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Properties

Label 585.2.bm.a.511.7
Level $585$
Weight $2$
Character 585.511
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 511.7
Character \(\chi\) \(=\) 585.511
Dual form 585.2.bm.a.166.7

$q$-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 \(\left(\mathbb{Z}/585\mathbb{Z}\right)^\times\).

\(n\) \(326\) \(352\) \(496\)
\(\chi(n)\) \(e\left(\frac{2}{3}\right)\) \(1\) \(e\left(\frac{1}{6}\right)\)

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 −0.992891 − 0.119029i \(-0.962022\pi\)
−0.393363 + 0.919383i \(0.628688\pi\)
\(3\) −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 \(0.439487\pi\)
−0.188963 + 0.981984i \(0.560513\pi\)
\(8\) 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 0.00411205 − 0.999992i \(-0.498691\pi\)
0.868074 + 0.496435i \(0.165358\pi\)
\(12\) −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 −0.881102 − 0.472927i \(-0.843198\pi\)
0.0309844 − 0.999520i \(-0.490136\pi\)
\(18\) −4.32727 + 5.23408i −1.01995 + 1.23368i
\(19\) 2.44419 − 1.41115i 0.560735 − 0.323740i −0.192706 − 0.981257i \(-0.561726\pi\)
0.753440 + 0.657516i \(0.228393\pi\)
\(20\) 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 0.967638 − 0.252344i \(-0.0812015\pi\)
−0.702355 + 0.711827i \(0.747868\pi\)
\(30\) −1.32759 + 3.68932i −0.242384 + 0.673575i
\(31\) 3.32101 − 1.91738i 0.596470 − 0.344372i −0.171181 − 0.985240i \(-0.554758\pi\)
0.767652 + 0.640867i \(0.221425\pi\)
\(32\) −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 0.999851 − 0.0172672i \(-0.00549658\pi\)
0.484972 + 0.874530i \(0.338830\pi\)
\(38\) −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 \(0.0874959\pi\)
−0.962459 + 0.271428i \(0.912504\pi\)
\(42\) 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 0.793457 − 0.608626i \(-0.208279\pi\)
−0.130357 + 0.991467i \(0.541612\pi\)
\(48\) −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 0.539623 − 0.841907i \(-0.318567\pi\)
−0.459301 + 0.888281i \(0.651900\pi\)
\(60\) −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 \(0.0454616\pi\)
−0.989818 + 0.142337i \(0.954538\pi\)
\(68\) −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 −0.680714 − 0.732549i \(-0.738330\pi\)
0.294049 + 0.955790i \(0.404997\pi\)
\(72\) 2.66425 + 7.15716i 0.313985 + 0.843480i
\(73\) − 5.50446i − 0.644248i −0.946697 − 0.322124i \(-0.895603\pi\)
0.946697 − 0.322124i \(-0.104397\pi\)
\(74\) −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 −0.971537 − 0.236888i \(-0.923872\pi\)
0.690920 + 0.722932i \(0.257206\pi\)
\(80\) −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 0.527264 − 0.849701i \(-0.323218\pi\)
−0.472231 + 0.881475i \(0.656551\pi\)
\(84\) −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 0.693398 − 0.720554i \(-0.256113\pi\)
−0.277319 + 0.960778i \(0.589446\pi\)
\(90\) 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 \(-0.827146\pi\)
0.856143 − 0.516738i \(-0.172854\pi\)
\(98\) 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.511.7 yes 112
9.4 even 3 585.2.ba.a.121.7 ✓ 112
13.10 even 6 585.2.ba.a.556.50 yes 112
117.49 even 6 inner 585.2.bm.a.166.7 yes 112
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
585.2.ba.a.121.7 ✓ 112 9.4 even 3
585.2.ba.a.556.50 yes 112 13.10 even 6
585.2.bm.a.166.7 yes 112 117.49 even 6 inner
585.2.bm.a.511.7 yes 112 1.1 even 1 trivial