LMFDB - Embedded newform 7440.2.a.bm.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7440,2,Mod(1,7440)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7440, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("7440.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7440 = 2^{4} \cdot 3 \cdot 5 \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7440.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:	$59.4086991038$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.148.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 3x + 1 $ x^3 - x^2 - 3*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 465)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$0.311108$ of defining polynomial
Character	$\chi$	$=$	7440.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} -1.00000 q^{5} +1.59210 q^{7} +1.00000 q^{9} -0.622216 q^{11} +0.214320 q^{13} +1.00000 q^{15} +3.52543 q^{17} -1.80642 q^{19} -1.59210 q^{21} -6.90321 q^{23} +1.00000 q^{25} -1.00000 q^{27} +9.73975 q^{29} +1.00000 q^{31} +0.622216 q^{33} -1.59210 q^{35} -4.83654 q^{37} -0.214320 q^{39} -7.47949 q^{41} -8.23506 q^{43} -1.00000 q^{45} -11.4652 q^{47} -4.46520 q^{49} -3.52543 q^{51} +13.7605 q^{53} +0.622216 q^{55} +1.80642 q^{57} +4.26025 q^{59} +2.85728 q^{61} +1.59210 q^{63} -0.214320 q^{65} +2.08097 q^{67} +6.90321 q^{69} -1.31111 q^{71} -1.65233 q^{73} -1.00000 q^{75} -0.990632 q^{77} +5.19850 q^{79} +1.00000 q^{81} +5.65878 q^{83} -3.52543 q^{85} -9.73975 q^{87} -1.93332 q^{89} +0.341219 q^{91} -1.00000 q^{93} +1.80642 q^{95} +6.91750 q^{97} -0.622216 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{3} - 3 q^{5} - 2 q^{7} + 3 q^{9} - 2 q^{11} - 6 q^{13} + 3 q^{15} + 4 q^{17} + 8 q^{19} + 2 q^{21} - 14 q^{23} + 3 q^{25} - 3 q^{27} + 16 q^{29} + 3 q^{31} + 2 q^{33} + 2 q^{35} - 8 q^{37} + 6 q^{39}+ \cdots - 2 q^{99}+O(q^{100}) $ 3 * q - 3 * q^3 - 3 * q^5 - 2 * q^7 + 3 * q^9 - 2 * q^11 - 6 * q^13 + 3 * q^15 + 4 * q^17 + 8 * q^19 + 2 * q^21 - 14 * q^23 + 3 * q^25 - 3 * q^27 + 16 * q^29 + 3 * q^31 + 2 * q^33 + 2 * q^35 - 8 * q^37 + 6 * q^39 + 4 * q^41 + 2 * q^43 - 3 * q^45 - 14 * q^47 + 7 * q^49 - 4 * q^51 + 8 * q^53 + 2 * q^55 - 8 * q^57 + 26 * q^59 - 18 * q^61 - 2 * q^63 + 6 * q^65 + 14 * q^69 - 4 * q^71 - 12 * q^73 - 3 * q^75 + 24 * q^77 - 4 * q^79 + 3 * q^81 + 10 * q^83 - 4 * q^85 - 16 * q^87 - 6 * q^89 + 8 * q^91 - 3 * q^93 - 8 * q^95 + 8 * q^97 - 2 * 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
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	1.59210	0.601759	0.300879	−	0.953662i	$-0.402720\pi$
$7$	1.59210	0.601759	0.300879	+	0.953662i	$0.402720\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−0.622216	−0.187605	−0.0938025	−	0.995591i	$-0.529902\pi$
$11$	−0.622216	−0.187605	−0.0938025	+	0.995591i	$0.529902\pi$
$12$	0	0
$13$	0.214320	0.0594416	0.0297208	−	0.999558i	$-0.490538\pi$
$13$	0.214320	0.0594416	0.0297208	+	0.999558i	$0.490538\pi$
$14$	0	0
$15$	1.00000	0.258199
$16$	0	0
$17$	3.52543	0.855042	0.427521	−	0.904005i	$-0.359387\pi$
$17$	3.52543	0.855042	0.427521	+	0.904005i	$0.359387\pi$
$18$	0	0
$19$	−1.80642	−0.414422	−0.207211	−	0.978296i	$-0.566439\pi$
$19$	−1.80642	−0.414422	−0.207211	+	0.978296i	$0.566439\pi$
$20$	0	0
$21$	−1.59210	−0.347426
$22$	0	0
$23$	−6.90321	−1.43942	−0.719710	−	0.694275i	$-0.755725\pi$
$23$	−6.90321	−1.43942	−0.719710	+	0.694275i	$0.755725\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	9.73975	1.80863	0.904313	−	0.426870i	$-0.140384\pi$
$29$	9.73975	1.80863	0.904313	+	0.426870i	$0.140384\pi$
$30$	0	0
$31$	1.00000	0.179605
$31$	1.00000	0.179605
$32$	0	0
$33$	0.622216	0.108314
$34$	0	0
$35$	−1.59210	−0.269115
$36$	0	0
$37$	−4.83654	−0.795122	−0.397561	−	0.917576i	$-0.630143\pi$
$37$	−4.83654	−0.795122	−0.397561	+	0.917576i	$0.630143\pi$
$38$	0	0
$39$	−0.214320	−0.0343186
$40$	0	0
$41$	−7.47949	−1.16810	−0.584050	−	0.811717i	$-0.698533\pi$
$41$	−7.47949	−1.16810	−0.584050	+	0.811717i	$0.698533\pi$
$42$	0	0
$43$	−8.23506	−1.25584	−0.627918	−	0.778280i	$-0.716093\pi$
$43$	−8.23506	−1.25584	−0.627918	+	0.778280i	$0.716093\pi$
$44$	0	0
$45$	−1.00000	−0.149071
$46$	0	0
$47$	−11.4652	−1.67237	−0.836186	−	0.548446i	$-0.815220\pi$
$47$	−11.4652	−1.67237	−0.836186	+	0.548446i	$0.815220\pi$
$48$	0	0
$49$	−4.46520	−0.637886
$50$	0	0
$51$	−3.52543	−0.493659
$52$	0	0
$53$	13.7605	1.89015	0.945074	−	0.326855i	$-0.105989\pi$
$53$	13.7605	1.89015	0.945074	+	0.326855i	$0.105989\pi$
$54$	0	0
$55$	0.622216	0.0838995
$56$	0	0
$57$	1.80642	0.239267
$58$	0	0
$59$	4.26025	0.554638	0.277319	−	0.960778i	$-0.410554\pi$
$59$	4.26025	0.554638	0.277319	+	0.960778i	$0.410554\pi$
$60$	0	0
$61$	2.85728	0.365837	0.182919	−	0.983128i	$-0.441446\pi$
$61$	2.85728	0.365837	0.182919	+	0.983128i	$0.441446\pi$
$62$	0	0
$63$	1.59210	0.200586
$64$	0	0
$65$	−0.214320	−0.0265831
$66$	0	0
$67$	2.08097	0.254231	0.127115	−	0.991888i	$-0.459428\pi$
$67$	2.08097	0.254231	0.127115	+	0.991888i	$0.459428\pi$
$68$	0	0
$69$	6.90321	0.831049
$70$	0	0
$71$	−1.31111	−0.155600	−0.0777999	−	0.996969i	$-0.524790\pi$
$71$	−1.31111	−0.155600	−0.0777999	+	0.996969i	$0.524790\pi$
$72$	0	0
$73$	−1.65233	−0.193390	−0.0966951	−	0.995314i	$-0.530827\pi$
$73$	−1.65233	−0.193390	−0.0966951	+	0.995314i	$0.530827\pi$
$74$	0	0
$75$	−1.00000	−0.115470
$76$	0	0
$77$	−0.990632	−0.112893
$78$	0	0
$79$	5.19850	0.584877	0.292438	−	0.956284i	$-0.405533\pi$
$79$	5.19850	0.584877	0.292438	+	0.956284i	$0.405533\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	5.65878	0.621132	0.310566	−	0.950552i	$-0.399481\pi$
$83$	5.65878	0.621132	0.310566	+	0.950552i	$0.399481\pi$
$84$	0	0
$85$	−3.52543	−0.382386
$86$	0	0
$87$	−9.73975	−1.04421
$88$	0	0
$89$	−1.93332	−0.204932	−0.102466	−	0.994737i	$-0.532673\pi$
$89$	−1.93332	−0.204932	−0.102466	+	0.994737i	$0.532673\pi$
$90$	0	0
$91$	0.341219	0.0357695
$92$	0	0
$93$	−1.00000	−0.103695
$94$	0	0
$95$	1.80642	0.185335
$96$	0	0
$97$	6.91750	0.702366	0.351183	−	0.936307i	$-0.385780\pi$
$97$	6.91750	0.702366	0.351183	+	0.936307i	$0.385780\pi$
$98$	0	0
$99$	−0.622216	−0.0625350
$100$	0	0
$101$	−9.28592	−0.923983	−0.461992	−	0.886884i	$-0.652865\pi$
$101$	−9.28592	−0.923983	−0.461992	+	0.886884i	$0.652865\pi$
$102$	0	0
$103$	−0.274543	−0.0270515	−0.0135258	−	0.999909i	$-0.504306\pi$
$103$	−0.274543	−0.0270515	−0.0135258	+	0.999909i	$0.504306\pi$
$104$	0	0
$105$	1.59210	0.155373
$106$	0	0
$107$	8.18913	0.791673	0.395837	−	0.918321i	$-0.370455\pi$
$107$	8.18913	0.791673	0.395837	+	0.918321i	$0.370455\pi$
$108$	0	0
$109$	−9.13828	−0.875288	−0.437644	−	0.899148i	$-0.644187\pi$
$109$	−9.13828	−0.875288	−0.437644	+	0.899148i	$0.644187\pi$
$110$	0	0
$111$	4.83654	0.459064
$112$	0	0
$113$	7.67307	0.721822	0.360911	−	0.932600i	$-0.382466\pi$
$113$	7.67307	0.721822	0.360911	+	0.932600i	$0.382466\pi$
$114$	0	0
$115$	6.90321	0.643728
$116$	0	0
$117$	0.214320	0.0198139
$118$	0	0
$119$	5.61285	0.514529
$120$	0	0
$121$	−10.6128	−0.964804
$122$	0	0
$123$	7.47949	0.674403
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−13.6731	−1.21329	−0.606644	−	0.794973i	$-0.707485\pi$
$127$	−13.6731	−1.21329	−0.606644	+	0.794973i	$0.707485\pi$
$128$	0	0
$129$	8.23506	0.725057
$130$	0	0
$131$	22.2701	1.94575	0.972874	−	0.231337i	$-0.0743100\pi$
$131$	22.2701	1.94575	0.972874	+	0.231337i	$0.0743100\pi$
$132$	0	0
$133$	−2.87601	−0.249382
$134$	0	0
$135$	1.00000	0.0860663
$136$	0	0
$137$	−18.1891	−1.55400	−0.777001	−	0.629499i	$-0.783260\pi$
$137$	−18.1891	−1.55400	−0.777001	+	0.629499i	$0.783260\pi$
$138$	0	0
$139$	4.13335	0.350586	0.175293	−	0.984516i	$-0.443913\pi$
$139$	4.13335	0.350586	0.175293	+	0.984516i	$0.443913\pi$
$140$	0	0
$141$	11.4652	0.965544
$142$	0	0
$143$	−0.133353	−0.0111515
$144$	0	0
$145$	−9.73975	−0.808842
$146$	0	0
$147$	4.46520	0.368284
$148$	0	0
$149$	6.23506	0.510796	0.255398	−	0.966836i	$-0.417793\pi$
$149$	6.23506	0.510796	0.255398	+	0.966836i	$0.417793\pi$
$150$	0	0
$151$	−8.38271	−0.682175	−0.341087	−	0.940032i	$-0.610795\pi$
$151$	−8.38271	−0.682175	−0.341087	+	0.940032i	$0.610795\pi$
$152$	0	0
$153$	3.52543	0.285014
$154$	0	0
$155$	−1.00000	−0.0803219
$156$	0	0
$157$	−10.2351	−0.816847	−0.408423	−	0.912793i	$-0.633921\pi$
$157$	−10.2351	−0.816847	−0.408423	+	0.912793i	$0.633921\pi$
$158$	0	0
$159$	−13.7605	−1.09128
$160$	0	0
$161$	−10.9906	−0.866183
$162$	0	0
$163$	−5.82717	−0.456419	−0.228209	−	0.973612i	$-0.573287\pi$
$163$	−5.82717	−0.456419	−0.228209	+	0.973612i	$0.573287\pi$
$164$	0	0
$165$	−0.622216	−0.0484394
$166$	0	0
$167$	−23.1842	−1.79405	−0.897024	−	0.441982i	$-0.854276\pi$
$167$	−23.1842	−1.79405	−0.897024	+	0.441982i	$0.854276\pi$
$168$	0	0
$169$	−12.9541	−0.996467
$170$	0	0
$171$	−1.80642	−0.138141
$172$	0	0
$173$	1.57136	0.119468	0.0597342	−	0.998214i	$-0.480975\pi$
$173$	1.57136	0.119468	0.0597342	+	0.998214i	$0.480975\pi$
$174$	0	0
$175$	1.59210	0.120352
$176$	0	0
$177$	−4.26025	−0.320220
$178$	0	0
$179$	−1.53972	−0.115084	−0.0575420	−	0.998343i	$-0.518326\pi$
$179$	−1.53972	−0.115084	−0.0575420	+	0.998343i	$0.518326\pi$
$180$	0	0
$181$	2.00000	0.148659	0.0743294	−	0.997234i	$-0.476318\pi$
$181$	2.00000	0.148659	0.0743294	+	0.997234i	$0.476318\pi$
$182$	0	0
$183$	−2.85728	−0.211216
$184$	0	0
$185$	4.83654	0.355589
$186$	0	0
$187$	−2.19358	−0.160410
$188$	0	0
$189$	−1.59210	−0.115809
$190$	0	0
$191$	−9.11753	−0.659721	−0.329861	−	0.944030i	$-0.607002\pi$
$191$	−9.11753	−0.659721	−0.329861	+	0.944030i	$0.607002\pi$
$192$	0	0
$193$	−4.72393	−0.340036	−0.170018	−	0.985441i	$-0.554383\pi$
$193$	−4.72393	−0.340036	−0.170018	+	0.985441i	$0.554383\pi$
$194$	0	0
$195$	0.214320	0.0153478
$196$	0	0
$197$	15.7003	1.11860	0.559299	−	0.828966i	$-0.311070\pi$
$197$	15.7003	1.11860	0.559299	+	0.828966i	$0.311070\pi$
$198$	0	0
$199$	−6.28100	−0.445248	−0.222624	−	0.974904i	$-0.571462\pi$
$199$	−6.28100	−0.445248	−0.222624	+	0.974904i	$0.571462\pi$
$200$	0	0
$201$	−2.08097	−0.146780
$202$	0	0
$203$	15.5067	1.08836
$204$	0	0
$205$	7.47949	0.522391
$206$	0	0
$207$	−6.90321	−0.479806
$208$	0	0
$209$	1.12399	0.0777477
$210$	0	0
$211$	−19.4795	−1.34102	−0.670512	−	0.741899i	$-0.733925\pi$
$211$	−19.4795	−1.34102	−0.670512	+	0.741899i	$0.733925\pi$
$212$	0	0
$213$	1.31111	0.0898356
$214$	0	0
$215$	8.23506	0.561627
$216$	0	0
$217$	1.59210	0.108079
$218$	0	0
$219$	1.65233	0.111654
$220$	0	0
$221$	0.755569	0.0508251
$222$	0	0
$223$	15.4193	1.03255	0.516275	−	0.856423i	$-0.327318\pi$
$223$	15.4193	1.03255	0.516275	+	0.856423i	$0.327318\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	12.8430	0.852419	0.426210	−	0.904624i	$-0.359849\pi$
$227$	12.8430	0.852419	0.426210	+	0.904624i	$0.359849\pi$
$228$	0	0
$229$	−3.08250	−0.203697	−0.101849	−	0.994800i	$-0.532476\pi$
$229$	−3.08250	−0.203697	−0.101849	+	0.994800i	$0.532476\pi$
$230$	0	0
$231$	0.990632	0.0651788
$232$	0	0
$233$	8.76986	0.574533	0.287266	−	0.957851i	$-0.407254\pi$
$233$	8.76986	0.574533	0.287266	+	0.957851i	$0.407254\pi$
$234$	0	0
$235$	11.4652	0.747907
$236$	0	0
$237$	−5.19850	−0.337679
$238$	0	0
$239$	−3.12399	−0.202074	−0.101037	−	0.994883i	$-0.532216\pi$
$239$	−3.12399	−0.202074	−0.101037	+	0.994883i	$0.532216\pi$
$240$	0	0
$241$	29.5625	1.90429	0.952143	−	0.305653i	$-0.0988747\pi$
$241$	29.5625	1.90429	0.952143	+	0.305653i	$0.0988747\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	4.46520	0.285271
$246$	0	0
$247$	−0.387152	−0.0246339
$248$	0	0
$249$	−5.65878	−0.358611
$250$	0	0
$251$	−21.4193	−1.35197	−0.675986	−	0.736914i	$-0.736282\pi$
$251$	−21.4193	−1.35197	−0.675986	+	0.736914i	$0.736282\pi$
$252$	0	0
$253$	4.29529	0.270042
$254$	0	0
$255$	3.52543	0.220771
$256$	0	0
$257$	−17.6271	−1.09955	−0.549775	−	0.835313i	$-0.685287\pi$
$257$	−17.6271	−1.09955	−0.549775	+	0.835313i	$0.685287\pi$
$258$	0	0
$259$	−7.70027	−0.478471
$260$	0	0
$261$	9.73975	0.602875
$262$	0	0
$263$	14.7052	0.906761	0.453380	−	0.891317i	$-0.350218\pi$
$263$	14.7052	0.906761	0.453380	+	0.891317i	$0.350218\pi$
$264$	0	0
$265$	−13.7605	−0.845300
$266$	0	0
$267$	1.93332	0.118317
$268$	0	0
$269$	1.35260	0.0824692	0.0412346	−	0.999149i	$-0.486871\pi$
$269$	1.35260	0.0824692	0.0412346	+	0.999149i	$0.486871\pi$
$270$	0	0
$271$	−12.1334	−0.737049	−0.368524	−	0.929618i	$-0.620137\pi$
$271$	−12.1334	−0.737049	−0.368524	+	0.929618i	$0.620137\pi$
$272$	0	0
$273$	−0.341219	−0.0206515
$274$	0	0
$275$	−0.622216	−0.0375210
$276$	0	0
$277$	18.9382	1.13789	0.568944	−	0.822376i	$-0.307352\pi$
$277$	18.9382	1.13789	0.568944	+	0.822376i	$0.307352\pi$
$278$	0	0
$279$	1.00000	0.0598684
$280$	0	0
$281$	12.9175	0.770594	0.385297	−	0.922793i	$-0.374099\pi$
$281$	12.9175	0.770594	0.385297	+	0.922793i	$0.374099\pi$
$282$	0	0
$283$	−26.5827	−1.58018	−0.790090	−	0.612991i	$-0.789966\pi$
$283$	−26.5827	−1.58018	−0.790090	+	0.612991i	$0.789966\pi$
$284$	0	0
$285$	−1.80642	−0.107003
$286$	0	0
$287$	−11.9081	−0.702915
$288$	0	0
$289$	−4.57136	−0.268904
$290$	0	0
$291$	−6.91750	−0.405511
$292$	0	0
$293$	−7.99555	−0.467105	−0.233553	−	0.972344i	$-0.575035\pi$
$293$	−7.99555	−0.467105	−0.233553	+	0.972344i	$0.575035\pi$
$294$	0	0
$295$	−4.26025	−0.248042
$296$	0	0
$297$	0.622216	0.0361046
$298$	0	0
$299$	−1.47949	−0.0855614
$300$	0	0
$301$	−13.1111	−0.755710
$302$	0	0
$303$	9.28592	0.533462
$304$	0	0
$305$	−2.85728	−0.163607
$306$	0	0
$307$	17.3669	0.991180	0.495590	−	0.868556i	$-0.334952\pi$
$307$	17.3669	0.991180	0.495590	+	0.868556i	$0.334952\pi$
$308$	0	0
$309$	0.274543	0.0156182
$310$	0	0
$311$	27.5877	1.56435	0.782176	−	0.623057i	$-0.214110\pi$
$311$	27.5877	1.56435	0.782176	+	0.623057i	$0.214110\pi$
$312$	0	0
$313$	−13.7255	−0.775809	−0.387904	−	0.921700i	$-0.626801\pi$
$313$	−13.7255	−0.775809	−0.387904	+	0.921700i	$0.626801\pi$
$314$	0	0
$315$	−1.59210	−0.0897049
$316$	0	0
$317$	−24.9447	−1.40103	−0.700517	−	0.713636i	$-0.747047\pi$
$317$	−24.9447	−1.40103	−0.700517	+	0.713636i	$0.747047\pi$
$318$	0	0
$319$	−6.06022	−0.339307
$320$	0	0
$321$	−8.18913	−0.457073
$322$	0	0
$323$	−6.36842	−0.354348
$324$	0	0
$325$	0.214320	0.0118883
$326$	0	0
$327$	9.13828	0.505348
$328$	0	0
$329$	−18.2538	−1.00636
$330$	0	0
$331$	−19.1798	−1.05422	−0.527108	−	0.849799i	$-0.676724\pi$
$331$	−19.1798	−1.05422	−0.527108	+	0.849799i	$0.676724\pi$
$332$	0	0
$333$	−4.83654	−0.265041
$334$	0	0
$335$	−2.08097	−0.113695
$336$	0	0
$337$	−25.3067	−1.37854	−0.689271	−	0.724504i	$-0.742069\pi$
$337$	−25.3067	−1.37854	−0.689271	+	0.724504i	$0.742069\pi$
$338$	0	0
$339$	−7.67307	−0.416744
$340$	0	0
$341$	−0.622216	−0.0336949
$342$	0	0
$343$	−18.2538	−0.985613
$344$	0	0
$345$	−6.90321	−0.371656
$346$	0	0
$347$	−19.0638	−1.02340	−0.511698	−	0.859165i	$-0.670983\pi$
$347$	−19.0638	−1.02340	−0.511698	+	0.859165i	$0.670983\pi$
$348$	0	0
$349$	−1.23014	−0.0658479	−0.0329240	−	0.999458i	$-0.510482\pi$
$349$	−1.23014	−0.0658479	−0.0329240	+	0.999458i	$0.510482\pi$
$350$	0	0
$351$	−0.214320	−0.0114395
$352$	0	0
$353$	4.50315	0.239679	0.119839	−	0.992793i	$-0.461762\pi$
$353$	4.50315	0.239679	0.119839	+	0.992793i	$0.461762\pi$
$354$	0	0
$355$	1.31111	0.0695864
$356$	0	0
$357$	−5.61285	−0.297063
$358$	0	0
$359$	−18.9240	−0.998768	−0.499384	−	0.866381i	$-0.666440\pi$
$359$	−18.9240	−0.998768	−0.499384	+	0.866381i	$0.666440\pi$
$360$	0	0
$361$	−15.7368	−0.828254
$362$	0	0
$363$	10.6128	0.557030
$364$	0	0
$365$	1.65233	0.0864868
$366$	0	0
$367$	9.71456	0.507096	0.253548	−	0.967323i	$-0.418402\pi$
$367$	9.71456	0.507096	0.253548	+	0.967323i	$0.418402\pi$
$368$	0	0
$369$	−7.47949	−0.389367
$370$	0	0
$371$	21.9081	1.13741
$372$	0	0
$373$	−35.1941	−1.82228	−0.911139	−	0.412098i	$-0.864796\pi$
$373$	−35.1941	−1.82228	−0.911139	+	0.412098i	$0.864796\pi$
$374$	0	0
$375$	1.00000	0.0516398
$376$	0	0
$377$	2.08742	0.107508
$378$	0	0
$379$	22.6637	1.16416	0.582078	−	0.813133i	$-0.302240\pi$
$379$	22.6637	1.16416	0.582078	+	0.813133i	$0.302240\pi$
$380$	0	0
$381$	13.6731	0.700493
$382$	0	0
$383$	−28.7926	−1.47123	−0.735617	−	0.677398i	$-0.763108\pi$
$383$	−28.7926	−1.47123	−0.735617	+	0.677398i	$0.763108\pi$
$384$	0	0
$385$	0.990632	0.0504873
$386$	0	0
$387$	−8.23506	−0.418612
$388$	0	0
$389$	−25.0672	−1.27096	−0.635478	−	0.772119i	$-0.719197\pi$
$389$	−25.0672	−1.27096	−0.635478	+	0.772119i	$0.719197\pi$
$390$	0	0
$391$	−24.3368	−1.23076
$392$	0	0
$393$	−22.2701	−1.12338
$394$	0	0
$395$	−5.19850	−0.261565
$396$	0	0
$397$	−29.0509	−1.45802	−0.729010	−	0.684503i	$-0.760019\pi$
$397$	−29.0509	−1.45802	−0.729010	+	0.684503i	$0.760019\pi$
$398$	0	0
$399$	2.87601	0.143981
$400$	0	0
$401$	11.2607	0.562334	0.281167	−	0.959659i	$-0.409279\pi$
$401$	11.2607	0.562334	0.281167	+	0.959659i	$0.409279\pi$
$402$	0	0
$403$	0.214320	0.0106760
$404$	0	0
$405$	−1.00000	−0.0496904
$406$	0	0
$407$	3.00937	0.149169
$408$	0	0
$409$	−25.6128	−1.26647	−0.633237	−	0.773958i	$-0.718274\pi$
$409$	−25.6128	−1.26647	−0.633237	+	0.773958i	$0.718274\pi$
$410$	0	0
$411$	18.1891	0.897204
$412$	0	0
$413$	6.78277	0.333758
$414$	0	0
$415$	−5.65878	−0.277779
$416$	0	0
$417$	−4.13335	−0.202411
$418$	0	0
$419$	4.36196	0.213096	0.106548	−	0.994308i	$-0.466020\pi$
$419$	4.36196	0.213096	0.106548	+	0.994308i	$0.466020\pi$
$420$	0	0
$421$	5.77923	0.281662	0.140831	−	0.990034i	$-0.455023\pi$
$421$	5.77923	0.281662	0.140831	+	0.990034i	$0.455023\pi$
$422$	0	0
$423$	−11.4652	−0.557457
$424$	0	0
$425$	3.52543	0.171008
$426$	0	0
$427$	4.54909	0.220146
$428$	0	0
$429$	0.133353	0.00643835
$430$	0	0
$431$	4.62867	0.222955	0.111478	−	0.993767i	$-0.464442\pi$
$431$	4.62867	0.222955	0.111478	+	0.993767i	$0.464442\pi$
$432$	0	0
$433$	6.81780	0.327643	0.163821	−	0.986490i	$-0.447618\pi$
$433$	6.81780	0.327643	0.163821	+	0.986490i	$0.447618\pi$
$434$	0	0
$435$	9.73975	0.466985
$436$	0	0
$437$	12.4701	0.596527
$438$	0	0
$439$	−9.71456	−0.463651	−0.231825	−	0.972757i	$-0.574470\pi$
$439$	−9.71456	−0.463651	−0.231825	+	0.972757i	$0.574470\pi$
$440$	0	0
$441$	−4.46520	−0.212629
$442$	0	0
$443$	−31.1481	−1.47989	−0.739946	−	0.672666i	$-0.765149\pi$
$443$	−31.1481	−1.47989	−0.739946	+	0.672666i	$0.765149\pi$
$444$	0	0
$445$	1.93332	0.0916483
$446$	0	0
$447$	−6.23506	−0.294908
$448$	0	0
$449$	28.1180	1.32697	0.663485	−	0.748189i	$-0.269076\pi$
$449$	28.1180	1.32697	0.663485	+	0.748189i	$0.269076\pi$
$450$	0	0
$451$	4.65386	0.219142
$452$	0	0
$453$	8.38271	0.393854
$454$	0	0
$455$	−0.341219	−0.0159966
$456$	0	0
$457$	−41.4400	−1.93848	−0.969241	−	0.246113i	$-0.920846\pi$
$457$	−41.4400	−1.93848	−0.969241	+	0.246113i	$0.920846\pi$
$458$	0	0
$459$	−3.52543	−0.164553
$460$	0	0
$461$	10.3432	0.481732	0.240866	−	0.970558i	$-0.422569\pi$
$461$	10.3432	0.481732	0.240866	+	0.970558i	$0.422569\pi$
$462$	0	0
$463$	−13.3047	−0.618320	−0.309160	−	0.951010i	$-0.600048\pi$
$463$	−13.3047	−0.618320	−0.309160	+	0.951010i	$0.600048\pi$
$464$	0	0
$465$	1.00000	0.0463739
$466$	0	0
$467$	6.22077	0.287863	0.143932	−	0.989588i	$-0.454025\pi$
$467$	6.22077	0.287863	0.143932	+	0.989588i	$0.454025\pi$
$468$	0	0
$469$	3.31312	0.152985
$470$	0	0
$471$	10.2351	0.471607
$472$	0	0
$473$	5.12399	0.235601
$474$	0	0
$475$	−1.80642	−0.0828844
$476$	0	0
$477$	13.7605	0.630050
$478$	0	0
$479$	24.8825	1.13691	0.568454	−	0.822715i	$-0.307542\pi$
$479$	24.8825	1.13691	0.568454	+	0.822715i	$0.307542\pi$
$480$	0	0
$481$	−1.03657	−0.0472633
$482$	0	0
$483$	10.9906	0.500091
$484$	0	0
$485$	−6.91750	−0.314108
$486$	0	0
$487$	9.84791	0.446251	0.223126	−	0.974790i	$-0.428374\pi$
$487$	9.84791	0.446251	0.223126	+	0.974790i	$0.428374\pi$
$488$	0	0
$489$	5.82717	0.263514
$490$	0	0
$491$	9.84791	0.444430	0.222215	−	0.974998i	$-0.428671\pi$
$491$	9.84791	0.444430	0.222215	+	0.974998i	$0.428671\pi$
$492$	0	0
$493$	34.3368	1.54645
$494$	0	0
$495$	0.622216	0.0279665
$496$	0	0
$497$	−2.08742	−0.0936336
$498$	0	0
$499$	−0.653858	−0.0292707	−0.0146354	−	0.999893i	$-0.504659\pi$
$499$	−0.653858	−0.0292707	−0.0146354	+	0.999893i	$0.504659\pi$
$500$	0	0
$501$	23.1842	1.03579
$502$	0	0
$503$	−13.9541	−0.622181	−0.311091	−	0.950380i	$-0.600694\pi$
$503$	−13.9541	−0.622181	−0.311091	+	0.950380i	$0.600694\pi$
$504$	0	0
$505$	9.28592	0.413218
$506$	0	0
$507$	12.9541	0.575310
$508$	0	0
$509$	−42.7116	−1.89316	−0.946580	−	0.322469i	$-0.895487\pi$
$509$	−42.7116	−1.89316	−0.946580	+	0.322469i	$0.895487\pi$
$510$	0	0
$511$	−2.63068	−0.116374
$512$	0	0
$513$	1.80642	0.0797556
$514$	0	0
$515$	0.274543	0.0120978
$516$	0	0
$517$	7.13383	0.313745
$518$	0	0
$519$	−1.57136	−0.0689751
$520$	0	0
$521$	−12.6035	−0.552168	−0.276084	−	0.961133i	$-0.589037\pi$
$521$	−12.6035	−0.552168	−0.276084	+	0.961133i	$0.589037\pi$
$522$	0	0
$523$	29.8479	1.30516	0.652579	−	0.757721i	$-0.273687\pi$
$523$	29.8479	1.30516	0.652579	+	0.757721i	$0.273687\pi$
$524$	0	0
$525$	−1.59210	−0.0694851
$526$	0	0
$527$	3.52543	0.153570
$528$	0	0
$529$	24.6543	1.07193
$530$	0	0
$531$	4.26025	0.184879
$532$	0	0
$533$	−1.60300	−0.0694338
$534$	0	0
$535$	−8.18913	−0.354047
$536$	0	0
$537$	1.53972	0.0664437
$538$	0	0
$539$	2.77832	0.119671
$540$	0	0
$541$	−13.9541	−0.599932	−0.299966	−	0.953950i	$-0.596975\pi$
$541$	−13.9541	−0.599932	−0.299966	+	0.953950i	$0.596975\pi$
$542$	0	0
$543$	−2.00000	−0.0858282
$544$	0	0
$545$	9.13828	0.391441
$546$	0	0
$547$	−5.74419	−0.245604	−0.122802	−	0.992431i	$-0.539188\pi$
$547$	−5.74419	−0.245604	−0.122802	+	0.992431i	$0.539188\pi$
$548$	0	0
$549$	2.85728	0.121946
$550$	0	0
$551$	−17.5941	−0.749534
$552$	0	0
$553$	8.27655	0.351955
$554$	0	0
$555$	−4.83654	−0.205299
$556$	0	0
$557$	−14.7511	−0.625025	−0.312513	−	0.949914i	$-0.601171\pi$
$557$	−14.7511	−0.625025	−0.312513	+	0.949914i	$0.601171\pi$
$558$	0	0
$559$	−1.76494	−0.0746489
$560$	0	0
$561$	2.19358	0.0926129
$562$	0	0
$563$	−32.9260	−1.38766	−0.693832	−	0.720137i	$-0.744079\pi$
$563$	−32.9260	−1.38766	−0.693832	+	0.720137i	$0.744079\pi$
$564$	0	0
$565$	−7.67307	−0.322809
$566$	0	0
$567$	1.59210	0.0668621
$568$	0	0
$569$	−3.07604	−0.128954	−0.0644772	−	0.997919i	$-0.520538\pi$
$569$	−3.07604	−0.128954	−0.0644772	+	0.997919i	$0.520538\pi$
$570$	0	0
$571$	−4.52051	−0.189177	−0.0945886	−	0.995516i	$-0.530154\pi$
$571$	−4.52051	−0.189177	−0.0945886	+	0.995516i	$0.530154\pi$
$572$	0	0
$573$	9.11753	0.380890
$574$	0	0
$575$	−6.90321	−0.287884
$576$	0	0
$577$	−30.6923	−1.27774	−0.638868	−	0.769316i	$-0.720597\pi$
$577$	−30.6923	−1.27774	−0.638868	+	0.769316i	$0.720597\pi$
$578$	0	0
$579$	4.72393	0.196320
$580$	0	0
$581$	9.00937	0.373772
$582$	0	0
$583$	−8.56199	−0.354602
$584$	0	0
$585$	−0.214320	−0.00886103
$586$	0	0
$587$	−14.8256	−0.611919	−0.305960	−	0.952044i	$-0.598977\pi$
$587$	−14.8256	−0.611919	−0.305960	+	0.952044i	$0.598977\pi$
$588$	0	0
$589$	−1.80642	−0.0744324
$590$	0	0
$591$	−15.7003	−0.645823
$592$	0	0
$593$	23.9684	0.984262	0.492131	−	0.870521i	$-0.336218\pi$
$593$	23.9684	0.984262	0.492131	+	0.870521i	$0.336218\pi$
$594$	0	0
$595$	−5.61285	−0.230104
$596$	0	0
$597$	6.28100	0.257064
$598$	0	0
$599$	39.1086	1.59794	0.798968	−	0.601374i	$-0.205380\pi$
$599$	39.1086	1.59794	0.798968	+	0.601374i	$0.205380\pi$
$600$	0	0
$601$	22.3082	0.909970	0.454985	−	0.890499i	$-0.349645\pi$
$601$	22.3082	0.909970	0.454985	+	0.890499i	$0.349645\pi$
$602$	0	0
$603$	2.08097	0.0847435
$604$	0	0
$605$	10.6128	0.431474
$606$	0	0
$607$	−2.82669	−0.114732	−0.0573659	−	0.998353i	$-0.518270\pi$
$607$	−2.82669	−0.114732	−0.0573659	+	0.998353i	$0.518270\pi$
$608$	0	0
$609$	−15.5067	−0.628363
$610$	0	0
$611$	−2.45722	−0.0994085
$612$	0	0
$613$	−36.6844	−1.48167	−0.740835	−	0.671687i	$-0.765570\pi$
$613$	−36.6844	−1.48167	−0.740835	+	0.671687i	$0.765570\pi$
$614$	0	0
$615$	−7.47949	−0.301602
$616$	0	0
$617$	1.46659	0.0590426	0.0295213	−	0.999564i	$-0.490602\pi$
$617$	1.46659	0.0590426	0.0295213	+	0.999564i	$0.490602\pi$
$618$	0	0
$619$	15.9541	0.641248	0.320624	−	0.947207i	$-0.396107\pi$
$619$	15.9541	0.641248	0.320624	+	0.947207i	$0.396107\pi$
$620$	0	0
$621$	6.90321	0.277016
$622$	0	0
$623$	−3.07805	−0.123320
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−1.12399	−0.0448876
$628$	0	0
$629$	−17.0509	−0.679862
$630$	0	0
$631$	32.0830	1.27720	0.638602	−	0.769538i	$-0.279513\pi$
$631$	32.0830	1.27720	0.638602	+	0.769538i	$0.279513\pi$
$632$	0	0
$633$	19.4795	0.774240
$634$	0	0
$635$	13.6731	0.542599
$636$	0	0
$637$	−0.956981	−0.0379170
$638$	0	0
$639$	−1.31111	−0.0518666
$640$	0	0
$641$	−14.8923	−0.588211	−0.294105	−	0.955773i	$-0.595022\pi$
$641$	−14.8923	−0.588211	−0.294105	+	0.955773i	$0.595022\pi$
$642$	0	0
$643$	−12.5205	−0.493761	−0.246880	−	0.969046i	$-0.579405\pi$
$643$	−12.5205	−0.493761	−0.246880	+	0.969046i	$0.579405\pi$
$644$	0	0
$645$	−8.23506	−0.324255
$646$	0	0
$647$	−45.6227	−1.79361	−0.896807	−	0.442423i	$-0.854119\pi$
$647$	−45.6227	−1.79361	−0.896807	+	0.442423i	$0.854119\pi$
$648$	0	0
$649$	−2.65080	−0.104053
$650$	0	0
$651$	−1.59210	−0.0623995
$652$	0	0
$653$	−18.7797	−0.734907	−0.367453	−	0.930042i	$-0.619770\pi$
$653$	−18.7797	−0.734907	−0.367453	+	0.930042i	$0.619770\pi$
$654$	0	0
$655$	−22.2701	−0.870165
$656$	0	0
$657$	−1.65233	−0.0644634
$658$	0	0
$659$	28.9240	1.12672	0.563359	−	0.826212i	$-0.309509\pi$
$659$	28.9240	1.12672	0.563359	+	0.826212i	$0.309509\pi$
$660$	0	0
$661$	−24.3269	−0.946208	−0.473104	−	0.881007i	$-0.656866\pi$
$661$	−24.3269	−0.946208	−0.473104	+	0.881007i	$0.656866\pi$
$662$	0	0
$663$	−0.755569	−0.0293439
$664$	0	0
$665$	2.87601	0.111527
$666$	0	0
$667$	−67.2355	−2.60337
$668$	0	0
$669$	−15.4193	−0.596143
$670$	0	0
$671$	−1.77784	−0.0686329
$672$	0	0
$673$	−36.5511	−1.40894	−0.704471	−	0.709733i	$-0.748815\pi$
$673$	−36.5511	−1.40894	−0.704471	+	0.709733i	$0.748815\pi$
$674$	0	0
$675$	−1.00000	−0.0384900
$676$	0	0
$677$	−32.1575	−1.23591	−0.617956	−	0.786212i	$-0.712039\pi$
$677$	−32.1575	−1.23591	−0.617956	+	0.786212i	$0.712039\pi$
$678$	0	0
$679$	11.0134	0.422655
$680$	0	0
$681$	−12.8430	−0.492144
$682$	0	0
$683$	38.2623	1.46406	0.732032	−	0.681270i	$-0.238572\pi$
$683$	38.2623	1.46406	0.732032	+	0.681270i	$0.238572\pi$
$684$	0	0
$685$	18.1891	0.694971
$686$	0	0
$687$	3.08250	0.117605
$688$	0	0
$689$	2.94914	0.112353
$690$	0	0
$691$	12.3082	0.468226	0.234113	−	0.972209i	$-0.424781\pi$
$691$	12.3082	0.468226	0.234113	+	0.972209i	$0.424781\pi$
$692$	0	0
$693$	−0.990632	−0.0376310
$694$	0	0
$695$	−4.13335	−0.156787
$696$	0	0
$697$	−26.3684	−0.998775
$698$	0	0
$699$	−8.76986	−0.331707
$700$	0	0
$701$	31.5210	1.19053	0.595266	−	0.803529i	$-0.297047\pi$
$701$	31.5210	1.19053	0.595266	+	0.803529i	$0.297047\pi$
$702$	0	0
$703$	8.73683	0.329516
$704$	0	0
$705$	−11.4652	−0.431805
$706$	0	0
$707$	−14.7841	−0.556015
$708$	0	0
$709$	−7.51114	−0.282087	−0.141043	−	0.990003i	$-0.545046\pi$
$709$	−7.51114	−0.282087	−0.141043	+	0.990003i	$0.545046\pi$
$710$	0	0
$711$	5.19850	0.194959
$712$	0	0
$713$	−6.90321	−0.258527
$714$	0	0
$715$	0.133353	0.00498712
$716$	0	0
$717$	3.12399	0.116667
$718$	0	0
$719$	−32.3180	−1.20526	−0.602630	−	0.798021i	$-0.705880\pi$
$719$	−32.3180	−1.20526	−0.602630	+	0.798021i	$0.705880\pi$
$720$	0	0
$721$	−0.437101	−0.0162785
$722$	0	0
$723$	−29.5625	−1.09944
$724$	0	0
$725$	9.73975	0.361725
$726$	0	0
$727$	−6.12245	−0.227069	−0.113535	−	0.993534i	$-0.536217\pi$
$727$	−6.12245	−0.227069	−0.113535	+	0.993534i	$0.536217\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−29.0321	−1.07379
$732$	0	0
$733$	−30.9403	−1.14280	−0.571402	−	0.820670i	$-0.693600\pi$
$733$	−30.9403	−1.14280	−0.571402	+	0.820670i	$0.693600\pi$
$734$	0	0
$735$	−4.46520	−0.164702
$736$	0	0
$737$	−1.29481	−0.0476949
$738$	0	0
$739$	12.6780	0.466368	0.233184	−	0.972433i	$-0.425086\pi$
$739$	12.6780	0.466368	0.233184	+	0.972433i	$0.425086\pi$
$740$	0	0
$741$	0.387152	0.0142224
$742$	0	0
$743$	−38.4385	−1.41017	−0.705086	−	0.709122i	$-0.749091\pi$
$743$	−38.4385	−1.41017	−0.705086	+	0.709122i	$0.749091\pi$
$744$	0	0
$745$	−6.23506	−0.228435
$746$	0	0
$747$	5.65878	0.207044
$748$	0	0
$749$	13.0379	0.476396
$750$	0	0
$751$	13.4608	0.491190	0.245595	−	0.969373i	$-0.421017\pi$
$751$	13.4608	0.491190	0.245595	+	0.969373i	$0.421017\pi$
$752$	0	0
$753$	21.4193	0.780562
$754$	0	0
$755$	8.38271	0.305078
$756$	0	0
$757$	−20.5511	−0.746942	−0.373471	−	0.927642i	$-0.621832\pi$
$757$	−20.5511	−0.746942	−0.373471	+	0.927642i	$0.621832\pi$
$758$	0	0
$759$	−4.29529	−0.155909
$760$	0	0
$761$	42.3432	1.53494	0.767470	−	0.641084i	$-0.221515\pi$
$761$	42.3432	1.53494	0.767470	+	0.641084i	$0.221515\pi$
$762$	0	0
$763$	−14.5491	−0.526712
$764$	0	0
$765$	−3.52543	−0.127462
$766$	0	0
$767$	0.913056	0.0329686
$768$	0	0
$769$	40.5388	1.46187	0.730933	−	0.682449i	$-0.239085\pi$
$769$	40.5388	1.46187	0.730933	+	0.682449i	$0.239085\pi$
$770$	0	0
$771$	17.6271	0.634826
$772$	0	0
$773$	26.2810	0.945262	0.472631	−	0.881260i	$-0.343304\pi$
$773$	26.2810	0.945262	0.472631	+	0.881260i	$0.343304\pi$
$774$	0	0
$775$	1.00000	0.0359211
$776$	0	0
$777$	7.70027	0.276246
$778$	0	0
$779$	13.5111	0.484087
$780$	0	0
$781$	0.815792	0.0291913
$782$	0	0
$783$	−9.73975	−0.348070
$784$	0	0
$785$	10.2351	0.365305
$786$	0	0
$787$	0.755569	0.0269331	0.0134666	−	0.999909i	$-0.495713\pi$
$787$	0.755569	0.0269331	0.0134666	+	0.999909i	$0.495713\pi$
$788$	0	0
$789$	−14.7052	−0.523519
$790$	0	0
$791$	12.2163	0.434363
$792$	0	0
$793$	0.612371	0.0217459
$794$	0	0
$795$	13.7605	0.488034
$796$	0	0
$797$	1.89384	0.0670834	0.0335417	−	0.999437i	$-0.489321\pi$
$797$	1.89384	0.0670834	0.0335417	+	0.999437i	$0.489321\pi$
$798$	0	0
$799$	−40.4197	−1.42995
$800$	0	0
$801$	−1.93332	−0.0683106
$802$	0	0
$803$	1.02810	0.0362810
$804$	0	0
$805$	10.9906	0.387369
$806$	0	0
$807$	−1.35260	−0.0476136
$808$	0	0
$809$	−28.6671	−1.00788	−0.503941	−	0.863738i	$-0.668117\pi$
$809$	−28.6671	−1.00788	−0.503941	+	0.863738i	$0.668117\pi$
$810$	0	0
$811$	38.0228	1.33516	0.667580	−	0.744538i	$-0.267330\pi$
$811$	38.0228	1.33516	0.667580	+	0.744538i	$0.267330\pi$
$812$	0	0
$813$	12.1334	0.425535
$814$	0	0
$815$	5.82717	0.204117
$816$	0	0
$817$	14.8760	0.520446
$818$	0	0
$819$	0.341219	0.0119232
$820$	0	0
$821$	−27.2321	−0.950409	−0.475204	−	0.879876i	$-0.657626\pi$
$821$	−27.2321	−0.950409	−0.475204	+	0.879876i	$0.657626\pi$
$822$	0	0
$823$	−14.5749	−0.508049	−0.254025	−	0.967198i	$-0.581754\pi$
$823$	−14.5749	−0.508049	−0.254025	+	0.967198i	$0.581754\pi$
$824$	0	0
$825$	0.622216	0.0216628
$826$	0	0
$827$	−35.0781	−1.21978	−0.609892	−	0.792485i	$-0.708787\pi$
$827$	−35.0781	−1.21978	−0.609892	+	0.792485i	$0.708787\pi$
$828$	0	0
$829$	18.9077	0.656690	0.328345	−	0.944558i	$-0.393509\pi$
$829$	18.9077	0.656690	0.328345	+	0.944558i	$0.393509\pi$
$830$	0	0
$831$	−18.9382	−0.656960
$832$	0	0
$833$	−15.7418	−0.545419
$834$	0	0
$835$	23.1842	0.802323
$836$	0	0
$837$	−1.00000	−0.0345651
$838$	0	0
$839$	39.5274	1.36464	0.682319	−	0.731054i	$-0.260971\pi$
$839$	39.5274	1.36464	0.682319	+	0.731054i	$0.260971\pi$
$840$	0	0
$841$	65.8627	2.27113
$842$	0	0
$843$	−12.9175	−0.444902
$844$	0	0
$845$	12.9541	0.445633
$846$	0	0
$847$	−16.8968	−0.580579
$848$	0	0
$849$	26.5827	0.912317
$850$	0	0
$851$	33.3876	1.14451
$852$	0	0
$853$	41.8894	1.43427	0.717133	−	0.696937i	$-0.245454\pi$
$853$	41.8894	1.43427	0.717133	+	0.696937i	$0.245454\pi$
$854$	0	0
$855$	1.80642	0.0617784
$856$	0	0
$857$	43.2859	1.47862	0.739309	−	0.673366i	$-0.235152\pi$
$857$	43.2859	1.47862	0.739309	+	0.673366i	$0.235152\pi$
$858$	0	0
$859$	49.6513	1.69408	0.847040	−	0.531530i	$-0.178383\pi$
$859$	49.6513	1.69408	0.847040	+	0.531530i	$0.178383\pi$
$860$	0	0
$861$	11.9081	0.405828
$862$	0	0
$863$	31.6958	1.07894	0.539469	−	0.842005i	$-0.318625\pi$
$863$	31.6958	1.07894	0.539469	+	0.842005i	$0.318625\pi$
$864$	0	0
$865$	−1.57136	−0.0534279
$866$	0	0
$867$	4.57136	0.155252
$868$	0	0
$869$	−3.23459	−0.109726
$870$	0	0
$871$	0.445992	0.0151119
$872$	0	0
$873$	6.91750	0.234122
$874$	0	0
$875$	−1.59210	−0.0538229
$876$	0	0
$877$	−24.0000	−0.810422	−0.405211	−	0.914223i	$-0.632802\pi$
$877$	−24.0000	−0.810422	−0.405211	+	0.914223i	$0.632802\pi$
$878$	0	0
$879$	7.99555	0.269683
$880$	0	0
$881$	−45.6577	−1.53825	−0.769124	−	0.639100i	$-0.779307\pi$
$881$	−45.6577	−1.53825	−0.769124	+	0.639100i	$0.779307\pi$
$882$	0	0
$883$	−7.63158	−0.256823	−0.128412	−	0.991721i	$-0.540988\pi$
$883$	−7.63158	−0.256823	−0.128412	+	0.991721i	$0.540988\pi$
$884$	0	0
$885$	4.26025	0.143207
$886$	0	0
$887$	−13.9541	−0.468532	−0.234266	−	0.972173i	$-0.575269\pi$
$887$	−13.9541	−0.468532	−0.234266	+	0.972173i	$0.575269\pi$
$888$	0	0
$889$	−21.7690	−0.730107
$890$	0	0
$891$	−0.622216	−0.0208450
$892$	0	0
$893$	20.7110	0.693068
$894$	0	0
$895$	1.53972	0.0514671
$896$	0	0
$897$	1.47949	0.0493989
$898$	0	0
$899$	9.73975	0.324839
$900$	0	0
$901$	48.5116	1.61616
$902$	0	0
$903$	13.1111	0.436309
$904$	0	0
$905$	−2.00000	−0.0664822
$906$	0	0
$907$	−50.6242	−1.68095	−0.840475	−	0.541851i	$-0.817724\pi$
$907$	−50.6242	−1.68095	−0.840475	+	0.541851i	$0.817724\pi$
$908$	0	0
$909$	−9.28592	−0.307994
$910$	0	0
$911$	−40.5847	−1.34463	−0.672316	−	0.740264i	$-0.734700\pi$
$911$	−40.5847	−1.34463	−0.672316	+	0.740264i	$0.734700\pi$
$912$	0	0
$913$	−3.52098	−0.116527
$914$	0	0
$915$	2.85728	0.0944587
$916$	0	0
$917$	35.4563	1.17087
$918$	0	0
$919$	−53.3274	−1.75911	−0.879554	−	0.475798i	$-0.842159\pi$
$919$	−53.3274	−1.75911	−0.879554	+	0.475798i	$0.842159\pi$
$920$	0	0
$921$	−17.3669	−0.572258
$922$	0	0
$923$	−0.280996	−0.00924911
$924$	0	0
$925$	−4.83654	−0.159024
$926$	0	0
$927$	−0.274543	−0.00901717
$928$	0	0
$929$	4.67905	0.153515	0.0767573	−	0.997050i	$-0.475543\pi$
$929$	4.67905	0.153515	0.0767573	+	0.997050i	$0.475543\pi$
$930$	0	0
$931$	8.06605	0.264354
$932$	0	0
$933$	−27.5877	−0.903179
$934$	0	0
$935$	2.19358	0.0717376
$936$	0	0
$937$	−19.8952	−0.649949	−0.324974	−	0.945723i	$-0.605356\pi$
$937$	−19.8952	−0.649949	−0.324974	+	0.945723i	$0.605356\pi$
$938$	0	0
$939$	13.7255	0.447913
$940$	0	0
$941$	−1.27946	−0.0417094	−0.0208547	−	0.999783i	$-0.506639\pi$
$941$	−1.27946	−0.0417094	−0.0208547	+	0.999783i	$0.506639\pi$
$942$	0	0
$943$	51.6325	1.68139
$944$	0	0
$945$	1.59210	0.0517912
$946$	0	0
$947$	35.6686	1.15907	0.579537	−	0.814946i	$-0.303233\pi$
$947$	35.6686	1.15907	0.579537	+	0.814946i	$0.303233\pi$
$948$	0	0
$949$	−0.354126	−0.0114954
$950$	0	0
$951$	24.9447	0.808887
$952$	0	0
$953$	39.7832	1.28871	0.644353	−	0.764728i	$-0.277127\pi$
$953$	39.7832	1.28871	0.644353	+	0.764728i	$0.277127\pi$
$954$	0	0
$955$	9.11753	0.295036
$956$	0	0
$957$	6.06022	0.195899
$958$	0	0
$959$	−28.9590	−0.935135
$960$	0	0
$961$	1.00000	0.0322581
$962$	0	0
$963$	8.18913	0.263891
$964$	0	0
$965$	4.72393	0.152069
$966$	0	0
$967$	52.9304	1.70213	0.851064	−	0.525063i	$-0.175958\pi$
$967$	52.9304	1.70213	0.851064	+	0.525063i	$0.175958\pi$
$968$	0	0
$969$	6.36842	0.204583
$970$	0	0
$971$	54.9624	1.76383	0.881913	−	0.471412i	$-0.156255\pi$
$971$	54.9624	1.76383	0.881913	+	0.471412i	$0.156255\pi$
$972$	0	0
$973$	6.58073	0.210968
$974$	0	0
$975$	−0.214320	−0.00686372
$976$	0	0
$977$	2.97773	0.0952659	0.0476329	−	0.998865i	$-0.484832\pi$
$977$	2.97773	0.0952659	0.0476329	+	0.998865i	$0.484832\pi$
$978$	0	0
$979$	1.20294	0.0384463
$980$	0	0
$981$	−9.13828	−0.291763
$982$	0	0
$983$	26.9876	0.860770	0.430385	−	0.902645i	$-0.358378\pi$
$983$	26.9876	0.860770	0.430385	+	0.902645i	$0.358378\pi$
$984$	0	0
$985$	−15.7003	−0.500252
$986$	0	0
$987$	18.2538	0.581025
$988$	0	0
$989$	56.8484	1.80767
$990$	0	0
$991$	−0.520505	−0.0165344	−0.00826720	−	0.999966i	$-0.502632\pi$
$991$	−0.520505	−0.0165344	−0.00826720	+	0.999966i	$0.502632\pi$
$992$	0	0
$993$	19.1798	0.608651
$994$	0	0
$995$	6.28100	0.199121
$996$	0	0
$997$	−25.2128	−0.798497	−0.399249	−	0.916843i	$-0.630729\pi$
$997$	−25.2128	−0.798497	−0.399249	+	0.916843i	$0.630729\pi$
$998$	0	0
$999$	4.83654	0.153021

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7440.2.a.bm.1.3		3
4.3	odd	2		465.2.a.g.1.1	✓	3
12.11	even	2		1395.2.a.h.1.3		3
20.3	even	4		2325.2.c.l.1024.4		6
20.7	even	4		2325.2.c.l.1024.3		6
20.19	odd	2		2325.2.a.p.1.3		3
60.59	even	2		6975.2.a.bi.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
465.2.a.g.1.1	✓	3	4.3	odd	2
1395.2.a.h.1.3		3	12.11	even	2
2325.2.a.p.1.3		3	20.19	odd	2
2325.2.c.l.1024.3		6	20.7	even	4
2325.2.c.l.1024.4		6	20.3	even	4
6975.2.a.bi.1.1		3	60.59	even	2
7440.2.a.bm.1.3		3	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 7440 = 2^{4} \cdot 3 \cdot 5 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7440.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -1.00000 q^{5} +1.59210 q^{7} +1.00000 q^{9} -0.622216 q^{11} +0.214320 q^{13} +1.00000 q^{15} +3.52543 q^{17} -1.80642 q^{19} -1.59210 q^{21} -6.90321 q^{23} +1.00000 q^{25} -1.00000 q^{27} +9.73975 q^{29} +1.00000 q^{31} +0.622216 q^{33} -1.59210 q^{35} -4.83654 q^{37} -0.214320 q^{39} -7.47949 q^{41} -8.23506 q^{43} -1.00000 q^{45} -11.4652 q^{47} -4.46520 q^{49} -3.52543 q^{51} +13.7605 q^{53} +0.622216 q^{55} +1.80642 q^{57} +4.26025 q^{59} +2.85728 q^{61} +1.59210 q^{63} -0.214320 q^{65} +2.08097 q^{67} +6.90321 q^{69} -1.31111 q^{71} -1.65233 q^{73} -1.00000 q^{75} -0.990632 q^{77} +5.19850 q^{79} +1.00000 q^{81} +5.65878 q^{83} -3.52543 q^{85} -9.73975 q^{87} -1.93332 q^{89} +0.341219 q^{91} -1.00000 q^{93} +1.80642 q^{95} +6.91750 q^{97} -0.622216 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{3} - 3 q^{5} - 2 q^{7} + 3 q^{9} - 2 q^{11} - 6 q^{13} + 3 q^{15} + 4 q^{17} + 8 q^{19} + 2 q^{21} - 14 q^{23} + 3 q^{25} - 3 q^{27} + 16 q^{29} + 3 q^{31} + 2 q^{33} + 2 q^{35} - 8 q^{37} + 6 q^{39}+ \cdots - 2 q^{99}+O(q^{100}) \) 3 * q - 3 * q^3 - 3 * q^5 - 2 * q^7 + 3 * q^9 - 2 * q^11 - 6 * q^13 + 3 * q^15 + 4 * q^17 + 8 * q^19 + 2 * q^21 - 14 * q^23 + 3 * q^25 - 3 * q^27 + 16 * q^29 + 3 * q^31 + 2 * q^33 + 2 * q^35 - 8 * q^37 + 6 * q^39 + 4 * q^41 + 2 * q^43 - 3 * q^45 - 14 * q^47 + 7 * q^49 - 4 * q^51 + 8 * q^53 + 2 * q^55 - 8 * q^57 + 26 * q^59 - 18 * q^61 - 2 * q^63 + 6 * q^65 + 14 * q^69 - 4 * q^71 - 12 * q^73 - 3 * q^75 + 24 * q^77 - 4 * q^79 + 3 * q^81 + 10 * q^83 - 4 * q^85 - 16 * q^87 - 6 * q^89 + 8 * q^91 - 3 * q^93 - 8 * q^95 + 8 * q^97 - 2 * q^99

Introduction

L-functions

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