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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 43.17
Character \(\chi\) \(=\) 112.43
Dual form 112.5.k.a.99.17

$q$-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+(-1.61373 - 3.66004i) q^{2} +(6.54673 + 6.54673i) q^{3} +(-10.7917 + 11.8126i) q^{4} +(-10.9039 - 10.9039i) q^{5} +(13.3966 - 34.5259i) q^{6} -18.5203 q^{7} +(60.6497 + 20.4357i) q^{8} +4.71923i q^{9} +(-22.3128 + 57.5049i) q^{10} +(-4.98403 + 4.98403i) q^{11} +(-147.985 + 6.68364i) q^{12} +(-55.7937 + 55.7937i) q^{13} +(29.8868 + 67.7848i) q^{14} -142.770i q^{15} +(-23.0771 - 254.958i) q^{16} -492.666 q^{17} +(17.2725 - 7.61557i) q^{18} +(-337.985 - 337.985i) q^{19} +(246.477 - 11.1320i) q^{20} +(-121.247 - 121.247i) q^{21} +(26.2846 + 10.1988i) q^{22} -290.895 q^{23} +(263.270 + 530.844i) q^{24} -387.208i q^{25} +(294.243 + 114.171i) q^{26} +(499.389 - 499.389i) q^{27} +(199.866 - 218.773i) q^{28} +(-1132.19 + 1132.19i) q^{29} +(-522.544 + 230.393i) q^{30} +998.354i q^{31} +(-895.914 + 495.897i) q^{32} -65.2582 q^{33} +(795.031 + 1803.18i) q^{34} +(201.944 + 201.944i) q^{35} +(-55.7465 - 50.9286i) q^{36} +(-319.046 - 319.046i) q^{37} +(-691.620 + 1782.45i) q^{38} -730.532 q^{39} +(-438.491 - 884.150i) q^{40} -433.353i q^{41} +(-248.108 + 639.429i) q^{42} +(499.804 - 499.804i) q^{43} +(-5.08826 - 112.661i) q^{44} +(51.4582 - 51.4582i) q^{45} +(469.427 + 1064.69i) q^{46} -1763.91i q^{47} +(1518.06 - 1820.22i) q^{48} +343.000 q^{49} +(-1417.19 + 624.850i) q^{50} +(-3225.35 - 3225.35i) q^{51} +(-56.9605 - 1261.18i) q^{52} +(3086.00 + 3086.00i) q^{53} +(-2633.66 - 1021.90i) q^{54} +108.691 q^{55} +(-1123.25 - 378.474i) q^{56} -4425.39i q^{57} +(5970.90 + 2316.80i) q^{58} +(1177.76 - 1177.76i) q^{59} +(1686.49 + 1540.74i) q^{60} +(3815.32 - 3815.32i) q^{61} +(3654.01 - 1611.08i) q^{62} -87.4013i q^{63} +(3260.77 + 2478.83i) q^{64} +1216.74 q^{65} +(105.309 + 238.847i) q^{66} +(-722.719 - 722.719i) q^{67} +(5316.72 - 5819.69i) q^{68} +(-1904.41 - 1904.41i) q^{69} +(413.238 - 1065.01i) q^{70} -6982.08 q^{71} +(-96.4406 + 286.220i) q^{72} +2567.95i q^{73} +(-652.866 + 1682.58i) q^{74} +(2534.94 - 2534.94i) q^{75} +(7639.94 - 345.053i) q^{76} +(92.3055 - 92.3055i) q^{77} +(1178.88 + 2673.77i) q^{78} +5338.40i q^{79} +(-2528.41 + 3031.68i) q^{80} +6920.99 q^{81} +(-1586.09 + 699.316i) q^{82} +(-7409.85 - 7409.85i) q^{83} +(2740.71 - 123.783i) q^{84} +(5372.00 + 5372.00i) q^{85} +(-2635.85 - 1022.75i) q^{86} -14824.2 q^{87} +(-404.132 + 200.428i) q^{88} +1263.43i q^{89} +(-271.379 - 105.299i) q^{90} +(1033.31 - 1033.31i) q^{91} +(3139.26 - 3436.24i) q^{92} +(-6535.95 + 6535.95i) q^{93} +(-6455.97 + 2846.48i) q^{94} +7370.74i q^{95} +(-9111.80 - 2618.80i) q^{96} +1211.89 q^{97} +(-553.510 - 1255.39i) q^{98} +(-23.5208 - 23.5208i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.61373 − 3.66004i −0.403433 − 0.915009i
\(3\) 6.54673 + 6.54673i 0.727414 + 0.727414i 0.970104 − 0.242690i \(-0.0780298\pi\)
−0.242690 + 0.970104i \(0.578030\pi\)
\(4\) −10.7917 + 11.8126i −0.674483 + 0.738290i
\(5\) −10.9039 − 10.9039i −0.436158 − 0.436158i 0.454559 − 0.890717i \(-0.349797\pi\)
−0.890717 + 0.454559i \(0.849797\pi\)
\(6\) 13.3966 − 34.5259i 0.372127 − 0.959053i
\(7\) −18.5203 −0.377964
\(8\) 60.6497 + 20.4357i 0.947651 + 0.319307i
\(9\) 4.71923i 0.0582621i
\(10\) −22.3128 + 57.5049i −0.223128 + 0.575049i
\(11\) −4.98403 + 4.98403i −0.0411903 + 0.0411903i −0.727402 − 0.686212i \(-0.759272\pi\)
0.686212 + 0.727402i \(0.259272\pi\)
\(12\) −147.985 + 6.68364i −1.02767 + 0.0464141i
\(13\) −55.7937 + 55.7937i −0.330140 + 0.330140i −0.852640 − 0.522500i \(-0.825000\pi\)
0.522500 + 0.852640i \(0.325000\pi\)
\(14\) 29.8868 + 67.7848i 0.152483 + 0.345841i
\(15\) − 142.770i − 0.634535i
\(16\) −23.0771 − 254.958i −0.0901449 − 0.995929i
\(17\) −492.666 −1.70473 −0.852363 − 0.522950i \(-0.824831\pi\)
−0.852363 + 0.522950i \(0.824831\pi\)
\(18\) 17.2725 − 7.61557i 0.0533103 − 0.0235049i
\(19\) −337.985 − 337.985i −0.936246 − 0.936246i 0.0618397 − 0.998086i \(-0.480303\pi\)
−0.998086 + 0.0618397i \(0.980303\pi\)
\(20\) 246.477 − 11.1320i 0.616192 − 0.0278300i
\(21\) −121.247 − 121.247i −0.274937 − 0.274937i
\(22\) 26.2846 + 10.1988i 0.0543071 + 0.0210720i
\(23\) −290.895 −0.549896 −0.274948 − 0.961459i \(-0.588661\pi\)
−0.274948 + 0.961459i \(0.588661\pi\)
\(24\) 263.270 + 530.844i 0.457066 + 0.921603i
\(25\) − 387.208i − 0.619533i
\(26\) 294.243 + 114.171i 0.435271 + 0.168892i
\(27\) 499.389 − 499.389i 0.685033 − 0.685033i
\(28\) 199.866 − 218.773i 0.254931 − 0.279047i
\(29\) −1132.19 + 1132.19i −1.34624 + 1.34624i −0.456533 + 0.889706i \(0.650909\pi\)
−0.889706 + 0.456533i \(0.849091\pi\)
\(30\) −522.544 + 230.393i −0.580605 + 0.255992i
\(31\) 998.354i 1.03887i 0.854510 + 0.519435i \(0.173857\pi\)
−0.854510 + 0.519435i \(0.826143\pi\)
\(32\) −895.914 + 495.897i −0.874916 + 0.484274i
\(33\) −65.2582 −0.0599248
\(34\) 795.031 + 1803.18i 0.687743 + 1.55984i
\(35\) 201.944 + 201.944i 0.164852 + 0.164852i
\(36\) −55.7465 − 50.9286i −0.0430143 − 0.0392968i
\(37\) −319.046 − 319.046i −0.233051 − 0.233051i 0.580914 − 0.813965i \(-0.302695\pi\)
−0.813965 + 0.580914i \(0.802695\pi\)
\(38\) −691.620 + 1782.45i −0.478961 + 1.23439i
\(39\) −730.532 −0.480297
\(40\) −438.491 − 884.150i −0.274057 − 0.552594i
\(41\) − 433.353i − 0.257795i −0.991658 − 0.128897i \(-0.958856\pi\)
0.991658 − 0.128897i \(-0.0411438\pi\)
\(42\) −248.108 + 639.429i −0.140651 + 0.362488i
\(43\) 499.804 − 499.804i 0.270310 − 0.270310i −0.558915 − 0.829225i \(-0.688782\pi\)
0.829225 + 0.558915i \(0.188782\pi\)
\(44\) −5.08826 − 112.661i −0.00262823 − 0.0581926i
\(45\) 51.4582 − 51.4582i 0.0254115 − 0.0254115i
\(46\) 469.427 + 1064.69i 0.221847 + 0.503160i
\(47\) − 1763.91i − 0.798510i −0.916840 − 0.399255i \(-0.869269\pi\)
0.916840 − 0.399255i \(-0.130731\pi\)
\(48\) 1518.06 − 1820.22i 0.658880 − 0.790025i
\(49\) 343.000 0.142857
\(50\) −1417.19 + 624.850i −0.566878 + 0.249940i
\(51\) −3225.35 − 3225.35i −1.24004 − 1.24004i
\(52\) −56.9605 − 1261.18i −0.0210653 − 0.466413i
\(53\) 3086.00 + 3086.00i 1.09861 + 1.09861i 0.994573 + 0.104039i \(0.0331767\pi\)
0.104039 + 0.994573i \(0.466823\pi\)
\(54\) −2633.66 − 1021.90i −0.903177 − 0.350446i
\(55\) 108.691 0.0359310
\(56\) −1123.25 − 378.474i −0.358178 − 0.120687i
\(57\) − 4425.39i − 1.36208i
\(58\) 5970.90 + 2316.80i 1.77494 + 0.688704i
\(59\) 1177.76 − 1177.76i 0.338339 − 0.338339i −0.517403 − 0.855742i \(-0.673101\pi\)
0.855742 + 0.517403i \(0.173101\pi\)
\(60\) 1686.49 + 1540.74i 0.468471 + 0.427983i
\(61\) 3815.32 − 3815.32i 1.02535 − 1.02535i 0.0256768 − 0.999670i \(-0.491826\pi\)
0.999670 − 0.0256768i \(-0.00817409\pi\)
\(62\) 3654.01 − 1611.08i 0.950575 − 0.419115i
\(63\) − 87.4013i − 0.0220210i
\(64\) 3260.77 + 2478.83i 0.796086 + 0.605184i
\(65\) 1216.74 0.287986
\(66\) 105.309 + 238.847i 0.0241757 + 0.0548318i
\(67\) −722.719 − 722.719i −0.160998 − 0.160998i 0.622011 − 0.783009i \(-0.286316\pi\)
−0.783009 + 0.622011i \(0.786316\pi\)
\(68\) 5316.72 − 5819.69i 1.14981 − 1.25858i
\(69\) −1904.41 − 1904.41i −0.400002 − 0.400002i
\(70\) 413.238 − 1065.01i 0.0843344 − 0.217348i
\(71\) −6982.08 −1.38506 −0.692529 − 0.721390i \(-0.743503\pi\)
−0.692529 + 0.721390i \(0.743503\pi\)
\(72\) −96.4406 + 286.220i −0.0186035 + 0.0552121i
\(73\) 2567.95i 0.481883i 0.970540 + 0.240941i \(0.0774562\pi\)
−0.970540 + 0.240941i \(0.922544\pi\)
\(74\) −652.866 + 1682.58i −0.119223 + 0.307264i
\(75\) 2534.94 − 2534.94i 0.450657 − 0.450657i
\(76\) 7639.94 − 345.053i 1.32270 − 0.0597391i
\(77\) 92.3055 − 92.3055i 0.0155685 − 0.0155685i
\(78\) 1178.88 + 2673.77i 0.193768 + 0.439476i
\(79\) 5338.40i 0.855375i 0.903927 + 0.427688i \(0.140672\pi\)
−0.903927 + 0.427688i \(0.859328\pi\)
\(80\) −2528.41 + 3031.68i −0.395065 + 0.473700i
\(81\) 6920.99 1.05487
\(82\) −1586.09 + 699.316i −0.235885 + 0.104003i
\(83\) −7409.85 − 7409.85i −1.07561 − 1.07561i −0.996898 − 0.0787086i \(-0.974920\pi\)
−0.0787086 − 0.996898i \(-0.525080\pi\)
\(84\) 2740.71 − 123.783i 0.388423 − 0.0175429i
\(85\) 5372.00 + 5372.00i 0.743530 + 0.743530i
\(86\) −2635.85 − 1022.75i −0.356388 − 0.138284i
\(87\) −14824.2 −1.95855
\(88\) −404.132 + 200.428i −0.0521864 + 0.0258817i
\(89\) 1263.43i 0.159503i 0.996815 + 0.0797517i \(0.0254127\pi\)
−0.996815 + 0.0797517i \(0.974587\pi\)
\(90\) −271.379 − 105.299i −0.0335035 − 0.0129999i
\(91\) 1033.31 − 1033.31i 0.124781 − 0.124781i
\(92\) 3139.26 − 3436.24i 0.370896 − 0.405983i
\(93\) −6535.95 + 6535.95i −0.755688 + 0.755688i
\(94\) −6455.97 + 2846.48i −0.730644 + 0.322146i
\(95\) 7370.74i 0.816703i
\(96\) −9111.80 − 2618.80i −0.988694 − 0.284159i
\(97\) 1211.89 0.128801 0.0644007 − 0.997924i \(-0.479486\pi\)
0.0644007 + 0.997924i \(0.479486\pi\)
\(98\) −553.510 − 1255.39i −0.0576333 − 0.130716i
\(99\) −23.5208 − 23.5208i −0.00239983 − 0.00239983i
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.43.17 ✓ 96
4.3 odd 2 448.5.k.a.15.14 96
16.3 odd 4 inner 112.5.k.a.99.17 yes 96
16.13 even 4 448.5.k.a.239.14 96
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
112.5.k.a.43.17 ✓ 96 1.1 even 1 trivial
112.5.k.a.99.17 yes 96 16.3 odd 4 inner
448.5.k.a.15.14 96 4.3 odd 2
448.5.k.a.239.14 96 16.13 even 4