LMFDB - Embedded newform 3856.2.a.h.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3856,2,Mod(1,3856)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3856, 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("3856.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3856 = 2^{4} \cdot 241 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3856.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:	$30.7903150194$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	6.6.131357120.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 10x^{4} + 16x^{3} + 26x^{2} - 30x - 13 $ x^6 - 2x^5 - 10x^4 + 16x^3 + 26x^2 - 30*x - 13
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 482)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-0.353400$ of defining polynomial
Character	$\chi$	$=$	3856.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.353400 q^{3} -2.12850 q^{5} +2.50020 q^{7} -2.87511 q^{9} +O(q^{10})$	$q+0.353400 q^{3} -2.12850 q^{5} +2.50020 q^{7} -2.87511 q^{9} -2.52171 q^{11} +5.05984 q^{13} -0.752214 q^{15} +4.65021 q^{17} -5.59797 q^{19} +0.883571 q^{21} +0.393203 q^{23} -0.469468 q^{25} -2.07627 q^{27} -3.27392 q^{29} -3.41645 q^{31} -0.891172 q^{33} -5.32169 q^{35} +11.9026 q^{37} +1.78815 q^{39} -6.50158 q^{41} -0.0565882 q^{43} +6.11968 q^{45} -11.9265 q^{47} -0.749001 q^{49} +1.64339 q^{51} +0.324051 q^{53} +5.36747 q^{55} -1.97833 q^{57} +9.42767 q^{59} +10.8153 q^{61} -7.18834 q^{63} -10.7699 q^{65} -2.67034 q^{67} +0.138958 q^{69} -6.56464 q^{71} -3.79563 q^{73} -0.165910 q^{75} -6.30477 q^{77} -4.39560 q^{79} +7.89157 q^{81} +9.14257 q^{83} -9.89800 q^{85} -1.15701 q^{87} -5.81491 q^{89} +12.6506 q^{91} -1.20738 q^{93} +11.9153 q^{95} +8.52085 q^{97} +7.25018 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 2 q^{3} - 5 q^{5} - 10 q^{7} + 6 q^{9}+O(q^{10}) $ 6 * q - 2 * q^3 - 5 * q^5 - 10 * q^7 + 6 * q^9	$ 6 q - 2 q^{3} - 5 q^{5} - 10 q^{7} + 6 q^{9} + 4 q^{11} + 9 q^{13} - 5 q^{15} + q^{17} - 10 q^{19} + 8 q^{21} - 9 q^{23} + 13 q^{25} - 8 q^{27} - q^{29} - 14 q^{31} + 10 q^{33} + 3 q^{35} + 20 q^{37} - 13 q^{39} - 4 q^{41} - 19 q^{43} - 6 q^{45} - q^{47} + 14 q^{49} - 5 q^{51} - 3 q^{53} - 11 q^{55} + 12 q^{57} + q^{59} + 4 q^{61} - 14 q^{63} + 5 q^{65} - 5 q^{67} - 15 q^{69} - 2 q^{71} + 15 q^{73} + 11 q^{75} - 6 q^{77} - 12 q^{79} - 18 q^{81} + 8 q^{83} - 32 q^{85} - 11 q^{87} - 24 q^{89} - q^{91} - 18 q^{93} - 9 q^{95} - 12 q^{97} + 40 q^{99}+O(q^{100}) $ 6 * q - 2 * q^3 - 5 * q^5 - 10 * q^7 + 6 * q^9 + 4 * q^11 + 9 * q^13 - 5 * q^15 + q^17 - 10 * q^19 + 8 * q^21 - 9 * q^23 + 13 * q^25 - 8 * q^27 - q^29 - 14 * q^31 + 10 * q^33 + 3 * q^35 + 20 * q^37 - 13 * q^39 - 4 * q^41 - 19 * q^43 - 6 * q^45 - q^47 + 14 * q^49 - 5 * q^51 - 3 * q^53 - 11 * q^55 + 12 * q^57 + q^59 + 4 * q^61 - 14 * q^63 + 5 * q^65 - 5 * q^67 - 15 * q^69 - 2 * q^71 + 15 * q^73 + 11 * q^75 - 6 * q^77 - 12 * q^79 - 18 * q^81 + 8 * q^83 - 32 * q^85 - 11 * q^87 - 24 * q^89 - q^91 - 18 * q^93 - 9 * q^95 - 12 * q^97 + 40 * 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
$3$	0.353400	0.204036	0.102018	−	0.994783i	$-0.467470\pi$
$3$	0.353400	0.204036	0.102018	+	0.994783i	$0.467470\pi$
$4$	0	0
$5$	−2.12850	−0.951896	−0.475948	−	0.879473i	$-0.657895\pi$
$5$	−2.12850	−0.951896	−0.475948	+	0.879473i	$0.657895\pi$
$6$	0	0
$7$	2.50020	0.944987	0.472493	−	0.881334i	$-0.343354\pi$
$7$	2.50020	0.944987	0.472493	+	0.881334i	$0.343354\pi$
$8$	0	0
$9$	−2.87511	−0.958369
$10$	0	0
$11$	−2.52171	−0.760324	−0.380162	−	0.924920i	$-0.624132\pi$
$11$	−2.52171	−0.760324	−0.380162	+	0.924920i	$0.624132\pi$
$12$	0	0
$13$	5.05984	1.40335	0.701674	−	0.712499i	$-0.252436\pi$
$13$	5.05984	1.40335	0.701674	+	0.712499i	$0.252436\pi$
$14$	0	0
$15$	−0.752214	−0.194221
$16$	0	0
$17$	4.65021	1.12784	0.563921	−	0.825829i	$-0.309292\pi$
$17$	4.65021	1.12784	0.563921	+	0.825829i	$0.309292\pi$
$18$	0	0
$19$	−5.59797	−1.28426	−0.642132	−	0.766594i	$-0.721950\pi$
$19$	−5.59797	−1.28426	−0.642132	+	0.766594i	$0.721950\pi$
$20$	0	0
$21$	0.883571	0.192811
$22$	0	0
$23$	0.393203	0.0819886	0.0409943	−	0.999159i	$-0.486947\pi$
$23$	0.393203	0.0819886	0.0409943	+	0.999159i	$0.486947\pi$
$24$	0	0
$25$	−0.469468	−0.0938937
$26$	0	0
$27$	−2.07627	−0.399577
$28$	0	0
$29$	−3.27392	−0.607952	−0.303976	−	0.952680i	$-0.598314\pi$
$29$	−3.27392	−0.607952	−0.303976	+	0.952680i	$0.598314\pi$
$30$	0	0
$31$	−3.41645	−0.613613	−0.306807	−	0.951772i	$-0.599260\pi$
$31$	−3.41645	−0.613613	−0.306807	+	0.951772i	$0.599260\pi$
$32$	0	0
$33$	−0.891172	−0.155133
$34$	0	0
$35$	−5.32169	−0.899529
$36$	0	0
$37$	11.9026	1.95678	0.978390	−	0.206769i	$-0.0662947\pi$
$37$	11.9026	1.95678	0.978390	+	0.206769i	$0.0662947\pi$
$38$	0	0
$39$	1.78815	0.286333
$40$	0	0
$41$	−6.50158	−1.01538	−0.507688	−	0.861541i	$-0.669500\pi$
$41$	−6.50158	−1.01538	−0.507688	+	0.861541i	$0.669500\pi$
$42$	0	0
$43$	−0.0565882	−0.00862962	−0.00431481	−	0.999991i	$-0.501373\pi$
$43$	−0.0565882	−0.00862962	−0.00431481	+	0.999991i	$0.501373\pi$
$44$	0	0
$45$	6.11968	0.912268
$46$	0	0
$47$	−11.9265	−1.73966	−0.869828	−	0.493354i	$-0.835771\pi$
$47$	−11.9265	−1.73966	−0.869828	+	0.493354i	$0.835771\pi$
$48$	0	0
$49$	−0.749001	−0.107000
$50$	0	0
$51$	1.64339	0.230120
$52$	0	0
$53$	0.324051	0.0445118	0.0222559	−	0.999752i	$-0.492915\pi$
$53$	0.324051	0.0445118	0.0222559	+	0.999752i	$0.492915\pi$
$54$	0	0
$55$	5.36747	0.723749
$56$	0	0
$57$	−1.97833	−0.262036
$58$	0	0
$59$	9.42767	1.22738	0.613689	−	0.789548i	$-0.289685\pi$
$59$	9.42767	1.22738	0.613689	+	0.789548i	$0.289685\pi$
$60$	0	0
$61$	10.8153	1.38476	0.692379	−	0.721534i	$-0.256563\pi$
$61$	10.8153	1.38476	0.692379	+	0.721534i	$0.256563\pi$
$62$	0	0
$63$	−7.18834	−0.905646
$64$	0	0
$65$	−10.7699	−1.33584
$66$	0	0
$67$	−2.67034	−0.326234	−0.163117	−	0.986607i	$-0.552155\pi$
$67$	−2.67034	−0.326234	−0.163117	+	0.986607i	$0.552155\pi$
$68$	0	0
$69$	0.138958	0.0167286
$70$	0	0
$71$	−6.56464	−0.779079	−0.389539	−	0.921010i	$-0.627366\pi$
$71$	−6.56464	−0.779079	−0.389539	+	0.921010i	$0.627366\pi$
$72$	0	0
$73$	−3.79563	−0.444245	−0.222122	−	0.975019i	$-0.571298\pi$
$73$	−3.79563	−0.444245	−0.222122	+	0.975019i	$0.571298\pi$
$74$	0	0
$75$	−0.165910	−0.0191577
$76$	0	0
$77$	−6.30477	−0.718496
$78$	0	0
$79$	−4.39560	−0.494544	−0.247272	−	0.968946i	$-0.579534\pi$
$79$	−4.39560	−0.494544	−0.247272	+	0.968946i	$0.579534\pi$
$80$	0	0
$81$	7.89157	0.876841
$82$	0	0
$83$	9.14257	1.00353	0.501764	−	0.865005i	$-0.332685\pi$
$83$	9.14257	1.00353	0.501764	+	0.865005i	$0.332685\pi$
$84$	0	0
$85$	−9.89800	−1.07359
$86$	0	0
$87$	−1.15701	−0.124044
$88$	0	0
$89$	−5.81491	−0.616379	−0.308189	−	0.951325i	$-0.599723\pi$
$89$	−5.81491	−0.616379	−0.308189	+	0.951325i	$0.599723\pi$
$90$	0	0
$91$	12.6506	1.32614
$92$	0	0
$93$	−1.20738	−0.125199
$94$	0	0
$95$	11.9153	1.22249
$96$	0	0
$97$	8.52085	0.865161	0.432580	−	0.901595i	$-0.357603\pi$
$97$	8.52085	0.865161	0.432580	+	0.901595i	$0.357603\pi$
$98$	0	0
$99$	7.25018	0.728671
$100$	0	0
$101$	−7.87373	−0.783465	−0.391733	−	0.920079i	$-0.628124\pi$
$101$	−7.87373	−0.783465	−0.391733	+	0.920079i	$0.628124\pi$
$102$	0	0
$103$	−18.5767	−1.83042	−0.915208	−	0.402981i	$-0.867974\pi$
$103$	−18.5767	−1.83042	−0.915208	+	0.402981i	$0.867974\pi$
$104$	0	0
$105$	−1.88069	−0.183536
$106$	0	0
$107$	−14.4257	−1.39459	−0.697293	−	0.716786i	$-0.745612\pi$
$107$	−14.4257	−1.39459	−0.697293	+	0.716786i	$0.745612\pi$
$108$	0	0
$109$	−12.1790	−1.16654	−0.583270	−	0.812278i	$-0.698227\pi$
$109$	−12.1790	−1.16654	−0.583270	+	0.812278i	$0.698227\pi$
$110$	0	0
$111$	4.20639	0.399253
$112$	0	0
$113$	−17.0290	−1.60196	−0.800978	−	0.598694i	$-0.795687\pi$
$113$	−17.0290	−1.60196	−0.800978	+	0.598694i	$0.795687\pi$
$114$	0	0
$115$	−0.836935	−0.0780446
$116$	0	0
$117$	−14.5476	−1.34493
$118$	0	0
$119$	11.6265	1.06580
$120$	0	0
$121$	−4.64099	−0.421908
$122$	0	0
$123$	−2.29766	−0.207173
$124$	0	0
$125$	11.6418	1.04127
$126$	0	0
$127$	−10.5721	−0.938124	−0.469062	−	0.883165i	$-0.655408\pi$
$127$	−10.5721	−0.938124	−0.469062	+	0.883165i	$0.655408\pi$
$128$	0	0
$129$	−0.0199983	−0.00176075
$130$	0	0
$131$	−16.1642	−1.41227	−0.706137	−	0.708075i	$-0.749564\pi$
$131$	−16.1642	−1.41227	−0.706137	+	0.708075i	$0.749564\pi$
$132$	0	0
$133$	−13.9961	−1.21361
$134$	0	0
$135$	4.41934	0.380356
$136$	0	0
$137$	−3.56378	−0.304474	−0.152237	−	0.988344i	$-0.548648\pi$
$137$	−3.56378	−0.304474	−0.152237	+	0.988344i	$0.548648\pi$
$138$	0	0
$139$	4.63976	0.393539	0.196770	−	0.980450i	$-0.436955\pi$
$139$	4.63976	0.393539	0.196770	+	0.980450i	$0.436955\pi$
$140$	0	0
$141$	−4.21482	−0.354952
$142$	0	0
$143$	−12.7594	−1.06700
$144$	0	0
$145$	6.96856	0.578707
$146$	0	0
$147$	−0.264697	−0.0218319
$148$	0	0
$149$	7.11248	0.582677	0.291339	−	0.956620i	$-0.405899\pi$
$149$	7.11248	0.582677	0.291339	+	0.956620i	$0.405899\pi$
$150$	0	0
$151$	10.2961	0.837883	0.418941	−	0.908013i	$-0.362401\pi$
$151$	10.2961	0.837883	0.418941	+	0.908013i	$0.362401\pi$
$152$	0	0
$153$	−13.3699	−1.08089
$154$	0	0
$155$	7.27194	0.584096
$156$	0	0
$157$	−13.1714	−1.05119	−0.525597	−	0.850734i	$-0.676158\pi$
$157$	−13.1714	−1.05119	−0.525597	+	0.850734i	$0.676158\pi$
$158$	0	0
$159$	0.114520	0.00908200
$160$	0	0
$161$	0.983087	0.0774781
$162$	0	0
$163$	−2.96133	−0.231949	−0.115975	−	0.993252i	$-0.536999\pi$
$163$	−2.96133	−0.231949	−0.115975	+	0.993252i	$0.536999\pi$
$164$	0	0
$165$	1.89686	0.147671
$166$	0	0
$167$	10.2365	0.792125	0.396062	−	0.918224i	$-0.370376\pi$
$167$	10.2365	0.792125	0.396062	+	0.918224i	$0.370376\pi$
$168$	0	0
$169$	12.6020	0.969383
$170$	0	0
$171$	16.0948	1.23080
$172$	0	0
$173$	−6.09578	−0.463453	−0.231727	−	0.972781i	$-0.574437\pi$
$173$	−6.09578	−0.463453	−0.231727	+	0.972781i	$0.574437\pi$
$174$	0	0
$175$	−1.17376	−0.0887283
$176$	0	0
$177$	3.33174	0.250429
$178$	0	0
$179$	−13.7361	−1.02668	−0.513342	−	0.858184i	$-0.671593\pi$
$179$	−13.7361	−1.02668	−0.513342	+	0.858184i	$0.671593\pi$
$180$	0	0
$181$	4.38316	0.325798	0.162899	−	0.986643i	$-0.447916\pi$
$181$	4.38316	0.325798	0.162899	+	0.986643i	$0.447916\pi$
$182$	0	0
$183$	3.82213	0.282540
$184$	0	0
$185$	−25.3348	−1.86265
$186$	0	0
$187$	−11.7265	−0.857525
$188$	0	0
$189$	−5.19108	−0.377595
$190$	0	0
$191$	−11.0285	−0.797994	−0.398997	−	0.916952i	$-0.630642\pi$
$191$	−11.0285	−0.797994	−0.398997	+	0.916952i	$0.630642\pi$
$192$	0	0
$193$	10.9494	0.788153	0.394077	−	0.919078i	$-0.371064\pi$
$193$	10.9494	0.788153	0.394077	+	0.919078i	$0.371064\pi$
$194$	0	0
$195$	−3.80608	−0.272559
$196$	0	0
$197$	−3.19880	−0.227905	−0.113952	−	0.993486i	$-0.536351\pi$
$197$	−3.19880	−0.227905	−0.113952	+	0.993486i	$0.536351\pi$
$198$	0	0
$199$	−17.6047	−1.24796	−0.623981	−	0.781440i	$-0.714486\pi$
$199$	−17.6047	−1.24796	−0.623981	+	0.781440i	$0.714486\pi$
$200$	0	0
$201$	−0.943698	−0.0665634
$202$	0	0
$203$	−8.18546	−0.574507
$204$	0	0
$205$	13.8386	0.966533
$206$	0	0
$207$	−1.13050	−0.0785753
$208$	0	0
$209$	14.1165	0.976455
$210$	0	0
$211$	19.6602	1.35346	0.676732	−	0.736229i	$-0.263396\pi$
$211$	19.6602	1.35346	0.676732	+	0.736229i	$0.263396\pi$
$212$	0	0
$213$	−2.31994	−0.158960
$214$	0	0
$215$	0.120448	0.00821450
$216$	0	0
$217$	−8.54182	−0.579856
$218$	0	0
$219$	−1.34138	−0.0906418
$220$	0	0
$221$	23.5293	1.58275
$222$	0	0
$223$	−24.8591	−1.66469	−0.832345	−	0.554258i	$-0.813002\pi$
$223$	−24.8591	−1.66469	−0.832345	+	0.554258i	$0.813002\pi$
$224$	0	0
$225$	1.34977	0.0899848
$226$	0	0
$227$	−18.8885	−1.25367	−0.626836	−	0.779151i	$-0.715651\pi$
$227$	−18.8885	−1.25367	−0.626836	+	0.779151i	$0.715651\pi$
$228$	0	0
$229$	−16.6787	−1.10216	−0.551080	−	0.834453i	$-0.685784\pi$
$229$	−16.6787	−1.10216	−0.551080	+	0.834453i	$0.685784\pi$
$230$	0	0
$231$	−2.22811	−0.146599
$232$	0	0
$233$	2.79678	0.183223	0.0916115	−	0.995795i	$-0.470798\pi$
$233$	2.79678	0.183223	0.0916115	+	0.995795i	$0.470798\pi$
$234$	0	0
$235$	25.3856	1.65597
$236$	0	0
$237$	−1.55341	−0.100905
$238$	0	0
$239$	13.7309	0.888181	0.444090	−	0.895982i	$-0.353527\pi$
$239$	13.7309	0.888181	0.444090	+	0.895982i	$0.353527\pi$
$240$	0	0
$241$	1.00000	0.0644157
$241$	1.00000	0.0644157
$242$	0	0
$243$	9.01768	0.578484
$244$	0	0
$245$	1.59425	0.101853
$246$	0	0
$247$	−28.3249	−1.80227
$248$	0	0
$249$	3.23099	0.204756
$250$	0	0
$251$	1.18757	0.0749589	0.0374794	−	0.999297i	$-0.488067\pi$
$251$	1.18757	0.0749589	0.0374794	+	0.999297i	$0.488067\pi$
$252$	0	0
$253$	−0.991544	−0.0623378
$254$	0	0
$255$	−3.49796	−0.219050
$256$	0	0
$257$	19.7847	1.23413	0.617067	−	0.786910i	$-0.288321\pi$
$257$	19.7847	1.23413	0.617067	+	0.786910i	$0.288321\pi$
$258$	0	0
$259$	29.7589	1.84913
$260$	0	0
$261$	9.41288	0.582643
$262$	0	0
$263$	−14.6704	−0.904614	−0.452307	−	0.891862i	$-0.649399\pi$
$263$	−14.6704	−0.904614	−0.452307	+	0.891862i	$0.649399\pi$
$264$	0	0
$265$	−0.689744	−0.0423706
$266$	0	0
$267$	−2.05499	−0.125763
$268$	0	0
$269$	−16.8314	−1.02623	−0.513113	−	0.858321i	$-0.671508\pi$
$269$	−16.8314	−1.02623	−0.513113	+	0.858321i	$0.671508\pi$
$270$	0	0
$271$	8.63365	0.524457	0.262229	−	0.965006i	$-0.415543\pi$
$271$	8.63365	0.524457	0.262229	+	0.965006i	$0.415543\pi$
$272$	0	0
$273$	4.47073	0.270581
$274$	0	0
$275$	1.18386	0.0713896
$276$	0	0
$277$	25.9188	1.55731	0.778655	−	0.627453i	$-0.215902\pi$
$277$	25.9188	1.55731	0.778655	+	0.627453i	$0.215902\pi$
$278$	0	0
$279$	9.82267	0.588068
$280$	0	0
$281$	0.273027	0.0162874	0.00814371	−	0.999967i	$-0.497408\pi$
$281$	0.273027	0.0162874	0.00814371	+	0.999967i	$0.497408\pi$
$282$	0	0
$283$	−0.514995	−0.0306133	−0.0153066	−	0.999883i	$-0.504872\pi$
$283$	−0.514995	−0.0306133	−0.0153066	+	0.999883i	$0.504872\pi$
$284$	0	0
$285$	4.21087	0.249431
$286$	0	0
$287$	−16.2553	−0.959517
$288$	0	0
$289$	4.62448	0.272028
$290$	0	0
$291$	3.01127	0.176524
$292$	0	0
$293$	−26.9687	−1.57553	−0.787765	−	0.615976i	$-0.788762\pi$
$293$	−26.9687	−1.57553	−0.787765	+	0.615976i	$0.788762\pi$
$294$	0	0
$295$	−20.0668	−1.16834
$296$	0	0
$297$	5.23573	0.303808
$298$	0	0
$299$	1.98955	0.115058
$300$	0	0
$301$	−0.141482	−0.00815487
$302$	0	0
$303$	−2.78258	−0.159855
$304$	0	0
$305$	−23.0204	−1.31815
$306$	0	0
$307$	29.6437	1.69186	0.845928	−	0.533297i	$-0.179047\pi$
$307$	29.6437	1.69186	0.845928	+	0.533297i	$0.179047\pi$
$308$	0	0
$309$	−6.56501	−0.373470
$310$	0	0
$311$	−24.3411	−1.38026	−0.690128	−	0.723688i	$-0.742446\pi$
$311$	−24.3411	−1.38026	−0.690128	+	0.723688i	$0.742446\pi$
$312$	0	0
$313$	−16.8506	−0.952451	−0.476226	−	0.879323i	$-0.657995\pi$
$313$	−16.8506	−0.952451	−0.476226	+	0.879323i	$0.657995\pi$
$314$	0	0
$315$	15.3004	0.862081
$316$	0	0
$317$	−29.9346	−1.68130	−0.840648	−	0.541582i	$-0.817826\pi$
$317$	−29.9346	−1.68130	−0.840648	+	0.541582i	$0.817826\pi$
$318$	0	0
$319$	8.25588	0.462240
$320$	0	0
$321$	−5.09805	−0.284546
$322$	0	0
$323$	−26.0318	−1.44845
$324$	0	0
$325$	−2.37544	−0.131765
$326$	0	0
$327$	−4.30407	−0.238016
$328$	0	0
$329$	−29.8186	−1.64395
$330$	0	0
$331$	24.1991	1.33010	0.665052	−	0.746797i	$-0.268409\pi$
$331$	24.1991	1.33010	0.665052	+	0.746797i	$0.268409\pi$
$332$	0	0
$333$	−34.2213	−1.87532
$334$	0	0
$335$	5.68383	0.310541
$336$	0	0
$337$	24.0554	1.31038	0.655190	−	0.755464i	$-0.272588\pi$
$337$	24.0554	1.31038	0.655190	+	0.755464i	$0.272588\pi$
$338$	0	0
$339$	−6.01806	−0.326856
$340$	0	0
$341$	8.61530	0.466545
$342$	0	0
$343$	−19.3741	−1.04610
$344$	0	0
$345$	−0.295773	−0.0159239
$346$	0	0
$347$	−24.2366	−1.30109	−0.650543	−	0.759469i	$-0.725459\pi$
$347$	−24.2366	−1.30109	−0.650543	+	0.759469i	$0.725459\pi$
$348$	0	0
$349$	20.8748	1.11740	0.558700	−	0.829370i	$-0.311300\pi$
$349$	20.8748	1.11740	0.558700	+	0.829370i	$0.311300\pi$
$350$	0	0
$351$	−10.5056	−0.560746
$352$	0	0
$353$	16.8655	0.897663	0.448831	−	0.893617i	$-0.351840\pi$
$353$	16.8655	0.897663	0.448831	+	0.893617i	$0.351840\pi$
$354$	0	0
$355$	13.9729	0.741602
$356$	0	0
$357$	4.10879	0.217460
$358$	0	0
$359$	−24.4996	−1.29304	−0.646519	−	0.762898i	$-0.723776\pi$
$359$	−24.4996	−1.29304	−0.646519	+	0.762898i	$0.723776\pi$
$360$	0	0
$361$	12.3373	0.649332
$362$	0	0
$363$	−1.64013	−0.0860843
$364$	0	0
$365$	8.07902	0.422875
$366$	0	0
$367$	17.6489	0.921267	0.460633	−	0.887590i	$-0.347622\pi$
$367$	17.6489	0.921267	0.460633	+	0.887590i	$0.347622\pi$
$368$	0	0
$369$	18.6928	0.973106
$370$	0	0
$371$	0.810192	0.0420631
$372$	0	0
$373$	27.6104	1.42961	0.714807	−	0.699321i	$-0.246514\pi$
$373$	27.6104	1.42961	0.714807	+	0.699321i	$0.246514\pi$
$374$	0	0
$375$	4.11421	0.212457
$376$	0	0
$377$	−16.5655	−0.853168
$378$	0	0
$379$	11.5761	0.594622	0.297311	−	0.954781i	$-0.403910\pi$
$379$	11.5761	0.594622	0.297311	+	0.954781i	$0.403910\pi$
$380$	0	0
$381$	−3.73619	−0.191411
$382$	0	0
$383$	34.8424	1.78037	0.890183	−	0.455604i	$-0.150577\pi$
$383$	34.8424	1.78037	0.890183	+	0.455604i	$0.150577\pi$
$384$	0	0
$385$	13.4197	0.683933
$386$	0	0
$387$	0.162697	0.00827036
$388$	0	0
$389$	27.5774	1.39823	0.699115	−	0.715009i	$-0.253577\pi$
$389$	27.5774	1.39823	0.699115	+	0.715009i	$0.253577\pi$
$390$	0	0
$391$	1.82848	0.0924702
$392$	0	0
$393$	−5.71244	−0.288154
$394$	0	0
$395$	9.35606	0.470754
$396$	0	0
$397$	−34.7402	−1.74356	−0.871780	−	0.489897i	$-0.837034\pi$
$397$	−34.7402	−1.74356	−0.871780	+	0.489897i	$0.837034\pi$
$398$	0	0
$399$	−4.94621	−0.247620
$400$	0	0
$401$	9.34627	0.466731	0.233365	−	0.972389i	$-0.425026\pi$
$401$	9.34627	0.466731	0.233365	+	0.972389i	$0.425026\pi$
$402$	0	0
$403$	−17.2867	−0.861112
$404$	0	0
$405$	−16.7972	−0.834662
$406$	0	0
$407$	−30.0149	−1.48779
$408$	0	0
$409$	−18.6410	−0.921737	−0.460868	−	0.887468i	$-0.652462\pi$
$409$	−18.6410	−0.921737	−0.460868	+	0.887468i	$0.652462\pi$
$410$	0	0
$411$	−1.25944	−0.0621236
$412$	0	0
$413$	23.5711	1.15986
$414$	0	0
$415$	−19.4600	−0.955254
$416$	0	0
$417$	1.63969	0.0802961
$418$	0	0
$419$	23.8188	1.16362	0.581812	−	0.813323i	$-0.302344\pi$
$419$	23.8188	1.16362	0.581812	+	0.813323i	$0.302344\pi$
$420$	0	0
$421$	−17.3129	−0.843780	−0.421890	−	0.906647i	$-0.638633\pi$
$421$	−17.3129	−0.843780	−0.421890	+	0.906647i	$0.638633\pi$
$422$	0	0
$423$	34.2899	1.66723
$424$	0	0
$425$	−2.18313	−0.105897
$426$	0	0
$427$	27.0404	1.30858
$428$	0	0
$429$	−4.50919	−0.217706
$430$	0	0
$431$	−12.1208	−0.583840	−0.291920	−	0.956443i	$-0.594294\pi$
$431$	−12.1208	−0.583840	−0.291920	+	0.956443i	$0.594294\pi$
$432$	0	0
$433$	−3.98023	−0.191278	−0.0956388	−	0.995416i	$-0.530489\pi$
$433$	−3.98023	−0.191278	−0.0956388	+	0.995416i	$0.530489\pi$
$434$	0	0
$435$	2.46269	0.118077
$436$	0	0
$437$	−2.20114	−0.105295
$438$	0	0
$439$	11.4715	0.547504	0.273752	−	0.961800i	$-0.411735\pi$
$439$	11.4715	0.547504	0.273752	+	0.961800i	$0.411735\pi$
$440$	0	0
$441$	2.15346	0.102546
$442$	0	0
$443$	8.06760	0.383303	0.191652	−	0.981463i	$-0.438616\pi$
$443$	8.06760	0.383303	0.191652	+	0.981463i	$0.438616\pi$
$444$	0	0
$445$	12.3771	0.586729
$446$	0	0
$447$	2.51355	0.118887
$448$	0	0
$449$	−2.77955	−0.131175	−0.0655875	−	0.997847i	$-0.520892\pi$
$449$	−2.77955	−0.131175	−0.0655875	+	0.997847i	$0.520892\pi$
$450$	0	0
$451$	16.3951	0.772015
$452$	0	0
$453$	3.63864	0.170958
$454$	0	0
$455$	−26.9269	−1.26235
$456$	0	0
$457$	−25.5469	−1.19503	−0.597516	−	0.801857i	$-0.703846\pi$
$457$	−25.5469	−1.19503	−0.597516	+	0.801857i	$0.703846\pi$
$458$	0	0
$459$	−9.65507	−0.450660
$460$	0	0
$461$	−3.40760	−0.158708	−0.0793539	−	0.996847i	$-0.525286\pi$
$461$	−3.40760	−0.158708	−0.0793539	+	0.996847i	$0.525286\pi$
$462$	0	0
$463$	−11.5667	−0.537551	−0.268775	−	0.963203i	$-0.586619\pi$
$463$	−11.5667	−0.537551	−0.268775	+	0.963203i	$0.586619\pi$
$464$	0	0
$465$	2.56991	0.119176
$466$	0	0
$467$	−10.0267	−0.463979	−0.231990	−	0.972718i	$-0.574524\pi$
$467$	−10.0267	−0.463979	−0.231990	+	0.972718i	$0.574524\pi$
$468$	0	0
$469$	−6.67638	−0.308287
$470$	0	0
$471$	−4.65478	−0.214481
$472$	0	0
$473$	0.142699	0.00656130
$474$	0	0
$475$	2.62807	0.120584
$476$	0	0
$477$	−0.931681	−0.0426588
$478$	0	0
$479$	16.1601	0.738375	0.369187	−	0.929355i	$-0.379636\pi$
$479$	16.1601	0.738375	0.369187	+	0.929355i	$0.379636\pi$
$480$	0	0
$481$	60.2254	2.74604
$482$	0	0
$483$	0.347423	0.0158083
$484$	0	0
$485$	−18.1367	−0.823543
$486$	0	0
$487$	36.0715	1.63456	0.817279	−	0.576243i	$-0.195482\pi$
$487$	36.0715	1.63456	0.817279	+	0.576243i	$0.195482\pi$
$488$	0	0
$489$	−1.04654	−0.0473260
$490$	0	0
$491$	18.3779	0.829381	0.414691	−	0.909963i	$-0.363890\pi$
$491$	18.3779	0.829381	0.414691	+	0.909963i	$0.363890\pi$
$492$	0	0
$493$	−15.2244	−0.685674
$494$	0	0
$495$	−15.4320	−0.693619
$496$	0	0
$497$	−16.4129	−0.736219
$498$	0	0
$499$	−26.1542	−1.17082	−0.585412	−	0.810736i	$-0.699067\pi$
$499$	−26.1542	−1.17082	−0.585412	+	0.810736i	$0.699067\pi$
$500$	0	0
$501$	3.61759	0.161622
$502$	0	0
$503$	−19.7899	−0.882389	−0.441194	−	0.897412i	$-0.645445\pi$
$503$	−19.7899	−0.882389	−0.441194	+	0.897412i	$0.645445\pi$
$504$	0	0
$505$	16.7593	0.745777
$506$	0	0
$507$	4.45355	0.197789
$508$	0	0
$509$	−34.4634	−1.52756	−0.763781	−	0.645475i	$-0.776659\pi$
$509$	−34.4634	−1.52756	−0.763781	+	0.645475i	$0.776659\pi$
$510$	0	0
$511$	−9.48983	−0.419806
$512$	0	0
$513$	11.6229	0.513162
$514$	0	0
$515$	39.5406	1.74237
$516$	0	0
$517$	30.0751	1.32270
$518$	0	0
$519$	−2.15425	−0.0945610
$520$	0	0
$521$	16.2715	0.712868	0.356434	−	0.934321i	$-0.383992\pi$
$521$	16.2715	0.712868	0.356434	+	0.934321i	$0.383992\pi$
$522$	0	0
$523$	4.59657	0.200994	0.100497	−	0.994937i	$-0.467957\pi$
$523$	4.59657	0.200994	0.100497	+	0.994937i	$0.467957\pi$
$524$	0	0
$525$	−0.414809	−0.0181037
$526$	0	0
$527$	−15.8872	−0.692059
$528$	0	0
$529$	−22.8454	−0.993278
$530$	0	0
$531$	−27.1056	−1.17628
$532$	0	0
$533$	−32.8970	−1.42493
$534$	0	0
$535$	30.7052	1.32750
$536$	0	0
$537$	−4.85434	−0.209480
$538$	0	0
$539$	1.88876	0.0813547
$540$	0	0
$541$	44.1029	1.89613	0.948065	−	0.318077i	$-0.103037\pi$
$541$	44.1029	1.89613	0.948065	+	0.318077i	$0.103037\pi$
$542$	0	0
$543$	1.54901	0.0664744
$544$	0	0
$545$	25.9231	1.11042
$546$	0	0
$547$	22.5046	0.962226	0.481113	−	0.876659i	$-0.340233\pi$
$547$	22.5046	0.962226	0.481113	+	0.876659i	$0.340233\pi$
$548$	0	0
$549$	−31.0952	−1.32711
$550$	0	0
$551$	18.3273	0.780770
$552$	0	0
$553$	−10.9899	−0.467337
$554$	0	0
$555$	−8.95332	−0.380047
$556$	0	0
$557$	24.0445	1.01880	0.509399	−	0.860531i	$-0.329868\pi$
$557$	24.0445	1.01880	0.509399	+	0.860531i	$0.329868\pi$
$558$	0	0
$559$	−0.286327	−0.0121103
$560$	0	0
$561$	−4.14414	−0.174966
$562$	0	0
$563$	0.851224	0.0358748	0.0179374	−	0.999839i	$-0.494290\pi$
$563$	0.851224	0.0358748	0.0179374	+	0.999839i	$0.494290\pi$
$564$	0	0
$565$	36.2464	1.52490
$566$	0	0
$567$	19.7305	0.828603
$568$	0	0
$569$	1.04696	0.0438909	0.0219455	−	0.999759i	$-0.493014\pi$
$569$	1.04696	0.0438909	0.0219455	+	0.999759i	$0.493014\pi$
$570$	0	0
$571$	−18.5352	−0.775675	−0.387838	−	0.921728i	$-0.626778\pi$
$571$	−18.5352	−0.775675	−0.387838	+	0.921728i	$0.626778\pi$
$572$	0	0
$573$	−3.89748	−0.162819
$574$	0	0
$575$	−0.184597	−0.00769821
$576$	0	0
$577$	37.4999	1.56114	0.780570	−	0.625068i	$-0.214929\pi$
$577$	37.4999	1.56114	0.780570	+	0.625068i	$0.214929\pi$
$578$	0	0
$579$	3.86951	0.160811
$580$	0	0
$581$	22.8583	0.948320
$582$	0	0
$583$	−0.817162	−0.0338434
$584$	0	0
$585$	30.9646	1.28023
$586$	0	0
$587$	19.1994	0.792446	0.396223	−	0.918154i	$-0.370321\pi$
$587$	19.1994	0.792446	0.396223	+	0.918154i	$0.370321\pi$
$588$	0	0
$589$	19.1252	0.788041
$590$	0	0
$591$	−1.13046	−0.0465008
$592$	0	0
$593$	−29.7156	−1.22027	−0.610137	−	0.792296i	$-0.708886\pi$
$593$	−29.7156	−1.22027	−0.610137	+	0.792296i	$0.708886\pi$
$594$	0	0
$595$	−24.7470	−1.01453
$596$	0	0
$597$	−6.22149	−0.254629
$598$	0	0
$599$	15.1679	0.619743	0.309871	−	0.950778i	$-0.399714\pi$
$599$	15.1679	0.619743	0.309871	+	0.950778i	$0.399714\pi$
$600$	0	0
$601$	21.4884	0.876531	0.438265	−	0.898846i	$-0.355593\pi$
$601$	21.4884	0.876531	0.438265	+	0.898846i	$0.355593\pi$
$602$	0	0
$603$	7.67751	0.312652
$604$	0	0
$605$	9.87837	0.401613
$606$	0	0
$607$	−8.41526	−0.341565	−0.170782	−	0.985309i	$-0.554630\pi$
$607$	−8.41526	−0.341565	−0.170782	+	0.985309i	$0.554630\pi$
$608$	0	0
$609$	−2.89274	−0.117220
$610$	0	0
$611$	−60.3461	−2.44134
$612$	0	0
$613$	20.7262	0.837122	0.418561	−	0.908189i	$-0.362535\pi$
$613$	20.7262	0.837122	0.418561	+	0.908189i	$0.362535\pi$
$614$	0	0
$615$	4.89058	0.197207
$616$	0	0
$617$	17.7218	0.713455	0.356727	−	0.934209i	$-0.383893\pi$
$617$	17.7218	0.713455	0.356727	+	0.934209i	$0.383893\pi$
$618$	0	0
$619$	25.2795	1.01607	0.508034	−	0.861337i	$-0.330372\pi$
$619$	25.2795	1.01607	0.508034	+	0.861337i	$0.330372\pi$
$620$	0	0
$621$	−0.816394	−0.0327608
$622$	0	0
$623$	−14.5384	−0.582470
$624$	0	0
$625$	−22.4323	−0.897290
$626$	0	0
$627$	4.98876	0.199232
$628$	0	0
$629$	55.3497	2.20694
$630$	0	0
$631$	1.21500	0.0483685	0.0241843	−	0.999708i	$-0.492301\pi$
$631$	1.21500	0.0483685	0.0241843	+	0.999708i	$0.492301\pi$
$632$	0	0
$633$	6.94792	0.276155
$634$	0	0
$635$	22.5028	0.892996
$636$	0	0
$637$	−3.78983	−0.150158
$638$	0	0
$639$	18.8740	0.746645
$640$	0	0
$641$	43.7799	1.72920	0.864600	−	0.502461i	$-0.167572\pi$
$641$	43.7799	1.72920	0.864600	+	0.502461i	$0.167572\pi$
$642$	0	0
$643$	−7.85863	−0.309914	−0.154957	−	0.987921i	$-0.549524\pi$
$643$	−7.85863	−0.309914	−0.154957	+	0.987921i	$0.549524\pi$
$644$	0	0
$645$	0.0425664	0.00167605
$646$	0	0
$647$	37.9648	1.49255	0.746275	−	0.665638i	$-0.231840\pi$
$647$	37.9648	1.49255	0.746275	+	0.665638i	$0.231840\pi$
$648$	0	0
$649$	−23.7738	−0.933204
$650$	0	0
$651$	−3.01868	−0.118311
$652$	0	0
$653$	40.6310	1.59001	0.795006	−	0.606601i	$-0.207467\pi$
$653$	40.6310	1.59001	0.795006	+	0.606601i	$0.207467\pi$
$654$	0	0
$655$	34.4056	1.34434
$656$	0	0
$657$	10.9128	0.425751
$658$	0	0
$659$	5.99805	0.233651	0.116825	−	0.993152i	$-0.462728\pi$
$659$	5.99805	0.233651	0.116825	+	0.993152i	$0.462728\pi$
$660$	0	0
$661$	−30.0600	−1.16920	−0.584600	−	0.811322i	$-0.698748\pi$
$661$	−30.0600	−1.16920	−0.584600	+	0.811322i	$0.698748\pi$
$662$	0	0
$663$	8.31527	0.322938
$664$	0	0
$665$	29.7907	1.15523
$666$	0	0
$667$	−1.28732	−0.0498451
$668$	0	0
$669$	−8.78522	−0.339656
$670$	0	0
$671$	−27.2730	−1.05286
$672$	0	0
$673$	−25.9606	−1.00071	−0.500354	−	0.865821i	$-0.666797\pi$
$673$	−25.9606	−1.00071	−0.500354	+	0.865821i	$0.666797\pi$
$674$	0	0
$675$	0.974741	0.0375178
$676$	0	0
$677$	−7.21350	−0.277237	−0.138619	−	0.990346i	$-0.544266\pi$
$677$	−7.21350	−0.277237	−0.138619	+	0.990346i	$0.544266\pi$
$678$	0	0
$679$	21.3038	0.817565
$680$	0	0
$681$	−6.67520	−0.255794
$682$	0	0
$683$	−30.7701	−1.17739	−0.588693	−	0.808357i	$-0.700357\pi$
$683$	−30.7701	−1.17739	−0.588693	+	0.808357i	$0.700357\pi$
$684$	0	0
$685$	7.58552	0.289828
$686$	0	0
$687$	−5.89426	−0.224880
$688$	0	0
$689$	1.63965	0.0624655
$690$	0	0
$691$	−15.6465	−0.595221	−0.297610	−	0.954687i	$-0.596190\pi$
$691$	−15.6465	−0.595221	−0.297610	+	0.954687i	$0.596190\pi$
$692$	0	0
$693$	18.1269	0.688584
$694$	0	0
$695$	−9.87575	−0.374608
$696$	0	0
$697$	−30.2337	−1.14518
$698$	0	0
$699$	0.988383	0.0373841
$700$	0	0
$701$	−46.2103	−1.74534	−0.872670	−	0.488311i	$-0.837613\pi$
$701$	−46.2103	−1.74534	−0.872670	+	0.488311i	$0.837613\pi$
$702$	0	0
$703$	−66.6306	−2.51302
$704$	0	0
$705$	8.97127	0.337878
$706$	0	0
$707$	−19.6859	−0.740364
$708$	0	0
$709$	−5.22166	−0.196104	−0.0980518	−	0.995181i	$-0.531261\pi$
$709$	−5.22166	−0.196104	−0.0980518	+	0.995181i	$0.531261\pi$
$710$	0	0
$711$	12.6378	0.473956
$712$	0	0
$713$	−1.34336	−0.0503093
$714$	0	0
$715$	27.1585	1.01567
$716$	0	0
$717$	4.85252	0.181221
$718$	0	0
$719$	31.5509	1.17665	0.588326	−	0.808624i	$-0.299787\pi$
$719$	31.5509	1.17665	0.588326	+	0.808624i	$0.299787\pi$
$720$	0	0
$721$	−46.4455	−1.72972
$722$	0	0
$723$	0.353400	0.0131431
$724$	0	0
$725$	1.53700	0.0570829
$726$	0	0
$727$	−10.5409	−0.390941	−0.195470	−	0.980710i	$-0.562623\pi$
$727$	−10.5409	−0.390941	−0.195470	+	0.980710i	$0.562623\pi$
$728$	0	0
$729$	−20.4879	−0.758810
$730$	0	0
$731$	−0.263147	−0.00973284
$732$	0	0
$733$	−35.2735	−1.30286	−0.651429	−	0.758710i	$-0.725830\pi$
$733$	−35.2735	−1.30286	−0.651429	+	0.758710i	$0.725830\pi$
$734$	0	0
$735$	0.563409	0.0207817
$736$	0	0
$737$	6.73381	0.248043
$738$	0	0
$739$	−6.52067	−0.239867	−0.119933	−	0.992782i	$-0.538268\pi$
$739$	−6.52067	−0.239867	−0.119933	+	0.992782i	$0.538268\pi$
$740$	0	0
$741$	−10.0100	−0.367727
$742$	0	0
$743$	−0.908293	−0.0333220	−0.0166610	−	0.999861i	$-0.505304\pi$
$743$	−0.908293	−0.0333220	−0.0166610	+	0.999861i	$0.505304\pi$
$744$	0	0
$745$	−15.1389	−0.554648
$746$	0	0
$747$	−26.2859	−0.961750
$748$	0	0
$749$	−36.0672	−1.31787
$750$	0	0
$751$	25.4559	0.928900	0.464450	−	0.885599i	$-0.346252\pi$
$751$	25.4559	0.928900	0.464450	+	0.885599i	$0.346252\pi$
$752$	0	0
$753$	0.419688	0.0152943
$754$	0	0
$755$	−21.9152	−0.797577
$756$	0	0
$757$	−8.83081	−0.320961	−0.160481	−	0.987039i	$-0.551304\pi$
$757$	−8.83081	−0.320961	−0.160481	+	0.987039i	$0.551304\pi$
$758$	0	0
$759$	−0.350412	−0.0127191
$760$	0	0
$761$	−35.1999	−1.27600	−0.637998	−	0.770038i	$-0.720237\pi$
$761$	−35.1999	−1.27600	−0.637998	+	0.770038i	$0.720237\pi$
$762$	0	0
$763$	−30.4500	−1.10236
$764$	0	0
$765$	28.4578	1.02889
$766$	0	0
$767$	47.7025	1.72244
$768$	0	0
$769$	−24.8776	−0.897108	−0.448554	−	0.893756i	$-0.648061\pi$
$769$	−24.8776	−0.897108	−0.448554	+	0.893756i	$0.648061\pi$
$770$	0	0
$771$	6.99191	0.251808
$772$	0	0
$773$	−7.64940	−0.275130	−0.137565	−	0.990493i	$-0.543928\pi$
$773$	−7.64940	−0.275130	−0.137565	+	0.990493i	$0.543928\pi$
$774$	0	0
$775$	1.60392	0.0576144
$776$	0	0
$777$	10.5168	0.377289
$778$	0	0
$779$	36.3957	1.30401
$780$	0	0
$781$	16.5541	0.592352
$782$	0	0
$783$	6.79753	0.242924
$784$	0	0
$785$	28.0354	1.00063
$786$	0	0
$787$	34.1467	1.21720	0.608599	−	0.793478i	$-0.291732\pi$
$787$	34.1467	1.21720	0.608599	+	0.793478i	$0.291732\pi$
$788$	0	0
$789$	−5.18452	−0.184574
$790$	0	0
$791$	−42.5760	−1.51383
$792$	0	0
$793$	54.7237	1.94330
$794$	0	0
$795$	−0.243756	−0.00864512
$796$	0	0
$797$	30.7836	1.09041	0.545206	−	0.838302i	$-0.316452\pi$
$797$	30.7836	1.09041	0.545206	+	0.838302i	$0.316452\pi$
$798$	0	0
$799$	−55.4607	−1.96206
$800$	0	0
$801$	16.7185	0.590719
$802$	0	0
$803$	9.57147	0.337770
$804$	0	0
$805$	−2.09251	−0.0737511
$806$	0	0
$807$	−5.94821	−0.209387
$808$	0	0
$809$	35.8170	1.25926	0.629629	−	0.776896i	$-0.283207\pi$
$809$	35.8170	1.25926	0.629629	+	0.776896i	$0.283207\pi$
$810$	0	0
$811$	5.94610	0.208796	0.104398	−	0.994536i	$-0.466708\pi$
$811$	5.94610	0.208796	0.104398	+	0.994536i	$0.466708\pi$
$812$	0	0
$813$	3.05114	0.107008
$814$	0	0
$815$	6.30321	0.220792
$816$	0	0
$817$	0.316779	0.0110827
$818$	0	0
$819$	−36.3719	−1.27094
$820$	0	0
$821$	28.6429	0.999643	0.499821	−	0.866129i	$-0.333399\pi$
$821$	28.6429	0.999643	0.499821	+	0.866129i	$0.333399\pi$
$822$	0	0
$823$	19.9544	0.695568	0.347784	−	0.937575i	$-0.386934\pi$
$823$	19.9544	0.695568	0.347784	+	0.937575i	$0.386934\pi$
$824$	0	0
$825$	0.418377	0.0145660
$826$	0	0
$827$	−18.0517	−0.627721	−0.313860	−	0.949469i	$-0.601622\pi$
$827$	−18.0517	−0.627721	−0.313860	+	0.949469i	$0.601622\pi$
$828$	0	0
$829$	−16.9709	−0.589423	−0.294711	−	0.955586i	$-0.595224\pi$
$829$	−16.9709	−0.589423	−0.294711	+	0.955586i	$0.595224\pi$
$830$	0	0
$831$	9.15971	0.317747
$832$	0	0
$833$	−3.48301	−0.120679
$834$	0	0
$835$	−21.7885	−0.754021
$836$	0	0
$837$	7.09346	0.245186
$838$	0	0
$839$	−16.9523	−0.585257	−0.292629	−	0.956226i	$-0.594530\pi$
$839$	−16.9523	−0.585257	−0.292629	+	0.956226i	$0.594530\pi$
$840$	0	0
$841$	−18.2814	−0.630394
$842$	0	0
$843$	0.0964878	0.00332322
$844$	0	0
$845$	−26.8234	−0.922752
$846$	0	0
$847$	−11.6034	−0.398698
$848$	0	0
$849$	−0.181999	−0.00624620
$850$	0	0
$851$	4.68015	0.160434
$852$	0	0
$853$	−39.2726	−1.34467	−0.672335	−	0.740247i	$-0.734708\pi$
$853$	−39.2726	−1.34467	−0.672335	+	0.740247i	$0.734708\pi$
$854$	0	0
$855$	−34.2578	−1.17159
$856$	0	0
$857$	−18.3927	−0.628282	−0.314141	−	0.949376i	$-0.601717\pi$
$857$	−18.3927	−0.628282	−0.314141	+	0.949376i	$0.601717\pi$
$858$	0	0
$859$	17.8753	0.609896	0.304948	−	0.952369i	$-0.401361\pi$
$859$	17.8753	0.609896	0.304948	+	0.952369i	$0.401361\pi$
$860$	0	0
$861$	−5.74461	−0.195776
$862$	0	0
$863$	−27.3927	−0.932458	−0.466229	−	0.884664i	$-0.654388\pi$
$863$	−27.3927	−0.932458	−0.466229	+	0.884664i	$0.654388\pi$
$864$	0	0
$865$	12.9749	0.441159
$866$	0	0
$867$	1.63429	0.0555034
$868$	0	0
$869$	11.0844	0.376013
$870$	0	0
$871$	−13.5115	−0.457819
$872$	0	0
$873$	−24.4984	−0.829144
$874$	0	0
$875$	29.1068	0.983989
$876$	0	0
$877$	−18.6533	−0.629876	−0.314938	−	0.949112i	$-0.601984\pi$
$877$	−18.6533	−0.629876	−0.314938	+	0.949112i	$0.601984\pi$
$878$	0	0
$879$	−9.53075	−0.321464
$880$	0	0
$881$	21.6705	0.730097	0.365049	−	0.930988i	$-0.381052\pi$
$881$	21.6705	0.730097	0.365049	+	0.930988i	$0.381052\pi$
$882$	0	0
$883$	44.9288	1.51198	0.755988	−	0.654586i	$-0.227157\pi$
$883$	44.9288	1.51198	0.755988	+	0.654586i	$0.227157\pi$
$884$	0	0
$885$	−7.09163	−0.238382
$886$	0	0
$887$	−51.5840	−1.73202	−0.866011	−	0.500025i	$-0.833324\pi$
$887$	−51.5840	−1.73202	−0.866011	+	0.500025i	$0.833324\pi$
$888$	0	0
$889$	−26.4324	−0.886514
$890$	0	0
$891$	−19.9002	−0.666683
$892$	0	0
$893$	66.7641	2.23418
$894$	0	0
$895$	29.2374	0.977297
$896$	0	0
$897$	0.703106	0.0234760
$898$	0	0
$899$	11.1852	0.373047
$900$	0	0
$901$	1.50691	0.0502023
$902$	0	0
$903$	−0.0499997	−0.00166389
$904$	0	0
$905$	−9.32958	−0.310126
$906$	0	0
$907$	20.8619	0.692709	0.346355	−	0.938104i	$-0.387419\pi$
$907$	20.8619	0.692709	0.346355	+	0.938104i	$0.387419\pi$
$908$	0	0
$909$	22.6378	0.750849
$910$	0	0
$911$	40.8440	1.35322	0.676611	−	0.736341i	$-0.263448\pi$
$911$	40.8440	1.35322	0.676611	+	0.736341i	$0.263448\pi$
$912$	0	0
$913$	−23.0549	−0.763006
$914$	0	0
$915$	−8.13543	−0.268949
$916$	0	0
$917$	−40.4138	−1.33458
$918$	0	0
$919$	−16.3664	−0.539878	−0.269939	−	0.962877i	$-0.587004\pi$
$919$	−16.3664	−0.539878	−0.269939	+	0.962877i	$0.587004\pi$
$920$	0	0
$921$	10.4761	0.345199
$922$	0	0
$923$	−33.2160	−1.09332
$924$	0	0
$925$	−5.58791	−0.183729
$926$	0	0
$927$	53.4100	1.75422
$928$	0	0
$929$	−30.7584	−1.00915	−0.504576	−	0.863367i	$-0.668351\pi$
$929$	−30.7584	−1.00915	−0.504576	+	0.863367i	$0.668351\pi$
$930$	0	0
$931$	4.19289	0.137416
$932$	0	0
$933$	−8.60214	−0.281621
$934$	0	0
$935$	24.9599	0.816275
$936$	0	0
$937$	−1.18830	−0.0388202	−0.0194101	−	0.999812i	$-0.506179\pi$
$937$	−1.18830	−0.0388202	−0.0194101	+	0.999812i	$0.506179\pi$
$938$	0	0
$939$	−5.95500	−0.194334
$940$	0	0
$941$	35.2371	1.14870	0.574348	−	0.818611i	$-0.305256\pi$
$941$	35.2371	1.14870	0.574348	+	0.818611i	$0.305256\pi$
$942$	0	0
$943$	−2.55644	−0.0832493
$944$	0	0
$945$	11.0492	0.359432
$946$	0	0
$947$	17.8807	0.581045	0.290522	−	0.956868i	$-0.406171\pi$
$947$	17.8807	0.581045	0.290522	+	0.956868i	$0.406171\pi$
$948$	0	0
$949$	−19.2053	−0.623430
$950$	0	0
$951$	−10.5789	−0.343045
$952$	0	0
$953$	35.4110	1.14708	0.573538	−	0.819179i	$-0.305571\pi$
$953$	35.4110	1.14708	0.573538	+	0.819179i	$0.305571\pi$
$954$	0	0
$955$	23.4742	0.759608
$956$	0	0
$957$	2.91763	0.0943135
$958$	0	0
$959$	−8.91016	−0.287724
$960$	0	0
$961$	−19.3278	−0.623479
$962$	0	0
$963$	41.4755	1.33653
$964$	0	0
$965$	−23.3058	−0.750240
$966$	0	0
$967$	2.90006	0.0932598	0.0466299	−	0.998912i	$-0.485152\pi$
$967$	2.90006	0.0932598	0.0466299	+	0.998912i	$0.485152\pi$
$968$	0	0
$969$	−9.19963	−0.295535
$970$	0	0
$971$	19.3462	0.620850	0.310425	−	0.950598i	$-0.399529\pi$
$971$	19.3462	0.620850	0.310425	+	0.950598i	$0.399529\pi$
$972$	0	0
$973$	11.6003	0.371889
$974$	0	0
$975$	−0.839480	−0.0268849
$976$	0	0
$977$	−3.76131	−0.120335	−0.0601675	−	0.998188i	$-0.519163\pi$
$977$	−3.76131	−0.120335	−0.0601675	+	0.998188i	$0.519163\pi$
$978$	0	0
$979$	14.6635	0.468647
$980$	0	0
$981$	35.0160	1.11798
$982$	0	0
$983$	49.3781	1.57492	0.787459	−	0.616367i	$-0.211396\pi$
$983$	49.3781	1.57492	0.787459	+	0.616367i	$0.211396\pi$
$984$	0	0
$985$	6.80866	0.216942
$986$	0	0
$987$	−10.5379	−0.335425
$988$	0	0
$989$	−0.0222507	−0.000707530 0
$990$	0	0
$991$	20.3997	0.648018	0.324009	−	0.946054i	$-0.394969\pi$
$991$	20.3997	0.648018	0.324009	+	0.946054i	$0.394969\pi$
$992$	0	0
$993$	8.55198	0.271389
$994$	0	0
$995$	37.4716	1.18793
$996$	0	0
$997$	8.50276	0.269285	0.134643	−	0.990894i	$-0.457011\pi$
$997$	8.50276	0.269285	0.134643	+	0.990894i	$0.457011\pi$
$998$	0	0
$999$	−24.7130	−0.781885

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3856.2.a.h.1.4		6
4.3	odd	2		482.2.a.d.1.3	✓	6
12.11	even	2		4338.2.a.u.1.4		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
482.2.a.d.1.3	✓	6	4.3	odd	2
3856.2.a.h.1.4		6	1.1	even	1	trivial
4338.2.a.u.1.4		6	12.11	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 3856 = 2^{4} \cdot 241 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3856.a (trivial)

\(f(q)\)	\(=\)	\(q+0.353400 q^{3} -2.12850 q^{5} +2.50020 q^{7} -2.87511 q^{9} +O(q^{10})\)	\(q+0.353400 q^{3} -2.12850 q^{5} +2.50020 q^{7} -2.87511 q^{9} -2.52171 q^{11} +5.05984 q^{13} -0.752214 q^{15} +4.65021 q^{17} -5.59797 q^{19} +0.883571 q^{21} +0.393203 q^{23} -0.469468 q^{25} -2.07627 q^{27} -3.27392 q^{29} -3.41645 q^{31} -0.891172 q^{33} -5.32169 q^{35} +11.9026 q^{37} +1.78815 q^{39} -6.50158 q^{41} -0.0565882 q^{43} +6.11968 q^{45} -11.9265 q^{47} -0.749001 q^{49} +1.64339 q^{51} +0.324051 q^{53} +5.36747 q^{55} -1.97833 q^{57} +9.42767 q^{59} +10.8153 q^{61} -7.18834 q^{63} -10.7699 q^{65} -2.67034 q^{67} +0.138958 q^{69} -6.56464 q^{71} -3.79563 q^{73} -0.165910 q^{75} -6.30477 q^{77} -4.39560 q^{79} +7.89157 q^{81} +9.14257 q^{83} -9.89800 q^{85} -1.15701 q^{87} -5.81491 q^{89} +12.6506 q^{91} -1.20738 q^{93} +11.9153 q^{95} +8.52085 q^{97} +7.25018 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 2 q^{3} - 5 q^{5} - 10 q^{7} + 6 q^{9}+O(q^{10}) \) 6 * q - 2 * q^3 - 5 * q^5 - 10 * q^7 + 6 * q^9	\( 6 q - 2 q^{3} - 5 q^{5} - 10 q^{7} + 6 q^{9} + 4 q^{11} + 9 q^{13} - 5 q^{15} + q^{17} - 10 q^{19} + 8 q^{21} - 9 q^{23} + 13 q^{25} - 8 q^{27} - q^{29} - 14 q^{31} + 10 q^{33} + 3 q^{35} + 20 q^{37} - 13 q^{39} - 4 q^{41} - 19 q^{43} - 6 q^{45} - q^{47} + 14 q^{49} - 5 q^{51} - 3 q^{53} - 11 q^{55} + 12 q^{57} + q^{59} + 4 q^{61} - 14 q^{63} + 5 q^{65} - 5 q^{67} - 15 q^{69} - 2 q^{71} + 15 q^{73} + 11 q^{75} - 6 q^{77} - 12 q^{79} - 18 q^{81} + 8 q^{83} - 32 q^{85} - 11 q^{87} - 24 q^{89} - q^{91} - 18 q^{93} - 9 q^{95} - 12 q^{97} + 40 q^{99}+O(q^{100}) \) 6 * q - 2 * q^3 - 5 * q^5 - 10 * q^7 + 6 * q^9 + 4 * q^11 + 9 * q^13 - 5 * q^15 + q^17 - 10 * q^19 + 8 * q^21 - 9 * q^23 + 13 * q^25 - 8 * q^27 - q^29 - 14 * q^31 + 10 * q^33 + 3 * q^35 + 20 * q^37 - 13 * q^39 - 4 * q^41 - 19 * q^43 - 6 * q^45 - q^47 + 14 * q^49 - 5 * q^51 - 3 * q^53 - 11 * q^55 + 12 * q^57 + q^59 + 4 * q^61 - 14 * q^63 + 5 * q^65 - 5 * q^67 - 15 * q^69 - 2 * q^71 + 15 * q^73 + 11 * q^75 - 6 * q^77 - 12 * q^79 - 18 * q^81 + 8 * q^83 - 32 * q^85 - 11 * q^87 - 24 * q^89 - q^91 - 18 * q^93 - 9 * q^95 - 12 * q^97 + 40 * q^99

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