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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 43.16
Character \(\chi\) \(=\) 112.43
Dual form 112.5.k.a.99.16

$q$-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+(-1.92494 + 3.50637i) q^{2} +(-4.29879 - 4.29879i) q^{3} +(-8.58925 - 13.4991i) q^{4} +(-23.0339 - 23.0339i) q^{5} +(23.3480 - 6.79825i) q^{6} +18.5203 q^{7} +(63.8665 - 4.13222i) q^{8} -44.0408i q^{9} +(125.104 - 36.4266i) q^{10} +(-36.6590 + 36.6590i) q^{11} +(-21.1063 + 94.9530i) q^{12} +(-134.321 + 134.321i) q^{13} +(-35.6503 + 64.9389i) q^{14} +198.036i q^{15} +(-108.450 + 231.894i) q^{16} +91.7028 q^{17} +(154.423 + 84.7758i) q^{18} +(49.2224 + 49.2224i) q^{19} +(-113.092 + 508.780i) q^{20} +(-79.6147 - 79.6147i) q^{21} +(-57.9738 - 199.106i) q^{22} +138.221 q^{23} +(-292.312 - 256.785i) q^{24} +436.123i q^{25} +(-212.420 - 729.539i) q^{26} +(-537.524 + 537.524i) q^{27} +(-159.075 - 250.006i) q^{28} +(-684.688 + 684.688i) q^{29} +(-694.387 - 381.206i) q^{30} -26.4269i q^{31} +(-604.346 - 826.645i) q^{32} +315.179 q^{33} +(-176.522 + 321.544i) q^{34} +(-426.594 - 426.594i) q^{35} +(-594.510 + 378.277i) q^{36} +(1504.45 + 1504.45i) q^{37} +(-267.342 + 77.8419i) q^{38} +1154.84 q^{39} +(-1566.28 - 1375.91i) q^{40} +916.482i q^{41} +(432.412 - 125.905i) q^{42} +(-1235.93 + 1235.93i) q^{43} +(809.736 + 179.989i) q^{44} +(-1014.43 + 1014.43i) q^{45} +(-266.067 + 484.655i) q^{46} -2991.77i q^{47} +(1463.06 - 530.659i) q^{48} +343.000 q^{49} +(-1529.21 - 839.508i) q^{50} +(-394.211 - 394.211i) q^{51} +(2966.93 + 659.493i) q^{52} +(3052.65 + 3052.65i) q^{53} +(-850.058 - 2919.46i) q^{54} +1688.80 q^{55} +(1182.82 - 76.5298i) q^{56} -423.193i q^{57} +(-1082.79 - 3718.75i) q^{58} +(2241.55 - 2241.55i) q^{59} +(2673.30 - 1700.98i) q^{60} +(-4003.52 + 4003.52i) q^{61} +(92.6625 + 50.8701i) q^{62} -815.648i q^{63} +(4061.85 - 527.821i) q^{64} +6187.89 q^{65} +(-606.699 + 1105.13i) q^{66} +(2473.08 + 2473.08i) q^{67} +(-787.658 - 1237.90i) q^{68} +(-594.184 - 594.184i) q^{69} +(2316.96 - 674.630i) q^{70} +1074.73 q^{71} +(-181.987 - 2812.73i) q^{72} +659.751i q^{73} +(-8171.11 + 2379.18i) q^{74} +(1874.80 - 1874.80i) q^{75} +(241.673 - 1087.24i) q^{76} +(-678.935 + 678.935i) q^{77} +(-2222.99 + 4049.28i) q^{78} +2279.53i q^{79} +(7839.44 - 2843.40i) q^{80} +1054.10 q^{81} +(-3213.53 - 1764.17i) q^{82} +(-5484.67 - 5484.67i) q^{83} +(-390.894 + 1758.55i) q^{84} +(-2112.27 - 2112.27i) q^{85} +(-1954.54 - 6712.72i) q^{86} +5886.66 q^{87} +(-2189.80 + 2492.77i) q^{88} +937.959i q^{89} +(-1604.26 - 5509.69i) q^{90} +(-2487.66 + 2487.66i) q^{91} +(-1187.22 - 1865.86i) q^{92} +(-113.604 + 113.604i) q^{93} +(10490.3 + 5758.97i) q^{94} -2267.57i q^{95} +(-955.617 + 6151.53i) q^{96} -5958.69 q^{97} +(-660.253 + 1202.68i) q^{98} +(1614.49 + 1614.49i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.92494 + 3.50637i −0.481234 + 0.876592i
\(3\) −4.29879 − 4.29879i −0.477643 − 0.477643i 0.426734 − 0.904377i \(-0.359664\pi\)
−0.904377 + 0.426734i \(0.859664\pi\)
\(4\) −8.58925 − 13.4991i −0.536828 − 0.843692i
\(5\) −23.0339 − 23.0339i −0.921357 − 0.921357i 0.0757687 − 0.997125i \(-0.475859\pi\)
−0.997125 + 0.0757687i \(0.975859\pi\)
\(6\) 23.3480 − 6.79825i 0.648556 − 0.188840i
\(7\) 18.5203 0.377964
\(8\) 63.8665 − 4.13222i 0.997913 − 0.0645660i
\(9\) − 44.0408i − 0.543714i
\(10\) 125.104 − 36.4266i 1.25104 − 0.364266i
\(11\) −36.6590 + 36.6590i −0.302967 + 0.302967i −0.842174 − 0.539206i \(-0.818724\pi\)
0.539206 + 0.842174i \(0.318724\pi\)
\(12\) −21.1063 + 94.9530i −0.146571 + 0.659396i
\(13\) −134.321 + 134.321i −0.794800 + 0.794800i −0.982270 − 0.187470i \(-0.939971\pi\)
0.187470 + 0.982270i \(0.439971\pi\)
\(14\) −35.6503 + 64.9389i −0.181889 + 0.331321i
\(15\) 198.036i 0.880160i
\(16\) −108.450 + 231.894i −0.423632 + 0.905834i
\(17\) 91.7028 0.317311 0.158655 − 0.987334i \(-0.449284\pi\)
0.158655 + 0.987334i \(0.449284\pi\)
\(18\) 154.423 + 84.7758i 0.476615 + 0.261654i
\(19\) 49.2224 + 49.2224i 0.136350 + 0.136350i 0.771988 − 0.635638i \(-0.219263\pi\)
−0.635638 + 0.771988i \(0.719263\pi\)
\(20\) −113.092 + 508.780i −0.282731 + 1.27195i
\(21\) −79.6147 − 79.6147i −0.180532 − 0.180532i
\(22\) −57.9738 − 199.106i −0.119781 − 0.411377i
\(23\) 138.221 0.261288 0.130644 − 0.991429i \(-0.458296\pi\)
0.130644 + 0.991429i \(0.458296\pi\)
\(24\) −292.312 − 256.785i −0.507486 − 0.445807i
\(25\) 436.123i 0.697796i
\(26\) −212.420 − 729.539i −0.314231 − 1.07920i
\(27\) −537.524 + 537.524i −0.737344 + 0.737344i
\(28\) −159.075 − 250.006i −0.202902 − 0.318886i
\(29\) −684.688 + 684.688i −0.814136 + 0.814136i −0.985251 − 0.171115i \(-0.945263\pi\)
0.171115 + 0.985251i \(0.445263\pi\)
\(30\) −694.387 − 381.206i −0.771541 − 0.423563i
\(31\) − 26.4269i − 0.0274994i −0.999905 − 0.0137497i \(-0.995623\pi\)
0.999905 − 0.0137497i \(-0.00437680\pi\)
\(32\) −604.346 − 826.645i −0.590181 − 0.807271i
\(33\) 315.179 0.289420
\(34\) −176.522 + 321.544i −0.152701 + 0.278152i
\(35\) −426.594 − 426.594i −0.348240 − 0.348240i
\(36\) −594.510 + 378.277i −0.458727 + 0.291881i
\(37\) 1504.45 + 1504.45i 1.09894 + 1.09894i 0.994535 + 0.104404i \(0.0332934\pi\)
0.104404 + 0.994535i \(0.466707\pi\)
\(38\) −267.342 + 77.8419i −0.185140 + 0.0539071i
\(39\) 1154.84 0.759261
\(40\) −1566.28 − 1375.91i −0.978923 − 0.859946i
\(41\) 916.482i 0.545201i 0.962127 + 0.272600i \(0.0878837\pi\)
−0.962127 + 0.272600i \(0.912116\pi\)
\(42\) 432.412 − 125.905i 0.245131 − 0.0713749i
\(43\) −1235.93 + 1235.93i −0.668432 + 0.668432i −0.957353 − 0.288921i \(-0.906704\pi\)
0.288921 + 0.957353i \(0.406704\pi\)
\(44\) 809.736 + 179.989i 0.418252 + 0.0929697i
\(45\) −1014.43 + 1014.43i −0.500954 + 0.500954i
\(46\) −266.067 + 484.655i −0.125741 + 0.229043i
\(47\) − 2991.77i − 1.35436i −0.735820 − 0.677178i \(-0.763203\pi\)
0.735820 − 0.677178i \(-0.236797\pi\)
\(48\) 1463.06 − 530.659i 0.635011 − 0.230321i
\(49\) 343.000 0.142857
\(50\) −1529.21 − 839.508i −0.611683 − 0.335803i
\(51\) −394.211 − 394.211i −0.151561 − 0.151561i
\(52\) 2966.93 + 659.493i 1.09724 + 0.243895i
\(53\) 3052.65 + 3052.65i 1.08674 + 1.08674i 0.995862 + 0.0908784i \(0.0289675\pi\)
0.0908784 + 0.995862i \(0.471033\pi\)
\(54\) −850.058 − 2919.46i −0.291515 − 1.00119i
\(55\) 1688.80 0.558282
\(56\) 1182.82 − 76.5298i 0.377176 − 0.0244036i
\(57\) − 423.193i − 0.130253i
\(58\) −1082.79 − 3718.75i −0.321875 − 1.10546i
\(59\) 2241.55 − 2241.55i 0.643938 − 0.643938i −0.307583 − 0.951521i \(-0.599520\pi\)
0.951521 + 0.307583i \(0.0995202\pi\)
\(60\) 2673.30 − 1700.98i 0.742583 − 0.472494i
\(61\) −4003.52 + 4003.52i −1.07593 + 1.07593i −0.0790563 + 0.996870i \(0.525191\pi\)
−0.996870 + 0.0790563i \(0.974809\pi\)
\(62\) 92.6625 + 50.8701i 0.0241057 + 0.0132336i
\(63\) − 815.648i − 0.205505i
\(64\) 4061.85 − 527.821i 0.991662 − 0.128863i
\(65\) 6187.89 1.46459
\(66\) −606.699 + 1105.13i −0.139279 + 0.253704i
\(67\) 2473.08 + 2473.08i 0.550920 + 0.550920i 0.926706 − 0.375787i \(-0.122627\pi\)
−0.375787 + 0.926706i \(0.622627\pi\)
\(68\) −787.658 − 1237.90i −0.170341 − 0.267712i
\(69\) −594.184 − 594.184i −0.124802 − 0.124802i
\(70\) 2316.96 − 674.630i 0.472850 − 0.137680i
\(71\) 1074.73 0.213197 0.106598 − 0.994302i \(-0.466004\pi\)
0.106598 + 0.994302i \(0.466004\pi\)
\(72\) −181.987 − 2812.73i −0.0351054 − 0.542579i
\(73\) 659.751i 0.123804i 0.998082 + 0.0619020i \(0.0197166\pi\)
−0.998082 + 0.0619020i \(0.980283\pi\)
\(74\) −8171.11 + 2379.18i −1.49217 + 0.434475i
\(75\) 1874.80 − 1874.80i 0.333298 − 0.333298i
\(76\) 241.673 − 1087.24i 0.0418409 − 0.188234i
\(77\) −678.935 + 678.935i −0.114511 + 0.114511i
\(78\) −2222.99 + 4049.28i −0.365382 + 0.665563i
\(79\) 2279.53i 0.365250i 0.983183 + 0.182625i \(0.0584595\pi\)
−0.983183 + 0.182625i \(0.941541\pi\)
\(80\) 7839.44 − 2843.40i 1.22491 − 0.444281i
\(81\) 1054.10 0.160661
\(82\) −3213.53 − 1764.17i −0.477919 − 0.262369i
\(83\) −5484.67 − 5484.67i −0.796150 − 0.796150i 0.186336 − 0.982486i \(-0.440339\pi\)
−0.982486 + 0.186336i \(0.940339\pi\)
\(84\) −390.894 + 1758.55i −0.0553988 + 0.249228i
\(85\) −2112.27 − 2112.27i −0.292356 − 0.292356i
\(86\) −1954.54 − 6712.72i −0.264270 − 0.907615i
\(87\) 5886.66 0.777733
\(88\) −2189.80 + 2492.77i −0.282774 + 0.321896i
\(89\) 937.959i 0.118414i 0.998246 + 0.0592071i \(0.0188572\pi\)
−0.998246 + 0.0592071i \(0.981143\pi\)
\(90\) −1604.26 − 5509.69i −0.198057 − 0.680209i
\(91\) −2487.66 + 2487.66i −0.300406 + 0.300406i
\(92\) −1187.22 − 1865.86i −0.140267 − 0.220446i
\(93\) −113.604 + 113.604i −0.0131349 + 0.0131349i
\(94\) 10490.3 + 5758.97i 1.18722 + 0.651762i
\(95\) − 2267.57i − 0.251254i
\(96\) −955.617 + 6151.53i −0.103691 + 0.667483i
\(97\) −5958.69 −0.633297 −0.316648 − 0.948543i \(-0.602557\pi\)
−0.316648 + 0.948543i \(0.602557\pi\)
\(98\) −660.253 + 1202.68i −0.0687477 + 0.125227i
\(99\) 1614.49 + 1614.49i 0.164727 + 0.164727i
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.43.16 ✓ 96
4.3 odd 2 448.5.k.a.15.31 96
16.3 odd 4 inner 112.5.k.a.99.16 yes 96
16.13 even 4 448.5.k.a.239.31 96
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
112.5.k.a.43.16 ✓ 96 1.1 even 1 trivial
112.5.k.a.99.16 yes 96 16.3 odd 4 inner
448.5.k.a.15.31 96 4.3 odd 2
448.5.k.a.239.31 96 16.13 even 4