Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [425,2,Mod(251,425)] 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("425.251"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(425, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 425 = 5^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 425.e (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [12,0,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(3.39364208590\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(i)\)
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: \( x^{12} + 18x^{10} + 83x^{8} + 152x^{6} + 111x^{4} + 22x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 85)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 251.6
Root \(0.254679i\) of defining polynomial
Character \(\chi\) \(=\) 425.251
Dual form 425.2.e.f.276.1

$q$-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)
 
\(f(q)\) \(=\) \(q+2.51230i q^{2} +(0.887192 + 0.887192i) q^{3} -4.31167 q^{4} +(-2.22889 + 2.22889i) q^{6} +(-1.14187 + 1.14187i) q^{7} -5.80761i q^{8} -1.42578i q^{9} +(-2.32404 + 2.32404i) q^{11} +(-3.82528 - 3.82528i) q^{12} -6.35524 q^{13} +(-2.86872 - 2.86872i) q^{14} +5.96715 q^{16} +(0.768287 + 4.05089i) q^{17} +3.58200 q^{18} +0.747167i q^{19} -2.02612 q^{21} +(-5.83869 - 5.83869i) q^{22} +(-0.101240 + 0.101240i) q^{23} +(5.15247 - 5.15247i) q^{24} -15.9663i q^{26} +(3.92652 - 3.92652i) q^{27} +(4.92337 - 4.92337i) q^{28} +(6.22477 + 6.22477i) q^{29} +(5.10259 + 5.10259i) q^{31} +3.37606i q^{32} -4.12374 q^{33} +(-10.1771 + 1.93017i) q^{34} +6.14750i q^{36} +(0.439195 + 0.439195i) q^{37} -1.87711 q^{38} +(-5.63832 - 5.63832i) q^{39} +(-4.49889 + 4.49889i) q^{41} -5.09022i q^{42} +2.74801i q^{43} +(10.0205 - 10.0205i) q^{44} +(-0.254346 - 0.254346i) q^{46} +11.9308 q^{47} +(5.29400 + 5.29400i) q^{48} +4.39226i q^{49} +(-2.91230 + 4.27554i) q^{51} +27.4017 q^{52} +2.71404i q^{53} +(9.86460 + 9.86460i) q^{54} +(6.63154 + 6.63154i) q^{56} +(-0.662880 + 0.662880i) q^{57} +(-15.6385 + 15.6385i) q^{58} -12.6815i q^{59} +(0.328162 - 0.328162i) q^{61} +(-12.8193 + 12.8193i) q^{62} +(1.62806 + 1.62806i) q^{63} +3.45261 q^{64} -10.3601i q^{66} -1.81686 q^{67} +(-3.31260 - 17.4661i) q^{68} -0.179639 q^{69} +(-3.01043 - 3.01043i) q^{71} -8.28039 q^{72} +(0.856488 + 0.856488i) q^{73} +(-1.10339 + 1.10339i) q^{74} -3.22154i q^{76} -5.30750i q^{77} +(14.1652 - 14.1652i) q^{78} +(-3.57000 + 3.57000i) q^{79} +2.68980 q^{81} +(-11.3026 - 11.3026i) q^{82} -3.58494i q^{83} +8.73594 q^{84} -6.90384 q^{86} +11.0451i q^{87} +(13.4971 + 13.4971i) q^{88} -10.5906 q^{89} +(7.25687 - 7.25687i) q^{91} +(0.436515 - 0.436515i) q^{92} +9.05395i q^{93} +29.9739i q^{94} +(-2.99521 + 2.99521i) q^{96} +(4.92762 + 4.92762i) q^{97} -11.0347 q^{98} +(3.31357 + 3.31357i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 4 q^{3} - 12 q^{4} - 4 q^{11} + 8 q^{12} - 4 q^{14} + 4 q^{16} - 12 q^{17} - 28 q^{18} - 16 q^{21} - 20 q^{22} - 12 q^{23} + 4 q^{24} + 4 q^{27} - 4 q^{28} - 12 q^{29} + 16 q^{33} - 12 q^{34} - 12 q^{37}+ \cdots - 44 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/425\mathbb{Z}\right)^\times\).

\(n\) \(52\) \(326\)
\(\chi(n)\) \(1\) \(e\left(\frac{1}{4}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.51230i 1.77647i 0.459392 + 0.888233i \(0.348067\pi\)
−0.459392 + 0.888233i \(0.651933\pi\)
\(3\) 0.887192 + 0.887192i 0.512220 + 0.512220i 0.915206 − 0.402986i \(-0.132028\pi\)
−0.402986 + 0.915206i \(0.632028\pi\)
\(4\) −4.31167 −2.15583
\(5\) 0 0
\(6\) −2.22889 + 2.22889i −0.909943 + 0.909943i
\(7\) −1.14187 + 1.14187i −0.431586 + 0.431586i −0.889168 − 0.457581i \(-0.848716\pi\)
0.457581 + 0.889168i \(0.348716\pi\)
\(8\) − 5.80761i − 2.05330i
\(9\) − 1.42578i − 0.475261i
\(10\) 0 0
\(11\) −2.32404 + 2.32404i −0.700724 + 0.700724i −0.964566 − 0.263842i \(-0.915010\pi\)
0.263842 + 0.964566i \(0.415010\pi\)
\(12\) −3.82528 − 3.82528i −1.10426 − 1.10426i
\(13\) −6.35524 −1.76263 −0.881314 − 0.472531i \(-0.843340\pi\)
−0.881314 + 0.472531i \(0.843340\pi\)
\(14\) −2.86872 − 2.86872i −0.766699 − 0.766699i
\(15\) 0 0
\(16\) 5.96715 1.49179
\(17\) 0.768287 + 4.05089i 0.186337 + 0.982486i
\(18\) 3.58200 0.844284
\(19\) 0.747167i 0.171412i 0.996320 + 0.0857059i \(0.0273145\pi\)
−0.996320 + 0.0857059i \(0.972685\pi\)
\(20\) 0 0
\(21\) −2.02612 −0.442135
\(22\) −5.83869 − 5.83869i −1.24481 − 1.24481i
\(23\) −0.101240 + 0.101240i −0.0211101 + 0.0211101i −0.717583 − 0.696473i \(-0.754752\pi\)
0.696473 + 0.717583i \(0.254752\pi\)
\(24\) 5.15247 − 5.15247i 1.05174 − 1.05174i
\(25\) 0 0
\(26\) − 15.9663i − 3.13125i
\(27\) 3.92652 − 3.92652i 0.755659 − 0.755659i
\(28\) 4.92337 − 4.92337i 0.930429 − 0.930429i
\(29\) 6.22477 + 6.22477i 1.15591 + 1.15591i 0.985346 + 0.170565i \(0.0545592\pi\)
0.170565 + 0.985346i \(0.445441\pi\)
\(30\) 0 0
\(31\) 5.10259 + 5.10259i 0.916452 + 0.916452i 0.996769 − 0.0803174i \(-0.0255934\pi\)
−0.0803174 + 0.996769i \(0.525593\pi\)
\(32\) 3.37606i 0.596809i
\(33\) −4.12374 −0.717850
\(34\) −10.1771 + 1.93017i −1.74535 + 0.331021i
\(35\) 0 0
\(36\) 6.14750i 1.02458i
\(37\) 0.439195 + 0.439195i 0.0722032 + 0.0722032i 0.742286 − 0.670083i \(-0.233742\pi\)
−0.670083 + 0.742286i \(0.733742\pi\)
\(38\) −1.87711 −0.304507
\(39\) −5.63832 − 5.63832i −0.902854 − 0.902854i
\(40\) 0 0
\(41\) −4.49889 + 4.49889i −0.702609 + 0.702609i −0.964970 − 0.262361i \(-0.915499\pi\)
0.262361 + 0.964970i \(0.415499\pi\)
\(42\) − 5.09022i − 0.785438i
\(43\) 2.74801i 0.419068i 0.977801 + 0.209534i \(0.0671947\pi\)
−0.977801 + 0.209534i \(0.932805\pi\)
\(44\) 10.0205 − 10.0205i 1.51064 − 1.51064i
\(45\) 0 0
\(46\) −0.254346 − 0.254346i −0.0375013 − 0.0375013i
\(47\) 11.9308 1.74029 0.870146 − 0.492794i \(-0.164024\pi\)
0.870146 + 0.492794i \(0.164024\pi\)
\(48\) 5.29400 + 5.29400i 0.764124 + 0.764124i
\(49\) 4.39226i 0.627466i
\(50\) 0 0
\(51\) −2.91230 + 4.27554i −0.407804 + 0.598695i
\(52\) 27.4017 3.79993
\(53\) 2.71404i 0.372802i 0.982474 + 0.186401i \(0.0596823\pi\)
−0.982474 + 0.186401i \(0.940318\pi\)
\(54\) 9.86460 + 9.86460i 1.34240 + 1.34240i
\(55\) 0 0
\(56\) 6.63154 + 6.63154i 0.886177 + 0.886177i
\(57\) −0.662880 + 0.662880i −0.0878006 + 0.0878006i
\(58\) −15.6385 + 15.6385i −2.05344 + 2.05344i
\(59\) − 12.6815i − 1.65099i −0.564413 − 0.825493i \(-0.690897\pi\)
0.564413 − 0.825493i \(-0.309103\pi\)
\(60\) 0 0
\(61\) 0.328162 − 0.328162i 0.0420168 − 0.0420168i −0.685786 − 0.727803i \(-0.740542\pi\)
0.727803 + 0.685786i \(0.240542\pi\)
\(62\) −12.8193 + 12.8193i −1.62805 + 1.62805i
\(63\) 1.62806 + 1.62806i 0.205116 + 0.205116i
\(64\) 3.45261 0.431576
\(65\) 0 0
\(66\) − 10.3601i − 1.27524i
\(67\) −1.81686 −0.221965 −0.110982 − 0.993822i \(-0.535400\pi\)
−0.110982 + 0.993822i \(0.535400\pi\)
\(68\) −3.31260 − 17.4661i −0.401712 − 2.11808i
\(69\) −0.179639 −0.0216260
\(70\) 0 0
\(71\) −3.01043 − 3.01043i −0.357272 − 0.357272i 0.505535 − 0.862806i \(-0.331295\pi\)
−0.862806 + 0.505535i \(0.831295\pi\)
\(72\) −8.28039 −0.975853
\(73\) 0.856488 + 0.856488i 0.100244 + 0.100244i 0.755450 − 0.655206i \(-0.227418\pi\)
−0.655206 + 0.755450i \(0.727418\pi\)
\(74\) −1.10339 + 1.10339i −0.128267 + 0.128267i
\(75\) 0 0
\(76\) − 3.22154i − 0.369535i
\(77\) − 5.30750i − 0.604846i
\(78\) 14.1652 − 14.1652i 1.60389 − 1.60389i
\(79\) −3.57000 + 3.57000i −0.401656 + 0.401656i −0.878816 − 0.477160i \(-0.841666\pi\)
0.477160 + 0.878816i \(0.341666\pi\)
\(80\) 0 0
\(81\) 2.68980 0.298867
\(82\) −11.3026 − 11.3026i −1.24816 − 1.24816i
\(83\) − 3.58494i − 0.393498i −0.980454 − 0.196749i \(-0.936962\pi\)
0.980454 − 0.196749i \(-0.0630385\pi\)
\(84\) 8.73594 0.953169
\(85\) 0 0
\(86\) −6.90384 −0.744460
\(87\) 11.0451i 1.18416i
\(88\) 13.4971 + 13.4971i 1.43880 + 1.43880i
\(89\) −10.5906 −1.12260 −0.561300 − 0.827613i \(-0.689698\pi\)
−0.561300 + 0.827613i \(0.689698\pi\)
\(90\) 0 0
\(91\) 7.25687 − 7.25687i 0.760726 − 0.760726i
\(92\) 0.436515 − 0.436515i 0.0455098 − 0.0455098i
\(93\) 9.05395i 0.938851i
\(94\) 29.9739i 3.09157i
\(95\) 0 0
\(96\) −2.99521 + 2.99521i −0.305698 + 0.305698i
\(97\) 4.92762 + 4.92762i 0.500324 + 0.500324i 0.911539 − 0.411215i \(-0.134895\pi\)
−0.411215 + 0.911539i \(0.634895\pi\)
\(98\) −11.0347 −1.11467
\(99\) 3.31357 + 3.31357i 0.333026 + 0.333026i
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 425.2.e.f.251.6 12
5.2 odd 4 425.2.j.b.149.1 12
5.3 odd 4 425.2.j.c.149.6 12
5.4 even 2 85.2.e.a.81.1 yes 12
15.14 odd 2 765.2.k.b.676.6 12
17.2 even 8 7225.2.a.bb.1.1 6
17.4 even 4 inner 425.2.e.f.276.1 12
17.15 even 8 7225.2.a.z.1.1 6
20.19 odd 2 1360.2.bt.d.81.4 12
85.4 even 4 85.2.e.a.21.6 ✓ 12
85.9 even 8 1445.2.d.g.866.1 12
85.19 even 8 1445.2.a.n.1.6 6
85.38 odd 4 425.2.j.b.174.1 12
85.49 even 8 1445.2.a.o.1.6 6
85.59 even 8 1445.2.d.g.866.2 12
85.72 odd 4 425.2.j.c.174.6 12
255.89 odd 4 765.2.k.b.361.1 12
340.259 odd 4 1360.2.bt.d.1041.4 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
85.2.e.a.21.6 ✓ 12 85.4 even 4
85.2.e.a.81.1 yes 12 5.4 even 2
425.2.e.f.251.6 12 1.1 even 1 trivial
425.2.e.f.276.1 12 17.4 even 4 inner
425.2.j.b.149.1 12 5.2 odd 4
425.2.j.b.174.1 12 85.38 odd 4
425.2.j.c.149.6 12 5.3 odd 4
425.2.j.c.174.6 12 85.72 odd 4
765.2.k.b.361.1 12 255.89 odd 4
765.2.k.b.676.6 12 15.14 odd 2
1360.2.bt.d.81.4 12 20.19 odd 2
1360.2.bt.d.1041.4 12 340.259 odd 4
1445.2.a.n.1.6 6 85.19 even 8
1445.2.a.o.1.6 6 85.49 even 8
1445.2.d.g.866.1 12 85.9 even 8
1445.2.d.g.866.2 12 85.59 even 8
7225.2.a.z.1.1 6 17.15 even 8
7225.2.a.bb.1.1 6 17.2 even 8