LMFDB - Embedded newform 6016.2.a.r.1.13

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:	$14$
Coefficient field:	$\mathbb{Q}[x]/(x^{14} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{14} - 2 x^{13} - 30 x^{12} + 56 x^{11} + 331 x^{10} - 562 x^{9} - 1630 x^{8} + 2458 x^{7} + \cdots + 128 $ x^14 - 2x^13 - 30x^12 + 56x^11 + 331x^10 - 562x^9 - 1630x^8 + 2458x^7 + 3313x^6 - 4512x^5 - 1612x^4 + 2960x^3 - 256x^2 - 512*x + 128
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.13
Root			$-2.72539$ 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)

$f(q)$	$=$	$q+2.72539 q^{3} -2.56607 q^{5} +3.53774 q^{7} +4.42776 q^{9} +O(q^{10})$	$q+2.72539 q^{3} -2.56607 q^{5} +3.53774 q^{7} +4.42776 q^{9} -5.54788 q^{11} +4.78129 q^{13} -6.99354 q^{15} +6.97106 q^{17} -2.28996 q^{19} +9.64174 q^{21} +6.72560 q^{23} +1.58471 q^{25} +3.89119 q^{27} +4.09896 q^{29} -0.135235 q^{31} -15.1201 q^{33} -9.07809 q^{35} -4.46485 q^{37} +13.0309 q^{39} +8.40282 q^{41} -9.23904 q^{43} -11.3619 q^{45} +1.00000 q^{47} +5.51564 q^{49} +18.9989 q^{51} +7.73312 q^{53} +14.2362 q^{55} -6.24103 q^{57} +1.18234 q^{59} -3.33829 q^{61} +15.6643 q^{63} -12.2691 q^{65} -13.6353 q^{67} +18.3299 q^{69} +4.48245 q^{71} -7.96265 q^{73} +4.31894 q^{75} -19.6270 q^{77} -2.23209 q^{79} -2.67824 q^{81} +9.32504 q^{83} -17.8882 q^{85} +11.1713 q^{87} -2.69205 q^{89} +16.9150 q^{91} -0.368568 q^{93} +5.87619 q^{95} +12.9071 q^{97} -24.5647 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 14 q - 2 q^{3} + 6 q^{5} + 2 q^{7} + 22 q^{9}+O(q^{10}) $ 14 * q - 2 * q^3 + 6 * q^5 + 2 * q^7 + 22 * q^9	$ 14 q - 2 q^{3} + 6 q^{5} + 2 q^{7} + 22 q^{9} - 14 q^{11} + 4 q^{13} + 6 q^{15} + 8 q^{17} - 8 q^{19} + 6 q^{21} + 18 q^{23} + 22 q^{25} - 8 q^{27} + 22 q^{29} + 4 q^{31} + 2 q^{33} - 26 q^{35} + 6 q^{37} + 20 q^{39} + 16 q^{41} - 12 q^{43} + 30 q^{45} + 14 q^{47} + 34 q^{49} - 18 q^{51} + 20 q^{53} + 2 q^{55} - 4 q^{57} - 32 q^{59} + 12 q^{61} + 40 q^{65} - 16 q^{67} + 46 q^{69} + 16 q^{71} - 12 q^{73} + 16 q^{75} + 10 q^{77} + 16 q^{79} + 74 q^{81} - 14 q^{83} + 12 q^{85} + 14 q^{87} - 28 q^{91} - 16 q^{93} + 52 q^{95} + 28 q^{97} - 50 q^{99}+O(q^{100}) $ 14 * q - 2 * q^3 + 6 * q^5 + 2 * q^7 + 22 * q^9 - 14 * q^11 + 4 * q^13 + 6 * q^15 + 8 * q^17 - 8 * q^19 + 6 * q^21 + 18 * q^23 + 22 * q^25 - 8 * q^27 + 22 * q^29 + 4 * q^31 + 2 * q^33 - 26 * q^35 + 6 * q^37 + 20 * q^39 + 16 * q^41 - 12 * q^43 + 30 * q^45 + 14 * q^47 + 34 * q^49 - 18 * q^51 + 20 * q^53 + 2 * q^55 - 4 * q^57 - 32 * q^59 + 12 * q^61 + 40 * q^65 - 16 * q^67 + 46 * q^69 + 16 * q^71 - 12 * q^73 + 16 * q^75 + 10 * q^77 + 16 * q^79 + 74 * q^81 - 14 * q^83 + 12 * q^85 + 14 * q^87 - 28 * q^91 - 16 * q^93 + 52 * q^95 + 28 * q^97 - 50 * 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
$3$	2.72539	1.57351	0.786753	−	0.617268i	$-0.211761\pi$
$3$	2.72539	1.57351	0.786753	+	0.617268i	$0.211761\pi$
$4$	0	0
$5$	−2.56607	−1.14758	−0.573790	−	0.819002i	$-0.694528\pi$
$5$	−2.56607	−1.14758	−0.573790	+	0.819002i	$0.694528\pi$
$6$	0	0
$7$	3.53774	1.33714	0.668571	−	0.743648i	$-0.266906\pi$
$7$	3.53774	1.33714	0.668571	+	0.743648i	$0.266906\pi$
$8$	0	0
$9$	4.42776	1.47592
$10$	0	0
$11$	−5.54788	−1.67275	−0.836375	−	0.548158i	$-0.815329\pi$
$11$	−5.54788	−1.67275	−0.836375	+	0.548158i	$0.815329\pi$
$12$	0	0
$13$	4.78129	1.32609	0.663046	−	0.748579i	$-0.269263\pi$
$13$	4.78129	1.32609	0.663046	+	0.748579i	$0.269263\pi$
$14$	0	0
$15$	−6.99354	−1.80572
$16$	0	0
$17$	6.97106	1.69073	0.845365	−	0.534190i	$-0.179383\pi$
$17$	6.97106	1.69073	0.845365	+	0.534190i	$0.179383\pi$
$18$	0	0
$19$	−2.28996	−0.525352	−0.262676	−	0.964884i	$-0.584605\pi$
$19$	−2.28996	−0.525352	−0.262676	+	0.964884i	$0.584605\pi$
$20$	0	0
$21$	9.64174	2.10400
$22$	0	0
$23$	6.72560	1.40239	0.701193	−	0.712972i	$-0.252651\pi$
$23$	6.72560	1.40239	0.701193	+	0.712972i	$0.252651\pi$
$24$	0	0
$25$	1.58471	0.316941
$26$	0	0
$27$	3.89119	0.748861
$28$	0	0
$29$	4.09896	0.761157	0.380579	−	0.924749i	$-0.375725\pi$
$29$	4.09896	0.761157	0.380579	+	0.924749i	$0.375725\pi$
$30$	0	0
$31$	−0.135235	−0.0242889	−0.0121444	−	0.999926i	$-0.503866\pi$
$31$	−0.135235	−0.0242889	−0.0121444	+	0.999926i	$0.503866\pi$
$32$	0	0
$33$	−15.1201	−2.63208
$34$	0	0
$35$	−9.07809	−1.53448
$36$	0	0
$37$	−4.46485	−0.734017	−0.367008	−	0.930218i	$-0.619618\pi$
$37$	−4.46485	−0.734017	−0.367008	+	0.930218i	$0.619618\pi$
$38$	0	0
$39$	13.0309	2.08661
$40$	0	0
$41$	8.40282	1.31230	0.656150	−	0.754631i	$-0.272184\pi$
$41$	8.40282	1.31230	0.656150	+	0.754631i	$0.272184\pi$
$42$	0	0
$43$	−9.23904	−1.40894	−0.704470	−	0.709734i	$-0.748815\pi$
$43$	−9.23904	−1.40894	−0.704470	+	0.709734i	$0.748815\pi$
$44$	0	0
$45$	−11.3619	−1.69374
$46$	0	0
$47$	1.00000	0.145865
$47$	1.00000	0.145865
$48$	0	0
$49$	5.51564	0.787948
$50$	0	0
$51$	18.9989	2.66037
$52$	0	0
$53$	7.73312	1.06223	0.531113	−	0.847301i	$-0.321774\pi$
$53$	7.73312	1.06223	0.531113	+	0.847301i	$0.321774\pi$
$54$	0	0
$55$	14.2362	1.91961
$56$	0	0
$57$	−6.24103	−0.826645
$58$	0	0
$59$	1.18234	0.153928	0.0769641	−	0.997034i	$-0.475477\pi$
$59$	1.18234	0.153928	0.0769641	+	0.997034i	$0.475477\pi$
$60$	0	0
$61$	−3.33829	−0.427424	−0.213712	−	0.976897i	$-0.568555\pi$
$61$	−3.33829	−0.427424	−0.213712	+	0.976897i	$0.568555\pi$
$62$	0	0
$63$	15.6643	1.97351
$64$	0	0
$65$	−12.2691	−1.52180
$66$	0	0
$67$	−13.6353	−1.66582	−0.832912	−	0.553406i	$-0.813328\pi$
$67$	−13.6353	−1.66582	−0.832912	+	0.553406i	$0.813328\pi$
$68$	0	0
$69$	18.3299	2.20666
$70$	0	0
$71$	4.48245	0.531968	0.265984	−	0.963977i	$-0.414303\pi$
$71$	4.48245	0.531968	0.265984	+	0.963977i	$0.414303\pi$
$72$	0	0
$73$	−7.96265	−0.931957	−0.465979	−	0.884796i	$-0.654298\pi$
$73$	−7.96265	−0.931957	−0.465979	+	0.884796i	$0.654298\pi$
$74$	0	0
$75$	4.31894	0.498709
$76$	0	0
$77$	−19.6270	−2.23670
$78$	0	0
$79$	−2.23209	−0.251129	−0.125565	−	0.992085i	$-0.540074\pi$
$79$	−2.23209	−0.251129	−0.125565	+	0.992085i	$0.540074\pi$
$80$	0	0
$81$	−2.67824	−0.297582
$82$	0	0
$83$	9.32504	1.02356	0.511778	−	0.859118i	$-0.328987\pi$
$83$	9.32504	1.02356	0.511778	+	0.859118i	$0.328987\pi$
$84$	0	0
$85$	−17.8882	−1.94025
$86$	0	0
$87$	11.1713	1.19768
$88$	0	0
$89$	−2.69205	−0.285356	−0.142678	−	0.989769i	$-0.545571\pi$
$89$	−2.69205	−0.285356	−0.142678	+	0.989769i	$0.545571\pi$
$90$	0	0
$91$	16.9150	1.77317
$92$	0	0
$93$	−0.368568	−0.0382187
$94$	0	0
$95$	5.87619	0.602884
$96$	0	0
$97$	12.9071	1.31051	0.655257	−	0.755406i	$-0.272560\pi$
$97$	12.9071	1.31051	0.655257	+	0.755406i	$0.272560\pi$
$98$	0	0
$99$	−24.5647	−2.46884
$100$	0	0
$101$	−6.54339	−0.651092	−0.325546	−	0.945526i	$-0.605548\pi$
$101$	−6.54339	−0.651092	−0.325546	+	0.945526i	$0.605548\pi$
$102$	0	0
$103$	7.92273	0.780650	0.390325	−	0.920677i	$-0.372363\pi$
$103$	7.92273	0.780650	0.390325	+	0.920677i	$0.372363\pi$
$104$	0	0
$105$	−24.7414	−2.41451
$106$	0	0
$107$	15.6584	1.51376	0.756878	−	0.653556i	$-0.226724\pi$
$107$	15.6584	1.51376	0.756878	+	0.653556i	$0.226724\pi$
$108$	0	0
$109$	18.9108	1.81133	0.905663	−	0.423999i	$-0.139374\pi$
$109$	18.9108	1.81133	0.905663	+	0.423999i	$0.139374\pi$
$110$	0	0
$111$	−12.1685	−1.15498
$112$	0	0
$113$	−9.89209	−0.930570	−0.465285	−	0.885161i	$-0.654048\pi$
$113$	−9.89209	−0.930570	−0.465285	+	0.885161i	$0.654048\pi$
$114$	0	0
$115$	−17.2584	−1.60935
$116$	0	0
$117$	21.1704	1.95720
$118$	0	0
$119$	24.6618	2.26074
$120$	0	0
$121$	19.7790	1.79809
$122$	0	0
$123$	22.9010	2.06491
$124$	0	0
$125$	8.76388	0.783865
$126$	0	0
$127$	−5.20816	−0.462149	−0.231075	−	0.972936i	$-0.574224\pi$
$127$	−5.20816	−0.462149	−0.231075	+	0.972936i	$0.574224\pi$
$128$	0	0
$129$	−25.1800	−2.21697
$130$	0	0
$131$	18.1585	1.58651	0.793257	−	0.608887i	$-0.208384\pi$
$131$	18.1585	1.58651	0.793257	+	0.608887i	$0.208384\pi$
$132$	0	0
$133$	−8.10129	−0.702471
$134$	0	0
$135$	−9.98507	−0.859378
$136$	0	0
$137$	−7.87606	−0.672897	−0.336449	−	0.941702i	$-0.609226\pi$
$137$	−7.87606	−0.672897	−0.336449	+	0.941702i	$0.609226\pi$
$138$	0	0
$139$	1.90929	0.161944	0.0809718	−	0.996716i	$-0.474198\pi$
$139$	1.90929	0.161944	0.0809718	+	0.996716i	$0.474198\pi$
$140$	0	0
$141$	2.72539	0.229519
$142$	0	0
$143$	−26.5261	−2.21822
$144$	0	0
$145$	−10.5182	−0.873489
$146$	0	0
$147$	15.0323	1.23984
$148$	0	0
$149$	11.4321	0.936555	0.468277	−	0.883582i	$-0.344875\pi$
$149$	11.4321	0.936555	0.468277	+	0.883582i	$0.344875\pi$
$150$	0	0
$151$	7.11991	0.579410	0.289705	−	0.957116i	$-0.406443\pi$
$151$	7.11991	0.579410	0.289705	+	0.957116i	$0.406443\pi$
$152$	0	0
$153$	30.8661	2.49538
$154$	0	0
$155$	0.347022	0.0278734
$156$	0	0
$157$	10.7678	0.859364	0.429682	−	0.902980i	$-0.358626\pi$
$157$	10.7678	0.859364	0.429682	+	0.902980i	$0.358626\pi$
$158$	0	0
$159$	21.0758	1.67142
$160$	0	0
$161$	23.7935	1.87519
$162$	0	0
$163$	10.2755	0.804838	0.402419	−	0.915456i	$-0.368170\pi$
$163$	10.2755	0.804838	0.402419	+	0.915456i	$0.368170\pi$
$164$	0	0
$165$	38.7993	3.02052
$166$	0	0
$167$	−12.2816	−0.950379	−0.475190	−	0.879883i	$-0.657621\pi$
$167$	−12.2816	−0.950379	−0.475190	+	0.879883i	$0.657621\pi$
$168$	0	0
$169$	9.86076	0.758520
$170$	0	0
$171$	−10.1394	−0.775378
$172$	0	0
$173$	8.28864	0.630173	0.315087	−	0.949063i	$-0.397966\pi$
$173$	8.28864	0.630173	0.315087	+	0.949063i	$0.397966\pi$
$174$	0	0
$175$	5.60629	0.423795
$176$	0	0
$177$	3.22235	0.242207
$178$	0	0
$179$	−10.5228	−0.786513	−0.393256	−	0.919429i	$-0.628652\pi$
$179$	−10.5228	−0.786513	−0.393256	+	0.919429i	$0.628652\pi$
$180$	0	0
$181$	24.7875	1.84244	0.921219	−	0.389044i	$-0.127195\pi$
$181$	24.7875	1.84244	0.921219	+	0.389044i	$0.127195\pi$
$182$	0	0
$183$	−9.09813	−0.672553
$184$	0	0
$185$	11.4571	0.842343
$186$	0	0
$187$	−38.6746	−2.82817
$188$	0	0
$189$	13.7661	1.00133
$190$	0	0
$191$	6.20753	0.449161	0.224581	−	0.974456i	$-0.427899\pi$
$191$	6.20753	0.449161	0.224581	+	0.974456i	$0.427899\pi$
$192$	0	0
$193$	−9.30335	−0.669670	−0.334835	−	0.942277i	$-0.608681\pi$
$193$	−9.30335	−0.669670	−0.334835	+	0.942277i	$0.608681\pi$
$194$	0	0
$195$	−33.4382	−2.39456
$196$	0	0
$197$	0.994536	0.0708578	0.0354289	−	0.999372i	$-0.488720\pi$
$197$	0.994536	0.0708578	0.0354289	+	0.999372i	$0.488720\pi$
$198$	0	0
$199$	1.08753	0.0770932	0.0385466	−	0.999257i	$-0.487727\pi$
$199$	1.08753	0.0770932	0.0385466	+	0.999257i	$0.487727\pi$
$200$	0	0
$201$	−37.1617	−2.62118
$202$	0	0
$203$	14.5011	1.01778
$204$	0	0
$205$	−21.5622	−1.50597
$206$	0	0
$207$	29.7793	2.06981
$208$	0	0
$209$	12.7044	0.878783
$210$	0	0
$211$	−21.7843	−1.49970	−0.749848	−	0.661610i	$-0.769873\pi$
$211$	−21.7843	−1.49970	−0.749848	+	0.661610i	$0.769873\pi$
$212$	0	0
$213$	12.2164	0.837055
$214$	0	0
$215$	23.7080	1.61687
$216$	0	0
$217$	−0.478426	−0.0324777
$218$	0	0
$219$	−21.7013	−1.46644
$220$	0	0
$221$	33.3307	2.24206
$222$	0	0
$223$	0.489121	0.0327540	0.0163770	−	0.999866i	$-0.494787\pi$
$223$	0.489121	0.0327540	0.0163770	+	0.999866i	$0.494787\pi$
$224$	0	0
$225$	7.01669	0.467779
$226$	0	0
$227$	11.6681	0.774440	0.387220	−	0.921987i	$-0.373435\pi$
$227$	11.6681	0.774440	0.387220	+	0.921987i	$0.373435\pi$
$228$	0	0
$229$	−21.9528	−1.45068	−0.725340	−	0.688391i	$-0.758317\pi$
$229$	−21.9528	−1.45068	−0.725340	+	0.688391i	$0.758317\pi$
$230$	0	0
$231$	−53.4912	−3.51946
$232$	0	0
$233$	−0.399357	−0.0261627	−0.0130814	−	0.999914i	$-0.504164\pi$
$233$	−0.399357	−0.0261627	−0.0130814	+	0.999914i	$0.504164\pi$
$234$	0	0
$235$	−2.56607	−0.167392
$236$	0	0
$237$	−6.08331	−0.395153
$238$	0	0
$239$	−22.9093	−1.48188	−0.740941	−	0.671570i	$-0.765620\pi$
$239$	−22.9093	−1.48188	−0.740941	+	0.671570i	$0.765620\pi$
$240$	0	0
$241$	0.202708	0.0130576	0.00652878	−	0.999979i	$-0.497922\pi$
$241$	0.202708	0.0130576	0.00652878	+	0.999979i	$0.497922\pi$
$242$	0	0
$243$	−18.9728	−1.21711
$244$	0	0
$245$	−14.1535	−0.904234
$246$	0	0
$247$	−10.9490	−0.696666
$248$	0	0
$249$	25.4144	1.61057
$250$	0	0
$251$	−16.1616	−1.02011	−0.510055	−	0.860142i	$-0.670375\pi$
$251$	−16.1616	−1.02011	−0.510055	+	0.860142i	$0.670375\pi$
$252$	0	0
$253$	−37.3129	−2.34584
$254$	0	0
$255$	−48.7524	−3.05299
$256$	0	0
$257$	21.4646	1.33893	0.669463	−	0.742846i	$-0.266524\pi$
$257$	21.4646	1.33893	0.669463	+	0.742846i	$0.266524\pi$
$258$	0	0
$259$	−15.7955	−0.981484
$260$	0	0
$261$	18.1492	1.12341
$262$	0	0
$263$	−24.5213	−1.51205	−0.756024	−	0.654544i	$-0.772861\pi$
$263$	−24.5213	−1.51205	−0.756024	+	0.654544i	$0.772861\pi$
$264$	0	0
$265$	−19.8437	−1.21899
$266$	0	0
$267$	−7.33688	−0.449010
$268$	0	0
$269$	29.3632	1.79031	0.895154	−	0.445756i	$-0.147065\pi$
$269$	29.3632	1.79031	0.895154	+	0.445756i	$0.147065\pi$
$270$	0	0
$271$	10.4095	0.632332	0.316166	−	0.948704i	$-0.397604\pi$
$271$	10.4095	0.632332	0.316166	+	0.948704i	$0.397604\pi$
$272$	0	0
$273$	46.1000	2.79010
$274$	0	0
$275$	−8.79176	−0.530163
$276$	0	0
$277$	−6.25297	−0.375704	−0.187852	−	0.982197i	$-0.560153\pi$
$277$	−6.25297	−0.375704	−0.187852	+	0.982197i	$0.560153\pi$
$278$	0	0
$279$	−0.598786	−0.0358484
$280$	0	0
$281$	12.5471	0.748498	0.374249	−	0.927328i	$-0.377901\pi$
$281$	12.5471	0.748498	0.374249	+	0.927328i	$0.377901\pi$
$282$	0	0
$283$	26.6254	1.58272	0.791359	−	0.611351i	$-0.209374\pi$
$283$	26.6254	1.58272	0.791359	+	0.611351i	$0.209374\pi$
$284$	0	0
$285$	16.0149	0.948642
$286$	0	0
$287$	29.7270	1.75473
$288$	0	0
$289$	31.5956	1.85857
$290$	0	0
$291$	35.1768	2.06210
$292$	0	0
$293$	−15.6147	−0.912223	−0.456112	−	0.889923i	$-0.650758\pi$
$293$	−15.6147	−0.912223	−0.456112	+	0.889923i	$0.650758\pi$
$294$	0	0
$295$	−3.03398	−0.176645
$296$	0	0
$297$	−21.5879	−1.25266
$298$	0	0
$299$	32.1571	1.85969
$300$	0	0
$301$	−32.6854	−1.88395
$302$	0	0
$303$	−17.8333	−1.02450
$304$	0	0
$305$	8.56627	0.490503
$306$	0	0
$307$	21.7343	1.24044	0.620220	−	0.784428i	$-0.287043\pi$
$307$	21.7343	1.24044	0.620220	+	0.784428i	$0.287043\pi$
$308$	0	0
$309$	21.5925	1.22836
$310$	0	0
$311$	−23.7606	−1.34734	−0.673671	−	0.739031i	$-0.735284\pi$
$311$	−23.7606	−1.34734	−0.673671	+	0.739031i	$0.735284\pi$
$312$	0	0
$313$	−27.1199	−1.53291	−0.766454	−	0.642299i	$-0.777981\pi$
$313$	−27.1199	−1.53291	−0.766454	+	0.642299i	$0.777981\pi$
$314$	0	0
$315$	−40.1956	−2.26476
$316$	0	0
$317$	−28.1646	−1.58188	−0.790941	−	0.611892i	$-0.790409\pi$
$317$	−28.1646	−1.58188	−0.790941	+	0.611892i	$0.790409\pi$
$318$	0	0
$319$	−22.7405	−1.27323
$320$	0	0
$321$	42.6753	2.38190
$322$	0	0
$323$	−15.9634	−0.888229
$324$	0	0
$325$	7.57694	0.420293
$326$	0	0
$327$	51.5393	2.85013
$328$	0	0
$329$	3.53774	0.195042
$330$	0	0
$331$	−19.0966	−1.04965	−0.524823	−	0.851211i	$-0.675868\pi$
$331$	−19.0966	−1.04965	−0.524823	+	0.851211i	$0.675868\pi$
$332$	0	0
$333$	−19.7693	−1.08335
$334$	0	0
$335$	34.9892	1.91167
$336$	0	0
$337$	−0.205636	−0.0112017	−0.00560086	−	0.999984i	$-0.501783\pi$
$337$	−0.205636	−0.0112017	−0.00560086	+	0.999984i	$0.501783\pi$
$338$	0	0
$339$	−26.9598	−1.46426
$340$	0	0
$341$	0.750266	0.0406292
$342$	0	0
$343$	−5.25130	−0.283544
$344$	0	0
$345$	−47.0358	−2.53232
$346$	0	0
$347$	4.19495	0.225197	0.112598	−	0.993641i	$-0.464083\pi$
$347$	4.19495	0.225197	0.112598	+	0.993641i	$0.464083\pi$
$348$	0	0
$349$	9.91525	0.530751	0.265376	−	0.964145i	$-0.414504\pi$
$349$	9.91525	0.530751	0.265376	+	0.964145i	$0.414504\pi$
$350$	0	0
$351$	18.6049	0.993058
$352$	0	0
$353$	6.78825	0.361302	0.180651	−	0.983547i	$-0.442180\pi$
$353$	6.78825	0.361302	0.180651	+	0.983547i	$0.442180\pi$
$354$	0	0
$355$	−11.5023	−0.610477
$356$	0	0
$357$	67.2131	3.55729
$358$	0	0
$359$	19.8314	1.04666	0.523332	−	0.852129i	$-0.324689\pi$
$359$	19.8314	1.04666	0.523332	+	0.852129i	$0.324689\pi$
$360$	0	0
$361$	−13.7561	−0.724005
$362$	0	0
$363$	53.9055	2.82931
$364$	0	0
$365$	20.4327	1.06950
$366$	0	0
$367$	29.9617	1.56399	0.781995	−	0.623284i	$-0.214202\pi$
$367$	29.9617	1.56399	0.781995	+	0.623284i	$0.214202\pi$
$368$	0	0
$369$	37.2056	1.93685
$370$	0	0
$371$	27.3578	1.42035
$372$	0	0
$373$	13.1238	0.679523	0.339761	−	0.940512i	$-0.389654\pi$
$373$	13.1238	0.679523	0.339761	+	0.940512i	$0.389654\pi$
$374$	0	0
$375$	23.8850	1.23342
$376$	0	0
$377$	19.5983	1.00936
$378$	0	0
$379$	−0.570183	−0.0292883	−0.0146442	−	0.999893i	$-0.504662\pi$
$379$	−0.570183	−0.0292883	−0.0146442	+	0.999893i	$0.504662\pi$
$380$	0	0
$381$	−14.1943	−0.727194
$382$	0	0
$383$	−19.0326	−0.972521	−0.486260	−	0.873814i	$-0.661639\pi$
$383$	−19.0326	−0.972521	−0.486260	+	0.873814i	$0.661639\pi$
$384$	0	0
$385$	50.3642	2.56680
$386$	0	0
$387$	−40.9082	−2.07948
$388$	0	0
$389$	−28.6465	−1.45243	−0.726217	−	0.687466i	$-0.758723\pi$
$389$	−28.6465	−1.45243	−0.726217	+	0.687466i	$0.758723\pi$
$390$	0	0
$391$	46.8846	2.37105
$392$	0	0
$393$	49.4890	2.49639
$394$	0	0
$395$	5.72769	0.288191
$396$	0	0
$397$	−28.7525	−1.44304	−0.721522	−	0.692391i	$-0.756557\pi$
$397$	−28.7525	−1.44304	−0.721522	+	0.692391i	$0.756557\pi$
$398$	0	0
$399$	−22.0792	−1.10534
$400$	0	0
$401$	36.2271	1.80910	0.904549	−	0.426371i	$-0.140208\pi$
$401$	36.2271	1.80910	0.904549	+	0.426371i	$0.140208\pi$
$402$	0	0
$403$	−0.646597	−0.0322093
$404$	0	0
$405$	6.87255	0.341500
$406$	0	0
$407$	24.7705	1.22783
$408$	0	0
$409$	−3.76158	−0.185998	−0.0929991	−	0.995666i	$-0.529645\pi$
$409$	−3.76158	−0.185998	−0.0929991	+	0.995666i	$0.529645\pi$
$410$	0	0
$411$	−21.4653	−1.05881
$412$	0	0
$413$	4.18283	0.205824
$414$	0	0
$415$	−23.9287	−1.17461
$416$	0	0
$417$	5.20356	0.254819
$418$	0	0
$419$	−7.94006	−0.387897	−0.193949	−	0.981012i	$-0.562130\pi$
$419$	−7.94006	−0.387897	−0.193949	+	0.981012i	$0.562130\pi$
$420$	0	0
$421$	−0.669568	−0.0326328	−0.0163164	−	0.999867i	$-0.505194\pi$
$421$	−0.669568	−0.0326328	−0.0163164	+	0.999867i	$0.505194\pi$
$422$	0	0
$423$	4.42776	0.215285
$424$	0	0
$425$	11.0471	0.535862
$426$	0	0
$427$	−11.8100	−0.571526
$428$	0	0
$429$	−72.2939	−3.49038
$430$	0	0
$431$	−9.44293	−0.454850	−0.227425	−	0.973796i	$-0.573031\pi$
$431$	−9.44293	−0.454850	−0.227425	+	0.973796i	$0.573031\pi$
$432$	0	0
$433$	−24.8676	−1.19506	−0.597529	−	0.801847i	$-0.703851\pi$
$433$	−24.8676	−1.19506	−0.597529	+	0.801847i	$0.703851\pi$
$434$	0	0
$435$	−28.6662	−1.37444
$436$	0	0
$437$	−15.4014	−0.736747
$438$	0	0
$439$	−27.4610	−1.31064	−0.655321	−	0.755350i	$-0.727467\pi$
$439$	−27.4610	−1.31064	−0.655321	+	0.755350i	$0.727467\pi$
$440$	0	0
$441$	24.4219	1.16295
$442$	0	0
$443$	−40.4394	−1.92133	−0.960666	−	0.277706i	$-0.910426\pi$
$443$	−40.4394	−1.92133	−0.960666	+	0.277706i	$0.910426\pi$
$444$	0	0
$445$	6.90797	0.327469
$446$	0	0
$447$	31.1570	1.47367
$448$	0	0
$449$	17.9249	0.845929	0.422965	−	0.906146i	$-0.360989\pi$
$449$	17.9249	0.845929	0.422965	+	0.906146i	$0.360989\pi$
$450$	0	0
$451$	−46.6178	−2.19515
$452$	0	0
$453$	19.4045	0.911705
$454$	0	0
$455$	−43.4050	−2.03486
$456$	0	0
$457$	−32.8408	−1.53623	−0.768114	−	0.640314i	$-0.778804\pi$
$457$	−32.8408	−1.53623	−0.768114	+	0.640314i	$0.778804\pi$
$458$	0	0
$459$	27.1257	1.26612
$460$	0	0
$461$	−28.0976	−1.30864	−0.654318	−	0.756219i	$-0.727044\pi$
$461$	−28.0976	−1.30864	−0.654318	+	0.756219i	$0.727044\pi$
$462$	0	0
$463$	−29.7699	−1.38353	−0.691763	−	0.722125i	$-0.743166\pi$
$463$	−29.7699	−1.38353	−0.691763	+	0.722125i	$0.743166\pi$
$464$	0	0
$465$	0.945769	0.0438590
$466$	0	0
$467$	38.9428	1.80206	0.901030	−	0.433758i	$-0.142813\pi$
$467$	38.9428	1.80206	0.901030	+	0.433758i	$0.142813\pi$
$468$	0	0
$469$	−48.2384	−2.22744
$470$	0	0
$471$	29.3465	1.35221
$472$	0	0
$473$	51.2571	2.35680
$474$	0	0
$475$	−3.62891	−0.166506
$476$	0	0
$477$	34.2404	1.56776
$478$	0	0
$479$	16.0847	0.734929	0.367465	−	0.930038i	$-0.380226\pi$
$479$	16.0847	0.734929	0.367465	+	0.930038i	$0.380226\pi$
$480$	0	0
$481$	−21.3478	−0.973374
$482$	0	0
$483$	64.8465	2.95062
$484$	0	0
$485$	−33.1204	−1.50392
$486$	0	0
$487$	8.85825	0.401406	0.200703	−	0.979652i	$-0.435677\pi$
$487$	8.85825	0.401406	0.200703	+	0.979652i	$0.435677\pi$
$488$	0	0
$489$	28.0047	1.26642
$490$	0	0
$491$	−17.8396	−0.805090	−0.402545	−	0.915400i	$-0.631874\pi$
$491$	−17.8396	−0.805090	−0.402545	+	0.915400i	$0.631874\pi$
$492$	0	0
$493$	28.5741	1.28691
$494$	0	0
$495$	63.0346	2.83320
$496$	0	0
$497$	15.8577	0.711317
$498$	0	0
$499$	−37.1749	−1.66418	−0.832089	−	0.554642i	$-0.812855\pi$
$499$	−37.1749	−1.66418	−0.832089	+	0.554642i	$0.812855\pi$
$500$	0	0
$501$	−33.4722	−1.49543
$502$	0	0
$503$	3.37075	0.150294	0.0751472	−	0.997172i	$-0.476057\pi$
$503$	3.37075	0.150294	0.0751472	+	0.997172i	$0.476057\pi$
$504$	0	0
$505$	16.7908	0.747180
$506$	0	0
$507$	26.8744	1.19354
$508$	0	0
$509$	−7.69801	−0.341208	−0.170604	−	0.985340i	$-0.554572\pi$
$509$	−7.69801	−0.341208	−0.170604	+	0.985340i	$0.554572\pi$
$510$	0	0
$511$	−28.1698	−1.24616
$512$	0	0
$513$	−8.91067	−0.393416
$514$	0	0
$515$	−20.3303	−0.895858
$516$	0	0
$517$	−5.54788	−0.243996
$518$	0	0
$519$	22.5898	0.991581
$520$	0	0
$521$	−32.3454	−1.41708	−0.708538	−	0.705672i	$-0.750645\pi$
$521$	−32.3454	−1.41708	−0.708538	+	0.705672i	$0.750645\pi$
$522$	0	0
$523$	−21.1393	−0.924357	−0.462178	−	0.886787i	$-0.652932\pi$
$523$	−21.1393	−0.924357	−0.462178	+	0.886787i	$0.652932\pi$
$524$	0	0
$525$	15.2793	0.666844
$526$	0	0
$527$	−0.942729	−0.0410659
$528$	0	0
$529$	22.2337	0.966684
$530$	0	0
$531$	5.23514	0.227186
$532$	0	0
$533$	40.1763	1.74023
$534$	0	0
$535$	−40.1806	−1.73716
$536$	0	0
$537$	−28.6788	−1.23758
$538$	0	0
$539$	−30.6001	−1.31804
$540$	0	0
$541$	−37.4431	−1.60980	−0.804902	−	0.593408i	$-0.797782\pi$
$541$	−37.4431	−1.60980	−0.804902	+	0.593408i	$0.797782\pi$
$542$	0	0
$543$	67.5556	2.89909
$544$	0	0
$545$	−48.5264	−2.07864
$546$	0	0
$547$	30.9271	1.32235	0.661174	−	0.750233i	$-0.270059\pi$
$547$	30.9271	1.32235	0.661174	+	0.750233i	$0.270059\pi$
$548$	0	0
$549$	−14.7811	−0.630843
$550$	0	0
$551$	−9.38644	−0.399876
$552$	0	0
$553$	−7.89655	−0.335796
$554$	0	0
$555$	31.2251	1.32543
$556$	0	0
$557$	7.10354	0.300987	0.150493	−	0.988611i	$-0.451914\pi$
$557$	7.10354	0.300987	0.150493	+	0.988611i	$0.451914\pi$
$558$	0	0
$559$	−44.1745	−1.86838
$560$	0	0
$561$	−105.403	−4.45014
$562$	0	0
$563$	29.2483	1.23267	0.616335	−	0.787484i	$-0.288617\pi$
$563$	29.2483	1.23267	0.616335	+	0.787484i	$0.288617\pi$
$564$	0	0
$565$	25.3838	1.06790
$566$	0	0
$567$	−9.47494	−0.397910
$568$	0	0
$569$	17.8378	0.747797	0.373899	−	0.927470i	$-0.378021\pi$
$569$	17.8378	0.747797	0.373899	+	0.927470i	$0.378021\pi$
$570$	0	0
$571$	14.3954	0.602427	0.301214	−	0.953557i	$-0.402608\pi$
$571$	14.3954	0.602427	0.301214	+	0.953557i	$0.402608\pi$
$572$	0	0
$573$	16.9180	0.706758
$574$	0	0
$575$	10.6581	0.444474
$576$	0	0
$577$	−41.4345	−1.72494	−0.862469	−	0.506109i	$-0.831083\pi$
$577$	−41.4345	−1.72494	−0.862469	+	0.506109i	$0.831083\pi$
$578$	0	0
$579$	−25.3553	−1.05373
$580$	0	0
$581$	32.9896	1.36864
$582$	0	0
$583$	−42.9024	−1.77684
$584$	0	0
$585$	−54.3247	−2.24605
$586$	0	0
$587$	−16.7858	−0.692823	−0.346411	−	0.938083i	$-0.612600\pi$
$587$	−16.7858	−0.692823	−0.346411	+	0.938083i	$0.612600\pi$
$588$	0	0
$589$	0.309682	0.0127602
$590$	0	0
$591$	2.71050	0.111495
$592$	0	0
$593$	12.4212	0.510076	0.255038	−	0.966931i	$-0.417912\pi$
$593$	12.4212	0.510076	0.255038	+	0.966931i	$0.417912\pi$
$594$	0	0
$595$	−63.2839	−2.59439
$596$	0	0
$597$	2.96395	0.121307
$598$	0	0
$599$	−27.7800	−1.13506	−0.567530	−	0.823353i	$-0.692101\pi$
$599$	−27.7800	−1.13506	−0.567530	+	0.823353i	$0.692101\pi$
$600$	0	0
$601$	8.30844	0.338908	0.169454	−	0.985538i	$-0.445800\pi$
$601$	8.30844	0.338908	0.169454	+	0.985538i	$0.445800\pi$
$602$	0	0
$603$	−60.3740	−2.45862
$604$	0	0
$605$	−50.7543	−2.06345
$606$	0	0
$607$	7.22740	0.293351	0.146675	−	0.989185i	$-0.453143\pi$
$607$	7.22740	0.293351	0.146675	+	0.989185i	$0.453143\pi$
$608$	0	0
$609$	39.5211	1.60147
$610$	0	0
$611$	4.78129	0.193430
$612$	0	0
$613$	28.9441	1.16904	0.584521	−	0.811379i	$-0.301283\pi$
$613$	28.9441	1.16904	0.584521	+	0.811379i	$0.301283\pi$
$614$	0	0
$615$	−58.7654	−2.36965
$616$	0	0
$617$	−0.993677	−0.0400039	−0.0200020	−	0.999800i	$-0.506367\pi$
$617$	−0.993677	−0.0400039	−0.0200020	+	0.999800i	$0.506367\pi$
$618$	0	0
$619$	17.0701	0.686107	0.343053	−	0.939316i	$-0.388539\pi$
$619$	17.0701	0.686107	0.343053	+	0.939316i	$0.388539\pi$
$620$	0	0
$621$	26.1706	1.05019
$622$	0	0
$623$	−9.52377	−0.381562
$624$	0	0
$625$	−30.4122	−1.21649
$626$	0	0
$627$	34.6245	1.38277
$628$	0	0
$629$	−31.1247	−1.24102
$630$	0	0
$631$	−12.2131	−0.486196	−0.243098	−	0.970002i	$-0.578164\pi$
$631$	−12.2131	−0.486196	−0.243098	+	0.970002i	$0.578164\pi$
$632$	0	0
$633$	−59.3709	−2.35978
$634$	0	0
$635$	13.3645	0.530354
$636$	0	0
$637$	26.3719	1.04489
$638$	0	0
$639$	19.8472	0.785142
$640$	0	0
$641$	5.15272	0.203520	0.101760	−	0.994809i	$-0.467553\pi$
$641$	5.15272	0.203520	0.101760	+	0.994809i	$0.467553\pi$
$642$	0	0
$643$	−0.348024	−0.0137247	−0.00686237	−	0.999976i	$-0.502184\pi$
$643$	−0.348024	−0.0137247	−0.00686237	+	0.999976i	$0.502184\pi$
$644$	0	0
$645$	64.6136	2.54416
$646$	0	0
$647$	12.2139	0.480179	0.240090	−	0.970751i	$-0.422823\pi$
$647$	12.2139	0.480179	0.240090	+	0.970751i	$0.422823\pi$
$648$	0	0
$649$	−6.55951	−0.257483
$650$	0	0
$651$	−1.30390	−0.0511038
$652$	0	0
$653$	−34.4277	−1.34726	−0.673631	−	0.739068i	$-0.735266\pi$
$653$	−34.4277	−1.34726	−0.673631	+	0.739068i	$0.735266\pi$
$654$	0	0
$655$	−46.5959	−1.82065
$656$	0	0
$657$	−35.2567	−1.37549
$658$	0	0
$659$	−17.8582	−0.695655	−0.347828	−	0.937559i	$-0.613081\pi$
$659$	−17.8582	−0.695655	−0.347828	+	0.937559i	$0.613081\pi$
$660$	0	0
$661$	−39.7602	−1.54649	−0.773246	−	0.634106i	$-0.781368\pi$
$661$	−39.7602	−1.54649	−0.773246	+	0.634106i	$0.781368\pi$
$662$	0	0
$663$	90.8391	3.52790
$664$	0	0
$665$	20.7885	0.806142
$666$	0	0
$667$	27.5680	1.06744
$668$	0	0
$669$	1.33305	0.0515385
$670$	0	0
$671$	18.5204	0.714973
$672$	0	0
$673$	−18.9134	−0.729056	−0.364528	−	0.931192i	$-0.618770\pi$
$673$	−18.9134	−0.729056	−0.364528	+	0.931192i	$0.618770\pi$
$674$	0	0
$675$	6.16640	0.237345
$676$	0	0
$677$	15.7499	0.605316	0.302658	−	0.953099i	$-0.402126\pi$
$677$	15.7499	0.605316	0.302658	+	0.953099i	$0.402126\pi$
$678$	0	0
$679$	45.6619	1.75234
$680$	0	0
$681$	31.8002	1.21858
$682$	0	0
$683$	−47.0032	−1.79853	−0.899263	−	0.437408i	$-0.855897\pi$
$683$	−47.0032	−1.79853	−0.899263	+	0.437408i	$0.855897\pi$
$684$	0	0
$685$	20.2105	0.772204
$686$	0	0
$687$	−59.8299	−2.28265
$688$	0	0
$689$	36.9743	1.40861
$690$	0	0
$691$	−18.4220	−0.700805	−0.350402	−	0.936599i	$-0.613955\pi$
$691$	−18.4220	−0.700805	−0.350402	+	0.936599i	$0.613955\pi$
$692$	0	0
$693$	−86.9035	−3.30119
$694$	0	0
$695$	−4.89936	−0.185843
$696$	0	0
$697$	58.5765	2.21874
$698$	0	0
$699$	−1.08840	−0.0411672
$700$	0	0
$701$	50.1385	1.89370	0.946852	−	0.321670i	$-0.104244\pi$
$701$	50.1385	1.89370	0.946852	+	0.321670i	$0.104244\pi$
$702$	0	0
$703$	10.2243	0.385617
$704$	0	0
$705$	−6.99354	−0.263392
$706$	0	0
$707$	−23.1488	−0.870602
$708$	0	0
$709$	−5.44951	−0.204661	−0.102330	−	0.994750i	$-0.532630\pi$
$709$	−5.44951	−0.204661	−0.102330	+	0.994750i	$0.532630\pi$
$710$	0	0
$711$	−9.88313	−0.370647
$712$	0	0
$713$	−0.909535	−0.0340624
$714$	0	0
$715$	68.0677	2.54559
$716$	0	0
$717$	−62.4369	−2.33175
$718$	0	0
$719$	39.9103	1.48840	0.744202	−	0.667954i	$-0.232830\pi$
$719$	39.9103	1.48840	0.744202	+	0.667954i	$0.232830\pi$
$720$	0	0
$721$	28.0286	1.04384
$722$	0	0
$723$	0.552458	0.0205461
$724$	0	0
$725$	6.49564	0.241242
$726$	0	0
$727$	−16.2208	−0.601597	−0.300798	−	0.953688i	$-0.597253\pi$
$727$	−16.2208	−0.601597	−0.300798	+	0.953688i	$0.597253\pi$
$728$	0	0
$729$	−43.6737	−1.61754
$730$	0	0
$731$	−64.4058	−2.38214
$732$	0	0
$733$	33.6784	1.24394	0.621970	−	0.783041i	$-0.286332\pi$
$733$	33.6784	1.24394	0.621970	+	0.783041i	$0.286332\pi$
$734$	0	0
$735$	−38.5738	−1.42282
$736$	0	0
$737$	75.6473	2.78650
$738$	0	0
$739$	1.88009	0.0691603	0.0345802	−	0.999402i	$-0.488991\pi$
$739$	1.88009	0.0691603	0.0345802	+	0.999402i	$0.488991\pi$
$740$	0	0
$741$	−29.8402	−1.09621
$742$	0	0
$743$	48.7595	1.78881	0.894406	−	0.447256i	$-0.147599\pi$
$743$	48.7595	1.78881	0.894406	+	0.447256i	$0.147599\pi$
$744$	0	0
$745$	−29.3356	−1.07477
$746$	0	0
$747$	41.2890	1.51068
$748$	0	0
$749$	55.3955	2.02411
$750$	0	0
$751$	−12.8423	−0.468621	−0.234311	−	0.972162i	$-0.575283\pi$
$751$	−12.8423	−0.468621	−0.234311	+	0.972162i	$0.575283\pi$
$752$	0	0
$753$	−44.0467	−1.60515
$754$	0	0
$755$	−18.2702	−0.664920
$756$	0	0
$757$	13.9009	0.505237	0.252618	−	0.967566i	$-0.418708\pi$
$757$	13.9009	0.505237	0.252618	+	0.967566i	$0.418708\pi$
$758$	0	0
$759$	−101.692	−3.69119
$760$	0	0
$761$	−26.8676	−0.973948	−0.486974	−	0.873416i	$-0.661899\pi$
$761$	−26.8676	−0.973948	−0.486974	+	0.873416i	$0.661899\pi$
$762$	0	0
$763$	66.9016	2.42200
$764$	0	0
$765$	−79.2046	−2.86365
$766$	0	0
$767$	5.65314	0.204123
$768$	0	0
$769$	31.9827	1.15333	0.576663	−	0.816982i	$-0.304355\pi$
$769$	31.9827	1.15333	0.576663	+	0.816982i	$0.304355\pi$
$770$	0	0
$771$	58.4994	2.10681
$772$	0	0
$773$	15.9661	0.574259	0.287130	−	0.957892i	$-0.407299\pi$
$773$	15.9661	0.574259	0.287130	+	0.957892i	$0.407299\pi$
$774$	0	0
$775$	−0.214307	−0.00769815
$776$	0	0
$777$	−43.0489	−1.54437
$778$	0	0
$779$	−19.2421	−0.689420
$780$	0	0
$781$	−24.8681	−0.889850
$782$	0	0
$783$	15.9498	0.570001
$784$	0	0
$785$	−27.6309	−0.986190
$786$	0	0
$787$	−6.56129	−0.233885	−0.116942	−	0.993139i	$-0.537309\pi$
$787$	−6.56129	−0.233885	−0.116942	+	0.993139i	$0.537309\pi$
$788$	0	0
$789$	−66.8301	−2.37922
$790$	0	0
$791$	−34.9957	−1.24430
$792$	0	0
$793$	−15.9613	−0.566803
$794$	0	0
$795$	−54.0819	−1.91809
$796$	0	0
$797$	23.7535	0.841392	0.420696	−	0.907202i	$-0.361786\pi$
$797$	23.7535	0.841392	0.420696	+	0.907202i	$0.361786\pi$
$798$	0	0
$799$	6.97106	0.246618
$800$	0	0
$801$	−11.9197	−0.421163
$802$	0	0
$803$	44.1758	1.55893
$804$	0	0
$805$	−61.0557	−2.15193
$806$	0	0
$807$	80.0263	2.81706
$808$	0	0
$809$	−27.1141	−0.953282	−0.476641	−	0.879098i	$-0.658146\pi$
$809$	−27.1141	−0.953282	−0.476641	+	0.879098i	$0.658146\pi$
$810$	0	0
$811$	−30.1414	−1.05841	−0.529204	−	0.848495i	$-0.677509\pi$
$811$	−30.1414	−1.05841	−0.529204	+	0.848495i	$0.677509\pi$
$812$	0	0
$813$	28.3699	0.994977
$814$	0	0
$815$	−26.3676	−0.923616
$816$	0	0
$817$	21.1570	0.740190
$818$	0	0
$819$	74.8955	2.61706
$820$	0	0
$821$	6.13912	0.214257	0.107128	−	0.994245i	$-0.465834\pi$
$821$	6.13912	0.214257	0.107128	+	0.994245i	$0.465834\pi$
$822$	0	0
$823$	23.9308	0.834176	0.417088	−	0.908866i	$-0.363051\pi$
$823$	23.9308	0.834176	0.417088	+	0.908866i	$0.363051\pi$
$824$	0	0
$825$	−23.9610	−0.834215
$826$	0	0
$827$	−12.5169	−0.435256	−0.217628	−	0.976032i	$-0.569832\pi$
$827$	−12.5169	−0.435256	−0.217628	+	0.976032i	$0.569832\pi$
$828$	0	0
$829$	−14.3610	−0.498780	−0.249390	−	0.968403i	$-0.580230\pi$
$829$	−14.3610	−0.498780	−0.249390	+	0.968403i	$0.580230\pi$
$830$	0	0
$831$	−17.0418	−0.591173
$832$	0	0
$833$	38.4498	1.33221
$834$	0	0
$835$	31.5154	1.09064
$836$	0	0
$837$	−0.526225	−0.0181890
$838$	0	0
$839$	−40.7592	−1.40716	−0.703582	−	0.710614i	$-0.748417\pi$
$839$	−40.7592	−1.40716	−0.703582	+	0.710614i	$0.748417\pi$
$840$	0	0
$841$	−12.1985	−0.420640
$842$	0	0
$843$	34.1958	1.17777
$844$	0	0
$845$	−25.3034	−0.870463
$846$	0	0
$847$	69.9730	2.40430
$848$	0	0
$849$	72.5648	2.49042
$850$	0	0
$851$	−30.0288	−1.02937
$852$	0	0
$853$	13.3611	0.457475	0.228737	−	0.973488i	$-0.426540\pi$
$853$	13.3611	0.457475	0.228737	+	0.973488i	$0.426540\pi$
$854$	0	0
$855$	26.0183	0.889808
$856$	0	0
$857$	−16.5375	−0.564911	−0.282455	−	0.959280i	$-0.591149\pi$
$857$	−16.5375	−0.564911	−0.282455	+	0.959280i	$0.591149\pi$
$858$	0	0
$859$	−57.8299	−1.97313	−0.986565	−	0.163368i	$-0.947764\pi$
$859$	−57.8299	−1.97313	−0.986565	+	0.163368i	$0.947764\pi$
$860$	0	0
$861$	81.0178	2.76108
$862$	0	0
$863$	−38.6137	−1.31443	−0.657214	−	0.753704i	$-0.728265\pi$
$863$	−38.6137	−1.31443	−0.657214	+	0.753704i	$0.728265\pi$
$864$	0	0
$865$	−21.2692	−0.723175
$866$	0	0
$867$	86.1104	2.92446
$868$	0	0
$869$	12.3834	0.420076
$870$	0	0
$871$	−65.1946	−2.20903
$872$	0	0
$873$	57.1493	1.93421
$874$	0	0
$875$	31.0044	1.04814
$876$	0	0
$877$	23.1158	0.780565	0.390282	−	0.920695i	$-0.372377\pi$
$877$	23.1158	0.780565	0.390282	+	0.920695i	$0.372377\pi$
$878$	0	0
$879$	−42.5563	−1.43539
$880$	0	0
$881$	7.97781	0.268779	0.134390	−	0.990929i	$-0.457093\pi$
$881$	7.97781	0.268779	0.134390	+	0.990929i	$0.457093\pi$
$882$	0	0
$883$	4.09531	0.137818	0.0689090	−	0.997623i	$-0.478048\pi$
$883$	4.09531	0.137818	0.0689090	+	0.997623i	$0.478048\pi$
$884$	0	0
$885$	−8.26878	−0.277952
$886$	0	0
$887$	−39.1984	−1.31615	−0.658077	−	0.752950i	$-0.728630\pi$
$887$	−39.1984	−1.31615	−0.658077	+	0.752950i	$0.728630\pi$
$888$	0	0
$889$	−18.4251	−0.617959
$890$	0	0
$891$	14.8586	0.497781
$892$	0	0
$893$	−2.28996	−0.0766305
$894$	0	0
$895$	27.0023	0.902587
$896$	0	0
$897$	87.6406	2.92624
$898$	0	0
$899$	−0.554321	−0.0184877
$900$	0	0
$901$	53.9080	1.79594
$902$	0	0
$903$	−89.0804	−2.96441
$904$	0	0
$905$	−63.6063	−2.11435
$906$	0	0
$907$	−13.2763	−0.440831	−0.220415	−	0.975406i	$-0.570741\pi$
$907$	−13.2763	−0.440831	−0.220415	+	0.975406i	$0.570741\pi$
$908$	0	0
$909$	−28.9725	−0.960959
$910$	0	0
$911$	25.2642	0.837041	0.418520	−	0.908207i	$-0.362549\pi$
$911$	25.2642	0.837041	0.418520	+	0.908207i	$0.362549\pi$
$912$	0	0
$913$	−51.7342	−1.71215
$914$	0	0
$915$	23.3464	0.771809
$916$	0	0
$917$	64.2401	2.12139
$918$	0	0
$919$	−24.8490	−0.819692	−0.409846	−	0.912155i	$-0.634418\pi$
$919$	−24.8490	−0.819692	−0.409846	+	0.912155i	$0.634418\pi$
$920$	0	0
$921$	59.2344	1.95184
$922$	0	0
$923$	21.4319	0.705439
$924$	0	0
$925$	−7.07547	−0.232640
$926$	0	0
$927$	35.0799	1.15218
$928$	0	0
$929$	−38.4717	−1.26222	−0.631108	−	0.775695i	$-0.717400\pi$
$929$	−38.4717	−1.26222	−0.631108	+	0.775695i	$0.717400\pi$
$930$	0	0
$931$	−12.6306	−0.413950
$932$	0	0
$933$	−64.7570	−2.12005
$934$	0	0
$935$	99.2417	3.24555
$936$	0	0
$937$	30.1618	0.985343	0.492671	−	0.870215i	$-0.336020\pi$
$937$	30.1618	0.985343	0.492671	+	0.870215i	$0.336020\pi$
$938$	0	0
$939$	−73.9124	−2.41204
$940$	0	0
$941$	44.5105	1.45100	0.725500	−	0.688222i	$-0.241609\pi$
$941$	44.5105	1.45100	0.725500	+	0.688222i	$0.241609\pi$
$942$	0	0
$943$	56.5140	1.84035
$944$	0	0
$945$	−35.3246	−1.14911
$946$	0	0
$947$	−19.4052	−0.630585	−0.315292	−	0.948995i	$-0.602103\pi$
$947$	−19.4052	−0.630585	−0.315292	+	0.948995i	$0.602103\pi$
$948$	0	0
$949$	−38.0717	−1.23586
$950$	0	0
$951$	−76.7596	−2.48910
$952$	0	0
$953$	44.3199	1.43566	0.717831	−	0.696218i	$-0.245135\pi$
$953$	44.3199	1.43566	0.717831	+	0.696218i	$0.245135\pi$
$954$	0	0
$955$	−15.9290	−0.515449
$956$	0	0
$957$	−61.9768	−2.00343
$958$	0	0
$959$	−27.8635	−0.899759
$960$	0	0
$961$	−30.9817	−0.999410
$962$	0	0
$963$	69.3317	2.23418
$964$	0	0
$965$	23.8730	0.768500
$966$	0	0
$967$	−28.9358	−0.930513	−0.465256	−	0.885176i	$-0.654038\pi$
$967$	−28.9358	−0.930513	−0.465256	+	0.885176i	$0.654038\pi$
$968$	0	0
$969$	−43.5066	−1.39763
$970$	0	0
$971$	14.4540	0.463850	0.231925	−	0.972734i	$-0.425498\pi$
$971$	14.4540	0.463850	0.231925	+	0.972734i	$0.425498\pi$
$972$	0	0
$973$	6.75457	0.216542
$974$	0	0
$975$	20.6501	0.661334
$976$	0	0
$977$	−36.0874	−1.15454	−0.577269	−	0.816554i	$-0.695882\pi$
$977$	−36.0874	−1.15454	−0.577269	+	0.816554i	$0.695882\pi$
$978$	0	0
$979$	14.9352	0.477330
$980$	0	0
$981$	83.7324	2.67337
$982$	0	0
$983$	38.4981	1.22790	0.613949	−	0.789346i	$-0.289580\pi$
$983$	38.4981	1.22790	0.613949	+	0.789346i	$0.289580\pi$
$984$	0	0
$985$	−2.55205	−0.0813150
$986$	0	0
$987$	9.64174	0.306900
$988$	0	0
$989$	−62.1381	−1.97588
$990$	0	0
$991$	−22.3506	−0.709989	−0.354995	−	0.934868i	$-0.615517\pi$
$991$	−22.3506	−0.709989	−0.354995	+	0.934868i	$0.615517\pi$
$992$	0	0
$993$	−52.0458	−1.65162
$994$	0	0
$995$	−2.79068	−0.0884706
$996$	0	0
$997$	33.1677	1.05043	0.525216	−	0.850969i	$-0.323985\pi$
$997$	33.1677	1.05043	0.525216	+	0.850969i	$0.323985\pi$
$998$	0	0
$999$	−17.3736	−0.549676

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6016.2.a.r.1.13	yes	14
4.3	odd	2		6016.2.a.t.1.2	yes	14
8.3	odd	2		6016.2.a.q.1.13	✓	14
8.5	even	2		6016.2.a.s.1.2	yes	14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6016.2.a.q.1.13	✓	14	8.3	odd	2
6016.2.a.r.1.13	yes	14	1.1	even	1	trivial
6016.2.a.s.1.2	yes	14	8.5	even	2
6016.2.a.t.1.2	yes	14	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:	\( N \)	\(=\)	\( 6016 = 2^{7} \cdot 47 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6016.a (trivial)

\(f(q)\)	\(=\)	\(q+2.72539 q^{3} -2.56607 q^{5} +3.53774 q^{7} +4.42776 q^{9} +O(q^{10})\)	\(q+2.72539 q^{3} -2.56607 q^{5} +3.53774 q^{7} +4.42776 q^{9} -5.54788 q^{11} +4.78129 q^{13} -6.99354 q^{15} +6.97106 q^{17} -2.28996 q^{19} +9.64174 q^{21} +6.72560 q^{23} +1.58471 q^{25} +3.89119 q^{27} +4.09896 q^{29} -0.135235 q^{31} -15.1201 q^{33} -9.07809 q^{35} -4.46485 q^{37} +13.0309 q^{39} +8.40282 q^{41} -9.23904 q^{43} -11.3619 q^{45} +1.00000 q^{47} +5.51564 q^{49} +18.9989 q^{51} +7.73312 q^{53} +14.2362 q^{55} -6.24103 q^{57} +1.18234 q^{59} -3.33829 q^{61} +15.6643 q^{63} -12.2691 q^{65} -13.6353 q^{67} +18.3299 q^{69} +4.48245 q^{71} -7.96265 q^{73} +4.31894 q^{75} -19.6270 q^{77} -2.23209 q^{79} -2.67824 q^{81} +9.32504 q^{83} -17.8882 q^{85} +11.1713 q^{87} -2.69205 q^{89} +16.9150 q^{91} -0.368568 q^{93} +5.87619 q^{95} +12.9071 q^{97} -24.5647 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q - 2 q^{3} + 6 q^{5} + 2 q^{7} + 22 q^{9}+O(q^{10}) \) 14 * q - 2 * q^3 + 6 * q^5 + 2 * q^7 + 22 * q^9	\( 14 q - 2 q^{3} + 6 q^{5} + 2 q^{7} + 22 q^{9} - 14 q^{11} + 4 q^{13} + 6 q^{15} + 8 q^{17} - 8 q^{19} + 6 q^{21} + 18 q^{23} + 22 q^{25} - 8 q^{27} + 22 q^{29} + 4 q^{31} + 2 q^{33} - 26 q^{35} + 6 q^{37} + 20 q^{39} + 16 q^{41} - 12 q^{43} + 30 q^{45} + 14 q^{47} + 34 q^{49} - 18 q^{51} + 20 q^{53} + 2 q^{55} - 4 q^{57} - 32 q^{59} + 12 q^{61} + 40 q^{65} - 16 q^{67} + 46 q^{69} + 16 q^{71} - 12 q^{73} + 16 q^{75} + 10 q^{77} + 16 q^{79} + 74 q^{81} - 14 q^{83} + 12 q^{85} + 14 q^{87} - 28 q^{91} - 16 q^{93} + 52 q^{95} + 28 q^{97} - 50 q^{99}+O(q^{100}) \) 14 * q - 2 * q^3 + 6 * q^5 + 2 * q^7 + 22 * q^9 - 14 * q^11 + 4 * q^13 + 6 * q^15 + 8 * q^17 - 8 * q^19 + 6 * q^21 + 18 * q^23 + 22 * q^25 - 8 * q^27 + 22 * q^29 + 4 * q^31 + 2 * q^33 - 26 * q^35 + 6 * q^37 + 20 * q^39 + 16 * q^41 - 12 * q^43 + 30 * q^45 + 14 * q^47 + 34 * q^49 - 18 * q^51 + 20 * q^53 + 2 * q^55 - 4 * q^57 - 32 * q^59 + 12 * q^61 + 40 * q^65 - 16 * q^67 + 46 * q^69 + 16 * q^71 - 12 * q^73 + 16 * q^75 + 10 * q^77 + 16 * q^79 + 74 * q^81 - 14 * q^83 + 12 * q^85 + 14 * q^87 - 28 * q^91 - 16 * q^93 + 52 * q^95 + 28 * q^97 - 50 * q^99

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