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Properties

Label 1950.2.bc.k
Level $1950$
Weight $2$
Character orbit 1950.bc
Analytic conductor $15.571$
Analytic rank $0$
Dimension $12$
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(751,1950)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1950, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1950.751");
 
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.bc (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: \(15.5708283941\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 6 x^{11} + 87 x^{10} - 380 x^{9} + 2556 x^{8} - 8010 x^{7} + 29687 x^{6} - 62556 x^{5} + 115386 x^{4} - 135130 x^{3} + 113253 x^{2} - 54888 x + 14089 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{8} q^{2} + \beta_{4} q^{3} + ( - \beta_{4} + 1) q^{4} + \beta_{9} q^{6} + (\beta_{9} - \beta_{3}) q^{7} + ( - \beta_{9} + \beta_{8}) q^{8} + (\beta_{4} - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{8} q^{2} + \beta_{4} q^{3} + ( - \beta_{4} + 1) q^{4} + \beta_{9} q^{6} + (\beta_{9} - \beta_{3}) q^{7} + ( - \beta_{9} + \beta_{8}) q^{8} + (\beta_{4} - 1) q^{9} + (\beta_{11} + \beta_{10} + \beta_{9} + \beta_{8} - \beta_{6} - \beta_{5} - \beta_{3} + \beta_{2} - 1) q^{11} + q^{12} + ( - \beta_{10} - \beta_{8} + \beta_{7} + \beta_{3} - \beta_1) q^{13} + (\beta_1 + 1) q^{14} - \beta_{4} q^{16} + (\beta_{11} + \beta_{9} + \beta_{7} - \beta_{6} - \beta_{5} + \beta_{2} - \beta_1) q^{17} + (\beta_{9} - \beta_{8}) q^{18} + (\beta_{10} - 2 \beta_{9} - \beta_{8} + \beta_{6} - \beta_{5} - \beta_{4} - \beta_{3} + 2 \beta_1) q^{19} + (\beta_{9} - \beta_{8} - \beta_{3} + \beta_{2} - 1) q^{21} + (\beta_{11} + \beta_{9} - \beta_{8} + \beta_{7} - \beta_{6} - \beta_{4} + \beta_{2} - \beta_1 + 1) q^{22} + (\beta_{9} + \beta_{5} - \beta_{4} - \beta_1) q^{23} + \beta_{8} q^{24} + ( - \beta_{11} - \beta_{9} + \beta_{4} - 1) q^{26} - q^{27} + (\beta_{8} - \beta_{2} + 1) q^{28} + ( - \beta_{11} - \beta_{9} + \beta_{7} + \beta_{5} - \beta_1) q^{29} + ( - \beta_{7} + \beta_{6} - 2 \beta_{5} - \beta_{3} + 2 \beta_1 - 1) q^{31} - \beta_{9} q^{32} + (\beta_{11} + \beta_{10} + 2 \beta_{9} - \beta_{7} - \beta_{5} - 2 \beta_{4} - 2 \beta_{3} + \beta_{2} + \cdots - 1) q^{33}+ \cdots + (\beta_{9} - \beta_{8} - \beta_{7} + \beta_{6} - 2 \beta_{4} - \beta_{3} + \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 6 q^{3} + 6 q^{4} + 6 q^{7} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 6 q^{3} + 6 q^{4} + 6 q^{7} - 6 q^{9} - 12 q^{11} + 12 q^{12} + 8 q^{14} - 6 q^{16} - 6 q^{19} + 4 q^{22} - 4 q^{23} - 4 q^{26} - 12 q^{27} + 6 q^{28} - 12 q^{33} + 6 q^{36} + 12 q^{37} - 24 q^{38} + 6 q^{39} + 4 q^{42} + 10 q^{43} + 12 q^{46} + 6 q^{48} + 32 q^{49} - 6 q^{52} + 16 q^{53} + 4 q^{56} + 24 q^{61} - 8 q^{62} - 6 q^{63} - 12 q^{64} + 8 q^{66} - 6 q^{67} + 4 q^{69} + 12 q^{71} - 12 q^{74} - 6 q^{76} + 48 q^{77} - 8 q^{78} + 52 q^{79} - 6 q^{81} + 6 q^{84} - 4 q^{88} + 24 q^{89} - 54 q^{91} - 8 q^{92} - 8 q^{94} - 12 q^{97} - 36 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 6 x^{11} + 87 x^{10} - 380 x^{9} + 2556 x^{8} - 8010 x^{7} + 29687 x^{6} - 62556 x^{5} + 115386 x^{4} - 135130 x^{3} + 113253 x^{2} - 54888 x + 14089 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 176 \nu^{10} + 880 \nu^{9} - 13279 \nu^{8} + 47836 \nu^{7} - 335904 \nu^{6} + 843982 \nu^{5} - 3291723 \nu^{4} + 5230858 \nu^{3} - 9800366 \nu^{2} + \cdots - 3607897 ) / 737586 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 1070 \nu^{11} - 9796 \nu^{10} + 107747 \nu^{9} - 581520 \nu^{8} + 3242004 \nu^{7} - 10618216 \nu^{6} + 34751213 \nu^{5} - 61642348 \nu^{4} + \cdots - 30150460 ) / 33157458 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 1070 \nu^{11} + 1974 \nu^{10} - 68637 \nu^{9} + 123933 \nu^{8} - 1646316 \nu^{7} + 3281180 \nu^{6} - 18160751 \nu^{5} + 37476279 \nu^{4} + \cdots + 8385745 ) / 33157458 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 2140 \nu^{11} - 11770 \nu^{10} + 176384 \nu^{9} - 705453 \nu^{8} + 4888320 \nu^{7} - 13899396 \nu^{6} + 52911964 \nu^{5} - 99118627 \nu^{4} + \cdots - 38536205 ) / 33157458 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 196298 \nu^{11} - 1423807 \nu^{10} + 17298796 \nu^{9} - 87970594 \nu^{8} + 505040292 \nu^{7} - 1815307513 \nu^{6} + 5860211676 \nu^{5} + \cdots - 8896864806 ) / 2884698846 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 199579 \nu^{11} + 43670 \nu^{10} - 10853189 \nu^{9} - 20210311 \nu^{8} - 114973215 \nu^{7} - 1069324210 \nu^{6} + 1898810217 \nu^{5} + \cdots - 34891536081 ) / 2884698846 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 292669 \nu^{11} - 3011773 \nu^{10} + 30806433 \nu^{9} - 211570737 \nu^{8} + 1058307265 \nu^{7} - 5033911301 \nu^{6} + 14392851385 \nu^{5} + \cdots - 39899132821 ) / 2884698846 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 370049 \nu^{11} - 2416592 \nu^{10} + 32929734 \nu^{9} - 154475736 \nu^{8} + 981881669 \nu^{7} - 3282046186 \nu^{6} + 11487231632 \nu^{5} + \cdots - 13717956122 ) / 2884698846 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 370049 \nu^{11} + 1653947 \nu^{10} - 29116509 \nu^{9} + 94203315 \nu^{8} - 763671335 \nu^{7} + 1695380863 \nu^{6} - 7474956287 \nu^{5} + \cdots - 3683244445 ) / 2884698846 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 1663873 \nu^{11} + 8187240 \nu^{10} - 136318210 \nu^{9} + 487832491 \nu^{8} - 3738605045 \nu^{7} + 9381467744 \nu^{6} + \cdots + 14213570375 ) / 2884698846 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 2230220 \nu^{11} - 13193117 \nu^{10} + 192374130 \nu^{9} - 827318553 \nu^{8} + 5578721594 \nu^{7} - 17140264765 \nu^{6} + \cdots - 52508450723 ) / 2884698846 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( -\beta_{4} - \beta_{3} + \beta_{2} \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{11} + \beta_{10} + 4 \beta_{9} + 3 \beta_{8} - 2 \beta_{7} - 2 \beta_{6} - \beta_{5} - 2 \beta_{4} - 4 \beta_{3} + 2 \beta_{2} + 2 \beta _1 - 12 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2 \beta_{11} + \beta_{10} + 12 \beta_{9} - \beta_{8} - 2 \beta_{7} - 4 \beta_{6} - 6 \beta_{5} + 32 \beta_{4} + 15 \beta_{3} - 17 \beta_{2} + 4 \beta _1 - 15 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 16 \beta_{11} - 18 \beta_{10} - 89 \beta_{9} - 95 \beta_{8} + 45 \beta_{7} + 41 \beta_{6} + 8 \beta_{5} + 85 \beta_{4} + 100 \beta_{3} - 55 \beta_{2} - 32 \beta _1 + 242 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 44 \beta_{11} - 46 \beta_{10} - 321 \beta_{9} - 158 \beta_{8} + 92 \beta_{7} + 133 \beta_{6} + 183 \beta_{5} - 684 \beta_{4} - 235 \beta_{3} + 327 \beta_{2} - 139 \beta _1 + 592 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 329 \beta_{11} + 328 \beta_{10} + 2130 \beta_{9} + 2213 \beta_{8} - 971 \beta_{7} - 838 \beta_{6} + 108 \beta_{5} - 2686 \beta_{4} - 2484 \beta_{3} + 1567 \beta_{2} + 352 \beta _1 - 4964 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 1001 \beta_{11} + 1617 \beta_{10} + 8482 \beta_{9} + 8101 \beta_{8} - 3297 \beta_{7} - 3829 \beta_{6} - 4846 \beta_{5} + 13692 \beta_{4} + 2733 \beta_{3} - 5841 \beta_{2} + 3739 \beta _1 - 19054 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 7402 \beta_{11} - 4938 \beta_{10} - 51821 \beta_{9} - 43913 \beta_{8} + 19371 \beta_{7} + 16613 \beta_{6} - 10036 \beta_{5} + 77335 \beta_{4} + 59344 \beta_{3} - 41809 \beta_{2} + 1372 \beta _1 + 96085 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 23870 \beta_{11} - 47752 \beta_{10} - 233457 \beta_{9} - 283244 \beta_{8} + 100886 \beta_{7} + 104755 \beta_{6} + 115911 \beta_{5} - 256271 \beta_{4} - 156 \beta_{3} + 91646 \beta_{2} + \cdots + 551212 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 175288 \beta_{11} + 37401 \beta_{10} + 1241888 \beta_{9} + 697514 \beta_{8} - 348537 \beta_{7} - 307556 \beta_{6} + 418801 \beta_{5} - 2117336 \beta_{4} - 1364810 \beta_{3} + \cdots - 1683590 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 602019 \beta_{11} + 1254110 \beta_{10} + 6570934 \beta_{9} + 8427566 \beta_{8} - 2812501 \beta_{7} - 2761947 \beta_{6} - 2508922 \beta_{5} + 4247176 \beta_{4} - 1432740 \beta_{3} + \cdots - 14885497 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(301\) \(1301\) \(1327\)
\(\chi(n)\) \(1 - \beta_{4}\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
751.1
0.500000 − 4.99624i
0.500000 + 0.414256i
0.500000 + 4.71596i
0.500000 − 4.41310i
0.500000 + 0.822735i
0.500000 + 1.72434i
0.500000 + 4.99624i
0.500000 − 0.414256i
0.500000 − 4.71596i
0.500000 + 4.41310i
0.500000 − 0.822735i
0.500000 − 1.72434i
−0.866025 + 0.500000i 0.500000 + 0.866025i 0.500000 − 0.866025i 0 −0.866025 − 0.500000i −4.44290 − 2.56511i 1.00000i −0.500000 + 0.866025i 0
751.2 −0.866025 + 0.500000i 0.500000 + 0.866025i 0.500000 − 0.866025i 0 −0.866025 − 0.500000i 0.242731 + 0.140141i 1.00000i −0.500000 + 0.866025i 0
751.3 −0.866025 + 0.500000i 0.500000 + 0.866025i 0.500000 − 0.866025i 0 −0.866025 − 0.500000i 3.96812 + 2.29099i 1.00000i −0.500000 + 0.866025i 0
751.4 0.866025 − 0.500000i 0.500000 + 0.866025i 0.500000 − 0.866025i 0 0.866025 + 0.500000i −2.20583 − 1.27354i − 1.00000i −0.500000 + 0.866025i 0
751.5 0.866025 − 0.500000i 0.500000 + 0.866025i 0.500000 − 0.866025i 0 0.866025 + 0.500000i 2.32854 + 1.34438i − 1.00000i −0.500000 + 0.866025i 0
751.6 0.866025 − 0.500000i 0.500000 + 0.866025i 0.500000 − 0.866025i 0 0.866025 + 0.500000i 3.10934 + 1.79518i − 1.00000i −0.500000 + 0.866025i 0
901.1 −0.866025 − 0.500000i 0.500000 − 0.866025i 0.500000 + 0.866025i 0 −0.866025 + 0.500000i −4.44290 + 2.56511i − 1.00000i −0.500000 − 0.866025i 0
901.2 −0.866025 − 0.500000i 0.500000 − 0.866025i 0.500000 + 0.866025i 0 −0.866025 + 0.500000i 0.242731 − 0.140141i − 1.00000i −0.500000 − 0.866025i 0
901.3 −0.866025 − 0.500000i 0.500000 − 0.866025i 0.500000 + 0.866025i 0 −0.866025 + 0.500000i 3.96812 − 2.29099i − 1.00000i −0.500000 − 0.866025i 0
901.4 0.866025 + 0.500000i 0.500000 − 0.866025i 0.500000 + 0.866025i 0 0.866025 − 0.500000i −2.20583 + 1.27354i 1.00000i −0.500000 − 0.866025i 0
901.5 0.866025 + 0.500000i 0.500000 − 0.866025i 0.500000 + 0.866025i 0 0.866025 − 0.500000i 2.32854 − 1.34438i 1.00000i −0.500000 − 0.866025i 0
901.6 0.866025 + 0.500000i 0.500000 − 0.866025i 0.500000 + 0.866025i 0 0.866025 − 0.500000i 3.10934 − 1.79518i 1.00000i −0.500000 − 0.866025i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 751.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.e even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1950.2.bc.k yes 12
5.b even 2 1 1950.2.bc.h ✓ 12
5.c odd 4 1 1950.2.y.m 12
5.c odd 4 1 1950.2.y.n 12
13.e even 6 1 inner 1950.2.bc.k yes 12
65.l even 6 1 1950.2.bc.h ✓ 12
65.r odd 12 1 1950.2.y.m 12
65.r odd 12 1 1950.2.y.n 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1950.2.y.m 12 5.c odd 4 1
1950.2.y.m 12 65.r odd 12 1
1950.2.y.n 12 5.c odd 4 1
1950.2.y.n 12 65.r odd 12 1
1950.2.bc.h ✓ 12 5.b even 2 1
1950.2.bc.h ✓ 12 65.l even 6 1
1950.2.bc.k yes 12 1.a even 1 1 trivial
1950.2.bc.k yes 12 13.e even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{12} - 6 T_{7}^{11} - 19 T_{7}^{10} + 186 T_{7}^{9} + 421 T_{7}^{8} - 5244 T_{7}^{7} + 7296 T_{7}^{6} + 27900 T_{7}^{5} - 50994 T_{7}^{4} - 142560 T_{7}^{3} + 410184 T_{7}^{2} - 174960 T_{7} + 26244 \) acting on \(S_{2}^{\mathrm{new}}(1950, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - T^{2} + 1)^{3} \) Copy content Toggle raw display
$3$ \( (T^{2} - T + 1)^{6} \) Copy content Toggle raw display
$5$ \( T^{12} \) Copy content Toggle raw display
$7$ \( T^{12} - 6 T^{11} - 19 T^{10} + \cdots + 26244 \) Copy content Toggle raw display
$11$ \( T^{12} + 12 T^{11} + 26 T^{10} + \cdots + 11664 \) Copy content Toggle raw display
$13$ \( T^{12} + 10 T^{10} + 60 T^{9} + \cdots + 4826809 \) Copy content Toggle raw display
$17$ \( T^{12} + 50 T^{10} - 168 T^{9} + \cdots + 876096 \) Copy content Toggle raw display
$19$ \( T^{12} + 6 T^{11} - 59 T^{10} + \cdots + 73513476 \) Copy content Toggle raw display
$23$ \( T^{12} + 4 T^{11} + 48 T^{10} + \cdots + 46656 \) Copy content Toggle raw display
$29$ \( T^{12} + 92 T^{10} + 480 T^{9} + \cdots + 56070144 \) Copy content Toggle raw display
$31$ \( T^{12} + 212 T^{10} + \cdots + 248629824 \) Copy content Toggle raw display
$37$ \( T^{12} - 12 T^{11} - 8 T^{10} + \cdots + 56070144 \) Copy content Toggle raw display
$41$ \( T^{12} - 50 T^{10} + 2118 T^{8} + \cdots + 876096 \) Copy content Toggle raw display
$43$ \( T^{12} - 10 T^{11} + 145 T^{10} + \cdots + 45077796 \) Copy content Toggle raw display
$47$ \( T^{12} + 200 T^{10} + 9976 T^{8} + \cdots + 2985984 \) Copy content Toggle raw display
$53$ \( (T^{6} - 8 T^{5} - 74 T^{4} + 612 T^{3} + \cdots + 15768)^{2} \) Copy content Toggle raw display
$59$ \( T^{12} - 104 T^{10} + 8700 T^{8} + \cdots + 746496 \) Copy content Toggle raw display
$61$ \( T^{12} - 24 T^{11} + 404 T^{10} + \cdots + 82955664 \) Copy content Toggle raw display
$67$ \( T^{12} + 6 T^{11} - 185 T^{10} + \cdots + 753831936 \) Copy content Toggle raw display
$71$ \( T^{12} - 12 T^{11} + \cdots + 438439973904 \) Copy content Toggle raw display
$73$ \( T^{12} + 458 T^{10} + 59239 T^{8} + \cdots + 2653641 \) Copy content Toggle raw display
$79$ \( (T^{6} - 26 T^{5} - 73 T^{4} + \cdots + 320086)^{2} \) Copy content Toggle raw display
$83$ \( T^{12} + 812 T^{10} + \cdots + 900388436544 \) Copy content Toggle raw display
$89$ \( T^{12} - 24 T^{11} + 106 T^{10} + \cdots + 77158656 \) Copy content Toggle raw display
$97$ \( T^{12} + 12 T^{11} - 134 T^{10} + \cdots + 21233664 \) Copy content Toggle raw display
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