LMFDB - Embedded newform 1840.2.a.l.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1840,2,Mod(1,1840)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1840, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("1840.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1840 = 2^{4} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1840.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:	$14.6924739719$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 230)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.618034$ of defining polynomial
Character	$\chi$	$=$	1840.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.618034 q^{3} +1.00000 q^{5} -1.61803 q^{7} -2.61803 q^{9} +O(q^{10})$	$q+0.618034 q^{3} +1.00000 q^{5} -1.61803 q^{7} -2.61803 q^{9} -3.85410 q^{11} +4.09017 q^{13} +0.618034 q^{15} -5.09017 q^{17} +4.85410 q^{19} -1.00000 q^{21} -1.00000 q^{23} +1.00000 q^{25} -3.47214 q^{27} -4.76393 q^{29} +2.09017 q^{31} -2.38197 q^{33} -1.61803 q^{35} -2.47214 q^{37} +2.52786 q^{39} -12.3262 q^{41} -2.61803 q^{45} -9.70820 q^{47} -4.38197 q^{49} -3.14590 q^{51} -8.47214 q^{53} -3.85410 q^{55} +3.00000 q^{57} +11.7082 q^{59} +6.32624 q^{61} +4.23607 q^{63} +4.09017 q^{65} -5.52786 q^{67} -0.618034 q^{69} -7.09017 q^{71} -1.23607 q^{73} +0.618034 q^{75} +6.23607 q^{77} -10.4721 q^{79} +5.70820 q^{81} -10.9443 q^{83} -5.09017 q^{85} -2.94427 q^{87} -1.52786 q^{89} -6.61803 q^{91} +1.29180 q^{93} +4.85410 q^{95} +14.6180 q^{97} +10.0902 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{3} + 2 q^{5} - q^{7} - 3 q^{9}+O(q^{10}) $ 2 * q - q^3 + 2 * q^5 - q^7 - 3 * q^9	$ 2 q - q^{3} + 2 q^{5} - q^{7} - 3 q^{9} - q^{11} - 3 q^{13} - q^{15} + q^{17} + 3 q^{19} - 2 q^{21} - 2 q^{23} + 2 q^{25} + 2 q^{27} - 14 q^{29} - 7 q^{31} - 7 q^{33} - q^{35} + 4 q^{37} + 14 q^{39} - 9 q^{41} - 3 q^{45} - 6 q^{47} - 11 q^{49} - 13 q^{51} - 8 q^{53} - q^{55} + 6 q^{57} + 10 q^{59} - 3 q^{61} + 4 q^{63} - 3 q^{65} - 20 q^{67} + q^{69} - 3 q^{71} + 2 q^{73} - q^{75} + 8 q^{77} - 12 q^{79} - 2 q^{81} - 4 q^{83} + q^{85} + 12 q^{87} - 12 q^{89} - 11 q^{91} + 16 q^{93} + 3 q^{95} + 27 q^{97} + 9 q^{99}+O(q^{100}) $ 2 * q - q^3 + 2 * q^5 - q^7 - 3 * q^9 - q^11 - 3 * q^13 - q^15 + q^17 + 3 * q^19 - 2 * q^21 - 2 * q^23 + 2 * q^25 + 2 * q^27 - 14 * q^29 - 7 * q^31 - 7 * q^33 - q^35 + 4 * q^37 + 14 * q^39 - 9 * q^41 - 3 * q^45 - 6 * q^47 - 11 * q^49 - 13 * q^51 - 8 * q^53 - q^55 + 6 * q^57 + 10 * q^59 - 3 * q^61 + 4 * q^63 - 3 * q^65 - 20 * q^67 + q^69 - 3 * q^71 + 2 * q^73 - q^75 + 8 * q^77 - 12 * q^79 - 2 * q^81 - 4 * q^83 + q^85 + 12 * q^87 - 12 * q^89 - 11 * q^91 + 16 * q^93 + 3 * q^95 + 27 * q^97 + 9 * 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
$3$	0.618034	0.356822	0.178411	−	0.983956i	$-0.442904\pi$
$3$	0.618034	0.356822	0.178411	+	0.983956i	$0.442904\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−1.61803	−0.611559	−0.305780	−	0.952102i	$-0.598917\pi$
$7$	−1.61803	−0.611559	−0.305780	+	0.952102i	$0.598917\pi$
$8$	0	0
$9$	−2.61803	−0.872678
$10$	0	0
$11$	−3.85410	−1.16206	−0.581028	−	0.813884i	$-0.697349\pi$
$11$	−3.85410	−1.16206	−0.581028	+	0.813884i	$0.697349\pi$
$12$	0	0
$13$	4.09017	1.13441	0.567205	−	0.823577i	$-0.308025\pi$
$13$	4.09017	1.13441	0.567205	+	0.823577i	$0.308025\pi$
$14$	0	0
$15$	0.618034	0.159576
$16$	0	0
$17$	−5.09017	−1.23455	−0.617274	−	0.786748i	$-0.711763\pi$
$17$	−5.09017	−1.23455	−0.617274	+	0.786748i	$0.711763\pi$
$18$	0	0
$19$	4.85410	1.11361	0.556804	−	0.830644i	$-0.312028\pi$
$19$	4.85410	1.11361	0.556804	+	0.830644i	$0.312028\pi$
$20$	0	0
$21$	−1.00000	−0.218218
$22$	0	0
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−3.47214	−0.668213
$28$	0	0
$29$	−4.76393	−0.884640	−0.442320	−	0.896857i	$-0.645844\pi$
$29$	−4.76393	−0.884640	−0.442320	+	0.896857i	$0.645844\pi$
$30$	0	0
$31$	2.09017	0.375406	0.187703	−	0.982226i	$-0.439896\pi$
$31$	2.09017	0.375406	0.187703	+	0.982226i	$0.439896\pi$
$32$	0	0
$33$	−2.38197	−0.414647
$34$	0	0
$35$	−1.61803	−0.273498
$36$	0	0
$37$	−2.47214	−0.406417	−0.203208	−	0.979136i	$-0.565137\pi$
$37$	−2.47214	−0.406417	−0.203208	+	0.979136i	$0.565137\pi$
$38$	0	0
$39$	2.52786	0.404782
$40$	0	0
$41$	−12.3262	−1.92503	−0.962517	−	0.271220i	$-0.912573\pi$
$41$	−12.3262	−1.92503	−0.962517	+	0.271220i	$0.912573\pi$
$42$	0	0
$43$	0	0		−	1.00000i	$-0.5\pi$
$43$	0	0			1.00000i	$0.5\pi$
$44$	0	0
$45$	−2.61803	−0.390273
$46$	0	0
$47$	−9.70820	−1.41609	−0.708044	−	0.706169i	$-0.750422\pi$
$47$	−9.70820	−1.41609	−0.708044	+	0.706169i	$0.750422\pi$
$48$	0	0
$49$	−4.38197	−0.625995
$50$	0	0
$51$	−3.14590	−0.440514
$52$	0	0
$53$	−8.47214	−1.16374	−0.581869	−	0.813283i	$-0.697678\pi$
$53$	−8.47214	−1.16374	−0.581869	+	0.813283i	$0.697678\pi$
$54$	0	0
$55$	−3.85410	−0.519687
$56$	0	0
$57$	3.00000	0.397360
$58$	0	0
$59$	11.7082	1.52428	0.762139	−	0.647413i	$-0.224149\pi$
$59$	11.7082	1.52428	0.762139	+	0.647413i	$0.224149\pi$
$60$	0	0
$61$	6.32624	0.809992	0.404996	−	0.914319i	$-0.367273\pi$
$61$	6.32624	0.809992	0.404996	+	0.914319i	$0.367273\pi$
$62$	0	0
$63$	4.23607	0.533694
$64$	0	0
$65$	4.09017	0.507323
$66$	0	0
$67$	−5.52786	−0.675336	−0.337668	−	0.941265i	$-0.609638\pi$
$67$	−5.52786	−0.675336	−0.337668	+	0.941265i	$0.609638\pi$
$68$	0	0
$69$	−0.618034	−0.0744025
$70$	0	0
$71$	−7.09017	−0.841448	−0.420724	−	0.907189i	$-0.638224\pi$
$71$	−7.09017	−0.841448	−0.420724	+	0.907189i	$0.638224\pi$
$72$	0	0
$73$	−1.23607	−0.144671	−0.0723354	−	0.997380i	$-0.523045\pi$
$73$	−1.23607	−0.144671	−0.0723354	+	0.997380i	$0.523045\pi$
$74$	0	0
$75$	0.618034	0.0713644
$76$	0	0
$77$	6.23607	0.710666
$78$	0	0
$79$	−10.4721	−1.17821	−0.589104	−	0.808057i	$-0.700519\pi$
$79$	−10.4721	−1.17821	−0.589104	+	0.808057i	$0.700519\pi$
$80$	0	0
$81$	5.70820	0.634245
$82$	0	0
$83$	−10.9443	−1.20129	−0.600645	−	0.799516i	$-0.705089\pi$
$83$	−10.9443	−1.20129	−0.600645	+	0.799516i	$0.705089\pi$
$84$	0	0
$85$	−5.09017	−0.552106
$86$	0	0
$87$	−2.94427	−0.315659
$88$	0	0
$89$	−1.52786	−0.161953	−0.0809766	−	0.996716i	$-0.525804\pi$
$89$	−1.52786	−0.161953	−0.0809766	+	0.996716i	$0.525804\pi$
$90$	0	0
$91$	−6.61803	−0.693758
$92$	0	0
$93$	1.29180	0.133953
$94$	0	0
$95$	4.85410	0.498020
$96$	0	0
$97$	14.6180	1.48424	0.742118	−	0.670269i	$-0.233821\pi$
$97$	14.6180	1.48424	0.742118	+	0.670269i	$0.233821\pi$
$98$	0	0
$99$	10.0902	1.01410
$100$	0	0
$101$	−13.7082	−1.36402	−0.682009	−	0.731344i	$-0.738893\pi$
$101$	−13.7082	−1.36402	−0.682009	+	0.731344i	$0.738893\pi$
$102$	0	0
$103$	3.56231	0.351004	0.175502	−	0.984479i	$-0.443845\pi$
$103$	3.56231	0.351004	0.175502	+	0.984479i	$0.443845\pi$
$104$	0	0
$105$	−1.00000	−0.0975900
$106$	0	0
$107$	4.18034	0.404129	0.202064	−	0.979372i	$-0.435235\pi$
$107$	4.18034	0.404129	0.202064	+	0.979372i	$0.435235\pi$
$108$	0	0
$109$	8.56231	0.820120	0.410060	−	0.912059i	$-0.365508\pi$
$109$	8.56231	0.820120	0.410060	+	0.912059i	$0.365508\pi$
$110$	0	0
$111$	−1.52786	−0.145018
$112$	0	0
$113$	18.9443	1.78213	0.891064	−	0.453878i	$-0.149960\pi$
$113$	18.9443	1.78213	0.891064	+	0.453878i	$0.149960\pi$
$114$	0	0
$115$	−1.00000	−0.0932505
$116$	0	0
$117$	−10.7082	−0.989974
$118$	0	0
$119$	8.23607	0.754999
$120$	0	0
$121$	3.85410	0.350373
$122$	0	0
$123$	−7.61803	−0.686895
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	6.18034	0.548416	0.274208	−	0.961670i	$-0.411584\pi$
$127$	6.18034	0.548416	0.274208	+	0.961670i	$0.411584\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	14.9443	1.30569	0.652844	−	0.757493i	$-0.273576\pi$
$131$	14.9443	1.30569	0.652844	+	0.757493i	$0.273576\pi$
$132$	0	0
$133$	−7.85410	−0.681037
$134$	0	0
$135$	−3.47214	−0.298834
$136$	0	0
$137$	5.32624	0.455051	0.227526	−	0.973772i	$-0.426936\pi$
$137$	5.32624	0.455051	0.227526	+	0.973772i	$0.426936\pi$
$138$	0	0
$139$	−17.2361	−1.46194	−0.730972	−	0.682407i	$-0.760933\pi$
$139$	−17.2361	−1.46194	−0.730972	+	0.682407i	$0.760933\pi$
$140$	0	0
$141$	−6.00000	−0.505291
$142$	0	0
$143$	−15.7639	−1.31825
$144$	0	0
$145$	−4.76393	−0.395623
$146$	0	0
$147$	−2.70820	−0.223369
$148$	0	0
$149$	−1.14590	−0.0938756	−0.0469378	−	0.998898i	$-0.514946\pi$
$149$	−1.14590	−0.0938756	−0.0469378	+	0.998898i	$0.514946\pi$
$150$	0	0
$151$	−17.5623	−1.42920	−0.714600	−	0.699533i	$-0.753391\pi$
$151$	−17.5623	−1.42920	−0.714600	+	0.699533i	$0.753391\pi$
$152$	0	0
$153$	13.3262	1.07736
$154$	0	0
$155$	2.09017	0.167886
$156$	0	0
$157$	9.70820	0.774799	0.387400	−	0.921912i	$-0.373373\pi$
$157$	9.70820	0.774799	0.387400	+	0.921912i	$0.373373\pi$
$158$	0	0
$159$	−5.23607	−0.415247
$160$	0	0
$161$	1.61803	0.127519
$162$	0	0
$163$	−3.61803	−0.283386	−0.141693	−	0.989911i	$-0.545255\pi$
$163$	−3.61803	−0.283386	−0.141693	+	0.989911i	$0.545255\pi$
$164$	0	0
$165$	−2.38197	−0.185436
$166$	0	0
$167$	8.00000	0.619059	0.309529	−	0.950890i	$-0.399829\pi$
$167$	8.00000	0.619059	0.309529	+	0.950890i	$0.399829\pi$
$168$	0	0
$169$	3.72949	0.286884
$170$	0	0
$171$	−12.7082	−0.971821
$172$	0	0
$173$	−21.5623	−1.63935	−0.819676	−	0.572828i	$-0.805846\pi$
$173$	−21.5623	−1.63935	−0.819676	+	0.572828i	$0.805846\pi$
$174$	0	0
$175$	−1.61803	−0.122312
$176$	0	0
$177$	7.23607	0.543896
$178$	0	0
$179$	20.1803	1.50835	0.754175	−	0.656674i	$-0.228037\pi$
$179$	20.1803	1.50835	0.754175	+	0.656674i	$0.228037\pi$
$180$	0	0
$181$	−18.8541	−1.40141	−0.700707	−	0.713449i	$-0.747132\pi$
$181$	−18.8541	−1.40141	−0.700707	+	0.713449i	$0.747132\pi$
$182$	0	0
$183$	3.90983	0.289023
$184$	0	0
$185$	−2.47214	−0.181755
$186$	0	0
$187$	19.6180	1.43461
$188$	0	0
$189$	5.61803	0.408652
$190$	0	0
$191$	0.291796	0.0211136	0.0105568	−	0.999944i	$-0.496640\pi$
$191$	0.291796	0.0211136	0.0105568	+	0.999944i	$0.496640\pi$
$192$	0	0
$193$	5.23607	0.376900	0.188450	−	0.982083i	$-0.439654\pi$
$193$	5.23607	0.376900	0.188450	+	0.982083i	$0.439654\pi$
$194$	0	0
$195$	2.52786	0.181024
$196$	0	0
$197$	−2.43769	−0.173679	−0.0868393	−	0.996222i	$-0.527677\pi$
$197$	−2.43769	−0.173679	−0.0868393	+	0.996222i	$0.527677\pi$
$198$	0	0
$199$	−2.00000	−0.141776	−0.0708881	−	0.997484i	$-0.522583\pi$
$199$	−2.00000	−0.141776	−0.0708881	+	0.997484i	$0.522583\pi$
$200$	0	0
$201$	−3.41641	−0.240975
$202$	0	0
$203$	7.70820	0.541010
$204$	0	0
$205$	−12.3262	−0.860902
$206$	0	0
$207$	2.61803	0.181966
$208$	0	0
$209$	−18.7082	−1.29407
$210$	0	0
$211$	−14.0000	−0.963800	−0.481900	−	0.876226i	$-0.660053\pi$
$211$	−14.0000	−0.963800	−0.481900	+	0.876226i	$0.660053\pi$
$212$	0	0
$213$	−4.38197	−0.300247
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−3.38197	−0.229583
$218$	0	0
$219$	−0.763932	−0.0516217
$220$	0	0
$221$	−20.8197	−1.40048
$222$	0	0
$223$	3.05573	0.204627	0.102313	−	0.994752i	$-0.467376\pi$
$223$	3.05573	0.204627	0.102313	+	0.994752i	$0.467376\pi$
$224$	0	0
$225$	−2.61803	−0.174536
$226$	0	0
$227$	23.2361	1.54223	0.771116	−	0.636695i	$-0.219699\pi$
$227$	23.2361	1.54223	0.771116	+	0.636695i	$0.219699\pi$
$228$	0	0
$229$	10.0000	0.660819	0.330409	−	0.943838i	$-0.392813\pi$
$229$	10.0000	0.660819	0.330409	+	0.943838i	$0.392813\pi$
$230$	0	0
$231$	3.85410	0.253581
$232$	0	0
$233$	19.7082	1.29113	0.645564	−	0.763706i	$-0.276622\pi$
$233$	19.7082	1.29113	0.645564	+	0.763706i	$0.276622\pi$
$234$	0	0
$235$	−9.70820	−0.633293
$236$	0	0
$237$	−6.47214	−0.420410
$238$	0	0
$239$	−24.3607	−1.57576	−0.787881	−	0.615828i	$-0.788822\pi$
$239$	−24.3607	−1.57576	−0.787881	+	0.615828i	$0.788822\pi$
$240$	0	0
$241$	0	0		−	1.00000i	$-0.5\pi$
$241$	0	0			1.00000i	$0.5\pi$
$242$	0	0
$243$	13.9443	0.894525
$244$	0	0
$245$	−4.38197	−0.279954
$246$	0	0
$247$	19.8541	1.26329
$248$	0	0
$249$	−6.76393	−0.428647
$250$	0	0
$251$	−12.8541	−0.811344	−0.405672	−	0.914019i	$-0.632962\pi$
$251$	−12.8541	−0.811344	−0.405672	+	0.914019i	$0.632962\pi$
$252$	0	0
$253$	3.85410	0.242305
$254$	0	0
$255$	−3.14590	−0.197004
$256$	0	0
$257$	−30.1803	−1.88260	−0.941299	−	0.337574i	$-0.890394\pi$
$257$	−30.1803	−1.88260	−0.941299	+	0.337574i	$0.890394\pi$
$258$	0	0
$259$	4.00000	0.248548
$260$	0	0
$261$	12.4721	0.772006
$262$	0	0
$263$	21.7426	1.34071	0.670354	−	0.742041i	$-0.266142\pi$
$263$	21.7426	1.34071	0.670354	+	0.742041i	$0.266142\pi$
$264$	0	0
$265$	−8.47214	−0.520439
$266$	0	0
$267$	−0.944272	−0.0577885
$268$	0	0
$269$	8.18034	0.498764	0.249382	−	0.968405i	$-0.419773\pi$
$269$	8.18034	0.498764	0.249382	+	0.968405i	$0.419773\pi$
$270$	0	0
$271$	14.6738	0.891368	0.445684	−	0.895190i	$-0.352961\pi$
$271$	14.6738	0.891368	0.445684	+	0.895190i	$0.352961\pi$
$272$	0	0
$273$	−4.09017	−0.247548
$274$	0	0
$275$	−3.85410	−0.232411
$276$	0	0
$277$	2.58359	0.155233	0.0776165	−	0.996983i	$-0.475269\pi$
$277$	2.58359	0.155233	0.0776165	+	0.996983i	$0.475269\pi$
$278$	0	0
$279$	−5.47214	−0.327608
$280$	0	0
$281$	27.2361	1.62477	0.812384	−	0.583123i	$-0.198169\pi$
$281$	27.2361	1.62477	0.812384	+	0.583123i	$0.198169\pi$
$282$	0	0
$283$	9.05573	0.538307	0.269154	−	0.963097i	$-0.413256\pi$
$283$	9.05573	0.538307	0.269154	+	0.963097i	$0.413256\pi$
$284$	0	0
$285$	3.00000	0.177705
$286$	0	0
$287$	19.9443	1.17727
$288$	0	0
$289$	8.90983	0.524108
$290$	0	0
$291$	9.03444	0.529608
$292$	0	0
$293$	15.8885	0.928219	0.464109	−	0.885778i	$-0.346374\pi$
$293$	15.8885	0.928219	0.464109	+	0.885778i	$0.346374\pi$
$294$	0	0
$295$	11.7082	0.681678
$296$	0	0
$297$	13.3820	0.776500
$298$	0	0
$299$	−4.09017	−0.236541
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−8.47214	−0.486711
$304$	0	0
$305$	6.32624	0.362239
$306$	0	0
$307$	−27.4508	−1.56670	−0.783351	−	0.621579i	$-0.786491\pi$
$307$	−27.4508	−1.56670	−0.783351	+	0.621579i	$0.786491\pi$
$308$	0	0
$309$	2.20163	0.125246
$310$	0	0
$311$	−4.00000	−0.226819	−0.113410	−	0.993548i	$-0.536177\pi$
$311$	−4.00000	−0.226819	−0.113410	+	0.993548i	$0.536177\pi$
$312$	0	0
$313$	−11.7984	−0.666884	−0.333442	−	0.942771i	$-0.608210\pi$
$313$	−11.7984	−0.666884	−0.333442	+	0.942771i	$0.608210\pi$
$314$	0	0
$315$	4.23607	0.238675
$316$	0	0
$317$	0.0901699	0.00506445	0.00253222	−	0.999997i	$-0.499194\pi$
$317$	0.0901699	0.00506445	0.00253222	+	0.999997i	$0.499194\pi$
$318$	0	0
$319$	18.3607	1.02800
$320$	0	0
$321$	2.58359	0.144202
$322$	0	0
$323$	−24.7082	−1.37480
$324$	0	0
$325$	4.09017	0.226882
$326$	0	0
$327$	5.29180	0.292637
$328$	0	0
$329$	15.7082	0.866021
$330$	0	0
$331$	−14.7639	−0.811499	−0.405750	−	0.913984i	$-0.632989\pi$
$331$	−14.7639	−0.811499	−0.405750	+	0.913984i	$0.632989\pi$
$332$	0	0
$333$	6.47214	0.354671
$334$	0	0
$335$	−5.52786	−0.302019
$336$	0	0
$337$	29.3262	1.59750	0.798751	−	0.601662i	$-0.205494\pi$
$337$	29.3262	1.59750	0.798751	+	0.601662i	$0.205494\pi$
$338$	0	0
$339$	11.7082	0.635902
$340$	0	0
$341$	−8.05573	−0.436242
$342$	0	0
$343$	18.4164	0.994393
$344$	0	0
$345$	−0.618034	−0.0332738
$346$	0	0
$347$	8.61803	0.462640	0.231320	−	0.972878i	$-0.425696\pi$
$347$	8.61803	0.462640	0.231320	+	0.972878i	$0.425696\pi$
$348$	0	0
$349$	−2.00000	−0.107058	−0.0535288	−	0.998566i	$-0.517047\pi$
$349$	−2.00000	−0.107058	−0.0535288	+	0.998566i	$0.517047\pi$
$350$	0	0
$351$	−14.2016	−0.758027
$352$	0	0
$353$	24.0000	1.27739	0.638696	−	0.769460i	$-0.279474\pi$
$353$	24.0000	1.27739	0.638696	+	0.769460i	$0.279474\pi$
$354$	0	0
$355$	−7.09017	−0.376307
$356$	0	0
$357$	5.09017	0.269400
$358$	0	0
$359$	18.3607	0.969040	0.484520	−	0.874780i	$-0.338994\pi$
$359$	18.3607	0.969040	0.484520	+	0.874780i	$0.338994\pi$
$360$	0	0
$361$	4.56231	0.240121
$362$	0	0
$363$	2.38197	0.125021
$364$	0	0
$365$	−1.23607	−0.0646988
$366$	0	0
$367$	2.47214	0.129044	0.0645222	−	0.997916i	$-0.479448\pi$
$367$	2.47214	0.129044	0.0645222	+	0.997916i	$0.479448\pi$
$368$	0	0
$369$	32.2705	1.67994
$370$	0	0
$371$	13.7082	0.711694
$372$	0	0
$373$	−2.18034	−0.112894	−0.0564469	−	0.998406i	$-0.517977\pi$
$373$	−2.18034	−0.112894	−0.0564469	+	0.998406i	$0.517977\pi$
$374$	0	0
$375$	0.618034	0.0319151
$376$	0	0
$377$	−19.4853	−1.00354
$378$	0	0
$379$	−33.4508	−1.71825	−0.859127	−	0.511762i	$-0.828993\pi$
$379$	−33.4508	−1.71825	−0.859127	+	0.511762i	$0.828993\pi$
$380$	0	0
$381$	3.81966	0.195687
$382$	0	0
$383$	−17.8885	−0.914062	−0.457031	−	0.889451i	$-0.651087\pi$
$383$	−17.8885	−0.914062	−0.457031	+	0.889451i	$0.651087\pi$
$384$	0	0
$385$	6.23607	0.317819
$386$	0	0
$387$	0	0
$388$	0	0
$389$	5.67376	0.287671	0.143836	−	0.989602i	$-0.454056\pi$
$389$	5.67376	0.287671	0.143836	+	0.989602i	$0.454056\pi$
$390$	0	0
$391$	5.09017	0.257421
$392$	0	0
$393$	9.23607	0.465898
$394$	0	0
$395$	−10.4721	−0.526910
$396$	0	0
$397$	−8.32624	−0.417882	−0.208941	−	0.977928i	$-0.567002\pi$
$397$	−8.32624	−0.417882	−0.208941	+	0.977928i	$0.567002\pi$
$398$	0	0
$399$	−4.85410	−0.243009
$400$	0	0
$401$	11.7082	0.584680	0.292340	−	0.956314i	$-0.405566\pi$
$401$	11.7082	0.584680	0.292340	+	0.956314i	$0.405566\pi$
$402$	0	0
$403$	8.54915	0.425864
$404$	0	0
$405$	5.70820	0.283643
$406$	0	0
$407$	9.52786	0.472279
$408$	0	0
$409$	21.2148	1.04900	0.524502	−	0.851409i	$-0.324252\pi$
$409$	21.2148	1.04900	0.524502	+	0.851409i	$0.324252\pi$
$410$	0	0
$411$	3.29180	0.162372
$412$	0	0
$413$	−18.9443	−0.932187
$414$	0	0
$415$	−10.9443	−0.537233
$416$	0	0
$417$	−10.6525	−0.521654
$418$	0	0
$419$	−5.52786	−0.270054	−0.135027	−	0.990842i	$-0.543112\pi$
$419$	−5.52786	−0.270054	−0.135027	+	0.990842i	$0.543112\pi$
$420$	0	0
$421$	−28.7426	−1.40083	−0.700415	−	0.713735i	$-0.747002\pi$
$421$	−28.7426	−1.40083	−0.700415	+	0.713735i	$0.747002\pi$
$422$	0	0
$423$	25.4164	1.23579
$424$	0	0
$425$	−5.09017	−0.246910
$426$	0	0
$427$	−10.2361	−0.495358
$428$	0	0
$429$	−9.74265	−0.470379
$430$	0	0
$431$	−34.6525	−1.66915	−0.834576	−	0.550894i	$-0.814287\pi$
$431$	−34.6525	−1.66915	−0.834576	+	0.550894i	$0.814287\pi$
$432$	0	0
$433$	−29.5066	−1.41800	−0.708998	−	0.705211i	$-0.750852\pi$
$433$	−29.5066	−1.41800	−0.708998	+	0.705211i	$0.750852\pi$
$434$	0	0
$435$	−2.94427	−0.141167
$436$	0	0
$437$	−4.85410	−0.232203
$438$	0	0
$439$	15.6180	0.745408	0.372704	−	0.927950i	$-0.378431\pi$
$439$	15.6180	0.745408	0.372704	+	0.927950i	$0.378431\pi$
$440$	0	0
$441$	11.4721	0.546292
$442$	0	0
$443$	−13.9098	−0.660876	−0.330438	−	0.943828i	$-0.607196\pi$
$443$	−13.9098	−0.660876	−0.330438	+	0.943828i	$0.607196\pi$
$444$	0	0
$445$	−1.52786	−0.0724277
$446$	0	0
$447$	−0.708204	−0.0334969
$448$	0	0
$449$	18.5623	0.876009	0.438005	−	0.898973i	$-0.355685\pi$
$449$	18.5623	0.876009	0.438005	+	0.898973i	$0.355685\pi$
$450$	0	0
$451$	47.5066	2.23700
$452$	0	0
$453$	−10.8541	−0.509970
$454$	0	0
$455$	−6.61803	−0.310258
$456$	0	0
$457$	33.7771	1.58003	0.790013	−	0.613090i	$-0.210074\pi$
$457$	33.7771	1.58003	0.790013	+	0.613090i	$0.210074\pi$
$458$	0	0
$459$	17.6738	0.824941
$460$	0	0
$461$	−34.7639	−1.61912	−0.809559	−	0.587039i	$-0.800294\pi$
$461$	−34.7639	−1.61912	−0.809559	+	0.587039i	$0.800294\pi$
$462$	0	0
$463$	−2.00000	−0.0929479	−0.0464739	−	0.998920i	$-0.514798\pi$
$463$	−2.00000	−0.0929479	−0.0464739	+	0.998920i	$0.514798\pi$
$464$	0	0
$465$	1.29180	0.0599056
$466$	0	0
$467$	−23.1246	−1.07008	−0.535040	−	0.844827i	$-0.679703\pi$
$467$	−23.1246	−1.07008	−0.535040	+	0.844827i	$0.679703\pi$
$468$	0	0
$469$	8.94427	0.413008
$470$	0	0
$471$	6.00000	0.276465
$472$	0	0
$473$	0	0
$474$	0	0
$475$	4.85410	0.222721
$476$	0	0
$477$	22.1803	1.01557
$478$	0	0
$479$	−3.88854	−0.177672	−0.0888361	−	0.996046i	$-0.528315\pi$
$479$	−3.88854	−0.177672	−0.0888361	+	0.996046i	$0.528315\pi$
$480$	0	0
$481$	−10.1115	−0.461043
$482$	0	0
$483$	1.00000	0.0455016
$484$	0	0
$485$	14.6180	0.663771
$486$	0	0
$487$	42.1803	1.91137	0.955687	−	0.294385i	$-0.0951149\pi$
$487$	42.1803	1.91137	0.955687	+	0.294385i	$0.0951149\pi$
$488$	0	0
$489$	−2.23607	−0.101118
$490$	0	0
$491$	16.1803	0.730209	0.365104	−	0.930967i	$-0.381033\pi$
$491$	16.1803	0.730209	0.365104	+	0.930967i	$0.381033\pi$
$492$	0	0
$493$	24.2492	1.09213
$494$	0	0
$495$	10.0902	0.453519
$496$	0	0
$497$	11.4721	0.514596
$498$	0	0
$499$	32.3607	1.44866	0.724331	−	0.689452i	$-0.242149\pi$
$499$	32.3607	1.44866	0.724331	+	0.689452i	$0.242149\pi$
$500$	0	0
$501$	4.94427	0.220894
$502$	0	0
$503$	−20.6738	−0.921797	−0.460899	−	0.887453i	$-0.652473\pi$
$503$	−20.6738	−0.921797	−0.460899	+	0.887453i	$0.652473\pi$
$504$	0	0
$505$	−13.7082	−0.610007
$506$	0	0
$507$	2.30495	0.102366
$508$	0	0
$509$	5.34752	0.237025	0.118512	−	0.992953i	$-0.462187\pi$
$509$	5.34752	0.237025	0.118512	+	0.992953i	$0.462187\pi$
$510$	0	0
$511$	2.00000	0.0884748
$512$	0	0
$513$	−16.8541	−0.744127
$514$	0	0
$515$	3.56231	0.156974
$516$	0	0
$517$	37.4164	1.64557
$518$	0	0
$519$	−13.3262	−0.584957
$520$	0	0
$521$	−24.4721	−1.07214	−0.536072	−	0.844172i	$-0.680092\pi$
$521$	−24.4721	−1.07214	−0.536072	+	0.844172i	$0.680092\pi$
$522$	0	0
$523$	26.0000	1.13690	0.568450	−	0.822718i	$-0.307543\pi$
$523$	26.0000	1.13690	0.568450	+	0.822718i	$0.307543\pi$
$524$	0	0
$525$	−1.00000	−0.0436436
$526$	0	0
$527$	−10.6393	−0.463456
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	−30.6525	−1.33020
$532$	0	0
$533$	−50.4164	−2.18378
$534$	0	0
$535$	4.18034	0.180732
$536$	0	0
$537$	12.4721	0.538212
$538$	0	0
$539$	16.8885	0.727441
$540$	0	0
$541$	−30.8328	−1.32561	−0.662803	−	0.748794i	$-0.730633\pi$
$541$	−30.8328	−1.32561	−0.662803	+	0.748794i	$0.730633\pi$
$542$	0	0
$543$	−11.6525	−0.500056
$544$	0	0
$545$	8.56231	0.366769
$546$	0	0
$547$	36.9230	1.57871	0.789356	−	0.613935i	$-0.210414\pi$
$547$	36.9230	1.57871	0.789356	+	0.613935i	$0.210414\pi$
$548$	0	0
$549$	−16.5623	−0.706862
$550$	0	0
$551$	−23.1246	−0.985142
$552$	0	0
$553$	16.9443	0.720544
$554$	0	0
$555$	−1.52786	−0.0648542
$556$	0	0
$557$	30.8328	1.30643	0.653214	−	0.757173i	$-0.273420\pi$
$557$	30.8328	1.30643	0.653214	+	0.757173i	$0.273420\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	12.1246	0.511902
$562$	0	0
$563$	−21.8885	−0.922492	−0.461246	−	0.887272i	$-0.652597\pi$
$563$	−21.8885	−0.922492	−0.461246	+	0.887272i	$0.652597\pi$
$564$	0	0
$565$	18.9443	0.796992
$566$	0	0
$567$	−9.23607	−0.387878
$568$	0	0
$569$	−2.00000	−0.0838444	−0.0419222	−	0.999121i	$-0.513348\pi$
$569$	−2.00000	−0.0838444	−0.0419222	+	0.999121i	$0.513348\pi$
$570$	0	0
$571$	30.9787	1.29642	0.648209	−	0.761462i	$-0.275518\pi$
$571$	30.9787	1.29642	0.648209	+	0.761462i	$0.275518\pi$
$572$	0	0
$573$	0.180340	0.00753381
$574$	0	0
$575$	−1.00000	−0.0417029
$576$	0	0
$577$	−12.4721	−0.519222	−0.259611	−	0.965713i	$-0.583594\pi$
$577$	−12.4721	−0.519222	−0.259611	+	0.965713i	$0.583594\pi$
$578$	0	0
$579$	3.23607	0.134486
$580$	0	0
$581$	17.7082	0.734660
$582$	0	0
$583$	32.6525	1.35233
$584$	0	0
$585$	−10.7082	−0.442730
$586$	0	0
$587$	−11.3820	−0.469784	−0.234892	−	0.972021i	$-0.575474\pi$
$587$	−11.3820	−0.469784	−0.234892	+	0.972021i	$0.575474\pi$
$588$	0	0
$589$	10.1459	0.418054
$590$	0	0
$591$	−1.50658	−0.0619723
$592$	0	0
$593$	34.7639	1.42758	0.713792	−	0.700358i	$-0.246976\pi$
$593$	34.7639	1.42758	0.713792	+	0.700358i	$0.246976\pi$
$594$	0	0
$595$	8.23607	0.337646
$596$	0	0
$597$	−1.23607	−0.0505889
$598$	0	0
$599$	20.6180	0.842430	0.421215	−	0.906961i	$-0.361604\pi$
$599$	20.6180	0.842430	0.421215	+	0.906961i	$0.361604\pi$
$600$	0	0
$601$	0.270510	0.0110343	0.00551716	−	0.999985i	$-0.498244\pi$
$601$	0.270510	0.0110343	0.00551716	+	0.999985i	$0.498244\pi$
$602$	0	0
$603$	14.4721	0.589351
$604$	0	0
$605$	3.85410	0.156692
$606$	0	0
$607$	−17.5279	−0.711434	−0.355717	−	0.934594i	$-0.615763\pi$
$607$	−17.5279	−0.711434	−0.355717	+	0.934594i	$0.615763\pi$
$608$	0	0
$609$	4.76393	0.193044
$610$	0	0
$611$	−39.7082	−1.60642
$612$	0	0
$613$	−43.3050	−1.74907	−0.874535	−	0.484962i	$-0.838833\pi$
$613$	−43.3050	−1.74907	−0.874535	+	0.484962i	$0.838833\pi$
$614$	0	0
$615$	−7.61803	−0.307189
$616$	0	0
$617$	22.9098	0.922315	0.461158	−	0.887318i	$-0.347434\pi$
$617$	22.9098	0.922315	0.461158	+	0.887318i	$0.347434\pi$
$618$	0	0
$619$	−21.7984	−0.876151	−0.438075	−	0.898938i	$-0.644340\pi$
$619$	−21.7984	−0.876151	−0.438075	+	0.898938i	$0.644340\pi$
$620$	0	0
$621$	3.47214	0.139332
$622$	0	0
$623$	2.47214	0.0990440
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−11.5623	−0.461754
$628$	0	0
$629$	12.5836	0.501741
$630$	0	0
$631$	16.0689	0.639692	0.319846	−	0.947470i	$-0.396369\pi$
$631$	16.0689	0.639692	0.319846	+	0.947470i	$0.396369\pi$
$632$	0	0
$633$	−8.65248	−0.343905
$634$	0	0
$635$	6.18034	0.245259
$636$	0	0
$637$	−17.9230	−0.710135
$638$	0	0
$639$	18.5623	0.734313
$640$	0	0
$641$	−44.3607	−1.75214	−0.876071	−	0.482183i	$-0.839844\pi$
$641$	−44.3607	−1.75214	−0.876071	+	0.482183i	$0.839844\pi$
$642$	0	0
$643$	21.7082	0.856088	0.428044	−	0.903758i	$-0.359203\pi$
$643$	21.7082	0.856088	0.428044	+	0.903758i	$0.359203\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−44.2492	−1.73962	−0.869808	−	0.493390i	$-0.835758\pi$
$647$	−44.2492	−1.73962	−0.869808	+	0.493390i	$0.835758\pi$
$648$	0	0
$649$	−45.1246	−1.77130
$650$	0	0
$651$	−2.09017	−0.0819202
$652$	0	0
$653$	21.0344	0.823141	0.411571	−	0.911378i	$-0.364980\pi$
$653$	21.0344	0.823141	0.411571	+	0.911378i	$0.364980\pi$
$654$	0	0
$655$	14.9443	0.583921
$656$	0	0
$657$	3.23607	0.126251
$658$	0	0
$659$	−34.2492	−1.33416	−0.667080	−	0.744986i	$-0.732456\pi$
$659$	−34.2492	−1.33416	−0.667080	+	0.744986i	$0.732456\pi$
$660$	0	0
$661$	34.3262	1.33514	0.667568	−	0.744549i	$-0.267335\pi$
$661$	34.3262	1.33514	0.667568	+	0.744549i	$0.267335\pi$
$662$	0	0
$663$	−12.8673	−0.499723
$664$	0	0
$665$	−7.85410	−0.304569
$666$	0	0
$667$	4.76393	0.184460
$668$	0	0
$669$	1.88854	0.0730153
$670$	0	0
$671$	−24.3820	−0.941255
$672$	0	0
$673$	6.94427	0.267682	0.133841	−	0.991003i	$-0.457269\pi$
$673$	6.94427	0.267682	0.133841	+	0.991003i	$0.457269\pi$
$674$	0	0
$675$	−3.47214	−0.133643
$676$	0	0
$677$	−33.0557	−1.27043	−0.635217	−	0.772333i	$-0.719090\pi$
$677$	−33.0557	−1.27043	−0.635217	+	0.772333i	$0.719090\pi$
$678$	0	0
$679$	−23.6525	−0.907699
$680$	0	0
$681$	14.3607	0.550302
$682$	0	0
$683$	11.4377	0.437651	0.218826	−	0.975764i	$-0.429777\pi$
$683$	11.4377	0.437651	0.218826	+	0.975764i	$0.429777\pi$
$684$	0	0
$685$	5.32624	0.203505
$686$	0	0
$687$	6.18034	0.235795
$688$	0	0
$689$	−34.6525	−1.32015
$690$	0	0
$691$	24.7639	0.942064	0.471032	−	0.882116i	$-0.343882\pi$
$691$	24.7639	0.942064	0.471032	+	0.882116i	$0.343882\pi$
$692$	0	0
$693$	−16.3262	−0.620182
$694$	0	0
$695$	−17.2361	−0.653801
$696$	0	0
$697$	62.7426	2.37655
$698$	0	0
$699$	12.1803	0.460703
$700$	0	0
$701$	−48.3394	−1.82575	−0.912877	−	0.408235i	$-0.866144\pi$
$701$	−48.3394	−1.82575	−0.912877	+	0.408235i	$0.866144\pi$
$702$	0	0
$703$	−12.0000	−0.452589
$704$	0	0
$705$	−6.00000	−0.225973
$706$	0	0
$707$	22.1803	0.834178
$708$	0	0
$709$	−14.9098	−0.559950	−0.279975	−	0.960007i	$-0.590326\pi$
$709$	−14.9098	−0.559950	−0.279975	+	0.960007i	$0.590326\pi$
$710$	0	0
$711$	27.4164	1.02820
$712$	0	0
$713$	−2.09017	−0.0782775
$714$	0	0
$715$	−15.7639	−0.589538
$716$	0	0
$717$	−15.0557	−0.562266
$718$	0	0
$719$	−1.72949	−0.0644991	−0.0322495	−	0.999480i	$-0.510267\pi$
$719$	−1.72949	−0.0644991	−0.0322495	+	0.999480i	$0.510267\pi$
$720$	0	0
$721$	−5.76393	−0.214660
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−4.76393	−0.176928
$726$	0	0
$727$	−52.7984	−1.95818	−0.979092	−	0.203420i	$-0.934794\pi$
$727$	−52.7984	−1.95818	−0.979092	+	0.203420i	$0.934794\pi$
$728$	0	0
$729$	−8.50658	−0.315058
$730$	0	0
$731$	0	0
$732$	0	0
$733$	−2.58359	−0.0954272	−0.0477136	−	0.998861i	$-0.515193\pi$
$733$	−2.58359	−0.0954272	−0.0477136	+	0.998861i	$0.515193\pi$
$734$	0	0
$735$	−2.70820	−0.0998936
$736$	0	0
$737$	21.3050	0.784778
$738$	0	0
$739$	−21.8885	−0.805183	−0.402592	−	0.915380i	$-0.631890\pi$
$739$	−21.8885	−0.805183	−0.402592	+	0.915380i	$0.631890\pi$
$740$	0	0
$741$	12.2705	0.450768
$742$	0	0
$743$	44.6312	1.63736	0.818680	−	0.574250i	$-0.194706\pi$
$743$	44.6312	1.63736	0.818680	+	0.574250i	$0.194706\pi$
$744$	0	0
$745$	−1.14590	−0.0419825
$746$	0	0
$747$	28.6525	1.04834
$748$	0	0
$749$	−6.76393	−0.247149
$750$	0	0
$751$	−29.0132	−1.05871	−0.529353	−	0.848402i	$-0.677565\pi$
$751$	−29.0132	−1.05871	−0.529353	+	0.848402i	$0.677565\pi$
$752$	0	0
$753$	−7.94427	−0.289505
$754$	0	0
$755$	−17.5623	−0.639158
$756$	0	0
$757$	17.8885	0.650170	0.325085	−	0.945685i	$-0.394607\pi$
$757$	17.8885	0.650170	0.325085	+	0.945685i	$0.394607\pi$
$758$	0	0
$759$	2.38197	0.0864599
$760$	0	0
$761$	−35.8673	−1.30019	−0.650094	−	0.759854i	$-0.725270\pi$
$761$	−35.8673	−1.30019	−0.650094	+	0.759854i	$0.725270\pi$
$762$	0	0
$763$	−13.8541	−0.501552
$764$	0	0
$765$	13.3262	0.481811
$766$	0	0
$767$	47.8885	1.72916
$768$	0	0
$769$	−33.4164	−1.20503	−0.602513	−	0.798109i	$-0.705834\pi$
$769$	−33.4164	−1.20503	−0.602513	+	0.798109i	$0.705834\pi$
$770$	0	0
$771$	−18.6525	−0.671753
$772$	0	0
$773$	11.0557	0.397647	0.198823	−	0.980035i	$-0.436288\pi$
$773$	11.0557	0.397647	0.198823	+	0.980035i	$0.436288\pi$
$774$	0	0
$775$	2.09017	0.0750811
$776$	0	0
$777$	2.47214	0.0886874
$778$	0	0
$779$	−59.8328	−2.14373
$780$	0	0
$781$	27.3262	0.977810
$782$	0	0
$783$	16.5410	0.591128
$784$	0	0
$785$	9.70820	0.346501
$786$	0	0
$787$	43.1246	1.53723	0.768613	−	0.639714i	$-0.220947\pi$
$787$	43.1246	1.53723	0.768613	+	0.639714i	$0.220947\pi$
$788$	0	0
$789$	13.4377	0.478395
$790$	0	0
$791$	−30.6525	−1.08988
$792$	0	0
$793$	25.8754	0.918862
$794$	0	0
$795$	−5.23607	−0.185704
$796$	0	0
$797$	0.291796	0.0103359	0.00516797	−	0.999987i	$-0.498355\pi$
$797$	0.291796	0.0103359	0.00516797	+	0.999987i	$0.498355\pi$
$798$	0	0
$799$	49.4164	1.74823
$800$	0	0
$801$	4.00000	0.141333
$802$	0	0
$803$	4.76393	0.168116
$804$	0	0
$805$	1.61803	0.0570282
$806$	0	0
$807$	5.05573	0.177970
$808$	0	0
$809$	−4.25735	−0.149681	−0.0748403	−	0.997196i	$-0.523845\pi$
$809$	−4.25735	−0.149681	−0.0748403	+	0.997196i	$0.523845\pi$
$810$	0	0
$811$	−44.1803	−1.55138	−0.775691	−	0.631113i	$-0.782598\pi$
$811$	−44.1803	−1.55138	−0.775691	+	0.631113i	$0.782598\pi$
$812$	0	0
$813$	9.06888	0.318060
$814$	0	0
$815$	−3.61803	−0.126734
$816$	0	0
$817$	0	0
$818$	0	0
$819$	17.3262	0.605428
$820$	0	0
$821$	−50.9443	−1.77797	−0.888984	−	0.457939i	$-0.848588\pi$
$821$	−50.9443	−1.77797	−0.888984	+	0.457939i	$0.848588\pi$
$822$	0	0
$823$	1.41641	0.0493729	0.0246864	−	0.999695i	$-0.492141\pi$
$823$	1.41641	0.0493729	0.0246864	+	0.999695i	$0.492141\pi$
$824$	0	0
$825$	−2.38197	−0.0829294
$826$	0	0
$827$	−8.29180	−0.288334	−0.144167	−	0.989553i	$-0.546050\pi$
$827$	−8.29180	−0.288334	−0.144167	+	0.989553i	$0.546050\pi$
$828$	0	0
$829$	1.05573	0.0366670	0.0183335	−	0.999832i	$-0.494164\pi$
$829$	1.05573	0.0366670	0.0183335	+	0.999832i	$0.494164\pi$
$830$	0	0
$831$	1.59675	0.0553906
$832$	0	0
$833$	22.3050	0.772821
$834$	0	0
$835$	8.00000	0.276851
$836$	0	0
$837$	−7.25735	−0.250851
$838$	0	0
$839$	43.0132	1.48498	0.742490	−	0.669858i	$-0.233645\pi$
$839$	43.0132	1.48498	0.742490	+	0.669858i	$0.233645\pi$
$840$	0	0
$841$	−6.30495	−0.217412
$842$	0	0
$843$	16.8328	0.579753
$844$	0	0
$845$	3.72949	0.128298
$846$	0	0
$847$	−6.23607	−0.214274
$848$	0	0
$849$	5.59675	0.192080
$850$	0	0
$851$	2.47214	0.0847437
$852$	0	0
$853$	13.7984	0.472447	0.236224	−	0.971699i	$-0.424090\pi$
$853$	13.7984	0.472447	0.236224	+	0.971699i	$0.424090\pi$
$854$	0	0
$855$	−12.7082	−0.434611
$856$	0	0
$857$	−33.4164	−1.14148	−0.570741	−	0.821130i	$-0.693344\pi$
$857$	−33.4164	−1.14148	−0.570741	+	0.821130i	$0.693344\pi$
$858$	0	0
$859$	−34.0689	−1.16242	−0.581208	−	0.813755i	$-0.697420\pi$
$859$	−34.0689	−1.16242	−0.581208	+	0.813755i	$0.697420\pi$
$860$	0	0
$861$	12.3262	0.420077
$862$	0	0
$863$	−37.2361	−1.26753	−0.633765	−	0.773525i	$-0.718491\pi$
$863$	−37.2361	−1.26753	−0.633765	+	0.773525i	$0.718491\pi$
$864$	0	0
$865$	−21.5623	−0.733140
$866$	0	0
$867$	5.50658	0.187013
$868$	0	0
$869$	40.3607	1.36914
$870$	0	0
$871$	−22.6099	−0.766107
$872$	0	0
$873$	−38.2705	−1.29526
$874$	0	0
$875$	−1.61803	−0.0546995
$876$	0	0
$877$	−23.7426	−0.801732	−0.400866	−	0.916137i	$-0.631291\pi$
$877$	−23.7426	−0.801732	−0.400866	+	0.916137i	$0.631291\pi$
$878$	0	0
$879$	9.81966	0.331209
$880$	0	0
$881$	35.4164	1.19321	0.596605	−	0.802535i	$-0.296516\pi$
$881$	35.4164	1.19321	0.596605	+	0.802535i	$0.296516\pi$
$882$	0	0
$883$	4.56231	0.153534	0.0767669	−	0.997049i	$-0.475540\pi$
$883$	4.56231	0.153534	0.0767669	+	0.997049i	$0.475540\pi$
$884$	0	0
$885$	7.23607	0.243238
$886$	0	0
$887$	58.8328	1.97541	0.987706	−	0.156321i	$-0.0499634\pi$
$887$	58.8328	1.97541	0.987706	+	0.156321i	$0.0499634\pi$
$888$	0	0
$889$	−10.0000	−0.335389
$890$	0	0
$891$	−22.0000	−0.737028
$892$	0	0
$893$	−47.1246	−1.57697
$894$	0	0
$895$	20.1803	0.674554
$896$	0	0
$897$	−2.52786	−0.0844029
$898$	0	0
$899$	−9.95743	−0.332099
$900$	0	0
$901$	43.1246	1.43669
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−18.8541	−0.626732
$906$	0	0
$907$	33.1246	1.09988	0.549942	−	0.835203i	$-0.314650\pi$
$907$	33.1246	1.09988	0.549942	+	0.835203i	$0.314650\pi$
$908$	0	0
$909$	35.8885	1.19035
$910$	0	0
$911$	22.0689	0.731175	0.365587	−	0.930777i	$-0.380868\pi$
$911$	22.0689	0.731175	0.365587	+	0.930777i	$0.380868\pi$
$912$	0	0
$913$	42.1803	1.39597
$914$	0	0
$915$	3.90983	0.129255
$916$	0	0
$917$	−24.1803	−0.798505
$918$	0	0
$919$	−40.0000	−1.31948	−0.659739	−	0.751495i	$-0.729333\pi$
$919$	−40.0000	−1.31948	−0.659739	+	0.751495i	$0.729333\pi$
$920$	0	0
$921$	−16.9656	−0.559034
$922$	0	0
$923$	−29.0000	−0.954547
$924$	0	0
$925$	−2.47214	−0.0812833
$926$	0	0
$927$	−9.32624	−0.306314
$928$	0	0
$929$	−12.4721	−0.409198	−0.204599	−	0.978846i	$-0.565589\pi$
$929$	−12.4721	−0.409198	−0.204599	+	0.978846i	$0.565589\pi$
$930$	0	0
$931$	−21.2705	−0.697113
$932$	0	0
$933$	−2.47214	−0.0809341
$934$	0	0
$935$	19.6180	0.641578
$936$	0	0
$937$	−12.2016	−0.398610	−0.199305	−	0.979938i	$-0.563868\pi$
$937$	−12.2016	−0.398610	−0.199305	+	0.979938i	$0.563868\pi$
$938$	0	0
$939$	−7.29180	−0.237959
$940$	0	0
$941$	−60.5066	−1.97246	−0.986229	−	0.165385i	$-0.947113\pi$
$941$	−60.5066	−1.97246	−0.986229	+	0.165385i	$0.947113\pi$
$942$	0	0
$943$	12.3262	0.401398
$944$	0	0
$945$	5.61803	0.182755
$946$	0	0
$947$	−5.68692	−0.184800	−0.0924000	−	0.995722i	$-0.529454\pi$
$947$	−5.68692	−0.184800	−0.0924000	+	0.995722i	$0.529454\pi$
$948$	0	0
$949$	−5.05573	−0.164116
$950$	0	0
$951$	0.0557281	0.00180711
$952$	0	0
$953$	−20.7984	−0.673725	−0.336863	−	0.941554i	$-0.609366\pi$
$953$	−20.7984	−0.673725	−0.336863	+	0.941554i	$0.609366\pi$
$954$	0	0
$955$	0.291796	0.00944230
$956$	0	0
$957$	11.3475	0.366813
$958$	0	0
$959$	−8.61803	−0.278291
$960$	0	0
$961$	−26.6312	−0.859071
$962$	0	0
$963$	−10.9443	−0.352674
$964$	0	0
$965$	5.23607	0.168555
$966$	0	0
$967$	−50.5410	−1.62529	−0.812645	−	0.582759i	$-0.801973\pi$
$967$	−50.5410	−1.62529	−0.812645	+	0.582759i	$0.801973\pi$
$968$	0	0
$969$	−15.2705	−0.490559
$970$	0	0
$971$	−0.729490	−0.0234105	−0.0117052	−	0.999931i	$-0.503726\pi$
$971$	−0.729490	−0.0234105	−0.0117052	+	0.999931i	$0.503726\pi$
$972$	0	0
$973$	27.8885	0.894066
$974$	0	0
$975$	2.52786	0.0809564
$976$	0	0
$977$	−3.43769	−0.109982	−0.0549908	−	0.998487i	$-0.517513\pi$
$977$	−3.43769	−0.109982	−0.0549908	+	0.998487i	$0.517513\pi$
$978$	0	0
$979$	5.88854	0.188199
$980$	0	0
$981$	−22.4164	−0.715701
$982$	0	0
$983$	−19.2705	−0.614634	−0.307317	−	0.951607i	$-0.599431\pi$
$983$	−19.2705	−0.614634	−0.307317	+	0.951607i	$0.599431\pi$
$984$	0	0
$985$	−2.43769	−0.0776714
$986$	0	0
$987$	9.70820	0.309016
$988$	0	0
$989$	0	0
$990$	0	0
$991$	10.5066	0.333752	0.166876	−	0.985978i	$-0.446632\pi$
$991$	10.5066	0.333752	0.166876	+	0.985978i	$0.446632\pi$
$992$	0	0
$993$	−9.12461	−0.289561
$994$	0	0
$995$	−2.00000	−0.0634043
$996$	0	0
$997$	−41.1935	−1.30461	−0.652306	−	0.757956i	$-0.726198\pi$
$997$	−41.1935	−1.30461	−0.652306	+	0.757956i	$0.726198\pi$
$998$	0	0
$999$	8.58359	0.271573

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1840.2.a.l.1.2		2
4.3	odd	2		230.2.a.c.1.1	✓	2
5.4	even	2		9200.2.a.bu.1.1		2
8.3	odd	2		7360.2.a.bh.1.2		2
8.5	even	2		7360.2.a.bn.1.1		2
12.11	even	2		2070.2.a.u.1.2		2
20.3	even	4		1150.2.b.i.599.1		4
20.7	even	4		1150.2.b.i.599.4		4
20.19	odd	2		1150.2.a.j.1.2		2
92.91	even	2		5290.2.a.o.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
230.2.a.c.1.1	✓	2	4.3	odd	2
1150.2.a.j.1.2		2	20.19	odd	2
1150.2.b.i.599.1		4	20.3	even	4
1150.2.b.i.599.4		4	20.7	even	4
1840.2.a.l.1.2		2	1.1	even	1	trivial
2070.2.a.u.1.2		2	12.11	even	2
5290.2.a.o.1.1		2	92.91	even	2
7360.2.a.bh.1.2		2	8.3	odd	2
7360.2.a.bn.1.1		2	8.5	even	2
9200.2.a.bu.1.1		2	5.4	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 1840 = 2^{4} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1840.a (trivial)

\(f(q)\)	\(=\)	\(q+0.618034 q^{3} +1.00000 q^{5} -1.61803 q^{7} -2.61803 q^{9} +O(q^{10})\)	\(q+0.618034 q^{3} +1.00000 q^{5} -1.61803 q^{7} -2.61803 q^{9} -3.85410 q^{11} +4.09017 q^{13} +0.618034 q^{15} -5.09017 q^{17} +4.85410 q^{19} -1.00000 q^{21} -1.00000 q^{23} +1.00000 q^{25} -3.47214 q^{27} -4.76393 q^{29} +2.09017 q^{31} -2.38197 q^{33} -1.61803 q^{35} -2.47214 q^{37} +2.52786 q^{39} -12.3262 q^{41} -2.61803 q^{45} -9.70820 q^{47} -4.38197 q^{49} -3.14590 q^{51} -8.47214 q^{53} -3.85410 q^{55} +3.00000 q^{57} +11.7082 q^{59} +6.32624 q^{61} +4.23607 q^{63} +4.09017 q^{65} -5.52786 q^{67} -0.618034 q^{69} -7.09017 q^{71} -1.23607 q^{73} +0.618034 q^{75} +6.23607 q^{77} -10.4721 q^{79} +5.70820 q^{81} -10.9443 q^{83} -5.09017 q^{85} -2.94427 q^{87} -1.52786 q^{89} -6.61803 q^{91} +1.29180 q^{93} +4.85410 q^{95} +14.6180 q^{97} +10.0902 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{3} + 2 q^{5} - q^{7} - 3 q^{9}+O(q^{10}) \) 2 * q - q^3 + 2 * q^5 - q^7 - 3 * q^9	\( 2 q - q^{3} + 2 q^{5} - q^{7} - 3 q^{9} - q^{11} - 3 q^{13} - q^{15} + q^{17} + 3 q^{19} - 2 q^{21} - 2 q^{23} + 2 q^{25} + 2 q^{27} - 14 q^{29} - 7 q^{31} - 7 q^{33} - q^{35} + 4 q^{37} + 14 q^{39} - 9 q^{41} - 3 q^{45} - 6 q^{47} - 11 q^{49} - 13 q^{51} - 8 q^{53} - q^{55} + 6 q^{57} + 10 q^{59} - 3 q^{61} + 4 q^{63} - 3 q^{65} - 20 q^{67} + q^{69} - 3 q^{71} + 2 q^{73} - q^{75} + 8 q^{77} - 12 q^{79} - 2 q^{81} - 4 q^{83} + q^{85} + 12 q^{87} - 12 q^{89} - 11 q^{91} + 16 q^{93} + 3 q^{95} + 27 q^{97} + 9 q^{99}+O(q^{100}) \) 2 * q - q^3 + 2 * q^5 - q^7 - 3 * q^9 - q^11 - 3 * q^13 - q^15 + q^17 + 3 * q^19 - 2 * q^21 - 2 * q^23 + 2 * q^25 + 2 * q^27 - 14 * q^29 - 7 * q^31 - 7 * q^33 - q^35 + 4 * q^37 + 14 * q^39 - 9 * q^41 - 3 * q^45 - 6 * q^47 - 11 * q^49 - 13 * q^51 - 8 * q^53 - q^55 + 6 * q^57 + 10 * q^59 - 3 * q^61 + 4 * q^63 - 3 * q^65 - 20 * q^67 + q^69 - 3 * q^71 + 2 * q^73 - q^75 + 8 * q^77 - 12 * q^79 - 2 * q^81 - 4 * q^83 + q^85 + 12 * q^87 - 12 * q^89 - 11 * q^91 + 16 * q^93 + 3 * q^95 + 27 * q^97 + 9 * q^99

Introduction

L-functions

Modular forms

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