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Properties

Label 285.10.a.g
Level $285$
Weight $10$
Character orbit 285.a
Self dual yes
Analytic conductor $146.785$
Analytic rank $0$
Dimension $14$
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] = [285,10,Mod(1,285)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(285, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("285.1");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 285 = 3 \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 285.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: \(146.785213307\)
Analytic rank: \(0\)
Dimension: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} - x^{13} - 5365 x^{12} - 189 x^{11} + 10977394 x^{10} + 7027376 x^{9} - 10830403360 x^{8} + \cdots - 25\!\cdots\!72 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{19}\cdot 3^{5}\cdot 5^{2} \)
Twist minimal: yes
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 \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + 81 q^{3} + (\beta_{2} + \beta_1 + 254) q^{4} - 625 q^{5} - 81 \beta_1 q^{6} + ( - \beta_{3} + \beta_{2} - 14 \beta_1 + 365) q^{7} + ( - \beta_{4} - \beta_{3} + \cdots - 1097) q^{8}+ \cdots + 6561 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + 81 q^{3} + (\beta_{2} + \beta_1 + 254) q^{4} - 625 q^{5} - 81 \beta_1 q^{6} + ( - \beta_{3} + \beta_{2} - 14 \beta_1 + 365) q^{7} + ( - \beta_{4} - \beta_{3} + \cdots - 1097) q^{8}+ \cdots + ( - 6561 \beta_{9} - 6561 \beta_{3} + \cdots + 24485652) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q - q^{2} + 1134 q^{3} + 3563 q^{4} - 8750 q^{5} - 81 q^{6} + 5100 q^{7} - 15639 q^{8} + 91854 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 14 q - q^{2} + 1134 q^{3} + 3563 q^{4} - 8750 q^{5} - 81 q^{6} + 5100 q^{7} - 15639 q^{8} + 91854 q^{9} + 625 q^{10} + 51948 q^{11} + 288603 q^{12} - 132486 q^{13} + 147524 q^{14} - 708750 q^{15} + 866291 q^{16} + 863534 q^{17} - 6561 q^{18} - 1824494 q^{19} - 2226875 q^{20} + 413100 q^{21} + 3720448 q^{22} + 2484890 q^{23} - 1266759 q^{24} + 5468750 q^{25} + 9411466 q^{26} + 7440174 q^{27} + 4972532 q^{28} + 2815068 q^{29} + 50625 q^{30} + 961874 q^{31} - 25650579 q^{32} + 4207788 q^{33} - 37246358 q^{34} - 3187500 q^{35} + 23376843 q^{36} - 37290998 q^{37} + 130321 q^{38} - 10731366 q^{39} + 9774375 q^{40} + 17423038 q^{41} + 11949444 q^{42} + 14382154 q^{43} + 49726676 q^{44} - 57408750 q^{45} + 52052020 q^{46} - 17335582 q^{47} + 70169571 q^{48} + 149672554 q^{49} - 390625 q^{50} + 69946254 q^{51} + 151153798 q^{52} + 99988350 q^{53} - 531441 q^{54} - 32467500 q^{55} + 370280048 q^{56} - 147784014 q^{57} + 295217470 q^{58} + 44522786 q^{59} - 180376875 q^{60} - 34966612 q^{61} - 148881328 q^{62} + 33461100 q^{63} + 608322695 q^{64} + 82803750 q^{65} + 301356288 q^{66} + 377614368 q^{67} + 1431668950 q^{68} + 201276090 q^{69} - 92202500 q^{70} + 591763088 q^{71} - 102607479 q^{72} + 1304021964 q^{73} + 2237746538 q^{74} + 442968750 q^{75} - 464333723 q^{76} + 742596220 q^{77} + 762328746 q^{78} + 420389862 q^{79} - 541431875 q^{80} + 602654094 q^{81} + 1164846810 q^{82} + 959406596 q^{83} + 402775092 q^{84} - 539708750 q^{85} + 687513696 q^{86} + 228020508 q^{87} - 56656640 q^{88} + 984155434 q^{89} + 4100625 q^{90} + 229519320 q^{91} + 4032637660 q^{92} + 77911794 q^{93} - 2102787716 q^{94} + 1140308750 q^{95} - 2077696899 q^{96} + 61321966 q^{97} + 26809371 q^{98} + 340830828 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} - x^{13} - 5365 x^{12} - 189 x^{11} + 10977394 x^{10} + 7027376 x^{9} - 10830403360 x^{8} + \cdots - 25\!\cdots\!72 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 766 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 92\!\cdots\!67 \nu^{13} + \cdots - 50\!\cdots\!36 ) / 67\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 92\!\cdots\!67 \nu^{13} + \cdots + 51\!\cdots\!56 ) / 67\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 15\!\cdots\!59 \nu^{13} + \cdots - 28\!\cdots\!72 ) / 29\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 41\!\cdots\!59 \nu^{13} + \cdots - 77\!\cdots\!72 ) / 29\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 87\!\cdots\!99 \nu^{13} + \cdots + 36\!\cdots\!00 ) / 44\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 23\!\cdots\!07 \nu^{13} + \cdots - 11\!\cdots\!56 ) / 88\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 12\!\cdots\!13 \nu^{13} + \cdots - 37\!\cdots\!24 ) / 44\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 63\!\cdots\!91 \nu^{13} + \cdots + 19\!\cdots\!28 ) / 22\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 30\!\cdots\!09 \nu^{13} + \cdots - 11\!\cdots\!32 ) / 88\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 90\!\cdots\!09 \nu^{13} + \cdots - 25\!\cdots\!72 ) / 22\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 49\!\cdots\!29 \nu^{13} + \cdots + 99\!\cdots\!92 ) / 88\!\cdots\!80 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 766 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{4} + \beta_{3} + 4\beta_{2} + 1272\beta _1 + 1097 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 3 \beta_{13} - 4 \beta_{12} + \beta_{11} - 3 \beta_{10} - 3 \beta_{9} + \beta_{8} + 2 \beta_{7} + \cdots + 975961 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 18 \beta_{13} - 9 \beta_{12} - 20 \beta_{11} - 69 \beta_{10} + 18 \beta_{9} - 46 \beta_{8} + \cdots + 4070264 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 7551 \beta_{13} - 9443 \beta_{12} + 2251 \beta_{11} - 7328 \beta_{10} - 7655 \beta_{9} + \cdots + 1471113174 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 57216 \beta_{13} - 40000 \beta_{12} - 21880 \beta_{11} - 212920 \beta_{10} + 60808 \beta_{9} + \cdots + 10453765233 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 14635811 \beta_{13} - 16520340 \beta_{12} + 4210449 \beta_{11} - 14138867 \beta_{10} + \cdots + 2407512808389 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 138036870 \beta_{13} - 103715401 \beta_{12} + 17896816 \beta_{11} - 504110241 \beta_{10} + \cdots + 23605652099208 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 26116605995 \beta_{13} - 25934945511 \beta_{12} + 7849292311 \beta_{11} - 25810546104 \beta_{10} + \cdots + 41\!\cdots\!74 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 297744446684 \beta_{13} - 220357948120 \beta_{12} + 146937351044 \beta_{11} - 1084561789652 \beta_{10} + \cdots + 50\!\cdots\!33 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 45140123970727 \beta_{13} - 38662515086744 \beta_{12} + 15057398937637 \beta_{11} + \cdots + 72\!\cdots\!41 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 604134054347034 \beta_{13} - 421938815464041 \beta_{12} + 463680705687124 \beta_{11} + \cdots + 10\!\cdots\!08 \) Copy content Toggle raw display

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

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
43.9239
39.6240
27.0243
26.1844
20.2533
9.90667
6.36356
−7.60109
−7.92573
−24.2459
−25.6600
−28.1197
−37.1018
−41.6258
−43.9239 81.0000 1417.31 −625.000 −3557.83 −8219.78 −39764.6 6561.00 27452.4
1.2 −39.6240 81.0000 1058.06 −625.000 −3209.55 7151.60 −21637.3 6561.00 24765.0
1.3 −27.0243 81.0000 218.311 −625.000 −2188.97 9011.08 7936.73 6561.00 16890.2
1.4 −26.1844 81.0000 173.621 −625.000 −2120.93 2253.78 8860.23 6561.00 16365.2
1.5 −20.2533 81.0000 −101.805 −625.000 −1640.51 −11379.8 12431.6 6561.00 12658.3
1.6 −9.90667 81.0000 −413.858 −625.000 −802.441 1711.96 9172.17 6561.00 6191.67
1.7 −6.36356 81.0000 −471.505 −625.000 −515.448 1042.69 6258.59 6561.00 3977.22
1.8 7.60109 81.0000 −454.223 −625.000 615.688 6316.77 −7344.35 6561.00 −4750.68
1.9 7.92573 81.0000 −449.183 −625.000 641.984 −3940.08 −7618.07 6561.00 −4953.58
1.10 24.2459 81.0000 75.8631 −625.000 1963.92 −7477.36 −10574.5 6561.00 −15153.7
1.11 25.6600 81.0000 146.436 −625.000 2078.46 7775.67 −9380.37 6561.00 −16037.5
1.12 28.1197 81.0000 278.715 −625.000 2277.69 −10314.1 −6559.90 6561.00 −17574.8
1.13 37.1018 81.0000 864.546 −625.000 3005.25 9801.55 13080.1 6561.00 −23188.6
1.14 41.6258 81.0000 1220.71 −625.000 3371.69 1366.05 29500.7 6561.00 −26016.2
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.14
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \( -1 \)
\(5\) \( +1 \)
\(19\) \( +1 \)

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 285.10.a.g ✓ 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
285.10.a.g ✓ 14 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{14} + T_{2}^{13} - 5365 T_{2}^{12} + 189 T_{2}^{11} + 10977394 T_{2}^{10} - 7027376 T_{2}^{9} + \cdots - 25\!\cdots\!72 \) acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(285))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{14} + \cdots - 25\!\cdots\!72 \) Copy content Toggle raw display
$3$ \( (T - 81)^{14} \) Copy content Toggle raw display
$5$ \( (T + 625)^{14} \) Copy content Toggle raw display
$7$ \( T^{14} + \cdots - 48\!\cdots\!00 \) Copy content Toggle raw display
$11$ \( T^{14} + \cdots + 38\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( T^{14} + \cdots + 17\!\cdots\!12 \) Copy content Toggle raw display
$17$ \( T^{14} + \cdots - 11\!\cdots\!36 \) Copy content Toggle raw display
$19$ \( (T + 130321)^{14} \) Copy content Toggle raw display
$23$ \( T^{14} + \cdots + 15\!\cdots\!08 \) Copy content Toggle raw display
$29$ \( T^{14} + \cdots - 42\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{14} + \cdots - 63\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( T^{14} + \cdots + 69\!\cdots\!00 \) Copy content Toggle raw display
$41$ \( T^{14} + \cdots - 10\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( T^{14} + \cdots - 20\!\cdots\!20 \) Copy content Toggle raw display
$47$ \( T^{14} + \cdots - 28\!\cdots\!64 \) Copy content Toggle raw display
$53$ \( T^{14} + \cdots + 30\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( T^{14} + \cdots - 28\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{14} + \cdots + 21\!\cdots\!72 \) Copy content Toggle raw display
$67$ \( T^{14} + \cdots + 19\!\cdots\!28 \) Copy content Toggle raw display
$71$ \( T^{14} + \cdots - 41\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( T^{14} + \cdots + 12\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( T^{14} + \cdots - 13\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{14} + \cdots - 60\!\cdots\!20 \) Copy content Toggle raw display
$89$ \( T^{14} + \cdots - 14\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{14} + \cdots - 30\!\cdots\!00 \) Copy content Toggle raw display
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