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Properties

Label 370.2.h.d.253.4
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 253.4
Root \(2.03431 + 0.602710i\) of defining polynomial
Character \(\chi\) \(=\) 370.253
Dual form 370.2.h.d.117.4

$q$-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} +(1.78347 - 1.78347i) q^{3} +1.00000 q^{4} +(-2.22195 + 0.250846i) q^{5} +(1.78347 - 1.78347i) q^{6} +(0.501691 - 0.501691i) q^{7} +1.00000 q^{8} -3.36152i q^{9} +(-2.22195 + 0.250846i) q^{10} -5.77574i q^{11} +(1.78347 - 1.78347i) q^{12} -0.334084 q^{13} +(0.501691 - 0.501691i) q^{14} +(-3.51541 + 4.41016i) q^{15} +1.00000 q^{16} +4.86321i q^{17} -3.36152i q^{18} +(5.60813 + 5.60813i) q^{19} +(-2.22195 + 0.250846i) q^{20} -1.78950i q^{21} -5.77574i q^{22} +1.33747 q^{23} +(1.78347 - 1.78347i) q^{24} +(4.87415 - 1.11473i) q^{25} -0.334084 q^{26} +(-0.644753 - 0.644753i) q^{27} +(0.501691 - 0.501691i) q^{28} +(-7.32635 + 7.32635i) q^{29} +(-3.51541 + 4.41016i) q^{30} +(-2.92991 - 2.92991i) q^{31} +1.00000 q^{32} +(-10.3008 - 10.3008i) q^{33} +4.86321i q^{34} +(-0.988888 + 1.24058i) q^{35} -3.36152i q^{36} +(-6.07201 + 0.361517i) q^{37} +(5.60813 + 5.60813i) q^{38} +(-0.595828 + 0.595828i) q^{39} +(-2.22195 + 0.250846i) q^{40} +7.31524i q^{41} -1.78950i q^{42} +1.15610 q^{43} -5.77574i q^{44} +(0.843222 + 7.46913i) q^{45} +1.33747 q^{46} +(4.83578 - 4.83578i) q^{47} +(1.78347 - 1.78347i) q^{48} +6.49661i q^{49} +(4.87415 - 1.11473i) q^{50} +(8.67338 + 8.67338i) q^{51} -0.334084 q^{52} +(-6.77574 - 6.77574i) q^{53} +(-0.644753 - 0.644753i) q^{54} +(1.44882 + 12.8334i) q^{55} +(0.501691 - 0.501691i) q^{56} +20.0038 q^{57} +(-7.32635 + 7.32635i) q^{58} +(-8.73793 - 8.73793i) q^{59} +(-3.51541 + 4.41016i) q^{60} +(0.472799 + 0.472799i) q^{61} +(-2.92991 - 2.92991i) q^{62} +(-1.68644 - 1.68644i) q^{63} +1.00000 q^{64} +(0.742319 - 0.0838036i) q^{65} +(-10.3008 - 10.3008i) q^{66} +(1.79312 + 1.79312i) q^{67} +4.86321i q^{68} +(2.38533 - 2.38533i) q^{69} +(-0.988888 + 1.24058i) q^{70} -3.33183 q^{71} -3.36152i q^{72} +(4.96073 - 4.96073i) q^{73} +(-6.07201 + 0.361517i) q^{74} +(6.70480 - 10.6810i) q^{75} +(5.60813 + 5.60813i) q^{76} +(-2.89764 - 2.89764i) q^{77} +(-0.595828 + 0.595828i) q^{78} +(5.53177 + 5.53177i) q^{79} +(-2.22195 + 0.250846i) q^{80} +7.78476 q^{81} +7.31524i q^{82} +(4.77799 + 4.77799i) q^{83} -1.78950i q^{84} +(-1.21991 - 10.8058i) q^{85} +1.15610 q^{86} +26.1326i q^{87} -5.77574i q^{88} +(3.08239 - 3.08239i) q^{89} +(0.843222 + 7.46913i) q^{90} +(-0.167607 + 0.167607i) q^{91} +1.33747 q^{92} -10.4508 q^{93} +(4.83578 - 4.83578i) q^{94} +(-13.8678 - 11.0542i) q^{95} +(1.78347 - 1.78347i) q^{96} +3.56694i q^{97} +6.49661i q^{98} -19.4152 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
\(3\) 1.78347 − 1.78347i 1.02969 − 1.02969i 0.0301401 − 0.999546i \(-0.490405\pi\)
0.999546 − 0.0301401i \(-0.00959534\pi\)
\(4\) 1.00000 0.500000
\(5\) −2.22195 + 0.250846i −0.993688 + 0.112182i
\(6\) 1.78347 − 1.78347i 0.728098 − 0.728098i
\(7\) 0.501691 − 0.501691i 0.189621 − 0.189621i −0.605911 − 0.795532i \(-0.707191\pi\)
0.795532 + 0.605911i \(0.207191\pi\)
\(8\) 1.00000 0.353553
\(9\) − 3.36152i − 1.12051i
\(10\) −2.22195 + 0.250846i −0.702643 + 0.0793244i
\(11\) − 5.77574i − 1.74145i −0.491770 − 0.870725i \(-0.663650\pi\)
0.491770 − 0.870725i \(-0.336350\pi\)
\(12\) 1.78347 − 1.78347i 0.514843 − 0.514843i
\(13\) −0.334084 −0.0926583 −0.0463291 − 0.998926i \(-0.514752\pi\)
−0.0463291 + 0.998926i \(0.514752\pi\)
\(14\) 0.501691 − 0.501691i 0.134083 − 0.134083i
\(15\) −3.51541 + 4.41016i −0.907674 + 1.13870i
\(16\) 1.00000 0.250000
\(17\) 4.86321i 1.17950i 0.807585 + 0.589751i \(0.200774\pi\)
−0.807585 + 0.589751i \(0.799226\pi\)
\(18\) − 3.36152i − 0.792317i
\(19\) 5.60813 + 5.60813i 1.28659 + 1.28659i 0.936843 + 0.349751i \(0.113734\pi\)
0.349751 + 0.936843i \(0.386266\pi\)
\(20\) −2.22195 + 0.250846i −0.496844 + 0.0560908i
\(21\) − 1.78950i − 0.390501i
\(22\) − 5.77574i − 1.23139i
\(23\) 1.33747 0.278881 0.139441 − 0.990230i \(-0.455470\pi\)
0.139441 + 0.990230i \(0.455470\pi\)
\(24\) 1.78347 − 1.78347i 0.364049 − 0.364049i
\(25\) 4.87415 − 1.11473i 0.974831 − 0.222947i
\(26\) −0.334084 −0.0655193
\(27\) −0.644753 − 0.644753i −0.124083 − 0.124083i
\(28\) 0.501691 − 0.501691i 0.0948107 − 0.0948107i
\(29\) −7.32635 + 7.32635i −1.36047 + 1.36047i −0.487153 + 0.873316i \(0.661965\pi\)
−0.873316 + 0.487153i \(0.838035\pi\)
\(30\) −3.51541 + 4.41016i −0.641823 + 0.805181i
\(31\) −2.92991 − 2.92991i −0.526228 − 0.526228i 0.393218 − 0.919445i \(-0.371362\pi\)
−0.919445 + 0.393218i \(0.871362\pi\)
\(32\) 1.00000 0.176777
\(33\) −10.3008 − 10.3008i −1.79315 − 1.79315i
\(34\) 4.86321i 0.834033i
\(35\) −0.988888 + 1.24058i −0.167153 + 0.209697i
\(36\) − 3.36152i − 0.560253i
\(37\) −6.07201 + 0.361517i −0.998232 + 0.0594330i
\(38\) 5.60813 + 5.60813i 0.909759 + 0.909759i
\(39\) −0.595828 + 0.595828i −0.0954089 + 0.0954089i
\(40\) −2.22195 + 0.250846i −0.351322 + 0.0396622i
\(41\) 7.31524i 1.14245i 0.820794 + 0.571224i \(0.193531\pi\)
−0.820794 + 0.571224i \(0.806469\pi\)
\(42\) − 1.78950i − 0.276126i
\(43\) 1.15610 0.176303 0.0881516 − 0.996107i \(-0.471904\pi\)
0.0881516 + 0.996107i \(0.471904\pi\)
\(44\) − 5.77574i − 0.870725i
\(45\) 0.843222 + 7.46913i 0.125700 + 1.11343i
\(46\) 1.33747 0.197199
\(47\) 4.83578 − 4.83578i 0.705370 − 0.705370i −0.260188 − 0.965558i \(-0.583784\pi\)
0.965558 + 0.260188i \(0.0837844\pi\)
\(48\) 1.78347 − 1.78347i 0.257421 − 0.257421i
\(49\) 6.49661i 0.928087i
\(50\) 4.87415 − 1.11473i 0.689309 − 0.157647i
\(51\) 8.67338 + 8.67338i 1.21452 + 1.21452i
\(52\) −0.334084 −0.0463291
\(53\) −6.77574 − 6.77574i −0.930719 − 0.930719i 0.0670316 − 0.997751i \(-0.478647\pi\)
−0.997751 + 0.0670316i \(0.978647\pi\)
\(54\) −0.644753 − 0.644753i −0.0877398 − 0.0877398i
\(55\) 1.44882 + 12.8334i 0.195359 + 1.73046i
\(56\) 0.501691 − 0.501691i 0.0670413 − 0.0670413i
\(57\) 20.0038 2.64957
\(58\) −7.32635 + 7.32635i −0.961997 + 0.961997i
\(59\) −8.73793 − 8.73793i −1.13758 − 1.13758i −0.988883 − 0.148698i \(-0.952492\pi\)
−0.148698 − 0.988883i \(-0.547508\pi\)
\(60\) −3.51541 + 4.41016i −0.453837 + 0.569349i
\(61\) 0.472799 + 0.472799i 0.0605357 + 0.0605357i 0.736727 − 0.676191i \(-0.236371\pi\)
−0.676191 + 0.736727i \(0.736371\pi\)
\(62\) −2.92991 − 2.92991i −0.372099 − 0.372099i
\(63\) −1.68644 − 1.68644i −0.212472 − 0.212472i
\(64\) 1.00000 0.125000
\(65\) 0.742319 − 0.0838036i 0.0920734 − 0.0103946i
\(66\) −10.3008 − 10.3008i −1.26795 − 1.26795i
\(67\) 1.79312 + 1.79312i 0.219065 + 0.219065i 0.808104 − 0.589040i \(-0.200494\pi\)
−0.589040 + 0.808104i \(0.700494\pi\)
\(68\) 4.86321i 0.589751i
\(69\) 2.38533 − 2.38533i 0.287160 − 0.287160i
\(70\) −0.988888 + 1.24058i −0.118195 + 0.148278i
\(71\) −3.33183 −0.395416 −0.197708 − 0.980261i \(-0.563350\pi\)
−0.197708 + 0.980261i \(0.563350\pi\)
\(72\) − 3.36152i − 0.396159i
\(73\) 4.96073 − 4.96073i 0.580609 − 0.580609i −0.354461 − 0.935071i \(-0.615336\pi\)
0.935071 + 0.354461i \(0.115336\pi\)
\(74\) −6.07201 + 0.361517i −0.705857 + 0.0420254i
\(75\) 6.70480 − 10.6810i 0.774204 − 1.23333i
\(76\) 5.60813 + 5.60813i 0.643297 + 0.643297i
\(77\) −2.89764 − 2.89764i −0.330216 − 0.330216i
\(78\) −0.595828 + 0.595828i −0.0674643 + 0.0674643i
\(79\) 5.53177 + 5.53177i 0.622373 + 0.622373i 0.946138 − 0.323765i \(-0.104949\pi\)
−0.323765 + 0.946138i \(0.604949\pi\)
\(80\) −2.22195 + 0.250846i −0.248422 + 0.0280454i
\(81\) 7.78476 0.864973
\(82\) 7.31524i 0.807833i
\(83\) 4.77799 + 4.77799i 0.524453 + 0.524453i 0.918913 − 0.394460i \(-0.129068\pi\)
−0.394460 + 0.918913i \(0.629068\pi\)
\(84\) − 1.78950i − 0.195251i
\(85\) −1.21991 − 10.8058i −0.132318 − 1.17206i
\(86\) 1.15610 0.124665
\(87\) 26.1326i 2.80171i
\(88\) − 5.77574i − 0.615696i
\(89\) 3.08239 − 3.08239i 0.326733 − 0.326733i −0.524610 − 0.851343i \(-0.675789\pi\)
0.851343 + 0.524610i \(0.175789\pi\)
\(90\) 0.843222 + 7.46913i 0.0888834 + 0.787316i
\(91\) −0.167607 + 0.167607i −0.0175700 + 0.0175700i
\(92\) 1.33747 0.139441
\(93\) −10.4508 −1.08370
\(94\) 4.83578 − 4.83578i 0.498772 − 0.498772i
\(95\) −13.8678 − 11.0542i −1.42280 − 1.13414i
\(96\) 1.78347 − 1.78347i 0.182024 − 0.182024i
\(97\) 3.56694i 0.362167i 0.983468 + 0.181084i \(0.0579605\pi\)
−0.983468 + 0.181084i \(0.942039\pi\)
\(98\) 6.49661i 0.656257i
\(99\) −19.4152 −1.95131
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.253.4 yes 10
5.2 odd 4 370.2.g.d.327.2 yes 10
37.6 odd 4 370.2.g.d.43.2 ✓ 10
185.117 even 4 inner 370.2.h.d.117.4 yes 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
370.2.g.d.43.2 ✓ 10 37.6 odd 4
370.2.g.d.327.2 yes 10 5.2 odd 4
370.2.h.d.117.4 yes 10 185.117 even 4 inner
370.2.h.d.253.4 yes 10 1.1 even 1 trivial