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Properties

Label 351.2.f.a.100.9
Level $351$
Weight $2$
Character 351.100
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(100,351)] 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("351.100"); 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, 4])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 351 = 3^{3} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 351.f (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 100.9
Character \(\chi\) \(=\) 351.100
Dual form 351.2.f.a.172.9

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.567922 + 0.983670i) q^{2} +(0.354929 - 0.614756i) q^{4} +(0.0587384 + 0.101738i) q^{5} +0.849445 q^{7} +3.07798 q^{8} +(-0.0667177 + 0.115558i) q^{10} +(-0.231971 - 0.401786i) q^{11} +(1.54170 - 3.25932i) q^{13} +(0.482419 + 0.835574i) q^{14} +(1.03819 + 1.79820i) q^{16} +(1.26296 + 2.18751i) q^{17} +(3.09113 + 5.35399i) q^{19} +0.0833920 q^{20} +(0.263483 - 0.456366i) q^{22} -6.65773 q^{23} +(2.49310 - 4.31818i) q^{25} +(4.08166 - 0.334515i) q^{26} +(0.301493 - 0.522201i) q^{28} +(0.952265 + 1.64937i) q^{29} +(0.657577 + 1.13896i) q^{31} +(1.89875 - 3.28874i) q^{32} +(-1.43452 + 2.48467i) q^{34} +(0.0498951 + 0.0864208i) q^{35} +(-2.01347 + 3.48743i) q^{37} +(-3.51104 + 6.08129i) q^{38} +(0.180795 + 0.313147i) q^{40} -9.68663 q^{41} -4.20953 q^{43} -0.329333 q^{44} +(-3.78107 - 6.54901i) q^{46} +(-1.34586 + 2.33109i) q^{47} -6.27844 q^{49} +5.66354 q^{50} +(-1.45649 - 2.10460i) q^{52} +0.389682 q^{53} +(0.0272512 - 0.0472005i) q^{55} +2.61457 q^{56} +(-1.08162 + 1.87343i) q^{58} +(5.53661 - 9.58969i) q^{59} -7.76759 q^{61} +(-0.746905 + 1.29368i) q^{62} +8.46614 q^{64} +(0.422153 - 0.0345978i) q^{65} -1.02270 q^{67} +1.79304 q^{68} +(-0.0566730 + 0.0981605i) q^{70} +(-3.61012 - 6.25291i) q^{71} -3.31321 q^{73} -4.57397 q^{74} +4.38853 q^{76} +(-0.197047 - 0.341295i) q^{77} +(-4.41302 + 7.64357i) q^{79} +(-0.121963 + 0.211247i) q^{80} +(-5.50125 - 9.52844i) q^{82} +(-1.75800 + 3.04495i) q^{83} +(-0.148368 + 0.256981i) q^{85} +(-2.39069 - 4.14079i) q^{86} +(-0.714002 - 1.23669i) q^{88} +(6.62760 - 11.4793i) q^{89} +(1.30959 - 2.76861i) q^{91} +(-2.36303 + 4.09288i) q^{92} -3.05736 q^{94} +(-0.363136 + 0.628970i) q^{95} +15.7455 q^{97} +(-3.56567 - 6.17591i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 24 q - q^{2} - 9 q^{4} + 2 q^{5} - 6 q^{7} + 18 q^{8} + 3 q^{11} - 2 q^{14} - 3 q^{16} - 6 q^{17} - 3 q^{19} - 22 q^{20} + 9 q^{22} + 34 q^{23} - 6 q^{25} + 12 q^{26} - 12 q^{29} - 6 q^{31} - 19 q^{32}+ \cdots + 61 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.567922 + 0.983670i 0.401581 + 0.695559i 0.993917 − 0.110132i \(-0.0351273\pi\)
−0.592336 + 0.805691i \(0.701794\pi\)
\(3\) 0 0
\(4\) 0.354929 − 0.614756i 0.177465 − 0.307378i
\(5\) 0.0587384 + 0.101738i 0.0262686 + 0.0454986i 0.878861 − 0.477078i \(-0.158304\pi\)
−0.852592 + 0.522577i \(0.824971\pi\)
\(6\) 0 0
\(7\) 0.849445 0.321060 0.160530 − 0.987031i \(-0.448680\pi\)
0.160530 + 0.987031i \(0.448680\pi\)
\(8\) 3.07798 1.08823
\(9\) 0 0
\(10\) −0.0667177 + 0.115558i −0.0210980 + 0.0365428i
\(11\) −0.231971 − 0.401786i −0.0699419 − 0.121143i 0.828934 − 0.559347i \(-0.188948\pi\)
−0.898876 + 0.438204i \(0.855615\pi\)
\(12\) 0 0
\(13\) 1.54170 − 3.25932i 0.427591 − 0.903972i
\(14\) 0.482419 + 0.835574i 0.128932 + 0.223316i
\(15\) 0 0
\(16\) 1.03819 + 1.79820i 0.259548 + 0.449550i
\(17\) 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 \(0.582411\pi\)
\(20\) 0.0833920 0.0186470
\(21\) 0 0
\(22\) 0.263483 − 0.456366i 0.0561748 − 0.0972975i
\(23\) −6.65773 −1.38823 −0.694117 − 0.719862i \(-0.744205\pi\)
−0.694117 + 0.719862i \(0.744205\pi\)
\(24\) 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
\(29\) 0.952265 + 1.64937i 0.176831 + 0.306281i 0.940793 − 0.338980i \(-0.110082\pi\)
−0.763962 + 0.645261i \(0.776749\pi\)
\(30\) 0 0
\(31\) 0.657577 + 1.13896i 0.118104 + 0.204563i 0.919016 − 0.394219i \(-0.128985\pi\)
−0.800912 + 0.598782i \(0.795652\pi\)
\(32\) 1.89875 − 3.28874i 0.335655 − 0.581372i
\(33\) 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
\(41\) −9.68663 −1.51280 −0.756399 − 0.654111i \(-0.773043\pi\)
−0.756399 + 0.654111i \(0.773043\pi\)
\(42\) 0 0
\(43\) −4.20953 −0.641948 −0.320974 − 0.947088i \(-0.604010\pi\)
−0.320974 + 0.947088i \(0.604010\pi\)
\(44\) −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 −0.947330 − 0.320258i \(-0.896230\pi\)
0.751017 + 0.660283i \(0.229564\pi\)
\(48\) 0 0
\(49\) −6.27844 −0.896920
\(50\) 5.66354 0.800946
\(51\) 0 0
\(52\) −1.45649 − 2.10460i −0.201979 − 0.291855i
\(53\) 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
\(56\) 2.61457 0.349387
\(57\) 0 0
\(58\) −1.08162 + 1.87343i −0.142024 + 0.245993i
\(59\) 5.53661 − 9.58969i 0.720805 − 1.24847i −0.239872 − 0.970805i \(-0.577105\pi\)
0.960677 − 0.277667i \(-0.0895612\pi\)
\(60\) 0 0
\(61\) −7.76759 −0.994538 −0.497269 − 0.867596i \(-0.665664\pi\)
−0.497269 + 0.867596i \(0.665664\pi\)
\(62\) −0.746905 + 1.29368i −0.0948570 + 0.164297i
\(63\) 0 0
\(64\) 8.46614 1.05827
\(65\) 0.422153 − 0.0345978i 0.0523617 − 0.00429133i
\(66\) 0 0
\(67\) −1.02270 −0.124943 −0.0624715 − 0.998047i \(-0.519898\pi\)
−0.0624715 + 0.998047i \(0.519898\pi\)
\(68\) 1.79304 0.217438
\(69\) 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 \(0.562111\pi\)
\(74\) −4.57397 −0.531713
\(75\) 0 0
\(76\) 4.38853 0.503398
\(77\) −0.197047 − 0.341295i −0.0224556 − 0.0388942i
\(78\) 0 0
\(79\) −4.41302 + 7.64357i −0.496503 + 0.859969i −0.999992 − 0.00403289i \(-0.998716\pi\)
0.503489 + 0.864002i \(0.332050\pi\)
\(80\) −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 −0.946232 − 0.323489i \(-0.895144\pi\)
0.753266 + 0.657716i \(0.228477\pi\)
\(84\) 0 0
\(85\) −0.148368 + 0.256981i −0.0160928 + 0.0278735i
\(86\) −2.39069 − 4.14079i −0.257794 − 0.446513i
\(87\) 0 0
\(88\) −0.714002 − 1.23669i −0.0761128 − 0.131831i
\(89\) 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
\(94\) −3.05736 −0.315343
\(95\) −0.363136 + 0.628970i −0.0372569 + 0.0645309i
\(96\) 0 0
\(97\) 15.7455 1.59871 0.799354 − 0.600860i \(-0.205175\pi\)
0.799354 + 0.600860i \(0.205175\pi\)
\(98\) −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.f.a.100.9 24
3.2 odd 2 117.2.f.a.61.4 ✓ 24
9.4 even 3 351.2.h.a.334.4 24
9.5 odd 6 117.2.h.a.22.9 yes 24
13.3 even 3 351.2.h.a.289.4 24
39.29 odd 6 117.2.h.a.16.9 yes 24
117.68 odd 6 117.2.f.a.94.4 yes 24
117.94 even 3 inner 351.2.f.a.172.9 24
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
117.2.f.a.61.4 ✓ 24 3.2 odd 2
117.2.f.a.94.4 yes 24 117.68 odd 6
117.2.h.a.16.9 yes 24 39.29 odd 6
117.2.h.a.22.9 yes 24 9.5 odd 6
351.2.f.a.100.9 24 1.1 even 1 trivial
351.2.f.a.172.9 24 117.94 even 3 inner
351.2.h.a.289.4 24 13.3 even 3
351.2.h.a.334.4 24 9.4 even 3