Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [112,5,Mod(43,112)] 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("112.43"); S:= CuspForms(chi, 5); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(112, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([2, 1, 0])) N = Newforms(chi, 5, names="a")
 
Level: \( N \) \(=\) \( 112 = 2^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 112.k (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(11.5774358654\)
Analytic rank: \(0\)
Dimension: \(96\)
Relative dimension: \(48\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 99.6
Character \(\chi\) \(=\) 112.99
Dual form 112.5.k.a.43.6

$q$-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+(-3.77066 + 1.33496i) q^{2} +(10.8491 - 10.8491i) q^{3} +(12.4357 - 10.0674i) q^{4} +(-31.7296 + 31.7296i) q^{5} +(-26.4251 + 55.3914i) q^{6} +18.5203 q^{7} +(-33.4513 + 54.5620i) q^{8} -154.406i q^{9} +(77.2836 - 161.999i) q^{10} +(-126.465 - 126.465i) q^{11} +(25.6944 - 244.139i) q^{12} +(-44.9589 - 44.9589i) q^{13} +(-69.8336 + 24.7239i) q^{14} +688.475i q^{15} +(53.2951 - 250.391i) q^{16} -110.269 q^{17} +(206.127 + 582.213i) q^{18} +(-125.611 + 125.611i) q^{19} +(-75.1466 + 714.015i) q^{20} +(200.928 - 200.928i) q^{21} +(645.681 + 308.029i) q^{22} -915.524 q^{23} +(229.032 + 954.865i) q^{24} -1388.54i q^{25} +(229.543 + 109.506i) q^{26} +(-796.390 - 796.390i) q^{27} +(230.313 - 186.451i) q^{28} +(-459.446 - 459.446i) q^{29} +(-919.090 - 2596.01i) q^{30} +1611.67i q^{31} +(133.305 + 1015.29i) q^{32} -2744.06 q^{33} +(415.788 - 147.206i) q^{34} +(-587.640 + 587.640i) q^{35} +(-1554.47 - 1920.15i) q^{36} +(964.893 - 964.893i) q^{37} +(305.950 - 641.323i) q^{38} -975.527 q^{39} +(-669.833 - 2792.63i) q^{40} +936.914i q^{41} +(-489.400 + 1025.86i) q^{42} +(-895.948 - 895.948i) q^{43} +(-2845.85 - 299.512i) q^{44} +(4899.24 + 4899.24i) q^{45} +(3452.13 - 1222.19i) q^{46} -1582.55i q^{47} +(-2138.31 - 3294.72i) q^{48} +343.000 q^{49} +(1853.65 + 5235.69i) q^{50} +(-1196.32 + 1196.32i) q^{51} +(-1011.72 - 106.478i) q^{52} +(-909.872 + 909.872i) q^{53} +(4066.07 + 1939.76i) q^{54} +8025.34 q^{55} +(-619.527 + 1010.50i) q^{56} +2725.53i q^{57} +(2345.76 + 1119.07i) q^{58} +(-950.591 - 950.591i) q^{59} +(6931.15 + 8561.70i) q^{60} +(-658.183 - 658.183i) q^{61} +(-2151.52 - 6077.05i) q^{62} -2859.64i q^{63} +(-1858.02 - 3650.34i) q^{64} +2853.05 q^{65} +(10346.9 - 3663.22i) q^{66} +(-2974.42 + 2974.42i) q^{67} +(-1371.28 + 1110.12i) q^{68} +(-9932.62 + 9932.62i) q^{69} +(1431.31 - 3000.27i) q^{70} +7205.11 q^{71} +(8424.70 + 5165.08i) q^{72} -2350.55i q^{73} +(-2350.19 + 4926.38i) q^{74} +(-15064.4 - 15064.4i) q^{75} +(-297.490 + 2826.64i) q^{76} +(-2342.16 - 2342.16i) q^{77} +(3678.38 - 1302.29i) q^{78} -341.141i q^{79} +(6253.77 + 9635.84i) q^{80} -4773.34 q^{81} +(-1250.75 - 3532.78i) q^{82} +(6304.16 - 6304.16i) q^{83} +(475.867 - 4521.51i) q^{84} +(3498.80 - 3498.80i) q^{85} +(4574.37 + 2182.25i) q^{86} -9969.15 q^{87} +(11130.6 - 2669.75i) q^{88} -4326.70i q^{89} +(-25013.7 - 11933.1i) q^{90} +(-832.650 - 832.650i) q^{91} +(-11385.2 + 9216.95i) q^{92} +(17485.2 + 17485.2i) q^{93} +(2112.64 + 5967.25i) q^{94} -7971.18i q^{95} +(12461.2 + 9568.70i) q^{96} +3632.54 q^{97} +(-1293.34 + 457.893i) q^{98} +(-19526.9 + 19526.9i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 96 q + 6 q^{4} - 132 q^{6} - 200 q^{10} - 96 q^{11} + 660 q^{12} - 294 q^{14} + 642 q^{16} - 810 q^{18} + 1408 q^{19} - 2604 q^{20} - 106 q^{22} - 1152 q^{23} + 3880 q^{24} + 6828 q^{26} + 3648 q^{27} - 864 q^{29}+ \cdots - 59552 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/112\mathbb{Z}\right)^\times\).

\(n\) \(15\) \(17\) \(85\)
\(\chi(n)\) \(-1\) \(1\) \(e\left(\frac{3}{4}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −3.77066 + 1.33496i −0.942665 + 0.333741i
\(3\) 10.8491 − 10.8491i 1.20546 − 1.20546i 0.232973 − 0.972483i \(-0.425155\pi\)
0.972483 − 0.232973i \(-0.0748452\pi\)
\(4\) 12.4357 − 10.0674i 0.777234 − 0.629212i
\(5\) −31.7296 + 31.7296i −1.26918 + 1.26918i −0.322674 + 0.946510i \(0.604582\pi\)
−0.946510 + 0.322674i \(0.895418\pi\)
\(6\) −26.4251 + 55.3914i −0.734030 + 1.53865i
\(7\) 18.5203 0.377964
\(8\) −33.4513 + 54.5620i −0.522677 + 0.852531i
\(9\) − 154.406i − 1.90625i
\(10\) 77.2836 − 161.999i 0.772836 − 1.61999i
\(11\) −126.465 − 126.465i −1.04516 − 1.04516i −0.998931 − 0.0462313i \(-0.985279\pi\)
−0.0462313 − 0.998931i \(-0.514721\pi\)
\(12\) 25.6944 − 244.139i 0.178433 − 1.69541i
\(13\) −44.9589 − 44.9589i −0.266029 − 0.266029i 0.561469 − 0.827498i \(-0.310236\pi\)
−0.827498 + 0.561469i \(0.810236\pi\)
\(14\) −69.8336 + 24.7239i −0.356294 + 0.126142i
\(15\) 688.475i 3.05989i
\(16\) 53.2951 − 250.391i 0.208184 − 0.978090i
\(17\) −110.269 −0.381555 −0.190777 − 0.981633i \(-0.561101\pi\)
−0.190777 + 0.981633i \(0.561101\pi\)
\(18\) 206.127 + 582.213i 0.636193 + 1.79695i
\(19\) −125.611 + 125.611i −0.347953 + 0.347953i −0.859347 − 0.511394i \(-0.829129\pi\)
0.511394 + 0.859347i \(0.329129\pi\)
\(20\) −75.1466 + 714.015i −0.187866 + 1.78504i
\(21\) 200.928 − 200.928i 0.455619 − 0.455619i
\(22\) 645.681 + 308.029i 1.33405 + 0.636424i
\(23\) −915.524 −1.73067 −0.865335 − 0.501194i \(-0.832894\pi\)
−0.865335 + 0.501194i \(0.832894\pi\)
\(24\) 229.032 + 954.865i 0.397625 + 1.65775i
\(25\) − 1388.54i − 2.22166i
\(26\) 229.543 + 109.506i 0.339561 + 0.161991i
\(27\) −796.390 − 796.390i −1.09244 − 1.09244i
\(28\) 230.313 − 186.451i 0.293767 − 0.237820i
\(29\) −459.446 − 459.446i −0.546309 − 0.546309i 0.379062 − 0.925371i \(-0.376247\pi\)
−0.925371 + 0.379062i \(0.876247\pi\)
\(30\) −919.090 − 2596.01i −1.02121 − 2.88445i
\(31\) 1611.67i 1.67707i 0.544844 + 0.838537i \(0.316589\pi\)
−0.544844 + 0.838537i \(0.683411\pi\)
\(32\) 133.305 + 1015.29i 0.130181 + 0.991490i
\(33\) −2744.06 −2.51979
\(34\) 415.788 − 147.206i 0.359678 − 0.127340i
\(35\) −587.640 + 587.640i −0.479706 + 0.479706i
\(36\) −1554.47 − 1920.15i −1.19943 − 1.48160i
\(37\) 964.893 − 964.893i 0.704816 − 0.704816i −0.260624 − 0.965440i \(-0.583928\pi\)
0.965440 + 0.260624i \(0.0839283\pi\)
\(38\) 305.950 − 641.323i 0.211877 − 0.444129i
\(39\) −975.527 −0.641372
\(40\) −669.833 − 2792.63i −0.418646 − 1.74539i
\(41\) 936.914i 0.557355i 0.960385 + 0.278678i \(0.0898961\pi\)
−0.960385 + 0.278678i \(0.910104\pi\)
\(42\) −489.400 + 1025.86i −0.277437 + 0.581555i
\(43\) −895.948 − 895.948i −0.484558 − 0.484558i 0.422026 − 0.906584i \(-0.361319\pi\)
−0.906584 + 0.422026i \(0.861319\pi\)
\(44\) −2845.85 − 299.512i −1.46996 − 0.154706i
\(45\) 4899.24 + 4899.24i 2.41938 + 2.41938i
\(46\) 3452.13 − 1222.19i 1.63144 − 0.577596i
\(47\) − 1582.55i − 0.716409i −0.933643 − 0.358204i \(-0.883389\pi\)
0.933643 − 0.358204i \(-0.116611\pi\)
\(48\) −2138.31 − 3294.72i −0.928087 − 1.43000i
\(49\) 343.000 0.142857
\(50\) 1853.65 + 5235.69i 0.741458 + 2.09428i
\(51\) −1196.32 + 1196.32i −0.459947 + 0.459947i
\(52\) −1011.72 − 106.478i −0.374155 − 0.0393780i
\(53\) −909.872 + 909.872i −0.323913 + 0.323913i −0.850266 − 0.526353i \(-0.823559\pi\)
0.526353 + 0.850266i \(0.323559\pi\)
\(54\) 4066.07 + 1939.76i 1.39440 + 0.665213i
\(55\) 8025.34 2.65301
\(56\) −619.527 + 1010.50i −0.197553 + 0.322226i
\(57\) 2725.53i 0.838884i
\(58\) 2345.76 + 1119.07i 0.697312 + 0.332660i
\(59\) −950.591 − 950.591i −0.273080 − 0.273080i 0.557259 − 0.830339i \(-0.311853\pi\)
−0.830339 + 0.557259i \(0.811853\pi\)
\(60\) 6931.15 + 8561.70i 1.92532 + 2.37825i
\(61\) −658.183 − 658.183i −0.176883 − 0.176883i 0.613112 − 0.789996i \(-0.289917\pi\)
−0.789996 + 0.613112i \(0.789917\pi\)
\(62\) −2151.52 − 6077.05i −0.559709 − 1.58092i
\(63\) − 2859.64i − 0.720494i
\(64\) −1858.02 − 3650.34i −0.453618 − 0.891196i
\(65\) 2853.05 0.675279
\(66\) 10346.9 − 3663.22i 2.37532 − 0.840959i
\(67\) −2974.42 + 2974.42i −0.662603 + 0.662603i −0.955993 − 0.293390i \(-0.905217\pi\)
0.293390 + 0.955993i \(0.405217\pi\)
\(68\) −1371.28 + 1110.12i −0.296557 + 0.240079i
\(69\) −9932.62 + 9932.62i −2.08625 + 2.08625i
\(70\) 1431.31 − 3000.27i 0.292105 − 0.612300i
\(71\) 7205.11 1.42930 0.714651 − 0.699481i \(-0.246586\pi\)
0.714651 + 0.699481i \(0.246586\pi\)
\(72\) 8424.70 + 5165.08i 1.62514 + 0.996351i
\(73\) − 2350.55i − 0.441087i −0.975377 − 0.220543i \(-0.929217\pi\)
0.975377 − 0.220543i \(-0.0707830\pi\)
\(74\) −2350.19 + 4926.38i −0.429179 + 0.899632i
\(75\) −15064.4 − 15064.4i −2.67811 − 2.67811i
\(76\) −297.490 + 2826.64i −0.0515045 + 0.489377i
\(77\) −2342.16 − 2342.16i −0.395034 − 0.395034i
\(78\) 3678.38 − 1302.29i 0.604599 − 0.214052i
\(79\) − 341.141i − 0.0546613i −0.999626 − 0.0273306i \(-0.991299\pi\)
0.999626 − 0.0273306i \(-0.00870069\pi\)
\(80\) 6253.77 + 9635.84i 0.977152 + 1.50560i
\(81\) −4773.34 −0.727533
\(82\) −1250.75 − 3532.78i −0.186012 − 0.525399i
\(83\) 6304.16 − 6304.16i 0.915105 − 0.915105i −0.0815627 − 0.996668i \(-0.525991\pi\)
0.996668 + 0.0815627i \(0.0259911\pi\)
\(84\) 475.867 − 4521.51i 0.0674415 − 0.640804i
\(85\) 3498.80 − 3498.80i 0.484263 − 0.484263i
\(86\) 4574.37 + 2182.25i 0.618493 + 0.295059i
\(87\) −9969.15 −1.31710
\(88\) 11130.6 − 2669.75i 1.43731 − 0.344751i
\(89\) − 4326.70i − 0.546231i −0.961981 − 0.273115i \(-0.911946\pi\)
0.961981 − 0.273115i \(-0.0880541\pi\)
\(90\) −25013.7 − 11933.1i −3.08811 − 1.47322i
\(91\) −832.650 − 832.650i −0.100549 − 0.100549i
\(92\) −11385.2 + 9216.95i −1.34513 + 1.08896i
\(93\) 17485.2 + 17485.2i 2.02164 + 2.02164i
\(94\) 2112.64 + 5967.25i 0.239095 + 0.675333i
\(95\) − 7971.18i − 0.883233i
\(96\) 12461.2 + 9568.70i 1.35213 + 1.03827i
\(97\) 3632.54 0.386070 0.193035 − 0.981192i \(-0.438167\pi\)
0.193035 + 0.981192i \(0.438167\pi\)
\(98\) −1293.34 + 457.893i −0.134666 + 0.0476773i
\(99\) −19526.9 + 19526.9i −1.99234 + 1.99234i
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 112.5.k.a.99.6 yes 96
4.3 odd 2 448.5.k.a.239.4 96
16.5 even 4 448.5.k.a.15.4 96
16.11 odd 4 inner 112.5.k.a.43.6 ✓ 96
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
112.5.k.a.43.6 ✓ 96 16.11 odd 4 inner
112.5.k.a.99.6 yes 96 1.1 even 1 trivial
448.5.k.a.15.4 96 16.5 even 4
448.5.k.a.239.4 96 4.3 odd 2