 q^{85} - 1152 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 144x^{10} + 12024x^{8} - 766296x^{6} + 11751192x^{4} + 565147728x^{2} + 9666232489 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 3318602 \nu^{10} - 1842496320 \nu^{8} + 175521061471 \nu^{6} - 14275504328400 \nu^{4} + \cdots + 68\!\cdots\!00 ) / 15\!\cdots\!99 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 3477275166 \nu^{10} - 286339368234 \nu^{8} + 19065411796560 \nu^{6} + \cdots + 52\!\cdots\!00 ) / 14\!\cdots\!61 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 4695177480 \nu^{10} - 608547921622 \nu^{8} + 41780235132384 \nu^{6} + \cdots + 48\!\cdots\!28 ) / 14\!\cdots\!61 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 18\!\cdots\!00 \nu^{10} + \cdots + 50\!\cdots\!92 ) / 17\!\cdots\!31 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 13\!\cdots\!24 \nu^{10} + \cdots - 17\!\cdots\!60 ) / 56\!\cdots\!77 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 15\!\cdots\!67 \nu^{11} + \cdots - 37\!\cdots\!51 \nu ) / 11\!\cdots\!71 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 128246966 \nu^{11} - 15627873193 \nu^{9} + 1347426163828 \nu^{7} + \cdots + 71\!\cdots\!72 \nu ) / 73\!\cdots\!29 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 12648632976 \nu^{11} + 2307062091369 \nu^{9} - 214686097236954 \nu^{7} + \cdots - 21\!\cdots\!98 \nu ) / 15\!\cdots\!77 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 12\!\cdots\!38 \nu^{11} + \cdots - 99\!\cdots\!40 \nu ) / 11\!\cdots\!71 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 25\!\cdots\!75 \nu^{11} + \cdots + 25\!\cdots\!67 \nu ) / 20\!\cdots\!69 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 44\!\cdots\!22 \nu^{11} + \cdots + 10\!\cdots\!32 \nu ) / 20\!\cdots\!69 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -15\beta_{11} + 6\beta_{10} - 21\beta_{9} - 7\beta_{8} - 21\beta_{7} ) / 168 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -31\beta_{5} + 29\beta_{4} - 63\beta_{3} - 63\beta_{2} - 323\beta _1 + 2016 ) / 84 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -189\beta_{11} - 378\beta_{10} - 819\beta_{9} - 943\beta_{8} - 3549\beta_{7} + 1344\beta_{6} ) / 84 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -320\beta_{5} + 151\beta_{4} + 1113\beta_{3} - 3500\beta_{2} - 5410\beta _1 - 15456 ) / 28 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -5523\beta_{11} - 119406\beta_{10} + 41811\beta_{9} - 155651\beta_{8} + 516831\beta_{7} + 65940\beta_{6} ) / 168 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 16315\beta_{5} - 54905\beta_{4} + 22302\beta_{3} - 142506\beta_{2} + 352172\beta _1 + 316764 ) / 21 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 3651243 \beta_{11} + 801114 \beta_{10} + 11212005 \beta_{9} - 4598419 \beta_{8} + \cdots - 774396 \beta_{6} ) / 168 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 103528\beta_{5} - 683687\beta_{4} - 2019847\beta_{3} + 471996\beta_{2} + 3010490\beta _1 + 77465280 ) / 4 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 243259767 \beta_{11} + 348689934 \beta_{10} - 65364579 \beta_{9} + 76904179 \beta_{8} + \cdots + 516097596 \beta_{6} ) / 84 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 1112282839 \beta_{5} + 670703741 \beta_{4} - 3211420401 \beta_{3} + 1177728615 \beta_{2} + \cdots + 120237568416 ) / 84 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 15230380335 \beta_{11} + 5562041406 \beta_{10} - 95856292203 \beta_{9} + \cdots + 168350377944 \beta_{6} ) / 168 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1008\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(577\) \(757\) \(785\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
575.1
−8.41502 − 0.655192i
8.41502 − 0.655192i
−1.44334 + 3.80934i
1.44334 + 3.80934i
−6.97168 − 5.87875i
6.97168 − 5.87875i
6.97168 + 5.87875i
−6.97168 + 5.87875i
1.44334 − 3.80934i
−1.44334 − 3.80934i
8.41502 + 0.655192i
−8.41502 + 0.655192i
0 0 0 − 14.1054i 0 − 7.00000i 0 0 0
575.2 0 0 0 − 14.1054i 0 7.00000i 0 0 0
575.3 0 0 0 − 11.9196i 0 − 7.00000i 0 0 0
575.4 0 0 0 − 11.9196i 0 7.00000i 0 0 0
575.5 0 0 0 − 3.60008i 0 − 7.00000i 0 0 0
575.6 0 0 0 − 3.60008i 0 7.00000i 0 0 0
575.7 0 0 0 3.60008i 0 − 7.00000i 0 0 0
575.8 0 0 0 3.60008i 0 7.00000i 0 0 0
575.9 0 0 0 11.9196i 0 − 7.00000i 0 0 0
575.10 0 0 0 11.9196i 0 7.00000i 0 0 0
575.11 0 0 0 14.1054i 0 − 7.00000i 0 0 0
575.12 0 0 0 14.1054i 0 7.00000i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 575.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
4.b odd 2 1 inner
12.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1008.4.h.a ✓ 12
3.b odd 2 1 inner 1008.4.h.a ✓ 12
4.b odd 2 1 inner 1008.4.h.a ✓ 12
12.b even 2 1 inner 1008.4.h.a ✓ 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1008.4.h.a ✓ 12 1.a even 1 1 trivial
1008.4.h.a ✓ 12 3.b odd 2 1 inner
1008.4.h.a ✓ 12 4.b odd 2 1 inner
1008.4.h.a ✓ 12 12.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{6} + 354T_{5}^{4} + 32688T_{5}^{2} + 366368 \) acting on \(S_{4}^{\mathrm{new}}(1008, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( (T^{6} + 354 T^{4} + \cdots + 366368)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} + 49)^{6} \) Copy content Toggle raw display
$11$ \( (T^{6} - 3900 T^{4} + \cdots - 568249472)^{2} \) Copy content Toggle raw display
$13$ \( (T^{3} - 48 T^{2} + \cdots + 31120)^{4} \) Copy content Toggle raw display
$17$ \( (T^{6} + 14994 T^{4} + \cdots + 99040982048)^{2} \) Copy content Toggle raw display
$19$ \( (T^{6} + 13920 T^{4} + \cdots + 3550253056)^{2} \) Copy content Toggle raw display
$23$ \( (T^{6} - 65340 T^{4} + \cdots - 41393700992)^{2} \) Copy content Toggle raw display
$29$ \( (T^{6} + 85206 T^{4} + \cdots + 12863438408)^{2} \) Copy content Toggle raw display
$31$ \( (T^{6} + \cdots + 62452983398400)^{2} \) Copy content Toggle raw display
$37$ \( (T^{3} - 18 T^{2} + \cdots + 19047232)^{4} \) Copy content Toggle raw display
$41$ \( (T^{6} + \cdots + 3879449415200)^{2} \) Copy content Toggle raw display
$43$ \( (T^{6} + \cdots + 180657685282816)^{2} \) Copy content Toggle raw display
$47$ \( (T^{6} + \cdots - 979262779686912)^{2} \) Copy content Toggle raw display
$53$ \( (T^{6} + \cdots + 80471408316488)^{2} \) Copy content Toggle raw display
$59$ \( (T^{6} + \cdots - 261036024209408)^{2} \) Copy content Toggle raw display
$61$ \( (T^{3} + 450 T^{2} + \cdots + 13277160)^{4} \) Copy content Toggle raw display
$67$ \( (T^{6} + \cdots + 12\!\cdots\!16)^{2} \) Copy content Toggle raw display
$71$ \( (T^{6} + \cdots - 595230436669568)^{2} \) Copy content Toggle raw display
$73$ \( (T^{3} - 473724 T + 58448672)^{4} \) Copy content Toggle raw display
$79$ \( (T^{6} + \cdots + 32\!\cdots\!44)^{2} \) Copy content Toggle raw display
$83$ \( (T^{6} + \cdots - 46\!\cdots\!68)^{2} \) Copy content Toggle raw display
$89$ \( (T^{6} + \cdots + 195937482286112)^{2} \) Copy content Toggle raw display
$97$ \( (T^{3} + 288 T^{2} + \cdots + 314729920)^{4} \) Copy content Toggle raw display
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