LMFDB - Embedded newform 1840.2.a.u.1.3

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:	$0$
Dimension:	$4$
Coefficient field:	4.4.15317.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 4x^{2} + 5x + 2 $ x^4 - 2x^3 - 4x^2 + 5*x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 115)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$2.69353$ 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)

$f(q)$	$=$	$q+2.56155 q^{3} +1.00000 q^{5} +0.819031 q^{7} +3.56155 q^{9} +O(q^{10})$	$q+2.56155 q^{3} +1.00000 q^{5} +0.819031 q^{7} +3.56155 q^{9} -5.38705 q^{11} +2.46356 q^{13} +2.56155 q^{15} +4.20608 q^{17} +5.38705 q^{19} +2.09799 q^{21} +1.00000 q^{23} +1.00000 q^{25} +1.43845 q^{27} +2.35547 q^{29} -6.66964 q^{31} -13.7992 q^{33} +0.819031 q^{35} +7.42718 q^{37} +6.31054 q^{39} +5.64453 q^{41} +1.90201 q^{43} +3.56155 q^{45} +9.33565 q^{47} -6.32919 q^{49} +10.7741 q^{51} +13.8571 q^{53} -5.38705 q^{55} +13.7992 q^{57} -5.27896 q^{59} -8.51016 q^{61} +2.91702 q^{63} +2.46356 q^{65} -4.10809 q^{67} +2.56155 q^{69} +11.7927 q^{71} -10.4636 q^{73} +2.56155 q^{75} -4.41216 q^{77} -13.7992 q^{79} -7.00000 q^{81} -3.32919 q^{83} +4.20608 q^{85} +6.03366 q^{87} -3.58304 q^{89} +2.01773 q^{91} -17.0846 q^{93} +5.38705 q^{95} -9.02511 q^{97} -19.1863 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{3} + 4 q^{5} + 3 q^{7} + 6 q^{9}+O(q^{10}) $ 4 * q + 2 * q^3 + 4 * q^5 + 3 * q^7 + 6 * q^9	$ 4 q + 2 q^{3} + 4 q^{5} + 3 q^{7} + 6 q^{9} - 4 q^{11} + 2 q^{15} - q^{17} + 4 q^{19} + 10 q^{21} + 4 q^{23} + 4 q^{25} + 14 q^{27} + 19 q^{29} + q^{31} - 2 q^{33} + 3 q^{35} - 3 q^{37} + 13 q^{41} + 6 q^{43} + 6 q^{45} - 6 q^{47} + 9 q^{49} + 8 q^{51} + 19 q^{53} - 4 q^{55} + 2 q^{57} - 23 q^{59} + 13 q^{63} + 3 q^{67} + 2 q^{69} + 3 q^{71} - 32 q^{73} + 2 q^{75} + 18 q^{77} - 2 q^{79} - 28 q^{81} + 21 q^{83} - q^{85} + 18 q^{87} + 40 q^{91} - 8 q^{93} + 4 q^{95} - 18 q^{97} - 6 q^{99}+O(q^{100}) $ 4 * q + 2 * q^3 + 4 * q^5 + 3 * q^7 + 6 * q^9 - 4 * q^11 + 2 * q^15 - q^17 + 4 * q^19 + 10 * q^21 + 4 * q^23 + 4 * q^25 + 14 * q^27 + 19 * q^29 + q^31 - 2 * q^33 + 3 * q^35 - 3 * q^37 + 13 * q^41 + 6 * q^43 + 6 * q^45 - 6 * q^47 + 9 * q^49 + 8 * q^51 + 19 * q^53 - 4 * q^55 + 2 * q^57 - 23 * q^59 + 13 * q^63 + 3 * q^67 + 2 * q^69 + 3 * q^71 - 32 * q^73 + 2 * q^75 + 18 * q^77 - 2 * q^79 - 28 * q^81 + 21 * q^83 - q^85 + 18 * q^87 + 40 * q^91 - 8 * q^93 + 4 * q^95 - 18 * q^97 - 6 * 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
$3$	2.56155	1.47891	0.739457	−	0.673204i	$-0.235083\pi$
$3$	2.56155	1.47891	0.739457	+	0.673204i	$0.235083\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	0.819031	0.309565	0.154782	−	0.987949i	$-0.450532\pi$
$7$	0.819031	0.309565	0.154782	+	0.987949i	$0.450532\pi$
$8$	0	0
$9$	3.56155	1.18718
$10$	0	0
$11$	−5.38705	−1.62426	−0.812128	−	0.583479i	$-0.801691\pi$
$11$	−5.38705	−1.62426	−0.812128	+	0.583479i	$0.801691\pi$
$12$	0	0
$13$	2.46356	0.683269	0.341634	−	0.939833i	$-0.389020\pi$
$13$	2.46356	0.683269	0.341634	+	0.939833i	$0.389020\pi$
$14$	0	0
$15$	2.56155	0.661390
$16$	0	0
$17$	4.20608	1.02012	0.510062	−	0.860137i	$-0.329622\pi$
$17$	4.20608	1.02012	0.510062	+	0.860137i	$0.329622\pi$
$18$	0	0
$19$	5.38705	1.23587	0.617937	−	0.786228i	$-0.287969\pi$
$19$	5.38705	1.23587	0.617937	+	0.786228i	$0.287969\pi$
$20$	0	0
$21$	2.09799	0.457819
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	1.43845	0.276829
$28$	0	0
$29$	2.35547	0.437400	0.218700	−	0.975792i	$-0.429818\pi$
$29$	2.35547	0.437400	0.218700	+	0.975792i	$0.429818\pi$
$30$	0	0
$31$	−6.66964	−1.19790	−0.598952	−	0.800785i	$-0.704416\pi$
$31$	−6.66964	−1.19790	−0.598952	+	0.800785i	$0.704416\pi$
$32$	0	0
$33$	−13.7992	−2.40213
$34$	0	0
$35$	0.819031	0.138442
$36$	0	0
$37$	7.42718	1.22102	0.610510	−	0.792008i	$-0.290964\pi$
$37$	7.42718	1.22102	0.610510	+	0.792008i	$0.290964\pi$
$38$	0	0
$39$	6.31054	1.01050
$40$	0	0
$41$	5.64453	0.881527	0.440764	−	0.897623i	$-0.354708\pi$
$41$	5.64453	0.881527	0.440764	+	0.897623i	$0.354708\pi$
$42$	0	0
$43$	1.90201	0.290053	0.145027	−	0.989428i	$-0.453673\pi$
$43$	1.90201	0.290053	0.145027	+	0.989428i	$0.453673\pi$
$44$	0	0
$45$	3.56155	0.530925
$46$	0	0
$47$	9.33565	1.36175	0.680873	−	0.732402i	$-0.261601\pi$
$47$	9.33565	1.36175	0.680873	+	0.732402i	$0.261601\pi$
$48$	0	0
$49$	−6.32919	−0.904170
$50$	0	0
$51$	10.7741	1.50868
$52$	0	0
$53$	13.8571	1.90342	0.951708	−	0.307005i	$-0.0993268\pi$
$53$	13.8571	1.90342	0.951708	+	0.307005i	$0.0993268\pi$
$54$	0	0
$55$	−5.38705	−0.726390
$56$	0	0
$57$	13.7992	1.82775
$58$	0	0
$59$	−5.27896	−0.687262	−0.343631	−	0.939105i	$-0.611657\pi$
$59$	−5.27896	−0.687262	−0.343631	+	0.939105i	$0.611657\pi$
$60$	0	0
$61$	−8.51016	−1.08961	−0.544807	−	0.838562i	$-0.683397\pi$
$61$	−8.51016	−1.08961	−0.544807	+	0.838562i	$0.683397\pi$
$62$	0	0
$63$	2.91702	0.367510
$64$	0	0
$65$	2.46356	0.305567
$66$	0	0
$67$	−4.10809	−0.501883	−0.250942	−	0.968002i	$-0.580740\pi$
$67$	−4.10809	−0.501883	−0.250942	+	0.968002i	$0.580740\pi$
$68$	0	0
$69$	2.56155	0.308375
$70$	0	0
$71$	11.7927	1.39954	0.699771	−	0.714367i	$-0.253285\pi$
$71$	11.7927	1.39954	0.699771	+	0.714367i	$0.253285\pi$
$72$	0	0
$73$	−10.4636	−1.22467	−0.612334	−	0.790600i	$-0.709769\pi$
$73$	−10.4636	−1.22467	−0.612334	+	0.790600i	$0.709769\pi$
$74$	0	0
$75$	2.56155	0.295783
$76$	0	0
$77$	−4.41216	−0.502813
$78$	0	0
$79$	−13.7992	−1.55253	−0.776266	−	0.630405i	$-0.782889\pi$
$79$	−13.7992	−1.55253	−0.776266	+	0.630405i	$0.782889\pi$
$80$	0	0
$81$	−7.00000	−0.777778
$82$	0	0
$83$	−3.32919	−0.365426	−0.182713	−	0.983166i	$-0.558488\pi$
$83$	−3.32919	−0.365426	−0.182713	+	0.983166i	$0.558488\pi$
$84$	0	0
$85$	4.20608	0.456214
$86$	0	0
$87$	6.03366	0.646877
$88$	0	0
$89$	−3.58304	−0.379801	−0.189900	−	0.981803i	$-0.560817\pi$
$89$	−3.58304	−0.379801	−0.189900	+	0.981803i	$0.560817\pi$
$90$	0	0
$91$	2.01773	0.211516
$92$	0	0
$93$	−17.0846	−1.77159
$94$	0	0
$95$	5.38705	0.552700
$96$	0	0
$97$	−9.02511	−0.916361	−0.458181	−	0.888859i	$-0.651499\pi$
$97$	−9.02511	−0.916361	−0.458181	+	0.888859i	$0.651499\pi$
$98$	0	0
$99$	−19.1863	−1.92829
$100$	0	0
$101$	−12.1033	−1.20432	−0.602161	−	0.798375i	$-0.705694\pi$
$101$	−12.1033	−1.20432	−0.602161	+	0.798375i	$0.705694\pi$
$102$	0	0
$103$	3.80402	0.374821	0.187410	−	0.982282i	$-0.439991\pi$
$103$	3.80402	0.374821	0.187410	+	0.982282i	$0.439991\pi$
$104$	0	0
$105$	2.09799	0.204743
$106$	0	0
$107$	2.47003	0.238786	0.119393	−	0.992847i	$-0.461905\pi$
$107$	2.47003	0.238786	0.119393	+	0.992847i	$0.461905\pi$
$108$	0	0
$109$	−4.49722	−0.430756	−0.215378	−	0.976531i	$-0.569098\pi$
$109$	−4.49722	−0.430756	−0.215378	+	0.976531i	$0.569098\pi$
$110$	0	0
$111$	19.0251	1.80578
$112$	0	0
$113$	6.30407	0.593037	0.296519	−	0.955027i	$-0.404174\pi$
$113$	6.30407	0.593037	0.296519	+	0.955027i	$0.404174\pi$
$114$	0	0
$115$	1.00000	0.0932505
$116$	0	0
$117$	8.77410	0.811166
$118$	0	0
$119$	3.44491	0.315795
$120$	0	0
$121$	18.0203	1.63821
$122$	0	0
$123$	14.4588	1.30370
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	0.910558	0.0807989	0.0403995	−	0.999184i	$-0.487137\pi$
$127$	0.910558	0.0807989	0.0403995	+	0.999184i	$0.487137\pi$
$128$	0	0
$129$	4.87209	0.428964
$130$	0	0
$131$	−18.6070	−1.62570	−0.812850	−	0.582474i	$-0.802085\pi$
$131$	−18.6070	−1.62570	−0.812850	+	0.582474i	$0.802085\pi$
$132$	0	0
$133$	4.41216	0.382583
$134$	0	0
$135$	1.43845	0.123802
$136$	0	0
$137$	−14.4251	−1.23242	−0.616210	−	0.787582i	$-0.711333\pi$
$137$	−14.4251	−1.23242	−0.616210	+	0.787582i	$0.711333\pi$
$138$	0	0
$139$	−18.3507	−1.55648	−0.778242	−	0.627965i	$-0.783888\pi$
$139$	−18.3507	−1.55648	−0.778242	+	0.627965i	$0.783888\pi$
$140$	0	0
$141$	23.9138	2.01390
$142$	0	0
$143$	−13.2713	−1.10980
$144$	0	0
$145$	2.35547	0.195611
$146$	0	0
$147$	−16.2125	−1.33719
$148$	0	0
$149$	7.63806	0.625734	0.312867	−	0.949797i	$-0.398711\pi$
$149$	7.63806	0.625734	0.312867	+	0.949797i	$0.398711\pi$
$150$	0	0
$151$	0.659545	0.0536730	0.0268365	−	0.999640i	$-0.491457\pi$
$151$	0.659545	0.0536730	0.0268365	+	0.999640i	$0.491457\pi$
$152$	0	0
$153$	14.9802	1.21108
$154$	0	0
$155$	−6.66964	−0.535719
$156$	0	0
$157$	3.94214	0.314617	0.157308	−	0.987550i	$-0.449718\pi$
$157$	3.94214	0.314617	0.157308	+	0.987550i	$0.449718\pi$
$158$	0	0
$159$	35.4956	2.81499
$160$	0	0
$161$	0.819031	0.0645487
$162$	0	0
$163$	3.80038	0.297669	0.148835	−	0.988862i	$-0.452448\pi$
$163$	3.80038	0.297669	0.148835	+	0.988862i	$0.452448\pi$
$164$	0	0
$165$	−13.7992	−1.07427
$166$	0	0
$167$	2.24621	0.173817	0.0869085	−	0.996216i	$-0.472301\pi$
$167$	2.24621	0.173817	0.0869085	+	0.996216i	$0.472301\pi$
$168$	0	0
$169$	−6.93087	−0.533144
$170$	0	0
$171$	19.1863	1.46721
$172$	0	0
$173$	−13.3822	−1.01743	−0.508717	−	0.860934i	$-0.669880\pi$
$173$	−13.3822	−1.01743	−0.508717	+	0.860934i	$0.669880\pi$
$174$	0	0
$175$	0.819031	0.0619130
$176$	0	0
$177$	−13.5223	−1.01640
$178$	0	0
$179$	−18.2077	−1.36091	−0.680455	−	0.732789i	$-0.738218\pi$
$179$	−18.2077	−1.36091	−0.680455	+	0.732789i	$0.738218\pi$
$180$	0	0
$181$	−1.01218	−0.0752348	−0.0376174	−	0.999292i	$-0.511977\pi$
$181$	−1.01218	−0.0752348	−0.0376174	+	0.999292i	$0.511977\pi$
$182$	0	0
$183$	−21.7992	−1.61144
$184$	0	0
$185$	7.42718	0.546057
$186$	0	0
$187$	−22.6584	−1.65694
$188$	0	0
$189$	1.17813	0.0856966
$190$	0	0
$191$	13.1231	0.949555	0.474777	−	0.880106i	$-0.342529\pi$
$191$	13.1231	0.949555	0.474777	+	0.880106i	$0.342529\pi$
$192$	0	0
$193$	−21.3179	−1.53450	−0.767249	−	0.641350i	$-0.778375\pi$
$193$	−21.3179	−1.53450	−0.767249	+	0.641350i	$0.778375\pi$
$194$	0	0
$195$	6.31054	0.451907
$196$	0	0
$197$	26.3608	1.87813	0.939063	−	0.343744i	$-0.111695\pi$
$197$	26.3608	1.87813	0.939063	+	0.343744i	$0.111695\pi$
$198$	0	0
$199$	4.16115	0.294976	0.147488	−	0.989064i	$-0.452881\pi$
$199$	4.16115	0.294976	0.147488	+	0.989064i	$0.452881\pi$
$200$	0	0
$201$	−10.5231	−0.742241
$202$	0	0
$203$	1.92920	0.135404
$204$	0	0
$205$	5.64453	0.394231
$206$	0	0
$207$	3.56155	0.247545
$208$	0	0
$209$	−29.0203	−2.00738
$210$	0	0
$211$	15.7414	1.08368	0.541840	−	0.840482i	$-0.317728\pi$
$211$	15.7414	1.08368	0.541840	+	0.840482i	$0.317728\pi$
$212$	0	0
$213$	30.2077	2.06980
$214$	0	0
$215$	1.90201	0.129716
$216$	0	0
$217$	−5.46265	−0.370829
$218$	0	0
$219$	−26.8030	−1.81118
$220$	0	0
$221$	10.3619	0.697019
$222$	0	0
$223$	−0.0802588	−0.00537453	−0.00268726	−	0.999996i	$-0.500855\pi$
$223$	−0.0802588	−0.00537453	−0.00268726	+	0.999996i	$0.500855\pi$
$224$	0	0
$225$	3.56155	0.237437
$226$	0	0
$227$	28.4705	1.88965	0.944827	−	0.327568i	$-0.106229\pi$
$227$	28.4705	1.88965	0.944827	+	0.327568i	$0.106229\pi$
$228$	0	0
$229$	−0.612950	−0.0405048	−0.0202524	−	0.999795i	$-0.506447\pi$
$229$	−0.612950	−0.0405048	−0.0202524	+	0.999795i	$0.506447\pi$
$230$	0	0
$231$	−11.3020	−0.743616
$232$	0	0
$233$	3.79558	0.248657	0.124328	−	0.992241i	$-0.460322\pi$
$233$	3.79558	0.248657	0.124328	+	0.992241i	$0.460322\pi$
$234$	0	0
$235$	9.33565	0.608991
$236$	0	0
$237$	−35.3474	−2.29606
$238$	0	0
$239$	−21.1118	−1.36561	−0.682806	−	0.730600i	$-0.739240\pi$
$239$	−21.1118	−1.36561	−0.682806	+	0.730600i	$0.739240\pi$
$240$	0	0
$241$	−0.774101	−0.0498642	−0.0249321	−	0.999689i	$-0.507937\pi$
$241$	−0.774101	−0.0498642	−0.0249321	+	0.999689i	$0.507937\pi$
$242$	0	0
$243$	−22.2462	−1.42710
$244$	0	0
$245$	−6.32919	−0.404357
$246$	0	0
$247$	13.2713	0.844434
$248$	0	0
$249$	−8.52789	−0.540433
$250$	0	0
$251$	16.8041	1.06067	0.530334	−	0.847789i	$-0.322067\pi$
$251$	16.8041	1.06067	0.530334	+	0.847789i	$0.322067\pi$
$252$	0	0
$253$	−5.38705	−0.338681
$254$	0	0
$255$	10.7741	0.674700
$256$	0	0
$257$	−5.12428	−0.319644	−0.159822	−	0.987146i	$-0.551092\pi$
$257$	−5.12428	−0.319644	−0.159822	+	0.987146i	$0.551092\pi$
$258$	0	0
$259$	6.08309	0.377985
$260$	0	0
$261$	8.38913	0.519274
$262$	0	0
$263$	−6.97205	−0.429915	−0.214958	−	0.976623i	$-0.568961\pi$
$263$	−6.97205	−0.429915	−0.214958	+	0.976623i	$0.568961\pi$
$264$	0	0
$265$	13.8571	0.851233
$266$	0	0
$267$	−9.17813	−0.561693
$268$	0	0
$269$	−21.3328	−1.30068	−0.650342	−	0.759641i	$-0.725375\pi$
$269$	−21.3328	−1.30068	−0.650342	+	0.759641i	$0.725375\pi$
$270$	0	0
$271$	0.751071	0.0456243	0.0228122	−	0.999740i	$-0.492738\pi$
$271$	0.751071	0.0456243	0.0228122	+	0.999740i	$0.492738\pi$
$272$	0	0
$273$	5.16853	0.312814
$274$	0	0
$275$	−5.38705	−0.324851
$276$	0	0
$277$	−8.20775	−0.493156	−0.246578	−	0.969123i	$-0.579306\pi$
$277$	−8.20775	−0.493156	−0.246578	+	0.969123i	$0.579306\pi$
$278$	0	0
$279$	−23.7543	−1.42213
$280$	0	0
$281$	14.9749	0.893327	0.446663	−	0.894702i	$-0.352612\pi$
$281$	14.9749	0.893327	0.446663	+	0.894702i	$0.352612\pi$
$282$	0	0
$283$	23.4224	1.39232	0.696158	−	0.717889i	$-0.254891\pi$
$283$	23.4224	1.39232	0.696158	+	0.717889i	$0.254891\pi$
$284$	0	0
$285$	13.7992	0.817395
$286$	0	0
$287$	4.62305	0.272890
$288$	0	0
$289$	0.691125	0.0406544
$290$	0	0
$291$	−23.1183	−1.35522
$292$	0	0
$293$	14.9600	0.873972	0.436986	−	0.899468i	$-0.356046\pi$
$293$	14.9600	0.873972	0.436986	+	0.899468i	$0.356046\pi$
$294$	0	0
$295$	−5.27896	−0.307353
$296$	0	0
$297$	−7.74899	−0.449642
$298$	0	0
$299$	2.46356	0.142471
$300$	0	0
$301$	1.55780	0.0897903
$302$	0	0
$303$	−31.0032	−1.78109
$304$	0	0
$305$	−8.51016	−0.487290
$306$	0	0
$307$	19.1863	1.09502	0.547509	−	0.836800i	$-0.315576\pi$
$307$	19.1863	1.09502	0.547509	+	0.836800i	$0.315576\pi$
$308$	0	0
$309$	9.74419	0.554327
$310$	0	0
$311$	19.4336	1.10198	0.550990	−	0.834512i	$-0.314250\pi$
$311$	19.4336	1.10198	0.550990	+	0.834512i	$0.314250\pi$
$312$	0	0
$313$	−20.9300	−1.18303	−0.591516	−	0.806294i	$-0.701470\pi$
$313$	−20.9300	−1.18303	−0.591516	+	0.806294i	$0.701470\pi$
$314$	0	0
$315$	2.91702	0.164356
$316$	0	0
$317$	10.8041	0.606821	0.303410	−	0.952860i	$-0.401875\pi$
$317$	10.8041	0.606821	0.303410	+	0.952860i	$0.401875\pi$
$318$	0	0
$319$	−12.6890	−0.710450
$320$	0	0
$321$	6.32710	0.353144
$322$	0	0
$323$	22.6584	1.26075
$324$	0	0
$325$	2.46356	0.136654
$326$	0	0
$327$	−11.5199	−0.637051
$328$	0	0
$329$	7.64619	0.421548
$330$	0	0
$331$	−22.7628	−1.25116	−0.625579	−	0.780161i	$-0.715137\pi$
$331$	−22.7628	−1.25116	−0.625579	+	0.780161i	$0.715137\pi$
$332$	0	0
$333$	26.4523	1.44958
$334$	0	0
$335$	−4.10809	−0.224449
$336$	0	0
$337$	−2.97489	−0.162052	−0.0810262	−	0.996712i	$-0.525820\pi$
$337$	−2.97489	−0.162052	−0.0810262	+	0.996712i	$0.525820\pi$
$338$	0	0
$339$	16.1482	0.877051
$340$	0	0
$341$	35.9297	1.94570
$342$	0	0
$343$	−10.9170	−0.589464
$344$	0	0
$345$	2.56155	0.137909
$346$	0	0
$347$	−35.4024	−1.90050	−0.950251	−	0.311484i	$-0.899174\pi$
$347$	−35.4024	−1.90050	−0.950251	+	0.311484i	$0.899174\pi$
$348$	0	0
$349$	22.4818	1.20342	0.601711	−	0.798714i	$-0.294486\pi$
$349$	22.4818	1.20342	0.601711	+	0.798714i	$0.294486\pi$
$350$	0	0
$351$	3.54370	0.189149
$352$	0	0
$353$	6.28051	0.334278	0.167139	−	0.985933i	$-0.446547\pi$
$353$	6.28051	0.334278	0.167139	+	0.985933i	$0.446547\pi$
$354$	0	0
$355$	11.7927	0.625894
$356$	0	0
$357$	8.82433	0.467033
$358$	0	0
$359$	−7.35702	−0.388289	−0.194144	−	0.980973i	$-0.562193\pi$
$359$	−7.35702	−0.388289	−0.194144	+	0.980973i	$0.562193\pi$
$360$	0	0
$361$	10.0203	0.527385
$362$	0	0
$363$	46.1600	2.42277
$364$	0	0
$365$	−10.4636	−0.547688
$366$	0	0
$367$	0.886992	0.0463006	0.0231503	−	0.999732i	$-0.492630\pi$
$367$	0.886992	0.0463006	0.0231503	+	0.999732i	$0.492630\pi$
$368$	0	0
$369$	20.1033	1.04654
$370$	0	0
$371$	11.3494	0.589231
$372$	0	0
$373$	−14.6283	−0.757427	−0.378713	−	0.925514i	$-0.623633\pi$
$373$	−14.6283	−0.757427	−0.378713	+	0.925514i	$0.623633\pi$
$374$	0	0
$375$	2.56155	0.132278
$376$	0	0
$377$	5.80285	0.298862
$378$	0	0
$379$	32.5183	1.67035	0.835176	−	0.549983i	$-0.185366\pi$
$379$	32.5183	1.67035	0.835176	+	0.549983i	$0.185366\pi$
$380$	0	0
$381$	2.33244	0.119495
$382$	0	0
$383$	−21.8393	−1.11594	−0.557969	−	0.829862i	$-0.688419\pi$
$383$	−21.8393	−1.11594	−0.557969	+	0.829862i	$0.688419\pi$
$384$	0	0
$385$	−4.41216	−0.224865
$386$	0	0
$387$	6.77410	0.344347
$388$	0	0
$389$	−15.1490	−0.768083	−0.384042	−	0.923316i	$-0.625468\pi$
$389$	−15.1490	−0.768083	−0.384042	+	0.923316i	$0.625468\pi$
$390$	0	0
$391$	4.20608	0.212711
$392$	0	0
$393$	−47.6628	−2.40427
$394$	0	0
$395$	−13.7992	−0.694314
$396$	0	0
$397$	−7.70239	−0.386572	−0.193286	−	0.981142i	$-0.561915\pi$
$397$	−7.70239	−0.386572	−0.193286	+	0.981142i	$0.561915\pi$
$398$	0	0
$399$	11.3020	0.565807
$400$	0	0
$401$	−23.7442	−1.18573	−0.592864	−	0.805303i	$-0.702003\pi$
$401$	−23.7442	−1.18573	−0.592864	+	0.805303i	$0.702003\pi$
$402$	0	0
$403$	−16.4311	−0.818490
$404$	0	0
$405$	−7.00000	−0.347833
$406$	0	0
$407$	−40.0106	−1.98325
$408$	0	0
$409$	10.2397	0.506323	0.253161	−	0.967424i	$-0.418530\pi$
$409$	10.2397	0.506323	0.253161	+	0.967424i	$0.418530\pi$
$410$	0	0
$411$	−36.9506	−1.82264
$412$	0	0
$413$	−4.32363	−0.212752
$414$	0	0
$415$	−3.32919	−0.163423
$416$	0	0
$417$	−47.0062	−2.30190
$418$	0	0
$419$	9.98227	0.487666	0.243833	−	0.969817i	$-0.421595\pi$
$419$	9.98227	0.487666	0.243833	+	0.969817i	$0.421595\pi$
$420$	0	0
$421$	−17.2470	−0.840566	−0.420283	−	0.907393i	$-0.638069\pi$
$421$	−17.2470	−0.840566	−0.420283	+	0.907393i	$0.638069\pi$
$422$	0	0
$423$	33.2494	1.61664
$424$	0	0
$425$	4.20608	0.204025
$426$	0	0
$427$	−6.97009	−0.337306
$428$	0	0
$429$	−33.9952	−1.64130
$430$	0	0
$431$	3.15377	0.151912	0.0759559	−	0.997111i	$-0.475799\pi$
$431$	3.15377	0.151912	0.0759559	+	0.997111i	$0.475799\pi$
$432$	0	0
$433$	−26.1333	−1.25589	−0.627944	−	0.778259i	$-0.716103\pi$
$433$	−26.1333	−1.25589	−0.627944	+	0.778259i	$0.716103\pi$
$434$	0	0
$435$	6.03366	0.289292
$436$	0	0
$437$	5.38705	0.257698
$438$	0	0
$439$	−2.08143	−0.0993412	−0.0496706	−	0.998766i	$-0.515817\pi$
$439$	−2.08143	−0.0993412	−0.0496706	+	0.998766i	$0.515817\pi$
$440$	0	0
$441$	−22.5417	−1.07342
$442$	0	0
$443$	12.2799	0.583434	0.291717	−	0.956505i	$-0.405773\pi$
$443$	12.2799	0.583434	0.291717	+	0.956505i	$0.405773\pi$
$444$	0	0
$445$	−3.58304	−0.169852
$446$	0	0
$447$	19.5653	0.925407
$448$	0	0
$449$	−19.3794	−0.914571	−0.457286	−	0.889320i	$-0.651178\pi$
$449$	−19.3794	−0.914571	−0.457286	+	0.889320i	$0.651178\pi$
$450$	0	0
$451$	−30.4074	−1.43183
$452$	0	0
$453$	1.68946	0.0793777
$454$	0	0
$455$	2.01773	0.0945928
$456$	0	0
$457$	−2.65788	−0.124330	−0.0621652	−	0.998066i	$-0.519801\pi$
$457$	−2.65788	−0.124330	−0.0621652	+	0.998066i	$0.519801\pi$
$458$	0	0
$459$	6.05023	0.282400
$460$	0	0
$461$	8.49423	0.395616	0.197808	−	0.980241i	$-0.436618\pi$
$461$	8.49423	0.395616	0.197808	+	0.980241i	$0.436618\pi$
$462$	0	0
$463$	−15.1604	−0.704564	−0.352282	−	0.935894i	$-0.614594\pi$
$463$	−15.1604	−0.704564	−0.352282	+	0.935894i	$0.614594\pi$
$464$	0	0
$465$	−17.0846	−0.792281
$466$	0	0
$467$	12.7039	0.587868	0.293934	−	0.955826i	$-0.405035\pi$
$467$	12.7039	0.587868	0.293934	+	0.955826i	$0.405035\pi$
$468$	0	0
$469$	−3.36465	−0.155365
$470$	0	0
$471$	10.0980	0.465291
$472$	0	0
$473$	−10.2462	−0.471121
$474$	0	0
$475$	5.38705	0.247175
$476$	0	0
$477$	49.3527	2.25971
$478$	0	0
$479$	11.6681	0.533129	0.266564	−	0.963817i	$-0.414111\pi$
$479$	11.6681	0.533129	0.266564	+	0.963817i	$0.414111\pi$
$480$	0	0
$481$	18.2973	0.834285
$482$	0	0
$483$	2.09799	0.0954620
$484$	0	0
$485$	−9.02511	−0.409809
$486$	0	0
$487$	−31.7778	−1.43999	−0.719996	−	0.693978i	$-0.755856\pi$
$487$	−31.7778	−1.43999	−0.719996	+	0.693978i	$0.755856\pi$
$488$	0	0
$489$	9.73489	0.440227
$490$	0	0
$491$	6.17722	0.278774	0.139387	−	0.990238i	$-0.455487\pi$
$491$	6.17722	0.278774	0.139387	+	0.990238i	$0.455487\pi$
$492$	0	0
$493$	9.90730	0.446203
$494$	0	0
$495$	−19.1863	−0.862358
$496$	0	0
$497$	9.65863	0.433249
$498$	0	0
$499$	37.3409	1.67161	0.835805	−	0.549026i	$-0.185001\pi$
$499$	37.3409	1.67161	0.835805	+	0.549026i	$0.185001\pi$
$500$	0	0
$501$	5.75379	0.257060
$502$	0	0
$503$	−9.36648	−0.417631	−0.208815	−	0.977955i	$-0.566961\pi$
$503$	−9.36648	−0.417631	−0.208815	+	0.977955i	$0.566961\pi$
$504$	0	0
$505$	−12.1033	−0.538589
$506$	0	0
$507$	−17.7538	−0.788473
$508$	0	0
$509$	−21.0798	−0.934347	−0.467174	−	0.884166i	$-0.654728\pi$
$509$	−21.0798	−0.934347	−0.467174	+	0.884166i	$0.654728\pi$
$510$	0	0
$511$	−8.56999	−0.379114
$512$	0	0
$513$	7.74899	0.342126
$514$	0	0
$515$	3.80402	0.167625
$516$	0	0
$517$	−50.2916	−2.21182
$518$	0	0
$519$	−34.2793	−1.50470
$520$	0	0
$521$	−8.87689	−0.388904	−0.194452	−	0.980912i	$-0.562293\pi$
$521$	−8.87689	−0.388904	−0.194452	+	0.980912i	$0.562293\pi$
$522$	0	0
$523$	2.82187	0.123392	0.0616958	−	0.998095i	$-0.480349\pi$
$523$	2.82187	0.123392	0.0616958	+	0.998095i	$0.480349\pi$
$524$	0	0
$525$	2.09799	0.0915639
$526$	0	0
$527$	−28.0531	−1.22201
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	−18.8013	−0.815907
$532$	0	0
$533$	13.9056	0.602320
$534$	0	0
$535$	2.47003	0.106789
$536$	0	0
$537$	−46.6401	−2.01267
$538$	0	0
$539$	34.0957	1.46860
$540$	0	0
$541$	21.3786	0.919139	0.459569	−	0.888142i	$-0.348004\pi$
$541$	21.3786	0.919139	0.459569	+	0.888142i	$0.348004\pi$
$542$	0	0
$543$	−2.59276	−0.111266
$544$	0	0
$545$	−4.49722	−0.192640
$546$	0	0
$547$	−7.28543	−0.311502	−0.155751	−	0.987796i	$-0.549780\pi$
$547$	−7.28543	−0.311502	−0.155751	+	0.987796i	$0.549780\pi$
$548$	0	0
$549$	−30.3094	−1.29357
$550$	0	0
$551$	12.6890	0.540571
$552$	0	0
$553$	−11.3020	−0.480610
$554$	0	0
$555$	19.0251	0.807571
$556$	0	0
$557$	41.4101	1.75460	0.877301	−	0.479941i	$-0.159342\pi$
$557$	41.4101	1.75460	0.877301	+	0.479941i	$0.159342\pi$
$558$	0	0
$559$	4.68571	0.198184
$560$	0	0
$561$	−58.0406	−2.45048
$562$	0	0
$563$	−21.5210	−0.907002	−0.453501	−	0.891256i	$-0.649825\pi$
$563$	−21.5210	−0.907002	−0.453501	+	0.891256i	$0.649825\pi$
$564$	0	0
$565$	6.30407	0.265214
$566$	0	0
$567$	−5.73322	−0.240773
$568$	0	0
$569$	21.9645	0.920801	0.460401	−	0.887711i	$-0.347706\pi$
$569$	21.9645	0.920801	0.460401	+	0.887711i	$0.347706\pi$
$570$	0	0
$571$	24.4924	1.02498	0.512488	−	0.858694i	$-0.328724\pi$
$571$	24.4924	1.02498	0.512488	+	0.858694i	$0.328724\pi$
$572$	0	0
$573$	33.6155	1.40431
$574$	0	0
$575$	1.00000	0.0417029
$576$	0	0
$577$	−10.0159	−0.416969	−0.208484	−	0.978026i	$-0.566853\pi$
$577$	−10.0159	−0.416969	−0.208484	+	0.978026i	$0.566853\pi$
$578$	0	0
$579$	−54.6070	−2.26939
$580$	0	0
$581$	−2.72671	−0.113123
$582$	0	0
$583$	−74.6488	−3.09164
$584$	0	0
$585$	8.77410	0.362764
$586$	0	0
$587$	5.64736	0.233092	0.116546	−	0.993185i	$-0.462818\pi$
$587$	5.64736	0.233092	0.116546	+	0.993185i	$0.462818\pi$
$588$	0	0
$589$	−35.9297	−1.48046
$590$	0	0
$591$	67.5245	2.77759
$592$	0	0
$593$	12.4624	0.511769	0.255885	−	0.966707i	$-0.417633\pi$
$593$	12.4624	0.511769	0.255885	+	0.966707i	$0.417633\pi$
$594$	0	0
$595$	3.44491	0.141228
$596$	0	0
$597$	10.6590	0.436244
$598$	0	0
$599$	−24.7985	−1.01324	−0.506619	−	0.862170i	$-0.669105\pi$
$599$	−24.7985	−1.01324	−0.506619	+	0.862170i	$0.669105\pi$
$600$	0	0
$601$	9.10371	0.371348	0.185674	−	0.982611i	$-0.440553\pi$
$601$	9.10371	0.371348	0.185674	+	0.982611i	$0.440553\pi$
$602$	0	0
$603$	−14.6312	−0.595828
$604$	0	0
$605$	18.0203	0.732630
$606$	0	0
$607$	−20.6283	−0.837279	−0.418639	−	0.908153i	$-0.637493\pi$
$607$	−20.6283	−0.837279	−0.418639	+	0.908153i	$0.637493\pi$
$608$	0	0
$609$	4.94176	0.200250
$610$	0	0
$611$	22.9989	0.930438
$612$	0	0
$613$	−11.7514	−0.474637	−0.237318	−	0.971432i	$-0.576268\pi$
$613$	−11.7514	−0.474637	−0.237318	+	0.971432i	$0.576268\pi$
$614$	0	0
$615$	14.4588	0.583033
$616$	0	0
$617$	0.462764	0.0186302	0.00931510	−	0.999957i	$-0.497035\pi$
$617$	0.462764	0.0186302	0.00931510	+	0.999957i	$0.497035\pi$
$618$	0	0
$619$	21.7183	0.872933	0.436467	−	0.899720i	$-0.356230\pi$
$619$	21.7183	0.872933	0.436467	+	0.899720i	$0.356230\pi$
$620$	0	0
$621$	1.43845	0.0577229
$622$	0	0
$623$	−2.93462	−0.117573
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−74.3371	−2.96874
$628$	0	0
$629$	31.2393	1.24559
$630$	0	0
$631$	4.18369	0.166550	0.0832750	−	0.996527i	$-0.473462\pi$
$631$	4.18369	0.166550	0.0832750	+	0.996527i	$0.473462\pi$
$632$	0	0
$633$	40.3223	1.60267
$634$	0	0
$635$	0.910558	0.0361344
$636$	0	0
$637$	−15.5923	−0.617791
$638$	0	0
$639$	42.0005	1.66151
$640$	0	0
$641$	−13.6535	−0.539279	−0.269640	−	0.962961i	$-0.586905\pi$
$641$	−13.6535	−0.539279	−0.269640	+	0.962961i	$0.586905\pi$
$642$	0	0
$643$	−39.5583	−1.56003	−0.780014	−	0.625763i	$-0.784788\pi$
$643$	−39.5583	−1.56003	−0.780014	+	0.625763i	$0.784788\pi$
$644$	0	0
$645$	4.87209	0.191838
$646$	0	0
$647$	−9.12674	−0.358809	−0.179405	−	0.983775i	$-0.557417\pi$
$647$	−9.12674	−0.358809	−0.179405	+	0.983775i	$0.557417\pi$
$648$	0	0
$649$	28.4380	1.11629
$650$	0	0
$651$	−13.9929	−0.548423
$652$	0	0
$653$	0.745239	0.0291634	0.0145817	−	0.999894i	$-0.495358\pi$
$653$	0.745239	0.0291634	0.0145817	+	0.999894i	$0.495358\pi$
$654$	0	0
$655$	−18.6070	−0.727035
$656$	0	0
$657$	−37.2665	−1.45391
$658$	0	0
$659$	33.6285	1.30998	0.654989	−	0.755638i	$-0.272673\pi$
$659$	33.6285	1.30998	0.654989	+	0.755638i	$0.272673\pi$
$660$	0	0
$661$	22.0000	0.855701	0.427850	−	0.903850i	$-0.359271\pi$
$661$	22.0000	0.855701	0.427850	+	0.903850i	$0.359271\pi$
$662$	0	0
$663$	26.5426	1.03083
$664$	0	0
$665$	4.41216	0.171096
$666$	0	0
$667$	2.35547	0.0912042
$668$	0	0
$669$	−0.205587	−0.00794846
$670$	0	0
$671$	45.8446	1.76981
$672$	0	0
$673$	19.5641	0.754142	0.377071	−	0.926184i	$-0.376931\pi$
$673$	19.5641	0.754142	0.377071	+	0.926184i	$0.376931\pi$
$674$	0	0
$675$	1.43845	0.0553659
$676$	0	0
$677$	−12.7412	−0.489685	−0.244843	−	0.969563i	$-0.578736\pi$
$677$	−12.7412	−0.489685	−0.244843	+	0.969563i	$0.578736\pi$
$678$	0	0
$679$	−7.39185	−0.283673
$680$	0	0
$681$	72.9287	2.79464
$682$	0	0
$683$	12.5186	0.479010	0.239505	−	0.970895i	$-0.423015\pi$
$683$	12.5186	0.479010	0.239505	+	0.970895i	$0.423015\pi$
$684$	0	0
$685$	−14.4251	−0.551155
$686$	0	0
$687$	−1.57010	−0.0599032
$688$	0	0
$689$	34.1377	1.30054
$690$	0	0
$691$	−3.70356	−0.140890	−0.0704451	−	0.997516i	$-0.522442\pi$
$691$	−3.70356	−0.140890	−0.0704451	+	0.997516i	$0.522442\pi$
$692$	0	0
$693$	−15.7142	−0.596931
$694$	0	0
$695$	−18.3507	−0.696081
$696$	0	0
$697$	23.7414	0.899268
$698$	0	0
$699$	9.72259	0.367742
$700$	0	0
$701$	12.6388	0.477361	0.238681	−	0.971098i	$-0.423285\pi$
$701$	12.6388	0.477361	0.238681	+	0.971098i	$0.423285\pi$
$702$	0	0
$703$	40.0106	1.50903
$704$	0	0
$705$	23.9138	0.900645
$706$	0	0
$707$	−9.91297	−0.372816
$708$	0	0
$709$	21.5978	0.811122	0.405561	−	0.914068i	$-0.367076\pi$
$709$	21.5978	0.811122	0.405561	+	0.914068i	$0.367076\pi$
$710$	0	0
$711$	−49.1466	−1.84314
$712$	0	0
$713$	−6.66964	−0.249780
$714$	0	0
$715$	−13.2713	−0.496319
$716$	0	0
$717$	−54.0791	−2.01962
$718$	0	0
$719$	31.8418	1.18750	0.593749	−	0.804650i	$-0.297647\pi$
$719$	31.8418	1.18750	0.593749	+	0.804650i	$0.297647\pi$
$720$	0	0
$721$	3.11561	0.116031
$722$	0	0
$723$	−1.98290	−0.0737448
$724$	0	0
$725$	2.35547	0.0874800
$726$	0	0
$727$	32.0507	1.18870	0.594348	−	0.804208i	$-0.297410\pi$
$727$	32.0507	1.18870	0.594348	+	0.804208i	$0.297410\pi$
$728$	0	0
$729$	−35.9848	−1.33277
$730$	0	0
$731$	8.00000	0.295891
$732$	0	0
$733$	−35.9202	−1.32674	−0.663372	−	0.748290i	$-0.730875\pi$
$733$	−35.9202	−1.32674	−0.663372	+	0.748290i	$0.730875\pi$
$734$	0	0
$735$	−16.2125	−0.598009
$736$	0	0
$737$	22.1305	0.815187
$738$	0	0
$739$	−1.74252	−0.0640997	−0.0320498	−	0.999486i	$-0.510204\pi$
$739$	−1.74252	−0.0640997	−0.0320498	+	0.999486i	$0.510204\pi$
$740$	0	0
$741$	33.9952	1.24884
$742$	0	0
$743$	3.80402	0.139556	0.0697779	−	0.997563i	$-0.477771\pi$
$743$	3.80402	0.139556	0.0697779	+	0.997563i	$0.477771\pi$
$744$	0	0
$745$	7.63806	0.279837
$746$	0	0
$747$	−11.8571	−0.433828
$748$	0	0
$749$	2.02303	0.0739199
$750$	0	0
$751$	−22.3046	−0.813905	−0.406953	−	0.913449i	$-0.633409\pi$
$751$	−22.3046	−0.813905	−0.406953	+	0.913449i	$0.633409\pi$
$752$	0	0
$753$	43.0447	1.56864
$754$	0	0
$755$	0.659545	0.0240033
$756$	0	0
$757$	−34.3973	−1.25019	−0.625095	−	0.780549i	$-0.714940\pi$
$757$	−34.3973	−1.25019	−0.625095	+	0.780549i	$0.714940\pi$
$758$	0	0
$759$	−13.7992	−0.500880
$760$	0	0
$761$	−28.1069	−1.01888	−0.509438	−	0.860508i	$-0.670147\pi$
$761$	−28.1069	−1.01888	−0.509438	+	0.860508i	$0.670147\pi$
$762$	0	0
$763$	−3.68337	−0.133347
$764$	0	0
$765$	14.9802	0.541610
$766$	0	0
$767$	−13.0050	−0.469585
$768$	0	0
$769$	5.49230	0.198058	0.0990288	−	0.995085i	$-0.468426\pi$
$769$	5.49230	0.198058	0.0990288	+	0.995085i	$0.468426\pi$
$770$	0	0
$771$	−13.1261	−0.472725
$772$	0	0
$773$	27.5678	0.991543	0.495772	−	0.868453i	$-0.334885\pi$
$773$	27.5678	0.991543	0.495772	+	0.868453i	$0.334885\pi$
$774$	0	0
$775$	−6.66964	−0.239581
$776$	0	0
$777$	15.5822	0.559007
$778$	0	0
$779$	30.4074	1.08946
$780$	0	0
$781$	−63.5281	−2.27322
$782$	0	0
$783$	3.38822	0.121085
$784$	0	0
$785$	3.94214	0.140701
$786$	0	0
$787$	−26.7316	−0.952880	−0.476440	−	0.879207i	$-0.658073\pi$
$787$	−26.7316	−0.952880	−0.476440	+	0.879207i	$0.658073\pi$
$788$	0	0
$789$	−17.8593	−0.635807
$790$	0	0
$791$	5.16324	0.183584
$792$	0	0
$793$	−20.9653	−0.744499
$794$	0	0
$795$	35.4956	1.25890
$796$	0	0
$797$	−34.3017	−1.21503	−0.607515	−	0.794308i	$-0.707833\pi$
$797$	−34.3017	−1.21503	−0.607515	+	0.794308i	$0.707833\pi$
$798$	0	0
$799$	39.2665	1.38915
$800$	0	0
$801$	−12.7612	−0.450894
$802$	0	0
$803$	56.3677	1.98917
$804$	0	0
$805$	0.819031	0.0288671
$806$	0	0
$807$	−54.6451	−1.92360
$808$	0	0
$809$	−25.9575	−0.912618	−0.456309	−	0.889821i	$-0.650829\pi$
$809$	−25.9575	−0.912618	−0.456309	+	0.889821i	$0.650829\pi$
$810$	0	0
$811$	25.2252	0.885777	0.442889	−	0.896577i	$-0.353954\pi$
$811$	25.2252	0.885777	0.442889	+	0.896577i	$0.353954\pi$
$812$	0	0
$813$	1.92391	0.0674744
$814$	0	0
$815$	3.80038	0.133122
$816$	0	0
$817$	10.2462	0.358470
$818$	0	0
$819$	7.18626	0.251108
$820$	0	0
$821$	25.2761	0.882143	0.441071	−	0.897472i	$-0.354599\pi$
$821$	25.2761	0.882143	0.441071	+	0.897472i	$0.354599\pi$
$822$	0	0
$823$	22.0337	0.768045	0.384023	−	0.923324i	$-0.374538\pi$
$823$	22.0337	0.768045	0.384023	+	0.923324i	$0.374538\pi$
$824$	0	0
$825$	−13.7992	−0.480427
$826$	0	0
$827$	16.0531	0.558220	0.279110	−	0.960259i	$-0.409961\pi$
$827$	16.0531	0.558220	0.279110	+	0.960259i	$0.409961\pi$
$828$	0	0
$829$	−20.5251	−0.712865	−0.356432	−	0.934321i	$-0.616007\pi$
$829$	−20.5251	−0.712865	−0.356432	+	0.934321i	$0.616007\pi$
$830$	0	0
$831$	−21.0246	−0.729334
$832$	0	0
$833$	−26.6211	−0.922366
$834$	0	0
$835$	2.24621	0.0777333
$836$	0	0
$837$	−9.59393	−0.331615
$838$	0	0
$839$	−43.0284	−1.48551	−0.742753	−	0.669565i	$-0.766481\pi$
$839$	−43.0284	−1.48551	−0.742753	+	0.669565i	$0.766481\pi$
$840$	0	0
$841$	−23.4518	−0.808681
$842$	0	0
$843$	38.3590	1.32115
$844$	0	0
$845$	−6.93087	−0.238429
$846$	0	0
$847$	14.7592	0.507132
$848$	0	0
$849$	59.9977	2.05911
$850$	0	0
$851$	7.42718	0.254600
$852$	0	0
$853$	29.8203	1.02103	0.510513	−	0.859870i	$-0.329455\pi$
$853$	29.8203	1.02103	0.510513	+	0.859870i	$0.329455\pi$
$854$	0	0
$855$	19.1863	0.656156
$856$	0	0
$857$	−20.4465	−0.698438	−0.349219	−	0.937041i	$-0.613553\pi$
$857$	−20.4465	−0.698438	−0.349219	+	0.937041i	$0.613553\pi$
$858$	0	0
$859$	−50.6000	−1.72645	−0.863224	−	0.504820i	$-0.831559\pi$
$859$	−50.6000	−1.72645	−0.863224	+	0.504820i	$0.831559\pi$
$860$	0	0
$861$	11.8422	0.403580
$862$	0	0
$863$	−28.1900	−0.959599	−0.479800	−	0.877378i	$-0.659291\pi$
$863$	−28.1900	−0.959599	−0.479800	+	0.877378i	$0.659291\pi$
$864$	0	0
$865$	−13.3822	−0.455010
$866$	0	0
$867$	1.77035	0.0601243
$868$	0	0
$869$	74.3371	2.52171
$870$	0	0
$871$	−10.1205	−0.342921
$872$	0	0
$873$	−32.1434	−1.08789
$874$	0	0
$875$	0.819031	0.0276883
$876$	0	0
$877$	3.25176	0.109804	0.0549021	−	0.998492i	$-0.482515\pi$
$877$	3.25176	0.109804	0.0549021	+	0.998492i	$0.482515\pi$
$878$	0	0
$879$	38.3208	1.29253
$880$	0	0
$881$	40.4455	1.36264	0.681322	−	0.731984i	$-0.261405\pi$
$881$	40.4455	1.36264	0.681322	+	0.731984i	$0.261405\pi$
$882$	0	0
$883$	12.4178	0.417893	0.208947	−	0.977927i	$-0.432996\pi$
$883$	12.4178	0.417893	0.208947	+	0.977927i	$0.432996\pi$
$884$	0	0
$885$	−13.5223	−0.454548
$886$	0	0
$887$	13.2199	0.443882	0.221941	−	0.975060i	$-0.428761\pi$
$887$	13.2199	0.443882	0.221941	+	0.975060i	$0.428761\pi$
$888$	0	0
$889$	0.745775	0.0250125
$890$	0	0
$891$	37.7094	1.26331
$892$	0	0
$893$	50.2916	1.68295
$894$	0	0
$895$	−18.2077	−0.608618
$896$	0	0
$897$	6.31054	0.210703
$898$	0	0
$899$	−15.7101	−0.523963
$900$	0	0
$901$	58.2840	1.94172
$902$	0	0
$903$	3.99040	0.132792
$904$	0	0
$905$	−1.01218	−0.0336460
$906$	0	0
$907$	58.2386	1.93378	0.966890	−	0.255193i	$-0.0821391\pi$
$907$	58.2386	1.93378	0.966890	+	0.255193i	$0.0821391\pi$
$908$	0	0
$909$	−43.1065	−1.42975
$910$	0	0
$911$	46.5977	1.54385	0.771925	−	0.635714i	$-0.219294\pi$
$911$	46.5977	1.54385	0.771925	+	0.635714i	$0.219294\pi$
$912$	0	0
$913$	17.9345	0.593545
$914$	0	0
$915$	−21.7992	−0.720660
$916$	0	0
$917$	−15.2397	−0.503259
$918$	0	0
$919$	−20.1386	−0.664312	−0.332156	−	0.943225i	$-0.607776\pi$
$919$	−20.1386	−0.664312	−0.332156	+	0.943225i	$0.607776\pi$
$920$	0	0
$921$	49.1466	1.61944
$922$	0	0
$923$	29.0521	0.956263
$924$	0	0
$925$	7.42718	0.244204
$926$	0	0
$927$	13.5482	0.444981
$928$	0	0
$929$	−1.01385	−0.0332632	−0.0166316	−	0.999862i	$-0.505294\pi$
$929$	−1.01385	−0.0332632	−0.0166316	+	0.999862i	$0.505294\pi$
$930$	0	0
$931$	−34.0957	−1.11744
$932$	0	0
$933$	49.7803	1.62973
$934$	0	0
$935$	−22.6584	−0.741008
$936$	0	0
$937$	32.5904	1.06468	0.532341	−	0.846530i	$-0.321312\pi$
$937$	32.5904	1.06468	0.532341	+	0.846530i	$0.321312\pi$
$938$	0	0
$939$	−53.6132	−1.74960
$940$	0	0
$941$	−1.01848	−0.0332017	−0.0166008	−	0.999862i	$-0.505284\pi$
$941$	−1.01848	−0.0332017	−0.0166008	+	0.999862i	$0.505284\pi$
$942$	0	0
$943$	5.64453	0.183811
$944$	0	0
$945$	1.17813	0.0383247
$946$	0	0
$947$	−0.212548	−0.00690688	−0.00345344	−	0.999994i	$-0.501099\pi$
$947$	−0.212548	−0.00690688	−0.00345344	+	0.999994i	$0.501099\pi$
$948$	0	0
$949$	−25.7776	−0.836777
$950$	0	0
$951$	27.6754	0.897435
$952$	0	0
$953$	47.2067	1.52917	0.764587	−	0.644520i	$-0.222943\pi$
$953$	47.2067	1.52917	0.764587	+	0.644520i	$0.222943\pi$
$954$	0	0
$955$	13.1231	0.424654
$956$	0	0
$957$	−32.5036	−1.05069
$958$	0	0
$959$	−11.8146	−0.381514
$960$	0	0
$961$	13.4841	0.434972
$962$	0	0
$963$	8.79713	0.283484
$964$	0	0
$965$	−21.3179	−0.686248
$966$	0	0
$967$	23.7851	0.764878	0.382439	−	0.923981i	$-0.375084\pi$
$967$	23.7851	0.764878	0.382439	+	0.923981i	$0.375084\pi$
$968$	0	0
$969$	58.0406	1.86453
$970$	0	0
$971$	47.7012	1.53081	0.765403	−	0.643552i	$-0.222540\pi$
$971$	47.7012	1.53081	0.765403	+	0.643552i	$0.222540\pi$
$972$	0	0
$973$	−15.0298	−0.481832
$974$	0	0
$975$	6.31054	0.202099
$976$	0	0
$977$	−5.04505	−0.161405	−0.0807027	−	0.996738i	$-0.525716\pi$
$977$	−5.04505	−0.161405	−0.0807027	+	0.996738i	$0.525716\pi$
$978$	0	0
$979$	19.3020	0.616894
$980$	0	0
$981$	−16.0171	−0.511387
$982$	0	0
$983$	3.52184	0.112329	0.0561647	−	0.998422i	$-0.482113\pi$
$983$	3.52184	0.112329	0.0561647	+	0.998422i	$0.482113\pi$
$984$	0	0
$985$	26.3608	0.839924
$986$	0	0
$987$	19.5861	0.623433
$988$	0	0
$989$	1.90201	0.0604803
$990$	0	0
$991$	−53.3794	−1.69565	−0.847826	−	0.530274i	$-0.822089\pi$
$991$	−53.3794	−1.69565	−0.847826	+	0.530274i	$0.822089\pi$
$992$	0	0
$993$	−58.3082	−1.85035
$994$	0	0
$995$	4.16115	0.131917
$996$	0	0
$997$	55.2269	1.74905	0.874527	−	0.484978i	$-0.161172\pi$
$997$	55.2269	1.74905	0.874527	+	0.484978i	$0.161172\pi$
$998$	0	0
$999$	10.6836	0.338014

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1840.2.a.u.1.3		4
4.3	odd	2		115.2.a.c.1.1	✓	4
5.4	even	2		9200.2.a.cl.1.2		4
8.3	odd	2		7360.2.a.cj.1.4		4
8.5	even	2		7360.2.a.cg.1.1		4
12.11	even	2		1035.2.a.o.1.4		4
20.3	even	4		575.2.b.e.24.7		8
20.7	even	4		575.2.b.e.24.2		8
20.19	odd	2		575.2.a.h.1.4		4
28.27	even	2		5635.2.a.v.1.1		4
60.59	even	2		5175.2.a.bx.1.1		4
92.91	even	2		2645.2.a.m.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
115.2.a.c.1.1	✓	4	4.3	odd	2
575.2.a.h.1.4		4	20.19	odd	2
575.2.b.e.24.2		8	20.7	even	4
575.2.b.e.24.7		8	20.3	even	4
1035.2.a.o.1.4		4	12.11	even	2
1840.2.a.u.1.3		4	1.1	even	1	trivial
2645.2.a.m.1.1		4	92.91	even	2
5175.2.a.bx.1.1		4	60.59	even	2
5635.2.a.v.1.1		4	28.27	even	2
7360.2.a.cg.1.1		4	8.5	even	2
7360.2.a.cj.1.4		4	8.3	odd	2
9200.2.a.cl.1.2		4	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)

\(f(q)\)	\(=\)	\(q+2.56155 q^{3} +1.00000 q^{5} +0.819031 q^{7} +3.56155 q^{9} +O(q^{10})\)	\(q+2.56155 q^{3} +1.00000 q^{5} +0.819031 q^{7} +3.56155 q^{9} -5.38705 q^{11} +2.46356 q^{13} +2.56155 q^{15} +4.20608 q^{17} +5.38705 q^{19} +2.09799 q^{21} +1.00000 q^{23} +1.00000 q^{25} +1.43845 q^{27} +2.35547 q^{29} -6.66964 q^{31} -13.7992 q^{33} +0.819031 q^{35} +7.42718 q^{37} +6.31054 q^{39} +5.64453 q^{41} +1.90201 q^{43} +3.56155 q^{45} +9.33565 q^{47} -6.32919 q^{49} +10.7741 q^{51} +13.8571 q^{53} -5.38705 q^{55} +13.7992 q^{57} -5.27896 q^{59} -8.51016 q^{61} +2.91702 q^{63} +2.46356 q^{65} -4.10809 q^{67} +2.56155 q^{69} +11.7927 q^{71} -10.4636 q^{73} +2.56155 q^{75} -4.41216 q^{77} -13.7992 q^{79} -7.00000 q^{81} -3.32919 q^{83} +4.20608 q^{85} +6.03366 q^{87} -3.58304 q^{89} +2.01773 q^{91} -17.0846 q^{93} +5.38705 q^{95} -9.02511 q^{97} -19.1863 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{3} + 4 q^{5} + 3 q^{7} + 6 q^{9}+O(q^{10}) \) 4 * q + 2 * q^3 + 4 * q^5 + 3 * q^7 + 6 * q^9	\( 4 q + 2 q^{3} + 4 q^{5} + 3 q^{7} + 6 q^{9} - 4 q^{11} + 2 q^{15} - q^{17} + 4 q^{19} + 10 q^{21} + 4 q^{23} + 4 q^{25} + 14 q^{27} + 19 q^{29} + q^{31} - 2 q^{33} + 3 q^{35} - 3 q^{37} + 13 q^{41} + 6 q^{43} + 6 q^{45} - 6 q^{47} + 9 q^{49} + 8 q^{51} + 19 q^{53} - 4 q^{55} + 2 q^{57} - 23 q^{59} + 13 q^{63} + 3 q^{67} + 2 q^{69} + 3 q^{71} - 32 q^{73} + 2 q^{75} + 18 q^{77} - 2 q^{79} - 28 q^{81} + 21 q^{83} - q^{85} + 18 q^{87} + 40 q^{91} - 8 q^{93} + 4 q^{95} - 18 q^{97} - 6 q^{99}+O(q^{100}) \) 4 * q + 2 * q^3 + 4 * q^5 + 3 * q^7 + 6 * q^9 - 4 * q^11 + 2 * q^15 - q^17 + 4 * q^19 + 10 * q^21 + 4 * q^23 + 4 * q^25 + 14 * q^27 + 19 * q^29 + q^31 - 2 * q^33 + 3 * q^35 - 3 * q^37 + 13 * q^41 + 6 * q^43 + 6 * q^45 - 6 * q^47 + 9 * q^49 + 8 * q^51 + 19 * q^53 - 4 * q^55 + 2 * q^57 - 23 * q^59 + 13 * q^63 + 3 * q^67 + 2 * q^69 + 3 * q^71 - 32 * q^73 + 2 * q^75 + 18 * q^77 - 2 * q^79 - 28 * q^81 + 21 * q^83 - q^85 + 18 * q^87 + 40 * q^91 - 8 * q^93 + 4 * q^95 - 18 * q^97 - 6 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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