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Properties

Label 750.2.h.e
Level $750$
Weight $2$
Character orbit 750.h
Analytic conductor $5.989$
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] = [750,2,Mod(49,750)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(750, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([0, 7]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("750.49");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 750 = 2 \cdot 3 \cdot 5^{3} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 750.h (of order \(10\), degree \(4\), not 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: \(5.98878015160\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(4\) over \(\Q(\zeta_{10})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 10x^{14} + 61x^{12} + 335x^{10} + 846x^{8} + 3445x^{6} + 14551x^{4} + 12635x^{2} + 130321 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 5^{2} \)
Twist minimal: no (minimal twist has level 150)
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{11} q^{2} + \beta_{7} q^{3} + \beta_{9} q^{4} + ( - \beta_{9} - \beta_{3} + \beta_1 + 1) q^{6} + ( - \beta_{14} + \beta_{11} + \cdots - \beta_{4}) q^{7}+ \cdots + \beta_{3} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{11} q^{2} + \beta_{7} q^{3} + \beta_{9} q^{4} + ( - \beta_{9} - \beta_{3} + \beta_1 + 1) q^{6} + ( - \beta_{14} + \beta_{11} + \cdots - \beta_{4}) q^{7}+ \cdots + (\beta_{6} + \beta_{5} - \beta_{2} - 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 4 q^{4} + 4 q^{6} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 4 q^{4} + 4 q^{6} + 4 q^{9} - 10 q^{11} - 4 q^{14} - 4 q^{16} - 16 q^{19} + 6 q^{21} + 16 q^{24} + 28 q^{26} + 36 q^{29} + 18 q^{31} + 6 q^{34} - 4 q^{36} - 12 q^{39} + 4 q^{41} - 40 q^{46} - 44 q^{49} - 4 q^{51} - 4 q^{54} - 6 q^{56} + 50 q^{59} + 20 q^{61} + 4 q^{64} + 10 q^{66} - 40 q^{69} + 52 q^{74} + 16 q^{76} + 12 q^{79} - 4 q^{81} + 4 q^{84} + 4 q^{86} - 18 q^{89} - 8 q^{91} - 20 q^{94} + 4 q^{96} - 20 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 10x^{14} + 61x^{12} + 335x^{10} + 846x^{8} + 3445x^{6} + 14551x^{4} + 12635x^{2} + 130321 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 1745235270 \nu^{14} + 19063543943 \nu^{12} + 109100471433 \nu^{10} + \cdots + 83795218251789 ) / 315676686340924 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 912701669 \nu^{14} - 14211195929 \nu^{12} - 567226550825 \nu^{10} + \cdots - 742484645310074 ) / 157838343170462 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 3811901982 \nu^{14} + 43280820291 \nu^{12} + 170361790217 \nu^{10} + 1431873055901 \nu^{8} + \cdots - 53310687773263 ) / 315676686340924 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 4260837575 \nu^{15} - 224170585399 \nu^{13} - 3588630391399 \nu^{11} + \cdots - 25\!\cdots\!02 \nu ) / 59\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 1477345796 \nu^{14} + 47345518314 \nu^{12} + 392180216258 \nu^{10} + 2095363055173 \nu^{8} + \cdots + 233908382128265 ) / 78919171585231 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 3828298547 \nu^{14} + 70899880941 \nu^{12} + 168444825961 \nu^{10} + \cdots - 248436426812116 ) / 157838343170462 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 7423048843 \nu^{15} - 632194946106 \nu^{13} - 3940915551408 \nu^{11} + \cdots - 874778988946617 \nu ) / 59\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 233315150 \nu^{15} + 17139402003 \nu^{13} + 140291562285 \nu^{11} + \cdots + 267704171697655 \nu ) / 157838343170462 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 12829703553 \nu^{14} - 84643137491 \nu^{12} - 480336484907 \nu^{10} + \cdots + 2185420420980 ) / 315676686340924 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 17060941697 \nu^{15} + 1038677077769 \nu^{13} + 6918969263957 \nu^{11} + \cdots + 37\!\cdots\!54 \nu ) / 59\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 22205725831 \nu^{15} + 145865842677 \nu^{13} - 1609186292607 \nu^{11} + \cdots - 31\!\cdots\!60 \nu ) / 59\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 18356511022 \nu^{14} + 124492087157 \nu^{12} + 484988953205 \nu^{10} + \cdots - 308819520995991 ) / 157838343170462 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 43337008993 \nu^{15} - 26931423167 \nu^{13} - 1296698956987 \nu^{11} + \cdots - 13\!\cdots\!80 \nu ) / 59\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 425143 \nu^{15} - 5025775 \nu^{13} - 29880175 \nu^{11} - 127755475 \nu^{9} + \cdots - 6517277400 \nu ) / 26886333458 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 303390401043 \nu^{15} - 1616048705488 \nu^{13} - 9032928354798 \nu^{11} + \cdots + 22\!\cdots\!69 \nu ) / 59\!\cdots\!56 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{15} - \beta_{14} - 4\beta_{13} - \beta_{11} + \beta_{10} - 3\beta_{8} - 3\beta_{7} + 2\beta_{4} ) / 5 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 3\beta_{12} + 4\beta_{9} - 3\beta_{6} - 2\beta_{5} + \beta_{3} - \beta_{2} + 17\beta _1 - 2 ) / 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 3\beta_{15} - 3\beta_{14} + 3\beta_{13} + 37\beta_{11} + 3\beta_{10} + 6\beta_{8} + 31\beta_{7} - 39\beta_{4} ) / 5 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -3\beta_{12} - 8\beta_{9} + 4\beta_{6} - 3\beta_{5} - \beta_{3} - 2\beta_{2} + 3\beta _1 - 1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 4 \beta_{15} - 46 \beta_{14} + 16 \beta_{13} - 191 \beta_{11} + 16 \beta_{10} + 2 \beta_{8} + \cdots + 207 \beta_{4} ) / 5 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -72\beta_{12} - 326\beta_{9} - 123\beta_{6} + 123\beta_{5} - 254\beta_{3} + 174\beta_{2} + 22\beta _1 + 73 ) / 5 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 242 \beta_{15} + 747 \beta_{14} + 113 \beta_{13} + 202 \beta_{11} - 242 \beta_{10} + \cdots - 129 \beta_{4} ) / 5 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 59\beta_{12} + 460\beta_{9} + 42\beta_{6} - 34\beta_{5} + 689\beta_{3} - 118\beta_{2} - 443\beta _1 - 42 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 666 \beta_{15} - 766 \beta_{14} - 1719 \beta_{13} - 666 \beta_{11} + 2106 \beta_{10} + \cdots + 287 \beta_{4} ) / 5 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 2168 \beta_{12} + 1084 \beta_{9} - 128 \beta_{6} - 1212 \beta_{5} - 11069 \beta_{3} - 2296 \beta_{2} + \cdots - 8017 ) / 5 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 2633 \beta_{15} - 12133 \beta_{14} + 9438 \beta_{13} + 16852 \beta_{11} - 7352 \beta_{10} + \cdots - 32944 \beta_{4} ) / 5 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 3843 \beta_{12} - 8766 \beta_{9} + 3843 \beta_{6} + 1263 \beta_{5} - 2580 \beta_{3} + 2580 \beta_{2} + \cdots + 7503 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 9364 \beta_{15} - 2351 \beta_{14} - 9364 \beta_{13} - 161161 \beta_{11} + 2351 \beta_{10} + \cdots + 266982 \beta_{4} ) / 5 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 45173 \beta_{12} - 76866 \beta_{9} - 160358 \beta_{6} + 45173 \beta_{5} - 9929 \beta_{3} + 80179 \beta_{2} + \cdots + 90108 ) / 5 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 175872 \beta_{15} + 479567 \beta_{14} - 63762 \beta_{13} + 816112 \beta_{11} + \cdots - 879874 \beta_{4} ) / 5 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/750\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(251\)
\(\chi(n)\) \(\beta_{9}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
49.1
−1.45788 + 1.86886i
−0.0809628 − 2.36886i
0.0809628 + 2.36886i
1.45788 − 1.86886i
−0.0809628 + 2.36886i
−1.45788 − 1.86886i
1.45788 + 1.86886i
0.0809628 − 2.36886i
−1.64827 − 0.815575i
1.28500 + 1.31557i
−1.28500 − 1.31557i
1.64827 + 0.815575i
1.28500 − 1.31557i
−1.64827 + 0.815575i
1.64827 − 0.815575i
−1.28500 + 1.31557i
−0.587785 + 0.809017i 0.951057 − 0.309017i −0.309017 − 0.951057i 0 −0.309017 + 0.951057i − 2.31003i 0.951057 + 0.309017i 0.809017 − 0.587785i 0
49.2 −0.587785 + 0.809017i 0.951057 − 0.309017i −0.309017 − 0.951057i 0 −0.309017 + 0.951057i 2.92807i 0.951057 + 0.309017i 0.809017 − 0.587785i 0
49.3 0.587785 − 0.809017i −0.951057 + 0.309017i −0.309017 − 0.951057i 0 −0.309017 + 0.951057i − 2.92807i −0.951057 − 0.309017i 0.809017 − 0.587785i 0
49.4 0.587785 − 0.809017i −0.951057 + 0.309017i −0.309017 − 0.951057i 0 −0.309017 + 0.951057i 2.31003i −0.951057 − 0.309017i 0.809017 − 0.587785i 0
199.1 −0.587785 − 0.809017i 0.951057 + 0.309017i −0.309017 + 0.951057i 0 −0.309017 − 0.951057i − 2.92807i 0.951057 − 0.309017i 0.809017 + 0.587785i 0
199.2 −0.587785 − 0.809017i 0.951057 + 0.309017i −0.309017 + 0.951057i 0 −0.309017 − 0.951057i 2.31003i 0.951057 − 0.309017i 0.809017 + 0.587785i 0
199.3 0.587785 + 0.809017i −0.951057 − 0.309017i −0.309017 + 0.951057i 0 −0.309017 − 0.951057i − 2.31003i −0.951057 + 0.309017i 0.809017 + 0.587785i 0
199.4 0.587785 + 0.809017i −0.951057 − 0.309017i −0.309017 + 0.951057i 0 −0.309017 − 0.951057i 2.92807i −0.951057 + 0.309017i 0.809017 + 0.587785i 0
349.1 −0.951057 + 0.309017i −0.587785 − 0.809017i 0.809017 − 0.587785i 0 0.809017 + 0.587785i − 2.63925i −0.587785 + 0.809017i −0.309017 + 0.951057i 0
349.2 −0.951057 + 0.309017i −0.587785 − 0.809017i 0.809017 − 0.587785i 0 0.809017 + 0.587785i 4.25729i −0.587785 + 0.809017i −0.309017 + 0.951057i 0
349.3 0.951057 − 0.309017i 0.587785 + 0.809017i 0.809017 − 0.587785i 0 0.809017 + 0.587785i − 4.25729i 0.587785 − 0.809017i −0.309017 + 0.951057i 0
349.4 0.951057 − 0.309017i 0.587785 + 0.809017i 0.809017 − 0.587785i 0 0.809017 + 0.587785i 2.63925i 0.587785 − 0.809017i −0.309017 + 0.951057i 0
649.1 −0.951057 − 0.309017i −0.587785 + 0.809017i 0.809017 + 0.587785i 0 0.809017 − 0.587785i − 4.25729i −0.587785 − 0.809017i −0.309017 − 0.951057i 0
649.2 −0.951057 − 0.309017i −0.587785 + 0.809017i 0.809017 + 0.587785i 0 0.809017 − 0.587785i 2.63925i −0.587785 − 0.809017i −0.309017 − 0.951057i 0
649.3 0.951057 + 0.309017i 0.587785 − 0.809017i 0.809017 + 0.587785i 0 0.809017 − 0.587785i − 2.63925i 0.587785 + 0.809017i −0.309017 − 0.951057i 0
649.4 0.951057 + 0.309017i 0.587785 − 0.809017i 0.809017 + 0.587785i 0 0.809017 − 0.587785i 4.25729i 0.587785 + 0.809017i −0.309017 − 0.951057i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 49.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
25.d even 5 1 inner
25.e even 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 750.2.h.e 16
5.b even 2 1 inner 750.2.h.e 16
5.c odd 4 1 150.2.g.c ✓ 8
5.c odd 4 1 750.2.g.d 8
15.e even 4 1 450.2.h.d 8
25.d even 5 1 inner 750.2.h.e 16
25.d even 5 1 3750.2.c.h 8
25.e even 10 1 inner 750.2.h.e 16
25.e even 10 1 3750.2.c.h 8
25.f odd 20 1 150.2.g.c ✓ 8
25.f odd 20 1 750.2.g.d 8
25.f odd 20 1 3750.2.a.l 4
25.f odd 20 1 3750.2.a.q 4
75.l even 20 1 450.2.h.d 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
150.2.g.c ✓ 8 5.c odd 4 1
150.2.g.c ✓ 8 25.f odd 20 1
450.2.h.d 8 15.e even 4 1
450.2.h.d 8 75.l even 20 1
750.2.g.d 8 5.c odd 4 1
750.2.g.d 8 25.f odd 20 1
750.2.h.e 16 1.a even 1 1 trivial
750.2.h.e 16 5.b even 2 1 inner
750.2.h.e 16 25.d even 5 1 inner
750.2.h.e 16 25.e even 10 1 inner
3750.2.a.l 4 25.f odd 20 1
3750.2.a.q 4 25.f odd 20 1
3750.2.c.h 8 25.d even 5 1
3750.2.c.h 8 25.e even 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{8} + 39T_{7}^{6} + 521T_{7}^{4} + 2904T_{7}^{2} + 5776 \) acting on \(S_{2}^{\mathrm{new}}(750, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{8} - T^{6} + T^{4} + \cdots + 1)^{2} \) Copy content Toggle raw display
$3$ \( (T^{8} - T^{6} + T^{4} + \cdots + 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{16} \) Copy content Toggle raw display
$7$ \( (T^{8} + 39 T^{6} + \cdots + 5776)^{2} \) Copy content Toggle raw display
$11$ \( (T^{8} + 5 T^{7} + \cdots + 400)^{2} \) Copy content Toggle raw display
$13$ \( T^{16} - 28 T^{14} + \cdots + 65536 \) Copy content Toggle raw display
$17$ \( T^{16} - 152 T^{14} + \cdots + 33362176 \) Copy content Toggle raw display
$19$ \( (T^{4} + 4 T^{3} + 16 T^{2} + \cdots + 16)^{4} \) Copy content Toggle raw display
$23$ \( (T^{8} - 20 T^{6} + \cdots + 160000)^{2} \) Copy content Toggle raw display
$29$ \( (T^{8} - 18 T^{7} + \cdots + 1860496)^{2} \) Copy content Toggle raw display
$31$ \( (T^{8} - 9 T^{7} + \cdots + 1296)^{2} \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 125044638416896 \) Copy content Toggle raw display
$41$ \( (T^{8} - 2 T^{7} + \cdots + 1936)^{2} \) Copy content Toggle raw display
$43$ \( (T^{8} + 264 T^{6} + \cdots + 2027776)^{2} \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 599695360000 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 72\!\cdots\!41 \) Copy content Toggle raw display
$59$ \( (T^{8} - 25 T^{7} + \cdots + 2310400)^{2} \) Copy content Toggle raw display
$61$ \( (T^{8} - 10 T^{7} + \cdots + 400)^{2} \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 47408402661376 \) Copy content Toggle raw display
$71$ \( (T^{8} - 40 T^{6} + \cdots + 102400)^{2} \) Copy content Toggle raw display
$73$ \( T^{16} - 168 T^{14} + \cdots + 1679616 \) Copy content Toggle raw display
$79$ \( (T^{8} - 6 T^{7} + \cdots + 331776)^{2} \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 1016096256256 \) Copy content Toggle raw display
$89$ \( (T^{8} + 9 T^{7} + \cdots + 1296)^{2} \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 16983563041 \) Copy content Toggle raw display
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