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Properties

Label 75.9.c.e
Level $75$
Weight $9$
Character orbit 75.c
Analytic conductor $30.553$
Analytic rank $0$
Dimension $10$
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] = [75,9,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, 9, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("75.26");
 
S:= CuspForms(chi, 9);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 75 = 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 9 \)
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: \(30.5533957546\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} + 1634x^{8} + 776307x^{6} + 148116566x^{4} + 10575941812x^{2} + 105274575720 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{8}\cdot 3^{11}\cdot 5^{5} \)
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 a basis \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{2} q^{2} + (\beta_{4} - \beta_{2} - 2) q^{3} + ( - \beta_1 - 155) q^{4} + (\beta_{6} + \beta_{5} - \beta_{4} + \cdots + 224) q^{6}+ \cdots + ( - \beta_{8} - \beta_{7} + \cdots - 1119) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{2} + (\beta_{4} - \beta_{2} - 2) q^{3} + ( - \beta_1 - 155) q^{4} + (\beta_{6} + \beta_{5} - \beta_{4} + \cdots + 224) q^{6}+ \cdots + ( - 20700 \beta_{9} + 43198 \beta_{8} + \cdots - 38563731) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 25 q^{3} - 1554 q^{4} + 2257 q^{6} + 1960 q^{7} - 11207 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - 25 q^{3} - 1554 q^{4} + 2257 q^{6} + 1960 q^{7} - 11207 q^{9} - 5915 q^{12} - 16920 q^{13} + 44634 q^{16} - 224875 q^{18} - 143934 q^{19} + 673428 q^{21} + 818990 q^{22} - 1016859 q^{24} - 260830 q^{27} + 3810100 q^{28} - 3014060 q^{31} - 4677515 q^{33} + 4977146 q^{34} + 4500527 q^{36} + 3016760 q^{37} - 7513282 q^{39} - 4001760 q^{42} + 11747340 q^{43} - 13938636 q^{46} - 14748755 q^{48} + 8953546 q^{49} + 6209287 q^{51} - 38918320 q^{52} - 8886272 q^{54} + 14759525 q^{57} - 48407900 q^{58} + 1520220 q^{61} + 74748240 q^{63} - 4536998 q^{64} + 10465295 q^{66} - 16269290 q^{67} + 11394978 q^{69} + 172231185 q^{72} - 52090170 q^{73} - 29529046 q^{76} + 198205810 q^{78} + 8549896 q^{79} + 22612945 q^{81} - 295714190 q^{82} - 136883292 q^{84} + 318901610 q^{87} - 310673250 q^{88} - 107224264 q^{91} + 79679130 q^{93} + 356118596 q^{94} + 525424001 q^{96} - 402167800 q^{97} - 382421335 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} + 1634x^{8} + 776307x^{6} + 148116566x^{4} + 10575941812x^{2} + 105274575720 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 872557 \nu^{8} - 1265751468 \nu^{6} - 444620825535 \nu^{4} - 46859095322612 \nu^{2} - 420291052310412 ) / 157481360352 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 8400715 \nu^{9} + 12128076852 \nu^{7} + 4207972253529 \nu^{5} + 435449159046332 \nu^{3} + 32\!\cdots\!04 \nu ) / 153386844982848 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 57764587 \nu^{9} + 16283051488 \nu^{8} + 95509278940 \nu^{7} + 23429801734080 \nu^{6} + \cdots + 85\!\cdots\!12 ) / 818063173241856 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 472531299 \nu^{9} - 5511595228 \nu^{8} + 683931862716 \nu^{7} - 7973245106928 \nu^{6} + \cdots - 31\!\cdots\!36 ) / 24\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 2093833615 \nu^{9} + 135239021452 \nu^{8} + 3031560854316 \nu^{7} + 195884281212336 \nu^{6} + \cdots + 78\!\cdots\!24 ) / 24\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 227599251 \nu^{9} - 4459205273 \nu^{8} - 326570626108 \nu^{7} - 6426141790740 \nu^{6} + \cdots - 23\!\cdots\!08 ) / 204515793310464 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 2367048033 \nu^{9} + 7788738074 \nu^{8} + 3397055634708 \nu^{7} + 11462785669512 \nu^{6} + \cdots + 61\!\cdots\!24 ) / 12\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 1932377350 \nu^{9} - 4685079769 \nu^{8} + 2788945929432 \nu^{7} - 6744083301684 \nu^{6} + \cdots - 23\!\cdots\!00 ) / 613547379931392 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 3538913459 \nu^{9} - 949770776 \nu^{8} + 5148112932156 \nu^{7} - 1377320091264 \nu^{6} + \cdots - 74\!\cdots\!32 ) / 613547379931392 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( 9 \beta_{9} - 17 \beta_{8} + 15 \beta_{7} - 18 \beta_{5} - 358 \beta_{4} - 2 \beta_{3} + 1043 \beta_{2} + \cdots - 182 ) / 4320 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 39\beta_{8} + 24\beta_{6} - 2\beta_{5} - 563\beta_{4} + 15\beta_{3} + 236\beta_{2} + 114\beta _1 - 88558 ) / 270 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 375 \beta_{9} + 3967 \beta_{8} - 3153 \beta_{7} - 3696 \beta_{6} + 3198 \beta_{5} + 63242 \beta_{4} + \cdots + 31114 ) / 1440 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 8805 \beta_{8} - 4944 \beta_{6} + 1894 \beta_{5} + 123217 \beta_{4} - 3861 \beta_{3} + \cdots + 12132530 ) / 54 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 587295 \beta_{9} - 10471033 \beta_{8} + 8476887 \beta_{7} + 13073904 \beta_{6} - 7623762 \beta_{5} + \cdots - 69764806 ) / 4320 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 4991021 \beta_{8} + 2823976 \beta_{6} - 1189238 \beta_{5} - 69115377 \beta_{4} + 2167045 \beta_{3} + \cdots - 6045548182 ) / 30 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 1002570867 \beta_{9} + 10039223749 \beta_{8} - 8231449035 \beta_{7} - 13312393200 \beta_{6} + \cdots + 63697009294 ) / 4320 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 8964366177 \beta_{8} - 5112252240 \beta_{6} + 2161618478 \beta_{5} + 123729099293 \beta_{4} + \cdots + 10528560319474 ) / 54 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 366131162877 \beta_{9} - 3286296411019 \beta_{8} + 2707361971365 \beta_{7} + 4414995739920 \beta_{6} + \cdots - 20593816306834 ) / 1440 \) Copy content Toggle raw display

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

Character values

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

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

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
3.43232i
13.5890i
− 31.4704i
− 13.7835i
− 16.0371i
16.0371i
13.7835i
31.4704i
− 13.5890i
− 3.43232i
− 29.4751i −29.7353 + 75.3446i −612.784 0 2220.79 + 876.451i −3164.62 10516.3i −4792.63 − 4480.79i 0
26.2 − 24.1370i 39.5746 − 70.6742i −326.594 0 −1705.86 − 955.212i −1042.22 1703.92i −3428.70 − 5593.81i 0
26.3 − 18.2182i 69.9805 + 40.7889i −75.9021 0 743.099 − 1274.92i 4676.68 − 3281.06i 3233.54 + 5708.85i 0
26.4 − 16.2639i −77.7167 − 22.8278i −8.51375 0 −371.269 + 1263.98i 569.895 − 4025.09i 5518.78 + 3548.20i 0
26.5 − 3.03414i −14.6031 + 79.6728i 246.794 0 241.739 + 44.3080i −59.7348 − 1525.55i −6134.50 − 2326.94i 0
26.6 3.03414i −14.6031 − 79.6728i 246.794 0 241.739 − 44.3080i −59.7348 1525.55i −6134.50 + 2326.94i 0
26.7 16.2639i −77.7167 + 22.8278i −8.51375 0 −371.269 − 1263.98i 569.895 4025.09i 5518.78 − 3548.20i 0
26.8 18.2182i 69.9805 − 40.7889i −75.9021 0 743.099 + 1274.92i 4676.68 3281.06i 3233.54 − 5708.85i 0
26.9 24.1370i 39.5746 + 70.6742i −326.594 0 −1705.86 + 955.212i −1042.22 − 1703.92i −3428.70 + 5593.81i 0
26.10 29.4751i −29.7353 − 75.3446i −612.784 0 2220.79 − 876.451i −3164.62 − 10516.3i −4792.63 + 4480.79i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 26.10
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 75.9.c.e ✓ 10
3.b odd 2 1 inner 75.9.c.e ✓ 10
5.b even 2 1 75.9.c.f yes 10
5.c odd 4 2 75.9.d.d 20
15.d odd 2 1 75.9.c.f yes 10
15.e even 4 2 75.9.d.d 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
75.9.c.e ✓ 10 1.a even 1 1 trivial
75.9.c.e ✓ 10 3.b odd 2 1 inner
75.9.c.f yes 10 5.b even 2 1
75.9.c.f yes 10 15.d odd 2 1
75.9.d.d 20 5.c odd 4 2
75.9.d.d 20 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 \(S_{9}^{\mathrm{new}}(75, [\chi])\):

\( T_{2}^{10} + 2057T_{2}^{8} + 1478418T_{2}^{6} + 442732064T_{2}^{4} + 48388216960T_{2}^{2} + 409079808000 \) Copy content Toggle raw display
\( T_{7}^{5} - 980T_{7}^{4} - 16170189T_{7}^{3} - 7054500600T_{7}^{2} + 8426583900000T_{7} + 525098700000000 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} + \cdots + 409079808000 \) Copy content Toggle raw display
$3$ \( T^{10} + \cdots + 12\!\cdots\!01 \) Copy content Toggle raw display
$5$ \( T^{10} \) Copy content Toggle raw display
$7$ \( (T^{5} + \cdots + 525098700000000)^{2} \) Copy content Toggle raw display
$11$ \( T^{10} + \cdots + 20\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( (T^{5} + \cdots + 59\!\cdots\!00)^{2} \) Copy content Toggle raw display
$17$ \( T^{10} + \cdots + 16\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( (T^{5} + \cdots + 52\!\cdots\!67)^{2} \) Copy content Toggle raw display
$23$ \( T^{10} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( T^{10} + \cdots + 85\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( (T^{5} + \cdots + 96\!\cdots\!08)^{2} \) Copy content Toggle raw display
$37$ \( (T^{5} + \cdots - 19\!\cdots\!00)^{2} \) Copy content Toggle raw display
$41$ \( T^{10} + \cdots + 40\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( (T^{5} + \cdots - 35\!\cdots\!00)^{2} \) Copy content Toggle raw display
$47$ \( T^{10} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{10} + \cdots + 15\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( T^{10} + \cdots + 84\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( (T^{5} + \cdots + 41\!\cdots\!68)^{2} \) Copy content Toggle raw display
$67$ \( (T^{5} + \cdots - 15\!\cdots\!75)^{2} \) Copy content Toggle raw display
$71$ \( T^{10} + \cdots + 61\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( (T^{5} + \cdots + 11\!\cdots\!00)^{2} \) Copy content Toggle raw display
$79$ \( (T^{5} + \cdots + 23\!\cdots\!52)^{2} \) Copy content Toggle raw display
$83$ \( T^{10} + \cdots + 20\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( T^{10} + \cdots + 63\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( (T^{5} + \cdots + 82\!\cdots\!00)^{2} \) Copy content Toggle raw display
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