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Properties

Label 1950.2.t.f
Level $1950$
Weight $2$
Character orbit 1950.t
Analytic conductor $15.571$
Analytic rank $0$
Dimension $16$
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] = [1950,2,Mod(307,1950)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1950, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1950.307");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1950 = 2 \cdot 3 \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1950.t (of order \(4\), 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: \(15.5708283941\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 24x^{14} + 218x^{12} + 968x^{10} + 2241x^{8} + 2672x^{6} + 1512x^{4} + 320x^{2} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{8} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{8} q^{2} + \beta_{7} q^{3} - q^{4} + \beta_{10} q^{6} + \beta_{14} q^{7} - \beta_{8} q^{8} - \beta_{8} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{8} q^{2} + \beta_{7} q^{3} - q^{4} + \beta_{10} q^{6} + \beta_{14} q^{7} - \beta_{8} q^{8} - \beta_{8} q^{9} + ( - \beta_{13} - \beta_{11} + \beta_{8} + \cdots - 1) q^{11}+ \cdots + ( - \beta_{12} - \beta_{10} + \beta_{8} + \cdots + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 16 q^{4} + 8 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 16 q^{4} + 8 q^{7} - 16 q^{11} + 8 q^{13} + 16 q^{16} - 8 q^{17} + 16 q^{18} - 8 q^{19} - 16 q^{22} + 24 q^{23} + 8 q^{26} - 8 q^{28} - 8 q^{31} - 8 q^{34} + 16 q^{37} - 8 q^{38} + 8 q^{39} - 8 q^{43} + 16 q^{44} - 24 q^{46} + 16 q^{47} - 40 q^{49} - 8 q^{52} + 16 q^{53} - 8 q^{58} - 16 q^{59} + 24 q^{61} + 8 q^{62} - 16 q^{64} + 16 q^{66} + 8 q^{68} + 16 q^{69} + 32 q^{71} - 16 q^{72} + 8 q^{76} + 8 q^{77} + 8 q^{78} - 16 q^{81} - 24 q^{83} + 8 q^{86} - 8 q^{87} + 16 q^{88} - 48 q^{91} - 24 q^{92} - 16 q^{93} + 16 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 24x^{14} + 218x^{12} + 968x^{10} + 2241x^{8} + 2672x^{6} + 1512x^{4} + 320x^{2} + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{15} - 22\nu^{13} - 174\nu^{11} - 612\nu^{9} - 865\nu^{7} + 26\nu^{5} + 1020\nu^{3} + 488\nu ) / 64 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{12} + 20\nu^{10} + 138\nu^{8} + 412\nu^{6} + 525\nu^{4} + 232\nu^{2} + 20 ) / 16 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 7 \nu^{15} + \nu^{14} - 152 \nu^{13} + 22 \nu^{12} - 1162 \nu^{11} + 182 \nu^{10} - 3752 \nu^{9} + \cdots - 40 ) / 128 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 18 \nu^{15} - 3 \nu^{14} + 410 \nu^{13} - 70 \nu^{12} + 3420 \nu^{11} - 610 \nu^{10} + 13180 \nu^{9} + \cdots - 344 ) / 128 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -11\nu^{15} - 250\nu^{13} - 2082\nu^{11} - 8036\nu^{9} - 14627\nu^{7} - 11186\nu^{5} - 2716\nu^{3} - 232\nu ) / 64 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 18 \nu^{15} + 3 \nu^{14} + 410 \nu^{13} + 70 \nu^{12} + 3420 \nu^{11} + 610 \nu^{10} + 13180 \nu^{9} + \cdots + 344 ) / 128 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 5 \nu^{15} - 11 \nu^{14} - 120 \nu^{13} - 250 \nu^{12} - 1086 \nu^{11} - 2082 \nu^{10} - 4760 \nu^{9} + \cdots - 232 ) / 128 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -11\nu^{15} - 254\nu^{13} - 2170\nu^{11} - 8740\nu^{9} - 17227\nu^{7} - 15646\nu^{5} - 5644\nu^{3} - 568\nu ) / 64 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( \nu^{15} + 6 \nu^{14} + 28 \nu^{13} + 136 \nu^{12} + 310 \nu^{11} + 1132 \nu^{10} + 1744 \nu^{9} + \cdots + 96 ) / 64 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 5 \nu^{15} - 11 \nu^{14} + 120 \nu^{13} - 250 \nu^{12} + 1086 \nu^{11} - 2082 \nu^{10} + 4760 \nu^{9} + \cdots - 232 ) / 128 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 3 \nu^{15} - 3 \nu^{14} - 68 \nu^{13} - 68 \nu^{12} - 562 \nu^{11} - 566 \nu^{10} - 2128 \nu^{9} + \cdots - 48 ) / 32 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( -5\nu^{14} - 114\nu^{12} - 954\nu^{10} - 3712\nu^{8} - 6873\nu^{6} - 5494\nu^{4} - 1476\nu^{2} + 32\nu - 88 ) / 32 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 5\nu^{14} + 114\nu^{12} + 954\nu^{10} + 3712\nu^{8} + 6873\nu^{6} + 5494\nu^{4} + 1476\nu^{2} + 32\nu + 88 ) / 32 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( -11\nu^{14} - 254\nu^{12} - 2162\nu^{10} - 8588\nu^{8} - 16275\nu^{6} - 13286\nu^{4} - 3580\nu^{2} - 120 ) / 64 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( -3\nu^{14} - 68\nu^{12} - 562\nu^{10} - 2128\nu^{8} - 3699\nu^{6} - 2492\nu^{4} - 324\nu^{2} + 16 ) / 16 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{13} + \beta_{12} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{14} - \beta_{10} - \beta_{7} + \beta_{2} - 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 5 \beta_{13} - 5 \beta_{12} - 2 \beta_{11} + 4 \beta_{10} + 2 \beta_{8} - 2 \beta_{7} + \cdots - 2 \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2 \beta_{15} - 7 \beta_{14} + 2 \beta_{13} - 2 \beta_{12} + 2 \beta_{11} + 8 \beta_{10} + 6 \beta_{7} + \cdots + 17 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 35 \beta_{13} + 35 \beta_{12} + 20 \beta_{11} - 54 \beta_{10} - 12 \beta_{9} - 38 \beta_{8} + \cdots + 22 \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 26 \beta_{15} + 53 \beta_{14} - 28 \beta_{13} + 28 \beta_{12} - 24 \beta_{11} - 69 \beta_{10} + \cdots - 123 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 279 \beta_{13} - 279 \beta_{12} - 182 \beta_{11} + 576 \beta_{10} + 176 \beta_{9} + 458 \beta_{8} + \cdots - 202 \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 270 \beta_{15} - 433 \beta_{14} + 306 \beta_{13} - 306 \beta_{12} + 242 \beta_{11} + 624 \beta_{10} + \cdots + 1003 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 2385 \beta_{13} + 2385 \beta_{12} + 1652 \beta_{11} - 5690 \beta_{10} - 1940 \beta_{9} + \cdots + 1834 \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 2622 \beta_{15} + 3735 \beta_{14} - 3068 \beta_{13} + 3068 \beta_{12} - 2332 \beta_{11} - 5751 \beta_{10} + \cdots - 8713 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 21205 \beta_{13} - 21205 \beta_{12} - 15114 \beta_{11} + 54448 \beta_{10} + 19456 \beta_{9} + \cdots - 16798 \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 24842 \beta_{15} - 33339 \beta_{14} + 29618 \beta_{13} - 29618 \beta_{12} + 22082 \beta_{11} + \cdots + 78273 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 192651 \beta_{13} + 192651 \beta_{12} + 139220 \beta_{11} - 513534 \beta_{10} - 187612 \beta_{9} + \cdots + 155006 \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 233026 \beta_{15} + 303493 \beta_{14} - 280796 \beta_{13} + 280796 \beta_{12} - 207392 \beta_{11} + \cdots - 715547 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 1770639 \beta_{13} - 1770639 \beta_{12} - 1288118 \beta_{11} + 4810512 \beta_{10} + \cdots - 1436554 \beta_1 ) / 2 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1950\mathbb{Z}\right)^\times\).

\(n\) \(301\) \(1301\) \(1327\)
\(\chi(n)\) \(\beta_{8}\) \(1\) \(\beta_{8}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
307.1
− 1.06676i
2.19832i
− 0.266470i
0.549125i
2.12462i
− 3.05093i
1.11907i
− 1.60698i
1.06676i
− 2.19832i
0.266470i
− 0.549125i
− 2.12462i
3.05093i
− 1.11907i
1.60698i
1.00000i −0.707107 + 0.707107i −1.00000 0 −0.707107 − 0.707107i −1.06083 − 1.00000i − 1.00000i 0
307.2 1.00000i −0.707107 + 0.707107i −1.00000 0 −0.707107 − 0.707107i −0.137935 − 1.00000i − 1.00000i 0
307.3 1.00000i −0.707107 + 0.707107i −1.00000 0 −0.707107 − 0.707107i 1.13793 − 1.00000i − 1.00000i 0
307.4 1.00000i −0.707107 + 0.707107i −1.00000 0 −0.707107 − 0.707107i 2.06083 − 1.00000i − 1.00000i 0
307.5 1.00000i 0.707107 − 0.707107i −1.00000 0 0.707107 + 0.707107i −3.10444 − 1.00000i − 1.00000i 0
307.6 1.00000i 0.707107 − 0.707107i −1.00000 0 0.707107 + 0.707107i −0.579276 − 1.00000i − 1.00000i 0
307.7 1.00000i 0.707107 − 0.707107i −1.00000 0 0.707107 + 0.707107i 1.57928 − 1.00000i − 1.00000i 0
307.8 1.00000i 0.707107 − 0.707107i −1.00000 0 0.707107 + 0.707107i 4.10444 − 1.00000i − 1.00000i 0
343.1 − 1.00000i −0.707107 − 0.707107i −1.00000 0 −0.707107 + 0.707107i −1.06083 1.00000i 1.00000i 0
343.2 − 1.00000i −0.707107 − 0.707107i −1.00000 0 −0.707107 + 0.707107i −0.137935 1.00000i 1.00000i 0
343.3 − 1.00000i −0.707107 − 0.707107i −1.00000 0 −0.707107 + 0.707107i 1.13793 1.00000i 1.00000i 0
343.4 − 1.00000i −0.707107 − 0.707107i −1.00000 0 −0.707107 + 0.707107i 2.06083 1.00000i 1.00000i 0
343.5 − 1.00000i 0.707107 + 0.707107i −1.00000 0 0.707107 − 0.707107i −3.10444 1.00000i 1.00000i 0
343.6 − 1.00000i 0.707107 + 0.707107i −1.00000 0 0.707107 − 0.707107i −0.579276 1.00000i 1.00000i 0
343.7 − 1.00000i 0.707107 + 0.707107i −1.00000 0 0.707107 − 0.707107i 1.57928 1.00000i 1.00000i 0
343.8 − 1.00000i 0.707107 + 0.707107i −1.00000 0 0.707107 − 0.707107i 4.10444 1.00000i 1.00000i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 307.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
65.f even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1950.2.t.f yes 16
5.b even 2 1 1950.2.t.d yes 16
5.c odd 4 1 1950.2.j.d ✓ 16
5.c odd 4 1 1950.2.j.f yes 16
13.d odd 4 1 1950.2.j.f yes 16
65.f even 4 1 inner 1950.2.t.f yes 16
65.g odd 4 1 1950.2.j.d ✓ 16
65.k even 4 1 1950.2.t.d yes 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1950.2.j.d ✓ 16 5.c odd 4 1
1950.2.j.d ✓ 16 65.g odd 4 1
1950.2.j.f yes 16 5.c odd 4 1
1950.2.j.f yes 16 13.d odd 4 1
1950.2.t.d yes 16 5.b even 2 1
1950.2.t.d yes 16 65.k even 4 1
1950.2.t.f yes 16 1.a even 1 1 trivial
1950.2.t.f yes 16 65.f even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{8} - 4T_{7}^{7} - 10T_{7}^{6} + 44T_{7}^{5} - 3T_{7}^{4} - 72T_{7}^{3} + 12T_{7}^{2} + 32T_{7} + 4 \) acting on \(S_{2}^{\mathrm{new}}(1950, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 1)^{8} \) Copy content Toggle raw display
$3$ \( (T^{4} + 1)^{4} \) Copy content Toggle raw display
$5$ \( T^{16} \) Copy content Toggle raw display
$7$ \( (T^{8} - 4 T^{7} - 10 T^{6} + \cdots + 4)^{2} \) Copy content Toggle raw display
$11$ \( T^{16} + 16 T^{15} + \cdots + 16384 \) Copy content Toggle raw display
$13$ \( T^{16} + \cdots + 815730721 \) Copy content Toggle raw display
$17$ \( T^{16} + 8 T^{15} + \cdots + 11316496 \) Copy content Toggle raw display
$19$ \( T^{16} + 8 T^{15} + \cdots + 4194304 \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 1321467904 \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots + 328298161 \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots + 1363677184 \) Copy content Toggle raw display
$37$ \( (T^{8} - 8 T^{7} + \cdots - 1532)^{2} \) Copy content Toggle raw display
$41$ \( T^{16} - 16 T^{13} + \cdots + 14622976 \) Copy content Toggle raw display
$43$ \( T^{16} + 8 T^{15} + \cdots + 58736896 \) Copy content Toggle raw display
$47$ \( (T^{8} - 8 T^{7} + \cdots - 4407167)^{2} \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 9018731089 \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 67033352464 \) Copy content Toggle raw display
$61$ \( (T^{8} - 12 T^{7} + \cdots + 512896)^{2} \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 67871191441 \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 2005645099264 \) Copy content Toggle raw display
$73$ \( T^{16} + 328 T^{14} + \cdots + 65536 \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 4426507024 \) Copy content Toggle raw display
$83$ \( (T^{8} + 12 T^{7} + \cdots + 16908772)^{2} \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 2578054119424 \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 335617123860736 \) Copy content Toggle raw display
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