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Properties

Label 350.2.g.b
Level $350$
Weight $2$
Character orbit 350.g
Analytic conductor $2.795$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [350,2,Mod(293,350)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(350, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([3, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("350.293");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 350 = 2 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 350.g (of order \(4\), 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: \(2.79476407074\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 4 x^{15} + 22 x^{14} - 52 x^{13} + 72 x^{12} - 32 x^{11} + 148 x^{10} + 268 x^{9} - 461 x^{8} + \cdots + 81 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{10} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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.

\(f(q)\) \(=\) \( q - \beta_{2} q^{2} + \beta_{3} q^{3} + \beta_{10} q^{4} + \beta_{15} q^{6} + (\beta_{12} + \beta_{6} + 2 \beta_{2}) q^{7} - \beta_{8} q^{8} + (4 \beta_{10} - 2 \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{2} q^{2} + \beta_{3} q^{3} + \beta_{10} q^{4} + \beta_{15} q^{6} + (\beta_{12} + \beta_{6} + 2 \beta_{2}) q^{7} - \beta_{8} q^{8} + (4 \beta_{10} - 2 \beta_1) q^{9} + \beta_{13} q^{11} - \beta_{14} q^{12} + ( - \beta_{11} + \beta_{8} + 2 \beta_{5}) q^{13} + ( - \beta_{10} + \beta_{9}) q^{14} - q^{16} + (\beta_{14} + 2 \beta_{12} + \cdots + \beta_{2}) q^{17}+ \cdots + ( - 6 \beta_{10} + 4 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 16 q^{16} - 8 q^{21} - 64 q^{36} + 48 q^{46} + 96 q^{51} + 24 q^{56} + 48 q^{71} - 112 q^{81} - 128 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 4 x^{15} + 22 x^{14} - 52 x^{13} + 72 x^{12} - 32 x^{11} + 148 x^{10} + 268 x^{9} - 461 x^{8} + \cdots + 81 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 38872 \nu^{15} - 2381588 \nu^{14} + 12272984 \nu^{13} - 69438992 \nu^{12} + 212969856 \nu^{11} + \cdots - 2919969 ) / 345215871 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 164256682774 \nu^{15} + 204536771036 \nu^{14} + 325719164833 \nu^{13} + \cdots - 284144487711858 ) / 549627118237197 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 43638140738 \nu^{15} - 108501139208 \nu^{14} + 1089181275053 \nu^{13} + \cdots - 82638681727491 ) / 78097150848717 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 163615281827 \nu^{15} - 1924117603277 \nu^{14} + 9820060318151 \nu^{13} + \cdots - 146731242111066 ) / 120872353803201 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 70174138296448 \nu^{15} + 577544903020459 \nu^{14} + \cdots - 46\!\cdots\!79 ) / 29\!\cdots\!41 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 2609331078951 \nu^{15} - 16870152732110 \nu^{14} + 88009506904826 \nu^{13} + \cdots + 743111712710100 ) / 746929160681319 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 54279338604888 \nu^{15} + 236515227234986 \nu^{14} + \cdots - 24\!\cdots\!22 ) / 97\!\cdots\!47 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 3142423666847 \nu^{15} - 16131464631308 \nu^{14} + 87273661483760 \nu^{13} + \cdots + 421448942618136 ) / 549627118237197 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 235914283929869 \nu^{15} + \cdots + 47\!\cdots\!20 ) / 29\!\cdots\!41 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 1220621110 \nu^{15} + 5538431044 \nu^{14} - 29916467584 \nu^{13} + 79884634924 \nu^{12} + \cdots - 104638235589 ) / 113348549853 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 9183033380386 \nu^{15} - 42272449001952 \nu^{14} + 226004099288307 \nu^{13} + \cdots + 614103236946774 ) / 746929160681319 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 369366743509877 \nu^{15} + \cdots - 15\!\cdots\!50 ) / 29\!\cdots\!41 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 126187710 \nu^{15} + 593368666 \nu^{14} - 3184365676 \nu^{13} + 8758095976 \nu^{12} + \cdots + 2771443701 ) / 9819835389 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 1457028738787 \nu^{15} + 7164351711370 \nu^{14} - 38812705709614 \nu^{13} + \cdots - 181099278294957 ) / 78097150848717 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 3123343272944 \nu^{15} - 14410051312652 \nu^{14} + 78052801664720 \nu^{13} + \cdots + 277416603956664 ) / 120872353803201 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( - \beta_{15} - \beta_{14} - \beta_{13} + \beta_{11} - 2 \beta_{8} + \beta_{7} - \beta_{6} + \cdots + \beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{14} + 2 \beta_{13} + 2 \beta_{11} - \beta_{9} + 3 \beta_{8} - \beta_{7} - 4 \beta_{6} - 2 \beta_{5} + \cdots - 4 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 9 \beta_{15} + 9 \beta_{14} + 3 \beta_{13} + 3 \beta_{12} + 4 \beta_{11} + 16 \beta_{10} - \beta_{9} + \cdots - 4 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 15 \beta_{15} - 4 \beta_{14} - 36 \beta_{13} - 12 \beta_{12} - 26 \beta_{11} + 51 \beta_{10} + 24 \beta_{9} + \cdots + 48 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 185 \beta_{15} - 221 \beta_{14} - 259 \beta_{13} - 194 \beta_{12} - 279 \beta_{11} - 282 \beta_{10} + \cdots + 142 ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 315 \beta_{15} + 245 \beta_{13} + 130 \beta_{11} - 798 \beta_{10} - 182 \beta_{9} - 130 \beta_{8} + \cdots - 798 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 2007 \beta_{15} + 3365 \beta_{14} + 4053 \beta_{13} + 2688 \beta_{12} + 4189 \beta_{11} + 2782 \beta_{10} + \cdots - 4640 ) / 4 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 5644 \beta_{15} + 1584 \beta_{14} - 1530 \beta_{13} + 1632 \beta_{12} + 12 \beta_{11} + 12546 \beta_{10} + \cdots + 9503 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 10188 \beta_{15} - 24621 \beta_{14} - 31146 \beta_{13} - 16820 \beta_{12} - 31146 \beta_{11} + \cdots + 47531 ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 98523 \beta_{15} - 51649 \beta_{14} - 2583 \beta_{13} - 36076 \beta_{12} - 25868 \beta_{11} + \cdots - 86674 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 109009 \beta_{15} + 637253 \beta_{14} + 981177 \beta_{13} + 464092 \beta_{12} + 936605 \beta_{11} + \cdots - 1581580 ) / 4 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 787347 \beta_{15} + 556710 \beta_{14} + 282051 \beta_{13} + 398860 \beta_{12} + 450940 \beta_{11} + \cdots + 282051 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 1632245 \beta_{15} - 7435165 \beta_{14} - 13915707 \beta_{13} - 5572470 \beta_{12} - 12497147 \beta_{11} + \cdots + 25389014 ) / 4 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 23172484 \beta_{15} - 20744243 \beta_{14} - 15986471 \beta_{13} - 15336412 \beta_{12} + \cdots + 4024043 ) / 2 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 35996985 \beta_{15} + 35996985 \beta_{14} + 89333392 \beta_{13} + 26174563 \beta_{12} + \cdots - 191583097 ) / 2 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(101\) \(127\)
\(\chi(n)\) \(-1\) \(\beta_{10}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
293.1
0.510680 + 3.87900i
2.11123 − 1.62000i
−0.645299 + 0.495156i
−0.269499 − 2.04705i
−0.847914 + 0.111630i
0.205100 + 0.267292i
−0.671026 − 0.874498i
1.60673 − 0.211530i
0.510680 − 3.87900i
2.11123 + 1.62000i
−0.645299 − 0.495156i
−0.269499 + 2.04705i
−0.847914 − 0.111630i
0.205100 − 0.267292i
−0.671026 + 0.874498i
1.60673 + 0.211530i
−0.707107 + 0.707107i −2.28737 + 2.28737i − 1.00000i 0 − 3.23483i 0.835798 − 2.51027i 0.707107 + 0.707107i − 7.46410i 0
293.2 −0.707107 + 0.707107i −1.32964 + 1.32964i − 1.00000i 0 − 1.88040i 2.26461 + 1.36804i 0.707107 + 0.707107i − 0.535898i 0
293.3 −0.707107 + 0.707107i 1.32964 − 1.32964i − 1.00000i 0 1.88040i −1.36804 − 2.26461i 0.707107 + 0.707107i − 0.535898i 0
293.4 −0.707107 + 0.707107i 2.28737 − 2.28737i − 1.00000i 0 3.23483i 2.51027 − 0.835798i 0.707107 + 0.707107i − 7.46410i 0
293.5 0.707107 − 0.707107i −2.28737 + 2.28737i − 1.00000i 0 3.23483i −2.51027 + 0.835798i −0.707107 − 0.707107i − 7.46410i 0
293.6 0.707107 − 0.707107i −1.32964 + 1.32964i − 1.00000i 0 1.88040i 1.36804 + 2.26461i −0.707107 − 0.707107i − 0.535898i 0
293.7 0.707107 − 0.707107i 1.32964 − 1.32964i − 1.00000i 0 − 1.88040i −2.26461 − 1.36804i −0.707107 − 0.707107i − 0.535898i 0
293.8 0.707107 − 0.707107i 2.28737 − 2.28737i − 1.00000i 0 − 3.23483i −0.835798 + 2.51027i −0.707107 − 0.707107i − 7.46410i 0
307.1 −0.707107 − 0.707107i −2.28737 − 2.28737i 1.00000i 0 3.23483i 0.835798 + 2.51027i 0.707107 − 0.707107i 7.46410i 0
307.2 −0.707107 − 0.707107i −1.32964 − 1.32964i 1.00000i 0 1.88040i 2.26461 − 1.36804i 0.707107 − 0.707107i 0.535898i 0
307.3 −0.707107 − 0.707107i 1.32964 + 1.32964i 1.00000i 0 − 1.88040i −1.36804 + 2.26461i 0.707107 − 0.707107i 0.535898i 0
307.4 −0.707107 − 0.707107i 2.28737 + 2.28737i 1.00000i 0 − 3.23483i 2.51027 + 0.835798i 0.707107 − 0.707107i 7.46410i 0
307.5 0.707107 + 0.707107i −2.28737 − 2.28737i 1.00000i 0 − 3.23483i −2.51027 − 0.835798i −0.707107 + 0.707107i 7.46410i 0
307.6 0.707107 + 0.707107i −1.32964 − 1.32964i 1.00000i 0 − 1.88040i 1.36804 − 2.26461i −0.707107 + 0.707107i 0.535898i 0
307.7 0.707107 + 0.707107i 1.32964 + 1.32964i 1.00000i 0 1.88040i −2.26461 + 1.36804i −0.707107 + 0.707107i 0.535898i 0
307.8 0.707107 + 0.707107i 2.28737 + 2.28737i 1.00000i 0 3.23483i −0.835798 − 2.51027i −0.707107 + 0.707107i 7.46410i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 293.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
5.c odd 4 2 inner
7.b odd 2 1 inner
35.c odd 2 1 inner
35.f even 4 2 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 350.2.g.b ✓ 16
5.b even 2 1 inner 350.2.g.b ✓ 16
5.c odd 4 2 inner 350.2.g.b ✓ 16
7.b odd 2 1 inner 350.2.g.b ✓ 16
35.c odd 2 1 inner 350.2.g.b ✓ 16
35.f even 4 2 inner 350.2.g.b ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
350.2.g.b ✓ 16 1.a even 1 1 trivial
350.2.g.b ✓ 16 5.b even 2 1 inner
350.2.g.b ✓ 16 5.c odd 4 2 inner
350.2.g.b ✓ 16 7.b odd 2 1 inner
350.2.g.b ✓ 16 35.c odd 2 1 inner
350.2.g.b ✓ 16 35.f even 4 2 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{8} + 122T_{3}^{4} + 1369 \) acting on \(S_{2}^{\mathrm{new}}(350, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} + 1)^{4} \) Copy content Toggle raw display
$3$ \( (T^{8} + 122 T^{4} + 1369)^{2} \) Copy content Toggle raw display
$5$ \( T^{16} \) Copy content Toggle raw display
$7$ \( T^{16} + 28 T^{12} + \cdots + 5764801 \) Copy content Toggle raw display
$11$ \( (T^{2} - 3)^{8} \) Copy content Toggle raw display
$13$ \( (T^{8} + 728 T^{4} + 21904)^{2} \) Copy content Toggle raw display
$17$ \( (T^{8} + 1098 T^{4} + 110889)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} - 42 T^{2} + 333)^{4} \) Copy content Toggle raw display
$23$ \( (T^{8} + 4536 T^{4} + 104976)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} + 24 T^{2} + 36)^{4} \) Copy content Toggle raw display
$31$ \( (T^{4} + 96 T^{2} + 1332)^{4} \) Copy content Toggle raw display
$37$ \( (T^{8} + 4536 T^{4} + 104976)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} + 126 T^{2} + 2997)^{4} \) Copy content Toggle raw display
$43$ \( (T^{4} + 144)^{4} \) Copy content Toggle raw display
$47$ \( (T^{8} + 6552 T^{4} + 1774224)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} + 40824 T^{4} + 8503056)^{2} \) Copy content Toggle raw display
$59$ \( T^{16} \) Copy content Toggle raw display
$61$ \( (T^{4} + 168 T^{2} + 5328)^{4} \) Copy content Toggle raw display
$67$ \( (T^{8} + 9954 T^{4} + 2313441)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} - 6 T - 18)^{8} \) Copy content Toggle raw display
$73$ \( (T^{8} + 16394 T^{4} + 39100009)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} + 4)^{8} \) Copy content Toggle raw display
$83$ \( (T^{8} + 9882 T^{4} + 8982009)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} - 126 T^{2} + 2997)^{4} \) Copy content Toggle raw display
$97$ \( (T^{8} + 11648 T^{4} + 5607424)^{2} \) Copy content Toggle raw display
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