Properties

Label 75.16.b.b
Level $75$
Weight $16$
Character orbit 75.b
Analytic conductor $107.020$
Analytic rank $0$
Dimension $2$
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] = [75,16,Mod(49,75)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(75, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 16, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("75.49");
 
S:= CuspForms(chi, 16);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 75 = 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 16 \)
Character orbit: \([\chi]\) \(=\) 75.b (of order \(2\), degree \(1\), 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: \(107.020128825\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 3)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 72 i q^{2} + 2187 i q^{3} + 27584 q^{4} - 157464 q^{6} + 2149000 i q^{7} + 4345344 i q^{8} - 4782969 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 72 i q^{2} + 2187 i q^{3} + 27584 q^{4} - 157464 q^{6} + 2149000 i q^{7} + 4345344 i q^{8} - 4782969 q^{9} + 37169316 q^{11} + 60326208 i q^{12} - 279974266 i q^{13} - 154728000 q^{14} + 591007744 q^{16} - 2492912754 i q^{17} - 344373768 i q^{18} + 4669782244 q^{19} - 4699863000 q^{21} + 2676190752 i q^{22} - 18467933400 i q^{23} - 9503267328 q^{24} + 20158147152 q^{26} - 10460353203 i q^{27} + 59278016000 i q^{28} + 115953449418 q^{29} - 56187023200 q^{31} + 184940789760 i q^{32} + 81289294092 i q^{33} + 179489718288 q^{34} - 131933416896 q^{36} - 614764926830 i q^{37} + 336224321568 i q^{38} + 612303719742 q^{39} + 549859792410 q^{41} - 338390136000 i q^{42} - 982884444028 i q^{43} + 1025278412544 q^{44} + 1329691204800 q^{46} - 2076144322896 i q^{47} + 1292533936128 i q^{48} + 129360509943 q^{49} + 5452000192998 q^{51} - 7722810153344 i q^{52} - 12048378188130 i q^{53} + 753145430616 q^{54} - 9338144256000 q^{56} + 10212813767628 i q^{57} + 8348648358096 i q^{58} - 23087905758324 q^{59} - 8505809142442 q^{61} - 4045465670400 i q^{62} - 10278600381000 i q^{63} + 6050404892672 q^{64} - 5852829174624 q^{66} + 12331010771476 i q^{67} - 68764505406336 i q^{68} + 40389370345800 q^{69} + 58989192692472 q^{71} - 20783645646336 i q^{72} - 5609828808070 i q^{73} + 44263074731760 q^{74} + 128811273418496 q^{76} + 79876860084000 i q^{77} + 44085867821424 i q^{78} - 159918683826800 q^{79} + 22876792454961 q^{81} + 39589905053520 i q^{82} + 57675894342876 i q^{83} - 129641020992000 q^{84} + 70767679970016 q^{86} + 253590193877166 i q^{87} + 161513464264704 i q^{88} + 362287610413974 q^{89} + 601664697634000 q^{91} - 509419474905600 i q^{92} - 122881019738400 i q^{93} + 149482391248512 q^{94} - 404465507205120 q^{96} + 539786645144926 i q^{97} + 9313956715896 i q^{98} - 177779686179204 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 55168 q^{4} - 314928 q^{6} - 9565938 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 55168 q^{4} - 314928 q^{6} - 9565938 q^{9} + 74338632 q^{11} - 309456000 q^{14} + 1182015488 q^{16} + 9339564488 q^{19} - 9399726000 q^{21} - 19006534656 q^{24} + 40316294304 q^{26} + 231906898836 q^{29} - 112374046400 q^{31} + 358979436576 q^{34} - 263866833792 q^{36} + 1224607439484 q^{39} + 1099719584820 q^{41} + 2050556825088 q^{44} + 2659382409600 q^{46} + 258721019886 q^{49} + 10904000385996 q^{51} + 1506290861232 q^{54} - 18676288512000 q^{56} - 46175811516648 q^{59} - 17011618284884 q^{61} + 12100809785344 q^{64} - 11705658349248 q^{66} + 80778740691600 q^{69} + 117978385384944 q^{71} + 88526149463520 q^{74} + 257622546836992 q^{76} - 319837367653600 q^{79} + 45753584909922 q^{81} - 259282041984000 q^{84} + 141535359940032 q^{86} + 724575220827948 q^{89} + 12\!\cdots\!00 q^{91}+ \cdots - 355559372358408 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/75\mathbb{Z}\right)^\times\).

\(n\) \(26\) \(52\)
\(\chi(n)\) \(1\) \(-1\)

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
1.00000i
1.00000i
72.0000i 2187.00i 27584.0 0 −157464. 2.14900e6i 4.34534e6i −4.78297e6 0
49.2 72.0000i 2187.00i 27584.0 0 −157464. 2.14900e6i 4.34534e6i −4.78297e6 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 75.16.b.b 2
5.b even 2 1 inner 75.16.b.b 2
5.c odd 4 1 3.16.a.b 1
5.c odd 4 1 75.16.a.a 1
15.e even 4 1 9.16.a.c 1
20.e even 4 1 48.16.a.a 1
35.f even 4 1 147.16.a.b 1
60.l odd 4 1 144.16.a.l 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3.16.a.b 1 5.c odd 4 1
9.16.a.c 1 15.e even 4 1
48.16.a.a 1 20.e even 4 1
75.16.a.a 1 5.c odd 4 1
75.16.b.b 2 1.a even 1 1 trivial
75.16.b.b 2 5.b even 2 1 inner
144.16.a.l 1 60.l odd 4 1
147.16.a.b 1 35.f even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} + 5184 \) acting on \(S_{16}^{\mathrm{new}}(75, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 5184 \) Copy content Toggle raw display
$3$ \( T^{2} + 4782969 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 4618201000000 \) Copy content Toggle raw display
$11$ \( (T - 37169316)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 78\!\cdots\!56 \) Copy content Toggle raw display
$17$ \( T^{2} + 62\!\cdots\!16 \) Copy content Toggle raw display
$19$ \( (T - 4669782244)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 34\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( (T - 115953449418)^{2} \) Copy content Toggle raw display
$31$ \( (T + 56187023200)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 37\!\cdots\!00 \) Copy content Toggle raw display
$41$ \( (T - 549859792410)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 96\!\cdots\!84 \) Copy content Toggle raw display
$47$ \( T^{2} + 43\!\cdots\!16 \) Copy content Toggle raw display
$53$ \( T^{2} + 14\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( (T + 23087905758324)^{2} \) Copy content Toggle raw display
$61$ \( (T + 8505809142442)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 15\!\cdots\!76 \) Copy content Toggle raw display
$71$ \( (T - 58989192692472)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 31\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( (T + 159918683826800)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 33\!\cdots\!76 \) Copy content Toggle raw display
$89$ \( (T - 362287610413974)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 29\!\cdots\!76 \) Copy content Toggle raw display
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