LMFDB - Embedded newform 3344.2.a.u.1.3

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:	$5$
Coefficient field:	5.5.3979184.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 9x^{3} + 18x^{2} + 5x - 12 $ x^5 - 2x^4 - 9x^3 + 18x^2 + 5x - 12
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			1.3
Root			$2.94407$ 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)

$f(q)$	$=$	$q+0.576583 q^{3} +3.94407 q^{5} +0.944071 q^{7} -2.66755 q^{9} +O(q^{10})$	$q+0.576583 q^{3} +3.94407 q^{5} +0.944071 q^{7} -2.66755 q^{9} +1.00000 q^{11} -1.84440 q^{13} +2.27409 q^{15} +2.00000 q^{17} -1.00000 q^{19} +0.544335 q^{21} -2.72348 q^{23} +10.5557 q^{25} -3.26782 q^{27} +7.54190 q^{29} -4.51592 q^{31} +0.576583 q^{33} +3.72348 q^{35} +6.61162 q^{37} -1.06345 q^{39} +11.4907 q^{41} -3.51438 q^{43} -10.5210 q^{45} +9.33510 q^{47} -6.10873 q^{49} +1.15317 q^{51} +2.00000 q^{53} +3.94407 q^{55} -0.576583 q^{57} +8.85346 q^{59} +13.9582 q^{61} -2.51836 q^{63} -7.27444 q^{65} -3.92039 q^{67} -1.57031 q^{69} +15.1303 q^{71} +6.54817 q^{73} +6.08624 q^{75} +0.944071 q^{77} +3.88814 q^{79} +6.11849 q^{81} -10.3044 q^{83} +7.88814 q^{85} +4.34854 q^{87} +6.23034 q^{89} -1.74124 q^{91} -2.60381 q^{93} -3.94407 q^{95} -16.4304 q^{97} -2.66755 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - q^{3} + 7 q^{5} - 8 q^{7} + 8 q^{9}+O(q^{10}) $ 5 * q - q^3 + 7 * q^5 - 8 * q^7 + 8 * q^9	$ 5 q - q^{3} + 7 q^{5} - 8 q^{7} + 8 q^{9} + 5 q^{11} + 4 q^{13} - q^{15} + 10 q^{17} - 5 q^{19} + 2 q^{21} - 5 q^{23} + 6 q^{25} - 7 q^{27} + 16 q^{29} - q^{31} - q^{33} + 10 q^{35} - q^{37} + 4 q^{39} + 28 q^{41} - 4 q^{43} + 12 q^{45} + 4 q^{47} - q^{49} - 2 q^{51} + 10 q^{53} + 7 q^{55} + q^{57} + q^{59} - 16 q^{61} - 12 q^{63} + 24 q^{65} + 9 q^{67} - 7 q^{69} - 7 q^{71} + 8 q^{73} + 8 q^{75} - 8 q^{77} - 6 q^{79} + 5 q^{81} - 4 q^{83} + 14 q^{85} + 2 q^{87} + 31 q^{89} + 12 q^{91} - 7 q^{93} - 7 q^{95} + 13 q^{97} + 8 q^{99}+O(q^{100}) $ 5 * q - q^3 + 7 * q^5 - 8 * q^7 + 8 * q^9 + 5 * q^11 + 4 * q^13 - q^15 + 10 * q^17 - 5 * q^19 + 2 * q^21 - 5 * q^23 + 6 * q^25 - 7 * q^27 + 16 * q^29 - q^31 - q^33 + 10 * q^35 - q^37 + 4 * q^39 + 28 * q^41 - 4 * q^43 + 12 * q^45 + 4 * q^47 - q^49 - 2 * q^51 + 10 * q^53 + 7 * q^55 + q^57 + q^59 - 16 * q^61 - 12 * q^63 + 24 * q^65 + 9 * q^67 - 7 * q^69 - 7 * q^71 + 8 * q^73 + 8 * q^75 - 8 * q^77 - 6 * q^79 + 5 * q^81 - 4 * q^83 + 14 * q^85 + 2 * q^87 + 31 * q^89 + 12 * q^91 - 7 * q^93 - 7 * q^95 + 13 * q^97 + 8 * 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
$3$	0.576583	0.332891	0.166445	−	0.986051i	$-0.446771\pi$
$3$	0.576583	0.332891	0.166445	+	0.986051i	$0.446771\pi$
$4$	0	0
$5$	3.94407	1.76384	0.881921	−	0.471397i	$-0.156250\pi$
$5$	3.94407	1.76384	0.881921	+	0.471397i	$0.156250\pi$
$6$	0	0
$7$	0.944071	0.356825	0.178413	−	0.983956i	$-0.442904\pi$
$7$	0.944071	0.356825	0.178413	+	0.983956i	$0.442904\pi$
$8$	0	0
$9$	−2.66755	−0.889184
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	−1.84440	−0.511544	−0.255772	−	0.966737i	$-0.582330\pi$
$13$	−1.84440	−0.511544	−0.255772	+	0.966737i	$0.582330\pi$
$14$	0	0
$15$	2.27409	0.587166
$16$	0	0
$17$	2.00000	0.485071	0.242536	−	0.970143i	$-0.422021\pi$
$17$	2.00000	0.485071	0.242536	+	0.970143i	$0.422021\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	0.544335	0.118784
$22$	0	0
$23$	−2.72348	−0.567885	−0.283943	−	0.958841i	$-0.591642\pi$
$23$	−2.72348	−0.567885	−0.283943	+	0.958841i	$0.591642\pi$
$24$	0	0
$25$	10.5557	2.11114
$26$	0	0
$27$	−3.26782	−0.628892
$28$	0	0
$29$	7.54190	1.40050	0.700248	−	0.713900i	$-0.253073\pi$
$29$	7.54190	1.40050	0.700248	+	0.713900i	$0.253073\pi$
$30$	0	0
$31$	−4.51592	−0.811084	−0.405542	−	0.914076i	$-0.632917\pi$
$31$	−4.51592	−0.811084	−0.405542	+	0.914076i	$0.632917\pi$
$32$	0	0
$33$	0.576583	0.100370
$34$	0	0
$35$	3.72348	0.629383
$36$	0	0
$37$	6.61162	1.08694	0.543472	−	0.839427i	$-0.317109\pi$
$37$	6.61162	1.08694	0.543472	+	0.839427i	$0.317109\pi$
$38$	0	0
$39$	−1.06345	−0.170288
$40$	0	0
$41$	11.4907	1.79455	0.897273	−	0.441476i	$-0.145545\pi$
$41$	11.4907	1.79455	0.897273	+	0.441476i	$0.145545\pi$
$42$	0	0
$43$	−3.51438	−0.535939	−0.267969	−	0.963427i	$-0.586353\pi$
$43$	−3.51438	−0.535939	−0.267969	+	0.963427i	$0.586353\pi$
$44$	0	0
$45$	−10.5210	−1.56838
$46$	0	0
$47$	9.33510	1.36166	0.680832	−	0.732439i	$-0.261618\pi$
$47$	9.33510	1.36166	0.680832	+	0.732439i	$0.261618\pi$
$48$	0	0
$49$	−6.10873	−0.872676
$50$	0	0
$51$	1.15317	0.161476
$52$	0	0
$53$	2.00000	0.274721	0.137361	−	0.990521i	$-0.456138\pi$
$53$	2.00000	0.274721	0.137361	+	0.990521i	$0.456138\pi$
$54$	0	0
$55$	3.94407	0.531818
$56$	0	0
$57$	−0.576583	−0.0763703
$58$	0	0
$59$	8.85346	1.15262	0.576311	−	0.817230i	$-0.304492\pi$
$59$	8.85346	1.15262	0.576311	+	0.817230i	$0.304492\pi$
$60$	0	0
$61$	13.9582	1.78717	0.893583	−	0.448897i	$-0.148183\pi$
$61$	13.9582	1.78717	0.893583	+	0.448897i	$0.148183\pi$
$62$	0	0
$63$	−2.51836	−0.317283
$64$	0	0
$65$	−7.27444	−0.902284
$66$	0	0
$67$	−3.92039	−0.478952	−0.239476	−	0.970902i	$-0.576976\pi$
$67$	−3.92039	−0.478952	−0.239476	+	0.970902i	$0.576976\pi$
$68$	0	0
$69$	−1.57031	−0.189044
$70$	0	0
$71$	15.1303	1.79564	0.897820	−	0.440362i	$-0.145150\pi$
$71$	15.1303	1.79564	0.897820	+	0.440362i	$0.145150\pi$
$72$	0	0
$73$	6.54817	0.766406	0.383203	−	0.923664i	$-0.374821\pi$
$73$	6.54817	0.766406	0.383203	+	0.923664i	$0.374821\pi$
$74$	0	0
$75$	6.08624	0.702778
$76$	0	0
$77$	0.944071	0.107587
$78$	0	0
$79$	3.88814	0.437450	0.218725	−	0.975787i	$-0.429810\pi$
$79$	3.88814	0.437450	0.218725	+	0.975787i	$0.429810\pi$
$80$	0	0
$81$	6.11849	0.679832
$82$	0	0
$83$	−10.3044	−1.13105	−0.565527	−	0.824730i	$-0.691327\pi$
$83$	−10.3044	−1.13105	−0.565527	+	0.824730i	$0.691327\pi$
$84$	0	0
$85$	7.88814	0.855589
$86$	0	0
$87$	4.34854	0.466212
$88$	0	0
$89$	6.23034	0.660415	0.330208	−	0.943908i	$-0.392881\pi$
$89$	6.23034	0.660415	0.330208	+	0.943908i	$0.392881\pi$
$90$	0	0
$91$	−1.74124	−0.182532
$92$	0	0
$93$	−2.60381	−0.270002
$94$	0	0
$95$	−3.94407	−0.404653
$96$	0	0
$97$	−16.4304	−1.66825	−0.834127	−	0.551572i	$-0.814028\pi$
$97$	−16.4304	−1.66825	−0.834127	+	0.551572i	$0.814028\pi$
$98$	0	0
$99$	−2.66755	−0.268099
$100$	0	0
$101$	−15.3532	−1.52770	−0.763851	−	0.645392i	$-0.776694\pi$
$101$	−15.3532	−1.52770	−0.763851	+	0.645392i	$0.776694\pi$
$102$	0	0
$103$	−4.80693	−0.473641	−0.236820	−	0.971553i	$-0.576105\pi$
$103$	−4.80693	−0.473641	−0.236820	+	0.971553i	$0.576105\pi$
$104$	0	0
$105$	2.14690	0.209516
$106$	0	0
$107$	−17.2232	−1.66503	−0.832517	−	0.554000i	$-0.813101\pi$
$107$	−17.2232	−1.66503	−0.832517	+	0.554000i	$0.813101\pi$
$108$	0	0
$109$	1.75816	0.168401	0.0842007	−	0.996449i	$-0.473166\pi$
$109$	1.75816	0.168401	0.0842007	+	0.996449i	$0.473166\pi$
$110$	0	0
$111$	3.81215	0.361833
$112$	0	0
$113$	6.53668	0.614919	0.307459	−	0.951561i	$-0.400521\pi$
$113$	6.53668	0.614919	0.307459	+	0.951561i	$0.400521\pi$
$114$	0	0
$115$	−10.7416	−1.00166
$116$	0	0
$117$	4.92003	0.454857
$118$	0	0
$119$	1.88814	0.173086
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	6.62535	0.597388
$124$	0	0
$125$	21.9120	1.95987
$126$	0	0
$127$	11.8833	1.05447	0.527235	−	0.849719i	$-0.323229\pi$
$127$	11.8833	1.05447	0.527235	+	0.849719i	$0.323229\pi$
$128$	0	0
$129$	−2.02634	−0.178409
$130$	0	0
$131$	18.4691	1.61365	0.806824	−	0.590791i	$-0.201184\pi$
$131$	18.4691	1.61365	0.806824	+	0.590791i	$0.201184\pi$
$132$	0	0
$133$	−0.944071	−0.0818613
$134$	0	0
$135$	−12.8885	−1.10927
$136$	0	0
$137$	−7.66755	−0.655083	−0.327542	−	0.944837i	$-0.606220\pi$
$137$	−7.66755	−0.655083	−0.327542	+	0.944837i	$0.606220\pi$
$138$	0	0
$139$	−14.1500	−1.20019	−0.600095	−	0.799929i	$-0.704871\pi$
$139$	−14.1500	−1.20019	−0.600095	+	0.799929i	$0.704871\pi$
$140$	0	0
$141$	5.38247	0.453285
$142$	0	0
$143$	−1.84440	−0.154236
$144$	0	0
$145$	29.7458	2.47025
$146$	0	0
$147$	−3.52219	−0.290506
$148$	0	0
$149$	−10.7301	−0.879045	−0.439522	−	0.898232i	$-0.644852\pi$
$149$	−10.7301	−0.879045	−0.439522	+	0.898232i	$0.644852\pi$
$150$	0	0
$151$	−10.7301	−0.873204	−0.436602	−	0.899655i	$-0.643818\pi$
$151$	−10.7301	−0.873204	−0.436602	+	0.899655i	$0.643818\pi$
$152$	0	0
$153$	−5.33510	−0.431318
$154$	0	0
$155$	−17.8111	−1.43062
$156$	0	0
$157$	−7.98657	−0.637397	−0.318699	−	0.947856i	$-0.603246\pi$
$157$	−7.98657	−0.637397	−0.318699	+	0.947856i	$0.603246\pi$
$158$	0	0
$159$	1.15317	0.0914521
$160$	0	0
$161$	−2.57116	−0.202636
$162$	0	0
$163$	6.97681	0.546466	0.273233	−	0.961948i	$-0.411907\pi$
$163$	6.97681	0.546466	0.273233	+	0.961948i	$0.411907\pi$
$164$	0	0
$165$	2.27409	0.177037
$166$	0	0
$167$	−3.63684	−0.281427	−0.140714	−	0.990050i	$-0.544940\pi$
$167$	−3.63684	−0.281427	−0.140714	+	0.990050i	$0.544940\pi$
$168$	0	0
$169$	−9.59819	−0.738322
$170$	0	0
$171$	2.66755	0.203993
$172$	0	0
$173$	−3.60640	−0.274189	−0.137095	−	0.990558i	$-0.543776\pi$
$173$	−3.60640	−0.274189	−0.137095	+	0.990558i	$0.543776\pi$
$174$	0	0
$175$	9.96532	0.753307
$176$	0	0
$177$	5.10476	0.383697
$178$	0	0
$179$	−19.3855	−1.44894	−0.724469	−	0.689307i	$-0.757915\pi$
$179$	−19.3855	−1.44894	−0.724469	+	0.689307i	$0.757915\pi$
$180$	0	0
$181$	−23.3543	−1.73591	−0.867955	−	0.496644i	$-0.834566\pi$
$181$	−23.3543	−1.73591	−0.867955	+	0.496644i	$0.834566\pi$
$182$	0	0
$183$	8.04808	0.594931
$184$	0	0
$185$	26.0767	1.91720
$186$	0	0
$187$	2.00000	0.146254
$188$	0	0
$189$	−3.08505	−0.224404
$190$	0	0
$191$	2.67152	0.193305	0.0966523	−	0.995318i	$-0.469187\pi$
$191$	2.67152	0.193305	0.0966523	+	0.995318i	$0.469187\pi$
$192$	0	0
$193$	9.79628	0.705152	0.352576	−	0.935783i	$-0.385306\pi$
$193$	9.79628	0.705152	0.352576	+	0.935783i	$0.385306\pi$
$194$	0	0
$195$	−4.19432	−0.300362
$196$	0	0
$197$	5.82365	0.414918	0.207459	−	0.978244i	$-0.433481\pi$
$197$	5.82365	0.414918	0.207459	+	0.978244i	$0.433481\pi$
$198$	0	0
$199$	−22.8127	−1.61715	−0.808576	−	0.588392i	$-0.799761\pi$
$199$	−22.8127	−1.61715	−0.808576	+	0.588392i	$0.799761\pi$
$200$	0	0
$201$	−2.26043	−0.159439
$202$	0	0
$203$	7.12009	0.499732
$204$	0	0
$205$	45.3201	3.16530
$206$	0	0
$207$	7.26503	0.504954
$208$	0	0
$209$	−1.00000	−0.0691714
$210$	0	0
$211$	−3.17129	−0.218320	−0.109160	−	0.994024i	$-0.534816\pi$
$211$	−3.17129	−0.218320	−0.109160	+	0.994024i	$0.534816\pi$
$212$	0	0
$213$	8.72390	0.597752
$214$	0	0
$215$	−13.8610	−0.945311
$216$	0	0
$217$	−4.26335	−0.289415
$218$	0	0
$219$	3.77557	0.255129
$220$	0	0
$221$	−3.68880	−0.248136
$222$	0	0
$223$	−23.0400	−1.54287	−0.771436	−	0.636307i	$-0.780461\pi$
$223$	−23.0400	−1.54287	−0.771436	+	0.636307i	$0.780461\pi$
$224$	0	0
$225$	−28.1579	−1.87719
$226$	0	0
$227$	14.9169	0.990070	0.495035	−	0.868873i	$-0.335155\pi$
$227$	14.9169	0.990070	0.495035	+	0.868873i	$0.335155\pi$
$228$	0	0
$229$	−24.7915	−1.63827	−0.819133	−	0.573603i	$-0.805545\pi$
$229$	−24.7915	−1.63827	−0.819133	+	0.573603i	$0.805545\pi$
$230$	0	0
$231$	0.544335	0.0358146
$232$	0	0
$233$	−18.4939	−1.21157	−0.605786	−	0.795627i	$-0.707141\pi$
$233$	−18.4939	−1.21157	−0.605786	+	0.795627i	$0.707141\pi$
$234$	0	0
$235$	36.8183	2.40176
$236$	0	0
$237$	2.24184	0.145623
$238$	0	0
$239$	12.7282	0.823317	0.411658	−	0.911338i	$-0.364950\pi$
$239$	12.7282	0.823317	0.411658	+	0.911338i	$0.364950\pi$
$240$	0	0
$241$	16.9425	1.09136	0.545681	−	0.837993i	$-0.316271\pi$
$241$	16.9425	1.09136	0.545681	+	0.837993i	$0.316271\pi$
$242$	0	0
$243$	13.3313	0.855201
$244$	0	0
$245$	−24.0933	−1.53926
$246$	0	0
$247$	1.84440	0.117356
$248$	0	0
$249$	−5.94134	−0.376517
$250$	0	0
$251$	−3.46153	−0.218490	−0.109245	−	0.994015i	$-0.534843\pi$
$251$	−3.46153	−0.218490	−0.109245	+	0.994015i	$0.534843\pi$
$252$	0	0
$253$	−2.72348	−0.171224
$254$	0	0
$255$	4.54817	0.284818
$256$	0	0
$257$	6.86495	0.428224	0.214112	−	0.976809i	$-0.431314\pi$
$257$	6.86495	0.428224	0.214112	+	0.976809i	$0.431314\pi$
$258$	0	0
$259$	6.24184	0.387849
$260$	0	0
$261$	−20.1184	−1.24530
$262$	0	0
$263$	15.1481	0.934075	0.467037	−	0.884238i	$-0.345321\pi$
$263$	15.1481	0.934075	0.467037	+	0.884238i	$0.345321\pi$
$264$	0	0
$265$	7.88814	0.484565
$266$	0	0
$267$	3.59231	0.219846
$268$	0	0
$269$	9.04617	0.551555	0.275777	−	0.961222i	$-0.411065\pi$
$269$	9.04617	0.551555	0.275777	+	0.961222i	$0.411065\pi$
$270$	0	0
$271$	−19.2040	−1.16656	−0.583281	−	0.812271i	$-0.698231\pi$
$271$	−19.2040	−1.16656	−0.583281	+	0.812271i	$0.698231\pi$
$272$	0	0
$273$	−1.00397	−0.0607632
$274$	0	0
$275$	10.5557	0.636532
$276$	0	0
$277$	−3.78115	−0.227187	−0.113594	−	0.993527i	$-0.536236\pi$
$277$	−3.78115	−0.227187	−0.113594	+	0.993527i	$0.536236\pi$
$278$	0	0
$279$	12.0465	0.721203
$280$	0	0
$281$	15.9106	0.949149	0.474575	−	0.880215i	$-0.342602\pi$
$281$	15.9106	0.949149	0.474575	+	0.880215i	$0.342602\pi$
$282$	0	0
$283$	22.0680	1.31181	0.655903	−	0.754845i	$-0.272288\pi$
$283$	22.0680	1.31181	0.655903	+	0.754845i	$0.272288\pi$
$284$	0	0
$285$	−2.27409	−0.134705
$286$	0	0
$287$	10.8480	0.640339
$288$	0	0
$289$	−13.0000	−0.764706
$290$	0	0
$291$	−9.47350	−0.555346
$292$	0	0
$293$	13.7420	0.802816	0.401408	−	0.915899i	$-0.368521\pi$
$293$	13.7420	0.802816	0.401408	+	0.915899i	$0.368521\pi$
$294$	0	0
$295$	34.9187	2.03304
$296$	0	0
$297$	−3.26782	−0.189618
$298$	0	0
$299$	5.02319	0.290498
$300$	0	0
$301$	−3.31783	−0.191236
$302$	0	0
$303$	−8.85241	−0.508558
$304$	0	0
$305$	55.0522	3.15228
$306$	0	0
$307$	18.9074	1.07911	0.539553	−	0.841952i	$-0.318593\pi$
$307$	18.9074	1.07911	0.539553	+	0.841952i	$0.318593\pi$
$308$	0	0
$309$	−2.77160	−0.157671
$310$	0	0
$311$	11.2232	0.636412	0.318206	−	0.948022i	$-0.396920\pi$
$311$	11.2232	0.636412	0.318206	+	0.948022i	$0.396920\pi$
$312$	0	0
$313$	−30.6972	−1.73511	−0.867553	−	0.497344i	$-0.834309\pi$
$313$	−30.6972	−1.73511	−0.867553	+	0.497344i	$0.834309\pi$
$314$	0	0
$315$	−9.93258	−0.559637
$316$	0	0
$317$	8.24539	0.463107	0.231554	−	0.972822i	$-0.425619\pi$
$317$	8.24539	0.463107	0.231554	+	0.972822i	$0.425619\pi$
$318$	0	0
$319$	7.54190	0.422265
$320$	0	0
$321$	−9.93064	−0.554274
$322$	0	0
$323$	−2.00000	−0.111283
$324$	0	0
$325$	−19.4689	−1.07994
$326$	0	0
$327$	1.01373	0.0560592
$328$	0	0
$329$	8.81300	0.485876
$330$	0	0
$331$	2.35773	0.129593	0.0647963	−	0.997899i	$-0.479360\pi$
$331$	2.35773	0.129593	0.0647963	+	0.997899i	$0.479360\pi$
$332$	0	0
$333$	−17.6368	−0.966493
$334$	0	0
$335$	−15.4623	−0.844795
$336$	0	0
$337$	−15.2025	−0.828132	−0.414066	−	0.910247i	$-0.635892\pi$
$337$	−15.2025	−0.828132	−0.414066	+	0.910247i	$0.635892\pi$
$338$	0	0
$339$	3.76894	0.204701
$340$	0	0
$341$	−4.51592	−0.244551
$342$	0	0
$343$	−12.3756	−0.668218
$344$	0	0
$345$	−6.19343	−0.333443
$346$	0	0
$347$	5.20026	0.279164	0.139582	−	0.990210i	$-0.455424\pi$
$347$	5.20026	0.279164	0.139582	+	0.990210i	$0.455424\pi$
$348$	0	0
$349$	0.350619	0.0187682	0.00938409	−	0.999956i	$-0.497013\pi$
$349$	0.350619	0.0187682	0.00938409	+	0.999956i	$0.497013\pi$
$350$	0	0
$351$	6.02716	0.321706
$352$	0	0
$353$	−10.9103	−0.580696	−0.290348	−	0.956921i	$-0.593771\pi$
$353$	−10.9103	−0.580696	−0.290348	+	0.956921i	$0.593771\pi$
$354$	0	0
$355$	59.6751	3.16723
$356$	0	0
$357$	1.08867	0.0576186
$358$	0	0
$359$	14.9441	0.788718	0.394359	−	0.918956i	$-0.370967\pi$
$359$	14.9441	0.788718	0.394359	+	0.918956i	$0.370967\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	0.576583	0.0302628
$364$	0	0
$365$	25.8265	1.35182
$366$	0	0
$367$	−24.3109	−1.26902	−0.634509	−	0.772916i	$-0.718797\pi$
$367$	−24.3109	−1.26902	−0.634509	+	0.772916i	$0.718797\pi$
$368$	0	0
$369$	−30.6520	−1.59568
$370$	0	0
$371$	1.88814	0.0980274
$372$	0	0
$373$	5.37014	0.278055	0.139028	−	0.990288i	$-0.455602\pi$
$373$	5.37014	0.278055	0.139028	+	0.990288i	$0.455602\pi$
$374$	0	0
$375$	12.6341	0.652423
$376$	0	0
$377$	−13.9103	−0.716416
$378$	0	0
$379$	32.8893	1.68941	0.844704	−	0.535234i	$-0.179777\pi$
$379$	32.8893	1.68941	0.844704	+	0.535234i	$0.179777\pi$
$380$	0	0
$381$	6.85170	0.351023
$382$	0	0
$383$	−8.01484	−0.409539	−0.204770	−	0.978810i	$-0.565645\pi$
$383$	−8.01484	−0.409539	−0.204770	+	0.978810i	$0.565645\pi$
$384$	0	0
$385$	3.72348	0.189766
$386$	0	0
$387$	9.37480	0.476548
$388$	0	0
$389$	−16.0016	−0.811314	−0.405657	−	0.914025i	$-0.632957\pi$
$389$	−16.0016	−0.811314	−0.405657	+	0.914025i	$0.632957\pi$
$390$	0	0
$391$	−5.44696	−0.275465
$392$	0	0
$393$	10.6490	0.537169
$394$	0	0
$395$	15.3351	0.771593
$396$	0	0
$397$	−11.7310	−0.588762	−0.294381	−	0.955688i	$-0.595113\pi$
$397$	−11.7310	−0.588762	−0.294381	+	0.955688i	$0.595113\pi$
$398$	0	0
$399$	−0.544335	−0.0272509
$400$	0	0
$401$	−1.98955	−0.0993535	−0.0496767	−	0.998765i	$-0.515819\pi$
$401$	−1.98955	−0.0993535	−0.0496767	+	0.998765i	$0.515819\pi$
$402$	0	0
$403$	8.32917	0.414905
$404$	0	0
$405$	24.1317	1.19912
$406$	0	0
$407$	6.61162	0.327726
$408$	0	0
$409$	−1.55931	−0.0771028	−0.0385514	−	0.999257i	$-0.512274\pi$
$409$	−1.55931	−0.0771028	−0.0385514	+	0.999257i	$0.512274\pi$
$410$	0	0
$411$	−4.42098	−0.218071
$412$	0	0
$413$	8.35829	0.411285
$414$	0	0
$415$	−40.6413	−1.99500
$416$	0	0
$417$	−8.15868	−0.399532
$418$	0	0
$419$	10.5781	0.516775	0.258388	−	0.966041i	$-0.416809\pi$
$419$	10.5781	0.516775	0.258388	+	0.966041i	$0.416809\pi$
$420$	0	0
$421$	−16.8570	−0.821560	−0.410780	−	0.911734i	$-0.634744\pi$
$421$	−16.8570	−0.821560	−0.410780	+	0.911734i	$0.634744\pi$
$422$	0	0
$423$	−24.9019	−1.21077
$424$	0	0
$425$	21.1114	1.02405
$426$	0	0
$427$	13.1775	0.637706
$428$	0	0
$429$	−1.06345	−0.0513439
$430$	0	0
$431$	−23.2107	−1.11802	−0.559010	−	0.829161i	$-0.688819\pi$
$431$	−23.2107	−1.11802	−0.559010	+	0.829161i	$0.688819\pi$
$432$	0	0
$433$	26.3454	1.26608	0.633040	−	0.774119i	$-0.281807\pi$
$433$	26.3454	1.26608	0.633040	+	0.774119i	$0.281807\pi$
$434$	0	0
$435$	17.1509	0.822324
$436$	0	0
$437$	2.72348	0.130282
$438$	0	0
$439$	−10.7946	−0.515198	−0.257599	−	0.966252i	$-0.582931\pi$
$439$	−10.7946	−0.515198	−0.257599	+	0.966252i	$0.582931\pi$
$440$	0	0
$441$	16.2954	0.775969
$442$	0	0
$443$	13.8978	0.660307	0.330153	−	0.943927i	$-0.392900\pi$
$443$	13.8978	0.660307	0.330153	+	0.943927i	$0.392900\pi$
$444$	0	0
$445$	24.5729	1.16487
$446$	0	0
$447$	−6.18680	−0.292626
$448$	0	0
$449$	−11.4536	−0.540528	−0.270264	−	0.962786i	$-0.587111\pi$
$449$	−11.4536	−0.540528	−0.270264	+	0.962786i	$0.587111\pi$
$450$	0	0
$451$	11.4907	0.541076
$452$	0	0
$453$	−6.18680	−0.290681
$454$	0	0
$455$	−6.86759	−0.321957
$456$	0	0
$457$	4.65236	0.217628	0.108814	−	0.994062i	$-0.465295\pi$
$457$	4.65236	0.217628	0.108814	+	0.994062i	$0.465295\pi$
$458$	0	0
$459$	−6.53563	−0.305057
$460$	0	0
$461$	33.2885	1.55040	0.775199	−	0.631717i	$-0.217650\pi$
$461$	33.2885	1.55040	0.775199	+	0.631717i	$0.217650\pi$
$462$	0	0
$463$	−32.8667	−1.52745	−0.763723	−	0.645544i	$-0.776631\pi$
$463$	−32.8667	−1.52745	−0.763723	+	0.645544i	$0.776631\pi$
$464$	0	0
$465$	−10.2696	−0.476241
$466$	0	0
$467$	−39.6237	−1.83357	−0.916783	−	0.399386i	$-0.869223\pi$
$467$	−39.6237	−1.83357	−0.916783	+	0.399386i	$0.869223\pi$
$468$	0	0
$469$	−3.70112	−0.170902
$470$	0	0
$471$	−4.60492	−0.212184
$472$	0	0
$473$	−3.51438	−0.161592
$474$	0	0
$475$	−10.5557	−0.484328
$476$	0	0
$477$	−5.33510	−0.244278
$478$	0	0
$479$	24.0750	1.10001	0.550007	−	0.835160i	$-0.314625\pi$
$479$	24.0750	1.10001	0.550007	+	0.835160i	$0.314625\pi$
$480$	0	0
$481$	−12.1945	−0.556020
$482$	0	0
$483$	−1.48249	−0.0674555
$484$	0	0
$485$	−64.8027	−2.94254
$486$	0	0
$487$	−5.45408	−0.247148	−0.123574	−	0.992335i	$-0.539436\pi$
$487$	−5.45408	−0.247148	−0.123574	+	0.992335i	$0.539436\pi$
$488$	0	0
$489$	4.02271	0.181913
$490$	0	0
$491$	2.31708	0.104568	0.0522841	−	0.998632i	$-0.483350\pi$
$491$	2.31708	0.104568	0.0522841	+	0.998632i	$0.483350\pi$
$492$	0	0
$493$	15.0838	0.679340
$494$	0	0
$495$	−10.5210	−0.472884
$496$	0	0
$497$	14.2841	0.640730
$498$	0	0
$499$	19.3256	0.865135	0.432567	−	0.901602i	$-0.357608\pi$
$499$	19.3256	0.865135	0.432567	+	0.901602i	$0.357608\pi$
$500$	0	0
$501$	−2.09694	−0.0936845
$502$	0	0
$503$	−39.9765	−1.78246	−0.891232	−	0.453548i	$-0.850158\pi$
$503$	−39.9765	−1.78246	−0.891232	+	0.453548i	$0.850158\pi$
$504$	0	0
$505$	−60.5542	−2.69463
$506$	0	0
$507$	−5.53416	−0.245781
$508$	0	0
$509$	10.6493	0.472020	0.236010	−	0.971751i	$-0.424160\pi$
$509$	10.6493	0.472020	0.236010	+	0.971751i	$0.424160\pi$
$510$	0	0
$511$	6.18194	0.273473
$512$	0	0
$513$	3.26782	0.144278
$514$	0	0
$515$	−18.9589	−0.835427
$516$	0	0
$517$	9.33510	0.410557
$518$	0	0
$519$	−2.07939	−0.0912751
$520$	0	0
$521$	8.31323	0.364209	0.182105	−	0.983279i	$-0.441709\pi$
$521$	8.31323	0.364209	0.182105	+	0.983279i	$0.441709\pi$
$522$	0	0
$523$	23.3446	1.02079	0.510393	−	0.859941i	$-0.329500\pi$
$523$	23.3446	1.02079	0.510393	+	0.859941i	$0.329500\pi$
$524$	0	0
$525$	5.74584	0.250769
$526$	0	0
$527$	−9.03185	−0.393433
$528$	0	0
$529$	−15.5827	−0.677507
$530$	0	0
$531$	−23.6171	−1.02489
$532$	0	0
$533$	−21.1935	−0.917990
$534$	0	0
$535$	−67.9297	−2.93686
$536$	0	0
$537$	−11.1773	−0.482338
$538$	0	0
$539$	−6.10873	−0.263122
$540$	0	0
$541$	−1.20611	−0.0518548	−0.0259274	−	0.999664i	$-0.508254\pi$
$541$	−1.20611	−0.0518548	−0.0259274	+	0.999664i	$0.508254\pi$
$542$	0	0
$543$	−13.4657	−0.577868
$544$	0	0
$545$	6.93432	0.297033
$546$	0	0
$547$	42.5521	1.81940	0.909698	−	0.415270i	$-0.136313\pi$
$547$	42.5521	1.81940	0.909698	+	0.415270i	$0.136313\pi$
$548$	0	0
$549$	−37.2343	−1.58912
$550$	0	0
$551$	−7.54190	−0.321296
$552$	0	0
$553$	3.67068	0.156093
$554$	0	0
$555$	15.0354	0.638217
$556$	0	0
$557$	−0.214383	−0.00908372	−0.00454186	−	0.999990i	$-0.501446\pi$
$557$	−0.214383	−0.00908372	−0.00454186	+	0.999990i	$0.501446\pi$
$558$	0	0
$559$	6.48193	0.274156
$560$	0	0
$561$	1.15317	0.0486867
$562$	0	0
$563$	−37.4226	−1.57717	−0.788587	−	0.614923i	$-0.789187\pi$
$563$	−37.4226	−1.57717	−0.788587	+	0.614923i	$0.789187\pi$
$564$	0	0
$565$	25.7811	1.08462
$566$	0	0
$567$	5.77628	0.242581
$568$	0	0
$569$	−11.5608	−0.484653	−0.242327	−	0.970195i	$-0.577911\pi$
$569$	−11.5608	−0.484653	−0.242327	+	0.970195i	$0.577911\pi$
$570$	0	0
$571$	4.43281	0.185507	0.0927537	−	0.995689i	$-0.470433\pi$
$571$	4.43281	0.185507	0.0927537	+	0.995689i	$0.470433\pi$
$572$	0	0
$573$	1.54036	0.0643493
$574$	0	0
$575$	−28.7482	−1.19888
$576$	0	0
$577$	19.5014	0.811853	0.405926	−	0.913906i	$-0.366949\pi$
$577$	19.5014	0.811853	0.405926	+	0.913906i	$0.366949\pi$
$578$	0	0
$579$	5.64837	0.234738
$580$	0	0
$581$	−9.72808	−0.403589
$582$	0	0
$583$	2.00000	0.0828315
$584$	0	0
$585$	19.4050	0.802296
$586$	0	0
$587$	−41.8164	−1.72595	−0.862974	−	0.505248i	$-0.831401\pi$
$587$	−41.8164	−1.72595	−0.862974	+	0.505248i	$0.831401\pi$
$588$	0	0
$589$	4.51592	0.186075
$590$	0	0
$591$	3.35782	0.138122
$592$	0	0
$593$	−38.8473	−1.59527	−0.797633	−	0.603143i	$-0.793915\pi$
$593$	−38.8473	−1.59527	−0.797633	+	0.603143i	$0.793915\pi$
$594$	0	0
$595$	7.44696	0.305296
$596$	0	0
$597$	−13.1534	−0.538334
$598$	0	0
$599$	−43.7294	−1.78674	−0.893368	−	0.449326i	$-0.851664\pi$
$599$	−43.7294	−1.78674	−0.893368	+	0.449326i	$0.851664\pi$
$600$	0	0
$601$	−26.4153	−1.07750	−0.538751	−	0.842465i	$-0.681104\pi$
$601$	−26.4153	−1.07750	−0.538751	+	0.842465i	$0.681104\pi$
$602$	0	0
$603$	10.4578	0.425876
$604$	0	0
$605$	3.94407	0.160349
$606$	0	0
$607$	28.7434	1.16666	0.583329	−	0.812236i	$-0.301750\pi$
$607$	28.7434	1.16666	0.583329	+	0.812236i	$0.301750\pi$
$608$	0	0
$609$	4.10532	0.166356
$610$	0	0
$611$	−17.2177	−0.696552
$612$	0	0
$613$	−16.2049	−0.654511	−0.327255	−	0.944936i	$-0.606124\pi$
$613$	−16.2049	−0.654511	−0.327255	+	0.944936i	$0.606124\pi$
$614$	0	0
$615$	26.1308	1.05370
$616$	0	0
$617$	37.9394	1.52738	0.763692	−	0.645580i	$-0.223384\pi$
$617$	37.9394	1.52738	0.763692	+	0.645580i	$0.223384\pi$
$618$	0	0
$619$	24.8511	0.998849	0.499425	−	0.866357i	$-0.333545\pi$
$619$	24.8511	0.998849	0.499425	+	0.866357i	$0.333545\pi$
$620$	0	0
$621$	8.89984	0.357138
$622$	0	0
$623$	5.88188	0.235653
$624$	0	0
$625$	33.6442	1.34577
$626$	0	0
$627$	−0.576583	−0.0230265
$628$	0	0
$629$	13.2232	0.527245
$630$	0	0
$631$	32.4531	1.29194	0.645969	−	0.763364i	$-0.276453\pi$
$631$	32.4531	1.29194	0.645969	+	0.763364i	$0.276453\pi$
$632$	0	0
$633$	−1.82851	−0.0726768
$634$	0	0
$635$	46.8685	1.85992
$636$	0	0
$637$	11.2669	0.446413
$638$	0	0
$639$	−40.3610	−1.59665
$640$	0	0
$641$	−30.6172	−1.20931	−0.604653	−	0.796489i	$-0.706688\pi$
$641$	−30.6172	−1.20931	−0.604653	+	0.796489i	$0.706688\pi$
$642$	0	0
$643$	−39.4693	−1.55652	−0.778259	−	0.627943i	$-0.783897\pi$
$643$	−39.4693	−1.55652	−0.778259	+	0.627943i	$0.783897\pi$
$644$	0	0
$645$	−7.99201	−0.314685
$646$	0	0
$647$	20.1182	0.790929	0.395464	−	0.918481i	$-0.370584\pi$
$647$	20.1182	0.790929	0.395464	+	0.918481i	$0.370584\pi$
$648$	0	0
$649$	8.85346	0.347529
$650$	0	0
$651$	−2.45818	−0.0963436
$652$	0	0
$653$	−30.8490	−1.20722	−0.603608	−	0.797282i	$-0.706271\pi$
$653$	−30.8490	−1.20722	−0.603608	+	0.797282i	$0.706271\pi$
$654$	0	0
$655$	72.8433	2.84622
$656$	0	0
$657$	−17.4676	−0.681475
$658$	0	0
$659$	2.86424	0.111575	0.0557874	−	0.998443i	$-0.482233\pi$
$659$	2.86424	0.111575	0.0557874	+	0.998443i	$0.482233\pi$
$660$	0	0
$661$	−23.7261	−0.922838	−0.461419	−	0.887182i	$-0.652659\pi$
$661$	−23.7261	−0.922838	−0.461419	+	0.887182i	$0.652659\pi$
$662$	0	0
$663$	−2.12690	−0.0826020
$664$	0	0
$665$	−3.72348	−0.144390
$666$	0	0
$667$	−20.5402	−0.795321
$668$	0	0
$669$	−13.2845	−0.513608
$670$	0	0
$671$	13.9582	0.538851
$672$	0	0
$673$	11.6576	0.449367	0.224684	−	0.974432i	$-0.427865\pi$
$673$	11.6576	0.449367	0.224684	+	0.974432i	$0.427865\pi$
$674$	0	0
$675$	−34.4941	−1.32768
$676$	0	0
$677$	−0.672481	−0.0258456	−0.0129228	−	0.999916i	$-0.504114\pi$
$677$	−0.672481	−0.0258456	−0.0129228	+	0.999916i	$0.504114\pi$
$678$	0	0
$679$	−15.5115	−0.595275
$680$	0	0
$681$	8.60084	0.329585
$682$	0	0
$683$	−14.8698	−0.568978	−0.284489	−	0.958679i	$-0.591824\pi$
$683$	−14.8698	−0.568978	−0.284489	+	0.958679i	$0.591824\pi$
$684$	0	0
$685$	−30.2414	−1.15546
$686$	0	0
$687$	−14.2944	−0.545364
$688$	0	0
$689$	−3.68880	−0.140532
$690$	0	0
$691$	32.8347	1.24909	0.624545	−	0.780989i	$-0.285285\pi$
$691$	32.8347	1.24909	0.624545	+	0.780989i	$0.285285\pi$
$692$	0	0
$693$	−2.51836	−0.0956645
$694$	0	0
$695$	−55.8088	−2.11695
$696$	0	0
$697$	22.9814	0.870483
$698$	0	0
$699$	−10.6632	−0.403321
$700$	0	0
$701$	−13.1901	−0.498183	−0.249091	−	0.968480i	$-0.580132\pi$
$701$	−13.1901	−0.498183	−0.249091	+	0.968480i	$0.580132\pi$
$702$	0	0
$703$	−6.61162	−0.249362
$704$	0	0
$705$	21.2288	0.799524
$706$	0	0
$707$	−14.4945	−0.545123
$708$	0	0
$709$	−41.6045	−1.56249	−0.781244	−	0.624225i	$-0.785415\pi$
$709$	−41.6045	−1.56249	−0.781244	+	0.624225i	$0.785415\pi$
$710$	0	0
$711$	−10.3718	−0.388974
$712$	0	0
$713$	12.2990	0.460602
$714$	0	0
$715$	−7.27444	−0.272049
$716$	0	0
$717$	7.33885	0.274074
$718$	0	0
$719$	−20.0217	−0.746682	−0.373341	−	0.927694i	$-0.621788\pi$
$719$	−20.0217	−0.746682	−0.373341	+	0.927694i	$0.621788\pi$
$720$	0	0
$721$	−4.53808	−0.169007
$722$	0	0
$723$	9.76876	0.363304
$724$	0	0
$725$	79.6100	2.95664
$726$	0	0
$727$	−9.40650	−0.348868	−0.174434	−	0.984669i	$-0.555810\pi$
$727$	−9.40650	−0.348868	−0.174434	+	0.984669i	$0.555810\pi$
$728$	0	0
$729$	−10.6689	−0.395143
$730$	0	0
$731$	−7.02877	−0.259968
$732$	0	0
$733$	23.4100	0.864670	0.432335	−	0.901713i	$-0.357690\pi$
$733$	23.4100	0.864670	0.432335	+	0.901713i	$0.357690\pi$
$734$	0	0
$735$	−13.8918	−0.512406
$736$	0	0
$737$	−3.92039	−0.144409
$738$	0	0
$739$	16.0421	0.590117	0.295058	−	0.955479i	$-0.404661\pi$
$739$	16.0421	0.590117	0.295058	+	0.955479i	$0.404661\pi$
$740$	0	0
$741$	1.06345	0.0390668
$742$	0	0
$743$	−34.8640	−1.27903	−0.639517	−	0.768777i	$-0.720866\pi$
$743$	−34.8640	−1.27903	−0.639517	+	0.768777i	$0.720866\pi$
$744$	0	0
$745$	−42.3203	−1.55050
$746$	0	0
$747$	27.4875	1.00572
$748$	0	0
$749$	−16.2600	−0.594126
$750$	0	0
$751$	38.9263	1.42044	0.710221	−	0.703979i	$-0.248595\pi$
$751$	38.9263	1.42044	0.710221	+	0.703979i	$0.248595\pi$
$752$	0	0
$753$	−1.99586	−0.0727333
$754$	0	0
$755$	−42.3203	−1.54019
$756$	0	0
$757$	−29.3914	−1.06825	−0.534124	−	0.845406i	$-0.679359\pi$
$757$	−29.3914	−1.06825	−0.534124	+	0.845406i	$0.679359\pi$
$758$	0	0
$759$	−1.57031	−0.0569988
$760$	0	0
$761$	23.1117	0.837797	0.418898	−	0.908033i	$-0.362416\pi$
$761$	23.1117	0.837797	0.418898	+	0.908033i	$0.362416\pi$
$762$	0	0
$763$	1.65983	0.0600898
$764$	0	0
$765$	−21.0420	−0.760776
$766$	0	0
$767$	−16.3293	−0.589618
$768$	0	0
$769$	−36.4514	−1.31447	−0.657235	−	0.753686i	$-0.728274\pi$
$769$	−36.4514	−1.31447	−0.657235	+	0.753686i	$0.728274\pi$
$770$	0	0
$771$	3.95822	0.142552
$772$	0	0
$773$	−7.48874	−0.269351	−0.134676	−	0.990890i	$-0.542999\pi$
$773$	−7.48874	−0.269351	−0.134676	+	0.990890i	$0.542999\pi$
$774$	0	0
$775$	−47.6687	−1.71231
$776$	0	0
$777$	3.59894	0.129111
$778$	0	0
$779$	−11.4907	−0.411697
$780$	0	0
$781$	15.1303	0.541406
$782$	0	0
$783$	−24.6456	−0.880760
$784$	0	0
$785$	−31.4996	−1.12427
$786$	0	0
$787$	41.0270	1.46245	0.731227	−	0.682134i	$-0.238948\pi$
$787$	41.0270	1.46245	0.731227	+	0.682134i	$0.238948\pi$
$788$	0	0
$789$	8.73417	0.310945
$790$	0	0
$791$	6.17109	0.219419
$792$	0	0
$793$	−25.7445	−0.914215
$794$	0	0
$795$	4.54817	0.161307
$796$	0	0
$797$	45.1503	1.59931	0.799653	−	0.600462i	$-0.205017\pi$
$797$	45.1503	1.59931	0.799653	+	0.600462i	$0.205017\pi$
$798$	0	0
$799$	18.6702	0.660504
$800$	0	0
$801$	−16.6198	−0.587231
$802$	0	0
$803$	6.54817	0.231080
$804$	0	0
$805$	−10.1408	−0.357417
$806$	0	0
$807$	5.21587	0.183607
$808$	0	0
$809$	18.5632	0.652648	0.326324	−	0.945258i	$-0.394190\pi$
$809$	18.5632	0.652648	0.326324	+	0.945258i	$0.394190\pi$
$810$	0	0
$811$	−23.8993	−0.839218	−0.419609	−	0.907705i	$-0.637833\pi$
$811$	−23.8993	−0.839218	−0.419609	+	0.907705i	$0.637833\pi$
$812$	0	0
$813$	−11.0727	−0.388337
$814$	0	0
$815$	27.5170	0.963880
$816$	0	0
$817$	3.51438	0.122953
$818$	0	0
$819$	4.64486	0.162304
$820$	0	0
$821$	−1.31768	−0.0459875	−0.0229937	−	0.999736i	$-0.507320\pi$
$821$	−1.31768	−0.0459875	−0.0229937	+	0.999736i	$0.507320\pi$
$822$	0	0
$823$	−0.536951	−0.0187169	−0.00935846	−	0.999956i	$-0.502979\pi$
$823$	−0.536951	−0.0187169	−0.00935846	+	0.999956i	$0.502979\pi$
$824$	0	0
$825$	6.08624	0.211896
$826$	0	0
$827$	20.0202	0.696171	0.348086	−	0.937463i	$-0.386832\pi$
$827$	20.0202	0.696171	0.348086	+	0.937463i	$0.386832\pi$
$828$	0	0
$829$	−11.2866	−0.391999	−0.195999	−	0.980604i	$-0.562795\pi$
$829$	−11.2866	−0.391999	−0.195999	+	0.980604i	$0.562795\pi$
$830$	0	0
$831$	−2.18015	−0.0756285
$832$	0	0
$833$	−12.2175	−0.423310
$834$	0	0
$835$	−14.3440	−0.496393
$836$	0	0
$837$	14.7572	0.510084
$838$	0	0
$839$	15.8075	0.545736	0.272868	−	0.962052i	$-0.412028\pi$
$839$	15.8075	0.545736	0.272868	+	0.962052i	$0.412028\pi$
$840$	0	0
$841$	27.8803	0.961389
$842$	0	0
$843$	9.17381	0.315963
$844$	0	0
$845$	−37.8559	−1.30228
$846$	0	0
$847$	0.944071	0.0324386
$848$	0	0
$849$	12.7240	0.436688
$850$	0	0
$851$	−18.0066	−0.617259
$852$	0	0
$853$	28.5189	0.976470	0.488235	−	0.872712i	$-0.337641\pi$
$853$	28.5189	0.976470	0.488235	+	0.872712i	$0.337641\pi$
$854$	0	0
$855$	10.5210	0.359811
$856$	0	0
$857$	−40.2246	−1.37405	−0.687023	−	0.726636i	$-0.741083\pi$
$857$	−40.2246	−1.37405	−0.687023	+	0.726636i	$0.741083\pi$
$858$	0	0
$859$	32.9825	1.12535	0.562673	−	0.826679i	$-0.309773\pi$
$859$	32.9825	1.12535	0.562673	+	0.826679i	$0.309773\pi$
$860$	0	0
$861$	6.25480	0.213163
$862$	0	0
$863$	−42.8864	−1.45987	−0.729935	−	0.683516i	$-0.760450\pi$
$863$	−42.8864	−1.45987	−0.729935	+	0.683516i	$0.760450\pi$
$864$	0	0
$865$	−14.2239	−0.483627
$866$	0	0
$867$	−7.49558	−0.254563
$868$	0	0
$869$	3.88814	0.131896
$870$	0	0
$871$	7.23077	0.245005
$872$	0	0
$873$	43.8289	1.48338
$874$	0	0
$875$	20.6865	0.699332
$876$	0	0
$877$	34.4895	1.16463	0.582313	−	0.812964i	$-0.302148\pi$
$877$	34.4895	1.16463	0.582313	+	0.812964i	$0.302148\pi$
$878$	0	0
$879$	7.92341	0.267250
$880$	0	0
$881$	22.6499	0.763096	0.381548	−	0.924349i	$-0.375391\pi$
$881$	22.6499	0.763096	0.381548	+	0.924349i	$0.375391\pi$
$882$	0	0
$883$	0.758636	0.0255301	0.0127651	−	0.999919i	$-0.495937\pi$
$883$	0.758636	0.0255301	0.0127651	+	0.999919i	$0.495937\pi$
$884$	0	0
$885$	20.1335	0.676781
$886$	0	0
$887$	0.899492	0.0302020	0.0151010	−	0.999886i	$-0.495193\pi$
$887$	0.899492	0.0302020	0.0151010	+	0.999886i	$0.495193\pi$
$888$	0	0
$889$	11.2186	0.376261
$890$	0	0
$891$	6.11849	0.204977
$892$	0	0
$893$	−9.33510	−0.312387
$894$	0	0
$895$	−76.4577	−2.55570
$896$	0	0
$897$	2.89629	0.0967042
$898$	0	0
$899$	−34.0587	−1.13592
$900$	0	0
$901$	4.00000	0.133259
$902$	0	0
$903$	−1.91300	−0.0636608
$904$	0	0
$905$	−92.1109	−3.06187
$906$	0	0
$907$	24.6276	0.817745	0.408873	−	0.912591i	$-0.365922\pi$
$907$	24.6276	0.817745	0.408873	+	0.912591i	$0.365922\pi$
$908$	0	0
$909$	40.9555	1.35841
$910$	0	0
$911$	−41.3163	−1.36887	−0.684435	−	0.729074i	$-0.739951\pi$
$911$	−41.3163	−1.36887	−0.684435	+	0.729074i	$0.739951\pi$
$912$	0	0
$913$	−10.3044	−0.341026
$914$	0	0
$915$	31.7422	1.04936
$916$	0	0
$917$	17.4361	0.575790
$918$	0	0
$919$	−31.4385	−1.03706	−0.518530	−	0.855059i	$-0.673521\pi$
$919$	−31.4385	−1.03706	−0.518530	+	0.855059i	$0.673521\pi$
$920$	0	0
$921$	10.9017	0.359224
$922$	0	0
$923$	−27.9064	−0.918550
$924$	0	0
$925$	69.7903	2.29469
$926$	0	0
$927$	12.8227	0.421154
$928$	0	0
$929$	−23.2834	−0.763905	−0.381952	−	0.924182i	$-0.624748\pi$
$929$	−23.2834	−0.763905	−0.381952	+	0.924182i	$0.624748\pi$
$930$	0	0
$931$	6.10873	0.200206
$932$	0	0
$933$	6.47114	0.211856
$934$	0	0
$935$	7.88814	0.257970
$936$	0	0
$937$	44.6180	1.45761	0.728803	−	0.684724i	$-0.240077\pi$
$937$	44.6180	1.45761	0.728803	+	0.684724i	$0.240077\pi$
$938$	0	0
$939$	−17.6995	−0.577601
$940$	0	0
$941$	−6.63079	−0.216158	−0.108079	−	0.994142i	$-0.534470\pi$
$941$	−6.63079	−0.216158	−0.108079	+	0.994142i	$0.534470\pi$
$942$	0	0
$943$	−31.2947	−1.01910
$944$	0	0
$945$	−12.1677	−0.395814
$946$	0	0
$947$	13.5679	0.440899	0.220449	−	0.975398i	$-0.429248\pi$
$947$	13.5679	0.440899	0.220449	+	0.975398i	$0.429248\pi$
$948$	0	0
$949$	−12.0774	−0.392051
$950$	0	0
$951$	4.75415	0.154164
$952$	0	0
$953$	36.4094	1.17942	0.589709	−	0.807616i	$-0.299243\pi$
$953$	36.4094	1.17942	0.589709	+	0.807616i	$0.299243\pi$
$954$	0	0
$955$	10.5367	0.340959
$956$	0	0
$957$	4.34854	0.140568
$958$	0	0
$959$	−7.23871	−0.233750
$960$	0	0
$961$	−10.6064	−0.342143
$962$	0	0
$963$	45.9439	1.48052
$964$	0	0
$965$	38.6372	1.24378
$966$	0	0
$967$	−33.7453	−1.08517	−0.542587	−	0.840000i	$-0.682555\pi$
$967$	−33.7453	−1.08517	−0.542587	+	0.840000i	$0.682555\pi$
$968$	0	0
$969$	−1.15317	−0.0370451
$970$	0	0
$971$	28.5954	0.917671	0.458835	−	0.888521i	$-0.348267\pi$
$971$	28.5954	0.917671	0.458835	+	0.888521i	$0.348267\pi$
$972$	0	0
$973$	−13.3586	−0.428258
$974$	0	0
$975$	−11.2255	−0.359502
$976$	0	0
$977$	3.98083	0.127358	0.0636791	−	0.997970i	$-0.479717\pi$
$977$	3.98083	0.127358	0.0636791	+	0.997970i	$0.479717\pi$
$978$	0	0
$979$	6.23034	0.199123
$980$	0	0
$981$	−4.68999	−0.149740
$982$	0	0
$983$	5.81649	0.185517	0.0927586	−	0.995689i	$-0.470432\pi$
$983$	5.81649	0.185517	0.0927586	+	0.995689i	$0.470432\pi$
$984$	0	0
$985$	22.9689	0.731849
$986$	0	0
$987$	5.08143	0.161744
$988$	0	0
$989$	9.57136	0.304352
$990$	0	0
$991$	42.3003	1.34371	0.671856	−	0.740682i	$-0.265497\pi$
$991$	42.3003	1.34371	0.671856	+	0.740682i	$0.265497\pi$
$992$	0	0
$993$	1.35943	0.0431402
$994$	0	0
$995$	−89.9750	−2.85240
$996$	0	0
$997$	4.07343	0.129007	0.0645034	−	0.997917i	$-0.479454\pi$
$997$	4.07343	0.129007	0.0645034	+	0.997917i	$0.479454\pi$
$998$	0	0
$999$	−21.6056	−0.683570

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3344.2.a.u.1.3		5
4.3	odd	2		1672.2.a.g.1.3	✓	5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1672.2.a.g.1.3	✓	5	4.3	odd	2
3344.2.a.u.1.3		5	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)

\(f(q)\)	\(=\)	\(q+0.576583 q^{3} +3.94407 q^{5} +0.944071 q^{7} -2.66755 q^{9} +O(q^{10})\)	\(q+0.576583 q^{3} +3.94407 q^{5} +0.944071 q^{7} -2.66755 q^{9} +1.00000 q^{11} -1.84440 q^{13} +2.27409 q^{15} +2.00000 q^{17} -1.00000 q^{19} +0.544335 q^{21} -2.72348 q^{23} +10.5557 q^{25} -3.26782 q^{27} +7.54190 q^{29} -4.51592 q^{31} +0.576583 q^{33} +3.72348 q^{35} +6.61162 q^{37} -1.06345 q^{39} +11.4907 q^{41} -3.51438 q^{43} -10.5210 q^{45} +9.33510 q^{47} -6.10873 q^{49} +1.15317 q^{51} +2.00000 q^{53} +3.94407 q^{55} -0.576583 q^{57} +8.85346 q^{59} +13.9582 q^{61} -2.51836 q^{63} -7.27444 q^{65} -3.92039 q^{67} -1.57031 q^{69} +15.1303 q^{71} +6.54817 q^{73} +6.08624 q^{75} +0.944071 q^{77} +3.88814 q^{79} +6.11849 q^{81} -10.3044 q^{83} +7.88814 q^{85} +4.34854 q^{87} +6.23034 q^{89} -1.74124 q^{91} -2.60381 q^{93} -3.94407 q^{95} -16.4304 q^{97} -2.66755 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - q^{3} + 7 q^{5} - 8 q^{7} + 8 q^{9}+O(q^{10}) \) 5 * q - q^3 + 7 * q^5 - 8 * q^7 + 8 * q^9	\( 5 q - q^{3} + 7 q^{5} - 8 q^{7} + 8 q^{9} + 5 q^{11} + 4 q^{13} - q^{15} + 10 q^{17} - 5 q^{19} + 2 q^{21} - 5 q^{23} + 6 q^{25} - 7 q^{27} + 16 q^{29} - q^{31} - q^{33} + 10 q^{35} - q^{37} + 4 q^{39} + 28 q^{41} - 4 q^{43} + 12 q^{45} + 4 q^{47} - q^{49} - 2 q^{51} + 10 q^{53} + 7 q^{55} + q^{57} + q^{59} - 16 q^{61} - 12 q^{63} + 24 q^{65} + 9 q^{67} - 7 q^{69} - 7 q^{71} + 8 q^{73} + 8 q^{75} - 8 q^{77} - 6 q^{79} + 5 q^{81} - 4 q^{83} + 14 q^{85} + 2 q^{87} + 31 q^{89} + 12 q^{91} - 7 q^{93} - 7 q^{95} + 13 q^{97} + 8 q^{99}+O(q^{100}) \) 5 * q - q^3 + 7 * q^5 - 8 * q^7 + 8 * q^9 + 5 * q^11 + 4 * q^13 - q^15 + 10 * q^17 - 5 * q^19 + 2 * q^21 - 5 * q^23 + 6 * q^25 - 7 * q^27 + 16 * q^29 - q^31 - q^33 + 10 * q^35 - q^37 + 4 * q^39 + 28 * q^41 - 4 * q^43 + 12 * q^45 + 4 * q^47 - q^49 - 2 * q^51 + 10 * q^53 + 7 * q^55 + q^57 + q^59 - 16 * q^61 - 12 * q^63 + 24 * q^65 + 9 * q^67 - 7 * q^69 - 7 * q^71 + 8 * q^73 + 8 * q^75 - 8 * q^77 - 6 * q^79 + 5 * q^81 - 4 * q^83 + 14 * q^85 + 2 * q^87 + 31 * q^89 + 12 * q^91 - 7 * q^93 - 7 * q^95 + 13 * q^97 + 8 * q^99

Introduction

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