LMFDB - Embedded newform 1860.2.a.i.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1860,2,Mod(1,1860)] 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("1860.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1860, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 1860 = 2^{2} \cdot 3 \cdot 5 \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1860.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,0,-4,0,4,0,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	yes
Analytic conductor:	$14.8521747760$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.224148.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 11x^{2} + 9x + 14 $ x^4 - x^3 - 11x^2 + 9x + 14
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.75476$ of defining polynomial
Character	$\chi$	$=$	1860.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} +1.00000 q^{5} -4.44005 q^{7} +1.00000 q^{9} +3.50952 q^{11} -5.94957 q^{13} -1.00000 q^{15} +4.67558 q^{17} +4.44005 q^{21} -4.67558 q^{23} +1.00000 q^{25} -1.00000 q^{27} +7.27398 q^{29} +1.00000 q^{31} -3.50952 q^{33} -4.44005 q^{35} -2.44005 q^{37} +5.94957 q^{39} -1.50952 q^{41} -3.50952 q^{43} +1.00000 q^{45} +11.5557 q^{47} +12.7140 q^{49} -4.67558 q^{51} +9.55568 q^{53} +3.50952 q^{55} +12.9573 q^{59} +4.33213 q^{61} -4.44005 q^{63} -5.94957 q^{65} -6.28170 q^{67} +4.67558 q^{69} -7.11563 q^{71} +13.4591 q^{73} -1.00000 q^{75} -15.5824 q^{77} -9.06520 q^{79} +1.00000 q^{81} +0.675582 q^{83} +4.67558 q^{85} -7.27398 q^{87} -1.11563 q^{89} +26.4164 q^{91} -1.00000 q^{93} +14.8607 q^{97} +3.50952 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{3} + 4 q^{5} - 4 q^{7} + 4 q^{9} + 2 q^{11} + 2 q^{13} - 4 q^{15} + 2 q^{17} + 4 q^{21} - 2 q^{23} + 4 q^{25} - 4 q^{27} + 20 q^{29} + 4 q^{31} - 2 q^{33} - 4 q^{35} + 4 q^{37} - 2 q^{39} + 6 q^{41}+ \cdots + 2 q^{99}+O(q^{100}) $ 4 * q - 4 * q^3 + 4 * q^5 - 4 * q^7 + 4 * q^9 + 2 * q^11 + 2 * q^13 - 4 * q^15 + 2 * q^17 + 4 * q^21 - 2 * q^23 + 4 * q^25 - 4 * q^27 + 20 * q^29 + 4 * q^31 - 2 * q^33 - 4 * q^35 + 4 * q^37 - 2 * q^39 + 6 * q^41 - 2 * q^43 + 4 * q^45 + 2 * q^47 + 28 * q^49 - 2 * q^51 - 6 * q^53 + 2 * q^55 + 8 * q^61 - 4 * q^63 + 2 * q^65 + 10 * q^67 + 2 * q^69 + 2 * q^71 + 16 * q^73 - 4 * q^75 + 32 * q^77 + 20 * q^79 + 4 * q^81 - 14 * q^83 + 2 * q^85 - 20 * q^87 + 26 * q^89 + 16 * q^91 - 4 * q^93 + 14 * q^97 + 2 * 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
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−4.44005	−1.67818	−0.839090	−	0.543992i	$-0.816912\pi$
$7$	−4.44005	−1.67818	−0.839090	+	0.543992i	$0.816912\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	3.50952	1.05816	0.529080	−	0.848572i	$-0.322537\pi$
$11$	3.50952	1.05816	0.529080	+	0.848572i	$0.322537\pi$
$12$	0	0
$13$	−5.94957	−1.65011	−0.825056	−	0.565051i	$-0.808857\pi$
$13$	−5.94957	−1.65011	−0.825056	+	0.565051i	$0.808857\pi$
$14$	0	0
$15$	−1.00000	−0.258199
$16$	0	0
$17$	4.67558	1.13400	0.566998	−	0.823719i	$-0.308105\pi$
$17$	4.67558	1.13400	0.566998	+	0.823719i	$0.308105\pi$
$18$	0	0
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	0	0
$21$	4.44005	0.968898
$22$	0	0
$23$	−4.67558	−0.974926	−0.487463	−	0.873144i	$-0.662078\pi$
$23$	−4.67558	−0.974926	−0.487463	+	0.873144i	$0.662078\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	7.27398	1.35074	0.675372	−	0.737477i	$-0.263983\pi$
$29$	7.27398	1.35074	0.675372	+	0.737477i	$0.263983\pi$
$30$	0	0
$31$	1.00000	0.179605
$31$	1.00000	0.179605
$32$	0	0
$33$	−3.50952	−0.610928
$34$	0	0
$35$	−4.44005	−0.750505
$36$	0	0
$37$	−2.44005	−0.401142	−0.200571	−	0.979679i	$-0.564280\pi$
$37$	−2.44005	−0.401142	−0.200571	+	0.979679i	$0.564280\pi$
$38$	0	0
$39$	5.94957	0.952693
$40$	0	0
$41$	−1.50952	−0.235747	−0.117873	−	0.993029i	$-0.537608\pi$
$41$	−1.50952	−0.235747	−0.117873	+	0.993029i	$0.537608\pi$
$42$	0	0
$43$	−3.50952	−0.535196	−0.267598	−	0.963531i	$-0.586230\pi$
$43$	−3.50952	−0.535196	−0.267598	+	0.963531i	$0.586230\pi$
$44$	0	0
$45$	1.00000	0.149071
$46$	0	0
$47$	11.5557	1.68557	0.842785	−	0.538251i	$-0.180915\pi$
$47$	11.5557	1.68557	0.842785	+	0.538251i	$0.180915\pi$
$48$	0	0
$49$	12.7140	1.81629
$50$	0	0
$51$	−4.67558	−0.654712
$52$	0	0
$53$	9.55568	1.31257	0.656287	−	0.754512i	$-0.272126\pi$
$53$	9.55568	1.31257	0.656287	+	0.754512i	$0.272126\pi$
$54$	0	0
$55$	3.50952	0.473223
$56$	0	0
$57$	0	0
$58$	0	0
$59$	12.9573	1.68689	0.843447	−	0.537213i	$-0.180523\pi$
$59$	12.9573	1.68689	0.843447	+	0.537213i	$0.180523\pi$
$60$	0	0
$61$	4.33213	0.554672	0.277336	−	0.960773i	$-0.410548\pi$
$61$	4.33213	0.554672	0.277336	+	0.960773i	$0.410548\pi$
$62$	0	0
$63$	−4.44005	−0.559394
$64$	0	0
$65$	−5.94957	−0.737953
$66$	0	0
$67$	−6.28170	−0.767431	−0.383716	−	0.923451i	$-0.625356\pi$
$67$	−6.28170	−0.767431	−0.383716	+	0.923451i	$0.625356\pi$
$68$	0	0
$69$	4.67558	0.562874
$70$	0	0
$71$	−7.11563	−0.844470	−0.422235	−	0.906486i	$-0.638754\pi$
$71$	−7.11563	−0.844470	−0.422235	+	0.906486i	$0.638754\pi$
$72$	0	0
$73$	13.4591	1.57527	0.787633	−	0.616144i	$-0.211306\pi$
$73$	13.4591	1.57527	0.787633	+	0.616144i	$0.211306\pi$
$74$	0	0
$75$	−1.00000	−0.115470
$76$	0	0
$77$	−15.5824	−1.77578
$78$	0	0
$79$	−9.06520	−1.01991	−0.509957	−	0.860200i	$-0.670339\pi$
$79$	−9.06520	−1.01991	−0.509957	+	0.860200i	$0.670339\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	0.675582	0.0741547	0.0370774	−	0.999312i	$-0.488195\pi$
$83$	0.675582	0.0741547	0.0370774	+	0.999312i	$0.488195\pi$
$84$	0	0
$85$	4.67558	0.507138
$86$	0	0
$87$	−7.27398	−0.779853
$88$	0	0
$89$	−1.11563	−0.118257	−0.0591283	−	0.998250i	$-0.518832\pi$
$89$	−1.11563	−0.118257	−0.0591283	+	0.998250i	$0.518832\pi$
$90$	0	0
$91$	26.4164	2.76919
$92$	0	0
$93$	−1.00000	−0.103695
$94$	0	0
$95$	0	0
$96$	0	0
$97$	14.8607	1.50887	0.754437	−	0.656373i	$-0.227910\pi$
$97$	14.8607	1.50887	0.754437	+	0.656373i	$0.227910\pi$
$98$	0	0
$99$	3.50952	0.352720
$100$	0	0
$101$	0.158353	0.0157567	0.00787836	−	0.999969i	$-0.497492\pi$
$101$	0.158353	0.0157567	0.00787836	+	0.999969i	$0.497492\pi$
$102$	0	0
$103$	0.930532	0.0916881	0.0458440	−	0.998949i	$-0.485402\pi$
$103$	0.930532	0.0916881	0.0458440	+	0.998949i	$0.485402\pi$
$104$	0	0
$105$	4.44005	0.433304
$106$	0	0
$107$	−2.67558	−0.258658	−0.129329	−	0.991602i	$-0.541282\pi$
$107$	−2.67558	−0.258658	−0.129329	+	0.991602i	$0.541282\pi$
$108$	0	0
$109$	8.83393	0.846137	0.423069	−	0.906098i	$-0.360953\pi$
$109$	8.83393	0.846137	0.423069	+	0.906098i	$0.360953\pi$
$110$	0	0
$111$	2.44005	0.231599
$112$	0	0
$113$	9.89913	0.931232	0.465616	−	0.884987i	$-0.345833\pi$
$113$	9.89913	0.931232	0.465616	+	0.884987i	$0.345833\pi$
$114$	0	0
$115$	−4.67558	−0.436000
$116$	0	0
$117$	−5.94957	−0.550037
$118$	0	0
$119$	−20.7598	−1.90305
$120$	0	0
$121$	1.31671	0.119701
$122$	0	0
$123$	1.50952	0.136109
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	15.8991	1.41082	0.705410	−	0.708800i	$-0.250763\pi$
$127$	15.8991	1.41082	0.705410	+	0.708800i	$0.250763\pi$
$128$	0	0
$129$	3.50952	0.308996
$130$	0	0
$131$	−14.1541	−1.23665	−0.618324	−	0.785923i	$-0.712188\pi$
$131$	−14.1541	−1.23665	−0.618324	+	0.785923i	$0.712188\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−1.00000	−0.0860663
$136$	0	0
$137$	11.0077	0.940452	0.470226	−	0.882546i	$-0.344172\pi$
$137$	11.0077	0.940452	0.470226	+	0.882546i	$0.344172\pi$
$138$	0	0
$139$	19.2122	1.62956	0.814780	−	0.579770i	$-0.196858\pi$
$139$	19.2122	1.62956	0.814780	+	0.579770i	$0.196858\pi$
$140$	0	0
$141$	−11.5557	−0.973164
$142$	0	0
$143$	−20.8801	−1.74608
$144$	0	0
$145$	7.27398	0.604071
$146$	0	0
$147$	−12.7140	−1.04864
$148$	0	0
$149$	13.1928	1.08080	0.540399	−	0.841409i	$-0.318273\pi$
$149$	13.1928	1.08080	0.540399	+	0.841409i	$0.318273\pi$
$150$	0	0
$151$	5.16607	0.420408	0.210204	−	0.977658i	$-0.432587\pi$
$151$	5.16607	0.420408	0.210204	+	0.977658i	$0.432587\pi$
$152$	0	0
$153$	4.67558	0.377998
$154$	0	0
$155$	1.00000	0.0803219
$156$	0	0
$157$	−15.2697	−1.21866	−0.609328	−	0.792918i	$-0.708561\pi$
$157$	−15.2697	−1.21866	−0.609328	+	0.792918i	$0.708561\pi$
$158$	0	0
$159$	−9.55568	−0.757815
$160$	0	0
$161$	20.7598	1.63610
$162$	0	0
$163$	−7.94957	−0.622658	−0.311329	−	0.950302i	$-0.600774\pi$
$163$	−7.94957	−0.622658	−0.311329	+	0.950302i	$0.600774\pi$
$164$	0	0
$165$	−3.50952	−0.273215
$166$	0	0
$167$	−2.13894	−0.165516	−0.0827579	−	0.996570i	$-0.526373\pi$
$167$	−2.13894	−0.165516	−0.0827579	+	0.996570i	$0.526373\pi$
$168$	0	0
$169$	22.3973	1.72287
$170$	0	0
$171$	0	0
$172$	0	0
$173$	6.88010	0.523084	0.261542	−	0.965192i	$-0.415769\pi$
$173$	6.88010	0.523084	0.261542	+	0.965192i	$0.415769\pi$
$174$	0	0
$175$	−4.44005	−0.335636
$176$	0	0
$177$	−12.9573	−0.973929
$178$	0	0
$179$	1.66787	0.124662	0.0623312	−	0.998056i	$-0.480146\pi$
$179$	1.66787	0.124662	0.0623312	+	0.998056i	$0.480146\pi$
$180$	0	0
$181$	−20.1304	−1.49628	−0.748140	−	0.663541i	$-0.769053\pi$
$181$	−20.1304	−1.49628	−0.748140	+	0.663541i	$0.769053\pi$
$182$	0	0
$183$	−4.33213	−0.320240
$184$	0	0
$185$	−2.44005	−0.179396
$186$	0	0
$187$	16.4090	1.19995
$188$	0	0
$189$	4.44005	0.322966
$190$	0	0
$191$	−16.2930	−1.17892	−0.589461	−	0.807797i	$-0.700660\pi$
$191$	−16.2930	−1.17892	−0.589461	+	0.807797i	$0.700660\pi$
$192$	0	0
$193$	22.2887	1.60438	0.802189	−	0.597070i	$-0.203668\pi$
$193$	22.2887	1.60438	0.802189	+	0.597070i	$0.203668\pi$
$194$	0	0
$195$	5.94957	0.426057
$196$	0	0
$197$	15.6946	1.11820	0.559098	−	0.829102i	$-0.311148\pi$
$197$	15.6946	1.11820	0.559098	+	0.829102i	$0.311148\pi$
$198$	0	0
$199$	6.14703	0.435752	0.217876	−	0.975977i	$-0.430087\pi$
$199$	6.14703	0.435752	0.217876	+	0.975977i	$0.430087\pi$
$200$	0	0
$201$	6.28170	0.443077
$202$	0	0
$203$	−32.2968	−2.26679
$204$	0	0
$205$	−1.50952	−0.105429
$206$	0	0
$207$	−4.67558	−0.324975
$208$	0	0
$209$	0	0
$210$	0	0
$211$	24.3082	1.67344	0.836721	−	0.547629i	$-0.184469\pi$
$211$	24.3082	1.67344	0.836721	+	0.547629i	$0.184469\pi$
$212$	0	0
$213$	7.11563	0.487555
$214$	0	0
$215$	−3.50952	−0.239347
$216$	0	0
$217$	−4.44005	−0.301410
$218$	0	0
$219$	−13.4591	−0.909480
$220$	0	0
$221$	−27.8177	−1.87122
$222$	0	0
$223$	−16.3896	−1.09753	−0.548765	−	0.835977i	$-0.684902\pi$
$223$	−16.3896	−1.09753	−0.548765	+	0.835977i	$0.684902\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	20.4358	1.35637	0.678185	−	0.734891i	$-0.262767\pi$
$227$	20.4358	1.35637	0.678185	+	0.734891i	$0.262767\pi$
$228$	0	0
$229$	−29.0105	−1.91707	−0.958534	−	0.284980i	$-0.908013\pi$
$229$	−29.0105	−1.91707	−0.958534	+	0.284980i	$0.908013\pi$
$230$	0	0
$231$	15.5824	1.02525
$232$	0	0
$233$	−4.12761	−0.270409	−0.135205	−	0.990818i	$-0.543169\pi$
$233$	−4.12761	−0.270409	−0.135205	+	0.990818i	$0.543169\pi$
$234$	0	0
$235$	11.5557	0.753809
$236$	0	0
$237$	9.06520	0.588848
$238$	0	0
$239$	−4.96155	−0.320936	−0.160468	−	0.987041i	$-0.551300\pi$
$239$	−4.96155	−0.320936	−0.160468	+	0.987041i	$0.551300\pi$
$240$	0	0
$241$	−20.9182	−1.34746	−0.673729	−	0.738979i	$-0.735308\pi$
$241$	−20.9182	−1.34746	−0.673729	+	0.738979i	$0.735308\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	12.7140	0.812270
$246$	0	0
$247$	0	0
$248$	0	0
$249$	−0.675582	−0.0428133
$250$	0	0
$251$	−7.31310	−0.461599	−0.230799	−	0.973001i	$-0.574134\pi$
$251$	−7.31310	−0.461599	−0.230799	+	0.973001i	$0.574134\pi$
$252$	0	0
$253$	−16.4090	−1.03163
$254$	0	0
$255$	−4.67558	−0.292796
$256$	0	0
$257$	6.20452	0.387027	0.193514	−	0.981098i	$-0.438012\pi$
$257$	6.20452	0.387027	0.193514	+	0.981098i	$0.438012\pi$
$258$	0	0
$259$	10.8339	0.673188
$260$	0	0
$261$	7.27398	0.450248
$262$	0	0
$263$	27.0105	1.66554	0.832769	−	0.553621i	$-0.186754\pi$
$263$	27.0105	1.66554	0.832769	+	0.553621i	$0.186754\pi$
$264$	0	0
$265$	9.55568	0.587001
$266$	0	0
$267$	1.11563	0.0682755
$268$	0	0
$269$	−7.46718	−0.455282	−0.227641	−	0.973745i	$-0.573101\pi$
$269$	−7.46718	−0.455282	−0.227641	+	0.973745i	$0.573101\pi$
$270$	0	0
$271$	9.56700	0.581154	0.290577	−	0.956852i	$-0.406153\pi$
$271$	9.56700	0.581154	0.290577	+	0.956852i	$0.406153\pi$
$272$	0	0
$273$	−26.4164	−1.59879
$274$	0	0
$275$	3.50952	0.211632
$276$	0	0
$277$	27.2193	1.63545	0.817724	−	0.575611i	$-0.195236\pi$
$277$	27.2193	1.63545	0.817724	+	0.575611i	$0.195236\pi$
$278$	0	0
$279$	1.00000	0.0598684
$280$	0	0
$281$	−3.86106	−0.230332	−0.115166	−	0.993346i	$-0.536740\pi$
$281$	−3.86106	−0.230332	−0.115166	+	0.993346i	$0.536740\pi$
$282$	0	0
$283$	−16.8102	−0.999265	−0.499633	−	0.866237i	$-0.666532\pi$
$283$	−16.8102	−0.999265	−0.499633	+	0.866237i	$0.666532\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	6.70233	0.395626
$288$	0	0
$289$	4.86106	0.285945
$290$	0	0
$291$	−14.8607	−0.871148
$292$	0	0
$293$	−30.5747	−1.78619	−0.893097	−	0.449864i	$-0.851472\pi$
$293$	−30.5747	−1.78619	−0.893097	+	0.449864i	$0.851472\pi$
$294$	0	0
$295$	12.9573	0.754402
$296$	0	0
$297$	−3.50952	−0.203643
$298$	0	0
$299$	27.8177	1.60874
$300$	0	0
$301$	15.5824	0.898156
$302$	0	0
$303$	−0.158353	−0.00909715
$304$	0	0
$305$	4.33213	0.248057
$306$	0	0
$307$	15.1424	0.864221	0.432111	−	0.901821i	$-0.357769\pi$
$307$	15.1424	0.864221	0.432111	+	0.901821i	$0.357769\pi$
$308$	0	0
$309$	−0.930532	−0.0529361
$310$	0	0
$311$	−22.2355	−1.26086	−0.630431	−	0.776246i	$-0.717122\pi$
$311$	−22.2355	−1.26086	−0.630431	+	0.776246i	$0.717122\pi$
$312$	0	0
$313$	−21.8487	−1.23496	−0.617481	−	0.786586i	$-0.711847\pi$
$313$	−21.8487	−1.23496	−0.617481	+	0.786586i	$0.711847\pi$
$314$	0	0
$315$	−4.44005	−0.250168
$316$	0	0
$317$	33.3159	1.87121	0.935603	−	0.353054i	$-0.114857\pi$
$317$	33.3159	1.87121	0.935603	+	0.353054i	$0.114857\pi$
$318$	0	0
$319$	25.5282	1.42930
$320$	0	0
$321$	2.67558	0.149336
$322$	0	0
$323$	0	0
$324$	0	0
$325$	−5.94957	−0.330022
$326$	0	0
$327$	−8.83393	−0.488518
$328$	0	0
$329$	−51.3078	−2.82869
$330$	0	0
$331$	2.36287	0.129875	0.0649375	−	0.997889i	$-0.479315\pi$
$331$	2.36287	0.129875	0.0649375	+	0.997889i	$0.479315\pi$
$332$	0	0
$333$	−2.44005	−0.133714
$334$	0	0
$335$	−6.28170	−0.343206
$336$	0	0
$337$	−27.2193	−1.48273	−0.741364	−	0.671103i	$-0.765821\pi$
$337$	−27.2193	−1.48273	−0.741364	+	0.671103i	$0.765821\pi$
$338$	0	0
$339$	−9.89913	−0.537647
$340$	0	0
$341$	3.50952	0.190051
$342$	0	0
$343$	−25.3706	−1.36988
$344$	0	0
$345$	4.67558	0.251725
$346$	0	0
$347$	4.21584	0.226318	0.113159	−	0.993577i	$-0.463903\pi$
$347$	4.21584	0.226318	0.113159	+	0.993577i	$0.463903\pi$
$348$	0	0
$349$	−1.49820	−0.0801966	−0.0400983	−	0.999196i	$-0.512767\pi$
$349$	−1.49820	−0.0801966	−0.0400983	+	0.999196i	$0.512767\pi$
$350$	0	0
$351$	5.94957	0.317564
$352$	0	0
$353$	−18.3435	−0.976323	−0.488162	−	0.872753i	$-0.662332\pi$
$353$	−18.3435	−0.976323	−0.488162	+	0.872753i	$0.662332\pi$
$354$	0	0
$355$	−7.11563	−0.377658
$356$	0	0
$357$	20.7598	1.09873
$358$	0	0
$359$	0.0771799	0.00407340	0.00203670	−	0.999998i	$-0.499352\pi$
$359$	0.0771799	0.00407340	0.00203670	+	0.999998i	$0.499352\pi$
$360$	0	0
$361$	−19.0000	−1.00000
$362$	0	0
$363$	−1.31671	−0.0691092
$364$	0	0
$365$	13.4591	0.704481
$366$	0	0
$367$	0	0		−	1.00000i	$-0.5\pi$
$367$	0	0			1.00000i	$0.5\pi$
$368$	0	0
$369$	−1.50952	−0.0785823
$370$	0	0
$371$	−42.4277	−2.20274
$372$	0	0
$373$	16.2313	0.840423	0.420211	−	0.907426i	$-0.361956\pi$
$373$	16.2313	0.840423	0.420211	+	0.907426i	$0.361956\pi$
$374$	0	0
$375$	−1.00000	−0.0516398
$376$	0	0
$377$	−43.2770	−2.22888
$378$	0	0
$379$	−6.74116	−0.346270	−0.173135	−	0.984898i	$-0.555390\pi$
$379$	−6.74116	−0.346270	−0.173135	+	0.984898i	$0.555390\pi$
$380$	0	0
$381$	−15.8991	−0.814537
$382$	0	0
$383$	−21.0846	−1.07737	−0.538687	−	0.842506i	$-0.681079\pi$
$383$	−21.0846	−1.07737	−0.538687	+	0.842506i	$0.681079\pi$
$384$	0	0
$385$	−15.5824	−0.794154
$386$	0	0
$387$	−3.50952	−0.178399
$388$	0	0
$389$	−7.76447	−0.393674	−0.196837	−	0.980436i	$-0.563067\pi$
$389$	−7.76447	−0.393674	−0.196837	+	0.980436i	$0.563067\pi$
$390$	0	0
$391$	−21.8611	−1.10556
$392$	0	0
$393$	14.1541	0.713979
$394$	0	0
$395$	−9.06520	−0.456119
$396$	0	0
$397$	−33.3892	−1.67576	−0.837879	−	0.545856i	$-0.816204\pi$
$397$	−33.3892	−1.67576	−0.837879	+	0.545856i	$0.816204\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	1.11563	0.0557119	0.0278560	−	0.999612i	$-0.491132\pi$
$401$	1.11563	0.0557119	0.0278560	+	0.999612i	$0.491132\pi$
$402$	0	0
$403$	−5.94957	−0.296369
$404$	0	0
$405$	1.00000	0.0496904
$406$	0	0
$407$	−8.56339	−0.424472
$408$	0	0
$409$	1.59097	0.0786683	0.0393342	−	0.999226i	$-0.487476\pi$
$409$	1.59097	0.0786683	0.0393342	+	0.999226i	$0.487476\pi$
$410$	0	0
$411$	−11.0077	−0.542970
$412$	0	0
$413$	−57.5309	−2.83091
$414$	0	0
$415$	0.675582	0.0331630
$416$	0	0
$417$	−19.2122	−0.940827
$418$	0	0
$419$	−13.2546	−0.647528	−0.323764	−	0.946138i	$-0.604948\pi$
$419$	−13.2546	−0.647528	−0.323764	+	0.946138i	$0.604948\pi$
$420$	0	0
$421$	9.71403	0.473433	0.236716	−	0.971579i	$-0.423929\pi$
$421$	9.71403	0.473433	0.236716	+	0.971579i	$0.423929\pi$
$422$	0	0
$423$	11.5557	0.561856
$424$	0	0
$425$	4.67558	0.226799
$426$	0	0
$427$	−19.2349	−0.930841
$428$	0	0
$429$	20.8801	1.00810
$430$	0	0
$431$	−15.3315	−0.738491	−0.369245	−	0.929332i	$-0.620384\pi$
$431$	−15.3315	−0.738491	−0.369245	+	0.929332i	$0.620384\pi$
$432$	0	0
$433$	−16.2623	−0.781515	−0.390758	−	0.920494i	$-0.627787\pi$
$433$	−16.2623	−0.781515	−0.390758	+	0.920494i	$0.627787\pi$
$434$	0	0
$435$	−7.27398	−0.348761
$436$	0	0
$437$	0	0
$438$	0	0
$439$	18.7023	0.892613	0.446307	−	0.894880i	$-0.352739\pi$
$439$	18.7023	0.892613	0.446307	+	0.894880i	$0.352739\pi$
$440$	0	0
$441$	12.7140	0.605430
$442$	0	0
$443$	−11.7715	−0.559282	−0.279641	−	0.960105i	$-0.590215\pi$
$443$	−11.7715	−0.559282	−0.279641	+	0.960105i	$0.590215\pi$
$444$	0	0
$445$	−1.11563	−0.0528860
$446$	0	0
$447$	−13.1928	−0.623999
$448$	0	0
$449$	26.8758	1.26835	0.634174	−	0.773190i	$-0.281340\pi$
$449$	26.8758	1.26835	0.634174	+	0.773190i	$0.281340\pi$
$450$	0	0
$451$	−5.29767	−0.249458
$452$	0	0
$453$	−5.16607	−0.242723
$454$	0	0
$455$	26.4164	1.23842
$456$	0	0
$457$	2.14598	0.100385	0.0501925	−	0.998740i	$-0.484017\pi$
$457$	2.14598	0.100385	0.0501925	+	0.998740i	$0.484017\pi$
$458$	0	0
$459$	−4.67558	−0.218237
$460$	0	0
$461$	21.8220	1.01635	0.508175	−	0.861254i	$-0.330320\pi$
$461$	21.8220	1.01635	0.508175	+	0.861254i	$0.330320\pi$
$462$	0	0
$463$	−5.52494	−0.256766	−0.128383	−	0.991725i	$-0.540979\pi$
$463$	−5.52494	−0.256766	−0.128383	+	0.991725i	$0.540979\pi$
$464$	0	0
$465$	−1.00000	−0.0463739
$466$	0	0
$467$	−21.6326	−1.00104	−0.500518	−	0.865726i	$-0.666857\pi$
$467$	−21.6326	−1.00104	−0.500518	+	0.865726i	$0.666857\pi$
$468$	0	0
$469$	27.8910	1.28789
$470$	0	0
$471$	15.2697	0.703591
$472$	0	0
$473$	−12.3167	−0.566323
$474$	0	0
$475$	0	0
$476$	0	0
$477$	9.55568	0.437524
$478$	0	0
$479$	15.3315	0.700513	0.350256	−	0.936654i	$-0.386094\pi$
$479$	15.3315	0.700513	0.350256	+	0.936654i	$0.386094\pi$
$480$	0	0
$481$	14.5172	0.661929
$482$	0	0
$483$	−20.7598	−0.944604
$484$	0	0
$485$	14.8607	0.674789
$486$	0	0
$487$	−26.2313	−1.18865	−0.594326	−	0.804224i	$-0.702581\pi$
$487$	−26.2313	−1.18865	−0.594326	+	0.804224i	$0.702581\pi$
$488$	0	0
$489$	7.94957	0.359492
$490$	0	0
$491$	35.9915	1.62427	0.812136	−	0.583468i	$-0.198305\pi$
$491$	35.9915	1.62427	0.812136	+	0.583468i	$0.198305\pi$
$492$	0	0
$493$	34.0101	1.53174
$494$	0	0
$495$	3.50952	0.157741
$496$	0	0
$497$	31.5937	1.41717
$498$	0	0
$499$	−11.7983	−0.528163	−0.264081	−	0.964500i	$-0.585069\pi$
$499$	−11.7983	−0.528163	−0.264081	+	0.964500i	$0.585069\pi$
$500$	0	0
$501$	2.13894	0.0955606
$502$	0	0
$503$	−19.1466	−0.853707	−0.426853	−	0.904321i	$-0.640378\pi$
$503$	−19.1466	−0.853707	−0.426853	+	0.904321i	$0.640378\pi$
$504$	0	0
$505$	0.158353	0.00704662
$506$	0	0
$507$	−22.3973	−0.994700
$508$	0	0
$509$	−2.97669	−0.131940	−0.0659698	−	0.997822i	$-0.521014\pi$
$509$	−2.97669	−0.131940	−0.0659698	+	0.997822i	$0.521014\pi$
$510$	0	0
$511$	−59.7590	−2.64358
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0.930532	0.0410041
$516$	0	0
$517$	40.5548	1.78360
$518$	0	0
$519$	−6.88010	−0.302003
$520$	0	0
$521$	30.2887	1.32697	0.663487	−	0.748188i	$-0.269076\pi$
$521$	30.2887	1.32697	0.663487	+	0.748188i	$0.269076\pi$
$522$	0	0
$523$	33.5670	1.46778	0.733891	−	0.679267i	$-0.237702\pi$
$523$	33.5670	1.46778	0.733891	+	0.679267i	$0.237702\pi$
$524$	0	0
$525$	4.44005	0.193780
$526$	0	0
$527$	4.67558	0.203672
$528$	0	0
$529$	−1.13894	−0.0495189
$530$	0	0
$531$	12.9573	0.562298
$532$	0	0
$533$	8.98097	0.389009
$534$	0	0
$535$	−2.67558	−0.115675
$536$	0	0
$537$	−1.66787	−0.0719739
$538$	0	0
$539$	44.6201	1.92192
$540$	0	0
$541$	−25.7675	−1.10783	−0.553916	−	0.832572i	$-0.686867\pi$
$541$	−25.7675	−1.10783	−0.553916	+	0.832572i	$0.686867\pi$
$542$	0	0
$543$	20.1304	0.863878
$544$	0	0
$545$	8.83393	0.378404
$546$	0	0
$547$	−29.7098	−1.27030	−0.635149	−	0.772390i	$-0.719061\pi$
$547$	−29.7098	−1.27030	−0.635149	+	0.772390i	$0.719061\pi$
$548$	0	0
$549$	4.33213	0.184891
$550$	0	0
$551$	0	0
$552$	0	0
$553$	40.2499	1.71160
$554$	0	0
$555$	2.44005	0.103574
$556$	0	0
$557$	11.0077	0.466412	0.233206	−	0.972427i	$-0.425078\pi$
$557$	11.0077	0.466412	0.233206	+	0.972427i	$0.425078\pi$
$558$	0	0
$559$	20.8801	0.883134
$560$	0	0
$561$	−16.4090	−0.692790
$562$	0	0
$563$	−8.34345	−0.351635	−0.175817	−	0.984423i	$-0.556257\pi$
$563$	−8.34345	−0.351635	−0.175817	+	0.984423i	$0.556257\pi$
$564$	0	0
$565$	9.89913	0.416460
$566$	0	0
$567$	−4.44005	−0.186465
$568$	0	0
$569$	23.8374	0.999315	0.499657	−	0.866223i	$-0.333459\pi$
$569$	23.8374	0.999315	0.499657	+	0.866223i	$0.333459\pi$
$570$	0	0
$571$	43.9915	1.84099	0.920493	−	0.390760i	$-0.127788\pi$
$571$	43.9915	1.84099	0.920493	+	0.390760i	$0.127788\pi$
$572$	0	0
$573$	16.2930	0.680651
$574$	0	0
$575$	−4.67558	−0.194985
$576$	0	0
$577$	−15.2783	−0.636042	−0.318021	−	0.948084i	$-0.603018\pi$
$577$	−15.2783	−0.636042	−0.318021	+	0.948084i	$0.603018\pi$
$578$	0	0
$579$	−22.2887	−0.926289
$580$	0	0
$581$	−2.99962	−0.124445
$582$	0	0
$583$	33.5358	1.38891
$584$	0	0
$585$	−5.94957	−0.245984
$586$	0	0
$587$	18.5480	0.765557	0.382778	−	0.923840i	$-0.374967\pi$
$587$	18.5480	0.765557	0.382778	+	0.923840i	$0.374967\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	−15.6946	−0.645590
$592$	0	0
$593$	−9.49010	−0.389712	−0.194856	−	0.980832i	$-0.562424\pi$
$593$	−9.49010	−0.389712	−0.194856	+	0.980832i	$0.562424\pi$
$594$	0	0
$595$	−20.7598	−0.851069
$596$	0	0
$597$	−6.14703	−0.251581
$598$	0	0
$599$	40.9908	1.67484	0.837419	−	0.546561i	$-0.184063\pi$
$599$	40.9908	1.67484	0.837419	+	0.546561i	$0.184063\pi$
$600$	0	0
$601$	−27.9760	−1.14117	−0.570583	−	0.821240i	$-0.693283\pi$
$601$	−27.9760	−1.14117	−0.570583	+	0.821240i	$0.693283\pi$
$602$	0	0
$603$	−6.28170	−0.255810
$604$	0	0
$605$	1.31671	0.0535317
$606$	0	0
$607$	15.9496	0.647373	0.323686	−	0.946164i	$-0.395078\pi$
$607$	15.9496	0.647373	0.323686	+	0.946164i	$0.395078\pi$
$608$	0	0
$609$	32.2968	1.30873
$610$	0	0
$611$	−68.7513	−2.78138
$612$	0	0
$613$	−19.7718	−0.798575	−0.399288	−	0.916826i	$-0.630743\pi$
$613$	−19.7718	−0.798575	−0.399288	+	0.916826i	$0.630743\pi$
$614$	0	0
$615$	1.50952	0.0608696
$616$	0	0
$617$	−20.8173	−0.838073	−0.419036	−	0.907969i	$-0.637632\pi$
$617$	−20.8173	−0.838073	−0.419036	+	0.907969i	$0.637632\pi$
$618$	0	0
$619$	32.8100	1.31874	0.659372	−	0.751817i	$-0.270822\pi$
$619$	32.8100	1.31874	0.659372	+	0.751817i	$0.270822\pi$
$620$	0	0
$621$	4.67558	0.187625
$622$	0	0
$623$	4.95345	0.198456
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−11.4086	−0.454893
$630$	0	0
$631$	−41.5585	−1.65442	−0.827208	−	0.561896i	$-0.810072\pi$
$631$	−41.5585	−1.65442	−0.827208	+	0.561896i	$0.810072\pi$
$632$	0	0
$633$	−24.3082	−0.966163
$634$	0	0
$635$	15.8991	0.630938
$636$	0	0
$637$	−75.6430	−2.99708
$638$	0	0
$639$	−7.11563	−0.281490
$640$	0	0
$641$	41.9143	1.65551	0.827757	−	0.561087i	$-0.189616\pi$
$641$	41.9143	1.65551	0.827757	+	0.561087i	$0.189616\pi$
$642$	0	0
$643$	26.8955	1.06066	0.530328	−	0.847793i	$-0.322069\pi$
$643$	26.8955	1.06066	0.530328	+	0.847793i	$0.322069\pi$
$644$	0	0
$645$	3.50952	0.138187
$646$	0	0
$647$	−15.1114	−0.594089	−0.297044	−	0.954864i	$-0.596001\pi$
$647$	−15.1114	−0.594089	−0.297044	+	0.954864i	$0.596001\pi$
$648$	0	0
$649$	45.4738	1.78500
$650$	0	0
$651$	4.44005	0.174019
$652$	0	0
$653$	41.7087	1.63219	0.816094	−	0.577919i	$-0.196135\pi$
$653$	41.7087	1.63219	0.816094	+	0.577919i	$0.196135\pi$
$654$	0	0
$655$	−14.1541	−0.553046
$656$	0	0
$657$	13.4591	0.525089
$658$	0	0
$659$	−13.9415	−0.543083	−0.271541	−	0.962427i	$-0.587533\pi$
$659$	−13.9415	−0.543083	−0.271541	+	0.962427i	$0.587533\pi$
$660$	0	0
$661$	45.3807	1.76510	0.882552	−	0.470215i	$-0.155824\pi$
$661$	45.3807	1.76510	0.882552	+	0.470215i	$0.155824\pi$
$662$	0	0
$663$	27.8177	1.08035
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−34.0101	−1.31688
$668$	0	0
$669$	16.3896	0.633659
$670$	0	0
$671$	15.2037	0.586932
$672$	0	0
$673$	44.6125	1.71969	0.859843	−	0.510559i	$-0.170562\pi$
$673$	44.6125	1.71969	0.859843	+	0.510559i	$0.170562\pi$
$674$	0	0
$675$	−1.00000	−0.0384900
$676$	0	0
$677$	35.7955	1.37573	0.687866	−	0.725838i	$-0.258548\pi$
$677$	35.7955	1.37573	0.687866	+	0.725838i	$0.258548\pi$
$678$	0	0
$679$	−65.9821	−2.53216
$680$	0	0
$681$	−20.4358	−0.783101
$682$	0	0
$683$	−40.2340	−1.53951	−0.769756	−	0.638338i	$-0.779622\pi$
$683$	−40.2340	−1.53951	−0.769756	+	0.638338i	$0.779622\pi$
$684$	0	0
$685$	11.0077	0.420583
$686$	0	0
$687$	29.0105	1.10682
$688$	0	0
$689$	−56.8521	−2.16589
$690$	0	0
$691$	19.4901	0.741438	0.370719	−	0.928745i	$-0.379111\pi$
$691$	19.4901	0.741438	0.370719	+	0.928745i	$0.379111\pi$
$692$	0	0
$693$	−15.5824	−0.591927
$694$	0	0
$695$	19.2122	0.728761
$696$	0	0
$697$	−7.05787	−0.267336
$698$	0	0
$699$	4.12761	0.156121
$700$	0	0
$701$	13.4086	0.506438	0.253219	−	0.967409i	$-0.418511\pi$
$701$	13.4086	0.506438	0.253219	+	0.967409i	$0.418511\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	−11.5557	−0.435212
$706$	0	0
$707$	−0.703096	−0.0264426
$708$	0	0
$709$	−17.1340	−0.643481	−0.321740	−	0.946828i	$-0.604268\pi$
$709$	−17.1340	−0.643481	−0.321740	+	0.946828i	$0.604268\pi$
$710$	0	0
$711$	−9.06520	−0.339971
$712$	0	0
$713$	−4.67558	−0.175102
$714$	0	0
$715$	−20.8801	−0.780871
$716$	0	0
$717$	4.96155	0.185292
$718$	0	0
$719$	13.1547	0.490589	0.245295	−	0.969449i	$-0.421115\pi$
$719$	13.1547	0.490589	0.245295	+	0.969449i	$0.421115\pi$
$720$	0	0
$721$	−4.13161	−0.153869
$722$	0	0
$723$	20.9182	0.777955
$724$	0	0
$725$	7.27398	0.270149
$726$	0	0
$727$	18.8451	0.698926	0.349463	−	0.936950i	$-0.386364\pi$
$727$	18.8451	0.698926	0.349463	+	0.936950i	$0.386364\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−16.4090	−0.606910
$732$	0	0
$733$	−48.8173	−1.80311	−0.901554	−	0.432667i	$-0.857573\pi$
$733$	−48.8173	−1.80311	−0.901554	+	0.432667i	$0.857573\pi$
$734$	0	0
$735$	−12.7140	−0.468964
$736$	0	0
$737$	−22.0457	−0.812064
$738$	0	0
$739$	41.9453	1.54298	0.771491	−	0.636240i	$-0.219511\pi$
$739$	41.9453	1.54298	0.771491	+	0.636240i	$0.219511\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−7.69184	−0.282186	−0.141093	−	0.989996i	$-0.545062\pi$
$743$	−7.69184	−0.282186	−0.141093	+	0.989996i	$0.545062\pi$
$744$	0	0
$745$	13.1928	0.483347
$746$	0	0
$747$	0.675582	0.0247182
$748$	0	0
$749$	11.8797	0.434075
$750$	0	0
$751$	−32.2158	−1.17557	−0.587786	−	0.809016i	$-0.700000\pi$
$751$	−32.2158	−1.17557	−0.587786	+	0.809016i	$0.700000\pi$
$752$	0	0
$753$	7.31310	0.266504
$754$	0	0
$755$	5.16607	0.188012
$756$	0	0
$757$	34.2577	1.24512	0.622559	−	0.782573i	$-0.286093\pi$
$757$	34.2577	1.24512	0.622559	+	0.782573i	$0.286093\pi$
$758$	0	0
$759$	16.4090	0.595610
$760$	0	0
$761$	−32.7175	−1.18601	−0.593004	−	0.805200i	$-0.702058\pi$
$761$	−32.7175	−1.18601	−0.593004	+	0.805200i	$0.702058\pi$
$762$	0	0
$763$	−39.2231	−1.41997
$764$	0	0
$765$	4.67558	0.169046
$766$	0	0
$767$	−77.0902	−2.78356
$768$	0	0
$769$	9.92987	0.358080	0.179040	−	0.983842i	$-0.442701\pi$
$769$	9.92987	0.358080	0.179040	+	0.983842i	$0.442701\pi$
$770$	0	0
$771$	−6.20452	−0.223450
$772$	0	0
$773$	4.01132	0.144277	0.0721386	−	0.997395i	$-0.477018\pi$
$773$	4.01132	0.144277	0.0721386	+	0.997395i	$0.477018\pi$
$774$	0	0
$775$	1.00000	0.0359211
$776$	0	0
$777$	−10.8339	−0.388665
$778$	0	0
$779$	0	0
$780$	0	0
$781$	−24.9724	−0.893584
$782$	0	0
$783$	−7.27398	−0.259951
$784$	0	0
$785$	−15.2697	−0.544999
$786$	0	0
$787$	−12.3281	−0.439451	−0.219725	−	0.975562i	$-0.570516\pi$
$787$	−12.3281	−0.439451	−0.219725	+	0.975562i	$0.570516\pi$
$788$	0	0
$789$	−27.0105	−0.961599
$790$	0	0
$791$	−43.9526	−1.56278
$792$	0	0
$793$	−25.7743	−0.915272
$794$	0	0
$795$	−9.55568	−0.338905
$796$	0	0
$797$	−26.7525	−0.947622	−0.473811	−	0.880627i	$-0.657122\pi$
$797$	−26.7525	−0.947622	−0.473811	+	0.880627i	$0.657122\pi$
$798$	0	0
$799$	54.0295	1.91143
$800$	0	0
$801$	−1.11563	−0.0394189
$802$	0	0
$803$	47.2349	1.66688
$804$	0	0
$805$	20.7598	0.731687
$806$	0	0
$807$	7.46718	0.262857
$808$	0	0
$809$	−46.7361	−1.64315	−0.821577	−	0.570097i	$-0.806906\pi$
$809$	−46.7361	−1.64315	−0.821577	+	0.570097i	$0.806906\pi$
$810$	0	0
$811$	−6.80320	−0.238893	−0.119446	−	0.992841i	$-0.538112\pi$
$811$	−6.80320	−0.238893	−0.119446	+	0.992841i	$0.538112\pi$
$812$	0	0
$813$	−9.56700	−0.335529
$814$	0	0
$815$	−7.94957	−0.278461
$816$	0	0
$817$	0	0
$818$	0	0
$819$	26.4164	0.923062
$820$	0	0
$821$	42.6600	1.48884	0.744422	−	0.667710i	$-0.232725\pi$
$821$	42.6600	1.48884	0.744422	+	0.667710i	$0.232725\pi$
$822$	0	0
$823$	34.4191	1.19978	0.599888	−	0.800084i	$-0.295212\pi$
$823$	34.4191	1.19978	0.599888	+	0.800084i	$0.295212\pi$
$824$	0	0
$825$	−3.50952	−0.122186
$826$	0	0
$827$	54.1191	1.88190	0.940952	−	0.338539i	$-0.109933\pi$
$827$	54.1191	1.88190	0.940952	+	0.338539i	$0.109933\pi$
$828$	0	0
$829$	30.7707	1.06871	0.534355	−	0.845260i	$-0.320555\pi$
$829$	30.7707	1.06871	0.534355	+	0.845260i	$0.320555\pi$
$830$	0	0
$831$	−27.2193	−0.944226
$832$	0	0
$833$	59.4455	2.05966
$834$	0	0
$835$	−2.13894	−0.0740209
$836$	0	0
$837$	−1.00000	−0.0345651
$838$	0	0
$839$	−23.8840	−0.824567	−0.412283	−	0.911056i	$-0.635269\pi$
$839$	−23.8840	−0.824567	−0.412283	+	0.911056i	$0.635269\pi$
$840$	0	0
$841$	23.9108	0.824512
$842$	0	0
$843$	3.86106	0.132982
$844$	0	0
$845$	22.3973	0.770491
$846$	0	0
$847$	−5.84624	−0.200879
$848$	0	0
$849$	16.8102	0.576926
$850$	0	0
$851$	11.4086	0.391083
$852$	0	0
$853$	−23.3706	−0.800193	−0.400097	−	0.916473i	$-0.631023\pi$
$853$	−23.3706	−0.800193	−0.400097	+	0.916473i	$0.631023\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−17.6213	−0.601931	−0.300965	−	0.953635i	$-0.597309\pi$
$857$	−17.6213	−0.601931	−0.300965	+	0.953635i	$0.597309\pi$
$858$	0	0
$859$	48.4625	1.65352	0.826760	−	0.562555i	$-0.190181\pi$
$859$	48.4625	1.65352	0.826760	+	0.562555i	$0.190181\pi$
$860$	0	0
$861$	−6.70233	−0.228415
$862$	0	0
$863$	46.8173	1.59368	0.796840	−	0.604191i	$-0.206504\pi$
$863$	46.8173	1.59368	0.796840	+	0.604191i	$0.206504\pi$
$864$	0	0
$865$	6.88010	0.233930
$866$	0	0
$867$	−4.86106	−0.165090
$868$	0	0
$869$	−31.8145	−1.07923
$870$	0	0
$871$	37.3734	1.26635
$872$	0	0
$873$	14.8607	0.502958
$874$	0	0
$875$	−4.44005	−0.150101
$876$	0	0
$877$	3.77961	0.127628	0.0638142	−	0.997962i	$-0.479673\pi$
$877$	3.77961	0.127628	0.0638142	+	0.997962i	$0.479673\pi$
$878$	0	0
$879$	30.5747	1.03126
$880$	0	0
$881$	12.5134	0.421587	0.210794	−	0.977531i	$-0.432395\pi$
$881$	12.5134	0.421587	0.210794	+	0.977531i	$0.432395\pi$
$882$	0	0
$883$	57.2223	1.92568	0.962842	−	0.270064i	$-0.0870448\pi$
$883$	57.2223	1.92568	0.962842	+	0.270064i	$0.0870448\pi$
$884$	0	0
$885$	−12.9573	−0.435554
$886$	0	0
$887$	54.4653	1.82877	0.914383	−	0.404851i	$-0.132676\pi$
$887$	54.4653	1.82877	0.914383	+	0.404851i	$0.132676\pi$
$888$	0	0
$889$	−70.5929	−2.36761
$890$	0	0
$891$	3.50952	0.117573
$892$	0	0
$893$	0	0
$894$	0	0
$895$	1.66787	0.0557507
$896$	0	0
$897$	−27.8177	−0.928805
$898$	0	0
$899$	7.27398	0.242601
$900$	0	0
$901$	44.6784	1.48845
$902$	0	0
$903$	−15.5824	−0.518551
$904$	0	0
$905$	−20.1304	−0.669157
$906$	0	0
$907$	7.75315	0.257439	0.128719	−	0.991681i	$-0.458913\pi$
$907$	7.75315	0.257439	0.128719	+	0.991681i	$0.458913\pi$
$908$	0	0
$909$	0.158353	0.00525224
$910$	0	0
$911$	8.28874	0.274618	0.137309	−	0.990528i	$-0.456155\pi$
$911$	8.28874	0.274618	0.137309	+	0.990528i	$0.456155\pi$
$912$	0	0
$913$	2.37097	0.0784675
$914$	0	0
$915$	−4.33213	−0.143216
$916$	0	0
$917$	62.8448	2.07532
$918$	0	0
$919$	−23.1114	−0.762373	−0.381187	−	0.924498i	$-0.624484\pi$
$919$	−23.1114	−0.762373	−0.381187	+	0.924498i	$0.624484\pi$
$920$	0	0
$921$	−15.1424	−0.498958
$922$	0	0
$923$	42.3349	1.39347
$924$	0	0
$925$	−2.44005	−0.0802283
$926$	0	0
$927$	0.930532	0.0305627
$928$	0	0
$929$	−59.0637	−1.93782	−0.968909	−	0.247419i	$-0.920418\pi$
$929$	−59.0637	−1.93782	−0.968909	+	0.247419i	$0.920418\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	22.2355	0.727959
$934$	0	0
$935$	16.4090	0.536633
$936$	0	0
$937$	−42.8607	−1.40020	−0.700099	−	0.714046i	$-0.746861\pi$
$937$	−42.8607	−1.40020	−0.700099	+	0.714046i	$0.746861\pi$
$938$	0	0
$939$	21.8487	0.713005
$940$	0	0
$941$	−4.45137	−0.145111	−0.0725553	−	0.997364i	$-0.523115\pi$
$941$	−4.45137	−0.145111	−0.0725553	+	0.997364i	$0.523115\pi$
$942$	0	0
$943$	7.05787	0.229836
$944$	0	0
$945$	4.44005	0.144435
$946$	0	0
$947$	50.1417	1.62939	0.814693	−	0.579892i	$-0.196905\pi$
$947$	50.1417	1.62939	0.814693	+	0.579892i	$0.196905\pi$
$948$	0	0
$949$	−80.0757	−2.59937
$950$	0	0
$951$	−33.3159	−1.08034
$952$	0	0
$953$	15.3018	0.495673	0.247837	−	0.968802i	$-0.420280\pi$
$953$	15.3018	0.495673	0.247837	+	0.968802i	$0.420280\pi$
$954$	0	0
$955$	−16.2930	−0.527230
$956$	0	0
$957$	−25.5282	−0.825208
$958$	0	0
$959$	−48.8748	−1.57825
$960$	0	0
$961$	1.00000	0.0322581
$962$	0	0
$963$	−2.67558	−0.0862194
$964$	0	0
$965$	22.2887	0.717500
$966$	0	0
$967$	−36.4657	−1.17266	−0.586330	−	0.810073i	$-0.699428\pi$
$967$	−36.4657	−1.17266	−0.586330	+	0.810073i	$0.699428\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	27.2306	0.873871	0.436936	−	0.899493i	$-0.356064\pi$
$971$	27.2306	0.873871	0.436936	+	0.899493i	$0.356064\pi$
$972$	0	0
$973$	−85.3032	−2.73470
$974$	0	0
$975$	5.94957	0.190539
$976$	0	0
$977$	−19.0959	−0.610933	−0.305467	−	0.952203i	$-0.598812\pi$
$977$	−19.0959	−0.610933	−0.305467	+	0.952203i	$0.598812\pi$
$978$	0	0
$979$	−3.91532	−0.125134
$980$	0	0
$981$	8.83393	0.282046
$982$	0	0
$983$	−19.4281	−0.619659	−0.309830	−	0.950792i	$-0.600272\pi$
$983$	−19.4281	−0.619659	−0.309830	+	0.950792i	$0.600272\pi$
$984$	0	0
$985$	15.6946	0.500072
$986$	0	0
$987$	51.3078	1.63314
$988$	0	0
$989$	16.4090	0.521777
$990$	0	0
$991$	−25.5670	−0.812163	−0.406081	−	0.913837i	$-0.633105\pi$
$991$	−25.5670	−0.812163	−0.406081	+	0.913837i	$0.633105\pi$
$992$	0	0
$993$	−2.36287	−0.0749834
$994$	0	0
$995$	6.14703	0.194874
$996$	0	0
$997$	25.7897	0.816769	0.408384	−	0.912810i	$-0.366092\pi$
$997$	25.7897	0.816769	0.408384	+	0.912810i	$0.366092\pi$
$998$	0	0
$999$	2.44005	0.0771997

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1860.2.a.i.1.2	✓	4
3.2	odd	2		5580.2.a.m.1.2		4
4.3	odd	2		7440.2.a.cb.1.3		4
5.2	odd	4		9300.2.g.s.3349.6		8
5.3	odd	4		9300.2.g.s.3349.3		8
5.4	even	2		9300.2.a.x.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1860.2.a.i.1.2	✓	4	1.1	even	1	trivial
5580.2.a.m.1.2		4	3.2	odd	2
7440.2.a.cb.1.3		4	4.3	odd	2
9300.2.a.x.1.3		4	5.4	even	2
9300.2.g.s.3349.3		8	5.3	odd	4
9300.2.g.s.3349.6		8	5.2	odd	4

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:	\( N \)	\(=\)	\( 1860 = 2^{2} \cdot 3 \cdot 5 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1860.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +1.00000 q^{5} -4.44005 q^{7} +1.00000 q^{9} +3.50952 q^{11} -5.94957 q^{13} -1.00000 q^{15} +4.67558 q^{17} +4.44005 q^{21} -4.67558 q^{23} +1.00000 q^{25} -1.00000 q^{27} +7.27398 q^{29} +1.00000 q^{31} -3.50952 q^{33} -4.44005 q^{35} -2.44005 q^{37} +5.94957 q^{39} -1.50952 q^{41} -3.50952 q^{43} +1.00000 q^{45} +11.5557 q^{47} +12.7140 q^{49} -4.67558 q^{51} +9.55568 q^{53} +3.50952 q^{55} +12.9573 q^{59} +4.33213 q^{61} -4.44005 q^{63} -5.94957 q^{65} -6.28170 q^{67} +4.67558 q^{69} -7.11563 q^{71} +13.4591 q^{73} -1.00000 q^{75} -15.5824 q^{77} -9.06520 q^{79} +1.00000 q^{81} +0.675582 q^{83} +4.67558 q^{85} -7.27398 q^{87} -1.11563 q^{89} +26.4164 q^{91} -1.00000 q^{93} +14.8607 q^{97} +3.50952 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{3} + 4 q^{5} - 4 q^{7} + 4 q^{9} + 2 q^{11} + 2 q^{13} - 4 q^{15} + 2 q^{17} + 4 q^{21} - 2 q^{23} + 4 q^{25} - 4 q^{27} + 20 q^{29} + 4 q^{31} - 2 q^{33} - 4 q^{35} + 4 q^{37} - 2 q^{39} + 6 q^{41}+ \cdots + 2 q^{99}+O(q^{100}) \) 4 * q - 4 * q^3 + 4 * q^5 - 4 * q^7 + 4 * q^9 + 2 * q^11 + 2 * q^13 - 4 * q^15 + 2 * q^17 + 4 * q^21 - 2 * q^23 + 4 * q^25 - 4 * q^27 + 20 * q^29 + 4 * q^31 - 2 * q^33 - 4 * q^35 + 4 * q^37 - 2 * q^39 + 6 * q^41 - 2 * q^43 + 4 * q^45 + 2 * q^47 + 28 * q^49 - 2 * q^51 - 6 * q^53 + 2 * q^55 + 8 * q^61 - 4 * q^63 + 2 * q^65 + 10 * q^67 + 2 * q^69 + 2 * q^71 + 16 * q^73 - 4 * q^75 + 32 * q^77 + 20 * q^79 + 4 * q^81 - 14 * q^83 + 2 * q^85 - 20 * q^87 + 26 * q^89 + 16 * q^91 - 4 * q^93 + 14 * q^97 + 2 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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