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Properties

Label 252.4.j.b
Level $252$
Weight $4$
Character orbit 252.j
Analytic conductor $14.868$
Analytic rank $0$
Dimension $18$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [252,4,Mod(85,252)]
 
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, 2, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("252.85");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 252 = 2^{2} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 252.j (of order \(3\), 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: \(18\)
Relative dimension: \(9\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{18} - 7 x^{17} + 81 x^{16} - 470 x^{15} + 2687 x^{14} - 12243 x^{13} + 43732 x^{12} + \cdots + 5500612092612 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{6}\cdot 3^{14} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{17}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_1 + 1) q^{3} + (\beta_{3} + \beta_{2} - 1) q^{5} - 7 \beta_{2} q^{7} + (\beta_{4} - \beta_{2} - \beta_1 - 5) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 + 1) q^{3} + (\beta_{3} + \beta_{2} - 1) q^{5} - 7 \beta_{2} q^{7} + (\beta_{4} - \beta_{2} - \beta_1 - 5) q^{9} + ( - \beta_{12} + \beta_{6} + \cdots - 10 \beta_{2}) q^{11}+ \cdots + ( - 9 \beta_{17} + 6 \beta_{16} + \cdots + 261) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q + 11 q^{3} - 6 q^{5} - 63 q^{7} - 109 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 18 q + 11 q^{3} - 6 q^{5} - 63 q^{7} - 109 q^{9} - 93 q^{11} - 18 q^{13} + 162 q^{15} - 54 q^{17} - 90 q^{19} - 49 q^{21} - 246 q^{23} - 315 q^{25} - 394 q^{27} - 318 q^{29} - 18 q^{31} + 33 q^{33} + 84 q^{35} - 72 q^{37} + 268 q^{39} - 57 q^{41} - 171 q^{43} - 318 q^{45} - 1056 q^{47} - 441 q^{49} + 705 q^{51} + 1512 q^{53} - 1800 q^{55} - 1271 q^{57} - 411 q^{59} - 198 q^{61} + 308 q^{63} - 1326 q^{65} - 441 q^{67} + 642 q^{69} + 3516 q^{71} + 54 q^{73} - 2497 q^{75} - 651 q^{77} + 72 q^{79} + 1163 q^{81} - 558 q^{83} - 1008 q^{85} + 2766 q^{87} + 2784 q^{89} + 252 q^{91} - 2618 q^{93} - 156 q^{95} + 909 q^{97} + 6318 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{18} - 7 x^{17} + 81 x^{16} - 470 x^{15} + 2687 x^{14} - 12243 x^{13} + 43732 x^{12} + \cdots + 5500612092612 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - \nu^{17} + 6 \nu^{16} - 75 \nu^{15} + 395 \nu^{14} - 2292 \nu^{13} + 9951 \nu^{12} + \cdots - 1842555855894 ) / 282429536481 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 13315800857 \nu^{17} - 468924789381 \nu^{16} + 3153303989877 \nu^{15} + \cdots - 11\!\cdots\!68 ) / 28\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 9988724986 \nu^{17} - 2313647830569 \nu^{16} + 16444663818747 \nu^{15} + \cdots - 46\!\cdots\!90 ) / 14\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 87164256743 \nu^{17} - 1343101290501 \nu^{16} + 12045789087477 \nu^{15} + \cdots - 15\!\cdots\!28 ) / 28\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 133776214024 \nu^{17} + 6794583925713 \nu^{16} - 48031849953471 \nu^{15} + \cdots + 12\!\cdots\!34 ) / 14\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 52446298994 \nu^{17} - 1450917493401 \nu^{16} + 10638858706899 \nu^{15} + \cdots - 23\!\cdots\!02 ) / 47\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 21375320078 \nu^{17} + 163188313455 \nu^{16} - 1262052587301 \nu^{15} + \cdots + 38\!\cdots\!02 ) / 10\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 541960125439 \nu^{17} + 4635014468847 \nu^{16} - 35551554518235 \nu^{15} + \cdots + 19\!\cdots\!48 ) / 14\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 1127380069015 \nu^{17} + 8981968251993 \nu^{16} - 70210606342593 \nu^{15} + \cdots + 21\!\cdots\!08 ) / 28\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 808888063876 \nu^{17} - 3731737888569 \nu^{16} + 24220753702287 \nu^{15} + \cdots - 19\!\cdots\!50 ) / 14\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 345092458312 \nu^{17} - 4149165254847 \nu^{16} + 29911777199121 \nu^{15} + \cdots - 37\!\cdots\!50 ) / 47\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 835642665467 \nu^{17} + 295562583375 \nu^{16} + 865211681409 \nu^{15} + \cdots + 13\!\cdots\!52 ) / 70\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 1808290611835 \nu^{17} + 24078182823471 \nu^{16} - 162934882103391 \nu^{15} + \cdots + 79\!\cdots\!16 ) / 14\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 985735662991 \nu^{17} - 2977298291364 \nu^{16} + 27222686370246 \nu^{15} + \cdots + 92\!\cdots\!02 ) / 70\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 4518126258467 \nu^{17} + 74078356177587 \nu^{16} - 553906196437551 \nu^{15} + \cdots + 87\!\cdots\!20 ) / 28\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 5301535780511 \nu^{17} + 24231887336811 \nu^{16} - 201658791233823 \nu^{15} + \cdots - 28\!\cdots\!20 ) / 28\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 1807606890121 \nu^{17} - 12687264062385 \nu^{16} + 99424585969077 \nu^{15} + \cdots - 32\!\cdots\!84 ) / 94\!\cdots\!20 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{9} - \beta_{8} - \beta_{2} - 3\beta _1 + 5 ) / 9 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - 6 \beta_{17} - 6 \beta_{16} + 3 \beta_{11} + 3 \beta_{10} + \beta_{9} + 2 \beta_{8} - 3 \beta_{7} + \cdots - 58 ) / 9 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 21 \beta_{17} - 30 \beta_{16} + 9 \beta_{15} + 18 \beta_{14} - 18 \beta_{13} + 9 \beta_{12} + \cdots - 13 ) / 9 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 72 \beta_{17} - 180 \beta_{16} + 27 \beta_{15} - 54 \beta_{14} - 108 \beta_{13} + 162 \beta_{12} + \cdots + 1217 ) / 9 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 456 \beta_{17} - 240 \beta_{16} + 27 \beta_{15} - 243 \beta_{14} - 162 \beta_{13} + 459 \beta_{12} + \cdots - 2449 ) / 9 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 1587 \beta_{17} + 1428 \beta_{16} - 666 \beta_{15} - 252 \beta_{14} + 576 \beta_{13} - 72 \beta_{12} + \cdots - 55219 ) / 9 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 7461 \beta_{17} + 4302 \beta_{16} - 3159 \beta_{15} + 162 \beta_{14} + 3888 \beta_{13} - 6885 \beta_{12} + \cdots - 384460 ) / 9 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 20982 \beta_{17} - 11013 \beta_{16} - 14175 \beta_{15} + 12042 \beta_{14} - 3618 \beta_{13} + \cdots - 103510 ) / 9 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 439068 \beta_{17} + 71520 \beta_{16} - 309060 \beta_{15} - 5706 \beta_{14} - 73107 \beta_{13} + \cdots - 3245269 ) / 9 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 3824217 \beta_{17} + 1765701 \beta_{16} - 3699297 \beta_{15} - 903258 \beta_{14} + 91395 \beta_{13} + \cdots - 61187920 ) / 9 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 11087304 \beta_{17} + 15749805 \beta_{16} - 20902563 \beta_{15} - 6591834 \beta_{14} + \cdots - 232243624 ) / 9 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 14676342 \beta_{17} + 89633085 \beta_{16} - 59481288 \beta_{15} - 44294508 \beta_{14} + \cdots - 2427640285 ) / 9 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 442774980 \beta_{17} + 605352321 \beta_{16} - 97572249 \beta_{15} - 214361937 \beta_{14} + \cdots - 32994122641 ) / 9 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 5315664396 \beta_{17} + 5932936452 \beta_{16} - 444688353 \beta_{15} - 344211282 \beta_{14} + \cdots - 235211094478 ) / 9 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 46755578436 \beta_{17} + 54143189529 \beta_{16} - 7754425749 \beta_{15} - 3121162803 \beta_{14} + \cdots - 879993183487 ) / 9 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( ( 307650105540 \beta_{17} + 421537425192 \beta_{16} - 87575030820 \beta_{15} - 42696660708 \beta_{14} + \cdots - 1342007900404 ) / 9 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( ( 1584764063562 \beta_{17} + 2562826191528 \beta_{16} - 637015741470 \beta_{15} - 221969986308 \beta_{14} + \cdots + 2952115076105 ) / 9 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(29\) \(73\) \(127\)
\(\chi(n)\) \(-1 + \beta_{2}\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
85.1
6.01410 + 1.36338i
4.38317 − 3.94388i
2.54993 + 4.95961i
0.0652281 + 5.11138i
−0.0672952 − 5.08536i
−0.208655 − 5.05363i
−2.16791 + 4.11878i
−2.97222 − 3.34985i
−4.09635 + 1.01354i
6.01410 − 1.36338i
4.38317 + 3.94388i
2.54993 − 4.95961i
0.0652281 − 5.11138i
−0.0672952 + 5.08536i
−0.208655 + 5.05363i
−2.16791 − 4.11878i
−2.97222 + 3.34985i
−4.09635 − 1.01354i
0 −5.01410 + 1.36338i 0 −0.333146 + 0.577025i 0 −3.50000 − 6.06218i 0 23.2824 − 13.6722i 0
85.2 0 −3.38317 − 3.94388i 0 −8.26221 + 14.3106i 0 −3.50000 − 6.06218i 0 −4.10831 + 26.6856i 0
85.3 0 −1.54993 + 4.95961i 0 3.85902 − 6.68401i 0 −3.50000 − 6.06218i 0 −22.1955 − 15.3741i 0
85.4 0 0.934772 + 5.11138i 0 −7.13443 + 12.3572i 0 −3.50000 − 6.06218i 0 −25.2524 + 9.55595i 0
85.5 0 1.06730 − 5.08536i 0 1.60891 − 2.78671i 0 −3.50000 − 6.06218i 0 −24.7218 − 10.8552i 0
85.6 0 1.20866 − 5.05363i 0 7.66838 − 13.2820i 0 −3.50000 − 6.06218i 0 −24.0783 − 12.2162i 0
85.7 0 3.16791 + 4.11878i 0 8.76630 − 15.1837i 0 −3.50000 − 6.06218i 0 −6.92866 + 26.0959i 0
85.8 0 3.97222 − 3.34985i 0 −9.35724 + 16.2072i 0 −3.50000 − 6.06218i 0 4.55703 − 26.6127i 0
85.9 0 5.09635 + 1.01354i 0 0.184423 − 0.319430i 0 −3.50000 − 6.06218i 0 24.9455 + 10.3307i 0
169.1 0 −5.01410 − 1.36338i 0 −0.333146 − 0.577025i 0 −3.50000 + 6.06218i 0 23.2824 + 13.6722i 0
169.2 0 −3.38317 + 3.94388i 0 −8.26221 − 14.3106i 0 −3.50000 + 6.06218i 0 −4.10831 − 26.6856i 0
169.3 0 −1.54993 − 4.95961i 0 3.85902 + 6.68401i 0 −3.50000 + 6.06218i 0 −22.1955 + 15.3741i 0
169.4 0 0.934772 − 5.11138i 0 −7.13443 − 12.3572i 0 −3.50000 + 6.06218i 0 −25.2524 − 9.55595i 0
169.5 0 1.06730 + 5.08536i 0 1.60891 + 2.78671i 0 −3.50000 + 6.06218i 0 −24.7218 + 10.8552i 0
169.6 0 1.20866 + 5.05363i 0 7.66838 + 13.2820i 0 −3.50000 + 6.06218i 0 −24.0783 + 12.2162i 0
169.7 0 3.16791 − 4.11878i 0 8.76630 + 15.1837i 0 −3.50000 + 6.06218i 0 −6.92866 − 26.0959i 0
169.8 0 3.97222 + 3.34985i 0 −9.35724 − 16.2072i 0 −3.50000 + 6.06218i 0 4.55703 + 26.6127i 0
169.9 0 5.09635 − 1.01354i 0 0.184423 + 0.319430i 0 −3.50000 + 6.06218i 0 24.9455 − 10.3307i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 85.9
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 252.4.j.b ✓ 18
3.b odd 2 1 756.4.j.b 18
9.c even 3 1 inner 252.4.j.b ✓ 18
9.c even 3 1 2268.4.a.i 9
9.d odd 6 1 756.4.j.b 18
9.d odd 6 1 2268.4.a.h 9
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
252.4.j.b ✓ 18 1.a even 1 1 trivial
252.4.j.b ✓ 18 9.c even 3 1 inner
756.4.j.b 18 3.b odd 2 1
756.4.j.b 18 9.d odd 6 1
2268.4.a.h 9 9.d odd 6 1
2268.4.a.i 9 9.c even 3 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{18} + 6 T_{5}^{17} + 738 T_{5}^{16} + 1368 T_{5}^{15} + 352323 T_{5}^{14} + 281502 T_{5}^{13} + \cdots + 52444912836996 \) acting on \(S_{4}^{\mathrm{new}}(252, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{18} \) Copy content Toggle raw display
$3$ \( T^{18} + \cdots + 7625597484987 \) Copy content Toggle raw display
$5$ \( T^{18} + \cdots + 52444912836996 \) Copy content Toggle raw display
$7$ \( (T^{2} + 7 T + 49)^{9} \) Copy content Toggle raw display
$11$ \( T^{18} + \cdots + 77\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( T^{18} + \cdots + 18\!\cdots\!16 \) Copy content Toggle raw display
$17$ \( (T^{9} + \cdots - 109582832291940)^{2} \) Copy content Toggle raw display
$19$ \( (T^{9} + \cdots - 20\!\cdots\!75)^{2} \) Copy content Toggle raw display
$23$ \( T^{18} + \cdots + 43\!\cdots\!96 \) Copy content Toggle raw display
$29$ \( T^{18} + \cdots + 32\!\cdots\!84 \) Copy content Toggle raw display
$31$ \( T^{18} + \cdots + 20\!\cdots\!76 \) Copy content Toggle raw display
$37$ \( (T^{9} + \cdots - 12\!\cdots\!48)^{2} \) Copy content Toggle raw display
$41$ \( T^{18} + \cdots + 19\!\cdots\!96 \) Copy content Toggle raw display
$43$ \( T^{18} + \cdots + 16\!\cdots\!84 \) Copy content Toggle raw display
$47$ \( T^{18} + \cdots + 20\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( (T^{9} + \cdots - 62\!\cdots\!32)^{2} \) Copy content Toggle raw display
$59$ \( T^{18} + \cdots + 51\!\cdots\!84 \) Copy content Toggle raw display
$61$ \( T^{18} + \cdots + 73\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( T^{18} + \cdots + 49\!\cdots\!44 \) Copy content Toggle raw display
$71$ \( (T^{9} + \cdots + 35\!\cdots\!40)^{2} \) Copy content Toggle raw display
$73$ \( (T^{9} + \cdots - 11\!\cdots\!96)^{2} \) Copy content Toggle raw display
$79$ \( T^{18} + \cdots + 62\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{18} + \cdots + 82\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( (T^{9} + \cdots + 28\!\cdots\!40)^{2} \) Copy content Toggle raw display
$97$ \( T^{18} + \cdots + 82\!\cdots\!00 \) Copy content Toggle raw display
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