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Properties

Label 888.2.o.a.517.18
Level $888$
Weight $2$
Character 888.517
Analytic conductor $7.091$
Analytic rank $0$
Dimension $76$
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] = [888,2,Mod(517,888)] 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("888.517"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(888, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 888 = 2^{3} \cdot 3 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 888.o (of order \(2\), degree \(1\), 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: \(7.09071569949\)
Analytic rank: \(0\)
Dimension: \(76\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 517.18
Character \(\chi\) \(=\) 888.517
Dual form 888.2.o.a.517.17

$q$-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+(-1.07422 + 0.919807i) q^{2} -1.00000i q^{3} +(0.307910 - 1.97616i) q^{4} +1.74292 q^{5} +(0.919807 + 1.07422i) q^{6} -1.43072 q^{7} +(1.48692 + 2.40605i) q^{8} -1.00000 q^{9} +(-1.87229 + 1.60315i) q^{10} -3.72034i q^{11} +(-1.97616 - 0.307910i) q^{12} -2.79208 q^{13} +(1.53691 - 1.31598i) q^{14} -1.74292i q^{15} +(-3.81038 - 1.21695i) q^{16} -1.26972i q^{17} +(1.07422 - 0.919807i) q^{18} +7.31134 q^{19} +(0.536663 - 3.44429i) q^{20} +1.43072i q^{21} +(3.42199 + 3.99647i) q^{22} -5.34305i q^{23} +(2.40605 - 1.48692i) q^{24} -1.96222 q^{25} +(2.99932 - 2.56818i) q^{26} +1.00000i q^{27} +(-0.440531 + 2.82732i) q^{28} -7.75927 q^{29} +(1.60315 + 1.87229i) q^{30} +2.62702i q^{31} +(5.21256 - 2.19754i) q^{32} -3.72034 q^{33} +(1.16790 + 1.36396i) q^{34} -2.49363 q^{35} +(-0.307910 + 1.97616i) q^{36} +(3.34888 - 5.07789i) q^{37} +(-7.85401 + 6.72503i) q^{38} +2.79208i q^{39} +(2.59159 + 4.19356i) q^{40} -4.94502 q^{41} +(-1.31598 - 1.53691i) q^{42} +7.69210 q^{43} +(-7.35196 - 1.14553i) q^{44} -1.74292 q^{45} +(4.91458 + 5.73963i) q^{46} -1.35969 q^{47} +(-1.21695 + 3.81038i) q^{48} -4.95305 q^{49} +(2.10786 - 1.80486i) q^{50} -1.26972 q^{51} +(-0.859709 + 5.51759i) q^{52} -9.73082i q^{53} +(-0.919807 - 1.07422i) q^{54} -6.48426i q^{55} +(-2.12736 - 3.44237i) q^{56} -7.31134i q^{57} +(8.33518 - 7.13703i) q^{58} -8.99586 q^{59} +(-3.44429 - 0.536663i) q^{60} -13.0860 q^{61} +(-2.41635 - 2.82200i) q^{62} +1.43072 q^{63} +(-3.57815 + 7.15520i) q^{64} -4.86639 q^{65} +(3.99647 - 3.42199i) q^{66} -14.2089i q^{67} +(-2.50916 - 0.390958i) q^{68} -5.34305 q^{69} +(2.67871 - 2.29366i) q^{70} +6.35199 q^{71} +(-1.48692 - 2.40605i) q^{72} +16.4014 q^{73} +(1.07323 + 8.53511i) q^{74} +1.96222i q^{75} +(2.25123 - 14.4484i) q^{76} +5.32274i q^{77} +(-2.56818 - 2.99932i) q^{78} +6.71894i q^{79} +(-6.64121 - 2.12106i) q^{80} +1.00000 q^{81} +(5.31205 - 4.54846i) q^{82} -13.8813i q^{83} +(2.82732 + 0.440531i) q^{84} -2.21302i q^{85} +(-8.26303 + 7.07525i) q^{86} +7.75927i q^{87} +(8.95131 - 5.53184i) q^{88} -3.67741i q^{89} +(1.87229 - 1.60315i) q^{90} +3.99468 q^{91} +(-10.5587 - 1.64518i) q^{92} +2.62702 q^{93} +(1.46061 - 1.25065i) q^{94} +12.7431 q^{95} +(-2.19754 - 5.21256i) q^{96} +7.40645i q^{97} +(5.32068 - 4.55585i) q^{98} +3.72034i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 76 q + 4 q^{4} + 8 q^{7} - 76 q^{9} + 4 q^{16} + 84 q^{25} + 12 q^{28} + 8 q^{30} - 12 q^{34} - 4 q^{36} + 8 q^{38} - 56 q^{40} - 8 q^{41} - 24 q^{44} - 44 q^{46} - 8 q^{48} + 60 q^{49} - 40 q^{58} + 32 q^{62}+ \cdots - 56 q^{86}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/888\mathbb{Z}\right)^\times\).

\(n\) \(223\) \(409\) \(445\) \(593\)
\(\chi(n)\) \(1\) \(-1\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.07422 + 0.919807i −0.759590 + 0.650402i
\(3\) − 1.00000i − 0.577350i
\(4\) 0.307910 − 1.97616i 0.153955 − 0.988078i
\(5\) 1.74292 0.779459 0.389730 − 0.920929i \(-0.372568\pi\)
0.389730 + 0.920929i \(0.372568\pi\)
\(6\) 0.919807 + 1.07422i 0.375510 + 0.438550i
\(7\) −1.43072 −0.540760 −0.270380 − 0.962754i \(-0.587149\pi\)
−0.270380 + 0.962754i \(0.587149\pi\)
\(8\) 1.48692 + 2.40605i 0.525705 + 0.850667i
\(9\) −1.00000 −0.333333
\(10\) −1.87229 + 1.60315i −0.592070 + 0.506962i
\(11\) − 3.72034i − 1.12172i −0.827909 − 0.560862i \(-0.810470\pi\)
0.827909 − 0.560862i \(-0.189530\pi\)
\(12\) −1.97616 − 0.307910i −0.570467 − 0.0888859i
\(13\) −2.79208 −0.774384 −0.387192 − 0.921999i \(-0.626555\pi\)
−0.387192 + 0.921999i \(0.626555\pi\)
\(14\) 1.53691 − 1.31598i 0.410756 − 0.351711i
\(15\) − 1.74292i − 0.450021i
\(16\) −3.81038 − 1.21695i −0.952596 − 0.304239i
\(17\) − 1.26972i − 0.307952i −0.988075 − 0.153976i \(-0.950792\pi\)
0.988075 − 0.153976i \(-0.0492078\pi\)
\(18\) 1.07422 − 0.919807i 0.253197 − 0.216801i
\(19\) 7.31134 1.67734 0.838669 − 0.544642i \(-0.183335\pi\)
0.838669 + 0.544642i \(0.183335\pi\)
\(20\) 0.536663 − 3.44429i 0.120001 − 0.770166i
\(21\) 1.43072i 0.312208i
\(22\) 3.42199 + 3.99647i 0.729571 + 0.852050i
\(23\) − 5.34305i − 1.11410i −0.830478 − 0.557052i \(-0.811932\pi\)
0.830478 − 0.557052i \(-0.188068\pi\)
\(24\) 2.40605 − 1.48692i 0.491133 − 0.303516i
\(25\) −1.96222 −0.392443
\(26\) 2.99932 − 2.56818i 0.588215 − 0.503661i
\(27\) 1.00000i 0.192450i
\(28\) −0.440531 + 2.82732i −0.0832526 + 0.534313i
\(29\) −7.75927 −1.44086 −0.720430 − 0.693528i \(-0.756055\pi\)
−0.720430 + 0.693528i \(0.756055\pi\)
\(30\) 1.60315 + 1.87229i 0.292694 + 0.341832i
\(31\) 2.62702i 0.471827i 0.971774 + 0.235913i \(0.0758081\pi\)
−0.971774 + 0.235913i \(0.924192\pi\)
\(32\) 5.21256 − 2.19754i 0.921460 − 0.388473i
\(33\) −3.72034 −0.647627
\(34\) 1.16790 + 1.36396i 0.200292 + 0.233917i
\(35\) −2.49363 −0.421500
\(36\) −0.307910 + 1.97616i −0.0513183 + 0.329359i
\(37\) 3.34888 − 5.07789i 0.550553 − 0.834800i
\(38\) −7.85401 + 6.72503i −1.27409 + 1.09094i
\(39\) 2.79208i 0.447091i
\(40\) 2.59159 + 4.19356i 0.409766 + 0.663060i
\(41\) −4.94502 −0.772282 −0.386141 − 0.922440i \(-0.626192\pi\)
−0.386141 + 0.922440i \(0.626192\pi\)
\(42\) −1.31598 − 1.53691i −0.203061 − 0.237150i
\(43\) 7.69210 1.17303 0.586517 − 0.809937i \(-0.300499\pi\)
0.586517 + 0.809937i \(0.300499\pi\)
\(44\) −7.35196 − 1.14553i −1.10835 − 0.172695i
\(45\) −1.74292 −0.259820
\(46\) 4.91458 + 5.73963i 0.724615 + 0.846262i
\(47\) −1.35969 −0.198332 −0.0991658 − 0.995071i \(-0.531617\pi\)
−0.0991658 + 0.995071i \(0.531617\pi\)
\(48\) −1.21695 + 3.81038i −0.175652 + 0.549981i
\(49\) −4.95305 −0.707579
\(50\) 2.10786 − 1.80486i 0.298096 − 0.255246i
\(51\) −1.26972 −0.177796
\(52\) −0.859709 + 5.51759i −0.119220 + 0.765152i
\(53\) − 9.73082i − 1.33663i −0.743878 − 0.668316i \(-0.767016\pi\)
0.743878 − 0.668316i \(-0.232984\pi\)
\(54\) −0.919807 − 1.07422i −0.125170 − 0.146183i
\(55\) − 6.48426i − 0.874338i
\(56\) −2.12736 − 3.44237i −0.284280 − 0.460006i
\(57\) − 7.31134i − 0.968411i
\(58\) 8.33518 − 7.13703i 1.09446 − 0.937138i
\(59\) −8.99586 −1.17116 −0.585581 − 0.810614i \(-0.699133\pi\)
−0.585581 + 0.810614i \(0.699133\pi\)
\(60\) −3.44429 − 0.536663i −0.444656 − 0.0692829i
\(61\) −13.0860 −1.67549 −0.837744 − 0.546063i \(-0.816126\pi\)
−0.837744 + 0.546063i \(0.816126\pi\)
\(62\) −2.41635 − 2.82200i −0.306877 − 0.358395i
\(63\) 1.43072 0.180253
\(64\) −3.57815 + 7.15520i −0.447268 + 0.894400i
\(65\) −4.86639 −0.603601
\(66\) 3.99647 − 3.42199i 0.491931 − 0.421218i
\(67\) − 14.2089i − 1.73590i −0.496653 − 0.867949i \(-0.665438\pi\)
0.496653 − 0.867949i \(-0.334562\pi\)
\(68\) −2.50916 − 0.390958i −0.304280 − 0.0474106i
\(69\) −5.34305 −0.643228
\(70\) 2.67871 − 2.29366i 0.320167 − 0.274144i
\(71\) 6.35199 0.753843 0.376921 − 0.926245i \(-0.376983\pi\)
0.376921 + 0.926245i \(0.376983\pi\)
\(72\) −1.48692 − 2.40605i −0.175235 − 0.283556i
\(73\) 16.4014 1.91963 0.959817 − 0.280626i \(-0.0905420\pi\)
0.959817 + 0.280626i \(0.0905420\pi\)
\(74\) 1.07323 + 8.53511i 0.124761 + 0.992187i
\(75\) 1.96222i 0.226577i
\(76\) 2.25123 − 14.4484i 0.258234 − 1.65734i
\(77\) 5.32274i 0.606583i
\(78\) −2.56818 − 2.99932i −0.290789 − 0.339606i
\(79\) 6.71894i 0.755940i 0.925818 + 0.377970i \(0.123378\pi\)
−0.925818 + 0.377970i \(0.876622\pi\)
\(80\) −6.64121 − 2.12106i −0.742510 − 0.237142i
\(81\) 1.00000 0.111111
\(82\) 5.31205 − 4.54846i 0.586618 − 0.502293i
\(83\) − 13.8813i − 1.52367i −0.647772 − 0.761834i \(-0.724299\pi\)
0.647772 − 0.761834i \(-0.275701\pi\)
\(84\) 2.82732 + 0.440531i 0.308486 + 0.0480659i
\(85\) − 2.21302i − 0.240036i
\(86\) −8.26303 + 7.07525i −0.891025 + 0.762944i
\(87\) 7.75927i 0.831881i
\(88\) 8.95131 − 5.53184i 0.954213 − 0.589696i
\(89\) − 3.67741i − 0.389805i −0.980823 − 0.194902i \(-0.937561\pi\)
0.980823 − 0.194902i \(-0.0624390\pi\)
\(90\) 1.87229 − 1.60315i 0.197357 − 0.168987i
\(91\) 3.99468 0.418756
\(92\) −10.5587 − 1.64518i −1.10082 − 0.171522i
\(93\) 2.62702 0.272409
\(94\) 1.46061 − 1.25065i 0.150651 − 0.128995i
\(95\) 12.7431 1.30742
\(96\) −2.19754 − 5.21256i −0.224285 − 0.532005i
\(97\) 7.40645i 0.752011i 0.926618 + 0.376005i \(0.122703\pi\)
−0.926618 + 0.376005i \(0.877297\pi\)
\(98\) 5.32068 − 4.55585i 0.537470 − 0.460211i
\(99\) 3.72034i 0.373908i
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 888.2.o.a.517.18 yes 76
4.3 odd 2 3552.2.o.a.2737.54 76
8.3 odd 2 3552.2.o.a.2737.45 76
8.5 even 2 inner 888.2.o.a.517.60 yes 76
37.36 even 2 inner 888.2.o.a.517.59 yes 76
148.147 odd 2 3552.2.o.a.2737.46 76
296.147 odd 2 3552.2.o.a.2737.53 76
296.221 even 2 inner 888.2.o.a.517.17 ✓ 76
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
888.2.o.a.517.17 ✓ 76 296.221 even 2 inner
888.2.o.a.517.18 yes 76 1.1 even 1 trivial
888.2.o.a.517.59 yes 76 37.36 even 2 inner
888.2.o.a.517.60 yes 76 8.5 even 2 inner
3552.2.o.a.2737.45 76 8.3 odd 2
3552.2.o.a.2737.46 76 148.147 odd 2
3552.2.o.a.2737.53 76 296.147 odd 2
3552.2.o.a.2737.54 76 4.3 odd 2