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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 43.8
Character \(\chi\) \(=\) 112.43
Dual form 112.5.k.a.99.8

$q$-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+(-3.71065 - 1.49368i) q^{2} +(-11.6411 - 11.6411i) q^{3} +(11.5379 + 11.0850i) q^{4} +(-10.4274 - 10.4274i) q^{5} +(25.8080 + 60.5840i) q^{6} -18.5203 q^{7} +(-26.2555 - 58.3665i) q^{8} +190.030i q^{9} +(23.1173 + 54.2677i) q^{10} +(-80.4876 + 80.4876i) q^{11} +(-5.27131 - 263.355i) q^{12} +(-103.627 + 103.627i) q^{13} +(68.7222 + 27.6633i) q^{14} +242.773i q^{15} +(10.2441 + 255.795i) q^{16} +311.148 q^{17} +(283.843 - 705.134i) q^{18} +(-482.191 - 482.191i) q^{19} +(-4.72173 - 235.898i) q^{20} +(215.596 + 215.596i) q^{21} +(418.884 - 178.439i) q^{22} +684.267 q^{23} +(-373.807 + 985.092i) q^{24} -407.538i q^{25} +(539.309 - 229.738i) q^{26} +(1269.22 - 1269.22i) q^{27} +(-213.684 - 205.298i) q^{28} +(-408.600 + 408.600i) q^{29} +(362.624 - 900.845i) q^{30} -745.113i q^{31} +(344.063 - 964.467i) q^{32} +1873.93 q^{33} +(-1154.56 - 464.755i) q^{34} +(193.118 + 193.118i) q^{35} +(-2106.49 + 2192.54i) q^{36} +(259.404 + 259.404i) q^{37} +(1069.00 + 2509.48i) q^{38} +2412.66 q^{39} +(-334.835 + 882.388i) q^{40} +2649.35i q^{41} +(-477.970 - 1122.03i) q^{42} +(-1342.55 + 1342.55i) q^{43} +(-1820.86 + 36.4464i) q^{44} +(1981.52 - 1981.52i) q^{45} +(-2539.07 - 1022.07i) q^{46} -2102.77i q^{47} +(2858.48 - 3096.98i) q^{48} +343.000 q^{49} +(-608.731 + 1512.23i) q^{50} +(-3622.10 - 3622.10i) q^{51} +(-2344.34 + 46.9243i) q^{52} +(605.948 + 605.948i) q^{53} +(-6605.46 + 2813.84i) q^{54} +1678.56 q^{55} +(486.258 + 1080.96i) q^{56} +11226.4i q^{57} +(2126.49 - 905.855i) q^{58} +(3306.61 - 3306.61i) q^{59} +(-2691.14 + 2801.08i) q^{60} +(1274.10 - 1274.10i) q^{61} +(-1112.96 + 2764.85i) q^{62} -3519.40i q^{63} +(-2717.30 + 3064.88i) q^{64} +2161.12 q^{65} +(-6953.49 - 2799.04i) q^{66} +(1902.79 + 1902.79i) q^{67} +(3589.98 + 3449.09i) q^{68} +(-7965.61 - 7965.61i) q^{69} +(-428.138 - 1005.05i) q^{70} -1552.68 q^{71} +(11091.4 - 4989.32i) q^{72} +544.771i q^{73} +(-575.091 - 1350.02i) q^{74} +(-4744.19 + 4744.19i) q^{75} +(-218.345 - 10908.5i) q^{76} +(1490.65 - 1490.65i) q^{77} +(-8952.54 - 3603.74i) q^{78} +8750.33i q^{79} +(2560.46 - 2774.10i) q^{80} -14157.9 q^{81} +(3957.28 - 9830.82i) q^{82} +(6932.13 + 6932.13i) q^{83} +(97.6260 + 4877.40i) q^{84} +(-3244.47 - 3244.47i) q^{85} +(6987.05 - 2976.39i) q^{86} +9513.10 q^{87} +(6811.02 + 2584.54i) q^{88} +5732.21i q^{89} +(-10312.5 + 4392.97i) q^{90} +(1919.20 - 1919.20i) q^{91} +(7894.97 + 7585.12i) q^{92} +(-8673.92 + 8673.92i) q^{93} +(-3140.86 + 7802.65i) q^{94} +10056.0i q^{95} +(-15232.7 + 7222.18i) q^{96} -3081.45 q^{97} +(-1272.75 - 512.332i) q^{98} +(-15295.0 - 15295.0i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −3.71065 − 1.49368i −0.927663 − 0.373419i
\(3\) −11.6411 − 11.6411i −1.29345 − 1.29345i −0.932636 − 0.360818i \(-0.882498\pi\)
−0.360818 − 0.932636i \(-0.617502\pi\)
\(4\) 11.5379 + 11.0850i 0.721116 + 0.692815i
\(5\) −10.4274 − 10.4274i −0.417096 − 0.417096i 0.467105 − 0.884202i \(-0.345297\pi\)
−0.884202 + 0.467105i \(0.845297\pi\)
\(6\) 25.8080 + 60.5840i 0.716888 + 1.68289i
\(7\) −18.5203 −0.377964
\(8\) −26.2555 − 58.3665i −0.410242 − 0.911977i
\(9\) 190.030i 2.34605i
\(10\) 23.1173 + 54.2677i 0.231173 + 0.542677i
\(11\) −80.4876 + 80.4876i −0.665187 + 0.665187i −0.956598 − 0.291411i \(-0.905875\pi\)
0.291411 + 0.956598i \(0.405875\pi\)
\(12\) −5.27131 − 263.355i −0.0366063 − 1.82885i
\(13\) −103.627 + 103.627i −0.613178 + 0.613178i −0.943773 − 0.330595i \(-0.892751\pi\)
0.330595 + 0.943773i \(0.392751\pi\)
\(14\) 68.7222 + 27.6633i 0.350624 + 0.141139i
\(15\) 242.773i 1.07899i
\(16\) 10.2441 + 255.795i 0.0400159 + 0.999199i
\(17\) 311.148 1.07664 0.538318 − 0.842742i \(-0.319060\pi\)
0.538318 + 0.842742i \(0.319060\pi\)
\(18\) 283.843 − 705.134i 0.876059 − 2.17634i
\(19\) −482.191 − 482.191i −1.33571 − 1.33571i −0.900170 − 0.435539i \(-0.856558\pi\)
−0.435539 − 0.900170i \(-0.643442\pi\)
\(20\) −4.72173 − 235.898i −0.0118043 − 0.589745i
\(21\) 215.596 + 215.596i 0.488880 + 0.488880i
\(22\) 418.884 − 178.439i 0.865463 − 0.368675i
\(23\) 684.267 1.29351 0.646755 − 0.762698i \(-0.276126\pi\)
0.646755 + 0.762698i \(0.276126\pi\)
\(24\) −373.807 + 985.092i −0.648971 + 1.71023i
\(25\) − 407.538i − 0.652061i
\(26\) 539.309 − 229.738i 0.797794 − 0.339849i
\(27\) 1269.22 − 1269.22i 1.74105 − 1.74105i
\(28\) −213.684 − 205.298i −0.272556 − 0.261859i
\(29\) −408.600 + 408.600i −0.485850 + 0.485850i −0.906994 − 0.421144i \(-0.861629\pi\)
0.421144 + 0.906994i \(0.361629\pi\)
\(30\) 362.624 − 900.845i 0.402916 − 1.00094i
\(31\) − 745.113i − 0.775351i −0.921796 − 0.387676i \(-0.873278\pi\)
0.921796 − 0.387676i \(-0.126722\pi\)
\(32\) 344.063 − 964.467i 0.335999 − 0.941862i
\(33\) 1873.93 1.72078
\(34\) −1154.56 − 464.755i −0.998756 − 0.402037i
\(35\) 193.118 + 193.118i 0.157648 + 0.157648i
\(36\) −2106.49 + 2192.54i −1.62538 + 1.69177i
\(37\) 259.404 + 259.404i 0.189484 + 0.189484i 0.795473 − 0.605989i \(-0.207222\pi\)
−0.605989 + 0.795473i \(0.707222\pi\)
\(38\) 1069.00 + 2509.48i 0.740307 + 1.73787i
\(39\) 2412.66 1.58623
\(40\) −334.835 + 882.388i −0.209272 + 0.551493i
\(41\) 2649.35i 1.57606i 0.615638 + 0.788029i \(0.288898\pi\)
−0.615638 + 0.788029i \(0.711102\pi\)
\(42\) −477.970 − 1122.03i −0.270958 − 0.636073i
\(43\) −1342.55 + 1342.55i −0.726093 + 0.726093i −0.969839 − 0.243746i \(-0.921624\pi\)
0.243746 + 0.969839i \(0.421624\pi\)
\(44\) −1820.86 + 36.4464i −0.940528 + 0.0188256i
\(45\) 1981.52 − 1981.52i 0.978528 − 0.978528i
\(46\) −2539.07 − 1022.07i −1.19994 − 0.483022i
\(47\) − 2102.77i − 0.951911i −0.879469 − 0.475955i \(-0.842102\pi\)
0.879469 − 0.475955i \(-0.157898\pi\)
\(48\) 2858.48 − 3096.98i 1.24066 − 1.34418i
\(49\) 343.000 0.142857
\(50\) −608.731 + 1512.23i −0.243492 + 0.604893i
\(51\) −3622.10 − 3622.10i −1.39258 − 1.39258i
\(52\) −2344.34 + 46.9243i −0.866990 + 0.0173537i
\(53\) 605.948 + 605.948i 0.215717 + 0.215717i 0.806691 − 0.590974i \(-0.201256\pi\)
−0.590974 + 0.806691i \(0.701256\pi\)
\(54\) −6605.46 + 2813.84i −2.26525 + 0.964964i
\(55\) 1678.56 0.554894
\(56\) 486.258 + 1080.96i 0.155057 + 0.344695i
\(57\) 11226.4i 3.45536i
\(58\) 2126.49 − 905.855i 0.632131 − 0.269279i
\(59\) 3306.61 − 3306.61i 0.949902 − 0.949902i −0.0489014 − 0.998804i \(-0.515572\pi\)
0.998804 + 0.0489014i \(0.0155720\pi\)
\(60\) −2691.14 + 2801.08i −0.747540 + 0.778077i
\(61\) 1274.10 − 1274.10i 0.342409 − 0.342409i −0.514863 − 0.857272i \(-0.672157\pi\)
0.857272 + 0.514863i \(0.172157\pi\)
\(62\) −1112.96 + 2764.85i −0.289531 + 0.719264i
\(63\) − 3519.40i − 0.886722i
\(64\) −2717.30 + 3064.88i −0.663404 + 0.748262i
\(65\) 2161.12 0.511508
\(66\) −6953.49 − 2799.04i −1.59630 − 0.642572i
\(67\) 1902.79 + 1902.79i 0.423879 + 0.423879i 0.886537 − 0.462658i \(-0.153104\pi\)
−0.462658 + 0.886537i \(0.653104\pi\)
\(68\) 3589.98 + 3449.09i 0.776380 + 0.745910i
\(69\) −7965.61 − 7965.61i −1.67309 − 1.67309i
\(70\) −428.138 − 1005.05i −0.0873751 − 0.205113i
\(71\) −1552.68 −0.308010 −0.154005 − 0.988070i \(-0.549217\pi\)
−0.154005 + 0.988070i \(0.549217\pi\)
\(72\) 11091.4 − 4989.32i 2.13954 − 0.962446i
\(73\) 544.771i 0.102228i 0.998693 + 0.0511138i \(0.0162771\pi\)
−0.998693 + 0.0511138i \(0.983723\pi\)
\(74\) −575.091 − 1350.02i −0.105020 − 0.246535i
\(75\) −4744.19 + 4744.19i −0.843411 + 0.843411i
\(76\) −218.345 − 10908.5i −0.0378022 − 1.88860i
\(77\) 1490.65 − 1490.65i 0.251417 − 0.251417i
\(78\) −8952.54 − 3603.74i −1.47149 − 0.592331i
\(79\) 8750.33i 1.40207i 0.713126 + 0.701036i \(0.247279\pi\)
−0.713126 + 0.701036i \(0.752721\pi\)
\(80\) 2560.46 − 2774.10i 0.400072 − 0.433453i
\(81\) −14157.9 −2.15789
\(82\) 3957.28 − 9830.82i 0.588531 − 1.46205i
\(83\) 6932.13 + 6932.13i 1.00626 + 1.00626i 0.999980 + 0.00628040i \(0.00199913\pi\)
0.00628040 + 0.999980i \(0.498001\pi\)
\(84\) 97.6260 + 4877.40i 0.0138359 + 0.691242i
\(85\) −3244.47 − 3244.47i −0.449061 − 0.449061i
\(86\) 6987.05 − 2976.39i 0.944706 − 0.402432i
\(87\) 9513.10 1.25685
\(88\) 6811.02 + 2584.54i 0.879523 + 0.333748i
\(89\) 5732.21i 0.723673i 0.932242 + 0.361836i \(0.117850\pi\)
−0.932242 + 0.361836i \(0.882150\pi\)
\(90\) −10312.5 + 4392.97i −1.27314 + 0.542342i
\(91\) 1919.20 − 1919.20i 0.231759 − 0.231759i
\(92\) 7894.97 + 7585.12i 0.932770 + 0.896162i
\(93\) −8673.92 + 8673.92i −1.00288 + 1.00288i
\(94\) −3140.86 + 7802.65i −0.355462 + 0.883052i
\(95\) 10056.0i 1.11424i
\(96\) −15232.7 + 7222.18i −1.65285 + 0.783656i
\(97\) −3081.45 −0.327500 −0.163750 − 0.986502i \(-0.552359\pi\)
−0.163750 + 0.986502i \(0.552359\pi\)
\(98\) −1272.75 − 512.332i −0.132523 − 0.0533456i
\(99\) −15295.0 − 15295.0i −1.56056 − 1.56056i
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.43.8 ✓ 96
4.3 odd 2 448.5.k.a.15.46 96
16.3 odd 4 inner 112.5.k.a.99.8 yes 96
16.13 even 4 448.5.k.a.239.46 96
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
112.5.k.a.43.8 ✓ 96 1.1 even 1 trivial
112.5.k.a.99.8 yes 96 16.3 odd 4 inner
448.5.k.a.15.46 96 4.3 odd 2
448.5.k.a.239.46 96 16.13 even 4