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Properties

Label 350.2.h.c
Level $350$
Weight $2$
Character orbit 350.h
Analytic conductor $2.795$
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] = [350,2,Mod(71,350)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(350, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([6, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("350.71");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 350 = 2 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 350.h (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: \(2.79476407074\)
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: \( x^{16} + 25x^{14} + 241x^{12} + 1145x^{10} + 2841x^{8} + 3600x^{6} + 2156x^{4} + 480x^{2} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 5 \)
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.

\(f(q)\) \(=\) \( q + \beta_{6} q^{2} - \beta_{10} q^{3} - \beta_1 q^{4} + (\beta_{14} - \beta_{12} + \cdots - \beta_1) q^{5}+ \cdots + (\beta_{15} - 2 \beta_{11} + \cdots + 2 \beta_{2}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{6} q^{2} - \beta_{10} q^{3} - \beta_1 q^{4} + (\beta_{14} - \beta_{12} + \cdots - \beta_1) q^{5}+ \cdots + (2 \beta_{14} + \beta_{13} - \beta_{12} + \cdots - 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 4 q^{2} + q^{3} - 4 q^{4} - 4 q^{5} + q^{6} - 16 q^{7} - 4 q^{8} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 4 q^{2} + q^{3} - 4 q^{4} - 4 q^{5} + q^{6} - 16 q^{7} - 4 q^{8} - q^{9} + q^{10} + 7 q^{11} - 4 q^{12} + 9 q^{13} + 4 q^{14} - 4 q^{16} - 2 q^{17} + 14 q^{18} - 24 q^{19} - 9 q^{20} - q^{21} - 8 q^{22} - 5 q^{23} + 6 q^{24} - 6 q^{25} - 6 q^{26} - 32 q^{27} + 4 q^{28} + 20 q^{29} + 15 q^{30} + 7 q^{31} + 16 q^{32} + 15 q^{33} + 3 q^{34} + 4 q^{35} - q^{36} - 6 q^{37} + 6 q^{38} + 34 q^{39} + 11 q^{40} + 9 q^{41} - q^{42} - 22 q^{43} - 8 q^{44} - 8 q^{45} - 40 q^{47} - 4 q^{48} + 16 q^{49} - q^{50} + 14 q^{51} + 9 q^{52} - 24 q^{53} + 23 q^{54} - 26 q^{55} + 4 q^{56} + 52 q^{57} + 20 q^{58} - 17 q^{59} - 5 q^{60} + 2 q^{61} - 23 q^{62} + q^{63} - 4 q^{64} - 16 q^{65} - 10 q^{66} - 14 q^{67} - 2 q^{68} - 35 q^{69} - q^{70} + 7 q^{71} - 6 q^{72} + 5 q^{73} - 36 q^{74} + 35 q^{75} + 36 q^{76} - 7 q^{77} - 46 q^{78} + 20 q^{79} + q^{80} + 49 q^{81} + 44 q^{82} + 17 q^{83} + 4 q^{84} - 13 q^{85} + 8 q^{86} - 66 q^{87} + 7 q^{88} + 27 q^{89} + 37 q^{90} - 9 q^{91} - 34 q^{93} - 40 q^{94} - 20 q^{95} + q^{96} + 6 q^{97} - 4 q^{98} - 16 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 25x^{14} + 241x^{12} + 1145x^{10} + 2841x^{8} + 3600x^{6} + 2156x^{4} + 480x^{2} + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 21 \nu^{15} + 32 \nu^{14} + 487 \nu^{13} + 784 \nu^{12} + 4215 \nu^{11} + 7248 \nu^{10} + 17143 \nu^{9} + \cdots + 448 ) / 128 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 9\nu^{14} + 223\nu^{12} + 2087\nu^{10} + 9191\nu^{8} + 19079\nu^{6} + 16022\nu^{4} + 4132\nu^{2} + 192 ) / 32 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 21 \nu^{15} - 32 \nu^{14} + 487 \nu^{13} - 784 \nu^{12} + 4215 \nu^{11} - 7248 \nu^{10} + 17143 \nu^{9} + \cdots - 448 ) / 128 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 10 \nu^{15} - 31 \nu^{14} - 228 \nu^{13} - 721 \nu^{12} - 1928 \nu^{11} - 6249 \nu^{10} - 7608 \nu^{9} + \cdots - 224 ) / 64 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 10 \nu^{15} + 31 \nu^{14} - 228 \nu^{13} + 721 \nu^{12} - 1928 \nu^{11} + 6249 \nu^{10} - 7608 \nu^{9} + \cdots + 224 ) / 64 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 51 \nu^{15} + 32 \nu^{14} + 1219 \nu^{13} + 784 \nu^{12} + 10943 \nu^{11} + 7248 \nu^{10} + 46191 \nu^{9} + \cdots + 384 ) / 128 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 19 \nu^{15} + 60 \nu^{14} + 467 \nu^{13} + 1484 \nu^{12} + 4335 \nu^{11} + 13884 \nu^{10} + \cdots + 1248 ) / 128 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 19 \nu^{15} - 60 \nu^{14} + 467 \nu^{13} - 1484 \nu^{12} + 4335 \nu^{11} - 13884 \nu^{10} + \cdots - 1248 ) / 128 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 3 \nu^{15} + 61 \nu^{14} - 53 \nu^{13} + 1447 \nu^{12} - 253 \nu^{11} + 12871 \nu^{10} + 163 \nu^{9} + \cdots + 944 ) / 64 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 65 \nu^{15} + 98 \nu^{14} + 1537 \nu^{13} + 2330 \nu^{12} + 13589 \nu^{11} + 20786 \nu^{10} + \cdots + 1456 ) / 128 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 65 \nu^{15} - 98 \nu^{14} + 1537 \nu^{13} - 2330 \nu^{12} + 13589 \nu^{11} - 20786 \nu^{10} + \cdots - 1456 ) / 128 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 51 \nu^{15} - 118 \nu^{14} + 1239 \nu^{13} - 2734 \nu^{12} + 11371 \nu^{11} - 23574 \nu^{10} + \cdots - 1232 ) / 128 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 64 \nu^{15} - 3 \nu^{14} + 1531 \nu^{13} - 59 \nu^{12} + 13757 \nu^{11} - 391 \nu^{10} + 58117 \nu^{9} + \cdots + 104 ) / 64 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 39 \nu^{15} - 61 \nu^{14} - 906 \nu^{13} - 1447 \nu^{12} - 7832 \nu^{11} - 12871 \nu^{10} + \cdots - 912 ) / 64 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 67 \nu^{15} - 58 \nu^{14} - 1590 \nu^{13} - 1388 \nu^{12} - 14148 \nu^{11} - 12480 \nu^{10} + \cdots - 920 ) / 64 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( 2 \beta_{15} + \beta_{14} + \beta_{13} + \beta_{12} + \beta_{11} + 3 \beta_{10} + 2 \beta_{9} - 2 \beta_{8} + \cdots - 1 ) / 5 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{13} - \beta_{12} + \beta_{11} - \beta_{10} + \beta_{9} - \beta_{8} - \beta_{6} - \beta_{5} - \beta_{3} - \beta_{2} - 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 2 \beta_{15} - 11 \beta_{14} - 6 \beta_{13} - 6 \beta_{12} - 16 \beta_{11} - 18 \beta_{10} + \cdots + 21 ) / 5 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2 \beta_{14} - 8 \beta_{13} + 8 \beta_{12} - 10 \beta_{11} + 10 \beta_{10} - 10 \beta_{9} + 7 \beta_{8} + \cdots + 29 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 38 \beta_{15} + 96 \beta_{14} + 41 \beta_{13} + 41 \beta_{12} + 151 \beta_{11} + 113 \beta_{10} + \cdots - 201 ) / 5 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 23 \beta_{14} + 66 \beta_{13} - 66 \beta_{12} + 86 \beta_{11} - 86 \beta_{10} + 89 \beta_{9} + \cdots - 231 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 598 \beta_{15} - 846 \beta_{14} - 281 \beta_{13} - 281 \beta_{12} - 1351 \beta_{11} - 753 \beta_{10} + \cdots + 1756 ) / 5 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 205 \beta_{14} - 564 \beta_{13} + 564 \beta_{12} - 720 \beta_{11} + 720 \beta_{10} - 769 \beta_{9} + \cdots + 1911 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 6618 \beta_{15} + 7526 \beta_{14} + 1926 \beta_{13} + 1926 \beta_{12} + 11936 \beta_{11} + \cdots - 15011 ) / 5 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 1702 \beta_{14} + 4893 \beta_{13} - 4893 \beta_{12} + 6019 \beta_{11} - 6019 \beta_{10} + \cdots - 16113 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 65408 \beta_{15} - 66901 \beta_{14} - 13346 \beta_{13} - 13346 \beta_{12} - 104836 \beta_{11} + \cdots + 127841 ) / 5 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 13890 \beta_{14} - 42651 \beta_{13} + 42651 \beta_{12} - 50591 \beta_{11} + 50591 \beta_{10} + \cdots + 137304 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 613478 \beta_{15} + 592041 \beta_{14} + 94356 \beta_{13} + 94356 \beta_{12} + 917336 \beta_{11} + \cdots - 1090801 ) / 5 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 113642 \beta_{14} + 372004 \beta_{13} - 372004 \beta_{12} + 428106 \beta_{11} - 428106 \beta_{10} + \cdots - 1177335 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 5592498 \beta_{15} - 5214016 \beta_{14} - 684591 \beta_{13} - 684591 \beta_{12} - 8006061 \beta_{11} + \cdots + 9337971 ) / 5 \) Copy content Toggle raw display

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

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
71.1
2.46620i
− 1.07895i
− 1.93398i
− 0.628834i
− 0.200203i
2.94531i
− 1.92944i
1.08645i
0.200203i
− 2.94531i
1.92944i
− 1.08645i
− 2.46620i
1.07895i
1.93398i
0.628834i
0.309017 − 0.951057i −1.78422 − 1.29631i −0.809017 − 0.587785i −0.298278 − 2.21608i −1.78422 + 1.29631i −1.00000 −0.809017 + 0.587785i 0.575968 + 1.77265i −2.19979 − 0.401128i
71.2 0.309017 − 0.951057i −1.46563 − 1.06484i −0.809017 − 0.587785i −1.01402 + 1.99293i −1.46563 + 1.06484i −1.00000 −0.809017 + 0.587785i 0.0871311 + 0.268162i 1.58203 + 1.58024i
71.3 0.309017 − 0.951057i 0.689113 + 0.500670i −0.809017 − 0.587785i −1.78357 − 1.34866i 0.689113 − 0.500670i −1.00000 −0.809017 + 0.587785i −0.702844 − 2.16313i −1.83381 + 1.27951i
71.4 0.309017 − 0.951057i 2.25172 + 1.63597i −0.809017 − 0.587785i 2.09587 − 0.779319i 2.25172 − 1.63597i −1.00000 −0.809017 + 0.587785i 1.46680 + 4.51433i −0.0935175 − 2.23411i
141.1 −0.809017 − 0.587785i −0.491708 − 1.51332i 0.309017 + 0.951057i 2.15788 + 0.586125i −0.491708 + 1.51332i −1.00000 0.309017 − 0.951057i 0.378688 − 0.275133i −1.40125 − 1.74256i
141.2 −0.809017 − 0.587785i −0.0305168 − 0.0939212i 0.309017 + 0.951057i −1.05684 + 1.97056i −0.0305168 + 0.0939212i −1.00000 0.309017 − 0.951057i 2.41916 − 1.75762i 2.01326 − 0.973023i
141.3 −0.809017 − 0.587785i 0.315945 + 0.972380i 0.309017 + 0.951057i −2.00668 − 0.986533i 0.315945 − 0.972380i −1.00000 0.309017 − 0.951057i 1.58135 − 1.14892i 1.04357 + 1.97762i
141.4 −0.809017 − 0.587785i 1.01530 + 3.12476i 0.309017 + 0.951057i −0.0943702 + 2.23408i 1.01530 − 3.12476i −1.00000 0.309017 − 0.951057i −6.30625 + 4.58176i 1.38950 − 1.75194i
211.1 −0.809017 + 0.587785i −0.491708 + 1.51332i 0.309017 − 0.951057i 2.15788 − 0.586125i −0.491708 − 1.51332i −1.00000 0.309017 + 0.951057i 0.378688 + 0.275133i −1.40125 + 1.74256i
211.2 −0.809017 + 0.587785i −0.0305168 + 0.0939212i 0.309017 − 0.951057i −1.05684 − 1.97056i −0.0305168 − 0.0939212i −1.00000 0.309017 + 0.951057i 2.41916 + 1.75762i 2.01326 + 0.973023i
211.3 −0.809017 + 0.587785i 0.315945 − 0.972380i 0.309017 − 0.951057i −2.00668 + 0.986533i 0.315945 + 0.972380i −1.00000 0.309017 + 0.951057i 1.58135 + 1.14892i 1.04357 − 1.97762i
211.4 −0.809017 + 0.587785i 1.01530 − 3.12476i 0.309017 − 0.951057i −0.0943702 − 2.23408i 1.01530 + 3.12476i −1.00000 0.309017 + 0.951057i −6.30625 − 4.58176i 1.38950 + 1.75194i
281.1 0.309017 + 0.951057i −1.78422 + 1.29631i −0.809017 + 0.587785i −0.298278 + 2.21608i −1.78422 − 1.29631i −1.00000 −0.809017 − 0.587785i 0.575968 − 1.77265i −2.19979 + 0.401128i
281.2 0.309017 + 0.951057i −1.46563 + 1.06484i −0.809017 + 0.587785i −1.01402 − 1.99293i −1.46563 − 1.06484i −1.00000 −0.809017 − 0.587785i 0.0871311 − 0.268162i 1.58203 − 1.58024i
281.3 0.309017 + 0.951057i 0.689113 − 0.500670i −0.809017 + 0.587785i −1.78357 + 1.34866i 0.689113 + 0.500670i −1.00000 −0.809017 − 0.587785i −0.702844 + 2.16313i −1.83381 − 1.27951i
281.4 0.309017 + 0.951057i 2.25172 − 1.63597i −0.809017 + 0.587785i 2.09587 + 0.779319i 2.25172 + 1.63597i −1.00000 −0.809017 − 0.587785i 1.46680 − 4.51433i −0.0935175 + 2.23411i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 71.4
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 350.2.h.c ✓ 16
25.d even 5 1 inner 350.2.h.c ✓ 16
25.d even 5 1 8750.2.a.u 8
25.e even 10 1 8750.2.a.s 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
350.2.h.c ✓ 16 1.a even 1 1 trivial
350.2.h.c ✓ 16 25.d even 5 1 inner
8750.2.a.s 8 25.e even 10 1
8750.2.a.u 8 25.d even 5 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{16} - T_{3}^{15} + 7 T_{3}^{14} + 12 T_{3}^{13} + 25 T_{3}^{12} - 22 T_{3}^{11} + 387 T_{3}^{10} + \cdots + 25 \) acting on \(S_{2}^{\mathrm{new}}(350, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} + T^{3} + T^{2} + \cdots + 1)^{4} \) Copy content Toggle raw display
$3$ \( T^{16} - T^{15} + \cdots + 25 \) Copy content Toggle raw display
$5$ \( T^{16} + 4 T^{15} + \cdots + 390625 \) Copy content Toggle raw display
$7$ \( (T + 1)^{16} \) Copy content Toggle raw display
$11$ \( T^{16} - 7 T^{15} + \cdots + 4000000 \) Copy content Toggle raw display
$13$ \( T^{16} + \cdots + 312193561 \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 2943713536 \) Copy content Toggle raw display
$19$ \( T^{16} + 24 T^{15} + \cdots + 22278400 \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 346890625 \) Copy content Toggle raw display
$29$ \( T^{16} - 20 T^{15} + \cdots + 4393216 \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots + 19461366016 \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 205062400 \) Copy content Toggle raw display
$41$ \( T^{16} - 9 T^{15} + \cdots + 63744256 \) Copy content Toggle raw display
$43$ \( (T^{8} + 11 T^{7} + \cdots - 704)^{2} \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 15862186045696 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 5147201536 \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 325982760601 \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 313864132196521 \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 161809889536 \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 210492852025 \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 11\!\cdots\!76 \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 7568826001 \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 29231082841 \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 377585670400 \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 726040326400 \) Copy content Toggle raw display
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