Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [91,2,Mod(22,91)] 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("91.22"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(91, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 4])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 91 = 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 91.f (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,-3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(0.726638658394\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{5})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} + 2x^{2} + x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 29.1
Root \(0.809017 + 1.40126i\) of defining polynomial
Character \(\chi\) \(=\) 91.29
Dual form 91.2.f.a.22.1

$q$-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+(-1.30902 - 2.26728i) q^{2} +(1.30902 + 2.26728i) q^{3} +(-2.42705 + 4.20378i) q^{4} +2.61803 q^{5} +(3.42705 - 5.93583i) q^{6} +(-0.500000 + 0.866025i) q^{7} +7.47214 q^{8} +(-1.92705 + 3.33775i) q^{9} +(-3.42705 - 5.93583i) q^{10} +(-0.927051 - 1.60570i) q^{11} -12.7082 q^{12} +(-2.50000 - 2.59808i) q^{13} +2.61803 q^{14} +(3.42705 + 5.93583i) q^{15} +(-4.92705 - 8.53390i) q^{16} +(0.736068 - 1.27491i) q^{17} +10.0902 q^{18} +(0.927051 - 1.60570i) q^{19} +(-6.35410 + 11.0056i) q^{20} -2.61803 q^{21} +(-2.42705 + 4.20378i) q^{22} +(2.23607 + 3.87298i) q^{23} +(9.78115 + 16.9415i) q^{24} +1.85410 q^{25} +(-2.61803 + 9.06914i) q^{26} -2.23607 q^{27} +(-2.42705 - 4.20378i) q^{28} +(-3.54508 - 6.14027i) q^{29} +(8.97214 - 15.5402i) q^{30} -4.70820 q^{31} +(-5.42705 + 9.39993i) q^{32} +(2.42705 - 4.20378i) q^{33} -3.85410 q^{34} +(-1.30902 + 2.26728i) q^{35} +(-9.35410 - 16.2018i) q^{36} +(-2.00000 - 3.46410i) q^{37} -4.85410 q^{38} +(2.61803 - 9.06914i) q^{39} +19.5623 q^{40} +(-0.381966 - 0.661585i) q^{41} +(3.42705 + 5.93583i) q^{42} +(-6.28115 + 10.8793i) q^{43} +9.00000 q^{44} +(-5.04508 + 8.73834i) q^{45} +(5.85410 - 10.1396i) q^{46} -2.23607 q^{47} +(12.8992 - 22.3420i) q^{48} +(-0.500000 - 0.866025i) q^{49} +(-2.42705 - 4.20378i) q^{50} +3.85410 q^{51} +(16.9894 - 4.20378i) q^{52} +3.76393 q^{53} +(2.92705 + 5.06980i) q^{54} +(-2.42705 - 4.20378i) q^{55} +(-3.73607 + 6.47106i) q^{56} +4.85410 q^{57} +(-9.28115 + 16.0754i) q^{58} +(1.11803 - 1.93649i) q^{59} -33.2705 q^{60} +(3.00000 - 5.19615i) q^{61} +(6.16312 + 10.6748i) q^{62} +(-1.92705 - 3.33775i) q^{63} +8.70820 q^{64} +(-6.54508 - 6.80185i) q^{65} -12.7082 q^{66} +(6.35410 + 11.0056i) q^{67} +(3.57295 + 6.18853i) q^{68} +(-5.85410 + 10.1396i) q^{69} +6.85410 q^{70} +(7.09017 - 12.2805i) q^{71} +(-14.3992 + 24.9401i) q^{72} -2.00000 q^{73} +(-5.23607 + 9.06914i) q^{74} +(2.42705 + 4.20378i) q^{75} +(4.50000 + 7.79423i) q^{76} +1.85410 q^{77} +(-23.9894 + 5.93583i) q^{78} +4.00000 q^{79} +(-12.8992 - 22.3420i) q^{80} +(2.85410 + 4.94345i) q^{81} +(-1.00000 + 1.73205i) q^{82} +6.70820 q^{83} +(6.35410 - 11.0056i) q^{84} +(1.92705 - 3.33775i) q^{85} +32.8885 q^{86} +(9.28115 - 16.0754i) q^{87} +(-6.92705 - 11.9980i) q^{88} +(-2.45492 - 4.25204i) q^{89} +26.4164 q^{90} +(3.50000 - 0.866025i) q^{91} -21.7082 q^{92} +(-6.16312 - 10.6748i) q^{93} +(2.92705 + 5.06980i) q^{94} +(2.42705 - 4.20378i) q^{95} -28.4164 q^{96} +(-9.42705 + 16.3281i) q^{97} +(-1.30902 + 2.26728i) q^{98} +7.14590 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 3 q^{2} + 3 q^{3} - 3 q^{4} + 6 q^{5} + 7 q^{6} - 2 q^{7} + 12 q^{8} - q^{9} - 7 q^{10} + 3 q^{11} - 24 q^{12} - 10 q^{13} + 6 q^{14} + 7 q^{15} - 13 q^{16} - 6 q^{17} + 18 q^{18} - 3 q^{19} - 12 q^{20}+ \cdots + 42 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/91\mathbb{Z}\right)^\times\).

\(n\) \(15\) \(66\)
\(\chi(n)\) \(e\left(\frac{1}{3}\right)\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.30902 − 2.26728i −0.925615 − 1.60321i −0.790569 − 0.612372i \(-0.790215\pi\)
−0.135045 − 0.990839i \(-0.543118\pi\)
\(3\) 1.30902 + 2.26728i 0.755761 + 1.30902i 0.944995 + 0.327085i \(0.106066\pi\)
−0.189234 + 0.981932i \(0.560600\pi\)
\(4\) −2.42705 + 4.20378i −1.21353 + 2.10189i
\(5\) 2.61803 1.17082 0.585410 − 0.810737i \(-0.300933\pi\)
0.585410 + 0.810737i \(0.300933\pi\)
\(6\) 3.42705 − 5.93583i 1.39909 − 2.42329i
\(7\) −0.500000 + 0.866025i −0.188982 + 0.327327i
\(8\) 7.47214 2.64180
\(9\) −1.92705 + 3.33775i −0.642350 + 1.11258i
\(10\) −3.42705 − 5.93583i −1.08373 − 1.87707i
\(11\) −0.927051 − 1.60570i −0.279516 − 0.484137i 0.691748 − 0.722139i \(-0.256841\pi\)
−0.971265 + 0.238002i \(0.923507\pi\)
\(12\) −12.7082 −3.66854
\(13\) −2.50000 − 2.59808i −0.693375 − 0.720577i
\(14\) 2.61803 0.699699
\(15\) 3.42705 + 5.93583i 0.884861 + 1.53262i
\(16\) −4.92705 − 8.53390i −1.23176 − 2.13348i
\(17\) 0.736068 − 1.27491i 0.178523 − 0.309210i −0.762852 − 0.646573i \(-0.776202\pi\)
0.941375 + 0.337363i \(0.109535\pi\)
\(18\) 10.0902 2.37828
\(19\) 0.927051 − 1.60570i 0.212680 − 0.368373i −0.739872 − 0.672747i \(-0.765114\pi\)
0.952552 + 0.304375i \(0.0984475\pi\)
\(20\) −6.35410 + 11.0056i −1.42082 + 2.46093i
\(21\) −2.61803 −0.571302
\(22\) −2.42705 + 4.20378i −0.517449 + 0.896248i
\(23\) 2.23607 + 3.87298i 0.466252 + 0.807573i 0.999257 − 0.0385394i \(-0.0122705\pi\)
−0.533005 + 0.846112i \(0.678937\pi\)
\(24\) 9.78115 + 16.9415i 1.99657 + 3.45816i
\(25\) 1.85410 0.370820
\(26\) −2.61803 + 9.06914i −0.513439 + 1.77860i
\(27\) −2.23607 −0.430331
\(28\) −2.42705 − 4.20378i −0.458670 − 0.794439i
\(29\) −3.54508 − 6.14027i −0.658306 − 1.14022i −0.981054 − 0.193734i \(-0.937940\pi\)
0.322748 − 0.946485i \(-0.395393\pi\)
\(30\) 8.97214 − 15.5402i 1.63808 − 2.83724i
\(31\) −4.70820 −0.845618 −0.422809 − 0.906219i \(-0.638956\pi\)
−0.422809 + 0.906219i \(0.638956\pi\)
\(32\) −5.42705 + 9.39993i −0.959376 + 1.66169i
\(33\) 2.42705 − 4.20378i 0.422495 − 0.731783i
\(34\) −3.85410 −0.660973
\(35\) −1.30902 + 2.26728i −0.221264 + 0.383241i
\(36\) −9.35410 − 16.2018i −1.55902 − 2.70030i
\(37\) −2.00000 − 3.46410i −0.328798 − 0.569495i 0.653476 − 0.756948i \(-0.273310\pi\)
−0.982274 + 0.187453i \(0.939977\pi\)
\(38\) −4.85410 −0.787439
\(39\) 2.61803 − 9.06914i 0.419221 − 1.45222i
\(40\) 19.5623 3.09307
\(41\) −0.381966 − 0.661585i −0.0596531 − 0.103322i 0.834657 − 0.550771i \(-0.185666\pi\)
−0.894310 + 0.447449i \(0.852333\pi\)
\(42\) 3.42705 + 5.93583i 0.528805 + 0.915918i
\(43\) −6.28115 + 10.8793i −0.957867 + 1.65907i −0.230200 + 0.973143i \(0.573938\pi\)
−0.727667 + 0.685931i \(0.759395\pi\)
\(44\) 9.00000 1.35680
\(45\) −5.04508 + 8.73834i −0.752077 + 1.30264i
\(46\) 5.85410 − 10.1396i 0.863140 − 1.49500i
\(47\) −2.23607 −0.326164 −0.163082 − 0.986613i \(-0.552144\pi\)
−0.163082 + 0.986613i \(0.552144\pi\)
\(48\) 12.8992 − 22.3420i 1.86184 − 3.22480i
\(49\) −0.500000 − 0.866025i −0.0714286 − 0.123718i
\(50\) −2.42705 − 4.20378i −0.343237 − 0.594504i
\(51\) 3.85410 0.539682
\(52\) 16.9894 − 4.20378i 2.35600 − 0.582959i
\(53\) 3.76393 0.517016 0.258508 − 0.966009i \(-0.416769\pi\)
0.258508 + 0.966009i \(0.416769\pi\)
\(54\) 2.92705 + 5.06980i 0.398321 + 0.689913i
\(55\) −2.42705 − 4.20378i −0.327263 − 0.566837i
\(56\) −3.73607 + 6.47106i −0.499253 + 0.864732i
\(57\) 4.85410 0.642942
\(58\) −9.28115 + 16.0754i −1.21868 + 2.11081i
\(59\) 1.11803 − 1.93649i 0.145556 − 0.252110i −0.784024 − 0.620730i \(-0.786836\pi\)
0.929580 + 0.368620i \(0.120170\pi\)
\(60\) −33.2705 −4.29520
\(61\) 3.00000 − 5.19615i 0.384111 − 0.665299i −0.607535 − 0.794293i \(-0.707841\pi\)
0.991645 + 0.128994i \(0.0411748\pi\)
\(62\) 6.16312 + 10.6748i 0.782717 + 1.35571i
\(63\) −1.92705 − 3.33775i −0.242786 − 0.420517i
\(64\) 8.70820 1.08853
\(65\) −6.54508 − 6.80185i −0.811818 − 0.843666i
\(66\) −12.7082 −1.56427
\(67\) 6.35410 + 11.0056i 0.776277 + 1.34455i 0.934074 + 0.357080i \(0.116228\pi\)
−0.157797 + 0.987472i \(0.550439\pi\)
\(68\) 3.57295 + 6.18853i 0.433284 + 0.750469i
\(69\) −5.85410 + 10.1396i −0.704751 + 1.22066i
\(70\) 6.85410 0.819222
\(71\) 7.09017 − 12.2805i 0.841448 − 1.45743i −0.0472218 − 0.998884i \(-0.515037\pi\)
0.888670 − 0.458547i \(-0.151630\pi\)
\(72\) −14.3992 + 24.9401i −1.69696 + 2.93922i
\(73\) −2.00000 −0.234082 −0.117041 − 0.993127i \(-0.537341\pi\)
−0.117041 + 0.993127i \(0.537341\pi\)
\(74\) −5.23607 + 9.06914i −0.608681 + 1.05427i
\(75\) 2.42705 + 4.20378i 0.280252 + 0.485410i
\(76\) 4.50000 + 7.79423i 0.516185 + 0.894059i
\(77\) 1.85410 0.211295
\(78\) −23.9894 + 5.93583i −2.71626 + 0.672100i
\(79\) 4.00000 0.450035 0.225018 − 0.974355i \(-0.427756\pi\)
0.225018 + 0.974355i \(0.427756\pi\)
\(80\) −12.8992 − 22.3420i −1.44217 − 2.49792i
\(81\) 2.85410 + 4.94345i 0.317122 + 0.549272i
\(82\) −1.00000 + 1.73205i −0.110432 + 0.191273i
\(83\) 6.70820 0.736321 0.368161 − 0.929762i \(-0.379988\pi\)
0.368161 + 0.929762i \(0.379988\pi\)
\(84\) 6.35410 − 11.0056i 0.693289 − 1.20081i
\(85\) 1.92705 − 3.33775i 0.209018 − 0.362030i
\(86\) 32.8885 3.54646
\(87\) 9.28115 − 16.0754i 0.995044 − 1.72347i
\(88\) −6.92705 − 11.9980i −0.738426 − 1.27899i
\(89\) −2.45492 − 4.25204i −0.260220 − 0.450715i 0.706080 − 0.708132i \(-0.250462\pi\)
−0.966300 + 0.257417i \(0.917129\pi\)
\(90\) 26.4164 2.78453
\(91\) 3.50000 − 0.866025i 0.366900 − 0.0907841i
\(92\) −21.7082 −2.26324
\(93\) −6.16312 − 10.6748i −0.639086 − 1.10693i
\(94\) 2.92705 + 5.06980i 0.301902 + 0.522910i
\(95\) 2.42705 − 4.20378i 0.249010 − 0.431298i
\(96\) −28.4164 −2.90024
\(97\) −9.42705 + 16.3281i −0.957172 + 1.65787i −0.227854 + 0.973695i \(0.573171\pi\)
−0.729318 + 0.684175i \(0.760162\pi\)
\(98\) −1.30902 + 2.26728i −0.132231 + 0.229030i
\(99\) 7.14590 0.718190
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 91.2.f.a.29.1 yes 4
3.2 odd 2 819.2.o.c.757.2 4
4.3 odd 2 1456.2.s.h.1121.1 4
7.2 even 3 637.2.g.b.263.1 4
7.3 odd 6 637.2.h.f.471.2 4
7.4 even 3 637.2.h.g.471.2 4
7.5 odd 6 637.2.g.c.263.1 4
7.6 odd 2 637.2.f.c.393.1 4
13.2 odd 12 1183.2.c.c.337.1 4
13.3 even 3 1183.2.a.g.1.2 2
13.9 even 3 inner 91.2.f.a.22.1 ✓ 4
13.10 even 6 1183.2.a.c.1.1 2
13.11 odd 12 1183.2.c.c.337.4 4
39.35 odd 6 819.2.o.c.568.2 4
52.35 odd 6 1456.2.s.h.113.1 4
91.9 even 3 637.2.h.g.165.2 4
91.48 odd 6 637.2.f.c.295.1 4
91.55 odd 6 8281.2.a.bb.1.2 2
91.61 odd 6 637.2.h.f.165.2 4
91.62 odd 6 8281.2.a.n.1.1 2
91.74 even 3 637.2.g.b.373.1 4
91.87 odd 6 637.2.g.c.373.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
91.2.f.a.22.1 ✓ 4 13.9 even 3 inner
91.2.f.a.29.1 yes 4 1.1 even 1 trivial
637.2.f.c.295.1 4 91.48 odd 6
637.2.f.c.393.1 4 7.6 odd 2
637.2.g.b.263.1 4 7.2 even 3
637.2.g.b.373.1 4 91.74 even 3
637.2.g.c.263.1 4 7.5 odd 6
637.2.g.c.373.1 4 91.87 odd 6
637.2.h.f.165.2 4 91.61 odd 6
637.2.h.f.471.2 4 7.3 odd 6
637.2.h.g.165.2 4 91.9 even 3
637.2.h.g.471.2 4 7.4 even 3
819.2.o.c.568.2 4 39.35 odd 6
819.2.o.c.757.2 4 3.2 odd 2
1183.2.a.c.1.1 2 13.10 even 6
1183.2.a.g.1.2 2 13.3 even 3
1183.2.c.c.337.1 4 13.2 odd 12
1183.2.c.c.337.4 4 13.11 odd 12
1456.2.s.h.113.1 4 52.35 odd 6
1456.2.s.h.1121.1 4 4.3 odd 2
8281.2.a.n.1.1 2 91.62 odd 6
8281.2.a.bb.1.2 2 91.55 odd 6