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Properties

Label 35.6.b.a
Level $35$
Weight $6$
Character orbit 35.b
Analytic conductor $5.613$
Analytic rank $0$
Dimension $14$
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [35,6,Mod(29,35)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(35, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0])) N = Newforms(chi, 6, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("35.29"); S:= CuspForms(chi, 6); N := Newforms(S);
 
Level: \( N \) \(=\) \( 35 = 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 35.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(5.61343369345\)
Analytic rank: \(0\)
Dimension: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} + \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} + 321 x^{12} + 38486 x^{10} + 2095502 x^{8} + 49076817 x^{6} + 346644641 x^{4} + 698155096 x^{2} + 308213136 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{4}\cdot 5^{3}\cdot 7^{8} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{4} q^{2} + \beta_{6} q^{3} + (\beta_1 - 15) q^{4} + ( - \beta_{9} + \beta_{6} + 2 \beta_{4} + 11) q^{5} + (\beta_{3} + \beta_{2} - 13) q^{6} + (\beta_{7} + \beta_{6} + \beta_{4}) q^{7} + (\beta_{13} + \beta_{12} + \cdots + 26 \beta_{4}) q^{8}+ \cdots + (178 \beta_{12} + 40 \beta_{10} + \cdots + 12542) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q - 212 q^{4} + 156 q^{5} - 180 q^{6} - 796 q^{9} + 904 q^{10} - 1158 q^{11} + 784 q^{14} - 2602 q^{15} + 9012 q^{16} - 1152 q^{19} - 3572 q^{20} - 2254 q^{21} - 24312 q^{24} - 494 q^{25} + 11908 q^{26}+ \cdots + 173220 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} + 321 x^{12} + 38486 x^{10} + 2095502 x^{8} + 49076817 x^{6} + 346644641 x^{4} + 698155096 x^{2} + 308213136 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 407 \nu^{12} + 196876 \nu^{10} + 31213590 \nu^{8} + 2049840764 \nu^{6} + 51006521955 \nu^{4} + \cdots + 3627490857008 ) / 79243521344 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 2849 \nu^{12} - 1378132 \nu^{10} - 218495130 \nu^{8} - 14348885348 \nu^{6} + \cdots - 34509168976 ) / 158487042688 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 1698797 \nu^{12} - 519099304 \nu^{10} - 57601755726 \nu^{8} - 2794371074680 \nu^{6} + \cdots - 385455624522288 ) / 9509222561280 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 252209 \nu^{13} - 82745412 \nu^{11} - 10570604338 \nu^{9} - 665500910428 \nu^{7} + \cdots - 17\!\cdots\!08 \nu ) / 695599630357632 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 8274251648 \nu^{13} + 441183010037 \nu^{12} - 2562215584396 \nu^{11} + \cdots + 79\!\cdots\!28 ) / 13\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 194194939 \nu^{13} + 61942782008 \nu^{11} + 7347234404322 \nu^{9} + \cdots + 77\!\cdots\!56 \nu ) / 21\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 3008775487 \nu^{13} - 964127357144 \nu^{11} - 115809607585146 \nu^{9} + \cdots - 30\!\cdots\!68 \nu ) / 27\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 38573241481 \nu^{13} - 184768834481 \nu^{12} - 12455677639712 \nu^{11} + \cdots - 81\!\cdots\!64 ) / 27\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 59964157577 \nu^{13} + 13073588913 \nu^{12} - 19168820718904 \nu^{11} + \cdots + 19\!\cdots\!72 ) / 27\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 43086325339 \nu^{13} + 4357862971 \nu^{12} - 13961291773928 \nu^{11} + \cdots + 65\!\cdots\!24 ) / 92\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 8195184873 \nu^{13} + 688083627 \nu^{12} - 2649249249496 \nu^{11} + \cdots + 10\!\cdots\!88 ) / 14\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 111109628293 \nu^{13} + 13073588913 \nu^{12} + 35634230106736 \nu^{11} + \cdots + 19\!\cdots\!72 ) / 13\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 12657297751 \nu^{13} - 396169361 \nu^{12} - 4035796842552 \nu^{11} + \cdots - 59\!\cdots\!84 ) / 84\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{7} + \beta_{6} - 48\beta_{4} ) / 49 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2\beta_{2} + 7\beta _1 - 320 ) / 7 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 28\beta_{13} + 28\beta_{12} + 21\beta_{11} - 49\beta_{9} - 78\beta_{7} + 153\beta_{6} + 4010\beta_{4} ) / 49 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 33 \beta_{12} + 11 \beta_{10} + 71 \beta_{9} + 28 \beta_{8} - 16 \beta_{7} - 38 \beta_{6} + 6 \beta_{5} + \cdots + 25976 ) / 7 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 4298 \beta_{13} - 4858 \beta_{12} - 2380 \beta_{11} - 1288 \beta_{10} + 9086 \beta_{9} + \cdots - 359400 \beta_{4} ) / 49 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 4932 \beta_{12} - 1580 \beta_{10} - 10412 \beta_{9} - 4576 \beta_{8} + 2128 \beta_{7} + \cdots - 2291620 ) / 7 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 520772 \beta_{13} + 602308 \beta_{12} + 231525 \beta_{11} + 226576 \beta_{10} + \cdots + 33427946 \beta_{4} ) / 49 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 574593 \beta_{12} + 182123 \beta_{10} + 1201719 \beta_{9} + 550284 \beta_{8} - 234656 \beta_{7} + \cdots + 210677000 ) / 7 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 57819818 \beta_{13} - 66751258 \beta_{12} - 21274652 \beta_{11} - 29700216 \beta_{10} + \cdots - 3184671496 \beta_{4} ) / 49 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 61260676 \beta_{12} - 19379244 \beta_{10} - 127305212 \beta_{9} - 59599760 \beta_{8} + \cdots - 19888958788 ) / 7 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 6167333956 \beta_{13} + 7053021444 \beta_{12} + 1889018565 \beta_{11} + 3482018624 \beta_{10} + \cdots + 308141386314 \beta_{4} ) / 49 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 6270874577 \beta_{12} + 1985926427 \beta_{10} + 12960333495 \beta_{9} + 6165385532 \beta_{8} + \cdots + 1910259452296 ) / 7 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 644511621866 \beta_{13} - 728443975322 \beta_{12} - 163229634092 \beta_{11} + \cdots - 30115071886248 \beta_{4} ) / 49 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/35\mathbb{Z}\right)^\times\).

\(n\) \(22\) \(31\)
\(\chi(n)\) \(-1\) \(1\)

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
29.1
− 10.0673i
− 8.06369i
− 9.75757i
− 7.03499i
− 1.52374i
− 0.781947i
− 2.64414i
2.64414i
0.781947i
1.52374i
7.03499i
9.75757i
8.06369i
10.0673i
− 11.0673i 19.8020i −90.4856 27.8943 + 48.4449i 219.156 49.0000i 647.279i −149.121 536.155 − 308.715i
29.2 − 9.06369i − 16.5461i −50.1505 27.0501 − 48.9213i −149.968 49.0000i 164.511i −30.7720 −443.408 − 245.174i
29.3 − 8.75757i − 14.0411i −44.6950 −44.1840 + 34.2458i −122.966 − 49.0000i 111.177i 45.8478 299.910 + 386.944i
29.4 − 6.03499i 6.92010i −4.42114 55.2784 − 8.32482i 41.7627 − 49.0000i − 166.438i 195.112 −50.2402 − 333.605i
29.5 − 2.52374i 0.880876i 25.6307 1.50070 + 55.8816i 2.22310 49.0000i − 145.445i 242.224 141.030 − 3.78739i
29.6 − 1.78195i − 17.7758i 28.8247 −40.8094 − 38.2046i −31.6756 49.0000i − 108.386i −72.9803 −68.0785 + 72.7203i
29.7 − 1.64414i − 29.5180i 29.2968 51.2700 + 22.2796i −48.5317 − 49.0000i − 100.781i −628.311 36.6308 − 84.2952i
29.8 1.64414i 29.5180i 29.2968 51.2700 − 22.2796i −48.5317 49.0000i 100.781i −628.311 36.6308 + 84.2952i
29.9 1.78195i 17.7758i 28.8247 −40.8094 + 38.2046i −31.6756 − 49.0000i 108.386i −72.9803 −68.0785 − 72.7203i
29.10 2.52374i − 0.880876i 25.6307 1.50070 − 55.8816i 2.22310 − 49.0000i 145.445i 242.224 141.030 + 3.78739i
29.11 6.03499i − 6.92010i −4.42114 55.2784 + 8.32482i 41.7627 49.0000i 166.438i 195.112 −50.2402 + 333.605i
29.12 8.75757i 14.0411i −44.6950 −44.1840 − 34.2458i −122.966 49.0000i − 111.177i 45.8478 299.910 − 386.944i
29.13 9.06369i 16.5461i −50.1505 27.0501 + 48.9213i −149.968 − 49.0000i − 164.511i −30.7720 −443.408 + 245.174i
29.14 11.0673i − 19.8020i −90.4856 27.8943 − 48.4449i 219.156 − 49.0000i − 647.279i −149.121 536.155 + 308.715i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 29.14
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 35.6.b.a ✓ 14
3.b odd 2 1 315.6.d.a 14
4.b odd 2 1 560.6.g.e 14
5.b even 2 1 inner 35.6.b.a ✓ 14
5.c odd 4 1 175.6.a.k 7
5.c odd 4 1 175.6.a.l 7
7.b odd 2 1 245.6.b.d 14
15.d odd 2 1 315.6.d.a 14
20.d odd 2 1 560.6.g.e 14
35.c odd 2 1 245.6.b.d 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.6.b.a ✓ 14 1.a even 1 1 trivial
35.6.b.a ✓ 14 5.b even 2 1 inner
175.6.a.k 7 5.c odd 4 1
175.6.a.l 7 5.c odd 4 1
245.6.b.d 14 7.b odd 2 1
245.6.b.d 14 35.c odd 2 1
315.6.d.a 14 3.b odd 2 1
315.6.d.a 14 15.d odd 2 1
560.6.g.e 14 4.b odd 2 1
560.6.g.e 14 20.d odd 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{6}^{\mathrm{new}}(35, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{14} + \cdots + 1536640000 \) Copy content Toggle raw display
$3$ \( T^{14} + \cdots + 216519570876816 \) Copy content Toggle raw display
$5$ \( T^{14} + \cdots + 29\!\cdots\!25 \) Copy content Toggle raw display
$7$ \( (T^{2} + 2401)^{7} \) Copy content Toggle raw display
$11$ \( (T^{7} + \cdots - 15\!\cdots\!92)^{2} \) Copy content Toggle raw display
$13$ \( T^{14} + \cdots + 15\!\cdots\!00 \) Copy content Toggle raw display
$17$ \( T^{14} + \cdots + 19\!\cdots\!76 \) Copy content Toggle raw display
$19$ \( (T^{7} + \cdots + 12\!\cdots\!00)^{2} \) Copy content Toggle raw display
$23$ \( T^{14} + \cdots + 22\!\cdots\!76 \) Copy content Toggle raw display
$29$ \( (T^{7} + \cdots - 11\!\cdots\!00)^{2} \) Copy content Toggle raw display
$31$ \( (T^{7} + \cdots + 14\!\cdots\!00)^{2} \) Copy content Toggle raw display
$37$ \( T^{14} + \cdots + 19\!\cdots\!24 \) Copy content Toggle raw display
$41$ \( (T^{7} + \cdots + 24\!\cdots\!68)^{2} \) Copy content Toggle raw display
$43$ \( T^{14} + \cdots + 19\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( T^{14} + \cdots + 56\!\cdots\!44 \) Copy content Toggle raw display
$53$ \( T^{14} + \cdots + 11\!\cdots\!84 \) Copy content Toggle raw display
$59$ \( (T^{7} + \cdots - 23\!\cdots\!00)^{2} \) Copy content Toggle raw display
$61$ \( (T^{7} + \cdots - 35\!\cdots\!08)^{2} \) Copy content Toggle raw display
$67$ \( T^{14} + \cdots + 39\!\cdots\!36 \) Copy content Toggle raw display
$71$ \( (T^{7} + \cdots - 23\!\cdots\!00)^{2} \) Copy content Toggle raw display
$73$ \( T^{14} + \cdots + 12\!\cdots\!76 \) Copy content Toggle raw display
$79$ \( (T^{7} + \cdots + 17\!\cdots\!00)^{2} \) Copy content Toggle raw display
$83$ \( T^{14} + \cdots + 62\!\cdots\!24 \) Copy content Toggle raw display
$89$ \( (T^{7} + \cdots + 11\!\cdots\!00)^{2} \) Copy content Toggle raw display
$97$ \( T^{14} + \cdots + 27\!\cdots\!76 \) Copy content Toggle raw display
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