LMFDB - Embedded newform 1840.2.a.p.1.1

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:	$\Z[a_1, \ldots, a_{7}]$
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			1.1
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)

$f(q)$	$=$	$q+1.00000 q^{3} -1.00000 q^{5} -1.23607 q^{7} -2.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -1.00000 q^{5} -1.23607 q^{7} -2.00000 q^{9} +3.23607 q^{11} -6.23607 q^{13} -1.00000 q^{15} +2.47214 q^{17} +5.70820 q^{19} -1.23607 q^{21} +1.00000 q^{23} +1.00000 q^{25} -5.00000 q^{27} -0.527864 q^{29} -4.23607 q^{31} +3.23607 q^{33} +1.23607 q^{35} -9.70820 q^{37} -6.23607 q^{39} -7.47214 q^{41} -3.70820 q^{43} +2.00000 q^{45} -9.47214 q^{47} -5.47214 q^{49} +2.47214 q^{51} -6.00000 q^{53} -3.23607 q^{55} +5.70820 q^{57} -8.94427 q^{59} +12.1803 q^{61} +2.47214 q^{63} +6.23607 q^{65} -9.70820 q^{67} +1.00000 q^{69} +6.23607 q^{71} -6.70820 q^{73} +1.00000 q^{75} -4.00000 q^{77} -8.76393 q^{79} +1.00000 q^{81} +6.47214 q^{83} -2.47214 q^{85} -0.527864 q^{87} +2.76393 q^{89} +7.70820 q^{91} -4.23607 q^{93} -5.70820 q^{95} -6.18034 q^{97} -6.47214 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} - 2 q^{5} + 2 q^{7} - 4 q^{9}+O(q^{10}) $ 2 * q + 2 * q^3 - 2 * q^5 + 2 * q^7 - 4 * q^9	$ 2 q + 2 q^{3} - 2 q^{5} + 2 q^{7} - 4 q^{9} + 2 q^{11} - 8 q^{13} - 2 q^{15} - 4 q^{17} - 2 q^{19} + 2 q^{21} + 2 q^{23} + 2 q^{25} - 10 q^{27} - 10 q^{29} - 4 q^{31} + 2 q^{33} - 2 q^{35} - 6 q^{37} - 8 q^{39} - 6 q^{41} + 6 q^{43} + 4 q^{45} - 10 q^{47} - 2 q^{49} - 4 q^{51} - 12 q^{53} - 2 q^{55} - 2 q^{57} + 2 q^{61} - 4 q^{63} + 8 q^{65} - 6 q^{67} + 2 q^{69} + 8 q^{71} + 2 q^{75} - 8 q^{77} - 22 q^{79} + 2 q^{81} + 4 q^{83} + 4 q^{85} - 10 q^{87} + 10 q^{89} + 2 q^{91} - 4 q^{93} + 2 q^{95} + 10 q^{97} - 4 q^{99}+O(q^{100}) $ 2 * q + 2 * q^3 - 2 * q^5 + 2 * q^7 - 4 * q^9 + 2 * q^11 - 8 * q^13 - 2 * q^15 - 4 * q^17 - 2 * q^19 + 2 * q^21 + 2 * q^23 + 2 * q^25 - 10 * q^27 - 10 * q^29 - 4 * q^31 + 2 * q^33 - 2 * q^35 - 6 * q^37 - 8 * q^39 - 6 * q^41 + 6 * q^43 + 4 * q^45 - 10 * q^47 - 2 * q^49 - 4 * q^51 - 12 * q^53 - 2 * q^55 - 2 * q^57 + 2 * q^61 - 4 * q^63 + 8 * q^65 - 6 * q^67 + 2 * q^69 + 8 * q^71 + 2 * q^75 - 8 * q^77 - 22 * q^79 + 2 * q^81 + 4 * q^83 + 4 * q^85 - 10 * q^87 + 10 * q^89 + 2 * q^91 - 4 * q^93 + 2 * q^95 + 10 * q^97 - 4 * 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
$3$	1.00000	0.577350	0.288675	−	0.957427i	$-0.406785\pi$
$3$	1.00000	0.577350	0.288675	+	0.957427i	$0.406785\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−1.23607	−0.467190	−0.233595	−	0.972334i	$-0.575049\pi$
$7$	−1.23607	−0.467190	−0.233595	+	0.972334i	$0.575049\pi$
$8$	0	0
$9$	−2.00000	−0.666667
$10$	0	0
$11$	3.23607	0.975711	0.487856	−	0.872924i	$-0.337779\pi$
$11$	3.23607	0.975711	0.487856	+	0.872924i	$0.337779\pi$
$12$	0	0
$13$	−6.23607	−1.72957	−0.864787	−	0.502139i	$-0.832547\pi$
$13$	−6.23607	−1.72957	−0.864787	+	0.502139i	$0.832547\pi$
$14$	0	0
$15$	−1.00000	−0.258199
$16$	0	0
$17$	2.47214	0.599581	0.299791	−	0.954005i	$-0.403083\pi$
$17$	2.47214	0.599581	0.299791	+	0.954005i	$0.403083\pi$
$18$	0	0
$19$	5.70820	1.30955	0.654776	−	0.755823i	$-0.272763\pi$
$19$	5.70820	1.30955	0.654776	+	0.755823i	$0.272763\pi$
$20$	0	0
$21$	−1.23607	−0.269732
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−5.00000	−0.962250
$28$	0	0
$29$	−0.527864	−0.0980219	−0.0490109	−	0.998798i	$-0.515607\pi$
$29$	−0.527864	−0.0980219	−0.0490109	+	0.998798i	$0.515607\pi$
$30$	0	0
$31$	−4.23607	−0.760820	−0.380410	−	0.924818i	$-0.624217\pi$
$31$	−4.23607	−0.760820	−0.380410	+	0.924818i	$0.624217\pi$
$32$	0	0
$33$	3.23607	0.563327
$34$	0	0
$35$	1.23607	0.208934
$36$	0	0
$37$	−9.70820	−1.59602	−0.798009	−	0.602645i	$-0.794114\pi$
$37$	−9.70820	−1.59602	−0.798009	+	0.602645i	$0.794114\pi$
$38$	0	0
$39$	−6.23607	−0.998570
$40$	0	0
$41$	−7.47214	−1.16695	−0.583476	−	0.812131i	$-0.698308\pi$
$41$	−7.47214	−1.16695	−0.583476	+	0.812131i	$0.698308\pi$
$42$	0	0
$43$	−3.70820	−0.565496	−0.282748	−	0.959194i	$-0.591246\pi$
$43$	−3.70820	−0.565496	−0.282748	+	0.959194i	$0.591246\pi$
$44$	0	0
$45$	2.00000	0.298142
$46$	0	0
$47$	−9.47214	−1.38165	−0.690827	−	0.723021i	$-0.742753\pi$
$47$	−9.47214	−1.38165	−0.690827	+	0.723021i	$0.742753\pi$
$48$	0	0
$49$	−5.47214	−0.781734
$50$	0	0
$51$	2.47214	0.346168
$52$	0	0
$53$	−6.00000	−0.824163	−0.412082	−	0.911147i	$-0.635198\pi$
$53$	−6.00000	−0.824163	−0.412082	+	0.911147i	$0.635198\pi$
$54$	0	0
$55$	−3.23607	−0.436351
$56$	0	0
$57$	5.70820	0.756070
$58$	0	0
$59$	−8.94427	−1.16445	−0.582223	−	0.813029i	$-0.697817\pi$
$59$	−8.94427	−1.16445	−0.582223	+	0.813029i	$0.697817\pi$
$60$	0	0
$61$	12.1803	1.55953	0.779766	−	0.626071i	$-0.215338\pi$
$61$	12.1803	1.55953	0.779766	+	0.626071i	$0.215338\pi$
$62$	0	0
$63$	2.47214	0.311460
$64$	0	0
$65$	6.23607	0.773489
$66$	0	0
$67$	−9.70820	−1.18605	−0.593023	−	0.805186i	$-0.702066\pi$
$67$	−9.70820	−1.18605	−0.593023	+	0.805186i	$0.702066\pi$
$68$	0	0
$69$	1.00000	0.120386
$70$	0	0
$71$	6.23607	0.740085	0.370043	−	0.929015i	$-0.379343\pi$
$71$	6.23607	0.740085	0.370043	+	0.929015i	$0.379343\pi$
$72$	0	0
$73$	−6.70820	−0.785136	−0.392568	−	0.919723i	$-0.628413\pi$
$73$	−6.70820	−0.785136	−0.392568	+	0.919723i	$0.628413\pi$
$74$	0	0
$75$	1.00000	0.115470
$76$	0	0
$77$	−4.00000	−0.455842
$78$	0	0
$79$	−8.76393	−0.986019	−0.493010	−	0.870024i	$-0.664103\pi$
$79$	−8.76393	−0.986019	−0.493010	+	0.870024i	$0.664103\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	6.47214	0.710409	0.355205	−	0.934789i	$-0.384411\pi$
$83$	6.47214	0.710409	0.355205	+	0.934789i	$0.384411\pi$
$84$	0	0
$85$	−2.47214	−0.268141
$86$	0	0
$87$	−0.527864	−0.0565930
$88$	0	0
$89$	2.76393	0.292976	0.146488	−	0.989212i	$-0.453203\pi$
$89$	2.76393	0.292976	0.146488	+	0.989212i	$0.453203\pi$
$90$	0	0
$91$	7.70820	0.808039
$92$	0	0
$93$	−4.23607	−0.439260
$94$	0	0
$95$	−5.70820	−0.585649
$96$	0	0
$97$	−6.18034	−0.627518	−0.313759	−	0.949503i	$-0.601588\pi$
$97$	−6.18034	−0.627518	−0.313759	+	0.949503i	$0.601588\pi$
$98$	0	0
$99$	−6.47214	−0.650474
$100$	0	0
$101$	2.94427	0.292966	0.146483	−	0.989213i	$-0.453205\pi$
$101$	2.94427	0.292966	0.146483	+	0.989213i	$0.453205\pi$
$102$	0	0
$103$	6.47214	0.637719	0.318859	−	0.947802i	$-0.396700\pi$
$103$	6.47214	0.637719	0.318859	+	0.947802i	$0.396700\pi$
$104$	0	0
$105$	1.23607	0.120628
$106$	0	0
$107$	8.18034	0.790823	0.395412	−	0.918504i	$-0.370602\pi$
$107$	8.18034	0.790823	0.395412	+	0.918504i	$0.370602\pi$
$108$	0	0
$109$	−11.7082	−1.12144	−0.560721	−	0.828005i	$-0.689476\pi$
$109$	−11.7082	−1.12144	−0.560721	+	0.828005i	$0.689476\pi$
$110$	0	0
$111$	−9.70820	−0.921462
$112$	0	0
$113$	5.23607	0.492568	0.246284	−	0.969198i	$-0.420790\pi$
$113$	5.23607	0.492568	0.246284	+	0.969198i	$0.420790\pi$
$114$	0	0
$115$	−1.00000	−0.0932505
$116$	0	0
$117$	12.4721	1.15305
$118$	0	0
$119$	−3.05573	−0.280118
$120$	0	0
$121$	−0.527864	−0.0479876
$122$	0	0
$123$	−7.47214	−0.673740
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	18.4164	1.63419	0.817096	−	0.576502i	$-0.195583\pi$
$127$	18.4164	1.63419	0.817096	+	0.576502i	$0.195583\pi$
$128$	0	0
$129$	−3.70820	−0.326489
$130$	0	0
$131$	−0.236068	−0.0206254	−0.0103127	−	0.999947i	$-0.503283\pi$
$131$	−0.236068	−0.0206254	−0.0103127	+	0.999947i	$0.503283\pi$
$132$	0	0
$133$	−7.05573	−0.611809
$134$	0	0
$135$	5.00000	0.430331
$136$	0	0
$137$	8.94427	0.764161	0.382080	−	0.924129i	$-0.375208\pi$
$137$	8.94427	0.764161	0.382080	+	0.924129i	$0.375208\pi$
$138$	0	0
$139$	2.70820	0.229707	0.114853	−	0.993382i	$-0.463360\pi$
$139$	2.70820	0.229707	0.114853	+	0.993382i	$0.463360\pi$
$140$	0	0
$141$	−9.47214	−0.797698
$142$	0	0
$143$	−20.1803	−1.68756
$144$	0	0
$145$	0.527864	0.0438367
$146$	0	0
$147$	−5.47214	−0.451334
$148$	0	0
$149$	−20.4721	−1.67714	−0.838571	−	0.544792i	$-0.816609\pi$
$149$	−20.4721	−1.67714	−0.838571	+	0.544792i	$0.816609\pi$
$150$	0	0
$151$	18.2361	1.48403	0.742015	−	0.670383i	$-0.233870\pi$
$151$	18.2361	1.48403	0.742015	+	0.670383i	$0.233870\pi$
$152$	0	0
$153$	−4.94427	−0.399721
$154$	0	0
$155$	4.23607	0.340249
$156$	0	0
$157$	7.70820	0.615182	0.307591	−	0.951519i	$-0.400477\pi$
$157$	7.70820	0.615182	0.307591	+	0.951519i	$0.400477\pi$
$158$	0	0
$159$	−6.00000	−0.475831
$160$	0	0
$161$	−1.23607	−0.0974158
$162$	0	0
$163$	−8.41641	−0.659224	−0.329612	−	0.944116i	$-0.606918\pi$
$163$	−8.41641	−0.659224	−0.329612	+	0.944116i	$0.606918\pi$
$164$	0	0
$165$	−3.23607	−0.251928
$166$	0	0
$167$	12.0000	0.928588	0.464294	−	0.885681i	$-0.346308\pi$
$167$	12.0000	0.928588	0.464294	+	0.885681i	$0.346308\pi$
$168$	0	0
$169$	25.8885	1.99143
$170$	0	0
$171$	−11.4164	−0.873035
$172$	0	0
$173$	−6.94427	−0.527963	−0.263982	−	0.964528i	$-0.585036\pi$
$173$	−6.94427	−0.527963	−0.263982	+	0.964528i	$0.585036\pi$
$174$	0	0
$175$	−1.23607	−0.0934380
$176$	0	0
$177$	−8.94427	−0.672293
$178$	0	0
$179$	−6.23607	−0.466106	−0.233053	−	0.972464i	$-0.574871\pi$
$179$	−6.23607	−0.466106	−0.233053	+	0.972464i	$0.574871\pi$
$180$	0	0
$181$	20.7639	1.54337	0.771685	−	0.636004i	$-0.219414\pi$
$181$	20.7639	1.54337	0.771685	+	0.636004i	$0.219414\pi$
$182$	0	0
$183$	12.1803	0.900397
$184$	0	0
$185$	9.70820	0.713761
$186$	0	0
$187$	8.00000	0.585018
$188$	0	0
$189$	6.18034	0.449554
$190$	0	0
$191$	6.00000	0.434145	0.217072	−	0.976156i	$-0.430349\pi$
$191$	6.00000	0.434145	0.217072	+	0.976156i	$0.430349\pi$
$192$	0	0
$193$	0.708204	0.0509776	0.0254888	−	0.999675i	$-0.491886\pi$
$193$	0.708204	0.0509776	0.0254888	+	0.999675i	$0.491886\pi$
$194$	0	0
$195$	6.23607	0.446574
$196$	0	0
$197$	−13.7639	−0.980640	−0.490320	−	0.871543i	$-0.663120\pi$
$197$	−13.7639	−0.980640	−0.490320	+	0.871543i	$0.663120\pi$
$198$	0	0
$199$	−5.70820	−0.404644	−0.202322	−	0.979319i	$-0.564849\pi$
$199$	−5.70820	−0.404644	−0.202322	+	0.979319i	$0.564849\pi$
$200$	0	0
$201$	−9.70820	−0.684764
$202$	0	0
$203$	0.652476	0.0457948
$204$	0	0
$205$	7.47214	0.521877
$206$	0	0
$207$	−2.00000	−0.139010
$208$	0	0
$209$	18.4721	1.27774
$210$	0	0
$211$	9.52786	0.655925	0.327963	−	0.944691i	$-0.393638\pi$
$211$	9.52786	0.655925	0.327963	+	0.944691i	$0.393638\pi$
$212$	0	0
$213$	6.23607	0.427288
$214$	0	0
$215$	3.70820	0.252897
$216$	0	0
$217$	5.23607	0.355447
$218$	0	0
$219$	−6.70820	−0.453298
$220$	0	0
$221$	−15.4164	−1.03702
$222$	0	0
$223$	−4.00000	−0.267860	−0.133930	−	0.990991i	$-0.542760\pi$
$223$	−4.00000	−0.267860	−0.133930	+	0.990991i	$0.542760\pi$
$224$	0	0
$225$	−2.00000	−0.133333
$226$	0	0
$227$	12.6525	0.839774	0.419887	−	0.907576i	$-0.362070\pi$
$227$	12.6525	0.839774	0.419887	+	0.907576i	$0.362070\pi$
$228$	0	0
$229$	−10.1803	−0.672736	−0.336368	−	0.941731i	$-0.609199\pi$
$229$	−10.1803	−0.672736	−0.336368	+	0.941731i	$0.609199\pi$
$230$	0	0
$231$	−4.00000	−0.263181
$232$	0	0
$233$	9.18034	0.601424	0.300712	−	0.953715i	$-0.402776\pi$
$233$	9.18034	0.601424	0.300712	+	0.953715i	$0.402776\pi$
$234$	0	0
$235$	9.47214	0.617894
$236$	0	0
$237$	−8.76393	−0.569279
$238$	0	0
$239$	2.23607	0.144639	0.0723196	−	0.997382i	$-0.476960\pi$
$239$	2.23607	0.144639	0.0723196	+	0.997382i	$0.476960\pi$
$240$	0	0
$241$	20.4721	1.31873	0.659363	−	0.751825i	$-0.270826\pi$
$241$	20.4721	1.31873	0.659363	+	0.751825i	$0.270826\pi$
$242$	0	0
$243$	16.0000	1.02640
$244$	0	0
$245$	5.47214	0.349602
$246$	0	0
$247$	−35.5967	−2.26497
$248$	0	0
$249$	6.47214	0.410155
$250$	0	0
$251$	−16.9443	−1.06951	−0.534756	−	0.845006i	$-0.679597\pi$
$251$	−16.9443	−1.06951	−0.534756	+	0.845006i	$0.679597\pi$
$252$	0	0
$253$	3.23607	0.203450
$254$	0	0
$255$	−2.47214	−0.154811
$256$	0	0
$257$	−11.1803	−0.697410	−0.348705	−	0.937232i	$-0.613379\pi$
$257$	−11.1803	−0.697410	−0.348705	+	0.937232i	$0.613379\pi$
$258$	0	0
$259$	12.0000	0.745644
$260$	0	0
$261$	1.05573	0.0653479
$262$	0	0
$263$	−21.7082	−1.33859	−0.669293	−	0.742999i	$-0.733403\pi$
$263$	−21.7082	−1.33859	−0.669293	+	0.742999i	$0.733403\pi$
$264$	0	0
$265$	6.00000	0.368577
$266$	0	0
$267$	2.76393	0.169150
$268$	0	0
$269$	−12.5279	−0.763837	−0.381919	−	0.924196i	$-0.624737\pi$
$269$	−12.5279	−0.763837	−0.381919	+	0.924196i	$0.624737\pi$
$270$	0	0
$271$	7.41641	0.450515	0.225257	−	0.974299i	$-0.427678\pi$
$271$	7.41641	0.450515	0.225257	+	0.974299i	$0.427678\pi$
$272$	0	0
$273$	7.70820	0.466522
$274$	0	0
$275$	3.23607	0.195142
$276$	0	0
$277$	−21.1803	−1.27260	−0.636302	−	0.771440i	$-0.719537\pi$
$277$	−21.1803	−1.27260	−0.636302	+	0.771440i	$0.719537\pi$
$278$	0	0
$279$	8.47214	0.507214
$280$	0	0
$281$	−1.23607	−0.0737376	−0.0368688	−	0.999320i	$-0.511738\pi$
$281$	−1.23607	−0.0737376	−0.0368688	+	0.999320i	$0.511738\pi$
$282$	0	0
$283$	−11.5279	−0.685260	−0.342630	−	0.939470i	$-0.611318\pi$
$283$	−11.5279	−0.685260	−0.342630	+	0.939470i	$0.611318\pi$
$284$	0	0
$285$	−5.70820	−0.338125
$286$	0	0
$287$	9.23607	0.545188
$288$	0	0
$289$	−10.8885	−0.640503
$290$	0	0
$291$	−6.18034	−0.362298
$292$	0	0
$293$	−12.9443	−0.756212	−0.378106	−	0.925762i	$-0.623425\pi$
$293$	−12.9443	−0.756212	−0.378106	+	0.925762i	$0.623425\pi$
$294$	0	0
$295$	8.94427	0.520756
$296$	0	0
$297$	−16.1803	−0.938879
$298$	0	0
$299$	−6.23607	−0.360641
$300$	0	0
$301$	4.58359	0.264194
$302$	0	0
$303$	2.94427	0.169144
$304$	0	0
$305$	−12.1803	−0.697444
$306$	0	0
$307$	4.58359	0.261599	0.130800	−	0.991409i	$-0.458246\pi$
$307$	4.58359	0.261599	0.130800	+	0.991409i	$0.458246\pi$
$308$	0	0
$309$	6.47214	0.368187
$310$	0	0
$311$	6.81966	0.386707	0.193354	−	0.981129i	$-0.438064\pi$
$311$	6.81966	0.386707	0.193354	+	0.981129i	$0.438064\pi$
$312$	0	0
$313$	7.41641	0.419200	0.209600	−	0.977787i	$-0.432784\pi$
$313$	7.41641	0.419200	0.209600	+	0.977787i	$0.432784\pi$
$314$	0	0
$315$	−2.47214	−0.139289
$316$	0	0
$317$	−34.9443	−1.96267	−0.981333	−	0.192317i	$-0.938400\pi$
$317$	−34.9443	−1.96267	−0.981333	+	0.192317i	$0.938400\pi$
$318$	0	0
$319$	−1.70820	−0.0956411
$320$	0	0
$321$	8.18034	0.456582
$322$	0	0
$323$	14.1115	0.785182
$324$	0	0
$325$	−6.23607	−0.345915
$326$	0	0
$327$	−11.7082	−0.647465
$328$	0	0
$329$	11.7082	0.645494
$330$	0	0
$331$	14.7082	0.808436	0.404218	−	0.914663i	$-0.367544\pi$
$331$	14.7082	0.808436	0.404218	+	0.914663i	$0.367544\pi$
$332$	0	0
$333$	19.4164	1.06401
$334$	0	0
$335$	9.70820	0.530416
$336$	0	0
$337$	19.1246	1.04178	0.520892	−	0.853623i	$-0.325599\pi$
$337$	19.1246	1.04178	0.520892	+	0.853623i	$0.325599\pi$
$338$	0	0
$339$	5.23607	0.284384
$340$	0	0
$341$	−13.7082	−0.742341
$342$	0	0
$343$	15.4164	0.832408
$344$	0	0
$345$	−1.00000	−0.0538382
$346$	0	0
$347$	−19.0557	−1.02297	−0.511483	−	0.859294i	$-0.670904\pi$
$347$	−19.0557	−1.02297	−0.511483	+	0.859294i	$0.670904\pi$
$348$	0	0
$349$	21.3607	1.14341	0.571705	−	0.820459i	$-0.306282\pi$
$349$	21.3607	1.14341	0.571705	+	0.820459i	$0.306282\pi$
$350$	0	0
$351$	31.1803	1.66428
$352$	0	0
$353$	0.347524	0.0184968	0.00924842	−	0.999957i	$-0.497056\pi$
$353$	0.347524	0.0184968	0.00924842	+	0.999957i	$0.497056\pi$
$354$	0	0
$355$	−6.23607	−0.330976
$356$	0	0
$357$	−3.05573	−0.161726
$358$	0	0
$359$	24.7639	1.30699	0.653495	−	0.756931i	$-0.273302\pi$
$359$	24.7639	1.30699	0.653495	+	0.756931i	$0.273302\pi$
$360$	0	0
$361$	13.5836	0.714926
$362$	0	0
$363$	−0.527864	−0.0277057
$364$	0	0
$365$	6.70820	0.351123
$366$	0	0
$367$	12.0000	0.626395	0.313197	−	0.949688i	$-0.398600\pi$
$367$	12.0000	0.626395	0.313197	+	0.949688i	$0.398600\pi$
$368$	0	0
$369$	14.9443	0.777968
$370$	0	0
$371$	7.41641	0.385041
$372$	0	0
$373$	−6.58359	−0.340885	−0.170443	−	0.985368i	$-0.554520\pi$
$373$	−6.58359	−0.340885	−0.170443	+	0.985368i	$0.554520\pi$
$374$	0	0
$375$	−1.00000	−0.0516398
$376$	0	0
$377$	3.29180	0.169536
$378$	0	0
$379$	−24.0000	−1.23280	−0.616399	−	0.787434i	$-0.711409\pi$
$379$	−24.0000	−1.23280	−0.616399	+	0.787434i	$0.711409\pi$
$380$	0	0
$381$	18.4164	0.943501
$382$	0	0
$383$	−18.1803	−0.928972	−0.464486	−	0.885580i	$-0.653761\pi$
$383$	−18.1803	−0.928972	−0.464486	+	0.885580i	$0.653761\pi$
$384$	0	0
$385$	4.00000	0.203859
$386$	0	0
$387$	7.41641	0.376997
$388$	0	0
$389$	−24.0000	−1.21685	−0.608424	−	0.793612i	$-0.708198\pi$
$389$	−24.0000	−1.21685	−0.608424	+	0.793612i	$0.708198\pi$
$390$	0	0
$391$	2.47214	0.125021
$392$	0	0
$393$	−0.236068	−0.0119081
$394$	0	0
$395$	8.76393	0.440961
$396$	0	0
$397$	17.7639	0.891546	0.445773	−	0.895146i	$-0.352929\pi$
$397$	17.7639	0.891546	0.445773	+	0.895146i	$0.352929\pi$
$398$	0	0
$399$	−7.05573	−0.353228
$400$	0	0
$401$	−4.47214	−0.223328	−0.111664	−	0.993746i	$-0.535618\pi$
$401$	−4.47214	−0.223328	−0.111664	+	0.993746i	$0.535618\pi$
$402$	0	0
$403$	26.4164	1.31590
$404$	0	0
$405$	−1.00000	−0.0496904
$406$	0	0
$407$	−31.4164	−1.55725
$408$	0	0
$409$	5.00000	0.247234	0.123617	−	0.992330i	$-0.460551\pi$
$409$	5.00000	0.247234	0.123617	+	0.992330i	$0.460551\pi$
$410$	0	0
$411$	8.94427	0.441188
$412$	0	0
$413$	11.0557	0.544017
$414$	0	0
$415$	−6.47214	−0.317705
$416$	0	0
$417$	2.70820	0.132621
$418$	0	0
$419$	5.23607	0.255799	0.127899	−	0.991787i	$-0.459177\pi$
$419$	5.23607	0.255799	0.127899	+	0.991787i	$0.459177\pi$
$420$	0	0
$421$	35.1246	1.71187	0.855934	−	0.517084i	$-0.172983\pi$
$421$	35.1246	1.71187	0.855934	+	0.517084i	$0.172983\pi$
$422$	0	0
$423$	18.9443	0.921102
$424$	0	0
$425$	2.47214	0.119916
$426$	0	0
$427$	−15.0557	−0.728598
$428$	0	0
$429$	−20.1803	−0.974316
$430$	0	0
$431$	20.6525	0.994795	0.497397	−	0.867523i	$-0.334289\pi$
$431$	20.6525	0.994795	0.497397	+	0.867523i	$0.334289\pi$
$432$	0	0
$433$	8.47214	0.407145	0.203572	−	0.979060i	$-0.434745\pi$
$433$	8.47214	0.407145	0.203572	+	0.979060i	$0.434745\pi$
$434$	0	0
$435$	0.527864	0.0253091
$436$	0	0
$437$	5.70820	0.273060
$438$	0	0
$439$	22.7082	1.08380	0.541902	−	0.840442i	$-0.317704\pi$
$439$	22.7082	1.08380	0.541902	+	0.840442i	$0.317704\pi$
$440$	0	0
$441$	10.9443	0.521156
$442$	0	0
$443$	−8.52786	−0.405171	−0.202586	−	0.979265i	$-0.564934\pi$
$443$	−8.52786	−0.405171	−0.202586	+	0.979265i	$0.564934\pi$
$444$	0	0
$445$	−2.76393	−0.131023
$446$	0	0
$447$	−20.4721	−0.968299
$448$	0	0
$449$	−20.8328	−0.983161	−0.491581	−	0.870832i	$-0.663581\pi$
$449$	−20.8328	−0.983161	−0.491581	+	0.870832i	$0.663581\pi$
$450$	0	0
$451$	−24.1803	−1.13861
$452$	0	0
$453$	18.2361	0.856805
$454$	0	0
$455$	−7.70820	−0.361366
$456$	0	0
$457$	−11.4164	−0.534037	−0.267019	−	0.963691i	$-0.586038\pi$
$457$	−11.4164	−0.534037	−0.267019	+	0.963691i	$0.586038\pi$
$458$	0	0
$459$	−12.3607	−0.576947
$460$	0	0
$461$	21.0000	0.978068	0.489034	−	0.872265i	$-0.337349\pi$
$461$	21.0000	0.978068	0.489034	+	0.872265i	$0.337349\pi$
$462$	0	0
$463$	6.47214	0.300786	0.150393	−	0.988626i	$-0.451946\pi$
$463$	6.47214	0.300786	0.150393	+	0.988626i	$0.451946\pi$
$464$	0	0
$465$	4.23607	0.196443
$466$	0	0
$467$	−15.5279	−0.718544	−0.359272	−	0.933233i	$-0.616975\pi$
$467$	−15.5279	−0.718544	−0.359272	+	0.933233i	$0.616975\pi$
$468$	0	0
$469$	12.0000	0.554109
$470$	0	0
$471$	7.70820	0.355175
$472$	0	0
$473$	−12.0000	−0.551761
$474$	0	0
$475$	5.70820	0.261910
$476$	0	0
$477$	12.0000	0.549442
$478$	0	0
$479$	−26.8328	−1.22602	−0.613011	−	0.790074i	$-0.710042\pi$
$479$	−26.8328	−1.22602	−0.613011	+	0.790074i	$0.710042\pi$
$480$	0	0
$481$	60.5410	2.76043
$482$	0	0
$483$	−1.23607	−0.0562430
$484$	0	0
$485$	6.18034	0.280635
$486$	0	0
$487$	−6.88854	−0.312150	−0.156075	−	0.987745i	$-0.549884\pi$
$487$	−6.88854	−0.312150	−0.156075	+	0.987745i	$0.549884\pi$
$488$	0	0
$489$	−8.41641	−0.380603
$490$	0	0
$491$	−1.65248	−0.0745752	−0.0372876	−	0.999305i	$-0.511872\pi$
$491$	−1.65248	−0.0745752	−0.0372876	+	0.999305i	$0.511872\pi$
$492$	0	0
$493$	−1.30495	−0.0587721
$494$	0	0
$495$	6.47214	0.290901
$496$	0	0
$497$	−7.70820	−0.345760
$498$	0	0
$499$	36.1246	1.61716	0.808580	−	0.588386i	$-0.200237\pi$
$499$	36.1246	1.61716	0.808580	+	0.588386i	$0.200237\pi$
$500$	0	0
$501$	12.0000	0.536120
$502$	0	0
$503$	−6.00000	−0.267527	−0.133763	−	0.991013i	$-0.542706\pi$
$503$	−6.00000	−0.267527	−0.133763	+	0.991013i	$0.542706\pi$
$504$	0	0
$505$	−2.94427	−0.131018
$506$	0	0
$507$	25.8885	1.14975
$508$	0	0
$509$	−11.9443	−0.529421	−0.264710	−	0.964328i	$-0.585276\pi$
$509$	−11.9443	−0.529421	−0.264710	+	0.964328i	$0.585276\pi$
$510$	0	0
$511$	8.29180	0.366807
$512$	0	0
$513$	−28.5410	−1.26012
$514$	0	0
$515$	−6.47214	−0.285196
$516$	0	0
$517$	−30.6525	−1.34809
$518$	0	0
$519$	−6.94427	−0.304820
$520$	0	0
$521$	−21.8885	−0.958955	−0.479477	−	0.877554i	$-0.659174\pi$
$521$	−21.8885	−0.958955	−0.479477	+	0.877554i	$0.659174\pi$
$522$	0	0
$523$	−23.1246	−1.01117	−0.505584	−	0.862777i	$-0.668723\pi$
$523$	−23.1246	−1.01117	−0.505584	+	0.862777i	$0.668723\pi$
$524$	0	0
$525$	−1.23607	−0.0539464
$526$	0	0
$527$	−10.4721	−0.456173
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	17.8885	0.776297
$532$	0	0
$533$	46.5967	2.01833
$534$	0	0
$535$	−8.18034	−0.353667
$536$	0	0
$537$	−6.23607	−0.269106
$538$	0	0
$539$	−17.7082	−0.762746
$540$	0	0
$541$	−1.11146	−0.0477852	−0.0238926	−	0.999715i	$-0.507606\pi$
$541$	−1.11146	−0.0477852	−0.0238926	+	0.999715i	$0.507606\pi$
$542$	0	0
$543$	20.7639	0.891066
$544$	0	0
$545$	11.7082	0.501524
$546$	0	0
$547$	−17.0000	−0.726868	−0.363434	−	0.931620i	$-0.618396\pi$
$547$	−17.0000	−0.726868	−0.363434	+	0.931620i	$0.618396\pi$
$548$	0	0
$549$	−24.3607	−1.03969
$550$	0	0
$551$	−3.01316	−0.128365
$552$	0	0
$553$	10.8328	0.460658
$554$	0	0
$555$	9.70820	0.412090
$556$	0	0
$557$	21.5967	0.915084	0.457542	−	0.889188i	$-0.348730\pi$
$557$	21.5967	0.915084	0.457542	+	0.889188i	$0.348730\pi$
$558$	0	0
$559$	23.1246	0.978067
$560$	0	0
$561$	8.00000	0.337760
$562$	0	0
$563$	−24.3607	−1.02668	−0.513340	−	0.858185i	$-0.671592\pi$
$563$	−24.3607	−1.02668	−0.513340	+	0.858185i	$0.671592\pi$
$564$	0	0
$565$	−5.23607	−0.220283
$566$	0	0
$567$	−1.23607	−0.0519100
$568$	0	0
$569$	2.36068	0.0989648	0.0494824	−	0.998775i	$-0.484243\pi$
$569$	2.36068	0.0989648	0.0494824	+	0.998775i	$0.484243\pi$
$570$	0	0
$571$	−30.8328	−1.29031	−0.645157	−	0.764050i	$-0.723208\pi$
$571$	−30.8328	−1.29031	−0.645157	+	0.764050i	$0.723208\pi$
$572$	0	0
$573$	6.00000	0.250654
$574$	0	0
$575$	1.00000	0.0417029
$576$	0	0
$577$	28.7082	1.19514	0.597569	−	0.801817i	$-0.296133\pi$
$577$	28.7082	1.19514	0.597569	+	0.801817i	$0.296133\pi$
$578$	0	0
$579$	0.708204	0.0294320
$580$	0	0
$581$	−8.00000	−0.331896
$582$	0	0
$583$	−19.4164	−0.804145
$584$	0	0
$585$	−12.4721	−0.515659
$586$	0	0
$587$	−40.3050	−1.66356	−0.831782	−	0.555103i	$-0.812679\pi$
$587$	−40.3050	−1.66356	−0.831782	+	0.555103i	$0.812679\pi$
$588$	0	0
$589$	−24.1803	−0.996334
$590$	0	0
$591$	−13.7639	−0.566173
$592$	0	0
$593$	−1.63932	−0.0673188	−0.0336594	−	0.999433i	$-0.510716\pi$
$593$	−1.63932	−0.0673188	−0.0336594	+	0.999433i	$0.510716\pi$
$594$	0	0
$595$	3.05573	0.125273
$596$	0	0
$597$	−5.70820	−0.233621
$598$	0	0
$599$	47.7771	1.95212	0.976059	−	0.217504i	$-0.0697915\pi$
$599$	47.7771	1.95212	0.976059	+	0.217504i	$0.0697915\pi$
$600$	0	0
$601$	−3.94427	−0.160890	−0.0804451	−	0.996759i	$-0.525634\pi$
$601$	−3.94427	−0.160890	−0.0804451	+	0.996759i	$0.525634\pi$
$602$	0	0
$603$	19.4164	0.790697
$604$	0	0
$605$	0.527864	0.0214607
$606$	0	0
$607$	18.4721	0.749761	0.374880	−	0.927073i	$-0.377684\pi$
$607$	18.4721	0.749761	0.374880	+	0.927073i	$0.377684\pi$
$608$	0	0
$609$	0.652476	0.0264397
$610$	0	0
$611$	59.0689	2.38967
$612$	0	0
$613$	−17.3050	−0.698940	−0.349470	−	0.936947i	$-0.613638\pi$
$613$	−17.3050	−0.698940	−0.349470	+	0.936947i	$0.613638\pi$
$614$	0	0
$615$	7.47214	0.301306
$616$	0	0
$617$	12.1803	0.490362	0.245181	−	0.969477i	$-0.421153\pi$
$617$	12.1803	0.490362	0.245181	+	0.969477i	$0.421153\pi$
$618$	0	0
$619$	36.3607	1.46146	0.730730	−	0.682667i	$-0.239180\pi$
$619$	36.3607	1.46146	0.730730	+	0.682667i	$0.239180\pi$
$620$	0	0
$621$	−5.00000	−0.200643
$622$	0	0
$623$	−3.41641	−0.136875
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	18.4721	0.737706
$628$	0	0
$629$	−24.0000	−0.956943
$630$	0	0
$631$	3.70820	0.147621	0.0738106	−	0.997272i	$-0.476484\pi$
$631$	3.70820	0.147621	0.0738106	+	0.997272i	$0.476484\pi$
$632$	0	0
$633$	9.52786	0.378699
$634$	0	0
$635$	−18.4164	−0.730833
$636$	0	0
$637$	34.1246	1.35207
$638$	0	0
$639$	−12.4721	−0.493390
$640$	0	0
$641$	50.1803	1.98200	0.991002	−	0.133846i	$-0.0427328\pi$
$641$	50.1803	1.98200	0.991002	+	0.133846i	$0.0427328\pi$
$642$	0	0
$643$	20.8328	0.821566	0.410783	−	0.911733i	$-0.365255\pi$
$643$	20.8328	0.821566	0.410783	+	0.911733i	$0.365255\pi$
$644$	0	0
$645$	3.70820	0.146010
$646$	0	0
$647$	23.8328	0.936965	0.468482	−	0.883473i	$-0.344801\pi$
$647$	23.8328	0.936965	0.468482	+	0.883473i	$0.344801\pi$
$648$	0	0
$649$	−28.9443	−1.13616
$650$	0	0
$651$	5.23607	0.205218
$652$	0	0
$653$	−1.65248	−0.0646664	−0.0323332	−	0.999477i	$-0.510294\pi$
$653$	−1.65248	−0.0646664	−0.0323332	+	0.999477i	$0.510294\pi$
$654$	0	0
$655$	0.236068	0.00922394
$656$	0	0
$657$	13.4164	0.523424
$658$	0	0
$659$	24.9443	0.971691	0.485845	−	0.874045i	$-0.338512\pi$
$659$	24.9443	0.971691	0.485845	+	0.874045i	$0.338512\pi$
$660$	0	0
$661$	−26.0000	−1.01128	−0.505641	−	0.862744i	$-0.668744\pi$
$661$	−26.0000	−1.01128	−0.505641	+	0.862744i	$0.668744\pi$
$662$	0	0
$663$	−15.4164	−0.598724
$664$	0	0
$665$	7.05573	0.273609
$666$	0	0
$667$	−0.527864	−0.0204390
$668$	0	0
$669$	−4.00000	−0.154649
$670$	0	0
$671$	39.4164	1.52165
$672$	0	0
$673$	−39.5410	−1.52419	−0.762097	−	0.647463i	$-0.775830\pi$
$673$	−39.5410	−1.52419	−0.762097	+	0.647463i	$0.775830\pi$
$674$	0	0
$675$	−5.00000	−0.192450
$676$	0	0
$677$	−6.00000	−0.230599	−0.115299	−	0.993331i	$-0.536783\pi$
$677$	−6.00000	−0.230599	−0.115299	+	0.993331i	$0.536783\pi$
$678$	0	0
$679$	7.63932	0.293170
$680$	0	0
$681$	12.6525	0.484844
$682$	0	0
$683$	22.4164	0.857740	0.428870	−	0.903366i	$-0.358912\pi$
$683$	22.4164	0.857740	0.428870	+	0.903366i	$0.358912\pi$
$684$	0	0
$685$	−8.94427	−0.341743
$686$	0	0
$687$	−10.1803	−0.388404
$688$	0	0
$689$	37.4164	1.42545
$690$	0	0
$691$	14.4721	0.550546	0.275273	−	0.961366i	$-0.411232\pi$
$691$	14.4721	0.550546	0.275273	+	0.961366i	$0.411232\pi$
$692$	0	0
$693$	8.00000	0.303895
$694$	0	0
$695$	−2.70820	−0.102728
$696$	0	0
$697$	−18.4721	−0.699682
$698$	0	0
$699$	9.18034	0.347232
$700$	0	0
$701$	−45.5967	−1.72217	−0.861083	−	0.508465i	$-0.830213\pi$
$701$	−45.5967	−1.72217	−0.861083	+	0.508465i	$0.830213\pi$
$702$	0	0
$703$	−55.4164	−2.09007
$704$	0	0
$705$	9.47214	0.356741
$706$	0	0
$707$	−3.63932	−0.136871
$708$	0	0
$709$	−23.7082	−0.890380	−0.445190	−	0.895436i	$-0.646864\pi$
$709$	−23.7082	−0.890380	−0.445190	+	0.895436i	$0.646864\pi$
$710$	0	0
$711$	17.5279	0.657346
$712$	0	0
$713$	−4.23607	−0.158642
$714$	0	0
$715$	20.1803	0.754702
$716$	0	0
$717$	2.23607	0.0835075
$718$	0	0
$719$	7.41641	0.276585	0.138293	−	0.990391i	$-0.455839\pi$
$719$	7.41641	0.276585	0.138293	+	0.990391i	$0.455839\pi$
$720$	0	0
$721$	−8.00000	−0.297936
$722$	0	0
$723$	20.4721	0.761367
$724$	0	0
$725$	−0.527864	−0.0196044
$726$	0	0
$727$	28.8328	1.06935	0.534675	−	0.845058i	$-0.320434\pi$
$727$	28.8328	1.06935	0.534675	+	0.845058i	$0.320434\pi$
$728$	0	0
$729$	13.0000	0.481481
$730$	0	0
$731$	−9.16718	−0.339061
$732$	0	0
$733$	7.41641	0.273931	0.136966	−	0.990576i	$-0.456265\pi$
$733$	7.41641	0.273931	0.136966	+	0.990576i	$0.456265\pi$
$734$	0	0
$735$	5.47214	0.201843
$736$	0	0
$737$	−31.4164	−1.15724
$738$	0	0
$739$	−44.4853	−1.63642	−0.818209	−	0.574921i	$-0.805033\pi$
$739$	−44.4853	−1.63642	−0.818209	+	0.574921i	$0.805033\pi$
$740$	0	0
$741$	−35.5967	−1.30768
$742$	0	0
$743$	−17.5279	−0.643035	−0.321517	−	0.946904i	$-0.604193\pi$
$743$	−17.5279	−0.643035	−0.321517	+	0.946904i	$0.604193\pi$
$744$	0	0
$745$	20.4721	0.750041
$746$	0	0
$747$	−12.9443	−0.473606
$748$	0	0
$749$	−10.1115	−0.369465
$750$	0	0
$751$	−47.7082	−1.74090	−0.870449	−	0.492259i	$-0.836171\pi$
$751$	−47.7082	−1.74090	−0.870449	+	0.492259i	$0.836171\pi$
$752$	0	0
$753$	−16.9443	−0.617484
$754$	0	0
$755$	−18.2361	−0.663678
$756$	0	0
$757$	−53.1246	−1.93085	−0.965423	−	0.260687i	$-0.916051\pi$
$757$	−53.1246	−1.93085	−0.965423	+	0.260687i	$0.916051\pi$
$758$	0	0
$759$	3.23607	0.117462
$760$	0	0
$761$	−32.8885	−1.19221	−0.596104	−	0.802907i	$-0.703286\pi$
$761$	−32.8885	−1.19221	−0.596104	+	0.802907i	$0.703286\pi$
$762$	0	0
$763$	14.4721	0.523926
$764$	0	0
$765$	4.94427	0.178761
$766$	0	0
$767$	55.7771	2.01399
$768$	0	0
$769$	54.3607	1.96030	0.980148	−	0.198267i	$-0.0635312\pi$
$769$	54.3607	1.96030	0.980148	+	0.198267i	$0.0635312\pi$
$770$	0	0
$771$	−11.1803	−0.402650
$772$	0	0
$773$	39.4853	1.42019	0.710094	−	0.704107i	$-0.248653\pi$
$773$	39.4853	1.42019	0.710094	+	0.704107i	$0.248653\pi$
$774$	0	0
$775$	−4.23607	−0.152164
$776$	0	0
$777$	12.0000	0.430498
$778$	0	0
$779$	−42.6525	−1.52818
$780$	0	0
$781$	20.1803	0.722109
$782$	0	0
$783$	2.63932	0.0943216
$784$	0	0
$785$	−7.70820	−0.275118
$786$	0	0
$787$	−45.3050	−1.61495	−0.807474	−	0.589904i	$-0.799166\pi$
$787$	−45.3050	−1.61495	−0.807474	+	0.589904i	$0.799166\pi$
$788$	0	0
$789$	−21.7082	−0.772833
$790$	0	0
$791$	−6.47214	−0.230123
$792$	0	0
$793$	−75.9574	−2.69733
$794$	0	0
$795$	6.00000	0.212798
$796$	0	0
$797$	−9.70820	−0.343882	−0.171941	−	0.985107i	$-0.555004\pi$
$797$	−9.70820	−0.343882	−0.171941	+	0.985107i	$0.555004\pi$
$798$	0	0
$799$	−23.4164	−0.828413
$800$	0	0
$801$	−5.52786	−0.195317
$802$	0	0
$803$	−21.7082	−0.766066
$804$	0	0
$805$	1.23607	0.0435657
$806$	0	0
$807$	−12.5279	−0.441002
$808$	0	0
$809$	12.1115	0.425816	0.212908	−	0.977072i	$-0.431707\pi$
$809$	12.1115	0.425816	0.212908	+	0.977072i	$0.431707\pi$
$810$	0	0
$811$	−43.0689	−1.51235	−0.756177	−	0.654368i	$-0.772935\pi$
$811$	−43.0689	−1.51235	−0.756177	+	0.654368i	$0.772935\pi$
$812$	0	0
$813$	7.41641	0.260105
$814$	0	0
$815$	8.41641	0.294814
$816$	0	0
$817$	−21.1672	−0.740546
$818$	0	0
$819$	−15.4164	−0.538693
$820$	0	0
$821$	−0.111456	−0.00388985	−0.00194492	−	0.999998i	$-0.500619\pi$
$821$	−0.111456	−0.00388985	−0.00194492	+	0.999998i	$0.500619\pi$
$822$	0	0
$823$	−5.47214	−0.190747	−0.0953733	−	0.995442i	$-0.530404\pi$
$823$	−5.47214	−0.190747	−0.0953733	+	0.995442i	$0.530404\pi$
$824$	0	0
$825$	3.23607	0.112665
$826$	0	0
$827$	−49.3050	−1.71450	−0.857251	−	0.514899i	$-0.827829\pi$
$827$	−49.3050	−1.71450	−0.857251	+	0.514899i	$0.827829\pi$
$828$	0	0
$829$	28.2492	0.981136	0.490568	−	0.871403i	$-0.336789\pi$
$829$	28.2492	0.981136	0.490568	+	0.871403i	$0.336789\pi$
$830$	0	0
$831$	−21.1803	−0.734738
$832$	0	0
$833$	−13.5279	−0.468713
$834$	0	0
$835$	−12.0000	−0.415277
$836$	0	0
$837$	21.1803	0.732100
$838$	0	0
$839$	−33.8197	−1.16758	−0.583792	−	0.811903i	$-0.698432\pi$
$839$	−33.8197	−1.16758	−0.583792	+	0.811903i	$0.698432\pi$
$840$	0	0
$841$	−28.7214	−0.990392
$842$	0	0
$843$	−1.23607	−0.0425724
$844$	0	0
$845$	−25.8885	−0.890593
$846$	0	0
$847$	0.652476	0.0224193
$848$	0	0
$849$	−11.5279	−0.395635
$850$	0	0
$851$	−9.70820	−0.332793
$852$	0	0
$853$	20.8328	0.713302	0.356651	−	0.934238i	$-0.383919\pi$
$853$	20.8328	0.713302	0.356651	+	0.934238i	$0.383919\pi$
$854$	0	0
$855$	11.4164	0.390433
$856$	0	0
$857$	48.5967	1.66003	0.830017	−	0.557739i	$-0.188331\pi$
$857$	48.5967	1.66003	0.830017	+	0.557739i	$0.188331\pi$
$858$	0	0
$859$	32.1246	1.09608	0.548039	−	0.836453i	$-0.315375\pi$
$859$	32.1246	1.09608	0.548039	+	0.836453i	$0.315375\pi$
$860$	0	0
$861$	9.23607	0.314764
$862$	0	0
$863$	18.7771	0.639179	0.319590	−	0.947556i	$-0.396455\pi$
$863$	18.7771	0.639179	0.319590	+	0.947556i	$0.396455\pi$
$864$	0	0
$865$	6.94427	0.236112
$866$	0	0
$867$	−10.8885	−0.369794
$868$	0	0
$869$	−28.3607	−0.962070
$870$	0	0
$871$	60.5410	2.05135
$872$	0	0
$873$	12.3607	0.418346
$874$	0	0
$875$	1.23607	0.0417867
$876$	0	0
$877$	−2.58359	−0.0872417	−0.0436209	−	0.999048i	$-0.513889\pi$
$877$	−2.58359	−0.0872417	−0.0436209	+	0.999048i	$0.513889\pi$
$878$	0	0
$879$	−12.9443	−0.436599
$880$	0	0
$881$	17.8885	0.602680	0.301340	−	0.953517i	$-0.402566\pi$
$881$	17.8885	0.602680	0.301340	+	0.953517i	$0.402566\pi$
$882$	0	0
$883$	33.8885	1.14044	0.570220	−	0.821492i	$-0.306858\pi$
$883$	33.8885	1.14044	0.570220	+	0.821492i	$0.306858\pi$
$884$	0	0
$885$	8.94427	0.300658
$886$	0	0
$887$	−31.9443	−1.07258	−0.536292	−	0.844033i	$-0.680175\pi$
$887$	−31.9443	−1.07258	−0.536292	+	0.844033i	$0.680175\pi$
$888$	0	0
$889$	−22.7639	−0.763478
$890$	0	0
$891$	3.23607	0.108412
$892$	0	0
$893$	−54.0689	−1.80935
$894$	0	0
$895$	6.23607	0.208449
$896$	0	0
$897$	−6.23607	−0.208216
$898$	0	0
$899$	2.23607	0.0745770
$900$	0	0
$901$	−14.8328	−0.494153
$902$	0	0
$903$	4.58359	0.152532
$904$	0	0
$905$	−20.7639	−0.690216
$906$	0	0
$907$	27.2361	0.904359	0.452179	−	0.891927i	$-0.350647\pi$
$907$	27.2361	0.904359	0.452179	+	0.891927i	$0.350647\pi$
$908$	0	0
$909$	−5.88854	−0.195311
$910$	0	0
$911$	−59.2361	−1.96258	−0.981289	−	0.192539i	$-0.938328\pi$
$911$	−59.2361	−1.96258	−0.981289	+	0.192539i	$0.938328\pi$
$912$	0	0
$913$	20.9443	0.693154
$914$	0	0
$915$	−12.1803	−0.402670
$916$	0	0
$917$	0.291796	0.00963596
$918$	0	0
$919$	12.2918	0.405469	0.202734	−	0.979234i	$-0.435017\pi$
$919$	12.2918	0.405469	0.202734	+	0.979234i	$0.435017\pi$
$920$	0	0
$921$	4.58359	0.151034
$922$	0	0
$923$	−38.8885	−1.28003
$924$	0	0
$925$	−9.70820	−0.319204
$926$	0	0
$927$	−12.9443	−0.425146
$928$	0	0
$929$	37.4721	1.22942	0.614710	−	0.788753i	$-0.289273\pi$
$929$	37.4721	1.22942	0.614710	+	0.788753i	$0.289273\pi$
$930$	0	0
$931$	−31.2361	−1.02372
$932$	0	0
$933$	6.81966	0.223266
$934$	0	0
$935$	−8.00000	−0.261628
$936$	0	0
$937$	−39.2361	−1.28179	−0.640893	−	0.767630i	$-0.721436\pi$
$937$	−39.2361	−1.28179	−0.640893	+	0.767630i	$0.721436\pi$
$938$	0	0
$939$	7.41641	0.242025
$940$	0	0
$941$	1.52786	0.0498069	0.0249035	−	0.999690i	$-0.492072\pi$
$941$	1.52786	0.0498069	0.0249035	+	0.999690i	$0.492072\pi$
$942$	0	0
$943$	−7.47214	−0.243326
$944$	0	0
$945$	−6.18034	−0.201046
$946$	0	0
$947$	−20.3050	−0.659822	−0.329911	−	0.944012i	$-0.607019\pi$
$947$	−20.3050	−0.659822	−0.329911	+	0.944012i	$0.607019\pi$
$948$	0	0
$949$	41.8328	1.35795
$950$	0	0
$951$	−34.9443	−1.13315
$952$	0	0
$953$	10.3607	0.335615	0.167808	−	0.985820i	$-0.446331\pi$
$953$	10.3607	0.335615	0.167808	+	0.985820i	$0.446331\pi$
$954$	0	0
$955$	−6.00000	−0.194155
$956$	0	0
$957$	−1.70820	−0.0552184
$958$	0	0
$959$	−11.0557	−0.357008
$960$	0	0
$961$	−13.0557	−0.421153
$962$	0	0
$963$	−16.3607	−0.527216
$964$	0	0
$965$	−0.708204	−0.0227979
$966$	0	0
$967$	−4.05573	−0.130423	−0.0652117	−	0.997871i	$-0.520772\pi$
$967$	−4.05573	−0.130423	−0.0652117	+	0.997871i	$0.520772\pi$
$968$	0	0
$969$	14.1115	0.453325
$970$	0	0
$971$	32.8328	1.05366	0.526828	−	0.849972i	$-0.323381\pi$
$971$	32.8328	1.05366	0.526828	+	0.849972i	$0.323381\pi$
$972$	0	0
$973$	−3.34752	−0.107317
$974$	0	0
$975$	−6.23607	−0.199714
$976$	0	0
$977$	−23.8197	−0.762058	−0.381029	−	0.924563i	$-0.624430\pi$
$977$	−23.8197	−0.762058	−0.381029	+	0.924563i	$0.624430\pi$
$978$	0	0
$979$	8.94427	0.285860
$980$	0	0
$981$	23.4164	0.747628
$982$	0	0
$983$	−23.5279	−0.750422	−0.375211	−	0.926939i	$-0.622430\pi$
$983$	−23.5279	−0.750422	−0.375211	+	0.926939i	$0.622430\pi$
$984$	0	0
$985$	13.7639	0.438555
$986$	0	0
$987$	11.7082	0.372676
$988$	0	0
$989$	−3.70820	−0.117914
$990$	0	0
$991$	11.0557	0.351197	0.175598	−	0.984462i	$-0.443814\pi$
$991$	11.0557	0.351197	0.175598	+	0.984462i	$0.443814\pi$
$992$	0	0
$993$	14.7082	0.466751
$994$	0	0
$995$	5.70820	0.180962
$996$	0	0
$997$	−36.4721	−1.15508	−0.577542	−	0.816361i	$-0.695988\pi$
$997$	−36.4721	−1.15508	−0.577542	+	0.816361i	$0.695988\pi$
$998$	0	0
$999$	48.5410	1.53577

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1840.2.a.p.1.1		2
4.3	odd	2		115.2.a.b.1.1	✓	2
5.4	even	2		9200.2.a.bm.1.2		2
8.3	odd	2		7360.2.a.bt.1.2		2
8.5	even	2		7360.2.a.bf.1.1		2
12.11	even	2		1035.2.a.m.1.2		2
20.3	even	4		575.2.b.c.24.4		4
20.7	even	4		575.2.b.c.24.1		4
20.19	odd	2		575.2.a.g.1.2		2
28.27	even	2		5635.2.a.n.1.1		2
60.59	even	2		5175.2.a.bb.1.1		2
92.91	even	2		2645.2.a.d.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
115.2.a.b.1.1	✓	2	4.3	odd	2
575.2.a.g.1.2		2	20.19	odd	2
575.2.b.c.24.1		4	20.7	even	4
575.2.b.c.24.4		4	20.3	even	4
1035.2.a.m.1.2		2	12.11	even	2
1840.2.a.p.1.1		2	1.1	even	1	trivial
2645.2.a.d.1.1		2	92.91	even	2
5175.2.a.bb.1.1		2	60.59	even	2
5635.2.a.n.1.1		2	28.27	even	2
7360.2.a.bf.1.1		2	8.5	even	2
7360.2.a.bt.1.2		2	8.3	odd	2
9200.2.a.bm.1.2		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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -1.00000 q^{5} -1.23607 q^{7} -2.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -1.00000 q^{5} -1.23607 q^{7} -2.00000 q^{9} +3.23607 q^{11} -6.23607 q^{13} -1.00000 q^{15} +2.47214 q^{17} +5.70820 q^{19} -1.23607 q^{21} +1.00000 q^{23} +1.00000 q^{25} -5.00000 q^{27} -0.527864 q^{29} -4.23607 q^{31} +3.23607 q^{33} +1.23607 q^{35} -9.70820 q^{37} -6.23607 q^{39} -7.47214 q^{41} -3.70820 q^{43} +2.00000 q^{45} -9.47214 q^{47} -5.47214 q^{49} +2.47214 q^{51} -6.00000 q^{53} -3.23607 q^{55} +5.70820 q^{57} -8.94427 q^{59} +12.1803 q^{61} +2.47214 q^{63} +6.23607 q^{65} -9.70820 q^{67} +1.00000 q^{69} +6.23607 q^{71} -6.70820 q^{73} +1.00000 q^{75} -4.00000 q^{77} -8.76393 q^{79} +1.00000 q^{81} +6.47214 q^{83} -2.47214 q^{85} -0.527864 q^{87} +2.76393 q^{89} +7.70820 q^{91} -4.23607 q^{93} -5.70820 q^{95} -6.18034 q^{97} -6.47214 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} - 2 q^{5} + 2 q^{7} - 4 q^{9}+O(q^{10}) \) 2 * q + 2 * q^3 - 2 * q^5 + 2 * q^7 - 4 * q^9	\( 2 q + 2 q^{3} - 2 q^{5} + 2 q^{7} - 4 q^{9} + 2 q^{11} - 8 q^{13} - 2 q^{15} - 4 q^{17} - 2 q^{19} + 2 q^{21} + 2 q^{23} + 2 q^{25} - 10 q^{27} - 10 q^{29} - 4 q^{31} + 2 q^{33} - 2 q^{35} - 6 q^{37} - 8 q^{39} - 6 q^{41} + 6 q^{43} + 4 q^{45} - 10 q^{47} - 2 q^{49} - 4 q^{51} - 12 q^{53} - 2 q^{55} - 2 q^{57} + 2 q^{61} - 4 q^{63} + 8 q^{65} - 6 q^{67} + 2 q^{69} + 8 q^{71} + 2 q^{75} - 8 q^{77} - 22 q^{79} + 2 q^{81} + 4 q^{83} + 4 q^{85} - 10 q^{87} + 10 q^{89} + 2 q^{91} - 4 q^{93} + 2 q^{95} + 10 q^{97} - 4 q^{99}+O(q^{100}) \) 2 * q + 2 * q^3 - 2 * q^5 + 2 * q^7 - 4 * q^9 + 2 * q^11 - 8 * q^13 - 2 * q^15 - 4 * q^17 - 2 * q^19 + 2 * q^21 + 2 * q^23 + 2 * q^25 - 10 * q^27 - 10 * q^29 - 4 * q^31 + 2 * q^33 - 2 * q^35 - 6 * q^37 - 8 * q^39 - 6 * q^41 + 6 * q^43 + 4 * q^45 - 10 * q^47 - 2 * q^49 - 4 * q^51 - 12 * q^53 - 2 * q^55 - 2 * q^57 + 2 * q^61 - 4 * q^63 + 8 * q^65 - 6 * q^67 + 2 * q^69 + 8 * q^71 + 2 * q^75 - 8 * q^77 - 22 * q^79 + 2 * q^81 + 4 * q^83 + 4 * q^85 - 10 * q^87 + 10 * q^89 + 2 * q^91 - 4 * q^93 + 2 * q^95 + 10 * q^97 - 4 * q^99

Introduction

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