LMFDB - Embedded newform 1449.2.a.o.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1449,2,Mod(1,1449)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1449, 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("1449.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1449 = 3^{2} \cdot 7 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1449.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:	$11.5703232529$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.15317.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 4x^{2} + 5x + 2 $ x^4 - 2x^3 - 4x^2 + 5*x + 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 483)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-0.329727$ of defining polynomial
Character	$\chi$	$=$	1449.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.329727 q^{2} -1.89128 q^{4} +2.73589 q^{5} -1.00000 q^{7} -1.28306 q^{8} +O(q^{10})$	\(q+0.329727 q^{2} -1.89128 q^{4} +2.73589 q^{5} -1.00000 q^{7} -1.28306 q^{8} +0.902098 q^{10} +2.50407 q^{11} +1.48511 q^{13} -0.329727 q^{14} +3.35950 q^{16} -0.902098 q^{17} +2.50407 q^{19} -5.17434 q^{20} +0.825659 q^{22} -1.00000 q^{23} +2.48511 q^{25} +0.489682 q^{26} +1.89128 q^{28} -6.68466 q^{29} +1.09790 q^{31} +3.67384 q^{32} -0.297446 q^{34} -2.73589 q^{35} +9.91023 q^{37} +0.825659 q^{38} -3.51032 q^{40} +2.30826 q^{41} +5.83380 q^{43} -4.73589 q^{44} -0.329727 q^{46} +6.74671 q^{47} +1.00000 q^{49} +0.819409 q^{50} -2.80877 q^{52} +4.91648 q^{53} +6.85086 q^{55} +1.28306 q^{56} -2.20411 q^{58} +7.32973 q^{59} -1.00625 q^{61} +0.362008 q^{62} -5.50764 q^{64} +4.06311 q^{65} +11.2929 q^{67} +1.70612 q^{68} -0.902098 q^{70} +0.362008 q^{71} -5.34411 q^{73} +3.26767 q^{74} -4.73589 q^{76} -2.50407 q^{77} +10.4672 q^{79} +9.19123 q^{80} +0.761098 q^{82} -7.59946 q^{83} -2.46805 q^{85} +1.92356 q^{86} -3.21287 q^{88} -10.6245 q^{89} -1.48511 q^{91} +1.89128 q^{92} +2.22457 q^{94} +6.85086 q^{95} +13.9102 q^{97} +0.329727 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 2 q^{2} + 4 q^{4} - 5 q^{5} - 4 q^{7} - 9 q^{8}+O(q^{10}) $ 4 * q - 2 * q^2 + 4 * q^4 - 5 * q^5 - 4 * q^7 - 9 * q^8	$ 4 q - 2 q^{2} + 4 q^{4} - 5 q^{5} - 4 q^{7} - 9 q^{8} + 2 q^{10} - q^{11} + 7 q^{13} + 2 q^{14} + 8 q^{16} - 2 q^{17} - q^{19} - 13 q^{20} + 11 q^{22} - 4 q^{23} + 11 q^{25} + 19 q^{26} - 4 q^{28} - 2 q^{29} + 6 q^{31} - 20 q^{32} + 23 q^{34} + 5 q^{35} + 16 q^{37} + 11 q^{38} + 3 q^{40} - 5 q^{41} + 9 q^{43} - 3 q^{44} + 2 q^{46} + 21 q^{47} + 4 q^{49} + 17 q^{50} - 24 q^{52} - 10 q^{53} + 17 q^{55} + 9 q^{56} + q^{58} + 26 q^{59} + 2 q^{61} + 19 q^{62} + 27 q^{64} - 26 q^{65} + 5 q^{67} - 7 q^{68} - 2 q^{70} + 19 q^{71} + 10 q^{73} - 9 q^{74} - 3 q^{76} + q^{77} - 6 q^{79} + 24 q^{80} - 31 q^{82} + 2 q^{83} - 7 q^{85} + 17 q^{86} - 20 q^{88} + 17 q^{89} - 7 q^{91} - 4 q^{92} - 44 q^{94} + 17 q^{95} + 32 q^{97} - 2 q^{98}+O(q^{100}) $ 4 * q - 2 * q^2 + 4 * q^4 - 5 * q^5 - 4 * q^7 - 9 * q^8 + 2 * q^10 - q^11 + 7 * q^13 + 2 * q^14 + 8 * q^16 - 2 * q^17 - q^19 - 13 * q^20 + 11 * q^22 - 4 * q^23 + 11 * q^25 + 19 * q^26 - 4 * q^28 - 2 * q^29 + 6 * q^31 - 20 * q^32 + 23 * q^34 + 5 * q^35 + 16 * q^37 + 11 * q^38 + 3 * q^40 - 5 * q^41 + 9 * q^43 - 3 * q^44 + 2 * q^46 + 21 * q^47 + 4 * q^49 + 17 * q^50 - 24 * q^52 - 10 * q^53 + 17 * q^55 + 9 * q^56 + q^58 + 26 * q^59 + 2 * q^61 + 19 * q^62 + 27 * q^64 - 26 * q^65 + 5 * q^67 - 7 * q^68 - 2 * q^70 + 19 * q^71 + 10 * q^73 - 9 * q^74 - 3 * q^76 + q^77 - 6 * q^79 + 24 * q^80 - 31 * q^82 + 2 * q^83 - 7 * q^85 + 17 * q^86 - 20 * q^88 + 17 * q^89 - 7 * q^91 - 4 * q^92 - 44 * q^94 + 17 * q^95 + 32 * q^97 - 2 * 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 $

$2$	0.329727	0.233152	0.116576	−	0.993182i	$-0.462808\pi$
$2$	0.329727	0.233152	0.116576	+	0.993182i	$0.462808\pi$
$3$	0	0
$3$	0	0
$4$	−1.89128	−0.945640
$5$	2.73589	1.22353	0.611764	−	0.791040i	$-0.290460\pi$
$5$	2.73589	1.22353	0.611764	+	0.791040i	$0.290460\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	−1.28306	−0.453630
$9$	0	0
$10$	0.902098	0.285269
$11$	2.50407	0.755005	0.377502	−	0.926009i	$-0.376783\pi$
$11$	2.50407	0.755005	0.377502	+	0.926009i	$0.376783\pi$
$12$	0	0
$13$	1.48511	0.411896	0.205948	−	0.978563i	$-0.433972\pi$
$13$	1.48511	0.411896	0.205948	+	0.978563i	$0.433972\pi$
$14$	−0.329727	−0.0881233
$15$	0	0
$16$	3.35950	0.839875
$17$	−0.902098	−0.218791	−0.109396	−	0.993998i	$-0.534892\pi$
$17$	−0.902098	−0.218791	−0.109396	+	0.993998i	$0.534892\pi$
$18$	0	0
$19$	2.50407	0.574473	0.287236	−	0.957860i	$-0.407264\pi$
$19$	2.50407	0.574473	0.287236	+	0.957860i	$0.407264\pi$
$20$	−5.17434	−1.15702
$21$	0	0
$22$	0.825659	0.176031
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	2.48511	0.497023
$26$	0.489682	0.0960346
$27$	0	0
$28$	1.89128	0.357418
$29$	−6.68466	−1.24131	−0.620655	−	0.784084i	$-0.713133\pi$
$29$	−6.68466	−1.24131	−0.620655	+	0.784084i	$0.713133\pi$
$30$	0	0
$31$	1.09790	0.197189	0.0985945	−	0.995128i	$-0.468565\pi$
$31$	1.09790	0.197189	0.0985945	+	0.995128i	$0.468565\pi$
$32$	3.67384	0.649449
$33$	0	0
$34$	−0.297446	−0.0510116
$35$	−2.73589	−0.462450
$36$	0	0
$37$	9.91023	1.62923	0.814616	−	0.580000i	$-0.196948\pi$
$37$	9.91023	1.62923	0.814616	+	0.580000i	$0.196948\pi$
$38$	0.825659	0.133940
$39$	0	0
$40$	−3.51032	−0.555030
$41$	2.30826	0.360490	0.180245	−	0.983622i	$-0.442311\pi$
$41$	2.30826	0.360490	0.180245	+	0.983622i	$0.442311\pi$
$42$	0	0
$43$	5.83380	0.889645	0.444823	−	0.895619i	$-0.353267\pi$
$43$	5.83380	0.889645	0.444823	+	0.895619i	$0.353267\pi$
$44$	−4.73589	−0.713963
$45$	0	0
$46$	−0.329727	−0.0486156
$47$	6.74671	0.984109	0.492055	−	0.870564i	$-0.336246\pi$
$47$	6.74671	0.984109	0.492055	+	0.870564i	$0.336246\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0.819409	0.115882
$51$	0	0
$52$	−2.80877	−0.389506
$53$	4.91648	0.675331	0.337666	−	0.941266i	$-0.390363\pi$
$53$	4.91648	0.675331	0.337666	+	0.941266i	$0.390363\pi$
$54$	0	0
$55$	6.85086	0.923770
$56$	1.28306	0.171456
$57$	0	0
$58$	−2.20411	−0.289414
$59$	7.32973	0.954249	0.477125	−	0.878836i	$-0.341679\pi$
$59$	7.32973	0.954249	0.477125	+	0.878836i	$0.341679\pi$
$60$	0	0
$61$	−1.00625	−0.128837	−0.0644185	−	0.997923i	$-0.520519\pi$
$61$	−1.00625	−0.128837	−0.0644185	+	0.997923i	$0.520519\pi$
$62$	0.362008	0.0459751
$63$	0	0
$64$	−5.50764	−0.688454
$65$	4.06311	0.503967
$66$	0	0
$67$	11.2929	1.37964	0.689822	−	0.723979i	$-0.257689\pi$
$67$	11.2929	1.37964	0.689822	+	0.723979i	$0.257689\pi$
$68$	1.70612	0.206898
$69$	0	0
$70$	−0.902098	−0.107821
$71$	0.362008	0.0429624	0.0214812	−	0.999769i	$-0.493162\pi$
$71$	0.362008	0.0429624	0.0214812	+	0.999769i	$0.493162\pi$
$72$	0	0
$73$	−5.34411	−0.625481	−0.312741	−	0.949839i	$-0.601247\pi$
$73$	−5.34411	−0.625481	−0.312741	+	0.949839i	$0.601247\pi$
$74$	3.26767	0.379859
$75$	0	0
$76$	−4.73589	−0.543244
$77$	−2.50407	−0.285365
$78$	0	0
$79$	10.4672	1.17765	0.588827	−	0.808259i	$-0.299590\pi$
$79$	10.4672	1.17765	0.588827	+	0.808259i	$0.299590\pi$
$80$	9.19123	1.02761
$81$	0	0
$82$	0.761098	0.0840492
$83$	−7.59946	−0.834149	−0.417075	−	0.908872i	$-0.636945\pi$
$83$	−7.59946	−0.834149	−0.417075	+	0.908872i	$0.636945\pi$
$84$	0	0
$85$	−2.46805	−0.267697
$86$	1.92356	0.207423
$87$	0	0
$88$	−3.21287	−0.342493
$89$	−10.6245	−1.12619	−0.563097	−	0.826391i	$-0.690390\pi$
$89$	−10.6245	−1.12619	−0.563097	+	0.826391i	$0.690390\pi$
$90$	0	0
$91$	−1.48511	−0.155682
$92$	1.89128	0.197180
$93$	0	0
$94$	2.22457	0.229447
$95$	6.85086	0.702884
$96$	0	0
$97$	13.9102	1.41237	0.706185	−	0.708027i	$-0.250415\pi$
$97$	13.9102	1.41237	0.706185	+	0.708027i	$0.250415\pi$
$98$	0.329727	0.0333075
$99$	0	0
$100$	−4.70005	−0.470005
$101$	−1.81502	−0.180601	−0.0903004	−	0.995915i	$-0.528783\pi$
$101$	−1.81502	−0.180601	−0.0903004	+	0.995915i	$0.528783\pi$
$102$	0	0
$103$	−16.6605	−1.64161	−0.820805	−	0.571209i	$-0.806475\pi$
$103$	−16.6605	−1.64161	−0.820805	+	0.571209i	$0.806475\pi$
$104$	−1.90549	−0.186849
$105$	0	0
$106$	1.62110	0.157455
$107$	−1.68092	−0.162500	−0.0812502	−	0.996694i	$-0.525891\pi$
$107$	−1.68092	−0.162500	−0.0812502	+	0.996694i	$0.525891\pi$
$108$	0	0
$109$	11.4333	1.09511	0.547554	−	0.836771i	$-0.315559\pi$
$109$	11.4333	1.09511	0.547554	+	0.836771i	$0.315559\pi$
$110$	2.25892	0.215379
$111$	0	0
$112$	−3.35950	−0.317443
$113$	−3.14914	−0.296246	−0.148123	−	0.988969i	$-0.547323\pi$
$113$	−3.14914	−0.296246	−0.148123	+	0.988969i	$0.547323\pi$
$114$	0	0
$115$	−2.73589	−0.255123
$116$	12.6426	1.17383
$117$	0	0
$118$	2.41681	0.222485
$119$	0.902098	0.0826952
$120$	0	0
$121$	−4.72964	−0.429968
$122$	−0.331788	−0.0300387
$123$	0	0
$124$	−2.07644	−0.186470
$125$	−6.88046	−0.615407
$126$	0	0
$127$	0.449266	0.0398659	0.0199329	−	0.999801i	$-0.493655\pi$
$127$	0.449266	0.0398659	0.0199329	+	0.999801i	$0.493655\pi$
$128$	−9.16370	−0.809964
$129$	0	0
$130$	1.33972	0.117501
$131$	19.9759	1.74530	0.872649	−	0.488347i	$-0.162400\pi$
$131$	19.9759	1.74530	0.872649	+	0.488347i	$0.162400\pi$
$132$	0	0
$133$	−2.50407	−0.217130
$134$	3.72357	0.321667
$135$	0	0
$136$	1.15745	0.0992503
$137$	−0.270356	−0.0230981	−0.0115490	−	0.999933i	$-0.503676\pi$
$137$	−0.270356	−0.0230981	−0.0115490	+	0.999933i	$0.503676\pi$
$138$	0	0
$139$	4.42512	0.375334	0.187667	−	0.982233i	$-0.439907\pi$
$139$	4.42512	0.375334	0.187667	+	0.982233i	$0.439907\pi$
$140$	5.17434	0.437312
$141$	0	0
$142$	0.119364	0.0100168
$143$	3.71883	0.310984
$144$	0	0
$145$	−18.2885	−1.51878
$146$	−1.76210	−0.145832
$147$	0	0
$148$	−18.7430	−1.54067
$149$	3.69530	0.302731	0.151365	−	0.988478i	$-0.451633\pi$
$149$	3.69530	0.302731	0.151365	+	0.988478i	$0.451633\pi$
$150$	0	0
$151$	8.63174	0.702441	0.351221	−	0.936293i	$-0.385767\pi$
$151$	8.63174	0.702441	0.351221	+	0.936293i	$0.385767\pi$
$152$	−3.21287	−0.260598
$153$	0	0
$154$	−0.825659	−0.0665335
$155$	3.00374	0.241266
$156$	0	0
$157$	9.93438	0.792850	0.396425	−	0.918067i	$-0.370251\pi$
$157$	9.93438	0.792850	0.396425	+	0.918067i	$0.370251\pi$
$158$	3.45133	0.274573
$159$	0	0
$160$	10.0512	0.794620
$161$	1.00000	0.0788110
$162$	0	0
$163$	14.7082	1.15203	0.576017	−	0.817438i	$-0.304606\pi$
$163$	14.7082	1.15203	0.576017	+	0.817438i	$0.304606\pi$
$164$	−4.36558	−0.340894
$165$	0	0
$166$	−2.50575	−0.194484
$167$	5.47430	0.423614	0.211807	−	0.977312i	$-0.432065\pi$
$167$	5.47430	0.423614	0.211807	+	0.977312i	$0.432065\pi$
$168$	0	0
$169$	−10.7944	−0.830341
$170$	−0.813782	−0.0624142
$171$	0	0
$172$	−11.0333	−0.841284
$173$	−18.1727	−1.38165	−0.690823	−	0.723024i	$-0.742752\pi$
$173$	−18.1727	−1.38165	−0.690823	+	0.723024i	$0.742752\pi$
$174$	0	0
$175$	−2.48511	−0.187857
$176$	8.41242	0.634110
$177$	0	0
$178$	−3.50318	−0.262575
$179$	−10.1860	−0.761341	−0.380670	−	0.924711i	$-0.624307\pi$
$179$	−10.1860	−0.761341	−0.380670	+	0.924711i	$0.624307\pi$
$180$	0	0
$181$	13.5186	1.00483	0.502416	−	0.864626i	$-0.332445\pi$
$181$	13.5186	1.00483	0.502416	+	0.864626i	$0.332445\pi$
$182$	−0.489682	−0.0362977
$183$	0	0
$184$	1.28306	0.0945885
$185$	27.1133	1.99341
$186$	0	0
$187$	−2.25892	−0.165188
$188$	−12.7599	−0.930613
$189$	0	0
$190$	2.25892	0.163879
$191$	0.631742	0.0457113	0.0228556	−	0.999739i	$-0.492724\pi$
$191$	0.631742	0.0457113	0.0228556	+	0.999739i	$0.492724\pi$
$192$	0	0
$193$	1.36826	0.0984893	0.0492447	−	0.998787i	$-0.484319\pi$
$193$	1.36826	0.0984893	0.0492447	+	0.998787i	$0.484319\pi$
$194$	4.58658	0.329297
$195$	0	0
$196$	−1.89128	−0.135091
$197$	−24.7611	−1.76416	−0.882078	−	0.471104i	$-0.843856\pi$
$197$	−24.7611	−1.76416	−0.882078	+	0.471104i	$0.843856\pi$
$198$	0	0
$199$	1.44470	0.102412	0.0512059	−	0.998688i	$-0.483694\pi$
$199$	1.44470	0.102412	0.0512059	+	0.998688i	$0.483694\pi$
$200$	−3.18855	−0.225465
$201$	0	0
$202$	−0.598460	−0.0421075
$203$	6.68466	0.469171
$204$	0	0
$205$	6.31517	0.441070
$206$	−5.49342	−0.382745
$207$	0	0
$208$	4.98924	0.345941
$209$	6.27036	0.433730
$210$	0	0
$211$	−23.0990	−1.59020	−0.795099	−	0.606480i	$-0.792581\pi$
$211$	−23.0990	−1.59020	−0.795099	+	0.606480i	$0.792581\pi$
$212$	−9.29845	−0.638620
$213$	0	0
$214$	−0.554244	−0.0378873
$215$	15.9606	1.08851
$216$	0	0
$217$	−1.09790	−0.0745304
$218$	3.76986	0.255327
$219$	0	0
$220$	−12.9569	−0.873554
$221$	−1.33972	−0.0901192
$222$	0	0
$223$	−18.6497	−1.24888	−0.624438	−	0.781074i	$-0.714672\pi$
$223$	−18.6497	−1.24888	−0.624438	+	0.781074i	$0.714672\pi$
$224$	−3.67384	−0.245469
$225$	0	0
$226$	−1.03836	−0.0690704
$227$	−20.6334	−1.36949	−0.684744	−	0.728783i	$-0.740086\pi$
$227$	−20.6334	−1.36949	−0.684744	+	0.728783i	$0.740086\pi$
$228$	0	0
$229$	14.9544	0.988214	0.494107	−	0.869401i	$-0.335495\pi$
$229$	14.9544	0.988214	0.494107	+	0.869401i	$0.335495\pi$
$230$	−0.902098	−0.0594826
$231$	0	0
$232$	8.57682	0.563096
$233$	−28.1052	−1.84123	−0.920617	−	0.390467i	$-0.872313\pi$
$233$	−28.1052	−1.84123	−0.920617	+	0.390467i	$0.872313\pi$
$234$	0	0
$235$	18.4583	1.20409
$236$	−13.8626	−0.902376
$237$	0	0
$238$	0.297446	0.0192806
$239$	−25.8536	−1.67233	−0.836166	−	0.548476i	$-0.815208\pi$
$239$	−25.8536	−1.67233	−0.836166	+	0.548476i	$0.815208\pi$
$240$	0	0
$241$	−14.8311	−0.955356	−0.477678	−	0.878535i	$-0.658521\pi$
$241$	−14.8311	−0.955356	−0.477678	+	0.878535i	$0.658521\pi$
$242$	−1.55949	−0.100248
$243$	0	0
$244$	1.90310	0.121833
$245$	2.73589	0.174790
$246$	0	0
$247$	3.71883	0.236623
$248$	−1.40867	−0.0894509
$249$	0	0
$250$	−2.26868	−0.143484
$251$	16.2841	1.02784	0.513922	−	0.857837i	$-0.328192\pi$
$251$	16.2841	1.02784	0.513922	+	0.857837i	$0.328192\pi$
$252$	0	0
$253$	−2.50407	−0.157429
$254$	0.148135	0.00929482
$255$	0	0
$256$	7.99375	0.499609
$257$	−22.9446	−1.43125	−0.715623	−	0.698486i	$-0.753857\pi$
$257$	−22.9446	−1.43125	−0.715623	+	0.698486i	$0.753857\pi$
$258$	0	0
$259$	−9.91023	−0.615792
$260$	−7.68448	−0.476571
$261$	0	0
$262$	6.58658	0.406920
$263$	0.776932	0.0479077	0.0239538	−	0.999713i	$-0.492375\pi$
$263$	0.776932	0.0479077	0.0239538	+	0.999713i	$0.492375\pi$
$264$	0	0
$265$	13.4510	0.826287
$266$	−0.825659	−0.0506244
$267$	0	0
$268$	−21.3580	−1.30465
$269$	−3.05937	−0.186533	−0.0932666	−	0.995641i	$-0.529731\pi$
$269$	−3.05937	−0.186533	−0.0932666	+	0.995641i	$0.529731\pi$
$270$	0	0
$271$	−16.0631	−0.975765	−0.487882	−	0.872909i	$-0.662230\pi$
$271$	−16.0631	−0.975765	−0.487882	+	0.872909i	$0.662230\pi$
$272$	−3.03060	−0.183757
$273$	0	0
$274$	−0.0891438	−0.00538537
$275$	6.22289	0.375255
$276$	0	0
$277$	−11.6103	−0.697594	−0.348797	−	0.937198i	$-0.613410\pi$
$277$	−11.6103	−0.697594	−0.348797	+	0.937198i	$0.613410\pi$
$278$	1.45908	0.0875100
$279$	0	0
$280$	3.51032	0.209782
$281$	−9.89753	−0.590437	−0.295219	−	0.955430i	$-0.595392\pi$
$281$	−9.89753	−0.590437	−0.295219	+	0.955430i	$0.595392\pi$
$282$	0	0
$283$	13.9994	0.832177	0.416088	−	0.909324i	$-0.363401\pi$
$283$	13.9994	0.832177	0.416088	+	0.909324i	$0.363401\pi$
$284$	−0.684658	−0.0406270
$285$	0	0
$286$	1.22620	0.0725066
$287$	−2.30826	−0.136253
$288$	0	0
$289$	−16.1862	−0.952130
$290$	−6.03022	−0.354107
$291$	0	0
$292$	10.1072	0.591480
$293$	8.14038	0.475566	0.237783	−	0.971318i	$-0.423579\pi$
$293$	8.14038	0.475566	0.237783	+	0.971318i	$0.423579\pi$
$294$	0	0
$295$	20.0534	1.16755
$296$	−12.7154	−0.739070
$297$	0	0
$298$	1.21844	0.0705824
$299$	−1.48511	−0.0858863
$300$	0	0
$301$	−5.83380	−0.336254
$302$	2.84612	0.163776
$303$	0	0
$304$	8.41242	0.482485
$305$	−2.75299	−0.157636
$306$	0	0
$307$	13.5095	0.771027	0.385514	−	0.922702i	$-0.374024\pi$
$307$	13.5095	0.771027	0.385514	+	0.922702i	$0.374024\pi$
$308$	4.73589	0.269853
$309$	0	0
$310$	0.990415	0.0562518
$311$	24.3035	1.37813	0.689063	−	0.724701i	$-0.258022\pi$
$311$	24.3035	1.37813	0.689063	+	0.724701i	$0.258022\pi$
$312$	0	0
$313$	−33.7836	−1.90956	−0.954782	−	0.297308i	$-0.903911\pi$
$313$	−33.7836	−1.90956	−0.954782	+	0.297308i	$0.903911\pi$
$314$	3.27563	0.184855
$315$	0	0
$316$	−19.7964	−1.11364
$317$	−32.1594	−1.80625	−0.903126	−	0.429376i	$-0.858733\pi$
$317$	−32.1594	−1.80625	−0.903126	+	0.429376i	$0.858733\pi$
$318$	0	0
$319$	−16.7388	−0.937195
$320$	−15.0683	−0.842344
$321$	0	0
$322$	0.329727	0.0183750
$323$	−2.25892	−0.125689
$324$	0	0
$325$	3.69068	0.204722
$326$	4.84969	0.268599
$327$	0	0
$328$	−2.96164	−0.163529
$329$	−6.74671	−0.371958
$330$	0	0
$331$	−4.84004	−0.266033	−0.133016	−	0.991114i	$-0.542466\pi$
$331$	−4.84004	−0.266033	−0.133016	+	0.991114i	$0.542466\pi$
$332$	14.3727	0.788805
$333$	0	0
$334$	1.80502	0.0987665
$335$	30.8961	1.68803
$336$	0	0
$337$	−2.59195	−0.141192	−0.0705962	−	0.997505i	$-0.522490\pi$
$337$	−2.59195	−0.141192	−0.0705962	+	0.997505i	$0.522490\pi$
$338$	−3.55922	−0.193596
$339$	0	0
$340$	4.66776	0.253145
$341$	2.74922	0.148879
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	−7.48511	−0.403570
$345$	0	0
$346$	−5.99204	−0.322134
$347$	15.9390	0.855651	0.427825	−	0.903861i	$-0.359280\pi$
$347$	15.9390	0.855651	0.427825	+	0.903861i	$0.359280\pi$
$348$	0	0
$349$	−8.02603	−0.429624	−0.214812	−	0.976655i	$-0.568914\pi$
$349$	−8.02603	−0.429624	−0.214812	+	0.976655i	$0.568914\pi$
$350$	−0.819409	−0.0437993
$351$	0	0
$352$	9.19954	0.490337
$353$	−10.7871	−0.574141	−0.287070	−	0.957909i	$-0.592681\pi$
$353$	−10.7871	−0.574141	−0.287070	+	0.957909i	$0.592681\pi$
$354$	0	0
$355$	0.990415	0.0525658
$356$	20.0939	1.06497
$357$	0	0
$358$	−3.35862	−0.177508
$359$	2.17891	0.114998	0.0574992	−	0.998346i	$-0.481687\pi$
$359$	2.17891	0.114998	0.0574992	+	0.998346i	$0.481687\pi$
$360$	0	0
$361$	−12.7296	−0.669981
$362$	4.45746	0.234279
$363$	0	0
$364$	2.80877	0.147219
$365$	−14.6209	−0.765294
$366$	0	0
$367$	27.0388	1.41141	0.705707	−	0.708504i	$-0.250630\pi$
$367$	27.0388	1.41141	0.705707	+	0.708504i	$0.250630\pi$
$368$	−3.35950	−0.175126
$369$	0	0
$370$	8.94001	0.464769
$371$	−4.91648	−0.255251
$372$	0	0
$373$	16.4189	0.850140	0.425070	−	0.905161i	$-0.360249\pi$
$373$	16.4189	0.850140	0.425070	+	0.905161i	$0.360249\pi$
$374$	−0.744826	−0.0385140
$375$	0	0
$376$	−8.65644	−0.446422
$377$	−9.92748	−0.511291
$378$	0	0
$379$	−34.6399	−1.77933	−0.889667	−	0.456610i	$-0.849064\pi$
$379$	−34.6399	−1.77933	−0.889667	+	0.456610i	$0.849064\pi$
$380$	−12.9569	−0.664675
$381$	0	0
$382$	0.208303	0.0106577
$383$	10.6722	0.545322	0.272661	−	0.962110i	$-0.412096\pi$
$383$	10.6722	0.545322	0.272661	+	0.962110i	$0.412096\pi$
$384$	0	0
$385$	−6.85086	−0.349152
$386$	0.451152	0.0229630
$387$	0	0
$388$	−26.3081	−1.33559
$389$	−0.0302201	−0.00153222	−0.000766110	−	1.00000i	$-0.500244\pi$
$389$	−0.0302201	−0.00153222	−0.000766110		1.00000i	$0.500244\pi$
$390$	0	0
$391$	0.902098	0.0456211
$392$	−1.28306	−0.0648044
$393$	0	0
$394$	−8.16441	−0.411317
$395$	28.6372	1.44089
$396$	0	0
$397$	20.4749	1.02761	0.513803	−	0.857908i	$-0.328236\pi$
$397$	20.4749	1.02761	0.513803	+	0.857908i	$0.328236\pi$
$398$	0.476356	0.0238776
$399$	0	0
$400$	8.34874	0.417437
$401$	36.4304	1.81925	0.909623	−	0.415435i	$-0.136371\pi$
$401$	36.4304	1.81925	0.909623	+	0.415435i	$0.136371\pi$
$402$	0	0
$403$	1.63051	0.0812214
$404$	3.43270	0.170783
$405$	0	0
$406$	2.20411	0.109388
$407$	24.8159	1.23008
$408$	0	0
$409$	37.4054	1.84958	0.924790	−	0.380479i	$-0.124241\pi$
$409$	37.4054	1.84958	0.924790	+	0.380479i	$0.124241\pi$
$410$	2.08228	0.102837
$411$	0	0
$412$	31.5097	1.55237
$413$	−7.32973	−0.360672
$414$	0	0
$415$	−20.7913	−1.02061
$416$	5.45607	0.267506
$417$	0	0
$418$	2.06751	0.101125
$419$	37.2017	1.81742	0.908710	−	0.417428i	$-0.137068\pi$
$419$	37.2017	1.81742	0.908710	+	0.417428i	$0.137068\pi$
$420$	0	0
$421$	27.6326	1.34673	0.673366	−	0.739309i	$-0.264848\pi$
$421$	27.6326	1.34673	0.673366	+	0.739309i	$0.264848\pi$
$422$	−7.61636	−0.370758
$423$	0	0
$424$	−6.30815	−0.306351
$425$	−2.24182	−0.108744
$426$	0	0
$427$	1.00625	0.0486958
$428$	3.17908	0.153667
$429$	0	0
$430$	5.26266	0.253788
$431$	−35.0676	−1.68915	−0.844573	−	0.535441i	$-0.820145\pi$
$431$	−35.0676	−1.68915	−0.844573	+	0.535441i	$0.820145\pi$
$432$	0	0
$433$	−1.00708	−0.0483970	−0.0241985	−	0.999707i	$-0.507703\pi$
$433$	−1.00708	−0.0483970	−0.0241985	+	0.999707i	$0.507703\pi$
$434$	−0.362008	−0.0173769
$435$	0	0
$436$	−21.6235	−1.03558
$437$	−2.50407	−0.119786
$438$	0	0
$439$	−4.04649	−0.193129	−0.0965643	−	0.995327i	$-0.530785\pi$
$439$	−4.04649	−0.193129	−0.0965643	+	0.995327i	$0.530785\pi$
$440$	−8.79007	−0.419050
$441$	0	0
$442$	−0.441742	−0.0210115
$443$	29.3947	1.39659	0.698293	−	0.715812i	$-0.253943\pi$
$443$	29.3947	1.39659	0.698293	+	0.715812i	$0.253943\pi$
$444$	0	0
$445$	−29.0675	−1.37793
$446$	−6.14931	−0.291178
$447$	0	0
$448$	5.50764	0.260211
$449$	4.87627	0.230126	0.115063	−	0.993358i	$-0.463293\pi$
$449$	4.87627	0.230126	0.115063	+	0.993358i	$0.463293\pi$
$450$	0	0
$451$	5.78005	0.272172
$452$	5.95590	0.280142
$453$	0	0
$454$	−6.80340	−0.319299
$455$	−4.06311	−0.190482
$456$	0	0
$457$	−35.5212	−1.66161	−0.830806	−	0.556562i	$-0.812120\pi$
$457$	−35.5212	−1.66161	−0.830806	+	0.556562i	$0.812120\pi$
$458$	4.93087	0.230404
$459$	0	0
$460$	5.17434	0.241255
$461$	18.9442	0.882319	0.441160	−	0.897429i	$-0.354567\pi$
$461$	18.9442	0.882319	0.441160	+	0.897429i	$0.354567\pi$
$462$	0	0
$463$	14.1933	0.659618	0.329809	−	0.944048i	$-0.393016\pi$
$463$	14.1933	0.659618	0.329809	+	0.944048i	$0.393016\pi$
$464$	−22.4571	−1.04255
$465$	0	0
$466$	−9.26705	−0.429288
$467$	31.0355	1.43615	0.718075	−	0.695966i	$-0.245024\pi$
$467$	31.0355	1.43615	0.718075	+	0.695966i	$0.245024\pi$
$468$	0	0
$469$	−11.2929	−0.521457
$470$	6.08620	0.280735
$471$	0	0
$472$	−9.40449	−0.432877
$473$	14.6082	0.671687
$474$	0	0
$475$	6.22289	0.285526
$476$	−1.70612	−0.0781999
$477$	0	0
$478$	−8.52465	−0.389908
$479$	−10.5214	−0.480735	−0.240367	−	0.970682i	$-0.577268\pi$
$479$	−10.5214	−0.480735	−0.240367	+	0.970682i	$0.577268\pi$
$480$	0	0
$481$	14.7178	0.671075
$482$	−4.89022	−0.222743
$483$	0	0
$484$	8.94508	0.406595
$485$	38.0569	1.72808
$486$	0	0
$487$	−31.7378	−1.43818	−0.719089	−	0.694918i	$-0.755441\pi$
$487$	−31.7378	−1.43818	−0.719089	+	0.694918i	$0.755441\pi$
$488$	1.29108	0.0584444
$489$	0	0
$490$	0.902098	0.0407527
$491$	31.8788	1.43867	0.719336	−	0.694662i	$-0.244446\pi$
$491$	31.8788	1.43867	0.719336	+	0.694662i	$0.244446\pi$
$492$	0	0
$493$	6.03022	0.271587
$494$	1.22620	0.0551692
$495$	0	0
$496$	3.68840	0.165614
$497$	−0.362008	−0.0162383
$498$	0	0
$499$	−39.4279	−1.76503	−0.882517	−	0.470280i	$-0.844153\pi$
$499$	−39.4279	−1.76503	−0.882517	+	0.470280i	$0.844153\pi$
$500$	13.0129	0.581954
$501$	0	0
$502$	5.36932	0.239644
$503$	−24.4589	−1.09057	−0.545284	−	0.838251i	$-0.683578\pi$
$503$	−24.4589	−1.09057	−0.545284	+	0.838251i	$0.683578\pi$
$504$	0	0
$505$	−4.96569	−0.220970
$506$	−0.825659	−0.0367050
$507$	0	0
$508$	−0.849687	−0.0376988
$509$	−20.3927	−0.903889	−0.451944	−	0.892046i	$-0.649269\pi$
$509$	−20.3927	−0.903889	−0.451944	+	0.892046i	$0.649269\pi$
$510$	0	0
$511$	5.34411	0.236410
$512$	20.9632	0.926449
$513$	0	0
$514$	−7.56547	−0.333699
$515$	−45.5814	−2.00856
$516$	0	0
$517$	16.8942	0.743007
$518$	−3.26767	−0.143573
$519$	0	0
$520$	−5.21322	−0.228615
$521$	−26.3058	−1.15248	−0.576238	−	0.817282i	$-0.695480\pi$
$521$	−26.3058	−1.15248	−0.576238	+	0.817282i	$0.695480\pi$
$522$	0	0
$523$	−40.4256	−1.76769	−0.883843	−	0.467783i	$-0.845053\pi$
$523$	−40.4256	−1.76769	−0.883843	+	0.467783i	$0.845053\pi$
$524$	−37.7799	−1.65042
$525$	0	0
$526$	0.256176	0.0111698
$527$	−0.990415	−0.0431432
$528$	0	0
$529$	1.00000	0.0434783
$530$	4.43515	0.192651
$531$	0	0
$532$	4.73589	0.205327
$533$	3.42804	0.148485
$534$	0	0
$535$	−4.59881	−0.198824
$536$	−14.4894	−0.625849
$537$	0	0
$538$	−1.00876	−0.0434906
$539$	2.50407	0.107858
$540$	0	0
$541$	0.360122	0.0154828	0.00774142	−	0.999970i	$-0.497536\pi$
$541$	0.360122	0.0154828	0.00774142	+	0.999970i	$0.497536\pi$
$542$	−5.29644	−0.227502
$543$	0	0
$544$	−3.31417	−0.142094
$545$	31.2802	1.33990
$546$	0	0
$547$	20.2023	0.863789	0.431894	−	0.901924i	$-0.357845\pi$
$547$	20.2023	0.863789	0.431894	+	0.901924i	$0.357845\pi$
$548$	0.511319	0.0218425
$549$	0	0
$550$	2.05186	0.0874915
$551$	−16.7388	−0.713099
$552$	0	0
$553$	−10.4672	−0.445111
$554$	−3.82822	−0.162646
$555$	0	0
$556$	−8.36914	−0.354931
$557$	−9.71086	−0.411463	−0.205731	−	0.978609i	$-0.565957\pi$
$557$	−9.71086	−0.411463	−0.205731	+	0.978609i	$0.565957\pi$
$558$	0	0
$559$	8.66385	0.366442
$560$	−9.19123	−0.388401
$561$	0	0
$562$	−3.26348	−0.137662
$563$	−27.2789	−1.14967	−0.574835	−	0.818269i	$-0.694934\pi$
$563$	−27.2789	−1.14967	−0.574835	+	0.818269i	$0.694934\pi$
$564$	0	0
$565$	−8.61570	−0.362465
$566$	4.61598	0.194024
$567$	0	0
$568$	−0.464478	−0.0194891
$569$	−19.6132	−0.822229	−0.411115	−	0.911584i	$-0.634860\pi$
$569$	−19.6132	−0.822229	−0.411115	+	0.911584i	$0.634860\pi$
$570$	0	0
$571$	21.1859	0.886601	0.443301	−	0.896373i	$-0.353807\pi$
$571$	21.1859	0.886601	0.443301	+	0.896373i	$0.353807\pi$
$572$	−7.03334	−0.294079
$573$	0	0
$574$	−0.761098	−0.0317676
$575$	−2.48511	−0.103636
$576$	0	0
$577$	−12.4079	−0.516547	−0.258273	−	0.966072i	$-0.583154\pi$
$577$	−12.4079	−0.516547	−0.258273	+	0.966072i	$0.583154\pi$
$578$	−5.33704	−0.221991
$579$	0	0
$580$	34.5887	1.43622
$581$	7.59946	0.315279
$582$	0	0
$583$	12.3112	0.509878
$584$	6.85682	0.283737
$585$	0	0
$586$	2.68410	0.110879
$587$	9.35493	0.386119	0.193060	−	0.981187i	$-0.438159\pi$
$587$	9.35493	0.386119	0.193060	+	0.981187i	$0.438159\pi$
$588$	0	0
$589$	2.74922	0.113280
$590$	6.61214	0.272217
$591$	0	0
$592$	33.2934	1.36835
$593$	16.7836	0.689218	0.344609	−	0.938746i	$-0.388011\pi$
$593$	16.7836	0.689218	0.344609	+	0.938746i	$0.388011\pi$
$594$	0	0
$595$	2.46805	0.101180
$596$	−6.98885	−0.286275
$597$	0	0
$598$	−0.489682	−0.0200246
$599$	16.0512	0.655836	0.327918	−	0.944706i	$-0.393653\pi$
$599$	16.0512	0.655836	0.327918	+	0.944706i	$0.393653\pi$
$600$	0	0
$601$	−24.4364	−0.996781	−0.498390	−	0.866953i	$-0.666075\pi$
$601$	−24.4364	−0.996781	−0.498390	+	0.866953i	$0.666075\pi$
$602$	−1.92356	−0.0783985
$603$	0	0
$604$	−16.3250	−0.664257
$605$	−12.9398	−0.526078
$606$	0	0
$607$	28.2329	1.14594	0.572969	−	0.819577i	$-0.305792\pi$
$607$	28.2329	1.14594	0.572969	+	0.819577i	$0.305792\pi$
$608$	9.19954	0.373091
$609$	0	0
$610$	−0.907736	−0.0367532
$611$	10.0196	0.405351
$612$	0	0
$613$	17.9873	0.726500	0.363250	−	0.931692i	$-0.381667\pi$
$613$	17.9873	0.726500	0.363250	+	0.931692i	$0.381667\pi$
$614$	4.45445	0.179767
$615$	0	0
$616$	3.21287	0.129450
$617$	−12.9998	−0.523353	−0.261677	−	0.965156i	$-0.584275\pi$
$617$	−12.9998	−0.523353	−0.261677	+	0.965156i	$0.584275\pi$
$618$	0	0
$619$	−1.26266	−0.0507505	−0.0253752	−	0.999678i	$-0.508078\pi$
$619$	−1.26266	−0.0507505	−0.0253752	+	0.999678i	$0.508078\pi$
$620$	−5.68092	−0.228151
$621$	0	0
$622$	8.01353	0.321313
$623$	10.6245	0.425661
$624$	0	0
$625$	−31.2498	−1.24999
$626$	−11.1394	−0.445219
$627$	0	0
$628$	−18.7887	−0.749750
$629$	−8.94001	−0.356461
$630$	0	0
$631$	−24.4568	−0.973611	−0.486806	−	0.873510i	$-0.661838\pi$
$631$	−24.4568	−0.973611	−0.486806	+	0.873510i	$0.661838\pi$
$632$	−13.4301	−0.534220
$633$	0	0
$634$	−10.6038	−0.421132
$635$	1.22914	0.0487771
$636$	0	0
$637$	1.48511	0.0588423
$638$	−5.51925	−0.218509
$639$	0	0
$640$	−25.0709	−0.991014
$641$	1.29037	0.0509665	0.0254833	−	0.999675i	$-0.491888\pi$
$641$	1.29037	0.0509665	0.0254833	+	0.999675i	$0.491888\pi$
$642$	0	0
$643$	34.3110	1.35309	0.676547	−	0.736399i	$-0.263476\pi$
$643$	34.3110	1.35309	0.676547	+	0.736399i	$0.263476\pi$
$644$	−1.89128	−0.0745269
$645$	0	0
$646$	−0.744826	−0.0293048
$647$	2.62092	0.103039	0.0515196	−	0.998672i	$-0.483594\pi$
$647$	2.62092	0.103039	0.0515196	+	0.998672i	$0.483594\pi$
$648$	0	0
$649$	18.3541	0.720463
$650$	1.21692	0.0477314
$651$	0	0
$652$	−27.8173	−1.08941
$653$	−7.53697	−0.294944	−0.147472	−	0.989066i	$-0.547114\pi$
$653$	−7.53697	−0.294944	−0.147472	+	0.989066i	$0.547114\pi$
$654$	0	0
$655$	54.6518	2.13542
$656$	7.75462	0.302767
$657$	0	0
$658$	−2.22457	−0.0867229
$659$	−22.8213	−0.888990	−0.444495	−	0.895781i	$-0.646617\pi$
$659$	−22.8213	−0.888990	−0.444495	+	0.895781i	$0.646617\pi$
$660$	0	0
$661$	27.3668	1.06445	0.532223	−	0.846604i	$-0.321357\pi$
$661$	27.3668	1.06445	0.532223	+	0.846604i	$0.321357\pi$
$662$	−1.59589	−0.0620262
$663$	0	0
$664$	9.75057	0.378396
$665$	−6.85086	−0.265665
$666$	0	0
$667$	6.68466	0.258831
$668$	−10.3534	−0.400586
$669$	0	0
$670$	10.1873	0.393569
$671$	−2.51972	−0.0972726
$672$	0	0
$673$	22.9790	0.885774	0.442887	−	0.896577i	$-0.353954\pi$
$673$	22.9790	0.885774	0.442887	+	0.896577i	$0.353954\pi$
$674$	−0.854636	−0.0329193
$675$	0	0
$676$	20.4153	0.785204
$677$	8.80814	0.338524	0.169262	−	0.985571i	$-0.445862\pi$
$677$	8.80814	0.338524	0.169262	+	0.985571i	$0.445862\pi$
$678$	0	0
$679$	−13.9102	−0.533826
$680$	3.16665	0.121436
$681$	0	0
$682$	0.906493	0.0347114
$683$	−34.4889	−1.31968	−0.659840	−	0.751406i	$-0.729376\pi$
$683$	−34.4889	−1.31968	−0.659840	+	0.751406i	$0.729376\pi$
$684$	0	0
$685$	−0.739666	−0.0282612
$686$	−0.329727	−0.0125890
$687$	0	0
$688$	19.5986	0.747191
$689$	7.30154	0.278166
$690$	0	0
$691$	37.6110	1.43079	0.715395	−	0.698721i	$-0.246247\pi$
$691$	37.6110	1.43079	0.715395	+	0.698721i	$0.246247\pi$
$692$	34.3697	1.30654
$693$	0	0
$694$	5.25552	0.199497
$695$	12.1067	0.459232
$696$	0	0
$697$	−2.08228	−0.0788721
$698$	−2.64640	−0.100168
$699$	0	0
$700$	4.70005	0.177645
$701$	47.5348	1.79536	0.897682	−	0.440644i	$-0.145250\pi$
$701$	47.5348	1.79536	0.897682	+	0.440644i	$0.145250\pi$
$702$	0	0
$703$	24.8159	0.935949
$704$	−13.7915	−0.519786
$705$	0	0
$706$	−3.55681	−0.133862
$707$	1.81502	0.0682607
$708$	0	0
$709$	−41.1332	−1.54479	−0.772395	−	0.635143i	$-0.780941\pi$
$709$	−41.1332	−1.54479	−0.772395	+	0.635143i	$0.780941\pi$
$710$	0.326567	0.0122558
$711$	0	0
$712$	13.6319	0.510876
$713$	−1.09790	−0.0411167
$714$	0	0
$715$	10.1743	0.380498
$716$	19.2647	0.719954
$717$	0	0
$718$	0.718446	0.0268122
$719$	−8.25857	−0.307993	−0.153996	−	0.988071i	$-0.549214\pi$
$719$	−8.25857	−0.307993	−0.153996	+	0.988071i	$0.549214\pi$
$720$	0	0
$721$	16.6605	0.620470
$722$	−4.19731	−0.156208
$723$	0	0
$724$	−25.5675	−0.950209
$725$	−16.6121	−0.616959
$726$	0	0
$727$	0.985235	0.0365403	0.0182702	−	0.999833i	$-0.494184\pi$
$727$	0.985235	0.0365403	0.0182702	+	0.999833i	$0.494184\pi$
$728$	1.90549	0.0706222
$729$	0	0
$730$	−4.82092	−0.178430
$731$	−5.26266	−0.194646
$732$	0	0
$733$	27.7847	1.02625	0.513125	−	0.858314i	$-0.328488\pi$
$733$	27.7847	1.02625	0.513125	+	0.858314i	$0.328488\pi$
$734$	8.91542	0.329074
$735$	0	0
$736$	−3.67384	−0.135420
$737$	28.2781	1.04164
$738$	0	0
$739$	30.0540	1.10555	0.552777	−	0.833329i	$-0.313568\pi$
$739$	30.0540	1.10555	0.552777	+	0.833329i	$0.313568\pi$
$740$	−51.2789	−1.88505
$741$	0	0
$742$	−1.62110	−0.0595124
$743$	−40.1617	−1.47339	−0.736695	−	0.676225i	$-0.763615\pi$
$743$	−40.1617	−1.47339	−0.736695	+	0.676225i	$0.763615\pi$
$744$	0	0
$745$	10.1100	0.370400
$746$	5.41377	0.198212
$747$	0	0
$748$	4.27224	0.156209
$749$	1.68092	0.0614194
$750$	0	0
$751$	−2.15350	−0.0785823	−0.0392912	−	0.999228i	$-0.512510\pi$
$751$	−2.15350	−0.0785823	−0.0392912	+	0.999228i	$0.512510\pi$
$752$	22.6656	0.826529
$753$	0	0
$754$	−3.27336	−0.119209
$755$	23.6155	0.859457
$756$	0	0
$757$	−32.0577	−1.16516	−0.582579	−	0.812774i	$-0.697956\pi$
$757$	−32.0577	−1.16516	−0.582579	+	0.812774i	$0.697956\pi$
$758$	−11.4217	−0.414856
$759$	0	0
$760$	−8.79007	−0.318849
$761$	−0.669651	−0.0242748	−0.0121374	−	0.999926i	$-0.503864\pi$
$761$	−0.669651	−0.0242748	−0.0121374	+	0.999926i	$0.503864\pi$
$762$	0	0
$763$	−11.4333	−0.413912
$764$	−1.19480	−0.0432264
$765$	0	0
$766$	3.51890	0.127143
$767$	10.8855	0.393052
$768$	0	0
$769$	43.5115	1.56906	0.784532	−	0.620089i	$-0.212903\pi$
$769$	43.5115	1.56906	0.784532	+	0.620089i	$0.212903\pi$
$770$	−2.25892	−0.0814057
$771$	0	0
$772$	−2.58776	−0.0931355
$773$	−21.0046	−0.755482	−0.377741	−	0.925911i	$-0.623299\pi$
$773$	−21.0046	−0.755482	−0.377741	+	0.925911i	$0.623299\pi$
$774$	0	0
$775$	2.72841	0.0980074
$776$	−17.8477	−0.640694
$777$	0	0
$778$	−0.00996438	−0.000357240 0
$779$	5.78005	0.207092
$780$	0	0
$781$	0.906493	0.0324369
$782$	0.297446	0.0106367
$783$	0	0
$784$	3.35950	0.119982
$785$	27.1794	0.970075
$786$	0	0
$787$	22.1975	0.791255	0.395627	−	0.918411i	$-0.370527\pi$
$787$	22.1975	0.791255	0.395627	+	0.918411i	$0.370527\pi$
$788$	46.8302	1.66826
$789$	0	0
$790$	9.44246	0.335948
$791$	3.14914	0.111970
$792$	0	0
$793$	−1.49440	−0.0530675
$794$	6.75113	0.239589
$795$	0	0
$796$	−2.73233	−0.0968447
$797$	−22.8502	−0.809397	−0.404699	−	0.914450i	$-0.632624\pi$
$797$	−22.8502	−0.809397	−0.404699	+	0.914450i	$0.632624\pi$
$798$	0	0
$799$	−6.08620	−0.215314
$800$	9.12991	0.322791
$801$	0	0
$802$	12.0121	0.424161
$803$	−13.3820	−0.472241
$804$	0	0
$805$	2.73589	0.0964276
$806$	0.537623	0.0189370
$807$	0	0
$808$	2.32878	0.0819260
$809$	38.0621	1.33819	0.669097	−	0.743175i	$-0.266681\pi$
$809$	38.0621	1.33819	0.669097	+	0.743175i	$0.266681\pi$
$810$	0	0
$811$	21.0208	0.738142	0.369071	−	0.929401i	$-0.379676\pi$
$811$	21.0208	0.738142	0.369071	+	0.929401i	$0.379676\pi$
$812$	−12.6426	−0.443667
$813$	0	0
$814$	8.18248	0.286796
$815$	40.2400	1.40955
$816$	0	0
$817$	14.6082	0.511077
$818$	12.3336	0.431234
$819$	0	0
$820$	−11.9437	−0.417094
$821$	10.4132	0.363425	0.181712	−	0.983352i	$-0.441836\pi$
$821$	10.4132	0.363425	0.181712	+	0.983352i	$0.441836\pi$
$822$	0	0
$823$	30.0396	1.04711	0.523557	−	0.851991i	$-0.324605\pi$
$823$	30.0396	1.04711	0.523557	+	0.851991i	$0.324605\pi$
$824$	21.3765	0.744684
$825$	0	0
$826$	−2.41681	−0.0840916
$827$	−28.6978	−0.997920	−0.498960	−	0.866625i	$-0.666285\pi$
$827$	−28.6978	−0.997920	−0.498960	+	0.866625i	$0.666285\pi$
$828$	0	0
$829$	16.4384	0.570931	0.285465	−	0.958389i	$-0.407852\pi$
$829$	16.4384	0.570931	0.285465	+	0.958389i	$0.407852\pi$
$830$	−6.85546	−0.237957
$831$	0	0
$832$	−8.17946	−0.283572
$833$	−0.902098	−0.0312559
$834$	0	0
$835$	14.9771	0.518304
$836$	−11.8590	−0.410152
$837$	0	0
$838$	12.2664	0.423736
$839$	12.5437	0.433055	0.216528	−	0.976277i	$-0.430527\pi$
$839$	12.5437	0.433055	0.216528	+	0.976277i	$0.430527\pi$
$840$	0	0
$841$	15.6847	0.540850
$842$	9.11123	0.313994
$843$	0	0
$844$	43.6866	1.50375
$845$	−29.5324	−1.01595
$846$	0	0
$847$	4.72964	0.162512
$848$	16.5169	0.567194
$849$	0	0
$850$	−0.739188	−0.0253539
$851$	−9.91023	−0.339718
$852$	0	0
$853$	−22.0613	−0.755365	−0.377683	−	0.925935i	$-0.623279\pi$
$853$	−22.0613	−0.755365	−0.377683	+	0.925935i	$0.623279\pi$
$854$	0.331788	0.0113535
$855$	0	0
$856$	2.15672	0.0737151
$857$	−6.15359	−0.210203	−0.105101	−	0.994462i	$-0.533517\pi$
$857$	−6.15359	−0.210203	−0.105101	+	0.994462i	$0.533517\pi$
$858$	0	0
$859$	23.0698	0.787131	0.393566	−	0.919296i	$-0.371241\pi$
$859$	23.0698	0.787131	0.393566	+	0.919296i	$0.371241\pi$
$860$	−30.1860	−1.02934
$861$	0	0
$862$	−11.5627	−0.393828
$863$	5.22145	0.177740	0.0888702	−	0.996043i	$-0.471674\pi$
$863$	5.22145	0.177740	0.0888702	+	0.996043i	$0.471674\pi$
$864$	0	0
$865$	−49.7186	−1.69048
$866$	−0.332061	−0.0112839
$867$	0	0
$868$	2.07644	0.0704789
$869$	26.2106	0.889135
$870$	0	0
$871$	16.7712	0.568271
$872$	−14.6696	−0.496774
$873$	0	0
$874$	−0.825659	−0.0279283
$875$	6.88046	0.232602
$876$	0	0
$877$	49.8716	1.68404	0.842022	−	0.539443i	$-0.181365\pi$
$877$	49.8716	1.68404	0.842022	+	0.539443i	$0.181365\pi$
$878$	−1.33424	−0.0450284
$879$	0	0
$880$	23.0155	0.775852
$881$	20.3054	0.684107	0.342053	−	0.939681i	$-0.388878\pi$
$881$	20.3054	0.684107	0.342053	+	0.939681i	$0.388878\pi$
$882$	0	0
$883$	−50.3500	−1.69441	−0.847206	−	0.531265i	$-0.821717\pi$
$883$	−50.3500	−1.69441	−0.847206	+	0.531265i	$0.821717\pi$
$884$	2.53378	0.0852203
$885$	0	0
$886$	9.69224	0.325617
$887$	−47.2694	−1.58715	−0.793576	−	0.608471i	$-0.791783\pi$
$887$	−47.2694	−1.58715	−0.793576	+	0.608471i	$0.791783\pi$
$888$	0	0
$889$	−0.449266	−0.0150679
$890$	−9.58434	−0.321268
$891$	0	0
$892$	35.2718	1.18099
$893$	16.8942	0.565344
$894$	0	0
$895$	−27.8679	−0.931522
$896$	9.16370	0.306138
$897$	0	0
$898$	1.60784	0.0536543
$899$	−7.33910	−0.244773
$900$	0	0
$901$	−4.43515	−0.147756
$902$	1.90584	0.0634575
$903$	0	0
$904$	4.04053	0.134386
$905$	36.9855	1.22944
$906$	0	0
$907$	−5.65035	−0.187617	−0.0938084	−	0.995590i	$-0.529904\pi$
$907$	−5.65035	−0.187617	−0.0938084	+	0.995590i	$0.529904\pi$
$908$	39.0236	1.29504
$909$	0	0
$910$	−1.33972	−0.0444112
$911$	−18.2196	−0.603641	−0.301820	−	0.953365i	$-0.597594\pi$
$911$	−18.2196	−0.603641	−0.301820	+	0.953365i	$0.597594\pi$
$912$	0	0
$913$	−19.0296	−0.629787
$914$	−11.7123	−0.387409
$915$	0	0
$916$	−28.2829	−0.934495
$917$	−19.9759	−0.659661
$918$	0	0
$919$	7.79561	0.257154	0.128577	−	0.991700i	$-0.458959\pi$
$919$	7.79561	0.257154	0.128577	+	0.991700i	$0.458959\pi$
$920$	3.51032	0.115732
$921$	0	0
$922$	6.24642	0.205715
$923$	0.537623	0.0176961
$924$	0	0
$925$	24.6281	0.809766
$926$	4.67992	0.153792
$927$	0	0
$928$	−24.5584	−0.806168
$929$	−11.5912	−0.380293	−0.190147	−	0.981756i	$-0.560896\pi$
$929$	−11.5912	−0.380293	−0.190147	+	0.981756i	$0.560896\pi$
$930$	0	0
$931$	2.50407	0.0820675
$932$	53.1548	1.74114
$933$	0	0
$934$	10.2332	0.334842
$935$	−6.18015	−0.202113
$936$	0	0
$937$	−32.0050	−1.04556	−0.522778	−	0.852469i	$-0.675105\pi$
$937$	−32.0050	−1.04556	−0.522778	+	0.852469i	$0.675105\pi$
$938$	−3.72357	−0.121579
$939$	0	0
$940$	−34.9098	−1.13863
$941$	−11.1090	−0.362141	−0.181071	−	0.983470i	$-0.557956\pi$
$941$	−11.1090	−0.362141	−0.181071	+	0.983470i	$0.557956\pi$
$942$	0	0
$943$	−2.30826	−0.0751674
$944$	24.6242	0.801450
$945$	0	0
$946$	4.81673	0.156605
$947$	12.1262	0.394049	0.197025	−	0.980399i	$-0.436872\pi$
$947$	12.1262	0.394049	0.197025	+	0.980399i	$0.436872\pi$
$948$	0	0
$949$	−7.93661	−0.257633
$950$	2.05186	0.0665710
$951$	0	0
$952$	−1.15745	−0.0375131
$953$	−27.6645	−0.896140	−0.448070	−	0.893999i	$-0.647888\pi$
$953$	−27.6645	−0.896140	−0.448070	+	0.893999i	$0.647888\pi$
$954$	0	0
$955$	1.72838	0.0559291
$956$	48.8965	1.58142
$957$	0	0
$958$	−3.46919	−0.112084
$959$	0.270356	0.00873026
$960$	0	0
$961$	−29.7946	−0.961117
$962$	4.85287	0.156463
$963$	0	0
$964$	28.0498	0.903423
$965$	3.74341	0.120505
$966$	0	0
$967$	−56.1838	−1.80675	−0.903374	−	0.428853i	$-0.858918\pi$
$967$	−56.1838	−1.80675	−0.903374	+	0.428853i	$0.858918\pi$
$968$	6.06842	0.195046
$969$	0	0
$970$	12.5484	0.402905
$971$	43.9884	1.41166	0.705828	−	0.708383i	$-0.250575\pi$
$971$	43.9884	1.41166	0.705828	+	0.708383i	$0.250575\pi$
$972$	0	0
$973$	−4.42512	−0.141863
$974$	−10.4648	−0.335315
$975$	0	0
$976$	−3.38050	−0.108207
$977$	6.47804	0.207251	0.103625	−	0.994616i	$-0.466956\pi$
$977$	6.47804	0.207251	0.103625	+	0.994616i	$0.466956\pi$
$978$	0	0
$979$	−26.6044	−0.850282
$980$	−5.17434	−0.165288
$981$	0	0
$982$	10.5113	0.335430
$983$	−37.8961	−1.20870	−0.604349	−	0.796720i	$-0.706567\pi$
$983$	−37.8961	−1.20870	−0.604349	+	0.796720i	$0.706567\pi$
$984$	0	0
$985$	−67.7437	−2.15849
$986$	1.98833	0.0633212
$987$	0	0
$988$	−7.03334	−0.223760
$989$	−5.83380	−0.185504
$990$	0	0
$991$	53.3499	1.69472	0.847358	−	0.531022i	$-0.178192\pi$
$991$	53.3499	1.69472	0.847358	+	0.531022i	$0.178192\pi$
$992$	4.03351	0.128064
$993$	0	0
$994$	−0.119364	−0.00378599
$995$	3.95254	0.125304
$996$	0	0
$997$	24.8754	0.787813	0.393907	−	0.919150i	$-0.371123\pi$
$997$	24.8754	0.787813	0.393907	+	0.919150i	$0.371123\pi$
$998$	−13.0004	−0.411522
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1449.2.a.o.1.3		4
3.2	odd	2		483.2.a.j.1.2	✓	4
12.11	even	2		7728.2.a.ce.1.1		4
21.20	even	2		3381.2.a.x.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
483.2.a.j.1.2	✓	4	3.2	odd	2
1449.2.a.o.1.3		4	1.1	even	1	trivial
3381.2.a.x.1.2		4	21.20	even	2
7728.2.a.ce.1.1		4	12.11	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:	\( N \)	\(=\)	\( 1449 = 3^{2} \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1449.a (trivial)

\(f(q)\)	\(=\)	\(q+0.329727 q^{2} -1.89128 q^{4} +2.73589 q^{5} -1.00000 q^{7} -1.28306 q^{8} +O(q^{10})\)	\(q+0.329727 q^{2} -1.89128 q^{4} +2.73589 q^{5} -1.00000 q^{7} -1.28306 q^{8} +0.902098 q^{10} +2.50407 q^{11} +1.48511 q^{13} -0.329727 q^{14} +3.35950 q^{16} -0.902098 q^{17} +2.50407 q^{19} -5.17434 q^{20} +0.825659 q^{22} -1.00000 q^{23} +2.48511 q^{25} +0.489682 q^{26} +1.89128 q^{28} -6.68466 q^{29} +1.09790 q^{31} +3.67384 q^{32} -0.297446 q^{34} -2.73589 q^{35} +9.91023 q^{37} +0.825659 q^{38} -3.51032 q^{40} +2.30826 q^{41} +5.83380 q^{43} -4.73589 q^{44} -0.329727 q^{46} +6.74671 q^{47} +1.00000 q^{49} +0.819409 q^{50} -2.80877 q^{52} +4.91648 q^{53} +6.85086 q^{55} +1.28306 q^{56} -2.20411 q^{58} +7.32973 q^{59} -1.00625 q^{61} +0.362008 q^{62} -5.50764 q^{64} +4.06311 q^{65} +11.2929 q^{67} +1.70612 q^{68} -0.902098 q^{70} +0.362008 q^{71} -5.34411 q^{73} +3.26767 q^{74} -4.73589 q^{76} -2.50407 q^{77} +10.4672 q^{79} +9.19123 q^{80} +0.761098 q^{82} -7.59946 q^{83} -2.46805 q^{85} +1.92356 q^{86} -3.21287 q^{88} -10.6245 q^{89} -1.48511 q^{91} +1.89128 q^{92} +2.22457 q^{94} +6.85086 q^{95} +13.9102 q^{97} +0.329727 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 2 q^{2} + 4 q^{4} - 5 q^{5} - 4 q^{7} - 9 q^{8}+O(q^{10}) \) 4 * q - 2 * q^2 + 4 * q^4 - 5 * q^5 - 4 * q^7 - 9 * q^8	\( 4 q - 2 q^{2} + 4 q^{4} - 5 q^{5} - 4 q^{7} - 9 q^{8} + 2 q^{10} - q^{11} + 7 q^{13} + 2 q^{14} + 8 q^{16} - 2 q^{17} - q^{19} - 13 q^{20} + 11 q^{22} - 4 q^{23} + 11 q^{25} + 19 q^{26} - 4 q^{28} - 2 q^{29} + 6 q^{31} - 20 q^{32} + 23 q^{34} + 5 q^{35} + 16 q^{37} + 11 q^{38} + 3 q^{40} - 5 q^{41} + 9 q^{43} - 3 q^{44} + 2 q^{46} + 21 q^{47} + 4 q^{49} + 17 q^{50} - 24 q^{52} - 10 q^{53} + 17 q^{55} + 9 q^{56} + q^{58} + 26 q^{59} + 2 q^{61} + 19 q^{62} + 27 q^{64} - 26 q^{65} + 5 q^{67} - 7 q^{68} - 2 q^{70} + 19 q^{71} + 10 q^{73} - 9 q^{74} - 3 q^{76} + q^{77} - 6 q^{79} + 24 q^{80} - 31 q^{82} + 2 q^{83} - 7 q^{85} + 17 q^{86} - 20 q^{88} + 17 q^{89} - 7 q^{91} - 4 q^{92} - 44 q^{94} + 17 q^{95} + 32 q^{97} - 2 q^{98}+O(q^{100}) \) 4 * q - 2 * q^2 + 4 * q^4 - 5 * q^5 - 4 * q^7 - 9 * q^8 + 2 * q^10 - q^11 + 7 * q^13 + 2 * q^14 + 8 * q^16 - 2 * q^17 - q^19 - 13 * q^20 + 11 * q^22 - 4 * q^23 + 11 * q^25 + 19 * q^26 - 4 * q^28 - 2 * q^29 + 6 * q^31 - 20 * q^32 + 23 * q^34 + 5 * q^35 + 16 * q^37 + 11 * q^38 + 3 * q^40 - 5 * q^41 + 9 * q^43 - 3 * q^44 + 2 * q^46 + 21 * q^47 + 4 * q^49 + 17 * q^50 - 24 * q^52 - 10 * q^53 + 17 * q^55 + 9 * q^56 + q^58 + 26 * q^59 + 2 * q^61 + 19 * q^62 + 27 * q^64 - 26 * q^65 + 5 * q^67 - 7 * q^68 - 2 * q^70 + 19 * q^71 + 10 * q^73 - 9 * q^74 - 3 * q^76 + q^77 - 6 * q^79 + 24 * q^80 - 31 * q^82 + 2 * q^83 - 7 * q^85 + 17 * q^86 - 20 * q^88 + 17 * q^89 - 7 * q^91 - 4 * q^92 - 44 * q^94 + 17 * q^95 + 32 * q^97 - 2 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more