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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [230,4,Mod(7,230)] 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("230.7"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(230, base_ring=CyclotomicField(44)) chi = DirichletCharacter(H, H._module([11, 38])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 230 = 2 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 230.l (of order \(44\), degree \(20\), 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: \(13.5704393013\)
Analytic rank: \(0\)
Dimension: \(720\)
Relative dimension: \(36\) over \(\Q(\zeta_{44})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{44}]$

Embedding invariants

Embedding label 33.14
Character \(\chi\) \(=\) 230.33
Dual form 230.4.l.a.7.14

$q$-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+(-1.99490 - 0.142678i) q^{2} +(1.08789 + 5.00093i) q^{3} +(3.95929 + 0.569259i) q^{4} +(3.50493 + 10.6168i) q^{5} +(-1.45670 - 10.1316i) q^{6} +(-3.80039 - 6.95990i) q^{7} +(-7.81717 - 1.70052i) q^{8} +(0.734227 - 0.335310i) q^{9} +(-5.47721 - 21.6795i) q^{10} +(31.8728 + 27.6180i) q^{11} +(1.46043 + 20.4194i) q^{12} +(80.9600 + 44.2075i) q^{13} +(6.58839 + 14.4266i) q^{14} +(-49.2807 + 29.0777i) q^{15} +(15.3519 + 4.50772i) q^{16} +(-23.8116 + 31.8086i) q^{17} +(-1.51255 + 0.564153i) q^{18} +(7.08724 - 49.2928i) q^{19} +(7.83332 + 44.0300i) q^{20} +(30.6716 - 26.5771i) q^{21} +(-59.6428 - 59.6428i) q^{22} +(-44.9587 - 100.726i) q^{23} -40.9431i q^{24} +(-100.431 + 74.4219i) q^{25} +(-155.200 - 99.7410i) q^{26} +(85.2857 + 113.928i) q^{27} +(-11.0849 - 29.7196i) q^{28} +(-198.773 + 28.5792i) q^{29} +(102.459 - 50.9760i) q^{30} +(162.337 - 104.328i) q^{31} +(-29.9824 - 11.1829i) q^{32} +(-103.442 + 189.439i) q^{33} +(52.0403 - 60.0577i) q^{34} +(60.5715 - 64.7418i) q^{35} +(3.09789 - 0.909623i) q^{36} +(-89.1022 + 238.892i) q^{37} +(-21.1714 + 97.3233i) q^{38} +(-133.004 + 452.968i) q^{39} +(-9.34460 - 88.9532i) q^{40} +(-53.7303 + 117.653i) q^{41} +(-64.9789 + 48.6426i) q^{42} +(-148.033 + 32.2027i) q^{43} +(110.472 + 127.491i) q^{44} +(6.13332 + 6.61987i) q^{45} +(75.3168 + 207.353i) q^{46} +(-433.707 + 433.707i) q^{47} +(-5.84170 + 81.6777i) q^{48} +(151.443 - 235.649i) q^{49} +(210.969 - 134.135i) q^{50} +(-184.977 - 84.4761i) q^{51} +(295.378 + 221.117i) q^{52} +(246.251 - 134.463i) q^{53} +(-153.882 - 239.445i) q^{54} +(-181.501 + 435.185i) q^{55} +(17.8729 + 60.8694i) q^{56} +(254.220 - 18.1822i) q^{57} +(400.610 - 28.6522i) q^{58} +(-143.975 - 490.335i) q^{59} +(-211.669 + 87.0735i) q^{60} +(-233.353 - 363.104i) q^{61} +(-338.733 + 184.962i) q^{62} +(-5.12407 - 3.83583i) q^{63} +(58.2164 + 26.5866i) q^{64} +(-185.581 + 1014.48i) q^{65} +(233.385 - 363.154i) q^{66} +(4.59708 - 64.2756i) q^{67} +(-112.384 + 112.384i) q^{68} +(454.814 - 334.414i) q^{69} +(-130.072 + 120.511i) q^{70} +(735.767 + 849.121i) q^{71} +(-6.30978 + 1.37261i) q^{72} +(-578.166 + 432.809i) q^{73} +(211.835 - 463.854i) q^{74} +(-481.437 - 421.286i) q^{75} +(56.1208 - 191.130i) q^{76} +(71.0890 - 326.791i) q^{77} +(329.958 - 884.652i) q^{78} +(769.807 - 226.036i) q^{79} +(5.94989 + 178.786i) q^{80} +(-462.695 + 533.979i) q^{81} +(123.973 - 227.040i) q^{82} +(565.718 + 211.002i) q^{83} +(136.567 - 87.7662i) q^{84} +(-421.162 - 141.315i) q^{85} +(299.907 - 43.1201i) q^{86} +(-359.165 - 962.959i) q^{87} +(-202.191 - 270.095i) q^{88} +(-1166.03 - 749.365i) q^{89} +(-11.2909 - 14.0811i) q^{90} -731.480i q^{91} +(-120.665 - 424.396i) q^{92} +(698.341 + 698.341i) q^{93} +(927.084 - 803.323i) q^{94} +(548.170 - 97.5243i) q^{95} +(23.3073 - 162.106i) q^{96} +(763.302 - 284.697i) q^{97} +(-335.735 + 448.490i) q^{98} +(32.6625 + 9.59056i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 720 q + 16 q^{3} + 16 q^{6} + 64 q^{12} - 192 q^{13} + 1152 q^{16} + 144 q^{18} + 1660 q^{23} - 880 q^{25} - 304 q^{26} - 728 q^{27} - 352 q^{28} - 608 q^{31} - 3872 q^{33} - 688 q^{35} - 2816 q^{36} - 2376 q^{37}+ \cdots + 14352 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(47\) \(51\)
\(\chi(n)\) \(e\left(\frac{3}{4}\right)\) \(e\left(\frac{3}{22}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.99490 − 0.142678i −0.705305 − 0.0504444i
\(3\) 1.08789 + 5.00093i 0.209364 + 0.962430i 0.955122 + 0.296213i \(0.0957239\pi\)
−0.745758 + 0.666217i \(0.767912\pi\)
\(4\) 3.95929 + 0.569259i 0.494911 + 0.0711574i
\(5\) 3.50493 + 10.6168i 0.313490 + 0.949591i
\(6\) −1.45670 − 10.1316i −0.0991162 − 0.689368i
\(7\) −3.80039 − 6.95990i −0.205202 − 0.375799i 0.754729 − 0.656037i \(-0.227769\pi\)
−0.959931 + 0.280238i \(0.909587\pi\)
\(8\) −7.81717 − 1.70052i −0.345474 − 0.0751532i
\(9\) 0.734227 − 0.335310i 0.0271936 − 0.0124189i
\(10\) −5.47721 − 21.6795i −0.173205 − 0.685566i
\(11\) 31.8728 + 27.6180i 0.873638 + 0.757012i 0.971212 − 0.238216i \(-0.0765628\pi\)
−0.0975738 + 0.995228i \(0.531108\pi\)
\(12\) 1.46043 + 20.4194i 0.0351324 + 0.491215i
\(13\) 80.9600 + 44.2075i 1.72725 + 0.943151i 0.949044 + 0.315144i \(0.102053\pi\)
0.778208 + 0.628007i \(0.216129\pi\)
\(14\) 6.58839 + 14.4266i 0.125773 + 0.275404i
\(15\) −49.2807 + 29.0777i −0.848282 + 0.500523i
\(16\) 15.3519 + 4.50772i 0.239873 + 0.0704331i
\(17\) −23.8116 + 31.8086i −0.339715 + 0.453807i −0.937569 − 0.347798i \(-0.886929\pi\)
0.597854 + 0.801605i \(0.296020\pi\)
\(18\) −1.51255 + 0.564153i −0.0198062 + 0.00738734i
\(19\) 7.08724 − 49.2928i 0.0855750 − 0.595187i −0.901238 − 0.433324i \(-0.857341\pi\)
0.986813 − 0.161863i \(-0.0517503\pi\)
\(20\) 7.83332 + 44.0300i 0.0875792 + 0.492270i
\(21\) 30.6716 − 26.5771i 0.318719 − 0.276171i
\(22\) −59.6428 − 59.6428i −0.577995 − 0.577995i
\(23\) −44.9587 − 100.726i −0.407588 − 0.913166i
\(24\) − 40.9431i − 0.348229i
\(25\) −100.431 + 74.4219i −0.803448 + 0.595375i
\(26\) −155.200 − 99.7410i −1.17066 − 0.752339i
\(27\) 85.2857 + 113.928i 0.607898 + 0.812056i
\(28\) −11.0849 − 29.7196i −0.0748157 − 0.200589i
\(29\) −198.773 + 28.5792i −1.27280 + 0.183001i −0.745401 − 0.666617i \(-0.767742\pi\)
−0.527399 + 0.849618i \(0.676833\pi\)
\(30\) 102.459 − 50.9760i 0.623546 − 0.310230i
\(31\) 162.337 − 104.328i 0.940537 − 0.604447i 0.0219901 − 0.999758i \(-0.493000\pi\)
0.918547 + 0.395312i \(0.129363\pi\)
\(32\) −29.9824 − 11.1829i −0.165631 − 0.0617771i
\(33\) −103.442 + 189.439i −0.545663 + 0.999307i
\(34\) 52.0403 − 60.0577i 0.262495 − 0.302935i
\(35\) 60.5715 − 64.7418i 0.292527 − 0.312667i
\(36\) 3.09789 − 0.909623i 0.0143421 − 0.00421122i
\(37\) −89.1022 + 238.892i −0.395900 + 1.06145i 0.574194 + 0.818720i \(0.305316\pi\)
−0.970094 + 0.242730i \(0.921957\pi\)
\(38\) −21.1714 + 97.3233i −0.0903803 + 0.415472i
\(39\) −133.004 + 452.968i −0.546093 + 1.85982i
\(40\) −9.34460 − 88.9532i −0.0369378 − 0.351619i
\(41\) −53.7303 + 117.653i −0.204665 + 0.448154i −0.983933 − 0.178537i \(-0.942864\pi\)
0.779268 + 0.626690i \(0.215591\pi\)
\(42\) −64.9789 + 48.6426i −0.238725 + 0.178707i
\(43\) −148.033 + 32.2027i −0.524997 + 0.114206i −0.467250 − 0.884125i \(-0.654755\pi\)
−0.0577467 + 0.998331i \(0.518392\pi\)
\(44\) 110.472 + 127.491i 0.378506 + 0.436819i
\(45\) 6.13332 + 6.61987i 0.0203178 + 0.0219296i
\(46\) 75.3168 + 207.353i 0.241410 + 0.664621i
\(47\) −433.707 + 433.707i −1.34601 + 1.34601i −0.456069 + 0.889945i \(0.650743\pi\)
−0.889945 + 0.456069i \(0.849257\pi\)
\(48\) −5.84170 + 81.6777i −0.0175662 + 0.245607i
\(49\) 151.443 − 235.649i 0.441524 − 0.687024i
\(50\) 210.969 − 134.135i 0.596709 − 0.379392i
\(51\) −184.977 − 84.4761i −0.507881 − 0.231942i
\(52\) 295.378 + 221.117i 0.787723 + 0.589682i
\(53\) 246.251 − 134.463i 0.638212 − 0.348490i −0.127345 − 0.991859i \(-0.540645\pi\)
0.765557 + 0.643368i \(0.222464\pi\)
\(54\) −153.882 − 239.445i −0.387790 − 0.603412i
\(55\) −181.501 + 435.185i −0.444975 + 1.06692i
\(56\) 17.8729 + 60.8694i 0.0426493 + 0.145250i
\(57\) 254.220 − 18.1822i 0.590742 − 0.0422507i
\(58\) 400.610 − 28.6522i 0.906943 − 0.0648659i
\(59\) −143.975 − 490.335i −0.317695 − 1.08197i −0.951288 − 0.308303i \(-0.900239\pi\)
0.633594 − 0.773666i \(-0.281579\pi\)
\(60\) −211.669 + 87.0735i −0.455440 + 0.187352i
\(61\) −233.353 − 363.104i −0.489799 − 0.762142i 0.505095 − 0.863064i \(-0.331457\pi\)
−0.994894 + 0.100921i \(0.967821\pi\)
\(62\) −338.733 + 184.962i −0.693857 + 0.378874i
\(63\) −5.12407 − 3.83583i −0.0102472 − 0.00767095i
\(64\) 58.2164 + 26.5866i 0.113704 + 0.0519269i
\(65\) −185.581 + 1014.48i −0.354131 + 1.93585i
\(66\) 233.385 − 363.154i 0.435268 − 0.677290i
\(67\) 4.59708 − 64.2756i 0.00838243 − 0.117202i −0.991538 − 0.129817i \(-0.958561\pi\)
0.999920 + 0.0126151i \(0.00401562\pi\)
\(68\) −112.384 + 112.384i −0.200421 + 0.200421i
\(69\) 454.814 − 334.414i 0.793524 − 0.583459i
\(70\) −130.072 + 120.511i −0.222093 + 0.205770i
\(71\) 735.767 + 849.121i 1.22985 + 1.41933i 0.874804 + 0.484478i \(0.160990\pi\)
0.355049 + 0.934848i \(0.384464\pi\)
\(72\) −6.30978 + 1.37261i −0.0103280 + 0.00224671i
\(73\) −578.166 + 432.809i −0.926975 + 0.693925i −0.952109 − 0.305759i \(-0.901090\pi\)
0.0251339 + 0.999684i \(0.491999\pi\)
\(74\) 211.835 − 463.854i 0.332775 − 0.728675i
\(75\) −481.437 − 421.286i −0.741220 − 0.648612i
\(76\) 56.1208 − 191.130i 0.0847039 − 0.288475i
\(77\) 71.0890 − 326.791i 0.105212 − 0.483653i
\(78\) 329.958 − 884.652i 0.478980 − 1.28419i
\(79\) 769.807 − 226.036i 1.09633 − 0.321911i 0.316936 − 0.948447i \(-0.397346\pi\)
0.779393 + 0.626535i \(0.215528\pi\)
\(80\) 5.94989 + 178.786i 0.00831522 + 0.249862i
\(81\) −462.695 + 533.979i −0.634698 + 0.732481i
\(82\) 123.973 − 227.040i 0.166958 − 0.305761i
\(83\) 565.718 + 211.002i 0.748140 + 0.279042i 0.694475 − 0.719517i \(-0.255637\pi\)
0.0536655 + 0.998559i \(0.482910\pi\)
\(84\) 136.567 − 87.7662i 0.177389 − 0.114001i
\(85\) −421.162 − 141.315i −0.537428 − 0.180327i
\(86\) 299.907 − 43.1201i 0.376044 − 0.0540670i
\(87\) −359.165 − 962.959i −0.442604 − 1.18667i
\(88\) −202.191 − 270.095i −0.244927 − 0.327184i
\(89\) −1166.03 − 749.365i −1.38876 − 0.892500i −0.389168 − 0.921167i \(-0.627238\pi\)
−0.999589 + 0.0286663i \(0.990874\pi\)
\(90\) −11.2909 − 14.0811i −0.0132240 − 0.0164920i
\(91\) − 731.480i − 0.842636i
\(92\) −120.665 − 424.396i −0.136741 − 0.480938i
\(93\) 698.341 + 698.341i 0.778652 + 0.778652i
\(94\) 927.084 − 803.323i 1.01725 − 0.881451i
\(95\) 548.170 − 97.5243i 0.592011 − 0.105324i
\(96\) 23.3073 − 162.106i 0.0247790 − 0.172342i
\(97\) 763.302 − 284.697i 0.798985 − 0.298006i 0.0833862 − 0.996517i \(-0.473427\pi\)
0.715599 + 0.698511i \(0.246154\pi\)
\(98\) −335.735 + 448.490i −0.346065 + 0.462289i
\(99\) 32.6625 + 9.59056i 0.0331586 + 0.00973624i
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.l.a.33.14 yes 720
5.2 odd 4 inner 230.4.l.a.217.32 yes 720
23.7 odd 22 inner 230.4.l.a.53.32 yes 720
115.7 even 44 inner 230.4.l.a.7.14 ✓ 720
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
230.4.l.a.7.14 ✓ 720 115.7 even 44 inner
230.4.l.a.33.14 yes 720 1.1 even 1 trivial
230.4.l.a.53.32 yes 720 23.7 odd 22 inner
230.4.l.a.217.32 yes 720 5.2 odd 4 inner