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Properties

Label 351.2.h.a.334.4
Level $351$
Weight $2$
Character 351.334
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 334.4
Character \(\chi\) \(=\) 351.334
Dual form 351.2.h.a.289.4

$q$-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)
 
\(f(q)\) \(=\) \(q-1.13584 q^{2} -0.709859 q^{4} +(0.0587384 - 0.101738i) q^{5} +(-0.424723 + 0.735641i) q^{7} +3.07798 q^{8} +(-0.0667177 + 0.115558i) q^{10} +0.463942 q^{11} +(-3.59350 + 0.294508i) q^{13} +(0.482419 - 0.835574i) q^{14} -2.07638 q^{16} +(1.26296 + 2.18751i) q^{17} +(3.09113 + 5.35399i) q^{19} +(-0.0416960 + 0.0722196i) q^{20} -0.526966 q^{22} +(3.32887 + 5.76577i) q^{23} +(2.49310 + 4.31818i) q^{25} +(4.08166 - 0.334515i) q^{26} +(0.301493 - 0.522201i) q^{28} -1.90453 q^{29} +(0.657577 - 1.13896i) q^{31} -3.79751 q^{32} +(-1.43452 - 2.48467i) q^{34} +(0.0498951 + 0.0864208i) q^{35} +(-2.01347 + 3.48743i) q^{37} +(-3.51104 - 6.08129i) q^{38} +(0.180795 - 0.313147i) q^{40} +(4.84331 + 8.38887i) q^{41} +(2.10477 - 3.64556i) q^{43} -0.329333 q^{44} +(-3.78107 - 6.54901i) q^{46} +(-1.34586 - 2.33109i) q^{47} +(3.13922 + 5.43729i) q^{49} +(-2.83177 - 4.90477i) q^{50} +(2.55088 - 0.209059i) q^{52} +0.389682 q^{53} +(0.0272512 - 0.0472005i) q^{55} +(-1.30729 + 2.26429i) q^{56} +2.16325 q^{58} -11.0732 q^{59} +(3.88380 - 6.72693i) q^{61} +(-0.746905 + 1.29368i) q^{62} +8.46614 q^{64} +(-0.181114 + 0.382895i) q^{65} +(0.511351 + 0.885686i) q^{67} +(-0.896521 - 1.55282i) q^{68} +(-0.0566730 - 0.0981605i) q^{70} +(-3.61012 - 6.25291i) q^{71} -3.31321 q^{73} +(2.28699 - 3.96117i) q^{74} +(-2.19426 - 3.80057i) q^{76} +(-0.197047 + 0.341295i) q^{77} +(-4.41302 - 7.64357i) q^{79} +(-0.121963 + 0.211247i) q^{80} +(-5.50125 - 9.52844i) q^{82} +(-1.75800 - 3.04495i) q^{83} +0.296736 q^{85} +(-2.39069 + 4.14079i) q^{86} +1.42800 q^{88} +(6.62760 - 11.4793i) q^{89} +(1.30959 - 2.76861i) q^{91} +(-2.36303 - 4.09288i) q^{92} +(1.52868 + 2.64776i) q^{94} +0.726272 q^{95} +(-7.87273 + 13.6360i) q^{97} +(-3.56567 - 6.17591i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 24 q + 2 q^{2} + 18 q^{4} + 2 q^{5} + 3 q^{7} + 18 q^{8} - 6 q^{11} - 2 q^{14} + 6 q^{16} - 6 q^{17} - 3 q^{19} + 11 q^{20} - 18 q^{22} - 17 q^{23} - 6 q^{25} + 12 q^{26} + 24 q^{29} - 6 q^{31} + 38 q^{32}+ \cdots + 61 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.13584 −0.803163 −0.401581 − 0.915823i \(-0.631539\pi\)
−0.401581 + 0.915823i \(0.631539\pi\)
\(3\) 0 0
\(4\) −0.709859 −0.354929
\(5\) 0.0587384 − 0.101738i 0.0262686 − 0.0454986i −0.852592 − 0.522577i \(-0.824971\pi\)
0.878861 + 0.477078i \(0.158304\pi\)
\(6\) 0 0
\(7\) −0.424723 + 0.735641i −0.160530 + 0.278046i −0.935059 − 0.354492i \(-0.884654\pi\)
0.774529 + 0.632539i \(0.217987\pi\)
\(8\) 3.07798 1.08823
\(9\) 0 0
\(10\) −0.0667177 + 0.115558i −0.0210980 + 0.0365428i
\(11\) 0.463942 0.139884 0.0699419 − 0.997551i \(-0.477719\pi\)
0.0699419 + 0.997551i \(0.477719\pi\)
\(12\) 0 0
\(13\) −3.59350 + 0.294508i −0.996658 + 0.0816817i
\(14\) 0.482419 − 0.835574i 0.128932 − 0.223316i
\(15\) 0 0
\(16\) −2.07638 −0.519096
\(17\) 1.26296 + 2.18751i 0.306312 + 0.530548i 0.977553 − 0.210692i \(-0.0675716\pi\)
−0.671240 + 0.741240i \(0.734238\pi\)
\(18\) 0 0
\(19\) 3.09113 + 5.35399i 0.709153 + 1.22829i 0.965172 + 0.261617i \(0.0842558\pi\)
−0.256019 + 0.966672i \(0.582411\pi\)
\(20\) −0.0416960 + 0.0722196i −0.00932350 + 0.0161488i
\(21\) 0 0
\(22\) −0.526966 −0.112350
\(23\) 3.32887 + 5.76577i 0.694117 + 1.20225i 0.970478 + 0.241192i \(0.0775382\pi\)
−0.276361 + 0.961054i \(0.589128\pi\)
\(24\) 0 0
\(25\) 2.49310 + 4.31818i 0.498620 + 0.863635i
\(26\) 4.08166 − 0.334515i 0.800479 − 0.0656037i
\(27\) 0 0
\(28\) 0.301493 − 0.522201i 0.0569768 − 0.0986868i
\(29\) −1.90453 −0.353662 −0.176831 − 0.984241i \(-0.556585\pi\)
−0.176831 + 0.984241i \(0.556585\pi\)
\(30\) 0 0
\(31\) 0.657577 − 1.13896i 0.118104 − 0.204563i −0.800912 − 0.598782i \(-0.795652\pi\)
0.919016 + 0.394219i \(0.128985\pi\)
\(32\) −3.79751 −0.671310
\(33\) 0 0
\(34\) −1.43452 − 2.48467i −0.246019 − 0.426117i
\(35\) 0.0498951 + 0.0864208i 0.00843381 + 0.0146078i
\(36\) 0 0
\(37\) −2.01347 + 3.48743i −0.331012 + 0.573330i −0.982711 − 0.185148i \(-0.940723\pi\)
0.651698 + 0.758478i \(0.274057\pi\)
\(38\) −3.51104 − 6.08129i −0.569565 − 0.986516i
\(39\) 0 0
\(40\) 0.180795 − 0.313147i 0.0285863 − 0.0495129i
\(41\) 4.84331 + 8.38887i 0.756399 + 1.31012i 0.944676 + 0.328005i \(0.106376\pi\)
−0.188277 + 0.982116i \(0.560290\pi\)
\(42\) 0 0
\(43\) 2.10477 − 3.64556i 0.320974 − 0.555943i −0.659715 − 0.751516i \(-0.729323\pi\)
0.980689 + 0.195573i \(0.0626565\pi\)
\(44\) −0.329333 −0.0496489
\(45\) 0 0
\(46\) −3.78107 − 6.54901i −0.557489 − 0.965599i
\(47\) −1.34586 − 2.33109i −0.196313 − 0.340025i 0.751017 − 0.660283i \(-0.229564\pi\)
−0.947330 + 0.320258i \(0.896230\pi\)
\(48\) 0 0
\(49\) 3.13922 + 5.43729i 0.448460 + 0.776756i
\(50\) −2.83177 − 4.90477i −0.400473 − 0.693640i
\(51\) 0 0
\(52\) 2.55088 − 0.209059i 0.353743 − 0.0289912i
\(53\) 0.389682 0.0535269 0.0267634 − 0.999642i \(-0.491480\pi\)
0.0267634 + 0.999642i \(0.491480\pi\)
\(54\) 0 0
\(55\) 0.0272512 − 0.0472005i 0.00367456 − 0.00636452i
\(56\) −1.30729 + 2.26429i −0.174693 + 0.302578i
\(57\) 0 0
\(58\) 2.16325 0.284049
\(59\) −11.0732 −1.44161 −0.720805 − 0.693137i \(-0.756228\pi\)
−0.720805 + 0.693137i \(0.756228\pi\)
\(60\) 0 0
\(61\) 3.88380 − 6.72693i 0.497269 − 0.861295i −0.502726 − 0.864446i \(-0.667669\pi\)
0.999995 + 0.00315044i \(0.00100282\pi\)
\(62\) −0.746905 + 1.29368i −0.0948570 + 0.164297i
\(63\) 0 0
\(64\) 8.46614 1.05827
\(65\) −0.181114 + 0.382895i −0.0224644 + 0.0474922i
\(66\) 0 0
\(67\) 0.511351 + 0.885686i 0.0624715 + 0.108204i 0.895570 − 0.444922i \(-0.146768\pi\)
−0.833098 + 0.553125i \(0.813435\pi\)
\(68\) −0.896521 − 1.55282i −0.108719 − 0.188307i
\(69\) 0 0
\(70\) −0.0566730 − 0.0981605i −0.00677372 − 0.0117324i
\(71\) −3.61012 − 6.25291i −0.428442 − 0.742083i 0.568293 − 0.822826i \(-0.307604\pi\)
−0.996735 + 0.0807430i \(0.974271\pi\)
\(72\) 0 0
\(73\) −3.31321 −0.387782 −0.193891 − 0.981023i \(-0.562111\pi\)
−0.193891 + 0.981023i \(0.562111\pi\)
\(74\) 2.28699 − 3.96117i 0.265857 − 0.460477i
\(75\) 0 0
\(76\) −2.19426 − 3.80057i −0.251699 − 0.435956i
\(77\) −0.197047 + 0.341295i −0.0224556 + 0.0388942i
\(78\) 0 0
\(79\) −4.41302 − 7.64357i −0.496503 − 0.859969i 0.503489 − 0.864002i \(-0.332050\pi\)
−0.999992 + 0.00403289i \(0.998716\pi\)
\(80\) −0.121963 + 0.211247i −0.0136359 + 0.0236181i
\(81\) 0 0
\(82\) −5.50125 − 9.52844i −0.607511 − 1.05224i
\(83\) −1.75800 − 3.04495i −0.192966 − 0.334227i 0.753266 − 0.657716i \(-0.228477\pi\)
−0.946232 + 0.323489i \(0.895144\pi\)
\(84\) 0 0
\(85\) 0.296736 0.0321856
\(86\) −2.39069 + 4.14079i −0.257794 + 0.446513i
\(87\) 0 0
\(88\) 1.42800 0.152226
\(89\) 6.62760 − 11.4793i 0.702525 − 1.21681i −0.265053 − 0.964234i \(-0.585389\pi\)
0.967577 − 0.252574i \(-0.0812773\pi\)
\(90\) 0 0
\(91\) 1.30959 − 2.76861i 0.137282 − 0.290230i
\(92\) −2.36303 − 4.09288i −0.246362 − 0.426712i
\(93\) 0 0
\(94\) 1.52868 + 2.64776i 0.157672 + 0.273095i
\(95\) 0.726272 0.0745139
\(96\) 0 0
\(97\) −7.87273 + 13.6360i −0.799354 + 1.38452i 0.120683 + 0.992691i \(0.461492\pi\)
−0.920037 + 0.391831i \(0.871842\pi\)
\(98\) −3.56567 − 6.17591i −0.360187 − 0.623861i
\(99\) 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.334.4 24
3.2 odd 2 117.2.h.a.22.9 yes 24
9.2 odd 6 117.2.f.a.61.4 ✓ 24
9.7 even 3 351.2.f.a.100.9 24
13.3 even 3 351.2.f.a.172.9 24
39.29 odd 6 117.2.f.a.94.4 yes 24
117.16 even 3 inner 351.2.h.a.289.4 24
117.29 odd 6 117.2.h.a.16.9 yes 24
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
117.2.f.a.61.4 ✓ 24 9.2 odd 6
117.2.f.a.94.4 yes 24 39.29 odd 6
117.2.h.a.16.9 yes 24 117.29 odd 6
117.2.h.a.22.9 yes 24 3.2 odd 2
351.2.f.a.100.9 24 9.7 even 3
351.2.f.a.172.9 24 13.3 even 3
351.2.h.a.289.4 24 117.16 even 3 inner
351.2.h.a.334.4 24 1.1 even 1 trivial