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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 43.6
Character \(\chi\) \(=\) 112.43
Dual form 112.5.k.a.99.6

$q$-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 \(\left(\mathbb{Z}/112\mathbb{Z}\right)^\times\).

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

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −3.77066 − 1.33496i −0.942665 − 0.333741i
\(3\) 10.8491 + 10.8491i 1.20546 + 1.20546i 0.972483 + 0.232973i \(0.0748452\pi\)
0.232973 + 0.972483i \(0.425155\pi\)
\(4\) 12.4357 + 10.0674i 0.777234 + 0.629212i
\(5\) −31.7296 − 31.7296i −1.26918 − 1.26918i −0.946510 − 0.322674i \(-0.895418\pi\)
−0.322674 − 0.946510i \(-0.604582\pi\)
\(6\) −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 −0.0462313 + 0.998931i \(0.514721\pi\)
−0.998931 + 0.0462313i \(0.985279\pi\)
\(12\) 25.6944 + 244.139i 0.178433 + 1.69541i
\(13\) −44.9589 + 44.9589i −0.266029 + 0.266029i −0.827498 − 0.561469i \(-0.810236\pi\)
0.561469 + 0.827498i \(0.310236\pi\)
\(14\) −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 0.511394 − 0.859347i \(-0.329129\pi\)
−0.859347 + 0.511394i \(0.829129\pi\)
\(20\) −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 −0.925371 − 0.379062i \(-0.876247\pi\)
0.379062 + 0.925371i \(0.376247\pi\)
\(30\) −919.090 + 2596.01i −1.02121 + 2.88445i
\(31\) − 1611.67i − 1.67707i −0.544844 − 0.838537i \(-0.683411\pi\)
0.544844 − 0.838537i \(-0.316589\pi\)
\(32\) 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 0.965440 − 0.260624i \(-0.0839283\pi\)
−0.260624 + 0.965440i \(0.583928\pi\)
\(38\) 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 \(-0.910104\pi\)
0.960385 − 0.278678i \(-0.0898961\pi\)
\(42\) −489.400 − 1025.86i −0.277437 − 0.581555i
\(43\) −895.948 + 895.948i −0.484558 + 0.484558i −0.906584 − 0.422026i \(-0.861319\pi\)
0.422026 + 0.906584i \(0.361319\pi\)
\(44\) −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 \(0.116611\pi\)
−0.933643 + 0.358204i \(0.883389\pi\)
\(48\) −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 0.526353 − 0.850266i \(-0.323559\pi\)
−0.850266 + 0.526353i \(0.823559\pi\)
\(54\) 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 −0.830339 − 0.557259i \(-0.811853\pi\)
0.557259 + 0.830339i \(0.311853\pi\)
\(60\) 6931.15 − 8561.70i 1.92532 − 2.37825i
\(61\) −658.183 + 658.183i −0.176883 + 0.176883i −0.789996 − 0.613112i \(-0.789917\pi\)
0.613112 + 0.789996i \(0.289917\pi\)
\(62\) −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 0.293390 − 0.955993i \(-0.405217\pi\)
−0.955993 + 0.293390i \(0.905217\pi\)
\(68\) −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 \(0.0707830\pi\)
−0.975377 + 0.220543i \(0.929217\pi\)
\(74\) −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 \(0.00870069\pi\)
−0.999626 + 0.0273306i \(0.991299\pi\)
\(80\) 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 0.996668 − 0.0815627i \(-0.0259911\pi\)
−0.0815627 + 0.996668i \(0.525991\pi\)
\(84\) 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 \(0.0880541\pi\)
−0.961981 + 0.273115i \(0.911946\pi\)
\(90\) −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.43.6 ✓ 96
4.3 odd 2 448.5.k.a.15.4 96
16.3 odd 4 inner 112.5.k.a.99.6 yes 96
16.13 even 4 448.5.k.a.239.4 96
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
112.5.k.a.43.6 ✓ 96 1.1 even 1 trivial
112.5.k.a.99.6 yes 96 16.3 odd 4 inner
448.5.k.a.15.4 96 4.3 odd 2
448.5.k.a.239.4 96 16.13 even 4