LMFDB - Embedded newform 3344.2.a.t.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:	$1$
Dimension:	$5$
Coefficient field:	5.5.246832.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 5x^{3} + 6x^{2} + 7x - 2 $ x^5 - 2x^4 - 5x^3 + 6x^2 + 7x - 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 209)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-1.51908$ 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)

$f(q)$	$=$	$q+0.563416 q^{3} +2.34577 q^{5} -1.69239 q^{7} -2.68256 q^{9} -1.00000 q^{11} +4.11168 q^{13} +1.32164 q^{15} -6.16499 q^{17} +1.00000 q^{19} -0.953520 q^{21} -3.52199 q^{23} +0.502638 q^{25} -3.20164 q^{27} -8.10336 q^{29} +2.30144 q^{31} -0.563416 q^{33} -3.96996 q^{35} -6.56016 q^{37} +2.31659 q^{39} +7.75013 q^{41} -7.75102 q^{43} -6.29268 q^{45} +10.8969 q^{47} -4.13581 q^{49} -3.47345 q^{51} +7.93511 q^{53} -2.34577 q^{55} +0.563416 q^{57} -10.9247 q^{59} -4.51162 q^{61} +4.53995 q^{63} +9.64506 q^{65} -14.7201 q^{67} -1.98435 q^{69} -3.12026 q^{71} +11.5827 q^{73} +0.283194 q^{75} +1.69239 q^{77} -4.96184 q^{79} +6.24383 q^{81} +1.82905 q^{83} -14.4617 q^{85} -4.56556 q^{87} -9.37496 q^{89} -6.95858 q^{91} +1.29666 q^{93} +2.34577 q^{95} -10.9937 q^{97} +2.68256 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - q^{3} - 5 q^{5} - 6 q^{7} + 4 q^{9} - 5 q^{11} + 4 q^{13} - 3 q^{15} - 4 q^{17} + 5 q^{19} + 10 q^{21} - 3 q^{23} + 6 q^{25} + 11 q^{27} + 10 q^{29} - 11 q^{31} + q^{33} + 8 q^{35} + q^{37} - 2 q^{39}+ \cdots - 4 q^{99}+O(q^{100}) $ 5 * q - q^3 - 5 * q^5 - 6 * q^7 + 4 * q^9 - 5 * q^11 + 4 * q^13 - 3 * q^15 - 4 * q^17 + 5 * q^19 + 10 * q^21 - 3 * q^23 + 6 * q^25 + 11 * q^27 + 10 * q^29 - 11 * q^31 + q^33 + 8 * q^35 + q^37 - 2 * q^39 + 2 * q^41 - 20 * q^43 - 28 * q^45 + 20 * q^47 + 3 * q^49 - 24 * q^51 - 14 * q^53 + 5 * q^55 - q^57 - 3 * q^59 - 10 * q^61 - 24 * q^63 - 9 * q^67 - 5 * q^69 - 23 * q^71 + 18 * q^75 + 6 * q^77 - 44 * q^79 + q^81 + 14 * q^83 - 12 * q^85 - 28 * q^87 - 27 * q^89 - 24 * q^91 - 27 * q^93 - 5 * q^95 + 15 * q^97 - 4 * q^99

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
$3$	0.563416	0.325288	0.162644	−	0.986685i	$-0.447998\pi$
$3$	0.563416	0.325288	0.162644	+	0.986685i	$0.447998\pi$
$4$	0	0
$5$	2.34577	1.04906	0.524530	−	0.851392i	$-0.324241\pi$
$5$	2.34577	1.04906	0.524530	+	0.851392i	$0.324241\pi$
$6$	0	0
$7$	−1.69239	−0.639664	−0.319832	−	0.947474i	$-0.603626\pi$
$7$	−1.69239	−0.639664	−0.319832	+	0.947474i	$0.603626\pi$
$8$	0	0
$9$	−2.68256	−0.894188
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	4.11168	1.14038	0.570188	−	0.821514i	$-0.306870\pi$
$13$	4.11168	1.14038	0.570188	+	0.821514i	$0.306870\pi$
$14$	0	0
$15$	1.32164	0.341247
$16$	0	0
$17$	−6.16499	−1.49523	−0.747615	−	0.664132i	$-0.768801\pi$
$17$	−6.16499	−1.49523	−0.747615	+	0.664132i	$0.768801\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	−0.953520	−0.208075
$22$	0	0
$23$	−3.52199	−0.734386	−0.367193	−	0.930145i	$-0.619681\pi$
$23$	−3.52199	−0.734386	−0.367193	+	0.930145i	$0.619681\pi$
$24$	0	0
$25$	0.502638	0.100528
$26$	0	0
$27$	−3.20164	−0.616157
$28$	0	0
$29$	−8.10336	−1.50476	−0.752378	−	0.658731i	$-0.771094\pi$
$29$	−8.10336	−1.50476	−0.752378	+	0.658731i	$0.771094\pi$
$30$	0	0
$31$	2.30144	0.413350	0.206675	−	0.978410i	$-0.433736\pi$
$31$	2.30144	0.413350	0.206675	+	0.978410i	$0.433736\pi$
$32$	0	0
$33$	−0.563416	−0.0980781
$34$	0	0
$35$	−3.96996	−0.671046
$36$	0	0
$37$	−6.56016	−1.07848	−0.539242	−	0.842151i	$-0.681289\pi$
$37$	−6.56016	−1.07848	−0.539242	+	0.842151i	$0.681289\pi$
$38$	0	0
$39$	2.31659	0.370951
$40$	0	0
$41$	7.75013	1.21037	0.605183	−	0.796086i	$-0.293100\pi$
$41$	7.75013	1.21037	0.605183	+	0.796086i	$0.293100\pi$
$42$	0	0
$43$	−7.75102	−1.18202	−0.591010	−	0.806664i	$-0.701271\pi$
$43$	−7.75102	−1.18202	−0.591010	+	0.806664i	$0.701271\pi$
$44$	0	0
$45$	−6.29268	−0.938057
$46$	0	0
$47$	10.8969	1.58948	0.794742	−	0.606948i	$-0.207606\pi$
$47$	10.8969	1.58948	0.794742	+	0.606948i	$0.207606\pi$
$48$	0	0
$49$	−4.13581	−0.590830
$50$	0	0
$51$	−3.47345	−0.486381
$52$	0	0
$53$	7.93511	1.08997	0.544986	−	0.838445i	$-0.316535\pi$
$53$	7.93511	1.08997	0.544986	+	0.838445i	$0.316535\pi$
$54$	0	0
$55$	−2.34577	−0.316304
$56$	0	0
$57$	0.563416	0.0746262
$58$	0	0
$59$	−10.9247	−1.42228	−0.711140	−	0.703051i	$-0.751821\pi$
$59$	−10.9247	−1.42228	−0.711140	+	0.703051i	$0.751821\pi$
$60$	0	0
$61$	−4.51162	−0.577653	−0.288827	−	0.957381i	$-0.593265\pi$
$61$	−4.51162	−0.577653	−0.288827	+	0.957381i	$0.593265\pi$
$62$	0	0
$63$	4.53995	0.571980
$64$	0	0
$65$	9.64506	1.19632
$66$	0	0
$67$	−14.7201	−1.79835	−0.899173	−	0.437594i	$-0.855831\pi$
$67$	−14.7201	−1.79835	−0.899173	+	0.437594i	$0.855831\pi$
$68$	0	0
$69$	−1.98435	−0.238887
$70$	0	0
$71$	−3.12026	−0.370307	−0.185153	−	0.982710i	$-0.559278\pi$
$71$	−3.12026	−0.370307	−0.185153	+	0.982710i	$0.559278\pi$
$72$	0	0
$73$	11.5827	1.35565	0.677824	−	0.735224i	$-0.262923\pi$
$73$	11.5827	1.35565	0.677824	+	0.735224i	$0.262923\pi$
$74$	0	0
$75$	0.283194	0.0327004
$76$	0	0
$77$	1.69239	0.192866
$78$	0	0
$79$	−4.96184	−0.558250	−0.279125	−	0.960255i	$-0.590044\pi$
$79$	−4.96184	−0.558250	−0.279125	+	0.960255i	$0.590044\pi$
$80$	0	0
$81$	6.24383	0.693759
$82$	0	0
$83$	1.82905	0.200765	0.100382	−	0.994949i	$-0.467993\pi$
$83$	1.82905	0.200765	0.100382	+	0.994949i	$0.467993\pi$
$84$	0	0
$85$	−14.4617	−1.56859
$86$	0	0
$87$	−4.56556	−0.489480
$88$	0	0
$89$	−9.37496	−0.993743	−0.496872	−	0.867824i	$-0.665518\pi$
$89$	−9.37496	−0.993743	−0.496872	+	0.867824i	$0.665518\pi$
$90$	0	0
$91$	−6.95858	−0.729457
$92$	0	0
$93$	1.29666	0.134458
$94$	0	0
$95$	2.34577	0.240671
$96$	0	0
$97$	−10.9937	−1.11625	−0.558123	−	0.829758i	$-0.688478\pi$
$97$	−10.9937	−1.11625	−0.558123	+	0.829758i	$0.688478\pi$
$98$	0	0
$99$	2.68256	0.269608
$100$	0	0
$101$	2.30621	0.229476	0.114738	−	0.993396i	$-0.463397\pi$
$101$	2.30621	0.229476	0.114738	+	0.993396i	$0.463397\pi$
$102$	0	0
$103$	−6.12000	−0.603021	−0.301511	−	0.953463i	$-0.597491\pi$
$103$	−6.12000	−0.603021	−0.301511	+	0.953463i	$0.597491\pi$
$104$	0	0
$105$	−2.23674	−0.218283
$106$	0	0
$107$	−0.422947	−0.0408878	−0.0204439	−	0.999791i	$-0.506508\pi$
$107$	−0.422947	−0.0408878	−0.0204439	+	0.999791i	$0.506508\pi$
$108$	0	0
$109$	14.0180	1.34268	0.671338	−	0.741151i	$-0.265720\pi$
$109$	14.0180	1.34268	0.671338	+	0.741151i	$0.265720\pi$
$110$	0	0
$111$	−3.69609	−0.350818
$112$	0	0
$113$	−2.53638	−0.238602	−0.119301	−	0.992858i	$-0.538065\pi$
$113$	−2.53638	−0.238602	−0.119301	+	0.992858i	$0.538065\pi$
$114$	0	0
$115$	−8.26179	−0.770416
$116$	0	0
$117$	−11.0298	−1.01971
$118$	0	0
$119$	10.4336	0.956445
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	4.36654	0.393718
$124$	0	0
$125$	−10.5498	−0.943601
$126$	0	0
$127$	−1.07275	−0.0951914	−0.0475957	−	0.998867i	$-0.515156\pi$
$127$	−1.07275	−0.0951914	−0.0475957	+	0.998867i	$0.515156\pi$
$128$	0	0
$129$	−4.36705	−0.384497
$130$	0	0
$131$	7.82709	0.683856	0.341928	−	0.939726i	$-0.388920\pi$
$131$	7.82709	0.683856	0.341928	+	0.939726i	$0.388920\pi$
$132$	0	0
$133$	−1.69239	−0.146749
$134$	0	0
$135$	−7.51032	−0.646386
$136$	0	0
$137$	−11.2473	−0.960919	−0.480460	−	0.877017i	$-0.659530\pi$
$137$	−11.2473	−0.960919	−0.480460	+	0.877017i	$0.659530\pi$
$138$	0	0
$139$	0.905926	0.0768397	0.0384198	−	0.999262i	$-0.487768\pi$
$139$	0.905926	0.0768397	0.0384198	+	0.999262i	$0.487768\pi$
$140$	0	0
$141$	6.13951	0.517040
$142$	0	0
$143$	−4.11168	−0.343836
$144$	0	0
$145$	−19.0086	−1.57858
$146$	0	0
$147$	−2.33018	−0.192190
$148$	0	0
$149$	10.5174	0.861622	0.430811	−	0.902442i	$-0.358228\pi$
$149$	10.5174	0.861622	0.430811	+	0.902442i	$0.358228\pi$
$150$	0	0
$151$	−13.2436	−1.07775	−0.538873	−	0.842387i	$-0.681150\pi$
$151$	−13.2436	−1.07775	−0.538873	+	0.842387i	$0.681150\pi$
$152$	0	0
$153$	16.5380	1.33702
$154$	0	0
$155$	5.39864	0.433629
$156$	0	0
$157$	1.99915	0.159549	0.0797747	−	0.996813i	$-0.474580\pi$
$157$	1.99915	0.159549	0.0797747	+	0.996813i	$0.474580\pi$
$158$	0	0
$159$	4.47076	0.354555
$160$	0	0
$161$	5.96059	0.469761
$162$	0	0
$163$	18.7557	1.46906	0.734531	−	0.678575i	$-0.237402\pi$
$163$	18.7557	1.46906	0.734531	+	0.678575i	$0.237402\pi$
$164$	0	0
$165$	−1.32164	−0.102890
$166$	0	0
$167$	6.31203	0.488440	0.244220	−	0.969720i	$-0.421468\pi$
$167$	6.31203	0.488440	0.244220	+	0.969720i	$0.421468\pi$
$168$	0	0
$169$	3.90593	0.300456
$170$	0	0
$171$	−2.68256	−0.205141
$172$	0	0
$173$	−21.5269	−1.63666	−0.818328	−	0.574751i	$-0.805099\pi$
$173$	−21.5269	−1.63666	−0.818328	+	0.574751i	$0.805099\pi$
$174$	0	0
$175$	−0.850661	−0.0643039
$176$	0	0
$177$	−6.15516	−0.462650
$178$	0	0
$179$	7.97018	0.595719	0.297860	−	0.954610i	$-0.403727\pi$
$179$	7.97018	0.595719	0.297860	+	0.954610i	$0.403727\pi$
$180$	0	0
$181$	−1.16955	−0.0869317	−0.0434659	−	0.999055i	$-0.513840\pi$
$181$	−1.16955	−0.0869317	−0.0434659	+	0.999055i	$0.513840\pi$
$182$	0	0
$183$	−2.54191	−0.187904
$184$	0	0
$185$	−15.3886	−1.13139
$186$	0	0
$187$	6.16499	0.450829
$188$	0	0
$189$	5.41844	0.394133
$190$	0	0
$191$	15.6673	1.13365	0.566824	−	0.823839i	$-0.308172\pi$
$191$	15.6673	1.13365	0.566824	+	0.823839i	$0.308172\pi$
$192$	0	0
$193$	21.6769	1.56034	0.780169	−	0.625568i	$-0.215133\pi$
$193$	21.6769	1.56034	0.780169	+	0.625568i	$0.215133\pi$
$194$	0	0
$195$	5.43418	0.389149
$196$	0	0
$197$	4.58794	0.326877	0.163439	−	0.986554i	$-0.447741\pi$
$197$	4.58794	0.326877	0.163439	+	0.986554i	$0.447741\pi$
$198$	0	0
$199$	−13.0619	−0.925937	−0.462968	−	0.886375i	$-0.653216\pi$
$199$	−13.0619	−0.925937	−0.462968	+	0.886375i	$0.653216\pi$
$200$	0	0
$201$	−8.29353	−0.584980
$202$	0	0
$203$	13.7141	0.962539
$204$	0	0
$205$	18.1800	1.26975
$206$	0	0
$207$	9.44797	0.656679
$208$	0	0
$209$	−1.00000	−0.0691714
$210$	0	0
$211$	−2.91188	−0.200462	−0.100231	−	0.994964i	$-0.531958\pi$
$211$	−2.91188	−0.200462	−0.100231	+	0.994964i	$0.531958\pi$
$212$	0	0
$213$	−1.75800	−0.120456
$214$	0	0
$215$	−18.1821	−1.24001
$216$	0	0
$217$	−3.89493	−0.264405
$218$	0	0
$219$	6.52585	0.440976
$220$	0	0
$221$	−25.3485	−1.70512
$222$	0	0
$223$	−6.34161	−0.424665	−0.212333	−	0.977197i	$-0.568106\pi$
$223$	−6.34161	−0.424665	−0.212333	+	0.977197i	$0.568106\pi$
$224$	0	0
$225$	−1.34836	−0.0898905
$226$	0	0
$227$	2.00429	0.133029	0.0665147	−	0.997785i	$-0.478812\pi$
$227$	2.00429	0.133029	0.0665147	+	0.997785i	$0.478812\pi$
$228$	0	0
$229$	−20.9893	−1.38701	−0.693507	−	0.720450i	$-0.743935\pi$
$229$	−20.9893	−1.38701	−0.693507	+	0.720450i	$0.743935\pi$
$230$	0	0
$231$	0.953520	0.0627370
$232$	0	0
$233$	−17.6006	−1.15305	−0.576527	−	0.817078i	$-0.695593\pi$
$233$	−17.6006	−1.15305	−0.576527	+	0.817078i	$0.695593\pi$
$234$	0	0
$235$	25.5617	1.66746
$236$	0	0
$237$	−2.79558	−0.181592
$238$	0	0
$239$	−26.6207	−1.72195	−0.860975	−	0.508647i	$-0.830146\pi$
$239$	−26.6207	−1.72195	−0.860975	+	0.508647i	$0.830146\pi$
$240$	0	0
$241$	13.1342	0.846049	0.423024	−	0.906118i	$-0.360968\pi$
$241$	13.1342	0.846049	0.423024	+	0.906118i	$0.360968\pi$
$242$	0	0
$243$	13.1228	0.841828
$244$	0	0
$245$	−9.70166	−0.619816
$246$	0	0
$247$	4.11168	0.261620
$248$	0	0
$249$	1.03052	0.0653063
$250$	0	0
$251$	26.1636	1.65143	0.825715	−	0.564087i	$-0.190772\pi$
$251$	26.1636	1.65143	0.825715	+	0.564087i	$0.190772\pi$
$252$	0	0
$253$	3.52199	0.221426
$254$	0	0
$255$	−8.14792	−0.510243
$256$	0	0
$257$	12.6117	0.786697	0.393349	−	0.919389i	$-0.371317\pi$
$257$	12.6117	0.786697	0.393349	+	0.919389i	$0.371317\pi$
$258$	0	0
$259$	11.1024	0.689867
$260$	0	0
$261$	21.7378	1.34554
$262$	0	0
$263$	20.0550	1.23664	0.618322	−	0.785925i	$-0.287813\pi$
$263$	20.0550	1.23664	0.618322	+	0.785925i	$0.287813\pi$
$264$	0	0
$265$	18.6139	1.14345
$266$	0	0
$267$	−5.28200	−0.323253
$268$	0	0
$269$	−20.1150	−1.22644	−0.613218	−	0.789914i	$-0.710125\pi$
$269$	−20.1150	−1.22644	−0.613218	+	0.789914i	$0.710125\pi$
$270$	0	0
$271$	−15.5586	−0.945117	−0.472558	−	0.881299i	$-0.656669\pi$
$271$	−15.5586	−0.945117	−0.472558	+	0.881299i	$0.656669\pi$
$272$	0	0
$273$	−3.92057	−0.237284
$274$	0	0
$275$	−0.502638	−0.0303102
$276$	0	0
$277$	1.89211	0.113686	0.0568429	−	0.998383i	$-0.481897\pi$
$277$	1.89211	0.113686	0.0568429	+	0.998383i	$0.481897\pi$
$278$	0	0
$279$	−6.17375	−0.369613
$280$	0	0
$281$	4.59299	0.273995	0.136997	−	0.990571i	$-0.456255\pi$
$281$	4.59299	0.273995	0.136997	+	0.990571i	$0.456255\pi$
$282$	0	0
$283$	1.17798	0.0700234	0.0350117	−	0.999387i	$-0.488853\pi$
$283$	1.17798	0.0700234	0.0350117	+	0.999387i	$0.488853\pi$
$284$	0	0
$285$	1.32164	0.0782874
$286$	0	0
$287$	−13.1163	−0.774228
$288$	0	0
$289$	21.0071	1.23571
$290$	0	0
$291$	−6.19405	−0.363101
$292$	0	0
$293$	−11.2606	−0.657851	−0.328925	−	0.944356i	$-0.606686\pi$
$293$	−11.2606	−0.657851	−0.328925	+	0.944356i	$0.606686\pi$
$294$	0	0
$295$	−25.6269	−1.49206
$296$	0	0
$297$	3.20164	0.185778
$298$	0	0
$299$	−14.4813	−0.837476
$300$	0	0
$301$	13.1178	0.756096
$302$	0	0
$303$	1.29935	0.0746459
$304$	0	0
$305$	−10.5832	−0.605993
$306$	0	0
$307$	−4.35245	−0.248407	−0.124204	−	0.992257i	$-0.539638\pi$
$307$	−4.35245	−0.248407	−0.124204	+	0.992257i	$0.539638\pi$
$308$	0	0
$309$	−3.44810	−0.196156
$310$	0	0
$311$	−7.61725	−0.431935	−0.215967	−	0.976401i	$-0.569291\pi$
$311$	−7.61725	−0.431935	−0.215967	+	0.976401i	$0.569291\pi$
$312$	0	0
$313$	−17.5654	−0.992855	−0.496427	−	0.868078i	$-0.665355\pi$
$313$	−17.5654	−0.992855	−0.496427	+	0.868078i	$0.665355\pi$
$314$	0	0
$315$	10.6497	0.600041
$316$	0	0
$317$	−1.94192	−0.109069	−0.0545345	−	0.998512i	$-0.517367\pi$
$317$	−1.94192	−0.109069	−0.0545345	+	0.998512i	$0.517367\pi$
$318$	0	0
$319$	8.10336	0.453701
$320$	0	0
$321$	−0.238295	−0.0133003
$322$	0	0
$323$	−6.16499	−0.343029
$324$	0	0
$325$	2.06669	0.114639
$326$	0	0
$327$	7.89793	0.436757
$328$	0	0
$329$	−18.4419	−1.01674
$330$	0	0
$331$	−13.3988	−0.736462	−0.368231	−	0.929734i	$-0.620036\pi$
$331$	−13.3988	−0.736462	−0.368231	+	0.929734i	$0.620036\pi$
$332$	0	0
$333$	17.5980	0.964366
$334$	0	0
$335$	−34.5300	−1.88657
$336$	0	0
$337$	33.1631	1.80651	0.903254	−	0.429107i	$-0.141172\pi$
$337$	33.1631	1.80651	0.903254	+	0.429107i	$0.141172\pi$
$338$	0	0
$339$	−1.42903	−0.0776145
$340$	0	0
$341$	−2.30144	−0.124630
$342$	0	0
$343$	18.8462	1.01760
$344$	0	0
$345$	−4.65482	−0.250607
$346$	0	0
$347$	−18.0268	−0.967730	−0.483865	−	0.875143i	$-0.660767\pi$
$347$	−18.0268	−0.967730	−0.483865	+	0.875143i	$0.660767\pi$
$348$	0	0
$349$	8.81411	0.471808	0.235904	−	0.971776i	$-0.424195\pi$
$349$	8.81411	0.471808	0.235904	+	0.971776i	$0.424195\pi$
$350$	0	0
$351$	−13.1641	−0.702650
$352$	0	0
$353$	−1.59249	−0.0847599	−0.0423799	−	0.999102i	$-0.513494\pi$
$353$	−1.59249	−0.0847599	−0.0423799	+	0.999102i	$0.513494\pi$
$354$	0	0
$355$	−7.31942	−0.388474
$356$	0	0
$357$	5.87844	0.311120
$358$	0	0
$359$	−36.3774	−1.91993	−0.959963	−	0.280126i	$-0.909624\pi$
$359$	−36.3774	−1.91993	−0.959963	+	0.280126i	$0.909624\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	0.563416	0.0295716
$364$	0	0
$365$	27.1703	1.42216
$366$	0	0
$367$	5.00276	0.261142	0.130571	−	0.991439i	$-0.458319\pi$
$367$	5.00276	0.261142	0.130571	+	0.991439i	$0.458319\pi$
$368$	0	0
$369$	−20.7902	−1.08229
$370$	0	0
$371$	−13.4293	−0.697216
$372$	0	0
$373$	−34.1313	−1.76725	−0.883625	−	0.468195i	$-0.844904\pi$
$373$	−34.1313	−1.76725	−0.883625	+	0.468195i	$0.844904\pi$
$374$	0	0
$375$	−5.94391	−0.306942
$376$	0	0
$377$	−33.3185	−1.71599
$378$	0	0
$379$	−15.7749	−0.810302	−0.405151	−	0.914250i	$-0.632781\pi$
$379$	−15.7749	−0.810302	−0.405151	+	0.914250i	$0.632781\pi$
$380$	0	0
$381$	−0.604405	−0.0309646
$382$	0	0
$383$	22.4260	1.14592	0.572958	−	0.819585i	$-0.305796\pi$
$383$	22.4260	1.14592	0.572958	+	0.819585i	$0.305796\pi$
$384$	0	0
$385$	3.96996	0.202328
$386$	0	0
$387$	20.7926	1.05695
$388$	0	0
$389$	−18.3488	−0.930321	−0.465160	−	0.885226i	$-0.654003\pi$
$389$	−18.3488	−0.930321	−0.465160	+	0.885226i	$0.654003\pi$
$390$	0	0
$391$	21.7131	1.09808
$392$	0	0
$393$	4.40990	0.222450
$394$	0	0
$395$	−11.6393	−0.585638
$396$	0	0
$397$	−0.511483	−0.0256706	−0.0128353	−	0.999918i	$-0.504086\pi$
$397$	−0.511483	−0.0256706	−0.0128353	+	0.999918i	$0.504086\pi$
$398$	0	0
$399$	−0.953520	−0.0477357
$400$	0	0
$401$	18.9824	0.947935	0.473967	−	0.880542i	$-0.342821\pi$
$401$	18.9824	0.947935	0.473967	+	0.880542i	$0.342821\pi$
$402$	0	0
$403$	9.46277	0.471374
$404$	0	0
$405$	14.6466	0.727795
$406$	0	0
$407$	6.56016	0.325175
$408$	0	0
$409$	1.10089	0.0544355	0.0272177	−	0.999630i	$-0.491335\pi$
$409$	1.10089	0.0544355	0.0272177	+	0.999630i	$0.491335\pi$
$410$	0	0
$411$	−6.33689	−0.312576
$412$	0	0
$413$	18.4889	0.909781
$414$	0	0
$415$	4.29054	0.210614
$416$	0	0
$417$	0.510413	0.0249950
$418$	0	0
$419$	18.7929	0.918094	0.459047	−	0.888412i	$-0.348191\pi$
$419$	18.7929	0.918094	0.459047	+	0.888412i	$0.348191\pi$
$420$	0	0
$421$	−24.7618	−1.20682	−0.603408	−	0.797433i	$-0.706191\pi$
$421$	−24.7618	−1.20682	−0.603408	+	0.797433i	$0.706191\pi$
$422$	0	0
$423$	−29.2318	−1.42130
$424$	0	0
$425$	−3.09876	−0.150312
$426$	0	0
$427$	7.63542	0.369504
$428$	0	0
$429$	−2.31659	−0.111846
$430$	0	0
$431$	32.1985	1.55095	0.775474	−	0.631380i	$-0.217511\pi$
$431$	32.1985	1.55095	0.775474	+	0.631380i	$0.217511\pi$
$432$	0	0
$433$	−15.3042	−0.735475	−0.367737	−	0.929930i	$-0.619867\pi$
$433$	−15.3042	−0.735475	−0.367737	+	0.929930i	$0.619867\pi$
$434$	0	0
$435$	−10.7098	−0.513494
$436$	0	0
$437$	−3.52199	−0.168480
$438$	0	0
$439$	−26.1812	−1.24956	−0.624781	−	0.780800i	$-0.714812\pi$
$439$	−26.1812	−1.24956	−0.624781	+	0.780800i	$0.714812\pi$
$440$	0	0
$441$	11.0946	0.528313
$442$	0	0
$443$	−21.5579	−1.02425	−0.512123	−	0.858912i	$-0.671141\pi$
$443$	−21.5579	−1.02425	−0.512123	+	0.858912i	$0.671141\pi$
$444$	0	0
$445$	−21.9915	−1.04250
$446$	0	0
$447$	5.92569	0.280275
$448$	0	0
$449$	11.7990	0.556829	0.278415	−	0.960461i	$-0.410191\pi$
$449$	11.7990	0.556829	0.278415	+	0.960461i	$0.410191\pi$
$450$	0	0
$451$	−7.75013	−0.364939
$452$	0	0
$453$	−7.46163	−0.350578
$454$	0	0
$455$	−16.3232	−0.765245
$456$	0	0
$457$	−15.6863	−0.733773	−0.366886	−	0.930266i	$-0.619576\pi$
$457$	−15.6863	−0.733773	−0.366886	+	0.930266i	$0.619576\pi$
$458$	0	0
$459$	19.7381	0.921296
$460$	0	0
$461$	27.2451	1.26893	0.634466	−	0.772951i	$-0.281220\pi$
$461$	27.2451	1.26893	0.634466	+	0.772951i	$0.281220\pi$
$462$	0	0
$463$	17.9364	0.833573	0.416787	−	0.909004i	$-0.363156\pi$
$463$	17.9364	0.833573	0.416787	+	0.909004i	$0.363156\pi$
$464$	0	0
$465$	3.04168	0.141054
$466$	0	0
$467$	−29.0100	−1.34242	−0.671211	−	0.741266i	$-0.734226\pi$
$467$	−29.0100	−1.34242	−0.671211	+	0.741266i	$0.734226\pi$
$468$	0	0
$469$	24.9122	1.15034
$470$	0	0
$471$	1.12635	0.0518995
$472$	0	0
$473$	7.75102	0.356392
$474$	0	0
$475$	0.502638	0.0230626
$476$	0	0
$477$	−21.2864	−0.974639
$478$	0	0
$479$	−34.0896	−1.55759	−0.778797	−	0.627276i	$-0.784170\pi$
$479$	−34.0896	−1.55759	−0.778797	+	0.627276i	$0.784170\pi$
$480$	0	0
$481$	−26.9733	−1.22988
$482$	0	0
$483$	3.35829	0.152808
$484$	0	0
$485$	−25.7888	−1.17101
$486$	0	0
$487$	−6.38349	−0.289264	−0.144632	−	0.989486i	$-0.546200\pi$
$487$	−6.38349	−0.289264	−0.144632	+	0.989486i	$0.546200\pi$
$488$	0	0
$489$	10.5673	0.477869
$490$	0	0
$491$	31.4552	1.41956	0.709778	−	0.704426i	$-0.248795\pi$
$491$	31.4552	1.41956	0.709778	+	0.704426i	$0.248795\pi$
$492$	0	0
$493$	49.9572	2.24996
$494$	0	0
$495$	6.29268	0.282835
$496$	0	0
$497$	5.28071	0.236872
$498$	0	0
$499$	31.9667	1.43103	0.715514	−	0.698599i	$-0.246193\pi$
$499$	31.9667	1.43103	0.715514	+	0.698599i	$0.246193\pi$
$500$	0	0
$501$	3.55630	0.158884
$502$	0	0
$503$	34.0522	1.51831	0.759157	−	0.650907i	$-0.225611\pi$
$503$	34.0522	1.51831	0.759157	+	0.650907i	$0.225611\pi$
$504$	0	0
$505$	5.40983	0.240734
$506$	0	0
$507$	2.20066	0.0977347
$508$	0	0
$509$	25.7904	1.14314	0.571569	−	0.820554i	$-0.306335\pi$
$509$	25.7904	1.14314	0.571569	+	0.820554i	$0.306335\pi$
$510$	0	0
$511$	−19.6024	−0.867160
$512$	0	0
$513$	−3.20164	−0.141356
$514$	0	0
$515$	−14.3561	−0.632606
$516$	0	0
$517$	−10.8969	−0.479247
$518$	0	0
$519$	−12.1286	−0.532385
$520$	0	0
$521$	31.3774	1.37467	0.687334	−	0.726342i	$-0.258781\pi$
$521$	31.3774	1.37467	0.687334	+	0.726342i	$0.258781\pi$
$522$	0	0
$523$	23.2388	1.01616	0.508082	−	0.861309i	$-0.330355\pi$
$523$	23.2388	1.01616	0.508082	+	0.861309i	$0.330355\pi$
$524$	0	0
$525$	−0.479275	−0.0209173
$526$	0	0
$527$	−14.1883	−0.618054
$528$	0	0
$529$	−10.5956	−0.460677
$530$	0	0
$531$	29.3063	1.27178
$532$	0	0
$533$	31.8661	1.38027
$534$	0	0
$535$	−0.992136	−0.0428938
$536$	0	0
$537$	4.49052	0.193780
$538$	0	0
$539$	4.13581	0.178142
$540$	0	0
$541$	36.0085	1.54812	0.774062	−	0.633109i	$-0.218222\pi$
$541$	36.0085	1.54812	0.774062	+	0.633109i	$0.218222\pi$
$542$	0	0
$543$	−0.658941	−0.0282779
$544$	0	0
$545$	32.8829	1.40855
$546$	0	0
$547$	28.1855	1.20513	0.602563	−	0.798071i	$-0.294146\pi$
$547$	28.1855	1.20513	0.602563	+	0.798071i	$0.294146\pi$
$548$	0	0
$549$	12.1027	0.516530
$550$	0	0
$551$	−8.10336	−0.345215
$552$	0	0
$553$	8.39738	0.357093
$554$	0	0
$555$	−8.67019	−0.368029
$556$	0	0
$557$	21.6248	0.916274	0.458137	−	0.888882i	$-0.348517\pi$
$557$	21.6248	0.916274	0.458137	+	0.888882i	$0.348517\pi$
$558$	0	0
$559$	−31.8697	−1.34795
$560$	0	0
$561$	3.47345	0.146649
$562$	0	0
$563$	−13.1791	−0.555434	−0.277717	−	0.960663i	$-0.589578\pi$
$563$	−13.1791	−0.555434	−0.277717	+	0.960663i	$0.589578\pi$
$564$	0	0
$565$	−5.94976	−0.250308
$566$	0	0
$567$	−10.5670	−0.443773
$568$	0	0
$569$	8.61143	0.361010	0.180505	−	0.983574i	$-0.442227\pi$
$569$	8.61143	0.361010	0.180505	+	0.983574i	$0.442227\pi$
$570$	0	0
$571$	17.6546	0.738821	0.369410	−	0.929266i	$-0.379560\pi$
$571$	17.6546	0.738821	0.369410	+	0.929266i	$0.379560\pi$
$572$	0	0
$573$	8.82722	0.368762
$574$	0	0
$575$	−1.77029	−0.0738261
$576$	0	0
$577$	−31.0907	−1.29432	−0.647162	−	0.762352i	$-0.724044\pi$
$577$	−31.0907	−1.29432	−0.647162	+	0.762352i	$0.724044\pi$
$578$	0	0
$579$	12.2131	0.507560
$580$	0	0
$581$	−3.09547	−0.128422
$582$	0	0
$583$	−7.93511	−0.328639
$584$	0	0
$585$	−25.8735	−1.06974
$586$	0	0
$587$	26.7211	1.10290	0.551450	−	0.834208i	$-0.314075\pi$
$587$	26.7211	1.10290	0.551450	+	0.834208i	$0.314075\pi$
$588$	0	0
$589$	2.30144	0.0948290
$590$	0	0
$591$	2.58492	0.106329
$592$	0	0
$593$	−24.2460	−0.995666	−0.497833	−	0.867273i	$-0.665871\pi$
$593$	−24.2460	−0.995666	−0.497833	+	0.867273i	$0.665871\pi$
$594$	0	0
$595$	24.4748	1.00337
$596$	0	0
$597$	−7.35930	−0.301196
$598$	0	0
$599$	18.2095	0.744019	0.372009	−	0.928229i	$-0.378669\pi$
$599$	18.2095	0.744019	0.372009	+	0.928229i	$0.378669\pi$
$600$	0	0
$601$	37.9824	1.54934	0.774668	−	0.632368i	$-0.217917\pi$
$601$	37.9824	1.54934	0.774668	+	0.632368i	$0.217917\pi$
$602$	0	0
$603$	39.4876	1.60806
$604$	0	0
$605$	2.34577	0.0953691
$606$	0	0
$607$	20.0130	0.812301	0.406151	−	0.913806i	$-0.366871\pi$
$607$	20.0130	0.812301	0.406151	+	0.913806i	$0.366871\pi$
$608$	0	0
$609$	7.72672	0.313103
$610$	0	0
$611$	44.8048	1.81261
$612$	0	0
$613$	−42.1414	−1.70208	−0.851038	−	0.525105i	$-0.824026\pi$
$613$	−42.1414	−1.70208	−0.851038	+	0.525105i	$0.824026\pi$
$614$	0	0
$615$	10.2429	0.413034
$616$	0	0
$617$	16.9044	0.680545	0.340273	−	0.940327i	$-0.389481\pi$
$617$	16.9044	0.680545	0.340273	+	0.940327i	$0.389481\pi$
$618$	0	0
$619$	−5.31177	−0.213498	−0.106749	−	0.994286i	$-0.534044\pi$
$619$	−5.31177	−0.213498	−0.106749	+	0.994286i	$0.534044\pi$
$620$	0	0
$621$	11.2762	0.452497
$622$	0	0
$623$	15.8661	0.635662
$624$	0	0
$625$	−27.2605	−1.09042
$626$	0	0
$627$	−0.563416	−0.0225006
$628$	0	0
$629$	40.4433	1.61258
$630$	0	0
$631$	−34.0425	−1.35521	−0.677604	−	0.735427i	$-0.736982\pi$
$631$	−34.0425	−1.35521	−0.677604	+	0.735427i	$0.736982\pi$
$632$	0	0
$633$	−1.64060	−0.0652079
$634$	0	0
$635$	−2.51643	−0.0998615
$636$	0	0
$637$	−17.0051	−0.673768
$638$	0	0
$639$	8.37030	0.331124
$640$	0	0
$641$	−15.0405	−0.594063	−0.297031	−	0.954868i	$-0.595997\pi$
$641$	−15.0405	−0.594063	−0.297031	+	0.954868i	$0.595997\pi$
$642$	0	0
$643$	−32.3618	−1.27622	−0.638112	−	0.769943i	$-0.720285\pi$
$643$	−32.3618	−1.27622	−0.638112	+	0.769943i	$0.720285\pi$
$644$	0	0
$645$	−10.2441	−0.403361
$646$	0	0
$647$	49.0042	1.92655	0.963276	−	0.268512i	$-0.0865319\pi$
$647$	49.0042	1.92655	0.963276	+	0.268512i	$0.0865319\pi$
$648$	0	0
$649$	10.9247	0.428833
$650$	0	0
$651$	−2.19447	−0.0860079
$652$	0	0
$653$	31.1992	1.22092	0.610460	−	0.792047i	$-0.290984\pi$
$653$	31.1992	1.22092	0.610460	+	0.792047i	$0.290984\pi$
$654$	0	0
$655$	18.3605	0.717406
$656$	0	0
$657$	−31.0712	−1.21220
$658$	0	0
$659$	35.7237	1.39160	0.695799	−	0.718237i	$-0.255050\pi$
$659$	35.7237	1.39160	0.695799	+	0.718237i	$0.255050\pi$
$660$	0	0
$661$	33.8677	1.31730	0.658651	−	0.752449i	$-0.271128\pi$
$661$	33.8677	1.31730	0.658651	+	0.752449i	$0.271128\pi$
$662$	0	0
$663$	−14.2817	−0.554657
$664$	0	0
$665$	−3.96996	−0.153949
$666$	0	0
$667$	28.5400	1.10507
$668$	0	0
$669$	−3.57296	−0.138139
$670$	0	0
$671$	4.51162	0.174169
$672$	0	0
$673$	21.1648	0.815842	0.407921	−	0.913017i	$-0.366254\pi$
$673$	21.1648	0.815842	0.407921	+	0.913017i	$0.366254\pi$
$674$	0	0
$675$	−1.60927	−0.0619408
$676$	0	0
$677$	−34.5239	−1.32686	−0.663431	−	0.748238i	$-0.730900\pi$
$677$	−34.5239	−1.32686	−0.663431	+	0.748238i	$0.730900\pi$
$678$	0	0
$679$	18.6057	0.714022
$680$	0	0
$681$	1.12925	0.0432729
$682$	0	0
$683$	−12.8022	−0.489862	−0.244931	−	0.969540i	$-0.578765\pi$
$683$	−12.8022	−0.489862	−0.244931	+	0.969540i	$0.578765\pi$
$684$	0	0
$685$	−26.3835	−1.00806
$686$	0	0
$687$	−11.8257	−0.451179
$688$	0	0
$689$	32.6267	1.24298
$690$	0	0
$691$	−39.5406	−1.50419	−0.752097	−	0.659052i	$-0.770958\pi$
$691$	−39.5406	−1.50419	−0.752097	+	0.659052i	$0.770958\pi$
$692$	0	0
$693$	−4.53995	−0.172458
$694$	0	0
$695$	2.12510	0.0806094
$696$	0	0
$697$	−47.7795	−1.80978
$698$	0	0
$699$	−9.91646	−0.375075
$700$	0	0
$701$	22.4546	0.848097	0.424049	−	0.905639i	$-0.360609\pi$
$701$	22.4546	0.848097	0.424049	+	0.905639i	$0.360609\pi$
$702$	0	0
$703$	−6.56016	−0.247421
$704$	0	0
$705$	14.4019	0.542406
$706$	0	0
$707$	−3.90301	−0.146788
$708$	0	0
$709$	−1.41244	−0.0530451	−0.0265226	−	0.999648i	$-0.508443\pi$
$709$	−1.41244	−0.0530451	−0.0265226	+	0.999648i	$0.508443\pi$
$710$	0	0
$711$	13.3104	0.499181
$712$	0	0
$713$	−8.10564	−0.303559
$714$	0	0
$715$	−9.64506	−0.360705
$716$	0	0
$717$	−14.9985	−0.560130
$718$	0	0
$719$	39.0879	1.45773	0.728867	−	0.684655i	$-0.240047\pi$
$719$	39.0879	1.45773	0.728867	+	0.684655i	$0.240047\pi$
$720$	0	0
$721$	10.3574	0.385731
$722$	0	0
$723$	7.40002	0.275210
$724$	0	0
$725$	−4.07306	−0.151270
$726$	0	0
$727$	−9.42098	−0.349405	−0.174702	−	0.984621i	$-0.555896\pi$
$727$	−9.42098	−0.349405	−0.174702	+	0.984621i	$0.555896\pi$
$728$	0	0
$729$	−11.3379	−0.419922
$730$	0	0
$731$	47.7850	1.76739
$732$	0	0
$733$	4.63361	0.171146	0.0855731	−	0.996332i	$-0.472728\pi$
$733$	4.63361	0.171146	0.0855731	+	0.996332i	$0.472728\pi$
$734$	0	0
$735$	−5.46606	−0.201619
$736$	0	0
$737$	14.7201	0.542222
$738$	0	0
$739$	13.1654	0.484298	0.242149	−	0.970239i	$-0.422148\pi$
$739$	13.1654	0.484298	0.242149	+	0.970239i	$0.422148\pi$
$740$	0	0
$741$	2.31659	0.0851019
$742$	0	0
$743$	14.7689	0.541817	0.270909	−	0.962605i	$-0.412676\pi$
$743$	14.7689	0.541817	0.270909	+	0.962605i	$0.412676\pi$
$744$	0	0
$745$	24.6715	0.903894
$746$	0	0
$747$	−4.90655	−0.179521
$748$	0	0
$749$	0.715792	0.0261545
$750$	0	0
$751$	−35.0415	−1.27868	−0.639341	−	0.768924i	$-0.720793\pi$
$751$	−35.0415	−1.27868	−0.639341	+	0.768924i	$0.720793\pi$
$752$	0	0
$753$	14.7410	0.537191
$754$	0	0
$755$	−31.0664	−1.13062
$756$	0	0
$757$	−23.8005	−0.865044	−0.432522	−	0.901623i	$-0.642376\pi$
$757$	−23.8005	−0.865044	−0.432522	+	0.901623i	$0.642376\pi$
$758$	0	0
$759$	1.98435	0.0720272
$760$	0	0
$761$	−39.0282	−1.41477	−0.707386	−	0.706827i	$-0.750126\pi$
$761$	−39.0282	−1.41477	−0.707386	+	0.706827i	$0.750126\pi$
$762$	0	0
$763$	−23.7239	−0.858862
$764$	0	0
$765$	38.7943	1.40261
$766$	0	0
$767$	−44.9190	−1.62193
$768$	0	0
$769$	9.84757	0.355113	0.177556	−	0.984111i	$-0.443181\pi$
$769$	9.84757	0.355113	0.177556	+	0.984111i	$0.443181\pi$
$770$	0	0
$771$	7.10564	0.255903
$772$	0	0
$773$	−42.9837	−1.54602	−0.773008	−	0.634396i	$-0.781249\pi$
$773$	−42.9837	−1.54602	−0.773008	+	0.634396i	$0.781249\pi$
$774$	0	0
$775$	1.15679	0.0415531
$776$	0	0
$777$	6.25524	0.224405
$778$	0	0
$779$	7.75013	0.277677
$780$	0	0
$781$	3.12026	0.111652
$782$	0	0
$783$	25.9441	0.927166
$784$	0	0
$785$	4.68954	0.167377
$786$	0	0
$787$	25.8709	0.922199	0.461099	−	0.887349i	$-0.347455\pi$
$787$	25.8709	0.922199	0.461099	+	0.887349i	$0.347455\pi$
$788$	0	0
$789$	11.2993	0.402266
$790$	0	0
$791$	4.29254	0.152625
$792$	0	0
$793$	−18.5503	−0.658741
$794$	0	0
$795$	10.4874	0.371949
$796$	0	0
$797$	3.62989	0.128577	0.0642886	−	0.997931i	$-0.479522\pi$
$797$	3.62989	0.128577	0.0642886	+	0.997931i	$0.479522\pi$
$798$	0	0
$799$	−67.1796	−2.37664
$800$	0	0
$801$	25.1489	0.888593
$802$	0	0
$803$	−11.5827	−0.408743
$804$	0	0
$805$	13.9822	0.492807
$806$	0	0
$807$	−11.3331	−0.398945
$808$	0	0
$809$	−13.1327	−0.461720	−0.230860	−	0.972987i	$-0.574154\pi$
$809$	−13.1327	−0.461720	−0.230860	+	0.972987i	$0.574154\pi$
$810$	0	0
$811$	−9.40230	−0.330160	−0.165080	−	0.986280i	$-0.552788\pi$
$811$	−9.40230	−0.330160	−0.165080	+	0.986280i	$0.552788\pi$
$812$	0	0
$813$	−8.76595	−0.307435
$814$	0	0
$815$	43.9966	1.54114
$816$	0	0
$817$	−7.75102	−0.271174
$818$	0	0
$819$	18.6668	0.652272
$820$	0	0
$821$	−21.1701	−0.738843	−0.369422	−	0.929262i	$-0.620444\pi$
$821$	−21.1701	−0.738843	−0.369422	+	0.929262i	$0.620444\pi$
$822$	0	0
$823$	−25.2277	−0.879384	−0.439692	−	0.898149i	$-0.644912\pi$
$823$	−25.2277	−0.879384	−0.439692	+	0.898149i	$0.644912\pi$
$824$	0	0
$825$	−0.283194	−0.00985955
$826$	0	0
$827$	39.4460	1.37167	0.685836	−	0.727756i	$-0.259437\pi$
$827$	39.4460	1.37167	0.685836	+	0.727756i	$0.259437\pi$
$828$	0	0
$829$	−36.5068	−1.26793	−0.633967	−	0.773360i	$-0.718574\pi$
$829$	−36.5068	−1.26793	−0.633967	+	0.773360i	$0.718574\pi$
$830$	0	0
$831$	1.06604	0.0369806
$832$	0	0
$833$	25.4972	0.883427
$834$	0	0
$835$	14.8066	0.512403
$836$	0	0
$837$	−7.36838	−0.254688
$838$	0	0
$839$	−42.4670	−1.46612	−0.733061	−	0.680163i	$-0.761909\pi$
$839$	−42.4670	−1.46612	−0.733061	+	0.680163i	$0.761909\pi$
$840$	0	0
$841$	36.6645	1.26429
$842$	0	0
$843$	2.58776	0.0891273
$844$	0	0
$845$	9.16241	0.315196
$846$	0	0
$847$	−1.69239	−0.0581513
$848$	0	0
$849$	0.663690	0.0227778
$850$	0	0
$851$	23.1048	0.792023
$852$	0	0
$853$	21.5088	0.736446	0.368223	−	0.929738i	$-0.379966\pi$
$853$	21.5088	0.736446	0.368223	+	0.929738i	$0.379966\pi$
$854$	0	0
$855$	−6.29268	−0.215205
$856$	0	0
$857$	−12.6370	−0.431670	−0.215835	−	0.976430i	$-0.569247\pi$
$857$	−12.6370	−0.431670	−0.215835	+	0.976430i	$0.569247\pi$
$858$	0	0
$859$	3.39720	0.115911	0.0579555	−	0.998319i	$-0.481542\pi$
$859$	3.39720	0.115911	0.0579555	+	0.998319i	$0.481542\pi$
$860$	0	0
$861$	−7.38990	−0.251847
$862$	0	0
$863$	10.9308	0.372088	0.186044	−	0.982541i	$-0.440433\pi$
$863$	10.9308	0.372088	0.186044	+	0.982541i	$0.440433\pi$
$864$	0	0
$865$	−50.4971	−1.71695
$866$	0	0
$867$	11.8358	0.401963
$868$	0	0
$869$	4.96184	0.168319
$870$	0	0
$871$	−60.5243	−2.05079
$872$	0	0
$873$	29.4914	0.998133
$874$	0	0
$875$	17.8544	0.603588
$876$	0	0
$877$	31.7230	1.07121	0.535605	−	0.844468i	$-0.320083\pi$
$877$	31.7230	1.07121	0.535605	+	0.844468i	$0.320083\pi$
$878$	0	0
$879$	−6.34439	−0.213991
$880$	0	0
$881$	−6.45839	−0.217589	−0.108794	−	0.994064i	$-0.534699\pi$
$881$	−6.45839	−0.217589	−0.108794	+	0.994064i	$0.534699\pi$
$882$	0	0
$883$	48.5242	1.63297	0.816485	−	0.577367i	$-0.195920\pi$
$883$	48.5242	1.63297	0.816485	+	0.577367i	$0.195920\pi$
$884$	0	0
$885$	−14.4386	−0.485348
$886$	0	0
$887$	44.0097	1.47770	0.738851	−	0.673869i	$-0.235369\pi$
$887$	44.0097	1.47770	0.738851	+	0.673869i	$0.235369\pi$
$888$	0	0
$889$	1.81552	0.0608905
$890$	0	0
$891$	−6.24383	−0.209176
$892$	0	0
$893$	10.8969	0.364652
$894$	0	0
$895$	18.6962	0.624946
$896$	0	0
$897$	−8.15900	−0.272421
$898$	0	0
$899$	−18.6494	−0.621991
$900$	0	0
$901$	−48.9199	−1.62976
$902$	0	0
$903$	7.39076	0.245949
$904$	0	0
$905$	−2.74349	−0.0911966
$906$	0	0
$907$	2.05084	0.0680971	0.0340485	−	0.999420i	$-0.489160\pi$
$907$	2.05084	0.0680971	0.0340485	+	0.999420i	$0.489160\pi$
$908$	0	0
$909$	−6.18655	−0.205195
$910$	0	0
$911$	16.8417	0.557990	0.278995	−	0.960293i	$-0.409999\pi$
$911$	16.8417	0.557990	0.278995	+	0.960293i	$0.409999\pi$
$912$	0	0
$913$	−1.82905	−0.0605328
$914$	0	0
$915$	−5.96275	−0.197122
$916$	0	0
$917$	−13.2465	−0.437438
$918$	0	0
$919$	−8.01221	−0.264299	−0.132149	−	0.991230i	$-0.542188\pi$
$919$	−8.01221	−0.264299	−0.132149	+	0.991230i	$0.542188\pi$
$920$	0	0
$921$	−2.45224	−0.0808039
$922$	0	0
$923$	−12.8295	−0.422289
$924$	0	0
$925$	−3.29738	−0.108417
$926$	0	0
$927$	16.4173	0.539214
$928$	0	0
$929$	16.7897	0.550853	0.275426	−	0.961322i	$-0.411181\pi$
$929$	16.7897	0.550853	0.275426	+	0.961322i	$0.411181\pi$
$930$	0	0
$931$	−4.13581	−0.135546
$932$	0	0
$933$	−4.29168	−0.140503
$934$	0	0
$935$	14.4617	0.472947
$936$	0	0
$937$	21.8471	0.713713	0.356856	−	0.934159i	$-0.383849\pi$
$937$	21.8471	0.713713	0.356856	+	0.934159i	$0.383849\pi$
$938$	0	0
$939$	−9.89661	−0.322964
$940$	0	0
$941$	17.8421	0.581635	0.290817	−	0.956779i	$-0.406073\pi$
$941$	17.8421	0.581635	0.290817	+	0.956779i	$0.406073\pi$
$942$	0	0
$943$	−27.2959	−0.888877
$944$	0	0
$945$	12.7104	0.413470
$946$	0	0
$947$	−41.9736	−1.36396	−0.681980	−	0.731371i	$-0.738881\pi$
$947$	−41.9736	−1.36396	−0.681980	+	0.731371i	$0.738881\pi$
$948$	0	0
$949$	47.6242	1.54595
$950$	0	0
$951$	−1.09411	−0.0354788
$952$	0	0
$953$	−8.06945	−0.261395	−0.130697	−	0.991422i	$-0.541722\pi$
$953$	−8.06945	−0.261395	−0.130697	+	0.991422i	$0.541722\pi$
$954$	0	0
$955$	36.7520	1.18927
$956$	0	0
$957$	4.56556	0.147584
$958$	0	0
$959$	19.0348	0.614666
$960$	0	0
$961$	−25.7034	−0.829142
$962$	0	0
$963$	1.13458	0.0365614
$964$	0	0
$965$	50.8491	1.63689
$966$	0	0
$967$	3.86360	0.124245	0.0621225	−	0.998069i	$-0.480213\pi$
$967$	3.86360	0.124245	0.0621225	+	0.998069i	$0.480213\pi$
$968$	0	0
$969$	−3.47345	−0.111583
$970$	0	0
$971$	−42.7787	−1.37283	−0.686417	−	0.727208i	$-0.740818\pi$
$971$	−42.7787	−1.37283	−0.686417	+	0.727208i	$0.740818\pi$
$972$	0	0
$973$	−1.53318	−0.0491516
$974$	0	0
$975$	1.16440	0.0372908
$976$	0	0
$977$	54.5146	1.74408	0.872039	−	0.489437i	$-0.162798\pi$
$977$	54.5146	1.74408	0.872039	+	0.489437i	$0.162798\pi$
$978$	0	0
$979$	9.37496	0.299625
$980$	0	0
$981$	−37.6040	−1.20060
$982$	0	0
$983$	−33.5422	−1.06983	−0.534916	−	0.844905i	$-0.679657\pi$
$983$	−33.5422	−1.06983	−0.534916	+	0.844905i	$0.679657\pi$
$984$	0	0
$985$	10.7623	0.342914
$986$	0	0
$987$	−10.3905	−0.330732
$988$	0	0
$989$	27.2991	0.868060
$990$	0	0
$991$	−31.7767	−1.00942	−0.504710	−	0.863289i	$-0.668400\pi$
$991$	−31.7767	−1.00942	−0.504710	+	0.863289i	$0.668400\pi$
$992$	0	0
$993$	−7.54907	−0.239562
$994$	0	0
$995$	−30.6403	−0.971363
$996$	0	0
$997$	−47.5668	−1.50646	−0.753228	−	0.657759i	$-0.771504\pi$
$997$	−47.5668	−1.50646	−0.753228	+	0.657759i	$0.771504\pi$
$998$	0	0
$999$	21.0033	0.664515

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3344.2.a.t.1.4		5
4.3	odd	2		209.2.a.c.1.4	✓	5
12.11	even	2		1881.2.a.k.1.2		5
20.19	odd	2		5225.2.a.h.1.2		5
44.43	even	2		2299.2.a.n.1.2		5
76.75	even	2		3971.2.a.h.1.2		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
209.2.a.c.1.4	✓	5	4.3	odd	2
1881.2.a.k.1.2		5	12.11	even	2
2299.2.a.n.1.2		5	44.43	even	2
3344.2.a.t.1.4		5	1.1	even	1	trivial
3971.2.a.h.1.2		5	76.75	even	2
5225.2.a.h.1.2		5	20.19	odd	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)

\(f(q)\)	\(=\)	\(q+0.563416 q^{3} +2.34577 q^{5} -1.69239 q^{7} -2.68256 q^{9} -1.00000 q^{11} +4.11168 q^{13} +1.32164 q^{15} -6.16499 q^{17} +1.00000 q^{19} -0.953520 q^{21} -3.52199 q^{23} +0.502638 q^{25} -3.20164 q^{27} -8.10336 q^{29} +2.30144 q^{31} -0.563416 q^{33} -3.96996 q^{35} -6.56016 q^{37} +2.31659 q^{39} +7.75013 q^{41} -7.75102 q^{43} -6.29268 q^{45} +10.8969 q^{47} -4.13581 q^{49} -3.47345 q^{51} +7.93511 q^{53} -2.34577 q^{55} +0.563416 q^{57} -10.9247 q^{59} -4.51162 q^{61} +4.53995 q^{63} +9.64506 q^{65} -14.7201 q^{67} -1.98435 q^{69} -3.12026 q^{71} +11.5827 q^{73} +0.283194 q^{75} +1.69239 q^{77} -4.96184 q^{79} +6.24383 q^{81} +1.82905 q^{83} -14.4617 q^{85} -4.56556 q^{87} -9.37496 q^{89} -6.95858 q^{91} +1.29666 q^{93} +2.34577 q^{95} -10.9937 q^{97} +2.68256 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - q^{3} - 5 q^{5} - 6 q^{7} + 4 q^{9} - 5 q^{11} + 4 q^{13} - 3 q^{15} - 4 q^{17} + 5 q^{19} + 10 q^{21} - 3 q^{23} + 6 q^{25} + 11 q^{27} + 10 q^{29} - 11 q^{31} + q^{33} + 8 q^{35} + q^{37} - 2 q^{39}+ \cdots - 4 q^{99}+O(q^{100}) \) 5 * q - q^3 - 5 * q^5 - 6 * q^7 + 4 * q^9 - 5 * q^11 + 4 * q^13 - 3 * q^15 - 4 * q^17 + 5 * q^19 + 10 * q^21 - 3 * q^23 + 6 * q^25 + 11 * q^27 + 10 * q^29 - 11 * q^31 + q^33 + 8 * q^35 + q^37 - 2 * q^39 + 2 * q^41 - 20 * q^43 - 28 * q^45 + 20 * q^47 + 3 * q^49 - 24 * q^51 - 14 * q^53 + 5 * q^55 - q^57 - 3 * q^59 - 10 * q^61 - 24 * q^63 - 9 * q^67 - 5 * q^69 - 23 * q^71 + 18 * q^75 + 6 * q^77 - 44 * q^79 + q^81 + 14 * q^83 - 12 * q^85 - 28 * q^87 - 27 * q^89 - 24 * q^91 - 27 * q^93 - 5 * q^95 + 15 * q^97 - 4 * q^99

Introduction

L-functions

Modular forms

Varieties

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Representations

Groups

Database

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