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Properties

Label 585.2.bm.a.166.7
Level $585$
Weight $2$
Character 585.166
Analytic conductor $4.671$
Analytic rank $0$
Dimension $112$
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] = [585,2,Mod(166,585)] 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("585.166"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(585, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([2, 0, 5])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 585 = 3^{2} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 585.bm (of order \(6\), degree \(2\), 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: \(4.67124851824\)
Analytic rank: \(0\)
Dimension: \(112\)
Relative dimension: \(56\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 166.7
Character \(\chi\) \(=\) 585.166
Dual form 585.2.bm.a.511.7

$q$-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+(-1.96046 - 1.13187i) q^{2} +(-1.70463 - 0.306984i) q^{3} +(1.56227 + 2.70593i) q^{4} +(-0.866025 - 0.500000i) q^{5} +(2.99439 + 2.53125i) q^{6} -5.19617i q^{7} -2.54566i q^{8} +(2.81152 + 1.04659i) q^{9} +(1.13187 + 1.96046i) q^{10} +(2.89271 + 1.67011i) q^{11} +(-1.83241 - 5.09219i) q^{12} +(2.19813 + 2.85801i) q^{13} +(-5.88140 + 10.1869i) q^{14} +(1.32276 + 1.11817i) q^{15} +(0.243179 - 0.421198i) q^{16} +(-3.50512 + 6.07105i) q^{17} +(-4.32727 - 5.23408i) q^{18} +(2.44419 + 1.41115i) q^{19} -3.12453i q^{20} +(-1.59514 + 8.85755i) q^{21} +(-3.78070 - 6.54836i) q^{22} +4.78317 q^{23} +(-0.781476 + 4.33940i) q^{24} +(0.500000 + 0.866025i) q^{25} +(-1.07443 - 8.09102i) q^{26} +(-4.47131 - 2.64714i) q^{27} +(14.0605 - 8.11781i) q^{28} +(1.42859 - 2.47439i) q^{29} +(-1.32759 - 3.68932i) q^{30} +(3.32101 + 1.91738i) q^{31} +(-5.36269 + 3.09615i) q^{32} +(-4.41831 - 3.73493i) q^{33} +(13.7433 - 7.93470i) q^{34} +(-2.59809 + 4.50002i) q^{35} +(1.56036 + 9.24282i) q^{36} +(9.03182 - 5.21453i) q^{37} +(-3.19448 - 5.53301i) q^{38} +(-2.86962 - 5.54664i) q^{39} +(-1.27283 + 2.20460i) q^{40} -3.47597i q^{41} +(13.1528 - 15.5594i) q^{42} +5.17333 q^{43} +10.4366i q^{44} +(-1.91155 - 2.31213i) q^{45} +(-9.37721 - 5.41394i) q^{46} +(4.54599 - 2.62463i) q^{47} +(-0.543831 + 0.643334i) q^{48} -20.0002 q^{49} -2.26374i q^{50} +(7.83865 - 9.27288i) q^{51} +(-4.29951 + 10.4129i) q^{52} -2.11156 q^{53} +(5.76961 + 10.2506i) q^{54} +(-1.67011 - 2.89271i) q^{55} -13.2277 q^{56} +(-3.73323 - 3.15582i) q^{57} +(-5.60138 + 3.23396i) q^{58} +(0.616967 - 0.356206i) q^{59} +(-0.959183 + 5.32617i) q^{60} -6.03375 q^{61} +(-4.34046 - 7.51790i) q^{62} +(5.43826 - 14.6091i) q^{63} +13.0451 q^{64} +(-0.474625 - 3.57418i) q^{65} +(4.43444 + 12.3231i) q^{66} -2.33015i q^{67} -21.9038 q^{68} +(-8.15353 - 1.46836i) q^{69} +(10.1869 - 5.88140i) q^{70} +(-3.25810 - 1.88106i) q^{71} +(2.66425 - 7.15716i) q^{72} +5.50446i q^{73} -23.6087 q^{74} +(-0.586458 - 1.62974i) q^{75} +8.81838i q^{76} +(8.67818 - 15.0310i) q^{77} +(-0.652310 + 14.1220i) q^{78} +(-2.49418 - 4.32005i) q^{79} +(-0.421198 + 0.243179i) q^{80} +(6.80930 + 5.88501i) q^{81} +(-3.93436 + 6.81450i) q^{82} +(0.501375 - 0.289469i) q^{83} +(-26.4599 + 9.52151i) q^{84} +(6.07105 - 3.50512i) q^{85} +(-10.1421 - 5.85554i) q^{86} +(-3.19481 + 3.77936i) q^{87} +(4.25152 - 7.36385i) q^{88} +(3.92528 - 2.26626i) q^{89} +(1.13049 + 6.69648i) q^{90} +(14.8507 - 11.4218i) q^{91} +(7.47259 + 12.9429i) q^{92} +(-5.07248 - 4.28792i) q^{93} -11.8830 q^{94} +(-1.41115 - 2.44419i) q^{95} +(10.0919 - 3.63153i) q^{96} +10.1786i q^{97} +(39.2096 + 22.6377i) q^{98} +(6.38501 + 7.72303i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 112 q + 56 q^{4} - 2 q^{9} + 12 q^{12} + 2 q^{13} - 56 q^{16} - 16 q^{17} + 24 q^{18} + 6 q^{19} - 6 q^{21} + 48 q^{23} - 60 q^{24} + 56 q^{25} - 12 q^{26} - 24 q^{27} + 10 q^{29} + 8 q^{30} - 24 q^{31}+ \cdots + 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/585\mathbb{Z}\right)^\times\).

\(n\) \(326\) \(352\) \(496\)
\(\chi(n)\) \(e\left(\frac{1}{3}\right)\) \(1\) \(e\left(\frac{5}{6}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.96046 − 1.13187i −1.38625 − 0.800354i −0.393363 − 0.919383i \(-0.628688\pi\)
−0.992891 + 0.119029i \(0.962022\pi\)
\(3\) −1.70463 − 0.306984i −0.984168 − 0.177237i
\(4\) 1.56227 + 2.70593i 0.781133 + 1.35296i
\(5\) −0.866025 − 0.500000i −0.387298 − 0.223607i
\(6\) 2.99439 + 2.53125i 1.22245 + 1.03338i
\(7\) − 5.19617i − 1.96397i −0.188963 − 0.981984i \(-0.560513\pi\)
0.188963 − 0.981984i \(-0.439487\pi\)
\(8\) − 2.54566i − 0.900025i
\(9\) 2.81152 + 1.04659i 0.937174 + 0.348863i
\(10\) 1.13187 + 1.96046i 0.357929 + 0.619952i
\(11\) 2.89271 + 1.67011i 0.872186 + 0.503557i 0.868074 − 0.496435i \(-0.165358\pi\)
0.00411205 + 0.999992i \(0.498691\pi\)
\(12\) −1.83241 − 5.09219i −0.528971 − 1.46999i
\(13\) 2.19813 + 2.85801i 0.609650 + 0.792670i
\(14\) −5.88140 + 10.1869i −1.57187 + 2.72256i
\(15\) 1.32276 + 1.11817i 0.341535 + 0.288710i
\(16\) 0.243179 − 0.421198i 0.0607947 − 0.105299i
\(17\) −3.50512 + 6.07105i −0.850117 + 1.47245i 0.0309844 + 0.999520i \(0.490136\pi\)
−0.881102 + 0.472927i \(0.843198\pi\)
\(18\) −4.32727 − 5.23408i −1.01995 − 1.23368i
\(19\) 2.44419 + 1.41115i 0.560735 + 0.323740i 0.753440 − 0.657516i \(-0.228393\pi\)
−0.192706 + 0.981257i \(0.561726\pi\)
\(20\) − 3.12453i − 0.698667i
\(21\) −1.59514 + 8.85755i −0.348089 + 1.93288i
\(22\) −3.78070 − 6.54836i −0.806048 − 1.39612i
\(23\) 4.78317 0.997360 0.498680 − 0.866786i \(-0.333818\pi\)
0.498680 + 0.866786i \(0.333818\pi\)
\(24\) −0.781476 + 4.33940i −0.159518 + 0.885776i
\(25\) 0.500000 + 0.866025i 0.100000 + 0.173205i
\(26\) −1.07443 − 8.09102i −0.210713 − 1.58678i
\(27\) −4.47131 − 2.64714i −0.860505 − 0.509442i
\(28\) 14.0605 − 8.11781i 2.65718 − 1.53412i
\(29\) 1.42859 − 2.47439i 0.265282 − 0.459483i −0.702355 − 0.711827i \(-0.747868\pi\)
0.967638 + 0.252344i \(0.0812015\pi\)
\(30\) −1.32759 − 3.68932i −0.242384 − 0.673575i
\(31\) 3.32101 + 1.91738i 0.596470 + 0.344372i 0.767652 − 0.640867i \(-0.221425\pi\)
−0.171181 + 0.985240i \(0.554758\pi\)
\(32\) −5.36269 + 3.09615i −0.947998 + 0.547327i
\(33\) −4.41831 − 3.73493i −0.769129 − 0.650169i
\(34\) 13.7433 − 7.93470i 2.35696 − 1.36079i
\(35\) −2.59809 + 4.50002i −0.439157 + 0.760642i
\(36\) 1.56036 + 9.24282i 0.260059 + 1.54047i
\(37\) 9.03182 − 5.21453i 1.48482 − 0.857263i 0.484972 − 0.874530i \(-0.338830\pi\)
0.999851 + 0.0172672i \(0.00549658\pi\)
\(38\) −3.19448 − 5.53301i −0.518214 − 0.897573i
\(39\) −2.86962 − 5.54664i −0.459507 − 0.888174i
\(40\) −1.27283 + 2.20460i −0.201252 + 0.348578i
\(41\) − 3.47597i − 0.542856i −0.962459 − 0.271428i \(-0.912504\pi\)
0.962459 − 0.271428i \(-0.0874959\pi\)
\(42\) 13.1528 − 15.5594i 2.02952 − 2.40086i
\(43\) 5.17333 0.788925 0.394463 − 0.918912i \(-0.370931\pi\)
0.394463 + 0.918912i \(0.370931\pi\)
\(44\) 10.4366i 1.57338i
\(45\) −1.91155 − 2.31213i −0.284958 − 0.344672i
\(46\) −9.37721 − 5.41394i −1.38259 − 0.798241i
\(47\) 4.54599 − 2.62463i 0.663101 − 0.382841i −0.130357 − 0.991467i \(-0.541612\pi\)
0.793457 + 0.608626i \(0.208279\pi\)
\(48\) −0.543831 + 0.643334i −0.0784952 + 0.0928573i
\(49\) −20.0002 −2.85717
\(50\) − 2.26374i − 0.320142i
\(51\) 7.83865 − 9.27288i 1.09763 − 1.29846i
\(52\) −4.29951 + 10.4129i −0.596235 + 1.44402i
\(53\) −2.11156 −0.290045 −0.145022 − 0.989428i \(-0.546325\pi\)
−0.145022 + 0.989428i \(0.546325\pi\)
\(54\) 5.76961 + 10.2506i 0.785144 + 1.39492i
\(55\) −1.67011 − 2.89271i −0.225197 − 0.390054i
\(56\) −13.2277 −1.76762
\(57\) −3.73323 − 3.15582i −0.494478 − 0.417998i
\(58\) −5.60138 + 3.23396i −0.735498 + 0.424640i
\(59\) 0.616967 − 0.356206i 0.0803222 − 0.0463740i −0.459301 − 0.888281i \(-0.651900\pi\)
0.539623 + 0.841907i \(0.318567\pi\)
\(60\) −0.959183 + 5.32617i −0.123830 + 0.687606i
\(61\) −6.03375 −0.772543 −0.386271 − 0.922385i \(-0.626237\pi\)
−0.386271 + 0.922385i \(0.626237\pi\)
\(62\) −4.34046 − 7.51790i −0.551239 − 0.954775i
\(63\) 5.43826 − 14.6091i 0.685156 − 1.84058i
\(64\) 13.0451 1.63063
\(65\) −0.474625 − 3.57418i −0.0588700 − 0.443322i
\(66\) 4.43444 + 12.3231i 0.545842 + 1.51687i
\(67\) − 2.33015i − 0.284673i −0.989818 − 0.142337i \(-0.954538\pi\)
0.989818 − 0.142337i \(-0.0454616\pi\)
\(68\) −21.9038 −2.65622
\(69\) −8.15353 − 1.46836i −0.981570 − 0.176770i
\(70\) 10.1869 − 5.88140i 1.21757 − 0.702962i
\(71\) −3.25810 − 1.88106i −0.386665 − 0.223241i 0.294049 − 0.955790i \(-0.404997\pi\)
−0.680714 + 0.732549i \(0.738330\pi\)
\(72\) 2.66425 − 7.15716i 0.313985 − 0.843480i
\(73\) 5.50446i 0.644248i 0.946697 + 0.322124i \(0.104397\pi\)
−0.946697 + 0.322124i \(0.895603\pi\)
\(74\) −23.6087 −2.74445
\(75\) −0.586458 − 1.62974i −0.0677184 − 0.188187i
\(76\) 8.81838i 1.01154i
\(77\) 8.67818 − 15.0310i 0.988970 − 1.71295i
\(78\) −0.652310 + 14.1220i −0.0738596 + 1.59900i
\(79\) −2.49418 − 4.32005i −0.280617 − 0.486043i 0.690920 − 0.722932i \(-0.257206\pi\)
−0.971537 + 0.236888i \(0.923872\pi\)
\(80\) −0.421198 + 0.243179i −0.0470914 + 0.0271882i
\(81\) 6.80930 + 5.88501i 0.756589 + 0.653890i
\(82\) −3.93436 + 6.81450i −0.434477 + 0.752536i
\(83\) 0.501375 − 0.289469i 0.0550331 − 0.0317734i −0.472231 − 0.881475i \(-0.656551\pi\)
0.527264 + 0.849701i \(0.323218\pi\)
\(84\) −26.4599 + 9.52151i −2.88701 + 1.03888i
\(85\) 6.07105 − 3.50512i 0.658498 − 0.380184i
\(86\) −10.1421 − 5.85554i −1.09365 − 0.631420i
\(87\) −3.19481 + 3.77936i −0.342520 + 0.405190i
\(88\) 4.25152 − 7.36385i 0.453214 − 0.784989i
\(89\) 3.92528 − 2.26626i 0.416079 − 0.240223i −0.277319 − 0.960778i \(-0.589446\pi\)
0.693398 + 0.720554i \(0.256113\pi\)
\(90\) 1.13049 + 6.69648i 0.119164 + 0.705871i
\(91\) 14.8507 − 11.4218i 1.55678 − 1.19733i
\(92\) 7.47259 + 12.9429i 0.779071 + 1.34939i
\(93\) −5.07248 − 4.28792i −0.525991 − 0.444637i
\(94\) −11.8830 −1.22563
\(95\) −1.41115 − 2.44419i −0.144781 − 0.250768i
\(96\) 10.0919 − 3.63153i 1.03000 − 0.370641i
\(97\) 10.1786i 1.03348i 0.856143 + 0.516738i \(0.172854\pi\)
−0.856143 + 0.516738i \(0.827146\pi\)
\(98\) 39.2096 + 22.6377i 3.96077 + 2.28675i
\(99\) 6.38501 + 7.72303i 0.641718 + 0.776194i
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 585.2.bm.a.166.7 yes 112
9.7 even 3 585.2.ba.a.556.50 yes 112
13.4 even 6 585.2.ba.a.121.7 ✓ 112
117.43 even 6 inner 585.2.bm.a.511.7 yes 112
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
585.2.ba.a.121.7 ✓ 112 13.4 even 6
585.2.ba.a.556.50 yes 112 9.7 even 3
585.2.bm.a.166.7 yes 112 1.1 even 1 trivial
585.2.bm.a.511.7 yes 112 117.43 even 6 inner