LMFDB - Embedded newform 6016.2.a.p.1.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6016,2,Mod(1,6016)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6016, 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("6016.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6016 = 2^{7} \cdot 47 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6016.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:	$48.0380018560$
Analytic rank:	$0$
Dimension:	$13$
Coefficient field:	$\mathbb{Q}[x]/(x^{13} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{13} - 4 x^{12} - 22 x^{11} + 96 x^{10} + 163 x^{9} - 840 x^{8} - 438 x^{7} + 3358 x^{6} + 5 x^{5} + \cdots - 832 $ x^13 - 4x^12 - 22x^11 + 96x^10 + 163x^9 - 840x^8 - 438x^7 + 3358x^6 + 5x^5 - 6230x^4 + 1488x^3 + 4720x^2 - 1440x - 832
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$0.904794$ of defining polynomial
Character	$\chi$	$=$	6016.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.904794 q^{3} +3.10347 q^{5} +1.70842 q^{7} -2.18135 q^{9} +O(q^{10})$	$q+0.904794 q^{3} +3.10347 q^{5} +1.70842 q^{7} -2.18135 q^{9} -6.05981 q^{11} +1.97561 q^{13} +2.80800 q^{15} +0.802172 q^{17} +2.24854 q^{19} +1.54576 q^{21} -2.79083 q^{23} +4.63153 q^{25} -4.68805 q^{27} +9.23320 q^{29} +7.61706 q^{31} -5.48288 q^{33} +5.30202 q^{35} +3.25627 q^{37} +1.78752 q^{39} +3.62554 q^{41} -0.635801 q^{43} -6.76975 q^{45} +1.00000 q^{47} -4.08131 q^{49} +0.725800 q^{51} -4.24499 q^{53} -18.8064 q^{55} +2.03446 q^{57} +2.01915 q^{59} +6.89875 q^{61} -3.72665 q^{63} +6.13125 q^{65} -6.92370 q^{67} -2.52512 q^{69} +11.3962 q^{71} +14.6621 q^{73} +4.19058 q^{75} -10.3527 q^{77} +12.3443 q^{79} +2.30233 q^{81} -0.818293 q^{83} +2.48952 q^{85} +8.35414 q^{87} +10.3807 q^{89} +3.37517 q^{91} +6.89187 q^{93} +6.97827 q^{95} -16.2464 q^{97} +13.2186 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 13 q + 4 q^{3} + 6 q^{5} + 2 q^{7} + 21 q^{9}+O(q^{10}) $ 13 * q + 4 * q^3 + 6 * q^5 + 2 * q^7 + 21 * q^9	$ 13 q + 4 q^{3} + 6 q^{5} + 2 q^{7} + 21 q^{9} + 10 q^{11} + 4 q^{13} - 14 q^{15} + 10 q^{17} + 8 q^{19} + 10 q^{21} - 18 q^{23} + 23 q^{25} + 16 q^{27} + 14 q^{29} - 4 q^{31} + 14 q^{33} + 14 q^{35} + 16 q^{37} - 12 q^{39} + 10 q^{41} + 12 q^{43} + 10 q^{45} + 13 q^{47} + 9 q^{49} + 22 q^{51} + 26 q^{53} + 2 q^{55} + 20 q^{57} + 30 q^{59} + 18 q^{61} - 12 q^{63} - 4 q^{65} + 4 q^{67} + 2 q^{69} - 36 q^{71} + 10 q^{73} + 38 q^{75} + 42 q^{77} + 21 q^{81} + 12 q^{83} + 4 q^{85} - 6 q^{87} + 50 q^{89} - 4 q^{91} + 52 q^{93} - 8 q^{95} - 10 q^{97} + 34 q^{99}+O(q^{100}) $ 13 * q + 4 * q^3 + 6 * q^5 + 2 * q^7 + 21 * q^9 + 10 * q^11 + 4 * q^13 - 14 * q^15 + 10 * q^17 + 8 * q^19 + 10 * q^21 - 18 * q^23 + 23 * q^25 + 16 * q^27 + 14 * q^29 - 4 * q^31 + 14 * q^33 + 14 * q^35 + 16 * q^37 - 12 * q^39 + 10 * q^41 + 12 * q^43 + 10 * q^45 + 13 * q^47 + 9 * q^49 + 22 * q^51 + 26 * q^53 + 2 * q^55 + 20 * q^57 + 30 * q^59 + 18 * q^61 - 12 * q^63 - 4 * q^65 + 4 * q^67 + 2 * q^69 - 36 * q^71 + 10 * q^73 + 38 * q^75 + 42 * q^77 + 21 * q^81 + 12 * q^83 + 4 * q^85 - 6 * q^87 + 50 * q^89 - 4 * q^91 + 52 * q^93 - 8 * q^95 - 10 * q^97 + 34 * 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
$3$	0.904794	0.522383	0.261191	−	0.965287i	$-0.415885\pi$
$3$	0.904794	0.522383	0.261191	+	0.965287i	$0.415885\pi$
$4$	0	0
$5$	3.10347	1.38791	0.693957	−	0.720016i	$-0.255866\pi$
$5$	3.10347	1.38791	0.693957	+	0.720016i	$0.255866\pi$
$6$	0	0
$7$	1.70842	0.645721	0.322860	−	0.946447i	$-0.395356\pi$
$7$	1.70842	0.645721	0.322860	+	0.946447i	$0.395356\pi$
$8$	0	0
$9$	−2.18135	−0.727116
$10$	0	0
$11$	−6.05981	−1.82710	−0.913551	−	0.406725i	$-0.866671\pi$
$11$	−6.05981	−1.82710	−0.913551	+	0.406725i	$0.866671\pi$
$12$	0	0
$13$	1.97561	0.547936	0.273968	−	0.961739i	$-0.411664\pi$
$13$	1.97561	0.547936	0.273968	+	0.961739i	$0.411664\pi$
$14$	0	0
$15$	2.80800	0.725023
$16$	0	0
$17$	0.802172	0.194555	0.0972777	−	0.995257i	$-0.468987\pi$
$17$	0.802172	0.194555	0.0972777	+	0.995257i	$0.468987\pi$
$18$	0	0
$19$	2.24854	0.515850	0.257925	−	0.966165i	$-0.416961\pi$
$19$	2.24854	0.515850	0.257925	+	0.966165i	$0.416961\pi$
$20$	0	0
$21$	1.54576	0.337314
$22$	0	0
$23$	−2.79083	−0.581928	−0.290964	−	0.956734i	$-0.593976\pi$
$23$	−2.79083	−0.581928	−0.290964	+	0.956734i	$0.593976\pi$
$24$	0	0
$25$	4.63153	0.926306
$26$	0	0
$27$	−4.68805	−0.902216
$28$	0	0
$29$	9.23320	1.71456	0.857281	−	0.514848i	$-0.172152\pi$
$29$	9.23320	1.71456	0.857281	+	0.514848i	$0.172152\pi$
$30$	0	0
$31$	7.61706	1.36806	0.684032	−	0.729452i	$-0.260225\pi$
$31$	7.61706	1.36806	0.684032	+	0.729452i	$0.260225\pi$
$32$	0	0
$33$	−5.48288	−0.954446
$34$	0	0
$35$	5.30202	0.896205
$36$	0	0
$37$	3.25627	0.535327	0.267664	−	0.963512i	$-0.413748\pi$
$37$	3.25627	0.535327	0.267664	+	0.963512i	$0.413748\pi$
$38$	0	0
$39$	1.78752	0.286232
$40$	0	0
$41$	3.62554	0.566214	0.283107	−	0.959088i	$-0.408635\pi$
$41$	3.62554	0.566214	0.283107	+	0.959088i	$0.408635\pi$
$42$	0	0
$43$	−0.635801	−0.0969587	−0.0484794	−	0.998824i	$-0.515438\pi$
$43$	−0.635801	−0.0969587	−0.0484794	+	0.998824i	$0.515438\pi$
$44$	0	0
$45$	−6.76975	−1.00917
$46$	0	0
$47$	1.00000	0.145865
$47$	1.00000	0.145865
$48$	0	0
$49$	−4.08131	−0.583044
$50$	0	0
$51$	0.725800	0.101632
$52$	0	0
$53$	−4.24499	−0.583094	−0.291547	−	0.956557i	$-0.594170\pi$
$53$	−4.24499	−0.583094	−0.291547	+	0.956557i	$0.594170\pi$
$54$	0	0
$55$	−18.8064	−2.53586
$56$	0	0
$57$	2.03446	0.269471
$58$	0	0
$59$	2.01915	0.262871	0.131436	−	0.991325i	$-0.458041\pi$
$59$	2.01915	0.262871	0.131436	+	0.991325i	$0.458041\pi$
$60$	0	0
$61$	6.89875	0.883294	0.441647	−	0.897189i	$-0.354394\pi$
$61$	6.89875	0.883294	0.441647	+	0.897189i	$0.354394\pi$
$62$	0	0
$63$	−3.72665	−0.469514
$64$	0	0
$65$	6.13125	0.760488
$66$	0	0
$67$	−6.92370	−0.845865	−0.422933	−	0.906161i	$-0.638999\pi$
$67$	−6.92370	−0.845865	−0.422933	+	0.906161i	$0.638999\pi$
$68$	0	0
$69$	−2.52512	−0.303989
$70$	0	0
$71$	11.3962	1.35248	0.676240	−	0.736682i	$-0.263608\pi$
$71$	11.3962	1.35248	0.676240	+	0.736682i	$0.263608\pi$
$72$	0	0
$73$	14.6621	1.71607	0.858034	−	0.513593i	$-0.171686\pi$
$73$	14.6621	1.71607	0.858034	+	0.513593i	$0.171686\pi$
$74$	0	0
$75$	4.19058	0.483886
$76$	0	0
$77$	−10.3527	−1.17980
$78$	0	0
$79$	12.3443	1.38885	0.694423	−	0.719567i	$-0.255660\pi$
$79$	12.3443	1.38885	0.694423	+	0.719567i	$0.255660\pi$
$80$	0	0
$81$	2.30233	0.255814
$82$	0	0
$83$	−0.818293	−0.0898194	−0.0449097	−	0.998991i	$-0.514300\pi$
$83$	−0.818293	−0.0898194	−0.0449097	+	0.998991i	$0.514300\pi$
$84$	0	0
$85$	2.48952	0.270026
$86$	0	0
$87$	8.35414	0.895658
$88$	0	0
$89$	10.3807	1.10035	0.550175	−	0.835050i	$-0.314561\pi$
$89$	10.3807	1.10035	0.550175	+	0.835050i	$0.314561\pi$
$90$	0	0
$91$	3.37517	0.353813
$92$	0	0
$93$	6.89187	0.714653
$94$	0	0
$95$	6.97827	0.715955
$96$	0	0
$97$	−16.2464	−1.64958	−0.824788	−	0.565442i	$-0.808705\pi$
$97$	−16.2464	−1.64958	−0.824788	+	0.565442i	$0.808705\pi$
$98$	0	0
$99$	13.2186	1.32851
$100$	0	0
$101$	6.19583	0.616508	0.308254	−	0.951304i	$-0.400255\pi$
$101$	6.19583	0.616508	0.308254	+	0.951304i	$0.400255\pi$
$102$	0	0
$103$	−5.69517	−0.561162	−0.280581	−	0.959830i	$-0.590527\pi$
$103$	−5.69517	−0.561162	−0.280581	+	0.959830i	$0.590527\pi$
$104$	0	0
$105$	4.79724	0.468162
$106$	0	0
$107$	18.8124	1.81867	0.909333	−	0.416069i	$-0.136592\pi$
$107$	18.8124	1.81867	0.909333	+	0.416069i	$0.136592\pi$
$108$	0	0
$109$	13.6970	1.31193	0.655966	−	0.754791i	$-0.272262\pi$
$109$	13.6970	1.31193	0.655966	+	0.754791i	$0.272262\pi$
$110$	0	0
$111$	2.94625	0.279646
$112$	0	0
$113$	0.463650	0.0436165	0.0218083	−	0.999762i	$-0.493058\pi$
$113$	0.463650	0.0436165	0.0218083	+	0.999762i	$0.493058\pi$
$114$	0	0
$115$	−8.66126	−0.807666
$116$	0	0
$117$	−4.30949	−0.398413
$118$	0	0
$119$	1.37044	0.125628
$120$	0	0
$121$	25.7213	2.33830
$122$	0	0
$123$	3.28036	0.295780
$124$	0	0
$125$	−1.14354	−0.102281
$126$	0	0
$127$	−12.4102	−1.10123	−0.550615	−	0.834759i	$-0.685607\pi$
$127$	−12.4102	−1.10123	−0.550615	+	0.834759i	$0.685607\pi$
$128$	0	0
$129$	−0.575269	−0.0506496
$130$	0	0
$131$	19.1000	1.66877	0.834386	−	0.551181i	$-0.185823\pi$
$131$	19.1000	1.66877	0.834386	+	0.551181i	$0.185823\pi$
$132$	0	0
$133$	3.84144	0.333095
$134$	0	0
$135$	−14.5492	−1.25220
$136$	0	0
$137$	16.1961	1.38372	0.691862	−	0.722030i	$-0.256791\pi$
$137$	16.1961	1.38372	0.691862	+	0.722030i	$0.256791\pi$
$138$	0	0
$139$	−15.2948	−1.29728	−0.648642	−	0.761093i	$-0.724663\pi$
$139$	−15.2948	−1.29728	−0.648642	+	0.761093i	$0.724663\pi$
$140$	0	0
$141$	0.904794	0.0761974
$142$	0	0
$143$	−11.9718	−1.00113
$144$	0	0
$145$	28.6550	2.37967
$146$	0	0
$147$	−3.69274	−0.304572
$148$	0	0
$149$	18.9005	1.54839	0.774194	−	0.632949i	$-0.218156\pi$
$149$	18.9005	1.54839	0.774194	+	0.632949i	$0.218156\pi$
$150$	0	0
$151$	−6.22346	−0.506458	−0.253229	−	0.967406i	$-0.581493\pi$
$151$	−6.22346	−0.506458	−0.253229	+	0.967406i	$0.581493\pi$
$152$	0	0
$153$	−1.74982	−0.141464
$154$	0	0
$155$	23.6393	1.89876
$156$	0	0
$157$	−18.3932	−1.46794	−0.733970	−	0.679182i	$-0.762335\pi$
$157$	−18.3932	−1.46794	−0.733970	+	0.679182i	$0.762335\pi$
$158$	0	0
$159$	−3.84084	−0.304598
$160$	0	0
$161$	−4.76790	−0.375763
$162$	0	0
$163$	−4.29391	−0.336325	−0.168162	−	0.985759i	$-0.553783\pi$
$163$	−4.29391	−0.336325	−0.168162	+	0.985759i	$0.553783\pi$
$164$	0	0
$165$	−17.0159	−1.32469
$166$	0	0
$167$	−15.3400	−1.18704	−0.593521	−	0.804818i	$-0.702263\pi$
$167$	−15.3400	−1.18704	−0.593521	+	0.804818i	$0.702263\pi$
$168$	0	0
$169$	−9.09697	−0.699767
$170$	0	0
$171$	−4.90484	−0.375083
$172$	0	0
$173$	−7.97616	−0.606417	−0.303208	−	0.952924i	$-0.598058\pi$
$173$	−7.97616	−0.606417	−0.303208	+	0.952924i	$0.598058\pi$
$174$	0	0
$175$	7.91258	0.598135
$176$	0	0
$177$	1.82691	0.137319
$178$	0	0
$179$	10.5311	0.787129	0.393564	−	0.919297i	$-0.371242\pi$
$179$	10.5311	0.787129	0.393564	+	0.919297i	$0.371242\pi$
$180$	0	0
$181$	−15.3224	−1.13891	−0.569453	−	0.822024i	$-0.692845\pi$
$181$	−15.3224	−1.13891	−0.569453	+	0.822024i	$0.692845\pi$
$182$	0	0
$183$	6.24195	0.461418
$184$	0	0
$185$	10.1057	0.742988
$186$	0	0
$187$	−4.86101	−0.355472
$188$	0	0
$189$	−8.00915	−0.582580
$190$	0	0
$191$	−16.6814	−1.20703	−0.603513	−	0.797353i	$-0.706233\pi$
$191$	−16.6814	−1.20703	−0.603513	+	0.797353i	$0.706233\pi$
$192$	0	0
$193$	4.75035	0.341938	0.170969	−	0.985276i	$-0.445310\pi$
$193$	4.75035	0.341938	0.170969	+	0.985276i	$0.445310\pi$
$194$	0	0
$195$	5.54751	0.397266
$196$	0	0
$197$	3.61648	0.257663	0.128832	−	0.991666i	$-0.458877\pi$
$197$	3.61648	0.257663	0.128832	+	0.991666i	$0.458877\pi$
$198$	0	0
$199$	−4.05163	−0.287212	−0.143606	−	0.989635i	$-0.545870\pi$
$199$	−4.05163	−0.287212	−0.143606	+	0.989635i	$0.545870\pi$
$200$	0	0
$201$	−6.26452	−0.441865
$202$	0	0
$203$	15.7742	1.10713
$204$	0	0
$205$	11.2518	0.785856
$206$	0	0
$207$	6.08777	0.423129
$208$	0	0
$209$	−13.6257	−0.942510
$210$	0	0
$211$	1.38418	0.0952910	0.0476455	−	0.998864i	$-0.484828\pi$
$211$	1.38418	0.0952910	0.0476455	+	0.998864i	$0.484828\pi$
$212$	0	0
$213$	10.3112	0.706512
$214$	0	0
$215$	−1.97319	−0.134570
$216$	0	0
$217$	13.0131	0.883388
$218$	0	0
$219$	13.2662	0.896445
$220$	0	0
$221$	1.58478	0.106604
$222$	0	0
$223$	−23.9003	−1.60048	−0.800242	−	0.599677i	$-0.795296\pi$
$223$	−23.9003	−1.60048	−0.800242	+	0.599677i	$0.795296\pi$
$224$	0	0
$225$	−10.1030	−0.673532
$226$	0	0
$227$	11.0611	0.734150	0.367075	−	0.930191i	$-0.380359\pi$
$227$	11.0611	0.734150	0.367075	+	0.930191i	$0.380359\pi$
$228$	0	0
$229$	26.8042	1.77127	0.885637	−	0.464379i	$-0.153722\pi$
$229$	26.8042	1.77127	0.885637	+	0.464379i	$0.153722\pi$
$230$	0	0
$231$	−9.36704	−0.616306
$232$	0	0
$233$	−20.2502	−1.32663	−0.663316	−	0.748340i	$-0.730851\pi$
$233$	−20.2502	−1.32663	−0.663316	+	0.748340i	$0.730851\pi$
$234$	0	0
$235$	3.10347	0.202448
$236$	0	0
$237$	11.1691	0.725510
$238$	0	0
$239$	−7.76035	−0.501975	−0.250988	−	0.967990i	$-0.580755\pi$
$239$	−7.76035	−0.501975	−0.250988	+	0.967990i	$0.580755\pi$
$240$	0	0
$241$	−11.5432	−0.743564	−0.371782	−	0.928320i	$-0.621253\pi$
$241$	−11.5432	−0.743564	−0.371782	+	0.928320i	$0.621253\pi$
$242$	0	0
$243$	16.1473	1.03585
$244$	0	0
$245$	−12.6662	−0.809216
$246$	0	0
$247$	4.44223	0.282652
$248$	0	0
$249$	−0.740387	−0.0469201
$250$	0	0
$251$	3.90956	0.246769	0.123385	−	0.992359i	$-0.460625\pi$
$251$	3.90956	0.246769	0.123385	+	0.992359i	$0.460625\pi$
$252$	0	0
$253$	16.9119	1.06324
$254$	0	0
$255$	2.25250	0.141057
$256$	0	0
$257$	−5.92556	−0.369626	−0.184813	−	0.982774i	$-0.559168\pi$
$257$	−5.92556	−0.369626	−0.184813	+	0.982774i	$0.559168\pi$
$258$	0	0
$259$	5.56307	0.345672
$260$	0	0
$261$	−20.1408	−1.24669
$262$	0	0
$263$	0.888642	0.0547960	0.0273980	−	0.999625i	$-0.491278\pi$
$263$	0.888642	0.0547960	0.0273980	+	0.999625i	$0.491278\pi$
$264$	0	0
$265$	−13.1742	−0.809284
$266$	0	0
$267$	9.39237	0.574804
$268$	0	0
$269$	14.2461	0.868598	0.434299	−	0.900769i	$-0.356996\pi$
$269$	14.2461	0.868598	0.434299	+	0.900769i	$0.356996\pi$
$270$	0	0
$271$	1.78285	0.108300	0.0541501	−	0.998533i	$-0.482755\pi$
$271$	1.78285	0.108300	0.0541501	+	0.998533i	$0.482755\pi$
$272$	0	0
$273$	3.05383	0.184826
$274$	0	0
$275$	−28.0662	−1.69245
$276$	0	0
$277$	10.3947	0.624557	0.312278	−	0.949991i	$-0.398908\pi$
$277$	10.3947	0.624557	0.312278	+	0.949991i	$0.398908\pi$
$278$	0	0
$279$	−16.6155	−0.994742
$280$	0	0
$281$	6.11705	0.364913	0.182456	−	0.983214i	$-0.441595\pi$
$281$	6.11705	0.364913	0.182456	+	0.983214i	$0.441595\pi$
$282$	0	0
$283$	−23.0070	−1.36763	−0.683813	−	0.729657i	$-0.739680\pi$
$283$	−23.0070	−1.36763	−0.683813	+	0.729657i	$0.739680\pi$
$284$	0	0
$285$	6.31389	0.374003
$286$	0	0
$287$	6.19393	0.365616
$288$	0	0
$289$	−16.3565	−0.962148
$290$	0	0
$291$	−14.6997	−0.861710
$292$	0	0
$293$	11.0586	0.646053	0.323027	−	0.946390i	$-0.395300\pi$
$293$	11.0586	0.646053	0.323027	+	0.946390i	$0.395300\pi$
$294$	0	0
$295$	6.26637	0.364842
$296$	0	0
$297$	28.4087	1.64844
$298$	0	0
$299$	−5.51359	−0.318859
$300$	0	0
$301$	−1.08621	−0.0626083
$302$	0	0
$303$	5.60594	0.322053
$304$	0	0
$305$	21.4101	1.22594
$306$	0	0
$307$	−6.42157	−0.366499	−0.183249	−	0.983066i	$-0.558662\pi$
$307$	−6.42157	−0.366499	−0.183249	+	0.983066i	$0.558662\pi$
$308$	0	0
$309$	−5.15296	−0.293141
$310$	0	0
$311$	17.1247	0.971053	0.485527	−	0.874222i	$-0.338628\pi$
$311$	17.1247	0.971053	0.485527	+	0.874222i	$0.338628\pi$
$312$	0	0
$313$	−31.8876	−1.80239	−0.901196	−	0.433412i	$-0.857309\pi$
$313$	−31.8876	−1.80239	−0.901196	+	0.433412i	$0.857309\pi$
$314$	0	0
$315$	−11.5656	−0.651645
$316$	0	0
$317$	15.2104	0.854303	0.427152	−	0.904180i	$-0.359517\pi$
$317$	15.2104	0.854303	0.427152	+	0.904180i	$0.359517\pi$
$318$	0	0
$319$	−55.9514	−3.13268
$320$	0	0
$321$	17.0214	0.950040
$322$	0	0
$323$	1.80371	0.100361
$324$	0	0
$325$	9.15010	0.507556
$326$	0	0
$327$	12.3929	0.685331
$328$	0	0
$329$	1.70842	0.0941881
$330$	0	0
$331$	−13.8340	−0.760384	−0.380192	−	0.924908i	$-0.624142\pi$
$331$	−13.8340	−0.760384	−0.380192	+	0.924908i	$0.624142\pi$
$332$	0	0
$333$	−7.10306	−0.389245
$334$	0	0
$335$	−21.4875	−1.17399
$336$	0	0
$337$	−10.4625	−0.569930	−0.284965	−	0.958538i	$-0.591982\pi$
$337$	−10.4625	−0.569930	−0.284965	+	0.958538i	$0.591982\pi$
$338$	0	0
$339$	0.419507	0.0227845
$340$	0	0
$341$	−46.1579	−2.49959
$342$	0	0
$343$	−18.9315	−1.02220
$344$	0	0
$345$	−7.83665	−0.421911
$346$	0	0
$347$	−7.13337	−0.382939	−0.191470	−	0.981499i	$-0.561325\pi$
$347$	−7.13337	−0.382939	−0.191470	+	0.981499i	$0.561325\pi$
$348$	0	0
$349$	21.7215	1.16272	0.581362	−	0.813645i	$-0.302520\pi$
$349$	21.7215	1.16272	0.581362	+	0.813645i	$0.302520\pi$
$350$	0	0
$351$	−9.26176	−0.494356
$352$	0	0
$353$	−7.88265	−0.419551	−0.209776	−	0.977750i	$-0.567273\pi$
$353$	−7.88265	−0.419551	−0.209776	+	0.977750i	$0.567273\pi$
$354$	0	0
$355$	35.3677	1.87713
$356$	0	0
$357$	1.23997	0.0656262
$358$	0	0
$359$	−20.7942	−1.09747	−0.548737	−	0.835995i	$-0.684891\pi$
$359$	−20.7942	−1.09747	−0.548737	+	0.835995i	$0.684891\pi$
$360$	0	0
$361$	−13.9441	−0.733899
$362$	0	0
$363$	23.2725	1.22149
$364$	0	0
$365$	45.5034	2.38176
$366$	0	0
$367$	3.84224	0.200563	0.100282	−	0.994959i	$-0.468026\pi$
$367$	3.84224	0.200563	0.100282	+	0.994959i	$0.468026\pi$
$368$	0	0
$369$	−7.90856	−0.411703
$370$	0	0
$371$	−7.25221	−0.376516
$372$	0	0
$373$	−4.67398	−0.242010	−0.121005	−	0.992652i	$-0.538612\pi$
$373$	−4.67398	−0.242010	−0.121005	+	0.992652i	$0.538612\pi$
$374$	0	0
$375$	−1.03466	−0.0534298
$376$	0	0
$377$	18.2412	0.939470
$378$	0	0
$379$	−33.5947	−1.72564	−0.862822	−	0.505508i	$-0.831305\pi$
$379$	−33.5947	−1.72564	−0.862822	+	0.505508i	$0.831305\pi$
$380$	0	0
$381$	−11.2287	−0.575264
$382$	0	0
$383$	−13.7061	−0.700350	−0.350175	−	0.936684i	$-0.613878\pi$
$383$	−13.7061	−0.700350	−0.350175	+	0.936684i	$0.613878\pi$
$384$	0	0
$385$	−32.1292	−1.63746
$386$	0	0
$387$	1.38690	0.0705003
$388$	0	0
$389$	5.21325	0.264322	0.132161	−	0.991228i	$-0.457808\pi$
$389$	5.21325	0.264322	0.132161	+	0.991228i	$0.457808\pi$
$390$	0	0
$391$	−2.23873	−0.113217
$392$	0	0
$393$	17.2815	0.871738
$394$	0	0
$395$	38.3103	1.92760
$396$	0	0
$397$	−36.2484	−1.81926	−0.909629	−	0.415422i	$-0.863634\pi$
$397$	−36.2484	−1.81926	−0.909629	+	0.415422i	$0.863634\pi$
$398$	0	0
$399$	3.47571	0.174003
$400$	0	0
$401$	−16.7780	−0.837852	−0.418926	−	0.908020i	$-0.637593\pi$
$401$	−16.7780	−0.837852	−0.418926	+	0.908020i	$0.637593\pi$
$402$	0	0
$403$	15.0483	0.749611
$404$	0	0
$405$	7.14520	0.355048
$406$	0	0
$407$	−19.7324	−0.978097
$408$	0	0
$409$	21.5321	1.06469	0.532346	−	0.846527i	$-0.321311\pi$
$409$	21.5321	1.06469	0.532346	+	0.846527i	$0.321311\pi$
$410$	0	0
$411$	14.6541	0.722833
$412$	0	0
$413$	3.44955	0.169741
$414$	0	0
$415$	−2.53955	−0.124662
$416$	0	0
$417$	−13.8386	−0.677679
$418$	0	0
$419$	5.99082	0.292671	0.146335	−	0.989235i	$-0.453252\pi$
$419$	5.99082	0.292671	0.146335	+	0.989235i	$0.453252\pi$
$420$	0	0
$421$	27.6128	1.34577	0.672884	−	0.739748i	$-0.265055\pi$
$421$	27.6128	1.34577	0.672884	+	0.739748i	$0.265055\pi$
$422$	0	0
$423$	−2.18135	−0.106061
$424$	0	0
$425$	3.71529	0.180218
$426$	0	0
$427$	11.7859	0.570362
$428$	0	0
$429$	−10.8320	−0.522975
$430$	0	0
$431$	14.2613	0.686940	0.343470	−	0.939164i	$-0.388398\pi$
$431$	14.2613	0.686940	0.343470	+	0.939164i	$0.388398\pi$
$432$	0	0
$433$	33.4057	1.60537	0.802687	−	0.596400i	$-0.203403\pi$
$433$	33.4057	1.60537	0.802687	+	0.596400i	$0.203403\pi$
$434$	0	0
$435$	25.9268	1.24310
$436$	0	0
$437$	−6.27528	−0.300188
$438$	0	0
$439$	11.5898	0.553151	0.276575	−	0.960992i	$-0.410800\pi$
$439$	11.5898	0.553151	0.276575	+	0.960992i	$0.410800\pi$
$440$	0	0
$441$	8.90276	0.423941
$442$	0	0
$443$	−30.6610	−1.45675	−0.728375	−	0.685179i	$-0.759724\pi$
$443$	−30.6610	−1.45675	−0.728375	+	0.685179i	$0.759724\pi$
$444$	0	0
$445$	32.2161	1.52719
$446$	0	0
$447$	17.1010	0.808851
$448$	0	0
$449$	2.55670	0.120658	0.0603290	−	0.998179i	$-0.480785\pi$
$449$	2.55670	0.120658	0.0603290	+	0.998179i	$0.480785\pi$
$450$	0	0
$451$	−21.9701	−1.03453
$452$	0	0
$453$	−5.63095	−0.264565
$454$	0	0
$455$	10.4747	0.491063
$456$	0	0
$457$	0.723510	0.0338444	0.0169222	−	0.999857i	$-0.494613\pi$
$457$	0.723510	0.0338444	0.0169222	+	0.999857i	$0.494613\pi$
$458$	0	0
$459$	−3.76062	−0.175531
$460$	0	0
$461$	3.29442	0.153436	0.0767182	−	0.997053i	$-0.475556\pi$
$461$	3.29442	0.153436	0.0767182	+	0.997053i	$0.475556\pi$
$462$	0	0
$463$	−0.386079	−0.0179426	−0.00897130	−	0.999960i	$-0.502856\pi$
$463$	−0.386079	−0.0179426	−0.00897130	+	0.999960i	$0.502856\pi$
$464$	0	0
$465$	21.3887	0.991878
$466$	0	0
$467$	18.8400	0.871813	0.435906	−	0.899992i	$-0.356428\pi$
$467$	18.8400	0.871813	0.435906	+	0.899992i	$0.356428\pi$
$468$	0	0
$469$	−11.8286	−0.546193
$470$	0	0
$471$	−16.6421	−0.766826
$472$	0	0
$473$	3.85283	0.177153
$474$	0	0
$475$	10.4142	0.477835
$476$	0	0
$477$	9.25980	0.423977
$478$	0	0
$479$	3.98968	0.182293	0.0911465	−	0.995837i	$-0.470947\pi$
$479$	3.98968	0.182293	0.0911465	+	0.995837i	$0.470947\pi$
$480$	0	0
$481$	6.43312	0.293325
$482$	0	0
$483$	−4.31397	−0.196292
$484$	0	0
$485$	−50.4203	−2.28947
$486$	0	0
$487$	6.84955	0.310383	0.155191	−	0.987884i	$-0.450401\pi$
$487$	6.84955	0.310383	0.155191	+	0.987884i	$0.450401\pi$
$488$	0	0
$489$	−3.88510	−0.175690
$490$	0	0
$491$	9.02535	0.407308	0.203654	−	0.979043i	$-0.434718\pi$
$491$	9.02535	0.407308	0.203654	+	0.979043i	$0.434718\pi$
$492$	0	0
$493$	7.40662	0.333577
$494$	0	0
$495$	41.0234	1.84386
$496$	0	0
$497$	19.4694	0.873324
$498$	0	0
$499$	−1.93588	−0.0866619	−0.0433309	−	0.999061i	$-0.513797\pi$
$499$	−1.93588	−0.0866619	−0.0433309	+	0.999061i	$0.513797\pi$
$500$	0	0
$501$	−13.8795	−0.620091
$502$	0	0
$503$	−20.4584	−0.912196	−0.456098	−	0.889929i	$-0.650753\pi$
$503$	−20.4584	−0.912196	−0.456098	+	0.889929i	$0.650753\pi$
$504$	0	0
$505$	19.2286	0.855660
$506$	0	0
$507$	−8.23088	−0.365546
$508$	0	0
$509$	13.5664	0.601319	0.300659	−	0.953732i	$-0.402793\pi$
$509$	13.5664	0.601319	0.300659	+	0.953732i	$0.402793\pi$
$510$	0	0
$511$	25.0490	1.10810
$512$	0	0
$513$	−10.5413	−0.465408
$514$	0	0
$515$	−17.6748	−0.778845
$516$	0	0
$517$	−6.05981	−0.266510
$518$	0	0
$519$	−7.21678	−0.316782
$520$	0	0
$521$	−25.7717	−1.12908	−0.564540	−	0.825406i	$-0.690946\pi$
$521$	−25.7717	−1.12908	−0.564540	+	0.825406i	$0.690946\pi$
$522$	0	0
$523$	1.82465	0.0797862	0.0398931	−	0.999204i	$-0.487298\pi$
$523$	1.82465	0.0797862	0.0398931	+	0.999204i	$0.487298\pi$
$524$	0	0
$525$	7.15926	0.312456
$526$	0	0
$527$	6.11020	0.266164
$528$	0	0
$529$	−15.2113	−0.661360
$530$	0	0
$531$	−4.40447	−0.191138
$532$	0	0
$533$	7.16265	0.310249
$534$	0	0
$535$	58.3838	2.52415
$536$	0	0
$537$	9.52844	0.411182
$538$	0	0
$539$	24.7320	1.06528
$540$	0	0
$541$	36.2752	1.55959	0.779796	−	0.626034i	$-0.215323\pi$
$541$	36.2752	1.55959	0.779796	+	0.626034i	$0.215323\pi$
$542$	0	0
$543$	−13.8636	−0.594945
$544$	0	0
$545$	42.5081	1.82085
$546$	0	0
$547$	38.4379	1.64349	0.821744	−	0.569857i	$-0.193001\pi$
$547$	38.4379	1.64349	0.821744	+	0.569857i	$0.193001\pi$
$548$	0	0
$549$	−15.0486	−0.642258
$550$	0	0
$551$	20.7612	0.884457
$552$	0	0
$553$	21.0893	0.896808
$554$	0	0
$555$	9.14360	0.388124
$556$	0	0
$557$	12.9869	0.550274	0.275137	−	0.961405i	$-0.411277\pi$
$557$	12.9869	0.550274	0.275137	+	0.961405i	$0.411277\pi$
$558$	0	0
$559$	−1.25609	−0.0531271
$560$	0	0
$561$	−4.39821	−0.185693
$562$	0	0
$563$	−18.6370	−0.785455	−0.392727	−	0.919655i	$-0.628468\pi$
$563$	−18.6370	−0.785455	−0.392727	+	0.919655i	$0.628468\pi$
$564$	0	0
$565$	1.43892	0.0605360
$566$	0	0
$567$	3.93333	0.165185
$568$	0	0
$569$	−10.6112	−0.444845	−0.222422	−	0.974950i	$-0.571396\pi$
$569$	−10.6112	−0.444845	−0.222422	+	0.974950i	$0.571396\pi$
$570$	0	0
$571$	−26.5317	−1.11032	−0.555158	−	0.831745i	$-0.687342\pi$
$571$	−26.5317	−1.11032	−0.555158	+	0.831745i	$0.687342\pi$
$572$	0	0
$573$	−15.0932	−0.630529
$574$	0	0
$575$	−12.9258	−0.539044
$576$	0	0
$577$	7.25089	0.301858	0.150929	−	0.988545i	$-0.451773\pi$
$577$	7.25089	0.301858	0.150929	+	0.988545i	$0.451773\pi$
$578$	0	0
$579$	4.29809	0.178623
$580$	0	0
$581$	−1.39799	−0.0579982
$582$	0	0
$583$	25.7238	1.06537
$584$	0	0
$585$	−13.3744	−0.552963
$586$	0	0
$587$	−15.5420	−0.641487	−0.320743	−	0.947166i	$-0.603933\pi$
$587$	−15.5420	−0.641487	−0.320743	+	0.947166i	$0.603933\pi$
$588$	0	0
$589$	17.1272	0.705716
$590$	0	0
$591$	3.27217	0.134599
$592$	0	0
$593$	−9.53631	−0.391609	−0.195805	−	0.980643i	$-0.562732\pi$
$593$	−9.53631	−0.391609	−0.195805	+	0.980643i	$0.562732\pi$
$594$	0	0
$595$	4.25314	0.174362
$596$	0	0
$597$	−3.66589	−0.150035
$598$	0	0
$599$	−28.8238	−1.17771	−0.588854	−	0.808239i	$-0.700421\pi$
$599$	−28.8238	−1.17771	−0.588854	+	0.808239i	$0.700421\pi$
$600$	0	0
$601$	25.6563	1.04654	0.523272	−	0.852166i	$-0.324711\pi$
$601$	25.6563	1.04654	0.523272	+	0.852166i	$0.324711\pi$
$602$	0	0
$603$	15.1030	0.615042
$604$	0	0
$605$	79.8253	3.24536
$606$	0	0
$607$	10.1171	0.410639	0.205319	−	0.978695i	$-0.434177\pi$
$607$	10.1171	0.410639	0.205319	+	0.978695i	$0.434177\pi$
$608$	0	0
$609$	14.2724	0.578345
$610$	0	0
$611$	1.97561	0.0799246
$612$	0	0
$613$	−33.2492	−1.34292	−0.671461	−	0.741040i	$-0.734333\pi$
$613$	−33.2492	−1.34292	−0.671461	+	0.741040i	$0.734333\pi$
$614$	0	0
$615$	10.1805	0.410518
$616$	0	0
$617$	−39.1115	−1.57457	−0.787285	−	0.616589i	$-0.788514\pi$
$617$	−39.1115	−1.57457	−0.787285	+	0.616589i	$0.788514\pi$
$618$	0	0
$619$	−19.7327	−0.793122	−0.396561	−	0.918008i	$-0.629797\pi$
$619$	−19.7327	−0.793122	−0.396561	+	0.918008i	$0.629797\pi$
$620$	0	0
$621$	13.0836	0.525025
$622$	0	0
$623$	17.7345	0.710519
$624$	0	0
$625$	−26.7066	−1.06826
$626$	0	0
$627$	−12.3285	−0.492351
$628$	0	0
$629$	2.61209	0.104151
$630$	0	0
$631$	−2.94916	−0.117404	−0.0587022	−	0.998276i	$-0.518696\pi$
$631$	−2.94916	−0.117404	−0.0587022	+	0.998276i	$0.518696\pi$
$632$	0	0
$633$	1.25240	0.0497784
$634$	0	0
$635$	−38.5148	−1.52841
$636$	0	0
$637$	−8.06308	−0.319471
$638$	0	0
$639$	−24.8591	−0.983409
$640$	0	0
$641$	40.8533	1.61361	0.806804	−	0.590819i	$-0.201195\pi$
$641$	40.8533	1.61361	0.806804	+	0.590819i	$0.201195\pi$
$642$	0	0
$643$	49.5313	1.95332	0.976662	−	0.214784i	$-0.0689047\pi$
$643$	49.5313	1.95332	0.976662	+	0.214784i	$0.0689047\pi$
$644$	0	0
$645$	−1.78533	−0.0702973
$646$	0	0
$647$	20.1422	0.791872	0.395936	−	0.918278i	$-0.370420\pi$
$647$	20.1422	0.791872	0.395936	+	0.918278i	$0.370420\pi$
$648$	0	0
$649$	−12.2357	−0.480292
$650$	0	0
$651$	11.7742	0.461467
$652$	0	0
$653$	31.5708	1.23546	0.617731	−	0.786390i	$-0.288052\pi$
$653$	31.5708	1.23546	0.617731	+	0.786390i	$0.288052\pi$
$654$	0	0
$655$	59.2762	2.31611
$656$	0	0
$657$	−31.9831	−1.24778
$658$	0	0
$659$	24.5255	0.955379	0.477689	−	0.878529i	$-0.341474\pi$
$659$	24.5255	0.955379	0.477689	+	0.878529i	$0.341474\pi$
$660$	0	0
$661$	19.1347	0.744254	0.372127	−	0.928182i	$-0.378629\pi$
$661$	19.1347	0.744254	0.372127	+	0.928182i	$0.378629\pi$
$662$	0	0
$663$	1.43390	0.0556880
$664$	0	0
$665$	11.9218	0.462307
$666$	0	0
$667$	−25.7683	−0.997753
$668$	0	0
$669$	−21.6249	−0.836065
$670$	0	0
$671$	−41.8051	−1.61387
$672$	0	0
$673$	15.0430	0.579865	0.289932	−	0.957047i	$-0.406367\pi$
$673$	15.0430	0.579865	0.289932	+	0.957047i	$0.406367\pi$
$674$	0	0
$675$	−21.7128	−0.835728
$676$	0	0
$677$	−36.2052	−1.39148	−0.695740	−	0.718294i	$-0.744923\pi$
$677$	−36.2052	−1.39148	−0.695740	+	0.718294i	$0.744923\pi$
$678$	0	0
$679$	−27.7557	−1.06517
$680$	0	0
$681$	10.0080	0.383508
$682$	0	0
$683$	15.5572	0.595278	0.297639	−	0.954678i	$-0.403801\pi$
$683$	15.5572	0.595278	0.297639	+	0.954678i	$0.403801\pi$
$684$	0	0
$685$	50.2640	1.92049
$686$	0	0
$687$	24.2523	0.925283
$688$	0	0
$689$	−8.38644	−0.319498
$690$	0	0
$691$	−44.9577	−1.71027	−0.855135	−	0.518405i	$-0.826526\pi$
$691$	−44.9577	−1.71027	−0.855135	+	0.518405i	$0.826526\pi$
$692$	0	0
$693$	22.5828	0.857850
$694$	0	0
$695$	−47.4668	−1.80052
$696$	0	0
$697$	2.90831	0.110160
$698$	0	0
$699$	−18.3222	−0.693010
$700$	0	0
$701$	9.29361	0.351015	0.175507	−	0.984478i	$-0.443843\pi$
$701$	9.29361	0.351015	0.175507	+	0.984478i	$0.443843\pi$
$702$	0	0
$703$	7.32184	0.276148
$704$	0	0
$705$	2.80800	0.105755
$706$	0	0
$707$	10.5851	0.398092
$708$	0	0
$709$	38.6450	1.45134	0.725672	−	0.688040i	$-0.241529\pi$
$709$	38.6450	1.45134	0.725672	+	0.688040i	$0.241529\pi$
$710$	0	0
$711$	−26.9273	−1.00985
$712$	0	0
$713$	−21.2579	−0.796115
$714$	0	0
$715$	−37.1542	−1.38949
$716$	0	0
$717$	−7.02152	−0.262223
$718$	0	0
$719$	−27.5083	−1.02588	−0.512942	−	0.858423i	$-0.671445\pi$
$719$	−27.5083	−1.02588	−0.512942	+	0.858423i	$0.671445\pi$
$720$	0	0
$721$	−9.72973	−0.362354
$722$	0	0
$723$	−10.4442	−0.388425
$724$	0	0
$725$	42.7639	1.58821
$726$	0	0
$727$	−11.4162	−0.423403	−0.211702	−	0.977334i	$-0.567900\pi$
$727$	−11.4162	−0.423403	−0.211702	+	0.977334i	$0.567900\pi$
$728$	0	0
$729$	7.70298	0.285295
$730$	0	0
$731$	−0.510022	−0.0188638
$732$	0	0
$733$	−4.02880	−0.148807	−0.0744035	−	0.997228i	$-0.523705\pi$
$733$	−4.02880	−0.148807	−0.0744035	+	0.997228i	$0.523705\pi$
$734$	0	0
$735$	−11.4603	−0.422720
$736$	0	0
$737$	41.9563	1.54548
$738$	0	0
$739$	−20.6451	−0.759441	−0.379720	−	0.925101i	$-0.623980\pi$
$739$	−20.6451	−0.759441	−0.379720	+	0.925101i	$0.623980\pi$
$740$	0	0
$741$	4.01930	0.147653
$742$	0	0
$743$	17.1888	0.630596	0.315298	−	0.948993i	$-0.397896\pi$
$743$	17.1888	0.630596	0.315298	+	0.948993i	$0.397896\pi$
$744$	0	0
$745$	58.6571	2.14903
$746$	0	0
$747$	1.78498	0.0653091
$748$	0	0
$749$	32.1395	1.17435
$750$	0	0
$751$	−31.8442	−1.16201	−0.581006	−	0.813899i	$-0.697341\pi$
$751$	−31.8442	−1.16201	−0.581006	+	0.813899i	$0.697341\pi$
$752$	0	0
$753$	3.53735	0.128908
$754$	0	0
$755$	−19.3143	−0.702921
$756$	0	0
$757$	−34.5104	−1.25430	−0.627151	−	0.778898i	$-0.715779\pi$
$757$	−34.5104	−1.25430	−0.627151	+	0.778898i	$0.715779\pi$
$758$	0	0
$759$	15.3018	0.555419
$760$	0	0
$761$	12.4728	0.452137	0.226068	−	0.974111i	$-0.427413\pi$
$761$	12.4728	0.452137	0.226068	+	0.974111i	$0.427413\pi$
$762$	0	0
$763$	23.4001	0.847142
$764$	0	0
$765$	−5.43051	−0.196340
$766$	0	0
$767$	3.98905	0.144036
$768$	0	0
$769$	11.7854	0.424993	0.212496	−	0.977162i	$-0.431841\pi$
$769$	11.7854	0.424993	0.212496	+	0.977162i	$0.431841\pi$
$770$	0	0
$771$	−5.36141	−0.193086
$772$	0	0
$773$	0.175957	0.00632874	0.00316437	−	0.999995i	$-0.498993\pi$
$773$	0.175957	0.00632874	0.00316437	+	0.999995i	$0.498993\pi$
$774$	0	0
$775$	35.2786	1.26725
$776$	0	0
$777$	5.03343	0.180573
$778$	0	0
$779$	8.15216	0.292081
$780$	0	0
$781$	−69.0587	−2.47112
$782$	0	0
$783$	−43.2857	−1.54691
$784$	0	0
$785$	−57.0828	−2.03737
$786$	0	0
$787$	−2.39218	−0.0852719	−0.0426359	−	0.999091i	$-0.513576\pi$
$787$	−2.39218	−0.0852719	−0.0426359	+	0.999091i	$0.513576\pi$
$788$	0	0
$789$	0.804038	0.0286245
$790$	0	0
$791$	0.792107	0.0281641
$792$	0	0
$793$	13.6292	0.483988
$794$	0	0
$795$	−11.9199	−0.422756
$796$	0	0
$797$	−19.8575	−0.703390	−0.351695	−	0.936115i	$-0.614395\pi$
$797$	−19.8575	−0.703390	−0.351695	+	0.936115i	$0.614395\pi$
$798$	0	0
$799$	0.802172	0.0283788
$800$	0	0
$801$	−22.6439	−0.800082
$802$	0	0
$803$	−88.8495	−3.13543
$804$	0	0
$805$	−14.7970	−0.521527
$806$	0	0
$807$	12.8897	0.453741
$808$	0	0
$809$	2.50199	0.0879652	0.0439826	−	0.999032i	$-0.485995\pi$
$809$	2.50199	0.0879652	0.0439826	+	0.999032i	$0.485995\pi$
$810$	0	0
$811$	34.4012	1.20799	0.603995	−	0.796988i	$-0.293575\pi$
$811$	34.4012	1.20799	0.603995	+	0.796988i	$0.293575\pi$
$812$	0	0
$813$	1.61311	0.0565741
$814$	0	0
$815$	−13.3260	−0.466790
$816$	0	0
$817$	−1.42962	−0.0500161
$818$	0	0
$819$	−7.36241	−0.257264
$820$	0	0
$821$	−37.3196	−1.30246	−0.651231	−	0.758879i	$-0.725747\pi$
$821$	−37.3196	−1.30246	−0.651231	+	0.758879i	$0.725747\pi$
$822$	0	0
$823$	−34.3942	−1.19891	−0.599454	−	0.800410i	$-0.704615\pi$
$823$	−34.3942	−1.19891	−0.599454	+	0.800410i	$0.704615\pi$
$824$	0	0
$825$	−25.3941	−0.884109
$826$	0	0
$827$	30.1413	1.04812	0.524058	−	0.851682i	$-0.324417\pi$
$827$	30.1413	1.04812	0.524058	+	0.851682i	$0.324417\pi$
$828$	0	0
$829$	−52.6152	−1.82740	−0.913701	−	0.406387i	$-0.866789\pi$
$829$	−52.6152	−1.82740	−0.913701	+	0.406387i	$0.866789\pi$
$830$	0	0
$831$	9.40506	0.326258
$832$	0	0
$833$	−3.27391	−0.113434
$834$	0	0
$835$	−47.6071	−1.64751
$836$	0	0
$837$	−35.7092	−1.23429
$838$	0	0
$839$	19.8543	0.685448	0.342724	−	0.939436i	$-0.388651\pi$
$839$	19.8543	0.685448	0.342724	+	0.939436i	$0.388651\pi$
$840$	0	0
$841$	56.2520	1.93973
$842$	0	0
$843$	5.53467	0.190624
$844$	0	0
$845$	−28.2322	−0.971216
$846$	0	0
$847$	43.9427	1.50989
$848$	0	0
$849$	−20.8166	−0.714425
$850$	0	0
$851$	−9.08769	−0.311522
$852$	0	0
$853$	27.6358	0.946230	0.473115	−	0.881001i	$-0.343129\pi$
$853$	27.6358	0.946230	0.473115	+	0.881001i	$0.343129\pi$
$854$	0	0
$855$	−15.2220	−0.520583
$856$	0	0
$857$	24.8177	0.847757	0.423879	−	0.905719i	$-0.360668\pi$
$857$	24.8177	0.847757	0.423879	+	0.905719i	$0.360668\pi$
$858$	0	0
$859$	22.0555	0.752525	0.376263	−	0.926513i	$-0.377209\pi$
$859$	22.0555	0.752525	0.376263	+	0.926513i	$0.377209\pi$
$860$	0	0
$861$	5.60423	0.190992
$862$	0	0
$863$	41.7334	1.42062	0.710310	−	0.703889i	$-0.248555\pi$
$863$	41.7334	1.42062	0.710310	+	0.703889i	$0.248555\pi$
$864$	0	0
$865$	−24.7538	−0.841654
$866$	0	0
$867$	−14.7993	−0.502610
$868$	0	0
$869$	−74.8043	−2.53756
$870$	0	0
$871$	−13.6785	−0.463480
$872$	0	0
$873$	35.4391	1.19943
$874$	0	0
$875$	−1.95364	−0.0660450
$876$	0	0
$877$	−14.1170	−0.476696	−0.238348	−	0.971180i	$-0.576606\pi$
$877$	−14.1170	−0.476696	−0.238348	+	0.971180i	$0.576606\pi$
$878$	0	0
$879$	10.0058	0.337487
$880$	0	0
$881$	−48.5767	−1.63659	−0.818296	−	0.574797i	$-0.805081\pi$
$881$	−48.5767	−1.63659	−0.818296	+	0.574797i	$0.805081\pi$
$882$	0	0
$883$	−45.5828	−1.53398	−0.766991	−	0.641658i	$-0.778247\pi$
$883$	−45.5828	−1.53398	−0.766991	+	0.641658i	$0.778247\pi$
$884$	0	0
$885$	5.66978	0.190587
$886$	0	0
$887$	2.54019	0.0852911	0.0426455	−	0.999090i	$-0.486421\pi$
$887$	2.54019	0.0852911	0.0426455	+	0.999090i	$0.486421\pi$
$888$	0	0
$889$	−21.2019	−0.711087
$890$	0	0
$891$	−13.9517	−0.467398
$892$	0	0
$893$	2.24854	0.0752444
$894$	0	0
$895$	32.6828	1.09247
$896$	0	0
$897$	−4.98866	−0.166567
$898$	0	0
$899$	70.3299	2.34563
$900$	0	0
$901$	−3.40521	−0.113444
$902$	0	0
$903$	−0.982799	−0.0327055
$904$	0	0
$905$	−47.5527	−1.58070
$906$	0	0
$907$	24.5434	0.814951	0.407476	−	0.913216i	$-0.366409\pi$
$907$	24.5434	0.814951	0.407476	+	0.913216i	$0.366409\pi$
$908$	0	0
$909$	−13.5153	−0.448273
$910$	0	0
$911$	−40.2560	−1.33374	−0.666870	−	0.745174i	$-0.732366\pi$
$911$	−40.2560	−1.33374	−0.666870	+	0.745174i	$0.732366\pi$
$912$	0	0
$913$	4.95870	0.164109
$914$	0	0
$915$	19.3717	0.640408
$916$	0	0
$917$	32.6307	1.07756
$918$	0	0
$919$	4.39048	0.144829	0.0724143	−	0.997375i	$-0.476930\pi$
$919$	4.39048	0.144829	0.0724143	+	0.997375i	$0.476930\pi$
$920$	0	0
$921$	−5.81020	−0.191453
$922$	0	0
$923$	22.5144	0.741071
$924$	0	0
$925$	15.0815	0.495877
$926$	0	0
$927$	12.4232	0.408030
$928$	0	0
$929$	−45.5577	−1.49470	−0.747350	−	0.664431i	$-0.768674\pi$
$929$	−45.5577	−1.49470	−0.747350	+	0.664431i	$0.768674\pi$
$930$	0	0
$931$	−9.17698	−0.300763
$932$	0	0
$933$	15.4943	0.507262
$934$	0	0
$935$	−15.0860	−0.493365
$936$	0	0
$937$	−39.7060	−1.29714	−0.648570	−	0.761155i	$-0.724632\pi$
$937$	−39.7060	−1.29714	−0.648570	+	0.761155i	$0.724632\pi$
$938$	0	0
$939$	−28.8517	−0.941538
$940$	0	0
$941$	36.3212	1.18404	0.592018	−	0.805925i	$-0.298331\pi$
$941$	36.3212	1.18404	0.592018	+	0.805925i	$0.298331\pi$
$942$	0	0
$943$	−10.1183	−0.329496
$944$	0	0
$945$	−24.8562	−0.808571
$946$	0	0
$947$	44.5317	1.44709	0.723543	−	0.690279i	$-0.242512\pi$
$947$	44.5317	1.44709	0.723543	+	0.690279i	$0.242512\pi$
$948$	0	0
$949$	28.9666	0.940295
$950$	0	0
$951$	13.7623	0.446273
$952$	0	0
$953$	3.29735	0.106812	0.0534058	−	0.998573i	$-0.482992\pi$
$953$	3.29735	0.106812	0.0534058	+	0.998573i	$0.482992\pi$
$954$	0	0
$955$	−51.7703	−1.67525
$956$	0	0
$957$	−50.6245	−1.63646
$958$	0	0
$959$	27.6696	0.893499
$960$	0	0
$961$	27.0196	0.871601
$962$	0	0
$963$	−41.0365	−1.32238
$964$	0	0
$965$	14.7426	0.474581
$966$	0	0
$967$	33.2862	1.07041	0.535207	−	0.844721i	$-0.320234\pi$
$967$	33.2862	1.07041	0.535207	+	0.844721i	$0.320234\pi$
$968$	0	0
$969$	1.63199	0.0524270
$970$	0	0
$971$	29.5093	0.947000	0.473500	−	0.880794i	$-0.342990\pi$
$971$	29.5093	0.947000	0.473500	+	0.880794i	$0.342990\pi$
$972$	0	0
$973$	−26.1298	−0.837684
$974$	0	0
$975$	8.27895	0.265139
$976$	0	0
$977$	−47.1102	−1.50719	−0.753594	−	0.657340i	$-0.771682\pi$
$977$	−47.1102	−1.50719	−0.753594	+	0.657340i	$0.771682\pi$
$978$	0	0
$979$	−62.9049	−2.01045
$980$	0	0
$981$	−29.8779	−0.953927
$982$	0	0
$983$	15.3825	0.490627	0.245313	−	0.969444i	$-0.421109\pi$
$983$	15.3825	0.490627	0.245313	+	0.969444i	$0.421109\pi$
$984$	0	0
$985$	11.2236	0.357615
$986$	0	0
$987$	1.54576	0.0492022
$988$	0	0
$989$	1.77441	0.0564230
$990$	0	0
$991$	37.5123	1.19162	0.595808	−	0.803127i	$-0.296832\pi$
$991$	37.5123	1.19162	0.595808	+	0.803127i	$0.296832\pi$
$992$	0	0
$993$	−12.5169	−0.397211
$994$	0	0
$995$	−12.5741	−0.398626
$996$	0	0
$997$	2.29299	0.0726197	0.0363099	−	0.999341i	$-0.488440\pi$
$997$	2.29299	0.0726197	0.0363099	+	0.999341i	$0.488440\pi$
$998$	0	0
$999$	−15.2656	−0.482981

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6016.2.a.p.1.7	yes	13
4.3	odd	2		6016.2.a.n.1.7	yes	13
8.3	odd	2		6016.2.a.o.1.7	yes	13
8.5	even	2		6016.2.a.m.1.7	✓	13

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6016.2.a.m.1.7	✓	13	8.5	even	2
6016.2.a.n.1.7	yes	13	4.3	odd	2
6016.2.a.o.1.7	yes	13	8.3	odd	2
6016.2.a.p.1.7	yes	13	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 6016 = 2^{7} \cdot 47 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6016.a (trivial)

\(f(q)\)	\(=\)	\(q+0.904794 q^{3} +3.10347 q^{5} +1.70842 q^{7} -2.18135 q^{9} +O(q^{10})\)	\(q+0.904794 q^{3} +3.10347 q^{5} +1.70842 q^{7} -2.18135 q^{9} -6.05981 q^{11} +1.97561 q^{13} +2.80800 q^{15} +0.802172 q^{17} +2.24854 q^{19} +1.54576 q^{21} -2.79083 q^{23} +4.63153 q^{25} -4.68805 q^{27} +9.23320 q^{29} +7.61706 q^{31} -5.48288 q^{33} +5.30202 q^{35} +3.25627 q^{37} +1.78752 q^{39} +3.62554 q^{41} -0.635801 q^{43} -6.76975 q^{45} +1.00000 q^{47} -4.08131 q^{49} +0.725800 q^{51} -4.24499 q^{53} -18.8064 q^{55} +2.03446 q^{57} +2.01915 q^{59} +6.89875 q^{61} -3.72665 q^{63} +6.13125 q^{65} -6.92370 q^{67} -2.52512 q^{69} +11.3962 q^{71} +14.6621 q^{73} +4.19058 q^{75} -10.3527 q^{77} +12.3443 q^{79} +2.30233 q^{81} -0.818293 q^{83} +2.48952 q^{85} +8.35414 q^{87} +10.3807 q^{89} +3.37517 q^{91} +6.89187 q^{93} +6.97827 q^{95} -16.2464 q^{97} +13.2186 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 13 q + 4 q^{3} + 6 q^{5} + 2 q^{7} + 21 q^{9}+O(q^{10}) \) 13 * q + 4 * q^3 + 6 * q^5 + 2 * q^7 + 21 * q^9	\( 13 q + 4 q^{3} + 6 q^{5} + 2 q^{7} + 21 q^{9} + 10 q^{11} + 4 q^{13} - 14 q^{15} + 10 q^{17} + 8 q^{19} + 10 q^{21} - 18 q^{23} + 23 q^{25} + 16 q^{27} + 14 q^{29} - 4 q^{31} + 14 q^{33} + 14 q^{35} + 16 q^{37} - 12 q^{39} + 10 q^{41} + 12 q^{43} + 10 q^{45} + 13 q^{47} + 9 q^{49} + 22 q^{51} + 26 q^{53} + 2 q^{55} + 20 q^{57} + 30 q^{59} + 18 q^{61} - 12 q^{63} - 4 q^{65} + 4 q^{67} + 2 q^{69} - 36 q^{71} + 10 q^{73} + 38 q^{75} + 42 q^{77} + 21 q^{81} + 12 q^{83} + 4 q^{85} - 6 q^{87} + 50 q^{89} - 4 q^{91} + 52 q^{93} - 8 q^{95} - 10 q^{97} + 34 q^{99}+O(q^{100}) \) 13 * q + 4 * q^3 + 6 * q^5 + 2 * q^7 + 21 * q^9 + 10 * q^11 + 4 * q^13 - 14 * q^15 + 10 * q^17 + 8 * q^19 + 10 * q^21 - 18 * q^23 + 23 * q^25 + 16 * q^27 + 14 * q^29 - 4 * q^31 + 14 * q^33 + 14 * q^35 + 16 * q^37 - 12 * q^39 + 10 * q^41 + 12 * q^43 + 10 * q^45 + 13 * q^47 + 9 * q^49 + 22 * q^51 + 26 * q^53 + 2 * q^55 + 20 * q^57 + 30 * q^59 + 18 * q^61 - 12 * q^63 - 4 * q^65 + 4 * q^67 + 2 * q^69 - 36 * q^71 + 10 * q^73 + 38 * q^75 + 42 * q^77 + 21 * q^81 + 12 * q^83 + 4 * q^85 - 6 * q^87 + 50 * q^89 - 4 * q^91 + 52 * q^93 - 8 * q^95 - 10 * q^97 + 34 * q^99

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