LMFDB - Embedded newform 4012.2.a.i.1.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4012,2,Mod(1,4012)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4012, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("4012.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4012 = 2^{2} \cdot 17 \cdot 59 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4012.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$32.0359812909$
Analytic rank:	$0$
Dimension:	$18$
Coefficient field:	$\mathbb{Q}[x]/(x^{18} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{18} - 8 x^{17} - 4 x^{16} + 178 x^{15} - 265 x^{14} - 1405 x^{13} + 3503 x^{12} + 4295 x^{11} + \cdots - 16 $ x^18 - 8x^17 - 4x^16 + 178x^15 - 265x^14 - 1405x^13 + 3503x^12 + 4295x^11 - 17374x^10 - 893x^9 + 38112x^8 - 18700x^7 - 32137x^6 + 26381x^5 + 3964x^4 - 5788x^3 - 108x^2 + 232*x - 16
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
Root			$0.154866$ of defining polynomial
Character	$\chi$	$=$	4012.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.154866 q^{3} -1.21070 q^{5} -0.213015 q^{7} -2.97602 q^{9} +O(q^{10})$	$q+0.154866 q^{3} -1.21070 q^{5} -0.213015 q^{7} -2.97602 q^{9} +6.43345 q^{11} +5.00862 q^{13} -0.187496 q^{15} -1.00000 q^{17} -5.48083 q^{19} -0.0329889 q^{21} +0.395346 q^{23} -3.53421 q^{25} -0.925484 q^{27} +0.149252 q^{29} +6.05516 q^{31} +0.996326 q^{33} +0.257897 q^{35} +9.39455 q^{37} +0.775666 q^{39} +3.27099 q^{41} -10.2298 q^{43} +3.60306 q^{45} +5.33452 q^{47} -6.95462 q^{49} -0.154866 q^{51} -1.89023 q^{53} -7.78897 q^{55} -0.848797 q^{57} -1.00000 q^{59} +12.6911 q^{61} +0.633937 q^{63} -6.06392 q^{65} +4.18405 q^{67} +0.0612258 q^{69} -13.4344 q^{71} +10.3569 q^{73} -0.547330 q^{75} -1.37042 q^{77} +12.1957 q^{79} +8.78472 q^{81} +14.4904 q^{83} +1.21070 q^{85} +0.0231141 q^{87} -13.0985 q^{89} -1.06691 q^{91} +0.937741 q^{93} +6.63564 q^{95} -12.7867 q^{97} -19.1461 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 18 q + 8 q^{3} + 4 q^{5} + 2 q^{7} + 18 q^{9}+O(q^{10}) $ 18 * q + 8 * q^3 + 4 * q^5 + 2 * q^7 + 18 * q^9	$ 18 q + 8 q^{3} + 4 q^{5} + 2 q^{7} + 18 q^{9} + 12 q^{11} + 2 q^{13} - 18 q^{17} + 5 q^{19} - 3 q^{21} + 21 q^{23} + 16 q^{25} + 26 q^{27} + 14 q^{29} + 15 q^{31} + 19 q^{33} + 20 q^{35} + 2 q^{37} - 14 q^{39} + 34 q^{41} + 21 q^{43} + 49 q^{45} + 69 q^{47} + 28 q^{49} - 8 q^{51} - 4 q^{53} + 18 q^{55} + 5 q^{57} - 18 q^{59} + 11 q^{61} + 35 q^{63} + 27 q^{65} + 34 q^{67} - 4 q^{69} + 37 q^{71} + 18 q^{73} + 72 q^{75} + 11 q^{77} + 11 q^{79} + 30 q^{81} + 28 q^{83} - 4 q^{85} + 7 q^{87} + 44 q^{89} - 23 q^{91} - 3 q^{93} - 11 q^{95} + 11 q^{97} + 56 q^{99}+O(q^{100}) $ 18 * q + 8 * q^3 + 4 * q^5 + 2 * q^7 + 18 * q^9 + 12 * q^11 + 2 * q^13 - 18 * q^17 + 5 * q^19 - 3 * q^21 + 21 * q^23 + 16 * q^25 + 26 * q^27 + 14 * q^29 + 15 * q^31 + 19 * q^33 + 20 * q^35 + 2 * q^37 - 14 * q^39 + 34 * q^41 + 21 * q^43 + 49 * q^45 + 69 * q^47 + 28 * q^49 - 8 * q^51 - 4 * q^53 + 18 * q^55 + 5 * q^57 - 18 * q^59 + 11 * q^61 + 35 * q^63 + 27 * q^65 + 34 * q^67 - 4 * q^69 + 37 * q^71 + 18 * q^73 + 72 * q^75 + 11 * q^77 + 11 * q^79 + 30 * q^81 + 28 * q^83 - 4 * q^85 + 7 * q^87 + 44 * q^89 - 23 * q^91 - 3 * q^93 - 11 * q^95 + 11 * q^97 + 56 * q^99

Display 100 coefficients 10 coefficients

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
$3$	0.154866	0.0894121	0.0447061	−	0.999000i	$-0.485765\pi$
$3$	0.154866	0.0894121	0.0447061	+	0.999000i	$0.485765\pi$
$4$	0	0
$5$	−1.21070	−0.541441	−0.270720	−	0.962658i	$-0.587262\pi$
$5$	−1.21070	−0.541441	−0.270720	+	0.962658i	$0.587262\pi$
$6$	0	0
$7$	−0.213015	−0.0805122	−0.0402561	−	0.999189i	$-0.512817\pi$
$7$	−0.213015	−0.0805122	−0.0402561	+	0.999189i	$0.512817\pi$
$8$	0	0
$9$	−2.97602	−0.992005
$10$	0	0
$11$	6.43345	1.93976	0.969880	−	0.243585i	$-0.0783235\pi$
$11$	6.43345	1.93976	0.969880	+	0.243585i	$0.0783235\pi$
$12$	0	0
$13$	5.00862	1.38914	0.694570	−	0.719425i	$-0.255595\pi$
$13$	5.00862	1.38914	0.694570	+	0.719425i	$0.255595\pi$
$14$	0	0
$15$	−0.187496	−0.0484114
$16$	0	0
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	0	0
$19$	−5.48083	−1.25739	−0.628695	−	0.777652i	$-0.716411\pi$
$19$	−5.48083	−1.25739	−0.628695	+	0.777652i	$0.716411\pi$
$20$	0	0
$21$	−0.0329889	−0.00719877
$22$	0	0
$23$	0.395346	0.0824353	0.0412177	−	0.999150i	$-0.486876\pi$
$23$	0.395346	0.0824353	0.0412177	+	0.999150i	$0.486876\pi$
$24$	0	0
$25$	−3.53421	−0.706842
$26$	0	0
$27$	−0.925484	−0.178109
$28$	0	0
$29$	0.149252	0.0277154	0.0138577	−	0.999904i	$-0.495589\pi$
$29$	0.149252	0.0277154	0.0138577	+	0.999904i	$0.495589\pi$
$30$	0	0
$31$	6.05516	1.08754	0.543770	−	0.839234i	$-0.316996\pi$
$31$	6.05516	1.08754	0.543770	+	0.839234i	$0.316996\pi$
$32$	0	0
$33$	0.996326	0.173438
$34$	0	0
$35$	0.257897	0.0435926
$36$	0	0
$37$	9.39455	1.54445	0.772227	−	0.635347i	$-0.219143\pi$
$37$	9.39455	1.54445	0.772227	+	0.635347i	$0.219143\pi$
$38$	0	0
$39$	0.775666	0.124206
$40$	0	0
$41$	3.27099	0.510843	0.255421	−	0.966830i	$-0.417786\pi$
$41$	3.27099	0.510843	0.255421	+	0.966830i	$0.417786\pi$
$42$	0	0
$43$	−10.2298	−1.56003	−0.780013	−	0.625763i	$-0.784788\pi$
$43$	−10.2298	−1.56003	−0.780013	+	0.625763i	$0.784788\pi$
$44$	0	0
$45$	3.60306	0.537112
$46$	0	0
$47$	5.33452	0.778119	0.389060	−	0.921213i	$-0.372800\pi$
$47$	5.33452	0.778119	0.389060	+	0.921213i	$0.372800\pi$
$48$	0	0
$49$	−6.95462	−0.993518
$50$	0	0
$51$	−0.154866	−0.0216856
$52$	0	0
$53$	−1.89023	−0.259644	−0.129822	−	0.991537i	$-0.541441\pi$
$53$	−1.89023	−0.259644	−0.129822	+	0.991537i	$0.541441\pi$
$54$	0	0
$55$	−7.78897	−1.05026
$56$	0	0
$57$	−0.848797	−0.112426
$58$	0	0
$59$	−1.00000	−0.130189
$59$	−1.00000	−0.130189
$60$	0	0
$61$	12.6911	1.62493	0.812465	−	0.583010i	$-0.198125\pi$
$61$	12.6911	1.62493	0.812465	+	0.583010i	$0.198125\pi$
$62$	0	0
$63$	0.633937	0.0798685
$64$	0	0
$65$	−6.06392	−0.752137
$66$	0	0
$67$	4.18405	0.511163	0.255581	−	0.966788i	$-0.417733\pi$
$67$	4.18405	0.511163	0.255581	+	0.966788i	$0.417733\pi$
$68$	0	0
$69$	0.0612258	0.00737072
$70$	0	0
$71$	−13.4344	−1.59437	−0.797184	−	0.603736i	$-0.793678\pi$
$71$	−13.4344	−1.59437	−0.797184	+	0.603736i	$0.793678\pi$
$72$	0	0
$73$	10.3569	1.21218	0.606091	−	0.795395i	$-0.292737\pi$
$73$	10.3569	1.21218	0.606091	+	0.795395i	$0.292737\pi$
$74$	0	0
$75$	−0.547330	−0.0632003
$76$	0	0
$77$	−1.37042	−0.156174
$78$	0	0
$79$	12.1957	1.37213	0.686064	−	0.727541i	$-0.259337\pi$
$79$	12.1957	1.37213	0.686064	+	0.727541i	$0.259337\pi$
$80$	0	0
$81$	8.78472	0.976080
$82$	0	0
$83$	14.4904	1.59053	0.795264	−	0.606263i	$-0.207332\pi$
$83$	14.4904	1.59053	0.795264	+	0.606263i	$0.207332\pi$
$84$	0	0
$85$	1.21070	0.131319
$86$	0	0
$87$	0.0231141	0.00247809
$88$	0	0
$89$	−13.0985	−1.38844	−0.694219	−	0.719764i	$-0.744250\pi$
$89$	−13.0985	−1.38844	−0.694219	+	0.719764i	$0.744250\pi$
$90$	0	0
$91$	−1.06691	−0.111843
$92$	0	0
$93$	0.937741	0.0972392
$94$	0	0
$95$	6.63564	0.680802
$96$	0	0
$97$	−12.7867	−1.29829	−0.649145	−	0.760664i	$-0.724873\pi$
$97$	−12.7867	−1.29829	−0.649145	+	0.760664i	$0.724873\pi$
$98$	0	0
$99$	−19.1461	−1.92425
$100$	0	0
$101$	−6.89097	−0.685677	−0.342839	−	0.939394i	$-0.611388\pi$
$101$	−6.89097	−0.685677	−0.342839	+	0.939394i	$0.611388\pi$
$102$	0	0
$103$	−12.0012	−1.18251	−0.591255	−	0.806484i	$-0.701367\pi$
$103$	−12.0012	−1.18251	−0.591255	+	0.806484i	$0.701367\pi$
$104$	0	0
$105$	0.0399396	0.00389770
$106$	0	0
$107$	−0.468199	−0.0452625	−0.0226313	−	0.999744i	$-0.507204\pi$
$107$	−0.468199	−0.0452625	−0.0226313	+	0.999744i	$0.507204\pi$
$108$	0	0
$109$	18.1376	1.73726	0.868632	−	0.495458i	$-0.165000\pi$
$109$	18.1376	1.73726	0.868632	+	0.495458i	$0.165000\pi$
$110$	0	0
$111$	1.45490	0.138093
$112$	0	0
$113$	2.31463	0.217742	0.108871	−	0.994056i	$-0.465277\pi$
$113$	2.31463	0.217742	0.108871	+	0.994056i	$0.465277\pi$
$114$	0	0
$115$	−0.478645	−0.0446338
$116$	0	0
$117$	−14.9057	−1.37803
$118$	0	0
$119$	0.213015	0.0195271
$120$	0	0
$121$	30.3893	2.76267
$122$	0	0
$123$	0.506566	0.0456755
$124$	0	0
$125$	10.3324	0.924154
$126$	0	0
$127$	−18.3425	−1.62763	−0.813815	−	0.581124i	$-0.802613\pi$
$127$	−18.3425	−1.62763	−0.813815	+	0.581124i	$0.802613\pi$
$128$	0	0
$129$	−1.58425	−0.139485
$130$	0	0
$131$	0.439482	0.0383977	0.0191989	−	0.999816i	$-0.493888\pi$
$131$	0.439482	0.0383977	0.0191989	+	0.999816i	$0.493888\pi$
$132$	0	0
$133$	1.16750	0.101235
$134$	0	0
$135$	1.12048	0.0964357
$136$	0	0
$137$	−0.0470298	−0.00401802	−0.00200901	−	0.999998i	$-0.500639\pi$
$137$	−0.0470298	−0.00401802	−0.00200901	+	0.999998i	$0.500639\pi$
$138$	0	0
$139$	8.24857	0.699635	0.349817	−	0.936818i	$-0.386244\pi$
$139$	8.24857	0.699635	0.349817	+	0.936818i	$0.386244\pi$
$140$	0	0
$141$	0.826137	0.0695733
$142$	0	0
$143$	32.2227	2.69460
$144$	0	0
$145$	−0.180699	−0.0150063
$146$	0	0
$147$	−1.07704	−0.0888326
$148$	0	0
$149$	13.0755	1.07119	0.535593	−	0.844476i	$-0.320088\pi$
$149$	13.0755	1.07119	0.535593	+	0.844476i	$0.320088\pi$
$150$	0	0
$151$	15.7179	1.27911	0.639553	−	0.768747i	$-0.279119\pi$
$151$	15.7179	1.27911	0.639553	+	0.768747i	$0.279119\pi$
$152$	0	0
$153$	2.97602	0.240597
$154$	0	0
$155$	−7.33098	−0.588838
$156$	0	0
$157$	8.55421	0.682700	0.341350	−	0.939936i	$-0.389116\pi$
$157$	8.55421	0.682700	0.341350	+	0.939936i	$0.389116\pi$
$158$	0	0
$159$	−0.292734	−0.0232153
$160$	0	0
$161$	−0.0842147	−0.00663705
$162$	0	0
$163$	20.7642	1.62638	0.813189	−	0.582000i	$-0.197730\pi$
$163$	20.7642	1.62638	0.813189	+	0.582000i	$0.197730\pi$
$164$	0	0
$165$	−1.20625	−0.0939064
$166$	0	0
$167$	10.4974	0.812312	0.406156	−	0.913804i	$-0.366869\pi$
$167$	10.4974	0.812312	0.406156	+	0.913804i	$0.366869\pi$
$168$	0	0
$169$	12.0862	0.929710
$170$	0	0
$171$	16.3111	1.24734
$172$	0	0
$173$	20.1047	1.52853	0.764264	−	0.644903i	$-0.223102\pi$
$173$	20.1047	1.52853	0.764264	+	0.644903i	$0.223102\pi$
$174$	0	0
$175$	0.752840	0.0569094
$176$	0	0
$177$	−0.154866	−0.0116405
$178$	0	0
$179$	22.8655	1.70905	0.854524	−	0.519412i	$-0.173849\pi$
$179$	22.8655	1.70905	0.854524	+	0.519412i	$0.173849\pi$
$180$	0	0
$181$	5.51996	0.410295	0.205148	−	0.978731i	$-0.434233\pi$
$181$	5.51996	0.410295	0.205148	+	0.978731i	$0.434233\pi$
$182$	0	0
$183$	1.96543	0.145288
$184$	0	0
$185$	−11.3740	−0.836230
$186$	0	0
$187$	−6.43345	−0.470461
$188$	0	0
$189$	0.197142	0.0143400
$190$	0	0
$191$	7.90896	0.572272	0.286136	−	0.958189i	$-0.407629\pi$
$191$	7.90896	0.572272	0.286136	+	0.958189i	$0.407629\pi$
$192$	0	0
$193$	−15.6563	−1.12696	−0.563482	−	0.826128i	$-0.690539\pi$
$193$	−15.6563	−1.12696	−0.563482	+	0.826128i	$0.690539\pi$
$194$	0	0
$195$	−0.939098	−0.0672502
$196$	0	0
$197$	14.4693	1.03089	0.515447	−	0.856922i	$-0.327626\pi$
$197$	14.4693	1.03089	0.515447	+	0.856922i	$0.327626\pi$
$198$	0	0
$199$	−4.43595	−0.314456	−0.157228	−	0.987562i	$-0.550256\pi$
$199$	−4.43595	−0.314456	−0.157228	+	0.987562i	$0.550256\pi$
$200$	0	0
$201$	0.647968	0.0457042
$202$	0	0
$203$	−0.0317930	−0.00223143
$204$	0	0
$205$	−3.96018	−0.276591
$206$	0	0
$207$	−1.17656	−0.0817763
$208$	0	0
$209$	−35.2607	−2.43903
$210$	0	0
$211$	11.8643	0.816775	0.408388	−	0.912809i	$-0.366091\pi$
$211$	11.8643	0.816775	0.408388	+	0.912809i	$0.366091\pi$
$212$	0	0
$213$	−2.08054	−0.142556
$214$	0	0
$215$	12.3852	0.844661
$216$	0	0
$217$	−1.28984	−0.0875602
$218$	0	0
$219$	1.60393	0.108384
$220$	0	0
$221$	−5.00862	−0.336916
$222$	0	0
$223$	6.32308	0.423425	0.211712	−	0.977332i	$-0.432096\pi$
$223$	6.32308	0.423425	0.211712	+	0.977332i	$0.432096\pi$
$224$	0	0
$225$	10.5179	0.701191
$226$	0	0
$227$	4.82472	0.320228	0.160114	−	0.987099i	$-0.448814\pi$
$227$	4.82472	0.320228	0.160114	+	0.987099i	$0.448814\pi$
$228$	0	0
$229$	−11.3737	−0.751598	−0.375799	−	0.926701i	$-0.622632\pi$
$229$	−11.3737	−0.751598	−0.375799	+	0.926701i	$0.622632\pi$
$230$	0	0
$231$	−0.212232	−0.0139639
$232$	0	0
$233$	23.4142	1.53392	0.766958	−	0.641697i	$-0.221769\pi$
$233$	23.4142	1.53392	0.766958	+	0.641697i	$0.221769\pi$
$234$	0	0
$235$	−6.45849	−0.421305
$236$	0	0
$237$	1.88871	0.122685
$238$	0	0
$239$	−21.8007	−1.41017	−0.705086	−	0.709121i	$-0.749092\pi$
$239$	−21.8007	−1.41017	−0.705086	+	0.709121i	$0.749092\pi$
$240$	0	0
$241$	−5.58383	−0.359686	−0.179843	−	0.983695i	$-0.557559\pi$
$241$	−5.58383	−0.359686	−0.179843	+	0.983695i	$0.557559\pi$
$242$	0	0
$243$	4.13691	0.265383
$244$	0	0
$245$	8.41995	0.537931
$246$	0	0
$247$	−27.4514	−1.74669
$248$	0	0
$249$	2.24408	0.142212
$250$	0	0
$251$	23.0835	1.45701	0.728507	−	0.685038i	$-0.240214\pi$
$251$	23.0835	1.45701	0.728507	+	0.685038i	$0.240214\pi$
$252$	0	0
$253$	2.54344	0.159905
$254$	0	0
$255$	0.187496	0.0117415
$256$	0	0
$257$	15.4097	0.961234	0.480617	−	0.876931i	$-0.340413\pi$
$257$	15.4097	0.961234	0.480617	+	0.876931i	$0.340413\pi$
$258$	0	0
$259$	−2.00118	−0.124347
$260$	0	0
$261$	−0.444177	−0.0274938
$262$	0	0
$263$	−2.89257	−0.178364	−0.0891818	−	0.996015i	$-0.528425\pi$
$263$	−2.89257	−0.178364	−0.0891818	+	0.996015i	$0.528425\pi$
$264$	0	0
$265$	2.28850	0.140582
$266$	0	0
$267$	−2.02852	−0.124143
$268$	0	0
$269$	−14.5423	−0.886662	−0.443331	−	0.896358i	$-0.646203\pi$
$269$	−14.5423	−0.886662	−0.443331	+	0.896358i	$0.646203\pi$
$270$	0	0
$271$	7.50613	0.455965	0.227982	−	0.973665i	$-0.426787\pi$
$271$	7.50613	0.455965	0.227982	+	0.973665i	$0.426787\pi$
$272$	0	0
$273$	−0.165229	−0.0100001
$274$	0	0
$275$	−22.7372	−1.37110
$276$	0	0
$277$	−17.9921	−1.08104	−0.540520	−	0.841331i	$-0.681773\pi$
$277$	−17.9921	−1.08104	−0.540520	+	0.841331i	$0.681773\pi$
$278$	0	0
$279$	−18.0203	−1.07885
$280$	0	0
$281$	11.2171	0.669159	0.334579	−	0.942368i	$-0.391406\pi$
$281$	11.2171	0.669159	0.334579	+	0.942368i	$0.391406\pi$
$282$	0	0
$283$	−18.5968	−1.10546	−0.552732	−	0.833359i	$-0.686415\pi$
$283$	−18.5968	−1.10546	−0.552732	+	0.833359i	$0.686415\pi$
$284$	0	0
$285$	1.02764	0.0608720
$286$	0	0
$287$	−0.696770	−0.0411290
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	−1.98023	−0.116083
$292$	0	0
$293$	−30.3210	−1.77137	−0.885685	−	0.464287i	$-0.846311\pi$
$293$	−30.3210	−1.77137	−0.885685	+	0.464287i	$0.846311\pi$
$294$	0	0
$295$	1.21070	0.0704896
$296$	0	0
$297$	−5.95406	−0.345489
$298$	0	0
$299$	1.98014	0.114514
$300$	0	0
$301$	2.17910	0.125601
$302$	0	0
$303$	−1.06718	−0.0613079
$304$	0	0
$305$	−15.3651	−0.879803
$306$	0	0
$307$	6.33261	0.361421	0.180710	−	0.983536i	$-0.442160\pi$
$307$	6.33261	0.361421	0.180710	+	0.983536i	$0.442160\pi$
$308$	0	0
$309$	−1.85858	−0.105731
$310$	0	0
$311$	18.1013	1.02643	0.513216	−	0.858260i	$-0.328454\pi$
$311$	18.1013	1.02643	0.513216	+	0.858260i	$0.328454\pi$
$312$	0	0
$313$	−14.8928	−0.841793	−0.420896	−	0.907109i	$-0.638284\pi$
$313$	−14.8928	−0.841793	−0.420896	+	0.907109i	$0.638284\pi$
$314$	0	0
$315$	−0.767506	−0.0432441
$316$	0	0
$317$	−5.08565	−0.285639	−0.142819	−	0.989749i	$-0.545617\pi$
$317$	−5.08565	−0.285639	−0.142819	+	0.989749i	$0.545617\pi$
$318$	0	0
$319$	0.960206	0.0537612
$320$	0	0
$321$	−0.0725084	−0.00404702
$322$	0	0
$323$	5.48083	0.304962
$324$	0	0
$325$	−17.7015	−0.981902
$326$	0	0
$327$	2.80890	0.155332
$328$	0	0
$329$	−1.13633	−0.0626480
$330$	0	0
$331$	−14.1994	−0.780471	−0.390236	−	0.920715i	$-0.627606\pi$
$331$	−14.1994	−0.780471	−0.390236	+	0.920715i	$0.627606\pi$
$332$	0	0
$333$	−27.9583	−1.53211
$334$	0	0
$335$	−5.06562	−0.276764
$336$	0	0
$337$	25.8799	1.40977	0.704884	−	0.709322i	$-0.250999\pi$
$337$	25.8799	1.40977	0.704884	+	0.709322i	$0.250999\pi$
$338$	0	0
$339$	0.358458	0.0194687
$340$	0	0
$341$	38.9556	2.10956
$342$	0	0
$343$	2.97255	0.160502
$344$	0	0
$345$	−0.0741259	−0.00399081
$346$	0	0
$347$	12.5536	0.673912	0.336956	−	0.941520i	$-0.390603\pi$
$347$	12.5536	0.673912	0.336956	+	0.941520i	$0.390603\pi$
$348$	0	0
$349$	−29.6085	−1.58491	−0.792454	−	0.609932i	$-0.791197\pi$
$349$	−29.6085	−1.58491	−0.792454	+	0.609932i	$0.791197\pi$
$350$	0	0
$351$	−4.63539	−0.247419
$352$	0	0
$353$	17.0172	0.905735	0.452868	−	0.891578i	$-0.350401\pi$
$353$	17.0172	0.905735	0.452868	+	0.891578i	$0.350401\pi$
$354$	0	0
$355$	16.2650	0.863256
$356$	0	0
$357$	0.0329889	0.00174596
$358$	0	0
$359$	14.7011	0.775894	0.387947	−	0.921682i	$-0.373184\pi$
$359$	14.7011	0.775894	0.387947	+	0.921682i	$0.373184\pi$
$360$	0	0
$361$	11.0395	0.581029
$362$	0	0
$363$	4.70628	0.247016
$364$	0	0
$365$	−12.5391	−0.656325
$366$	0	0
$367$	−9.68067	−0.505327	−0.252664	−	0.967554i	$-0.581307\pi$
$367$	−9.68067	−0.505327	−0.252664	+	0.967554i	$0.581307\pi$
$368$	0	0
$369$	−9.73452	−0.506759
$370$	0	0
$371$	0.402649	0.0209045
$372$	0	0
$373$	14.3874	0.744953	0.372477	−	0.928042i	$-0.378509\pi$
$373$	14.3874	0.744953	0.372477	+	0.928042i	$0.378509\pi$
$374$	0	0
$375$	1.60013	0.0826306
$376$	0	0
$377$	0.747546	0.0385006
$378$	0	0
$379$	22.1104	1.13573	0.567867	−	0.823120i	$-0.307769\pi$
$379$	22.1104	1.13573	0.567867	+	0.823120i	$0.307769\pi$
$380$	0	0
$381$	−2.84063	−0.145530
$382$	0	0
$383$	−33.8662	−1.73048	−0.865242	−	0.501354i	$-0.832835\pi$
$383$	−33.8662	−1.73048	−0.865242	+	0.501354i	$0.832835\pi$
$384$	0	0
$385$	1.65917	0.0845591
$386$	0	0
$387$	30.4440	1.54755
$388$	0	0
$389$	−8.42082	−0.426952	−0.213476	−	0.976948i	$-0.568479\pi$
$389$	−8.42082	−0.426952	−0.213476	+	0.976948i	$0.568479\pi$
$390$	0	0
$391$	−0.395346	−0.0199935
$392$	0	0
$393$	0.0680610	0.00343322
$394$	0	0
$395$	−14.7654	−0.742926
$396$	0	0
$397$	−8.97061	−0.450222	−0.225111	−	0.974333i	$-0.572274\pi$
$397$	−8.97061	−0.450222	−0.225111	+	0.974333i	$0.572274\pi$
$398$	0	0
$399$	0.180807	0.00905165
$400$	0	0
$401$	−38.7783	−1.93650	−0.968248	−	0.249990i	$-0.919573\pi$
$401$	−38.7783	−1.93650	−0.968248	+	0.249990i	$0.919573\pi$
$402$	0	0
$403$	30.3280	1.51074
$404$	0	0
$405$	−10.6356	−0.528490
$406$	0	0
$407$	60.4394	2.99587
$408$	0	0
$409$	−19.3626	−0.957419	−0.478710	−	0.877973i	$-0.658895\pi$
$409$	−19.3626	−0.957419	−0.478710	+	0.877973i	$0.658895\pi$
$410$	0	0
$411$	−0.00728333	−0.000359260 0
$412$	0	0
$413$	0.213015	0.0104818
$414$	0	0
$415$	−17.5435	−0.861176
$416$	0	0
$417$	1.27743	0.0625559
$418$	0	0
$419$	22.9122	1.11933	0.559666	−	0.828718i	$-0.310929\pi$
$419$	22.9122	1.11933	0.559666	+	0.828718i	$0.310929\pi$
$420$	0	0
$421$	−19.5818	−0.954358	−0.477179	−	0.878806i	$-0.658341\pi$
$421$	−19.5818	−0.954358	−0.477179	+	0.878806i	$0.658341\pi$
$422$	0	0
$423$	−15.8756	−0.771898
$424$	0	0
$425$	3.53421	0.171434
$426$	0	0
$427$	−2.70340	−0.130827
$428$	0	0
$429$	4.99021	0.240930
$430$	0	0
$431$	−36.5109	−1.75867	−0.879334	−	0.476205i	$-0.842012\pi$
$431$	−36.5109	−1.75867	−0.879334	+	0.476205i	$0.842012\pi$
$432$	0	0
$433$	−5.34427	−0.256829	−0.128415	−	0.991721i	$-0.540989\pi$
$433$	−5.34427	−0.256829	−0.128415	+	0.991721i	$0.540989\pi$
$434$	0	0
$435$	−0.0279842	−0.00134174
$436$	0	0
$437$	−2.16683	−0.103653
$438$	0	0
$439$	−33.4058	−1.59437	−0.797186	−	0.603734i	$-0.793679\pi$
$439$	−33.4058	−1.59437	−0.797186	+	0.603734i	$0.793679\pi$
$440$	0	0
$441$	20.6971	0.985575
$442$	0	0
$443$	27.1676	1.29077	0.645385	−	0.763857i	$-0.276697\pi$
$443$	27.1676	1.29077	0.645385	+	0.763857i	$0.276697\pi$
$444$	0	0
$445$	15.8583	0.751757
$446$	0	0
$447$	2.02496	0.0957771
$448$	0	0
$449$	−23.1360	−1.09185	−0.545927	−	0.837832i	$-0.683823\pi$
$449$	−23.1360	−1.09185	−0.545927	+	0.837832i	$0.683823\pi$
$450$	0	0
$451$	21.0438	0.990912
$452$	0	0
$453$	2.43418	0.114368
$454$	0	0
$455$	1.29171	0.0605562
$456$	0	0
$457$	−9.46572	−0.442788	−0.221394	−	0.975184i	$-0.571061\pi$
$457$	−9.46572	−0.442788	−0.221394	+	0.975184i	$0.571061\pi$
$458$	0	0
$459$	0.925484	0.0431979
$460$	0	0
$461$	−6.97720	−0.324961	−0.162480	−	0.986712i	$-0.551949\pi$
$461$	−6.97720	−0.324961	−0.162480	+	0.986712i	$0.551949\pi$
$462$	0	0
$463$	−2.64737	−0.123034	−0.0615168	−	0.998106i	$-0.519594\pi$
$463$	−2.64737	−0.123034	−0.0615168	+	0.998106i	$0.519594\pi$
$464$	0	0
$465$	−1.13532	−0.0526493
$466$	0	0
$467$	17.4991	0.809760	0.404880	−	0.914370i	$-0.367313\pi$
$467$	17.4991	0.809760	0.404880	+	0.914370i	$0.367313\pi$
$468$	0	0
$469$	−0.891266	−0.0411548
$470$	0	0
$471$	1.32476	0.0610417
$472$	0	0
$473$	−65.8128	−3.02607
$474$	0	0
$475$	19.3704	0.888776
$476$	0	0
$477$	5.62537	0.257568
$478$	0	0
$479$	24.2628	1.10860	0.554298	−	0.832318i	$-0.312987\pi$
$479$	24.2628	1.10860	0.554298	+	0.832318i	$0.312987\pi$
$480$	0	0
$481$	47.0537	2.14546
$482$	0	0
$483$	−0.0130420	−0.000593433 0
$484$	0	0
$485$	15.4808	0.702947
$486$	0	0
$487$	−0.570679	−0.0258599	−0.0129300	−	0.999916i	$-0.504116\pi$
$487$	−0.570679	−0.0258599	−0.0129300	+	0.999916i	$0.504116\pi$
$488$	0	0
$489$	3.21568	0.145418
$490$	0	0
$491$	34.3222	1.54894	0.774469	−	0.632612i	$-0.218017\pi$
$491$	34.3222	1.54894	0.774469	+	0.632612i	$0.218017\pi$
$492$	0	0
$493$	−0.149252	−0.00672198
$494$	0	0
$495$	23.1801	1.04187
$496$	0	0
$497$	2.86173	0.128366
$498$	0	0
$499$	−4.32912	−0.193798	−0.0968990	−	0.995294i	$-0.530892\pi$
$499$	−4.32912	−0.193798	−0.0968990	+	0.995294i	$0.530892\pi$
$500$	0	0
$501$	1.62569	0.0726306
$502$	0	0
$503$	27.1245	1.20942	0.604711	−	0.796445i	$-0.293288\pi$
$503$	27.1245	1.20942	0.604711	+	0.796445i	$0.293288\pi$
$504$	0	0
$505$	8.34288	0.371253
$506$	0	0
$507$	1.87175	0.0831273
$508$	0	0
$509$	−19.7922	−0.877274	−0.438637	−	0.898664i	$-0.644539\pi$
$509$	−19.7922	−0.877274	−0.438637	+	0.898664i	$0.644539\pi$
$510$	0	0
$511$	−2.20617	−0.0975954
$512$	0	0
$513$	5.07242	0.223953
$514$	0	0
$515$	14.5298	0.640259
$516$	0	0
$517$	34.3194	1.50936
$518$	0	0
$519$	3.11353	0.136669
$520$	0	0
$521$	24.5041	1.07354	0.536771	−	0.843728i	$-0.319644\pi$
$521$	24.5041	1.07354	0.536771	+	0.843728i	$0.319644\pi$
$522$	0	0
$523$	27.8968	1.21984	0.609922	−	0.792462i	$-0.291201\pi$
$523$	27.8968	1.21984	0.609922	+	0.792462i	$0.291201\pi$
$524$	0	0
$525$	0.116590	0.00508839
$526$	0	0
$527$	−6.05516	−0.263767
$528$	0	0
$529$	−22.8437	−0.993204
$530$	0	0
$531$	2.97602	0.129148
$532$	0	0
$533$	16.3831	0.709632
$534$	0	0
$535$	0.566848	0.0245070
$536$	0	0
$537$	3.54110	0.152810
$538$	0	0
$539$	−44.7423	−1.92719
$540$	0	0
$541$	4.75991	0.204645	0.102322	−	0.994751i	$-0.467373\pi$
$541$	4.75991	0.204645	0.102322	+	0.994751i	$0.467373\pi$
$542$	0	0
$543$	0.854856	0.0366854
$544$	0	0
$545$	−21.9591	−0.940625
$546$	0	0
$547$	−34.9974	−1.49638	−0.748191	−	0.663483i	$-0.769077\pi$
$547$	−34.9974	−1.49638	−0.748191	+	0.663483i	$0.769077\pi$
$548$	0	0
$549$	−37.7689	−1.61194
$550$	0	0
$551$	−0.818026	−0.0348491
$552$	0	0
$553$	−2.59788	−0.110473
$554$	0	0
$555$	−1.76144	−0.0747691
$556$	0	0
$557$	32.7072	1.38585	0.692924	−	0.721011i	$-0.256322\pi$
$557$	32.7072	1.38585	0.692924	+	0.721011i	$0.256322\pi$
$558$	0	0
$559$	−51.2370	−2.16709
$560$	0	0
$561$	−0.996326	−0.0420649
$562$	0	0
$563$	−4.12852	−0.173996	−0.0869981	−	0.996208i	$-0.527727\pi$
$563$	−4.12852	−0.173996	−0.0869981	+	0.996208i	$0.527727\pi$
$564$	0	0
$565$	−2.80231	−0.117894
$566$	0	0
$567$	−1.87128	−0.0785863
$568$	0	0
$569$	7.21268	0.302371	0.151186	−	0.988505i	$-0.451691\pi$
$569$	7.21268	0.302371	0.151186	+	0.988505i	$0.451691\pi$
$570$	0	0
$571$	−14.1763	−0.593259	−0.296630	−	0.954993i	$-0.595863\pi$
$571$	−14.1763	−0.593259	−0.296630	+	0.954993i	$0.595863\pi$
$572$	0	0
$573$	1.22483	0.0511681
$574$	0	0
$575$	−1.39724	−0.0582687
$576$	0	0
$577$	−21.1637	−0.881057	−0.440528	−	0.897739i	$-0.645209\pi$
$577$	−21.1637	−0.881057	−0.440528	+	0.897739i	$0.645209\pi$
$578$	0	0
$579$	−2.42463	−0.100764
$580$	0	0
$581$	−3.08667	−0.128057
$582$	0	0
$583$	−12.1607	−0.503646
$584$	0	0
$585$	18.0463	0.746124
$586$	0	0
$587$	−4.64727	−0.191814	−0.0959068	−	0.995390i	$-0.530575\pi$
$587$	−4.64727	−0.191814	−0.0959068	+	0.995390i	$0.530575\pi$
$588$	0	0
$589$	−33.1874	−1.36746
$590$	0	0
$591$	2.24080	0.0921744
$592$	0	0
$593$	7.52020	0.308818	0.154409	−	0.988007i	$-0.450653\pi$
$593$	7.52020	0.308818	0.154409	+	0.988007i	$0.450653\pi$
$594$	0	0
$595$	−0.257897	−0.0105727
$596$	0	0
$597$	−0.686979	−0.0281162
$598$	0	0
$599$	21.7693	0.889469	0.444735	−	0.895662i	$-0.353298\pi$
$599$	21.7693	0.889469	0.444735	+	0.895662i	$0.353298\pi$
$600$	0	0
$601$	30.5961	1.24804	0.624019	−	0.781409i	$-0.285499\pi$
$601$	30.5961	1.24804	0.624019	+	0.781409i	$0.285499\pi$
$602$	0	0
$603$	−12.4518	−0.507076
$604$	0	0
$605$	−36.7923	−1.49582
$606$	0	0
$607$	1.99631	0.0810277	0.0405138	−	0.999179i	$-0.487101\pi$
$607$	1.99631	0.0810277	0.0405138	+	0.999179i	$0.487101\pi$
$608$	0	0
$609$	−0.00492366	−0.000199517 0
$610$	0	0
$611$	26.7185	1.08092
$612$	0	0
$613$	−40.1358	−1.62107	−0.810535	−	0.585690i	$-0.800824\pi$
$613$	−40.1358	−1.62107	−0.810535	+	0.585690i	$0.800824\pi$
$614$	0	0
$615$	−0.613299	−0.0247306
$616$	0	0
$617$	−0.546845	−0.0220151	−0.0110076	−	0.999939i	$-0.503504\pi$
$617$	−0.546845	−0.0220151	−0.0110076	+	0.999939i	$0.503504\pi$
$618$	0	0
$619$	39.1881	1.57510	0.787552	−	0.616248i	$-0.211348\pi$
$619$	39.1881	1.57510	0.787552	+	0.616248i	$0.211348\pi$
$620$	0	0
$621$	−0.365886	−0.0146825
$622$	0	0
$623$	2.79018	0.111786
$624$	0	0
$625$	5.16169	0.206468
$626$	0	0
$627$	−5.46070	−0.218079
$628$	0	0
$629$	−9.39455	−0.374585
$630$	0	0
$631$	10.0356	0.399512	0.199756	−	0.979846i	$-0.435985\pi$
$631$	10.0356	0.399512	0.199756	+	0.979846i	$0.435985\pi$
$632$	0	0
$633$	1.83739	0.0730296
$634$	0	0
$635$	22.2072	0.881265
$636$	0	0
$637$	−34.8330	−1.38014
$638$	0	0
$639$	39.9810	1.58162
$640$	0	0
$641$	−32.4071	−1.28000	−0.640002	−	0.768373i	$-0.721067\pi$
$641$	−32.4071	−1.28000	−0.640002	+	0.768373i	$0.721067\pi$
$642$	0	0
$643$	−23.1113	−0.911421	−0.455710	−	0.890128i	$-0.650615\pi$
$643$	−23.1113	−0.911421	−0.455710	+	0.890128i	$0.650615\pi$
$644$	0	0
$645$	1.91805	0.0755230
$646$	0	0
$647$	−12.3387	−0.485085	−0.242543	−	0.970141i	$-0.577981\pi$
$647$	−12.3387	−0.485085	−0.242543	+	0.970141i	$0.577981\pi$
$648$	0	0
$649$	−6.43345	−0.252535
$650$	0	0
$651$	−0.199753	−0.00782894
$652$	0	0
$653$	−21.1235	−0.826626	−0.413313	−	0.910589i	$-0.635628\pi$
$653$	−21.1235	−0.826626	−0.413313	+	0.910589i	$0.635628\pi$
$654$	0	0
$655$	−0.532080	−0.0207901
$656$	0	0
$657$	−30.8223	−1.20249
$658$	0	0
$659$	−20.2877	−0.790295	−0.395148	−	0.918618i	$-0.629307\pi$
$659$	−20.2877	−0.790295	−0.395148	+	0.918618i	$0.629307\pi$
$660$	0	0
$661$	−24.6495	−0.958753	−0.479377	−	0.877609i	$-0.659137\pi$
$661$	−24.6495	−0.958753	−0.479377	+	0.877609i	$0.659137\pi$
$662$	0	0
$663$	−0.775666	−0.0301244
$664$	0	0
$665$	−1.41349	−0.0548128
$666$	0	0
$667$	0.0590062	0.00228473
$668$	0	0
$669$	0.979233	0.0378593
$670$	0	0
$671$	81.6476	3.15197
$672$	0	0
$673$	−42.8333	−1.65110	−0.825551	−	0.564327i	$-0.809136\pi$
$673$	−42.8333	−1.65110	−0.825551	+	0.564327i	$0.809136\pi$
$674$	0	0
$675$	3.27085	0.125895
$676$	0	0
$677$	47.8241	1.83803	0.919015	−	0.394223i	$-0.128986\pi$
$677$	47.8241	1.83803	0.919015	+	0.394223i	$0.128986\pi$
$678$	0	0
$679$	2.72376	0.104528
$680$	0	0
$681$	0.747186	0.0286322
$682$	0	0
$683$	−16.2235	−0.620775	−0.310388	−	0.950610i	$-0.600459\pi$
$683$	−16.2235	−0.620775	−0.310388	+	0.950610i	$0.600459\pi$
$684$	0	0
$685$	0.0569388	0.00217552
$686$	0	0
$687$	−1.76141	−0.0672020
$688$	0	0
$689$	−9.46745	−0.360681
$690$	0	0
$691$	2.40804	0.0916063	0.0458032	−	0.998950i	$-0.485415\pi$
$691$	2.40804	0.0916063	0.0458032	+	0.998950i	$0.485415\pi$
$692$	0	0
$693$	4.07840	0.154926
$694$	0	0
$695$	−9.98653	−0.378811
$696$	0	0
$697$	−3.27099	−0.123898
$698$	0	0
$699$	3.62608	0.137151
$700$	0	0
$701$	−46.7914	−1.76729	−0.883643	−	0.468161i	$-0.844917\pi$
$701$	−46.7914	−1.76729	−0.883643	+	0.468161i	$0.844917\pi$
$702$	0	0
$703$	−51.4900	−1.94198
$704$	0	0
$705$	−1.00020	−0.0376698
$706$	0	0
$707$	1.46788	0.0552053
$708$	0	0
$709$	21.7339	0.816232	0.408116	−	0.912930i	$-0.366186\pi$
$709$	21.7339	0.816232	0.408116	+	0.912930i	$0.366186\pi$
$710$	0	0
$711$	−36.2947	−1.36116
$712$	0	0
$713$	2.39388	0.0896517
$714$	0	0
$715$	−39.0120	−1.45896
$716$	0	0
$717$	−3.37620	−0.126087
$718$	0	0
$719$	−6.90123	−0.257372	−0.128686	−	0.991685i	$-0.541076\pi$
$719$	−6.90123	−0.257372	−0.128686	+	0.991685i	$0.541076\pi$
$720$	0	0
$721$	2.55643	0.0952065
$722$	0	0
$723$	−0.864748	−0.0321603
$724$	0	0
$725$	−0.527488	−0.0195904
$726$	0	0
$727$	−28.9722	−1.07452	−0.537259	−	0.843417i	$-0.680540\pi$
$727$	−28.9722	−1.07452	−0.537259	+	0.843417i	$0.680540\pi$
$728$	0	0
$729$	−25.7135	−0.952352
$730$	0	0
$731$	10.2298	0.378362
$732$	0	0
$733$	−31.4382	−1.16120	−0.580599	−	0.814190i	$-0.697181\pi$
$733$	−31.4382	−1.16120	−0.580599	+	0.814190i	$0.697181\pi$
$734$	0	0
$735$	1.30397	0.0480976
$736$	0	0
$737$	26.9179	0.991533
$738$	0	0
$739$	32.7559	1.20495	0.602473	−	0.798139i	$-0.294182\pi$
$739$	32.7559	1.20495	0.602473	+	0.798139i	$0.294182\pi$
$740$	0	0
$741$	−4.25130	−0.156175
$742$	0	0
$743$	27.7621	1.01849	0.509247	−	0.860620i	$-0.329924\pi$
$743$	27.7621	1.01849	0.509247	+	0.860620i	$0.329924\pi$
$744$	0	0
$745$	−15.8305	−0.579984
$746$	0	0
$747$	−43.1237	−1.57781
$748$	0	0
$749$	0.0997336	0.00364419
$750$	0	0
$751$	2.41491	0.0881212	0.0440606	−	0.999029i	$-0.485971\pi$
$751$	2.41491	0.0881212	0.0440606	+	0.999029i	$0.485971\pi$
$752$	0	0
$753$	3.57485	0.130275
$754$	0	0
$755$	−19.0297	−0.692560
$756$	0	0
$757$	−26.9623	−0.979961	−0.489980	−	0.871733i	$-0.662996\pi$
$757$	−26.9623	−0.979961	−0.489980	+	0.871733i	$0.662996\pi$
$758$	0	0
$759$	0.393893	0.0142974
$760$	0	0
$761$	4.12547	0.149548	0.0747741	−	0.997200i	$-0.476176\pi$
$761$	4.12547	0.149548	0.0747741	+	0.997200i	$0.476176\pi$
$762$	0	0
$763$	−3.86358	−0.139871
$764$	0	0
$765$	−3.60306	−0.130269
$766$	0	0
$767$	−5.00862	−0.180851
$768$	0	0
$769$	40.0856	1.44553	0.722763	−	0.691096i	$-0.242872\pi$
$769$	40.0856	1.44553	0.722763	+	0.691096i	$0.242872\pi$
$770$	0	0
$771$	2.38645	0.0859460
$772$	0	0
$773$	8.99613	0.323568	0.161784	−	0.986826i	$-0.448275\pi$
$773$	8.99613	0.323568	0.161784	+	0.986826i	$0.448275\pi$
$774$	0	0
$775$	−21.4002	−0.768719
$776$	0	0
$777$	−0.309916	−0.0111182
$778$	0	0
$779$	−17.9277	−0.642328
$780$	0	0
$781$	−86.4295	−3.09269
$782$	0	0
$783$	−0.138130	−0.00493638
$784$	0	0
$785$	−10.3566	−0.369642
$786$	0	0
$787$	−47.3484	−1.68779	−0.843894	−	0.536509i	$-0.819743\pi$
$787$	−47.3484	−1.68779	−0.843894	+	0.536509i	$0.819743\pi$
$788$	0	0
$789$	−0.447962	−0.0159479
$790$	0	0
$791$	−0.493050	−0.0175309
$792$	0	0
$793$	63.5649	2.25725
$794$	0	0
$795$	0.354412	0.0125697
$796$	0	0
$797$	7.45209	0.263966	0.131983	−	0.991252i	$-0.457866\pi$
$797$	7.45209	0.263966	0.131983	+	0.991252i	$0.457866\pi$
$798$	0	0
$799$	−5.33452	−0.188722
$800$	0	0
$801$	38.9814	1.37734
$802$	0	0
$803$	66.6306	2.35134
$804$	0	0
$805$	0.101959	0.00359357
$806$	0	0
$807$	−2.25212	−0.0792783
$808$	0	0
$809$	−25.5647	−0.898806	−0.449403	−	0.893329i	$-0.648363\pi$
$809$	−25.5647	−0.898806	−0.449403	+	0.893329i	$0.648363\pi$
$810$	0	0
$811$	13.3449	0.468602	0.234301	−	0.972164i	$-0.424720\pi$
$811$	13.3449	0.468602	0.234301	+	0.972164i	$0.424720\pi$
$812$	0	0
$813$	1.16245	0.0407688
$814$	0	0
$815$	−25.1392	−0.880587
$816$	0	0
$817$	56.0677	1.96156
$818$	0	0
$819$	3.17514	0.110949
$820$	0	0
$821$	−34.2068	−1.19383	−0.596913	−	0.802306i	$-0.703606\pi$
$821$	−34.2068	−1.19383	−0.596913	+	0.802306i	$0.703606\pi$
$822$	0	0
$823$	−35.5017	−1.23751	−0.618755	−	0.785584i	$-0.712363\pi$
$823$	−35.5017	−1.23751	−0.618755	+	0.785584i	$0.712363\pi$
$824$	0	0
$825$	−3.52122	−0.122593
$826$	0	0
$827$	−52.8143	−1.83653	−0.918267	−	0.395962i	$-0.870411\pi$
$827$	−52.8143	−1.83653	−0.918267	+	0.395962i	$0.870411\pi$
$828$	0	0
$829$	−15.8624	−0.550925	−0.275463	−	0.961312i	$-0.588831\pi$
$829$	−15.8624	−0.550925	−0.275463	+	0.961312i	$0.588831\pi$
$830$	0	0
$831$	−2.78637	−0.0966582
$832$	0	0
$833$	6.95462	0.240963
$834$	0	0
$835$	−12.7092	−0.439819
$836$	0	0
$837$	−5.60396	−0.193701
$838$	0	0
$839$	54.2127	1.87163	0.935815	−	0.352491i	$-0.114665\pi$
$839$	54.2127	1.87163	0.935815	+	0.352491i	$0.114665\pi$
$840$	0	0
$841$	−28.9777	−0.999232
$842$	0	0
$843$	1.73716	0.0598309
$844$	0	0
$845$	−14.6328	−0.503383
$846$	0	0
$847$	−6.47339	−0.222428
$848$	0	0
$849$	−2.88002	−0.0988419
$850$	0	0
$851$	3.71410	0.127318
$852$	0	0
$853$	33.7651	1.15610	0.578048	−	0.816003i	$-0.303815\pi$
$853$	33.7651	1.15610	0.578048	+	0.816003i	$0.303815\pi$
$854$	0	0
$855$	−19.7478	−0.675359
$856$	0	0
$857$	25.5332	0.872198	0.436099	−	0.899899i	$-0.356360\pi$
$857$	25.5332	0.872198	0.436099	+	0.899899i	$0.356360\pi$
$858$	0	0
$859$	−54.0964	−1.84575	−0.922873	−	0.385104i	$-0.874166\pi$
$859$	−54.0964	−1.84575	−0.922873	+	0.385104i	$0.874166\pi$
$860$	0	0
$861$	−0.107906	−0.00367744
$862$	0	0
$863$	−8.06227	−0.274443	−0.137221	−	0.990540i	$-0.543817\pi$
$863$	−8.06227	−0.274443	−0.137221	+	0.990540i	$0.543817\pi$
$864$	0	0
$865$	−24.3407	−0.827608
$866$	0	0
$867$	0.154866	0.00525954
$868$	0	0
$869$	78.4607	2.66160
$870$	0	0
$871$	20.9563	0.710077
$872$	0	0
$873$	38.0534	1.28791
$874$	0	0
$875$	−2.20095	−0.0744056
$876$	0	0
$877$	43.0088	1.45230	0.726152	−	0.687534i	$-0.241307\pi$
$877$	43.0088	1.45230	0.726152	+	0.687534i	$0.241307\pi$
$878$	0	0
$879$	−4.69570	−0.158382
$880$	0	0
$881$	6.83018	0.230114	0.115057	−	0.993359i	$-0.463295\pi$
$881$	6.83018	0.230114	0.115057	+	0.993359i	$0.463295\pi$
$882$	0	0
$883$	−24.1117	−0.811423	−0.405711	−	0.914001i	$-0.632976\pi$
$883$	−24.1117	−0.811423	−0.405711	+	0.914001i	$0.632976\pi$
$884$	0	0
$885$	0.187496	0.00630262
$886$	0	0
$887$	26.0569	0.874906	0.437453	−	0.899241i	$-0.355881\pi$
$887$	26.0569	0.874906	0.437453	+	0.899241i	$0.355881\pi$
$888$	0	0
$889$	3.90722	0.131044
$890$	0	0
$891$	56.5161	1.89336
$892$	0	0
$893$	−29.2376	−0.978399
$894$	0	0
$895$	−27.6832	−0.925348
$896$	0	0
$897$	0.306656	0.0102390
$898$	0	0
$899$	0.903746	0.0301416
$900$	0	0
$901$	1.89023	0.0629728
$902$	0	0
$903$	0.337469	0.0112303
$904$	0	0
$905$	−6.68301	−0.222151
$906$	0	0
$907$	−23.4776	−0.779561	−0.389781	−	0.920908i	$-0.627449\pi$
$907$	−23.4776	−0.779561	−0.389781	+	0.920908i	$0.627449\pi$
$908$	0	0
$909$	20.5076	0.680195
$910$	0	0
$911$	30.9245	1.02457	0.512287	−	0.858814i	$-0.328798\pi$
$911$	30.9245	1.02457	0.512287	+	0.858814i	$0.328798\pi$
$912$	0	0
$913$	93.2233	3.08524
$914$	0	0
$915$	−2.37954	−0.0786651
$916$	0	0
$917$	−0.0936164	−0.00309149
$918$	0	0
$919$	4.50836	0.148717	0.0743585	−	0.997232i	$-0.476309\pi$
$919$	4.50836	0.148717	0.0743585	+	0.997232i	$0.476309\pi$
$920$	0	0
$921$	0.980708	0.0323154
$922$	0	0
$923$	−67.2877	−2.21480
$924$	0	0
$925$	−33.2023	−1.09169
$926$	0	0
$927$	35.7157	1.17306
$928$	0	0
$929$	−5.71398	−0.187470	−0.0937348	−	0.995597i	$-0.529881\pi$
$929$	−5.71398	−0.187470	−0.0937348	+	0.995597i	$0.529881\pi$
$930$	0	0
$931$	38.1171	1.24924
$932$	0	0
$933$	2.80329	0.0917755
$934$	0	0
$935$	7.78897	0.254727
$936$	0	0
$937$	11.6320	0.380002	0.190001	−	0.981784i	$-0.439151\pi$
$937$	11.6320	0.380002	0.190001	+	0.981784i	$0.439151\pi$
$938$	0	0
$939$	−2.30640	−0.0752665
$940$	0	0
$941$	19.5658	0.637826	0.318913	−	0.947784i	$-0.396682\pi$
$941$	19.5658	0.637826	0.318913	+	0.947784i	$0.396682\pi$
$942$	0	0
$943$	1.29317	0.0421115
$944$	0	0
$945$	−0.238680	−0.00776425
$946$	0	0
$947$	7.38112	0.239854	0.119927	−	0.992783i	$-0.461734\pi$
$947$	7.38112	0.239854	0.119927	+	0.992783i	$0.461734\pi$
$948$	0	0
$949$	51.8737	1.68389
$950$	0	0
$951$	−0.787596	−0.0255396
$952$	0	0
$953$	−47.6349	−1.54304	−0.771522	−	0.636202i	$-0.780504\pi$
$953$	−47.6349	−1.54304	−0.771522	+	0.636202i	$0.780504\pi$
$954$	0	0
$955$	−9.57536	−0.309851
$956$	0	0
$957$	0.148704	0.00480691
$958$	0	0
$959$	0.0100181	0.000323500 0
$960$	0	0
$961$	5.66501	0.182742
$962$	0	0
$963$	1.39337	0.0449007
$964$	0	0
$965$	18.9550	0.610184
$966$	0	0
$967$	−0.267700	−0.00860864	−0.00430432	−	0.999991i	$-0.501370\pi$
$967$	−0.267700	−0.00860864	−0.00430432	+	0.999991i	$0.501370\pi$
$968$	0	0
$969$	0.848797	0.0272673
$970$	0	0
$971$	−7.09136	−0.227573	−0.113786	−	0.993505i	$-0.536298\pi$
$971$	−7.09136	−0.227573	−0.113786	+	0.993505i	$0.536298\pi$
$972$	0	0
$973$	−1.75707	−0.0563291
$974$	0	0
$975$	−2.74137	−0.0877940
$976$	0	0
$977$	−17.5581	−0.561735	−0.280867	−	0.959747i	$-0.590622\pi$
$977$	−17.5581	−0.561735	−0.280867	+	0.959747i	$0.590622\pi$
$978$	0	0
$979$	−84.2686	−2.69324
$980$	0	0
$981$	−53.9777	−1.72337
$982$	0	0
$983$	21.8426	0.696670	0.348335	−	0.937370i	$-0.386747\pi$
$983$	21.8426	0.696670	0.348335	+	0.937370i	$0.386747\pi$
$984$	0	0
$985$	−17.5179	−0.558168
$986$	0	0
$987$	−0.175980	−0.00560150
$988$	0	0
$989$	−4.04430	−0.128601
$990$	0	0
$991$	3.21841	0.102236	0.0511181	−	0.998693i	$-0.483721\pi$
$991$	3.21841	0.102236	0.0511181	+	0.998693i	$0.483721\pi$
$992$	0	0
$993$	−2.19901	−0.0697836
$994$	0	0
$995$	5.37059	0.170259
$996$	0	0
$997$	−52.9056	−1.67554	−0.837769	−	0.546025i	$-0.816141\pi$
$997$	−52.9056	−1.67554	−0.837769	+	0.546025i	$0.816141\pi$
$998$	0	0
$999$	−8.69450	−0.275082

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4012.2.a.i.1.9	✓	18

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4012.2.a.i.1.9	✓	18	1.1	even	1	trivial

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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 4012 = 2^{2} \cdot 17 \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4012.a (trivial)

\(f(q)\)	\(=\)	\(q+0.154866 q^{3} -1.21070 q^{5} -0.213015 q^{7} -2.97602 q^{9} +O(q^{10})\)	\(q+0.154866 q^{3} -1.21070 q^{5} -0.213015 q^{7} -2.97602 q^{9} +6.43345 q^{11} +5.00862 q^{13} -0.187496 q^{15} -1.00000 q^{17} -5.48083 q^{19} -0.0329889 q^{21} +0.395346 q^{23} -3.53421 q^{25} -0.925484 q^{27} +0.149252 q^{29} +6.05516 q^{31} +0.996326 q^{33} +0.257897 q^{35} +9.39455 q^{37} +0.775666 q^{39} +3.27099 q^{41} -10.2298 q^{43} +3.60306 q^{45} +5.33452 q^{47} -6.95462 q^{49} -0.154866 q^{51} -1.89023 q^{53} -7.78897 q^{55} -0.848797 q^{57} -1.00000 q^{59} +12.6911 q^{61} +0.633937 q^{63} -6.06392 q^{65} +4.18405 q^{67} +0.0612258 q^{69} -13.4344 q^{71} +10.3569 q^{73} -0.547330 q^{75} -1.37042 q^{77} +12.1957 q^{79} +8.78472 q^{81} +14.4904 q^{83} +1.21070 q^{85} +0.0231141 q^{87} -13.0985 q^{89} -1.06691 q^{91} +0.937741 q^{93} +6.63564 q^{95} -12.7867 q^{97} -19.1461 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q + 8 q^{3} + 4 q^{5} + 2 q^{7} + 18 q^{9}+O(q^{10}) \) 18 * q + 8 * q^3 + 4 * q^5 + 2 * q^7 + 18 * q^9	\( 18 q + 8 q^{3} + 4 q^{5} + 2 q^{7} + 18 q^{9} + 12 q^{11} + 2 q^{13} - 18 q^{17} + 5 q^{19} - 3 q^{21} + 21 q^{23} + 16 q^{25} + 26 q^{27} + 14 q^{29} + 15 q^{31} + 19 q^{33} + 20 q^{35} + 2 q^{37} - 14 q^{39} + 34 q^{41} + 21 q^{43} + 49 q^{45} + 69 q^{47} + 28 q^{49} - 8 q^{51} - 4 q^{53} + 18 q^{55} + 5 q^{57} - 18 q^{59} + 11 q^{61} + 35 q^{63} + 27 q^{65} + 34 q^{67} - 4 q^{69} + 37 q^{71} + 18 q^{73} + 72 q^{75} + 11 q^{77} + 11 q^{79} + 30 q^{81} + 28 q^{83} - 4 q^{85} + 7 q^{87} + 44 q^{89} - 23 q^{91} - 3 q^{93} - 11 q^{95} + 11 q^{97} + 56 q^{99}+O(q^{100}) \) 18 * q + 8 * q^3 + 4 * q^5 + 2 * q^7 + 18 * q^9 + 12 * q^11 + 2 * q^13 - 18 * q^17 + 5 * q^19 - 3 * q^21 + 21 * q^23 + 16 * q^25 + 26 * q^27 + 14 * q^29 + 15 * q^31 + 19 * q^33 + 20 * q^35 + 2 * q^37 - 14 * q^39 + 34 * q^41 + 21 * q^43 + 49 * q^45 + 69 * q^47 + 28 * q^49 - 8 * q^51 - 4 * q^53 + 18 * q^55 + 5 * q^57 - 18 * q^59 + 11 * q^61 + 35 * q^63 + 27 * q^65 + 34 * q^67 - 4 * q^69 + 37 * q^71 + 18 * q^73 + 72 * q^75 + 11 * q^77 + 11 * q^79 + 30 * q^81 + 28 * q^83 - 4 * q^85 + 7 * q^87 + 44 * q^89 - 23 * q^91 - 3 * q^93 - 11 * q^95 + 11 * q^97 + 56 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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