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Properties

Label 555.2.bp.a.4.10
Level $555$
Weight $2$
Character 555.4
Analytic conductor $4.432$
Analytic rank $0$
Dimension $240$
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] = [555,2,Mod(4,555)] 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("555.4"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(555, base_ring=CyclotomicField(18)) chi = DirichletCharacter(H, H._module([0, 9, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 555 = 3 \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 555.bp (of order \(18\), degree \(6\), 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: \(4.43169731218\)
Analytic rank: \(0\)
Dimension: \(240\)
Relative dimension: \(40\) over \(\Q(\zeta_{18})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label 4.10
Character \(\chi\) \(=\) 555.4
Dual form 555.2.bp.a.139.10

$q$-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+(-0.315810 + 1.79105i) q^{2} +(0.984808 - 0.173648i) q^{3} +(-1.22873 - 0.447221i) q^{4} +(1.61367 - 1.54792i) q^{5} +1.81868i q^{6} +(-1.35469 - 1.61445i) q^{7} +(-0.629637 + 1.09056i) q^{8} +(0.939693 - 0.342020i) q^{9} +(2.26279 + 3.37901i) q^{10} +(2.46547 - 4.27032i) q^{11} +(-1.28772 - 0.227060i) q^{12} +(5.08309 + 1.85009i) q^{13} +(3.31939 - 1.91645i) q^{14} +(1.32036 - 1.80462i) q^{15} +(-3.75775 - 3.15312i) q^{16} +(-4.78314 + 1.74092i) q^{17} +(0.315810 + 1.79105i) q^{18} +(2.74415 - 0.483868i) q^{19} +(-2.67503 + 1.18031i) q^{20} +(-1.61445 - 1.35469i) q^{21} +(6.86973 + 5.76439i) q^{22} +(3.60921 + 6.25134i) q^{23} +(-0.430697 + 1.18333i) q^{24} +(0.207883 - 4.99568i) q^{25} +(-4.91890 + 8.51978i) q^{26} +(0.866025 - 0.500000i) q^{27} +(0.942528 + 2.58957i) q^{28} +(4.39764 + 2.53898i) q^{29} +(2.81517 + 2.93475i) q^{30} -3.16923i q^{31} +(4.90481 - 4.11562i) q^{32} +(1.68648 - 4.63357i) q^{33} +(-1.60751 - 9.11662i) q^{34} +(-4.68507 - 0.508252i) q^{35} -1.30759 q^{36} +(4.37931 - 4.22157i) q^{37} +5.06772i q^{38} +(5.32713 + 0.939318i) q^{39} +(0.672077 + 2.73444i) q^{40} +(-1.10910 - 0.403678i) q^{41} +(2.93617 - 2.46374i) q^{42} -1.68817 q^{43} +(-4.93918 + 4.14446i) q^{44} +(0.986937 - 2.00648i) q^{45} +(-12.3363 + 4.49004i) q^{46} +(-8.50226 + 4.90878i) q^{47} +(-4.24819 - 2.45269i) q^{48} +(0.444255 - 2.51950i) q^{49} +(8.88184 + 1.95001i) q^{50} +(-4.40816 + 2.54505i) q^{51} +(-5.41835 - 4.54654i) q^{52} +(-4.41526 + 5.26190i) q^{53} +(0.622024 + 1.70900i) q^{54} +(-2.63165 - 10.7073i) q^{55} +(2.61362 - 0.460853i) q^{56} +(2.61844 - 0.953034i) q^{57} +(-5.93625 + 7.07455i) q^{58} +(-6.03368 + 7.19066i) q^{59} +(-2.42944 + 1.62689i) q^{60} +(-4.86192 + 13.3580i) q^{61} +(5.67625 + 1.00088i) q^{62} +(-1.82517 - 1.05376i) q^{63} +(0.916901 + 1.58812i) q^{64} +(11.0663 - 4.88278i) q^{65} +(7.76634 + 4.48390i) q^{66} +(-6.36879 - 7.59003i) q^{67} +6.65576 q^{68} +(4.63992 + 5.52964i) q^{69} +(2.38990 - 8.23067i) q^{70} +(-1.00436 - 5.69603i) q^{71} +(-0.218671 + 1.24014i) q^{72} +11.5949i q^{73} +(6.17800 + 9.17677i) q^{74} +(-0.662765 - 4.95588i) q^{75} +(-3.58822 - 0.632700i) q^{76} +(-10.2342 + 1.80456i) q^{77} +(-3.36473 + 9.24451i) q^{78} +(-3.45194 - 4.11386i) q^{79} +(-10.9446 + 0.728582i) q^{80} +(0.766044 - 0.642788i) q^{81} +(1.07327 - 1.85896i) q^{82} +(-1.75507 - 4.82200i) q^{83} +(1.37788 + 2.38656i) q^{84} +(-5.02361 + 10.2132i) q^{85} +(0.533140 - 3.02359i) q^{86} +(4.77172 + 1.73676i) q^{87} +(3.10470 + 5.37750i) q^{88} +(0.330800 - 0.394232i) q^{89} +(3.28201 + 2.40132i) q^{90} +(-3.89911 - 10.7127i) q^{91} +(-1.63902 - 9.29533i) q^{92} +(-0.550332 - 3.12109i) q^{93} +(-6.10676 - 16.7782i) q^{94} +(3.67918 - 5.02854i) q^{95} +(4.11562 - 4.90481i) q^{96} +(-6.36745 - 11.0287i) q^{97} +(4.37224 + 1.59136i) q^{98} +(0.856249 - 4.85603i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 240 q + 12 q^{5} + 36 q^{14} + 12 q^{19} - 30 q^{20} + 12 q^{21} - 48 q^{25} - 36 q^{26} + 24 q^{30} + 48 q^{34} + 12 q^{35} - 264 q^{36} + 36 q^{39} + 66 q^{40} - 144 q^{41} + 192 q^{44} + 24 q^{46} - 12 q^{49}+ \cdots + 12 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/555\mathbb{Z}\right)^\times\).

\(n\) \(76\) \(112\) \(371\)
\(\chi(n)\) \(e\left(\frac{1}{18}\right)\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.315810 + 1.79105i −0.223311 + 1.26646i 0.642576 + 0.766222i \(0.277866\pi\)
−0.865887 + 0.500240i \(0.833245\pi\)
\(3\) 0.984808 − 0.173648i 0.568579 − 0.100256i
\(4\) −1.22873 − 0.447221i −0.614365 − 0.223611i
\(5\) 1.61367 − 1.54792i 0.721657 − 0.692251i
\(6\) 1.81868i 0.742472i
\(7\) −1.35469 − 1.61445i −0.512024 − 0.610206i 0.446652 − 0.894708i \(-0.352616\pi\)
−0.958675 + 0.284502i \(0.908172\pi\)
\(8\) −0.629637 + 1.09056i −0.222610 + 0.385572i
\(9\) 0.939693 − 0.342020i 0.313231 − 0.114007i
\(10\) 2.26279 + 3.37901i 0.715556 + 1.06854i
\(11\) 2.46547 − 4.27032i 0.743367 − 1.28755i −0.207586 − 0.978217i \(-0.566561\pi\)
0.950954 − 0.309333i \(-0.100106\pi\)
\(12\) −1.28772 − 0.227060i −0.371734 − 0.0655467i
\(13\) 5.08309 + 1.85009i 1.40980 + 0.513124i 0.931070 − 0.364840i \(-0.118876\pi\)
0.478726 + 0.877964i \(0.341099\pi\)
\(14\) 3.31939 − 1.91645i 0.887143 − 0.512193i
\(15\) 1.32036 − 1.80462i 0.340917 − 0.465950i
\(16\) −3.75775 − 3.15312i −0.939437 − 0.788281i
\(17\) −4.78314 + 1.74092i −1.16008 + 0.422235i −0.849126 − 0.528190i \(-0.822871\pi\)
−0.310954 + 0.950425i \(0.600649\pi\)
\(18\) 0.315810 + 1.79105i 0.0744371 + 0.422154i
\(19\) 2.74415 − 0.483868i 0.629552 − 0.111007i 0.150237 − 0.988650i \(-0.451996\pi\)
0.479315 + 0.877643i \(0.340885\pi\)
\(20\) −2.67503 + 1.18031i −0.598156 + 0.263925i
\(21\) −1.61445 − 1.35469i −0.352303 − 0.295617i
\(22\) 6.86973 + 5.76439i 1.46463 + 1.22897i
\(23\) 3.60921 + 6.25134i 0.752573 + 1.30349i 0.946572 + 0.322493i \(0.104521\pi\)
−0.193999 + 0.981002i \(0.562146\pi\)
\(24\) −0.430697 + 1.18333i −0.0879157 + 0.241546i
\(25\) 0.207883 − 4.99568i 0.0415767 − 0.999135i
\(26\) −4.91890 + 8.51978i −0.964676 + 1.67087i
\(27\) 0.866025 − 0.500000i 0.166667 − 0.0962250i
\(28\) 0.942528 + 2.58957i 0.178121 + 0.489383i
\(29\) 4.39764 + 2.53898i 0.816621 + 0.471476i 0.849250 − 0.527991i \(-0.177055\pi\)
−0.0326289 + 0.999468i \(0.510388\pi\)
\(30\) 2.81517 + 2.93475i 0.513977 + 0.535810i
\(31\) − 3.16923i − 0.569211i −0.958645 − 0.284606i \(-0.908137\pi\)
0.958645 − 0.284606i \(-0.0918627\pi\)
\(32\) 4.90481 − 4.11562i 0.867056 − 0.727546i
\(33\) 1.68648 − 4.63357i 0.293579 − 0.806601i
\(34\) −1.60751 − 9.11662i −0.275685 − 1.56349i
\(35\) −4.68507 − 0.508252i −0.791921 − 0.0859103i
\(36\) −1.30759 −0.217931
\(37\) 4.37931 − 4.22157i 0.719954 − 0.694022i
\(38\) 5.06772i 0.822093i
\(39\) 5.32713 + 0.939318i 0.853024 + 0.150411i
\(40\) 0.672077 + 2.73444i 0.106265 + 0.432353i
\(41\) −1.10910 − 0.403678i −0.173212 − 0.0630439i 0.253959 − 0.967215i \(-0.418267\pi\)
−0.427170 + 0.904171i \(0.640490\pi\)
\(42\) 2.93617 − 2.46374i 0.453061 − 0.380163i
\(43\) −1.68817 −0.257443 −0.128721 − 0.991681i \(-0.541087\pi\)
−0.128721 + 0.991681i \(0.541087\pi\)
\(44\) −4.93918 + 4.14446i −0.744609 + 0.624801i
\(45\) 0.986937 − 2.00648i 0.147124 − 0.299108i
\(46\) −12.3363 + 4.49004i −1.81888 + 0.662020i
\(47\) −8.50226 + 4.90878i −1.24018 + 0.716019i −0.969131 − 0.246546i \(-0.920704\pi\)
−0.271051 + 0.962565i \(0.587371\pi\)
\(48\) −4.24819 − 2.45269i −0.613174 − 0.354016i
\(49\) 0.444255 − 2.51950i 0.0634650 − 0.359928i
\(50\) 8.88184 + 1.95001i 1.25608 + 0.275774i
\(51\) −4.40816 + 2.54505i −0.617266 + 0.356379i
\(52\) −5.41835 − 4.54654i −0.751390 − 0.630491i
\(53\) −4.41526 + 5.26190i −0.606482 + 0.722777i −0.978683 − 0.205375i \(-0.934159\pi\)
0.372201 + 0.928152i \(0.378603\pi\)
\(54\) 0.622024 + 1.70900i 0.0846468 + 0.232565i
\(55\) −2.63165 − 10.7073i −0.354852 − 1.44377i
\(56\) 2.61362 − 0.460853i 0.349260 − 0.0615840i
\(57\) 2.61844 − 0.953034i 0.346821 − 0.126232i
\(58\) −5.93625 + 7.07455i −0.779468 + 0.928933i
\(59\) −6.03368 + 7.19066i −0.785519 + 0.936145i −0.999169 − 0.0407657i \(-0.987020\pi\)
0.213650 + 0.976910i \(0.431465\pi\)
\(60\) −2.42944 + 1.62689i −0.313639 + 0.210031i
\(61\) −4.86192 + 13.3580i −0.622505 + 1.71032i 0.0782665 + 0.996932i \(0.475062\pi\)
−0.700771 + 0.713386i \(0.747161\pi\)
\(62\) 5.67625 + 1.00088i 0.720885 + 0.127111i
\(63\) −1.82517 − 1.05376i −0.229949 − 0.132761i
\(64\) 0.916901 + 1.58812i 0.114613 + 0.198515i
\(65\) 11.0663 − 4.88278i 1.37260 − 0.605634i
\(66\) 7.76634 + 4.48390i 0.955970 + 0.551929i
\(67\) −6.36879 − 7.59003i −0.778072 − 0.927270i 0.220773 − 0.975325i \(-0.429142\pi\)
−0.998845 + 0.0480556i \(0.984698\pi\)
\(68\) 6.65576 0.807130
\(69\) 4.63992 + 5.52964i 0.558580 + 0.665690i
\(70\) 2.38990 − 8.23067i 0.285647 − 0.983753i
\(71\) −1.00436 − 5.69603i −0.119196 − 0.675994i −0.984587 − 0.174897i \(-0.944041\pi\)
0.865391 − 0.501098i \(-0.167070\pi\)
\(72\) −0.218671 + 1.24014i −0.0257706 + 0.146152i
\(73\) 11.5949i 1.35708i 0.734564 + 0.678539i \(0.237387\pi\)
−0.734564 + 0.678539i \(0.762613\pi\)
\(74\) 6.17800 + 9.17677i 0.718178 + 1.06678i
\(75\) −0.662765 − 4.95588i −0.0765295 − 0.572256i
\(76\) −3.58822 − 0.632700i −0.411597 − 0.0725757i
\(77\) −10.2342 + 1.80456i −1.16629 + 0.205649i
\(78\) −3.36473 + 9.24451i −0.380980 + 1.04673i
\(79\) −3.45194 − 4.11386i −0.388374 − 0.462846i 0.536065 − 0.844177i \(-0.319910\pi\)
−0.924439 + 0.381331i \(0.875466\pi\)
\(80\) −10.9446 + 0.728582i −1.22364 + 0.0814579i
\(81\) 0.766044 − 0.642788i 0.0851160 − 0.0714208i
\(82\) 1.07327 − 1.85896i 0.118523 − 0.205288i
\(83\) −1.75507 − 4.82200i −0.192643 − 0.529284i 0.805336 − 0.592818i \(-0.201985\pi\)
−0.997980 + 0.0635348i \(0.979763\pi\)
\(84\) 1.37788 + 2.38656i 0.150339 + 0.260396i
\(85\) −5.02361 + 10.2132i −0.544887 + 1.10778i
\(86\) 0.533140 − 3.02359i 0.0574900 − 0.326042i
\(87\) 4.77172 + 1.73676i 0.511582 + 0.186201i
\(88\) 3.10470 + 5.37750i 0.330962 + 0.573244i
\(89\) 0.330800 − 0.394232i 0.0350647 − 0.0417885i −0.748227 − 0.663443i \(-0.769095\pi\)
0.783292 + 0.621654i \(0.213539\pi\)
\(90\) 3.28201 + 2.40132i 0.345955 + 0.253121i
\(91\) −3.89911 − 10.7127i −0.408738 − 1.12300i
\(92\) −1.63902 − 9.29533i −0.170879 − 0.969106i
\(93\) −0.550332 − 3.12109i −0.0570668 − 0.323642i
\(94\) −6.10676 − 16.7782i −0.629864 − 1.73054i
\(95\) 3.67918 − 5.02854i 0.377476 − 0.515917i
\(96\) 4.11562 − 4.90481i 0.420049 − 0.500595i
\(97\) −6.36745 − 11.0287i −0.646516 − 1.11980i −0.983949 − 0.178449i \(-0.942892\pi\)
0.337433 − 0.941350i \(-0.390441\pi\)
\(98\) 4.37224 + 1.59136i 0.441663 + 0.160752i
\(99\) 0.856249 − 4.85603i 0.0860563 − 0.488049i
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 555.2.bp.a.4.10 ✓ 240
5.4 even 2 inner 555.2.bp.a.4.31 yes 240
37.28 even 18 inner 555.2.bp.a.139.31 yes 240
185.139 even 18 inner 555.2.bp.a.139.10 yes 240
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
555.2.bp.a.4.10 ✓ 240 1.1 even 1 trivial
555.2.bp.a.4.31 yes 240 5.4 even 2 inner
555.2.bp.a.139.10 yes 240 185.139 even 18 inner
555.2.bp.a.139.31 yes 240 37.28 even 18 inner