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Properties

Label 351.2.h.a.289.3
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 289.3
Character \(\chi\) \(=\) 351.289
Dual form 351.2.h.a.334.3

$q$-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)
 
\(f(q)\) \(=\) \(q-1.80162 q^{2} +1.24582 q^{4} +(-1.73153 - 2.99909i) q^{5} +(-1.62239 - 2.81005i) q^{7} +1.35875 q^{8} +(3.11954 + 5.40321i) q^{10} -0.609131 q^{11} +(0.536447 + 3.56542i) q^{13} +(2.92291 + 5.06264i) q^{14} -4.93957 q^{16} +(-1.20773 + 2.09186i) q^{17} +(-0.877705 + 1.52023i) q^{19} +(-2.15717 - 3.73632i) q^{20} +1.09742 q^{22} +(0.0812242 - 0.140684i) q^{23} +(-3.49636 + 6.05587i) q^{25} +(-0.966470 - 6.42352i) q^{26} +(-2.02120 - 3.50082i) q^{28} -2.90723 q^{29} +(-4.62084 - 8.00352i) q^{31} +6.18172 q^{32} +(2.17587 - 3.76872i) q^{34} +(-5.61840 + 9.73136i) q^{35} +(-0.826776 - 1.43202i) q^{37} +(1.58129 - 2.73887i) q^{38} +(-2.35270 - 4.07500i) q^{40} +(-4.11123 + 7.12085i) q^{41} +(6.00278 + 10.3971i) q^{43} -0.758866 q^{44} +(-0.146335 + 0.253459i) q^{46} +(-4.38799 + 7.60023i) q^{47} +(-1.76427 + 3.05580i) q^{49} +(6.29909 - 10.9103i) q^{50} +(0.668315 + 4.44187i) q^{52} -9.43717 q^{53} +(1.05473 + 1.82684i) q^{55} +(-2.20441 - 3.81815i) q^{56} +5.23772 q^{58} +8.26434 q^{59} +(-4.96418 - 8.59821i) q^{61} +(8.32497 + 14.4193i) q^{62} -1.25794 q^{64} +(9.76414 - 7.78247i) q^{65} +(-1.14763 + 1.98775i) q^{67} +(-1.50462 + 2.60607i) q^{68} +(10.1222 - 17.5322i) q^{70} +(4.87460 - 8.44305i) q^{71} -2.59634 q^{73} +(1.48953 + 2.57994i) q^{74} +(-1.09346 + 1.89393i) q^{76} +(0.988245 + 1.71169i) q^{77} +(-2.53739 + 4.39489i) q^{79} +(8.55300 + 14.8142i) q^{80} +(7.40685 - 12.8290i) q^{82} +(4.14134 - 7.17301i) q^{83} +8.36489 q^{85} +(-10.8147 - 18.7316i) q^{86} -0.827654 q^{88} +(-4.93949 - 8.55544i) q^{89} +(9.14870 - 7.29193i) q^{91} +(0.101191 - 0.175267i) q^{92} +(7.90547 - 13.6927i) q^{94} +6.07908 q^{95} +(3.73905 + 6.47622i) q^{97} +(3.17853 - 5.50538i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 24 q + 2 q^{2} + 18 q^{4} + 2 q^{5} + 3 q^{7} + 18 q^{8} - 6 q^{11} - 2 q^{14} + 6 q^{16} - 6 q^{17} - 3 q^{19} + 11 q^{20} - 18 q^{22} - 17 q^{23} - 6 q^{25} + 12 q^{26} + 24 q^{29} - 6 q^{31} + 38 q^{32}+ \cdots + 61 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.80162 −1.27393 −0.636967 − 0.770891i \(-0.719811\pi\)
−0.636967 + 0.770891i \(0.719811\pi\)
\(3\) 0 0
\(4\) 1.24582 0.622909
\(5\) −1.73153 − 2.99909i −0.774362 − 1.34123i −0.935153 − 0.354245i \(-0.884738\pi\)
0.160791 − 0.986988i \(-0.448595\pi\)
\(6\) 0 0
\(7\) −1.62239 − 2.81005i −0.613204 − 1.06210i −0.990697 − 0.136088i \(-0.956547\pi\)
0.377493 − 0.926013i \(-0.376786\pi\)
\(8\) 1.35875 0.480389
\(9\) 0 0
\(10\) 3.11954 + 5.40321i 0.986486 + 1.70864i
\(11\) −0.609131 −0.183660 −0.0918300 − 0.995775i \(-0.529272\pi\)
−0.0918300 + 0.995775i \(0.529272\pi\)
\(12\) 0 0
\(13\) 0.536447 + 3.56542i 0.148783 + 0.988870i
\(14\) 2.92291 + 5.06264i 0.781182 + 1.35305i
\(15\) 0 0
\(16\) −4.93957 −1.23489
\(17\) −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 \(0.231203\pi\)
\(20\) −2.15717 − 3.73632i −0.482357 − 0.835466i
\(21\) 0 0
\(22\) 1.09742 0.233971
\(23\) 0.0812242 − 0.140684i 0.0169364 − 0.0293347i −0.857433 − 0.514596i \(-0.827942\pi\)
0.874369 + 0.485261i \(0.161275\pi\)
\(24\) 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
\(29\) −2.90723 −0.539860 −0.269930 − 0.962880i \(-0.587000\pi\)
−0.269930 + 0.962880i \(0.587000\pi\)
\(30\) 0 0
\(31\) −4.62084 − 8.00352i −0.829927 − 1.43748i −0.898095 − 0.439802i \(-0.855049\pi\)
0.0681682 − 0.997674i \(-0.478285\pi\)
\(32\) 6.18172 1.09278
\(33\) 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
\(41\) −4.11123 + 7.12085i −0.642065 + 1.11209i 0.342905 + 0.939370i \(0.388589\pi\)
−0.984971 + 0.172720i \(0.944744\pi\)
\(42\) 0 0
\(43\) 6.00278 + 10.3971i 0.915415 + 1.58554i 0.806293 + 0.591516i \(0.201470\pi\)
0.109122 + 0.994028i \(0.465196\pi\)
\(44\) −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 0.345366 + 0.938468i \(0.387755\pi\)
−0.985420 + 0.170139i \(0.945578\pi\)
\(48\) 0 0
\(49\) −1.76427 + 3.05580i −0.252038 + 0.436543i
\(50\) 6.29909 − 10.9103i 0.890826 − 1.54296i
\(51\) 0 0
\(52\) 0.668315 + 4.44187i 0.0926786 + 0.615976i
\(53\) −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
\(56\) −2.20441 − 3.81815i −0.294577 − 0.510222i
\(57\) 0 0
\(58\) 5.23772 0.687746
\(59\) 8.26434 1.07593 0.537963 − 0.842968i \(-0.319194\pi\)
0.537963 + 0.842968i \(0.319194\pi\)
\(60\) 0 0
\(61\) −4.96418 − 8.59821i −0.635598 − 1.10089i −0.986388 − 0.164434i \(-0.947420\pi\)
0.350790 − 0.936454i \(-0.385913\pi\)
\(62\) 8.32497 + 14.4193i 1.05727 + 1.83125i
\(63\) 0 0
\(64\) −1.25794 −0.157242
\(65\) 9.76414 − 7.78247i 1.21109 − 0.965296i
\(66\) 0 0
\(67\) −1.14763 + 1.98775i −0.140205 + 0.242842i −0.927574 − 0.373640i \(-0.878109\pi\)
0.787369 + 0.616482i \(0.211443\pi\)
\(68\) −1.50462 + 2.60607i −0.182462 + 0.316033i
\(69\) 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 \(0.548552\pi\)
\(74\) 1.48953 + 2.57994i 0.173155 + 0.299912i
\(75\) 0 0
\(76\) −1.09346 + 1.89393i −0.125429 + 0.217249i
\(77\) 0.988245 + 1.71169i 0.112621 + 0.195065i
\(78\) 0 0
\(79\) −2.53739 + 4.39489i −0.285479 + 0.494464i −0.972725 − 0.231961i \(-0.925486\pi\)
0.687246 + 0.726424i \(0.258819\pi\)
\(80\) 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 −0.544093 − 0.839025i \(-0.683126\pi\)
0.998663 + 0.0516855i \(0.0164593\pi\)
\(84\) 0 0
\(85\) 8.36489 0.907300
\(86\) −10.8147 − 18.7316i −1.16618 − 2.01988i
\(87\) 0 0
\(88\) −0.827654 −0.0882282
\(89\) −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
\(94\) 7.90547 − 13.6927i 0.815387 − 1.41229i
\(95\) 6.07908 0.623700
\(96\) 0 0
\(97\) 3.73905 + 6.47622i 0.379643 + 0.657561i 0.991010 − 0.133786i \(-0.0427136\pi\)
−0.611367 + 0.791347i \(0.709380\pi\)
\(98\) 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.h.a.289.3 24
3.2 odd 2 117.2.h.a.16.10 yes 24
9.4 even 3 351.2.f.a.172.10 24
9.5 odd 6 117.2.f.a.94.3 yes 24
13.9 even 3 351.2.f.a.100.10 24
39.35 odd 6 117.2.f.a.61.3 ✓ 24
117.22 even 3 inner 351.2.h.a.334.3 24
117.113 odd 6 117.2.h.a.22.10 yes 24
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
117.2.f.a.61.3 ✓ 24 39.35 odd 6
117.2.f.a.94.3 yes 24 9.5 odd 6
117.2.h.a.16.10 yes 24 3.2 odd 2
117.2.h.a.22.10 yes 24 117.113 odd 6
351.2.f.a.100.10 24 13.9 even 3
351.2.f.a.172.10 24 9.4 even 3
351.2.h.a.289.3 24 1.1 even 1 trivial
351.2.h.a.334.3 24 117.22 even 3 inner