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Properties

Label 150.4.g.c
Level $150$
Weight $4$
Character orbit 150.g
Analytic conductor $8.850$
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] = [150,4,Mod(31,150)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(150, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([0, 4]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("150.31");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 150 = 2 \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 150.g (of order \(5\), 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: \(8.85028650086\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(4\) over \(\Q(\zeta_{5})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 289 x^{14} + 30961 x^{12} + 1537059 x^{10} + 36752711 x^{8} + 389532130 x^{6} + \cdots + 282912400 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4}\cdot 5^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( - 2 \beta_{3} + 2 \beta_{2} + \cdots + 2) q^{2}+ \cdots - 9 \beta_{3} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - 2 \beta_{3} + 2 \beta_{2} + \cdots + 2) q^{2}+ \cdots + (9 \beta_{13} - 9 \beta_{12} + \cdots - 36) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 8 q^{2} - 12 q^{3} - 16 q^{4} + 5 q^{5} + 24 q^{6} + 12 q^{7} + 32 q^{8} - 36 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 8 q^{2} - 12 q^{3} - 16 q^{4} + 5 q^{5} + 24 q^{6} + 12 q^{7} + 32 q^{8} - 36 q^{9} - 30 q^{10} + 60 q^{11} - 48 q^{12} - 44 q^{13} + 66 q^{14} - 15 q^{15} - 64 q^{16} - 113 q^{17} - 288 q^{18} - 71 q^{19} - 20 q^{20} + 81 q^{21} + 90 q^{22} + 45 q^{23} - 384 q^{24} - 85 q^{25} - 292 q^{26} - 108 q^{27} + 108 q^{28} + 501 q^{29} + 63 q^{31} - 512 q^{32} - 135 q^{33} - 334 q^{34} - 190 q^{35} - 144 q^{36} + 479 q^{37} - 168 q^{38} - 132 q^{39} + 344 q^{41} - 162 q^{42} + 948 q^{43} - 180 q^{44} - 135 q^{45} + 140 q^{46} + 545 q^{47} - 192 q^{48} + 616 q^{49} - 620 q^{50} - 324 q^{51} - 176 q^{52} + 437 q^{53} + 216 q^{54} + 2265 q^{55} - 216 q^{56} - 78 q^{57} - 1002 q^{58} - 150 q^{59} - 180 q^{60} + 1850 q^{61} + 164 q^{62} - 297 q^{63} - 256 q^{64} + 295 q^{65} - 360 q^{66} - 268 q^{67} - 432 q^{68} - 210 q^{69} + 50 q^{70} - 115 q^{71} + 288 q^{72} + 2666 q^{73} + 1032 q^{74} - 1995 q^{75} - 104 q^{76} - 845 q^{77} + 174 q^{78} + 3702 q^{79} - 240 q^{80} - 324 q^{81} + 2032 q^{82} - 2414 q^{83} - 396 q^{84} + 590 q^{85} + 2184 q^{86} - 207 q^{87} - 480 q^{88} - 3008 q^{89} - 90 q^{90} + 2857 q^{91} - 280 q^{92} + 114 q^{93} - 1090 q^{94} - 2085 q^{95} + 384 q^{96} + 3134 q^{97} + 4098 q^{98} - 270 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 289 x^{14} + 30961 x^{12} + 1537059 x^{10} + 36752711 x^{8} + 389532130 x^{6} + \cdots + 282912400 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 10\!\cdots\!51 \nu^{15} + \cdots + 20\!\cdots\!20 ) / 14\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 10\!\cdots\!51 \nu^{15} + \cdots - 20\!\cdots\!20 ) / 14\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 28\!\cdots\!18 \nu^{15} + \cdots - 68\!\cdots\!40 ) / 70\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 76\!\cdots\!81 \nu^{15} + \cdots - 33\!\cdots\!20 ) / 12\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 19\!\cdots\!95 \nu^{15} + \cdots + 32\!\cdots\!40 ) / 24\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 19\!\cdots\!95 \nu^{15} + \cdots - 57\!\cdots\!60 ) / 24\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 91\!\cdots\!41 \nu^{15} + \cdots + 28\!\cdots\!40 ) / 70\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 53\!\cdots\!01 \nu^{15} + \cdots + 78\!\cdots\!80 ) / 35\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 29\!\cdots\!41 \nu^{15} + \cdots - 82\!\cdots\!40 ) / 14\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 43\!\cdots\!87 \nu^{15} + \cdots - 20\!\cdots\!00 ) / 14\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 22\!\cdots\!79 \nu^{15} + \cdots - 32\!\cdots\!60 ) / 48\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 81\!\cdots\!21 \nu^{15} + \cdots - 16\!\cdots\!80 ) / 14\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 81\!\cdots\!21 \nu^{15} + \cdots - 29\!\cdots\!80 ) / 14\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 84\!\cdots\!01 \nu^{15} + \cdots - 28\!\cdots\!00 ) / 14\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 46\!\cdots\!35 \nu^{15} + \cdots + 10\!\cdots\!40 ) / 70\!\cdots\!40 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{15} + 2 \beta_{14} + 3 \beta_{12} - \beta_{11} - 3 \beta_{10} + 3 \beta_{9} - \beta_{8} + \cdots + 2 ) / 10 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 7 \beta_{15} - 8 \beta_{13} + 15 \beta_{12} - 15 \beta_{11} + 15 \beta_{10} - \beta_{9} + 15 \beta_{8} + \cdots - 340 ) / 10 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 71 \beta_{15} - 122 \beta_{14} + 30 \beta_{13} - 163 \beta_{12} + 91 \beta_{11} + 213 \beta_{10} + \cdots - 1122 ) / 10 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 333 \beta_{15} + 1146 \beta_{13} - 1479 \beta_{12} + 1735 \beta_{11} - 1735 \beta_{10} + \cdots + 26192 ) / 10 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 5737 \beta_{15} + 10162 \beta_{14} - 4068 \beta_{13} + 11807 \beta_{12} - 8371 \beta_{11} + \cdots + 137290 ) / 10 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 23469 \beta_{15} - 116464 \beta_{13} + 139933 \beta_{12} - 177895 \beta_{11} + 177895 \beta_{10} + \cdots - 2311024 ) / 10 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 562553 \beta_{15} - 1035822 \beta_{14} + 465116 \beta_{13} - 938491 \beta_{12} + 721741 \beta_{11} + \cdots - 14487548 ) / 10 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 2348225 \beta_{15} + 11068662 \beta_{13} - 13416887 \beta_{12} + 17636315 \beta_{11} + \cdots + 212449306 ) / 10 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 60474249 \beta_{15} + 112990622 \beta_{14} - 49944324 \beta_{13} + 76646335 \beta_{12} + \cdots + 1472769286 ) / 10 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 269623531 \beta_{15} - 1038006300 \beta_{13} + 1307629831 \beta_{12} - 1731833415 \beta_{11} + \cdots - 19858640448 ) / 10 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 6682095285 \beta_{15} - 12469028322 \beta_{14} + 5224307282 \beta_{13} - 6270371689 \beta_{12} + \cdots - 148227744454 ) / 10 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 31589159237 \beta_{15} + 97357775748 \beta_{13} - 128946934985 \beta_{12} + 169924368065 \beta_{11} + \cdots + 1873699673750 ) / 10 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 738804356521 \beta_{15} + 1368712345722 \beta_{14} - 540547368870 \beta_{13} + 507245931233 \beta_{12} + \cdots + 14895662830082 ) / 10 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 3648725951913 \beta_{15} - 9167066032926 \beta_{13} + 12815791984839 \beta_{12} + \cdots - 178023113521172 ) / 10 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 81047782120467 \beta_{15} - 148905087575162 \beta_{14} + 55685794269758 \beta_{13} + \cdots - 14\!\cdots\!80 ) / 10 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/150\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(127\)
\(\chi(n)\) \(1\) \(-\beta_{1}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
31.1
5.32239i
− 10.1515i
− 0.552839i
6.55750i
1.93564i
4.55041i
− 9.42282i
1.03466i
− 1.93564i
− 4.55041i
9.42282i
− 1.03466i
− 5.32239i
10.1515i
0.552839i
− 6.55750i
−0.618034 − 1.90211i −2.42705 + 1.76336i −3.23607 + 2.35114i −10.7402 − 3.10626i 4.85410 + 3.52671i −2.17344 6.47214 + 4.70228i 2.78115 − 8.55951i 0.729339 + 22.3488i
31.2 −0.618034 − 1.90211i −2.42705 + 1.76336i −3.23607 + 2.35114i 0.165586 − 11.1791i 4.85410 + 3.52671i 17.1962 6.47214 + 4.70228i 2.78115 − 8.55951i −21.3663 + 6.59411i
31.3 −0.618034 − 1.90211i −2.42705 + 1.76336i −3.23607 + 2.35114i 1.37095 + 11.0960i 4.85410 + 3.52671i −7.49851 6.47214 + 4.70228i 2.78115 − 8.55951i 20.2585 − 9.46539i
31.4 −0.618034 − 1.90211i −2.42705 + 1.76336i −3.23607 + 2.35114i 11.0126 − 1.92915i 4.85410 + 3.52671i −17.9407 6.47214 + 4.70228i 2.78115 − 8.55951i −10.4757 − 19.7550i
61.1 1.61803 − 1.17557i 0.927051 − 2.85317i 1.23607 − 3.80423i −11.0948 + 1.38060i −1.85410 − 5.70634i −16.5191 −2.47214 − 7.60845i −7.28115 − 5.29007i −16.3287 + 15.2765i
61.2 1.61803 − 1.17557i 0.927051 − 2.85317i 1.23607 − 3.80423i −2.82602 + 10.8173i −1.85410 − 5.70634i 27.7579 −2.47214 − 7.60845i −7.28115 − 5.29007i 8.14388 + 20.8249i
61.3 1.61803 − 1.17557i 0.927051 − 2.85317i 1.23607 − 3.80423i 6.21475 − 9.29392i −1.85410 − 5.70634i 28.2469 −2.47214 − 7.60845i −7.28115 − 5.29007i −0.869977 − 22.3437i
61.4 1.61803 − 1.17557i 0.927051 − 2.85317i 1.23607 − 3.80423i 8.39703 − 7.38173i −1.85410 − 5.70634i −23.0694 −2.47214 − 7.60845i −7.28115 − 5.29007i 4.90893 − 21.8152i
91.1 1.61803 + 1.17557i 0.927051 + 2.85317i 1.23607 + 3.80423i −11.0948 − 1.38060i −1.85410 + 5.70634i −16.5191 −2.47214 + 7.60845i −7.28115 + 5.29007i −16.3287 − 15.2765i
91.2 1.61803 + 1.17557i 0.927051 + 2.85317i 1.23607 + 3.80423i −2.82602 − 10.8173i −1.85410 + 5.70634i 27.7579 −2.47214 + 7.60845i −7.28115 + 5.29007i 8.14388 − 20.8249i
91.3 1.61803 + 1.17557i 0.927051 + 2.85317i 1.23607 + 3.80423i 6.21475 + 9.29392i −1.85410 + 5.70634i 28.2469 −2.47214 + 7.60845i −7.28115 + 5.29007i −0.869977 + 22.3437i
91.4 1.61803 + 1.17557i 0.927051 + 2.85317i 1.23607 + 3.80423i 8.39703 + 7.38173i −1.85410 + 5.70634i −23.0694 −2.47214 + 7.60845i −7.28115 + 5.29007i 4.90893 + 21.8152i
121.1 −0.618034 + 1.90211i −2.42705 − 1.76336i −3.23607 − 2.35114i −10.7402 + 3.10626i 4.85410 − 3.52671i −2.17344 6.47214 − 4.70228i 2.78115 + 8.55951i 0.729339 − 22.3488i
121.2 −0.618034 + 1.90211i −2.42705 − 1.76336i −3.23607 − 2.35114i 0.165586 + 11.1791i 4.85410 − 3.52671i 17.1962 6.47214 − 4.70228i 2.78115 + 8.55951i −21.3663 − 6.59411i
121.3 −0.618034 + 1.90211i −2.42705 − 1.76336i −3.23607 − 2.35114i 1.37095 − 11.0960i 4.85410 − 3.52671i −7.49851 6.47214 − 4.70228i 2.78115 + 8.55951i 20.2585 + 9.46539i
121.4 −0.618034 + 1.90211i −2.42705 − 1.76336i −3.23607 − 2.35114i 11.0126 + 1.92915i 4.85410 − 3.52671i −17.9407 6.47214 − 4.70228i 2.78115 + 8.55951i −10.4757 + 19.7550i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 31.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
25.d even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 150.4.g.c ✓ 16
25.d even 5 1 inner 150.4.g.c ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
150.4.g.c ✓ 16 1.a even 1 1 trivial
150.4.g.c ✓ 16 25.d even 5 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{8} - 6 T_{7}^{7} - 1508 T_{7}^{6} + 447 T_{7}^{5} + 743405 T_{7}^{4} + 3557228 T_{7}^{3} + \cdots - 1502362224 \) acting on \(S_{4}^{\mathrm{new}}(150, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - 2 T^{3} + 4 T^{2} + \cdots + 16)^{4} \) Copy content Toggle raw display
$3$ \( (T^{4} + 3 T^{3} + 9 T^{2} + \cdots + 81)^{4} \) Copy content Toggle raw display
$5$ \( T^{16} + \cdots + 59\!\cdots\!25 \) Copy content Toggle raw display
$7$ \( (T^{8} - 6 T^{7} + \cdots - 1502362224)^{2} \) Copy content Toggle raw display
$11$ \( T^{16} + \cdots + 22\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( T^{16} + \cdots + 81\!\cdots\!21 \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 27\!\cdots\!01 \) Copy content Toggle raw display
$19$ \( T^{16} + \cdots + 14\!\cdots\!36 \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 16\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots + 79\!\cdots\!01 \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots + 52\!\cdots\!16 \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 28\!\cdots\!21 \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 11\!\cdots\!61 \) Copy content Toggle raw display
$43$ \( (T^{8} + \cdots + 136344920115536)^{2} \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 62\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 52\!\cdots\!41 \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 32\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 50\!\cdots\!25 \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 18\!\cdots\!96 \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 27\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 32\!\cdots\!41 \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 14\!\cdots\!16 \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 10\!\cdots\!56 \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 90\!\cdots\!01 \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 84\!\cdots\!21 \) Copy content Toggle raw display
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