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Properties

Label 309.4.a.c
Level $309$
Weight $4$
Character orbit 309.a
Self dual yes
Analytic conductor $18.232$
Analytic rank $0$
Dimension $12$
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] = [309,4,Mod(1,309)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(309, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("309.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 309 = 3 \cdot 103 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 309.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: \(18.2315901918\)
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: \( x^{12} - 4 x^{11} - 65 x^{10} + 246 x^{9} + 1479 x^{8} - 5396 x^{7} - 13483 x^{6} + 51490 x^{5} + \cdots - 31232 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} - 3 q^{3} + (\beta_{2} + 4) q^{4} + ( - \beta_{7} + 1) q^{5} - 3 \beta_1 q^{6} + ( - \beta_{8} - \beta_{5} + \cdots + \beta_1) q^{7}+ \cdots + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} - 3 q^{3} + (\beta_{2} + 4) q^{4} + ( - \beta_{7} + 1) q^{5} - 3 \beta_1 q^{6} + ( - \beta_{8} - \beta_{5} + \cdots + \beta_1) q^{7}+ \cdots + ( - 9 \beta_{10} - 9 \beta_{5} + \cdots + 63) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 4 q^{2} - 36 q^{3} + 50 q^{4} + 10 q^{5} - 12 q^{6} + 5 q^{7} + 42 q^{8} + 108 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 4 q^{2} - 36 q^{3} + 50 q^{4} + 10 q^{5} - 12 q^{6} + 5 q^{7} + 42 q^{8} + 108 q^{9} - 23 q^{10} + 75 q^{11} - 150 q^{12} - 51 q^{13} + 90 q^{14} - 30 q^{15} + 278 q^{16} + 145 q^{17} + 36 q^{18} + 278 q^{19} + 215 q^{20} - 15 q^{21} - 197 q^{22} + 122 q^{23} - 126 q^{24} - 30 q^{25} + 311 q^{26} - 324 q^{27} + 322 q^{28} + 528 q^{29} + 69 q^{30} + 473 q^{31} + 1470 q^{32} - 225 q^{33} + 750 q^{34} + 1022 q^{35} + 450 q^{36} - 57 q^{37} + 1641 q^{38} + 153 q^{39} + 913 q^{40} + 927 q^{41} - 270 q^{42} - 374 q^{43} + 1573 q^{44} + 90 q^{45} + 376 q^{46} + 1218 q^{47} - 834 q^{48} + 915 q^{49} + 1041 q^{50} - 435 q^{51} + 83 q^{52} + 748 q^{53} - 108 q^{54} + 330 q^{55} + 2806 q^{56} - 834 q^{57} + 1227 q^{58} + 2611 q^{59} - 645 q^{60} + 677 q^{61} + 2296 q^{62} + 45 q^{63} + 2118 q^{64} + 20 q^{65} + 591 q^{66} + 870 q^{67} + 2554 q^{68} - 366 q^{69} + 2 q^{70} + 676 q^{71} + 378 q^{72} - 1156 q^{73} + 1103 q^{74} + 90 q^{75} + 813 q^{76} - 674 q^{77} - 933 q^{78} + 685 q^{79} + 1867 q^{80} + 972 q^{81} - 3278 q^{82} + 101 q^{83} - 966 q^{84} - 1758 q^{85} - 2385 q^{86} - 1584 q^{87} - 3265 q^{88} - 751 q^{89} - 207 q^{90} + 414 q^{91} - 3282 q^{92} - 1419 q^{93} - 1308 q^{94} + 1752 q^{95} - 4410 q^{96} - 3520 q^{97} - 3054 q^{98} + 675 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 4 x^{11} - 65 x^{10} + 246 x^{9} + 1479 x^{8} - 5396 x^{7} - 13483 x^{6} + 51490 x^{5} + \cdots - 31232 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 12 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 65423 \nu^{11} + 77741 \nu^{10} + 3546958 \nu^{9} + 879536 \nu^{8} - 56688591 \nu^{7} + \cdots - 6320454976 ) / 546984512 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 344303 \nu^{11} - 62338 \nu^{10} + 28871061 \nu^{9} - 785086 \nu^{8} - 850298789 \nu^{7} + \cdots + 10796178432 ) / 1093969024 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 271413 \nu^{11} + 381563 \nu^{10} + 18684646 \nu^{9} - 16702248 \nu^{8} - 457405941 \nu^{7} + \cdots + 1039745472 ) / 546984512 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 378513 \nu^{11} + 546582 \nu^{10} + 28088231 \nu^{9} - 35232818 \nu^{8} - 741457931 \nu^{7} + \cdots - 21774909568 ) / 546984512 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 919001 \nu^{11} + 1173158 \nu^{10} + 65715883 \nu^{9} - 59775534 \nu^{8} - 1679015771 \nu^{7} + \cdots - 5734761600 ) / 1093969024 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 676725 \nu^{11} - 1754913 \nu^{10} - 46304534 \nu^{9} + 100916198 \nu^{8} + 1128097537 \nu^{7} + \cdots + 10947807872 ) / 546984512 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 1448491 \nu^{11} + 3467696 \nu^{10} + 97774199 \nu^{9} - 191280794 \nu^{8} + \cdots - 12218449152 ) / 1093969024 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 1773871 \nu^{11} + 5027194 \nu^{10} + 121052413 \nu^{9} - 290718578 \nu^{8} + \cdots - 47279894784 ) / 1093969024 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 964074 \nu^{11} + 2372661 \nu^{10} + 64726945 \nu^{9} - 130411422 \nu^{8} + \cdots - 13743432768 ) / 546984512 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 12 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{11} + \beta_{8} - \beta_{7} + \beta_{4} + \beta_{2} + 20\beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{11} + 2\beta_{9} + 3\beta_{8} - 3\beta_{7} + 4\beta_{5} + \beta_{4} - 4\beta_{3} + 28\beta_{2} + 9\beta _1 + 244 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 30 \beta_{11} + 4 \beta_{10} + 10 \beta_{9} + 44 \beta_{8} - 32 \beta_{7} - 4 \beta_{5} + 34 \beta_{4} + \cdots + 190 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 34 \beta_{11} + 8 \beta_{10} + 88 \beta_{9} + 134 \beta_{8} - 134 \beta_{7} + 12 \beta_{6} + 176 \beta_{5} + \cdots + 5824 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 771 \beta_{11} + 172 \beta_{10} + 504 \beta_{9} + 1423 \beta_{8} - 939 \beta_{7} + 4 \beta_{6} + \cdots + 8058 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 903 \beta_{11} + 484 \beta_{10} + 3058 \beta_{9} + 4597 \beta_{8} - 4681 \beta_{7} + 524 \beta_{6} + \cdots + 150020 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 18816 \beta_{11} + 6040 \beta_{10} + 18802 \beta_{9} + 42186 \beta_{8} - 27506 \beta_{7} + 76 \beta_{6} + \cdots + 281310 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 20280 \beta_{11} + 21364 \beta_{10} + 100064 \beta_{9} + 144632 \beta_{8} - 150484 \beta_{7} + \cdots + 4024736 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 444125 \beta_{11} + 201704 \beta_{10} + 631400 \beta_{9} + 1214213 \beta_{8} - 814493 \beta_{7} + \cdots + 9124010 \) 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
−5.08429
−3.97775
−3.63890
−2.97628
−0.690684
0.302797
1.42608
1.86896
2.47280
3.65974
5.16808
5.46944
−5.08429 −3.00000 17.8500 1.61828 15.2529 −28.8292 −50.0801 9.00000 −8.22779
1.2 −3.97775 −3.00000 7.82249 −3.33248 11.9332 18.9340 0.706106 9.00000 13.2558
1.3 −3.63890 −3.00000 5.24161 12.4774 10.9167 22.8725 10.0375 9.00000 −45.4042
1.4 −2.97628 −3.00000 0.858238 −3.90594 8.92884 −6.74338 21.2559 9.00000 11.6252
1.5 −0.690684 −3.00000 −7.52296 17.3484 2.07205 20.8748 10.7215 9.00000 −11.9823
1.6 0.302797 −3.00000 −7.90831 −14.6682 −0.908392 −30.0334 −4.81700 9.00000 −4.44151
1.7 1.42608 −3.00000 −5.96628 −7.36012 −4.27825 −12.9411 −19.9171 9.00000 −10.4962
1.8 1.86896 −3.00000 −4.50698 14.8870 −5.60689 −21.8370 −23.3751 9.00000 27.8232
1.9 2.47280 −3.00000 −1.88527 −16.2348 −7.41840 2.67676 −24.4443 9.00000 −40.1453
1.10 3.65974 −3.00000 5.39370 4.09718 −10.9792 34.2802 −9.53837 9.00000 14.9946
1.11 5.16808 −3.00000 18.7091 −7.46812 −15.5042 −3.90427 55.3452 9.00000 −38.5958
1.12 5.46944 −3.00000 21.9147 12.5414 −16.4083 9.65016 76.1057 9.00000 68.5943
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.12
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(103\) \(1\)

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 309.4.a.c ✓ 12
3.b odd 2 1 927.4.a.d 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
309.4.a.c ✓ 12 1.a even 1 1 trivial
927.4.a.d 12 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{12} - 4 T_{2}^{11} - 65 T_{2}^{10} + 246 T_{2}^{9} + 1479 T_{2}^{8} - 5396 T_{2}^{7} + \cdots - 31232 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(309))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} - 4 T^{11} + \cdots - 31232 \) Copy content Toggle raw display
$3$ \( (T + 3)^{12} \) Copy content Toggle raw display
$5$ \( T^{12} + \cdots + 45655145344 \) Copy content Toggle raw display
$7$ \( T^{12} + \cdots + 51568789313536 \) Copy content Toggle raw display
$11$ \( T^{12} + \cdots + 92\!\cdots\!36 \) Copy content Toggle raw display
$13$ \( T^{12} + \cdots - 46\!\cdots\!60 \) Copy content Toggle raw display
$17$ \( T^{12} + \cdots + 94\!\cdots\!80 \) Copy content Toggle raw display
$19$ \( T^{12} + \cdots - 39\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 29\!\cdots\!54 \) Copy content Toggle raw display
$29$ \( T^{12} + \cdots - 41\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{12} + \cdots - 29\!\cdots\!20 \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 60\!\cdots\!52 \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots + 81\!\cdots\!68 \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 18\!\cdots\!56 \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 15\!\cdots\!72 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 65\!\cdots\!60 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 31\!\cdots\!40 \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots - 16\!\cdots\!40 \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 23\!\cdots\!84 \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 66\!\cdots\!56 \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 45\!\cdots\!96 \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots - 31\!\cdots\!80 \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 27\!\cdots\!68 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 47\!\cdots\!40 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 79\!\cdots\!02 \) Copy content Toggle raw display
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