LMFDB - Embedded newform 1336.2.a.d.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1336,2,Mod(1,1336)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1336, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1336.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1336 = 2^{3} \cdot 167 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1336.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:	$10.6680137100$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - x^{11} - 25 x^{10} + 20 x^{9} + 224 x^{8} - 135 x^{7} - 865 x^{6} + 330 x^{5} + 1341 x^{4} + \cdots + 64 $ x^12 - x^11 - 25x^10 + 20x^9 + 224x^8 - 135x^7 - 865x^6 + 330x^5 + 1341x^4 - 127x^3 - 652x^2 - 48x + 64
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$0.830621$ of defining polynomial
Character	$\chi$	$=$	1336.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.830621 q^{3} -1.03423 q^{5} -1.08177 q^{7} -2.31007 q^{9} +O(q^{10})$	$q-0.830621 q^{3} -1.03423 q^{5} -1.08177 q^{7} -2.31007 q^{9} -5.19449 q^{11} -0.733510 q^{13} +0.859056 q^{15} +5.46167 q^{17} +7.16217 q^{19} +0.898545 q^{21} -3.63888 q^{23} -3.93036 q^{25} +4.41066 q^{27} +7.44035 q^{29} +0.434043 q^{31} +4.31465 q^{33} +1.11881 q^{35} +6.82241 q^{37} +0.609269 q^{39} +9.91593 q^{41} -8.12915 q^{43} +2.38915 q^{45} +9.13889 q^{47} -5.82976 q^{49} -4.53658 q^{51} +7.48428 q^{53} +5.37231 q^{55} -5.94905 q^{57} +6.12085 q^{59} -3.72131 q^{61} +2.49897 q^{63} +0.758620 q^{65} -1.15978 q^{67} +3.02253 q^{69} -11.2004 q^{71} -5.23145 q^{73} +3.26464 q^{75} +5.61926 q^{77} -10.0054 q^{79} +3.26662 q^{81} +3.84939 q^{83} -5.64864 q^{85} -6.18011 q^{87} -2.20747 q^{89} +0.793492 q^{91} -0.360526 q^{93} -7.40735 q^{95} +19.2778 q^{97} +11.9996 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - q^{3} + 8 q^{5} - 4 q^{7} + 15 q^{9}+O(q^{10}) $ 12 * q - q^3 + 8 * q^5 - 4 * q^7 + 15 * q^9	$ 12 q - q^{3} + 8 q^{5} - 4 q^{7} + 15 q^{9} + 6 q^{11} + 13 q^{13} + 4 q^{15} + 10 q^{17} - q^{19} + 13 q^{21} + 3 q^{23} + 18 q^{25} - 10 q^{27} + 29 q^{29} - 3 q^{31} + 3 q^{35} + 41 q^{37} + 10 q^{39} + 20 q^{41} - q^{43} + 42 q^{45} + 5 q^{47} + 20 q^{49} + 14 q^{51} + 39 q^{53} + 3 q^{55} + 3 q^{57} + 8 q^{59} + 30 q^{61} - 2 q^{63} + 21 q^{65} - 9 q^{67} + 33 q^{69} + 29 q^{71} + 12 q^{73} + q^{75} + 19 q^{77} - 2 q^{79} + 24 q^{81} - 5 q^{83} + 44 q^{85} - 2 q^{87} + 7 q^{89} - 4 q^{91} + 37 q^{93} + 18 q^{95} - 14 q^{97} + 5 q^{99}+O(q^{100}) $ 12 * q - q^3 + 8 * q^5 - 4 * q^7 + 15 * q^9 + 6 * q^11 + 13 * q^13 + 4 * q^15 + 10 * q^17 - q^19 + 13 * q^21 + 3 * q^23 + 18 * q^25 - 10 * q^27 + 29 * q^29 - 3 * q^31 + 3 * q^35 + 41 * q^37 + 10 * q^39 + 20 * q^41 - q^43 + 42 * q^45 + 5 * q^47 + 20 * q^49 + 14 * q^51 + 39 * q^53 + 3 * q^55 + 3 * q^57 + 8 * q^59 + 30 * q^61 - 2 * q^63 + 21 * q^65 - 9 * q^67 + 33 * q^69 + 29 * q^71 + 12 * q^73 + q^75 + 19 * q^77 - 2 * q^79 + 24 * q^81 - 5 * q^83 + 44 * q^85 - 2 * q^87 + 7 * q^89 - 4 * q^91 + 37 * q^93 + 18 * q^95 - 14 * q^97 + 5 * 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
$3$	−0.830621	−0.479559	−0.239780	−	0.970827i	$-0.577075\pi$
$3$	−0.830621	−0.479559	−0.239780	+	0.970827i	$0.577075\pi$
$4$	0	0
$5$	−1.03423	−0.462523	−0.231261	−	0.972892i	$-0.574285\pi$
$5$	−1.03423	−0.462523	−0.231261	+	0.972892i	$0.574285\pi$
$6$	0	0
$7$	−1.08177	−0.408872	−0.204436	−	0.978880i	$-0.565536\pi$
$7$	−1.08177	−0.408872	−0.204436	+	0.978880i	$0.565536\pi$
$8$	0	0
$9$	−2.31007	−0.770023
$10$	0	0
$11$	−5.19449	−1.56620	−0.783099	−	0.621898i	$-0.786362\pi$
$11$	−5.19449	−1.56620	−0.783099	+	0.621898i	$0.786362\pi$
$12$	0	0
$13$	−0.733510	−0.203439	−0.101719	−	0.994813i	$-0.532434\pi$
$13$	−0.733510	−0.203439	−0.101719	+	0.994813i	$0.532434\pi$
$14$	0	0
$15$	0.859056	0.221807
$16$	0	0
$17$	5.46167	1.32465	0.662325	−	0.749216i	$-0.269570\pi$
$17$	5.46167	1.32465	0.662325	+	0.749216i	$0.269570\pi$
$18$	0	0
$19$	7.16217	1.64311	0.821557	−	0.570127i	$-0.193106\pi$
$19$	7.16217	1.64311	0.821557	+	0.570127i	$0.193106\pi$
$20$	0	0
$21$	0.898545	0.196078
$22$	0	0
$23$	−3.63888	−0.758760	−0.379380	−	0.925241i	$-0.623863\pi$
$23$	−3.63888	−0.758760	−0.379380	+	0.925241i	$0.623863\pi$
$24$	0	0
$25$	−3.93036	−0.786073
$26$	0	0
$27$	4.41066	0.848831
$28$	0	0
$29$	7.44035	1.38164	0.690819	−	0.723028i	$-0.257250\pi$
$29$	7.44035	1.38164	0.690819	+	0.723028i	$0.257250\pi$
$30$	0	0
$31$	0.434043	0.0779565	0.0389782	−	0.999240i	$-0.487590\pi$
$31$	0.434043	0.0779565	0.0389782	+	0.999240i	$0.487590\pi$
$32$	0	0
$33$	4.31465	0.751085
$34$	0	0
$35$	1.11881	0.189113
$36$	0	0
$37$	6.82241	1.12160	0.560799	−	0.827952i	$-0.310494\pi$
$37$	6.82241	1.12160	0.560799	+	0.827952i	$0.310494\pi$
$38$	0	0
$39$	0.609269	0.0975611
$40$	0	0
$41$	9.91593	1.54861	0.774304	−	0.632813i	$-0.218100\pi$
$41$	9.91593	1.54861	0.774304	+	0.632813i	$0.218100\pi$
$42$	0	0
$43$	−8.12915	−1.23968	−0.619842	−	0.784727i	$-0.712803\pi$
$43$	−8.12915	−1.23968	−0.619842	+	0.784727i	$0.712803\pi$
$44$	0	0
$45$	2.38915	0.356153
$46$	0	0
$47$	9.13889	1.33304	0.666522	−	0.745485i	$-0.267782\pi$
$47$	9.13889	1.33304	0.666522	+	0.745485i	$0.267782\pi$
$48$	0	0
$49$	−5.82976	−0.832824
$50$	0	0
$51$	−4.53658	−0.635248
$52$	0	0
$53$	7.48428	1.02805	0.514023	−	0.857777i	$-0.328155\pi$
$53$	7.48428	1.02805	0.514023	+	0.857777i	$0.328155\pi$
$54$	0	0
$55$	5.37231	0.724402
$56$	0	0
$57$	−5.94905	−0.787970
$58$	0	0
$59$	6.12085	0.796867	0.398433	−	0.917197i	$-0.369554\pi$
$59$	6.12085	0.796867	0.398433	+	0.917197i	$0.369554\pi$
$60$	0	0
$61$	−3.72131	−0.476465	−0.238233	−	0.971208i	$-0.576568\pi$
$61$	−3.72131	−0.476465	−0.238233	+	0.971208i	$0.576568\pi$
$62$	0	0
$63$	2.49897	0.314841
$64$	0	0
$65$	0.758620	0.0940952
$66$	0	0
$67$	−1.15978	−0.141690	−0.0708448	−	0.997487i	$-0.522569\pi$
$67$	−1.15978	−0.141690	−0.0708448	+	0.997487i	$0.522569\pi$
$68$	0	0
$69$	3.02253	0.363870
$70$	0	0
$71$	−11.2004	−1.32924	−0.664621	−	0.747180i	$-0.731407\pi$
$71$	−11.2004	−1.32924	−0.664621	+	0.747180i	$0.731407\pi$
$72$	0	0
$73$	−5.23145	−0.612295	−0.306147	−	0.951984i	$-0.599040\pi$
$73$	−5.23145	−0.612295	−0.306147	+	0.951984i	$0.599040\pi$
$74$	0	0
$75$	3.26464	0.376968
$76$	0	0
$77$	5.61926	0.640375
$78$	0	0
$79$	−10.0054	−1.12569	−0.562847	−	0.826561i	$-0.690294\pi$
$79$	−10.0054	−1.12569	−0.562847	+	0.826561i	$0.690294\pi$
$80$	0	0
$81$	3.26662	0.362958
$82$	0	0
$83$	3.84939	0.422526	0.211263	−	0.977429i	$-0.432242\pi$
$83$	3.84939	0.422526	0.211263	+	0.977429i	$0.432242\pi$
$84$	0	0
$85$	−5.64864	−0.612681
$86$	0	0
$87$	−6.18011	−0.662577
$88$	0	0
$89$	−2.20747	−0.233991	−0.116996	−	0.993132i	$-0.537326\pi$
$89$	−2.20747	−0.233991	−0.116996	+	0.993132i	$0.537326\pi$
$90$	0	0
$91$	0.793492	0.0831806
$92$	0	0
$93$	−0.360526	−0.0373848
$94$	0	0
$95$	−7.40735	−0.759978
$96$	0	0
$97$	19.2778	1.95736	0.978681	−	0.205386i	$-0.0658448\pi$
$97$	19.2778	1.95736	0.978681	+	0.205386i	$0.0658448\pi$
$98$	0	0
$99$	11.9996	1.20601
$100$	0	0
$101$	16.0673	1.59875	0.799377	−	0.600830i	$-0.205163\pi$
$101$	16.0673	1.59875	0.799377	+	0.600830i	$0.205163\pi$
$102$	0	0
$103$	−1.04112	−0.102584	−0.0512922	−	0.998684i	$-0.516334\pi$
$103$	−1.04112	−0.102584	−0.0512922	+	0.998684i	$0.516334\pi$
$104$	0	0
$105$	−0.929304	−0.0906908
$106$	0	0
$107$	−0.545538	−0.0527391	−0.0263696	−	0.999652i	$-0.508395\pi$
$107$	−0.545538	−0.0527391	−0.0263696	+	0.999652i	$0.508395\pi$
$108$	0	0
$109$	14.8074	1.41829	0.709144	−	0.705064i	$-0.249082\pi$
$109$	14.8074	1.41829	0.709144	+	0.705064i	$0.249082\pi$
$110$	0	0
$111$	−5.66684	−0.537872
$112$	0	0
$113$	5.35835	0.504072	0.252036	−	0.967718i	$-0.418900\pi$
$113$	5.35835	0.504072	0.252036	+	0.967718i	$0.418900\pi$
$114$	0	0
$115$	3.76345	0.350944
$116$	0	0
$117$	1.69446	0.156653
$118$	0	0
$119$	−5.90830	−0.541613
$120$	0	0
$121$	15.9827	1.45297
$122$	0	0
$123$	−8.23639	−0.742650
$124$	0	0
$125$	9.23607	0.826100
$126$	0	0
$127$	11.5656	1.02629	0.513143	−	0.858303i	$-0.328481\pi$
$127$	11.5656	1.02629	0.513143	+	0.858303i	$0.328481\pi$
$128$	0	0
$129$	6.75224	0.594502
$130$	0	0
$131$	7.88443	0.688865	0.344433	−	0.938811i	$-0.388071\pi$
$131$	7.88443	0.688865	0.344433	+	0.938811i	$0.388071\pi$
$132$	0	0
$133$	−7.74785	−0.671823
$134$	0	0
$135$	−4.56164	−0.392604
$136$	0	0
$137$	−18.9629	−1.62011	−0.810055	−	0.586354i	$-0.800563\pi$
$137$	−18.9629	−1.62011	−0.810055	+	0.586354i	$0.800563\pi$
$138$	0	0
$139$	10.4787	0.888791	0.444395	−	0.895831i	$-0.353419\pi$
$139$	10.4787	0.888791	0.444395	+	0.895831i	$0.353419\pi$
$140$	0	0
$141$	−7.59095	−0.639274
$142$	0	0
$143$	3.81021	0.318626
$144$	0	0
$145$	−7.69505	−0.639039
$146$	0	0
$147$	4.84233	0.399388
$148$	0	0
$149$	11.0335	0.903897	0.451948	−	0.892044i	$-0.350729\pi$
$149$	11.0335	0.903897	0.451948	+	0.892044i	$0.350729\pi$
$150$	0	0
$151$	−23.3978	−1.90408	−0.952041	−	0.305970i	$-0.901019\pi$
$151$	−23.3978	−1.90408	−0.952041	+	0.305970i	$0.901019\pi$
$152$	0	0
$153$	−12.6168	−1.02001
$154$	0	0
$155$	−0.448902	−0.0360567
$156$	0	0
$157$	−3.86681	−0.308605	−0.154302	−	0.988024i	$-0.549313\pi$
$157$	−3.86681	−0.308605	−0.154302	+	0.988024i	$0.549313\pi$
$158$	0	0
$159$	−6.21660	−0.493009
$160$	0	0
$161$	3.93645	0.310236
$162$	0	0
$163$	−6.51420	−0.510231	−0.255116	−	0.966911i	$-0.582114\pi$
$163$	−6.51420	−0.510231	−0.255116	+	0.966911i	$0.582114\pi$
$164$	0	0
$165$	−4.46235	−0.347394
$166$	0	0
$167$	−1.00000	−0.0773823
$167$	−1.00000	−0.0773823
$168$	0	0
$169$	−12.4620	−0.958613
$170$	0	0
$171$	−16.5451	−1.26523
$172$	0	0
$173$	−14.5058	−1.10286	−0.551429	−	0.834222i	$-0.685917\pi$
$173$	−14.5058	−1.10286	−0.551429	+	0.834222i	$0.685917\pi$
$174$	0	0
$175$	4.25176	0.321403
$176$	0	0
$177$	−5.08411	−0.382145
$178$	0	0
$179$	−6.35369	−0.474897	−0.237449	−	0.971400i	$-0.576311\pi$
$179$	−6.35369	−0.474897	−0.237449	+	0.971400i	$0.576311\pi$
$180$	0	0
$181$	18.4840	1.37391	0.686954	−	0.726701i	$-0.258947\pi$
$181$	18.4840	1.37391	0.686954	+	0.726701i	$0.258947\pi$
$182$	0	0
$183$	3.09100	0.228493
$184$	0	0
$185$	−7.05596	−0.518764
$186$	0	0
$187$	−28.3706	−2.07466
$188$	0	0
$189$	−4.77133	−0.347063
$190$	0	0
$191$	21.6023	1.56308	0.781542	−	0.623852i	$-0.214433\pi$
$191$	21.6023	1.56308	0.781542	+	0.623852i	$0.214433\pi$
$192$	0	0
$193$	19.4755	1.40188	0.700938	−	0.713223i	$-0.252765\pi$
$193$	19.4755	1.40188	0.700938	+	0.713223i	$0.252765\pi$
$194$	0	0
$195$	−0.630126	−0.0451242
$196$	0	0
$197$	1.79687	0.128022	0.0640108	−	0.997949i	$-0.479611\pi$
$197$	1.79687	0.128022	0.0640108	+	0.997949i	$0.479611\pi$
$198$	0	0
$199$	−2.03506	−0.144261	−0.0721307	−	0.997395i	$-0.522980\pi$
$199$	−2.03506	−0.144261	−0.0721307	+	0.997395i	$0.522980\pi$
$200$	0	0
$201$	0.963337	0.0679485
$202$	0	0
$203$	−8.04878	−0.564913
$204$	0	0
$205$	−10.2554	−0.716267
$206$	0	0
$207$	8.40607	0.584262
$208$	0	0
$209$	−37.2038	−2.57344
$210$	0	0
$211$	−24.4701	−1.68459	−0.842297	−	0.539014i	$-0.818797\pi$
$211$	−24.4701	−1.68459	−0.842297	+	0.539014i	$0.818797\pi$
$212$	0	0
$213$	9.30329	0.637451
$214$	0	0
$215$	8.40743	0.573382
$216$	0	0
$217$	−0.469537	−0.0318742
$218$	0	0
$219$	4.34535	0.293632
$220$	0	0
$221$	−4.00619	−0.269486
$222$	0	0
$223$	28.1464	1.88482	0.942411	−	0.334458i	$-0.108553\pi$
$223$	28.1464	1.88482	0.942411	+	0.334458i	$0.108553\pi$
$224$	0	0
$225$	9.07941	0.605294
$226$	0	0
$227$	16.4044	1.08880	0.544399	−	0.838826i	$-0.316758\pi$
$227$	16.4044	1.08880	0.544399	+	0.838826i	$0.316758\pi$
$228$	0	0
$229$	−6.58294	−0.435013	−0.217507	−	0.976059i	$-0.569792\pi$
$229$	−6.58294	−0.435013	−0.217507	+	0.976059i	$0.569792\pi$
$230$	0	0
$231$	−4.66748	−0.307098
$232$	0	0
$233$	21.8552	1.43178	0.715892	−	0.698211i	$-0.246020\pi$
$233$	21.8552	1.43178	0.715892	+	0.698211i	$0.246020\pi$
$234$	0	0
$235$	−9.45174	−0.616563
$236$	0	0
$237$	8.31069	0.539837
$238$	0	0
$239$	5.47698	0.354276	0.177138	−	0.984186i	$-0.443316\pi$
$239$	5.47698	0.354276	0.177138	+	0.984186i	$0.443316\pi$
$240$	0	0
$241$	−11.2718	−0.726079	−0.363039	−	0.931774i	$-0.618261\pi$
$241$	−11.2718	−0.726079	−0.363039	+	0.931774i	$0.618261\pi$
$242$	0	0
$243$	−15.9453	−1.02289
$244$	0	0
$245$	6.02933	0.385200
$246$	0	0
$247$	−5.25352	−0.334273
$248$	0	0
$249$	−3.19739	−0.202626
$250$	0	0
$251$	12.5496	0.792121	0.396061	−	0.918224i	$-0.370377\pi$
$251$	12.5496	0.792121	0.396061	+	0.918224i	$0.370377\pi$
$252$	0	0
$253$	18.9021	1.18837
$254$	0	0
$255$	4.69188	0.293817
$256$	0	0
$257$	−24.0529	−1.50038	−0.750189	−	0.661224i	$-0.770037\pi$
$257$	−24.0529	−1.50038	−0.750189	+	0.661224i	$0.770037\pi$
$258$	0	0
$259$	−7.38031	−0.458590
$260$	0	0
$261$	−17.1877	−1.06389
$262$	0	0
$263$	−28.0441	−1.72928	−0.864638	−	0.502396i	$-0.832452\pi$
$263$	−28.0441	−1.72928	−0.864638	+	0.502396i	$0.832452\pi$
$264$	0	0
$265$	−7.74049	−0.475494
$266$	0	0
$267$	1.83357	0.112213
$268$	0	0
$269$	−7.48293	−0.456242	−0.228121	−	0.973633i	$-0.573258\pi$
$269$	−7.48293	−0.456242	−0.228121	+	0.973633i	$0.573258\pi$
$270$	0	0
$271$	2.61679	0.158958	0.0794792	−	0.996837i	$-0.474674\pi$
$271$	2.61679	0.158958	0.0794792	+	0.996837i	$0.474674\pi$
$272$	0	0
$273$	−0.659091	−0.0398900
$274$	0	0
$275$	20.4162	1.23114
$276$	0	0
$277$	−19.2844	−1.15869	−0.579343	−	0.815084i	$-0.696691\pi$
$277$	−19.2844	−1.15869	−0.579343	+	0.815084i	$0.696691\pi$
$278$	0	0
$279$	−1.00267	−0.0600283
$280$	0	0
$281$	−17.5920	−1.04945	−0.524724	−	0.851272i	$-0.675831\pi$
$281$	−17.5920	−1.04945	−0.524724	+	0.851272i	$0.675831\pi$
$282$	0	0
$283$	−12.0387	−0.715626	−0.357813	−	0.933793i	$-0.616477\pi$
$283$	−12.0387	−0.715626	−0.357813	+	0.933793i	$0.616477\pi$
$284$	0	0
$285$	6.15270	0.364454
$286$	0	0
$287$	−10.7268	−0.633183
$288$	0	0
$289$	12.8299	0.754699
$290$	0	0
$291$	−16.0125	−0.938671
$292$	0	0
$293$	15.0896	0.881545	0.440772	−	0.897619i	$-0.354705\pi$
$293$	15.0896	0.881545	0.440772	+	0.897619i	$0.354705\pi$
$294$	0	0
$295$	−6.33038	−0.368569
$296$	0	0
$297$	−22.9111	−1.32944
$298$	0	0
$299$	2.66916	0.154361
$300$	0	0
$301$	8.79390	0.506872
$302$	0	0
$303$	−13.3458	−0.766697
$304$	0	0
$305$	3.84870	0.220376
$306$	0	0
$307$	−12.3513	−0.704925	−0.352462	−	0.935826i	$-0.614656\pi$
$307$	−12.3513	−0.704925	−0.352462	+	0.935826i	$0.614656\pi$
$308$	0	0
$309$	0.864774	0.0491953
$310$	0	0
$311$	28.9333	1.64066	0.820328	−	0.571893i	$-0.193791\pi$
$311$	28.9333	1.64066	0.820328	+	0.571893i	$0.193791\pi$
$312$	0	0
$313$	−2.38177	−0.134625	−0.0673127	−	0.997732i	$-0.521443\pi$
$313$	−2.38177	−0.134625	−0.0673127	+	0.997732i	$0.521443\pi$
$314$	0	0
$315$	−2.58452	−0.145621
$316$	0	0
$317$	3.66976	0.206114	0.103057	−	0.994675i	$-0.467138\pi$
$317$	3.66976	0.206114	0.103057	+	0.994675i	$0.467138\pi$
$318$	0	0
$319$	−38.6488	−2.16392
$320$	0	0
$321$	0.453135	0.0252915
$322$	0	0
$323$	39.1174	2.17655
$324$	0	0
$325$	2.88296	0.159918
$326$	0	0
$327$	−12.2993	−0.680153
$328$	0	0
$329$	−9.88621	−0.545045
$330$	0	0
$331$	0.184961	0.0101664	0.00508318	−	0.999987i	$-0.498382\pi$
$331$	0.184961	0.0101664	0.00508318	+	0.999987i	$0.498382\pi$
$332$	0	0
$333$	−15.7602	−0.863656
$334$	0	0
$335$	1.19948	0.0655346
$336$	0	0
$337$	−30.1126	−1.64034	−0.820169	−	0.572122i	$-0.806121\pi$
$337$	−30.1126	−1.64034	−0.820169	+	0.572122i	$0.806121\pi$
$338$	0	0
$339$	−4.45076	−0.241732
$340$	0	0
$341$	−2.25463	−0.122095
$342$	0	0
$343$	13.8789	0.749391
$344$	0	0
$345$	−3.12600	−0.168298
$346$	0	0
$347$	1.67354	0.0898402	0.0449201	−	0.998991i	$-0.485697\pi$
$347$	1.67354	0.0898402	0.0449201	+	0.998991i	$0.485697\pi$
$348$	0	0
$349$	33.0874	1.77113	0.885564	−	0.464517i	$-0.153772\pi$
$349$	33.0874	1.77113	0.885564	+	0.464517i	$0.153772\pi$
$350$	0	0
$351$	−3.23526	−0.172685
$352$	0	0
$353$	−6.19491	−0.329722	−0.164861	−	0.986317i	$-0.552718\pi$
$353$	−6.19491	−0.329722	−0.164861	+	0.986317i	$0.552718\pi$
$354$	0	0
$355$	11.5838	0.614805
$356$	0	0
$357$	4.90756	0.259735
$358$	0	0
$359$	34.9233	1.84318	0.921591	−	0.388162i	$-0.126890\pi$
$359$	34.9233	1.84318	0.921591	+	0.388162i	$0.126890\pi$
$360$	0	0
$361$	32.2966	1.69982
$362$	0	0
$363$	−13.2756	−0.696787
$364$	0	0
$365$	5.41054	0.283200
$366$	0	0
$367$	5.99095	0.312725	0.156363	−	0.987700i	$-0.450023\pi$
$367$	5.99095	0.312725	0.156363	+	0.987700i	$0.450023\pi$
$368$	0	0
$369$	−22.9065	−1.19246
$370$	0	0
$371$	−8.09630	−0.420339
$372$	0	0
$373$	14.3731	0.744209	0.372105	−	0.928191i	$-0.378636\pi$
$373$	14.3731	0.744209	0.372105	+	0.928191i	$0.378636\pi$
$374$	0	0
$375$	−7.67168	−0.396164
$376$	0	0
$377$	−5.45757	−0.281079
$378$	0	0
$379$	−12.7508	−0.654964	−0.327482	−	0.944857i	$-0.606200\pi$
$379$	−12.7508	−0.654964	−0.327482	+	0.944857i	$0.606200\pi$
$380$	0	0
$381$	−9.60667	−0.492165
$382$	0	0
$383$	−24.6239	−1.25822	−0.629111	−	0.777315i	$-0.716581\pi$
$383$	−24.6239	−1.25822	−0.629111	+	0.777315i	$0.716581\pi$
$384$	0	0
$385$	−5.81163	−0.296188
$386$	0	0
$387$	18.7789	0.954584
$388$	0	0
$389$	8.61953	0.437027	0.218514	−	0.975834i	$-0.429879\pi$
$389$	8.61953	0.437027	0.218514	+	0.975834i	$0.429879\pi$
$390$	0	0
$391$	−19.8744	−1.00509
$392$	0	0
$393$	−6.54897	−0.330352
$394$	0	0
$395$	10.3479	0.520660
$396$	0	0
$397$	22.0289	1.10560	0.552800	−	0.833314i	$-0.313560\pi$
$397$	22.0289	1.10560	0.552800	+	0.833314i	$0.313560\pi$
$398$	0	0
$399$	6.43552	0.322179
$400$	0	0
$401$	−23.4949	−1.17328	−0.586639	−	0.809849i	$-0.699549\pi$
$401$	−23.4949	−1.17328	−0.586639	+	0.809849i	$0.699549\pi$
$402$	0	0
$403$	−0.318375	−0.0158594
$404$	0	0
$405$	−3.37845	−0.167876
$406$	0	0
$407$	−35.4389	−1.75664
$408$	0	0
$409$	1.68171	0.0831552	0.0415776	−	0.999135i	$-0.486762\pi$
$409$	1.68171	0.0831552	0.0415776	+	0.999135i	$0.486762\pi$
$410$	0	0
$411$	15.7510	0.776939
$412$	0	0
$413$	−6.62138	−0.325817
$414$	0	0
$415$	−3.98117	−0.195428
$416$	0	0
$417$	−8.70382	−0.426228
$418$	0	0
$419$	25.6411	1.25265	0.626326	−	0.779562i	$-0.284558\pi$
$419$	25.6411	1.25265	0.626326	+	0.779562i	$0.284558\pi$
$420$	0	0
$421$	−4.45167	−0.216961	−0.108481	−	0.994099i	$-0.534599\pi$
$421$	−4.45167	−0.216961	−0.108481	+	0.994099i	$0.534599\pi$
$422$	0	0
$423$	−21.1115	−1.02647
$424$	0	0
$425$	−21.4664	−1.04127
$426$	0	0
$427$	4.02562	0.194813
$428$	0	0
$429$	−3.16484	−0.152800
$430$	0	0
$431$	9.28120	0.447060	0.223530	−	0.974697i	$-0.428242\pi$
$431$	9.28120	0.447060	0.223530	+	0.974697i	$0.428242\pi$
$432$	0	0
$433$	−27.6559	−1.32906	−0.664528	−	0.747263i	$-0.731367\pi$
$433$	−27.6559	−1.32906	−0.664528	+	0.747263i	$0.731367\pi$
$434$	0	0
$435$	6.39167	0.306457
$436$	0	0
$437$	−26.0623	−1.24673
$438$	0	0
$439$	23.6543	1.12896	0.564480	−	0.825447i	$-0.309077\pi$
$439$	23.6543	1.12896	0.564480	+	0.825447i	$0.309077\pi$
$440$	0	0
$441$	13.4672	0.641293
$442$	0	0
$443$	22.3493	1.06185	0.530924	−	0.847419i	$-0.321845\pi$
$443$	22.3493	1.06185	0.530924	+	0.847419i	$0.321845\pi$
$444$	0	0
$445$	2.28304	0.108226
$446$	0	0
$447$	−9.16463	−0.433472
$448$	0	0
$449$	30.2514	1.42765	0.713827	−	0.700322i	$-0.246960\pi$
$449$	30.2514	1.42765	0.713827	+	0.700322i	$0.246960\pi$
$450$	0	0
$451$	−51.5082	−2.42543
$452$	0	0
$453$	19.4347	0.913120
$454$	0	0
$455$	−0.820655	−0.0384729
$456$	0	0
$457$	−41.2635	−1.93022	−0.965112	−	0.261838i	$-0.915671\pi$
$457$	−41.2635	−1.93022	−0.965112	+	0.261838i	$0.915671\pi$
$458$	0	0
$459$	24.0896	1.12440
$460$	0	0
$461$	26.1155	1.21632	0.608160	−	0.793815i	$-0.291908\pi$
$461$	26.1155	1.21632	0.608160	+	0.793815i	$0.291908\pi$
$462$	0	0
$463$	42.3969	1.97035	0.985175	−	0.171552i	$-0.0548781\pi$
$463$	42.3969	1.97035	0.985175	+	0.171552i	$0.0548781\pi$
$464$	0	0
$465$	0.372867	0.0172913
$466$	0	0
$467$	25.7545	1.19178	0.595889	−	0.803067i	$-0.296800\pi$
$467$	25.7545	1.19178	0.595889	+	0.803067i	$0.296800\pi$
$468$	0	0
$469$	1.25462	0.0579329
$470$	0	0
$471$	3.21185	0.147994
$472$	0	0
$473$	42.2268	1.94159
$474$	0	0
$475$	−28.1499	−1.29161
$476$	0	0
$477$	−17.2892	−0.791618
$478$	0	0
$479$	−20.7591	−0.948505	−0.474253	−	0.880389i	$-0.657282\pi$
$479$	−20.7591	−0.948505	−0.474253	+	0.880389i	$0.657282\pi$
$480$	0	0
$481$	−5.00430	−0.228177
$482$	0	0
$483$	−3.26970	−0.148776
$484$	0	0
$485$	−19.9377	−0.905325
$486$	0	0
$487$	13.0364	0.590735	0.295368	−	0.955384i	$-0.404558\pi$
$487$	13.0364	0.590735	0.295368	+	0.955384i	$0.404558\pi$
$488$	0	0
$489$	5.41083	0.244686
$490$	0	0
$491$	−3.58494	−0.161786	−0.0808930	−	0.996723i	$-0.525777\pi$
$491$	−3.58494	−0.161786	−0.0808930	+	0.996723i	$0.525777\pi$
$492$	0	0
$493$	40.6367	1.83019
$494$	0	0
$495$	−12.4104	−0.557806
$496$	0	0
$497$	12.1163	0.543490
$498$	0	0
$499$	26.8807	1.20335	0.601673	−	0.798742i	$-0.294501\pi$
$499$	26.8807	1.20335	0.601673	+	0.798742i	$0.294501\pi$
$500$	0	0
$501$	0.830621	0.0371094
$502$	0	0
$503$	−1.55688	−0.0694176	−0.0347088	−	0.999397i	$-0.511050\pi$
$503$	−1.55688	−0.0694176	−0.0347088	+	0.999397i	$0.511050\pi$
$504$	0	0
$505$	−16.6173	−0.739460
$506$	0	0
$507$	10.3512	0.459712
$508$	0	0
$509$	17.2289	0.763656	0.381828	−	0.924233i	$-0.375295\pi$
$509$	17.2289	0.763656	0.381828	+	0.924233i	$0.375295\pi$
$510$	0	0
$511$	5.65925	0.250350
$512$	0	0
$513$	31.5898	1.39473
$514$	0	0
$515$	1.07676	0.0474476
$516$	0	0
$517$	−47.4718	−2.08781
$518$	0	0
$519$	12.0488	0.528885
$520$	0	0
$521$	19.5957	0.858502	0.429251	−	0.903185i	$-0.358778\pi$
$521$	19.5957	0.858502	0.429251	+	0.903185i	$0.358778\pi$
$522$	0	0
$523$	35.3535	1.54590	0.772950	−	0.634467i	$-0.218780\pi$
$523$	35.3535	1.54590	0.772950	+	0.634467i	$0.218780\pi$
$524$	0	0
$525$	−3.53161	−0.154132
$526$	0	0
$527$	2.37060	0.103265
$528$	0	0
$529$	−9.75852	−0.424283
$530$	0	0
$531$	−14.1396	−0.613606
$532$	0	0
$533$	−7.27343	−0.315047
$534$	0	0
$535$	0.564213	0.0243930
$536$	0	0
$537$	5.27751	0.227741
$538$	0	0
$539$	30.2826	1.30437
$540$	0	0
$541$	−7.18029	−0.308705	−0.154352	−	0.988016i	$-0.549329\pi$
$541$	−7.18029	−0.308705	−0.154352	+	0.988016i	$0.549329\pi$
$542$	0	0
$543$	−15.3532	−0.658871
$544$	0	0
$545$	−15.3143	−0.655991
$546$	0	0
$547$	−21.3404	−0.912447	−0.456224	−	0.889865i	$-0.650798\pi$
$547$	−21.3404	−0.912447	−0.456224	+	0.889865i	$0.650798\pi$
$548$	0	0
$549$	8.59648	0.366889
$550$	0	0
$551$	53.2890	2.27019
$552$	0	0
$553$	10.8236	0.460265
$554$	0	0
$555$	5.86083	0.248778
$556$	0	0
$557$	−4.26602	−0.180757	−0.0903785	−	0.995907i	$-0.528808\pi$
$557$	−4.26602	−0.180757	−0.0903785	+	0.995907i	$0.528808\pi$
$558$	0	0
$559$	5.96281	0.252200
$560$	0	0
$561$	23.5652	0.994924
$562$	0	0
$563$	−31.7112	−1.33647	−0.668233	−	0.743952i	$-0.732949\pi$
$563$	−31.7112	−1.33647	−0.668233	+	0.743952i	$0.732949\pi$
$564$	0	0
$565$	−5.54178	−0.233145
$566$	0	0
$567$	−3.53375	−0.148403
$568$	0	0
$569$	5.11910	0.214604	0.107302	−	0.994226i	$-0.465779\pi$
$569$	5.11910	0.214604	0.107302	+	0.994226i	$0.465779\pi$
$570$	0	0
$571$	28.2476	1.18213	0.591063	−	0.806625i	$-0.298708\pi$
$571$	28.2476	1.18213	0.591063	+	0.806625i	$0.298708\pi$
$572$	0	0
$573$	−17.9433	−0.749592
$574$	0	0
$575$	14.3021	0.596440
$576$	0	0
$577$	−15.9293	−0.663146	−0.331573	−	0.943430i	$-0.607579\pi$
$577$	−15.9293	−0.663146	−0.331573	+	0.943430i	$0.607579\pi$
$578$	0	0
$579$	−16.1767	−0.672282
$580$	0	0
$581$	−4.16417	−0.172759
$582$	0	0
$583$	−38.8770	−1.61012
$584$	0	0
$585$	−1.75246	−0.0724555
$586$	0	0
$587$	−17.8954	−0.738622	−0.369311	−	0.929306i	$-0.620406\pi$
$587$	−17.8954	−0.738622	−0.369311	+	0.929306i	$0.620406\pi$
$588$	0	0
$589$	3.10869	0.128091
$590$	0	0
$591$	−1.49252	−0.0613940
$592$	0	0
$593$	−31.3747	−1.28840	−0.644202	−	0.764855i	$-0.722810\pi$
$593$	−31.3747	−1.28840	−0.644202	+	0.764855i	$0.722810\pi$
$594$	0	0
$595$	6.11055	0.250508
$596$	0	0
$597$	1.69036	0.0691819
$598$	0	0
$599$	36.3230	1.48412	0.742058	−	0.670336i	$-0.233850\pi$
$599$	36.3230	1.48412	0.742058	+	0.670336i	$0.233850\pi$
$600$	0	0
$601$	−5.70041	−0.232524	−0.116262	−	0.993219i	$-0.537091\pi$
$601$	−5.70041	−0.232524	−0.116262	+	0.993219i	$0.537091\pi$
$602$	0	0
$603$	2.67917	0.109104
$604$	0	0
$605$	−16.5298	−0.672034
$606$	0	0
$607$	−24.6830	−1.00185	−0.500927	−	0.865490i	$-0.667007\pi$
$607$	−24.6830	−1.00185	−0.500927	+	0.865490i	$0.667007\pi$
$608$	0	0
$609$	6.68548	0.270909
$610$	0	0
$611$	−6.70346	−0.271193
$612$	0	0
$613$	−18.2621	−0.737600	−0.368800	−	0.929509i	$-0.620231\pi$
$613$	−18.2621	−0.737600	−0.368800	+	0.929509i	$0.620231\pi$
$614$	0	0
$615$	8.51834	0.343493
$616$	0	0
$617$	42.8756	1.72611	0.863053	−	0.505113i	$-0.168549\pi$
$617$	42.8756	1.72611	0.863053	+	0.505113i	$0.168549\pi$
$618$	0	0
$619$	34.3522	1.38073	0.690366	−	0.723460i	$-0.257449\pi$
$619$	34.3522	1.38073	0.690366	+	0.723460i	$0.257449\pi$
$620$	0	0
$621$	−16.0499	−0.644059
$622$	0	0
$623$	2.38798	0.0956725
$624$	0	0
$625$	10.0996	0.403983
$626$	0	0
$627$	30.9023	1.23412
$628$	0	0
$629$	37.2618	1.48572
$630$	0	0
$631$	−17.6344	−0.702013	−0.351006	−	0.936373i	$-0.614160\pi$
$631$	−17.6344	−0.702013	−0.351006	+	0.936373i	$0.614160\pi$
$632$	0	0
$633$	20.3254	0.807862
$634$	0	0
$635$	−11.9616	−0.474681
$636$	0	0
$637$	4.27619	0.169429
$638$	0	0
$639$	25.8737	1.02355
$640$	0	0
$641$	−18.3646	−0.725357	−0.362679	−	0.931914i	$-0.618138\pi$
$641$	−18.3646	−0.725357	−0.362679	+	0.931914i	$0.618138\pi$
$642$	0	0
$643$	10.9040	0.430011	0.215006	−	0.976613i	$-0.431023\pi$
$643$	10.9040	0.430011	0.215006	+	0.976613i	$0.431023\pi$
$644$	0	0
$645$	−6.98339	−0.274971
$646$	0	0
$647$	−26.8485	−1.05552	−0.527762	−	0.849392i	$-0.676969\pi$
$647$	−26.8485	−1.05552	−0.527762	+	0.849392i	$0.676969\pi$
$648$	0	0
$649$	−31.7947	−1.24805
$650$	0	0
$651$	0.390007	0.0152856
$652$	0	0
$653$	−12.2927	−0.481049	−0.240524	−	0.970643i	$-0.577319\pi$
$653$	−12.2927	−0.481049	−0.240524	+	0.970643i	$0.577319\pi$
$654$	0	0
$655$	−8.15433	−0.318616
$656$	0	0
$657$	12.0850	0.471481
$658$	0	0
$659$	47.6173	1.85491	0.927453	−	0.373940i	$-0.121993\pi$
$659$	47.6173	1.85491	0.927453	+	0.373940i	$0.121993\pi$
$660$	0	0
$661$	28.7386	1.11780	0.558901	−	0.829235i	$-0.311223\pi$
$661$	28.7386	1.11780	0.558901	+	0.829235i	$0.311223\pi$
$662$	0	0
$663$	3.32763	0.129234
$664$	0	0
$665$	8.01308	0.310734
$666$	0	0
$667$	−27.0746	−1.04833
$668$	0	0
$669$	−23.3790	−0.903884
$670$	0	0
$671$	19.3303	0.746238
$672$	0	0
$673$	−3.76705	−0.145209	−0.0726045	−	0.997361i	$-0.523131\pi$
$673$	−3.76705	−0.145209	−0.0726045	+	0.997361i	$0.523131\pi$
$674$	0	0
$675$	−17.3355	−0.667243
$676$	0	0
$677$	38.7622	1.48975	0.744876	−	0.667203i	$-0.232508\pi$
$677$	38.7622	1.48975	0.744876	+	0.667203i	$0.232508\pi$
$678$	0	0
$679$	−20.8542	−0.800311
$680$	0	0
$681$	−13.6258	−0.522143
$682$	0	0
$683$	−23.0200	−0.880836	−0.440418	−	0.897793i	$-0.645170\pi$
$683$	−23.0200	−0.880836	−0.440418	+	0.897793i	$0.645170\pi$
$684$	0	0
$685$	19.6120	0.749338
$686$	0	0
$687$	5.46793	0.208615
$688$	0	0
$689$	−5.48979	−0.209144
$690$	0	0
$691$	30.7984	1.17163	0.585813	−	0.810447i	$-0.300775\pi$
$691$	30.7984	1.17163	0.585813	+	0.810447i	$0.300775\pi$
$692$	0	0
$693$	−12.9809	−0.493103
$694$	0	0
$695$	−10.8374	−0.411086
$696$	0	0
$697$	54.1576	2.05137
$698$	0	0
$699$	−18.1534	−0.686625
$700$	0	0
$701$	21.9764	0.830038	0.415019	−	0.909813i	$-0.363775\pi$
$701$	21.9764	0.830038	0.415019	+	0.909813i	$0.363775\pi$
$702$	0	0
$703$	48.8632	1.84291
$704$	0	0
$705$	7.85081	0.295679
$706$	0	0
$707$	−17.3812	−0.653686
$708$	0	0
$709$	−13.3721	−0.502198	−0.251099	−	0.967961i	$-0.580792\pi$
$709$	−13.3721	−0.502198	−0.251099	+	0.967961i	$0.580792\pi$
$710$	0	0
$711$	23.1131	0.866810
$712$	0	0
$713$	−1.57943	−0.0591502
$714$	0	0
$715$	−3.94064	−0.147372
$716$	0	0
$717$	−4.54930	−0.169897
$718$	0	0
$719$	25.4278	0.948296	0.474148	−	0.880445i	$-0.342756\pi$
$719$	25.4278	0.948296	0.474148	+	0.880445i	$0.342756\pi$
$720$	0	0
$721$	1.12625	0.0419439
$722$	0	0
$723$	9.36257	0.348198
$724$	0	0
$725$	−29.2433	−1.08607
$726$	0	0
$727$	27.4398	1.01769	0.508844	−	0.860859i	$-0.330073\pi$
$727$	27.4398	1.01769	0.508844	+	0.860859i	$0.330073\pi$
$728$	0	0
$729$	3.44463	0.127579
$730$	0	0
$731$	−44.3987	−1.64215
$732$	0	0
$733$	−22.4326	−0.828566	−0.414283	−	0.910148i	$-0.635968\pi$
$733$	−22.4326	−0.828566	−0.414283	+	0.910148i	$0.635968\pi$
$734$	0	0
$735$	−5.00809	−0.184726
$736$	0	0
$737$	6.02446	0.221914
$738$	0	0
$739$	−40.5554	−1.49186	−0.745928	−	0.666027i	$-0.767994\pi$
$739$	−40.5554	−1.49186	−0.745928	+	0.666027i	$0.767994\pi$
$740$	0	0
$741$	4.36368	0.160304
$742$	0	0
$743$	5.37531	0.197201	0.0986005	−	0.995127i	$-0.468563\pi$
$743$	5.37531	0.197201	0.0986005	+	0.995127i	$0.468563\pi$
$744$	0	0
$745$	−11.4112	−0.418073
$746$	0	0
$747$	−8.89236	−0.325354
$748$	0	0
$749$	0.590149	0.0215636
$750$	0	0
$751$	42.3923	1.54692	0.773459	−	0.633847i	$-0.218525\pi$
$751$	42.3923	1.54692	0.773459	+	0.633847i	$0.218525\pi$
$752$	0	0
$753$	−10.4239	−0.379869
$754$	0	0
$755$	24.1987	0.880682
$756$	0	0
$757$	−8.76775	−0.318669	−0.159335	−	0.987225i	$-0.550935\pi$
$757$	−8.76775	−0.318669	−0.159335	+	0.987225i	$0.550935\pi$
$758$	0	0
$759$	−15.7005	−0.569893
$760$	0	0
$761$	−2.51304	−0.0910977	−0.0455489	−	0.998962i	$-0.514504\pi$
$761$	−2.51304	−0.0910977	−0.0455489	+	0.998962i	$0.514504\pi$
$762$	0	0
$763$	−16.0182	−0.579898
$764$	0	0
$765$	13.0487	0.471779
$766$	0	0
$767$	−4.48970	−0.162114
$768$	0	0
$769$	−48.0753	−1.73364	−0.866821	−	0.498620i	$-0.833840\pi$
$769$	−48.0753	−1.73364	−0.866821	+	0.498620i	$0.833840\pi$
$770$	0	0
$771$	19.9788	0.719520
$772$	0	0
$773$	−31.8450	−1.14539	−0.572693	−	0.819770i	$-0.694101\pi$
$773$	−31.8450	−1.14539	−0.572693	+	0.819770i	$0.694101\pi$
$774$	0	0
$775$	−1.70595	−0.0612794
$776$	0	0
$777$	6.13024	0.219921
$778$	0	0
$779$	71.0196	2.54454
$780$	0	0
$781$	58.1803	2.08186
$782$	0	0
$783$	32.8168	1.17278
$784$	0	0
$785$	3.99918	0.142737
$786$	0	0
$787$	−15.9313	−0.567889	−0.283945	−	0.958841i	$-0.591643\pi$
$787$	−15.9313	−0.567889	−0.283945	+	0.958841i	$0.591643\pi$
$788$	0	0
$789$	23.2941	0.829290
$790$	0	0
$791$	−5.79653	−0.206101
$792$	0	0
$793$	2.72962	0.0969316
$794$	0	0
$795$	6.42941	0.228028
$796$	0	0
$797$	19.7613	0.699980	0.349990	−	0.936754i	$-0.386185\pi$
$797$	19.7613	0.699980	0.349990	+	0.936754i	$0.386185\pi$
$798$	0	0
$799$	49.9136	1.76582
$800$	0	0
$801$	5.09940	0.180179
$802$	0	0
$803$	27.1747	0.958975
$804$	0	0
$805$	−4.07121	−0.143491
$806$	0	0
$807$	6.21548	0.218795
$808$	0	0
$809$	−5.77954	−0.203198	−0.101599	−	0.994825i	$-0.532396\pi$
$809$	−5.77954	−0.203198	−0.101599	+	0.994825i	$0.532396\pi$
$810$	0	0
$811$	−25.5001	−0.895428	−0.447714	−	0.894177i	$-0.647762\pi$
$811$	−25.5001	−0.895428	−0.447714	+	0.894177i	$0.647762\pi$
$812$	0	0
$813$	−2.17356	−0.0762300
$814$	0	0
$815$	6.73719	0.235994
$816$	0	0
$817$	−58.2223	−2.03694
$818$	0	0
$819$	−1.83302	−0.0640509
$820$	0	0
$821$	−37.7594	−1.31781	−0.658905	−	0.752226i	$-0.728980\pi$
$821$	−37.7594	−1.31781	−0.658905	+	0.752226i	$0.728980\pi$
$822$	0	0
$823$	−3.30358	−0.115156	−0.0575778	−	0.998341i	$-0.518338\pi$
$823$	−3.30358	−0.115156	−0.0575778	+	0.998341i	$0.518338\pi$
$824$	0	0
$825$	−16.9581	−0.590407
$826$	0	0
$827$	−28.6912	−0.997689	−0.498845	−	0.866691i	$-0.666242\pi$
$827$	−28.6912	−0.997689	−0.498845	+	0.866691i	$0.666242\pi$
$828$	0	0
$829$	33.5625	1.16568	0.582838	−	0.812589i	$-0.301942\pi$
$829$	33.5625	1.16568	0.582838	+	0.812589i	$0.301942\pi$
$830$	0	0
$831$	16.0180	0.555659
$832$	0	0
$833$	−31.8403	−1.10320
$834$	0	0
$835$	1.03423	0.0357911
$836$	0	0
$837$	1.91442	0.0661719
$838$	0	0
$839$	−13.2539	−0.457577	−0.228788	−	0.973476i	$-0.573476\pi$
$839$	−13.2539	−0.457577	−0.228788	+	0.973476i	$0.573476\pi$
$840$	0	0
$841$	26.3588	0.908923
$842$	0	0
$843$	14.6122	0.503272
$844$	0	0
$845$	12.8886	0.443380
$846$	0	0
$847$	−17.2897	−0.594081
$848$	0	0
$849$	9.99959	0.343185
$850$	0	0
$851$	−24.8260	−0.851023
$852$	0	0
$853$	−14.2608	−0.488281	−0.244141	−	0.969740i	$-0.578506\pi$
$853$	−14.2608	−0.488281	−0.244141	+	0.969740i	$0.578506\pi$
$854$	0	0
$855$	17.1115	0.585200
$856$	0	0
$857$	−25.8027	−0.881402	−0.440701	−	0.897654i	$-0.645270\pi$
$857$	−25.8027	−0.881402	−0.440701	+	0.897654i	$0.645270\pi$
$858$	0	0
$859$	1.82468	0.0622574	0.0311287	−	0.999515i	$-0.490090\pi$
$859$	1.82468	0.0622574	0.0311287	+	0.999515i	$0.490090\pi$
$860$	0	0
$861$	8.90991	0.303649
$862$	0	0
$863$	19.5867	0.666739	0.333370	−	0.942796i	$-0.391814\pi$
$863$	19.5867	0.666739	0.333370	+	0.942796i	$0.391814\pi$
$864$	0	0
$865$	15.0024	0.510097
$866$	0	0
$867$	−10.6568	−0.361923
$868$	0	0
$869$	51.9729	1.76306
$870$	0	0
$871$	0.850709	0.0288252
$872$	0	0
$873$	−44.5330	−1.50721
$874$	0	0
$875$	−9.99135	−0.337769
$876$	0	0
$877$	10.7448	0.362825	0.181413	−	0.983407i	$-0.441933\pi$
$877$	10.7448	0.362825	0.181413	+	0.983407i	$0.441933\pi$
$878$	0	0
$879$	−12.5338	−0.422753
$880$	0	0
$881$	−23.0010	−0.774924	−0.387462	−	0.921886i	$-0.626648\pi$
$881$	−23.0010	−0.774924	−0.387462	+	0.921886i	$0.626648\pi$
$882$	0	0
$883$	−33.1211	−1.11461	−0.557307	−	0.830307i	$-0.688165\pi$
$883$	−33.1211	−1.11461	−0.557307	+	0.830307i	$0.688165\pi$
$884$	0	0
$885$	5.25815	0.176751
$886$	0	0
$887$	−31.6073	−1.06127	−0.530635	−	0.847600i	$-0.678047\pi$
$887$	−31.6073	−1.06127	−0.530635	+	0.847600i	$0.678047\pi$
$888$	0	0
$889$	−12.5114	−0.419620
$890$	0	0
$891$	−16.9684	−0.568464
$892$	0	0
$893$	65.4542	2.19034
$894$	0	0
$895$	6.57120	0.219651
$896$	0	0
$897$	−2.21706	−0.0740254
$898$	0	0
$899$	3.22943	0.107708
$900$	0	0
$901$	40.8767	1.36180
$902$	0	0
$903$	−7.30440	−0.243075
$904$	0	0
$905$	−19.1168	−0.635464
$906$	0	0
$907$	5.51606	0.183158	0.0915789	−	0.995798i	$-0.470809\pi$
$907$	5.51606	0.183158	0.0915789	+	0.995798i	$0.470809\pi$
$908$	0	0
$909$	−37.1165	−1.23108
$910$	0	0
$911$	54.7115	1.81267	0.906337	−	0.422556i	$-0.138867\pi$
$911$	54.7115	1.81267	0.906337	+	0.422556i	$0.138867\pi$
$912$	0	0
$913$	−19.9956	−0.661759
$914$	0	0
$915$	−3.19681	−0.105683
$916$	0	0
$917$	−8.52917	−0.281658
$918$	0	0
$919$	27.8582	0.918957	0.459479	−	0.888189i	$-0.348036\pi$
$919$	27.8582	0.918957	0.459479	+	0.888189i	$0.348036\pi$
$920$	0	0
$921$	10.2592	0.338053
$922$	0	0
$923$	8.21560	0.270420
$924$	0	0
$925$	−26.8145	−0.881657
$926$	0	0
$927$	2.40505	0.0789923
$928$	0	0
$929$	28.3003	0.928501	0.464251	−	0.885704i	$-0.346324\pi$
$929$	28.3003	0.928501	0.464251	+	0.885704i	$0.346324\pi$
$930$	0	0
$931$	−41.7537	−1.36842
$932$	0	0
$933$	−24.0326	−0.786792
$934$	0	0
$935$	29.3418	0.959580
$936$	0	0
$937$	39.1562	1.27918	0.639589	−	0.768717i	$-0.279105\pi$
$937$	39.1562	1.27918	0.639589	+	0.768717i	$0.279105\pi$
$938$	0	0
$939$	1.97835	0.0645609
$940$	0	0
$941$	19.7044	0.642346	0.321173	−	0.947020i	$-0.395923\pi$
$941$	19.7044	0.642346	0.321173	+	0.947020i	$0.395923\pi$
$942$	0	0
$943$	−36.0829	−1.17502
$944$	0	0
$945$	4.93467	0.160525
$946$	0	0
$947$	44.1629	1.43510	0.717551	−	0.696506i	$-0.245263\pi$
$947$	44.1629	1.43510	0.717551	+	0.696506i	$0.245263\pi$
$948$	0	0
$949$	3.83732	0.124565
$950$	0	0
$951$	−3.04818	−0.0988440
$952$	0	0
$953$	24.3926	0.790153	0.395076	−	0.918648i	$-0.370718\pi$
$953$	24.3926	0.790153	0.395076	+	0.918648i	$0.370718\pi$
$954$	0	0
$955$	−22.3418	−0.722962
$956$	0	0
$957$	32.1025	1.03773
$958$	0	0
$959$	20.5136	0.662418
$960$	0	0
$961$	−30.8116	−0.993923
$962$	0	0
$963$	1.26023	0.0406103
$964$	0	0
$965$	−20.1422	−0.648400
$966$	0	0
$967$	−25.7176	−0.827024	−0.413512	−	0.910499i	$-0.635698\pi$
$967$	−25.7176	−0.827024	−0.413512	+	0.910499i	$0.635698\pi$
$968$	0	0
$969$	−32.4917	−1.04379
$970$	0	0
$971$	−13.5706	−0.435501	−0.217751	−	0.976004i	$-0.569872\pi$
$971$	−13.5706	−0.435501	−0.217751	+	0.976004i	$0.569872\pi$
$972$	0	0
$973$	−11.3356	−0.363402
$974$	0	0
$975$	−2.39465	−0.0766901
$976$	0	0
$977$	−46.9937	−1.50346	−0.751730	−	0.659471i	$-0.770780\pi$
$977$	−46.9937	−1.50346	−0.751730	+	0.659471i	$0.770780\pi$
$978$	0	0
$979$	11.4667	0.366476
$980$	0	0
$981$	−34.2060	−1.09211
$982$	0	0
$983$	−4.46330	−0.142357	−0.0711787	−	0.997464i	$-0.522676\pi$
$983$	−4.46330	−0.142357	−0.0711787	+	0.997464i	$0.522676\pi$
$984$	0	0
$985$	−1.85838	−0.0592129
$986$	0	0
$987$	8.21170	0.261381
$988$	0	0
$989$	29.5810	0.940622
$990$	0	0
$991$	4.83984	0.153743	0.0768713	−	0.997041i	$-0.475507\pi$
$991$	4.83984	0.153743	0.0768713	+	0.997041i	$0.475507\pi$
$992$	0	0
$993$	−0.153632	−0.00487537
$994$	0	0
$995$	2.10472	0.0667242
$996$	0	0
$997$	−23.1605	−0.733499	−0.366749	−	0.930320i	$-0.619529\pi$
$997$	−23.1605	−0.733499	−0.366749	+	0.930320i	$0.619529\pi$
$998$	0	0
$999$	30.0913	0.952047

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1336.2.a.d.1.5	✓	12
4.3	odd	2		2672.2.a.p.1.8		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1336.2.a.d.1.5	✓	12	1.1	even	1	trivial
2672.2.a.p.1.8		12	4.3	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:	\( N \)	\(=\)	\( 1336 = 2^{3} \cdot 167 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1336.a (trivial)

\(f(q)\)	\(=\)	\(q-0.830621 q^{3} -1.03423 q^{5} -1.08177 q^{7} -2.31007 q^{9} +O(q^{10})\)	\(q-0.830621 q^{3} -1.03423 q^{5} -1.08177 q^{7} -2.31007 q^{9} -5.19449 q^{11} -0.733510 q^{13} +0.859056 q^{15} +5.46167 q^{17} +7.16217 q^{19} +0.898545 q^{21} -3.63888 q^{23} -3.93036 q^{25} +4.41066 q^{27} +7.44035 q^{29} +0.434043 q^{31} +4.31465 q^{33} +1.11881 q^{35} +6.82241 q^{37} +0.609269 q^{39} +9.91593 q^{41} -8.12915 q^{43} +2.38915 q^{45} +9.13889 q^{47} -5.82976 q^{49} -4.53658 q^{51} +7.48428 q^{53} +5.37231 q^{55} -5.94905 q^{57} +6.12085 q^{59} -3.72131 q^{61} +2.49897 q^{63} +0.758620 q^{65} -1.15978 q^{67} +3.02253 q^{69} -11.2004 q^{71} -5.23145 q^{73} +3.26464 q^{75} +5.61926 q^{77} -10.0054 q^{79} +3.26662 q^{81} +3.84939 q^{83} -5.64864 q^{85} -6.18011 q^{87} -2.20747 q^{89} +0.793492 q^{91} -0.360526 q^{93} -7.40735 q^{95} +19.2778 q^{97} +11.9996 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - q^{3} + 8 q^{5} - 4 q^{7} + 15 q^{9}+O(q^{10}) \) 12 * q - q^3 + 8 * q^5 - 4 * q^7 + 15 * q^9	\( 12 q - q^{3} + 8 q^{5} - 4 q^{7} + 15 q^{9} + 6 q^{11} + 13 q^{13} + 4 q^{15} + 10 q^{17} - q^{19} + 13 q^{21} + 3 q^{23} + 18 q^{25} - 10 q^{27} + 29 q^{29} - 3 q^{31} + 3 q^{35} + 41 q^{37} + 10 q^{39} + 20 q^{41} - q^{43} + 42 q^{45} + 5 q^{47} + 20 q^{49} + 14 q^{51} + 39 q^{53} + 3 q^{55} + 3 q^{57} + 8 q^{59} + 30 q^{61} - 2 q^{63} + 21 q^{65} - 9 q^{67} + 33 q^{69} + 29 q^{71} + 12 q^{73} + q^{75} + 19 q^{77} - 2 q^{79} + 24 q^{81} - 5 q^{83} + 44 q^{85} - 2 q^{87} + 7 q^{89} - 4 q^{91} + 37 q^{93} + 18 q^{95} - 14 q^{97} + 5 q^{99}+O(q^{100}) \) 12 * q - q^3 + 8 * q^5 - 4 * q^7 + 15 * q^9 + 6 * q^11 + 13 * q^13 + 4 * q^15 + 10 * q^17 - q^19 + 13 * q^21 + 3 * q^23 + 18 * q^25 - 10 * q^27 + 29 * q^29 - 3 * q^31 + 3 * q^35 + 41 * q^37 + 10 * q^39 + 20 * q^41 - q^43 + 42 * q^45 + 5 * q^47 + 20 * q^49 + 14 * q^51 + 39 * q^53 + 3 * q^55 + 3 * q^57 + 8 * q^59 + 30 * q^61 - 2 * q^63 + 21 * q^65 - 9 * q^67 + 33 * q^69 + 29 * q^71 + 12 * q^73 + q^75 + 19 * q^77 - 2 * q^79 + 24 * q^81 - 5 * q^83 + 44 * q^85 - 2 * q^87 + 7 * q^89 - 4 * q^91 + 37 * q^93 + 18 * q^95 - 14 * q^97 + 5 * q^99

Introduction

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