LMFDB - Embedded newform 1449.2.a.p.1.2

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:	4.4.24197.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 6x^{2} - x + 2 $ x^4 - 6*x^2 - 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			1.2
Root			$-0.700017$ 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)

$f(q)$	$=$	$q-0.700017 q^{2} -1.50998 q^{4} +1.15706 q^{5} +1.00000 q^{7} +2.45704 q^{8} +O(q^{10})$	\(q-0.700017 q^{2} -1.50998 q^{4} +1.15706 q^{5} +1.00000 q^{7} +2.45704 q^{8} -0.809960 q^{10} -3.66704 q^{11} +2.34710 q^{13} -0.700017 q^{14} +1.29998 q^{16} -4.80996 q^{17} +7.06707 q^{19} -1.74713 q^{20} +2.56699 q^{22} +1.00000 q^{23} -3.66122 q^{25} -1.64301 q^{26} -1.50998 q^{28} +6.52411 q^{29} +2.80996 q^{31} -5.82409 q^{32} +3.36705 q^{34} +1.15706 q^{35} +5.40993 q^{37} -4.94707 q^{38} +2.84294 q^{40} -0.647082 q^{41} -4.76709 q^{43} +5.53714 q^{44} -0.700017 q^{46} -4.85708 q^{47} +1.00000 q^{49} +2.56291 q^{50} -3.54406 q^{52} -7.63405 q^{53} -4.24297 q^{55} +2.45704 q^{56} -4.56699 q^{58} +10.2142 q^{59} +9.91001 q^{61} -1.96702 q^{62} +1.47700 q^{64} +2.71573 q^{65} +6.07114 q^{67} +7.26293 q^{68} -0.809960 q^{70} +3.65290 q^{71} +11.8183 q^{73} -3.78704 q^{74} -10.6711 q^{76} -3.66704 q^{77} -9.12408 q^{79} +1.50416 q^{80} +0.452968 q^{82} +15.8582 q^{83} -5.56541 q^{85} +3.33704 q^{86} -9.01006 q^{88} -4.66122 q^{89} +2.34710 q^{91} -1.50998 q^{92} +3.40003 q^{94} +8.17701 q^{95} -4.90419 q^{97} -0.700017 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{4} - 5 q^{5} + 4 q^{7} + 3 q^{8}+O(q^{10}) $ 4 * q + 4 * q^4 - 5 * q^5 + 4 * q^7 + 3 * q^8	$ 4 q + 4 q^{4} - 5 q^{5} + 4 q^{7} + 3 q^{8} + 4 q^{10} + 5 q^{11} + 7 q^{13} + 8 q^{16} - 12 q^{17} + 3 q^{19} + q^{20} - q^{22} + 4 q^{23} + 7 q^{25} - 5 q^{26} + 4 q^{28} - 6 q^{29} + 4 q^{31} + 6 q^{32} - 9 q^{34} - 5 q^{35} + 20 q^{37} - 23 q^{38} + 21 q^{40} - 3 q^{41} + 9 q^{43} + 27 q^{44} - 7 q^{47} + 4 q^{49} - 3 q^{50} + 38 q^{52} + 6 q^{53} - 21 q^{55} + 3 q^{56} - 7 q^{58} + 2 q^{59} + 24 q^{61} + 9 q^{62} - 21 q^{64} + 14 q^{65} + q^{67} + 13 q^{68} + 4 q^{70} + 17 q^{71} + 16 q^{73} + 33 q^{74} - 25 q^{76} + 5 q^{77} - 10 q^{79} - 6 q^{80} - 7 q^{82} - 8 q^{83} + 17 q^{85} + 35 q^{86} - 12 q^{88} + 3 q^{89} + 7 q^{91} + 4 q^{92} + 8 q^{94} + 3 q^{95} - 2 q^{97}+O(q^{100}) $ 4 * q + 4 * q^4 - 5 * q^5 + 4 * q^7 + 3 * q^8 + 4 * q^10 + 5 * q^11 + 7 * q^13 + 8 * q^16 - 12 * q^17 + 3 * q^19 + q^20 - q^22 + 4 * q^23 + 7 * q^25 - 5 * q^26 + 4 * q^28 - 6 * q^29 + 4 * q^31 + 6 * q^32 - 9 * q^34 - 5 * q^35 + 20 * q^37 - 23 * q^38 + 21 * q^40 - 3 * q^41 + 9 * q^43 + 27 * q^44 - 7 * q^47 + 4 * q^49 - 3 * q^50 + 38 * q^52 + 6 * q^53 - 21 * q^55 + 3 * q^56 - 7 * q^58 + 2 * q^59 + 24 * q^61 + 9 * q^62 - 21 * q^64 + 14 * q^65 + q^67 + 13 * q^68 + 4 * q^70 + 17 * q^71 + 16 * q^73 + 33 * q^74 - 25 * q^76 + 5 * q^77 - 10 * q^79 - 6 * q^80 - 7 * q^82 - 8 * q^83 + 17 * q^85 + 35 * q^86 - 12 * q^88 + 3 * q^89 + 7 * q^91 + 4 * q^92 + 8 * q^94 + 3 * q^95 - 2 * 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 $

$2$	−0.700017	−0.494986	−0.247493	−	0.968890i	$-0.579607\pi$
$2$	−0.700017	−0.494986	−0.247493	+	0.968890i	$0.579607\pi$
$3$	0	0
$3$	0	0
$4$	−1.50998	−0.754988
$5$	1.15706	0.517452	0.258726	−	0.965951i	$-0.416697\pi$
$5$	1.15706	0.517452	0.258726	+	0.965951i	$0.416697\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	2.45704	0.868696
$9$	0	0
$10$	−0.809960	−0.256132
$11$	−3.66704	−1.10565	−0.552826	−	0.833296i	$-0.686451\pi$
$11$	−3.66704	−1.10565	−0.552826	+	0.833296i	$0.686451\pi$
$12$	0	0
$13$	2.34710	0.650968	0.325484	−	0.945548i	$-0.394473\pi$
$13$	2.34710	0.650968	0.325484	+	0.945548i	$0.394473\pi$
$14$	−0.700017	−0.187087
$15$	0	0
$16$	1.29998	0.324996
$17$	−4.80996	−1.16659	−0.583293	−	0.812262i	$-0.698236\pi$
$17$	−4.80996	−1.16659	−0.583293	+	0.812262i	$0.698236\pi$
$18$	0	0
$19$	7.06707	1.62130	0.810648	−	0.585533i	$-0.199115\pi$
$19$	7.06707	1.62130	0.810648	+	0.585533i	$0.199115\pi$
$20$	−1.74713	−0.390670
$21$	0	0
$22$	2.56699	0.547283
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	−3.66122	−0.732243
$26$	−1.64301	−0.322220
$27$	0	0
$28$	−1.50998	−0.285359
$29$	6.52411	1.21150	0.605748	−	0.795656i	$-0.292874\pi$
$29$	6.52411	1.21150	0.605748	+	0.795656i	$0.292874\pi$
$30$	0	0
$31$	2.80996	0.504684	0.252342	−	0.967638i	$-0.418799\pi$
$31$	2.80996	0.504684	0.252342	+	0.967638i	$0.418799\pi$
$32$	−5.82409	−1.02956
$33$	0	0
$34$	3.36705	0.577445
$35$	1.15706	0.195579
$36$	0	0
$37$	5.40993	0.889387	0.444693	−	0.895683i	$-0.353313\pi$
$37$	5.40993	0.889387	0.444693	+	0.895683i	$0.353313\pi$
$38$	−4.94707	−0.802520
$39$	0	0
$40$	2.84294	0.449509
$41$	−0.647082	−0.101057	−0.0505286	−	0.998723i	$-0.516091\pi$
$41$	−0.647082	−0.101057	−0.0505286	+	0.998723i	$0.516091\pi$
$42$	0	0
$43$	−4.76709	−0.726974	−0.363487	−	0.931599i	$-0.618414\pi$
$43$	−4.76709	−0.726974	−0.363487	+	0.931599i	$0.618414\pi$
$44$	5.53714	0.834755
$45$	0	0
$46$	−0.700017	−0.103212
$47$	−4.85708	−0.708477	−0.354239	−	0.935155i	$-0.615260\pi$
$47$	−4.85708	−0.708477	−0.354239	+	0.935155i	$0.615260\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	2.56291	0.362450
$51$	0	0
$52$	−3.54406	−0.491473
$53$	−7.63405	−1.04862	−0.524309	−	0.851528i	$-0.675676\pi$
$53$	−7.63405	−1.04862	−0.524309	+	0.851528i	$0.675676\pi$
$54$	0	0
$55$	−4.24297	−0.572123
$56$	2.45704	0.328336
$57$	0	0
$58$	−4.56699	−0.599675
$59$	10.2142	1.32978	0.664890	−	0.746941i	$-0.268478\pi$
$59$	10.2142	1.32978	0.664890	+	0.746941i	$0.268478\pi$
$60$	0	0
$61$	9.91001	1.26885	0.634423	−	0.772986i	$-0.281238\pi$
$61$	9.91001	1.26885	0.634423	+	0.772986i	$0.281238\pi$
$62$	−1.96702	−0.249812
$63$	0	0
$64$	1.47700	0.184624
$65$	2.71573	0.336845
$66$	0	0
$67$	6.07114	0.741708	0.370854	−	0.928691i	$-0.379065\pi$
$67$	6.07114	0.741708	0.370854	+	0.928691i	$0.379065\pi$
$68$	7.26293	0.880759
$69$	0	0
$70$	−0.809960	−0.0968088
$71$	3.65290	0.433520	0.216760	−	0.976225i	$-0.430451\pi$
$71$	3.65290	0.433520	0.216760	+	0.976225i	$0.430451\pi$
$72$	0	0
$73$	11.8183	1.38322	0.691612	−	0.722269i	$-0.256901\pi$
$73$	11.8183	1.38322	0.691612	+	0.722269i	$0.256901\pi$
$74$	−3.78704	−0.440234
$75$	0	0
$76$	−10.6711	−1.22406
$77$	−3.66704	−0.417897
$78$	0	0
$79$	−9.12408	−1.02654	−0.513269	−	0.858228i	$-0.671566\pi$
$79$	−9.12408	−1.02654	−0.513269	+	0.858228i	$0.671566\pi$
$80$	1.50416	0.168170
$81$	0	0
$82$	0.452968	0.0500219
$83$	15.8582	1.74066	0.870331	−	0.492468i	$-0.163905\pi$
$83$	15.8582	1.74066	0.870331	+	0.492468i	$0.163905\pi$
$84$	0	0
$85$	−5.56541	−0.603653
$86$	3.33704	0.359842
$87$	0	0
$88$	−9.01006	−0.960476
$89$	−4.66122	−0.494088	−0.247044	−	0.969004i	$-0.579459\pi$
$89$	−4.66122	−0.494088	−0.247044	+	0.969004i	$0.579459\pi$
$90$	0	0
$91$	2.34710	0.246043
$92$	−1.50998	−0.157426
$93$	0	0
$94$	3.40003	0.350687
$95$	8.17701	0.838944
$96$	0	0
$97$	−4.90419	−0.497945	−0.248973	−	0.968511i	$-0.580093\pi$
$97$	−4.90419	−0.497945	−0.248973	+	0.968511i	$0.580093\pi$
$98$	−0.700017	−0.0707124
$99$	0	0
$100$	5.52835	0.552835
$101$	5.80589	0.577707	0.288854	−	0.957373i	$-0.406726\pi$
$101$	5.80589	0.577707	0.288854	+	0.957373i	$0.406726\pi$
$102$	0	0
$103$	17.1169	1.68658	0.843288	−	0.537463i	$-0.180617\pi$
$103$	17.1169	1.68658	0.843288	+	0.537463i	$0.180617\pi$
$104$	5.76692	0.565493
$105$	0	0
$106$	5.34396	0.519052
$107$	8.34710	0.806944	0.403472	−	0.914992i	$-0.367803\pi$
$107$	8.34710	0.806944	0.403472	+	0.914992i	$0.367803\pi$
$108$	0	0
$109$	16.3771	1.56864	0.784321	−	0.620355i	$-0.213011\pi$
$109$	16.3771	1.56864	0.784321	+	0.620355i	$0.213011\pi$
$110$	2.97015	0.283193
$111$	0	0
$112$	1.29998	0.122837
$113$	14.1770	1.33366	0.666831	−	0.745209i	$-0.267650\pi$
$113$	14.1770	1.33366	0.666831	+	0.745209i	$0.267650\pi$
$114$	0	0
$115$	1.15706	0.107896
$116$	−9.85126	−0.914666
$117$	0	0
$118$	−7.15013	−0.658223
$119$	−4.80996	−0.440928
$120$	0	0
$121$	2.44715	0.222468
$122$	−6.93717	−0.628062
$123$	0	0
$124$	−4.24297	−0.381030
$125$	−10.0215	−0.896353
$126$	0	0
$127$	13.1184	1.16407	0.582036	−	0.813163i	$-0.302256\pi$
$127$	13.1184	1.16407	0.582036	+	0.813163i	$0.302256\pi$
$128$	10.6143	0.938177
$129$	0	0
$130$	−1.90106	−0.166734
$131$	1.28696	0.112442	0.0562209	−	0.998418i	$-0.482095\pi$
$131$	1.28696	0.112442	0.0562209	+	0.998418i	$0.482095\pi$
$132$	0	0
$133$	7.06707	0.612793
$134$	−4.24990	−0.367135
$135$	0	0
$136$	−11.8183	−1.01341
$137$	11.3152	0.966725	0.483362	−	0.875420i	$-0.339415\pi$
$137$	11.3152	0.966725	0.483362	+	0.875420i	$0.339415\pi$
$138$	0	0
$139$	8.93717	0.758041	0.379020	−	0.925388i	$-0.376261\pi$
$139$	8.93717	0.758041	0.379020	+	0.925388i	$0.376261\pi$
$140$	−1.74713	−0.147660
$141$	0	0
$142$	−2.55709	−0.214586
$143$	−8.60689	−0.719745
$144$	0	0
$145$	7.54878	0.626892
$146$	−8.27299	−0.684677
$147$	0	0
$148$	−8.16886	−0.671476
$149$	−1.67693	−0.137379	−0.0686897	−	0.997638i	$-0.521882\pi$
$149$	−1.67693	−0.137379	−0.0686897	+	0.997638i	$0.521882\pi$
$150$	0	0
$151$	−19.0110	−1.54709	−0.773547	−	0.633739i	$-0.781519\pi$
$151$	−19.0110	−1.54709	−0.773547	+	0.633739i	$0.781519\pi$
$152$	17.3641	1.40841
$153$	0	0
$154$	2.56699	0.206854
$155$	3.25129	0.261150
$156$	0	0
$157$	12.5712	1.00329	0.501647	−	0.865073i	$-0.332728\pi$
$157$	12.5712	1.00329	0.501647	+	0.865073i	$0.332728\pi$
$158$	6.38701	0.508123
$159$	0	0
$160$	−6.73882	−0.532750
$161$	1.00000	0.0788110
$162$	0	0
$163$	−15.6740	−1.22768	−0.613840	−	0.789431i	$-0.710376\pi$
$163$	−15.6740	−1.22768	−0.613840	+	0.789431i	$0.710376\pi$
$164$	0.977078	0.0762970
$165$	0	0
$166$	−11.1010	−0.861604
$167$	−4.96120	−0.383909	−0.191955	−	0.981404i	$-0.561483\pi$
$167$	−4.96120	−0.383909	−0.191955	+	0.981404i	$0.561483\pi$
$168$	0	0
$169$	−7.49113	−0.576241
$170$	3.89588	0.298800
$171$	0	0
$172$	7.19819	0.548857
$173$	−12.9042	−0.981087	−0.490544	−	0.871417i	$-0.663202\pi$
$173$	−12.9042	−0.981087	−0.490544	+	0.871417i	$0.663202\pi$
$174$	0	0
$175$	−3.66122	−0.276762
$176$	−4.76709	−0.359333
$177$	0	0
$178$	3.26293	0.244567
$179$	0.262762	0.0196398	0.00981989	−	0.999952i	$-0.496874\pi$
$179$	0.262762	0.0196398	0.00981989	+	0.999952i	$0.496874\pi$
$180$	0	0
$181$	−12.1440	−0.902659	−0.451329	−	0.892357i	$-0.649050\pi$
$181$	−12.1440	−0.902659	−0.451329	+	0.892357i	$0.649050\pi$
$182$	−1.64301	−0.121788
$183$	0	0
$184$	2.45704	0.181136
$185$	6.25960	0.460215
$186$	0	0
$187$	17.6383	1.28984
$188$	7.33407	0.534892
$189$	0	0
$190$	−5.72404	−0.415266
$191$	−2.68524	−0.194297	−0.0971487	−	0.995270i	$-0.530972\pi$
$191$	−2.68524	−0.194297	−0.0971487	+	0.995270i	$0.530972\pi$
$192$	0	0
$193$	−18.7252	−1.34787	−0.673933	−	0.738793i	$-0.735396\pi$
$193$	−18.7252	−1.34787	−0.673933	+	0.738793i	$0.735396\pi$
$194$	3.43301	0.246476
$195$	0	0
$196$	−1.50998	−0.107855
$197$	−19.3953	−1.38186	−0.690930	−	0.722922i	$-0.742799\pi$
$197$	−19.3953	−1.38186	−0.690930	+	0.722922i	$0.742799\pi$
$198$	0	0
$199$	1.10005	0.0779805	0.0389902	−	0.999240i	$-0.487586\pi$
$199$	1.10005	0.0779805	0.0389902	+	0.999240i	$0.487586\pi$
$200$	−8.99576	−0.636096
$201$	0	0
$202$	−4.06422	−0.285957
$203$	6.52411	0.457903
$204$	0	0
$205$	−0.748712	−0.0522923
$206$	−11.9821	−0.834832
$207$	0	0
$208$	3.05119	0.211562
$209$	−25.9152	−1.79259
$210$	0	0
$211$	25.5235	1.75711	0.878554	−	0.477643i	$-0.158509\pi$
$211$	25.5235	1.75711	0.878554	+	0.477643i	$0.158509\pi$
$212$	11.5272	0.791694
$213$	0	0
$214$	−5.84311	−0.399427
$215$	−5.51580	−0.376174
$216$	0	0
$217$	2.80996	0.190753
$218$	−11.4642	−0.776457
$219$	0	0
$220$	6.40679	0.431946
$221$	−11.2894	−0.759411
$222$	0	0
$223$	−17.1772	−1.15027	−0.575134	−	0.818059i	$-0.695050\pi$
$223$	−17.1772	−1.15027	−0.575134	+	0.818059i	$0.695050\pi$
$224$	−5.82409	−0.389139
$225$	0	0
$226$	−9.92414	−0.660144
$227$	26.0334	1.72790	0.863950	−	0.503577i	$-0.167983\pi$
$227$	26.0334	1.72790	0.863950	+	0.503577i	$0.167983\pi$
$228$	0	0
$229$	−7.01430	−0.463518	−0.231759	−	0.972773i	$-0.574448\pi$
$229$	−7.01430	−0.463518	−0.231759	+	0.972773i	$0.574448\pi$
$230$	−0.809960	−0.0534072
$231$	0	0
$232$	16.0300	1.05242
$233$	−0.671109	−0.0439658	−0.0219829	−	0.999758i	$-0.506998\pi$
$233$	−0.671109	−0.0439658	−0.0219829	+	0.999758i	$0.506998\pi$
$234$	0	0
$235$	−5.61992	−0.366603
$236$	−15.4233	−1.00397
$237$	0	0
$238$	3.36705	0.218254
$239$	26.5571	1.71784	0.858918	−	0.512114i	$-0.171137\pi$
$239$	26.5571	1.71784	0.858918	+	0.512114i	$0.171137\pi$
$240$	0	0
$241$	−4.97284	−0.320329	−0.160164	−	0.987090i	$-0.551202\pi$
$241$	−4.97284	−0.320329	−0.160164	+	0.987090i	$0.551202\pi$
$242$	−1.71304	−0.110119
$243$	0	0
$244$	−14.9639	−0.957965
$245$	1.15706	0.0739218
$246$	0	0
$247$	16.5871	1.05541
$248$	6.90419	0.438417
$249$	0	0
$250$	7.01524	0.443683
$251$	14.4281	0.910696	0.455348	−	0.890314i	$-0.349515\pi$
$251$	14.4281	0.910696	0.455348	+	0.890314i	$0.349515\pi$
$252$	0	0
$253$	−3.66704	−0.230545
$254$	−9.18311	−0.576200
$255$	0	0
$256$	−10.3842	−0.649010
$257$	−5.44889	−0.339893	−0.169946	−	0.985453i	$-0.554359\pi$
$257$	−5.44889	−0.339893	−0.169946	+	0.985453i	$0.554359\pi$
$258$	0	0
$259$	5.40993	0.336157
$260$	−4.10069	−0.254314
$261$	0	0
$262$	−0.900890	−0.0556572
$263$	−15.8128	−0.975060	−0.487530	−	0.873106i	$-0.662102\pi$
$263$	−15.8128	−0.975060	−0.487530	+	0.873106i	$0.662102\pi$
$264$	0	0
$265$	−8.83305	−0.542610
$266$	−4.94707	−0.303324
$267$	0	0
$268$	−9.16728	−0.559981
$269$	−11.9011	−0.725620	−0.362810	−	0.931863i	$-0.618183\pi$
$269$	−11.9011	−0.725620	−0.362810	+	0.931863i	$0.618183\pi$
$270$	0	0
$271$	−16.6664	−1.01241	−0.506206	−	0.862413i	$-0.668952\pi$
$271$	−16.6664	−1.01241	−0.506206	+	0.862413i	$0.668952\pi$
$272$	−6.25287	−0.379136
$273$	0	0
$274$	−7.92084	−0.478516
$275$	13.4258	0.809607
$276$	0	0
$277$	20.1299	1.20949	0.604744	−	0.796420i	$-0.293275\pi$
$277$	20.1299	1.20949	0.604744	+	0.796420i	$0.293275\pi$
$278$	−6.25617	−0.375220
$279$	0	0
$280$	2.84294	0.169898
$281$	−19.4022	−1.15744	−0.578720	−	0.815526i	$-0.696448\pi$
$281$	−19.4022	−1.15744	−0.578720	+	0.815526i	$0.696448\pi$
$282$	0	0
$283$	−10.8313	−0.643854	−0.321927	−	0.946764i	$-0.604331\pi$
$283$	−10.8313	−0.643854	−0.321927	+	0.946764i	$0.604331\pi$
$284$	−5.51580	−0.327302
$285$	0	0
$286$	6.02497	0.356264
$287$	−0.647082	−0.0381960
$288$	0	0
$289$	6.13572	0.360925
$290$	−5.28427	−0.310303
$291$	0	0
$292$	−17.8453	−1.04432
$293$	−8.89430	−0.519610	−0.259805	−	0.965661i	$-0.583658\pi$
$293$	−8.89430	−0.519610	−0.259805	+	0.965661i	$0.583658\pi$
$294$	0	0
$295$	11.8185	0.688098
$296$	13.2924	0.772606
$297$	0	0
$298$	1.17388	0.0680009
$299$	2.34710	0.135736
$300$	0	0
$301$	−4.76709	−0.274770
$302$	13.3080	0.765790
$303$	0	0
$304$	9.18707	0.526915
$305$	11.4665	0.656568
$306$	0	0
$307$	19.7442	1.12686	0.563429	−	0.826164i	$-0.309482\pi$
$307$	19.7442	1.12686	0.563429	+	0.826164i	$0.309482\pi$
$308$	5.53714	0.315508
$309$	0	0
$310$	−2.27596	−0.129266
$311$	28.5898	1.62118	0.810589	−	0.585615i	$-0.199147\pi$
$311$	28.5898	1.62118	0.810589	+	0.585615i	$0.199147\pi$
$312$	0	0
$313$	12.6570	0.715415	0.357707	−	0.933834i	$-0.383559\pi$
$313$	12.6570	0.715415	0.357707	+	0.933834i	$0.383559\pi$
$314$	−8.80007	−0.496616
$315$	0	0
$316$	13.7771	0.775025
$317$	−4.83130	−0.271353	−0.135676	−	0.990753i	$-0.543321\pi$
$317$	−4.83130	−0.271353	−0.135676	+	0.990753i	$0.543321\pi$
$318$	0	0
$319$	−23.9241	−1.33949
$320$	1.70897	0.0955343
$321$	0	0
$322$	−0.700017	−0.0390104
$323$	−33.9923	−1.89138
$324$	0	0
$325$	−8.59323	−0.476667
$326$	10.9720	0.607685
$327$	0	0
$328$	−1.58991	−0.0877879
$329$	−4.85708	−0.267779
$330$	0	0
$331$	−6.84544	−0.376259	−0.188130	−	0.982144i	$-0.560242\pi$
$331$	−6.84544	−0.376259	−0.188130	+	0.982144i	$0.560242\pi$
$332$	−23.9455	−1.31418
$333$	0	0
$334$	3.47292	0.190030
$335$	7.02467	0.383799
$336$	0	0
$337$	−8.17527	−0.445335	−0.222668	−	0.974894i	$-0.571476\pi$
$337$	−8.17527	−0.445335	−0.222668	+	0.974894i	$0.571476\pi$
$338$	5.24391	0.285231
$339$	0	0
$340$	8.40363	0.455751
$341$	−10.3042	−0.558005
$342$	0	0
$343$	1.00000	0.0539949
$344$	−11.7129	−0.631519
$345$	0	0
$346$	9.03315	0.485625
$347$	−21.2699	−1.14183	−0.570913	−	0.821011i	$-0.693411\pi$
$347$	−21.2699	−1.14183	−0.570913	+	0.821011i	$0.693411\pi$
$348$	0	0
$349$	1.96527	0.105199	0.0525993	−	0.998616i	$-0.483249\pi$
$349$	1.96527	0.105199	0.0525993	+	0.998616i	$0.483249\pi$
$350$	2.56291	0.136993
$351$	0	0
$352$	21.3572	1.13834
$353$	−10.3124	−0.548872	−0.274436	−	0.961605i	$-0.588491\pi$
$353$	−10.3124	−0.548872	−0.274436	+	0.961605i	$0.588491\pi$
$354$	0	0
$355$	4.22662	0.224326
$356$	7.03833	0.373031
$357$	0	0
$358$	−0.183938	−0.00972142
$359$	26.3856	1.39258	0.696289	−	0.717761i	$-0.254833\pi$
$359$	26.3856	1.39258	0.696289	+	0.717761i	$0.254833\pi$
$360$	0	0
$361$	30.9435	1.62860
$362$	8.50102	0.446804
$363$	0	0
$364$	−3.54406	−0.185759
$365$	13.6744	0.715753
$366$	0	0
$367$	−33.5667	−1.75217	−0.876083	−	0.482160i	$-0.839852\pi$
$367$	−33.5667	−1.75217	−0.876083	+	0.482160i	$0.839852\pi$
$368$	1.29998	0.0677663
$369$	0	0
$370$	−4.38183	−0.227800
$371$	−7.63405	−0.396340
$372$	0	0
$373$	−17.0698	−0.883838	−0.441919	−	0.897055i	$-0.645702\pi$
$373$	−17.0698	−0.883838	−0.441919	+	0.897055i	$0.645702\pi$
$374$	−12.3471	−0.638453
$375$	0	0
$376$	−11.9340	−0.615451
$377$	15.3127	0.788646
$378$	0	0
$379$	−15.7541	−0.809232	−0.404616	−	0.914487i	$-0.632595\pi$
$379$	−15.7541	−0.809232	−0.404616	+	0.914487i	$0.632595\pi$
$380$	−12.3471	−0.633393
$381$	0	0
$382$	1.87971	0.0961746
$383$	−3.85818	−0.197144	−0.0985719	−	0.995130i	$-0.531427\pi$
$383$	−3.85818	−0.197144	−0.0985719	+	0.995130i	$0.531427\pi$
$384$	0	0
$385$	−4.24297	−0.216242
$386$	13.1079	0.667175
$387$	0	0
$388$	7.40521	0.375943
$389$	−35.0666	−1.77795	−0.888973	−	0.457959i	$-0.848581\pi$
$389$	−35.0666	−1.77795	−0.888973	+	0.457959i	$0.848581\pi$
$390$	0	0
$391$	−4.80996	−0.243250
$392$	2.45704	0.124099
$393$	0	0
$394$	13.5770	0.684002
$395$	−10.5571	−0.531185
$396$	0	0
$397$	−15.3422	−0.770004	−0.385002	−	0.922916i	$-0.625799\pi$
$397$	−15.3422	−0.770004	−0.385002	+	0.922916i	$0.625799\pi$
$398$	−0.770053	−0.0385993
$399$	0	0
$400$	−4.75952	−0.237976
$401$	9.24705	0.461776	0.230888	−	0.972980i	$-0.425837\pi$
$401$	9.24705	0.461776	0.230888	+	0.972980i	$0.425837\pi$
$402$	0	0
$403$	6.59525	0.328533
$404$	−8.76675	−0.436162
$405$	0	0
$406$	−4.56699	−0.226656
$407$	−19.8384	−0.983353
$408$	0	0
$409$	−16.6498	−0.823278	−0.411639	−	0.911347i	$-0.635044\pi$
$409$	−16.6498	−0.823278	−0.411639	+	0.911347i	$0.635044\pi$
$410$	0.524110	0.0258840
$411$	0	0
$412$	−25.8461	−1.27334
$413$	10.2142	0.502610
$414$	0	0
$415$	18.3488	0.900709
$416$	−13.6697	−0.670213
$417$	0	0
$418$	18.1411	0.887308
$419$	6.86306	0.335282	0.167641	−	0.985848i	$-0.446385\pi$
$419$	6.86306	0.335282	0.167641	+	0.985848i	$0.446385\pi$
$420$	0	0
$421$	−13.1754	−0.642131	−0.321066	−	0.947057i	$-0.604041\pi$
$421$	−13.1754	−0.642131	−0.321066	+	0.947057i	$0.604041\pi$
$422$	−17.8669	−0.869745
$423$	0	0
$424$	−18.7572	−0.910930
$425$	17.6103	0.854225
$426$	0	0
$427$	9.91001	0.479579
$428$	−12.6039	−0.609234
$429$	0	0
$430$	3.86115	0.186201
$431$	26.0837	1.25641	0.628204	−	0.778048i	$-0.283790\pi$
$431$	26.0837	1.25641	0.628204	+	0.778048i	$0.283790\pi$
$432$	0	0
$433$	−39.8676	−1.91591	−0.957957	−	0.286911i	$-0.907372\pi$
$433$	−39.8676	−1.91591	−0.957957	+	0.286911i	$0.907372\pi$
$434$	−1.96702	−0.0944199
$435$	0	0
$436$	−24.7291	−1.18431
$437$	7.06707	0.338064
$438$	0	0
$439$	−17.9160	−0.855084	−0.427542	−	0.903996i	$-0.640620\pi$
$439$	−17.9160	−0.855084	−0.427542	+	0.903996i	$0.640620\pi$
$440$	−10.4252	−0.497000
$441$	0	0
$442$	7.90280	0.375898
$443$	3.02810	0.143869	0.0719347	−	0.997409i	$-0.477083\pi$
$443$	3.02810	0.143869	0.0719347	+	0.997409i	$0.477083\pi$
$444$	0	0
$445$	−5.39330	−0.255667
$446$	12.0243	0.569368
$447$	0	0
$448$	1.47700	0.0697815
$449$	30.0253	1.41698	0.708491	−	0.705720i	$-0.249376\pi$
$449$	30.0253	1.41698	0.708491	+	0.705720i	$0.249376\pi$
$450$	0	0
$451$	2.37287	0.111734
$452$	−21.4070	−1.00690
$453$	0	0
$454$	−18.2238	−0.855287
$455$	2.71573	0.127315
$456$	0	0
$457$	12.2169	0.571483	0.285742	−	0.958307i	$-0.407760\pi$
$457$	12.2169	0.571483	0.285742	+	0.958307i	$0.407760\pi$
$458$	4.91013	0.229435
$459$	0	0
$460$	−1.74713	−0.0814604
$461$	−33.7511	−1.57194	−0.785972	−	0.618261i	$-0.787837\pi$
$461$	−33.7511	−1.57194	−0.785972	+	0.618261i	$0.787837\pi$
$462$	0	0
$463$	−21.9187	−1.01865	−0.509324	−	0.860575i	$-0.670104\pi$
$463$	−21.9187	−1.01865	−0.509324	+	0.860575i	$0.670104\pi$
$464$	8.48124	0.393731
$465$	0	0
$466$	0.469788	0.0217625
$467$	15.6499	0.724193	0.362096	−	0.932141i	$-0.382061\pi$
$467$	15.6499	0.724193	0.362096	+	0.932141i	$0.382061\pi$
$468$	0	0
$469$	6.07114	0.280339
$470$	3.93404	0.181464
$471$	0	0
$472$	25.0968	1.15517
$473$	17.4811	0.803780
$474$	0	0
$475$	−25.8741	−1.18718
$476$	7.26293	0.332896
$477$	0	0
$478$	−18.5904	−0.850305
$479$	−6.20999	−0.283742	−0.141871	−	0.989885i	$-0.545312\pi$
$479$	−6.20999	−0.283742	−0.141871	+	0.989885i	$0.545312\pi$
$480$	0	0
$481$	12.6976	0.578962
$482$	3.48107	0.158558
$483$	0	0
$484$	−3.69514	−0.167961
$485$	−5.67444	−0.257663
$486$	0	0
$487$	−7.82642	−0.354649	−0.177325	−	0.984152i	$-0.556744\pi$
$487$	−7.82642	−0.354649	−0.177325	+	0.984152i	$0.556744\pi$
$488$	24.3493	1.10224
$489$	0	0
$490$	−0.809960	−0.0365903
$491$	32.0550	1.44662	0.723311	−	0.690523i	$-0.242620\pi$
$491$	32.0550	1.44662	0.723311	+	0.690523i	$0.242620\pi$
$492$	0	0
$493$	−31.3807	−1.41332
$494$	−11.6112	−0.522415
$495$	0	0
$496$	3.65290	0.164020
$497$	3.65290	0.163855
$498$	0	0
$499$	42.3592	1.89626	0.948129	−	0.317885i	$-0.102973\pi$
$499$	42.3592	1.89626	0.948129	+	0.317885i	$0.102973\pi$
$500$	15.1323	0.676736
$501$	0	0
$502$	−10.0999	−0.450782
$503$	−11.6053	−0.517455	−0.258728	−	0.965950i	$-0.583303\pi$
$503$	−11.6053	−0.517455	−0.258728	+	0.965950i	$0.583303\pi$
$504$	0	0
$505$	6.71775	0.298936
$506$	2.56699	0.114116
$507$	0	0
$508$	−19.8085	−0.878861
$509$	12.1512	0.538594	0.269297	−	0.963057i	$-0.413209\pi$
$509$	12.1512	0.538594	0.269297	+	0.963057i	$0.413209\pi$
$510$	0	0
$511$	11.8183	0.522810
$512$	−13.9595	−0.616926
$513$	0	0
$514$	3.81432	0.168242
$515$	19.8052	0.872722
$516$	0	0
$517$	17.8111	0.783330
$518$	−3.78704	−0.166393
$519$	0	0
$520$	6.67266	0.292616
$521$	15.3884	0.674178	0.337089	−	0.941473i	$-0.390558\pi$
$521$	15.3884	0.674178	0.337089	+	0.941473i	$0.390558\pi$
$522$	0	0
$523$	−14.3536	−0.627637	−0.313818	−	0.949483i	$-0.601608\pi$
$523$	−14.3536	−0.627637	−0.313818	+	0.949483i	$0.601608\pi$
$524$	−1.94327	−0.0848923
$525$	0	0
$526$	11.0692	0.482641
$527$	−13.5158	−0.588757
$528$	0	0
$529$	1.00000	0.0434783
$530$	6.18328	0.268585
$531$	0	0
$532$	−10.6711	−0.462651
$533$	−1.51876	−0.0657850
$534$	0	0
$535$	9.65808	0.417555
$536$	14.9171	0.644319
$537$	0	0
$538$	8.33094	0.359172
$539$	−3.66704	−0.157950
$540$	0	0
$541$	31.1451	1.33903	0.669517	−	0.742797i	$-0.266501\pi$
$541$	31.1451	1.33903	0.669517	+	0.742797i	$0.266501\pi$
$542$	11.6668	0.501130
$543$	0	0
$544$	28.0137	1.20108
$545$	18.9493	0.811698
$546$	0	0
$547$	12.6630	0.541429	0.270715	−	0.962660i	$-0.412740\pi$
$547$	12.6630	0.541429	0.270715	+	0.962660i	$0.412740\pi$
$548$	−17.0857	−0.729866
$549$	0	0
$550$	−9.39829	−0.400744
$551$	46.1063	1.96420
$552$	0	0
$553$	−9.12408	−0.387995
$554$	−14.0913	−0.598680
$555$	0	0
$556$	−13.4949	−0.572312
$557$	30.7740	1.30394	0.651968	−	0.758246i	$-0.273943\pi$
$557$	30.7740	1.30394	0.651968	+	0.758246i	$0.273943\pi$
$558$	0	0
$559$	−11.1888	−0.473237
$560$	1.50416	0.0635622
$561$	0	0
$562$	13.5819	0.572918
$563$	−35.4352	−1.49342	−0.746708	−	0.665152i	$-0.768367\pi$
$563$	−35.4352	−1.49342	−0.746708	+	0.665152i	$0.768367\pi$
$564$	0	0
$565$	16.4036	0.690106
$566$	7.58209	0.318699
$567$	0	0
$568$	8.97533	0.376597
$569$	−2.15939	−0.0905262	−0.0452631	−	0.998975i	$-0.514413\pi$
$569$	−2.15939	−0.0905262	−0.0452631	+	0.998975i	$0.514413\pi$
$570$	0	0
$571$	−26.4182	−1.10557	−0.552784	−	0.833324i	$-0.686435\pi$
$571$	−26.4182	−1.10557	−0.552784	+	0.833324i	$0.686435\pi$
$572$	12.9962	0.543399
$573$	0	0
$574$	0.452968	0.0189065
$575$	−3.66122	−0.152683
$576$	0	0
$577$	−12.9885	−0.540719	−0.270360	−	0.962759i	$-0.587143\pi$
$577$	−12.9885	−0.540719	−0.270360	+	0.962759i	$0.587143\pi$
$578$	−4.29510	−0.178653
$579$	0	0
$580$	−11.3985	−0.473296
$581$	15.8582	0.657908
$582$	0	0
$583$	27.9943	1.15941
$584$	29.0380	1.20160
$585$	0	0
$586$	6.22615	0.257200
$587$	−42.1437	−1.73946	−0.869729	−	0.493529i	$-0.835707\pi$
$587$	−42.1437	−1.73946	−0.869729	+	0.493529i	$0.835707\pi$
$588$	0	0
$589$	19.8582	0.818242
$590$	−8.27312	−0.340599
$591$	0	0
$592$	7.03282	0.289047
$593$	−10.2777	−0.422055	−0.211027	−	0.977480i	$-0.567681\pi$
$593$	−10.2777	−0.422055	−0.211027	+	0.977480i	$0.567681\pi$
$594$	0	0
$595$	−5.56541	−0.228159
$596$	2.53212	0.103720
$597$	0	0
$598$	−1.64301	−0.0671876
$599$	6.73566	0.275212	0.137606	−	0.990487i	$-0.456059\pi$
$599$	6.73566	0.275212	0.137606	+	0.990487i	$0.456059\pi$
$600$	0	0
$601$	−45.6995	−1.86412	−0.932062	−	0.362300i	$-0.881992\pi$
$601$	−45.6995	−1.86412	−0.932062	+	0.362300i	$0.881992\pi$
$602$	3.33704	0.136008
$603$	0	0
$604$	28.7062	1.16804
$605$	2.83149	0.115117
$606$	0	0
$607$	46.6518	1.89354	0.946769	−	0.321914i	$-0.104326\pi$
$607$	46.6518	1.89354	0.946769	+	0.321914i	$0.104326\pi$
$608$	−41.1593	−1.66923
$609$	0	0
$610$	−8.02671	−0.324992
$611$	−11.4000	−0.461196
$612$	0	0
$613$	40.2204	1.62448	0.812242	−	0.583320i	$-0.198247\pi$
$613$	40.2204	1.62448	0.812242	+	0.583320i	$0.198247\pi$
$614$	−13.8212	−0.557780
$615$	0	0
$616$	−9.01006	−0.363026
$617$	−11.1211	−0.447719	−0.223860	−	0.974621i	$-0.571866\pi$
$617$	−11.1211	−0.447719	−0.223860	+	0.974621i	$0.571866\pi$
$618$	0	0
$619$	−45.1974	−1.81664	−0.908319	−	0.418278i	$-0.862634\pi$
$619$	−45.1974	−1.81664	−0.908319	+	0.418278i	$0.862634\pi$
$620$	−4.90937	−0.197165
$621$	0	0
$622$	−20.0133	−0.802461
$623$	−4.66122	−0.186748
$624$	0	0
$625$	6.71058	0.268423
$626$	−8.86009	−0.354121
$627$	0	0
$628$	−18.9823	−0.757475
$629$	−26.0215	−1.03755
$630$	0	0
$631$	−22.9463	−0.913478	−0.456739	−	0.889601i	$-0.650983\pi$
$631$	−22.9463	−0.913478	−0.456739	+	0.889601i	$0.650983\pi$
$632$	−22.4182	−0.891750
$633$	0	0
$634$	3.38199	0.134316
$635$	15.1788	0.602352
$636$	0	0
$637$	2.34710	0.0929954
$638$	16.7473	0.663032
$639$	0	0
$640$	12.2813	0.485462
$641$	12.5876	0.497179	0.248590	−	0.968609i	$-0.420033\pi$
$641$	12.5876	0.497179	0.248590	+	0.968609i	$0.420033\pi$
$642$	0	0
$643$	0.547672	0.0215981	0.0107990	−	0.999942i	$-0.496562\pi$
$643$	0.547672	0.0215981	0.0107990	+	0.999942i	$0.496562\pi$
$644$	−1.50998	−0.0595014
$645$	0	0
$646$	23.7952	0.936209
$647$	16.7393	0.658089	0.329045	−	0.944314i	$-0.393273\pi$
$647$	16.7393	0.658089	0.329045	+	0.944314i	$0.393273\pi$
$648$	0	0
$649$	−37.4560	−1.47027
$650$	6.01541	0.235944
$651$	0	0
$652$	23.6673	0.926884
$653$	−2.63120	−0.102967	−0.0514834	−	0.998674i	$-0.516395\pi$
$653$	−2.63120	−0.102967	−0.0514834	+	0.998674i	$0.516395\pi$
$654$	0	0
$655$	1.48908	0.0581833
$656$	−0.841196	−0.0328432
$657$	0	0
$658$	3.40003	0.132547
$659$	28.7983	1.12182	0.560912	−	0.827876i	$-0.310451\pi$
$659$	28.7983	1.12182	0.560912	+	0.827876i	$0.310451\pi$
$660$	0	0
$661$	2.76143	0.107407	0.0537036	−	0.998557i	$-0.482897\pi$
$661$	2.76143	0.107407	0.0537036	+	0.998557i	$0.482897\pi$
$662$	4.79192	0.186243
$663$	0	0
$664$	38.9642	1.51210
$665$	8.17701	0.317091
$666$	0	0
$667$	6.52411	0.252615
$668$	7.49130	0.289847
$669$	0	0
$670$	−4.91738	−0.189975
$671$	−36.3404	−1.40290
$672$	0	0
$673$	32.8777	1.26734	0.633670	−	0.773603i	$-0.281548\pi$
$673$	32.8777	1.26734	0.633670	+	0.773603i	$0.281548\pi$
$674$	5.72282	0.220435
$675$	0	0
$676$	11.3114	0.435055
$677$	−34.5014	−1.32599	−0.662997	−	0.748622i	$-0.730716\pi$
$677$	−34.5014	−1.32599	−0.662997	+	0.748622i	$0.730716\pi$
$678$	0	0
$679$	−4.90419	−0.188206
$680$	−13.6744	−0.524391
$681$	0	0
$682$	7.21313	0.276205
$683$	−34.9805	−1.33849	−0.669246	−	0.743041i	$-0.733383\pi$
$683$	−34.9805	−1.33849	−0.669246	+	0.743041i	$0.733383\pi$
$684$	0	0
$685$	13.0924	0.500234
$686$	−0.700017	−0.0267268
$687$	0	0
$688$	−6.19713	−0.236263
$689$	−17.9179	−0.682617
$690$	0	0
$691$	34.0895	1.29683	0.648413	−	0.761289i	$-0.275433\pi$
$691$	34.0895	1.29683	0.648413	+	0.761289i	$0.275433\pi$
$692$	19.4850	0.740710
$693$	0	0
$694$	14.8893	0.565188
$695$	10.3408	0.392250
$696$	0	0
$697$	3.11244	0.117892
$698$	−1.37572	−0.0520719
$699$	0	0
$700$	5.52835	0.208952
$701$	43.6564	1.64888	0.824439	−	0.565950i	$-0.191491\pi$
$701$	43.6564	1.64888	0.824439	+	0.565950i	$0.191491\pi$
$702$	0	0
$703$	38.2323	1.44196
$704$	−5.41619	−0.204131
$705$	0	0
$706$	7.21883	0.271684
$707$	5.80589	0.218353
$708$	0	0
$709$	11.8247	0.444087	0.222044	−	0.975037i	$-0.428727\pi$
$709$	11.8247	0.444087	0.222044	+	0.975037i	$0.428727\pi$
$710$	−2.95870	−0.111038
$711$	0	0
$712$	−11.4528	−0.429212
$713$	2.80996	0.105234
$714$	0	0
$715$	−9.95868	−0.372433
$716$	−0.396765	−0.0148278
$717$	0	0
$718$	−18.4704	−0.689307
$719$	−33.8406	−1.26204	−0.631021	−	0.775766i	$-0.717364\pi$
$719$	−33.8406	−1.26204	−0.631021	+	0.775766i	$0.717364\pi$
$720$	0	0
$721$	17.1169	0.637466
$722$	−21.6609	−0.806136
$723$	0	0
$724$	18.3372	0.681497
$725$	−23.8862	−0.887110
$726$	0	0
$727$	43.3730	1.60862	0.804308	−	0.594212i	$-0.202536\pi$
$727$	43.3730	1.60862	0.804308	+	0.594212i	$0.202536\pi$
$728$	5.76692	0.213736
$729$	0	0
$730$	−9.57233	−0.354288
$731$	22.9295	0.848078
$732$	0	0
$733$	−35.3627	−1.30615	−0.653075	−	0.757293i	$-0.726521\pi$
$733$	−35.3627	−1.30615	−0.653075	+	0.757293i	$0.726521\pi$
$734$	23.4972	0.867299
$735$	0	0
$736$	−5.82409	−0.214679
$737$	−22.2631	−0.820072
$738$	0	0
$739$	−20.9639	−0.771169	−0.385584	−	0.922673i	$-0.626000\pi$
$739$	−20.9639	−0.771169	−0.385584	+	0.922673i	$0.626000\pi$
$740$	−9.45185	−0.347457
$741$	0	0
$742$	5.34396	0.196183
$743$	3.66859	0.134587	0.0672937	−	0.997733i	$-0.478564\pi$
$743$	3.66859	0.134587	0.0672937	+	0.997733i	$0.478564\pi$
$744$	0	0
$745$	−1.94031	−0.0710873
$746$	11.9491	0.437488
$747$	0	0
$748$	−26.6334	−0.973814
$749$	8.34710	0.304996
$750$	0	0
$751$	26.4450	0.964990	0.482495	−	0.875899i	$-0.339731\pi$
$751$	26.4450	0.964990	0.482495	+	0.875899i	$0.339731\pi$
$752$	−6.31412	−0.230252
$753$	0	0
$754$	−10.7192	−0.390369
$755$	−21.9968	−0.800547
$756$	0	0
$757$	31.6239	1.14939	0.574695	−	0.818368i	$-0.305121\pi$
$757$	31.6239	1.14939	0.574695	+	0.818368i	$0.305121\pi$
$758$	11.0281	0.400559
$759$	0	0
$760$	20.0913	0.728787
$761$	−13.8173	−0.500878	−0.250439	−	0.968132i	$-0.580575\pi$
$761$	−13.8173	−0.500878	−0.250439	+	0.968132i	$0.580575\pi$
$762$	0	0
$763$	16.3771	0.592891
$764$	4.05465	0.146692
$765$	0	0
$766$	2.70079	0.0975835
$767$	23.9738	0.865644
$768$	0	0
$769$	24.1624	0.871319	0.435659	−	0.900112i	$-0.356515\pi$
$769$	24.1624	0.871319	0.435659	+	0.900112i	$0.356515\pi$
$770$	2.97015	0.107037
$771$	0	0
$772$	28.2745	1.01762
$773$	−24.9557	−0.897595	−0.448798	−	0.893633i	$-0.648148\pi$
$773$	−24.9557	−0.897595	−0.448798	+	0.893633i	$0.648148\pi$
$774$	0	0
$775$	−10.2879	−0.369551
$776$	−12.0498	−0.432563
$777$	0	0
$778$	24.5472	0.880060
$779$	−4.57297	−0.163844
$780$	0	0
$781$	−13.3953	−0.479322
$782$	3.36705	0.120406
$783$	0	0
$784$	1.29998	0.0464280
$785$	14.5456	0.519156
$786$	0	0
$787$	27.6309	0.984935	0.492468	−	0.870331i	$-0.336095\pi$
$787$	27.6309	0.984935	0.492468	+	0.870331i	$0.336095\pi$
$788$	29.2865	1.04329
$789$	0	0
$790$	7.39014	0.262929
$791$	14.1770	0.504077
$792$	0	0
$793$	23.2598	0.825979
$794$	10.7398	0.381142
$795$	0	0
$796$	−1.66105	−0.0588743
$797$	47.8222	1.69395	0.846975	−	0.531632i	$-0.178421\pi$
$797$	47.8222	1.69395	0.846975	+	0.531632i	$0.178421\pi$
$798$	0	0
$799$	23.3623	0.826500
$800$	21.3233	0.753891
$801$	0	0
$802$	−6.47309	−0.228573
$803$	−43.3380	−1.52937
$804$	0	0
$805$	1.15706	0.0407810
$806$	−4.61679	−0.162619
$807$	0	0
$808$	14.2653	0.501852
$809$	53.9350	1.89625	0.948127	−	0.317891i	$-0.102975\pi$
$809$	53.9350	1.89625	0.948127	+	0.317891i	$0.102975\pi$
$810$	0	0
$811$	29.8321	1.04755	0.523774	−	0.851857i	$-0.324524\pi$
$811$	29.8321	1.04755	0.523774	+	0.851857i	$0.324524\pi$
$812$	−9.85126	−0.345711
$813$	0	0
$814$	13.8872	0.486746
$815$	−18.1357	−0.635266
$816$	0	0
$817$	−33.6893	−1.17864
$818$	11.6551	0.407511
$819$	0	0
$820$	1.13054	0.0394801
$821$	−38.4517	−1.34198	−0.670988	−	0.741469i	$-0.734130\pi$
$821$	−38.4517	−1.34198	−0.670988	+	0.741469i	$0.734130\pi$
$822$	0	0
$823$	22.4705	0.783274	0.391637	−	0.920120i	$-0.371909\pi$
$823$	22.4705	0.783274	0.391637	+	0.920120i	$0.371909\pi$
$824$	42.0569	1.46512
$825$	0	0
$826$	−7.15013	−0.248785
$827$	−15.5282	−0.539968	−0.269984	−	0.962865i	$-0.587018\pi$
$827$	−15.5282	−0.539968	−0.269984	+	0.962865i	$0.587018\pi$
$828$	0	0
$829$	−35.6581	−1.23846	−0.619228	−	0.785211i	$-0.712554\pi$
$829$	−35.6581	−1.23846	−0.619228	+	0.785211i	$0.712554\pi$
$830$	−12.8445	−0.445839
$831$	0	0
$832$	3.46665	0.120185
$833$	−4.80996	−0.166655
$834$	0	0
$835$	−5.74040	−0.198655
$836$	39.1313	1.35339
$837$	0	0
$838$	−4.80426	−0.165960
$839$	−16.9411	−0.584873	−0.292436	−	0.956285i	$-0.594466\pi$
$839$	−16.9411	−0.584873	−0.292436	+	0.956285i	$0.594466\pi$
$840$	0	0
$841$	13.5640	0.467725
$842$	9.22302	0.317846
$843$	0	0
$844$	−38.5398	−1.32660
$845$	−8.66768	−0.298177
$846$	0	0
$847$	2.44715	0.0840850
$848$	−9.92414	−0.340796
$849$	0	0
$850$	−12.3275	−0.422830
$851$	5.40993	0.185450
$852$	0	0
$853$	−2.83149	−0.0969485	−0.0484742	−	0.998824i	$-0.515436\pi$
$853$	−2.83149	−0.0969485	−0.0484742	+	0.998824i	$0.515436\pi$
$854$	−6.93717	−0.237385
$855$	0	0
$856$	20.5092	0.700989
$857$	3.73125	0.127457	0.0637286	−	0.997967i	$-0.479701\pi$
$857$	3.73125	0.127457	0.0637286	+	0.997967i	$0.479701\pi$
$858$	0	0
$859$	3.83133	0.130723	0.0653616	−	0.997862i	$-0.479180\pi$
$859$	3.83133	0.130723	0.0653616	+	0.997862i	$0.479180\pi$
$860$	8.32872	0.284007
$861$	0	0
$862$	−18.2590	−0.621905
$863$	21.3080	0.725333	0.362667	−	0.931919i	$-0.381866\pi$
$863$	21.3080	0.725333	0.362667	+	0.931919i	$0.381866\pi$
$864$	0	0
$865$	−14.9309	−0.507666
$866$	27.9080	0.948352
$867$	0	0
$868$	−4.24297	−0.144016
$869$	33.4583	1.13500
$870$	0	0
$871$	14.2496	0.482828
$872$	40.2393	1.36267
$873$	0	0
$874$	−4.94707	−0.167337
$875$	−10.0215	−0.338790
$876$	0	0
$877$	−34.4456	−1.16314	−0.581572	−	0.813495i	$-0.697562\pi$
$877$	−34.4456	−1.16314	−0.581572	+	0.813495i	$0.697562\pi$
$878$	12.5415	0.423255
$879$	0	0
$880$	−5.51580	−0.185937
$881$	−5.70600	−0.192240	−0.0961201	−	0.995370i	$-0.530643\pi$
$881$	−5.70600	−0.192240	−0.0961201	+	0.995370i	$0.530643\pi$
$882$	0	0
$883$	5.89756	0.198469	0.0992344	−	0.995064i	$-0.468361\pi$
$883$	5.89756	0.198469	0.0992344	+	0.995064i	$0.468361\pi$
$884$	17.0468	0.573346
$885$	0	0
$886$	−2.11972	−0.0712134
$887$	41.8153	1.40402	0.702010	−	0.712167i	$-0.252286\pi$
$887$	41.8153	1.40402	0.702010	+	0.712167i	$0.252286\pi$
$888$	0	0
$889$	13.1184	0.439978
$890$	3.77540	0.126552
$891$	0	0
$892$	25.9371	0.868440
$893$	−34.3253	−1.14865
$894$	0	0
$895$	0.304031	0.0101626
$896$	10.6143	0.354598
$897$	0	0
$898$	−21.0182	−0.701387
$899$	18.3325	0.611423
$900$	0	0
$901$	36.7195	1.22330
$902$	−1.66105	−0.0553069
$903$	0	0
$904$	34.8335	1.15855
$905$	−14.0514	−0.467083
$906$	0	0
$907$	−39.9593	−1.32683	−0.663414	−	0.748253i	$-0.730893\pi$
$907$	−39.9593	−1.32683	−0.663414	+	0.748253i	$0.730893\pi$
$908$	−39.3099	−1.30454
$909$	0	0
$910$	−1.90106	−0.0630194
$911$	25.1107	0.831954	0.415977	−	0.909375i	$-0.363440\pi$
$911$	25.1107	0.831954	0.415977	+	0.909375i	$0.363440\pi$
$912$	0	0
$913$	−58.1525	−1.92457
$914$	−8.55205	−0.282877
$915$	0	0
$916$	10.5914	0.349951
$917$	1.28696	0.0424990
$918$	0	0
$919$	37.6701	1.24262	0.621310	−	0.783565i	$-0.286601\pi$
$919$	37.6701	1.24262	0.621310	+	0.783565i	$0.286601\pi$
$920$	2.84294	0.0937290
$921$	0	0
$922$	23.6263	0.778091
$923$	8.57372	0.282207
$924$	0	0
$925$	−19.8069	−0.651247
$926$	15.3434	0.504217
$927$	0	0
$928$	−37.9970	−1.24731
$929$	15.5425	0.509933	0.254967	−	0.966950i	$-0.417936\pi$
$929$	15.5425	0.509933	0.254967	+	0.966950i	$0.417936\pi$
$930$	0	0
$931$	7.06707	0.231614
$932$	1.01336	0.0331937
$933$	0	0
$934$	−10.9552	−0.358466
$935$	20.4085	0.667431
$936$	0	0
$937$	18.2576	0.596449	0.298224	−	0.954496i	$-0.403606\pi$
$937$	18.2576	0.596449	0.298224	+	0.954496i	$0.403606\pi$
$938$	−4.24990	−0.138764
$939$	0	0
$940$	8.48595	0.276781
$941$	0.0776022	0.00252976	0.00126488	−	0.999999i	$-0.499597\pi$
$941$	0.0776022	0.00252976	0.00126488	+	0.999999i	$0.499597\pi$
$942$	0	0
$943$	−0.647082	−0.0210719
$944$	13.2783	0.432173
$945$	0	0
$946$	−12.2370	−0.397860
$947$	−10.0364	−0.326140	−0.163070	−	0.986615i	$-0.552140\pi$
$947$	−10.0364	−0.326140	−0.163070	+	0.986615i	$0.552140\pi$
$948$	0	0
$949$	27.7387	0.900435
$950$	18.1123	0.587640
$951$	0	0
$952$	−11.8183	−0.383032
$953$	−51.2265	−1.65939	−0.829695	−	0.558218i	$-0.811485\pi$
$953$	−51.2265	−1.65939	−0.829695	+	0.558218i	$0.811485\pi$
$954$	0	0
$955$	−3.10698	−0.100540
$956$	−40.1006	−1.29695
$957$	0	0
$958$	4.34710	0.140448
$959$	11.3152	0.365388
$960$	0	0
$961$	−23.1041	−0.745294
$962$	−8.88855	−0.286578
$963$	0	0
$964$	7.50887	0.241844
$965$	−21.6661	−0.697456
$966$	0	0
$967$	12.2315	0.393339	0.196670	−	0.980470i	$-0.436987\pi$
$967$	12.2315	0.393339	0.196670	+	0.980470i	$0.436987\pi$
$968$	6.01275	0.193257
$969$	0	0
$970$	3.97220	0.127540
$971$	−33.5753	−1.07748	−0.538741	−	0.842471i	$-0.681100\pi$
$971$	−33.5753	−1.07748	−0.538741	+	0.842471i	$0.681100\pi$
$972$	0	0
$973$	8.93717	0.286513
$974$	5.47863	0.175546
$975$	0	0
$976$	12.8828	0.412370
$977$	−23.7149	−0.758707	−0.379353	−	0.925252i	$-0.623854\pi$
$977$	−23.7149	−0.758707	−0.379353	+	0.925252i	$0.623854\pi$
$978$	0	0
$979$	17.0928	0.546290
$980$	−1.74713	−0.0558101
$981$	0	0
$982$	−22.4390	−0.716058
$983$	1.47810	0.0471441	0.0235721	−	0.999722i	$-0.492496\pi$
$983$	1.47810	0.0471441	0.0235721	+	0.999722i	$0.492496\pi$
$984$	0	0
$985$	−22.4415	−0.715046
$986$	21.9670	0.699572
$987$	0	0
$988$	−25.0461	−0.796824
$989$	−4.76709	−0.151584
$990$	0	0
$991$	20.5203	0.651850	0.325925	−	0.945396i	$-0.394324\pi$
$991$	20.5203	0.651850	0.325925	+	0.945396i	$0.394324\pi$
$992$	−16.3655	−0.519604
$993$	0	0
$994$	−2.55709	−0.0811060
$995$	1.27282	0.0403512
$996$	0	0
$997$	−22.7211	−0.719583	−0.359791	−	0.933033i	$-0.617152\pi$
$997$	−22.7211	−0.719583	−0.359791	+	0.933033i	$0.617152\pi$
$998$	−29.6521	−0.938622
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1449.2.a.p.1.2		4
3.2	odd	2		483.2.a.i.1.3	✓	4
12.11	even	2		7728.2.a.cd.1.1		4
21.20	even	2		3381.2.a.w.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
483.2.a.i.1.3	✓	4	3.2	odd	2
1449.2.a.p.1.2		4	1.1	even	1	trivial
3381.2.a.w.1.3		4	21.20	even	2
7728.2.a.cd.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)

\(f(q)\)	\(=\)	\(q-0.700017 q^{2} -1.50998 q^{4} +1.15706 q^{5} +1.00000 q^{7} +2.45704 q^{8} +O(q^{10})\)	\(q-0.700017 q^{2} -1.50998 q^{4} +1.15706 q^{5} +1.00000 q^{7} +2.45704 q^{8} -0.809960 q^{10} -3.66704 q^{11} +2.34710 q^{13} -0.700017 q^{14} +1.29998 q^{16} -4.80996 q^{17} +7.06707 q^{19} -1.74713 q^{20} +2.56699 q^{22} +1.00000 q^{23} -3.66122 q^{25} -1.64301 q^{26} -1.50998 q^{28} +6.52411 q^{29} +2.80996 q^{31} -5.82409 q^{32} +3.36705 q^{34} +1.15706 q^{35} +5.40993 q^{37} -4.94707 q^{38} +2.84294 q^{40} -0.647082 q^{41} -4.76709 q^{43} +5.53714 q^{44} -0.700017 q^{46} -4.85708 q^{47} +1.00000 q^{49} +2.56291 q^{50} -3.54406 q^{52} -7.63405 q^{53} -4.24297 q^{55} +2.45704 q^{56} -4.56699 q^{58} +10.2142 q^{59} +9.91001 q^{61} -1.96702 q^{62} +1.47700 q^{64} +2.71573 q^{65} +6.07114 q^{67} +7.26293 q^{68} -0.809960 q^{70} +3.65290 q^{71} +11.8183 q^{73} -3.78704 q^{74} -10.6711 q^{76} -3.66704 q^{77} -9.12408 q^{79} +1.50416 q^{80} +0.452968 q^{82} +15.8582 q^{83} -5.56541 q^{85} +3.33704 q^{86} -9.01006 q^{88} -4.66122 q^{89} +2.34710 q^{91} -1.50998 q^{92} +3.40003 q^{94} +8.17701 q^{95} -4.90419 q^{97} -0.700017 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{4} - 5 q^{5} + 4 q^{7} + 3 q^{8}+O(q^{10}) \) 4 * q + 4 * q^4 - 5 * q^5 + 4 * q^7 + 3 * q^8	\( 4 q + 4 q^{4} - 5 q^{5} + 4 q^{7} + 3 q^{8} + 4 q^{10} + 5 q^{11} + 7 q^{13} + 8 q^{16} - 12 q^{17} + 3 q^{19} + q^{20} - q^{22} + 4 q^{23} + 7 q^{25} - 5 q^{26} + 4 q^{28} - 6 q^{29} + 4 q^{31} + 6 q^{32} - 9 q^{34} - 5 q^{35} + 20 q^{37} - 23 q^{38} + 21 q^{40} - 3 q^{41} + 9 q^{43} + 27 q^{44} - 7 q^{47} + 4 q^{49} - 3 q^{50} + 38 q^{52} + 6 q^{53} - 21 q^{55} + 3 q^{56} - 7 q^{58} + 2 q^{59} + 24 q^{61} + 9 q^{62} - 21 q^{64} + 14 q^{65} + q^{67} + 13 q^{68} + 4 q^{70} + 17 q^{71} + 16 q^{73} + 33 q^{74} - 25 q^{76} + 5 q^{77} - 10 q^{79} - 6 q^{80} - 7 q^{82} - 8 q^{83} + 17 q^{85} + 35 q^{86} - 12 q^{88} + 3 q^{89} + 7 q^{91} + 4 q^{92} + 8 q^{94} + 3 q^{95} - 2 q^{97}+O(q^{100}) \) 4 * q + 4 * q^4 - 5 * q^5 + 4 * q^7 + 3 * q^8 + 4 * q^10 + 5 * q^11 + 7 * q^13 + 8 * q^16 - 12 * q^17 + 3 * q^19 + q^20 - q^22 + 4 * q^23 + 7 * q^25 - 5 * q^26 + 4 * q^28 - 6 * q^29 + 4 * q^31 + 6 * q^32 - 9 * q^34 - 5 * q^35 + 20 * q^37 - 23 * q^38 + 21 * q^40 - 3 * q^41 + 9 * q^43 + 27 * q^44 - 7 * q^47 + 4 * q^49 - 3 * q^50 + 38 * q^52 + 6 * q^53 - 21 * q^55 + 3 * q^56 - 7 * q^58 + 2 * q^59 + 24 * q^61 + 9 * q^62 - 21 * q^64 + 14 * q^65 + q^67 + 13 * q^68 + 4 * q^70 + 17 * q^71 + 16 * q^73 + 33 * q^74 - 25 * q^76 + 5 * q^77 - 10 * q^79 - 6 * q^80 - 7 * q^82 - 8 * q^83 + 17 * q^85 + 35 * q^86 - 12 * q^88 + 3 * q^89 + 7 * q^91 + 4 * q^92 + 8 * q^94 + 3 * q^95 - 2 * q^97

Introduction

L-functions

Modular forms

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Database

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