Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [230,4,Mod(31,230)] 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("230.31"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(230, base_ring=CyclotomicField(22)) chi = DirichletCharacter(H, H._module([0, 6])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 230 = 2 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 230.g (of order \(11\), degree \(10\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [70,-14,3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(13.5704393013\)
Analytic rank: \(0\)
Dimension: \(70\)
Relative dimension: \(7\) over \(\Q(\zeta_{11})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{11}]$

Embedding invariants

Embedding label 71.5
Character \(\chi\) \(=\) 230.71
Dual form 230.4.g.d.81.5

$q$-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+(1.68251 + 1.08128i) q^{2} +(0.442628 + 3.07854i) q^{3} +(1.66166 + 3.63853i) q^{4} +(-4.79746 - 1.40866i) q^{5} +(-2.58405 + 5.65828i) q^{6} +(21.5377 - 24.8559i) q^{7} +(-1.13852 + 7.91857i) q^{8} +(16.6248 - 4.88148i) q^{9} +(-6.54861 - 7.55750i) q^{10} +(48.3816 - 31.0930i) q^{11} +(-10.4659 + 6.72601i) q^{12} +(-35.6549 - 41.1479i) q^{13} +(63.1135 - 18.5318i) q^{14} +(2.21314 - 15.3927i) q^{15} +(-10.4778 + 12.0920i) q^{16} +(-33.2004 + 72.6987i) q^{17} +(33.2496 + 9.76296i) q^{18} +(58.9512 + 129.085i) q^{19} +(-2.84630 - 19.7964i) q^{20} +(86.0530 + 55.3029i) q^{21} +115.023 q^{22} +(37.6771 - 103.670i) q^{23} -24.8816 q^{24} +(21.0313 + 13.5160i) q^{25} +(-15.4971 - 107.785i) q^{26} +(57.2711 + 125.406i) q^{27} +(126.227 + 37.0636i) q^{28} +(-88.4129 + 193.597i) q^{29} +(20.3675 - 23.5053i) q^{30} +(34.9958 - 243.401i) q^{31} +(-30.7038 + 9.01544i) q^{32} +(117.136 + 135.182i) q^{33} +(-134.468 + 86.4171i) q^{34} +(-138.340 + 88.9057i) q^{35} +(45.3862 + 52.3784i) q^{36} +(-29.4316 + 8.64189i) q^{37} +(-40.3916 + 280.930i) q^{38} +(110.894 - 127.978i) q^{39} +(16.6166 - 36.3853i) q^{40} +(-241.436 - 70.8921i) q^{41} +(84.9868 + 186.095i) q^{42} +(1.82292 + 12.6787i) q^{43} +(193.527 + 124.372i) q^{44} -86.6332 q^{45} +(175.488 - 133.686i) q^{46} +243.478 q^{47} +(-41.8635 - 26.9040i) q^{48} +(-105.126 - 731.168i) q^{49} +(20.7708 + 45.4816i) q^{50} +(-238.502 - 70.0304i) q^{51} +(90.4716 - 198.105i) q^{52} +(305.332 - 352.372i) q^{53} +(-39.2404 + 272.923i) q^{54} +(-275.909 + 81.0141i) q^{55} +(172.302 + 198.847i) q^{56} +(-371.301 + 238.621i) q^{57} +(-358.089 + 230.130i) q^{58} +(327.089 + 377.481i) q^{59} +(59.6843 - 17.5249i) q^{60} +(-34.7351 + 241.588i) q^{61} +(322.066 - 371.684i) q^{62} +(236.727 - 518.359i) q^{63} +(-61.4076 - 18.0309i) q^{64} +(113.089 + 247.631i) q^{65} +(50.9123 + 354.103i) q^{66} +(-224.508 - 144.282i) q^{67} -319.684 q^{68} +(335.829 + 70.1036i) q^{69} -328.890 q^{70} +(260.367 + 167.328i) q^{71} +(19.7267 + 137.202i) q^{72} +(163.084 + 357.103i) q^{73} +(-58.8631 - 17.2838i) q^{74} +(-32.3006 + 70.7285i) q^{75} +(-371.723 + 428.991i) q^{76} +(269.187 - 1872.24i) q^{77} +(324.960 - 95.4170i) q^{78} +(-417.116 - 481.377i) q^{79} +(67.3003 - 43.2513i) q^{80} +(32.8359 - 21.1023i) q^{81} +(-329.564 - 380.337i) q^{82} +(-839.984 + 246.641i) q^{83} +(-58.2304 + 405.001i) q^{84} +(261.686 - 302.001i) q^{85} +(-10.6421 + 23.3030i) q^{86} +(-635.132 - 186.492i) q^{87} +(191.129 + 418.513i) q^{88} +(210.421 + 1463.51i) q^{89} +(-145.761 - 93.6749i) q^{90} -1790.69 q^{91} +(439.812 - 35.1747i) q^{92} +764.811 q^{93} +(409.653 + 263.268i) q^{94} +(-100.979 - 702.324i) q^{95} +(-41.3448 - 90.5324i) q^{96} +(-455.835 - 133.845i) q^{97} +(613.723 - 1343.87i) q^{98} +(652.555 - 753.088i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 70 q - 14 q^{2} + 3 q^{3} - 28 q^{4} - 35 q^{5} + 6 q^{6} - 78 q^{7} - 56 q^{8} - 24 q^{9} - 70 q^{10} - 15 q^{11} - 120 q^{12} - 270 q^{13} + 64 q^{14} + 15 q^{15} - 112 q^{16} + 114 q^{17} - 48 q^{18}+ \cdots + 11285 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/230\mathbb{Z}\right)^\times\).

\(n\) \(47\) \(51\)
\(\chi(n)\) \(1\) \(e\left(\frac{1}{11}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.68251 + 1.08128i 0.594856 + 0.382291i
\(3\) 0.442628 + 3.07854i 0.0851838 + 0.592466i 0.987045 + 0.160441i \(0.0512917\pi\)
−0.901862 + 0.432025i \(0.857799\pi\)
\(4\) 1.66166 + 3.63853i 0.207708 + 0.454816i
\(5\) −4.79746 − 1.40866i −0.429098 − 0.125995i
\(6\) −2.58405 + 5.65828i −0.175822 + 0.384997i
\(7\) 21.5377 − 24.8559i 1.16293 − 1.34209i 0.233820 − 0.972280i \(-0.424877\pi\)
0.929107 − 0.369810i \(-0.120577\pi\)
\(8\) −1.13852 + 7.91857i −0.0503159 + 0.349955i
\(9\) 16.6248 − 4.88148i 0.615733 − 0.180796i
\(10\) −6.54861 − 7.55750i −0.207085 − 0.238989i
\(11\) 48.3816 − 31.0930i 1.32615 − 0.852263i 0.330350 − 0.943858i \(-0.392833\pi\)
0.995796 + 0.0915959i \(0.0291968\pi\)
\(12\) −10.4659 + 6.72601i −0.251770 + 0.161803i
\(13\) −35.6549 − 41.1479i −0.760683 − 0.877875i 0.234875 − 0.972026i \(-0.424532\pi\)
−0.995558 + 0.0941503i \(0.969987\pi\)
\(14\) 63.1135 − 18.5318i 1.20484 − 0.353774i
\(15\) 2.21314 − 15.3927i 0.0380953 − 0.264959i
\(16\) −10.4778 + 12.0920i −0.163715 + 0.188937i
\(17\) −33.2004 + 72.6987i −0.473663 + 1.03718i 0.510494 + 0.859881i \(0.329463\pi\)
−0.984157 + 0.177297i \(0.943265\pi\)
\(18\) 33.2496 + 9.76296i 0.435389 + 0.127842i
\(19\) 58.9512 + 129.085i 0.711807 + 1.55864i 0.825041 + 0.565072i \(0.191152\pi\)
−0.113234 + 0.993568i \(0.536121\pi\)
\(20\) −2.84630 − 19.7964i −0.0318226 − 0.221331i
\(21\) 86.0530 + 55.3029i 0.894205 + 0.574671i
\(22\) 115.023 1.11468
\(23\) 37.6771 − 103.670i 0.341575 − 0.939854i
\(24\) −24.8816 −0.211622
\(25\) 21.0313 + 13.5160i 0.168251 + 0.108128i
\(26\) −15.4971 − 107.785i −0.116893 − 0.813012i
\(27\) 57.2711 + 125.406i 0.408216 + 0.893868i
\(28\) 126.227 + 37.0636i 0.851953 + 0.250156i
\(29\) −88.4129 + 193.597i −0.566133 + 1.23966i 0.382697 + 0.923874i \(0.374995\pi\)
−0.948831 + 0.315786i \(0.897732\pi\)
\(30\) 20.3675 − 23.5053i 0.123953 − 0.143049i
\(31\) 34.9958 − 243.401i 0.202756 − 1.41020i −0.593304 − 0.804979i \(-0.702177\pi\)
0.796059 − 0.605219i \(-0.206914\pi\)
\(32\) −30.7038 + 9.01544i −0.169616 + 0.0498038i
\(33\) 117.136 + 135.182i 0.617903 + 0.713098i
\(34\) −134.468 + 86.4171i −0.678265 + 0.435895i
\(35\) −138.340 + 88.9057i −0.668106 + 0.429366i
\(36\) 45.3862 + 52.3784i 0.210121 + 0.242493i
\(37\) −29.4316 + 8.64189i −0.130771 + 0.0383978i −0.346463 − 0.938064i \(-0.612617\pi\)
0.215692 + 0.976461i \(0.430799\pi\)
\(38\) −40.3916 + 280.930i −0.172431 + 1.19928i
\(39\) 110.894 − 127.978i 0.455314 − 0.525460i
\(40\) 16.6166 − 36.3853i 0.0656829 − 0.143825i
\(41\) −241.436 − 70.8921i −0.919659 − 0.270036i −0.212558 − 0.977148i \(-0.568179\pi\)
−0.707101 + 0.707112i \(0.749998\pi\)
\(42\) 84.9868 + 186.095i 0.312232 + 0.683693i
\(43\) 1.82292 + 12.6787i 0.00646493 + 0.0449646i 0.992799 − 0.119796i \(-0.0382240\pi\)
−0.986334 + 0.164760i \(0.947315\pi\)
\(44\) 193.527 + 124.372i 0.663073 + 0.426131i
\(45\) −86.6332 −0.286989
\(46\) 175.488 − 133.686i 0.562486 − 0.428497i
\(47\) 243.478 0.755636 0.377818 − 0.925880i \(-0.376675\pi\)
0.377818 + 0.925880i \(0.376675\pi\)
\(48\) −41.8635 − 26.9040i −0.125885 − 0.0809013i
\(49\) −105.126 − 731.168i −0.306490 − 2.13169i
\(50\) 20.7708 + 45.4816i 0.0587486 + 0.128641i
\(51\) −238.502 − 70.0304i −0.654841 − 0.192279i
\(52\) 90.4716 − 198.105i 0.241272 − 0.528312i
\(53\) 305.332 − 352.372i 0.791333 − 0.913247i −0.206540 − 0.978438i \(-0.566220\pi\)
0.997873 + 0.0651916i \(0.0207659\pi\)
\(54\) −39.2404 + 272.923i −0.0988879 + 0.687780i
\(55\) −275.909 + 81.0141i −0.676428 + 0.198617i
\(56\) 172.302 + 198.847i 0.411157 + 0.474500i
\(57\) −371.301 + 238.621i −0.862807 + 0.554493i
\(58\) −358.089 + 230.130i −0.810678 + 0.520991i
\(59\) 327.089 + 377.481i 0.721753 + 0.832947i 0.991517 − 0.129980i \(-0.0414912\pi\)
−0.269764 + 0.962926i \(0.586946\pi\)
\(60\) 59.6843 − 17.5249i 0.128420 − 0.0377076i
\(61\) −34.7351 + 241.588i −0.0729078 + 0.507085i 0.920344 + 0.391109i \(0.127909\pi\)
−0.993252 + 0.115976i \(0.963001\pi\)
\(62\) 322.066 − 371.684i 0.659716 − 0.761353i
\(63\) 236.727 − 518.359i 0.473409 − 1.03662i
\(64\) −61.4076 − 18.0309i −0.119937 − 0.0352166i
\(65\) 113.089 + 247.631i 0.215800 + 0.472537i
\(66\) 50.9123 + 354.103i 0.0949525 + 0.660409i
\(67\) −224.508 − 144.282i −0.409373 − 0.263088i 0.319707 − 0.947516i \(-0.396416\pi\)
−0.729080 + 0.684428i \(0.760052\pi\)
\(68\) −319.684 −0.570109
\(69\) 335.829 + 70.1036i 0.585929 + 0.122311i
\(70\) −328.890 −0.561570
\(71\) 260.367 + 167.328i 0.435210 + 0.279692i 0.739846 − 0.672777i \(-0.234898\pi\)
−0.304636 + 0.952469i \(0.598535\pi\)
\(72\) 19.7267 + 137.202i 0.0322891 + 0.224576i
\(73\) 163.084 + 357.103i 0.261472 + 0.572545i 0.994147 − 0.108034i \(-0.0344556\pi\)
−0.732675 + 0.680579i \(0.761728\pi\)
\(74\) −58.8631 − 17.2838i −0.0924689 − 0.0271513i
\(75\) −32.3006 + 70.7285i −0.0497300 + 0.108894i
\(76\) −371.723 + 428.991i −0.561047 + 0.647483i
\(77\) 269.187 − 1872.24i 0.398399 − 2.77093i
\(78\) 324.960 − 95.4170i 0.471724 − 0.138511i
\(79\) −417.116 − 481.377i −0.594040 − 0.685559i 0.376522 − 0.926408i \(-0.377120\pi\)
−0.970563 + 0.240848i \(0.922574\pi\)
\(80\) 67.3003 − 43.2513i 0.0940550 − 0.0604455i
\(81\) 32.8359 − 21.1023i 0.0450423 − 0.0289469i
\(82\) −329.564 − 380.337i −0.443832 − 0.512210i
\(83\) −839.984 + 246.641i −1.11085 + 0.326174i −0.785153 − 0.619302i \(-0.787416\pi\)
−0.325693 + 0.945476i \(0.605597\pi\)
\(84\) −58.2304 + 405.001i −0.0756364 + 0.526062i
\(85\) 261.686 − 302.001i 0.333927 − 0.385372i
\(86\) −10.6421 + 23.3030i −0.0133438 + 0.0292189i
\(87\) −635.132 − 186.492i −0.782682 − 0.229816i
\(88\) 191.129 + 418.513i 0.231527 + 0.506974i
\(89\) 210.421 + 1463.51i 0.250613 + 1.74305i 0.594550 + 0.804058i \(0.297330\pi\)
−0.343937 + 0.938993i \(0.611761\pi\)
\(90\) −145.761 − 93.6749i −0.170717 − 0.109713i
\(91\) −1790.69 −2.06281
\(92\) 439.812 − 35.1747i 0.498409 − 0.0398610i
\(93\) 764.811 0.852766
\(94\) 409.653 + 263.268i 0.449494 + 0.288872i
\(95\) −100.979 − 702.324i −0.109055 − 0.758494i
\(96\) −41.3448 − 90.5324i −0.0439556 − 0.0962493i
\(97\) −455.835 − 133.845i −0.477144 − 0.140102i 0.0343125 − 0.999411i \(-0.489076\pi\)
−0.511457 + 0.859309i \(0.670894\pi\)
\(98\) 613.723 − 1343.87i 0.632606 − 1.38521i
\(99\) 652.555 − 753.088i 0.662467 − 0.764528i
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 230.4.g.d.71.5 ✓ 70
23.12 even 11 inner 230.4.g.d.81.5 yes 70
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
230.4.g.d.71.5 ✓ 70 1.1 even 1 trivial
230.4.g.d.81.5 yes 70 23.12 even 11 inner