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Properties

Label 35.5.h.a
Level $35$
Weight $5$
Character orbit 35.h
Analytic conductor $3.618$
Analytic rank $0$
Dimension $20$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [35,5,Mod(26,35)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(35, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 5]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("35.26");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 35 = 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 35.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: \(3.61794870793\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(10\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - 4 x^{19} + 116 x^{18} - 264 x^{17} + 7945 x^{16} - 15716 x^{15} + 322816 x^{14} + \cdots + 48948907536 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 5^{4}\cdot 7^{3} \)
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 \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{4} + \beta_{3} + \beta_1 + 1) q^{2} + (\beta_{14} - \beta_{4} - 1) q^{3} + ( - \beta_{14} + \beta_{9} + \cdots + \beta_1) q^{4} + \beta_{2} q^{5} + (\beta_{19} - \beta_{15} + \cdots - 7 \beta_1) q^{6}+ \cdots + (149 \beta_{19} + 122 \beta_{18} + \cdots - 2192) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 6 q^{2} - 18 q^{3} - 58 q^{4} - 54 q^{7} - 372 q^{8} + 266 q^{9} + 90 q^{11} + 1266 q^{12} - 690 q^{14} + 100 q^{15} - 126 q^{16} - 864 q^{17} - 848 q^{18} - 1662 q^{19} + 1680 q^{21} + 2584 q^{22}+ \cdots - 36716 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - 4 x^{19} + 116 x^{18} - 264 x^{17} + 7945 x^{16} - 15716 x^{15} + 322816 x^{14} + \cdots + 48948907536 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 40\!\cdots\!81 \nu^{19} + \cdots + 27\!\cdots\!48 ) / 29\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 11\!\cdots\!75 \nu^{19} + \cdots + 32\!\cdots\!76 ) / 35\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 12\!\cdots\!67 \nu^{19} + \cdots - 30\!\cdots\!24 ) / 65\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 10\!\cdots\!75 \nu^{19} + \cdots + 78\!\cdots\!96 ) / 29\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 51\!\cdots\!01 \nu^{19} + \cdots + 65\!\cdots\!60 ) / 97\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 96\!\cdots\!59 \nu^{19} + \cdots - 63\!\cdots\!24 ) / 14\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 18\!\cdots\!62 \nu^{19} + \cdots + 55\!\cdots\!12 ) / 14\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 56\!\cdots\!05 \nu^{19} + \cdots + 38\!\cdots\!12 ) / 29\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 21\!\cdots\!85 \nu^{19} + \cdots + 56\!\cdots\!44 ) / 97\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 76\!\cdots\!99 \nu^{19} + \cdots + 10\!\cdots\!52 ) / 29\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 29\!\cdots\!24 \nu^{19} + \cdots + 61\!\cdots\!00 ) / 97\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 12\!\cdots\!90 \nu^{19} + \cdots - 14\!\cdots\!16 ) / 29\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 17\!\cdots\!13 \nu^{19} + \cdots + 67\!\cdots\!52 ) / 29\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 82\!\cdots\!58 \nu^{19} + \cdots - 90\!\cdots\!76 ) / 97\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 85\!\cdots\!44 \nu^{19} + \cdots - 10\!\cdots\!44 ) / 97\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 15\!\cdots\!70 \nu^{19} + \cdots - 45\!\cdots\!32 ) / 16\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( - 30\!\cdots\!01 \nu^{19} + \cdots - 27\!\cdots\!96 ) / 29\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( - 10\!\cdots\!49 \nu^{19} + \cdots - 37\!\cdots\!80 ) / 97\!\cdots\!40 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{14} - \beta_{9} - 22\beta_{4} + \beta_{3} + \beta _1 - 21 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( - \beta_{18} + \beta_{17} + \beta_{16} - 3 \beta_{14} - \beta_{13} - 2 \beta_{10} + 2 \beta_{9} + \cdots - 13 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 7 \beta_{19} - 9 \beta_{18} + 3 \beta_{17} - 3 \beta_{16} + \beta_{15} - 56 \beta_{14} + 3 \beta_{13} + \cdots - 8 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 74 \beta_{19} - 11 \beta_{18} - 70 \beta_{17} - 132 \beta_{16} + 2 \beta_{15} - 33 \beta_{14} + \cdots + 1387 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 180 \beta_{19} + 447 \beta_{18} - 556 \beta_{17} - 421 \beta_{16} + 6 \beta_{15} + 1185 \beta_{14} + \cdots + 31892 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 5475 \beta_{19} + 5879 \beta_{18} - 1031 \beta_{17} + 3770 \beta_{16} - 214 \beta_{15} + 17549 \beta_{14} + \cdots + 6670 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 30169 \beta_{19} + 18670 \beta_{18} + 24199 \beta_{17} + 50216 \beta_{16} - 4695 \beta_{15} + \cdots - 1526718 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 70051 \beta_{19} - 283938 \beta_{18} + 336461 \beta_{17} + 250362 \beta_{16} - 3044 \beta_{15} + \cdots - 8271555 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 2702857 \beta_{19} - 3130397 \beta_{18} + 903937 \beta_{17} - 1446745 \beta_{16} + 234225 \beta_{15} + \cdots - 3022258 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 16708088 \beta_{19} - 6641419 \beta_{18} - 15205308 \beta_{17} - 28281796 \beta_{16} + 1516654 \beta_{15} + \cdots + 514872347 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 45510762 \beta_{19} + 127990069 \beta_{18} - 160018254 \beta_{17} - 114438923 \beta_{16} + \cdots + 4888464096 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 1327783653 \beta_{19} + 1478630793 \beta_{18} - 366932143 \beta_{17} + 787392466 \beta_{16} + \cdots + 1552634552 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 7608620461 \beta_{19} + 3829164800 \beta_{18} + 6694345523 \beta_{17} + 12885732664 \beta_{16} + \cdots - 277258692548 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 20159515471 \beta_{19} - 63606203192 \beta_{18} + 78327441717 \beta_{17} + 56042989760 \beta_{16} + \cdots - 2131989068231 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 635213119771 \beta_{19} - 719092435903 \beta_{18} + 192926001167 \beta_{17} - 360584533817 \beta_{16} + \cdots - 724988370594 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( 3724297344530 \beta_{19} - 1709397798639 \beta_{18} - 3345985534874 \beta_{17} - 6308094738524 \beta_{16} + \cdots + 125938383099303 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( - 10106625991764 \beta_{19} + 30044391955061 \beta_{18} - 37318802941868 \beta_{17} + \cdots + 10\!\cdots\!14 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( - 304530812896503 \beta_{19} + 342260211735283 \beta_{18} - 89137309464753 \beta_{17} + \cdots + 351336733427532 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(22\) \(31\)
\(\chi(n)\) \(1\) \(1 + \beta_{4}\)

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
3.91002 + 6.77236i
2.53453 + 4.38993i
2.46009 + 4.26100i
1.56222 + 2.70585i
0.863703 + 1.49598i
−0.481645 − 0.834233i
−0.681265 − 1.17999i
−2.44602 − 4.23664i
−2.83171 − 4.90466i
−2.88992 − 5.00549i
3.91002 − 6.77236i
2.53453 − 4.38993i
2.46009 − 4.26100i
1.56222 − 2.70585i
0.863703 − 1.49598i
−0.481645 + 0.834233i
−0.681265 + 1.17999i
−2.44602 + 4.23664i
−2.83171 + 4.90466i
−2.88992 + 5.00549i
−3.41002 + 5.90633i −7.02310 + 4.05479i −15.2565 − 26.4250i −9.68246 − 5.59017i − 55.3077i 41.9906 + 25.2546i 98.9794 −7.61740 + 13.1937i 66.0348 − 38.1252i
26.2 −2.03453 + 3.52390i −12.7316 + 7.35058i −0.278590 − 0.482532i 9.68246 + 5.59017i − 59.8198i −11.2754 − 47.6851i −62.8376 67.5621 − 117.021i −39.3984 + 22.7467i
26.3 −1.96009 + 3.39498i 4.26025 − 2.45966i 0.316095 + 0.547493i 9.68246 + 5.59017i 19.2846i 34.8977 + 34.3969i −65.2012 −28.4002 + 49.1905i −37.9570 + 21.9145i
26.4 −1.06222 + 1.83982i −3.12759 + 1.80571i 5.74337 + 9.94782i −9.68246 − 5.59017i − 7.67226i −47.4339 + 12.2893i −58.3940 −33.9788 + 58.8530i 20.5698 − 11.8760i
26.5 −0.363703 + 0.629951i 10.6613 − 6.15528i 7.73544 + 13.3982i −9.68246 − 5.59017i 8.95477i 40.3461 − 27.8063i −22.8921 35.2750 − 61.0981i 7.04307 − 4.06632i
26.6 0.981645 − 1.70026i 11.1418 − 6.43269i 6.07275 + 10.5183i 9.68246 + 5.59017i − 25.2585i −48.9898 + 0.997416i 55.2577 42.2591 − 73.1949i 19.0095 − 10.9751i
26.7 1.18127 − 2.04601i −7.64511 + 4.41391i 5.20922 + 9.02264i 9.68246 + 5.59017i 20.8560i 1.69012 + 48.9708i 62.4144 −1.53484 + 2.65843i 22.8751 − 13.2069i
26.8 2.94602 − 5.10266i 7.51414 − 4.33829i −9.35812 − 16.2087i −9.68246 − 5.59017i − 51.1228i 17.5876 + 45.7349i −16.0042 −2.85850 + 4.95108i −57.0495 + 32.9376i
26.9 3.33171 − 5.77068i 2.41118 − 1.39210i −14.2005 − 24.5960i 9.68246 + 5.59017i − 18.5522i −7.25097 − 48.4605i −82.6333 −36.6241 + 63.4349i 64.5182 − 37.2496i
26.10 3.38992 − 5.87152i −14.4612 + 8.34918i −14.9831 − 25.9515i −9.68246 − 5.59017i 113.212i −48.5619 + 6.53743i −94.6891 98.9176 − 171.330i −65.6455 + 37.9005i
31.1 −3.41002 − 5.90633i −7.02310 − 4.05479i −15.2565 + 26.4250i −9.68246 + 5.59017i 55.3077i 41.9906 − 25.2546i 98.9794 −7.61740 − 13.1937i 66.0348 + 38.1252i
31.2 −2.03453 − 3.52390i −12.7316 − 7.35058i −0.278590 + 0.482532i 9.68246 − 5.59017i 59.8198i −11.2754 + 47.6851i −62.8376 67.5621 + 117.021i −39.3984 − 22.7467i
31.3 −1.96009 − 3.39498i 4.26025 + 2.45966i 0.316095 − 0.547493i 9.68246 − 5.59017i − 19.2846i 34.8977 − 34.3969i −65.2012 −28.4002 − 49.1905i −37.9570 − 21.9145i
31.4 −1.06222 − 1.83982i −3.12759 − 1.80571i 5.74337 − 9.94782i −9.68246 + 5.59017i 7.67226i −47.4339 − 12.2893i −58.3940 −33.9788 − 58.8530i 20.5698 + 11.8760i
31.5 −0.363703 − 0.629951i 10.6613 + 6.15528i 7.73544 − 13.3982i −9.68246 + 5.59017i − 8.95477i 40.3461 + 27.8063i −22.8921 35.2750 + 61.0981i 7.04307 + 4.06632i
31.6 0.981645 + 1.70026i 11.1418 + 6.43269i 6.07275 − 10.5183i 9.68246 − 5.59017i 25.2585i −48.9898 − 0.997416i 55.2577 42.2591 + 73.1949i 19.0095 + 10.9751i
31.7 1.18127 + 2.04601i −7.64511 − 4.41391i 5.20922 − 9.02264i 9.68246 − 5.59017i − 20.8560i 1.69012 − 48.9708i 62.4144 −1.53484 − 2.65843i 22.8751 + 13.2069i
31.8 2.94602 + 5.10266i 7.51414 + 4.33829i −9.35812 + 16.2087i −9.68246 + 5.59017i 51.1228i 17.5876 − 45.7349i −16.0042 −2.85850 − 4.95108i −57.0495 − 32.9376i
31.9 3.33171 + 5.77068i 2.41118 + 1.39210i −14.2005 + 24.5960i 9.68246 − 5.59017i 18.5522i −7.25097 + 48.4605i −82.6333 −36.6241 − 63.4349i 64.5182 + 37.2496i
31.10 3.38992 + 5.87152i −14.4612 − 8.34918i −14.9831 + 25.9515i −9.68246 + 5.59017i − 113.212i −48.5619 − 6.53743i −94.6891 98.9176 + 171.330i −65.6455 − 37.9005i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 26.10
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 35.5.h.a ✓ 20
5.b even 2 1 175.5.i.b 20
5.c odd 4 2 175.5.j.b 40
7.c even 3 1 245.5.d.a 20
7.d odd 6 1 inner 35.5.h.a ✓ 20
7.d odd 6 1 245.5.d.a 20
35.i odd 6 1 175.5.i.b 20
35.k even 12 2 175.5.j.b 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.5.h.a ✓ 20 1.a even 1 1 trivial
35.5.h.a ✓ 20 7.d odd 6 1 inner
175.5.i.b 20 5.b even 2 1
175.5.i.b 20 35.i odd 6 1
175.5.j.b 40 5.c odd 4 2
175.5.j.b 40 35.k even 12 2
245.5.d.a 20 7.c even 3 1
245.5.d.a 20 7.d odd 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{5}^{\mathrm{new}}(35, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} + \cdots + 43082814096 \) Copy content Toggle raw display
$3$ \( T^{20} + \cdots + 14\!\cdots\!84 \) Copy content Toggle raw display
$5$ \( (T^{4} - 125 T^{2} + 15625)^{5} \) Copy content Toggle raw display
$7$ \( T^{20} + \cdots + 63\!\cdots\!01 \) Copy content Toggle raw display
$11$ \( T^{20} + \cdots + 33\!\cdots\!96 \) Copy content Toggle raw display
$13$ \( T^{20} + \cdots + 25\!\cdots\!00 \) Copy content Toggle raw display
$17$ \( T^{20} + \cdots + 10\!\cdots\!24 \) Copy content Toggle raw display
$19$ \( T^{20} + \cdots + 18\!\cdots\!24 \) Copy content Toggle raw display
$23$ \( T^{20} + \cdots + 41\!\cdots\!41 \) Copy content Toggle raw display
$29$ \( (T^{10} + \cdots - 20\!\cdots\!96)^{2} \) Copy content Toggle raw display
$31$ \( T^{20} + \cdots + 31\!\cdots\!24 \) Copy content Toggle raw display
$37$ \( T^{20} + \cdots + 23\!\cdots\!16 \) Copy content Toggle raw display
$41$ \( T^{20} + \cdots + 14\!\cdots\!25 \) Copy content Toggle raw display
$43$ \( (T^{10} + \cdots - 23\!\cdots\!00)^{2} \) Copy content Toggle raw display
$47$ \( T^{20} + \cdots + 54\!\cdots\!44 \) Copy content Toggle raw display
$53$ \( T^{20} + \cdots + 55\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( T^{20} + \cdots + 42\!\cdots\!84 \) Copy content Toggle raw display
$61$ \( T^{20} + \cdots + 63\!\cdots\!24 \) Copy content Toggle raw display
$67$ \( T^{20} + \cdots + 32\!\cdots\!56 \) Copy content Toggle raw display
$71$ \( (T^{10} + \cdots - 20\!\cdots\!56)^{2} \) Copy content Toggle raw display
$73$ \( T^{20} + \cdots + 39\!\cdots\!44 \) Copy content Toggle raw display
$79$ \( T^{20} + \cdots + 53\!\cdots\!76 \) Copy content Toggle raw display
$83$ \( T^{20} + \cdots + 32\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( T^{20} + \cdots + 46\!\cdots\!04 \) Copy content Toggle raw display
$97$ \( T^{20} + \cdots + 34\!\cdots\!00 \) Copy content Toggle raw display
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