LMFDB - Embedded newform 2576.2.a.y.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2576,2,Mod(1,2576)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2576, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("2576.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2576 = 2^{4} \cdot 7 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2576.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:	$20.5694635607$
Analytic rank:	$0$
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:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1288)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.17009$ of defining polynomial
Character	$\chi$	$=$	2576.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.539189 q^{3} -2.87936 q^{5} +1.00000 q^{7} -2.70928 q^{9} +O(q^{10})$	$q+0.539189 q^{3} -2.87936 q^{5} +1.00000 q^{7} -2.70928 q^{9} +2.63090 q^{11} -4.34017 q^{13} -1.55252 q^{15} +7.80098 q^{17} +0.539189 q^{21} -1.00000 q^{23} +3.29072 q^{25} -3.07838 q^{27} +5.70928 q^{29} -10.1412 q^{31} +1.41855 q^{33} -2.87936 q^{35} +3.65983 q^{37} -2.34017 q^{39} -8.83710 q^{41} +8.49693 q^{43} +7.80098 q^{45} -3.80098 q^{47} +1.00000 q^{49} +4.20620 q^{51} +4.83710 q^{53} -7.57531 q^{55} +8.72261 q^{59} +12.4547 q^{61} -2.70928 q^{63} +12.4969 q^{65} +11.3112 q^{67} -0.539189 q^{69} -2.15676 q^{71} +5.23513 q^{73} +1.77432 q^{75} +2.63090 q^{77} +2.04945 q^{79} +6.46800 q^{81} +6.34017 q^{83} -22.4619 q^{85} +3.07838 q^{87} +8.69594 q^{89} -4.34017 q^{91} -5.46800 q^{93} +11.1929 q^{97} -7.12783 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 4 q^{5} + 3 q^{7} - q^{9}+O(q^{10}) $ 3 * q + 4 * q^5 + 3 * q^7 - q^9	$ 3 q + 4 q^{5} + 3 q^{7} - q^{9} + 4 q^{11} - 2 q^{13} - 4 q^{15} + 14 q^{17} - 3 q^{23} + 17 q^{25} - 6 q^{27} + 10 q^{29} - 10 q^{31} - 10 q^{33} + 4 q^{35} + 22 q^{37} + 4 q^{39} + 2 q^{41} + 8 q^{43} + 14 q^{45} - 2 q^{47} + 3 q^{49} - 12 q^{51} - 14 q^{53} - 2 q^{55} + 20 q^{59} + 4 q^{61} - q^{63} + 20 q^{65} + 8 q^{67} + 6 q^{73} - 6 q^{75} + 4 q^{77} - 12 q^{79} - 13 q^{81} + 8 q^{83} + 6 q^{87} + 18 q^{89} - 2 q^{91} + 16 q^{93} + 8 q^{97}+O(q^{100}) $ 3 * q + 4 * q^5 + 3 * q^7 - q^9 + 4 * q^11 - 2 * q^13 - 4 * q^15 + 14 * q^17 - 3 * q^23 + 17 * q^25 - 6 * q^27 + 10 * q^29 - 10 * q^31 - 10 * q^33 + 4 * q^35 + 22 * q^37 + 4 * q^39 + 2 * q^41 + 8 * q^43 + 14 * q^45 - 2 * q^47 + 3 * q^49 - 12 * q^51 - 14 * q^53 - 2 * q^55 + 20 * q^59 + 4 * q^61 - q^63 + 20 * q^65 + 8 * q^67 + 6 * q^73 - 6 * q^75 + 4 * q^77 - 12 * q^79 - 13 * q^81 + 8 * q^83 + 6 * q^87 + 18 * q^89 - 2 * q^91 + 16 * q^93 + 8 * q^97

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	0.539189	0.311301	0.155650	−	0.987812i	$-0.450253\pi$
$3$	0.539189	0.311301	0.155650	+	0.987812i	$0.450253\pi$
$4$	0	0
$5$	−2.87936	−1.28769	−0.643845	−	0.765156i	$-0.722662\pi$
$5$	−2.87936	−1.28769	−0.643845	+	0.765156i	$0.722662\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	−2.70928	−0.903092
$10$	0	0
$11$	2.63090	0.793245	0.396623	−	0.917982i	$-0.370182\pi$
$11$	2.63090	0.793245	0.396623	+	0.917982i	$0.370182\pi$
$12$	0	0
$13$	−4.34017	−1.20375	−0.601874	−	0.798591i	$-0.705579\pi$
$13$	−4.34017	−1.20375	−0.601874	+	0.798591i	$0.705579\pi$
$14$	0	0
$15$	−1.55252	−0.400859
$16$	0	0
$17$	7.80098	1.89202	0.946008	−	0.324142i	$-0.105076\pi$
$17$	7.80098	1.89202	0.946008	+	0.324142i	$0.105076\pi$
$18$	0	0
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	0	0
$21$	0.539189	0.117661
$22$	0	0
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	3.29072	0.658145
$26$	0	0
$27$	−3.07838	−0.592434
$28$	0	0
$29$	5.70928	1.06019	0.530093	−	0.847940i	$-0.322157\pi$
$29$	5.70928	1.06019	0.530093	+	0.847940i	$0.322157\pi$
$30$	0	0
$31$	−10.1412	−1.82141	−0.910703	−	0.413062i	$-0.864459\pi$
$31$	−10.1412	−1.82141	−0.910703	+	0.413062i	$0.864459\pi$
$32$	0	0
$33$	1.41855	0.246938
$34$	0	0
$35$	−2.87936	−0.486701
$36$	0	0
$37$	3.65983	0.601672	0.300836	−	0.953676i	$-0.402734\pi$
$37$	3.65983	0.601672	0.300836	+	0.953676i	$0.402734\pi$
$38$	0	0
$39$	−2.34017	−0.374728
$40$	0	0
$41$	−8.83710	−1.38012	−0.690062	−	0.723751i	$-0.742417\pi$
$41$	−8.83710	−1.38012	−0.690062	+	0.723751i	$0.742417\pi$
$42$	0	0
$43$	8.49693	1.29577	0.647885	−	0.761738i	$-0.275654\pi$
$43$	8.49693	1.29577	0.647885	+	0.761738i	$0.275654\pi$
$44$	0	0
$45$	7.80098	1.16290
$46$	0	0
$47$	−3.80098	−0.554431	−0.277215	−	0.960808i	$-0.589412\pi$
$47$	−3.80098	−0.554431	−0.277215	+	0.960808i	$0.589412\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	4.20620	0.588986
$52$	0	0
$53$	4.83710	0.664427	0.332213	−	0.943204i	$-0.392205\pi$
$53$	4.83710	0.664427	0.332213	+	0.943204i	$0.392205\pi$
$54$	0	0
$55$	−7.57531	−1.02145
$56$	0	0
$57$	0	0
$58$	0	0
$59$	8.72261	1.13559	0.567793	−	0.823171i	$-0.307797\pi$
$59$	8.72261	1.13559	0.567793	+	0.823171i	$0.307797\pi$
$60$	0	0
$61$	12.4547	1.59466	0.797328	−	0.603546i	$-0.206246\pi$
$61$	12.4547	1.59466	0.797328	+	0.603546i	$0.206246\pi$
$62$	0	0
$63$	−2.70928	−0.341337
$64$	0	0
$65$	12.4969	1.55005
$66$	0	0
$67$	11.3112	1.38189	0.690944	−	0.722908i	$-0.257195\pi$
$67$	11.3112	1.38189	0.690944	+	0.722908i	$0.257195\pi$
$68$	0	0
$69$	−0.539189	−0.0649107
$70$	0	0
$71$	−2.15676	−0.255960	−0.127980	−	0.991777i	$-0.540849\pi$
$71$	−2.15676	−0.255960	−0.127980	+	0.991777i	$0.540849\pi$
$72$	0	0
$73$	5.23513	0.612726	0.306363	−	0.951915i	$-0.400888\pi$
$73$	5.23513	0.612726	0.306363	+	0.951915i	$0.400888\pi$
$74$	0	0
$75$	1.77432	0.204881
$76$	0	0
$77$	2.63090	0.299819
$78$	0	0
$79$	2.04945	0.230581	0.115290	−	0.993332i	$-0.463220\pi$
$79$	2.04945	0.230581	0.115290	+	0.993332i	$0.463220\pi$
$80$	0	0
$81$	6.46800	0.718667
$82$	0	0
$83$	6.34017	0.695924	0.347962	−	0.937509i	$-0.386874\pi$
$83$	6.34017	0.695924	0.347962	+	0.937509i	$0.386874\pi$
$84$	0	0
$85$	−22.4619	−2.43633
$86$	0	0
$87$	3.07838	0.330037
$88$	0	0
$89$	8.69594	0.921768	0.460884	−	0.887460i	$-0.347532\pi$
$89$	8.69594	0.921768	0.460884	+	0.887460i	$0.347532\pi$
$90$	0	0
$91$	−4.34017	−0.454974
$92$	0	0
$93$	−5.46800	−0.567005
$94$	0	0
$95$	0	0
$96$	0	0
$97$	11.1929	1.13646	0.568232	−	0.822868i	$-0.307628\pi$
$97$	11.1929	1.13646	0.568232	+	0.822868i	$0.307628\pi$
$98$	0	0
$99$	−7.12783	−0.716373
$100$	0	0
$101$	11.7587	1.17004	0.585018	−	0.811020i	$-0.301087\pi$
$101$	11.7587	1.17004	0.585018	+	0.811020i	$0.301087\pi$
$102$	0	0
$103$	8.68035	0.855300	0.427650	−	0.903944i	$-0.359342\pi$
$103$	8.68035	0.855300	0.427650	+	0.903944i	$0.359342\pi$
$104$	0	0
$105$	−1.55252	−0.151510
$106$	0	0
$107$	−10.6537	−1.02993	−0.514965	−	0.857211i	$-0.672195\pi$
$107$	−10.6537	−1.02993	−0.514965	+	0.857211i	$0.672195\pi$
$108$	0	0
$109$	3.44521	0.329992	0.164996	−	0.986294i	$-0.447239\pi$
$109$	3.44521	0.329992	0.164996	+	0.986294i	$0.447239\pi$
$110$	0	0
$111$	1.97334	0.187301
$112$	0	0
$113$	−4.15676	−0.391035	−0.195517	−	0.980700i	$-0.562639\pi$
$113$	−4.15676	−0.391035	−0.195517	+	0.980700i	$0.562639\pi$
$114$	0	0
$115$	2.87936	0.268502
$116$	0	0
$117$	11.7587	1.08709
$118$	0	0
$119$	7.80098	0.715115
$120$	0	0
$121$	−4.07838	−0.370762
$122$	0	0
$123$	−4.76487	−0.429634
$124$	0	0
$125$	4.92162	0.440203
$126$	0	0
$127$	1.65983	0.147286	0.0736429	−	0.997285i	$-0.476537\pi$
$127$	1.65983	0.147286	0.0736429	+	0.997285i	$0.476537\pi$
$128$	0	0
$129$	4.58145	0.403374
$130$	0	0
$131$	−17.3763	−1.51817	−0.759087	−	0.650989i	$-0.774354\pi$
$131$	−17.3763	−1.51817	−0.759087	+	0.650989i	$0.774354\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	8.86376	0.762871
$136$	0	0
$137$	9.02052	0.770675	0.385337	−	0.922776i	$-0.374085\pi$
$137$	9.02052	0.770675	0.385337	+	0.922776i	$0.374085\pi$
$138$	0	0
$139$	0.722606	0.0612907	0.0306453	−	0.999530i	$-0.490244\pi$
$139$	0.722606	0.0612907	0.0306453	+	0.999530i	$0.490244\pi$
$140$	0	0
$141$	−2.04945	−0.172595
$142$	0	0
$143$	−11.4186	−0.954867
$144$	0	0
$145$	−16.4391	−1.36519
$146$	0	0
$147$	0.539189	0.0444715
$148$	0	0
$149$	−6.99386	−0.572959	−0.286480	−	0.958086i	$-0.592485\pi$
$149$	−6.99386	−0.572959	−0.286480	+	0.958086i	$0.592485\pi$
$150$	0	0
$151$	21.8576	1.77875	0.889374	−	0.457180i	$-0.151141\pi$
$151$	21.8576	1.77875	0.889374	+	0.457180i	$0.151141\pi$
$152$	0	0
$153$	−21.1350	−1.70866
$154$	0	0
$155$	29.2001	2.34541
$156$	0	0
$157$	−2.53919	−0.202649	−0.101325	−	0.994853i	$-0.532308\pi$
$157$	−2.53919	−0.202649	−0.101325	+	0.994853i	$0.532308\pi$
$158$	0	0
$159$	2.60811	0.206837
$160$	0	0
$161$	−1.00000	−0.0788110
$162$	0	0
$163$	18.6225	1.45863	0.729313	−	0.684180i	$-0.239840\pi$
$163$	18.6225	1.45863	0.729313	+	0.684180i	$0.239840\pi$
$164$	0	0
$165$	−4.08452	−0.317980
$166$	0	0
$167$	−16.6114	−1.28543	−0.642715	−	0.766105i	$-0.722192\pi$
$167$	−16.6114	−1.28543	−0.642715	+	0.766105i	$0.722192\pi$
$168$	0	0
$169$	5.83710	0.449008
$170$	0	0
$171$	0	0
$172$	0	0
$173$	22.6803	1.72435	0.862177	−	0.506607i	$-0.169100\pi$
$173$	22.6803	1.72435	0.862177	+	0.506607i	$0.169100\pi$
$174$	0	0
$175$	3.29072	0.248755
$176$	0	0
$177$	4.70313	0.353509
$178$	0	0
$179$	−0.183417	−0.0137093	−0.00685463	−	0.999977i	$-0.502182\pi$
$179$	−0.183417	−0.0137093	−0.00685463	+	0.999977i	$0.502182\pi$
$180$	0	0
$181$	−21.8732	−1.62582	−0.812912	−	0.582387i	$-0.802119\pi$
$181$	−21.8732	−1.62582	−0.812912	+	0.582387i	$0.802119\pi$
$182$	0	0
$183$	6.71542	0.496418
$184$	0	0
$185$	−10.5380	−0.774767
$186$	0	0
$187$	20.5236	1.50083
$188$	0	0
$189$	−3.07838	−0.223919
$190$	0	0
$191$	−7.02052	−0.507987	−0.253993	−	0.967206i	$-0.581744\pi$
$191$	−7.02052	−0.507987	−0.253993	+	0.967206i	$0.581744\pi$
$192$	0	0
$193$	13.6020	0.979091	0.489546	−	0.871978i	$-0.337163\pi$
$193$	13.6020	0.979091	0.489546	+	0.871978i	$0.337163\pi$
$194$	0	0
$195$	6.73820	0.482533
$196$	0	0
$197$	0.523590	0.0373043	0.0186521	−	0.999826i	$-0.494062\pi$
$197$	0.523590	0.0373043	0.0186521	+	0.999826i	$0.494062\pi$
$198$	0	0
$199$	−12.6803	−0.898886	−0.449443	−	0.893309i	$-0.648377\pi$
$199$	−12.6803	−0.898886	−0.449443	+	0.893309i	$0.648377\pi$
$200$	0	0
$201$	6.09890	0.430183
$202$	0	0
$203$	5.70928	0.400713
$204$	0	0
$205$	25.4452	1.77717
$206$	0	0
$207$	2.70928	0.188308
$208$	0	0
$209$	0	0
$210$	0	0
$211$	4.97948	0.342802	0.171401	−	0.985201i	$-0.445171\pi$
$211$	4.97948	0.342802	0.171401	+	0.985201i	$0.445171\pi$
$212$	0	0
$213$	−1.16290	−0.0796805
$214$	0	0
$215$	−24.4657	−1.66855
$216$	0	0
$217$	−10.1412	−0.688427
$218$	0	0
$219$	2.82273	0.190742
$220$	0	0
$221$	−33.8576	−2.27751
$222$	0	0
$223$	−8.72261	−0.584109	−0.292054	−	0.956402i	$-0.594339\pi$
$223$	−8.72261	−0.584109	−0.292054	+	0.956402i	$0.594339\pi$
$224$	0	0
$225$	−8.91548	−0.594365
$226$	0	0
$227$	−13.2618	−0.880216	−0.440108	−	0.897945i	$-0.645060\pi$
$227$	−13.2618	−0.880216	−0.440108	+	0.897945i	$0.645060\pi$
$228$	0	0
$229$	22.7370	1.50250	0.751251	−	0.660017i	$-0.229451\pi$
$229$	22.7370	1.50250	0.751251	+	0.660017i	$0.229451\pi$
$230$	0	0
$231$	1.41855	0.0933338
$232$	0	0
$233$	10.3896	0.680647	0.340323	−	0.940308i	$-0.389463\pi$
$233$	10.3896	0.680647	0.340323	+	0.940308i	$0.389463\pi$
$234$	0	0
$235$	10.9444	0.713934
$236$	0	0
$237$	1.10504	0.0717800
$238$	0	0
$239$	7.02052	0.454120	0.227060	−	0.973881i	$-0.427089\pi$
$239$	7.02052	0.454120	0.227060	+	0.973881i	$0.427089\pi$
$240$	0	0
$241$	−2.35577	−0.151749	−0.0758743	−	0.997117i	$-0.524175\pi$
$241$	−2.35577	−0.151749	−0.0758743	+	0.997117i	$0.524175\pi$
$242$	0	0
$243$	12.7226	0.816156
$244$	0	0
$245$	−2.87936	−0.183956
$246$	0	0
$247$	0	0
$248$	0	0
$249$	3.41855	0.216642
$250$	0	0
$251$	−22.3545	−1.41101	−0.705503	−	0.708707i	$-0.749279\pi$
$251$	−22.3545	−1.41101	−0.705503	+	0.708707i	$0.749279\pi$
$252$	0	0
$253$	−2.63090	−0.165403
$254$	0	0
$255$	−12.1112	−0.758432
$256$	0	0
$257$	−8.52359	−0.531687	−0.265843	−	0.964016i	$-0.585650\pi$
$257$	−8.52359	−0.531687	−0.265843	+	0.964016i	$0.585650\pi$
$258$	0	0
$259$	3.65983	0.227411
$260$	0	0
$261$	−15.4680	−0.957445
$262$	0	0
$263$	−31.4101	−1.93683	−0.968416	−	0.249340i	$-0.919786\pi$
$263$	−31.4101	−1.93683	−0.968416	+	0.249340i	$0.919786\pi$
$264$	0	0
$265$	−13.9278	−0.855576
$266$	0	0
$267$	4.68876	0.286947
$268$	0	0
$269$	−8.83710	−0.538808	−0.269404	−	0.963027i	$-0.586827\pi$
$269$	−8.83710	−0.538808	−0.269404	+	0.963027i	$0.586827\pi$
$270$	0	0
$271$	26.7103	1.62254	0.811268	−	0.584674i	$-0.198778\pi$
$271$	26.7103	1.62254	0.811268	+	0.584674i	$0.198778\pi$
$272$	0	0
$273$	−2.34017	−0.141634
$274$	0	0
$275$	8.65756	0.522070
$276$	0	0
$277$	−14.0761	−0.845752	−0.422876	−	0.906188i	$-0.638979\pi$
$277$	−14.0761	−0.845752	−0.422876	+	0.906188i	$0.638979\pi$
$278$	0	0
$279$	27.4752	1.64490
$280$	0	0
$281$	−3.39189	−0.202343	−0.101172	−	0.994869i	$-0.532259\pi$
$281$	−3.39189	−0.202343	−0.101172	+	0.994869i	$0.532259\pi$
$282$	0	0
$283$	0.214614	0.0127575	0.00637875	−	0.999980i	$-0.497970\pi$
$283$	0.214614	0.0127575	0.00637875	+	0.999980i	$0.497970\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−8.83710	−0.521638
$288$	0	0
$289$	43.8554	2.57973
$290$	0	0
$291$	6.03507	0.353782
$292$	0	0
$293$	9.53305	0.556926	0.278463	−	0.960447i	$-0.410175\pi$
$293$	9.53305	0.556926	0.278463	+	0.960447i	$0.410175\pi$
$294$	0	0
$295$	−25.1155	−1.46228
$296$	0	0
$297$	−8.09890	−0.469946
$298$	0	0
$299$	4.34017	0.250999
$300$	0	0
$301$	8.49693	0.489755
$302$	0	0
$303$	6.34017	0.364233
$304$	0	0
$305$	−35.8615	−2.05342
$306$	0	0
$307$	12.9639	0.739888	0.369944	−	0.929054i	$-0.379377\pi$
$307$	12.9639	0.739888	0.369944	+	0.929054i	$0.379377\pi$
$308$	0	0
$309$	4.68035	0.266256
$310$	0	0
$311$	−20.2401	−1.14771	−0.573854	−	0.818958i	$-0.694552\pi$
$311$	−20.2401	−1.14771	−0.573854	+	0.818958i	$0.694552\pi$
$312$	0	0
$313$	−34.6525	−1.95867	−0.979336	−	0.202238i	$-0.935179\pi$
$313$	−34.6525	−1.95867	−0.979336	+	0.202238i	$0.935179\pi$
$314$	0	0
$315$	7.80098	0.439536
$316$	0	0
$317$	−29.5402	−1.65914	−0.829572	−	0.558399i	$-0.811416\pi$
$317$	−29.5402	−1.65914	−0.829572	+	0.558399i	$0.811416\pi$
$318$	0	0
$319$	15.0205	0.840988
$320$	0	0
$321$	−5.74435	−0.320618
$322$	0	0
$323$	0	0
$324$	0	0
$325$	−14.2823	−0.792240
$326$	0	0
$327$	1.85762	0.102727
$328$	0	0
$329$	−3.80098	−0.209555
$330$	0	0
$331$	4.08452	0.224506	0.112253	−	0.993680i	$-0.464193\pi$
$331$	4.08452	0.224506	0.112253	+	0.993680i	$0.464193\pi$
$332$	0	0
$333$	−9.91548	−0.543365
$334$	0	0
$335$	−32.5692	−1.77944
$336$	0	0
$337$	29.9155	1.62960	0.814800	−	0.579742i	$-0.196847\pi$
$337$	29.9155	1.62960	0.814800	+	0.579742i	$0.196847\pi$
$338$	0	0
$339$	−2.24128	−0.121729
$340$	0	0
$341$	−26.6803	−1.44482
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	1.55252	0.0835849
$346$	0	0
$347$	12.6947	0.681488	0.340744	−	0.940156i	$-0.389321\pi$
$347$	12.6947	0.681488	0.340744	+	0.940156i	$0.389321\pi$
$348$	0	0
$349$	8.62249	0.461551	0.230776	−	0.973007i	$-0.425874\pi$
$349$	8.62249	0.461551	0.230776	+	0.973007i	$0.425874\pi$
$350$	0	0
$351$	13.3607	0.713141
$352$	0	0
$353$	15.9733	0.850175	0.425087	−	0.905152i	$-0.360243\pi$
$353$	15.9733	0.850175	0.425087	+	0.905152i	$0.360243\pi$
$354$	0	0
$355$	6.21008	0.329597
$356$	0	0
$357$	4.20620	0.222616
$358$	0	0
$359$	6.63090	0.349965	0.174983	−	0.984572i	$-0.444013\pi$
$359$	6.63090	0.349965	0.174983	+	0.984572i	$0.444013\pi$
$360$	0	0
$361$	−19.0000	−1.00000
$362$	0	0
$363$	−2.19902	−0.115418
$364$	0	0
$365$	−15.0738	−0.789001
$366$	0	0
$367$	−30.3402	−1.58374	−0.791872	−	0.610687i	$-0.790893\pi$
$367$	−30.3402	−1.58374	−0.791872	+	0.610687i	$0.790893\pi$
$368$	0	0
$369$	23.9421	1.24638
$370$	0	0
$371$	4.83710	0.251130
$372$	0	0
$373$	32.0410	1.65902	0.829511	−	0.558490i	$-0.188619\pi$
$373$	32.0410	1.65902	0.829511	+	0.558490i	$0.188619\pi$
$374$	0	0
$375$	2.65368	0.137036
$376$	0	0
$377$	−24.7792	−1.27620
$378$	0	0
$379$	−11.0966	−0.569996	−0.284998	−	0.958528i	$-0.591993\pi$
$379$	−11.0966	−0.569996	−0.284998	+	0.958528i	$0.591993\pi$
$380$	0	0
$381$	0.894960	0.0458502
$382$	0	0
$383$	−12.5503	−0.641288	−0.320644	−	0.947200i	$-0.603899\pi$
$383$	−12.5503	−0.641288	−0.320644	+	0.947200i	$0.603899\pi$
$384$	0	0
$385$	−7.57531	−0.386073
$386$	0	0
$387$	−23.0205	−1.17020
$388$	0	0
$389$	−0.0722347	−0.00366244	−0.00183122	−	0.999998i	$-0.500583\pi$
$389$	−0.0722347	−0.00366244	−0.00183122	+	0.999998i	$0.500583\pi$
$390$	0	0
$391$	−7.80098	−0.394513
$392$	0	0
$393$	−9.36910	−0.472609
$394$	0	0
$395$	−5.90110	−0.296917
$396$	0	0
$397$	−0.707008	−0.0354837	−0.0177419	−	0.999843i	$-0.505648\pi$
$397$	−0.707008	−0.0354837	−0.0177419	+	0.999843i	$0.505648\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	22.5958	1.12838	0.564191	−	0.825644i	$-0.309188\pi$
$401$	22.5958	1.12838	0.564191	+	0.825644i	$0.309188\pi$
$402$	0	0
$403$	44.0144	2.19251
$404$	0	0
$405$	−18.6237	−0.925420
$406$	0	0
$407$	9.62863	0.477273
$408$	0	0
$409$	−10.4969	−0.519040	−0.259520	−	0.965738i	$-0.583564\pi$
$409$	−10.4969	−0.519040	−0.259520	+	0.965738i	$0.583564\pi$
$410$	0	0
$411$	4.86376	0.239912
$412$	0	0
$413$	8.72261	0.429211
$414$	0	0
$415$	−18.2557	−0.896135
$416$	0	0
$417$	0.389621	0.0190798
$418$	0	0
$419$	39.7152	1.94022	0.970108	−	0.242673i	$-0.0780241\pi$
$419$	39.7152	1.94022	0.970108	+	0.242673i	$0.0780241\pi$
$420$	0	0
$421$	18.8781	0.920064	0.460032	−	0.887902i	$-0.347838\pi$
$421$	18.8781	0.920064	0.460032	+	0.887902i	$0.347838\pi$
$422$	0	0
$423$	10.2979	0.500702
$424$	0	0
$425$	25.6709	1.24522
$426$	0	0
$427$	12.4547	0.602724
$428$	0	0
$429$	−6.15676	−0.297251
$430$	0	0
$431$	15.2039	0.732348	0.366174	−	0.930546i	$-0.380668\pi$
$431$	15.2039	0.732348	0.366174	+	0.930546i	$0.380668\pi$
$432$	0	0
$433$	−3.17850	−0.152749	−0.0763744	−	0.997079i	$-0.524334\pi$
$433$	−3.17850	−0.152749	−0.0763744	+	0.997079i	$0.524334\pi$
$434$	0	0
$435$	−8.86376	−0.424985
$436$	0	0
$437$	0	0
$438$	0	0
$439$	10.8937	0.519930	0.259965	−	0.965618i	$-0.416289\pi$
$439$	10.8937	0.519930	0.259965	+	0.965618i	$0.416289\pi$
$440$	0	0
$441$	−2.70928	−0.129013
$442$	0	0
$443$	−24.7792	−1.17730	−0.588649	−	0.808389i	$-0.700340\pi$
$443$	−24.7792	−1.17730	−0.588649	+	0.808389i	$0.700340\pi$
$444$	0	0
$445$	−25.0388	−1.18695
$446$	0	0
$447$	−3.77101	−0.178363
$448$	0	0
$449$	−19.8660	−0.937536	−0.468768	−	0.883321i	$-0.655302\pi$
$449$	−19.8660	−0.937536	−0.468768	+	0.883321i	$0.655302\pi$
$450$	0	0
$451$	−23.2495	−1.09478
$452$	0	0
$453$	11.7854	0.553726
$454$	0	0
$455$	12.4969	0.585865
$456$	0	0
$457$	27.0928	1.26735	0.633673	−	0.773601i	$-0.281547\pi$
$457$	27.0928	1.26735	0.633673	+	0.773601i	$0.281547\pi$
$458$	0	0
$459$	−24.0144	−1.12090
$460$	0	0
$461$	−5.88428	−0.274058	−0.137029	−	0.990567i	$-0.543755\pi$
$461$	−5.88428	−0.274058	−0.137029	+	0.990567i	$0.543755\pi$
$462$	0	0
$463$	34.4391	1.60052	0.800260	−	0.599654i	$-0.204695\pi$
$463$	34.4391	1.60052	0.800260	+	0.599654i	$0.204695\pi$
$464$	0	0
$465$	15.7443	0.730127
$466$	0	0
$467$	−15.2039	−0.703554	−0.351777	−	0.936084i	$-0.614423\pi$
$467$	−15.2039	−0.703554	−0.351777	+	0.936084i	$0.614423\pi$
$468$	0	0
$469$	11.3112	0.522305
$470$	0	0
$471$	−1.36910	−0.0630849
$472$	0	0
$473$	22.3545	1.02786
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−13.1050	−0.600039
$478$	0	0
$479$	26.4534	1.20869	0.604344	−	0.796723i	$-0.293435\pi$
$479$	26.4534	1.20869	0.604344	+	0.796723i	$0.293435\pi$
$480$	0	0
$481$	−15.8843	−0.724261
$482$	0	0
$483$	−0.539189	−0.0245339
$484$	0	0
$485$	−32.2283	−1.46341
$486$	0	0
$487$	−35.8310	−1.62366	−0.811828	−	0.583897i	$-0.801527\pi$
$487$	−35.8310	−1.62366	−0.811828	+	0.583897i	$0.801527\pi$
$488$	0	0
$489$	10.0410	0.454071
$490$	0	0
$491$	24.1978	1.09203	0.546016	−	0.837775i	$-0.316144\pi$
$491$	24.1978	1.09203	0.546016	+	0.837775i	$0.316144\pi$
$492$	0	0
$493$	44.5380	2.00589
$494$	0	0
$495$	20.5236	0.922467
$496$	0	0
$497$	−2.15676	−0.0967437
$498$	0	0
$499$	−22.7526	−1.01855	−0.509273	−	0.860605i	$-0.670086\pi$
$499$	−22.7526	−1.01855	−0.509273	+	0.860605i	$0.670086\pi$
$500$	0	0
$501$	−8.95669	−0.400156
$502$	0	0
$503$	−19.1050	−0.851852	−0.425926	−	0.904758i	$-0.640052\pi$
$503$	−19.1050	−0.851852	−0.425926	+	0.904758i	$0.640052\pi$
$504$	0	0
$505$	−33.8576	−1.50664
$506$	0	0
$507$	3.14730	0.139777
$508$	0	0
$509$	−6.58145	−0.291718	−0.145859	−	0.989305i	$-0.546595\pi$
$509$	−6.58145	−0.291718	−0.145859	+	0.989305i	$0.546595\pi$
$510$	0	0
$511$	5.23513	0.231589
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−24.9939	−1.10136
$516$	0	0
$517$	−10.0000	−0.439799
$518$	0	0
$519$	12.2290	0.536793
$520$	0	0
$521$	10.1990	0.446827	0.223413	−	0.974724i	$-0.428280\pi$
$521$	10.1990	0.446827	0.223413	+	0.974724i	$0.428280\pi$
$522$	0	0
$523$	6.37137	0.278601	0.139300	−	0.990250i	$-0.455515\pi$
$523$	6.37137	0.278601	0.139300	+	0.990250i	$0.455515\pi$
$524$	0	0
$525$	1.77432	0.0774378
$526$	0	0
$527$	−79.1110	−3.44613
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	−23.6319	−1.02554
$532$	0	0
$533$	38.3545	1.66132
$534$	0	0
$535$	30.6758	1.32623
$536$	0	0
$537$	−0.0988967	−0.00426771
$538$	0	0
$539$	2.63090	0.113321
$540$	0	0
$541$	−12.9132	−0.555182	−0.277591	−	0.960699i	$-0.589536\pi$
$541$	−12.9132	−0.555182	−0.277591	+	0.960699i	$0.589536\pi$
$542$	0	0
$543$	−11.7938	−0.506120
$544$	0	0
$545$	−9.92001	−0.424927
$546$	0	0
$547$	13.6598	0.584052	0.292026	−	0.956410i	$-0.405671\pi$
$547$	13.6598	0.584052	0.292026	+	0.956410i	$0.405671\pi$
$548$	0	0
$549$	−33.7431	−1.44012
$550$	0	0
$551$	0	0
$552$	0	0
$553$	2.04945	0.0871514
$554$	0	0
$555$	−5.68195	−0.241186
$556$	0	0
$557$	36.3545	1.54039	0.770196	−	0.637807i	$-0.220158\pi$
$557$	36.3545	1.54039	0.770196	+	0.637807i	$0.220158\pi$
$558$	0	0
$559$	−36.8781	−1.55978
$560$	0	0
$561$	11.0661	0.467211
$562$	0	0
$563$	−18.4247	−0.776508	−0.388254	−	0.921552i	$-0.626922\pi$
$563$	−18.4247	−0.776508	−0.388254	+	0.921552i	$0.626922\pi$
$564$	0	0
$565$	11.9688	0.503531
$566$	0	0
$567$	6.46800	0.271630
$568$	0	0
$569$	31.8576	1.33554	0.667770	−	0.744367i	$-0.267249\pi$
$569$	31.8576	1.33554	0.667770	+	0.744367i	$0.267249\pi$
$570$	0	0
$571$	16.1301	0.675023	0.337512	−	0.941321i	$-0.390415\pi$
$571$	16.1301	0.675023	0.337512	+	0.941321i	$0.390415\pi$
$572$	0	0
$573$	−3.78539	−0.158137
$574$	0	0
$575$	−3.29072	−0.137233
$576$	0	0
$577$	35.6886	1.48573	0.742867	−	0.669438i	$-0.233465\pi$
$577$	35.6886	1.48573	0.742867	+	0.669438i	$0.233465\pi$
$578$	0	0
$579$	7.33403	0.304792
$580$	0	0
$581$	6.34017	0.263035
$582$	0	0
$583$	12.7259	0.527054
$584$	0	0
$585$	−33.8576	−1.39984
$586$	0	0
$587$	−21.1206	−0.871742	−0.435871	−	0.900009i	$-0.643560\pi$
$587$	−21.1206	−0.871742	−0.435871	+	0.900009i	$0.643560\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	0.282314	0.0116129
$592$	0	0
$593$	3.57531	0.146820	0.0734101	−	0.997302i	$-0.476612\pi$
$593$	3.57531	0.146820	0.0734101	+	0.997302i	$0.476612\pi$
$594$	0	0
$595$	−22.4619	−0.920846
$596$	0	0
$597$	−6.83710	−0.279824
$598$	0	0
$599$	−36.0144	−1.47151	−0.735754	−	0.677249i	$-0.763172\pi$
$599$	−36.0144	−1.47151	−0.735754	+	0.677249i	$0.763172\pi$
$600$	0	0
$601$	−35.1773	−1.43491	−0.717455	−	0.696604i	$-0.754693\pi$
$601$	−35.1773	−1.43491	−0.717455	+	0.696604i	$0.754693\pi$
$602$	0	0
$603$	−30.6453	−1.24797
$604$	0	0
$605$	11.7431	0.477426
$606$	0	0
$607$	−21.1617	−0.858926	−0.429463	−	0.903084i	$-0.641297\pi$
$607$	−21.1617	−0.858926	−0.429463	+	0.903084i	$0.641297\pi$
$608$	0	0
$609$	3.07838	0.124742
$610$	0	0
$611$	16.4969	0.667394
$612$	0	0
$613$	37.4863	1.51406	0.757028	−	0.653383i	$-0.226651\pi$
$613$	37.4863	1.51406	0.757028	+	0.653383i	$0.226651\pi$
$614$	0	0
$615$	13.7198	0.553235
$616$	0	0
$617$	−10.8638	−0.437359	−0.218679	−	0.975797i	$-0.570175\pi$
$617$	−10.8638	−0.437359	−0.218679	+	0.975797i	$0.570175\pi$
$618$	0	0
$619$	−24.0144	−0.965219	−0.482610	−	0.875836i	$-0.660311\pi$
$619$	−24.0144	−0.965219	−0.482610	+	0.875836i	$0.660311\pi$
$620$	0	0
$621$	3.07838	0.123531
$622$	0	0
$623$	8.69594	0.348396
$624$	0	0
$625$	−30.6248	−1.22499
$626$	0	0
$627$	0	0
$628$	0	0
$629$	28.5503	1.13837
$630$	0	0
$631$	−23.2579	−0.925883	−0.462942	−	0.886389i	$-0.653206\pi$
$631$	−23.2579	−0.925883	−0.462942	+	0.886389i	$0.653206\pi$
$632$	0	0
$633$	2.68488	0.106714
$634$	0	0
$635$	−4.77924	−0.189658
$636$	0	0
$637$	−4.34017	−0.171964
$638$	0	0
$639$	5.84324	0.231155
$640$	0	0
$641$	10.9171	0.431199	0.215600	−	0.976482i	$-0.430829\pi$
$641$	10.9171	0.431199	0.215600	+	0.976482i	$0.430829\pi$
$642$	0	0
$643$	−16.0456	−0.632776	−0.316388	−	0.948630i	$-0.602470\pi$
$643$	−16.0456	−0.632776	−0.316388	+	0.948630i	$0.602470\pi$
$644$	0	0
$645$	−13.1917	−0.519421
$646$	0	0
$647$	−40.8794	−1.60713	−0.803567	−	0.595215i	$-0.797067\pi$
$647$	−40.8794	−1.60713	−0.803567	+	0.595215i	$0.797067\pi$
$648$	0	0
$649$	22.9483	0.900799
$650$	0	0
$651$	−5.46800	−0.214308
$652$	0	0
$653$	−45.7152	−1.78898	−0.894488	−	0.447092i	$-0.852460\pi$
$653$	−45.7152	−1.78898	−0.894488	+	0.447092i	$0.852460\pi$
$654$	0	0
$655$	50.0326	1.95494
$656$	0	0
$657$	−14.1834	−0.553348
$658$	0	0
$659$	−4.00000	−0.155818	−0.0779089	−	0.996960i	$-0.524824\pi$
$659$	−4.00000	−0.155818	−0.0779089	+	0.996960i	$0.524824\pi$
$660$	0	0
$661$	−11.2774	−0.438640	−0.219320	−	0.975653i	$-0.570384\pi$
$661$	−11.2774	−0.438640	−0.219320	+	0.975653i	$0.570384\pi$
$662$	0	0
$663$	−18.2557	−0.708991
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−5.70928	−0.221064
$668$	0	0
$669$	−4.70313	−0.181834
$670$	0	0
$671$	32.7670	1.26495
$672$	0	0
$673$	30.2206	1.16492	0.582459	−	0.812860i	$-0.302091\pi$
$673$	30.2206	1.16492	0.582459	+	0.812860i	$0.302091\pi$
$674$	0	0
$675$	−10.1301	−0.389907
$676$	0	0
$677$	1.30406	0.0501189	0.0250595	−	0.999686i	$-0.492022\pi$
$677$	1.30406	0.0501189	0.0250595	+	0.999686i	$0.492022\pi$
$678$	0	0
$679$	11.1929	0.429543
$680$	0	0
$681$	−7.15061	−0.274012
$682$	0	0
$683$	−13.1461	−0.503021	−0.251510	−	0.967855i	$-0.580927\pi$
$683$	−13.1461	−0.503021	−0.251510	+	0.967855i	$0.580927\pi$
$684$	0	0
$685$	−25.9733	−0.992390
$686$	0	0
$687$	12.2595	0.467730
$688$	0	0
$689$	−20.9939	−0.799802
$690$	0	0
$691$	8.11450	0.308690	0.154345	−	0.988017i	$-0.450673\pi$
$691$	8.11450	0.308690	0.154345	+	0.988017i	$0.450673\pi$
$692$	0	0
$693$	−7.12783	−0.270764
$694$	0	0
$695$	−2.08065	−0.0789234
$696$	0	0
$697$	−68.9381	−2.61122
$698$	0	0
$699$	5.60197	0.211886
$700$	0	0
$701$	23.2306	0.877408	0.438704	−	0.898632i	$-0.355438\pi$
$701$	23.2306	0.877408	0.438704	+	0.898632i	$0.355438\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	5.90110	0.222248
$706$	0	0
$707$	11.7587	0.442232
$708$	0	0
$709$	27.0784	1.01695	0.508475	−	0.861077i	$-0.330209\pi$
$709$	27.0784	1.01695	0.508475	+	0.861077i	$0.330209\pi$
$710$	0	0
$711$	−5.55252	−0.208236
$712$	0	0
$713$	10.1412	0.379789
$714$	0	0
$715$	32.8781	1.22957
$716$	0	0
$717$	3.78539	0.141368
$718$	0	0
$719$	0.650372	0.0242548	0.0121274	−	0.999926i	$-0.496140\pi$
$719$	0.650372	0.0242548	0.0121274	+	0.999926i	$0.496140\pi$
$720$	0	0
$721$	8.68035	0.323273
$722$	0	0
$723$	−1.27021	−0.0472395
$724$	0	0
$725$	18.7877	0.697756
$726$	0	0
$727$	19.2039	0.712235	0.356117	−	0.934441i	$-0.384100\pi$
$727$	19.2039	0.712235	0.356117	+	0.934441i	$0.384100\pi$
$728$	0	0
$729$	−12.5441	−0.464597
$730$	0	0
$731$	66.2844	2.45162
$732$	0	0
$733$	−28.6069	−1.05662	−0.528310	−	0.849052i	$-0.677174\pi$
$733$	−28.6069	−1.05662	−0.528310	+	0.849052i	$0.677174\pi$
$734$	0	0
$735$	−1.55252	−0.0572656
$736$	0	0
$737$	29.7587	1.09618
$738$	0	0
$739$	−46.5523	−1.71246	−0.856228	−	0.516598i	$-0.827198\pi$
$739$	−46.5523	−1.71246	−0.856228	+	0.516598i	$0.827198\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	37.0843	1.36049	0.680246	−	0.732984i	$-0.261873\pi$
$743$	37.0843	1.36049	0.680246	+	0.732984i	$0.261873\pi$
$744$	0	0
$745$	20.1378	0.737794
$746$	0	0
$747$	−17.1773	−0.628484
$748$	0	0
$749$	−10.6537	−0.389277
$750$	0	0
$751$	34.5152	1.25948	0.629738	−	0.776807i	$-0.283162\pi$
$751$	34.5152	1.25948	0.629738	+	0.776807i	$0.283162\pi$
$752$	0	0
$753$	−12.0533	−0.439248
$754$	0	0
$755$	−62.9360	−2.29048
$756$	0	0
$757$	35.1240	1.27660	0.638301	−	0.769787i	$-0.279638\pi$
$757$	35.1240	1.27660	0.638301	+	0.769787i	$0.279638\pi$
$758$	0	0
$759$	−1.41855	−0.0514901
$760$	0	0
$761$	−17.7464	−0.643308	−0.321654	−	0.946857i	$-0.604239\pi$
$761$	−17.7464	−0.643308	−0.321654	+	0.946857i	$0.604239\pi$
$762$	0	0
$763$	3.44521	0.124725
$764$	0	0
$765$	60.8554	2.20023
$766$	0	0
$767$	−37.8576	−1.36696
$768$	0	0
$769$	13.9122	0.501686	0.250843	−	0.968028i	$-0.419292\pi$
$769$	13.9122	0.501686	0.250843	+	0.968028i	$0.419292\pi$
$770$	0	0
$771$	−4.59583	−0.165515
$772$	0	0
$773$	34.4378	1.23864	0.619322	−	0.785137i	$-0.287408\pi$
$773$	34.4378	1.23864	0.619322	+	0.785137i	$0.287408\pi$
$774$	0	0
$775$	−33.3718	−1.19875
$776$	0	0
$777$	1.97334	0.0707931
$778$	0	0
$779$	0	0
$780$	0	0
$781$	−5.67420	−0.203039
$782$	0	0
$783$	−17.5753	−0.628090
$784$	0	0
$785$	7.31124	0.260949
$786$	0	0
$787$	−3.65529	−0.130297	−0.0651485	−	0.997876i	$-0.520752\pi$
$787$	−3.65529	−0.130297	−0.0651485	+	0.997876i	$0.520752\pi$
$788$	0	0
$789$	−16.9360	−0.602937
$790$	0	0
$791$	−4.15676	−0.147797
$792$	0	0
$793$	−54.0554	−1.91956
$794$	0	0
$795$	−7.50970	−0.266341
$796$	0	0
$797$	−37.8108	−1.33933	−0.669664	−	0.742664i	$-0.733562\pi$
$797$	−37.8108	−1.33933	−0.669664	+	0.742664i	$0.733562\pi$
$798$	0	0
$799$	−29.6514	−1.04899
$800$	0	0
$801$	−23.5597	−0.832441
$802$	0	0
$803$	13.7731	0.486042
$804$	0	0
$805$	2.87936	0.101484
$806$	0	0
$807$	−4.76487	−0.167731
$808$	0	0
$809$	−27.7587	−0.975945	−0.487972	−	0.872859i	$-0.662263\pi$
$809$	−27.7587	−0.975945	−0.487972	+	0.872859i	$0.662263\pi$
$810$	0	0
$811$	36.2979	1.27459	0.637296	−	0.770619i	$-0.280053\pi$
$811$	36.2979	1.27459	0.637296	+	0.770619i	$0.280053\pi$
$812$	0	0
$813$	14.4019	0.505097
$814$	0	0
$815$	−53.6209	−1.87826
$816$	0	0
$817$	0	0
$818$	0	0
$819$	11.7587	0.410883
$820$	0	0
$821$	−45.0121	−1.57093	−0.785467	−	0.618904i	$-0.787577\pi$
$821$	−45.0121	−1.57093	−0.785467	+	0.618904i	$0.787577\pi$
$822$	0	0
$823$	−43.5006	−1.51634	−0.758168	−	0.652059i	$-0.773905\pi$
$823$	−43.5006	−1.51634	−0.758168	+	0.652059i	$0.773905\pi$
$824$	0	0
$825$	4.66806	0.162521
$826$	0	0
$827$	36.9315	1.28423	0.642116	−	0.766607i	$-0.278057\pi$
$827$	36.9315	1.28423	0.642116	+	0.766607i	$0.278057\pi$
$828$	0	0
$829$	−0.255652	−0.00887917	−0.00443958	−	0.999990i	$-0.501413\pi$
$829$	−0.255652	−0.00887917	−0.00443958	+	0.999990i	$0.501413\pi$
$830$	0	0
$831$	−7.58968	−0.263283
$832$	0	0
$833$	7.80098	0.270288
$834$	0	0
$835$	47.8303	1.65524
$836$	0	0
$837$	31.2183	1.07906
$838$	0	0
$839$	−8.01438	−0.276687	−0.138343	−	0.990384i	$-0.544178\pi$
$839$	−8.01438	−0.276687	−0.138343	+	0.990384i	$0.544178\pi$
$840$	0	0
$841$	3.59583	0.123994
$842$	0	0
$843$	−1.82887	−0.0629896
$844$	0	0
$845$	−16.8071	−0.578183
$846$	0	0
$847$	−4.07838	−0.140135
$848$	0	0
$849$	0.115718	0.00397142
$850$	0	0
$851$	−3.65983	−0.125457
$852$	0	0
$853$	−13.1362	−0.449776	−0.224888	−	0.974385i	$-0.572202\pi$
$853$	−13.1362	−0.449776	−0.224888	+	0.974385i	$0.572202\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	31.5585	1.07802	0.539009	−	0.842300i	$-0.318799\pi$
$857$	31.5585	1.07802	0.539009	+	0.842300i	$0.318799\pi$
$858$	0	0
$859$	−50.2667	−1.71508	−0.857539	−	0.514419i	$-0.828008\pi$
$859$	−50.2667	−1.71508	−0.857539	+	0.514419i	$0.828008\pi$
$860$	0	0
$861$	−4.76487	−0.162386
$862$	0	0
$863$	39.7776	1.35405	0.677023	−	0.735962i	$-0.263270\pi$
$863$	39.7776	1.35405	0.677023	+	0.735962i	$0.263270\pi$
$864$	0	0
$865$	−65.3049	−2.22043
$866$	0	0
$867$	23.6463	0.803071
$868$	0	0
$869$	5.39189	0.182907
$870$	0	0
$871$	−49.0928	−1.66344
$872$	0	0
$873$	−30.3246	−1.02633
$874$	0	0
$875$	4.92162	0.166381
$876$	0	0
$877$	−25.3256	−0.855185	−0.427593	−	0.903972i	$-0.640638\pi$
$877$	−25.3256	−0.855185	−0.427593	+	0.903972i	$0.640638\pi$
$878$	0	0
$879$	5.14011	0.173372
$880$	0	0
$881$	−8.01560	−0.270052	−0.135026	−	0.990842i	$-0.543112\pi$
$881$	−8.01560	−0.270052	−0.135026	+	0.990842i	$0.543112\pi$
$882$	0	0
$883$	3.81658	0.128438	0.0642191	−	0.997936i	$-0.479544\pi$
$883$	3.81658	0.128438	0.0642191	+	0.997936i	$0.479544\pi$
$884$	0	0
$885$	−13.5420	−0.455210
$886$	0	0
$887$	−8.82150	−0.296197	−0.148099	−	0.988973i	$-0.547315\pi$
$887$	−8.82150	−0.296197	−0.148099	+	0.988973i	$0.547315\pi$
$888$	0	0
$889$	1.65983	0.0556688
$890$	0	0
$891$	17.0166	0.570079
$892$	0	0
$893$	0	0
$894$	0	0
$895$	0.528125	0.0176533
$896$	0	0
$897$	2.34017	0.0781361
$898$	0	0
$899$	−57.8987	−1.93103
$900$	0	0
$901$	37.7342	1.25711
$902$	0	0
$903$	4.58145	0.152461
$904$	0	0
$905$	62.9809	2.09356
$906$	0	0
$907$	−29.1629	−0.968338	−0.484169	−	0.874974i	$-0.660878\pi$
$907$	−29.1629	−0.968338	−0.484169	+	0.874974i	$0.660878\pi$
$908$	0	0
$909$	−31.8576	−1.05665
$910$	0	0
$911$	−31.3400	−1.03834	−0.519170	−	0.854671i	$-0.673759\pi$
$911$	−31.3400	−1.03834	−0.519170	+	0.854671i	$0.673759\pi$
$912$	0	0
$913$	16.6803	0.552039
$914$	0	0
$915$	−19.3361	−0.639232
$916$	0	0
$917$	−17.3763	−0.573816
$918$	0	0
$919$	14.0950	0.464952	0.232476	−	0.972602i	$-0.425317\pi$
$919$	14.0950	0.464952	0.232476	+	0.972602i	$0.425317\pi$
$920$	0	0
$921$	6.98998	0.230328
$922$	0	0
$923$	9.36069	0.308111
$924$	0	0
$925$	12.0435	0.395987
$926$	0	0
$927$	−23.5174	−0.772414
$928$	0	0
$929$	−56.6057	−1.85717	−0.928586	−	0.371118i	$-0.878975\pi$
$929$	−56.6057	−1.85717	−0.928586	+	0.371118i	$0.878975\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−10.9132	−0.357283
$934$	0	0
$935$	−59.0948	−1.93261
$936$	0	0
$937$	−8.65037	−0.282595	−0.141298	−	0.989967i	$-0.545127\pi$
$937$	−8.65037	−0.282595	−0.141298	+	0.989967i	$0.545127\pi$
$938$	0	0
$939$	−18.6842	−0.609737
$940$	0	0
$941$	55.9286	1.82322	0.911611	−	0.411055i	$-0.134840\pi$
$941$	55.9286	1.82322	0.911611	+	0.411055i	$0.134840\pi$
$942$	0	0
$943$	8.83710	0.287776
$944$	0	0
$945$	8.86376	0.288338
$946$	0	0
$947$	23.4863	0.763201	0.381600	−	0.924327i	$-0.375373\pi$
$947$	23.4863	0.763201	0.381600	+	0.924327i	$0.375373\pi$
$948$	0	0
$949$	−22.7214	−0.737567
$950$	0	0
$951$	−15.9278	−0.516493
$952$	0	0
$953$	−10.5814	−0.342767	−0.171383	−	0.985204i	$-0.554824\pi$
$953$	−10.5814	−0.342767	−0.171383	+	0.985204i	$0.554824\pi$
$954$	0	0
$955$	20.2146	0.654130
$956$	0	0
$957$	8.09890	0.261800
$958$	0	0
$959$	9.02052	0.291288
$960$	0	0
$961$	71.8431	2.31752
$962$	0	0
$963$	28.8638	0.930122
$964$	0	0
$965$	−39.1650	−1.26077
$966$	0	0
$967$	−24.2967	−0.781329	−0.390664	−	0.920533i	$-0.627755\pi$
$967$	−24.2967	−0.781329	−0.390664	+	0.920533i	$0.627755\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	3.40417	0.109245	0.0546226	−	0.998507i	$-0.482604\pi$
$971$	3.40417	0.109245	0.0546226	+	0.998507i	$0.482604\pi$
$972$	0	0
$973$	0.722606	0.0231657
$974$	0	0
$975$	−7.70086	−0.246625
$976$	0	0
$977$	24.6081	0.787283	0.393642	−	0.919264i	$-0.371215\pi$
$977$	24.6081	0.787283	0.393642	+	0.919264i	$0.371215\pi$
$978$	0	0
$979$	22.8781	0.731189
$980$	0	0
$981$	−9.33403	−0.298013
$982$	0	0
$983$	36.5503	1.16577	0.582886	−	0.812554i	$-0.301923\pi$
$983$	36.5503	1.16577	0.582886	+	0.812554i	$0.301923\pi$
$984$	0	0
$985$	−1.50761	−0.0480363
$986$	0	0
$987$	−2.04945	−0.0652347
$988$	0	0
$989$	−8.49693	−0.270187
$990$	0	0
$991$	19.4641	0.618298	0.309149	−	0.951014i	$-0.399956\pi$
$991$	19.4641	0.618298	0.309149	+	0.951014i	$0.399956\pi$
$992$	0	0
$993$	2.20233	0.0698888
$994$	0	0
$995$	36.5113	1.15749
$996$	0	0
$997$	5.68649	0.180093	0.0900465	−	0.995938i	$-0.471298\pi$
$997$	5.68649	0.180093	0.0900465	+	0.995938i	$0.471298\pi$
$998$	0	0
$999$	−11.2663	−0.356451

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2576.2.a.y.1.2		3
4.3	odd	2		1288.2.a.l.1.2	✓	3
28.27	even	2		9016.2.a.z.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1288.2.a.l.1.2	✓	3	4.3	odd	2
2576.2.a.y.1.2		3	1.1	even	1	trivial
9016.2.a.z.1.2		3	28.27	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:	\( N \)	\(=\)	\( 2576 = 2^{4} \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2576.a (trivial)

\(f(q)\)	\(=\)	\(q+0.539189 q^{3} -2.87936 q^{5} +1.00000 q^{7} -2.70928 q^{9} +O(q^{10})\)	\(q+0.539189 q^{3} -2.87936 q^{5} +1.00000 q^{7} -2.70928 q^{9} +2.63090 q^{11} -4.34017 q^{13} -1.55252 q^{15} +7.80098 q^{17} +0.539189 q^{21} -1.00000 q^{23} +3.29072 q^{25} -3.07838 q^{27} +5.70928 q^{29} -10.1412 q^{31} +1.41855 q^{33} -2.87936 q^{35} +3.65983 q^{37} -2.34017 q^{39} -8.83710 q^{41} +8.49693 q^{43} +7.80098 q^{45} -3.80098 q^{47} +1.00000 q^{49} +4.20620 q^{51} +4.83710 q^{53} -7.57531 q^{55} +8.72261 q^{59} +12.4547 q^{61} -2.70928 q^{63} +12.4969 q^{65} +11.3112 q^{67} -0.539189 q^{69} -2.15676 q^{71} +5.23513 q^{73} +1.77432 q^{75} +2.63090 q^{77} +2.04945 q^{79} +6.46800 q^{81} +6.34017 q^{83} -22.4619 q^{85} +3.07838 q^{87} +8.69594 q^{89} -4.34017 q^{91} -5.46800 q^{93} +11.1929 q^{97} -7.12783 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 4 q^{5} + 3 q^{7} - q^{9}+O(q^{10}) \) 3 * q + 4 * q^5 + 3 * q^7 - q^9	\( 3 q + 4 q^{5} + 3 q^{7} - q^{9} + 4 q^{11} - 2 q^{13} - 4 q^{15} + 14 q^{17} - 3 q^{23} + 17 q^{25} - 6 q^{27} + 10 q^{29} - 10 q^{31} - 10 q^{33} + 4 q^{35} + 22 q^{37} + 4 q^{39} + 2 q^{41} + 8 q^{43} + 14 q^{45} - 2 q^{47} + 3 q^{49} - 12 q^{51} - 14 q^{53} - 2 q^{55} + 20 q^{59} + 4 q^{61} - q^{63} + 20 q^{65} + 8 q^{67} + 6 q^{73} - 6 q^{75} + 4 q^{77} - 12 q^{79} - 13 q^{81} + 8 q^{83} + 6 q^{87} + 18 q^{89} - 2 q^{91} + 16 q^{93} + 8 q^{97}+O(q^{100}) \) 3 * q + 4 * q^5 + 3 * q^7 - q^9 + 4 * q^11 - 2 * q^13 - 4 * q^15 + 14 * q^17 - 3 * q^23 + 17 * q^25 - 6 * q^27 + 10 * q^29 - 10 * q^31 - 10 * q^33 + 4 * q^35 + 22 * q^37 + 4 * q^39 + 2 * q^41 + 8 * q^43 + 14 * q^45 - 2 * q^47 + 3 * q^49 - 12 * q^51 - 14 * q^53 - 2 * q^55 + 20 * q^59 + 4 * q^61 - q^63 + 20 * q^65 + 8 * q^67 + 6 * q^73 - 6 * q^75 + 4 * q^77 - 12 * q^79 - 13 * q^81 + 8 * q^83 + 6 * q^87 + 18 * q^89 - 2 * q^91 + 16 * q^93 + 8 * q^97

Introduction

L-functions

Modular forms

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Representations

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Database

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