LMFDB - Embedded newform 6016.2.a.m.1.8

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:	$1$
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			1.8
Root			$-0.319482$ 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)

$f(q)$	$=$	$q+0.319482 q^{3} +2.22340 q^{5} -0.484368 q^{7} -2.89793 q^{9} +O(q^{10})$	$q+0.319482 q^{3} +2.22340 q^{5} -0.484368 q^{7} -2.89793 q^{9} +3.07431 q^{11} -5.65430 q^{13} +0.710336 q^{15} +1.35477 q^{17} +2.21910 q^{19} -0.154747 q^{21} -1.53410 q^{23} -0.0565035 q^{25} -1.88428 q^{27} -3.86238 q^{29} -5.33192 q^{31} +0.982186 q^{33} -1.07694 q^{35} +6.61936 q^{37} -1.80645 q^{39} -0.811395 q^{41} -0.404417 q^{43} -6.44325 q^{45} +1.00000 q^{47} -6.76539 q^{49} +0.432825 q^{51} +9.23122 q^{53} +6.83541 q^{55} +0.708963 q^{57} -1.47489 q^{59} +2.46683 q^{61} +1.40366 q^{63} -12.5718 q^{65} -13.7368 q^{67} -0.490119 q^{69} -11.9263 q^{71} -3.68046 q^{73} -0.0180519 q^{75} -1.48910 q^{77} -15.1950 q^{79} +8.09180 q^{81} -5.17022 q^{83} +3.01219 q^{85} -1.23396 q^{87} +13.6589 q^{89} +2.73876 q^{91} -1.70345 q^{93} +4.93394 q^{95} +5.17590 q^{97} -8.90913 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	0.319482	0.184453	0.0922266	−	0.995738i	$-0.470602\pi$
$3$	0.319482	0.184453	0.0922266	+	0.995738i	$0.470602\pi$
$4$	0	0
$5$	2.22340	0.994334	0.497167	−	0.867655i	$-0.334374\pi$
$5$	2.22340	0.994334	0.497167	+	0.867655i	$0.334374\pi$
$6$	0	0
$7$	−0.484368	−0.183074	−0.0915369	−	0.995802i	$-0.529178\pi$
$7$	−0.484368	−0.183074	−0.0915369	+	0.995802i	$0.529178\pi$
$8$	0	0
$9$	−2.89793	−0.965977
$10$	0	0
$11$	3.07431	0.926938	0.463469	−	0.886113i	$-0.346604\pi$
$11$	3.07431	0.926938	0.463469	+	0.886113i	$0.346604\pi$
$12$	0	0
$13$	−5.65430	−1.56822	−0.784111	−	0.620621i	$-0.786881\pi$
$13$	−5.65430	−1.56822	−0.784111	+	0.620621i	$0.786881\pi$
$14$	0	0
$15$	0.710336	0.183408
$16$	0	0
$17$	1.35477	0.328580	0.164290	−	0.986412i	$-0.447467\pi$
$17$	1.35477	0.328580	0.164290	+	0.986412i	$0.447467\pi$
$18$	0	0
$19$	2.21910	0.509097	0.254548	−	0.967060i	$-0.418073\pi$
$19$	2.21910	0.509097	0.254548	+	0.967060i	$0.418073\pi$
$20$	0	0
$21$	−0.154747	−0.0337686
$22$	0	0
$23$	−1.53410	−0.319883	−0.159941	−	0.987127i	$-0.551131\pi$
$23$	−1.53410	−0.319883	−0.159941	+	0.987127i	$0.551131\pi$
$24$	0	0
$25$	−0.0565035	−0.0113007
$26$	0	0
$27$	−1.88428	−0.362631
$28$	0	0
$29$	−3.86238	−0.717226	−0.358613	−	0.933486i	$-0.616750\pi$
$29$	−3.86238	−0.717226	−0.358613	+	0.933486i	$0.616750\pi$
$30$	0	0
$31$	−5.33192	−0.957641	−0.478820	−	0.877913i	$-0.658935\pi$
$31$	−5.33192	−0.957641	−0.478820	+	0.877913i	$0.658935\pi$
$32$	0	0
$33$	0.982186	0.170977
$34$	0	0
$35$	−1.07694	−0.182036
$36$	0	0
$37$	6.61936	1.08822	0.544108	−	0.839015i	$-0.316868\pi$
$37$	6.61936	1.08822	0.544108	+	0.839015i	$0.316868\pi$
$38$	0	0
$39$	−1.80645	−0.289264
$40$	0	0
$41$	−0.811395	−0.126719	−0.0633593	−	0.997991i	$-0.520181\pi$
$41$	−0.811395	−0.126719	−0.0633593	+	0.997991i	$0.520181\pi$
$42$	0	0
$43$	−0.404417	−0.0616731	−0.0308365	−	0.999524i	$-0.509817\pi$
$43$	−0.404417	−0.0616731	−0.0308365	+	0.999524i	$0.509817\pi$
$44$	0	0
$45$	−6.44325	−0.960503
$46$	0	0
$47$	1.00000	0.145865
$47$	1.00000	0.145865
$48$	0	0
$49$	−6.76539	−0.966484
$50$	0	0
$51$	0.432825	0.0606077
$52$	0	0
$53$	9.23122	1.26801	0.634003	−	0.773331i	$-0.281411\pi$
$53$	9.23122	1.26801	0.634003	+	0.773331i	$0.281411\pi$
$54$	0	0
$55$	6.83541	0.921686
$56$	0	0
$57$	0.708963	0.0939045
$58$	0	0
$59$	−1.47489	−0.192014	−0.0960069	−	0.995381i	$-0.530607\pi$
$59$	−1.47489	−0.192014	−0.0960069	+	0.995381i	$0.530607\pi$
$60$	0	0
$61$	2.46683	0.315845	0.157923	−	0.987451i	$-0.449520\pi$
$61$	2.46683	0.315845	0.157923	+	0.987451i	$0.449520\pi$
$62$	0	0
$63$	1.40366	0.176845
$64$	0	0
$65$	−12.5718	−1.55934
$66$	0	0
$67$	−13.7368	−1.67822	−0.839108	−	0.543965i	$-0.816923\pi$
$67$	−13.7368	−1.67822	−0.839108	+	0.543965i	$0.816923\pi$
$68$	0	0
$69$	−0.490119	−0.0590034
$70$	0	0
$71$	−11.9263	−1.41539	−0.707694	−	0.706519i	$-0.750264\pi$
$71$	−11.9263	−1.41539	−0.707694	+	0.706519i	$0.750264\pi$
$72$	0	0
$73$	−3.68046	−0.430766	−0.215383	−	0.976530i	$-0.569100\pi$
$73$	−3.68046	−0.430766	−0.215383	+	0.976530i	$0.569100\pi$
$74$	0	0
$75$	−0.0180519	−0.00208445
$76$	0	0
$77$	−1.48910	−0.169698
$78$	0	0
$79$	−15.1950	−1.70957	−0.854787	−	0.518980i	$-0.826312\pi$
$79$	−15.1950	−1.70957	−0.854787	+	0.518980i	$0.826312\pi$
$80$	0	0
$81$	8.09180	0.899089
$82$	0	0
$83$	−5.17022	−0.567505	−0.283753	−	0.958898i	$-0.591579\pi$
$83$	−5.17022	−0.567505	−0.283753	+	0.958898i	$0.591579\pi$
$84$	0	0
$85$	3.01219	0.326718
$86$	0	0
$87$	−1.23396	−0.132295
$88$	0	0
$89$	13.6589	1.44784	0.723922	−	0.689882i	$-0.242337\pi$
$89$	13.6589	1.44784	0.723922	+	0.689882i	$0.242337\pi$
$90$	0	0
$91$	2.73876	0.287100
$92$	0	0
$93$	−1.70345	−0.176640
$94$	0	0
$95$	4.93394	0.506212
$96$	0	0
$97$	5.17590	0.525533	0.262766	−	0.964859i	$-0.415365\pi$
$97$	5.17590	0.525533	0.262766	+	0.964859i	$0.415365\pi$
$98$	0	0
$99$	−8.90913	−0.895401
$100$	0	0
$101$	−9.62759	−0.957981	−0.478990	−	0.877820i	$-0.658997\pi$
$101$	−9.62759	−0.957981	−0.478990	+	0.877820i	$0.658997\pi$
$102$	0	0
$103$	−15.9258	−1.56922	−0.784608	−	0.619992i	$-0.787136\pi$
$103$	−15.9258	−1.56922	−0.784608	+	0.619992i	$0.787136\pi$
$104$	0	0
$105$	−0.344064	−0.0335772
$106$	0	0
$107$	−14.9979	−1.44990	−0.724949	−	0.688803i	$-0.758137\pi$
$107$	−14.9979	−1.44990	−0.724949	+	0.688803i	$0.758137\pi$
$108$	0	0
$109$	3.54676	0.339718	0.169859	−	0.985468i	$-0.445669\pi$
$109$	3.54676	0.339718	0.169859	+	0.985468i	$0.445669\pi$
$110$	0	0
$111$	2.11477	0.200725
$112$	0	0
$113$	20.4175	1.92072	0.960358	−	0.278769i	$-0.0899264\pi$
$113$	20.4175	1.92072	0.960358	+	0.278769i	$0.0899264\pi$
$114$	0	0
$115$	−3.41092	−0.318070
$116$	0	0
$117$	16.3858	1.51487
$118$	0	0
$119$	−0.656208	−0.0601545
$120$	0	0
$121$	−1.54864	−0.140785
$122$	0	0
$123$	−0.259226	−0.0233736
$124$	0	0
$125$	−11.2426	−1.00557
$126$	0	0
$127$	−3.36589	−0.298674	−0.149337	−	0.988786i	$-0.547714\pi$
$127$	−3.36589	−0.298674	−0.149337	+	0.988786i	$0.547714\pi$
$128$	0	0
$129$	−0.129204	−0.0113758
$130$	0	0
$131$	18.2049	1.59057	0.795285	−	0.606236i	$-0.207321\pi$
$131$	18.2049	1.59057	0.795285	+	0.606236i	$0.207321\pi$
$132$	0	0
$133$	−1.07486	−0.0932023
$134$	0	0
$135$	−4.18951	−0.360576
$136$	0	0
$137$	2.24716	0.191988	0.0959939	−	0.995382i	$-0.469397\pi$
$137$	2.24716	0.191988	0.0959939	+	0.995382i	$0.469397\pi$
$138$	0	0
$139$	−5.07267	−0.430258	−0.215129	−	0.976586i	$-0.569017\pi$
$139$	−5.07267	−0.430258	−0.215129	+	0.976586i	$0.569017\pi$
$140$	0	0
$141$	0.319482	0.0269053
$142$	0	0
$143$	−17.3831	−1.45364
$144$	0	0
$145$	−8.58761	−0.713162
$146$	0	0
$147$	−2.16142	−0.178271
$148$	0	0
$149$	−0.0464165	−0.00380259	−0.00190129	−	0.999998i	$-0.500605\pi$
$149$	−0.0464165	−0.00380259	−0.00190129	+	0.999998i	$0.500605\pi$
$150$	0	0
$151$	8.31165	0.676392	0.338196	−	0.941076i	$-0.390183\pi$
$151$	8.31165	0.676392	0.338196	+	0.941076i	$0.390183\pi$
$152$	0	0
$153$	−3.92603	−0.317401
$154$	0	0
$155$	−11.8550	−0.952214
$156$	0	0
$157$	−17.4540	−1.39298	−0.696489	−	0.717568i	$-0.745255\pi$
$157$	−17.4540	−1.39298	−0.696489	+	0.717568i	$0.745255\pi$
$158$	0	0
$159$	2.94921	0.233888
$160$	0	0
$161$	0.743071	0.0585622
$162$	0	0
$163$	−23.1395	−1.81243	−0.906213	−	0.422821i	$-0.861040\pi$
$163$	−23.1395	−1.81243	−0.906213	+	0.422821i	$0.861040\pi$
$164$	0	0
$165$	2.18379	0.170008
$166$	0	0
$167$	15.2533	1.18034	0.590168	−	0.807281i	$-0.299062\pi$
$167$	15.2533	1.18034	0.590168	+	0.807281i	$0.299062\pi$
$168$	0	0
$169$	18.9712	1.45932
$170$	0	0
$171$	−6.43080	−0.491776
$172$	0	0
$173$	−22.7991	−1.73339	−0.866693	−	0.498843i	$-0.833759\pi$
$173$	−22.7991	−1.73339	−0.866693	+	0.498843i	$0.833759\pi$
$174$	0	0
$175$	0.0273685	0.00206886
$176$	0	0
$177$	−0.471200	−0.0354175
$178$	0	0
$179$	3.28783	0.245744	0.122872	−	0.992423i	$-0.460790\pi$
$179$	3.28783	0.245744	0.122872	+	0.992423i	$0.460790\pi$
$180$	0	0
$181$	−20.7882	−1.54517	−0.772586	−	0.634910i	$-0.781037\pi$
$181$	−20.7882	−1.54517	−0.772586	+	0.634910i	$0.781037\pi$
$182$	0	0
$183$	0.788108	0.0582587
$184$	0	0
$185$	14.7175	1.08205
$186$	0	0
$187$	4.16498	0.304574
$188$	0	0
$189$	0.912687	0.0663882
$190$	0	0
$191$	−0.671922	−0.0486186	−0.0243093	−	0.999704i	$-0.507739\pi$
$191$	−0.671922	−0.0486186	−0.0243093	+	0.999704i	$0.507739\pi$
$192$	0	0
$193$	−4.70443	−0.338632	−0.169316	−	0.985562i	$-0.554156\pi$
$193$	−4.70443	−0.338632	−0.169316	+	0.985562i	$0.554156\pi$
$194$	0	0
$195$	−4.01646	−0.287624
$196$	0	0
$197$	−16.1862	−1.15322	−0.576610	−	0.817020i	$-0.695625\pi$
$197$	−16.1862	−1.15322	−0.576610	+	0.817020i	$0.695625\pi$
$198$	0	0
$199$	−19.0137	−1.34785	−0.673923	−	0.738801i	$-0.735392\pi$
$199$	−19.0137	−1.34785	−0.673923	+	0.738801i	$0.735392\pi$
$200$	0	0
$201$	−4.38866	−0.309552
$202$	0	0
$203$	1.87081	0.131305
$204$	0	0
$205$	−1.80405	−0.126001
$206$	0	0
$207$	4.44573	0.308999
$208$	0	0
$209$	6.82220	0.471901
$210$	0	0
$211$	17.2828	1.18980	0.594898	−	0.803801i	$-0.297192\pi$
$211$	17.2828	1.18980	0.594898	+	0.803801i	$0.297192\pi$
$212$	0	0
$213$	−3.81023	−0.261073
$214$	0	0
$215$	−0.899181	−0.0613236
$216$	0	0
$217$	2.58261	0.175319
$218$	0	0
$219$	−1.17584	−0.0794561
$220$	0	0
$221$	−7.66029	−0.515287
$222$	0	0
$223$	−5.52092	−0.369708	−0.184854	−	0.982766i	$-0.559181\pi$
$223$	−5.52092	−0.369708	−0.184854	+	0.982766i	$0.559181\pi$
$224$	0	0
$225$	0.163743	0.0109162
$226$	0	0
$227$	10.7768	0.715284	0.357642	−	0.933859i	$-0.383581\pi$
$227$	10.7768	0.715284	0.357642	+	0.933859i	$0.383581\pi$
$228$	0	0
$229$	−1.62920	−0.107661	−0.0538303	−	0.998550i	$-0.517143\pi$
$229$	−1.62920	−0.107661	−0.0538303	+	0.998550i	$0.517143\pi$
$230$	0	0
$231$	−0.475740	−0.0313014
$232$	0	0
$233$	20.9418	1.37194	0.685970	−	0.727630i	$-0.259378\pi$
$233$	20.9418	1.37194	0.685970	+	0.727630i	$0.259378\pi$
$234$	0	0
$235$	2.22340	0.145038
$236$	0	0
$237$	−4.85454	−0.315336
$238$	0	0
$239$	−5.74445	−0.371578	−0.185789	−	0.982590i	$-0.559484\pi$
$239$	−5.74445	−0.371578	−0.185789	+	0.982590i	$0.559484\pi$
$240$	0	0
$241$	6.31475	0.406769	0.203384	−	0.979099i	$-0.434806\pi$
$241$	6.31475	0.406769	0.203384	+	0.979099i	$0.434806\pi$
$242$	0	0
$243$	8.23804	0.528470
$244$	0	0
$245$	−15.0421	−0.961007
$246$	0	0
$247$	−12.5475	−0.798376
$248$	0	0
$249$	−1.65179	−0.104678
$250$	0	0
$251$	−1.66570	−0.105138	−0.0525690	−	0.998617i	$-0.516741\pi$
$251$	−1.66570	−0.105138	−0.0525690	+	0.998617i	$0.516741\pi$
$252$	0	0
$253$	−4.71631	−0.296512
$254$	0	0
$255$	0.962343	0.0602642
$256$	0	0
$257$	6.19750	0.386589	0.193295	−	0.981141i	$-0.438083\pi$
$257$	6.19750	0.386589	0.193295	+	0.981141i	$0.438083\pi$
$258$	0	0
$259$	−3.20620	−0.199224
$260$	0	0
$261$	11.1929	0.692824
$262$	0	0
$263$	17.3028	1.06694	0.533469	−	0.845819i	$-0.320888\pi$
$263$	17.3028	1.06694	0.533469	+	0.845819i	$0.320888\pi$
$264$	0	0
$265$	20.5247	1.26082
$266$	0	0
$267$	4.36379	0.267060
$268$	0	0
$269$	6.30797	0.384604	0.192302	−	0.981336i	$-0.438405\pi$
$269$	6.30797	0.384604	0.192302	+	0.981336i	$0.438405\pi$
$270$	0	0
$271$	10.6993	0.649938	0.324969	−	0.945725i	$-0.394646\pi$
$271$	10.6993	0.649938	0.324969	+	0.945725i	$0.394646\pi$
$272$	0	0
$273$	0.874986	0.0529566
$274$	0	0
$275$	−0.173709	−0.0104751
$276$	0	0
$277$	−3.81187	−0.229033	−0.114517	−	0.993421i	$-0.536532\pi$
$277$	−3.81187	−0.229033	−0.114517	+	0.993421i	$0.536532\pi$
$278$	0	0
$279$	15.4515	0.925059
$280$	0	0
$281$	15.0178	0.895889	0.447944	−	0.894061i	$-0.352156\pi$
$281$	15.0178	0.895889	0.447944	+	0.894061i	$0.352156\pi$
$282$	0	0
$283$	−26.5823	−1.58015	−0.790076	−	0.613009i	$-0.789959\pi$
$283$	−26.5823	−1.58015	−0.790076	+	0.613009i	$0.789959\pi$
$284$	0	0
$285$	1.57631	0.0933724
$286$	0	0
$287$	0.393014	0.0231989
$288$	0	0
$289$	−15.1646	−0.892035
$290$	0	0
$291$	1.65361	0.0969362
$292$	0	0
$293$	−1.92391	−0.112396	−0.0561979	−	0.998420i	$-0.517898\pi$
$293$	−1.92391	−0.112396	−0.0561979	+	0.998420i	$0.517898\pi$
$294$	0	0
$295$	−3.27926	−0.190926
$296$	0	0
$297$	−5.79287	−0.336136
$298$	0	0
$299$	8.67429	0.501647
$300$	0	0
$301$	0.195887	0.0112907
$302$	0	0
$303$	−3.07584	−0.176703
$304$	0	0
$305$	5.48474	0.314056
$306$	0	0
$307$	5.02012	0.286513	0.143257	−	0.989686i	$-0.454243\pi$
$307$	5.02012	0.286513	0.143257	+	0.989686i	$0.454243\pi$
$308$	0	0
$309$	−5.08801	−0.289447
$310$	0	0
$311$	20.4391	1.15900	0.579498	−	0.814974i	$-0.303249\pi$
$311$	20.4391	1.15900	0.579498	+	0.814974i	$0.303249\pi$
$312$	0	0
$313$	9.35860	0.528979	0.264490	−	0.964389i	$-0.414797\pi$
$313$	9.35860	0.528979	0.264490	+	0.964389i	$0.414797\pi$
$314$	0	0
$315$	3.12090	0.175843
$316$	0	0
$317$	3.50828	0.197045	0.0985224	−	0.995135i	$-0.468588\pi$
$317$	3.50828	0.197045	0.0985224	+	0.995135i	$0.468588\pi$
$318$	0	0
$319$	−11.8741	−0.664824
$320$	0	0
$321$	−4.79155	−0.267438
$322$	0	0
$323$	3.00637	0.167279
$324$	0	0
$325$	0.319488	0.0177220
$326$	0	0
$327$	1.13313	0.0626620
$328$	0	0
$329$	−0.484368	−0.0267041
$330$	0	0
$331$	6.33727	0.348328	0.174164	−	0.984717i	$-0.444278\pi$
$331$	6.33727	0.348328	0.174164	+	0.984717i	$0.444278\pi$
$332$	0	0
$333$	−19.1824	−1.05119
$334$	0	0
$335$	−30.5423	−1.66871
$336$	0	0
$337$	−20.3821	−1.11028	−0.555142	−	0.831756i	$-0.687336\pi$
$337$	−20.3821	−1.11028	−0.555142	+	0.831756i	$0.687336\pi$
$338$	0	0
$339$	6.52303	0.354282
$340$	0	0
$341$	−16.3919	−0.887674
$342$	0	0
$343$	6.66751	0.360012
$344$	0	0
$345$	−1.08973	−0.0586691
$346$	0	0
$347$	17.1246	0.919299	0.459649	−	0.888100i	$-0.347975\pi$
$347$	17.1246	0.919299	0.459649	+	0.888100i	$0.347975\pi$
$348$	0	0
$349$	−5.94264	−0.318102	−0.159051	−	0.987270i	$-0.550843\pi$
$349$	−5.94264	−0.318102	−0.159051	+	0.987270i	$0.550843\pi$
$350$	0	0
$351$	10.6543	0.568685
$352$	0	0
$353$	21.4991	1.14428	0.572141	−	0.820155i	$-0.306113\pi$
$353$	21.4991	1.14428	0.572141	+	0.820155i	$0.306113\pi$
$354$	0	0
$355$	−26.5168	−1.40737
$356$	0	0
$357$	−0.209647	−0.0110957
$358$	0	0
$359$	−33.4888	−1.76747	−0.883735	−	0.467987i	$-0.844979\pi$
$359$	−33.4888	−1.76747	−0.883735	+	0.467987i	$0.844979\pi$
$360$	0	0
$361$	−14.0756	−0.740821
$362$	0	0
$363$	−0.494763	−0.0259683
$364$	0	0
$365$	−8.18314	−0.428325
$366$	0	0
$367$	15.5899	0.813786	0.406893	−	0.913476i	$-0.366612\pi$
$367$	15.5899	0.813786	0.406893	+	0.913476i	$0.366612\pi$
$368$	0	0
$369$	2.35137	0.122407
$370$	0	0
$371$	−4.47131	−0.232139
$372$	0	0
$373$	4.96106	0.256874	0.128437	−	0.991718i	$-0.459004\pi$
$373$	4.96106	0.256874	0.128437	+	0.991718i	$0.459004\pi$
$374$	0	0
$375$	−3.59182	−0.185481
$376$	0	0
$377$	21.8391	1.12477
$378$	0	0
$379$	−6.00049	−0.308224	−0.154112	−	0.988053i	$-0.549252\pi$
$379$	−6.00049	−0.308224	−0.154112	+	0.988053i	$0.549252\pi$
$380$	0	0
$381$	−1.07534	−0.0550914
$382$	0	0
$383$	0.332863	0.0170085	0.00850426	−	0.999964i	$-0.497293\pi$
$383$	0.332863	0.0170085	0.00850426	+	0.999964i	$0.497293\pi$
$384$	0	0
$385$	−3.31085	−0.168737
$386$	0	0
$387$	1.17197	0.0595748
$388$	0	0
$389$	13.0435	0.661333	0.330666	−	0.943748i	$-0.392727\pi$
$389$	13.0435	0.661333	0.330666	+	0.943748i	$0.392727\pi$
$390$	0	0
$391$	−2.07836	−0.105107
$392$	0	0
$393$	5.81614	0.293386
$394$	0	0
$395$	−33.7846	−1.69989
$396$	0	0
$397$	16.7275	0.839528	0.419764	−	0.907633i	$-0.362113\pi$
$397$	16.7275	0.839528	0.419764	+	0.907633i	$0.362113\pi$
$398$	0	0
$399$	−0.343399	−0.0171915
$400$	0	0
$401$	−9.79357	−0.489068	−0.244534	−	0.969641i	$-0.578635\pi$
$401$	−9.79357	−0.489068	−0.244534	+	0.969641i	$0.578635\pi$
$402$	0	0
$403$	30.1483	1.50179
$404$	0	0
$405$	17.9913	0.893994
$406$	0	0
$407$	20.3499	1.00871
$408$	0	0
$409$	−10.1041	−0.499614	−0.249807	−	0.968296i	$-0.580367\pi$
$409$	−10.1041	−0.499614	−0.249807	+	0.968296i	$0.580367\pi$
$410$	0	0
$411$	0.717927	0.0354127
$412$	0	0
$413$	0.714387	0.0351527
$414$	0	0
$415$	−11.4954	−0.564289
$416$	0	0
$417$	−1.62063	−0.0793625
$418$	0	0
$419$	−11.5777	−0.565607	−0.282804	−	0.959178i	$-0.591264\pi$
$419$	−11.5777	−0.565607	−0.282804	+	0.959178i	$0.591264\pi$
$420$	0	0
$421$	35.4182	1.72618	0.863088	−	0.505053i	$-0.168527\pi$
$421$	35.4182	1.72618	0.863088	+	0.505053i	$0.168527\pi$
$422$	0	0
$423$	−2.89793	−0.140902
$424$	0	0
$425$	−0.0765494	−0.00371319
$426$	0	0
$427$	−1.19485	−0.0578230
$428$	0	0
$429$	−5.55358	−0.268129
$430$	0	0
$431$	3.17684	0.153023	0.0765115	−	0.997069i	$-0.475622\pi$
$431$	3.17684	0.153023	0.0765115	+	0.997069i	$0.475622\pi$
$432$	0	0
$433$	12.8991	0.619891	0.309945	−	0.950754i	$-0.399689\pi$
$433$	12.8991	0.619891	0.309945	+	0.950754i	$0.399689\pi$
$434$	0	0
$435$	−2.74359	−0.131545
$436$	0	0
$437$	−3.40433	−0.162851
$438$	0	0
$439$	−21.3953	−1.02114	−0.510571	−	0.859836i	$-0.670566\pi$
$439$	−21.3953	−1.02114	−0.510571	+	0.859836i	$0.670566\pi$
$440$	0	0
$441$	19.6056	0.933601
$442$	0	0
$443$	22.6642	1.07681	0.538404	−	0.842687i	$-0.319028\pi$
$443$	22.6642	1.07681	0.538404	+	0.842687i	$0.319028\pi$
$444$	0	0
$445$	30.3692	1.43964
$446$	0	0
$447$	−0.0148292	−0.000701399 0
$448$	0	0
$449$	−11.1457	−0.525997	−0.262999	−	0.964796i	$-0.584711\pi$
$449$	−11.1457	−0.525997	−0.262999	+	0.964796i	$0.584711\pi$
$450$	0	0
$451$	−2.49448	−0.117460
$452$	0	0
$453$	2.65542	0.124763
$454$	0	0
$455$	6.08936	0.285474
$456$	0	0
$457$	9.75212	0.456185	0.228092	−	0.973639i	$-0.426751\pi$
$457$	9.75212	0.456185	0.228092	+	0.973639i	$0.426751\pi$
$458$	0	0
$459$	−2.55277	−0.119153
$460$	0	0
$461$	−24.1185	−1.12331	−0.561656	−	0.827371i	$-0.689836\pi$
$461$	−24.1185	−1.12331	−0.561656	+	0.827371i	$0.689836\pi$
$462$	0	0
$463$	−14.8910	−0.692042	−0.346021	−	0.938227i	$-0.612467\pi$
$463$	−14.8910	−0.692042	−0.346021	+	0.938227i	$0.612467\pi$
$464$	0	0
$465$	−3.78745	−0.175639
$466$	0	0
$467$	−31.2381	−1.44552	−0.722762	−	0.691097i	$-0.757128\pi$
$467$	−31.2381	−1.44552	−0.722762	+	0.691097i	$0.757128\pi$
$468$	0	0
$469$	6.65366	0.307238
$470$	0	0
$471$	−5.57623	−0.256939
$472$	0	0
$473$	−1.24330	−0.0571671
$474$	0	0
$475$	−0.125387	−0.00575315
$476$	0	0
$477$	−26.7514	−1.22486
$478$	0	0
$479$	30.5583	1.39624	0.698122	−	0.715979i	$-0.254020\pi$
$479$	30.5583	1.39624	0.698122	+	0.715979i	$0.254020\pi$
$480$	0	0
$481$	−37.4279	−1.70656
$482$	0	0
$483$	0.237398	0.0108020
$484$	0	0
$485$	11.5081	0.522555
$486$	0	0
$487$	−32.0918	−1.45422	−0.727108	−	0.686523i	$-0.759136\pi$
$487$	−32.0918	−1.45422	−0.727108	+	0.686523i	$0.759136\pi$
$488$	0	0
$489$	−7.39266	−0.334308
$490$	0	0
$491$	−24.3060	−1.09692	−0.548458	−	0.836178i	$-0.684785\pi$
$491$	−24.3060	−1.09692	−0.548458	+	0.836178i	$0.684785\pi$
$492$	0	0
$493$	−5.23264	−0.235666
$494$	0	0
$495$	−19.8085	−0.890327
$496$	0	0
$497$	5.77670	0.259121
$498$	0	0
$499$	−33.4247	−1.49630	−0.748148	−	0.663532i	$-0.769057\pi$
$499$	−33.4247	−1.49630	−0.748148	+	0.663532i	$0.769057\pi$
$500$	0	0
$501$	4.87316	0.217717
$502$	0	0
$503$	2.00080	0.0892114	0.0446057	−	0.999005i	$-0.485797\pi$
$503$	2.00080	0.0892114	0.0446057	+	0.999005i	$0.485797\pi$
$504$	0	0
$505$	−21.4060	−0.952553
$506$	0	0
$507$	6.06095	0.269176
$508$	0	0
$509$	−18.4667	−0.818524	−0.409262	−	0.912417i	$-0.634214\pi$
$509$	−18.4667	−0.818524	−0.409262	+	0.912417i	$0.634214\pi$
$510$	0	0
$511$	1.78270	0.0788620
$512$	0	0
$513$	−4.18142	−0.184614
$514$	0	0
$515$	−35.4094	−1.56032
$516$	0	0
$517$	3.07431	0.135208
$518$	0	0
$519$	−7.28392	−0.319728
$520$	0	0
$521$	35.7052	1.56427	0.782136	−	0.623108i	$-0.214130\pi$
$521$	35.7052	1.56427	0.782136	+	0.623108i	$0.214130\pi$
$522$	0	0
$523$	−31.2754	−1.36758	−0.683788	−	0.729681i	$-0.739669\pi$
$523$	−31.2754	−1.36758	−0.683788	+	0.729681i	$0.739669\pi$
$524$	0	0
$525$	0.00874375	0.000381609 0
$526$	0	0
$527$	−7.22353	−0.314662
$528$	0	0
$529$	−20.6465	−0.897675
$530$	0	0
$531$	4.27412	0.185481
$532$	0	0
$533$	4.58787	0.198723
$534$	0	0
$535$	−33.3462	−1.44168
$536$	0	0
$537$	1.05040	0.0453282
$538$	0	0
$539$	−20.7989	−0.895871
$540$	0	0
$541$	−2.59469	−0.111554	−0.0557771	−	0.998443i	$-0.517764\pi$
$541$	−2.59469	−0.111554	−0.0557771	+	0.998443i	$0.517764\pi$
$542$	0	0
$543$	−6.64145	−0.285012
$544$	0	0
$545$	7.88585	0.337793
$546$	0	0
$547$	−2.38576	−0.102008	−0.0510039	−	0.998698i	$-0.516242\pi$
$547$	−2.38576	−0.102008	−0.0510039	+	0.998698i	$0.516242\pi$
$548$	0	0
$549$	−7.14870	−0.305099
$550$	0	0
$551$	−8.57101	−0.365137
$552$	0	0
$553$	7.35998	0.312978
$554$	0	0
$555$	4.70197	0.199587
$556$	0	0
$557$	15.6704	0.663975	0.331987	−	0.943284i	$-0.392281\pi$
$557$	15.6704	0.663975	0.331987	+	0.943284i	$0.392281\pi$
$558$	0	0
$559$	2.28670	0.0967171
$560$	0	0
$561$	1.33064	0.0561796
$562$	0	0
$563$	−7.36541	−0.310415	−0.155208	−	0.987882i	$-0.549605\pi$
$563$	−7.36541	−0.310415	−0.155208	+	0.987882i	$0.549605\pi$
$564$	0	0
$565$	45.3962	1.90983
$566$	0	0
$567$	−3.91941	−0.164600
$568$	0	0
$569$	−10.3538	−0.434055	−0.217027	−	0.976166i	$-0.569636\pi$
$569$	−10.3538	−0.434055	−0.217027	+	0.976166i	$0.569636\pi$
$570$	0	0
$571$	35.0150	1.46533	0.732667	−	0.680587i	$-0.238275\pi$
$571$	35.0150	1.46533	0.732667	+	0.680587i	$0.238275\pi$
$572$	0	0
$573$	−0.214667	−0.00896785
$574$	0	0
$575$	0.0866823	0.00361490
$576$	0	0
$577$	−2.45390	−0.102157	−0.0510787	−	0.998695i	$-0.516266\pi$
$577$	−2.45390	−0.102157	−0.0510787	+	0.998695i	$0.516266\pi$
$578$	0	0
$579$	−1.50298	−0.0624618
$580$	0	0
$581$	2.50429	0.103895
$582$	0	0
$583$	28.3796	1.17536
$584$	0	0
$585$	36.4321	1.50628
$586$	0	0
$587$	−11.7076	−0.483223	−0.241612	−	0.970373i	$-0.577676\pi$
$587$	−11.7076	−0.483223	−0.241612	+	0.970373i	$0.577676\pi$
$588$	0	0
$589$	−11.8321	−0.487532
$590$	0	0
$591$	−5.17121	−0.212715
$592$	0	0
$593$	11.3762	0.467164	0.233582	−	0.972337i	$-0.424955\pi$
$593$	11.3762	0.467164	0.233582	+	0.972337i	$0.424955\pi$
$594$	0	0
$595$	−1.45901	−0.0598136
$596$	0	0
$597$	−6.07455	−0.248615
$598$	0	0
$599$	−20.1049	−0.821462	−0.410731	−	0.911757i	$-0.634727\pi$
$599$	−20.1049	−0.821462	−0.410731	+	0.911757i	$0.634727\pi$
$600$	0	0
$601$	19.1573	0.781442	0.390721	−	0.920509i	$-0.372226\pi$
$601$	19.1573	0.781442	0.390721	+	0.920509i	$0.372226\pi$
$602$	0	0
$603$	39.8083	1.62112
$604$	0	0
$605$	−3.44324	−0.139988
$606$	0	0
$607$	7.28066	0.295513	0.147756	−	0.989024i	$-0.452795\pi$
$607$	7.28066	0.295513	0.147756	+	0.989024i	$0.452795\pi$
$608$	0	0
$609$	0.597692	0.0242197
$610$	0	0
$611$	−5.65430	−0.228749
$612$	0	0
$613$	−46.2456	−1.86784	−0.933922	−	0.357478i	$-0.883637\pi$
$613$	−46.2456	−1.86784	−0.933922	+	0.357478i	$0.883637\pi$
$614$	0	0
$615$	−0.576363	−0.0232412
$616$	0	0
$617$	27.3903	1.10269	0.551346	−	0.834276i	$-0.314114\pi$
$617$	27.3903	1.10269	0.551346	+	0.834276i	$0.314114\pi$
$618$	0	0
$619$	−6.45368	−0.259395	−0.129698	−	0.991554i	$-0.541401\pi$
$619$	−6.45368	−0.259395	−0.129698	+	0.991554i	$0.541401\pi$
$620$	0	0
$621$	2.89069	0.115999
$622$	0	0
$623$	−6.61595	−0.265062
$624$	0	0
$625$	−24.7143	−0.988572
$626$	0	0
$627$	2.17957	0.0870437
$628$	0	0
$629$	8.96771	0.357566
$630$	0	0
$631$	3.54519	0.141132	0.0705658	−	0.997507i	$-0.477520\pi$
$631$	3.54519	0.141132	0.0705658	+	0.997507i	$0.477520\pi$
$632$	0	0
$633$	5.52154	0.219462
$634$	0	0
$635$	−7.48371	−0.296982
$636$	0	0
$637$	38.2536	1.51566
$638$	0	0
$639$	34.5615	1.36723
$640$	0	0
$641$	−13.8851	−0.548430	−0.274215	−	0.961668i	$-0.588418\pi$
$641$	−13.8851	−0.548430	−0.274215	+	0.961668i	$0.588418\pi$
$642$	0	0
$643$	26.5735	1.04796	0.523979	−	0.851731i	$-0.324447\pi$
$643$	26.5735	1.04796	0.523979	+	0.851731i	$0.324447\pi$
$644$	0	0
$645$	−0.287272	−0.0113113
$646$	0	0
$647$	25.2966	0.994513	0.497257	−	0.867604i	$-0.334341\pi$
$647$	25.2966	0.994513	0.497257	+	0.867604i	$0.334341\pi$
$648$	0	0
$649$	−4.53425	−0.177985
$650$	0	0
$651$	0.825098	0.0323381
$652$	0	0
$653$	−36.8365	−1.44152	−0.720762	−	0.693183i	$-0.756208\pi$
$653$	−36.8365	−1.44152	−0.720762	+	0.693183i	$0.756208\pi$
$654$	0	0
$655$	40.4767	1.58156
$656$	0	0
$657$	10.6657	0.416110
$658$	0	0
$659$	35.2527	1.37325	0.686625	−	0.727011i	$-0.259091\pi$
$659$	35.2527	1.37325	0.686625	+	0.727011i	$0.259091\pi$
$660$	0	0
$661$	−21.6035	−0.840277	−0.420139	−	0.907460i	$-0.638019\pi$
$661$	−21.6035	−0.840277	−0.420139	+	0.907460i	$0.638019\pi$
$662$	0	0
$663$	−2.44733	−0.0950463
$664$	0	0
$665$	−2.38984	−0.0926742
$666$	0	0
$667$	5.92529	0.229428
$668$	0	0
$669$	−1.76384	−0.0681938
$670$	0	0
$671$	7.58379	0.292769
$672$	0	0
$673$	50.6048	1.95067	0.975336	−	0.220725i	$-0.0708422\pi$
$673$	50.6048	1.95067	0.975336	+	0.220725i	$0.0708422\pi$
$674$	0	0
$675$	0.106469	0.00409798
$676$	0	0
$677$	19.4981	0.749372	0.374686	−	0.927152i	$-0.377751\pi$
$677$	19.4981	0.749372	0.374686	+	0.927152i	$0.377751\pi$
$678$	0	0
$679$	−2.50704	−0.0962113
$680$	0	0
$681$	3.44301	0.131936
$682$	0	0
$683$	−15.7769	−0.603687	−0.301843	−	0.953358i	$-0.597602\pi$
$683$	−15.7769	−0.603687	−0.301843	+	0.953358i	$0.597602\pi$
$684$	0	0
$685$	4.99633	0.190900
$686$	0	0
$687$	−0.520501	−0.0198583
$688$	0	0
$689$	−52.1961	−1.98851
$690$	0	0
$691$	−3.13237	−0.119161	−0.0595804	−	0.998224i	$-0.518976\pi$
$691$	−3.13237	−0.119161	−0.0595804	+	0.998224i	$0.518976\pi$
$692$	0	0
$693$	4.31530	0.163925
$694$	0	0
$695$	−11.2786	−0.427820
$696$	0	0
$697$	−1.09925	−0.0416372
$698$	0	0
$699$	6.69052	0.253059
$700$	0	0
$701$	−21.3411	−0.806041	−0.403020	−	0.915191i	$-0.632040\pi$
$701$	−21.3411	−0.806041	−0.403020	+	0.915191i	$0.632040\pi$
$702$	0	0
$703$	14.6890	0.554007
$704$	0	0
$705$	0.710336	0.0267528
$706$	0	0
$707$	4.66330	0.175381
$708$	0	0
$709$	1.62825	0.0611504	0.0305752	−	0.999532i	$-0.490266\pi$
$709$	1.62825	0.0611504	0.0305752	+	0.999532i	$0.490266\pi$
$710$	0	0
$711$	44.0341	1.65141
$712$	0	0
$713$	8.17972	0.306333
$714$	0	0
$715$	−38.6495	−1.44541
$716$	0	0
$717$	−1.83525	−0.0685387
$718$	0	0
$719$	−27.7237	−1.03392	−0.516959	−	0.856010i	$-0.672936\pi$
$719$	−27.7237	−1.03392	−0.516959	+	0.856010i	$0.672936\pi$
$720$	0	0
$721$	7.71395	0.287283
$722$	0	0
$723$	2.01745	0.0750298
$724$	0	0
$725$	0.218238	0.00810516
$726$	0	0
$727$	1.03885	0.0385289	0.0192645	−	0.999814i	$-0.493868\pi$
$727$	1.03885	0.0385289	0.0192645	+	0.999814i	$0.493868\pi$
$728$	0	0
$729$	−21.6435	−0.801611
$730$	0	0
$731$	−0.547893	−0.0202646
$732$	0	0
$733$	−14.7895	−0.546264	−0.273132	−	0.961977i	$-0.588060\pi$
$733$	−14.7895	−0.546264	−0.273132	+	0.961977i	$0.588060\pi$
$734$	0	0
$735$	−4.80570	−0.177261
$736$	0	0
$737$	−42.2311	−1.55560
$738$	0	0
$739$	−51.0700	−1.87864	−0.939320	−	0.343042i	$-0.888543\pi$
$739$	−51.0700	−1.87864	−0.939320	+	0.343042i	$0.888543\pi$
$740$	0	0
$741$	−4.00869	−0.147263
$742$	0	0
$743$	−50.3206	−1.84609	−0.923043	−	0.384698i	$-0.874306\pi$
$743$	−50.3206	−1.84609	−0.923043	+	0.384698i	$0.874306\pi$
$744$	0	0
$745$	−0.103202	−0.00378104
$746$	0	0
$747$	14.9829	0.548197
$748$	0	0
$749$	7.26448	0.265438
$750$	0	0
$751$	−20.6518	−0.753593	−0.376797	−	0.926296i	$-0.622974\pi$
$751$	−20.6518	−0.753593	−0.376797	+	0.926296i	$0.622974\pi$
$752$	0	0
$753$	−0.532161	−0.0193930
$754$	0	0
$755$	18.4801	0.672559
$756$	0	0
$757$	15.2835	0.555490	0.277745	−	0.960655i	$-0.410413\pi$
$757$	15.2835	0.555490	0.277745	+	0.960655i	$0.410413\pi$
$758$	0	0
$759$	−1.50678	−0.0546925
$760$	0	0
$761$	17.4154	0.631307	0.315653	−	0.948875i	$-0.397776\pi$
$761$	17.4154	0.631307	0.315653	+	0.948875i	$0.397776\pi$
$762$	0	0
$763$	−1.71793	−0.0621934
$764$	0	0
$765$	−8.72913	−0.315602
$766$	0	0
$767$	8.33945	0.301120
$768$	0	0
$769$	20.3812	0.734964	0.367482	−	0.930031i	$-0.380220\pi$
$769$	20.3812	0.734964	0.367482	+	0.930031i	$0.380220\pi$
$770$	0	0
$771$	1.97999	0.0713076
$772$	0	0
$773$	27.7751	0.999001	0.499501	−	0.866314i	$-0.333517\pi$
$773$	27.7751	0.999001	0.499501	+	0.866314i	$0.333517\pi$
$774$	0	0
$775$	0.301272	0.0108220
$776$	0	0
$777$	−1.02433	−0.0367475
$778$	0	0
$779$	−1.80057	−0.0645120
$780$	0	0
$781$	−36.6650	−1.31198
$782$	0	0
$783$	7.27782	0.260088
$784$	0	0
$785$	−38.8071	−1.38508
$786$	0	0
$787$	21.2889	0.758869	0.379434	−	0.925219i	$-0.376119\pi$
$787$	21.2889	0.758869	0.379434	+	0.925219i	$0.376119\pi$
$788$	0	0
$789$	5.52795	0.196800
$790$	0	0
$791$	−9.88958	−0.351633
$792$	0	0
$793$	−13.9482	−0.495315
$794$	0	0
$795$	6.55727	0.232562
$796$	0	0
$797$	4.85844	0.172095	0.0860474	−	0.996291i	$-0.472576\pi$
$797$	4.85844	0.172095	0.0860474	+	0.996291i	$0.472576\pi$
$798$	0	0
$799$	1.35477	0.0479284
$800$	0	0
$801$	−39.5827	−1.39858
$802$	0	0
$803$	−11.3149	−0.399293
$804$	0	0
$805$	1.65214	0.0582303
$806$	0	0
$807$	2.01528	0.0709414
$808$	0	0
$809$	29.1902	1.02627	0.513137	−	0.858307i	$-0.328483\pi$
$809$	29.1902	1.02627	0.513137	+	0.858307i	$0.328483\pi$
$810$	0	0
$811$	27.0426	0.949593	0.474796	−	0.880096i	$-0.342522\pi$
$811$	27.0426	0.949593	0.474796	+	0.880096i	$0.342522\pi$
$812$	0	0
$813$	3.41825	0.119883
$814$	0	0
$815$	−51.4483	−1.80216
$816$	0	0
$817$	−0.897443	−0.0313976
$818$	0	0
$819$	−7.93675	−0.277332
$820$	0	0
$821$	44.3932	1.54933	0.774667	−	0.632369i	$-0.217917\pi$
$821$	44.3932	1.54933	0.774667	+	0.632369i	$0.217917\pi$
$822$	0	0
$823$	41.1924	1.43588	0.717939	−	0.696106i	$-0.245086\pi$
$823$	41.1924	1.43588	0.717939	+	0.696106i	$0.245086\pi$
$824$	0	0
$825$	−0.0554970	−0.00193216
$826$	0	0
$827$	16.2906	0.566478	0.283239	−	0.959049i	$-0.408591\pi$
$827$	16.2906	0.566478	0.283239	+	0.959049i	$0.408591\pi$
$828$	0	0
$829$	0.662232	0.0230003	0.0115001	−	0.999934i	$-0.496339\pi$
$829$	0.662232	0.0230003	0.0115001	+	0.999934i	$0.496339\pi$
$830$	0	0
$831$	−1.21783	−0.0422459
$832$	0	0
$833$	−9.16555	−0.317568
$834$	0	0
$835$	33.9141	1.17365
$836$	0	0
$837$	10.0469	0.347270
$838$	0	0
$839$	−17.7298	−0.612100	−0.306050	−	0.952015i	$-0.599007\pi$
$839$	−17.7298	−0.612100	−0.306050	+	0.952015i	$0.599007\pi$
$840$	0	0
$841$	−14.0820	−0.485587
$842$	0	0
$843$	4.79793	0.165250
$844$	0	0
$845$	42.1804	1.45105
$846$	0	0
$847$	0.750112	0.0257741
$848$	0	0
$849$	−8.49256	−0.291464
$850$	0	0
$851$	−10.1548	−0.348101
$852$	0	0
$853$	−43.0506	−1.47403	−0.737013	−	0.675879i	$-0.763764\pi$
$853$	−43.0506	−1.47403	−0.737013	+	0.675879i	$0.763764\pi$
$854$	0	0
$855$	−14.2982	−0.488989
$856$	0	0
$857$	39.2426	1.34050	0.670250	−	0.742135i	$-0.266187\pi$
$857$	39.2426	1.34050	0.670250	+	0.742135i	$0.266187\pi$
$858$	0	0
$859$	20.7175	0.706871	0.353436	−	0.935459i	$-0.385013\pi$
$859$	20.7175	0.706871	0.353436	+	0.935459i	$0.385013\pi$
$860$	0	0
$861$	0.125561	0.00427910
$862$	0	0
$863$	−6.56027	−0.223314	−0.111657	−	0.993747i	$-0.535616\pi$
$863$	−6.56027	−0.223314	−0.111657	+	0.993747i	$0.535616\pi$
$864$	0	0
$865$	−50.6915	−1.72356
$866$	0	0
$867$	−4.84482	−0.164539
$868$	0	0
$869$	−46.7141	−1.58467
$870$	0	0
$871$	77.6720	2.63182
$872$	0	0
$873$	−14.9994	−0.507653
$874$	0	0
$875$	5.44556	0.184094
$876$	0	0
$877$	31.7489	1.07208	0.536042	−	0.844191i	$-0.319919\pi$
$877$	31.7489	1.07208	0.536042	+	0.844191i	$0.319919\pi$
$878$	0	0
$879$	−0.614654	−0.0207318
$880$	0	0
$881$	−59.0699	−1.99012	−0.995058	−	0.0992929i	$-0.968342\pi$
$881$	−59.0699	−1.99012	−0.995058	+	0.0992929i	$0.968342\pi$
$882$	0	0
$883$	−35.7421	−1.20282	−0.601409	−	0.798941i	$-0.705394\pi$
$883$	−35.7421	−1.20282	−0.601409	+	0.798941i	$0.705394\pi$
$884$	0	0
$885$	−1.04766	−0.0352168
$886$	0	0
$887$	−6.45708	−0.216808	−0.108404	−	0.994107i	$-0.534574\pi$
$887$	−6.45708	−0.216808	−0.108404	+	0.994107i	$0.534574\pi$
$888$	0	0
$889$	1.63033	0.0546795
$890$	0	0
$891$	24.8767	0.833400
$892$	0	0
$893$	2.21910	0.0742594
$894$	0	0
$895$	7.31015	0.244351
$896$	0	0
$897$	2.77128	0.0925304
$898$	0	0
$899$	20.5939	0.686845
$900$	0	0
$901$	12.5062	0.416641
$902$	0	0
$903$	0.0625824	0.00208261
$904$	0	0
$905$	−46.2203	−1.53642
$906$	0	0
$907$	−8.85759	−0.294111	−0.147056	−	0.989128i	$-0.546980\pi$
$907$	−8.85759	−0.294111	−0.147056	+	0.989128i	$0.546980\pi$
$908$	0	0
$909$	27.9001	0.925388
$910$	0	0
$911$	−54.6382	−1.81024	−0.905122	−	0.425153i	$-0.860220\pi$
$911$	−54.6382	−1.81024	−0.905122	+	0.425153i	$0.860220\pi$
$912$	0	0
$913$	−15.8948	−0.526042
$914$	0	0
$915$	1.75228	0.0579285
$916$	0	0
$917$	−8.81787	−0.291192
$918$	0	0
$919$	−10.8715	−0.358617	−0.179308	−	0.983793i	$-0.557386\pi$
$919$	−10.8715	−0.358617	−0.179308	+	0.983793i	$0.557386\pi$
$920$	0	0
$921$	1.60384	0.0528483
$922$	0	0
$923$	67.4348	2.21964
$924$	0	0
$925$	−0.374017	−0.0122976
$926$	0	0
$927$	46.1519	1.51583
$928$	0	0
$929$	23.4837	0.770475	0.385238	−	0.922817i	$-0.374120\pi$
$929$	23.4837	0.770475	0.385238	+	0.922817i	$0.374120\pi$
$930$	0	0
$931$	−15.0131	−0.492034
$932$	0	0
$933$	6.52993	0.213780
$934$	0	0
$935$	9.26041	0.302848
$936$	0	0
$937$	6.18294	0.201988	0.100994	−	0.994887i	$-0.467798\pi$
$937$	6.18294	0.201988	0.100994	+	0.994887i	$0.467798\pi$
$938$	0	0
$939$	2.98991	0.0975719
$940$	0	0
$941$	8.11689	0.264603	0.132302	−	0.991210i	$-0.457763\pi$
$941$	8.11689	0.264603	0.132302	+	0.991210i	$0.457763\pi$
$942$	0	0
$943$	1.24476	0.0405351
$944$	0	0
$945$	2.02927	0.0660120
$946$	0	0
$947$	16.0127	0.520343	0.260172	−	0.965562i	$-0.416221\pi$
$947$	16.0127	0.520343	0.260172	+	0.965562i	$0.416221\pi$
$948$	0	0
$949$	20.8105	0.675536
$950$	0	0
$951$	1.12083	0.0363455
$952$	0	0
$953$	8.03372	0.260238	0.130119	−	0.991498i	$-0.458464\pi$
$953$	8.03372	0.260238	0.130119	+	0.991498i	$0.458464\pi$
$954$	0	0
$955$	−1.49395	−0.0483431
$956$	0	0
$957$	−3.79358	−0.122629
$958$	0	0
$959$	−1.08845	−0.0351479
$960$	0	0
$961$	−2.57065	−0.0829242
$962$	0	0
$963$	43.4628	1.40057
$964$	0	0
$965$	−10.4598	−0.336714
$966$	0	0
$967$	58.9584	1.89597	0.947986	−	0.318311i	$-0.103115\pi$
$967$	58.9584	1.89597	0.947986	+	0.318311i	$0.103115\pi$
$968$	0	0
$969$	0.960483	0.0308552
$970$	0	0
$971$	−53.8290	−1.72745	−0.863727	−	0.503961i	$-0.831876\pi$
$971$	−53.8290	−1.72745	−0.863727	+	0.503961i	$0.831876\pi$
$972$	0	0
$973$	2.45704	0.0787690
$974$	0	0
$975$	0.102071	0.00326888
$976$	0	0
$977$	31.0563	0.993578	0.496789	−	0.867871i	$-0.334512\pi$
$977$	31.0563	0.993578	0.496789	+	0.867871i	$0.334512\pi$
$978$	0	0
$979$	41.9918	1.34206
$980$	0	0
$981$	−10.2783	−0.328159
$982$	0	0
$983$	−19.8434	−0.632907	−0.316454	−	0.948608i	$-0.602492\pi$
$983$	−19.8434	−0.632907	−0.316454	+	0.948608i	$0.602492\pi$
$984$	0	0
$985$	−35.9884	−1.14668
$986$	0	0
$987$	−0.154747	−0.00492565
$988$	0	0
$989$	0.620418	0.0197282
$990$	0	0
$991$	25.5528	0.811711	0.405856	−	0.913937i	$-0.366974\pi$
$991$	25.5528	0.811711	0.405856	+	0.913937i	$0.366974\pi$
$992$	0	0
$993$	2.02465	0.0642502
$994$	0	0
$995$	−42.2751	−1.34021
$996$	0	0
$997$	45.6882	1.44696	0.723481	−	0.690345i	$-0.242541\pi$
$997$	45.6882	1.44696	0.723481	+	0.690345i	$0.242541\pi$
$998$	0	0
$999$	−12.4728	−0.394620

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6016.2.a.m.1.8	✓	13
4.3	odd	2		6016.2.a.o.1.6	yes	13
8.3	odd	2		6016.2.a.n.1.8	yes	13
8.5	even	2		6016.2.a.p.1.6	yes	13

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6016.2.a.m.1.8	✓	13	1.1	even	1	trivial
6016.2.a.n.1.8	yes	13	8.3	odd	2
6016.2.a.o.1.6	yes	13	4.3	odd	2
6016.2.a.p.1.6	yes	13	8.5	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:	\( N \)	\(=\)	\( 6016 = 2^{7} \cdot 47 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6016.a (trivial)

\(f(q)\)	\(=\)	\(q+0.319482 q^{3} +2.22340 q^{5} -0.484368 q^{7} -2.89793 q^{9} +O(q^{10})\)	\(q+0.319482 q^{3} +2.22340 q^{5} -0.484368 q^{7} -2.89793 q^{9} +3.07431 q^{11} -5.65430 q^{13} +0.710336 q^{15} +1.35477 q^{17} +2.21910 q^{19} -0.154747 q^{21} -1.53410 q^{23} -0.0565035 q^{25} -1.88428 q^{27} -3.86238 q^{29} -5.33192 q^{31} +0.982186 q^{33} -1.07694 q^{35} +6.61936 q^{37} -1.80645 q^{39} -0.811395 q^{41} -0.404417 q^{43} -6.44325 q^{45} +1.00000 q^{47} -6.76539 q^{49} +0.432825 q^{51} +9.23122 q^{53} +6.83541 q^{55} +0.708963 q^{57} -1.47489 q^{59} +2.46683 q^{61} +1.40366 q^{63} -12.5718 q^{65} -13.7368 q^{67} -0.490119 q^{69} -11.9263 q^{71} -3.68046 q^{73} -0.0180519 q^{75} -1.48910 q^{77} -15.1950 q^{79} +8.09180 q^{81} -5.17022 q^{83} +3.01219 q^{85} -1.23396 q^{87} +13.6589 q^{89} +2.73876 q^{91} -1.70345 q^{93} +4.93394 q^{95} +5.17590 q^{97} -8.90913 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 13 q - 4 q^{3} - 6 q^{5} + 2 q^{7} + 21 q^{9}+O(q^{10}) \) 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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