Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [176,4,Mod(1,176)] 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("176.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(176, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 176 = 2^{4} \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 176.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,-9,0,11] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(10.3843361610\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{97}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 24 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 44)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(5.42443\) of defining polynomial
Character \(\chi\) \(=\) 176.1

$q$-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)
 
\(f(q)\) \(=\) \(q-9.42443 q^{3} +0.575571 q^{5} +24.5466 q^{7} +61.8199 q^{9} -11.0000 q^{11} -20.8489 q^{13} -5.42443 q^{15} -99.6977 q^{17} -104.489 q^{19} -231.337 q^{21} +172.762 q^{23} -124.669 q^{25} -328.157 q^{27} +137.826 q^{29} -205.424 q^{31} +103.669 q^{33} +14.1283 q^{35} +57.2381 q^{37} +196.489 q^{39} -247.640 q^{41} -353.035 q^{43} +35.5817 q^{45} -198.559 q^{47} +259.534 q^{49} +939.594 q^{51} -101.511 q^{53} -6.33128 q^{55} +984.745 q^{57} -240.646 q^{59} -43.8717 q^{61} +1517.47 q^{63} -12.0000 q^{65} -631.321 q^{67} -1628.18 q^{69} +38.2856 q^{71} +388.617 q^{73} +1174.93 q^{75} -270.012 q^{77} -1219.83 q^{79} +1423.56 q^{81} +119.197 q^{83} -57.3831 q^{85} -1298.93 q^{87} -520.973 q^{89} -511.768 q^{91} +1936.01 q^{93} -60.1406 q^{95} -335.101 q^{97} -680.018 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 9 q^{3} + 11 q^{5} - 10 q^{7} + 35 q^{9} - 22 q^{11} - 22 q^{13} - q^{15} - 160 q^{17} - 12 q^{19} - 246 q^{21} + 119 q^{23} - 141 q^{25} - 351 q^{27} - 138 q^{29} - 401 q^{31} + 99 q^{33} - 346 q^{35}+ \cdots - 385 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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
\(3\) −9.42443 −1.81373 −0.906866 − 0.421419i \(-0.861532\pi\)
−0.906866 + 0.421419i \(0.861532\pi\)
\(4\) 0 0
\(5\) 0.575571 0.0514806 0.0257403 − 0.999669i \(-0.491806\pi\)
0.0257403 + 0.999669i \(0.491806\pi\)
\(6\) 0 0
\(7\) 24.5466 1.32539 0.662695 − 0.748889i \(-0.269413\pi\)
0.662695 + 0.748889i \(0.269413\pi\)
\(8\) 0 0
\(9\) 61.8199 2.28962
\(10\) 0 0
\(11\) −11.0000 −0.301511
\(12\) 0 0
\(13\) −20.8489 −0.444803 −0.222401 − 0.974955i \(-0.571389\pi\)
−0.222401 + 0.974955i \(0.571389\pi\)
\(14\) 0 0
\(15\) −5.42443 −0.0933721
\(16\) 0 0
\(17\) −99.6977 −1.42237 −0.711184 − 0.703006i \(-0.751841\pi\)
−0.711184 + 0.703006i \(0.751841\pi\)
\(18\) 0 0
\(19\) −104.489 −1.26165 −0.630824 − 0.775926i \(-0.717283\pi\)
−0.630824 + 0.775926i \(0.717283\pi\)
\(20\) 0 0
\(21\) −231.337 −2.40390
\(22\) 0 0
\(23\) 172.762 1.56623 0.783116 − 0.621876i \(-0.213629\pi\)
0.783116 + 0.621876i \(0.213629\pi\)
\(24\) 0 0
\(25\) −124.669 −0.997350
\(26\) 0 0
\(27\) −328.157 −2.33903
\(28\) 0 0
\(29\) 137.826 0.882540 0.441270 − 0.897374i \(-0.354528\pi\)
0.441270 + 0.897374i \(0.354528\pi\)
\(30\) 0 0
\(31\) −205.424 −1.19017 −0.595086 − 0.803662i \(-0.702882\pi\)
−0.595086 + 0.803662i \(0.702882\pi\)
\(32\) 0 0
\(33\) 103.669 0.546861
\(34\) 0 0
\(35\) 14.1283 0.0682320
\(36\) 0 0
\(37\) 57.2381 0.254321 0.127161 − 0.991882i \(-0.459414\pi\)
0.127161 + 0.991882i \(0.459414\pi\)
\(38\) 0 0
\(39\) 196.489 0.806753
\(40\) 0 0
\(41\) −247.640 −0.943288 −0.471644 − 0.881789i \(-0.656339\pi\)
−0.471644 + 0.881789i \(0.656339\pi\)
\(42\) 0 0
\(43\) −353.035 −1.25203 −0.626016 − 0.779810i \(-0.715315\pi\)
−0.626016 + 0.779810i \(0.715315\pi\)
\(44\) 0 0
\(45\) 35.5817 0.117871
\(46\) 0 0
\(47\) −198.559 −0.616230 −0.308115 − 0.951349i \(-0.599698\pi\)
−0.308115 + 0.951349i \(0.599698\pi\)
\(48\) 0 0
\(49\) 259.534 0.756660
\(50\) 0 0
\(51\) 939.594 2.57979
\(52\) 0 0
\(53\) −101.511 −0.263088 −0.131544 − 0.991310i \(-0.541993\pi\)
−0.131544 + 0.991310i \(0.541993\pi\)
\(54\) 0 0
\(55\) −6.33128 −0.0155220
\(56\) 0 0
\(57\) 984.745 2.28829
\(58\) 0 0
\(59\) −240.646 −0.531007 −0.265504 − 0.964110i \(-0.585538\pi\)
−0.265504 + 0.964110i \(0.585538\pi\)
\(60\) 0 0
\(61\) −43.8717 −0.0920852 −0.0460426 − 0.998939i \(-0.514661\pi\)
−0.0460426 + 0.998939i \(0.514661\pi\)
\(62\) 0 0
\(63\) 1517.47 3.03465
\(64\) 0 0
\(65\) −12.0000 −0.0228987
\(66\) 0 0
\(67\) −631.321 −1.15117 −0.575583 − 0.817743i \(-0.695225\pi\)
−0.575583 + 0.817743i \(0.695225\pi\)
\(68\) 0 0
\(69\) −1628.18 −2.84073
\(70\) 0 0
\(71\) 38.2856 0.0639953 0.0319976 − 0.999488i \(-0.489813\pi\)
0.0319976 + 0.999488i \(0.489813\pi\)
\(72\) 0 0
\(73\) 388.617 0.623071 0.311535 − 0.950235i \(-0.399157\pi\)
0.311535 + 0.950235i \(0.399157\pi\)
\(74\) 0 0
\(75\) 1174.93 1.80893
\(76\) 0 0
\(77\) −270.012 −0.399620
\(78\) 0 0
\(79\) −1219.83 −1.73723 −0.868615 − 0.495488i \(-0.834989\pi\)
−0.868615 + 0.495488i \(0.834989\pi\)
\(80\) 0 0
\(81\) 1423.56 1.95276
\(82\) 0 0
\(83\) 119.197 0.157633 0.0788166 − 0.996889i \(-0.474886\pi\)
0.0788166 + 0.996889i \(0.474886\pi\)
\(84\) 0 0
\(85\) −57.3831 −0.0732244
\(86\) 0 0
\(87\) −1298.93 −1.60069
\(88\) 0 0
\(89\) −520.973 −0.620483 −0.310242 − 0.950658i \(-0.600410\pi\)
−0.310242 + 0.950658i \(0.600410\pi\)
\(90\) 0 0
\(91\) −511.768 −0.589537
\(92\) 0 0
\(93\) 1936.01 2.15865
\(94\) 0 0
\(95\) −60.1406 −0.0649505
\(96\) 0 0
\(97\) −335.101 −0.350767 −0.175383 − 0.984500i \(-0.556116\pi\)
−0.175383 + 0.984500i \(0.556116\pi\)
\(98\) 0 0
\(99\) −680.018 −0.690348
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 176.4.a.g.1.1 2
3.2 odd 2 1584.4.a.z.1.2 2
4.3 odd 2 44.4.a.b.1.2 ✓ 2
8.3 odd 2 704.4.a.m.1.1 2
8.5 even 2 704.4.a.r.1.2 2
11.10 odd 2 1936.4.a.o.1.1 2
12.11 even 2 396.4.a.i.1.2 2
20.3 even 4 1100.4.b.e.749.4 4
20.7 even 4 1100.4.b.e.749.1 4
20.19 odd 2 1100.4.a.e.1.1 2
28.27 even 2 2156.4.a.d.1.1 2
44.43 even 2 484.4.a.e.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
44.4.a.b.1.2 ✓ 2 4.3 odd 2
176.4.a.g.1.1 2 1.1 even 1 trivial
396.4.a.i.1.2 2 12.11 even 2
484.4.a.e.1.2 2 44.43 even 2
704.4.a.m.1.1 2 8.3 odd 2
704.4.a.r.1.2 2 8.5 even 2
1100.4.a.e.1.1 2 20.19 odd 2
1100.4.b.e.749.1 4 20.7 even 4
1100.4.b.e.749.4 4 20.3 even 4
1584.4.a.z.1.2 2 3.2 odd 2
1936.4.a.o.1.1 2 11.10 odd 2
2156.4.a.d.1.1 2 28.27 even 2