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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [880,2,Mod(81,880)] 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("880.81"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(880, base_ring=CyclotomicField(10)) chi = DirichletCharacter(H, H._module([0, 0, 0, 2])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 880 = 2^{4} \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 880.bo (of order \(5\), degree \(4\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [12,0,1,0,3,0,8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(7.02683537787\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(3\) over \(\Q(\zeta_{5})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 2 x^{11} + 15 x^{10} - 22 x^{9} + 89 x^{8} - 118 x^{7} + 205 x^{6} - 68 x^{5} + 1061 x^{4} + \cdots + 400 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 440)
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

Embedding invariants

Embedding label 641.2
Root \(0.421568 + 1.29745i\) of defining polynomial
Character \(\chi\) \(=\) 880.641
Dual form 880.2.bo.j.81.2

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.260543 - 0.801870i) q^{3} +(0.809017 + 0.587785i) q^{5} +(-1.46997 + 4.52411i) q^{7} +(1.85194 - 1.34551i) q^{9} +(-0.231864 + 3.30851i) q^{11} +(-3.06058 + 2.22364i) q^{13} +(0.260543 - 0.801870i) q^{15} +(-4.03055 - 2.92836i) q^{17} +(-0.964489 - 2.96839i) q^{19} +4.01074 q^{21} -9.39001 q^{23} +(0.309017 + 0.951057i) q^{25} +(-3.60777 - 2.62120i) q^{27} +(-0.805585 + 2.47934i) q^{29} +(-2.98421 + 2.16816i) q^{31} +(2.71341 - 0.676086i) q^{33} +(-3.84844 + 2.79605i) q^{35} +(-1.06536 + 3.27883i) q^{37} +(2.58048 + 1.87483i) q^{39} +(2.88263 + 8.87181i) q^{41} +11.7051 q^{43} +2.28912 q^{45} +(0.513451 + 1.58024i) q^{47} +(-12.6436 - 9.18613i) q^{49} +(-1.29804 + 3.99494i) q^{51} +(1.56945 - 1.14027i) q^{53} +(-2.13228 + 2.54035i) q^{55} +(-2.12897 + 1.54679i) q^{57} +(-0.0321114 + 0.0988288i) q^{59} +(8.21258 + 5.96679i) q^{61} +(3.36494 + 10.3562i) q^{63} -3.78308 q^{65} +4.99277 q^{67} +(2.44651 + 7.52957i) q^{69} +(-5.03960 - 3.66149i) q^{71} +(-2.25783 + 6.94889i) q^{73} +(0.682112 - 0.495583i) q^{75} +(-14.6272 - 5.91239i) q^{77} +(8.53069 - 6.19791i) q^{79} +(0.960251 - 2.95535i) q^{81} +(4.60338 + 3.34455i) q^{83} +(-1.53953 - 4.73819i) q^{85} +2.19800 q^{87} +13.6371 q^{89} +(-5.56102 - 17.1151i) q^{91} +(2.51610 + 1.82805i) q^{93} +(0.964489 - 2.96839i) q^{95} +(-11.0809 + 8.05072i) q^{97} +(4.02224 + 6.43913i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + q^{3} + 3 q^{5} + 8 q^{7} + 10 q^{9} + 4 q^{11} - 7 q^{13} - q^{15} + 7 q^{17} - 3 q^{19} + 4 q^{21} - 36 q^{23} - 3 q^{25} - 8 q^{27} + 13 q^{29} - 2 q^{31} - 19 q^{33} + 2 q^{35} - 22 q^{37}+ \cdots + 79 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/880\mathbb{Z}\right)^\times\).

\(n\) \(111\) \(177\) \(321\) \(661\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{4}{5}\right)\) \(1\)

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 0
\(3\) −0.260543 − 0.801870i −0.150425 − 0.462960i 0.847244 − 0.531204i \(-0.178260\pi\)
−0.997669 + 0.0682442i \(0.978260\pi\)
\(4\) 0 0
\(5\) 0.809017 + 0.587785i 0.361803 + 0.262866i
\(6\) 0 0
\(7\) −1.46997 + 4.52411i −0.555597 + 1.70995i 0.138764 + 0.990325i \(0.455687\pi\)
−0.694361 + 0.719627i \(0.744313\pi\)
\(8\) 0 0
\(9\) 1.85194 − 1.34551i 0.617313 − 0.448504i
\(10\) 0 0
\(11\) −0.231864 + 3.30851i −0.0699096 + 0.997553i
\(12\) 0 0
\(13\) −3.06058 + 2.22364i −0.848851 + 0.616726i −0.924829 − 0.380383i \(-0.875792\pi\)
0.0759779 + 0.997109i \(0.475792\pi\)
\(14\) 0 0
\(15\) 0.260543 − 0.801870i 0.0672720 − 0.207042i
\(16\) 0 0
\(17\) −4.03055 − 2.92836i −0.977551 − 0.710233i −0.0203914 − 0.999792i \(-0.506491\pi\)
−0.957160 + 0.289559i \(0.906491\pi\)
\(18\) 0 0
\(19\) −0.964489 − 2.96839i −0.221269 − 0.680996i −0.998649 − 0.0519645i \(-0.983452\pi\)
0.777380 − 0.629031i \(-0.216548\pi\)
\(20\) 0 0
\(21\) 4.01074 0.875215
\(22\) 0 0
\(23\) −9.39001 −1.95795 −0.978976 − 0.203975i \(-0.934614\pi\)
−0.978976 + 0.203975i \(0.934614\pi\)
\(24\) 0 0
\(25\) 0.309017 + 0.951057i 0.0618034 + 0.190211i
\(26\) 0 0
\(27\) −3.60777 − 2.62120i −0.694316 − 0.504450i
\(28\) 0 0
\(29\) −0.805585 + 2.47934i −0.149593 + 0.460401i −0.997573 − 0.0696276i \(-0.977819\pi\)
0.847980 + 0.530029i \(0.177819\pi\)
\(30\) 0 0
\(31\) −2.98421 + 2.16816i −0.535980 + 0.389413i −0.822590 − 0.568635i \(-0.807472\pi\)
0.286610 + 0.958047i \(0.407472\pi\)
\(32\) 0 0
\(33\) 2.71341 − 0.676086i 0.472343 − 0.117691i
\(34\) 0 0
\(35\) −3.84844 + 2.79605i −0.650504 + 0.472619i
\(36\) 0 0
\(37\) −1.06536 + 3.27883i −0.175143 + 0.539036i −0.999640 − 0.0268308i \(-0.991458\pi\)
0.824497 + 0.565867i \(0.191458\pi\)
\(38\) 0 0
\(39\) 2.58048 + 1.87483i 0.413208 + 0.300213i
\(40\) 0 0
\(41\) 2.88263 + 8.87181i 0.450190 + 1.38554i 0.876690 + 0.481056i \(0.159747\pi\)
−0.426499 + 0.904488i \(0.640253\pi\)
\(42\) 0 0
\(43\) 11.7051 1.78500 0.892502 − 0.451043i \(-0.148948\pi\)
0.892502 + 0.451043i \(0.148948\pi\)
\(44\) 0 0
\(45\) 2.28912 0.341242
\(46\) 0 0
\(47\) 0.513451 + 1.58024i 0.0748945 + 0.230502i 0.981495 − 0.191489i \(-0.0613316\pi\)
−0.906600 + 0.421990i \(0.861332\pi\)
\(48\) 0 0
\(49\) −12.6436 − 9.18613i −1.80623 − 1.31230i
\(50\) 0 0
\(51\) −1.29804 + 3.99494i −0.181761 + 0.559404i
\(52\) 0 0
\(53\) 1.56945 − 1.14027i 0.215580 − 0.156628i −0.474755 − 0.880118i \(-0.657463\pi\)
0.690335 + 0.723490i \(0.257463\pi\)
\(54\) 0 0
\(55\) −2.13228 + 2.54035i −0.287516 + 0.342541i
\(56\) 0 0
\(57\) −2.12897 + 1.54679i −0.281990 + 0.204877i
\(58\) 0 0
\(59\) −0.0321114 + 0.0988288i −0.00418055 + 0.0128664i −0.953125 − 0.302577i \(-0.902153\pi\)
0.948944 + 0.315443i \(0.102153\pi\)
\(60\) 0 0
\(61\) 8.21258 + 5.96679i 1.05151 + 0.763969i 0.972499 − 0.232905i \(-0.0748231\pi\)
0.0790130 + 0.996874i \(0.474823\pi\)
\(62\) 0 0
\(63\) 3.36494 + 10.3562i 0.423943 + 1.30476i
\(64\) 0 0
\(65\) −3.78308 −0.469233
\(66\) 0 0
\(67\) 4.99277 0.609964 0.304982 − 0.952358i \(-0.401350\pi\)
0.304982 + 0.952358i \(0.401350\pi\)
\(68\) 0 0
\(69\) 2.44651 + 7.52957i 0.294525 + 0.906454i
\(70\) 0 0
\(71\) −5.03960 − 3.66149i −0.598091 − 0.434538i 0.247110 − 0.968987i \(-0.420519\pi\)
−0.845201 + 0.534449i \(0.820519\pi\)
\(72\) 0 0
\(73\) −2.25783 + 6.94889i −0.264259 + 0.813306i 0.727604 + 0.685998i \(0.240634\pi\)
−0.991863 + 0.127309i \(0.959366\pi\)
\(74\) 0 0
\(75\) 0.682112 − 0.495583i 0.0787635 − 0.0572250i
\(76\) 0 0
\(77\) −14.6272 − 5.91239i −1.66693 − 0.673780i
\(78\) 0 0
\(79\) 8.53069 − 6.19791i 0.959778 − 0.697319i 0.00667849 − 0.999978i \(-0.497874\pi\)
0.953099 + 0.302658i \(0.0978742\pi\)
\(80\) 0 0
\(81\) 0.960251 − 2.95535i 0.106695 − 0.328372i
\(82\) 0 0
\(83\) 4.60338 + 3.34455i 0.505287 + 0.367113i 0.811033 − 0.585001i \(-0.198906\pi\)
−0.305746 + 0.952113i \(0.598906\pi\)
\(84\) 0 0
\(85\) −1.53953 − 4.73819i −0.166986 − 0.513929i
\(86\) 0 0
\(87\) 2.19800 0.235650
\(88\) 0 0
\(89\) 13.6371 1.44553 0.722767 − 0.691091i \(-0.242870\pi\)
0.722767 + 0.691091i \(0.242870\pi\)
\(90\) 0 0
\(91\) −5.56102 − 17.1151i −0.582953 − 1.79415i
\(92\) 0 0
\(93\) 2.51610 + 1.82805i 0.260907 + 0.189560i
\(94\) 0 0
\(95\) 0.964489 − 2.96839i 0.0989545 − 0.304551i
\(96\) 0 0
\(97\) −11.0809 + 8.05072i −1.12509 + 0.817427i −0.984973 − 0.172708i \(-0.944748\pi\)
−0.140118 + 0.990135i \(0.544748\pi\)
\(98\) 0 0
\(99\) 4.02224 + 6.43913i 0.404250 + 0.647157i
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 880.2.bo.j.641.2 12
4.3 odd 2 440.2.y.b.201.2 yes 12
11.2 odd 10 9680.2.a.cy.1.3 6
11.4 even 5 inner 880.2.bo.j.81.2 12
11.9 even 5 9680.2.a.cx.1.3 6
44.15 odd 10 440.2.y.b.81.2 ✓ 12
44.31 odd 10 4840.2.a.bf.1.4 6
44.35 even 10 4840.2.a.be.1.4 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
440.2.y.b.81.2 ✓ 12 44.15 odd 10
440.2.y.b.201.2 yes 12 4.3 odd 2
880.2.bo.j.81.2 12 11.4 even 5 inner
880.2.bo.j.641.2 12 1.1 even 1 trivial
4840.2.a.be.1.4 6 44.35 even 10
4840.2.a.bf.1.4 6 44.31 odd 10
9680.2.a.cx.1.3 6 11.9 even 5
9680.2.a.cy.1.3 6 11.2 odd 10