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Properties

Label 75.13.c.e
Level $75$
Weight $13$
Character orbit 75.c
Analytic conductor $68.550$
Analytic rank $0$
Dimension $16$
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [75,13,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, 13, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("75.26");
 
S:= CuspForms(chi, 13);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 75 = 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 13 \)
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: \(68.5495362957\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 48921 x^{14} + 954361746 x^{12} + 9474039334576 x^{10} + \cdots + 33\!\cdots\!00 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{21}\cdot 3^{32}\cdot 5^{14} \)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{2} - 29) q^{3} + (\beta_{3} + \beta_{2} - 2019) q^{4} + ( - \beta_{5} - 73 \beta_1 - 2025) q^{6} + (\beta_{6} + \beta_{3} + 13 \beta_{2} + \cdots - 5008) q^{7}+ \cdots + ( - \beta_{11} - \beta_{6} + \cdots - 10352) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{2} - 29) q^{3} + (\beta_{3} + \beta_{2} - 2019) q^{4} + ( - \beta_{5} - 73 \beta_1 - 2025) q^{6} + (\beta_{6} + \beta_{3} + 13 \beta_{2} + \cdots - 5008) q^{7}+ \cdots + (8929 \beta_{15} + 47432 \beta_{14} + \cdots + 5751141791) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 460 q^{3} - 32306 q^{4} - 32399 q^{6} - 80080 q^{7} - 165740 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 460 q^{3} - 32306 q^{4} - 32399 q^{6} - 80080 q^{7} - 165740 q^{9} + 5271925 q^{12} - 2621080 q^{13} + 35234458 q^{16} + 32345285 q^{18} + 81892040 q^{19} + 108334056 q^{21} - 209116050 q^{22} - 56353947 q^{24} - 500540680 q^{27} + 661282100 q^{28} + 683606912 q^{31} - 2199262340 q^{33} + 667646778 q^{34} - 605088433 q^{36} + 78641360 q^{37} - 3252211864 q^{39} - 1117219920 q^{42} + 6515758880 q^{43} - 39021810348 q^{46} - 48414628835 q^{48} + 41978607576 q^{49} + 33056884756 q^{51} + 1585610480 q^{52} - 80344038896 q^{54} - 102415535980 q^{57} + 259201050660 q^{58} - 97224654088 q^{61} - 298527097920 q^{63} + 221192480602 q^{64} + 334914771455 q^{66} + 195891603800 q^{67} - 321003738216 q^{69} - 311869350975 q^{72} + 755437359800 q^{73} - 1028091315286 q^{76} - 1165232450510 q^{78} + 451290012320 q^{79} + 944941696036 q^{81} + 439746664050 q^{82} - 1668995110764 q^{84} - 125136293560 q^{87} + 1905931601310 q^{88} - 817056288848 q^{91} - 959022596160 q^{93} - 519350182332 q^{94} + 112887266273 q^{96} + 309746342120 q^{97} + 91555198460 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 48921 x^{14} + 954361746 x^{12} + 9474039334576 x^{10} + \cdots + 33\!\cdots\!00 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 96\!\cdots\!25 \nu^{15} + \cdots + 57\!\cdots\!00 ) / 20\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 96\!\cdots\!25 \nu^{15} + \cdots - 45\!\cdots\!00 ) / 20\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 49\!\cdots\!79 \nu^{15} + \cdots - 13\!\cdots\!00 ) / 68\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 78\!\cdots\!99 \nu^{15} + \cdots - 22\!\cdots\!00 ) / 14\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 19\!\cdots\!25 \nu^{15} + \cdots + 45\!\cdots\!00 ) / 34\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 97\!\cdots\!13 \nu^{15} + \cdots - 15\!\cdots\!00 ) / 20\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 45\!\cdots\!91 \nu^{15} + \cdots - 19\!\cdots\!00 ) / 51\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 76\!\cdots\!75 \nu^{15} + \cdots - 47\!\cdots\!00 ) / 68\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 55\!\cdots\!53 \nu^{15} + \cdots - 76\!\cdots\!00 ) / 22\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 13\!\cdots\!01 \nu^{15} + \cdots - 26\!\cdots\!00 ) / 51\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 42\!\cdots\!75 \nu^{15} + \cdots + 37\!\cdots\!00 ) / 17\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 12\!\cdots\!97 \nu^{15} + \cdots - 21\!\cdots\!00 ) / 41\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 37\!\cdots\!91 \nu^{15} + \cdots + 21\!\cdots\!00 ) / 51\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 52\!\cdots\!31 \nu^{15} + \cdots + 90\!\cdots\!00 ) / 68\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + \beta_{2} - 6115 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{5} + \beta_{4} + 13\beta_{2} - 9909\beta _1 - 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{14} + 3 \beta_{12} + 2 \beta_{11} + \beta_{10} + 21 \beta_{9} - \beta_{8} - \beta_{7} + \cdots + 60567198 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 21 \beta_{15} - 13 \beta_{14} - 18 \beta_{13} + 39 \beta_{12} + 566 \beta_{11} - 123 \beta_{9} + \cdots + 52409 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 1023 \beta_{15} - 20713 \beta_{14} - 62139 \beta_{12} - 7128 \beta_{11} - 18128 \beta_{10} + \cdots - 679767133799 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 443571 \beta_{15} + 203151 \beta_{14} + 625780 \beta_{13} - 609453 \beta_{12} - 11843668 \beta_{11} + \cdots - 769747015 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 30407415 \beta_{15} + 335053167 \beta_{14} + 1005159501 \beta_{12} - 350652396 \beta_{11} + \cdots + 80\!\cdots\!79 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 7330454241 \beta_{15} - 1945783725 \beta_{14} - 14029714416 \beta_{13} + 5837351175 \beta_{12} + \cdots + 9014125564989 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 627138009201 \beta_{15} - 4986020927909 \beta_{14} - 14958062783727 \beta_{12} + \cdots - 97\!\cdots\!53 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 111676521282003 \beta_{15} + 7573313701319 \beta_{14} + 260660285222184 \beta_{13} + \cdots - 79\!\cdots\!99 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 11\!\cdots\!79 \beta_{15} + \cdots + 12\!\cdots\!55 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 16\!\cdots\!97 \beta_{15} + \cdots + 31\!\cdots\!21 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 18\!\cdots\!45 \beta_{15} + \cdots - 15\!\cdots\!97 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 23\!\cdots\!59 \beta_{15} + \cdots + 70\!\cdots\!49 \) 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
− 116.687i
− 107.354i
− 97.7240i
− 88.3498i
− 54.6723i
− 45.6637i
− 26.2870i
− 25.6772i
25.6772i
26.2870i
45.6637i
54.6723i
88.3498i
97.7240i
107.354i
116.687i
− 116.687i −695.919 − 217.113i −9519.86 0 −25334.3 + 81204.7i −159101. 632894.i 437165. + 302187.i 0
26.2 − 107.354i 84.5279 + 724.083i −7428.92 0 77733.3 − 9074.43i 76606.4 357803.i −517151. + 122410.i 0
26.3 − 97.7240i 178.872 − 706.715i −5453.97 0 −69063.0 − 17480.1i 195864. 132707.i −467451. − 252823.i 0
26.4 − 88.3498i 725.902 − 67.1399i −3709.68 0 −5931.79 − 64133.3i −144393. − 34131.0i 522425. − 97473.9i 0
26.5 − 54.6723i −502.251 + 528.380i 1106.94 0 28887.8 + 27459.2i −26114.9 − 284457.i −26929.8 − 530758.i 0
26.6 − 45.6637i −657.408 − 315.050i 2010.83 0 −14386.3 + 30019.7i 82178.2 − 278860.i 332928. + 414232.i 0
26.7 − 26.2870i 46.9464 − 727.487i 3404.99 0 −19123.5 − 1234.08i −158582. − 197179.i −527033. − 68305.8i 0
26.8 − 25.6772i 589.329 + 429.106i 3436.68 0 11018.3 − 15132.3i 93502.2 − 193418.i 163176. + 505770.i 0
26.9 25.6772i 589.329 − 429.106i 3436.68 0 11018.3 + 15132.3i 93502.2 193418.i 163176. − 505770.i 0
26.10 26.2870i 46.9464 + 727.487i 3404.99 0 −19123.5 + 1234.08i −158582. 197179.i −527033. + 68305.8i 0
26.11 45.6637i −657.408 + 315.050i 2010.83 0 −14386.3 − 30019.7i 82178.2 278860.i 332928. − 414232.i 0
26.12 54.6723i −502.251 − 528.380i 1106.94 0 28887.8 − 27459.2i −26114.9 284457.i −26929.8 + 530758.i 0
26.13 88.3498i 725.902 + 67.1399i −3709.68 0 −5931.79 + 64133.3i −144393. 34131.0i 522425. + 97473.9i 0
26.14 97.7240i 178.872 + 706.715i −5453.97 0 −69063.0 + 17480.1i 195864. − 132707.i −467451. + 252823.i 0
26.15 107.354i 84.5279 − 724.083i −7428.92 0 77733.3 + 9074.43i 76606.4 − 357803.i −517151. − 122410.i 0
26.16 116.687i −695.919 + 217.113i −9519.86 0 −25334.3 − 81204.7i −159101. − 632894.i 437165. − 302187.i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 26.16
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 75.13.c.e ✓ 16
3.b odd 2 1 inner 75.13.c.e ✓ 16
5.b even 2 1 75.13.c.f yes 16
5.c odd 4 2 75.13.d.d 32
15.d odd 2 1 75.13.c.f yes 16
15.e even 4 2 75.13.d.d 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
75.13.c.e ✓ 16 1.a even 1 1 trivial
75.13.c.e ✓ 16 3.b odd 2 1 inner
75.13.c.f yes 16 5.b even 2 1
75.13.c.f yes 16 15.d odd 2 1
75.13.d.d 32 5.c odd 4 2
75.13.d.d 32 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 \(S_{13}^{\mathrm{new}}(75, [\chi])\):

\( T_{2}^{16} + 48921 T_{2}^{14} + 954361746 T_{2}^{12} + 9474039334576 T_{2}^{10} + \cdots + 33\!\cdots\!00 \) Copy content Toggle raw display
\( T_{7}^{8} + 40040 T_{7}^{7} - 65058199898 T_{7}^{6} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} + \cdots + 33\!\cdots\!00 \) Copy content Toggle raw display
$3$ \( T^{16} + \cdots + 63\!\cdots\!21 \) Copy content Toggle raw display
$5$ \( T^{16} \) Copy content Toggle raw display
$7$ \( (T^{8} + \cdots + 10\!\cdots\!00)^{2} \) Copy content Toggle raw display
$11$ \( T^{16} + \cdots + 74\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( (T^{8} + \cdots + 52\!\cdots\!00)^{2} \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( (T^{8} + \cdots - 29\!\cdots\!19)^{2} \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 79\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots + 12\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( (T^{8} + \cdots + 47\!\cdots\!76)^{2} \) Copy content Toggle raw display
$37$ \( (T^{8} + \cdots + 42\!\cdots\!00)^{2} \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 77\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( (T^{8} + \cdots + 11\!\cdots\!00)^{2} \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 20\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 51\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 40\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( (T^{8} + \cdots + 21\!\cdots\!76)^{2} \) Copy content Toggle raw display
$67$ \( (T^{8} + \cdots - 70\!\cdots\!75)^{2} \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 34\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( (T^{8} + \cdots + 73\!\cdots\!00)^{2} \) Copy content Toggle raw display
$79$ \( (T^{8} + \cdots + 10\!\cdots\!96)^{2} \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 45\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 17\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( (T^{8} + \cdots + 45\!\cdots\!00)^{2} \) Copy content Toggle raw display
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