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Properties

Label 351.2.f.a.172.10
Level $351$
Weight $2$
Character 351.172
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 172.10
Character \(\chi\) \(=\) 351.172
Dual form 351.2.f.a.100.10

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.900808 - 1.56024i) q^{2} +(-0.622909 - 1.07891i) q^{4} +(-1.73153 + 2.99909i) q^{5} +3.24477 q^{7} +1.35875 q^{8} +(3.11954 + 5.40321i) q^{10} +(0.304566 - 0.527523i) q^{11} +(2.81952 - 2.24729i) q^{13} +(2.92291 - 5.06264i) q^{14} +(2.46979 - 4.27780i) q^{16} +(-1.20773 + 2.09186i) q^{17} +(-0.877705 + 1.52023i) q^{19} +4.31433 q^{20} +(-0.548710 - 0.950393i) q^{22} -0.162448 q^{23} +(-3.49636 - 6.05587i) q^{25} +(-0.966470 - 6.42352i) q^{26} +(-2.02120 - 3.50082i) q^{28} +(1.45362 - 2.51774i) q^{29} +(-4.62084 + 8.00352i) q^{31} +(-3.09086 - 5.35353i) q^{32} +(2.17587 + 3.76872i) q^{34} +(-5.61840 + 9.73136i) q^{35} +(-0.826776 - 1.43202i) q^{37} +(1.58129 + 2.73887i) q^{38} +(-2.35270 + 4.07500i) q^{40} +8.22245 q^{41} -12.0056 q^{43} -0.758866 q^{44} +(-0.146335 + 0.253459i) q^{46} +(-4.38799 - 7.60023i) q^{47} +3.52854 q^{49} -12.5982 q^{50} +(-4.18093 - 1.64216i) q^{52} -9.43717 q^{53} +(1.05473 + 1.82684i) q^{55} +4.40882 q^{56} +(-2.61886 - 4.53599i) q^{58} +(-4.13217 - 7.15713i) q^{59} +9.92836 q^{61} +(8.32497 + 14.4193i) q^{62} -1.25794 q^{64} +(1.85774 + 12.3472i) q^{65} +2.29525 q^{67} +3.00924 q^{68} +(10.1222 + 17.5322i) q^{70} +(4.87460 - 8.44305i) q^{71} -2.59634 q^{73} -2.97906 q^{74} +2.18692 q^{76} +(0.988245 - 1.71169i) q^{77} +(-2.53739 - 4.39489i) q^{79} +(8.55300 + 14.8142i) q^{80} +(7.40685 - 12.8290i) q^{82} +(4.14134 + 7.17301i) q^{83} +(-4.18245 - 7.24421i) q^{85} +(-10.8147 + 18.7316i) q^{86} +(0.413827 - 0.716770i) q^{88} +(-4.93949 - 8.55544i) q^{89} +(9.14870 - 7.29193i) q^{91} +(0.101191 + 0.175267i) q^{92} -15.8109 q^{94} +(-3.03954 - 5.26463i) q^{95} -7.47810 q^{97} +(3.17853 - 5.50538i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 24 q - q^{2} - 9 q^{4} + 2 q^{5} - 6 q^{7} + 18 q^{8} + 3 q^{11} - 2 q^{14} - 3 q^{16} - 6 q^{17} - 3 q^{19} - 22 q^{20} + 9 q^{22} + 34 q^{23} - 6 q^{25} + 12 q^{26} - 12 q^{29} - 6 q^{31} - 19 q^{32}+ \cdots + 61 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.900808 − 1.56024i 0.636967 − 1.10326i −0.349127 − 0.937075i \(-0.613522\pi\)
0.986095 − 0.166184i \(-0.0531447\pi\)
\(3\) 0 0
\(4\) −0.622909 − 1.07891i −0.311455 − 0.539455i
\(5\) −1.73153 + 2.99909i −0.774362 + 1.34123i 0.160791 + 0.986988i \(0.448595\pi\)
−0.935153 + 0.354245i \(0.884738\pi\)
\(6\) 0 0
\(7\) 3.24477 1.22641 0.613204 − 0.789925i \(-0.289880\pi\)
0.613204 + 0.789925i \(0.289880\pi\)
\(8\) 1.35875 0.480389
\(9\) 0 0
\(10\) 3.11954 + 5.40321i 0.986486 + 1.70864i
\(11\) 0.304566 − 0.527523i 0.0918300 − 0.159054i −0.816451 − 0.577414i \(-0.804062\pi\)
0.908281 + 0.418360i \(0.137395\pi\)
\(12\) 0 0
\(13\) 2.81952 − 2.24729i 0.781995 − 0.623285i
\(14\) 2.92291 − 5.06264i 0.781182 − 1.35305i
\(15\) 0 0
\(16\) 2.46979 − 4.27780i 0.617447 − 1.06945i
\(17\) −1.20773 + 2.09186i −0.292919 + 0.507350i −0.974499 − 0.224394i \(-0.927960\pi\)
0.681580 + 0.731744i \(0.261293\pi\)
\(18\) 0 0
\(19\) −0.877705 + 1.52023i −0.201359 + 0.348765i −0.948967 − 0.315376i \(-0.897869\pi\)
0.747607 + 0.664141i \(0.231203\pi\)
\(20\) 4.31433 0.964714
\(21\) 0 0
\(22\) −0.548710 − 0.950393i −0.116985 − 0.202625i
\(23\) −0.162448 −0.0338728 −0.0169364 − 0.999857i \(-0.505391\pi\)
−0.0169364 + 0.999857i \(0.505391\pi\)
\(24\) 0 0
\(25\) −3.49636 − 6.05587i −0.699272 − 1.21117i
\(26\) −0.966470 − 6.42352i −0.189540 − 1.25976i
\(27\) 0 0
\(28\) −2.02120 − 3.50082i −0.381970 − 0.661592i
\(29\) 1.45362 − 2.51774i 0.269930 − 0.467532i −0.698914 − 0.715206i \(-0.746333\pi\)
0.968843 + 0.247674i \(0.0796662\pi\)
\(30\) 0 0
\(31\) −4.62084 + 8.00352i −0.829927 + 1.43748i 0.0681682 + 0.997674i \(0.478285\pi\)
−0.898095 + 0.439802i \(0.855049\pi\)
\(32\) −3.09086 − 5.35353i −0.546392 − 0.946379i
\(33\) 0 0
\(34\) 2.17587 + 3.76872i 0.373159 + 0.646331i
\(35\) −5.61840 + 9.73136i −0.949683 + 1.64490i
\(36\) 0 0
\(37\) −0.826776 − 1.43202i −0.135921 − 0.235422i 0.790028 − 0.613071i \(-0.210066\pi\)
−0.925949 + 0.377649i \(0.876733\pi\)
\(38\) 1.58129 + 2.73887i 0.256519 + 0.444303i
\(39\) 0 0
\(40\) −2.35270 + 4.07500i −0.371995 + 0.644314i
\(41\) 8.22245 1.28413 0.642065 − 0.766650i \(-0.278078\pi\)
0.642065 + 0.766650i \(0.278078\pi\)
\(42\) 0 0
\(43\) −12.0056 −1.83083 −0.915415 − 0.402512i \(-0.868137\pi\)
−0.915415 + 0.402512i \(0.868137\pi\)
\(44\) −0.758866 −0.114403
\(45\) 0 0
\(46\) −0.146335 + 0.253459i −0.0215759 + 0.0373705i
\(47\) −4.38799 − 7.60023i −0.640054 − 1.10861i −0.985420 − 0.170139i \(-0.945578\pi\)
0.345366 − 0.938468i \(-0.387755\pi\)
\(48\) 0 0
\(49\) 3.52854 0.504077
\(50\) −12.5982 −1.78165
\(51\) 0 0
\(52\) −4.18093 − 1.64216i −0.579790 − 0.227726i
\(53\) −9.43717 −1.29630 −0.648148 − 0.761515i \(-0.724456\pi\)
−0.648148 + 0.761515i \(0.724456\pi\)
\(54\) 0 0
\(55\) 1.05473 + 1.82684i 0.142219 + 0.246331i
\(56\) 4.40882 0.589153
\(57\) 0 0
\(58\) −2.61886 − 4.53599i −0.343873 − 0.595605i
\(59\) −4.13217 − 7.15713i −0.537963 − 0.931779i −0.999014 − 0.0444054i \(-0.985861\pi\)
0.461051 − 0.887374i \(-0.347473\pi\)
\(60\) 0 0
\(61\) 9.92836 1.27120 0.635598 − 0.772020i \(-0.280753\pi\)
0.635598 + 0.772020i \(0.280753\pi\)
\(62\) 8.32497 + 14.4193i 1.05727 + 1.83125i
\(63\) 0 0
\(64\) −1.25794 −0.157242
\(65\) 1.85774 + 12.3472i 0.230424 + 1.53149i
\(66\) 0 0
\(67\) 2.29525 0.280410 0.140205 − 0.990123i \(-0.455224\pi\)
0.140205 + 0.990123i \(0.455224\pi\)
\(68\) 3.00924 0.364923
\(69\) 0 0
\(70\) 10.1222 + 17.5322i 1.20983 + 2.09549i
\(71\) 4.87460 − 8.44305i 0.578508 − 1.00201i −0.417143 − 0.908841i \(-0.636968\pi\)
0.995651 − 0.0931644i \(-0.0296982\pi\)
\(72\) 0 0
\(73\) −2.59634 −0.303878 −0.151939 − 0.988390i \(-0.548552\pi\)
−0.151939 + 0.988390i \(0.548552\pi\)
\(74\) −2.97906 −0.346309
\(75\) 0 0
\(76\) 2.18692 0.250857
\(77\) 0.988245 − 1.71169i 0.112621 − 0.195065i
\(78\) 0 0
\(79\) −2.53739 − 4.39489i −0.285479 − 0.494464i 0.687246 − 0.726424i \(-0.258819\pi\)
−0.972725 + 0.231961i \(0.925486\pi\)
\(80\) 8.55300 + 14.8142i 0.956254 + 1.65628i
\(81\) 0 0
\(82\) 7.40685 − 12.8290i 0.817949 − 1.41673i
\(83\) 4.14134 + 7.17301i 0.454571 + 0.787340i 0.998663 − 0.0516855i \(-0.0164593\pi\)
−0.544093 + 0.839025i \(0.683126\pi\)
\(84\) 0 0
\(85\) −4.18245 − 7.24421i −0.453650 − 0.785745i
\(86\) −10.8147 + 18.7316i −1.16618 + 2.01988i
\(87\) 0 0
\(88\) 0.413827 − 0.716770i 0.0441141 − 0.0764079i
\(89\) −4.93949 − 8.55544i −0.523584 − 0.906875i −0.999623 − 0.0274506i \(-0.991261\pi\)
0.476039 − 0.879424i \(-0.342072\pi\)
\(90\) 0 0
\(91\) 9.14870 − 7.29193i 0.959044 − 0.764402i
\(92\) 0.101191 + 0.175267i 0.0105498 + 0.0182729i
\(93\) 0 0
\(94\) −15.8109 −1.63077
\(95\) −3.03954 − 5.26463i −0.311850 − 0.540140i
\(96\) 0 0
\(97\) −7.47810 −0.759286 −0.379643 − 0.925133i \(-0.623953\pi\)
−0.379643 + 0.925133i \(0.623953\pi\)
\(98\) 3.17853 − 5.50538i 0.321080 − 0.556127i
\(99\) 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.172.10 24
3.2 odd 2 117.2.f.a.94.3 yes 24
9.2 odd 6 117.2.h.a.16.10 yes 24
9.7 even 3 351.2.h.a.289.3 24
13.9 even 3 351.2.h.a.334.3 24
39.35 odd 6 117.2.h.a.22.10 yes 24
117.61 even 3 inner 351.2.f.a.100.10 24
117.74 odd 6 117.2.f.a.61.3 ✓ 24
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
117.2.f.a.61.3 ✓ 24 117.74 odd 6
117.2.f.a.94.3 yes 24 3.2 odd 2
117.2.h.a.16.10 yes 24 9.2 odd 6
117.2.h.a.22.10 yes 24 39.35 odd 6
351.2.f.a.100.10 24 117.61 even 3 inner
351.2.f.a.172.10 24 1.1 even 1 trivial
351.2.h.a.289.3 24 9.7 even 3
351.2.h.a.334.3 24 13.9 even 3