LMFDB - Embedded newform 3344.2.a.w.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3344, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("3344.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3344 = 2^{4} \cdot 11 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3344.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$26.7019744359$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.6.744786576.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} - 17x^{4} + 13x^{3} + 69x^{2} - 21x - 30 $ x^6 - x^5 - 17x^4 + 13x^3 + 69x^2 - 21x - 30
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 836)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-0.566270$ of defining polynomial
Character	$\chi$	$=$	3344.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.566270 q^{3} -3.92909 q^{5} +2.44546 q^{7} -2.67934 q^{9} +O(q^{10})$	$q+0.566270 q^{3} -3.92909 q^{5} +2.44546 q^{7} -2.67934 q^{9} +1.00000 q^{11} +6.76733 q^{13} -2.22493 q^{15} -0.175988 q^{17} +1.00000 q^{19} +1.38479 q^{21} -8.30079 q^{23} +10.4377 q^{25} -3.21604 q^{27} -0.843595 q^{29} +0.224925 q^{31} +0.566270 q^{33} -9.60842 q^{35} -3.49175 q^{37} +3.83214 q^{39} +5.09084 q^{41} -9.48647 q^{43} +10.5273 q^{45} +9.68218 q^{47} -1.01972 q^{49} -0.0996567 q^{51} -4.72563 q^{53} -3.92909 q^{55} +0.566270 q^{57} +7.88318 q^{59} +14.9907 q^{61} -6.55221 q^{63} -26.5894 q^{65} -2.09858 q^{67} -4.70049 q^{69} +7.53352 q^{71} -1.36696 q^{73} +5.91057 q^{75} +2.44546 q^{77} +1.04345 q^{79} +6.21686 q^{81} -2.37170 q^{83} +0.691471 q^{85} -0.477703 q^{87} +3.38140 q^{89} +16.5492 q^{91} +0.127369 q^{93} -3.92909 q^{95} +19.0920 q^{97} -2.67934 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - q^{3} + 5 q^{5} + 2 q^{7} + 17 q^{9}+O(q^{10}) $ 6 * q - q^3 + 5 * q^5 + 2 * q^7 + 17 * q^9	$ 6 q - q^{3} + 5 q^{5} + 2 q^{7} + 17 q^{9} + 6 q^{11} + 8 q^{13} - 9 q^{15} - 2 q^{17} + 6 q^{19} + 18 q^{21} - 5 q^{23} + 13 q^{25} - 7 q^{27} + 10 q^{29} - 3 q^{31} - q^{33} + 4 q^{35} + 7 q^{37} + 10 q^{39} + 6 q^{41} - 16 q^{43} + 42 q^{45} + 12 q^{49} - 8 q^{51} + 20 q^{53} + 5 q^{55} - q^{57} - 15 q^{59} + 24 q^{61} + 20 q^{63} - 28 q^{65} - 25 q^{67} - 33 q^{69} + 9 q^{71} - 26 q^{73} - 28 q^{75} + 2 q^{77} + 16 q^{79} + 58 q^{81} + 2 q^{83} + 12 q^{85} + 36 q^{87} + 7 q^{89} - 8 q^{91} - 55 q^{93} + 5 q^{95} + 37 q^{97} + 17 q^{99}+O(q^{100}) $ 6 * q - q^3 + 5 * q^5 + 2 * q^7 + 17 * q^9 + 6 * q^11 + 8 * q^13 - 9 * q^15 - 2 * q^17 + 6 * q^19 + 18 * q^21 - 5 * q^23 + 13 * q^25 - 7 * q^27 + 10 * q^29 - 3 * q^31 - q^33 + 4 * q^35 + 7 * q^37 + 10 * q^39 + 6 * q^41 - 16 * q^43 + 42 * q^45 + 12 * q^49 - 8 * q^51 + 20 * q^53 + 5 * q^55 - q^57 - 15 * q^59 + 24 * q^61 + 20 * q^63 - 28 * q^65 - 25 * q^67 - 33 * q^69 + 9 * q^71 - 26 * q^73 - 28 * q^75 + 2 * q^77 + 16 * q^79 + 58 * q^81 + 2 * q^83 + 12 * q^85 + 36 * q^87 + 7 * q^89 - 8 * q^91 - 55 * q^93 + 5 * q^95 + 37 * q^97 + 17 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	0.566270	0.326936	0.163468	−	0.986549i	$-0.447732\pi$
$3$	0.566270	0.326936	0.163468	+	0.986549i	$0.447732\pi$
$4$	0	0
$5$	−3.92909	−1.75714	−0.878570	−	0.477613i	$-0.841502\pi$
$5$	−3.92909	−1.75714	−0.878570	+	0.477613i	$0.841502\pi$
$6$	0	0
$7$	2.44546	0.924297	0.462149	−	0.886803i	$-0.347079\pi$
$7$	2.44546	0.924297	0.462149	+	0.886803i	$0.347079\pi$
$8$	0	0
$9$	−2.67934	−0.893113
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	6.76733	1.87692	0.938460	−	0.345388i	$-0.112253\pi$
$13$	6.76733	1.87692	0.938460	+	0.345388i	$0.112253\pi$
$14$	0	0
$15$	−2.22493	−0.574473
$16$	0	0
$17$	−0.175988	−0.0426833	−0.0213417	−	0.999772i	$-0.506794\pi$
$17$	−0.175988	−0.0426833	−0.0213417	+	0.999772i	$0.506794\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	1.38479	0.302186
$22$	0	0
$23$	−8.30079	−1.73083	−0.865417	−	0.501053i	$-0.832946\pi$
$23$	−8.30079	−1.73083	−0.865417	+	0.501053i	$0.832946\pi$
$24$	0	0
$25$	10.4377	2.08754
$26$	0	0
$27$	−3.21604	−0.618927
$28$	0	0
$29$	−0.843595	−0.156652	−0.0783258	−	0.996928i	$-0.524957\pi$
$29$	−0.843595	−0.156652	−0.0783258	+	0.996928i	$0.524957\pi$
$30$	0	0
$31$	0.224925	0.0403978	0.0201989	−	0.999796i	$-0.493570\pi$
$31$	0.224925	0.0403978	0.0201989	+	0.999796i	$0.493570\pi$
$32$	0	0
$33$	0.566270	0.0985750
$34$	0	0
$35$	−9.60842	−1.62412
$36$	0	0
$37$	−3.49175	−0.574041	−0.287020	−	0.957924i	$-0.592665\pi$
$37$	−3.49175	−0.574041	−0.287020	+	0.957924i	$0.592665\pi$
$38$	0	0
$39$	3.83214	0.613634
$40$	0	0
$41$	5.09084	0.795056	0.397528	−	0.917590i	$-0.369868\pi$
$41$	5.09084	0.795056	0.397528	+	0.917590i	$0.369868\pi$
$42$	0	0
$43$	−9.48647	−1.44667	−0.723337	−	0.690495i	$-0.757393\pi$
$43$	−9.48647	−1.44667	−0.723337	+	0.690495i	$0.757393\pi$
$44$	0	0
$45$	10.5273	1.56932
$46$	0	0
$47$	9.68218	1.41229	0.706146	−	0.708066i	$-0.250432\pi$
$47$	9.68218	1.41229	0.706146	+	0.708066i	$0.250432\pi$
$48$	0	0
$49$	−1.01972	−0.145675
$50$	0	0
$51$	−0.0996567	−0.0139547
$52$	0	0
$53$	−4.72563	−0.649115	−0.324558	−	0.945866i	$-0.605215\pi$
$53$	−4.72563	−0.649115	−0.324558	+	0.945866i	$0.605215\pi$
$54$	0	0
$55$	−3.92909	−0.529798
$56$	0	0
$57$	0.566270	0.0750044
$58$	0	0
$59$	7.88318	1.02630	0.513151	−	0.858298i	$-0.328478\pi$
$59$	7.88318	1.02630	0.513151	+	0.858298i	$0.328478\pi$
$60$	0	0
$61$	14.9907	1.91936	0.959682	−	0.281088i	$-0.0906952\pi$
$61$	14.9907	1.91936	0.959682	+	0.281088i	$0.0906952\pi$
$62$	0	0
$63$	−6.55221	−0.825501
$64$	0	0
$65$	−26.5894	−3.29801
$66$	0	0
$67$	−2.09858	−0.256383	−0.128191	−	0.991749i	$-0.540917\pi$
$67$	−2.09858	−0.256383	−0.128191	+	0.991749i	$0.540917\pi$
$68$	0	0
$69$	−4.70049	−0.565872
$70$	0	0
$71$	7.53352	0.894065	0.447032	−	0.894518i	$-0.352481\pi$
$71$	7.53352	0.894065	0.447032	+	0.894518i	$0.352481\pi$
$72$	0	0
$73$	−1.36696	−0.159990	−0.0799950	−	0.996795i	$-0.525490\pi$
$73$	−1.36696	−0.159990	−0.0799950	+	0.996795i	$0.525490\pi$
$74$	0	0
$75$	5.91057	0.682494
$76$	0	0
$77$	2.44546	0.278686
$78$	0	0
$79$	1.04345	0.117397	0.0586985	−	0.998276i	$-0.481305\pi$
$79$	1.04345	0.117397	0.0586985	+	0.998276i	$0.481305\pi$
$80$	0	0
$81$	6.21686	0.690763
$82$	0	0
$83$	−2.37170	−0.260328	−0.130164	−	0.991492i	$-0.541550\pi$
$83$	−2.37170	−0.260328	−0.130164	+	0.991492i	$0.541550\pi$
$84$	0	0
$85$	0.691471	0.0750006
$86$	0	0
$87$	−0.477703	−0.0512151
$88$	0	0
$89$	3.38140	0.358428	0.179214	−	0.983810i	$-0.442645\pi$
$89$	3.38140	0.358428	0.179214	+	0.983810i	$0.442645\pi$
$90$	0	0
$91$	16.5492	1.73483
$92$	0	0
$93$	0.127369	0.0132075
$94$	0	0
$95$	−3.92909	−0.403116
$96$	0	0
$97$	19.0920	1.93850	0.969252	−	0.246070i	$-0.0791394\pi$
$97$	19.0920	1.93850	0.969252	+	0.246070i	$0.0791394\pi$
$98$	0	0
$99$	−2.67934	−0.269284
$100$	0	0
$101$	10.5496	1.04973	0.524864	−	0.851186i	$-0.324116\pi$
$101$	10.5496	1.04973	0.524864	+	0.851186i	$0.324116\pi$
$102$	0	0
$103$	10.9384	1.07779	0.538895	−	0.842373i	$-0.318842\pi$
$103$	10.9384	1.07779	0.538895	+	0.842373i	$0.318842\pi$
$104$	0	0
$105$	−5.44097	−0.530984
$106$	0	0
$107$	−14.6835	−1.41950	−0.709752	−	0.704452i	$-0.751193\pi$
$107$	−14.6835	−1.41950	−0.709752	+	0.704452i	$0.751193\pi$
$108$	0	0
$109$	17.9401	1.71835	0.859173	−	0.511685i	$-0.170979\pi$
$109$	17.9401	1.71835	0.859173	+	0.511685i	$0.170979\pi$
$110$	0	0
$111$	−1.97728	−0.187675
$112$	0	0
$113$	12.9266	1.21603	0.608017	−	0.793924i	$-0.291965\pi$
$113$	12.9266	1.21603	0.608017	+	0.793924i	$0.291965\pi$
$114$	0	0
$115$	32.6145	3.04132
$116$	0	0
$117$	−18.1320	−1.67630
$118$	0	0
$119$	−0.430371	−0.0394521
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	2.88279	0.259933
$124$	0	0
$125$	−21.3653	−1.91097
$126$	0	0
$127$	18.3365	1.62710	0.813550	−	0.581495i	$-0.197532\pi$
$127$	18.3365	1.62710	0.813550	+	0.581495i	$0.197532\pi$
$128$	0	0
$129$	−5.37191	−0.472970
$130$	0	0
$131$	2.97871	0.260251	0.130126	−	0.991498i	$-0.458462\pi$
$131$	2.97871	0.260251	0.130126	+	0.991498i	$0.458462\pi$
$132$	0	0
$133$	2.44546	0.212048
$134$	0	0
$135$	12.6361	1.08754
$136$	0	0
$137$	8.90447	0.760760	0.380380	−	0.924830i	$-0.375793\pi$
$137$	8.90447	0.760760	0.380380	+	0.924830i	$0.375793\pi$
$138$	0	0
$139$	−5.34451	−0.453315	−0.226658	−	0.973974i	$-0.572780\pi$
$139$	−5.34451	−0.453315	−0.226658	+	0.973974i	$0.572780\pi$
$140$	0	0
$141$	5.48274	0.461730
$142$	0	0
$143$	6.76733	0.565913
$144$	0	0
$145$	3.31456	0.275259
$146$	0	0
$147$	−0.577440	−0.0476264
$148$	0	0
$149$	−9.26958	−0.759394	−0.379697	−	0.925111i	$-0.623972\pi$
$149$	−9.26958	−0.759394	−0.379697	+	0.925111i	$0.623972\pi$
$150$	0	0
$151$	−2.51548	−0.204707	−0.102354	−	0.994748i	$-0.532637\pi$
$151$	−2.51548	−0.204707	−0.102354	+	0.994748i	$0.532637\pi$
$152$	0	0
$153$	0.471531	0.0381210
$154$	0	0
$155$	−0.883751	−0.0709846
$156$	0	0
$157$	−1.12005	−0.0893900	−0.0446950	−	0.999001i	$-0.514232\pi$
$157$	−1.12005	−0.0893900	−0.0446950	+	0.999001i	$0.514232\pi$
$158$	0	0
$159$	−2.67599	−0.212219
$160$	0	0
$161$	−20.2992	−1.59980
$162$	0	0
$163$	−16.7598	−1.31273	−0.656364	−	0.754444i	$-0.727906\pi$
$163$	−16.7598	−1.31273	−0.656364	+	0.754444i	$0.727906\pi$
$164$	0	0
$165$	−2.22493	−0.173210
$166$	0	0
$167$	16.0057	1.23856	0.619279	−	0.785171i	$-0.287425\pi$
$167$	16.0057	1.23856	0.619279	+	0.785171i	$0.287425\pi$
$168$	0	0
$169$	32.7968	2.52283
$170$	0	0
$171$	−2.67934	−0.204894
$172$	0	0
$173$	−2.12366	−0.161459	−0.0807293	−	0.996736i	$-0.525725\pi$
$173$	−2.12366	−0.161459	−0.0807293	+	0.996736i	$0.525725\pi$
$174$	0	0
$175$	25.5250	1.92951
$176$	0	0
$177$	4.46401	0.335536
$178$	0	0
$179$	−9.36816	−0.700210	−0.350105	−	0.936711i	$-0.613854\pi$
$179$	−9.36816	−0.700210	−0.350105	+	0.936711i	$0.613854\pi$
$180$	0	0
$181$	14.6585	1.08955	0.544777	−	0.838581i	$-0.316614\pi$
$181$	14.6585	1.08955	0.544777	+	0.838581i	$0.316614\pi$
$182$	0	0
$183$	8.48880	0.627510
$184$	0	0
$185$	13.7194	1.00867
$186$	0	0
$187$	−0.175988	−0.0128695
$188$	0	0
$189$	−7.86470	−0.572073
$190$	0	0
$191$	−2.41984	−0.175093	−0.0875467	−	0.996160i	$-0.527903\pi$
$191$	−2.41984	−0.175093	−0.0875467	+	0.996160i	$0.527903\pi$
$192$	0	0
$193$	16.4050	1.18086	0.590429	−	0.807090i	$-0.298959\pi$
$193$	16.4050	1.18086	0.590429	+	0.807090i	$0.298959\pi$
$194$	0	0
$195$	−15.0568	−1.07824
$196$	0	0
$197$	−7.67149	−0.546571	−0.273285	−	0.961933i	$-0.588110\pi$
$197$	−7.67149	−0.546571	−0.273285	+	0.961933i	$0.588110\pi$
$198$	0	0
$199$	6.97781	0.494644	0.247322	−	0.968933i	$-0.420449\pi$
$199$	6.97781	0.494644	0.247322	+	0.968933i	$0.420449\pi$
$200$	0	0
$201$	−1.18837	−0.0838209
$202$	0	0
$203$	−2.06298	−0.144793
$204$	0	0
$205$	−20.0024	−1.39702
$206$	0	0
$207$	22.2406	1.54583
$208$	0	0
$209$	1.00000	0.0691714
$210$	0	0
$211$	−11.7584	−0.809480	−0.404740	−	0.914432i	$-0.632638\pi$
$211$	−11.7584	−0.809480	−0.404740	+	0.914432i	$0.632638\pi$
$212$	0	0
$213$	4.26601	0.292302
$214$	0	0
$215$	37.2732	2.54201
$216$	0	0
$217$	0.550046	0.0373396
$218$	0	0
$219$	−0.774067	−0.0523066
$220$	0	0
$221$	−1.19097	−0.0801132
$222$	0	0
$223$	20.7798	1.39152	0.695759	−	0.718275i	$-0.255068\pi$
$223$	20.7798	1.39152	0.695759	+	0.718275i	$0.255068\pi$
$224$	0	0
$225$	−27.9662	−1.86441
$226$	0	0
$227$	18.2424	1.21079	0.605395	−	0.795925i	$-0.293015\pi$
$227$	18.2424	1.21079	0.605395	+	0.795925i	$0.293015\pi$
$228$	0	0
$229$	3.93873	0.260278	0.130139	−	0.991496i	$-0.458458\pi$
$229$	3.93873	0.260278	0.130139	+	0.991496i	$0.458458\pi$
$230$	0	0
$231$	1.38479	0.0911126
$232$	0	0
$233$	−9.86045	−0.645980	−0.322990	−	0.946402i	$-0.604688\pi$
$233$	−9.86045	−0.645980	−0.322990	+	0.946402i	$0.604688\pi$
$234$	0	0
$235$	−38.0421	−2.48160
$236$	0	0
$237$	0.590873	0.0383813
$238$	0	0
$239$	−3.81226	−0.246595	−0.123297	−	0.992370i	$-0.539347\pi$
$239$	−3.81226	−0.246595	−0.123297	+	0.992370i	$0.539347\pi$
$240$	0	0
$241$	−17.4880	−1.12650	−0.563251	−	0.826286i	$-0.690450\pi$
$241$	−17.4880	−1.12650	−0.563251	+	0.826286i	$0.690450\pi$
$242$	0	0
$243$	13.1686	0.844763
$244$	0	0
$245$	4.00658	0.255971
$246$	0	0
$247$	6.76733	0.430595
$248$	0	0
$249$	−1.34302	−0.0851107
$250$	0	0
$251$	−29.9887	−1.89287	−0.946434	−	0.322898i	$-0.895343\pi$
$251$	−29.9887	−1.89287	−0.946434	+	0.322898i	$0.895343\pi$
$252$	0	0
$253$	−8.30079	−0.521866
$254$	0	0
$255$	0.391560	0.0245204
$256$	0	0
$257$	−20.9251	−1.30527	−0.652635	−	0.757672i	$-0.726337\pi$
$257$	−20.9251	−1.30527	−0.652635	+	0.757672i	$0.726337\pi$
$258$	0	0
$259$	−8.53894	−0.530584
$260$	0	0
$261$	2.26028	0.139908
$262$	0	0
$263$	−5.29450	−0.326473	−0.163236	−	0.986587i	$-0.552193\pi$
$263$	−5.29450	−0.326473	−0.163236	+	0.986587i	$0.552193\pi$
$264$	0	0
$265$	18.5674	1.14059
$266$	0	0
$267$	1.91479	0.117183
$268$	0	0
$269$	17.9602	1.09506	0.547528	−	0.836787i	$-0.315569\pi$
$269$	17.9602	1.09506	0.547528	+	0.836787i	$0.315569\pi$
$270$	0	0
$271$	−0.957856	−0.0581856	−0.0290928	−	0.999577i	$-0.509262\pi$
$271$	−0.957856	−0.0581856	−0.0290928	+	0.999577i	$0.509262\pi$
$272$	0	0
$273$	9.37135	0.567180
$274$	0	0
$275$	10.4377	0.629418
$276$	0	0
$277$	−16.3422	−0.981907	−0.490953	−	0.871186i	$-0.663351\pi$
$277$	−16.3422	−0.981907	−0.490953	+	0.871186i	$0.663351\pi$
$278$	0	0
$279$	−0.602651	−0.0360798
$280$	0	0
$281$	8.36035	0.498737	0.249368	−	0.968409i	$-0.419777\pi$
$281$	8.36035	0.498737	0.249368	+	0.968409i	$0.419777\pi$
$282$	0	0
$283$	9.02926	0.536734	0.268367	−	0.963317i	$-0.413516\pi$
$283$	9.02926	0.536734	0.268367	+	0.963317i	$0.413516\pi$
$284$	0	0
$285$	−2.22493	−0.131793
$286$	0	0
$287$	12.4494	0.734868
$288$	0	0
$289$	−16.9690	−0.998178
$290$	0	0
$291$	10.8113	0.633768
$292$	0	0
$293$	7.42733	0.433909	0.216955	−	0.976182i	$-0.430388\pi$
$293$	7.42733	0.433909	0.216955	+	0.976182i	$0.430388\pi$
$294$	0	0
$295$	−30.9737	−1.80336
$296$	0	0
$297$	−3.21604	−0.186614
$298$	0	0
$299$	−56.1742	−3.24864
$300$	0	0
$301$	−23.1988	−1.33716
$302$	0	0
$303$	5.97395	0.343195
$304$	0	0
$305$	−58.8998	−3.37259
$306$	0	0
$307$	−17.1405	−0.978261	−0.489130	−	0.872211i	$-0.662686\pi$
$307$	−17.1405	−0.978261	−0.489130	+	0.872211i	$0.662686\pi$
$308$	0	0
$309$	6.19408	0.352369
$310$	0	0
$311$	−6.52618	−0.370066	−0.185033	−	0.982732i	$-0.559239\pi$
$311$	−6.52618	−0.370066	−0.185033	+	0.982732i	$0.559239\pi$
$312$	0	0
$313$	−17.3089	−0.978359	−0.489180	−	0.872183i	$-0.662704\pi$
$313$	−17.3089	−0.978359	−0.489180	+	0.872183i	$0.662704\pi$
$314$	0	0
$315$	25.7442	1.45052
$316$	0	0
$317$	33.6814	1.89174	0.945868	−	0.324551i	$-0.105213\pi$
$317$	33.6814	1.89174	0.945868	+	0.324551i	$0.105213\pi$
$318$	0	0
$319$	−0.843595	−0.0472322
$320$	0	0
$321$	−8.31481	−0.464087
$322$	0	0
$323$	−0.175988	−0.00979222
$324$	0	0
$325$	70.6355	3.91815
$326$	0	0
$327$	10.1589	0.561790
$328$	0	0
$329$	23.6774	1.30538
$330$	0	0
$331$	6.51703	0.358208	0.179104	−	0.983830i	$-0.442680\pi$
$331$	6.51703	0.358208	0.179104	+	0.983830i	$0.442680\pi$
$332$	0	0
$333$	9.35559	0.512683
$334$	0	0
$335$	8.24551	0.450501
$336$	0	0
$337$	29.8951	1.62849	0.814245	−	0.580522i	$-0.197151\pi$
$337$	29.8951	1.62849	0.814245	+	0.580522i	$0.197151\pi$
$338$	0	0
$339$	7.31997	0.397566
$340$	0	0
$341$	0.224925	0.0121804
$342$	0	0
$343$	−19.6119	−1.05894
$344$	0	0
$345$	18.4686	0.994318
$346$	0	0
$347$	−8.24960	−0.442862	−0.221431	−	0.975176i	$-0.571073\pi$
$347$	−8.24960	−0.442862	−0.221431	+	0.975176i	$0.571073\pi$
$348$	0	0
$349$	−8.18396	−0.438078	−0.219039	−	0.975716i	$-0.570292\pi$
$349$	−8.18396	−0.438078	−0.219039	+	0.975716i	$0.570292\pi$
$350$	0	0
$351$	−21.7640	−1.16168
$352$	0	0
$353$	−14.4897	−0.771207	−0.385604	−	0.922664i	$-0.626007\pi$
$353$	−14.4897	−0.771207	−0.385604	+	0.922664i	$0.626007\pi$
$354$	0	0
$355$	−29.5999	−1.57100
$356$	0	0
$357$	−0.243706	−0.0128983
$358$	0	0
$359$	15.7807	0.832872	0.416436	−	0.909165i	$-0.363279\pi$
$359$	15.7807	0.832872	0.416436	+	0.909165i	$0.363279\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	0.566270	0.0297215
$364$	0	0
$365$	5.37089	0.281125
$366$	0	0
$367$	−17.8387	−0.931174	−0.465587	−	0.885002i	$-0.654157\pi$
$367$	−17.8387	−0.931174	−0.465587	+	0.885002i	$0.654157\pi$
$368$	0	0
$369$	−13.6401	−0.710074
$370$	0	0
$371$	−11.5563	−0.599975
$372$	0	0
$373$	−30.0896	−1.55798	−0.778990	−	0.627036i	$-0.784268\pi$
$373$	−30.0896	−1.55798	−0.778990	+	0.627036i	$0.784268\pi$
$374$	0	0
$375$	−12.0985	−0.624765
$376$	0	0
$377$	−5.70889	−0.294023
$378$	0	0
$379$	−12.2583	−0.629667	−0.314834	−	0.949147i	$-0.601949\pi$
$379$	−12.2583	−0.629667	−0.314834	+	0.949147i	$0.601949\pi$
$380$	0	0
$381$	10.3834	0.531958
$382$	0	0
$383$	22.5978	1.15469	0.577346	−	0.816499i	$-0.304088\pi$
$383$	22.5978	1.15469	0.577346	+	0.816499i	$0.304088\pi$
$384$	0	0
$385$	−9.60842	−0.489691
$386$	0	0
$387$	25.4175	1.29204
$388$	0	0
$389$	29.7216	1.50695	0.753473	−	0.657479i	$-0.228377\pi$
$389$	29.7216	1.50695	0.753473	+	0.657479i	$0.228377\pi$
$390$	0	0
$391$	1.46084	0.0738777
$392$	0	0
$393$	1.68676	0.0850856
$394$	0	0
$395$	−4.09979	−0.206283
$396$	0	0
$397$	24.3750	1.22335	0.611674	−	0.791110i	$-0.290496\pi$
$397$	24.3750	1.22335	0.611674	+	0.791110i	$0.290496\pi$
$398$	0	0
$399$	1.38479	0.0693263
$400$	0	0
$401$	−28.9859	−1.44749	−0.723744	−	0.690069i	$-0.757580\pi$
$401$	−28.9859	−1.44749	−0.723744	+	0.690069i	$0.757580\pi$
$402$	0	0
$403$	1.52214	0.0758234
$404$	0	0
$405$	−24.4266	−1.21377
$406$	0	0
$407$	−3.49175	−0.173080
$408$	0	0
$409$	21.0718	1.04193	0.520967	−	0.853577i	$-0.325572\pi$
$409$	21.0718	1.04193	0.520967	+	0.853577i	$0.325572\pi$
$410$	0	0
$411$	5.04234	0.248720
$412$	0	0
$413$	19.2780	0.948608
$414$	0	0
$415$	9.31861	0.457433
$416$	0	0
$417$	−3.02644	−0.148205
$418$	0	0
$419$	−8.59929	−0.420103	−0.210051	−	0.977690i	$-0.567363\pi$
$419$	−8.59929	−0.420103	−0.210051	+	0.977690i	$0.567363\pi$
$420$	0	0
$421$	4.07441	0.198575	0.0992874	−	0.995059i	$-0.468344\pi$
$421$	4.07441	0.198575	0.0992874	+	0.995059i	$0.468344\pi$
$422$	0	0
$423$	−25.9418	−1.26134
$424$	0	0
$425$	−1.83691	−0.0891033
$426$	0	0
$427$	36.6592	1.77406
$428$	0	0
$429$	3.83214	0.185017
$430$	0	0
$431$	12.8572	0.619308	0.309654	−	0.950849i	$-0.399787\pi$
$431$	12.8572	0.619308	0.309654	+	0.950849i	$0.399787\pi$
$432$	0	0
$433$	5.97021	0.286910	0.143455	−	0.989657i	$-0.454179\pi$
$433$	5.97021	0.286910	0.143455	+	0.989657i	$0.454179\pi$
$434$	0	0
$435$	1.87694	0.0899922
$436$	0	0
$437$	−8.30079	−0.397080
$438$	0	0
$439$	−34.9965	−1.67029	−0.835146	−	0.550029i	$-0.814617\pi$
$439$	−34.9965	−1.67029	−0.835146	+	0.550029i	$0.814617\pi$
$440$	0	0
$441$	2.73219	0.130104
$442$	0	0
$443$	−31.3469	−1.48934	−0.744669	−	0.667434i	$-0.767393\pi$
$443$	−31.3469	−1.48934	−0.744669	+	0.667434i	$0.767393\pi$
$444$	0	0
$445$	−13.2858	−0.629808
$446$	0	0
$447$	−5.24909	−0.248273
$448$	0	0
$449$	29.7198	1.40256	0.701282	−	0.712884i	$-0.252611\pi$
$449$	29.7198	1.40256	0.701282	+	0.712884i	$0.252611\pi$
$450$	0	0
$451$	5.09084	0.239718
$452$	0	0
$453$	−1.42444	−0.0669262
$454$	0	0
$455$	−65.0234	−3.04834
$456$	0	0
$457$	24.4809	1.14517	0.572585	−	0.819846i	$-0.305941\pi$
$457$	24.4809	1.14517	0.572585	+	0.819846i	$0.305941\pi$
$458$	0	0
$459$	0.565984	0.0264179
$460$	0	0
$461$	−21.5138	−1.00200	−0.500999	−	0.865448i	$-0.667034\pi$
$461$	−21.5138	−1.00200	−0.500999	+	0.865448i	$0.667034\pi$
$462$	0	0
$463$	2.04695	0.0951300	0.0475650	−	0.998868i	$-0.484854\pi$
$463$	2.04695	0.0951300	0.0475650	+	0.998868i	$0.484854\pi$
$464$	0	0
$465$	−0.500442	−0.0232075
$466$	0	0
$467$	23.7411	1.09861	0.549303	−	0.835623i	$-0.314893\pi$
$467$	23.7411	1.09861	0.549303	+	0.835623i	$0.314893\pi$
$468$	0	0
$469$	−5.13200	−0.236974
$470$	0	0
$471$	−0.634253	−0.0292249
$472$	0	0
$473$	−9.48647	−0.436188
$474$	0	0
$475$	10.4377	0.478915
$476$	0	0
$477$	12.6616	0.579733
$478$	0	0
$479$	28.4414	1.29952	0.649759	−	0.760140i	$-0.274870\pi$
$479$	28.4414	1.29952	0.649759	+	0.760140i	$0.274870\pi$
$480$	0	0
$481$	−23.6299	−1.07743
$482$	0	0
$483$	−11.4949	−0.523034
$484$	0	0
$485$	−75.0143	−3.40622
$486$	0	0
$487$	1.40149	0.0635074	0.0317537	−	0.999496i	$-0.489891\pi$
$487$	1.40149	0.0635074	0.0317537	+	0.999496i	$0.489891\pi$
$488$	0	0
$489$	−9.49057	−0.429179
$490$	0	0
$491$	−30.2244	−1.36401	−0.682005	−	0.731348i	$-0.738892\pi$
$491$	−30.2244	−1.36401	−0.682005	+	0.731348i	$0.738892\pi$
$492$	0	0
$493$	0.148462	0.00668641
$494$	0	0
$495$	10.5273	0.473169
$496$	0	0
$497$	18.4229	0.826381
$498$	0	0
$499$	−24.5835	−1.10051	−0.550255	−	0.834997i	$-0.685470\pi$
$499$	−24.5835	−1.10051	−0.550255	+	0.834997i	$0.685470\pi$
$500$	0	0
$501$	9.06355	0.404930
$502$	0	0
$503$	−12.9000	−0.575184	−0.287592	−	0.957753i	$-0.592855\pi$
$503$	−12.9000	−0.575184	−0.287592	+	0.957753i	$0.592855\pi$
$504$	0	0
$505$	−41.4505	−1.84452
$506$	0	0
$507$	18.5718	0.824805
$508$	0	0
$509$	−15.2890	−0.677674	−0.338837	−	0.940845i	$-0.610034\pi$
$509$	−15.2890	−0.677674	−0.338837	+	0.940845i	$0.610034\pi$
$510$	0	0
$511$	−3.34284	−0.147878
$512$	0	0
$513$	−3.21604	−0.141992
$514$	0	0
$515$	−42.9778	−1.89383
$516$	0	0
$517$	9.68218	0.425822
$518$	0	0
$519$	−1.20256	−0.0527867
$520$	0	0
$521$	38.6252	1.69220	0.846100	−	0.533024i	$-0.178944\pi$
$521$	38.6252	1.69220	0.846100	+	0.533024i	$0.178944\pi$
$522$	0	0
$523$	1.26936	0.0555054	0.0277527	−	0.999615i	$-0.491165\pi$
$523$	1.26936	0.0555054	0.0277527	+	0.999615i	$0.491165\pi$
$524$	0	0
$525$	14.4541	0.630827
$526$	0	0
$527$	−0.0395841	−0.00172431
$528$	0	0
$529$	45.9030	1.99578
$530$	0	0
$531$	−21.1217	−0.916604
$532$	0	0
$533$	34.4514	1.49226
$534$	0	0
$535$	57.6926	2.49427
$536$	0	0
$537$	−5.30492	−0.228924
$538$	0	0
$539$	−1.01972	−0.0439226
$540$	0	0
$541$	24.5110	1.05381	0.526904	−	0.849925i	$-0.323353\pi$
$541$	24.5110	1.05381	0.526904	+	0.849925i	$0.323353\pi$
$542$	0	0
$543$	8.30065	0.356215
$544$	0	0
$545$	−70.4880	−3.01938
$546$	0	0
$547$	27.2833	1.16655	0.583274	−	0.812276i	$-0.301771\pi$
$547$	27.2833	1.16655	0.583274	+	0.812276i	$0.301771\pi$
$548$	0	0
$549$	−40.1652	−1.71421
$550$	0	0
$551$	−0.843595	−0.0359383
$552$	0	0
$553$	2.55171	0.108510
$554$	0	0
$555$	7.76889	0.329771
$556$	0	0
$557$	−10.7434	−0.455212	−0.227606	−	0.973753i	$-0.573090\pi$
$557$	−10.7434	−0.455212	−0.227606	+	0.973753i	$0.573090\pi$
$558$	0	0
$559$	−64.1981	−2.71529
$560$	0	0
$561$	−0.0996567	−0.00420751
$562$	0	0
$563$	30.0868	1.26801	0.634003	−	0.773330i	$-0.281411\pi$
$563$	30.0868	1.26801	0.634003	+	0.773330i	$0.281411\pi$
$564$	0	0
$565$	−50.7898	−2.13674
$566$	0	0
$567$	15.2031	0.638470
$568$	0	0
$569$	22.0838	0.925803	0.462901	−	0.886410i	$-0.346808\pi$
$569$	22.0838	0.925803	0.462901	+	0.886410i	$0.346808\pi$
$570$	0	0
$571$	36.7811	1.53924	0.769621	−	0.638500i	$-0.220445\pi$
$571$	36.7811	1.53924	0.769621	+	0.638500i	$0.220445\pi$
$572$	0	0
$573$	−1.37028	−0.0572444
$574$	0	0
$575$	−86.6413	−3.61319
$576$	0	0
$577$	−32.2242	−1.34151	−0.670755	−	0.741679i	$-0.734030\pi$
$577$	−32.2242	−1.34151	−0.670755	+	0.741679i	$0.734030\pi$
$578$	0	0
$579$	9.28966	0.386065
$580$	0	0
$581$	−5.79990	−0.240620
$582$	0	0
$583$	−4.72563	−0.195716
$584$	0	0
$585$	71.2421	2.94550
$586$	0	0
$587$	16.3940	0.676654	0.338327	−	0.941029i	$-0.390139\pi$
$587$	16.3940	0.676654	0.338327	+	0.941029i	$0.390139\pi$
$588$	0	0
$589$	0.224925	0.00926789
$590$	0	0
$591$	−4.34414	−0.178694
$592$	0	0
$593$	−7.33700	−0.301294	−0.150647	−	0.988588i	$-0.548136\pi$
$593$	−7.33700	−0.301294	−0.150647	+	0.988588i	$0.548136\pi$
$594$	0	0
$595$	1.69097	0.0693228
$596$	0	0
$597$	3.95133	0.161717
$598$	0	0
$599$	38.0717	1.55557	0.777784	−	0.628532i	$-0.216344\pi$
$599$	38.0717	1.55557	0.777784	+	0.628532i	$0.216344\pi$
$600$	0	0
$601$	−13.8342	−0.564310	−0.282155	−	0.959369i	$-0.591049\pi$
$601$	−13.8342	−0.564310	−0.282155	+	0.959369i	$0.591049\pi$
$602$	0	0
$603$	5.62281	0.228979
$604$	0	0
$605$	−3.92909	−0.159740
$606$	0	0
$607$	−20.2603	−0.822340	−0.411170	−	0.911559i	$-0.634880\pi$
$607$	−20.2603	−0.822340	−0.411170	+	0.911559i	$0.634880\pi$
$608$	0	0
$609$	−1.16820	−0.0473380
$610$	0	0
$611$	65.5226	2.65076
$612$	0	0
$613$	34.9954	1.41345	0.706726	−	0.707488i	$-0.250171\pi$
$613$	34.9954	1.41345	0.706726	+	0.707488i	$0.250171\pi$
$614$	0	0
$615$	−11.3267	−0.456738
$616$	0	0
$617$	3.87381	0.155954	0.0779769	−	0.996955i	$-0.475154\pi$
$617$	3.87381	0.155954	0.0779769	+	0.996955i	$0.475154\pi$
$618$	0	0
$619$	5.80250	0.233222	0.116611	−	0.993178i	$-0.462797\pi$
$619$	5.80250	0.233222	0.116611	+	0.993178i	$0.462797\pi$
$620$	0	0
$621$	26.6957	1.07126
$622$	0	0
$623$	8.26908	0.331294
$624$	0	0
$625$	31.7574	1.27029
$626$	0	0
$627$	0.566270	0.0226147
$628$	0	0
$629$	0.614506	0.0245020
$630$	0	0
$631$	−18.5664	−0.739115	−0.369558	−	0.929208i	$-0.620491\pi$
$631$	−18.5664	−0.739115	−0.369558	+	0.929208i	$0.620491\pi$
$632$	0	0
$633$	−6.65842	−0.264649
$634$	0	0
$635$	−72.0456	−2.85904
$636$	0	0
$637$	−6.90081	−0.273420
$638$	0	0
$639$	−20.1849	−0.798500
$640$	0	0
$641$	8.91056	0.351946	0.175973	−	0.984395i	$-0.443693\pi$
$641$	8.91056	0.351946	0.175973	+	0.984395i	$0.443693\pi$
$642$	0	0
$643$	36.9688	1.45791	0.728953	−	0.684564i	$-0.240007\pi$
$643$	36.9688	1.45791	0.728953	+	0.684564i	$0.240007\pi$
$644$	0	0
$645$	21.1067	0.831075
$646$	0	0
$647$	−21.3503	−0.839367	−0.419683	−	0.907671i	$-0.637859\pi$
$647$	−21.3503	−0.839367	−0.419683	+	0.907671i	$0.637859\pi$
$648$	0	0
$649$	7.88318	0.309442
$650$	0	0
$651$	0.311475	0.0122077
$652$	0	0
$653$	−9.01791	−0.352898	−0.176449	−	0.984310i	$-0.556461\pi$
$653$	−9.01791	−0.352898	−0.176449	+	0.984310i	$0.556461\pi$
$654$	0	0
$655$	−11.7036	−0.457298
$656$	0	0
$657$	3.66254	0.142889
$658$	0	0
$659$	−0.402928	−0.0156958	−0.00784792	−	0.999969i	$-0.502498\pi$
$659$	−0.402928	−0.0156958	−0.00784792	+	0.999969i	$0.502498\pi$
$660$	0	0
$661$	−0.400038	−0.0155597	−0.00777983	−	0.999970i	$-0.502476\pi$
$661$	−0.400038	−0.0155597	−0.00777983	+	0.999970i	$0.502476\pi$
$662$	0	0
$663$	−0.674410	−0.0261919
$664$	0	0
$665$	−9.60842	−0.372599
$666$	0	0
$667$	7.00250	0.271138
$668$	0	0
$669$	11.7670	0.454938
$670$	0	0
$671$	14.9907	0.578710
$672$	0	0
$673$	−42.9949	−1.65733	−0.828666	−	0.559743i	$-0.810900\pi$
$673$	−42.9949	−1.65733	−0.828666	+	0.559743i	$0.810900\pi$
$674$	0	0
$675$	−33.5681	−1.29204
$676$	0	0
$677$	12.7272	0.489148	0.244574	−	0.969631i	$-0.421352\pi$
$677$	12.7272	0.489148	0.244574	+	0.969631i	$0.421352\pi$
$678$	0	0
$679$	46.6888	1.79175
$680$	0	0
$681$	10.3301	0.395851
$682$	0	0
$683$	−47.4876	−1.81706	−0.908532	−	0.417816i	$-0.862796\pi$
$683$	−47.4876	−1.81706	−0.908532	+	0.417816i	$0.862796\pi$
$684$	0	0
$685$	−34.9864	−1.33676
$686$	0	0
$687$	2.23038	0.0850945
$688$	0	0
$689$	−31.9799	−1.21834
$690$	0	0
$691$	15.9991	0.608636	0.304318	−	0.952570i	$-0.401571\pi$
$691$	15.9991	0.608636	0.304318	+	0.952570i	$0.401571\pi$
$692$	0	0
$693$	−6.55221	−0.248898
$694$	0	0
$695$	20.9990	0.796539
$696$	0	0
$697$	−0.895926	−0.0339356
$698$	0	0
$699$	−5.58368	−0.211194
$700$	0	0
$701$	−2.12285	−0.0801789	−0.0400895	−	0.999196i	$-0.512764\pi$
$701$	−2.12285	−0.0801789	−0.0400895	+	0.999196i	$0.512764\pi$
$702$	0	0
$703$	−3.49175	−0.131694
$704$	0	0
$705$	−21.5421	−0.811324
$706$	0	0
$707$	25.7987	0.970261
$708$	0	0
$709$	25.4377	0.955334	0.477667	−	0.878541i	$-0.341483\pi$
$709$	25.4377	0.955334	0.477667	+	0.878541i	$0.341483\pi$
$710$	0	0
$711$	−2.79575	−0.104849
$712$	0	0
$713$	−1.86706	−0.0699219
$714$	0	0
$715$	−26.5894	−0.994388
$716$	0	0
$717$	−2.15877	−0.0806208
$718$	0	0
$719$	−4.59583	−0.171395	−0.0856977	−	0.996321i	$-0.527312\pi$
$719$	−4.59583	−0.171395	−0.0856977	+	0.996321i	$0.527312\pi$
$720$	0	0
$721$	26.7494	0.996199
$722$	0	0
$723$	−9.90295	−0.368295
$724$	0	0
$725$	−8.80520	−0.327017
$726$	0	0
$727$	−11.8175	−0.438288	−0.219144	−	0.975692i	$-0.570327\pi$
$727$	−11.8175	−0.438288	−0.219144	+	0.975692i	$0.570327\pi$
$728$	0	0
$729$	−11.1936	−0.414579
$730$	0	0
$731$	1.66950	0.0617488
$732$	0	0
$733$	6.16798	0.227819	0.113910	−	0.993491i	$-0.463663\pi$
$733$	6.16798	0.227819	0.113910	+	0.993491i	$0.463663\pi$
$734$	0	0
$735$	2.26881	0.0836863
$736$	0	0
$737$	−2.09858	−0.0773023
$738$	0	0
$739$	1.40065	0.0515239	0.0257619	−	0.999668i	$-0.491799\pi$
$739$	1.40065	0.0515239	0.0257619	+	0.999668i	$0.491799\pi$
$740$	0	0
$741$	3.83214	0.140777
$742$	0	0
$743$	−2.10767	−0.0773228	−0.0386614	−	0.999252i	$-0.512309\pi$
$743$	−2.10767	−0.0773228	−0.0386614	+	0.999252i	$0.512309\pi$
$744$	0	0
$745$	36.4210	1.33436
$746$	0	0
$747$	6.35458	0.232502
$748$	0	0
$749$	−35.9078	−1.31204
$750$	0	0
$751$	48.6988	1.77704	0.888522	−	0.458833i	$-0.151732\pi$
$751$	48.6988	1.77704	0.888522	+	0.458833i	$0.151732\pi$
$752$	0	0
$753$	−16.9817	−0.618847
$754$	0	0
$755$	9.88355	0.359699
$756$	0	0
$757$	−12.4787	−0.453545	−0.226773	−	0.973948i	$-0.572817\pi$
$757$	−12.4787	−0.453545	−0.226773	+	0.973948i	$0.572817\pi$
$758$	0	0
$759$	−4.70049	−0.170617
$760$	0	0
$761$	−40.0185	−1.45067	−0.725334	−	0.688397i	$-0.758315\pi$
$761$	−40.0185	−1.45067	−0.725334	+	0.688397i	$0.758315\pi$
$762$	0	0
$763$	43.8717	1.58826
$764$	0	0
$765$	−1.85269	−0.0669840
$766$	0	0
$767$	53.3481	1.92629
$768$	0	0
$769$	−17.9730	−0.648125	−0.324062	−	0.946036i	$-0.605049\pi$
$769$	−17.9730	−0.648125	−0.324062	+	0.946036i	$0.605049\pi$
$770$	0	0
$771$	−11.8493	−0.426740
$772$	0	0
$773$	13.8605	0.498526	0.249263	−	0.968436i	$-0.419812\pi$
$773$	13.8605	0.498526	0.249263	+	0.968436i	$0.419812\pi$
$774$	0	0
$775$	2.34771	0.0843322
$776$	0	0
$777$	−4.83535	−0.173467
$778$	0	0
$779$	5.09084	0.182398
$780$	0	0
$781$	7.53352	0.269571
$782$	0	0
$783$	2.71304	0.0969560
$784$	0	0
$785$	4.40079	0.157071
$786$	0	0
$787$	36.9065	1.31557	0.657787	−	0.753204i	$-0.271493\pi$
$787$	36.9065	1.31557	0.657787	+	0.753204i	$0.271493\pi$
$788$	0	0
$789$	−2.99812	−0.106736
$790$	0	0
$791$	31.6115	1.12398
$792$	0	0
$793$	101.447	3.60249
$794$	0	0
$795$	10.5142	0.372899
$796$	0	0
$797$	−31.6732	−1.12192	−0.560961	−	0.827843i	$-0.689568\pi$
$797$	−31.6732	−1.12192	−0.560961	+	0.827843i	$0.689568\pi$
$798$	0	0
$799$	−1.70395	−0.0602813
$800$	0	0
$801$	−9.05991	−0.320116
$802$	0	0
$803$	−1.36696	−0.0482388
$804$	0	0
$805$	79.7575	2.81108
$806$	0	0
$807$	10.1704	0.358014
$808$	0	0
$809$	−55.5256	−1.95218	−0.976089	−	0.217372i	$-0.930251\pi$
$809$	−55.5256	−1.95218	−0.976089	+	0.217372i	$0.930251\pi$
$810$	0	0
$811$	−30.7281	−1.07901	−0.539505	−	0.841983i	$-0.681389\pi$
$811$	−30.7281	−1.07901	−0.539505	+	0.841983i	$0.681389\pi$
$812$	0	0
$813$	−0.542405	−0.0190230
$814$	0	0
$815$	65.8507	2.30665
$816$	0	0
$817$	−9.48647	−0.331890
$818$	0	0
$819$	−44.3410	−1.54940
$820$	0	0
$821$	26.6887	0.931444	0.465722	−	0.884931i	$-0.345795\pi$
$821$	26.6887	0.931444	0.465722	+	0.884931i	$0.345795\pi$
$822$	0	0
$823$	37.6630	1.31285	0.656424	−	0.754392i	$-0.272068\pi$
$823$	37.6630	1.31285	0.656424	+	0.754392i	$0.272068\pi$
$824$	0	0
$825$	5.91057	0.205780
$826$	0	0
$827$	50.4754	1.75520	0.877601	−	0.479391i	$-0.159143\pi$
$827$	50.4754	1.75520	0.877601	+	0.479391i	$0.159143\pi$
$828$	0	0
$829$	−49.8921	−1.73282	−0.866411	−	0.499331i	$-0.833579\pi$
$829$	−49.8921	−1.73282	−0.866411	+	0.499331i	$0.833579\pi$
$830$	0	0
$831$	−9.25410	−0.321021
$832$	0	0
$833$	0.179459	0.00621789
$834$	0	0
$835$	−62.8877	−2.17632
$836$	0	0
$837$	−0.723369	−0.0250033
$838$	0	0
$839$	36.7818	1.26985	0.634924	−	0.772574i	$-0.281031\pi$
$839$	36.7818	1.26985	0.634924	+	0.772574i	$0.281031\pi$
$840$	0	0
$841$	−28.2883	−0.975460
$842$	0	0
$843$	4.73422	0.163055
$844$	0	0
$845$	−128.861	−4.43297
$846$	0	0
$847$	2.44546	0.0840270
$848$	0	0
$849$	5.11301	0.175478
$850$	0	0
$851$	28.9843	0.993569
$852$	0	0
$853$	−27.1023	−0.927964	−0.463982	−	0.885845i	$-0.653580\pi$
$853$	−27.1023	−0.927964	−0.463982	+	0.885845i	$0.653580\pi$
$854$	0	0
$855$	10.5273	0.360028
$856$	0	0
$857$	−9.71381	−0.331817	−0.165909	−	0.986141i	$-0.553056\pi$
$857$	−9.71381	−0.331817	−0.165909	+	0.986141i	$0.553056\pi$
$858$	0	0
$859$	3.38681	0.115556	0.0577782	−	0.998329i	$-0.481598\pi$
$859$	3.38681	0.115556	0.0577782	+	0.998329i	$0.481598\pi$
$860$	0	0
$861$	7.04975	0.240255
$862$	0	0
$863$	−44.8663	−1.52727	−0.763634	−	0.645650i	$-0.776587\pi$
$863$	−44.8663	−1.52727	−0.763634	+	0.645650i	$0.776587\pi$
$864$	0	0
$865$	8.34403	0.283706
$866$	0	0
$867$	−9.60906	−0.326341
$868$	0	0
$869$	1.04345	0.0353965
$870$	0	0
$871$	−14.2018	−0.481210
$872$	0	0
$873$	−51.1540	−1.73130
$874$	0	0
$875$	−52.2479	−1.76630
$876$	0	0
$877$	5.09806	0.172149	0.0860746	−	0.996289i	$-0.472568\pi$
$877$	5.09806	0.172149	0.0860746	+	0.996289i	$0.472568\pi$
$878$	0	0
$879$	4.20588	0.141861
$880$	0	0
$881$	35.7531	1.20455	0.602276	−	0.798288i	$-0.294261\pi$
$881$	35.7531	1.20455	0.602276	+	0.798288i	$0.294261\pi$
$882$	0	0
$883$	−36.1681	−1.21715	−0.608577	−	0.793495i	$-0.708259\pi$
$883$	−36.1681	−1.21715	−0.608577	+	0.793495i	$0.708259\pi$
$884$	0	0
$885$	−17.5395	−0.589583
$886$	0	0
$887$	8.68127	0.291489	0.145744	−	0.989322i	$-0.453442\pi$
$887$	8.68127	0.291489	0.145744	+	0.989322i	$0.453442\pi$
$888$	0	0
$889$	44.8412	1.50392
$890$	0	0
$891$	6.21686	0.208273
$892$	0	0
$893$	9.68218	0.324002
$894$	0	0
$895$	36.8083	1.23037
$896$	0	0
$897$	−31.8098	−1.06210
$898$	0	0
$899$	−0.189746	−0.00632838
$900$	0	0
$901$	0.831653	0.0277064
$902$	0	0
$903$	−13.1368	−0.437165
$904$	0	0
$905$	−57.5943	−1.91450
$906$	0	0
$907$	−23.1887	−0.769970	−0.384985	−	0.922923i	$-0.625793\pi$
$907$	−23.1887	−0.769970	−0.384985	+	0.922923i	$0.625793\pi$
$908$	0	0
$909$	−28.2661	−0.937526
$910$	0	0
$911$	−12.1119	−0.401284	−0.200642	−	0.979665i	$-0.564303\pi$
$911$	−12.1119	−0.401284	−0.200642	+	0.979665i	$0.564303\pi$
$912$	0	0
$913$	−2.37170	−0.0784918
$914$	0	0
$915$	−33.3532	−1.10262
$916$	0	0
$917$	7.28432	0.240549
$918$	0	0
$919$	−41.2716	−1.36142	−0.680711	−	0.732552i	$-0.738329\pi$
$919$	−41.2716	−1.36142	−0.680711	+	0.732552i	$0.738329\pi$
$920$	0	0
$921$	−9.70617	−0.319829
$922$	0	0
$923$	50.9818	1.67809
$924$	0	0
$925$	−36.4459	−1.19834
$926$	0	0
$927$	−29.3076	−0.962589
$928$	0	0
$929$	33.3106	1.09288	0.546442	−	0.837497i	$-0.315982\pi$
$929$	33.3106	1.09288	0.546442	+	0.837497i	$0.315982\pi$
$930$	0	0
$931$	−1.01972	−0.0334201
$932$	0	0
$933$	−3.69558	−0.120988
$934$	0	0
$935$	0.691471	0.0226135
$936$	0	0
$937$	−16.4558	−0.537586	−0.268793	−	0.963198i	$-0.586625\pi$
$937$	−16.4558	−0.537586	−0.268793	+	0.963198i	$0.586625\pi$
$938$	0	0
$939$	−9.80154	−0.319861
$940$	0	0
$941$	36.1773	1.17934	0.589672	−	0.807642i	$-0.299257\pi$
$941$	36.1773	1.17934	0.589672	+	0.807642i	$0.299257\pi$
$942$	0	0
$943$	−42.2580	−1.37611
$944$	0	0
$945$	30.9011	1.00521
$946$	0	0
$947$	−53.5210	−1.73920	−0.869600	−	0.493758i	$-0.835623\pi$
$947$	−53.5210	−1.73920	−0.869600	+	0.493758i	$0.835623\pi$
$948$	0	0
$949$	−9.25064	−0.300289
$950$	0	0
$951$	19.0728	0.618478
$952$	0	0
$953$	3.55665	0.115211	0.0576056	−	0.998339i	$-0.481653\pi$
$953$	3.55665	0.115211	0.0576056	+	0.998339i	$0.481653\pi$
$954$	0	0
$955$	9.50776	0.307664
$956$	0	0
$957$	−0.477703	−0.0154419
$958$	0	0
$959$	21.7755	0.703168
$960$	0	0
$961$	−30.9494	−0.998368
$962$	0	0
$963$	39.3419	1.26778
$964$	0	0
$965$	−64.4566	−2.07493
$966$	0	0
$967$	−29.3275	−0.943108	−0.471554	−	0.881837i	$-0.656307\pi$
$967$	−29.3275	−0.943108	−0.471554	+	0.881837i	$0.656307\pi$
$968$	0	0
$969$	−0.0996567	−0.00320143
$970$	0	0
$971$	27.8992	0.895327	0.447663	−	0.894202i	$-0.352256\pi$
$971$	27.8992	0.895327	0.447663	+	0.894202i	$0.352256\pi$
$972$	0	0
$973$	−13.0698	−0.418998
$974$	0	0
$975$	39.9988	1.28099
$976$	0	0
$977$	38.7061	1.23832	0.619158	−	0.785266i	$-0.287474\pi$
$977$	38.7061	1.23832	0.619158	+	0.785266i	$0.287474\pi$
$978$	0	0
$979$	3.38140	0.108070
$980$	0	0
$981$	−48.0675	−1.53468
$982$	0	0
$983$	−54.6223	−1.74218	−0.871090	−	0.491124i	$-0.836586\pi$
$983$	−54.6223	−1.74218	−0.871090	+	0.491124i	$0.836586\pi$
$984$	0	0
$985$	30.1419	0.960402
$986$	0	0
$987$	13.4078	0.426775
$988$	0	0
$989$	78.7452	2.50395
$990$	0	0
$991$	−25.9988	−0.825879	−0.412939	−	0.910759i	$-0.635498\pi$
$991$	−25.9988	−0.825879	−0.412939	+	0.910759i	$0.635498\pi$
$992$	0	0
$993$	3.69040	0.117111
$994$	0	0
$995$	−27.4164	−0.869159
$996$	0	0
$997$	−28.6019	−0.905830	−0.452915	−	0.891554i	$-0.649616\pi$
$997$	−28.6019	−0.905830	−0.452915	+	0.891554i	$0.649616\pi$
$998$	0	0
$999$	11.2296	0.355290

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3344.2.a.w.1.4		6
4.3	odd	2		836.2.a.e.1.3	✓	6
12.11	even	2		7524.2.a.q.1.6		6
44.43	even	2		9196.2.a.n.1.3		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
836.2.a.e.1.3	✓	6	4.3	odd	2
3344.2.a.w.1.4		6	1.1	even	1	trivial
7524.2.a.q.1.6		6	12.11	even	2
9196.2.a.n.1.3		6	44.43	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:	\( N \)	\(=\)	\( 3344 = 2^{4} \cdot 11 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3344.a (trivial)

\(f(q)\)	\(=\)	\(q+0.566270 q^{3} -3.92909 q^{5} +2.44546 q^{7} -2.67934 q^{9} +O(q^{10})\)	\(q+0.566270 q^{3} -3.92909 q^{5} +2.44546 q^{7} -2.67934 q^{9} +1.00000 q^{11} +6.76733 q^{13} -2.22493 q^{15} -0.175988 q^{17} +1.00000 q^{19} +1.38479 q^{21} -8.30079 q^{23} +10.4377 q^{25} -3.21604 q^{27} -0.843595 q^{29} +0.224925 q^{31} +0.566270 q^{33} -9.60842 q^{35} -3.49175 q^{37} +3.83214 q^{39} +5.09084 q^{41} -9.48647 q^{43} +10.5273 q^{45} +9.68218 q^{47} -1.01972 q^{49} -0.0996567 q^{51} -4.72563 q^{53} -3.92909 q^{55} +0.566270 q^{57} +7.88318 q^{59} +14.9907 q^{61} -6.55221 q^{63} -26.5894 q^{65} -2.09858 q^{67} -4.70049 q^{69} +7.53352 q^{71} -1.36696 q^{73} +5.91057 q^{75} +2.44546 q^{77} +1.04345 q^{79} +6.21686 q^{81} -2.37170 q^{83} +0.691471 q^{85} -0.477703 q^{87} +3.38140 q^{89} +16.5492 q^{91} +0.127369 q^{93} -3.92909 q^{95} +19.0920 q^{97} -2.67934 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - q^{3} + 5 q^{5} + 2 q^{7} + 17 q^{9}+O(q^{10}) \) 6 * q - q^3 + 5 * q^5 + 2 * q^7 + 17 * q^9	\( 6 q - q^{3} + 5 q^{5} + 2 q^{7} + 17 q^{9} + 6 q^{11} + 8 q^{13} - 9 q^{15} - 2 q^{17} + 6 q^{19} + 18 q^{21} - 5 q^{23} + 13 q^{25} - 7 q^{27} + 10 q^{29} - 3 q^{31} - q^{33} + 4 q^{35} + 7 q^{37} + 10 q^{39} + 6 q^{41} - 16 q^{43} + 42 q^{45} + 12 q^{49} - 8 q^{51} + 20 q^{53} + 5 q^{55} - q^{57} - 15 q^{59} + 24 q^{61} + 20 q^{63} - 28 q^{65} - 25 q^{67} - 33 q^{69} + 9 q^{71} - 26 q^{73} - 28 q^{75} + 2 q^{77} + 16 q^{79} + 58 q^{81} + 2 q^{83} + 12 q^{85} + 36 q^{87} + 7 q^{89} - 8 q^{91} - 55 q^{93} + 5 q^{95} + 37 q^{97} + 17 q^{99}+O(q^{100}) \) 6 * q - q^3 + 5 * q^5 + 2 * q^7 + 17 * q^9 + 6 * q^11 + 8 * q^13 - 9 * q^15 - 2 * q^17 + 6 * q^19 + 18 * q^21 - 5 * q^23 + 13 * q^25 - 7 * q^27 + 10 * q^29 - 3 * q^31 - q^33 + 4 * q^35 + 7 * q^37 + 10 * q^39 + 6 * q^41 - 16 * q^43 + 42 * q^45 + 12 * q^49 - 8 * q^51 + 20 * q^53 + 5 * q^55 - q^57 - 15 * q^59 + 24 * q^61 + 20 * q^63 - 28 * q^65 - 25 * q^67 - 33 * q^69 + 9 * q^71 - 26 * q^73 - 28 * q^75 + 2 * q^77 + 16 * q^79 + 58 * q^81 + 2 * q^83 + 12 * q^85 + 36 * q^87 + 7 * q^89 - 8 * q^91 - 55 * q^93 + 5 * q^95 + 37 * q^97 + 17 * q^99

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