Properties

Label 75.5.c.f
Level $75$
Weight $5$
Character orbit 75.c
Analytic conductor $7.753$
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] = [75,5,Mod(26,75)]
 
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([1, 0]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("75.26");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 75 = 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 75.c (of order \(2\), degree \(1\), 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: \(7.75274723129\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-35}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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 \(\beta = \frac{1}{2}(1 + \sqrt{-35})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - 2 \beta + 1) q^{2} + 3 \beta q^{3} - 19 q^{4} + ( - 3 \beta + 54) q^{6} - 72 q^{7} + (6 \beta - 3) q^{8} + (9 \beta - 81) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - 2 \beta + 1) q^{2} + 3 \beta q^{3} - 19 q^{4} + ( - 3 \beta + 54) q^{6} - 72 q^{7} + (6 \beta - 3) q^{8} + (9 \beta - 81) q^{9} + (62 \beta - 31) q^{11} - 57 \beta q^{12} - 202 q^{13} + (144 \beta - 72) q^{14} - 199 q^{16} + ( - 26 \beta + 13) q^{17} + (153 \beta + 81) q^{18} + 367 q^{19} - 216 \beta q^{21} + 1085 q^{22} + ( - 12 \beta + 6) q^{23} + (9 \beta - 162) q^{24} + (404 \beta - 202) q^{26} + ( - 216 \beta - 243) q^{27} + 1368 q^{28} + (124 \beta - 62) q^{29} - 318 q^{31} + (494 \beta - 247) q^{32} + (93 \beta - 1674) q^{33} - 455 q^{34} + ( - 171 \beta + 1539) q^{36} - 752 q^{37} + ( - 734 \beta + 367) q^{38} - 606 \beta q^{39} + ( - 542 \beta + 271) q^{41} + (216 \beta - 3888) q^{42} - 982 q^{43} + ( - 1178 \beta + 589) q^{44} - 210 q^{46} + ( - 404 \beta + 202) q^{47} - 597 \beta q^{48} + 2783 q^{49} + ( - 39 \beta + 702) q^{51} + 3838 q^{52} + (796 \beta - 398) q^{53} + (702 \beta - 4131) q^{54} + ( - 432 \beta + 216) q^{56} + 1101 \beta q^{57} + 2170 q^{58} + (8 \beta - 4) q^{59} - 3728 q^{61} + (636 \beta - 318) q^{62} + ( - 648 \beta + 5832) q^{63} + 5461 q^{64} + 3255 \beta q^{66} + 7443 q^{67} + (494 \beta - 247) q^{68} + ( - 18 \beta + 324) q^{69} + ( - 2276 \beta + 1138) q^{71} + ( - 459 \beta - 243) q^{72} - 397 q^{73} + (1504 \beta - 752) q^{74} - 6973 q^{76} + ( - 4464 \beta + 2232) q^{77} + (606 \beta - 10908) q^{78} - 6108 q^{79} + ( - 1377 \beta + 5832) q^{81} - 9485 q^{82} + (4494 \beta - 2247) q^{83} + 4104 \beta q^{84} + (1964 \beta - 982) q^{86} + (186 \beta - 3348) q^{87} - 3255 q^{88} + (1602 \beta - 801) q^{89} + 14544 q^{91} + (228 \beta - 114) q^{92} - 954 \beta q^{93} - 7070 q^{94} + (741 \beta - 13338) q^{96} - 8042 q^{97} + ( - 5566 \beta + 2783) q^{98} + ( - 4743 \beta - 2511) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 3 q^{3} - 38 q^{4} + 105 q^{6} - 144 q^{7} - 153 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 3 q^{3} - 38 q^{4} + 105 q^{6} - 144 q^{7} - 153 q^{9} - 57 q^{12} - 404 q^{13} - 398 q^{16} + 315 q^{18} + 734 q^{19} - 216 q^{21} + 2170 q^{22} - 315 q^{24} - 702 q^{27} + 2736 q^{28} - 636 q^{31} - 3255 q^{33} - 910 q^{34} + 2907 q^{36} - 1504 q^{37} - 606 q^{39} - 7560 q^{42} - 1964 q^{43} - 420 q^{46} - 597 q^{48} + 5566 q^{49} + 1365 q^{51} + 7676 q^{52} - 7560 q^{54} + 1101 q^{57} + 4340 q^{58} - 7456 q^{61} + 11016 q^{63} + 10922 q^{64} + 3255 q^{66} + 14886 q^{67} + 630 q^{69} - 945 q^{72} - 794 q^{73} - 13946 q^{76} - 21210 q^{78} - 12216 q^{79} + 10287 q^{81} - 18970 q^{82} + 4104 q^{84} - 6510 q^{87} - 6510 q^{88} + 29088 q^{91} - 954 q^{93} - 14140 q^{94} - 25935 q^{96} - 16084 q^{97} - 9765 q^{99}+O(q^{100}) \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
26.1
0.500000 + 2.95804i
0.500000 2.95804i
5.91608i 1.50000 + 8.87412i −19.0000 0 52.5000 8.87412i −72.0000 17.7482i −76.5000 + 26.6224i 0
26.2 5.91608i 1.50000 8.87412i −19.0000 0 52.5000 + 8.87412i −72.0000 17.7482i −76.5000 26.6224i 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
3.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 75.5.c.f yes 2
3.b odd 2 1 inner 75.5.c.f yes 2
5.b even 2 1 75.5.c.d 2
5.c odd 4 2 75.5.d.c 4
15.d odd 2 1 75.5.c.d 2
15.e even 4 2 75.5.d.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
75.5.c.d 2 5.b even 2 1
75.5.c.d 2 15.d odd 2 1
75.5.c.f yes 2 1.a even 1 1 trivial
75.5.c.f yes 2 3.b odd 2 1 inner
75.5.d.c 4 5.c odd 4 2
75.5.d.c 4 15.e even 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{5}^{\mathrm{new}}(75, [\chi])\):

\( T_{2}^{2} + 35 \) Copy content Toggle raw display
\( T_{7} + 72 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 35 \) Copy content Toggle raw display
$3$ \( T^{2} - 3T + 81 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( (T + 72)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 33635 \) Copy content Toggle raw display
$13$ \( (T + 202)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 5915 \) Copy content Toggle raw display
$19$ \( (T - 367)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 1260 \) Copy content Toggle raw display
$29$ \( T^{2} + 134540 \) Copy content Toggle raw display
$31$ \( (T + 318)^{2} \) Copy content Toggle raw display
$37$ \( (T + 752)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + 2570435 \) Copy content Toggle raw display
$43$ \( (T + 982)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 1428140 \) Copy content Toggle raw display
$53$ \( T^{2} + 5544140 \) Copy content Toggle raw display
$59$ \( T^{2} + 560 \) Copy content Toggle raw display
$61$ \( (T + 3728)^{2} \) Copy content Toggle raw display
$67$ \( (T - 7443)^{2} \) Copy content Toggle raw display
$71$ \( T^{2} + 45326540 \) Copy content Toggle raw display
$73$ \( (T + 397)^{2} \) Copy content Toggle raw display
$79$ \( (T + 6108)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 176715315 \) Copy content Toggle raw display
$89$ \( T^{2} + 22456035 \) Copy content Toggle raw display
$97$ \( (T + 8042)^{2} \) Copy content Toggle raw display
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