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Properties

Label 975.2.bl.f
Level $975$
Weight $2$
Character orbit 975.bl
Analytic conductor $7.785$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [975,2,Mod(193,975)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(975, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([0, 9, 7]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("975.193");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 975 = 3 \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 975.bl (of order \(12\), degree \(4\), 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: \(7.78541419707\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(4\) over \(\Q(\zeta_{12})\)
Coefficient field: 16.0.12877254853348294656.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 4x^{14} + 26x^{12} + 12x^{10} + 35x^{8} + 180x^{6} + 686x^{4} + 632x^{2} + 169 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( - \beta_{13} + \beta_{7} + \cdots - \beta_1) q^{2}+ \cdots + \beta_{10} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{13} + \beta_{7} + \cdots - \beta_1) q^{2}+ \cdots + ( - 2 \beta_{15} + 2 \beta_{11} + \cdots + 2 \beta_{4}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 8 q^{4} + 4 q^{6}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 8 q^{4} + 4 q^{6} - 24 q^{16} + 8 q^{19} - 16 q^{21} + 52 q^{24} - 4 q^{26} + 24 q^{29} - 24 q^{36} - 16 q^{39} + 56 q^{41} - 16 q^{44} + 72 q^{46} - 8 q^{49} - 4 q^{54} - 60 q^{56} + 104 q^{59} - 16 q^{61} + 176 q^{64} + 16 q^{66} - 32 q^{69} - 32 q^{71} + 24 q^{76} + 8 q^{81} - 8 q^{84} + 80 q^{86} - 32 q^{89} - 40 q^{91} + 24 q^{94} - 56 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 4x^{14} + 26x^{12} + 12x^{10} + 35x^{8} + 180x^{6} + 686x^{4} + 632x^{2} + 169 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 2535 \nu^{15} - 10159 \nu^{13} + 68531 \nu^{11} + 17346 \nu^{9} + 164653 \nu^{7} + \cdots + 1819121 \nu ) / 882024 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 7944 \nu^{15} + 39895 \nu^{13} - 244988 \nu^{11} + 140334 \nu^{9} - 343708 \nu^{7} + \cdots - 921431 \nu ) / 882024 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 5087 \nu^{14} - 26978 \nu^{12} + 162746 \nu^{10} - 123410 \nu^{8} + 206398 \nu^{6} + 869936 \nu^{4} + \cdots + 559676 ) / 882024 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 9980 \nu^{15} - 47132 \nu^{13} + 293474 \nu^{11} - 101381 \nu^{9} + 457768 \nu^{7} + \cdots + 2149787 \nu ) / 882024 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 12190 \nu^{15} - 53524 \nu^{13} + 336112 \nu^{11} + 22037 \nu^{9} + 361814 \nu^{7} + \cdots + 4608433 \nu ) / 882024 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 12621 \nu^{15} + 55249 \nu^{13} - 346712 \nu^{11} - 30687 \nu^{9} - 386698 \nu^{7} + \cdots - 4825886 \nu ) / 882024 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 8119 \nu^{14} - 38444 \nu^{12} + 235662 \nu^{10} - 65668 \nu^{8} + 220254 \nu^{6} + 1268122 \nu^{4} + \cdots + 1342536 ) / 882024 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 8572 \nu^{14} + 43799 \nu^{12} - 271518 \nu^{10} + 204748 \nu^{8} - 555030 \nu^{6} + \cdots - 1042197 ) / 882024 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 12621 \nu^{14} - 55249 \nu^{12} + 346712 \nu^{10} + 30687 \nu^{8} + 386698 \nu^{6} + 2094579 \nu^{4} + \cdots + 4825886 ) / 882024 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 13317 \nu^{14} - 60350 \nu^{12} + 382687 \nu^{10} - 63984 \nu^{8} + 623729 \nu^{6} + 2103120 \nu^{4} + \cdots + 4437940 ) / 882024 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 13336 \nu^{14} - 62971 \nu^{12} + 395761 \nu^{10} - 139912 \nu^{8} + 645671 \nu^{6} + \cdots + 3102307 ) / 882024 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 23422 \nu^{15} + 108061 \nu^{13} - 673942 \nu^{11} + 126571 \nu^{9} - 867716 \nu^{7} + \cdots - 6109072 \nu ) / 882024 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 26260 \nu^{15} - 118357 \nu^{13} + 743110 \nu^{11} - 67567 \nu^{9} + 983084 \nu^{7} + \cdots + 8688484 \nu ) / 882024 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 3028 \nu^{14} - 13890 \nu^{12} + 86828 \nu^{10} - 13315 \nu^{8} + 108628 \nu^{6} + 503662 \nu^{4} + \cdots + 822263 ) / 110253 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -2\beta_{15} + 3\beta_{12} - \beta_{11} + 2\beta_{10} + \beta_{9} + \beta_{4} - 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -4\beta_{14} - 4\beta_{13} + \beta_{6} + \beta_{5} + 2\beta_{3} + 2\beta_{2} + 2\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -7\beta_{15} + 4\beta_{12} + 4\beta_{11} + 4\beta_{10} + 4\beta_{9} + 2\beta_{8} + 6\beta_{4} - 6 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -11\beta_{14} - 2\beta_{13} - 4\beta_{7} + 8\beta_{6} + 5\beta_{5} - 3\beta_{3} + 8\beta_{2} - \beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 9\beta_{15} - 36\beta_{12} + 20\beta_{11} - 6\beta_{10} - 12\beta_{9} - 7\beta_{8} + 5\beta_{4} - 15 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 55\beta_{14} + 87\beta_{13} - 42\beta_{7} - 26\beta_{6} - 4\beta_{5} - 58\beta_{3} - 16\beta_{2} - 51\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 227\beta_{15} - 228\beta_{12} - 20\beta_{11} - 144\beta_{10} - 108\beta_{9} - 100\beta_{8} - 96\beta_{4} + 83 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 515\beta_{14} + 420\beta_{13} - 88\beta_{7} - 308\beta_{6} - 219\beta_{5} - 223\beta_{3} - 248\beta_{2} - 224\beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 737\beta_{15} + 33\beta_{12} - 539\beta_{11} - 536\beta_{10} - 117\beta_{9} - 311\beta_{8} - 556\beta_{4} + 720 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 937 \beta_{14} - 389 \beta_{13} + 814 \beta_{7} - 739 \beta_{6} - 931 \beta_{5} + 486 \beta_{3} + \cdots + 211 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 2556 \beta_{15} + 5874 \beta_{12} - 1654 \beta_{11} + 1236 \beta_{10} + 2270 \beta_{9} + 1300 \beta_{8} + \cdots + 801 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 9080 \beta_{14} - 12612 \beta_{13} + 6152 \beta_{7} + 4870 \beta_{6} + 1586 \beta_{5} + \cdots + 6311 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 29800 \beta_{15} + 23849 \beta_{12} + 6989 \beta_{11} + 18774 \beta_{10} + 12467 \beta_{9} + \cdots - 14751 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 63998 \beta_{14} - 45276 \beta_{13} + 6664 \beta_{7} + 41187 \beta_{6} + 31693 \beta_{5} + \cdots + 22420 \beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/975\mathbb{Z}\right)^\times\).

\(n\) \(301\) \(326\) \(352\)
\(\chi(n)\) \(\beta_{10}\) \(1\) \(\beta_{10} + \beta_{12}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
193.1
−0.489978 + 1.19709i
−1.90419 + 1.19709i
1.90419 − 1.19709i
0.489978 − 1.19709i
1.46009 + 0.752986i
0.0458788 + 0.752986i
−0.0458788 − 0.752986i
−1.46009 − 0.752986i
−0.489978 − 1.19709i
−1.90419 − 1.19709i
1.90419 + 1.19709i
0.489978 + 1.19709i
1.46009 − 0.752986i
0.0458788 − 0.752986i
−0.0458788 + 0.752986i
−1.46009 + 0.752986i
−0.791717 − 1.37129i −0.258819 + 0.965926i −0.253631 + 0.439303i 0 1.52948 − 0.409823i 3.29816 + 1.90419i −2.36365 −0.866025 − 0.500000i 0
193.2 −0.0846102 − 0.146549i 0.258819 − 0.965926i 0.985682 − 1.70725i 0 −0.163454 + 0.0437974i 0.848667 + 0.489978i −0.672036 −0.866025 − 0.500000i 0
193.3 0.0846102 + 0.146549i −0.258819 + 0.965926i 0.985682 − 1.70725i 0 −0.163454 + 0.0437974i −0.848667 − 0.489978i 0.672036 −0.866025 − 0.500000i 0
193.4 0.791717 + 1.37129i 0.258819 − 0.965926i −0.253631 + 0.439303i 0 1.52948 − 0.409823i −3.29816 − 1.90419i 2.36365 −0.866025 − 0.500000i 0
457.1 −1.38215 − 2.39396i 0.965926 + 0.258819i −2.82068 + 4.88556i 0 −0.715454 − 2.67011i 0.0794644 + 0.0458788i 10.0658 0.866025 + 0.500000i 0
457.2 −0.675044 − 1.16921i −0.965926 − 0.258819i 0.0886311 − 0.153513i 0 0.349429 + 1.30408i 2.52895 + 1.46009i −2.93950 0.866025 + 0.500000i 0
457.3 0.675044 + 1.16921i 0.965926 + 0.258819i 0.0886311 − 0.153513i 0 0.349429 + 1.30408i −2.52895 − 1.46009i 2.93950 0.866025 + 0.500000i 0
457.4 1.38215 + 2.39396i −0.965926 − 0.258819i −2.82068 + 4.88556i 0 −0.715454 − 2.67011i −0.0794644 − 0.0458788i −10.0658 0.866025 + 0.500000i 0
682.1 −0.791717 + 1.37129i −0.258819 − 0.965926i −0.253631 − 0.439303i 0 1.52948 + 0.409823i 3.29816 − 1.90419i −2.36365 −0.866025 + 0.500000i 0
682.2 −0.0846102 + 0.146549i 0.258819 + 0.965926i 0.985682 + 1.70725i 0 −0.163454 − 0.0437974i 0.848667 − 0.489978i −0.672036 −0.866025 + 0.500000i 0
682.3 0.0846102 − 0.146549i −0.258819 − 0.965926i 0.985682 + 1.70725i 0 −0.163454 − 0.0437974i −0.848667 + 0.489978i 0.672036 −0.866025 + 0.500000i 0
682.4 0.791717 − 1.37129i 0.258819 + 0.965926i −0.253631 − 0.439303i 0 1.52948 + 0.409823i −3.29816 + 1.90419i 2.36365 −0.866025 + 0.500000i 0
943.1 −1.38215 + 2.39396i 0.965926 − 0.258819i −2.82068 − 4.88556i 0 −0.715454 + 2.67011i 0.0794644 − 0.0458788i 10.0658 0.866025 − 0.500000i 0
943.2 −0.675044 + 1.16921i −0.965926 + 0.258819i 0.0886311 + 0.153513i 0 0.349429 − 1.30408i 2.52895 − 1.46009i −2.93950 0.866025 − 0.500000i 0
943.3 0.675044 − 1.16921i 0.965926 − 0.258819i 0.0886311 + 0.153513i 0 0.349429 − 1.30408i −2.52895 + 1.46009i 2.93950 0.866025 − 0.500000i 0
943.4 1.38215 − 2.39396i −0.965926 + 0.258819i −2.82068 − 4.88556i 0 −0.715454 + 2.67011i −0.0794644 + 0.0458788i −10.0658 0.866025 − 0.500000i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 193.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
65.o even 12 1 inner
65.t even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 975.2.bl.f ✓ 16
5.b even 2 1 inner 975.2.bl.f ✓ 16
5.c odd 4 2 975.2.bu.g yes 16
13.f odd 12 1 975.2.bu.g yes 16
65.o even 12 1 inner 975.2.bl.f ✓ 16
65.s odd 12 1 975.2.bu.g yes 16
65.t even 12 1 inner 975.2.bl.f ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
975.2.bl.f ✓ 16 1.a even 1 1 trivial
975.2.bl.f ✓ 16 5.b even 2 1 inner
975.2.bl.f ✓ 16 65.o even 12 1 inner
975.2.bl.f ✓ 16 65.t even 12 1 inner
975.2.bu.g yes 16 5.c odd 4 2
975.2.bu.g yes 16 13.f odd 12 1
975.2.bu.g yes 16 65.s odd 12 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(975, [\chi])\):

\( T_{2}^{16} + 12T_{2}^{14} + 106T_{2}^{12} + 384T_{2}^{10} + 1011T_{2}^{8} + 1344T_{2}^{6} + 1258T_{2}^{4} + 36T_{2}^{2} + 1 \) Copy content Toggle raw display
\( T_{7}^{16} - 24T_{7}^{14} + 430T_{7}^{12} - 3264T_{7}^{10} + 18435T_{7}^{8} - 17472T_{7}^{6} + 14254T_{7}^{4} - 120T_{7}^{2} + 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} + 12 T^{14} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( (T^{8} - T^{4} + 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{16} \) Copy content Toggle raw display
$7$ \( T^{16} - 24 T^{14} + \cdots + 1 \) Copy content Toggle raw display
$11$ \( (T^{8} - 24 T^{6} + \cdots + 43264)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} + 24 T^{2} + 169)^{4} \) Copy content Toggle raw display
$17$ \( T^{16} - 48 T^{14} + \cdots + 65536 \) Copy content Toggle raw display
$19$ \( (T^{8} - 4 T^{7} + \cdots + 2209)^{2} \) Copy content Toggle raw display
$23$ \( T^{16} + 120 T^{14} + \cdots + 59969536 \) Copy content Toggle raw display
$29$ \( (T^{8} - 12 T^{7} + \cdots + 732736)^{2} \) Copy content Toggle raw display
$31$ \( (T^{8} - 192 T^{5} + \cdots + 21904)^{2} \) Copy content Toggle raw display
$37$ \( T^{16} - 248 T^{14} + \cdots + 3748096 \) Copy content Toggle raw display
$41$ \( (T^{8} - 28 T^{7} + \cdots + 4393216)^{2} \) Copy content Toggle raw display
$43$ \( T^{16} + 96 T^{14} + \cdots + 88529281 \) Copy content Toggle raw display
$47$ \( (T^{8} + 96 T^{6} + \cdots + 1024)^{2} \) Copy content Toggle raw display
$53$ \( T^{16} + 12832 T^{12} + \cdots + 4096 \) Copy content Toggle raw display
$59$ \( (T^{8} - 52 T^{7} + \cdots + 10601536)^{2} \) Copy content Toggle raw display
$61$ \( (T^{8} + 8 T^{7} + \cdots + 59969536)^{2} \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 2743558264161 \) Copy content Toggle raw display
$71$ \( (T^{8} + 16 T^{7} + \cdots + 19642624)^{2} \) Copy content Toggle raw display
$73$ \( (T^{8} - 360 T^{6} + \cdots + 1656369)^{2} \) Copy content Toggle raw display
$79$ \( (T^{8} + 60 T^{6} + \cdots + 20449)^{2} \) Copy content Toggle raw display
$83$ \( (T^{8} + 264 T^{6} + \cdots + 6270016)^{2} \) Copy content Toggle raw display
$89$ \( (T^{8} + 16 T^{7} + \cdots + 760384)^{2} \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 1507158896896 \) Copy content Toggle raw display
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