LMFDB - Embedded newform 32.16.a.c.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [32,16,Mod(1,32)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("32.1"); S:= CuspForms(chi, 16); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(32, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 16, names="a")

Level:	$ N $	$=$	$ 32 = 2^{5} $
Weight:	$ k $	$=$	$ 16 $
Character orbit:	$[\chi]$	$=$	32.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,0,-2912] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$45.6619216320$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	$\mathbb{Q}[x]/(x^{4} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 10174x^{2} - 369720x - 3191805 $ x^4 - 10174x^2 - 369720x - 3191805
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{27}\cdot 3\cdot 5 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$116.540$ of defining polynomial
Character	$\chi$	$=$	32.1

$q$-expansion

comment:q-expansion

sage:f.q_expansion() # note that sage often uses an isomorphic number field

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+634.991 q^{3} +134146. q^{5} +1.14384e6 q^{7} -1.39457e7 q^{9} -9.38825e7 q^{11} +1.44001e8 q^{13} +8.51813e7 q^{15} -1.07455e9 q^{17} +1.01588e9 q^{19} +7.26327e8 q^{21} +3.73320e9 q^{23} -1.25225e10 q^{25} -1.79668e10 q^{27} +5.59549e10 q^{29} -1.38114e11 q^{31} -5.96145e10 q^{33} +1.53441e11 q^{35} -1.62068e11 q^{37} +9.14394e10 q^{39} +1.27301e12 q^{41} -2.57947e12 q^{43} -1.87075e12 q^{45} -5.19038e12 q^{47} -3.43919e12 q^{49} -6.82329e11 q^{51} +8.52305e12 q^{53} -1.25939e13 q^{55} +6.45076e11 q^{57} -3.13853e13 q^{59} -1.54479e13 q^{61} -1.59516e13 q^{63} +1.93171e13 q^{65} -6.71953e13 q^{67} +2.37055e12 q^{69} -1.14103e14 q^{71} +2.31840e12 q^{73} -7.95168e12 q^{75} -1.07386e14 q^{77} +9.87720e13 q^{79} +1.88697e14 q^{81} +3.67908e13 q^{83} -1.44146e14 q^{85} +3.55309e13 q^{87} -5.46013e13 q^{89} +1.64714e14 q^{91} -8.77013e13 q^{93} +1.36276e14 q^{95} +8.64627e14 q^{97} +1.30926e15 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 2912 q^{3} - 98280 q^{5} - 2755776 q^{7} + 26457940 q^{9} + 110853856 q^{11} - 187741448 q^{13} - 442991680 q^{15} + 2121294984 q^{17} + 419203872 q^{19} + 8397459968 q^{21} - 5330808384 q^{23} + 8564955740 q^{25}+ \cdots + 26\!\cdots\!84 q^{99}+O(q^{100}) $ 4 * q - 2912 * q^3 - 98280 * q^5 - 2755776 * q^7 + 26457940 * q^9 + 110853856 * q^11 - 187741448 * q^13 - 442991680 * q^15 + 2121294984 * q^17 + 419203872 * q^19 + 8397459968 * q^21 - 5330808384 * q^23 + 8564955740 * q^25 - 41191036352 * q^27 - 86832359880 * q^29 + 96274977024 * q^31 + 41399413504 * q^33 - 929253215360 * q^35 - 112979590376 * q^37 - 2001683116352 * q^39 - 1534010535960 * q^41 - 3567397370400 * q^43 + 3155482037240 * q^45 - 9031658937472 * q^47 + 8694343447332 * q^49 - 37558794281664 * q^51 - 14370240161064 * q^53 - 46434781152960 * q^55 - 5813065407232 * q^57 - 68701112336800 * q^59 + 4806363233592 * q^61 - 192971404565440 * q^63 + 7651382051280 * q^65 - 167433288699744 * q^67 - 80410496862720 * q^69 - 180096642956992 * q^71 + 173676000667560 * q^73 - 227575010162080 * q^75 - 133277064067584 * q^77 - 232741753133952 * q^79 + 263803331371492 * q^81 + 20415834690336 * q^83 - 342055719314640 * q^85 + 870209244139712 * q^87 + 938354283775848 * q^89 + 1478922417113472 * q^91 - 2098341777250304 * q^93 + 2700969259238080 * q^95 + 1510271716715080 * q^97 + 2629725177868384 * q^99

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	634.991	0.167632	0.0838162	−	0.996481i	$-0.473289\pi$
$3$	634.991	0.167632	0.0838162	+	0.996481i	$0.473289\pi$
$4$	0	0
$5$	134146.	0.767895	0.383947	−	0.923355i	$-0.374564\pi$
$5$	134146.	0.767895	0.383947	+	0.923355i	$0.374564\pi$
$6$	0	0
$7$	1.14384e6	0.524964	0.262482	−	0.964937i	$-0.415459\pi$
$7$	1.14384e6	0.524964	0.262482	+	0.964937i	$0.415459\pi$
$8$	0	0
$9$	−1.39457e7	−0.971899
$10$	0	0
$11$	−9.38825e7	−1.45258	−0.726289	−	0.687390i	$-0.758756\pi$
$11$	−9.38825e7	−1.45258	−0.726289	+	0.687390i	$0.758756\pi$
$12$	0	0
$13$	1.44001e8	0.636489	0.318245	−	0.948009i	$-0.396907\pi$
$13$	1.44001e8	0.636489	0.318245	+	0.948009i	$0.396907\pi$
$14$	0	0
$15$	8.51813e7	0.128724
$16$	0	0
$17$	−1.07455e9	−0.635125	−0.317563	−	0.948237i	$-0.602864\pi$
$17$	−1.07455e9	−0.635125	−0.317563	+	0.948237i	$0.602864\pi$
$18$	0	0
$19$	1.01588e9	0.260730	0.130365	−	0.991466i	$-0.458385\pi$
$19$	1.01588e9	0.260730	0.130365	+	0.991466i	$0.458385\pi$
$20$	0	0
$21$	7.26327e8	0.0880010
$22$	0	0
$23$	3.73320e9	0.228625	0.114312	−	0.993445i	$-0.463534\pi$
$23$	3.73320e9	0.228625	0.114312	+	0.993445i	$0.463534\pi$
$24$	0	0
$25$	−1.25225e10	−0.410338
$26$	0	0
$27$	−1.79668e10	−0.330554
$28$	0	0
$29$	5.59549e10	0.602356	0.301178	−	0.953568i	$-0.402620\pi$
$29$	5.59549e10	0.602356	0.301178	+	0.953568i	$0.402620\pi$
$30$	0	0
$31$	−1.38114e11	−0.901624	−0.450812	−	0.892619i	$-0.648866\pi$
$31$	−1.38114e11	−0.901624	−0.450812	+	0.892619i	$0.648866\pi$
$32$	0	0
$33$	−5.96145e10	−0.243499
$34$	0	0
$35$	1.53441e11	0.403117
$36$	0	0
$37$	−1.62068e11	−0.280662	−0.140331	−	0.990105i	$-0.544817\pi$
$37$	−1.62068e11	−0.280662	−0.140331	+	0.990105i	$0.544817\pi$
$38$	0	0
$39$	9.14394e10	0.106696
$40$	0	0
$41$	1.27301e12	1.02083	0.510415	−	0.859928i	$-0.329492\pi$
$41$	1.27301e12	1.02083	0.510415	+	0.859928i	$0.329492\pi$
$42$	0	0
$43$	−2.57947e12	−1.44716	−0.723582	−	0.690238i	$-0.757506\pi$
$43$	−2.57947e12	−1.44716	−0.723582	+	0.690238i	$0.757506\pi$
$44$	0	0
$45$	−1.87075e12	−0.746316
$46$	0	0
$47$	−5.19038e12	−1.49440	−0.747198	−	0.664602i	$-0.768601\pi$
$47$	−5.19038e12	−1.49440	−0.747198	+	0.664602i	$0.768601\pi$
$48$	0	0
$49$	−3.43919e12	−0.724413
$50$	0	0
$51$	−6.82329e11	−0.106468
$52$	0	0
$53$	8.52305e12	0.996611	0.498306	−	0.867001i	$-0.333956\pi$
$53$	8.52305e12	0.996611	0.498306	+	0.867001i	$0.333956\pi$
$54$	0	0
$55$	−1.25939e13	−1.11543
$56$	0	0
$57$	6.45076e11	0.0437069
$58$	0	0
$59$	−3.13853e13	−1.64186	−0.820931	−	0.571028i	$-0.806545\pi$
$59$	−3.13853e13	−1.64186	−0.820931	+	0.571028i	$0.806545\pi$
$60$	0	0
$61$	−1.54479e13	−0.629355	−0.314677	−	0.949199i	$-0.601896\pi$
$61$	−1.54479e13	−0.629355	−0.314677	+	0.949199i	$0.601896\pi$
$62$	0	0
$63$	−1.59516e13	−0.510212
$64$	0	0
$65$	1.93171e13	0.488757
$66$	0	0
$67$	−6.71953e13	−1.35450	−0.677248	−	0.735755i	$-0.736828\pi$
$67$	−6.71953e13	−1.35450	−0.677248	+	0.735755i	$0.736828\pi$
$68$	0	0
$69$	2.37055e12	0.0383249
$70$	0	0
$71$	−1.14103e14	−1.48888	−0.744439	−	0.667690i	$-0.767283\pi$
$71$	−1.14103e14	−1.48888	−0.744439	+	0.667690i	$0.767283\pi$
$72$	0	0
$73$	2.31840e12	0.0245622	0.0122811	−	0.999925i	$-0.496091\pi$
$73$	2.31840e12	0.0245622	0.0122811	+	0.999925i	$0.496091\pi$
$74$	0	0
$75$	−7.95168e12	−0.0687859
$76$	0	0
$77$	−1.07386e14	−0.762551
$78$	0	0
$79$	9.87720e13	0.578669	0.289335	−	0.957228i	$-0.406566\pi$
$79$	9.87720e13	0.578669	0.289335	+	0.957228i	$0.406566\pi$
$80$	0	0
$81$	1.88697e14	0.916488
$82$	0	0
$83$	3.67908e13	0.148817	0.0744087	−	0.997228i	$-0.476293\pi$
$83$	3.67908e13	0.148817	0.0744087	+	0.997228i	$0.476293\pi$
$84$	0	0
$85$	−1.44146e14	−0.487709
$86$	0	0
$87$	3.55309e13	0.100974
$88$	0	0
$89$	−5.46013e13	−0.130851	−0.0654256	−	0.997857i	$-0.520841\pi$
$89$	−5.46013e13	−0.130851	−0.0654256	+	0.997857i	$0.520841\pi$
$90$	0	0
$91$	1.64714e14	0.334134
$92$	0	0
$93$	−8.77013e13	−0.151142
$94$	0	0
$95$	1.36276e14	0.200214
$96$	0	0
$97$	8.64627e14	1.08653	0.543264	−	0.839562i	$-0.317188\pi$
$97$	8.64627e14	1.08653	0.543264	+	0.839562i	$0.317188\pi$
$98$	0	0
$99$	1.30926e15	1.41176
$100$	0	0
$101$	−3.33628e14	−0.309636	−0.154818	−	0.987943i	$-0.549479\pi$
$101$	−3.33628e14	−0.309636	−0.154818	+	0.987943i	$0.549479\pi$
$102$	0	0
$103$	2.36802e15	1.89717	0.948585	−	0.316521i	$-0.102515\pi$
$103$	2.36802e15	1.89717	0.948585	+	0.316521i	$0.102515\pi$
$104$	0	0
$105$	9.74337e13	0.0675755
$106$	0	0
$107$	7.63016e14	0.459362	0.229681	−	0.973266i	$-0.426232\pi$
$107$	7.63016e14	0.459362	0.229681	+	0.973266i	$0.426232\pi$
$108$	0	0
$109$	−3.39386e15	−1.77826	−0.889130	−	0.457655i	$-0.848689\pi$
$109$	−3.39386e15	−1.77826	−0.889130	+	0.457655i	$0.848689\pi$
$110$	0	0
$111$	−1.02911e14	−0.0470480
$112$	0	0
$113$	4.21910e15	1.68706	0.843531	−	0.537080i	$-0.180473\pi$
$113$	4.21910e15	1.68706	0.843531	+	0.537080i	$0.180473\pi$
$114$	0	0
$115$	5.00793e14	0.175560
$116$	0	0
$117$	−2.00820e15	−0.618603
$118$	0	0
$119$	−1.22911e15	−0.333418
$120$	0	0
$121$	4.63667e15	1.10998
$122$	0	0
$123$	8.08351e14	0.171124
$124$	0	0
$125$	−5.77364e15	−1.08299
$126$	0	0
$127$	1.97771e14	0.0329332	0.0164666	−	0.999864i	$-0.494758\pi$
$127$	1.97771e14	0.0329332	0.0164666	+	0.999864i	$0.494758\pi$
$128$	0	0
$129$	−1.63794e15	−0.242592
$130$	0	0
$131$	−1.01974e16	−1.34572	−0.672858	−	0.739772i	$-0.734934\pi$
$131$	−1.01974e16	−1.34572	−0.672858	+	0.739772i	$0.734934\pi$
$132$	0	0
$133$	1.16201e15	0.136874
$134$	0	0
$135$	−2.41017e15	−0.253831
$136$	0	0
$137$	9.54155e15	0.899942	0.449971	−	0.893043i	$-0.351434\pi$
$137$	9.54155e15	0.899942	0.449971	+	0.893043i	$0.351434\pi$
$138$	0	0
$139$	−6.69800e15	−0.566675	−0.283338	−	0.959020i	$-0.591442\pi$
$139$	−6.69800e15	−0.566675	−0.283338	+	0.959020i	$0.591442\pi$
$140$	0	0
$141$	−3.29585e15	−0.250509
$142$	0	0
$143$	−1.35192e16	−0.924550
$144$	0	0
$145$	7.50611e15	0.462546
$146$	0	0
$147$	−2.18386e15	−0.121435
$148$	0	0
$149$	−3.24398e16	−1.62998	−0.814988	−	0.579478i	$-0.803256\pi$
$149$	−3.24398e16	−1.62998	−0.814988	+	0.579478i	$0.803256\pi$
$150$	0	0
$151$	−3.37588e16	−1.53483	−0.767414	−	0.641152i	$-0.778457\pi$
$151$	−3.37588e16	−1.53483	−0.767414	+	0.641152i	$0.778457\pi$
$152$	0	0
$153$	1.49853e16	0.617278
$154$	0	0
$155$	−1.85274e16	−0.692353
$156$	0	0
$157$	8.10693e15	0.275175	0.137588	−	0.990490i	$-0.456065\pi$
$157$	8.10693e15	0.275175	0.137588	+	0.990490i	$0.456065\pi$
$158$	0	0
$159$	5.41206e15	0.167064
$160$	0	0
$161$	4.27018e15	0.120020
$162$	0	0
$163$	2.95667e16	0.757522	0.378761	−	0.925494i	$-0.376350\pi$
$163$	2.95667e16	0.757522	0.378761	+	0.925494i	$0.376350\pi$
$164$	0	0
$165$	−7.99703e15	−0.186982
$166$	0	0
$167$	8.40284e16	1.79495	0.897475	−	0.441065i	$-0.145399\pi$
$167$	8.40284e16	1.79495	0.897475	+	0.441065i	$0.145399\pi$
$168$	0	0
$169$	−3.04495e16	−0.594882
$170$	0	0
$171$	−1.41672e16	−0.253404
$172$	0	0
$173$	6.13670e16	1.00598	0.502990	−	0.864292i	$-0.332233\pi$
$173$	6.13670e16	1.00598	0.502990	+	0.864292i	$0.332233\pi$
$174$	0	0
$175$	−1.43237e16	−0.215413
$176$	0	0
$177$	−1.99294e16	−0.275229
$178$	0	0
$179$	6.99725e16	0.888238	0.444119	−	0.895968i	$-0.353517\pi$
$179$	6.99725e16	0.888238	0.444119	+	0.895968i	$0.353517\pi$
$180$	0	0
$181$	3.86157e16	0.450998	0.225499	−	0.974243i	$-0.427599\pi$
$181$	3.86157e16	0.450998	0.225499	+	0.974243i	$0.427599\pi$
$182$	0	0
$183$	−9.80927e15	−0.105500
$184$	0	0
$185$	−2.17407e16	−0.215519
$186$	0	0
$187$	1.00881e17	0.922569
$188$	0	0
$189$	−2.05511e16	−0.173529
$190$	0	0
$191$	1.48426e17	1.15813	0.579067	−	0.815280i	$-0.303417\pi$
$191$	1.48426e17	1.15813	0.579067	+	0.815280i	$0.303417\pi$
$192$	0	0
$193$	−1.92668e17	−1.39036	−0.695182	−	0.718833i	$-0.744676\pi$
$193$	−1.92668e17	−1.39036	−0.695182	+	0.718833i	$0.744676\pi$
$194$	0	0
$195$	1.22662e16	0.0819315
$196$	0	0
$197$	2.12947e17	1.31758	0.658788	−	0.752328i	$-0.271069\pi$
$197$	2.12947e17	1.31758	0.658788	+	0.752328i	$0.271069\pi$
$198$	0	0
$199$	1.41451e17	0.811352	0.405676	−	0.914017i	$-0.367036\pi$
$199$	1.41451e17	0.811352	0.405676	+	0.914017i	$0.367036\pi$
$200$	0	0
$201$	−4.26684e16	−0.227057
$202$	0	0
$203$	6.40034e16	0.316215
$204$	0	0
$205$	1.70769e17	0.783890
$206$	0	0
$207$	−5.20621e16	−0.222200
$208$	0	0
$209$	−9.53736e16	−0.378731
$210$	0	0
$211$	−2.30928e17	−0.853803	−0.426902	−	0.904298i	$-0.640395\pi$
$211$	−2.30928e17	−0.853803	−0.426902	+	0.904298i	$0.640395\pi$
$212$	0	0
$213$	−7.24543e16	−0.249584
$214$	0	0
$215$	−3.46025e17	−1.11127
$216$	0	0
$217$	−1.57980e17	−0.473321
$218$	0	0
$219$	1.47216e15	0.00411742
$220$	0	0
$221$	−1.54736e17	−0.404250
$222$	0	0
$223$	−2.92734e17	−0.714802	−0.357401	−	0.933951i	$-0.616337\pi$
$223$	−2.92734e17	−0.714802	−0.357401	+	0.933951i	$0.616337\pi$
$224$	0	0
$225$	1.74635e17	0.398807
$226$	0	0
$227$	−2.75305e17	−0.588329	−0.294165	−	0.955755i	$-0.595041\pi$
$227$	−2.75305e17	−0.588329	−0.294165	+	0.955755i	$0.595041\pi$
$228$	0	0
$229$	8.96352e17	1.79355	0.896773	−	0.442491i	$-0.145905\pi$
$229$	8.96352e17	1.79355	0.896773	+	0.442491i	$0.145905\pi$
$230$	0	0
$231$	−6.81894e16	−0.127828
$232$	0	0
$233$	−4.86086e17	−0.854169	−0.427085	−	0.904212i	$-0.640459\pi$
$233$	−4.86086e17	−0.854169	−0.427085	+	0.904212i	$0.640459\pi$
$234$	0	0
$235$	−6.96267e17	−1.14754
$236$	0	0
$237$	6.27193e16	0.0970037
$238$	0	0
$239$	−5.19083e17	−0.753794	−0.376897	−	0.926255i	$-0.623009\pi$
$239$	−5.19083e17	−0.753794	−0.376897	+	0.926255i	$0.623009\pi$
$240$	0	0
$241$	−4.01529e17	−0.547759	−0.273880	−	0.961764i	$-0.588307\pi$
$241$	−4.01529e17	−0.547759	−0.273880	+	0.961764i	$0.588307\pi$
$242$	0	0
$243$	3.77625e17	0.484187
$244$	0	0
$245$	−4.61353e17	−0.556273
$246$	0	0
$247$	1.46288e17	0.165952
$248$	0	0
$249$	2.33618e16	0.0249466
$250$	0	0
$251$	−9.05569e16	−0.0910686	−0.0455343	−	0.998963i	$-0.514499\pi$
$251$	−9.05569e16	−0.0910686	−0.0455343	+	0.998963i	$0.514499\pi$
$252$	0	0
$253$	−3.50482e17	−0.332095
$254$	0	0
$255$	−9.15315e16	−0.0817559
$256$	0	0
$257$	1.55330e18	1.30845	0.654224	−	0.756301i	$-0.272995\pi$
$257$	1.55330e18	1.30845	0.654224	+	0.756301i	$0.272995\pi$
$258$	0	0
$259$	−1.85379e17	−0.147337
$260$	0	0
$261$	−7.80330e17	−0.585429
$262$	0	0
$263$	−4.03350e17	−0.285768	−0.142884	−	0.989739i	$-0.545638\pi$
$263$	−4.03350e17	−0.285768	−0.142884	+	0.989739i	$0.545638\pi$
$264$	0	0
$265$	1.14333e18	0.765293
$266$	0	0
$267$	−3.46713e16	−0.0219349
$268$	0	0
$269$	−1.16667e18	−0.697922	−0.348961	−	0.937137i	$-0.613465\pi$
$269$	−1.16667e18	−0.697922	−0.348961	+	0.937137i	$0.613465\pi$
$270$	0	0
$271$	2.85152e18	1.61364	0.806821	−	0.590796i	$-0.201186\pi$
$271$	2.85152e18	1.61364	0.806821	+	0.590796i	$0.201186\pi$
$272$	0	0
$273$	1.04592e17	0.0560117
$274$	0	0
$275$	1.17564e18	0.596047
$276$	0	0
$277$	−1.70828e18	−0.820278	−0.410139	−	0.912023i	$-0.634520\pi$
$277$	−1.70828e18	−0.820278	−0.410139	+	0.912023i	$0.634520\pi$
$278$	0	0
$279$	1.92610e18	0.876288
$280$	0	0
$281$	3.12673e18	1.34832	0.674161	−	0.738585i	$-0.264505\pi$
$281$	3.12673e18	1.34832	0.674161	+	0.738585i	$0.264505\pi$
$282$	0	0
$283$	3.93221e18	1.60782	0.803912	−	0.594748i	$-0.202748\pi$
$283$	3.93221e18	1.60782	0.803912	+	0.594748i	$0.202748\pi$
$284$	0	0
$285$	8.65342e16	0.0335623
$286$	0	0
$287$	1.45612e18	0.535899
$288$	0	0
$289$	−1.70777e18	−0.596616
$290$	0	0
$291$	5.49030e17	0.182137
$292$	0	0
$293$	−9.78910e15	−0.00308486	−0.00154243	−	0.999999i	$-0.500491\pi$
$293$	−9.78910e15	−0.00308486	−0.00154243	+	0.999999i	$0.500491\pi$
$294$	0	0
$295$	−4.21021e18	−1.26078
$296$	0	0
$297$	1.68677e18	0.480156
$298$	0	0
$299$	5.37586e17	0.145517
$300$	0	0
$301$	−2.95050e18	−0.759709
$302$	0	0
$303$	−2.11851e17	−0.0519051
$304$	0	0
$305$	−2.07227e18	−0.483278
$306$	0	0
$307$	−2.66489e18	−0.591754	−0.295877	−	0.955226i	$-0.595612\pi$
$307$	−2.66489e18	−0.591754	−0.295877	+	0.955226i	$0.595612\pi$
$308$	0	0
$309$	1.50367e18	0.318027
$310$	0	0
$311$	6.21474e18	1.25233	0.626167	−	0.779689i	$-0.284623\pi$
$311$	6.21474e18	1.25233	0.626167	+	0.779689i	$0.284623\pi$
$312$	0	0
$313$	7.37565e18	1.41650	0.708252	−	0.705960i	$-0.249484\pi$
$313$	7.37565e18	1.41650	0.708252	+	0.705960i	$0.249484\pi$
$314$	0	0
$315$	−2.13984e18	−0.391789
$316$	0	0
$317$	−4.51003e18	−0.787472	−0.393736	−	0.919224i	$-0.628818\pi$
$317$	−4.51003e18	−0.787472	−0.393736	+	0.919224i	$0.628818\pi$
$318$	0	0
$319$	−5.25319e18	−0.874969
$320$	0	0
$321$	4.84508e17	0.0770040
$322$	0	0
$323$	−1.09162e18	−0.165597
$324$	0	0
$325$	−1.80326e18	−0.261175
$326$	0	0
$327$	−2.15507e18	−0.298094
$328$	0	0
$329$	−5.93696e18	−0.784504
$330$	0	0
$331$	−7.62828e18	−0.963200	−0.481600	−	0.876391i	$-0.659944\pi$
$331$	−7.62828e18	−0.963200	−0.481600	+	0.876391i	$0.659944\pi$
$332$	0	0
$333$	2.26014e18	0.272775
$334$	0	0
$335$	−9.01396e18	−1.04011
$336$	0	0
$337$	−3.59960e17	−0.0397219	−0.0198609	−	0.999803i	$-0.506322\pi$
$337$	−3.59960e17	−0.0397219	−0.0198609	+	0.999803i	$0.506322\pi$
$338$	0	0
$339$	2.67909e18	0.282806
$340$	0	0
$341$	1.29665e19	1.30968
$342$	0	0
$343$	−9.36433e18	−0.905255
$344$	0	0
$345$	3.17999e17	0.0294295
$346$	0	0
$347$	−3.93125e18	−0.348385	−0.174192	−	0.984712i	$-0.555731\pi$
$347$	−3.93125e18	−0.348385	−0.174192	+	0.984712i	$0.555731\pi$
$348$	0	0
$349$	3.15117e18	0.267474	0.133737	−	0.991017i	$-0.457302\pi$
$349$	3.15117e18	0.267474	0.133737	+	0.991017i	$0.457302\pi$
$350$	0	0
$351$	−2.58724e18	−0.210394
$352$	0	0
$353$	−8.44905e18	−0.658411	−0.329206	−	0.944258i	$-0.606781\pi$
$353$	−8.44905e18	−0.658411	−0.329206	+	0.944258i	$0.606781\pi$
$354$	0	0
$355$	−1.53064e19	−1.14330
$356$	0	0
$357$	−7.80475e17	−0.0558917
$358$	0	0
$359$	9.74186e18	0.669011	0.334506	−	0.942394i	$-0.391431\pi$
$359$	9.74186e18	0.669011	0.334506	+	0.942394i	$0.391431\pi$
$360$	0	0
$361$	−1.41491e19	−0.932020
$362$	0	0
$363$	2.94424e18	0.186069
$364$	0	0
$365$	3.11003e17	0.0188612
$366$	0	0
$367$	1.38111e19	0.803957	0.401979	−	0.915649i	$-0.368323\pi$
$367$	1.38111e19	0.803957	0.401979	+	0.915649i	$0.368323\pi$
$368$	0	0
$369$	−1.77530e19	−0.992144
$370$	0	0
$371$	9.74899e18	0.523185
$372$	0	0
$373$	3.07694e19	1.58600	0.792999	−	0.609222i	$-0.208518\pi$
$373$	3.07694e19	1.58600	0.792999	+	0.609222i	$0.208518\pi$
$374$	0	0
$375$	−3.66621e18	−0.181544
$376$	0	0
$377$	8.05758e18	0.383393
$378$	0	0
$379$	1.19711e19	0.547442	0.273721	−	0.961809i	$-0.411745\pi$
$379$	1.19711e19	0.547442	0.273721	+	0.961809i	$0.411745\pi$
$380$	0	0
$381$	1.25583e17	0.00552068
$382$	0	0
$383$	2.18781e19	0.924740	0.462370	−	0.886687i	$-0.346999\pi$
$383$	2.18781e19	0.924740	0.462370	+	0.886687i	$0.346999\pi$
$384$	0	0
$385$	−1.44054e19	−0.585559
$386$	0	0
$387$	3.59726e19	1.40650
$388$	0	0
$389$	3.37297e19	1.26879	0.634396	−	0.773008i	$-0.281249\pi$
$389$	3.37297e19	1.26879	0.634396	+	0.773008i	$0.281249\pi$
$390$	0	0
$391$	−4.01151e18	−0.145205
$392$	0	0
$393$	−6.47524e18	−0.225586
$394$	0	0
$395$	1.32498e19	0.444357
$396$	0	0
$397$	−4.30571e19	−1.39032	−0.695162	−	0.718853i	$-0.744667\pi$
$397$	−4.30571e19	−1.39032	−0.695162	+	0.718853i	$0.744667\pi$
$398$	0	0
$399$	7.37864e17	0.0229445
$400$	0	0
$401$	−2.66863e19	−0.799292	−0.399646	−	0.916670i	$-0.630867\pi$
$401$	−2.66863e19	−0.799292	−0.399646	+	0.916670i	$0.630867\pi$
$402$	0	0
$403$	−1.98886e19	−0.573874
$404$	0	0
$405$	2.53128e19	0.703766
$406$	0	0
$407$	1.52153e19	0.407683
$408$	0	0
$409$	6.77946e19	1.75094	0.875468	−	0.483276i	$-0.160553\pi$
$409$	6.77946e19	1.75094	0.875468	+	0.483276i	$0.160553\pi$
$410$	0	0
$411$	6.05880e18	0.150860
$412$	0	0
$413$	−3.58998e19	−0.861919
$414$	0	0
$415$	4.93533e18	0.114276
$416$	0	0
$417$	−4.25317e18	−0.0949931
$418$	0	0
$419$	3.00699e19	0.647928	0.323964	−	0.946069i	$-0.394984\pi$
$419$	3.00699e19	0.647928	0.323964	+	0.946069i	$0.394984\pi$
$420$	0	0
$421$	5.20870e18	0.108296	0.0541481	−	0.998533i	$-0.482756\pi$
$421$	5.20870e18	0.108296	0.0541481	+	0.998533i	$0.482756\pi$
$422$	0	0
$423$	7.23835e19	1.45240
$424$	0	0
$425$	1.34561e19	0.260616
$426$	0	0
$427$	−1.76699e19	−0.330389
$428$	0	0
$429$	−8.58456e18	−0.154985
$430$	0	0
$431$	−5.23618e19	−0.912924	−0.456462	−	0.889743i	$-0.650884\pi$
$431$	−5.23618e19	−0.912924	−0.456462	+	0.889743i	$0.650884\pi$
$432$	0	0
$433$	−6.72276e19	−1.13211	−0.566054	−	0.824368i	$-0.691531\pi$
$433$	−6.72276e19	−1.13211	−0.566054	+	0.824368i	$0.691531\pi$
$434$	0	0
$435$	4.76631e18	0.0775377
$436$	0	0
$437$	3.79250e18	0.0596094
$438$	0	0
$439$	−1.08865e20	−1.65350	−0.826752	−	0.562567i	$-0.809814\pi$
$439$	−1.08865e20	−1.65350	−0.826752	+	0.562567i	$0.809814\pi$
$440$	0	0
$441$	4.79619e19	0.704056
$442$	0	0
$443$	−7.96094e19	−1.12963	−0.564815	−	0.825217i	$-0.691053\pi$
$443$	−7.96094e19	−1.12963	−0.564815	+	0.825217i	$0.691053\pi$
$444$	0	0
$445$	−7.32453e18	−0.100480
$446$	0	0
$447$	−2.05990e19	−0.273237
$448$	0	0
$449$	4.82462e18	0.0618893	0.0309446	−	0.999521i	$-0.490148\pi$
$449$	4.82462e18	0.0618893	0.0309446	+	0.999521i	$0.490148\pi$
$450$	0	0
$451$	−1.19513e20	−1.48283
$452$	0	0
$453$	−2.14365e19	−0.257287
$454$	0	0
$455$	2.20957e19	0.256580
$456$	0	0
$457$	1.10838e20	1.24542	0.622712	−	0.782451i	$-0.286031\pi$
$457$	1.10838e20	1.24542	0.622712	+	0.782451i	$0.286031\pi$
$458$	0	0
$459$	1.93062e19	0.209943
$460$	0	0
$461$	−1.11041e20	−1.16876	−0.584382	−	0.811479i	$-0.698663\pi$
$461$	−1.11041e20	−1.16876	−0.584382	+	0.811479i	$0.698663\pi$
$462$	0	0
$463$	−7.48407e19	−0.762572	−0.381286	−	0.924457i	$-0.624519\pi$
$463$	−7.48407e19	−0.762572	−0.381286	+	0.924457i	$0.624519\pi$
$464$	0	0
$465$	−1.17647e19	−0.116061
$466$	0	0
$467$	−1.10986e19	−0.106021	−0.0530106	−	0.998594i	$-0.516882\pi$
$467$	−1.10986e19	−0.106021	−0.0530106	+	0.998594i	$0.516882\pi$
$468$	0	0
$469$	−7.68606e19	−0.711062
$470$	0	0
$471$	5.14783e18	0.0461283
$472$	0	0
$473$	2.42167e20	2.10212
$474$	0	0
$475$	−1.27214e19	−0.106988
$476$	0	0
$477$	−1.18860e20	−0.968606
$478$	0	0
$479$	5.04298e19	0.398264	0.199132	−	0.979973i	$-0.436188\pi$
$479$	5.04298e19	0.398264	0.199132	+	0.979973i	$0.436188\pi$
$480$	0	0
$481$	−2.33379e19	−0.178638
$482$	0	0
$483$	2.71153e18	0.0201192
$484$	0	0
$485$	1.15986e20	0.834339
$486$	0	0
$487$	−2.44347e20	−1.70427	−0.852137	−	0.523319i	$-0.824693\pi$
$487$	−2.44347e20	−1.70427	−0.852137	+	0.523319i	$0.824693\pi$
$488$	0	0
$489$	1.87746e19	0.126985
$490$	0	0
$491$	2.19942e20	1.44277	0.721385	−	0.692534i	$-0.243506\pi$
$491$	2.19942e20	1.44277	0.721385	+	0.692534i	$0.243506\pi$
$492$	0	0
$493$	−6.01263e19	−0.382572
$494$	0	0
$495$	1.75631e20	1.08408
$496$	0	0
$497$	−1.30515e20	−0.781608
$498$	0	0
$499$	−1.54659e20	−0.898716	−0.449358	−	0.893352i	$-0.648347\pi$
$499$	−1.54659e20	−0.898716	−0.449358	+	0.893352i	$0.648347\pi$
$500$	0	0
$501$	5.33573e19	0.300892
$502$	0	0
$503$	2.48198e20	1.35844	0.679218	−	0.733937i	$-0.262319\pi$
$503$	2.48198e20	1.35844	0.679218	+	0.733937i	$0.262319\pi$
$504$	0	0
$505$	−4.47548e19	−0.237768
$506$	0	0
$507$	−1.93352e19	−0.0997214
$508$	0	0
$509$	−2.89865e20	−1.45148	−0.725742	−	0.687967i	$-0.758503\pi$
$509$	−2.89865e20	−1.45148	−0.725742	+	0.687967i	$0.758503\pi$
$510$	0	0
$511$	2.65188e18	0.0128943
$512$	0	0
$513$	−1.82522e19	−0.0861856
$514$	0	0
$515$	3.17660e20	1.45683
$516$	0	0
$517$	4.87286e20	2.17072
$518$	0	0
$519$	3.89675e19	0.168635
$520$	0	0
$521$	1.33511e20	0.561348	0.280674	−	0.959803i	$-0.409442\pi$
$521$	1.33511e20	0.561348	0.280674	+	0.959803i	$0.409442\pi$
$522$	0	0
$523$	−2.18247e20	−0.891630	−0.445815	−	0.895125i	$-0.647086\pi$
$523$	−2.18247e20	−0.891630	−0.445815	+	0.895125i	$0.647086\pi$
$524$	0	0
$525$	−9.09544e18	−0.0361101
$526$	0	0
$527$	1.48411e20	0.572645
$528$	0	0
$529$	−2.52698e20	−0.947731
$530$	0	0
$531$	4.37690e20	1.59572
$532$	0	0
$533$	1.83315e20	0.649747
$534$	0	0
$535$	1.02355e20	0.352742
$536$	0	0
$537$	4.44319e19	0.148898
$538$	0	0
$539$	3.22880e20	1.05227
$540$	0	0
$541$	−2.08153e20	−0.659787	−0.329894	−	0.944018i	$-0.607013\pi$
$541$	−2.08153e20	−0.659787	−0.329894	+	0.944018i	$0.607013\pi$
$542$	0	0
$543$	2.45206e19	0.0756019
$544$	0	0
$545$	−4.55272e20	−1.36552
$546$	0	0
$547$	−1.78808e20	−0.521775	−0.260887	−	0.965369i	$-0.584015\pi$
$547$	−1.78808e20	−0.521775	−0.260887	+	0.965369i	$0.584015\pi$
$548$	0	0
$549$	2.15432e20	0.611670
$550$	0	0
$551$	5.68437e19	0.157053
$552$	0	0
$553$	1.12979e20	0.303781
$554$	0	0
$555$	−1.38051e19	−0.0361279
$556$	0	0
$557$	−8.66937e18	−0.0220838	−0.0110419	−	0.999939i	$-0.503515\pi$
$557$	−8.66937e18	−0.0220838	−0.0110419	+	0.999939i	$0.503515\pi$
$558$	0	0
$559$	−3.71447e20	−0.921104
$560$	0	0
$561$	6.40587e19	0.154652
$562$	0	0
$563$	−3.90636e20	−0.918247	−0.459124	−	0.888372i	$-0.651836\pi$
$563$	−3.90636e20	−0.918247	−0.459124	+	0.888372i	$0.651836\pi$
$564$	0	0
$565$	5.65974e20	1.29549
$566$	0	0
$567$	2.15839e20	0.481123
$568$	0	0
$569$	−2.40917e18	−0.00523028	−0.00261514	−	0.999997i	$-0.500832\pi$
$569$	−2.40917e18	−0.00523028	−0.00261514	+	0.999997i	$0.500832\pi$
$570$	0	0
$571$	−8.00598e20	−1.69295	−0.846474	−	0.532430i	$-0.821279\pi$
$571$	−8.00598e20	−1.69295	−0.846474	+	0.532430i	$0.821279\pi$
$572$	0	0
$573$	9.42490e19	0.194141
$574$	0	0
$575$	−4.67491e19	−0.0938133
$576$	0	0
$577$	−7.41944e20	−1.45062	−0.725308	−	0.688424i	$-0.758303\pi$
$577$	−7.41944e20	−1.45062	−0.725308	+	0.688424i	$0.758303\pi$
$578$	0	0
$579$	−1.22342e20	−0.233070
$580$	0	0
$581$	4.20828e19	0.0781238
$582$	0	0
$583$	−8.00165e20	−1.44766
$584$	0	0
$585$	−2.69391e20	−0.475022
$586$	0	0
$587$	2.34877e20	0.403696	0.201848	−	0.979417i	$-0.435305\pi$
$587$	2.34877e20	0.403696	0.201848	+	0.979417i	$0.435305\pi$
$588$	0	0
$589$	−1.40308e20	−0.235081
$590$	0	0
$591$	1.35220e20	0.220869
$592$	0	0
$593$	1.12197e20	0.178678	0.0893392	−	0.996001i	$-0.471524\pi$
$593$	1.12197e20	0.178678	0.0893392	+	0.996001i	$0.471524\pi$
$594$	0	0
$595$	−1.64880e20	−0.256030
$596$	0	0
$597$	8.98204e19	0.136009
$598$	0	0
$599$	−2.72137e20	−0.401871	−0.200935	−	0.979604i	$-0.564398\pi$
$599$	−2.72137e20	−0.401871	−0.200935	+	0.979604i	$0.564398\pi$
$600$	0	0
$601$	−9.54015e19	−0.137403	−0.0687016	−	0.997637i	$-0.521886\pi$
$601$	−9.54015e19	−0.137403	−0.0687016	+	0.997637i	$0.521886\pi$
$602$	0	0
$603$	9.37085e20	1.31643
$604$	0	0
$605$	6.21989e20	0.852349
$606$	0	0
$607$	−8.53048e20	−1.14040	−0.570201	−	0.821505i	$-0.693135\pi$
$607$	−8.53048e20	−1.14040	−0.570201	+	0.821505i	$0.693135\pi$
$608$	0	0
$609$	4.06416e19	0.0530079
$610$	0	0
$611$	−7.47421e20	−0.951166
$612$	0	0
$613$	−2.69183e20	−0.334267	−0.167134	−	0.985934i	$-0.553451\pi$
$613$	−2.69183e20	−0.334267	−0.167134	+	0.985934i	$0.553451\pi$
$614$	0	0
$615$	1.08437e20	0.131405
$616$	0	0
$617$	5.48639e20	0.648855	0.324428	−	0.945911i	$-0.394828\pi$
$617$	5.48639e20	0.648855	0.324428	+	0.945911i	$0.394828\pi$
$618$	0	0
$619$	−8.74983e20	−1.01000	−0.504998	−	0.863120i	$-0.668507\pi$
$619$	−8.74983e20	−1.01000	−0.504998	+	0.863120i	$0.668507\pi$
$620$	0	0
$621$	−6.70738e19	−0.0755728
$622$	0	0
$623$	−6.24551e19	−0.0686922
$624$	0	0
$625$	−3.92353e20	−0.421285
$626$	0	0
$627$	−6.05614e19	−0.0634876
$628$	0	0
$629$	1.74150e20	0.178255
$630$	0	0
$631$	−2.36778e20	−0.236658	−0.118329	−	0.992974i	$-0.537754\pi$
$631$	−2.36778e20	−0.236658	−0.118329	+	0.992974i	$0.537754\pi$
$632$	0	0
$633$	−1.46637e20	−0.143125
$634$	0	0
$635$	2.65301e19	0.0252892
$636$	0	0
$637$	−4.95248e20	−0.461081
$638$	0	0
$639$	1.59124e21	1.44704
$640$	0	0
$641$	−5.34458e19	−0.0474764	−0.0237382	−	0.999718i	$-0.507557\pi$
$641$	−5.34458e19	−0.0474764	−0.0237382	+	0.999718i	$0.507557\pi$
$642$	0	0
$643$	1.27376e21	1.10536	0.552682	−	0.833392i	$-0.313605\pi$
$643$	1.27376e21	1.10536	0.552682	+	0.833392i	$0.313605\pi$
$644$	0	0
$645$	−2.19723e20	−0.186285
$646$	0	0
$647$	2.01963e21	1.67297	0.836486	−	0.547988i	$-0.184606\pi$
$647$	2.01963e21	1.67297	0.836486	+	0.547988i	$0.184606\pi$
$648$	0	0
$649$	2.94653e21	2.38493
$650$	0	0
$651$	−1.00316e20	−0.0793439
$652$	0	0
$653$	−2.00632e21	−1.55078	−0.775392	−	0.631480i	$-0.782448\pi$
$653$	−2.00632e21	−1.55078	−0.775392	+	0.631480i	$0.782448\pi$
$654$	0	0
$655$	−1.36793e21	−1.03337
$656$	0	0
$657$	−3.23317e19	−0.0238720
$658$	0	0
$659$	2.07879e21	1.50027	0.750135	−	0.661285i	$-0.229989\pi$
$659$	2.07879e21	1.50027	0.750135	+	0.661285i	$0.229989\pi$
$660$	0	0
$661$	−1.14859e21	−0.810317	−0.405158	−	0.914247i	$-0.632784\pi$
$661$	−1.14859e21	−0.810317	−0.405158	+	0.914247i	$0.632784\pi$
$662$	0	0
$663$	−9.82562e19	−0.0677655
$664$	0	0
$665$	1.55878e20	0.105105
$666$	0	0
$667$	2.08891e20	0.137713
$668$	0	0
$669$	−1.85883e20	−0.119824
$670$	0	0
$671$	1.45029e21	0.914187
$672$	0	0
$673$	1.06762e21	0.658120	0.329060	−	0.944309i	$-0.393268\pi$
$673$	1.06762e21	0.658120	0.329060	+	0.944309i	$0.393268\pi$
$674$	0	0
$675$	2.24990e20	0.135639
$676$	0	0
$677$	−1.46442e21	−0.863479	−0.431739	−	0.901998i	$-0.642100\pi$
$677$	−1.46442e21	−0.863479	−0.431739	+	0.901998i	$0.642100\pi$
$678$	0	0
$679$	9.88994e20	0.570388
$680$	0	0
$681$	−1.74816e20	−0.0986231
$682$	0	0
$683$	9.87991e20	0.545253	0.272626	−	0.962120i	$-0.412108\pi$
$683$	9.87991e20	0.545253	0.272626	+	0.962120i	$0.412108\pi$
$684$	0	0
$685$	1.27996e21	0.691061
$686$	0	0
$687$	5.69175e20	0.300656
$688$	0	0
$689$	1.22733e21	0.634332
$690$	0	0
$691$	−1.14670e21	−0.579917	−0.289959	−	0.957039i	$-0.593642\pi$
$691$	−1.14670e21	−0.579917	−0.289959	+	0.957039i	$0.593642\pi$
$692$	0	0
$693$	1.49758e21	0.741123
$694$	0	0
$695$	−8.98508e20	−0.435147
$696$	0	0
$697$	−1.36791e21	−0.648355
$698$	0	0
$699$	−3.08660e20	−0.143187
$700$	0	0
$701$	9.61458e19	0.0436561	0.0218281	−	0.999762i	$-0.493051\pi$
$701$	9.61458e19	0.0436561	0.0218281	+	0.999762i	$0.493051\pi$
$702$	0	0
$703$	−1.64642e20	−0.0731770
$704$	0	0
$705$	−4.42123e20	−0.192365
$706$	0	0
$707$	−3.81617e20	−0.162548
$708$	0	0
$709$	−2.01099e21	−0.838616	−0.419308	−	0.907844i	$-0.637727\pi$
$709$	−2.01099e21	−0.838616	−0.419308	+	0.907844i	$0.637727\pi$
$710$	0	0
$711$	−1.37744e21	−0.562408
$712$	0	0
$713$	−5.15609e20	−0.206134
$714$	0	0
$715$	−1.81354e21	−0.709957
$716$	0	0
$717$	−3.29613e20	−0.126360
$718$	0	0
$719$	−5.47206e20	−0.205440	−0.102720	−	0.994710i	$-0.532755\pi$
$719$	−5.47206e20	−0.205440	−0.102720	+	0.994710i	$0.532755\pi$
$720$	0	0
$721$	2.70864e21	0.995947
$722$	0	0
$723$	−2.54968e20	−0.0918222
$724$	0	0
$725$	−7.00696e20	−0.247169
$726$	0	0
$727$	−1.21154e21	−0.418630	−0.209315	−	0.977848i	$-0.567123\pi$
$727$	−1.21154e21	−0.418630	−0.209315	+	0.977848i	$0.567123\pi$
$728$	0	0
$729$	−2.46780e21	−0.835322
$730$	0	0
$731$	2.77177e21	0.919131
$732$	0	0
$733$	1.21006e21	0.393121	0.196561	−	0.980492i	$-0.437023\pi$
$733$	1.21006e21	0.393121	0.196561	+	0.980492i	$0.437023\pi$
$734$	0	0
$735$	−2.92955e20	−0.0932493
$736$	0	0
$737$	6.30846e21	1.96751
$738$	0	0
$739$	−3.33450e21	−1.01905	−0.509527	−	0.860455i	$-0.670180\pi$
$739$	−3.33450e21	−1.01905	−0.509527	+	0.860455i	$0.670180\pi$
$740$	0	0
$741$	9.28918e19	0.0278190
$742$	0	0
$743$	−2.93853e21	−0.862410	−0.431205	−	0.902254i	$-0.641911\pi$
$743$	−2.93853e21	−0.862410	−0.431205	+	0.902254i	$0.641911\pi$
$744$	0	0
$745$	−4.35166e21	−1.25165
$746$	0	0
$747$	−5.13074e20	−0.144636
$748$	0	0
$749$	8.72768e20	0.241149
$750$	0	0
$751$	−5.03046e20	−0.136241	−0.0681206	−	0.997677i	$-0.521700\pi$
$751$	−5.03046e20	−0.136241	−0.0681206	+	0.997677i	$0.521700\pi$
$752$	0	0
$753$	−5.75028e19	−0.0152660
$754$	0	0
$755$	−4.52860e21	−1.17859
$756$	0	0
$757$	−1.61904e21	−0.413083	−0.206542	−	0.978438i	$-0.566221\pi$
$757$	−1.61904e21	−0.413083	−0.206542	+	0.978438i	$0.566221\pi$
$758$	0	0
$759$	−2.22553e20	−0.0556699
$760$	0	0
$761$	−3.90363e21	−0.957378	−0.478689	−	0.877985i	$-0.658888\pi$
$761$	−3.90363e21	−0.957378	−0.478689	+	0.877985i	$0.658888\pi$
$762$	0	0
$763$	−3.88203e21	−0.933522
$764$	0	0
$765$	2.01022e21	0.474004
$766$	0	0
$767$	−4.51953e21	−1.04503
$768$	0	0
$769$	−6.51753e21	−1.47787	−0.738934	−	0.673778i	$-0.764670\pi$
$769$	−6.51753e21	−1.47787	−0.738934	+	0.673778i	$0.764670\pi$
$770$	0	0
$771$	9.86330e20	0.219338
$772$	0	0
$773$	2.11533e21	0.461352	0.230676	−	0.973031i	$-0.425906\pi$
$773$	2.11533e21	0.461352	0.230676	+	0.973031i	$0.425906\pi$
$774$	0	0
$775$	1.72954e21	0.369970
$776$	0	0
$777$	−1.17714e20	−0.0246985
$778$	0	0
$779$	1.29323e21	0.266161
$780$	0	0
$781$	1.07123e22	2.16271
$782$	0	0
$783$	−1.00533e21	−0.199111
$784$	0	0
$785$	1.08751e21	0.211306
$786$	0	0
$787$	7.66416e21	1.46101	0.730507	−	0.682906i	$-0.239284\pi$
$787$	7.66416e21	1.46101	0.730507	+	0.682906i	$0.239284\pi$
$788$	0	0
$789$	−2.56124e20	−0.0479040
$790$	0	0
$791$	4.82597e21	0.885647
$792$	0	0
$793$	−2.22452e21	−0.400578
$794$	0	0
$795$	7.26004e20	0.128288
$796$	0	0
$797$	7.50732e20	0.130181	0.0650904	−	0.997879i	$-0.479266\pi$
$797$	7.50732e20	0.130181	0.0650904	+	0.997879i	$0.479266\pi$
$798$	0	0
$799$	5.57732e21	0.949128
$800$	0	0
$801$	7.61453e20	0.127174
$802$	0	0
$803$	−2.17657e20	−0.0356785
$804$	0	0
$805$	5.72827e20	0.0921625
$806$	0	0
$807$	−7.40826e20	−0.116994
$808$	0	0
$809$	−5.65783e21	−0.877073	−0.438536	−	0.898713i	$-0.644503\pi$
$809$	−5.65783e21	−0.877073	−0.438536	+	0.898713i	$0.644503\pi$
$810$	0	0
$811$	6.21577e21	0.945886	0.472943	−	0.881093i	$-0.343192\pi$
$811$	6.21577e21	0.945886	0.472943	+	0.881093i	$0.343192\pi$
$812$	0	0
$813$	1.81069e21	0.270499
$814$	0	0
$815$	3.96624e21	0.581698
$816$	0	0
$817$	−2.62044e21	−0.377320
$818$	0	0
$819$	−2.29705e21	−0.324745
$820$	0	0
$821$	−9.52351e21	−1.32198	−0.660988	−	0.750397i	$-0.729863\pi$
$821$	−9.52351e21	−1.32198	−0.660988	+	0.750397i	$0.729863\pi$
$822$	0	0
$823$	8.61991e21	1.17491	0.587454	−	0.809258i	$-0.300130\pi$
$823$	8.61991e21	1.17491	0.587454	+	0.809258i	$0.300130\pi$
$824$	0	0
$825$	7.46523e20	0.0999169
$826$	0	0
$827$	−8.11120e21	−1.06609	−0.533045	−	0.846087i	$-0.678952\pi$
$827$	−8.11120e21	−1.06609	−0.533045	+	0.846087i	$0.678952\pi$
$828$	0	0
$829$	1.26605e22	1.63414	0.817072	−	0.576536i	$-0.195596\pi$
$829$	1.26605e22	1.63414	0.817072	+	0.576536i	$0.195596\pi$
$830$	0	0
$831$	−1.08474e21	−0.137505
$832$	0	0
$833$	3.69558e21	0.460093
$834$	0	0
$835$	1.12720e22	1.37833
$836$	0	0
$837$	2.48147e21	0.298036
$838$	0	0
$839$	−3.77471e21	−0.445317	−0.222658	−	0.974897i	$-0.571473\pi$
$839$	−3.77471e21	−0.445317	−0.222658	+	0.974897i	$0.571473\pi$
$840$	0	0
$841$	−5.49824e21	−0.637167
$842$	0	0
$843$	1.98545e21	0.226022
$844$	0	0
$845$	−4.08468e21	−0.456806
$846$	0	0
$847$	5.30360e21	0.582701
$848$	0	0
$849$	2.49692e21	0.269524
$850$	0	0
$851$	−6.05031e20	−0.0641662
$852$	0	0
$853$	−1.11887e22	−1.16590	−0.582948	−	0.812509i	$-0.698101\pi$
$853$	−1.11887e22	−1.16590	−0.582948	+	0.812509i	$0.698101\pi$
$854$	0	0
$855$	−1.90047e21	−0.194587
$856$	0	0
$857$	6.99179e21	0.703449	0.351724	−	0.936104i	$-0.385595\pi$
$857$	6.99179e21	0.703449	0.351724	+	0.936104i	$0.385595\pi$
$858$	0	0
$859$	1.77065e22	1.75058	0.875292	−	0.483595i	$-0.160669\pi$
$859$	1.77065e22	1.75058	0.875292	+	0.483595i	$0.160669\pi$
$860$	0	0
$861$	9.24623e20	0.0898341
$862$	0	0
$863$	−1.67429e21	−0.159864	−0.0799319	−	0.996800i	$-0.525470\pi$
$863$	−1.67429e21	−0.159864	−0.0799319	+	0.996800i	$0.525470\pi$
$864$	0	0
$865$	8.23212e21	0.772487
$866$	0	0
$867$	−1.08442e21	−0.100012
$868$	0	0
$869$	−9.27295e21	−0.840562
$870$	0	0
$871$	−9.67620e21	−0.862122
$872$	0	0
$873$	−1.20578e22	−1.05600
$874$	0	0
$875$	−6.60412e21	−0.568531
$876$	0	0
$877$	6.20926e21	0.525464	0.262732	−	0.964869i	$-0.415376\pi$
$877$	6.20926e21	0.525464	0.262732	+	0.964869i	$0.415376\pi$
$878$	0	0
$879$	−6.21599e18	−0.000517123 0
$880$	0	0
$881$	1.96275e22	1.60526	0.802629	−	0.596478i	$-0.203434\pi$
$881$	1.96275e22	1.60526	0.802629	+	0.596478i	$0.203434\pi$
$882$	0	0
$883$	−1.04753e22	−0.842292	−0.421146	−	0.906993i	$-0.638372\pi$
$883$	−1.04753e22	−0.842292	−0.421146	+	0.906993i	$0.638372\pi$
$884$	0	0
$885$	−2.67344e21	−0.211347
$886$	0	0
$887$	−3.27875e21	−0.254848	−0.127424	−	0.991848i	$-0.540671\pi$
$887$	−3.27875e21	−0.254848	−0.127424	+	0.991848i	$0.540671\pi$
$888$	0	0
$889$	2.26218e20	0.0172888
$890$	0	0
$891$	−1.77153e22	−1.33127
$892$	0	0
$893$	−5.27282e21	−0.389634
$894$	0	0
$895$	9.38650e21	0.682073
$896$	0	0
$897$	3.41362e20	0.0243934
$898$	0	0
$899$	−7.72817e21	−0.543099
$900$	0	0
$901$	−9.15843e21	−0.632973
$902$	0	0
$903$	−1.87354e21	−0.127352
$904$	0	0
$905$	5.18013e21	0.346319
$906$	0	0
$907$	−3.71072e20	−0.0244008	−0.0122004	−	0.999926i	$-0.503884\pi$
$907$	−3.71072e20	−0.0244008	−0.0122004	+	0.999926i	$0.503884\pi$
$908$	0	0
$909$	4.65267e21	0.300936
$910$	0	0
$911$	−1.97479e22	−1.25641	−0.628206	−	0.778047i	$-0.716210\pi$
$911$	−1.97479e22	−1.25641	−0.628206	+	0.778047i	$0.716210\pi$
$912$	0	0
$913$	−3.45401e21	−0.216169
$914$	0	0
$915$	−1.31587e21	−0.0810131
$916$	0	0
$917$	−1.16641e22	−0.706453
$918$	0	0
$919$	−1.89055e21	−0.112648	−0.0563239	−	0.998413i	$-0.517938\pi$
$919$	−1.89055e21	−0.112648	−0.0563239	+	0.998413i	$0.517938\pi$
$920$	0	0
$921$	−1.69218e21	−0.0991971
$922$	0	0
$923$	−1.64310e22	−0.947655
$924$	0	0
$925$	2.02949e21	0.115166
$926$	0	0
$927$	−3.30237e22	−1.84386
$928$	0	0
$929$	3.34951e22	1.84020	0.920098	−	0.391689i	$-0.128109\pi$
$929$	3.34951e22	1.84020	0.920098	+	0.391689i	$0.128109\pi$
$930$	0	0
$931$	−3.49382e21	−0.188876
$932$	0	0
$933$	3.94630e21	0.209932
$934$	0	0
$935$	1.35328e22	0.708436
$936$	0	0
$937$	−7.69977e21	−0.396671	−0.198336	−	0.980134i	$-0.563554\pi$
$937$	−7.69977e21	−0.396671	−0.198336	+	0.980134i	$0.563554\pi$
$938$	0	0
$939$	4.68347e21	0.237452
$940$	0	0
$941$	−2.32873e22	−1.16198	−0.580988	−	0.813912i	$-0.697334\pi$
$941$	−2.32873e22	−1.16198	−0.580988	+	0.813912i	$0.697334\pi$
$942$	0	0
$943$	4.75241e21	0.233387
$944$	0	0
$945$	−2.75685e21	−0.133252
$946$	0	0
$947$	1.94058e22	0.923225	0.461613	−	0.887082i	$-0.347271\pi$
$947$	1.94058e22	0.923225	0.461613	+	0.887082i	$0.347271\pi$
$948$	0	0
$949$	3.33852e20	0.0156336
$950$	0	0
$951$	−2.86383e21	−0.132006
$952$	0	0
$953$	9.32103e21	0.422929	0.211464	−	0.977386i	$-0.432177\pi$
$953$	9.32103e21	0.422929	0.211464	+	0.977386i	$0.432177\pi$
$954$	0	0
$955$	1.99107e22	0.889326
$956$	0	0
$957$	−3.33572e21	−0.146673
$958$	0	0
$959$	1.09140e22	0.472438
$960$	0	0
$961$	−4.38972e21	−0.187073
$962$	0	0
$963$	−1.06408e22	−0.446454
$964$	0	0
$965$	−2.58455e22	−1.06765
$966$	0	0
$967$	4.38828e22	1.78482	0.892412	−	0.451222i	$-0.149012\pi$
$967$	4.38828e22	1.78482	0.892412	+	0.451222i	$0.149012\pi$
$968$	0	0
$969$	−6.93166e20	−0.0277593
$970$	0	0
$971$	−1.35870e22	−0.535772	−0.267886	−	0.963451i	$-0.586325\pi$
$971$	−1.35870e22	−0.535772	−0.267886	+	0.963451i	$0.586325\pi$
$972$	0	0
$973$	−7.66143e21	−0.297484
$974$	0	0
$975$	−1.14505e21	−0.0437815
$976$	0	0
$977$	1.14769e21	0.0432132	0.0216066	−	0.999767i	$-0.493122\pi$
$977$	1.14769e21	0.0432132	0.0216066	+	0.999767i	$0.493122\pi$
$978$	0	0
$979$	5.12610e21	0.190072
$980$	0	0
$981$	4.73297e22	1.72829
$982$	0	0
$983$	−4.01501e22	−1.44389	−0.721946	−	0.691949i	$-0.756752\pi$
$983$	−4.01501e22	−1.44389	−0.721946	+	0.691949i	$0.756752\pi$
$984$	0	0
$985$	2.85660e22	1.01176
$986$	0	0
$987$	−3.76992e21	−0.131508
$988$	0	0
$989$	−9.62971e21	−0.330857
$990$	0	0
$991$	4.39824e22	1.48842	0.744212	−	0.667944i	$-0.232825\pi$
$991$	4.39824e22	1.48842	0.744212	+	0.667944i	$0.232825\pi$
$992$	0	0
$993$	−4.84389e21	−0.161464
$994$	0	0
$995$	1.89751e22	0.623033
$996$	0	0
$997$	3.15841e22	1.02154	0.510769	−	0.859718i	$-0.329361\pi$
$997$	3.15841e22	1.02154	0.510769	+	0.859718i	$0.329361\pi$
$998$	0	0
$999$	2.91184e21	0.0927739

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	32.16.a.c.1.3	✓	4
4.3	odd	2		32.16.a.e.1.2	yes	4
8.3	odd	2		64.16.a.o.1.3		4
8.5	even	2		64.16.a.q.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
32.16.a.c.1.3	✓	4	1.1	even	1	trivial
32.16.a.e.1.2	yes	4	4.3	odd	2
64.16.a.o.1.3		4	8.3	odd	2
64.16.a.q.1.2		4	8.5	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 32 = 2^{5} \)
Weight:	\( k \)	\(=\)	\( 16 \)
Character orbit:	\([\chi]\)	\(=\)	32.a (trivial)

\(f(q)\)	\(=\)	\(q+634.991 q^{3} +134146. q^{5} +1.14384e6 q^{7} -1.39457e7 q^{9} -9.38825e7 q^{11} +1.44001e8 q^{13} +8.51813e7 q^{15} -1.07455e9 q^{17} +1.01588e9 q^{19} +7.26327e8 q^{21} +3.73320e9 q^{23} -1.25225e10 q^{25} -1.79668e10 q^{27} +5.59549e10 q^{29} -1.38114e11 q^{31} -5.96145e10 q^{33} +1.53441e11 q^{35} -1.62068e11 q^{37} +9.14394e10 q^{39} +1.27301e12 q^{41} -2.57947e12 q^{43} -1.87075e12 q^{45} -5.19038e12 q^{47} -3.43919e12 q^{49} -6.82329e11 q^{51} +8.52305e12 q^{53} -1.25939e13 q^{55} +6.45076e11 q^{57} -3.13853e13 q^{59} -1.54479e13 q^{61} -1.59516e13 q^{63} +1.93171e13 q^{65} -6.71953e13 q^{67} +2.37055e12 q^{69} -1.14103e14 q^{71} +2.31840e12 q^{73} -7.95168e12 q^{75} -1.07386e14 q^{77} +9.87720e13 q^{79} +1.88697e14 q^{81} +3.67908e13 q^{83} -1.44146e14 q^{85} +3.55309e13 q^{87} -5.46013e13 q^{89} +1.64714e14 q^{91} -8.77013e13 q^{93} +1.36276e14 q^{95} +8.64627e14 q^{97} +1.30926e15 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 2912 q^{3} - 98280 q^{5} - 2755776 q^{7} + 26457940 q^{9} + 110853856 q^{11} - 187741448 q^{13} - 442991680 q^{15} + 2121294984 q^{17} + 419203872 q^{19} + 8397459968 q^{21} - 5330808384 q^{23} + 8564955740 q^{25}+ \cdots + 26\!\cdots\!84 q^{99}+O(q^{100}) \) 4 * q - 2912 * q^3 - 98280 * q^5 - 2755776 * q^7 + 26457940 * q^9 + 110853856 * q^11 - 187741448 * q^13 - 442991680 * q^15 + 2121294984 * q^17 + 419203872 * q^19 + 8397459968 * q^21 - 5330808384 * q^23 + 8564955740 * q^25 - 41191036352 * q^27 - 86832359880 * q^29 + 96274977024 * q^31 + 41399413504 * q^33 - 929253215360 * q^35 - 112979590376 * q^37 - 2001683116352 * q^39 - 1534010535960 * q^41 - 3567397370400 * q^43 + 3155482037240 * q^45 - 9031658937472 * q^47 + 8694343447332 * q^49 - 37558794281664 * q^51 - 14370240161064 * q^53 - 46434781152960 * q^55 - 5813065407232 * q^57 - 68701112336800 * q^59 + 4806363233592 * q^61 - 192971404565440 * q^63 + 7651382051280 * q^65 - 167433288699744 * q^67 - 80410496862720 * q^69 - 180096642956992 * q^71 + 173676000667560 * q^73 - 227575010162080 * q^75 - 133277064067584 * q^77 - 232741753133952 * q^79 + 263803331371492 * q^81 + 20415834690336 * q^83 - 342055719314640 * q^85 + 870209244139712 * q^87 + 938354283775848 * q^89 + 1478922417113472 * q^91 - 2098341777250304 * q^93 + 2700969259238080 * q^95 + 1510271716715080 * q^97 + 2629725177868384 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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