LMFDB - Embedded newform 1449.2.a.p.1.4

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			1.4
Root			$2.46506$ 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)

$f(q)$	$=$	$q+2.46506 q^{2} +4.07653 q^{4} +0.653724 q^{5} +1.00000 q^{7} +5.11879 q^{8} +O(q^{10})$	\(q+2.46506 q^{2} +4.07653 q^{4} +0.653724 q^{5} +1.00000 q^{7} +5.11879 q^{8} +1.61147 q^{10} +2.42281 q^{11} +4.26520 q^{13} +2.46506 q^{14} +4.46506 q^{16} -2.38853 q^{17} -5.35294 q^{19} +2.66493 q^{20} +5.97238 q^{22} +1.00000 q^{23} -4.57264 q^{25} +10.5140 q^{26} +4.07653 q^{28} -3.23415 q^{29} +0.388529 q^{31} +0.769086 q^{32} -5.88787 q^{34} +0.653724 q^{35} +9.31865 q^{37} -13.1953 q^{38} +3.34628 q^{40} -5.73026 q^{41} +10.8180 q^{43} +9.87667 q^{44} +2.46506 q^{46} -1.18866 q^{47} +1.00000 q^{49} -11.2719 q^{50} +17.3872 q^{52} +1.38056 q^{53} +1.58385 q^{55} +5.11879 q^{56} -7.97238 q^{58} -12.9480 q^{59} -2.00666 q^{61} +0.957747 q^{62} -7.03428 q^{64} +2.78826 q^{65} +10.8913 q^{67} -9.73692 q^{68} +1.61147 q^{70} +1.73480 q^{71} +12.2264 q^{73} +22.9711 q^{74} -21.8214 q^{76} +2.42281 q^{77} -5.69598 q^{79} +2.91892 q^{80} -14.1254 q^{82} -6.07977 q^{83} -1.56144 q^{85} +26.6670 q^{86} +12.4018 q^{88} -5.57264 q^{89} +4.26520 q^{91} +4.07653 q^{92} -2.93013 q^{94} -3.49934 q^{95} +0.0112053 q^{97} +2.46506 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	2.46506	1.74306	0.871531	−	0.490340i	$-0.163127\pi$
$2$	2.46506	1.74306	0.871531	+	0.490340i	$0.163127\pi$
$3$	0	0
$3$	0	0
$4$	4.07653	2.03827
$5$	0.653724	0.292354	0.146177	−	0.989258i	$-0.453303\pi$
$5$	0.653724	0.292354	0.146177	+	0.989258i	$0.453303\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	5.11879	1.80976
$9$	0	0
$10$	1.61147	0.509592
$11$	2.42281	0.730505	0.365252	−	0.930909i	$-0.380983\pi$
$11$	2.42281	0.730505	0.365252	+	0.930909i	$0.380983\pi$
$12$	0	0
$13$	4.26520	1.18295	0.591476	−	0.806322i	$-0.298545\pi$
$13$	4.26520	1.18295	0.591476	+	0.806322i	$0.298545\pi$
$14$	2.46506	0.658816
$15$	0	0
$16$	4.46506	1.11627
$17$	−2.38853	−0.579303	−0.289652	−	0.957132i	$-0.593539\pi$
$17$	−2.38853	−0.579303	−0.289652	+	0.957132i	$0.593539\pi$
$18$	0	0
$19$	−5.35294	−1.22805	−0.614024	−	0.789288i	$-0.710450\pi$
$19$	−5.35294	−1.22805	−0.614024	+	0.789288i	$0.710450\pi$
$20$	2.66493	0.595896
$21$	0	0
$22$	5.97238	1.27332
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	−4.57264	−0.914529
$26$	10.5140	2.06196
$27$	0	0
$28$	4.07653	0.770393
$29$	−3.23415	−0.600566	−0.300283	−	0.953850i	$-0.597081\pi$
$29$	−3.23415	−0.600566	−0.300283	+	0.953850i	$0.597081\pi$
$30$	0	0
$31$	0.388529	0.0697818	0.0348909	−	0.999391i	$-0.488892\pi$
$31$	0.388529	0.0697818	0.0348909	+	0.999391i	$0.488892\pi$
$32$	0.769086	0.135956
$33$	0	0
$34$	−5.88787	−1.00976
$35$	0.653724	0.110500
$36$	0	0
$37$	9.31865	1.53198	0.765989	−	0.642854i	$-0.222250\pi$
$37$	9.31865	1.53198	0.765989	+	0.642854i	$0.222250\pi$
$38$	−13.1953	−2.14056
$39$	0	0
$40$	3.34628	0.529093
$41$	−5.73026	−0.894916	−0.447458	−	0.894305i	$-0.647671\pi$
$41$	−5.73026	−0.894916	−0.447458	+	0.894305i	$0.647671\pi$
$42$	0	0
$43$	10.8180	1.64973	0.824865	−	0.565330i	$-0.191251\pi$
$43$	10.8180	1.64973	0.824865	+	0.565330i	$0.191251\pi$
$44$	9.87667	1.48896
$45$	0	0
$46$	2.46506	0.363454
$47$	−1.18866	−0.173384	−0.0866921	−	0.996235i	$-0.527630\pi$
$47$	−1.18866	−0.173384	−0.0866921	+	0.996235i	$0.527630\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−11.2719	−1.59408
$51$	0	0
$52$	17.3872	2.41117
$53$	1.38056	0.189634	0.0948170	−	0.995495i	$-0.469773\pi$
$53$	1.38056	0.189634	0.0948170	+	0.995495i	$0.469773\pi$
$54$	0	0
$55$	1.58385	0.213566
$56$	5.11879	0.684027
$57$	0	0
$58$	−7.97238	−1.04682
$59$	−12.9480	−1.68568	−0.842842	−	0.538160i	$-0.819119\pi$
$59$	−12.9480	−1.68568	−0.842842	+	0.538160i	$0.819119\pi$
$60$	0	0
$61$	−2.00666	−0.256926	−0.128463	−	0.991714i	$-0.541004\pi$
$61$	−2.00666	−0.256926	−0.128463	+	0.991714i	$0.541004\pi$
$62$	0.957747	0.121634
$63$	0	0
$64$	−7.03428	−0.879285
$65$	2.78826	0.345841
$66$	0	0
$67$	10.8913	1.33058	0.665292	−	0.746583i	$-0.268307\pi$
$67$	10.8913	1.33058	0.665292	+	0.746583i	$0.268307\pi$
$68$	−9.73692	−1.18077
$69$	0	0
$70$	1.61147	0.192608
$71$	1.73480	0.205883	0.102942	−	0.994687i	$-0.467174\pi$
$71$	1.73480	0.205883	0.102942	+	0.994687i	$0.467174\pi$
$72$	0	0
$73$	12.2264	1.43099	0.715494	−	0.698619i	$-0.246202\pi$
$73$	12.2264	1.43099	0.715494	+	0.698619i	$0.246202\pi$
$74$	22.9711	2.67033
$75$	0	0
$76$	−21.8214	−2.50309
$77$	2.42281	0.276105
$78$	0	0
$79$	−5.69598	−0.640848	−0.320424	−	0.947274i	$-0.603825\pi$
$79$	−5.69598	−0.640848	−0.320424	+	0.947274i	$0.603825\pi$
$80$	2.91892	0.326345
$81$	0	0
$82$	−14.1254	−1.55989
$83$	−6.07977	−0.667341	−0.333671	−	0.942690i	$-0.608287\pi$
$83$	−6.07977	−0.667341	−0.333671	+	0.942690i	$0.608287\pi$
$84$	0	0
$85$	−1.56144	−0.169362
$86$	26.6670	2.87558
$87$	0	0
$88$	12.4018	1.32204
$89$	−5.57264	−0.590699	−0.295350	−	0.955389i	$-0.595436\pi$
$89$	−5.57264	−0.590699	−0.295350	+	0.955389i	$0.595436\pi$
$90$	0	0
$91$	4.26520	0.447114
$92$	4.07653	0.425008
$93$	0	0
$94$	−2.93013	−0.302219
$95$	−3.49934	−0.359025
$96$	0	0
$97$	0.0112053	0.00113772	0.000568862	−	1.00000i	$-0.499819\pi$
$97$	0.0112053	0.00113772	0.000568862		1.00000i	$0.499819\pi$
$98$	2.46506	0.249009
$99$	0	0
$100$	−18.6405	−1.86405
$101$	−13.8557	−1.37869	−0.689347	−	0.724431i	$-0.742102\pi$
$101$	−13.8557	−1.37869	−0.689347	+	0.724431i	$0.742102\pi$
$102$	0	0
$103$	−7.41029	−0.730158	−0.365079	−	0.930977i	$-0.618958\pi$
$103$	−7.41029	−0.730158	−0.365079	+	0.930977i	$0.618958\pi$
$104$	21.8326	2.14087
$105$	0	0
$106$	3.40316	0.330544
$107$	10.2652	0.992374	0.496187	−	0.868216i	$-0.334733\pi$
$107$	10.2652	0.992374	0.496187	+	0.868216i	$0.334733\pi$
$108$	0	0
$109$	−14.2897	−1.36871	−0.684353	−	0.729150i	$-0.739915\pi$
$109$	−14.2897	−1.36871	−0.684353	+	0.729150i	$0.739915\pi$
$110$	3.90429	0.372259
$111$	0	0
$112$	4.46506	0.421909
$113$	2.50066	0.235242	0.117621	−	0.993059i	$-0.462473\pi$
$113$	2.50066	0.235242	0.117621	+	0.993059i	$0.462473\pi$
$114$	0	0
$115$	0.653724	0.0609601
$116$	−13.1841	−1.22411
$117$	0	0
$118$	−31.9176	−2.93825
$119$	−2.38853	−0.218956
$120$	0	0
$121$	−5.12999	−0.466363
$122$	−4.94654	−0.447839
$123$	0	0
$124$	1.58385	0.142234
$125$	−6.25787	−0.559721
$126$	0	0
$127$	−14.9592	−1.32741	−0.663707	−	0.747993i	$-0.731018\pi$
$127$	−14.9592	−1.32741	−0.663707	+	0.747993i	$0.731018\pi$
$128$	−18.8781	−1.66861
$129$	0	0
$130$	6.87324	0.602823
$131$	−9.64575	−0.842753	−0.421377	−	0.906886i	$-0.638453\pi$
$131$	−9.64575	−0.842753	−0.421377	+	0.906886i	$0.638453\pi$
$132$	0	0
$133$	−5.35294	−0.464158
$134$	26.8477	2.31929
$135$	0	0
$136$	−12.2264	−1.04840
$137$	−7.96098	−0.680153	−0.340076	−	0.940398i	$-0.610453\pi$
$137$	−7.96098	−0.680153	−0.340076	+	0.940398i	$0.610453\pi$
$138$	0	0
$139$	6.94654	0.589198	0.294599	−	0.955621i	$-0.404814\pi$
$139$	6.94654	0.589198	0.294599	+	0.955621i	$0.404814\pi$
$140$	2.66493	0.225228
$141$	0	0
$142$	4.27640	0.358868
$143$	10.3338	0.864152
$144$	0	0
$145$	−2.11424	−0.175578
$146$	30.1388	2.49430
$147$	0	0
$148$	37.9878	3.12258
$149$	−5.82597	−0.477282	−0.238641	−	0.971108i	$-0.576702\pi$
$149$	−5.82597	−0.477282	−0.238641	+	0.971108i	$0.576702\pi$
$150$	0	0
$151$	−10.9803	−0.893568	−0.446784	−	0.894642i	$-0.647431\pi$
$151$	−10.9803	−0.893568	−0.446784	+	0.894642i	$0.647431\pi$
$152$	−27.4005	−2.22248
$153$	0	0
$154$	5.97238	0.481268
$155$	0.253991	0.0204010
$156$	0	0
$157$	1.56598	0.124979	0.0624896	−	0.998046i	$-0.480096\pi$
$157$	1.56598	0.124979	0.0624896	+	0.998046i	$0.480096\pi$
$158$	−14.0409	−1.11704
$159$	0	0
$160$	0.502770	0.0397475
$161$	1.00000	0.0788110
$162$	0	0
$163$	15.6867	1.22868	0.614338	−	0.789043i	$-0.289423\pi$
$163$	15.6867	1.22868	0.614338	+	0.789043i	$0.289423\pi$
$164$	−23.3596	−1.82408
$165$	0	0
$166$	−14.9870	−1.16322
$167$	−9.03771	−0.699359	−0.349679	−	0.936869i	$-0.613709\pi$
$167$	−9.03771	−0.699359	−0.349679	+	0.936869i	$0.613709\pi$
$168$	0	0
$169$	5.19190	0.399377
$170$	−3.84905	−0.295208
$171$	0	0
$172$	44.0999	3.36259
$173$	−7.98879	−0.607377	−0.303688	−	0.952771i	$-0.598218\pi$
$173$	−7.98879	−0.607377	−0.303688	+	0.952771i	$0.598218\pi$
$174$	0	0
$175$	−4.57264	−0.345659
$176$	10.8180	0.815437
$177$	0	0
$178$	−13.7369	−1.02963
$179$	14.9137	1.11470	0.557351	−	0.830277i	$-0.311818\pi$
$179$	14.9137	1.11470	0.557351	+	0.830277i	$0.311818\pi$
$180$	0	0
$181$	2.45709	0.182634	0.0913171	−	0.995822i	$-0.470892\pi$
$181$	2.45709	0.182634	0.0913171	+	0.995822i	$0.470892\pi$
$182$	10.5140	0.779348
$183$	0	0
$184$	5.11879	0.377362
$185$	6.09183	0.447880
$186$	0	0
$187$	−5.78695	−0.423184
$188$	−4.84562	−0.353403
$189$	0	0
$190$	−8.62610	−0.625803
$191$	−9.66381	−0.699249	−0.349624	−	0.936890i	$-0.613691\pi$
$191$	−9.66381	−0.699249	−0.349624	+	0.936890i	$0.613691\pi$
$192$	0	0
$193$	−3.35767	−0.241691	−0.120845	−	0.992671i	$-0.538560\pi$
$193$	−3.35767	−0.241691	−0.120845	+	0.992671i	$0.538560\pi$
$194$	0.0276217	0.00198312
$195$	0	0
$196$	4.07653	0.291181
$197$	−1.79690	−0.128024	−0.0640119	−	0.997949i	$-0.520390\pi$
$197$	−1.79690	−0.128024	−0.0640119	+	0.997949i	$0.520390\pi$
$198$	0	0
$199$	−8.39519	−0.595119	−0.297560	−	0.954703i	$-0.596173\pi$
$199$	−8.39519	−0.595119	−0.297560	+	0.954703i	$0.596173\pi$
$200$	−23.4064	−1.65508
$201$	0	0
$202$	−34.1552	−2.40315
$203$	−3.23415	−0.226993
$204$	0	0
$205$	−3.74601	−0.261633
$206$	−18.2668	−1.27271
$207$	0	0
$208$	19.0444	1.32049
$209$	−12.9691	−0.897094
$210$	0	0
$211$	21.7371	1.49644	0.748222	−	0.663448i	$-0.230908\pi$
$211$	21.7371	1.49644	0.748222	+	0.663448i	$0.230908\pi$
$212$	5.62789	0.386525
$213$	0	0
$214$	25.3044	1.72977
$215$	7.07199	0.482306
$216$	0	0
$217$	0.388529	0.0263750
$218$	−35.2251	−2.38574
$219$	0	0
$220$	6.45662	0.435305
$221$	−10.1875	−0.685288
$222$	0	0
$223$	26.1500	1.75113	0.875566	−	0.483099i	$-0.160489\pi$
$223$	26.1500	1.75113	0.875566	+	0.483099i	$0.160489\pi$
$224$	0.769086	0.0513867
$225$	0	0
$226$	6.16427	0.410041
$227$	16.6606	1.10580	0.552901	−	0.833247i	$-0.313521\pi$
$227$	16.6606	1.10580	0.552901	+	0.833247i	$0.313521\pi$
$228$	0	0
$229$	28.8082	1.90370	0.951851	−	0.306561i	$-0.0991783\pi$
$229$	28.8082	1.90370	0.951851	+	0.306561i	$0.0991783\pi$
$230$	1.61147	0.106257
$231$	0	0
$232$	−16.5549	−1.08688
$233$	−11.8214	−0.774447	−0.387224	−	0.921986i	$-0.626566\pi$
$233$	−11.8214	−0.774447	−0.387224	+	0.921986i	$0.626566\pi$
$234$	0	0
$235$	−0.777057	−0.0506896
$236$	−52.7829	−3.43588
$237$	0	0
$238$	−5.88787	−0.381654
$239$	19.7236	1.27581	0.637907	−	0.770114i	$-0.279801\pi$
$239$	19.7236	1.27581	0.637907	+	0.770114i	$0.279801\pi$
$240$	0	0
$241$	4.95320	0.319064	0.159532	−	0.987193i	$-0.449002\pi$
$241$	4.95320	0.319064	0.159532	+	0.987193i	$0.449002\pi$
$242$	−12.6458	−0.812900
$243$	0	0
$244$	−8.18022	−0.523685
$245$	0.653724	0.0417649
$246$	0	0
$247$	−22.8313	−1.45272
$248$	1.98879	0.126289
$249$	0	0
$250$	−15.4260	−0.975629
$251$	31.4053	1.98228	0.991142	−	0.132809i	$-0.0423997\pi$
$251$	31.4053	1.98228	0.991142	+	0.132809i	$0.0423997\pi$
$252$	0	0
$253$	2.42281	0.152321
$254$	−36.8754	−2.31377
$255$	0	0
$256$	−32.4672	−2.02920
$257$	26.3697	1.64490	0.822448	−	0.568841i	$-0.192608\pi$
$257$	26.3697	1.64490	0.822448	+	0.568841i	$0.192608\pi$
$258$	0	0
$259$	9.31865	0.579033
$260$	11.3664	0.704917
$261$	0	0
$262$	−23.7774	−1.46897
$263$	29.1196	1.79559	0.897795	−	0.440413i	$-0.145168\pi$
$263$	29.1196	1.79559	0.897795	+	0.440413i	$0.145168\pi$
$264$	0	0
$265$	0.902504	0.0554404
$266$	−13.1953	−0.809057
$267$	0	0
$268$	44.3988	2.71209
$269$	−3.12676	−0.190642	−0.0953209	−	0.995447i	$-0.530388\pi$
$269$	−3.12676	−0.190642	−0.0953209	+	0.995447i	$0.530388\pi$
$270$	0	0
$271$	−16.5484	−1.00525	−0.502623	−	0.864506i	$-0.667632\pi$
$271$	−16.5484	−1.00525	−0.502623	+	0.864506i	$0.667632\pi$
$272$	−10.6649	−0.646656
$273$	0	0
$274$	−19.6243	−1.18555
$275$	−11.0786	−0.668068
$276$	0	0
$277$	9.70052	0.582848	0.291424	−	0.956594i	$-0.405871\pi$
$277$	9.70052	0.582848	0.291424	+	0.956594i	$0.405871\pi$
$278$	17.1237	1.02701
$279$	0	0
$280$	3.34628	0.199978
$281$	23.4670	1.39992	0.699961	−	0.714181i	$-0.253200\pi$
$281$	23.4670	1.39992	0.699961	+	0.714181i	$0.253200\pi$
$282$	0	0
$283$	−25.3372	−1.50614	−0.753070	−	0.657941i	$-0.771428\pi$
$283$	−25.3372	−1.50614	−0.753070	+	0.657941i	$0.771428\pi$
$284$	7.07199	0.419645
$285$	0	0
$286$	25.4734	1.50627
$287$	−5.73026	−0.338246
$288$	0	0
$289$	−11.2949	−0.664408
$290$	−5.21174	−0.306044
$291$	0	0
$292$	49.8412	2.91674
$293$	6.25998	0.365712	0.182856	−	0.983140i	$-0.441466\pi$
$293$	6.25998	0.365712	0.182856	+	0.983140i	$0.441466\pi$
$294$	0	0
$295$	−8.46442	−0.492817
$296$	47.7002	2.77252
$297$	0	0
$298$	−14.3614	−0.831932
$299$	4.26520	0.246663
$300$	0	0
$301$	10.8180	0.623539
$302$	−27.0673	−1.55755
$303$	0	0
$304$	−23.9012	−1.37083
$305$	−1.31180	−0.0751136
$306$	0	0
$307$	−20.1776	−1.15160	−0.575798	−	0.817592i	$-0.695309\pi$
$307$	−20.1776	−1.15160	−0.575798	+	0.817592i	$0.695309\pi$
$308$	9.87667	0.562775
$309$	0	0
$310$	0.626103	0.0355602
$311$	−21.6888	−1.22986	−0.614930	−	0.788582i	$-0.710816\pi$
$311$	−21.6888	−1.22986	−0.614930	+	0.788582i	$0.710816\pi$
$312$	0	0
$313$	27.9790	1.58147	0.790734	−	0.612159i	$-0.209699\pi$
$313$	27.9790	1.58147	0.790734	+	0.612159i	$0.209699\pi$
$314$	3.86025	0.217847
$315$	0	0
$316$	−23.2198	−1.30622
$317$	−19.3372	−1.08608	−0.543042	−	0.839705i	$-0.682728\pi$
$317$	−19.3372	−1.08608	−0.543042	+	0.839705i	$0.682728\pi$
$318$	0	0
$319$	−7.83573	−0.438716
$320$	−4.59848	−0.257063
$321$	0	0
$322$	2.46506	0.137373
$323$	12.7856	0.711412
$324$	0	0
$325$	−19.5032	−1.08184
$326$	38.6687	2.14166
$327$	0	0
$328$	−29.3320	−1.61959
$329$	−1.18866	−0.0655330
$330$	0	0
$331$	−17.1796	−0.944275	−0.472137	−	0.881525i	$-0.656517\pi$
$331$	−17.1796	−0.944275	−0.472137	+	0.881525i	$0.656517\pi$
$332$	−24.7844	−1.36022
$333$	0	0
$334$	−22.2785	−1.21903
$335$	7.11991	0.389002
$336$	0	0
$337$	−20.7403	−1.12980	−0.564899	−	0.825160i	$-0.691085\pi$
$337$	−20.7403	−1.12980	−0.564899	+	0.825160i	$0.691085\pi$
$338$	12.7983	0.696138
$339$	0	0
$340$	−6.36526	−0.345205
$341$	0.941331	0.0509759
$342$	0	0
$343$	1.00000	0.0539949
$344$	55.3750	2.98562
$345$	0	0
$346$	−19.6929	−1.05870
$347$	21.0008	1.12738	0.563691	−	0.825986i	$-0.309381\pi$
$347$	21.0008	1.12738	0.563691	+	0.825986i	$0.309381\pi$
$348$	0	0
$349$	23.2819	1.24625	0.623127	−	0.782121i	$-0.285862\pi$
$349$	23.2819	1.24625	0.623127	+	0.782121i	$0.285862\pi$
$350$	−11.2719	−0.602506
$351$	0	0
$352$	1.86335	0.0993168
$353$	−33.5471	−1.78553	−0.892767	−	0.450519i	$-0.851239\pi$
$353$	−33.5471	−1.78553	−0.892767	+	0.450519i	$0.851239\pi$
$354$	0	0
$355$	1.13408	0.0601909
$356$	−22.7171	−1.20400
$357$	0	0
$358$	36.7632	1.94300
$359$	−33.1025	−1.74708	−0.873542	−	0.486749i	$-0.838183\pi$
$359$	−33.1025	−1.74708	−0.873542	+	0.486749i	$0.838183\pi$
$360$	0	0
$361$	9.65392	0.508101
$362$	6.05688	0.318343
$363$	0	0
$364$	17.3872	0.911338
$365$	7.99267	0.418356
$366$	0	0
$367$	20.3576	1.06266	0.531329	−	0.847165i	$-0.321693\pi$
$367$	20.3576	1.06266	0.531329	+	0.847165i	$0.321693\pi$
$368$	4.46506	0.232757
$369$	0	0
$370$	15.0167	0.780683
$371$	1.38056	0.0716749
$372$	0	0
$373$	6.21043	0.321564	0.160782	−	0.986990i	$-0.448598\pi$
$373$	6.21043	0.321564	0.160782	+	0.986990i	$0.448598\pi$
$374$	−14.2652	−0.737636
$375$	0	0
$376$	−6.08451	−0.313784
$377$	−13.7943	−0.710441
$378$	0	0
$379$	13.9288	0.715475	0.357738	−	0.933822i	$-0.383548\pi$
$379$	13.9288	0.715475	0.357738	+	0.933822i	$0.383548\pi$
$380$	−14.2652	−0.731789
$381$	0	0
$382$	−23.8219	−1.21883
$383$	18.0798	0.923833	0.461916	−	0.886923i	$-0.347162\pi$
$383$	18.0798	0.923833	0.461916	+	0.886923i	$0.347162\pi$
$384$	0	0
$385$	1.58385	0.0807205
$386$	−8.27687	−0.421282
$387$	0	0
$388$	0.0456787	0.00231898
$389$	3.03230	0.153744	0.0768720	−	0.997041i	$-0.475507\pi$
$389$	3.03230	0.153744	0.0768720	+	0.997041i	$0.475507\pi$
$390$	0	0
$391$	−2.38853	−0.120793
$392$	5.11879	0.258538
$393$	0	0
$394$	−4.42947	−0.223153
$395$	−3.72360	−0.187355
$396$	0	0
$397$	−37.6428	−1.88924	−0.944620	−	0.328166i	$-0.893570\pi$
$397$	−37.6428	−1.88924	−0.944620	+	0.328166i	$0.893570\pi$
$398$	−20.6947	−1.03733
$399$	0	0
$400$	−20.4171	−1.02086
$401$	20.6604	1.03173	0.515865	−	0.856670i	$-0.327471\pi$
$401$	20.6604	1.03173	0.515865	+	0.856670i	$0.327471\pi$
$402$	0	0
$403$	1.65715	0.0825485
$404$	−56.4833	−2.81015
$405$	0	0
$406$	−7.97238	−0.395662
$407$	22.5773	1.11912
$408$	0	0
$409$	−10.8728	−0.537624	−0.268812	−	0.963193i	$-0.586631\pi$
$409$	−10.8728	−0.537624	−0.268812	+	0.963193i	$0.586631\pi$
$410$	−9.23415	−0.456042
$411$	0	0
$412$	−30.2083	−1.48826
$413$	−12.9480	−0.637129
$414$	0	0
$415$	−3.97449	−0.195100
$416$	3.28030	0.160830
$417$	0	0
$418$	−31.9698	−1.56369
$419$	−35.4574	−1.73221	−0.866104	−	0.499864i	$-0.833383\pi$
$419$	−35.4574	−1.73221	−0.866104	+	0.499864i	$0.833383\pi$
$420$	0	0
$421$	5.91028	0.288050	0.144025	−	0.989574i	$-0.453995\pi$
$421$	5.91028	0.288050	0.144025	+	0.989574i	$0.453995\pi$
$422$	53.5833	2.60840
$423$	0	0
$424$	7.06678	0.343193
$425$	10.9219	0.529790
$426$	0	0
$427$	−2.00666	−0.0971091
$428$	41.8464	2.02272
$429$	0	0
$430$	17.4329	0.840689
$431$	30.2826	1.45866	0.729330	−	0.684162i	$-0.239832\pi$
$431$	30.2826	1.45866	0.729330	+	0.684162i	$0.239832\pi$
$432$	0	0
$433$	−2.48964	−0.119645	−0.0598223	−	0.998209i	$-0.519053\pi$
$433$	−2.48964	−0.119645	−0.0598223	+	0.998209i	$0.519053\pi$
$434$	0.957747	0.0459733
$435$	0	0
$436$	−58.2525	−2.78979
$437$	−5.35294	−0.256066
$438$	0	0
$439$	32.6527	1.55843	0.779215	−	0.626757i	$-0.215618\pi$
$439$	32.6527	1.55843	0.779215	+	0.626757i	$0.215618\pi$
$440$	8.10739	0.386505
$441$	0	0
$442$	−25.1129	−1.19450
$443$	26.3354	1.25123	0.625616	−	0.780131i	$-0.284848\pi$
$443$	26.3354	1.25123	0.625616	+	0.780131i	$0.284848\pi$
$444$	0	0
$445$	−3.64297	−0.172693
$446$	64.4613	3.05233
$447$	0	0
$448$	−7.03428	−0.332339
$449$	−13.8279	−0.652579	−0.326289	−	0.945270i	$-0.605798\pi$
$449$	−13.8279	−0.652579	−0.326289	+	0.945270i	$0.605798\pi$
$450$	0	0
$451$	−13.8833	−0.653740
$452$	10.1940	0.479486
$453$	0	0
$454$	41.0694	1.92748
$455$	2.78826	0.130716
$456$	0	0
$457$	−21.8055	−1.02002	−0.510009	−	0.860169i	$-0.670358\pi$
$457$	−21.8055	−1.02002	−0.510009	+	0.860169i	$0.670358\pi$
$458$	71.0141	3.31827
$459$	0	0
$460$	2.66493	0.124253
$461$	31.4415	1.46438	0.732188	−	0.681103i	$-0.238499\pi$
$461$	31.4415	1.46438	0.732188	+	0.681103i	$0.238499\pi$
$462$	0	0
$463$	39.5102	1.83620	0.918098	−	0.396353i	$-0.129724\pi$
$463$	39.5102	1.83620	0.918098	+	0.396353i	$0.129724\pi$
$464$	−14.4407	−0.670391
$465$	0	0
$466$	−29.1406	−1.34991
$467$	−21.7779	−1.00776	−0.503880	−	0.863774i	$-0.668094\pi$
$467$	−21.7779	−1.00776	−0.503880	+	0.863774i	$0.668094\pi$
$468$	0	0
$469$	10.8913	0.502913
$470$	−1.91549	−0.0883552
$471$	0	0
$472$	−66.2780	−3.05069
$473$	26.2100	1.20513
$474$	0	0
$475$	24.4771	1.12309
$476$	−9.73692	−0.446291
$477$	0	0
$478$	48.6199	2.22382
$479$	2.54160	0.116129	0.0580643	−	0.998313i	$-0.481507\pi$
$479$	2.54160	0.116129	0.0580643	+	0.998313i	$0.481507\pi$
$480$	0	0
$481$	39.7459	1.81226
$482$	12.2100	0.556148
$483$	0	0
$484$	−20.9126	−0.950572
$485$	0.00732516	0.000332619 0
$486$	0	0
$487$	−42.7148	−1.93559	−0.967797	−	0.251732i	$-0.919000\pi$
$487$	−42.7148	−1.93559	−0.967797	+	0.251732i	$0.919000\pi$
$488$	−10.2717	−0.464976
$489$	0	0
$490$	1.61147	0.0727989
$491$	18.9184	0.853778	0.426889	−	0.904304i	$-0.359610\pi$
$491$	18.9184	0.853778	0.426889	+	0.904304i	$0.359610\pi$
$492$	0	0
$493$	7.72486	0.347910
$494$	−56.2806	−2.53219
$495$	0	0
$496$	1.73480	0.0778950
$497$	1.73480	0.0778166
$498$	0	0
$499$	17.9771	0.804766	0.402383	−	0.915471i	$-0.368182\pi$
$499$	17.9771	0.804766	0.402383	+	0.915471i	$0.368182\pi$
$500$	−25.5104	−1.14086
$501$	0	0
$502$	77.4160	3.45524
$503$	14.7447	0.657434	0.328717	−	0.944429i	$-0.393384\pi$
$503$	14.7447	0.657434	0.328717	+	0.944429i	$0.393384\pi$
$504$	0	0
$505$	−9.05781	−0.403067
$506$	5.97238	0.265505
$507$	0	0
$508$	−60.9817	−2.70562
$509$	18.6492	0.826610	0.413305	−	0.910593i	$-0.364374\pi$
$509$	18.6492	0.826610	0.413305	+	0.910593i	$0.364374\pi$
$510$	0	0
$511$	12.2264	0.540863
$512$	−42.2774	−1.86841
$513$	0	0
$514$	65.0029	2.86716
$515$	−4.84429	−0.213465
$516$	0	0
$517$	−2.87990	−0.126658
$518$	22.9711	1.00929
$519$	0	0
$520$	14.2725	0.625891
$521$	23.0608	1.01031	0.505156	−	0.863028i	$-0.331435\pi$
$521$	23.0608	1.01031	0.505156	+	0.863028i	$0.331435\pi$
$522$	0	0
$523$	34.6781	1.51637	0.758183	−	0.652042i	$-0.226087\pi$
$523$	34.6781	1.51637	0.758183	+	0.652042i	$0.226087\pi$
$524$	−39.3212	−1.71776
$525$	0	0
$526$	71.7816	3.12983
$527$	−0.928011	−0.0404248
$528$	0	0
$529$	1.00000	0.0434783
$530$	2.22473	0.0966360
$531$	0	0
$532$	−21.8214	−0.946079
$533$	−24.4407	−1.05864
$534$	0	0
$535$	6.71061	0.290125
$536$	55.7502	2.40804
$537$	0	0
$538$	−7.70766	−0.332301
$539$	2.42281	0.104358
$540$	0	0
$541$	−1.72552	−0.0741859	−0.0370930	−	0.999312i	$-0.511810\pi$
$541$	−1.72552	−0.0741859	−0.0370930	+	0.999312i	$0.511810\pi$
$542$	−40.7930	−1.75221
$543$	0	0
$544$	−1.83698	−0.0787600
$545$	−9.34154	−0.400148
$546$	0	0
$547$	−10.6670	−0.456090	−0.228045	−	0.973651i	$-0.573233\pi$
$547$	−10.6670	−0.456090	−0.228045	+	0.973651i	$0.573233\pi$
$548$	−32.4532	−1.38633
$549$	0	0
$550$	−27.3096	−1.16448
$551$	17.3122	0.737524
$552$	0	0
$553$	−5.69598	−0.242218
$554$	23.9124	1.01594
$555$	0	0
$556$	28.3178	1.20094
$557$	−10.0819	−0.427183	−0.213592	−	0.976923i	$-0.568516\pi$
$557$	−10.0819	−0.427183	−0.213592	+	0.976923i	$0.568516\pi$
$558$	0	0
$559$	46.1409	1.95155
$560$	2.91892	0.123347
$561$	0	0
$562$	57.8476	2.44015
$563$	4.50924	0.190042	0.0950209	−	0.995475i	$-0.469708\pi$
$563$	4.50924	0.190042	0.0950209	+	0.995475i	$0.469708\pi$
$564$	0	0
$565$	1.63474	0.0687740
$566$	−62.4577	−2.62529
$567$	0	0
$568$	8.88009	0.372600
$569$	−43.1376	−1.80842	−0.904212	−	0.427084i	$-0.859541\pi$
$569$	−43.1376	−1.80842	−0.904212	+	0.427084i	$0.859541\pi$
$570$	0	0
$571$	−33.1565	−1.38756	−0.693778	−	0.720189i	$-0.744055\pi$
$571$	−33.1565	−1.38756	−0.693778	+	0.720189i	$0.744055\pi$
$572$	42.1259	1.76137
$573$	0	0
$574$	−14.1254	−0.589585
$575$	−4.57264	−0.190692
$576$	0	0
$577$	4.65972	0.193987	0.0969933	−	0.995285i	$-0.469077\pi$
$577$	4.65972	0.193987	0.0969933	+	0.995285i	$0.469077\pi$
$578$	−27.8427	−1.15810
$579$	0	0
$580$	−8.61878	−0.357875
$581$	−6.07977	−0.252231
$582$	0	0
$583$	3.34483	0.138529
$584$	62.5842	2.58975
$585$	0	0
$586$	15.4313	0.637459
$587$	18.8272	0.777084	0.388542	−	0.921431i	$-0.372979\pi$
$587$	18.8272	0.777084	0.388542	+	0.921431i	$0.372979\pi$
$588$	0	0
$589$	−2.07977	−0.0856953
$590$	−20.8653	−0.859012
$591$	0	0
$592$	41.6084	1.71009
$593$	16.8658	0.692595	0.346298	−	0.938125i	$-0.387439\pi$
$593$	16.8658	0.692595	0.346298	+	0.938125i	$0.387439\pi$
$594$	0	0
$595$	−1.56144	−0.0640128
$596$	−23.7498	−0.972828
$597$	0	0
$598$	10.5140	0.429948
$599$	−15.3246	−0.626148	−0.313074	−	0.949729i	$-0.601359\pi$
$599$	−15.3246	−0.626148	−0.313074	+	0.949729i	$0.601359\pi$
$600$	0	0
$601$	−16.8556	−0.687553	−0.343776	−	0.939052i	$-0.611706\pi$
$601$	−16.8556	−0.687553	−0.343776	+	0.939052i	$0.611706\pi$
$602$	26.6670	1.08687
$603$	0	0
$604$	−44.7618	−1.82133
$605$	−3.35360	−0.136343
$606$	0	0
$607$	25.0267	1.01580	0.507901	−	0.861415i	$-0.330422\pi$
$607$	25.0267	1.01580	0.507901	+	0.861415i	$0.330422\pi$
$608$	−4.11687	−0.166961
$609$	0	0
$610$	−3.23367	−0.130928
$611$	−5.06987	−0.205105
$612$	0	0
$613$	29.4100	1.18786	0.593930	−	0.804517i	$-0.297576\pi$
$613$	29.4100	1.18786	0.593930	+	0.804517i	$0.297576\pi$
$614$	−49.7390	−2.00730
$615$	0	0
$616$	12.4018	0.499685
$617$	27.8167	1.11986	0.559929	−	0.828541i	$-0.310828\pi$
$617$	27.8167	1.11986	0.559929	+	0.828541i	$0.310828\pi$
$618$	0	0
$619$	−10.0504	−0.403960	−0.201980	−	0.979390i	$-0.564738\pi$
$619$	−10.0504	−0.403960	−0.201980	+	0.979390i	$0.564738\pi$
$620$	1.03540	0.0415827
$621$	0	0
$622$	−53.4643	−2.14372
$623$	−5.57264	−0.223263
$624$	0	0
$625$	18.7723	0.750892
$626$	68.9701	2.75660
$627$	0	0
$628$	6.38379	0.254741
$629$	−22.2579	−0.887479
$630$	0	0
$631$	40.8542	1.62638	0.813190	−	0.581998i	$-0.197729\pi$
$631$	40.8542	1.62638	0.813190	+	0.581998i	$0.197729\pi$
$632$	−29.1565	−1.15978
$633$	0	0
$634$	−47.6674	−1.89311
$635$	−9.77919	−0.388075
$636$	0	0
$637$	4.26520	0.168993
$638$	−19.3156	−0.764710
$639$	0	0
$640$	−12.3411	−0.487824
$641$	−1.15195	−0.0454992	−0.0227496	−	0.999741i	$-0.507242\pi$
$641$	−1.15195	−0.0454992	−0.0227496	+	0.999741i	$0.507242\pi$
$642$	0	0
$643$	9.15419	0.361006	0.180503	−	0.983574i	$-0.442227\pi$
$643$	9.15419	0.361006	0.180503	+	0.983574i	$0.442227\pi$
$644$	4.07653	0.160638
$645$	0	0
$646$	31.5174	1.24004
$647$	−2.79994	−0.110077	−0.0550385	−	0.998484i	$-0.517528\pi$
$647$	−2.79994	−0.110077	−0.0550385	+	0.998484i	$0.517528\pi$
$648$	0	0
$649$	−31.3705	−1.23140
$650$	−48.0767	−1.88572
$651$	0	0
$652$	63.9473	2.50437
$653$	−36.1276	−1.41378	−0.706890	−	0.707323i	$-0.749903\pi$
$653$	−36.1276	−1.41378	−0.706890	+	0.707323i	$0.749903\pi$
$654$	0	0
$655$	−6.30566	−0.246383
$656$	−25.5860	−0.998964
$657$	0	0
$658$	−2.93013	−0.114228
$659$	40.3794	1.57296	0.786480	−	0.617616i	$-0.211901\pi$
$659$	40.3794	1.57296	0.786480	+	0.617616i	$0.211901\pi$
$660$	0	0
$661$	−37.4732	−1.45754	−0.728769	−	0.684760i	$-0.759907\pi$
$661$	−37.4732	−1.45754	−0.728769	+	0.684760i	$0.759907\pi$
$662$	−42.3487	−1.64593
$663$	0	0
$664$	−31.1210	−1.20773
$665$	−3.49934	−0.135699
$666$	0	0
$667$	−3.23415	−0.125227
$668$	−36.8425	−1.42548
$669$	0	0
$670$	17.5510	0.678055
$671$	−4.86175	−0.187686
$672$	0	0
$673$	−25.9122	−0.998842	−0.499421	−	0.866359i	$-0.666454\pi$
$673$	−25.9122	−0.998842	−0.499421	+	0.866359i	$0.666454\pi$
$674$	−51.1262	−1.96931
$675$	0	0
$676$	21.1649	0.814036
$677$	31.2444	1.20082	0.600409	−	0.799693i	$-0.295004\pi$
$677$	31.2444	1.20082	0.600409	+	0.799693i	$0.295004\pi$
$678$	0	0
$679$	0.0112053	0.000430019 0
$680$	−7.99267	−0.306505
$681$	0	0
$682$	2.32044	0.0888542
$683$	−33.8559	−1.29546	−0.647730	−	0.761870i	$-0.724281\pi$
$683$	−33.8559	−1.29546	−0.647730	+	0.761870i	$0.724281\pi$
$684$	0	0
$685$	−5.20429	−0.198846
$686$	2.46506	0.0941165
$687$	0	0
$688$	48.3030	1.84154
$689$	5.88835	0.224328
$690$	0	0
$691$	20.3273	0.773287	0.386643	−	0.922229i	$-0.373634\pi$
$691$	20.3273	0.773287	0.386643	+	0.922229i	$0.373634\pi$
$692$	−32.5666	−1.23800
$693$	0	0
$694$	51.7683	1.96510
$695$	4.54112	0.172255
$696$	0	0
$697$	13.6869	0.518428
$698$	57.3914	2.17230
$699$	0	0
$700$	−18.6405	−0.704546
$701$	−44.7211	−1.68909	−0.844547	−	0.535482i	$-0.820130\pi$
$701$	−44.7211	−1.68909	−0.844547	+	0.535482i	$0.820130\pi$
$702$	0	0
$703$	−49.8822	−1.88134
$704$	−17.0427	−0.642322
$705$	0	0
$706$	−82.6958	−3.11230
$707$	−13.8557	−0.521097
$708$	0	0
$709$	−0.740344	−0.0278042	−0.0139021	−	0.999903i	$-0.504425\pi$
$709$	−0.740344	−0.0278042	−0.0139021	+	0.999903i	$0.504425\pi$
$710$	2.79559	0.104917
$711$	0	0
$712$	−28.5252	−1.06903
$713$	0.388529	0.0145505
$714$	0	0
$715$	6.75543	0.252639
$716$	60.7962	2.27206
$717$	0	0
$718$	−81.5998	−3.04528
$719$	45.1142	1.68248	0.841238	−	0.540666i	$-0.181828\pi$
$719$	45.1142	1.68248	0.841238	+	0.540666i	$0.181828\pi$
$720$	0	0
$721$	−7.41029	−0.275974
$722$	23.7975	0.885652
$723$	0	0
$724$	10.0164	0.372257
$725$	14.7886	0.549235
$726$	0	0
$727$	−4.53395	−0.168155	−0.0840775	−	0.996459i	$-0.526794\pi$
$727$	−4.53395	−0.168155	−0.0840775	+	0.996459i	$0.526794\pi$
$728$	21.8326	0.809171
$729$	0	0
$730$	19.7024	0.729220
$731$	−25.8391	−0.955694
$732$	0	0
$733$	48.4621	1.78999	0.894994	−	0.446078i	$-0.147179\pi$
$733$	48.4621	1.78999	0.894994	+	0.446078i	$0.147179\pi$
$734$	50.1828	1.85228
$735$	0	0
$736$	0.769086	0.0283489
$737$	26.3875	0.971998
$738$	0	0
$739$	−14.1802	−0.521628	−0.260814	−	0.965389i	$-0.583991\pi$
$739$	−14.1802	−0.521628	−0.260814	+	0.965389i	$0.583991\pi$
$740$	24.8336	0.912900
$741$	0	0
$742$	3.40316	0.124934
$743$	−5.97171	−0.219081	−0.109540	−	0.993982i	$-0.534938\pi$
$743$	−5.97171	−0.219081	−0.109540	+	0.993982i	$0.534938\pi$
$744$	0	0
$745$	−3.80858	−0.139536
$746$	15.3091	0.560506
$747$	0	0
$748$	−23.5907	−0.862561
$749$	10.2652	0.375082
$750$	0	0
$751$	28.3902	1.03597	0.517986	−	0.855389i	$-0.326682\pi$
$751$	28.3902	1.03597	0.517986	+	0.855389i	$0.326682\pi$
$752$	−5.30745	−0.193543
$753$	0	0
$754$	−34.0038	−1.23834
$755$	−7.17812	−0.261239
$756$	0	0
$757$	−33.9995	−1.23573	−0.617866	−	0.786283i	$-0.712003\pi$
$757$	−33.9995	−1.23573	−0.617866	+	0.786283i	$0.712003\pi$
$758$	34.3354	1.24712
$759$	0	0
$760$	−17.9124	−0.649751
$761$	−0.844171	−0.0306012	−0.0153006	−	0.999883i	$-0.504871\pi$
$761$	−0.844171	−0.0306012	−0.0153006	+	0.999883i	$0.504871\pi$
$762$	0	0
$763$	−14.2897	−0.517323
$764$	−39.3949	−1.42526
$765$	0	0
$766$	44.5678	1.61030
$767$	−55.2257	−1.99408
$768$	0	0
$769$	−9.02110	−0.325309	−0.162655	−	0.986683i	$-0.552006\pi$
$769$	−9.02110	−0.325309	−0.162655	+	0.986683i	$0.552006\pi$
$770$	3.90429	0.140701
$771$	0	0
$772$	−13.6877	−0.492630
$773$	16.3083	0.586567	0.293284	−	0.956026i	$-0.405252\pi$
$773$	16.3083	0.586567	0.293284	+	0.956026i	$0.405252\pi$
$774$	0	0
$775$	−1.77660	−0.0638175
$776$	0.0573574	0.00205901
$777$	0	0
$778$	7.47482	0.267985
$779$	30.6737	1.09900
$780$	0	0
$781$	4.20310	0.150399
$782$	−5.88787	−0.210550
$783$	0	0
$784$	4.46506	0.159467
$785$	1.02372	0.0365382
$786$	0	0
$787$	3.79756	0.135369	0.0676843	−	0.997707i	$-0.478439\pi$
$787$	3.79756	0.135369	0.0676843	+	0.997707i	$0.478439\pi$
$788$	−7.32512	−0.260947
$789$	0	0
$790$	−9.17890	−0.326571
$791$	2.50066	0.0889131
$792$	0	0
$793$	−8.55880	−0.303932
$794$	−92.7920	−3.29306
$795$	0	0
$796$	−34.2233	−1.21301
$797$	−12.5502	−0.444550	−0.222275	−	0.974984i	$-0.571348\pi$
$797$	−12.5502	−0.444550	−0.222275	+	0.974984i	$0.571348\pi$
$798$	0	0
$799$	2.83915	0.100442
$800$	−3.51675	−0.124336
$801$	0	0
$802$	50.9291	1.79837
$803$	29.6222	1.04534
$804$	0	0
$805$	0.653724	0.0230408
$806$	4.08498	0.143887
$807$	0	0
$808$	−70.9244	−2.49511
$809$	−48.2047	−1.69479	−0.847394	−	0.530965i	$-0.821829\pi$
$809$	−48.2047	−1.69479	−0.847394	+	0.530965i	$0.821829\pi$
$810$	0	0
$811$	−20.3014	−0.712879	−0.356439	−	0.934318i	$-0.616009\pi$
$811$	−20.3014	−0.712879	−0.356439	+	0.934318i	$0.616009\pi$
$812$	−13.1841	−0.462672
$813$	0	0
$814$	55.6545	1.95069
$815$	10.2548	0.359209
$816$	0	0
$817$	−57.9080	−2.02595
$818$	−26.8021	−0.937112
$819$	0	0
$820$	−15.2707	−0.533277
$821$	35.8778	1.25214	0.626072	−	0.779765i	$-0.284662\pi$
$821$	35.8778	1.25214	0.626072	+	0.779765i	$0.284662\pi$
$822$	0	0
$823$	26.9324	0.938806	0.469403	−	0.882984i	$-0.344469\pi$
$823$	26.9324	0.938806	0.469403	+	0.882984i	$0.344469\pi$
$824$	−37.9317	−1.32141
$825$	0	0
$826$	−31.9176	−1.11056
$827$	−23.0101	−0.800139	−0.400070	−	0.916485i	$-0.631014\pi$
$827$	−23.0101	−0.800139	−0.400070	+	0.916485i	$0.631014\pi$
$828$	0	0
$829$	−32.7106	−1.13609	−0.568043	−	0.822999i	$-0.692299\pi$
$829$	−32.7106	−1.13609	−0.568043	+	0.822999i	$0.692299\pi$
$830$	−9.79737	−0.340072
$831$	0	0
$832$	−30.0026	−1.04015
$833$	−2.38853	−0.0827576
$834$	0	0
$835$	−5.90817	−0.204461
$836$	−52.8692	−1.82852
$837$	0	0
$838$	−87.4048	−3.01935
$839$	45.8300	1.58223	0.791114	−	0.611669i	$-0.209502\pi$
$839$	45.8300	1.58223	0.791114	+	0.611669i	$0.209502\pi$
$840$	0	0
$841$	−18.5403	−0.639320
$842$	14.5692	0.502088
$843$	0	0
$844$	88.6121	3.05015
$845$	3.39407	0.116760
$846$	0	0
$847$	−5.12999	−0.176269
$848$	6.16427	0.211682
$849$	0	0
$850$	26.9231	0.923456
$851$	9.31865	0.319439
$852$	0	0
$853$	3.35360	0.114825	0.0574126	−	0.998351i	$-0.481715\pi$
$853$	3.35360	0.114825	0.0574126	+	0.998351i	$0.481715\pi$
$854$	−4.94654	−0.169267
$855$	0	0
$856$	52.5454	1.79596
$857$	27.7324	0.947320	0.473660	−	0.880708i	$-0.342933\pi$
$857$	27.7324	0.947320	0.473660	+	0.880708i	$0.342933\pi$
$858$	0	0
$859$	29.2970	0.999602	0.499801	−	0.866140i	$-0.333407\pi$
$859$	29.2970	0.999602	0.499801	+	0.866140i	$0.333407\pi$
$860$	28.8292	0.983068
$861$	0	0
$862$	74.6485	2.54254
$863$	−19.0673	−0.649057	−0.324528	−	0.945876i	$-0.605206\pi$
$863$	−19.0673	−0.649057	−0.324528	+	0.945876i	$0.605206\pi$
$864$	0	0
$865$	−5.22247	−0.177569
$866$	−6.13713	−0.208548
$867$	0	0
$868$	1.58385	0.0537594
$869$	−13.8003	−0.468142
$870$	0	0
$871$	46.4535	1.57402
$872$	−73.1460	−2.47704
$873$	0	0
$874$	−13.1953	−0.446338
$875$	−6.25787	−0.211555
$876$	0	0
$877$	−19.4591	−0.657086	−0.328543	−	0.944489i	$-0.606558\pi$
$877$	−19.4591	−0.657086	−0.328543	+	0.944489i	$0.606558\pi$
$878$	80.4911	2.71644
$879$	0	0
$880$	7.07199	0.238397
$881$	36.1111	1.21662	0.608308	−	0.793701i	$-0.291849\pi$
$881$	36.1111	1.21662	0.608308	+	0.793701i	$0.291849\pi$
$882$	0	0
$883$	45.6061	1.53477	0.767384	−	0.641187i	$-0.221558\pi$
$883$	45.6061	1.53477	0.767384	+	0.641187i	$0.221558\pi$
$884$	−41.5299	−1.39680
$885$	0	0
$886$	64.9184	2.18098
$887$	6.71371	0.225424	0.112712	−	0.993628i	$-0.464046\pi$
$887$	6.71371	0.225424	0.112712	+	0.993628i	$0.464046\pi$
$888$	0	0
$889$	−14.9592	−0.501715
$890$	−8.98016	−0.301016
$891$	0	0
$892$	106.601	3.56927
$893$	6.36283	0.212924
$894$	0	0
$895$	9.74945	0.325888
$896$	−18.8781	−0.630674
$897$	0	0
$898$	−34.0866	−1.13749
$899$	−1.25656	−0.0419086
$900$	0	0
$901$	−3.29750	−0.109856
$902$	−34.2233	−1.13951
$903$	0	0
$904$	12.8003	0.425732
$905$	1.60626	0.0533939
$906$	0	0
$907$	9.74339	0.323524	0.161762	−	0.986830i	$-0.448282\pi$
$907$	9.74339	0.323524	0.161762	+	0.986830i	$0.448282\pi$
$908$	67.9174	2.25392
$909$	0	0
$910$	6.87324	0.227846
$911$	59.9266	1.98546	0.992728	−	0.120379i	$-0.0384110\pi$
$911$	59.9266	1.98546	0.992728	+	0.120379i	$0.0384110\pi$
$912$	0	0
$913$	−14.7301	−0.487496
$914$	−53.7519	−1.77795
$915$	0	0
$916$	117.438	3.88025
$917$	−9.64575	−0.318531
$918$	0	0
$919$	−42.5816	−1.40464	−0.702318	−	0.711863i	$-0.747851\pi$
$919$	−42.5816	−1.40464	−0.702318	+	0.711863i	$0.747851\pi$
$920$	3.34628	0.110323
$921$	0	0
$922$	77.5052	2.55250
$923$	7.39928	0.243550
$924$	0	0
$925$	−42.6109	−1.40104
$926$	97.3952	3.20060
$927$	0	0
$928$	−2.48734	−0.0816508
$929$	−1.83831	−0.0603131	−0.0301566	−	0.999545i	$-0.509601\pi$
$929$	−1.83831	−0.0603131	−0.0301566	+	0.999545i	$0.509601\pi$
$930$	0	0
$931$	−5.35294	−0.175435
$932$	−48.1904	−1.57853
$933$	0	0
$934$	−53.6838	−1.75659
$935$	−3.78307	−0.123720
$936$	0	0
$937$	−4.03863	−0.131936	−0.0659682	−	0.997822i	$-0.521014\pi$
$937$	−4.03863	−0.131936	−0.0659682	+	0.997822i	$0.521014\pi$
$938$	26.8477	0.876610
$939$	0	0
$940$	−3.16770	−0.103319
$941$	−8.07541	−0.263251	−0.131625	−	0.991300i	$-0.542020\pi$
$941$	−8.07541	−0.263251	−0.131625	+	0.991300i	$0.542020\pi$
$942$	0	0
$943$	−5.73026	−0.186603
$944$	−57.8136	−1.88167
$945$	0	0
$946$	64.6092	2.10063
$947$	−36.1732	−1.17547	−0.587736	−	0.809053i	$-0.699981\pi$
$947$	−36.1732	−1.17547	−0.587736	+	0.809053i	$0.699981\pi$
$948$	0	0
$949$	52.1479	1.69279
$950$	60.3375	1.95761
$951$	0	0
$952$	−12.2264	−0.396259
$953$	29.8867	0.968125	0.484063	−	0.875033i	$-0.339161\pi$
$953$	29.8867	0.968125	0.484063	+	0.875033i	$0.339161\pi$
$954$	0	0
$955$	−6.31747	−0.204428
$956$	80.4039	2.60045
$957$	0	0
$958$	6.26520	0.202419
$959$	−7.96098	−0.257073
$960$	0	0
$961$	−30.8490	−0.995131
$962$	97.9761	3.15888
$963$	0	0
$964$	20.1919	0.650337
$965$	−2.19499	−0.0706593
$966$	0	0
$967$	−0.283726	−0.00912402	−0.00456201	−	0.999990i	$-0.501452\pi$
$967$	−0.283726	−0.00912402	−0.00456201	+	0.999990i	$0.501452\pi$
$968$	−26.2593	−0.844007
$969$	0	0
$970$	0.0180570	0.000579775 0
$971$	−39.8102	−1.27757	−0.638785	−	0.769385i	$-0.720563\pi$
$971$	−39.8102	−1.27757	−0.638785	+	0.769385i	$0.720563\pi$
$972$	0	0
$973$	6.94654	0.222696
$974$	−105.295	−3.37386
$975$	0	0
$976$	−8.95986	−0.286798
$977$	−50.4503	−1.61405	−0.807024	−	0.590518i	$-0.798923\pi$
$977$	−50.4503	−1.61405	−0.807024	+	0.590518i	$0.798923\pi$
$978$	0	0
$979$	−13.5015	−0.431508
$980$	2.66493	0.0851281
$981$	0	0
$982$	46.6352	1.48819
$983$	−25.3027	−0.807031	−0.403516	−	0.914973i	$-0.632212\pi$
$983$	−25.3027	−0.807031	−0.403516	+	0.914973i	$0.632212\pi$
$984$	0	0
$985$	−1.17468	−0.0374283
$986$	19.0423	0.606429
$987$	0	0
$988$	−93.0726	−2.96104
$989$	10.8180	0.343992
$990$	0	0
$991$	12.8751	0.408990	0.204495	−	0.978868i	$-0.434445\pi$
$991$	12.8751	0.408990	0.204495	+	0.978868i	$0.434445\pi$
$992$	0.298812	0.00948728
$993$	0	0
$994$	4.27640	0.135639
$995$	−5.48814	−0.173986
$996$	0	0
$997$	20.8464	0.660212	0.330106	−	0.943944i	$-0.392915\pi$
$997$	20.8464	0.660212	0.330106	+	0.943944i	$0.392915\pi$
$998$	44.3147	1.40276
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1449.2.a.p.1.4		4
3.2	odd	2		483.2.a.i.1.1	✓	4
12.11	even	2		7728.2.a.cd.1.2		4
21.20	even	2		3381.2.a.w.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
483.2.a.i.1.1	✓	4	3.2	odd	2
1449.2.a.p.1.4		4	1.1	even	1	trivial
3381.2.a.w.1.1		4	21.20	even	2
7728.2.a.cd.1.2		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)

\(f(q)\)	\(=\)	\(q+2.46506 q^{2} +4.07653 q^{4} +0.653724 q^{5} +1.00000 q^{7} +5.11879 q^{8} +O(q^{10})\)	\(q+2.46506 q^{2} +4.07653 q^{4} +0.653724 q^{5} +1.00000 q^{7} +5.11879 q^{8} +1.61147 q^{10} +2.42281 q^{11} +4.26520 q^{13} +2.46506 q^{14} +4.46506 q^{16} -2.38853 q^{17} -5.35294 q^{19} +2.66493 q^{20} +5.97238 q^{22} +1.00000 q^{23} -4.57264 q^{25} +10.5140 q^{26} +4.07653 q^{28} -3.23415 q^{29} +0.388529 q^{31} +0.769086 q^{32} -5.88787 q^{34} +0.653724 q^{35} +9.31865 q^{37} -13.1953 q^{38} +3.34628 q^{40} -5.73026 q^{41} +10.8180 q^{43} +9.87667 q^{44} +2.46506 q^{46} -1.18866 q^{47} +1.00000 q^{49} -11.2719 q^{50} +17.3872 q^{52} +1.38056 q^{53} +1.58385 q^{55} +5.11879 q^{56} -7.97238 q^{58} -12.9480 q^{59} -2.00666 q^{61} +0.957747 q^{62} -7.03428 q^{64} +2.78826 q^{65} +10.8913 q^{67} -9.73692 q^{68} +1.61147 q^{70} +1.73480 q^{71} +12.2264 q^{73} +22.9711 q^{74} -21.8214 q^{76} +2.42281 q^{77} -5.69598 q^{79} +2.91892 q^{80} -14.1254 q^{82} -6.07977 q^{83} -1.56144 q^{85} +26.6670 q^{86} +12.4018 q^{88} -5.57264 q^{89} +4.26520 q^{91} +4.07653 q^{92} -2.93013 q^{94} -3.49934 q^{95} +0.0112053 q^{97} +2.46506 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{4} - 5 q^{5} + 4 q^{7} + 3 q^{8}+O(q^{10}) \) 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

Varieties

Fields

Representations

Groups

Database

Properties

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