LMFDB - Embedded newform 3344.2.a.s.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3344,2,Mod(1,3344)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3344, 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("3344.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3344 = 2^{4} \cdot 11 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3344.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:	$26.7019744359$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.13676.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 6x^{2} + 7x + 1 $ x^4 - x^3 - 6x^2 + 7x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1672)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$1.42957$ of defining polynomial
Character	$\chi$	$=$	3344.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.42957 q^{3} -3.22628 q^{5} -0.429567 q^{7} -0.956338 q^{9} +O(q^{10})$	$q+1.42957 q^{3} -3.22628 q^{5} -0.429567 q^{7} -0.956338 q^{9} -1.00000 q^{11} +4.88213 q^{13} -4.61219 q^{15} +3.70939 q^{17} -1.00000 q^{19} -0.614095 q^{21} +3.32349 q^{23} +5.40890 q^{25} -5.65585 q^{27} -2.75305 q^{29} -0.699512 q^{31} -1.42957 q^{33} +1.38591 q^{35} -1.61409 q^{37} +6.97934 q^{39} -8.99809 q^{41} -7.40890 q^{43} +3.08542 q^{45} +10.9051 q^{47} -6.81547 q^{49} +5.30283 q^{51} +0.452567 q^{53} +3.22628 q^{55} -1.42957 q^{57} -2.38591 q^{59} -7.62207 q^{61} +0.410811 q^{63} -15.7511 q^{65} -7.47132 q^{67} +4.75115 q^{69} +1.32248 q^{71} -13.7643 q^{73} +7.73239 q^{75} +0.429567 q^{77} +2.11596 q^{79} -5.21640 q^{81} +8.64597 q^{83} -11.9676 q^{85} -3.93568 q^{87} +3.89201 q^{89} -2.09720 q^{91} -1.00000 q^{93} +3.22628 q^{95} -13.9793 q^{97} +0.956338 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + q^{3} - 3 q^{5} + 3 q^{7} + q^{9}+O(q^{10}) $ 4 * q + q^3 - 3 * q^5 + 3 * q^7 + q^9	$ 4 q + q^{3} - 3 q^{5} + 3 q^{7} + q^{9} - 4 q^{11} - 5 q^{13} + q^{15} - 4 q^{19} - 12 q^{21} + 8 q^{23} - 3 q^{25} - 8 q^{27} - q^{29} + 7 q^{31} - q^{33} - 4 q^{35} - 16 q^{37} + 8 q^{39} - 7 q^{41} - 5 q^{43} - 7 q^{45} + 4 q^{47} - 13 q^{49} - 4 q^{51} - 18 q^{53} + 3 q^{55} - q^{57} - 6 q^{61} + 6 q^{63} - 24 q^{65} - q^{67} - 20 q^{69} + q^{71} - 6 q^{73} + q^{75} - 3 q^{77} + 4 q^{79} - 16 q^{81} + 25 q^{83} - 4 q^{85} + 9 q^{87} - 14 q^{89} - 13 q^{91} - 4 q^{93} + 3 q^{95} - 36 q^{97} - q^{99}+O(q^{100}) $ 4 * q + q^3 - 3 * q^5 + 3 * q^7 + q^9 - 4 * q^11 - 5 * q^13 + q^15 - 4 * q^19 - 12 * q^21 + 8 * q^23 - 3 * q^25 - 8 * q^27 - q^29 + 7 * q^31 - q^33 - 4 * q^35 - 16 * q^37 + 8 * q^39 - 7 * q^41 - 5 * q^43 - 7 * q^45 + 4 * q^47 - 13 * q^49 - 4 * q^51 - 18 * q^53 + 3 * q^55 - q^57 - 6 * q^61 + 6 * q^63 - 24 * q^65 - q^67 - 20 * q^69 + q^71 - 6 * q^73 + q^75 - 3 * q^77 + 4 * q^79 - 16 * q^81 + 25 * q^83 - 4 * q^85 + 9 * q^87 - 14 * q^89 - 13 * q^91 - 4 * q^93 + 3 * q^95 - 36 * q^97 - q^99

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
$3$	1.42957	0.825361	0.412680	−	0.910876i	$-0.364593\pi$
$3$	1.42957	0.825361	0.412680	+	0.910876i	$0.364593\pi$
$4$	0	0
$5$	−3.22628	−1.44284	−0.721419	−	0.692499i	$-0.756510\pi$
$5$	−3.22628	−1.44284	−0.721419	+	0.692499i	$0.756510\pi$
$6$	0	0
$7$	−0.429567	−0.162361	−0.0811805	−	0.996699i	$-0.525869\pi$
$7$	−0.429567	−0.162361	−0.0811805	+	0.996699i	$0.525869\pi$
$8$	0	0
$9$	−0.956338	−0.318779
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	4.88213	1.35406	0.677030	−	0.735955i	$-0.263267\pi$
$13$	4.88213	1.35406	0.677030	+	0.735955i	$0.263267\pi$
$14$	0	0
$15$	−4.61219	−1.19086
$16$	0	0
$17$	3.70939	0.899660	0.449830	−	0.893114i	$-0.351485\pi$
$17$	3.70939	0.899660	0.449830	+	0.893114i	$0.351485\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	−0.614095	−0.134006
$22$	0	0
$23$	3.32349	0.692995	0.346498	−	0.938051i	$-0.387371\pi$
$23$	3.32349	0.692995	0.346498	+	0.938051i	$0.387371\pi$
$24$	0	0
$25$	5.40890	1.08178
$26$	0	0
$27$	−5.65585	−1.08847
$28$	0	0
$29$	−2.75305	−0.511229	−0.255615	−	0.966779i	$-0.582278\pi$
$29$	−2.75305	−0.511229	−0.255615	+	0.966779i	$0.582278\pi$
$30$	0	0
$31$	−0.699512	−0.125636	−0.0628181	−	0.998025i	$-0.520009\pi$
$31$	−0.699512	−0.125636	−0.0628181	+	0.998025i	$0.520009\pi$
$32$	0	0
$33$	−1.42957	−0.248856
$34$	0	0
$35$	1.38591	0.234261
$36$	0	0
$37$	−1.61409	−0.265356	−0.132678	−	0.991159i	$-0.542358\pi$
$37$	−1.61409	−0.265356	−0.132678	+	0.991159i	$0.542358\pi$
$38$	0	0
$39$	6.97934	1.11759
$40$	0	0
$41$	−8.99809	−1.40527	−0.702633	−	0.711552i	$-0.747992\pi$
$41$	−8.99809	−1.40527	−0.702633	+	0.711552i	$0.747992\pi$
$42$	0	0
$43$	−7.40890	−1.12985	−0.564924	−	0.825143i	$-0.691094\pi$
$43$	−7.40890	−1.12985	−0.564924	+	0.825143i	$0.691094\pi$
$44$	0	0
$45$	3.08542	0.459947
$46$	0	0
$47$	10.9051	1.59068	0.795339	−	0.606165i	$-0.207293\pi$
$47$	10.9051	1.59068	0.795339	+	0.606165i	$0.207293\pi$
$48$	0	0
$49$	−6.81547	−0.973639
$50$	0	0
$51$	5.30283	0.742544
$52$	0	0
$53$	0.452567	0.0621648	0.0310824	−	0.999517i	$-0.490105\pi$
$53$	0.452567	0.0621648	0.0310824	+	0.999517i	$0.490105\pi$
$54$	0	0
$55$	3.22628	0.435032
$56$	0	0
$57$	−1.42957	−0.189351
$58$	0	0
$59$	−2.38591	−0.310618	−0.155309	−	0.987866i	$-0.549637\pi$
$59$	−2.38591	−0.310618	−0.155309	+	0.987866i	$0.549637\pi$
$60$	0	0
$61$	−7.62207	−0.975906	−0.487953	−	0.872870i	$-0.662256\pi$
$61$	−7.62207	−0.975906	−0.487953	+	0.872870i	$0.662256\pi$
$62$	0	0
$63$	0.410811	0.0517574
$64$	0	0
$65$	−15.7511	−1.95369
$66$	0	0
$67$	−7.47132	−0.912767	−0.456384	−	0.889783i	$-0.650856\pi$
$67$	−7.47132	−0.912767	−0.456384	+	0.889783i	$0.650856\pi$
$68$	0	0
$69$	4.75115	0.571971
$70$	0	0
$71$	1.32248	0.156950	0.0784749	−	0.996916i	$-0.474995\pi$
$71$	1.32248	0.156950	0.0784749	+	0.996916i	$0.474995\pi$
$72$	0	0
$73$	−13.7643	−1.61099	−0.805493	−	0.592605i	$-0.798099\pi$
$73$	−13.7643	−1.61099	−0.805493	+	0.592605i	$0.798099\pi$
$74$	0	0
$75$	7.73239	0.892860
$76$	0	0
$77$	0.429567	0.0489537
$78$	0	0
$79$	2.11596	0.238064	0.119032	−	0.992890i	$-0.462021\pi$
$79$	2.11596	0.238064	0.119032	+	0.992890i	$0.462021\pi$
$80$	0	0
$81$	−5.21640	−0.579600
$82$	0	0
$83$	8.64597	0.949019	0.474509	−	0.880251i	$-0.342626\pi$
$83$	8.64597	0.949019	0.474509	+	0.880251i	$0.342626\pi$
$84$	0	0
$85$	−11.9676	−1.29806
$86$	0	0
$87$	−3.93568	−0.421949
$88$	0	0
$89$	3.89201	0.412553	0.206276	−	0.978494i	$-0.433865\pi$
$89$	3.89201	0.412553	0.206276	+	0.978494i	$0.433865\pi$
$90$	0	0
$91$	−2.09720	−0.219847
$92$	0	0
$93$	−1.00000	−0.103695
$94$	0	0
$95$	3.22628	0.331010
$96$	0	0
$97$	−13.9793	−1.41939	−0.709693	−	0.704511i	$-0.751167\pi$
$97$	−13.9793	−1.41939	−0.709693	+	0.704511i	$0.751167\pi$
$98$	0	0
$99$	0.956338	0.0961156
$100$	0	0
$101$	6.65585	0.662282	0.331141	−	0.943581i	$-0.392566\pi$
$101$	6.65585	0.662282	0.331141	+	0.943581i	$0.392566\pi$
$102$	0	0
$103$	−16.4014	−1.61607	−0.808037	−	0.589132i	$-0.799470\pi$
$103$	−16.4014	−1.61607	−0.808037	+	0.589132i	$0.799470\pi$
$104$	0	0
$105$	1.98124	0.193350
$106$	0	0
$107$	−9.01222	−0.871244	−0.435622	−	0.900130i	$-0.643472\pi$
$107$	−9.01222	−0.871244	−0.435622	+	0.900130i	$0.643472\pi$
$108$	0	0
$109$	−7.71827	−0.739276	−0.369638	−	0.929176i	$-0.620518\pi$
$109$	−7.71827	−0.739276	−0.369638	+	0.929176i	$0.620518\pi$
$110$	0	0
$111$	−2.30746	−0.219014
$112$	0	0
$113$	−18.7511	−1.76396	−0.881980	−	0.471287i	$-0.843790\pi$
$113$	−18.7511	−1.76396	−0.881980	+	0.471287i	$0.843790\pi$
$114$	0	0
$115$	−10.7225	−0.999879
$116$	0	0
$117$	−4.66897	−0.431646
$118$	0	0
$119$	−1.59343	−0.146070
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−12.8634	−1.15985
$124$	0	0
$125$	−1.31924	−0.117997
$126$	0	0
$127$	−0.762934	−0.0676995	−0.0338497	−	0.999427i	$-0.510777\pi$
$127$	−0.762934	−0.0676995	−0.0338497	+	0.999427i	$0.510777\pi$
$128$	0	0
$129$	−10.5915	−0.932532
$130$	0	0
$131$	16.9568	1.48152	0.740760	−	0.671770i	$-0.234466\pi$
$131$	16.9568	1.48152	0.740760	+	0.671770i	$0.234466\pi$
$132$	0	0
$133$	0.429567	0.0372482
$134$	0	0
$135$	18.2474	1.57048
$136$	0	0
$137$	16.9075	1.44450	0.722251	−	0.691631i	$-0.243107\pi$
$137$	16.9075	1.44450	0.722251	+	0.691631i	$0.243107\pi$
$138$	0	0
$139$	−17.9577	−1.52315	−0.761575	−	0.648077i	$-0.775574\pi$
$139$	−17.9577	−1.52315	−0.761575	+	0.648077i	$0.775574\pi$
$140$	0	0
$141$	15.5896	1.31288
$142$	0	0
$143$	−4.88213	−0.408265
$144$	0	0
$145$	8.88213	0.737621
$146$	0	0
$147$	−9.74317	−0.803603
$148$	0	0
$149$	−19.1633	−1.56992	−0.784959	−	0.619548i	$-0.787316\pi$
$149$	−19.1633	−1.56992	−0.784959	+	0.619548i	$0.787316\pi$
$150$	0	0
$151$	9.02997	0.734848	0.367424	−	0.930053i	$-0.380240\pi$
$151$	9.02997	0.734848	0.367424	+	0.930053i	$0.380240\pi$
$152$	0	0
$153$	−3.54743	−0.286793
$154$	0	0
$155$	2.25683	0.181273
$156$	0	0
$157$	−6.47423	−0.516700	−0.258350	−	0.966051i	$-0.583179\pi$
$157$	−6.47423	−0.516700	−0.258350	+	0.966051i	$0.583179\pi$
$158$	0	0
$159$	0.646975	0.0513084
$160$	0	0
$161$	−1.42766	−0.112515
$162$	0	0
$163$	15.9962	1.25292	0.626459	−	0.779454i	$-0.284504\pi$
$163$	15.9962	1.25292	0.626459	+	0.779454i	$0.284504\pi$
$164$	0	0
$165$	4.61219	0.359058
$166$	0	0
$167$	−2.85026	−0.220560	−0.110280	−	0.993901i	$-0.535175\pi$
$167$	−2.85026	−0.220560	−0.110280	+	0.993901i	$0.535175\pi$
$168$	0	0
$169$	10.8352	0.833479
$170$	0	0
$171$	0.956338	0.0731330
$172$	0	0
$173$	−20.7991	−1.58132	−0.790661	−	0.612254i	$-0.790263\pi$
$173$	−20.7991	−1.58132	−0.790661	+	0.612254i	$0.790263\pi$
$174$	0	0
$175$	−2.32349	−0.175639
$176$	0	0
$177$	−3.41081	−0.256372
$178$	0	0
$179$	−1.84038	−0.137556	−0.0687782	−	0.997632i	$-0.521910\pi$
$179$	−1.84038	−0.137556	−0.0687782	+	0.997632i	$0.521910\pi$
$180$	0	0
$181$	−2.94556	−0.218941	−0.109471	−	0.993990i	$-0.534916\pi$
$181$	−2.94556	−0.218941	−0.109471	+	0.993990i	$0.534916\pi$
$182$	0	0
$183$	−10.8963	−0.805475
$184$	0	0
$185$	5.20753	0.382865
$186$	0	0
$187$	−3.70939	−0.271258
$188$	0	0
$189$	2.42957	0.176725
$190$	0	0
$191$	14.3732	1.04001	0.520005	−	0.854163i	$-0.325930\pi$
$191$	14.3732	1.04001	0.520005	+	0.854163i	$0.325930\pi$
$192$	0	0
$193$	−9.62106	−0.692539	−0.346270	−	0.938135i	$-0.612552\pi$
$193$	−9.62106	−0.692539	−0.346270	+	0.938135i	$0.612552\pi$
$194$	0	0
$195$	−22.5173	−1.61250
$196$	0	0
$197$	−21.0671	−1.50097	−0.750484	−	0.660888i	$-0.770180\pi$
$197$	−21.0671	−1.50097	−0.750484	+	0.660888i	$0.770180\pi$
$198$	0	0
$199$	−15.0911	−1.06978	−0.534888	−	0.844923i	$-0.679646\pi$
$199$	−15.0911	−1.06978	−0.534888	+	0.844923i	$0.679646\pi$
$200$	0	0
$201$	−10.6808	−0.753362
$202$	0	0
$203$	1.18262	0.0830038
$204$	0	0
$205$	29.0304	2.02757
$206$	0	0
$207$	−3.17838	−0.220912
$208$	0	0
$209$	1.00000	0.0691714
$210$	0	0
$211$	−0.272856	−0.0187842	−0.00939209	−	0.999956i	$-0.502990\pi$
$211$	−0.272856	−0.0187842	−0.00939209	+	0.999956i	$0.502990\pi$
$212$	0	0
$213$	1.89058	0.129540
$214$	0	0
$215$	23.9032	1.63019
$216$	0	0
$217$	0.300488	0.0203984
$218$	0	0
$219$	−19.6769	−1.32964
$220$	0	0
$221$	18.1098	1.21819
$222$	0	0
$223$	12.8760	0.862240	0.431120	−	0.902295i	$-0.358119\pi$
$223$	12.8760	0.862240	0.431120	+	0.902295i	$0.358119\pi$
$224$	0	0
$225$	−5.17274	−0.344849
$226$	0	0
$227$	−23.1046	−1.53351	−0.766753	−	0.641942i	$-0.778129\pi$
$227$	−23.1046	−1.53351	−0.766753	+	0.641942i	$0.778129\pi$
$228$	0	0
$229$	18.9187	1.25019	0.625093	−	0.780550i	$-0.285061\pi$
$229$	18.9187	1.25019	0.625093	+	0.780550i	$0.285061\pi$
$230$	0	0
$231$	0.614095	0.0404045
$232$	0	0
$233$	15.8802	1.04035	0.520174	−	0.854060i	$-0.325867\pi$
$233$	15.8802	1.04035	0.520174	+	0.854060i	$0.325867\pi$
$234$	0	0
$235$	−35.1831	−2.29509
$236$	0	0
$237$	3.02491	0.196489
$238$	0	0
$239$	26.6793	1.72574	0.862870	−	0.505427i	$-0.168665\pi$
$239$	26.6793	1.72574	0.862870	+	0.505427i	$0.168665\pi$
$240$	0	0
$241$	−2.78550	−0.179430	−0.0897150	−	0.995967i	$-0.528596\pi$
$241$	−2.78550	−0.179430	−0.0897150	+	0.995967i	$0.528596\pi$
$242$	0	0
$243$	9.51035	0.610089
$244$	0	0
$245$	21.9886	1.40480
$246$	0	0
$247$	−4.88213	−0.310643
$248$	0	0
$249$	12.3600	0.783283
$250$	0	0
$251$	−11.6014	−0.732274	−0.366137	−	0.930561i	$-0.619320\pi$
$251$	−11.6014	−0.732274	−0.366137	+	0.930561i	$0.619320\pi$
$252$	0	0
$253$	−3.32349	−0.208946
$254$	0	0
$255$	−17.1084	−1.07137
$256$	0	0
$257$	−2.04600	−0.127626	−0.0638130	−	0.997962i	$-0.520326\pi$
$257$	−2.04600	−0.127626	−0.0638130	+	0.997962i	$0.520326\pi$
$258$	0	0
$259$	0.693362	0.0430834
$260$	0	0
$261$	2.63285	0.162969
$262$	0	0
$263$	26.1807	1.61437	0.807186	−	0.590297i	$-0.200989\pi$
$263$	26.1807	1.61437	0.807186	+	0.590297i	$0.200989\pi$
$264$	0	0
$265$	−1.46011	−0.0896938
$266$	0	0
$267$	5.56389	0.340505
$268$	0	0
$269$	14.2864	0.871058	0.435529	−	0.900175i	$-0.356561\pi$
$269$	14.2864	0.871058	0.435529	+	0.900175i	$0.356561\pi$
$270$	0	0
$271$	−11.4920	−0.698088	−0.349044	−	0.937106i	$-0.613494\pi$
$271$	−11.4920	−0.698088	−0.349044	+	0.937106i	$0.613494\pi$
$272$	0	0
$273$	−2.99809	−0.181453
$274$	0	0
$275$	−5.40890	−0.326169
$276$	0	0
$277$	−20.6057	−1.23807	−0.619037	−	0.785362i	$-0.712477\pi$
$277$	−20.6057	−1.23807	−0.619037	+	0.785362i	$0.712477\pi$
$278$	0	0
$279$	0.668970	0.0400502
$280$	0	0
$281$	17.4051	1.03830	0.519150	−	0.854683i	$-0.326248\pi$
$281$	17.4051	1.03830	0.519150	+	0.854683i	$0.326248\pi$
$282$	0	0
$283$	2.92813	0.174059	0.0870297	−	0.996206i	$-0.472262\pi$
$283$	2.92813	0.174059	0.0870297	+	0.996206i	$0.472262\pi$
$284$	0	0
$285$	4.61219	0.273202
$286$	0	0
$287$	3.86528	0.228161
$288$	0	0
$289$	−3.24041	−0.190612
$290$	0	0
$291$	−19.9844	−1.17151
$292$	0	0
$293$	9.97319	0.582640	0.291320	−	0.956626i	$-0.405906\pi$
$293$	9.97319	0.582640	0.291320	+	0.956626i	$0.405906\pi$
$294$	0	0
$295$	7.69761	0.448172
$296$	0	0
$297$	5.65585	0.328186
$298$	0	0
$299$	16.2257	0.938357
$300$	0	0
$301$	3.18262	0.183443
$302$	0	0
$303$	9.51498	0.546622
$304$	0	0
$305$	24.5910	1.40807
$306$	0	0
$307$	10.9140	0.622895	0.311448	−	0.950263i	$-0.399186\pi$
$307$	10.9140	0.622895	0.311448	+	0.950263i	$0.399186\pi$
$308$	0	0
$309$	−23.4468	−1.33384
$310$	0	0
$311$	9.14087	0.518331	0.259165	−	0.965833i	$-0.416552\pi$
$311$	9.14087	0.518331	0.259165	+	0.965833i	$0.416552\pi$
$312$	0	0
$313$	13.9193	0.786763	0.393381	−	0.919375i	$-0.371305\pi$
$313$	13.9193	0.786763	0.393381	+	0.919375i	$0.371305\pi$
$314$	0	0
$315$	−1.32539	−0.0746775
$316$	0	0
$317$	−28.2159	−1.58476	−0.792382	−	0.610025i	$-0.791159\pi$
$317$	−28.2159	−1.58476	−0.792382	+	0.610025i	$0.791159\pi$
$318$	0	0
$319$	2.75305	0.154141
$320$	0	0
$321$	−12.8836	−0.719091
$322$	0	0
$323$	−3.70939	−0.206396
$324$	0	0
$325$	26.4070	1.46480
$326$	0	0
$327$	−11.0338	−0.610170
$328$	0	0
$329$	−4.68449	−0.258264
$330$	0	0
$331$	6.29338	0.345915	0.172958	−	0.984929i	$-0.444668\pi$
$331$	6.29338	0.345915	0.172958	+	0.984929i	$0.444668\pi$
$332$	0	0
$333$	1.54362	0.0845899
$334$	0	0
$335$	24.1046	1.31698
$336$	0	0
$337$	9.57043	0.521335	0.260667	−	0.965429i	$-0.416057\pi$
$337$	9.57043	0.521335	0.260667	+	0.965429i	$0.416057\pi$
$338$	0	0
$339$	−26.8060	−1.45590
$340$	0	0
$341$	0.699512	0.0378807
$342$	0	0
$343$	5.93467	0.320442
$344$	0	0
$345$	−15.3286	−0.825261
$346$	0	0
$347$	−4.46144	−0.239503	−0.119751	−	0.992804i	$-0.538210\pi$
$347$	−4.46144	−0.239503	−0.119751	+	0.992804i	$0.538210\pi$
$348$	0	0
$349$	13.1320	0.702939	0.351470	−	0.936199i	$-0.385682\pi$
$349$	13.1320	0.702939	0.351470	+	0.936199i	$0.385682\pi$
$350$	0	0
$351$	−27.6126	−1.47385
$352$	0	0
$353$	−21.1140	−1.12378	−0.561892	−	0.827210i	$-0.689926\pi$
$353$	−21.1140	−1.12378	−0.561892	+	0.827210i	$0.689926\pi$
$354$	0	0
$355$	−4.26671	−0.226453
$356$	0	0
$357$	−2.27792	−0.120560
$358$	0	0
$359$	8.78593	0.463704	0.231852	−	0.972751i	$-0.425522\pi$
$359$	8.78593	0.463704	0.231852	+	0.972751i	$0.425522\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	1.42957	0.0750328
$364$	0	0
$365$	44.4074	2.32439
$366$	0	0
$367$	−7.68672	−0.401244	−0.200622	−	0.979669i	$-0.564296\pi$
$367$	−7.68672	−0.401244	−0.200622	+	0.979669i	$0.564296\pi$
$368$	0	0
$369$	8.60522	0.447970
$370$	0	0
$371$	−0.194408	−0.0100932
$372$	0	0
$373$	1.71040	0.0885610	0.0442805	−	0.999019i	$-0.485900\pi$
$373$	1.71040	0.0885610	0.0442805	+	0.999019i	$0.485900\pi$
$374$	0	0
$375$	−1.88595	−0.0973899
$376$	0	0
$377$	−13.4408	−0.692235
$378$	0	0
$379$	12.7417	0.654497	0.327249	−	0.944938i	$-0.393879\pi$
$379$	12.7417	0.654497	0.327249	+	0.944938i	$0.393879\pi$
$380$	0	0
$381$	−1.09067	−0.0558765
$382$	0	0
$383$	25.9652	1.32676	0.663380	−	0.748282i	$-0.269121\pi$
$383$	25.9652	1.32676	0.663380	+	0.748282i	$0.269121\pi$
$384$	0	0
$385$	−1.38591	−0.0706323
$386$	0	0
$387$	7.08542	0.360172
$388$	0	0
$389$	−2.83747	−0.143865	−0.0719327	−	0.997409i	$-0.522917\pi$
$389$	−2.83747	−0.143865	−0.0719327	+	0.997409i	$0.522917\pi$
$390$	0	0
$391$	12.3281	0.623460
$392$	0	0
$393$	24.2408	1.22279
$394$	0	0
$395$	−6.82669	−0.343488
$396$	0	0
$397$	2.44226	0.122573	0.0612866	−	0.998120i	$-0.480480\pi$
$397$	2.44226	0.122573	0.0612866	+	0.998120i	$0.480480\pi$
$398$	0	0
$399$	0.614095	0.0307432
$400$	0	0
$401$	23.2666	1.16188	0.580938	−	0.813948i	$-0.302686\pi$
$401$	23.2666	1.16188	0.580938	+	0.813948i	$0.302686\pi$
$402$	0	0
$403$	−3.41511	−0.170119
$404$	0	0
$405$	16.8296	0.836269
$406$	0	0
$407$	1.61409	0.0800077
$408$	0	0
$409$	11.6910	0.578084	0.289042	−	0.957316i	$-0.406663\pi$
$409$	11.6910	0.578084	0.289042	+	0.957316i	$0.406663\pi$
$410$	0	0
$411$	24.1704	1.19224
$412$	0	0
$413$	1.02491	0.0504323
$414$	0	0
$415$	−27.8944	−1.36928
$416$	0	0
$417$	−25.6717	−1.25715
$418$	0	0
$419$	−33.5285	−1.63798	−0.818988	−	0.573811i	$-0.805465\pi$
$419$	−33.5285	−1.63798	−0.818988	+	0.573811i	$0.805465\pi$
$420$	0	0
$421$	−6.60098	−0.321712	−0.160856	−	0.986978i	$-0.551425\pi$
$421$	−6.60098	−0.321712	−0.160856	+	0.986978i	$0.551425\pi$
$422$	0	0
$423$	−10.4290	−0.507075
$424$	0	0
$425$	20.0638	0.973235
$426$	0	0
$427$	3.27419	0.158449
$428$	0	0
$429$	−6.97934	−0.336966
$430$	0	0
$431$	6.24041	0.300590	0.150295	−	0.988641i	$-0.451978\pi$
$431$	6.24041	0.300590	0.150295	+	0.988641i	$0.451978\pi$
$432$	0	0
$433$	−25.2610	−1.21397	−0.606983	−	0.794715i	$-0.707620\pi$
$433$	−25.2610	−1.21397	−0.606983	+	0.794715i	$0.707620\pi$
$434$	0	0
$435$	12.6976	0.608804
$436$	0	0
$437$	−3.32349	−0.158984
$438$	0	0
$439$	0.948854	0.0452863	0.0226432	−	0.999744i	$-0.492792\pi$
$439$	0.948854	0.0452863	0.0226432	+	0.999744i	$0.492792\pi$
$440$	0	0
$441$	6.51790	0.310376
$442$	0	0
$443$	−15.1713	−0.720809	−0.360404	−	0.932796i	$-0.617361\pi$
$443$	−15.1713	−0.720809	−0.360404	+	0.932796i	$0.617361\pi$
$444$	0	0
$445$	−12.5567	−0.595247
$446$	0	0
$447$	−27.3952	−1.29575
$448$	0	0
$449$	−35.6787	−1.68378	−0.841891	−	0.539647i	$-0.818558\pi$
$449$	−35.6787	−1.68378	−0.841891	+	0.539647i	$0.818558\pi$
$450$	0	0
$451$	8.99809	0.423704
$452$	0	0
$453$	12.9089	0.606515
$454$	0	0
$455$	6.76617	0.317203
$456$	0	0
$457$	22.8904	1.07077	0.535385	−	0.844608i	$-0.320167\pi$
$457$	22.8904	1.07077	0.535385	+	0.844608i	$0.320167\pi$
$458$	0	0
$459$	−20.9798	−0.979252
$460$	0	0
$461$	−26.7980	−1.24811	−0.624055	−	0.781381i	$-0.714516\pi$
$461$	−26.7980	−1.24811	−0.624055	+	0.781381i	$0.714516\pi$
$462$	0	0
$463$	−14.6314	−0.679978	−0.339989	−	0.940429i	$-0.610423\pi$
$463$	−14.6314	−0.679978	−0.339989	+	0.940429i	$0.610423\pi$
$464$	0	0
$465$	3.22628	0.149615
$466$	0	0
$467$	−15.7460	−0.728638	−0.364319	−	0.931274i	$-0.618698\pi$
$467$	−15.7460	−0.728638	−0.364319	+	0.931274i	$0.618698\pi$
$468$	0	0
$469$	3.20943	0.148198
$470$	0	0
$471$	−9.25535	−0.426464
$472$	0	0
$473$	7.40890	0.340662
$474$	0	0
$475$	−5.40890	−0.248178
$476$	0	0
$477$	−0.432807	−0.0198169
$478$	0	0
$479$	8.90571	0.406912	0.203456	−	0.979084i	$-0.434783\pi$
$479$	8.90571	0.406912	0.203456	+	0.979084i	$0.434783\pi$
$480$	0	0
$481$	−7.88023	−0.359307
$482$	0	0
$483$	−2.04094	−0.0928658
$484$	0	0
$485$	45.1013	2.04794
$486$	0	0
$487$	−3.35073	−0.151836	−0.0759181	−	0.997114i	$-0.524189\pi$
$487$	−3.35073	−0.151836	−0.0759181	+	0.997114i	$0.524189\pi$
$488$	0	0
$489$	22.8676	1.03411
$490$	0	0
$491$	19.1601	0.864681	0.432341	−	0.901710i	$-0.357688\pi$
$491$	19.1601	0.864681	0.432341	+	0.901710i	$0.357688\pi$
$492$	0	0
$493$	−10.2122	−0.459933
$494$	0	0
$495$	−3.08542	−0.138679
$496$	0	0
$497$	−0.568095	−0.0254826
$498$	0	0
$499$	−28.4665	−1.27434	−0.637168	−	0.770725i	$-0.719894\pi$
$499$	−28.4665	−1.27434	−0.637168	+	0.770725i	$0.719894\pi$
$500$	0	0
$501$	−4.07464	−0.182041
$502$	0	0
$503$	−23.2407	−1.03625	−0.518127	−	0.855304i	$-0.673370\pi$
$503$	−23.2407	−1.03625	−0.518127	+	0.855304i	$0.673370\pi$
$504$	0	0
$505$	−21.4737	−0.955565
$506$	0	0
$507$	15.4897	0.687921
$508$	0	0
$509$	20.3295	0.901089	0.450545	−	0.892754i	$-0.351230\pi$
$509$	20.3295	0.901089	0.450545	+	0.892754i	$0.351230\pi$
$510$	0	0
$511$	5.91268	0.261561
$512$	0	0
$513$	5.65585	0.249712
$514$	0	0
$515$	52.9154	2.33173
$516$	0	0
$517$	−10.9051	−0.479607
$518$	0	0
$519$	−29.7336	−1.30516
$520$	0	0
$521$	6.68539	0.292892	0.146446	−	0.989219i	$-0.453217\pi$
$521$	6.68539	0.292892	0.146446	+	0.989219i	$0.453217\pi$
$522$	0	0
$523$	26.3463	1.15205	0.576023	−	0.817434i	$-0.304604\pi$
$523$	26.3463	1.15205	0.576023	+	0.817434i	$0.304604\pi$
$524$	0	0
$525$	−3.32158	−0.144966
$526$	0	0
$527$	−2.59477	−0.113030
$528$	0	0
$529$	−11.9544	−0.519758
$530$	0	0
$531$	2.28173	0.0990187
$532$	0	0
$533$	−43.9299	−1.90282
$534$	0	0
$535$	29.0760	1.25706
$536$	0	0
$537$	−2.63094	−0.113534
$538$	0	0
$539$	6.81547	0.293563
$540$	0	0
$541$	3.65958	0.157338	0.0786688	−	0.996901i	$-0.474933\pi$
$541$	3.65958	0.157338	0.0786688	+	0.996901i	$0.474933\pi$
$542$	0	0
$543$	−4.21087	−0.180706
$544$	0	0
$545$	24.9013	1.06666
$546$	0	0
$547$	33.7064	1.44118	0.720592	−	0.693360i	$-0.243870\pi$
$547$	33.7064	1.44118	0.720592	+	0.693360i	$0.243870\pi$
$548$	0	0
$549$	7.28927	0.311099
$550$	0	0
$551$	2.75305	0.117284
$552$	0	0
$553$	−0.908947	−0.0386523
$554$	0	0
$555$	7.44451	0.316002
$556$	0	0
$557$	26.3897	1.11817	0.559083	−	0.829112i	$-0.311153\pi$
$557$	26.3897	1.11817	0.559083	+	0.829112i	$0.311153\pi$
$558$	0	0
$559$	−36.1713	−1.52988
$560$	0	0
$561$	−5.30283	−0.223885
$562$	0	0
$563$	−0.209105	−0.00881272	−0.00440636	−	0.999990i	$-0.501403\pi$
$563$	−0.209105	−0.00881272	−0.00440636	+	0.999990i	$0.501403\pi$
$564$	0	0
$565$	60.4965	2.54511
$566$	0	0
$567$	2.24080	0.0941045
$568$	0	0
$569$	−42.3233	−1.77429	−0.887143	−	0.461494i	$-0.847314\pi$
$569$	−42.3233	−1.77429	−0.887143	+	0.461494i	$0.847314\pi$
$570$	0	0
$571$	−33.5628	−1.40456	−0.702280	−	0.711901i	$-0.747834\pi$
$571$	−33.5628	−1.40456	−0.702280	+	0.711901i	$0.747834\pi$
$572$	0	0
$573$	20.5475	0.858383
$574$	0	0
$575$	17.9764	0.749669
$576$	0	0
$577$	10.4287	0.434151	0.217076	−	0.976155i	$-0.430348\pi$
$577$	10.4287	0.434151	0.217076	+	0.976155i	$0.430348\pi$
$578$	0	0
$579$	−13.7540	−0.571595
$580$	0	0
$581$	−3.71402	−0.154084
$582$	0	0
$583$	−0.452567	−0.0187434
$584$	0	0
$585$	15.0634	0.622796
$586$	0	0
$587$	−37.5023	−1.54789	−0.773943	−	0.633256i	$-0.781718\pi$
$587$	−37.5023	−1.54789	−0.773943	+	0.633256i	$0.781718\pi$
$588$	0	0
$589$	0.699512	0.0288229
$590$	0	0
$591$	−30.1168	−1.23884
$592$	0	0
$593$	23.8262	0.978425	0.489213	−	0.872165i	$-0.337284\pi$
$593$	23.8262	0.978425	0.489213	+	0.872165i	$0.337284\pi$
$594$	0	0
$595$	5.14087	0.210755
$596$	0	0
$597$	−21.5737	−0.882952
$598$	0	0
$599$	2.89144	0.118141	0.0590705	−	0.998254i	$-0.481186\pi$
$599$	2.89144	0.118141	0.0590705	+	0.998254i	$0.481186\pi$
$600$	0	0
$601$	30.3469	1.23788	0.618938	−	0.785440i	$-0.287563\pi$
$601$	30.3469	1.23788	0.618938	+	0.785440i	$0.287563\pi$
$602$	0	0
$603$	7.14511	0.290971
$604$	0	0
$605$	−3.22628	−0.131167
$606$	0	0
$607$	21.2851	0.863934	0.431967	−	0.901889i	$-0.357820\pi$
$607$	21.2851	0.863934	0.431967	+	0.901889i	$0.357820\pi$
$608$	0	0
$609$	1.69064	0.0685081
$610$	0	0
$611$	53.2403	2.15387
$612$	0	0
$613$	−32.1384	−1.29806	−0.649029	−	0.760764i	$-0.724824\pi$
$613$	−32.1384	−1.29806	−0.649029	+	0.760764i	$0.724824\pi$
$614$	0	0
$615$	41.5009	1.67348
$616$	0	0
$617$	−13.4507	−0.541503	−0.270752	−	0.962649i	$-0.587272\pi$
$617$	−13.4507	−0.541503	−0.270752	+	0.962649i	$0.587272\pi$
$618$	0	0
$619$	−15.6849	−0.630430	−0.315215	−	0.949020i	$-0.602077\pi$
$619$	−15.6849	−0.630430	−0.315215	+	0.949020i	$0.602077\pi$
$620$	0	0
$621$	−18.7971	−0.754304
$622$	0	0
$623$	−1.67188	−0.0669825
$624$	0	0
$625$	−22.7883	−0.911531
$626$	0	0
$627$	1.42957	0.0570914
$628$	0	0
$629$	−5.98731	−0.238730
$630$	0	0
$631$	0.0557776	0.00222047	0.00111024	−	0.999999i	$-0.499647\pi$
$631$	0.0557776	0.00222047	0.00111024	+	0.999999i	$0.499647\pi$
$632$	0	0
$633$	−0.390066	−0.0155037
$634$	0	0
$635$	2.46144	0.0976794
$636$	0	0
$637$	−33.2740	−1.31837
$638$	0	0
$639$	−1.26474	−0.0500324
$640$	0	0
$641$	−18.3071	−0.723086	−0.361543	−	0.932355i	$-0.617750\pi$
$641$	−18.3071	−0.723086	−0.361543	+	0.932355i	$0.617750\pi$
$642$	0	0
$643$	48.8246	1.92546	0.962728	−	0.270472i	$-0.0871797\pi$
$643$	48.8246	1.92546	0.962728	+	0.270472i	$0.0871797\pi$
$644$	0	0
$645$	34.1713	1.34549
$646$	0	0
$647$	−17.3919	−0.683747	−0.341873	−	0.939746i	$-0.611061\pi$
$647$	−17.3919	−0.683747	−0.341873	+	0.939746i	$0.611061\pi$
$648$	0	0
$649$	2.38591	0.0936550
$650$	0	0
$651$	0.429567	0.0168361
$652$	0	0
$653$	−27.1216	−1.06135	−0.530676	−	0.847575i	$-0.678062\pi$
$653$	−27.1216	−1.06135	−0.530676	+	0.847575i	$0.678062\pi$
$654$	0	0
$655$	−54.7073	−2.13759
$656$	0	0
$657$	13.1633	0.513549
$658$	0	0
$659$	16.0696	0.625982	0.312991	−	0.949756i	$-0.398669\pi$
$659$	16.0696	0.625982	0.312991	+	0.949756i	$0.398669\pi$
$660$	0	0
$661$	2.36233	0.0918841	0.0459420	−	0.998944i	$-0.485371\pi$
$661$	2.36233	0.0918841	0.0459420	+	0.998944i	$0.485371\pi$
$662$	0	0
$663$	25.8891	1.00545
$664$	0	0
$665$	−1.38591	−0.0537431
$666$	0	0
$667$	−9.14974	−0.354279
$668$	0	0
$669$	18.4071	0.711659
$670$	0	0
$671$	7.62207	0.294247
$672$	0	0
$673$	−10.1474	−0.391155	−0.195578	−	0.980688i	$-0.562658\pi$
$673$	−10.1474	−0.391155	−0.195578	+	0.980688i	$0.562658\pi$
$674$	0	0
$675$	−30.5920	−1.17749
$676$	0	0
$677$	−35.8623	−1.37830	−0.689151	−	0.724618i	$-0.742016\pi$
$677$	−35.8623	−1.37830	−0.689151	+	0.724618i	$0.742016\pi$
$678$	0	0
$679$	6.00506	0.230453
$680$	0	0
$681$	−33.0296	−1.26570
$682$	0	0
$683$	−11.5108	−0.440448	−0.220224	−	0.975449i	$-0.570679\pi$
$683$	−11.5108	−0.440448	−0.220224	+	0.975449i	$0.570679\pi$
$684$	0	0
$685$	−54.5483	−2.08418
$686$	0	0
$687$	27.0456	1.03185
$688$	0	0
$689$	2.20949	0.0841750
$690$	0	0
$691$	7.57517	0.288173	0.144086	−	0.989565i	$-0.453976\pi$
$691$	7.57517	0.288173	0.144086	+	0.989565i	$0.453976\pi$
$692$	0	0
$693$	−0.410811	−0.0156054
$694$	0	0
$695$	57.9365	2.19766
$696$	0	0
$697$	−33.3775	−1.26426
$698$	0	0
$699$	22.7019	0.858663
$700$	0	0
$701$	32.9211	1.24341	0.621706	−	0.783251i	$-0.286440\pi$
$701$	32.9211	1.24341	0.621706	+	0.783251i	$0.286440\pi$
$702$	0	0
$703$	1.61409	0.0608767
$704$	0	0
$705$	−50.2965	−1.89428
$706$	0	0
$707$	−2.85913	−0.107529
$708$	0	0
$709$	−38.2867	−1.43789	−0.718944	−	0.695068i	$-0.755374\pi$
$709$	−38.2867	−1.43789	−0.718944	+	0.695068i	$0.755374\pi$
$710$	0	0
$711$	−2.02357	−0.0758899
$712$	0	0
$713$	−2.32482	−0.0870652
$714$	0	0
$715$	15.7511	0.589060
$716$	0	0
$717$	38.1398	1.42436
$718$	0	0
$719$	−27.2146	−1.01493	−0.507467	−	0.861671i	$-0.669418\pi$
$719$	−27.2146	−1.01493	−0.507467	+	0.861671i	$0.669418\pi$
$720$	0	0
$721$	7.04549	0.262388
$722$	0	0
$723$	−3.98206	−0.148095
$724$	0	0
$725$	−14.8910	−0.553038
$726$	0	0
$727$	−2.44078	−0.0905235	−0.0452618	−	0.998975i	$-0.514412\pi$
$727$	−2.44078	−0.0905235	−0.0452618	+	0.998975i	$0.514412\pi$
$728$	0	0
$729$	29.2449	1.08314
$730$	0	0
$731$	−27.4825	−1.01648
$732$	0	0
$733$	−31.3825	−1.15914	−0.579570	−	0.814922i	$-0.696779\pi$
$733$	−31.3825	−1.15914	−0.579570	+	0.814922i	$0.696779\pi$
$734$	0	0
$735$	31.4342	1.15947
$736$	0	0
$737$	7.47132	0.275210
$738$	0	0
$739$	−14.9277	−0.549124	−0.274562	−	0.961569i	$-0.588533\pi$
$739$	−14.9277	−0.549124	−0.274562	+	0.961569i	$0.588533\pi$
$740$	0	0
$741$	−6.97934	−0.256392
$742$	0	0
$743$	46.2868	1.69810	0.849049	−	0.528314i	$-0.177176\pi$
$743$	46.2868	1.69810	0.849049	+	0.528314i	$0.177176\pi$
$744$	0	0
$745$	61.8262	2.26514
$746$	0	0
$747$	−8.26847	−0.302528
$748$	0	0
$749$	3.87135	0.141456
$750$	0	0
$751$	−15.2102	−0.555028	−0.277514	−	0.960722i	$-0.589510\pi$
$751$	−15.2102	−0.555028	−0.277514	+	0.960722i	$0.589510\pi$
$752$	0	0
$753$	−16.5850	−0.604391
$754$	0	0
$755$	−29.1332	−1.06027
$756$	0	0
$757$	35.6145	1.29443	0.647215	−	0.762308i	$-0.275934\pi$
$757$	35.6145	1.29443	0.647215	+	0.762308i	$0.275934\pi$
$758$	0	0
$759$	−4.75115	−0.172456
$760$	0	0
$761$	37.0701	1.34379	0.671896	−	0.740646i	$-0.265480\pi$
$761$	37.0701	1.34379	0.671896	+	0.740646i	$0.265480\pi$
$762$	0	0
$763$	3.31551	0.120030
$764$	0	0
$765$	11.4450	0.413796
$766$	0	0
$767$	−11.6483	−0.420596
$768$	0	0
$769$	7.21311	0.260111	0.130056	−	0.991507i	$-0.458484\pi$
$769$	7.21311	0.260111	0.130056	+	0.991507i	$0.458484\pi$
$770$	0	0
$771$	−2.92489	−0.105337
$772$	0	0
$773$	−52.8299	−1.90016	−0.950081	−	0.312005i	$-0.898999\pi$
$773$	−52.8299	−1.90016	−0.950081	+	0.312005i	$0.898999\pi$
$774$	0	0
$775$	−3.78360	−0.135911
$776$	0	0
$777$	0.991207	0.0355594
$778$	0	0
$779$	8.99809	0.322390
$780$	0	0
$781$	−1.32248	−0.0473222
$782$	0	0
$783$	15.5709	0.556457
$784$	0	0
$785$	20.8877	0.745514
$786$	0	0
$787$	−43.5012	−1.55065	−0.775324	−	0.631563i	$-0.782414\pi$
$787$	−43.5012	−1.55065	−0.775324	+	0.631563i	$0.782414\pi$
$788$	0	0
$789$	37.4271	1.33244
$790$	0	0
$791$	8.05488	0.286398
$792$	0	0
$793$	−37.2120	−1.32144
$794$	0	0
$795$	−2.08732	−0.0740298
$796$	0	0
$797$	−20.5718	−0.728691	−0.364346	−	0.931264i	$-0.618707\pi$
$797$	−20.5718	−0.728691	−0.364346	+	0.931264i	$0.618707\pi$
$798$	0	0
$799$	40.4514	1.43107
$800$	0	0
$801$	−3.72208	−0.131513
$802$	0	0
$803$	13.7643	0.485730
$804$	0	0
$805$	4.60604	0.162342
$806$	0	0
$807$	20.4234	0.718937
$808$	0	0
$809$	47.2365	1.66075	0.830374	−	0.557207i	$-0.188127\pi$
$809$	47.2365	1.66075	0.830374	+	0.557207i	$0.188127\pi$
$810$	0	0
$811$	2.86621	0.100646	0.0503230	−	0.998733i	$-0.483975\pi$
$811$	2.86621	0.100646	0.0503230	+	0.998733i	$0.483975\pi$
$812$	0	0
$813$	−16.4286	−0.576175
$814$	0	0
$815$	−51.6082	−1.80776
$816$	0	0
$817$	7.40890	0.259205
$818$	0	0
$819$	2.00564	0.0700826
$820$	0	0
$821$	−10.3986	−0.362912	−0.181456	−	0.983399i	$-0.558081\pi$
$821$	−10.3986	−0.362912	−0.181456	+	0.983399i	$0.558081\pi$
$822$	0	0
$823$	55.6014	1.93814	0.969072	−	0.246780i	$-0.0793724\pi$
$823$	55.6014	1.93814	0.969072	+	0.246780i	$0.0793724\pi$
$824$	0	0
$825$	−7.73239	−0.269207
$826$	0	0
$827$	25.2728	0.878820	0.439410	−	0.898287i	$-0.355187\pi$
$827$	25.2728	0.878820	0.439410	+	0.898287i	$0.355187\pi$
$828$	0	0
$829$	−16.8845	−0.586422	−0.293211	−	0.956048i	$-0.594724\pi$
$829$	−16.8845	−0.586422	−0.293211	+	0.956048i	$0.594724\pi$
$830$	0	0
$831$	−29.4572	−1.02186
$832$	0	0
$833$	−25.2813	−0.875944
$834$	0	0
$835$	9.19574	0.318232
$836$	0	0
$837$	3.95634	0.136751
$838$	0	0
$839$	35.6042	1.22919	0.614597	−	0.788841i	$-0.289319\pi$
$839$	35.6042	1.22919	0.614597	+	0.788841i	$0.289319\pi$
$840$	0	0
$841$	−21.4207	−0.738645
$842$	0	0
$843$	24.8817	0.856973
$844$	0	0
$845$	−34.9575	−1.20258
$846$	0	0
$847$	−0.429567	−0.0147601
$848$	0	0
$849$	4.18596	0.143662
$850$	0	0
$851$	−5.36442	−0.183890
$852$	0	0
$853$	37.6592	1.28943	0.644713	−	0.764425i	$-0.276977\pi$
$853$	37.6592	1.28943	0.644713	+	0.764425i	$0.276977\pi$
$854$	0	0
$855$	−3.08542	−0.105519
$856$	0	0
$857$	−4.25693	−0.145414	−0.0727069	−	0.997353i	$-0.523164\pi$
$857$	−4.25693	−0.145414	−0.0727069	+	0.997353i	$0.523164\pi$
$858$	0	0
$859$	46.4679	1.58546	0.792732	−	0.609571i	$-0.208658\pi$
$859$	46.4679	1.58546	0.792732	+	0.609571i	$0.208658\pi$
$860$	0	0
$861$	5.52568	0.188315
$862$	0	0
$863$	53.8291	1.83236	0.916181	−	0.400764i	$-0.131255\pi$
$863$	53.8291	1.83236	0.916181	+	0.400764i	$0.131255\pi$
$864$	0	0
$865$	67.1036	2.28159
$866$	0	0
$867$	−4.63238	−0.157324
$868$	0	0
$869$	−2.11596	−0.0717790
$870$	0	0
$871$	−36.4760	−1.23594
$872$	0	0
$873$	13.3690	0.452471
$874$	0	0
$875$	0.566703	0.0191581
$876$	0	0
$877$	22.8239	0.770708	0.385354	−	0.922769i	$-0.374079\pi$
$877$	22.8239	0.770708	0.385354	+	0.922769i	$0.374079\pi$
$878$	0	0
$879$	14.2573	0.480888
$880$	0	0
$881$	44.0631	1.48452	0.742262	−	0.670109i	$-0.233753\pi$
$881$	44.0631	1.48452	0.742262	+	0.670109i	$0.233753\pi$
$882$	0	0
$883$	11.8215	0.397826	0.198913	−	0.980017i	$-0.436259\pi$
$883$	11.8215	0.397826	0.198913	+	0.980017i	$0.436259\pi$
$884$	0	0
$885$	11.0042	0.369904
$886$	0	0
$887$	−24.1079	−0.809466	−0.404733	−	0.914435i	$-0.632635\pi$
$887$	−24.1079	−0.809466	−0.404733	+	0.914435i	$0.632635\pi$
$888$	0	0
$889$	0.327731	0.0109918
$890$	0	0
$891$	5.21640	0.174756
$892$	0	0
$893$	−10.9051	−0.364926
$894$	0	0
$895$	5.93758	0.198471
$896$	0	0
$897$	23.1957	0.774483
$898$	0	0
$899$	1.92580	0.0642289
$900$	0	0
$901$	1.67875	0.0559272
$902$	0	0
$903$	4.54977	0.151407
$904$	0	0
$905$	9.50320	0.315897
$906$	0	0
$907$	−17.1371	−0.569027	−0.284513	−	0.958672i	$-0.591832\pi$
$907$	−17.1371	−0.569027	−0.284513	+	0.958672i	$0.591832\pi$
$908$	0	0
$909$	−6.36524	−0.211122
$910$	0	0
$911$	2.22446	0.0736997	0.0368498	−	0.999321i	$-0.488268\pi$
$911$	2.22446	0.0736997	0.0368498	+	0.999321i	$0.488268\pi$
$912$	0	0
$913$	−8.64597	−0.286140
$914$	0	0
$915$	35.1544	1.16217
$916$	0	0
$917$	−7.28407	−0.240541
$918$	0	0
$919$	8.86662	0.292483	0.146241	−	0.989249i	$-0.453282\pi$
$919$	8.86662	0.292483	0.146241	+	0.989249i	$0.453282\pi$
$920$	0	0
$921$	15.6023	0.514114
$922$	0	0
$923$	6.45654	0.212520
$924$	0	0
$925$	−8.73049	−0.287057
$926$	0	0
$927$	15.6852	0.515171
$928$	0	0
$929$	27.8252	0.912914	0.456457	−	0.889745i	$-0.349118\pi$
$929$	27.8252	0.912914	0.456457	+	0.889745i	$0.349118\pi$
$930$	0	0
$931$	6.81547	0.223368
$932$	0	0
$933$	13.0675	0.427810
$934$	0	0
$935$	11.9676	0.391381
$936$	0	0
$937$	−5.45009	−0.178047	−0.0890233	−	0.996030i	$-0.528375\pi$
$937$	−5.45009	−0.178047	−0.0890233	+	0.996030i	$0.528375\pi$
$938$	0	0
$939$	19.8985	0.649363
$940$	0	0
$941$	−0.858747	−0.0279943	−0.0139972	−	0.999902i	$-0.504456\pi$
$941$	−0.858747	−0.0279943	−0.0139972	+	0.999902i	$0.504456\pi$
$942$	0	0
$943$	−29.9051	−0.973842
$944$	0	0
$945$	−7.83847	−0.254986
$946$	0	0
$947$	−35.6323	−1.15789	−0.578947	−	0.815365i	$-0.696536\pi$
$947$	−35.6323	−1.15789	−0.578947	+	0.815365i	$0.696536\pi$
$948$	0	0
$949$	−67.1990	−2.18137
$950$	0	0
$951$	−40.3366	−1.30800
$952$	0	0
$953$	−11.2728	−0.365161	−0.182580	−	0.983191i	$-0.558445\pi$
$953$	−11.2728	−0.365161	−0.182580	+	0.983191i	$0.558445\pi$
$954$	0	0
$955$	−46.3721	−1.50056
$956$	0	0
$957$	3.93568	0.127222
$958$	0	0
$959$	−7.26289	−0.234531
$960$	0	0
$961$	−30.5107	−0.984216
$962$	0	0
$963$	8.61873	0.277735
$964$	0	0
$965$	31.0403	0.999222
$966$	0	0
$967$	−9.59850	−0.308667	−0.154333	−	0.988019i	$-0.549323\pi$
$967$	−9.59850	−0.308667	−0.154333	+	0.988019i	$0.549323\pi$
$968$	0	0
$969$	−5.30283	−0.170351
$970$	0	0
$971$	−25.8632	−0.829991	−0.414995	−	0.909824i	$-0.636217\pi$
$971$	−25.8632	−0.829991	−0.414995	+	0.909824i	$0.636217\pi$
$972$	0	0
$973$	7.71402	0.247300
$974$	0	0
$975$	37.7506	1.20899
$976$	0	0
$977$	13.3510	0.427136	0.213568	−	0.976928i	$-0.431492\pi$
$977$	13.3510	0.427136	0.213568	+	0.976928i	$0.431492\pi$
$978$	0	0
$979$	−3.89201	−0.124389
$980$	0	0
$981$	7.38127	0.235666
$982$	0	0
$983$	47.6233	1.51895	0.759473	−	0.650539i	$-0.225457\pi$
$983$	47.6233	1.51895	0.759473	+	0.650539i	$0.225457\pi$
$984$	0	0
$985$	67.9684	2.16565
$986$	0	0
$987$	−6.69679	−0.213161
$988$	0	0
$989$	−24.6234	−0.782979
$990$	0	0
$991$	47.5586	1.51075	0.755374	−	0.655293i	$-0.227455\pi$
$991$	47.5586	1.51075	0.755374	+	0.655293i	$0.227455\pi$
$992$	0	0
$993$	8.99680	0.285505
$994$	0	0
$995$	48.6880	1.54351
$996$	0	0
$997$	17.7152	0.561046	0.280523	−	0.959847i	$-0.409492\pi$
$997$	17.7152	0.561046	0.280523	+	0.959847i	$0.409492\pi$
$998$	0	0
$999$	9.12908	0.288831

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3344.2.a.s.1.3		4
4.3	odd	2		1672.2.a.f.1.2	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1672.2.a.f.1.2	✓	4	4.3	odd	2
3344.2.a.s.1.3		4	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:	\( N \)	\(=\)	\( 3344 = 2^{4} \cdot 11 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3344.a (trivial)

\(f(q)\)	\(=\)	\(q+1.42957 q^{3} -3.22628 q^{5} -0.429567 q^{7} -0.956338 q^{9} +O(q^{10})\)	\(q+1.42957 q^{3} -3.22628 q^{5} -0.429567 q^{7} -0.956338 q^{9} -1.00000 q^{11} +4.88213 q^{13} -4.61219 q^{15} +3.70939 q^{17} -1.00000 q^{19} -0.614095 q^{21} +3.32349 q^{23} +5.40890 q^{25} -5.65585 q^{27} -2.75305 q^{29} -0.699512 q^{31} -1.42957 q^{33} +1.38591 q^{35} -1.61409 q^{37} +6.97934 q^{39} -8.99809 q^{41} -7.40890 q^{43} +3.08542 q^{45} +10.9051 q^{47} -6.81547 q^{49} +5.30283 q^{51} +0.452567 q^{53} +3.22628 q^{55} -1.42957 q^{57} -2.38591 q^{59} -7.62207 q^{61} +0.410811 q^{63} -15.7511 q^{65} -7.47132 q^{67} +4.75115 q^{69} +1.32248 q^{71} -13.7643 q^{73} +7.73239 q^{75} +0.429567 q^{77} +2.11596 q^{79} -5.21640 q^{81} +8.64597 q^{83} -11.9676 q^{85} -3.93568 q^{87} +3.89201 q^{89} -2.09720 q^{91} -1.00000 q^{93} +3.22628 q^{95} -13.9793 q^{97} +0.956338 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + q^{3} - 3 q^{5} + 3 q^{7} + q^{9}+O(q^{10}) \) 4 * q + q^3 - 3 * q^5 + 3 * q^7 + q^9	\( 4 q + q^{3} - 3 q^{5} + 3 q^{7} + q^{9} - 4 q^{11} - 5 q^{13} + q^{15} - 4 q^{19} - 12 q^{21} + 8 q^{23} - 3 q^{25} - 8 q^{27} - q^{29} + 7 q^{31} - q^{33} - 4 q^{35} - 16 q^{37} + 8 q^{39} - 7 q^{41} - 5 q^{43} - 7 q^{45} + 4 q^{47} - 13 q^{49} - 4 q^{51} - 18 q^{53} + 3 q^{55} - q^{57} - 6 q^{61} + 6 q^{63} - 24 q^{65} - q^{67} - 20 q^{69} + q^{71} - 6 q^{73} + q^{75} - 3 q^{77} + 4 q^{79} - 16 q^{81} + 25 q^{83} - 4 q^{85} + 9 q^{87} - 14 q^{89} - 13 q^{91} - 4 q^{93} + 3 q^{95} - 36 q^{97} - q^{99}+O(q^{100}) \) 4 * q + q^3 - 3 * q^5 + 3 * q^7 + q^9 - 4 * q^11 - 5 * q^13 + q^15 - 4 * q^19 - 12 * q^21 + 8 * q^23 - 3 * q^25 - 8 * q^27 - q^29 + 7 * q^31 - q^33 - 4 * q^35 - 16 * q^37 + 8 * q^39 - 7 * q^41 - 5 * q^43 - 7 * q^45 + 4 * q^47 - 13 * q^49 - 4 * q^51 - 18 * q^53 + 3 * q^55 - q^57 - 6 * q^61 + 6 * q^63 - 24 * q^65 - q^67 - 20 * q^69 + q^71 - 6 * q^73 + q^75 - 3 * q^77 + 4 * q^79 - 16 * q^81 + 25 * q^83 - 4 * q^85 + 9 * q^87 - 14 * q^89 - 13 * q^91 - 4 * q^93 + 3 * q^95 - 36 * q^97 - q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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