LMFDB - Embedded newform 3344.2.a.y.1.1

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:	6.6.57500224.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 7x^{4} + 12x^{3} + 11x^{2} - 18x + 2 $ x^6 - 2x^5 - 7x^4 + 12x^3 + 11x^2 - 18*x + 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1672)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.72058$ of defining polynomial
Character	$\chi$	$=$	3344.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.72058 q^{3} -2.22719 q^{5} +1.96772 q^{7} -0.0396114 q^{9} +O(q^{10})$	$q-1.72058 q^{3} -2.22719 q^{5} +1.96772 q^{7} -0.0396114 q^{9} -1.00000 q^{11} +7.04754 q^{13} +3.83206 q^{15} -7.60817 q^{17} +1.00000 q^{19} -3.38562 q^{21} +6.60084 q^{23} -0.0396114 q^{25} +5.22989 q^{27} -6.25147 q^{29} -5.36962 q^{31} +1.72058 q^{33} -4.38250 q^{35} +7.61473 q^{37} -12.1258 q^{39} -9.40485 q^{41} -6.26919 q^{43} +0.0882223 q^{45} +10.4279 q^{47} -3.12807 q^{49} +13.0905 q^{51} +4.34870 q^{53} +2.22719 q^{55} -1.72058 q^{57} -2.10518 q^{59} -8.77442 q^{61} -0.0779443 q^{63} -15.6962 q^{65} +13.7675 q^{67} -11.3573 q^{69} -1.14707 q^{71} -6.04471 q^{73} +0.0681546 q^{75} -1.96772 q^{77} +5.68740 q^{79} -8.87960 q^{81} -9.02439 q^{83} +16.9449 q^{85} +10.7561 q^{87} -4.83170 q^{89} +13.8676 q^{91} +9.23885 q^{93} -2.22719 q^{95} +12.0691 q^{97} +0.0396114 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 4 q^{3} - 4 q^{5} + 4 q^{7} + 2 q^{9}+O(q^{10}) $ 6 * q + 4 * q^3 - 4 * q^5 + 4 * q^7 + 2 * q^9	$ 6 q + 4 q^{3} - 4 q^{5} + 4 q^{7} + 2 q^{9} - 6 q^{11} + 2 q^{13} - 10 q^{17} + 6 q^{19} + 10 q^{21} + 14 q^{23} + 2 q^{25} + 16 q^{27} + 6 q^{29} - 4 q^{31} - 4 q^{33} - 2 q^{35} - 4 q^{37} + 2 q^{39} - 8 q^{41} + 26 q^{43} - 2 q^{45} + 24 q^{47} + 20 q^{49} + 14 q^{51} - 8 q^{53} + 4 q^{55} + 4 q^{57} + 4 q^{59} - 6 q^{61} + 40 q^{63} - 34 q^{65} + 44 q^{67} + 8 q^{69} + 16 q^{71} + 12 q^{73} + 28 q^{75} - 4 q^{77} - 6 q^{79} + 10 q^{81} + 28 q^{83} - 10 q^{85} + 24 q^{87} + 64 q^{91} - 14 q^{93} - 4 q^{95} + 4 q^{97} - 2 q^{99}+O(q^{100}) $ 6 * q + 4 * q^3 - 4 * q^5 + 4 * q^7 + 2 * q^9 - 6 * q^11 + 2 * q^13 - 10 * q^17 + 6 * q^19 + 10 * q^21 + 14 * q^23 + 2 * q^25 + 16 * q^27 + 6 * q^29 - 4 * q^31 - 4 * q^33 - 2 * q^35 - 4 * q^37 + 2 * q^39 - 8 * q^41 + 26 * q^43 - 2 * q^45 + 24 * q^47 + 20 * q^49 + 14 * q^51 - 8 * q^53 + 4 * q^55 + 4 * q^57 + 4 * q^59 - 6 * q^61 + 40 * q^63 - 34 * q^65 + 44 * q^67 + 8 * q^69 + 16 * q^71 + 12 * q^73 + 28 * q^75 - 4 * q^77 - 6 * q^79 + 10 * q^81 + 28 * q^83 - 10 * q^85 + 24 * q^87 + 64 * q^91 - 14 * q^93 - 4 * q^95 + 4 * q^97 - 2 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	−1.72058	−0.993376	−0.496688	−	0.867929i	$-0.665451\pi$
$3$	−1.72058	−0.993376	−0.496688	+	0.867929i	$0.665451\pi$
$4$	0	0
$5$	−2.22719	−0.996031	−0.498015	−	0.867168i	$-0.665938\pi$
$5$	−2.22719	−0.996031	−0.498015	+	0.867168i	$0.665938\pi$
$6$	0	0
$7$	1.96772	0.743729	0.371864	−	0.928287i	$-0.378719\pi$
$7$	1.96772	0.743729	0.371864	+	0.928287i	$0.378719\pi$
$8$	0	0
$9$	−0.0396114	−0.0132038
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	7.04754	1.95464	0.977318	−	0.211779i	$-0.0679257\pi$
$13$	7.04754	1.95464	0.977318	+	0.211779i	$0.0679257\pi$
$14$	0	0
$15$	3.83206	0.989433
$16$	0	0
$17$	−7.60817	−1.84525	−0.922627	−	0.385694i	$-0.873962\pi$
$17$	−7.60817	−1.84525	−0.922627	+	0.385694i	$0.873962\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	−3.38562	−0.738803
$22$	0	0
$23$	6.60084	1.37637	0.688185	−	0.725535i	$-0.258408\pi$
$23$	6.60084	1.37637	0.688185	+	0.725535i	$0.258408\pi$
$24$	0	0
$25$	−0.0396114	−0.00792229
$26$	0	0
$27$	5.22989	1.00649
$28$	0	0
$29$	−6.25147	−1.16087	−0.580435	−	0.814307i	$-0.697117\pi$
$29$	−6.25147	−1.16087	−0.580435	+	0.814307i	$0.697117\pi$
$30$	0	0
$31$	−5.36962	−0.964413	−0.482206	−	0.876058i	$-0.660164\pi$
$31$	−5.36962	−0.964413	−0.482206	+	0.876058i	$0.660164\pi$
$32$	0	0
$33$	1.72058	0.299514
$34$	0	0
$35$	−4.38250	−0.740777
$36$	0	0
$37$	7.61473	1.25185	0.625927	−	0.779881i	$-0.284721\pi$
$37$	7.61473	1.25185	0.625927	+	0.779881i	$0.284721\pi$
$38$	0	0
$39$	−12.1258	−1.94169
$40$	0	0
$41$	−9.40485	−1.46879	−0.734396	−	0.678722i	$-0.762534\pi$
$41$	−9.40485	−1.46879	−0.734396	+	0.678722i	$0.762534\pi$
$42$	0	0
$43$	−6.26919	−0.956042	−0.478021	−	0.878348i	$-0.658646\pi$
$43$	−6.26919	−0.956042	−0.478021	+	0.878348i	$0.658646\pi$
$44$	0	0
$45$	0.0882223	0.0131514
$46$	0	0
$47$	10.4279	1.52107	0.760535	−	0.649297i	$-0.224937\pi$
$47$	10.4279	1.52107	0.760535	+	0.649297i	$0.224937\pi$
$48$	0	0
$49$	−3.12807	−0.446867
$50$	0	0
$51$	13.0905	1.83303
$52$	0	0
$53$	4.34870	0.597340	0.298670	−	0.954356i	$-0.403457\pi$
$53$	4.34870	0.597340	0.298670	+	0.954356i	$0.403457\pi$
$54$	0	0
$55$	2.22719	0.300315
$56$	0	0
$57$	−1.72058	−0.227896
$58$	0	0
$59$	−2.10518	−0.274071	−0.137035	−	0.990566i	$-0.543757\pi$
$59$	−2.10518	−0.274071	−0.137035	+	0.990566i	$0.543757\pi$
$60$	0	0
$61$	−8.77442	−1.12345	−0.561725	−	0.827324i	$-0.689862\pi$
$61$	−8.77442	−1.12345	−0.561725	+	0.827324i	$0.689862\pi$
$62$	0	0
$63$	−0.0779443	−0.00982006
$64$	0	0
$65$	−15.6962	−1.94688
$66$	0	0
$67$	13.7675	1.68197	0.840984	−	0.541060i	$-0.181977\pi$
$67$	13.7675	1.68197	0.840984	+	0.541060i	$0.181977\pi$
$68$	0	0
$69$	−11.3573	−1.36725
$70$	0	0
$71$	−1.14707	−0.136132	−0.0680659	−	0.997681i	$-0.521683\pi$
$71$	−1.14707	−0.136132	−0.0680659	+	0.997681i	$0.521683\pi$
$72$	0	0
$73$	−6.04471	−0.707480	−0.353740	−	0.935344i	$-0.615090\pi$
$73$	−6.04471	−0.707480	−0.353740	+	0.935344i	$0.615090\pi$
$74$	0	0
$75$	0.0681546	0.00786981
$76$	0	0
$77$	−1.96772	−0.224243
$78$	0	0
$79$	5.68740	0.639882	0.319941	−	0.947437i	$-0.396337\pi$
$79$	5.68740	0.639882	0.319941	+	0.947437i	$0.396337\pi$
$80$	0	0
$81$	−8.87960	−0.986622
$82$	0	0
$83$	−9.02439	−0.990555	−0.495277	−	0.868735i	$-0.664934\pi$
$83$	−9.02439	−0.990555	−0.495277	+	0.868735i	$0.664934\pi$
$84$	0	0
$85$	16.9449	1.83793
$86$	0	0
$87$	10.7561	1.15318
$88$	0	0
$89$	−4.83170	−0.512160	−0.256080	−	0.966656i	$-0.582431\pi$
$89$	−4.83170	−0.512160	−0.256080	+	0.966656i	$0.582431\pi$
$90$	0	0
$91$	13.8676	1.45372
$92$	0	0
$93$	9.23885	0.958025
$94$	0	0
$95$	−2.22719	−0.228505
$96$	0	0
$97$	12.0691	1.22543	0.612717	−	0.790303i	$-0.290077\pi$
$97$	12.0691	1.22543	0.612717	+	0.790303i	$0.290077\pi$
$98$	0	0
$99$	0.0396114	0.00398110
$100$	0	0
$101$	17.1105	1.70256	0.851280	−	0.524712i	$-0.175827\pi$
$101$	17.1105	1.70256	0.851280	+	0.524712i	$0.175827\pi$
$102$	0	0
$103$	11.4053	1.12379	0.561897	−	0.827207i	$-0.310072\pi$
$103$	11.4053	1.12379	0.561897	+	0.827207i	$0.310072\pi$
$104$	0	0
$105$	7.54043	0.735870
$106$	0	0
$107$	6.56007	0.634186	0.317093	−	0.948394i	$-0.397293\pi$
$107$	6.56007	0.634186	0.317093	+	0.948394i	$0.397293\pi$
$108$	0	0
$109$	−1.49644	−0.143333	−0.0716663	−	0.997429i	$-0.522832\pi$
$109$	−1.49644	−0.143333	−0.0716663	+	0.997429i	$0.522832\pi$
$110$	0	0
$111$	−13.1017	−1.24356
$112$	0	0
$113$	3.54633	0.333611	0.166805	−	0.985990i	$-0.446655\pi$
$113$	3.54633	0.333611	0.166805	+	0.985990i	$0.446655\pi$
$114$	0	0
$115$	−14.7013	−1.37091
$116$	0	0
$117$	−0.279163	−0.0258086
$118$	0	0
$119$	−14.9708	−1.37237
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	16.1818	1.45906
$124$	0	0
$125$	11.2242	1.00392
$126$	0	0
$127$	4.70447	0.417454	0.208727	−	0.977974i	$-0.433068\pi$
$127$	4.70447	0.417454	0.208727	+	0.977974i	$0.433068\pi$
$128$	0	0
$129$	10.7866	0.949710
$130$	0	0
$131$	1.53122	0.133784	0.0668918	−	0.997760i	$-0.478692\pi$
$131$	1.53122	0.133784	0.0668918	+	0.997760i	$0.478692\pi$
$132$	0	0
$133$	1.96772	0.170623
$134$	0	0
$135$	−11.6480	−1.00250
$136$	0	0
$137$	−16.4781	−1.40782	−0.703910	−	0.710289i	$-0.748564\pi$
$137$	−16.4781	−1.40782	−0.703910	+	0.710289i	$0.748564\pi$
$138$	0	0
$139$	−2.61989	−0.222216	−0.111108	−	0.993808i	$-0.535440\pi$
$139$	−2.61989	−0.222216	−0.111108	+	0.993808i	$0.535440\pi$
$140$	0	0
$141$	−17.9421	−1.51099
$142$	0	0
$143$	−7.04754	−0.589345
$144$	0	0
$145$	13.9232	1.15626
$146$	0	0
$147$	5.38209	0.443907
$148$	0	0
$149$	11.0818	0.907860	0.453930	−	0.891037i	$-0.350022\pi$
$149$	11.0818	0.907860	0.453930	+	0.891037i	$0.350022\pi$
$150$	0	0
$151$	1.83904	0.149659	0.0748293	−	0.997196i	$-0.476159\pi$
$151$	1.83904	0.149659	0.0748293	+	0.997196i	$0.476159\pi$
$152$	0	0
$153$	0.301371	0.0243644
$154$	0	0
$155$	11.9592	0.960585
$156$	0	0
$157$	−2.36554	−0.188791	−0.0943954	−	0.995535i	$-0.530092\pi$
$157$	−2.36554	−0.188791	−0.0943954	+	0.995535i	$0.530092\pi$
$158$	0	0
$159$	−7.48228	−0.593384
$160$	0	0
$161$	12.9886	1.02365
$162$	0	0
$163$	19.2217	1.50556	0.752781	−	0.658271i	$-0.228712\pi$
$163$	19.2217	1.50556	0.752781	+	0.658271i	$0.228712\pi$
$164$	0	0
$165$	−3.83206	−0.298325
$166$	0	0
$167$	−13.0347	−1.00865	−0.504326	−	0.863513i	$-0.668259\pi$
$167$	−13.0347	−1.00865	−0.504326	+	0.863513i	$0.668259\pi$
$168$	0	0
$169$	36.6678	2.82060
$170$	0	0
$171$	−0.0396114	−0.00302916
$172$	0	0
$173$	12.9243	0.982614	0.491307	−	0.870987i	$-0.336519\pi$
$173$	12.9243	0.982614	0.491307	+	0.870987i	$0.336519\pi$
$174$	0	0
$175$	−0.0779443	−0.00589204
$176$	0	0
$177$	3.62212	0.272255
$178$	0	0
$179$	7.25316	0.542126	0.271063	−	0.962562i	$-0.412625\pi$
$179$	7.25316	0.542126	0.271063	+	0.962562i	$0.412625\pi$
$180$	0	0
$181$	22.4890	1.67159	0.835797	−	0.549039i	$-0.185006\pi$
$181$	22.4890	1.67159	0.835797	+	0.549039i	$0.185006\pi$
$182$	0	0
$183$	15.0971	1.11601
$184$	0	0
$185$	−16.9595	−1.24689
$186$	0	0
$187$	7.60817	0.556365
$188$	0	0
$189$	10.2910	0.748558
$190$	0	0
$191$	−1.63679	−0.118434	−0.0592169	−	0.998245i	$-0.518860\pi$
$191$	−1.63679	−0.118434	−0.0592169	+	0.998245i	$0.518860\pi$
$192$	0	0
$193$	8.67334	0.624321	0.312160	−	0.950029i	$-0.398947\pi$
$193$	8.67334	0.624321	0.312160	+	0.950029i	$0.398947\pi$
$194$	0	0
$195$	27.0066	1.93398
$196$	0	0
$197$	14.4678	1.03079	0.515393	−	0.856954i	$-0.327646\pi$
$197$	14.4678	1.03079	0.515393	+	0.856954i	$0.327646\pi$
$198$	0	0
$199$	8.92883	0.632948	0.316474	−	0.948601i	$-0.397501\pi$
$199$	8.92883	0.632948	0.316474	+	0.948601i	$0.397501\pi$
$200$	0	0
$201$	−23.6881	−1.67083
$202$	0	0
$203$	−12.3012	−0.863372
$204$	0	0
$205$	20.9464	1.46296
$206$	0	0
$207$	−0.261469	−0.0181733
$208$	0	0
$209$	−1.00000	−0.0691714
$210$	0	0
$211$	13.9296	0.958953	0.479477	−	0.877555i	$-0.340827\pi$
$211$	13.9296	0.958953	0.479477	+	0.877555i	$0.340827\pi$
$212$	0	0
$213$	1.97362	0.135230
$214$	0	0
$215$	13.9627	0.952248
$216$	0	0
$217$	−10.5659	−0.717262
$218$	0	0
$219$	10.4004	0.702794
$220$	0	0
$221$	−53.6189	−3.60680
$222$	0	0
$223$	1.23753	0.0828713	0.0414356	−	0.999141i	$-0.486807\pi$
$223$	1.23753	0.0828713	0.0414356	+	0.999141i	$0.486807\pi$
$224$	0	0
$225$	0.00156907	0.000104604 0
$226$	0	0
$227$	19.4085	1.28819	0.644093	−	0.764947i	$-0.277235\pi$
$227$	19.4085	1.28819	0.644093	+	0.764947i	$0.277235\pi$
$228$	0	0
$229$	−13.6297	−0.900675	−0.450338	−	0.892858i	$-0.648696\pi$
$229$	−13.6297	−0.900675	−0.450338	+	0.892858i	$0.648696\pi$
$230$	0	0
$231$	3.38562	0.222757
$232$	0	0
$233$	6.87370	0.450311	0.225156	−	0.974323i	$-0.427711\pi$
$233$	6.87370	0.450311	0.225156	+	0.974323i	$0.427711\pi$
$234$	0	0
$235$	−23.2250	−1.51503
$236$	0	0
$237$	−9.78561	−0.635644
$238$	0	0
$239$	−26.9021	−1.74015	−0.870075	−	0.492919i	$-0.835930\pi$
$239$	−26.9021	−1.74015	−0.870075	+	0.492919i	$0.835930\pi$
$240$	0	0
$241$	10.2375	0.659453	0.329727	−	0.944076i	$-0.393043\pi$
$241$	10.2375	0.659453	0.329727	+	0.944076i	$0.393043\pi$
$242$	0	0
$243$	−0.411627	−0.0264059
$244$	0	0
$245$	6.96682	0.445094
$246$	0	0
$247$	7.04754	0.448424
$248$	0	0
$249$	15.5272	0.983994
$250$	0	0
$251$	14.8369	0.936498	0.468249	−	0.883596i	$-0.344885\pi$
$251$	14.8369	0.936498	0.468249	+	0.883596i	$0.344885\pi$
$252$	0	0
$253$	−6.60084	−0.414991
$254$	0	0
$255$	−29.1550	−1.82576
$256$	0	0
$257$	5.88893	0.367341	0.183671	−	0.982988i	$-0.441202\pi$
$257$	5.88893	0.367341	0.183671	+	0.982988i	$0.441202\pi$
$258$	0	0
$259$	14.9837	0.931041
$260$	0	0
$261$	0.247630	0.0153279
$262$	0	0
$263$	−16.7421	−1.03236	−0.516181	−	0.856480i	$-0.672647\pi$
$263$	−16.7421	−1.03236	−0.516181	+	0.856480i	$0.672647\pi$
$264$	0	0
$265$	−9.68540	−0.594969
$266$	0	0
$267$	8.31332	0.508767
$268$	0	0
$269$	9.63411	0.587402	0.293701	−	0.955897i	$-0.405113\pi$
$269$	9.63411	0.587402	0.293701	+	0.955897i	$0.405113\pi$
$270$	0	0
$271$	−26.2625	−1.59533	−0.797667	−	0.603098i	$-0.793933\pi$
$271$	−26.2625	−1.59533	−0.797667	+	0.603098i	$0.793933\pi$
$272$	0	0
$273$	−23.8603	−1.44409
$274$	0	0
$275$	0.0396114	0.00238866
$276$	0	0
$277$	27.1065	1.62867	0.814336	−	0.580394i	$-0.197101\pi$
$277$	27.1065	1.62867	0.814336	+	0.580394i	$0.197101\pi$
$278$	0	0
$279$	0.212698	0.0127339
$280$	0	0
$281$	−25.2426	−1.50585	−0.752925	−	0.658106i	$-0.771358\pi$
$281$	−25.2426	−1.50585	−0.752925	+	0.658106i	$0.771358\pi$
$282$	0	0
$283$	−3.88126	−0.230717	−0.115359	−	0.993324i	$-0.536802\pi$
$283$	−3.88126	−0.230717	−0.115359	+	0.993324i	$0.536802\pi$
$284$	0	0
$285$	3.83206	0.226992
$286$	0	0
$287$	−18.5061	−1.09238
$288$	0	0
$289$	40.8843	2.40496
$290$	0	0
$291$	−20.7659	−1.21732
$292$	0	0
$293$	9.16065	0.535171	0.267585	−	0.963534i	$-0.413774\pi$
$293$	9.16065	0.535171	0.267585	+	0.963534i	$0.413774\pi$
$294$	0	0
$295$	4.68864	0.272983
$296$	0	0
$297$	−5.22989	−0.303469
$298$	0	0
$299$	46.5197	2.69030
$300$	0	0
$301$	−12.3360	−0.711036
$302$	0	0
$303$	−29.4400	−1.69128
$304$	0	0
$305$	19.5423	1.11899
$306$	0	0
$307$	7.45110	0.425256	0.212628	−	0.977133i	$-0.431798\pi$
$307$	7.45110	0.425256	0.212628	+	0.977133i	$0.431798\pi$
$308$	0	0
$309$	−19.6236	−1.11635
$310$	0	0
$311$	28.1280	1.59499	0.797496	−	0.603324i	$-0.206157\pi$
$311$	28.1280	1.59499	0.797496	+	0.603324i	$0.206157\pi$
$312$	0	0
$313$	−3.71407	−0.209932	−0.104966	−	0.994476i	$-0.533473\pi$
$313$	−3.71407	−0.209932	−0.104966	+	0.994476i	$0.533473\pi$
$314$	0	0
$315$	0.173597	0.00978108
$316$	0	0
$317$	4.79790	0.269477	0.134739	−	0.990881i	$-0.456981\pi$
$317$	4.79790	0.269477	0.134739	+	0.990881i	$0.456981\pi$
$318$	0	0
$319$	6.25147	0.350015
$320$	0	0
$321$	−11.2871	−0.629985
$322$	0	0
$323$	−7.60817	−0.423330
$324$	0	0
$325$	−0.279163	−0.0154852
$326$	0	0
$327$	2.57474	0.142383
$328$	0	0
$329$	20.5193	1.13126
$330$	0	0
$331$	−9.44840	−0.519331	−0.259665	−	0.965699i	$-0.583612\pi$
$331$	−9.44840	−0.519331	−0.259665	+	0.965699i	$0.583612\pi$
$332$	0	0
$333$	−0.301631	−0.0165293
$334$	0	0
$335$	−30.6629	−1.67529
$336$	0	0
$337$	−27.1381	−1.47831	−0.739155	−	0.673536i	$-0.764775\pi$
$337$	−27.1381	−1.47831	−0.739155	+	0.673536i	$0.764775\pi$
$338$	0	0
$339$	−6.10174	−0.331401
$340$	0	0
$341$	5.36962	0.290781
$342$	0	0
$343$	−19.9292	−1.07608
$344$	0	0
$345$	25.2948	1.36183
$346$	0	0
$347$	−5.46828	−0.293553	−0.146776	−	0.989170i	$-0.546890\pi$
$347$	−5.46828	−0.293553	−0.146776	+	0.989170i	$0.546890\pi$
$348$	0	0
$349$	3.27557	0.175337	0.0876687	−	0.996150i	$-0.472058\pi$
$349$	3.27557	0.175337	0.0876687	+	0.996150i	$0.472058\pi$
$350$	0	0
$351$	36.8578	1.96733
$352$	0	0
$353$	16.6340	0.885340	0.442670	−	0.896685i	$-0.354031\pi$
$353$	16.6340	0.885340	0.442670	+	0.896685i	$0.354031\pi$
$354$	0	0
$355$	2.55474	0.135592
$356$	0	0
$357$	25.7584	1.36328
$358$	0	0
$359$	29.1556	1.53877	0.769387	−	0.638783i	$-0.220562\pi$
$359$	29.1556	1.53877	0.769387	+	0.638783i	$0.220562\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	−1.72058	−0.0903069
$364$	0	0
$365$	13.4627	0.704672
$366$	0	0
$367$	27.9369	1.45830	0.729148	−	0.684356i	$-0.239917\pi$
$367$	27.9369	1.45830	0.729148	+	0.684356i	$0.239917\pi$
$368$	0	0
$369$	0.372540	0.0193936
$370$	0	0
$371$	8.55704	0.444259
$372$	0	0
$373$	14.7198	0.762163	0.381082	−	0.924541i	$-0.375552\pi$
$373$	14.7198	0.762163	0.381082	+	0.924541i	$0.375552\pi$
$374$	0	0
$375$	−19.3121	−0.997272
$376$	0	0
$377$	−44.0575	−2.26908
$378$	0	0
$379$	−0.342859	−0.0176115	−0.00880574	−	0.999961i	$-0.502803\pi$
$379$	−0.342859	−0.0176115	−0.00880574	+	0.999961i	$0.502803\pi$
$380$	0	0
$381$	−8.09441	−0.414689
$382$	0	0
$383$	31.3669	1.60277	0.801386	−	0.598148i	$-0.204096\pi$
$383$	31.3669	1.60277	0.801386	+	0.598148i	$0.204096\pi$
$384$	0	0
$385$	4.38250	0.223353
$386$	0	0
$387$	0.248332	0.0126234
$388$	0	0
$389$	−32.0391	−1.62445	−0.812225	−	0.583345i	$-0.801744\pi$
$389$	−32.0391	−1.62445	−0.812225	+	0.583345i	$0.801744\pi$
$390$	0	0
$391$	−50.2203	−2.53975
$392$	0	0
$393$	−2.63459	−0.132897
$394$	0	0
$395$	−12.6669	−0.637342
$396$	0	0
$397$	−33.4621	−1.67941	−0.839706	−	0.543041i	$-0.817273\pi$
$397$	−33.4621	−1.67941	−0.839706	+	0.543041i	$0.817273\pi$
$398$	0	0
$399$	−3.38562	−0.169493
$400$	0	0
$401$	9.41337	0.470081	0.235041	−	0.971986i	$-0.424478\pi$
$401$	9.41337	0.470081	0.235041	+	0.971986i	$0.424478\pi$
$402$	0	0
$403$	−37.8426	−1.88507
$404$	0	0
$405$	19.7766	0.982706
$406$	0	0
$407$	−7.61473	−0.377448
$408$	0	0
$409$	−10.0101	−0.494966	−0.247483	−	0.968892i	$-0.579603\pi$
$409$	−10.0101	−0.494966	−0.247483	+	0.968892i	$0.579603\pi$
$410$	0	0
$411$	28.3519	1.39849
$412$	0	0
$413$	−4.14240	−0.203834
$414$	0	0
$415$	20.0990	0.986623
$416$	0	0
$417$	4.50772	0.220744
$418$	0	0
$419$	−6.41377	−0.313333	−0.156667	−	0.987652i	$-0.550075\pi$
$419$	−6.41377	−0.313333	−0.156667	+	0.987652i	$0.550075\pi$
$420$	0	0
$421$	10.1476	0.494565	0.247282	−	0.968943i	$-0.420462\pi$
$421$	10.1476	0.494565	0.247282	+	0.968943i	$0.420462\pi$
$422$	0	0
$423$	−0.413065	−0.0200839
$424$	0	0
$425$	0.301371	0.0146186
$426$	0	0
$427$	−17.2656	−0.835542
$428$	0	0
$429$	12.1258	0.585441
$430$	0	0
$431$	−5.59530	−0.269516	−0.134758	−	0.990879i	$-0.543026\pi$
$431$	−5.59530	−0.269516	−0.134758	+	0.990879i	$0.543026\pi$
$432$	0	0
$433$	−34.1228	−1.63984	−0.819918	−	0.572481i	$-0.805981\pi$
$433$	−34.1228	−1.63984	−0.819918	+	0.572481i	$0.805981\pi$
$434$	0	0
$435$	−23.9560	−1.14860
$436$	0	0
$437$	6.60084	0.315761
$438$	0	0
$439$	−6.28891	−0.300153	−0.150077	−	0.988674i	$-0.547952\pi$
$439$	−6.28891	−0.300153	−0.150077	+	0.988674i	$0.547952\pi$
$440$	0	0
$441$	0.123907	0.00590035
$442$	0	0
$443$	22.1169	1.05081	0.525403	−	0.850853i	$-0.323914\pi$
$443$	22.1169	1.05081	0.525403	+	0.850853i	$0.323914\pi$
$444$	0	0
$445$	10.7611	0.510127
$446$	0	0
$447$	−19.0672	−0.901847
$448$	0	0
$449$	3.54479	0.167289	0.0836444	−	0.996496i	$-0.473344\pi$
$449$	3.54479	0.167289	0.0836444	+	0.996496i	$0.473344\pi$
$450$	0	0
$451$	9.40485	0.442857
$452$	0	0
$453$	−3.16421	−0.148667
$454$	0	0
$455$	−30.8858	−1.44795
$456$	0	0
$457$	29.0898	1.36076	0.680382	−	0.732858i	$-0.261814\pi$
$457$	29.0898	1.36076	0.680382	+	0.732858i	$0.261814\pi$
$458$	0	0
$459$	−39.7899	−1.85723
$460$	0	0
$461$	14.5291	0.676688	0.338344	−	0.941023i	$-0.390133\pi$
$461$	14.5291	0.676688	0.338344	+	0.941023i	$0.390133\pi$
$462$	0	0
$463$	20.6005	0.957386	0.478693	−	0.877982i	$-0.341111\pi$
$463$	20.6005	0.957386	0.478693	+	0.877982i	$0.341111\pi$
$464$	0	0
$465$	−20.5767	−0.954222
$466$	0	0
$467$	−11.8174	−0.546846	−0.273423	−	0.961894i	$-0.588156\pi$
$467$	−11.8174	−0.546846	−0.273423	+	0.961894i	$0.588156\pi$
$468$	0	0
$469$	27.0906	1.25093
$470$	0	0
$471$	4.07010	0.187540
$472$	0	0
$473$	6.26919	0.288258
$474$	0	0
$475$	−0.0396114	−0.00181750
$476$	0	0
$477$	−0.172258	−0.00788717
$478$	0	0
$479$	39.0255	1.78312	0.891561	−	0.452901i	$-0.149611\pi$
$479$	39.0255	1.78312	0.891561	+	0.452901i	$0.149611\pi$
$480$	0	0
$481$	53.6651	2.44692
$482$	0	0
$483$	−22.3479	−1.01687
$484$	0	0
$485$	−26.8803	−1.22057
$486$	0	0
$487$	29.3964	1.33208	0.666039	−	0.745917i	$-0.267989\pi$
$487$	29.3964	1.33208	0.666039	+	0.745917i	$0.267989\pi$
$488$	0	0
$489$	−33.0725	−1.49559
$490$	0	0
$491$	−38.5585	−1.74012	−0.870061	−	0.492943i	$-0.835921\pi$
$491$	−38.5585	−1.74012	−0.870061	+	0.492943i	$0.835921\pi$
$492$	0	0
$493$	47.5623	2.14210
$494$	0	0
$495$	−0.0882223	−0.00396530
$496$	0	0
$497$	−2.25711	−0.101245
$498$	0	0
$499$	−13.4393	−0.601626	−0.300813	−	0.953683i	$-0.597258\pi$
$499$	−13.4393	−0.601626	−0.300813	+	0.953683i	$0.597258\pi$
$500$	0	0
$501$	22.4272	1.00197
$502$	0	0
$503$	16.4840	0.734984	0.367492	−	0.930027i	$-0.380217\pi$
$503$	16.4840	0.734984	0.367492	+	0.930027i	$0.380217\pi$
$504$	0	0
$505$	−38.1084	−1.69580
$506$	0	0
$507$	−63.0898	−2.80192
$508$	0	0
$509$	−20.6969	−0.917374	−0.458687	−	0.888598i	$-0.651680\pi$
$509$	−20.6969	−0.917374	−0.458687	+	0.888598i	$0.651680\pi$
$510$	0	0
$511$	−11.8943	−0.526173
$512$	0	0
$513$	5.22989	0.230905
$514$	0	0
$515$	−25.4017	−1.11933
$516$	0	0
$517$	−10.4279	−0.458620
$518$	0	0
$519$	−22.2372	−0.976105
$520$	0	0
$521$	12.5616	0.550334	0.275167	−	0.961396i	$-0.411267\pi$
$521$	12.5616	0.550334	0.275167	+	0.961396i	$0.411267\pi$
$522$	0	0
$523$	−40.5981	−1.77523	−0.887616	−	0.460584i	$-0.847640\pi$
$523$	−40.5981	−1.77523	−0.887616	+	0.460584i	$0.847640\pi$
$524$	0	0
$525$	0.134109	0.00585301
$526$	0	0
$527$	40.8530	1.77959
$528$	0	0
$529$	20.5711	0.894396
$530$	0	0
$531$	0.0833891	0.00361878
$532$	0	0
$533$	−66.2810	−2.87095
$534$	0	0
$535$	−14.6105	−0.631669
$536$	0	0
$537$	−12.4796	−0.538535
$538$	0	0
$539$	3.12807	0.134736
$540$	0	0
$541$	−16.6583	−0.716197	−0.358099	−	0.933684i	$-0.616575\pi$
$541$	−16.6583	−0.716197	−0.358099	+	0.933684i	$0.616575\pi$
$542$	0	0
$543$	−38.6941	−1.66052
$544$	0	0
$545$	3.33285	0.142764
$546$	0	0
$547$	−24.5756	−1.05078	−0.525389	−	0.850862i	$-0.676080\pi$
$547$	−24.5756	−1.05078	−0.525389	+	0.850862i	$0.676080\pi$
$548$	0	0
$549$	0.347567	0.0148338
$550$	0	0
$551$	−6.25147	−0.266322
$552$	0	0
$553$	11.1912	0.475899
$554$	0	0
$555$	29.1801	1.23863
$556$	0	0
$557$	−21.4752	−0.909934	−0.454967	−	0.890508i	$-0.650349\pi$
$557$	−21.4752	−0.909934	−0.454967	+	0.890508i	$0.650349\pi$
$558$	0	0
$559$	−44.1823	−1.86871
$560$	0	0
$561$	−13.0905	−0.552680
$562$	0	0
$563$	−20.9440	−0.882686	−0.441343	−	0.897338i	$-0.645498\pi$
$563$	−20.9440	−0.882686	−0.441343	+	0.897338i	$0.645498\pi$
$564$	0	0
$565$	−7.89837	−0.332287
$566$	0	0
$567$	−17.4726	−0.733779
$568$	0	0
$569$	−21.1031	−0.884687	−0.442343	−	0.896846i	$-0.645853\pi$
$569$	−21.1031	−0.884687	−0.442343	+	0.896846i	$0.645853\pi$
$570$	0	0
$571$	27.6827	1.15848	0.579242	−	0.815156i	$-0.303349\pi$
$571$	27.6827	1.15848	0.579242	+	0.815156i	$0.303349\pi$
$572$	0	0
$573$	2.81622	0.117649
$574$	0	0
$575$	−0.261469	−0.0109040
$576$	0	0
$577$	15.9430	0.663714	0.331857	−	0.943330i	$-0.392325\pi$
$577$	15.9430	0.663714	0.331857	+	0.943330i	$0.392325\pi$
$578$	0	0
$579$	−14.9232	−0.620185
$580$	0	0
$581$	−17.7575	−0.736704
$582$	0	0
$583$	−4.34870	−0.180105
$584$	0	0
$585$	0.621750	0.0257062
$586$	0	0
$587$	−6.93504	−0.286240	−0.143120	−	0.989705i	$-0.545713\pi$
$587$	−6.93504	−0.286240	−0.143120	+	0.989705i	$0.545713\pi$
$588$	0	0
$589$	−5.36962	−0.221251
$590$	0	0
$591$	−24.8929	−1.02396
$592$	0	0
$593$	−13.0536	−0.536049	−0.268024	−	0.963412i	$-0.586371\pi$
$593$	−13.0536	−0.536049	−0.268024	+	0.963412i	$0.586371\pi$
$594$	0	0
$595$	33.3428	1.36692
$596$	0	0
$597$	−15.3627	−0.628755
$598$	0	0
$599$	−21.5165	−0.879142	−0.439571	−	0.898208i	$-0.644870\pi$
$599$	−21.5165	−0.879142	−0.439571	+	0.898208i	$0.644870\pi$
$600$	0	0
$601$	−16.9862	−0.692883	−0.346442	−	0.938072i	$-0.612610\pi$
$601$	−16.9862	−0.692883	−0.346442	+	0.938072i	$0.612610\pi$
$602$	0	0
$603$	−0.545351	−0.0222084
$604$	0	0
$605$	−2.22719	−0.0905483
$606$	0	0
$607$	8.55167	0.347102	0.173551	−	0.984825i	$-0.444476\pi$
$607$	8.55167	0.347102	0.173551	+	0.984825i	$0.444476\pi$
$608$	0	0
$609$	21.1651	0.857653
$610$	0	0
$611$	73.4912	2.97314
$612$	0	0
$613$	2.17578	0.0878790	0.0439395	−	0.999034i	$-0.486009\pi$
$613$	2.17578	0.0878790	0.0439395	+	0.999034i	$0.486009\pi$
$614$	0	0
$615$	−36.0400	−1.45327
$616$	0	0
$617$	39.5625	1.59273	0.796364	−	0.604818i	$-0.206754\pi$
$617$	39.5625	1.59273	0.796364	+	0.604818i	$0.206754\pi$
$618$	0	0
$619$	2.42787	0.0975844	0.0487922	−	0.998809i	$-0.484463\pi$
$619$	2.42787	0.0975844	0.0487922	+	0.998809i	$0.484463\pi$
$620$	0	0
$621$	34.5217	1.38531
$622$	0	0
$623$	−9.50745	−0.380908
$624$	0	0
$625$	−24.8004	−0.992015
$626$	0	0
$627$	1.72058	0.0687133
$628$	0	0
$629$	−57.9342	−2.30999
$630$	0	0
$631$	11.9436	0.475467	0.237733	−	0.971330i	$-0.423596\pi$
$631$	11.9436	0.475467	0.237733	+	0.971330i	$0.423596\pi$
$632$	0	0
$633$	−23.9670	−0.952601
$634$	0	0
$635$	−10.4778	−0.415798
$636$	0	0
$637$	−22.0452	−0.873462
$638$	0	0
$639$	0.0454370	0.00179746
$640$	0	0
$641$	26.0294	1.02810	0.514050	−	0.857760i	$-0.328145\pi$
$641$	26.0294	1.02810	0.514050	+	0.857760i	$0.328145\pi$
$642$	0	0
$643$	37.6765	1.48582	0.742909	−	0.669393i	$-0.233446\pi$
$643$	37.6765	1.48582	0.742909	+	0.669393i	$0.233446\pi$
$644$	0	0
$645$	−24.0239	−0.945940
$646$	0	0
$647$	−37.3030	−1.46653	−0.733266	−	0.679941i	$-0.762005\pi$
$647$	−37.3030	−1.46653	−0.733266	+	0.679941i	$0.762005\pi$
$648$	0	0
$649$	2.10518	0.0826354
$650$	0	0
$651$	18.1795	0.712511
$652$	0	0
$653$	2.67401	0.104642	0.0523210	−	0.998630i	$-0.483338\pi$
$653$	2.67401	0.104642	0.0523210	+	0.998630i	$0.483338\pi$
$654$	0	0
$655$	−3.41033	−0.133253
$656$	0	0
$657$	0.239440	0.00934143
$658$	0	0
$659$	43.7451	1.70407	0.852034	−	0.523486i	$-0.175369\pi$
$659$	43.7451	1.70407	0.852034	+	0.523486i	$0.175369\pi$
$660$	0	0
$661$	−43.6277	−1.69692	−0.848460	−	0.529260i	$-0.822470\pi$
$661$	−43.6277	−1.69692	−0.848460	+	0.529260i	$0.822470\pi$
$662$	0	0
$663$	92.2555	3.58291
$664$	0	0
$665$	−4.38250	−0.169946
$666$	0	0
$667$	−41.2650	−1.59779
$668$	0	0
$669$	−2.12927	−0.0823224
$670$	0	0
$671$	8.77442	0.338733
$672$	0	0
$673$	−35.3184	−1.36142	−0.680712	−	0.732551i	$-0.738329\pi$
$673$	−35.3184	−1.36142	−0.680712	+	0.732551i	$0.738329\pi$
$674$	0	0
$675$	−0.207163	−0.00797372
$676$	0	0
$677$	−17.8219	−0.684952	−0.342476	−	0.939527i	$-0.611265\pi$
$677$	−17.8219	−0.684952	−0.342476	+	0.939527i	$0.611265\pi$
$678$	0	0
$679$	23.7487	0.911390
$680$	0	0
$681$	−33.3938	−1.27965
$682$	0	0
$683$	18.3781	0.703220	0.351610	−	0.936147i	$-0.385634\pi$
$683$	18.3781	0.703220	0.351610	+	0.936147i	$0.385634\pi$
$684$	0	0
$685$	36.6999	1.40223
$686$	0	0
$687$	23.4509	0.894709
$688$	0	0
$689$	30.6476	1.16758
$690$	0	0
$691$	2.93938	0.111819	0.0559097	−	0.998436i	$-0.482194\pi$
$691$	2.93938	0.111819	0.0559097	+	0.998436i	$0.482194\pi$
$692$	0	0
$693$	0.0779443	0.00296086
$694$	0	0
$695$	5.83500	0.221334
$696$	0	0
$697$	71.5538	2.71029
$698$	0	0
$699$	−11.8267	−0.447328
$700$	0	0
$701$	46.4261	1.75349	0.876744	−	0.480956i	$-0.159710\pi$
$701$	46.4261	1.75349	0.876744	+	0.480956i	$0.159710\pi$
$702$	0	0
$703$	7.61473	0.287195
$704$	0	0
$705$	39.9604	1.50500
$706$	0	0
$707$	33.6687	1.26624
$708$	0	0
$709$	8.55669	0.321353	0.160677	−	0.987007i	$-0.448632\pi$
$709$	8.55669	0.321353	0.160677	+	0.987007i	$0.448632\pi$
$710$	0	0
$711$	−0.225286	−0.00844888
$712$	0	0
$713$	−35.4440	−1.32739
$714$	0	0
$715$	15.6962	0.587006
$716$	0	0
$717$	46.2871	1.72862
$718$	0	0
$719$	−1.29599	−0.0483324	−0.0241662	−	0.999708i	$-0.507693\pi$
$719$	−1.29599	−0.0483324	−0.0241662	+	0.999708i	$0.507693\pi$
$720$	0	0
$721$	22.4424	0.835798
$722$	0	0
$723$	−17.6144	−0.655085
$724$	0	0
$725$	0.247630	0.00919674
$726$	0	0
$727$	−12.2174	−0.453119	−0.226559	−	0.973997i	$-0.572748\pi$
$727$	−12.2174	−0.453119	−0.226559	+	0.973997i	$0.572748\pi$
$728$	0	0
$729$	27.3470	1.01285
$730$	0	0
$731$	47.6971	1.76414
$732$	0	0
$733$	−19.2105	−0.709556	−0.354778	−	0.934951i	$-0.615444\pi$
$733$	−19.2105	−0.709556	−0.354778	+	0.934951i	$0.615444\pi$
$734$	0	0
$735$	−11.9870	−0.442145
$736$	0	0
$737$	−13.7675	−0.507132
$738$	0	0
$739$	29.7110	1.09294	0.546469	−	0.837480i	$-0.315972\pi$
$739$	29.7110	1.09294	0.546469	+	0.837480i	$0.315972\pi$
$740$	0	0
$741$	−12.1258	−0.445454
$742$	0	0
$743$	7.71120	0.282896	0.141448	−	0.989946i	$-0.454824\pi$
$743$	7.71120	0.282896	0.141448	+	0.989946i	$0.454824\pi$
$744$	0	0
$745$	−24.6814	−0.904257
$746$	0	0
$747$	0.357469	0.0130791
$748$	0	0
$749$	12.9084	0.471662
$750$	0	0
$751$	33.3681	1.21762	0.608809	−	0.793317i	$-0.291647\pi$
$751$	33.3681	1.21762	0.608809	+	0.793317i	$0.291647\pi$
$752$	0	0
$753$	−25.5281	−0.930295
$754$	0	0
$755$	−4.09589	−0.149065
$756$	0	0
$757$	−45.7593	−1.66315	−0.831575	−	0.555412i	$-0.812561\pi$
$757$	−45.7593	−1.66315	−0.831575	+	0.555412i	$0.812561\pi$
$758$	0	0
$759$	11.3573	0.412242
$760$	0	0
$761$	35.9488	1.30314	0.651572	−	0.758587i	$-0.274110\pi$
$761$	35.9488	1.30314	0.651572	+	0.758587i	$0.274110\pi$
$762$	0	0
$763$	−2.94457	−0.106601
$764$	0	0
$765$	−0.671211	−0.0242677
$766$	0	0
$767$	−14.8363	−0.535708
$768$	0	0
$769$	−20.5476	−0.740964	−0.370482	−	0.928840i	$-0.620807\pi$
$769$	−20.5476	−0.740964	−0.370482	+	0.928840i	$0.620807\pi$
$770$	0	0
$771$	−10.1324	−0.364908
$772$	0	0
$773$	−11.1888	−0.402434	−0.201217	−	0.979547i	$-0.564490\pi$
$773$	−11.1888	−0.402434	−0.201217	+	0.979547i	$0.564490\pi$
$774$	0	0
$775$	0.212698	0.00764036
$776$	0	0
$777$	−25.7806	−0.924874
$778$	0	0
$779$	−9.40485	−0.336964
$780$	0	0
$781$	1.14707	0.0410453
$782$	0	0
$783$	−32.6945	−1.16841
$784$	0	0
$785$	5.26852	0.188041
$786$	0	0
$787$	24.3619	0.868409	0.434205	−	0.900814i	$-0.357029\pi$
$787$	24.3619	0.868409	0.434205	+	0.900814i	$0.357029\pi$
$788$	0	0
$789$	28.8061	1.02552
$790$	0	0
$791$	6.97820	0.248116
$792$	0	0
$793$	−61.8380	−2.19593
$794$	0	0
$795$	16.6645	0.591028
$796$	0	0
$797$	−20.6680	−0.732097	−0.366048	−	0.930596i	$-0.619290\pi$
$797$	−20.6680	−0.732097	−0.366048	+	0.930596i	$0.619290\pi$
$798$	0	0
$799$	−79.3375	−2.80676
$800$	0	0
$801$	0.191391	0.00676246
$802$	0	0
$803$	6.04471	0.213313
$804$	0	0
$805$	−28.9282	−1.01958
$806$	0	0
$807$	−16.5762	−0.583511
$808$	0	0
$809$	−16.4564	−0.578577	−0.289289	−	0.957242i	$-0.593419\pi$
$809$	−16.4564	−0.578577	−0.289289	+	0.957242i	$0.593419\pi$
$810$	0	0
$811$	21.7377	0.763312	0.381656	−	0.924304i	$-0.375354\pi$
$811$	21.7377	0.763312	0.381656	+	0.924304i	$0.375354\pi$
$812$	0	0
$813$	45.1867	1.58477
$814$	0	0
$815$	−42.8105	−1.49959
$816$	0	0
$817$	−6.26919	−0.219331
$818$	0	0
$819$	−0.549315	−0.0191946
$820$	0	0
$821$	2.84811	0.0993997	0.0496998	−	0.998764i	$-0.484174\pi$
$821$	2.84811	0.0993997	0.0496998	+	0.998764i	$0.484174\pi$
$822$	0	0
$823$	−2.30793	−0.0804494	−0.0402247	−	0.999191i	$-0.512807\pi$
$823$	−2.30793	−0.0804494	−0.0402247	+	0.999191i	$0.512807\pi$
$824$	0	0
$825$	−0.0681546	−0.00237284
$826$	0	0
$827$	−40.2836	−1.40080	−0.700398	−	0.713752i	$-0.746994\pi$
$827$	−40.2836	−1.40080	−0.700398	+	0.713752i	$0.746994\pi$
$828$	0	0
$829$	−12.6009	−0.437647	−0.218824	−	0.975764i	$-0.570222\pi$
$829$	−12.6009	−0.437647	−0.218824	+	0.975764i	$0.570222\pi$
$830$	0	0
$831$	−46.6388	−1.61788
$832$	0	0
$833$	23.7989	0.824583
$834$	0	0
$835$	29.0307	1.00465
$836$	0	0
$837$	−28.0825	−0.970674
$838$	0	0
$839$	−10.5744	−0.365068	−0.182534	−	0.983200i	$-0.558430\pi$
$839$	−10.5744	−0.365068	−0.182534	+	0.983200i	$0.558430\pi$
$840$	0	0
$841$	10.0809	0.347617
$842$	0	0
$843$	43.4319	1.49588
$844$	0	0
$845$	−81.6662	−2.80940
$846$	0	0
$847$	1.96772	0.0676117
$848$	0	0
$849$	6.67802	0.229189
$850$	0	0
$851$	50.2636	1.72302
$852$	0	0
$853$	−4.84636	−0.165936	−0.0829681	−	0.996552i	$-0.526440\pi$
$853$	−4.84636	−0.165936	−0.0829681	+	0.996552i	$0.526440\pi$
$854$	0	0
$855$	0.0882223	0.00301714
$856$	0	0
$857$	−40.1185	−1.37042	−0.685211	−	0.728344i	$-0.740290\pi$
$857$	−40.1185	−1.37042	−0.685211	+	0.728344i	$0.740290\pi$
$858$	0	0
$859$	−35.7826	−1.22089	−0.610443	−	0.792060i	$-0.709008\pi$
$859$	−35.7826	−1.22089	−0.610443	+	0.792060i	$0.709008\pi$
$860$	0	0
$861$	31.8412	1.08515
$862$	0	0
$863$	37.8507	1.28845	0.644226	−	0.764835i	$-0.277180\pi$
$863$	37.8507	1.28845	0.644226	+	0.764835i	$0.277180\pi$
$864$	0	0
$865$	−28.7848	−0.978714
$866$	0	0
$867$	−70.3446	−2.38903
$868$	0	0
$869$	−5.68740	−0.192932
$870$	0	0
$871$	97.0270	3.28763
$872$	0	0
$873$	−0.478075	−0.0161804
$874$	0	0
$875$	22.0861	0.746646
$876$	0	0
$877$	19.9970	0.675250	0.337625	−	0.941281i	$-0.390377\pi$
$877$	19.9970	0.675250	0.337625	+	0.941281i	$0.390377\pi$
$878$	0	0
$879$	−15.7616	−0.531626
$880$	0	0
$881$	29.6003	0.997261	0.498631	−	0.866815i	$-0.333836\pi$
$881$	29.6003	0.997261	0.498631	+	0.866815i	$0.333836\pi$
$882$	0	0
$883$	13.3489	0.449225	0.224612	−	0.974448i	$-0.427888\pi$
$883$	13.3489	0.449225	0.224612	+	0.974448i	$0.427888\pi$
$884$	0	0
$885$	−8.06716	−0.271175
$886$	0	0
$887$	25.8492	0.867931	0.433966	−	0.900929i	$-0.357114\pi$
$887$	25.8492	0.867931	0.433966	+	0.900929i	$0.357114\pi$
$888$	0	0
$889$	9.25709	0.310473
$890$	0	0
$891$	8.87960	0.297478
$892$	0	0
$893$	10.4279	0.348957
$894$	0	0
$895$	−16.1542	−0.539975
$896$	0	0
$897$	−80.0407	−2.67248
$898$	0	0
$899$	33.5680	1.11956
$900$	0	0
$901$	−33.0857	−1.10224
$902$	0	0
$903$	21.2251	0.706327
$904$	0	0
$905$	−50.0873	−1.66496
$906$	0	0
$907$	37.2165	1.23575	0.617876	−	0.786276i	$-0.287993\pi$
$907$	37.2165	1.23575	0.617876	+	0.786276i	$0.287993\pi$
$908$	0	0
$909$	−0.677772	−0.0224803
$910$	0	0
$911$	−26.0666	−0.863624	−0.431812	−	0.901964i	$-0.642126\pi$
$911$	−26.0666	−0.863624	−0.431812	+	0.901964i	$0.642126\pi$
$912$	0	0
$913$	9.02439	0.298664
$914$	0	0
$915$	−33.6241	−1.11158
$916$	0	0
$917$	3.01302	0.0994988
$918$	0	0
$919$	−23.4079	−0.772154	−0.386077	−	0.922467i	$-0.626170\pi$
$919$	−23.4079	−0.772154	−0.386077	+	0.922467i	$0.626170\pi$
$920$	0	0
$921$	−12.8202	−0.422440
$922$	0	0
$923$	−8.08400	−0.266088
$924$	0	0
$925$	−0.301631	−0.00991755
$926$	0	0
$927$	−0.451779	−0.0148384
$928$	0	0
$929$	−25.4027	−0.833436	−0.416718	−	0.909036i	$-0.636820\pi$
$929$	−25.4027	−0.833436	−0.416718	+	0.909036i	$0.636820\pi$
$930$	0	0
$931$	−3.12807	−0.102518
$932$	0	0
$933$	−48.3964	−1.58443
$934$	0	0
$935$	−16.9449	−0.554157
$936$	0	0
$937$	47.2309	1.54297	0.771483	−	0.636249i	$-0.219515\pi$
$937$	47.2309	1.54297	0.771483	+	0.636249i	$0.219515\pi$
$938$	0	0
$939$	6.39035	0.208541
$940$	0	0
$941$	−9.11795	−0.297237	−0.148618	−	0.988895i	$-0.547483\pi$
$941$	−9.11795	−0.297237	−0.148618	+	0.988895i	$0.547483\pi$
$942$	0	0
$943$	−62.0799	−2.02160
$944$	0	0
$945$	−22.9200	−0.745587
$946$	0	0
$947$	51.4205	1.67094	0.835471	−	0.549535i	$-0.185195\pi$
$947$	51.4205	1.67094	0.835471	+	0.549535i	$0.185195\pi$
$948$	0	0
$949$	−42.6003	−1.38287
$950$	0	0
$951$	−8.25517	−0.267692
$952$	0	0
$953$	20.6292	0.668247	0.334123	−	0.942529i	$-0.391560\pi$
$953$	20.6292	0.668247	0.334123	+	0.942529i	$0.391560\pi$
$954$	0	0
$955$	3.64544	0.117964
$956$	0	0
$957$	−10.7561	−0.347697
$958$	0	0
$959$	−32.4243	−1.04704
$960$	0	0
$961$	−2.16716	−0.0699083
$962$	0	0
$963$	−0.259854	−0.00837367
$964$	0	0
$965$	−19.3172	−0.621843
$966$	0	0
$967$	31.7119	1.01978	0.509892	−	0.860238i	$-0.329685\pi$
$967$	31.7119	1.01978	0.509892	+	0.860238i	$0.329685\pi$
$968$	0	0
$969$	13.0905	0.420526
$970$	0	0
$971$	−20.6220	−0.661791	−0.330896	−	0.943667i	$-0.607351\pi$
$971$	−20.6220	−0.661791	−0.330896	+	0.943667i	$0.607351\pi$
$972$	0	0
$973$	−5.15521	−0.165269
$974$	0	0
$975$	0.480322	0.0153826
$976$	0	0
$977$	39.1819	1.25354	0.626771	−	0.779204i	$-0.284376\pi$
$977$	39.1819	1.25354	0.626771	+	0.779204i	$0.284376\pi$
$978$	0	0
$979$	4.83170	0.154422
$980$	0	0
$981$	0.0592760	0.00189254
$982$	0	0
$983$	0.917754	0.0292718	0.0146359	−	0.999893i	$-0.495341\pi$
$983$	0.917754	0.0292718	0.0146359	+	0.999893i	$0.495341\pi$
$984$	0	0
$985$	−32.2225	−1.02669
$986$	0	0
$987$	−35.3050	−1.12377
$988$	0	0
$989$	−41.3819	−1.31587
$990$	0	0
$991$	−2.64141	−0.0839070	−0.0419535	−	0.999120i	$-0.513358\pi$
$991$	−2.64141	−0.0839070	−0.0419535	+	0.999120i	$0.513358\pi$
$992$	0	0
$993$	16.2567	0.515891
$994$	0	0
$995$	−19.8862	−0.630436
$996$	0	0
$997$	0.656840	0.0208023	0.0104012	−	0.999946i	$-0.496689\pi$
$997$	0.656840	0.0208023	0.0104012	+	0.999946i	$0.496689\pi$
$998$	0	0
$999$	39.8242	1.25998

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3344.2.a.y.1.1		6
4.3	odd	2		1672.2.a.h.1.6	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1672.2.a.h.1.6	✓	6	4.3	odd	2
3344.2.a.y.1.1		6	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

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

\(f(q)\)	\(=\)	\(q-1.72058 q^{3} -2.22719 q^{5} +1.96772 q^{7} -0.0396114 q^{9} +O(q^{10})\)	\(q-1.72058 q^{3} -2.22719 q^{5} +1.96772 q^{7} -0.0396114 q^{9} -1.00000 q^{11} +7.04754 q^{13} +3.83206 q^{15} -7.60817 q^{17} +1.00000 q^{19} -3.38562 q^{21} +6.60084 q^{23} -0.0396114 q^{25} +5.22989 q^{27} -6.25147 q^{29} -5.36962 q^{31} +1.72058 q^{33} -4.38250 q^{35} +7.61473 q^{37} -12.1258 q^{39} -9.40485 q^{41} -6.26919 q^{43} +0.0882223 q^{45} +10.4279 q^{47} -3.12807 q^{49} +13.0905 q^{51} +4.34870 q^{53} +2.22719 q^{55} -1.72058 q^{57} -2.10518 q^{59} -8.77442 q^{61} -0.0779443 q^{63} -15.6962 q^{65} +13.7675 q^{67} -11.3573 q^{69} -1.14707 q^{71} -6.04471 q^{73} +0.0681546 q^{75} -1.96772 q^{77} +5.68740 q^{79} -8.87960 q^{81} -9.02439 q^{83} +16.9449 q^{85} +10.7561 q^{87} -4.83170 q^{89} +13.8676 q^{91} +9.23885 q^{93} -2.22719 q^{95} +12.0691 q^{97} +0.0396114 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 4 q^{3} - 4 q^{5} + 4 q^{7} + 2 q^{9}+O(q^{10}) \) 6 * q + 4 * q^3 - 4 * q^5 + 4 * q^7 + 2 * q^9	\( 6 q + 4 q^{3} - 4 q^{5} + 4 q^{7} + 2 q^{9} - 6 q^{11} + 2 q^{13} - 10 q^{17} + 6 q^{19} + 10 q^{21} + 14 q^{23} + 2 q^{25} + 16 q^{27} + 6 q^{29} - 4 q^{31} - 4 q^{33} - 2 q^{35} - 4 q^{37} + 2 q^{39} - 8 q^{41} + 26 q^{43} - 2 q^{45} + 24 q^{47} + 20 q^{49} + 14 q^{51} - 8 q^{53} + 4 q^{55} + 4 q^{57} + 4 q^{59} - 6 q^{61} + 40 q^{63} - 34 q^{65} + 44 q^{67} + 8 q^{69} + 16 q^{71} + 12 q^{73} + 28 q^{75} - 4 q^{77} - 6 q^{79} + 10 q^{81} + 28 q^{83} - 10 q^{85} + 24 q^{87} + 64 q^{91} - 14 q^{93} - 4 q^{95} + 4 q^{97} - 2 q^{99}+O(q^{100}) \) 6 * q + 4 * q^3 - 4 * q^5 + 4 * q^7 + 2 * q^9 - 6 * q^11 + 2 * q^13 - 10 * q^17 + 6 * q^19 + 10 * q^21 + 14 * q^23 + 2 * q^25 + 16 * q^27 + 6 * q^29 - 4 * q^31 - 4 * q^33 - 2 * q^35 - 4 * q^37 + 2 * q^39 - 8 * q^41 + 26 * q^43 - 2 * q^45 + 24 * q^47 + 20 * q^49 + 14 * q^51 - 8 * q^53 + 4 * q^55 + 4 * q^57 + 4 * q^59 - 6 * q^61 + 40 * q^63 - 34 * q^65 + 44 * q^67 + 8 * q^69 + 16 * q^71 + 12 * q^73 + 28 * q^75 - 4 * q^77 - 6 * q^79 + 10 * q^81 + 28 * q^83 - 10 * q^85 + 24 * q^87 + 64 * q^91 - 14 * q^93 - 4 * q^95 + 4 * q^97 - 2 * q^99

Introduction

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