LMFDB - Embedded newform 4400.2.a.bz.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [4400,2,Mod(1,4400)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("4400.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4400, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 4400 = 2^{4} \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4400.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,0,1,0,0,0,3,0,6,0,-3,0,-3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$35.1341768894$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.1229.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 7x + 6 $ x^3 - x^2 - 7*x + 6
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 2200)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.59261$ of defining polynomial
Character	$\chi$	$=$	4400.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-2.59261 q^{3} +2.72165 q^{7} +3.72165 q^{9} -1.00000 q^{11} -1.00000 q^{13} +4.59261 q^{17} -8.18523 q^{19} -7.05619 q^{21} +0.407385 q^{23} -1.87096 q^{27} +7.46358 q^{29} +4.44330 q^{31} +2.59261 q^{33} +7.31427 q^{37} +2.59261 q^{39} -3.31427 q^{41} -7.49950 q^{43} +7.05619 q^{47} +0.407385 q^{49} -11.9069 q^{51} -0.979724 q^{53} +21.2211 q^{57} -7.05619 q^{59} -4.46358 q^{61} +10.1290 q^{63} -2.25807 q^{67} -1.05619 q^{69} -10.3143 q^{71} +12.1650 q^{73} -2.72165 q^{77} -3.14931 q^{79} -6.31427 q^{81} +16.6488 q^{83} -19.3502 q^{87} +8.27835 q^{89} -2.72165 q^{91} -11.5198 q^{93} -3.03592 q^{97} -3.72165 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + q^{3} + 3 q^{7} + 6 q^{9} - 3 q^{11} - 3 q^{13} + 5 q^{17} - 7 q^{19} + 10 q^{23} - 2 q^{27} + 10 q^{29} + 3 q^{31} - q^{33} + 8 q^{37} - q^{39} + 4 q^{41} + 9 q^{43} + 10 q^{49} - 13 q^{51} - 5 q^{53}+ \cdots - 6 q^{99}+O(q^{100}) $ 3 * q + q^3 + 3 * q^7 + 6 * q^9 - 3 * q^11 - 3 * q^13 + 5 * q^17 - 7 * q^19 + 10 * q^23 - 2 * q^27 + 10 * q^29 + 3 * q^31 - q^33 + 8 * q^37 - q^39 + 4 * q^41 + 9 * q^43 + 10 * q^49 - 13 * q^51 - 5 * q^53 + 27 * q^57 - q^61 + 34 * q^63 - 14 * q^67 + 18 * q^69 - 17 * q^71 + 21 * q^73 - 3 * q^77 - 11 * q^79 - 5 * q^81 + 20 * q^83 - 25 * q^87 + 30 * q^89 - 3 * q^91 - q^93 + 10 * q^97 - 6 * q^99

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
$3$	−2.59261	−1.49685	−0.748423	−	0.663221i	$-0.769189\pi$
$3$	−2.59261	−1.49685	−0.748423	+	0.663221i	$0.769189\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	2.72165	1.02869	0.514344	−	0.857584i	$-0.328036\pi$
$7$	2.72165	1.02869	0.514344	+	0.857584i	$0.328036\pi$
$8$	0	0
$9$	3.72165	1.24055
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	−1.00000	−0.277350	−0.138675	−	0.990338i	$-0.544284\pi$
$13$	−1.00000	−0.277350	−0.138675	+	0.990338i	$0.544284\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	4.59261	1.11387	0.556936	−	0.830555i	$-0.311977\pi$
$17$	4.59261	1.11387	0.556936	+	0.830555i	$0.311977\pi$
$18$	0	0
$19$	−8.18523	−1.87782	−0.938910	−	0.344162i	$-0.888163\pi$
$19$	−8.18523	−1.87782	−0.938910	+	0.344162i	$0.888163\pi$
$20$	0	0
$21$	−7.05619	−1.53979
$22$	0	0
$23$	0.407385	0.0849457	0.0424728	−	0.999098i	$-0.486476\pi$
$23$	0.407385	0.0849457	0.0424728	+	0.999098i	$0.486476\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−1.87096	−0.360067
$28$	0	0
$29$	7.46358	1.38595	0.692976	−	0.720961i	$-0.256299\pi$
$29$	7.46358	1.38595	0.692976	+	0.720961i	$0.256299\pi$
$30$	0	0
$31$	4.44330	0.798041	0.399020	−	0.916942i	$-0.369350\pi$
$31$	4.44330	0.798041	0.399020	+	0.916942i	$0.369350\pi$
$32$	0	0
$33$	2.59261	0.451316
$34$	0	0
$35$	0	0
$36$	0	0
$37$	7.31427	1.20246	0.601229	−	0.799077i	$-0.294678\pi$
$37$	7.31427	1.20246	0.601229	+	0.799077i	$0.294678\pi$
$38$	0	0
$39$	2.59261	0.415151
$40$	0	0
$41$	−3.31427	−0.517601	−0.258801	−	0.965931i	$-0.583327\pi$
$41$	−3.31427	−0.517601	−0.258801	+	0.965931i	$0.583327\pi$
$42$	0	0
$43$	−7.49950	−1.14366	−0.571831	−	0.820371i	$-0.693767\pi$
$43$	−7.49950	−1.14366	−0.571831	+	0.820371i	$0.693767\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	7.05619	1.02925	0.514626	−	0.857415i	$-0.327931\pi$
$47$	7.05619	1.02925	0.514626	+	0.857415i	$0.327931\pi$
$48$	0	0
$49$	0.407385	0.0581979
$50$	0	0
$51$	−11.9069	−1.66730
$52$	0	0
$53$	−0.979724	−0.134575	−0.0672877	−	0.997734i	$-0.521435\pi$
$53$	−0.979724	−0.134575	−0.0672877	+	0.997734i	$0.521435\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	21.2211	2.81081
$58$	0	0
$59$	−7.05619	−0.918638	−0.459319	−	0.888271i	$-0.651907\pi$
$59$	−7.05619	−0.918638	−0.459319	+	0.888271i	$0.651907\pi$
$60$	0	0
$61$	−4.46358	−0.571503	−0.285751	−	0.958304i	$-0.592243\pi$
$61$	−4.46358	−0.571503	−0.285751	+	0.958304i	$0.592243\pi$
$62$	0	0
$63$	10.1290	1.27614
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−2.25807	−0.275868	−0.137934	−	0.990441i	$-0.544046\pi$
$67$	−2.25807	−0.275868	−0.137934	+	0.990441i	$0.544046\pi$
$68$	0	0
$69$	−1.05619	−0.127151
$70$	0	0
$71$	−10.3143	−1.22408	−0.612039	−	0.790828i	$-0.709650\pi$
$71$	−10.3143	−1.22408	−0.612039	+	0.790828i	$0.709650\pi$
$72$	0	0
$73$	12.1650	1.42380	0.711900	−	0.702281i	$-0.247835\pi$
$73$	12.1650	1.42380	0.711900	+	0.702281i	$0.247835\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−2.72165	−0.310161
$78$	0	0
$79$	−3.14931	−0.354325	−0.177163	−	0.984182i	$-0.556692\pi$
$79$	−3.14931	−0.354325	−0.177163	+	0.984182i	$0.556692\pi$
$80$	0	0
$81$	−6.31427	−0.701585
$82$	0	0
$83$	16.6488	1.82744	0.913722	−	0.406340i	$-0.133195\pi$
$83$	16.6488	1.82744	0.913722	+	0.406340i	$0.133195\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−19.3502	−2.07456
$88$	0	0
$89$	8.27835	0.877503	0.438752	−	0.898608i	$-0.355421\pi$
$89$	8.27835	0.877503	0.438752	+	0.898608i	$0.355421\pi$
$90$	0	0
$91$	−2.72165	−0.285307
$92$	0	0
$93$	−11.5198	−1.19454
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−3.03592	−0.308251	−0.154125	−	0.988051i	$-0.549256\pi$
$97$	−3.03592	−0.308251	−0.154125	+	0.988051i	$0.549256\pi$
$98$	0	0
$99$	−3.72165	−0.374040
$100$	0	0
$101$	5.59261	0.556486	0.278243	−	0.960511i	$-0.410248\pi$
$101$	5.59261	0.556486	0.278243	+	0.960511i	$0.410248\pi$
$102$	0	0
$103$	−4.64881	−0.458061	−0.229030	−	0.973419i	$-0.573555\pi$
$103$	−4.64881	−0.458061	−0.229030	+	0.973419i	$0.573555\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	0.701375	0.0678045	0.0339023	−	0.999425i	$-0.489207\pi$
$107$	0.701375	0.0678045	0.0339023	+	0.999425i	$0.489207\pi$
$108$	0	0
$109$	4.27835	0.409791	0.204896	−	0.978784i	$-0.434314\pi$
$109$	4.27835	0.409791	0.204896	+	0.978784i	$0.434314\pi$
$110$	0	0
$111$	−18.9631	−1.79990
$112$	0	0
$113$	−11.6847	−1.09921	−0.549603	−	0.835426i	$-0.685221\pi$
$113$	−11.6847	−1.09921	−0.549603	+	0.835426i	$0.685221\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−3.72165	−0.344067
$118$	0	0
$119$	12.4995	1.14583
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	8.59261	0.774770
$124$	0	0
$125$	0	0
$126$	0	0
$127$	9.51977	0.844743	0.422372	−	0.906423i	$-0.361198\pi$
$127$	9.51977	0.844743	0.422372	+	0.906423i	$0.361198\pi$
$128$	0	0
$129$	19.4433	1.71189
$130$	0	0
$131$	−22.4792	−1.96402	−0.982009	−	0.188833i	$-0.939530\pi$
$131$	−22.4792	−1.96402	−0.982009	+	0.188833i	$0.939530\pi$
$132$	0	0
$133$	−22.2773	−1.93169
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−15.5354	−1.32728	−0.663640	−	0.748052i	$-0.730989\pi$
$137$	−15.5354	−1.32728	−0.663640	+	0.748052i	$0.730989\pi$
$138$	0	0
$139$	−13.3299	−1.13063	−0.565314	−	0.824876i	$-0.691245\pi$
$139$	−13.3299	−1.13063	−0.565314	+	0.824876i	$0.691245\pi$
$140$	0	0
$141$	−18.2940	−1.54063
$142$	0	0
$143$	1.00000	0.0836242
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−1.05619	−0.0871133
$148$	0	0
$149$	−0.628532	−0.0514913	−0.0257457	−	0.999669i	$-0.508196\pi$
$149$	−0.628532	−0.0514913	−0.0257457	+	0.999669i	$0.508196\pi$
$150$	0	0
$151$	19.1280	1.55662	0.778308	−	0.627882i	$-0.216078\pi$
$151$	19.1280	1.55662	0.778308	+	0.627882i	$0.216078\pi$
$152$	0	0
$153$	17.0921	1.38182
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−8.88660	−0.709228	−0.354614	−	0.935013i	$-0.615388\pi$
$157$	−8.88660	−0.709228	−0.354614	+	0.935013i	$0.615388\pi$
$158$	0	0
$159$	2.54005	0.201439
$160$	0	0
$161$	1.10876	0.0873826
$162$	0	0
$163$	20.8340	1.63185	0.815924	−	0.578159i	$-0.196229\pi$
$163$	20.8340	1.63185	0.815924	+	0.578159i	$0.196229\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	17.4267	1.34851	0.674257	−	0.738496i	$-0.264464\pi$
$167$	17.4267	1.34851	0.674257	+	0.738496i	$0.264464\pi$
$168$	0	0
$169$	−12.0000	−0.923077
$170$	0	0
$171$	−30.4626	−2.32953
$172$	0	0
$173$	8.18523	0.622311	0.311156	−	0.950359i	$-0.399284\pi$
$173$	8.18523	0.622311	0.311156	+	0.950359i	$0.399284\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	18.2940	1.37506
$178$	0	0
$179$	10.7217	0.801374	0.400687	−	0.916215i	$-0.368772\pi$
$179$	10.7217	0.801374	0.400687	+	0.916215i	$0.368772\pi$
$180$	0	0
$181$	23.4626	1.74396	0.871980	−	0.489542i	$-0.162836\pi$
$181$	23.4626	1.74396	0.871980	+	0.489542i	$0.162836\pi$
$182$	0	0
$183$	11.5723	0.855452
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−4.59261	−0.335845
$188$	0	0
$189$	−5.09211	−0.370397
$190$	0	0
$191$	15.5354	1.12410	0.562052	−	0.827102i	$-0.310012\pi$
$191$	15.5354	1.12410	0.562052	+	0.827102i	$0.310012\pi$
$192$	0	0
$193$	1.62954	0.117297	0.0586485	−	0.998279i	$-0.481321\pi$
$193$	1.62954	0.117297	0.0586485	+	0.998279i	$0.481321\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	2.77784	0.197913	0.0989566	−	0.995092i	$-0.468449\pi$
$197$	2.77784	0.197913	0.0989566	+	0.995092i	$0.468449\pi$
$198$	0	0
$199$	26.1639	1.85471	0.927356	−	0.374179i	$-0.122076\pi$
$199$	26.1639	1.85471	0.927356	+	0.374179i	$0.122076\pi$
$200$	0	0
$201$	5.85431	0.412931
$202$	0	0
$203$	20.3133	1.42571
$204$	0	0
$205$	0	0
$206$	0	0
$207$	1.51615	0.105379
$208$	0	0
$209$	8.18523	0.566184
$210$	0	0
$211$	2.38711	0.164335	0.0821677	−	0.996619i	$-0.473816\pi$
$211$	2.38711	0.164335	0.0821677	+	0.996619i	$0.473816\pi$
$212$	0	0
$213$	26.7409	1.83226
$214$	0	0
$215$	0	0
$216$	0	0
$217$	12.0931	0.820934
$218$	0	0
$219$	−31.5390	−2.13121
$220$	0	0
$221$	−4.59261	−0.308933
$222$	0	0
$223$	25.4267	1.70269	0.851347	−	0.524603i	$-0.175786\pi$
$223$	25.4267	1.70269	0.851347	+	0.524603i	$0.175786\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	10.2783	0.682198	0.341099	−	0.940027i	$-0.389201\pi$
$227$	10.2783	0.682198	0.341099	+	0.940027i	$0.389201\pi$
$228$	0	0
$229$	29.0349	1.91868	0.959340	−	0.282252i	$-0.0910814\pi$
$229$	29.0349	1.91868	0.959340	+	0.282252i	$0.0910814\pi$
$230$	0	0
$231$	7.05619	0.464263
$232$	0	0
$233$	−5.90688	−0.386973	−0.193486	−	0.981103i	$-0.561980\pi$
$233$	−5.90688	−0.386973	−0.193486	+	0.981103i	$0.561980\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	8.16495	0.530371
$238$	0	0
$239$	−3.79449	−0.245445	−0.122723	−	0.992441i	$-0.539163\pi$
$239$	−3.79449	−0.245445	−0.122723	+	0.992441i	$0.539163\pi$
$240$	0	0
$241$	−23.5916	−1.51967	−0.759834	−	0.650117i	$-0.774720\pi$
$241$	−23.5916	−1.51967	−0.759834	+	0.650117i	$0.774720\pi$
$242$	0	0
$243$	21.9833	1.41023
$244$	0	0
$245$	0	0
$246$	0	0
$247$	8.18523	0.520814
$248$	0	0
$249$	−43.1639	−2.73540
$250$	0	0
$251$	21.5916	1.36285	0.681425	−	0.731888i	$-0.261361\pi$
$251$	21.5916	1.36285	0.681425	+	0.731888i	$0.261361\pi$
$252$	0	0
$253$	−0.407385	−0.0256121
$254$	0	0
$255$	0	0
$256$	0	0
$257$	4.01564	0.250489	0.125244	−	0.992126i	$-0.460028\pi$
$257$	4.01564	0.250489	0.125244	+	0.992126i	$0.460028\pi$
$258$	0	0
$259$	19.9069	1.23695
$260$	0	0
$261$	27.7768	1.71934
$262$	0	0
$263$	−29.2404	−1.80304	−0.901521	−	0.432736i	$-0.857548\pi$
$263$	−29.2404	−1.80304	−0.901521	+	0.432736i	$0.857548\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−21.4626	−1.31349
$268$	0	0
$269$	−8.03592	−0.489959	−0.244979	−	0.969528i	$-0.578781\pi$
$269$	−8.03592	−0.489959	−0.244979	+	0.969528i	$0.578781\pi$
$270$	0	0
$271$	19.4267	1.18009	0.590043	−	0.807372i	$-0.299111\pi$
$271$	19.4267	1.18009	0.590043	+	0.807372i	$0.299111\pi$
$272$	0	0
$273$	7.05619	0.427060
$274$	0	0
$275$	0	0
$276$	0	0
$277$	1.83041	0.109979	0.0549894	−	0.998487i	$-0.482487\pi$
$277$	1.83041	0.109979	0.0549894	+	0.998487i	$0.482487\pi$
$278$	0	0
$279$	16.5364	0.990010
$280$	0	0
$281$	17.6847	1.05498	0.527491	−	0.849561i	$-0.323133\pi$
$281$	17.6847	1.05498	0.527491	+	0.849561i	$0.323133\pi$
$282$	0	0
$283$	16.8507	1.00167	0.500835	−	0.865543i	$-0.333026\pi$
$283$	16.8507	1.00167	0.500835	+	0.865543i	$0.333026\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−9.02028	−0.532450
$288$	0	0
$289$	4.09211	0.240712
$290$	0	0
$291$	7.87096	0.461404
$292$	0	0
$293$	28.3705	1.65742	0.828710	−	0.559678i	$-0.189075\pi$
$293$	28.3705	1.65742	0.828710	+	0.559678i	$0.189075\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	1.87096	0.108564
$298$	0	0
$299$	−0.407385	−0.0235597
$300$	0	0
$301$	−20.4110	−1.17647
$302$	0	0
$303$	−14.4995	−0.832974
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−2.29399	−0.130925	−0.0654625	−	0.997855i	$-0.520852\pi$
$307$	−2.29399	−0.130925	−0.0654625	+	0.997855i	$0.520852\pi$
$308$	0	0
$309$	12.0526	0.685647
$310$	0	0
$311$	−11.2581	−0.638387	−0.319193	−	0.947690i	$-0.603412\pi$
$311$	−11.2581	−0.638387	−0.319193	+	0.947690i	$0.603412\pi$
$312$	0	0
$313$	−7.18523	−0.406133	−0.203067	−	0.979165i	$-0.565091\pi$
$313$	−7.18523	−0.406133	−0.203067	+	0.979165i	$0.565091\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−29.5198	−1.65800	−0.828998	−	0.559252i	$-0.811088\pi$
$317$	−29.5198	−1.65800	−0.828998	+	0.559252i	$0.811088\pi$
$318$	0	0
$319$	−7.46358	−0.417880
$320$	0	0
$321$	−1.81840	−0.101493
$322$	0	0
$323$	−37.5916	−2.09165
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−11.0921	−0.613395
$328$	0	0
$329$	19.2045	1.05878
$330$	0	0
$331$	6.12904	0.336882	0.168441	−	0.985712i	$-0.446127\pi$
$331$	6.12904	0.336882	0.168441	+	0.985712i	$0.446127\pi$
$332$	0	0
$333$	27.2211	1.49171
$334$	0	0
$335$	0	0
$336$	0	0
$337$	10.0000	0.544735	0.272367	−	0.962193i	$-0.412193\pi$
$337$	10.0000	0.544735	0.272367	+	0.962193i	$0.412193\pi$
$338$	0	0
$339$	30.2940	1.64534
$340$	0	0
$341$	−4.44330	−0.240618
$342$	0	0
$343$	−17.9428	−0.968820
$344$	0	0
$345$	0	0
$346$	0	0
$347$	14.7217	0.790300	0.395150	−	0.918617i	$-0.370693\pi$
$347$	14.7217	0.790300	0.395150	+	0.918617i	$0.370693\pi$
$348$	0	0
$349$	32.2976	1.72885	0.864426	−	0.502760i	$-0.167682\pi$
$349$	32.2976	1.72885	0.864426	+	0.502760i	$0.167682\pi$
$350$	0	0
$351$	1.87096	0.0998646
$352$	0	0
$353$	−3.37046	−0.179391	−0.0896957	−	0.995969i	$-0.528589\pi$
$353$	−3.37046	−0.179391	−0.0896957	+	0.995969i	$0.528589\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−32.4064	−1.71513
$358$	0	0
$359$	19.7419	1.04194	0.520970	−	0.853575i	$-0.325571\pi$
$359$	19.7419	1.04194	0.520970	+	0.853575i	$0.325571\pi$
$360$	0	0
$361$	47.9980	2.52621
$362$	0	0
$363$	−2.59261	−0.136077
$364$	0	0
$365$	0	0
$366$	0	0
$367$	30.4792	1.59100	0.795501	−	0.605952i	$-0.207208\pi$
$367$	30.4792	1.59100	0.795501	+	0.605952i	$0.207208\pi$
$368$	0	0
$369$	−12.3345	−0.642111
$370$	0	0
$371$	−2.66647	−0.138436
$372$	0	0
$373$	27.4985	1.42382	0.711909	−	0.702272i	$-0.247831\pi$
$373$	27.4985	1.42382	0.711909	+	0.702272i	$0.247831\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−7.46358	−0.384394
$378$	0	0
$379$	−3.87096	−0.198838	−0.0994190	−	0.995046i	$-0.531698\pi$
$379$	−3.87096	−0.198838	−0.0994190	+	0.995046i	$0.531698\pi$
$380$	0	0
$381$	−24.6811	−1.26445
$382$	0	0
$383$	−33.2414	−1.69856	−0.849279	−	0.527945i	$-0.822963\pi$
$383$	−33.2414	−1.69856	−0.849279	+	0.527945i	$0.822963\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−27.9105	−1.41877
$388$	0	0
$389$	33.9980	1.72377	0.861883	−	0.507107i	$-0.169285\pi$
$389$	33.9980	1.72377	0.861883	+	0.507107i	$0.169285\pi$
$390$	0	0
$391$	1.87096	0.0946187
$392$	0	0
$393$	58.2800	2.93983
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−2.85069	−0.143072	−0.0715360	−	0.997438i	$-0.522790\pi$
$397$	−2.85069	−0.143072	−0.0715360	+	0.997438i	$0.522790\pi$
$398$	0	0
$399$	57.7566	2.89144
$400$	0	0
$401$	16.0967	0.803833	0.401917	−	0.915676i	$-0.368344\pi$
$401$	16.0967	0.803833	0.401917	+	0.915676i	$0.368344\pi$
$402$	0	0
$403$	−4.44330	−0.221337
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−7.31427	−0.362555
$408$	0	0
$409$	0.757568	0.0374593	0.0187297	−	0.999825i	$-0.494038\pi$
$409$	0.757568	0.0374593	0.0187297	+	0.999825i	$0.494038\pi$
$410$	0	0
$411$	40.2773	1.98673
$412$	0	0
$413$	−19.2045	−0.944991
$414$	0	0
$415$	0	0
$416$	0	0
$417$	34.5593	1.69238
$418$	0	0
$419$	−36.8700	−1.80122	−0.900608	−	0.434633i	$-0.856878\pi$
$419$	−36.8700	−1.80122	−0.900608	+	0.434633i	$0.856878\pi$
$420$	0	0
$421$	−32.2055	−1.56960	−0.784800	−	0.619749i	$-0.787234\pi$
$421$	−32.2055	−1.56960	−0.784800	+	0.619749i	$0.787234\pi$
$422$	0	0
$423$	26.2607	1.27684
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−12.1483	−0.587898
$428$	0	0
$429$	−2.59261	−0.125173
$430$	0	0
$431$	−17.9428	−0.864274	−0.432137	−	0.901808i	$-0.642240\pi$
$431$	−17.9428	−0.864274	−0.432137	+	0.901808i	$0.642240\pi$
$432$	0	0
$433$	14.2258	0.683647	0.341824	−	0.939764i	$-0.388955\pi$
$433$	14.2258	0.683647	0.341824	+	0.939764i	$0.388955\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−3.33454	−0.159513
$438$	0	0
$439$	9.70601	0.463243	0.231621	−	0.972806i	$-0.425597\pi$
$439$	9.70601	0.463243	0.231621	+	0.972806i	$0.425597\pi$
$440$	0	0
$441$	1.51615	0.0721974
$442$	0	0
$443$	−4.44431	−0.211156	−0.105578	−	0.994411i	$-0.533669\pi$
$443$	−4.44431	−0.211156	−0.105578	+	0.994411i	$0.533669\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	1.62954	0.0770746
$448$	0	0
$449$	−12.4479	−0.587454	−0.293727	−	0.955889i	$-0.594896\pi$
$449$	−12.4479	−0.587454	−0.293727	+	0.955889i	$0.594896\pi$
$450$	0	0
$451$	3.31427	0.156063
$452$	0	0
$453$	−49.5916	−2.33002
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−16.2368	−0.759525	−0.379762	−	0.925084i	$-0.623994\pi$
$457$	−16.2368	−0.759525	−0.379762	+	0.925084i	$0.623994\pi$
$458$	0	0
$459$	−8.59261	−0.401069
$460$	0	0
$461$	−41.8689	−1.95003	−0.975016	−	0.222136i	$-0.928697\pi$
$461$	−41.8689	−1.95003	−0.975016	+	0.222136i	$0.928697\pi$
$462$	0	0
$463$	18.7014	0.869127	0.434563	−	0.900641i	$-0.356903\pi$
$463$	18.7014	0.869127	0.434563	+	0.900641i	$0.356903\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	12.2820	0.568342	0.284171	−	0.958774i	$-0.408282\pi$
$467$	12.2820	0.568342	0.284171	+	0.958774i	$0.408282\pi$
$468$	0	0
$469$	−6.14569	−0.283781
$470$	0	0
$471$	23.0395	1.06161
$472$	0	0
$473$	7.49950	0.344827
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−3.64619	−0.166948
$478$	0	0
$479$	−6.75757	−0.308761	−0.154381	−	0.988011i	$-0.549338\pi$
$479$	−6.75757	−0.308761	−0.154381	+	0.988011i	$0.549338\pi$
$480$	0	0
$481$	−7.31427	−0.333502
$482$	0	0
$483$	−2.87459	−0.130798
$484$	0	0
$485$	0	0
$486$	0	0
$487$	34.0395	1.54248	0.771239	−	0.636545i	$-0.219637\pi$
$487$	34.0395	1.54248	0.771239	+	0.636545i	$0.219637\pi$
$488$	0	0
$489$	−54.0146	−2.44263
$490$	0	0
$491$	17.3299	0.782088	0.391044	−	0.920372i	$-0.372114\pi$
$491$	17.3299	0.782088	0.391044	+	0.920372i	$0.372114\pi$
$492$	0	0
$493$	34.2773	1.54377
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−28.0718	−1.25919
$498$	0	0
$499$	14.0931	0.630895	0.315447	−	0.948943i	$-0.397845\pi$
$499$	14.0931	0.630895	0.315447	+	0.948943i	$0.397845\pi$
$500$	0	0
$501$	−45.1806	−2.01852
$502$	0	0
$503$	15.7825	0.703706	0.351853	−	0.936055i	$-0.385552\pi$
$503$	15.7825	0.703706	0.351853	+	0.936055i	$0.385552\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	31.1114	1.38170
$508$	0	0
$509$	−13.3907	−0.593534	−0.296767	−	0.954950i	$-0.595908\pi$
$509$	−13.3907	−0.593534	−0.296767	+	0.954950i	$0.595908\pi$
$510$	0	0
$511$	33.1088	1.46465
$512$	0	0
$513$	15.3143	0.676141
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−7.05619	−0.310331
$518$	0	0
$519$	−21.2211	−0.931505
$520$	0	0
$521$	11.6857	0.511961	0.255981	−	0.966682i	$-0.417602\pi$
$521$	11.6857	0.511961	0.255981	+	0.966682i	$0.417602\pi$
$522$	0	0
$523$	18.9022	0.826538	0.413269	−	0.910609i	$-0.364387\pi$
$523$	18.9022	0.826538	0.413269	+	0.910609i	$0.364387\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	20.4064	0.888916
$528$	0	0
$529$	−22.8340	−0.992784
$530$	0	0
$531$	−26.2607	−1.13962
$532$	0	0
$533$	3.31427	0.143557
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−27.7971	−1.19953
$538$	0	0
$539$	−0.407385	−0.0175473
$540$	0	0
$541$	−10.8497	−0.466464	−0.233232	−	0.972421i	$-0.574930\pi$
$541$	−10.8497	−0.466464	−0.233232	+	0.972421i	$0.574930\pi$
$542$	0	0
$543$	−60.8294	−2.61044
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−44.9621	−1.92244	−0.961220	−	0.275784i	$-0.911062\pi$
$547$	−44.9621	−1.92244	−0.961220	+	0.275784i	$0.911062\pi$
$548$	0	0
$549$	−16.6119	−0.708978
$550$	0	0
$551$	−61.0911	−2.60257
$552$	0	0
$553$	−8.57133	−0.364490
$554$	0	0
$555$	0	0
$556$	0	0
$557$	1.99899	0.0847000	0.0423500	−	0.999103i	$-0.486516\pi$
$557$	1.99899	0.0847000	0.0423500	+	0.999103i	$0.486516\pi$
$558$	0	0
$559$	7.49950	0.317195
$560$	0	0
$561$	11.9069	0.502709
$562$	0	0
$563$	−28.8340	−1.21521	−0.607605	−	0.794239i	$-0.707870\pi$
$563$	−28.8340	−1.21521	−0.607605	+	0.794239i	$0.707870\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−17.1852	−0.721712
$568$	0	0
$569$	25.6894	1.07695	0.538477	−	0.842640i	$-0.319000\pi$
$569$	25.6894	1.07695	0.538477	+	0.842640i	$0.319000\pi$
$570$	0	0
$571$	36.4792	1.52661	0.763304	−	0.646040i	$-0.223576\pi$
$571$	36.4792	1.52661	0.763304	+	0.646040i	$0.223576\pi$
$572$	0	0
$573$	−40.2773	−1.68261
$574$	0	0
$575$	0	0
$576$	0	0
$577$	36.1078	1.50319	0.751593	−	0.659628i	$-0.229286\pi$
$577$	36.1078	1.50319	0.751593	+	0.659628i	$0.229286\pi$
$578$	0	0
$579$	−4.22477	−0.175576
$580$	0	0
$581$	45.3122	1.87987
$582$	0	0
$583$	0.979724	0.0405760
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−5.77784	−0.238477	−0.119239	−	0.992866i	$-0.538045\pi$
$587$	−5.77784	−0.238477	−0.119239	+	0.992866i	$0.538045\pi$
$588$	0	0
$589$	−36.3694	−1.49858
$590$	0	0
$591$	−7.20188	−0.296246
$592$	0	0
$593$	−32.9418	−1.35276	−0.676379	−	0.736554i	$-0.736452\pi$
$593$	−32.9418	−1.35276	−0.676379	+	0.736554i	$0.736452\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−67.8330	−2.77622
$598$	0	0
$599$	8.14830	0.332931	0.166465	−	0.986047i	$-0.446765\pi$
$599$	8.14830	0.332931	0.166465	+	0.986047i	$0.446765\pi$
$600$	0	0
$601$	7.39073	0.301474	0.150737	−	0.988574i	$-0.451835\pi$
$601$	7.39073	0.301474	0.150737	+	0.988574i	$0.451835\pi$
$602$	0	0
$603$	−8.40376	−0.342228
$604$	0	0
$605$	0	0
$606$	0	0
$607$	27.9980	1.13640	0.568202	−	0.822889i	$-0.307639\pi$
$607$	27.9980	1.13640	0.568202	+	0.822889i	$0.307639\pi$
$608$	0	0
$609$	−52.6644	−2.13407
$610$	0	0
$611$	−7.05619	−0.285463
$612$	0	0
$613$	−39.0036	−1.57534	−0.787671	−	0.616096i	$-0.788713\pi$
$613$	−39.0036	−1.57534	−0.787671	+	0.616096i	$0.788713\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	26.1686	1.05351	0.526754	−	0.850018i	$-0.323409\pi$
$617$	26.1686	1.05351	0.526754	+	0.850018i	$0.323409\pi$
$618$	0	0
$619$	22.2820	0.895588	0.447794	−	0.894137i	$-0.352210\pi$
$619$	22.2820	0.895588	0.447794	+	0.894137i	$0.352210\pi$
$620$	0	0
$621$	−0.762203	−0.0305862
$622$	0	0
$623$	22.5308	0.902677
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−21.2211	−0.847491
$628$	0	0
$629$	33.5916	1.33939
$630$	0	0
$631$	−34.1629	−1.36000	−0.680002	−	0.733210i	$-0.738021\pi$
$631$	−34.1629	−1.36000	−0.680002	+	0.733210i	$0.738021\pi$
$632$	0	0
$633$	−6.18885	−0.245985
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−0.407385	−0.0161412
$638$	0	0
$639$	−38.3861	−1.51853
$640$	0	0
$641$	−5.37046	−0.212120	−0.106060	−	0.994360i	$-0.533824\pi$
$641$	−5.37046	−0.212120	−0.106060	+	0.994360i	$0.533824\pi$
$642$	0	0
$643$	15.8856	0.626467	0.313233	−	0.949676i	$-0.398588\pi$
$643$	15.8856	0.626467	0.313233	+	0.949676i	$0.398588\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	4.87197	0.191537	0.0957685	−	0.995404i	$-0.469469\pi$
$647$	4.87197	0.191537	0.0957685	+	0.995404i	$0.469469\pi$
$648$	0	0
$649$	7.05619	0.276980
$650$	0	0
$651$	−31.3528	−1.22881
$652$	0	0
$653$	16.8101	0.657831	0.328916	−	0.944359i	$-0.393317\pi$
$653$	16.8101	0.657831	0.328916	+	0.944359i	$0.393317\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	45.2737	1.76630
$658$	0	0
$659$	−2.79449	−0.108858	−0.0544290	−	0.998518i	$-0.517334\pi$
$659$	−2.79449	−0.108858	−0.0544290	+	0.998518i	$0.517334\pi$
$660$	0	0
$661$	33.4267	1.30015	0.650073	−	0.759872i	$-0.274738\pi$
$661$	33.4267	1.30015	0.650073	+	0.759872i	$0.274738\pi$
$662$	0	0
$663$	11.9069	0.462425
$664$	0	0
$665$	0	0
$666$	0	0
$667$	3.04055	0.117731
$668$	0	0
$669$	−65.9215	−2.54867
$670$	0	0
$671$	4.46358	0.172315
$672$	0	0
$673$	−11.9115	−0.459155	−0.229578	−	0.973290i	$-0.573734\pi$
$673$	−11.9115	−0.459155	−0.229578	+	0.973290i	$0.573734\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−13.4277	−0.516067	−0.258033	−	0.966136i	$-0.583074\pi$
$677$	−13.4277	−0.516067	−0.258033	+	0.966136i	$0.583074\pi$
$678$	0	0
$679$	−8.26271	−0.317094
$680$	0	0
$681$	−26.6478	−1.02115
$682$	0	0
$683$	−47.7160	−1.82580	−0.912901	−	0.408181i	$-0.866163\pi$
$683$	−47.7160	−1.82580	−0.912901	+	0.408181i	$0.866163\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−75.2763	−2.87197
$688$	0	0
$689$	0.979724	0.0373245
$690$	0	0
$691$	29.2450	1.11253	0.556267	−	0.831004i	$-0.312233\pi$
$691$	29.2450	1.11253	0.556267	+	0.831004i	$0.312233\pi$
$692$	0	0
$693$	−10.1290	−0.384770
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−15.2211	−0.576542
$698$	0	0
$699$	15.3143	0.579239
$700$	0	0
$701$	18.6442	0.704181	0.352090	−	0.935966i	$-0.385471\pi$
$701$	18.6442	0.704181	0.352090	+	0.935966i	$0.385471\pi$
$702$	0	0
$703$	−59.8689	−2.25800
$704$	0	0
$705$	0	0
$706$	0	0
$707$	15.2211	0.572450
$708$	0	0
$709$	−30.3705	−1.14059	−0.570293	−	0.821441i	$-0.693170\pi$
$709$	−30.3705	−1.14059	−0.570293	+	0.821441i	$0.693170\pi$
$710$	0	0
$711$	−11.7206	−0.439558
$712$	0	0
$713$	1.81014	0.0677901
$714$	0	0
$715$	0	0
$716$	0	0
$717$	9.83766	0.367394
$718$	0	0
$719$	−4.49950	−0.167803	−0.0839014	−	0.996474i	$-0.526738\pi$
$719$	−4.49950	−0.167803	−0.0839014	+	0.996474i	$0.526738\pi$
$720$	0	0
$721$	−12.6524	−0.471201
$722$	0	0
$723$	61.1639	2.27471
$724$	0	0
$725$	0	0
$726$	0	0
$727$	38.7925	1.43873	0.719367	−	0.694631i	$-0.244432\pi$
$727$	38.7925	1.43873	0.719367	+	0.694631i	$0.244432\pi$
$728$	0	0
$729$	−38.0516	−1.40932
$730$	0	0
$731$	−34.4423	−1.27389
$732$	0	0
$733$	−29.5547	−1.09163	−0.545813	−	0.837907i	$-0.683779\pi$
$733$	−29.5547	−1.09163	−0.545813	+	0.837907i	$0.683779\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	2.25807	0.0831772
$738$	0	0
$739$	−10.9079	−0.401253	−0.200627	−	0.979668i	$-0.564298\pi$
$739$	−10.9079	−0.401253	−0.200627	+	0.979668i	$0.564298\pi$
$740$	0	0
$741$	−21.2211	−0.779578
$742$	0	0
$743$	7.94743	0.291563	0.145782	−	0.989317i	$-0.453430\pi$
$743$	7.94743	0.291563	0.145782	+	0.989317i	$0.453430\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	61.9611	2.26704
$748$	0	0
$749$	1.90890	0.0697497
$750$	0	0
$751$	18.2783	0.666986	0.333493	−	0.942753i	$-0.391773\pi$
$751$	18.2783	0.666986	0.333493	+	0.942753i	$0.391773\pi$
$752$	0	0
$753$	−55.9787	−2.03998
$754$	0	0
$755$	0	0
$756$	0	0
$757$	30.2727	1.10028	0.550140	−	0.835072i	$-0.314574\pi$
$757$	30.2727	1.10028	0.550140	+	0.835072i	$0.314574\pi$
$758$	0	0
$759$	1.05619	0.0383374
$760$	0	0
$761$	14.7429	0.534431	0.267216	−	0.963637i	$-0.413896\pi$
$761$	14.7429	0.534431	0.267216	+	0.963637i	$0.413896\pi$
$762$	0	0
$763$	11.6442	0.421547
$764$	0	0
$765$	0	0
$766$	0	0
$767$	7.05619	0.254784
$768$	0	0
$769$	3.94280	0.142181	0.0710905	−	0.997470i	$-0.477352\pi$
$769$	3.94280	0.142181	0.0710905	+	0.997470i	$0.477352\pi$
$770$	0	0
$771$	−10.4110	−0.374943
$772$	0	0
$773$	52.6478	1.89361	0.946805	−	0.321808i	$-0.104291\pi$
$773$	52.6478	1.89361	0.946805	+	0.321808i	$0.104291\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−51.6109	−1.85153
$778$	0	0
$779$	27.1280	0.971962
$780$	0	0
$781$	10.3143	0.369073
$782$	0	0
$783$	−13.9641	−0.499036
$784$	0	0
$785$	0	0
$786$	0	0
$787$	44.5713	1.58880	0.794398	−	0.607397i	$-0.207786\pi$
$787$	44.5713	1.58880	0.794398	+	0.607397i	$0.207786\pi$
$788$	0	0
$789$	75.8091	2.69888
$790$	0	0
$791$	−31.8017	−1.13074
$792$	0	0
$793$	4.46358	0.158506
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−9.03693	−0.320104	−0.160052	−	0.987109i	$-0.551166\pi$
$797$	−9.03693	−0.320104	−0.160052	+	0.987109i	$0.551166\pi$
$798$	0	0
$799$	32.4064	1.14646
$800$	0	0
$801$	30.8091	1.08859
$802$	0	0
$803$	−12.1650	−0.429292
$804$	0	0
$805$	0	0
$806$	0	0
$807$	20.8340	0.733393
$808$	0	0
$809$	9.14468	0.321510	0.160755	−	0.986994i	$-0.448607\pi$
$809$	9.14468	0.321510	0.160755	+	0.986994i	$0.448607\pi$
$810$	0	0
$811$	1.35482	0.0475741	0.0237870	−	0.999717i	$-0.492428\pi$
$811$	1.35482	0.0475741	0.0237870	+	0.999717i	$0.492428\pi$
$812$	0	0
$813$	−50.3658	−1.76641
$814$	0	0
$815$	0	0
$816$	0	0
$817$	61.3851	2.14759
$818$	0	0
$819$	−10.1290	−0.353937
$820$	0	0
$821$	−27.2414	−0.950732	−0.475366	−	0.879788i	$-0.657684\pi$
$821$	−27.2414	−0.950732	−0.475366	+	0.879788i	$0.657684\pi$
$822$	0	0
$823$	6.96771	0.242879	0.121440	−	0.992599i	$-0.461249\pi$
$823$	6.96771	0.242879	0.121440	+	0.992599i	$0.461249\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	2.68472	0.0933570	0.0466785	−	0.998910i	$-0.485136\pi$
$827$	2.68472	0.0933570	0.0466785	+	0.998910i	$0.485136\pi$
$828$	0	0
$829$	−20.2222	−0.702345	−0.351172	−	0.936311i	$-0.614217\pi$
$829$	−20.2222	−0.702345	−0.351172	+	0.936311i	$0.614217\pi$
$830$	0	0
$831$	−4.74555	−0.164621
$832$	0	0
$833$	1.87096	0.0648250
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−8.31326	−0.287348
$838$	0	0
$839$	−35.5344	−1.22678	−0.613392	−	0.789779i	$-0.710195\pi$
$839$	−35.5344	−1.22678	−0.613392	+	0.789779i	$0.710195\pi$
$840$	0	0
$841$	26.7050	0.920862
$842$	0	0
$843$	−45.8497	−1.57915
$844$	0	0
$845$	0	0
$846$	0	0
$847$	2.72165	0.0935170
$848$	0	0
$849$	−43.6873	−1.49935
$850$	0	0
$851$	2.97972	0.102144
$852$	0	0
$853$	−26.3253	−0.901360	−0.450680	−	0.892686i	$-0.648818\pi$
$853$	−26.3253	−0.901360	−0.450680	+	0.892686i	$0.648818\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	48.1270	1.64399	0.821994	−	0.569496i	$-0.192862\pi$
$857$	48.1270	1.64399	0.821994	+	0.569496i	$0.192862\pi$
$858$	0	0
$859$	−40.8148	−1.39258	−0.696291	−	0.717760i	$-0.745168\pi$
$859$	−40.8148	−1.39258	−0.696291	+	0.717760i	$0.745168\pi$
$860$	0	0
$861$	23.3861	0.796996
$862$	0	0
$863$	−39.8866	−1.35776	−0.678878	−	0.734251i	$-0.737533\pi$
$863$	−39.8866	−1.35776	−0.678878	+	0.734251i	$0.737533\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−10.6093	−0.360310
$868$	0	0
$869$	3.14931	0.106833
$870$	0	0
$871$	2.25807	0.0765119
$872$	0	0
$873$	−11.2986	−0.382401
$874$	0	0
$875$	0	0
$876$	0	0
$877$	11.7263	0.395969	0.197984	−	0.980205i	$-0.436561\pi$
$877$	11.7263	0.395969	0.197984	+	0.980205i	$0.436561\pi$
$878$	0	0
$879$	−73.5537	−2.48090
$880$	0	0
$881$	−11.7363	−0.395405	−0.197703	−	0.980262i	$-0.563348\pi$
$881$	−11.7363	−0.395405	−0.197703	+	0.980262i	$0.563348\pi$
$882$	0	0
$883$	−14.0146	−0.471630	−0.235815	−	0.971798i	$-0.575776\pi$
$883$	−14.0146	−0.471630	−0.235815	+	0.971798i	$0.575776\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−4.12904	−0.138639	−0.0693197	−	0.997594i	$-0.522083\pi$
$887$	−4.12904	−0.138639	−0.0693197	+	0.997594i	$0.522083\pi$
$888$	0	0
$889$	25.9095	0.868977
$890$	0	0
$891$	6.31427	0.211536
$892$	0	0
$893$	−57.7566	−1.93275
$894$	0	0
$895$	0	0
$896$	0	0
$897$	1.05619	0.0352653
$898$	0	0
$899$	33.1629	1.10605
$900$	0	0
$901$	−4.49950	−0.149900
$902$	0	0
$903$	52.9179	1.76100
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−25.2258	−0.837608	−0.418804	−	0.908077i	$-0.637551\pi$
$907$	−25.2258	−0.837608	−0.418804	+	0.908077i	$0.637551\pi$
$908$	0	0
$909$	20.8138	0.690349
$910$	0	0
$911$	19.7419	0.654079	0.327040	−	0.945011i	$-0.393949\pi$
$911$	19.7419	0.654079	0.327040	+	0.945011i	$0.393949\pi$
$912$	0	0
$913$	−16.6488	−0.550995
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−61.1806	−2.02036
$918$	0	0
$919$	−39.1114	−1.29017	−0.645083	−	0.764113i	$-0.723177\pi$
$919$	−39.1114	−1.29017	−0.645083	+	0.764113i	$0.723177\pi$
$920$	0	0
$921$	5.94743	0.195975
$922$	0	0
$923$	10.3143	0.339498
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−17.3012	−0.568247
$928$	0	0
$929$	−0.670093	−0.0219850	−0.0109925	−	0.999940i	$-0.503499\pi$
$929$	−0.670093	−0.0219850	−0.0109925	+	0.999940i	$0.503499\pi$
$930$	0	0
$931$	−3.33454	−0.109285
$932$	0	0
$933$	29.1878	0.955567
$934$	0	0
$935$	0	0
$936$	0	0
$937$	31.3715	1.02486	0.512431	−	0.858729i	$-0.328745\pi$
$937$	31.3715	1.02486	0.512431	+	0.858729i	$0.328745\pi$
$938$	0	0
$939$	18.6285	0.607919
$940$	0	0
$941$	−39.5703	−1.28996	−0.644978	−	0.764201i	$-0.723133\pi$
$941$	−39.5703	−1.28996	−0.644978	+	0.764201i	$0.723133\pi$
$942$	0	0
$943$	−1.35018	−0.0439680
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−9.31427	−0.302673	−0.151336	−	0.988482i	$-0.548358\pi$
$947$	−9.31427	−0.302673	−0.151336	+	0.988482i	$0.548358\pi$
$948$	0	0
$949$	−12.1650	−0.394891
$950$	0	0
$951$	76.5334	2.48177
$952$	0	0
$953$	26.7409	0.866223	0.433112	−	0.901340i	$-0.357416\pi$
$953$	26.7409	0.866223	0.433112	+	0.901340i	$0.357416\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	19.3502	0.625503
$958$	0	0
$959$	−42.2820	−1.36536
$960$	0	0
$961$	−11.2571	−0.363131
$962$	0	0
$963$	2.61027	0.0841149
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−29.0442	−0.933998	−0.466999	−	0.884258i	$-0.654665\pi$
$967$	−29.0442	−0.933998	−0.466999	+	0.884258i	$0.654665\pi$
$968$	0	0
$969$	97.4606	3.13088
$970$	0	0
$971$	41.3482	1.32693	0.663463	−	0.748209i	$-0.269086\pi$
$971$	41.3482	1.32693	0.663463	+	0.748209i	$0.269086\pi$
$972$	0	0
$973$	−36.2794	−1.16306
$974$	0	0
$975$	0	0
$976$	0	0
$977$	14.9105	0.477029	0.238515	−	0.971139i	$-0.423339\pi$
$977$	14.9105	0.477029	0.238515	+	0.971139i	$0.423339\pi$
$978$	0	0
$979$	−8.27835	−0.264577
$980$	0	0
$981$	15.9225	0.508367
$982$	0	0
$983$	−21.4682	−0.684730	−0.342365	−	0.939567i	$-0.611228\pi$
$983$	−21.4682	−0.684730	−0.342365	+	0.939567i	$0.611228\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−49.7899	−1.58483
$988$	0	0
$989$	−3.05518	−0.0971492
$990$	0	0
$991$	−3.47922	−0.110521	−0.0552605	−	0.998472i	$-0.517599\pi$
$991$	−3.47922	−0.110521	−0.0552605	+	0.998472i	$0.517599\pi$
$992$	0	0
$993$	−15.8902	−0.504261
$994$	0	0
$995$	0	0
$996$	0	0
$997$	23.4054	0.741255	0.370628	−	0.928782i	$-0.379143\pi$
$997$	23.4054	0.741255	0.370628	+	0.928782i	$0.379143\pi$
$998$	0	0
$999$	−13.6847	−0.432966

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4400.2.a.bz.1.1		3
4.3	odd	2		2200.2.a.u.1.3	✓	3
5.2	odd	4		4400.2.b.bb.4049.5		6
5.3	odd	4		4400.2.b.bb.4049.2		6
5.4	even	2		4400.2.a.by.1.3		3
20.3	even	4		2200.2.b.m.1849.5		6
20.7	even	4		2200.2.b.m.1849.2		6
20.19	odd	2		2200.2.a.v.1.1	yes	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2200.2.a.u.1.3	✓	3	4.3	odd	2
2200.2.a.v.1.1	yes	3	20.19	odd	2
2200.2.b.m.1849.2		6	20.7	even	4
2200.2.b.m.1849.5		6	20.3	even	4
4400.2.a.by.1.3		3	5.4	even	2
4400.2.a.bz.1.1		3	1.1	even	1	trivial
4400.2.b.bb.4049.2		6	5.3	odd	4
4400.2.b.bb.4049.5		6	5.2	odd	4

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 4400 = 2^{4} \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4400.a (trivial)

\(f(q)\)	\(=\)	\(q-2.59261 q^{3} +2.72165 q^{7} +3.72165 q^{9} -1.00000 q^{11} -1.00000 q^{13} +4.59261 q^{17} -8.18523 q^{19} -7.05619 q^{21} +0.407385 q^{23} -1.87096 q^{27} +7.46358 q^{29} +4.44330 q^{31} +2.59261 q^{33} +7.31427 q^{37} +2.59261 q^{39} -3.31427 q^{41} -7.49950 q^{43} +7.05619 q^{47} +0.407385 q^{49} -11.9069 q^{51} -0.979724 q^{53} +21.2211 q^{57} -7.05619 q^{59} -4.46358 q^{61} +10.1290 q^{63} -2.25807 q^{67} -1.05619 q^{69} -10.3143 q^{71} +12.1650 q^{73} -2.72165 q^{77} -3.14931 q^{79} -6.31427 q^{81} +16.6488 q^{83} -19.3502 q^{87} +8.27835 q^{89} -2.72165 q^{91} -11.5198 q^{93} -3.03592 q^{97} -3.72165 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + q^{3} + 3 q^{7} + 6 q^{9} - 3 q^{11} - 3 q^{13} + 5 q^{17} - 7 q^{19} + 10 q^{23} - 2 q^{27} + 10 q^{29} + 3 q^{31} - q^{33} + 8 q^{37} - q^{39} + 4 q^{41} + 9 q^{43} + 10 q^{49} - 13 q^{51} - 5 q^{53}+ \cdots - 6 q^{99}+O(q^{100}) \) 3 * q + q^3 + 3 * q^7 + 6 * q^9 - 3 * q^11 - 3 * q^13 + 5 * q^17 - 7 * q^19 + 10 * q^23 - 2 * q^27 + 10 * q^29 + 3 * q^31 - q^33 + 8 * q^37 - q^39 + 4 * q^41 + 9 * q^43 + 10 * q^49 - 13 * q^51 - 5 * q^53 + 27 * q^57 - q^61 + 34 * q^63 - 14 * q^67 + 18 * q^69 - 17 * q^71 + 21 * q^73 - 3 * q^77 - 11 * q^79 - 5 * q^81 + 20 * q^83 - 25 * q^87 + 30 * q^89 - 3 * q^91 - q^93 + 10 * q^97 - 6 * q^99

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