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Properties

Label 370.2.h.d.253.3
Level $370$
Weight $2$
Character 370.253
Analytic conductor $2.954$
Analytic rank $0$
Dimension $10$
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [370,2,Mod(117,370)] 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("370.117"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(370, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([1, 3])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 370 = 2 \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 370.h (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [10,10,2] 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: \(2.95446487479\)
Analytic rank: \(0\)
Dimension: \(10\)
Relative dimension: \(5\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} + \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} + 3x^{8} - 8x^{7} - 26x^{6} + 12x^{5} + 24x^{4} + 166x^{3} + 113x^{2} - 152x + 160 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 253.3
Root \(-0.551861 + 1.73844i\) of defining polynomial
Character \(\chi\) \(=\) 370.253
Dual form 370.2.h.d.117.3

$q$-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+1.00000 q^{2} +(-0.215231 + 0.215231i) q^{3} +1.00000 q^{4} +(2.21058 - 0.336630i) q^{5} +(-0.215231 + 0.215231i) q^{6} +(-0.673260 + 0.673260i) q^{7} +1.00000 q^{8} +2.90735i q^{9} +(2.21058 - 0.336630i) q^{10} -1.69991i q^{11} +(-0.215231 + 0.215231i) q^{12} +2.06054 q^{13} +(-0.673260 + 0.673260i) q^{14} +(-0.403333 + 0.548239i) q^{15} +1.00000 q^{16} -2.58061i q^{17} +2.90735i q^{18} +(0.312630 + 0.312630i) q^{19} +(2.21058 - 0.336630i) q^{20} -0.289813i q^{21} -1.69991i q^{22} -3.40706 q^{23} +(-0.215231 + 0.215231i) q^{24} +(4.77336 - 1.48830i) q^{25} +2.06054 q^{26} +(-1.27145 - 1.27145i) q^{27} +(-0.673260 + 0.673260i) q^{28} +(-2.85460 + 2.85460i) q^{29} +(-0.403333 + 0.548239i) q^{30} +(-0.382954 - 0.382954i) q^{31} +1.00000 q^{32} +(0.365873 + 0.365873i) q^{33} -2.58061i q^{34} +(-1.26166 + 1.71494i) q^{35} +2.90735i q^{36} +(1.45024 - 5.90735i) q^{37} +(0.312630 + 0.312630i) q^{38} +(-0.443492 + 0.443492i) q^{39} +(2.21058 - 0.336630i) q^{40} +3.11626i q^{41} -0.289813i q^{42} -7.38424 q^{43} -1.69991i q^{44} +(0.978702 + 6.42694i) q^{45} -3.40706 q^{46} +(1.26620 - 1.26620i) q^{47} +(-0.215231 + 0.215231i) q^{48} +6.09344i q^{49} +(4.77336 - 1.48830i) q^{50} +(0.555428 + 0.555428i) q^{51} +2.06054 q^{52} +(-2.69991 - 2.69991i) q^{53} +(-1.27145 - 1.27145i) q^{54} +(-0.572240 - 3.75779i) q^{55} +(-0.673260 + 0.673260i) q^{56} -0.134575 q^{57} +(-2.85460 + 2.85460i) q^{58} +(-3.61030 - 3.61030i) q^{59} +(-0.403333 + 0.548239i) q^{60} +(-4.54721 - 4.54721i) q^{61} +(-0.382954 - 0.382954i) q^{62} +(-1.95740 - 1.95740i) q^{63} +1.00000 q^{64} +(4.55499 - 0.693639i) q^{65} +(0.365873 + 0.365873i) q^{66} +(-8.19766 - 8.19766i) q^{67} -2.58061i q^{68} +(0.733305 - 0.733305i) q^{69} +(-1.26166 + 1.71494i) q^{70} -8.12107 q^{71} +2.90735i q^{72} +(-3.81038 + 3.81038i) q^{73} +(1.45024 - 5.90735i) q^{74} +(-0.707048 + 1.34770i) q^{75} +(0.312630 + 0.312630i) q^{76} +(1.14448 + 1.14448i) q^{77} +(-0.443492 + 0.443492i) q^{78} +(3.33149 + 3.33149i) q^{79} +(2.21058 - 0.336630i) q^{80} -8.17474 q^{81} +3.11626i q^{82} +(-6.48170 - 6.48170i) q^{83} -0.289813i q^{84} +(-0.868711 - 5.70466i) q^{85} -7.38424 q^{86} -1.22880i q^{87} -1.69991i q^{88} +(0.486184 - 0.486184i) q^{89} +(0.978702 + 6.42694i) q^{90} +(-1.38728 + 1.38728i) q^{91} -3.40706 q^{92} +0.164847 q^{93} +(1.26620 - 1.26620i) q^{94} +(0.796335 + 0.585854i) q^{95} +(-0.215231 + 0.215231i) q^{96} -0.430462i q^{97} +6.09344i q^{98} +4.94223 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 10 q^{2} + 2 q^{3} + 10 q^{4} + 2 q^{5} + 2 q^{6} - 4 q^{7} + 10 q^{8} + 2 q^{10} + 2 q^{12} - 12 q^{13} - 4 q^{14} - 14 q^{15} + 10 q^{16} + 8 q^{19} + 2 q^{20} + 4 q^{23} + 2 q^{24} + 28 q^{25}+ \cdots + 28 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/370\mathbb{Z}\right)^\times\).

\(n\) \(261\) \(297\)
\(\chi(n)\) \(e\left(\frac{1}{4}\right)\) \(e\left(\frac{3}{4}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) −0.215231 + 0.215231i −0.124264 + 0.124264i −0.766504 − 0.642240i \(-0.778005\pi\)
0.642240 + 0.766504i \(0.278005\pi\)
\(4\) 1.00000 0.500000
\(5\) 2.21058 − 0.336630i 0.988603 − 0.150546i
\(6\) −0.215231 + 0.215231i −0.0878677 + 0.0878677i
\(7\) −0.673260 + 0.673260i −0.254468 + 0.254468i −0.822800 − 0.568331i \(-0.807589\pi\)
0.568331 + 0.822800i \(0.307589\pi\)
\(8\) 1.00000 0.353553
\(9\) 2.90735i 0.969117i
\(10\) 2.21058 − 0.336630i 0.699048 − 0.106452i
\(11\) − 1.69991i − 0.512541i −0.966605 − 0.256271i \(-0.917506\pi\)
0.966605 − 0.256271i \(-0.0824938\pi\)
\(12\) −0.215231 + 0.215231i −0.0621319 + 0.0621319i
\(13\) 2.06054 0.571490 0.285745 − 0.958306i \(-0.407759\pi\)
0.285745 + 0.958306i \(0.407759\pi\)
\(14\) −0.673260 + 0.673260i −0.179936 + 0.179936i
\(15\) −0.403333 + 0.548239i −0.104140 + 0.141555i
\(16\) 1.00000 0.250000
\(17\) − 2.58061i − 0.625890i −0.949771 − 0.312945i \(-0.898684\pi\)
0.949771 − 0.312945i \(-0.101316\pi\)
\(18\) 2.90735i 0.685269i
\(19\) 0.312630 + 0.312630i 0.0717222 + 0.0717222i 0.742058 − 0.670336i \(-0.233850\pi\)
−0.670336 + 0.742058i \(0.733850\pi\)
\(20\) 2.21058 − 0.336630i 0.494302 − 0.0752728i
\(21\) − 0.289813i − 0.0632424i
\(22\) − 1.69991i − 0.362421i
\(23\) −3.40706 −0.710421 −0.355210 − 0.934786i \(-0.615591\pi\)
−0.355210 + 0.934786i \(0.615591\pi\)
\(24\) −0.215231 + 0.215231i −0.0439339 + 0.0439339i
\(25\) 4.77336 − 1.48830i 0.954672 − 0.297660i
\(26\) 2.06054 0.404105
\(27\) −1.27145 − 1.27145i −0.244690 − 0.244690i
\(28\) −0.673260 + 0.673260i −0.127234 + 0.127234i
\(29\) −2.85460 + 2.85460i −0.530086 + 0.530086i −0.920598 − 0.390512i \(-0.872298\pi\)
0.390512 + 0.920598i \(0.372298\pi\)
\(30\) −0.403333 + 0.548239i −0.0736382 + 0.100094i
\(31\) −0.382954 − 0.382954i −0.0687806 − 0.0687806i 0.671880 − 0.740660i \(-0.265487\pi\)
−0.740660 + 0.671880i \(0.765487\pi\)
\(32\) 1.00000 0.176777
\(33\) 0.365873 + 0.365873i 0.0636903 + 0.0636903i
\(34\) − 2.58061i − 0.442571i
\(35\) −1.26166 + 1.71494i −0.213259 + 0.289877i
\(36\) 2.90735i 0.484559i
\(37\) 1.45024 − 5.90735i 0.238418 − 0.971163i
\(38\) 0.312630 + 0.312630i 0.0507152 + 0.0507152i
\(39\) −0.443492 + 0.443492i −0.0710155 + 0.0710155i
\(40\) 2.21058 − 0.336630i 0.349524 − 0.0532259i
\(41\) 3.11626i 0.486678i 0.969941 + 0.243339i \(0.0782427\pi\)
−0.969941 + 0.243339i \(0.921757\pi\)
\(42\) − 0.289813i − 0.0447191i
\(43\) −7.38424 −1.12609 −0.563043 − 0.826428i \(-0.690369\pi\)
−0.563043 + 0.826428i \(0.690369\pi\)
\(44\) − 1.69991i − 0.256271i
\(45\) 0.978702 + 6.42694i 0.145896 + 0.958072i
\(46\) −3.40706 −0.502343
\(47\) 1.26620 − 1.26620i 0.184695 − 0.184695i −0.608703 − 0.793398i \(-0.708310\pi\)
0.793398 + 0.608703i \(0.208310\pi\)
\(48\) −0.215231 + 0.215231i −0.0310659 + 0.0310659i
\(49\) 6.09344i 0.870492i
\(50\) 4.77336 − 1.48830i 0.675055 − 0.210477i
\(51\) 0.555428 + 0.555428i 0.0777754 + 0.0777754i
\(52\) 2.06054 0.285745
\(53\) −2.69991 − 2.69991i −0.370861 − 0.370861i 0.496930 − 0.867791i \(-0.334461\pi\)
−0.867791 + 0.496930i \(0.834461\pi\)
\(54\) −1.27145 − 1.27145i −0.173022 − 0.173022i
\(55\) −0.572240 − 3.75779i −0.0771608 − 0.506700i
\(56\) −0.673260 + 0.673260i −0.0899681 + 0.0899681i
\(57\) −0.134575 −0.0178249
\(58\) −2.85460 + 2.85460i −0.374827 + 0.374827i
\(59\) −3.61030 − 3.61030i −0.470020 − 0.470020i 0.431901 − 0.901921i \(-0.357843\pi\)
−0.901921 + 0.431901i \(0.857843\pi\)
\(60\) −0.403333 + 0.548239i −0.0520701 + 0.0707774i
\(61\) −4.54721 − 4.54721i −0.582211 − 0.582211i 0.353299 − 0.935510i \(-0.385060\pi\)
−0.935510 + 0.353299i \(0.885060\pi\)
\(62\) −0.382954 − 0.382954i −0.0486353 − 0.0486353i
\(63\) −1.95740 − 1.95740i −0.246610 − 0.246610i
\(64\) 1.00000 0.125000
\(65\) 4.55499 − 0.693639i 0.564977 − 0.0860353i
\(66\) 0.365873 + 0.365873i 0.0450358 + 0.0450358i
\(67\) −8.19766 − 8.19766i −1.00150 − 1.00150i −0.999999 − 0.00150430i \(-0.999521\pi\)
−0.00150430 − 0.999999i \(-0.500479\pi\)
\(68\) − 2.58061i − 0.312945i
\(69\) 0.733305 − 0.733305i 0.0882795 − 0.0882795i
\(70\) −1.26166 + 1.71494i −0.150797 + 0.204974i
\(71\) −8.12107 −0.963794 −0.481897 − 0.876228i \(-0.660052\pi\)
−0.481897 + 0.876228i \(0.660052\pi\)
\(72\) 2.90735i 0.342635i
\(73\) −3.81038 + 3.81038i −0.445971 + 0.445971i −0.894013 − 0.448042i \(-0.852122\pi\)
0.448042 + 0.894013i \(0.352122\pi\)
\(74\) 1.45024 − 5.90735i 0.168587 − 0.686716i
\(75\) −0.707048 + 1.34770i −0.0816428 + 0.155619i
\(76\) 0.312630 + 0.312630i 0.0358611 + 0.0358611i
\(77\) 1.14448 + 1.14448i 0.130426 + 0.130426i
\(78\) −0.443492 + 0.443492i −0.0502155 + 0.0502155i
\(79\) 3.33149 + 3.33149i 0.374822 + 0.374822i 0.869230 − 0.494408i \(-0.164615\pi\)
−0.494408 + 0.869230i \(0.664615\pi\)
\(80\) 2.21058 − 0.336630i 0.247151 − 0.0376364i
\(81\) −8.17474 −0.908305
\(82\) 3.11626i 0.344133i
\(83\) −6.48170 − 6.48170i −0.711460 − 0.711460i 0.255381 − 0.966840i \(-0.417799\pi\)
−0.966840 + 0.255381i \(0.917799\pi\)
\(84\) − 0.289813i − 0.0316212i
\(85\) −0.868711 − 5.70466i −0.0942249 − 0.618757i
\(86\) −7.38424 −0.796263
\(87\) − 1.22880i − 0.131741i
\(88\) − 1.69991i − 0.181211i
\(89\) 0.486184 − 0.486184i 0.0515354 − 0.0515354i −0.680869 − 0.732405i \(-0.738398\pi\)
0.732405 + 0.680869i \(0.238398\pi\)
\(90\) 0.978702 + 6.42694i 0.103164 + 0.677459i
\(91\) −1.38728 + 1.38728i −0.145426 + 0.145426i
\(92\) −3.40706 −0.355210
\(93\) 0.164847 0.0170939
\(94\) 1.26620 − 1.26620i 0.130599 − 0.130599i
\(95\) 0.796335 + 0.585854i 0.0817022 + 0.0601073i
\(96\) −0.215231 + 0.215231i −0.0219669 + 0.0219669i
\(97\) − 0.430462i − 0.0437068i −0.999761 − 0.0218534i \(-0.993043\pi\)
0.999761 − 0.0218534i \(-0.00695671\pi\)
\(98\) 6.09344i 0.615531i
\(99\) 4.94223 0.496712
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 370.2.h.d.253.3 yes 10
5.2 odd 4 370.2.g.d.327.3 yes 10
37.6 odd 4 370.2.g.d.43.3 ✓ 10
185.117 even 4 inner 370.2.h.d.117.3 yes 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
370.2.g.d.43.3 ✓ 10 37.6 odd 4
370.2.g.d.327.3 yes 10 5.2 odd 4
370.2.h.d.117.3 yes 10 185.117 even 4 inner
370.2.h.d.253.3 yes 10 1.1 even 1 trivial