LMFDB - Embedded newform 3344.2.a.w.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:	$6$
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			1.3
Root			$0.837309$ 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)

$f(q)$	$=$	$q-0.837309 q^{3} +3.44987 q^{5} -0.535976 q^{7} -2.29891 q^{9} +O(q^{10})$	$q-0.837309 q^{3} +3.44987 q^{5} -0.535976 q^{7} -2.29891 q^{9} +1.00000 q^{11} +5.54895 q^{13} -2.88861 q^{15} +1.49993 q^{17} +1.00000 q^{19} +0.448778 q^{21} -3.26301 q^{23} +6.90158 q^{25} +4.43683 q^{27} -3.17227 q^{29} +0.888605 q^{31} -0.837309 q^{33} -1.84905 q^{35} +11.0600 q^{37} -4.64619 q^{39} -8.44868 q^{41} +7.61261 q^{43} -7.93094 q^{45} -3.39981 q^{47} -6.71273 q^{49} -1.25590 q^{51} +7.22511 q^{53} +3.44987 q^{55} -0.837309 q^{57} +1.36560 q^{59} -2.57435 q^{61} +1.23216 q^{63} +19.1431 q^{65} +10.8862 q^{67} +2.73215 q^{69} -2.66509 q^{71} +9.82294 q^{73} -5.77876 q^{75} -0.535976 q^{77} +2.17469 q^{79} +3.18174 q^{81} -4.71288 q^{83} +5.17454 q^{85} +2.65617 q^{87} -4.66282 q^{89} -2.97411 q^{91} -0.744038 q^{93} +3.44987 q^{95} +6.93516 q^{97} -2.29891 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−0.837309	−0.483421	−0.241710	−	0.970348i	$-0.577708\pi$
$3$	−0.837309	−0.483421	−0.241710	+	0.970348i	$0.577708\pi$
$4$	0	0
$5$	3.44987	1.54283	0.771414	−	0.636334i	$-0.219550\pi$
$5$	3.44987	1.54283	0.771414	+	0.636334i	$0.219550\pi$
$6$	0	0
$7$	−0.535976	−0.202580	−0.101290	−	0.994857i	$-0.532297\pi$
$7$	−0.535976	−0.202580	−0.101290	+	0.994857i	$0.532297\pi$
$8$	0	0
$9$	−2.29891	−0.766304
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	5.54895	1.53900	0.769501	−	0.638646i	$-0.220505\pi$
$13$	5.54895	1.53900	0.769501	+	0.638646i	$0.220505\pi$
$14$	0	0
$15$	−2.88861	−0.745835
$16$	0	0
$17$	1.49993	0.363785	0.181893	−	0.983318i	$-0.441778\pi$
$17$	1.49993	0.363785	0.181893	+	0.983318i	$0.441778\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	0.448778	0.0979314
$22$	0	0
$23$	−3.26301	−0.680385	−0.340192	−	0.940356i	$-0.610492\pi$
$23$	−3.26301	−0.680385	−0.340192	+	0.940356i	$0.610492\pi$
$24$	0	0
$25$	6.90158	1.38032
$26$	0	0
$27$	4.43683	0.853868
$28$	0	0
$29$	−3.17227	−0.589076	−0.294538	−	0.955640i	$-0.595166\pi$
$29$	−3.17227	−0.589076	−0.294538	+	0.955640i	$0.595166\pi$
$30$	0	0
$31$	0.888605	0.159598	0.0797991	−	0.996811i	$-0.474572\pi$
$31$	0.888605	0.159598	0.0797991	+	0.996811i	$0.474572\pi$
$32$	0	0
$33$	−0.837309	−0.145757
$34$	0	0
$35$	−1.84905	−0.312546
$36$	0	0
$37$	11.0600	1.81825	0.909127	−	0.416520i	$-0.136750\pi$
$37$	11.0600	1.81825	0.909127	+	0.416520i	$0.136750\pi$
$38$	0	0
$39$	−4.64619	−0.743985
$40$	0	0
$41$	−8.44868	−1.31946	−0.659731	−	0.751502i	$-0.729330\pi$
$41$	−8.44868	−1.31946	−0.659731	+	0.751502i	$0.729330\pi$
$42$	0	0
$43$	7.61261	1.16091	0.580456	−	0.814292i	$-0.302874\pi$
$43$	7.61261	1.16091	0.580456	+	0.814292i	$0.302874\pi$
$44$	0	0
$45$	−7.93094	−1.18228
$46$	0	0
$47$	−3.39981	−0.495913	−0.247956	−	0.968771i	$-0.579759\pi$
$47$	−3.39981	−0.495913	−0.247956	+	0.968771i	$0.579759\pi$
$48$	0	0
$49$	−6.71273	−0.958961
$50$	0	0
$51$	−1.25590	−0.175861
$52$	0	0
$53$	7.22511	0.992446	0.496223	−	0.868195i	$-0.334720\pi$
$53$	7.22511	0.992446	0.496223	+	0.868195i	$0.334720\pi$
$54$	0	0
$55$	3.44987	0.465180
$56$	0	0
$57$	−0.837309	−0.110904
$58$	0	0
$59$	1.36560	0.177786	0.0888932	−	0.996041i	$-0.471667\pi$
$59$	1.36560	0.177786	0.0888932	+	0.996041i	$0.471667\pi$
$60$	0	0
$61$	−2.57435	−0.329612	−0.164806	−	0.986326i	$-0.552700\pi$
$61$	−2.57435	−0.329612	−0.164806	+	0.986326i	$0.552700\pi$
$62$	0	0
$63$	1.23216	0.155238
$64$	0	0
$65$	19.1431	2.37441
$66$	0	0
$67$	10.8862	1.32997	0.664983	−	0.746859i	$-0.268439\pi$
$67$	10.8862	1.32997	0.664983	+	0.746859i	$0.268439\pi$
$68$	0	0
$69$	2.73215	0.328912
$70$	0	0
$71$	−2.66509	−0.316288	−0.158144	−	0.987416i	$-0.550551\pi$
$71$	−2.66509	−0.316288	−0.158144	+	0.987416i	$0.550551\pi$
$72$	0	0
$73$	9.82294	1.14969	0.574844	−	0.818263i	$-0.305063\pi$
$73$	9.82294	1.14969	0.574844	+	0.818263i	$0.305063\pi$
$74$	0	0
$75$	−5.77876	−0.667273
$76$	0	0
$77$	−0.535976	−0.0610802
$78$	0	0
$79$	2.17469	0.244672	0.122336	−	0.992489i	$-0.460961\pi$
$79$	2.17469	0.244672	0.122336	+	0.992489i	$0.460961\pi$
$80$	0	0
$81$	3.18174	0.353527
$82$	0	0
$83$	−4.71288	−0.517305	−0.258653	−	0.965970i	$-0.583279\pi$
$83$	−4.71288	−0.517305	−0.258653	+	0.965970i	$0.583279\pi$
$84$	0	0
$85$	5.17454	0.561258
$86$	0	0
$87$	2.65617	0.284772
$88$	0	0
$89$	−4.66282	−0.494258	−0.247129	−	0.968983i	$-0.579487\pi$
$89$	−4.66282	−0.494258	−0.247129	+	0.968983i	$0.579487\pi$
$90$	0	0
$91$	−2.97411	−0.311771
$92$	0	0
$93$	−0.744038	−0.0771531
$94$	0	0
$95$	3.44987	0.353949
$96$	0	0
$97$	6.93516	0.704158	0.352079	−	0.935970i	$-0.385475\pi$
$97$	6.93516	0.704158	0.352079	+	0.935970i	$0.385475\pi$
$98$	0	0
$99$	−2.29891	−0.231049
$100$	0	0
$101$	0.274812	0.0273448	0.0136724	−	0.999907i	$-0.495648\pi$
$101$	0.274812	0.0273448	0.0136724	+	0.999907i	$0.495648\pi$
$102$	0	0
$103$	−16.0745	−1.58387	−0.791933	−	0.610608i	$-0.790925\pi$
$103$	−16.0745	−1.58387	−0.791933	+	0.610608i	$0.790925\pi$
$104$	0	0
$105$	1.54822	0.151091
$106$	0	0
$107$	10.8689	1.05074	0.525370	−	0.850874i	$-0.323927\pi$
$107$	10.8689	1.05074	0.525370	+	0.850874i	$0.323927\pi$
$108$	0	0
$109$	−12.2947	−1.17762	−0.588809	−	0.808272i	$-0.700403\pi$
$109$	−12.2947	−1.17762	−0.588809	+	0.808272i	$0.700403\pi$
$110$	0	0
$111$	−9.26064	−0.878981
$112$	0	0
$113$	7.54030	0.709331	0.354666	−	0.934993i	$-0.384595\pi$
$113$	7.54030	0.709331	0.354666	+	0.934993i	$0.384595\pi$
$114$	0	0
$115$	−11.2570	−1.04972
$116$	0	0
$117$	−12.7566	−1.17934
$118$	0	0
$119$	−0.803925	−0.0736957
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	7.07416	0.637856
$124$	0	0
$125$	6.56019	0.586762
$126$	0	0
$127$	13.8753	1.23123	0.615615	−	0.788047i	$-0.288908\pi$
$127$	13.8753	1.23123	0.615615	+	0.788047i	$0.288908\pi$
$128$	0	0
$129$	−6.37411	−0.561209
$130$	0	0
$131$	22.7894	1.99112	0.995558	−	0.0941505i	$-0.0300135\pi$
$131$	22.7894	1.99112	0.995558	+	0.0941505i	$0.0300135\pi$
$132$	0	0
$133$	−0.535976	−0.0464750
$134$	0	0
$135$	15.3065	1.31737
$136$	0	0
$137$	−17.4238	−1.48861	−0.744306	−	0.667839i	$-0.767220\pi$
$137$	−17.4238	−1.48861	−0.744306	+	0.667839i	$0.767220\pi$
$138$	0	0
$139$	13.7540	1.16660	0.583301	−	0.812256i	$-0.301761\pi$
$139$	13.7540	1.16660	0.583301	+	0.812256i	$0.301761\pi$
$140$	0	0
$141$	2.84669	0.239735
$142$	0	0
$143$	5.54895	0.464027
$144$	0	0
$145$	−10.9439	−0.908843
$146$	0	0
$147$	5.62063	0.463582
$148$	0	0
$149$	−12.4471	−1.01971	−0.509855	−	0.860261i	$-0.670301\pi$
$149$	−12.4471	−1.01971	−0.509855	+	0.860261i	$0.670301\pi$
$150$	0	0
$151$	−8.67447	−0.705918	−0.352959	−	0.935639i	$-0.614825\pi$
$151$	−8.67447	−0.705918	−0.352959	+	0.935639i	$0.614825\pi$
$152$	0	0
$153$	−3.44820	−0.278770
$154$	0	0
$155$	3.06557	0.246233
$156$	0	0
$157$	15.7729	1.25881	0.629406	−	0.777076i	$-0.283298\pi$
$157$	15.7729	1.25881	0.629406	+	0.777076i	$0.283298\pi$
$158$	0	0
$159$	−6.04966	−0.479769
$160$	0	0
$161$	1.74890	0.137832
$162$	0	0
$163$	11.6248	0.910523	0.455261	−	0.890358i	$-0.349546\pi$
$163$	11.6248	0.910523	0.455261	+	0.890358i	$0.349546\pi$
$164$	0	0
$165$	−2.88861	−0.224878
$166$	0	0
$167$	−9.39744	−0.727196	−0.363598	−	0.931556i	$-0.618452\pi$
$167$	−9.39744	−0.727196	−0.363598	+	0.931556i	$0.618452\pi$
$168$	0	0
$169$	17.7908	1.36853
$170$	0	0
$171$	−2.29891	−0.175802
$172$	0	0
$173$	9.00005	0.684261	0.342131	−	0.939652i	$-0.388851\pi$
$173$	9.00005	0.684261	0.342131	+	0.939652i	$0.388851\pi$
$174$	0	0
$175$	−3.69908	−0.279624
$176$	0	0
$177$	−1.14343	−0.0859456
$178$	0	0
$179$	0.439101	0.0328200	0.0164100	−	0.999865i	$-0.494776\pi$
$179$	0.439101	0.0328200	0.0164100	+	0.999865i	$0.494776\pi$
$180$	0	0
$181$	−19.1343	−1.42224	−0.711120	−	0.703071i	$-0.751812\pi$
$181$	−19.1343	−1.42224	−0.711120	+	0.703071i	$0.751812\pi$
$182$	0	0
$183$	2.15553	0.159341
$184$	0	0
$185$	38.1555	2.80525
$186$	0	0
$187$	1.49993	0.109685
$188$	0	0
$189$	−2.37804	−0.172977
$190$	0	0
$191$	22.2325	1.60869	0.804343	−	0.594165i	$-0.202518\pi$
$191$	22.2325	1.60869	0.804343	+	0.594165i	$0.202518\pi$
$192$	0	0
$193$	−20.6415	−1.48581	−0.742904	−	0.669397i	$-0.766552\pi$
$193$	−20.6415	−1.48581	−0.742904	+	0.669397i	$0.766552\pi$
$194$	0	0
$195$	−16.0287	−1.14784
$196$	0	0
$197$	−2.25328	−0.160540	−0.0802698	−	0.996773i	$-0.525578\pi$
$197$	−2.25328	−0.160540	−0.0802698	+	0.996773i	$0.525578\pi$
$198$	0	0
$199$	3.27743	0.232331	0.116166	−	0.993230i	$-0.462940\pi$
$199$	3.27743	0.232331	0.116166	+	0.993230i	$0.462940\pi$
$200$	0	0
$201$	−9.11515	−0.642933
$202$	0	0
$203$	1.70026	0.119335
$204$	0	0
$205$	−29.1468	−2.03570
$206$	0	0
$207$	7.50138	0.521382
$208$	0	0
$209$	1.00000	0.0691714
$210$	0	0
$211$	−8.60267	−0.592232	−0.296116	−	0.955152i	$-0.595691\pi$
$211$	−8.60267	−0.592232	−0.296116	+	0.955152i	$0.595691\pi$
$212$	0	0
$213$	2.23150	0.152900
$214$	0	0
$215$	26.2625	1.79109
$216$	0	0
$217$	−0.476271	−0.0323314
$218$	0	0
$219$	−8.22484	−0.555783
$220$	0	0
$221$	8.32301	0.559867
$222$	0	0
$223$	−17.1038	−1.14535	−0.572677	−	0.819781i	$-0.694095\pi$
$223$	−17.1038	−1.14535	−0.572677	+	0.819781i	$0.694095\pi$
$224$	0	0
$225$	−15.8661	−1.05774
$226$	0	0
$227$	−0.0197823	−0.00131300	−0.000656499	−	1.00000i	$-0.500209\pi$
$227$	−0.0197823	−0.00131300	−0.000656499		1.00000i	$0.500209\pi$
$228$	0	0
$229$	23.3471	1.54282	0.771410	−	0.636338i	$-0.219552\pi$
$229$	23.3471	1.54282	0.771410	+	0.636338i	$0.219552\pi$
$230$	0	0
$231$	0.448778	0.0295274
$232$	0	0
$233$	−10.6262	−0.696149	−0.348074	−	0.937467i	$-0.613164\pi$
$233$	−10.6262	−0.696149	−0.348074	+	0.937467i	$0.613164\pi$
$234$	0	0
$235$	−11.7289	−0.765108
$236$	0	0
$237$	−1.82089	−0.118280
$238$	0	0
$239$	10.0843	0.652297	0.326148	−	0.945319i	$-0.394249\pi$
$239$	10.0843	0.652297	0.326148	+	0.945319i	$0.394249\pi$
$240$	0	0
$241$	19.7997	1.27541	0.637704	−	0.770281i	$-0.279884\pi$
$241$	19.7997	1.27541	0.637704	+	0.770281i	$0.279884\pi$
$242$	0	0
$243$	−15.9746	−1.02477
$244$	0	0
$245$	−23.1580	−1.47951
$246$	0	0
$247$	5.54895	0.353071
$248$	0	0
$249$	3.94614	0.250076
$250$	0	0
$251$	13.5342	0.854273	0.427137	−	0.904187i	$-0.359522\pi$
$251$	13.5342	0.854273	0.427137	+	0.904187i	$0.359522\pi$
$252$	0	0
$253$	−3.26301	−0.205144
$254$	0	0
$255$	−4.33269	−0.271324
$256$	0	0
$257$	1.47161	0.0917966	0.0458983	−	0.998946i	$-0.485385\pi$
$257$	1.47161	0.0917966	0.0458983	+	0.998946i	$0.485385\pi$
$258$	0	0
$259$	−5.92790	−0.368342
$260$	0	0
$261$	7.29278	0.451412
$262$	0	0
$263$	30.2847	1.86744	0.933718	−	0.358010i	$-0.116545\pi$
$263$	30.2847	1.86744	0.933718	+	0.358010i	$0.116545\pi$
$264$	0	0
$265$	24.9257	1.53117
$266$	0	0
$267$	3.90422	0.238934
$268$	0	0
$269$	7.12385	0.434349	0.217174	−	0.976133i	$-0.430316\pi$
$269$	7.12385	0.434349	0.217174	+	0.976133i	$0.430316\pi$
$270$	0	0
$271$	29.8853	1.81540	0.907702	−	0.419616i	$-0.137835\pi$
$271$	29.8853	1.81540	0.907702	+	0.419616i	$0.137835\pi$
$272$	0	0
$273$	2.49025	0.150717
$274$	0	0
$275$	6.90158	0.416181
$276$	0	0
$277$	13.5222	0.812469	0.406235	−	0.913769i	$-0.366842\pi$
$277$	13.5222	0.812469	0.406235	+	0.913769i	$0.366842\pi$
$278$	0	0
$279$	−2.04283	−0.122301
$280$	0	0
$281$	4.37760	0.261146	0.130573	−	0.991439i	$-0.458318\pi$
$281$	4.37760	0.261146	0.130573	+	0.991439i	$0.458318\pi$
$282$	0	0
$283$	−20.6608	−1.22816	−0.614079	−	0.789244i	$-0.710472\pi$
$283$	−20.6608	−1.22816	−0.614079	+	0.789244i	$0.710472\pi$
$284$	0	0
$285$	−2.88861	−0.171106
$286$	0	0
$287$	4.52829	0.267297
$288$	0	0
$289$	−14.7502	−0.867660
$290$	0	0
$291$	−5.80687	−0.340405
$292$	0	0
$293$	−10.5734	−0.617706	−0.308853	−	0.951110i	$-0.599945\pi$
$293$	−10.5734	−0.617706	−0.308853	+	0.951110i	$0.599945\pi$
$294$	0	0
$295$	4.71115	0.274294
$296$	0	0
$297$	4.43683	0.257451
$298$	0	0
$299$	−18.1063	−1.04711
$300$	0	0
$301$	−4.08018	−0.235178
$302$	0	0
$303$	−0.230102	−0.0132190
$304$	0	0
$305$	−8.88117	−0.508534
$306$	0	0
$307$	−4.45392	−0.254199	−0.127099	−	0.991890i	$-0.540567\pi$
$307$	−4.45392	−0.254199	−0.127099	+	0.991890i	$0.540567\pi$
$308$	0	0
$309$	13.4593	0.765673
$310$	0	0
$311$	−5.02138	−0.284736	−0.142368	−	0.989814i	$-0.545472\pi$
$311$	−5.02138	−0.284736	−0.142368	+	0.989814i	$0.545472\pi$
$312$	0	0
$313$	15.1246	0.854892	0.427446	−	0.904041i	$-0.359413\pi$
$313$	15.1246	0.854892	0.427446	+	0.904041i	$0.359413\pi$
$314$	0	0
$315$	4.25080	0.239505
$316$	0	0
$317$	−17.8288	−1.00137	−0.500683	−	0.865630i	$-0.666918\pi$
$317$	−17.8288	−1.00137	−0.500683	+	0.865630i	$0.666918\pi$
$318$	0	0
$319$	−3.17227	−0.177613
$320$	0	0
$321$	−9.10067	−0.507950
$322$	0	0
$323$	1.49993	0.0834581
$324$	0	0
$325$	38.2965	2.12431
$326$	0	0
$327$	10.2945	0.569285
$328$	0	0
$329$	1.82222	0.100462
$330$	0	0
$331$	−32.7851	−1.80203	−0.901016	−	0.433786i	$-0.857177\pi$
$331$	−32.7851	−1.80203	−0.901016	+	0.433786i	$0.857177\pi$
$332$	0	0
$333$	−25.4260	−1.39334
$334$	0	0
$335$	37.5561	2.05191
$336$	0	0
$337$	17.0964	0.931298	0.465649	−	0.884970i	$-0.345821\pi$
$337$	17.0964	0.931298	0.465649	+	0.884970i	$0.345821\pi$
$338$	0	0
$339$	−6.31356	−0.342905
$340$	0	0
$341$	0.888605	0.0481207
$342$	0	0
$343$	7.34970	0.396846
$344$	0	0
$345$	9.42555	0.507455
$346$	0	0
$347$	−1.52587	−0.0819132	−0.0409566	−	0.999161i	$-0.513041\pi$
$347$	−1.52587	−0.0819132	−0.0409566	+	0.999161i	$0.513041\pi$
$348$	0	0
$349$	3.37139	0.180466	0.0902331	−	0.995921i	$-0.471239\pi$
$349$	3.37139	0.180466	0.0902331	+	0.995921i	$0.471239\pi$
$350$	0	0
$351$	24.6197	1.31410
$352$	0	0
$353$	6.08870	0.324069	0.162034	−	0.986785i	$-0.448194\pi$
$353$	6.08870	0.324069	0.162034	+	0.986785i	$0.448194\pi$
$354$	0	0
$355$	−9.19420	−0.487978
$356$	0	0
$357$	0.673134	0.0356260
$358$	0	0
$359$	20.8084	1.09823	0.549113	−	0.835748i	$-0.314966\pi$
$359$	20.8084	1.09823	0.549113	+	0.835748i	$0.314966\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	−0.837309	−0.0439473
$364$	0	0
$365$	33.8878	1.77377
$366$	0	0
$367$	17.8042	0.929372	0.464686	−	0.885476i	$-0.346167\pi$
$367$	17.8042	0.929372	0.464686	+	0.885476i	$0.346167\pi$
$368$	0	0
$369$	19.4228	1.01111
$370$	0	0
$371$	−3.87249	−0.201050
$372$	0	0
$373$	17.2736	0.894395	0.447198	−	0.894435i	$-0.352422\pi$
$373$	17.2736	0.894395	0.447198	+	0.894435i	$0.352422\pi$
$374$	0	0
$375$	−5.49291	−0.283653
$376$	0	0
$377$	−17.6028	−0.906589
$378$	0	0
$379$	−17.9061	−0.919773	−0.459886	−	0.887978i	$-0.652110\pi$
$379$	−17.9061	−0.919773	−0.459886	+	0.887978i	$0.652110\pi$
$380$	0	0
$381$	−11.6179	−0.595202
$382$	0	0
$383$	−3.83449	−0.195933	−0.0979667	−	0.995190i	$-0.531234\pi$
$383$	−3.83449	−0.195933	−0.0979667	+	0.995190i	$0.531234\pi$
$384$	0	0
$385$	−1.84905	−0.0942361
$386$	0	0
$387$	−17.5007	−0.889612
$388$	0	0
$389$	2.75314	0.139590	0.0697949	−	0.997561i	$-0.477766\pi$
$389$	2.75314	0.139590	0.0697949	+	0.997561i	$0.477766\pi$
$390$	0	0
$391$	−4.89427	−0.247514
$392$	0	0
$393$	−19.0817	−0.962547
$394$	0	0
$395$	7.50240	0.377487
$396$	0	0
$397$	7.78879	0.390908	0.195454	−	0.980713i	$-0.437382\pi$
$397$	7.78879	0.390908	0.195454	+	0.980713i	$0.437382\pi$
$398$	0	0
$399$	0.448778	0.0224670
$400$	0	0
$401$	−2.64767	−0.132218	−0.0661092	−	0.997812i	$-0.521059\pi$
$401$	−2.64767	−0.132218	−0.0661092	+	0.997812i	$0.521059\pi$
$402$	0	0
$403$	4.93083	0.245622
$404$	0	0
$405$	10.9766	0.545431
$406$	0	0
$407$	11.0600	0.548224
$408$	0	0
$409$	32.0043	1.58251	0.791256	−	0.611485i	$-0.209428\pi$
$409$	32.0043	1.58251	0.791256	+	0.611485i	$0.209428\pi$
$410$	0	0
$411$	14.5891	0.719626
$412$	0	0
$413$	−0.731931	−0.0360160
$414$	0	0
$415$	−16.2588	−0.798113
$416$	0	0
$417$	−11.5164	−0.563960
$418$	0	0
$419$	17.0000	0.830502	0.415251	−	0.909707i	$-0.363694\pi$
$419$	17.0000	0.830502	0.415251	+	0.909707i	$0.363694\pi$
$420$	0	0
$421$	17.5236	0.854050	0.427025	−	0.904240i	$-0.359562\pi$
$421$	17.5236	0.854050	0.427025	+	0.904240i	$0.359562\pi$
$422$	0	0
$423$	7.81586	0.380020
$424$	0	0
$425$	10.3519	0.502139
$426$	0	0
$427$	1.37979	0.0667728
$428$	0	0
$429$	−4.64619	−0.224320
$430$	0	0
$431$	−29.8949	−1.43998	−0.719992	−	0.693982i	$-0.755855\pi$
$431$	−29.8949	−1.43998	−0.719992	+	0.693982i	$0.755855\pi$
$432$	0	0
$433$	−11.0433	−0.530708	−0.265354	−	0.964151i	$-0.585489\pi$
$433$	−11.0433	−0.530708	−0.265354	+	0.964151i	$0.585489\pi$
$434$	0	0
$435$	9.16344	0.439354
$436$	0	0
$437$	−3.26301	−0.156091
$438$	0	0
$439$	−13.7529	−0.656389	−0.328195	−	0.944610i	$-0.606440\pi$
$439$	−13.7529	−0.656389	−0.328195	+	0.944610i	$0.606440\pi$
$440$	0	0
$441$	15.4320	0.734856
$442$	0	0
$443$	−40.2861	−1.91405	−0.957026	−	0.290001i	$-0.906344\pi$
$443$	−40.2861	−1.91405	−0.957026	+	0.290001i	$0.906344\pi$
$444$	0	0
$445$	−16.0861	−0.762554
$446$	0	0
$447$	10.4221	0.492949
$448$	0	0
$449$	0.932979	0.0440300	0.0220150	−	0.999758i	$-0.492992\pi$
$449$	0.932979	0.0440300	0.0220150	+	0.999758i	$0.492992\pi$
$450$	0	0
$451$	−8.44868	−0.397833
$452$	0	0
$453$	7.26322	0.341256
$454$	0	0
$455$	−10.2603	−0.481009
$456$	0	0
$457$	−24.6463	−1.15290	−0.576452	−	0.817131i	$-0.695563\pi$
$457$	−24.6463	−1.15290	−0.576452	+	0.817131i	$0.695563\pi$
$458$	0	0
$459$	6.65492	0.310625
$460$	0	0
$461$	31.5768	1.47068	0.735339	−	0.677700i	$-0.237023\pi$
$461$	31.5768	1.47068	0.735339	+	0.677700i	$0.237023\pi$
$462$	0	0
$463$	−35.4243	−1.64631	−0.823154	−	0.567818i	$-0.807788\pi$
$463$	−35.4243	−1.64631	−0.823154	+	0.567818i	$0.807788\pi$
$464$	0	0
$465$	−2.56683	−0.119034
$466$	0	0
$467$	42.9301	1.98657	0.993285	−	0.115696i	$-0.0369098\pi$
$467$	42.9301	1.98657	0.993285	+	0.115696i	$0.0369098\pi$
$468$	0	0
$469$	−5.83477	−0.269424
$470$	0	0
$471$	−13.2068	−0.608536
$472$	0	0
$473$	7.61261	0.350028
$474$	0	0
$475$	6.90158	0.316666
$476$	0	0
$477$	−16.6099	−0.760516
$478$	0	0
$479$	−26.4557	−1.20879	−0.604397	−	0.796683i	$-0.706586\pi$
$479$	−26.4557	−1.20879	−0.604397	+	0.796683i	$0.706586\pi$
$480$	0	0
$481$	61.3714	2.79830
$482$	0	0
$483$	−1.46437	−0.0666310
$484$	0	0
$485$	23.9254	1.08639
$486$	0	0
$487$	2.95751	0.134017	0.0670087	−	0.997752i	$-0.478654\pi$
$487$	2.95751	0.134017	0.0670087	+	0.997752i	$0.478654\pi$
$488$	0	0
$489$	−9.73353	−0.440165
$490$	0	0
$491$	9.73732	0.439439	0.219720	−	0.975563i	$-0.429486\pi$
$491$	9.73732	0.439439	0.219720	+	0.975563i	$0.429486\pi$
$492$	0	0
$493$	−4.75817	−0.214297
$494$	0	0
$495$	−7.93094	−0.356469
$496$	0	0
$497$	1.42842	0.0640736
$498$	0	0
$499$	−38.3394	−1.71631	−0.858154	−	0.513393i	$-0.828388\pi$
$499$	−38.3394	−1.71631	−0.858154	+	0.513393i	$0.828388\pi$
$500$	0	0
$501$	7.86856	0.351542
$502$	0	0
$503$	−42.0185	−1.87351	−0.936757	−	0.349980i	$-0.886188\pi$
$503$	−42.0185	−1.87351	−0.936757	+	0.349980i	$0.886188\pi$
$504$	0	0
$505$	0.948063	0.0421883
$506$	0	0
$507$	−14.8964	−0.661574
$508$	0	0
$509$	−39.3516	−1.74423	−0.872115	−	0.489301i	$-0.837252\pi$
$509$	−39.3516	−1.74423	−0.872115	+	0.489301i	$0.837252\pi$
$510$	0	0
$511$	−5.26486	−0.232904
$512$	0	0
$513$	4.43683	0.195891
$514$	0	0
$515$	−55.4548	−2.44363
$516$	0	0
$517$	−3.39981	−0.149523
$518$	0	0
$519$	−7.53583	−0.330786
$520$	0	0
$521$	−12.9100	−0.565599	−0.282800	−	0.959179i	$-0.591263\pi$
$521$	−12.9100	−0.565599	−0.282800	+	0.959179i	$0.591263\pi$
$522$	0	0
$523$	−4.51924	−0.197612	−0.0988062	−	0.995107i	$-0.531502\pi$
$523$	−4.51924	−0.197612	−0.0988062	+	0.995107i	$0.531502\pi$
$524$	0	0
$525$	3.09728	0.135176
$526$	0	0
$527$	1.33284	0.0580595
$528$	0	0
$529$	−12.3528	−0.537077
$530$	0	0
$531$	−3.13940	−0.136238
$532$	0	0
$533$	−46.8813	−2.03066
$534$	0	0
$535$	37.4964	1.62111
$536$	0	0
$537$	−0.367664	−0.0158659
$538$	0	0
$539$	−6.71273	−0.289138
$540$	0	0
$541$	−18.5678	−0.798291	−0.399145	−	0.916888i	$-0.630693\pi$
$541$	−18.5678	−0.798291	−0.399145	+	0.916888i	$0.630693\pi$
$542$	0	0
$543$	16.0213	0.687540
$544$	0	0
$545$	−42.4151	−1.81686
$546$	0	0
$547$	−4.82176	−0.206164	−0.103082	−	0.994673i	$-0.532870\pi$
$547$	−4.82176	−0.206164	−0.103082	+	0.994673i	$0.532870\pi$
$548$	0	0
$549$	5.91821	0.252583
$550$	0	0
$551$	−3.17227	−0.135143
$552$	0	0
$553$	−1.16558	−0.0495657
$554$	0	0
$555$	−31.9480	−1.35612
$556$	0	0
$557$	−15.4258	−0.653610	−0.326805	−	0.945092i	$-0.605972\pi$
$557$	−15.4258	−0.653610	−0.326805	+	0.945092i	$0.605972\pi$
$558$	0	0
$559$	42.2420	1.78665
$560$	0	0
$561$	−1.25590	−0.0530242
$562$	0	0
$563$	−29.2902	−1.23444	−0.617218	−	0.786792i	$-0.711740\pi$
$563$	−29.2902	−1.23444	−0.617218	+	0.786792i	$0.711740\pi$
$564$	0	0
$565$	26.0130	1.09438
$566$	0	0
$567$	−1.70534	−0.0716174
$568$	0	0
$569$	6.50295	0.272618	0.136309	−	0.990666i	$-0.456476\pi$
$569$	6.50295	0.272618	0.136309	+	0.990666i	$0.456476\pi$
$570$	0	0
$571$	10.8414	0.453700	0.226850	−	0.973930i	$-0.427157\pi$
$571$	10.8414	0.453700	0.226850	+	0.973930i	$0.427157\pi$
$572$	0	0
$573$	−18.6155	−0.777672
$574$	0	0
$575$	−22.5199	−0.939146
$576$	0	0
$577$	−12.7957	−0.532694	−0.266347	−	0.963877i	$-0.585817\pi$
$577$	−12.7957	−0.532694	−0.266347	+	0.963877i	$0.585817\pi$
$578$	0	0
$579$	17.2833	0.718271
$580$	0	0
$581$	2.52599	0.104796
$582$	0	0
$583$	7.22511	0.299234
$584$	0	0
$585$	−44.0084	−1.81952
$586$	0	0
$587$	−34.9956	−1.44442	−0.722212	−	0.691671i	$-0.756875\pi$
$587$	−34.9956	−1.44442	−0.722212	+	0.691671i	$0.756875\pi$
$588$	0	0
$589$	0.888605	0.0366143
$590$	0	0
$591$	1.88669	0.0776082
$592$	0	0
$593$	−11.8232	−0.485522	−0.242761	−	0.970086i	$-0.578053\pi$
$593$	−11.8232	−0.485522	−0.242761	+	0.970086i	$0.578053\pi$
$594$	0	0
$595$	−2.77343	−0.113700
$596$	0	0
$597$	−2.74422	−0.112314
$598$	0	0
$599$	−13.1662	−0.537957	−0.268978	−	0.963146i	$-0.586686\pi$
$599$	−13.1662	−0.537957	−0.268978	+	0.963146i	$0.586686\pi$
$600$	0	0
$601$	−4.97707	−0.203019	−0.101509	−	0.994835i	$-0.532367\pi$
$601$	−4.97707	−0.203019	−0.101509	+	0.994835i	$0.532367\pi$
$602$	0	0
$603$	−25.0265	−1.01916
$604$	0	0
$605$	3.44987	0.140257
$606$	0	0
$607$	−5.87279	−0.238369	−0.119185	−	0.992872i	$-0.538028\pi$
$607$	−5.87279	−0.238369	−0.119185	+	0.992872i	$0.538028\pi$
$608$	0	0
$609$	−1.42365	−0.0576890
$610$	0	0
$611$	−18.8654	−0.763211
$612$	0	0
$613$	35.7193	1.44269	0.721345	−	0.692576i	$-0.243524\pi$
$613$	35.7193	1.44269	0.721345	+	0.692576i	$0.243524\pi$
$614$	0	0
$615$	24.4049	0.984101
$616$	0	0
$617$	−44.6231	−1.79646	−0.898230	−	0.439526i	$-0.855146\pi$
$617$	−44.6231	−1.79646	−0.898230	+	0.439526i	$0.855146\pi$
$618$	0	0
$619$	−19.9642	−0.802427	−0.401214	−	0.915985i	$-0.631411\pi$
$619$	−19.9642	−0.802427	−0.401214	+	0.915985i	$0.631411\pi$
$620$	0	0
$621$	−14.4774	−0.580959
$622$	0	0
$623$	2.49916	0.100127
$624$	0	0
$625$	−11.8761	−0.475044
$626$	0	0
$627$	−0.837309	−0.0334389
$628$	0	0
$629$	16.5892	0.661454
$630$	0	0
$631$	−26.9609	−1.07330	−0.536649	−	0.843806i	$-0.680310\pi$
$631$	−26.9609	−1.07330	−0.536649	+	0.843806i	$0.680310\pi$
$632$	0	0
$633$	7.20309	0.286297
$634$	0	0
$635$	47.8678	1.89958
$636$	0	0
$637$	−37.2486	−1.47584
$638$	0	0
$639$	6.12681	0.242373
$640$	0	0
$641$	−38.3564	−1.51499	−0.757494	−	0.652843i	$-0.773576\pi$
$641$	−38.3564	−1.51499	−0.757494	+	0.652843i	$0.773576\pi$
$642$	0	0
$643$	5.26918	0.207796	0.103898	−	0.994588i	$-0.466868\pi$
$643$	5.26918	0.207796	0.103898	+	0.994588i	$0.466868\pi$
$644$	0	0
$645$	−21.9898	−0.865849
$646$	0	0
$647$	−23.8457	−0.937471	−0.468736	−	0.883338i	$-0.655290\pi$
$647$	−23.8457	−0.937471	−0.468736	+	0.883338i	$0.655290\pi$
$648$	0	0
$649$	1.36560	0.0536046
$650$	0	0
$651$	0.398786	0.0156297
$652$	0	0
$653$	−10.2029	−0.399269	−0.199635	−	0.979870i	$-0.563975\pi$
$653$	−10.2029	−0.399269	−0.199635	+	0.979870i	$0.563975\pi$
$654$	0	0
$655$	78.6203	3.07195
$656$	0	0
$657$	−22.5821	−0.881011
$658$	0	0
$659$	−19.2704	−0.750666	−0.375333	−	0.926890i	$-0.622472\pi$
$659$	−19.2704	−0.750666	−0.375333	+	0.926890i	$0.622472\pi$
$660$	0	0
$661$	−49.7415	−1.93472	−0.967360	−	0.253408i	$-0.918449\pi$
$661$	−49.7415	−1.93472	−0.967360	+	0.253408i	$0.918449\pi$
$662$	0	0
$663$	−6.96894	−0.270651
$664$	0	0
$665$	−1.84905	−0.0717030
$666$	0	0
$667$	10.3512	0.400799
$668$	0	0
$669$	14.3212	0.553688
$670$	0	0
$671$	−2.57435	−0.0993817
$672$	0	0
$673$	20.7375	0.799372	0.399686	−	0.916652i	$-0.369119\pi$
$673$	20.7375	0.799372	0.399686	+	0.916652i	$0.369119\pi$
$674$	0	0
$675$	30.6211	1.17861
$676$	0	0
$677$	47.5162	1.82620	0.913098	−	0.407740i	$-0.133683\pi$
$677$	47.5162	1.82620	0.913098	+	0.407740i	$0.133683\pi$
$678$	0	0
$679$	−3.71708	−0.142648
$680$	0	0
$681$	0.0165639	0.000634731 0
$682$	0	0
$683$	−0.951416	−0.0364049	−0.0182025	−	0.999834i	$-0.505794\pi$
$683$	−0.951416	−0.0364049	−0.0182025	+	0.999834i	$0.505794\pi$
$684$	0	0
$685$	−60.1096	−2.29667
$686$	0	0
$687$	−19.5488	−0.745832
$688$	0	0
$689$	40.0918	1.52738
$690$	0	0
$691$	−44.1537	−1.67969	−0.839843	−	0.542829i	$-0.817353\pi$
$691$	−44.1537	−1.67969	−0.839843	+	0.542829i	$0.817353\pi$
$692$	0	0
$693$	1.23216	0.0468060
$694$	0	0
$695$	47.4496	1.79987
$696$	0	0
$697$	−12.6724	−0.480001
$698$	0	0
$699$	8.89746	0.336533
$700$	0	0
$701$	−34.9736	−1.32093	−0.660467	−	0.750855i	$-0.729642\pi$
$701$	−34.9736	−1.32093	−0.660467	+	0.750855i	$0.729642\pi$
$702$	0	0
$703$	11.0600	0.417136
$704$	0	0
$705$	9.82070	0.369869
$706$	0	0
$707$	−0.147292	−0.00553950
$708$	0	0
$709$	21.9016	0.822531	0.411266	−	0.911516i	$-0.365087\pi$
$709$	21.9016	0.822531	0.411266	+	0.911516i	$0.365087\pi$
$710$	0	0
$711$	−4.99943	−0.187493
$712$	0	0
$713$	−2.89953	−0.108588
$714$	0	0
$715$	19.1431	0.715913
$716$	0	0
$717$	−8.44365	−0.315334
$718$	0	0
$719$	21.7324	0.810482	0.405241	−	0.914210i	$-0.367188\pi$
$719$	21.7324	0.810482	0.405241	+	0.914210i	$0.367188\pi$
$720$	0	0
$721$	8.61554	0.320859
$722$	0	0
$723$	−16.5784	−0.616559
$724$	0	0
$725$	−21.8937	−0.813111
$726$	0	0
$727$	−0.468343	−0.0173699	−0.00868494	−	0.999962i	$-0.502765\pi$
$727$	−0.468343	−0.0173699	−0.00868494	+	0.999962i	$0.502765\pi$
$728$	0	0
$729$	3.83045	0.141868
$730$	0	0
$731$	11.4184	0.422323
$732$	0	0
$733$	−25.5434	−0.943468	−0.471734	−	0.881741i	$-0.656372\pi$
$733$	−25.5434	−0.943468	−0.471734	+	0.881741i	$0.656372\pi$
$734$	0	0
$735$	19.3904	0.715227
$736$	0	0
$737$	10.8862	0.401000
$738$	0	0
$739$	−39.9284	−1.46879	−0.734395	−	0.678722i	$-0.762534\pi$
$739$	−39.9284	−1.46879	−0.734395	+	0.678722i	$0.762534\pi$
$740$	0	0
$741$	−4.64619	−0.170682
$742$	0	0
$743$	−33.2994	−1.22164	−0.610818	−	0.791771i	$-0.709159\pi$
$743$	−33.2994	−1.22164	−0.610818	+	0.791771i	$0.709159\pi$
$744$	0	0
$745$	−42.9410	−1.57324
$746$	0	0
$747$	10.8345	0.396413
$748$	0	0
$749$	−5.82550	−0.212859
$750$	0	0
$751$	−7.88424	−0.287700	−0.143850	−	0.989600i	$-0.545948\pi$
$751$	−7.88424	−0.287700	−0.143850	+	0.989600i	$0.545948\pi$
$752$	0	0
$753$	−11.3323	−0.412974
$754$	0	0
$755$	−29.9258	−1.08911
$756$	0	0
$757$	46.8309	1.70210	0.851049	−	0.525086i	$-0.175967\pi$
$757$	46.8309	1.70210	0.851049	+	0.525086i	$0.175967\pi$
$758$	0	0
$759$	2.73215	0.0991707
$760$	0	0
$761$	−13.5091	−0.489704	−0.244852	−	0.969561i	$-0.578739\pi$
$761$	−13.5091	−0.489704	−0.244852	+	0.969561i	$0.578739\pi$
$762$	0	0
$763$	6.58967	0.238562
$764$	0	0
$765$	−11.8958	−0.430095
$766$	0	0
$767$	7.57766	0.273614
$768$	0	0
$769$	−45.9453	−1.65683	−0.828415	−	0.560114i	$-0.810757\pi$
$769$	−45.9453	−1.65683	−0.828415	+	0.560114i	$0.810757\pi$
$770$	0	0
$771$	−1.23219	−0.0443764
$772$	0	0
$773$	14.6262	0.526070	0.263035	−	0.964786i	$-0.415277\pi$
$773$	14.6262	0.526070	0.263035	+	0.964786i	$0.415277\pi$
$774$	0	0
$775$	6.13278	0.220296
$776$	0	0
$777$	4.96349	0.178064
$778$	0	0
$779$	−8.44868	−0.302705
$780$	0	0
$781$	−2.66509	−0.0953644
$782$	0	0
$783$	−14.0748	−0.502993
$784$	0	0
$785$	54.4143	1.94213
$786$	0	0
$787$	−20.6203	−0.735035	−0.367517	−	0.930017i	$-0.619792\pi$
$787$	−20.6203	−0.735035	−0.367517	+	0.930017i	$0.619792\pi$
$788$	0	0
$789$	−25.3577	−0.902757
$790$	0	0
$791$	−4.04142	−0.143696
$792$	0	0
$793$	−14.2849	−0.507273
$794$	0	0
$795$	−20.8705	−0.740201
$796$	0	0
$797$	−30.5069	−1.08061	−0.540305	−	0.841469i	$-0.681691\pi$
$797$	−30.5069	−1.08061	−0.540305	+	0.841469i	$0.681691\pi$
$798$	0	0
$799$	−5.09946	−0.180406
$800$	0	0
$801$	10.7194	0.378752
$802$	0	0
$803$	9.82294	0.346644
$804$	0	0
$805$	6.03346	0.212651
$806$	0	0
$807$	−5.96486	−0.209973
$808$	0	0
$809$	−5.15845	−0.181362	−0.0906808	−	0.995880i	$-0.528904\pi$
$809$	−5.15845	−0.181362	−0.0906808	+	0.995880i	$0.528904\pi$
$810$	0	0
$811$	27.8881	0.979283	0.489641	−	0.871924i	$-0.337128\pi$
$811$	27.8881	0.979283	0.489641	+	0.871924i	$0.337128\pi$
$812$	0	0
$813$	−25.0233	−0.877604
$814$	0	0
$815$	40.1039	1.40478
$816$	0	0
$817$	7.61261	0.266332
$818$	0	0
$819$	6.83721	0.238911
$820$	0	0
$821$	28.9562	1.01058	0.505289	−	0.862950i	$-0.331386\pi$
$821$	28.9562	1.01058	0.505289	+	0.862950i	$0.331386\pi$
$822$	0	0
$823$	12.6621	0.441373	0.220686	−	0.975345i	$-0.429170\pi$
$823$	12.6621	0.441373	0.220686	+	0.975345i	$0.429170\pi$
$824$	0	0
$825$	−5.77876	−0.201190
$826$	0	0
$827$	46.3476	1.61166	0.805832	−	0.592144i	$-0.201718\pi$
$827$	46.3476	1.61166	0.805832	+	0.592144i	$0.201718\pi$
$828$	0	0
$829$	0.681275	0.0236616	0.0118308	−	0.999930i	$-0.496234\pi$
$829$	0.681275	0.0236616	0.0118308	+	0.999930i	$0.496234\pi$
$830$	0	0
$831$	−11.3222	−0.392764
$832$	0	0
$833$	−10.0686	−0.348856
$834$	0	0
$835$	−32.4199	−1.12194
$836$	0	0
$837$	3.94259	0.136276
$838$	0	0
$839$	−7.58503	−0.261864	−0.130932	−	0.991391i	$-0.541797\pi$
$839$	−7.58503	−0.261864	−0.130932	+	0.991391i	$0.541797\pi$
$840$	0	0
$841$	−18.9367	−0.652989
$842$	0	0
$843$	−3.66541	−0.126243
$844$	0	0
$845$	61.3760	2.11140
$846$	0	0
$847$	−0.535976	−0.0184164
$848$	0	0
$849$	17.2995	0.593717
$850$	0	0
$851$	−36.0889	−1.23711
$852$	0	0
$853$	21.0662	0.721293	0.360646	−	0.932703i	$-0.382556\pi$
$853$	21.0662	0.721293	0.360646	+	0.932703i	$0.382556\pi$
$854$	0	0
$855$	−7.93094	−0.271233
$856$	0	0
$857$	29.2288	0.998438	0.499219	−	0.866476i	$-0.333620\pi$
$857$	29.2288	0.998438	0.499219	+	0.866476i	$0.333620\pi$
$858$	0	0
$859$	−40.1410	−1.36959	−0.684797	−	0.728734i	$-0.740109\pi$
$859$	−40.1410	−1.36959	−0.684797	+	0.728734i	$0.740109\pi$
$860$	0	0
$861$	−3.79158	−0.129217
$862$	0	0
$863$	−13.5487	−0.461204	−0.230602	−	0.973048i	$-0.574069\pi$
$863$	−13.5487	−0.461204	−0.230602	+	0.973048i	$0.574069\pi$
$864$	0	0
$865$	31.0490	1.05570
$866$	0	0
$867$	12.3505	0.419445
$868$	0	0
$869$	2.17469	0.0737714
$870$	0	0
$871$	60.4072	2.04682
$872$	0	0
$873$	−15.9433	−0.539600
$874$	0	0
$875$	−3.51611	−0.118866
$876$	0	0
$877$	31.2008	1.05358	0.526788	−	0.849997i	$-0.323396\pi$
$877$	31.2008	1.05358	0.526788	+	0.849997i	$0.323396\pi$
$878$	0	0
$879$	8.85323	0.298612
$880$	0	0
$881$	17.5664	0.591827	0.295914	−	0.955215i	$-0.404376\pi$
$881$	17.5664	0.591827	0.295914	+	0.955215i	$0.404376\pi$
$882$	0	0
$883$	8.30172	0.279375	0.139688	−	0.990196i	$-0.455390\pi$
$883$	8.30172	0.279375	0.139688	+	0.990196i	$0.455390\pi$
$884$	0	0
$885$	−3.94469	−0.132599
$886$	0	0
$887$	39.8748	1.33886	0.669432	−	0.742873i	$-0.266538\pi$
$887$	39.8748	1.33886	0.669432	+	0.742873i	$0.266538\pi$
$888$	0	0
$889$	−7.43681	−0.249423
$890$	0	0
$891$	3.18174	0.106592
$892$	0	0
$893$	−3.39981	−0.113770
$894$	0	0
$895$	1.51484	0.0506355
$896$	0	0
$897$	15.1606	0.506196
$898$	0	0
$899$	−2.81890	−0.0940155
$900$	0	0
$901$	10.8371	0.361037
$902$	0	0
$903$	3.41637	0.113690
$904$	0	0
$905$	−66.0107	−2.19427
$906$	0	0
$907$	−30.5058	−1.01293	−0.506463	−	0.862261i	$-0.669047\pi$
$907$	−30.5058	−1.01293	−0.506463	+	0.862261i	$0.669047\pi$
$908$	0	0
$909$	−0.631768	−0.0209544
$910$	0	0
$911$	−42.5459	−1.40961	−0.704804	−	0.709402i	$-0.748965\pi$
$911$	−42.5459	−1.40961	−0.704804	+	0.709402i	$0.748965\pi$
$912$	0	0
$913$	−4.71288	−0.155973
$914$	0	0
$915$	7.43629	0.245836
$916$	0	0
$917$	−12.2146	−0.403360
$918$	0	0
$919$	55.8366	1.84188	0.920939	−	0.389706i	$-0.127423\pi$
$919$	55.8366	1.84188	0.920939	+	0.389706i	$0.127423\pi$
$920$	0	0
$921$	3.72931	0.122885
$922$	0	0
$923$	−14.7885	−0.486768
$924$	0	0
$925$	76.3315	2.50976
$926$	0	0
$927$	36.9538	1.21372
$928$	0	0
$929$	27.1283	0.890052	0.445026	−	0.895518i	$-0.353194\pi$
$929$	27.1283	0.890052	0.445026	+	0.895518i	$0.353194\pi$
$930$	0	0
$931$	−6.71273	−0.220001
$932$	0	0
$933$	4.20445	0.137648
$934$	0	0
$935$	5.17454	0.169226
$936$	0	0
$937$	−1.34615	−0.0439767	−0.0219883	−	0.999758i	$-0.507000\pi$
$937$	−1.34615	−0.0439767	−0.0219883	+	0.999758i	$0.507000\pi$
$938$	0	0
$939$	−12.6640	−0.413273
$940$	0	0
$941$	32.1301	1.04741	0.523706	−	0.851899i	$-0.324549\pi$
$941$	32.1301	1.04741	0.523706	+	0.851899i	$0.324549\pi$
$942$	0	0
$943$	27.5681	0.897742
$944$	0	0
$945$	−8.20390	−0.266873
$946$	0	0
$947$	47.8773	1.55580	0.777901	−	0.628387i	$-0.216284\pi$
$947$	47.8773	1.55580	0.777901	+	0.628387i	$0.216284\pi$
$948$	0	0
$949$	54.5070	1.76937
$950$	0	0
$951$	14.9282	0.484082
$952$	0	0
$953$	−4.67682	−0.151497	−0.0757485	−	0.997127i	$-0.524135\pi$
$953$	−4.67682	−0.151497	−0.0757485	+	0.997127i	$0.524135\pi$
$954$	0	0
$955$	76.6991	2.48192
$956$	0	0
$957$	2.65617	0.0858619
$958$	0	0
$959$	9.33872	0.301563
$960$	0	0
$961$	−30.2104	−0.974528
$962$	0	0
$963$	−24.9868	−0.805187
$964$	0	0
$965$	−71.2105	−2.29235
$966$	0	0
$967$	−17.3817	−0.558957	−0.279478	−	0.960152i	$-0.590162\pi$
$967$	−17.3817	−0.558957	−0.279478	+	0.960152i	$0.590162\pi$
$968$	0	0
$969$	−1.25590	−0.0403454
$970$	0	0
$971$	−20.9338	−0.671799	−0.335899	−	0.941898i	$-0.609040\pi$
$971$	−20.9338	−0.671799	−0.335899	+	0.941898i	$0.609040\pi$
$972$	0	0
$973$	−7.37184	−0.236330
$974$	0	0
$975$	−32.0660	−1.02693
$976$	0	0
$977$	−55.7692	−1.78422	−0.892108	−	0.451823i	$-0.850774\pi$
$977$	−55.7692	−1.78422	−0.892108	+	0.451823i	$0.850774\pi$
$978$	0	0
$979$	−4.66282	−0.149024
$980$	0	0
$981$	28.2644	0.902414
$982$	0	0
$983$	−46.8831	−1.49534	−0.747670	−	0.664070i	$-0.768828\pi$
$983$	−46.8831	−1.49534	−0.747670	+	0.664070i	$0.768828\pi$
$984$	0	0
$985$	−7.77352	−0.247685
$986$	0	0
$987$	−1.52576	−0.0485654
$988$	0	0
$989$	−24.8400	−0.789867
$990$	0	0
$991$	35.2300	1.11912	0.559559	−	0.828791i	$-0.310971\pi$
$991$	35.2300	1.11912	0.559559	+	0.828791i	$0.310971\pi$
$992$	0	0
$993$	27.4513	0.871140
$994$	0	0
$995$	11.3067	0.358447
$996$	0	0
$997$	−28.6068	−0.905986	−0.452993	−	0.891514i	$-0.649644\pi$
$997$	−28.6068	−0.905986	−0.452993	+	0.891514i	$0.649644\pi$
$998$	0	0
$999$	49.0713	1.55255

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3344.2.a.w.1.3		6
4.3	odd	2		836.2.a.e.1.4	✓	6
12.11	even	2		7524.2.a.q.1.2		6
44.43	even	2		9196.2.a.n.1.4		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
836.2.a.e.1.4	✓	6	4.3	odd	2
3344.2.a.w.1.3		6	1.1	even	1	trivial
7524.2.a.q.1.2		6	12.11	even	2
9196.2.a.n.1.4		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)

\(f(q)\)	\(=\)	\(q-0.837309 q^{3} +3.44987 q^{5} -0.535976 q^{7} -2.29891 q^{9} +O(q^{10})\)	\(q-0.837309 q^{3} +3.44987 q^{5} -0.535976 q^{7} -2.29891 q^{9} +1.00000 q^{11} +5.54895 q^{13} -2.88861 q^{15} +1.49993 q^{17} +1.00000 q^{19} +0.448778 q^{21} -3.26301 q^{23} +6.90158 q^{25} +4.43683 q^{27} -3.17227 q^{29} +0.888605 q^{31} -0.837309 q^{33} -1.84905 q^{35} +11.0600 q^{37} -4.64619 q^{39} -8.44868 q^{41} +7.61261 q^{43} -7.93094 q^{45} -3.39981 q^{47} -6.71273 q^{49} -1.25590 q^{51} +7.22511 q^{53} +3.44987 q^{55} -0.837309 q^{57} +1.36560 q^{59} -2.57435 q^{61} +1.23216 q^{63} +19.1431 q^{65} +10.8862 q^{67} +2.73215 q^{69} -2.66509 q^{71} +9.82294 q^{73} -5.77876 q^{75} -0.535976 q^{77} +2.17469 q^{79} +3.18174 q^{81} -4.71288 q^{83} +5.17454 q^{85} +2.65617 q^{87} -4.66282 q^{89} -2.97411 q^{91} -0.744038 q^{93} +3.44987 q^{95} +6.93516 q^{97} -2.29891 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - q^{3} + 5 q^{5} + 2 q^{7} + 17 q^{9}+O(q^{10}) \) 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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