LMFDB - Embedded newform 2070.2.a.w.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2070,2,Mod(1,2070)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2070, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("2070.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2070 = 2 \cdot 3^{2} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2070.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:	$16.5290332184$
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:	$\Z[a_1, \ldots, a_{7}]$
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			$2.30278$ of defining polynomial
Character	$\chi$	$=$	2070.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} +3.30278 q^{7} +1.00000 q^{8} +O(q^{10})$	\(q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} +3.30278 q^{7} +1.00000 q^{8} -1.00000 q^{10} +1.69722 q^{11} +3.30278 q^{13} +3.30278 q^{14} +1.00000 q^{16} -6.90833 q^{17} +5.90833 q^{19} -1.00000 q^{20} +1.69722 q^{22} +1.00000 q^{23} +1.00000 q^{25} +3.30278 q^{26} +3.30278 q^{28} +2.60555 q^{29} -7.90833 q^{31} +1.00000 q^{32} -6.90833 q^{34} -3.30278 q^{35} +8.00000 q^{37} +5.90833 q^{38} -1.00000 q^{40} -0.908327 q^{41} -9.21110 q^{43} +1.69722 q^{44} +1.00000 q^{46} +2.60555 q^{47} +3.90833 q^{49} +1.00000 q^{50} +3.30278 q^{52} +11.2111 q^{53} -1.69722 q^{55} +3.30278 q^{56} +2.60555 q^{58} +3.39445 q^{59} +11.5139 q^{61} -7.90833 q^{62} +1.00000 q^{64} -3.30278 q^{65} -4.00000 q^{67} -6.90833 q^{68} -3.30278 q^{70} +16.3028 q^{71} -5.81665 q^{73} +8.00000 q^{74} +5.90833 q^{76} +5.60555 q^{77} -14.4222 q^{79} -1.00000 q^{80} -0.908327 q^{82} -11.2111 q^{83} +6.90833 q^{85} -9.21110 q^{86} +1.69722 q^{88} +10.9083 q^{91} +1.00000 q^{92} +2.60555 q^{94} -5.90833 q^{95} +6.30278 q^{97} +3.90833 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} + 2 q^{4} - 2 q^{5} + 3 q^{7} + 2 q^{8}+O(q^{10}) $ 2 * q + 2 * q^2 + 2 * q^4 - 2 * q^5 + 3 * q^7 + 2 * q^8	$ 2 q + 2 q^{2} + 2 q^{4} - 2 q^{5} + 3 q^{7} + 2 q^{8} - 2 q^{10} + 7 q^{11} + 3 q^{13} + 3 q^{14} + 2 q^{16} - 3 q^{17} + q^{19} - 2 q^{20} + 7 q^{22} + 2 q^{23} + 2 q^{25} + 3 q^{26} + 3 q^{28} - 2 q^{29} - 5 q^{31} + 2 q^{32} - 3 q^{34} - 3 q^{35} + 16 q^{37} + q^{38} - 2 q^{40} + 9 q^{41} - 4 q^{43} + 7 q^{44} + 2 q^{46} - 2 q^{47} - 3 q^{49} + 2 q^{50} + 3 q^{52} + 8 q^{53} - 7 q^{55} + 3 q^{56} - 2 q^{58} + 14 q^{59} + 5 q^{61} - 5 q^{62} + 2 q^{64} - 3 q^{65} - 8 q^{67} - 3 q^{68} - 3 q^{70} + 29 q^{71} + 10 q^{73} + 16 q^{74} + q^{76} + 4 q^{77} - 2 q^{80} + 9 q^{82} - 8 q^{83} + 3 q^{85} - 4 q^{86} + 7 q^{88} + 11 q^{91} + 2 q^{92} - 2 q^{94} - q^{95} + 9 q^{97} - 3 q^{98}+O(q^{100}) $ 2 * q + 2 * q^2 + 2 * q^4 - 2 * q^5 + 3 * q^7 + 2 * q^8 - 2 * q^10 + 7 * q^11 + 3 * q^13 + 3 * q^14 + 2 * q^16 - 3 * q^17 + q^19 - 2 * q^20 + 7 * q^22 + 2 * q^23 + 2 * q^25 + 3 * q^26 + 3 * q^28 - 2 * q^29 - 5 * q^31 + 2 * q^32 - 3 * q^34 - 3 * q^35 + 16 * q^37 + q^38 - 2 * q^40 + 9 * q^41 - 4 * q^43 + 7 * q^44 + 2 * q^46 - 2 * q^47 - 3 * q^49 + 2 * q^50 + 3 * q^52 + 8 * q^53 - 7 * q^55 + 3 * q^56 - 2 * q^58 + 14 * q^59 + 5 * q^61 - 5 * q^62 + 2 * q^64 - 3 * q^65 - 8 * q^67 - 3 * q^68 - 3 * q^70 + 29 * q^71 + 10 * q^73 + 16 * q^74 + q^76 + 4 * q^77 - 2 * q^80 + 9 * q^82 - 8 * q^83 + 3 * q^85 - 4 * q^86 + 7 * q^88 + 11 * q^91 + 2 * q^92 - 2 * q^94 - q^95 + 9 * q^97 - 3 * q^98

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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	3.30278	1.24833	0.624166	−	0.781292i	$-0.285439\pi$
$7$	3.30278	1.24833	0.624166	+	0.781292i	$0.285439\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	−1.00000	−0.316228
$11$	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	$0.348561\pi$
$14$	3.30278	0.882704
$15$	0	0
$16$	1.00000	0.250000
$17$	−6.90833	−1.67552	−0.837758	−	0.546042i	$-0.816134\pi$
$17$	−6.90833	−1.67552	−0.837758	+	0.546042i	$0.816134\pi$
$18$	0	0
$19$	5.90833	1.35546	0.677732	−	0.735309i	$-0.262963\pi$
$19$	5.90833	1.35546	0.677732	+	0.735309i	$0.262963\pi$
$20$	−1.00000	−0.223607
$21$	0	0
$22$	1.69722	0.361849
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	1.00000	0.200000
$26$	3.30278	0.647728
$27$	0	0
$28$	3.30278	0.624166
$29$	2.60555	0.483839	0.241919	−	0.970296i	$-0.422223\pi$
$29$	2.60555	0.483839	0.241919	+	0.970296i	$0.422223\pi$
$30$	0	0
$31$	−7.90833	−1.42038	−0.710189	−	0.704011i	$-0.751390\pi$
$31$	−7.90833	−1.42038	−0.710189	+	0.704011i	$0.751390\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	−6.90833	−1.18477
$35$	−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	$0.271573\pi$
$38$	5.90833	0.958457
$39$	0	0
$40$	−1.00000	−0.158114
$41$	−0.908327	−0.141857	−0.0709284	−	0.997481i	$-0.522596\pi$
$41$	−0.908327	−0.141857	−0.0709284	+	0.997481i	$0.522596\pi$
$42$	0	0
$43$	−9.21110	−1.40468	−0.702340	−	0.711842i	$-0.747861\pi$
$43$	−9.21110	−1.40468	−0.702340	+	0.711842i	$0.747861\pi$
$44$	1.69722	0.255866
$45$	0	0
$46$	1.00000	0.147442
$47$	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
$50$	1.00000	0.141421
$51$	0	0
$52$	3.30278	0.458013
$53$	11.2111	1.53996	0.769982	−	0.638066i	$-0.220265\pi$
$53$	11.2111	1.53996	0.769982	+	0.638066i	$0.220265\pi$
$54$	0	0
$55$	−1.69722	−0.228854
$56$	3.30278	0.441352
$57$	0	0
$58$	2.60555	0.342126
$59$	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	$0.236194\pi$
$62$	−7.90833	−1.00436
$63$	0	0
$64$	1.00000	0.125000
$65$	−3.30278	−0.409659
$66$	0	0
$67$	−4.00000	−0.488678	−0.244339	−	0.969690i	$-0.578571\pi$
$67$	−4.00000	−0.488678	−0.244339	+	0.969690i	$0.578571\pi$
$68$	−6.90833	−0.837758
$69$	0	0
$70$	−3.30278	−0.394757
$71$	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	$0.610560\pi$
$74$	8.00000	0.929981
$75$	0	0
$76$	5.90833	0.677732
$77$	5.60555	0.638812
$78$	0	0
$79$	−14.4222	−1.62262	−0.811312	−	0.584613i	$-0.801246\pi$
$79$	−14.4222	−1.62262	−0.811312	+	0.584613i	$0.801246\pi$
$80$	−1.00000	−0.111803
$81$	0	0
$82$	−0.908327	−0.100308
$83$	−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
$86$	−9.21110	−0.993259
$87$	0	0
$88$	1.69722	0.180925
$89$	0	0		−	1.00000i	$-0.5\pi$
$89$	0	0			1.00000i	$0.5\pi$
$90$	0	0
$91$	10.9083	1.14350
$92$	1.00000	0.104257
$93$	0	0
$94$	2.60555	0.268742
$95$	−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	$0.396325\pi$
$98$	3.90833	0.394801
$99$	0	0
$100$	1.00000	0.100000
$101$	−2.60555	−0.259262	−0.129631	−	0.991562i	$-0.541379\pi$
$101$	−2.60555	−0.259262	−0.129631	+	0.991562i	$0.541379\pi$
$102$	0	0
$103$	8.11943	0.800031	0.400016	−	0.916508i	$-0.369005\pi$
$103$	8.11943	0.800031	0.400016	+	0.916508i	$0.369005\pi$
$104$	3.30278	0.323864
$105$	0	0
$106$	11.2111	1.08892
$107$	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	$0.477326\pi$
$110$	−1.69722	−0.161824
$111$	0	0
$112$	3.30278	0.312083
$113$	16.4222	1.54487	0.772436	−	0.635093i	$-0.219038\pi$
$113$	16.4222	1.54487	0.772436	+	0.635093i	$0.219038\pi$
$114$	0	0
$115$	−1.00000	−0.0932505
$116$	2.60555	0.241919
$117$	0	0
$118$	3.39445	0.312484
$119$	−22.8167	−2.09160
$120$	0	0
$121$	−8.11943	−0.738130
$122$	11.5139	1.04242
$123$	0	0
$124$	−7.90833	−0.710189
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	9.81665	0.871087	0.435544	−	0.900168i	$-0.356556\pi$
$127$	9.81665	0.871087	0.435544	+	0.900168i	$0.356556\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	−3.30278	−0.289673
$131$	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
$134$	−4.00000	−0.345547
$135$	0	0
$136$	−6.90833	−0.592384
$137$	−3.90833	−0.333911	−0.166955	−	0.985964i	$-0.553394\pi$
$137$	−3.90833	−0.333911	−0.166955	+	0.985964i	$0.553394\pi$
$138$	0	0
$139$	−12.6056	−1.06919	−0.534594	−	0.845109i	$-0.679536\pi$
$139$	−12.6056	−1.06919	−0.534594	+	0.845109i	$0.679536\pi$
$140$	−3.30278	−0.279135
$141$	0	0
$142$	16.3028	1.36810
$143$	5.60555	0.468760
$144$	0	0
$145$	−2.60555	−0.216379
$146$	−5.81665	−0.481390
$147$	0	0
$148$	8.00000	0.657596
$149$	−13.3028	−1.08981	−0.544903	−	0.838499i	$-0.683433\pi$
$149$	−13.3028	−1.08981	−0.544903	+	0.838499i	$0.683433\pi$
$150$	0	0
$151$	8.90833	0.724949	0.362475	−	0.931994i	$-0.381932\pi$
$151$	8.90833	0.724949	0.362475	+	0.931994i	$0.381932\pi$
$152$	5.90833	0.479229
$153$	0	0
$154$	5.60555	0.451708
$155$	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	$0.766333\pi$
$158$	−14.4222	−1.14737
$159$	0	0
$160$	−1.00000	−0.0790569
$161$	3.30278	0.260295
$162$	0	0
$163$	9.30278	0.728650	0.364325	−	0.931272i	$-0.381300\pi$
$163$	9.30278	0.728650	0.364325	+	0.931272i	$0.381300\pi$
$164$	−0.908327	−0.0709284
$165$	0	0
$166$	−11.2111	−0.870150
$167$	−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
$170$	6.90833	0.529844
$171$	0	0
$172$	−9.21110	−0.702340
$173$	−19.6972	−1.49755	−0.748776	−	0.662823i	$-0.769358\pi$
$173$	−19.6972	−1.49755	−0.748776	+	0.662823i	$0.769358\pi$
$174$	0	0
$175$	3.30278	0.249666
$176$	1.69722	0.127933
$177$	0	0
$178$	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	$0.280489\pi$
$182$	10.9083	0.808579
$183$	0	0
$184$	1.00000	0.0737210
$185$	−8.00000	−0.588172
$186$	0	0
$187$	−11.7250	−0.857416
$188$	2.60555	0.190029
$189$	0	0
$190$	−5.90833	−0.428635
$191$	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	$0.721578\pi$
$194$	6.30278	0.452513
$195$	0	0
$196$	3.90833	0.279166
$197$	−4.30278	−0.306560	−0.153280	−	0.988183i	$-0.548984\pi$
$197$	−4.30278	−0.306560	−0.153280	+	0.988183i	$0.548984\pi$
$198$	0	0
$199$	−20.4222	−1.44769	−0.723846	−	0.689962i	$-0.757627\pi$
$199$	−20.4222	−1.44769	−0.723846	+	0.689962i	$0.757627\pi$
$200$	1.00000	0.0707107
$201$	0	0
$202$	−2.60555	−0.183326
$203$	8.60555	0.603991
$204$	0	0
$205$	0.908327	0.0634403
$206$	8.11943	0.565707
$207$	0	0
$208$	3.30278	0.229006
$209$	10.0278	0.693634
$210$	0	0
$211$	7.21110	0.496433	0.248216	−	0.968705i	$-0.420156\pi$
$211$	7.21110	0.496433	0.248216	+	0.968705i	$0.420156\pi$
$212$	11.2111	0.769982
$213$	0	0
$214$	2.60555	0.178112
$215$	9.21110	0.628192
$216$	0	0
$217$	−26.1194	−1.77310
$218$	1.48612	0.100653
$219$	0	0
$220$	−1.69722	−0.114427
$221$	−22.8167	−1.53481
$222$	0	0
$223$	−4.00000	−0.267860	−0.133930	−	0.990991i	$-0.542760\pi$
$223$	−4.00000	−0.267860	−0.133930	+	0.990991i	$0.542760\pi$
$224$	3.30278	0.220676
$225$	0	0
$226$	16.4222	1.09239
$227$	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	$0.478950\pi$
$230$	−1.00000	−0.0659380
$231$	0	0
$232$	2.60555	0.171063
$233$	−25.8167	−1.69131	−0.845653	−	0.533734i	$-0.820788\pi$
$233$	−25.8167	−1.69131	−0.845653	+	0.533734i	$0.820788\pi$
$234$	0	0
$235$	−2.60555	−0.169967
$236$	3.39445	0.220960
$237$	0	0
$238$	−22.8167	−1.47898
$239$	−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	$0.653769\pi$
$242$	−8.11943	−0.521937
$243$	0	0
$244$	11.5139	0.737101
$245$	−3.90833	−0.249694
$246$	0	0
$247$	19.5139	1.24164
$248$	−7.90833	−0.502179
$249$	0	0
$250$	−1.00000	−0.0632456
$251$	−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
$254$	9.81665	0.615952
$255$	0	0
$256$	1.00000	0.0625000
$257$	1.81665	0.113320	0.0566599	−	0.998394i	$-0.481955\pi$
$257$	1.81665	0.113320	0.0566599	+	0.998394i	$0.481955\pi$
$258$	0	0
$259$	26.4222	1.64180
$260$	−3.30278	−0.204829
$261$	0	0
$262$	11.2111	0.692624
$263$	−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
$266$	19.5139	1.19647
$267$	0	0
$268$	−4.00000	−0.244339
$269$	−4.18335	−0.255063	−0.127532	−	0.991835i	$-0.540705\pi$
$269$	−4.18335	−0.255063	−0.127532	+	0.991835i	$0.540705\pi$
$270$	0	0
$271$	−2.69722	−0.163845	−0.0819224	−	0.996639i	$-0.526106\pi$
$271$	−2.69722	−0.163845	−0.0819224	+	0.996639i	$0.526106\pi$
$272$	−6.90833	−0.418879
$273$	0	0
$274$	−3.90833	−0.236111
$275$	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	$0.804629\pi$
$278$	−12.6056	−0.756031
$279$	0	0
$280$	−3.30278	−0.197379
$281$	26.6056	1.58715	0.793577	−	0.608470i	$-0.208216\pi$
$281$	26.6056	1.58715	0.793577	+	0.608470i	$0.208216\pi$
$282$	0	0
$283$	2.00000	0.118888	0.0594438	−	0.998232i	$-0.481067\pi$
$283$	2.00000	0.118888	0.0594438	+	0.998232i	$0.481067\pi$
$284$	16.3028	0.967392
$285$	0	0
$286$	5.60555	0.331463
$287$	−3.00000	−0.177084
$288$	0	0
$289$	30.7250	1.80735
$290$	−2.60555	−0.153003
$291$	0	0
$292$	−5.81665	−0.340394
$293$	−23.2111	−1.35601	−0.678004	−	0.735059i	$-0.737155\pi$
$293$	−23.2111	−1.35601	−0.678004	+	0.735059i	$0.737155\pi$
$294$	0	0
$295$	−3.39445	−0.197632
$296$	8.00000	0.464991
$297$	0	0
$298$	−13.3028	−0.770609
$299$	3.30278	0.191004
$300$	0	0
$301$	−30.4222	−1.75351
$302$	8.90833	0.512617
$303$	0	0
$304$	5.90833	0.338866
$305$	−11.5139	−0.659283
$306$	0	0
$307$	−11.6972	−0.667596	−0.333798	−	0.942645i	$-0.608330\pi$
$307$	−11.6972	−0.667596	−0.333798	+	0.942645i	$0.608330\pi$
$308$	5.60555	0.319406
$309$	0	0
$310$	7.90833	0.449163
$311$	−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	$0.311776\pi$
$314$	−18.6056	−1.04997
$315$	0	0
$316$	−14.4222	−0.811312
$317$	−17.7250	−0.995534	−0.497767	−	0.867311i	$-0.665847\pi$
$317$	−17.7250	−0.995534	−0.497767	+	0.867311i	$0.665847\pi$
$318$	0	0
$319$	4.42221	0.247596
$320$	−1.00000	−0.0559017
$321$	0	0
$322$	3.30278	0.184056
$323$	−40.8167	−2.27110
$324$	0	0
$325$	3.30278	0.183205
$326$	9.30278	0.515233
$327$	0	0
$328$	−0.908327	−0.0501540
$329$	8.60555	0.474439
$330$	0	0
$331$	16.6056	0.912724	0.456362	−	0.889794i	$-0.349152\pi$
$331$	16.6056	0.912724	0.456362	+	0.889794i	$0.349152\pi$
$332$	−11.2111	−0.615289
$333$	0	0
$334$	−6.78890	−0.371472
$335$	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	$0.710120\pi$
$338$	−2.09167	−0.113772
$339$	0	0
$340$	6.90833	0.374657
$341$	−13.4222	−0.726853
$342$	0	0
$343$	−10.2111	−0.551348
$344$	−9.21110	−0.496629
$345$	0	0
$346$	−19.6972	−1.05893
$347$	−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	$0.759684\pi$
$350$	3.30278	0.176541
$351$	0	0
$352$	1.69722	0.0904624
$353$	10.4222	0.554718	0.277359	−	0.960766i	$-0.410541\pi$
$353$	10.4222	0.554718	0.277359	+	0.960766i	$0.410541\pi$
$354$	0	0
$355$	−16.3028	−0.865261
$356$	0	0
$357$	0	0
$358$	−9.39445	−0.496512
$359$	−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
$362$	17.1194	0.899777
$363$	0	0
$364$	10.9083	0.571752
$365$	5.81665	0.304458
$366$	0	0
$367$	14.7889	0.771974	0.385987	−	0.922504i	$-0.373861\pi$
$367$	14.7889	0.771974	0.385987	+	0.922504i	$0.373861\pi$
$368$	1.00000	0.0521286
$369$	0	0
$370$	−8.00000	−0.415900
$371$	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	$0.461956\pi$
$374$	−11.7250	−0.606284
$375$	0	0
$376$	2.60555	0.134371
$377$	8.60555	0.443208
$378$	0	0
$379$	14.9083	0.765789	0.382895	−	0.923792i	$-0.374927\pi$
$379$	14.9083	0.765789	0.382895	+	0.923792i	$0.374927\pi$
$380$	−5.90833	−0.303091
$381$	0	0
$382$	8.60555	0.440298
$383$	0	0		−	1.00000i	$-0.5\pi$
$383$	0	0			1.00000i	$0.5\pi$
$384$	0	0
$385$	−5.60555	−0.285685
$386$	−17.8167	−0.906844
$387$	0	0
$388$	6.30278	0.319975
$389$	25.9361	1.31501	0.657506	−	0.753449i	$-0.271612\pi$
$389$	25.9361	1.31501	0.657506	+	0.753449i	$0.271612\pi$
$390$	0	0
$391$	−6.90833	−0.349369
$392$	3.90833	0.197400
$393$	0	0
$394$	−4.30278	−0.216771
$395$	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	$0.413262\pi$
$398$	−20.4222	−1.02367
$399$	0	0
$400$	1.00000	0.0500000
$401$	8.60555	0.429741	0.214870	−	0.976643i	$-0.431067\pi$
$401$	8.60555	0.429741	0.214870	+	0.976643i	$0.431067\pi$
$402$	0	0
$403$	−26.1194	−1.30110
$404$	−2.60555	−0.129631
$405$	0	0
$406$	8.60555	0.427086
$407$	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	$0.721290\pi$
$410$	0.908327	0.0448591
$411$	0	0
$412$	8.11943	0.400016
$413$	11.2111	0.551662
$414$	0	0
$415$	11.2111	0.550331
$416$	3.30278	0.161932
$417$	0	0
$418$	10.0278	0.490474
$419$	−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	$0.230992\pi$
$422$	7.21110	0.351031
$423$	0	0
$424$	11.2111	0.544459
$425$	−6.90833	−0.335103
$426$	0	0
$427$	38.0278	1.84029
$428$	2.60555	0.125944
$429$	0	0
$430$	9.21110	0.444199
$431$	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	$0.696512\pi$
$434$	−26.1194	−1.25377
$435$	0	0
$436$	1.48612	0.0711723
$437$	5.90833	0.282634
$438$	0	0
$439$	−14.6972	−0.701460	−0.350730	−	0.936477i	$-0.614067\pi$
$439$	−14.6972	−0.701460	−0.350730	+	0.936477i	$0.614067\pi$
$440$	−1.69722	−0.0809120
$441$	0	0
$442$	−22.8167	−1.08528
$443$	−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
$446$	−4.00000	−0.189405
$447$	0	0
$448$	3.30278	0.156041
$449$	2.09167	0.0987122	0.0493561	−	0.998781i	$-0.484283\pi$
$449$	2.09167	0.0987122	0.0493561	+	0.998781i	$0.484283\pi$
$450$	0	0
$451$	−1.54163	−0.0725927
$452$	16.4222	0.772436
$453$	0	0
$454$	14.6056	0.685472
$455$	−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	$0.773981\pi$
$458$	2.00000	0.0934539
$459$	0	0
$460$	−1.00000	−0.0466252
$461$	−10.1833	−0.474286	−0.237143	−	0.971475i	$-0.576211\pi$
$461$	−10.1833	−0.474286	−0.237143	+	0.971475i	$0.576211\pi$
$462$	0	0
$463$	17.6333	0.819489	0.409745	−	0.912200i	$-0.365618\pi$
$463$	17.6333	0.819489	0.409745	+	0.912200i	$0.365618\pi$
$464$	2.60555	0.120960
$465$	0	0
$466$	−25.8167	−1.19593
$467$	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
$470$	−2.60555	−0.120185
$471$	0	0
$472$	3.39445	0.156242
$473$	−15.6333	−0.718820
$474$	0	0
$475$	5.90833	0.271093
$476$	−22.8167	−1.04580
$477$	0	0
$478$	−5.21110	−0.238350
$479$	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
$482$	−14.4222	−0.656913
$483$	0	0
$484$	−8.11943	−0.369065
$485$	−6.30278	−0.286194
$486$	0	0
$487$	9.81665	0.444835	0.222418	−	0.974952i	$-0.428605\pi$
$487$	9.81665	0.444835	0.222418	+	0.974952i	$0.428605\pi$
$488$	11.5139	0.521209
$489$	0	0
$490$	−3.90833	−0.176560
$491$	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
$494$	19.5139	0.877971
$495$	0	0
$496$	−7.90833	−0.355094
$497$	53.8444	2.41525
$498$	0	0
$499$	−31.6333	−1.41610	−0.708051	−	0.706162i	$-0.750425\pi$
$499$	−31.6333	−1.41610	−0.708051	+	0.706162i	$0.750425\pi$
$500$	−1.00000	−0.0447214
$501$	0	0
$502$	−12.5139	−0.558522
$503$	−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
$506$	1.69722	0.0754508
$507$	0	0
$508$	9.81665	0.435544
$509$	35.4500	1.57129	0.785646	−	0.618676i	$-0.212331\pi$
$509$	35.4500	1.57129	0.785646	+	0.618676i	$0.212331\pi$
$510$	0	0
$511$	−19.2111	−0.849849
$512$	1.00000	0.0441942
$513$	0	0
$514$	1.81665	0.0801292
$515$	−8.11943	−0.357785
$516$	0	0
$517$	4.42221	0.194488
$518$	26.4222	1.16093
$519$	0	0
$520$	−3.30278	−0.144836
$521$	−6.00000	−0.262865	−0.131432	−	0.991325i	$-0.541958\pi$
$521$	−6.00000	−0.262865	−0.131432	+	0.991325i	$0.541958\pi$
$522$	0	0
$523$	−20.4222	−0.893001	−0.446500	−	0.894783i	$-0.647330\pi$
$523$	−20.4222	−0.893001	−0.446500	+	0.894783i	$0.647330\pi$
$524$	11.2111	0.489759
$525$	0	0
$526$	−3.51388	−0.153212
$527$	54.6333	2.37986
$528$	0	0
$529$	1.00000	0.0434783
$530$	−11.2111	−0.486979
$531$	0	0
$532$	19.5139	0.846034
$533$	−3.00000	−0.129944
$534$	0	0
$535$	−2.60555	−0.112648
$536$	−4.00000	−0.172774
$537$	0	0
$538$	−4.18335	−0.180357
$539$	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	$0.287109\pi$
$542$	−2.69722	−0.115856
$543$	0	0
$544$	−6.90833	−0.296192
$545$	−1.48612	−0.0636585
$546$	0	0
$547$	−10.5139	−0.449541	−0.224770	−	0.974412i	$-0.572163\pi$
$547$	−10.5139	−0.449541	−0.224770	+	0.974412i	$0.572163\pi$
$548$	−3.90833	−0.166955
$549$	0	0
$550$	1.69722	0.0723699
$551$	15.3944	0.655826
$552$	0	0
$553$	−47.6333	−2.02557
$554$	−27.2111	−1.15609
$555$	0	0
$556$	−12.6056	−0.534594
$557$	−22.4222	−0.950059	−0.475030	−	0.879970i	$-0.657563\pi$
$557$	−22.4222	−0.950059	−0.475030	+	0.879970i	$0.657563\pi$
$558$	0	0
$559$	−30.4222	−1.28672
$560$	−3.30278	−0.139568
$561$	0	0
$562$	26.6056	1.12229
$563$	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
$566$	2.00000	0.0840663
$567$	0	0
$568$	16.3028	0.684049
$569$	−28.4222	−1.19152	−0.595760	−	0.803162i	$-0.703149\pi$
$569$	−28.4222	−1.19152	−0.595760	+	0.803162i	$0.703149\pi$
$570$	0	0
$571$	−16.1194	−0.674577	−0.337289	−	0.941401i	$-0.609510\pi$
$571$	−16.1194	−0.674577	−0.337289	+	0.941401i	$0.609510\pi$
$572$	5.60555	0.234380
$573$	0	0
$574$	−3.00000	−0.125218
$575$	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	$0.486745\pi$
$578$	30.7250	1.27799
$579$	0	0
$580$	−2.60555	−0.108190
$581$	−37.0278	−1.53617
$582$	0	0
$583$	19.0278	0.788049
$584$	−5.81665	−0.240695
$585$	0	0
$586$	−23.2111	−0.958842
$587$	−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
$590$	−3.39445	−0.139747
$591$	0	0
$592$	8.00000	0.328798
$593$	1.81665	0.0746010	0.0373005	−	0.999304i	$-0.488124\pi$
$593$	1.81665	0.0746010	0.0373005	+	0.999304i	$0.488124\pi$
$594$	0	0
$595$	22.8167	0.935392
$596$	−13.3028	−0.544903
$597$	0	0
$598$	3.30278	0.135061
$599$	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	$0.160401\pi$
$602$	−30.4222	−1.23992
$603$	0	0
$604$	8.90833	0.362475
$605$	8.11943	0.330102
$606$	0	0
$607$	46.0555	1.86934	0.934668	−	0.355522i	$-0.115697\pi$
$607$	46.0555	1.86934	0.934668	+	0.355522i	$0.115697\pi$
$608$	5.90833	0.239614
$609$	0	0
$610$	−11.5139	−0.466183
$611$	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	$0.476981\pi$
$614$	−11.6972	−0.472062
$615$	0	0
$616$	5.60555	0.225854
$617$	−18.9083	−0.761221	−0.380610	−	0.924736i	$-0.624286\pi$
$617$	−18.9083	−0.761221	−0.380610	+	0.924736i	$0.624286\pi$
$618$	0	0
$619$	−12.3305	−0.495606	−0.247803	−	0.968810i	$-0.579709\pi$
$619$	−12.3305	−0.495606	−0.247803	+	0.968810i	$0.579709\pi$
$620$	7.90833	0.317606
$621$	0	0
$622$	−22.4222	−0.899049
$623$	0	0
$624$	0	0
$625$	1.00000	0.0400000
$626$	19.7250	0.788369
$627$	0	0
$628$	−18.6056	−0.742442
$629$	−55.2666	−2.20362
$630$	0	0
$631$	23.3944	0.931318	0.465659	−	0.884964i	$-0.345817\pi$
$631$	23.3944	0.931318	0.465659	+	0.884964i	$0.345817\pi$
$632$	−14.4222	−0.573685
$633$	0	0
$634$	−17.7250	−0.703949
$635$	−9.81665	−0.389562
$636$	0	0
$637$	12.9083	0.511447
$638$	4.42221	0.175077
$639$	0	0
$640$	−1.00000	−0.0395285
$641$	36.0000	1.42191	0.710957	−	0.703235i	$-0.248262\pi$
$641$	36.0000	1.42191	0.710957	+	0.703235i	$0.248262\pi$
$642$	0	0
$643$	−34.2389	−1.35025	−0.675124	−	0.737704i	$-0.735910\pi$
$643$	−34.2389	−1.35025	−0.675124	+	0.737704i	$0.735910\pi$
$644$	3.30278	0.130148
$645$	0	0
$646$	−40.8167	−1.60591
$647$	−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
$650$	3.30278	0.129546
$651$	0	0
$652$	9.30278	0.364325
$653$	−41.7250	−1.63282	−0.816412	−	0.577469i	$-0.804040\pi$
$653$	−41.7250	−1.63282	−0.816412	+	0.577469i	$0.804040\pi$
$654$	0	0
$655$	−11.2111	−0.438054
$656$	−0.908327	−0.0354642
$657$	0	0
$658$	8.60555	0.335479
$659$	−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	$0.737538\pi$
$662$	16.6056	0.645393
$663$	0	0
$664$	−11.2111	−0.435075
$665$	−19.5139	−0.756716
$666$	0	0
$667$	2.60555	0.100887
$668$	−6.78890	−0.262670
$669$	0	0
$670$	4.00000	0.154533
$671$	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	$0.758312\pi$
$674$	−22.5139	−0.867202
$675$	0	0
$676$	−2.09167	−0.0804490
$677$	−16.4222	−0.631157	−0.315578	−	0.948900i	$-0.602198\pi$
$677$	−16.4222	−0.631157	−0.315578	+	0.948900i	$0.602198\pi$
$678$	0	0
$679$	20.8167	0.798870
$680$	6.90833	0.264922
$681$	0	0
$682$	−13.4222	−0.513963
$683$	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
$686$	−10.2111	−0.389862
$687$	0	0
$688$	−9.21110	−0.351170
$689$	37.0278	1.41065
$690$	0	0
$691$	51.8167	1.97120	0.985599	−	0.169098i	$-0.0540855\pi$
$691$	51.8167	1.97120	0.985599	+	0.169098i	$0.0540855\pi$
$692$	−19.6972	−0.748776
$693$	0	0
$694$	−28.5416	−1.08343
$695$	12.6056	0.478156
$696$	0	0
$697$	6.27502	0.237683
$698$	−27.2111	−1.02996
$699$	0	0
$700$	3.30278	0.124833
$701$	−32.0917	−1.21209	−0.606043	−	0.795432i	$-0.707244\pi$
$701$	−32.0917	−1.21209	−0.606043	+	0.795432i	$0.707244\pi$
$702$	0	0
$703$	47.2666	1.78269
$704$	1.69722	0.0639665
$705$	0	0
$706$	10.4222	0.392245
$707$	−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	$0.596387\pi$
$710$	−16.3028	−0.611832
$711$	0	0
$712$	0	0
$713$	−7.90833	−0.296169
$714$	0	0
$715$	−5.60555	−0.209636
$716$	−9.39445	−0.351087
$717$	0	0
$718$	−11.2111	−0.418395
$719$	−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
$722$	15.9083	0.592047
$723$	0	0
$724$	17.1194	0.636239
$725$	2.60555	0.0967677
$726$	0	0
$727$	2.90833	0.107864	0.0539319	−	0.998545i	$-0.482825\pi$
$727$	2.90833	0.107864	0.0539319	+	0.998545i	$0.482825\pi$
$728$	10.9083	0.404289
$729$	0	0
$730$	5.81665	0.215284
$731$	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	$0.315669\pi$
$734$	14.7889	0.545868
$735$	0	0
$736$	1.00000	0.0368605
$737$	−6.78890	−0.250072
$738$	0	0
$739$	35.6333	1.31079	0.655396	−	0.755285i	$-0.272502\pi$
$739$	35.6333	1.31079	0.655396	+	0.755285i	$0.272502\pi$
$740$	−8.00000	−0.294086
$741$	0	0
$742$	37.0278	1.35933
$743$	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
$746$	4.60555	0.168621
$747$	0	0
$748$	−11.7250	−0.428708
$749$	8.60555	0.314440
$750$	0	0
$751$	21.8167	0.796101	0.398051	−	0.917364i	$-0.369687\pi$
$751$	21.8167	0.796101	0.398051	+	0.917364i	$0.369687\pi$
$752$	2.60555	0.0950147
$753$	0	0
$754$	8.60555	0.313396
$755$	−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	$0.422825\pi$
$758$	14.9083	0.541495
$759$	0	0
$760$	−5.90833	−0.214318
$761$	49.5416	1.79588	0.897941	−	0.440115i	$-0.145062\pi$
$761$	49.5416	1.79588	0.897941	+	0.440115i	$0.145062\pi$
$762$	0	0
$763$	4.90833	0.177693
$764$	8.60555	0.311338
$765$	0	0
$766$	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	$0.196089\pi$
$770$	−5.60555	−0.202010
$771$	0	0
$772$	−17.8167	−0.641235
$773$	−12.0000	−0.431610	−0.215805	−	0.976436i	$-0.569238\pi$
$773$	−12.0000	−0.431610	−0.215805	+	0.976436i	$0.569238\pi$
$774$	0	0
$775$	−7.90833	−0.284075
$776$	6.30278	0.226256
$777$	0	0
$778$	25.9361	0.929854
$779$	−5.36669	−0.192282
$780$	0	0
$781$	27.6695	0.990091
$782$	−6.90833	−0.247041
$783$	0	0
$784$	3.90833	0.139583
$785$	18.6056	0.664061
$786$	0	0
$787$	37.4500	1.33495	0.667473	−	0.744634i	$-0.267376\pi$
$787$	37.4500	1.33495	0.667473	+	0.744634i	$0.267376\pi$
$788$	−4.30278	−0.153280
$789$	0	0
$790$	14.4222	0.513119
$791$	54.2389	1.92851
$792$	0	0
$793$	38.0278	1.35041
$794$	10.7250	0.380615
$795$	0	0
$796$	−20.4222	−0.723846
$797$	−1.81665	−0.0643492	−0.0321746	−	0.999482i	$-0.510243\pi$
$797$	−1.81665	−0.0643492	−0.0321746	+	0.999482i	$0.510243\pi$
$798$	0	0
$799$	−18.0000	−0.636794
$800$	1.00000	0.0353553
$801$	0	0
$802$	8.60555	0.303873
$803$	−9.87217	−0.348381
$804$	0	0
$805$	−3.30278	−0.116408
$806$	−26.1194	−0.920018
$807$	0	0
$808$	−2.60555	−0.0916630
$809$	−50.7250	−1.78340	−0.891698	−	0.452631i	$-0.850485\pi$
$809$	−50.7250	−1.78340	−0.891698	+	0.452631i	$0.850485\pi$
$810$	0	0
$811$	−41.0278	−1.44068	−0.720340	−	0.693621i	$-0.756014\pi$
$811$	−41.0278	−1.44068	−0.720340	+	0.693621i	$0.756014\pi$
$812$	8.60555	0.301996
$813$	0	0
$814$	13.5778	0.475901
$815$	−9.30278	−0.325862
$816$	0	0
$817$	−54.4222	−1.90399
$818$	−25.9083	−0.905863
$819$	0	0
$820$	0.908327	0.0317202
$821$	30.0000	1.04701	0.523504	−	0.852023i	$-0.324625\pi$
$821$	30.0000	1.04701	0.523504	+	0.852023i	$0.324625\pi$
$822$	0	0
$823$	−15.2111	−0.530226	−0.265113	−	0.964217i	$-0.585409\pi$
$823$	−15.2111	−0.530226	−0.265113	+	0.964217i	$0.585409\pi$
$824$	8.11943	0.282854
$825$	0	0
$826$	11.2111	0.390084
$827$	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	$0.317666\pi$
$830$	11.2111	0.389143
$831$	0	0
$832$	3.30278	0.114503
$833$	−27.0000	−0.935495
$834$	0	0
$835$	6.78890	0.234939
$836$	10.0278	0.346817
$837$	0	0
$838$	−3.63331	−0.125511
$839$	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
$842$	30.6972	1.05790
$843$	0	0
$844$	7.21110	0.248216
$845$	2.09167	0.0719557
$846$	0	0
$847$	−26.8167	−0.921431
$848$	11.2111	0.384991
$849$	0	0
$850$	−6.90833	−0.236954
$851$	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	$0.621302\pi$
$854$	38.0278	1.30128
$855$	0	0
$856$	2.60555	0.0890559
$857$	−9.63331	−0.329068	−0.164534	−	0.986371i	$-0.552612\pi$
$857$	−9.63331	−0.329068	−0.164534	+	0.986371i	$0.552612\pi$
$858$	0	0
$859$	−35.8167	−1.22205	−0.611024	−	0.791612i	$-0.709242\pi$
$859$	−35.8167	−1.22205	−0.611024	+	0.791612i	$0.709242\pi$
$860$	9.21110	0.314096
$861$	0	0
$862$	30.2389	1.02994
$863$	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
$866$	−24.0917	−0.818668
$867$	0	0
$868$	−26.1194	−0.886551
$869$	−24.4777	−0.830350
$870$	0	0
$871$	−13.2111	−0.447641
$872$	1.48612	0.0503264
$873$	0	0
$874$	5.90833	0.199852
$875$	−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	$0.802373\pi$
$878$	−14.6972	−0.496007
$879$	0	0
$880$	−1.69722	−0.0572134
$881$	−55.2666	−1.86198	−0.930990	−	0.365045i	$-0.881054\pi$
$881$	−55.2666	−1.86198	−0.930990	+	0.365045i	$0.881054\pi$
$882$	0	0
$883$	8.27502	0.278477	0.139238	−	0.990259i	$-0.455535\pi$
$883$	8.27502	0.278477	0.139238	+	0.990259i	$0.455535\pi$
$884$	−22.8167	−0.767407
$885$	0	0
$886$	−17.4861	−0.587458
$887$	−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
$891$	0	0
$892$	−4.00000	−0.133930
$893$	15.3944	0.515156
$894$	0	0
$895$	9.39445	0.314022
$896$	3.30278	0.110338
$897$	0	0
$898$	2.09167	0.0698000
$899$	−20.6056	−0.687234
$900$	0	0
$901$	−77.4500	−2.58023
$902$	−1.54163	−0.0513308
$903$	0	0
$904$	16.4222	0.546194
$905$	−17.1194	−0.569069
$906$	0	0
$907$	48.6611	1.61576	0.807882	−	0.589344i	$-0.200614\pi$
$907$	48.6611	1.61576	0.807882	+	0.589344i	$0.200614\pi$
$908$	14.6056	0.484702
$909$	0	0
$910$	−10.9083	−0.361608
$911$	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
$914$	−32.4222	−1.07243
$915$	0	0
$916$	2.00000	0.0660819
$917$	37.0278	1.22276
$918$	0	0
$919$	44.0000	1.45143	0.725713	−	0.687998i	$-0.241510\pi$
$919$	44.0000	1.45143	0.725713	+	0.687998i	$0.241510\pi$
$920$	−1.00000	−0.0329690
$921$	0	0
$922$	−10.1833	−0.335371
$923$	53.8444	1.77231
$924$	0	0
$925$	8.00000	0.263038
$926$	17.6333	0.579466
$927$	0	0
$928$	2.60555	0.0855314
$929$	14.3667	0.471356	0.235678	−	0.971831i	$-0.424269\pi$
$929$	14.3667	0.471356	0.235678	+	0.971831i	$0.424269\pi$
$930$	0	0
$931$	23.0917	0.756799
$932$	−25.8167	−0.845653
$933$	0	0
$934$	1.81665	0.0594427
$935$	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	$0.287087\pi$
$938$	−13.2111	−0.431358
$939$	0	0
$940$	−2.60555	−0.0849837
$941$	−25.9361	−0.845492	−0.422746	−	0.906248i	$-0.638934\pi$
$941$	−25.9361	−0.845492	−0.422746	+	0.906248i	$0.638934\pi$
$942$	0	0
$943$	−0.908327	−0.0295792
$944$	3.39445	0.110480
$945$	0	0
$946$	−15.6333	−0.508283
$947$	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
$950$	5.90833	0.191691
$951$	0	0
$952$	−22.8167	−0.739492
$953$	41.3305	1.33883	0.669414	−	0.742890i	$-0.266545\pi$
$953$	41.3305	1.33883	0.669414	+	0.742890i	$0.266545\pi$
$954$	0	0
$955$	−8.60555	−0.278469
$956$	−5.21110	−0.168539
$957$	0	0
$958$	30.0000	0.969256
$959$	−12.9083	−0.416832
$960$	0	0
$961$	31.5416	1.01747
$962$	26.4222	0.851886
$963$	0	0
$964$	−14.4222	−0.464508
$965$	17.8167	0.573538
$966$	0	0
$967$	−12.6056	−0.405367	−0.202684	−	0.979244i	$-0.564966\pi$
$967$	−12.6056	−0.405367	−0.202684	+	0.979244i	$0.564966\pi$
$968$	−8.11943	−0.260968
$969$	0	0
$970$	−6.30278	−0.202370
$971$	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
$974$	9.81665	0.314546
$975$	0	0
$976$	11.5139	0.368550
$977$	−6.51388	−0.208397	−0.104199	−	0.994556i	$-0.533228\pi$
$977$	−6.51388	−0.208397	−0.104199	+	0.994556i	$0.533228\pi$
$978$	0	0
$979$	0	0
$980$	−3.90833	−0.124847
$981$	0	0
$982$	4.18335	0.133496
$983$	−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
$986$	−18.0000	−0.573237
$987$	0	0
$988$	19.5139	0.620819
$989$	−9.21110	−0.292896
$990$	0	0
$991$	−15.3305	−0.486990	−0.243495	−	0.969902i	$-0.578294\pi$
$991$	−15.3305	−0.486990	−0.243495	+	0.969902i	$0.578294\pi$
$992$	−7.90833	−0.251090
$993$	0	0
$994$	53.8444	1.70784
$995$	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	$0.585654\pi$
$998$	−31.6333	−1.00133
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2070.2.a.w.1.2		2
3.2	odd	2		230.2.a.b.1.1	✓	2
12.11	even	2		1840.2.a.j.1.2		2
15.2	even	4		1150.2.b.f.599.2		4
15.8	even	4		1150.2.b.f.599.3		4
15.14	odd	2		1150.2.a.m.1.2		2
24.5	odd	2		7360.2.a.bc.1.2		2
24.11	even	2		7360.2.a.bu.1.1		2
60.59	even	2		9200.2.a.ca.1.1		2
69.68	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	3.2	odd	2
1150.2.a.m.1.2		2	15.14	odd	2
1150.2.b.f.599.2		4	15.2	even	4
1150.2.b.f.599.3		4	15.8	even	4
1840.2.a.j.1.2		2	12.11	even	2
2070.2.a.w.1.2		2	1.1	even	1	trivial
5290.2.a.j.1.1		2	69.68	even	2
7360.2.a.bc.1.2		2	24.5	odd	2
7360.2.a.bu.1.1		2	24.11	even	2
9200.2.a.ca.1.1		2	60.59	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:	\( N \)	\(=\)	\( 2070 = 2 \cdot 3^{2} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2070.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} +3.30278 q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} +3.30278 q^{7} +1.00000 q^{8} -1.00000 q^{10} +1.69722 q^{11} +3.30278 q^{13} +3.30278 q^{14} +1.00000 q^{16} -6.90833 q^{17} +5.90833 q^{19} -1.00000 q^{20} +1.69722 q^{22} +1.00000 q^{23} +1.00000 q^{25} +3.30278 q^{26} +3.30278 q^{28} +2.60555 q^{29} -7.90833 q^{31} +1.00000 q^{32} -6.90833 q^{34} -3.30278 q^{35} +8.00000 q^{37} +5.90833 q^{38} -1.00000 q^{40} -0.908327 q^{41} -9.21110 q^{43} +1.69722 q^{44} +1.00000 q^{46} +2.60555 q^{47} +3.90833 q^{49} +1.00000 q^{50} +3.30278 q^{52} +11.2111 q^{53} -1.69722 q^{55} +3.30278 q^{56} +2.60555 q^{58} +3.39445 q^{59} +11.5139 q^{61} -7.90833 q^{62} +1.00000 q^{64} -3.30278 q^{65} -4.00000 q^{67} -6.90833 q^{68} -3.30278 q^{70} +16.3028 q^{71} -5.81665 q^{73} +8.00000 q^{74} +5.90833 q^{76} +5.60555 q^{77} -14.4222 q^{79} -1.00000 q^{80} -0.908327 q^{82} -11.2111 q^{83} +6.90833 q^{85} -9.21110 q^{86} +1.69722 q^{88} +10.9083 q^{91} +1.00000 q^{92} +2.60555 q^{94} -5.90833 q^{95} +6.30278 q^{97} +3.90833 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 2 q^{4} - 2 q^{5} + 3 q^{7} + 2 q^{8}+O(q^{10}) \) 2 * q + 2 * q^2 + 2 * q^4 - 2 * q^5 + 3 * q^7 + 2 * q^8	\( 2 q + 2 q^{2} + 2 q^{4} - 2 q^{5} + 3 q^{7} + 2 q^{8} - 2 q^{10} + 7 q^{11} + 3 q^{13} + 3 q^{14} + 2 q^{16} - 3 q^{17} + q^{19} - 2 q^{20} + 7 q^{22} + 2 q^{23} + 2 q^{25} + 3 q^{26} + 3 q^{28} - 2 q^{29} - 5 q^{31} + 2 q^{32} - 3 q^{34} - 3 q^{35} + 16 q^{37} + q^{38} - 2 q^{40} + 9 q^{41} - 4 q^{43} + 7 q^{44} + 2 q^{46} - 2 q^{47} - 3 q^{49} + 2 q^{50} + 3 q^{52} + 8 q^{53} - 7 q^{55} + 3 q^{56} - 2 q^{58} + 14 q^{59} + 5 q^{61} - 5 q^{62} + 2 q^{64} - 3 q^{65} - 8 q^{67} - 3 q^{68} - 3 q^{70} + 29 q^{71} + 10 q^{73} + 16 q^{74} + q^{76} + 4 q^{77} - 2 q^{80} + 9 q^{82} - 8 q^{83} + 3 q^{85} - 4 q^{86} + 7 q^{88} + 11 q^{91} + 2 q^{92} - 2 q^{94} - q^{95} + 9 q^{97} - 3 q^{98}+O(q^{100}) \) 2 * q + 2 * q^2 + 2 * q^4 - 2 * q^5 + 3 * q^7 + 2 * q^8 - 2 * q^10 + 7 * q^11 + 3 * q^13 + 3 * q^14 + 2 * q^16 - 3 * q^17 + q^19 - 2 * q^20 + 7 * q^22 + 2 * q^23 + 2 * q^25 + 3 * q^26 + 3 * q^28 - 2 * q^29 - 5 * q^31 + 2 * q^32 - 3 * q^34 - 3 * q^35 + 16 * q^37 + q^38 - 2 * q^40 + 9 * q^41 - 4 * q^43 + 7 * q^44 + 2 * q^46 - 2 * q^47 - 3 * q^49 + 2 * q^50 + 3 * q^52 + 8 * q^53 - 7 * q^55 + 3 * q^56 - 2 * q^58 + 14 * q^59 + 5 * q^61 - 5 * q^62 + 2 * q^64 - 3 * q^65 - 8 * q^67 - 3 * q^68 - 3 * q^70 + 29 * q^71 + 10 * q^73 + 16 * q^74 + q^76 + 4 * q^77 - 2 * q^80 + 9 * q^82 - 8 * q^83 + 3 * q^85 - 4 * q^86 + 7 * q^88 + 11 * q^91 + 2 * q^92 - 2 * q^94 - q^95 + 9 * q^97 - 3 * q^98

Introduction

L-functions

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