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Properties

Label 585.2.bt.b.571.31
Level $585$
Weight $2$
Character 585.571
Analytic conductor $4.671$
Analytic rank $0$
Dimension $108$
Inner twists $4$

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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(376,585)] 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("585.376"); 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([4, 0, 3])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 585 = 3^{2} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 585.bt (of order \(6\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [108] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(4.67124851824\)
Analytic rank: \(0\)
Dimension: \(108\)
Relative dimension: \(54\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 571.31
Character \(\chi\) \(=\) 585.571
Dual form 585.2.bt.b.376.31

$q$-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+(0.355151 - 0.205047i) q^{2} +(0.841866 + 1.51369i) q^{3} +(-0.915912 + 1.58641i) q^{4} +(0.866025 + 0.500000i) q^{5} +(0.609367 + 0.364968i) q^{6} +(0.661508 - 0.381922i) q^{7} +1.57141i q^{8} +(-1.58252 + 2.54865i) q^{9} +0.410093 q^{10} +(-0.384678 + 0.222094i) q^{11} +(-3.17240 - 0.0508663i) q^{12} +(0.944367 + 3.47968i) q^{13} +(0.156624 - 0.271280i) q^{14} +(-0.0277681 + 1.73183i) q^{15} +(-1.50961 - 2.61472i) q^{16} -1.99449 q^{17} +(-0.0394425 + 1.22965i) q^{18} -3.58544i q^{19} +(-1.58641 + 0.915912i) q^{20} +(1.13501 + 0.679791i) q^{21} +(-0.0910792 + 0.157754i) q^{22} +(0.803282 - 1.39133i) q^{23} +(-2.37862 + 1.32291i) q^{24} +(0.500000 + 0.866025i) q^{25} +(1.04889 + 1.04217i) q^{26} +(-5.19014 - 0.249828i) q^{27} +1.39923i q^{28} +(0.298185 + 0.516472i) q^{29} +(0.345244 + 0.620755i) q^{30} +(6.94569 + 4.01010i) q^{31} +(-3.79404 - 2.19049i) q^{32} +(-0.660028 - 0.395310i) q^{33} +(-0.708345 + 0.408963i) q^{34} +0.763843 q^{35} +(-2.59374 - 4.84486i) q^{36} +0.218284i q^{37} +(-0.735183 - 1.27338i) q^{38} +(-4.47213 + 4.35891i) q^{39} +(-0.785703 + 1.36088i) q^{40} +(0.971758 + 0.561045i) q^{41} +(0.542490 + 0.00869829i) q^{42} +(-0.412762 - 0.714925i) q^{43} -0.813673i q^{44} +(-2.64483 + 1.41594i) q^{45} -0.658842i q^{46} +(-3.55589 + 2.05299i) q^{47} +(2.68699 - 4.48633i) q^{48} +(-3.20827 + 5.55689i) q^{49} +(0.355151 + 0.205047i) q^{50} +(-1.67909 - 3.01904i) q^{51} +(-6.38514 - 1.68893i) q^{52} +5.08548 q^{53} +(-1.89451 + 0.975495i) q^{54} -0.444187 q^{55} +(0.600154 + 1.03950i) q^{56} +(5.42725 - 3.01846i) q^{57} +(0.211802 + 0.122284i) q^{58} +(-3.52561 - 2.03551i) q^{59} +(-2.72195 - 1.63025i) q^{60} +(1.88333 + 3.26202i) q^{61} +3.28903 q^{62} +(-0.0734659 + 2.29035i) q^{63} +4.24184 q^{64} +(-0.921994 + 3.48567i) q^{65} +(-0.315467 - 0.00505820i) q^{66} +(12.0106 + 6.93435i) q^{67} +(1.82677 - 3.16406i) q^{68} +(2.78229 + 0.0446113i) q^{69} +(0.271280 - 0.156624i) q^{70} -12.1545i q^{71} +(-4.00497 - 2.48679i) q^{72} -3.80995i q^{73} +(0.0447585 + 0.0775240i) q^{74} +(-0.889962 + 1.48592i) q^{75} +(5.68797 + 3.28395i) q^{76} +(-0.169645 + 0.293833i) q^{77} +(-0.694504 + 2.46507i) q^{78} +(4.87017 + 8.43538i) q^{79} -3.01922i q^{80} +(-3.99124 - 8.06660i) q^{81} +0.460162 q^{82} +(13.4267 - 7.75194i) q^{83} +(-2.11800 + 1.17796i) q^{84} +(-1.72728 - 0.997243i) q^{85} +(-0.293186 - 0.169271i) q^{86} +(-0.530747 + 0.886160i) q^{87} +(-0.348999 - 0.604485i) q^{88} -12.2951i q^{89} +(-0.648982 + 1.04519i) q^{90} +(1.95367 + 1.94116i) q^{91} +(1.47147 + 2.54866i) q^{92} +(-0.222706 + 13.8896i) q^{93} +(-0.841918 + 1.45825i) q^{94} +(1.79272 - 3.10508i) q^{95} +(0.121652 - 7.58710i) q^{96} +(12.3546 - 7.13294i) q^{97} +2.63138i q^{98} +(0.0427216 - 1.33188i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 108 q - 6 q^{3} + 52 q^{4} - 14 q^{9} - 8 q^{10} - 36 q^{12} - 4 q^{13} - 8 q^{14} - 64 q^{16} + 36 q^{17} + 24 q^{22} - 22 q^{23} + 54 q^{25} + 40 q^{26} + 48 q^{27} - 16 q^{29} + 20 q^{30} - 40 q^{35}+ \cdots - 56 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.355151 − 0.205047i 0.251130 − 0.144990i −0.369152 − 0.929369i \(-0.620352\pi\)
0.620282 + 0.784379i \(0.287018\pi\)
\(3\) 0.841866 + 1.51369i 0.486052 + 0.873930i
\(4\) −0.915912 + 1.58641i −0.457956 + 0.793203i
\(5\) 0.866025 + 0.500000i 0.387298 + 0.223607i
\(6\) 0.609367 + 0.364968i 0.248773 + 0.148997i
\(7\) 0.661508 − 0.381922i 0.250026 − 0.144353i −0.369750 − 0.929131i \(-0.620557\pi\)
0.619776 + 0.784779i \(0.287223\pi\)
\(8\) 1.57141i 0.555576i
\(9\) −1.58252 + 2.54865i −0.527508 + 0.849550i
\(10\) 0.410093 0.129683
\(11\) −0.384678 + 0.222094i −0.115985 + 0.0669638i −0.556870 − 0.830600i \(-0.687998\pi\)
0.440885 + 0.897563i \(0.354665\pi\)
\(12\) −3.17240 − 0.0508663i −0.915794 − 0.0146838i
\(13\) 0.944367 + 3.47968i 0.261920 + 0.965089i
\(14\) 0.156624 − 0.271280i 0.0418594 − 0.0725026i
\(15\) −0.0277681 + 1.73183i −0.00716970 + 0.447156i
\(16\) −1.50961 − 2.61472i −0.377403 − 0.653681i
\(17\) −1.99449 −0.483734 −0.241867 − 0.970309i \(-0.577760\pi\)
−0.241867 + 0.970309i \(0.577760\pi\)
\(18\) −0.0394425 + 1.22965i −0.00929669 + 0.289831i
\(19\) − 3.58544i − 0.822557i −0.911510 − 0.411279i \(-0.865082\pi\)
0.911510 − 0.411279i \(-0.134918\pi\)
\(20\) −1.58641 + 0.915912i −0.354731 + 0.204804i
\(21\) 1.13501 + 0.679791i 0.247680 + 0.148343i
\(22\) −0.0910792 + 0.157754i −0.0194181 + 0.0336332i
\(23\) 0.803282 − 1.39133i 0.167496 − 0.290112i −0.770043 − 0.637992i \(-0.779765\pi\)
0.937539 + 0.347881i \(0.113099\pi\)
\(24\) −2.37862 + 1.32291i −0.485534 + 0.270039i
\(25\) 0.500000 + 0.866025i 0.100000 + 0.173205i
\(26\) 1.04889 + 1.04217i 0.205704 + 0.204387i
\(27\) −5.19014 − 0.249828i −0.998844 − 0.0480793i
\(28\) 1.39923i 0.264429i
\(29\) 0.298185 + 0.516472i 0.0553716 + 0.0959064i 0.892383 − 0.451280i \(-0.149032\pi\)
−0.837011 + 0.547186i \(0.815699\pi\)
\(30\) 0.345244 + 0.620755i 0.0630326 + 0.113334i
\(31\) 6.94569 + 4.01010i 1.24748 + 0.720235i 0.970607 − 0.240671i \(-0.0773674\pi\)
0.276876 + 0.960906i \(0.410701\pi\)
\(32\) −3.79404 − 2.19049i −0.670697 − 0.387227i
\(33\) −0.660028 − 0.395310i −0.114896 − 0.0688146i
\(34\) −0.708345 + 0.408963i −0.121480 + 0.0701366i
\(35\) 0.763843 0.129113
\(36\) −2.59374 − 4.84486i −0.432291 − 0.807477i
\(37\) 0.218284i 0.0358857i 0.999839 + 0.0179429i \(0.00571170\pi\)
−0.999839 + 0.0179429i \(0.994288\pi\)
\(38\) −0.735183 − 1.27338i −0.119263 − 0.206569i
\(39\) −4.47213 + 4.35891i −0.716114 + 0.697983i
\(40\) −0.785703 + 1.36088i −0.124231 + 0.215174i
\(41\) 0.971758 + 0.561045i 0.151763 + 0.0876205i 0.573959 − 0.818884i \(-0.305407\pi\)
−0.422195 + 0.906505i \(0.638740\pi\)
\(42\) 0.542490 + 0.00869829i 0.0837081 + 0.00134218i
\(43\) −0.412762 − 0.714925i −0.0629456 − 0.109025i 0.832835 − 0.553521i \(-0.186716\pi\)
−0.895781 + 0.444496i \(0.853383\pi\)
\(44\) − 0.813673i − 0.122666i
\(45\) −2.64483 + 1.41594i −0.394268 + 0.211075i
\(46\) − 0.658842i − 0.0971409i
\(47\) −3.55589 + 2.05299i −0.518679 + 0.299460i −0.736394 − 0.676553i \(-0.763473\pi\)
0.217715 + 0.976012i \(0.430140\pi\)
\(48\) 2.68699 − 4.48633i 0.387834 − 0.647547i
\(49\) −3.20827 + 5.55689i −0.458325 + 0.793841i
\(50\) 0.355151 + 0.205047i 0.0502260 + 0.0289980i
\(51\) −1.67909 − 3.01904i −0.235120 − 0.422750i
\(52\) −6.38514 − 1.68893i −0.885460 − 0.234212i
\(53\) 5.08548 0.698544 0.349272 − 0.937021i \(-0.386429\pi\)
0.349272 + 0.937021i \(0.386429\pi\)
\(54\) −1.89451 + 0.975495i −0.257811 + 0.132748i
\(55\) −0.444187 −0.0598942
\(56\) 0.600154 + 1.03950i 0.0801989 + 0.138909i
\(57\) 5.42725 − 3.01846i 0.718857 − 0.399805i
\(58\) 0.211802 + 0.122284i 0.0278109 + 0.0160566i
\(59\) −3.52561 − 2.03551i −0.458996 − 0.265001i 0.252626 − 0.967564i \(-0.418706\pi\)
−0.711622 + 0.702563i \(0.752039\pi\)
\(60\) −2.72195 − 1.63025i −0.351402 − 0.210465i
\(61\) 1.88333 + 3.26202i 0.241136 + 0.417659i 0.961038 − 0.276416i \(-0.0891468\pi\)
−0.719902 + 0.694075i \(0.755814\pi\)
\(62\) 3.28903 0.417707
\(63\) −0.0734659 + 2.29035i −0.00925583 + 0.288557i
\(64\) 4.24184 0.530230
\(65\) −0.921994 + 3.48567i −0.114359 + 0.432345i
\(66\) −0.315467 − 0.00505820i −0.0388313 − 0.000622621i
\(67\) 12.0106 + 6.93435i 1.46733 + 0.847165i 0.999331 − 0.0365620i \(-0.0116407\pi\)
0.468002 + 0.883727i \(0.344974\pi\)
\(68\) 1.82677 − 3.16406i 0.221529 − 0.383699i
\(69\) 2.78229 + 0.0446113i 0.334949 + 0.00537057i
\(70\) 0.271280 − 0.156624i 0.0324242 − 0.0187201i
\(71\) − 12.1545i − 1.44248i −0.692687 − 0.721238i \(-0.743573\pi\)
0.692687 − 0.721238i \(-0.256427\pi\)
\(72\) −4.00497 − 2.48679i −0.471990 − 0.293070i
\(73\) − 3.80995i − 0.445921i −0.974827 − 0.222961i \(-0.928428\pi\)
0.974827 − 0.222961i \(-0.0715721\pi\)
\(74\) 0.0447585 + 0.0775240i 0.00520307 + 0.00901198i
\(75\) −0.889962 + 1.48592i −0.102764 + 0.171580i
\(76\) 5.68797 + 3.28395i 0.652455 + 0.376695i
\(77\) −0.169645 + 0.293833i −0.0193328 + 0.0334854i
\(78\) −0.694504 + 2.46507i −0.0786371 + 0.279114i
\(79\) 4.87017 + 8.43538i 0.547937 + 0.949055i 0.998416 + 0.0562674i \(0.0179199\pi\)
−0.450479 + 0.892787i \(0.648747\pi\)
\(80\) − 3.01922i − 0.337559i
\(81\) −3.99124 − 8.06660i −0.443472 − 0.896288i
\(82\) 0.460162 0.0508164
\(83\) 13.4267 − 7.75194i 1.47378 − 0.850885i 0.474213 − 0.880410i \(-0.342733\pi\)
0.999564 + 0.0295248i \(0.00939940\pi\)
\(84\) −2.11800 + 1.17796i −0.231092 + 0.128526i
\(85\) −1.72728 − 0.997243i −0.187349 − 0.108166i
\(86\) −0.293186 − 0.169271i −0.0316151 − 0.0182530i
\(87\) −0.530747 + 0.886160i −0.0569020 + 0.0950064i
\(88\) −0.348999 − 0.604485i −0.0372035 − 0.0644383i
\(89\) − 12.2951i − 1.30328i −0.758527 − 0.651641i \(-0.774081\pi\)
0.758527 − 0.651641i \(-0.225919\pi\)
\(90\) −0.648982 + 1.04519i −0.0684087 + 0.110172i
\(91\) 1.95367 + 1.94116i 0.204800 + 0.203489i
\(92\) 1.47147 + 2.54866i 0.153412 + 0.265717i
\(93\) −0.222706 + 13.8896i −0.0230935 + 1.44028i
\(94\) −0.841918 + 1.45825i −0.0868372 + 0.150407i
\(95\) 1.79272 − 3.10508i 0.183929 − 0.318575i
\(96\) 0.121652 − 7.58710i 0.0124160 − 0.774355i
\(97\) 12.3546 − 7.13294i 1.25442 − 0.724241i 0.282437 − 0.959286i \(-0.408857\pi\)
0.971984 + 0.235045i \(0.0755238\pi\)
\(98\) 2.63138i 0.265810i
\(99\) 0.0427216 − 1.33188i 0.00429369 − 0.133859i
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.bt.b.571.31 yes 108
9.7 even 3 inner 585.2.bt.b.376.24 ✓ 108
13.12 even 2 inner 585.2.bt.b.571.24 yes 108
117.25 even 6 inner 585.2.bt.b.376.31 yes 108
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
585.2.bt.b.376.24 ✓ 108 9.7 even 3 inner
585.2.bt.b.376.31 yes 108 117.25 even 6 inner
585.2.bt.b.571.24 yes 108 13.12 even 2 inner
585.2.bt.b.571.31 yes 108 1.1 even 1 trivial