Properties

Label 7.15.d.a
Level $7$
Weight $15$
Character orbit 7.d
Analytic conductor $8.703$
Analytic rank $0$
Dimension $16$
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] = [7,15,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, 15, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("7.3");
 
S:= CuspForms(chi, 15);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 7 \)
Weight: \( k \) \(=\) \( 15 \)
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: \(8.70302777063\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 4 x^{15} + 85634 x^{14} - 1484856 x^{13} + 5100567568 x^{12} - 86634496244 x^{11} + 158667006353068 x^{10} + \cdots + 39\!\cdots\!81 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{24}\cdot 3^{8}\cdot 7^{14} \)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{3} + 11 \beta_{2} + \beta_1 - 1) q^{2} + (\beta_{6} - \beta_{5} - 2 \beta_{3} - 91 \beta_{2} - \beta_1 + 92) q^{3} + ( - \beta_{9} - \beta_{6} - 5148 \beta_{2} - 12 \beta_1 - 5149) q^{4} + (\beta_{11} - \beta_{9} - \beta_{8} + \beta_{7} + 4 \beta_{5} + 46 \beta_{3} - 49 \beta_{2} + \cdots - 140) q^{5}+ \cdots + (\beta_{15} + 3 \beta_{14} + 4 \beta_{13} + \beta_{12} - 3 \beta_{11} + 4 \beta_{10} - 24 \beta_{9} + \cdots + 2308) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{3} + 11 \beta_{2} + \beta_1 - 1) q^{2} + (\beta_{6} - \beta_{5} - 2 \beta_{3} - 91 \beta_{2} - \beta_1 + 92) q^{3} + ( - \beta_{9} - \beta_{6} - 5148 \beta_{2} - 12 \beta_1 - 5149) q^{4} + (\beta_{11} - \beta_{9} - \beta_{8} + \beta_{7} + 4 \beta_{5} + 46 \beta_{3} - 49 \beta_{2} + \cdots - 140) q^{5}+ \cdots + ( - 1584543 \beta_{15} + 13286812 \beta_{14} + \cdots - 40222062855063) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 92 q^{2} + 2184 q^{3} - 41236 q^{4} - 1680 q^{5} - 1237432 q^{7} + 1870496 q^{8} + 7242264 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 92 q^{2} + 2184 q^{3} - 41236 q^{4} - 1680 q^{5} - 1237432 q^{7} + 1870496 q^{8} + 7242264 q^{9} + 23132340 q^{10} - 4587872 q^{11} + 69092268 q^{12} + 55670944 q^{14} - 223427040 q^{15} + 555097336 q^{16} - 1336830432 q^{17} + 475650384 q^{18} + 1925554176 q^{19} - 6528716712 q^{21} - 2649001728 q^{22} + 5352602296 q^{23} - 8686607832 q^{24} + 4550509840 q^{25} + 10636248168 q^{26} + 14250310732 q^{28} + 15354837712 q^{29} + 12847273680 q^{30} + 38066141568 q^{31} + 27950170656 q^{32} - 170902903752 q^{33} + 71627246880 q^{35} - 139843951104 q^{36} - 168016938776 q^{37} + 83559246960 q^{38} + 3136606200 q^{39} + 268041382560 q^{40} + 117398174712 q^{42} - 722232426464 q^{43} + 410527899972 q^{44} - 517892581080 q^{45} + 914356199712 q^{46} + 2038558764144 q^{47} - 1045889186720 q^{49} - 4711294150160 q^{50} + 1635252469056 q^{51} + 1710632683032 q^{52} - 129224247224 q^{53} + 8837224278624 q^{54} - 4518465261520 q^{56} - 16456353306768 q^{57} - 468045611952 q^{58} + 6357829627128 q^{59} + 5550712719300 q^{60} + 9622511463768 q^{61} - 8740195548504 q^{63} - 44332488949216 q^{64} + 2985448032840 q^{65} + 44228462203548 q^{66} + 8708745371912 q^{67} + 48964599457452 q^{68} - 40892055741600 q^{70} - 77950624797440 q^{71} + 11978163920784 q^{72} + 56185530197832 q^{73} + 13051980699356 q^{74} + 132383042747760 q^{75} - 65818138480976 q^{77} - 332756612190912 q^{78} + 31180109483480 q^{79} + 67254236074200 q^{80} + 57312114745392 q^{81} + 294813463771512 q^{82} - 157122163346820 q^{84} - 385028327095920 q^{85} + 93251953307840 q^{86} + 133056855170568 q^{87} + 103912717897464 q^{88} + 303521059372440 q^{89} - 61817753170992 q^{91} - 591270018604728 q^{92} - 8125955668296 q^{93} + 297888933759984 q^{94} + 32985525486360 q^{95} + 540224642064240 q^{96} - 206078649040292 q^{98} - 642933959965104 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 4 x^{15} + 85634 x^{14} - 1484856 x^{13} + 5100567568 x^{12} - 86634496244 x^{11} + 158667006353068 x^{10} + \cdots + 39\!\cdots\!81 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 55\!\cdots\!08 \nu^{15} + \cdots - 92\!\cdots\!52 ) / 50\!\cdots\!15 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 19\!\cdots\!58 \nu^{15} + \cdots - 26\!\cdots\!81 ) / 78\!\cdots\!55 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 27\!\cdots\!70 \nu^{15} + \cdots + 73\!\cdots\!61 ) / 20\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 45\!\cdots\!97 \nu^{15} + \cdots + 78\!\cdots\!02 ) / 42\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 22\!\cdots\!07 \nu^{15} + \cdots + 11\!\cdots\!27 ) / 60\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 19\!\cdots\!54 \nu^{15} + \cdots + 96\!\cdots\!09 ) / 42\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 48\!\cdots\!86 \nu^{15} + \cdots + 11\!\cdots\!11 ) / 29\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 79\!\cdots\!63 \nu^{15} + \cdots - 51\!\cdots\!59 ) / 29\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 15\!\cdots\!30 \nu^{15} + \cdots + 17\!\cdots\!57 ) / 29\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 20\!\cdots\!15 \nu^{15} + \cdots - 85\!\cdots\!83 ) / 29\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 45\!\cdots\!67 \nu^{15} + \cdots + 57\!\cdots\!51 ) / 29\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 20\!\cdots\!97 \nu^{15} + \cdots - 78\!\cdots\!73 ) / 29\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 63\!\cdots\!02 \nu^{15} + \cdots + 22\!\cdots\!93 ) / 29\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 83\!\cdots\!55 \nu^{15} + \cdots - 78\!\cdots\!31 ) / 29\!\cdots\!60 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{9} + \beta_{7} + 2\beta_{6} - \beta_{5} - 11\beta_{3} + 21411\beta_{2} - 10\beta _1 + 11 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{14} - \beta_{13} - \beta_{12} - 2 \beta_{11} - \beta_{10} + \beta_{8} - 33 \beta_{7} + 34 \beta_{6} - 104 \beta_{5} - 2 \beta_{4} + 33331 \beta_{3} + 2 \beta_{2} - 69 \beta _1 + 197489 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 38 \beta_{15} - 38 \beta_{14} - 8 \beta_{13} + 8 \beta_{12} + 766 \beta_{11} - 368 \beta_{10} - 40984 \beta_{9} - 1532 \beta_{8} + 582 \beta_{7} - 59192 \beta_{6} - 17388 \beta_{5} - 184 \beta_{4} - 715126423 \beta_{2} + \cdots - 715185025 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 58018 \beta_{15} + 824 \beta_{14} + 58842 \beta_{13} + 58018 \beta_{12} + 51170 \beta_{11} - 47792 \beta_{10} + 1868434 \beta_{9} + 51170 \beta_{8} + 1820642 \beta_{7} - 9518202 \beta_{6} + \cdots + 1214584433 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 146632 \beta_{15} + 2193570 \beta_{14} - 2193570 \beta_{13} - 2340202 \beta_{12} - 92780164 \beta_{11} + 13431380 \beta_{10} + 59821462 \beta_{9} + 46390082 \beta_{8} + \cdots + 25967439445301 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 2812412231 \beta_{15} - 2812412231 \beta_{14} + 6957416 \beta_{13} - 6957416 \beta_{12} + 2592249519 \beta_{11} + 3520630410 \beta_{10} - 92033279232 \beta_{9} + \cdots - 11\!\cdots\!14 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 136258041276 \beta_{15} - 4533204720 \beta_{14} + 131724836556 \beta_{13} + 136258041276 \beta_{12} + 2271049362300 \beta_{11} + 728650629552 \beta_{10} + \cdots + 25\!\cdots\!60 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 2494215902352 \beta_{15} + 129121236637012 \beta_{14} - 129121236637012 \beta_{13} - 126627020734660 \beta_{12} - 274945717745576 \beta_{11} + \cdots + 53\!\cdots\!59 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 71\!\cdots\!20 \beta_{15} + \cdots - 39\!\cdots\!84 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 57\!\cdots\!97 \beta_{15} + \cdots + 72\!\cdots\!69 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 40\!\cdots\!48 \beta_{15} + \cdots + 15\!\cdots\!45 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 24\!\cdots\!58 \beta_{15} + \cdots - 11\!\cdots\!12 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 20\!\cdots\!90 \beta_{15} + \cdots + 23\!\cdots\!31 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 58\!\cdots\!88 \beta_{15} + \cdots + 48\!\cdots\!09 \) Copy content Toggle raw display

Character values

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

\(n\) \(3\)
\(\chi(n)\) \(-\beta_{2}\)

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
99.6079 172.526i
75.4148 130.622i
73.8531 127.917i
4.32091 7.48404i
−7.15615 + 12.3948i
−47.0463 + 81.4867i
−92.8856 + 160.882i
−104.109 + 180.322i
99.6079 + 172.526i
75.4148 + 130.622i
73.8531 + 127.917i
4.32091 + 7.48404i
−7.15615 12.3948i
−47.0463 81.4867i
−92.8856 160.882i
−104.109 180.322i
−105.108 182.052i −2073.47 1197.12i −13903.4 + 24081.3i −42557.3 + 24570.5i 503307.i −425296. 705228.i 2.40123e6 474710. + 822222.i 8.94621e6 + 5.16510e6i
3.2 −80.9148 140.148i 3151.06 + 1819.27i −4902.40 + 8491.21i −92594.8 + 53459.6i 588822.i −746488. + 347821.i −1.06471e6 4.22798e6 + 7.32307e6i 1.49846e7 + 8.65135e6i
3.3 −79.3531 137.444i 619.565 + 357.706i −4401.84 + 7624.22i 110713. 63919.9i 113540.i 784145. + 251675.i −1.20304e6 −2.13558e6 3.69893e6i −1.75708e7 1.01445e7i
3.4 −9.82091 17.0103i −2131.70 1230.74i 7999.10 13854.8i 1808.28 1044.01i 48347.8i −454468. + 686791.i −636045. 637942. + 1.10495e6i −35517.9 20506.3i
3.5 1.65615 + 2.86854i 527.809 + 304.731i 8186.51 14179.5i −65578.3 + 37861.7i 2018.72i 652201. 502849.i 108501. −2.20576e6 3.82049e6i −217216. 125409.i
3.6 41.5463 + 71.9604i 2597.81 + 1499.84i 4739.80 8209.58i 95097.7 54904.7i 249252.i −660543. 491840.i 2.14908e6 2.10758e6 + 3.65044e6i 7.90192e6 + 4.56218e6i
3.7 87.3856 + 151.356i −2625.37 1515.76i −7080.47 + 12263.7i 40938.7 23636.0i 529821.i 156750. 808488.i 388527. 2.20356e6 + 3.81667e6i 7.15491e6 + 4.13089e6i
3.8 98.6087 + 170.795i 1026.30 + 592.534i −11255.4 + 19494.8i −48666.8 + 28097.8i 233716.i 74982.8 + 820122.i −1.20829e6 −1.68929e6 2.92594e6i −9.59794e6 5.54137e6i
5.1 −105.108 + 182.052i −2073.47 + 1197.12i −13903.4 24081.3i −42557.3 24570.5i 503307.i −425296. + 705228.i 2.40123e6 474710. 822222.i 8.94621e6 5.16510e6i
5.2 −80.9148 + 140.148i 3151.06 1819.27i −4902.40 8491.21i −92594.8 53459.6i 588822.i −746488. 347821.i −1.06471e6 4.22798e6 7.32307e6i 1.49846e7 8.65135e6i
5.3 −79.3531 + 137.444i 619.565 357.706i −4401.84 7624.22i 110713. + 63919.9i 113540.i 784145. 251675.i −1.20304e6 −2.13558e6 + 3.69893e6i −1.75708e7 + 1.01445e7i
5.4 −9.82091 + 17.0103i −2131.70 + 1230.74i 7999.10 + 13854.8i 1808.28 + 1044.01i 48347.8i −454468. 686791.i −636045. 637942. 1.10495e6i −35517.9 + 20506.3i
5.5 1.65615 2.86854i 527.809 304.731i 8186.51 + 14179.5i −65578.3 37861.7i 2018.72i 652201. + 502849.i 108501. −2.20576e6 + 3.82049e6i −217216. + 125409.i
5.6 41.5463 71.9604i 2597.81 1499.84i 4739.80 + 8209.58i 95097.7 + 54904.7i 249252.i −660543. + 491840.i 2.14908e6 2.10758e6 3.65044e6i 7.90192e6 4.56218e6i
5.7 87.3856 151.356i −2625.37 + 1515.76i −7080.47 12263.7i 40938.7 + 23636.0i 529821.i 156750. + 808488.i 388527. 2.20356e6 3.81667e6i 7.15491e6 4.13089e6i
5.8 98.6087 170.795i 1026.30 592.534i −11255.4 19494.8i −48666.8 28097.8i 233716.i 74982.8 820122.i −1.20829e6 −1.68929e6 + 2.92594e6i −9.59794e6 + 5.54137e6i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3.8
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 7.15.d.a 16
3.b odd 2 1 63.15.m.b 16
7.b odd 2 1 49.15.d.c 16
7.c even 3 1 49.15.b.a 16
7.c even 3 1 49.15.d.c 16
7.d odd 6 1 inner 7.15.d.a 16
7.d odd 6 1 49.15.b.a 16
21.g even 6 1 63.15.m.b 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7.15.d.a 16 1.a even 1 1 trivial
7.15.d.a 16 7.d odd 6 1 inner
49.15.b.a 16 7.c even 3 1
49.15.b.a 16 7.d odd 6 1
49.15.d.c 16 7.b odd 2 1
49.15.d.c 16 7.c even 3 1
63.15.m.b 16 3.b odd 2 1
63.15.m.b 16 21.g even 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{15}^{\mathrm{new}}(7, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} + 92 T^{15} + \cdots + 10\!\cdots\!64 \) Copy content Toggle raw display
$3$ \( T^{16} - 2184 T^{15} + \cdots + 10\!\cdots\!49 \) Copy content Toggle raw display
$5$ \( T^{16} + 1680 T^{15} + \cdots + 95\!\cdots\!25 \) Copy content Toggle raw display
$7$ \( T^{16} + 1237432 T^{15} + \cdots + 44\!\cdots\!01 \) Copy content Toggle raw display
$11$ \( T^{16} + 4587872 T^{15} + \cdots + 26\!\cdots\!61 \) Copy content Toggle raw display
$13$ \( T^{16} + \cdots + 21\!\cdots\!00 \) Copy content Toggle raw display
$17$ \( T^{16} + 1336830432 T^{15} + \cdots + 14\!\cdots\!49 \) Copy content Toggle raw display
$19$ \( T^{16} - 1925554176 T^{15} + \cdots + 16\!\cdots\!21 \) Copy content Toggle raw display
$23$ \( T^{16} - 5352602296 T^{15} + \cdots + 13\!\cdots\!49 \) Copy content Toggle raw display
$29$ \( (T^{8} - 7677418856 T^{7} + \cdots - 97\!\cdots\!64)^{2} \) Copy content Toggle raw display
$31$ \( T^{16} - 38066141568 T^{15} + \cdots + 72\!\cdots\!01 \) Copy content Toggle raw display
$37$ \( T^{16} + 168016938776 T^{15} + \cdots + 15\!\cdots\!49 \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 79\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( (T^{8} + 361116213232 T^{7} + \cdots - 22\!\cdots\!00)^{2} \) Copy content Toggle raw display
$47$ \( T^{16} - 2038558764144 T^{15} + \cdots + 88\!\cdots\!29 \) Copy content Toggle raw display
$53$ \( T^{16} + 129224247224 T^{15} + \cdots + 67\!\cdots\!25 \) Copy content Toggle raw display
$59$ \( T^{16} - 6357829627128 T^{15} + \cdots + 49\!\cdots\!81 \) Copy content Toggle raw display
$61$ \( T^{16} - 9622511463768 T^{15} + \cdots + 52\!\cdots\!01 \) Copy content Toggle raw display
$67$ \( T^{16} - 8708745371912 T^{15} + \cdots + 11\!\cdots\!09 \) Copy content Toggle raw display
$71$ \( (T^{8} + 38975312398720 T^{7} + \cdots - 10\!\cdots\!24)^{2} \) Copy content Toggle raw display
$73$ \( T^{16} - 56185530197832 T^{15} + \cdots + 10\!\cdots\!29 \) Copy content Toggle raw display
$79$ \( T^{16} - 31180109483480 T^{15} + \cdots + 32\!\cdots\!61 \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 38\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( T^{16} - 303521059372440 T^{15} + \cdots + 60\!\cdots\!41 \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 25\!\cdots\!00 \) Copy content Toggle raw display
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