Properties

Label 252.4.t.a
Level $252$
Weight $4$
Character orbit 252.t
Analytic conductor $14.868$
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] = [252,4,Mod(17,252)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(252, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 3, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("252.17");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 252 = 2^{2} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 252.t (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: \(14.8684813214\)
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: \( x^{16} - 4 x^{15} - 290 x^{14} + 1728 x^{13} + 29275 x^{12} - 246984 x^{11} - 955194 x^{10} + 14344616 x^{9} - 18123280 x^{8} - 273588032 x^{7} + \cdots + 7375227456 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{6}\cdot 3^{18} \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q + \beta_{9} q^{5} + ( - \beta_{8} - \beta_{6} + 2 \beta_1 + 1) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{9} q^{5} + ( - \beta_{8} - \beta_{6} + 2 \beta_1 + 1) q^{7} + ( - \beta_{5} - \beta_{3}) q^{11} + (\beta_{8} + \beta_{2} + \beta_1 + 1) q^{13} + ( - \beta_{14} + \beta_{11} - 2 \beta_{9} + 3 \beta_{5} - \beta_{3}) q^{17} + (\beta_{13} - \beta_{12} + 3 \beta_{6} + 2 \beta_{2} - 2 \beta_1 - 6) q^{19} + (2 \beta_{14} + 2 \beta_{11} - \beta_{10} + \beta_{9} - 2 \beta_{5} - \beta_{3}) q^{23} + ( - 2 \beta_{13} + \beta_{12} + 6 \beta_{8} - 2 \beta_{7} + 4 \beta_{6} + 5 \beta_{2} + \cdots - 26) q^{25}+ \cdots + ( - \beta_{13} - 42 \beta_{8} - 10 \beta_{7} - 10 \beta_{6} - 42 \beta_{2} + \cdots + 294) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 4 q^{7} - 72 q^{19} - 212 q^{25} - 708 q^{31} + 76 q^{37} + 1408 q^{43} + 400 q^{49} - 1632 q^{61} - 1528 q^{67} - 2700 q^{73} - 364 q^{79} + 7392 q^{85} + 2472 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 4 x^{15} - 290 x^{14} + 1728 x^{13} + 29275 x^{12} - 246984 x^{11} - 955194 x^{10} + 14344616 x^{9} - 18123280 x^{8} - 273588032 x^{7} + \cdots + 7375227456 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 1229561931 \nu^{15} + 2498779578 \nu^{14} + 394581872608 \nu^{13} - 1317945813970 \nu^{12} - 48388978127485 \nu^{11} + \cdots - 24\!\cdots\!28 ) / 13\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 61\!\cdots\!81 \nu^{15} + \cdots - 20\!\cdots\!52 ) / 70\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 11\!\cdots\!52 \nu^{15} + \cdots - 56\!\cdots\!12 ) / 70\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 24\!\cdots\!91 \nu^{15} + \cdots - 18\!\cdots\!08 ) / 70\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 13\!\cdots\!96 \nu^{15} + \cdots + 88\!\cdots\!72 ) / 35\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 39\!\cdots\!72 \nu^{15} + \cdots + 23\!\cdots\!48 ) / 70\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 59\!\cdots\!29 \nu^{15} + \cdots + 59\!\cdots\!04 ) / 70\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 10\!\cdots\!68 \nu^{15} + \cdots - 37\!\cdots\!96 ) / 70\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 12\!\cdots\!66 \nu^{15} + \cdots - 11\!\cdots\!36 ) / 70\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 13\!\cdots\!89 \nu^{15} + \cdots - 16\!\cdots\!72 ) / 70\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 14\!\cdots\!19 \nu^{15} + \cdots - 16\!\cdots\!68 ) / 70\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 15\!\cdots\!47 \nu^{15} + \cdots - 21\!\cdots\!56 ) / 70\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 10\!\cdots\!21 \nu^{15} + \cdots + 63\!\cdots\!76 ) / 35\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 22\!\cdots\!43 \nu^{15} + \cdots + 11\!\cdots\!96 ) / 70\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 28\!\cdots\!32 \nu^{15} + \cdots - 13\!\cdots\!52 ) / 70\!\cdots\!88 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( 2 \beta_{15} + 3 \beta_{13} - 6 \beta_{12} + 2 \beta_{9} + 6 \beta_{8} - 3 \beta_{7} + 3 \beta_{6} - 4 \beta_{5} - 2 \beta_{4} - 12 \beta_{2} - 9 \beta _1 + 9 ) / 54 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - 4 \beta_{15} + 6 \beta_{14} - 3 \beta_{13} + 6 \beta_{12} + 12 \beta_{11} - 6 \beta_{10} + 32 \beta_{9} - 60 \beta_{8} - 15 \beta_{7} + 15 \beta_{6} - 58 \beta_{5} + 4 \beta_{4} - 6 \beta_{3} + 66 \beta_{2} + 171 \beta _1 + 2097 ) / 54 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 262 \beta_{15} + 153 \beta_{14} + 207 \beta_{13} - 360 \beta_{12} - 126 \beta_{11} + 63 \beta_{10} - 8 \beta_{9} + 594 \beta_{8} - 333 \beta_{7} + 279 \beta_{6} - 259 \beta_{5} - 258 \beta_{4} + 63 \beta_{3} - 954 \beta_{2} + \cdots - 5841 ) / 54 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 1512 \beta_{15} + 612 \beta_{14} - 747 \beta_{13} + 1386 \beta_{12} + 2016 \beta_{11} - 972 \beta_{10} + 3240 \beta_{9} - 4824 \beta_{8} - 27 \beta_{7} + 1647 \beta_{6} - 4732 \beta_{5} + 1496 \beta_{4} + \cdots + 156969 ) / 54 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 32630 \beta_{15} + 16515 \beta_{14} + 19083 \beta_{13} - 27726 \beta_{12} - 19350 \beta_{11} + 9045 \beta_{10} - 15340 \beta_{9} + 54780 \beta_{8} - 30477 \beta_{7} + 13197 \beta_{6} - 6157 \beta_{5} + \cdots - 816273 ) / 54 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 247460 \beta_{15} + 30918 \beta_{14} - 112923 \beta_{13} + 164556 \beta_{12} + 254220 \beta_{11} - 112062 \beta_{10} + 359104 \beta_{9} - 403758 \beta_{8} + 144729 \beta_{7} + \cdots + 13960017 ) / 54 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 3810126 \beta_{15} + 1493289 \beta_{14} + 1989039 \beta_{13} - 2371200 \beta_{12} - 2477664 \beta_{11} + 1037421 \beta_{10} - 2684670 \beta_{9} + 4879122 \beta_{8} - 3297981 \beta_{7} + \cdots - 96370029 ) / 54 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 33101968 \beta_{15} + 310632 \beta_{14} - 15078675 \beta_{13} + 17453526 \beta_{12} + 29246880 \beta_{11} - 11119128 \beta_{10} + 41540144 \beta_{9} - 34834524 \beta_{8} + \cdots + 1341758745 ) / 54 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 433006054 \beta_{15} + 126418959 \beta_{14} + 217504551 \beta_{13} - 213140382 \beta_{12} - 290064474 \beta_{11} + 101356605 \beta_{10} - 374035880 \beta_{9} + \cdots - 10853159733 ) / 54 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 4062381420 \beta_{15} - 138233742 \beta_{14} - 1892854743 \beta_{13} + 1767898236 \beta_{12} + 3216969492 \beta_{11} - 969690954 \beta_{10} + \cdots + 133822224693 ) / 54 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 48499514030 \beta_{15} + 10112577321 \beta_{14} + 24136734963 \beta_{13} - 19515342264 \beta_{12} - 32215401768 \beta_{11} + 8414179125 \beta_{10} + \cdots - 1187441814825 ) / 54 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 475514212088 \beta_{15} - 19107195348 \beta_{14} - 227182483671 \beta_{13} + 172645902462 \beta_{12} + 343680414960 \beta_{11} - 70338926292 \beta_{10} + \cdots + 13532011977573 ) / 54 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 5371594977942 \beta_{15} + 739350340911 \beta_{14} + 2677131100059 \beta_{13} - 1777856438046 \beta_{12} - 3450978812034 \beta_{11} + \cdots - 126521367686001 ) / 54 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 53952898642708 \beta_{15} - 1366278539058 \beta_{14} - 26310946432011 \beta_{13} + 16173600573612 \beta_{12} + 35853220709676 \beta_{11} + \cdots + 13\!\cdots\!61 ) / 54 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 588210720499726 \beta_{15} + 44260971175041 \beta_{14} + 294499803340167 \beta_{13} - 157561454266608 \beta_{12} - 359035155404016 \beta_{11} + \cdots - 13\!\cdots\!21 ) / 54 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/252\mathbb{Z}\right)^\times\).

\(n\) \(29\) \(73\) \(127\)
\(\chi(n)\) \(-1\) \(-\beta_{1}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
17.1
4.65022 0.707107i
−10.4548 + 0.707107i
−1.35249 + 0.707107i
8.15703 0.707107i
5.70754 + 0.707107i
1.09700 0.707107i
−8.00527 0.707107i
2.20073 + 0.707107i
4.65022 + 0.707107i
−10.4548 0.707107i
−1.35249 0.707107i
8.15703 + 0.707107i
5.70754 0.707107i
1.09700 + 0.707107i
−8.00527 + 0.707107i
2.20073 0.707107i
0 0 0 −10.1259 17.5386i 0 12.0269 14.0838i 0 0 0
17.2 0 0 0 −4.36813 7.56582i 0 14.4904 + 11.5338i 0 0 0
17.3 0 0 0 −4.27106 7.39769i 0 −12.5787 13.5933i 0 0 0
17.4 0 0 0 −3.41226 5.91021i 0 −14.9386 + 10.9471i 0 0 0
17.5 0 0 0 3.41226 + 5.91021i 0 −14.9386 + 10.9471i 0 0 0
17.6 0 0 0 4.27106 + 7.39769i 0 −12.5787 13.5933i 0 0 0
17.7 0 0 0 4.36813 + 7.56582i 0 14.4904 + 11.5338i 0 0 0
17.8 0 0 0 10.1259 + 17.5386i 0 12.0269 14.0838i 0 0 0
89.1 0 0 0 −10.1259 + 17.5386i 0 12.0269 + 14.0838i 0 0 0
89.2 0 0 0 −4.36813 + 7.56582i 0 14.4904 11.5338i 0 0 0
89.3 0 0 0 −4.27106 + 7.39769i 0 −12.5787 + 13.5933i 0 0 0
89.4 0 0 0 −3.41226 + 5.91021i 0 −14.9386 10.9471i 0 0 0
89.5 0 0 0 3.41226 5.91021i 0 −14.9386 10.9471i 0 0 0
89.6 0 0 0 4.27106 7.39769i 0 −12.5787 + 13.5933i 0 0 0
89.7 0 0 0 4.36813 7.56582i 0 14.4904 11.5338i 0 0 0
89.8 0 0 0 10.1259 17.5386i 0 12.0269 + 14.0838i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 17.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.d odd 6 1 inner
21.g even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 252.4.t.a 16
3.b odd 2 1 inner 252.4.t.a 16
4.b odd 2 1 1008.4.bt.b 16
7.b odd 2 1 1764.4.t.b 16
7.c even 3 1 1764.4.f.a 16
7.c even 3 1 1764.4.t.b 16
7.d odd 6 1 inner 252.4.t.a 16
7.d odd 6 1 1764.4.f.a 16
12.b even 2 1 1008.4.bt.b 16
21.c even 2 1 1764.4.t.b 16
21.g even 6 1 inner 252.4.t.a 16
21.g even 6 1 1764.4.f.a 16
21.h odd 6 1 1764.4.f.a 16
21.h odd 6 1 1764.4.t.b 16
28.f even 6 1 1008.4.bt.b 16
84.j odd 6 1 1008.4.bt.b 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
252.4.t.a 16 1.a even 1 1 trivial
252.4.t.a 16 3.b odd 2 1 inner
252.4.t.a 16 7.d odd 6 1 inner
252.4.t.a 16 21.g even 6 1 inner
1008.4.bt.b 16 4.b odd 2 1
1008.4.bt.b 16 12.b even 2 1
1008.4.bt.b 16 28.f even 6 1
1008.4.bt.b 16 84.j odd 6 1
1764.4.f.a 16 7.c even 3 1
1764.4.f.a 16 7.d odd 6 1
1764.4.f.a 16 21.g even 6 1
1764.4.f.a 16 21.h odd 6 1
1764.4.t.b 16 7.b odd 2 1
1764.4.t.b 16 7.c even 3 1
1764.4.t.b 16 21.c even 2 1
1764.4.t.b 16 21.h odd 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(252, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( T^{16} + 606 T^{14} + \cdots + 11\!\cdots\!16 \) Copy content Toggle raw display
$7$ \( (T^{8} + 2 T^{7} - 98 T^{6} + \cdots + 13841287201)^{2} \) Copy content Toggle raw display
$11$ \( T^{16} - 4806 T^{14} + \cdots + 70\!\cdots\!96 \) Copy content Toggle raw display
$13$ \( (T^{8} + 2826 T^{6} + \cdots + 22573860516)^{2} \) Copy content Toggle raw display
$17$ \( T^{16} + 21396 T^{14} + \cdots + 49\!\cdots\!76 \) Copy content Toggle raw display
$19$ \( (T^{8} + 36 T^{7} + \cdots + 46677071171844)^{2} \) Copy content Toggle raw display
$23$ \( T^{16} - 71460 T^{14} + \cdots + 49\!\cdots\!36 \) Copy content Toggle raw display
$29$ \( (T^{8} + 133110 T^{6} + \cdots + 41\!\cdots\!36)^{2} \) Copy content Toggle raw display
$31$ \( (T^{8} + 354 T^{7} + \cdots + 17\!\cdots\!61)^{2} \) Copy content Toggle raw display
$37$ \( (T^{8} - 38 T^{7} + \cdots + 17\!\cdots\!16)^{2} \) Copy content Toggle raw display
$41$ \( (T^{8} - 240060 T^{6} + \cdots + 34\!\cdots\!24)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} - 352 T^{3} - 146451 T^{2} + \cdots - 6254809742)^{4} \) Copy content Toggle raw display
$47$ \( T^{16} + 522912 T^{14} + \cdots + 47\!\cdots\!36 \) Copy content Toggle raw display
$53$ \( T^{16} - 252054 T^{14} + \cdots + 39\!\cdots\!76 \) Copy content Toggle raw display
$59$ \( T^{16} + 705414 T^{14} + \cdots + 56\!\cdots\!76 \) Copy content Toggle raw display
$61$ \( (T^{8} + 816 T^{7} + \cdots + 10\!\cdots\!44)^{2} \) Copy content Toggle raw display
$67$ \( (T^{8} + 764 T^{7} + \cdots + 90\!\cdots\!56)^{2} \) Copy content Toggle raw display
$71$ \( (T^{8} + 1006848 T^{6} + \cdots + 17\!\cdots\!44)^{2} \) Copy content Toggle raw display
$73$ \( (T^{8} + 1350 T^{7} + \cdots + 11\!\cdots\!84)^{2} \) Copy content Toggle raw display
$79$ \( (T^{8} + 182 T^{7} + \cdots + 15\!\cdots\!01)^{2} \) Copy content Toggle raw display
$83$ \( (T^{8} - 1589970 T^{6} + \cdots + 10\!\cdots\!84)^{2} \) Copy content Toggle raw display
$89$ \( T^{16} + 3076488 T^{14} + \cdots + 10\!\cdots\!96 \) Copy content Toggle raw display
$97$ \( (T^{8} + 5135454 T^{6} + \cdots + 47\!\cdots\!36)^{2} \) Copy content Toggle raw display
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