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Properties

Label 700.2.bj.a.323.71
Level $700$
Weight $2$
Character 700.323
Analytic conductor $5.590$
Analytic rank $0$
Dimension $720$
Inner twists $4$

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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] = [700,2,Mod(127,700)] 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("700.127"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(700, base_ring=CyclotomicField(20)) chi = DirichletCharacter(H, H._module([10, 1, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 700 = 2^{2} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 700.bj (of order \(20\), degree \(8\), 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: \(5.58952814149\)
Analytic rank: \(0\)
Dimension: \(720\)
Relative dimension: \(90\) over \(\Q(\zeta_{20})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{20}]$

Embedding invariants

Embedding label 323.71
Character \(\chi\) \(=\) 700.323
Dual form 700.2.bj.a.687.71

$q$-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+(1.11271 + 0.872854i) q^{2} +(1.16698 + 2.29033i) q^{3} +(0.476251 + 1.94247i) q^{4} +(-1.22764 - 1.86893i) q^{5} +(-0.700611 + 3.56708i) q^{6} +(-0.707107 - 0.707107i) q^{7} +(-1.16556 + 2.57710i) q^{8} +(-2.12042 + 2.91850i) q^{9} +(0.265287 - 3.15113i) q^{10} +(1.52142 + 2.09405i) q^{11} +(-3.89312 + 3.35760i) q^{12} +(1.01891 + 6.43312i) q^{13} +(-0.169604 - 1.40401i) q^{14} +(2.84782 - 4.99272i) q^{15} +(-3.54637 + 1.85021i) q^{16} +(-6.85573 - 3.49317i) q^{17} +(-4.90684 + 1.39664i) q^{18} +(1.13155 + 3.48255i) q^{19} +(3.04566 - 3.27474i) q^{20} +(0.794328 - 2.44469i) q^{21} +(-0.134904 + 3.65805i) q^{22} +(0.788484 - 4.97829i) q^{23} +(-7.26261 + 0.337913i) q^{24} +(-1.98578 + 4.58876i) q^{25} +(-4.48143 + 8.04757i) q^{26} +(-1.54227 - 0.244272i) q^{27} +(1.03677 - 1.71029i) q^{28} +(3.64389 + 1.18397i) q^{29} +(7.52672 - 3.06972i) q^{30} +(3.56893 - 1.15962i) q^{31} +(-5.56104 - 1.03672i) q^{32} +(-3.02061 + 5.92828i) q^{33} +(-4.57942 - 9.87094i) q^{34} +(-0.453455 + 2.18961i) q^{35} +(-6.67895 - 2.72890i) q^{36} +(8.69766 - 1.37757i) q^{37} +(-1.78067 + 4.86274i) q^{38} +(-13.5449 + 9.84098i) q^{39} +(6.24732 - 0.985419i) q^{40} +(1.06920 + 0.776820i) q^{41} +(3.01772 - 2.02690i) q^{42} +(3.97309 - 3.97309i) q^{43} +(-3.34305 + 3.95260i) q^{44} +(8.05759 + 0.380020i) q^{45} +(5.22268 - 4.85117i) q^{46} +(6.64051 - 3.38351i) q^{47} +(-8.37614 - 5.96320i) q^{48} +1.00000i q^{49} +(-6.21491 + 3.37267i) q^{50} -19.7784i q^{51} +(-12.0109 + 5.04298i) q^{52} +(9.62486 - 4.90411i) q^{53} +(-1.50289 - 1.61798i) q^{54} +(2.04587 - 5.41417i) q^{55} +(2.64646 - 0.998112i) q^{56} +(-6.65569 + 6.65569i) q^{57} +(3.02116 + 4.49800i) q^{58} +(-6.33527 - 4.60284i) q^{59} +(11.0545 + 3.15402i) q^{60} +(-1.96751 + 1.42948i) q^{61} +(4.98336 + 1.82484i) q^{62} +(3.56306 - 0.564332i) q^{63} +(-5.28293 - 6.00755i) q^{64} +(10.7722 - 9.80185i) q^{65} +(-8.53558 + 3.95991i) q^{66} +(2.71027 - 5.31921i) q^{67} +(3.52032 - 14.9807i) q^{68} +(12.3221 - 4.00369i) q^{69} +(-2.41577 + 2.04060i) q^{70} +(-2.81026 - 0.913110i) q^{71} +(-5.04981 - 8.86623i) q^{72} +(-3.73321 - 0.591283i) q^{73} +(10.8804 + 6.05895i) q^{74} +(-12.8271 + 0.806909i) q^{75} +(-6.22584 + 3.85656i) q^{76} +(0.404914 - 2.55652i) q^{77} +(-23.6613 - 0.872596i) q^{78} +(-1.32582 + 4.08046i) q^{79} +(7.81158 + 4.35651i) q^{80} +(2.10397 + 6.47534i) q^{81} +(0.511661 + 1.79763i) q^{82} +(6.96042 + 3.54651i) q^{83} +(5.12703 + 0.378670i) q^{84} +(1.88792 + 17.1012i) q^{85} +(7.88884 - 0.952974i) q^{86} +(1.54067 + 9.72738i) q^{87} +(-7.16990 + 1.48011i) q^{88} +(0.173868 + 0.239309i) q^{89} +(8.63407 + 7.45595i) q^{90} +(3.82843 - 5.26938i) q^{91} +(10.0457 - 0.839314i) q^{92} +(6.82078 + 6.82078i) q^{93} +(10.3423 + 2.03133i) q^{94} +(5.11949 - 6.39011i) q^{95} +(-4.11522 - 13.9465i) q^{96} +(-1.19096 - 2.33739i) q^{97} +(-0.872854 + 1.11271i) q^{98} -9.33754 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 720 q + 16 q^{10} + 16 q^{12} + 4 q^{13} + 20 q^{17} - 28 q^{18} - 20 q^{20} - 4 q^{22} + 20 q^{25} + 4 q^{30} - 20 q^{37} - 64 q^{40} - 80 q^{42} - 140 q^{44} - 20 q^{45} - 236 q^{48} - 40 q^{50} - 16 q^{52}+ \cdots - 20 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/700\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(351\) \(477\)
\(\chi(n)\) \(1\) \(-1\) \(e\left(\frac{11}{20}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.11271 + 0.872854i 0.786805 + 0.617201i
\(3\) 1.16698 + 2.29033i 0.673758 + 1.32232i 0.934171 + 0.356827i \(0.116141\pi\)
−0.260413 + 0.965497i \(0.583859\pi\)
\(4\) 0.476251 + 1.94247i 0.238126 + 0.971234i
\(5\) −1.22764 − 1.86893i −0.549019 − 0.835810i
\(6\) −0.700611 + 3.56708i −0.286023 + 1.45626i
\(7\) −0.707107 − 0.707107i −0.267261 − 0.267261i
\(8\) −1.16556 + 2.57710i −0.412088 + 0.911144i
\(9\) −2.12042 + 2.91850i −0.706806 + 0.972835i
\(10\) 0.265287 − 3.15113i 0.0838911 − 0.996475i
\(11\) 1.52142 + 2.09405i 0.458725 + 0.631381i 0.974244 − 0.225498i \(-0.0724008\pi\)
−0.515519 + 0.856878i \(0.672401\pi\)
\(12\) −3.89312 + 3.35760i −1.12385 + 0.969256i
\(13\) 1.01891 + 6.43312i 0.282594 + 1.78423i 0.565161 + 0.824980i \(0.308814\pi\)
−0.282568 + 0.959247i \(0.591186\pi\)
\(14\) −0.169604 − 1.40401i −0.0453287 − 0.375237i
\(15\) 2.84782 − 4.99272i 0.735305 − 1.28911i
\(16\) −3.54637 + 1.85021i −0.886592 + 0.462552i
\(17\) −6.85573 − 3.49317i −1.66276 − 0.847218i −0.994667 − 0.103140i \(-0.967111\pi\)
−0.668092 − 0.744078i \(-0.732889\pi\)
\(18\) −4.90684 + 1.39664i −1.15655 + 0.329190i
\(19\) 1.13155 + 3.48255i 0.259595 + 0.798951i 0.992889 + 0.119040i \(0.0379816\pi\)
−0.733295 + 0.679911i \(0.762018\pi\)
\(20\) 3.04566 − 3.27474i 0.681031 − 0.732254i
\(21\) 0.794328 − 2.44469i 0.173337 − 0.533475i
\(22\) −0.134904 + 3.65805i −0.0287615 + 0.779899i
\(23\) 0.788484 − 4.97829i 0.164410 − 1.03805i −0.758118 − 0.652117i \(-0.773881\pi\)
0.922528 − 0.385929i \(-0.126119\pi\)
\(24\) −7.26261 + 0.337913i −1.48247 + 0.0689762i
\(25\) −1.98578 + 4.58876i −0.397156 + 0.917751i
\(26\) −4.48143 + 8.04757i −0.878881 + 1.57826i
\(27\) −1.54227 − 0.244272i −0.296811 − 0.0470102i
\(28\) 1.03677 − 1.71029i 0.195932 − 0.323215i
\(29\) 3.64389 + 1.18397i 0.676653 + 0.219858i 0.627129 − 0.778915i \(-0.284230\pi\)
0.0495235 + 0.998773i \(0.484230\pi\)
\(30\) 7.52672 − 3.06972i 1.37418 − 0.560451i
\(31\) 3.56893 − 1.15962i 0.640999 − 0.208273i 0.0295576 − 0.999563i \(-0.490590\pi\)
0.611441 + 0.791290i \(0.290590\pi\)
\(32\) −5.56104 − 1.03672i −0.983063 − 0.183267i
\(33\) −3.02061 + 5.92828i −0.525820 + 1.03198i
\(34\) −4.57942 − 9.87094i −0.785364 − 1.69285i
\(35\) −0.453455 + 2.18961i −0.0766479 + 0.370111i
\(36\) −6.67895 − 2.72890i −1.11316 − 0.454817i
\(37\) 8.69766 − 1.37757i 1.42989 − 0.226472i 0.607011 − 0.794693i \(-0.292368\pi\)
0.822875 + 0.568222i \(0.192368\pi\)
\(38\) −1.78067 + 4.86274i −0.288863 + 0.788841i
\(39\) −13.5449 + 9.84098i −2.16893 + 1.57582i
\(40\) 6.24732 − 0.985419i 0.987787 − 0.155808i
\(41\) 1.06920 + 0.776820i 0.166981 + 0.121319i 0.668137 − 0.744038i \(-0.267092\pi\)
−0.501156 + 0.865357i \(0.667092\pi\)
\(42\) 3.01772 − 2.02690i 0.465644 − 0.312758i
\(43\) 3.97309 − 3.97309i 0.605891 − 0.605891i −0.335978 − 0.941870i \(-0.609067\pi\)
0.941870 + 0.335978i \(0.109067\pi\)
\(44\) −3.34305 + 3.95260i −0.503984 + 0.595877i
\(45\) 8.05759 + 0.380020i 1.20115 + 0.0566500i
\(46\) 5.22268 − 4.85117i 0.770042 − 0.715266i
\(47\) 6.64051 − 3.38351i 0.968617 − 0.493535i 0.103240 − 0.994656i \(-0.467079\pi\)
0.865377 + 0.501121i \(0.167079\pi\)
\(48\) −8.37614 − 5.96320i −1.20899 − 0.860714i
\(49\) 1.00000i 0.142857i
\(50\) −6.21491 + 3.37267i −0.878921 + 0.476967i
\(51\) − 19.7784i − 2.76953i
\(52\) −12.0109 + 5.04298i −1.66561 + 0.699335i
\(53\) 9.62486 − 4.90411i 1.32208 − 0.673632i 0.356633 − 0.934244i \(-0.383925\pi\)
0.965443 + 0.260613i \(0.0839245\pi\)
\(54\) −1.50289 − 1.61798i −0.204518 − 0.220180i
\(55\) 2.04587 − 5.41417i 0.275865 − 0.730047i
\(56\) 2.64646 − 0.998112i 0.353649 − 0.133378i
\(57\) −6.65569 + 6.65569i −0.881568 + 0.881568i
\(58\) 3.02116 + 4.49800i 0.396698 + 0.590616i
\(59\) −6.33527 − 4.60284i −0.824782 − 0.599239i 0.0932962 − 0.995638i \(-0.470260\pi\)
−0.918078 + 0.396399i \(0.870260\pi\)
\(60\) 11.0545 + 3.15402i 1.42713 + 0.407182i
\(61\) −1.96751 + 1.42948i −0.251914 + 0.183026i −0.706574 − 0.707639i \(-0.749760\pi\)
0.454661 + 0.890665i \(0.349760\pi\)
\(62\) 4.98336 + 1.82484i 0.632888 + 0.231755i
\(63\) 3.56306 − 0.564332i 0.448903 − 0.0710992i
\(64\) −5.28293 − 6.00755i −0.660367 − 0.750943i
\(65\) 10.7722 − 9.80185i 1.33613 − 1.21577i
\(66\) −8.53558 + 3.95991i −1.05066 + 0.487431i
\(67\) 2.71027 − 5.31921i 0.331112 − 0.649845i −0.664093 − 0.747650i \(-0.731182\pi\)
0.995206 + 0.0978052i \(0.0311822\pi\)
\(68\) 3.52032 − 14.9807i 0.426902 − 1.81667i
\(69\) 12.3221 − 4.00369i 1.48341 − 0.481988i
\(70\) −2.41577 + 2.04060i −0.288740 + 0.243898i
\(71\) −2.81026 − 0.913110i −0.333517 − 0.108366i 0.137471 − 0.990506i \(-0.456103\pi\)
−0.470988 + 0.882140i \(0.656103\pi\)
\(72\) −5.04981 − 8.86623i −0.595126 − 1.04490i
\(73\) −3.73321 − 0.591283i −0.436940 − 0.0692045i −0.0659103 − 0.997826i \(-0.520995\pi\)
−0.371029 + 0.928621i \(0.620995\pi\)
\(74\) 10.8804 + 6.05895i 1.26482 + 0.704338i
\(75\) −12.8271 + 0.806909i −1.48115 + 0.0931738i
\(76\) −6.22584 + 3.85656i −0.714152 + 0.442378i
\(77\) 0.404914 − 2.55652i 0.0461442 − 0.291343i
\(78\) −23.6613 − 0.872596i −2.67912 − 0.0988020i
\(79\) −1.32582 + 4.08046i −0.149167 + 0.459087i −0.997523 − 0.0703377i \(-0.977592\pi\)
0.848357 + 0.529425i \(0.177592\pi\)
\(80\) 7.81158 + 4.35651i 0.873362 + 0.487072i
\(81\) 2.10397 + 6.47534i 0.233774 + 0.719482i
\(82\) 0.511661 + 1.79763i 0.0565035 + 0.198515i
\(83\) 6.96042 + 3.54651i 0.764005 + 0.389280i 0.792152 − 0.610324i \(-0.208961\pi\)
−0.0281468 + 0.999604i \(0.508961\pi\)
\(84\) 5.12703 + 0.378670i 0.559405 + 0.0413163i
\(85\) 1.88792 + 17.1012i 0.204774 + 1.85489i
\(86\) 7.88884 − 0.952974i 0.850675 − 0.102762i
\(87\) 1.54067 + 9.72738i 0.165177 + 1.04289i
\(88\) −7.16990 + 1.48011i −0.764314 + 0.157780i
\(89\) 0.173868 + 0.239309i 0.0184300 + 0.0253667i 0.818133 − 0.575030i \(-0.195009\pi\)
−0.799703 + 0.600396i \(0.795009\pi\)
\(90\) 8.63407 + 7.45595i 0.910111 + 0.785926i
\(91\) 3.82843 − 5.26938i 0.401329 − 0.552381i
\(92\) 10.0457 − 0.839314i 1.04734 − 0.0875045i
\(93\) 6.82078 + 6.82078i 0.707282 + 0.707282i
\(94\) 10.3423 + 2.03133i 1.06672 + 0.209515i
\(95\) 5.11949 − 6.39011i 0.525248 − 0.655611i
\(96\) −4.11522 − 13.9465i −0.420007 − 1.42341i
\(97\) −1.19096 − 2.33739i −0.120924 − 0.237326i 0.822608 − 0.568610i \(-0.192518\pi\)
−0.943531 + 0.331283i \(0.892518\pi\)
\(98\) −0.872854 + 1.11271i −0.0881716 + 0.112401i
\(99\) −9.33754 −0.938458
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 700.2.bj.a.323.71 yes 720
4.3 odd 2 inner 700.2.bj.a.323.12 ✓ 720
25.12 odd 20 inner 700.2.bj.a.687.12 yes 720
100.87 even 20 inner 700.2.bj.a.687.71 yes 720
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
700.2.bj.a.323.12 ✓ 720 4.3 odd 2 inner
700.2.bj.a.323.71 yes 720 1.1 even 1 trivial
700.2.bj.a.687.12 yes 720 25.12 odd 20 inner
700.2.bj.a.687.71 yes 720 100.87 even 20 inner