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Properties

Label 35.3.h.a
Level $35$
Weight $3$
Character orbit 35.h
Analytic conductor $0.954$
Analytic rank $0$
Dimension $12$
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] = [35,3,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, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("35.26");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 35 = 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
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: \(0.953680925261\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 2 x^{11} + 19 x^{10} - 26 x^{9} + 244 x^{8} - 338 x^{7} + 1249 x^{6} - 986 x^{5} + 3532 x^{4} + \cdots + 441 \) 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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{2} + \beta_1) q^{2} + (\beta_{7} - 1) q^{3} + (2 \beta_{10} + \beta_{8} - \beta_{6} + \cdots - 2) q^{4}+ \cdots + (\beta_{11} - 2 \beta_{10} + \beta_{9} + \cdots + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{2} + \beta_1) q^{2} + (\beta_{7} - 1) q^{3} + (2 \beta_{10} + \beta_{8} - \beta_{6} + \cdots - 2) q^{4}+ \cdots + ( - 15 \beta_{11} - 3 \beta_{10} + \cdots + 9) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 2 q^{2} - 6 q^{3} - 10 q^{4} - 2 q^{7} - 4 q^{8} + 14 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 2 q^{2} - 6 q^{3} - 10 q^{4} - 2 q^{7} - 4 q^{8} + 14 q^{9} - 14 q^{11} + 18 q^{12} - 2 q^{14} - 20 q^{15} - 22 q^{16} + 48 q^{17} + 64 q^{18} - 30 q^{19} - 84 q^{21} - 88 q^{22} - 14 q^{23} - 36 q^{24} + 30 q^{25} + 66 q^{26} + 202 q^{28} + 64 q^{29} + 20 q^{30} + 132 q^{31} - 54 q^{32} - 192 q^{33} + 30 q^{35} + 156 q^{36} + 44 q^{37} - 300 q^{38} - 24 q^{39} - 138 q^{42} - 4 q^{43} + 6 q^{44} - 180 q^{45} - 214 q^{46} + 204 q^{47} - 24 q^{49} - 20 q^{50} - 132 q^{51} + 252 q^{52} + 196 q^{53} + 168 q^{54} - 460 q^{56} - 48 q^{57} + 158 q^{58} + 72 q^{59} + 150 q^{60} + 72 q^{61} + 536 q^{63} - 140 q^{64} + 30 q^{65} + 744 q^{66} - 138 q^{67} - 348 q^{68} + 240 q^{70} - 8 q^{71} - 196 q^{72} - 528 q^{73} + 50 q^{74} - 30 q^{75} - 176 q^{77} - 312 q^{78} - 12 q^{79} - 240 q^{80} - 310 q^{81} - 378 q^{82} - 276 q^{84} - 40 q^{86} + 138 q^{87} + 604 q^{88} + 204 q^{89} - 480 q^{91} + 732 q^{92} + 84 q^{93} - 42 q^{94} + 60 q^{95} + 540 q^{96} + 898 q^{98} - 44 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 2 x^{11} + 19 x^{10} - 26 x^{9} + 244 x^{8} - 338 x^{7} + 1249 x^{6} - 986 x^{5} + 3532 x^{4} + \cdots + 441 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 115287976 \nu^{11} + 155566808 \nu^{10} + 1678212789 \nu^{9} + 3282242448 \nu^{8} + \cdots - 423574271316 ) / 1182297257421 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 960485876 \nu^{11} + 1805683776 \nu^{10} - 18404798452 \nu^{9} + 23294419987 \nu^{8} + \cdots + 1461932499888 ) / 1182297257421 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 6740525471 \nu^{11} + 28434054878 \nu^{10} - 139462837212 \nu^{9} + 440457052261 \nu^{8} + \cdots + 26654089921641 ) / 7093783544526 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 12593073845 \nu^{11} + 52875316152 \nu^{10} - 270584479006 \nu^{9} + \cdots + 41846762790189 ) / 7093783544526 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 20099706409 \nu^{11} - 38300982339 \nu^{10} + 346926560957 \nu^{9} - 453614466560 \nu^{8} + \cdots + 20528318534472 ) / 7093783544526 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 22979355133 \nu^{11} - 77959111894 \nu^{10} + 482194495722 \nu^{9} - 1167796047734 \nu^{8} + \cdots - 71882680944312 ) / 7093783544526 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 9058064565 \nu^{11} - 1839528853 \nu^{10} - 143550656091 \nu^{9} - 113877990722 \nu^{8} + \cdots - 3111541612074 ) / 2364594514842 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 27689168462 \nu^{11} + 31316075951 \nu^{10} - 479491314879 \nu^{9} + 276064172287 \nu^{8} + \cdots - 5553545565645 ) / 7093783544526 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 42084124100 \nu^{11} - 50430282123 \nu^{10} + 738914826223 \nu^{9} - 485302351243 \nu^{8} + \cdots + 9745511328693 ) / 7093783544526 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 65452485346 \nu^{11} + 87736926802 \nu^{10} - 1160252919867 \nu^{9} + \cdots - 14009629724496 ) / 7093783544526 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{10} + \beta_{6} + \beta_{5} - 6\beta_{3} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{9} + \beta_{8} + \beta_{4} + 9\beta_{2} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 4\beta_{11} + 28\beta_{10} + 12\beta_{8} - 2\beta_{7} - 12\beta_{6} - 2\beta_{5} + 57\beta_{3} - 2\beta _1 - 59 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 2 \beta_{11} + 2 \beta_{10} - 16 \beta_{9} - 4 \beta_{7} - 14 \beta_{6} + 10 \beta_{5} - 32 \beta_{4} + \cdots + 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 34 \beta_{11} - 177 \beta_{10} + 4 \beta_{9} - 143 \beta_{8} - 34 \beta_{7} - 177 \beta_{5} + \cdots + 692 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 84 \beta_{11} - 116 \beta_{10} + 422 \beta_{9} - 189 \beta_{8} + 42 \beta_{7} + 189 \beta_{6} + \cdots + 180 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 464 \beta_{11} - 2176 \beta_{10} + 100 \beta_{9} + 928 \beta_{7} + 1732 \beta_{6} + 2620 \beta_{5} + \cdots - 464 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 664 \beta_{11} + 1116 \beta_{10} - 2640 \beta_{9} + 2548 \beta_{8} + 664 \beta_{7} + 1116 \beta_{5} + \cdots - 3932 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 11888 \beta_{11} + 53074 \beta_{10} - 3560 \beta_{9} + 21165 \beta_{8} - 5944 \beta_{7} - 21165 \beta_{6} + \cdots - 91894 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 9504 \beta_{11} + 18380 \beta_{10} - 32481 \beta_{9} - 19008 \beta_{7} - 34205 \beta_{6} - 2555 \beta_{5} + \cdots + 9504 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(22\) \(31\)
\(\chi(n)\) \(1\) \(\beta_{3}\)

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
1.77870 + 3.08079i
1.18241 + 2.04800i
0.410701 + 0.711354i
0.242987 + 0.420865i
−0.925400 − 1.60284i
−1.68940 − 2.92612i
1.77870 − 3.08079i
1.18241 − 2.04800i
0.410701 − 0.711354i
0.242987 − 0.420865i
−0.925400 + 1.60284i
−1.68940 + 2.92612i
−1.77870 + 3.08079i −2.10717 + 1.21658i −4.32752 − 7.49548i 1.93649 + 1.11803i − 8.65567i −5.39402 + 4.46146i 16.5598 −1.53989 + 2.66717i −6.88886 + 3.97728i
26.2 −1.18241 + 2.04800i 4.45439 − 2.57174i −0.796202 − 1.37906i −1.93649 − 1.11803i 12.1634i −1.42520 + 6.85338i −5.69355 8.72772 − 15.1168i 4.57947 − 2.64396i
26.3 −0.410701 + 0.711354i 0.507487 − 0.292998i 1.66265 + 2.87979i 1.93649 + 1.11803i 0.481337i 1.91172 − 6.73389i −6.01701 −4.32830 + 7.49684i −1.59064 + 0.918354i
26.4 −0.242987 + 0.420865i −4.74894 + 2.74180i 1.88192 + 3.25957i −1.93649 − 1.11803i − 2.66488i 5.87843 + 3.80053i −3.77301 10.5350 − 18.2471i 0.941083 − 0.543334i
26.5 0.925400 − 1.60284i 0.731043 − 0.422068i 0.287270 + 0.497567i −1.93649 − 1.11803i − 1.56233i −6.88972 − 1.23763i 8.46656 −4.14372 + 7.17713i −3.58406 + 2.06926i
26.6 1.68940 − 2.92612i −1.83681 + 1.06048i −3.70812 − 6.42265i 1.93649 + 1.11803i 7.16630i 4.91879 + 4.98051i −11.5428 −2.25076 + 3.89842i 6.54300 − 3.77760i
31.1 −1.77870 − 3.08079i −2.10717 − 1.21658i −4.32752 + 7.49548i 1.93649 − 1.11803i 8.65567i −5.39402 − 4.46146i 16.5598 −1.53989 − 2.66717i −6.88886 − 3.97728i
31.2 −1.18241 − 2.04800i 4.45439 + 2.57174i −0.796202 + 1.37906i −1.93649 + 1.11803i − 12.1634i −1.42520 − 6.85338i −5.69355 8.72772 + 15.1168i 4.57947 + 2.64396i
31.3 −0.410701 − 0.711354i 0.507487 + 0.292998i 1.66265 − 2.87979i 1.93649 − 1.11803i − 0.481337i 1.91172 + 6.73389i −6.01701 −4.32830 − 7.49684i −1.59064 − 0.918354i
31.4 −0.242987 − 0.420865i −4.74894 − 2.74180i 1.88192 − 3.25957i −1.93649 + 1.11803i 2.66488i 5.87843 − 3.80053i −3.77301 10.5350 + 18.2471i 0.941083 + 0.543334i
31.5 0.925400 + 1.60284i 0.731043 + 0.422068i 0.287270 − 0.497567i −1.93649 + 1.11803i 1.56233i −6.88972 + 1.23763i 8.46656 −4.14372 − 7.17713i −3.58406 − 2.06926i
31.6 1.68940 + 2.92612i −1.83681 − 1.06048i −3.70812 + 6.42265i 1.93649 − 1.11803i − 7.16630i 4.91879 − 4.98051i −11.5428 −2.25076 − 3.89842i 6.54300 + 3.77760i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 26.6
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 35.3.h.a ✓ 12
3.b odd 2 1 315.3.w.c 12
4.b odd 2 1 560.3.bx.c 12
5.b even 2 1 175.3.i.d 12
5.c odd 4 2 175.3.j.b 24
7.b odd 2 1 245.3.h.c 12
7.c even 3 1 245.3.d.a 12
7.c even 3 1 245.3.h.c 12
7.d odd 6 1 inner 35.3.h.a ✓ 12
7.d odd 6 1 245.3.d.a 12
21.g even 6 1 315.3.w.c 12
28.f even 6 1 560.3.bx.c 12
35.i odd 6 1 175.3.i.d 12
35.k even 12 2 175.3.j.b 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.3.h.a ✓ 12 1.a even 1 1 trivial
35.3.h.a ✓ 12 7.d odd 6 1 inner
175.3.i.d 12 5.b even 2 1
175.3.i.d 12 35.i odd 6 1
175.3.j.b 24 5.c odd 4 2
175.3.j.b 24 35.k even 12 2
245.3.d.a 12 7.c even 3 1
245.3.d.a 12 7.d odd 6 1
245.3.h.c 12 7.b odd 2 1
245.3.h.c 12 7.c even 3 1
315.3.w.c 12 3.b odd 2 1
315.3.w.c 12 21.g even 6 1
560.3.bx.c 12 4.b odd 2 1
560.3.bx.c 12 28.f even 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(35, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} + 2 T^{11} + \cdots + 441 \) Copy content Toggle raw display
$3$ \( T^{12} + 6 T^{11} + \cdots + 5184 \) Copy content Toggle raw display
$5$ \( (T^{4} - 5 T^{2} + 25)^{3} \) Copy content Toggle raw display
$7$ \( T^{12} + \cdots + 13841287201 \) Copy content Toggle raw display
$11$ \( T^{12} + \cdots + 4134617957376 \) Copy content Toggle raw display
$13$ \( T^{12} + 546 T^{10} + \cdots + 77792400 \) Copy content Toggle raw display
$17$ \( T^{12} + \cdots + 8707129344 \) Copy content Toggle raw display
$19$ \( T^{12} + \cdots + 1590595171344 \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 44219321357121 \) Copy content Toggle raw display
$29$ \( (T^{6} - 32 T^{5} + \cdots + 486804)^{2} \) Copy content Toggle raw display
$31$ \( T^{12} + \cdots + 76\!\cdots\!04 \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 110961448062976 \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots + 77\!\cdots\!25 \) Copy content Toggle raw display
$43$ \( (T^{6} + 2 T^{5} + \cdots - 9258464)^{2} \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 11\!\cdots\!04 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 18\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 92\!\cdots\!84 \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 23\!\cdots\!24 \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 28\!\cdots\!56 \) Copy content Toggle raw display
$71$ \( (T^{6} + 4 T^{5} + \cdots + 46762703904)^{2} \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 15\!\cdots\!64 \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 560965048576 \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 26\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 81\!\cdots\!44 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
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