Properties

Label 7.5.d.a
Level $7$
Weight $5$
Character orbit 7.d
Analytic conductor $0.724$
Analytic rank $0$
Dimension $4$
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] = [7,5,Mod(3,7)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(7, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("7.3");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 7 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 7.d (of order \(6\), degree \(2\), 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: \(0.723589741587\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{22})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 22x^{2} + 484 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - 2 \beta_{2} + \beta_1 - 2) q^{2} + (\beta_{3} + \beta_{2} - \beta_1 + 2) q^{3} + ( - 4 \beta_{3} + 10 \beta_{2} - 4 \beta_1) q^{4} + (4 \beta_{3} + 5 \beta_{2} + 2 \beta_1 - 5) q^{5} + (3 \beta_{3} - 48 \beta_{2} + 6 \beta_1 - 24) q^{6} + ( - 7 \beta_{3} + 42 \beta_{2} + 21) q^{7} + (2 \beta_{3} + 76) q^{8} + ( - 12 \beta_{2} - 6 \beta_1 - 12) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - 2 \beta_{2} + \beta_1 - 2) q^{2} + (\beta_{3} + \beta_{2} - \beta_1 + 2) q^{3} + ( - 4 \beta_{3} + 10 \beta_{2} - 4 \beta_1) q^{4} + (4 \beta_{3} + 5 \beta_{2} + 2 \beta_1 - 5) q^{5} + (3 \beta_{3} - 48 \beta_{2} + 6 \beta_1 - 24) q^{6} + ( - 7 \beta_{3} + 42 \beta_{2} + 21) q^{7} + (2 \beta_{3} + 76) q^{8} + ( - 12 \beta_{2} - 6 \beta_1 - 12) q^{9} + (\beta_{3} - 34 \beta_{2} - \beta_1 - 68) q^{10} + (17 \beta_{3} + 29 \beta_{2} + 17 \beta_1) q^{11} + ( - 28 \beta_{3} + 98 \beta_{2} - 14 \beta_1 - 98) q^{12} + ( - 14 \beta_{3} - 140 \beta_{2} - 28 \beta_1 - 70) q^{13} + (42 \beta_{3} + 112 \beta_{2} + 7 \beta_1 + 196) q^{14} + ( - 9 \beta_{3} + 117) q^{15} + ( - 36 \beta_{2} + 16 \beta_1 - 36) q^{16} + ( - 34 \beta_{3} - 41 \beta_{2} + 34 \beta_1 - 82) q^{17} - 108 \beta_{2} q^{18} + (74 \beta_{3} - 107 \beta_{2} + 37 \beta_1 + 107) q^{19} + (252 \beta_{2} + 126) q^{20} + ( - 70 \beta_{3} - 91 \beta_{2} - 56 \beta_1 - 308) q^{21} + ( - 5 \beta_{3} - 316) q^{22} + (145 \beta_{2} - 41 \beta_1 + 145) q^{23} + (78 \beta_{3} + 120 \beta_{2} - 78 \beta_1 + 240) q^{24} + ( - 60 \beta_{3} + 286 \beta_{2} - 60 \beta_1) q^{25} + ( - 84 \beta_{3} - 168 \beta_{2} - 42 \beta_1 + 168) q^{26} + (87 \beta_{3} + 78 \beta_{2} + 174 \beta_1 + 39) q^{27} + ( - 14 \beta_{3} - 826 \beta_{2} + 154 \beta_1 - 420) q^{28} + (70 \beta_{3} - 544) q^{29} + ( - 36 \beta_{2} + 99 \beta_1 - 36) q^{30} + (29 \beta_{3} + 603 \beta_{2} - 29 \beta_1 + 1206) q^{31} + ( - 36 \beta_{3} - 792 \beta_{2} - 36 \beta_1) q^{32} + ( - 24 \beta_{3} - 345 \beta_{2} - 12 \beta_1 + 345) q^{33} + ( - 109 \beta_{3} + 1660 \beta_{2} - 218 \beta_1 + 830) q^{34} + (70 \beta_{3} - 7 \beta_{2} - 91 \beta_1 - 623) q^{35} + ( - 12 \beta_{3} - 408) q^{36} + ( - 135 \beta_{2} - 104 \beta_1 - 135) q^{37} + ( - 181 \beta_{3} - 1028 \beta_{2} + 181 \beta_1 - 2056) q^{38} + (168 \beta_{3} + 714 \beta_{2} + 168 \beta_1) q^{39} + (284 \beta_{3} + 292 \beta_{2} + 142 \beta_1 - 292) q^{40} + ( - 42 \beta_{3} - 1596 \beta_{2} - 84 \beta_1 - 798) q^{41} + (21 \beta_{3} + 924 \beta_{2} - 336 \beta_1 + 1974) q^{42} + ( - 350 \beta_{3} + 618) q^{43} + (1206 \beta_{2} - 54 \beta_1 + 1206) q^{44} + ( - 54 \beta_{3} + 324 \beta_{2} + 54 \beta_1 + 648) q^{45} + (227 \beta_{3} - 1192 \beta_{2} + 227 \beta_1) q^{46} + ( - 374 \beta_{3} + 257 \beta_{2} - 187 \beta_1 - 257) q^{47} + (52 \beta_{3} - 776 \beta_{2} + 104 \beta_1 - 388) q^{48} + (294 \beta_{3} + 588 \beta_1 - 245) q^{49} + (406 \beta_{3} + 1892) q^{50} + ( - 2367 \beta_{2} + 225 \beta_1 - 2367) q^{51} + (140 \beta_{3} - 532 \beta_{2} - 140 \beta_1 - 1064) q^{52} + ( - 340 \beta_{3} + 2255 \beta_{2} - 340 \beta_1) q^{53} + ( - 270 \beta_{3} + 1836 \beta_{2} - 135 \beta_1 - 1836) q^{54} + ( - 143 \beta_{3} - 1786 \beta_{2} - 286 \beta_1 - 893) q^{55} + ( - 574 \beta_{3} + 3192 \beta_{2} - 84 \beta_1 + 1288) q^{56} + (432 \beta_{3} + 2763) q^{57} + ( - 452 \beta_{2} - 404 \beta_1 - 452) q^{58} + (449 \beta_{3} + 421 \beta_{2} - 449 \beta_1 + 842) q^{59} + ( - 378 \beta_{3} + 378 \beta_{2} - 378 \beta_1) q^{60} + (480 \beta_{3} + 47 \beta_{2} + 240 \beta_1 - 47) q^{61} + (661 \beta_{3} - 3688 \beta_{2} + 1322 \beta_1 - 1844) q^{62} + ( - 252 \beta_{3} - 1176 \beta_{2} - 210 \beta_1 - 672) q^{63} + ( - 976 \beta_{3} - 1368) q^{64} + (2898 \beta_{2} + 630 \beta_1 + 2898) q^{65} + ( - 321 \beta_{3} - 426 \beta_{2} + 321 \beta_1 - 852) q^{66} + (45 \beta_{3} + 659 \beta_{2} + 45 \beta_1) q^{67} + (1008 \beta_{3} - 3402 \beta_{2} + 504 \beta_1 + 3402) q^{68} + ( - 186 \beta_{3} + 2094 \beta_{2} - 372 \beta_1 + 1047) q^{69} + (175 \beta_{3} - 2296 \beta_{2} - 301 \beta_1 - 308) q^{70} + (238 \beta_{3} - 2602) q^{71} + ( - 648 \beta_{2} - 432 \beta_1 - 648) q^{72} + ( - 272 \beta_{3} + 869 \beta_{2} + 272 \beta_1 + 1738) q^{73} + (73 \beta_{3} - 2018 \beta_{2} + 73 \beta_1) q^{74} + ( - 692 \beta_{3} + 1606 \beta_{2} - 346 \beta_1 - 1606) q^{75} + ( - 798 \beta_{3} + 8652 \beta_{2} - 1596 \beta_1 + 4326) q^{76} + (560 \beta_{3} + 2009 \beta_{2} - 154 \beta_1 - 1218) q^{77} + (378 \beta_{3} - 2268) q^{78} + ( - 4055 \beta_{2} + 351 \beta_1 - 4055) q^{79} + (8 \beta_{3} - 524 \beta_{2} - 8 \beta_1 - 1048) q^{80} + (630 \beta_{3} - 4653 \beta_{2} + 630 \beta_1) q^{81} + ( - 1428 \beta_{3} + 672 \beta_{2} - 714 \beta_1 - 672) q^{82} + ( - 84 \beta_{3} + 3864 \beta_{2} - 168 \beta_1 + 1932) q^{83} + (1372 \beta_{3} - 8330 \beta_{2} + 1568 \beta_1 - 4018) q^{84} + (264 \beta_{3} - 3873) q^{85} + (6464 \beta_{2} - 82 \beta_1 + 6464) q^{86} + ( - 474 \beta_{3} + 996 \beta_{2} + 474 \beta_1 + 1992) q^{87} + (1234 \beta_{3} + 1456 \beta_{2} + 1234 \beta_1) q^{88} + ( - 752 \beta_{3} - 5665 \beta_{2} - 376 \beta_1 + 5665) q^{89} + (216 \beta_{3} + 1080 \beta_{2} + 432 \beta_1 + 540) q^{90} + ( - 1372 \beta_{3} - 4312 \beta_{2} - 980 \beta_1 + 2254) q^{91} + ( - 990 \beta_{3} - 5058) q^{92} + (3723 \beta_{2} - 1896 \beta_1 + 3723) q^{93} + (631 \beta_{3} + 4628 \beta_{2} - 631 \beta_1 + 9256) q^{94} + (87 \beta_{3} - 3279 \beta_{2} + 87 \beta_1) q^{95} + (1512 \beta_{3} + 756 \beta_1) q^{96} + (1274 \beta_{3} - 1372 \beta_{2} + 2548 \beta_1 - 686) q^{97} + ( - 1176 \beta_{3} + 6958 \beta_{2} - 833 \beta_1 - 5978) q^{98} + ( - 378 \beta_{3} + 2592) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{2} + 6 q^{3} - 20 q^{4} - 30 q^{5} + 304 q^{8} - 24 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{2} + 6 q^{3} - 20 q^{4} - 30 q^{5} + 304 q^{8} - 24 q^{9} - 204 q^{10} - 58 q^{11} - 588 q^{12} + 560 q^{14} + 468 q^{15} - 72 q^{16} - 246 q^{17} + 216 q^{18} + 642 q^{19} - 1050 q^{21} - 1264 q^{22} + 290 q^{23} + 720 q^{24} - 572 q^{25} + 1008 q^{26} - 28 q^{28} - 2176 q^{29} - 72 q^{30} + 3618 q^{31} + 1584 q^{32} + 2070 q^{33} - 2478 q^{35} - 1632 q^{36} - 270 q^{37} - 6168 q^{38} - 1428 q^{39} - 1752 q^{40} + 6048 q^{42} + 2472 q^{43} + 2412 q^{44} + 1944 q^{45} + 2384 q^{46} - 1542 q^{47} - 980 q^{49} + 7568 q^{50} - 4734 q^{51} - 3192 q^{52} - 4510 q^{53} - 11016 q^{54} - 1232 q^{56} + 11052 q^{57} - 904 q^{58} + 2526 q^{59} - 756 q^{60} - 282 q^{61} - 336 q^{63} - 5472 q^{64} + 5796 q^{65} - 2556 q^{66} - 1318 q^{67} + 20412 q^{68} + 3360 q^{70} - 10408 q^{71} - 1296 q^{72} + 5214 q^{73} + 4036 q^{74} - 9636 q^{75} - 8890 q^{77} - 9072 q^{78} - 8110 q^{79} - 3144 q^{80} + 9306 q^{81} - 4032 q^{82} + 588 q^{84} - 15492 q^{85} + 12928 q^{86} + 5976 q^{87} - 2912 q^{88} + 33990 q^{89} + 17640 q^{91} - 20232 q^{92} + 7446 q^{93} + 27768 q^{94} + 6558 q^{95} - 37828 q^{98} + 10368 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 22x^{2} + 484 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} ) / 22 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} ) / 22 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 22\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 22\beta_{3} \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/7\mathbb{Z}\right)^\times\).

\(n\) \(3\)
\(\chi(n)\) \(1 + \beta_{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3.1
−2.34521 4.06202i
2.34521 + 4.06202i
−2.34521 + 4.06202i
2.34521 4.06202i
−3.34521 5.79407i 8.53562 + 4.92804i −14.3808 + 24.9083i 6.57125 3.79391i 65.9413i −32.8329 + 36.3731i 85.3808 8.07125 + 13.9798i −43.9644 25.3828i
3.2 1.34521 + 2.32997i −5.53562 3.19599i 4.38083 7.58782i −21.5712 + 12.4542i 17.1971i 32.8329 + 36.3731i 66.6192 −20.0712 34.7644i −58.0356 33.5069i
5.1 −3.34521 + 5.79407i 8.53562 4.92804i −14.3808 24.9083i 6.57125 + 3.79391i 65.9413i −32.8329 36.3731i 85.3808 8.07125 13.9798i −43.9644 + 25.3828i
5.2 1.34521 2.32997i −5.53562 + 3.19599i 4.38083 + 7.58782i −21.5712 12.4542i 17.1971i 32.8329 36.3731i 66.6192 −20.0712 + 34.7644i −58.0356 + 33.5069i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 7.5.d.a 4
3.b odd 2 1 63.5.m.d 4
4.b odd 2 1 112.5.s.a 4
5.b even 2 1 175.5.i.a 4
5.c odd 4 2 175.5.j.a 8
7.b odd 2 1 49.5.d.b 4
7.c even 3 1 49.5.b.a 4
7.c even 3 1 49.5.d.b 4
7.d odd 6 1 inner 7.5.d.a 4
7.d odd 6 1 49.5.b.a 4
21.g even 6 1 63.5.m.d 4
21.g even 6 1 441.5.d.d 4
21.h odd 6 1 441.5.d.d 4
28.f even 6 1 112.5.s.a 4
28.f even 6 1 784.5.c.c 4
28.g odd 6 1 784.5.c.c 4
35.i odd 6 1 175.5.i.a 4
35.k even 12 2 175.5.j.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7.5.d.a 4 1.a even 1 1 trivial
7.5.d.a 4 7.d odd 6 1 inner
49.5.b.a 4 7.c even 3 1
49.5.b.a 4 7.d odd 6 1
49.5.d.b 4 7.b odd 2 1
49.5.d.b 4 7.c even 3 1
63.5.m.d 4 3.b odd 2 1
63.5.m.d 4 21.g even 6 1
112.5.s.a 4 4.b odd 2 1
112.5.s.a 4 28.f even 6 1
175.5.i.a 4 5.b even 2 1
175.5.i.a 4 35.i odd 6 1
175.5.j.a 8 5.c odd 4 2
175.5.j.a 8 35.k even 12 2
441.5.d.d 4 21.g even 6 1
441.5.d.d 4 21.h odd 6 1
784.5.c.c 4 28.f even 6 1
784.5.c.c 4 28.g odd 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{5}^{\mathrm{new}}(7, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 4 T^{3} + 34 T^{2} - 72 T + 324 \) Copy content Toggle raw display
$3$ \( T^{4} - 6 T^{3} - 51 T^{2} + \cdots + 3969 \) Copy content Toggle raw display
$5$ \( T^{4} + 30 T^{3} + 111 T^{2} + \cdots + 35721 \) Copy content Toggle raw display
$7$ \( T^{4} + 490 T^{2} + 5764801 \) Copy content Toggle raw display
$11$ \( T^{4} + 58 T^{3} + 8881 T^{2} + \cdots + 30437289 \) Copy content Toggle raw display
$13$ \( T^{4} + 55272 T^{2} + \cdots + 3111696 \) Copy content Toggle raw display
$17$ \( T^{4} + 246 T^{3} + \cdots + 5076990009 \) Copy content Toggle raw display
$19$ \( T^{4} - 642 T^{3} + \cdots + 3136784049 \) Copy content Toggle raw display
$23$ \( T^{4} - 290 T^{3} + \cdots + 254625849 \) Copy content Toggle raw display
$29$ \( (T^{2} + 1088 T + 188136)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} - 3618 T^{3} + \cdots + 1071889573041 \) Copy content Toggle raw display
$37$ \( T^{4} + 270 T^{3} + \cdots + 48279954529 \) Copy content Toggle raw display
$41$ \( T^{4} + 4053672 T^{2} + \cdots + 3218392944144 \) Copy content Toggle raw display
$43$ \( (T^{2} - 1236 T - 2313076)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + 1542 T^{3} + \cdots + 4451285577249 \) Copy content Toggle raw display
$53$ \( T^{4} + 4510 T^{3} + \cdots + 6460874330625 \) Copy content Toggle raw display
$59$ \( T^{4} + \cdots + 163173619767249 \) Copy content Toggle raw display
$61$ \( T^{4} + 282 T^{3} + \cdots + 14401820070729 \) Copy content Toggle raw display
$67$ \( T^{4} + 1318 T^{3} + \cdots + 151890252361 \) Copy content Toggle raw display
$71$ \( (T^{2} + 5204 T + 5524236)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} - 5214 T^{3} + \cdots + 6851102086521 \) Copy content Toggle raw display
$79$ \( T^{4} + \cdots + 188584385155609 \) Copy content Toggle raw display
$83$ \( T^{4} + \cdots + 115179601694976 \) Copy content Toggle raw display
$89$ \( T^{4} - 33990 T^{3} + \cdots + 75\!\cdots\!81 \) Copy content Toggle raw display
$97$ \( T^{4} + 217069608 T^{2} + \cdots + 11\!\cdots\!84 \) Copy content Toggle raw display
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