LMFDB - Embedded newform 3344.2.a.bb.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("3344.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$26.7019744359$
Analytic rank:	$0$
Dimension:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - x^{8} - 22x^{7} + 22x^{6} + 152x^{5} - 136x^{4} - 341x^{3} + 169x^{2} + 196x + 32 $ x^9 - x^8 - 22x^7 + 22x^6 + 152x^5 - 136x^4 - 341x^3 + 169x^2 + 196*x + 32
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 1672)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$1.21469$ 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)

$f(q)$	$=$	$q-1.21469 q^{3} -1.18050 q^{5} -1.93448 q^{7} -1.52452 q^{9} +O(q^{10})$	$q-1.21469 q^{3} -1.18050 q^{5} -1.93448 q^{7} -1.52452 q^{9} -1.00000 q^{11} +0.786058 q^{13} +1.43395 q^{15} +7.49665 q^{17} +1.00000 q^{19} +2.34980 q^{21} -5.07727 q^{23} -3.60641 q^{25} +5.49590 q^{27} -7.06039 q^{29} -9.73002 q^{31} +1.21469 q^{33} +2.28366 q^{35} -2.45901 q^{37} -0.954818 q^{39} +8.04604 q^{41} -8.59832 q^{43} +1.79971 q^{45} -6.36101 q^{47} -3.25778 q^{49} -9.10612 q^{51} +0.239492 q^{53} +1.18050 q^{55} -1.21469 q^{57} +0.984896 q^{59} +13.1888 q^{61} +2.94917 q^{63} -0.927945 q^{65} +1.22103 q^{67} +6.16732 q^{69} -3.03786 q^{71} +7.19781 q^{73} +4.38068 q^{75} +1.93448 q^{77} +3.63428 q^{79} -2.10225 q^{81} +9.82280 q^{83} -8.84982 q^{85} +8.57620 q^{87} +0.592321 q^{89} -1.52062 q^{91} +11.8190 q^{93} -1.18050 q^{95} -4.59010 q^{97} +1.52452 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q - q^{3} + 6 q^{5} - 3 q^{7} + 18 q^{9}+O(q^{10}) $ 9 * q - q^3 + 6 * q^5 - 3 * q^7 + 18 * q^9	$ 9 q - q^{3} + 6 q^{5} - 3 q^{7} + 18 q^{9} - 9 q^{11} + q^{13} - 6 q^{15} + 7 q^{17} + 9 q^{19} + 3 q^{21} - 13 q^{23} + 31 q^{25} + 5 q^{27} + 9 q^{29} + 4 q^{31} + q^{33} + 4 q^{35} + 24 q^{37} - 13 q^{39} - 6 q^{41} + 14 q^{43} + 26 q^{45} - 24 q^{47} + 20 q^{49} + 33 q^{51} + 19 q^{53} - 6 q^{55} - q^{57} + 19 q^{59} + 28 q^{61} - 16 q^{63} + 16 q^{65} - 5 q^{67} + 35 q^{69} - 16 q^{71} + 15 q^{73} - 3 q^{75} + 3 q^{77} - 2 q^{79} + 37 q^{81} - 8 q^{83} + 20 q^{85} - 23 q^{87} + 12 q^{89} + 29 q^{91} + 44 q^{93} + 6 q^{95} - 4 q^{97} - 18 q^{99}+O(q^{100}) $ 9 * q - q^3 + 6 * q^5 - 3 * q^7 + 18 * q^9 - 9 * q^11 + q^13 - 6 * q^15 + 7 * q^17 + 9 * q^19 + 3 * q^21 - 13 * q^23 + 31 * q^25 + 5 * q^27 + 9 * q^29 + 4 * q^31 + q^33 + 4 * q^35 + 24 * q^37 - 13 * q^39 - 6 * q^41 + 14 * q^43 + 26 * q^45 - 24 * q^47 + 20 * q^49 + 33 * q^51 + 19 * q^53 - 6 * q^55 - q^57 + 19 * q^59 + 28 * q^61 - 16 * q^63 + 16 * q^65 - 5 * q^67 + 35 * q^69 - 16 * q^71 + 15 * q^73 - 3 * q^75 + 3 * q^77 - 2 * q^79 + 37 * q^81 - 8 * q^83 + 20 * q^85 - 23 * q^87 + 12 * q^89 + 29 * q^91 + 44 * q^93 + 6 * q^95 - 4 * q^97 - 18 * 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
$3$	−1.21469	−0.701302	−0.350651	−	0.936506i	$-0.614040\pi$
$3$	−1.21469	−0.701302	−0.350651	+	0.936506i	$0.614040\pi$
$4$	0	0
$5$	−1.18050	−0.527937	−0.263969	−	0.964531i	$-0.585032\pi$
$5$	−1.18050	−0.527937	−0.263969	+	0.964531i	$0.585032\pi$
$6$	0	0
$7$	−1.93448	−0.731166	−0.365583	−	0.930779i	$-0.619130\pi$
$7$	−1.93448	−0.731166	−0.365583	+	0.930779i	$0.619130\pi$
$8$	0	0
$9$	−1.52452	−0.508175
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	0.786058	0.218013	0.109007	−	0.994041i	$-0.465233\pi$
$13$	0.786058	0.218013	0.109007	+	0.994041i	$0.465233\pi$
$14$	0	0
$15$	1.43395	0.370244
$16$	0	0
$17$	7.49665	1.81820	0.909102	−	0.416573i	$-0.136769\pi$
$17$	7.49665	1.81820	0.909102	+	0.416573i	$0.136769\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	2.34980	0.512768
$22$	0	0
$23$	−5.07727	−1.05868	−0.529342	−	0.848408i	$-0.677561\pi$
$23$	−5.07727	−1.05868	−0.529342	+	0.848408i	$0.677561\pi$
$24$	0	0
$25$	−3.60641	−0.721282
$26$	0	0
$27$	5.49590	1.05769
$28$	0	0
$29$	−7.06039	−1.31108	−0.655541	−	0.755160i	$-0.727559\pi$
$29$	−7.06039	−1.31108	−0.655541	+	0.755160i	$0.727559\pi$
$30$	0	0
$31$	−9.73002	−1.74756	−0.873781	−	0.486319i	$-0.838339\pi$
$31$	−9.73002	−1.74756	−0.873781	+	0.486319i	$0.838339\pi$
$32$	0	0
$33$	1.21469	0.211451
$34$	0	0
$35$	2.28366	0.386010
$36$	0	0
$37$	−2.45901	−0.404258	−0.202129	−	0.979359i	$-0.564786\pi$
$37$	−2.45901	−0.404258	−0.202129	+	0.979359i	$0.564786\pi$
$38$	0	0
$39$	−0.954818	−0.152893
$40$	0	0
$41$	8.04604	1.25658	0.628290	−	0.777979i	$-0.283755\pi$
$41$	8.04604	1.25658	0.628290	+	0.777979i	$0.283755\pi$
$42$	0	0
$43$	−8.59832	−1.31123	−0.655616	−	0.755095i	$-0.727591\pi$
$43$	−8.59832	−1.31123	−0.655616	+	0.755095i	$0.727591\pi$
$44$	0	0
$45$	1.79971	0.268285
$46$	0	0
$47$	−6.36101	−0.927848	−0.463924	−	0.885875i	$-0.653559\pi$
$47$	−6.36101	−0.927848	−0.463924	+	0.885875i	$0.653559\pi$
$48$	0	0
$49$	−3.25778	−0.465397
$50$	0	0
$51$	−9.10612	−1.27511
$52$	0	0
$53$	0.239492	0.0328968	0.0164484	−	0.999865i	$-0.494764\pi$
$53$	0.239492	0.0328968	0.0164484	+	0.999865i	$0.494764\pi$
$54$	0	0
$55$	1.18050	0.159179
$56$	0	0
$57$	−1.21469	−0.160890
$58$	0	0
$59$	0.984896	0.128223	0.0641113	−	0.997943i	$-0.479579\pi$
$59$	0.984896	0.128223	0.0641113	+	0.997943i	$0.479579\pi$
$60$	0	0
$61$	13.1888	1.68865	0.844325	−	0.535831i	$-0.180002\pi$
$61$	13.1888	1.68865	0.844325	+	0.535831i	$0.180002\pi$
$62$	0	0
$63$	2.94917	0.371560
$64$	0	0
$65$	−0.927945	−0.115097
$66$	0	0
$67$	1.22103	0.149172	0.0745861	−	0.997215i	$-0.476236\pi$
$67$	1.22103	0.149172	0.0745861	+	0.997215i	$0.476236\pi$
$68$	0	0
$69$	6.16732	0.742458
$70$	0	0
$71$	−3.03786	−0.360527	−0.180264	−	0.983618i	$-0.557695\pi$
$71$	−3.03786	−0.360527	−0.180264	+	0.983618i	$0.557695\pi$
$72$	0	0
$73$	7.19781	0.842440	0.421220	−	0.906958i	$-0.361602\pi$
$73$	7.19781	0.842440	0.421220	+	0.906958i	$0.361602\pi$
$74$	0	0
$75$	4.38068	0.505837
$76$	0	0
$77$	1.93448	0.220455
$78$	0	0
$79$	3.63428	0.408889	0.204444	−	0.978878i	$-0.434461\pi$
$79$	3.63428	0.408889	0.204444	+	0.978878i	$0.434461\pi$
$80$	0	0
$81$	−2.10225	−0.233583
$82$	0	0
$83$	9.82280	1.07819	0.539096	−	0.842244i	$-0.318766\pi$
$83$	9.82280	1.07819	0.539096	+	0.842244i	$0.318766\pi$
$84$	0	0
$85$	−8.84982	−0.959898
$86$	0	0
$87$	8.57620	0.919465
$88$	0	0
$89$	0.592321	0.0627859	0.0313929	−	0.999507i	$-0.490006\pi$
$89$	0.592321	0.0627859	0.0313929	+	0.999507i	$0.490006\pi$
$90$	0	0
$91$	−1.52062	−0.159404
$92$	0	0
$93$	11.8190	1.22557
$94$	0	0
$95$	−1.18050	−0.121117
$96$	0	0
$97$	−4.59010	−0.466054	−0.233027	−	0.972470i	$-0.574863\pi$
$97$	−4.59010	−0.466054	−0.233027	+	0.972470i	$0.574863\pi$
$98$	0	0
$99$	1.52452	0.153221
$100$	0	0
$101$	16.7341	1.66510	0.832552	−	0.553947i	$-0.186879\pi$
$101$	16.7341	1.66510	0.832552	+	0.553947i	$0.186879\pi$
$102$	0	0
$103$	13.8986	1.36947	0.684734	−	0.728794i	$-0.259919\pi$
$103$	13.8986	1.36947	0.684734	+	0.728794i	$0.259919\pi$
$104$	0	0
$105$	−2.77395	−0.270710
$106$	0	0
$107$	6.76993	0.654474	0.327237	−	0.944942i	$-0.393883\pi$
$107$	6.76993	0.654474	0.327237	+	0.944942i	$0.393883\pi$
$108$	0	0
$109$	9.68985	0.928120	0.464060	−	0.885804i	$-0.346392\pi$
$109$	9.68985	0.928120	0.464060	+	0.885804i	$0.346392\pi$
$110$	0	0
$111$	2.98694	0.283507
$112$	0	0
$113$	12.2999	1.15708	0.578539	−	0.815655i	$-0.303623\pi$
$113$	12.2999	1.15708	0.578539	+	0.815655i	$0.303623\pi$
$114$	0	0
$115$	5.99374	0.558919
$116$	0	0
$117$	−1.19837	−0.110789
$118$	0	0
$119$	−14.5021	−1.32941
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−9.77345	−0.881243
$124$	0	0
$125$	10.1599	0.908729
$126$	0	0
$127$	9.02657	0.800979	0.400489	−	0.916301i	$-0.368840\pi$
$127$	9.02657	0.800979	0.400489	+	0.916301i	$0.368840\pi$
$128$	0	0
$129$	10.4443	0.919570
$130$	0	0
$131$	3.34314	0.292091	0.146046	−	0.989278i	$-0.453345\pi$
$131$	3.34314	0.292091	0.146046	+	0.989278i	$0.453345\pi$
$132$	0	0
$133$	−1.93448	−0.167741
$134$	0	0
$135$	−6.48793	−0.558392
$136$	0	0
$137$	−14.2710	−1.21925	−0.609625	−	0.792690i	$-0.708680\pi$
$137$	−14.2710	−1.21925	−0.609625	+	0.792690i	$0.708680\pi$
$138$	0	0
$139$	−0.833003	−0.0706544	−0.0353272	−	0.999376i	$-0.511247\pi$
$139$	−0.833003	−0.0706544	−0.0353272	+	0.999376i	$0.511247\pi$
$140$	0	0
$141$	7.72666	0.650702
$142$	0	0
$143$	−0.786058	−0.0657335
$144$	0	0
$145$	8.33482	0.692169
$146$	0	0
$147$	3.95719	0.326384
$148$	0	0
$149$	−1.15201	−0.0943765	−0.0471883	−	0.998886i	$-0.515026\pi$
$149$	−1.15201	−0.0943765	−0.0471883	+	0.998886i	$0.515026\pi$
$150$	0	0
$151$	19.2502	1.56656	0.783278	−	0.621671i	$-0.213546\pi$
$151$	19.2502	1.56656	0.783278	+	0.621671i	$0.213546\pi$
$152$	0	0
$153$	−11.4288	−0.923966
$154$	0	0
$155$	11.4863	0.922604
$156$	0	0
$157$	16.5659	1.32210	0.661052	−	0.750340i	$-0.270110\pi$
$157$	16.5659	1.32210	0.661052	+	0.750340i	$0.270110\pi$
$158$	0	0
$159$	−0.290909	−0.0230706
$160$	0	0
$161$	9.82190	0.774074
$162$	0	0
$163$	19.3659	1.51685	0.758426	−	0.651759i	$-0.225969\pi$
$163$	19.3659	1.51685	0.758426	+	0.651759i	$0.225969\pi$
$164$	0	0
$165$	−1.43395	−0.111633
$166$	0	0
$167$	−15.4695	−1.19707	−0.598533	−	0.801098i	$-0.704249\pi$
$167$	−15.4695	−1.19707	−0.598533	+	0.801098i	$0.704249\pi$
$168$	0	0
$169$	−12.3821	−0.952470
$170$	0	0
$171$	−1.52452	−0.116583
$172$	0	0
$173$	4.62605	0.351712	0.175856	−	0.984416i	$-0.443731\pi$
$173$	4.62605	0.351712	0.175856	+	0.984416i	$0.443731\pi$
$174$	0	0
$175$	6.97654	0.527377
$176$	0	0
$177$	−1.19634	−0.0899228
$178$	0	0
$179$	−10.4526	−0.781268	−0.390634	−	0.920546i	$-0.627744\pi$
$179$	−10.4526	−0.781268	−0.390634	+	0.920546i	$0.627744\pi$
$180$	0	0
$181$	−20.2543	−1.50549	−0.752744	−	0.658314i	$-0.771270\pi$
$181$	−20.2543	−1.50549	−0.752744	+	0.658314i	$0.771270\pi$
$182$	0	0
$183$	−16.0203	−1.18425
$184$	0	0
$185$	2.90287	0.213423
$186$	0	0
$187$	−7.49665	−0.548209
$188$	0	0
$189$	−10.6317	−0.773344
$190$	0	0
$191$	−13.1077	−0.948438	−0.474219	−	0.880407i	$-0.657269\pi$
$191$	−13.1077	−0.948438	−0.474219	+	0.880407i	$0.657269\pi$
$192$	0	0
$193$	−16.0363	−1.15432	−0.577159	−	0.816632i	$-0.695839\pi$
$193$	−16.0363	−1.15432	−0.577159	+	0.816632i	$0.695839\pi$
$194$	0	0
$195$	1.12717	0.0807181
$196$	0	0
$197$	−12.0198	−0.856376	−0.428188	−	0.903690i	$-0.640848\pi$
$197$	−12.0198	−0.856376	−0.428188	+	0.903690i	$0.640848\pi$
$198$	0	0
$199$	−13.9882	−0.991597	−0.495799	−	0.868437i	$-0.665125\pi$
$199$	−13.9882	−0.991597	−0.495799	+	0.868437i	$0.665125\pi$
$200$	0	0
$201$	−1.48317	−0.104615
$202$	0	0
$203$	13.6582	0.958618
$204$	0	0
$205$	−9.49838	−0.663395
$206$	0	0
$207$	7.74043	0.537997
$208$	0	0
$209$	−1.00000	−0.0691714
$210$	0	0
$211$	25.6885	1.76847	0.884236	−	0.467040i	$-0.154680\pi$
$211$	25.6885	1.76847	0.884236	+	0.467040i	$0.154680\pi$
$212$	0	0
$213$	3.69006	0.252839
$214$	0	0
$215$	10.1504	0.692248
$216$	0	0
$217$	18.8226	1.27776
$218$	0	0
$219$	−8.74312	−0.590805
$220$	0	0
$221$	5.89280	0.396393
$222$	0	0
$223$	4.82069	0.322817	0.161409	−	0.986888i	$-0.448396\pi$
$223$	4.82069	0.322817	0.161409	+	0.986888i	$0.448396\pi$
$224$	0	0
$225$	5.49806	0.366538
$226$	0	0
$227$	18.2570	1.21176	0.605880	−	0.795556i	$-0.292821\pi$
$227$	18.2570	1.21176	0.605880	+	0.795556i	$0.292821\pi$
$228$	0	0
$229$	−11.3387	−0.749280	−0.374640	−	0.927170i	$-0.622234\pi$
$229$	−11.3387	−0.749280	−0.374640	+	0.927170i	$0.622234\pi$
$230$	0	0
$231$	−2.34980	−0.154605
$232$	0	0
$233$	−4.57761	−0.299889	−0.149945	−	0.988694i	$-0.547910\pi$
$233$	−4.57761	−0.299889	−0.149945	+	0.988694i	$0.547910\pi$
$234$	0	0
$235$	7.50919	0.489846
$236$	0	0
$237$	−4.41453	−0.286755
$238$	0	0
$239$	−11.8589	−0.767085	−0.383543	−	0.923523i	$-0.625296\pi$
$239$	−11.8589	−0.767085	−0.383543	+	0.923523i	$0.625296\pi$
$240$	0	0
$241$	26.3965	1.70035	0.850173	−	0.526503i	$-0.176497\pi$
$241$	26.3965	1.70035	0.850173	+	0.526503i	$0.176497\pi$
$242$	0	0
$243$	−13.9341	−0.893874
$244$	0	0
$245$	3.84582	0.245700
$246$	0	0
$247$	0.786058	0.0500157
$248$	0	0
$249$	−11.9317	−0.756139
$250$	0	0
$251$	3.84570	0.242739	0.121369	−	0.992607i	$-0.461271\pi$
$251$	3.84570	0.242739	0.121369	+	0.992607i	$0.461271\pi$
$252$	0	0
$253$	5.07727	0.319205
$254$	0	0
$255$	10.7498	0.673179
$256$	0	0
$257$	25.5065	1.59105	0.795526	−	0.605919i	$-0.207194\pi$
$257$	25.5065	1.59105	0.795526	+	0.605919i	$0.207194\pi$
$258$	0	0
$259$	4.75691	0.295580
$260$	0	0
$261$	10.7637	0.666259
$262$	0	0
$263$	−13.1212	−0.809085	−0.404542	−	0.914519i	$-0.632569\pi$
$263$	−13.1212	−0.809085	−0.404542	+	0.914519i	$0.632569\pi$
$264$	0	0
$265$	−0.282722	−0.0173675
$266$	0	0
$267$	−0.719487	−0.0440319
$268$	0	0
$269$	9.74871	0.594389	0.297195	−	0.954817i	$-0.403949\pi$
$269$	9.74871	0.594389	0.297195	+	0.954817i	$0.403949\pi$
$270$	0	0
$271$	22.1707	1.34677	0.673387	−	0.739290i	$-0.264839\pi$
$271$	22.1707	1.34677	0.673387	+	0.739290i	$0.264839\pi$
$272$	0	0
$273$	1.84708	0.111790
$274$	0	0
$275$	3.60641	0.217475
$276$	0	0
$277$	−24.2488	−1.45697	−0.728484	−	0.685063i	$-0.759775\pi$
$277$	−24.2488	−1.45697	−0.728484	+	0.685063i	$0.759775\pi$
$278$	0	0
$279$	14.8337	0.888068
$280$	0	0
$281$	−20.8172	−1.24185	−0.620925	−	0.783870i	$-0.713243\pi$
$281$	−20.8172	−1.24185	−0.620925	+	0.783870i	$0.713243\pi$
$282$	0	0
$283$	24.9778	1.48478	0.742389	−	0.669970i	$-0.233693\pi$
$283$	24.9778	1.48478	0.742389	+	0.669970i	$0.233693\pi$
$284$	0	0
$285$	1.43395	0.0849397
$286$	0	0
$287$	−15.5649	−0.918768
$288$	0	0
$289$	39.1998	2.30587
$290$	0	0
$291$	5.57556	0.326845
$292$	0	0
$293$	13.1398	0.767635	0.383818	−	0.923409i	$-0.374609\pi$
$293$	13.1398	0.767635	0.383818	+	0.923409i	$0.374609\pi$
$294$	0	0
$295$	−1.16267	−0.0676935
$296$	0	0
$297$	−5.49590	−0.318905
$298$	0	0
$299$	−3.99103	−0.230807
$300$	0	0
$301$	16.6333	0.958728
$302$	0	0
$303$	−20.3268	−1.16774
$304$	0	0
$305$	−15.5694	−0.891502
$306$	0	0
$307$	25.1019	1.43264	0.716322	−	0.697770i	$-0.245824\pi$
$307$	25.1019	1.43264	0.716322	+	0.697770i	$0.245824\pi$
$308$	0	0
$309$	−16.8825	−0.960411
$310$	0	0
$311$	−2.84536	−0.161345	−0.0806727	−	0.996741i	$-0.525707\pi$
$311$	−2.84536	−0.161345	−0.0806727	+	0.996741i	$0.525707\pi$
$312$	0	0
$313$	−1.21125	−0.0684641	−0.0342321	−	0.999414i	$-0.510899\pi$
$313$	−1.21125	−0.0684641	−0.0342321	+	0.999414i	$0.510899\pi$
$314$	0	0
$315$	−3.48150	−0.196160
$316$	0	0
$317$	−11.5130	−0.646635	−0.323318	−	0.946290i	$-0.604798\pi$
$317$	−11.5130	−0.646635	−0.323318	+	0.946290i	$0.604798\pi$
$318$	0	0
$319$	7.06039	0.395306
$320$	0	0
$321$	−8.22337	−0.458984
$322$	0	0
$323$	7.49665	0.417125
$324$	0	0
$325$	−2.83485	−0.157249
$326$	0	0
$327$	−11.7702	−0.650893
$328$	0	0
$329$	12.3053	0.678411
$330$	0	0
$331$	−15.0011	−0.824536	−0.412268	−	0.911063i	$-0.635263\pi$
$331$	−15.0011	−0.824536	−0.412268	+	0.911063i	$0.635263\pi$
$332$	0	0
$333$	3.74882	0.205434
$334$	0	0
$335$	−1.44143	−0.0787536
$336$	0	0
$337$	−22.5828	−1.23017	−0.615083	−	0.788463i	$-0.710877\pi$
$337$	−22.5828	−1.23017	−0.615083	+	0.788463i	$0.710877\pi$
$338$	0	0
$339$	−14.9406	−0.811461
$340$	0	0
$341$	9.73002	0.526910
$342$	0	0
$343$	19.8435	1.07145
$344$	0	0
$345$	−7.28054	−0.391971
$346$	0	0
$347$	7.04142	0.378003	0.189002	−	0.981977i	$-0.439475\pi$
$347$	7.04142	0.378003	0.189002	+	0.981977i	$0.439475\pi$
$348$	0	0
$349$	−30.2722	−1.62043	−0.810216	−	0.586132i	$-0.800650\pi$
$349$	−30.2722	−1.62043	−0.810216	+	0.586132i	$0.800650\pi$
$350$	0	0
$351$	4.32010	0.230590
$352$	0	0
$353$	4.17934	0.222444	0.111222	−	0.993796i	$-0.464524\pi$
$353$	4.17934	0.222444	0.111222	+	0.993796i	$0.464524\pi$
$354$	0	0
$355$	3.58620	0.190336
$356$	0	0
$357$	17.6156	0.932318
$358$	0	0
$359$	10.9787	0.579433	0.289716	−	0.957113i	$-0.406439\pi$
$359$	10.9787	0.579433	0.289716	+	0.957113i	$0.406439\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	−1.21469	−0.0637548
$364$	0	0
$365$	−8.49704	−0.444756
$366$	0	0
$367$	18.0910	0.944343	0.472172	−	0.881507i	$-0.343470\pi$
$367$	18.0910	0.944343	0.472172	+	0.881507i	$0.343470\pi$
$368$	0	0
$369$	−12.2664	−0.638562
$370$	0	0
$371$	−0.463294	−0.0240530
$372$	0	0
$373$	21.3770	1.10686	0.553430	−	0.832896i	$-0.313319\pi$
$373$	21.3770	1.10686	0.553430	+	0.832896i	$0.313319\pi$
$374$	0	0
$375$	−12.3411	−0.637294
$376$	0	0
$377$	−5.54988	−0.285833
$378$	0	0
$379$	−20.5700	−1.05661	−0.528305	−	0.849055i	$-0.677172\pi$
$379$	−20.5700	−1.05661	−0.528305	+	0.849055i	$0.677172\pi$
$380$	0	0
$381$	−10.9645	−0.561728
$382$	0	0
$383$	−27.2503	−1.39243	−0.696213	−	0.717835i	$-0.745133\pi$
$383$	−27.2503	−1.39243	−0.696213	+	0.717835i	$0.745133\pi$
$384$	0	0
$385$	−2.28366	−0.116386
$386$	0	0
$387$	13.1084	0.666335
$388$	0	0
$389$	24.5825	1.24638	0.623191	−	0.782070i	$-0.285836\pi$
$389$	24.5825	1.24638	0.623191	+	0.782070i	$0.285836\pi$
$390$	0	0
$391$	−38.0625	−1.92491
$392$	0	0
$393$	−4.06088	−0.204844
$394$	0	0
$395$	−4.29029	−0.215868
$396$	0	0
$397$	24.8553	1.24745	0.623725	−	0.781644i	$-0.285619\pi$
$397$	24.8553	1.24745	0.623725	+	0.781644i	$0.285619\pi$
$398$	0	0
$399$	2.34980	0.117637
$400$	0	0
$401$	−32.6460	−1.63026	−0.815131	−	0.579277i	$-0.803335\pi$
$401$	−32.6460	−1.63026	−0.815131	+	0.579277i	$0.803335\pi$
$402$	0	0
$403$	−7.64836	−0.380992
$404$	0	0
$405$	2.48171	0.123317
$406$	0	0
$407$	2.45901	0.121888
$408$	0	0
$409$	2.08901	0.103295	0.0516475	−	0.998665i	$-0.483553\pi$
$409$	2.08901	0.103295	0.0516475	+	0.998665i	$0.483553\pi$
$410$	0	0
$411$	17.3348	0.855063
$412$	0	0
$413$	−1.90526	−0.0937519
$414$	0	0
$415$	−11.5959	−0.569218
$416$	0	0
$417$	1.01184	0.0495501
$418$	0	0
$419$	38.4733	1.87954	0.939772	−	0.341802i	$-0.111037\pi$
$419$	38.4733	1.87954	0.939772	+	0.341802i	$0.111037\pi$
$420$	0	0
$421$	26.0630	1.27023	0.635116	−	0.772417i	$-0.280952\pi$
$421$	26.0630	1.27023	0.635116	+	0.772417i	$0.280952\pi$
$422$	0	0
$423$	9.69751	0.471509
$424$	0	0
$425$	−27.0360	−1.31144
$426$	0	0
$427$	−25.5135	−1.23468
$428$	0	0
$429$	0.954818	0.0460991
$430$	0	0
$431$	37.2173	1.79269	0.896346	−	0.443355i	$-0.146212\pi$
$431$	37.2173	1.79269	0.896346	+	0.443355i	$0.146212\pi$
$432$	0	0
$433$	−24.8180	−1.19268	−0.596339	−	0.802733i	$-0.703378\pi$
$433$	−24.8180	−1.19268	−0.596339	+	0.802733i	$0.703378\pi$
$434$	0	0
$435$	−10.1242	−0.485420
$436$	0	0
$437$	−5.07727	−0.242879
$438$	0	0
$439$	8.86740	0.423218	0.211609	−	0.977354i	$-0.432130\pi$
$439$	8.86740	0.423218	0.211609	+	0.977354i	$0.432130\pi$
$440$	0	0
$441$	4.96656	0.236503
$442$	0	0
$443$	8.96547	0.425962	0.212981	−	0.977056i	$-0.431683\pi$
$443$	8.96547	0.425962	0.212981	+	0.977056i	$0.431683\pi$
$444$	0	0
$445$	−0.699237	−0.0331470
$446$	0	0
$447$	1.39934	0.0661865
$448$	0	0
$449$	−0.587763	−0.0277383	−0.0138691	−	0.999904i	$-0.504415\pi$
$449$	−0.587763	−0.0277383	−0.0138691	+	0.999904i	$0.504415\pi$
$450$	0	0
$451$	−8.04604	−0.378873
$452$	0	0
$453$	−23.3830	−1.09863
$454$	0	0
$455$	1.79509	0.0841553
$456$	0	0
$457$	−18.1363	−0.848382	−0.424191	−	0.905573i	$-0.639441\pi$
$457$	−18.1363	−0.848382	−0.424191	+	0.905573i	$0.639441\pi$
$458$	0	0
$459$	41.2009	1.92309
$460$	0	0
$461$	−26.1946	−1.22000	−0.610002	−	0.792400i	$-0.708832\pi$
$461$	−26.1946	−1.22000	−0.610002	+	0.792400i	$0.708832\pi$
$462$	0	0
$463$	−19.0428	−0.884993	−0.442497	−	0.896770i	$-0.645907\pi$
$463$	−19.0428	−0.884993	−0.442497	+	0.896770i	$0.645907\pi$
$464$	0	0
$465$	−13.9523	−0.647024
$466$	0	0
$467$	22.7644	1.05341	0.526705	−	0.850048i	$-0.323427\pi$
$467$	22.7644	1.05341	0.526705	+	0.850048i	$0.323427\pi$
$468$	0	0
$469$	−2.36206	−0.109070
$470$	0	0
$471$	−20.1225	−0.927195
$472$	0	0
$473$	8.59832	0.395351
$474$	0	0
$475$	−3.60641	−0.165473
$476$	0	0
$477$	−0.365112	−0.0167173
$478$	0	0
$479$	−16.7847	−0.766910	−0.383455	−	0.923560i	$-0.625266\pi$
$479$	−16.7847	−0.766910	−0.383455	+	0.923560i	$0.625266\pi$
$480$	0	0
$481$	−1.93292	−0.0881337
$482$	0	0
$483$	−11.9306	−0.542860
$484$	0	0
$485$	5.41863	0.246047
$486$	0	0
$487$	−9.98318	−0.452381	−0.226191	−	0.974083i	$-0.572627\pi$
$487$	−9.98318	−0.452381	−0.226191	+	0.974083i	$0.572627\pi$
$488$	0	0
$489$	−23.5236	−1.06377
$490$	0	0
$491$	3.62090	0.163409	0.0817044	−	0.996657i	$-0.473964\pi$
$491$	3.62090	0.163409	0.0817044	+	0.996657i	$0.473964\pi$
$492$	0	0
$493$	−52.9293	−2.38382
$494$	0	0
$495$	−1.79971	−0.0808908
$496$	0	0
$497$	5.87668	0.263605
$498$	0	0
$499$	32.1814	1.44064	0.720320	−	0.693642i	$-0.243995\pi$
$499$	32.1814	1.44064	0.720320	+	0.693642i	$0.243995\pi$
$500$	0	0
$501$	18.7907	0.839505
$502$	0	0
$503$	5.98450	0.266836	0.133418	−	0.991060i	$-0.457405\pi$
$503$	5.98450	0.266836	0.133418	+	0.991060i	$0.457405\pi$
$504$	0	0
$505$	−19.7547	−0.879071
$506$	0	0
$507$	15.0404	0.667970
$508$	0	0
$509$	4.45458	0.197446	0.0987228	−	0.995115i	$-0.468524\pi$
$509$	4.45458	0.197446	0.0987228	+	0.995115i	$0.468524\pi$
$510$	0	0
$511$	−13.9240	−0.615963
$512$	0	0
$513$	5.49590	0.242650
$514$	0	0
$515$	−16.4073	−0.722993
$516$	0	0
$517$	6.36101	0.279757
$518$	0	0
$519$	−5.61923	−0.246657
$520$	0	0
$521$	−38.5195	−1.68757	−0.843784	−	0.536683i	$-0.819677\pi$
$521$	−38.5195	−1.68757	−0.843784	+	0.536683i	$0.819677\pi$
$522$	0	0
$523$	7.59684	0.332187	0.166093	−	0.986110i	$-0.446885\pi$
$523$	7.59684	0.332187	0.166093	+	0.986110i	$0.446885\pi$
$524$	0	0
$525$	−8.47434	−0.369851
$526$	0	0
$527$	−72.9425	−3.17743
$528$	0	0
$529$	2.77869	0.120813
$530$	0	0
$531$	−1.50150	−0.0651595
$532$	0	0
$533$	6.32466	0.273951
$534$	0	0
$535$	−7.99193	−0.345521
$536$	0	0
$537$	12.6967	0.547905
$538$	0	0
$539$	3.25778	0.140322
$540$	0	0
$541$	37.4307	1.60927	0.804635	−	0.593770i	$-0.202361\pi$
$541$	37.4307	1.60927	0.804635	+	0.593770i	$0.202361\pi$
$542$	0	0
$543$	24.6027	1.05580
$544$	0	0
$545$	−11.4389	−0.489989
$546$	0	0
$547$	27.1984	1.16292	0.581459	−	0.813576i	$-0.302482\pi$
$547$	27.1984	1.16292	0.581459	+	0.813576i	$0.302482\pi$
$548$	0	0
$549$	−20.1066	−0.858130
$550$	0	0
$551$	−7.06039	−0.300783
$552$	0	0
$553$	−7.03046	−0.298966
$554$	0	0
$555$	−3.52609	−0.149674
$556$	0	0
$557$	−16.8377	−0.713438	−0.356719	−	0.934212i	$-0.616105\pi$
$557$	−16.8377	−0.713438	−0.356719	+	0.934212i	$0.616105\pi$
$558$	0	0
$559$	−6.75878	−0.285866
$560$	0	0
$561$	9.10612	0.384461
$562$	0	0
$563$	−4.39952	−0.185418	−0.0927089	−	0.995693i	$-0.529553\pi$
$563$	−4.39952	−0.185418	−0.0927089	+	0.995693i	$0.529553\pi$
$564$	0	0
$565$	−14.5201	−0.610864
$566$	0	0
$567$	4.06677	0.170788
$568$	0	0
$569$	−19.6806	−0.825054	−0.412527	−	0.910945i	$-0.635354\pi$
$569$	−19.6806	−0.825054	−0.412527	+	0.910945i	$0.635354\pi$
$570$	0	0
$571$	0.587623	0.0245913	0.0122956	−	0.999924i	$-0.496086\pi$
$571$	0.587623	0.0245913	0.0122956	+	0.999924i	$0.496086\pi$
$572$	0	0
$573$	15.9218	0.665142
$574$	0	0
$575$	18.3107	0.763610
$576$	0	0
$577$	28.4328	1.18367	0.591837	−	0.806057i	$-0.298403\pi$
$577$	28.4328	1.18367	0.591837	+	0.806057i	$0.298403\pi$
$578$	0	0
$579$	19.4791	0.809526
$580$	0	0
$581$	−19.0020	−0.788338
$582$	0	0
$583$	−0.239492	−0.00991876
$584$	0	0
$585$	1.41467	0.0584896
$586$	0	0
$587$	−41.1284	−1.69755	−0.848774	−	0.528755i	$-0.822659\pi$
$587$	−41.1284	−1.69755	−0.848774	+	0.528755i	$0.822659\pi$
$588$	0	0
$589$	−9.73002	−0.400918
$590$	0	0
$591$	14.6004	0.600579
$592$	0	0
$593$	37.2932	1.53145	0.765725	−	0.643168i	$-0.222380\pi$
$593$	37.2932	1.53145	0.765725	+	0.643168i	$0.222380\pi$
$594$	0	0
$595$	17.1198	0.701845
$596$	0	0
$597$	16.9913	0.695410
$598$	0	0
$599$	16.7266	0.683430	0.341715	−	0.939804i	$-0.388992\pi$
$599$	16.7266	0.683430	0.341715	+	0.939804i	$0.388992\pi$
$600$	0	0
$601$	32.6145	1.33037	0.665186	−	0.746678i	$-0.268352\pi$
$601$	32.6145	1.33037	0.665186	+	0.746678i	$0.268352\pi$
$602$	0	0
$603$	−1.86149	−0.0758056
$604$	0	0
$605$	−1.18050	−0.0479943
$606$	0	0
$607$	−21.9849	−0.892339	−0.446169	−	0.894949i	$-0.647212\pi$
$607$	−21.9849	−0.892339	−0.446169	+	0.894949i	$0.647212\pi$
$608$	0	0
$609$	−16.5905	−0.672281
$610$	0	0
$611$	−5.00012	−0.202283
$612$	0	0
$613$	−20.9044	−0.844322	−0.422161	−	0.906521i	$-0.638728\pi$
$613$	−20.9044	−0.844322	−0.422161	+	0.906521i	$0.638728\pi$
$614$	0	0
$615$	11.5376	0.465241
$616$	0	0
$617$	−11.5896	−0.466582	−0.233291	−	0.972407i	$-0.574949\pi$
$617$	−11.5896	−0.466582	−0.233291	+	0.972407i	$0.574949\pi$
$618$	0	0
$619$	26.3681	1.05982	0.529911	−	0.848053i	$-0.322225\pi$
$619$	26.3681	1.05982	0.529911	+	0.848053i	$0.322225\pi$
$620$	0	0
$621$	−27.9042	−1.11976
$622$	0	0
$623$	−1.14583	−0.0459069
$624$	0	0
$625$	6.03826	0.241530
$626$	0	0
$627$	1.21469	0.0485101
$628$	0	0
$629$	−18.4343	−0.735025
$630$	0	0
$631$	22.1053	0.879998	0.439999	−	0.897998i	$-0.354979\pi$
$631$	22.1053	0.879998	0.439999	+	0.897998i	$0.354979\pi$
$632$	0	0
$633$	−31.2037	−1.24023
$634$	0	0
$635$	−10.6559	−0.422867
$636$	0	0
$637$	−2.56080	−0.101463
$638$	0	0
$639$	4.63129	0.183211
$640$	0	0
$641$	27.7807	1.09727	0.548635	−	0.836062i	$-0.315148\pi$
$641$	27.7807	1.09727	0.548635	+	0.836062i	$0.315148\pi$
$642$	0	0
$643$	−13.7188	−0.541017	−0.270509	−	0.962718i	$-0.587192\pi$
$643$	−13.7188	−0.541017	−0.270509	+	0.962718i	$0.587192\pi$
$644$	0	0
$645$	−12.3295	−0.485475
$646$	0	0
$647$	−5.13914	−0.202040	−0.101020	−	0.994884i	$-0.532211\pi$
$647$	−5.13914	−0.202040	−0.101020	+	0.994884i	$0.532211\pi$
$648$	0	0
$649$	−0.984896	−0.0386605
$650$	0	0
$651$	−22.8636	−0.896095
$652$	0	0
$653$	−8.43851	−0.330224	−0.165112	−	0.986275i	$-0.552799\pi$
$653$	−8.43851	−0.330224	−0.165112	+	0.986275i	$0.552799\pi$
$654$	0	0
$655$	−3.94659	−0.154206
$656$	0	0
$657$	−10.9732	−0.428107
$658$	0	0
$659$	−37.7537	−1.47068	−0.735338	−	0.677701i	$-0.762977\pi$
$659$	−37.7537	−1.47068	−0.735338	+	0.677701i	$0.762977\pi$
$660$	0	0
$661$	−0.463752	−0.0180379	−0.00901894	−	0.999959i	$-0.502871\pi$
$661$	−0.463752	−0.0180379	−0.00901894	+	0.999959i	$0.502871\pi$
$662$	0	0
$663$	−7.15794	−0.277991
$664$	0	0
$665$	2.28366	0.0885567
$666$	0	0
$667$	35.8475	1.38802
$668$	0	0
$669$	−5.85565	−0.226393
$670$	0	0
$671$	−13.1888	−0.509147
$672$	0	0
$673$	−13.7221	−0.528949	−0.264475	−	0.964393i	$-0.585199\pi$
$673$	−13.7221	−0.528949	−0.264475	+	0.964393i	$0.585199\pi$
$674$	0	0
$675$	−19.8205	−0.762891
$676$	0	0
$677$	6.72462	0.258448	0.129224	−	0.991615i	$-0.458751\pi$
$677$	6.72462	0.258448	0.129224	+	0.991615i	$0.458751\pi$
$678$	0	0
$679$	8.87947	0.340763
$680$	0	0
$681$	−22.1766	−0.849810
$682$	0	0
$683$	−29.0691	−1.11230	−0.556149	−	0.831083i	$-0.687722\pi$
$683$	−29.0691	−1.11230	−0.556149	+	0.831083i	$0.687722\pi$
$684$	0	0
$685$	16.8469	0.643688
$686$	0	0
$687$	13.7730	0.525472
$688$	0	0
$689$	0.188255	0.00717194
$690$	0	0
$691$	−41.1286	−1.56461	−0.782303	−	0.622898i	$-0.785955\pi$
$691$	−41.1286	−1.56461	−0.782303	+	0.622898i	$0.785955\pi$
$692$	0	0
$693$	−2.94917	−0.112030
$694$	0	0
$695$	0.983363	0.0373011
$696$	0	0
$697$	60.3183	2.28472
$698$	0	0
$699$	5.56038	0.210313
$700$	0	0
$701$	37.0169	1.39811	0.699054	−	0.715069i	$-0.253605\pi$
$701$	37.0169	1.39811	0.699054	+	0.715069i	$0.253605\pi$
$702$	0	0
$703$	−2.45901	−0.0927432
$704$	0	0
$705$	−9.12135	−0.343530
$706$	0	0
$707$	−32.3718	−1.21747
$708$	0	0
$709$	28.2338	1.06034	0.530171	−	0.847891i	$-0.322128\pi$
$709$	28.2338	1.06034	0.530171	+	0.847891i	$0.322128\pi$
$710$	0	0
$711$	−5.54056	−0.207787
$712$	0	0
$713$	49.4019	1.85012
$714$	0	0
$715$	0.927945	0.0347032
$716$	0	0
$717$	14.4048	0.537959
$718$	0	0
$719$	−37.9993	−1.41713	−0.708567	−	0.705643i	$-0.750658\pi$
$719$	−37.9993	−1.41713	−0.708567	+	0.705643i	$0.750658\pi$
$720$	0	0
$721$	−26.8866	−1.00131
$722$	0	0
$723$	−32.0636	−1.19246
$724$	0	0
$725$	25.4627	0.945660
$726$	0	0
$727$	−40.1120	−1.48767	−0.743835	−	0.668363i	$-0.766995\pi$
$727$	−40.1120	−1.48767	−0.743835	+	0.668363i	$0.766995\pi$
$728$	0	0
$729$	23.2324	0.860459
$730$	0	0
$731$	−64.4586	−2.38409
$732$	0	0
$733$	−35.8896	−1.32561	−0.662807	−	0.748790i	$-0.730635\pi$
$733$	−35.8896	−1.32561	−0.662807	+	0.748790i	$0.730635\pi$
$734$	0	0
$735$	−4.67148	−0.172310
$736$	0	0
$737$	−1.22103	−0.0449771
$738$	0	0
$739$	47.0185	1.72960	0.864801	−	0.502115i	$-0.167444\pi$
$739$	47.0185	1.72960	0.864801	+	0.502115i	$0.167444\pi$
$740$	0	0
$741$	−0.954818	−0.0350761
$742$	0	0
$743$	−39.7543	−1.45844	−0.729222	−	0.684277i	$-0.760118\pi$
$743$	−39.7543	−1.45844	−0.729222	+	0.684277i	$0.760118\pi$
$744$	0	0
$745$	1.35995	0.0498249
$746$	0	0
$747$	−14.9751	−0.547910
$748$	0	0
$749$	−13.0963	−0.478529
$750$	0	0
$751$	6.02468	0.219844	0.109922	−	0.993940i	$-0.464940\pi$
$751$	6.02468	0.219844	0.109922	+	0.993940i	$0.464940\pi$
$752$	0	0
$753$	−4.67134	−0.170233
$754$	0	0
$755$	−22.7249	−0.827044
$756$	0	0
$757$	−18.7148	−0.680202	−0.340101	−	0.940389i	$-0.610461\pi$
$757$	−18.7148	−0.680202	−0.340101	+	0.940389i	$0.610461\pi$
$758$	0	0
$759$	−6.16732	−0.223859
$760$	0	0
$761$	17.6065	0.638236	0.319118	−	0.947715i	$-0.396613\pi$
$761$	17.6065	0.638236	0.319118	+	0.947715i	$0.396613\pi$
$762$	0	0
$763$	−18.7449	−0.678609
$764$	0	0
$765$	13.4918	0.487796
$766$	0	0
$767$	0.774186	0.0279542
$768$	0	0
$769$	54.4149	1.96225	0.981125	−	0.193375i	$-0.0619433\pi$
$769$	54.4149	1.96225	0.981125	+	0.193375i	$0.0619433\pi$
$770$	0	0
$771$	−30.9825	−1.11581
$772$	0	0
$773$	−28.4366	−1.02279	−0.511397	−	0.859345i	$-0.670872\pi$
$773$	−28.4366	−1.02279	−0.511397	+	0.859345i	$0.670872\pi$
$774$	0	0
$775$	35.0904	1.26049
$776$	0	0
$777$	−5.77818	−0.207291
$778$	0	0
$779$	8.04604	0.288279
$780$	0	0
$781$	3.03786	0.108703
$782$	0	0
$783$	−38.8032	−1.38671
$784$	0	0
$785$	−19.5561	−0.697989
$786$	0	0
$787$	8.42060	0.300162	0.150081	−	0.988674i	$-0.452047\pi$
$787$	8.42060	0.300162	0.150081	+	0.988674i	$0.452047\pi$
$788$	0	0
$789$	15.9382	0.567413
$790$	0	0
$791$	−23.7940	−0.846015
$792$	0	0
$793$	10.3672	0.368148
$794$	0	0
$795$	0.343420	0.0121798
$796$	0	0
$797$	−6.58933	−0.233406	−0.116703	−	0.993167i	$-0.537233\pi$
$797$	−6.58933	−0.233406	−0.116703	+	0.993167i	$0.537233\pi$
$798$	0	0
$799$	−47.6863	−1.68702
$800$	0	0
$801$	−0.903008	−0.0319062
$802$	0	0
$803$	−7.19781	−0.254005
$804$	0	0
$805$	−11.5948	−0.408662
$806$	0	0
$807$	−11.8417	−0.416847
$808$	0	0
$809$	25.0473	0.880616	0.440308	−	0.897847i	$-0.354869\pi$
$809$	25.0473	0.880616	0.440308	+	0.897847i	$0.354869\pi$
$810$	0	0
$811$	33.9452	1.19198	0.595988	−	0.802993i	$-0.296761\pi$
$811$	33.9452	1.19198	0.595988	+	0.802993i	$0.296761\pi$
$812$	0	0
$813$	−26.9306	−0.944496
$814$	0	0
$815$	−22.8615	−0.800803
$816$	0	0
$817$	−8.59832	−0.300817
$818$	0	0
$819$	2.31822	0.0810051
$820$	0	0
$821$	20.4398	0.713353	0.356676	−	0.934228i	$-0.383910\pi$
$821$	20.4398	0.713353	0.356676	+	0.934228i	$0.383910\pi$
$822$	0	0
$823$	28.0439	0.977548	0.488774	−	0.872410i	$-0.337444\pi$
$823$	28.0439	0.977548	0.488774	+	0.872410i	$0.337444\pi$
$824$	0	0
$825$	−4.38068	−0.152516
$826$	0	0
$827$	−4.03478	−0.140303	−0.0701516	−	0.997536i	$-0.522348\pi$
$827$	−4.03478	−0.140303	−0.0701516	+	0.997536i	$0.522348\pi$
$828$	0	0
$829$	−31.5704	−1.09648	−0.548242	−	0.836320i	$-0.684703\pi$
$829$	−31.5704	−1.09648	−0.548242	+	0.836320i	$0.684703\pi$
$830$	0	0
$831$	29.4548	1.02178
$832$	0	0
$833$	−24.4224	−0.846186
$834$	0	0
$835$	18.2618	0.631975
$836$	0	0
$837$	−53.4752	−1.84837
$838$	0	0
$839$	47.6222	1.64410	0.822050	−	0.569415i	$-0.192830\pi$
$839$	47.6222	1.64410	0.822050	+	0.569415i	$0.192830\pi$
$840$	0	0
$841$	20.8491	0.718936
$842$	0	0
$843$	25.2865	0.870913
$844$	0	0
$845$	14.6171	0.502845
$846$	0	0
$847$	−1.93448	−0.0664696
$848$	0	0
$849$	−30.3403	−1.04128
$850$	0	0
$851$	12.4851	0.427982
$852$	0	0
$853$	22.0686	0.755615	0.377808	−	0.925884i	$-0.376678\pi$
$853$	22.0686	0.755615	0.377808	+	0.925884i	$0.376678\pi$
$854$	0	0
$855$	1.79971	0.0615487
$856$	0	0
$857$	14.8070	0.505799	0.252899	−	0.967493i	$-0.418616\pi$
$857$	14.8070	0.505799	0.252899	+	0.967493i	$0.418616\pi$
$858$	0	0
$859$	34.9245	1.19161	0.595804	−	0.803130i	$-0.296834\pi$
$859$	34.9245	1.19161	0.595804	+	0.803130i	$0.296834\pi$
$860$	0	0
$861$	18.9066	0.644334
$862$	0	0
$863$	−19.3317	−0.658059	−0.329029	−	0.944320i	$-0.606722\pi$
$863$	−19.3317	−0.658059	−0.329029	+	0.944320i	$0.606722\pi$
$864$	0	0
$865$	−5.46107	−0.185682
$866$	0	0
$867$	−47.6156	−1.61711
$868$	0	0
$869$	−3.63428	−0.123285
$870$	0	0
$871$	0.959799	0.0325215
$872$	0	0
$873$	6.99772	0.236837
$874$	0	0
$875$	−19.6542	−0.664432
$876$	0	0
$877$	−1.86592	−0.0630077	−0.0315038	−	0.999504i	$-0.510030\pi$
$877$	−1.86592	−0.0630077	−0.0315038	+	0.999504i	$0.510030\pi$
$878$	0	0
$879$	−15.9608	−0.538344
$880$	0	0
$881$	31.7649	1.07019	0.535094	−	0.844793i	$-0.320276\pi$
$881$	31.7649	1.07019	0.535094	+	0.844793i	$0.320276\pi$
$882$	0	0
$883$	−27.8371	−0.936793	−0.468396	−	0.883518i	$-0.655168\pi$
$883$	−27.8371	−0.936793	−0.468396	+	0.883518i	$0.655168\pi$
$884$	0	0
$885$	1.41229	0.0474736
$886$	0	0
$887$	19.5057	0.654939	0.327469	−	0.944862i	$-0.393804\pi$
$887$	19.5057	0.654939	0.327469	+	0.944862i	$0.393804\pi$
$888$	0	0
$889$	−17.4617	−0.585648
$890$	0	0
$891$	2.10225	0.0704280
$892$	0	0
$893$	−6.36101	−0.212863
$894$	0	0
$895$	12.3394	0.412460
$896$	0	0
$897$	4.84787	0.161866
$898$	0	0
$899$	68.6977	2.29120
$900$	0	0
$901$	1.79539	0.0598131
$902$	0	0
$903$	−20.2043	−0.672358
$904$	0	0
$905$	23.9102	0.794803
$906$	0	0
$907$	−49.8378	−1.65484	−0.827418	−	0.561587i	$-0.810191\pi$
$907$	−49.8378	−1.65484	−0.827418	+	0.561587i	$0.810191\pi$
$908$	0	0
$909$	−25.5115	−0.846164
$910$	0	0
$911$	7.09487	0.235064	0.117532	−	0.993069i	$-0.462502\pi$
$911$	7.09487	0.235064	0.117532	+	0.993069i	$0.462502\pi$
$912$	0	0
$913$	−9.82280	−0.325087
$914$	0	0
$915$	18.9120	0.625212
$916$	0	0
$917$	−6.46724	−0.213567
$918$	0	0
$919$	−2.93251	−0.0967345	−0.0483672	−	0.998830i	$-0.515402\pi$
$919$	−2.93251	−0.0967345	−0.0483672	+	0.998830i	$0.515402\pi$
$920$	0	0
$921$	−30.4911	−1.00472
$922$	0	0
$923$	−2.38793	−0.0785998
$924$	0	0
$925$	8.86819	0.291584
$926$	0	0
$927$	−21.1887	−0.695929
$928$	0	0
$929$	45.9550	1.50774	0.753868	−	0.657026i	$-0.228186\pi$
$929$	45.9550	1.50774	0.753868	+	0.657026i	$0.228186\pi$
$930$	0	0
$931$	−3.25778	−0.106769
$932$	0	0
$933$	3.45623	0.113152
$934$	0	0
$935$	8.84982	0.289420
$936$	0	0
$937$	16.8686	0.551073	0.275536	−	0.961291i	$-0.411145\pi$
$937$	16.8686	0.551073	0.275536	+	0.961291i	$0.411145\pi$
$938$	0	0
$939$	1.47130	0.0480141
$940$	0	0
$941$	−6.22586	−0.202957	−0.101479	−	0.994838i	$-0.532357\pi$
$941$	−6.22586	−0.202957	−0.101479	+	0.994838i	$0.532357\pi$
$942$	0	0
$943$	−40.8519	−1.33032
$944$	0	0
$945$	12.5508	0.408277
$946$	0	0
$947$	−25.9812	−0.844274	−0.422137	−	0.906532i	$-0.638720\pi$
$947$	−25.9812	−0.844274	−0.422137	+	0.906532i	$0.638720\pi$
$948$	0	0
$949$	5.65790	0.183663
$950$	0	0
$951$	13.9848	0.453487
$952$	0	0
$953$	−54.1693	−1.75472	−0.877359	−	0.479835i	$-0.840696\pi$
$953$	−54.1693	−1.75472	−0.877359	+	0.479835i	$0.840696\pi$
$954$	0	0
$955$	15.4737	0.500716
$956$	0	0
$957$	−8.57620	−0.277229
$958$	0	0
$959$	27.6069	0.891474
$960$	0	0
$961$	63.6732	2.05398
$962$	0	0
$963$	−10.3209	−0.332587
$964$	0	0
$965$	18.9309	0.609407
$966$	0	0
$967$	5.49212	0.176615	0.0883073	−	0.996093i	$-0.471854\pi$
$967$	5.49212	0.176615	0.0883073	+	0.996093i	$0.471854\pi$
$968$	0	0
$969$	−9.10612	−0.292531
$970$	0	0
$971$	20.7404	0.665592	0.332796	−	0.942999i	$-0.392008\pi$
$971$	20.7404	0.665592	0.332796	+	0.942999i	$0.392008\pi$
$972$	0	0
$973$	1.61143	0.0516601
$974$	0	0
$975$	3.44347	0.110279
$976$	0	0
$977$	18.7217	0.598962	0.299481	−	0.954102i	$-0.403187\pi$
$977$	18.7217	0.598962	0.299481	+	0.954102i	$0.403187\pi$
$978$	0	0
$979$	−0.592321	−0.0189307
$980$	0	0
$981$	−14.7724	−0.471647
$982$	0	0
$983$	3.60008	0.114825	0.0574123	−	0.998351i	$-0.481715\pi$
$983$	3.60008	0.114825	0.0574123	+	0.998351i	$0.481715\pi$
$984$	0	0
$985$	14.1894	0.452113
$986$	0	0
$987$	−14.9471	−0.475771
$988$	0	0
$989$	43.6560	1.38818
$990$	0	0
$991$	59.1996	1.88054	0.940269	−	0.340433i	$-0.110574\pi$
$991$	59.1996	1.88054	0.940269	+	0.340433i	$0.110574\pi$
$992$	0	0
$993$	18.2217	0.578249
$994$	0	0
$995$	16.5131	0.523501
$996$	0	0
$997$	−8.05033	−0.254957	−0.127478	−	0.991841i	$-0.540688\pi$
$997$	−8.05033	−0.254957	−0.127478	+	0.991841i	$0.540688\pi$
$998$	0	0
$999$	−13.5145	−0.427579

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3344.2.a.bb.1.4		9
4.3	odd	2		1672.2.a.k.1.6	✓	9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1672.2.a.k.1.6	✓	9	4.3	odd	2
3344.2.a.bb.1.4		9	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)

\(f(q)\)	\(=\)	\(q-1.21469 q^{3} -1.18050 q^{5} -1.93448 q^{7} -1.52452 q^{9} +O(q^{10})\)	\(q-1.21469 q^{3} -1.18050 q^{5} -1.93448 q^{7} -1.52452 q^{9} -1.00000 q^{11} +0.786058 q^{13} +1.43395 q^{15} +7.49665 q^{17} +1.00000 q^{19} +2.34980 q^{21} -5.07727 q^{23} -3.60641 q^{25} +5.49590 q^{27} -7.06039 q^{29} -9.73002 q^{31} +1.21469 q^{33} +2.28366 q^{35} -2.45901 q^{37} -0.954818 q^{39} +8.04604 q^{41} -8.59832 q^{43} +1.79971 q^{45} -6.36101 q^{47} -3.25778 q^{49} -9.10612 q^{51} +0.239492 q^{53} +1.18050 q^{55} -1.21469 q^{57} +0.984896 q^{59} +13.1888 q^{61} +2.94917 q^{63} -0.927945 q^{65} +1.22103 q^{67} +6.16732 q^{69} -3.03786 q^{71} +7.19781 q^{73} +4.38068 q^{75} +1.93448 q^{77} +3.63428 q^{79} -2.10225 q^{81} +9.82280 q^{83} -8.84982 q^{85} +8.57620 q^{87} +0.592321 q^{89} -1.52062 q^{91} +11.8190 q^{93} -1.18050 q^{95} -4.59010 q^{97} +1.52452 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q - q^{3} + 6 q^{5} - 3 q^{7} + 18 q^{9}+O(q^{10}) \) 9 * q - q^3 + 6 * q^5 - 3 * q^7 + 18 * q^9	\( 9 q - q^{3} + 6 q^{5} - 3 q^{7} + 18 q^{9} - 9 q^{11} + q^{13} - 6 q^{15} + 7 q^{17} + 9 q^{19} + 3 q^{21} - 13 q^{23} + 31 q^{25} + 5 q^{27} + 9 q^{29} + 4 q^{31} + q^{33} + 4 q^{35} + 24 q^{37} - 13 q^{39} - 6 q^{41} + 14 q^{43} + 26 q^{45} - 24 q^{47} + 20 q^{49} + 33 q^{51} + 19 q^{53} - 6 q^{55} - q^{57} + 19 q^{59} + 28 q^{61} - 16 q^{63} + 16 q^{65} - 5 q^{67} + 35 q^{69} - 16 q^{71} + 15 q^{73} - 3 q^{75} + 3 q^{77} - 2 q^{79} + 37 q^{81} - 8 q^{83} + 20 q^{85} - 23 q^{87} + 12 q^{89} + 29 q^{91} + 44 q^{93} + 6 q^{95} - 4 q^{97} - 18 q^{99}+O(q^{100}) \) 9 * q - q^3 + 6 * q^5 - 3 * q^7 + 18 * q^9 - 9 * q^11 + q^13 - 6 * q^15 + 7 * q^17 + 9 * q^19 + 3 * q^21 - 13 * q^23 + 31 * q^25 + 5 * q^27 + 9 * q^29 + 4 * q^31 + q^33 + 4 * q^35 + 24 * q^37 - 13 * q^39 - 6 * q^41 + 14 * q^43 + 26 * q^45 - 24 * q^47 + 20 * q^49 + 33 * q^51 + 19 * q^53 - 6 * q^55 - q^57 + 19 * q^59 + 28 * q^61 - 16 * q^63 + 16 * q^65 - 5 * q^67 + 35 * q^69 - 16 * q^71 + 15 * q^73 - 3 * q^75 + 3 * q^77 - 2 * q^79 + 37 * q^81 - 8 * q^83 + 20 * q^85 - 23 * q^87 + 12 * q^89 + 29 * q^91 + 44 * q^93 + 6 * q^95 - 4 * q^97 - 18 * q^99

Introduction

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Modular forms

Varieties

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Groups

Database

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