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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [84,8,Mod(25,84)] 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("84.25"); S:= CuspForms(chi, 8); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(84, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 4])) N = Newforms(chi, 8, names="a")
 
Level: \( N \) \(=\) \( 84 = 2^{2} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 84.i (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8] 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: \(26.2403421407\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2x^{7} + 659x^{6} + 12718x^{5} + 417701x^{4} + 3735784x^{3} + 32480596x^{2} + 479136x + 7056 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{6}\cdot 3^{5}\cdot 7^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 25.4
Root \(-0.00737575 + 0.0127752i\) of defining polynomial
Character \(\chi\) \(=\) 84.25
Dual form 84.8.i.a.37.4

$q$-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)
 
\(f(q)\) \(=\) \(q+(13.5000 - 23.3827i) q^{3} +(80.0854 + 138.712i) q^{5} +(-254.682 - 871.022i) q^{7} +(-364.500 - 631.333i) q^{9} +(-1581.93 + 2739.98i) q^{11} -4771.53 q^{13} +4324.61 q^{15} +(6504.73 - 11266.5i) q^{17} +(-22058.1 - 38205.8i) q^{19} +(-23805.1 - 5803.65i) q^{21} +(32533.0 + 56348.8i) q^{23} +(26235.2 - 45440.6i) q^{25} -19683.0 q^{27} -246115. q^{29} +(-150328. + 260376. i) q^{31} +(42712.1 + 73979.5i) q^{33} +(100425. - 105084. i) q^{35} +(-258479. - 447698. i) q^{37} +(-64415.7 + 111571. i) q^{39} -377844. q^{41} -71420.3 q^{43} +(58382.3 - 101121. i) q^{45} +(-558378. - 967140. i) q^{47} +(-693817. + 443667. i) q^{49} +(-175628. - 304196. i) q^{51} +(184365. - 319330. i) q^{53} -506758. q^{55} -1.19114e6 q^{57} +(-515977. + 893698. i) q^{59} +(161341. + 279450. i) q^{61} +(-457073. + 478277. i) q^{63} +(-382130. - 661869. i) q^{65} +(11763.8 - 20375.5i) q^{67} +1.75678e6 q^{69} +2.84199e6 q^{71} +(-336523. + 582875. i) q^{73} +(-708349. - 1.22690e6i) q^{75} +(2.78947e6 + 680072. i) q^{77} +(429448. + 743826. i) q^{79} +(-265720. + 460241. i) q^{81} +7.32276e6 q^{83} +2.08373e6 q^{85} +(-3.32256e6 + 5.75484e6i) q^{87} +(-2.31400e6 - 4.00797e6i) q^{89} +(1.21522e6 + 4.15611e6i) q^{91} +(4.05886e6 + 7.03016e6i) q^{93} +(3.53307e6 - 6.11946e6i) q^{95} +951821. q^{97} +2.30645e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 108 q^{3} - 196 q^{5} - 434 q^{7} - 2916 q^{9} + 406 q^{11} + 3948 q^{13} - 10584 q^{15} - 7436 q^{17} + 15874 q^{19} - 39312 q^{21} - 6788 q^{23} + 69898 q^{25} - 157464 q^{27} - 189088 q^{29} - 55890 q^{31}+ \cdots - 591948 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/84\mathbb{Z}\right)^\times\).

\(n\) \(29\) \(43\) \(73\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{2}{3}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 13.5000 − 23.3827i 0.288675 − 0.500000i
\(4\) 0 0
\(5\) 80.0854 + 138.712i 0.286522 + 0.496271i 0.972977 − 0.230901i \(-0.0741674\pi\)
−0.686455 + 0.727172i \(0.740834\pi\)
\(6\) 0 0
\(7\) −254.682 − 871.022i −0.280644 − 0.959812i
\(8\) 0 0
\(9\) −364.500 − 631.333i −0.166667 − 0.288675i
\(10\) 0 0
\(11\) −1581.93 + 2739.98i −0.358354 + 0.620688i −0.987686 − 0.156449i \(-0.949995\pi\)
0.629332 + 0.777137i \(0.283329\pi\)
\(12\) 0 0
\(13\) −4771.53 −0.602360 −0.301180 − 0.953567i \(-0.597380\pi\)
−0.301180 + 0.953567i \(0.597380\pi\)
\(14\) 0 0
\(15\) 4324.61 0.330847
\(16\) 0 0
\(17\) 6504.73 − 11266.5i 0.321113 − 0.556184i −0.659605 − 0.751612i \(-0.729276\pi\)
0.980718 + 0.195429i \(0.0626098\pi\)
\(18\) 0 0
\(19\) −22058.1 − 38205.8i −0.737787 − 1.27789i −0.953490 − 0.301426i \(-0.902537\pi\)
0.215702 − 0.976459i \(-0.430796\pi\)
\(20\) 0 0
\(21\) −23805.1 − 5803.65i −0.560921 − 0.136752i
\(22\) 0 0
\(23\) 32533.0 + 56348.8i 0.557541 + 0.965690i 0.997701 + 0.0677701i \(0.0215885\pi\)
−0.440160 + 0.897919i \(0.645078\pi\)
\(24\) 0 0
\(25\) 26235.2 − 45440.6i 0.335810 − 0.581640i
\(26\) 0 0
\(27\) −19683.0 −0.192450
\(28\) 0 0
\(29\) −246115. −1.87390 −0.936949 − 0.349466i \(-0.886363\pi\)
−0.936949 + 0.349466i \(0.886363\pi\)
\(30\) 0 0
\(31\) −150328. + 260376.i −0.906306 + 1.56977i −0.0871514 + 0.996195i \(0.527776\pi\)
−0.819155 + 0.573573i \(0.805557\pi\)
\(32\) 0 0
\(33\) 42712.1 + 73979.5i 0.206896 + 0.358354i
\(34\) 0 0
\(35\) 100425. − 105084.i 0.395916 − 0.414283i
\(36\) 0 0
\(37\) −258479. − 447698.i −0.838916 − 1.45305i −0.890801 − 0.454393i \(-0.849856\pi\)
0.0518851 − 0.998653i \(-0.483477\pi\)
\(38\) 0 0
\(39\) −64415.7 + 111571.i −0.173886 + 0.301180i
\(40\) 0 0
\(41\) −377844. −0.856188 −0.428094 − 0.903734i \(-0.640815\pi\)
−0.428094 + 0.903734i \(0.640815\pi\)
\(42\) 0 0
\(43\) −71420.3 −0.136988 −0.0684940 − 0.997652i \(-0.521819\pi\)
−0.0684940 + 0.997652i \(0.521819\pi\)
\(44\) 0 0
\(45\) 58382.3 − 101121.i 0.0955074 − 0.165424i
\(46\) 0 0
\(47\) −558378. − 967140.i −0.784488 − 1.35877i −0.929305 − 0.369314i \(-0.879593\pi\)
0.144817 − 0.989458i \(-0.453741\pi\)
\(48\) 0 0
\(49\) −693817. + 443667.i −0.842478 + 0.538730i
\(50\) 0 0
\(51\) −175628. − 304196.i −0.185395 − 0.321113i
\(52\) 0 0
\(53\) 184365. − 319330.i 0.170104 − 0.294628i −0.768352 − 0.640027i \(-0.778923\pi\)
0.938456 + 0.345399i \(0.112256\pi\)
\(54\) 0 0
\(55\) −506758. −0.410706
\(56\) 0 0
\(57\) −1.19114e6 −0.851924
\(58\) 0 0
\(59\) −515977. + 893698.i −0.327076 + 0.566512i −0.981930 − 0.189243i \(-0.939397\pi\)
0.654855 + 0.755755i \(0.272730\pi\)
\(60\) 0 0
\(61\) 161341. + 279450.i 0.0910101 + 0.157634i 0.907936 − 0.419108i \(-0.137657\pi\)
−0.816926 + 0.576742i \(0.804324\pi\)
\(62\) 0 0
\(63\) −457073. + 478277.i −0.230300 + 0.240984i
\(64\) 0 0
\(65\) −382130. − 661869.i −0.172590 − 0.298934i
\(66\) 0 0
\(67\) 11763.8 − 20375.5i 0.00477844 − 0.00827650i −0.863626 − 0.504133i \(-0.831812\pi\)
0.868405 + 0.495856i \(0.165146\pi\)
\(68\) 0 0
\(69\) 1.75678e6 0.643793
\(70\) 0 0
\(71\) 2.84199e6 0.942363 0.471181 − 0.882036i \(-0.343828\pi\)
0.471181 + 0.882036i \(0.343828\pi\)
\(72\) 0 0
\(73\) −336523. + 582875.i −0.101248 + 0.175366i −0.912199 − 0.409748i \(-0.865617\pi\)
0.810951 + 0.585114i \(0.198950\pi\)
\(74\) 0 0
\(75\) −708349. − 1.22690e6i −0.193880 − 0.335810i
\(76\) 0 0
\(77\) 2.78947e6 + 680072.i 0.696314 + 0.169761i
\(78\) 0 0
\(79\) 429448. + 743826.i 0.0979977 + 0.169737i 0.910856 − 0.412725i \(-0.135423\pi\)
−0.812858 + 0.582462i \(0.802090\pi\)
\(80\) 0 0
\(81\) −265720. + 460241.i −0.0555556 + 0.0962250i
\(82\) 0 0
\(83\) 7.32276e6 1.40573 0.702864 − 0.711324i \(-0.251904\pi\)
0.702864 + 0.711324i \(0.251904\pi\)
\(84\) 0 0
\(85\) 2.08373e6 0.368024
\(86\) 0 0
\(87\) −3.32256e6 + 5.75484e6i −0.540948 + 0.936949i
\(88\) 0 0
\(89\) −2.31400e6 − 4.00797e6i −0.347935 − 0.602641i 0.637947 − 0.770080i \(-0.279784\pi\)
−0.985883 + 0.167439i \(0.946450\pi\)
\(90\) 0 0
\(91\) 1.21522e6 + 4.15611e6i 0.169048 + 0.578152i
\(92\) 0 0
\(93\) 4.05886e6 + 7.03016e6i 0.523256 + 0.906306i
\(94\) 0 0
\(95\) 3.53307e6 − 6.11946e6i 0.422785 − 0.732285i
\(96\) 0 0
\(97\) 951821. 0.105890 0.0529449 − 0.998597i \(-0.483139\pi\)
0.0529449 + 0.998597i \(0.483139\pi\)
\(98\) 0 0
\(99\) 2.30645e6 0.238903
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 84.8.i.a.25.4 ✓ 8
3.2 odd 2 252.8.k.b.109.1 8
7.2 even 3 inner 84.8.i.a.37.4 yes 8
7.3 odd 6 588.8.a.j.1.4 4
7.4 even 3 588.8.a.i.1.1 4
7.5 odd 6 588.8.i.o.373.1 8
7.6 odd 2 588.8.i.o.361.1 8
21.2 odd 6 252.8.k.b.37.1 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
84.8.i.a.25.4 ✓ 8 1.1 even 1 trivial
84.8.i.a.37.4 yes 8 7.2 even 3 inner
252.8.k.b.37.1 8 21.2 odd 6
252.8.k.b.109.1 8 3.2 odd 2
588.8.a.i.1.1 4 7.4 even 3
588.8.a.j.1.4 4 7.3 odd 6
588.8.i.o.361.1 8 7.6 odd 2
588.8.i.o.373.1 8 7.5 odd 6