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Properties

Label 351.2.h.a.289.8
Level $351$
Weight $2$
Character 351.289
Analytic conductor $2.803$
Analytic rank $0$
Dimension $24$
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] = [351,2,Mod(289,351)] 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("351.289"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(351, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([4, 2])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 351 = 3^{3} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 351.h (of order \(3\), degree \(2\), not 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: \(2.80274911095\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(12\) over \(\Q(\zeta_{3})\)
Twist minimal: no (minimal twist has level 117)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 289.8
Character \(\chi\) \(=\) 351.289
Dual form 351.2.h.a.334.8

$q$-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)
 
\(f(q)\) \(=\) \(q+0.867378 q^{2} -1.24766 q^{4} +(0.0324057 + 0.0561283i) q^{5} +(1.96209 + 3.39845i) q^{7} -2.81694 q^{8} +(0.0281080 + 0.0486844i) q^{10} +5.29315 q^{11} +(3.21261 + 1.63680i) q^{13} +(1.70188 + 2.94774i) q^{14} +0.0519555 q^{16} +(-2.28144 + 3.95157i) q^{17} +(0.281137 - 0.486944i) q^{19} +(-0.0404311 - 0.0700288i) q^{20} +4.59116 q^{22} +(-1.42839 + 2.47404i) q^{23} +(2.49790 - 4.32649i) q^{25} +(2.78655 + 1.41973i) q^{26} +(-2.44802 - 4.24009i) q^{28} -6.00595 q^{29} +(-4.23254 - 7.33098i) q^{31} +5.67895 q^{32} +(-1.97887 + 3.42750i) q^{34} +(-0.127166 + 0.220258i) q^{35} +(-0.506751 - 0.877718i) q^{37} +(0.243852 - 0.422364i) q^{38} +(-0.0912850 - 0.158110i) q^{40} +(0.674907 - 1.16897i) q^{41} +(3.45051 + 5.97645i) q^{43} -6.60403 q^{44} +(-1.23895 + 2.14593i) q^{46} +(2.22815 - 3.85927i) q^{47} +(-4.19963 + 7.27396i) q^{49} +(2.16662 - 3.75270i) q^{50} +(-4.00823 - 2.04217i) q^{52} -1.68875 q^{53} +(0.171528 + 0.297096i) q^{55} +(-5.52711 - 9.57324i) q^{56} -5.20943 q^{58} -9.14878 q^{59} +(-3.17857 - 5.50545i) q^{61} +(-3.67121 - 6.35873i) q^{62} +4.82189 q^{64} +(0.0122360 + 0.233360i) q^{65} +(-1.76507 + 3.05718i) q^{67} +(2.84645 - 4.93019i) q^{68} +(-0.110301 + 0.191047i) q^{70} +(5.02865 - 8.70987i) q^{71} +3.39546 q^{73} +(-0.439545 - 0.761314i) q^{74} +(-0.350762 + 0.607538i) q^{76} +(10.3857 + 17.9885i) q^{77} +(5.67573 - 9.83065i) q^{79} +(0.00168365 + 0.00291617i) q^{80} +(0.585399 - 1.01394i) q^{82} +(1.87243 - 3.24315i) q^{83} -0.295726 q^{85} +(2.99289 + 5.18384i) q^{86} -14.9105 q^{88} +(2.00609 + 3.47464i) q^{89} +(0.740862 + 14.1295i) q^{91} +(1.78214 - 3.08675i) q^{92} +(1.93265 - 3.34745i) q^{94} +0.0364418 q^{95} +(-1.67726 - 2.90510i) q^{97} +(-3.64266 + 6.30928i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 24 q + 2 q^{2} + 18 q^{4} + 2 q^{5} + 3 q^{7} + 18 q^{8} - 6 q^{11} - 2 q^{14} + 6 q^{16} - 6 q^{17} - 3 q^{19} + 11 q^{20} - 18 q^{22} - 17 q^{23} - 6 q^{25} + 12 q^{26} + 24 q^{29} - 6 q^{31} + 38 q^{32}+ \cdots + 61 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/351\mathbb{Z}\right)^\times\).

\(n\) \(28\) \(326\)
\(\chi(n)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{2}{3}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.867378 0.613329 0.306664 − 0.951818i \(-0.400787\pi\)
0.306664 + 0.951818i \(0.400787\pi\)
\(3\) 0 0
\(4\) −1.24766 −0.623828
\(5\) 0.0324057 + 0.0561283i 0.0144923 + 0.0251013i 0.873181 − 0.487397i \(-0.162053\pi\)
−0.858688 + 0.512498i \(0.828720\pi\)
\(6\) 0 0
\(7\) 1.96209 + 3.39845i 0.741602 + 1.28449i 0.951766 + 0.306826i \(0.0992669\pi\)
−0.210164 + 0.977666i \(0.567400\pi\)
\(8\) −2.81694 −0.995940
\(9\) 0 0
\(10\) 0.0281080 + 0.0486844i 0.00888852 + 0.0153954i
\(11\) 5.29315 1.59595 0.797973 − 0.602694i \(-0.205906\pi\)
0.797973 + 0.602694i \(0.205906\pi\)
\(12\) 0 0
\(13\) 3.21261 + 1.63680i 0.891018 + 0.453967i
\(14\) 1.70188 + 2.94774i 0.454846 + 0.787816i
\(15\) 0 0
\(16\) 0.0519555 0.0129889
\(17\) −2.28144 + 3.95157i −0.553330 + 0.958395i 0.444702 + 0.895679i \(0.353310\pi\)
−0.998031 + 0.0627166i \(0.980024\pi\)
\(18\) 0 0
\(19\) 0.281137 − 0.486944i 0.0644973 − 0.111713i −0.831974 − 0.554815i \(-0.812789\pi\)
0.896471 + 0.443103i \(0.146122\pi\)
\(20\) −0.0404311 − 0.0700288i −0.00904068 − 0.0156589i
\(21\) 0 0
\(22\) 4.59116 0.978839
\(23\) −1.42839 + 2.47404i −0.297840 + 0.515873i −0.975641 − 0.219371i \(-0.929599\pi\)
0.677802 + 0.735245i \(0.262933\pi\)
\(24\) 0 0
\(25\) 2.49790 − 4.32649i 0.499580 − 0.865298i
\(26\) 2.78655 + 1.41973i 0.546487 + 0.278431i
\(27\) 0 0
\(28\) −2.44802 − 4.24009i −0.462632 − 0.801302i
\(29\) −6.00595 −1.11528 −0.557639 − 0.830084i \(-0.688292\pi\)
−0.557639 + 0.830084i \(0.688292\pi\)
\(30\) 0 0
\(31\) −4.23254 − 7.33098i −0.760187 − 1.31668i −0.942754 − 0.333489i \(-0.891774\pi\)
0.182567 − 0.983193i \(-0.441559\pi\)
\(32\) 5.67895 1.00391
\(33\) 0 0
\(34\) −1.97887 + 3.42750i −0.339373 + 0.587812i
\(35\) −0.127166 + 0.220258i −0.0214950 + 0.0372304i
\(36\) 0 0
\(37\) −0.506751 − 0.877718i −0.0833094 − 0.144296i 0.821360 − 0.570410i \(-0.193216\pi\)
−0.904670 + 0.426114i \(0.859882\pi\)
\(38\) 0.243852 − 0.422364i 0.0395580 − 0.0685165i
\(39\) 0 0
\(40\) −0.0912850 − 0.158110i −0.0144334 − 0.0249994i
\(41\) 0.674907 − 1.16897i 0.105403 − 0.182563i −0.808500 − 0.588496i \(-0.799720\pi\)
0.913903 + 0.405933i \(0.133053\pi\)
\(42\) 0 0
\(43\) 3.45051 + 5.97645i 0.526197 + 0.911400i 0.999534 + 0.0305189i \(0.00971599\pi\)
−0.473337 + 0.880881i \(0.656951\pi\)
\(44\) −6.60403 −0.995595
\(45\) 0 0
\(46\) −1.23895 + 2.14593i −0.182674 + 0.316400i
\(47\) 2.22815 − 3.85927i 0.325009 − 0.562933i −0.656505 − 0.754322i \(-0.727966\pi\)
0.981514 + 0.191389i \(0.0612992\pi\)
\(48\) 0 0
\(49\) −4.19963 + 7.27396i −0.599946 + 1.03914i
\(50\) 2.16662 − 3.75270i 0.306407 − 0.530712i
\(51\) 0 0
\(52\) −4.00823 − 2.04217i −0.555842 − 0.283197i
\(53\) −1.68875 −0.231968 −0.115984 − 0.993251i \(-0.537002\pi\)
−0.115984 + 0.993251i \(0.537002\pi\)
\(54\) 0 0
\(55\) 0.171528 + 0.297096i 0.0231289 + 0.0400604i
\(56\) −5.52711 − 9.57324i −0.738591 − 1.27928i
\(57\) 0 0
\(58\) −5.20943 −0.684032
\(59\) −9.14878 −1.19107 −0.595535 − 0.803330i \(-0.703060\pi\)
−0.595535 + 0.803330i \(0.703060\pi\)
\(60\) 0 0
\(61\) −3.17857 − 5.50545i −0.406974 − 0.704900i 0.587575 − 0.809170i \(-0.300083\pi\)
−0.994549 + 0.104270i \(0.966750\pi\)
\(62\) −3.67121 − 6.35873i −0.466245 − 0.807559i
\(63\) 0 0
\(64\) 4.82189 0.602736
\(65\) 0.0122360 + 0.233360i 0.00151769 + 0.0289448i
\(66\) 0 0
\(67\) −1.76507 + 3.05718i −0.215637 + 0.373494i −0.953469 − 0.301490i \(-0.902516\pi\)
0.737832 + 0.674984i \(0.235849\pi\)
\(68\) 2.84645 − 4.93019i 0.345183 − 0.597874i
\(69\) 0 0
\(70\) −0.110301 + 0.191047i −0.0131835 + 0.0228345i
\(71\) 5.02865 − 8.70987i 0.596790 − 1.03367i −0.396501 − 0.918034i \(-0.629776\pi\)
0.993292 − 0.115637i \(-0.0368910\pi\)
\(72\) 0 0
\(73\) 3.39546 0.397409 0.198704 − 0.980059i \(-0.436327\pi\)
0.198704 + 0.980059i \(0.436327\pi\)
\(74\) −0.439545 − 0.761314i −0.0510960 − 0.0885009i
\(75\) 0 0
\(76\) −0.350762 + 0.607538i −0.0402352 + 0.0696894i
\(77\) 10.3857 + 17.9885i 1.18356 + 2.04998i
\(78\) 0 0
\(79\) 5.67573 − 9.83065i 0.638570 − 1.10604i −0.347177 − 0.937800i \(-0.612860\pi\)
0.985747 − 0.168235i \(-0.0538069\pi\)
\(80\) 0.00168365 + 0.00291617i 0.000188238 + 0.000326038i
\(81\) 0 0
\(82\) 0.585399 − 1.01394i 0.0646465 − 0.111971i
\(83\) 1.87243 − 3.24315i 0.205526 − 0.355982i −0.744774 − 0.667317i \(-0.767443\pi\)
0.950300 + 0.311335i \(0.100776\pi\)
\(84\) 0 0
\(85\) −0.295726 −0.0320760
\(86\) 2.99289 + 5.18384i 0.322732 + 0.558988i
\(87\) 0 0
\(88\) −14.9105 −1.58947
\(89\) 2.00609 + 3.47464i 0.212645 + 0.368312i 0.952541 − 0.304409i \(-0.0984589\pi\)
−0.739897 + 0.672721i \(0.765126\pi\)
\(90\) 0 0
\(91\) 0.740862 + 14.1295i 0.0776634 + 1.48117i
\(92\) 1.78214 − 3.08675i 0.185801 − 0.321816i
\(93\) 0 0
\(94\) 1.93265 − 3.34745i 0.199338 − 0.345263i
\(95\) 0.0364418 0.00373885
\(96\) 0 0
\(97\) −1.67726 − 2.90510i −0.170300 − 0.294968i 0.768225 − 0.640180i \(-0.221140\pi\)
−0.938525 + 0.345212i \(0.887807\pi\)
\(98\) −3.64266 + 6.30928i −0.367964 + 0.637333i
\(99\) 0 0
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 351.2.h.a.289.8 24
3.2 odd 2 117.2.h.a.16.5 yes 24
9.4 even 3 351.2.f.a.172.5 24
9.5 odd 6 117.2.f.a.94.8 yes 24
13.9 even 3 351.2.f.a.100.5 24
39.35 odd 6 117.2.f.a.61.8 ✓ 24
117.22 even 3 inner 351.2.h.a.334.8 24
117.113 odd 6 117.2.h.a.22.5 yes 24
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
117.2.f.a.61.8 ✓ 24 39.35 odd 6
117.2.f.a.94.8 yes 24 9.5 odd 6
117.2.h.a.16.5 yes 24 3.2 odd 2
117.2.h.a.22.5 yes 24 117.113 odd 6
351.2.f.a.100.5 24 13.9 even 3
351.2.f.a.172.5 24 9.4 even 3
351.2.h.a.289.8 24 1.1 even 1 trivial
351.2.h.a.334.8 24 117.22 even 3 inner