LMFDB - Embedded newform 1840.2.a.j.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:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{13}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 3 $ x^2 - x - 3
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			$-1.30278$ 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)

$f(q)$	$=$	$q+0.302776 q^{3} +1.00000 q^{5} -3.30278 q^{7} -2.90833 q^{9} +1.69722 q^{11} +3.30278 q^{13} +0.302776 q^{15} +6.90833 q^{17} -5.90833 q^{19} -1.00000 q^{21} +1.00000 q^{23} +1.00000 q^{25} -1.78890 q^{27} -2.60555 q^{29} +7.90833 q^{31} +0.513878 q^{33} -3.30278 q^{35} +8.00000 q^{37} +1.00000 q^{39} +0.908327 q^{41} +9.21110 q^{43} -2.90833 q^{45} +2.60555 q^{47} +3.90833 q^{49} +2.09167 q^{51} -11.2111 q^{53} +1.69722 q^{55} -1.78890 q^{57} +3.39445 q^{59} +11.5139 q^{61} +9.60555 q^{63} +3.30278 q^{65} +4.00000 q^{67} +0.302776 q^{69} +16.3028 q^{71} -5.81665 q^{73} +0.302776 q^{75} -5.60555 q^{77} +14.4222 q^{79} +8.18335 q^{81} -11.2111 q^{83} +6.90833 q^{85} -0.788897 q^{87} -10.9083 q^{91} +2.39445 q^{93} -5.90833 q^{95} +6.30278 q^{97} -4.93608 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 3 q^{3} + 2 q^{5} - 3 q^{7} + 5 q^{9} + 7 q^{11} + 3 q^{13} - 3 q^{15} + 3 q^{17} - q^{19} - 2 q^{21} + 2 q^{23} + 2 q^{25} - 18 q^{27} + 2 q^{29} + 5 q^{31} - 17 q^{33} - 3 q^{35} + 16 q^{37} + 2 q^{39}+ \cdots + 37 q^{99}+O(q^{100}) $ 2 * q - 3 * q^3 + 2 * q^5 - 3 * q^7 + 5 * q^9 + 7 * q^11 + 3 * q^13 - 3 * q^15 + 3 * q^17 - q^19 - 2 * q^21 + 2 * q^23 + 2 * q^25 - 18 * q^27 + 2 * q^29 + 5 * q^31 - 17 * q^33 - 3 * q^35 + 16 * q^37 + 2 * q^39 - 9 * q^41 + 4 * q^43 + 5 * q^45 - 2 * q^47 - 3 * q^49 + 15 * q^51 - 8 * q^53 + 7 * q^55 - 18 * q^57 + 14 * q^59 + 5 * q^61 + 12 * q^63 + 3 * q^65 + 8 * q^67 - 3 * q^69 + 29 * q^71 + 10 * q^73 - 3 * q^75 - 4 * q^77 + 38 * q^81 - 8 * q^83 + 3 * q^85 - 16 * q^87 - 11 * q^91 + 12 * q^93 - q^95 + 9 * q^97 + 37 * q^99

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	0.302776	0.174808	0.0874038	−	0.996173i	$-0.472143\pi$
$3$	0.302776	0.174808	0.0874038	+	0.996173i	$0.472143\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−3.30278	−1.24833	−0.624166	−	0.781292i	$-0.714561\pi$
$7$	−3.30278	−1.24833	−0.624166	+	0.781292i	$0.714561\pi$
$8$	0	0
$9$	−2.90833	−0.969442
$10$	0	0
$11$	1.69722	0.511732	0.255866	−	0.966712i	$-0.417639\pi$
$11$	1.69722	0.511732	0.255866	+	0.966712i	$0.417639\pi$
$12$	0	0
$13$	3.30278	0.916025	0.458013	−	0.888946i	$-0.348561\pi$
$13$	3.30278	0.916025	0.458013	+	0.888946i	$0.348561\pi$
$14$	0	0
$15$	0.302776	0.0781763
$16$	0	0
$17$	6.90833	1.67552	0.837758	−	0.546042i	$-0.183866\pi$
$17$	6.90833	1.67552	0.837758	+	0.546042i	$0.183866\pi$
$18$	0	0
$19$	−5.90833	−1.35546	−0.677732	−	0.735309i	$-0.737037\pi$
$19$	−5.90833	−1.35546	−0.677732	+	0.735309i	$0.737037\pi$
$20$	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
$27$	−1.78890	−0.344273
$28$	0	0
$29$	−2.60555	−0.483839	−0.241919	−	0.970296i	$-0.577777\pi$
$29$	−2.60555	−0.483839	−0.241919	+	0.970296i	$0.577777\pi$
$30$	0	0
$31$	7.90833	1.42038	0.710189	−	0.704011i	$-0.248610\pi$
$31$	7.90833	1.42038	0.710189	+	0.704011i	$0.248610\pi$
$32$	0	0
$33$	0.513878	0.0894547
$34$	0	0
$35$	−3.30278	−0.558271
$36$	0	0
$37$	8.00000	1.31519	0.657596	−	0.753371i	$-0.271573\pi$
$37$	8.00000	1.31519	0.657596	+	0.753371i	$0.271573\pi$
$38$	0	0
$39$	1.00000	0.160128
$40$	0	0
$41$	0.908327	0.141857	0.0709284	−	0.997481i	$-0.477404\pi$
$41$	0.908327	0.141857	0.0709284	+	0.997481i	$0.477404\pi$
$42$	0	0
$43$	9.21110	1.40468	0.702340	−	0.711842i	$-0.252139\pi$
$43$	9.21110	1.40468	0.702340	+	0.711842i	$0.252139\pi$
$44$	0	0
$45$	−2.90833	−0.433548
$46$	0	0
$47$	2.60555	0.380059	0.190029	−	0.981778i	$-0.439142\pi$
$47$	2.60555	0.380059	0.190029	+	0.981778i	$0.439142\pi$
$48$	0	0
$49$	3.90833	0.558332
$50$	0	0
$51$	2.09167	0.292893
$52$	0	0
$53$	−11.2111	−1.53996	−0.769982	−	0.638066i	$-0.779735\pi$
$53$	−11.2111	−1.53996	−0.769982	+	0.638066i	$0.779735\pi$
$54$	0	0
$55$	1.69722	0.228854
$56$	0	0
$57$	−1.78890	−0.236945
$58$	0	0
$59$	3.39445	0.441920	0.220960	−	0.975283i	$-0.429081\pi$
$59$	3.39445	0.441920	0.220960	+	0.975283i	$0.429081\pi$
$60$	0	0
$61$	11.5139	1.47420	0.737101	−	0.675783i	$-0.236194\pi$
$61$	11.5139	1.47420	0.737101	+	0.675783i	$0.236194\pi$
$62$	0	0
$63$	9.60555	1.21019
$64$	0	0
$65$	3.30278	0.409659
$66$	0	0
$67$	4.00000	0.488678	0.244339	−	0.969690i	$-0.421429\pi$
$67$	4.00000	0.488678	0.244339	+	0.969690i	$0.421429\pi$
$68$	0	0
$69$	0.302776	0.0364499
$70$	0	0
$71$	16.3028	1.93478	0.967392	−	0.253285i	$-0.0815110\pi$
$71$	16.3028	1.93478	0.967392	+	0.253285i	$0.0815110\pi$
$72$	0	0
$73$	−5.81665	−0.680788	−0.340394	−	0.940283i	$-0.610560\pi$
$73$	−5.81665	−0.680788	−0.340394	+	0.940283i	$0.610560\pi$
$74$	0	0
$75$	0.302776	0.0349615
$76$	0	0
$77$	−5.60555	−0.638812
$78$	0	0
$79$	14.4222	1.62262	0.811312	−	0.584613i	$-0.198754\pi$
$79$	14.4222	1.62262	0.811312	+	0.584613i	$0.198754\pi$
$80$	0	0
$81$	8.18335	0.909261
$82$	0	0
$83$	−11.2111	−1.23058	−0.615289	−	0.788301i	$-0.710961\pi$
$83$	−11.2111	−1.23058	−0.615289	+	0.788301i	$0.710961\pi$
$84$	0	0
$85$	6.90833	0.749313
$86$	0	0
$87$	−0.788897	−0.0845787
$88$	0	0
$89$	0	0		−	1.00000i	$-0.5\pi$
$89$	0	0			1.00000i	$0.5\pi$
$90$	0	0
$91$	−10.9083	−1.14350
$92$	0	0
$93$	2.39445	0.248293
$94$	0	0
$95$	−5.90833	−0.606182
$96$	0	0
$97$	6.30278	0.639950	0.319975	−	0.947426i	$-0.396325\pi$
$97$	6.30278	0.639950	0.319975	+	0.947426i	$0.396325\pi$
$98$	0	0
$99$	−4.93608	−0.496095
$100$	0	0
$101$	2.60555	0.259262	0.129631	−	0.991562i	$-0.458621\pi$
$101$	2.60555	0.259262	0.129631	+	0.991562i	$0.458621\pi$
$102$	0	0
$103$	−8.11943	−0.800031	−0.400016	−	0.916508i	$-0.630995\pi$
$103$	−8.11943	−0.800031	−0.400016	+	0.916508i	$0.630995\pi$
$104$	0	0
$105$	−1.00000	−0.0975900
$106$	0	0
$107$	2.60555	0.251888	0.125944	−	0.992037i	$-0.459804\pi$
$107$	2.60555	0.251888	0.125944	+	0.992037i	$0.459804\pi$
$108$	0	0
$109$	1.48612	0.142345	0.0711723	−	0.997464i	$-0.477326\pi$
$109$	1.48612	0.142345	0.0711723	+	0.997464i	$0.477326\pi$
$110$	0	0
$111$	2.42221	0.229906
$112$	0	0
$113$	−16.4222	−1.54487	−0.772436	−	0.635093i	$-0.780962\pi$
$113$	−16.4222	−1.54487	−0.772436	+	0.635093i	$0.780962\pi$
$114$	0	0
$115$	1.00000	0.0932505
$116$	0	0
$117$	−9.60555	−0.888034
$118$	0	0
$119$	−22.8167	−2.09160
$120$	0	0
$121$	−8.11943	−0.738130
$122$	0	0
$123$	0.275019	0.0247977
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−9.81665	−0.871087	−0.435544	−	0.900168i	$-0.643444\pi$
$127$	−9.81665	−0.871087	−0.435544	+	0.900168i	$0.643444\pi$
$128$	0	0
$129$	2.78890	0.245549
$130$	0	0
$131$	11.2111	0.979519	0.489759	−	0.871858i	$-0.337085\pi$
$131$	11.2111	0.979519	0.489759	+	0.871858i	$0.337085\pi$
$132$	0	0
$133$	19.5139	1.69207
$134$	0	0
$135$	−1.78890	−0.153964
$136$	0	0
$137$	3.90833	0.333911	0.166955	−	0.985964i	$-0.446606\pi$
$137$	3.90833	0.333911	0.166955	+	0.985964i	$0.446606\pi$
$138$	0	0
$139$	12.6056	1.06919	0.534594	−	0.845109i	$-0.320464\pi$
$139$	12.6056	1.06919	0.534594	+	0.845109i	$0.320464\pi$
$140$	0	0
$141$	0.788897	0.0664372
$142$	0	0
$143$	5.60555	0.468760
$144$	0	0
$145$	−2.60555	−0.216379
$146$	0	0
$147$	1.18335	0.0976007
$148$	0	0
$149$	13.3028	1.08981	0.544903	−	0.838499i	$-0.316567\pi$
$149$	13.3028	1.08981	0.544903	+	0.838499i	$0.316567\pi$
$150$	0	0
$151$	−8.90833	−0.724949	−0.362475	−	0.931994i	$-0.618068\pi$
$151$	−8.90833	−0.724949	−0.362475	+	0.931994i	$0.618068\pi$
$152$	0	0
$153$	−20.0917	−1.62432
$154$	0	0
$155$	7.90833	0.635212
$156$	0	0
$157$	−18.6056	−1.48488	−0.742442	−	0.669910i	$-0.766333\pi$
$157$	−18.6056	−1.48488	−0.742442	+	0.669910i	$0.766333\pi$
$158$	0	0
$159$	−3.39445	−0.269197
$160$	0	0
$161$	−3.30278	−0.260295
$162$	0	0
$163$	−9.30278	−0.728650	−0.364325	−	0.931272i	$-0.618700\pi$
$163$	−9.30278	−0.728650	−0.364325	+	0.931272i	$0.618700\pi$
$164$	0	0
$165$	0.513878	0.0400054
$166$	0	0
$167$	−6.78890	−0.525341	−0.262670	−	0.964886i	$-0.584603\pi$
$167$	−6.78890	−0.525341	−0.262670	+	0.964886i	$0.584603\pi$
$168$	0	0
$169$	−2.09167	−0.160898
$170$	0	0
$171$	17.1833	1.31404
$172$	0	0
$173$	19.6972	1.49755	0.748776	−	0.662823i	$-0.230642\pi$
$173$	19.6972	1.49755	0.748776	+	0.662823i	$0.230642\pi$
$174$	0	0
$175$	−3.30278	−0.249666
$176$	0	0
$177$	1.02776	0.0772509
$178$	0	0
$179$	−9.39445	−0.702174	−0.351087	−	0.936343i	$-0.614188\pi$
$179$	−9.39445	−0.702174	−0.351087	+	0.936343i	$0.614188\pi$
$180$	0	0
$181$	17.1194	1.27248	0.636239	−	0.771492i	$-0.280489\pi$
$181$	17.1194	1.27248	0.636239	+	0.771492i	$0.280489\pi$
$182$	0	0
$183$	3.48612	0.257702
$184$	0	0
$185$	8.00000	0.588172
$186$	0	0
$187$	11.7250	0.857416
$188$	0	0
$189$	5.90833	0.429768
$190$	0	0
$191$	8.60555	0.622676	0.311338	−	0.950299i	$-0.399223\pi$
$191$	8.60555	0.622676	0.311338	+	0.950299i	$0.399223\pi$
$192$	0	0
$193$	−17.8167	−1.28247	−0.641235	−	0.767344i	$-0.721578\pi$
$193$	−17.8167	−1.28247	−0.641235	+	0.767344i	$0.721578\pi$
$194$	0	0
$195$	1.00000	0.0716115
$196$	0	0
$197$	4.30278	0.306560	0.153280	−	0.988183i	$-0.451016\pi$
$197$	4.30278	0.306560	0.153280	+	0.988183i	$0.451016\pi$
$198$	0	0
$199$	20.4222	1.44769	0.723846	−	0.689962i	$-0.242373\pi$
$199$	20.4222	1.44769	0.723846	+	0.689962i	$0.242373\pi$
$200$	0	0
$201$	1.21110	0.0854246
$202$	0	0
$203$	8.60555	0.603991
$204$	0	0
$205$	0.908327	0.0634403
$206$	0	0
$207$	−2.90833	−0.202143
$208$	0	0
$209$	−10.0278	−0.693634
$210$	0	0
$211$	−7.21110	−0.496433	−0.248216	−	0.968705i	$-0.579844\pi$
$211$	−7.21110	−0.496433	−0.248216	+	0.968705i	$0.579844\pi$
$212$	0	0
$213$	4.93608	0.338215
$214$	0	0
$215$	9.21110	0.628192
$216$	0	0
$217$	−26.1194	−1.77310
$218$	0	0
$219$	−1.76114	−0.119007
$220$	0	0
$221$	22.8167	1.53481
$222$	0	0
$223$	4.00000	0.267860	0.133930	−	0.990991i	$-0.457240\pi$
$223$	4.00000	0.267860	0.133930	+	0.990991i	$0.457240\pi$
$224$	0	0
$225$	−2.90833	−0.193888
$226$	0	0
$227$	14.6056	0.969404	0.484702	−	0.874679i	$-0.338928\pi$
$227$	14.6056	0.969404	0.484702	+	0.874679i	$0.338928\pi$
$228$	0	0
$229$	2.00000	0.132164	0.0660819	−	0.997814i	$-0.478950\pi$
$229$	2.00000	0.132164	0.0660819	+	0.997814i	$0.478950\pi$
$230$	0	0
$231$	−1.69722	−0.111669
$232$	0	0
$233$	25.8167	1.69131	0.845653	−	0.533734i	$-0.179212\pi$
$233$	25.8167	1.69131	0.845653	+	0.533734i	$0.179212\pi$
$234$	0	0
$235$	2.60555	0.169967
$236$	0	0
$237$	4.36669	0.283647
$238$	0	0
$239$	−5.21110	−0.337078	−0.168539	−	0.985695i	$-0.553905\pi$
$239$	−5.21110	−0.337078	−0.168539	+	0.985695i	$0.553905\pi$
$240$	0	0
$241$	−14.4222	−0.929016	−0.464508	−	0.885569i	$-0.653769\pi$
$241$	−14.4222	−0.929016	−0.464508	+	0.885569i	$0.653769\pi$
$242$	0	0
$243$	7.84441	0.503219
$244$	0	0
$245$	3.90833	0.249694
$246$	0	0
$247$	−19.5139	−1.24164
$248$	0	0
$249$	−3.39445	−0.215114
$250$	0	0
$251$	−12.5139	−0.789869	−0.394934	−	0.918709i	$-0.629233\pi$
$251$	−12.5139	−0.789869	−0.394934	+	0.918709i	$0.629233\pi$
$252$	0	0
$253$	1.69722	0.106704
$254$	0	0
$255$	2.09167	0.130986
$256$	0	0
$257$	−1.81665	−0.113320	−0.0566599	−	0.998394i	$-0.518045\pi$
$257$	−1.81665	−0.113320	−0.0566599	+	0.998394i	$0.518045\pi$
$258$	0	0
$259$	−26.4222	−1.64180
$260$	0	0
$261$	7.57779	0.469054
$262$	0	0
$263$	−3.51388	−0.216675	−0.108338	−	0.994114i	$-0.534553\pi$
$263$	−3.51388	−0.216675	−0.108338	+	0.994114i	$0.534553\pi$
$264$	0	0
$265$	−11.2111	−0.688693
$266$	0	0
$267$	0	0
$268$	0	0
$269$	4.18335	0.255063	0.127532	−	0.991835i	$-0.459295\pi$
$269$	4.18335	0.255063	0.127532	+	0.991835i	$0.459295\pi$
$270$	0	0
$271$	2.69722	0.163845	0.0819224	−	0.996639i	$-0.473894\pi$
$271$	2.69722	0.163845	0.0819224	+	0.996639i	$0.473894\pi$
$272$	0	0
$273$	−3.30278	−0.199893
$274$	0	0
$275$	1.69722	0.102346
$276$	0	0
$277$	−27.2111	−1.63496	−0.817478	−	0.575959i	$-0.804629\pi$
$277$	−27.2111	−1.63496	−0.817478	+	0.575959i	$0.804629\pi$
$278$	0	0
$279$	−23.0000	−1.37697
$280$	0	0
$281$	−26.6056	−1.58715	−0.793577	−	0.608470i	$-0.791784\pi$
$281$	−26.6056	−1.58715	−0.793577	+	0.608470i	$0.791784\pi$
$282$	0	0
$283$	−2.00000	−0.118888	−0.0594438	−	0.998232i	$-0.518933\pi$
$283$	−2.00000	−0.118888	−0.0594438	+	0.998232i	$0.518933\pi$
$284$	0	0
$285$	−1.78890	−0.105965
$286$	0	0
$287$	−3.00000	−0.177084
$288$	0	0
$289$	30.7250	1.80735
$290$	0	0
$291$	1.90833	0.111868
$292$	0	0
$293$	23.2111	1.35601	0.678004	−	0.735059i	$-0.262845\pi$
$293$	23.2111	1.35601	0.678004	+	0.735059i	$0.262845\pi$
$294$	0	0
$295$	3.39445	0.197632
$296$	0	0
$297$	−3.03616	−0.176176
$298$	0	0
$299$	3.30278	0.191004
$300$	0	0
$301$	−30.4222	−1.75351
$302$	0	0
$303$	0.788897	0.0453210
$304$	0	0
$305$	11.5139	0.659283
$306$	0	0
$307$	11.6972	0.667596	0.333798	−	0.942645i	$-0.391670\pi$
$307$	11.6972	0.667596	0.333798	+	0.942645i	$0.391670\pi$
$308$	0	0
$309$	−2.45837	−0.139852
$310$	0	0
$311$	−22.4222	−1.27145	−0.635723	−	0.771917i	$-0.719298\pi$
$311$	−22.4222	−1.27145	−0.635723	+	0.771917i	$0.719298\pi$
$312$	0	0
$313$	19.7250	1.11492	0.557461	−	0.830203i	$-0.311776\pi$
$313$	19.7250	1.11492	0.557461	+	0.830203i	$0.311776\pi$
$314$	0	0
$315$	9.60555	0.541212
$316$	0	0
$317$	17.7250	0.995534	0.497767	−	0.867311i	$-0.334153\pi$
$317$	17.7250	0.995534	0.497767	+	0.867311i	$0.334153\pi$
$318$	0	0
$319$	−4.42221	−0.247596
$320$	0	0
$321$	0.788897	0.0440320
$322$	0	0
$323$	−40.8167	−2.27110
$324$	0	0
$325$	3.30278	0.183205
$326$	0	0
$327$	0.449961	0.0248829
$328$	0	0
$329$	−8.60555	−0.474439
$330$	0	0
$331$	−16.6056	−0.912724	−0.456362	−	0.889794i	$-0.650848\pi$
$331$	−16.6056	−0.912724	−0.456362	+	0.889794i	$0.650848\pi$
$332$	0	0
$333$	−23.2666	−1.27500
$334$	0	0
$335$	4.00000	0.218543
$336$	0	0
$337$	−22.5139	−1.22641	−0.613205	−	0.789924i	$-0.710120\pi$
$337$	−22.5139	−1.22641	−0.613205	+	0.789924i	$0.710120\pi$
$338$	0	0
$339$	−4.97224	−0.270055
$340$	0	0
$341$	13.4222	0.726853
$342$	0	0
$343$	10.2111	0.551348
$344$	0	0
$345$	0.302776	0.0163009
$346$	0	0
$347$	−28.5416	−1.53220	−0.766098	−	0.642724i	$-0.777804\pi$
$347$	−28.5416	−1.53220	−0.766098	+	0.642724i	$0.777804\pi$
$348$	0	0
$349$	−27.2111	−1.45658	−0.728288	−	0.685271i	$-0.759684\pi$
$349$	−27.2111	−1.45658	−0.728288	+	0.685271i	$0.759684\pi$
$350$	0	0
$351$	−5.90833	−0.315363
$352$	0	0
$353$	−10.4222	−0.554718	−0.277359	−	0.960766i	$-0.589459\pi$
$353$	−10.4222	−0.554718	−0.277359	+	0.960766i	$0.589459\pi$
$354$	0	0
$355$	16.3028	0.865261
$356$	0	0
$357$	−6.90833	−0.365627
$358$	0	0
$359$	−11.2111	−0.591699	−0.295850	−	0.955235i	$-0.595603\pi$
$359$	−11.2111	−0.591699	−0.295850	+	0.955235i	$0.595603\pi$
$360$	0	0
$361$	15.9083	0.837280
$362$	0	0
$363$	−2.45837	−0.129031
$364$	0	0
$365$	−5.81665	−0.304458
$366$	0	0
$367$	−14.7889	−0.771974	−0.385987	−	0.922504i	$-0.626139\pi$
$367$	−14.7889	−0.771974	−0.385987	+	0.922504i	$0.626139\pi$
$368$	0	0
$369$	−2.64171	−0.137522
$370$	0	0
$371$	37.0278	1.92239
$372$	0	0
$373$	4.60555	0.238466	0.119233	−	0.992866i	$-0.461956\pi$
$373$	4.60555	0.238466	0.119233	+	0.992866i	$0.461956\pi$
$374$	0	0
$375$	0.302776	0.0156353
$376$	0	0
$377$	−8.60555	−0.443208
$378$	0	0
$379$	−14.9083	−0.765789	−0.382895	−	0.923792i	$-0.625073\pi$
$379$	−14.9083	−0.765789	−0.382895	+	0.923792i	$0.625073\pi$
$380$	0	0
$381$	−2.97224	−0.152273
$382$	0	0
$383$	0	0		−	1.00000i	$-0.5\pi$
$383$	0	0			1.00000i	$0.5\pi$
$384$	0	0
$385$	−5.60555	−0.285685
$386$	0	0
$387$	−26.7889	−1.36176
$388$	0	0
$389$	−25.9361	−1.31501	−0.657506	−	0.753449i	$-0.728388\pi$
$389$	−25.9361	−1.31501	−0.657506	+	0.753449i	$0.728388\pi$
$390$	0	0
$391$	6.90833	0.349369
$392$	0	0
$393$	3.39445	0.171227
$394$	0	0
$395$	14.4222	0.725660
$396$	0	0
$397$	10.7250	0.538271	0.269136	−	0.963102i	$-0.413262\pi$
$397$	10.7250	0.538271	0.269136	+	0.963102i	$0.413262\pi$
$398$	0	0
$399$	5.90833	0.295786
$400$	0	0
$401$	−8.60555	−0.429741	−0.214870	−	0.976643i	$-0.568933\pi$
$401$	−8.60555	−0.429741	−0.214870	+	0.976643i	$0.568933\pi$
$402$	0	0
$403$	26.1194	1.30110
$404$	0	0
$405$	8.18335	0.406634
$406$	0	0
$407$	13.5778	0.673026
$408$	0	0
$409$	−25.9083	−1.28108	−0.640542	−	0.767923i	$-0.721290\pi$
$409$	−25.9083	−1.28108	−0.640542	+	0.767923i	$0.721290\pi$
$410$	0	0
$411$	1.18335	0.0583702
$412$	0	0
$413$	−11.2111	−0.551662
$414$	0	0
$415$	−11.2111	−0.550331
$416$	0	0
$417$	3.81665	0.186902
$418$	0	0
$419$	−3.63331	−0.177499	−0.0887493	−	0.996054i	$-0.528287\pi$
$419$	−3.63331	−0.177499	−0.0887493	+	0.996054i	$0.528287\pi$
$420$	0	0
$421$	30.6972	1.49609	0.748046	−	0.663647i	$-0.230992\pi$
$421$	30.6972	1.49609	0.748046	+	0.663647i	$0.230992\pi$
$422$	0	0
$423$	−7.57779	−0.368445
$424$	0	0
$425$	6.90833	0.335103
$426$	0	0
$427$	−38.0278	−1.84029
$428$	0	0
$429$	1.69722	0.0819428
$430$	0	0
$431$	30.2389	1.45655	0.728277	−	0.685283i	$-0.240321\pi$
$431$	30.2389	1.45655	0.728277	+	0.685283i	$0.240321\pi$
$432$	0	0
$433$	−24.0917	−1.15777	−0.578886	−	0.815409i	$-0.696512\pi$
$433$	−24.0917	−1.15777	−0.578886	+	0.815409i	$0.696512\pi$
$434$	0	0
$435$	−0.788897	−0.0378247
$436$	0	0
$437$	−5.90833	−0.282634
$438$	0	0
$439$	14.6972	0.701460	0.350730	−	0.936477i	$-0.385933\pi$
$439$	14.6972	0.701460	0.350730	+	0.936477i	$0.385933\pi$
$440$	0	0
$441$	−11.3667	−0.541271
$442$	0	0
$443$	−17.4861	−0.830791	−0.415395	−	0.909641i	$-0.636357\pi$
$443$	−17.4861	−0.830791	−0.415395	+	0.909641i	$0.636357\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	4.02776	0.190506
$448$	0	0
$449$	−2.09167	−0.0987122	−0.0493561	−	0.998781i	$-0.515717\pi$
$449$	−2.09167	−0.0987122	−0.0493561	+	0.998781i	$0.515717\pi$
$450$	0	0
$451$	1.54163	0.0725927
$452$	0	0
$453$	−2.69722	−0.126727
$454$	0	0
$455$	−10.9083	−0.511390
$456$	0	0
$457$	−32.4222	−1.51665	−0.758323	−	0.651879i	$-0.773981\pi$
$457$	−32.4222	−1.51665	−0.758323	+	0.651879i	$0.773981\pi$
$458$	0	0
$459$	−12.3583	−0.576836
$460$	0	0
$461$	10.1833	0.474286	0.237143	−	0.971475i	$-0.423789\pi$
$461$	10.1833	0.474286	0.237143	+	0.971475i	$0.423789\pi$
$462$	0	0
$463$	−17.6333	−0.819489	−0.409745	−	0.912200i	$-0.634382\pi$
$463$	−17.6333	−0.819489	−0.409745	+	0.912200i	$0.634382\pi$
$464$	0	0
$465$	2.39445	0.111040
$466$	0	0
$467$	1.81665	0.0840647	0.0420324	−	0.999116i	$-0.486617\pi$
$467$	1.81665	0.0840647	0.0420324	+	0.999116i	$0.486617\pi$
$468$	0	0
$469$	−13.2111	−0.610032
$470$	0	0
$471$	−5.63331	−0.259569
$472$	0	0
$473$	15.6333	0.718820
$474$	0	0
$475$	−5.90833	−0.271093
$476$	0	0
$477$	32.6056	1.49291
$478$	0	0
$479$	30.0000	1.37073	0.685367	−	0.728197i	$-0.259642\pi$
$479$	30.0000	1.37073	0.685367	+	0.728197i	$0.259642\pi$
$480$	0	0
$481$	26.4222	1.20475
$482$	0	0
$483$	−1.00000	−0.0455016
$484$	0	0
$485$	6.30278	0.286194
$486$	0	0
$487$	−9.81665	−0.444835	−0.222418	−	0.974952i	$-0.571395\pi$
$487$	−9.81665	−0.444835	−0.222418	+	0.974952i	$0.571395\pi$
$488$	0	0
$489$	−2.81665	−0.127373
$490$	0	0
$491$	4.18335	0.188792	0.0943959	−	0.995535i	$-0.469908\pi$
$491$	4.18335	0.188792	0.0943959	+	0.995535i	$0.469908\pi$
$492$	0	0
$493$	−18.0000	−0.810679
$494$	0	0
$495$	−4.93608	−0.221860
$496$	0	0
$497$	−53.8444	−2.41525
$498$	0	0
$499$	31.6333	1.41610	0.708051	−	0.706162i	$-0.249575\pi$
$499$	31.6333	1.41610	0.708051	+	0.706162i	$0.249575\pi$
$500$	0	0
$501$	−2.05551	−0.0918335
$502$	0	0
$503$	−29.7250	−1.32537	−0.662686	−	0.748898i	$-0.730583\pi$
$503$	−29.7250	−1.32537	−0.662686	+	0.748898i	$0.730583\pi$
$504$	0	0
$505$	2.60555	0.115946
$506$	0	0
$507$	−0.633308	−0.0281262
$508$	0	0
$509$	−35.4500	−1.57129	−0.785646	−	0.618676i	$-0.787669\pi$
$509$	−35.4500	−1.57129	−0.785646	+	0.618676i	$0.787669\pi$
$510$	0	0
$511$	19.2111	0.849849
$512$	0	0
$513$	10.5694	0.466650
$514$	0	0
$515$	−8.11943	−0.357785
$516$	0	0
$517$	4.42221	0.194488
$518$	0	0
$519$	5.96384	0.261784
$520$	0	0
$521$	6.00000	0.262865	0.131432	−	0.991325i	$-0.458042\pi$
$521$	6.00000	0.262865	0.131432	+	0.991325i	$0.458042\pi$
$522$	0	0
$523$	20.4222	0.893001	0.446500	−	0.894783i	$-0.352670\pi$
$523$	20.4222	0.893001	0.446500	+	0.894783i	$0.352670\pi$
$524$	0	0
$525$	−1.00000	−0.0436436
$526$	0	0
$527$	54.6333	2.37986
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	−9.87217	−0.428416
$532$	0	0
$533$	3.00000	0.129944
$534$	0	0
$535$	2.60555	0.112648
$536$	0	0
$537$	−2.84441	−0.122745
$538$	0	0
$539$	6.63331	0.285717
$540$	0	0
$541$	28.8444	1.24012	0.620059	−	0.784555i	$-0.287109\pi$
$541$	28.8444	1.24012	0.620059	+	0.784555i	$0.287109\pi$
$542$	0	0
$543$	5.18335	0.222439
$544$	0	0
$545$	1.48612	0.0636585
$546$	0	0
$547$	10.5139	0.449541	0.224770	−	0.974412i	$-0.427837\pi$
$547$	10.5139	0.449541	0.224770	+	0.974412i	$0.427837\pi$
$548$	0	0
$549$	−33.4861	−1.42915
$550$	0	0
$551$	15.3944	0.655826
$552$	0	0
$553$	−47.6333	−2.02557
$554$	0	0
$555$	2.42221	0.102817
$556$	0	0
$557$	22.4222	0.950059	0.475030	−	0.879970i	$-0.342437\pi$
$557$	22.4222	0.950059	0.475030	+	0.879970i	$0.342437\pi$
$558$	0	0
$559$	30.4222	1.28672
$560$	0	0
$561$	3.55004	0.149883
$562$	0	0
$563$	3.63331	0.153126	0.0765628	−	0.997065i	$-0.475605\pi$
$563$	3.63331	0.153126	0.0765628	+	0.997065i	$0.475605\pi$
$564$	0	0
$565$	−16.4222	−0.690887
$566$	0	0
$567$	−27.0278	−1.13506
$568$	0	0
$569$	28.4222	1.19152	0.595760	−	0.803162i	$-0.296851\pi$
$569$	28.4222	1.19152	0.595760	+	0.803162i	$0.296851\pi$
$570$	0	0
$571$	16.1194	0.674577	0.337289	−	0.941401i	$-0.390490\pi$
$571$	16.1194	0.674577	0.337289	+	0.941401i	$0.390490\pi$
$572$	0	0
$573$	2.60555	0.108848
$574$	0	0
$575$	1.00000	0.0417029
$576$	0	0
$577$	2.00000	0.0832611	0.0416305	−	0.999133i	$-0.486745\pi$
$577$	2.00000	0.0832611	0.0416305	+	0.999133i	$0.486745\pi$
$578$	0	0
$579$	−5.39445	−0.224186
$580$	0	0
$581$	37.0278	1.53617
$582$	0	0
$583$	−19.0278	−0.788049
$584$	0	0
$585$	−9.60555	−0.397141
$586$	0	0
$587$	−16.5416	−0.682746	−0.341373	−	0.939928i	$-0.610892\pi$
$587$	−16.5416	−0.682746	−0.341373	+	0.939928i	$0.610892\pi$
$588$	0	0
$589$	−46.7250	−1.92527
$590$	0	0
$591$	1.30278	0.0535890
$592$	0	0
$593$	−1.81665	−0.0746010	−0.0373005	−	0.999304i	$-0.511876\pi$
$593$	−1.81665	−0.0746010	−0.0373005	+	0.999304i	$0.511876\pi$
$594$	0	0
$595$	−22.8167	−0.935392
$596$	0	0
$597$	6.18335	0.253068
$598$	0	0
$599$	35.3305	1.44357	0.721783	−	0.692119i	$-0.243323\pi$
$599$	35.3305	1.44357	0.721783	+	0.692119i	$0.243323\pi$
$600$	0	0
$601$	42.9361	1.75140	0.875700	−	0.482856i	$-0.160401\pi$
$601$	42.9361	1.75140	0.875700	+	0.482856i	$0.160401\pi$
$602$	0	0
$603$	−11.6333	−0.473745
$604$	0	0
$605$	−8.11943	−0.330102
$606$	0	0
$607$	−46.0555	−1.86934	−0.934668	−	0.355522i	$-0.884303\pi$
$607$	−46.0555	−1.86934	−0.934668	+	0.355522i	$0.884303\pi$
$608$	0	0
$609$	2.60555	0.105582
$610$	0	0
$611$	8.60555	0.348143
$612$	0	0
$613$	3.57779	0.144506	0.0722529	−	0.997386i	$-0.476981\pi$
$613$	3.57779	0.144506	0.0722529	+	0.997386i	$0.476981\pi$
$614$	0	0
$615$	0.275019	0.0110898
$616$	0	0
$617$	18.9083	0.761221	0.380610	−	0.924736i	$-0.375714\pi$
$617$	18.9083	0.761221	0.380610	+	0.924736i	$0.375714\pi$
$618$	0	0
$619$	12.3305	0.495606	0.247803	−	0.968810i	$-0.420291\pi$
$619$	12.3305	0.495606	0.247803	+	0.968810i	$0.420291\pi$
$620$	0	0
$621$	−1.78890	−0.0717860
$622$	0	0
$623$	0	0
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−3.03616	−0.121253
$628$	0	0
$629$	55.2666	2.20362
$630$	0	0
$631$	−23.3944	−0.931318	−0.465659	−	0.884964i	$-0.654183\pi$
$631$	−23.3944	−0.931318	−0.465659	+	0.884964i	$0.654183\pi$
$632$	0	0
$633$	−2.18335	−0.0867802
$634$	0	0
$635$	−9.81665	−0.389562
$636$	0	0
$637$	12.9083	0.511447
$638$	0	0
$639$	−47.4138	−1.87566
$640$	0	0
$641$	−36.0000	−1.42191	−0.710957	−	0.703235i	$-0.751738\pi$
$641$	−36.0000	−1.42191	−0.710957	+	0.703235i	$0.751738\pi$
$642$	0	0
$643$	34.2389	1.35025	0.675124	−	0.737704i	$-0.264090\pi$
$643$	34.2389	1.35025	0.675124	+	0.737704i	$0.264090\pi$
$644$	0	0
$645$	2.78890	0.109813
$646$	0	0
$647$	−26.8444	−1.05536	−0.527681	−	0.849442i	$-0.676938\pi$
$647$	−26.8444	−1.05536	−0.527681	+	0.849442i	$0.676938\pi$
$648$	0	0
$649$	5.76114	0.226145
$650$	0	0
$651$	−7.90833	−0.309952
$652$	0	0
$653$	41.7250	1.63282	0.816412	−	0.577469i	$-0.195960\pi$
$653$	41.7250	1.63282	0.816412	+	0.577469i	$0.195960\pi$
$654$	0	0
$655$	11.2111	0.438054
$656$	0	0
$657$	16.9167	0.659985
$658$	0	0
$659$	−15.6333	−0.608987	−0.304494	−	0.952514i	$-0.598487\pi$
$659$	−15.6333	−0.608987	−0.304494	+	0.952514i	$0.598487\pi$
$660$	0	0
$661$	−34.9083	−1.35778	−0.678888	−	0.734242i	$-0.737538\pi$
$661$	−34.9083	−1.35778	−0.678888	+	0.734242i	$0.737538\pi$
$662$	0	0
$663$	6.90833	0.268297
$664$	0	0
$665$	19.5139	0.756716
$666$	0	0
$667$	−2.60555	−0.100887
$668$	0	0
$669$	1.21110	0.0468239
$670$	0	0
$671$	19.5416	0.754396
$672$	0	0
$673$	−37.6333	−1.45066	−0.725329	−	0.688403i	$-0.758312\pi$
$673$	−37.6333	−1.45066	−0.725329	+	0.688403i	$0.758312\pi$
$674$	0	0
$675$	−1.78890	−0.0688547
$676$	0	0
$677$	16.4222	0.631157	0.315578	−	0.948900i	$-0.397802\pi$
$677$	16.4222	0.631157	0.315578	+	0.948900i	$0.397802\pi$
$678$	0	0
$679$	−20.8167	−0.798870
$680$	0	0
$681$	4.42221	0.169459
$682$	0	0
$683$	0.275019	0.0105233	0.00526166	−	0.999986i	$-0.498325\pi$
$683$	0.275019	0.0105233	0.00526166	+	0.999986i	$0.498325\pi$
$684$	0	0
$685$	3.90833	0.149329
$686$	0	0
$687$	0.605551	0.0231032
$688$	0	0
$689$	−37.0278	−1.41065
$690$	0	0
$691$	−51.8167	−1.97120	−0.985599	−	0.169098i	$-0.945914\pi$
$691$	−51.8167	−1.97120	−0.985599	+	0.169098i	$0.945914\pi$
$692$	0	0
$693$	16.3028	0.619291
$694$	0	0
$695$	12.6056	0.478156
$696$	0	0
$697$	6.27502	0.237683
$698$	0	0
$699$	7.81665	0.295653
$700$	0	0
$701$	32.0917	1.21209	0.606043	−	0.795432i	$-0.292756\pi$
$701$	32.0917	1.21209	0.606043	+	0.795432i	$0.292756\pi$
$702$	0	0
$703$	−47.2666	−1.78269
$704$	0	0
$705$	0.788897	0.0297116
$706$	0	0
$707$	−8.60555	−0.323645
$708$	0	0
$709$	−15.8806	−0.596407	−0.298204	−	0.954502i	$-0.596387\pi$
$709$	−15.8806	−0.596407	−0.298204	+	0.954502i	$0.596387\pi$
$710$	0	0
$711$	−41.9445	−1.57304
$712$	0	0
$713$	7.90833	0.296169
$714$	0	0
$715$	5.60555	0.209636
$716$	0	0
$717$	−1.57779	−0.0589238
$718$	0	0
$719$	−10.6972	−0.398939	−0.199470	−	0.979904i	$-0.563922\pi$
$719$	−10.6972	−0.398939	−0.199470	+	0.979904i	$0.563922\pi$
$720$	0	0
$721$	26.8167	0.998704
$722$	0	0
$723$	−4.36669	−0.162399
$724$	0	0
$725$	−2.60555	−0.0967677
$726$	0	0
$727$	−2.90833	−0.107864	−0.0539319	−	0.998545i	$-0.517175\pi$
$727$	−2.90833	−0.107864	−0.0539319	+	0.998545i	$0.517175\pi$
$728$	0	0
$729$	−22.1749	−0.821294
$730$	0	0
$731$	63.6333	2.35356
$732$	0	0
$733$	29.6333	1.09453	0.547266	−	0.836959i	$-0.315669\pi$
$733$	29.6333	1.09453	0.547266	+	0.836959i	$0.315669\pi$
$734$	0	0
$735$	1.18335	0.0436484
$736$	0	0
$737$	6.78890	0.250072
$738$	0	0
$739$	−35.6333	−1.31079	−0.655396	−	0.755285i	$-0.727498\pi$
$739$	−35.6333	−1.31079	−0.655396	+	0.755285i	$0.727498\pi$
$740$	0	0
$741$	−5.90833	−0.217048
$742$	0	0
$743$	32.3305	1.18609	0.593046	−	0.805169i	$-0.297925\pi$
$743$	32.3305	1.18609	0.593046	+	0.805169i	$0.297925\pi$
$744$	0	0
$745$	13.3028	0.487376
$746$	0	0
$747$	32.6056	1.19297
$748$	0	0
$749$	−8.60555	−0.314440
$750$	0	0
$751$	−21.8167	−0.796101	−0.398051	−	0.917364i	$-0.630313\pi$
$751$	−21.8167	−0.796101	−0.398051	+	0.917364i	$0.630313\pi$
$752$	0	0
$753$	−3.78890	−0.138075
$754$	0	0
$755$	−8.90833	−0.324207
$756$	0	0
$757$	13.2111	0.480166	0.240083	−	0.970752i	$-0.422825\pi$
$757$	13.2111	0.480166	0.240083	+	0.970752i	$0.422825\pi$
$758$	0	0
$759$	0.513878	0.0186526
$760$	0	0
$761$	−49.5416	−1.79588	−0.897941	−	0.440115i	$-0.854938\pi$
$761$	−49.5416	−1.79588	−0.897941	+	0.440115i	$0.854938\pi$
$762$	0	0
$763$	−4.90833	−0.177693
$764$	0	0
$765$	−20.0917	−0.726416
$766$	0	0
$767$	11.2111	0.404809
$768$	0	0
$769$	45.2666	1.63236	0.816178	−	0.577801i	$-0.196089\pi$
$769$	45.2666	1.63236	0.816178	+	0.577801i	$0.196089\pi$
$770$	0	0
$771$	−0.550039	−0.0198092
$772$	0	0
$773$	12.0000	0.431610	0.215805	−	0.976436i	$-0.430762\pi$
$773$	12.0000	0.431610	0.215805	+	0.976436i	$0.430762\pi$
$774$	0	0
$775$	7.90833	0.284075
$776$	0	0
$777$	−8.00000	−0.286998
$778$	0	0
$779$	−5.36669	−0.192282
$780$	0	0
$781$	27.6695	0.990091
$782$	0	0
$783$	4.66106	0.166573
$784$	0	0
$785$	−18.6056	−0.664061
$786$	0	0
$787$	−37.4500	−1.33495	−0.667473	−	0.744634i	$-0.732624\pi$
$787$	−37.4500	−1.33495	−0.667473	+	0.744634i	$0.732624\pi$
$788$	0	0
$789$	−1.06392	−0.0378764
$790$	0	0
$791$	54.2389	1.92851
$792$	0	0
$793$	38.0278	1.35041
$794$	0	0
$795$	−3.39445	−0.120389
$796$	0	0
$797$	1.81665	0.0643492	0.0321746	−	0.999482i	$-0.489757\pi$
$797$	1.81665	0.0643492	0.0321746	+	0.999482i	$0.489757\pi$
$798$	0	0
$799$	18.0000	0.636794
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−9.87217	−0.348381
$804$	0	0
$805$	−3.30278	−0.116408
$806$	0	0
$807$	1.26662	0.0445870
$808$	0	0
$809$	50.7250	1.78340	0.891698	−	0.452631i	$-0.149515\pi$
$809$	50.7250	1.78340	0.891698	+	0.452631i	$0.149515\pi$
$810$	0	0
$811$	41.0278	1.44068	0.720340	−	0.693621i	$-0.243986\pi$
$811$	41.0278	1.44068	0.720340	+	0.693621i	$0.243986\pi$
$812$	0	0
$813$	0.816654	0.0286413
$814$	0	0
$815$	−9.30278	−0.325862
$816$	0	0
$817$	−54.4222	−1.90399
$818$	0	0
$819$	31.7250	1.10856
$820$	0	0
$821$	−30.0000	−1.04701	−0.523504	−	0.852023i	$-0.675375\pi$
$821$	−30.0000	−1.04701	−0.523504	+	0.852023i	$0.675375\pi$
$822$	0	0
$823$	15.2111	0.530226	0.265113	−	0.964217i	$-0.414591\pi$
$823$	15.2111	0.530226	0.265113	+	0.964217i	$0.414591\pi$
$824$	0	0
$825$	0.513878	0.0178909
$826$	0	0
$827$	29.4500	1.02408	0.512038	−	0.858963i	$-0.328891\pi$
$827$	29.4500	1.02408	0.512038	+	0.858963i	$0.328891\pi$
$828$	0	0
$829$	31.2111	1.08401	0.542003	−	0.840376i	$-0.317666\pi$
$829$	31.2111	1.08401	0.542003	+	0.840376i	$0.317666\pi$
$830$	0	0
$831$	−8.23886	−0.285803
$832$	0	0
$833$	27.0000	0.935495
$834$	0	0
$835$	−6.78890	−0.234939
$836$	0	0
$837$	−14.1472	−0.488998
$838$	0	0
$839$	43.8167	1.51272	0.756359	−	0.654156i	$-0.226976\pi$
$839$	43.8167	1.51272	0.756359	+	0.654156i	$0.226976\pi$
$840$	0	0
$841$	−22.2111	−0.765900
$842$	0	0
$843$	−8.05551	−0.277447
$844$	0	0
$845$	−2.09167	−0.0719557
$846$	0	0
$847$	26.8167	0.921431
$848$	0	0
$849$	−0.605551	−0.0207825
$850$	0	0
$851$	8.00000	0.274236
$852$	0	0
$853$	−21.7250	−0.743849	−0.371925	−	0.928263i	$-0.621302\pi$
$853$	−21.7250	−0.743849	−0.371925	+	0.928263i	$0.621302\pi$
$854$	0	0
$855$	17.1833	0.587658
$856$	0	0
$857$	9.63331	0.329068	0.164534	−	0.986371i	$-0.447388\pi$
$857$	9.63331	0.329068	0.164534	+	0.986371i	$0.447388\pi$
$858$	0	0
$859$	35.8167	1.22205	0.611024	−	0.791612i	$-0.290758\pi$
$859$	35.8167	1.22205	0.611024	+	0.791612i	$0.290758\pi$
$860$	0	0
$861$	−0.908327	−0.0309557
$862$	0	0
$863$	41.4500	1.41097	0.705487	−	0.708723i	$-0.250729\pi$
$863$	41.4500	1.41097	0.705487	+	0.708723i	$0.250729\pi$
$864$	0	0
$865$	19.6972	0.669726
$866$	0	0
$867$	9.30278	0.315939
$868$	0	0
$869$	24.4777	0.830350
$870$	0	0
$871$	13.2111	0.447641
$872$	0	0
$873$	−18.3305	−0.620395
$874$	0	0
$875$	−3.30278	−0.111654
$876$	0	0
$877$	−48.1749	−1.62675	−0.813376	−	0.581738i	$-0.802373\pi$
$877$	−48.1749	−1.62675	−0.813376	+	0.581738i	$0.802373\pi$
$878$	0	0
$879$	7.02776	0.237040
$880$	0	0
$881$	55.2666	1.86198	0.930990	−	0.365045i	$-0.118946\pi$
$881$	55.2666	1.86198	0.930990	+	0.365045i	$0.118946\pi$
$882$	0	0
$883$	−8.27502	−0.278477	−0.139238	−	0.990259i	$-0.544465\pi$
$883$	−8.27502	−0.278477	−0.139238	+	0.990259i	$0.544465\pi$
$884$	0	0
$885$	1.02776	0.0345477
$886$	0	0
$887$	−27.6333	−0.927836	−0.463918	−	0.885878i	$-0.653557\pi$
$887$	−27.6333	−0.927836	−0.463918	+	0.885878i	$0.653557\pi$
$888$	0	0
$889$	32.4222	1.08741
$890$	0	0
$891$	13.8890	0.465298
$892$	0	0
$893$	−15.3944	−0.515156
$894$	0	0
$895$	−9.39445	−0.314022
$896$	0	0
$897$	1.00000	0.0333890
$898$	0	0
$899$	−20.6056	−0.687234
$900$	0	0
$901$	−77.4500	−2.58023
$902$	0	0
$903$	−9.21110	−0.306526
$904$	0	0
$905$	17.1194	0.569069
$906$	0	0
$907$	−48.6611	−1.61576	−0.807882	−	0.589344i	$-0.799386\pi$
$907$	−48.6611	−1.61576	−0.807882	+	0.589344i	$0.799386\pi$
$908$	0	0
$909$	−7.57779	−0.251340
$910$	0	0
$911$	4.18335	0.138600	0.0693002	−	0.997596i	$-0.477923\pi$
$911$	4.18335	0.138600	0.0693002	+	0.997596i	$0.477923\pi$
$912$	0	0
$913$	−19.0278	−0.629727
$914$	0	0
$915$	3.48612	0.115248
$916$	0	0
$917$	−37.0278	−1.22276
$918$	0	0
$919$	−44.0000	−1.45143	−0.725713	−	0.687998i	$-0.758490\pi$
$919$	−44.0000	−1.45143	−0.725713	+	0.687998i	$0.758490\pi$
$920$	0	0
$921$	3.54163	0.116701
$922$	0	0
$923$	53.8444	1.77231
$924$	0	0
$925$	8.00000	0.263038
$926$	0	0
$927$	23.6140	0.775584
$928$	0	0
$929$	−14.3667	−0.471356	−0.235678	−	0.971831i	$-0.575731\pi$
$929$	−14.3667	−0.471356	−0.235678	+	0.971831i	$0.575731\pi$
$930$	0	0
$931$	−23.0917	−0.756799
$932$	0	0
$933$	−6.78890	−0.222259
$934$	0	0
$935$	11.7250	0.383448
$936$	0	0
$937$	37.9638	1.24022	0.620112	−	0.784513i	$-0.287087\pi$
$937$	37.9638	1.24022	0.620112	+	0.784513i	$0.287087\pi$
$938$	0	0
$939$	5.97224	0.194897
$940$	0	0
$941$	25.9361	0.845492	0.422746	−	0.906248i	$-0.361066\pi$
$941$	25.9361	0.845492	0.422746	+	0.906248i	$0.361066\pi$
$942$	0	0
$943$	0.908327	0.0295792
$944$	0	0
$945$	5.90833	0.192198
$946$	0	0
$947$	4.93608	0.160401	0.0802006	−	0.996779i	$-0.474444\pi$
$947$	4.93608	0.160401	0.0802006	+	0.996779i	$0.474444\pi$
$948$	0	0
$949$	−19.2111	−0.623619
$950$	0	0
$951$	5.36669	0.174027
$952$	0	0
$953$	−41.3305	−1.33883	−0.669414	−	0.742890i	$-0.733455\pi$
$953$	−41.3305	−1.33883	−0.669414	+	0.742890i	$0.733455\pi$
$954$	0	0
$955$	8.60555	0.278469
$956$	0	0
$957$	−1.33894	−0.0432817
$958$	0	0
$959$	−12.9083	−0.416832
$960$	0	0
$961$	31.5416	1.01747
$962$	0	0
$963$	−7.57779	−0.244191
$964$	0	0
$965$	−17.8167	−0.573538
$966$	0	0
$967$	12.6056	0.405367	0.202684	−	0.979244i	$-0.435034\pi$
$967$	12.6056	0.405367	0.202684	+	0.979244i	$0.435034\pi$
$968$	0	0
$969$	−12.3583	−0.397005
$970$	0	0
$971$	17.0917	0.548498	0.274249	−	0.961659i	$-0.411571\pi$
$971$	17.0917	0.548498	0.274249	+	0.961659i	$0.411571\pi$
$972$	0	0
$973$	−41.6333	−1.33470
$974$	0	0
$975$	1.00000	0.0320256
$976$	0	0
$977$	6.51388	0.208397	0.104199	−	0.994556i	$-0.466772\pi$
$977$	6.51388	0.208397	0.104199	+	0.994556i	$0.466772\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	−4.32213	−0.137995
$982$	0	0
$983$	−34.5416	−1.10171	−0.550854	−	0.834602i	$-0.685698\pi$
$983$	−34.5416	−1.10171	−0.550854	+	0.834602i	$0.685698\pi$
$984$	0	0
$985$	4.30278	0.137098
$986$	0	0
$987$	−2.60555	−0.0829356
$988$	0	0
$989$	9.21110	0.292896
$990$	0	0
$991$	15.3305	0.486990	0.243495	−	0.969902i	$-0.421706\pi$
$991$	15.3305	0.486990	0.243495	+	0.969902i	$0.421706\pi$
$992$	0	0
$993$	−5.02776	−0.159551
$994$	0	0
$995$	20.4222	0.647427
$996$	0	0
$997$	−16.7889	−0.531710	−0.265855	−	0.964013i	$-0.585654\pi$
$997$	−16.7889	−0.531710	−0.265855	+	0.964013i	$0.585654\pi$
$998$	0	0
$999$	−14.3112	−0.452786

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1840.2.a.j.1.2		2
4.3	odd	2		230.2.a.b.1.1	✓	2
5.4	even	2		9200.2.a.ca.1.1		2
8.3	odd	2		7360.2.a.bc.1.2		2
8.5	even	2		7360.2.a.bu.1.1		2
12.11	even	2		2070.2.a.w.1.2		2
20.3	even	4		1150.2.b.f.599.3		4
20.7	even	4		1150.2.b.f.599.2		4
20.19	odd	2		1150.2.a.m.1.2		2
92.91	even	2		5290.2.a.j.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
230.2.a.b.1.1	✓	2	4.3	odd	2
1150.2.a.m.1.2		2	20.19	odd	2
1150.2.b.f.599.2		4	20.7	even	4
1150.2.b.f.599.3		4	20.3	even	4
1840.2.a.j.1.2		2	1.1	even	1	trivial
2070.2.a.w.1.2		2	12.11	even	2
5290.2.a.j.1.1		2	92.91	even	2
7360.2.a.bc.1.2		2	8.3	odd	2
7360.2.a.bu.1.1		2	8.5	even	2
9200.2.a.ca.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)

\(f(q)\)	\(=\)	\(q+0.302776 q^{3} +1.00000 q^{5} -3.30278 q^{7} -2.90833 q^{9} +1.69722 q^{11} +3.30278 q^{13} +0.302776 q^{15} +6.90833 q^{17} -5.90833 q^{19} -1.00000 q^{21} +1.00000 q^{23} +1.00000 q^{25} -1.78890 q^{27} -2.60555 q^{29} +7.90833 q^{31} +0.513878 q^{33} -3.30278 q^{35} +8.00000 q^{37} +1.00000 q^{39} +0.908327 q^{41} +9.21110 q^{43} -2.90833 q^{45} +2.60555 q^{47} +3.90833 q^{49} +2.09167 q^{51} -11.2111 q^{53} +1.69722 q^{55} -1.78890 q^{57} +3.39445 q^{59} +11.5139 q^{61} +9.60555 q^{63} +3.30278 q^{65} +4.00000 q^{67} +0.302776 q^{69} +16.3028 q^{71} -5.81665 q^{73} +0.302776 q^{75} -5.60555 q^{77} +14.4222 q^{79} +8.18335 q^{81} -11.2111 q^{83} +6.90833 q^{85} -0.788897 q^{87} -10.9083 q^{91} +2.39445 q^{93} -5.90833 q^{95} +6.30278 q^{97} -4.93608 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 3 q^{3} + 2 q^{5} - 3 q^{7} + 5 q^{9} + 7 q^{11} + 3 q^{13} - 3 q^{15} + 3 q^{17} - q^{19} - 2 q^{21} + 2 q^{23} + 2 q^{25} - 18 q^{27} + 2 q^{29} + 5 q^{31} - 17 q^{33} - 3 q^{35} + 16 q^{37} + 2 q^{39}+ \cdots + 37 q^{99}+O(q^{100}) \) 2 * q - 3 * q^3 + 2 * q^5 - 3 * q^7 + 5 * q^9 + 7 * q^11 + 3 * q^13 - 3 * q^15 + 3 * q^17 - q^19 - 2 * q^21 + 2 * q^23 + 2 * q^25 - 18 * q^27 + 2 * q^29 + 5 * q^31 - 17 * q^33 - 3 * q^35 + 16 * q^37 + 2 * q^39 - 9 * q^41 + 4 * q^43 + 5 * q^45 - 2 * q^47 - 3 * q^49 + 15 * q^51 - 8 * q^53 + 7 * q^55 - 18 * q^57 + 14 * q^59 + 5 * q^61 + 12 * q^63 + 3 * q^65 + 8 * q^67 - 3 * q^69 + 29 * q^71 + 10 * q^73 - 3 * q^75 - 4 * q^77 + 38 * q^81 - 8 * q^83 + 3 * q^85 - 16 * q^87 - 11 * q^91 + 12 * q^93 - q^95 + 9 * q^97 + 37 * q^99

Introduction

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