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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 43.15
Character \(\chi\) \(=\) 112.43
Dual form 112.5.k.a.99.15

$q$-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+(-2.35239 - 3.23516i) q^{2} +(-5.21415 - 5.21415i) q^{3} +(-4.93249 + 15.2207i) q^{4} +(19.8037 + 19.8037i) q^{5} +(-4.60286 + 29.1343i) q^{6} -18.5203 q^{7} +(60.8446 - 19.8478i) q^{8} -26.6252i q^{9} +(17.4820 - 110.654i) q^{10} +(15.4917 - 15.4917i) q^{11} +(105.082 - 53.6445i) q^{12} +(94.3650 - 94.3650i) q^{13} +(43.5669 + 59.9160i) q^{14} -206.519i q^{15} +(-207.341 - 150.152i) q^{16} +148.210 q^{17} +(-86.1368 + 62.6330i) q^{18} +(-146.925 - 146.925i) q^{19} +(-399.108 + 203.745i) q^{20} +(96.5675 + 96.5675i) q^{21} +(-86.5609 - 13.6755i) q^{22} -802.059 q^{23} +(-420.742 - 213.764i) q^{24} +159.374i q^{25} +(-527.269 - 83.3020i) q^{26} +(-561.174 + 561.174i) q^{27} +(91.3509 - 281.892i) q^{28} +(567.760 - 567.760i) q^{29} +(-668.122 + 485.814i) q^{30} -1001.93i q^{31} +(1.98257 + 1024.00i) q^{32} -161.553 q^{33} +(-348.649 - 479.483i) q^{34} +(-366.770 - 366.770i) q^{35} +(405.255 + 131.328i) q^{36} +(-545.834 - 545.834i) q^{37} +(-129.700 + 820.950i) q^{38} -984.067 q^{39} +(1598.01 + 811.889i) q^{40} +12.8009i q^{41} +(85.2462 - 539.576i) q^{42} +(806.021 - 806.021i) q^{43} +(159.383 + 312.208i) q^{44} +(527.278 - 527.278i) q^{45} +(1886.76 + 2594.79i) q^{46} -3727.93i q^{47} +(298.193 + 1864.02i) q^{48} +343.000 q^{49} +(515.599 - 374.910i) q^{50} +(-772.790 - 772.790i) q^{51} +(970.850 + 1901.76i) q^{52} +(-2716.07 - 2716.07i) q^{53} +(3135.59 + 495.384i) q^{54} +613.588 q^{55} +(-1126.86 + 367.586i) q^{56} +1532.18i q^{57} +(-3172.39 - 501.198i) q^{58} +(196.912 - 196.912i) q^{59} +(3143.37 + 1018.65i) q^{60} +(1174.86 - 1174.86i) q^{61} +(-3241.41 + 2356.94i) q^{62} +493.106i q^{63} +(3308.13 - 2415.26i) q^{64} +3737.55 q^{65} +(380.035 + 522.648i) q^{66} +(-2275.58 - 2275.58i) q^{67} +(-731.045 + 2255.87i) q^{68} +(4182.06 + 4182.06i) q^{69} +(-323.771 + 2049.35i) q^{70} +9117.59 q^{71} +(-528.451 - 1620.00i) q^{72} +5965.10i q^{73} +(-481.842 + 3049.88i) q^{74} +(830.999 - 830.999i) q^{75} +(2961.01 - 1511.60i) q^{76} +(-286.911 + 286.911i) q^{77} +(2314.91 + 3183.61i) q^{78} +3103.75i q^{79} +(-1132.56 - 7079.69i) q^{80} +3695.46 q^{81} +(41.4130 - 30.1128i) q^{82} +(-3934.02 - 3934.02i) q^{83} +(-1946.14 + 993.510i) q^{84} +(2935.11 + 2935.11i) q^{85} +(-4503.68 - 711.526i) q^{86} -5920.78 q^{87} +(635.112 - 1250.07i) q^{88} +1130.82i q^{89} +(-2946.19 - 465.462i) q^{90} +(-1747.66 + 1747.66i) q^{91} +(3956.14 - 12207.9i) q^{92} +(-5224.23 + 5224.23i) q^{93} +(-12060.4 + 8769.56i) q^{94} -5819.31i q^{95} +(5328.95 - 5349.62i) q^{96} +14246.2 q^{97} +(-806.871 - 1109.66i) q^{98} +(-412.471 - 412.471i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.35239 − 3.23516i −0.588098 − 0.808789i
\(3\) −5.21415 − 5.21415i −0.579350 − 0.579350i 0.355374 − 0.934724i \(-0.384354\pi\)
−0.934724 + 0.355374i \(0.884354\pi\)
\(4\) −4.93249 + 15.2207i −0.308280 + 0.951296i
\(5\) 19.8037 + 19.8037i 0.792148 + 0.792148i 0.981843 − 0.189695i \(-0.0607498\pi\)
−0.189695 + 0.981843i \(0.560750\pi\)
\(6\) −4.60286 + 29.1343i −0.127857 + 0.809287i
\(7\) −18.5203 −0.377964
\(8\) 60.8446 − 19.8478i 0.950697 − 0.310122i
\(9\) − 26.6252i − 0.328706i
\(10\) 17.4820 − 110.654i 0.174820 − 1.10654i
\(11\) 15.4917 − 15.4917i 0.128031 − 0.128031i −0.640188 − 0.768219i \(-0.721143\pi\)
0.768219 + 0.640188i \(0.221143\pi\)
\(12\) 105.082 − 53.6445i 0.729736 − 0.372531i
\(13\) 94.3650 − 94.3650i 0.558373 − 0.558373i −0.370471 − 0.928844i \(-0.620804\pi\)
0.928844 + 0.370471i \(0.120804\pi\)
\(14\) 43.5669 + 59.9160i 0.222280 + 0.305694i
\(15\) − 206.519i − 0.917863i
\(16\) −207.341 − 150.152i −0.809926 − 0.586531i
\(17\) 148.210 0.512838 0.256419 − 0.966566i \(-0.417457\pi\)
0.256419 + 0.966566i \(0.417457\pi\)
\(18\) −86.1368 + 62.6330i −0.265854 + 0.193312i
\(19\) −146.925 − 146.925i −0.406994 − 0.406994i 0.473695 − 0.880689i \(-0.342920\pi\)
−0.880689 + 0.473695i \(0.842920\pi\)
\(20\) −399.108 + 203.745i −0.997771 + 0.509363i
\(21\) 96.5675 + 96.5675i 0.218974 + 0.218974i
\(22\) −86.5609 − 13.6755i −0.178845 − 0.0282552i
\(23\) −802.059 −1.51618 −0.758090 − 0.652150i \(-0.773867\pi\)
−0.758090 + 0.652150i \(0.773867\pi\)
\(24\) −420.742 − 213.764i −0.730456 − 0.371118i
\(25\) 159.374i 0.254998i
\(26\) −527.269 − 83.3020i −0.779984 − 0.123228i
\(27\) −561.174 + 561.174i −0.769786 + 0.769786i
\(28\) 91.3509 − 281.892i 0.116519 − 0.359556i
\(29\) 567.760 − 567.760i 0.675102 − 0.675102i −0.283786 − 0.958888i \(-0.591591\pi\)
0.958888 + 0.283786i \(0.0915906\pi\)
\(30\) −668.122 + 485.814i −0.742358 + 0.539794i
\(31\) − 1001.93i − 1.04259i −0.853375 − 0.521297i \(-0.825448\pi\)
0.853375 − 0.521297i \(-0.174552\pi\)
\(32\) 1.98257 + 1024.00i 0.00193611 + 0.999998i
\(33\) −161.553 −0.148349
\(34\) −348.649 − 479.483i −0.301599 − 0.414778i
\(35\) −366.770 − 366.770i −0.299404 − 0.299404i
\(36\) 405.255 + 131.328i 0.312697 + 0.101334i
\(37\) −545.834 − 545.834i −0.398710 − 0.398710i 0.479068 − 0.877778i \(-0.340975\pi\)
−0.877778 + 0.479068i \(0.840975\pi\)
\(38\) −129.700 + 820.950i −0.0898198 + 0.568525i
\(39\) −984.067 −0.646987
\(40\) 1598.01 + 811.889i 0.998755 + 0.507431i
\(41\) 12.8009i 0.00761506i 0.999993 + 0.00380753i \(0.00121198\pi\)
−0.999993 + 0.00380753i \(0.998788\pi\)
\(42\) 85.2462 − 539.576i 0.0483255 − 0.305882i
\(43\) 806.021 − 806.021i 0.435923 − 0.435923i −0.454715 − 0.890637i \(-0.650259\pi\)
0.890637 + 0.454715i \(0.150259\pi\)
\(44\) 159.383 + 312.208i 0.0823258 + 0.161265i
\(45\) 527.278 − 527.278i 0.260384 − 0.260384i
\(46\) 1886.76 + 2594.79i 0.891663 + 1.22627i
\(47\) − 3727.93i − 1.68761i −0.536650 − 0.843805i \(-0.680310\pi\)
0.536650 − 0.843805i \(-0.319690\pi\)
\(48\) 298.193 + 1864.02i 0.129424 + 0.809038i
\(49\) 343.000 0.142857
\(50\) 515.599 − 374.910i 0.206240 − 0.149964i
\(51\) −772.790 − 772.790i −0.297113 − 0.297113i
\(52\) 970.850 + 1901.76i 0.359042 + 0.703313i
\(53\) −2716.07 − 2716.07i −0.966918 − 0.966918i 0.0325516 − 0.999470i \(-0.489637\pi\)
−0.999470 + 0.0325516i \(0.989637\pi\)
\(54\) 3135.59 + 495.384i 1.07531 + 0.169885i
\(55\) 613.588 0.202839
\(56\) −1126.86 + 367.586i −0.359330 + 0.117215i
\(57\) 1532.18i 0.471584i
\(58\) −3172.39 − 501.198i −0.943041 − 0.148989i
\(59\) 196.912 − 196.912i 0.0565676 − 0.0565676i −0.678257 − 0.734825i \(-0.737264\pi\)
0.734825 + 0.678257i \(0.237264\pi\)
\(60\) 3143.37 + 1018.65i 0.873159 + 0.282959i
\(61\) 1174.86 − 1174.86i 0.315737 − 0.315737i −0.531390 − 0.847127i \(-0.678330\pi\)
0.847127 + 0.531390i \(0.178330\pi\)
\(62\) −3241.41 + 2356.94i −0.843238 + 0.613148i
\(63\) 493.106i 0.124239i
\(64\) 3308.13 − 2415.26i 0.807649 − 0.589663i
\(65\) 3737.55 0.884628
\(66\) 380.035 + 522.648i 0.0872441 + 0.119983i
\(67\) −2275.58 − 2275.58i −0.506925 − 0.506925i 0.406657 − 0.913581i \(-0.366695\pi\)
−0.913581 + 0.406657i \(0.866695\pi\)
\(68\) −731.045 + 2255.87i −0.158098 + 0.487860i
\(69\) 4182.06 + 4182.06i 0.878399 + 0.878399i
\(70\) −323.771 + 2049.35i −0.0660757 + 0.418234i
\(71\) 9117.59 1.80869 0.904344 − 0.426805i \(-0.140361\pi\)
0.904344 + 0.426805i \(0.140361\pi\)
\(72\) −528.451 − 1620.00i −0.101939 − 0.312500i
\(73\) 5965.10i 1.11937i 0.828707 + 0.559683i \(0.189077\pi\)
−0.828707 + 0.559683i \(0.810923\pi\)
\(74\) −481.842 + 3049.88i −0.0879917 + 0.556953i
\(75\) 830.999 − 830.999i 0.147733 − 0.147733i
\(76\) 2961.01 − 1511.60i 0.512640 − 0.261703i
\(77\) −286.911 + 286.911i −0.0483911 + 0.0483911i
\(78\) 2314.91 + 3183.61i 0.380492 + 0.523276i
\(79\) 3103.75i 0.497316i 0.968591 + 0.248658i \(0.0799895\pi\)
−0.968591 + 0.248658i \(0.920010\pi\)
\(80\) −1132.56 − 7079.69i −0.176962 − 1.10620i
\(81\) 3695.46 0.563246
\(82\) 41.4130 − 30.1128i 0.00615898 − 0.00447840i
\(83\) −3934.02 − 3934.02i −0.571058 − 0.571058i 0.361366 − 0.932424i \(-0.382310\pi\)
−0.932424 + 0.361366i \(0.882310\pi\)
\(84\) −1946.14 + 993.510i −0.275814 + 0.140804i
\(85\) 2935.11 + 2935.11i 0.406244 + 0.406244i
\(86\) −4503.68 − 711.526i −0.608935 − 0.0962041i
\(87\) −5920.78 −0.782241
\(88\) 635.112 − 1250.07i 0.0820134 − 0.161424i
\(89\) 1130.82i 0.142762i 0.997449 + 0.0713809i \(0.0227406\pi\)
−0.997449 + 0.0713809i \(0.977259\pi\)
\(90\) −2946.19 − 465.462i −0.363728 − 0.0574644i
\(91\) −1747.66 + 1747.66i −0.211045 + 0.211045i
\(92\) 3956.14 − 12207.9i 0.467408 − 1.44233i
\(93\) −5224.23 + 5224.23i −0.604027 + 0.604027i
\(94\) −12060.4 + 8769.56i −1.36492 + 0.992481i
\(95\) − 5819.31i − 0.644799i
\(96\) 5328.95 − 5349.62i 0.578228 − 0.580471i
\(97\) 14246.2 1.51410 0.757050 − 0.653357i \(-0.226640\pi\)
0.757050 + 0.653357i \(0.226640\pi\)
\(98\) −806.871 − 1109.66i −0.0840141 − 0.115541i
\(99\) −412.471 − 412.471i −0.0420846 − 0.0420846i
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.43.15 ✓ 96
4.3 odd 2 448.5.k.a.15.35 96
16.3 odd 4 inner 112.5.k.a.99.15 yes 96
16.13 even 4 448.5.k.a.239.35 96
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
112.5.k.a.43.15 ✓ 96 1.1 even 1 trivial
112.5.k.a.99.15 yes 96 16.3 odd 4 inner
448.5.k.a.15.35 96 4.3 odd 2
448.5.k.a.239.35 96 16.13 even 4