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Properties

Label 70.3.h.a
Level $70$
Weight $3$
Character orbit 70.h
Analytic conductor $1.907$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [70,3,Mod(19,70)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(70, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 5]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("70.19");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 70 = 2 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 70.h (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: \(1.90736185052\)
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: \( x^{16} + 42x^{14} + 1322x^{12} + 17616x^{10} + 175407x^{8} + 205392x^{6} + 203018x^{4} + 23226x^{2} + 2401 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{10} q^{2} + \beta_1 q^{3} + 2 \beta_{4} q^{4} + (\beta_{10} + \beta_{9} - \beta_{6}) q^{5} + (\beta_{4} + \beta_{3}) q^{6} + ( - \beta_{15} + \beta_{14} + \cdots - \beta_1) q^{7}+ \cdots + (\beta_{15} + \beta_{14} - \beta_{12} + \cdots - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{10} q^{2} + \beta_1 q^{3} + 2 \beta_{4} q^{4} + (\beta_{10} + \beta_{9} - \beta_{6}) q^{5} + (\beta_{4} + \beta_{3}) q^{6} + ( - \beta_{15} + \beta_{14} + \cdots - \beta_1) q^{7}+ \cdots + ( - 12 \beta_{13} - 2 \beta_{12} + \cdots + 12) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 16 q^{4} - 6 q^{5} - 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 16 q^{4} - 6 q^{5} - 12 q^{9} + 24 q^{10} - 12 q^{11} + 32 q^{14} - 28 q^{15} - 32 q^{16} - 12 q^{19} - 8 q^{21} - 24 q^{24} - 42 q^{25} - 48 q^{26} - 136 q^{29} + 32 q^{30} + 84 q^{31} - 190 q^{35} - 48 q^{36} + 312 q^{39} + 48 q^{40} + 24 q^{44} + 384 q^{45} - 68 q^{46} + 296 q^{49} - 96 q^{50} - 76 q^{51} - 108 q^{54} + 56 q^{56} - 372 q^{59} - 28 q^{60} + 348 q^{61} - 128 q^{64} - 104 q^{65} + 480 q^{66} - 296 q^{70} - 384 q^{71} + 208 q^{74} - 150 q^{75} + 148 q^{79} + 24 q^{80} - 140 q^{81} + 256 q^{84} + 580 q^{85} - 188 q^{86} - 912 q^{89} - 616 q^{91} - 600 q^{94} - 310 q^{95} - 48 q^{96} + 176 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 42x^{14} + 1322x^{12} + 17616x^{10} + 175407x^{8} + 205392x^{6} + 203018x^{4} + 23226x^{2} + 2401 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 8513 \nu^{14} + 359634 \nu^{12} + 11357045 \nu^{10} + 153331333 \nu^{8} + 1546485419 \nu^{6} + \cdots + 936322387 ) / 1257206580 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 14453580128 \nu^{14} - 607935008947 \nu^{12} - 19141328049185 \nu^{10} + \cdots - 459431336884611 ) / 585273665220300 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 7057106506 \nu^{14} + 295583461054 \nu^{12} + 9296166084420 \nu^{10} + \cdots + 162114054724727 ) / 146318416305075 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 7057106506 \nu^{15} + 295583461054 \nu^{13} + 9296166084420 \nu^{11} + \cdots + 162114054724727 \nu ) / 146318416305075 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 414649037697 \nu^{15} - 897176459200 \nu^{14} + 17831963178078 \nu^{13} + \cdots - 45\!\cdots\!40 ) / 81\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 584450652928 \nu^{15} - 1322157532080 \nu^{14} + 24908885259307 \nu^{13} + \cdots + 18\!\cdots\!70 ) / 81\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 584450652928 \nu^{15} + 1322157532080 \nu^{14} + 24908885259307 \nu^{13} + \cdots - 18\!\cdots\!70 ) / 81\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 6548657 \nu^{15} - 275460731 \nu^{13} - 8674946620 \nu^{11} - 115917636917 \nu^{9} + \cdots - 312949197288 \nu ) / 61603122420 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 551092950479 \nu^{15} - 23049479998846 \nu^{13} - 724524088686355 \nu^{11} + \cdots + 86\!\cdots\!27 \nu ) / 40\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 167119591406 \nu^{15} - 145202960288 \nu^{14} + 7111449957199 \nu^{13} + \cdots - 15\!\cdots\!41 ) / 11\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 932384285727 \nu^{15} + 2614531955616 \nu^{14} + 39022037916463 \nu^{13} + \cdots + 35\!\cdots\!82 ) / 81\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 241081769696 \nu^{15} + 43676687152 \nu^{14} - 10138603491254 \nu^{13} + \cdots - 42\!\cdots\!51 ) / 11\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 301793351536 \nu^{15} + 261683272705 \nu^{14} + 12623362736544 \nu^{13} + \cdots + 902525146627580 ) / 11\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 642947872342 \nu^{15} + 366356581787 \nu^{14} - 26892499830700 \nu^{13} + \cdots + 12\!\cdots\!12 ) / 16\!\cdots\!40 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{15} + \beta_{14} - \beta_{12} - 2 \beta_{10} - 2 \beta_{8} + \beta_{7} + \beta_{6} + 9 \beta_{4} + \cdots - 10 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2 \beta_{15} - 2 \beta_{14} + \beta_{13} + \beta_{12} - \beta_{11} - 2 \beta_{10} - \beta_{9} + \cdots - 23 \beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 24 \beta_{15} - 24 \beta_{14} + 24 \beta_{13} + 44 \beta_{12} + 24 \beta_{11} + 44 \beta_{10} + \cdots + 16 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 38 \beta_{15} + 38 \beta_{14} - 76 \beta_{13} - 24 \beta_{12} + 76 \beta_{11} + 18 \beta_{10} + \cdots + 24 \beta_{4} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 559 \beta_{13} - 497 \beta_{12} - 559 \beta_{11} + 62 \beta_{10} + 497 \beta_{8} - 435 \beta_{7} + \cdots + 4375 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 1161 \beta_{15} + 1161 \beta_{14} + 1161 \beta_{13} - 1161 \beta_{11} + 5205 \beta_{10} + \cdots + 12785 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 13344 \beta_{15} + 13344 \beta_{14} - 11624 \beta_{12} - 24968 \beta_{10} - 23248 \beta_{8} + \cdots - 106481 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 64880 \beta_{15} - 64880 \beta_{14} + 32440 \beta_{13} + 13344 \beta_{12} - 32440 \beta_{11} + \cdots - 311353 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 324697 \beta_{15} - 324697 \beta_{14} + 324697 \beta_{13} + 557826 \beta_{12} + 324697 \beta_{11} + \cdots + 30513 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 867281 \beta_{15} + 867281 \beta_{14} - 1734562 \beta_{13} - 324697 \beta_{12} + \cdots + 324697 \beta_{4} \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 8003016 \beta_{13} - 6811038 \beta_{12} - 8003016 \beta_{11} + 1191978 \beta_{10} + 6811038 \beta_{8} + \cdots + 61517287 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 22635774 \beta_{15} + 22635774 \beta_{14} + 22635774 \beta_{13} - 22635774 \beta_{11} + \cdots + 190905247 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 198908263 \beta_{15} + 198908263 \beta_{14} - 168269473 \beta_{12} - 367177736 \beta_{10} + \cdots - 1516980034 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 1165367954 \beta_{15} - 1165367954 \beta_{14} + 582683977 \beta_{13} + 198908263 \beta_{12} + \cdots - 4771158233 \beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/70\mathbb{Z}\right)^\times\).

\(n\) \(31\) \(57\)
\(\chi(n)\) \(1 - \beta_{4}\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1
−2.51048 − 4.34828i
−0.518845 − 0.898665i
−0.170094 − 0.294611i
1.97468 + 3.42024i
−1.97468 − 3.42024i
0.170094 + 0.294611i
0.518845 + 0.898665i
2.51048 + 4.34828i
−2.51048 + 4.34828i
−0.518845 + 0.898665i
−0.170094 + 0.294611i
1.97468 − 3.42024i
−1.97468 + 3.42024i
0.170094 − 0.294611i
0.518845 − 0.898665i
2.51048 − 4.34828i
−1.22474 − 0.707107i −2.51048 − 4.34828i 1.00000 + 1.73205i −4.93710 + 0.790609i 7.10072i 2.74755 + 6.43824i − 2.82843i −8.10505 + 14.0384i 6.60573 + 2.52276i
19.2 −1.22474 − 0.707107i −0.518845 − 0.898665i 1.00000 + 1.73205i 1.98086 − 4.59088i 1.46751i −6.67941 − 2.09417i − 2.82843i 3.96160 − 6.86169i −5.67229 + 4.22198i
19.3 −1.22474 − 0.707107i −0.170094 − 0.294611i 1.00000 + 1.73205i 2.91706 + 4.06088i 0.481097i 6.56322 − 2.43396i − 2.82843i 4.44214 − 7.69401i −0.701175 − 7.03622i
19.4 −1.22474 − 0.707107i 1.97468 + 3.42024i 1.00000 + 1.73205i −3.91031 + 3.11601i − 5.58523i −6.30560 + 3.03963i − 2.82843i −3.29869 + 5.71350i 6.99248 − 1.05131i
19.5 1.22474 + 0.707107i −1.97468 − 3.42024i 1.00000 + 1.73205i 0.743387 − 4.94443i − 5.58523i 6.30560 − 3.03963i 2.82843i −3.29869 + 5.71350i 4.40670 − 5.53001i
19.6 1.22474 + 0.707107i 0.170094 + 0.294611i 1.00000 + 1.73205i 4.97536 + 0.495806i 0.481097i −6.56322 + 2.43396i 2.82843i 4.44214 − 7.69401i 5.74296 + 4.12534i
19.7 1.22474 + 0.707107i 0.518845 + 0.898665i 1.00000 + 1.73205i −2.98539 + 4.01091i 1.46751i 6.67941 + 2.09417i 2.82843i 3.96160 − 6.86169i −6.49249 + 2.80135i
19.8 1.22474 + 0.707107i 2.51048 + 4.34828i 1.00000 + 1.73205i −1.78386 − 4.67096i 7.10072i −2.74755 − 6.43824i 2.82843i −8.10505 + 14.0384i 1.11809 − 6.98211i
59.1 −1.22474 + 0.707107i −2.51048 + 4.34828i 1.00000 − 1.73205i −4.93710 − 0.790609i − 7.10072i 2.74755 − 6.43824i 2.82843i −8.10505 − 14.0384i 6.60573 − 2.52276i
59.2 −1.22474 + 0.707107i −0.518845 + 0.898665i 1.00000 − 1.73205i 1.98086 + 4.59088i − 1.46751i −6.67941 + 2.09417i 2.82843i 3.96160 + 6.86169i −5.67229 − 4.22198i
59.3 −1.22474 + 0.707107i −0.170094 + 0.294611i 1.00000 − 1.73205i 2.91706 − 4.06088i − 0.481097i 6.56322 + 2.43396i 2.82843i 4.44214 + 7.69401i −0.701175 + 7.03622i
59.4 −1.22474 + 0.707107i 1.97468 − 3.42024i 1.00000 − 1.73205i −3.91031 − 3.11601i 5.58523i −6.30560 − 3.03963i 2.82843i −3.29869 − 5.71350i 6.99248 + 1.05131i
59.5 1.22474 − 0.707107i −1.97468 + 3.42024i 1.00000 − 1.73205i 0.743387 + 4.94443i 5.58523i 6.30560 + 3.03963i − 2.82843i −3.29869 − 5.71350i 4.40670 + 5.53001i
59.6 1.22474 − 0.707107i 0.170094 − 0.294611i 1.00000 − 1.73205i 4.97536 − 0.495806i − 0.481097i −6.56322 − 2.43396i − 2.82843i 4.44214 + 7.69401i 5.74296 − 4.12534i
59.7 1.22474 − 0.707107i 0.518845 − 0.898665i 1.00000 − 1.73205i −2.98539 − 4.01091i − 1.46751i 6.67941 − 2.09417i − 2.82843i 3.96160 + 6.86169i −6.49249 − 2.80135i
59.8 1.22474 − 0.707107i 2.51048 − 4.34828i 1.00000 − 1.73205i −1.78386 + 4.67096i − 7.10072i −2.74755 + 6.43824i − 2.82843i −8.10505 − 14.0384i 1.11809 + 6.98211i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 19.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
7.d odd 6 1 inner
35.i odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 70.3.h.a ✓ 16
3.b odd 2 1 630.3.bc.a 16
4.b odd 2 1 560.3.br.b 16
5.b even 2 1 inner 70.3.h.a ✓ 16
5.c odd 4 2 350.3.k.e 16
7.b odd 2 1 490.3.h.b 16
7.c even 3 1 490.3.d.a 16
7.c even 3 1 490.3.h.b 16
7.d odd 6 1 inner 70.3.h.a ✓ 16
7.d odd 6 1 490.3.d.a 16
15.d odd 2 1 630.3.bc.a 16
20.d odd 2 1 560.3.br.b 16
21.g even 6 1 630.3.bc.a 16
28.f even 6 1 560.3.br.b 16
35.c odd 2 1 490.3.h.b 16
35.i odd 6 1 inner 70.3.h.a ✓ 16
35.i odd 6 1 490.3.d.a 16
35.j even 6 1 490.3.d.a 16
35.j even 6 1 490.3.h.b 16
35.k even 12 2 350.3.k.e 16
105.p even 6 1 630.3.bc.a 16
140.s even 6 1 560.3.br.b 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
70.3.h.a ✓ 16 1.a even 1 1 trivial
70.3.h.a ✓ 16 5.b even 2 1 inner
70.3.h.a ✓ 16 7.d odd 6 1 inner
70.3.h.a ✓ 16 35.i odd 6 1 inner
350.3.k.e 16 5.c odd 4 2
350.3.k.e 16 35.k even 12 2
490.3.d.a 16 7.c even 3 1
490.3.d.a 16 7.d odd 6 1
490.3.d.a 16 35.i odd 6 1
490.3.d.a 16 35.j even 6 1
490.3.h.b 16 7.b odd 2 1
490.3.h.b 16 7.c even 3 1
490.3.h.b 16 35.c odd 2 1
490.3.h.b 16 35.j even 6 1
560.3.br.b 16 4.b odd 2 1
560.3.br.b 16 20.d odd 2 1
560.3.br.b 16 28.f even 6 1
560.3.br.b 16 140.s even 6 1
630.3.bc.a 16 3.b odd 2 1
630.3.bc.a 16 15.d odd 2 1
630.3.bc.a 16 21.g even 6 1
630.3.bc.a 16 105.p even 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(70, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - 2 T^{2} + 4)^{4} \) Copy content Toggle raw display
$3$ \( T^{16} + 42 T^{14} + \cdots + 2401 \) Copy content Toggle raw display
$5$ \( T^{16} + \cdots + 152587890625 \) Copy content Toggle raw display
$7$ \( T^{16} + \cdots + 33232930569601 \) Copy content Toggle raw display
$11$ \( (T^{8} + 6 T^{7} + \cdots + 51380224)^{2} \) Copy content Toggle raw display
$13$ \( (T^{8} - 672 T^{6} + \cdots + 692224)^{2} \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 12\!\cdots\!56 \) Copy content Toggle raw display
$19$ \( (T^{8} + 6 T^{7} + \cdots + 14256360000)^{2} \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 19\!\cdots\!01 \) Copy content Toggle raw display
$29$ \( (T^{4} + 34 T^{3} + \cdots - 911816)^{4} \) Copy content Toggle raw display
$31$ \( (T^{8} - 42 T^{7} + \cdots + 295266171456)^{2} \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 11\!\cdots\!76 \) Copy content Toggle raw display
$41$ \( (T^{8} + 5210 T^{6} + \cdots + 758592224784)^{2} \) Copy content Toggle raw display
$43$ \( (T^{8} + \cdots + 2602827235584)^{2} \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 14\!\cdots\!16 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 35\!\cdots\!96 \) Copy content Toggle raw display
$59$ \( (T^{8} + 186 T^{7} + \cdots + 1810672704)^{2} \) Copy content Toggle raw display
$61$ \( (T^{8} + \cdots + 445729169065401)^{2} \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 36\!\cdots\!61 \) Copy content Toggle raw display
$71$ \( (T^{4} + 96 T^{3} + \cdots + 283936)^{4} \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 10\!\cdots\!96 \) Copy content Toggle raw display
$79$ \( (T^{8} + \cdots + 1840363560000)^{2} \) Copy content Toggle raw display
$83$ \( (T^{8} + \cdots + 48\!\cdots\!44)^{2} \) Copy content Toggle raw display
$89$ \( (T^{8} + \cdots + 1228087292481)^{2} \) Copy content Toggle raw display
$97$ \( (T^{8} + \cdots + 302178979823616)^{2} \) Copy content Toggle raw display
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