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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 43.1
Character \(\chi\) \(=\) 112.43
Dual form 112.5.k.a.99.1

$q$-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+(-3.99491 + 0.201653i) q^{2} +(-6.16471 - 6.16471i) q^{3} +(15.9187 - 1.61118i) q^{4} +(31.8098 + 31.8098i) q^{5} +(25.8706 + 23.3843i) q^{6} -18.5203 q^{7} +(-63.2688 + 9.64656i) q^{8} -4.99279i q^{9} +(-133.492 - 120.663i) q^{10} +(66.4675 - 66.4675i) q^{11} +(-108.066 - 88.2015i) q^{12} +(-152.869 + 152.869i) q^{13} +(73.9868 - 3.73467i) q^{14} -392.196i q^{15} +(250.808 - 51.2956i) q^{16} -404.613 q^{17} +(1.00681 + 19.9458i) q^{18} +(352.626 + 352.626i) q^{19} +(557.621 + 455.119i) q^{20} +(114.172 + 114.172i) q^{21} +(-252.128 + 278.935i) q^{22} +587.458 q^{23} +(449.502 + 330.565i) q^{24} +1398.73i q^{25} +(579.871 - 641.524i) q^{26} +(-530.120 + 530.120i) q^{27} +(-294.818 + 29.8394i) q^{28} +(-333.142 + 333.142i) q^{29} +(79.0877 + 1566.79i) q^{30} -274.371i q^{31} +(-991.613 + 255.498i) q^{32} -819.505 q^{33} +(1616.39 - 81.5915i) q^{34} +(-589.126 - 589.126i) q^{35} +(-8.04426 - 79.4786i) q^{36} +(1254.06 + 1254.06i) q^{37} +(-1479.82 - 1337.60i) q^{38} +1884.78 q^{39} +(-2319.42 - 1705.71i) q^{40} +1516.05i q^{41} +(-479.130 - 433.084i) q^{42} +(-310.844 + 310.844i) q^{43} +(950.983 - 1165.16i) q^{44} +(158.820 - 158.820i) q^{45} +(-2346.85 + 118.463i) q^{46} +1914.63i q^{47} +(-1862.38 - 1229.94i) q^{48} +343.000 q^{49} +(-282.058 - 5587.79i) q^{50} +(2494.32 + 2494.32i) q^{51} +(-2187.17 + 2679.77i) q^{52} +(1598.23 + 1598.23i) q^{53} +(2010.88 - 2224.69i) q^{54} +4228.63 q^{55} +(1171.75 - 178.657i) q^{56} -4347.67i q^{57} +(1263.69 - 1398.05i) q^{58} +(-2331.14 + 2331.14i) q^{59} +(-631.897 - 6243.24i) q^{60} +(3090.49 - 3090.49i) q^{61} +(55.3278 + 1096.09i) q^{62} +92.4677i q^{63} +(3909.89 - 1220.65i) q^{64} -9725.45 q^{65} +(3273.85 - 165.256i) q^{66} +(143.983 + 143.983i) q^{67} +(-6440.90 + 651.902i) q^{68} +(-3621.51 - 3621.51i) q^{69} +(2472.31 + 2234.71i) q^{70} -6224.94 q^{71} +(48.1632 + 315.888i) q^{72} +8217.51i q^{73} +(-5262.76 - 4756.99i) q^{74} +(8622.74 - 8622.74i) q^{75} +(6181.48 + 5045.20i) q^{76} +(-1230.99 + 1230.99i) q^{77} +(-7529.54 + 380.073i) q^{78} -4147.17i q^{79} +(9609.86 + 6346.46i) q^{80} +6131.66 q^{81} +(-305.717 - 6056.50i) q^{82} +(-6829.91 - 6829.91i) q^{83} +(2001.42 + 1633.51i) q^{84} +(-12870.6 - 12870.6i) q^{85} +(1179.11 - 1304.48i) q^{86} +4107.44 q^{87} +(-3564.13 + 4846.50i) q^{88} +12554.9i q^{89} +(-602.444 + 666.497i) q^{90} +(2831.17 - 2831.17i) q^{91} +(9351.56 - 946.499i) q^{92} +(-1691.42 + 1691.42i) q^{93} +(-386.092 - 7648.78i) q^{94} +22433.9i q^{95} +(7688.07 + 4537.94i) q^{96} -5547.08 q^{97} +(-1370.26 + 69.1671i) q^{98} +(-331.858 - 331.858i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −3.99491 + 0.201653i −0.998728 + 0.0504133i
\(3\) −6.16471 − 6.16471i −0.684967 − 0.684967i 0.276148 − 0.961115i \(-0.410942\pi\)
−0.961115 + 0.276148i \(0.910942\pi\)
\(4\) 15.9187 − 1.61118i 0.994917 − 0.100698i
\(5\) 31.8098 + 31.8098i 1.27239 + 1.27239i 0.944829 + 0.327563i \(0.106227\pi\)
0.327563 + 0.944829i \(0.393773\pi\)
\(6\) 25.8706 + 23.3843i 0.718628 + 0.649565i
\(7\) −18.5203 −0.377964
\(8\) −63.2688 + 9.64656i −0.988575 + 0.150728i
\(9\) − 4.99279i − 0.0616394i
\(10\) −133.492 − 120.663i −1.33492 − 1.20663i
\(11\) 66.4675 − 66.4675i 0.549318 − 0.549318i −0.376926 − 0.926243i \(-0.623019\pi\)
0.926243 + 0.376926i \(0.123019\pi\)
\(12\) −108.066 − 88.2015i −0.750461 − 0.612511i
\(13\) −152.869 + 152.869i −0.904549 + 0.904549i −0.995826 − 0.0912770i \(-0.970905\pi\)
0.0912770 + 0.995826i \(0.470905\pi\)
\(14\) 73.9868 − 3.73467i 0.377484 − 0.0190545i
\(15\) − 392.196i − 1.74309i
\(16\) 250.808 − 51.2956i 0.979720 − 0.200373i
\(17\) −404.613 −1.40004 −0.700022 − 0.714121i \(-0.746826\pi\)
−0.700022 + 0.714121i \(0.746826\pi\)
\(18\) 1.00681 + 19.9458i 0.00310745 + 0.0615610i
\(19\) 352.626 + 352.626i 0.976804 + 0.976804i 0.999737 − 0.0229330i \(-0.00730043\pi\)
−0.0229330 + 0.999737i \(0.507300\pi\)
\(20\) 557.621 + 455.119i 1.39405 + 1.13780i
\(21\) 114.172 + 114.172i 0.258893 + 0.258893i
\(22\) −252.128 + 278.935i −0.520926 + 0.576312i
\(23\) 587.458 1.11051 0.555254 − 0.831681i \(-0.312621\pi\)
0.555254 + 0.831681i \(0.312621\pi\)
\(24\) 449.502 + 330.565i 0.780385 + 0.573898i
\(25\) 1398.73i 2.23796i
\(26\) 579.871 − 641.524i 0.857797 − 0.949000i
\(27\) −530.120 + 530.120i −0.727188 + 0.727188i
\(28\) −294.818 + 29.8394i −0.376043 + 0.0380604i
\(29\) −333.142 + 333.142i −0.396126 + 0.396126i −0.876864 − 0.480738i \(-0.840369\pi\)
0.480738 + 0.876864i \(0.340369\pi\)
\(30\) 79.0877 + 1566.79i 0.0878752 + 1.74088i
\(31\) − 274.371i − 0.285506i −0.989758 − 0.142753i \(-0.954405\pi\)
0.989758 − 0.142753i \(-0.0455954\pi\)
\(32\) −991.613 + 255.498i −0.968372 + 0.249509i
\(33\) −819.505 −0.752530
\(34\) 1616.39 − 81.5915i 1.39826 − 0.0705809i
\(35\) −589.126 − 589.126i −0.480919 − 0.480919i
\(36\) −8.04426 − 79.4786i −0.00620699 − 0.0613260i
\(37\) 1254.06 + 1254.06i 0.916043 + 0.916043i 0.996739 − 0.0806958i \(-0.0257142\pi\)
−0.0806958 + 0.996739i \(0.525714\pi\)
\(38\) −1479.82 − 1337.60i −1.02481 − 0.926318i
\(39\) 1884.78 1.23917
\(40\) −2319.42 − 1705.71i −1.44964 − 1.06607i
\(41\) 1516.05i 0.901875i 0.892555 + 0.450938i \(0.148910\pi\)
−0.892555 + 0.450938i \(0.851090\pi\)
\(42\) −479.130 − 433.084i −0.271616 − 0.245512i
\(43\) −310.844 + 310.844i −0.168115 + 0.168115i −0.786150 − 0.618036i \(-0.787929\pi\)
0.618036 + 0.786150i \(0.287929\pi\)
\(44\) 950.983 − 1165.16i 0.491210 − 0.601841i
\(45\) 158.820 − 158.820i 0.0784294 − 0.0784294i
\(46\) −2346.85 + 118.463i −1.10910 + 0.0559844i
\(47\) 1914.63i 0.866741i 0.901216 + 0.433370i \(0.142676\pi\)
−0.901216 + 0.433370i \(0.857324\pi\)
\(48\) −1862.38 − 1229.94i −0.808325 − 0.533827i
\(49\) 343.000 0.142857
\(50\) −282.058 − 5587.79i −0.112823 − 2.23512i
\(51\) 2494.32 + 2494.32i 0.958984 + 0.958984i
\(52\) −2187.17 + 2679.77i −0.808864 + 0.991037i
\(53\) 1598.23 + 1598.23i 0.568967 + 0.568967i 0.931839 − 0.362872i \(-0.118204\pi\)
−0.362872 + 0.931839i \(0.618204\pi\)
\(54\) 2010.88 − 2224.69i 0.689604 − 0.762924i
\(55\) 4228.63 1.39790
\(56\) 1171.75 − 178.657i 0.373646 − 0.0569697i
\(57\) − 4347.67i − 1.33816i
\(58\) 1263.69 − 1398.05i 0.375652 − 0.415592i
\(59\) −2331.14 + 2331.14i −0.669676 + 0.669676i −0.957641 − 0.287965i \(-0.907021\pi\)
0.287965 + 0.957641i \(0.407021\pi\)
\(60\) −631.897 − 6243.24i −0.175527 − 1.73423i
\(61\) 3090.49 − 3090.49i 0.830553 − 0.830553i −0.157040 − 0.987592i \(-0.550195\pi\)
0.987592 + 0.157040i \(0.0501950\pi\)
\(62\) 55.3278 + 1096.09i 0.0143933 + 0.285143i
\(63\) 92.4677i 0.0232975i
\(64\) 3909.89 − 1220.65i 0.954562 − 0.298011i
\(65\) −9725.45 −2.30188
\(66\) 3273.85 − 165.256i 0.751573 − 0.0379375i
\(67\) 143.983 + 143.983i 0.0320747 + 0.0320747i 0.722962 − 0.690888i \(-0.242780\pi\)
−0.690888 + 0.722962i \(0.742780\pi\)
\(68\) −6440.90 + 651.902i −1.39293 + 0.140982i
\(69\) −3621.51 − 3621.51i −0.760661 − 0.760661i
\(70\) 2472.31 + 2234.71i 0.504552 + 0.456063i
\(71\) −6224.94 −1.23486 −0.617431 − 0.786625i \(-0.711827\pi\)
−0.617431 + 0.786625i \(0.711827\pi\)
\(72\) 48.1632 + 315.888i 0.00929075 + 0.0609351i
\(73\) 8217.51i 1.54204i 0.636814 + 0.771018i \(0.280252\pi\)
−0.636814 + 0.771018i \(0.719748\pi\)
\(74\) −5262.76 − 4756.99i −0.961059 − 0.868697i
\(75\) 8622.74 − 8622.74i 1.53293 − 1.53293i
\(76\) 6181.48 + 5045.20i 1.07020 + 0.873476i
\(77\) −1230.99 + 1230.99i −0.207623 + 0.207623i
\(78\) −7529.54 + 380.073i −1.23760 + 0.0624708i
\(79\) − 4147.17i − 0.664505i −0.943191 − 0.332252i \(-0.892191\pi\)
0.943191 − 0.332252i \(-0.107809\pi\)
\(80\) 9609.86 + 6346.46i 1.50154 + 0.991634i
\(81\) 6131.66 0.934561
\(82\) −305.717 − 6056.50i −0.0454665 − 0.900728i
\(83\) −6829.91 − 6829.91i −0.991423 − 0.991423i 0.00854094 − 0.999964i \(-0.497281\pi\)
−0.999964 + 0.00854094i \(0.997281\pi\)
\(84\) 2001.42 + 1633.51i 0.283648 + 0.231507i
\(85\) −12870.6 − 12870.6i −1.78140 − 1.78140i
\(86\) 1179.11 − 1304.48i 0.159426 − 0.176376i
\(87\) 4107.44 0.542667
\(88\) −3564.13 + 4846.50i −0.460245 + 0.625839i
\(89\) 12554.9i 1.58502i 0.609862 + 0.792508i \(0.291225\pi\)
−0.609862 + 0.792508i \(0.708775\pi\)
\(90\) −602.444 + 666.497i −0.0743758 + 0.0822836i
\(91\) 2831.17 − 2831.17i 0.341887 − 0.341887i
\(92\) 9351.56 − 946.499i 1.10486 − 0.111826i
\(93\) −1691.42 + 1691.42i −0.195562 + 0.195562i
\(94\) −386.092 − 7648.78i −0.0436953 − 0.865638i
\(95\) 22433.9i 2.48576i
\(96\) 7688.07 + 4537.94i 0.834209 + 0.492398i
\(97\) −5547.08 −0.589550 −0.294775 − 0.955567i \(-0.595245\pi\)
−0.294775 + 0.955567i \(0.595245\pi\)
\(98\) −1370.26 + 69.1671i −0.142675 + 0.00720191i
\(99\) −331.858 − 331.858i −0.0338596 − 0.0338596i
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.43.1 ✓ 96
4.3 odd 2 448.5.k.a.15.37 96
16.3 odd 4 inner 112.5.k.a.99.1 yes 96
16.13 even 4 448.5.k.a.239.37 96
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
112.5.k.a.43.1 ✓ 96 1.1 even 1 trivial
112.5.k.a.99.1 yes 96 16.3 odd 4 inner
448.5.k.a.15.37 96 4.3 odd 2
448.5.k.a.239.37 96 16.13 even 4