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Properties

Label 336.3.bn.f.65.1
Level $336$
Weight $3$
Character 336.65
Analytic conductor $9.155$
Analytic rank $0$
Dimension $4$
Inner twists $4$

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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] = [336,3,Mod(65,336)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("336.65"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(336, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 3, 2])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 336 = 2^{4} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 336.bn (of order \(6\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,4,0,0,0,-14] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(9.15533688251\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{-5})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 5x^{2} + 25 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 21)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 65.1
Root \(1.93649 + 1.11803i\) of defining polynomial
Character \(\chi\) \(=\) 336.65
Dual form 336.3.bn.f.305.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.936492 + 2.85008i) q^{3} +(1.93649 + 1.11803i) q^{5} +(-3.50000 - 6.06218i) q^{7} +(-7.24597 - 5.33816i) q^{9} +(-9.68246 + 5.59017i) q^{11} -2.00000 q^{13} +(-5.00000 + 4.47214i) q^{15} +(-23.2379 + 13.4164i) q^{17} +(8.00000 - 13.8564i) q^{19} +(20.5554 - 4.29812i) q^{21} +(-11.6190 - 6.70820i) q^{23} +(-10.0000 - 17.3205i) q^{25} +(22.0000 - 15.6525i) q^{27} -15.6525i q^{29} +(-1.50000 - 2.59808i) q^{31} +(-6.86492 - 32.8310i) q^{33} -15.6525i q^{35} +(-6.00000 + 10.3923i) q^{37} +(1.87298 - 5.70017i) q^{39} -31.3050i q^{41} -44.0000 q^{43} +(-8.06351 - 18.4385i) q^{45} +(11.6190 + 6.70820i) q^{47} +(-24.5000 + 42.4352i) q^{49} +(-16.4758 - 78.7943i) q^{51} +(-17.4284 + 10.0623i) q^{53} -25.0000 q^{55} +(32.0000 + 35.7771i) q^{57} +(-17.4284 + 10.0623i) q^{59} +(13.0000 - 22.5167i) q^{61} +(-7.00000 + 62.6099i) q^{63} +(-3.87298 - 2.23607i) q^{65} +(26.0000 + 45.0333i) q^{67} +(30.0000 - 26.8328i) q^{69} +93.9149i q^{71} +(-9.00000 - 15.5885i) q^{73} +(58.7298 - 12.2803i) q^{75} +(67.7772 + 39.1312i) q^{77} +(-39.5000 + 68.4160i) q^{79} +(24.0081 + 77.3603i) q^{81} -140.872i q^{83} -60.0000 q^{85} +(44.6109 + 14.6584i) q^{87} +(42.6028 + 24.5967i) q^{89} +(7.00000 + 12.1244i) q^{91} +(8.80948 - 1.84205i) q^{93} +(30.9839 - 17.8885i) q^{95} -93.0000 q^{97} +(100.000 + 11.1803i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{3} - 14 q^{7} + 2 q^{9} - 8 q^{13} - 20 q^{15} + 32 q^{19} + 28 q^{21} - 40 q^{25} + 88 q^{27} - 6 q^{31} + 50 q^{33} - 24 q^{37} - 8 q^{39} - 176 q^{43} - 40 q^{45} - 98 q^{49} + 120 q^{51}+ \cdots + 400 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/336\mathbb{Z}\right)^\times\).

\(n\) \(85\) \(113\) \(127\) \(241\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(e\left(\frac{1}{3}\right)\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\). You can download additional coefficients here.



Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −0.936492 + 2.85008i −0.312164 + 0.950028i
\(4\) 0 0
\(5\) 1.93649 + 1.11803i 0.387298 + 0.223607i 0.680989 − 0.732294i \(-0.261550\pi\)
−0.293691 + 0.955901i \(0.594884\pi\)
\(6\) 0 0
\(7\) −3.50000 − 6.06218i −0.500000 − 0.866025i
\(8\) 0 0
\(9\) −7.24597 − 5.33816i −0.805107 − 0.593129i
\(10\) 0 0
\(11\) −9.68246 + 5.59017i −0.880223 + 0.508197i −0.870732 − 0.491758i \(-0.836355\pi\)
−0.00949140 + 0.999955i \(0.503021\pi\)
\(12\) 0 0
\(13\) −2.00000 −0.153846 −0.0769231 − 0.997037i \(-0.524510\pi\)
−0.0769231 + 0.997037i \(0.524510\pi\)
\(14\) 0 0
\(15\) −5.00000 + 4.47214i −0.333333 + 0.298142i
\(16\) 0 0
\(17\) −23.2379 + 13.4164i −1.36694 + 0.789200i −0.990535 − 0.137257i \(-0.956171\pi\)
−0.376400 + 0.926457i \(0.622838\pi\)
\(18\) 0 0
\(19\) 8.00000 − 13.8564i 0.421053 − 0.729285i −0.574990 − 0.818160i \(-0.694994\pi\)
0.996043 + 0.0888758i \(0.0283274\pi\)
\(20\) 0 0
\(21\) 20.5554 − 4.29812i 0.978831 − 0.204672i
\(22\) 0 0
\(23\) −11.6190 − 6.70820i −0.505172 − 0.291661i 0.225675 − 0.974203i \(-0.427541\pi\)
−0.730847 + 0.682542i \(0.760875\pi\)
\(24\) 0 0
\(25\) −10.0000 − 17.3205i −0.400000 − 0.692820i
\(26\) 0 0
\(27\) 22.0000 − 15.6525i 0.814815 − 0.579721i
\(28\) 0 0
\(29\) − 15.6525i − 0.539741i −0.962897 − 0.269870i \(-0.913019\pi\)
0.962897 − 0.269870i \(-0.0869808\pi\)
\(30\) 0 0
\(31\) −1.50000 − 2.59808i −0.0483871 − 0.0838089i 0.840817 − 0.541319i \(-0.182075\pi\)
−0.889205 + 0.457510i \(0.848741\pi\)
\(32\) 0 0
\(33\) −6.86492 − 32.8310i −0.208028 − 0.994878i
\(34\) 0 0
\(35\) − 15.6525i − 0.447214i
\(36\) 0 0
\(37\) −6.00000 + 10.3923i −0.162162 + 0.280873i −0.935644 − 0.352946i \(-0.885180\pi\)
0.773482 + 0.633819i \(0.218513\pi\)
\(38\) 0 0
\(39\) 1.87298 − 5.70017i 0.0480252 − 0.146158i
\(40\) 0 0
\(41\) − 31.3050i − 0.763535i −0.924258 − 0.381768i \(-0.875315\pi\)
0.924258 − 0.381768i \(-0.124685\pi\)
\(42\) 0 0
\(43\) −44.0000 −1.02326 −0.511628 − 0.859207i \(-0.670957\pi\)
−0.511628 + 0.859207i \(0.670957\pi\)
\(44\) 0 0
\(45\) −8.06351 − 18.4385i −0.179189 − 0.409745i
\(46\) 0 0
\(47\) 11.6190 + 6.70820i 0.247212 + 0.142728i 0.618487 − 0.785795i \(-0.287746\pi\)
−0.371275 + 0.928523i \(0.621079\pi\)
\(48\) 0 0
\(49\) −24.5000 + 42.4352i −0.500000 + 0.866025i
\(50\) 0 0
\(51\) −16.4758 − 78.7943i −0.323055 − 1.54499i
\(52\) 0 0
\(53\) −17.4284 + 10.0623i −0.328838 + 0.189855i −0.655325 − 0.755347i \(-0.727468\pi\)
0.326487 + 0.945202i \(0.394135\pi\)
\(54\) 0 0
\(55\) −25.0000 −0.454545
\(56\) 0 0
\(57\) 32.0000 + 35.7771i 0.561404 + 0.627668i
\(58\) 0 0
\(59\) −17.4284 + 10.0623i −0.295397 + 0.170548i −0.640373 − 0.768064i \(-0.721220\pi\)
0.344976 + 0.938611i \(0.387887\pi\)
\(60\) 0 0
\(61\) 13.0000 − 22.5167i 0.213115 − 0.369126i −0.739573 − 0.673076i \(-0.764973\pi\)
0.952688 + 0.303951i \(0.0983058\pi\)
\(62\) 0 0
\(63\) −7.00000 + 62.6099i −0.111111 + 0.993808i
\(64\) 0 0
\(65\) −3.87298 − 2.23607i −0.0595844 − 0.0344010i
\(66\) 0 0
\(67\) 26.0000 + 45.0333i 0.388060 + 0.672139i 0.992189 − 0.124748i \(-0.0398121\pi\)
−0.604129 + 0.796887i \(0.706479\pi\)
\(68\) 0 0
\(69\) 30.0000 − 26.8328i 0.434783 − 0.388881i
\(70\) 0 0
\(71\) 93.9149i 1.32274i 0.750058 + 0.661372i \(0.230026\pi\)
−0.750058 + 0.661372i \(0.769974\pi\)
\(72\) 0 0
\(73\) −9.00000 − 15.5885i −0.123288 − 0.213541i 0.797775 − 0.602956i \(-0.206010\pi\)
−0.921062 + 0.389415i \(0.872677\pi\)
\(74\) 0 0
\(75\) 58.7298 − 12.2803i 0.783064 − 0.163738i
\(76\) 0 0
\(77\) 67.7772 + 39.1312i 0.880223 + 0.508197i
\(78\) 0 0
\(79\) −39.5000 + 68.4160i −0.500000 + 0.866025i 0.500000 + 0.866025i \(0.333333\pi\)
−1.00000 \(\pi\)
\(80\) 0 0
\(81\) 24.0081 + 77.3603i 0.296396 + 0.955065i
\(82\) 0 0
\(83\) − 140.872i − 1.69726i −0.528990 − 0.848628i \(-0.677429\pi\)
0.528990 − 0.848628i \(-0.322571\pi\)
\(84\) 0 0
\(85\) −60.0000 −0.705882
\(86\) 0 0
\(87\) 44.6109 + 14.6584i 0.512769 + 0.168488i
\(88\) 0 0
\(89\) 42.6028 + 24.5967i 0.478683 + 0.276368i 0.719868 − 0.694111i \(-0.244202\pi\)
−0.241184 + 0.970479i \(0.577536\pi\)
\(90\) 0 0
\(91\) 7.00000 + 12.1244i 0.0769231 + 0.133235i
\(92\) 0 0
\(93\) 8.80948 − 1.84205i 0.0947255 − 0.0198070i
\(94\) 0 0
\(95\) 30.9839 − 17.8885i 0.326146 − 0.188300i
\(96\) 0 0
\(97\) −93.0000 −0.958763 −0.479381 − 0.877607i \(-0.659139\pi\)
−0.479381 + 0.877607i \(0.659139\pi\)
\(98\) 0 0
\(99\) 100.000 + 11.1803i 1.01010 + 0.112933i
Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 336.3.bn.f.65.1 4
3.2 odd 2 inner 336.3.bn.f.65.2 4
4.3 odd 2 21.3.h.b.2.1 ✓ 4
7.4 even 3 inner 336.3.bn.f.305.2 4
12.11 even 2 21.3.h.b.2.2 yes 4
21.11 odd 6 inner 336.3.bn.f.305.1 4
28.3 even 6 147.3.h.b.116.2 4
28.11 odd 6 21.3.h.b.11.2 yes 4
28.19 even 6 147.3.b.c.50.2 2
28.23 odd 6 147.3.b.d.50.2 2
28.27 even 2 147.3.h.b.128.1 4
84.11 even 6 21.3.h.b.11.1 yes 4
84.23 even 6 147.3.b.d.50.1 2
84.47 odd 6 147.3.b.c.50.1 2
84.59 odd 6 147.3.h.b.116.1 4
84.83 odd 2 147.3.h.b.128.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
21.3.h.b.2.1 ✓ 4 4.3 odd 2
21.3.h.b.2.2 yes 4 12.11 even 2
21.3.h.b.11.1 yes 4 84.11 even 6
21.3.h.b.11.2 yes 4 28.11 odd 6
147.3.b.c.50.1 2 84.47 odd 6
147.3.b.c.50.2 2 28.19 even 6
147.3.b.d.50.1 2 84.23 even 6
147.3.b.d.50.2 2 28.23 odd 6
147.3.h.b.116.1 4 84.59 odd 6
147.3.h.b.116.2 4 28.3 even 6
147.3.h.b.128.1 4 28.27 even 2
147.3.h.b.128.2 4 84.83 odd 2
336.3.bn.f.65.1 4 1.1 even 1 trivial
336.3.bn.f.65.2 4 3.2 odd 2 inner
336.3.bn.f.305.1 4 21.11 odd 6 inner
336.3.bn.f.305.2 4 7.4 even 3 inner