↑

Properties

Label 350.3.p.e
Level $350$
Weight $3$
Character orbit 350.p
Analytic conductor $9.537$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [350,3,Mod(93,350)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(350, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([9, 4]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("350.93");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 350 = 2 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 350.p (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: \(9.53680925261\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(4\) over \(\Q(\zeta_{12})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 2 x^{15} + 2 x^{14} - 8 x^{13} - 722 x^{12} + 1354 x^{11} - 1232 x^{10} + 9306 x^{9} + \cdots + 52200625 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 70)
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.

\(f(q)\) \(=\) \( q + ( - \beta_{8} + \beta_{6} + \beta_{4} + 1) q^{2} + ( - \beta_{7} + \beta_1) q^{3} + 2 \beta_{4} q^{4} + ( - \beta_{7} - \beta_{3}) q^{6} + (\beta_{14} - \beta_{9} + \cdots + \beta_{3}) q^{7}+ \cdots + (\beta_{14} - 2 \beta_{13} + \cdots - \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{8} + \beta_{6} + \beta_{4} + 1) q^{2} + ( - \beta_{7} + \beta_1) q^{3} + 2 \beta_{4} q^{4} + ( - \beta_{7} - \beta_{3}) q^{6} + (\beta_{14} - \beta_{9} + \cdots + \beta_{3}) q^{7}+ \cdots + ( - 4 \beta_{13} - 5 \beta_{11} + \cdots - 11 \beta_{3}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 8 q^{2} - 2 q^{3} - 8 q^{6} - 12 q^{7} + 32 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 8 q^{2} - 2 q^{3} - 8 q^{6} - 12 q^{7} + 32 q^{8} + 40 q^{11} - 4 q^{12} - 16 q^{13} + 32 q^{16} - 46 q^{17} + 52 q^{18} - 20 q^{21} + 80 q^{22} - 54 q^{23} - 16 q^{26} + 52 q^{27} + 36 q^{28} - 208 q^{31} - 32 q^{32} + 22 q^{33} + 208 q^{36} + 38 q^{37} - 36 q^{38} - 72 q^{41} - 184 q^{42} - 144 q^{43} + 108 q^{46} - 46 q^{47} - 16 q^{48} - 136 q^{51} + 16 q^{52} - 30 q^{53} - 48 q^{56} + 492 q^{57} - 132 q^{58} - 120 q^{61} - 416 q^{62} + 292 q^{63} - 44 q^{66} + 74 q^{67} + 92 q^{68} + 16 q^{71} + 104 q^{72} + 54 q^{73} - 144 q^{76} - 570 q^{77} - 168 q^{78} + 244 q^{81} - 36 q^{82} - 64 q^{83} - 144 q^{86} + 236 q^{87} + 80 q^{88} + 336 q^{91} + 216 q^{92} - 142 q^{93} - 16 q^{96} - 136 q^{97} + 268 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 2 x^{15} + 2 x^{14} - 8 x^{13} - 722 x^{12} + 1354 x^{11} - 1232 x^{10} + 9306 x^{9} + \cdots + 52200625 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 86\!\cdots\!42 \nu^{15} + \cdots + 16\!\cdots\!25 ) / 46\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 48\!\cdots\!09 \nu^{15} + \cdots - 30\!\cdots\!75 ) / 79\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 57\!\cdots\!79 \nu^{15} + \cdots + 49\!\cdots\!50 ) / 79\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 38\!\cdots\!71 \nu^{15} + \cdots - 45\!\cdots\!95 ) / 49\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 57\!\cdots\!33 \nu^{15} + \cdots - 24\!\cdots\!75 ) / 62\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 37\!\cdots\!34 \nu^{15} + \cdots - 38\!\cdots\!50 ) / 37\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 62\!\cdots\!22 \nu^{15} + \cdots + 40\!\cdots\!00 ) / 31\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 19\!\cdots\!91 \nu^{15} + \cdots - 12\!\cdots\!75 ) / 67\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 24\!\cdots\!69 \nu^{15} + \cdots - 67\!\cdots\!50 ) / 79\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 29\!\cdots\!11 \nu^{15} + \cdots + 31\!\cdots\!75 ) / 67\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 70\!\cdots\!61 \nu^{15} + \cdots - 60\!\cdots\!50 ) / 79\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 61\!\cdots\!41 \nu^{15} + \cdots - 25\!\cdots\!75 ) / 67\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 90\!\cdots\!91 \nu^{15} + \cdots - 26\!\cdots\!50 ) / 67\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 12\!\cdots\!46 \nu^{15} + \cdots + 16\!\cdots\!75 ) / 61\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{14} - 2\beta_{12} - \beta_{11} + \beta_{10} - \beta_{7} - 15\beta_{4} + \beta_{2} + \beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{13} - 4\beta_{9} - 5\beta_{6} - \beta_{5} + 25\beta_{3} - 5 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 58 \beta_{15} + 34 \beta_{14} - 26 \beta_{10} - 34 \beta_{9} - 355 \beta_{8} + 26 \beta_{3} + 34 \beta_{2} + \cdots + 355 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 30 \beta_{15} + 30 \beta_{12} - 130 \beta_{11} + 230 \beta_{8} - 659 \beta_{7} + 30 \beta_{5} + \cdots + 659 \beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -1568\beta_{13} - 949\beta_{11} - 949\beta_{9} - 629\beta_{7} + 9205\beta_{6} + 629\beta_{3} \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 929 \beta_{15} + 3486 \beta_{14} + 929 \beta_{13} - 929 \beta_{12} - 17395 \beta_{10} - 3486 \beta_{9} + \cdots - 7565 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 41992 \beta_{15} + 25296 \beta_{14} - 25296 \beta_{11} - 14944 \beta_{10} - 244185 \beta_{8} + \cdots + 14944 \beta_1 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 28800\beta_{13} - 89080\beta_{11} - 459241\beta_{7} - 230160\beta_{6} + 28800\beta_{5} + 230160 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 666201 \beta_{14} - 1122762 \beta_{13} + 1122762 \beta_{12} - 349641 \beta_{10} - 666201 \beta_{9} + \cdots - 349641 \beta_1 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 880041 \beta_{15} + 2241324 \beta_{14} - 880041 \beta_{12} - 12131185 \beta_{10} + \cdots - 6821565 \beta_{4} \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( -17493874\beta_{11} + 17493874\beta_{9} - 8032186\beta_{7} - 30023858\beta_{5} - 8032186\beta_{3} - 174597675 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 26489470 \beta_{15} + 26489470 \beta_{13} - 26489470 \beta_{12} - 199611350 \beta_{8} - 199611350 \beta_{6} + \cdots + 199611350 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 459206749 \beta_{14} + 803344408 \beta_{12} + 459206749 \beta_{11} - 180126989 \beta_{10} + \cdots - 180126989 \beta_1 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 787228089 \beta_{13} + 1393736566 \beta_{9} + 5791815605 \beta_{6} + 787228089 \beta_{5} + \cdots + 5791815605 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/350\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(127\)
\(\chi(n)\) \(-1 + \beta_{8}\) \(\beta_{6}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
93.1
−5.05060 + 1.35330i
−1.13099 + 0.303047i
2.63893 − 0.707100i
4.90868 − 1.31528i
−1.31528 − 4.90868i
−0.707100 − 2.63893i
0.303047 + 1.13099i
1.35330 + 5.05060i
−1.31528 + 4.90868i
−0.707100 + 2.63893i
0.303047 − 1.13099i
1.35330 − 5.05060i
−5.05060 − 1.35330i
−1.13099 − 0.303047i
2.63893 + 0.707100i
4.90868 + 1.31528i
−0.366025 − 1.36603i −1.35330 + 5.05060i −1.73205 + 1.00000i 0 7.39459 −6.88437 + 1.26709i 2.00000 + 2.00000i −15.8829 − 9.16999i 0
93.2 −0.366025 − 1.36603i −0.303047 + 1.13099i −1.73205 + 1.00000i 0 1.65588 −4.91991 − 4.97940i 2.00000 + 2.00000i 6.60693 + 3.81451i 0
93.3 −0.366025 − 1.36603i 0.707100 − 2.63893i −1.73205 + 1.00000i 0 −3.86367 1.73096 + 6.78261i 2.00000 + 2.00000i 1.33025 + 0.768021i 0
93.4 −0.366025 − 1.36603i 1.31528 − 4.90868i −1.73205 + 1.00000i 0 −7.18681 0.145113 − 6.99850i 2.00000 + 2.00000i −14.5710 − 8.41254i 0
107.1 1.36603 − 0.366025i −4.90868 − 1.31528i 1.73205 − 1.00000i 0 −7.18681 6.99850 + 0.145113i 2.00000 − 2.00000i 14.5710 + 8.41254i 0
107.2 1.36603 − 0.366025i −2.63893 − 0.707100i 1.73205 − 1.00000i 0 −3.86367 −6.78261 + 1.73096i 2.00000 − 2.00000i −1.33025 − 0.768021i 0
107.3 1.36603 − 0.366025i 1.13099 + 0.303047i 1.73205 − 1.00000i 0 1.65588 4.97940 − 4.91991i 2.00000 − 2.00000i −6.60693 − 3.81451i 0
107.4 1.36603 − 0.366025i 5.05060 + 1.35330i 1.73205 − 1.00000i 0 7.39459 −1.26709 − 6.88437i 2.00000 − 2.00000i 15.8829 + 9.16999i 0
193.1 1.36603 + 0.366025i −4.90868 + 1.31528i 1.73205 + 1.00000i 0 −7.18681 6.99850 − 0.145113i 2.00000 + 2.00000i 14.5710 − 8.41254i 0
193.2 1.36603 + 0.366025i −2.63893 + 0.707100i 1.73205 + 1.00000i 0 −3.86367 −6.78261 − 1.73096i 2.00000 + 2.00000i −1.33025 + 0.768021i 0
193.3 1.36603 + 0.366025i 1.13099 − 0.303047i 1.73205 + 1.00000i 0 1.65588 4.97940 + 4.91991i 2.00000 + 2.00000i −6.60693 + 3.81451i 0
193.4 1.36603 + 0.366025i 5.05060 − 1.35330i 1.73205 + 1.00000i 0 7.39459 −1.26709 + 6.88437i 2.00000 + 2.00000i 15.8829 − 9.16999i 0
207.1 −0.366025 + 1.36603i −1.35330 − 5.05060i −1.73205 − 1.00000i 0 7.39459 −6.88437 − 1.26709i 2.00000 − 2.00000i −15.8829 + 9.16999i 0
207.2 −0.366025 + 1.36603i −0.303047 − 1.13099i −1.73205 − 1.00000i 0 1.65588 −4.91991 + 4.97940i 2.00000 − 2.00000i 6.60693 − 3.81451i 0
207.3 −0.366025 + 1.36603i 0.707100 + 2.63893i −1.73205 − 1.00000i 0 −3.86367 1.73096 − 6.78261i 2.00000 − 2.00000i 1.33025 − 0.768021i 0
207.4 −0.366025 + 1.36603i 1.31528 + 4.90868i −1.73205 − 1.00000i 0 −7.18681 0.145113 + 6.99850i 2.00000 − 2.00000i −14.5710 + 8.41254i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 93.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
7.c even 3 1 inner
35.l odd 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 350.3.p.e 16
5.b even 2 1 70.3.l.c ✓ 16
5.c odd 4 1 70.3.l.c ✓ 16
5.c odd 4 1 inner 350.3.p.e 16
7.c even 3 1 inner 350.3.p.e 16
35.i odd 6 1 490.3.f.p 8
35.j even 6 1 70.3.l.c ✓ 16
35.j even 6 1 490.3.f.o 8
35.k even 12 1 490.3.f.p 8
35.l odd 12 1 70.3.l.c ✓ 16
35.l odd 12 1 inner 350.3.p.e 16
35.l odd 12 1 490.3.f.o 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
70.3.l.c ✓ 16 5.b even 2 1
70.3.l.c ✓ 16 5.c odd 4 1
70.3.l.c ✓ 16 35.j even 6 1
70.3.l.c ✓ 16 35.l odd 12 1
350.3.p.e 16 1.a even 1 1 trivial
350.3.p.e 16 5.c odd 4 1 inner
350.3.p.e 16 7.c even 3 1 inner
350.3.p.e 16 35.l odd 12 1 inner
490.3.f.o 8 35.j even 6 1
490.3.f.o 8 35.l odd 12 1
490.3.f.p 8 35.i odd 6 1
490.3.f.p 8 35.k even 12 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(350, [\chi])\):

\( T_{3}^{16} + 2 T_{3}^{15} + 2 T_{3}^{14} + 8 T_{3}^{13} - 722 T_{3}^{12} - 1354 T_{3}^{11} + \cdots + 52200625 \) Copy content Toggle raw display
\( T_{11}^{8} - 20 T_{11}^{7} + 537 T_{11}^{6} - 2972 T_{11}^{5} + 78459 T_{11}^{4} - 494072 T_{11}^{3} + \cdots + 6604900 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - 2 T^{3} + 2 T^{2} + \cdots + 4)^{4} \) Copy content Toggle raw display
$3$ \( T^{16} + 2 T^{15} + \cdots + 52200625 \) Copy content Toggle raw display
$5$ \( T^{16} \) Copy content Toggle raw display
$7$ \( T^{16} + \cdots + 33232930569601 \) Copy content Toggle raw display
$11$ \( (T^{8} - 20 T^{7} + \cdots + 6604900)^{2} \) Copy content Toggle raw display
$13$ \( (T^{8} + 8 T^{7} + \cdots + 409600)^{2} \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 12\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( T^{16} + \cdots + 70\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 18\!\cdots\!81 \) Copy content Toggle raw display
$29$ \( (T^{8} + 1898 T^{6} + \cdots + 10488217744)^{2} \) Copy content Toggle raw display
$31$ \( (T^{8} + 104 T^{7} + \cdots + 35705881600)^{2} \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 60\!\cdots\!16 \) Copy content Toggle raw display
$41$ \( (T^{4} + 18 T^{3} + \cdots - 367408)^{4} \) Copy content Toggle raw display
$43$ \( (T^{8} + 72 T^{7} + \cdots + 321477660100)^{2} \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 12\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 13\!\cdots\!96 \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 23\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( (T^{8} + 60 T^{7} + \cdots + 389070300025)^{2} \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 28\!\cdots\!25 \) Copy content Toggle raw display
$71$ \( (T^{4} - 4 T^{3} + \cdots + 2549080)^{4} \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 60\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 28\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( (T^{8} + \cdots + 468971508062500)^{2} \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 25\!\cdots\!25 \) Copy content Toggle raw display
$97$ \( (T^{8} + \cdots + 25\!\cdots\!00)^{2} \) Copy content Toggle raw display
show more
show less